diff --git a/.github/workflows/build.yml.in b/.github/build.in.yml
similarity index 90%
rename from .github/workflows/build.yml.in
rename to .github/build.in.yml
index 15ea57bab971f..3facc8b60a92b 100644
--- a/.github/workflows/build.yml.in
+++ b/.github/build.in.yml
@@ -1,3 +1,8 @@
+### NB: This is the master file for autogenerating
+### NB: `.github/workflows/{bors, build_fork, build}.yml`.
+### NB: If you need to edit any of those files, you should edit this file instead,
+### NB: and regenerate those files by manually running
+### NB: .github/workflows/mk_build_yml.sh
 
 jobs:
   # Cancels previous runs of jobs in this file
@@ -68,25 +73,6 @@ jobs:
         run: |
           ./scripts/lint-bib.sh
 
-  check_workflows:
-    if: github.repository MAIN_OR_FORK 'leanprover-community/mathlib4'
-    name: check workflowsJOB_NAME
-    runs-on: ubuntu-latest
-    steps:
-      - name: cleanup
-        run: |
-          find . -name . -o -prune -exec rm -rf -- {} +
-
-      - uses: actions/checkout@v4
-
-      - name: update workflows
-        run: |
-          cd .github/workflows/
-          ./mk_build_yml.sh
-
-      - name: check that workflows were consistent
-        run: git diff --exit-code
-
   build:
     if: github.repository MAIN_OR_FORK 'leanprover-community/mathlib4'
     name: BuildJOB_NAME
@@ -129,9 +115,6 @@ jobs:
           lean --version
           lake --version
 
-      - name: check {Mathlib, Tactic, Counterexamples, Archive}.lean are up to date
-        run: lake exe mk_all --check
-
       - name: build cache
         run: |
           lake build cache
@@ -148,10 +131,22 @@ jobs:
       - name: get cache
         id: get
         run: |
-          lake exe cache clean
+          rm -rf .lake/build/lib/Mathlib/
           # Fail quickly if the cache is completely cold, by checking for Mathlib.Init
-          lake exe cache get Mathlib.Init
-          ! test -e .lake/build/lib/Mathlib/Init.olean || lake exe cache get
+          lake exe cache get #Mathlib.Init
+          #lake build --no-build Mathlib.Init && lake exe cache get || echo "No cache for 'Mathlib.Init' available"
+
+      - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
+        id: mk_all
+        run: |
+
+          if ! lake exe mk_all --check
+          then
+            echo "Not all lean files are in the import all files"
+            echo "mk_all=false" >> "${GITHUB_OUTPUT}"
+          else
+            echo "mk_all=true" >> "${GITHUB_OUTPUT}"
+          fi
 
       - name: build mathlib
         id: build
@@ -215,6 +210,14 @@ jobs:
           MATHLIB_CACHE_SAS: ${{ secrets.MATHLIB_CACHE_SAS }}
           MATHLIB_CACHE_S3_TOKEN: ${{ secrets.MATHLIB_CACHE_S3_TOKEN }}
 
+      - name: Check {Mathlib, Tactic, Counterexamples, Archive}.lean
+        run: |
+          if [ ${{ steps.mk_all.outputs.mk_all }} == "false" ]
+          then
+            echo "Please run 'lake exe mk_all' to regenerate the import all files"
+            exit 1
+          fi
+
       - name: check for noisy stdout lines
         id: noisy
         run: |
@@ -275,7 +278,7 @@ jobs:
       # Output is posted to the zulip topic
       # https://leanprover.zulipchat.com/#narrow/stream/345428-mathlib-reviewers/topic/lean4checker
 
-      - name: Post comments for lean-pr-testing branch
+      - name: Post comments for lean-pr-testing-NNNN and batteries-pr-testing-NNNN branches
         if: always()
         env:
           TOKEN: ${{ secrets.LEAN_PR_TESTING }}
@@ -288,7 +291,8 @@ jobs:
           LINT_OUTCOME: ${{ steps.lint.outcome }}
           TEST_OUTCOME: ${{ steps.test.outcome }}
         run: |
-          scripts/lean-pr-testing-comments.sh
+          scripts/lean-pr-testing-comments.sh lean
+          scripts/lean-pr-testing-comments.sh batteries
 
   final:
     name: Post-CI jobJOB_NAME
@@ -318,7 +322,7 @@ jobs:
 
       - if: contains(steps.PR.outputs.pr_labels, 'auto-merge-after-CI')
         name: If `auto-merge-after-CI` is present, add a `bors merge` comment.
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           token: ${{ secrets.AUTO_MERGE_TOKEN }}
           issue-number: ${{ steps.PR.outputs.number }}
diff --git a/.github/workflows/PR_summary.yml b/.github/workflows/PR_summary.yml
index 25529d869f64b..bbde4ace8c7e5 100644
--- a/.github/workflows/PR_summary.yml
+++ b/.github/workflows/PR_summary.yml
@@ -55,12 +55,27 @@ jobs:
         PR="${{ github.event.pull_request.number }}"
         title="### PR summary"
 
+        graphAndHighPercentReports=$(python ./scripts/import-graph-report.py base.json head.json changed_files.txt)
+
         ## Import count comment
         importCount=$(
-          python ./scripts/import-graph-report.py base.json head.json changed_files.txt
+          printf '%s\n' "${graphAndHighPercentReports}" | sed '/^Import changes exceeding/Q'
           ./scripts/import_trans_difference.sh
         )
 
+        ## High percentage imports
+        high_percentages=$(
+          printf '%s\n' "${graphAndHighPercentReports}" | sed -n '/^Import changes exceeding/,$p'
+        )
+        # if there are files with large increase in transitive imports, then we add the `large-import` label
+        if [ -n "${high_percentages}" ]
+        then
+          high_percentages=$'\n\n'"${high_percentages}"
+          gh pr edit "${PR}" --add-label large-import
+        else # otherwise, we remove the label
+          gh pr edit "${PR}" --remove-label large-import
+        fi
+
         if [ "$(printf '%s' "${importCount}" | wc -l)" -gt 12 ]
         then
           importCount="$(printf '<details><summary>\n\n%s\n\n</summary>\n\n%s\n\n</details>\n' "#### Import changes for modified files" "${importCount}")"
@@ -80,6 +95,6 @@ jobs:
         currentHash="$(git rev-parse HEAD)"
         hashURL="https://github.com/${{ github.repository }}/pull/${{ github.event.pull_request.number }}/commits/${currentHash}"
 
-        message="$(printf '%s [%s](%s)\n\n%s\n\n---\n\n%s\n' "${title}" "$(git rev-parse --short HEAD)" "${hashURL}" "${importCount}" "${declDiff}")"
+        message="$(printf '%s [%s](%s)%s\n\n%s\n\n---\n\n%s\n' "${title}" "$(git rev-parse --short HEAD)" "${hashURL}" "${high_percentages}" "${importCount}" "${declDiff}")"
 
         ./scripts/update_PR_comment.sh "${message}" "${title}" "${PR}"
diff --git a/.github/workflows/actionlint.yml b/.github/workflows/actionlint.yml
index dd9a83c59a2b3..5d422196fdf43 100644
--- a/.github/workflows/actionlint.yml
+++ b/.github/workflows/actionlint.yml
@@ -1,10 +1,5 @@
-name: Actionlint
+name: Check workflows
 on:
-  push:
-    branches:
-      - 'master'
-    paths:
-      - '.github/**'
   pull_request:
     paths:
       - '.github/**'
@@ -14,7 +9,28 @@ jobs:
   actionlint:
     runs-on: ubuntu-latest
     steps:
-    - name: Checkout
-      uses: actions/checkout@v4
-    - name: actionlint
-      uses: raven-actions/actionlint@v1
+      - name: Checkout
+        uses: actions/checkout@v4
+
+      - name: suggester / actionlint
+        uses: reviewdog/action-actionlint@v1
+        with:
+          tool_name: actionlint
+          fail_on_error: true
+
+  check_build_yml:
+    name: check workflows generated by build.in.yml
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: update workflows
+        run: |
+          cd .github/workflows/
+          ./mk_build_yml.sh
+
+      - name: suggester / build.in.yml
+        uses: reviewdog/action-suggester@v1
+        with:
+          tool_name: mk_build_yml.sh
+          fail_on_error: true
diff --git a/.github/workflows/add_label_from_comment.yml b/.github/workflows/add_label_from_comment.yml
index 12e9585a4639d..30a161785134f 100644
--- a/.github/workflows/add_label_from_comment.yml
+++ b/.github/workflows/add_label_from_comment.yml
@@ -7,50 +7,15 @@ on:
 jobs:
   add_ready_to_merge_label:
     name: Add ready-to-merge label
-    if: (toJSON(github.event.issue.pull_request) != 'null') && (startsWith(github.event.comment.body, 'bors r+') || contains(toJSON(github.event.comment.body), '\r\nbors r+') || startsWith(github.event.comment.body, 'bors merge') || contains(toJSON(github.event.comment.body), '\r\nbors merge'))
+    if: github.event.issue.pull_request && (startsWith(github.event.comment.body, 'bors r+') || contains(toJSON(github.event.comment.body), '\nbors r+') || startsWith(github.event.comment.body, 'bors merge') || contains(toJSON(github.event.comment.body), '\nbors merge'))
     runs-on: ubuntu-latest
     steps:
-      - uses: octokit/request-action@v2.x
-        name: Get PR head
-        id: get_pr_head
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
         with:
-          route: GET /repos/:repository/pulls/:pull_number
-          repository: ${{ github.repository }}
-          pull_number: ${{ github.event.issue.number }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
-
-      # Parse steps.get_pr_head.outputs.data, since it is a string
-      - id: parse_pr_head
-        name: Parse PR head
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_pr_head.outputs.data }}
-          head_user: 'head.user.login'
-
-      # we skip the rest if this PR is from a fork,
-      # since the GITHUB_TOKEN doesn't have write perms
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        uses: octokit/request-action@v2.x
-        name: Get comment author
-        id: get_user
-        with:
-          route: GET /repos/:repository/collaborators/:username/permission
-          repository: ${{ github.repository }}
-          username: ${{ github.event.comment.user.login }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
-
-      # Parse steps.get_user.outputs.data, since it is a string
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        id: parse_user
-        name: Parse comment author permission
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_user.outputs.data }}
-          permission: 'permission'
+          require: 'admin'
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
         uses: octokit/request-action@v2.x
         id: add_label
         name: Add label
@@ -62,7 +27,7 @@ jobs:
         env:
           GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
         id: remove_labels
         name: Remove "awaiting-author"
         # we use curl rather than octokit/request-action so that the job won't fail
@@ -74,50 +39,15 @@ jobs:
 
   add_delegated_label:
     name: Add delegated label
-    if: (toJSON(github.event.issue.pull_request) != 'null') && (startsWith(github.event.comment.body, 'bors d') || contains(toJSON(github.event.comment.body), '\r\nbors d'))
+    if: github.event.issue.pull_request && (startsWith(github.event.comment.body, 'bors d') || contains(toJSON(github.event.comment.body), '\nbors d'))
     runs-on: ubuntu-latest
     steps:
-      - uses: octokit/request-action@v2.x
-        name: Get PR head
-        id: get_pr_head
-        with:
-          route: GET /repos/:repository/pulls/:pull_number
-          repository: ${{ github.repository }}
-          pull_number: ${{ github.event.issue.number }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
-
-      # Parse steps.get_pr_head.outputs.data, since it is a string
-      - id: parse_pr_head
-        name: Parse PR head
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_pr_head.outputs.data }}
-          head_user: 'head.user.login'
-
-      # we skip the rest if this PR is from a fork,
-      # since the GITHUB_TOKEN doesn't have write perms
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        uses: octokit/request-action@v2.x
-        name: Get comment author
-        id: get_user
-        with:
-          route: GET /repos/:repository/collaborators/:username/permission
-          repository: ${{ github.repository }}
-          username: ${{ github.event.comment.user.login }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
-
-      # Parse steps.get_user.outputs.data, since it is a string
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        id: parse_user
-        name: Parse comment author permission
-        uses: gr2m/get-json-paths-action@v1.x
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
         with:
-          json: ${{ steps.get_user.outputs.data }}
-          permission: 'permission'
+          require: 'admin'
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
         uses: octokit/request-action@v2.x
         id: add_label
         name: Add label
diff --git a/.github/workflows/add_label_from_diff.yaml b/.github/workflows/add_label_from_diff.yaml
new file mode 100644
index 0000000000000..271ac1b95ce59
--- /dev/null
+++ b/.github/workflows/add_label_from_diff.yaml
@@ -0,0 +1,42 @@
+name: Autolabel PRs
+
+on:
+  pull_request:
+    types: [opened]
+  push:
+    paths:
+      - scripts/autolabel.lean
+      - .github/workflows/add_label_from_diff.yaml
+
+jobs:
+  add_topic_label:
+    name: Add topic label
+    runs-on: ubuntu-latest
+    # Don't run on forks, where we wouldn't have permissions to add the label anyway.
+    if: github.repository == 'leanprover-community/mathlib4'
+    permissions:
+      issues: write
+      checks: write
+      pull-requests: write
+      contents: read
+    steps:
+      - name: Checkout code
+        uses: actions/checkout@v4
+        with:
+          fetch-depth: 0
+      - name: install elan
+        run: |
+          set -o pipefail
+          curl -sSfL https://github.com/leanprover/elan/releases/download/v3.1.1/elan-x86_64-unknown-linux-gnu.tar.gz | tar xz
+          ./elan-init -y --default-toolchain none
+          echo "$HOME/.elan/bin" >> "${GITHUB_PATH}"
+      - name: lake exe autolabel
+        run: |
+          # the checkout dance, to avoid a detached head
+          git checkout master
+          git checkout -
+          lake exe autolabel "$NUMBER"
+        env:
+          GH_TOKEN: ${{ secrets.GITHUB_TOKEN }}
+          GH_REPO: ${{ github.repository }}
+          NUMBER: ${{ github.event.number }}
diff --git a/.github/workflows/add_label_from_review.yml b/.github/workflows/add_label_from_review.yml
index 62f047b912236..9406866ff214e 100644
--- a/.github/workflows/add_label_from_review.yml
+++ b/.github/workflows/add_label_from_review.yml
@@ -7,50 +7,16 @@ on:
 jobs:
   add_ready_to_merge_label:
     name: Add ready-to-merge label
-    if: (startsWith(github.event.review.body, 'bors r+') || contains(toJSON(github.event.review.body), '\r\nbors r+') || startsWith(github.event.review.body, 'bors merge') || contains(toJSON(github.event.review.body), '\r\nbors merge'))
+    if: (startsWith(github.event.review.body, 'bors r+') || contains(toJSON(github.event.review.body), '\nbors r+') || startsWith(github.event.review.body, 'bors merge') || contains(toJSON(github.event.review.body), '\nbors merge'))
     runs-on: ubuntu-latest
     steps:
-      - uses: octokit/request-action@v2.x
-        name: Get PR head
-        id: get_pr_head
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
         with:
-          route: GET /repos/:repository/pulls/:pull_number
-          repository: ${{ github.repository }}
-          pull_number: ${{ github.event.pull_request.number }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
-
-      # Parse steps.get_pr_head.outputs.data, since it is a string
-      - id: parse_pr_head
-        name: Parse PR head
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_pr_head.outputs.data }}
-          head_user: 'head.user.login'
-
-      # we skip the rest if this PR is from a fork,
-      # since the GITHUB_TOKEN doesn't have write perms
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        uses: octokit/request-action@v2.x
-        name: Get comment author
-        id: get_user
-        with:
-          route: GET /repos/:repository/collaborators/:username/permission
-          repository: ${{ github.repository }}
-          username: ${{ github.event.review.user.login }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
-
-      # Parse steps.get_user.outputs.data, since it is a string
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        id: parse_user
-        name: Parse comment author permission
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_user.outputs.data }}
-          permission: 'permission'
+          require: 'write'
+          token: ${{ secrets.TRIAGE_TOKEN }}
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.review.user.login == 'leanprover-community-mathlib4-bot') || (github.event.review.user.login == 'leanprover-community-bot-assistant')
         uses: octokit/request-action@v2.x
         id: add_label
         name: Add label
@@ -62,7 +28,7 @@ jobs:
         env:
           GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.review.user.login == 'leanprover-community-mathlib4-bot') || (github.event.review.user.login == 'leanprover-community-bot-assistant')
         id: remove_labels
         name: Remove "awaiting-author"
         # we use curl rather than octokit/request-action so that the job won't fail
@@ -74,50 +40,16 @@ jobs:
 
   add_delegated_label:
     name: Add delegated label
-    if: (startsWith(github.event.review.body, 'bors d') || contains(toJSON(github.event.review.body), '\r\nbors d'))
+    if: (startsWith(github.event.review.body, 'bors d') || contains(toJSON(github.event.review.body), '\nbors d'))
     runs-on: ubuntu-latest
     steps:
-      - uses: octokit/request-action@v2.x
-        name: Get PR head
-        id: get_pr_head
-        with:
-          route: GET /repos/:repository/pulls/:pull_number
-          repository: ${{ github.repository }}
-          pull_number: ${{ github.event.pull_request.number }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
-
-      # Parse steps.get_pr_head.outputs.data, since it is a string
-      - id: parse_pr_head
-        name: Parse PR head
-        uses: gr2m/get-json-paths-action@v1.x
-        with:
-          json: ${{ steps.get_pr_head.outputs.data }}
-          head_user: 'head.user.login'
-
-      # we skip the rest if this PR is from a fork,
-      # since the GITHUB_TOKEN doesn't have write perms
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        uses: octokit/request-action@v2.x
-        name: Get comment author
-        id: get_user
-        with:
-          route: GET /repos/:repository/collaborators/:username/permission
-          repository: ${{ github.repository }}
-          username: ${{ github.event.review.user.login }}
-        env:
-          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
-
-      # Parse steps.get_user.outputs.data, since it is a string
-      - if: steps.parse_pr_head.outputs.head_user == 'leanprover-community'
-        id: parse_user
-        name: Parse comment author permission
-        uses: gr2m/get-json-paths-action@v1.x
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
         with:
-          json: ${{ steps.get_user.outputs.data }}
-          permission: 'permission'
+          require: 'write'
+          token: ${{ secrets.TRIAGE_TOKEN }}
 
-      - if: (steps.parse_pr_head.outputs.head_user == 'leanprover-community') && (steps.parse_user.outputs.permission == 'admin')
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.review.user.login == 'leanprover-community-mathlib4-bot') || (github.event.review.user.login == 'leanprover-community-bot-assistant')
         uses: octokit/request-action@v2.x
         id: add_label
         name: Add label
diff --git a/.github/workflows/add_label_from_review_comment.yml b/.github/workflows/add_label_from_review_comment.yml
new file mode 100644
index 0000000000000..da8b643fb0ee4
--- /dev/null
+++ b/.github/workflows/add_label_from_review_comment.yml
@@ -0,0 +1,62 @@
+name: Add "ready-to-merge" and "delegated" label from PR review comment
+
+on:
+  pull_request_review_comment:
+    types: [created]
+
+jobs:
+  add_ready_to_merge_label:
+    name: Add ready-to-merge label
+    if: (startsWith(github.event.comment.body, 'bors r+') || contains(toJSON(github.event.comment.body), '\nbors r+') || startsWith(github.event.comment.body, 'bors merge') || contains(toJSON(github.event.comment.body), '\nbors merge'))
+    runs-on: ubuntu-latest
+    steps:
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
+        with:
+          require: 'write'
+          token: ${{ secrets.TRIAGE_TOKEN }}
+
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
+        uses: octokit/request-action@v2.x
+        id: add_label
+        name: Add label
+        with:
+          route: POST /repos/:repository/issues/:issue_number/labels
+          repository: ${{ github.repository }}
+          issue_number: ${{ github.event.pull_request.number }}
+          labels: '["ready-to-merge"]'
+        env:
+          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
+
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
+        id: remove_labels
+        name: Remove "awaiting-author"
+        # we use curl rather than octokit/request-action so that the job won't fail
+        # (and send an annoying email) if the labels don't exist
+        run: |
+          curl --request DELETE \
+            --url https://api.github.com/repos/${{ github.repository }}/issues/${{ github.event.pull_request.number }}/labels/awaiting-author \
+            --header 'authorization: Bearer ${{ secrets.TRIAGE_TOKEN }}'
+
+  add_delegated_label:
+    name: Add delegated label
+    if: (startsWith(github.event.comment.body, 'bors d') || contains(toJSON(github.event.comment.body), '\nbors d'))
+    runs-on: ubuntu-latest
+    steps:
+      - id: user_permission
+        uses: actions-cool/check-user-permission@v2
+        with:
+          require: 'write'
+          token: ${{ secrets.TRIAGE_TOKEN }}
+
+      - if: (steps.user_permission.outputs.require-result == 'true') || (github.event.comment.user.login == 'leanprover-community-mathlib4-bot') || (github.event.comment.user.login == 'leanprover-community-bot-assistant')
+        uses: octokit/request-action@v2.x
+        id: add_label
+        name: Add label
+        with:
+          route: POST /repos/:repository/issues/:issue_number/labels
+          repository: ${{ github.repository }}
+          issue_number: ${{ github.event.pull_request.number }}
+          labels: '["delegated"]'
+        env:
+          GITHUB_TOKEN: ${{ secrets.TRIAGE_TOKEN }}
diff --git a/.github/workflows/bors.yml b/.github/workflows/bors.yml
index 1e7646d7ee20b..207aee8482616 100644
--- a/.github/workflows/bors.yml
+++ b/.github/workflows/bors.yml
@@ -1,7 +1,8 @@
 # DO NOT EDIT THIS FILE!!!
 
 # This file is automatically generated by mk_build_yml.sh
-# Edit build.yml.in instead and run mk_build_yml.sh to update.
+# Edit .github/build.in.yml instead and run
+# .github/workflows/mk_build_yml.sh to update.
 
 # Forks of mathlib and other projects should be able to use build_fork.yml directly
 # The jobs in this file run on self-hosted workers and will not be run from external forks
@@ -82,25 +83,6 @@ jobs:
         run: |
           ./scripts/lint-bib.sh
 
-  check_workflows:
-    if: github.repository == 'leanprover-community/mathlib4'
-    name: check workflows
-    runs-on: ubuntu-latest
-    steps:
-      - name: cleanup
-        run: |
-          find . -name . -o -prune -exec rm -rf -- {} +
-
-      - uses: actions/checkout@v4
-
-      - name: update workflows
-        run: |
-          cd .github/workflows/
-          ./mk_build_yml.sh
-
-      - name: check that workflows were consistent
-        run: git diff --exit-code
-
   build:
     if: github.repository == 'leanprover-community/mathlib4'
     name: Build
@@ -143,9 +125,6 @@ jobs:
           lean --version
           lake --version
 
-      - name: check {Mathlib, Tactic, Counterexamples, Archive}.lean are up to date
-        run: lake exe mk_all --check
-
       - name: build cache
         run: |
           lake build cache
@@ -162,10 +141,22 @@ jobs:
       - name: get cache
         id: get
         run: |
-          lake exe cache clean
+          rm -rf .lake/build/lib/Mathlib/
           # Fail quickly if the cache is completely cold, by checking for Mathlib.Init
-          lake exe cache get Mathlib.Init
-          ! test -e .lake/build/lib/Mathlib/Init.olean || lake exe cache get
+          lake exe cache get #Mathlib.Init
+          #lake build --no-build Mathlib.Init && lake exe cache get || echo "No cache for 'Mathlib.Init' available"
+
+      - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
+        id: mk_all
+        run: |
+
+          if ! lake exe mk_all --check
+          then
+            echo "Not all lean files are in the import all files"
+            echo "mk_all=false" >> "${GITHUB_OUTPUT}"
+          else
+            echo "mk_all=true" >> "${GITHUB_OUTPUT}"
+          fi
 
       - name: build mathlib
         id: build
@@ -229,6 +220,14 @@ jobs:
           MATHLIB_CACHE_SAS: ${{ secrets.MATHLIB_CACHE_SAS }}
           MATHLIB_CACHE_S3_TOKEN: ${{ secrets.MATHLIB_CACHE_S3_TOKEN }}
 
+      - name: Check {Mathlib, Tactic, Counterexamples, Archive}.lean
+        run: |
+          if [ ${{ steps.mk_all.outputs.mk_all }} == "false" ]
+          then
+            echo "Please run 'lake exe mk_all' to regenerate the import all files"
+            exit 1
+          fi
+
       - name: check for noisy stdout lines
         id: noisy
         run: |
@@ -289,7 +288,7 @@ jobs:
       # Output is posted to the zulip topic
       # https://leanprover.zulipchat.com/#narrow/stream/345428-mathlib-reviewers/topic/lean4checker
 
-      - name: Post comments for lean-pr-testing branch
+      - name: Post comments for lean-pr-testing-NNNN and batteries-pr-testing-NNNN branches
         if: always()
         env:
           TOKEN: ${{ secrets.LEAN_PR_TESTING }}
@@ -302,7 +301,8 @@ jobs:
           LINT_OUTCOME: ${{ steps.lint.outcome }}
           TEST_OUTCOME: ${{ steps.test.outcome }}
         run: |
-          scripts/lean-pr-testing-comments.sh
+          scripts/lean-pr-testing-comments.sh lean
+          scripts/lean-pr-testing-comments.sh batteries
 
   final:
     name: Post-CI job
@@ -332,7 +332,7 @@ jobs:
 
       - if: contains(steps.PR.outputs.pr_labels, 'auto-merge-after-CI')
         name: If `auto-merge-after-CI` is present, add a `bors merge` comment.
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           token: ${{ secrets.AUTO_MERGE_TOKEN }}
           issue-number: ${{ steps.PR.outputs.number }}
diff --git a/.github/workflows/bot_fix_style_comment.yaml b/.github/workflows/bot_fix_style_comment.yaml
index d63c2d39771e3..4414eb4b576b5 100644
--- a/.github/workflows/bot_fix_style_comment.yaml
+++ b/.github/workflows/bot_fix_style_comment.yaml
@@ -9,7 +9,6 @@ jobs:
     name: Fix style issues from lint
     if: (github.event.issue.pull_request) && (startsWith(github.event.comment.body, 'bot fix style') || contains(toJSON(github.event.comment.body), '\nbot fix style'))
     runs-on: ubuntu-latest
-    continue-on-error: true  # do not annoy the user when linting fails, as expected
     steps:
       - id: user_permission
         uses: actions-cool/check-user-permission@v2
@@ -58,7 +57,6 @@ jobs:
 
       - name: lint
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
           lake exe lint-style --fix
 
@@ -70,14 +68,15 @@ jobs:
 
       - name: lint references.bib
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
-          ./scripts/lint-bib.sh
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          ./scripts/lint-bib.sh || true
 
       - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
-        run: lake exe mk_all
+        run: |
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          lake exe mk_all || true
 
       - name: Commit and push changes
         if: steps.user_permission.outputs.require-result == 'true'
diff --git a/.github/workflows/bot_fix_style_review.yaml b/.github/workflows/bot_fix_style_review.yaml
index 3d99da3c124a6..99fa4ae9a7dab 100644
--- a/.github/workflows/bot_fix_style_review.yaml
+++ b/.github/workflows/bot_fix_style_review.yaml
@@ -10,7 +10,6 @@ jobs:
     name: Fix style issues from lint
     if: (startsWith(github.event.review.body, 'bot fix style') || contains(toJSON(github.event.review.body), '\nbot fix style'))
     runs-on: ubuntu-latest
-    continue-on-error: true  # do not annoy the user when linting fails, as expected
     steps:
       - id: user_permission
         uses: actions-cool/check-user-permission@v2
@@ -64,7 +63,6 @@ jobs:
 
       - name: lint
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
           lake exe lint-style --fix
 
@@ -76,14 +74,15 @@ jobs:
 
       - name: lint references.bib
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
-          ./scripts/lint-bib.sh
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          ./scripts/lint-bib.sh || true
 
       - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
-        run: lake exe mk_all
+        run: |
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          lake exe mk_all || true
 
       - name: Commit and push changes
         if: steps.user_permission.outputs.require-result == 'true'
diff --git a/.github/workflows/bot_fix_style_review_comment.yaml b/.github/workflows/bot_fix_style_review_comment.yaml
index 49d0ca91cb8c2..097b24ba35954 100644
--- a/.github/workflows/bot_fix_style_review_comment.yaml
+++ b/.github/workflows/bot_fix_style_review_comment.yaml
@@ -9,7 +9,6 @@ jobs:
     name: Fix style issues from lint
     if: (startsWith(github.event.comment.body, 'bot fix style') || contains(toJSON(github.event.comment.body), '\nbot fix style'))
     runs-on: ubuntu-latest
-    continue-on-error: true  # do not annoy the user when linting fails, as expected
     steps:
       - id: user_permission
         uses: actions-cool/check-user-permission@v2
@@ -62,7 +61,6 @@ jobs:
 
       - name: lint
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
           lake exe lint-style --fix
 
@@ -74,14 +72,15 @@ jobs:
 
       - name: lint references.bib
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
-          ./scripts/lint-bib.sh
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          ./scripts/lint-bib.sh || true
 
       - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
         if: steps.user_permission.outputs.require-result == 'true'
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
-        run: lake exe mk_all
+        run: |
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          lake exe mk_all || true
 
       - name: Commit and push changes
         if: steps.user_permission.outputs.require-result == 'true'
diff --git a/.github/workflows/build.yml b/.github/workflows/build.yml
index 539dca108bb3f..c47cd306ed4da 100644
--- a/.github/workflows/build.yml
+++ b/.github/workflows/build.yml
@@ -1,7 +1,8 @@
 # DO NOT EDIT THIS FILE!!!
 
 # This file is automatically generated by mk_build_yml.sh
-# Edit build.yml.in instead and run mk_build_yml.sh to update.
+# Edit .github/build.in.yml instead and run
+# .github/workflows/mk_build_yml.sh to update.
 
 # Forks of mathlib and other projects should be able to use build_fork.yml directly
 # The jobs in this file run on self-hosted workers and will not be run from external forks
@@ -89,25 +90,6 @@ jobs:
         run: |
           ./scripts/lint-bib.sh
 
-  check_workflows:
-    if: github.repository == 'leanprover-community/mathlib4'
-    name: check workflows
-    runs-on: ubuntu-latest
-    steps:
-      - name: cleanup
-        run: |
-          find . -name . -o -prune -exec rm -rf -- {} +
-
-      - uses: actions/checkout@v4
-
-      - name: update workflows
-        run: |
-          cd .github/workflows/
-          ./mk_build_yml.sh
-
-      - name: check that workflows were consistent
-        run: git diff --exit-code
-
   build:
     if: github.repository == 'leanprover-community/mathlib4'
     name: Build
@@ -150,9 +132,6 @@ jobs:
           lean --version
           lake --version
 
-      - name: check {Mathlib, Tactic, Counterexamples, Archive}.lean are up to date
-        run: lake exe mk_all --check
-
       - name: build cache
         run: |
           lake build cache
@@ -169,10 +148,22 @@ jobs:
       - name: get cache
         id: get
         run: |
-          lake exe cache clean
+          rm -rf .lake/build/lib/Mathlib/
           # Fail quickly if the cache is completely cold, by checking for Mathlib.Init
-          lake exe cache get Mathlib.Init
-          ! test -e .lake/build/lib/Mathlib/Init.olean || lake exe cache get
+          lake exe cache get #Mathlib.Init
+          #lake build --no-build Mathlib.Init && lake exe cache get || echo "No cache for 'Mathlib.Init' available"
+
+      - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
+        id: mk_all
+        run: |
+
+          if ! lake exe mk_all --check
+          then
+            echo "Not all lean files are in the import all files"
+            echo "mk_all=false" >> "${GITHUB_OUTPUT}"
+          else
+            echo "mk_all=true" >> "${GITHUB_OUTPUT}"
+          fi
 
       - name: build mathlib
         id: build
@@ -236,6 +227,14 @@ jobs:
           MATHLIB_CACHE_SAS: ${{ secrets.MATHLIB_CACHE_SAS }}
           MATHLIB_CACHE_S3_TOKEN: ${{ secrets.MATHLIB_CACHE_S3_TOKEN }}
 
+      - name: Check {Mathlib, Tactic, Counterexamples, Archive}.lean
+        run: |
+          if [ ${{ steps.mk_all.outputs.mk_all }} == "false" ]
+          then
+            echo "Please run 'lake exe mk_all' to regenerate the import all files"
+            exit 1
+          fi
+
       - name: check for noisy stdout lines
         id: noisy
         run: |
@@ -296,7 +295,7 @@ jobs:
       # Output is posted to the zulip topic
       # https://leanprover.zulipchat.com/#narrow/stream/345428-mathlib-reviewers/topic/lean4checker
 
-      - name: Post comments for lean-pr-testing branch
+      - name: Post comments for lean-pr-testing-NNNN and batteries-pr-testing-NNNN branches
         if: always()
         env:
           TOKEN: ${{ secrets.LEAN_PR_TESTING }}
@@ -309,7 +308,8 @@ jobs:
           LINT_OUTCOME: ${{ steps.lint.outcome }}
           TEST_OUTCOME: ${{ steps.test.outcome }}
         run: |
-          scripts/lean-pr-testing-comments.sh
+          scripts/lean-pr-testing-comments.sh lean
+          scripts/lean-pr-testing-comments.sh batteries
 
   final:
     name: Post-CI job
@@ -339,7 +339,7 @@ jobs:
 
       - if: contains(steps.PR.outputs.pr_labels, 'auto-merge-after-CI')
         name: If `auto-merge-after-CI` is present, add a `bors merge` comment.
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           token: ${{ secrets.AUTO_MERGE_TOKEN }}
           issue-number: ${{ steps.PR.outputs.number }}
diff --git a/.github/workflows/build_fork.yml b/.github/workflows/build_fork.yml
index 78c386218fa75..c431ad5e7d2f2 100644
--- a/.github/workflows/build_fork.yml
+++ b/.github/workflows/build_fork.yml
@@ -1,7 +1,8 @@
 # DO NOT EDIT THIS FILE!!!
 
 # This file is automatically generated by mk_build_yml.sh
-# Edit build.yml.in instead and run mk_build_yml.sh to update.
+# Edit .github/build.in.yml instead and run
+# .github/workflows/mk_build_yml.sh to update.
 
 # Forks of mathlib and other projects should be able to use build_fork.yml directly
 # The jobs in this file run on GitHub-hosted workers and will only be run from external forks
@@ -86,25 +87,6 @@ jobs:
         run: |
           ./scripts/lint-bib.sh
 
-  check_workflows:
-    if: github.repository != 'leanprover-community/mathlib4'
-    name: check workflows (fork)
-    runs-on: ubuntu-latest
-    steps:
-      - name: cleanup
-        run: |
-          find . -name . -o -prune -exec rm -rf -- {} +
-
-      - uses: actions/checkout@v4
-
-      - name: update workflows
-        run: |
-          cd .github/workflows/
-          ./mk_build_yml.sh
-
-      - name: check that workflows were consistent
-        run: git diff --exit-code
-
   build:
     if: github.repository != 'leanprover-community/mathlib4'
     name: Build (fork)
@@ -147,9 +129,6 @@ jobs:
           lean --version
           lake --version
 
-      - name: check {Mathlib, Tactic, Counterexamples, Archive}.lean are up to date
-        run: lake exe mk_all --check
-
       - name: build cache
         run: |
           lake build cache
@@ -166,10 +145,22 @@ jobs:
       - name: get cache
         id: get
         run: |
-          lake exe cache clean
+          rm -rf .lake/build/lib/Mathlib/
           # Fail quickly if the cache is completely cold, by checking for Mathlib.Init
-          lake exe cache get Mathlib.Init
-          ! test -e .lake/build/lib/Mathlib/Init.olean || lake exe cache get
+          lake exe cache get #Mathlib.Init
+          #lake build --no-build Mathlib.Init && lake exe cache get || echo "No cache for 'Mathlib.Init' available"
+
+      - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
+        id: mk_all
+        run: |
+
+          if ! lake exe mk_all --check
+          then
+            echo "Not all lean files are in the import all files"
+            echo "mk_all=false" >> "${GITHUB_OUTPUT}"
+          else
+            echo "mk_all=true" >> "${GITHUB_OUTPUT}"
+          fi
 
       - name: build mathlib
         id: build
@@ -233,6 +224,14 @@ jobs:
           MATHLIB_CACHE_SAS: ${{ secrets.MATHLIB_CACHE_SAS }}
           MATHLIB_CACHE_S3_TOKEN: ${{ secrets.MATHLIB_CACHE_S3_TOKEN }}
 
+      - name: Check {Mathlib, Tactic, Counterexamples, Archive}.lean
+        run: |
+          if [ ${{ steps.mk_all.outputs.mk_all }} == "false" ]
+          then
+            echo "Please run 'lake exe mk_all' to regenerate the import all files"
+            exit 1
+          fi
+
       - name: check for noisy stdout lines
         id: noisy
         run: |
@@ -293,7 +292,7 @@ jobs:
       # Output is posted to the zulip topic
       # https://leanprover.zulipchat.com/#narrow/stream/345428-mathlib-reviewers/topic/lean4checker
 
-      - name: Post comments for lean-pr-testing branch
+      - name: Post comments for lean-pr-testing-NNNN and batteries-pr-testing-NNNN branches
         if: always()
         env:
           TOKEN: ${{ secrets.LEAN_PR_TESTING }}
@@ -306,7 +305,8 @@ jobs:
           LINT_OUTCOME: ${{ steps.lint.outcome }}
           TEST_OUTCOME: ${{ steps.test.outcome }}
         run: |
-          scripts/lean-pr-testing-comments.sh
+          scripts/lean-pr-testing-comments.sh lean
+          scripts/lean-pr-testing-comments.sh batteries
 
   final:
     name: Post-CI job (fork)
@@ -336,7 +336,7 @@ jobs:
 
       - if: contains(steps.PR.outputs.pr_labels, 'auto-merge-after-CI')
         name: If `auto-merge-after-CI` is present, add a `bors merge` comment.
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           token: ${{ secrets.AUTO_MERGE_TOKEN }}
           issue-number: ${{ steps.PR.outputs.number }}
diff --git a/.github/workflows/dependent-issues.yml b/.github/workflows/dependent-issues.yml
index 682c5311f52b0..9e0b3d2da4b71 100644
--- a/.github/workflows/dependent-issues.yml
+++ b/.github/workflows/dependent-issues.yml
@@ -3,6 +3,7 @@ name: Dependent Issues
 on:
   schedule:
     - cron: '*/15 * * * *' # run every 15 minutes
+  workflow_dispatch:
 
 jobs:
   cancel:
@@ -18,7 +19,7 @@ jobs:
   check:
     runs-on: ubuntu-latest
     steps:
-      - uses: z0al/dependent-issues@v1
+      - uses: z0al/dependent-issues@75d554cd9494b6e1766bc9d08a81c26444ad5c5a
         env:
           # (Required) The token to use to make API calls to GitHub.
           GITHUB_TOKEN: ${{ secrets.DEPENDENT_ISSUES_TOKEN }}
diff --git a/.github/workflows/labels_from_comment.yml b/.github/workflows/labels_from_comment.yml
index 5b62e2ce226d6..b40b8d4c4daf6 100644
--- a/.github/workflows/labels_from_comment.yml
+++ b/.github/workflows/labels_from_comment.yml
@@ -10,7 +10,7 @@ on:
 
 jobs:
   update-label:
-    if: github.event.issue.pull_request != null && (contains(github.event.comment.body, 'awaiting-author') || contains(github.event.comment.body, 'WIP'))
+    if: github.event.issue.pull_request && (contains(github.event.comment.body, 'awaiting-author') || contains(github.event.comment.body, 'WIP'))
     runs-on: ubuntu-latest
 
     steps:
@@ -20,7 +20,7 @@ jobs:
         github-token: ${{ secrets.GITHUB_TOKEN }}
         script: |
           const { owner, repo, number: issue_number  } = context.issue;
-          const commentLines = context.payload.comment.body.split('\r\n');
+          const commentLines = context.payload.comment.body.split(/\r?\n/);
 
           const awaitingAuthor = commentLines.includes('awaiting-author');
           const wip = commentLines.includes('WIP');
diff --git a/.github/workflows/lean4checker.yml b/.github/workflows/lean4checker.yml
index 3152d364350df..8976cc19badb6 100644
--- a/.github/workflows/lean4checker.yml
+++ b/.github/workflows/lean4checker.yml
@@ -4,6 +4,7 @@ name: lean4checker Workflow
 on:
   schedule:
     - cron: '0 0 * * *'   # Runs at 00:00 UTC every day
+  workflow_dispatch:
 
 jobs:
   check-lean4checker:
@@ -69,7 +70,7 @@ jobs:
         run: |
           git clone https://github.com/leanprover/lean4checker
           cd lean4checker
-          git checkout v4.11.0-rc1
+          git checkout v4.13.0-rc3
           # Now that the git hash is embedded in each olean,
           # we need to compile lean4checker on the same toolchain
           cp ../lean-toolchain .
diff --git a/.github/workflows/lint_and_suggest_pr.yml b/.github/workflows/lint_and_suggest_pr.yml
index 6c8f932fc70a1..409e78ebc223e 100644
--- a/.github/workflows/lint_and_suggest_pr.yml
+++ b/.github/workflows/lint_and_suggest_pr.yml
@@ -30,14 +30,13 @@ jobs:
 
       # if you update this step (or its dependencies), please also update them in bot_fix_style_comment.yaml
       - name: lint
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
           lake exe lint-style --fix
 
       - name: suggester / lint-style
         uses: reviewdog/action-suggester@v1
         with:
-          tool_name: lint-style
+          tool_name: lint-style (comment with "bot fix style" to have the bot commit all style suggestions)
 
       - name: Install bibtool
         run: |
@@ -46,14 +45,14 @@ jobs:
 
       # if you update this step (or its dependencies), please also update them in bot_fix_style_comment.yaml
       - name: lint references.bib
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
         run: |
-          ./scripts/lint-bib.sh
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          ./scripts/lint-bib.sh || true
 
       - name: suggester / lint-bib
         uses: reviewdog/action-suggester@v1
         with:
-          tool_name: lint-bib
+          tool_name: lint-bib (comment with "bot fix style" to have the bot commit all style suggestions)
 
   check_imported:
     if: github.repository == 'leanprover-community/mathlib4' && github.event.pull_request.draft == false
@@ -75,10 +74,11 @@ jobs:
 
       # if you update this step (or its dependencies), please also update them in bot_fix_style_comment.yaml
       - name: update {Mathlib, Tactic, Counterexamples, Archive}.lean
-        continue-on-error: true  # allows the following `reviewdog` step to add GitHub suggestions
-        run: lake exe mk_all
+        run: 
+          # ignoring the return code allows the following `reviewdog` step to add GitHub suggestions
+          lake exe mk_all || true
 
       - name: suggester / import list
         uses: reviewdog/action-suggester@v1
         with:
-          tool_name: imports
+          tool_name: imports (comment with "bot fix style" to have the bot commit all style suggestions)
diff --git a/.github/workflows/maintainer_merge_comment.yml b/.github/workflows/maintainer_merge_comment.yml
index 61d41d9ac1375..db769c6592ab6 100644
--- a/.github/workflows/maintainer_merge_comment.yml
+++ b/.github/workflows/maintainer_merge_comment.yml
@@ -7,7 +7,7 @@ on:
 jobs:
   ping_zulip:
     name: Ping maintainers on Zulip
-    if: (github.event.issue.pull_request != 'null') && (startsWith(github.event.comment.body, 'maintainer merge') || contains(toJSON(github.event.comment.body), '\nmaintainer merge'))
+    if: github.event.issue.pull_request && (startsWith(github.event.comment.body, 'maintainer merge') || contains(toJSON(github.event.comment.body), '\nmaintainer merge'))
     runs-on: ubuntu-latest
     steps:
       - name: Check whether user is part of mathlib-reviewers team
@@ -20,6 +20,17 @@ jobs:
                                                   # This feature is only applicable in an issue (or PR) context
           exit: true # optional. If the action should exit if the user is not part of the team. Defaults to true.
 
+      - uses: actions/checkout@v4
+        with:
+          ref: master
+      - name: Determine Zulip topic
+        id: determine_topic
+        run: |
+          ./scripts/get_tlabel.sh "${PR}" >> "$GITHUB_OUTPUT"
+        env:
+          PR:  /repos/leanprover-community/mathlib4/issues/${{ github.event.issue.number }}
+          GH_TOKEN: ${{secrets.GITHUB_TOKEN}}
+
       - name: Send message on Zulip
         uses: zulip/github-actions-zulip/send-message@v1
         with:
@@ -28,14 +39,14 @@ jobs:
           organization-url: 'https://leanprover.zulipchat.com'
           to: 'mathlib reviewers'
           type: 'stream'
-          topic: 'maintainer merge'
+          topic: ${{ steps.determine_topic.outputs.topic }}
           content: |
             ${{ format('{0} requested a maintainer merge from comment on PR [#{1}]({2}):', github.event.comment.user.login, github.event.issue.number, github.event.issue.html_url ) }}
 
             > ${{ github.event.issue.title }}
 
       - name: Add comment to PR
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           issue-number: ${{ github.event.issue.number }}
           body: |
diff --git a/.github/workflows/maintainer_merge_review.yml b/.github/workflows/maintainer_merge_review.yml
index aaefc4ecbdcb2..c7d8357bbfa64 100644
--- a/.github/workflows/maintainer_merge_review.yml
+++ b/.github/workflows/maintainer_merge_review.yml
@@ -19,6 +19,17 @@ jobs:
           token: ${{ secrets.MATHLIB_REVIEWERS_TEAM_KEY }} # required. Personal Access Token with the `read:org` permission
           exit: true # optional. If the action should exit if the user is not part of the team. Defaults to true.
 
+      - uses: actions/checkout@v4
+        with:
+          ref: master
+      - name: Determine Zulip topic
+        id: determine_topic
+        run: |
+          ./scripts/get_tlabel.sh "${PR}" >> "$GITHUB_OUTPUT"
+        env:
+          PR:  /repos/leanprover-community/mathlib4/issues/${{ github.event.pull_request.number }}
+          GH_TOKEN: ${{secrets.GITHUB_TOKEN}}
+
       - name: Send message on Zulip
         uses: zulip/github-actions-zulip/send-message@v1
         with:
@@ -27,14 +38,14 @@ jobs:
           organization-url: 'https://leanprover.zulipchat.com'
           to: 'mathlib reviewers'
           type: 'stream'
-          topic: 'maintainer merge'
+          topic: ${{ steps.determine_topic.outputs.topic }}
           content: |
             ${{ format('{0} requested a maintainer merge from review on PR [#{1}]({2}):', github.event.review.user.login, github.event.pull_request.number, github.event.pull_request.html_url ) }}
 
             > ${{ github.event.pull_request.title }}
 
       - name: Add comment to PR
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           issue-number: ${{ github.event.pull_request.number }}
           body: |
diff --git a/.github/workflows/maintainer_merge_review_comment.yml b/.github/workflows/maintainer_merge_review_comment.yml
index 48ccc466ac1ea..79a5ea176d9d9 100644
--- a/.github/workflows/maintainer_merge_review_comment.yml
+++ b/.github/workflows/maintainer_merge_review_comment.yml
@@ -18,6 +18,17 @@ jobs:
           token: ${{ secrets.MATHLIB_REVIEWERS_TEAM_KEY }} # required. Personal Access Token with the `read:org` permission
           exit: true # optional. If the action should exit if the user is not part of the team. Defaults to true.
 
+      - uses: actions/checkout@v4
+        with:
+          ref: master
+      - name: Determine Zulip topic
+        id: determine_topic
+        run: |
+          ./scripts/get_tlabel.sh "${PR}" >> "$GITHUB_OUTPUT"
+        env:
+          PR:  /repos/leanprover-community/mathlib4/issues/${{ github.event.pull_request.number }}
+          GH_TOKEN: ${{secrets.GITHUB_TOKEN}}
+
       - name: Send message on Zulip
         uses: zulip/github-actions-zulip/send-message@v1
         with:
@@ -26,14 +37,14 @@ jobs:
           organization-url: 'https://leanprover.zulipchat.com'
           to: 'mathlib reviewers'
           type: 'stream'
-          topic: 'maintainer merge'
+          topic: ${{ steps.determine_topic.outputs.topic }}
           content: |
             ${{ format('{0} requested a maintainer merge from review comment on PR [#{1}]({2}):', github.event.comment.user.login, github.event.pull_request.number, github.event.pull_request.html_url ) }}
 
             > ${{ github.event.pull_request.title }}
 
       - name: Add comment to PR
-        uses: GrantBirki/comment@v2.0.1
+        uses: GrantBirki/comment@v2
         with:
           issue-number: ${{ github.event.pull_request.number }}
           body: |
diff --git a/.github/workflows/merge_conflicts.yml b/.github/workflows/merge_conflicts.yml
index 68b53db526dad..d10c0233ceaad 100644
--- a/.github/workflows/merge_conflicts.yml
+++ b/.github/workflows/merge_conflicts.yml
@@ -3,6 +3,7 @@ name: Merge conflicts
 on:
   schedule:
     - cron: '*/15 * * * *' # run every 15 minutes
+  workflow_dispatch:
 
 jobs:
   main:
diff --git a/.github/workflows/mk_build_yml.sh b/.github/workflows/mk_build_yml.sh
index 96cf301b33a26..435e7b4e4c6dc 100755
--- a/.github/workflows/mk_build_yml.sh
+++ b/.github/workflows/mk_build_yml.sh
@@ -1,5 +1,11 @@
 #!/usr/bin/env bash
-set -ex
+
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
+set -x
 cd $(dirname "$(realpath "$0")")
 
 header() {
@@ -7,7 +13,8 @@ header() {
 # DO NOT EDIT THIS FILE!!!
 
 # This file is automatically generated by mk_build_yml.sh
-# Edit build.yml.in instead and run mk_build_yml.sh to update.
+# Edit .github/build.in.yml instead and run
+# .github/workflows/mk_build_yml.sh to update.
 
 # Forks of mathlib and other projects should be able to use build_fork.yml directly
 EOF
@@ -76,7 +83,8 @@ include() {
     s/MAIN_OR_FORK/$3/g;
     s/JOB_NAME/$4/g;
     s/STYLE_LINT_RUNNER/$5/g;
-  " build.yml.in
+    /^### NB/d
+  " ../build.in.yml
 }
 
 build_yml > build.yml
diff --git a/.github/workflows/nightly_bump_toolchain.yml b/.github/workflows/nightly_bump_toolchain.yml
index ed8cd271a71b1..1d53a2c33a3da 100644
--- a/.github/workflows/nightly_bump_toolchain.yml
+++ b/.github/workflows/nightly_bump_toolchain.yml
@@ -1,9 +1,9 @@
 name: Bump lean-toolchain on nightly-testing
 
 on:
-  workflow_dispatch:
   schedule:
     - cron: '0 10/3 * * *'  # Run every three hours, starting at 11AM CET/2AM PT.
+  workflow_dispatch:
 
 jobs:
   update-toolchain:
diff --git a/.github/workflows/nightly_detect_failure.yml b/.github/workflows/nightly_detect_failure.yml
index e1cfd9147086b..25835698e99ab 100644
--- a/.github/workflows/nightly_detect_failure.yml
+++ b/.github/workflows/nightly_detect_failure.yml
@@ -23,6 +23,7 @@ jobs:
         topic: 'Mathlib status updates'
         content: |
           ❌ The latest CI for Mathlib's branch#nightly-testing has [failed](https://github.com/${{ github.repository }}/actions/runs/${{ github.event.workflow_run.id }}) ([${{ github.sha }}](https://github.com/${{ github.repository }}/commit/${{ github.sha }})).
+          You can `git fetch; git checkout nightly-testing` and push a fix.
 
   handle_success:
     if: ${{ github.event.workflow_run.conclusion == 'success' && github.event.workflow_run.head_branch == 'nightly-testing' }}
@@ -210,26 +211,15 @@ jobs:
             bump_branch_suffix = bump_branch.replace('bump/', '')
             payload = f"🛠️: it looks like it's time to create a new bump/nightly-{current_version} branch from nightly-testing (specifically {sha}), and then PR that to {bump_branch}. "
             payload += "To do so semi-automatically, run the following script from mathlib root:\n\n"
-            payload += f"```bash\n./scripts/create-adaptation-pr.sh {bump_branch_suffix} {current_version}\n```\n"
-            # Only post if the message is different
-            # We compare the first 160 characters, since that includes the date and bump version
-            if not messages or messages[0]['content'][:160] != payload[:160]:
-                # Log messages, because the bot seems to repeat itself...
-                if messages:
-                    print("###### Last message:")
-                    print(messages[0]['content'])
-                    print("###### Current message:")
-                    print(payload)
-                else:
-                    print('The strings match!')
-                # Post the reminder message
-                request = {
-                    'type': 'stream',
-                    'to': 'nightly-testing',
-                    'topic': 'Mathlib bump branch reminders',
-                    'content': payload
-                }
-                result = client.send_message(request)
-                print(result)
+            payload += f"```bash\n./scripts/create-adaptation-pr.sh --bumpversion={bump_branch_suffix} --nightlydate={current_version} --nightlysha={sha}\n```\n"
+            # Post the reminder message
+            request = {
+                'type': 'stream',
+                'to': 'nightly-testing',
+                'topic': 'Mathlib bump branch reminders',
+                'content': payload
+            }
+            result = client.send_message(request)
+            print(result)
         else:
             print('No action needed.')
diff --git a/.github/workflows/nightly_merge_master.yml b/.github/workflows/nightly_merge_master.yml
index 1321c9c54ca3b..d6f2480542294 100644
--- a/.github/workflows/nightly_merge_master.yml
+++ b/.github/workflows/nightly_merge_master.yml
@@ -4,7 +4,8 @@ name: Merge master to nightly
 
 on:
   schedule:
-    - cron: '30 */3 * * *'  # 8AM CET/11PM PT
+    - cron: '30 */3 * * *'  # At minute 30 past every 3rd hour.
+  workflow_dispatch:
 
 jobs:
   merge-to-nightly:
diff --git a/.github/workflows/nolints.yml b/.github/workflows/nolints.yml
index ed9b666b0d7eb..847e24854e855 100644
--- a/.github/workflows/nolints.yml
+++ b/.github/workflows/nolints.yml
@@ -2,7 +2,8 @@ name: update nolints
 
 on:
   schedule:
-    - cron: "0 0 * * 0"
+    - cron: "0 0 * * 0" # At 00:00 UTC on Sunday.
+  workflow_dispatch:
 
 jobs:
   build:
@@ -25,6 +26,9 @@ jobs:
           echo "$HOME/.elan/bin" >> "${GITHUB_PATH}"
 
       - uses: actions/checkout@v4
+        with:
+          ## fetch the whole repository, as we want to push to it later
+          fetch-depth: 0
 
       - name: print lean and lake versions
         run: |
@@ -53,15 +57,14 @@ jobs:
           run: |
             bash -o pipefail -c "env LEAN_ABORT_ON_PANIC=1 lake build --wfail -KCI"
 
-      - name: update nolints.json and style-exceptions.txt
+      - name: update nolints.json
         shell: bash
         run: |
           env LEAN_ABORT_ON_PANIC=1 lake exe runLinter --update Mathlib
-          lake exe lint-style --update
 
       - name: configure git setup
         run: |
-          git remote add origin-bot "https://leanprover-community-bot:${{ secrets.UPDATE_NOLINTS_TOKEN }}@github.com/leanprover-community/mathlib.git"
+          git remote add origin-bot "https://leanprover-community-bot:${{ secrets.UPDATE_NOLINTS_TOKEN }}@github.com/leanprover-community/mathlib4.git"
           git config user.email "leanprover.community@gmail.com"
           git config user.name "leanprover-community-bot"
 
@@ -70,7 +73,7 @@ jobs:
           # github using a different username.
           git config --unset http.https://github.com/.extraheader
 
-      - name: file a new PR to update nolints.json and style-exceptions.txt
+      - name: file a new PR to update nolints.json
         run: ./scripts/update_nolints_CI.sh
         env:
           DEPLOY_GITHUB_TOKEN: ${{ secrets.UPDATE_NOLINTS_TOKEN }}
diff --git a/.github/workflows/technical_debt_metrics.yml b/.github/workflows/technical_debt_metrics.yml
index d7055c85e59ae..2a83b1d682cec 100644
--- a/.github/workflows/technical_debt_metrics.yml
+++ b/.github/workflows/technical_debt_metrics.yml
@@ -2,7 +2,8 @@ name: Weekly Technical Debt Counters
 
 on:
   schedule:
-    - cron: '0 4 * * 1'  # Run at 04:00 every Monday
+    - cron: '0 4 * * 1'  # Run at 04:00 UTC every Monday
+  workflow_dispatch:
 
 jobs:
   run-script:
diff --git a/.github/workflows/update_dependencies.yml b/.github/workflows/update_dependencies.yml
index 885bef03a2be2..b415c88c6a8c8 100644
--- a/.github/workflows/update_dependencies.yml
+++ b/.github/workflows/update_dependencies.yml
@@ -3,6 +3,7 @@ name: Update Mathlib Dependencies
 on:
   schedule:
     - cron: '0 * * * *'  # This will run every hour
+  workflow_dispatch:
 
 jobs:
   update-dependencies:
@@ -19,25 +20,46 @@ jobs:
         uses: actions/checkout@v4
         with:
           fetch-depth: 0
-          token: "${{ secrets.NIGHTLY_TESTING }}"
+          token: "${{ secrets.UPDATE_DEPENDENCIES_TOKEN }}"
+
+      - name: Get sha of branch
+        id: sha
+        run: |
+          echo "sha=$(git rev-parse --verify origin/update-dependencies-bot-use-only)" >> "$GITHUB_OUTPUT"
+
+      - name: Get PR and labels
+        if: ${{ steps.sha.outputs.sha }}
+        id: PR # all the steps below are skipped if 'ready-to-merge' is in the list of labels found here
+        uses: 8BitJonny/gh-get-current-pr@3.0.0
+        # TODO: this may not work properly if the same commit is pushed to multiple branches:
+        # https://github.com/8BitJonny/gh-get-current-pr/issues/8
+        with:
+          github-token: ${{ secrets.GITHUB_TOKEN }}
+          sha: ${{ steps.sha.outputs.sha }}
+          # Only return if PR is still open
+          filterOutClosed: true
 
       - name: Configure Git User
+        if: ${{ !contains(steps.PR.outputs.pr_labels, 'ready-to-merge') }}
         run: |
           git config user.name "leanprover-community-mathlib4-bot"
           git config user.email "leanprover-community-mathlib4-bot@users.noreply.github.com"
 
       - name: Update dependencies
+        if: ${{ !contains(steps.PR.outputs.pr_labels, 'ready-to-merge') }}
         run: lake update
 
       - name: Generate PR title
+        if: ${{ !contains(steps.PR.outputs.pr_labels, 'ready-to-merge') }}
         run: |
           echo "timestamp=$(date -u +"%Y-%m-%d-%H-%M")" >> "$GITHUB_ENV"
           echo "pr_title=chore: update Mathlib dependencies $(date -u +"%Y-%m-%d")" >> "$GITHUB_ENV"
 
       - name: Create Pull Request
+        if: ${{ !contains(steps.PR.outputs.pr_labels, 'ready-to-merge') }}
         uses: peter-evans/create-pull-request@v6
         with:
-          token: "${{ secrets.NIGHTLY_TESTING }}"
+          token: "${{ secrets.UPDATE_DEPENDENCIES_TOKEN }}"
           commit-message: "chore: update Mathlib dependencies ${{ env.timestamp }}"
           # this branch is referenced in update_dependencies_zulip.yml
           branch: "update-dependencies-bot-use-only"
diff --git a/.github/workflows/update_dependencies_zulip.yml b/.github/workflows/update_dependencies_zulip.yml
index 466b06668f31f..f4e72b01d7034 100644
--- a/.github/workflows/update_dependencies_zulip.yml
+++ b/.github/workflows/update_dependencies_zulip.yml
@@ -1,7 +1,5 @@
 name: Monitor Dependency Update Failures
 
-# This action currently uses the NIGHTLY_TESTING secret, but could be moved to a separate secret.
-
 on:
   workflow_run:
     workflows: ["continuous integration"]
@@ -20,7 +18,7 @@ jobs:
         uses: actions/github-script@v7
         id: construct_message
         with:
-          github-token: ${{ secrets.NIGHTLY_TESTING }}
+          github-token: ${{ secrets.UPDATE_DEPENDENCIES_TOKEN }}
           result-encoding: string
           script: |
             const owner = context.repo.owner, repo = context.repo.repo;
@@ -42,7 +40,7 @@ jobs:
                 });
               }
             } else {
-              output += "No PR found for this run!";
+              output += "No PR found for this run! If you are feeling impatient and have write access, please go to the following page and click the "Run workflow" button!\nhttps://github.com/leanprover-community/mathlib4/actions/workflows/update_dependencies.yml";
             }
             return output;
 
diff --git a/.vscode/deprecated-alias.code-snippets b/.vscode/deprecated.code-snippets
similarity index 53%
rename from .vscode/deprecated-alias.code-snippets
rename to .vscode/deprecated.code-snippets
index c760b46045763..8be8e537cd5f0 100644
--- a/.vscode/deprecated-alias.code-snippets
+++ b/.vscode/deprecated.code-snippets
@@ -1,4 +1,11 @@
 {
+	"Deprecation for mathlib": {
+		"scope": "lean4",
+		"prefix": "deprecated",
+		"body": [
+			"@[deprecated $1 (since := \"${CURRENT_YEAR}-${CURRENT_MONTH}-${CURRENT_DATE}\")]"
+		]
+	},
 	"Deprecated alias for mathlib": {
 		"scope": "lean4",
 		"prefix": "deprecated alias",
diff --git a/Archive/Arithcc.lean b/Archive/Arithcc.lean
index 45f394d134fee..7f9601849cfa4 100644
--- a/Archive/Arithcc.lean
+++ b/Archive/Arithcc.lean
@@ -3,7 +3,6 @@ Copyright (c) 2020 Xi Wang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Xi Wang
 -/
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Data.Nat.Defs
 import Mathlib.Order.Basic
 import Mathlib.Tactic.Common
@@ -145,9 +144,9 @@ def outcome : List Instruction → State → State
 @[simp]
 theorem outcome_append (p₁ p₂ : List Instruction) (η : State) :
     outcome (p₁ ++ p₂) η = outcome p₂ (outcome p₁ η) := by
-  revert η
-  induction' p₁ with _ _ p₁_ih <;> intros <;> simp
-  apply p₁_ih
+  induction p₁ generalizing η with
+  | nil => simp
+  | cons _ _ p₁_ih => simp [p₁_ih]
 
 end Target
 
@@ -280,10 +279,7 @@ theorem compiler_correctness
   | const => simp [StateEq, step]; rfl
   -- 5.II
   | var =>
-    simp [hmap, StateEq, step] -- Porting note: was `finish [hmap, StateEq, step]`
-    constructor
-    · simp_all only [read, loc]
-    · rfl
+    simp_all [StateEq, StateEqRs, step]
   -- 5.III
   | sum =>
     rename_i e_s₁ e_s₂ e_ih_s₁ e_ih_s₂
diff --git a/Archive/Examples/IfNormalization/Statement.lean b/Archive/Examples/IfNormalization/Statement.lean
index dd642ec3af911..283f4eabf91e3 100644
--- a/Archive/Examples/IfNormalization/Statement.lean
+++ b/Archive/Examples/IfNormalization/Statement.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Lean FRO LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 
 /-!
diff --git a/Archive/Examples/IfNormalization/WithoutAesop.lean b/Archive/Examples/IfNormalization/WithoutAesop.lean
index 98e57cfe1575c..b3e94e0a5653e 100644
--- a/Archive/Examples/IfNormalization/WithoutAesop.lean
+++ b/Archive/Examples/IfNormalization/WithoutAesop.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Scott Morrison
+Authors: Chris Hughes, Kim Morrison
 -/
 import Archive.Examples.IfNormalization.Statement
 import Mathlib.Algebra.Order.Monoid.Canonical.Defs
@@ -92,22 +92,20 @@ def normalize' (l : AList (fun _ : ℕ => Bool)) :
               · simp_all
         · have := ht₃ v
           have := he₃ v
-          simp_all? says simp_all only [normalized, Bool.and_eq_true, Bool.not_eq_true',
-            AList.lookup_insert_eq_none, ne_eq, AList.lookup_insert, imp_false]
+          simp_all? says simp_all only [normalized, Bool.and_eq_true, Bool.not_eq_eq_eq_not,
+              Bool.not_true, AList.lookup_insert_eq_none, ne_eq, AList.lookup_insert]
           obtain ⟨⟨⟨tn, tc⟩, tr⟩, td⟩ := ht₂
           split <;> rename_i h'
           · subst h'
             simp_all
-          · simp_all? says simp_all only [hasNestedIf, Bool.or_self, hasConstantIf, and_self,
-              hasRedundantIf, Bool.or_false, beq_eq_false_iff_ne, ne_eq, not_false_eq_true,
-              disjoint, List.disjoint, decide_True, Bool.and_self]
+          · simp_all
         · have := ht₃ w
           have := he₃ w
           by_cases h : w = v
           · subst h; simp_all
-          · simp_all? says simp_all only [normalized, Bool.and_eq_true, Bool.not_eq_true',
-              AList.lookup_insert_eq_none, ne_eq, not_false_eq_true, AList.lookup_insert_ne,
-              implies_true]
+          · simp_all? says simp_all only [normalized, Bool.and_eq_true, Bool.not_eq_eq_eq_not,
+              Bool.not_true, AList.lookup_insert_eq_none, ne_eq, not_false_eq_true,
+              AList.lookup_insert_ne, implies_true]
             obtain ⟨⟨⟨en, ec⟩, er⟩, ed⟩ := he₂
             split at b <;> rename_i h'
             · subst h'; simp_all
diff --git a/Archive/Examples/MersennePrimes.lean b/Archive/Examples/MersennePrimes.lean
index e10d955eb51ee..a19f23f8431d4 100644
--- a/Archive/Examples/MersennePrimes.lean
+++ b/Archive/Examples/MersennePrimes.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.NumberTheory.LucasLehmer
 
diff --git a/Archive/Hairer.lean b/Archive/Hairer.lean
index acd326c0b13bf..e9b91c32132a8 100644
--- a/Archive/Hairer.lean
+++ b/Archive/Hairer.lean
@@ -100,7 +100,7 @@ lemma inj_L : Injective (L ι) :=
         fun g hg _h2g g_supp ↦ by
           simpa [mul_comm (g _), L] using congr($hp ⟨g, g_supp.trans ball_subset_closedBall, hg⟩)
     simp_rw [MvPolynomial.funext_iff, map_zero]
-    refine fun x ↦ AnalyticOn.eval_linearMap (EuclideanSpace.equiv ι ℝ).toLinearMap p
+    refine fun x ↦ AnalyticOnNhd.eval_linearMap (EuclideanSpace.equiv ι ℝ).toLinearMap p
       |>.eqOn_zero_of_preconnected_of_eventuallyEq_zero
       (preconnectedSpace_iff_univ.mp inferInstance) (z₀ := 0) trivial
       (Filter.mem_of_superset (Metric.ball_mem_nhds 0 zero_lt_one) ?_) trivial
diff --git a/Archive/Imo/Imo1962Q1.lean b/Archive/Imo/Imo1962Q1.lean
index 169a0d8dcaf98..d151c6df95fc3 100644
--- a/Archive/Imo/Imo1962Q1.lean
+++ b/Archive/Imo/Imo1962Q1.lean
@@ -32,110 +32,104 @@ def ProblemPredicate (n : ℕ) : Prop :=
 First, it's inconvenient to work with digits, so let's simplify them out of the problem.
 -/
 
-
 abbrev ProblemPredicate' (c n : ℕ) : Prop :=
   n = 10 * c + 6 ∧ 6 * 10 ^ (digits 10 c).length + c = 4 * n
 
-theorem without_digits {n : ℕ} (h1 : ProblemPredicate n) : ∃ c : ℕ, ProblemPredicate' c n := by
+lemma without_digits {n : ℕ} (hn : ProblemPredicate n) : ∃ c : ℕ, ProblemPredicate' c n := by
   use n / 10
   cases' n with n
-  · have h2 : ¬ProblemPredicate 0 := by norm_num [ProblemPredicate]
+  · have hpp : ¬ProblemPredicate 0 := by norm_num [ProblemPredicate]
     contradiction
   · rw [ProblemPredicate, digits_def' (by decide : 2 ≤ 10) n.succ_pos, List.headI, List.tail_cons,
-      List.concat_eq_append] at h1
+      List.concat_eq_append] at hn
     constructor
-    · rw [← h1.left, div_add_mod (n + 1) 10]
-    · rw [← h1.right, ofDigits_append, ofDigits_digits, ofDigits_singleton, add_comm, mul_comm]
+    · rw [← hn.left, div_add_mod (n + 1) 10]
+    · rw [← hn.right, ofDigits_append, ofDigits_digits, ofDigits_singleton, add_comm, mul_comm]
 
 /-!
 Now we can eliminate possibilities for `(digits 10 c).length` until we get to the one that works.
 -/
 
-
-theorem case_0_digit {c n : ℕ} (h1 : (digits 10 c).length = 0) : ¬ProblemPredicate' c n := by
-  intro h2
-  have h3 : 6 * 10 ^ 0 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 0 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
+lemma case_0_digits {c n : ℕ} (hc : (digits 10 c).length = 0) : ¬ProblemPredicate' c n := by
+  intro hpp
+  have hpow : 6 * 10 ^ 0 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 0 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
   linarith
 
-theorem case_1_digit {c n : ℕ} (h1 : (digits 10 c).length = 1) : ¬ProblemPredicate' c n := by
-  intro h2
-  have h3 : 6 * 10 ^ 1 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 1 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
-  have h6 : c > 0 := by linarith
+lemma case_1_digits {c n : ℕ} (hc : (digits 10 c).length = 1) : ¬ProblemPredicate' c n := by
+  intro hpp
+  have hpow : 6 * 10 ^ 1 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 1 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
+  have hpos : c > 0 := by linarith
   linarith
 
-theorem case_2_digit {c n : ℕ} (h1 : (digits 10 c).length = 2) : ¬ProblemPredicate' c n := by
-  intro h2
-  have h3 : 6 * 10 ^ 2 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 2 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
-  have h5 : c > 14 := by linarith
+lemma case_2_digits {c n : ℕ} (hc : (digits 10 c).length = 2) : ¬ProblemPredicate' c n := by
+  intro hpp
+  have hpow : 6 * 10 ^ 2 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 2 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
+  have hgt : c > 14 := by linarith
   linarith
 
-theorem case_3_digit {c n : ℕ} (h1 : (digits 10 c).length = 3) : ¬ProblemPredicate' c n := by
-  intro h2
-  have h3 : 6 * 10 ^ 3 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 3 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
-  have h5 : c > 153 := by linarith
+lemma case_3_digits {c n : ℕ} (hc : (digits 10 c).length = 3) : ¬ProblemPredicate' c n := by
+  intro hpp
+  have hpow : 6 * 10 ^ 3 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 3 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
+  have hgt : c > 153 := by linarith
   linarith
 
-theorem case_4_digit {c n : ℕ} (h1 : (digits 10 c).length = 4) : ¬ProblemPredicate' c n := by
-  intro h2
-  have h3 : 6 * 10 ^ 4 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 4 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
-  have h5 : c > 1537 := by linarith
+lemma case_4_digits {c n : ℕ} (hc : (digits 10 c).length = 4) : ¬ProblemPredicate' c n := by
+  intro hpp
+  have hpow : 6 * 10 ^ 4 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 4 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
+  have hgt : c > 1537 := by linarith
   linarith
 
 /-- Putting this inline causes a deep recursion error, so we separate it out. -/
-theorem helper_5_digit {c : ℤ} (h : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)) : c = 15384 := by linarith
+private lemma helper_5_digits {c : ℤ} (hc : 6 * 10 ^ 5 + c = 4 * (10 * c + 6)) : c = 15384 := by
+  linarith
 
-theorem case_5_digit {c n : ℕ} (h1 : (digits 10 c).length = 5) (h2 : ProblemPredicate' c n) :
+lemma case_5_digits {c n : ℕ} (hc : (digits 10 c).length = 5) (hpp : ProblemPredicate' c n) :
     c = 15384 := by
-  have h3 : 6 * 10 ^ 5 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [h1]
-  have h4 : 6 * 10 ^ 5 + c = 4 * (10 * c + 6) := by rw [h3, h2.right, h2.left]
+  have hpow : 6 * 10 ^ 5 + c = 6 * 10 ^ (digits 10 c).length + c := by rw [hc]
+  have hmul : 6 * 10 ^ 5 + c = 4 * (10 * c + 6) := by rw [hpow, hpp.right, hpp.left]
   zify at *
-  exact helper_5_digit h4
+  exact helper_5_digits hmul
 
 /-- `linarith` fails on numbers this large, so this lemma spells out some of the arithmetic
 that normally would be automated.
 -/
-theorem case_more_digits {c n : ℕ} (h1 : (digits 10 c).length ≥ 6) (h2 : ProblemPredicate' c n) :
+lemma case_more_digits {c n : ℕ} (hc : (digits 10 c).length ≥ 6) (hpp : ProblemPredicate' c n) :
     n ≥ 153846 := by
-  have h3 : c ≠ 0 := by
-    intro h4
-    have h5 : (digits 10 c).length = 0 := by simp [h4]
-    exact case_0_digit h5 h2
+  have hnz : c ≠ 0 := by
+    intro hc0
+    have hcl : (digits 10 c).length = 0 := by simp [hc0]
+    exact case_0_digits hcl hpp
   calc
-    n ≥ 10 * c := le.intro h2.left.symm
-    _ ≥ 10 ^ (digits 10 c).length := base_pow_length_digits_le 10 c (by decide) h3
-    _ ≥ 10 ^ 6 := pow_le_pow_right (by decide) h1
+    n ≥ 10 * c := le.intro hpp.left.symm
+    _ ≥ 10 ^ (digits 10 c).length := base_pow_length_digits_le 10 c (by decide) hnz
+    _ ≥ 10 ^ 6 := pow_right_mono₀ (by decide) hc
     _ ≥ 153846 := by norm_num
 
 /-!
 Now we combine these cases to show that 153846 is the smallest solution.
 -/
 
-
-theorem satisfied_by_153846 : ProblemPredicate 153846 := by
+lemma satisfied_by_153846 : ProblemPredicate 153846 := by
   norm_num [ProblemPredicate]
   decide
 
-theorem no_smaller_solutions (n : ℕ) (h1 : ProblemPredicate n) : n ≥ 153846 := by
-  have ⟨c, h2⟩ := without_digits h1
-  have h3 : (digits 10 c).length < 6 ∨ (digits 10 c).length ≥ 6 := by apply lt_or_ge
-  cases h3 with
-  | inr h3 => exact case_more_digits h3 h2
-  | inl h3 =>
-    interval_cases h : (digits 10 c).length
-    · exfalso; exact case_0_digit h h2
-    · exfalso; exact case_1_digit h h2
-    · exfalso; exact case_2_digit h h2
-    · exfalso; exact case_3_digit h h2
-    · exfalso; exact case_4_digit h h2
-    · have h4 : c = 15384 := case_5_digit h h2
-      have h5 : n = 10 * 15384 + 6 := h4 ▸ h2.left
-      norm_num at h5
-      exact h5.ge
+lemma no_smaller_solutions (n : ℕ) (hn : ProblemPredicate n) : n ≥ 153846 := by
+  have ⟨c, hcn⟩ := without_digits hn
+  cases lt_or_ge (digits 10 c).length 6 with
+  | inl =>
+    interval_cases hc : (digits 10 c).length
+    · exfalso; exact case_0_digits hc hcn
+    · exfalso; exact case_1_digits hc hcn
+    · exfalso; exact case_2_digits hc hcn
+    · exfalso; exact case_3_digits hc hcn
+    · exfalso; exact case_4_digits hc hcn
+    · exact (case_5_digits hc hcn ▸ hcn.left).ge
+  | inr hge => exact case_more_digits hge hcn
 
 end Imo1962Q1
 
diff --git a/Archive/Imo/Imo1972Q5.lean b/Archive/Imo/Imo1972Q5.lean
index f96965f9d6a75..ba2e4d8c6a0a0 100644
--- a/Archive/Imo/Imo1972Q5.lean
+++ b/Archive/Imo/Imo1972Q5.lean
@@ -50,7 +50,7 @@ theorem imo1972_q5 (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x - y
       calc
         0 < ‖f x‖ := norm_pos_iff.mpr hx
         _ ≤ k := hk₁ x
-    rw [div_lt_iff]
+    rw [div_lt_iff₀]
     · apply lt_mul_of_one_lt_right h₁ hneg
     · exact zero_lt_one.trans hneg
   -- Demonstrate that `k ≤ k'` using `hk₂`.
@@ -87,7 +87,7 @@ theorem imo1972_q5' (f g : ℝ → ℝ) (hf1 : ∀ x, ∀ y, f (x + y) + f (x -
   have h : ∀ x, ‖f x‖ ≤ k := le_ciSup hf2
   have hgy : 0 < ‖g y‖ := by linarith
   have k_pos : 0 < k := lt_of_lt_of_le (norm_pos_iff.mpr hx) (h x)
-  have : k / ‖g y‖ < k := (div_lt_iff hgy).mpr (lt_mul_of_one_lt_right k_pos H)
+  have : k / ‖g y‖ < k := (div_lt_iff₀ hgy).mpr (lt_mul_of_one_lt_right k_pos H)
   have : k ≤ k / ‖g y‖ := by
     suffices ∀ x, ‖f x‖ ≤ k / ‖g y‖ from ciSup_le this
     intro x
diff --git a/Archive/Imo/Imo1986Q5.lean b/Archive/Imo/Imo1986Q5.lean
index 6789efcbb837d..82fe5961c1647 100644
--- a/Archive/Imo/Imo1986Q5.lean
+++ b/Archive/Imo/Imo1986Q5.lean
@@ -54,7 +54,7 @@ theorem map_of_lt_two (hx : x < 2) : f x = 2 / (2 - x) := by
   have hx' : 0 < 2 - x := tsub_pos_of_lt hx
   have hfx : f x ≠ 0 := hf.map_ne_zero_iff.2 hx
   apply le_antisymm
-  · rw [le_div_iff₀ hx', ← NNReal.le_div_iff' hfx, tsub_le_iff_right, ← hf.map_eq_zero,
+  · rw [le_div_iff₀ hx', ← le_div_iff₀' hfx.bot_lt, tsub_le_iff_right, ← hf.map_eq_zero,
      hf.map_add, div_mul_cancel₀ _ hfx, hf.map_two, zero_mul]
   · rw [div_le_iff₀' hx', ← hf.map_eq_zero]
     refine (mul_eq_zero.1 ?_).resolve_right hfx
diff --git a/Archive/Imo/Imo1987Q1.lean b/Archive/Imo/Imo1987Q1.lean
index 717900655234a..d10995da9a7a4 100644
--- a/Archive/Imo/Imo1987Q1.lean
+++ b/Archive/Imo/Imo1987Q1.lean
@@ -46,8 +46,10 @@ def fixedPointsEquiv : { σx : α × Perm α // σx.2 σx.1 = σx.1 } ≃ Σ x :
 
 theorem card_fixed_points :
     card { σx : α × Perm α // σx.2 σx.1 = σx.1 } = card α * (card α - 1)! := by
-  simp [card_congr (fixedPointsEquiv α), card_perm, Finset.filter_not, Finset.card_sdiff,
-    Finset.filter_eq', Finset.card_univ]
+  simp only [card_congr (fixedPointsEquiv α), card_sigma, card_perm]
+  have (x) : ({x}ᶜ : Set α) = Finset.filter (· ≠ x) Finset.univ := by
+    ext; simp
+  simp [this]
 
 /-- Given `α : Type*` and `k : ℕ`, `fiber α k` is the set of permutations of `α` with exactly `k`
 fixed points. -/
diff --git a/Archive/Imo/Imo1994Q1.lean b/Archive/Imo/Imo1994Q1.lean
index e8d62532d713d..6ea9e04e8862f 100644
--- a/Archive/Imo/Imo1994Q1.lean
+++ b/Archive/Imo/Imo1994Q1.lean
@@ -77,7 +77,7 @@ theorem imo1994_q1 (n : ℕ) (m : ℕ) (A : Finset ℕ) (hm : A.card = m + 1)
   have hf : map f (Icc 0 k) ⊆ map a.toEmbedding (Ioc (rev k) (Fin.last m)) := by
     intro x hx
     simp only [Equiv.subLeft_apply, a, rev] at h
-    simp only [mem_map, mem_Icc, mem_Ioc, Fin.zero_le, true_and_iff, Equiv.subLeft_apply,
+    simp only [mem_map, mem_Icc, mem_Ioc, Fin.zero_le, true_and, Equiv.subLeft_apply,
       Function.Embedding.coeFn_mk, exists_prop, RelEmbedding.coe_toEmbedding, f, rev] at hx ⊢
     rcases hx with ⟨i, ⟨hi, rfl⟩⟩
     have h1 : a i + a (Fin.last m - k) ≤ n := by unfold_let; linarith only [h, a.monotone hi]
diff --git a/Archive/Imo/Imo1998Q2.lean b/Archive/Imo/Imo1998Q2.lean
index ac4e0dfb0f196..367aec35fd5fc 100644
--- a/Archive/Imo/Imo1998Q2.lean
+++ b/Archive/Imo/Imo1998Q2.lean
@@ -104,9 +104,9 @@ theorem A_fibre_over_contestant (c : C) :
     (Finset.univ.filter fun p : JudgePair J => p.Agree r c ∧ p.Distinct) =
       ((A r).filter fun a : AgreedTriple C J => a.contestant = c).image Prod.snd := by
   ext p
-  simp only [A, Finset.mem_univ, Finset.mem_filter, Finset.mem_image, true_and_iff, exists_prop]
+  simp only [A, Finset.mem_univ, Finset.mem_filter, Finset.mem_image, exists_prop]
   constructor
-  · rintro ⟨h₁, h₂⟩; refine ⟨(c, p), ?_⟩; tauto
+  · rintro ⟨_, h₂⟩; refine ⟨(c, p), ?_⟩; tauto
   · intro h; aesop
 
 theorem A_fibre_over_contestant_card (c : C) :
diff --git a/Archive/Imo/Imo2006Q5.lean b/Archive/Imo/Imo2006Q5.lean
index 0e2398d7919ba..d317993a8c618 100644
--- a/Archive/Imo/Imo2006Q5.lean
+++ b/Archive/Imo/Imo2006Q5.lean
@@ -122,7 +122,7 @@ theorem Polynomial.iterate_comp_sub_X_ne {P : Polynomial ℤ} (hP : 1 < P.natDeg
     (hk : 0 < k) : P.comp^[k] X - X ≠ 0 := by
   rw [sub_ne_zero]
   apply_fun natDegree
-  simpa using (one_lt_pow hP hk.ne').ne'
+  simpa using (one_lt_pow₀ hP hk.ne').ne'
 
 /-- We solve the problem for the specific case k = 2 first. -/
 theorem imo2006_q5' {P : Polynomial ℤ} (hP : 1 < P.natDegree) :
diff --git a/Archive/Imo/Imo2011Q5.lean b/Archive/Imo/Imo2011Q5.lean
index 8d7639aea087f..2fdc4ed7d8394 100644
--- a/Archive/Imo/Imo2011Q5.lean
+++ b/Archive/Imo/Imo2011Q5.lean
@@ -12,9 +12,9 @@ import Mathlib.Algebra.Ring.Int
 
 Let `f` be a function from the set of integers to the set
 of positive integers.  Suppose that, for any two integers
-`m` and `n`, the difference `f(m) - f(n)` is divisible by
-`f(m - n)`.  Prove that, for all integers `m` and `n` with
-`f(m) ≤ f(n)`, the number `f(n)` is divisible by `f(m)`.
+`m` and `n`, the difference `f m - f n` is divisible by
+`f (m - n)`.  Prove that, for all integers `m` and `n` with
+`f m ≤ f n`, the number `f n` is divisible by `f m`.
 -/
 
 
diff --git a/Archive/Imo/Imo2013Q5.lean b/Archive/Imo/Imo2013Q5.lean
index 20cccc01e5ed8..225e64ae35b02 100644
--- a/Archive/Imo/Imo2013Q5.lean
+++ b/Archive/Imo/Imo2013Q5.lean
@@ -38,9 +38,9 @@ theorem le_of_all_pow_lt_succ {x y : ℝ} (hx : 1 < x) (hy : 1 < y)
     have hterm : ∀ i : ℕ, i ∈ Finset.range n → 1 ≤ x ^ i * y ^ (n - 1 - i) := by
       intro i _
       calc
-        1 ≤ x ^ i := one_le_pow_of_one_le hx.le i
+        1 ≤ x ^ i := one_le_pow₀ hx.le
         _ = x ^ i * 1 := by ring
-        _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow_of_one_le hy.le
+        _ ≤ x ^ i * y ^ (n - 1 - i) := by gcongr; apply one_le_pow₀ hy.le
     calc
       (x - y) * (n : ℝ) = (n : ℝ) * (x - y) := by ring
       _ = (∑ _i ∈ Finset.range n, (1 : ℝ)) * (x - y) := by
@@ -58,7 +58,7 @@ theorem le_of_all_pow_lt_succ {x y : ℝ} (hx : 1 < x) (hy : 1 < y)
       _ ≤ x ^ N - y ^ N := hn N hNp
   linarith [h N hNp]
 
-/-- Like `le_of_all_pow_lt_succ`, but with a weaker assumption for y.
+/-- Like `le_of_all_pow_lt_succ`, but with a weaker assumption for `y`.
 -/
 theorem le_of_all_pow_lt_succ' {x y : ℝ} (hx : 1 < x) (hy : 0 < y)
     (h : ∀ n : ℕ, 0 < n → x ^ n - 1 < y ^ n) : x ≤ y := by
@@ -134,7 +134,7 @@ theorem fixed_point_of_pos_nat_pow {f : ℚ → ℝ} {n : ℕ} (hn : 0 < n)
     (H1 : ∀ x y, 0 < x → 0 < y → f (x * y) ≤ f x * f y) (H4 : ∀ n : ℕ, 0 < n → (n : ℝ) ≤ f n)
     (H5 : ∀ x : ℚ, 1 < x → (x : ℝ) ≤ f x) {a : ℚ} (ha1 : 1 < a) (hae : f a = a) :
     f (a ^ n) = a ^ n := by
-  have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow ha1 hn.ne')
+  have hh0 : (a : ℝ) ^ n ≤ f (a ^ n) := mod_cast H5 (a ^ n) (one_lt_pow₀ ha1 hn.ne')
   have hh1 :=
     calc
       f (a ^ n) ≤ f a ^ n := pow_f_le_f_pow hn ha1 H1 H4
@@ -206,7 +206,7 @@ theorem imo2013_q5 (f : ℚ → ℝ) (H1 : ∀ x y, 0 < x → 0 < y → f (x * y
       intro n hn
       calc
         (x : ℝ) ^ n - 1 < f (x ^ n) :=
-            mod_cast fx_gt_xm1 (one_le_pow_of_one_le hx.le n) H1 H2 H4
+            mod_cast fx_gt_xm1 (one_le_pow₀ hx.le) H1 H2 H4
         _ ≤ f x ^ n := pow_f_le_f_pow hn hx H1 H4
     have hx' : 1 < (x : ℝ) := mod_cast hx
     have hxp : 0 < x := by positivity
@@ -234,15 +234,14 @@ theorem imo2013_q5 (f : ℚ → ℝ) (H1 : ∀ x y, 0 < x → 0 < y → f (x * y
   have H : x * (↑(2 * x.den) : ℚ) = (↑(2 * x.num) : ℚ) := by push_cast; linear_combination 2 * H₀
   set x2denom := 2 * x.den
   set x2num := 2 * x.num
-  have := x.pos
   have hx2pos : 0 < 2 * x.den := by positivity
   have hx2cnezr : (x2denom : ℝ) ≠ (0 : ℝ) := by positivity
   have : 0 < x.num := by rwa [Rat.num_pos]
   have hx2num_gt_one : (1 : ℚ) < (2 * x.num : ℤ) := by norm_cast; linarith
   apply mul_left_cancel₀ hx2cnezr
   calc
-    x2denom * f x = f (x2denom * x) :=
-        (h_f_commutes_with_pos_nat_mul x2denom hx2pos x hx).symm
+    x2denom * f x
+      = f (x2denom * x) := (h_f_commutes_with_pos_nat_mul x2denom hx2pos x hx).symm
     _ = f x2num := by congr; linear_combination H
     _ = x2num := fixed_point_of_gt_1 hx2num_gt_one H1 H2 H4 H5 ha1 hae
     _ = ((x2num : ℚ) : ℝ) := by norm_cast
diff --git a/Archive/Imo/Imo2019Q2.lean b/Archive/Imo/Imo2019Q2.lean
index db1650a221391..86353bd5b6d91 100644
--- a/Archive/Imo/Imo2019Q2.lean
+++ b/Archive/Imo/Imo2019Q2.lean
@@ -57,7 +57,7 @@ rather than more literally with `affineSegment`.
 -/
 
 
-open Affine Affine.Simplex EuclideanGeometry FiniteDimensional
+open Affine Affine.Simplex EuclideanGeometry Module
 
 open scoped Affine EuclideanGeometry Real
 
diff --git a/Archive/Imo/Imo2024Q1.lean b/Archive/Imo/Imo2024Q1.lean
index 0e94469d1e0d9..00575c6abc94a 100644
--- a/Archive/Imo/Imo2024Q1.lean
+++ b/Archive/Imo/Imo2024Q1.lean
@@ -71,7 +71,7 @@ lemma mem_Ico_one_of_mem_Ioo (h : α ∈ Set.Ioo 0 2) : α ∈ Set.Ico 1 2 := by
   by_contra! hn
   have hr : 1 < ⌈α⁻¹⌉₊ := by
     rw [Nat.lt_ceil]
-    exact_mod_cast one_lt_inv h0 hn
+    exact_mod_cast (one_lt_inv₀ h0).2 hn
   apply hr.ne'
   suffices ⌈α⁻¹⌉₊ = (1 : ℤ) from mod_cast this
   apply Int.eq_one_of_dvd_one (Int.zero_le_ofNat _)
@@ -153,7 +153,7 @@ lemma not_condition_of_mem_Ioo {α : ℝ} (h : α ∈ Set.Ioo 0 2) : ¬Condition
     convert hna using 1
     field_simp
   rw [sub_eq_add_neg, ← le_sub_iff_add_le', neg_le, neg_sub] at hna'
-  rw [le_inv (by linarith) (mod_cast hn), ← not_lt] at hna'
+  rw [le_inv_comm₀ (by linarith) (mod_cast hn), ← not_lt] at hna'
   apply hna'
   exact_mod_cast Nat.lt_floor_add_one (_ : ℝ)
 
diff --git a/Archive/MiuLanguage/DecisionNec.lean b/Archive/MiuLanguage/DecisionNec.lean
index ba3cc0bc34669..694b599ac899d 100644
--- a/Archive/MiuLanguage/DecisionNec.lean
+++ b/Archive/MiuLanguage/DecisionNec.lean
@@ -67,18 +67,24 @@ is 1 or 2 modulo 3.
 theorem count_equiv_one_or_two_mod3_of_derivable (en : Miustr) :
     Derivable en → count I en % 3 = 1 ∨ count I en % 3 = 2 := by
   intro h
-  induction' h with _ _ h_ih _ _ h_ih _ _ _ h_ih _ _ _ h_ih
-  · left; rfl
-  any_goals apply mod3_eq_1_or_mod3_eq_2 h_ih
-  -- Porting note: `simp_rw [count_append]` usually doesn't work
-  · left; rw [count_append, count_append]; rfl
-  · right; simp_rw [count_append, count_cons, beq_iff_eq, ite_false, add_zero, two_mul]
-  · left; rw [count_append, count_append, count_append]
-    simp_rw [count_cons_self, count_nil, count_cons, beq_iff_eq, ite_false, add_right_comm,
-      add_mod_right]
+  induction h with
+  | mk => left; rfl
+  | r1 _ h_ih =>
+    apply mod3_eq_1_or_mod3_eq_2 h_ih; left
+    rw [count_append, count_append]; rfl
+  | r2 _ h_ih =>
+    apply mod3_eq_1_or_mod3_eq_2 h_ih; right
+    simp_rw [count_append, count_cons, beq_iff_eq, reduceCtorEq, ite_false, add_zero, two_mul]
+  | r3 _ h_ih =>
+    apply mod3_eq_1_or_mod3_eq_2 h_ih; left
+    rw [count_append, count_append, count_append]
+    simp_rw [count_cons_self, count_nil, count_cons, beq_iff_eq, reduceCtorEq, ite_false,
+      add_right_comm, add_mod_right]
     simp
-  · left; rw [count_append, count_append, count_append]
-    simp only [ne_eq, not_false_eq_true, count_cons_of_ne, count_nil, add_zero]
+  | r4 _ h_ih =>
+    apply mod3_eq_1_or_mod3_eq_2 h_ih; left
+    rw [count_append, count_append, count_append]
+    simp only [ne_eq, not_false_eq_true, count_cons_of_ne, count_nil, add_zero, reduceCtorEq]
 
 /-- Using the above theorem, we solve the MU puzzle, showing that `"MU"` is not derivable.
 Once we have proved that `Derivable` is an instance of `DecidablePred`, this will follow
@@ -128,7 +134,7 @@ theorem goodm_of_rule1 (xs : Miustr) (h₁ : Derivable (xs ++ ↑[I])) (h₂ : G
     exact mhead
   · change ¬M ∈ tail (xs ++ ↑([I] ++ [U]))
     rw [← append_assoc, tail_append_singleton_of_ne_nil]
-    · simp_rw [mem_append, mem_singleton, or_false]; exact nmtail
+    · simp_rw [mem_append, mem_singleton, reduceCtorEq, or_false]; exact nmtail
     · exact append_ne_nil_of_left_ne_nil this _
 
 theorem goodm_of_rule2 (xs : Miustr) (_ : Derivable (M :: xs)) (h₂ : Goodm (M :: xs)) :
diff --git a/Archive/MiuLanguage/DecisionSuf.lean b/Archive/MiuLanguage/DecisionSuf.lean
index 82a444848d684..2f1f8bbf87750 100644
--- a/Archive/MiuLanguage/DecisionSuf.lean
+++ b/Archive/MiuLanguage/DecisionSuf.lean
@@ -256,7 +256,8 @@ theorem count_I_eq_length_of_count_U_zero_and_neg_mem {ys : Miustr} (hu : count
       · simpa only [count]
       · rw [mem_cons, not_or] at hm; exact hm.2
     · -- case `x = U` gives a contradiction.
-      exfalso; simp only [count, countP_cons_of_pos (· == U) _ (rfl : U == U)] at hu
+      exfalso
+      simp only [count, countP_cons_of_pos (· == U) _ (rfl : U == U), reduceCtorEq] at hu
 
 /-- `base_case_suf` is the base case of the sufficiency result.
 -/
@@ -264,7 +265,8 @@ theorem base_case_suf (en : Miustr) (h : Decstr en) (hu : count U en = 0) : Deri
   rcases h with ⟨⟨mhead, nmtail⟩, hi⟩
   have : en ≠ nil := by
     intro k
-    simp only [k, count, countP, countP.go, if_false, zero_mod, zero_ne_one, false_or_iff] at hi
+    simp only [k, count, countP, countP.go, if_false, zero_mod, zero_ne_one, false_or,
+      reduceCtorEq] at hi
   rcases exists_cons_of_ne_nil this with ⟨y, ys, rfl⟩
   rcases mhead
   rsuffices ⟨c, rfl, hc⟩ : ∃ c, replicate c I = ys ∧ (c % 3 = 1 ∨ c % 3 = 2)
@@ -308,16 +310,17 @@ theorem ind_hyp_suf (k : ℕ) (ys : Miustr) (hu : count U ys = succ k) (hdec : D
   refine ⟨rfl, ?_, ?_, ?_⟩
   · -- Porting note: `simp_rw [count_append]` didn't work
     rw [count_append] at hu
-    simp_rw [count_cons, beq_self_eq_true, if_true, add_succ, beq_iff_eq, reduceIte, add_zero,
-      succ_inj'] at hu
+    simp_rw [count_cons, beq_self_eq_true, if_true, add_succ, beq_iff_eq, reduceCtorEq, reduceIte,
+      add_zero, succ_inj'] at hu
     rwa [count_append, count_append]
   · apply And.intro rfl
     rw [cons_append, cons_append]
     dsimp [tail] at nmtail ⊢
     rw [mem_append] at nmtail
-    simpa only [append_assoc, cons_append, nil_append, mem_append, mem_cons, false_or] using nmtail
+    simpa only [append_assoc, cons_append, nil_append, mem_append, mem_cons, reduceCtorEq,
+      false_or] using nmtail
   · rw [count_append, count_append]; rw [← cons_append, count_append] at hic
-    simp only [count_cons_self, count_nil, count_cons, if_false] at hic ⊢
+    simp only [count_cons_self, count_nil, count_cons, if_false, reduceCtorEq] at hic ⊢
     rw [add_right_comm, add_mod_right]; exact hic
 
 /-- `der_of_decstr` states that `Derivable en` follows from `Decstr en`.
diff --git a/Archive/Sensitivity.lean b/Archive/Sensitivity.lean
index ef2b1fd4be2f0..5cae8d062537c 100644
--- a/Archive/Sensitivity.lean
+++ b/Archive/Sensitivity.lean
@@ -41,7 +41,7 @@ noncomputable section
 
 local notation "√" => Real.sqrt
 
-open Function Bool LinearMap Fintype FiniteDimensional Module.DualBases
+open Function Bool LinearMap Fintype Module Module.DualBases
 
 /-!
 ### The hypercube
@@ -374,7 +374,7 @@ theorem exists_eigenvalue (H : Set (Q m.succ)) (hH : Card H ≥ 2 ^ m + 1) :
   suffices 0 < dim (W ⊓ img) by
     exact mod_cast exists_mem_ne_zero_of_rank_pos this
   have dim_le : dim (W ⊔ img) ≤ 2 ^ (m + 1 : Cardinal) := by
-    convert ← rank_submodule_le (W ⊔ img)
+    convert ← Submodule.rank_le (W ⊔ img)
     rw [← Nat.cast_succ]
     apply dim_V
   have dim_add : dim (W ⊔ img) + dim (W ⊓ img) = dim W + 2 ^ m := by
@@ -425,7 +425,8 @@ theorem huang_degree_theorem (H : Set (Q m.succ)) (hH : Card H ≥ 2 ^ m + 1) :
     _ =
         |(coeffs y).sum fun (i : Q m.succ) (a : ℝ) =>
             a • (ε q ∘ f m.succ ∘ fun i : Q m.succ => e i) i| := by
-      erw [(f m.succ).map_finsupp_total, (ε q).map_finsupp_total, Finsupp.total_apply]
+      erw [(f m.succ).map_finsupp_linearCombination, (ε q).map_finsupp_linearCombination,
+           Finsupp.linearCombination_apply]
     _ ≤ ∑ p ∈ (coeffs y).support, |coeffs y p * (ε q <| f m.succ <| e p)| :=
       (norm_sum_le _ fun p => coeffs y p * _)
     _ = ∑ p ∈ (coeffs y).support, |coeffs y p| * ite (p ∈ q.adjacent) 1 0 := by
diff --git a/Archive/Wiedijk100Theorems/AbelRuffini.lean b/Archive/Wiedijk100Theorems/AbelRuffini.lean
index df179eb859924..36026dc2ddd76 100644
--- a/Archive/Wiedijk100Theorems/AbelRuffini.lean
+++ b/Archive/Wiedijk100Theorems/AbelRuffini.lean
@@ -115,7 +115,7 @@ theorem real_roots_Phi_ge_aux (hab : b < a) :
   · have hf1 : f 1 < 0 := by simp [hf, hb]
     have hfa : 0 ≤ f a := by
       simp_rw [hf, ← sq]
-      refine add_nonneg (sub_nonneg.mpr (pow_le_pow_right ha ?_)) ?_ <;> norm_num
+      refine add_nonneg (sub_nonneg.mpr (pow_right_mono₀ ha ?_)) ?_ <;> norm_num
     obtain ⟨x, ⟨-, hx1⟩, hx2⟩ := intermediate_value_Ico' hle (hc _) (Set.mem_Ioc.mpr ⟨hf1, hf0⟩)
     obtain ⟨y, ⟨hy1, -⟩, hy2⟩ := intermediate_value_Ioc ha (hc _) (Set.mem_Ioc.mpr ⟨hf1, hfa⟩)
     exact ⟨x, y, (hx1.trans hy1).ne, hx2, hy2⟩
@@ -126,7 +126,7 @@ theorem real_roots_Phi_ge_aux (hab : b < a) :
         f (-a) = (a : ℝ) ^ 2 - (a : ℝ) ^ 5 + b := by
           norm_num [hf, ← sq, sub_eq_add_neg, add_comm, Odd.neg_pow (by decide : Odd 5)]
         _ ≤ (a : ℝ) ^ 2 - (a : ℝ) ^ 3 + (a - 1) := by
-          refine add_le_add (sub_le_sub_left (pow_le_pow_right ha ?_) _) ?_ <;> linarith
+          refine add_le_add (sub_le_sub_left (pow_right_mono₀ ha ?_) _) ?_ <;> linarith
         _ = -((a : ℝ) - 1) ^ 2 * (a + 1) := by ring
         _ ≤ 0 := by nlinarith
     have ha' := neg_nonpos.mpr (hle.trans ha)
diff --git a/Archive/Wiedijk100Theorems/AscendingDescendingSequences.lean b/Archive/Wiedijk100Theorems/AscendingDescendingSequences.lean
index 19e2a5834b237..df05d9256c5a5 100644
--- a/Archive/Wiedijk100Theorems/AscendingDescendingSequences.lean
+++ b/Archive/Wiedijk100Theorems/AscendingDescendingSequences.lean
@@ -3,6 +3,7 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta
 -/
+import Mathlib.Data.Finset.Max
 import Mathlib.Data.Fintype.Powerset
 
 /-!
@@ -147,7 +148,7 @@ theorem erdos_szekeres {r s n : ℕ} {f : Fin n → α} (hn : r * s < n) (hf : I
   have : image ab univ ⊆ ran := by
     -- First some logical shuffling
     rintro ⟨x₁, x₂⟩
-    simp only [ran, mem_image, exists_prop, mem_range, mem_univ, mem_product, true_and_iff,
+    simp only [ran, mem_image, exists_prop, mem_range, mem_univ, mem_product, true_and,
       Prod.ext_iff]
     rintro ⟨i, rfl, rfl⟩
     specialize q i
diff --git a/Archive/Wiedijk100Theorems/BallotProblem.lean b/Archive/Wiedijk100Theorems/BallotProblem.lean
index cb4e68cb8b8a6..8fbd5b2ecf7d0 100644
--- a/Archive/Wiedijk100Theorems/BallotProblem.lean
+++ b/Archive/Wiedijk100Theorems/BallotProblem.lean
@@ -52,7 +52,7 @@ theorem staysPositive_cons {x : ℤ} {l : List ℤ} :
 
 theorem sum_nonneg_of_staysPositive : ∀ {l : List ℤ}, l ∈ staysPositive → 0 ≤ l.sum
   | [], _ => le_rfl
-  | (_::_), h => (h _ (List.cons_ne_nil _ _) (List.suffix_refl _)).le
+  | (_::_), h => (h _ (List.cons_ne_nil _ _) List.suffix_rfl).le
 
 theorem staysPositive_cons_pos (x : ℤ) (hx : 0 < x) (l : List ℤ) :
     (x::l) ∈ staysPositive ↔ l ∈ staysPositive := by
@@ -206,7 +206,7 @@ theorem first_vote_pos :
         ((countedSequence_nonempty _ _).image _)]
     · have : List.cons (-1) '' countedSequence (p + 1) q ∩ {l : List ℤ | l.headI = 1} = ∅ := by
         ext
-        simp only [mem_inter_iff, mem_image, mem_setOf_eq, mem_empty_iff_false, iff_false_iff,
+        simp only [mem_inter_iff, mem_image, mem_setOf_eq, mem_empty_iff_false, iff_false,
           not_and, forall_exists_index, and_imp]
         rintro l _ rfl
         norm_num
@@ -235,7 +235,7 @@ theorem first_vote_neg (p q : ℕ) (h : 0 < p + q) :
   have := condCount_compl
     {l : List ℤ | l.headI = 1}ᶜ (countedSequence_finite p q) (countedSequence_nonempty p q)
   rw [compl_compl, first_vote_pos _ _ h] at this
-  rw [← ENNReal.sub_eq_of_add_eq _ this, ENNReal.eq_div_iff, ENNReal.mul_sub, mul_one,
+  rw [ENNReal.eq_sub_of_add_eq _ this, ENNReal.eq_div_iff, ENNReal.mul_sub, mul_one,
     ENNReal.mul_div_cancel', ENNReal.add_sub_cancel_left]
   all_goals simp_all [ENNReal.div_eq_top]
 
@@ -252,7 +252,7 @@ theorem ballot_edge (p : ℕ) : condCount (countedSequence (p + 1) 0) staysPosit
   rw [counted_right_zero]
   refine condCount_eq_one_of (finite_singleton _) (singleton_nonempty _) ?_
   refine singleton_subset_iff.2 fun l hl₁ hl₂ => List.sum_pos _ (fun x hx => ?_) hl₁
-  rw [List.eq_of_mem_replicate (List.mem_of_mem_suffix hx hl₂)]
+  rw [List.eq_of_mem_replicate (hl₂.mem hx)]
   norm_num
 
 theorem countedSequence_int_pos_counted_succ_succ (p q : ℕ) :
@@ -262,7 +262,7 @@ theorem countedSequence_int_pos_counted_succ_succ (p q : ℕ) :
       (_ : List.cons (-1) '' countedSequence (p + 1) q ∩ {l | l.headI = 1} = ∅), union_empty] <;>
     · ext
       simp only [mem_inter_iff, mem_image, mem_setOf_eq, and_iff_left_iff_imp, mem_empty_iff_false,
-        iff_false_iff, not_and, forall_exists_index, and_imp]
+        iff_false, not_and, forall_exists_index, and_imp]
       rintro y _ rfl
       norm_num
 
@@ -289,7 +289,7 @@ theorem countedSequence_int_neg_counted_succ_succ (p q : ℕ) :
       empty_union] <;>
     · ext
       simp only [mem_inter_iff, mem_image, mem_setOf_eq, and_iff_left_iff_imp, mem_empty_iff_false,
-        iff_false_iff, not_and, forall_exists_index, and_imp]
+        iff_false, not_and, forall_exists_index, and_imp]
       rintro y _ rfl
       norm_num
 
@@ -363,7 +363,7 @@ theorem ballot_problem :
   rwa [ENNReal.toReal_eq_toReal (measure_lt_top _ _).ne] at this
   simp only [Ne, ENNReal.div_eq_top, tsub_eq_zero_iff_le, Nat.cast_le, not_le,
     add_eq_zero, Nat.cast_eq_zero, ENNReal.add_eq_top, ENNReal.natCast_ne_top, or_self_iff,
-    not_false_iff, and_true_iff]
+    not_false_iff, and_true]
   push_neg
   exact ⟨fun _ _ => by linarith, (tsub_le_self.trans_lt (ENNReal.natCast_ne_top p).lt_top).ne⟩
 
diff --git a/Archive/Wiedijk100Theorems/CubingACube.lean b/Archive/Wiedijk100Theorems/CubingACube.lean
index 05b1712152b67..d7acf6143b041 100644
--- a/Archive/Wiedijk100Theorems/CubingACube.lean
+++ b/Archive/Wiedijk100Theorems/CubingACube.lean
@@ -71,7 +71,7 @@ theorem univ_pi_side (c : Cube n) : pi univ (side c) = c.toSet :=
 
 theorem toSet_subset {c c' : Cube n} : c.toSet ⊆ c'.toSet ↔ ∀ j, c.side j ⊆ c'.side j := by
   simp only [← univ_pi_side, univ_pi_subset_univ_pi_iff, (c.side_nonempty _).ne_empty, exists_false,
-    or_false_iff]
+    or_false]
 
 theorem toSet_disjoint {c c' : Cube n} :
     Disjoint c.toSet c'.toSet ↔ ∃ j, Disjoint (c.side j) (c'.side j) := by
@@ -361,7 +361,7 @@ theorem smallest_onBoundary {j} (bi : OnBoundary (mi_mem_bcubes : mi h v ∈ _)
     dsimp only [x]; rw [← bi, add_sub_assoc, add_lt_iff_neg_left, sub_lt_zero]
     apply mi_strict_minimal (Ne.symm h2i') hi'
   refine ⟨x, ⟨?_, ?_⟩, ?_⟩
-  · simp only [side, neg_lt_zero, hw, add_lt_iff_neg_left, and_true_iff, mem_Ico, sub_eq_add_neg, x]
+  · simp only [side, neg_lt_zero, hw, add_lt_iff_neg_left, and_true, mem_Ico, sub_eq_add_neg, x]
     rw [add_assoc, le_add_iff_nonneg_right, ← sub_eq_add_neg, sub_nonneg]
     apply le_of_lt (w_lt_w h v hi')
   · simp only [side, not_and_or, not_lt, not_le, mem_Ico]; left; exact hx
@@ -408,7 +408,7 @@ theorem mi_not_onBoundary (j : Fin n) : ¬OnBoundary (mi_mem_bcubes : mi h v ∈
   have i'_i'' : i' ≠ i'' := by
     rintro ⟨⟩
     have : (cs i).b ∈ (cs i').toSet := by
-      simp only [toSet, forall_iff_succ, hi.1, bottom_mem_side h2i', true_and_iff, mem_setOf_eq]
+      simp only [toSet, forall_iff_succ, hi.1, bottom_mem_side h2i', true_and, mem_setOf_eq]
       intro j₂; by_cases hj₂ : j₂ = j
       · simpa [p', side_tail, hj'.symm, hj₂] using hi''.2 j
       · simpa [p, hj₂] using hi'.2 j₂
diff --git a/Archive/Wiedijk100Theorems/FriendshipGraphs.lean b/Archive/Wiedijk100Theorems/FriendshipGraphs.lean
index 0674e74b0aa31..860afa94c8e2e 100644
--- a/Archive/Wiedijk100Theorems/FriendshipGraphs.lean
+++ b/Archive/Wiedijk100Theorems/FriendshipGraphs.lean
@@ -81,8 +81,10 @@ include hG in
 theorem adjMatrix_sq_of_ne {v w : V} (hvw : v ≠ w) :
     (G.adjMatrix R ^ 2 : Matrix V V R) v w = 1 := by
   rw [sq, ← Nat.cast_one, ← hG hvw]
-  simp [commonNeighbors, neighborFinset_eq_filter, Finset.filter_filter, and_comm,
-    ← neighborFinset_def]
+  simp only [mul_adjMatrix_apply, neighborFinset_eq_filter, adjMatrix_apply,
+    sum_boole, filter_filter, and_comm, commonNeighbors,
+    Fintype.card_ofFinset (s := filter (fun x ↦ x ∈ G.neighborSet v ∩ G.neighborSet w) univ),
+    Set.mem_inter_iff, mem_neighborSet]
 
 include hG in
 /-- This calculation amounts to counting the number of length 3 walks between nonadjacent vertices.
@@ -177,7 +179,7 @@ theorem card_of_regular (hd : G.IsRegularOfDegree d) : d + (Fintype.card V - 1)
   trans ((G.adjMatrix ℕ ^ 2) *ᵥ (fun _ => 1)) v
   · rw [adjMatrix_sq_of_regular hG hd, mulVec, dotProduct, ← insert_erase (mem_univ v)]
     simp only [sum_insert, mul_one, if_true, Nat.cast_id, eq_self_iff_true, mem_erase, not_true,
-      Ne, not_false_iff, add_right_inj, false_and_iff, of_apply]
+      Ne, not_false_iff, add_right_inj, false_and, of_apply]
     rw [Finset.sum_const_nat, card_erase_of_mem (mem_univ v), mul_one]; · rfl
     intro x hx; simp [(ne_of_mem_erase hx).symm]
   · rw [sq, ← mulVec_mulVec]
diff --git a/Archive/Wiedijk100Theorems/Konigsberg.lean b/Archive/Wiedijk100Theorems/Konigsberg.lean
index 8ae0509ffb4d6..80eff930911c5 100644
--- a/Archive/Wiedijk100Theorems/Konigsberg.lean
+++ b/Archive/Wiedijk100Theorems/Konigsberg.lean
@@ -81,6 +81,6 @@ theorem not_isEulerian {u v : Verts} (p : graph.Walk u v) (h : p.IsEulerian) : F
   have h' := setOf_odd_degree_eq
   apply_fun Fintype.card at h'
   rw [h'] at h
-  norm_num at h
+  simp at h
 
 end Konigsberg
diff --git a/Archive/Wiedijk100Theorems/Partition.lean b/Archive/Wiedijk100Theorems/Partition.lean
index 38232baf5bdab..0ced772d9f68f 100644
--- a/Archive/Wiedijk100Theorems/Partition.lean
+++ b/Archive/Wiedijk100Theorems/Partition.lean
@@ -127,7 +127,7 @@ theorem num_series' [Field α] (i : ℕ) :
     | zero => simp [mul_sub, zero_pow, constantCoeff_indicator]
     | succ n =>
       simp only [coeff_one, if_false, mul_sub, mul_one, coeff_indicator,
-        LinearMap.map_sub]
+        LinearMap.map_sub, reduceCtorEq]
       simp_rw [coeff_mul, coeff_X_pow, coeff_indicator, @boole_mul _ _ _ _]
       erw [sum_ite, sum_ite]
       simp_rw [@filter_filter _ _ _ _ _, sum_const_zero, add_zero, sum_const, nsmul_eq_mul, mul_one,
@@ -206,7 +206,7 @@ theorem partialGF_prop (α : Type*) [CommSemiring α] (n : ℕ) (s : Finset ℕ)
   · dsimp only
     intro p₁ hp₁ p₂ hp₂ h
     apply Nat.Partition.ext
-    simp only [true_and_iff, mem_univ, mem_filter] at hp₁ hp₂
+    simp only [true_and, mem_univ, mem_filter] at hp₁ hp₂
     ext i
     simp only [φ, ne_eq, Multiset.mem_toFinset, not_not, smul_eq_mul, Finsupp.mk.injEq] at h
     by_cases hi : i = 0
@@ -218,7 +218,7 @@ theorem partialGF_prop (α : Type*) [CommSemiring α] (n : ℕ) (s : Finset ℕ)
     · rw [← mul_left_inj' hi]
       rw [Function.funext_iff] at h
       exact h.2 i
-  · simp only [φ, mem_filter, mem_finsuppAntidiag, mem_univ, exists_prop, true_and_iff, and_assoc]
+  · simp only [φ, mem_filter, mem_finsuppAntidiag, mem_univ, exists_prop, true_and, and_assoc]
     rintro f ⟨hf, hf₃, hf₄⟩
     have hf' : f ∈ finsuppAntidiag s n := mem_finsuppAntidiag.mpr ⟨hf, hf₃⟩
     simp only [mem_finsuppAntidiag] at hf'
@@ -266,7 +266,7 @@ theorem partialOddGF_prop [Field α] (n m : ℕ) :
   convert partialGF_prop α n
     ((range m).map mkOdd) _ (fun _ => Set.univ) (fun _ _ => trivial) using 2
   · congr
-    simp only [true_and_iff, forall_const, Set.mem_univ]
+    simp only [true_and, forall_const, Set.mem_univ]
   · rw [Finset.prod_map]
     simp_rw [num_series']
     congr! 2 with x
@@ -314,10 +314,9 @@ theorem partialDistinctGF_prop [CommSemiring α] (n m : ℕ) :
   convert partialGF_prop α n
     ((range m).map ⟨Nat.succ, Nat.succ_injective⟩) _ (fun _ => {0, 1}) (fun _ _ => Or.inl rfl)
     using 2
-  · congr
-    congr! with p
+  · congr! with p
     rw [Multiset.nodup_iff_count_le_one]
-    congr! with i
+    congr! 1 with i
     rcases Multiset.count i p.parts with (_ | _ | ms) <;> simp
   · simp_rw [Finset.prod_map, two_series]
     congr with i
diff --git a/Archive/Wiedijk100Theorems/SumOfPrimeReciprocalsDiverges.lean b/Archive/Wiedijk100Theorems/SumOfPrimeReciprocalsDiverges.lean
index 74017293ce14b..e30881c10059f 100644
--- a/Archive/Wiedijk100Theorems/SumOfPrimeReciprocalsDiverges.lean
+++ b/Archive/Wiedijk100Theorems/SumOfPrimeReciprocalsDiverges.lean
@@ -154,7 +154,7 @@ theorem card_le_two_pow {x k : ℕ} :
     card M₁ ≤ card (image f K) := card_le_card h
     _ ≤ card K := card_image_le
     _ ≤ 2 ^ card (image Nat.succ (range k)) := by simp only [K, card_powerset]; rfl
-    _ ≤ 2 ^ card (range k) := pow_le_pow_right one_le_two card_image_le
+    _ ≤ 2 ^ card (range k) := pow_right_mono₀ one_le_two card_image_le
     _ = 2 ^ k := by rw [card_range k]
 
 /--
diff --git a/Archive/ZagierTwoSquares.lean b/Archive/ZagierTwoSquares.lean
index 894cf667b4e0c..f4846d4fea64c 100644
--- a/Archive/ZagierTwoSquares.lean
+++ b/Archive/ZagierTwoSquares.lean
@@ -50,7 +50,7 @@ lemma zagierSet_lower_bound {x y z : ℕ} (h : (x, y, z) ∈ zagierSet k) : 0 <
     cases' (Nat.dvd_prime hk.out).1 (dvd_of_mul_left_eq _ h) with e e
     all_goals
       simp only [e, self_eq_add_left, ne_eq, add_eq_zero, and_false, not_false_eq_true,
-        mul_eq_left₀] at h
+        mul_eq_left₀, reduceCtorEq] at h
       simp only [h, zero_add] at hk
       exact Nat.not_prime_one hk.out
 
@@ -148,7 +148,7 @@ theorem eq_of_mem_fixedPoints {t : zagierSet k} (mem : t ∈ fixedPoints (comple
   split_ifs at mem with less more <;>
     -- less (completely handled by the pre-applied `simp_all only`)
     simp_all only [not_lt, Prod.mk.injEq, add_right_eq_self, mul_eq_zero, false_or,
-      lt_self_iff_false]
+      lt_self_iff_false, reduceCtorEq]
   · -- more
     obtain ⟨_, _, _⟩ := mem; simp_all
   · -- middle (the one fixed point falls under this case)
@@ -161,7 +161,7 @@ theorem eq_of_mem_fixedPoints {t : zagierSet k} (mem : t ∈ fixedPoints (comple
     cases' (Nat.dvd_prime hk.out).1 (dvd_of_mul_left_eq _ h) with e e
     · rw [e, mul_one] at h
       simp_all [h, show z = 0 by linarith [e]]
-    · simp only [e, mul_left_eq_self₀, add_eq_zero, and_false, or_false] at h
+    · simp only [e, mul_left_eq_self₀, add_eq_zero, and_false, or_false, reduceCtorEq] at h
       simp only [h, true_and]
       linarith [e]
 
diff --git a/Cache/IO.lean b/Cache/IO.lean
index a2c34ef9a0e7e..43e3f70652623 100644
--- a/Cache/IO.lean
+++ b/Cache/IO.lean
@@ -137,7 +137,8 @@ private def CacheM.getContext : IO CacheM.Context := do
     ("Cli", LAKEPACKAGESDIR / "Cli"),
     ("ProofWidgets", LAKEPACKAGESDIR / "proofwidgets"),
     ("Qq", LAKEPACKAGESDIR / "Qq"),
-    ("ImportGraph", LAKEPACKAGESDIR / "importGraph")
+    ("ImportGraph", LAKEPACKAGESDIR / "importGraph"),
+    ("LeanSearchClient", LAKEPACKAGESDIR / "LeanSearchClient")
   ]⟩
 
 def CacheM.run (f : CacheM α) : IO α := do ReaderT.run f (← getContext)
@@ -337,7 +338,7 @@ def packCache (hashMap : HashMap) (overwrite verbose unpackedOnly : Bool)
 /-- Gets the set of all cached files -/
 def getLocalCacheSet : IO <| Lean.RBTree String compare := do
   let paths ← getFilesWithExtension CACHEDIR "ltar"
-  return .fromList (paths.data.map (·.withoutParent CACHEDIR |>.toString)) _
+  return .fromList (paths.toList.map (·.withoutParent CACHEDIR |>.toString)) _
 
 def isPathFromMathlib (path : FilePath) : Bool :=
   match path.components with
diff --git a/Cache/Requests.lean b/Cache/Requests.lean
index 5c4c6038aecaa..71b74cb4b051d 100644
--- a/Cache/Requests.lean
+++ b/Cache/Requests.lean
@@ -184,7 +184,7 @@ def UPLOAD_URL : String :=
 /-- Formats the config file for `curl`, containing the list of files to be uploaded -/
 def mkPutConfigContent (fileNames : Array String) (token : String) : IO String := do
   let token := if useFROCache then "" else s!"?{token}" -- the FRO cache doesn't pass the token here
-  let l ← fileNames.data.mapM fun fileName : String => do
+  let l ← fileNames.toList.mapM fun fileName : String => do
     pure s!"-T {(IO.CACHEDIR / fileName).toString}\nurl = {mkFileURL UPLOAD_URL fileName}{token}"
   return "\n".intercalate l
 
diff --git a/Counterexamples.lean b/Counterexamples.lean
index 3cc04e2e7faa1..85f441c3cd5d5 100644
--- a/Counterexamples.lean
+++ b/Counterexamples.lean
@@ -3,6 +3,7 @@ import Counterexamples.CharPZeroNeCharZero
 import Counterexamples.CliffordAlgebraNotInjective
 import Counterexamples.Cyclotomic105
 import Counterexamples.DirectSumIsInternal
+import Counterexamples.GameMultiplication
 import Counterexamples.Girard
 import Counterexamples.HomogeneousPrimeNotPrime
 import Counterexamples.LinearOrderWithPosMulPosEqZero
diff --git a/Counterexamples/CliffordAlgebraNotInjective.lean b/Counterexamples/CliffordAlgebraNotInjective.lean
index 357542c2f4108..8670d40f77954 100644
--- a/Counterexamples/CliffordAlgebraNotInjective.lean
+++ b/Counterexamples/CliffordAlgebraNotInjective.lean
@@ -52,7 +52,7 @@ theorem mem_kIdeal_iff (x : MvPolynomial (Fin 3) (ZMod 2)) :
   have :
       kIdeal = Ideal.span ((monomial · (1 : ZMod 2)) '' Set.range (Finsupp.single · 2)) := by
     simp_rw [kIdeal, X, monomial_mul, one_mul, ← Finsupp.single_add, ← Set.range_comp,
-      Function.comp]
+      Function.comp_def]
   rw [this, mem_ideal_span_monomial_image]
   simp
 
@@ -132,7 +132,7 @@ theorem sq_zero_of_αβγ_mul {x : K} : α * β * γ * x = 0 → x * x = 0 := by
   rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.Quotient.eq_zero_iff_mem]
   exact mul_self_mem_kIdeal_of_X0_X1_X2_mul_mem
 
-/-- Though `αβγ` is not itself zero-/
+/-- Though `αβγ` is not itself zero -/
 theorem αβγ_ne_zero : α * β * γ ≠ 0 := fun h =>
   X0_X1_X2_not_mem_kIdeal <| Ideal.Quotient.eq_zero_iff_mem.1 h
 
diff --git a/Counterexamples/GameMultiplication.lean b/Counterexamples/GameMultiplication.lean
new file mode 100644
index 0000000000000..f28687f92e547
--- /dev/null
+++ b/Counterexamples/GameMultiplication.lean
@@ -0,0 +1,81 @@
+/-
+Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+-/
+
+import Mathlib.SetTheory.Game.Basic
+import Mathlib.Tactic.FinCases
+
+/-!
+# Multiplication of pre-games can't be lifted to the quotient
+
+We show that there exist equivalent pregames `x₁ ≈ x₂` and `y` such that `x₁ * y ≉ x₂ * y`. In
+particular, we cannot define the multiplication of games in general.
+
+The specific counterexample we use is `x₁ = y = {0 | 0}` and `x₂ = {-1, 0 | 0, 1}`. The first game
+is colloquially known as `star`, so we use the name `star'` for the second. We prove that
+`star ≈ star'` and `star * star ≈ star`, but `star' * star ≉ star`.
+-/
+
+namespace Counterexample
+
+namespace PGame
+
+open SetTheory PGame
+
+/-- The game `{-1, 0 | 0, 1}`, which is equivalent but not identical to `*`. -/
+def star' : PGame := ofLists [0, -1] [0, 1]
+
+/-- `*'` is its own negative. -/
+theorem neg_star' : -star' = star' := by
+  simp [star']
+
+/-- `*'` is equivalent to `*`. -/
+theorem star'_equiv_star : star' ≈ star := by
+  have le : star' ≤ star := by
+    apply PGame.le_of_forall_lf
+    · rintro ⟨i⟩
+      fin_cases i
+      · exact zero_lf_star
+      · exact (neg_lt_zero_iff.2 PGame.zero_lt_one).trans_lf zero_lf_star
+    · exact fun _ => lf_zero_le.2 ⟨⟨0, Nat.zero_lt_two⟩, le_rfl⟩
+  constructor
+  case' right => rw [← neg_le_neg_iff, neg_star, neg_star']
+  assumption'
+
+/-- The equation `** = *` is an identity, though not a relabelling. -/
+theorem star_sq : star * star ≈ star := by
+  have le : star * star ≤ star := by
+    rw [le_iff_forall_lf]
+    constructor <;>
+    intro i
+    · apply leftMoves_mul_cases i <;>
+      intro _ _
+      case' hl => rw [mul_moveLeft_inl]
+      case' hr => rw [mul_moveLeft_inr]
+      all_goals rw [lf_iff_game_lf]; simpa using zero_lf_star
+    · refine lf_zero.2 ⟨toRightMovesMul (Sum.inl default), ?_⟩
+      rintro (j | j) <;> -- Instance can't be inferred otherwise.
+      exact isEmptyElim j
+  constructor
+  case' right =>
+    rw [← neg_le_neg_iff];
+    apply (negMulRelabelling _ _).symm.equiv.1.trans;
+    rw [neg_star]
+  assumption'
+
+/-- `*'* ⧏ *` implies `*'* ≉ *`.-/
+theorem star'_mul_star_lf : star' * star ⧏ star := by
+  rw [lf_iff_exists_le]
+  refine Or.inr ⟨toRightMovesMul (Sum.inr ⟨⟨1, Nat.one_lt_two⟩, default⟩), ?_⟩
+  rw [mul_moveRight_inr, le_iff_game_le]
+  simp [star']
+
+/-- Pre-game multiplication cannot be lifted to games. -/
+theorem mul_not_lift : ∃ x₁ x₂ y : PGame, x₁ ≈ x₂ ∧ ¬ x₁ * y ≈ x₂ * y :=
+  ⟨_, _, _, ⟨star'_equiv_star, fun h ↦ (PGame.Equiv.trans h star_sq).ge.not_gf star'_mul_star_lf⟩⟩
+
+end PGame
+
+end Counterexample
diff --git a/Counterexamples/MapFloor.lean b/Counterexamples/MapFloor.lean
index 08097ff2b0b83..3263f315ca27f 100644
--- a/Counterexamples/MapFloor.lean
+++ b/Counterexamples/MapFloor.lean
@@ -125,7 +125,7 @@ theorem forgetEpsilons_apply (p : ℤ[ε]) : forgetEpsilons p = coeff p 0 :=
 itself. -/
 theorem forgetEpsilons_floor_lt (n : ℤ) :
     forgetEpsilons ⌊(n - ↑ε : ℤ[ε])⌋ < ⌊forgetEpsilons (n - ↑ε)⌋ := by
-  suffices ⌊(n - ↑ε : ℤ[ε])⌋ = n - 1 by simp [this]
+  suffices ⌊(n - ↑ε : ℤ[ε])⌋ = n - 1 by simp [map_sub, this]
   have : (0 : ℤ[ε]) < ε := ⟨1, by simp⟩
   exact (if_neg <| by rw [coeff_sub, intCast_coeff_zero]; simp [this]).trans (by
     rw [coeff_sub, intCast_coeff_zero]; simp)
diff --git a/Counterexamples/MonicNonRegular.lean b/Counterexamples/MonicNonRegular.lean
index 80b870310e2b9..954e6c22a84c6 100644
--- a/Counterexamples/MonicNonRegular.lean
+++ b/Counterexamples/MonicNonRegular.lean
@@ -88,7 +88,8 @@ theorem monic_X_add_two : Monic (X + C 2 : N₃[X]) := by
   unfold Monic leadingCoeff
   nontriviality
   rw [natDegree_X_add_C 2]
-  simp only [natDegree_X_add_C 2, coeff_C, coeff_add, coeff_X_one, ite_false, add_zero]
+  simp only [natDegree_X_add_C 2, coeff_C, coeff_add, coeff_X_one, ite_false, add_zero,
+    reduceCtorEq]
 
 theorem not_isLeftRegular_X_add_two : ¬ IsLeftRegular (X + C 2 : N₃[X]) := by
   intro h
diff --git a/Counterexamples/Phillips.lean b/Counterexamples/Phillips.lean
index 755cb6b94dd69..cf03d0dc7c250 100644
--- a/Counterexamples/Phillips.lean
+++ b/Counterexamples/Phillips.lean
@@ -6,7 +6,7 @@ Authors: Sébastien Gouëzel
 import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
 import Mathlib.MeasureTheory.Integral.SetIntegral
 import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 
 /-!
 # A counterexample on Pettis integrability
@@ -467,14 +467,14 @@ theorem sierpinski_pathological_family (Hcont : #ℝ = aleph 1) :
   refine ⟨fun x => {y | r x y}, fun x => ?_, fun y => ?_⟩
   · have : univ \ {y | r x y} = {y | r y x} ∪ {x} := by
       ext y
-      simp only [true_and_iff, mem_univ, mem_setOf_eq, mem_insert_iff, union_singleton, mem_diff]
+      simp only [true_and, mem_univ, mem_setOf_eq, mem_insert_iff, union_singleton, mem_diff]
       rcases trichotomous_of r x y with (h | rfl | h)
-      · simp only [h, not_or, false_iff_iff, not_true]
+      · simp only [h, not_or, false_iff, not_true]
         constructor
         · rintro rfl; exact irrefl_of r y h
         · exact asymm h
-      · simp only [true_or_iff, eq_self_iff_true, iff_true_iff]; exact irrefl x
-      · simp only [h, iff_true_iff, or_true_iff]; exact asymm h
+      · simp only [true_or, eq_self_iff_true, iff_true]; exact irrefl x
+      · simp only [h, iff_true, or_true]; exact asymm h
     rw [this]
     apply Countable.union _ (countable_singleton _)
     rw [Cardinal.countable_iff_lt_aleph_one, ← Hcont]
diff --git a/Counterexamples/SeminormLatticeNotDistrib.lean b/Counterexamples/SeminormLatticeNotDistrib.lean
index 770184c227ba7..e547175d5a7d5 100644
--- a/Counterexamples/SeminormLatticeNotDistrib.lean
+++ b/Counterexamples/SeminormLatticeNotDistrib.lean
@@ -57,14 +57,14 @@ theorem not_distrib : ¬(p ⊔ q1) ⊓ (p ⊔ q2) ≤ p ⊔ q1 ⊓ q2 := by
         4 / 3 = 4 * (1 - 2 / 3) := by norm_num
         _ ≤ 4 * (1 - x.snd) := by gcongr
         _ ≤ 4 * |1 - x.snd| := by gcongr; apply le_abs_self
-        _ = q2 ((1, 1) - x) := by simp; rfl
+        _ = q2 ((1, 1) - x) := rfl
         _ ≤ (p ⊔ q2) ((1, 1) - x) := le_sup_right
         _ ≤ (p ⊔ q1) x + (p ⊔ q2) ((1, 1) - x) := le_add_of_nonneg_left (apply_nonneg _ _)
     · calc
         4 / 3 = 2 / 3 + (1 - 1 / 3) := by norm_num
         _ ≤ x.snd + (1 - x.fst) := by gcongr
         _ ≤ |x.snd| + |1 - x.fst| := add_le_add (le_abs_self _) (le_abs_self _)
-        _ ≤ p x + p ((1, 1) - x) := by exact add_le_add le_sup_right le_sup_left
+        _ ≤ p x + p ((1, 1) - x) := add_le_add le_sup_right le_sup_left
         _ ≤ (p ⊔ q1) x + (p ⊔ q2) ((1, 1) - x) := add_le_add le_sup_left le_sup_left
   · calc
       4 / 3 = 4 * (1 / 3) := by norm_num
diff --git a/Counterexamples/SorgenfreyLine.lean b/Counterexamples/SorgenfreyLine.lean
index 6015b4d98307e..bdebf6085e105 100644
--- a/Counterexamples/SorgenfreyLine.lean
+++ b/Counterexamples/SorgenfreyLine.lean
@@ -110,7 +110,7 @@ theorem nhds_countable_basis_Ico_inv_pnat (a : ℝₗ) :
 theorem nhds_antitone_basis_Ico_inv_pnat (a : ℝₗ) :
     (𝓝 a).HasAntitoneBasis fun n : ℕ+ => Ico a (a + (n : ℝₗ)⁻¹) :=
   ⟨nhds_basis_Ico_inv_pnat a, monotone_const.Ico <| Antitone.const_add
-    (fun k _l hkl => inv_le_inv_of_le (Nat.cast_pos.2 k.2)
+    (fun k _l hkl => inv_anti₀ (Nat.cast_pos.2 k.2)
       (Nat.mono_cast <| Subtype.coe_le_coe.2 hkl)) _⟩
 
 theorem isOpen_iff {s : Set ℝₗ} : IsOpen s ↔ ∀ x ∈ s, ∃ y > x, Ico x y ⊆ s :=
@@ -129,7 +129,7 @@ theorem map_toReal_nhds (a : ℝₗ) : map toReal (𝓝 a) = 𝓝[≥] toReal a
   simpa only [toReal.image_eq_preimage] using nhdsWithin_Ici_basis_Ico (toReal a)
 
 theorem nhds_eq_map (a : ℝₗ) : 𝓝 a = map toReal.symm (𝓝[≥] (toReal a)) := by
-  simp_rw [← map_toReal_nhds, map_map, (· ∘ ·), toReal.symm_apply_apply, map_id']
+  simp_rw [← map_toReal_nhds, map_map, Function.comp_def, toReal.symm_apply_apply, map_id']
 
 theorem nhds_eq_comap (a : ℝₗ) : 𝓝 a = comap toReal (𝓝[≥] (toReal a)) := by
   rw [← map_toReal_nhds, comap_map toReal.injective]
@@ -141,7 +141,7 @@ theorem continuous_toReal : Continuous toReal :=
     exact inf_le_left
 
 instance : OrderClosedTopology ℝₗ :=
-  ⟨isClosed_le_prod.preimage (continuous_toReal.prod_map continuous_toReal)⟩
+  ⟨isClosed_le_prod.preimage (continuous_toReal.prodMap continuous_toReal)⟩
 
 instance : ContinuousAdd ℝₗ := by
   refine ⟨continuous_iff_continuousAt.2 ?_⟩
diff --git a/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean b/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
index f12b01482968d..703b8f0eded4e 100644
--- a/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
+++ b/Counterexamples/ZeroDivisorsInAddMonoidAlgebras.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
 import Mathlib.Algebra.GeomSum
-import Mathlib.Algebra.Group.UniqueProds
+import Mathlib.Algebra.Group.UniqueProds.Basic
 import Mathlib.Algebra.MonoidAlgebra.Basic
 import Mathlib.Data.Finsupp.Lex
 import Mathlib.Data.ZMod.Basic
diff --git a/LongestPole/Main.lean b/LongestPole/Main.lean
index 4c21dc471afe4..b4abb23680785 100644
--- a/LongestPole/Main.lean
+++ b/LongestPole/Main.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import ImportGraph
 import Mathlib.Data.String.Defs
diff --git a/LongestPole/SpeedCenterJson.lean b/LongestPole/SpeedCenterJson.lean
index 89bfb54a07ffa..20a39f872dd0c 100644
--- a/LongestPole/SpeedCenterJson.lean
+++ b/LongestPole/SpeedCenterJson.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Lean.Data.Json
 open Lean
@@ -47,7 +47,7 @@ structure RunResponse where
   run : Run
 deriving ToJson, FromJson
 
-/-- The error response-/
+/-- The error response -/
 structure ErrorMessage where
   repo_id : String
   message : String
diff --git a/Mathlib.lean b/Mathlib.lean
index 09a18cb75f380..7ba04a7c6a7f7 100644
--- a/Mathlib.lean
+++ b/Mathlib.lean
@@ -21,6 +21,7 @@ import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Algebra.Spectrum
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Algebra.Algebra.Subalgebra.Directed
+import Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder
 import Mathlib.Algebra.Algebra.Subalgebra.MulOpposite
 import Mathlib.Algebra.Algebra.Subalgebra.Operations
 import Mathlib.Algebra.Algebra.Subalgebra.Order
@@ -31,16 +32,20 @@ import Mathlib.Algebra.Algebra.Subalgebra.Tower
 import Mathlib.Algebra.Algebra.Subalgebra.Unitization
 import Mathlib.Algebra.Algebra.Tower
 import Mathlib.Algebra.Algebra.Unitization
+import Mathlib.Algebra.Algebra.ZMod
 import Mathlib.Algebra.AlgebraicCard
 import Mathlib.Algebra.Associated.Basic
 import Mathlib.Algebra.Associated.OrderedCommMonoid
 import Mathlib.Algebra.BigOperators.Associated
+import Mathlib.Algebra.BigOperators.Balance
+import Mathlib.Algebra.BigOperators.Expect
 import Mathlib.Algebra.BigOperators.Fin
 import Mathlib.Algebra.BigOperators.Finprod
 import Mathlib.Algebra.BigOperators.Finsupp
 import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Algebra.BigOperators.Group.List
 import Mathlib.Algebra.BigOperators.Group.Multiset
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
 import Mathlib.Algebra.BigOperators.Intervals
 import Mathlib.Algebra.BigOperators.Module
@@ -53,7 +58,6 @@ import Mathlib.Algebra.BigOperators.Ring.Multiset
 import Mathlib.Algebra.BigOperators.Ring.Nat
 import Mathlib.Algebra.BigOperators.RingEquiv
 import Mathlib.Algebra.BigOperators.WithTop
-import Mathlib.Algebra.Bounds
 import Mathlib.Algebra.Category.AlgebraCat.Basic
 import Mathlib.Algebra.Category.AlgebraCat.Limits
 import Mathlib.Algebra.Category.AlgebraCat.Monoidal
@@ -62,6 +66,7 @@ import Mathlib.Algebra.Category.BialgebraCat.Basic
 import Mathlib.Algebra.Category.BoolRing
 import Mathlib.Algebra.Category.CoalgebraCat.Basic
 import Mathlib.Algebra.Category.CoalgebraCat.ComonEquivalence
+import Mathlib.Algebra.Category.CoalgebraCat.Monoidal
 import Mathlib.Algebra.Category.FGModuleCat.Basic
 import Mathlib.Algebra.Category.FGModuleCat.Limits
 import Mathlib.Algebra.Category.Grp.AB5
@@ -74,6 +79,7 @@ import Mathlib.Algebra.Category.Grp.EnoughInjectives
 import Mathlib.Algebra.Category.Grp.EpiMono
 import Mathlib.Algebra.Category.Grp.EquivalenceGroupAddGroup
 import Mathlib.Algebra.Category.Grp.FilteredColimits
+import Mathlib.Algebra.Category.Grp.FiniteGrp
 import Mathlib.Algebra.Category.Grp.ForgetCorepresentable
 import Mathlib.Algebra.Category.Grp.Images
 import Mathlib.Algebra.Category.Grp.Injective
@@ -239,6 +245,7 @@ import Mathlib.Algebra.Group.Embedding
 import Mathlib.Algebra.Group.Equiv.Basic
 import Mathlib.Algebra.Group.Equiv.TypeTags
 import Mathlib.Algebra.Group.Even
+import Mathlib.Algebra.Group.EvenFunction
 import Mathlib.Algebra.Group.Ext
 import Mathlib.Algebra.Group.Fin.Basic
 import Mathlib.Algebra.Group.Fin.Tuple
@@ -260,6 +267,10 @@ import Mathlib.Algebra.Group.Opposite
 import Mathlib.Algebra.Group.PNatPowAssoc
 import Mathlib.Algebra.Group.Pi.Basic
 import Mathlib.Algebra.Group.Pi.Lemmas
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Interval
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Card
 import Mathlib.Algebra.Group.Prod
 import Mathlib.Algebra.Group.Semiconj.Basic
 import Mathlib.Algebra.Group.Semiconj.Defs
@@ -284,7 +295,8 @@ import Mathlib.Algebra.Group.Subsemigroup.Operations
 import Mathlib.Algebra.Group.Support
 import Mathlib.Algebra.Group.TypeTags
 import Mathlib.Algebra.Group.ULift
-import Mathlib.Algebra.Group.UniqueProds
+import Mathlib.Algebra.Group.UniqueProds.Basic
+import Mathlib.Algebra.Group.UniqueProds.VectorSpace
 import Mathlib.Algebra.Group.Units
 import Mathlib.Algebra.Group.Units.Equiv
 import Mathlib.Algebra.Group.Units.Hom
@@ -312,6 +324,7 @@ import Mathlib.Algebra.GroupWithZero.NeZero
 import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
 import Mathlib.Algebra.GroupWithZero.Opposite
 import Mathlib.Algebra.GroupWithZero.Pi
+import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
 import Mathlib.Algebra.GroupWithZero.Prod
 import Mathlib.Algebra.GroupWithZero.Semiconj
 import Mathlib.Algebra.GroupWithZero.ULift
@@ -334,6 +347,7 @@ import Mathlib.Algebra.Homology.ConcreteCategory
 import Mathlib.Algebra.Homology.DerivedCategory.Basic
 import Mathlib.Algebra.Homology.DerivedCategory.ExactFunctor
 import Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic
+import Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences
 import Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
 import Mathlib.Algebra.Homology.DerivedCategory.HomologySequence
 import Mathlib.Algebra.Homology.DerivedCategory.ShortExact
@@ -342,6 +356,7 @@ import Mathlib.Algebra.Homology.DifferentialObject
 import Mathlib.Algebra.Homology.Embedding.Basic
 import Mathlib.Algebra.Homology.Embedding.Boundary
 import Mathlib.Algebra.Homology.Embedding.Extend
+import Mathlib.Algebra.Homology.Embedding.HomEquiv
 import Mathlib.Algebra.Homology.Embedding.IsSupported
 import Mathlib.Algebra.Homology.Embedding.Restriction
 import Mathlib.Algebra.Homology.Embedding.TruncGE
@@ -492,9 +507,11 @@ import Mathlib.Algebra.Module.Submodule.Range
 import Mathlib.Algebra.Module.Submodule.RestrictScalars
 import Mathlib.Algebra.Module.Torsion
 import Mathlib.Algebra.Module.ULift
-import Mathlib.Algebra.Module.Zlattice.Basic
-import Mathlib.Algebra.Module.Zlattice.Covolume
+import Mathlib.Algebra.Module.ZLattice.Basic
+import Mathlib.Algebra.Module.ZLattice.Covolume
+import Mathlib.Algebra.Module.ZMod
 import Mathlib.Algebra.MonoidAlgebra.Basic
+import Mathlib.Algebra.MonoidAlgebra.Defs
 import Mathlib.Algebra.MonoidAlgebra.Degree
 import Mathlib.Algebra.MonoidAlgebra.Division
 import Mathlib.Algebra.MonoidAlgebra.Grading
@@ -531,9 +548,14 @@ import Mathlib.Algebra.Order.Antidiag.Pi
 import Mathlib.Algebra.Order.Antidiag.Prod
 import Mathlib.Algebra.Order.Archimedean.Basic
 import Mathlib.Algebra.Order.Archimedean.Hom
+import Mathlib.Algebra.Order.Archimedean.Submonoid
+import Mathlib.Algebra.Order.BigOperators.Expect
 import Mathlib.Algebra.Order.BigOperators.Group.Finset
 import Mathlib.Algebra.Order.BigOperators.Group.List
+import Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite
 import Mathlib.Algebra.Order.BigOperators.Group.Multiset
+import Mathlib.Algebra.Order.BigOperators.GroupWithZero.List
+import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Multiset
 import Mathlib.Algebra.Order.BigOperators.Ring.Finset
 import Mathlib.Algebra.Order.BigOperators.Ring.List
 import Mathlib.Algebra.Order.BigOperators.Ring.Multiset
@@ -549,6 +571,7 @@ import Mathlib.Algebra.Order.Field.Canonical.Defs
 import Mathlib.Algebra.Order.Field.Defs
 import Mathlib.Algebra.Order.Field.InjSurj
 import Mathlib.Algebra.Order.Field.Pi
+import Mathlib.Algebra.Order.Field.Pointwise
 import Mathlib.Algebra.Order.Field.Power
 import Mathlib.Algebra.Order.Field.Rat
 import Mathlib.Algebra.Order.Field.Subfield
@@ -560,6 +583,7 @@ import Mathlib.Algebra.Order.Group.Action
 import Mathlib.Algebra.Order.Group.Action.Synonym
 import Mathlib.Algebra.Order.Group.Basic
 import Mathlib.Algebra.Order.Group.Bounds
+import Mathlib.Algebra.Order.Group.CompleteLattice
 import Mathlib.Algebra.Order.Group.Cone
 import Mathlib.Algebra.Order.Group.Defs
 import Mathlib.Algebra.Order.Group.DenselyOrdered
@@ -570,8 +594,11 @@ import Mathlib.Algebra.Order.Group.Int
 import Mathlib.Algebra.Order.Group.Lattice
 import Mathlib.Algebra.Order.Group.MinMax
 import Mathlib.Algebra.Order.Group.Nat
+import Mathlib.Algebra.Order.Group.Opposite
 import Mathlib.Algebra.Order.Group.OrderIso
 import Mathlib.Algebra.Order.Group.PiLex
+import Mathlib.Algebra.Order.Group.Pointwise.Bounds
+import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
 import Mathlib.Algebra.Order.Group.PosPart
 import Mathlib.Algebra.Order.Group.Prod
 import Mathlib.Algebra.Order.Group.Synonym
@@ -582,8 +609,10 @@ import Mathlib.Algebra.Order.Group.Unbundled.Int
 import Mathlib.Algebra.Order.Group.Units
 import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
 import Mathlib.Algebra.Order.GroupWithZero.Canonical
+import Mathlib.Algebra.Order.GroupWithZero.Submonoid
 import Mathlib.Algebra.Order.GroupWithZero.Synonym
 import Mathlib.Algebra.Order.GroupWithZero.Unbundled
+import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Lemmas
 import Mathlib.Algebra.Order.GroupWithZero.WithZero
 import Mathlib.Algebra.Order.Hom.Basic
 import Mathlib.Algebra.Order.Hom.Monoid
@@ -603,6 +632,7 @@ import Mathlib.Algebra.Order.Module.Pointwise
 import Mathlib.Algebra.Order.Module.Rat
 import Mathlib.Algebra.Order.Module.Synonym
 import Mathlib.Algebra.Order.Monoid.Basic
+import Mathlib.Algebra.Order.Monoid.Canonical.Basic
 import Mathlib.Algebra.Order.Monoid.Canonical.Defs
 import Mathlib.Algebra.Order.Monoid.Defs
 import Mathlib.Algebra.Order.Monoid.NatCast
@@ -626,7 +656,6 @@ import Mathlib.Algebra.Order.Nonneg.Floor
 import Mathlib.Algebra.Order.Nonneg.Module
 import Mathlib.Algebra.Order.Nonneg.Ring
 import Mathlib.Algebra.Order.Pi
-import Mathlib.Algebra.Order.Pointwise
 import Mathlib.Algebra.Order.Positive.Field
 import Mathlib.Algebra.Order.Positive.Ring
 import Mathlib.Algebra.Order.Rearrangement
@@ -640,6 +669,7 @@ import Mathlib.Algebra.Order.Ring.Finset
 import Mathlib.Algebra.Order.Ring.InjSurj
 import Mathlib.Algebra.Order.Ring.Int
 import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Algebra.Order.Ring.Opposite
 import Mathlib.Algebra.Order.Ring.Pow
 import Mathlib.Algebra.Order.Ring.Prod
 import Mathlib.Algebra.Order.Ring.Rat
@@ -649,12 +679,16 @@ import Mathlib.Algebra.Order.Ring.Unbundled.Basic
 import Mathlib.Algebra.Order.Ring.Unbundled.Nonneg
 import Mathlib.Algebra.Order.Ring.Unbundled.Rat
 import Mathlib.Algebra.Order.Ring.WithTop
+import Mathlib.Algebra.Order.Star.Basic
+import Mathlib.Algebra.Order.Star.Conjneg
 import Mathlib.Algebra.Order.Sub.Basic
 import Mathlib.Algebra.Order.Sub.Defs
 import Mathlib.Algebra.Order.Sub.Prod
 import Mathlib.Algebra.Order.Sub.Unbundled.Basic
 import Mathlib.Algebra.Order.Sub.Unbundled.Hom
 import Mathlib.Algebra.Order.Sub.WithTop
+import Mathlib.Algebra.Order.SuccPred
+import Mathlib.Algebra.Order.SuccPred.TypeTags
 import Mathlib.Algebra.Order.Sum
 import Mathlib.Algebra.Order.ToIntervalMod
 import Mathlib.Algebra.Order.UpperLower
@@ -747,12 +781,14 @@ import Mathlib.Algebra.Ring.NegOnePow
 import Mathlib.Algebra.Ring.Opposite
 import Mathlib.Algebra.Ring.Parity
 import Mathlib.Algebra.Ring.Pi
+import Mathlib.Algebra.Ring.Pointwise.Set
 import Mathlib.Algebra.Ring.Prod
 import Mathlib.Algebra.Ring.Rat
 import Mathlib.Algebra.Ring.Regular
 import Mathlib.Algebra.Ring.Semiconj
 import Mathlib.Algebra.Ring.Semireal.Defs
 import Mathlib.Algebra.Ring.Subring.Basic
+import Mathlib.Algebra.Ring.Subring.IntPolynomial
 import Mathlib.Algebra.Ring.Subring.MulOpposite
 import Mathlib.Algebra.Ring.Subring.Order
 import Mathlib.Algebra.Ring.Subring.Pointwise
@@ -773,11 +809,11 @@ import Mathlib.Algebra.Star.BigOperators
 import Mathlib.Algebra.Star.CHSH
 import Mathlib.Algebra.Star.Center
 import Mathlib.Algebra.Star.CentroidHom
+import Mathlib.Algebra.Star.Conjneg
 import Mathlib.Algebra.Star.Free
 import Mathlib.Algebra.Star.Module
 import Mathlib.Algebra.Star.NonUnitalSubalgebra
 import Mathlib.Algebra.Star.NonUnitalSubsemiring
-import Mathlib.Algebra.Star.Order
 import Mathlib.Algebra.Star.Pi
 import Mathlib.Algebra.Star.Pointwise
 import Mathlib.Algebra.Star.Prod
@@ -802,7 +838,9 @@ import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
 import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree
 import Mathlib.AlgebraicGeometry.EllipticCurve.Group
 import Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian
+import Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms
 import Mathlib.AlgebraicGeometry.EllipticCurve.Projective
+import Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange
 import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
 import Mathlib.AlgebraicGeometry.FunctionField
 import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
@@ -816,6 +854,7 @@ import Mathlib.AlgebraicGeometry.Morphisms.Affine
 import Mathlib.AlgebraicGeometry.Morphisms.Basic
 import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
 import Mathlib.AlgebraicGeometry.Morphisms.Constructors
+import Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation
 import Mathlib.AlgebraicGeometry.Morphisms.FiniteType
 import Mathlib.AlgebraicGeometry.Morphisms.IsIso
 import Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion
@@ -839,6 +878,7 @@ import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
 import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology
 import Mathlib.AlgebraicGeometry.Properties
 import Mathlib.AlgebraicGeometry.Pullbacks
+import Mathlib.AlgebraicGeometry.ResidueField
 import Mathlib.AlgebraicGeometry.Restrict
 import Mathlib.AlgebraicGeometry.Scheme
 import Mathlib.AlgebraicGeometry.Sites.BigZariski
@@ -879,6 +919,7 @@ import Mathlib.AlgebraicTopology.Nerve
 import Mathlib.AlgebraicTopology.Quasicategory
 import Mathlib.AlgebraicTopology.SimplexCategory
 import Mathlib.AlgebraicTopology.SimplicialCategory.Basic
+import Mathlib.AlgebraicTopology.SimplicialCategory.SimplicialObject
 import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.AlgebraicTopology.SimplicialSet
 import Mathlib.AlgebraicTopology.SimplicialSet.Monoidal
@@ -887,6 +928,7 @@ import Mathlib.AlgebraicTopology.SplitSimplicialObject
 import Mathlib.AlgebraicTopology.TopologicalSimplex
 import Mathlib.Analysis.Analytic.Basic
 import Mathlib.Analysis.Analytic.CPolynomial
+import Mathlib.Analysis.Analytic.ChangeOrigin
 import Mathlib.Analysis.Analytic.Composition
 import Mathlib.Analysis.Analytic.Constructions
 import Mathlib.Analysis.Analytic.Inverse
@@ -927,6 +969,7 @@ import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unitary
 import Mathlib.Analysis.CStarAlgebra.Exponential
 import Mathlib.Analysis.CStarAlgebra.GelfandDuality
+import Mathlib.Analysis.CStarAlgebra.Hom
 import Mathlib.Analysis.CStarAlgebra.Matrix
 import Mathlib.Analysis.CStarAlgebra.Module.Constructions
 import Mathlib.Analysis.CStarAlgebra.Module.Defs
@@ -946,9 +989,11 @@ import Mathlib.Analysis.Calculus.Conformal.NormedSpace
 import Mathlib.Analysis.Calculus.ContDiff.Basic
 import Mathlib.Analysis.Calculus.ContDiff.Bounds
 import Mathlib.Analysis.Calculus.ContDiff.Defs
+import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
 import Mathlib.Analysis.Calculus.ContDiff.FiniteDimension
 import Mathlib.Analysis.Calculus.ContDiff.RCLike
 import Mathlib.Analysis.Calculus.Darboux
+import Mathlib.Analysis.Calculus.Deriv.Abs
 import Mathlib.Analysis.Calculus.Deriv.Add
 import Mathlib.Analysis.Calculus.Deriv.AffineMap
 import Mathlib.Analysis.Calculus.Deriv.Basic
@@ -978,6 +1023,7 @@ import Mathlib.Analysis.Calculus.FDeriv.Extend
 import Mathlib.Analysis.Calculus.FDeriv.Linear
 import Mathlib.Analysis.Calculus.FDeriv.Measurable
 import Mathlib.Analysis.Calculus.FDeriv.Mul
+import Mathlib.Analysis.Calculus.FDeriv.Norm
 import Mathlib.Analysis.Calculus.FDeriv.Pi
 import Mathlib.Analysis.Calculus.FDeriv.Prod
 import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
@@ -1025,6 +1071,7 @@ import Mathlib.Analysis.Complex.Conformal
 import Mathlib.Analysis.Complex.Convex
 import Mathlib.Analysis.Complex.Hadamard
 import Mathlib.Analysis.Complex.HalfPlane
+import Mathlib.Analysis.Complex.IsIntegral
 import Mathlib.Analysis.Complex.Isometry
 import Mathlib.Analysis.Complex.Liouville
 import Mathlib.Analysis.Complex.LocallyUniformLimit
@@ -1050,6 +1097,7 @@ import Mathlib.Analysis.ConstantSpeed
 import Mathlib.Analysis.Convex.AmpleSet
 import Mathlib.Analysis.Convex.Basic
 import Mathlib.Analysis.Convex.Between
+import Mathlib.Analysis.Convex.Birkhoff
 import Mathlib.Analysis.Convex.Body
 import Mathlib.Analysis.Convex.Caratheodory
 import Mathlib.Analysis.Convex.Combination
@@ -1060,6 +1108,7 @@ import Mathlib.Analysis.Convex.Cone.Extension
 import Mathlib.Analysis.Convex.Cone.InnerDual
 import Mathlib.Analysis.Convex.Cone.Pointed
 import Mathlib.Analysis.Convex.Cone.Proper
+import Mathlib.Analysis.Convex.Continuous
 import Mathlib.Analysis.Convex.Contractible
 import Mathlib.Analysis.Convex.Deriv
 import Mathlib.Analysis.Convex.EGauge
@@ -1117,6 +1166,7 @@ import Mathlib.Analysis.InnerProductSpace.ConformalLinearMap
 import Mathlib.Analysis.InnerProductSpace.Dual
 import Mathlib.Analysis.InnerProductSpace.EuclideanDist
 import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
+import Mathlib.Analysis.InnerProductSpace.JointEigenspace
 import Mathlib.Analysis.InnerProductSpace.LaxMilgram
 import Mathlib.Analysis.InnerProductSpace.LinearPMap
 import Mathlib.Analysis.InnerProductSpace.MeanErgodic
@@ -1130,6 +1180,7 @@ import Mathlib.Analysis.InnerProductSpace.ProdL2
 import Mathlib.Analysis.InnerProductSpace.Projection
 import Mathlib.Analysis.InnerProductSpace.Rayleigh
 import Mathlib.Analysis.InnerProductSpace.Spectrum
+import Mathlib.Analysis.InnerProductSpace.StarOrder
 import Mathlib.Analysis.InnerProductSpace.Symmetric
 import Mathlib.Analysis.InnerProductSpace.TwoDim
 import Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology
@@ -1143,6 +1194,7 @@ import Mathlib.Analysis.LocallyConvex.ContinuousOfBounded
 import Mathlib.Analysis.LocallyConvex.Polar
 import Mathlib.Analysis.LocallyConvex.StrongTopology
 import Mathlib.Analysis.LocallyConvex.WeakDual
+import Mathlib.Analysis.LocallyConvex.WeakSpace
 import Mathlib.Analysis.LocallyConvex.WithSeminorms
 import Mathlib.Analysis.Matrix
 import Mathlib.Analysis.MeanInequalities
@@ -1157,6 +1209,7 @@ import Mathlib.Analysis.Normed.Affine.MazurUlam
 import Mathlib.Analysis.Normed.Algebra.Basic
 import Mathlib.Analysis.Normed.Algebra.Exponential
 import Mathlib.Analysis.Normed.Algebra.MatrixExponential
+import Mathlib.Analysis.Normed.Algebra.Norm
 import Mathlib.Analysis.Normed.Algebra.QuaternionExponential
 import Mathlib.Analysis.Normed.Algebra.Spectrum
 import Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt
@@ -1164,6 +1217,8 @@ import Mathlib.Analysis.Normed.Algebra.Unitization
 import Mathlib.Analysis.Normed.Algebra.UnitizationL1
 import Mathlib.Analysis.Normed.Field.Basic
 import Mathlib.Analysis.Normed.Field.InfiniteSum
+import Mathlib.Analysis.Normed.Field.Lemmas
+import Mathlib.Analysis.Normed.Field.ProperSpace
 import Mathlib.Analysis.Normed.Field.UnitBall
 import Mathlib.Analysis.Normed.Group.AddCircle
 import Mathlib.Analysis.Normed.Group.AddTorsor
@@ -1189,6 +1244,7 @@ import Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels
 import Mathlib.Analysis.Normed.Group.Seminorm
 import Mathlib.Analysis.Normed.Group.Submodule
 import Mathlib.Analysis.Normed.Group.Tannery
+import Mathlib.Analysis.Normed.Group.Ultra
 import Mathlib.Analysis.Normed.Group.Uniform
 import Mathlib.Analysis.Normed.Group.ZeroAtInfty
 import Mathlib.Analysis.Normed.Lp.LpEquiv
@@ -1215,9 +1271,11 @@ import Mathlib.Analysis.Normed.Operator.WeakOperatorTopology
 import Mathlib.Analysis.Normed.Order.Basic
 import Mathlib.Analysis.Normed.Order.Lattice
 import Mathlib.Analysis.Normed.Order.UpperLower
+import Mathlib.Analysis.Normed.Ring.IsPowMulFaithful
 import Mathlib.Analysis.Normed.Ring.Seminorm
 import Mathlib.Analysis.Normed.Ring.SeminormFromBounded
 import Mathlib.Analysis.Normed.Ring.SeminormFromConst
+import Mathlib.Analysis.Normed.Ring.Ultra
 import Mathlib.Analysis.Normed.Ring.Units
 import Mathlib.Analysis.NormedSpace.BallAction
 import Mathlib.Analysis.NormedSpace.ConformalLinearMap
@@ -1338,6 +1396,7 @@ import Mathlib.CategoryTheory.Abelian.Exact
 import Mathlib.CategoryTheory.Abelian.Ext
 import Mathlib.CategoryTheory.Abelian.FunctorCategory
 import Mathlib.CategoryTheory.Abelian.Generator
+import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms
 import Mathlib.CategoryTheory.Abelian.Images
 import Mathlib.CategoryTheory.Abelian.Injective
 import Mathlib.CategoryTheory.Abelian.InjectiveResolution
@@ -1358,13 +1417,15 @@ import Mathlib.CategoryTheory.Adjunction.Basic
 import Mathlib.CategoryTheory.Adjunction.Comma
 import Mathlib.CategoryTheory.Adjunction.Evaluation
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
-import Mathlib.CategoryTheory.Adjunction.Lifting
+import Mathlib.CategoryTheory.Adjunction.Lifting.Left
+import Mathlib.CategoryTheory.Adjunction.Lifting.Right
 import Mathlib.CategoryTheory.Adjunction.Limits
 import Mathlib.CategoryTheory.Adjunction.Mates
 import Mathlib.CategoryTheory.Adjunction.Opposites
 import Mathlib.CategoryTheory.Adjunction.Over
 import Mathlib.CategoryTheory.Adjunction.Reflective
 import Mathlib.CategoryTheory.Adjunction.Restrict
+import Mathlib.CategoryTheory.Adjunction.Triple
 import Mathlib.CategoryTheory.Adjunction.Unique
 import Mathlib.CategoryTheory.Adjunction.Whiskering
 import Mathlib.CategoryTheory.Balanced
@@ -1403,6 +1464,7 @@ import Mathlib.CategoryTheory.Category.PartialFun
 import Mathlib.CategoryTheory.Category.Pointed
 import Mathlib.CategoryTheory.Category.Preorder
 import Mathlib.CategoryTheory.Category.Quiv
+import Mathlib.CategoryTheory.Category.ReflQuiv
 import Mathlib.CategoryTheory.Category.RelCat
 import Mathlib.CategoryTheory.Category.TwoP
 import Mathlib.CategoryTheory.Category.ULift
@@ -1412,6 +1474,7 @@ import Mathlib.CategoryTheory.ChosenFiniteProducts.FunctorCategory
 import Mathlib.CategoryTheory.Closed.Cartesian
 import Mathlib.CategoryTheory.Closed.Functor
 import Mathlib.CategoryTheory.Closed.FunctorCategory
+import Mathlib.CategoryTheory.Closed.FunctorToTypes
 import Mathlib.CategoryTheory.Closed.Ideal
 import Mathlib.CategoryTheory.Closed.Monoidal
 import Mathlib.CategoryTheory.Closed.Types
@@ -1461,6 +1524,7 @@ import Mathlib.CategoryTheory.Extensive
 import Mathlib.CategoryTheory.FiberedCategory.BasedCategory
 import Mathlib.CategoryTheory.FiberedCategory.Cartesian
 import Mathlib.CategoryTheory.FiberedCategory.Cocartesian
+import Mathlib.CategoryTheory.FiberedCategory.Fibered
 import Mathlib.CategoryTheory.FiberedCategory.HomLift
 import Mathlib.CategoryTheory.Filtered.Basic
 import Mathlib.CategoryTheory.Filtered.Connected
@@ -1478,6 +1542,7 @@ import Mathlib.CategoryTheory.Functor.Derived.RightDerived
 import Mathlib.CategoryTheory.Functor.EpiMono
 import Mathlib.CategoryTheory.Functor.Flat
 import Mathlib.CategoryTheory.Functor.FullyFaithful
+import Mathlib.CategoryTheory.Functor.FunctorHom
 import Mathlib.CategoryTheory.Functor.Functorial
 import Mathlib.CategoryTheory.Functor.Hom
 import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
@@ -1486,11 +1551,16 @@ import Mathlib.CategoryTheory.Functor.KanExtension.Pointwise
 import Mathlib.CategoryTheory.Functor.OfSequence
 import Mathlib.CategoryTheory.Functor.ReflectsIso
 import Mathlib.CategoryTheory.Functor.Trifunctor
+import Mathlib.CategoryTheory.Galois.Action
 import Mathlib.CategoryTheory.Galois.Basic
 import Mathlib.CategoryTheory.Galois.Decomposition
+import Mathlib.CategoryTheory.Galois.EssSurj
 import Mathlib.CategoryTheory.Galois.Examples
+import Mathlib.CategoryTheory.Galois.Full
 import Mathlib.CategoryTheory.Galois.GaloisObjects
+import Mathlib.CategoryTheory.Galois.IsFundamentalgroup
 import Mathlib.CategoryTheory.Galois.Prorepresentability
+import Mathlib.CategoryTheory.Galois.Topology
 import Mathlib.CategoryTheory.Generator
 import Mathlib.CategoryTheory.GlueData
 import Mathlib.CategoryTheory.GradedObject
@@ -1807,7 +1877,6 @@ import Mathlib.CategoryTheory.Sites.CoverPreserving
 import Mathlib.CategoryTheory.Sites.Coverage
 import Mathlib.CategoryTheory.Sites.CoversTop
 import Mathlib.CategoryTheory.Sites.DenseSubsite
-import Mathlib.CategoryTheory.Sites.Discrete
 import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
 import Mathlib.CategoryTheory.Sites.EpiMono
 import Mathlib.CategoryTheory.Sites.EqualizerSheafCondition
@@ -1912,6 +1981,7 @@ import Mathlib.Combinatorics.Quiver.ConnectedComponent
 import Mathlib.Combinatorics.Quiver.Covering
 import Mathlib.Combinatorics.Quiver.Path
 import Mathlib.Combinatorics.Quiver.Push
+import Mathlib.Combinatorics.Quiver.ReflQuiver
 import Mathlib.Combinatorics.Quiver.SingleObj
 import Mathlib.Combinatorics.Quiver.Subquiver
 import Mathlib.Combinatorics.Quiver.Symmetric
@@ -1951,6 +2021,7 @@ import Mathlib.Combinatorics.SimpleGraph.Hasse
 import Mathlib.Combinatorics.SimpleGraph.IncMatrix
 import Mathlib.Combinatorics.SimpleGraph.Init
 import Mathlib.Combinatorics.SimpleGraph.LapMatrix
+import Mathlib.Combinatorics.SimpleGraph.LineGraph
 import Mathlib.Combinatorics.SimpleGraph.Maps
 import Mathlib.Combinatorics.SimpleGraph.Matching
 import Mathlib.Combinatorics.SimpleGraph.Metric
@@ -1997,6 +2068,8 @@ import Mathlib.Computability.TuringMachine
 import Mathlib.Condensed.Basic
 import Mathlib.Condensed.CartesianClosed
 import Mathlib.Condensed.Discrete.Basic
+import Mathlib.Condensed.Discrete.LocallyConstant
+import Mathlib.Condensed.Discrete.Module
 import Mathlib.Condensed.Epi
 import Mathlib.Condensed.Equivalence
 import Mathlib.Condensed.Explicit
@@ -2118,6 +2191,7 @@ import Mathlib.Data.Finset.Grade
 import Mathlib.Data.Finset.Image
 import Mathlib.Data.Finset.Interval
 import Mathlib.Data.Finset.Lattice
+import Mathlib.Data.Finset.Max
 import Mathlib.Data.Finset.MulAntidiagonal
 import Mathlib.Data.Finset.NAry
 import Mathlib.Data.Finset.NatAntidiagonal
@@ -2130,9 +2204,6 @@ import Mathlib.Data.Finset.Pairwise
 import Mathlib.Data.Finset.Pi
 import Mathlib.Data.Finset.PiInduction
 import Mathlib.Data.Finset.Piecewise
-import Mathlib.Data.Finset.Pointwise.Basic
-import Mathlib.Data.Finset.Pointwise.Card
-import Mathlib.Data.Finset.Pointwise.Interval
 import Mathlib.Data.Finset.Powerset
 import Mathlib.Data.Finset.Preimage
 import Mathlib.Data.Finset.Prod
@@ -2204,6 +2275,7 @@ import Mathlib.Data.Int.Cast.Prod
 import Mathlib.Data.Int.CharZero
 import Mathlib.Data.Int.ConditionallyCompleteOrder
 import Mathlib.Data.Int.Defs
+import Mathlib.Data.Int.DivMod
 import Mathlib.Data.Int.GCD
 import Mathlib.Data.Int.Interval
 import Mathlib.Data.Int.LeastGreatest
@@ -2219,6 +2291,7 @@ import Mathlib.Data.Int.Range
 import Mathlib.Data.Int.Sqrt
 import Mathlib.Data.Int.Star
 import Mathlib.Data.Int.SuccPred
+import Mathlib.Data.Int.WithZero
 import Mathlib.Data.LazyList.Basic
 import Mathlib.Data.List.AList
 import Mathlib.Data.List.Basic
@@ -2283,6 +2356,7 @@ import Mathlib.Data.Matrix.CharP
 import Mathlib.Data.Matrix.ColumnRowPartitioned
 import Mathlib.Data.Matrix.Composition
 import Mathlib.Data.Matrix.DMatrix
+import Mathlib.Data.Matrix.DoublyStochastic
 import Mathlib.Data.Matrix.DualNumber
 import Mathlib.Data.Matrix.Hadamard
 import Mathlib.Data.Matrix.Invertible
@@ -2465,6 +2539,7 @@ import Mathlib.Data.Real.GoldenRatio
 import Mathlib.Data.Real.Hyperreal
 import Mathlib.Data.Real.Irrational
 import Mathlib.Data.Real.Pi.Bounds
+import Mathlib.Data.Real.Pi.Irrational
 import Mathlib.Data.Real.Pi.Leibniz
 import Mathlib.Data.Real.Pi.Wallis
 import Mathlib.Data.Real.Pointwise
@@ -2495,6 +2570,7 @@ import Mathlib.Data.Set.Image
 import Mathlib.Data.Set.Lattice
 import Mathlib.Data.Set.List
 import Mathlib.Data.Set.MemPartition
+import Mathlib.Data.Set.Monotone
 import Mathlib.Data.Set.MulAntidiagonal
 import Mathlib.Data.Set.NAry
 import Mathlib.Data.Set.Notation
@@ -2502,7 +2578,6 @@ import Mathlib.Data.Set.Operations
 import Mathlib.Data.Set.Opposite
 import Mathlib.Data.Set.Pairwise.Basic
 import Mathlib.Data.Set.Pairwise.Lattice
-import Mathlib.Data.Set.Pointwise.Basic
 import Mathlib.Data.Set.Pointwise.BigOperators
 import Mathlib.Data.Set.Pointwise.BoundedMul
 import Mathlib.Data.Set.Pointwise.Finite
@@ -2560,14 +2635,11 @@ import Mathlib.Data.Vector3
 import Mathlib.Data.W.Basic
 import Mathlib.Data.W.Cardinal
 import Mathlib.Data.W.Constructions
-import Mathlib.Data.ZMod.Algebra
 import Mathlib.Data.ZMod.Basic
 import Mathlib.Data.ZMod.Coprime
 import Mathlib.Data.ZMod.Defs
 import Mathlib.Data.ZMod.Factorial
 import Mathlib.Data.ZMod.IntUnitsPower
-import Mathlib.Data.ZMod.Module
-import Mathlib.Data.ZMod.Parity
 import Mathlib.Data.ZMod.Quotient
 import Mathlib.Data.ZMod.Units
 import Mathlib.Deprecated.Aliases
@@ -2584,6 +2656,7 @@ import Mathlib.Dynamics.BirkhoffSum.Basic
 import Mathlib.Dynamics.BirkhoffSum.NormedSpace
 import Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber
 import Mathlib.Dynamics.Ergodic.Action.Basic
+import Mathlib.Dynamics.Ergodic.Action.OfMinimal
 import Mathlib.Dynamics.Ergodic.Action.Regular
 import Mathlib.Dynamics.Ergodic.AddCircle
 import Mathlib.Dynamics.Ergodic.Conservative
@@ -2599,12 +2672,14 @@ import Mathlib.Dynamics.OmegaLimit
 import Mathlib.Dynamics.PeriodicPts
 import Mathlib.Dynamics.TopologicalEntropy.CoverEntropy
 import Mathlib.Dynamics.TopologicalEntropy.DynamicalEntourage
+import Mathlib.Dynamics.TopologicalEntropy.NetEntropy
 import Mathlib.FieldTheory.AbelRuffini
 import Mathlib.FieldTheory.AbsoluteGaloisGroup
 import Mathlib.FieldTheory.Adjoin
 import Mathlib.FieldTheory.AxGrothendieck
 import Mathlib.FieldTheory.Cardinality
 import Mathlib.FieldTheory.ChevalleyWarning
+import Mathlib.FieldTheory.Differential.Basic
 import Mathlib.FieldTheory.Extension
 import Mathlib.FieldTheory.Finite.Basic
 import Mathlib.FieldTheory.Finite.GaloisField
@@ -2612,7 +2687,7 @@ import Mathlib.FieldTheory.Finite.Polynomial
 import Mathlib.FieldTheory.Finite.Trace
 import Mathlib.FieldTheory.Finiteness
 import Mathlib.FieldTheory.Fixed
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.FieldTheory.IntermediateField.Algebraic
 import Mathlib.FieldTheory.IntermediateField.Basic
 import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
@@ -2725,6 +2800,7 @@ import Mathlib.Geometry.RingedSpace.SheafedSpace
 import Mathlib.Geometry.RingedSpace.Stalks
 import Mathlib.GroupTheory.Abelianization
 import Mathlib.GroupTheory.Archimedean
+import Mathlib.GroupTheory.ArchimedeanDensely
 import Mathlib.GroupTheory.ClassEquation
 import Mathlib.GroupTheory.Commensurable
 import Mathlib.GroupTheory.Commutator.Basic
@@ -2757,7 +2833,6 @@ import Mathlib.GroupTheory.FreeGroup.Basic
 import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
 import Mathlib.GroupTheory.FreeGroup.NielsenSchreier
 import Mathlib.GroupTheory.GroupAction.Basic
-import Mathlib.GroupTheory.GroupAction.BigOperators
 import Mathlib.GroupTheory.GroupAction.Blocks
 import Mathlib.GroupTheory.GroupAction.CardCommute
 import Mathlib.GroupTheory.GroupAction.ConjAct
@@ -2790,6 +2865,7 @@ import Mathlib.GroupTheory.PGroup
 import Mathlib.GroupTheory.Perm.Basic
 import Mathlib.GroupTheory.Perm.Closure
 import Mathlib.GroupTheory.Perm.ClosureSwap
+import Mathlib.GroupTheory.Perm.ConjAct
 import Mathlib.GroupTheory.Perm.Cycle.Basic
 import Mathlib.GroupTheory.Perm.Cycle.Concrete
 import Mathlib.GroupTheory.Perm.Cycle.Factors
@@ -2832,9 +2908,7 @@ import Mathlib.GroupTheory.Transfer
 import Mathlib.InformationTheory.Hamming
 import Mathlib.Init
 import Mathlib.Init.Algebra.Classes
-import Mathlib.Init.Data.List.Lemmas
 import Mathlib.Init.Data.Nat.Lemmas
-import Mathlib.Init.Data.Quot
 import Mathlib.Init.Logic
 import Mathlib.Lean.CoreM
 import Mathlib.Lean.Elab.Tactic.Basic
@@ -3016,6 +3090,7 @@ import Mathlib.LinearAlgebra.PiTensorProduct
 import Mathlib.LinearAlgebra.Prod
 import Mathlib.LinearAlgebra.Projection
 import Mathlib.LinearAlgebra.Projectivization.Basic
+import Mathlib.LinearAlgebra.Projectivization.Constructions
 import Mathlib.LinearAlgebra.Projectivization.Independence
 import Mathlib.LinearAlgebra.Projectivization.Subspace
 import Mathlib.LinearAlgebra.QuadraticForm.Basic
@@ -3037,6 +3112,10 @@ import Mathlib.LinearAlgebra.Ray
 import Mathlib.LinearAlgebra.Reflection
 import Mathlib.LinearAlgebra.RootSystem.Basic
 import Mathlib.LinearAlgebra.RootSystem.Defs
+import Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear
+import Mathlib.LinearAlgebra.RootSystem.Hom
+import Mathlib.LinearAlgebra.RootSystem.OfBilinear
+import Mathlib.LinearAlgebra.RootSystem.RootPairingCat
 import Mathlib.LinearAlgebra.RootSystem.RootPositive
 import Mathlib.LinearAlgebra.SModEq
 import Mathlib.LinearAlgebra.Semisimple
@@ -3087,10 +3166,12 @@ import Mathlib.Logic.Equiv.Pairwise
 import Mathlib.Logic.Equiv.PartialEquiv
 import Mathlib.Logic.Equiv.Set
 import Mathlib.Logic.Equiv.TransferInstance
+import Mathlib.Logic.ExistsUnique
 import Mathlib.Logic.Function.Basic
 import Mathlib.Logic.Function.CompTypeclasses
 import Mathlib.Logic.Function.Conjugate
 import Mathlib.Logic.Function.Defs
+import Mathlib.Logic.Function.FiberPartition
 import Mathlib.Logic.Function.FromTypes
 import Mathlib.Logic.Function.Iterate
 import Mathlib.Logic.Function.OfArity
@@ -3102,6 +3183,7 @@ import Mathlib.Logic.Lemmas
 import Mathlib.Logic.Nonempty
 import Mathlib.Logic.Nontrivial.Basic
 import Mathlib.Logic.Nontrivial.Defs
+import Mathlib.Logic.OpClass
 import Mathlib.Logic.Pairwise
 import Mathlib.Logic.Relation
 import Mathlib.Logic.Relator
@@ -3242,6 +3324,8 @@ import Mathlib.MeasureTheory.MeasurableSpace.Defs
 import Mathlib.MeasureTheory.MeasurableSpace.Embedding
 import Mathlib.MeasureTheory.MeasurableSpace.Instances
 import Mathlib.MeasureTheory.MeasurableSpace.Invariants
+import Mathlib.MeasureTheory.MeasurableSpace.NCard
+import Mathlib.MeasureTheory.MeasurableSpace.PreorderRestrict
 import Mathlib.MeasureTheory.Measure.AEDisjoint
 import Mathlib.MeasureTheory.Measure.AEMeasurable
 import Mathlib.MeasureTheory.Measure.AddContent
@@ -3281,6 +3365,7 @@ import Mathlib.MeasureTheory.Measure.Portmanteau
 import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
 import Mathlib.MeasureTheory.Measure.Regular
 import Mathlib.MeasureTheory.Measure.Restrict
+import Mathlib.MeasureTheory.Measure.SeparableMeasure
 import Mathlib.MeasureTheory.Measure.Stieltjes
 import Mathlib.MeasureTheory.Measure.Sub
 import Mathlib.MeasureTheory.Measure.Tilted
@@ -3295,6 +3380,7 @@ import Mathlib.MeasureTheory.Order.Lattice
 import Mathlib.MeasureTheory.Order.UpperLower
 import Mathlib.MeasureTheory.OuterMeasure.AE
 import Mathlib.MeasureTheory.OuterMeasure.Basic
+import Mathlib.MeasureTheory.OuterMeasure.BorelCantelli
 import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
 import Mathlib.MeasureTheory.OuterMeasure.Defs
 import Mathlib.MeasureTheory.OuterMeasure.Induced
@@ -3305,6 +3391,7 @@ import Mathlib.MeasureTheory.SetAlgebra
 import Mathlib.MeasureTheory.SetSemiring
 import Mathlib.ModelTheory.Algebra.Field.Basic
 import Mathlib.ModelTheory.Algebra.Field.CharP
+import Mathlib.ModelTheory.Algebra.Field.IsAlgClosed
 import Mathlib.ModelTheory.Algebra.Ring.Basic
 import Mathlib.ModelTheory.Algebra.Ring.FreeCommRing
 import Mathlib.ModelTheory.Basic
@@ -3315,6 +3402,7 @@ import Mathlib.ModelTheory.DirectLimit
 import Mathlib.ModelTheory.ElementaryMaps
 import Mathlib.ModelTheory.ElementarySubstructures
 import Mathlib.ModelTheory.Encoding
+import Mathlib.ModelTheory.Equivalence
 import Mathlib.ModelTheory.FinitelyGenerated
 import Mathlib.ModelTheory.Fraisse
 import Mathlib.ModelTheory.Graph
@@ -3361,6 +3449,7 @@ import Mathlib.NumberTheory.EulerProduct.DirichletLSeries
 import Mathlib.NumberTheory.FLT.Basic
 import Mathlib.NumberTheory.FLT.Four
 import Mathlib.NumberTheory.FLT.Three
+import Mathlib.NumberTheory.FactorisationProperties
 import Mathlib.NumberTheory.FermatPsp
 import Mathlib.NumberTheory.FrobeniusNumber
 import Mathlib.NumberTheory.FunctionField
@@ -3379,6 +3468,7 @@ import Mathlib.NumberTheory.LSeries.Convergence
 import Mathlib.NumberTheory.LSeries.Convolution
 import Mathlib.NumberTheory.LSeries.Deriv
 import Mathlib.NumberTheory.LSeries.Dirichlet
+import Mathlib.NumberTheory.LSeries.DirichletContinuation
 import Mathlib.NumberTheory.LSeries.HurwitzZeta
 import Mathlib.NumberTheory.LSeries.HurwitzZetaEven
 import Mathlib.NumberTheory.LSeries.HurwitzZetaOdd
@@ -3386,6 +3476,7 @@ import Mathlib.NumberTheory.LSeries.HurwitzZetaValues
 import Mathlib.NumberTheory.LSeries.Linearity
 import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
 import Mathlib.NumberTheory.LSeries.RiemannZeta
+import Mathlib.NumberTheory.LSeries.ZMod
 import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
 import Mathlib.NumberTheory.LegendreSymbol.Basic
 import Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas
@@ -3423,6 +3514,7 @@ import Mathlib.NumberTheory.Multiplicity
 import Mathlib.NumberTheory.NumberField.Basic
 import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
 import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
+import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone
 import Mathlib.NumberTheory.NumberField.ClassNumber
 import Mathlib.NumberTheory.NumberField.Discriminant
 import Mathlib.NumberTheory.NumberField.Embeddings
@@ -3440,6 +3532,7 @@ import Mathlib.NumberTheory.Padics.PadicNorm
 import Mathlib.NumberTheory.Padics.PadicNumbers
 import Mathlib.NumberTheory.Padics.PadicVal.Basic
 import Mathlib.NumberTheory.Padics.PadicVal.Defs
+import Mathlib.NumberTheory.Padics.ProperSpace
 import Mathlib.NumberTheory.Padics.RingHoms
 import Mathlib.NumberTheory.Pell
 import Mathlib.NumberTheory.PellMatiyasevic
@@ -3501,6 +3594,7 @@ import Mathlib.Order.CompactlyGenerated.Intervals
 import Mathlib.Order.Compare
 import Mathlib.Order.CompleteBooleanAlgebra
 import Mathlib.Order.CompleteLattice
+import Mathlib.Order.CompleteLattice.Finset
 import Mathlib.Order.CompleteLatticeIntervals
 import Mathlib.Order.CompletePartialOrder
 import Mathlib.Order.CompleteSublattice
@@ -3518,8 +3612,15 @@ import Mathlib.Order.Disjointed
 import Mathlib.Order.Estimator
 import Mathlib.Order.Extension.Linear
 import Mathlib.Order.Extension.Well
-import Mathlib.Order.Filter.Archimedean
 import Mathlib.Order.Filter.AtTopBot
+import Mathlib.Order.Filter.AtTopBot.Archimedean
+import Mathlib.Order.Filter.AtTopBot.BigOperators
+import Mathlib.Order.Filter.AtTopBot.Field
+import Mathlib.Order.Filter.AtTopBot.Floor
+import Mathlib.Order.Filter.AtTopBot.Group
+import Mathlib.Order.Filter.AtTopBot.ModEq
+import Mathlib.Order.Filter.AtTopBot.Monoid
+import Mathlib.Order.Filter.AtTopBot.Ring
 import Mathlib.Order.Filter.Bases
 import Mathlib.Order.Filter.Basic
 import Mathlib.Order.Filter.CardinalInter
@@ -3528,6 +3629,7 @@ import Mathlib.Order.Filter.Cofinite
 import Mathlib.Order.Filter.CountableInter
 import Mathlib.Order.Filter.CountableSeparatingOn
 import Mathlib.Order.Filter.Curry
+import Mathlib.Order.Filter.Defs
 import Mathlib.Order.Filter.ENNReal
 import Mathlib.Order.Filter.EventuallyConst
 import Mathlib.Order.Filter.Extr
@@ -3539,7 +3641,6 @@ import Mathlib.Order.Filter.Interval
 import Mathlib.Order.Filter.Ker
 import Mathlib.Order.Filter.Lift
 import Mathlib.Order.Filter.ListTraverse
-import Mathlib.Order.Filter.ModEq
 import Mathlib.Order.Filter.NAry
 import Mathlib.Order.Filter.Partial
 import Mathlib.Order.Filter.Pi
@@ -3630,9 +3731,12 @@ import Mathlib.Order.RelIso.Basic
 import Mathlib.Order.RelIso.Group
 import Mathlib.Order.RelIso.Set
 import Mathlib.Order.RelSeries
+import Mathlib.Order.Restriction
+import Mathlib.Order.ScottContinuity
 import Mathlib.Order.SemiconjSup
 import Mathlib.Order.SetNotation
 import Mathlib.Order.Sublattice
+import Mathlib.Order.SuccPred.Archimedean
 import Mathlib.Order.SuccPred.Basic
 import Mathlib.Order.SuccPred.CompleteLinearOrder
 import Mathlib.Order.SuccPred.IntervalSucc
@@ -3663,11 +3767,13 @@ import Mathlib.Probability.Distributions.Exponential
 import Mathlib.Probability.Distributions.Gamma
 import Mathlib.Probability.Distributions.Gaussian
 import Mathlib.Probability.Distributions.Geometric
+import Mathlib.Probability.Distributions.Pareto
 import Mathlib.Probability.Distributions.Poisson
 import Mathlib.Probability.Distributions.Uniform
 import Mathlib.Probability.IdentDistrib
 import Mathlib.Probability.Independence.Basic
 import Mathlib.Probability.Independence.Conditional
+import Mathlib.Probability.Independence.Integrable
 import Mathlib.Probability.Independence.Kernel
 import Mathlib.Probability.Independence.ZeroOne
 import Mathlib.Probability.Integration
@@ -3675,6 +3781,7 @@ import Mathlib.Probability.Kernel.Basic
 import Mathlib.Probability.Kernel.Composition
 import Mathlib.Probability.Kernel.CondDistrib
 import Mathlib.Probability.Kernel.Condexp
+import Mathlib.Probability.Kernel.Defs
 import Mathlib.Probability.Kernel.Disintegration.Basic
 import Mathlib.Probability.Kernel.Disintegration.CDFToKernel
 import Mathlib.Probability.Kernel.Disintegration.CondCDF
@@ -3683,6 +3790,7 @@ import Mathlib.Probability.Kernel.Disintegration.Integral
 import Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes
 import Mathlib.Probability.Kernel.Disintegration.StandardBorel
 import Mathlib.Probability.Kernel.Disintegration.Unique
+import Mathlib.Probability.Kernel.Integral
 import Mathlib.Probability.Kernel.IntegralCompProd
 import Mathlib.Probability.Kernel.Invariance
 import Mathlib.Probability.Kernel.MeasurableIntegral
@@ -3750,6 +3858,7 @@ import Mathlib.RingTheory.ClassGroup
 import Mathlib.RingTheory.Coalgebra.Basic
 import Mathlib.RingTheory.Coalgebra.Equiv
 import Mathlib.RingTheory.Coalgebra.Hom
+import Mathlib.RingTheory.Coalgebra.TensorProduct
 import Mathlib.RingTheory.Complex
 import Mathlib.RingTheory.Congruence.Basic
 import Mathlib.RingTheory.Congruence.BigOperators
@@ -3771,15 +3880,18 @@ import Mathlib.RingTheory.DedekindDomain.SelmerGroup
 import Mathlib.RingTheory.Derivation.Basic
 import Mathlib.RingTheory.Derivation.DifferentialRing
 import Mathlib.RingTheory.Derivation.Lie
+import Mathlib.RingTheory.Derivation.MapCoeffs
 import Mathlib.RingTheory.Derivation.ToSquareZero
 import Mathlib.RingTheory.DiscreteValuationRing.Basic
 import Mathlib.RingTheory.DiscreteValuationRing.TFAE
 import Mathlib.RingTheory.Discriminant
+import Mathlib.RingTheory.DualNumber
 import Mathlib.RingTheory.EisensteinCriterion
 import Mathlib.RingTheory.EssentialFiniteness
 import Mathlib.RingTheory.Etale.Basic
 import Mathlib.RingTheory.EuclideanDomain
 import Mathlib.RingTheory.Filtration
+import Mathlib.RingTheory.FiniteLength
 import Mathlib.RingTheory.FinitePresentation
 import Mathlib.RingTheory.FiniteStability
 import Mathlib.RingTheory.FiniteType
@@ -3845,9 +3957,13 @@ import Mathlib.RingTheory.JacobsonIdeal
 import Mathlib.RingTheory.Kaehler.Basic
 import Mathlib.RingTheory.Kaehler.CotangentComplex
 import Mathlib.RingTheory.Kaehler.Polynomial
+import Mathlib.RingTheory.KrullDimension.Basic
+import Mathlib.RingTheory.KrullDimension.Field
 import Mathlib.RingTheory.LaurentSeries
 import Mathlib.RingTheory.LittleWedderburn
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.LocalProperties.Basic
+import Mathlib.RingTheory.LocalProperties.IntegrallyClosed
+import Mathlib.RingTheory.LocalProperties.Reduced
 import Mathlib.RingTheory.LocalRing.Basic
 import Mathlib.RingTheory.LocalRing.Defs
 import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
@@ -3856,12 +3972,12 @@ import Mathlib.RingTheory.LocalRing.Module
 import Mathlib.RingTheory.LocalRing.ResidueField.Basic
 import Mathlib.RingTheory.LocalRing.ResidueField.Defs
 import Mathlib.RingTheory.LocalRing.RingHom.Basic
-import Mathlib.RingTheory.LocalRing.RingHom.Defs
 import Mathlib.RingTheory.Localization.Algebra
 import Mathlib.RingTheory.Localization.AsSubring
 import Mathlib.RingTheory.Localization.AtPrime
 import Mathlib.RingTheory.Localization.Away.AdjoinRoot
 import Mathlib.RingTheory.Localization.Away.Basic
+import Mathlib.RingTheory.Localization.Away.Lemmas
 import Mathlib.RingTheory.Localization.BaseChange
 import Mathlib.RingTheory.Localization.Basic
 import Mathlib.RingTheory.Localization.Cardinality
@@ -3883,12 +3999,14 @@ import Mathlib.RingTheory.MvPolynomial.Basic
 import Mathlib.RingTheory.MvPolynomial.Homogeneous
 import Mathlib.RingTheory.MvPolynomial.Ideal
 import Mathlib.RingTheory.MvPolynomial.Localization
-import Mathlib.RingTheory.MvPolynomial.NewtonIdentities
-import Mathlib.RingTheory.MvPolynomial.Symmetric
+import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
+import Mathlib.RingTheory.MvPolynomial.Symmetric.FundamentalTheorem
+import Mathlib.RingTheory.MvPolynomial.Symmetric.NewtonIdentities
 import Mathlib.RingTheory.MvPolynomial.Tower
 import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
 import Mathlib.RingTheory.MvPowerSeries.Basic
 import Mathlib.RingTheory.MvPowerSeries.Inverse
+import Mathlib.RingTheory.MvPowerSeries.LexOrder
 import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
 import Mathlib.RingTheory.MvPowerSeries.Trunc
 import Mathlib.RingTheory.Nakayama
@@ -3958,6 +4076,7 @@ import Mathlib.RingTheory.RingHom.Finite
 import Mathlib.RingTheory.RingHom.FinitePresentation
 import Mathlib.RingTheory.RingHom.FiniteType
 import Mathlib.RingTheory.RingHom.Integral
+import Mathlib.RingTheory.RingHom.Locally
 import Mathlib.RingTheory.RingHom.Surjective
 import Mathlib.RingTheory.RingHomProperties
 import Mathlib.RingTheory.RingInvo
@@ -3966,6 +4085,8 @@ import Mathlib.RingTheory.RootsOfUnity.Complex
 import Mathlib.RingTheory.RootsOfUnity.Lemmas
 import Mathlib.RingTheory.RootsOfUnity.Minpoly
 import Mathlib.RingTheory.SimpleModule
+import Mathlib.RingTheory.SimpleRing.Basic
+import Mathlib.RingTheory.SimpleRing.Defs
 import Mathlib.RingTheory.Smooth.Basic
 import Mathlib.RingTheory.Smooth.Kaehler
 import Mathlib.RingTheory.Smooth.StandardSmooth
@@ -3976,11 +4097,16 @@ import Mathlib.RingTheory.TensorProduct.MvPolynomial
 import Mathlib.RingTheory.Trace.Basic
 import Mathlib.RingTheory.Trace.Defs
 import Mathlib.RingTheory.TwoSidedIdeal.Basic
+import Mathlib.RingTheory.TwoSidedIdeal.BigOperators
+import Mathlib.RingTheory.TwoSidedIdeal.Instances
 import Mathlib.RingTheory.TwoSidedIdeal.Lattice
+import Mathlib.RingTheory.TwoSidedIdeal.Operations
 import Mathlib.RingTheory.UniqueFactorizationDomain
 import Mathlib.RingTheory.Unramified.Basic
 import Mathlib.RingTheory.Unramified.Derivations
+import Mathlib.RingTheory.Unramified.Field
 import Mathlib.RingTheory.Unramified.Finite
+import Mathlib.RingTheory.Unramified.Pi
 import Mathlib.RingTheory.Valuation.AlgebraInstances
 import Mathlib.RingTheory.Valuation.Basic
 import Mathlib.RingTheory.Valuation.ExtendToLocalization
@@ -3991,6 +4117,7 @@ import Mathlib.RingTheory.Valuation.PrimeMultiplicity
 import Mathlib.RingTheory.Valuation.Quotient
 import Mathlib.RingTheory.Valuation.RamificationGroup
 import Mathlib.RingTheory.Valuation.RankOne
+import Mathlib.RingTheory.Valuation.ValExtension
 import Mathlib.RingTheory.Valuation.ValuationRing
 import Mathlib.RingTheory.Valuation.ValuationSubring
 import Mathlib.RingTheory.WittVector.Basic
@@ -4039,10 +4166,12 @@ import Mathlib.SetTheory.Lists
 import Mathlib.SetTheory.Ordinal.Arithmetic
 import Mathlib.SetTheory.Ordinal.Basic
 import Mathlib.SetTheory.Ordinal.CantorNormalForm
+import Mathlib.SetTheory.Ordinal.Enum
 import Mathlib.SetTheory.Ordinal.Exponential
 import Mathlib.SetTheory.Ordinal.FixedPoint
 import Mathlib.SetTheory.Ordinal.FixedPointApproximants
 import Mathlib.SetTheory.Ordinal.NaturalOps
+import Mathlib.SetTheory.Ordinal.Nimber
 import Mathlib.SetTheory.Ordinal.Notation
 import Mathlib.SetTheory.Ordinal.Principal
 import Mathlib.SetTheory.Ordinal.Topology
@@ -4051,9 +4180,12 @@ import Mathlib.SetTheory.Surreal.Dyadic
 import Mathlib.SetTheory.Surreal.Multiplication
 import Mathlib.SetTheory.ZFC.Basic
 import Mathlib.SetTheory.ZFC.Ordinal
+import Mathlib.SetTheory.ZFC.Rank
+import Mathlib.Std.Data.HashMap
 import Mathlib.Tactic
 import Mathlib.Tactic.Abel
 import Mathlib.Tactic.AdaptationNote
+import Mathlib.Tactic.Algebraize
 import Mathlib.Tactic.ApplyAt
 import Mathlib.Tactic.ApplyCongr
 import Mathlib.Tactic.ApplyFun
@@ -4076,13 +4208,25 @@ import Mathlib.Tactic.CancelDenoms.Core
 import Mathlib.Tactic.Cases
 import Mathlib.Tactic.CasesM
 import Mathlib.Tactic.CategoryTheory.BicategoricalComp
+import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
+import Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes
+import Mathlib.Tactic.CategoryTheory.Bicategory.Normalize
+import Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence
 import Mathlib.Tactic.CategoryTheory.BicategoryCoherence
 import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Coherence.Basic
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+import Mathlib.Tactic.CategoryTheory.Coherence.Normalize
+import Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence
 import Mathlib.Tactic.CategoryTheory.Elementwise
-import Mathlib.Tactic.CategoryTheory.Monoidal
+import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
+import Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes
+import Mathlib.Tactic.CategoryTheory.Monoidal.Normalize
+import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
 import Mathlib.Tactic.CategoryTheory.MonoidalComp
 import Mathlib.Tactic.CategoryTheory.Reassoc
 import Mathlib.Tactic.CategoryTheory.Slice
+import Mathlib.Tactic.CategoryTheory.ToApp
 import Mathlib.Tactic.Change
 import Mathlib.Tactic.Check
 import Mathlib.Tactic.Choose
@@ -4138,6 +4282,7 @@ import Mathlib.Tactic.FunProp.ToBatteries
 import Mathlib.Tactic.FunProp.Types
 import Mathlib.Tactic.GCongr
 import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 import Mathlib.Tactic.GCongr.ForwardAttr
 import Mathlib.Tactic.Generalize
 import Mathlib.Tactic.GeneralizeProofs
@@ -4172,14 +4317,18 @@ import Mathlib.Tactic.Linarith.Preprocessing
 import Mathlib.Tactic.Linarith.Verification
 import Mathlib.Tactic.LinearCombination
 import Mathlib.Tactic.LinearCombination'
+import Mathlib.Tactic.LinearCombination.Lemmas
 import Mathlib.Tactic.Linter
 import Mathlib.Tactic.Linter.AdmitLinter
+import Mathlib.Tactic.Linter.DocPrime
+import Mathlib.Tactic.Linter.FlexibleLinter
 import Mathlib.Tactic.Linter.GlobalAttributeIn
 import Mathlib.Tactic.Linter.HashCommandLinter
 import Mathlib.Tactic.Linter.HaveLetLinter
 import Mathlib.Tactic.Linter.Lint
 import Mathlib.Tactic.Linter.MinImports
 import Mathlib.Tactic.Linter.OldObtain
+import Mathlib.Tactic.Linter.PPRoundtrip
 import Mathlib.Tactic.Linter.RefineLinter
 import Mathlib.Tactic.Linter.Style
 import Mathlib.Tactic.Linter.TextBased
@@ -4189,6 +4338,7 @@ import Mathlib.Tactic.Measurability.Init
 import Mathlib.Tactic.MinImports
 import Mathlib.Tactic.MkIffOfInductiveProp
 import Mathlib.Tactic.ModCases
+import Mathlib.Tactic.Module
 import Mathlib.Tactic.Monotonicity
 import Mathlib.Tactic.Monotonicity.Attr
 import Mathlib.Tactic.Monotonicity.Basic
@@ -4244,6 +4394,7 @@ import Mathlib.Tactic.RewriteSearch
 import Mathlib.Tactic.Rify
 import Mathlib.Tactic.Ring
 import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.Compare
 import Mathlib.Tactic.Ring.PNat
 import Mathlib.Tactic.Ring.RingNF
 import Mathlib.Tactic.Sat.FromLRAT
@@ -4298,6 +4449,8 @@ import Mathlib.Topology.AlexandrovDiscrete
 import Mathlib.Topology.Algebra.Affine
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Topology.Algebra.Algebra.Rat
+import Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic
+import Mathlib.Topology.Algebra.ClosedSubgroup
 import Mathlib.Topology.Algebra.ConstMulAction
 import Mathlib.Topology.Algebra.Constructions
 import Mathlib.Topology.Algebra.Constructions.DomMulAct
@@ -4309,6 +4462,7 @@ import Mathlib.Topology.Algebra.FilterBasis
 import Mathlib.Topology.Algebra.Group.Basic
 import Mathlib.Topology.Algebra.Group.Compact
 import Mathlib.Topology.Algebra.Group.OpenMapping
+import Mathlib.Topology.Algebra.Group.SubmonoidClosure
 import Mathlib.Topology.Algebra.Group.TopologicalAbelianization
 import Mathlib.Topology.Algebra.GroupCompletion
 import Mathlib.Topology.Algebra.GroupWithZero
@@ -4326,6 +4480,7 @@ import Mathlib.Topology.Algebra.InfiniteSum.Real
 import Mathlib.Topology.Algebra.InfiniteSum.Ring
 import Mathlib.Topology.Algebra.Localization
 import Mathlib.Topology.Algebra.Module.Alternating.Basic
+import Mathlib.Topology.Algebra.Module.Alternating.Topology
 import Mathlib.Topology.Algebra.Module.Basic
 import Mathlib.Topology.Algebra.Module.Cardinality
 import Mathlib.Topology.Algebra.Module.CharacterSpace
@@ -4340,6 +4495,7 @@ import Mathlib.Topology.Algebra.Module.Simple
 import Mathlib.Topology.Algebra.Module.Star
 import Mathlib.Topology.Algebra.Module.StrongTopology
 import Mathlib.Topology.Algebra.Module.UniformConvergence
+import Mathlib.Topology.Algebra.Module.WeakBilin
 import Mathlib.Topology.Algebra.Module.WeakDual
 import Mathlib.Topology.Algebra.Monoid
 import Mathlib.Topology.Algebra.MulAction
@@ -4377,6 +4533,7 @@ import Mathlib.Topology.Algebra.Valued.NormedValued
 import Mathlib.Topology.Algebra.Valued.ValuationTopology
 import Mathlib.Topology.Algebra.Valued.ValuedField
 import Mathlib.Topology.Algebra.WithZeroTopology
+import Mathlib.Topology.Baire.BaireMeasurable
 import Mathlib.Topology.Baire.CompleteMetrizable
 import Mathlib.Topology.Baire.Lemmas
 import Mathlib.Topology.Baire.LocallyCompactRegular
@@ -4387,6 +4544,7 @@ import Mathlib.Topology.Bornology.Basic
 import Mathlib.Topology.Bornology.BoundedOperation
 import Mathlib.Topology.Bornology.Constructions
 import Mathlib.Topology.Bornology.Hom
+import Mathlib.Topology.CWComplex
 import Mathlib.Topology.Category.Born
 import Mathlib.Topology.Category.CompHaus.Basic
 import Mathlib.Topology.Category.CompHaus.EffectiveEpi
@@ -4395,12 +4553,14 @@ import Mathlib.Topology.Category.CompHaus.Projective
 import Mathlib.Topology.Category.CompHausLike.Basic
 import Mathlib.Topology.Category.CompHausLike.EffectiveEpi
 import Mathlib.Topology.Category.CompHausLike.Limits
+import Mathlib.Topology.Category.CompHausLike.SigmaComparison
 import Mathlib.Topology.Category.CompactlyGenerated
 import Mathlib.Topology.Category.Compactum
 import Mathlib.Topology.Category.FinTopCat
 import Mathlib.Topology.Category.LightProfinite.AsLimit
 import Mathlib.Topology.Category.LightProfinite.Basic
 import Mathlib.Topology.Category.LightProfinite.EffectiveEpi
+import Mathlib.Topology.Category.LightProfinite.Extend
 import Mathlib.Topology.Category.LightProfinite.Limits
 import Mathlib.Topology.Category.LightProfinite.Sequence
 import Mathlib.Topology.Category.Locale
@@ -4408,6 +4568,7 @@ import Mathlib.Topology.Category.Profinite.AsLimit
 import Mathlib.Topology.Category.Profinite.Basic
 import Mathlib.Topology.Category.Profinite.CofilteredLimit
 import Mathlib.Topology.Category.Profinite.EffectiveEpi
+import Mathlib.Topology.Category.Profinite.Extend
 import Mathlib.Topology.Category.Profinite.Limits
 import Mathlib.Topology.Category.Profinite.Nobeling
 import Mathlib.Topology.Category.Profinite.Product
@@ -4428,6 +4589,7 @@ import Mathlib.Topology.Category.TopCat.Limits.Products
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 import Mathlib.Topology.Category.TopCat.OpenNhds
 import Mathlib.Topology.Category.TopCat.Opens
+import Mathlib.Topology.Category.TopCat.Sphere
 import Mathlib.Topology.Category.TopCat.Yoneda
 import Mathlib.Topology.Category.TopCommRingCat
 import Mathlib.Topology.Category.UniformSpace
@@ -4437,6 +4599,7 @@ import Mathlib.Topology.CompactOpen
 import Mathlib.Topology.Compactification.OnePoint
 import Mathlib.Topology.Compactness.Compact
 import Mathlib.Topology.Compactness.CompactlyGeneratedSpace
+import Mathlib.Topology.Compactness.Exterior
 import Mathlib.Topology.Compactness.Lindelof
 import Mathlib.Topology.Compactness.LocallyCompact
 import Mathlib.Topology.Compactness.Paracompact
@@ -4450,24 +4613,26 @@ import Mathlib.Topology.Connected.PathConnected
 import Mathlib.Topology.Connected.Separation
 import Mathlib.Topology.Connected.TotallyDisconnected
 import Mathlib.Topology.Constructions
-import Mathlib.Topology.ContinuousFunction.Algebra
-import Mathlib.Topology.ContinuousFunction.Basic
-import Mathlib.Topology.ContinuousFunction.Bounded
-import Mathlib.Topology.ContinuousFunction.CocompactMap
-import Mathlib.Topology.ContinuousFunction.Compact
-import Mathlib.Topology.ContinuousFunction.CompactlySupported
-import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
-import Mathlib.Topology.ContinuousFunction.Ideals
-import Mathlib.Topology.ContinuousFunction.LocallyConstant
-import Mathlib.Topology.ContinuousFunction.Ordered
-import Mathlib.Topology.ContinuousFunction.Polynomial
-import Mathlib.Topology.ContinuousFunction.Sigma
-import Mathlib.Topology.ContinuousFunction.StarOrdered
-import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
-import Mathlib.Topology.ContinuousFunction.T0Sierpinski
-import Mathlib.Topology.ContinuousFunction.Units
-import Mathlib.Topology.ContinuousFunction.Weierstrass
-import Mathlib.Topology.ContinuousFunction.ZeroAtInfty
+import Mathlib.Topology.ContinuousMap.Algebra
+import Mathlib.Topology.ContinuousMap.Basic
+import Mathlib.Topology.ContinuousMap.Bounded
+import Mathlib.Topology.ContinuousMap.BoundedCompactlySupported
+import Mathlib.Topology.ContinuousMap.CocompactMap
+import Mathlib.Topology.ContinuousMap.Compact
+import Mathlib.Topology.ContinuousMap.CompactlySupported
+import Mathlib.Topology.ContinuousMap.ContinuousMapZero
+import Mathlib.Topology.ContinuousMap.Defs
+import Mathlib.Topology.ContinuousMap.Ideals
+import Mathlib.Topology.ContinuousMap.LocallyConstant
+import Mathlib.Topology.ContinuousMap.Ordered
+import Mathlib.Topology.ContinuousMap.Polynomial
+import Mathlib.Topology.ContinuousMap.Sigma
+import Mathlib.Topology.ContinuousMap.StarOrdered
+import Mathlib.Topology.ContinuousMap.StoneWeierstrass
+import Mathlib.Topology.ContinuousMap.T0Sierpinski
+import Mathlib.Topology.ContinuousMap.Units
+import Mathlib.Topology.ContinuousMap.Weierstrass
+import Mathlib.Topology.ContinuousMap.ZeroAtInfty
 import Mathlib.Topology.ContinuousOn
 import Mathlib.Topology.CountableSeparatingOn
 import Mathlib.Topology.Covering
@@ -4484,12 +4649,15 @@ import Mathlib.Topology.EMetricSpace.Defs
 import Mathlib.Topology.EMetricSpace.Diam
 import Mathlib.Topology.EMetricSpace.Lipschitz
 import Mathlib.Topology.EMetricSpace.Paracompact
+import Mathlib.Topology.EMetricSpace.Pi
 import Mathlib.Topology.ExtendFrom
+import Mathlib.Topology.Exterior
 import Mathlib.Topology.ExtremallyDisconnected
 import Mathlib.Topology.FiberBundle.Basic
 import Mathlib.Topology.FiberBundle.Constructions
 import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
 import Mathlib.Topology.FiberBundle.Trivialization
+import Mathlib.Topology.FiberPartition
 import Mathlib.Topology.Filter
 import Mathlib.Topology.GDelta
 import Mathlib.Topology.Germ
@@ -4535,6 +4703,7 @@ import Mathlib.Topology.LocallyConstant.Algebra
 import Mathlib.Topology.LocallyConstant.Basic
 import Mathlib.Topology.LocallyFinite
 import Mathlib.Topology.Maps.Basic
+import Mathlib.Topology.Maps.OpenQuotient
 import Mathlib.Topology.Maps.Proper.Basic
 import Mathlib.Topology.Maps.Proper.UniversallyClosed
 import Mathlib.Topology.MetricSpace.Algebra
@@ -4581,6 +4750,8 @@ import Mathlib.Topology.MetricSpace.ShrinkingLemma
 import Mathlib.Topology.MetricSpace.ThickenedIndicator
 import Mathlib.Topology.MetricSpace.Thickening
 import Mathlib.Topology.MetricSpace.Ultra.Basic
+import Mathlib.Topology.MetricSpace.Ultra.ContinuousMaps
+import Mathlib.Topology.MetricSpace.Ultra.TotallySeparated
 import Mathlib.Topology.Metrizable.Basic
 import Mathlib.Topology.Metrizable.ContinuousMap
 import Mathlib.Topology.Metrizable.Uniformity
@@ -4624,8 +4795,9 @@ import Mathlib.Topology.Partial
 import Mathlib.Topology.PartialHomeomorph
 import Mathlib.Topology.PartitionOfUnity
 import Mathlib.Topology.Perfect
+import Mathlib.Topology.PreorderRestrict
 import Mathlib.Topology.QuasiSeparated
-import Mathlib.Topology.RestrictGenTopology
+import Mathlib.Topology.RestrictGen
 import Mathlib.Topology.Semicontinuous
 import Mathlib.Topology.SeparatedMap
 import Mathlib.Topology.Separation
@@ -4676,6 +4848,7 @@ import Mathlib.Topology.UniformSpace.Completion
 import Mathlib.Topology.UniformSpace.Equicontinuity
 import Mathlib.Topology.UniformSpace.Equiv
 import Mathlib.Topology.UniformSpace.Matrix
+import Mathlib.Topology.UniformSpace.OfFun
 import Mathlib.Topology.UniformSpace.Pi
 import Mathlib.Topology.UniformSpace.Separation
 import Mathlib.Topology.UniformSpace.UniformConvergence
@@ -4709,6 +4882,5 @@ import Mathlib.Util.Superscript
 import Mathlib.Util.SynthesizeUsing
 import Mathlib.Util.Tactic
 import Mathlib.Util.TermBeta
-import Mathlib.Util.Time
 import Mathlib.Util.WhatsNew
 import Mathlib.Util.WithWeakNamespace
diff --git a/Mathlib/Algebra/AddTorsor.lean b/Mathlib/Algebra/AddTorsor.lean
index c0ef691f61fae..ed65a54d0f15f 100644
--- a/Mathlib/Algebra/AddTorsor.lean
+++ b/Mathlib/Algebra/AddTorsor.lean
@@ -3,7 +3,8 @@ Copyright (c) 2020 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers, Yury Kudryashov
 -/
-import Mathlib.Data.Set.Pointwise.SMul
+import Mathlib.Algebra.Group.Action.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 
 /-!
 # Torsors of additive group actions
@@ -247,7 +248,6 @@ instance instAddTorsor : AddTorsor (G × G') (P × P') where
   zero_vadd _ := Prod.ext (zero_vadd _ _) (zero_vadd _ _)
   add_vadd _ _ _ := Prod.ext (add_vadd _ _ _) (add_vadd _ _ _)
   vsub p₁ p₂ := (p₁.1 -ᵥ p₂.1, p₁.2 -ᵥ p₂.2)
-  nonempty := Prod.instNonempty
   vsub_vadd' _ _ := Prod.ext (vsub_vadd _ _) (vsub_vadd _ _)
   vadd_vsub' _ _ := Prod.ext (vadd_vsub _ _) (vadd_vsub _ _)
 
diff --git a/Mathlib/Algebra/Algebra/Basic.lean b/Mathlib/Algebra/Algebra/Basic.lean
index 95c18defa9bdb..f4a24f5977006 100644
--- a/Mathlib/Algebra/Algebra/Basic.lean
+++ b/Mathlib/Algebra/Algebra/Basic.lean
@@ -23,12 +23,12 @@ universe u v w u₁ v₁
 
 namespace Algebra
 
-variable {R : Type u} {S : Type v} {A : Type w} {B : Type*}
+variable {R : Type u} {A : Type w}
 
 section Semiring
 
-variable [CommSemiring R] [CommSemiring S]
-variable [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
+variable [CommSemiring R]
+variable [Semiring A] [Algebra R A]
 
 section PUnit
 
@@ -185,8 +185,7 @@ theorem End_algebraMap_isUnit_inv_apply_eq_iff {x : R}
     mpr := fun H =>
       H.symm ▸ by
         apply_fun ⇑h.unit.val using ((Module.End_isUnit_iff _).mp h).injective
-        erw [End_isUnit_apply_inv_apply_of_isUnit]
-        rfl }
+        simpa using End_isUnit_apply_inv_apply_of_isUnit h (x • m') }
 
 theorem End_algebraMap_isUnit_inv_apply_eq_iff' {x : R}
     (h : IsUnit (algebraMap R (Module.End S M) x)) (m m' : M) :
@@ -195,8 +194,7 @@ theorem End_algebraMap_isUnit_inv_apply_eq_iff' {x : R}
     mpr := fun H =>
       H.symm ▸ by
         apply_fun (↑h.unit : M → M) using ((Module.End_isUnit_iff _).mp h).injective
-        erw [End_isUnit_apply_inv_apply_of_isUnit]
-        rfl }
+        simpa using End_isUnit_apply_inv_apply_of_isUnit h (x • m') |>.symm }
 
 end
 
@@ -284,6 +282,16 @@ theorem algebraMap_injective [CommRing R] [Ring A] [Nontrivial A] [Algebra R A]
     [NoZeroSMulDivisors R A] : Function.Injective (algebraMap R A) := by
   simpa only [algebraMap_eq_smul_one'] using smul_left_injective R one_ne_zero
 
+@[simp]
+lemma algebraMap_eq_zero_iff [CommRing R] [Ring A] [Nontrivial A] [Algebra R A]
+    [NoZeroSMulDivisors R A] {r : R} : algebraMap R A r = 0 ↔ r = 0 :=
+  map_eq_zero_iff _ <| algebraMap_injective R A
+
+@[simp]
+lemma algebraMap_eq_one_iff [CommRing R] [Ring A] [Nontrivial A] [Algebra R A]
+    [NoZeroSMulDivisors R A] {r : R} : algebraMap R A r = 1 ↔ r = 1 :=
+  map_eq_one_iff _ <| algebraMap_injective R A
+
 theorem _root_.NeZero.of_noZeroSMulDivisors (n : ℕ) [CommRing R] [NeZero (n : R)] [Ring A]
     [Nontrivial A] [Algebra R A] [NoZeroSMulDivisors R A] : NeZero (n : A) :=
   NeZero.nat_of_injective <| NoZeroSMulDivisors.algebraMap_injective R A
@@ -304,17 +312,6 @@ instance (priority := 100) CharZero.noZeroSMulDivisors_int [Ring R] [NoZeroDivis
     [CharZero R] : NoZeroSMulDivisors ℤ R :=
   NoZeroSMulDivisors.of_algebraMap_injective <| (algebraMap ℤ R).injective_int
 
-section Field
-
-variable [Field R] [Semiring A] [Algebra R A]
-
--- see note [lower instance priority]
-instance (priority := 100) Algebra.noZeroSMulDivisors [Nontrivial A] [NoZeroDivisors A] :
-    NoZeroSMulDivisors R A :=
-  NoZeroSMulDivisors.of_algebraMap_injective (algebraMap R A).injective
-
-end Field
-
 end NoZeroSMulDivisors
 
 section IsScalarTower
@@ -322,7 +319,6 @@ section IsScalarTower
 variable {R : Type*} [CommSemiring R]
 variable (A : Type*) [Semiring A] [Algebra R A]
 variable {M : Type*} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M]
-variable {N : Type*} [AddCommMonoid N] [Module A N] [Module R N] [IsScalarTower R A N]
 
 theorem algebra_compatible_smul (r : R) (m : M) : r • m = (algebraMap R A) r • m := by
   rw [← one_smul A m, ← smul_assoc, Algebra.smul_def, mul_one, one_smul]
@@ -512,3 +508,21 @@ lemma LinearEquiv.extendScalarsOfSurjective_symm (f : M ≃ₗ[R] N) :
     (f.extendScalarsOfSurjective h).symm = f.symm.extendScalarsOfSurjective h := rfl
 
 end surjective
+
+namespace algebraMap
+
+section CommSemiringCommSemiring
+
+variable {R A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A] {ι : Type*} {s : Finset ι}
+
+@[norm_cast]
+theorem coe_prod (a : ι → R) : (↑(∏ i ∈ s, a i : R) : A) = ∏ i ∈ s, (↑(a i) : A) :=
+  map_prod (algebraMap R A) a s
+
+@[norm_cast]
+theorem coe_sum (a : ι → R) : ↑(∑ i ∈ s, a i) = ∑ i ∈ s, (↑(a i) : A) :=
+  map_sum (algebraMap R A) a s
+
+end CommSemiringCommSemiring
+
+end algebraMap
diff --git a/Mathlib/Algebra/Algebra/Defs.lean b/Mathlib/Algebra/Algebra/Defs.lean
index 10be082a45ea6..0393439d7bb1f 100644
--- a/Mathlib/Algebra/Algebra/Defs.lean
+++ b/Mathlib/Algebra/Algebra/Defs.lean
@@ -3,7 +3,6 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Yury Kudryashov
 -/
-import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Algebra.Module.LinearMap.Defs
 
 /-!
@@ -82,6 +81,7 @@ the second approach only when you need to weaken a condition on either `R` or `A
 -/
 
 assert_not_exists Field
+assert_not_exists Finset
 assert_not_exists Module.End
 
 universe u v w u₁ v₁
@@ -165,26 +165,6 @@ theorem coe_sub (a b : R) :
 
 end CommRingRing
 
-section CommSemiringCommSemiring
-
-variable {R A : Type*} [CommSemiring R] [CommSemiring A] [Algebra R A]
-
--- direct to_additive fails because of some mix-up with polynomials
-@[norm_cast]
-theorem coe_prod {ι : Type*} {s : Finset ι} (a : ι → R) :
-    (↑(∏ i ∈ s, a i : R) : A) = ∏ i ∈ s, (↑(a i) : A) :=
-  map_prod (algebraMap R A) a s
-
--- to_additive fails for some reason
-@[norm_cast]
-theorem coe_sum {ι : Type*} {s : Finset ι} (a : ι → R) :
-    ↑(∑ i ∈ s, a i) = ∑ i ∈ s, (↑(a i) : A) :=
-  map_sum (algebraMap R A) a s
-
--- Porting note: removed attribute [to_additive] coe_prod; why should this be a `to_additive`?
-
-end CommSemiringCommSemiring
-
 end algebraMap
 
 /-- Creating an algebra from a morphism to the center of a semiring. -/
diff --git a/Mathlib/Algebra/Algebra/Equiv.lean b/Mathlib/Algebra/Algebra/Equiv.lean
index 1336dd54b5ede..d19412446121d 100644
--- a/Mathlib/Algebra/Algebra/Equiv.lean
+++ b/Mathlib/Algebra/Algebra/Equiv.lean
@@ -86,6 +86,8 @@ variable [Algebra R A₁] [Algebra R A₂] [Algebra R A₃]
 variable [Algebra R A₁'] [Algebra R A₂'] [Algebra R A₃']
 variable (e : A₁ ≃ₐ[R] A₂)
 
+section coe
+
 instance : EquivLike (A₁ ≃ₐ[R] A₂) A₁ A₂ where
   coe f := f.toFun
   inv f := f.invFun
@@ -106,22 +108,6 @@ instance : AlgEquivClass (A₁ ≃ₐ[R] A₂) R A₁ A₂ where
   map_mul f := f.map_mul'
   commutes f := f.commutes'
 
--- Porting note: the default simps projection was `e.toEquiv.toFun`, it should be `FunLike.coe`
-/-- See Note [custom simps projection] -/
-def Simps.apply (e : A₁ ≃ₐ[R] A₂) : A₁ → A₂ :=
-  e
-
--- Porting note: the default simps projection was `e.toEquiv`, it should be `EquivLike.toEquiv`
-/-- See Note [custom simps projection] -/
-def Simps.toEquiv (e : A₁ ≃ₐ[R] A₂) : A₁ ≃ A₂ :=
-  e
-
--- Porting note: `protected` used to be an attribute below
-@[simp]
-protected theorem coe_coe {F : Type*} [EquivLike F A₁ A₂] [AlgEquivClass F R A₁ A₂] (f : F) :
-    ⇑(f : A₁ ≃ₐ[R] A₂) = f :=
-  rfl
-
 @[ext]
 theorem ext {f g : A₁ ≃ₐ[R] A₂} (h : ∀ a, f a = g a) : f = g :=
   DFunLike.ext f g h
@@ -132,13 +118,6 @@ protected theorem congr_arg {f : A₁ ≃ₐ[R] A₂} {x x' : A₁} : x = x' →
 protected theorem congr_fun {f g : A₁ ≃ₐ[R] A₂} (h : f = g) (x : A₁) : f x = g x :=
   DFunLike.congr_fun h x
 
-theorem coe_fun_injective : @Function.Injective (A₁ ≃ₐ[R] A₂) (A₁ → A₂) fun e => (e : A₁ → A₂) :=
-  DFunLike.coe_injective
-
--- Porting note: Made to CoeOut instance from Coe, not dangerous anymore
-instance hasCoeToRingEquiv : CoeOut (A₁ ≃ₐ[R] A₂) (A₁ ≃+* A₂) :=
-  ⟨AlgEquiv.toRingEquiv⟩
-
 @[simp]
 theorem coe_mk {toEquiv map_mul map_add commutes} :
     ⇑(⟨toEquiv, map_mul, map_add, commutes⟩ : A₁ ≃ₐ[R] A₂) = toEquiv :=
@@ -149,12 +128,25 @@ theorem mk_coe (e : A₁ ≃ₐ[R] A₂) (e' h₁ h₂ h₃ h₄ h₅) :
     (⟨⟨e, e', h₁, h₂⟩, h₃, h₄, h₅⟩ : A₁ ≃ₐ[R] A₂) = e :=
   ext fun _ => rfl
 
--- Porting note: `toFun_eq_coe` no longer needed in Lean4
-
 @[simp]
 theorem toEquiv_eq_coe : e.toEquiv = e :=
   rfl
 
+-- Porting note: `protected` used to be an attribute below
+@[simp]
+protected theorem coe_coe {F : Type*} [EquivLike F A₁ A₂] [AlgEquivClass F R A₁ A₂] (f : F) :
+    ⇑(f : A₁ ≃ₐ[R] A₂) = f :=
+  rfl
+
+theorem coe_fun_injective : @Function.Injective (A₁ ≃ₐ[R] A₂) (A₁ → A₂) fun e => (e : A₁ → A₂) :=
+  DFunLike.coe_injective
+
+-- Porting note: Made to CoeOut instance from Coe, not dangerous anymore
+instance hasCoeToRingEquiv : CoeOut (A₁ ≃ₐ[R] A₂) (A₁ ≃+* A₂) :=
+  ⟨AlgEquiv.toRingEquiv⟩
+
+-- Porting note: `toFun_eq_coe` no longer needed in Lean4
+
 @[simp]
 theorem toRingEquiv_eq_coe : e.toRingEquiv = e :=
   rfl
@@ -173,41 +165,6 @@ theorem coe_ringEquiv' : (e.toRingEquiv : A₁ → A₂) = e :=
 theorem coe_ringEquiv_injective : Function.Injective ((↑) : (A₁ ≃ₐ[R] A₂) → A₁ ≃+* A₂) :=
   fun _ _ h => ext <| RingEquiv.congr_fun h
 
-@[deprecated map_add (since := "2024-06-20")]
-protected theorem map_add : ∀ x y, e (x + y) = e x + e y :=
-  map_add e
-
-@[deprecated map_zero (since := "2024-06-20")]
-protected theorem map_zero : e 0 = 0 :=
-  map_zero e
-
-@[deprecated map_mul (since := "2024-06-20")]
-protected theorem map_mul : ∀ x y, e (x * y) = e x * e y :=
-  map_mul e
-
-@[deprecated map_one (since := "2024-06-20")]
-protected theorem map_one : e 1 = 1 :=
-  map_one e
-
-@[simp]
-theorem commutes : ∀ r : R, e (algebraMap R A₁ r) = algebraMap R A₂ r :=
-  e.commutes'
-
--- @[simp] -- Porting note (#10618): simp can prove this
-@[deprecated map_smul (since := "2024-06-20")]
-protected theorem map_smul (r : R) (x : A₁) : e (r • x) = r • e x :=
-  map_smul _ _ _
-
-@[deprecated map_sum (since := "2023-12-26")]
-protected theorem map_sum {ι : Type*} (f : ι → A₁) (s : Finset ι) :
-    e (∑ x ∈ s, f x) = ∑ x ∈ s, e (f x) :=
-  map_sum e f s
-
-@[deprecated map_finsupp_sum (since := "2024-06-20")]
-protected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A₁) :
-    e (f.sum g) = f.sum fun i b => e (g i b) :=
-  map_finsupp_sum _ _ _
-
 -- Porting note: Added [coe] attribute
 /-- Interpret an algebra equivalence as an algebra homomorphism.
 
@@ -238,18 +195,65 @@ lemma toAlgHom_toRingHom : ((e : A₁ →ₐ[R] A₂) : A₁ →+* A₂) = e :=
 theorem coe_ringHom_commutes : ((e : A₁ →ₐ[R] A₂) : A₁ →+* A₂) = ((e : A₁ ≃+* A₂) : A₁ →+* A₂) :=
   rfl
 
+@[simp]
+theorem commutes : ∀ r : R, e (algebraMap R A₁ r) = algebraMap R A₂ r :=
+  e.commutes'
+
+end coe
+
+section map
+
+@[deprecated map_add (since := "2024-06-20")]
+protected theorem map_add : ∀ x y, e (x + y) = e x + e y :=
+  map_add e
+
+@[deprecated map_zero (since := "2024-06-20")]
+protected theorem map_zero : e 0 = 0 :=
+  map_zero e
+
+@[deprecated map_mul (since := "2024-06-20")]
+protected theorem map_mul : ∀ x y, e (x * y) = e x * e y :=
+  map_mul e
+
+@[deprecated map_one (since := "2024-06-20")]
+protected theorem map_one : e 1 = 1 :=
+  map_one e
+
+-- @[simp] -- Porting note (#10618): simp can prove this
+@[deprecated map_smul (since := "2024-06-20")]
+protected theorem map_smul (r : R) (x : A₁) : e (r • x) = r • e x :=
+  map_smul _ _ _
+
 @[deprecated map_pow (since := "2024-06-20")]
 protected theorem map_pow : ∀ (x : A₁) (n : ℕ), e (x ^ n) = e x ^ n :=
   map_pow _
 
+@[deprecated map_sum (since := "2023-12-26")]
+protected theorem map_sum {ι : Type*} (f : ι → A₁) (s : Finset ι) :
+    e (∑ x ∈ s, f x) = ∑ x ∈ s, e (f x) :=
+  map_sum e f s
+
+@[deprecated map_finsupp_sum (since := "2024-06-20")]
+protected theorem map_finsupp_sum {α : Type*} [Zero α] {ι : Type*} (f : ι →₀ α) (g : ι → α → A₁) :
+    e (f.sum g) = f.sum fun i b => e (g i b) :=
+  map_finsupp_sum _ _ _
+
+end map
+
+section bijective
+
+protected theorem bijective : Function.Bijective e :=
+  EquivLike.bijective e
+
 protected theorem injective : Function.Injective e :=
   EquivLike.injective e
 
 protected theorem surjective : Function.Surjective e :=
   EquivLike.surjective e
 
-protected theorem bijective : Function.Bijective e :=
-  EquivLike.bijective e
+end bijective
+
+section refl
 
 /-- Algebra equivalences are reflexive. -/
 @[refl]
@@ -267,6 +271,10 @@ theorem refl_toAlgHom : ↑(refl : A₁ ≃ₐ[R] A₁) = AlgHom.id R A₁ :=
 theorem coe_refl : ⇑(refl : A₁ ≃ₐ[R] A₁) = id :=
   rfl
 
+end refl
+
+section symm
+
 /-- Algebra equivalences are symmetric. -/
 @[symm]
 def symm (e : A₁ ≃ₐ[R] A₂) : A₂ ≃ₐ[R] A₁ :=
@@ -277,11 +285,8 @@ def symm (e : A₁ ≃ₐ[R] A₂) : A₂ ≃ₐ[R] A₁ :=
       change _ = e _
       rw [e.commutes] }
 
-/-- See Note [custom simps projection] -/
-def Simps.symm_apply (e : A₁ ≃ₐ[R] A₂) : A₂ → A₁ :=
-  e.symm
-
-initialize_simps_projections AlgEquiv (toFun → apply, invFun → symm_apply)
+theorem invFun_eq_symm {e : A₁ ≃ₐ[R] A₂} : e.invFun = e.symm :=
+  rfl
 
 --@[simp] -- Porting note (#10618): simp can prove this once symm_mk is introduced
 theorem coe_apply_coe_coe_symm_apply {F : Type*} [EquivLike F A₁ A₂] [AlgEquivClass F R A₁ A₂]
@@ -300,9 +305,6 @@ theorem coe_coe_symm_apply_coe_apply {F : Type*} [EquivLike F A₁ A₂] [AlgEqu
 theorem symm_toEquiv_eq_symm {e : A₁ ≃ₐ[R] A₂} : (e : A₁ ≃ A₂).symm = e.symm :=
   rfl
 
-theorem invFun_eq_symm {e : A₁ ≃ₐ[R] A₂} : e.invFun = e.symm :=
-  rfl
-
 @[simp]
 theorem symm_symm (e : A₁ ≃ₐ[R] A₂) : e.symm.symm = e := rfl
 
@@ -338,12 +340,6 @@ theorem toRingEquiv_symm (f : A₁ ≃ₐ[R] A₁) : (f : A₁ ≃+* A₁).symm
 theorem symm_toRingEquiv : (e.symm : A₂ ≃+* A₁) = (e : A₁ ≃+* A₂).symm :=
   rfl
 
-/-- Algebra equivalences are transitive. -/
-@[trans]
-def trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : A₁ ≃ₐ[R] A₃ :=
-  { e₁.toRingEquiv.trans e₂.toRingEquiv with
-    commutes' := fun r => show e₂.toFun (e₁.toFun _) = _ by rw [e₁.commutes', e₂.commutes'] }
-
 @[simp]
 theorem apply_symm_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e (e.symm x) = x :=
   e.toEquiv.apply_symm_apply
@@ -352,18 +348,11 @@ theorem apply_symm_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e (e.symm x) = x :=
 theorem symm_apply_apply (e : A₁ ≃ₐ[R] A₂) : ∀ x, e.symm (e x) = x :=
   e.toEquiv.symm_apply_apply
 
-@[simp]
-theorem symm_trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₃) :
-    (e₁.trans e₂).symm x = e₁.symm (e₂.symm x) :=
-  rfl
+theorem symm_apply_eq (e : A₁ ≃ₐ[R] A₂) {x y} : e.symm x = y ↔ x = e y :=
+  e.toEquiv.symm_apply_eq
 
-@[simp]
-theorem coe_trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ :=
-  rfl
-
-@[simp]
-theorem trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₁) : (e₁.trans e₂) x = e₂ (e₁ x) :=
-  rfl
+theorem eq_symm_apply (e : A₁ ≃ₐ[R] A₂) {x y} : y = e.symm x ↔ e y = x :=
+  e.toEquiv.eq_symm_apply
 
 @[simp]
 theorem comp_symm (e : A₁ ≃ₐ[R] A₂) : AlgHom.comp (e : A₁ →ₐ[R] A₂) ↑e.symm = AlgHom.id R A₂ := by
@@ -381,6 +370,51 @@ theorem leftInverse_symm (e : A₁ ≃ₐ[R] A₂) : Function.LeftInverse e.symm
 theorem rightInverse_symm (e : A₁ ≃ₐ[R] A₂) : Function.RightInverse e.symm e :=
   e.right_inv
 
+end symm
+
+section simps
+
+-- Porting note: the default simps projection was `e.toEquiv.toFun`, it should be `FunLike.coe`
+/-- See Note [custom simps projection] -/
+def Simps.apply (e : A₁ ≃ₐ[R] A₂) : A₁ → A₂ :=
+  e
+
+-- Porting note: the default simps projection was `e.toEquiv`, it should be `EquivLike.toEquiv`
+/-- See Note [custom simps projection] -/
+def Simps.toEquiv (e : A₁ ≃ₐ[R] A₂) : A₁ ≃ A₂ :=
+  e
+
+/-- See Note [custom simps projection] -/
+def Simps.symm_apply (e : A₁ ≃ₐ[R] A₂) : A₂ → A₁ :=
+  e.symm
+
+initialize_simps_projections AlgEquiv (toFun → apply, invFun → symm_apply)
+
+end simps
+
+section trans
+
+/-- Algebra equivalences are transitive. -/
+@[trans]
+def trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : A₁ ≃ₐ[R] A₃ :=
+  { e₁.toRingEquiv.trans e₂.toRingEquiv with
+    commutes' := fun r => show e₂.toFun (e₁.toFun _) = _ by rw [e₁.commutes', e₂.commutes'] }
+
+@[simp]
+theorem coe_trans (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ :=
+  rfl
+
+@[simp]
+theorem trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₁) : (e₁.trans e₂) x = e₂ (e₁ x) :=
+  rfl
+
+@[simp]
+theorem symm_trans_apply (e₁ : A₁ ≃ₐ[R] A₂) (e₂ : A₂ ≃ₐ[R] A₃) (x : A₃) :
+    (e₁.trans e₂).symm x = e₁.symm (e₂.symm x) :=
+  rfl
+
+end trans
+
 /-- If `A₁` is equivalent to `A₁'` and `A₂` is equivalent to `A₂'`, then the type of maps
 `A₁ →ₐ[R] A₂` is equivalent to the type of maps `A₁' →ₐ[R] A₂'`. -/
 @[simps apply]
diff --git a/Mathlib/Algebra/Algebra/Hom.lean b/Mathlib/Algebra/Algebra/Hom.lean
index 0f526f05a2fe1..0942b526635de 100644
--- a/Mathlib/Algebra/Algebra/Hom.lean
+++ b/Mathlib/Algebra/Algebra/Hom.lean
@@ -39,7 +39,7 @@ infixr:25 " →ₐ " => AlgHom _
 notation:25 A " →ₐ[" R "] " B => AlgHom R A B
 
 /-- `AlgHomClass F R A B` asserts `F` is a type of bundled algebra homomorphisms
-from `A` to `B`.  -/
+from `A` to `B`. -/
 class AlgHomClass (F : Type*) (R A B : outParam Type*)
   [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
   [FunLike F A B] extends RingHomClass F A B : Prop where
@@ -421,11 +421,25 @@ def toNatAlgHom [Semiring R] [Semiring S] (f : R →+* S) : R →ₐ[ℕ] S :=
     toFun := f
     commutes' := fun n => by simp }
 
+@[simp]
+lemma toNatAlgHom_coe [Semiring R] [Semiring S] (f : R →+* S) :
+    ⇑f.toNatAlgHom = ⇑f := rfl
+
+lemma toNatAlgHom_apply [Semiring R] [Semiring S] (f : R →+* S) (x : R) :
+    f.toNatAlgHom x = f x := rfl
+
 /-- Reinterpret a `RingHom` as a `ℤ`-algebra homomorphism. -/
-def toIntAlgHom [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] (f : R →+* S) : R →ₐ[ℤ] S :=
+def toIntAlgHom [Ring R] [Ring S] (f : R →+* S) : R →ₐ[ℤ] S :=
   { f with commutes' := fun n => by simp }
 
-lemma toIntAlgHom_injective [Ring R] [Ring S] [Algebra ℤ R] [Algebra ℤ S] :
+@[simp]
+lemma toIntAlgHom_coe [Ring R] [Ring S] (f : R →+* S) :
+    ⇑f.toIntAlgHom = ⇑f := rfl
+
+lemma toIntAlgHom_apply [Ring R] [Ring S] (f : R →+* S) (x : R) :
+    f.toIntAlgHom x = f x := rfl
+
+lemma toIntAlgHom_injective [Ring R] [Ring S] :
     Function.Injective (RingHom.toIntAlgHom : (R →+* S) → _) :=
   fun _ _ e ↦ DFunLike.ext _ _ (fun x ↦ DFunLike.congr_fun e x)
 
diff --git a/Mathlib/Algebra/Algebra/NonUnitalHom.lean b/Mathlib/Algebra/Algebra/NonUnitalHom.lean
index b991f1ec71416..34a5518dc4de3 100644
--- a/Mathlib/Algebra/Algebra/NonUnitalHom.lean
+++ b/Mathlib/Algebra/Algebra/NonUnitalHom.lean
@@ -64,7 +64,7 @@ notation:25 A " →ₙₐ[" R "] " B => NonUnitalAlgHom (MonoidHom.id R) A B
 attribute [nolint docBlame] NonUnitalAlgHom.toMulHom
 
 /-- `NonUnitalAlgSemiHomClass F φ A B` asserts `F` is a type of bundled algebra homomorphisms
-from `A` to `B` which are equivariant with respect to `φ`.  -/
+from `A` to `B` which are equivariant with respect to `φ`. -/
 class NonUnitalAlgSemiHomClass (F : Type*) {R S : outParam Type*} [Monoid R] [Monoid S]
     (φ : outParam (R →* S)) (A B : outParam Type*)
     [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B]
@@ -483,7 +483,7 @@ variable (R : Type*) {S A B : Type*} [Monoid R] [Monoid S]
     [IsScalarTower R S A] [IsScalarTower R S B]
 
 /-- If a monoid `R` acts on another monoid `S`, then a non-unital algebra homomorphism
-over `S` can be viewed as a non-unital algebra homomorphism over `R`.  -/
+over `S` can be viewed as a non-unital algebra homomorphism over `R`. -/
 def restrictScalars (f : A →ₙₐ[S] B) : A →ₙₐ[R] B :=
   { (f : A →ₙ+* B) with
     map_smul' := fun r x ↦ by have := map_smul f (r • 1) x; simpa }
diff --git a/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean b/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
index ce534b664d2fa..d7beda182c549 100644
--- a/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
+++ b/Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean
@@ -726,6 +726,11 @@ theorem map_sup [IsScalarTower R B B] [SMulCommClass R B B]
     ((S ⊔ T).map f : NonUnitalSubalgebra R B) = S.map f ⊔ T.map f :=
   (NonUnitalSubalgebra.gc_map_comap f).l_sup
 
+theorem map_inf [IsScalarTower R B B] [SMulCommClass R B B]
+    (f : F) (hf : Function.Injective f) (S T : NonUnitalSubalgebra R A) :
+    ((S ⊓ T).map f : NonUnitalSubalgebra R B) = S.map f ⊓ T.map f :=
+  SetLike.coe_injective (Set.image_inter hf)
+
 @[simp, norm_cast]
 theorem coe_inf (S T : NonUnitalSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=
   rfl
@@ -768,6 +773,13 @@ theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} :
 theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubalgebra R A} {x : A} :
     (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range]
 
+theorem map_iInf {ι : Sort*} [Nonempty ι]
+    [IsScalarTower R B B] [SMulCommClass R B B] (f : F)
+    (hf : Function.Injective f) (S : ι → NonUnitalSubalgebra R A) :
+    ((⨅ i, S i).map f : NonUnitalSubalgebra R B) = ⨅ i, (S i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ S)
+
 @[simp]
 theorem iInf_toSubmodule {ι : Sort*} (S : ι → NonUnitalSubalgebra R A) :
     (⨅ i, S i).toSubmodule = ⨅ i, (S i).toSubmodule :=
diff --git a/Mathlib/Algebra/Algebra/Operations.lean b/Mathlib/Algebra/Algebra/Operations.lean
index d0bb76295683d..15f0ec29f05ab 100644
--- a/Mathlib/Algebra/Algebra/Operations.lean
+++ b/Mathlib/Algebra/Algebra/Operations.lean
@@ -6,12 +6,12 @@ Authors: Kenny Lau
 import Mathlib.Algebra.Algebra.Bilinear
 import Mathlib.Algebra.Algebra.Equiv
 import Mathlib.Algebra.Algebra.Opposite
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
 import Mathlib.Algebra.Module.Opposites
 import Mathlib.Algebra.Module.Submodule.Bilinear
 import Mathlib.Algebra.Module.Submodule.Pointwise
 import Mathlib.Algebra.Order.Kleene
-import Mathlib.Data.Finset.Pointwise.Basic
 import Mathlib.Data.Set.Pointwise.BigOperators
 import Mathlib.Data.Set.Semiring
 import Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise
@@ -359,12 +359,12 @@ theorem mul_smul_mul_eq_smul_mul_smul (x y : R) : (x * y) • (M * N) = (x • M
   · rintro ⟨_, hx, rfl⟩
     rw [DistribMulAction.toLinearMap_apply]
     refine Submodule.mul_induction_on hx (fun m hm n hn ↦ ?_) (fun _ _ hn hm ↦ ?_)
-    · rw [← smul_mul_smul x y m n]
+    · rw [mul_smul_mul_comm]
       exact mul_mem_mul (smul_mem_pointwise_smul m x M hm) (smul_mem_pointwise_smul n y N hn)
     · rw [smul_add]
       exact Submodule.add_mem _ hn hm
   · rintro _ ⟨m, hm, rfl⟩ _ ⟨n, hn, rfl⟩
-    erw [smul_mul_smul x y m n]
+    simp_rw [DistribMulAction.toLinearMap_apply, smul_mul_smul_comm]
     exact smul_mem_pointwise_smul _ _ _ (mul_mem_mul hm hn)
 
 /-- Sub-R-modules of an R-algebra form an idempotent semiring. -/
diff --git a/Mathlib/Algebra/Algebra/Quasispectrum.lean b/Mathlib/Algebra/Algebra/Quasispectrum.lean
index cb01e4b60ed6e..33a5807e877b8 100644
--- a/Mathlib/Algebra/Algebra/Quasispectrum.lean
+++ b/Mathlib/Algebra/Algebra/Quasispectrum.lean
@@ -259,6 +259,30 @@ instance quasispectrum.instZero [Nontrivial R] (a : A) : Zero (quasispectrum R a
 
 variable {R}
 
+/-- A version of `NonUnitalAlgHom.quasispectrum_apply_subset` which allows for `quasispectrum R`,
+where `R` is a *semi*ring, but `φ` must still function over a scalar ring `S`. In this case, we
+need `S` to be explicit. The primary use case is, for instance, `R := ℝ≥0` and `S := ℝ` or
+`S := ℂ`. -/
+lemma NonUnitalAlgHom.quasispectrum_apply_subset' {F R : Type*} (S : Type*) {A B : Type*}
+    [CommSemiring R] [CommRing S] [NonUnitalRing A] [NonUnitalRing B] [Module R S]
+    [Module S A] [Module R A] [Module S B] [Module R B] [IsScalarTower R S A] [IsScalarTower R S B]
+    [FunLike F A B] [NonUnitalAlgHomClass F S A B] (φ : F) (a : A) :
+    quasispectrum R (φ a) ⊆ quasispectrum R a := by
+  refine Set.compl_subset_compl.mp fun x ↦ ?_
+  simp only [quasispectrum, Set.mem_compl_iff, Set.mem_setOf_eq, not_forall, not_not,
+    forall_exists_index]
+  refine fun hx this ↦ ⟨hx, ?_⟩
+  rw [Units.smul_def, ← smul_one_smul S] at this ⊢
+  simpa [- smul_assoc] using this.map φ
+
+/-- If `φ` is non-unital algebra homomorphism over a scalar ring `R`, then
+`quasispectrum R (φ a) ⊆ quasispectrum R a`. -/
+lemma NonUnitalAlgHom.quasispectrum_apply_subset {F R A B : Type*}
+    [CommRing R] [NonUnitalRing A] [NonUnitalRing B] [Module R A] [Module R B]
+    [FunLike F A B] [NonUnitalAlgHomClass F R A B] (φ : F) (a : A) :
+    quasispectrum R (φ a) ⊆ quasispectrum R a :=
+  NonUnitalAlgHom.quasispectrum_apply_subset' R φ a
+
 @[simp]
 lemma quasispectrum.coe_zero [Nontrivial R] (a : A) : (0 : quasispectrum R a) = (0 : R) := rfl
 
@@ -312,7 +336,7 @@ lemma zero_mem_spectrum_inr (R S : Type*) {A : Type*} [CommSemiring R]
 lemma mem_spectrum_inr_of_not_isUnit {R A : Type*} [CommRing R]
     [NonUnitalRing A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A]
     (a : A) (r : R) (hr : ¬ IsUnit r) : r ∈ spectrum R (a : Unitization R A) :=
-  fun h ↦ hr <| by simpa using h.map (fstHom R A)
+  fun h ↦ hr <| by simpa [map_sub] using h.map (fstHom R A)
 
 lemma quasispectrum_eq_spectrum_inr (R : Type*) {A : Type*} [CommRing R] [Ring A]
     [Algebra R A] (a : A) : quasispectrum R a = spectrum R (a : Unitization R A) := by
diff --git a/Mathlib/Algebra/Algebra/Spectrum.lean b/Mathlib/Algebra/Algebra/Spectrum.lean
index 2919c52895656..65c69f706fb00 100644
--- a/Mathlib/Algebra/Algebra/Spectrum.lean
+++ b/Mathlib/Algebra/Algebra/Spectrum.lean
@@ -53,7 +53,7 @@ local notation "↑ₐ" => algebraMap R A
 -- definition and basic properties
 /-- Given a commutative ring `R` and an `R`-algebra `A`, the *resolvent set* of `a : A`
 is the `Set R` consisting of those `r : R` for which `r•1 - a` is a unit of the
-algebra `A`.  -/
+algebra `A`. -/
 def resolventSet (a : A) : Set R :=
   {r : R | IsUnit (↑ₐ r - a)}
 
@@ -61,7 +61,7 @@ def resolventSet (a : A) : Set R :=
 is the `Set R` consisting of those `r : R` for which `r•1 - a` is not a unit of the
 algebra `A`.
 
-The spectrum is simply the complement of the resolvent set.  -/
+The spectrum is simply the complement of the resolvent set. -/
 def spectrum (a : A) : Set R :=
   (resolventSet R a)ᶜ
 
@@ -113,6 +113,10 @@ theorem zero_not_mem_iff {a : A} : (0 : R) ∉ σ a ↔ IsUnit a := by
 
 alias ⟨isUnit_of_zero_not_mem, zero_not_mem⟩ := spectrum.zero_not_mem_iff
 
+@[simp]
+lemma _root_.Units.zero_not_mem_spectrum (a : Aˣ) : 0 ∉ spectrum R (a : A) :=
+  spectrum.zero_not_mem R a.isUnit
+
 lemma subset_singleton_zero_compl {a : A} (ha : IsUnit a) : spectrum R a ⊆ {0}ᶜ :=
   Set.subset_compl_singleton_iff.mpr <| spectrum.zero_not_mem R ha
 
@@ -396,7 +400,7 @@ end CommSemiring
 
 section CommRing
 
-variable {F R A B : Type*} [CommRing R] [Ring A] [Algebra R A] [Ring B] [Algebra R B]
+variable {F R A : Type*} [CommRing R] [Ring A] [Algebra R A]
 variable [FunLike F A R] [AlgHomClass F R A R]
 
 local notation "σ" => spectrum R
diff --git a/Mathlib/Algebra/Algebra/Subalgebra/Basic.lean b/Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
index 854c9922cd13d..311a49067acad 100644
--- a/Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
+++ b/Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
@@ -678,6 +678,9 @@ theorem mul_mem_sup {S T : Subalgebra R A} {x y : A} (hx : x ∈ S) (hy : y ∈
 theorem map_sup (f : A →ₐ[R] B) (S T : Subalgebra R A) : (S ⊔ T).map f = S.map f ⊔ T.map f :=
   (Subalgebra.gc_map_comap f).l_sup
 
+theorem map_inf (f : A →ₐ[R] B) (hf : Function.Injective f) (S T : Subalgebra R A) :
+    (S ⊓ T).map f = S.map f ⊓ T.map f := SetLike.coe_injective (Set.image_inter hf)
+
 @[simp, norm_cast]
 theorem coe_inf (S T : Subalgebra R A) : (↑(S ⊓ T) : Set A) = (S ∩ T : Set A) := rfl
 
@@ -718,6 +721,11 @@ theorem coe_iInf {ι : Sort*} {S : ι → Subalgebra R A} : (↑(⨅ i, S i) : S
 theorem mem_iInf {ι : Sort*} {S : ι → Subalgebra R A} {x : A} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by
   simp only [iInf, mem_sInf, Set.forall_mem_range]
 
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : A →ₐ[R] B) (hf : Function.Injective f)
+    (s : ι → Subalgebra R A) : (iInf s).map f = ⨅ (i : ι), (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 open Subalgebra in
 @[simp]
 theorem iInf_toSubmodule {ι : Sort*} (S : ι → Subalgebra R A) :
diff --git a/Mathlib/Algebra/Algebra/Subalgebra/IsSimpleOrder.lean b/Mathlib/Algebra/Algebra/Subalgebra/IsSimpleOrder.lean
new file mode 100644
index 0000000000000..2c9016caf3b6c
--- /dev/null
+++ b/Mathlib/Algebra/Algebra/Subalgebra/IsSimpleOrder.lean
@@ -0,0 +1,33 @@
+/-
+Copyright (c) 2020 Kenny Lau. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kenny Lau
+-/
+import Mathlib.LinearAlgebra.FiniteDimensional.Defs
+import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
+import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
+
+/-!
+If `A` is a domain, and a finite-dimensional algebra over a field `F`, with prime dimension,
+then there are no non-trivial `F`-subalgebras.
+-/
+
+open Module Submodule
+
+theorem Subalgebra.isSimpleOrder_of_finrank_prime (F A) [Field F] [Ring A] [IsDomain A]
+    [Algebra F A] (hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
+  { toNontrivial :=
+      ⟨⟨⊥, ⊤, fun he =>
+          Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
+    eq_bot_or_eq_top := fun K => by
+      haveI : FiniteDimensional _ _ := .of_finrank_pos hp.pos
+      letI := divisionRingOfFiniteDimensional F K
+      refine (hp.eq_one_or_self_of_dvd _ ⟨_, (finrank_mul_finrank F K A).symm⟩).imp ?_ fun h => ?_
+      · exact fun h' => Subalgebra.eq_bot_of_finrank_one h'
+      · exact
+          Algebra.toSubmodule_eq_top.1 (eq_top_of_finrank_eq <| K.finrank_toSubmodule.trans h) }
+-- TODO: `IntermediateField` version
+
+@[deprecated (since := "2024-08-11")]
+alias FiniteDimensional.Subalgebra.is_simple_order_of_finrank_prime :=
+  Subalgebra.isSimpleOrder_of_finrank_prime
diff --git a/Mathlib/Algebra/Algebra/Subalgebra/Operations.lean b/Mathlib/Algebra/Algebra/Subalgebra/Operations.lean
index c1fd4e6de44d8..b28a1c2c1dc76 100644
--- a/Mathlib/Algebra/Algebra/Subalgebra/Operations.lean
+++ b/Mathlib/Algebra/Algebra/Subalgebra/Operations.lean
@@ -67,7 +67,7 @@ theorem mem_of_finset_sum_eq_one_of_pow_smul_mem
   exact ⟨⟨_, hn i⟩, rfl⟩
 
 theorem mem_of_span_eq_top_of_smul_pow_mem
-    (s : Set S) (l : s →₀ S) (hs : Finsupp.total S ((↑) : s → S) l = 1)
+    (s : Set S) (l : s →₀ S) (hs : Finsupp.linearCombination S ((↑) : s → S) l = 1)
     (hs' : s ⊆ S') (hl : ∀ i, l i ∈ S') (x : S) (H : ∀ r : s, ∃ n : ℕ, (r : S) ^ n • x ∈ S') :
     x ∈ S' :=
   mem_of_finset_sum_eq_one_of_pow_smul_mem S' l.support (↑) l hs (fun x => hs' x.2) hl x H
diff --git a/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean b/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean
index 1dac252bbf591..e4b3c91e4d894 100644
--- a/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean
+++ b/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean
@@ -18,7 +18,7 @@ satisfies strong rank condition, we put them into a separate file.
 
 -/
 
-open FiniteDimensional
+open Module
 
 namespace Subalgebra
 
diff --git a/Mathlib/Algebra/Algebra/Unitization.lean b/Mathlib/Algebra/Algebra/Unitization.lean
index 410c45a19b938..42a25ab336b19 100644
--- a/Mathlib/Algebra/Algebra/Unitization.lean
+++ b/Mathlib/Algebra/Algebra/Unitization.lean
@@ -751,6 +751,66 @@ theorem starLift_symm_apply_apply (φ : Unitization R A →⋆ₐ[R] C) (a : A)
 
 end StarAlgHom
 
+section StarMap
+
+variable {R A B C : Type*} [CommSemiring R] [StarRing R]
+variable [NonUnitalSemiring A] [StarRing A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A]
+variable [NonUnitalSemiring B] [StarRing B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B]
+variable [NonUnitalSemiring C] [StarRing C] [Module R C] [SMulCommClass R C C] [IsScalarTower R C C]
+variable [StarModule R B] [StarModule R C]
+
+/-- The functorial map on morphisms between the category of non-unital C⋆-algebras with non-unital
+star homomorphisms and unital C⋆-algebras with unital star homomorphisms.
+
+This sends `φ : A →⋆ₙₐ[R] B` to a map `Unitization R A →⋆ₐ[R] Unitization R B` given by the formula
+`(r, a) ↦ (r, φ a)` (or perhaps more precisely,
+`algebraMap R _ r + ↑a ↦ algebraMap R _ r + ↑(φ a)`). -/
+@[simps!]
+def starMap (φ : A →⋆ₙₐ[R] B) : Unitization R A →⋆ₐ[R] Unitization R B :=
+  Unitization.starLift <| (Unitization.inrNonUnitalStarAlgHom R B).comp φ
+
+@[simp high]
+lemma starMap_inr (φ : A →⋆ₙₐ[R] B) (a : A) :
+    starMap φ (inr a) = inr (φ a) := by
+  simp
+
+@[simp high]
+lemma starMap_inl (φ : A →⋆ₙₐ[R] B) (r : R) :
+    starMap φ (inl r) = algebraMap R (Unitization R B) r := by
+  simp
+
+/-- If `φ : A →⋆ₙₐ[R] B` is injective, the lift `starMap φ : Unitization R A →⋆ₐ[R] Unitization R B`
+is also injective. -/
+lemma starMap_injective {φ : A →⋆ₙₐ[R] B} (hφ : Function.Injective φ) :
+    Function.Injective (starMap φ) := by
+  intro x y h
+  ext
+  · simpa using congr(fst $(h))
+  · exact hφ <| by simpa [algebraMap_eq_inl] using congr(snd $(h))
+
+/-- If `φ : A →⋆ₙₐ[R] B` is surjective, the lift
+`starMap φ : Unitization R A →⋆ₐ[R] Unitization R B` is also surjective. -/
+lemma starMap_surjective {φ : A →⋆ₙₐ[R] B} (hφ : Function.Surjective φ) :
+    Function.Surjective (starMap φ) := by
+  intro x
+  induction x using Unitization.ind with
+  | inl_add_inr r b =>
+    obtain ⟨a, rfl⟩ := hφ b
+    exact ⟨(r, a), by rfl⟩
+
+/-- `starMap` is functorial: `starMap (ψ.comp φ) = (starMap ψ).comp (starMap φ)`. -/
+lemma starMap_comp {φ : A →⋆ₙₐ[R] B} {ψ : B →⋆ₙₐ[R] C} :
+    starMap (ψ.comp φ) = (starMap ψ).comp (starMap φ) := by
+  ext; all_goals simp
+
+/-- `starMap` is functorial:
+`starMap (NonUnitalStarAlgHom.id R B) = StarAlgHom.id R (Unitization R B)`. -/
+@[simp]
+lemma starMap_id : starMap (NonUnitalStarAlgHom.id R B) = StarAlgHom.id R (Unitization R B) := by
+  ext; all_goals simp
+
+end StarMap
+
 section StarNormal
 
 variable {R A : Type*} [Semiring R]
diff --git a/Mathlib/Data/ZMod/Algebra.lean b/Mathlib/Algebra/Algebra/ZMod.lean
similarity index 100%
rename from Mathlib/Data/ZMod/Algebra.lean
rename to Mathlib/Algebra/Algebra/ZMod.lean
index 2f5737b396fd4..a04244c46ce31 100644
--- a/Mathlib/Data/ZMod/Algebra.lean
+++ b/Mathlib/Algebra/Algebra/ZMod.lean
@@ -3,8 +3,8 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Data.ZMod.Basic
 import Mathlib.Algebra.Algebra.Defs
+import Mathlib.Data.ZMod.Basic
 
 /-!
 # The `ZMod n`-algebra structure on rings whose characteristic divides `n`
diff --git a/Mathlib/Algebra/AlgebraicCard.lean b/Mathlib/Algebra/AlgebraicCard.lean
index 85eee288ffe90..f8be8db361367 100644
--- a/Mathlib/Algebra/AlgebraicCard.lean
+++ b/Mathlib/Algebra/AlgebraicCard.lean
@@ -10,10 +10,10 @@ import Mathlib.RingTheory.Algebraic
 ### Cardinality of algebraic numbers
 
 In this file, we prove variants of the following result: the cardinality of algebraic numbers under
-an R-algebra is at most `# R[X] * ℵ₀`.
+an R-algebra is at most `#R[X] * ℵ₀`.
 
 Although this can be used to prove that real or complex transcendental numbers exist, a more direct
-proof is given by `Liouville.is_transcendental`.
+proof is given by `Liouville.transcendental`.
 -/
 
 
diff --git a/Mathlib/Algebra/Associated/Basic.lean b/Mathlib/Algebra/Associated/Basic.lean
index 70a806de2380a..47308730de6b6 100644
--- a/Mathlib/Algebra/Associated/Basic.lean
+++ b/Mathlib/Algebra/Associated/Basic.lean
@@ -31,20 +31,20 @@ and prove basic properties of this quotient.
 assert_not_exists OrderedCommMonoid
 assert_not_exists Multiset
 
-variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
+variable {M N : Type*}
 
 section Prime
 
-variable [CommMonoidWithZero α]
+variable [CommMonoidWithZero M]
 
 /-- An element `p` of a commutative monoid with zero (e.g., a ring) is called *prime*,
 if it's not zero, not a unit, and `p ∣ a * b → p ∣ a ∨ p ∣ b` for all `a`, `b`. -/
-def Prime (p : α) : Prop :=
+def Prime (p : M) : Prop :=
   p ≠ 0 ∧ ¬IsUnit p ∧ ∀ a b, p ∣ a * b → p ∣ a ∨ p ∣ b
 
 namespace Prime
 
-variable {p : α} (hp : Prime p)
+variable {p : M} (hp : Prime p)
 include hp
 
 theorem ne_zero : p ≠ 0 :=
@@ -58,19 +58,19 @@ theorem not_dvd_one : ¬p ∣ 1 :=
 
 theorem ne_one : p ≠ 1 := fun h => hp.2.1 (h.symm ▸ isUnit_one)
 
-theorem dvd_or_dvd {a b : α} (h : p ∣ a * b) : p ∣ a ∨ p ∣ b :=
+theorem dvd_or_dvd {a b : M} (h : p ∣ a * b) : p ∣ a ∨ p ∣ b :=
   hp.2.2 a b h
 
-theorem dvd_mul {a b : α} : p ∣ a * b ↔ p ∣ a ∨ p ∣ b :=
+theorem dvd_mul {a b : M} : p ∣ a * b ↔ p ∣ a ∨ p ∣ b :=
   ⟨hp.dvd_or_dvd, (Or.elim · (dvd_mul_of_dvd_left · _) (dvd_mul_of_dvd_right · _))⟩
 
 theorem isPrimal : IsPrimal p := fun _a _b dvd ↦ (hp.dvd_or_dvd dvd).elim
   (fun h ↦ ⟨p, 1, h, one_dvd _, (mul_one p).symm⟩) fun h ↦ ⟨1, p, one_dvd _, h, (one_mul p).symm⟩
 
-theorem not_dvd_mul {a b : α} (ha : ¬ p ∣ a) (hb : ¬ p ∣ b) : ¬ p ∣ a * b :=
+theorem not_dvd_mul {a b : M} (ha : ¬ p ∣ a) (hb : ¬ p ∣ b) : ¬ p ∣ a * b :=
   hp.dvd_mul.not.mpr <| not_or.mpr ⟨ha, hb⟩
 
-theorem dvd_of_dvd_pow {a : α} {n : ℕ} (h : p ∣ a ^ n) : p ∣ a := by
+theorem dvd_of_dvd_pow {a : M} {n : ℕ} (h : p ∣ a ^ n) : p ∣ a := by
   induction n with
   | zero =>
     rw [pow_zero] at h
@@ -83,24 +83,24 @@ theorem dvd_of_dvd_pow {a : α} {n : ℕ} (h : p ∣ a ^ n) : p ∣ a := by
     · assumption
     · exact ih dvd_pow
 
-theorem dvd_pow_iff_dvd {a : α} {n : ℕ} (hn : n ≠ 0) : p ∣ a ^ n ↔ p ∣ a :=
+theorem dvd_pow_iff_dvd {a : M} {n : ℕ} (hn : n ≠ 0) : p ∣ a ^ n ↔ p ∣ a :=
   ⟨hp.dvd_of_dvd_pow, (dvd_pow · hn)⟩
 
 end Prime
 
 @[simp]
-theorem not_prime_zero : ¬Prime (0 : α) := fun h => h.ne_zero rfl
+theorem not_prime_zero : ¬Prime (0 : M) := fun h => h.ne_zero rfl
 
 @[simp]
-theorem not_prime_one : ¬Prime (1 : α) := fun h => h.not_unit isUnit_one
+theorem not_prime_one : ¬Prime (1 : M) := fun h => h.not_unit isUnit_one
 
 section Map
 
-variable [CommMonoidWithZero β] {F : Type*} {G : Type*} [FunLike F α β]
-variable [MonoidWithZeroHomClass F α β] [FunLike G β α] [MulHomClass G β α]
-variable (f : F) (g : G) {p : α}
+variable [CommMonoidWithZero N] {F : Type*} {G : Type*} [FunLike F M N]
+variable [MonoidWithZeroHomClass F M N] [FunLike G N M] [MulHomClass G N M]
+variable (f : F) (g : G) {p : M}
 
-theorem comap_prime (hinv : ∀ a, g (f a : β) = a) (hp : Prime (f p)) : Prime p :=
+theorem comap_prime (hinv : ∀ a, g (f a : N) = a) (hp : Prime (f p)) : Prime p :=
   ⟨fun h => hp.1 <| by simp [h], fun h => hp.2.1 <| h.map f, fun a b h => by
     refine
         (hp.2.2 (f a) (f b) <| by
@@ -110,7 +110,7 @@ theorem comap_prime (hinv : ∀ a, g (f a : β) = a) (hp : Prime (f p)) : Prime
       · intro h
         convert ← map_dvd g h <;> apply hinv⟩
 
-theorem MulEquiv.prime_iff (e : α ≃* β) : Prime p ↔ Prime (e p) :=
+theorem MulEquiv.prime_iff (e : M ≃* N) : Prime p ↔ Prime (e p) :=
   ⟨fun h => (comap_prime e.symm e fun a => by simp) <| (e.symm_apply_apply p).substr h,
     comap_prime e e.symm fun a => by simp⟩
 
@@ -118,15 +118,15 @@ end Map
 
 end Prime
 
-theorem Prime.left_dvd_or_dvd_right_of_dvd_mul [CancelCommMonoidWithZero α] {p : α} (hp : Prime p)
-    {a b : α} : a ∣ p * b → p ∣ a ∨ a ∣ b := by
+theorem Prime.left_dvd_or_dvd_right_of_dvd_mul [CancelCommMonoidWithZero M] {p : M} (hp : Prime p)
+    {a b : M} : a ∣ p * b → p ∣ a ∨ a ∣ b := by
   rintro ⟨c, hc⟩
   rcases hp.2.2 a c (hc ▸ dvd_mul_right _ _) with (h | ⟨x, rfl⟩)
   · exact Or.inl h
   · rw [mul_left_comm, mul_right_inj' hp.ne_zero] at hc
     exact Or.inr (hc.symm ▸ dvd_mul_right _ _)
 
-theorem Prime.pow_dvd_of_dvd_mul_left [CancelCommMonoidWithZero α] {p a b : α} (hp : Prime p)
+theorem Prime.pow_dvd_of_dvd_mul_left [CancelCommMonoidWithZero M] {p a b : M} (hp : Prime p)
     (n : ℕ) (h : ¬p ∣ a) (h' : p ^ n ∣ a * b) : p ^ n ∣ b := by
   induction n with
   | zero =>
@@ -138,12 +138,12 @@ theorem Prime.pow_dvd_of_dvd_mul_left [CancelCommMonoidWithZero α] {p a b : α}
     apply mul_dvd_mul_left _ ((hp.dvd_or_dvd _).resolve_left h)
     rwa [← mul_dvd_mul_iff_left (pow_ne_zero n hp.ne_zero), ← pow_succ, mul_left_comm]
 
-theorem Prime.pow_dvd_of_dvd_mul_right [CancelCommMonoidWithZero α] {p a b : α} (hp : Prime p)
+theorem Prime.pow_dvd_of_dvd_mul_right [CancelCommMonoidWithZero M] {p a b : M} (hp : Prime p)
     (n : ℕ) (h : ¬p ∣ b) (h' : p ^ n ∣ a * b) : p ^ n ∣ a := by
   rw [mul_comm] at h'
   exact hp.pow_dvd_of_dvd_mul_left n h h'
 
-theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [CancelCommMonoidWithZero α] {p a b : α}
+theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [CancelCommMonoidWithZero M] {p a b : M}
     {n : ℕ} (hp : Prime p) (hpow : p ^ n.succ ∣ a ^ n.succ * b ^ n) (hb : ¬p ^ 2 ∣ b) : p ∣ a := by
   -- Suppose `p ∣ b`, write `b = p * x` and `hy : a ^ n.succ * b ^ n = p ^ n.succ * y`.
   rcases hp.dvd_or_dvd ((dvd_pow_self p (Nat.succ_ne_zero n)).trans hpow) with H | hbdiv
@@ -161,7 +161,7 @@ theorem Prime.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd [CancelCommMonoidWith
   rw [pow_two, ← mul_assoc]
   exact dvd_mul_right _ _
 
-theorem prime_pow_succ_dvd_mul {α : Type*} [CancelCommMonoidWithZero α] {p x y : α} (h : Prime p)
+theorem prime_pow_succ_dvd_mul {M : Type*} [CancelCommMonoidWithZero M] {p x y : M} (h : Prime p)
     {i : ℕ} (hxy : p ^ (i + 1) ∣ x * y) : p ^ (i + 1) ∣ x ∨ p ∣ y := by
   rw [or_iff_not_imp_right]
   intro hy
@@ -178,7 +178,7 @@ theorem prime_pow_succ_dvd_mul {α : Type*} [CancelCommMonoidWithZero α] {p x y
 We explicitly avoid stating that `p` is non-zero, this would require a semiring. Assuming only a
 monoid allows us to reuse irreducible for associated elements.
 -/
-structure Irreducible [Monoid α] (p : α) : Prop where
+structure Irreducible [Monoid M] (p : M) : Prop where
   /-- `p` is not a unit -/
   not_unit : ¬IsUnit p
   /-- if `p` factors then one factor is a unit -/
@@ -186,38 +186,38 @@ structure Irreducible [Monoid α] (p : α) : Prop where
 
 namespace Irreducible
 
-theorem not_dvd_one [CommMonoid α] {p : α} (hp : Irreducible p) : ¬p ∣ 1 :=
+theorem not_dvd_one [CommMonoid M] {p : M} (hp : Irreducible p) : ¬p ∣ 1 :=
   mt (isUnit_of_dvd_one ·) hp.not_unit
 
-theorem isUnit_or_isUnit [Monoid α] {p : α} (hp : Irreducible p) {a b : α} (h : p = a * b) :
+theorem isUnit_or_isUnit [Monoid M] {p : M} (hp : Irreducible p) {a b : M} (h : p = a * b) :
     IsUnit a ∨ IsUnit b :=
   hp.isUnit_or_isUnit' a b h
 
 end Irreducible
 
-theorem irreducible_iff [Monoid α] {p : α} :
+theorem irreducible_iff [Monoid M] {p : M} :
     Irreducible p ↔ ¬IsUnit p ∧ ∀ a b, p = a * b → IsUnit a ∨ IsUnit b :=
   ⟨fun h => ⟨h.1, h.2⟩, fun h => ⟨h.1, h.2⟩⟩
 
 @[simp]
-theorem not_irreducible_one [Monoid α] : ¬Irreducible (1 : α) := by simp [irreducible_iff]
+theorem not_irreducible_one [Monoid M] : ¬Irreducible (1 : M) := by simp [irreducible_iff]
 
-theorem Irreducible.ne_one [Monoid α] : ∀ {p : α}, Irreducible p → p ≠ 1
+theorem Irreducible.ne_one [Monoid M] : ∀ {p : M}, Irreducible p → p ≠ 1
   | _, hp, rfl => not_irreducible_one hp
 
 @[simp]
-theorem not_irreducible_zero [MonoidWithZero α] : ¬Irreducible (0 : α)
+theorem not_irreducible_zero [MonoidWithZero M] : ¬Irreducible (0 : M)
   | ⟨hn0, h⟩ =>
-    have : IsUnit (0 : α) ∨ IsUnit (0 : α) := h 0 0 (mul_zero 0).symm
+    have : IsUnit (0 : M) ∨ IsUnit (0 : M) := h 0 0 (mul_zero 0).symm
     this.elim hn0 hn0
 
-theorem Irreducible.ne_zero [MonoidWithZero α] : ∀ {p : α}, Irreducible p → p ≠ 0
+theorem Irreducible.ne_zero [MonoidWithZero M] : ∀ {p : M}, Irreducible p → p ≠ 0
   | _, hp, rfl => not_irreducible_zero hp
 
-theorem of_irreducible_mul {α} [Monoid α] {x y : α} : Irreducible (x * y) → IsUnit x ∨ IsUnit y
+theorem of_irreducible_mul {M} [Monoid M] {x y : M} : Irreducible (x * y) → IsUnit x ∨ IsUnit y
   | ⟨_, h⟩ => h _ _ rfl
 
-theorem not_irreducible_pow {α} [Monoid α] {x : α} {n : ℕ} (hn : n ≠ 1) :
+theorem not_irreducible_pow {M} [Monoid M] {x : M} {n : ℕ} (hn : n ≠ 1) :
     ¬ Irreducible (x ^ n) := by
   cases n with
   | zero => simp
@@ -227,7 +227,7 @@ theorem not_irreducible_pow {α} [Monoid α] {x : α} {n : ℕ} (hn : n ≠ 1) :
     rw [isUnit_pow_iff (Nat.succ_ne_succ.mp hn), or_self] at this
     exact h₁ (this.pow _)
 
-theorem irreducible_or_factor {α} [Monoid α] (x : α) (h : ¬IsUnit x) :
+theorem irreducible_or_factor {M} [Monoid M] (x : M) (h : ¬IsUnit x) :
     Irreducible x ∨ ∃ a b, ¬IsUnit a ∧ ¬IsUnit b ∧ a * b = x := by
   haveI := Classical.dec
   refine or_iff_not_imp_right.2 fun H => ?_
@@ -239,20 +239,26 @@ theorem irreducible_or_factor {α} [Monoid α] (x : α) (h : ¬IsUnit x) :
   exact H _ o.1 _ o.2 h.symm
 
 /-- If `p` and `q` are irreducible, then `p ∣ q` implies `q ∣ p`. -/
-theorem Irreducible.dvd_symm [Monoid α] {p q : α} (hp : Irreducible p) (hq : Irreducible q) :
+theorem Irreducible.dvd_symm [Monoid M] {p q : M} (hp : Irreducible p) (hq : Irreducible q) :
     p ∣ q → q ∣ p := by
   rintro ⟨q', rfl⟩
   rw [IsUnit.mul_right_dvd (Or.resolve_left (of_irreducible_mul hq) hp.not_unit)]
 
-theorem Irreducible.dvd_comm [Monoid α] {p q : α} (hp : Irreducible p) (hq : Irreducible q) :
+theorem Irreducible.dvd_comm [Monoid M] {p q : M} (hp : Irreducible p) (hq : Irreducible q) :
     p ∣ q ↔ q ∣ p :=
   ⟨hp.dvd_symm hq, hq.dvd_symm hp⟩
 
+theorem Irreducible.of_map {F : Type*} [Monoid M] [Monoid N] [FunLike F M N] [MonoidHomClass F M N]
+    {f : F} [IsLocalRingHom f] {x} (hfx : Irreducible (f x)) : Irreducible x :=
+  ⟨fun hu ↦ hfx.not_unit <| hu.map f,
+   by rintro p q rfl
+      exact (hfx.isUnit_or_isUnit <| map_mul f p q).imp (.of_map f _) (.of_map f _)⟩
+
 section
 
-variable [Monoid α]
+variable [Monoid M]
 
-theorem irreducible_units_mul (a : αˣ) (b : α) : Irreducible (↑a * b) ↔ Irreducible b := by
+theorem irreducible_units_mul (a : Mˣ) (b : M) : Irreducible (↑a * b) ↔ Irreducible b := by
   simp only [irreducible_iff, Units.isUnit_units_mul, and_congr_right_iff]
   refine fun _ => ⟨fun h A B HAB => ?_, fun h A B HAB => ?_⟩
   · rw [← a.isUnit_units_mul]
@@ -262,11 +268,11 @@ theorem irreducible_units_mul (a : αˣ) (b : α) : Irreducible (↑a * b) ↔ I
     apply h
     rw [mul_assoc, ← HAB, Units.inv_mul_cancel_left]
 
-theorem irreducible_isUnit_mul {a b : α} (h : IsUnit a) : Irreducible (a * b) ↔ Irreducible b :=
+theorem irreducible_isUnit_mul {a b : M} (h : IsUnit a) : Irreducible (a * b) ↔ Irreducible b :=
   let ⟨a, ha⟩ := h
   ha ▸ irreducible_units_mul a b
 
-theorem irreducible_mul_units (a : αˣ) (b : α) : Irreducible (b * ↑a) ↔ Irreducible b := by
+theorem irreducible_mul_units (a : Mˣ) (b : M) : Irreducible (b * ↑a) ↔ Irreducible b := by
   simp only [irreducible_iff, Units.isUnit_mul_units, and_congr_right_iff]
   refine fun _ => ⟨fun h A B HAB => ?_, fun h A B HAB => ?_⟩
   · rw [← Units.isUnit_mul_units B a]
@@ -276,11 +282,11 @@ theorem irreducible_mul_units (a : αˣ) (b : α) : Irreducible (b * ↑a) ↔ I
     apply h
     rw [← mul_assoc, ← HAB, Units.mul_inv_cancel_right]
 
-theorem irreducible_mul_isUnit {a b : α} (h : IsUnit a) : Irreducible (b * a) ↔ Irreducible b :=
+theorem irreducible_mul_isUnit {a b : M} (h : IsUnit a) : Irreducible (b * a) ↔ Irreducible b :=
   let ⟨a, ha⟩ := h
   ha ▸ irreducible_mul_units a b
 
-theorem irreducible_mul_iff {a b : α} :
+theorem irreducible_mul_iff {a b : M} :
     Irreducible (a * b) ↔ Irreducible a ∧ IsUnit b ∨ Irreducible b ∧ IsUnit a := by
   constructor
   · refine fun h => Or.imp (fun h' => ⟨?_, h'⟩) (fun h' => ⟨?_, h'⟩) (h.isUnit_or_isUnit rfl).symm
@@ -294,7 +300,7 @@ end
 
 section CommMonoid
 
-variable [CommMonoid α] {a : α}
+variable [CommMonoid M] {a : M}
 
 theorem Irreducible.not_square (ha : Irreducible a) : ¬IsSquare a := by
   rw [isSquare_iff_exists_sq]
@@ -307,22 +313,22 @@ end CommMonoid
 
 section CommMonoidWithZero
 
-variable [CommMonoidWithZero α]
+variable [CommMonoidWithZero M]
 
-theorem Irreducible.prime_of_isPrimal {a : α}
+theorem Irreducible.prime_of_isPrimal {a : M}
     (irr : Irreducible a) (primal : IsPrimal a) : Prime a :=
   ⟨irr.ne_zero, irr.not_unit, fun a b dvd ↦ by
     obtain ⟨d₁, d₂, h₁, h₂, rfl⟩ := primal dvd
     exact (of_irreducible_mul irr).symm.imp (·.mul_right_dvd.mpr h₁) (·.mul_left_dvd.mpr h₂)⟩
 
-theorem Irreducible.prime [DecompositionMonoid α] {a : α} (irr : Irreducible a) : Prime a :=
+theorem Irreducible.prime [DecompositionMonoid M] {a : M} (irr : Irreducible a) : Prime a :=
   irr.prime_of_isPrimal (DecompositionMonoid.primal a)
 
 end CommMonoidWithZero
 
 section CancelCommMonoidWithZero
 
-variable [CancelCommMonoidWithZero α] {a p : α}
+variable [CancelCommMonoidWithZero M] {a p : M}
 
 protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
   ⟨hp.not_unit, fun a b ↦ by
@@ -333,10 +339,10 @@ protected theorem Prime.irreducible (hp : Prime p) : Irreducible p :=
       (isUnit_of_dvd_one <| (mul_dvd_mul_iff_left <| left_ne_zero_of_mul hp.ne_zero).mp <|
         dvd_mul_of_dvd_left · _)⟩
 
-theorem irreducible_iff_prime [DecompositionMonoid α] {a : α} : Irreducible a ↔ Prime a :=
+theorem irreducible_iff_prime [DecompositionMonoid M] {a : M} : Irreducible a ↔ Prime a :=
   ⟨Irreducible.prime, Prime.irreducible⟩
 
-theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : α} {k l : ℕ} :
+theorem succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul (hp : Prime p) {a b : M} {k l : ℕ} :
     p ^ k ∣ a → p ^ l ∣ b → p ^ (k + l + 1) ∣ a * b → p ^ (k + 1) ∣ a ∨ p ^ (l + 1) ∣ b :=
   fun ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩ =>
   have h : p ^ (k + l) * (x * y) = p ^ (k + l) * (p * z) := by
@@ -359,8 +365,8 @@ end CancelCommMonoidWithZero
 
 /-- Two elements of a `Monoid` are `Associated` if one of them is another one
 multiplied by a unit on the right. -/
-def Associated [Monoid α] (x y : α) : Prop :=
-  ∃ u : αˣ, x * u = y
+def Associated [Monoid M] (x y : M) : Prop :=
+  ∃ u : Mˣ, x * u = y
 
 /-- Notation for two elements of a monoid are associated, i.e.
 if one of them is another one multiplied by a unit on the right. -/
@@ -369,35 +375,35 @@ local infixl:50 " ~ᵤ " => Associated
 namespace Associated
 
 @[refl]
-protected theorem refl [Monoid α] (x : α) : x ~ᵤ x :=
+protected theorem refl [Monoid M] (x : M) : x ~ᵤ x :=
   ⟨1, by simp⟩
 
-protected theorem rfl [Monoid α] {x : α} : x ~ᵤ x :=
+protected theorem rfl [Monoid M] {x : M} : x ~ᵤ x :=
   .refl x
 
-instance [Monoid α] : IsRefl α Associated :=
+instance [Monoid M] : IsRefl M Associated :=
   ⟨Associated.refl⟩
 
 @[symm]
-protected theorem symm [Monoid α] : ∀ {x y : α}, x ~ᵤ y → y ~ᵤ x
+protected theorem symm [Monoid M] : ∀ {x y : M}, x ~ᵤ y → y ~ᵤ x
   | x, _, ⟨u, rfl⟩ => ⟨u⁻¹, by rw [mul_assoc, Units.mul_inv, mul_one]⟩
 
-instance [Monoid α] : IsSymm α Associated :=
+instance [Monoid M] : IsSymm M Associated :=
   ⟨fun _ _ => Associated.symm⟩
 
-protected theorem comm [Monoid α] {x y : α} : x ~ᵤ y ↔ y ~ᵤ x :=
+protected theorem comm [Monoid M] {x y : M} : x ~ᵤ y ↔ y ~ᵤ x :=
   ⟨Associated.symm, Associated.symm⟩
 
 @[trans]
-protected theorem trans [Monoid α] : ∀ {x y z : α}, x ~ᵤ y → y ~ᵤ z → x ~ᵤ z
+protected theorem trans [Monoid M] : ∀ {x y z : M}, x ~ᵤ y → y ~ᵤ z → x ~ᵤ z
   | x, _, _, ⟨u, rfl⟩, ⟨v, rfl⟩ => ⟨u * v, by rw [Units.val_mul, mul_assoc]⟩
 
-instance [Monoid α] : IsTrans α Associated :=
+instance [Monoid M] : IsTrans M Associated :=
   ⟨fun _ _ _ => Associated.trans⟩
 
 /-- The setoid of the relation `x ~ᵤ y` iff there is a unit `u` such that `x * u = y` -/
-protected def setoid (α : Type*) [Monoid α] :
-    Setoid α where
+protected def setoid (M : Type*) [Monoid M] :
+    Setoid M where
   r := Associated
   iseqv := ⟨Associated.refl, Associated.symm, Associated.trans⟩
 
@@ -410,11 +416,11 @@ end Associated
 
 attribute [local instance] Associated.setoid
 
-theorem unit_associated_one [Monoid α] {u : αˣ} : (u : α) ~ᵤ 1 :=
+theorem unit_associated_one [Monoid M] {u : Mˣ} : (u : M) ~ᵤ 1 :=
   ⟨u⁻¹, Units.mul_inv u⟩
 
 @[simp]
-theorem associated_one_iff_isUnit [Monoid α] {a : α} : (a : α) ~ᵤ 1 ↔ IsUnit a :=
+theorem associated_one_iff_isUnit [Monoid M] {a : M} : (a : M) ~ᵤ 1 ↔ IsUnit a :=
   Iff.intro
     (fun h =>
       let ⟨c, h⟩ := h.symm
@@ -422,98 +428,98 @@ theorem associated_one_iff_isUnit [Monoid α] {a : α} : (a : α) ~ᵤ 1 ↔ IsU
     fun ⟨c, h⟩ => Associated.symm ⟨c, by simp [h]⟩
 
 @[simp]
-theorem associated_zero_iff_eq_zero [MonoidWithZero α] (a : α) : a ~ᵤ 0 ↔ a = 0 :=
+theorem associated_zero_iff_eq_zero [MonoidWithZero M] (a : M) : a ~ᵤ 0 ↔ a = 0 :=
   Iff.intro
     (fun h => by
       let ⟨u, h⟩ := h.symm
       simpa using h.symm)
     fun h => h ▸ Associated.refl a
 
-theorem associated_one_of_mul_eq_one [CommMonoid α] {a : α} (b : α) (hab : a * b = 1) : a ~ᵤ 1 :=
-  show (Units.mkOfMulEqOne a b hab : α) ~ᵤ 1 from unit_associated_one
+theorem associated_one_of_mul_eq_one [CommMonoid M] {a : M} (b : M) (hab : a * b = 1) : a ~ᵤ 1 :=
+  show (Units.mkOfMulEqOne a b hab : M) ~ᵤ 1 from unit_associated_one
 
-theorem associated_one_of_associated_mul_one [CommMonoid α] {a b : α} : a * b ~ᵤ 1 → a ~ᵤ 1
+theorem associated_one_of_associated_mul_one [CommMonoid M] {a b : M} : a * b ~ᵤ 1 → a ~ᵤ 1
   | ⟨u, h⟩ => associated_one_of_mul_eq_one (b * u) <| by simpa [mul_assoc] using h
 
-theorem associated_mul_unit_left {β : Type*} [Monoid β] (a u : β) (hu : IsUnit u) :
+theorem associated_mul_unit_left {N : Type*} [Monoid N] (a u : N) (hu : IsUnit u) :
     Associated (a * u) a :=
   let ⟨u', hu⟩ := hu
   ⟨u'⁻¹, hu ▸ Units.mul_inv_cancel_right _ _⟩
 
-theorem associated_unit_mul_left {β : Type*} [CommMonoid β] (a u : β) (hu : IsUnit u) :
+theorem associated_unit_mul_left {N : Type*} [CommMonoid N] (a u : N) (hu : IsUnit u) :
     Associated (u * a) a := by
   rw [mul_comm]
   exact associated_mul_unit_left _ _ hu
 
-theorem associated_mul_unit_right {β : Type*} [Monoid β] (a u : β) (hu : IsUnit u) :
+theorem associated_mul_unit_right {N : Type*} [Monoid N] (a u : N) (hu : IsUnit u) :
     Associated a (a * u) :=
   (associated_mul_unit_left a u hu).symm
 
-theorem associated_unit_mul_right {β : Type*} [CommMonoid β] (a u : β) (hu : IsUnit u) :
+theorem associated_unit_mul_right {N : Type*} [CommMonoid N] (a u : N) (hu : IsUnit u) :
     Associated a (u * a) :=
   (associated_unit_mul_left a u hu).symm
 
-theorem associated_mul_isUnit_left_iff {β : Type*} [Monoid β] {a u b : β} (hu : IsUnit u) :
+theorem associated_mul_isUnit_left_iff {N : Type*} [Monoid N] {a u b : N} (hu : IsUnit u) :
     Associated (a * u) b ↔ Associated a b :=
   ⟨(associated_mul_unit_right _ _ hu).trans, (associated_mul_unit_left _ _ hu).trans⟩
 
-theorem associated_isUnit_mul_left_iff {β : Type*} [CommMonoid β] {u a b : β} (hu : IsUnit u) :
+theorem associated_isUnit_mul_left_iff {N : Type*} [CommMonoid N] {u a b : N} (hu : IsUnit u) :
     Associated (u * a) b ↔ Associated a b := by
   rw [mul_comm]
   exact associated_mul_isUnit_left_iff hu
 
-theorem associated_mul_isUnit_right_iff {β : Type*} [Monoid β] {a b u : β} (hu : IsUnit u) :
+theorem associated_mul_isUnit_right_iff {N : Type*} [Monoid N] {a b u : N} (hu : IsUnit u) :
     Associated a (b * u) ↔ Associated a b :=
   Associated.comm.trans <| (associated_mul_isUnit_left_iff hu).trans Associated.comm
 
-theorem associated_isUnit_mul_right_iff {β : Type*} [CommMonoid β] {a u b : β} (hu : IsUnit u) :
+theorem associated_isUnit_mul_right_iff {N : Type*} [CommMonoid N] {a u b : N} (hu : IsUnit u) :
     Associated a (u * b) ↔ Associated a b :=
   Associated.comm.trans <| (associated_isUnit_mul_left_iff hu).trans Associated.comm
 
 @[simp]
-theorem associated_mul_unit_left_iff {β : Type*} [Monoid β] {a b : β} {u : Units β} :
+theorem associated_mul_unit_left_iff {N : Type*} [Monoid N] {a b : N} {u : Units N} :
     Associated (a * u) b ↔ Associated a b :=
   associated_mul_isUnit_left_iff u.isUnit
 
 @[simp]
-theorem associated_unit_mul_left_iff {β : Type*} [CommMonoid β] {a b : β} {u : Units β} :
+theorem associated_unit_mul_left_iff {N : Type*} [CommMonoid N] {a b : N} {u : Units N} :
     Associated (↑u * a) b ↔ Associated a b :=
   associated_isUnit_mul_left_iff u.isUnit
 
 @[simp]
-theorem associated_mul_unit_right_iff {β : Type*} [Monoid β] {a b : β} {u : Units β} :
+theorem associated_mul_unit_right_iff {N : Type*} [Monoid N] {a b : N} {u : Units N} :
     Associated a (b * u) ↔ Associated a b :=
   associated_mul_isUnit_right_iff u.isUnit
 
 @[simp]
-theorem associated_unit_mul_right_iff {β : Type*} [CommMonoid β] {a b : β} {u : Units β} :
+theorem associated_unit_mul_right_iff {N : Type*} [CommMonoid N] {a b : N} {u : Units N} :
     Associated a (↑u * b) ↔ Associated a b :=
   associated_isUnit_mul_right_iff u.isUnit
 
-theorem Associated.mul_left [Monoid α] (a : α) {b c : α} (h : b ~ᵤ c) : a * b ~ᵤ a * c := by
+theorem Associated.mul_left [Monoid M] (a : M) {b c : M} (h : b ~ᵤ c) : a * b ~ᵤ a * c := by
   obtain ⟨d, rfl⟩ := h; exact ⟨d, mul_assoc _ _ _⟩
 
-theorem Associated.mul_right [CommMonoid α] {a b : α} (h : a ~ᵤ b) (c : α) : a * c ~ᵤ b * c := by
+theorem Associated.mul_right [CommMonoid M] {a b : M} (h : a ~ᵤ b) (c : M) : a * c ~ᵤ b * c := by
   obtain ⟨d, rfl⟩ := h; exact ⟨d, mul_right_comm _ _ _⟩
 
-theorem Associated.mul_mul [CommMonoid α] {a₁ a₂ b₁ b₂ : α}
+theorem Associated.mul_mul [CommMonoid M] {a₁ a₂ b₁ b₂ : M}
     (h₁ : a₁ ~ᵤ b₁) (h₂ : a₂ ~ᵤ b₂) : a₁ * a₂ ~ᵤ b₁ * b₂ := (h₁.mul_right _).trans (h₂.mul_left _)
 
-theorem Associated.pow_pow [CommMonoid α] {a b : α} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
+theorem Associated.pow_pow [CommMonoid M] {a b : M} {n : ℕ} (h : a ~ᵤ b) : a ^ n ~ᵤ b ^ n := by
   induction n with
   | zero => simp [Associated.refl]
   | succ n ih => convert h.mul_mul ih <;> rw [pow_succ']
 
-protected theorem Associated.dvd [Monoid α] {a b : α} : a ~ᵤ b → a ∣ b := fun ⟨u, hu⟩ =>
+protected theorem Associated.dvd [Monoid M] {a b : M} : a ~ᵤ b → a ∣ b := fun ⟨u, hu⟩ =>
   ⟨u, hu.symm⟩
 
-protected theorem Associated.dvd' [Monoid α] {a b : α} (h : a ~ᵤ b) : b ∣ a :=
+protected theorem Associated.dvd' [Monoid M] {a b : M} (h : a ~ᵤ b) : b ∣ a :=
   h.symm.dvd
 
-protected theorem Associated.dvd_dvd [Monoid α] {a b : α} (h : a ~ᵤ b) : a ∣ b ∧ b ∣ a :=
+protected theorem Associated.dvd_dvd [Monoid M] {a b : M} (h : a ~ᵤ b) : a ∣ b ∧ b ∣ a :=
   ⟨h.dvd, h.symm.dvd⟩
 
-theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣ b) (hba : b ∣ a) :
+theorem associated_of_dvd_dvd [CancelMonoidWithZero M] {a b : M} (hab : a ∣ b) (hba : b ∣ a) :
     a ~ᵤ b := by
   rcases hab with ⟨c, rfl⟩
   rcases hba with ⟨d, a_eq⟩
@@ -529,40 +535,40 @@ theorem associated_of_dvd_dvd [CancelMonoidWithZero α] {a b : α} (hab : a ∣
   have hdc : d * c = 1 := mul_left_cancel₀ hac0 this
   exact ⟨⟨c, d, hcd, hdc⟩, rfl⟩
 
-theorem dvd_dvd_iff_associated [CancelMonoidWithZero α] {a b : α} : a ∣ b ∧ b ∣ a ↔ a ~ᵤ b :=
+theorem dvd_dvd_iff_associated [CancelMonoidWithZero M] {a b : M} : a ∣ b ∧ b ∣ a ↔ a ~ᵤ b :=
   ⟨fun ⟨h1, h2⟩ => associated_of_dvd_dvd h1 h2, Associated.dvd_dvd⟩
 
-instance [CancelMonoidWithZero α] [DecidableRel ((· ∣ ·) : α → α → Prop)] :
-    DecidableRel ((· ~ᵤ ·) : α → α → Prop) := fun _ _ => decidable_of_iff _ dvd_dvd_iff_associated
+instance [CancelMonoidWithZero M] [DecidableRel ((· ∣ ·) : M → M → Prop)] :
+    DecidableRel ((· ~ᵤ ·) : M → M → Prop) := fun _ _ => decidable_of_iff _ dvd_dvd_iff_associated
 
-theorem Associated.dvd_iff_dvd_left [Monoid α] {a b c : α} (h : a ~ᵤ b) : a ∣ c ↔ b ∣ c :=
+theorem Associated.dvd_iff_dvd_left [Monoid M] {a b c : M} (h : a ~ᵤ b) : a ∣ c ↔ b ∣ c :=
   let ⟨_, hu⟩ := h
   hu ▸ Units.mul_right_dvd.symm
 
-theorem Associated.dvd_iff_dvd_right [Monoid α] {a b c : α} (h : b ~ᵤ c) : a ∣ b ↔ a ∣ c :=
+theorem Associated.dvd_iff_dvd_right [Monoid M] {a b c : M} (h : b ~ᵤ c) : a ∣ b ↔ a ∣ c :=
   let ⟨_, hu⟩ := h
   hu ▸ Units.dvd_mul_right.symm
 
-theorem Associated.eq_zero_iff [MonoidWithZero α] {a b : α} (h : a ~ᵤ b) : a = 0 ↔ b = 0 := by
+theorem Associated.eq_zero_iff [MonoidWithZero M] {a b : M} (h : a ~ᵤ b) : a = 0 ↔ b = 0 := by
   obtain ⟨u, rfl⟩ := h
   rw [← Units.eq_mul_inv_iff_mul_eq, zero_mul]
 
-theorem Associated.ne_zero_iff [MonoidWithZero α] {a b : α} (h : a ~ᵤ b) : a ≠ 0 ↔ b ≠ 0 :=
+theorem Associated.ne_zero_iff [MonoidWithZero M] {a b : M} (h : a ~ᵤ b) : a ≠ 0 ↔ b ≠ 0 :=
   not_congr h.eq_zero_iff
 
-theorem Associated.neg_left [Monoid α] [HasDistribNeg α] {a b : α} (h : Associated a b) :
+theorem Associated.neg_left [Monoid M] [HasDistribNeg M] {a b : M} (h : Associated a b) :
     Associated (-a) b :=
   let ⟨u, hu⟩ := h; ⟨-u, by simp [hu]⟩
 
-theorem Associated.neg_right [Monoid α] [HasDistribNeg α] {a b : α} (h : Associated a b) :
+theorem Associated.neg_right [Monoid M] [HasDistribNeg M] {a b : M} (h : Associated a b) :
     Associated a (-b) :=
   h.symm.neg_left.symm
 
-theorem Associated.neg_neg [Monoid α] [HasDistribNeg α] {a b : α} (h : Associated a b) :
+theorem Associated.neg_neg [Monoid M] [HasDistribNeg M] {a b : M} (h : Associated a b) :
     Associated (-a) (-b) :=
   h.neg_left.neg_right
 
-protected theorem Associated.prime [CommMonoidWithZero α] {p q : α} (h : p ~ᵤ q) (hp : Prime p) :
+protected theorem Associated.prime [CommMonoidWithZero M] {p q : M} (h : p ~ᵤ q) (hp : Prime p) :
     Prime q :=
   ⟨h.ne_zero_iff.1 hp.ne_zero,
     let ⟨u, hu⟩ := h
@@ -572,7 +578,7 @@ protected theorem Associated.prime [CommMonoidWithZero α] {p q : α} (h : p ~
         intro a b
         exact hp.dvd_or_dvd⟩⟩
 
-theorem prime_mul_iff [CancelCommMonoidWithZero α] {x y : α} :
+theorem prime_mul_iff [CancelCommMonoidWithZero M] {x y : M} :
     Prime (x * y) ↔ (Prime x ∧ IsUnit y) ∨ (IsUnit x ∧ Prime y) := by
   refine ⟨fun h ↦ ?_, ?_⟩
   · rcases of_irreducible_mul h.irreducible with hx | hy
@@ -583,7 +589,7 @@ theorem prime_mul_iff [CancelCommMonoidWithZero α] {x y : α} :
     · exact (associated_unit_mul_right y x hx).prime hy
 
 @[simp]
-lemma prime_pow_iff [CancelCommMonoidWithZero α] {p : α} {n : ℕ} :
+lemma prime_pow_iff [CancelCommMonoidWithZero M] {p : M} {n : ℕ} :
     Prime (p ^ n) ↔ Prime p ∧ n = 1 := by
   refine ⟨fun hp ↦ ?_, fun ⟨hp, hn⟩ ↦ by simpa [hn]⟩
   suffices n = 1 by aesop
@@ -598,7 +604,7 @@ lemma prime_pow_iff [CancelCommMonoidWithZero α] {p : α} {n : ℕ} :
     · exfalso
       exact hpn.not_unit (hp.pow n)
 
-theorem Irreducible.dvd_iff [Monoid α] {x y : α} (hx : Irreducible x) :
+theorem Irreducible.dvd_iff [Monoid M] {x y : M} (hx : Irreducible x) :
     y ∣ x ↔ IsUnit y ∨ Associated x y := by
   constructor
   · rintro ⟨z, hz⟩
@@ -610,67 +616,67 @@ theorem Irreducible.dvd_iff [Monoid α] {x y : α} (hx : Irreducible x) :
     · exact hy.dvd
     · exact h.symm.dvd
 
-theorem Irreducible.associated_of_dvd [Monoid α] {p q : α} (p_irr : Irreducible p)
+theorem Irreducible.associated_of_dvd [Monoid M] {p q : M} (p_irr : Irreducible p)
     (q_irr : Irreducible q) (dvd : p ∣ q) : Associated p q :=
   ((q_irr.dvd_iff.mp dvd).resolve_left p_irr.not_unit).symm
 
-theorem Irreducible.dvd_irreducible_iff_associated [Monoid α] {p q : α}
+theorem Irreducible.dvd_irreducible_iff_associated [Monoid M] {p q : M}
     (pp : Irreducible p) (qp : Irreducible q) : p ∣ q ↔ Associated p q :=
   ⟨Irreducible.associated_of_dvd pp qp, Associated.dvd⟩
 
-theorem Prime.associated_of_dvd [CancelCommMonoidWithZero α] {p q : α} (p_prime : Prime p)
+theorem Prime.associated_of_dvd [CancelCommMonoidWithZero M] {p q : M} (p_prime : Prime p)
     (q_prime : Prime q) (dvd : p ∣ q) : Associated p q :=
   p_prime.irreducible.associated_of_dvd q_prime.irreducible dvd
 
-theorem Prime.dvd_prime_iff_associated [CancelCommMonoidWithZero α] {p q : α} (pp : Prime p)
+theorem Prime.dvd_prime_iff_associated [CancelCommMonoidWithZero M] {p q : M} (pp : Prime p)
     (qp : Prime q) : p ∣ q ↔ Associated p q :=
   pp.irreducible.dvd_irreducible_iff_associated qp.irreducible
 
-theorem Associated.prime_iff [CommMonoidWithZero α] {p q : α} (h : p ~ᵤ q) : Prime p ↔ Prime q :=
+theorem Associated.prime_iff [CommMonoidWithZero M] {p q : M} (h : p ~ᵤ q) : Prime p ↔ Prime q :=
   ⟨h.prime, h.symm.prime⟩
 
-protected theorem Associated.isUnit [Monoid α] {a b : α} (h : a ~ᵤ b) : IsUnit a → IsUnit b :=
+protected theorem Associated.isUnit [Monoid M] {a b : M} (h : a ~ᵤ b) : IsUnit a → IsUnit b :=
   let ⟨u, hu⟩ := h
   fun ⟨v, hv⟩ => ⟨v * u, by simp [hv, hu.symm]⟩
 
-theorem Associated.isUnit_iff [Monoid α] {a b : α} (h : a ~ᵤ b) : IsUnit a ↔ IsUnit b :=
+theorem Associated.isUnit_iff [Monoid M] {a b : M} (h : a ~ᵤ b) : IsUnit a ↔ IsUnit b :=
   ⟨h.isUnit, h.symm.isUnit⟩
 
-theorem Irreducible.isUnit_iff_not_associated_of_dvd [Monoid α]
-    {x y : α} (hx : Irreducible x) (hy : y ∣ x) : IsUnit y ↔ ¬ Associated x y :=
+theorem Irreducible.isUnit_iff_not_associated_of_dvd [Monoid M]
+    {x y : M} (hx : Irreducible x) (hy : y ∣ x) : IsUnit y ↔ ¬ Associated x y :=
   ⟨fun hy hxy => hx.1 (hxy.symm.isUnit hy), (hx.dvd_iff.mp hy).resolve_right⟩
 
-protected theorem Associated.irreducible [Monoid α] {p q : α} (h : p ~ᵤ q) (hp : Irreducible p) :
+protected theorem Associated.irreducible [Monoid M] {p q : M} (h : p ~ᵤ q) (hp : Irreducible p) :
     Irreducible q :=
   ⟨mt h.symm.isUnit hp.1,
     let ⟨u, hu⟩ := h
     fun a b hab =>
-    have hpab : p = a * (b * (u⁻¹ : αˣ)) :=
+    have hpab : p = a * (b * (u⁻¹ : Mˣ)) :=
       calc
-        p = p * u * (u⁻¹ : αˣ) := by simp
+        p = p * u * (u⁻¹ : Mˣ) := by simp
         _ = _ := by rw [hu]; simp [hab, mul_assoc]
 
     (hp.isUnit_or_isUnit hpab).elim Or.inl fun ⟨v, hv⟩ => Or.inr ⟨v * u, by simp [hv]⟩⟩
 
-protected theorem Associated.irreducible_iff [Monoid α] {p q : α} (h : p ~ᵤ q) :
+protected theorem Associated.irreducible_iff [Monoid M] {p q : M} (h : p ~ᵤ q) :
     Irreducible p ↔ Irreducible q :=
   ⟨h.irreducible, h.symm.irreducible⟩
 
-theorem Associated.of_mul_left [CancelCommMonoidWithZero α] {a b c d : α} (h : a * b ~ᵤ c * d)
+theorem Associated.of_mul_left [CancelCommMonoidWithZero M] {a b c d : M} (h : a * b ~ᵤ c * d)
     (h₁ : a ~ᵤ c) (ha : a ≠ 0) : b ~ᵤ d :=
   let ⟨u, hu⟩ := h
   let ⟨v, hv⟩ := Associated.symm h₁
-  ⟨u * (v : αˣ),
+  ⟨u * (v : Mˣ),
     mul_left_cancel₀ ha
       (by
-        rw [← hv, mul_assoc c (v : α) d, mul_left_comm c, ← hu]
+        rw [← hv, mul_assoc c (v : M) d, mul_left_comm c, ← hu]
         simp [hv.symm, mul_assoc, mul_comm, mul_left_comm])⟩
 
-theorem Associated.of_mul_right [CancelCommMonoidWithZero α] {a b c d : α} :
+theorem Associated.of_mul_right [CancelCommMonoidWithZero M] {a b c d : M} :
     a * b ~ᵤ c * d → b ~ᵤ d → b ≠ 0 → a ~ᵤ c := by
   rw [mul_comm a, mul_comm c]; exact Associated.of_mul_left
 
-theorem Associated.of_pow_associated_of_prime [CancelCommMonoidWithZero α] {p₁ p₂ : α} {k₁ k₂ : ℕ}
+theorem Associated.of_pow_associated_of_prime [CancelCommMonoidWithZero M] {p₁ p₂ : M} {k₁ k₂ : ℕ}
     (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) : p₁ ~ᵤ p₂ := by
   have : p₁ ∣ p₂ ^ k₂ := by
     rw [← h.dvd_iff_dvd_right]
@@ -678,37 +684,37 @@ theorem Associated.of_pow_associated_of_prime [CancelCommMonoidWithZero α] {p
   rw [← hp₁.dvd_prime_iff_associated hp₂]
   exact hp₁.dvd_of_dvd_pow this
 
-theorem Associated.of_pow_associated_of_prime' [CancelCommMonoidWithZero α] {p₁ p₂ : α} {k₁ k₂ : ℕ}
+theorem Associated.of_pow_associated_of_prime' [CancelCommMonoidWithZero M] {p₁ p₂ : M} {k₁ k₂ : ℕ}
     (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₂ : 0 < k₂) (h : p₁ ^ k₁ ~ᵤ p₂ ^ k₂) : p₁ ~ᵤ p₂ :=
   (h.symm.of_pow_associated_of_prime hp₂ hp₁ hk₂).symm
 
 /-- See also `Irreducible.coprime_iff_not_dvd`. -/
-lemma Irreducible.isRelPrime_iff_not_dvd [Monoid α] {p n : α} (hp : Irreducible p) :
+lemma Irreducible.isRelPrime_iff_not_dvd [Monoid M] {p n : M} (hp : Irreducible p) :
     IsRelPrime p n ↔ ¬ p ∣ n := by
   refine ⟨fun h contra ↦ hp.not_unit (h dvd_rfl contra), fun hpn d hdp hdn ↦ ?_⟩
   contrapose! hpn
   suffices Associated p d from this.dvd.trans hdn
   exact (hp.dvd_iff.mp hdp).resolve_left hpn
 
-lemma Irreducible.dvd_or_isRelPrime [Monoid α] {p n : α} (hp : Irreducible p) :
+lemma Irreducible.dvd_or_isRelPrime [Monoid M] {p n : M} (hp : Irreducible p) :
     p ∣ n ∨ IsRelPrime p n := Classical.or_iff_not_imp_left.mpr hp.isRelPrime_iff_not_dvd.2
 
 section UniqueUnits
 
-variable [Monoid α] [Unique αˣ]
+variable [Monoid M] [Subsingleton Mˣ]
 
-theorem associated_iff_eq {x y : α} : x ~ᵤ y ↔ x = y := by
+theorem associated_iff_eq {x y : M} : x ~ᵤ y ↔ x = y := by
   constructor
   · rintro ⟨c, rfl⟩
     rw [units_eq_one c, Units.val_one, mul_one]
   · rintro rfl
     rfl
 
-theorem associated_eq_eq : (Associated : α → α → Prop) = Eq := by
+theorem associated_eq_eq : (Associated : M → M → Prop) = Eq := by
   ext
   rw [associated_iff_eq]
 
-theorem prime_dvd_prime_iff_eq {M : Type*} [CancelCommMonoidWithZero M] [Unique Mˣ] {p q : M}
+theorem prime_dvd_prime_iff_eq {M : Type*} [CancelCommMonoidWithZero M] [Subsingleton Mˣ] {p q : M}
     (pp : Prime p) (qp : Prime q) : p ∣ q ↔ p = q := by
   rw [pp.dvd_prime_iff_associated qp, ← associated_eq_eq]
 
@@ -716,7 +722,7 @@ end UniqueUnits
 
 section UniqueUnits₀
 
-variable {R : Type*} [CancelCommMonoidWithZero R] [Unique Rˣ] {p₁ p₂ : R} {k₁ k₂ : ℕ}
+variable {R : Type*} [CancelCommMonoidWithZero R] [Subsingleton Rˣ] {p₁ p₂ : R} {k₁ k₂ : ℕ}
 
 theorem eq_of_prime_pow_eq (hp₁ : Prime p₁) (hp₂ : Prime p₂) (hk₁ : 0 < k₁)
     (h : p₁ ^ k₁ = p₂ ^ k₂) : p₁ = p₂ := by
@@ -732,86 +738,86 @@ end UniqueUnits₀
 
 /-- The quotient of a monoid by the `Associated` relation. Two elements `x` and `y`
   are associated iff there is a unit `u` such that `x * u = y`. There is a natural
-  monoid structure on `Associates α`. -/
-abbrev Associates (α : Type*) [Monoid α] : Type _ :=
-  Quotient (Associated.setoid α)
+  monoid structure on `Associates M`. -/
+abbrev Associates (M : Type*) [Monoid M] : Type _ :=
+  Quotient (Associated.setoid M)
 
 namespace Associates
 
 open Associated
 
-/-- The canonical quotient map from a monoid `α` into the `Associates` of `α` -/
-protected abbrev mk {α : Type*} [Monoid α] (a : α) : Associates α :=
+/-- The canonical quotient map from a monoid `M` into the `Associates` of `M` -/
+protected abbrev mk {M : Type*} [Monoid M] (a : M) : Associates M :=
   ⟦a⟧
 
-instance [Monoid α] : Inhabited (Associates α) :=
+instance [Monoid M] : Inhabited (Associates M) :=
   ⟨⟦1⟧⟩
 
-theorem mk_eq_mk_iff_associated [Monoid α] {a b : α} : Associates.mk a = Associates.mk b ↔ a ~ᵤ b :=
+theorem mk_eq_mk_iff_associated [Monoid M] {a b : M} : Associates.mk a = Associates.mk b ↔ a ~ᵤ b :=
   Iff.intro Quotient.exact Quot.sound
 
-theorem quotient_mk_eq_mk [Monoid α] (a : α) : ⟦a⟧ = Associates.mk a :=
+theorem quotient_mk_eq_mk [Monoid M] (a : M) : ⟦a⟧ = Associates.mk a :=
   rfl
 
-theorem quot_mk_eq_mk [Monoid α] (a : α) : Quot.mk Setoid.r a = Associates.mk a :=
+theorem quot_mk_eq_mk [Monoid M] (a : M) : Quot.mk Setoid.r a = Associates.mk a :=
   rfl
 
 @[simp]
-theorem quot_out [Monoid α] (a : Associates α) : Associates.mk (Quot.out a) = a := by
+theorem quot_out [Monoid M] (a : Associates M) : Associates.mk (Quot.out a) = a := by
   rw [← quot_mk_eq_mk, Quot.out_eq]
 
-theorem mk_quot_out [Monoid α] (a : α) : Quot.out (Associates.mk a) ~ᵤ a := by
+theorem mk_quot_out [Monoid M] (a : M) : Quot.out (Associates.mk a) ~ᵤ a := by
   rw [← Associates.mk_eq_mk_iff_associated, Associates.quot_out]
 
-theorem forall_associated [Monoid α] {p : Associates α → Prop} :
+theorem forall_associated [Monoid M] {p : Associates M → Prop} :
     (∀ a, p a) ↔ ∀ a, p (Associates.mk a) :=
   Iff.intro (fun h _ => h _) fun h a => Quotient.inductionOn a h
 
-theorem mk_surjective [Monoid α] : Function.Surjective (@Associates.mk α _) :=
+theorem mk_surjective [Monoid M] : Function.Surjective (@Associates.mk M _) :=
   forall_associated.2 fun a => ⟨a, rfl⟩
 
-instance [Monoid α] : One (Associates α) :=
+instance [Monoid M] : One (Associates M) :=
   ⟨⟦1⟧⟩
 
 @[simp]
-theorem mk_one [Monoid α] : Associates.mk (1 : α) = 1 :=
+theorem mk_one [Monoid M] : Associates.mk (1 : M) = 1 :=
   rfl
 
-theorem one_eq_mk_one [Monoid α] : (1 : Associates α) = Associates.mk 1 :=
+theorem one_eq_mk_one [Monoid M] : (1 : Associates M) = Associates.mk 1 :=
   rfl
 
 @[simp]
-theorem mk_eq_one [Monoid α] {a : α} : Associates.mk a = 1 ↔ IsUnit a := by
+theorem mk_eq_one [Monoid M] {a : M} : Associates.mk a = 1 ↔ IsUnit a := by
   rw [← mk_one, mk_eq_mk_iff_associated, associated_one_iff_isUnit]
 
-instance [Monoid α] : Bot (Associates α) :=
+instance [Monoid M] : Bot (Associates M) :=
   ⟨1⟩
 
-theorem bot_eq_one [Monoid α] : (⊥ : Associates α) = 1 :=
+theorem bot_eq_one [Monoid M] : (⊥ : Associates M) = 1 :=
   rfl
 
-theorem exists_rep [Monoid α] (a : Associates α) : ∃ a0 : α, Associates.mk a0 = a :=
+theorem exists_rep [Monoid M] (a : Associates M) : ∃ a0 : M, Associates.mk a0 = a :=
   Quot.exists_rep a
 
-instance [Monoid α] [Subsingleton α] :
-    Unique (Associates α) where
+instance [Monoid M] [Subsingleton M] :
+    Unique (Associates M) where
   default := 1
   uniq := forall_associated.2 fun _ ↦ mk_eq_one.2 <| isUnit_of_subsingleton _
 
-theorem mk_injective [Monoid α] [Unique (Units α)] : Function.Injective (@Associates.mk α _) :=
+theorem mk_injective [Monoid M] [Subsingleton Mˣ] : Function.Injective (@Associates.mk M _) :=
   fun _ _ h => associated_iff_eq.mp (Associates.mk_eq_mk_iff_associated.mp h)
 
 section CommMonoid
 
-variable [CommMonoid α]
+variable [CommMonoid M]
 
-instance instMul : Mul (Associates α) :=
+instance instMul : Mul (Associates M) :=
   ⟨Quotient.map₂ (· * ·) fun _ _ h₁ _ _ h₂ ↦ h₁.mul_mul h₂⟩
 
-theorem mk_mul_mk {x y : α} : Associates.mk x * Associates.mk y = Associates.mk (x * y) :=
+theorem mk_mul_mk {x y : M} : Associates.mk x * Associates.mk y = Associates.mk (x * y) :=
   rfl
 
-instance instCommMonoid : CommMonoid (Associates α) where
+instance instCommMonoid : CommMonoid (Associates M) where
   one := 1
   mul := (· * ·)
   mul_one a' := Quotient.inductionOn a' fun a => show ⟦a * 1⟧ = ⟦a⟧ by simp
@@ -822,32 +828,32 @@ instance instCommMonoid : CommMonoid (Associates α) where
   mul_comm a' b' :=
     Quotient.inductionOn₂ a' b' fun a b => show ⟦a * b⟧ = ⟦b * a⟧ by rw [mul_comm]
 
-instance instPreorder : Preorder (Associates α) where
+instance instPreorder : Preorder (Associates M) where
   le := Dvd.dvd
   le_refl := dvd_refl
   le_trans a b c := dvd_trans
 
 /-- `Associates.mk` as a `MonoidHom`. -/
-protected def mkMonoidHom : α →* Associates α where
+protected def mkMonoidHom : M →* Associates M where
   toFun := Associates.mk
   map_one' := mk_one
   map_mul' _ _ := mk_mul_mk
 
 @[simp]
-theorem mkMonoidHom_apply (a : α) : Associates.mkMonoidHom a = Associates.mk a :=
+theorem mkMonoidHom_apply (a : M) : Associates.mkMonoidHom a = Associates.mk a :=
   rfl
 
-theorem associated_map_mk {f : Associates α →* α} (hinv : Function.RightInverse f Associates.mk)
-    (a : α) : a ~ᵤ f (Associates.mk a) :=
+theorem associated_map_mk {f : Associates M →* M} (hinv : Function.RightInverse f Associates.mk)
+    (a : M) : a ~ᵤ f (Associates.mk a) :=
   Associates.mk_eq_mk_iff_associated.1 (hinv (Associates.mk a)).symm
 
-theorem mk_pow (a : α) (n : ℕ) : Associates.mk (a ^ n) = Associates.mk a ^ n := by
+theorem mk_pow (a : M) (n : ℕ) : Associates.mk (a ^ n) = Associates.mk a ^ n := by
   induction n <;> simp [*, pow_succ, Associates.mk_mul_mk.symm]
 
-theorem dvd_eq_le : ((· ∣ ·) : Associates α → Associates α → Prop) = (· ≤ ·) :=
+theorem dvd_eq_le : ((· ∣ ·) : Associates M → Associates M → Prop) = (· ≤ ·) :=
   rfl
 
-instance uniqueUnits : Unique (Associates α)ˣ where
+instance uniqueUnits : Unique (Associates M)ˣ where
   uniq := by
     rintro ⟨a, b, hab, hba⟩
     revert hab hba
@@ -858,16 +864,16 @@ instance uniqueUnits : Unique (Associates α)ˣ where
 @[deprecated (since := "2024-07-22")] protected alias units_eq_one := Subsingleton.elim
 
 @[simp]
-theorem coe_unit_eq_one (u : (Associates α)ˣ) : (u : Associates α) = 1 := by
+theorem coe_unit_eq_one (u : (Associates M)ˣ) : (u : Associates M) = 1 := by
   simp [eq_iff_true_of_subsingleton]
 
-theorem isUnit_iff_eq_one (a : Associates α) : IsUnit a ↔ a = 1 :=
+theorem isUnit_iff_eq_one (a : Associates M) : IsUnit a ↔ a = 1 :=
   Iff.intro (fun ⟨_, h⟩ => h ▸ coe_unit_eq_one _) fun h => h.symm ▸ isUnit_one
 
-theorem isUnit_iff_eq_bot {a : Associates α} : IsUnit a ↔ a = ⊥ := by
+theorem isUnit_iff_eq_bot {a : Associates M} : IsUnit a ↔ a = ⊥ := by
   rw [Associates.isUnit_iff_eq_one, bot_eq_one]
 
-theorem isUnit_mk {a : α} : IsUnit (Associates.mk a) ↔ IsUnit a :=
+theorem isUnit_mk {a : M} : IsUnit (Associates.mk a) ↔ IsUnit a :=
   calc
     IsUnit (Associates.mk a) ↔ a ~ᵤ 1 := by
       rw [isUnit_iff_eq_one, one_eq_mk_one, mk_eq_mk_iff_associated]
@@ -875,27 +881,27 @@ theorem isUnit_mk {a : α} : IsUnit (Associates.mk a) ↔ IsUnit a :=
 
 section Order
 
-theorem mul_mono {a b c d : Associates α} (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d :=
+theorem mul_mono {a b c d : Associates M} (h₁ : a ≤ b) (h₂ : c ≤ d) : a * c ≤ b * d :=
   let ⟨x, hx⟩ := h₁
   let ⟨y, hy⟩ := h₂
   ⟨x * y, by simp [hx, hy, mul_comm, mul_assoc, mul_left_comm]⟩
 
-theorem one_le {a : Associates α} : 1 ≤ a :=
+theorem one_le {a : Associates M} : 1 ≤ a :=
   Dvd.intro _ (one_mul a)
 
-theorem le_mul_right {a b : Associates α} : a ≤ a * b :=
+theorem le_mul_right {a b : Associates M} : a ≤ a * b :=
   ⟨b, rfl⟩
 
-theorem le_mul_left {a b : Associates α} : a ≤ b * a := by rw [mul_comm]; exact le_mul_right
+theorem le_mul_left {a b : Associates M} : a ≤ b * a := by rw [mul_comm]; exact le_mul_right
 
-instance instOrderBot : OrderBot (Associates α) where
+instance instOrderBot : OrderBot (Associates M) where
   bot := 1
   bot_le _ := one_le
 
 end Order
 
 @[simp]
-theorem mk_dvd_mk {a b : α} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b := by
+theorem mk_dvd_mk {a b : M} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b := by
   simp only [dvd_def, mk_surjective.exists, mk_mul_mk, mk_eq_mk_iff_associated,
     Associated.comm (x := b)]
   constructor
@@ -904,18 +910,18 @@ theorem mk_dvd_mk {a b : α} : Associates.mk a ∣ Associates.mk b ↔ a ∣ b :
   · rintro ⟨c, rfl⟩
     use c
 
-theorem dvd_of_mk_le_mk {a b : α} : Associates.mk a ≤ Associates.mk b → a ∣ b :=
+theorem dvd_of_mk_le_mk {a b : M} : Associates.mk a ≤ Associates.mk b → a ∣ b :=
   mk_dvd_mk.mp
 
-theorem mk_le_mk_of_dvd {a b : α} : a ∣ b → Associates.mk a ≤ Associates.mk b :=
+theorem mk_le_mk_of_dvd {a b : M} : a ∣ b → Associates.mk a ≤ Associates.mk b :=
   mk_dvd_mk.mpr
 
-theorem mk_le_mk_iff_dvd {a b : α} : Associates.mk a ≤ Associates.mk b ↔ a ∣ b := mk_dvd_mk
+theorem mk_le_mk_iff_dvd {a b : M} : Associates.mk a ≤ Associates.mk b ↔ a ∣ b := mk_dvd_mk
 
 @[deprecated (since := "2024-03-16")] alias mk_le_mk_iff_dvd_iff := mk_le_mk_iff_dvd
 
 @[simp]
-theorem isPrimal_mk {a : α} : IsPrimal (Associates.mk a) ↔ IsPrimal a := by
+theorem isPrimal_mk {a : M} : IsPrimal (Associates.mk a) ↔ IsPrimal a := by
   simp_rw [IsPrimal, forall_associated, mk_surjective.exists, mk_mul_mk, mk_dvd_mk]
   constructor <;> intro h b c dvd <;> obtain ⟨a₁, a₂, h₁, h₂, eq⟩ := @h b c dvd
   · obtain ⟨u, rfl⟩ := mk_eq_mk_iff_associated.mp eq.symm
@@ -925,80 +931,80 @@ theorem isPrimal_mk {a : α} : IsPrimal (Associates.mk a) ↔ IsPrimal a := by
 @[deprecated (since := "2024-03-16")] alias isPrimal_iff := isPrimal_mk
 
 @[simp]
-theorem decompositionMonoid_iff : DecompositionMonoid (Associates α) ↔ DecompositionMonoid α := by
+theorem decompositionMonoid_iff : DecompositionMonoid (Associates M) ↔ DecompositionMonoid M := by
   simp_rw [_root_.decompositionMonoid_iff, forall_associated, isPrimal_mk]
 
-instance instDecompositionMonoid [DecompositionMonoid α] : DecompositionMonoid (Associates α) :=
+instance instDecompositionMonoid [DecompositionMonoid M] : DecompositionMonoid (Associates M) :=
   decompositionMonoid_iff.mpr ‹_›
 
 @[simp]
-theorem mk_isRelPrime_iff {a b : α} :
+theorem mk_isRelPrime_iff {a b : M} :
     IsRelPrime (Associates.mk a) (Associates.mk b) ↔ IsRelPrime a b := by
   simp_rw [IsRelPrime, forall_associated, mk_dvd_mk, isUnit_mk]
 
 end CommMonoid
 
-instance [Zero α] [Monoid α] : Zero (Associates α) :=
+instance [Zero M] [Monoid M] : Zero (Associates M) :=
   ⟨⟦0⟧⟩
 
-instance [Zero α] [Monoid α] : Top (Associates α) :=
+instance [Zero M] [Monoid M] : Top (Associates M) :=
   ⟨0⟩
 
-@[simp] theorem mk_zero [Zero α] [Monoid α] : Associates.mk (0 : α) = 0 := rfl
+@[simp] theorem mk_zero [Zero M] [Monoid M] : Associates.mk (0 : M) = 0 := rfl
 
 section MonoidWithZero
 
-variable [MonoidWithZero α]
+variable [MonoidWithZero M]
 
 @[simp]
-theorem mk_eq_zero {a : α} : Associates.mk a = 0 ↔ a = 0 :=
+theorem mk_eq_zero {a : M} : Associates.mk a = 0 ↔ a = 0 :=
   ⟨fun h => (associated_zero_iff_eq_zero a).1 <| Quotient.exact h, fun h => h.symm ▸ rfl⟩
 
 @[simp]
-theorem quot_out_zero : Quot.out (0 : Associates α) = 0 := by rw [← mk_eq_zero, quot_out]
+theorem quot_out_zero : Quot.out (0 : Associates M) = 0 := by rw [← mk_eq_zero, quot_out]
 
-theorem mk_ne_zero {a : α} : Associates.mk a ≠ 0 ↔ a ≠ 0 :=
+theorem mk_ne_zero {a : M} : Associates.mk a ≠ 0 ↔ a ≠ 0 :=
   not_congr mk_eq_zero
 
-instance [Nontrivial α] : Nontrivial (Associates α) :=
+instance [Nontrivial M] : Nontrivial (Associates M) :=
   ⟨⟨1, 0, mk_ne_zero.2 one_ne_zero⟩⟩
 
-theorem exists_non_zero_rep {a : Associates α} : a ≠ 0 → ∃ a0 : α, a0 ≠ 0 ∧ Associates.mk a0 = a :=
+theorem exists_non_zero_rep {a : Associates M} : a ≠ 0 → ∃ a0 : M, a0 ≠ 0 ∧ Associates.mk a0 = a :=
   Quotient.inductionOn a fun b nz => ⟨b, mt (congr_arg Quotient.mk'') nz, rfl⟩
 
 end MonoidWithZero
 
 section CommMonoidWithZero
 
-variable [CommMonoidWithZero α]
+variable [CommMonoidWithZero M]
 
-instance instCommMonoidWithZero : CommMonoidWithZero (Associates α) where
+instance instCommMonoidWithZero : CommMonoidWithZero (Associates M) where
     zero_mul := forall_associated.2 fun a ↦ by rw [← mk_zero, mk_mul_mk, zero_mul]
     mul_zero := forall_associated.2 fun a ↦ by rw [← mk_zero, mk_mul_mk, mul_zero]
 
-instance instOrderTop : OrderTop (Associates α) where
+instance instOrderTop : OrderTop (Associates M) where
   top := 0
   le_top := dvd_zero
 
-@[simp] protected theorem le_zero (a : Associates α) : a ≤ 0 := le_top
+@[simp] protected theorem le_zero (a : Associates M) : a ≤ 0 := le_top
 
-instance instBoundedOrder : BoundedOrder (Associates α) where
+instance instBoundedOrder : BoundedOrder (Associates M) where
 
-instance [DecidableRel ((· ∣ ·) : α → α → Prop)] :
-    DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop) := fun a b =>
+instance [DecidableRel ((· ∣ ·) : M → M → Prop)] :
+    DecidableRel ((· ∣ ·) : Associates M → Associates M → Prop) := fun a b =>
   Quotient.recOnSubsingleton₂ a b fun _ _ => decidable_of_iff' _ mk_dvd_mk
 
-theorem Prime.le_or_le {p : Associates α} (hp : Prime p) {a b : Associates α} (h : p ≤ a * b) :
+theorem Prime.le_or_le {p : Associates M} (hp : Prime p) {a b : Associates M} (h : p ≤ a * b) :
     p ≤ a ∨ p ≤ b :=
   hp.2.2 a b h
 
 @[simp]
-theorem prime_mk {p : α} : Prime (Associates.mk p) ↔ Prime p := by
+theorem prime_mk {p : M} : Prime (Associates.mk p) ↔ Prime p := by
   rw [Prime, _root_.Prime, forall_associated]
   simp only [forall_associated, mk_ne_zero, isUnit_mk, mk_mul_mk, mk_dvd_mk]
 
 @[simp]
-theorem irreducible_mk {a : α} : Irreducible (Associates.mk a) ↔ Irreducible a := by
+theorem irreducible_mk {a : M} : Irreducible (Associates.mk a) ↔ Irreducible a := by
   simp only [irreducible_iff, isUnit_mk, forall_associated, isUnit_mk, mk_mul_mk,
     mk_eq_mk_iff_associated, Associated.comm (x := a)]
   apply Iff.rfl.and
@@ -1009,7 +1015,7 @@ theorem irreducible_mk {a : α} : Irreducible (Associates.mk a) ↔ Irreducible
     simpa using h x (y * u) (mul_assoc _ _ _)
 
 @[simp]
-theorem mk_dvdNotUnit_mk_iff {a b : α} :
+theorem mk_dvdNotUnit_mk_iff {a b : M} :
     DvdNotUnit (Associates.mk a) (Associates.mk b) ↔ DvdNotUnit a b := by
   simp only [DvdNotUnit, mk_ne_zero, mk_surjective.exists, isUnit_mk, mk_mul_mk,
     mk_eq_mk_iff_associated, Associated.comm (x := b)]
@@ -1021,7 +1027,7 @@ theorem mk_dvdNotUnit_mk_iff {a b : α} :
   · rintro ⟨x, ⟨hx, rfl⟩⟩
     use x
 
-theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b := by
+theorem dvdNotUnit_of_lt {a b : Associates M} (hlt : a < b) : DvdNotUnit a b := by
   constructor
   · rintro rfl
     apply not_lt_of_le _ hlt
@@ -1033,46 +1039,46 @@ theorem dvdNotUnit_of_lt {a b : Associates α} (hlt : a < b) : DvdNotUnit a b :=
   simp
 
 theorem irreducible_iff_prime_iff :
-    (∀ a : α, Irreducible a ↔ Prime a) ↔ ∀ a : Associates α, Irreducible a ↔ Prime a := by
+    (∀ a : M, Irreducible a ↔ Prime a) ↔ ∀ a : Associates M, Irreducible a ↔ Prime a := by
   simp_rw [forall_associated, irreducible_mk, prime_mk]
 
 end CommMonoidWithZero
 
 section CancelCommMonoidWithZero
 
-variable [CancelCommMonoidWithZero α]
+variable [CancelCommMonoidWithZero M]
 
-instance instPartialOrder : PartialOrder (Associates α) where
+instance instPartialOrder : PartialOrder (Associates M) where
   le_antisymm := mk_surjective.forall₂.2 fun _a _b hab hba => mk_eq_mk_iff_associated.2 <|
     associated_of_dvd_dvd (dvd_of_mk_le_mk hab) (dvd_of_mk_le_mk hba)
 
-instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates α) :=
-  { (by infer_instance : CommMonoidWithZero (Associates α)) with
+instance instCancelCommMonoidWithZero : CancelCommMonoidWithZero (Associates M) :=
+  { (by infer_instance : CommMonoidWithZero (Associates M)) with
     mul_left_cancel_of_ne_zero := by
       rintro ⟨a⟩ ⟨b⟩ ⟨c⟩ ha h
       rcases Quotient.exact' h with ⟨u, hu⟩
       have hu : a * (b * ↑u) = a * c := by rwa [← mul_assoc]
       exact Quotient.sound' ⟨u, mul_left_cancel₀ (mk_ne_zero.1 ha) hu⟩ }
 
-theorem _root_.associates_irreducible_iff_prime [DecompositionMonoid α] {p : Associates α} :
+theorem _root_.associates_irreducible_iff_prime [DecompositionMonoid M] {p : Associates M} :
     Irreducible p ↔ Prime p := irreducible_iff_prime
 
-instance : NoZeroDivisors (Associates α) := by infer_instance
+instance : NoZeroDivisors (Associates M) := by infer_instance
 
-theorem le_of_mul_le_mul_left (a b c : Associates α) (ha : a ≠ 0) : a * b ≤ a * c → b ≤ c
+theorem le_of_mul_le_mul_left (a b c : Associates M) (ha : a ≠ 0) : a * b ≤ a * c → b ≤ c
   | ⟨d, hd⟩ => ⟨d, mul_left_cancel₀ ha <| by rwa [← mul_assoc]⟩
 
-theorem one_or_eq_of_le_of_prime {p m : Associates α} (hp : Prime p) (hle : m ≤ p) :
+theorem one_or_eq_of_le_of_prime {p m : Associates M} (hp : Prime p) (hle : m ≤ p) :
     m = 1 ∨ m = p := by
   rcases mk_surjective p with ⟨p, rfl⟩
   rcases mk_surjective m with ⟨m, rfl⟩
   simpa [mk_eq_mk_iff_associated, Associated.comm, -Quotient.eq]
     using (prime_mk.1 hp).irreducible.dvd_iff.mp (mk_le_mk_iff_dvd.1 hle)
 
-theorem dvdNotUnit_iff_lt {a b : Associates α} : DvdNotUnit a b ↔ a < b :=
+theorem dvdNotUnit_iff_lt {a b : Associates M} : DvdNotUnit a b ↔ a < b :=
   dvd_and_not_dvd_iff.symm
 
-theorem le_one_iff {p : Associates α} : p ≤ 1 ↔ p = 1 := by rw [← Associates.bot_eq_one, le_bot_iff]
+theorem le_one_iff {p : Associates M} : p ≤ 1 ↔ p = 1 := by rw [← Associates.bot_eq_one, le_bot_iff]
 
 end CancelCommMonoidWithZero
 
@@ -1080,20 +1086,20 @@ end Associates
 
 section CommMonoidWithZero
 
-theorem DvdNotUnit.isUnit_of_irreducible_right [CommMonoidWithZero α] {p q : α}
+theorem DvdNotUnit.isUnit_of_irreducible_right [CommMonoidWithZero M] {p q : M}
     (h : DvdNotUnit p q) (hq : Irreducible q) : IsUnit p := by
   obtain ⟨_, x, hx, hx'⟩ := h
   exact Or.resolve_right ((irreducible_iff.1 hq).right p x hx') hx
 
-theorem not_irreducible_of_not_unit_dvdNotUnit [CommMonoidWithZero α] {p q : α} (hp : ¬IsUnit p)
+theorem not_irreducible_of_not_unit_dvdNotUnit [CommMonoidWithZero M] {p q : M} (hp : ¬IsUnit p)
     (h : DvdNotUnit p q) : ¬Irreducible q :=
   mt h.isUnit_of_irreducible_right hp
 
-theorem DvdNotUnit.not_unit [CommMonoidWithZero α] {p q : α} (hp : DvdNotUnit p q) : ¬IsUnit q := by
+theorem DvdNotUnit.not_unit [CommMonoidWithZero M] {p q : M} (hp : DvdNotUnit p q) : ¬IsUnit q := by
   obtain ⟨-, x, hx, rfl⟩ := hp
   exact fun hc => hx (isUnit_iff_dvd_one.mpr (dvd_of_mul_left_dvd (isUnit_iff_dvd_one.mp hc)))
 
-theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero α] [Nontrivial α] {p q r : α}
+theorem dvdNotUnit_of_dvdNotUnit_associated [CommMonoidWithZero M] [Nontrivial M] {p q r : M}
     (h : DvdNotUnit p q) (h' : Associated q r) : DvdNotUnit p r := by
   obtain ⟨u, rfl⟩ := Associated.symm h'
   obtain ⟨hp, x, hx⟩ := h
@@ -1104,33 +1110,40 @@ end CommMonoidWithZero
 
 section CancelCommMonoidWithZero
 
-theorem isUnit_of_associated_mul [CancelCommMonoidWithZero α] {p b : α} (h : Associated (p * b) p)
+theorem isUnit_of_associated_mul [CancelCommMonoidWithZero M] {p b : M} (h : Associated (p * b) p)
     (hp : p ≠ 0) : IsUnit b := by
   obtain ⟨a, ha⟩ := h
   refine isUnit_of_mul_eq_one b a ((mul_right_inj' hp).mp ?_)
   rwa [← mul_assoc, mul_one]
 
-theorem DvdNotUnit.not_associated [CancelCommMonoidWithZero α] {p q : α} (h : DvdNotUnit p q) :
+theorem DvdNotUnit.not_associated [CancelCommMonoidWithZero M] {p q : M} (h : DvdNotUnit p q) :
     ¬Associated p q := by
   rintro ⟨a, rfl⟩
   obtain ⟨hp, x, hx, hx'⟩ := h
   rcases (mul_right_inj' hp).mp hx' with rfl
   exact hx a.isUnit
 
-theorem DvdNotUnit.ne [CancelCommMonoidWithZero α] {p q : α} (h : DvdNotUnit p q) : p ≠ q := by
+theorem DvdNotUnit.ne [CancelCommMonoidWithZero M] {p q : M} (h : DvdNotUnit p q) : p ≠ q := by
   by_contra hcontra
   obtain ⟨hp, x, hx', hx''⟩ := h
   conv_lhs at hx'' => rw [← hcontra, ← mul_one p]
   rw [(mul_left_cancel₀ hp hx'').symm] at hx'
   exact hx' isUnit_one
 
-theorem pow_injective_of_not_unit [CancelCommMonoidWithZero α] {q : α} (hq : ¬IsUnit q)
+theorem pow_injective_of_not_isUnit [CancelCommMonoidWithZero M] {q : M} (hq : ¬IsUnit q)
     (hq' : q ≠ 0) : Function.Injective fun n : ℕ => q ^ n := by
   refine injective_of_lt_imp_ne fun n m h => DvdNotUnit.ne ⟨pow_ne_zero n hq', q ^ (m - n), ?_, ?_⟩
   · exact not_isUnit_of_not_isUnit_dvd hq (dvd_pow (dvd_refl _) (Nat.sub_pos_of_lt h).ne')
   · exact (pow_mul_pow_sub q h.le).symm
 
-theorem dvd_prime_pow [CancelCommMonoidWithZero α] {p q : α} (hp : Prime p) (n : ℕ) :
+@[deprecated (since := "2024-09-22")]
+alias pow_injective_of_not_unit := pow_injective_of_not_isUnit
+
+theorem pow_inj_of_not_isUnit [CancelCommMonoidWithZero M] {q : M} (hq : ¬IsUnit q)
+    (hq' : q ≠ 0) {m n : ℕ} : q ^ m = q ^ n ↔ m = n :=
+  (pow_injective_of_not_isUnit hq hq').eq_iff
+
+theorem dvd_prime_pow [CancelCommMonoidWithZero M] {p q : M} (hp : Prime p) (n : ℕ) :
     q ∣ p ^ n ↔ ∃ i ≤ n, Associated q (p ^ i) := by
   induction n generalizing q with
   | zero =>
diff --git a/Mathlib/Algebra/Associated/OrderedCommMonoid.lean b/Mathlib/Algebra/Associated/OrderedCommMonoid.lean
index 16c1b23b72bf2..ab1745e4359de 100644
--- a/Mathlib/Algebra/Associated/OrderedCommMonoid.lean
+++ b/Mathlib/Algebra/Associated/OrderedCommMonoid.lean
@@ -23,14 +23,14 @@ Then we show that the quotient type `Associates` is a monoid
 and prove basic properties of this quotient.
 -/
 
-variable {α : Type*} [CancelCommMonoidWithZero α]
+variable {M : Type*} [CancelCommMonoidWithZero M]
 
 namespace Associates
 
-instance instOrderedCommMonoid : OrderedCommMonoid (Associates α) where
-  mul_le_mul_left := fun a _ ⟨d, hd⟩ c => hd.symm ▸ mul_assoc c a d ▸ le_mul_right (α := α)
+instance instOrderedCommMonoid : OrderedCommMonoid (Associates M) where
+  mul_le_mul_left := fun a _ ⟨d, hd⟩ c => hd.symm ▸ mul_assoc c a d ▸ le_mul_right
 
-instance : CanonicallyOrderedCommMonoid (Associates α) where
+instance : CanonicallyOrderedCommMonoid (Associates M) where
   exists_mul_of_le h := h
   le_self_mul _ b := ⟨b, rfl⟩
   bot_le _ := one_le
diff --git a/Mathlib/Algebra/BigOperators/Associated.lean b/Mathlib/Algebra/BigOperators/Associated.lean
index b8b8178cc0aa7..ebe6b44cb5983 100644
--- a/Mathlib/Algebra/BigOperators/Associated.lean
+++ b/Mathlib/Algebra/BigOperators/Associated.lean
@@ -74,6 +74,46 @@ theorem exists_associated_mem_of_dvd_prod [CancelCommMonoidWithZero α] {p : α}
     · rcases ih (fun r hr => hs _ (Multiset.mem_cons.2 (Or.inr hr))) h with ⟨q, hq₁, hq₂⟩
       exact ⟨q, Multiset.mem_cons.2 (Or.inr hq₁), hq₂⟩
 
+open Submonoid in
+/-- Let x, y ∈ α. If x * y can be written as a product of units and prime elements, then x can be
+written as a product of units and prime elements. -/
+theorem divisor_closure_eq_closure [CancelCommMonoidWithZero α]
+    (x y : α) (hxy : x * y ∈ closure { r : α | IsUnit r ∨ Prime r}) :
+    x ∈ closure { r : α | IsUnit r ∨ Prime r} := by
+  obtain ⟨m, hm, hprod⟩ := exists_multiset_of_mem_closure hxy
+  induction m using Multiset.induction generalizing x y with
+  | empty =>
+    apply subset_closure
+    simp only [Set.mem_setOf]
+    simp only [Multiset.prod_zero] at hprod
+    left; exact isUnit_of_mul_eq_one _ _ hprod.symm
+  | @cons c s hind =>
+    simp only [Multiset.mem_cons, forall_eq_or_imp, Set.mem_setOf] at hm
+    simp only [Multiset.prod_cons] at hprod
+    simp only [Set.mem_setOf_eq] at hind
+    obtain ⟨ha₁ | ha₂, hs⟩ := hm
+    · rcases ha₁.exists_right_inv with ⟨k, hk⟩
+      refine hind x (y*k) ?_ hs ?_
+      · simp only [← mul_assoc, ← hprod, ← Multiset.prod_cons, mul_comm]
+        refine multiset_prod_mem _ _ (Multiset.forall_mem_cons.2 ⟨subset_closure (Set.mem_def.2 ?_),
+          Multiset.forall_mem_cons.2 ⟨subset_closure (Set.mem_def.2 ?_), (fun t ht =>
+          subset_closure (hs t ht))⟩⟩)
+        · left; exact isUnit_of_mul_eq_one_right _ _ hk
+        · left; exact ha₁
+      · rw [← mul_one s.prod, ← hk, ← mul_assoc, ← mul_assoc, mul_eq_mul_right_iff, mul_comm]
+        left; exact hprod
+    · rcases ha₂.dvd_mul.1 (Dvd.intro _ hprod) with ⟨c, hc⟩ | ⟨c, hc⟩
+      · rw [hc]; rw [hc, mul_assoc] at hprod
+        refine Submonoid.mul_mem _ (subset_closure (Set.mem_def.2 ?_))
+          (hind _ _ ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod))
+        · right; exact ha₂
+        rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
+        exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
+      rw [hc, mul_comm x _, mul_assoc, mul_comm c _] at hprod
+      refine hind x c ?_ hs (mul_left_cancel₀ ha₂.ne_zero hprod)
+      rw [← mul_left_cancel₀ ha₂.ne_zero hprod]
+      exact multiset_prod_mem _ _ (fun t ht => subset_closure (hs t ht))
+
 theorem Multiset.prod_primes_dvd [CancelCommMonoidWithZero α]
     [∀ a : α, DecidablePred (Associated a)] {s : Multiset α} (n : α) (h : ∀ a ∈ s, Prime a)
     (div : ∀ a ∈ s, a ∣ n) (uniq : ∀ a, s.countP (Associated a) ≤ 1) : s.prod ∣ n := by
@@ -95,7 +135,7 @@ theorem Multiset.prod_primes_dvd [CancelCommMonoidWithZero α]
       Multiset.countP_pos] at this
     exact this ⟨b, b_in_s, assoc.symm⟩
 
-theorem Finset.prod_primes_dvd [CancelCommMonoidWithZero α] [Unique αˣ] {s : Finset α} (n : α)
+theorem Finset.prod_primes_dvd [CancelCommMonoidWithZero α] [Subsingleton αˣ] {s : Finset α} (n : α)
     (h : ∀ a ∈ s, Prime a) (div : ∀ a ∈ s, a ∣ n) : (∏ p ∈ s, p) ∣ n := by
   classical
     exact
diff --git a/Mathlib/Algebra/BigOperators/Balance.lean b/Mathlib/Algebra/BigOperators/Balance.lean
new file mode 100644
index 0000000000000..1b2b2ca767a64
--- /dev/null
+++ b/Mathlib/Algebra/BigOperators/Balance.lean
@@ -0,0 +1,56 @@
+/-
+Copyright (c) 2023 Yaël Dillies, Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies, Bhavik Mehta
+-/
+import Mathlib.Algebra.BigOperators.Expect
+
+/-!
+# Balancing a function
+
+This file defines the balancing of a function `f`, defined as `f` minus its average.
+
+This is the unique function `g` such that `f a - f b = g a - g b` for all `a` and `b`, and
+`∑ a, g a = 0`. This is particularly useful in Fourier analysis as `f` and `g` then have the same
+Fourier transform, except in the `0`-th frequency where the Fourier transform of `g` vanishes.
+-/
+
+open Finset Function
+open scoped BigOperators
+
+variable {ι H F G : Type*}
+
+namespace Fintype
+
+section AddCommGroup
+variable [Fintype ι] [AddCommGroup G] [Module ℚ≥0 G] [AddCommGroup H] [Module ℚ≥0 H]
+
+/-- The balancing of a function, namely the function minus its average. -/
+def balance (f : ι → G) : ι → G := f - Function.const _ (𝔼 y, f y)
+
+lemma balance_apply (f : ι → G) (x : ι) : balance f x = f x - 𝔼 y, f y := rfl
+
+@[simp] lemma balance_zero : balance (0 : ι → G) = 0 := by simp [balance]
+
+@[simp] lemma balance_add (f g : ι → G) : balance (f + g) = balance f + balance g := by
+  simp only [balance, expect_add_distrib, ← const_add, add_sub_add_comm, Pi.add_apply]
+
+@[simp] lemma balance_sub (f g : ι → G) : balance (f - g) = balance f - balance g := by
+  simp only [balance, expect_sub_distrib, const_sub, sub_sub_sub_comm, Pi.sub_apply]
+
+@[simp] lemma balance_neg (f : ι → G) : balance (-f) = -balance f := by
+  simp only [balance, expect_neg_distrib, const_neg, neg_sub', Pi.neg_apply]
+
+@[simp] lemma sum_balance (f : ι → G) : ∑ x, balance f x = 0 := by
+  cases isEmpty_or_nonempty ι <;> simp [balance_apply]
+
+@[simp] lemma expect_balance (f : ι → G) : 𝔼 x, balance f x = 0 := by simp [expect]
+
+@[simp] lemma balance_idem (f : ι → G) : balance (balance f) = balance f := by
+  cases isEmpty_or_nonempty ι <;> ext x <;> simp [balance, expect_sub_distrib, univ_nonempty]
+
+@[simp] lemma map_balance [FunLike F G H] [LinearMapClass F ℚ≥0 G H] (g : F) (f : ι → G) (a : ι) :
+    g (balance f a) = balance (g ∘ f) a := by simp [balance, map_expect]
+
+end AddCommGroup
+end Fintype
diff --git a/Mathlib/Algebra/BigOperators/Expect.lean b/Mathlib/Algebra/BigOperators/Expect.lean
new file mode 100644
index 0000000000000..cf6a0e4d72664
--- /dev/null
+++ b/Mathlib/Algebra/BigOperators/Expect.lean
@@ -0,0 +1,444 @@
+/-
+Copyright (c) 2024 Yaël Dillies, Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies, Bhavik Mehta
+-/
+import Mathlib.Algebra.Algebra.Rat
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
+import Mathlib.Algebra.BigOperators.Pi
+import Mathlib.Algebra.BigOperators.Ring
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
+import Mathlib.Algebra.Module.Pi
+import Mathlib.Data.Finset.Density
+import Mathlib.Data.Fintype.BigOperators
+
+/-!
+# Average over a finset
+
+This file defines `Finset.expect`, the average (aka expectation) of a function over a finset.
+
+## Notation
+
+* `𝔼 i ∈ s, f i` is notation for `Finset.expect s f`. It is the expectation of `f i` where `i`
+  ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance).
+* `𝔼 i, f i` is notation for `Finset.expect Finset.univ f`. It is the expectation of `f i` where `i`
+  ranges over the finite domain of `f`.
+* `𝔼 i ∈ s with p i, f i` is notation for `Finset.expect (Finset.filter p s) f`. This is referred to
+  as `expectWith` in lemma names.
+* `𝔼 (i ∈ s) (j ∈ t), f i j` is notation for `Finset.expect (s ×ˢ t) (fun ⟨i, j⟩ ↦ f i j)`.
+
+## Implementation notes
+
+This definition is a special case of the general convex comnination operator in a convex space.
+However:
+1. We don't yet have general convex spaces.
+2. The uniform weights case is a overwhelmingly useful special case which should have its own API.
+
+When convex spaces are finally defined, we should redefine `Finset.expect` in terms of that convex
+combination operator.
+
+## TODO
+
+* Connect `Finset.expect` with the expectation over `s` in the probability theory sense.
+* Give a formulation of Jensen's inequality in this language.
+-/
+
+open Finset Function
+open Fintype (card)
+open scoped Pointwise
+
+variable {ι κ M N : Type*}
+
+local notation a " /ℚ " q => (q : ℚ≥0)⁻¹ • a
+
+/-- Average of a function over a finset. If the finset is empty, this is equal to zero. -/
+def Finset.expect [AddCommMonoid M] [Module ℚ≥0 M] (s : Finset ι) (f : ι → M) : M :=
+  (s.card : ℚ≥0)⁻¹ • ∑ i ∈ s, f i
+
+namespace BigOperators
+open Batteries.ExtendedBinder Lean Meta
+
+/--
+* `𝔼 i ∈ s, f i` is notation for `Finset.expect s f`. It is the expectation of `f i` where `i`
+  ranges over the finite set `s` (either a `Finset` or a `Set` with a `Fintype` instance).
+* `𝔼 i, f i` is notation for `Finset.expect Finset.univ f`. It is the expectation of `f i` where `i`
+  ranges over the finite domain of `f`.
+* `𝔼 i ∈ s with p i, f i` is notation for `Finset.expect (Finset.filter p s) f`.
+* `𝔼 (i ∈ s) (j ∈ t), f i j` is notation for `Finset.expect (s ×ˢ t) (fun ⟨i, j⟩ ↦ f i j)`.
+
+These support destructuring, for example `𝔼 ⟨i, j⟩ ∈ s ×ˢ t, f i j`.
+
+Notation: `"𝔼" bigOpBinders* ("with" term)? "," term` -/
+scoped syntax (name := bigexpect) "𝔼 " bigOpBinders ("with " term)? ", " term:67 : term
+
+scoped macro_rules (kind := bigexpect)
+  | `(𝔼 $bs:bigOpBinders $[with $p?]?, $v) => do
+    let processed ← processBigOpBinders bs
+    let i ← bigOpBindersPattern processed
+    let s ← bigOpBindersProd processed
+    match p? with
+    | some p => `(Finset.expect (Finset.filter (fun $i ↦ $p) $s) (fun $i ↦ $v))
+    | none => `(Finset.expect $s (fun $i ↦ $v))
+
+open Lean Meta Parser.Term PrettyPrinter.Delaborator SubExpr
+open Batteries.ExtendedBinder
+
+/-- Delaborator for `Finset.expect`. The `pp.piBinderTypes` option controls whether
+to show the domain type when the expect is over `Finset.univ`. -/
+@[scoped delab app.Finset.expect] def delabFinsetExpect : Delab :=
+  whenPPOption getPPNotation <| withOverApp 6 <| do
+  let #[_, _, _, _, s, f] := (← getExpr).getAppArgs | failure
+  guard <| f.isLambda
+  let ppDomain ← getPPOption getPPPiBinderTypes
+  let (i, body) ← withAppArg <| withBindingBodyUnusedName fun i => do
+    return (i, ← delab)
+  if s.isAppOfArity ``Finset.univ 2 then
+    let binder ←
+      if ppDomain then
+        let ty ← withNaryArg 0 delab
+        `(bigOpBinder| $(.mk i):ident : $ty)
+      else
+        `(bigOpBinder| $(.mk i):ident)
+    `(𝔼 $binder:bigOpBinder, $body)
+  else
+    let ss ← withNaryArg 4 <| delab
+    `(𝔼 $(.mk i):ident ∈ $ss, $body)
+
+end BigOperators
+
+open scoped BigOperators
+
+namespace Finset
+section AddCommMonoid
+variable [AddCommMonoid M] [Module ℚ≥0 M] [AddCommMonoid N] [Module ℚ≥0 N] {s t : Finset ι}
+  {f g : ι → M} {m : N → M} {p q : ι → Prop} [DecidablePred p] [DecidablePred q]
+
+lemma expect_univ [Fintype ι] : 𝔼 i, f i = (∑ i, f i) /ℚ Fintype.card ι := by
+  rw [expect, card_univ]
+
+@[simp] lemma expect_empty (f : ι → M) : 𝔼 i ∈ ∅, f i = 0 := by simp [expect]
+@[simp] lemma expect_singleton (f : ι → M) (i : ι) : 𝔼 j ∈ {i}, f j = f i := by simp [expect]
+@[simp] lemma expect_const_zero (s : Finset ι) : 𝔼 _i ∈ s, (0 : M) = 0 := by simp [expect]
+
+@[congr]
+lemma expect_congr {t : Finset ι} (hst : s = t) (h : ∀ i ∈ t, f i = g i) :
+    𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by rw [expect, expect, sum_congr hst h, hst]
+
+lemma expectWith_congr (hst : s = t) (hpq : ∀ i ∈ t, p i ↔ q i) (h : ∀ i ∈ t, q i → f i = g i) :
+    𝔼 i ∈ s with p i, f i = 𝔼 i ∈ t with q i, g i :=
+  expect_congr (by rw [hst, filter_inj'.2 hpq]) <| by simpa using h
+
+lemma expect_sum_comm (s : Finset ι) (t : Finset κ) (f : ι → κ → M) :
+    𝔼 i ∈ s, ∑ j ∈ t, f i j = ∑ j ∈ t, 𝔼 i ∈ s, f i j := by
+  simpa only [expect, smul_sum] using sum_comm
+
+lemma expect_comm (s : Finset ι) (t : Finset κ) (f : ι → κ → M) :
+    𝔼 i ∈ s, 𝔼 j ∈ t, f i j = 𝔼 j ∈ t, 𝔼 i ∈ s, f i j := by
+  rw [expect, expect, ← expect_sum_comm, ← expect_sum_comm, expect, expect, smul_comm, sum_comm]
+
+lemma expect_eq_zero (h : ∀ i ∈ s, f i = 0) : 𝔼 i ∈ s, f i = 0 :=
+  (expect_congr rfl h).trans s.expect_const_zero
+
+lemma exists_ne_zero_of_expect_ne_zero (h : 𝔼 i ∈ s, f i ≠ 0) : ∃ i ∈ s, f i ≠ 0 := by
+  contrapose! h; exact expect_eq_zero h
+
+lemma expect_add_distrib (s : Finset ι) (f g : ι → M) :
+    𝔼 i ∈ s, (f i + g i) = 𝔼 i ∈ s, f i + 𝔼 i ∈ s, g i := by
+  simp [expect, sum_add_distrib]
+
+lemma expect_add_expect_comm (f₁ f₂ g₁ g₂ : ι → M) :
+    𝔼 i ∈ s, (f₁ i + f₂ i) + 𝔼 i ∈ s, (g₁ i + g₂ i) =
+      𝔼 i ∈ s, (f₁ i + g₁ i) + 𝔼 i ∈ s, (f₂ i + g₂ i) := by
+  simp_rw [expect_add_distrib, add_add_add_comm]
+
+lemma expect_eq_single_of_mem (i : ι) (hi : i ∈ s) (h : ∀ j ∈ s, j ≠ i → f j = 0) :
+    𝔼 i ∈ s, f i = f i /ℚ s.card := by rw [expect, sum_eq_single_of_mem _ hi h]
+
+lemma expect_ite_zero (s : Finset ι) (p : ι → Prop) [DecidablePred p]
+    (h : ∀ i ∈ s, ∀ j ∈ s, p i → p j → i = j) (a : M) :
+    𝔼 i ∈ s, ite (p i) a 0 = ite (∃ i ∈ s, p i) (a /ℚ s.card) 0 := by
+  split_ifs <;> simp [expect, sum_ite_zero _ _ h, *]
+
+section DecidableEq
+variable [DecidableEq ι]
+
+lemma expect_ite_mem (s t : Finset ι) (f : ι → M) :
+    𝔼 i ∈ s, (if i ∈ t then f i else 0) = ((s ∩ t).card / s.card : ℚ≥0) • 𝔼 i ∈ s ∩ t, f i := by
+  obtain hst | hst := (s ∩ t).eq_empty_or_nonempty
+  · simp [expect, hst]
+  · simp [expect, smul_smul, ← inv_mul_eq_div, hst.card_ne_zero]
+
+@[simp] lemma expect_dite_eq (i : ι) (f : ∀ j, i = j → M) :
+    𝔼 j ∈ s, (if h : i = j then f j h else 0) = if i ∈ s then f i rfl /ℚ s.card else 0 := by
+  split_ifs <;> simp [expect, *]
+
+@[simp] lemma expect_dite_eq' (i : ι) (f : ∀ j, j = i → M) :
+    𝔼 j ∈ s, (if h : j = i then f j h else 0) = if i ∈ s then f i rfl /ℚ s.card else 0 := by
+  split_ifs <;> simp [expect, *]
+
+@[simp] lemma expect_ite_eq (i : ι) (f : ι → M) :
+    𝔼 j ∈ s, (if i = j then f j else 0) = if i ∈ s then f i /ℚ s.card else 0 := by
+  split_ifs <;> simp [expect, *]
+
+@[simp] lemma expect_ite_eq' (i : ι) (f : ι → M) :
+    𝔼 j ∈ s, (if j = i then f j else 0) = if i ∈ s then f i /ℚ s.card else 0 := by
+  split_ifs <;> simp [expect, *]
+
+end DecidableEq
+
+section bij
+variable {t : Finset κ} {g : κ → M}
+
+/-- Reorder an average.
+
+The difference with `Finset.expect_bij'` is that the bijection is specified as a surjective
+injection, rather than by an inverse function.
+
+The difference with `Finset.expect_nbij` is that the bijection is allowed to use membership of the
+domain of the average, rather than being a non-dependent function. -/
+lemma expect_bij (i : ∀ a ∈ s, κ) (hi : ∀ a ha, i a ha ∈ t) (h : ∀ a ha, f a = g (i a ha))
+    (i_inj : ∀ a₁ ha₁ a₂ ha₂, i a₁ ha₁ = i a₂ ha₂ → a₁ = a₂)
+    (i_surj : ∀ b ∈ t, ∃ a ha, i a ha = b) : 𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by
+  simp_rw [expect, card_bij i hi i_inj i_surj, sum_bij i hi i_inj i_surj h]
+
+/-- Reorder an average.
+
+The difference with `Finset.expect_bij` is that the bijection is specified with an inverse, rather
+than as a surjective injection.
+
+The difference with `Finset.expect_nbij'` is that the bijection and its inverse are allowed to use
+membership of the domains of the averages, rather than being non-dependent functions. -/
+lemma expect_bij' (i : ∀ a ∈ s, κ) (j : ∀ a ∈ t, ι) (hi : ∀ a ha, i a ha ∈ t)
+    (hj : ∀ a ha, j a ha ∈ s) (left_inv : ∀ a ha, j (i a ha) (hi a ha) = a)
+    (right_inv : ∀ a ha, i (j a ha) (hj a ha) = a) (h : ∀ a ha, f a = g (i a ha)) :
+    𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by
+  simp_rw [expect, card_bij' i j hi hj left_inv right_inv, sum_bij' i j hi hj left_inv right_inv h]
+
+/-- Reorder an average.
+
+The difference with `Finset.expect_nbij'` is that the bijection is specified as a surjective
+injection, rather than by an inverse function.
+
+The difference with `Finset.expect_bij` is that the bijection is a non-dependent function, rather
+than being allowed to use membership of the domain of the average. -/
+lemma expect_nbij (i : ι → κ) (hi : ∀ a ∈ s, i a ∈ t) (h : ∀ a ∈ s, f a = g (i a))
+    (i_inj : (s : Set ι).InjOn i) (i_surj : (s : Set ι).SurjOn i t) :
+    𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by
+  simp_rw [expect, card_nbij i hi i_inj i_surj, sum_nbij i hi i_inj i_surj h]
+
+/-- Reorder an average.
+
+The difference with `Finset.expect_nbij` is that the bijection is specified with an inverse, rather
+than as a surjective injection.
+
+The difference with `Finset.expect_bij'` is that the bijection and its inverse are non-dependent
+functions, rather than being allowed to use membership of the domains of the averages.
+
+The difference with `Finset.expect_equiv` is that bijectivity is only required to hold on the
+domains of the averages, rather than on the entire types. -/
+lemma expect_nbij' (i : ι → κ) (j : κ → ι) (hi : ∀ a ∈ s, i a ∈ t) (hj : ∀ a ∈ t, j a ∈ s)
+    (left_inv : ∀ a ∈ s, j (i a) = a) (right_inv : ∀ a ∈ t, i (j a) = a)
+    (h : ∀ a ∈ s, f a = g (i a)) : 𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by
+  simp_rw [expect, card_nbij' i j hi hj left_inv right_inv,
+    sum_nbij' i j hi hj left_inv right_inv h]
+
+/-- `Finset.expect_equiv` is a specialization of `Finset.expect_bij` that automatically fills in
+most arguments. -/
+lemma expect_equiv (e : ι ≃ κ) (hst : ∀ i, i ∈ s ↔ e i ∈ t) (hfg : ∀ i ∈ s, f i = g (e i)) :
+    𝔼 i ∈ s, f i = 𝔼 i ∈ t, g i := by simp_rw [expect, card_equiv e hst, sum_equiv e hst hfg]
+
+/-- Expectation over a product set equals the expectation of the fiberwise expectations.
+
+For rewriting in the reverse direction, use `Finset.expect_product'`. -/
+lemma expect_product (s : Finset ι) (t : Finset κ) (f : ι × κ → M) :
+    𝔼 x ∈ s ×ˢ t, f x = 𝔼 i ∈ s, 𝔼 j ∈ t, f (i, j) := by
+  simp only [expect, card_product, sum_product, smul_sum, mul_inv, mul_smul, Nat.cast_mul]
+
+/-- Expectation over a product set equals the expectation of the fiberwise expectations.
+
+For rewriting in the reverse direction, use `Finset.expect_product`. -/
+lemma expect_product' (s : Finset ι) (t : Finset κ) (f : ι → κ → M) :
+    𝔼 i ∈ s ×ˢ t, f i.1 i.2 = 𝔼 i ∈ s, 𝔼 j ∈ t, f i j := by
+  simp only [expect, card_product, sum_product', smul_sum, mul_inv, mul_smul, Nat.cast_mul]
+
+@[simp]
+lemma expect_image [DecidableEq ι] {m : κ → ι} (hm : (t : Set κ).InjOn m) :
+    𝔼 i ∈ t.image m, f i = 𝔼 i ∈ t, f (m i) := by
+  simp_rw [expect, card_image_of_injOn hm, sum_image hm]
+
+end bij
+
+@[simp] lemma expect_inv_index [DecidableEq ι] [InvolutiveInv ι] (s : Finset ι) (f : ι → M) :
+    𝔼 i ∈ s⁻¹, f i = 𝔼 i ∈ s, f i⁻¹ := expect_image inv_injective.injOn
+
+@[simp] lemma expect_neg_index [DecidableEq ι] [InvolutiveNeg ι] (s : Finset ι) (f : ι → M) :
+    𝔼 i ∈ -s, f i = 𝔼 i ∈ s, f (-i) := expect_image neg_injective.injOn
+
+lemma _root_.map_expect {F : Type*} [FunLike F M N] [LinearMapClass F ℚ≥0 M N]
+    (g : F) (f : ι → M) (s : Finset ι) :
+    g (𝔼 i ∈ s, f i) = 𝔼 i ∈ s, g (f i) := by simp only [expect, map_smul, map_natCast, map_sum]
+
+@[simp]
+lemma card_smul_expect (s : Finset ι) (f : ι → M) : s.card • 𝔼 i ∈ s, f i = ∑ i ∈ s, f i := by
+  obtain rfl | hs := s.eq_empty_or_nonempty
+  · simp
+  · rw [expect, ← Nat.cast_smul_eq_nsmul ℚ≥0, smul_inv_smul₀]
+    exact mod_cast hs.card_ne_zero
+
+@[simp] lemma _root_.Fintype.card_smul_expect [Fintype ι] (f : ι → M) :
+    Fintype.card ι • 𝔼 i, f i = ∑ i, f i := Finset.card_smul_expect _ _
+
+@[simp] lemma expect_const (hs : s.Nonempty) (a : M) : 𝔼 _i ∈ s, a = a := by
+  rw [expect, sum_const, ← Nat.cast_smul_eq_nsmul ℚ≥0, inv_smul_smul₀]
+  exact mod_cast hs.card_ne_zero
+
+lemma smul_expect {G : Type*} [DistribSMul G M] [SMulCommClass G ℚ≥0 M] (a : G)
+    (s : Finset ι) (f : ι → M) : a • 𝔼 i ∈ s, f i = 𝔼 i ∈ s, a • f i := by
+  simp only [expect, smul_sum, smul_comm]
+
+end AddCommMonoid
+
+section AddCommGroup
+variable [AddCommGroup M] [Module ℚ≥0 M] [Field N] [Module ℚ≥0 N] {s : Finset ι}
+
+lemma expect_sub_distrib (s : Finset ι) (f g : ι → M) :
+    𝔼 i ∈ s, (f i - g i) = 𝔼 i ∈ s, f i - 𝔼 i ∈ s, g i := by
+  simp only [expect, sum_sub_distrib, smul_sub]
+
+@[simp]
+lemma expect_neg_distrib (s : Finset ι) (f : ι → M) : 𝔼 i ∈ s, -f i = -𝔼 i ∈ s, f i := by
+  simp [expect]
+
+end AddCommGroup
+
+section Semiring
+variable [Semiring M] [Module ℚ≥0 M] {s : Finset ι} {f g : ι → M} {m : N → M}
+
+@[simp] lemma card_mul_expect (s : Finset ι) (f : ι → M) :
+    s.card * 𝔼 i ∈ s, f i = ∑ i ∈ s, f i := by rw [← nsmul_eq_mul, card_smul_expect]
+
+@[simp] lemma _root_.Fintype.card_mul_expect [Fintype ι] (f : ι → M) :
+    Fintype.card ι * 𝔼 i, f i = ∑ i, f i := Finset.card_mul_expect _ _
+
+lemma expect_mul [IsScalarTower ℚ≥0 M M] (s : Finset ι) (f : ι → M) (a : M) :
+    (𝔼 i ∈ s, f i) * a = 𝔼 i ∈ s, f i * a := by rw [expect, expect, smul_mul_assoc, sum_mul]
+
+lemma mul_expect [SMulCommClass ℚ≥0 M M] (s : Finset ι) (f : ι → M) (a : M) :
+    a * 𝔼 i ∈ s, f i = 𝔼 i ∈ s, a * f i := by rw [expect, expect, mul_smul_comm, mul_sum]
+
+lemma expect_mul_expect [IsScalarTower ℚ≥0 M M] [SMulCommClass ℚ≥0 M M] (s : Finset ι)
+    (t : Finset κ) (f : ι → M) (g : κ → M) :
+    (𝔼 i ∈ s, f i) * 𝔼 j ∈ t, g j = 𝔼 i ∈ s, 𝔼 j ∈ t, f i * g j := by
+  simp_rw [expect_mul, mul_expect]
+
+end Semiring
+
+section CommSemiring
+variable [CommSemiring M] [Module ℚ≥0 M] [IsScalarTower ℚ≥0 M M] [SMulCommClass ℚ≥0 M M]
+
+lemma expect_pow (s : Finset ι) (f : ι → M) (n : ℕ) :
+    (𝔼 i ∈ s, f i) ^ n = 𝔼 p ∈ Fintype.piFinset fun _ : Fin n ↦ s, ∏ i, f (p i) := by
+  classical
+  rw [expect, smul_pow, sum_pow', expect, Fintype.card_piFinset_const, inv_pow, Nat.cast_pow]
+
+end CommSemiring
+
+section Semifield
+variable [Semifield M] [CharZero M] {s : Finset ι} {f g : ι → M} {m : N → M}
+
+lemma expect_boole_mul [Fintype ι] [Nonempty ι] [DecidableEq ι] (f : ι → M) (i : ι) :
+    𝔼 j, ite (i = j) (Fintype.card ι : M) 0 * f j = f i := by
+  simp_rw [expect_univ, ite_mul, zero_mul, sum_ite_eq, if_pos (mem_univ _)]
+  rw [← @NNRat.cast_natCast M, ← NNRat.smul_def, inv_smul_smul₀]
+  simp [Fintype.card_ne_zero]
+
+lemma expect_boole_mul' [Fintype ι] [Nonempty ι] [DecidableEq ι] (f : ι → M) (i : ι) :
+    𝔼 j, ite (j = i) (Fintype.card ι : M) 0 * f j = f i := by
+  simp_rw [@eq_comm _ _ i, expect_boole_mul]
+
+lemma expect_eq_sum_div_card (s : Finset ι) (f : ι → M) :
+    𝔼 i ∈ s, f i = (∑ i ∈ s, f i) / s.card := by
+  rw [expect, NNRat.smul_def, div_eq_inv_mul, NNRat.cast_inv, NNRat.cast_natCast]
+
+lemma _root_.Fintype.expect_eq_sum_div_card [Fintype ι] (f : ι → M) :
+    𝔼 i, f i = (∑ i, f i) / Fintype.card ι := Finset.expect_eq_sum_div_card _ _
+
+lemma expect_div (s : Finset ι) (f : ι → M) (a : M) : (𝔼 i ∈ s, f i) / a = 𝔼 i ∈ s, f i / a := by
+  simp_rw [div_eq_mul_inv, expect_mul]
+
+end Semifield
+
+@[simp] lemma expect_apply {α : Type*} {π : α → Type*} [∀ a, CommSemiring (π a)]
+    [∀ a, Module ℚ≥0 (π a)] (s : Finset ι) (f : ι → ∀ a, π a) (a : α) :
+    (𝔼 i ∈ s, f i) a = 𝔼 i ∈ s, f i a := by simp [expect]
+
+end Finset
+
+namespace algebraMap
+variable [Semifield M] [CharZero M] [Semifield N] [CharZero N] [Algebra M N]
+
+@[simp, norm_cast]
+lemma coe_expect (s : Finset ι) (f : ι → M) : 𝔼 i ∈ s, f i = 𝔼 i ∈ s, (f i : N) :=
+  map_expect (algebraMap _ _) _ _
+
+end algebraMap
+
+namespace Fintype
+variable [Fintype ι] [Fintype κ]
+
+section AddCommMonoid
+variable [AddCommMonoid M] [Module ℚ≥0 M] {f : ι → M}
+
+/-- `Fintype.expect_bijective` is a variant of `Finset.expect_bij` that accepts
+`Function.Bijective`.
+
+See `Function.Bijective.expect_comp` for a version without `h`. -/
+lemma expect_bijective (e : ι → κ) (he : Bijective e) (f : ι → M) (g : κ → M)
+    (h : ∀ i, f i = g (e i)) : 𝔼 i, f i = 𝔼 i, g i :=
+  expect_nbij e (fun _ _ ↦ mem_univ _) (fun i _ ↦ h i) he.injective.injOn <| by
+    simpa using he.surjective.surjOn _
+
+/-- `Fintype.expect_equiv` is a specialization of `Finset.expect_bij` that automatically fills in
+most arguments.
+
+See `Equiv.expect_comp` for a version without `h`. -/
+lemma expect_equiv (e : ι ≃ κ) (f : ι → M) (g : κ → M) (h : ∀ i, f i = g (e i)) :
+    𝔼 i, f i = 𝔼 i, g i := expect_bijective _ e.bijective f g h
+
+lemma expect_const [Nonempty ι] (a : M) : 𝔼 _i : ι, a = a := Finset.expect_const univ_nonempty _
+
+lemma expect_ite_zero (p : ι → Prop) [DecidablePred p] (h : ∀ i j, p i → p j → i = j) (a : M) :
+    𝔼 i, ite (p i) a 0 = ite (∃ i, p i) (a /ℚ Fintype.card ι) 0 := by
+  simp [univ.expect_ite_zero p (by simpa using h), card_univ]
+
+variable [DecidableEq ι]
+
+@[simp] lemma expect_ite_mem (s : Finset ι) (f : ι → M) :
+    𝔼 i, (if i ∈ s then f i else 0) = s.dens • 𝔼 i ∈ s, f i := by
+  simp [Finset.expect_ite_mem, dens]
+
+lemma expect_dite_eq (i : ι) (f : ∀ j, i = j → M) :
+    𝔼 j, (if h : i = j then f j h else 0) = f i rfl /ℚ card ι := by simp [card_univ]
+
+lemma expect_dite_eq' (i : ι) (f : ∀ j, j = i → M) :
+    𝔼 j, (if h : j = i then f j h else 0) = f i rfl /ℚ card ι := by simp [card_univ]
+
+lemma expect_ite_eq (i : ι) (f : ι → M) :
+    𝔼 j, (if i = j then f j else 0) = f i /ℚ card ι := by simp [card_univ]
+
+lemma expect_ite_eq' (i : ι) (f : ι → M) :
+    𝔼 j, (if j = i then f j else 0) = f i /ℚ card ι := by simp [card_univ]
+
+end AddCommMonoid
+
+section Semiring
+variable [Semiring M] [Module ℚ≥0 M]
+
+lemma expect_one [Nonempty ι] : 𝔼 _i : ι, (1 : M) = 1 := expect_const _
+
+lemma expect_mul_expect [IsScalarTower ℚ≥0 M M] [SMulCommClass ℚ≥0 M M] (f : ι → M)
+    (g : κ → M) : (𝔼 i, f i) * 𝔼 j, g j = 𝔼 i, 𝔼 j, f i * g j :=
+  Finset.expect_mul_expect ..
+
+end Semiring
+end Fintype
diff --git a/Mathlib/Algebra/BigOperators/Fin.lean b/Mathlib/Algebra/BigOperators/Fin.lean
index 42ddb80a5d4b1..f336a553c7f57 100644
--- a/Mathlib/Algebra/BigOperators/Fin.lean
+++ b/Mathlib/Algebra/BigOperators/Fin.lean
@@ -86,11 +86,16 @@ theorem prod_univ_get' [CommMonoid β] (l : List α) (f : α → β) :
     ∏ i : Fin l.length, f l[i.1] = (l.map f).prod := by
   simp [Finset.prod_eq_multiset_prod]
 
-@[to_additive]
+@[to_additive (attr := simp)]
 theorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :
     (∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by
   simp_rw [prod_univ_succ, cons_zero, cons_succ]
 
+@[to_additive (attr := simp)]
+theorem prod_snoc [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :
+    (∏ i : Fin n.succ, (snoc f x : Fin n.succ → β) i) = (∏ i : Fin n, f i) * x := by
+  simp [prod_univ_castSucc]
+
 @[to_additive sum_univ_one]
 theorem prod_univ_one [CommMonoid β] (f : Fin 1 → β) : ∏ i, f i = f 0 := by simp
 
diff --git a/Mathlib/Algebra/BigOperators/Finprod.lean b/Mathlib/Algebra/BigOperators/Finprod.lean
index 9a2dc40b1b77f..fbfabaaebfca3 100644
--- a/Mathlib/Algebra/BigOperators/Finprod.lean
+++ b/Mathlib/Algebra/BigOperators/Finprod.lean
@@ -750,7 +750,7 @@ theorem finprod_mem_insert_one (h : f a = 1) : ∏ᶠ i ∈ insert a s, f i = 
   finprod_mem_insert_of_eq_one_if_not_mem fun _ => h
 
 /-- If the multiplicative support of `f` is finite, then for every `x` in the domain of `f`, `f x`
-divides `finprod f`.  -/
+divides `finprod f`. -/
 theorem finprod_mem_dvd {f : α → N} (a : α) (hf : (mulSupport f).Finite) : f a ∣ finprod f := by
   by_cases ha : a ∈ mulSupport f
   · rw [finprod_eq_prod_of_mulSupport_toFinset_subset f hf (Set.Subset.refl _)]
@@ -1011,7 +1011,7 @@ theorem Finset.mulSupport_of_fiberwise_prod_subset_image [DecidableEq β] (s : F
   simp only [Finset.coe_image, Set.mem_image, Finset.mem_coe, Function.support_subset_iff]
   intro b h
   suffices (s.filter fun a : α => g a = b).Nonempty by
-    simpa only [s.fiber_nonempty_iff_mem_image g b, Finset.mem_image, exists_prop]
+    simpa only [fiber_nonempty_iff_mem_image, Finset.mem_image, exists_prop]
   exact Finset.nonempty_of_prod_ne_one h
 
 /-- Note that `b ∈ (s.filter (fun ab => Prod.fst ab = a)).image Prod.snd` iff `(a, b) ∈ s` so
diff --git a/Mathlib/Algebra/BigOperators/Finsupp.lean b/Mathlib/Algebra/BigOperators/Finsupp.lean
index bfada8aee913c..10fdcfb6979ea 100644
--- a/Mathlib/Algebra/BigOperators/Finsupp.lean
+++ b/Mathlib/Algebra/BigOperators/Finsupp.lean
@@ -249,12 +249,12 @@ theorem sum_apply [Zero M] [AddCommMonoid N] {f : α →₀ M} {g : α → M →
   finset_sum_apply _ _ _
 
 -- Porting note: inserted ⇑ on the rhs
-theorem coe_finset_sum [AddCommMonoid N] (S : Finset ι) (f : ι → α →₀ N) :
+@[simp, norm_cast] theorem coe_finset_sum [AddCommMonoid N] (S : Finset ι) (f : ι → α →₀ N) :
     ⇑(∑ i ∈ S, f i) = ∑ i ∈ S, ⇑(f i) :=
   map_sum (coeFnAddHom : (α →₀ N) →+ _) _ _
 
 -- Porting note: inserted ⇑ on the rhs
-theorem coe_sum [Zero M] [AddCommMonoid N] (f : α →₀ M) (g : α → M → β →₀ N) :
+@[simp, norm_cast] theorem coe_sum [Zero M] [AddCommMonoid N] (f : α →₀ M) (g : α → M → β →₀ N) :
     ⇑(f.sum g) = f.sum fun a₁ b => ⇑(g a₁ b) :=
   coe_finset_sum _ _
 
@@ -346,9 +346,8 @@ def liftAddHom [AddZeroClass M] [AddCommMonoid N] : (α → M →+ N) ≃+ ((α
     ext
     simp [singleAddHom]
   right_inv F := by
-  -- Porting note: This was `ext` and used the wrong lemma
-    apply Finsupp.addHom_ext'
-    simp [singleAddHom, AddMonoidHom.comp, Function.comp]
+    ext
+    simp [singleAddHom, AddMonoidHom.comp, Function.comp_def]
   map_add' F G := by
     ext x
     exact sum_add
diff --git a/Mathlib/Algebra/BigOperators/Group/Finset.lean b/Mathlib/Algebra/BigOperators/Group/Finset.lean
index e528ac15b6c6b..eb06a95f3115d 100644
--- a/Mathlib/Algebra/BigOperators/Group/Finset.lean
+++ b/Mathlib/Algebra/BigOperators/Group/Finset.lean
@@ -539,15 +539,18 @@ theorem prod_biUnion [DecidableEq α] {s : Finset γ} {t : γ → Finset α}
     (hs : Set.PairwiseDisjoint (↑s) t) : ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i := by
   rw [← disjiUnion_eq_biUnion _ _ hs, prod_disjiUnion]
 
-/-- Product over a sigma type equals the product of fiberwise products. For rewriting
-in the reverse direction, use `Finset.prod_sigma'`.  -/
-@[to_additive "Sum over a sigma type equals the sum of fiberwise sums. For rewriting
+/-- The product over a sigma type equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Finset.prod_sigma'`. -/
+@[to_additive "The sum over a sigma type equals the sum of the fiberwise sums. For rewriting
 in the reverse direction, use `Finset.sum_sigma'`"]
 theorem prod_sigma {σ : α → Type*} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : Sigma σ → β) :
     ∏ x ∈ s.sigma t, f x = ∏ a ∈ s, ∏ s ∈ t a, f ⟨a, s⟩ := by
   simp_rw [← disjiUnion_map_sigma_mk, prod_disjiUnion, prod_map, Function.Embedding.sigmaMk_apply]
 
-@[to_additive]
+/-- The product over a sigma type equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Finset.prod_sigma`. -/
+@[to_additive "The sum over a sigma type equals the sum of the fiberwise sums. For rewriting
+in the reverse direction, use `Finset.sum_sigma`"]
 theorem prod_sigma' {σ : α → Type*} (s : Finset α) (t : ∀ a, Finset (σ a)) (f : ∀ a, σ a → β) :
     (∏ a ∈ s, ∏ s ∈ t a, f a s) = ∏ x ∈ s.sigma t, f x.1 x.2 :=
   Eq.symm <| prod_sigma s t fun x => f x.1 x.2
@@ -775,27 +778,32 @@ lemma prod_mul_prod_comm (f g h i : α → β) :
     (∏ a ∈ s, f a * g a) * ∏ a ∈ s, h a * i a = (∏ a ∈ s, f a * h a) * ∏ a ∈ s, g a * i a := by
   simp_rw [prod_mul_distrib, mul_mul_mul_comm]
 
-@[to_additive]
-theorem prod_product {s : Finset γ} {t : Finset α} {f : γ × α → β} :
+/-- The product over a product set equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Finset.prod_product'`. -/
+@[to_additive "The sum over a product set equals the sum of the fiberwise sums. For rewriting
+in the reverse direction, use `Finset.sum_product'`"]
+theorem prod_product (s : Finset γ) (t : Finset α) (f : γ × α → β) :
     ∏ x ∈ s ×ˢ t, f x = ∏ x ∈ s, ∏ y ∈ t, f (x, y) :=
   prod_finset_product (s ×ˢ t) s (fun _a => t) fun _p => mem_product
 
-/-- An uncurried version of `Finset.prod_product`. -/
-@[to_additive "An uncurried version of `Finset.sum_product`"]
-theorem prod_product' {s : Finset γ} {t : Finset α} {f : γ → α → β} :
+/-- The product over a product set equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Finset.prod_product`. -/
+@[to_additive "The sum over a product set equals the sum of the fiberwise sums. For rewriting
+in the reverse direction, use `Finset.sum_product`"]
+theorem prod_product' (s : Finset γ) (t : Finset α) (f : γ → α → β) :
     ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ x ∈ s, ∏ y ∈ t, f x y :=
-  prod_product
+  prod_product ..
 
 @[to_additive]
-theorem prod_product_right {s : Finset γ} {t : Finset α} {f : γ × α → β} :
+theorem prod_product_right (s : Finset γ) (t : Finset α) (f : γ × α → β) :
     ∏ x ∈ s ×ˢ t, f x = ∏ y ∈ t, ∏ x ∈ s, f (x, y) :=
   prod_finset_product_right (s ×ˢ t) t (fun _a => s) fun _p => mem_product.trans and_comm
 
 /-- An uncurried version of `Finset.prod_product_right`. -/
 @[to_additive "An uncurried version of `Finset.sum_product_right`"]
-theorem prod_product_right' {s : Finset γ} {t : Finset α} {f : γ → α → β} :
+theorem prod_product_right' (s : Finset γ) (t : Finset α) (f : γ → α → β) :
     ∏ x ∈ s ×ˢ t, f x.1 x.2 = ∏ y ∈ t, ∏ x ∈ s, f x y :=
-  prod_product_right
+  prod_product_right ..
 
 /-- Generalization of `Finset.prod_comm` to the case when the inner `Finset`s depend on the outer
 variable. -/
@@ -1494,7 +1502,7 @@ theorem prod_pow (s : Finset α) (n : ℕ) (f : α → β) : ∏ x ∈ s, f x ^
   Multiset.prod_map_pow
 
 @[to_additive sum_nsmul_assoc]
-lemma prod_pow_eq_pow_sum  (s : Finset ι) (f : ι → ℕ) (a : β) :
+lemma prod_pow_eq_pow_sum (s : Finset ι) (f : ι → ℕ) (a : β) :
     ∏ i ∈ s, a ^ f i = a ^ ∑ i ∈ s, f i :=
   cons_induction (by simp) (fun _ _ _ _ ↦ by simp [prod_cons, sum_cons, pow_add, *]) s
 
@@ -2000,6 +2008,10 @@ lemma prod_ite_eq_ite_exists (p : ι → Prop) [DecidablePred p] (h : ∀ i j, p
 
 variable [DecidableEq ι]
 
+@[to_additive]
+lemma prod_ite_mem (s : Finset ι) (f : ι → α) : ∏ i, (if i ∈ s then f i else 1) = ∏ i ∈ s, f i := by
+  simp
+
 /-- See also `Finset.prod_dite_eq`. -/
 @[to_additive "See also `Finset.sum_dite_eq`."] lemma prod_dite_eq (i : ι) (f : ∀ j, i = j → α) :
     ∏ j, (if h : i = j then f j h else 1) = f i rfl := by
@@ -2074,7 +2086,7 @@ theorem disjoint_list_sum_left {a : Multiset α} {l : List (Multiset α)} :
     simp only [zero_disjoint, List.not_mem_nil, IsEmpty.forall_iff, forall_const, List.sum_nil]
   | cons b bs ih =>
     simp_rw [List.sum_cons, disjoint_add_left, List.mem_cons, forall_eq_or_imp]
-    simp [and_congr_left_iff, iff_self_iff, ih]
+    simp [and_congr_left_iff, ih]
 
 theorem disjoint_list_sum_right {a : Multiset α} {l : List (Multiset α)} :
     Multiset.Disjoint a l.sum ↔ ∀ b ∈ l, Multiset.Disjoint a b := by
@@ -2093,7 +2105,7 @@ theorem disjoint_sum_right {a : Multiset α} {i : Multiset (Multiset α)} :
 theorem disjoint_finset_sum_left {β : Type*} {i : Finset β} {f : β → Multiset α} {a : Multiset α} :
     Multiset.Disjoint (i.sum f) a ↔ ∀ b ∈ i, Multiset.Disjoint (f b) a := by
   convert @disjoint_sum_left _ a (map f i.val)
-  simp [and_congr_left_iff, iff_self_iff]
+  simp [and_congr_left_iff]
 
 theorem disjoint_finset_sum_right {β : Type*} {i : Finset β} {f : β → Multiset α}
     {a : Multiset α} : Multiset.Disjoint a (i.sum f) ↔ ∀ b ∈ i, Multiset.Disjoint a (f b) := by
@@ -2246,9 +2258,6 @@ theorem toAdd_prod (s : Finset ι) (f : ι → Multiplicative α) :
 
 end AddCommMonoid
 
-@[deprecated (since := "2023-12-23")] alias Equiv.prod_comp' := Fintype.prod_equiv
-@[deprecated (since := "2023-12-23")] alias Equiv.sum_comp' := Fintype.sum_equiv
-
 theorem Finset.sum_sym2_filter_not_isDiag {ι α} [LinearOrder ι] [AddCommMonoid α]
     (s : Finset ι) (p : Sym2 ι → α) :
     ∑ i in s.sym2.filter (¬ ·.IsDiag), p i =
@@ -2260,3 +2269,5 @@ theorem Finset.sum_sym2_filter_not_isDiag {ι α} [LinearOrder ι] [AddCommMonoi
     simp [and_assoc]
   · rintro ⟨⟨i₁, j₁⟩, hij₁⟩
     simp
+
+set_option linter.style.longFile 2400
diff --git a/Mathlib/Algebra/BigOperators/Group/List.lean b/Mathlib/Algebra/BigOperators/Group/List.lean
index 70dee290ddb9f..6a1a5fb8af68e 100644
--- a/Mathlib/Algebra/BigOperators/Group/List.lean
+++ b/Mathlib/Algebra/BigOperators/Group/List.lean
@@ -12,6 +12,7 @@ import Mathlib.Data.List.Perm
 import Mathlib.Data.List.ProdSigma
 import Mathlib.Data.List.Range
 import Mathlib.Data.List.Rotate
+import Mathlib.Data.List.Pairwise
 
 /-!
 # Sums and products from lists
@@ -124,7 +125,7 @@ theorem prod_replicate (n : ℕ) (a : M) : (replicate n a).prod = a ^ n := by
 
 @[to_additive sum_eq_card_nsmul]
 theorem prod_eq_pow_card (l : List M) (m : M) (h : ∀ x ∈ l, x = m) : l.prod = m ^ l.length := by
-  rw [← prod_replicate, ← List.eq_replicate.mpr ⟨rfl, h⟩]
+  rw [← prod_replicate, ← List.eq_replicate_iff.mpr ⟨rfl, h⟩]
 
 @[to_additive]
 theorem prod_hom_rel (l : List ι) {r : M → N → Prop} {f : ι → M} {g : ι → N} (h₁ : r 1 1)
@@ -136,24 +137,29 @@ theorem rel_prod {R : M → N → Prop} (h : R 1 1) (hf : (R ⇒ R ⇒ R) (· *
     (Forall₂ R ⇒ R) prod prod :=
   rel_foldl hf h
 
+@[to_additive]
+theorem prod_hom_nonempty {l : List M} {F : Type*} [FunLike F M N] [MulHomClass F M N] (f : F)
+    (hl : l ≠ []) : (l.map f).prod = f l.prod :=
+  match l, hl with | x :: xs, hl => by induction xs generalizing x <;> aesop
+
 @[to_additive]
 theorem prod_hom (l : List M) {F : Type*} [FunLike F M N] [MonoidHomClass F M N] (f : F) :
     (l.map f).prod = f l.prod := by
   simp only [prod, foldl_map, ← map_one f]
   exact l.foldl_hom f (· * ·) (· * f ·) 1 (fun x y => (map_mul f x y).symm)
 
+@[to_additive]
+theorem prod_hom₂_nonempty {l : List ι} (f : M → N → P)
+    (hf : ∀ a b c d, f (a * b) (c * d) = f a c * f b d) (f₁ : ι → M) (f₂ : ι → N) (hl : l ≠ []) :
+    (l.map fun i => f (f₁ i) (f₂ i)).prod = f (l.map f₁).prod (l.map f₂).prod := by
+  match l, hl with | x :: xs, hl => induction xs generalizing x <;> aesop
+
 @[to_additive]
 theorem prod_hom₂ (l : List ι) (f : M → N → P) (hf : ∀ a b c d, f (a * b) (c * d) = f a c * f b d)
     (hf' : f 1 1 = 1) (f₁ : ι → M) (f₂ : ι → N) :
     (l.map fun i => f (f₁ i) (f₂ i)).prod = f (l.map f₁).prod (l.map f₂).prod := by
-  simp only [prod, foldl_map]
-  -- Porting note: next 3 lines used to be
-  -- convert l.foldl_hom₂ (fun a b => f a b) _ _ _ _ _ fun a b i => _
-  -- · exact hf'.symm
-  -- · exact hf _ _ _ _
-  rw [← l.foldl_hom₂ (fun a b => f a b), hf']
-  intros
-  exact hf _ _ _ _
+  rw [prod, prod, prod, foldl_map, foldl_map, foldl_map,
+    ← l.foldl_hom₂ f _ _ (fun x y => x * f (f₁ y) (f₂ y)) _ _ (by simp [hf]), hf']
 
 @[to_additive (attr := simp)]
 theorem prod_map_mul {α : Type*} [CommMonoid α] {l : List ι} {f g : ι → α} :
@@ -183,21 +189,14 @@ theorem prod_isUnit_iff {α : Type*} [CommMonoid α] {L : List α} :
     exact fun m' h' ↦ Or.elim (eq_or_mem_of_mem_cons h') (fun H => H.substr h.1) fun H => ih h.2 _ H
 
 @[to_additive (attr := simp)]
-theorem prod_take_mul_prod_drop : ∀ (L : List M) (i : ℕ), (L.take i).prod * (L.drop i).prod = L.prod
-  | [], i => by simp [Nat.zero_le]
-  | L, 0 => by simp
-  | h :: t, n + 1 => by
-    dsimp
-    rw [prod_cons, prod_cons, mul_assoc, prod_take_mul_prod_drop t]
+theorem prod_take_mul_prod_drop (L : List M) (i : ℕ) :
+    (L.take i).prod * (L.drop i).prod = L.prod := by
+  simp [← prod_append]
 
 @[to_additive (attr := simp)]
-theorem prod_take_succ :
-    ∀ (L : List M) (i : ℕ) (p : i < L.length), (L.take (i + 1)).prod = (L.take i).prod * L[i]
-  | [], i, p => by cases p
-  | h :: t, 0, _ => rfl
-  | h :: t, n + 1, p => by
-    dsimp
-    rw [prod_cons, prod_cons, prod_take_succ t n (Nat.lt_of_succ_lt_succ p), mul_assoc]
+theorem prod_take_succ (L : List M) (i : ℕ) (p : i < L.length) :
+    (L.take (i + 1)).prod = (L.take i).prod * L[i] := by
+  simp [take_succ, p]
 
 /-- A list with product not one must have positive length. -/
 @[to_additive "A list with sum not zero must have positive length."]
@@ -267,9 +266,9 @@ last. -/
 @[to_additive
 "A variant of `sum_range_succ` which pulls off the first term in the sum rather than the last."]
 lemma prod_range_succ' (f : ℕ → M) (n : ℕ) :
-    ((range n.succ).map f).prod = f 0 * ((range n).map fun i ↦ f i.succ).prod :=
-  Nat.recOn n (show 1 * f 0 = f 0 * 1 by rw [one_mul, mul_one]) fun _ hd => by
-    rw [List.prod_range_succ, hd, mul_assoc, ← List.prod_range_succ]
+    ((range n.succ).map f).prod = f 0 * ((range n).map fun i ↦ f i.succ).prod := by
+  rw [range_succ_eq_map]
+  simp [Function.comp_def]
 
 @[to_additive] lemma prod_eq_one (hl : ∀ x ∈ l, x = 1) : l.prod = 1 := by
   induction l with
@@ -343,7 +342,7 @@ lemma prod_map_erase [DecidableEq α] (f : α → M) {a} :
     · simp only [map, erase_cons_tail (not_beq_of_ne ne.symm), prod_cons, prod_map_erase _ h,
         mul_left_comm (f a) (f b)]
 
-@[to_additive] lemma Perm.prod_eq (h : Perm l₁ l₂) : prod l₁ = prod l₂ := h.fold_op_eq
+@[to_additive] lemma Perm.prod_eq (h : Perm l₁ l₂) : prod l₁ = prod l₂ := h.foldl_op_eq
 
 @[to_additive] lemma prod_reverse (l : List M) : prod l.reverse = prod l := (reverse_perm l).prod_eq
 
@@ -595,7 +594,7 @@ theorem sum_map_count_dedup_filter_eq_countP (p : α → Bool) (l : List α) :
         obtain ⟨a', ha'⟩ := List.mem_map.1 hn
         split_ifs at ha' with ha
         · simp only [ha.symm, mem_filter, mem_dedup, find?, mem_cons, true_or, hp,
-            and_false, false_and] at ha'
+            and_false, false_and, reduceCtorEq] at ha'
         · exact ha'.2.symm
 
 theorem sum_map_count_dedup_eq_length (l : List α) :
@@ -627,7 +626,7 @@ end MonoidHom
 end MonoidHom
 
 @[simp] lemma Nat.sum_eq_listSum (l : List ℕ) : Nat.sum l = l.sum :=
-  (List.foldl_eq_foldr Nat.add_comm Nat.add_assoc _ _).symm
+  (List.foldl_eq_foldr _ _).symm
 
 namespace List
 
@@ -640,20 +639,15 @@ lemma ranges_join (l : List ℕ) : l.ranges.join = range l.sum := by simp [range
 lemma mem_mem_ranges_iff_lt_sum (l : List ℕ) {n : ℕ} :
     (∃ s ∈ l.ranges, n ∈ s) ↔ n < l.sum := by simp [mem_mem_ranges_iff_lt_natSum]
 
-lemma countP_join (p : α → Bool) : ∀ L : List (List α), countP p L.join = (L.map (countP p)).sum
-  | [] => rfl
-  | a :: l => by rw [join, countP_append, map_cons, sum_cons, countP_join _ l]
-
-lemma count_join [BEq α] (L : List (List α)) (a : α) : L.join.count a = (L.map (count a)).sum :=
-  countP_join _ _
-
 @[simp]
 theorem length_bind (l : List α) (f : α → List β) :
     length (List.bind l f) = sum (map (length ∘ f) l) := by
   rw [List.bind, length_join, map_map, Nat.sum_eq_listSum]
 
 lemma countP_bind (p : β → Bool) (l : List α) (f : α → List β) :
-    countP p (l.bind f) = sum (map (countP p ∘ f) l) := by rw [List.bind, countP_join, map_map]
+    countP p (l.bind f) = sum (map (countP p ∘ f) l) := by
+  rw [List.bind, countP_join, map_map]
+  simp
 
 lemma count_bind [BEq β] (l : List α) (f : α → List β) (x : β) :
     count x (l.bind f) = sum (map (count x ∘ f) l) := countP_bind _ _ _
diff --git a/Mathlib/Algebra/BigOperators/Group/Multiset.lean b/Mathlib/Algebra/BigOperators/Group/Multiset.lean
index 0d5d2fc6e95ea..0a0e9b28703cd 100644
--- a/Mathlib/Algebra/BigOperators/Group/Multiset.lean
+++ b/Mathlib/Algebra/BigOperators/Group/Multiset.lean
@@ -36,17 +36,17 @@ variable [CommMonoid α] [CommMonoid β] {s t : Multiset α} {a : α} {m : Multi
       "Sum of a multiset given a commutative additive monoid structure on `α`.
       `sum {a, b, c} = a + b + c`"]
 def prod : Multiset α → α :=
-  foldr (· * ·) (fun x y z => by simp [mul_left_comm]) 1
+  foldr (· * ·) 1
 
 @[to_additive]
 theorem prod_eq_foldr (s : Multiset α) :
-    prod s = foldr (· * ·) (fun x y z => by simp [mul_left_comm]) 1 s :=
+    prod s = foldr (· * ·) 1 s :=
   rfl
 
 @[to_additive]
 theorem prod_eq_foldl (s : Multiset α) :
-    prod s = foldl (· * ·) (fun x y z => by simp [mul_right_comm]) 1 s :=
-  (foldr_swap _ _ _ _).trans (by simp [mul_comm])
+    prod s = foldl (· * ·) 1 s :=
+  (foldr_swap _ _ _).trans (by simp [mul_comm])
 
 @[to_additive (attr := simp, norm_cast)]
 theorem prod_coe (l : List α) : prod ↑l = l.prod :=
@@ -63,7 +63,7 @@ theorem prod_zero : @prod α _ 0 = 1 :=
 
 @[to_additive (attr := simp)]
 theorem prod_cons (a : α) (s) : prod (a ::ₘ s) = a * prod s :=
-  foldr_cons _ _ _ _ _
+  foldr_cons _ _ _ _
 
 @[to_additive (attr := simp)]
 theorem prod_erase [DecidableEq α] (h : a ∈ s) : a * (s.erase a).prod = s.prod := by
@@ -122,6 +122,12 @@ lemma prod_eq_one (h : ∀ x ∈ s, x = (1 : α)) : s.prod = 1 := by
 theorem pow_count [DecidableEq α] (a : α) : a ^ s.count a = (s.filter (Eq a)).prod := by
   rw [filter_eq, prod_replicate]
 
+@[to_additive]
+theorem prod_hom_ne_zero {s : Multiset α} (hs : s ≠ 0) {F : Type*} [FunLike F α β]
+    [MulHomClass F α β] (f : F) :
+    (s.map f).prod = f s.prod := by
+  induction s using Quot.inductionOn; aesop (add simp List.prod_hom_nonempty)
+
 @[to_additive]
 theorem prod_hom (s : Multiset α) {F : Type*} [FunLike F α β]
     [MonoidHomClass F α β] (f : F) :
@@ -135,6 +141,12 @@ theorem prod_hom' (s : Multiset ι) {F : Type*} [FunLike F α β]
   convert (s.map g).prod_hom f
   exact (map_map _ _ _).symm
 
+@[to_additive]
+theorem prod_hom₂_ne_zero [CommMonoid γ] {s : Multiset ι} (hs : s ≠ 0) (f : α → β → γ)
+    (hf : ∀ a b c d, f (a * b) (c * d) = f a c * f b d) (f₁ : ι → α) (f₂ : ι → β) :
+    (s.map fun i => f (f₁ i) (f₂ i)).prod = f (s.map f₁).prod (s.map f₂).prod := by
+  induction s using Quotient.inductionOn; aesop (add simp List.prod_hom₂_nonempty)
+
 @[to_additive]
 theorem prod_hom₂ [CommMonoid γ] (s : Multiset ι) (f : α → β → γ)
     (hf : ∀ a b c d, f (a * b) (c * d) = f a c * f b d) (hf' : f 1 1 = 1) (f₁ : ι → α)
@@ -171,7 +183,7 @@ theorem prod_map_prod_map (m : Multiset β') (n : Multiset γ) {f : β' → γ 
 theorem prod_induction (p : α → Prop) (s : Multiset α) (p_mul : ∀ a b, p a → p b → p (a * b))
     (p_one : p 1) (p_s : ∀ a ∈ s, p a) : p s.prod := by
   rw [prod_eq_foldr]
-  exact foldr_induction (· * ·) (fun x y z => by simp [mul_left_comm]) 1 p s p_mul p_one p_s
+  exact foldr_induction (· * ·) 1 p s p_mul p_one p_s
 
 @[to_additive]
 theorem prod_induction_nonempty (p : α → Prop) (p_mul : ∀ a b, p a → p b → p (a * b)) (hs : s ≠ ∅)
@@ -194,10 +206,19 @@ theorem prod_dvd_prod_of_le (h : s ≤ t) : s.prod ∣ t.prod := by
 lemma _root_.map_multiset_prod [FunLike F α β] [MonoidHomClass F α β] (f : F) (s : Multiset α) :
     f s.prod = (s.map f).prod := (s.prod_hom f).symm
 
+@[to_additive]
+lemma _root_.map_multiset_ne_zero_prod [FunLike F α β] [MulHomClass F α β] (f : F)
+    {s : Multiset α} (hs : s ≠ 0):
+    f s.prod = (s.map f).prod := (s.prod_hom_ne_zero hs f).symm
+
 @[to_additive]
 protected lemma _root_.MonoidHom.map_multiset_prod (f : α →* β) (s : Multiset α) :
     f s.prod = (s.map f).prod := (s.prod_hom f).symm
 
+@[to_additive]
+protected lemma _root_.MulHom.map_multiset_ne_zero_prod (f : α →ₙ* β) (s : Multiset α)
+    (hs : s ≠ 0) : f s.prod = (s.map f).prod := (s.prod_hom_ne_zero hs f).symm
+
 lemma dvd_prod : a ∈ s → a ∣ s.prod :=
   Quotient.inductionOn s (fun l a h ↦ by simpa using List.dvd_prod h) a
 
diff --git a/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean b/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean
new file mode 100644
index 0000000000000..71d86a299d148
--- /dev/null
+++ b/Mathlib/Algebra/BigOperators/GroupWithZero/Action.lean
@@ -0,0 +1,119 @@
+/-
+Copyright (c) 2020 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Algebra.GroupWithZero.Action.Defs
+import Mathlib.Data.Finset.Basic
+import Mathlib.Data.Multiset.Basic
+import Mathlib.Algebra.BigOperators.Finprod
+
+/-!
+# Lemmas about group actions on big operators
+
+This file contains results about two kinds of actions:
+
+* sums over `DistribSMul`: `r • ∑ x ∈ s, f x = ∑ x ∈ s, r • f x`
+* products over `MulDistribMulAction` (with primed name): `r • ∏ x ∈ s, f x = ∏ x ∈ s, r • f x`
+* products over `SMulCommClass` (with unprimed name):
+  `b ^ s.card • ∏ x in s, f x = ∏ x in s, b • f x`
+
+Note that analogous lemmas for `Module`s like `Finset.sum_smul` appear in other files.
+-/
+
+
+variable {α β γ : Type*}
+
+section
+
+variable [AddMonoid β] [DistribSMul α β]
+
+theorem List.smul_sum {r : α} {l : List β} : r • l.sum = (l.map (r • ·)).sum :=
+  map_list_sum (DistribSMul.toAddMonoidHom β r) l
+
+end
+
+section
+
+variable [Monoid α] [Monoid β] [MulDistribMulAction α β]
+
+theorem List.smul_prod' {r : α} {l : List β} : r • l.prod = (l.map (r • ·)).prod :=
+  map_list_prod (MulDistribMulAction.toMonoidHom β r) l
+
+end
+
+section
+
+variable [AddCommMonoid β] [DistribSMul α β]
+
+theorem Multiset.smul_sum {r : α} {s : Multiset β} : r • s.sum = (s.map (r • ·)).sum :=
+  (DistribSMul.toAddMonoidHom β r).map_multiset_sum s
+
+theorem Finset.smul_sum {r : α} {f : γ → β} {s : Finset γ} :
+    (r • ∑ x ∈ s, f x) = ∑ x ∈ s, r • f x :=
+  map_sum (DistribSMul.toAddMonoidHom β r) f s
+
+end
+
+section
+
+variable [Monoid α] [CommMonoid β] [MulDistribMulAction α β]
+
+theorem Multiset.smul_prod' {r : α} {s : Multiset β} : r • s.prod = (s.map (r • ·)).prod :=
+  (MulDistribMulAction.toMonoidHom β r).map_multiset_prod s
+
+theorem Finset.smul_prod' {r : α} {f : γ → β} {s : Finset γ} :
+    (r • ∏ x ∈ s, f x) = ∏ x ∈ s, r • f x :=
+  map_prod (MulDistribMulAction.toMonoidHom β r) f s
+
+theorem smul_finprod' {ι : Sort*} [Finite ι] {f : ι → β} (r : α) :
+    r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x) := by
+  cases nonempty_fintype (PLift ι)
+  simp only [finprod_eq_prod_plift_of_mulSupport_subset (s := Finset.univ) (by simp),
+    finprod_eq_prod_of_fintype, Finset.smul_prod']
+
+end
+
+namespace List
+
+@[to_additive]
+theorem smul_prod [Monoid α] [Monoid β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β]
+    (l : List β) (m : α) :
+    m ^ l.length • l.prod = (l.map (m • ·)).prod := by
+  induction l with
+  | nil => simp
+  | cons head tail ih => simp [← ih, smul_mul_smul_comm, pow_succ']
+
+end List
+
+namespace Multiset
+
+@[to_additive]
+theorem smul_prod [Monoid α] [CommMonoid β] [MulAction α β] [IsScalarTower α β β]
+    [SMulCommClass α β β] (s : Multiset β) (b : α) :
+    b ^ card s • s.prod = (s.map (b • ·)).prod :=
+  Quot.induction_on s <| by simp [List.smul_prod]
+
+end Multiset
+
+namespace Finset
+
+theorem smul_prod
+    [CommMonoid β] [Monoid α] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β]
+    (s : Finset β) (b : α) (f : β → β) :
+    b ^ s.card • ∏ x in s, f x = ∏ x in s, b • f x := by
+  have : Multiset.map (fun (x : β) ↦ b • f x) s.val =
+      Multiset.map (fun x ↦ b • x) (Multiset.map (fun x ↦ f x) s.val) := by
+    simp only [Multiset.map_map, Function.comp_apply]
+  simp_rw [prod_eq_multiset_prod, card_def, this, ← Multiset.smul_prod _ b, Multiset.card_map]
+
+theorem prod_smul
+    [CommMonoid β] [CommMonoid α] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β]
+    (s : Finset β) (b : β → α) (f : β → β) :
+    ∏ i in s, b i • f i = (∏ i in s, b i) • ∏ i in s, f i := by
+  induction s using Finset.cons_induction_on with
+  | h₁ =>  simp
+  | h₂ hj ih => rw [prod_cons, ih, smul_mul_smul_comm, ← prod_cons hj, ← prod_cons hj]
+
+end Finset
diff --git a/Mathlib/Algebra/BigOperators/Intervals.lean b/Mathlib/Algebra/BigOperators/Intervals.lean
index fffc916fc80ef..3193db4f355ef 100644
--- a/Mathlib/Algebra/BigOperators/Intervals.lean
+++ b/Mathlib/Algebra/BigOperators/Intervals.lean
@@ -99,6 +99,12 @@ theorem prod_Ico_add [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [Locally
   convert prod_Ico_add' f a b c using 2
   rw [add_comm]
 
+@[to_additive (attr := simp)]
+theorem prod_Ico_add_right_sub_eq [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α]
+    [LocallyFiniteOrder α] [Sub α] [OrderedSub α] (a b c : α) :
+    ∏ x ∈ Ico (a + c) (b + c), f (x - c) = ∏ x ∈ Ico a b, f x := by
+  simp only [← map_add_right_Ico, prod_map, addRightEmbedding_apply, add_tsub_cancel_right]
+
 @[to_additive]
 theorem prod_Ico_succ_top {a b : ℕ} (hab : a ≤ b) (f : ℕ → M) :
     (∏ k ∈ Ico a (b + 1), f k) = (∏ k ∈ Ico a b, f k) * f b := by
@@ -138,6 +144,11 @@ theorem prod_range_mul_prod_Ico (f : ℕ → M) {m n : ℕ} (h : m ≤ n) :
     ((∏ k ∈ range m, f k) * ∏ k ∈ Ico m n, f k) = ∏ k ∈ range n, f k :=
   Nat.Ico_zero_eq_range ▸ Nat.Ico_zero_eq_range ▸ prod_Ico_consecutive f m.zero_le h
 
+@[to_additive]
+theorem prod_range_eq_mul_Ico (f : ℕ → M) {n : ℕ} (hn : 0 < n) :
+    ∏ x ∈ Finset.range n, f x = f 0 * ∏ x ∈ Ico 1 n, f x :=
+  Finset.range_eq_Ico ▸ Finset.prod_eq_prod_Ico_succ_bot hn f
+
 @[to_additive]
 theorem prod_Ico_eq_mul_inv {δ : Type*} [CommGroup δ] (f : ℕ → δ) {m n : ℕ} (h : m ≤ n) :
     ∏ k ∈ Ico m n, f k = (∏ k ∈ range n, f k) * (∏ k ∈ range m, f k)⁻¹ :=
diff --git a/Mathlib/Algebra/BigOperators/Module.lean b/Mathlib/Algebra/BigOperators/Module.lean
index fd9ddbeeb76ee..2329df72948f4 100644
--- a/Mathlib/Algebra/BigOperators/Module.lean
+++ b/Mathlib/Algebra/BigOperators/Module.lean
@@ -31,24 +31,9 @@ theorem sum_Ico_by_parts (hmn : m < n) :
     rw [← sum_Ico_sub_bot _ hmn, ← sum_Ico_succ_sub_top _ (Nat.le_sub_one_of_lt hmn),
       Nat.sub_add_cancel (pos_of_gt hmn), sub_add_cancel]
   rw [sum_eq_sum_Ico_succ_bot hmn]
-  -- Porting note: the following used to be done with `conv`
-  have h₃ : (Finset.sum (Ico (m + 1) n) fun i => f i • g i) =
-             (Finset.sum (Ico (m + 1) n) fun i =>
-                f i • ((Finset.sum (Finset.range (i + 1)) g) -
-                        (Finset.sum (Finset.range i) g))) := by
-    congr; funext; rw [← sum_range_succ_sub_sum g]
-  rw [h₃]
+  conv in (occs := 3) (f _ • g _) => rw [← sum_range_succ_sub_sum g]
   simp_rw [smul_sub, sum_sub_distrib, h₂, h₁]
-  -- Porting note: the following used to be done with `conv`
-  have h₄ : ((((Finset.sum (Ico m (n - 1)) fun i => f i • Finset.sum (range (i + 1)) fun i => g i) +
-      f (n - 1) • Finset.sum (range n) fun i => g i) -
-      f m • Finset.sum (range (m + 1)) fun i => g i) -
-      Finset.sum (Ico m (n - 1)) fun i => f (i + 1) • Finset.sum (range (i + 1)) fun i => g i) =
-      f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g +
-      Finset.sum (Ico m (n - 1)) (fun i => f i • (range (i + 1)).sum g -
-      f (i + 1) • (range (i + 1)).sum g) := by
-    rw [← add_sub, add_comm, ← add_sub, ← sum_sub_distrib]
-  rw [h₄]
+  conv_lhs => congr; rfl; rw [← add_sub, add_comm, ← add_sub, ← sum_sub_distrib]
   have : ∀ i, f i • G (i + 1) - f (i + 1) • G (i + 1) = -((f (i + 1) - f i) • G (i + 1)) := by
     intro i
     rw [sub_smul]
diff --git a/Mathlib/Algebra/BigOperators/Ring.lean b/Mathlib/Algebra/BigOperators/Ring.lean
index 2253d0bb0465c..8ffa27de019be 100644
--- a/Mathlib/Algebra/BigOperators/Ring.lean
+++ b/Mathlib/Algebra/BigOperators/Ring.lean
@@ -110,7 +110,7 @@ lemma prod_sum (s : Finset ι) (t : ∀ i, Finset (κ i)) (f : ∀ i, κ i → 
     ∏ a ∈ s, ∑ b ∈ t a, f a b = ∑ p ∈ s.pi t, ∏ x ∈ s.attach, f x.1 (p x.1 x.2) := by
   classical
   induction s using Finset.induction with
-  | empty => rw [pi_empty, sum_singleton]; rfl
+  | empty => simp
   | insert ha ih =>
     rename_i a s
     have h₁ : ∀ x ∈ t a, ∀ y ∈ t a, x ≠ y →
@@ -239,7 +239,7 @@ lemma prod_sub_ordered [LinearOrder ι] (s : Finset ι) (f g : ι → α) :
   simp
 
 /-- `∏ i, (1 - f i) = 1 - ∑ i, f i * (∏ j < i, 1 - f j)`. This formula is useful in construction of
-a partition of unity from a collection of “bump” functions.  -/
+a partition of unity from a collection of “bump” functions. -/
 theorem prod_one_sub_ordered [LinearOrder ι] (s : Finset ι) (f : ι → α) :
     ∏ i ∈ s, (1 - f i) = 1 - ∑ i ∈ s, f i * ∏ j ∈ s.filter (· < i), (1 - f j) := by
   rw [prod_sub_ordered]
diff --git a/Mathlib/Algebra/Bounds.lean b/Mathlib/Algebra/Bounds.lean
deleted file mode 100644
index 95219c22b4fa9..0000000000000
--- a/Mathlib/Algebra/Bounds.lean
+++ /dev/null
@@ -1,167 +0,0 @@
-/-
-Copyright (c) 2021 Yury Kudryashov. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov
--/
-import Mathlib.Algebra.Order.Group.OrderIso
-import Mathlib.Data.Set.Pointwise.Basic
-import Mathlib.Order.Bounds.OrderIso
-import Mathlib.Order.ConditionallyCompleteLattice.Basic
-import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
-
-/-!
-# Upper/lower bounds in ordered monoids and groups
-
-In this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”
-(`bddAbove_neg`).
--/
-
-
-open Function Set
-
-open Pointwise
-
-section InvNeg
-
-variable {G : Type*} [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]
-  [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}
-
-@[to_additive (attr := simp)]
-theorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=
-  (OrderIso.inv G).bddAbove_preimage
-
-@[to_additive (attr := simp)]
-theorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=
-  (OrderIso.inv G).bddBelow_preimage
-
-@[to_additive]
-theorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=
-  bddBelow_inv.2 h
-
-@[to_additive]
-theorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=
-  bddAbove_inv.2 h
-
-@[to_additive (attr := simp)]
-theorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=
-  (OrderIso.inv G).isLUB_preimage
-
-@[to_additive]
-theorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=
-  (OrderIso.inv G).isLUB_preimage'
-
-@[to_additive]
-theorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=
-  isLUB_inv'.2 h
-
-@[to_additive (attr := simp)]
-theorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=
-  (OrderIso.inv G).isGLB_preimage
-
-@[to_additive]
-theorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=
-  (OrderIso.inv G).isGLB_preimage'
-
-@[to_additive]
-theorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=
-  isGLB_inv'.2 h
-
-@[to_additive]
-lemma BddBelow.range_inv {α : Type*} {f : α → G} (hf : BddBelow (range f)) :
-    BddAbove (range (fun x => (f x)⁻¹)) :=
-  hf.range_comp (OrderIso.inv G).monotone
-
-@[to_additive]
-lemma BddAbove.range_inv {α : Type*} {f : α → G} (hf : BddAbove (range f)) :
-    BddBelow (range (fun x => (f x)⁻¹)) :=
-  BddBelow.range_inv (G := Gᵒᵈ) hf
-
-end InvNeg
-
-section mul_add
-
-variable {M : Type*} [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]
-  [CovariantClass M M (swap (· * ·)) (· ≤ ·)]
-
-@[to_additive]
-theorem mul_mem_upperBounds_mul {s t : Set M} {a b : M} (ha : a ∈ upperBounds s)
-    (hb : b ∈ upperBounds t) : a * b ∈ upperBounds (s * t) :=
-  forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)
-
-@[to_additive]
-theorem subset_upperBounds_mul (s t : Set M) :
-    upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=
-  image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy
-
-@[to_additive]
-theorem mul_mem_lowerBounds_mul {s t : Set M} {a b : M} (ha : a ∈ lowerBounds s)
-    (hb : b ∈ lowerBounds t) : a * b ∈ lowerBounds (s * t) :=
-  mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb
-
-@[to_additive]
-theorem subset_lowerBounds_mul (s t : Set M) :
-    lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=
-  subset_upperBounds_mul (M := Mᵒᵈ) _ _
-
-@[to_additive]
-theorem BddAbove.mul {s t : Set M} (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=
-  (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)
-
-@[to_additive]
-theorem BddBelow.mul {s t : Set M} (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=
-  (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)
-
-@[to_additive]
-lemma BddAbove.range_mul {α : Type*} {f g : α → M} (hf : BddAbove (range f))
-    (hg : BddAbove (range g)) : BddAbove (range (fun x => f x * g x)) :=
-  BddAbove.range_comp (f := fun x => (⟨f x, g x⟩ : M × M))
-    (bddAbove_range_prod.mpr ⟨hf, hg⟩) (Monotone.mul' monotone_fst monotone_snd)
-
-@[to_additive]
-lemma BddBelow.range_mul {α : Type*} {f g : α → M} (hf : BddBelow (range f))
-    (hg : BddBelow (range g)) : BddBelow (range (fun x => f x * g x)) :=
-  BddAbove.range_mul (M := Mᵒᵈ) hf hg
-
-end mul_add
-
-section ConditionallyCompleteLattice
-
-section Right
-
-variable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G]
-  [CovariantClass G G (Function.swap (· * ·)) (· ≤ ·)] [Nonempty ι] {f : ι → G}
-
-@[to_additive]
-theorem ciSup_mul (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=
-  (OrderIso.mulRight a).map_ciSup hf
-
-@[to_additive]
-theorem ciSup_div (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by
-  simp only [div_eq_mul_inv, ciSup_mul hf]
-
-@[to_additive]
-theorem ciInf_mul (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) * a = ⨅ i, f i * a :=
-  (OrderIso.mulRight a).map_ciInf hf
-
-@[to_additive]
-theorem ciInf_div (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by
-  simp only [div_eq_mul_inv, ciInf_mul hf]
-
-end Right
-
-section Left
-
-variable {ι : Sort*} {G : Type*} [Group G] [ConditionallyCompleteLattice G]
-  [CovariantClass G G (· * ·) (· ≤ ·)] [Nonempty ι] {f : ι → G}
-
-@[to_additive]
-theorem mul_ciSup (hf : BddAbove (range f)) (a : G) : (a * ⨆ i, f i) = ⨆ i, a * f i :=
-  (OrderIso.mulLeft a).map_ciSup hf
-
-@[to_additive]
-theorem mul_ciInf (hf : BddBelow (range f)) (a : G) : (a * ⨅ i, f i) = ⨅ i, a * f i :=
-  (OrderIso.mulLeft a).map_ciInf hf
-
-end Left
-
-end ConditionallyCompleteLattice
diff --git a/Mathlib/Algebra/Category/AlgebraCat/Basic.lean b/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
index 6d3c2563e4879..36546a9af0623 100644
--- a/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
+++ b/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Algebra.FreeAlgebra
@@ -77,7 +77,7 @@ instance hasForgetToRing : HasForget₂ (AlgebraCat.{v} R) RingCat.{v} where
 instance hasForgetToModule : HasForget₂ (AlgebraCat.{v} R) (ModuleCat.{v} R) where
   forget₂ :=
     { obj := fun M => ModuleCat.of R M
-      map := fun f => ModuleCat.ofHom f.toLinearMap }
+      map := fun f => ModuleCat.asHom f.toLinearMap }
 
 @[simp]
 lemma forget₂_module_obj (X : AlgebraCat.{v} R) :
@@ -86,7 +86,7 @@ lemma forget₂_module_obj (X : AlgebraCat.{v} R) :
 
 @[simp]
 lemma forget₂_module_map {X Y : AlgebraCat.{v} R} (f : X ⟶ Y) :
-    (forget₂ (AlgebraCat.{v} R) (ModuleCat.{v} R)).map f = ModuleCat.ofHom f.toLinearMap :=
+    (forget₂ (AlgebraCat.{v} R) (ModuleCat.{v} R)).map f = ModuleCat.asHom f.toLinearMap :=
   rfl
 
 /-- The object in the category of R-algebras associated to a type equipped with the appropriate
diff --git a/Mathlib/Algebra/Category/AlgebraCat/Limits.lean b/Mathlib/Algebra/Category/AlgebraCat/Limits.lean
index bdbde8808325a..a2f702f99e183 100644
--- a/Mathlib/Algebra/Category/AlgebraCat/Limits.lean
+++ b/Mathlib/Algebra/Category/AlgebraCat/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.AlgebraCat.Basic
 import Mathlib.Algebra.Category.ModuleCat.Basic
diff --git a/Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean b/Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean
index 48be5957eed6d..9dbed63a629b2 100644
--- a/Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean
+++ b/Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean
@@ -55,43 +55,14 @@ instance : MonoidalCategoryStruct (AlgebraCat.{u} R) where
   leftUnitor X := (Algebra.TensorProduct.lid R X).toAlgebraIso
   rightUnitor X := (Algebra.TensorProduct.rid R R X).toAlgebraIso
 
-theorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :
-    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =
-      (α_
-        (forget₂ _ (ModuleCat R) |>.obj X)
-        (forget₂ _ (ModuleCat R) |>.obj Y)
-        (forget₂ _ (ModuleCat R) |>.obj Z)).hom := by
-  rfl
-
-theorem forget₂_map_associator_inv (X Y Z : AlgebraCat.{u} R) :
-    (forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =
-      (α_
-        (forget₂ _ (ModuleCat R) |>.obj X)
-        (forget₂ _ (ModuleCat R) |>.obj Y)
-        (forget₂ _ (ModuleCat R) |>.obj Z)).inv := by
-  rfl
-
-set_option maxHeartbeats 800000 in
 noncomputable instance instMonoidalCategory : MonoidalCategory (AlgebraCat.{u} R) :=
   Monoidal.induced
     (forget₂ (AlgebraCat R) (ModuleCat R))
     { μIso := fun X Y => Iso.refl _
       εIso := Iso.refl _
-      associator_eq := fun X Y Z => by
-        dsimp only [forget₂_module_obj, forget₂_map_associator_hom]
-        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,
-          Iso.refl_hom, MonoidalCategory.tensor_id]
-        erw [Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id, Category.id_comp]
-      leftUnitor_eq := fun X => by
-        dsimp only [forget₂_module_obj, forget₂_module_map, Iso.refl_symm, Iso.trans_hom,
-          Iso.refl_hom, tensorIso_hom]
-        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]
-        rfl
-      rightUnitor_eq := fun X => by
-        dsimp
-        erw [Category.id_comp, MonoidalCategory.tensor_id, Category.id_comp]
-        exact congr_arg LinearEquiv.toLinearMap <|
-          TensorProduct.AlgebraTensorModule.rid_eq_rid R X }
+      associator_eq := fun X Y Z => TensorProduct.ext₃ (fun x y z => rfl)
+      leftUnitor_eq := fun X => TensorProduct.ext' (fun x y => rfl)
+      rightUnitor_eq := fun X => TensorProduct.ext' (fun x y => rfl) }
 
 variable (R) in
 /-- `forget₂ (AlgebraCat R) (ModuleCat R)` as a monoidal functor. -/
diff --git a/Mathlib/Algebra/Category/BialgebraCat/Basic.lean b/Mathlib/Algebra/Category/BialgebraCat/Basic.lean
index 3cf9cd5c6fcbe..0d3b3d8260dad 100644
--- a/Mathlib/Algebra/Category/BialgebraCat/Basic.lean
+++ b/Mathlib/Algebra/Category/BialgebraCat/Basic.lean
@@ -59,7 +59,7 @@ lemma of_counit {X : Type v} [Ring X] [Bialgebra R X] :
 /-- A type alias for `BialgHom` to avoid confusion between the categorical and
 algebraic spellings of composition. -/
 @[ext]
-structure Hom (V W : BialgebraCat.{v} R) :=
+structure Hom (V W : BialgebraCat.{v} R) where
   /-- The underlying `BialgHom` -/
   toBialgHom : V →ₐc[R] W
 
diff --git a/Mathlib/Algebra/Category/CoalgebraCat/Basic.lean b/Mathlib/Algebra/Category/CoalgebraCat/Basic.lean
index 2e3227b072f57..cbf7f0ebab743 100644
--- a/Mathlib/Algebra/Category/CoalgebraCat/Basic.lean
+++ b/Mathlib/Algebra/Category/CoalgebraCat/Basic.lean
@@ -62,7 +62,7 @@ lemma of_counit {X : Type v} [AddCommGroup X] [Module R X] [Coalgebra R X] :
 /-- A type alias for `CoalgHom` to avoid confusion between the categorical and
 algebraic spellings of composition. -/
 @[ext]
-structure Hom (V W : CoalgebraCat.{v} R) :=
+structure Hom (V W : CoalgebraCat.{v} R) where
   /-- The underlying `CoalgHom` -/
   toCoalgHom : V →ₗc[R] W
 
diff --git a/Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean b/Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
index 9936f06b55a0f..0a138d78803f1 100644
--- a/Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
+++ b/Mathlib/Algebra/Category/CoalgebraCat/ComonEquivalence.lean
@@ -15,7 +15,7 @@ Given a commutative ring `R`, this file defines the equivalence of categories be
 `R`-coalgebras and comonoid objects in the category of `R`-modules.
 
 We then use this to set up boilerplate for the `Coalgebra` instance on a tensor product of
-coalgebras defined in `Mathlib.RingTheory.Coalgebra.TensorProduct` in #11975.
+coalgebras defined in `Mathlib.RingTheory.Coalgebra.TensorProduct`.
 
 ## Implementation notes
 
@@ -23,7 +23,7 @@ We make the definition `CoalgebraCat.instMonoidalCategoryAux` in this file, whic
 monoidal structure on `CoalgebraCat` induced by the equivalence with `Comon(R-Mod)`. We
 use this to show the comultiplication and counit on a tensor product of coalgebras satisfy
 the coalgebra axioms, but our actual `MonoidalCategory` instance on `CoalgebraCat` is
-constructed in `Mathlib.Algebra.Category.CoalgebraCat.Monoidal` in #11976 to have better
+constructed in `Mathlib.Algebra.Category.CoalgebraCat.Monoidal` to have better
 definitional equalities.
 
 -/
@@ -38,8 +38,8 @@ variable {R : Type u} [CommRing R]
 /-- An `R`-coalgebra is a comonoid object in the category of `R`-modules. -/
 @[simps] def toComonObj (X : CoalgebraCat R) : Comon_ (ModuleCat R) where
   X := ModuleCat.of R X
-  counit := ModuleCat.ofHom Coalgebra.counit
-  comul := ModuleCat.ofHom Coalgebra.comul
+  counit := ModuleCat.asHom Coalgebra.counit
+  comul := ModuleCat.asHom Coalgebra.comul
   counit_comul := by simpa only [ModuleCat.of_coe] using Coalgebra.rTensor_counit_comp_comul
   comul_counit := by simpa only [ModuleCat.of_coe] using Coalgebra.lTensor_counit_comp_comul
   comul_assoc := by simp_rw [ModuleCat.of_coe]; exact Coalgebra.coassoc.symm
@@ -50,7 +50,7 @@ variable (R) in
 def toComon : CoalgebraCat R ⥤ Comon_ (ModuleCat R) where
   obj X := toComonObj X
   map f :=
-    { hom := ModuleCat.ofHom f.1
+    { hom := ModuleCat.asHom f.1
       hom_counit := f.1.counit_comp
       hom_comul := f.1.map_comp_comul.symm }
 
@@ -96,7 +96,7 @@ variable {R}
 
 /-- The monoidal category structure on the category of `R`-coalgebras induced by the
 equivalence with `Comon(R-Mod)`. This is just an auxiliary definition; the `MonoidalCategory`
-instance we make in `Mathlib.Algebra.Category.CoalgebraCat.Monoidal` in #11976 will have better
+instance we make in `Mathlib.Algebra.Category.CoalgebraCat.Monoidal` has better
 definitional equalities. -/
 noncomputable def instMonoidalCategoryAux : MonoidalCategory (CoalgebraCat R) :=
   Monoidal.transport (comonEquivalence R).symm
@@ -116,9 +116,10 @@ theorem tensorObj_comul (K L : CoalgebraCat R) :
       = (TensorProduct.tensorTensorTensorComm R K K L L).toLinearMap
       ∘ₗ TensorProduct.map Coalgebra.comul Coalgebra.comul := by
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [ModuleCat.ofHom, -Mon_.monMonoidalStruct_tensorObj_X,
-    instMonoidalCategoryStruct_tensorHom, ModuleCat.comp_def]
-  simp only [BraidedCategory.unop_tensor_μ, tensor_μ_eq_tensorTensorTensorComm]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj]
+  simp only [Comon_.monoidal_tensorObj_comul, toComonObj_X, ModuleCat.of_coe, toComonObj_comul,
+    tensor_μ_eq_tensorTensorTensorComm]
+  rfl
 
 theorem tensorHom_toLinearMap (f : M →ₗc[R] N) (g : P →ₗc[R] Q) :
     (CoalgebraCat.ofHom f ⊗ CoalgebraCat.ofHom g).1.toLinearMap
@@ -143,22 +144,26 @@ theorem comul_tensorObj :
     Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗ CoalgebraCat.of R N : CoalgebraCat R))
       = Coalgebra.comul (A := M ⊗[R] N) := by
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [- Mon_.monMonoidalStruct_tensorObj_X, instMonoidalCategoryStruct_tensorHom,
-    ModuleCat.comp_def, ModuleCat.ofHom, ModuleCat.of]
-  simp only [BraidedCategory.unop_tensor_μ, tensor_μ_eq_tensorTensorTensorComm]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
+    instCoalgebraStruct_comul]
+  simp only [Comon_.monoidal_tensorObj_comul, toComonObj_X, of_carrier, of_isAddCommGroup,
+    of_isModule, toComonObj_comul, of_instCoalgebra, tensor_μ_eq_tensorTensorTensorComm]
+  rfl
 
 theorem comul_tensorObj_tensorObj_right :
     Coalgebra.comul (R := R) (A := (CoalgebraCat.of R M ⊗
       (CoalgebraCat.of R N ⊗ CoalgebraCat.of R P) : CoalgebraCat R))
       = Coalgebra.comul (A := M ⊗[R] N ⊗[R] P) := by
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [- Mon_.monMonoidalStruct_tensorObj_X, instMonoidalCategoryStruct_tensorHom,
-    ModuleCat.comp_def, ModuleCat.ofHom, ModuleCat.of]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
+    instCoalgebraStruct_comul]
+  simp only [Comon_.monoidal_tensorObj_comul, toComonObj_X, of_carrier, of_isAddCommGroup,
+    of_isModule, ModuleCat.of_coe, toComonObj_comul, of_instCoalgebra]
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [- Mon_.monMonoidalStruct_tensorObj_X, instMonoidalCategoryStruct_tensorHom,
-    ModuleCat.comp_def, ModuleCat.ofHom, ModuleCat.of]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj]
   simp only [instMonoidalCategoryStruct_tensorObj, ModuleCat.MonoidalCategory.tensorObj,
-    ModuleCat.coe_of, BraidedCategory.unop_tensor_μ, tensor_μ_eq_tensorTensorTensorComm]
+    ModuleCat.coe_of, Comon_.monoidal_tensorObj_comul, toComonObj_X, of_carrier, of_isAddCommGroup,
+    of_isModule, toComonObj_comul, of_instCoalgebra, tensor_μ_eq_tensorTensorTensorComm]
   rfl
 
 theorem comul_tensorObj_tensorObj_left :
@@ -166,13 +171,15 @@ theorem comul_tensorObj_tensorObj_left :
       (A := ((CoalgebraCat.of R M ⊗ CoalgebraCat.of R N) ⊗ CoalgebraCat.of R P : CoalgebraCat R))
       = Coalgebra.comul (A := (M ⊗[R] N) ⊗[R] P) := by
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [- Mon_.monMonoidalStruct_tensorObj_X, instMonoidalCategoryStruct_tensorHom,
-    ModuleCat.comp_def, ModuleCat.ofHom, ModuleCat.of]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj,
+    instCoalgebraStruct_comul]
+  simp only [Comon_.monoidal_tensorObj_comul, toComonObj_X, ModuleCat.of_coe, of_carrier,
+    of_isAddCommGroup, of_isModule, toComonObj_comul, of_instCoalgebra]
   rw [ofComonObjCoalgebraStruct_comul]
-  dsimp [- Mon_.monMonoidalStruct_tensorObj_X, instMonoidalCategoryStruct_tensorHom,
-    ModuleCat.comp_def, ModuleCat.ofHom, ModuleCat.of]
+  dsimp only [Equivalence.symm_inverse, comonEquivalence_functor, toComon_obj]
   simp only [instMonoidalCategoryStruct_tensorObj, ModuleCat.MonoidalCategory.tensorObj,
-    ModuleCat.coe_of, BraidedCategory.unop_tensor_μ, tensor_μ_eq_tensorTensorTensorComm]
+    ModuleCat.coe_of, Comon_.monoidal_tensorObj_comul, toComonObj_X, of_carrier, of_isAddCommGroup,
+    of_isModule, toComonObj_comul, of_instCoalgebra, tensor_μ_eq_tensorTensorTensorComm]
   rfl
 
 theorem counit_tensorObj :
diff --git a/Mathlib/Algebra/Category/CoalgebraCat/Monoidal.lean b/Mathlib/Algebra/Category/CoalgebraCat/Monoidal.lean
new file mode 100644
index 0000000000000..6eb6644109de4
--- /dev/null
+++ b/Mathlib/Algebra/Category/CoalgebraCat/Monoidal.lean
@@ -0,0 +1,60 @@
+/-
+Copyright (c) 2024 Amelia Livingston. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Amelia Livingston
+-/
+
+import Mathlib.RingTheory.Coalgebra.TensorProduct
+
+/-!
+# The monoidal category structure on `R`-coalgebras
+
+In `Mathlib.RingTheory.Coalgebra.TensorProduct`, given two `R`-coalgebras `M, N`, we define a
+coalgebra instance on `M ⊗[R] N`, as well as the tensor product of two `CoalgHom`s as a
+`CoalgHom`, and the associator and left/right unitors for coalgebras as `CoalgEquiv`s.
+
+In this file, we declare a `MonoidalCategory` instance on the category of coalgebras, with data
+fields given by the definitions in `Mathlib.RingTheory.Coalgebra.TensorProduct`, and Prop
+fields proved by pulling back the `MonoidalCategory` instance on the category of modules,
+using `Monoidal.induced`.
+
+-/
+
+universe v u
+
+namespace CoalgebraCat
+variable (R : Type u) [CommRing R]
+
+open CategoryTheory Coalgebra
+open scoped TensorProduct MonoidalCategory
+
+@[simps]
+noncomputable instance instMonoidalCategoryStruct :
+    MonoidalCategoryStruct.{u} (CoalgebraCat R) where
+  tensorObj X Y := of R (X ⊗[R] Y)
+  whiskerLeft X _ _ f := ofHom (f.1.lTensor X)
+  whiskerRight f X := ofHom (f.1.rTensor X)
+  tensorHom f g := ofHom (Coalgebra.TensorProduct.map f.1 g.1)
+  tensorUnit := CoalgebraCat.of R R
+  associator X Y Z := (Coalgebra.TensorProduct.assoc R X Y Z).toCoalgebraCatIso
+  leftUnitor X := (Coalgebra.TensorProduct.lid R X).toCoalgebraCatIso
+  rightUnitor X := (Coalgebra.TensorProduct.rid R X).toCoalgebraCatIso
+
+/-- The data needed to induce a `MonoidalCategory` structure via
+`CoalgebraCat.instMonoidalCategoryStruct` and the forgetful functor to modules. -/
+@[simps]
+noncomputable def MonoidalCategory.inducingFunctorData :
+    Monoidal.InducingFunctorData (forget₂ (CoalgebraCat R) (ModuleCat R)) where
+  μIso X Y := Iso.refl _
+  whiskerLeft_eq X Y Z f := by ext; rfl
+  whiskerRight_eq X f := by ext; rfl
+  tensorHom_eq f g := by ext; rfl
+  εIso := Iso.refl _
+  associator_eq X Y Z := TensorProduct.ext <| TensorProduct.ext <| by ext; rfl
+  leftUnitor_eq X := TensorProduct.ext <| by ext; rfl
+  rightUnitor_eq X := TensorProduct.ext <| by ext; rfl
+
+noncomputable instance instMonoidalCategory : MonoidalCategory (CoalgebraCat R) :=
+  Monoidal.induced (forget₂ _ (ModuleCat R)) (MonoidalCategory.inducingFunctorData R)
+
+end CoalgebraCat
diff --git a/Mathlib/Algebra/Category/FGModuleCat/Basic.lean b/Mathlib/Algebra/Category/FGModuleCat/Basic.lean
index 01ae8aab6bb4d..50f9e8d2c035a 100644
--- a/Mathlib/Algebra/Category/FGModuleCat/Basic.lean
+++ b/Mathlib/Algebra/Category/FGModuleCat/Basic.lean
@@ -278,8 +278,8 @@ end FGModuleCat
 @[simp] theorem LinearMap.comp_id_fgModuleCat
     {R} [Ring R] {G : FGModuleCat.{u} R} {H : Type u} [AddCommGroup H] [Module R H]
     (f : G →ₗ[R] H) : f.comp (𝟙 G) = f :=
-  Category.id_comp (ModuleCat.ofHom f)
+  Category.id_comp (ModuleCat.asHom f)
 @[simp] theorem LinearMap.id_fgModuleCat_comp
     {R} [Ring R] {G : Type u} [AddCommGroup G] [Module R G] {H : FGModuleCat.{u} R}
     (f : G →ₗ[R] H) : LinearMap.comp (𝟙 H) f = f :=
-  Category.comp_id (ModuleCat.ofHom f)
+  Category.comp_id (ModuleCat.asHom f)
diff --git a/Mathlib/Algebra/Category/FGModuleCat/Limits.lean b/Mathlib/Algebra/Category/FGModuleCat/Limits.lean
index 8faee2b976d4c..3e5af1918343b 100644
--- a/Mathlib/Algebra/Category/FGModuleCat/Limits.lean
+++ b/Mathlib/Algebra/Category/FGModuleCat/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.FGModuleCat.Basic
 import Mathlib.Algebra.Category.ModuleCat.Limits
diff --git a/Mathlib/Algebra/Category/Grp/AB5.lean b/Mathlib/Algebra/Category/Grp/AB5.lean
index b3408f6ec5a77..2edc3d04999e0 100644
--- a/Mathlib/Algebra/Category/Grp/AB5.lean
+++ b/Mathlib/Algebra/Category/Grp/AB5.lean
@@ -9,6 +9,7 @@ import Mathlib.Algebra.Homology.ShortComplex.ExactFunctor
 import Mathlib.CategoryTheory.Abelian.Exact
 import Mathlib.Algebra.Category.Grp.FilteredColimits
 import Mathlib.CategoryTheory.Abelian.FunctorCategory
+import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms
 
 /-!
 # The category of abelian groups satisfies Grothendieck's axiom AB5
@@ -45,3 +46,9 @@ noncomputable instance :
 noncomputable instance :
     PreservesFiniteLimits <| colim (J := J) (C := AddCommGrp.{u}) := by
   apply Functor.preservesFiniteLimitsOfPreservesHomology
+
+instance : HasFilteredColimits (AddCommGrp.{u}) where
+  HasColimitsOfShape := inferInstance
+
+noncomputable instance : AB5 (AddCommGrp.{u}) where
+  preservesFiniteLimits := fun _ => inferInstance
diff --git a/Mathlib/Algebra/Category/Grp/Adjunctions.lean b/Mathlib/Algebra/Category/Grp/Adjunctions.lean
index 0ae1a7d27e135..d3443fcb9cfc5 100644
--- a/Mathlib/Algebra/Category/Grp/Adjunctions.lean
+++ b/Mathlib/Algebra/Category/Grp/Adjunctions.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johannes Hölzl
+Authors: Kim Morrison, Johannes Hölzl
 -/
 import Mathlib.Algebra.Category.Grp.Preadditive
 import Mathlib.GroupTheory.FreeAbelianGroup
@@ -177,8 +177,8 @@ def MonCat.units : MonCat.{u} ⥤ Grp.{u} where
   map_comp _ _ := MonoidHom.ext fun _ => Units.ext rfl
 
 /-- The forgetful-units adjunction between `Grp` and `MonCat`. -/
-def Grp.forget₂MonAdj : forget₂ Grp MonCat ⊣ MonCat.units.{u} where
-  homEquiv X Y :=
+def Grp.forget₂MonAdj : forget₂ Grp MonCat ⊣ MonCat.units.{u} := Adjunction.mk' {
+  homEquiv := fun X Y ↦
     { toFun := fun f => MonoidHom.toHomUnits f
       invFun := fun f => (Units.coeHom Y).comp f
       left_inv := fun f => MonoidHom.ext fun _ => rfl
@@ -188,9 +188,7 @@ def Grp.forget₂MonAdj : forget₂ Grp MonCat ⊣ MonCat.units.{u} where
       naturality := fun X Y f => MonoidHom.ext fun x => Units.ext rfl }
   counit :=
     { app := fun X => Units.coeHom X
-      naturality := by intros; exact MonoidHom.ext fun x => rfl }
-  homEquiv_unit := MonoidHom.ext fun _ => Units.ext rfl
-  homEquiv_counit := MonoidHom.ext fun _ => rfl
+      naturality := by intros; exact MonoidHom.ext fun x => rfl } }
 
 instance : MonCat.units.{u}.IsRightAdjoint :=
   ⟨_, ⟨Grp.forget₂MonAdj⟩⟩
@@ -204,20 +202,19 @@ def CommMonCat.units : CommMonCat.{u} ⥤ CommGrp.{u} where
   map_comp _ _ := MonoidHom.ext fun _ => Units.ext rfl
 
 /-- The forgetful-units adjunction between `CommGrp` and `CommMonCat`. -/
-def CommGrp.forget₂CommMonAdj : forget₂ CommGrp CommMonCat ⊣ CommMonCat.units.{u} where
-  homEquiv X Y :=
-    { toFun := fun f => MonoidHom.toHomUnits f
-      invFun := fun f => (Units.coeHom Y).comp f
-      left_inv := fun f => MonoidHom.ext fun _ => rfl
-      right_inv := fun f => MonoidHom.ext fun _ => Units.ext rfl }
-  unit :=
-    { app := fun X => { (@toUnits X _).toMonoidHom with }
-      naturality := fun X Y f => MonoidHom.ext fun x => Units.ext rfl }
-  counit :=
-    { app := fun X => Units.coeHom X
-      naturality := by intros; exact MonoidHom.ext fun x => rfl }
-  homEquiv_unit := MonoidHom.ext fun _ => Units.ext rfl
-  homEquiv_counit := MonoidHom.ext fun _ => rfl
+def CommGrp.forget₂CommMonAdj : forget₂ CommGrp CommMonCat ⊣ CommMonCat.units.{u} :=
+  Adjunction.mk' {
+    homEquiv := fun X Y ↦
+      { toFun := fun f => MonoidHom.toHomUnits f
+        invFun := fun f => (Units.coeHom Y).comp f
+        left_inv := fun f => MonoidHom.ext fun _ => rfl
+        right_inv := fun f => MonoidHom.ext fun _ => Units.ext rfl }
+    unit :=
+      { app := fun X => { (@toUnits X _).toMonoidHom with }
+        naturality := fun X Y f => MonoidHom.ext fun x => Units.ext rfl }
+    counit :=
+      { app := fun X => Units.coeHom X
+        naturality := by intros; exact MonoidHom.ext fun x => rfl } }
 
 instance : CommMonCat.units.{u}.IsRightAdjoint :=
   ⟨_, ⟨CommGrp.forget₂CommMonAdj⟩⟩
diff --git a/Mathlib/Algebra/Category/Grp/Biproducts.lean b/Mathlib/Algebra/Category/Grp/Biproducts.lean
index 343c5d74d0527..a45ba714db4a0 100644
--- a/Mathlib/Algebra/Category/Grp/Biproducts.lean
+++ b/Mathlib/Algebra/Category/Grp/Biproducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Pi.Lemmas
 import Mathlib.Algebra.Category.Grp.Preadditive
diff --git a/Mathlib/Algebra/Category/Grp/Colimits.lean b/Mathlib/Algebra/Category/Grp/Colimits.lean
index a53daac3791a2..52e9b257663ec 100644
--- a/Mathlib/Algebra/Category/Grp/Colimits.lean
+++ b/Mathlib/Algebra/Category/Grp/Colimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Grp.Preadditive
 import Mathlib.CategoryTheory.Limits.Shapes.Kernels
diff --git a/Mathlib/Algebra/Category/Grp/FilteredColimits.lean b/Mathlib/Algebra/Category/Grp/FilteredColimits.lean
index cd3489812cf51..b79cc51e0f2ed 100644
--- a/Mathlib/Algebra/Category/Grp/FilteredColimits.lean
+++ b/Mathlib/Algebra/Category/Grp/FilteredColimits.lean
@@ -58,7 +58,7 @@ abbrev G.mk : (Σ j, F.obj j) → G.{v, u} F :=
 theorem G.mk_eq (x y : Σ j, F.obj j)
     (h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) :
     G.mk.{v, u} F x = G.mk F y :=
-  Quot.EqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget Grp) x y h)
+  Quot.eqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget Grp) x y h)
 
 /-- The "unlifted" version of taking inverses in the colimit. -/
 @[to_additive "The \"unlifted\" version of negation in the colimit."]
diff --git a/Mathlib/Algebra/Category/Grp/FiniteGrp.lean b/Mathlib/Algebra/Category/Grp/FiniteGrp.lean
new file mode 100644
index 0000000000000..b1f4d7ead61ee
--- /dev/null
+++ b/Mathlib/Algebra/Category/Grp/FiniteGrp.lean
@@ -0,0 +1,79 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang, Nailin Guan, Yuyang Zhao
+-/
+import Mathlib.Algebra.Category.Grp.Basic
+
+/-!
+
+## Main definitions and results
+
+* `FiniteGrp` is the category of finite groups.
+
+-/
+
+universe u v
+
+open CategoryTheory
+
+/-- The category of finite groups. -/
+@[pp_with_univ]
+structure FiniteGrp where
+  /-- A group that is finite -/
+  toGrp : Grp
+  [isFinite : Finite toGrp]
+
+/-- The category of finite additive groups. -/
+@[pp_with_univ]
+structure FiniteAddGrp where
+  /-- An add group that is finite -/
+  toAddGrp : AddGrp
+  [isFinite : Finite toAddGrp]
+
+attribute [to_additive] FiniteGrp
+
+namespace FiniteGrp
+
+@[to_additive]
+instance : CoeSort FiniteGrp.{u} (Type u) where
+  coe G := G.toGrp
+
+@[to_additive]
+instance : Category FiniteGrp := InducedCategory.category FiniteGrp.toGrp
+
+@[to_additive]
+instance : ConcreteCategory FiniteGrp := InducedCategory.concreteCategory FiniteGrp.toGrp
+
+@[to_additive]
+instance (G : FiniteGrp) : Group G := inferInstanceAs <| Group G.toGrp
+
+@[to_additive]
+instance (G : FiniteGrp) : Finite G := G.isFinite
+
+@[to_additive]
+instance (G H : FiniteGrp) : FunLike (G ⟶ H) G H :=
+  inferInstanceAs <| FunLike (G →* H) G H
+
+@[to_additive]
+instance (G H : FiniteGrp) : MonoidHomClass (G ⟶ H) G H :=
+  inferInstanceAs <| MonoidHomClass (G →* H) G H
+
+/-- Construct a term of `FiniteGrp` from a type endowed with the structure of a finite group. -/
+@[to_additive "Construct a term of `FiniteAddGrp` from a type endowed with the structure of a
+finite additive group."]
+def of (G : Type u) [Group G] [Finite G] : FiniteGrp where
+  toGrp := Grp.of G
+  isFinite := ‹_›
+
+/-- The morphism in `FiniteGrp`, induced from a morphism of the category `Grp`. -/
+@[to_additive "The morphism in `FiniteAddGrp`, induced from a morphism of the category `AddGrp`"]
+def ofHom {X Y : Type u} [Group X] [Finite X] [Group Y] [Finite Y] (f : X →* Y) : of X ⟶ of Y :=
+  Grp.ofHom f
+
+@[to_additive]
+lemma ofHom_apply {X Y : Type u} [Group X] [Finite X] [Group Y] [Finite Y] (f : X →* Y) (x : X) :
+    ofHom f x = f x :=
+  rfl
+
+end FiniteGrp
diff --git a/Mathlib/Algebra/Category/Grp/ForgetCorepresentable.lean b/Mathlib/Algebra/Category/Grp/ForgetCorepresentable.lean
index 7f0cd7d6755d1..c7caa5ef7b7ad 100644
--- a/Mathlib/Algebra/Category/Grp/ForgetCorepresentable.lean
+++ b/Mathlib/Algebra/Category/Grp/ForgetCorepresentable.lean
@@ -95,18 +95,18 @@ def AddCommGrp.coyonedaObjIsoForget :
     coyoneda.obj (op (of (ULift.{u} ℤ))) ≅ forget AddCommGrp.{u} :=
   (NatIso.ofComponents (fun M => (AddMonoidHom.fromULiftIntEquiv M.α).toIso))
 
-instance Grp.forget_corepresentable :
-    (forget Grp.{u}).Corepresentable where
-  has_corepresentation := ⟨_, ⟨Grp.coyonedaObjIsoForget⟩⟩
+instance Grp.forget_isCorepresentable :
+    (forget Grp.{u}).IsCorepresentable :=
+  Functor.IsCorepresentable.mk' Grp.coyonedaObjIsoForget
 
-instance CommGrp.forget_corepresentable :
-    (forget CommGrp.{u}).Corepresentable where
-  has_corepresentation := ⟨_, ⟨CommGrp.coyonedaObjIsoForget⟩⟩
+instance CommGrp.forget_isCorepresentable :
+    (forget CommGrp.{u}).IsCorepresentable :=
+  Functor.IsCorepresentable.mk' CommGrp.coyonedaObjIsoForget
 
-instance AddGrp.forget_corepresentable :
-    (forget AddGrp.{u}).Corepresentable where
-  has_corepresentation := ⟨_, ⟨AddGrp.coyonedaObjIsoForget⟩⟩
+instance AddGrp.forget_isCorepresentable :
+    (forget AddGrp.{u}).IsCorepresentable :=
+  Functor.IsCorepresentable.mk' AddGrp.coyonedaObjIsoForget
 
-instance AddCommGrp.forget_corepresentable :
-    (forget AddCommGrp.{u}).Corepresentable where
-  has_corepresentation := ⟨_, ⟨AddCommGrp.coyonedaObjIsoForget⟩⟩
+instance AddCommGrp.forget_isCorepresentable :
+    (forget AddCommGrp.{u}).IsCorepresentable :=
+  Functor.IsCorepresentable.mk' AddCommGrp.coyonedaObjIsoForget
diff --git a/Mathlib/Algebra/Category/Grp/Images.lean b/Mathlib/Algebra/Category/Grp/Images.lean
index 5dd6fd3f51b21..e3f6641b97ec8 100644
--- a/Mathlib/Algebra/Category/Grp/Images.lean
+++ b/Mathlib/Algebra/Category/Grp/Images.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Grp.Abelian
 import Mathlib.CategoryTheory.Limits.Shapes.Images
diff --git a/Mathlib/Algebra/Category/Grp/Kernels.lean b/Mathlib/Algebra/Category/Grp/Kernels.lean
index 8f367b2c6f460..577c21325b183 100644
--- a/Mathlib/Algebra/Category/Grp/Kernels.lean
+++ b/Mathlib/Algebra/Category/Grp/Kernels.lean
@@ -26,7 +26,7 @@ def kernelCone : KernelFork f :=
 /-- The kernel of a group homomorphism is a kernel in the categorical sense. -/
 def kernelIsLimit : IsLimit <| kernelCone f :=
   Fork.IsLimit.mk _
-    (fun s => (by exact Fork.ι s : _ →+ G).codRestrict _ fun c => f.mem_ker.mpr <|
+    (fun s => (by exact Fork.ι s : _ →+ G).codRestrict _ fun c => mem_ker.mpr <|
       by exact DFunLike.congr_fun s.condition c)
     (fun _ => by rfl)
     (fun _ _ h => ext fun x => Subtype.ext_iff_val.mpr <| by exact DFunLike.congr_fun h x)
diff --git a/Mathlib/Algebra/Category/Grp/Limits.lean b/Mathlib/Algebra/Category/Grp/Limits.lean
index f605284f69e1f..0763ecc4606e9 100644
--- a/Mathlib/Algebra/Category/Grp/Limits.lean
+++ b/Mathlib/Algebra/Category/Grp/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.MonCat.Limits
 import Mathlib.Algebra.Category.Grp.ForgetCorepresentable
@@ -126,7 +126,7 @@ instance hasLimit : HasLimit F :=
 end
 
 /-- A functor `F : J ⥤ Grp.{u}` has a limit iff `(F ⋙ forget Grp).sections` is
-`u`-small.  -/
+`u`-small. -/
 @[to_additive "A functor `F : J ⥤ AddGrp.{u}` has a limit iff
 `(F ⋙ forget AddGrp).sections` is `u`-small."]
 lemma hasLimit_iff_small_sections :
@@ -281,7 +281,7 @@ instance hasLimit : HasLimit F :=
 end
 
 /-- A functor `F : J ⥤ CommGrp.{u}` has a limit iff `(F ⋙ forget CommGrp).sections` is
-`u`-small.  -/
+`u`-small. -/
 @[to_additive "A functor `F : J ⥤ AddCommGrp.{u}` has a limit iff
 `(F ⋙ forget AddCommGrp).sections` is `u`-small."]
 lemma hasLimit_iff_small_sections :
diff --git a/Mathlib/Algebra/Category/Grp/ZModuleEquivalence.lean b/Mathlib/Algebra/Category/Grp/ZModuleEquivalence.lean
index fd9a529d37985..242349db0fa0c 100644
--- a/Mathlib/Algebra/Category/Grp/ZModuleEquivalence.lean
+++ b/Mathlib/Algebra/Category/Grp/ZModuleEquivalence.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 
diff --git a/Mathlib/Algebra/Category/Grp/Zero.lean b/Mathlib/Algebra/Category/Grp/Zero.lean
index 7ecca001e70ea..188eb7bd2216a 100644
--- a/Mathlib/Algebra/Category/Grp/Zero.lean
+++ b/Mathlib/Algebra/Category/Grp/Zero.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Grp.Basic
 import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
diff --git a/Mathlib/Algebra/Category/HopfAlgebraCat/Basic.lean b/Mathlib/Algebra/Category/HopfAlgebraCat/Basic.lean
index b690a7f5f0bb5..413f5f5b36aba 100644
--- a/Mathlib/Algebra/Category/HopfAlgebraCat/Basic.lean
+++ b/Mathlib/Algebra/Category/HopfAlgebraCat/Basic.lean
@@ -58,7 +58,7 @@ lemma of_counit {X : Type v} [Ring X] [HopfAlgebra R X] :
 /-- A type alias for `BialgHom` to avoid confusion between the categorical and
 algebraic spellings of composition. -/
 @[ext]
-structure Hom (V W : HopfAlgebraCat.{v} R) :=
+structure Hom (V W : HopfAlgebraCat.{v} R) where
   /-- The underlying `BialgHom`. -/
   toBialgHom : V →ₐc[R] W
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean b/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
index 9035d34b23389..5fbd14b8fb7c0 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin
+Authors: Kim Morrison, Johan Commelin
 -/
 import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
 import Mathlib.CategoryTheory.Monoidal.Functorial
diff --git a/Mathlib/Algebra/Category/ModuleCat/Algebra.lean b/Mathlib/Algebra/Category/ModuleCat/Algebra.lean
index c350e298ae22e..96c3070180a88 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Algebra.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.CategoryTheory.Linear.Basic
diff --git a/Mathlib/Algebra/Category/ModuleCat/Basic.lean b/Mathlib/Algebra/Category/ModuleCat/Basic.lean
index 48187e0d55415..ec69a10b56fc3 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Basic.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Basic.lean
@@ -151,18 +151,6 @@ theorem forget₂_map (X Y : ModuleCat R) (f : X ⟶ Y) :
     (forget₂ (ModuleCat R) AddCommGrp).map f = LinearMap.toAddMonoidHom f :=
   rfl
 
--- Porting note (#11215): TODO: `ofHom` and `asHom` are duplicates!
-
-/-- Typecheck a `LinearMap` as a morphism in `Module R`. -/
-def ofHom {R : Type u} [Ring R] {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y]
-    [Module R Y] (f : X →ₗ[R] Y) : of R X ⟶ of R Y :=
-  f
-
-@[simp 1100]
-theorem ofHom_apply {R : Type u} [Ring R] {X Y : Type v} [AddCommGroup X] [Module R X]
-    [AddCommGroup Y] [Module R Y] (f : X →ₗ[R] Y) (x : X) : ofHom f x = f x :=
-  rfl
-
 instance : Inhabited (ModuleCat R) :=
   ⟨of R PUnit⟩
 
@@ -218,14 +206,25 @@ end ModuleCat
 variable {R}
 variable {X₁ X₂ : Type v}
 
+open ModuleCat
+
 /-- Reinterpreting a linear map in the category of `R`-modules. -/
 def ModuleCat.asHom [AddCommGroup X₁] [Module R X₁] [AddCommGroup X₂] [Module R X₂] :
     (X₁ →ₗ[R] X₂) → (ModuleCat.of R X₁ ⟶ ModuleCat.of R X₂) :=
   id
 
+@[deprecated (since := "2024-10-06")] alias ModuleCat.ofHom := ModuleCat.asHom
+
 /-- Reinterpreting a linear map in the category of `R`-modules -/
 scoped[ModuleCat] notation "↟" f:1024 => ModuleCat.asHom f
 
+@[simp 1100]
+theorem ModuleCat.asHom_apply {R : Type u} [Ring R] {X Y : Type v} [AddCommGroup X] [Module R X]
+    [AddCommGroup Y] [Module R Y] (f : X →ₗ[R] Y) (x : X) : (↟ f) x = f x :=
+  rfl
+
+@[deprecated (since := "2024-10-06")] alias ModuleCat.ofHom_apply := ModuleCat.asHom_apply
+
 /-- Reinterpreting a linear map in the category of `R`-modules. -/
 def ModuleCat.asHomRight [AddCommGroup X₁] [Module R X₁] {X₂ : ModuleCat.{v} R} :
     (X₁ →ₗ[R] X₂) → (ModuleCat.of R X₁ ⟶ X₂) :=
@@ -441,8 +440,9 @@ end ModuleCat
 @[simp] theorem LinearMap.comp_id_moduleCat
     {R} [Ring R] {G : ModuleCat.{u} R} {H : Type u} [AddCommGroup H] [Module R H] (f : G →ₗ[R] H) :
     f.comp (𝟙 G) = f :=
-  Category.id_comp (ModuleCat.ofHom f)
+  Category.id_comp (ModuleCat.asHom f)
 @[simp] theorem LinearMap.id_moduleCat_comp
     {R} [Ring R] {G : Type u} [AddCommGroup G] [Module R G] {H : ModuleCat.{u} R} (f : G →ₗ[R] H) :
     LinearMap.comp (𝟙 H) f = f :=
-  Category.comp_id (ModuleCat.ofHom f)
+  Category.comp_id (ModuleCat.asHom f)
+
diff --git a/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean b/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
index 276eafcbb77e3..4bb37bb584fd6 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Pi.Lemmas
 import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
@@ -152,8 +152,8 @@ of modules. -/
 noncomputable def lequivProdOfRightSplitExact {f : B →ₗ[R] M} (hj : Function.Injective j)
     (exac : LinearMap.range j = LinearMap.ker g) (h : g.comp f = LinearMap.id) : (A × B) ≃ₗ[R] M :=
   ((ShortComplex.Splitting.ofExactOfSection _
-    (ShortComplex.Exact.moduleCat_of_range_eq_ker (ModuleCat.ofHom j)
-    (ModuleCat.ofHom g) exac) (asHom f) h
+    (ShortComplex.Exact.moduleCat_of_range_eq_ker (ModuleCat.asHom j)
+    (ModuleCat.asHom g) exac) (asHom f) h
     (by simpa only [ModuleCat.mono_iff_injective])).isoBinaryBiproduct ≪≫
     biprodIsoProd _ _ ).symm.toLinearEquiv
 
@@ -162,8 +162,8 @@ of modules. -/
 noncomputable def lequivProdOfLeftSplitExact {f : M →ₗ[R] A} (hg : Function.Surjective g)
     (exac : LinearMap.range j = LinearMap.ker g) (h : f.comp j = LinearMap.id) : (A × B) ≃ₗ[R] M :=
   ((ShortComplex.Splitting.ofExactOfRetraction _
-    (ShortComplex.Exact.moduleCat_of_range_eq_ker (ModuleCat.ofHom j)
-    (ModuleCat.ofHom g) exac) (ModuleCat.ofHom f) h
+    (ShortComplex.Exact.moduleCat_of_range_eq_ker (ModuleCat.asHom j)
+    (ModuleCat.asHom g) exac) (ModuleCat.asHom f) h
     (by simpa only [ModuleCat.epi_iff_surjective] using hg)).isoBinaryBiproduct ≪≫
     biprodIsoProd _ _).symm.toLinearEquiv
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean b/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
index 0417f8ed023f7..42e62db55eedd 100644
--- a/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
@@ -155,11 +155,11 @@ def restrictScalarsId'App (hf : f = RingHom.id R) (M : ModuleCat R) :
 
 variable (hf : f = RingHom.id R)
 
-lemma restrictScalarsId'App_hom_apply (M : ModuleCat R) (x : M) :
+@[simp] lemma restrictScalarsId'App_hom_apply (M : ModuleCat R) (x : M) :
     (restrictScalarsId'App f hf M).hom x = x :=
   rfl
 
-lemma restrictScalarsId'App_inv_apply (M : ModuleCat R) (x : M) :
+@[simp] lemma restrictScalarsId'App_inv_apply (M : ModuleCat R) (x : M) :
     (restrictScalarsId'App f hf M).inv x = x :=
   rfl
 
@@ -202,11 +202,11 @@ def restrictScalarsComp'App (hgf : gf = g.comp f) (M : ModuleCat R₃) :
 
 variable (hgf : gf = g.comp f)
 
-lemma restrictScalarsComp'App_hom_apply (M : ModuleCat R₃) (x : M) :
+@[simp] lemma restrictScalarsComp'App_hom_apply (M : ModuleCat R₃) (x : M) :
     (restrictScalarsComp'App f g gf hgf M).hom x = x :=
   rfl
 
-lemma restrictScalarsComp'App_inv_apply (M : ModuleCat R₃) (x : M) :
+@[simp] lemma restrictScalarsComp'App_inv_apply (M : ModuleCat R₃) (x : M) :
     (restrictScalarsComp'App f g gf hgf M).inv x = x :=
   rfl
 
@@ -564,30 +564,31 @@ end RestrictionCoextensionAdj
 /-- Restriction of scalars is left adjoint to coextension of scalars. -/
 -- @[simps] Porting note: not in normal form and not used
 def restrictCoextendScalarsAdj {R : Type u₁} {S : Type u₂} [Ring R] [Ring S] (f : R →+* S) :
-    restrictScalars.{max v u₂,u₁,u₂} f ⊣ coextendScalars f where
-  homEquiv X Y :=
-    { toFun := RestrictionCoextensionAdj.HomEquiv.fromRestriction.{u₁,u₂,v} f
-      invFun := RestrictionCoextensionAdj.HomEquiv.toRestriction.{u₁,u₂,v} f
-      left_inv := fun g => LinearMap.ext fun x : X => by
-        -- Porting note (#10745): once just simp
-        rw [RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe,
-          LinearMap.coe_toAddHom, RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply,
-          one_smul]
-      right_inv := fun g => LinearMap.ext fun x => LinearMap.ext fun s : S => by
-        -- Porting note (#10745): once just simp
-        rw [RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply,
-          RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe,
-          LinearMap.coe_toAddHom, LinearMap.map_smulₛₗ, RingHom.id_apply,
-          CoextendScalars.smul_apply', one_mul] }
-  unit := RestrictionCoextensionAdj.unit'.{u₁,u₂,v} f
-  counit := RestrictionCoextensionAdj.counit'.{u₁,u₂,v} f
-  homEquiv_unit := LinearMap.ext fun y => rfl
-  homEquiv_counit := fun {X Y g} => LinearMap.ext <| by
-    -- Porting note (#10745): previously simp [RestrictionCoextensionAdj.counit']
-    intro x; dsimp
-    rw [coe_comp, Function.comp]
-    change _ = (((restrictScalars f).map g) x).toFun (1 : S)
-    rw [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, restrictScalars.map_apply]
+    restrictScalars.{max v u₂,u₁,u₂} f ⊣ coextendScalars f :=
+  Adjunction.mk' {
+    homEquiv := fun X Y ↦
+      { toFun := RestrictionCoextensionAdj.HomEquiv.fromRestriction.{u₁,u₂,v} f
+        invFun := RestrictionCoextensionAdj.HomEquiv.toRestriction.{u₁,u₂,v} f
+        left_inv := fun g => LinearMap.ext fun x : X => by
+          -- Porting note (#10745): once just simp
+          rw [RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe,
+            LinearMap.coe_toAddHom, RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply,
+            one_smul]
+        right_inv := fun g => LinearMap.ext fun x => LinearMap.ext fun s : S => by
+          -- Porting note (#10745): once just simp
+          rw [RestrictionCoextensionAdj.HomEquiv.fromRestriction_apply_apply,
+            RestrictionCoextensionAdj.HomEquiv.toRestriction_apply, AddHom.toFun_eq_coe,
+            LinearMap.coe_toAddHom, LinearMap.map_smulₛₗ, RingHom.id_apply,
+            CoextendScalars.smul_apply', one_mul] }
+    unit := RestrictionCoextensionAdj.unit'.{u₁,u₂,v} f
+    counit := RestrictionCoextensionAdj.counit'.{u₁,u₂,v} f
+    homEquiv_unit := LinearMap.ext fun y => rfl
+    homEquiv_counit := fun {X Y g} => LinearMap.ext <| by
+      -- Porting note (#10745): previously simp [RestrictionCoextensionAdj.counit']
+      intro x; dsimp
+      rw [coe_comp, Function.comp]
+      change _ = (((restrictScalars f).map g) x).toFun (1 : S)
+      rw [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, restrictScalars.map_apply] }
 
 instance {R : Type u₁} {S : Type u₂} [Ring R] [Ring S] (f : R →+* S) :
     (restrictScalars.{max u₂ w} f).IsLeftAdjoint  :=
@@ -797,31 +798,32 @@ scalars by `f` are adjoint to each other.
 -/
 -- @[simps] -- Porting note: removed not in normal form and not used
 def extendRestrictScalarsAdj {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] (f : R →+* S) :
-    extendScalars.{u₁,u₂,max v u₂} f ⊣ restrictScalars.{max v u₂,u₁,u₂} f where
-  homEquiv _ _ := ExtendRestrictScalarsAdj.homEquiv.{v,u₁,u₂} f
-  unit := ExtendRestrictScalarsAdj.unit.{v,u₁,u₂} f
-  counit := ExtendRestrictScalarsAdj.counit.{v,u₁,u₂} f
-  homEquiv_unit {X Y g} := LinearMap.ext fun x => by
-    dsimp
-    rw [ModuleCat.coe_comp, Function.comp_apply, restrictScalars.map_apply]
-    rfl
-  homEquiv_counit {X Y g} := LinearMap.ext fun x => by
-      -- Porting note: once again reminding Lean of the instances
-      letI m1 : Module R S := Module.compHom S f
-      letI m2 : Module R Y := Module.compHom Y f
-      induction x using TensorProduct.induction_on with
-      | zero => rw [map_zero, map_zero]
-      | tmul =>
-        rw [ExtendRestrictScalarsAdj.homEquiv_symm_apply]
-        -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-        erw [ModuleCat.coe_comp]
-        rw [Function.comp_apply, ExtendRestrictScalarsAdj.counit_app]
-        -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-        erw [ExtendRestrictScalarsAdj.Counit.map_apply]
-        set_option tactic.skipAssignedInstances false in dsimp
-        rw [TensorProduct.lift.tmul]
-        rfl
-      | add => rw [map_add, map_add]; congr 1
+    extendScalars.{u₁,u₂,max v u₂} f ⊣ restrictScalars.{max v u₂,u₁,u₂} f :=
+  Adjunction.mk' {
+    homEquiv := fun _ _ ↦ ExtendRestrictScalarsAdj.homEquiv.{v,u₁,u₂} f
+    unit := ExtendRestrictScalarsAdj.unit.{v,u₁,u₂} f
+    counit := ExtendRestrictScalarsAdj.counit.{v,u₁,u₂} f
+    homEquiv_unit := fun {X Y g} ↦ LinearMap.ext fun x => by
+      dsimp
+      rw [ModuleCat.coe_comp, Function.comp_apply, restrictScalars.map_apply]
+      rfl
+    homEquiv_counit := fun {X Y g} ↦ LinearMap.ext fun x => by
+        -- Porting note: once again reminding Lean of the instances
+        letI m1 : Module R S := Module.compHom S f
+        letI m2 : Module R Y := Module.compHom Y f
+        induction x using TensorProduct.induction_on with
+        | zero => rw [map_zero, map_zero]
+        | tmul =>
+          rw [ExtendRestrictScalarsAdj.homEquiv_symm_apply]
+          -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+          erw [ModuleCat.coe_comp]
+          rw [Function.comp_apply, ExtendRestrictScalarsAdj.counit_app]
+          -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+          erw [ExtendRestrictScalarsAdj.Counit.map_apply]
+          set_option tactic.skipAssignedInstances false in dsimp
+          rw [TensorProduct.lift.tmul]
+          rfl
+        | add => rw [map_add, map_add]; congr 1 }
 
 instance {R : Type u₁} {S : Type u₂} [CommRing R] [CommRing S] (f : R →+* S) :
     (extendScalars.{u₁, u₂, max u₂ w} f).IsLeftAdjoint :=
diff --git a/Mathlib/Algebra/Category/ModuleCat/Colimits.lean b/Mathlib/Algebra/Category/ModuleCat/Colimits.lean
index ae030fd209955..dd77562189b3e 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Colimits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Colimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
diff --git a/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
index b826d73a22b10..eef6ac3891d20 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
@@ -78,7 +78,7 @@ variable (d : M.Derivation φ)
 @[simps! d_apply]
 def postcomp (f : M ⟶ N) : N.Derivation φ where
   d := (f.app _).toAddMonoidHom.comp d.d
-  d_map _ _ := by simp [naturality_apply]
+  d_map {X Y} g x := by simpa using naturality_apply f g (d.d x)
   d_app {X} a := by
     dsimp
     erw [d_app, map_zero]
@@ -175,50 +175,44 @@ end Derivation'
 
 namespace DifferentialsConstruction
 
-/-- Auxiliary definition for `relativeDifferentials'`. -/
-noncomputable def relativeDifferentials'BundledCore :
-    BundledCorePresheafOfModules.{u} (R ⋙ forget₂ _ _) where
-  obj X := CommRingCat.KaehlerDifferential (φ'.app X)
-  map f := CommRingCat.KaehlerDifferential.map (φ'.naturality f)
-
 /-- The presheaf of relative differentials of a morphism of presheaves of
 commutative rings. -/
+@[simps (config := .lemmasOnly)]
 noncomputable def relativeDifferentials' :
-    PresheafOfModules.{u} (R ⋙ forget₂ _ _) :=
-  (relativeDifferentials'BundledCore φ').toPresheafOfModules
-
-@[simp]
-lemma relativeDifferentials'_obj (X : Dᵒᵖ) :
-    (relativeDifferentials' φ').obj X =
-      CommRingCat.KaehlerDifferential (φ'.app X) := rfl
+    PresheafOfModules.{u} (R ⋙ forget₂ _ _) where
+  obj X := CommRingCat.KaehlerDifferential (φ'.app X)
+  map f := CommRingCat.KaehlerDifferential.map (φ'.naturality f)
+  map_id _ := by ext; simp; rfl
+  map_comp _ _ := by ext; simp; rfl
 
--- Note: this cannot be a simp lemma because `dsimp` would
--- simplify the composition of functors `R ⋙ forget₂ _ _`
-lemma relativeDifferentials'_map_apply {X Y : Dᵒᵖ} (f : X ⟶ Y)
-    (x : CommRingCat.KaehlerDifferential (φ'.app X)) :
-    (relativeDifferentials' φ').map f x =
-        CommRingCat.KaehlerDifferential.map (φ'.naturality f) x := rfl
+attribute [simp] relativeDifferentials'_obj
 
-lemma relativeDifferentials'_map_d {X Y : Dᵒᵖ} (f : X ⟶ Y)
-    (x : R.obj X) :
-    (relativeDifferentials' φ').map f (CommRingCat.KaehlerDifferential.d x) =
-      CommRingCat.KaehlerDifferential.d (R.map f x) := by
-  rw [relativeDifferentials'_map_apply, CommRingCat.KaehlerDifferential.map_d]
+@[simp]
+lemma relativeDifferentials'_map_d {X Y : Dᵒᵖ} (f : X ⟶ Y) (x : R.obj X) :
+    DFunLike.coe (α := CommRingCat.KaehlerDifferential (φ'.app X))
+      (β := fun _ ↦ CommRingCat.KaehlerDifferential (φ'.app Y))
+      ((relativeDifferentials' φ').map f) (CommRingCat.KaehlerDifferential.d x) =
+        CommRingCat.KaehlerDifferential.d (R.map f x) :=
+  CommRingCat.KaehlerDifferential.map_d (φ'.naturality f) _
 
 /-- The universal derivation. -/
 noncomputable def derivation' : (relativeDifferentials' φ').Derivation' φ' :=
-  Derivation'.mk (fun X ↦ CommRingCat.KaehlerDifferential.D (φ'.app X)) (fun X Y f x ↦ by
-    rw [relativeDifferentials'_map_apply, CommRingCat.KaehlerDifferential.map_d])
+  Derivation'.mk (fun X ↦ CommRingCat.KaehlerDifferential.D (φ'.app X))
+    (fun _ _ f x ↦ (relativeDifferentials'_map_d φ' f x).symm)
 
 /-- The derivation `Derivation' φ'` is universal. -/
 noncomputable def isUniversal' : (derivation' φ').Universal :=
   Derivation'.Universal.mk
-    (fun {M'} d' ↦ Hom.mk'' (fun X ↦ (d'.app X).desc) (fun X Y f ↦
-      CommRingCat.KaehlerDifferential.ext (fun b ↦ by
-      dsimp [ModuleCat.ofHom]
-      erw [restrictionApp_apply, restrictionApp_apply]
-      simp only [relativeDifferentials'_map_d, ModuleCat.Derivation.desc_d,
-        d'.app_apply, d'.d_map])))
+    (fun {M'} d' ↦
+      { app := fun X ↦ (d'.app X).desc
+        naturality := fun {X Y} f ↦ CommRingCat.KaehlerDifferential.ext (fun b ↦ by
+          dsimp
+          rw [ModuleCat.Derivation.desc_d, Derivation'.app_apply]
+          erw [relativeDifferentials'_map_d φ' f]
+          rw [ModuleCat.Derivation.desc_d]
+          dsimp
+          rw [Derivation.d_map]
+          dsimp) })
     (fun {M'} d' ↦ by
       ext X b
       apply ModuleCat.Derivation.desc_d)
diff --git a/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean b/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
index d6f7847ecf6bc..ce1d6cca433f8 100644
--- a/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.LinearAlgebra.Quotient
 import Mathlib.Algebra.Category.ModuleCat.Basic
diff --git a/Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean b/Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean
index 2d0de3a1c067e..16abcd3247283 100644
--- a/Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean
@@ -51,7 +51,7 @@ abbrev M.mk : (Σ j, F.obj j) → M F :=
 
 theorem M.mk_eq (x y : Σ j, F.obj j)
     (h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) : M.mk F x = M.mk F y :=
-  Quot.EqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget (ModuleCat R)) x y h)
+  Quot.eqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget (ModuleCat R)) x y h)
 
 /-- The "unlifted" version of scalar multiplication in the colimit. -/
 def colimitSMulAux (r : R) (x : Σ j, F.obj j) : M F :=
diff --git a/Mathlib/Algebra/Category/ModuleCat/Free.lean b/Mathlib/Algebra/Category/ModuleCat/Free.lean
index 7c67bb3e86b4f..9f822325b9425 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Free.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Free.lean
@@ -14,19 +14,19 @@ This file proves results about linear independence and span in exact sequences o
 ## Main theorems
 
 * `linearIndependent_shortExact`: Given a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of
-  `R`-modules and linearly independent families `v : ι → X₁` and `w : ι' → X₃`, we get a linearly
+  `R`-modules and linearly independent families `v : ι → X₁` and `w : ι' → X₃`, we get a linearly
   independent family `ι ⊕ ι' → X₂`
 * `span_rightExact`: Given an exact sequence `X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of `R`-modules and spanning
-  families `v : ι → X₁` and `w : ι' → X₃`, we get a spanning family `ι ⊕ ι' → X₂`
+  families `v : ι → X₁` and `w : ι' → X₃`, we get a spanning family `ι ⊕ ι' → X₂`
 * Using `linearIndependent_shortExact` and `span_rightExact`, we prove `free_shortExact`: In a
-  short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` where `X₁` and `X₃` are free, `X₂` is free as well.
+  short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` where `X₁` and `X₃` are free, `X₂` is free as well.
 
 ## Tags
 linear algebra, module, free
 
 -/
 
-open CategoryTheory
+open CategoryTheory Module
 
 namespace ModuleCat
 
@@ -61,8 +61,8 @@ include hv hm in
          v|     u|     w|
           ι  → ι ⊕ ι' ← ι'
 ```
-where the top row is an exact sequence of modules and the maps on the bottom are `Sum.inl` and
-`Sum.inr`. If `u` is injective and `v` and `w` are linearly independent, then `u` is linearly
+where the top row is an exact sequence of modules and the maps on the bottom are `Sum.inl` and
+`Sum.inr`. If `u` is injective and `v` and `w` are linearly independent, then `u` is linearly
 independent. -/
 theorem linearIndependent_leftExact : LinearIndependent R u := by
   rw [linearIndependent_sum]
@@ -76,7 +76,7 @@ end
 
 include hS' hv in
 /-- Given a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of `R`-modules and linearly independent
-    families `v : ι → N` and `w : ι' → P`, we get a linearly independent family `ι ⊕ ι' → M` -/
+    families `v : ι → N` and `w : ι' → P`, we get a linearly independent family `ι ⊕ ι' → M` -/
 theorem linearIndependent_shortExact {w : ι' → S.X₃} (hw : LinearIndependent R w) :
     LinearIndependent R (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w)) := by
   apply linearIndependent_leftExact hS'.exact hv _ hS'.mono_f rfl
@@ -98,8 +98,8 @@ include hS in
 v|     u|     w|
  ι  → ι ⊕ ι' ← ι'
 ```
-where the top row is an exact sequence of modules and the maps on the bottom are `Sum.inl` and
-`Sum.inr`. If `v` spans `X₁` and `w` spans `X₃`, then `u` spans `X₂`. -/
+where the top row is an exact sequence of modules and the maps on the bottom are `Sum.inl` and
+`Sum.inr`. If `v` spans `X₁` and `w` spans `X₃`, then `u` spans `X₂`. -/
 theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl = S.f ∘ v)
     (hv : ⊤ ≤ span R (range v))
     (hw : ⊤ ≤ span R (range (S.g ∘ u ∘ Sum.inr))) :
@@ -135,7 +135,7 @@ theorem span_exact {β : Type*} {u : ι ⊕ β → S.X₂} (huv : u ∘ Sum.inl
 
 include hS in
 /-- Given an exact sequence `X₁ ⟶ X₂ ⟶ X₃ ⟶ 0` of `R`-modules and spanning
-    families `v : ι → X₁` and `w : ι' → X₃`, we get a spanning family `ι ⊕ ι' → X₂` -/
+    families `v : ι → X₁` and `w : ι' → X₃`, we get a spanning family `ι ⊕ ι' → X₂` -/
 theorem span_rightExact {w : ι' → S.X₃} (hv : ⊤ ≤ span R (range v))
     (hw : ⊤ ≤ span R (range w)) (hE : Epi S.g) :
     ⊤ ≤ span R (range (Sum.elim (S.f ∘ v) (S.g.toFun.invFun ∘ w))) := by
@@ -144,13 +144,13 @@ theorem span_rightExact {w : ι' → S.X₃} (hv : ⊤ ≤ span R (range v))
   · convert hw
     simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Sum.elim_comp_inr]
     rw [ModuleCat.epi_iff_surjective] at hE
-    rw [← Function.comp.assoc, Function.RightInverse.comp_eq_id (Function.rightInverse_invFun hE),
+    rw [← Function.comp_assoc, Function.RightInverse.comp_eq_id (Function.rightInverse_invFun hE),
       Function.id_comp]
 
 end Span
 
-/-- In a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`, given bases for `X₁` and `X₃`
-indexed by `ι` and `ι'` respectively, we get a basis for `X₂` indexed by `ι ⊕ ι'`. -/
+/-- In a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`, given bases for `X₁` and `X₃`
+indexed by `ι` and `ι'` respectively, we get a basis for `X₂` indexed by `ι ⊕ ι'`. -/
 noncomputable
 def Basis.ofShortExact
     (bN : Basis ι R S.X₁) (bP : Basis ι' R S.X₃) : Basis (ι ⊕ ι') R S.X₂ :=
@@ -159,7 +159,7 @@ def Basis.ofShortExact
 
 include hS'
 
-/-- In a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`, if `X₁` and `X₃` are free,
+/-- In a short exact sequence `0 ⟶ X₁ ⟶ X₂ ⟶ X₃ ⟶ 0`, if `X₁` and `X₃` are free,
 then `X₂` is free. -/
 theorem free_shortExact [Module.Free R S.X₁] [Module.Free R S.X₃] :
     Module.Free R S.X₂ :=
@@ -177,11 +177,11 @@ theorem free_shortExact_rank_add [Module.Free R S.X₁] [Module.Free R S.X₃]
 
 theorem free_shortExact_finrank_add {n p : ℕ} [Module.Free R S.X₁] [Module.Free R S.X₃]
     [Module.Finite R S.X₁] [Module.Finite R S.X₃]
-    (hN : FiniteDimensional.finrank R S.X₁ = n)
-    (hP : FiniteDimensional.finrank R S.X₃ = p)
+    (hN : Module.finrank R S.X₁ = n)
+    (hP : Module.finrank R S.X₃ = p)
     [StrongRankCondition R] :
-    FiniteDimensional.finrank R S.X₂ = n + p := by
-  apply FiniteDimensional.finrank_eq_of_rank_eq
+    finrank R S.X₂ = n + p := by
+  apply finrank_eq_of_rank_eq
   rw [free_shortExact_rank_add hS', ← hN, ← hP]
   simp only [Nat.cast_add, finrank_eq_rank]
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Images.lean b/Mathlib/Algebra/Category/ModuleCat/Images.lean
index 78b8b35d12247..cf1d2081b789e 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Images.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Images.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Abelian
 import Mathlib.CategoryTheory.Limits.Shapes.Images
@@ -97,12 +97,12 @@ noncomputable def imageIsoRange {G H : ModuleCat.{v} R} (f : G ⟶ H) :
 
 @[simp, reassoc, elementwise]
 theorem imageIsoRange_inv_image_ι {G H : ModuleCat.{v} R} (f : G ⟶ H) :
-    (imageIsoRange f).inv ≫ Limits.image.ι f = ModuleCat.ofHom f.range.subtype :=
+    (imageIsoRange f).inv ≫ Limits.image.ι f = ModuleCat.asHom f.range.subtype :=
   IsImage.isoExt_inv_m _ _
 
 @[simp, reassoc, elementwise]
 theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
-    (imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by
+    (imageIsoRange f).hom ≫ ModuleCat.asHom f.range.subtype = Limits.image.ι f := by
   erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
 
 end ModuleCat
diff --git a/Mathlib/Algebra/Category/ModuleCat/Limits.lean b/Mathlib/Algebra/Category/ModuleCat/Limits.lean
index 27b1d3ca656ff..5ad9e49af4722 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Limits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.Algebra.Category.Grp.Limits
diff --git a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
index 666c67174d69b..cad418c48ab2a 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
+Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.LinearAlgebra.TensorProduct.Basic
diff --git a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean
index e741ce18b6323..55bbfb7b8390c 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Closed.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
+Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
 -/
 import Mathlib.CategoryTheory.Closed.Monoidal
 import Mathlib.CategoryTheory.Linear.Yoneda
@@ -77,8 +77,9 @@ should give a map `M ⊗ Hom(M, N) ⟶ N`, so we flip the order of the arguments
 `Hom(M, N) ⟶ (M ⟶ N)` and uncurry the resulting map `M ⟶ Hom(M, N) ⟶ N.` -/
 theorem ihom_ev_app (M N : ModuleCat.{u} R) :
     (ihom.ev M).app N = TensorProduct.uncurry _ _ _ _ LinearMap.id.flip := by
+  rw [← MonoidalClosed.uncurry_id_eq_ev]
   apply TensorProduct.ext'
-  apply ModuleCat.monoidalClosed_uncurry
+  apply monoidalClosed_uncurry
 
 /-- Describes the unit of the adjunction `M ⊗ - ⊣ Hom(M, -)`. Given an `R`-module `N` this should
 define a map `N ⟶ Hom(M, M ⊗ N)`, which is given by flipping the arguments in the natural
diff --git a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
index 673a3f1d577d1..75e4669e16efc 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
+Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
@@ -88,14 +88,14 @@ theorem braiding_inv_apply {M N : ModuleCat.{u} R} (m : M) (n : N) :
   rfl
 
 theorem tensor_μ_eq_tensorTensorTensorComm {A B C D : ModuleCat R} :
-    tensor_μ _ (A, B) (C, D) = (TensorProduct.tensorTensorTensorComm R A B C D).toLinearMap :=
+    tensor_μ A B C D = (TensorProduct.tensorTensorTensorComm R A B C D).toLinearMap :=
   TensorProduct.ext <| TensorProduct.ext <| LinearMap.ext₂ fun _ _ =>
     TensorProduct.ext <| LinearMap.ext₂ fun _ _ => rfl
 
 @[simp]
 theorem tensor_μ_apply
     {A B C D : ModuleCat R} (x : A) (y : B) (z : C) (w : D) :
-    tensor_μ _ (A, B) (C, D) ((x ⊗ₜ y) ⊗ₜ (z ⊗ₜ w)) = (x ⊗ₜ z) ⊗ₜ (y ⊗ₜ w) := rfl
+    tensor_μ A B C D ((x ⊗ₜ y) ⊗ₜ (z ⊗ₜ w)) = (x ⊗ₜ z) ⊗ₜ (y ⊗ₜ w) := rfl
 
 end MonoidalCategory
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
index 5a9b76b43035b..caf2f760e93ea 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
 import Mathlib.Algebra.Category.Ring.Basic
@@ -9,22 +9,18 @@ import Mathlib.Algebra.Category.Ring.Basic
 /-!
 # Presheaves of modules over a presheaf of rings.
 
-We give a hands-on description of a presheaf of modules over a fixed presheaf of rings `R`,
-as a presheaf of abelian groups with additional data.
+Given a presheaf of rings `R : Cᵒᵖ ⥤ RingCat`, we define the category `PresheafOfModules R`.
+An object `M : PresheafOfModules R` consists of a family of modules
+`M.obj X : ModuleCat (R.obj X)` for all `X : Cᵒᵖ`, together with the data, for all `f : X ⟶ Y`,
+of a functorial linear map `M.map f` from `M.obj X` to the restriction
+of scalars of `M.obj Y` via `R.map f`.
 
-We also provide two alternative constructors :
-* When `M : CorePresheafOfModules R` consists of a family of unbundled modules over `R.obj X`
-for all `X`, the corresponding presheaf of modules is `M.toPresheafOfModules`.
-* When `M : BundledCorePresheafOfModules R` consists of a family of objects in
-`ModuleCat (R.obj X)` for all `X`, the corresponding presheaf of modules
-is `M.toPresheafOfModules`.
 
 ## Future work
 
 * Compare this to the definition as a presheaf of pairs `(R, M)` with specified first part.
 * Compare this to the definition as a module object of the presheaf of rings
   thought of as a monoid object.
-* (Pre)sheaves of modules over a given sheaf of rings are an abelian category.
 * Presheaves of modules over a presheaf of commutative rings form a monoidal category.
 * Pushforward and pullback.
 -/
@@ -33,148 +29,191 @@ universe v v₁ u₁ u
 
 open CategoryTheory LinearMap Opposite
 
-variable {C : Type u₁} [Category.{v₁} C]
+variable {C : Type u₁} [Category.{v₁} C] {R : Cᵒᵖ ⥤ RingCat.{u}}
 
-/-- A presheaf of modules over a given presheaf of rings,
-described as a presheaf of abelian groups, and the extra data of the action at each object,
-and a condition relating functoriality and scalar multiplication. -/
-structure PresheafOfModules (R : Cᵒᵖ ⥤ RingCat.{u}) where
-  presheaf : Cᵒᵖ ⥤ AddCommGrp.{v}
-  module : ∀ X : Cᵒᵖ, Module (R.obj X) (presheaf.obj X) := by infer_instance
-  map_smul : ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y) (r : R.obj X) (x : presheaf.obj X),
-    presheaf.map f (r • x) = R.map f r • presheaf.map f x := by aesop_cat
-
-variable {R : Cᵒᵖ ⥤ RingCat.{u}}
+variable (R) in
+/-- A presheaf of modules over `R : Cᵒᵖ ⥤ RingCat` consists of family of
+objects `obj X : ModuleCat (R.obj X)` for all `X : Cᵒᵖ` together with
+functorial maps `obj X ⟶ (ModuleCat.restrictScalars (R.map f)).obj (obj Y)`
+for all `f : X ⟶ Y` in `Cᵒᵖ`. -/
+structure PresheafOfModules where
+  /-- a family of modules over `R.obj X` for all `X` -/
+  obj (X : Cᵒᵖ) : ModuleCat.{v} (R.obj X)
+  /-- the restriction maps of a presheaf of modules -/
+  map {X Y : Cᵒᵖ} (f : X ⟶ Y) : obj X ⟶ (ModuleCat.restrictScalars (R.map f)).obj (obj Y)
+  map_id (X : Cᵒᵖ) :
+    map (𝟙 X) = (ModuleCat.restrictScalarsId' _ (R.map_id X)).inv.app _ := by aesop_cat
+  map_comp {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
+    map (f ≫ g) = map f ≫ (ModuleCat.restrictScalars _).map (map g) ≫
+      (ModuleCat.restrictScalarsComp' _ _ _ (R.map_comp f g)).inv.app _ := by aesop_cat
 
 namespace PresheafOfModules
 
-attribute [instance] PresheafOfModules.module
-
-/-- The bundled module over an object `X`. -/
-def obj (P : PresheafOfModules R) (X : Cᵒᵖ) : ModuleCat (R.obj X) :=
-  ModuleCat.of _ (P.presheaf.obj X)
-
-/--
-If `P` is a presheaf of modules over a presheaf of rings `R`, both over some category `C`,
-and `f : X ⟶ Y` is a morphism in `Cᵒᵖ`, we construct the `R.map f`-semilinear map
-from the `R.obj X`-module `P.presheaf.obj X` to the `R.obj Y`-module `P.presheaf.obj Y`.
- -/
-def map (P : PresheafOfModules R) {X Y : Cᵒᵖ} (f : X ⟶ Y) :
-    P.obj X →ₛₗ[R.map f] P.obj Y :=
-  { toAddHom := (P.presheaf.map f).toAddHom,
-    map_smul' := P.map_smul f, }
-
-theorem map_apply (P : PresheafOfModules R) {X Y : Cᵒᵖ} (f : X ⟶ Y) (x) :
-    P.map f x = (P.presheaf.map f) x :=
-  rfl
+attribute [simp] map_id map_comp
+attribute [reassoc] map_comp
 
-instance (X : Cᵒᵖ) : RingHomId (R.map (𝟙 X)) where
-  eq_id := R.map_id X
+variable (M M₁ M₂ : PresheafOfModules.{v} R)
 
-instance {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
-    RingHomCompTriple (R.map f) (R.map g) (R.map (f ≫ g)) where
-  comp_eq := (R.map_comp f g).symm
+lemma map_smul {X Y : Cᵒᵖ} (f : X ⟶ Y) (r : R.obj X) (m : M.obj X) :
+    M.map f (r • m) = R.map f r • M.map f m := by simp
 
-@[simp]
-theorem map_id (P : PresheafOfModules R) (X : Cᵒᵖ) :
-    P.map (𝟙 X) = LinearMap.id' := by
-  ext
-  simp [map_apply]
+lemma congr_map_apply {X Y : Cᵒᵖ} {f g : X ⟶ Y} (h : f = g) (m : M.obj X) :
+    M.map f m = M.map g m := by rw [h]
 
-@[simp]
-theorem map_comp (P : PresheafOfModules R) {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
-    P.map (f ≫ g) = (P.map g).comp (P.map f) := by
-  ext
-  simp [map_apply]
+/-- A morphism of presheaves of modules consists of a family of linear maps which
+satisfy the naturality condition. -/
+@[ext]
+structure Hom where
+  /-- a family of linear maps `M₁.obj X ⟶ M₂.obj X` for all `X`. -/
+  app (X : Cᵒᵖ) : M₁.obj X ⟶ M₂.obj X
+  naturality {X Y : Cᵒᵖ} (f : X ⟶ Y) :
+      M₁.map f ≫ (ModuleCat.restrictScalars (R.map f)).map (app Y) =
+        app X ≫ M₂.map f := by aesop_cat
 
-/-- A morphism of presheaves of modules. -/
-structure Hom (P Q : PresheafOfModules R) where
-  hom : P.presheaf ⟶ Q.presheaf
-  map_smul : ∀ (X : Cᵒᵖ) (r : R.obj X) (x : P.presheaf.obj X), hom.app X (r • x) = r • hom.app X x
+attribute [reassoc (attr := simp)] Hom.naturality
 
-namespace Hom
+instance : Category (PresheafOfModules.{v} R) where
+  Hom := Hom
+  id _ := { app := fun _ ↦ 𝟙 _ }
+  comp f g := { app := fun _ ↦ f.app _ ≫ g.app _ }
 
-/-- The identity morphism on a presheaf of modules. -/
-def id (P : PresheafOfModules R) : Hom P P where
-  hom := 𝟙 _
-  map_smul _ _ _ := rfl
+variable {M₁ M₂}
 
-/-- Composition of morphisms of presheaves of modules. -/
-def comp {P Q R : PresheafOfModules R} (f : Hom P Q) (g : Hom Q R) : Hom P R where
-  hom := f.hom ≫ g.hom
-  map_smul _ _ _ := by simp [Hom.map_smul]
+@[ext]
+lemma hom_ext {f g : M₁ ⟶ M₂} (h : ∀ (X : Cᵒᵖ), f.app X = g.app X) :
+    f = g := Hom.ext (by ext1; apply h)
 
-end Hom
+@[simp]
+lemma id_app (M : PresheafOfModules R) (X : Cᵒᵖ) : Hom.app (𝟙 M) X = 𝟙 _ := by
+  rfl
 
-instance : Category (PresheafOfModules R) where
-  Hom := Hom
-  id := Hom.id
-  comp f g := Hom.comp f g
+@[simp]
+lemma comp_app {M₁ M₂ M₃ : PresheafOfModules R} (f : M₁ ⟶ M₂) (g : M₂ ⟶ M₃) (X : Cᵒᵖ) :
+    (f ≫ g).app X = f.app X ≫ g.app X := by
+  rfl
 
-namespace Hom
+lemma naturality_apply (f : M₁ ⟶ M₂) {X Y : Cᵒᵖ} (g : X ⟶ Y) (x : M₁.obj X) :
+    Hom.app f Y (M₁.map g x) = M₂.map g (Hom.app f X x) :=
+  congr_fun ((forget _).congr_map (Hom.naturality f g)) x
 
-variable {P Q T : PresheafOfModules R}
+/-- The underlying presheaf of abelian groups of a presheaf of modules. -/
+def presheaf : Cᵒᵖ ⥤ Ab where
+  obj X := (forget₂ _ _).obj (M.obj X)
+  map f := AddMonoidHom.mk' (M.map f) (by simp)
 
-variable (P) in
 @[simp]
-lemma id_hom : Hom.hom (𝟙 P) = 𝟙 _ := rfl
-
-@[simp, reassoc]
-lemma comp_hom (f : P ⟶ Q) (g : Q ⟶ T) : (f ≫ g).hom = f.hom ≫ g.hom := rfl
-
-/--
-The `(X : Cᵒᵖ)`-component of morphism between presheaves of modules
-over a presheaf of rings `R`, as an `R.obj X`-linear map. -/
-def app (f : Hom P Q) (X : Cᵒᵖ) : P.obj X →ₗ[R.obj X] Q.obj X :=
-  { toAddHom := (f.hom.app X).toAddHom
-    map_smul' := f.map_smul X }
+lemma presheaf_obj_coe (X : Cᵒᵖ) :
+    (M.presheaf.obj X : Type _) = M.obj X := rfl
 
 @[simp]
-lemma comp_app (f : P ⟶ Q) (g : Q ⟶ T) (X : Cᵒᵖ) :
-    (f ≫ g).app X = (g.app X).comp (f.app X) := rfl
+lemma presheaf_map_apply_coe {X Y : Cᵒᵖ} (f : X ⟶ Y) (x : M.obj X) :
+    DFunLike.coe (α := M.obj X) (β := fun _ ↦ M.obj Y) (M.presheaf.map f) x = M.map f x := rfl
 
-@[ext]
-theorem ext {f g : P ⟶ Q} (w : ∀ X, f.app X = g.app X) : f = g := by
-  cases f; cases g
-  congr
-  ext X x
-  exact LinearMap.congr_fun (w X) x
+instance (M : PresheafOfModules R) (X : Cᵒᵖ) :
+    Module (R.obj X) (M.presheaf.obj X) :=
+  inferInstanceAs (Module (R.obj X) (M.obj X))
 
-instance : Zero (P ⟶ Q) := ⟨mk 0 (by
-  intros
-  simp only [Limits.zero_app, AddMonoidHom.zero_apply, smul_zero])⟩
+variable (R) in
+/-- The forgetful functor `PresheafOfModules R ⥤ Cᵒᵖ ⥤ Ab`. -/
+def toPresheaf : PresheafOfModules.{v} R ⥤ Cᵒᵖ ⥤ Ab where
+  obj M := M.presheaf
+  map f :=
+    { app := fun X ↦ AddMonoidHom.mk' (Hom.app f X) (by simp)
+      naturality := fun X Y g ↦ by ext x; exact naturality_apply f g x }
 
-variable (P Q)
+@[simp]
+lemma toPresheaf_obj_coe (X : Cᵒᵖ) :
+    (((toPresheaf R).obj M).obj X : Type _) = M.obj X := rfl
 
 @[simp]
-lemma zero_app (X : Cᵒᵖ) : (0 : P ⟶ Q).app X = 0 := rfl
+lemma toPresheaf_map_app_apply (f : M₁ ⟶ M₂) (X : Cᵒᵖ) (x : M₁.obj X) :
+    DFunLike.coe (α := M₁.obj X) (β := fun _ ↦ M₂.obj X)
+      (((toPresheaf R).map f).app X) x = f.app X x := rfl
 
-variable {P Q}
+instance : (toPresheaf R).Faithful where
+  map_injective {_ _ f g} h := by
+    ext X x
+    exact congr_fun (((evaluation _ _).obj X ⋙ forget _).congr_map h) x
 
-instance : Add (P ⟶ Q) := ⟨fun f g => mk (f.hom + g.hom) (by
-  intros
-  simp only [NatTrans.app_add, AddCommGrp.hom_add_apply, map_smul, smul_add])⟩
+section
 
-@[simp]
-lemma add_app (f g : P ⟶ Q) (X : Cᵒᵖ) : (f + g).app X = f.app X + g.app X := rfl
+variable (M : Cᵒᵖ ⥤ Ab.{v}) [∀ X, Module (R.obj X) (M.obj X)]
+  (map_smul : ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y) (r : R.obj X) (m : M.obj X),
+    M.map f (r • m) = R.map f r • M.map f m)
 
-instance : Sub (P ⟶ Q) := ⟨fun f g => mk (f.hom - g.hom) (by
-  intros
-  rw [NatTrans.app_sub, AddMonoidHom.sub_apply, AddMonoidHom.sub_apply,
-    smul_sub, map_smul, map_smul])⟩
+/-- The object in `PresheafOfModules R` that is obtained from `M : Cᵒᵖ ⥤ Ab.{v}` such
+that for all `X : Cᵒᵖ`, `M.obj X` is a `R.obj X` module, in such a way that the
+restriction maps are semilinear. (This constructor should be used only in cases
+when the preferred constructor `PresheafOfModules.mk` is not as convenient as this one.) -/
+@[simps]
+def ofPresheaf : PresheafOfModules.{v} R where
+  obj X := ModuleCat.of _ (M.obj X)
+  map f :=
+    { toFun := fun x ↦ M.map f x
+      map_add' := by simp
+      map_smul' := fun r m ↦ map_smul f r m }
 
 @[simp]
-lemma sub_app (f g : P ⟶ Q) (X : Cᵒᵖ) : (f - g).app X = f.app X - g.app X := rfl
+lemma ofPresheaf_presheaf : (ofPresheaf M map_smul).presheaf = M := rfl
 
-instance : Neg (P ⟶ Q) := ⟨fun f => mk (-f.hom) (by
-  intros
-  rw [NatTrans.app_neg, AddMonoidHom.neg_apply, AddMonoidHom.neg_apply,
-    map_smul, smul_neg])⟩
-
-@[simp]
-lemma neg_app (f : P ⟶ Q) (X : Cᵒᵖ) : (-f).app X = -f.app X := rfl
+end
 
-instance : AddCommGroup (P ⟶ Q) where
+/-- The morphism of presheaves of modules `M₁ ⟶ M₂` given by a morphism
+of abelian presheaves `M₁.presheaf ⟶ M₂.presheaf`
+which satisfy a suitable linearity condition. -/
+@[simps]
+def homMk (φ : M₁.presheaf ⟶ M₂.presheaf)
+    (hφ : ∀ (X : Cᵒᵖ) (r : R.obj X) (m : M₁.obj X), φ.app X (r • m) = r • φ.app X m) :
+    M₁ ⟶ M₂ where
+  app X :=
+    { toFun := φ.app X
+      map_add' := by simp
+      map_smul' := hφ X }
+  naturality := fun f ↦ by
+    ext x
+    exact congr_fun ((forget _).congr_map (φ.naturality f)) x
+
+instance : Zero (M₁ ⟶ M₂) where
+  zero := { app := fun _ ↦ 0 }
+
+variable (M₁ M₂) in
+@[simp] lemma zero_app (X : Cᵒᵖ) : (0 : M₁ ⟶ M₂).app X = 0 := rfl
+
+instance : Neg (M₁ ⟶ M₂) where
+  neg f :=
+    { app := fun X ↦ -f.app X
+      naturality := fun {X Y} h ↦ by
+        ext x
+        dsimp
+        erw [map_neg]
+        rw [← naturality_apply]
+        rfl }
+
+instance : Add (M₁ ⟶ M₂) where
+  add f g :=
+    { app := fun X ↦ f.app X + g.app X
+      naturality := fun {X Y} h ↦ by
+        ext x
+        dsimp
+        erw [map_add]
+        rw [← naturality_apply, ← naturality_apply]
+        rfl }
+
+instance : Sub (M₁ ⟶ M₂) where
+  sub f g :=
+    { app := fun X ↦ f.app X - g.app X
+      naturality := fun {X Y} h ↦ by
+        ext x
+        dsimp
+        erw [map_sub]
+        rw [← naturality_apply, ← naturality_apply]
+        rfl }
+
+@[simp] lemma neg_app (f : M₁ ⟶ M₂) (X : Cᵒᵖ) : (-f).app X = -f.app X := rfl
+@[simp] lemma add_app (f g : M₁ ⟶ M₂) (X : Cᵒᵖ) : (f + g).app X = f.app X + g.app X := rfl
+@[simp] lemma sub_app (f g : M₁ ⟶ M₂) (X : Cᵒᵖ) : (f - g).app X = f.app X - g.app X := rfl
+
+instance : AddCommGroup (M₁ ⟶ M₂) where
   add_assoc := by intros; ext1; simp only [add_app, add_assoc]
   zero_add := by intros; ext1; simp only [add_app, zero_app, zero_add]
   neg_add_cancel := by intros; ext1; simp only [add_app, neg_app, neg_add_cancel, zero_app]
@@ -185,41 +224,14 @@ instance : AddCommGroup (P ⟶ Q) where
   zsmul := zsmulRec
 
 instance : Preadditive (PresheafOfModules R) where
-  add_comp := by intros; ext1; simp only [comp_app, add_app, comp_add]
-  comp_add := by intros; ext1; simp only [comp_app, add_app, add_comp]
-
-end Hom
-
-lemma naturality_apply {P Q : PresheafOfModules R} (f : P ⟶ Q)
-    {X Y : Cᵒᵖ} (g : X ⟶ Y) (x : P.obj X) :
-    f.app Y (P.map g x) = Q.map g (f.app X x) :=
-  congr_fun ((forget _).congr_map (f.hom.naturality g)) x
-
-variable (R)
-
-/-- The functor from presheaves of modules over a specified presheaf of rings,
-to presheaves of abelian groups.
--/
-@[simps obj]
-def toPresheaf : PresheafOfModules.{v} R ⥤ (Cᵒᵖ ⥤ AddCommGrp.{v}) where
-  obj P := P.presheaf
-  map f := f.hom
-
-variable {R}
-
-@[simp]
-lemma toPresheaf_map_app {P Q : PresheafOfModules R}
-    (f : P ⟶ Q) (X : Cᵒᵖ) :
-    ((toPresheaf R).map f).app X = (f.app X).toAddMonoidHom := rfl
 
 instance : (toPresheaf R).Additive where
 
-instance : (toPresheaf R).Faithful where
-  map_injective {P Q} f g h := by
-    ext X x
-    have eq := congr_app h X
-    simp only [toPresheaf_obj, toPresheaf_map_app] at eq
-    simp only [← toAddMonoidHom_coe, eq]
+lemma zsmul_app (n : ℤ) (f : M₁ ⟶ M₂) (X : Cᵒᵖ) : (n • f).app X = n • f.app X := by
+  ext x
+  change (toPresheaf R ⋙ (evaluation _ _).obj X).map (n • f) x = _
+  rw [Functor.map_zsmul]
+  rfl
 
 variable (R)
 
@@ -230,215 +242,22 @@ def evaluation (X : Cᵒᵖ) : PresheafOfModules.{v} R ⥤ ModuleCat (R.obj X) w
   obj M := M.obj X
   map f := f.app X
 
-instance (X : Cᵒᵖ) : (evaluation R X).Additive where
-
-variable {R}
-
-/-- Given a presheaf of modules `M` on a category `C` and `f : X ⟶ Y` in `Cᵒᵖ`, this
-is the restriction map `M.obj X ⟶ M.obj Y`, considered as a linear map to
-the restriction of scalars of `M.obj Y`. -/
-noncomputable def restrictionApp {X Y : Cᵒᵖ} (f : X ⟶ Y) (M : PresheafOfModules.{v} R) :
-    M.obj X ⟶ (ModuleCat.restrictScalars (R.map f)).obj (M.obj Y) :=
-  ModuleCat.semilinearMapAddEquiv (R.map f) _ _ (M.map f)
-
-lemma restrictionApp_apply {X Y : Cᵒᵖ} (f : X ⟶ Y) (M : PresheafOfModules R) (x : M.obj X) :
-    restrictionApp f M x = M.map f x := by
-  rfl
-
-variable (R)
+instance (X : Cᵒᵖ) : (evaluation.{v} R X).Additive where
 
 /-- The restriction natural transformation on presheaves of modules, considered as linear maps
 to restriction of scalars. -/
 @[simps]
 noncomputable def restriction {X Y : Cᵒᵖ} (f : X ⟶ Y) :
     evaluation R X ⟶ evaluation R Y ⋙ ModuleCat.restrictScalars (R.map f) where
-  app := restrictionApp f
-  naturality := fun M N φ => by
-    ext x
-    exact (congr_hom (φ.hom.naturality f) x).symm
-
-variable {R}
-
-@[reassoc (attr := simp)]
-lemma restrictionApp_naturality {X Y : Cᵒᵖ} (f : X ⟶ Y)
-    {M N : PresheafOfModules R} (φ : M ⟶ N) :
-    restrictionApp f M ≫ (ModuleCat.restrictScalars (R.map f)).map (Hom.app φ Y) =
-      ModuleCat.ofHom (Hom.app φ X) ≫ restrictionApp f N :=
-  ((restriction R f).naturality φ).symm
-
-attribute [local simp] restrictionApp_apply
-
-lemma restrictionApp_id (M : PresheafOfModules R) (X : Cᵒᵖ) :
-    restrictionApp (𝟙 X) M =
-      (ModuleCat.restrictScalarsId' (R.map (𝟙 X)) (R.map_id X)).inv.app (M.obj X) := by aesop
-
-lemma restrictionApp_comp (M : PresheafOfModules R) {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
-    restrictionApp (f ≫ g) M =
-      restrictionApp f M ≫
-        (ModuleCat.restrictScalars (R.map f)).map (restrictionApp g M) ≫
-        (ModuleCat.restrictScalarsComp' _ _ _ (R.map_comp f g)).inv.app (M.obj Z) := by aesop
-
-namespace Hom
-
-variable {P Q : PresheafOfModules R} (app : ∀ X, P.obj X →ₗ[R.obj X] Q.obj X)
-
-section
-
-variable (naturality : ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y) (x : P.obj X),
-  app Y (P.map f x) = Q.map f (app X x))
-
-/-- A constructor for morphisms in `PresheafOfModules R` that is based on the data
-of a family of linear maps over the various rings `R.obj X`. -/
-def mk' : P ⟶ Q where
-  hom :=
-    { app := fun X => (app X).toAddMonoidHom
-      naturality := fun _ _ f => AddCommGrp.ext (naturality f) }
-  map_smul X := (app X).map_smul
-
-@[simp]
-lemma mk'_app : (mk' app naturality).app = app := rfl
-
-end
-
-/-- A constructor for morphisms in `PresheafOfModules R` that is based on the data
-of a family of linear maps over the various rings `R.obj X`, and for which the
-naturality condition is stated using the restriction of scalars. -/
-abbrev mk''
-    (naturality : ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y),
-      restrictionApp f P ≫ (ModuleCat.restrictScalars (R.map f)).map (app Y) =
-        ModuleCat.ofHom (app X) ≫ restrictionApp f Q) :
-    P ⟶ Q :=
-  mk' app (fun _ _ f x => congr_hom (naturality f) x)
-
-end Hom
-
-end PresheafOfModules
-
-variable (R) in
-/-- This structure contains the data and axioms in order to
-produce a `PresheafOfModules R` from a collection of types
-equipped with module structures over the various rings `R.obj X`.
-(See the constructor `PresheafOfModules.mk'`.) -/
-structure CorePresheafOfModules where
-  /-- the datum of a type for each object in `Cᵒᵖ` -/
-  obj (X : Cᵒᵖ) : Type v
-  /-- the abelian group structure on the types `obj X` -/
-  addCommGroup (X : Cᵒᵖ) : AddCommGroup (obj X) := by infer_instance
-  /-- the module structure on the types `obj X` over the various rings `R.obj X` -/
-  module (X : Cᵒᵖ) : Module (R.obj X) (obj X) := by infer_instance
-  /-- the semi-linear restriction maps -/
-  map {X Y : Cᵒᵖ} (f : X ⟶ Y) : obj X →ₛₗ[R.map f] obj Y
-  /-- `map` is compatible with the identities -/
-  map_id (X : Cᵒᵖ) (x : obj X) : map (𝟙 X) x = x := by aesop_cat
-  /-- `map` is compatible with the composition -/
-  map_comp {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (x : obj X) :
-    map (f ≫ g) x = map g (map f x) := by aesop_cat
-
--- this example is meant to test automation: the axioms for `CorePresheafOfModules` are
--- automatically found if we use the data from `M : PresheafOfModules R`
-example (M : PresheafOfModules R) : CorePresheafOfModules R where
-  obj X := M.obj X
-  map f := M.map f
-
-namespace CorePresheafOfModules
-
-attribute [instance] addCommGroup module
-attribute [simp] map_id map_comp
-
-variable (M : CorePresheafOfModules R)
-
-/-- The presheaf of abelian groups attached to a `CorePresheafOfModules R`. -/
-@[simps]
-def presheaf : Cᵒᵖ ⥤ AddCommGrp.{v} where
-  obj X := AddCommGrp.of (M.obj X)
-  map f := AddCommGrp.ofHom (M.map f).toAddMonoidHom
-
-instance (X : Cᵒᵖ) : Module (R.obj X) (M.presheaf.obj X) := M.module X
-
-/-- Constructor for `PresheafOfModules R` based on a collection of types
-equipped with module structures over the various rings `R.obj X`, see
-the structure `CorePresheafOfModules`. -/
-def toPresheafOfModules : PresheafOfModules R where
-  presheaf := M.presheaf
-
-@[simp]
-lemma toPresheafOfModules_obj (X : Cᵒᵖ) :
-    M.toPresheafOfModules.obj X = ModuleCat.of _ (M.obj X) := rfl
-
-@[simp]
-lemma toPresheafOfModules_presheaf_map_apply {X Y : Cᵒᵖ} (f : X ⟶ Y) (x : M.obj X) :
-    M.toPresheafOfModules.presheaf.map f x = M.map f x := rfl
-
-end CorePresheafOfModules
-
-variable (R) in
-/-- This structure contains the data and axioms in order to
-produce a `PresheafOfModules R` from a collection of objects
-of type `ModuleCat (R.obj X)` for all `X`, and restriction
-maps expressed as linear maps to restriction of scalars.
-(See the constructor `PresheafOfModules.mk''`.) -/
-structure BundledCorePresheafOfModules where
-  /-- the datum of a `ModuleCat (R.obj X)` for each object in `Cᵒᵖ` -/
-  obj (X : Cᵒᵖ) : ModuleCat.{v} (R.obj X)
-  /-- the restriction maps as linear maps to restriction of scalars -/
-  map {X Y : Cᵒᵖ} (f : X ⟶ Y) : obj X ⟶ (ModuleCat.restrictScalars (R.map f)).obj (obj Y)
-  /-- `map` is compatible with the identities -/
-  map_id (X : Cᵒᵖ) :
-    map (𝟙 X) = (ModuleCat.restrictScalarsId' (R.map (𝟙 X)) (R.map_id X)).inv.app (obj X) := by
-      aesop
-  /-- `map` is compatible with the composition -/
-  map_comp {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
-    map (f ≫ g) = map f ≫ (ModuleCat.restrictScalars (R.map f)).map (map g) ≫
-      (ModuleCat.restrictScalarsComp' (R.map f) (R.map g) (R.map (f ≫ g))
-        (R.map_comp f g)).inv.app (obj Z) := by aesop
-
-namespace BundledCorePresheafOfModules
-
-variable (M : BundledCorePresheafOfModules R)
-
-attribute [local simp] map_id map_comp
-
-/-- The obvious map `BundledCorePresheafOfModules R → CorePresheafOfModules R`. -/
-noncomputable def toCorePresheafOfModules : CorePresheafOfModules R where
-  obj X := (M.obj X).carrier
-  map {X Y} f := (ModuleCat.semilinearMapAddEquiv (R.map f) (M.obj X) (M.obj Y)).symm (M.map f)
-
-/-- Constructor for `PresheafOfModules R` based on a collection of objects
-of type `ModuleCat (R.obj X)` for all `X`, and restriction maps expressed
-as linear maps to restriction of scalars, see
-the structure `BundledCorePresheafOfModules`. -/
-noncomputable def toPresheafOfModules : PresheafOfModules R :=
-  M.toCorePresheafOfModules.toPresheafOfModules
-
-@[simp]
-lemma toPresheafOfModules_obj (X : Cᵒᵖ) :
-    M.toPresheafOfModules.obj X = (M.obj X).carrier := rfl
-
-@[simp]
-lemma toPresheafOfModules_presheaf_map_apply {X Y : Cᵒᵖ} (f : X ⟶ Y) (x : M.obj X) :
-    M.toPresheafOfModules.presheaf.map f x = M.map f x := rfl
-
-@[simp]
-lemma restrictionApp_toPresheafOfModules {X Y : Cᵒᵖ} (f : X ⟶ Y) :
-    PresheafOfModules.restrictionApp f M.toPresheafOfModules = M.map f := rfl
-
-end BundledCorePresheafOfModules
-
-namespace PresheafOfModules
-
-variable (R)
-
-/-- Auxiliary definition for `unit`. -/
-def unitCore : CorePresheafOfModules R where
-  obj X := R.obj X
-  map {X Y} f := by
-    exact
-      { toFun := (R.map f).toFun
-        map_add' := by simp
-        map_smul' := by simp }
+  app M := M.map f
 
 /-- The obvious free presheaf of modules of rank `1`. -/
-abbrev unit : PresheafOfModules R := (unitCore R).toPresheafOfModules
+def unit : PresheafOfModules R where
+  obj X := ModuleCat.of _ (R.obj X)
+  map {X Y } f :=
+    { toFun := fun x ↦ R.map f x
+      map_add' := by simp
+      map_smul' := by aesop_cat }
 
 lemma unit_map_one {X Y : Cᵒᵖ} (f : X ⟶ Y) : (unit R).map f (1 : R.obj X) = (1 : R.obj Y) :=
   (R.map f).map_one
@@ -448,6 +267,10 @@ variable {R}
 /-- The type of sections of a presheaf of modules. -/
 def sections (M : PresheafOfModules.{v} R) : Type _ := (M.presheaf ⋙ forget _).sections
 
+/-- Given a presheaf of modules `M`, `s : M.sections` and `X : Cᵒᵖ`, this is the induced
+element in `M.obj X`. -/
+abbrev sections.eval {M : PresheafOfModules.{v} R} (s : M.sections) (X : Cᵒᵖ) : M.obj X := s.1 X
+
 @[simp]
 lemma sections_property {M : PresheafOfModules.{v} R} (s : M.sections)
     {X Y : Cᵒᵖ} (f : X ⟶ Y) : M.map f (s.1 X) = s.1 Y := s.2 f
@@ -484,12 +307,12 @@ def unitHomEquiv (M : PresheafOfModules R) :
     (unit R ⟶ M) ≃ M.sections where
   toFun f := sectionsMk (fun X ↦ Hom.app f X (1 : R.obj X))
     (by intros; rw [← naturality_apply, unit_map_one])
-  invFun s := Hom.mk'
-    (fun X => (LinearMap.ringLmapEquivSelf (R.obj X) ℤ (M.obj X)).symm (s.val X)) (by
-      intro X Y p (x : R.obj X)
-      dsimp
-      rw [map_apply, M.map_smul, ← s.2 p]
-      rfl)
+  invFun s :=
+    { app := fun X ↦ (LinearMap.ringLmapEquivSelf (R.obj X) ℤ (M.obj X)).symm (s.val X)
+      naturality := fun {X Y} f ↦ by
+        ext (x : R.obj X)
+        change R.map f x • s.eval Y = M.map f (x • s.eval X)
+        simp }
   left_inv f := by
     ext1 X
     exact (LinearMap.ringLmapEquivSelf (R.obj X) ℤ (M.obj X)).symm_apply_apply (f.app X)
@@ -499,6 +322,8 @@ def unitHomEquiv (M : PresheafOfModules R) :
 
 section module_over_initial
 
+variable (X : Cᵒᵖ) (hX : Limits.IsInitial X)
+
 /-!
 ## `PresheafOfModules R ⥤ Cᵒᵖ ⥤ ModuleCat (R.obj X)` when `X` is initial
 
@@ -506,6 +331,36 @@ When `X` is initial, we have `Module (R.obj X) (M.obj c)` for any `c : Cᵒᵖ`.
 
 -/
 
+section
+
+variable (M : PresheafOfModules.{v} R)
+
+/-- Auxiliary definition for `forgetToPresheafModuleCatObj`. -/
+noncomputable abbrev forgetToPresheafModuleCatObjObj (Y : Cᵒᵖ) : ModuleCat (R.obj X) :=
+  (ModuleCat.restrictScalars (R.map (hX.to Y))).obj (M.obj Y)
+
+@[simp]
+lemma forgetToPresheafModuleCatObjObj_coe (Y : Cᵒᵖ) :
+    (forgetToPresheafModuleCatObjObj X hX M Y : Type _) = M.obj Y := rfl
+
+/-- Auxiliary definition for `forgetToPresheafModuleCatObj`. -/
+def forgetToPresheafModuleCatObjMap {Y Z : Cᵒᵖ} (f : Y ⟶ Z) :
+    forgetToPresheafModuleCatObjObj X hX M Y ⟶
+      forgetToPresheafModuleCatObjObj X hX M Z where
+  toFun x := M.map f x
+  map_add' := by simp
+  map_smul' r x := by
+    simp only [ModuleCat.restrictScalars.smul_def, AddHom.toFun_eq_coe, AddHom.coe_mk,
+      RingHom.id_apply, M.map_smul]
+    rw [← CategoryTheory.comp_apply, ← R.map_comp]
+    congr
+    apply hX.hom_ext
+
+@[simp]
+lemma forgetToPresheafModuleCatObjMap_apply {Y Z : Cᵒᵖ} (f : Y ⟶ Z) (m : M.obj Y) :
+    DFunLike.coe (α := M.obj Y) (β := fun _ ↦ M.obj Z)
+      (forgetToPresheafModuleCatObjMap X hX M f) m = M.map f m := rfl
+
 /--
 Implementation of the functor `PresheafOfModules R ⥤ Cᵒᵖ ⥤ ModuleCat (R.obj X)`
 when `X` is initial.
@@ -517,20 +372,11 @@ morphism level `(f : M ⟶ N) ↦ (c ↦ f(c))`.
 @[simps]
 noncomputable def forgetToPresheafModuleCatObj
     (X : Cᵒᵖ) (hX : Limits.IsInitial X) (M : PresheafOfModules.{v} R) :
-    Cᵒᵖ ⥤ ModuleCat (R.1.obj X) where
-  obj c :=
-    ModuleCat.restrictScalars (R.1.map (hX.to c)) |>.obj <| M.obj c
-  map := fun {c₁ c₂} f =>
-    { toFun := fun x => M.presheaf.map f x
-      map_add' := M.presheaf.map f |>.map_add
-      map_smul' := fun r (m :  ModuleCat.restrictScalars _ |>.obj _) => by
-        simp only [ModuleCat.restrictScalars.smul_def, RingHom.id_apply, M.map_smul]
-        rw [← CategoryTheory.comp_apply, ← R.map_comp]
-        congr
-        apply hX.hom_ext }
-  map_id := fun c => by ext; simp_rw [M.presheaf.map_id]; rfl
-  map_comp := fun {c₁ c₂ c₃} f g => by
-      ext x; simp_rw [M.presheaf.map_comp]; rfl
+    Cᵒᵖ ⥤ ModuleCat (R.obj X) where
+  obj Y := forgetToPresheafModuleCatObjObj X hX M Y
+  map f := forgetToPresheafModuleCatObjMap X hX M f
+
+end
 
 /--
 Implementation of the functor `PresheafOfModules R ⥤ Cᵒᵖ ⥤ ModuleCat (R.obj X)`
@@ -541,15 +387,15 @@ on `M(c)` is given by restriction of scalars along the unique morphism `R(c) ⟶
 morphism level `(f : M ⟶ N) ↦ (c ↦ f(c))`.
 -/
 noncomputable def forgetToPresheafModuleCatMap
-    (X : Cᵒᵖ) (hX : Limits.IsInitial X) {M N : PresheafOfModules.{v} R}
-    (f : M ⟶ N) :
-    forgetToPresheafModuleCatObj X hX M ⟶
-    forgetToPresheafModuleCatObj X hX N :=
-  { app := fun c =>
-    { toFun := f.app c
-      map_add' := (f.app c).map_add
-      map_smul' := fun r (m : M.presheaf.obj c) => (f.app c).map_smul (R.1.map (hX.to c) _) m }
-    naturality := fun {c₁ c₂} i => by ext x; exact congr($(f.hom.naturality i) x) }
+    (X : Cᵒᵖ) (hX : Limits.IsInitial X) {M N : PresheafOfModules.{v} R} (f : M ⟶ N) :
+    forgetToPresheafModuleCatObj X hX M ⟶ forgetToPresheafModuleCatObj X hX N where
+  app Y :=
+    { toFun := f.app Y
+      map_add' := by simp
+      map_smul' := fun r ↦ (f.app Y).map_smul (R.1.map (hX.to Y) _) }
+  naturality Y Z g := by
+    ext x
+    exact naturality_apply f g x
 
 /--
 The forgetful functor from presheaves of modules over a presheaf of rings `R` to presheaves of
@@ -561,7 +407,7 @@ morphism level `(f : M ⟶ N) ↦ (c ↦ f(c))`.
 -/
 @[simps]
 noncomputable def forgetToPresheafModuleCat (X : Cᵒᵖ) (hX : Limits.IsInitial X) :
-    PresheafOfModules.{v} R ⥤ Cᵒᵖ ⥤ ModuleCat (R.1.obj X) where
+    PresheafOfModules.{v} R ⥤ Cᵒᵖ ⥤ ModuleCat (R.obj X) where
   obj M := forgetToPresheafModuleCatObj X hX M
   map f := forgetToPresheafModuleCatMap X hX f
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Abelian.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Abelian.lean
index 80d80075b9e8a..42747bed79af5 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Abelian.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Abelian.lean
@@ -22,7 +22,7 @@ namespace PresheafOfModules
 variable {C : Type u₁} [Category.{v₁} C] (R : Cᵒᵖ ⥤ RingCat.{u})
 
 noncomputable instance : NormalEpiCategory (PresheafOfModules.{v} R) where
-  normalEpiOfEpi p _ := NormalEpi.mk _ (kernel.ι p)  (kernel.condition _)
+  normalEpiOfEpi p _ := NormalEpi.mk _ (kernel.ι p) (kernel.condition _)
     (evaluationJointlyReflectsColimits _ _ (fun _ =>
       Abelian.isColimitMapCoconeOfCokernelCoforkOfπ _ _))
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/ChangeOfRings.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/ChangeOfRings.lean
index c71d1bdc5ca09..987578da8c709 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/ChangeOfRings.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/ChangeOfRings.lean
@@ -25,34 +25,29 @@ variable {C : Type u'} [Category.{v'} C] {R R' : Cᵒᵖ ⥤ RingCat.{u}}
 
 /-- The restriction of scalars of presheaves of modules, on objects. -/
 @[simps]
-noncomputable def restrictScalarsBundledCore (M' : PresheafOfModules R') (α : R ⟶ R') :
-    BundledCorePresheafOfModules R where
-  obj X := (ModuleCat.restrictScalars (α.app X)).obj (M'.obj X)
-  map {X Y} f :=
+noncomputable def restrictScalarsObj (M' : PresheafOfModules.{v} R') (α : R ⟶ R') :
+    PresheafOfModules R where
+  obj := fun X ↦ (ModuleCat.restrictScalars (α.app X)).obj (M'.obj X)
+  map := fun {X Y} f ↦
     { toFun := M'.map f
       map_add' := map_add _
-      map_smul' := fun r x ↦ by
+      map_smul' := fun r x ↦ (M'.map_smul f (α.app _ r) x).trans (by
         have eq := RingHom.congr_fun (α.naturality f) r
-        apply (M'.map_smul f (α.app _ r) x).trans
-        dsimp at eq ⊢
+        dsimp at eq
         rw [← eq]
-        rfl }
-  map_id X := by
-    ext x
-    exact LinearMap.congr_fun (M'.map_id X) x
-  map_comp f g := by
-    ext x
-    exact LinearMap.congr_fun (M'.map_comp f g) x
+        rfl ) }
 
 /-- The restriction of scalars functor `PresheafOfModules R' ⥤ PresheafOfModules R`
 induced by a morphism of presheaves of rings `R ⟶ R'`. -/
 @[simps]
 noncomputable def restrictScalars (α : R ⟶ R') :
     PresheafOfModules.{v} R' ⥤ PresheafOfModules.{v} R where
-  obj M' := (M'.restrictScalarsBundledCore α).toPresheafOfModules
-  map {M₁' M₂'} φ :=
-    { hom := φ.hom
-      map_smul := fun X r ↦ φ.map_smul X (α.app _ r) }
+  obj M' := M'.restrictScalarsObj α
+  map φ' :=
+    { app := fun X ↦ (ModuleCat.restrictScalars (α.app X)).map (Hom.app φ' X)
+      naturality := fun {X Y} f ↦ by
+        ext x
+        exact naturality_apply φ' f x }
 
 instance (α : R ⟶ R') : (restrictScalars.{v} α).Additive where
 
@@ -61,4 +56,9 @@ instance : (restrictScalars (𝟙 R)).Full := inferInstanceAs (𝟭 _).Full
 instance (α : R ⟶ R') : (restrictScalars α).Faithful where
   map_injective h := (toPresheaf R').map_injective ((toPresheaf R).congr_map h)
 
+/-- The isomorphism `restrictScalars α ⋙ toPresheaf R ≅ toPresheaf R'` for any
+morphism of presheaves of rings `α : R ⟶ R'`. -/
+noncomputable def restrictScalarsCompToPresheaf (α : R ⟶ R') :
+    restrictScalars.{v} α ⋙ toPresheaf R ≅ toPresheaf R' := Iso.refl _
+
 end PresheafOfModules
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.lean
index ac5a7ef32dde8..26b9df8025f49 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Colimits.lean
@@ -32,13 +32,14 @@ variable [∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), PreservesColimit (F ⋙ evaluation
 of the functors `evaluation R X` for all `X`. -/
 def evaluationJointlyReflectsColimits (c : Cocone F)
     (hc : ∀ (X : Cᵒᵖ), IsColimit ((evaluation R X).mapCocone c)) : IsColimit c where
-  desc s := Hom.mk'' (fun X => (hc X).desc ((evaluation R X).mapCocone s)) (fun X Y f => by
-    apply (hc X).hom_ext
-    intro j
-    erw [(hc X).fac_assoc ((evaluation R X).mapCocone s) j, ← restrictionApp_naturality_assoc]
-    rw [← Functor.map_comp]
-    erw [(hc Y).fac ((evaluation R Y).mapCocone s), restrictionApp_naturality]
-    rfl)
+  desc s :=
+    { app := fun X => (hc X).desc ((evaluation R X).mapCocone s)
+      naturality := fun {X Y} f ↦ (hc X).hom_ext (fun j ↦ by
+        rw [(hc X).fac_assoc ((evaluation R X).mapCocone s) j]
+        have h₁ := (c.ι.app j).naturality f
+        have h₂ := (hc Y).fac ((evaluation R Y).mapCocone s)
+        dsimp at h₁ h₂ ⊢
+        simp only [← reassoc_of% h₁, ← Functor.map_comp, h₂, Hom.naturality]) }
   fac s j := by
     ext1 X
     exact (hc X).fac ((evaluation R X).mapCocone s) j
@@ -57,10 +58,10 @@ instance {X Y : Cᵒᵖ} (f : X ⟶ Y) :
   ⟨_, isColimitOfPreserves (ModuleCat.restrictScalars (R.map f))
     (colimit.isColimit (F ⋙ evaluation R Y))⟩
 
-/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the `BundledCorePresheafOfModules R` which
-corresponds to the presheaf of modules which sends `X` to the colimit of `F ⋙ evaluation R X`. -/
+/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the presheaf of modules obtained by
+taking a colimit in the category of modules over `R.obj X` for all `X`. -/
 @[simps]
-noncomputable def colimitBundledCore : BundledCorePresheafOfModules R where
+noncomputable def colimitPresheafOfModules : PresheafOfModules R where
   obj X := colimit (F ⋙ evaluation R X)
   map {X Y} f := colimMap (whiskerLeft F (restriction R f)) ≫
     (preservesColimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv
@@ -69,8 +70,8 @@ noncomputable def colimitBundledCore : BundledCorePresheafOfModules R where
     rw [ι_colimMap_assoc, whiskerLeft_app, restriction_app]
     erw [ι_preservesColimitsIso_inv (G := ModuleCat.restrictScalars (R.map (𝟙 X))),
       ModuleCat.restrictScalarsId'App_inv_naturality]
-    rw [restrictionApp_id]
-    rfl)
+    rw [map_id]
+    dsimp)
   map_comp {X Y Z} f g := colimit.hom_ext (fun j => by
     dsimp
     rw [ι_colimMap_assoc, whiskerLeft_app, restriction_app, assoc, ι_colimMap_assoc]
@@ -78,32 +79,26 @@ noncomputable def colimitBundledCore : BundledCorePresheafOfModules R where
       ι_preservesColimitsIso_inv_assoc (G := ModuleCat.restrictScalars (R.map f))]
     rw [← Functor.map_comp_assoc, ι_colimMap_assoc]
     erw [ι_preservesColimitsIso_inv (G := ModuleCat.restrictScalars (R.map g))]
-    rw [restrictionApp_comp, ModuleCat.restrictScalarsComp'_inv_app, assoc, assoc,
+    rw [map_comp, ModuleCat.restrictScalarsComp'_inv_app, assoc, assoc,
       whiskerLeft_app, whiskerLeft_app, restriction_app, restriction_app]
     simp only [Functor.map_comp, assoc]
     rfl)
 
-/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the canonical map
-`F.obj j ⟶ (colimitBundledCore F).toPresheafOfModules` for all `j : J`. -/
-noncomputable def colimitCoconeιApp (j : J) :
-    F.obj j ⟶ (colimitBundledCore F).toPresheafOfModules :=
-  PresheafOfModules.Hom.mk'' (fun X => colimit.ι (F ⋙ evaluation R X) j) (fun X Y f => by
-    dsimp
-    erw [colimit.ι_desc_assoc, assoc, ← ι_preservesColimitsIso_inv]
-    rfl)
-
-@[reassoc (attr := simp)]
-lemma colimitCoconeιApp_naturality {i j : J} (f : i ⟶ j) :
-    F.map f ≫ colimitCoconeιApp F j = colimitCoconeιApp F i := by
-  ext1 X
-  exact colimit.w (F ⋙ evaluation R X) f
-
 /-- The (colimit) cocone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed from
 the colimit of `F ⋙ evaluation R X` for all `X`. -/
 @[simps]
 noncomputable def colimitCocone : Cocone F where
-  pt := (colimitBundledCore F).toPresheafOfModules
-  ι := { app := colimitCoconeιApp F }
+  pt := colimitPresheafOfModules F
+  ι :=
+    { app := fun j ↦
+        { app := fun X ↦ colimit.ι (F ⋙ evaluation R X) j
+          naturality := fun {X Y} f ↦ by
+            dsimp
+            erw [colimit.ι_desc_assoc, assoc, ← ι_preservesColimitsIso_inv]
+            rfl }
+      naturality := fun {X Y} f ↦ by
+        ext1 X
+        simpa using colimit.w (F ⋙ evaluation R X) f }
 
 /-- The cocone `colimitCocone F` is colimit for any `F : J ⥤ PresheafOfModules.{v} R`. -/
 noncomputable def isColimitColimitCocone : IsColimit (colimitCocone F) :=
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
index d547a5c07236e..a8c8858cd5bfe 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Limits.lean
@@ -33,12 +33,16 @@ variable [∀ X, Small.{v} ((F ⋙ evaluation R X) ⋙ forget _).sections]
 of the functors `evaluation R X` for all `X`. -/
 def evaluationJointlyReflectsLimits (c : Cone F)
     (hc : ∀ (X : Cᵒᵖ), IsLimit ((evaluation R X).mapCone c)) : IsLimit c where
-  lift s := Hom.mk'' (fun X => (hc X).lift ((evaluation R X).mapCone s)) (fun X Y f => by
-    apply (isLimitOfPreserves (ModuleCat.restrictScalars (R.map f)) (hc Y)).hom_ext
-    intro j
-    rw [Functor.mapCone_π_app, assoc, assoc, ← Functor.map_comp]
-    erw [restrictionApp_naturality, IsLimit.fac, restrictionApp_naturality, IsLimit.fac_assoc]
-    rfl)
+  lift s :=
+    { app := fun X => (hc X).lift ((evaluation R X).mapCone s)
+      naturality := fun {X Y} f ↦ by
+        apply (isLimitOfPreserves (ModuleCat.restrictScalars (R.map f)) (hc Y)).hom_ext
+        intro j
+        have h₁ := (c.π.app j).naturality f
+        have h₂ := (hc X).fac ((evaluation R X).mapCone s) j
+        rw [Functor.mapCone_π_app, assoc, assoc, ← Functor.map_comp, IsLimit.fac]
+        dsimp at h₁ h₂ ⊢
+        rw [h₁, reassoc_of% h₂, Hom.naturality] }
   fac s j := by
     ext1 X
     exact (hc X).fac ((evaluation R X).mapCone s) j
@@ -47,19 +51,17 @@ def evaluationJointlyReflectsLimits (c : Cone F)
     apply (hc X).uniq ((evaluation R X).mapCone s)
     intro j
     dsimp
-    rw [← hm]
-    rfl
+    rw [← hm, comp_app]
 
 instance {X Y : Cᵒᵖ} (f : X ⟶ Y) :
     HasLimit (F ⋙ evaluation R Y ⋙ ModuleCat.restrictScalars (R.map f)) := by
   change HasLimit ((F ⋙ evaluation R Y) ⋙ ModuleCat.restrictScalars (R.map f))
   infer_instance
 
-set_option backward.isDefEq.lazyWhnfCore false in -- See https://github.com/leanprover-community/mathlib4/issues/12534
-/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the `BundledCorePresheafOfModules R` which
-corresponds to the presheaf of modules which sends `X` to the limit of `F ⋙ evaluation R X`. -/
+/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the presheaf of modules obtained by
+taking a limit in the category of modules over `R.obj X` for all `X`. -/
 @[simps]
-noncomputable def limitBundledCore : BundledCorePresheafOfModules R where
+noncomputable def limitPresheafOfModules : PresheafOfModules R where
   obj X := limit (F ⋙ evaluation R X)
   map {X Y} f := limMap (whiskerLeft F (restriction R f)) ≫
     (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv
@@ -72,7 +74,8 @@ noncomputable def limitBundledCore : BundledCorePresheafOfModules R where
     simp only [limMap_π, Functor.comp_obj, evaluation_obj, whiskerLeft_app,
       restriction_app, assoc]
     erw [preservesLimitsIso_hom_π]
-    rw [← ModuleCat.restrictScalarsId'App_inv_naturality, restrictionApp_id]
+    rw [← ModuleCat.restrictScalarsId'App_inv_naturality, map_id,
+      ModuleCat.restrictScalarsId'_inv_app]
     dsimp
   map_comp {X Y Z} f g := by
     dsimp
@@ -81,8 +84,9 @@ noncomputable def limitBundledCore : BundledCorePresheafOfModules R where
     apply limit.hom_ext
     intro j
     simp only [Functor.comp_obj, evaluation_obj, limMap_π, whiskerLeft_app, restriction_app,
-      Functor.map_comp, assoc, restrictionApp_comp]
-    erw [preservesLimitsIso_hom_π, ← ModuleCat.restrictScalarsComp'App_inv_naturality]
+      map_comp, ModuleCat.restrictScalarsComp'_inv_app, Functor.map_comp, assoc]
+    erw [preservesLimitsIso_hom_π]
+    rw [← ModuleCat.restrictScalarsComp'App_inv_naturality]
     dsimp
     rw [← Functor.map_comp_assoc, ← Functor.map_comp_assoc, assoc,
       preservesLimitsIso_inv_π]
@@ -92,27 +96,21 @@ noncomputable def limitBundledCore : BundledCorePresheafOfModules R where
     erw [limMap_π_assoc]
     dsimp
 
-/-- Given `F : J ⥤ PresheafOfModules.{v} R`, this is the canonical map
-`(limitBundledCore F).toPresheafOfModules ⟶ F.obj j` for all `j : J`. -/
-noncomputable def limitConeπApp (j : J) :
-    (limitBundledCore F).toPresheafOfModules ⟶ F.obj j :=
-  PresheafOfModules.Hom.mk'' (fun X => limit.π (F ⋙ evaluation R X) j) (fun X Y f => by
-    dsimp
-    simp only [assoc, preservesLimitsIso_inv_π]
-    apply limMap_π)
-
-@[reassoc (attr := simp)]
-lemma limitConeπApp_naturality {i j : J} (f : i ⟶ j) :
-    limitConeπApp F i ≫ F.map f = limitConeπApp F j := by
-  ext1 X
-  exact limit.w (F ⋙ evaluation R X) f
-
-/-- The (limit) cone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed for the limit
+/-- The (limit) cone for `F : J ⥤ PresheafOfModules.{v} R` that is constructed from the limit
 of `F ⋙ evaluation R X` for all `X`. -/
 @[simps]
 noncomputable def limitCone : Cone F where
-  pt := (limitBundledCore F).toPresheafOfModules
-  π := { app := limitConeπApp F }
+  pt := limitPresheafOfModules F
+  π :=
+    { app := fun j ↦
+        { app := fun X ↦ limit.π (F ⋙ evaluation R X) j
+          naturality := fun {X Y} f ↦ by
+            dsimp
+            simp only [assoc, preservesLimitsIso_inv_π]
+            apply limMap_π }
+      naturality := fun {j j'} f ↦ by
+        ext1 X
+        simpa using (limit.w (F ⋙ evaluation R X) f).symm }
 
 /-- The cone `limitCone F` is limit for any `F : J ⥤ PresheafOfModules.{v} R`. -/
 noncomputable def isLimitLimitCone : IsLimit (limitCone F) :=
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
index dd5f7c91a4e3b..ae8b9e272a34a 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
@@ -26,10 +26,6 @@ variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
 
 namespace PresheafOfModules
 
-instance {R : Dᵒᵖ ⥤ RingCat.{u}} (P : PresheafOfModules.{v} R) (F : C ⥤ D) (X : Cᵒᵖ) :
-    Module ((F.op ⋙ R).obj X) ((F.op ⋙ P.presheaf).obj X) :=
-  inferInstanceAs (Module (R.obj (F.op.obj X)) (P.presheaf.obj (F.op.obj X)))
-
 variable (F : C ⥤ D)
 
 /-- The pushforward functor on presheaves of modules for a functor `F : C ⥤ D` and
@@ -37,12 +33,16 @@ variable (F : C ⥤ D)
 by the precomposition with `F.op`. -/
 def pushforward₀ (R : Dᵒᵖ ⥤ RingCat.{u}) :
     PresheafOfModules.{v} R ⥤ PresheafOfModules.{v} (F.op ⋙ R) where
-  obj P :=
-    { presheaf := F.op ⋙ P.presheaf
-      map_smul := by intros; apply P.map_smul }
-  map {P Q} φ :=
-    { hom := whiskerLeft F.op φ.hom
-      map_smul := by intros; apply φ.map_smul }
+  obj M :=
+    { obj := fun X ↦ ModuleCat.of _ (M.obj (F.op.obj X))
+      map := fun {X Y} f ↦ M.map (F.op.map f)
+      map_id := fun X ↦ by
+        ext x
+        exact (M.congr_map_apply (F.op.map_id X) x).trans (by simp)
+      map_comp := fun f g ↦ by
+        ext x
+        exact (M.congr_map_apply (F.op.map_comp f g) x).trans (by simp) }
+  map {M₁ M₂} φ := { app := fun X ↦ φ.app _ }
 
 /-- The pushforward of presheaves of modules commutes with the forgetful functor
 to presheaves of abelian groups. -/
@@ -53,8 +53,10 @@ def pushforward₀CompToPresheaf (R : Dᵒᵖ ⥤ RingCat.{u}) :
 variable {F}
 variable {R : Dᵒᵖ ⥤ RingCat.{u}} {S : Cᵒᵖ ⥤ RingCat.{u}} (φ : S ⟶ F.op ⋙ R)
 
+attribute [local simp] pushforward₀ in
 /-- The pushforward functor `PresheafOfModules R ⥤ PresheafOfModules S` induced by
 a morphism of presheaves of rings `S ⟶ F.op ⋙ R`. -/
+@[simps! obj_obj]
 noncomputable def pushforward : PresheafOfModules.{v} R ⥤ PresheafOfModules.{v} S :=
   pushforward₀ F R ⋙ restrictScalars φ
 
@@ -64,21 +66,22 @@ noncomputable def pushforwardCompToPresheaf :
     pushforward.{v} φ ⋙ toPresheaf _ ≅ toPresheaf _ ⋙ (whiskeringLeft _ _ _).obj F.op :=
   Iso.refl _
 
--- unfortunately, `pushforward_obj_obj` and `pushforward_obj_map` cannot be both simp lemmas
-lemma pushforward_obj_obj (M : PresheafOfModules.{v} R) (X : Cᵒᵖ) :
-    ((pushforward φ).obj M).obj X =
-      (ModuleCat.restrictScalars (φ.app X)).obj (M.obj (Opposite.op (F.obj X.unop))) := rfl
-
 @[simp]
 lemma pushforward_obj_map_apply (M : PresheafOfModules.{v} R) {X Y : Cᵒᵖ} (f : X ⟶ Y)
     (m : (ModuleCat.restrictScalars (φ.app X)).obj (M.obj (Opposite.op (F.obj X.unop)))) :
-      ((pushforward φ).obj M).map f m = M.map (F.map f.unop).op m := by
-  rfl
+      DFunLike.coe
+        (α := (ModuleCat.restrictScalars (φ.app X)).obj (M.obj (Opposite.op (F.obj X.unop))))
+        (β := fun _ ↦ (ModuleCat.restrictScalars (φ.app Y)).obj
+          (M.obj (Opposite.op (F.obj Y.unop)))) (((pushforward φ).obj M).map f) m =
+        M.map (F.map f.unop).op m := rfl
 
 @[simp]
 lemma pushforward_map_app_apply {M N : PresheafOfModules.{v} R} (α : M ⟶ N) (X : Cᵒᵖ)
     (m : (ModuleCat.restrictScalars (φ.app X)).obj (M.obj (Opposite.op (F.obj X.unop)))) :
-    ((pushforward φ).map α).app X m = α.app (Opposite.op (F.obj X.unop)) m := by
-  rfl
+    DFunLike.coe
+      (α := (ModuleCat.restrictScalars (φ.app X)).obj (M.obj (Opposite.op (F.obj X.unop))))
+      (β := fun _ ↦ (ModuleCat.restrictScalars (φ.app X)).obj
+        (N.obj (Opposite.op (F.obj X.unop))))
+      (((pushforward φ).map α).app X) m = α.app (Opposite.op (F.obj X.unop)) m := rfl
 
 end PresheafOfModules
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean
index 8ec02b10ada53..db55107701398 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafification.lean
@@ -43,15 +43,19 @@ the associated sheaf of modules functor `PresheafOfModules.{v} R₀ ⥤ SheafOfM
 @[simps! (config := .lemmasOnly) map]
 noncomputable def sheafification : PresheafOfModules.{v} R₀ ⥤ SheafOfModules.{v} R where
   obj M₀ := sheafify α (CategoryTheory.toSheafify J M₀.presheaf)
-  map f := sheafifyMap _ _ _ f ((presheafToSheaf J AddCommGrp).map f.hom) (by simp)
+  map f := sheafifyMap _ _ _ f
+    ((toPresheaf R₀ ⋙ presheafToSheaf J AddCommGrp).map f)
+      (by apply toSheafify_naturality)
   map_id M₀ := by
     ext1
     apply (toPresheaf _).map_injective
-    simp [toPresheaf, sheafify]
+    simp
+    rfl
   map_comp _ _ := by
     ext1
     apply (toPresheaf _).map_injective
-    simp [toPresheaf, sheafify]
+    simp
+    rfl
 
 /-- The sheafification of presheaves of modules commutes with the functor which
 forgets the module structures. -/
@@ -75,19 +79,19 @@ noncomputable def sheafificationHomEquiv
       (P ⟶ (restrictScalars α).obj ((SheafOfModules.forget _).obj F)) := by
   apply sheafifyHomEquiv
 
-lemma sheafificationHomEquiv_hom'
+lemma toPresheaf_map_sheafificationHomEquiv_def
     {P : PresheafOfModules.{v} R₀} {F : SheafOfModules.{v} R}
     (f : (sheafification α).obj P ⟶ F) :
-    (sheafificationHomEquiv α f).hom =
-      CategoryTheory.toSheafify J P.presheaf ≫ f.val.hom := rfl
+    (toPresheaf R₀).map (sheafificationHomEquiv α f) =
+      CategoryTheory.toSheafify J P.presheaf ≫ (toPresheaf R.val).map f.val := rfl
 
-lemma sheafificationHomEquiv_hom
+lemma toPresheaf_map_sheafificationHomEquiv
     {P : PresheafOfModules.{v} R₀} {F : SheafOfModules.{v} R}
     (f : (sheafification α).obj P ⟶ F) :
-    (sheafificationHomEquiv α f).hom =
+    (toPresheaf R₀).map (sheafificationHomEquiv α f) =
       (sheafificationAdjunction J AddCommGrp).homEquiv P.presheaf
         ((SheafOfModules.toSheaf _).obj F) ((SheafOfModules.toSheaf _).map f) := by
-  rw [sheafificationHomEquiv_hom', Adjunction.homEquiv_unit]
+  rw [toPresheaf_map_sheafificationHomEquiv_def, Adjunction.homEquiv_unit]
   dsimp
 
 lemma toSheaf_map_sheafificationHomEquiv_symm
@@ -95,7 +99,7 @@ lemma toSheaf_map_sheafificationHomEquiv_symm
     (g : P ⟶ (restrictScalars α).obj ((SheafOfModules.forget _).obj F)) :
     (SheafOfModules.toSheaf _).map ((sheafificationHomEquiv α).symm g) =
       (((sheafificationAdjunction J AddCommGrp).homEquiv
-        P.presheaf ((SheafOfModules.toSheaf R).obj F)).symm g.hom) := by
+        P.presheaf ((SheafOfModules.toSheaf R).obj F)).symm ((toPresheaf R₀).map g)) := by
   obtain ⟨f, rfl⟩ := (sheafificationHomEquiv α).surjective g
   apply ((sheafificationAdjunction J AddCommGrp).homEquiv _ _).injective
   rw [Equiv.apply_symm_apply, Adjunction.homEquiv_unit, Equiv.symm_apply_apply]
@@ -104,7 +108,6 @@ lemma toSheaf_map_sheafificationHomEquiv_symm
 /-- Given a locally bijective morphism `α : R₀ ⟶ R.val` where `R₀` is a presheaf of rings
 and `R` a sheaf of rings, this is the adjunction
 `sheafification.{v} α ⊣ SheafOfModules.forget R ⋙ restrictScalars α`. -/
-@[simps! (config := .lemmasOnly) homEquiv_apply]
 noncomputable def sheafificationAdjunction :
     sheafification.{v} α ⊣ SheafOfModules.forget R ⋙ restrictScalars α :=
   Adjunction.mkOfHomEquiv
@@ -113,19 +116,22 @@ noncomputable def sheafificationAdjunction :
         apply (SheafOfModules.toSheaf _).map_injective
         rw [Functor.map_comp]
         erw [toSheaf_map_sheafificationHomEquiv_symm,
-          toSheaf_map_sheafificationHomEquiv_symm]
-        apply Adjunction.homEquiv_naturality_left_symm
+          toSheaf_map_sheafificationHomEquiv_symm α g]
+        rw [Functor.map_comp]
+        apply (CategoryTheory.sheafificationAdjunction J
+          AddCommGrp.{v}).homEquiv_naturality_left_symm
       homEquiv_naturality_right := fun {P₀ M N} f g ↦ by
         apply (toPresheaf _).map_injective
-        dsimp [toPresheaf]
-        erw [sheafificationHomEquiv_hom, sheafificationHomEquiv_hom]
-        rw [Functor.map_comp]
-        apply Adjunction.homEquiv_naturality_right }
+        erw [toPresheaf_map_sheafificationHomEquiv] }
+
+lemma sheaififcationAdjunction_homEquiv_apply {P : PresheafOfModules.{v} R₀}
+    {F : SheafOfModules.{v} R} (f : (sheafification α).obj P ⟶ F) :
+    (sheafificationAdjunction α).homEquiv P F f = sheafificationHomEquiv α f := rfl
 
 @[simp]
-lemma sheafificationAdjunction_unit_app_hom (M₀ : PresheafOfModules.{v} R₀) :
-    ((sheafificationAdjunction α).unit.app M₀).hom = CategoryTheory.toSheafify J M₀.presheaf := by
-  rfl
+lemma toPresheaf_map_sheafificationAdjunction_unit_app (M₀ : PresheafOfModules.{v} R₀) :
+    (toPresheaf _).map ((sheafificationAdjunction α).unit.app M₀) =
+      CategoryTheory.toSheafify J M₀.presheaf := rfl
 
 instance : (sheafification.{v} α).IsLeftAdjoint :=
   (sheafificationAdjunction α).isLeftAdjoint
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
index 0d10d6d1547c9..60d937b2076ae 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
@@ -16,7 +16,7 @@ of the underlying presheaf of abelian groups of `M₀`, i.e. we have a locally b
 map `φ : M₀.presheaf ⟶ A.val`, then we endow `A` with the structure of a
 sheaf of modules over `R`: this is `PresheafOfModules.sheafify α φ`.
 
-In many application, the morphism `α` shall be the identity, but this more
+In many applications, the morphism `α` shall be the identity, but this more
 general construction allows the sheafification of both the presheaf of rings
 and the presheaf of modules.
 
@@ -42,7 +42,7 @@ variable {R : Cᵒᵖ ⥤ RingCat.{u}} {M : PresheafOfModules.{v} R} {X : C} {P
 /-- The scalar multiplication of family of elements of a presheaf of modules `M` over `R`
 by a family of elements of `R`. -/
 def smul : FamilyOfElements (M.presheaf ⋙ forget _) P := fun Y f hf =>
-  HSMul.hSMul (α := R.obj (Opposite.op Y)) (β := M.presheaf.obj (Opposite.op Y)) (r f hf) (m f hf)
+  HSMul.hSMul (α := R.obj (Opposite.op Y)) (β := M.obj (Opposite.op Y)) (r f hf) (m f hf)
 
 end smul
 
@@ -57,7 +57,7 @@ include hA
 
 lemma _root_.PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective
     {Y : C} (r₀ r₀' : R₀.obj (Opposite.op Y))
-    (m₀ m₀' : M₀.presheaf.obj (Opposite.op Y))
+    (m₀ m₀' : M₀.obj (Opposite.op Y))
     (hr₀ : α.app _ r₀ = α.app _ r₀')
     (hm₀ : φ.app _ m₀ = φ.app _ m₀') :
     φ.app _ (r₀ • m₀) = φ.app _ (r₀' • m₀') := by
@@ -72,12 +72,12 @@ lemma _root_.PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective
 
 lemma isCompatible_map_smul_aux {Y Z : C} (f : Y ⟶ X) (g : Z ⟶ Y)
     (r₀ : R₀.obj (Opposite.op Y)) (r₀' : R₀.obj (Opposite.op Z))
-    (m₀ : M₀.presheaf.obj (Opposite.op Y)) (m₀' : M₀.presheaf.obj (Opposite.op Z))
+    (m₀ : M₀.obj (Opposite.op Y)) (m₀' : M₀.obj (Opposite.op Z))
     (hr₀ : α.app _ r₀ = R.map f.op r) (hr₀' : α.app _ r₀' = R.map (f.op ≫ g.op) r)
     (hm₀ : φ.app _ m₀ = A.map f.op m) (hm₀' : φ.app _ m₀' = A.map (f.op ≫ g.op) m) :
-    φ.app _ (M₀.presheaf.map g.op (r₀ • m₀)) = φ.app _ (r₀' • m₀') := by
+    φ.app _ (M₀.map g.op (r₀ • m₀)) = φ.app _ (r₀' • m₀') := by
   rw [← PresheafOfModules.Sheafify.app_eq_of_isLocallyInjective α φ hA (R₀.map g.op r₀) r₀'
-    (M₀.presheaf.map g.op m₀) m₀', M₀.map_smul]
+    (M₀.map g.op m₀) m₀', M₀.map_smul]
   · rw [hr₀', R.map_comp, comp_apply, ← hr₀, NatTrans.naturality_apply]
   · rw [hm₀', A.map_comp, AddCommGrp.coe_comp, Function.comp_apply, ← hm₀]
     erw [NatTrans.naturality_apply]
@@ -244,7 +244,7 @@ protected lemma smul_add : smul α φ r (m + m') = smul α φ r m + smul α φ r
     refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_
     all_goals apply Presheaf.imageSieve_mem
   apply A.isSeparated _ _ hS
-  rintro Y f ⟨⟨⟨r₀, hr₀⟩, ⟨m₀ : M₀.presheaf.obj _, hm₀⟩⟩, ⟨m₀' : M₀.presheaf.obj _, hm₀'⟩⟩
+  rintro Y f ⟨⟨⟨r₀, hr₀⟩, ⟨m₀ : M₀.obj _, hm₀⟩⟩, ⟨m₀' : M₀.obj _, hm₀'⟩⟩
   erw [(A.val.map f.op).map_add, map_smul_eq α φ r m f.op r₀ hr₀ m₀ hm₀,
     map_smul_eq α φ r m' f.op r₀ hr₀ m₀' hm₀',
     map_smul_eq α φ r (m + m') f.op r₀ hr₀ (m₀ + m₀')
@@ -269,7 +269,7 @@ protected lemma mul_smul : smul α φ (r * r') m = smul α φ r (smul α φ r' m
     refine J.intersection_covering (J.intersection_covering ?_ ?_) ?_
     all_goals apply Presheaf.imageSieve_mem
   apply A.isSeparated _ _ hS
-  rintro Y f ⟨⟨⟨r₀ : R₀.obj _, hr₀⟩, ⟨r₀' : R₀.obj _, hr₀'⟩⟩, ⟨m₀ : M₀.presheaf.obj _, hm₀⟩⟩
+  rintro Y f ⟨⟨⟨r₀ : R₀.obj _, hr₀⟩, ⟨r₀' : R₀.obj _, hr₀'⟩⟩, ⟨m₀ : M₀.obj _, hm₀⟩⟩
   erw [map_smul_eq α φ (r * r') m f.op (r₀ * r₀')
     (by rw [map_mul, map_mul, hr₀, hr₀']) m₀ hm₀, mul_smul,
     map_smul_eq α φ r (smul α φ r' m) f.op r₀ hr₀ (r₀' • m₀)
@@ -309,25 +309,28 @@ sheaf of abelian groups of a presheaf of modules `M₀` over `R₀`, this is
 the sheaf of modules over `R` which is obtained by endowing the sections of
 `A.val` with a scalar multiplication. -/
 noncomputable def sheafify : SheafOfModules.{v} R where
-  val :=
-    { presheaf := A.val
-      module := Sheafify.module α φ
-      map_smul := fun _ _ _ => by apply Sheafify.map_smul }
+  val := letI := Sheafify.module α φ; ofPresheaf A.val (Sheafify.map_smul _ _)
   isSheaf := A.cond
 
 /-- The canonical morphism from a presheaf of modules to its associated sheaf. -/
-@[simps]
-def toSheafify : M₀ ⟶ (restrictScalars α).obj (sheafify α φ).val where
-  hom := φ
-  map_smul X r₀ m₀ := by
+def toSheafify : M₀ ⟶ (restrictScalars α).obj (sheafify α φ).val :=
+  homMk φ (fun X r₀ m₀ ↦ by
     simpa using (Sheafify.map_smul_eq α φ (α.app _ r₀) (φ.app _ m₀) (𝟙 _)
-      r₀ (by aesop) m₀ (by simp)).symm
+      r₀ (by aesop) m₀ (by simp)).symm)
+
+@[simp]
+lemma toSheafify_app_apply (X : Cᵒᵖ) (x : M₀.obj X) :
+    DFunLike.coe (α := M₀.obj X) (β := fun _ ↦ A.val.obj X)
+      ((toSheafify α φ).app X) x = φ.app X x := rfl
+
+@[simp]
+lemma toPresheaf_map_toSheafify : (toPresheaf R₀).map (toSheafify α φ) = φ := rfl
 
-instance : Presheaf.IsLocallyInjective J (toSheafify α φ).hom := by
-  dsimp; infer_instance
+instance : IsLocallyInjective J (toSheafify α φ) := by
+  dsimp [IsLocallyInjective]; infer_instance
 
-instance : Presheaf.IsLocallySurjective J (toSheafify α φ).hom := by
-  dsimp; infer_instance
+instance : IsLocallySurjective J (toSheafify α φ) := by
+  dsimp [IsLocallySurjective]; infer_instance
 
 variable [J.WEqualsLocallyBijective AddCommGrp.{v}]
 
@@ -341,10 +344,10 @@ noncomputable def sheafifyHomEquiv' {F : PresheafOfModules.{v} R.val}
     (homEquivOfIsLocallyBijective (f := toSheafify α φ)
       (N := (restrictScalars α).obj F) hF)
 
-lemma comp_sheafifyHomEquiv'_symm_hom {F : PresheafOfModules.{v} R.val}
+lemma comp_toPresheaf_map_sheafifyHomEquiv'_symm_hom {F : PresheafOfModules.{v} R.val}
     (hF : Presheaf.IsSheaf J F.presheaf) (f : M₀ ⟶ (restrictScalars α).obj F) :
-    φ ≫ ((sheafifyHomEquiv' α φ hF).symm f).hom = f.hom :=
-  congr_arg Hom.hom ((sheafifyHomEquiv' α φ hF).apply_symm_apply f)
+    φ ≫ (toPresheaf R.val).map ((sheafifyHomEquiv' α φ hF).symm f) = (toPresheaf R₀).map f :=
+  (toPresheaf _).congr_map ((sheafifyHomEquiv' α φ hF).apply_symm_apply f)
 
 /-- The bijection
 `(sheafify α φ ⟶ F) ≃ (M₀ ⟶ (restrictScalars α).obj ((SheafOfModules.forget _).obj F))`
@@ -367,17 +370,15 @@ variable {M₀' : PresheafOfModules.{v} R₀} {A' : Sheaf J AddCommGrp.{v}}
 induced by morphisms `τ₀ : M₀ ⟶ M₀'` and `τ : A ⟶ A'`
 which satisfy `τ₀.hom ≫ φ' = φ ≫ τ.val`. -/
 @[simps]
-def sheafifyMap (fac : τ₀.hom ≫ φ' = φ ≫ τ.val) : sheafify α φ ⟶ sheafify α φ' where
-  val :=
-    { hom := τ.val
-      map_smul := by
-        let f := (sheafifyHomEquiv' α φ (by exact A'.cond)).symm (τ₀ ≫ toSheafify α φ')
-        have eq : τ.val = f.hom := ((J.W_of_isLocallyBijective φ).homEquiv _ A'.cond).injective
-          (by
-            dsimp [f]
-            erw [comp_sheafifyHomEquiv'_symm_hom]
-            simp only [← fac, toSheafify_hom, Hom.comp_hom])
-        convert f.map_smul }
+def sheafifyMap (fac : (toPresheaf R₀).map τ₀ ≫ φ' = φ ≫ τ.val) :
+    sheafify α φ ⟶ sheafify α φ' where
+  val := homMk τ.val (fun X r m ↦ by
+    let f := (sheafifyHomEquiv' α φ (by exact A'.cond)).symm (τ₀ ≫ toSheafify α φ')
+    suffices τ.val = (toPresheaf _).map f by simpa only [this] using (f.app X).map_smul r m
+    apply ((J.W_of_isLocallyBijective φ).homEquiv _ A'.cond).injective
+    dsimp [f]
+    erw [comp_toPresheaf_map_sheafifyHomEquiv'_symm_hom]
+    rw [← fac, Functor.map_comp, toPresheaf_map_toSheafify])
 
 end
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Products.lean b/Mathlib/Algebra/Category/ModuleCat/Products.lean
index b4aa2e92eb57f..b986cea15a7f5 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Products.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Products.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.LinearAlgebra.Pi
diff --git a/Mathlib/Algebra/Category/ModuleCat/Projective.lean b/Mathlib/Algebra/Category/ModuleCat/Projective.lean
index 80b2ec54bb07d..7cc29fe3e1107 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Projective.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Projective.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.EpiMono
 import Mathlib.Algebra.Module.Projective
@@ -59,11 +59,11 @@ instance moduleCat_enoughProjectives : EnoughProjectives (ModuleCat.{max u v} R)
         f := Finsupp.basisSingleOne.constr ℕ _root_.id
         epi := (epi_iff_range_eq_top _).mpr
             (range_eq_top.2 fun m => ⟨Finsupp.single m (1 : R), by
-              -- Porting note: simp [Finsupp.total_single] fails but rw succeeds
+              -- Porting note: simp [Finsupp.linearCombination_single] fails but rw succeeds
               dsimp [Basis.constr]
               simp only [Finsupp.lmapDomain_id, comp_id]
               -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-              erw [Finsupp.total_single]
+              erw [Finsupp.linearCombination_single]
               rw [one_smul]
               rfl ⟩) }⟩
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Sheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Sheaf.lean
index fb6b24bc7f604..3a23ba4b4c02e 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Sheaf.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Sheaf.lean
@@ -15,12 +15,6 @@ import Mathlib.CategoryTheory.Sites.Whiskering
 In this file, we define the category `SheafOfModules R` when `R : Sheaf J RingCat`
 is a sheaf of rings on a category `C` equipped with a Grothendieck topology `J`.
 
-## TODO
-* construct the associated sheaf: more precisely, given a morphism of `α : P ⟶ R.val`
-where `P` is a presheaf of rings and `R` a sheaf of rings such that `α` identifies
-`R` to the associated sheaf of `P`, then construct a sheafification functor
-`PresheafOfModules P ⥤ SheafOfModules R`.
-
 -/
 
 universe v v₁ u₁ u w
@@ -91,7 +85,7 @@ def evaluation (X : Cᵒᵖ) : SheafOfModules.{v} R ⥤ ModuleCat.{v} (R.val.obj
 @[simps]
 def toSheaf : SheafOfModules.{v} R ⥤ Sheaf J AddCommGrp.{v} where
   obj M := ⟨_, M.isSheaf⟩
-  map f := { val := f.val.hom }
+  map f := { val := (forget R ⋙ PresheafOfModules.toPresheaf R.val).map f }
 
 /--
 The forgetful functor from sheaves of modules over sheaf of ring `R` to sheaves of `R(X)`-module
@@ -185,36 +179,55 @@ end SheafOfModules
 
 namespace PresheafOfModules
 
-variable {R : Cᵒᵖ ⥤ RingCat.{u}} {M₁ M₂ : PresheafOfModules.{v} R}
-    (f : M₁ ⟶ M₂) {N : PresheafOfModules.{v} R}
-    (hN : Presheaf.IsSheaf J N.presheaf)
-    [J.WEqualsLocallyBijective AddCommGrp.{v}]
-    [Presheaf.IsLocallySurjective J f.hom]
-    [Presheaf.IsLocallyInjective J f.hom]
+variable (J)
+variable {R : Cᵒᵖ ⥤ RingCat.{u}} {M₁ M₂ : PresheafOfModules.{v} R} (f : M₁ ⟶ M₂)
+
+/-- A morphism of presheaves of modules is locally surjective
+if the underlying morphism of presheaves of abelian groups is. -/
+abbrev IsLocallySurjective : Prop :=
+  Presheaf.IsLocallySurjective J ((PresheafOfModules.toPresheaf R).map f)
+
+/-- A morphism of presheaves of modules is locally injective
+if the underlying morphism of presheaves of abelian groups is. -/
+abbrev IsLocallyInjective : Prop :=
+  Presheaf.IsLocallyInjective J ((PresheafOfModules.toPresheaf R).map f)
+
+variable {N : PresheafOfModules.{v} R} (hN : Presheaf.IsSheaf J N.presheaf)
+  [J.WEqualsLocallyBijective AddCommGrp.{v}]
+  [IsLocallySurjective J f] [IsLocallyInjective J f]
+
+variable {J}
 
 /-- The bijection `(M₂ ⟶ N) ≃ (M₁ ⟶ N)` induced by a locally bijective morphism
 `f : M₁ ⟶ M₂` of presheaves of modules, when `N` is a sheaf. -/
 @[simps]
 noncomputable def homEquivOfIsLocallyBijective : (M₂ ⟶ N) ≃ (M₁ ⟶ N) where
   toFun φ := f ≫ φ
-  invFun ψ :=
-    { hom := ((J.W_of_isLocallyBijective f.hom).homEquiv _ hN).symm ψ.hom
-      map_smul := by
-        obtain ⟨φ, hφ⟩ := ((J.W_of_isLocallyBijective f.hom).homEquiv _ hN).surjective ψ.hom
+  invFun ψ := homMk (((J.W_of_isLocallyBijective
+      ((PresheafOfModules.toPresheaf R).map f)).homEquiv _ hN).symm
+      ((PresheafOfModules.toPresheaf R).map ψ)) (by
+        obtain ⟨φ, hφ⟩ := ((J.W_of_isLocallyBijective
+          ((PresheafOfModules.toPresheaf R).map f)).homEquiv _ hN).surjective
+          ((PresheafOfModules.toPresheaf R).map ψ)
         simp only [← hφ, Equiv.symm_apply_apply]
-        dsimp at hφ
+        replace hφ : ∀ (Z : Cᵒᵖ) (x : M₁.obj Z), φ.app Z (f.app Z x) = ψ.app Z x :=
+          fun Z x ↦ congr_fun ((forget _).congr_map (congr_app hφ Z)) x
         intro X r y
-        apply hN.isSeparated _ _ (Presheaf.imageSieve_mem J f.hom y)
-        rintro Y p ⟨x, hx⟩
-        have eq := ψ.map_smul _ (R.map p.op r) x
-        simp only [← hφ] at eq
-        dsimp at eq
-        erw [← NatTrans.naturality_apply φ p.op (r • y), N.map_smul, M₂.map_smul,
-          ← NatTrans.naturality_apply φ p.op y, ← hx, ← eq, f.map_smul]
-        rfl }
+        apply hN.isSeparated _ _
+          (Presheaf.imageSieve_mem J ((toPresheaf R).map f) y)
+        rintro Y p ⟨x : M₁.obj _, hx : f.app _ x = M₂.map p.op y⟩
+        have hφ' : ∀ (z : M₂.obj X), φ.app _ (M₂.map p.op z) =
+            N.map p.op (φ.app _ z) := congr_fun ((forget _).congr_map (φ.naturality p.op))
+        change N.map p.op (φ.app X (r • y)) = N.map p.op (r • φ.app X y)
+        rw [← hφ', M₂.map_smul, ← hx, ← (f.app _).map_smul, hφ, (ψ.app _).map_smul,
+          ← hφ, hx, N.map_smul, hφ'])
   left_inv φ := (toPresheaf _).map_injective
-    (((J.W_of_isLocallyBijective f.hom).homEquiv _ hN).left_inv φ.hom)
+    (((J.W_of_isLocallyBijective
+      ((PresheafOfModules.toPresheaf R).map f)).homEquiv _ hN).left_inv
+      ((PresheafOfModules.toPresheaf R).map φ))
   right_inv ψ := (toPresheaf _).map_injective
-    (((J.W_of_isLocallyBijective f.hom).homEquiv _ hN).right_inv ψ.hom)
+    (((J.W_of_isLocallyBijective
+      ((PresheafOfModules.toPresheaf R).map f)).homEquiv _ hN).right_inv
+      ((PresheafOfModules.toPresheaf R).map ψ))
 
 end PresheafOfModules
diff --git a/Mathlib/Algebra/Category/ModuleCat/Sheaf/ChangeOfRings.lean b/Mathlib/Algebra/Category/ModuleCat/Sheaf/ChangeOfRings.lean
index 906a646c6048c..53503eb352661 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Sheaf/ChangeOfRings.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Sheaf/ChangeOfRings.lean
@@ -45,22 +45,23 @@ namespace PresheafOfModules
 variable {R R' : Cᵒᵖ ⥤ RingCat.{u}} (α : R ⟶ R')
   {M₁ M₂ : PresheafOfModules.{v} R'}
 
-/-- The functor `PresheafOfModules.restrictScalars α` induces bijection on
+/-- The functor `PresheafOfModules.restrictScalars α` induces bijections on
 morphisms if `α` is locally surjective and the target presheaf is a sheaf. -/
 noncomputable def restrictHomEquivOfIsLocallySurjective
     (hM₂ : Presheaf.IsSheaf J M₂.presheaf) [Presheaf.IsLocallySurjective J α] :
     (M₁ ⟶ M₂) ≃ ((restrictScalars α).obj M₁ ⟶ (restrictScalars α).obj M₂) where
   toFun f := (restrictScalars α).map f
-  invFun g :=
-    { hom := g.hom
-      map_smul := fun X r' m => by
-        apply hM₂.isSeparated _ _ (Presheaf.imageSieve_mem J α r')
-        rintro Y p ⟨r : R.obj _, hr⟩
-        erw [M₂.map_smul, ← NatTrans.naturality_apply g.hom p.op m,
-          ← hr, ← g.map_smul _ r (M₁.presheaf.map p.op m),
-          ← NatTrans.naturality_apply g.hom p.op (r' • m),
-          M₁.map_smul p.op r' m, ← hr]
-        rfl }
+  invFun g := homMk ((toPresheaf R).map g) (fun X r' m ↦ by
+    apply hM₂.isSeparated _ _ (Presheaf.imageSieve_mem J α r')
+    rintro Y p ⟨r : R.obj _, hr⟩
+    have hg : ∀ (z : M₁.obj X), g.app _ (M₁.map p.op z) = M₂.map p.op (g.app X z) :=
+      fun z ↦ congr_fun ((forget _).congr_map (g.naturality p.op)) z
+    change M₂.map p.op (g.app X (r' • m)) = M₂.map p.op (r' • show M₂.obj X from g.app X m)
+    dsimp at hg ⊢
+    rw [← hg, M₂.map_smul, ← hg, ← hr]
+    erw [← (g.app _).map_smul]
+    rw [M₁.map_smul, ← hr]
+    rfl)
   left_inv _ := rfl
   right_inv _ := rfl
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Sheaf/Generators.lean b/Mathlib/Algebra/Category/ModuleCat/Sheaf/Generators.lean
index ef7b06211193e..fdd0855f1ac33 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Sheaf/Generators.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Sheaf/Generators.lean
@@ -66,8 +66,6 @@ def ofEpi (σ : M.GeneratingSections) (p : M ⟶ N) [Epi p] :
     rw [← freeHomEquiv_symm_comp]
     apply epi_comp
 
-attribute [local instance] epi_comp
-
 lemma opEpi_id (σ : M.GeneratingSections) :
     σ.ofEpi (𝟙 M) = σ := rfl
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Simple.lean b/Mathlib/Algebra/Category/ModuleCat/Simple.lean
index f693c5fcbc18c..a36bd8151ab84 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Simple.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Simple.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Pierre-Alexandre Bazin, Scott Morrison
+Authors: Pierre-Alexandre Bazin, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Simple
 import Mathlib.Algebra.Category.ModuleCat.Subobject
@@ -34,7 +34,7 @@ instance simple_of_isSimpleModule [IsSimpleModule R M] : Simple (of R M) :=
 instance isSimpleModule_of_simple (M : ModuleCat R) [Simple M] : IsSimpleModule R M :=
   simple_iff_isSimpleModule.mp (Simple.of_iso (ofSelfIso M))
 
-open FiniteDimensional
+open Module
 
 attribute [local instance] moduleOfAlgebraModule isScalarTower_of_algebra_moduleCat
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Tannaka.lean b/Mathlib/Algebra/Category/ModuleCat/Tannaka.lean
index 747def447779c..3d2e15121ab7e 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Tannaka.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Tannaka.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.LinearAlgebra.Span
diff --git a/Mathlib/Algebra/Category/MonCat/Basic.lean b/Mathlib/Algebra/Category/MonCat/Basic.lean
index d4510713aa6c0..22bdae9208498 100644
--- a/Mathlib/Algebra/Category/MonCat/Basic.lean
+++ b/Mathlib/Algebra/Category/MonCat/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
 import Mathlib.Algebra.PUnitInstances.Algebra
diff --git a/Mathlib/Algebra/Category/MonCat/Colimits.lean b/Mathlib/Algebra/Category/MonCat/Colimits.lean
index 98bd6071b109b..bfb19dcc5a1bc 100644
--- a/Mathlib/Algebra/Category/MonCat/Colimits.lean
+++ b/Mathlib/Algebra/Category/MonCat/Colimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.MonCat.Basic
 import Mathlib.CategoryTheory.Limits.HasLimits
diff --git a/Mathlib/Algebra/Category/MonCat/FilteredColimits.lean b/Mathlib/Algebra/Category/MonCat/FilteredColimits.lean
index f565afec21772..9624ff56fed29 100644
--- a/Mathlib/Algebra/Category/MonCat/FilteredColimits.lean
+++ b/Mathlib/Algebra/Category/MonCat/FilteredColimits.lean
@@ -56,7 +56,7 @@ noncomputable abbrev M.mk : (Σ j, F.obj j) → M.{v, u} F :=
 theorem M.mk_eq (x y : Σ j, F.obj j)
     (h : ∃ (k : J) (f : x.1 ⟶ k) (g : y.1 ⟶ k), F.map f x.2 = F.map g y.2) :
     M.mk.{v, u} F x = M.mk F y :=
-  Quot.EqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget MonCat) x y h)
+  Quot.eqvGen_sound (Types.FilteredColimit.eqvGen_quot_rel_of_rel (F ⋙ forget MonCat) x y h)
 
 variable [IsFiltered J]
 
diff --git a/Mathlib/Algebra/Category/MonCat/Limits.lean b/Mathlib/Algebra/Category/MonCat/Limits.lean
index 0de870438eedb..77a05ad7f091f 100644
--- a/Mathlib/Algebra/Category/MonCat/Limits.lean
+++ b/Mathlib/Algebra/Category/MonCat/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.MonCat.Basic
 import Mathlib.Algebra.Group.Pi.Lemmas
diff --git a/Mathlib/Algebra/Category/Ring/Adjunctions.lean b/Mathlib/Algebra/Category/Ring/Adjunctions.lean
index 49cbc29469c4e..7e9619ecea87c 100644
--- a/Mathlib/Algebra/Category/Ring/Adjunctions.lean
+++ b/Mathlib/Algebra/Category/Ring/Adjunctions.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johannes Hölzl
+Authors: Kim Morrison, Johannes Hölzl
 -/
 import Mathlib.Algebra.Category.Ring.Basic
 import Mathlib.Algebra.MvPolynomial.CommRing
diff --git a/Mathlib/Algebra/Category/Ring/Basic.lean b/Mathlib/Algebra/Category/Ring/Basic.lean
index f195911a46493..4542d373913f0 100644
--- a/Mathlib/Algebra/Category/Ring/Basic.lean
+++ b/Mathlib/Algebra/Category/Ring/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johannes Hölzl, Yury Kudryashov
+Authors: Kim Morrison, Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Category.Grp.Basic
 import Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso
@@ -546,6 +546,12 @@ theorem coe_of (R : Type u) [CommRing R] : (CommRingCat.of R : Type u) = R :=
 instance hasForgetToRingCat : HasForget₂ CommRingCat RingCat :=
   BundledHom.forget₂ _ _
 
+@[simp] lemma forgetToRingCat_obj (A : CommRingCat.{u}) :
+    ((forget₂ _ RingCat).obj A : Type _) = A := rfl
+
+@[simp] lemma forgetToRingCat_map_apply {A B : CommRingCat.{u}} (f : A ⟶ B) (a : A) :
+    DFunLike.coe (α := A) (β := fun _ ↦ B) ((forget₂ _ RingCat).map f) a = f a := rfl
+
 /-- The forgetful functor from commutative rings to (multiplicative) commutative monoids. -/
 instance hasForgetToCommSemiRingCat : HasForget₂ CommRingCat CommSemiRingCat :=
   HasForget₂.mk' (fun R : CommRingCat => CommSemiRingCat.of R) (fun R => rfl)
diff --git a/Mathlib/Algebra/Category/Ring/Colimits.lean b/Mathlib/Algebra/Category/Ring/Colimits.lean
index 5d7cc51769fd3..b2c1204a2af62 100644
--- a/Mathlib/Algebra/Category/Ring/Colimits.lean
+++ b/Mathlib/Algebra/Category/Ring/Colimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Ring.Basic
 import Mathlib.CategoryTheory.Limits.HasLimits
diff --git a/Mathlib/Algebra/Category/Ring/Instances.lean b/Mathlib/Algebra/Category/Ring/Instances.lean
index c1e005f022104..fc588d27c8248 100644
--- a/Mathlib/Algebra/Category/Ring/Instances.lean
+++ b/Mathlib/Algebra/Category/Ring/Instances.lean
@@ -38,7 +38,7 @@ instance Localization.epi' {R : CommRingCat} (M : Submonoid R) :
 
 instance CommRingCat.isLocalRingHom_comp {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T)
     [IsLocalRingHom g] [IsLocalRingHom f] : IsLocalRingHom (f ≫ g) :=
-  _root_.isLocalRingHom_comp _ _
+  RingHom.isLocalRingHom_comp _ _
 
 theorem isLocalRingHom_of_iso {R S : CommRingCat} (f : R ≅ S) : IsLocalRingHom f.hom :=
   { map_nonunit := fun a ha => by
diff --git a/Mathlib/Algebra/Category/Ring/Limits.lean b/Mathlib/Algebra/Category/Ring/Limits.lean
index a7f813b611915..a0c514581878e 100644
--- a/Mathlib/Algebra/Category/Ring/Limits.lean
+++ b/Mathlib/Algebra/Category/Ring/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Ring.Pi
 import Mathlib.Algebra.Category.Ring.Basic
diff --git a/Mathlib/Algebra/CharP/CharAndCard.lean b/Mathlib/Algebra/CharP/CharAndCard.lean
index bbd284531afbf..75671724d2c94 100644
--- a/Mathlib/Algebra/CharP/CharAndCard.lean
+++ b/Mathlib/Algebra/CharP/CharAndCard.lean
@@ -70,8 +70,8 @@ theorem prime_dvd_char_iff_dvd_card {R : Type*} [CommRing R] [Fintype R] (p : 
   rw [mul_zero, ← mul_assoc, hu, one_mul] at hr₁
   exact mt AddMonoid.addOrderOf_eq_one_iff.mpr (ne_of_eq_of_ne hr (Nat.Prime.ne_one Fact.out)) hr₁
 
-/-- A prime that does not divide the cardinality of a finite commutative ring `R`
-is a unit in `R`. -/
+/-- A prime that divides the cardinality of a finite commutative ring `R`
+isn't a unit in `R`. -/
 theorem not_isUnit_prime_of_dvd_card {R : Type*} [CommRing R] [Fintype R] (p : ℕ) [Fact p.Prime]
     (hp : p ∣ Fintype.card R) : ¬IsUnit (p : R) :=
   mt (isUnit_iff_not_dvd_char R p).mp
diff --git a/Mathlib/Algebra/CharP/Defs.lean b/Mathlib/Algebra/CharP/Defs.lean
index b9c236c16d197..34b448a97de44 100644
--- a/Mathlib/Algebra/CharP/Defs.lean
+++ b/Mathlib/Algebra/CharP/Defs.lean
@@ -11,6 +11,7 @@ import Mathlib.Data.Nat.Cast.Prod
 import Mathlib.Data.Nat.Find
 import Mathlib.Data.Nat.Prime.Defs
 import Mathlib.Data.ULift
+import Mathlib.Tactic.NormNum.Basic
 
 /-!
 # Characteristic of semirings
@@ -18,6 +19,8 @@ import Mathlib.Data.ULift
 
 assert_not_exists Finset
 
+open Set
+
 variable (R : Type*)
 
 namespace CharP
@@ -91,12 +94,19 @@ end AddMonoidWithOne
 section AddGroupWithOne
 variable [AddGroupWithOne R] (p : ℕ) [CharP R p] {a b : ℤ}
 
+lemma intCast_injOn_Ico [IsRightCancelAdd R] : InjOn (Int.cast : ℤ → R) (Ico 0 p) := by
+  rintro a ⟨ha₀, ha⟩ b ⟨hb₀, hb⟩ hab
+  lift a to ℕ using ha₀
+  lift b to ℕ using hb₀
+  norm_cast at *
+  exact natCast_injOn_Iio _ _ ha hb hab
+
 lemma intCast_eq_zero_iff (a : ℤ) : (a : R) = 0 ↔ (p : ℤ) ∣ a := by
   rcases lt_trichotomy a 0 with (h | rfl | h)
-  · rw [← neg_eq_zero, ← Int.cast_neg, ← dvd_neg]
+  · rw [← neg_eq_zero, ← Int.cast_neg, ← Int.dvd_neg]
     lift -a to ℕ using neg_nonneg.mpr (le_of_lt h) with b
     rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
-  · simp only [Int.cast_zero, eq_self_iff_true, dvd_zero]
+  · simp only [Int.cast_zero, eq_self_iff_true, Int.dvd_zero]
   · lift a to ℕ using le_of_lt h with b
     rw [Int.cast_natCast, CharP.cast_eq_zero_iff R p, Int.natCast_dvd_natCast]
 
@@ -411,7 +421,6 @@ end CharZero
 
 namespace Fin
 
-instance charP (n : ℕ) : CharP (Fin (n + 1)) (n + 1) where
-    cast_eq_zero_iff' := by simp [Fin.ext_iff, Nat.dvd_iff_mod_eq_zero]
+instance charP (n : ℕ) [NeZero n] : CharP (Fin n) n where cast_eq_zero_iff' _ := natCast_eq_zero
 
 end Fin
diff --git a/Mathlib/Algebra/CharP/IntermediateField.lean b/Mathlib/Algebra/CharP/IntermediateField.lean
index 4424731754203..a577f00bd38d8 100644
--- a/Mathlib/Algebra/CharP/IntermediateField.lean
+++ b/Mathlib/Algebra/CharP/IntermediateField.lean
@@ -5,6 +5,7 @@ Authors: Jz Pan
 -/
 import Mathlib.Algebra.CharP.ExpChar
 import Mathlib.FieldTheory.IntermediateField.Basic
+import Mathlib.Algebra.EuclideanDomain.Field
 
 /-!
 
diff --git a/Mathlib/Algebra/CharZero/Defs.lean b/Mathlib/Algebra/CharZero/Defs.lean
index 8f2cf74d4cef2..c82356e6fa6e7 100644
--- a/Mathlib/Algebra/CharZero/Defs.lean
+++ b/Mathlib/Algebra/CharZero/Defs.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 import Mathlib.Data.Int.Cast.Defs
-import Mathlib.Algebra.NeZero
 import Mathlib.Logic.Function.Basic
 
 /-!
diff --git a/Mathlib/Algebra/CharZero/Lemmas.lean b/Mathlib/Algebra/CharZero/Lemmas.lean
index 4a3176488e297..15bdc16d9f97e 100644
--- a/Mathlib/Algebra/CharZero/Lemmas.lean
+++ b/Mathlib/Algebra/CharZero/Lemmas.lean
@@ -79,7 +79,7 @@ variable {R : Type*} [NonAssocSemiring R] [NoZeroDivisors R] [CharZero R] {a : R
 
 @[simp]
 theorem add_self_eq_zero {a : R} : a + a = 0 ↔ a = 0 := by
-  simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or_iff]
+  simp only [(two_mul a).symm, mul_eq_zero, two_ne_zero, false_or]
 
 end
 
diff --git a/Mathlib/Algebra/ContinuedFractions/Basic.lean b/Mathlib/Algebra/ContinuedFractions/Basic.lean
index cdf31ebeb15da..6a34dbcebe789 100644
--- a/Mathlib/Algebra/ContinuedFractions/Basic.lean
+++ b/Mathlib/Algebra/ContinuedFractions/Basic.lean
@@ -41,7 +41,7 @@ numerics, number theory, approximations, fractions
 -- Fix a carrier `α`.
 variable (α : Type*)
 
-/-!### Definitions-/
+/-!### Definitions -/
 
 -- Porting note: Originally `protected structure GenContFract.Pair`
 /-- We collect a partial numerator `aᵢ` and partial denominator `bᵢ` in a pair `⟨aᵢ, bᵢ⟩`. -/
diff --git a/Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean b/Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean
index 34d81f490bea6..73f695ecae794 100644
--- a/Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean
+++ b/Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean
@@ -97,9 +97,9 @@ theorem of_convergence_epsilon :
     have zero_lt_B : 0 < B := B_ineq.trans_lt' <| mod_cast fib_pos.2 n.succ_pos
     have nB_pos : 0 < nB := nB_ineq.trans_lt' <| mod_cast fib_pos.2 <| succ_pos _
     have zero_lt_mul_conts : 0 < B * nB := by positivity
-    suffices 1 < ε * (B * nB) from (div_lt_iff zero_lt_mul_conts).mpr this
+    suffices 1 < ε * (B * nB) from (div_lt_iff₀ zero_lt_mul_conts).mpr this
     -- use that `N' ≥ n` was obtained from the archimedean property to show the following
-    calc 1 < ε * (N' : K) := (div_lt_iff' ε_pos).mp one_div_ε_lt_N'
+    calc 1 < ε * (N' : K) := (div_lt_iff₀' ε_pos).mp one_div_ε_lt_N'
       _ ≤ ε * (B * nB) := ?_
     -- cancel `ε`
     gcongr
diff --git a/Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean b/Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean
index be6b7195e1964..e2e525d76dcdc 100644
--- a/Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean
+++ b/Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean
@@ -94,12 +94,9 @@ theorem one_le_succ_nth_stream_b {ifp_succ_n : IntFractPair K}
       ∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0
         ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n :=
     succ_nth_stream_eq_some_iff.1 succ_nth_stream_eq
-  suffices 1 ≤ ifp_n.fr⁻¹ by rwa [IntFractPair.of, le_floor, cast_one]
-  suffices ifp_n.fr ≤ 1 by
-    have h : 0 < ifp_n.fr :=
-      lt_of_le_of_ne (nth_stream_fr_nonneg nth_stream_eq) stream_nth_fr_ne_zero.symm
-    apply one_le_inv h this
-  simp only [le_of_lt (nth_stream_fr_lt_one nth_stream_eq)]
+  rw [IntFractPair.of, le_floor, cast_one, one_le_inv₀
+    ((nth_stream_fr_nonneg nth_stream_eq).lt_of_ne' stream_nth_fr_ne_zero)]
+  exact (nth_stream_fr_lt_one nth_stream_eq).le
 
 /--
 Shows that the `n + 1`th integer part `bₙ₊₁` of the stream is smaller or equal than the inverse of
diff --git a/Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean b/Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
index 33ad6a6ff6c3e..d732ff25b6a7a 100644
--- a/Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
+++ b/Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
@@ -249,7 +249,7 @@ theorem of_s_head_aux (v : K) : (of v).s.get? 0 = (IntFractPair.stream v 1).bind
   rw [of, IntFractPair.seq1]
   simp only [of, Stream'.Seq.map_tail, Stream'.Seq.map, Stream'.Seq.tail, Stream'.Seq.head,
     Stream'.Seq.get?, Stream'.map]
-  rw [← Stream'.get_succ, Stream'.get, Option.map]
+  rw [← Stream'.get_succ, Stream'.get, Option.map.eq_def]
   split <;> simp_all only [Option.some_bind, Option.none_bind, Function.comp_apply]
 
 /-- This gives the first pair of coefficients of the continued fraction of a non-integer `v`.
diff --git a/Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean b/Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
index 2adf582909701..99e76fa64a5f5 100644
--- a/Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
+++ b/Mathlib/Algebra/ContinuedFractions/ConvergentsEquiv.lean
@@ -166,9 +166,6 @@ theorem succ_succ_nth_conv'Aux_eq_succ_nth_conv'Aux_squashSeq :
         gp_head.a / (gp_head.b + convs'Aux s.tail (m + 2)) =
           convs'Aux (squashSeq s (m + 1)) (m + 2)
         by simpa only [convs'Aux, s_head_eq]
-      have : convs'Aux s.tail (m + 2) = convs'Aux (squashSeq s.tail m) (m + 1) := by
-        refine IH gp_succ_n ?_
-        simpa [Stream'.Seq.get?_tail] using s_succ_nth_eq
       have : (squashSeq s (m + 1)).head = some gp_head :=
         (squashSeq_nth_of_lt m.succ_pos).trans s_head_eq
       simp_all [convs'Aux, squashSeq_succ_n_tail_eq_squashSeq_tail_n]
diff --git a/Mathlib/Algebra/DirectLimit.lean b/Mathlib/Algebra/DirectLimit.lean
index f1aa6c7623464..f077a686dc355 100644
--- a/Mathlib/Algebra/DirectLimit.lean
+++ b/Mathlib/Algebra/DirectLimit.lean
@@ -316,12 +316,8 @@ theorem of.zero_exact_aux [∀ i (k : G i), Decidable (k ≠ 0)] [Nonempty ι] [
         ⟨k, fun l hl =>
           (Finset.mem_union.1 (DFinsupp.support_add hl)).elim (fun hl => le_trans (hi _ hl) hik)
             fun hl => le_trans (hj _ hl) hjk, by
-          -- Porting note: this had been
-          -- simp [LinearMap.map_add, hxi, hyj, toModule_totalize_of_le hik hi,
-          --   toModule_totalize_of_le hjk hj]
-          simp only [map_add]
-          rw [toModule_totalize_of_le hik hi, toModule_totalize_of_le hjk hj]
-          simp [hxi, hyj]⟩)
+          simp [LinearMap.map_add, hxi, hyj, toModule_totalize_of_le hik hi,
+             toModule_totalize_of_le hjk hj]⟩)
       fun a x ⟨i, hi, hxi⟩ =>
       ⟨i, fun k hk => hi k (DirectSum.support_smul _ _ hk), by simp [LinearMap.map_smul, hxi]⟩
 
@@ -721,9 +717,7 @@ theorem of.zero_exact_aux [Nonempty ι] [IsDirected ι (· ≤ ·)] {x : FreeCom
           dsimp only
           rw [(f' i i _).map_mul]
           · exact sub_self _
-        all_goals tauto
-        -- Porting note: was
-        --exacts [sub_self _, Or.inl rfl, Or.inr (Or.inr rfl), Or.inr (Or.inl rfl)]
+        exacts [Or.inl rfl, Or.inr (Or.inr rfl), Or.inr (Or.inl rfl)]
   · refine Nonempty.elim (by infer_instance) fun ind : ι => ?_
     refine ⟨ind, ∅, fun _ => False.elim, isSupported_zero, fun [_] => ?_⟩
     -- Porting note: `RingHom.map_zero` was `(restriction _).map_zero`
diff --git a/Mathlib/Algebra/DirectSum/LinearMap.lean b/Mathlib/Algebra/DirectSum/LinearMap.lean
index c3b5c570dd966..25ffc82193de4 100644
--- a/Mathlib/Algebra/DirectSum/LinearMap.lean
+++ b/Mathlib/Algebra/DirectSum/LinearMap.lean
@@ -78,12 +78,12 @@ lemma trace_eq_sum_trace_restrict' (h : IsInternal N) (hN : {i | N i ≠ ⊥}.Fi
   rw [← Finset.sum_coe_sort, trace_eq_sum_trace_restrict (isInternal_ne_bot_iff.mpr h) _]
   exact Fintype.sum_equiv hN.subtypeEquivToFinset _ _ (fun i ↦ rfl)
 
-lemma trace_eq_zero_of_mapsTo_ne (h : IsInternal N) [hn : IsNoetherian R M]
+lemma trace_eq_zero_of_mapsTo_ne (h : IsInternal N) [IsNoetherian R M]
     (σ : ι → ι) (hσ : ∀ i, σ i ≠ i) {f : Module.End R M}
     (hf : ∀ i, MapsTo f (N i) (N <| σ i)) :
     trace R M f = 0 := by
-  have hN : {i | N i ≠ ⊥}.Finite := CompleteLattice.WellFounded.finite_ne_bot_of_independent
-    hn.wf h.submodule_independent
+  have hN : {i | N i ≠ ⊥}.Finite := CompleteLattice.WellFoundedGT.finite_ne_bot_of_independent
+    h.submodule_independent
   let s := hN.toFinset
   let κ := fun i ↦ Module.Free.ChooseBasisIndex R (N i)
   let b : (i : s) → Basis (κ i) R (N i) := fun i ↦ Module.Free.chooseBasis R (N i)
@@ -100,27 +100,26 @@ lemma trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero
     [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M]
     {f g : Module.End R M}
     (h_comm : Commute f g)
-    (hf : ⨆ μ, ⨆ k, f.genEigenspace μ k = ⊤)
-    (hg : ∀ μ, trace R _ (g.restrict (f.mapsTo_iSup_genEigenspace_of_comm h_comm μ)) = 0) :
+    (hf : ⨆ μ, f.maxGenEigenspace μ = ⊤)
+    (hg : ∀ μ, trace R _ (g.restrict (f.mapsTo_maxGenEigenspace_of_comm h_comm μ)) = 0) :
     trace R _ (g ∘ₗ f) = 0 := by
   have hfg : ∀ μ,
-      MapsTo (g ∘ₗ f) ↑(⨆ k, f.genEigenspace μ k) ↑(⨆ k, f.genEigenspace μ k) :=
-    fun μ ↦ (f.mapsTo_iSup_genEigenspace_of_comm h_comm μ).comp
-      (f.mapsTo_iSup_genEigenspace_of_comm rfl μ)
+      MapsTo (g ∘ₗ f) ↑(f.maxGenEigenspace μ) ↑(f.maxGenEigenspace μ) :=
+    fun μ ↦ (f.mapsTo_maxGenEigenspace_of_comm h_comm μ).comp
+      (f.mapsTo_maxGenEigenspace_of_comm rfl μ)
   suffices ∀ μ, trace R _ ((g ∘ₗ f).restrict (hfg μ)) = 0 by
     classical
     have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
-      f.independent_genEigenspace hf
-    have h_fin : {μ | ⨆ k, f.genEigenspace μ k ≠ ⊥}.Finite :=
-      CompleteLattice.WellFounded.finite_ne_bot_of_independent IsWellFounded.wf
-        f.independent_genEigenspace
+      f.independent_maxGenEigenspace hf
+    have h_fin : {μ | f.maxGenEigenspace μ ≠ ⊥}.Finite :=
+      CompleteLattice.WellFoundedGT.finite_ne_bot_of_independent f.independent_maxGenEigenspace
     simp [trace_eq_sum_trace_restrict' hds h_fin hfg, this]
   intro μ
-  replace h_comm : Commute (g.restrict (f.mapsTo_iSup_genEigenspace_of_comm h_comm μ))
-      (f.restrict (f.mapsTo_iSup_genEigenspace_of_comm rfl μ)) :=
+  replace h_comm : Commute (g.restrict (f.mapsTo_maxGenEigenspace_of_comm h_comm μ))
+      (f.restrict (f.mapsTo_maxGenEigenspace_of_comm rfl μ)) :=
     restrict_commute h_comm.symm _ _
   rw [restrict_comp, trace_comp_eq_mul_of_commute_of_isNilpotent μ h_comm
-    (f.isNilpotent_restrict_iSup_sub_algebraMap μ), hg, mul_zero]
+    (f.isNilpotent_restrict_maxGenEigenspace_sub_algebraMap μ), hg, mul_zero]
 
 lemma mapsTo_biSup_of_mapsTo {ι : Type*} {N : ι → Submodule R M}
     (s : Set ι) {f : Module.End R M} (hf : ∀ i, MapsTo f (N i) (N i)) :
diff --git a/Mathlib/Algebra/DirectSum/Module.lean b/Mathlib/Algebra/DirectSum/Module.lean
index 5d0336e5ed1e4..291bd79759b1b 100644
--- a/Mathlib/Algebra/DirectSum/Module.lean
+++ b/Mathlib/Algebra/DirectSum/Module.lean
@@ -28,7 +28,7 @@ universe u v w u₁
 
 namespace DirectSum
 
-open DirectSum
+open DirectSum Finsupp
 
 section General
 
@@ -279,7 +279,7 @@ theorem coeLinearMap_eq_dfinsupp_sum [DecidableEq M] (x : DirectSum ι fun i =>
   simp only [coeLinearMap, toModule, DFinsupp.lsum, LinearEquiv.coe_mk, LinearMap.coe_mk,
     AddHom.coe_mk]
   rw [DFinsupp.sumAddHom_apply]
-  simp only [LinearMap.toAddMonoidHom_coe, Submodule.coeSubtype]
+  simp only [LinearMap.toAddMonoidHom_coe, Submodule.coe_subtype]
 
 @[simp]
 theorem coeLinearMap_of (i : ι) (x : A i) : DirectSum.coeLinearMap A (of (fun i ↦ A i) i x) = x :=
@@ -335,7 +335,7 @@ theorem IsInternal.collectedBasis_coe (h : IsInternal A) {α : ι → Type*}
   -- Porting note: was
   -- simp only [IsInternal.collectedBasis, toModule, coeLinearMap, Basis.coe_ofRepr,
   --   Basis.repr_symm_apply, DFinsupp.lsum_apply_apply, DFinsupp.mapRange.linearEquiv_apply,
-  --   DFinsupp.mapRange.linearEquiv_symm, DFinsupp.mapRange_single, Finsupp.total_single,
+  --   DFinsupp.mapRange.linearEquiv_symm, DFinsupp.mapRange_single, linearCombination_single,
   --   LinearEquiv.ofBijective_apply, LinearEquiv.symm_symm, LinearEquiv.symm_trans_apply, one_smul,
   --   sigmaFinsuppAddEquivDFinsupp_apply, sigmaFinsuppEquivDFinsupp_single,
   --   sigmaFinsuppLequivDFinsupp_apply]
@@ -346,10 +346,10 @@ theorem IsInternal.collectedBasis_coe (h : IsInternal A) {α : ι → Type*}
     sigmaFinsuppAddEquivDFinsupp_apply]
   rw [DFinsupp.mapRange.linearEquiv_symm]
   erw [DFinsupp.mapRange.linearEquiv_apply]
-  simp only [DFinsupp.mapRange_single, Basis.repr_symm_apply, Finsupp.total_single, one_smul,
+  simp only [DFinsupp.mapRange_single, Basis.repr_symm_apply, linearCombination_single, one_smul,
     toModule]
   erw [DFinsupp.lsum_single]
-  simp only [Submodule.coeSubtype]
+  simp only [Submodule.coe_subtype]
 
 theorem IsInternal.collectedBasis_mem (h : IsInternal A) {α : ι → Type*}
     (v : ∀ i, Basis (α i) R (A i)) (a : Σi, α i) : h.collectedBasis v a ∈ A a.1 := by simp
diff --git a/Mathlib/Algebra/EuclideanDomain/Defs.lean b/Mathlib/Algebra/EuclideanDomain/Defs.lean
index 81b62ab8bf111..429e46451dffe 100644
--- a/Mathlib/Algebra/EuclideanDomain/Defs.lean
+++ b/Mathlib/Algebra/EuclideanDomain/Defs.lean
@@ -6,6 +6,7 @@ Authors: Louis Carlin, Mario Carneiro
 import Mathlib.Algebra.Divisibility.Basic
 import Mathlib.Algebra.Group.Basic
 import Mathlib.Algebra.Ring.Defs
+import Mathlib.Order.RelClasses
 
 /-!
 # Euclidean domains
@@ -67,7 +68,7 @@ universe u
   satisfying `b * (a / b) + a % b = a`.
   The definition of a Euclidean domain usually includes a valuation function `R → ℕ`.
   This definition is slightly generalised to include a well founded relation
-  `r` with the property that `r (a % b) b`, instead of a valuation.  -/
+  `r` with the property that `r (a % b) b`, instead of a valuation. -/
 class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R where
   /-- A division function (denoted `/`) on `R`.
     This satisfies the property `b * (a / b) + a % b = a`, where `%` denotes `remainder`. -/
@@ -101,6 +102,9 @@ local infixl:50 " ≺ " => EuclideanDomain.r
 local instance wellFoundedRelation : WellFoundedRelation R where
   wf := r_wellFounded
 
+instance isWellFounded : IsWellFounded R (· ≺ ·) where
+  wf := r_wellFounded
+
 -- see Note [lower instance priority]
 instance (priority := 70) : Div R :=
   ⟨EuclideanDomain.quotient⟩
diff --git a/Mathlib/Algebra/Exact.lean b/Mathlib/Algebra/Exact.lean
index 8a023b47a692f..e3eb8a8116504 100644
--- a/Mathlib/Algebra/Exact.lean
+++ b/Mathlib/Algebra/Exact.lean
@@ -323,19 +323,17 @@ theorem Exact.split_tfae' (h : Function.Exact f g) :
       Function.Surjective g ∧ ∃ l, l ∘ₗ f = LinearMap.id,
       ∃ e : N ≃ₗ[R] M × P, f = e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ e] := by
   tfae_have 1 → 3
-  · rintro ⟨hf, l, hl⟩
-    exact ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩
+  | ⟨hf, l, hl⟩ => ⟨_, (h.splitSurjectiveEquiv hf ⟨l, hl⟩).2⟩
   tfae_have 2 → 3
-  · rintro ⟨hg, l, hl⟩
-    exact ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩
+  | ⟨hg, l, hl⟩ => ⟨_, (h.splitInjectiveEquiv hg ⟨l, hl⟩).2⟩
   tfae_have 3 → 1
-  · rintro ⟨e, e₁, e₂⟩
+  | ⟨e, e₁, e₂⟩ => by
     have : Function.Injective f := e₁ ▸ e.symm.injective.comp LinearMap.inl_injective
-    refine ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
+    exact ⟨this, ⟨_, ((h.splitSurjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
   tfae_have 3 → 2
-  · rintro ⟨e, e₁, e₂⟩
+  | ⟨e, e₁, e₂⟩ => by
     have : Function.Surjective g := e₂ ▸ Prod.snd_surjective.comp e.surjective
-    refine ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
+    exact ⟨this, ⟨_, ((h.splitInjectiveEquiv this).symm ⟨e, e₁, e₂⟩).2⟩⟩
   tfae_finish
 
 /-- Equivalent characterizations of split exact sequences. Also known as the **Splitting lemma**. -/
@@ -347,10 +345,10 @@ theorem Exact.split_tfae
       ∃ l, g ∘ₗ l = LinearMap.id,
       ∃ l, l ∘ₗ f = LinearMap.id,
       ∃ e : N ≃ₗ[R] M × P, f = e.symm ∘ₗ LinearMap.inl R M P ∧ g = LinearMap.snd R M P ∘ₗ e] := by
-  tfae_have 1 ↔ 3
-  · simpa using (h.splitSurjectiveEquiv hf).nonempty_congr
-  tfae_have 2 ↔ 3
-  · simpa using (h.splitInjectiveEquiv hg).nonempty_congr
+  tfae_have 1 ↔ 3 := by
+    simpa using (h.splitSurjectiveEquiv hf).nonempty_congr
+  tfae_have 2 ↔ 3 := by
+    simpa using (h.splitInjectiveEquiv hg).nonempty_congr
   tfae_finish
 
 end split
diff --git a/Mathlib/Algebra/Field/Basic.lean b/Mathlib/Algebra/Field/Basic.lean
index fa1a788fb74a7..fe9961bb0230a 100644
--- a/Mathlib/Algebra/Field/Basic.lean
+++ b/Mathlib/Algebra/Field/Basic.lean
@@ -18,16 +18,16 @@ open Function OrderDual Set
 
 universe u
 
-variable {α β K : Type*}
+variable {K L : Type*}
 
 section DivisionSemiring
 
-variable [DivisionSemiring α] {a b c d : α}
+variable [DivisionSemiring K] {a b c d : K}
 
-theorem add_div (a b c : α) : (a + b) / c = a / c + b / c := by simp_rw [div_eq_mul_inv, add_mul]
+theorem add_div (a b c : K) : (a + b) / c = a / c + b / c := by simp_rw [div_eq_mul_inv, add_mul]
 
 @[field_simps]
-theorem div_add_div_same (a b c : α) : a / c + b / c = (a + b) / c :=
+theorem div_add_div_same (a b c : K) : a / c + b / c = (a + b) / c :=
   (add_div _ _ _).symm
 
 theorem same_add_div (h : b ≠ 0) : (b + a) / b = 1 + a / b := by rw [← div_self h, add_div]
@@ -49,15 +49,15 @@ theorem one_div_mul_add_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb :
     1 / a * (a + b) * (1 / b) = 1 / a + 1 / b := by
   simpa only [one_div] using (inv_add_inv' ha hb).symm
 
-theorem add_div_eq_mul_add_div (a b : α) (hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
+theorem add_div_eq_mul_add_div (a b : K) (hc : c ≠ 0) : a + b / c = (a * c + b) / c :=
   (eq_div_iff_mul_eq hc).2 <| by rw [right_distrib, div_mul_cancel₀ _ hc]
 
 @[field_simps]
-theorem add_div' (a b c : α) (hc : c ≠ 0) : b + a / c = (b * c + a) / c := by
+theorem add_div' (a b c : K) (hc : c ≠ 0) : b + a / c = (b * c + a) / c := by
   rw [add_div, mul_div_cancel_right₀ _ hc]
 
 @[field_simps]
-theorem div_add' (a b c : α) (hc : c ≠ 0) : a / c + b = (a + b * c) / c := by
+theorem div_add' (a b c : K) (hc : c ≠ 0) : a / c + b = (a + b * c) / c := by
   rwa [add_comm, add_div', add_comm]
 
 protected theorem Commute.div_add_div (hbc : Commute b c) (hbd : Commute b d) (hb : b ≠ 0)
@@ -167,9 +167,9 @@ end DivisionRing
 
 section Semifield
 
-variable [Semifield α] {a b c d : α}
+variable [Semifield K] {a b d : K}
 
-theorem div_add_div (a : α) (c : α) (hb : b ≠ 0) (hd : d ≠ 0) :
+theorem div_add_div (a : K) (c : K) (hb : b ≠ 0) (hd : d ≠ 0) :
     a / b + c / d = (a * d + b * c) / (b * d) :=
   (Commute.all b _).div_add_div (Commute.all _ _) hb hd
 
@@ -211,7 +211,7 @@ end Field
 
 namespace RingHom
 
-protected theorem injective [DivisionRing α] [Semiring β] [Nontrivial β] (f : α →+* β) :
+protected theorem injective [DivisionRing K] [Semiring L] [Nontrivial L] (f : K →+* L) :
     Injective f :=
   (injective_iff_map_eq_zero f).2 fun _ ↦ (map_eq_zero f).1
 
@@ -242,18 +242,18 @@ noncomputable abbrev Field.ofIsUnitOrEqZero [CommRing R] (h : ∀ a : R, IsUnit
 end NoncomputableDefs
 
 namespace Function.Injective
-variable [Zero α] [Add α] [Neg α] [Sub α] [One α] [Mul α] [Inv α] [Div α] [SMul ℕ α] [SMul ℤ α]
-  [SMul ℚ≥0 α] [SMul ℚ α] [Pow α ℕ] [Pow α ℤ] [NatCast α] [IntCast α] [NNRatCast α] [RatCast α]
-  (f : α → β) (hf : Injective f)
+variable [Zero K] [Add K] [Neg K] [Sub K] [One K] [Mul K] [Inv K] [Div K] [SMul ℕ K] [SMul ℤ K]
+  [SMul ℚ≥0 K] [SMul ℚ K] [Pow K ℕ] [Pow K ℤ] [NatCast K] [IntCast K] [NNRatCast K] [RatCast K]
+  (f : K → L) (hf : Injective f)
 
 /-- Pullback a `DivisionSemiring` along an injective function. -/
 -- See note [reducible non-instances]
-protected abbrev divisionSemiring [DivisionSemiring β] (zero : f 0 = 0) (one : f 1 = 1)
+protected abbrev divisionSemiring [DivisionSemiring L] (zero : f 0 = 0) (one : f 1 = 1)
     (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
     (inv : ∀ x, f x⁻¹ = (f x)⁻¹) (div : ∀ x y, f (x / y) = f x / f y)
     (nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x), f (q • x) = q • f x)
     (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n)
-    (natCast : ∀ n : ℕ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q) : DivisionSemiring α where
+    (natCast : ∀ n : ℕ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q) : DivisionSemiring K where
   toSemiring := hf.semiring f zero one add mul nsmul npow natCast
   __ := hf.groupWithZero f zero one mul inv div npow zpow
   nnratCast_def q := hf <| by rw [nnratCast, NNRat.cast_def, div, natCast, natCast]
@@ -262,7 +262,7 @@ protected abbrev divisionSemiring [DivisionSemiring β] (zero : f 0 = 0) (one :
 
 /-- Pullback a `DivisionSemiring` along an injective function. -/
 -- See note [reducible non-instances]
-protected abbrev divisionRing [DivisionRing β] (zero : f 0 = 0) (one : f 1 = 1)
+protected abbrev divisionRing [DivisionRing L] (zero : f 0 = 0) (one : f 1 = 1)
     (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
     (neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
     (div : ∀ x y, f (x / y) = f x / f y)
@@ -270,7 +270,7 @@ protected abbrev divisionRing [DivisionRing β] (zero : f 0 = 0) (one : f 1 = 1)
     (nnqsmul : ∀ (q : ℚ≥0) (x), f (q • x) = q • f x) (qsmul : ∀ (q : ℚ) (x), f (q • x) = q • f x)
     (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n)
     (natCast : ∀ n : ℕ, f n = n) (intCast : ∀ n : ℤ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q)
-    (ratCast : ∀ q : ℚ, f q = q) : DivisionRing α where
+    (ratCast : ∀ q : ℚ, f q = q) : DivisionRing K where
   toRing := hf.ring f zero one add mul neg sub nsmul zsmul npow natCast intCast
   __ := hf.groupWithZero f zero one mul inv div npow zpow
   __ := hf.divisionSemiring f zero one add mul inv div nsmul nnqsmul npow zpow natCast nnratCast
@@ -280,19 +280,19 @@ protected abbrev divisionRing [DivisionRing β] (zero : f 0 = 0) (one : f 1 = 1)
 
 /-- Pullback a `Field` along an injective function. -/
 -- See note [reducible non-instances]
-protected abbrev semifield [Semifield β] (zero : f 0 = 0) (one : f 1 = 1)
+protected abbrev semifield [Semifield L] (zero : f 0 = 0) (one : f 1 = 1)
     (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
     (inv : ∀ x, f x⁻¹ = (f x)⁻¹) (div : ∀ x y, f (x / y) = f x / f y)
     (nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) (nnqsmul : ∀ (q : ℚ≥0) (x), f (q • x) = q • f x)
     (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n)
-    (natCast : ∀ n : ℕ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q) : Semifield α where
+    (natCast : ∀ n : ℕ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q) : Semifield K where
   toCommSemiring := hf.commSemiring f zero one add mul nsmul npow natCast
   __ := hf.commGroupWithZero f zero one mul inv div npow zpow
   __ := hf.divisionSemiring f zero one add mul inv div nsmul nnqsmul npow zpow natCast nnratCast
 
 /-- Pullback a `Field` along an injective function. -/
 -- See note [reducible non-instances]
-protected abbrev field [Field β] (zero : f 0 = 0) (one : f 1 = 1)
+protected abbrev field [Field L] (zero : f 0 = 0) (one : f 1 = 1)
     (add : ∀ x y, f (x + y) = f x + f y) (mul : ∀ x y, f (x * y) = f x * f y)
     (neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
     (div : ∀ x y, f (x / y) = f x / f y)
@@ -301,7 +301,7 @@ protected abbrev field [Field β] (zero : f 0 = 0) (one : f 1 = 1)
     (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n)
     (natCast : ∀ n : ℕ, f n = n) (intCast : ∀ n : ℤ, f n = n) (nnratCast : ∀ q : ℚ≥0, f q = q)
     (ratCast : ∀ q : ℚ, f q = q) :
-    Field α where
+    Field K where
   toCommRing := hf.commRing f zero one add mul neg sub nsmul zsmul npow natCast intCast
   __ := hf.divisionRing f zero one add mul neg sub inv div nsmul zsmul nnqsmul qsmul npow zpow
     natCast intCast nnratCast ratCast
@@ -312,30 +312,30 @@ end Function.Injective
 
 namespace OrderDual
 
-instance instRatCast [RatCast α] : RatCast αᵒᵈ := ‹_›
-instance instDivisionSemiring [DivisionSemiring α] : DivisionSemiring αᵒᵈ := ‹_›
-instance instDivisionRing [DivisionRing α] : DivisionRing αᵒᵈ := ‹_›
-instance instSemifield [Semifield α] : Semifield αᵒᵈ := ‹_›
-instance instField [Field α] : Field αᵒᵈ := ‹_›
+instance instRatCast [RatCast K] : RatCast Kᵒᵈ := ‹_›
+instance instDivisionSemiring [DivisionSemiring K] : DivisionSemiring Kᵒᵈ := ‹_›
+instance instDivisionRing [DivisionRing K] : DivisionRing Kᵒᵈ := ‹_›
+instance instSemifield [Semifield K] : Semifield Kᵒᵈ := ‹_›
+instance instField [Field K] : Field Kᵒᵈ := ‹_›
 
 end OrderDual
 
-@[simp] lemma toDual_ratCast [RatCast α] (n : ℚ) : toDual (n : α) = n := rfl
+@[simp] lemma toDual_ratCast [RatCast K] (n : ℚ) : toDual (n : K) = n := rfl
 
-@[simp] lemma ofDual_ratCast [RatCast α] (n : ℚ) : (ofDual n : α) = n := rfl
+@[simp] lemma ofDual_ratCast [RatCast K] (n : ℚ) : (ofDual n : K) = n := rfl
 
 /-! ### Lexicographic order -/
 
 namespace Lex
 
-instance instRatCast [RatCast α] : RatCast (Lex α) := ‹_›
-instance instDivisionSemiring [DivisionSemiring α] : DivisionSemiring (Lex α) := ‹_›
-instance instDivisionRing [DivisionRing α] : DivisionRing (Lex α) := ‹_›
-instance instSemifield [Semifield α] : Semifield (Lex α) := ‹_›
-instance instField [Field α] : Field (Lex α) := ‹_›
+instance instRatCast [RatCast K] : RatCast (Lex K) := ‹_›
+instance instDivisionSemiring [DivisionSemiring K] : DivisionSemiring (Lex K) := ‹_›
+instance instDivisionRing [DivisionRing K] : DivisionRing (Lex K) := ‹_›
+instance instSemifield [Semifield K] : Semifield (Lex K) := ‹_›
+instance instField [Field K] : Field (Lex K) := ‹_›
 
 end Lex
 
-@[simp] lemma toLex_ratCast [RatCast α] (n : ℚ) : toLex (n : α) = n := rfl
+@[simp] lemma toLex_ratCast [RatCast K] (n : ℚ) : toLex (n : K) = n := rfl
 
-@[simp] lemma ofLex_ratCast [RatCast α] (n : ℚ) : (ofLex n : α) = n := rfl
+@[simp] lemma ofLex_ratCast [RatCast K] (n : ℚ) : (ofLex n : K) = n := rfl
diff --git a/Mathlib/Algebra/Field/Defs.lean b/Mathlib/Algebra/Field/Defs.lean
index 4943b798adf4c..4bb07df74d908 100644
--- a/Mathlib/Algebra/Field/Defs.lean
+++ b/Mathlib/Algebra/Field/Defs.lean
@@ -45,8 +45,8 @@ field, division ring, skew field, skew-field, skewfield
 
 assert_not_imported Mathlib.Tactic.Common
 
--- `NeZero` should not be needed in the basic algebraic hierarchy.
-assert_not_exists NeZero
+-- `NeZero` theory should not be needed in the basic algebraic hierarchy
+assert_not_imported Mathlib.Algebra.NeZero
 
 assert_not_exists MonoidHom
 
@@ -54,7 +54,7 @@ open Function Set
 
 universe u
 
-variable {α β K : Type*}
+variable {K : Type*}
 
 /-- The default definition of the coercion `ℚ≥0 → K` for a division semiring `K`.
 
@@ -81,23 +81,23 @@ itself). See also note [forgetful inheritance].
 
 If the division semiring has positive characteristic `p`, our division by zero convention forces
 `nnratCast (1 / p) = 1 / 0 = 0`. -/
-class DivisionSemiring (α : Type*) extends Semiring α, GroupWithZero α, NNRatCast α where
+class DivisionSemiring (K : Type*) extends Semiring K, GroupWithZero K, NNRatCast K where
   protected nnratCast := NNRat.castRec
   /-- However `NNRat.cast` is defined, it must be propositionally equal to `a / b`.
 
   Do not use this lemma directly. Use `NNRat.cast_def` instead. -/
-  protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
+  protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : K) = q.num / q.den := by intros; rfl
   /-- Scalar multiplication by a nonnegative rational number.
 
   Unless there is a risk of a `Module ℚ≥0 _` instance diamond, write `nnqsmul := _`. This will set
   `nnqsmul` to `(NNRat.cast · * ·)` thanks to unification in the default proof of `nnqsmul_def`.
 
   Do not use directly. Instead use the `•` notation. -/
-  protected nnqsmul : ℚ≥0 → α → α
+  protected nnqsmul : ℚ≥0 → K → K
   /-- However `qsmul` is defined, it must be propositionally equal to multiplication by `Rat.cast`.
 
   Do not use this lemma directly. Use `NNRat.smul_def` instead. -/
-  protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
+  protected nnqsmul_def (q : ℚ≥0) (a : K) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
 
 /-- A `DivisionRing` is a `Ring` with multiplicative inverses for nonzero elements.
 
@@ -109,48 +109,48 @@ See also note [forgetful inheritance]. Similarly, there are maps `nnratCast ℚ
 
 If the division ring has positive characteristic `p`, our division by zero convention forces
 `ratCast (1 / p) = 1 / 0 = 0`. -/
-class DivisionRing (α : Type*)
-  extends Ring α, DivInvMonoid α, Nontrivial α, NNRatCast α, RatCast α where
+class DivisionRing (K : Type*)
+  extends Ring K, DivInvMonoid K, Nontrivial K, NNRatCast K, RatCast K where
   /-- For a nonzero `a`, `a⁻¹` is a right multiplicative inverse. -/
-  protected mul_inv_cancel : ∀ (a : α), a ≠ 0 → a * a⁻¹ = 1
+  protected mul_inv_cancel : ∀ (a : K), a ≠ 0 → a * a⁻¹ = 1
   /-- The inverse of `0` is `0` by convention. -/
-  protected inv_zero : (0 : α)⁻¹ = 0
+  protected inv_zero : (0 : K)⁻¹ = 0
   protected nnratCast := NNRat.castRec
   /-- However `NNRat.cast` is defined, it must be equal to `a / b`.
 
   Do not use this lemma directly. Use `NNRat.cast_def` instead. -/
-  protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : α) = q.num / q.den := by intros; rfl
+  protected nnratCast_def (q : ℚ≥0) : (NNRat.cast q : K) = q.num / q.den := by intros; rfl
   /-- Scalar multiplication by a nonnegative rational number.
 
   Unless there is a risk of a `Module ℚ≥0 _` instance diamond, write `nnqsmul := _`. This will set
   `nnqsmul` to `(NNRat.cast · * ·)` thanks to unification in the default proof of `nnqsmul_def`.
 
   Do not use directly. Instead use the `•` notation. -/
-  protected nnqsmul : ℚ≥0 → α → α
+  protected nnqsmul : ℚ≥0 → K → K
   /-- However `qsmul` is defined, it must be propositionally equal to multiplication by `Rat.cast`.
 
   Do not use this lemma directly. Use `NNRat.smul_def` instead. -/
-  protected nnqsmul_def (q : ℚ≥0) (a : α) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
+  protected nnqsmul_def (q : ℚ≥0) (a : K) : nnqsmul q a = NNRat.cast q * a := by intros; rfl
   protected ratCast := Rat.castRec
   /-- However `Rat.cast q` is defined, it must be propositionally equal to `q.num / q.den`.
 
   Do not use this lemma directly. Use `Rat.cast_def` instead. -/
-  protected ratCast_def (q : ℚ) : (Rat.cast q : α) = q.num / q.den := by intros; rfl
+  protected ratCast_def (q : ℚ) : (Rat.cast q : K) = q.num / q.den := by intros; rfl
   /-- Scalar multiplication by a rational number.
 
   Unless there is a risk of a `Module ℚ _` instance diamond, write `qsmul := _`. This will set
   `qsmul` to `(Rat.cast · * ·)` thanks to unification in the default proof of `qsmul_def`.
 
   Do not use directly. Instead use the `•` notation. -/
-  protected qsmul : ℚ → α → α
+  protected qsmul : ℚ → K → K
   /-- However `qsmul` is defined, it must be propositionally equal to multiplication by `Rat.cast`.
 
   Do not use this lemma directly. Use `Rat.cast_def` instead. -/
-  protected qsmul_def (a : ℚ) (x : α) : qsmul a x = Rat.cast a * x := by intros; rfl
+  protected qsmul_def (a : ℚ) (x : K) : qsmul a x = Rat.cast a * x := by intros; rfl
 
 -- see Note [lower instance priority]
-instance (priority := 100) DivisionRing.toDivisionSemiring [DivisionRing α] : DivisionSemiring α :=
-  { ‹DivisionRing α› with }
+instance (priority := 100) DivisionRing.toDivisionSemiring [DivisionRing K] : DivisionSemiring K :=
+  { ‹DivisionRing K› with }
 
 /-- A `Semifield` is a `CommSemiring` with multiplicative inverses for nonzero elements.
 
@@ -161,7 +161,7 @@ itself). See also note [forgetful inheritance].
 
 If the semifield has positive characteristic `p`, our division by zero convention forces
 `nnratCast (1 / p) = 1 / 0 = 0`. -/
-class Semifield (α : Type*) extends CommSemiring α, DivisionSemiring α, CommGroupWithZero α
+class Semifield (K : Type*) extends CommSemiring K, DivisionSemiring K, CommGroupWithZero K
 
 /-- A `Field` is a `CommRing` with multiplicative inverses for nonzero elements.
 
@@ -175,26 +175,26 @@ If the field has positive characteristic `p`, our division by zero convention fo
 class Field (K : Type u) extends CommRing K, DivisionRing K
 
 -- see Note [lower instance priority]
-instance (priority := 100) Field.toSemifield [Field α] : Semifield α := { ‹Field α› with }
+instance (priority := 100) Field.toSemifield [Field K] : Semifield K := { ‹Field K› with }
 
 namespace NNRat
-variable [DivisionSemiring α]
+variable [DivisionSemiring K]
 
-instance (priority := 100) smulDivisionSemiring : SMul ℚ≥0 α := ⟨DivisionSemiring.nnqsmul⟩
+instance (priority := 100) smulDivisionSemiring : SMul ℚ≥0 K := ⟨DivisionSemiring.nnqsmul⟩
 
-lemma cast_def (q : ℚ≥0) : (q : α) = q.num / q.den := DivisionSemiring.nnratCast_def _
-lemma smul_def (q : ℚ≥0) (a : α) : q • a = q * a := DivisionSemiring.nnqsmul_def q a
+lemma cast_def (q : ℚ≥0) : (q : K) = q.num / q.den := DivisionSemiring.nnratCast_def _
+lemma smul_def (q : ℚ≥0) (a : K) : q • a = q * a := DivisionSemiring.nnqsmul_def q a
 
-variable (α)
+variable (K)
 
-@[simp] lemma smul_one_eq_cast (q : ℚ≥0) : q • (1 : α) = q := by rw [NNRat.smul_def, mul_one]
+@[simp] lemma smul_one_eq_cast (q : ℚ≥0) : q • (1 : K) = q := by rw [NNRat.smul_def, mul_one]
 
 @[deprecated (since := "2024-05-03")] alias smul_one_eq_coe := smul_one_eq_cast
 
 end NNRat
 
 namespace Rat
-variable [DivisionRing K] {a b : K}
+variable [DivisionRing K]
 
 lemma cast_def (q : ℚ) : (q : K) = q.num / q.den := DivisionRing.ratCast_def _
 
diff --git a/Mathlib/Algebra/Field/Subfield.lean b/Mathlib/Algebra/Field/Subfield.lean
index b304f131e2f96..30b6731d7f52c 100644
--- a/Mathlib/Algebra/Field/Subfield.lean
+++ b/Mathlib/Algebra/Field/Subfield.lean
@@ -126,20 +126,20 @@ variable (S)
 /-- A subfield inherits a division ring structure -/
 instance (priority := 75) toDivisionRing (s : S) : DivisionRing s :=
   Subtype.coe_injective.divisionRing ((↑) : s → K)
-    (by rfl) (by rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl)
-    (by intros; rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl)
-    (by intros; rfl) (coe_nnqsmul _) (coe_qsmul _) (by intros; rfl) (by intros; rfl)
-    (by intros; rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl)
+    rfl rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl)
+    (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
+    (fun _ _ ↦ rfl) (coe_nnqsmul _) (coe_qsmul _) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
+    (fun _ ↦ rfl) (fun _ ↦ rfl) (fun _ ↦ rfl) (fun _ ↦ rfl)
 
 -- Prefer subclasses of `Field` over subclasses of `SubfieldClass`.
 /-- A subfield of a field inherits a field structure -/
 instance (priority := 75) toField {K} [Field K] [SetLike S K] [SubfieldClass S K] (s : S) :
     Field s :=
   Subtype.coe_injective.field ((↑) : s → K)
-    (by rfl) (by rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl)
-    (by intros; rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl) (by intros; rfl)
-    (coe_nnqsmul _) (coe_qsmul _) (by intros; rfl) (by intros; rfl) (by intros; rfl)
-    (by intros; rfl) (by intros; rfl) (by intros; rfl)
+    rfl rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl)
+    (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
+    (coe_nnqsmul _) (coe_qsmul _) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl)
+    (fun _ ↦ rfl) (fun _ ↦ rfl) (fun _ ↦ rfl)
 
 end SubfieldClass
 
@@ -313,15 +313,15 @@ instance : Pow s ℤ :=
 instance toDivisionRing (s : Subfield K) : DivisionRing s :=
   Subtype.coe_injective.divisionRing ((↑) : s → K) rfl rfl (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
     (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl)
-    (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (by intros; rfl) (fun _ ↦ rfl) (fun _ ↦ rfl)
-    (by intros; rfl) fun _ ↦ rfl
+    (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ _ ↦ rfl) (fun _ ↦ rfl) (fun _ ↦ rfl)
+    (fun _ ↦ rfl) fun _ ↦ rfl
 
 /-- A subfield inherits a field structure -/
 instance toField {K} [Field K] (s : Subfield K) : Field s :=
   Subtype.coe_injective.field ((↑) : s → K) rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl)
     (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl)
-    (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (by intros; rfl)  (fun _ => rfl)
-    (fun _ => rfl) (by intros; rfl) fun _ => rfl
+    (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ ↦ rfl) (fun _ => rfl)
+    (fun _ => rfl) (fun _ ↦ rfl) fun _ => rfl
 
 @[simp, norm_cast]
 theorem coe_add (x y : s) : (↑(x + y) : K) = ↑x + ↑y :=
@@ -545,6 +545,13 @@ theorem mem_sInf {S : Set (Subfield K)} {x : K} : x ∈ sInf S ↔ ∀ p ∈ S,
   Subring.mem_sInf.trans
     ⟨fun h p hp => h p.toSubring ⟨p, hp, rfl⟩, fun h _ ⟨p', hp', p_eq⟩ => p_eq ▸ h p' hp'⟩
 
+@[simp, norm_cast]
+theorem coe_iInf {ι : Sort*} {S : ι → Subfield K} : (↑(⨅ i, S i) : Set K) = ⋂ i, S i := by
+  simp only [iInf, coe_sInf, Set.biInter_range]
+
+theorem mem_iInf {ι : Sort*} {S : ι → Subfield K} {x : K} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by
+  simp only [iInf, mem_sInf, Set.forall_mem_range]
+
 @[simp]
 theorem sInf_toSubring (s : Set (Subfield K)) :
     (sInf s).toSubring = ⨅ t ∈ s, Subfield.toSubring t := by
@@ -659,6 +666,14 @@ theorem map_iSup {ι : Sort*} (f : K →+* L) (s : ι → Subfield K) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+theorem map_inf (s t : Subfield K) (f : K →+* L) : (s ⊓ t).map f = s.map f ⊓ t.map f :=
+  SetLike.coe_injective (Set.image_inter f.injective)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : K →+* L) (s : ι → Subfield K) :
+    (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective f.injective).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem comap_inf (s t : Subfield L) (f : K →+* L) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
   (gc_map_comap f).u_inf
 
diff --git a/Mathlib/Algebra/Free.lean b/Mathlib/Algebra/Free.lean
index 68fa9ba945b69..134a4e81ed9ff 100644
--- a/Mathlib/Algebra/Free.lean
+++ b/Mathlib/Algebra/Free.lean
@@ -6,7 +6,6 @@ Authors: Kenny Lau
 import Mathlib.Algebra.Group.Equiv.Basic
 import Mathlib.Control.Applicative
 import Mathlib.Control.Traversable.Basic
-import Mathlib.Data.List.Basic
 import Mathlib.Logic.Equiv.Defs
 import Mathlib.Tactic.AdaptationNote
 
@@ -244,10 +243,10 @@ instance : LawfulTraversable FreeMagma.{u} :=
         rw [traverse_mul, ih1, ih2, mul_map_seq]
     comp_traverse := fun f g x ↦
       FreeMagma.recOnPure x
-        (fun x ↦ by simp only [(· ∘ ·), traverse_pure, traverse_pure', functor_norm])
+        (fun x ↦ by simp only [Function.comp_def, traverse_pure, traverse_pure', functor_norm])
         (fun x y ih1 ih2 ↦ by
           rw [traverse_mul, ih1, ih2, traverse_mul]
-          simp [Functor.Comp.map_mk, Functor.map_map, (· ∘ ·), Comp.seq_mk, seq_map_assoc,
+          simp [Functor.Comp.map_mk, Functor.map_map, Function.comp_def, Comp.seq_mk, seq_map_assoc,
             map_seq, traverse_mul])
     naturality := fun η α β f x ↦
       FreeMagma.recOnPure x
@@ -592,7 +591,7 @@ theorem traverse_mul (x y : FreeSemigroup α) :
     (fun hd tl ih x ↦ show
         (· * ·) <$> pure <$> F x <*> traverse F (mk hd tl * mk y L2) =
           (· * ·) <$> ((· * ·) <$> pure <$> F x <*> traverse F (mk hd tl)) <*> traverse F (mk y L2)
-        by rw [ih]; simp only [(· ∘ ·), (mul_assoc _ _ _).symm, functor_norm])
+        by rw [ih]; simp only [Function.comp_def, (mul_assoc _ _ _).symm, functor_norm])
     x
 
 @[to_additive (attr := simp)]
@@ -618,9 +617,10 @@ instance : LawfulTraversable FreeSemigroup.{u} :=
       FreeSemigroup.recOnMul x (fun x ↦ rfl) fun x y ih1 ih2 ↦ by
         rw [traverse_mul, ih1, ih2, mul_map_seq]
     comp_traverse := fun f g x ↦
-      recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, (· ∘ ·)])
-        fun x y ih1 ih2 ↦ by (rw [traverse_mul, ih1, ih2,
-          traverse_mul, Functor.Comp.map_mk]; simp only [Function.comp, functor_norm, traverse_mul])
+      recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, Function.comp_def])
+        fun x y ih1 ih2 ↦ by
+          rw [traverse_mul, ih1, ih2, traverse_mul, Functor.Comp.map_mk]
+          simp only [Function.comp_def, functor_norm, traverse_mul]
     naturality := fun η α β f x ↦
       recOnPure x (fun x ↦ by simp only [traverse_pure, functor_norm, Function.comp])
           (fun x y ih1 ih2 ↦ by simp only [traverse_mul, functor_norm, ih1, ih2])
diff --git a/Mathlib/Algebra/FreeAlgebra.lean b/Mathlib/Algebra/FreeAlgebra.lean
index b10a44ea077c2..e844d11c4c136 100644
--- a/Mathlib/Algebra/FreeAlgebra.lean
+++ b/Mathlib/Algebra/FreeAlgebra.lean
@@ -1,12 +1,13 @@
 /-
 Copyright (c) 2020 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Adam Topaz, Eric Wieser
+Authors: Kim Morrison, Adam Topaz, Eric Wieser
 -/
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Algebra.Algebra.Tower
 import Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors
 import Mathlib.RingTheory.Adjoin.Basic
+import Mathlib.Algebra.MonoidAlgebra.Basic
 
 /-!
 # Free Algebras
@@ -151,7 +152,7 @@ namespace FreeAlgebra
 attribute [local instance] Pre.hasCoeGenerator Pre.hasCoeSemiring Pre.hasMul Pre.hasAdd
   Pre.hasZero Pre.hasOne Pre.hasSMul
 
-/-! Define the basic operations-/
+/-! Define the basic operations -/
 
 instance instSMul {A} [CommSemiring A] [Algebra R A] : SMul R (FreeAlgebra A X) where
   smul r := Quot.map (HMul.hMul (algebraMap R A r : Pre A X)) fun _ _ ↦ Rel.mul_compat_right
@@ -567,7 +568,7 @@ variable {A : Type*} [Semiring A] [Algebra R A]
 theorem _root_.Algebra.adjoin_range_eq_range_freeAlgebra_lift (f : X → A) :
     Algebra.adjoin R (Set.range f) = (FreeAlgebra.lift R f).range := by
   simp only [← Algebra.map_top, ← adjoin_range_ι, AlgHom.map_adjoin, ← Set.range_comp,
-    (· ∘ ·), lift_ι_apply]
+    Function.comp_def, lift_ι_apply]
 
 /-- Noncommutative version of `Algebra.adjoin_range_eq_range`. -/
 theorem _root_.Algebra.adjoin_eq_range_freeAlgebra_lift (s : Set A) :
diff --git a/Mathlib/Algebra/GCDMonoid/Basic.lean b/Mathlib/Algebra/GCDMonoid/Basic.lean
index 71a5cdc6fa600..fee781f192238 100644
--- a/Mathlib/Algebra/GCDMonoid/Basic.lean
+++ b/Mathlib/Algebra/GCDMonoid/Basic.lean
@@ -400,7 +400,7 @@ theorem gcd_mul_left [NormalizedGCDMonoid α] (a b c : α) :
     gcd (a * b) (a * c) = normalize a * gcd b c :=
   (by_cases (by rintro rfl; simp only [zero_mul, gcd_zero_left, normalize_zero]))
     fun ha : a ≠ 0 =>
-    suffices gcd (a * b) (a * c) = normalize (a * gcd b c) by simpa
+    suffices gcd (a * b) (a * c) = normalize (a * gcd b c) by simpa [- normalize_apply]
     let ⟨d, eq⟩ := dvd_gcd (dvd_mul_right a b) (dvd_mul_right a c)
     gcd_eq_normalize
       (eq.symm ▸ mul_dvd_mul_left a
@@ -478,7 +478,7 @@ theorem dvd_mul_gcd_iff_dvd_mul [GCDMonoid α] {m n k : α} : k ∣ m * gcd k n
 
 Note: In general, this representation is highly non-unique.
 
-See `Nat.prodDvdAndDvdOfDvdProd` for a constructive version on `ℕ`.  -/
+See `Nat.prodDvdAndDvdOfDvdProd` for a constructive version on `ℕ`. -/
 instance [h : Nonempty (GCDMonoid α)] : DecompositionMonoid α where
   primal k m n H := by
     cases h
@@ -589,8 +589,8 @@ theorem exists_associated_pow_of_mul_eq_pow [GCDMonoid α] {a b c : α} (hab : I
   use Units.mkOfMulEqOne _ _ h'
   rw [Units.val_mkOfMulEqOne, ha']
 
-theorem exists_eq_pow_of_mul_eq_pow [GCDMonoid α] [Unique αˣ] {a b c : α} (hab : IsUnit (gcd a b))
-    {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, a = d ^ k :=
+theorem exists_eq_pow_of_mul_eq_pow [GCDMonoid α] [Subsingleton αˣ]
+    {a b c : α} (hab : IsUnit (gcd a b)) {k : ℕ} (h : a * b = c ^ k) : ∃ d : α, a = d ^ k :=
   let ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow hab h
   ⟨d, (associated_iff_eq.mp hd).symm⟩
 
@@ -666,7 +666,7 @@ theorem lcm_dvd_iff [GCDMonoid α] {a b c : α} : lcm a b ∣ c ↔ a ∣ c ∧
   by_cases h : a = 0 ∨ b = 0
   · rcases h with (rfl | rfl) <;>
       simp (config := { contextual := true }) only [iff_def, lcm_zero_left, lcm_zero_right,
-        zero_dvd_iff, dvd_zero, eq_self_iff_true, and_true_iff, imp_true_iff]
+        zero_dvd_iff, dvd_zero, eq_self_iff_true, and_true, imp_true_iff]
   · obtain ⟨h1, h2⟩ := not_or.1 h
     have h : gcd a b ≠ 0 := fun H => h1 ((gcd_eq_zero_iff _ _).1 H).1
     rw [← mul_dvd_mul_iff_left h, (gcd_mul_lcm a b).dvd_iff_dvd_left, ←
@@ -767,7 +767,7 @@ theorem lcm_mul_left [NormalizedGCDMonoid α] (a b c : α) :
     lcm (a * b) (a * c) = normalize a * lcm b c :=
   (by_cases (by rintro rfl; simp only [zero_mul, lcm_zero_left, normalize_zero]))
     fun ha : a ≠ 0 =>
-    suffices lcm (a * b) (a * c) = normalize (a * lcm b c) by simpa
+    suffices lcm (a * b) (a * c) = normalize (a * lcm b c) by simpa [- normalize_apply]
     have : a ∣ lcm (a * b) (a * c) := (dvd_mul_right _ _).trans (dvd_lcm_left _ _)
     let ⟨d, eq⟩ := this
     lcm_eq_normalize
@@ -820,7 +820,7 @@ end GCDMonoid
 
 section UniqueUnit
 
-variable [CancelCommMonoidWithZero α] [Unique αˣ]
+variable [CancelCommMonoidWithZero α] [Subsingleton αˣ]
 
 -- see Note [lower instance priority]
 instance (priority := 100) normalizationMonoidOfUniqueUnits : NormalizationMonoid α where
@@ -1306,7 +1306,7 @@ instance (priority := 100) : NormalizedGCDMonoid G₀ where
         exact Associated.refl _
       -- Porting note(#12129): additional beta reduction needed
       · beta_reduce
-        rw [if_neg (not_and_of_not_left _ ha), one_mul, if_neg (not_or_of_not ha hb)]
+        rw [if_neg (not_and_of_not_left _ ha), one_mul, if_neg (not_or_intro ha hb)]
         exact (associated_one_iff_isUnit.mpr ((IsUnit.mk0 _ ha).mul (IsUnit.mk0 _ hb))).symm
   lcm_zero_left b := if_pos (Or.inl rfl)
   lcm_zero_right a := if_pos (Or.inr rfl)
diff --git a/Mathlib/Algebra/GeomSum.lean b/Mathlib/Algebra/GeomSum.lean
index 62bd098ce1152..70a08526df06e 100644
--- a/Mathlib/Algebra/GeomSum.lean
+++ b/Mathlib/Algebra/GeomSum.lean
@@ -482,7 +482,7 @@ theorem geom_sum_pos' [LinearOrderedRing α] (hx : 0 < x + 1) (hn : n ≠ 0) :
 
 theorem Odd.geom_sum_pos [LinearOrderedRing α] (h : Odd n) : 0 < ∑ i ∈ range n, x ^ i := by
   rcases n with (_ | _ | k)
-  · exact ((show ¬Odd 0 by decide) h).elim
+  · exact (Nat.not_odd_zero h).elim
   · simp only [zero_add, range_one, sum_singleton, pow_zero, zero_lt_one]
   rw [← Nat.not_even_iff_odd] at h
   rcases lt_trichotomy (x + 1) 0 with (hx | hx | hx)
diff --git a/Mathlib/Algebra/GradedMonoid.lean b/Mathlib/Algebra/GradedMonoid.lean
index 0179a285f612a..5f4ad73f043d1 100644
--- a/Mathlib/Algebra/GradedMonoid.lean
+++ b/Mathlib/Algebra/GradedMonoid.lean
@@ -302,7 +302,7 @@ variable {A}
 
 @[simp]
 theorem mk_zero_smul {i} (a : A 0) (b : A i) : mk _ (a • b) = mk _ a * mk _ b :=
-  Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
+  Sigma.ext (zero_add _).symm <| eqRec_heq _ _
 
 @[scoped simp]
 theorem GradeZero.smul_eq_mul (a b : A 0) : a • b = a * b :=
@@ -321,7 +321,7 @@ variable {A}
 
 @[simp]
 theorem mk_zero_pow (a : A 0) (n : ℕ) : mk _ (a ^ n) = mk _ a ^ n :=
-  Sigma.ext (nsmul_zero n).symm <| eq_rec_heq _ _
+  Sigma.ext (nsmul_zero n).symm <| eqRec_heq _ _
 
 variable (A)
 
diff --git a/Mathlib/Algebra/Group/Action/Defs.lean b/Mathlib/Algebra/Group/Action/Defs.lean
index 220aa2d2f5e5c..733f6bd366afc 100644
--- a/Mathlib/Algebra/Group/Action/Defs.lean
+++ b/Mathlib/Algebra/Group/Action/Defs.lean
@@ -48,7 +48,7 @@ assert_not_exists MonoidWithZero
 
 open Function (Injective Surjective)
 
-variable {M N G H A B α β γ δ : Type*}
+variable {M N G H α β γ δ : Type*}
 
 /-! ### Faithful actions -/
 
@@ -354,6 +354,24 @@ lemma smul_smul_smul_comm [SMul α β] [SMul α γ] [SMul β δ] [SMul α δ] [S
     [IsScalarTower α β δ] [IsScalarTower α γ δ] [SMulCommClass β γ δ] (a : α) (b : β) (c : γ)
     (d : δ) : (a • b) • c • d = (a • c) • b • d := by rw [smul_assoc, smul_assoc, smul_comm b]
 
+/-- Note that the `IsScalarTower α β β` and `SMulCommClass α β β` typeclass arguments are usually
+satisfied by `Algebra α β`. -/
+@[to_additive]
+lemma smul_mul_smul_comm [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β]
+    [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : β) (c : α) (d : β) :
+    (a • b) * (c • d) = (a * c) • (b * d) := by
+  have : SMulCommClass β α β := .symm ..; exact smul_smul_smul_comm a b c d
+
+@[to_additive (attr := deprecated (since := "2024-08-29"))]
+alias smul_mul_smul := smul_mul_smul_comm
+
+/-- Note that the `IsScalarTower α β β` and `SMulCommClass α β β` typeclass arguments are usually
+satisfied by `Algebra α β`. -/
+@[to_additive]
+lemma mul_smul_mul_comm [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β]
+    [IsScalarTower α α β] [SMulCommClass α β β] (a b : α) (c d : β) :
+    (a * b) • (c * d) = (a • c) * (b • d) := smul_smul_smul_comm a b c d
+
 variable [SMul M α]
 
 @[to_additive]
@@ -444,19 +462,12 @@ instance IsScalarTower.left : IsScalarTower M M α where
 
 variable {M}
 
-/-- Note that the `IsScalarTower M α α` and `SMulCommClass M α α` typeclass arguments are
-usually satisfied by `Algebra M α`. -/
-@[to_additive] -- Porting note: nolint to_additive_doc
-lemma smul_mul_smul [Mul α] (r s : M) (x y : α) [IsScalarTower M α α] [SMulCommClass M α α] :
-    r • x * s • y = (r * s) • (x * y) := by
-  rw [smul_mul_assoc, mul_smul_comm, ← smul_assoc, smul_eq_mul]
-
 section Monoid
 variable [Monoid N] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N]
 
 lemma smul_pow (r : M) (x : N) : ∀ n, (r • x) ^ n = r ^ n • x ^ n
   | 0 => by simp
-  | n + 1 => by rw [pow_succ', smul_pow _ _ n, smul_mul_smul, ← pow_succ', ← pow_succ']
+  | n + 1 => by rw [pow_succ', smul_pow _ _ n, smul_mul_smul_comm, ← pow_succ', ← pow_succ']
 
 end Monoid
 
@@ -476,7 +487,7 @@ lemma smul_inv_smul (g : G) (a : α) : g • g⁻¹ • a = a := by rw [smul_smu
   ⟨fun h ↦ by rw [h, smul_inv_smul], fun h ↦ by rw [← h, inv_smul_smul]⟩
 
 section Mul
-variable [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {g : G} {a b : H}
+variable [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {a b : H}
 
 @[simp] lemma Commute.smul_right_iff : Commute a (g • b) ↔ Commute a b :=
   ⟨fun h ↦ inv_smul_smul g b ▸ h.smul_right g⁻¹, fun h ↦ h.smul_right g⟩
@@ -489,7 +500,7 @@ end Mul
 variable [Group H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H]
 
 lemma smul_inv (g : G) (a : H) : (g • a)⁻¹ = g⁻¹ • a⁻¹ :=
-  inv_eq_of_mul_eq_one_right <| by rw [smul_mul_smul, mul_inv_cancel, mul_inv_cancel, one_smul]
+  inv_eq_of_mul_eq_one_right <| by rw [smul_mul_smul_comm, mul_inv_cancel, mul_inv_cancel, one_smul]
 
 lemma smul_zpow (g : G) (a : H) (n : ℤ) : (g • a) ^ n = g ^ n • a ^ n := by
   cases n <;> simp [smul_pow, smul_inv]
diff --git a/Mathlib/Algebra/Group/Action/Opposite.lean b/Mathlib/Algebra/Group/Action/Opposite.lean
index 8092a68ff91c7..1217487ac1eda 100644
--- a/Mathlib/Algebra/Group/Action/Opposite.lean
+++ b/Mathlib/Algebra/Group/Action/Opposite.lean
@@ -27,7 +27,7 @@ With `open scoped RightActions`, this provides:
 
 assert_not_exists MonoidWithZero
 
-variable {R M N α : Type*}
+variable {M N α β : Type*}
 
 /-!
 ### Actions _on_ the opposite type
@@ -97,7 +97,7 @@ In lemma names this is still called `op_vadd`. -/
 scoped notation3:73 m:73 " <+ᵥ " r:74 => AddOpposite.op r +ᵥ m
 
 section examples
-variable {α β : Type*} [SMul α β] [SMul αᵐᵒᵖ β] [VAdd α β] [VAdd αᵃᵒᵖ β] {a a₁ a₂ a₃ a₄ : α} {b : β}
+variable [SMul α β] [SMul αᵐᵒᵖ β] [VAdd α β] [VAdd αᵃᵒᵖ β] {a a₁ a₂ a₃ a₄ : α} {b : β}
 
 -- Left and right actions are just notation around the general `•` and `+ᵥ` notations
 example : a •> b = a • b := rfl
@@ -124,7 +124,7 @@ end examples
 end RightActions
 
 section
-variable {α β : Type*} [Monoid α] [MulAction αᵐᵒᵖ β]
+variable [Monoid α] [MulAction αᵐᵒᵖ β]
 
 open scoped RightActions
 
diff --git a/Mathlib/Algebra/Group/Action/Prod.lean b/Mathlib/Algebra/Group/Action/Prod.lean
index 0ce4d03aff4d1..e4434083c689b 100644
--- a/Mathlib/Algebra/Group/Action/Prod.lean
+++ b/Mathlib/Algebra/Group/Action/Prod.lean
@@ -132,7 +132,7 @@ section BundledSMul
 def smulMulHom [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] :
     α × β →ₙ* β where
   toFun a := a.1 • a.2
-  map_mul' _ _ := (smul_mul_smul _ _ _ _).symm
+  map_mul' _ _ := (smul_mul_smul_comm _ _ _ _).symm
 
 /-- Scalar multiplication as a monoid homomorphism. -/
 @[simps]
@@ -184,6 +184,6 @@ def MulAction.prodEquiv :
     congr 1
     · funext; congr; ext m a; (conv_rhs => rw [← hN.one_smul a]); rfl
     · ext n a; (conv_rhs => rw [← hM.one_smul (SMul.smul n a)]); rfl
-    · apply heq_prop
+    · exact proof_irrel_heq ..
 
 end Action_by_Prod
diff --git a/Mathlib/Algebra/Group/Action/Sum.lean b/Mathlib/Algebra/Group/Action/Sum.lean
index 5bc1602680cfc..0286b8e19b6e9 100644
--- a/Mathlib/Algebra/Group/Action/Sum.lean
+++ b/Mathlib/Algebra/Group/Action/Sum.lean
@@ -20,7 +20,7 @@ This file defines instances for additive and multiplicative actions on the binar
 
 assert_not_exists MonoidWithZero
 
-variable {M N P α β γ : Type*}
+variable {M N α β : Type*}
 
 namespace Sum
 
diff --git a/Mathlib/Algebra/Group/Action/Units.lean b/Mathlib/Algebra/Group/Action/Units.lean
index d7ad6e43cfdf4..b1b817a1fb920 100644
--- a/Mathlib/Algebra/Group/Action/Units.lean
+++ b/Mathlib/Algebra/Group/Action/Units.lean
@@ -69,8 +69,8 @@ instance mulAction' [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M]
     [IsScalarTower G M M] : MulAction G Mˣ where
   smul g m :=
     ⟨g • (m : M), (g⁻¹ • ((m⁻¹ : Mˣ) : M)),
-      by rw [smul_mul_smul, Units.mul_inv, mul_inv_cancel, one_smul],
-      by rw [smul_mul_smul, Units.inv_mul, inv_mul_cancel, one_smul]⟩
+      by rw [smul_mul_smul_comm, Units.mul_inv, mul_inv_cancel, one_smul],
+      by rw [smul_mul_smul_comm, Units.inv_mul, inv_mul_cancel, one_smul]⟩
   one_smul m := Units.ext <| one_smul _ _
   mul_smul g₁ g₂ m := Units.ext <| mul_smul _ _ _
 
diff --git a/Mathlib/Algebra/Group/AddChar.lean b/Mathlib/Algebra/Group/AddChar.lean
index 94647d455db1d..f9a1d383c185c 100644
--- a/Mathlib/Algebra/Group/AddChar.lean
+++ b/Mathlib/Algebra/Group/AddChar.lean
@@ -23,6 +23,15 @@ We also include some constructions specific to the case when `A = R` is a ring;
 For more refined results of a number-theoretic nature (primitive characters, Gauss sums, etc)
 see `Mathlib.NumberTheory.LegendreSymbol.AddCharacter`.
 
+# Implementation notes
+
+Due to their role as the dual of an additive group, additive characters must themselves be an
+additive group. This contrasts to their pointwise operations which make them a multiplicative group.
+We simply define both the additive and multiplicative group structures and prove them equal.
+
+For more information on this design decision, see the following zulip thread:
+https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Additive.20characters
+
 ## Tags
 
 additive character
@@ -33,6 +42,8 @@ additive character
 -/
 
 open Function Multiplicative
+open Finset hiding card
+open Fintype (card)
 
 section AddCharDef
 
@@ -171,11 +182,28 @@ lemma coe_toAddMonoidHomEquiv (ψ : AddChar A M) :
 @[simp] lemma toAddMonoidHomEquiv_symm_apply (ψ : A →+ Additive M) (a : A) :
     toAddMonoidHomEquiv.symm ψ a = Additive.toMul (ψ a) := rfl
 
-/-- The trivial additive character (sending everything to `1`) is `(1 : AddChar A M).` -/
+/-- The trivial additive character (sending everything to `1`). -/
 instance instOne : One (AddChar A M) := toMonoidHomEquiv.one
 
+/-- The trivial additive character (sending everything to `1`). -/
+instance instZero : Zero (AddChar A M) := ⟨1⟩
+
 @[simp, norm_cast] lemma coe_one : ⇑(1 : AddChar A M) = 1 := rfl
+@[simp, norm_cast] lemma coe_zero : ⇑(0 : AddChar A M) = 1 := rfl
 @[simp] lemma one_apply (a : A) : (1 : AddChar A M) a = 1 := rfl
+@[simp] lemma zero_apply (a : A) : (0 : AddChar A M) a = 1 := rfl
+
+lemma one_eq_zero : (1 : AddChar A M) = (0 : AddChar A M) := rfl
+
+@[simp, norm_cast] lemma coe_eq_one : ⇑ψ = 1 ↔ ψ = 0 := by rw [← coe_zero, DFunLike.coe_fn_eq]
+
+@[simp] lemma toMonoidHomEquiv_zero : toMonoidHomEquiv (0 : AddChar A M) = 1 := rfl
+@[simp] lemma toMonoidHomEquiv_symm_one :
+    toMonoidHomEquiv.symm (1 : Multiplicative A →* M) = 0 := rfl
+
+@[simp] lemma toAddMonoidHomEquiv_zero : toAddMonoidHomEquiv (0 : AddChar A M) = 0 := rfl
+@[simp] lemma toAddMonoidHomEquiv_symm_zero :
+    toAddMonoidHomEquiv.symm (0 : A →+ Additive M) = 0 := rfl
 
 instance instInhabited : Inhabited (AddChar A M) := ⟨1⟩
 
@@ -220,7 +248,9 @@ lemma compAddMonoidHom_injective_right (ψ : AddChar B M) (hψ : Injective ψ) :
   rw [DFunLike.ext'_iff] at h ⊢; exact hψ.comp_left h
 
 lemma eq_one_iff : ψ = 1 ↔ ∀ x, ψ x = 1 := DFunLike.ext_iff
+lemma eq_zero_iff : ψ = 0 ↔ ∀ x, ψ x = 1 := DFunLike.ext_iff
 lemma ne_one_iff : ψ ≠ 1 ↔ ∃ x, ψ x ≠ 1 := DFunLike.ne_iff
+lemma ne_zero_iff : ψ ≠ 0 ↔ ∃ x, ψ x ≠ 1 := DFunLike.ne_iff
 
 /-- An additive character is *nontrivial* if it takes a value `≠ 1`. -/
 @[deprecated (since := "2024-06-06")]
@@ -232,19 +262,51 @@ set_option linter.deprecated false in
 lemma isNontrivial_iff_ne_trivial (ψ : AddChar A M) : IsNontrivial ψ ↔ ψ ≠ 1 :=
   not_forall.symm.trans (DFunLike.ext_iff (f := ψ) (g := 1)).symm.not
 
+noncomputable instance : DecidableEq (AddChar A M) := Classical.decEq _
+
 end Basic
 
 section toCommMonoid
 
-variable {A M : Type*} [AddMonoid A] [CommMonoid M]
+variable {ι A M : Type*} [AddMonoid A] [CommMonoid M]
 
 /-- When `M` is commutative, `AddChar A M` is a commutative monoid. -/
 instance instCommMonoid : CommMonoid (AddChar A M) := toMonoidHomEquiv.commMonoid
+/-- When `M` is commutative, `AddChar A M` is an additive commutative monoid. -/
+instance instAddCommMonoid : AddCommMonoid (AddChar A M) := Additive.addCommMonoid
 
 @[simp, norm_cast] lemma coe_mul (ψ χ : AddChar A M) : ⇑(ψ * χ) = ψ * χ := rfl
+@[simp, norm_cast] lemma coe_add (ψ χ : AddChar A M) : ⇑(ψ + χ) = ψ * χ := rfl
 @[simp, norm_cast] lemma coe_pow (ψ : AddChar A M) (n : ℕ) : ⇑(ψ ^ n) = ψ ^ n := rfl
-@[simp, norm_cast] lemma mul_apply (ψ φ : AddChar A M) (a : A) : (ψ * φ) a = ψ a * φ a := rfl
-@[simp, norm_cast] lemma pow_apply (ψ : AddChar A M) (n : ℕ) (a : A) : (ψ ^ n) a = (ψ a) ^ n := rfl
+@[simp, norm_cast] lemma coe_nsmul (n : ℕ) (ψ : AddChar A M) : ⇑(n • ψ) = ψ ^ n := rfl
+
+@[simp, norm_cast]
+lemma coe_prod (s : Finset ι) (ψ : ι → AddChar A M) : ∏ i in s, ψ i = ∏ i in s, ⇑(ψ i) := by
+  induction s using Finset.cons_induction <;> simp [*]
+
+@[simp, norm_cast]
+lemma coe_sum (s : Finset ι) (ψ : ι → AddChar A M) : ∑ i in s, ψ i = ∏ i in s, ⇑(ψ i) := by
+  induction s using Finset.cons_induction <;> simp [*]
+
+@[simp] lemma mul_apply (ψ φ : AddChar A M) (a : A) : (ψ * φ) a = ψ a * φ a := rfl
+@[simp] lemma add_apply (ψ φ : AddChar A M) (a : A) : (ψ + φ) a = ψ a * φ a := rfl
+@[simp] lemma pow_apply (ψ : AddChar A M) (n : ℕ) (a : A) : (ψ ^ n) a = (ψ a) ^ n := rfl
+@[simp] lemma nsmul_apply (ψ : AddChar A M) (n : ℕ) (a : A) : (n • ψ) a = (ψ a) ^ n := rfl
+
+lemma prod_apply (s : Finset ι) (ψ : ι → AddChar A M) (a : A) :
+    (∏ i in s, ψ i) a = ∏ i in s, ψ i a := by rw [coe_prod, Finset.prod_apply]
+
+lemma sum_apply (s : Finset ι) (ψ : ι → AddChar A M) (a : A) :
+    (∑ i in s, ψ i) a = ∏ i in s, ψ i a := by rw [coe_sum, Finset.prod_apply]
+
+lemma mul_eq_add (ψ χ : AddChar A M) : ψ * χ = ψ + χ := rfl
+lemma pow_eq_nsmul (ψ : AddChar A M) (n : ℕ) : ψ ^ n = n • ψ := rfl
+lemma prod_eq_sum (s : Finset ι) (ψ : ι → AddChar A M) : ∏ i in s, ψ i = ∑ i in s, ψ i := rfl
+
+@[simp] lemma toMonoidHomEquiv_add (ψ φ : AddChar A M) :
+    toMonoidHomEquiv (ψ + φ) = toMonoidHomEquiv ψ * toMonoidHomEquiv φ := rfl
+@[simp] lemma toMonoidHomEquiv_symm_mul (ψ φ : Multiplicative A →* M) :
+    toMonoidHomEquiv.symm (ψ * φ) = toMonoidHomEquiv.symm ψ + toMonoidHomEquiv.symm φ := rfl
 
 /-- The natural equivalence to `(Multiplicative A →* M)` is a monoid isomorphism. -/
 def toMonoidHomMulEquiv : AddChar A M ≃* (Multiplicative A →* M) :=
@@ -255,8 +317,41 @@ def toMonoidHomMulEquiv : AddChar A M ≃* (Multiplicative A →* M) :=
 def toAddMonoidAddEquiv : Additive (AddChar A M) ≃+ (A →+ Additive M) :=
   { toAddMonoidHomEquiv with map_add' := fun φ ψ ↦ by rfl }
 
+/-- The double dual embedding. -/
+def doubleDualEmb : A →+ AddChar (AddChar A M) M where
+  toFun a := { toFun := fun ψ ↦ ψ a
+               map_zero_eq_one' := by simp
+               map_add_eq_mul' := by simp }
+  map_zero' := by ext; simp
+  map_add' _ _ := by ext; simp [map_add_eq_mul]
+
+@[simp] lemma doubleDualEmb_apply (a : A) (ψ : AddChar A M) : doubleDualEmb a ψ = ψ a := rfl
+
 end toCommMonoid
 
+section CommSemiring
+variable {A R : Type*} [AddGroup A] [Fintype A] [CommSemiring R] [IsDomain R]
+  {ψ : AddChar A R}
+
+lemma sum_eq_ite (ψ : AddChar A R) [Decidable (ψ = 0)] :
+    ∑ a, ψ a = if ψ = 0 then ↑(card A) else 0 := by
+  split_ifs with h
+  · simp [h, card_univ]
+  obtain ⟨x, hx⟩ := ne_one_iff.1 h
+  refine eq_zero_of_mul_eq_self_left hx ?_
+  rw [Finset.mul_sum]
+  exact Fintype.sum_equiv (Equiv.addLeft x) _ _ fun y ↦ (map_add_eq_mul ..).symm
+
+variable [CharZero R]
+
+lemma sum_eq_zero_iff_ne_zero : ∑ x, ψ x = 0 ↔ ψ ≠ 0 := by
+  classical
+  rw [sum_eq_ite, Ne.ite_eq_right_iff]; exact Nat.cast_ne_zero.2 Fintype.card_ne_zero
+
+lemma sum_ne_zero_iff_eq_zero : ∑ x, ψ x ≠ 0 ↔ ψ = 0 := sum_eq_zero_iff_ne_zero.not_left
+
+end CommSemiring
+
 /-!
 ## Additive characters of additive abelian groups
 -/
@@ -273,7 +368,13 @@ instance instCommGroup : CommGroup (AddChar A M) :=
     inv := fun ψ ↦ ψ.compAddMonoidHom negAddMonoidHom
     inv_mul_cancel := fun ψ ↦ by ext1 x; simp [negAddMonoidHom, ← map_add_eq_mul]}
 
-@[simp] lemma inv_apply (ψ : AddChar A M) (x : A) : ψ⁻¹ x = ψ (-x) := rfl
+/-- The additive characters on a commutative additive group form a commutative group. -/
+instance : AddCommGroup (AddChar A M) := Additive.addCommGroup
+
+@[simp] lemma inv_apply (ψ : AddChar A M) (a : A) : ψ⁻¹ a = ψ (-a) := rfl
+@[simp] lemma neg_apply (ψ : AddChar A M) (a : A) : (-ψ) a = ψ (-a) := rfl
+lemma div_apply (ψ χ : AddChar A M) (a : A) : (ψ / χ) a = ψ a * χ (-a) := rfl
+lemma sub_apply (ψ χ : AddChar A M) (a : A) : (ψ - χ) a = ψ a * χ (-a) := rfl
 
 end fromAddCommGroup
 
@@ -303,11 +404,22 @@ lemma map_zsmul_eq_zpow (ψ : AddChar A M) (n : ℤ) (a : A) : ψ (n • a) = (
 
 end fromAddGrouptoDivisionMonoid
 
-section fromAddGrouptoDivisionCommMonoid
-
+section fromAddCommGrouptoDivisionCommMonoid
 variable {A M : Type*} [AddCommGroup A] [DivisionCommMonoid M]
 
-lemma inv_apply' (ψ : AddChar A M) (x : A) : ψ⁻¹ x = (ψ x)⁻¹ := by rw [inv_apply, map_neg_eq_inv]
+lemma inv_apply' (ψ : AddChar A M) (a : A) : ψ⁻¹ a = (ψ a)⁻¹ := by rw [inv_apply, map_neg_eq_inv]
+lemma neg_apply' (ψ : AddChar A M) (a : A) : (-ψ) a = (ψ a)⁻¹ := map_neg_eq_inv _ _
+
+lemma div_apply' (ψ χ : AddChar A M) (a : A) : (ψ / χ) a = ψ a / χ a := by
+  rw [div_apply, map_neg_eq_inv, div_eq_mul_inv]
+
+lemma sub_apply' (ψ χ : AddChar A M) (a : A) : (ψ - χ) a = ψ a / χ a := by
+  rw [sub_apply, map_neg_eq_inv, div_eq_mul_inv]
+
+@[simp] lemma zsmul_apply (n : ℤ) (ψ : AddChar A M) (a : A) : (n • ψ) a = ψ a ^ n := by
+  cases n <;> simp [-neg_apply, neg_apply']
+
+@[simp] lemma zpow_apply (ψ : AddChar A M) (n : ℤ) (a : A) : (ψ ^ n) a = ψ a ^ n := zsmul_apply ..
 
 lemma map_sub_eq_div (ψ : AddChar A M) (a b : A) : ψ (a - b) = ψ a / ψ b :=
   ψ.toMonoidHom.map_div _ _
@@ -315,7 +427,7 @@ lemma map_sub_eq_div (ψ : AddChar A M) (a b : A) : ψ (a - b) = ψ a / ψ b :=
 lemma injective_iff {ψ : AddChar A M} : Injective ψ ↔ ∀ ⦃x⦄, ψ x = 1 → x = 0 :=
   ψ.toMonoidHom.ker_eq_bot_iff.symm.trans eq_bot_iff
 
-end fromAddGrouptoDivisionCommMonoid
+end fromAddCommGrouptoDivisionCommMonoid
 
 section MonoidWithZero
 variable {A M₀ : Type*} [AddGroup A] [MonoidWithZero M₀] [Nontrivial M₀]
diff --git a/Mathlib/Algebra/Group/Basic.lean b/Mathlib/Algebra/Group/Basic.lean
index be38ab0a3a6b8..244c63da588ad 100644
--- a/Mathlib/Algebra/Group/Basic.lean
+++ b/Mathlib/Algebra/Group/Basic.lean
@@ -90,7 +90,7 @@ section Semigroup
 variable [Semigroup α]
 
 @[to_additive]
-instance Semigroup.to_isAssociative : Std.Associative (α := α)  (· * ·) := ⟨mul_assoc⟩
+instance Semigroup.to_isAssociative : Std.Associative (α := α) (· * ·) := ⟨mul_assoc⟩
 
 /-- Composing two multiplications on the left by `y` then `x`
 is equal to a multiplication on the left by `x * y`.
@@ -148,12 +148,12 @@ section CommSemigroup
 variable [CommSemigroup G]
 
 @[to_additive]
-theorem mul_left_comm : ∀ a b c : G, a * (b * c) = b * (a * c) :=
-  left_comm Mul.mul mul_comm mul_assoc
+theorem mul_left_comm (a b c : G) : a * (b * c) = b * (a * c) := by
+  rw [← mul_assoc, mul_comm a, mul_assoc]
 
 @[to_additive]
-theorem mul_right_comm : ∀ a b c : G, a * b * c = a * c * b :=
-  right_comm Mul.mul mul_comm mul_assoc
+theorem mul_right_comm (a b c : G) : a * b * c = a * c * b := by
+  rw [mul_assoc, mul_comm b, mul_assoc]
 
 @[to_additive]
 theorem mul_mul_mul_comm (a b c d : G) : a * b * (c * d) = a * c * (b * d) := by
@@ -172,7 +172,7 @@ end CommSemigroup
 attribute [local simp] mul_assoc sub_eq_add_neg
 
 section Monoid
-variable [Monoid M] {a b c : M} {m n : ℕ}
+variable [Monoid M] {a b : M} {m n : ℕ}
 
 @[to_additive boole_nsmul]
 lemma pow_boole (P : Prop) [Decidable P] (a : M) :
@@ -316,7 +316,7 @@ end InvolutiveInv
 
 section DivInvMonoid
 
-variable [DivInvMonoid G] {a b c : G}
+variable [DivInvMonoid G]
 
 @[to_additive, field_simps] -- The attributes are out of order on purpose
 theorem inv_eq_one_div (x : G) : x⁻¹ = 1 / x := by rw [div_eq_mul_inv, one_mul]
@@ -586,14 +586,16 @@ theorem mul_div_mul_comm : a * b / (c * d) = a / c * (b / d) := by simp
   | (n : ℕ) => by simp_rw [zpow_natCast, mul_pow]
   | .negSucc n => by simp_rw [zpow_negSucc, ← inv_pow, mul_inv, mul_pow]
 
-@[to_additive (attr := simp) nsmul_sub]
+@[to_additive nsmul_sub]
 lemma div_pow (a b : α) (n : ℕ) : (a / b) ^ n = a ^ n / b ^ n := by
   simp only [div_eq_mul_inv, mul_pow, inv_pow]
 
-@[to_additive (attr := simp) zsmul_sub]
+@[to_additive zsmul_sub]
 lemma div_zpow (a b : α) (n : ℤ) : (a / b) ^ n = a ^ n / b ^ n := by
   simp only [div_eq_mul_inv, mul_zpow, inv_zpow]
 
+attribute [field_simps] div_pow div_zpow
+
 end DivisionCommMonoid
 
 section Group
@@ -643,6 +645,16 @@ theorem mul_eq_one_iff_eq_inv : a * b = 1 ↔ a = b⁻¹ :=
 theorem mul_eq_one_iff_inv_eq : a * b = 1 ↔ a⁻¹ = b := by
   rw [mul_eq_one_iff_eq_inv, inv_eq_iff_eq_inv]
 
+/-- Variant of `mul_eq_one_iff_eq_inv` with swapped equality. -/
+@[to_additive]
+theorem mul_eq_one_iff_eq_inv' : a * b = 1 ↔ b = a⁻¹ := by
+  rw [mul_eq_one_iff_inv_eq, eq_comm]
+
+/-- Variant of `mul_eq_one_iff_inv_eq` with swapped equality. -/
+@[to_additive]
+theorem mul_eq_one_iff_inv_eq' : a * b = 1 ↔ b⁻¹ = a := by
+  rw [mul_eq_one_iff_eq_inv, eq_comm]
+
 @[to_additive]
 theorem eq_inv_iff_mul_eq_one : a = b⁻¹ ↔ a * b = 1 :=
   mul_eq_one_iff_eq_inv.symm
diff --git a/Mathlib/Algebra/Group/Commute/Defs.lean b/Mathlib/Algebra/Group/Commute/Defs.lean
index b7bcb09ff56a4..324484366c26a 100644
--- a/Mathlib/Algebra/Group/Commute/Defs.lean
+++ b/Mathlib/Algebra/Group/Commute/Defs.lean
@@ -181,7 +181,7 @@ end Monoid
 
 section DivisionMonoid
 
-variable [DivisionMonoid G] {a b c d : G}
+variable [DivisionMonoid G] {a b : G}
 
 @[to_additive]
 protected theorem mul_inv (hab : Commute a b) : (a * b)⁻¹ = a⁻¹ * b⁻¹ := by rw [hab.eq, mul_inv_rev]
diff --git a/Mathlib/Algebra/Group/Commute/Hom.lean b/Mathlib/Algebra/Group/Commute/Hom.lean
index e4279c1534a47..f95402d1287eb 100644
--- a/Mathlib/Algebra/Group/Commute/Hom.lean
+++ b/Mathlib/Algebra/Group/Commute/Hom.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
+Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
   Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Commute.Defs
diff --git a/Mathlib/Algebra/Group/Defs.lean b/Mathlib/Algebra/Group/Defs.lean
index d1d9eaf700eef..ea928af610f81 100644
--- a/Mathlib/Algebra/Group/Defs.lean
+++ b/Mathlib/Algebra/Group/Defs.lean
@@ -35,7 +35,7 @@ actions and register the following instances:
 - `SMul ℕ M` for additive monoids `M`, and `SMul ℤ G` for additive groups `G`.
 
 `SMul` is typically, but not exclusively, used for scalar multiplication-like operators.
-See the module `Algebra.AddTorsor` for a motivating example for the name `VAdd` (vector addition)`.
+See the module `Algebra.AddTorsor` for a motivating example for the name `VAdd` (vector addition).
 
 ## Notation
 
@@ -49,7 +49,6 @@ See the module `Algebra.AddTorsor` for a motivating example for the name `VAdd`
 assert_not_exists MonoidWithZero
 assert_not_exists DenselyOrdered
 assert_not_exists Function.Injective.eq_iff
-assert_not_exists IsCommutative
 
 universe u v w
 
@@ -553,7 +552,7 @@ instance AddMonoid.toNatSMul {M : Type*} [AddMonoid M] : SMul ℕ M :=
 attribute [to_additive existing toNatSMul] Monoid.toNatPow
 
 section Monoid
-variable {M : Type*} [Monoid M] {a b c : M} {m n : ℕ}
+variable {M : Type*} [Monoid M] {a b c : M}
 
 @[to_additive (attr := simp) nsmul_eq_smul]
 theorem npow_eq_pow (n : ℕ) (x : M) : Monoid.npow n x = x ^ n :=
@@ -633,17 +632,17 @@ section LeftCancelMonoid
 /-- An additive monoid in which addition is left-cancellative.
 Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
 is useful to define the sum over the empty set, so `AddLeftCancelSemigroup` is not enough. -/
-class AddLeftCancelMonoid (M : Type u) extends AddLeftCancelSemigroup M, AddMonoid M
+class AddLeftCancelMonoid (M : Type u) extends AddMonoid M, AddLeftCancelSemigroup M
 
 attribute [instance 75] AddLeftCancelMonoid.toAddMonoid -- See note [lower cancel priority]
 
 /-- A monoid in which multiplication is left-cancellative. -/
 @[to_additive]
-class LeftCancelMonoid (M : Type u) extends LeftCancelSemigroup M, Monoid M
+class LeftCancelMonoid (M : Type u) extends Monoid M, LeftCancelSemigroup M
 
 attribute [instance 75] LeftCancelMonoid.toMonoid -- See note [lower cancel priority]
 
-attribute [to_additive existing] LeftCancelMonoid.toMonoid
+attribute [to_additive existing] LeftCancelMonoid.toLeftCancelSemigroup
 
 end LeftCancelMonoid
 
@@ -652,17 +651,17 @@ section RightCancelMonoid
 /-- An additive monoid in which addition is right-cancellative.
 Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
 is useful to define the sum over the empty set, so `AddRightCancelSemigroup` is not enough. -/
-class AddRightCancelMonoid (M : Type u) extends AddRightCancelSemigroup M, AddMonoid M
+class AddRightCancelMonoid (M : Type u) extends AddMonoid M, AddRightCancelSemigroup M
 
 attribute [instance 75] AddRightCancelMonoid.toAddMonoid -- See note [lower cancel priority]
 
 /-- A monoid in which multiplication is right-cancellative. -/
 @[to_additive]
-class RightCancelMonoid (M : Type u) extends RightCancelSemigroup M, Monoid M
+class RightCancelMonoid (M : Type u) extends Monoid M, RightCancelSemigroup M
 
 attribute [instance 75] RightCancelMonoid.toMonoid -- See note [lower cancel priority]
 
-attribute [to_additive existing] RightCancelMonoid.toMonoid
+attribute [to_additive existing] RightCancelMonoid.toRightCancelSemigroup
 
 end RightCancelMonoid
 
@@ -680,17 +679,17 @@ class CancelMonoid (M : Type u) extends LeftCancelMonoid M, RightCancelMonoid M
 attribute [to_additive existing] CancelMonoid.toRightCancelMonoid
 
 /-- Commutative version of `AddCancelMonoid`. -/
-class AddCancelCommMonoid (M : Type u) extends AddLeftCancelMonoid M, AddCommMonoid M
+class AddCancelCommMonoid (M : Type u) extends AddCommMonoid M, AddLeftCancelMonoid M
 
 attribute [instance 75] AddCancelCommMonoid.toAddCommMonoid -- See note [lower cancel priority]
 
 /-- Commutative version of `CancelMonoid`. -/
 @[to_additive]
-class CancelCommMonoid (M : Type u) extends LeftCancelMonoid M, CommMonoid M
+class CancelCommMonoid (M : Type u) extends CommMonoid M, LeftCancelMonoid M
 
 attribute [instance 75] CancelCommMonoid.toCommMonoid -- See note [lower cancel priority]
 
-attribute [to_additive existing] CancelCommMonoid.toCommMonoid
+attribute [to_additive existing] CancelCommMonoid.toLeftCancelMonoid
 
 -- see Note [lower instance priority]
 @[to_additive]
@@ -808,7 +807,7 @@ class DivInvMonoid (G : Type u) extends Monoid G, Inv G, Div G where
   /-- `a ^ 0 = 1` -/
   protected zpow_zero' : ∀ a : G, zpow 0 a = 1 := by intros; rfl
   /-- `a ^ (n + 1) = a ^ n * a` -/
-  protected zpow_succ' (n : ℕ) (a : G) : zpow (Int.ofNat n.succ) a = zpow (Int.ofNat n) a  * a := by
+  protected zpow_succ' (n : ℕ) (a : G) : zpow (Int.ofNat n.succ) a = zpow (Int.ofNat n) a * a := by
     intros; rfl
   /-- `a ^ -(n + 1) = (a ^ (n + 1))⁻¹` -/
   protected zpow_neg' (n : ℕ) (a : G) : zpow (Int.negSucc n) a = (zpow n.succ a)⁻¹ := by intros; rfl
@@ -880,7 +879,7 @@ theorem exists_zpow_surjective (G : Type*) [Pow G ℤ] [IsCyclic G] :
 
 section DivInvMonoid
 
-variable [DivInvMonoid G] {a b : G}
+variable [DivInvMonoid G]
 
 @[to_additive (attr := simp) zsmul_eq_smul] theorem zpow_eq_pow (n : ℤ) (x : G) :
     DivInvMonoid.zpow n x = x ^ n :=
@@ -1051,7 +1050,7 @@ attribute [to_additive] Group
 
 section Group
 
-variable [Group G] {a b c : G}
+variable [Group G] {a b : G}
 
 @[to_additive (attr := simp)]
 theorem inv_mul_cancel (a : G) : a⁻¹ * a = 1 :=
diff --git a/Mathlib/Algebra/Group/Equiv/Basic.lean b/Mathlib/Algebra/Group/Equiv/Basic.lean
index 6e39d0838d7ec..cb72fdfe7db40 100644
--- a/Mathlib/Algebra/Group/Equiv/Basic.lean
+++ b/Mathlib/Algebra/Group/Equiv/Basic.lean
@@ -28,7 +28,7 @@ Equiv, MulEquiv, AddEquiv
 
 open Function
 
-variable {F α β A B M N P Q G H : Type*}
+variable {F α β M N P G H : Type*}
 
 /-- Makes a `OneHom` inverse from the bijective inverse of a `OneHom` -/
 @[to_additive (attr := simps)
@@ -90,6 +90,12 @@ add_decl_doc MulEquiv.toEquiv
 /-- The `MulHom` underlying a `MulEquiv`. -/
 add_decl_doc MulEquiv.toMulHom
 
+/-- Notation for a `MulEquiv`. -/
+infixl:25 " ≃* " => MulEquiv
+
+/-- Notation for an `AddEquiv`. -/
+infixl:25 " ≃+ " => AddEquiv
+
 /-- `MulEquivClass F A B` states that `F` is a type of multiplication-preserving morphisms.
 You should extend this class when you extend `MulEquiv`. -/
 -- TODO: make this a synonym for MulHomClass?
@@ -99,12 +105,6 @@ class MulEquivClass (F : Type*) (A B : outParam Type*) [Mul A] [Mul B] [EquivLik
   /-- Preserves multiplication. -/
   map_mul : ∀ (f : F) (a b), f (a * b) = f a * f b
 
-/-- Notation for a `MulEquiv`. -/
-infixl:25 " ≃* " => MulEquiv
-
-/-- Notation for an `AddEquiv`. -/
-infixl:25 " ≃+ " => AddEquiv
-
 namespace MulEquivClass
 
 variable (F)
@@ -130,7 +130,6 @@ instance (priority := 100) instMonoidHomClass
         _ = e (EquivLike.inv e (1 : N)) := by rw [← map_mul, one_mul]
         _ = 1 := EquivLike.right_inv e 1 }
 
-variable [EquivLike F α β]
 variable {F}
 
 @[to_additive (attr := simp)]
@@ -169,7 +168,9 @@ theorem MulEquivClass.toMulEquiv_injective [Mul α] [Mul β] [MulEquivClass F α
 
 namespace MulEquiv
 section Mul
-variable [Mul M] [Mul N] [Mul P] [Mul Q]
+variable [Mul M] [Mul N] [Mul P]
+
+section coe
 
 @[to_additive]
 instance : EquivLike (M ≃* N) M N where
@@ -183,13 +184,35 @@ instance : EquivLike (M ≃* N) M N where
     congr
     apply Equiv.coe_fn_injective h₁
 
+@[to_additive] -- shortcut instance that doesn't generate any subgoals
+instance : CoeFun (M ≃* N) fun _ ↦ M → N where
+  coe f := f
+
 @[to_additive]
 instance : MulEquivClass (M ≃* N) M N where
   map_mul f := f.map_mul'
 
-@[to_additive] -- shortcut instance that doesn't generate any subgoals
-instance : CoeFun (M ≃* N) fun _ ↦ M → N where
-  coe f := f
+/-- Two multiplicative isomorphisms agree if they are defined by the
+same underlying function. -/
+@[to_additive (attr := ext)
+  "Two additive isomorphisms agree if they are defined by the same underlying function."]
+theorem ext {f g : MulEquiv M N} (h : ∀ x, f x = g x) : f = g :=
+  DFunLike.ext f g h
+
+@[to_additive]
+protected theorem congr_arg {f : MulEquiv M N} {x x' : M} : x = x' → f x = f x' :=
+  DFunLike.congr_arg f
+
+@[to_additive]
+protected theorem congr_fun {f g : MulEquiv M N} (h : f = g) (x : M) : f x = g x :=
+  DFunLike.congr_fun h x
+
+@[to_additive (attr := simp)]
+theorem coe_mk (f : M ≃ N) (hf : ∀ x y, f (x * y) = f x * f y) : (mk f hf : M → N) = f := rfl
+
+@[to_additive (attr := simp)]
+theorem mk_coe (e : M ≃* N) (e' h₁ h₂ h₃) : (⟨⟨e, e', h₁, h₂⟩, h₃⟩ : M ≃* N) = e :=
+  ext fun _ => rfl
 
 @[to_additive (attr := simp)]
 theorem toEquiv_eq_coe (f : M ≃* N) : f.toEquiv = f :=
@@ -210,6 +233,14 @@ theorem coe_toEquiv (f : M ≃* N) : ⇑(f : M ≃ N) = f := rfl
 @[to_additive (attr := simp 1100)]
 theorem coe_toMulHom {f : M ≃* N} : (f.toMulHom : M → N) = f := rfl
 
+/-- Makes a multiplicative isomorphism from a bijection which preserves multiplication. -/
+@[to_additive "Makes an additive isomorphism from a bijection which preserves addition."]
+def mk' (f : M ≃ N) (h : ∀ x y, f (x * y) = f x * f y) : M ≃* N := ⟨f, h⟩
+
+end coe
+
+section map
+
 /-- A multiplicative isomorphism preserves multiplication. -/
 @[to_additive "An additive isomorphism preserves addition."]
 protected theorem map_mul (f : M ≃* N) : ∀ x y, f (x * y) = f x * f y :=
@@ -218,9 +249,9 @@ protected theorem map_mul (f : M ≃* N) : ∀ x y, f (x * y) = f x * f y :=
 attribute [deprecated map_mul (since := "2024-08-08")] MulEquiv.map_mul
 attribute [deprecated map_add (since := "2024-08-08")] AddEquiv.map_add
 
-/-- Makes a multiplicative isomorphism from a bijection which preserves multiplication. -/
-@[to_additive "Makes an additive isomorphism from a bijection which preserves addition."]
-def mk' (f : M ≃ N) (h : ∀ x y, f (x * y) = f x * f y) : M ≃* N := ⟨f, h⟩
+end map
+
+section bijective
 
 @[to_additive]
 protected theorem bijective (e : M ≃* N) : Function.Bijective e :=
@@ -234,6 +265,15 @@ protected theorem injective (e : M ≃* N) : Function.Injective e :=
 protected theorem surjective (e : M ≃* N) : Function.Surjective e :=
   EquivLike.surjective e
 
+-- Porting note (#10618): `simp` can prove this
+@[to_additive]
+theorem apply_eq_iff_eq (e : M ≃* N) {x y : M} : e x = e y ↔ x = y :=
+  e.injective.eq_iff
+
+end bijective
+
+section refl
+
 /-- The identity map is a multiplicative isomorphism. -/
 @[to_additive (attr := refl) "The identity map is an additive isomorphism."]
 def refl (M : Type*) [Mul M] : M ≃* M :=
@@ -242,6 +282,16 @@ def refl (M : Type*) [Mul M] : M ≃* M :=
 @[to_additive]
 instance : Inhabited (M ≃* M) := ⟨refl M⟩
 
+@[to_additive (attr := simp)]
+theorem coe_refl : ↑(refl M) = id := rfl
+
+@[to_additive (attr := simp)]
+theorem refl_apply (m : M) : refl M m = m := rfl
+
+end refl
+
+section symm
+
 /-- An alias for `h.symm.map_mul`. Introduced to fix the issue in
 https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.234183.20.60simps.60.20maximum.20recursion.20depth
 -/
@@ -265,24 +315,9 @@ theorem coe_toEquiv_symm (f : M ≃* N) : ((f : M ≃ N).symm : N → M) = f.sym
 @[to_additive (attr := simp)]
 theorem equivLike_inv_eq_symm (f : M ≃* N) : EquivLike.inv f = f.symm := rfl
 
--- we don't hyperlink the note in the additive version, since that breaks syntax highlighting
--- in the whole file.
-
-/-- See Note [custom simps projection] -/
-@[to_additive "See Note [custom simps projection]"] -- this comment fixes the syntax highlighting "
-def Simps.symm_apply (e : M ≃* N) : N → M :=
-  e.symm
-
-initialize_simps_projections AddEquiv (toFun → apply, invFun → symm_apply)
-
-initialize_simps_projections MulEquiv (toFun → apply, invFun → symm_apply)
-
 @[to_additive (attr := simp)]
 theorem toEquiv_symm (f : M ≃* N) : (f.symm : N ≃ M) = (f : M ≃ N).symm := rfl
 
-@[to_additive (attr := simp)]
-theorem coe_mk (f : M ≃ N) (hf : ∀ x y, f (x * y) = f x * f y) : (mk f hf : M → N) = f := rfl
-
 -- Porting note: `toEquiv_mk` no longer needed in Lean4
 
 @[to_additive (attr := simp)]
@@ -292,6 +327,10 @@ theorem symm_symm (f : M ≃* N) : f.symm.symm = f := rfl
 theorem symm_bijective : Function.Bijective (symm : (M ≃* N) → N ≃* M) :=
   Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
 
+@[to_additive (attr := simp)]
+theorem mk_coe' (e : M ≃* N) (f h₁ h₂ h₃) : (MulEquiv.mk ⟨f, e, h₁, h₂⟩ h₃ : N ≃* M) = e.symm :=
+  symm_bijective.injective <| ext fun _ => rfl
+
 @[to_additive (attr := simp)]
 theorem symm_mk (f : M ≃ N) (h) :
     (MulEquiv.mk f h).symm = ⟨f.symm, (MulEquiv.mk f h).symm_map_mul⟩ := rfl
@@ -299,13 +338,6 @@ theorem symm_mk (f : M ≃ N) (h) :
 @[to_additive (attr := simp)]
 theorem refl_symm : (refl M).symm = refl M := rfl
 
-/-- Transitivity of multiplication-preserving isomorphisms -/
-@[to_additive (attr := trans) "Transitivity of addition-preserving isomorphisms"]
-def trans (h1 : M ≃* N) (h2 : N ≃* P) : M ≃* P :=
-  { h1.toEquiv.trans h2.toEquiv with
-    map_mul' := fun x y => show h2 (h1 (x * y)) = h2 (h1 x) * h2 (h1 y) by
-      rw [map_mul, map_mul] }
-
 /-- `e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`. -/
 @[to_additive (attr := simp) "`e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`."]
 theorem apply_symm_apply (e : M ≃* N) (y : N) : e (e.symm y) = y :=
@@ -324,27 +356,6 @@ theorem symm_comp_self (e : M ≃* N) : e.symm ∘ e = id :=
 theorem self_comp_symm (e : M ≃* N) : e ∘ e.symm = id :=
   funext e.apply_symm_apply
 
-@[to_additive (attr := simp)]
-theorem coe_refl : ↑(refl M) = id := rfl
-
-@[to_additive (attr := simp)]
-theorem refl_apply (m : M) : refl M m = m := rfl
-
-@[to_additive (attr := simp)]
-theorem coe_trans (e₁ : M ≃* N) (e₂ : N ≃* P) : ↑(e₁.trans e₂) = e₂ ∘ e₁ := rfl
-
-@[to_additive (attr := simp)]
-theorem trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (m : M) : e₁.trans e₂ m = e₂ (e₁ m) := rfl
-
-@[to_additive (attr := simp)]
-theorem symm_trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (p : P) :
-    (e₁.trans e₂).symm p = e₁.symm (e₂.symm p) := rfl
-
--- Porting note (#10618): `simp` can prove this
-@[to_additive]
-theorem apply_eq_iff_eq (e : M ≃* N) {x y : M} : e x = e y ↔ x = y :=
-  e.injective.eq_iff
-
 @[to_additive]
 theorem apply_eq_iff_symm_apply (e : M ≃* N) {x : M} {y : N} : e x = y ↔ x = e.symm y :=
   e.toEquiv.apply_eq_iff_eq_symm_apply
@@ -378,35 +389,65 @@ theorem symm_comp_eq {α : Type*} (e : M ≃* N) (f : α → M) (g : α → N) :
   e.toEquiv.symm_comp_eq f g
 
 @[to_additive (attr := simp)]
-theorem symm_trans_self (e : M ≃* N) : e.symm.trans e = refl N :=
-  DFunLike.ext _ _ e.apply_symm_apply
+theorem _root_.MulEquivClass.apply_coe_symm_apply {α β} [Mul α] [Mul β] {F} [EquivLike F α β]
+    [MulEquivClass F α β] (e : F) (x : β) :
+    e ((e : α ≃* β).symm x) = x :=
+  (e : α ≃* β).right_inv x
 
 @[to_additive (attr := simp)]
-theorem self_trans_symm (e : M ≃* N) : e.trans e.symm = refl M :=
-  DFunLike.ext _ _ e.symm_apply_apply
+theorem _root_.MulEquivClass.coe_symm_apply_apply {α β} [Mul α] [Mul β] {F} [EquivLike F α β]
+    [MulEquivClass F α β] (e : F) (x : α) :
+    (e : α ≃* β).symm (e x) = x :=
+  (e : α ≃* β).left_inv x
 
-/-- Two multiplicative isomorphisms agree if they are defined by the
-same underlying function. -/
-@[to_additive (attr := ext)
-  "Two additive isomorphisms agree if they are defined by the same underlying function."]
-theorem ext {f g : MulEquiv M N} (h : ∀ x, f x = g x) : f = g :=
-  DFunLike.ext f g h
+end symm
+
+section simps
+
+-- we don't hyperlink the note in the additive version, since that breaks syntax highlighting
+-- in the whole file.
+
+/-- See Note [custom simps projection] -/
+@[to_additive "See Note [custom simps projection]"] -- this comment fixes the syntax highlighting "
+def Simps.symm_apply (e : M ≃* N) : N → M :=
+  e.symm
+
+initialize_simps_projections AddEquiv (toFun → apply, invFun → symm_apply)
+
+initialize_simps_projections MulEquiv (toFun → apply, invFun → symm_apply)
+
+end simps
+
+section trans
+
+/-- Transitivity of multiplication-preserving isomorphisms -/
+@[to_additive (attr := trans) "Transitivity of addition-preserving isomorphisms"]
+def trans (h1 : M ≃* N) (h2 : N ≃* P) : M ≃* P :=
+  { h1.toEquiv.trans h2.toEquiv with
+    map_mul' := fun x y => show h2 (h1 (x * y)) = h2 (h1 x) * h2 (h1 y) by
+      rw [map_mul, map_mul] }
 
 @[to_additive (attr := simp)]
-theorem mk_coe (e : M ≃* N) (e' h₁ h₂ h₃) : (⟨⟨e, e', h₁, h₂⟩, h₃⟩ : M ≃* N) = e :=
-  ext fun _ => rfl
+theorem coe_trans (e₁ : M ≃* N) (e₂ : N ≃* P) : ↑(e₁.trans e₂) = e₂ ∘ e₁ := rfl
 
 @[to_additive (attr := simp)]
-theorem mk_coe' (e : M ≃* N) (f h₁ h₂ h₃) : (MulEquiv.mk ⟨f, e, h₁, h₂⟩ h₃ : N ≃* M) = e.symm :=
-  symm_bijective.injective <| ext fun _ => rfl
+theorem trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (m : M) : e₁.trans e₂ m = e₂ (e₁ m) := rfl
 
-@[to_additive]
-protected theorem congr_arg {f : MulEquiv M N} {x x' : M} : x = x' → f x = f x' :=
-  DFunLike.congr_arg f
+@[to_additive (attr := simp)]
+theorem symm_trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (p : P) :
+    (e₁.trans e₂).symm p = e₁.symm (e₂.symm p) := rfl
 
-@[to_additive]
-protected theorem congr_fun {f g : MulEquiv M N} (h : f = g) (x : M) : f x = g x :=
-  DFunLike.congr_fun h x
+@[to_additive (attr := simp)]
+theorem symm_trans_self (e : M ≃* N) : e.symm.trans e = refl N :=
+  DFunLike.ext _ _ e.apply_symm_apply
+
+@[to_additive (attr := simp)]
+theorem self_trans_symm (e : M ≃* N) : e.trans e.symm = refl M :=
+  DFunLike.ext _ _ e.symm_apply_apply
+
+end trans
+
+section unique
 
 /-- The `MulEquiv` between two monoids with a unique element. -/
 @[to_additive "The `AddEquiv` between two `AddMonoid`s with a unique element."]
@@ -420,6 +461,8 @@ instance {M N} [Unique M] [Unique N] [Mul M] [Mul N] : Unique (M ≃* N) where
   default := mulEquivOfUnique
   uniq _ := ext fun _ => Subsingleton.elim _ _
 
+end unique
+
 end Mul
 
 /-!
diff --git a/Mathlib/Algebra/Group/Equiv/TypeTags.lean b/Mathlib/Algebra/Group/Equiv/TypeTags.lean
index 812c4308708bd..78c085f074666 100644
--- a/Mathlib/Algebra/Group/Equiv/TypeTags.lean
+++ b/Mathlib/Algebra/Group/Equiv/TypeTags.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Equiv.Basic
+import Mathlib.Algebra.Group.Prod
 import Mathlib.Algebra.Group.TypeTags
 
 /-!
@@ -11,7 +12,7 @@ import Mathlib.Algebra.Group.TypeTags
 -/
 
 
-variable {G H : Type*}
+variable {ι G H : Type*}
 
 /-- Reinterpret `G ≃+ H` as `Multiplicative G ≃* Multiplicative H`. -/
 @[simps]
@@ -113,6 +114,38 @@ and multiplicative endomorphisms of `Multiplicative A`. -/
   { AddMonoidHom.toMultiplicative with
     map_mul' := fun _ _ => rfl }
 
+/-- `Multiplicative (∀ i : ι, K i)` is equivalent to `∀ i : ι, Multiplicative (K i)`. -/
+@[simps]
+def MulEquiv.piMultiplicative (K : ι → Type*) [∀ i, Add (K i)] :
+    Multiplicative (∀ i : ι, K i) ≃* (∀ i : ι, Multiplicative (K i)) where
+  toFun x := fun i ↦ Multiplicative.ofAdd <| Multiplicative.toAdd x i
+  invFun x := Multiplicative.ofAdd fun i ↦ Multiplicative.toAdd (x i)
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_mul' _ _ := rfl
+
+variable (ι) (G) in
+/-- `Multiplicative (ι → G)` is equivalent to `ι → Multiplicative G`. -/
+abbrev MulEquiv.funMultiplicative [Add G] :
+    Multiplicative (ι → G) ≃* (ι → Multiplicative G) :=
+  MulEquiv.piMultiplicative fun _ ↦ G
+
+/-- `Additive (∀ i : ι, K i)` is equivalent to `∀ i : ι, Additive (K i)`. -/
+@[simps]
+def AddEquiv.piAdditive (K : ι → Type*) [∀ i, Mul (K i)] :
+    Additive (∀ i : ι, K i) ≃+ (∀ i : ι, Additive (K i)) where
+  toFun x := fun i ↦ Additive.ofMul <| Additive.toMul x i
+  invFun x := Additive.ofMul fun i ↦ Additive.toMul (x i)
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+
+variable (ι) (G) in
+/-- `Additive (ι → G)` is equivalent to `ι → Additive G`. -/
+abbrev AddEquiv.funAdditive [Mul G] :
+    Additive (ι → G) ≃+ (ι → Additive G) :=
+  AddEquiv.piAdditive fun _ ↦ G
+
 section
 
 variable (G) (H)
@@ -127,4 +160,26 @@ def AddEquiv.additiveMultiplicative [AddZeroClass G] : Additive (Multiplicative
 def MulEquiv.multiplicativeAdditive [MulOneClass H] : Multiplicative (Additive H) ≃* H :=
   AddEquiv.toMultiplicative'' (AddEquiv.refl (Additive H))
 
+/-- `Multiplicative (G × H)` is equivalent to `Multiplicative G × Multiplicative H`. -/
+@[simps]
+def MulEquiv.prodMultiplicative [Add G] [Add H] :
+    Multiplicative (G × H) ≃* Multiplicative G × Multiplicative H where
+  toFun x := (Multiplicative.ofAdd (Multiplicative.toAdd x).1,
+    Multiplicative.ofAdd (Multiplicative.toAdd x).2)
+  invFun := fun (x, y) ↦ Multiplicative.ofAdd (Multiplicative.toAdd x, Multiplicative.toAdd y)
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_mul' _ _ := rfl
+
+/-- `Additive (G × H)` is equivalent to `Additive G × Additive H`. -/
+@[simps]
+def AddEquiv.prodAdditive [Mul G] [Mul H] :
+    Additive (G × H) ≃+ Additive G × Additive H where
+  toFun x := (Additive.ofMul (Additive.toMul x).1,
+    Additive.ofMul (Additive.toMul x).2)
+  invFun := fun (x, y) ↦ Additive.ofMul (Additive.toMul x, Additive.toMul y)
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+
 end
diff --git a/Mathlib/Algebra/Group/Even.lean b/Mathlib/Algebra/Group/Even.lean
index 428970edc8bf0..5bd24647c2f65 100644
--- a/Mathlib/Algebra/Group/Even.lean
+++ b/Mathlib/Algebra/Group/Even.lean
@@ -34,7 +34,7 @@ assert_not_exists DenselyOrdered
 
 open MulOpposite
 
-variable {F α β R : Type*}
+variable {F α β : Type*}
 
 section Mul
 variable [Mul α]
diff --git a/Mathlib/Algebra/Group/EvenFunction.lean b/Mathlib/Algebra/Group/EvenFunction.lean
new file mode 100644
index 0000000000000..3896969ce6795
--- /dev/null
+++ b/Mathlib/Algebra/Group/EvenFunction.lean
@@ -0,0 +1,152 @@
+/-
+Copyright (c) 2024 David Loeffler. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Loeffler
+-/
+import Mathlib.Algebra.Group.Action.Pi
+import Mathlib.Algebra.Module.Defs
+import Mathlib.Algebra.BigOperators.Group.Finset
+
+/-!
+# Even and odd functions
+
+We define even functions `α → β` assuming `α` has a negation, and odd functions assuming both `α`
+and `β` have negation.
+
+These definitions are `Function.Even` and `Function.Odd`; and they are `protected`, to avoid
+conflicting with the root-level definitions `Even` and `Odd` (which, for functions, mean that the
+function takes even resp. odd _values_, a wholly different concept).
+-/
+
+open scoped BigOperators
+
+namespace Function
+
+variable {α β : Type*} [Neg α]
+
+/-- A function `f` is _even_ if it satisfies `f (-x) = f x` for all `x`. -/
+protected def Even (f : α → β) : Prop := ∀ a, f (-a) = f a
+
+/-- A function `f` is _odd_ if it satisfies `f (-x) = -f x` for all `x`. -/
+protected def Odd [Neg β] (f : α → β) : Prop := ∀ a, f (-a) = -(f a)
+
+/-- Any constant function is even. -/
+lemma Even.const (b : β) : Function.Even (fun _ : α ↦ b) := fun _ ↦ rfl
+
+/-- The zero function is even. -/
+lemma Even.zero [Zero β] : Function.Even (fun (_ : α) ↦ (0 : β)) := Even.const 0
+
+/-- The zero function is odd. -/
+lemma Odd.zero [NegZeroClass β] : Function.Odd (fun (_ : α) ↦ (0 : β)) := fun _ ↦ neg_zero.symm
+
+section composition
+
+variable {γ : Type*}
+
+/-- If `f` is arbitrary and `g` is even, then `f ∘ g` is even. -/
+lemma Even.left_comp {g : α → β} (hg : g.Even) (f : β → γ) : (f ∘ g).Even :=
+  (congr_arg f <| hg ·)
+
+/-- If `f` is even and `g` is odd, then `f ∘ g` is even. -/
+lemma Even.comp_odd [Neg β] {f : β → γ} (hf : f.Even) {g : α → β} (hg : g.Odd) :
+    (f ∘ g).Even := by
+  intro a
+  simp only [comp_apply, hg a, hf _]
+
+/-- If `f` and `g` are odd, then `f ∘ g` is odd. -/
+lemma Odd.comp_odd [Neg β] [Neg γ] {f : β → γ} (hf : f.Odd) {g : α → β} (hg : g.Odd) :
+    (f ∘ g).Odd := by
+  intro a
+  simp only [comp_apply, hg a, hf _]
+
+end composition
+
+lemma Even.add [Add β] {f g : α → β} (hf : f.Even) (hg : g.Even) : (f + g).Even := by
+  intro a
+  simp only [hf a, hg a, Pi.add_apply]
+
+lemma Odd.add [SubtractionCommMonoid β] {f g : α → β} (hf : f.Odd) (hg : g.Odd) : (f + g).Odd := by
+  intro a
+  simp only [hf a, hg a, Pi.add_apply, neg_add]
+
+section smul
+
+variable {γ : Type*} {f : α → β} {g : α → γ}
+
+lemma Even.smul_even [SMul β γ] (hf : f.Even) (hg : g.Even) : (f • g).Even := by
+  intro a
+  simp only [Pi.smul_apply', hf a, hg a]
+
+lemma Even.smul_odd [Monoid β] [AddGroup γ] [DistribMulAction β γ] (hf : f.Even) (hg : g.Odd) :
+    (f • g).Odd := by
+  intro a
+  simp only [Pi.smul_apply', hf a, hg a, smul_neg]
+
+lemma Odd.smul_even [Ring β] [AddCommGroup γ] [Module β γ] (hf : f.Odd) (hg : g.Even) :
+    (f • g).Odd := by
+  intro a
+  simp only [Pi.smul_apply', hf a, hg a, neg_smul]
+
+lemma Odd.smul_odd [Ring β] [AddCommGroup γ] [Module β γ] (hf : f.Odd) (hg : g.Odd) :
+    (f • g).Even := by
+  intro a
+  simp only [Pi.smul_apply', hf a, hg a, smul_neg, neg_smul, neg_neg]
+
+lemma Even.const_smul [SMul β γ] (hg : g.Even) (r : β) : (r • g).Even := by
+  intro a
+  simp only [Pi.smul_apply, hg a]
+
+lemma Odd.const_smul [Monoid β] [AddGroup γ] [DistribMulAction β γ] (hg : g.Odd) (r : β) :
+    (r • g).Odd := by
+  intro a
+  simp only [Pi.smul_apply, hg a, smul_neg]
+
+end smul
+
+section mul
+
+variable {R : Type*} [Mul R] {f g : α → R}
+
+lemma Even.mul_even (hf : f.Even) (hg : g.Even) : (f * g).Even := by
+  intro a
+  simp only [Pi.mul_apply, hf a, hg a]
+
+lemma Even.mul_odd [HasDistribNeg R] (hf : f.Even) (hg : g.Odd) : (f * g).Odd := by
+  intro a
+  simp only [Pi.mul_apply, hf a, hg a, mul_neg]
+
+lemma Odd.mul_even [HasDistribNeg R] (hf : f.Odd) (hg : g.Even) : (f * g).Odd := by
+  intro a
+  simp only [Pi.mul_apply, hf a, hg a, neg_mul]
+
+lemma Odd.mul_odd [HasDistribNeg R] (hf : f.Odd) (hg : g.Odd) : (f * g).Even := by
+  intro a
+  simp only [Pi.mul_apply, hf a, hg a, mul_neg, neg_mul, neg_neg]
+
+end mul
+
+section torsionfree
+
+-- need to redeclare variables since `InvolutiveNeg α` conflicts with `Neg α`
+variable {α β : Type*} [AddCommGroup β] [NoZeroSMulDivisors ℕ β] {f : α → β}
+
+/--
+If `f` is both even and odd, and its target is a torsion-free commutative additive group,
+then `f = 0`.
+-/
+lemma zero_of_even_and_odd [Neg α] (he : f.Even) (ho : f.Odd) : f = 0 := by
+  ext r
+  rw [Pi.zero_apply, ← neg_eq_self ℕ, ← ho, he]
+
+/-- The sum of the values of an odd function is 0. -/
+lemma Odd.sum_eq_zero [Fintype α] [InvolutiveNeg α] {f : α → β} (hf : f.Odd) : ∑ a, f a = 0 := by
+  simpa only [neg_eq_self ℕ, Finset.sum_neg_distrib, funext hf, Equiv.neg_apply] using
+    Equiv.sum_comp (.neg α) f
+
+/-- An odd function vanishes at zero. -/
+lemma Odd.map_zero [NegZeroClass α] (hf : f.Odd) : f 0 = 0 := by
+  simp only [← neg_eq_self ℕ, ← hf 0, neg_zero]
+
+end torsionfree
+
+end Function
diff --git a/Mathlib/Algebra/Group/Fin/Basic.lean b/Mathlib/Algebra/Group/Fin/Basic.lean
index 27b58e867117f..d9710b7aedc23 100644
--- a/Mathlib/Algebra/Group/Fin/Basic.lean
+++ b/Mathlib/Algebra/Group/Fin/Basic.lean
@@ -89,17 +89,41 @@ lemma coe_sub_one (a : Fin (n + 1)) : ↑(a - 1) = if a = 0 then n else a - 1 :=
   rwa [Fin.ext_iff] at h
 
 @[simp]
+lemma lt_sub_iff {n : ℕ} {a b : Fin n} : a < a - b ↔ a < b := by
+  cases' n with n
+  · exact a.elim0
+  constructor
+  · contrapose!
+    intro h
+    obtain ⟨l, hl⟩ := Nat.exists_eq_add_of_le (Fin.not_lt.mp h)
+    simpa only [Fin.not_lt, le_iff_val_le_val, sub_def, hl, ← Nat.add_assoc, Nat.add_mod_left,
+      Nat.mod_eq_of_lt, Nat.sub_add_cancel b.is_lt.le] using
+        (le_trans (mod_le _ _) (le_add_left _ _))
+  · intro h
+    rw [lt_iff_val_lt_val, sub_def]
+    simp only
+    obtain ⟨k, hk⟩ := Nat.exists_eq_add_of_lt b.is_lt
+    have : n + 1 - b = k + 1 := by
+      simp_rw [hk, Nat.add_assoc, Nat.add_sub_cancel_left]
+      -- simp_rw because, otherwise, rw tries to rewrite inside `b : Fin (n + 1)`
+    rw [this, Nat.mod_eq_of_lt (hk.ge.trans_lt' ?_), Nat.lt_add_left_iff_pos] <;>
+    omega
+
+@[simp]
+lemma sub_le_iff {n : ℕ} {a b : Fin n} : a - b ≤ a ↔ b ≤ a := by
+  rw [← not_iff_not, Fin.not_le, Fin.not_le, lt_sub_iff]
+
+@[simp]
+lemma lt_one_iff {n : ℕ} (x : Fin (n + 2)) : x < 1 ↔ x = 0 := by
+  simp [lt_iff_val_lt_val, Fin.ext_iff]
+
 lemma lt_sub_one_iff {k : Fin (n + 2)} : k < k - 1 ↔ k = 0 := by
-  rcases k with ⟨_ | k, hk⟩
-  · simp only [zero_eta, zero_sub, lt_iff_val_lt_val, val_zero, coe_neg_one, zero_lt_succ]
-  have : (n + 1 + (k + 1)) % (n + 2) = k % (n + 2) := by
-    rw [Nat.add_comm, Nat.add_right_comm, Nat.add_assoc, Nat.add_assoc, add_mod_right]
-  simp [lt_iff_val_lt_val, Fin.ext_iff, Fin.coe_sub, this, mod_eq_of_lt ((lt_succ_self _).trans hk)]
+  simp
 
 @[simp] lemma le_sub_one_iff {k : Fin (n + 1)} : k ≤ k - 1 ↔ k = 0 := by
   cases n
   · simp [fin_one_eq_zero k]
-  simp [-val_fin_le, le_def]
+  simp only [le_def]
   rw [← lt_sub_one_iff, le_iff_lt_or_eq, val_fin_lt, val_inj, lt_sub_one_iff, or_iff_left_iff_imp,
     eq_comm, sub_eq_iff_eq_add]
   simp
@@ -112,4 +136,15 @@ lemma sub_one_lt_iff {k : Fin (n + 1)} : k - 1 < k ↔ 0 < k :=
 lemma neg_natCast_eq_one (n : ℕ) : -(n : Fin (n + 1)) = 1 := by
   simp only [natCast_eq_last, neg_last]
 
+lemma rev_add (a b : Fin n) : rev (a + b) = rev a - b := by
+  cases' n
+  · exact a.elim0
+  rw [← last_sub, ← last_sub, sub_add_eq_sub_sub]
+
+lemma rev_sub (a b : Fin n) : rev (a - b) = rev a + b := by
+  rw [rev_eq_iff, rev_add, rev_rev]
+
+lemma add_lt_left_iff {n : ℕ} {a b : Fin n} : a + b < a ↔ rev b < a := by
+  rw [← rev_lt_rev, Iff.comm, ← rev_lt_rev, rev_add, lt_sub_iff, rev_rev]
+
 end Fin
diff --git a/Mathlib/Algebra/Group/Hom/Basic.lean b/Mathlib/Algebra/Group/Hom/Basic.lean
index e1a97f26ed9b8..9eb317e8d7b71 100644
--- a/Mathlib/Algebra/Group/Hom/Basic.lean
+++ b/Mathlib/Algebra/Group/Hom/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
+Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
   Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Basic
@@ -14,9 +14,9 @@ import Mathlib.Algebra.Group.Hom.Defs
 
 -- `NeZero` cannot be additivised, hence its theory should be developed outside of the
 -- `Algebra.Group` folder.
-assert_not_exists NeZero
+assert_not_imported Mathlib.Algebra.NeZero
 
-variable {α β M N P : Type*}
+variable {α M N P : Type*}
 
 -- monoids
 variable {G : Type*} {H : Type*}
@@ -98,10 +98,10 @@ end MulHom
 
 namespace MonoidHom
 section Group
-variable [Group G] [CommGroup H]
+variable [Group G]
 
 /-- A homomorphism from a group to a monoid is injective iff its kernel is trivial.
-For the iff statement on the triviality of the kernel, see `injective_iff_map_eq_one'`.  -/
+For the iff statement on the triviality of the kernel, see `injective_iff_map_eq_one'`. -/
 @[to_additive
   "A homomorphism from an additive group to an additive monoid is injective iff
   its kernel is trivial. For the iff statement on the triviality of the kernel,
@@ -125,8 +125,6 @@ theorem _root_.injective_iff_map_eq_one' {G H} [Group G] [MulOneClass H]
   (injective_iff_map_eq_one f).trans <|
     forall_congr' fun _ => ⟨fun h => ⟨h, fun H => H.symm ▸ map_one f⟩, Iff.mp⟩
 
-variable [MulOneClass M]
-
 /-- Makes a group homomorphism from a proof that the map preserves right division
 `fun x y => x * y⁻¹`. See also `MonoidHom.of_map_div` for a version using `fun x y => x / y`.
 -/
diff --git a/Mathlib/Algebra/Group/Hom/CompTypeclasses.lean b/Mathlib/Algebra/Group/Hom/CompTypeclasses.lean
index 5f9f1cea90e7a..0bdd4b963dd4a 100644
--- a/Mathlib/Algebra/Group/Hom/CompTypeclasses.lean
+++ b/Mathlib/Algebra/Group/Hom/CompTypeclasses.lean
@@ -39,7 +39,7 @@ section MonoidHomCompTriple
 namespace MonoidHom
 
 /-- Class of composing triples -/
-class CompTriple  {M N P : Type*} [Monoid M] [Monoid N] [Monoid P]
+class CompTriple {M N P : Type*} [Monoid M] [Monoid N] [Monoid P]
   (φ : M →* N) (ψ : N →* P) (χ : outParam (M →* P)) : Prop where
   /-- The maps form a commuting triangle -/
   comp_eq : ψ.comp φ = χ
@@ -48,7 +48,6 @@ attribute [simp] CompTriple.comp_eq
 
 namespace CompTriple
 
-variable {M' : Type*} [Monoid M']
 variable {M N P : Type*} [Monoid M] [Monoid N] [Monoid P]
 
 /-- Class of Id maps -/
diff --git a/Mathlib/Algebra/Group/Hom/Defs.lean b/Mathlib/Algebra/Group/Hom/Defs.lean
index b75baec7c85f7..1a3bf932f98fd 100644
--- a/Mathlib/Algebra/Group/Hom/Defs.lean
+++ b/Mathlib/Algebra/Group/Hom/Defs.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
+Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
   Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Pi.Basic
@@ -143,8 +143,9 @@ homomorphisms.
 
 You should also extend this typeclass when you extend `AddMonoidHom`.
 -/
-class AddMonoidHomClass (F M N : Type*) [AddZeroClass M] [AddZeroClass N] [FunLike F M N]
-  extends AddHomClass F M N, ZeroHomClass F M N : Prop
+class AddMonoidHomClass (F : Type*) (M N : outParam Type*)
+    [AddZeroClass M] [AddZeroClass N] [FunLike F M N]
+    extends AddHomClass F M N, ZeroHomClass F M N : Prop
 
 -- Instances and lemmas are defined below through `@[to_additive]`.
 end add_zero
@@ -184,9 +185,29 @@ instance OneHom.funLike : FunLike (OneHom M N) M N where
 instance OneHom.oneHomClass : OneHomClass (OneHom M N) M N where
   map_one := OneHom.map_one'
 
+library_note "low priority simp lemmas"
+/--
+The hom class hierarchy allows for a single lemma, such as `map_one`, to apply to a large variety
+of morphism types, so long as they have an instance of `OneHomClass`. For example, this applies to
+to `MonoidHom`, `RingHom`, `AlgHom`, `StarAlgHom`, as well as their `Equiv` variants, etc. However,
+precisely because these lemmas are so widely applicable, they keys in the `simp` discrimination tree
+are necessarily highly non-specific. For example, the key for `map_one` is
+`@DFunLike.coe _ _ _ _ _ 1`.
+
+Consequently, whenever lean sees `⇑f 1`, for some `f : F`, it will attempt to synthesize a
+`OneHomClass F ?A ?B` instance. If no such instance exists, then Lean will need to traverse (almost)
+the entirety of the `FunLike` hierarchy in order to determine this because so many classes have a
+`OneHomClass` instance (in fact, this problem is likely worse for `ZeroHomClass`). This can lead to
+a significant performance hit when `map_one` fails to apply.
+
+To avoid this problem, we mark these widely applicable simp lemmas with key discimination tree keys
+with `low` priority in order to ensure that they are not tried first.
+-/
+
 variable [FunLike F M N]
 
-@[to_additive (attr := simp)]
+/-- See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low)]
 theorem map_one [OneHomClass F M N] (f : F) : f 1 = 1 :=
   OneHomClass.map_one f
 
@@ -277,7 +298,8 @@ instance MulHom.mulHomClass : MulHomClass (M →ₙ* N) M N where
 
 variable [FunLike F M N]
 
-@[to_additive (attr := simp)]
+/-- See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low)]
 theorem map_mul [MulHomClass F M N] (f : F) (x y : M) : f (x * y) = f x * f y :=
   MulHomClass.map_mul f x y
 
@@ -391,8 +413,10 @@ lemma map_comp_div' [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f
     (hf : ∀ a, f a⁻¹ = (f a)⁻¹) (g h : ι → G) : f ∘ (g / h) = f ∘ g / f ∘ h := by
   ext; simp [map_div' f hf]
 
-/-- Group homomorphisms preserve inverse. -/
-@[to_additive (attr := simp) "Additive group homomorphisms preserve negation."]
+/-- Group homomorphisms preserve inverse.
+
+See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low) "Additive group homomorphisms preserve negation."]
 theorem map_inv [Group G] [DivisionMonoid H] [MonoidHomClass F G H]
     (f : F) (a : G) : f a⁻¹ = (f a)⁻¹ :=
   eq_inv_of_mul_eq_one_left <| map_mul_eq_one f <| inv_mul_cancel _
@@ -410,8 +434,10 @@ theorem map_mul_inv [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F)
 lemma map_comp_mul_inv [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g h : ι → G) :
     f ∘ (g * h⁻¹) = f ∘ g * (f ∘ h)⁻¹ := by simp
 
-/-- Group homomorphisms preserve division. -/
-@[to_additive (attr := simp) "Additive group homomorphisms preserve subtraction."]
+/-- Group homomorphisms preserve division.
+
+See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low) "Additive group homomorphisms preserve subtraction."]
 theorem map_div [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) :
     ∀ a b, f (a / b) = f a / f b := map_div' _ <| map_inv f
 
@@ -419,14 +445,15 @@ theorem map_div [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) :
 lemma map_comp_div [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g h : ι → G) :
     f ∘ (g / h) = f ∘ g / f ∘ h := by ext; simp
 
-@[to_additive (attr := simp) (reorder := 9 10)]
+/-- See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low) (reorder := 9 10)]
 theorem map_pow [Monoid G] [Monoid H] [MonoidHomClass F G H] (f : F) (a : G) :
     ∀ n : ℕ, f (a ^ n) = f a ^ n
   | 0 => by rw [pow_zero, pow_zero, map_one]
   | n + 1 => by rw [pow_succ, pow_succ, map_mul, map_pow f a n]
 
 @[to_additive (attr := simp)]
-lemma map_comp_pow [Group G] [DivisionMonoid H] [MonoidHomClass F G H] (f : F) (g : ι → G) (n : ℕ) :
+lemma map_comp_pow [Monoid G] [Monoid H] [MonoidHomClass F G H] (f : F) (g : ι → G) (n : ℕ) :
     f ∘ (g ^ n) = f ∘ g ^ n := by ext; simp
 
 @[to_additive]
@@ -440,8 +467,10 @@ lemma map_comp_zpow' [DivInvMonoid G] [DivInvMonoid H] [MonoidHomClass F G H] (f
     (hf : ∀ x : G, f x⁻¹ = (f x)⁻¹) (g : ι → G) (n : ℤ) : f ∘ (g ^ n) = f ∘ g ^ n := by
   ext; simp [map_zpow' f hf]
 
-/-- Group homomorphisms preserve integer power. -/
-@[to_additive (attr := simp) (reorder := 9 10)
+/-- Group homomorphisms preserve integer power.
+
+See note [low priority simp lemmas] -/
+@[to_additive (attr := simp low) (reorder := 9 10)
 "Additive group homomorphisms preserve integer scaling."]
 theorem map_zpow [Group G] [DivisionMonoid H] [MonoidHomClass F G H]
     (f : F) (g : G) (n : ℤ) : f (g ^ n) = f g ^ n := map_zpow' f (map_inv f) g n
@@ -930,8 +959,6 @@ instance [MulOneClass M] [MulOneClass N] : Inhabited (M →* N) := ⟨1⟩
 
 namespace MonoidHom
 
-variable [Group G] [CommGroup H]
-
 @[to_additive (attr := simp)]
 theorem one_comp [MulOneClass M] [MulOneClass N] [MulOneClass P] (f : M →* N) :
     (1 : N →* P).comp f = 1 := rfl
diff --git a/Mathlib/Algebra/Group/Hom/End.lean b/Mathlib/Algebra/Group/Hom/End.lean
index 207791e8b527c..caa7de5333f3d 100644
--- a/Mathlib/Algebra/Group/Hom/End.lean
+++ b/Mathlib/Algebra/Group/Hom/End.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
+Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
   Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Commute.Defs
@@ -19,9 +19,9 @@ They are separate, and if someone would like to split this file in two that may
 -/
 
 
-universe uM uN uP uQ
+universe uM
 
-variable {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ}
+variable {M : Type uM}
 
 namespace AddMonoid.End
 
@@ -121,7 +121,7 @@ end Semiring
 
 section CommSemiring
 
-variable {R S : Type*} [NonUnitalNonAssocCommSemiring R]
+variable {R : Type*} [NonUnitalNonAssocCommSemiring R]
 
 namespace AddMonoid.End
 
diff --git a/Mathlib/Algebra/Group/Hom/Instances.lean b/Mathlib/Algebra/Group/Hom/Instances.lean
index 3b1f5e56a0684..f3371cbd05280 100644
--- a/Mathlib/Algebra/Group/Hom/Instances.lean
+++ b/Mathlib/Algebra/Group/Hom/Instances.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
+Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
   Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Hom.Basic
diff --git a/Mathlib/Algebra/Group/Indicator.lean b/Mathlib/Algebra/Group/Indicator.lean
index 68214b8258570..ccb3633809bca 100644
--- a/Mathlib/Algebra/Group/Indicator.lean
+++ b/Mathlib/Algebra/Group/Indicator.lean
@@ -41,7 +41,7 @@ section One
 
 variable [One M] [One N] {s t : Set α} {f g : α → M} {a : α}
 
-/-- `Set.mulIndicator s f a` is `f a` if `a ∈ s`, `1` otherwise.  -/
+/-- `Set.mulIndicator s f a` is `f a` if `a ∈ s`, `1` otherwise. -/
 @[to_additive "`Set.indicator s f a` is `f a` if `a ∈ s`, `0` otherwise."]
 noncomputable def mulIndicator (s : Set α) (f : α → M) (x : α) : M :=
   haveI := Classical.decPred (· ∈ s)
diff --git a/Mathlib/Algebra/Group/InjSurj.lean b/Mathlib/Algebra/Group/InjSurj.lean
index a47bbde22c75a..61cbae422d607 100644
--- a/Mathlib/Algebra/Group/InjSurj.lean
+++ b/Mathlib/Algebra/Group/InjSurj.lean
@@ -55,7 +55,7 @@ a semigroup. See note [reducible non-instances]. -/
 injective map that preserves `+` to an additive semigroup."]
 protected abbrev semigroup [Semigroup M₂] (f : M₁ → M₂) (hf : Injective f)
     (mul : ∀ x y, f (x * y) = f x * f y) : Semigroup M₁ :=
-  { ‹Mul M₁› with mul_assoc := fun x y z => hf <| by erw [mul, mul, mul, mul, mul_assoc] }
+  { ‹Mul M₁› with mul_assoc := fun x y z => hf <| by rw [mul, mul, mul, mul, mul_assoc] }
 
 /-- A type endowed with `*` is a commutative magma, if it admits a surjective map that preserves
 `*` from a commutative magma. -/
@@ -83,7 +83,7 @@ semigroup, if it admits an injective map that preserves `+` to an additive left
 protected abbrev leftCancelSemigroup [LeftCancelSemigroup M₂] (f : M₁ → M₂) (hf : Injective f)
     (mul : ∀ x y, f (x * y) = f x * f y) : LeftCancelSemigroup M₁ :=
   { hf.semigroup f mul with
-    mul_left_cancel := fun x y z H => hf <| (mul_right_inj (f x)).1 <| by erw [← mul, ← mul, H] }
+    mul_left_cancel := fun x y z H => hf <| (mul_right_inj (f x)).1 <| by rw [← mul, ← mul, H] }
 
 /-- A type endowed with `*` is a right cancel semigroup, if it admits an injective map that
 preserves `*` to a right cancel semigroup.  See note [reducible non-instances]. -/
@@ -93,7 +93,7 @@ semigroup."]
 protected abbrev rightCancelSemigroup [RightCancelSemigroup M₂] (f : M₁ → M₂) (hf : Injective f)
     (mul : ∀ x y, f (x * y) = f x * f y) : RightCancelSemigroup M₁ :=
   { hf.semigroup f mul with
-    mul_right_cancel := fun x y z H => hf <| (mul_left_inj (f y)).1 <| by erw [← mul, ← mul, H] }
+    mul_right_cancel := fun x y z H => hf <| (mul_left_inj (f y)).1 <| by rw [← mul, ← mul, H] }
 
 variable [One M₁]
 
@@ -105,8 +105,8 @@ injective map that preserves `0` and `+` to an `AddZeroClass`."]
 protected abbrev mulOneClass [MulOneClass M₂] (f : M₁ → M₂) (hf : Injective f) (one : f 1 = 1)
     (mul : ∀ x y, f (x * y) = f x * f y) : MulOneClass M₁ :=
   { ‹One M₁›, ‹Mul M₁› with
-    one_mul := fun x => hf <| by erw [mul, one, one_mul],
-    mul_one := fun x => hf <| by erw [mul, one, mul_one] }
+    one_mul := fun x => hf <| by rw [mul, one, one_mul],
+    mul_one := fun x => hf <| by rw [mul, one, mul_one] }
 
 variable [Pow M₁ ℕ]
 
@@ -120,8 +120,8 @@ protected abbrev monoid [Monoid M₂] (f : M₁ → M₂) (hf : Injective f) (on
     (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) : Monoid M₁ :=
   { hf.mulOneClass f one mul, hf.semigroup f mul with
     npow := fun n x => x ^ n,
-    npow_zero := fun x => hf <| by erw [npow, one, pow_zero],
-    npow_succ := fun n x => hf <| by erw [npow, pow_succ, mul, npow] }
+    npow_zero := fun x => hf <| by rw [npow, one, pow_zero],
+    npow_succ := fun n x => hf <| by rw [npow, pow_succ, mul, npow] }
 
 /-- A type endowed with `0`, `1` and `+` is an additive monoid with one,
 if it admits an injective map that preserves `0`, `1` and `+` to an additive monoid with one.
@@ -132,8 +132,8 @@ protected abbrev addMonoidWithOne {M₁} [Zero M₁] [One M₁] [Add M₁] [SMul
     (natCast : ∀ n : ℕ, f n = n) : AddMonoidWithOne M₁ :=
   { hf.addMonoid f zero add (swap nsmul) with
     natCast := Nat.cast,
-    natCast_zero := hf (by erw [natCast, Nat.cast_zero, zero]),
-    natCast_succ := fun n => hf (by erw [natCast, Nat.cast_succ, add, one, natCast]), one := 1 }
+    natCast_zero := hf (by rw [natCast, Nat.cast_zero, zero]),
+    natCast_succ := fun n => hf (by rw [natCast, Nat.cast_succ, add, one, natCast]), one := 1 }
 
 /-- A type endowed with `1` and `*` is a left cancel monoid, if it admits an injective map that
 preserves `1` and `*` to a left cancel monoid. See note [reducible non-instances]. -/
@@ -215,7 +215,7 @@ injective map that preserves `0` and unary `-` to an `NegZeroClass`."]
 protected abbrev invOneClass [InvOneClass M₂] (f : M₁ → M₂) (hf : Injective f) (one : f 1 = 1)
     (inv : ∀ x, f (x⁻¹) = (f x)⁻¹) : InvOneClass M₁ :=
   { ‹One M₁›, ‹Inv M₁› with
-    inv_one := hf <| by erw [inv, one, inv_one] }
+    inv_one := hf <| by rw [inv, one, inv_one] }
 
 variable [Div M₁] [Pow M₁ ℤ]
 
@@ -232,10 +232,11 @@ protected abbrev divInvMonoid [DivInvMonoid M₂] (f : M₁ → M₂) (hf : Inje
     (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : DivInvMonoid M₁ :=
   { hf.monoid f one mul npow, ‹Inv M₁›, ‹Div M₁› with
     zpow := fun n x => x ^ n,
-    zpow_zero' := fun x => hf <| by erw [zpow, zpow_zero, one],
-    zpow_succ' := fun n x => hf <| by erw [zpow, mul, zpow_natCast, pow_succ, zpow, zpow_natCast],
-    zpow_neg' := fun n x => hf <| by erw [zpow, zpow_negSucc, inv, zpow, zpow_natCast],
-    div_eq_mul_inv := fun x y => hf <| by erw [div, mul, inv, div_eq_mul_inv] }
+    zpow_zero' := fun x => hf <| by rw [zpow, zpow_zero, one],
+    zpow_succ' := fun n x => hf <| by rw [zpow, mul, Int.ofNat_eq_coe, zpow_natCast, pow_succ, zpow,
+      Int.ofNat_eq_coe, zpow_natCast],
+    zpow_neg' := fun n x => hf <| by rw [zpow, zpow_negSucc, inv, zpow, zpow_natCast],
+    div_eq_mul_inv := fun x y => hf <| by rw [div, mul, inv, div_eq_mul_inv] }
 
 /-- A type endowed with `1`, `*`, `⁻¹`, and `/` is a `DivInvOneMonoid` if it admits an injective
 map that preserves `1`, `*`, `⁻¹`, and `/` to a `DivInvOneMonoid`. See note
@@ -263,9 +264,9 @@ protected abbrev divisionMonoid [DivisionMonoid M₂] (f : M₁ → M₂) (hf :
     (div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
     (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : DivisionMonoid M₁ :=
   { hf.divInvMonoid f one mul inv div npow zpow, hf.involutiveInv f inv with
-    mul_inv_rev := fun x y => hf <| by erw [inv, mul, mul_inv_rev, mul, inv, inv],
+    mul_inv_rev := fun x y => hf <| by rw [inv, mul, mul_inv_rev, mul, inv, inv],
     inv_eq_of_mul := fun x y h => hf <| by
-      erw [inv, inv_eq_of_mul_eq_one_right (by erw [← mul, h, one])] }
+      rw [inv, inv_eq_of_mul_eq_one_right (by rw [← mul, h, one])] }
 
 /-- A type endowed with `1`, `*`, `⁻¹`, and `/` is a `DivisionCommMonoid` if it admits an
 injective map that preserves `1`, `*`, `⁻¹`, and `/` to a `DivisionCommMonoid`.
@@ -291,7 +292,7 @@ protected abbrev group [Group M₂] (f : M₁ → M₂) (hf : Injective f) (one
     (div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
     (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : Group M₁ :=
   { hf.divInvMonoid f one mul inv div npow zpow with
-    inv_mul_cancel := fun x => hf <| by erw [mul, inv, inv_mul_cancel, one] }
+    inv_mul_cancel := fun x => hf <| by rw [mul, inv, inv_mul_cancel, one] }
 
 /-- A type endowed with `0`, `1` and `+` is an additive group with one, if it admits an injective
 map that preserves `0`, `1` and `+` to an additive group with one.  See note
@@ -306,7 +307,7 @@ protected abbrev addGroupWithOne {M₁} [Zero M₁] [One M₁] [Add M₁] [SMul
     hf.addMonoidWithOne f zero one add nsmul natCast with
     intCast := Int.cast,
     intCast_ofNat := fun n => hf (by rw [natCast, ← Int.cast, intCast, Int.cast_natCast]),
-    intCast_negSucc := fun n => hf (by erw [intCast, neg, natCast, Int.cast_negSucc] ) }
+    intCast_negSucc := fun n => hf (by rw [intCast, neg, natCast, Int.cast_negSucc] ) }
 
 /-- A type endowed with `1`, `*` and `⁻¹` is a commutative group, if it admits an injective map that
 preserves `1`, `*` and `⁻¹` to a commutative group. See note [reducible non-instances]. -/
@@ -358,7 +359,7 @@ protected abbrev semigroup [Semigroup M₁] (f : M₁ → M₂) (hf : Surjective
 a surjective map that preserves `+` from an additive commutative semigroup."]
 protected abbrev commMagma [CommMagma M₁] (f : M₁ → M₂) (hf : Surjective f)
     (mul : ∀ x y, f (x * y) = f x * f y) : CommMagma M₂ where
-  mul_comm := hf.forall₂.2 fun x y => by erw [← mul, ← mul, mul_comm]
+  mul_comm := hf.forall₂.2 fun x y => by rw [← mul, ← mul, mul_comm]
 
 /-- A type endowed with `*` is a commutative semigroup, if it admits a surjective map that preserves
 `*` from a commutative semigroup. See note [reducible non-instances]. -/
@@ -380,8 +381,8 @@ surjective map that preserves `0` and `+` to an `AddZeroClass`."]
 protected abbrev mulOneClass [MulOneClass M₁] (f : M₁ → M₂) (hf : Surjective f) (one : f 1 = 1)
     (mul : ∀ x y, f (x * y) = f x * f y) : MulOneClass M₂ :=
   { ‹One M₂›, ‹Mul M₂› with
-    one_mul := hf.forall.2 fun x => by erw [← one, ← mul, one_mul],
-    mul_one := hf.forall.2 fun x => by erw [← one, ← mul, mul_one] }
+    one_mul := hf.forall.2 fun x => by rw [← one, ← mul, one_mul],
+    mul_one := hf.forall.2 fun x => by rw [← one, ← mul, mul_one] }
 
 variable [Pow M₂ ℕ]
 
@@ -395,10 +396,10 @@ protected abbrev monoid [Monoid M₁] (f : M₁ → M₂) (hf : Surjective f) (o
     (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) : Monoid M₂ :=
   { hf.semigroup f mul, hf.mulOneClass f one mul with
     npow := fun n x => x ^ n,
-    npow_zero := hf.forall.2 fun x => by dsimp only; erw [← npow, pow_zero, ← one],
+    npow_zero := hf.forall.2 fun x => by dsimp only; rw [← npow, pow_zero, ← one],
     npow_succ := fun n => hf.forall.2 fun x => by
       dsimp only
-      erw [← npow, pow_succ, ← npow, ← mul] }
+      rw [← npow, pow_succ, ← npow, ← mul] }
 
 /-- A type endowed with `0`, `1` and `+` is an additive monoid with one, if it admits a surjective
 map that preserves `0`, `1` and `*` from an additive monoid with one. See note
@@ -441,7 +442,7 @@ preserves `-` to a type which has an involutive negation."]
 protected abbrev involutiveInv {M₂ : Type*} [Inv M₂] [InvolutiveInv M₁] (f : M₁ → M₂)
     (hf : Surjective f) (inv : ∀ x, f x⁻¹ = (f x)⁻¹) : InvolutiveInv M₂ where
   inv := Inv.inv
-  inv_inv := hf.forall.2 fun x => by erw [← inv, ← inv, inv_inv]
+  inv_inv := hf.forall.2 fun x => by rw [← inv, ← inv, inv_inv]
 
 variable [Inv M₂] [Div M₂] [Pow M₂ ℤ]
 
@@ -457,14 +458,15 @@ protected abbrev divInvMonoid [DivInvMonoid M₁] (f : M₁ → M₂) (hf : Surj
     (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : DivInvMonoid M₂ :=
   { hf.monoid f one mul npow, ‹Div M₂›, ‹Inv M₂› with
     zpow := fun n x => x ^ n,
-    zpow_zero' := hf.forall.2 fun x => by dsimp only; erw [← zpow, zpow_zero, ← one],
+    zpow_zero' := hf.forall.2 fun x => by dsimp only; rw [← zpow, zpow_zero, ← one],
     zpow_succ' := fun n => hf.forall.2 fun x => by
       dsimp only
-      erw [← zpow, ← zpow, zpow_natCast, zpow_natCast, pow_succ, ← mul],
+      rw [← zpow, ← zpow, Int.ofNat_eq_coe, zpow_natCast, Int.ofNat_eq_coe, zpow_natCast, pow_succ,
+        ← mul],
     zpow_neg' := fun n => hf.forall.2 fun x => by
       dsimp only
-      erw [← zpow, ← zpow, zpow_negSucc, zpow_natCast, inv],
-    div_eq_mul_inv := hf.forall₂.2 fun x y => by erw [← inv, ← mul, ← div, div_eq_mul_inv] }
+      rw [← zpow, ← zpow, zpow_negSucc, zpow_natCast, inv],
+    div_eq_mul_inv := hf.forall₂.2 fun x y => by rw [← inv, ← mul, ← div, div_eq_mul_inv] }
 
 /-- A type endowed with `1`, `*` and `⁻¹` is a group, if it admits a surjective map that preserves
 `1`, `*` and `⁻¹` to a group. See note [reducible non-instances]. -/
@@ -476,7 +478,7 @@ protected abbrev group [Group M₁] (f : M₁ → M₂) (hf : Surjective f) (one
     (div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
     (zpow : ∀ (x) (n : ℤ), f (x ^ n) = f x ^ n) : Group M₂ :=
   { hf.divInvMonoid f one mul inv div npow zpow with
-    inv_mul_cancel := hf.forall.2 fun x => by erw [← inv, ← mul, inv_mul_cancel, one] }
+    inv_mul_cancel := hf.forall.2 fun x => by rw [← inv, ← mul, inv_mul_cancel, one] }
 
 /-- A type endowed with `0`, `1`, `+` is an additive group with one,
 if it admits a surjective map that preserves `0`, `1`, and `+` to an additive group with one.
diff --git a/Mathlib/Algebra/Group/Invertible/Defs.lean b/Mathlib/Algebra/Group/Invertible/Defs.lean
index a44dee8152e49..58a3021aca712 100644
--- a/Mathlib/Algebra/Group/Invertible/Defs.lean
+++ b/Mathlib/Algebra/Group/Invertible/Defs.lean
@@ -90,51 +90,64 @@ class Invertible [Mul α] [One α] (a : α) : Type u where
   mul_invOf_self : a * invOf = 1
 
 /-- The inverse of an `Invertible` element -/
-prefix:max
-  "⅟" =>-- This notation has the same precedence as `Inv.inv`.
-  Invertible.invOf
+-- This notation has the same precedence as `Inv.inv`.
+prefix:max "⅟" => Invertible.invOf
 
 @[simp]
 theorem invOf_mul_self' [Mul α] [One α] (a : α) {_ : Invertible a} : ⅟ a * a = 1 :=
   Invertible.invOf_mul_self
 
-theorem invOf_mul_self [Mul α] [One α] (a : α) [Invertible a] : ⅟ a * a = 1 :=
-  Invertible.invOf_mul_self
+theorem invOf_mul_self [Mul α] [One α] (a : α) [Invertible a] : ⅟ a * a = 1 := invOf_mul_self' _
 
 @[simp]
 theorem mul_invOf_self' [Mul α] [One α] (a : α) {_ : Invertible a} : a * ⅟ a = 1 :=
   Invertible.mul_invOf_self
 
-theorem mul_invOf_self [Mul α] [One α] (a : α) [Invertible a] : a * ⅟ a = 1 :=
-  Invertible.mul_invOf_self
+theorem mul_invOf_self [Mul α] [One α] (a : α) [Invertible a] : a * ⅟ a = 1 := mul_invOf_self' _
 
 @[simp]
-theorem invOf_mul_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : ⅟ a * (a * b) = b := by
+theorem invOf_mul_cancel_left' [Monoid α] (a b : α) {_ : Invertible a} : ⅟ a * (a * b) = b := by
   rw [← mul_assoc, invOf_mul_self, one_mul]
+example {G} [Group G] (a b : G) : a⁻¹ * (a * b) = b := inv_mul_cancel_left a b
 
-theorem invOf_mul_self_assoc [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b := by
-  rw [← mul_assoc, invOf_mul_self, one_mul]
+theorem invOf_mul_cancel_left [Monoid α] (a b : α) [Invertible a] : ⅟ a * (a * b) = b :=
+  invOf_mul_cancel_left' _ _
+
+@[deprecated (since := "2024-09-07")] alias invOf_mul_self_assoc' := invOf_mul_cancel_left'
+@[deprecated (since := "2024-09-07")] alias invOf_mul_self_assoc := invOf_mul_cancel_left
 
 @[simp]
-theorem mul_invOf_self_assoc' [Monoid α] (a b : α) {_ : Invertible a} : a * (⅟ a * b) = b := by
+theorem mul_invOf_cancel_left' [Monoid α] (a b : α) {_ : Invertible a} : a * (⅟ a * b) = b := by
   rw [← mul_assoc, mul_invOf_self, one_mul]
+example {G} [Group G] (a b : G) : a * (a⁻¹ * b) = b := mul_inv_cancel_left a b
 
-theorem mul_invOf_self_assoc [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b := by
-  rw [← mul_assoc, mul_invOf_self, one_mul]
+theorem mul_invOf_cancel_left [Monoid α] (a b : α) [Invertible a] : a * (⅟ a * b) = b :=
+  mul_invOf_cancel_left' a b
+
+@[deprecated (since := "2024-09-07")] alias mul_invOf_self_assoc' := mul_invOf_cancel_left'
+@[deprecated (since := "2024-09-07")] alias mul_invOf_self_assoc := mul_invOf_cancel_left
 
 @[simp]
-theorem mul_invOf_mul_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a := by
+theorem invOf_mul_cancel_right' [Monoid α] (a b : α) {_ : Invertible b} : a * ⅟ b * b = a := by
   simp [mul_assoc]
+example {G} [Group G] (a b : G) : a * b⁻¹ * b = a := inv_mul_cancel_right a b
 
-theorem mul_invOf_mul_self_cancel [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a := by
-  simp [mul_assoc]
+theorem invOf_mul_cancel_right [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a :=
+  invOf_mul_cancel_right' _ _
+
+@[deprecated (since := "2024-09-07")] alias mul_invOf_mul_self_cancel' := invOf_mul_cancel_right'
+@[deprecated (since := "2024-09-07")] alias mul_invOf_mul_self_cancel := invOf_mul_cancel_right
 
 @[simp]
-theorem mul_mul_invOf_self_cancel' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a := by
+theorem mul_invOf_cancel_right' [Monoid α] (a b : α) {_ : Invertible b} : a * b * ⅟ b = a := by
   simp [mul_assoc]
+example {G} [Group G] (a b : G) : a * b * b⁻¹ = a := mul_inv_cancel_right a b
 
-theorem mul_mul_invOf_self_cancel [Monoid α] (a b : α) [Invertible b] : a * b * ⅟ b = a := by
-  simp [mul_assoc]
+theorem mul_invOf_cancel_right [Monoid α] (a b : α) [Invertible b] : a * b * ⅟ b = a :=
+  mul_invOf_cancel_right' _ _
+
+@[deprecated (since := "2024-09-07")] alias mul_mul_invOf_self_cancel' := mul_invOf_cancel_right'
+@[deprecated (since := "2024-09-07")] alias mul_mul_invOf_self_cancel := mul_invOf_cancel_right
 
 theorem invOf_eq_right_inv [Monoid α] {a b : α} [Invertible a] (hac : a * b = 1) : ⅟ a = b :=
   left_inv_eq_right_inv (invOf_mul_self _) hac
@@ -185,8 +198,7 @@ def invertibleOne [Monoid α] : Invertible (1 : α) :=
 theorem invOf_one' [Monoid α] {_ : Invertible (1 : α)} : ⅟ (1 : α) = 1 :=
   invOf_eq_right_inv (mul_one _)
 
-theorem invOf_one [Monoid α] [Invertible (1 : α)] : ⅟ (1 : α) = 1 :=
-  invOf_eq_right_inv (mul_one _)
+theorem invOf_one [Monoid α] [Invertible (1 : α)] : ⅟ (1 : α) = 1 := invOf_one'
 
 /-- `a` is the inverse of `⅟a`. -/
 instance invertibleInvOf [One α] [Mul α] {a : α} [Invertible a] : Invertible (⅟ a) :=
@@ -226,15 +238,15 @@ theorem mul_left_inj_of_invertible : c * a = c * b ↔ a = b :=
   ⟨fun h => by simpa using congr_arg (⅟c * ·) h, congr_arg (_ * ·)⟩
 
 theorem invOf_mul_eq_iff_eq_mul_left : ⅟c * a = b ↔ a = c * b := by
-  rw [← mul_left_inj_of_invertible (c := c), mul_invOf_self_assoc]
+  rw [← mul_left_inj_of_invertible (c := c), mul_invOf_cancel_left]
 
 theorem mul_left_eq_iff_eq_invOf_mul : c * a = b ↔ a = ⅟c * b := by
-  rw [← mul_left_inj_of_invertible (c := ⅟c), invOf_mul_self_assoc]
+  rw [← mul_left_inj_of_invertible (c := ⅟c), invOf_mul_cancel_left]
 
 theorem mul_invOf_eq_iff_eq_mul_right : a * ⅟c = b ↔ a = b * c := by
-  rw [← mul_right_inj_of_invertible (c := c), mul_invOf_mul_self_cancel]
+  rw [← mul_right_inj_of_invertible (c := c), invOf_mul_cancel_right]
 
 theorem mul_right_eq_iff_eq_mul_invOf : a * c = b ↔ a = b * ⅟c := by
-  rw [← mul_right_inj_of_invertible (c := ⅟c), mul_mul_invOf_self_cancel]
+  rw [← mul_right_inj_of_invertible (c := ⅟c), mul_invOf_cancel_right]
 
 end
diff --git a/Mathlib/Algebra/Group/Opposite.lean b/Mathlib/Algebra/Group/Opposite.lean
index 07ef939bb18eb..d7ea15a86adfe 100644
--- a/Mathlib/Algebra/Group/Opposite.lean
+++ b/Mathlib/Algebra/Group/Opposite.lean
@@ -131,13 +131,13 @@ instance instMonoid [Monoid α] : Monoid αᵐᵒᵖ where
 
 @[to_additive]
 instance instLeftCancelMonoid [RightCancelMonoid α] : LeftCancelMonoid αᵐᵒᵖ where
-  toLeftCancelSemigroup := instLeftCancelSemigroup
-  __ := instMonoid
+  toMonoid := instMonoid
+  __ := instLeftCancelSemigroup
 
 @[to_additive]
 instance instRightCancelMonoid [LeftCancelMonoid α] : RightCancelMonoid αᵐᵒᵖ where
-  toRightCancelSemigroup := instRightCancelSemigroup
-  __ := instMonoid
+  toMonoid := instMonoid
+  __ := instRightCancelSemigroup
 
 @[to_additive]
 instance instCancelMonoid [CancelMonoid α] : CancelMonoid αᵐᵒᵖ where
@@ -151,8 +151,8 @@ instance instCommMonoid [CommMonoid α] : CommMonoid αᵐᵒᵖ where
 
 @[to_additive]
 instance instCancelCommMonoid [CancelCommMonoid α] : CancelCommMonoid αᵐᵒᵖ where
-  toLeftCancelMonoid := instLeftCancelMonoid
-  __ := instCommMonoid
+  toCommMonoid := instCommMonoid
+  __ := instLeftCancelMonoid
 
 @[to_additive AddOpposite.instSubNegMonoid]
 instance instDivInvMonoid [DivInvMonoid α] : DivInvMonoid αᵐᵒᵖ where
diff --git a/Mathlib/Algebra/Group/Pi/Basic.lean b/Mathlib/Algebra/Group/Pi/Basic.lean
index 480759b083256..7a88ae8220a12 100644
--- a/Mathlib/Algebra/Group/Pi/Basic.lean
+++ b/Mathlib/Algebra/Group/Pi/Basic.lean
@@ -400,18 +400,6 @@ theorem extend_div [Div γ] (f : α → β) (g₁ g₂ : α → γ) (e₁ e₂ :
 
 end Extend
 
-theorem surjective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Surjective (F i)) :
-    Surjective fun x : ∀ i, f i => fun i => F i (x i) := fun y =>
-  ⟨fun i => (hF i (y i)).choose, funext fun i => (hF i (y i)).choose_spec⟩
-
-theorem injective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Injective (F i)) :
-    Injective fun x : ∀ i, f i => fun i => F i (x i) :=
-  fun _ _ h => funext fun i => hF i <| (congr_fun h i : _)
-
-theorem bijective_pi_map {F : ∀ i, f i → g i} (hF : ∀ i, Bijective (F i)) :
-    Bijective fun x : ∀ i, f i => fun i => F i (x i) :=
-  ⟨injective_pi_map fun i => (hF i).injective, surjective_pi_map fun i => (hF i).surjective⟩
-
 lemma comp_eq_const_iff (b : β) (f : α → β) {g : β → γ} (hg : Injective g) :
     g ∘ f = Function.const _ (g b) ↔ f = Function.const _ b :=
   hg.comp_left.eq_iff' rfl
diff --git a/Mathlib/Algebra/Group/Pi/Lemmas.lean b/Mathlib/Algebra/Group/Pi/Lemmas.lean
index f3bc2c5b9eb5b..672442bbd487d 100644
--- a/Mathlib/Algebra/Group/Pi/Lemmas.lean
+++ b/Mathlib/Algebra/Group/Pi/Lemmas.lean
@@ -38,6 +38,15 @@ theorem Set.preimage_one {α β : Type*} [One β] (s : Set β) [Decidable ((1 :
     (1 : α → β) ⁻¹' s = if (1 : β) ∈ s then Set.univ else ∅ :=
   Set.preimage_const 1 s
 
+namespace Pi
+
+variable {α β : Type*} [Preorder α] [Preorder β]
+
+@[to_additive] lemma one_mono [One β] : Monotone (1 : α → β) := monotone_const
+@[to_additive] lemma one_anti [One β] : Antitone (1 : α → β) := antitone_const
+
+end Pi
+
 namespace MulHom
 
 @[to_additive]
@@ -239,6 +248,15 @@ theorem Pi.mulSingle_div [∀ i, Group <| f i] (i : I) (x y : f i) :
     mulSingle i (x / y) = mulSingle i x / mulSingle i y :=
   (MonoidHom.mulSingle f i).map_div x y
 
+@[to_additive]
+theorem Pi.mulSingle_pow [∀ i, Monoid (f i)] (i : I) (x : f i) (n : ℕ) :
+    mulSingle i (x ^ n) = mulSingle i x ^ n :=
+  (MonoidHom.mulSingle f i).map_pow x n
+
+@[to_additive]
+theorem Pi.mulSingle_zpow [∀ i, Group (f i)] (i : I) (x : f i) (n : ℤ) :
+    mulSingle i (x ^ n) = mulSingle i x ^ n :=
+  (MonoidHom.mulSingle f i).map_zpow x n
 
 /-- The injection into a pi group at different indices commutes.
 
diff --git a/Mathlib/Data/Finset/Pointwise/Basic.lean b/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean
similarity index 95%
rename from Mathlib/Data/Finset/Pointwise/Basic.lean
rename to Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean
index ffbddd2367556..5e5ebeefca3c2 100644
--- a/Mathlib/Data/Finset/Pointwise/Basic.lean
+++ b/Mathlib/Algebra/Group/Pointwise/Finset/Basic.lean
@@ -3,15 +3,15 @@ Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Yaël Dillies
 -/
+import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Algebra.Group.Action.Pi
+import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Algebra.Ring.Pointwise.Set
+import Mathlib.Data.Finset.Density
 import Mathlib.Data.Finset.NAry
-import Mathlib.Data.Finset.Preimage
 import Mathlib.Data.Set.Pointwise.Finite
-import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.Data.Set.Pointwise.ListOfFn
-import Mathlib.Data.ULift
-import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Data.Set.Pointwise.SMul
 
 /-!
 # Pointwise operations of finsets
@@ -210,17 +210,18 @@ theorem card_inv_le : s⁻¹.card ≤ s.card :=
 theorem inv_empty : (∅ : Finset α)⁻¹ = ∅ :=
   image_empty _
 
-@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
+@[to_additive (attr := simp)]
 theorem inv_nonempty_iff : s⁻¹.Nonempty ↔ s.Nonempty := image_nonempty
 
 alias ⟨Nonempty.of_inv, Nonempty.inv⟩ := inv_nonempty_iff
 
 attribute [to_additive] Nonempty.inv Nonempty.of_inv
+attribute [aesop safe apply (rule_sets := [finsetNonempty])] Nonempty.inv Nonempty.neg
 
 @[to_additive (attr := simp)]
 theorem inv_eq_empty : s⁻¹ = ∅ ↔ s = ∅ := image_eq_empty
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem inv_subset_inv (h : s ⊆ t) : s⁻¹ ⊆ t⁻¹ :=
   image_subset_image h
 
@@ -271,6 +272,9 @@ theorem coe_inv (s : Finset α) : ↑s⁻¹ = (s : Set α)⁻¹ := coe_image.tra
 @[to_additive (attr := simp)]
 theorem card_inv (s : Finset α) : s⁻¹.card = s.card := card_image_of_injective _ inv_injective
 
+@[to_additive (attr := simp)]
+lemma dens_inv [Fintype α] (s : Finset α) : s⁻¹.dens = s.dens := by simp [dens]
+
 @[to_additive (attr := simp)]
 theorem preimage_inv (s : Finset α) : s.preimage (·⁻¹) inv_injective.injOn = s⁻¹ :=
   coe_injective <| by rw [coe_preimage, Set.inv_preimage, coe_inv]
@@ -341,11 +345,11 @@ theorem mul_empty (s : Finset α) : s * ∅ = ∅ :=
 theorem mul_eq_empty : s * t = ∅ ↔ s = ∅ ∨ t = ∅ :=
   image₂_eq_empty_iff
 
-@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
+@[to_additive (attr := simp)]
 theorem mul_nonempty : (s * t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
-@[to_additive]
+@[to_additive (attr := aesop safe apply (rule_sets := [finsetNonempty]))]
 theorem Nonempty.mul : s.Nonempty → t.Nonempty → (s * t).Nonempty :=
   Nonempty.image₂
 
@@ -369,7 +373,7 @@ theorem singleton_mul (a : α) : {a} * s = s.image (a * ·) :=
 theorem singleton_mul_singleton (a b : α) : ({a} : Finset α) * {b} = {a * b} :=
   image₂_singleton
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem mul_subset_mul : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ * t₁ ⊆ s₂ * t₂ :=
   image₂_subset
 
@@ -416,7 +420,7 @@ theorem union_mul_inter_subset_union : (s₁ ∪ s₂) * (t₁ ∩ t₂) ⊆ s
       `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' + t'`."]
 theorem subset_mul {s t : Set α} :
     ↑u ⊆ s * t → ∃ s' t' : Finset α, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' * t' :=
-  subset_image₂
+  subset_set_image₂
 
 @[to_additive]
 theorem image_mul [DecidableEq β] : (s * t).image (f : α → β) = s.image f * t.image f :=
@@ -525,11 +529,11 @@ theorem div_empty (s : Finset α) : s / ∅ = ∅ :=
 theorem div_eq_empty : s / t = ∅ ↔ s = ∅ ∨ t = ∅ :=
   image₂_eq_empty_iff
 
-@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
+@[to_additive (attr := simp)]
 theorem div_nonempty : (s / t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
-@[to_additive]
+@[to_additive (attr := aesop safe apply (rule_sets := [finsetNonempty]))]
 theorem Nonempty.div : s.Nonempty → t.Nonempty → (s / t).Nonempty :=
   Nonempty.image₂
 
@@ -555,7 +559,7 @@ theorem singleton_div (a : α) : {a} / s = s.image (a / ·) :=
 theorem singleton_div_singleton (a b : α) : ({a} : Finset α) / {b} = {a / b} :=
   image₂_singleton
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem div_subset_div : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ / t₁ ⊆ s₂ / t₂ :=
   image₂_subset
 
@@ -602,7 +606,7 @@ theorem union_div_inter_subset_union : (s₁ ∪ s₂) / (t₁ ∩ t₂) ⊆ s
       `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' - t'`."]
 theorem subset_div {s t : Set α} :
     ↑u ⊆ s / t → ∃ s' t' : Finset α, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' / t' :=
-  subset_image₂
+  subset_set_image₂
 
 @[to_additive (attr := simp (default + 1))]
 lemma sup_div_le [SemilatticeSup β] [OrderBot β] {s t : Finset α} {f : α → β} {a : β} :
@@ -772,10 +776,9 @@ scoped[Pointwise] attribute [instance] Finset.monoid Finset.addMonoid
 @[to_additive]
 theorem pow_mem_pow (ha : a ∈ s) : ∀ n : ℕ, a ^ n ∈ s ^ n
   | 0 => by
-    rw [pow_zero]
-    exact one_mem_one
+    simp only [pow_zero, mem_one]
   | n + 1 => by
-    rw [pow_succ]
+    simp only [pow_succ]
     exact mul_mem_mul (pow_mem_pow ha n) ha
 
 @[to_additive]
@@ -898,6 +901,12 @@ theorem isUnit_coe : IsUnit (s : Set α) ↔ IsUnit s := by
 @[to_additive (attr := simp)]
 lemma univ_div_univ [Fintype α] : (univ / univ : Finset α) = univ := by simp [div_eq_mul_inv]
 
+@[to_additive] lemma subset_div_left (ht : 1 ∈ t) : s ⊆ s / t := by
+  rw [div_eq_mul_inv]; exact subset_mul_left _ <| by simpa
+
+@[to_additive] lemma inv_subset_div_right (hs : 1 ∈ s) : t⁻¹ ⊆ s / t := by
+  rw [div_eq_mul_inv]; exact subset_mul_right _ hs
+
 end DivisionMonoid
 
 /-- `Finset α` is a commutative division monoid under pointwise operations if `α` is. -/
@@ -990,12 +999,12 @@ to
   ∃ a, s = {a} ∧ IsUnit a -/
 -- @[simp]
 theorem isUnit_iff_singleton : IsUnit s ↔ ∃ a, s = {a} := by
-  simp only [isUnit_iff, Group.isUnit, and_true_iff]
+  simp only [isUnit_iff, Group.isUnit, and_true]
 
 @[simp]
 theorem isUnit_iff_singleton_aux {α} [Group α] {s : Finset α} :
     (∃ a, s = {a} ∧ IsUnit a) ↔ ∃ a, s = {a} := by
-  simp only [Group.isUnit, and_true_iff]
+  simp only [Group.isUnit, and_true]
 
 @[to_additive (attr := simp)]
 theorem image_mul_left :
@@ -1120,11 +1129,11 @@ theorem smul_empty (s : Finset α) : s • (∅ : Finset β) = ∅ :=
 theorem smul_eq_empty : s • t = ∅ ↔ s = ∅ ∨ t = ∅ :=
   image₂_eq_empty_iff
 
-@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
+@[to_additive (attr := simp)]
 theorem smul_nonempty_iff : (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
-@[to_additive]
+@[to_additive (attr := aesop safe apply (rule_sets := [finsetNonempty]))]
 theorem Nonempty.smul : s.Nonempty → t.Nonempty → (s • t).Nonempty :=
   Nonempty.image₂
 
@@ -1144,7 +1153,7 @@ theorem smul_singleton (b : β) : s • ({b} : Finset β) = s.image (· • b) :
 theorem singleton_smul_singleton (a : α) (b : β) : ({a} : Finset α) • ({b} : Finset β) = {a • b} :=
   image₂_singleton
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem smul_subset_smul : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ • t₁ ⊆ s₂ • t₂ :=
   image₂_subset
 
@@ -1191,7 +1200,7 @@ theorem union_smul_inter_subset_union [DecidableEq α] : (s₁ ∪ s₂) • (t
       finsets `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' +ᵥ t'`."]
 theorem subset_smul {s : Set α} {t : Set β} :
     ↑u ⊆ s • t → ∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' • t' :=
-  subset_image₂
+  subset_set_image₂
 
 end SMul
 
@@ -1241,10 +1250,11 @@ theorem vsub_empty (s : Finset β) : s -ᵥ (∅ : Finset β) = ∅ :=
 theorem vsub_eq_empty : s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅ :=
   image₂_eq_empty_iff
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem vsub_nonempty : (s -ᵥ t : Finset α).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 theorem Nonempty.vsub : s.Nonempty → t.Nonempty → (s -ᵥ t : Finset α).Nonempty :=
   Nonempty.image₂
 
@@ -1265,7 +1275,7 @@ theorem singleton_vsub (a : β) : ({a} : Finset β) -ᵥ t = t.image (a -ᵥ ·)
 theorem singleton_vsub_singleton (a b : β) : ({a} : Finset β) -ᵥ {b} = {a -ᵥ b} :=
   image₂_singleton
 
-@[mono]
+@[mono, gcongr]
 theorem vsub_subset_vsub : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ -ᵥ t₁ ⊆ s₂ -ᵥ t₂ :=
   image₂_subset
 
@@ -1300,7 +1310,7 @@ end
 finsets `s'`, `t'` such that `s' ⊆ s`, `t' ⊆ t` and `u ⊆ s' -ᵥ t'`. -/
 theorem subset_vsub {s t : Set β} :
     ↑u ⊆ s -ᵥ t → ∃ s' t' : Finset β, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' -ᵥ t' :=
-  subset_image₂
+  subset_set_image₂
 
 end VSub
 
@@ -1352,11 +1362,11 @@ theorem smul_finset_empty (a : α) : a • (∅ : Finset β) = ∅ :=
 theorem smul_finset_eq_empty : a • s = ∅ ↔ s = ∅ :=
   image_eq_empty
 
-@[to_additive (attr := simp, aesop safe apply (rule_sets := [finsetNonempty]))]
+@[to_additive (attr := simp)]
 theorem smul_finset_nonempty : (a • s).Nonempty ↔ s.Nonempty :=
   image_nonempty
 
-@[to_additive]
+@[to_additive (attr := aesop safe apply (rule_sets := [finsetNonempty]))]
 theorem Nonempty.smul_finset (hs : s.Nonempty) : (a • s).Nonempty :=
   hs.image _
 
@@ -1364,7 +1374,7 @@ theorem Nonempty.smul_finset (hs : s.Nonempty) : (a • s).Nonempty :=
 theorem singleton_smul (a : α) : ({a} : Finset α) • t = a • t :=
   image₂_singleton_left
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem smul_finset_subset_smul_finset : s ⊆ t → a • s ⊆ a • t :=
   image_subset_image
 
@@ -1376,6 +1386,10 @@ theorem smul_finset_singleton (b : β) : a • ({b} : Finset β) = {a • b} :=
 theorem smul_finset_union : a • (s₁ ∪ s₂) = a • s₁ ∪ a • s₂ :=
   image_union _ _
 
+@[to_additive]
+lemma smul_finset_insert (a : α) (b : β) (s : Finset β) : a • insert b s = insert (a • b) (a • s) :=
+  image_insert ..
+
 @[to_additive]
 theorem smul_finset_inter_subset : a • (s₁ ∩ s₂) ⊆ a • s₁ ∩ a • s₂ :=
   image_inter_subset _ _ _
@@ -1420,7 +1434,7 @@ instance smulCommClass [SMul α γ] [SMul β γ] [SMulCommClass α β γ] :
 @[to_additive vaddAssocClass]
 instance isScalarTower [SMul α β] [SMul α γ] [SMul β γ] [IsScalarTower α β γ] :
     IsScalarTower α β (Finset γ) :=
-  ⟨fun a b s => by simp only [← image_smul, image_image, smul_assoc, Function.comp]⟩
+  ⟨fun a b s => by simp only [← image_smul, image_image, smul_assoc, Function.comp_def]⟩
 
 variable [DecidableEq β]
 
@@ -1572,6 +1586,17 @@ theorem singleton_mul_inter : {a} * (s ∩ t) = {a} * s ∩ ({a} * t) :=
 theorem card_le_card_mul_left {s : Finset α} (hs : s.Nonempty) : t.card ≤ (s * t).card :=
   card_le_card_image₂_left _ hs mul_right_injective
 
+/--
+The size of `s * s` is at least the size of `s`, version with left-cancellative multiplication.
+See `card_le_card_mul_self'` for the version with right-cancellative multiplication.
+-/
+@[to_additive
+"The size of `s + s` is at least the size of `s`, version with left-cancellative addition.
+See `card_le_card_add_self'` for the version with right-cancellative addition."
+]
+theorem card_le_card_mul_self {s : Finset α} : s.card ≤ (s * s).card := by
+  cases s.eq_empty_or_nonempty <;> simp [card_le_card_mul_left, *]
+
 end IsLeftCancelMul
 
 section
@@ -1590,6 +1615,17 @@ theorem inter_mul_singleton : s ∩ t * {a} = s * {a} ∩ (t * {a}) :=
 theorem card_le_card_mul_right {t : Finset α} (ht : t.Nonempty) : s.card ≤ (s * t).card :=
   card_le_card_image₂_right _ ht mul_left_injective
 
+/--
+The size of `s * s` is at least the size of `s`, version with right-cancellative multiplication.
+See `card_le_card_mul_self` for the version with left-cancellative multiplication.
+-/
+@[to_additive
+"The size of `s + s` is at least the size of `s`, version with right-cancellative addition.
+See `card_le_card_add_self` for the version with left-cancellative addition."
+]
+theorem card_le_card_mul_self' {s : Finset α} : s.card ≤ (s * s).card := by
+  cases s.eq_empty_or_nonempty <;> simp [card_le_card_mul_right, *]
+
 end
 
 section Group
@@ -1601,6 +1637,9 @@ variable [Group α] [DecidableEq α] {s t : Finset α}
 @[to_additive] lemma card_le_card_div_right (ht : t.Nonempty) : s.card ≤ (s / t).card :=
   card_le_card_image₂_right _ ht fun _ ↦ div_left_injective
 
+@[to_additive] lemma card_le_card_div_self : s.card ≤ (s / s).card := by
+  cases s.eq_empty_or_nonempty <;> simp [card_le_card_div_left, *]
+
 end Group
 
 open Pointwise
@@ -1674,6 +1713,9 @@ theorem smul_univ [Fintype β] {s : Finset α} (hs : s.Nonempty) : s • (univ :
 theorem card_smul_finset (a : α) (s : Finset β) : (a • s).card = s.card :=
   card_image_of_injective _ <| MulAction.injective _
 
+@[to_additive (attr := simp)]
+lemma dens_smul_finset [Fintype β] (a : α) (s : Finset β) : (a • s).dens = s.dens := by simp [dens]
+
 /-- If the left cosets of `t` by elements of `s` are disjoint (but not necessarily distinct!), then
 the size of `t` divides the size of `s • t`. -/
 @[to_additive "If the left cosets of `t` by elements of `s` are disjoint (but not necessarily
@@ -1828,7 +1870,7 @@ variable [Monoid α] [AddGroup β] [DistribMulAction α β] [DecidableEq β] (a
 
 @[simp]
 theorem smul_finset_neg : a • -t = -(a • t) := by
-  simp only [← image_smul, ← image_neg, Function.comp, image_image, smul_neg]
+  simp only [← image_smul, ← image_neg, Function.comp_def, image_image, smul_neg]
 
 @[simp]
 protected theorem smul_neg : s • -t = -(s • t) := by
@@ -1844,7 +1886,7 @@ variable [Ring α] [AddCommGroup β] [Module α β] [DecidableEq β] {s : Finset
 
 @[simp]
 theorem neg_smul_finset : -a • t = -(a • t) := by
-  simp only [← image_smul, ← image_neg, image_image, neg_smul, Function.comp]
+  simp only [← image_smul, ← image_neg, image_image, neg_smul, Function.comp_def]
 
 @[simp]
 protected theorem neg_smul [DecidableEq α] : -s • t = -(s • t) := by
@@ -1913,6 +1955,10 @@ namespace Set
 
 section One
 
+-- Redeclaring an instance for better keys
+@[to_additive]
+instance instFintypeOne [One α] : Fintype (1 : Set α) := Set.fintypeSingleton _
+
 variable [One α]
 
 @[to_additive (attr := simp)]
@@ -1995,3 +2041,5 @@ instance Nat.decidablePred_mem_vadd_set {s : Set ℕ} [DecidablePred (· ∈ s)]
     DecidablePred (· ∈ a +ᵥ s) :=
   fun n ↦ decidable_of_iff' (a ≤ n ∧ n - a ∈ s) <| by
     simp only [Set.mem_vadd_set, vadd_eq_add]; aesop
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/Data/Finset/Pointwise/Interval.lean b/Mathlib/Algebra/Group/Pointwise/Finset/Interval.lean
similarity index 98%
rename from Mathlib/Data/Finset/Pointwise/Interval.lean
rename to Mathlib/Algebra/Group/Pointwise/Finset/Interval.lean
index 9178e9e993251..fd9cc57c0dd5b 100644
--- a/Mathlib/Data/Finset/Pointwise/Interval.lean
+++ b/Mathlib/Algebra/Group/Pointwise/Finset/Interval.lean
@@ -3,7 +3,7 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Data.Set.Pointwise.Interval
 import Mathlib.Order.Interval.Finset.Defs
 
diff --git a/Mathlib/Data/Set/Pointwise/Basic.lean b/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean
similarity index 65%
rename from Mathlib/Data/Set/Pointwise/Basic.lean
rename to Mathlib/Algebra/Group/Pointwise/Set/Basic.lean
index 81be1a2ed1658..bb8aba1b06f3a 100644
--- a/Mathlib/Data/Set/Pointwise/Basic.lean
+++ b/Mathlib/Algebra/Group/Pointwise/Set/Basic.lean
@@ -6,10 +6,7 @@ Authors: Johan Commelin, Floris van Doorn
 import Mathlib.Algebra.Group.Equiv.Basic
 import Mathlib.Algebra.Group.Units.Hom
 import Mathlib.Algebra.Opposites
-import Mathlib.Algebra.Ring.Defs
-import Mathlib.Algebra.GroupWithZero.Basic
 import Mathlib.Data.Set.Lattice
-import Mathlib.Tactic.Common
 
 /-!
 # Pointwise operations of sets
@@ -25,6 +22,11 @@ For sets `s` and `t` and scalar `a`:
 * `-s`: Negation, set of all `-x` where `x ∈ s`.
 * `s / t`: Division, set of all `x / y` where `x ∈ s` and `y ∈ t`.
 * `s - t`: Subtraction, set of all `x - y` where `x ∈ s` and `y ∈ t`.
+* `s • t`: Scalar multiplication, set of all `x • y` where `x ∈ s` and `y ∈ t`.
+* `s +ᵥ t`: Scalar addition, set of all `x +ᵥ y` where `x ∈ s` and `y ∈ t`.
+* `s -ᵥ t`: Scalar subtraction, set of all `x -ᵥ y` where `x ∈ s` and `y ∈ t`.
+* `a • s`: Scaling, set of all `a • x` where `x ∈ s`.
+* `a +ᵥ s`: Translation, set of all `a +ᵥ x` where `x ∈ s`.
 
 For `α` a semigroup/monoid, `Set α` is a semigroup/monoid.
 As an unfortunate side effect, this means that `n • s`, where `n : ℕ`, is ambiguous between
@@ -50,7 +52,7 @@ set multiplication, set addition, pointwise addition, pointwise multiplication,
 pointwise subtraction
 -/
 
-
+assert_not_exists MonoidWithZero
 assert_not_exists OrderedAddCommMonoid
 
 library_note "pointwise nat action"/--
@@ -179,10 +181,18 @@ theorem union_inv : (s ∪ t)⁻¹ = s⁻¹ ∪ t⁻¹ :=
 theorem iInter_inv (s : ι → Set α) : (⋂ i, s i)⁻¹ = ⋂ i, (s i)⁻¹ :=
   preimage_iInter
 
+@[to_additive (attr := simp)]
+theorem sInter_inv (S : Set (Set α)) : (⋂₀ S)⁻¹ = ⋂ s ∈ S, s⁻¹ :=
+  preimage_sInter
+
 @[to_additive (attr := simp)]
 theorem iUnion_inv (s : ι → Set α) : (⋃ i, s i)⁻¹ = ⋃ i, (s i)⁻¹ :=
   preimage_iUnion
 
+@[to_additive (attr := simp)]
+theorem sUnion_inv (S : Set (Set α)) : (⋃₀ S)⁻¹ = ⋃ s ∈ S, s⁻¹ :=
+  preimage_sUnion
+
 @[to_additive (attr := simp)]
 theorem compl_inv : sᶜ⁻¹ = s⁻¹ᶜ :=
   preimage_compl
@@ -234,7 +244,7 @@ theorem inv_insert (a : α) (s : Set α) : (insert a s)⁻¹ = insert a⁻¹ s
 @[to_additive]
 theorem inv_range {ι : Sort*} {f : ι → α} : (range f)⁻¹ = range fun i => (f i)⁻¹ := by
   rw [← image_inv]
-  exact (range_comp _ _).symm
+  exact (range_comp ..).symm
 
 open MulOpposite
 
@@ -322,7 +332,7 @@ theorem singleton_mul : {a} * t = (a * ·) '' t :=
 theorem singleton_mul_singleton : ({a} : Set α) * {b} = {a * b} :=
   image2_singleton
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem mul_subset_mul : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ * s₂ ⊆ t₁ * t₂ :=
   image2_subset
 
@@ -372,47 +382,63 @@ theorem iUnion_mul_right_image : ⋃ a ∈ t, (· * a) '' s = s * t :=
 
 @[to_additive]
 theorem iUnion_mul (s : ι → Set α) (t : Set α) : (⋃ i, s i) * t = ⋃ i, s i * t :=
-  image2_iUnion_left _ _ _
+  image2_iUnion_left ..
 
 @[to_additive]
 theorem mul_iUnion (s : Set α) (t : ι → Set α) : (s * ⋃ i, t i) = ⋃ i, s * t i :=
-  image2_iUnion_right _ _ _
+  image2_iUnion_right ..
+
+@[to_additive]
+theorem sUnion_mul (S : Set (Set α)) (t : Set α) : ⋃₀ S * t = ⋃ s ∈ S, s * t :=
+  image2_sUnion_left ..
+
+@[to_additive]
+theorem mul_sUnion (s : Set α) (T : Set (Set α)) : s * ⋃₀ T = ⋃ t ∈ T, s * t :=
+  image2_sUnion_right ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem iUnion₂_mul (s : ∀ i, κ i → Set α) (t : Set α) :
     (⋃ (i) (j), s i j) * t = ⋃ (i) (j), s i j * t :=
-  image2_iUnion₂_left _ _ _
+  image2_iUnion₂_left ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem mul_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
     (s * ⋃ (i) (j), t i j) = ⋃ (i) (j), s * t i j :=
-  image2_iUnion₂_right _ _ _
+  image2_iUnion₂_right ..
 
 @[to_additive]
 theorem iInter_mul_subset (s : ι → Set α) (t : Set α) : (⋂ i, s i) * t ⊆ ⋂ i, s i * t :=
-  Set.image2_iInter_subset_left _ _ _
+  Set.image2_iInter_subset_left ..
 
 @[to_additive]
 theorem mul_iInter_subset (s : Set α) (t : ι → Set α) : (s * ⋂ i, t i) ⊆ ⋂ i, s * t i :=
-  image2_iInter_subset_right _ _ _
+  image2_iInter_subset_right ..
+
+@[to_additive]
+lemma mul_sInter_subset (s : Set α) (T : Set (Set α)) :
+    s * ⋂₀ T ⊆ ⋂ t ∈ T, s * t := image2_sInter_right_subset s T (fun a b => a * b)
+
+@[to_additive]
+lemma sInter_mul_subset (S : Set (Set α)) (t : Set α) :
+    ⋂₀ S * t ⊆ ⋂ s ∈ S, s * t := image2_sInter_left_subset S t (fun a b => a * b)
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem iInter₂_mul_subset (s : ∀ i, κ i → Set α) (t : Set α) :
     (⋂ (i) (j), s i j) * t ⊆ ⋂ (i) (j), s i j * t :=
-  image2_iInter₂_subset_left _ _ _
+  image2_iInter₂_subset_left ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem mul_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set α) :
     (s * ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s * t i j :=
-  image2_iInter₂_subset_right _ _ _
+  image2_iInter₂_subset_right ..
 
 /-- The singleton operation as a `MulHom`. -/
 @[to_additive "The singleton operation as an `AddHom`."]
@@ -510,7 +536,7 @@ theorem singleton_div : {a} / t = (· / ·) a '' t :=
 theorem singleton_div_singleton : ({a} : Set α) / {b} = {a / b} :=
   image2_singleton
 
-@[to_additive (attr := mono)]
+@[to_additive (attr := mono, gcongr)]
 theorem div_subset_div : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ / s₂ ⊆ t₁ / t₂ :=
   image2_subset
 
@@ -560,50 +586,390 @@ theorem iUnion_div_right_image : ⋃ a ∈ t, (· / a) '' s = s / t :=
 
 @[to_additive]
 theorem iUnion_div (s : ι → Set α) (t : Set α) : (⋃ i, s i) / t = ⋃ i, s i / t :=
-  image2_iUnion_left _ _ _
+  image2_iUnion_left ..
 
 @[to_additive]
 theorem div_iUnion (s : Set α) (t : ι → Set α) : (s / ⋃ i, t i) = ⋃ i, s / t i :=
-  image2_iUnion_right _ _ _
+  image2_iUnion_right ..
+
+@[to_additive]
+theorem sUnion_div (S : Set (Set α)) (t : Set α) : ⋃₀ S / t = ⋃ s ∈ S, s / t :=
+  image2_sUnion_left ..
+
+@[to_additive]
+theorem div_sUnion (s : Set α) (T : Set (Set α)) : s / ⋃₀ T = ⋃ t ∈ T, s / t :=
+  image2_sUnion_right ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem iUnion₂_div (s : ∀ i, κ i → Set α) (t : Set α) :
     (⋃ (i) (j), s i j) / t = ⋃ (i) (j), s i j / t :=
-  image2_iUnion₂_left _ _ _
+  image2_iUnion₂_left ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem div_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set α) :
     (s / ⋃ (i) (j), t i j) = ⋃ (i) (j), s / t i j :=
-  image2_iUnion₂_right _ _ _
+  image2_iUnion₂_right ..
 
 @[to_additive]
 theorem iInter_div_subset (s : ι → Set α) (t : Set α) : (⋂ i, s i) / t ⊆ ⋂ i, s i / t :=
-  image2_iInter_subset_left _ _ _
+  image2_iInter_subset_left ..
 
 @[to_additive]
 theorem div_iInter_subset (s : Set α) (t : ι → Set α) : (s / ⋂ i, t i) ⊆ ⋂ i, s / t i :=
-  image2_iInter_subset_right _ _ _
+  image2_iInter_subset_right ..
+
+@[to_additive]
+theorem sInter_div_subset (S : Set (Set α)) (t : Set α) : ⋂₀ S / t ⊆ ⋂ s ∈ S, s / t :=
+  image2_sInter_subset_left ..
+
+@[to_additive]
+theorem div_sInter_subset (s : Set α) (T : Set (Set α)) : s / ⋂₀ T ⊆ ⋂ t ∈ T, s / t :=
+  image2_sInter_subset_right ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem iInter₂_div_subset (s : ∀ i, κ i → Set α) (t : Set α) :
     (⋂ (i) (j), s i j) / t ⊆ ⋂ (i) (j), s i j / t :=
-  image2_iInter₂_subset_left _ _ _
+  image2_iInter₂_subset_left ..
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 @[to_additive]
 theorem div_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set α) :
     (s / ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s / t i j :=
-  image2_iInter₂_subset_right _ _ _
+  image2_iInter₂_subset_right ..
 
 end Div
 
+/-! ### Translation/scaling of sets -/
+
+section SMul
+
+/-- The dilation of set `x • s` is defined as `{x • y | y ∈ s}` in locale `Pointwise`. -/
+@[to_additive
+"The translation of set `x +ᵥ s` is defined as `{x +ᵥ y | y ∈ s}` in locale `Pointwise`."]
+protected def smulSet [SMul α β] : SMul α (Set β) where smul a := image (a • ·)
+
+/-- The pointwise scalar multiplication of sets `s • t` is defined as `{x • y | x ∈ s, y ∈ t}` in
+locale `Pointwise`. -/
+@[to_additive
+"The pointwise scalar addition of sets `s +ᵥ t` is defined as `{x +ᵥ y | x ∈ s, y ∈ t}` in locale
+`Pointwise`."]
+protected def smul [SMul α β] : SMul (Set α) (Set β) where smul := image2 (· • ·)
+
+scoped[Pointwise] attribute [instance] Set.smulSet Set.smul
+scoped[Pointwise] attribute [instance] Set.vaddSet Set.vadd
+
+section SMul
+variable {ι : Sort*} {κ : ι → Sort*} [SMul α β] {s s₁ s₂ : Set α} {t t₁ t₂ u : Set β} {a : α}
+  {b : β}
+
+@[to_additive (attr := simp)] lemma image2_smul : image2 SMul.smul s t = s • t := rfl
+
+@[to_additive vadd_image_prod]
+lemma image_smul_prod : (fun x : α × β ↦ x.fst • x.snd) '' s ×ˢ t = s • t := image_prod _
+
+@[to_additive] lemma mem_smul : b ∈ s • t ↔ ∃ x ∈ s, ∃ y ∈ t, x • y = b := Iff.rfl
+
+@[to_additive] lemma smul_mem_smul : a ∈ s → b ∈ t → a • b ∈ s • t := mem_image2_of_mem
+
+@[to_additive (attr := simp)] lemma empty_smul : (∅ : Set α) • t = ∅ := image2_empty_left
+@[to_additive (attr := simp)] lemma smul_empty : s • (∅ : Set β) = ∅ := image2_empty_right
+
+@[to_additive (attr := simp)] lemma smul_eq_empty : s • t = ∅ ↔ s = ∅ ∨ t = ∅ := image2_eq_empty_iff
+
+@[to_additive (attr := simp)]
+lemma smul_nonempty : (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := image2_nonempty_iff
+
+@[to_additive] lemma Nonempty.smul : s.Nonempty → t.Nonempty → (s • t).Nonempty := .image2
+@[to_additive] lemma Nonempty.of_smul_left : (s • t).Nonempty → s.Nonempty := .of_image2_left
+@[to_additive] lemma Nonempty.of_smul_right : (s • t).Nonempty → t.Nonempty := .of_image2_right
+
+@[to_additive (attr := simp low+1)]
+lemma smul_singleton : s • ({b} : Set β) = (· • b) '' s := image2_singleton_right
+
+@[to_additive (attr := simp low+1)]
+lemma singleton_smul : ({a} : Set α) • t = a • t := image2_singleton_left
+
+@[to_additive (attr := simp high)]
+lemma singleton_smul_singleton : ({a} : Set α) • ({b} : Set β) = {a • b} := image2_singleton
+
+@[to_additive (attr := mono, gcongr)]
+lemma smul_subset_smul : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ • t₁ ⊆ s₂ • t₂ := image2_subset
+
+@[to_additive] lemma smul_subset_smul_left : t₁ ⊆ t₂ → s • t₁ ⊆ s • t₂ := image2_subset_left
+@[to_additive] lemma smul_subset_smul_right : s₁ ⊆ s₂ → s₁ • t ⊆ s₂ • t := image2_subset_right
+
+@[to_additive] lemma smul_subset_iff : s • t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a • b ∈ u := image2_subset_iff
+
+@[to_additive] lemma union_smul : (s₁ ∪ s₂) • t = s₁ • t ∪ s₂ • t := image2_union_left
+@[to_additive] lemma smul_union : s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂ := image2_union_right
+
+@[to_additive]
+lemma inter_smul_subset : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t := image2_inter_subset_left
+
+@[to_additive]
+lemma smul_inter_subset : s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂ := image2_inter_subset_right
+
+@[to_additive]
+lemma inter_smul_union_subset_union : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ :=
+  image2_inter_union_subset_union
+
+@[to_additive]
+lemma union_smul_inter_subset_union : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ :=
+  image2_union_inter_subset_union
+
+@[to_additive] lemma iUnion_smul_left_image : ⋃ a ∈ s, a • t = s • t := iUnion_image_left _
+
+@[to_additive]
+lemma iUnion_smul_right_image : ⋃ a ∈ t, (· • a) '' s = s • t := iUnion_image_right _
+
+@[to_additive]
+lemma iUnion_smul (s : ι → Set α) (t : Set β) : (⋃ i, s i) • t = ⋃ i, s i • t :=
+  image2_iUnion_left ..
+
+@[to_additive]
+lemma smul_iUnion (s : Set α) (t : ι → Set β) : (s • ⋃ i, t i) = ⋃ i, s • t i :=
+  image2_iUnion_right ..
+
+@[to_additive]
+lemma sUnion_smul (S : Set (Set α)) (t : Set β) : ⋃₀ S • t = ⋃ s ∈ S, s • t :=
+  image2_sUnion_left ..
+
+@[to_additive]
+lemma smul_sUnion (s : Set α) (T : Set (Set β)) : s • ⋃₀ T = ⋃ t ∈ T, s • t :=
+  image2_sUnion_right ..
+
+@[to_additive]
+lemma iUnion₂_smul (s : ∀ i, κ i → Set α) (t : Set β) :
+    (⋃ i, ⋃ j, s i j) • t = ⋃ i, ⋃ j, s i j • t := image2_iUnion₂_left ..
+
+@[to_additive]
+lemma smul_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set β) :
+    (s • ⋃ i, ⋃ j, t i j) = ⋃ i, ⋃ j, s • t i j := image2_iUnion₂_right ..
+
+@[to_additive]
+lemma iInter_smul_subset (s : ι → Set α) (t : Set β) : (⋂ i, s i) • t ⊆ ⋂ i, s i • t :=
+  image2_iInter_subset_left ..
+
+@[to_additive]
+lemma smul_iInter_subset (s : Set α) (t : ι → Set β) : (s • ⋂ i, t i) ⊆ ⋂ i, s • t i :=
+  image2_iInter_subset_right ..
+
+@[to_additive]
+lemma sInter_smul_subset (S : Set (Set α)) (t : Set β) : ⋂₀ S • t ⊆ ⋂ s ∈ S, s • t :=
+  image2_sInter_left_subset S t (fun a x => a • x)
+
+@[to_additive]
+lemma smul_sInter_subset (s : Set α) (T : Set (Set β)) : s • ⋂₀ T ⊆ ⋂ t ∈ T, s • t :=
+  image2_sInter_right_subset s T (fun a x => a • x)
+
+@[to_additive]
+lemma iInter₂_smul_subset (s : ∀ i, κ i → Set α) (t : Set β) :
+    (⋂ i, ⋂ j, s i j) • t ⊆ ⋂ i, ⋂ j, s i j • t := image2_iInter₂_subset_left ..
+
+@[to_additive]
+lemma smul_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set β) :
+    (s • ⋂ i, ⋂ j, t i j) ⊆ ⋂ i, ⋂ j, s • t i j := image2_iInter₂_subset_right ..
+
+@[to_additive]
+lemma smul_set_subset_smul {s : Set α} : a ∈ s → a • t ⊆ s • t := image_subset_image2_right
+
+@[to_additive (attr := simp)]
+lemma iUnion_smul_set (s : Set α) (t : Set β) : ⋃ a ∈ s, a • t = s • t := iUnion_image_left _
+
+end SMul
+
+section SMulSet
+variable {ι : Sort*} {κ : ι → Sort*} [SMul α β] {s t t₁ t₂ : Set β} {a : α} {b : β} {x y : β}
+
+@[to_additive] lemma image_smul : (fun x ↦ a • x) '' t = a • t := rfl
+
+scoped[Pointwise] attribute [simp] Set.image_smul Set.image_vadd
+
+@[to_additive] lemma mem_smul_set : x ∈ a • t ↔ ∃ y, y ∈ t ∧ a • y = x := Iff.rfl
+
+@[to_additive] lemma smul_mem_smul_set : b ∈ s → a • b ∈ a • s := mem_image_of_mem _
+
+@[to_additive (attr := simp)] lemma smul_set_empty : a • (∅ : Set β) = ∅ := image_empty _
+@[to_additive (attr := simp)] lemma smul_set_eq_empty : a • s = ∅ ↔ s = ∅ := image_eq_empty
+
+@[to_additive (attr := simp)]
+lemma smul_set_nonempty : (a • s).Nonempty ↔ s.Nonempty := image_nonempty
+
+@[to_additive (attr := simp)]
+lemma smul_set_singleton : a • ({b} : Set β) = {a • b} := image_singleton
+
+@[to_additive (attr := gcongr)] lemma smul_set_mono : s ⊆ t → a • s ⊆ a • t := image_subset _
+
+@[to_additive]
+lemma smul_set_subset_iff : a • s ⊆ t ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ t :=
+  image_subset_iff
+
+@[to_additive]
+lemma smul_set_union : a • (t₁ ∪ t₂) = a • t₁ ∪ a • t₂ :=
+  image_union ..
+
+@[to_additive]
+lemma smul_set_insert (a : α) (b : β) (s : Set β) : a • insert b s = insert (a • b) (a • s) :=
+  image_insert_eq ..
+
+@[to_additive]
+lemma smul_set_inter_subset : a • (t₁ ∩ t₂) ⊆ a • t₁ ∩ a • t₂ :=
+  image_inter_subset ..
+
+@[to_additive]
+lemma smul_set_iUnion (a : α) (s : ι → Set β) : a • ⋃ i, s i = ⋃ i, a • s i :=
+  image_iUnion
+
+@[to_additive]
+lemma smul_set_iUnion₂ (a : α) (s : ∀ i, κ i → Set β) :
+    a • ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, a • s i j := image_iUnion₂ ..
+
+@[to_additive]
+lemma smul_set_sUnion (a : α) (S : Set (Set β)) : a • ⋃₀ S = ⋃ s ∈ S, a • s := by
+  rw [sUnion_eq_biUnion, smul_set_iUnion₂]
+
+@[to_additive]
+lemma smul_set_iInter_subset (a : α) (t : ι → Set β) : a • ⋂ i, t i ⊆ ⋂ i, a • t i :=
+  image_iInter_subset ..
+
+@[to_additive]
+lemma smul_set_sInter_subset (a : α) (S : Set (Set β)) :
+    a • ⋂₀ S ⊆ ⋂ s ∈ S, a • s := image_sInter_subset ..
+
+@[to_additive]
+lemma smul_set_iInter₂_subset (a : α) (t : ∀ i, κ i → Set β) :
+    a • ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, a • t i j := image_iInter₂_subset ..
+
+@[to_additive] lemma Nonempty.smul_set : s.Nonempty → (a • s).Nonempty := Nonempty.image _
+
+end SMulSet
+
+variable {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
+
+@[to_additive]
+lemma range_smul_range {ι κ : Type*} [SMul α β] (b : ι → α) (c : κ → β) :
+    range b • range c = range fun p : ι × κ ↦ b p.1 • c p.2 :=
+  image2_range ..
+
+@[to_additive]
+lemma smul_set_range [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
+    a • range f = range fun i ↦ a • f i :=
+  (range_comp ..).symm
+
+@[to_additive] lemma range_smul [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
+    range (fun i ↦ a • f i) = a • range f := (smul_set_range ..).symm
+
+end SMul
+
+section VSub
+variable {ι : Sort*} {κ : ι → Sort*} [VSub α β] {s s₁ s₂ t t₁ t₂ : Set β} {u : Set α} {a : α}
+  {b c : β}
+
+instance vsub : VSub (Set α) (Set β) where vsub := image2 (· -ᵥ ·)
+
+@[simp] lemma image2_vsub : (image2 VSub.vsub s t : Set α) = s -ᵥ t := rfl
+
+lemma image_vsub_prod : (fun x : β × β ↦ x.fst -ᵥ x.snd) '' s ×ˢ t = s -ᵥ t := image_prod _
+
+lemma mem_vsub : a ∈ s -ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x -ᵥ y = a := Iff.rfl
+
+lemma vsub_mem_vsub (hb : b ∈ s) (hc : c ∈ t) : b -ᵥ c ∈ s -ᵥ t := mem_image2_of_mem hb hc
+
+@[simp] lemma empty_vsub (t : Set β) : ∅ -ᵥ t = ∅ := image2_empty_left
+@[simp] lemma vsub_empty (s : Set β) : s -ᵥ ∅ = ∅ := image2_empty_right
+
+@[simp] lemma vsub_eq_empty : s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅ := image2_eq_empty_iff
+
+@[simp]
+lemma vsub_nonempty : (s -ᵥ t : Set α).Nonempty ↔ s.Nonempty ∧ t.Nonempty := image2_nonempty_iff
+
+lemma Nonempty.vsub : s.Nonempty → t.Nonempty → (s -ᵥ t : Set α).Nonempty := .image2
+lemma Nonempty.of_vsub_left : (s -ᵥ t : Set α).Nonempty → s.Nonempty := .of_image2_left
+lemma Nonempty.of_vsub_right : (s -ᵥ t : Set α).Nonempty → t.Nonempty := .of_image2_right
+
+@[simp low+1]
+lemma vsub_singleton (s : Set β) (b : β) : s -ᵥ {b} = (· -ᵥ b) '' s := image2_singleton_right
+
+@[simp low+1]
+lemma singleton_vsub (t : Set β) (b : β) : {b} -ᵥ t = (b -ᵥ ·) '' t := image2_singleton_left
+
+@[simp high] lemma singleton_vsub_singleton : ({b} : Set β) -ᵥ {c} = {b -ᵥ c} := image2_singleton
+
+@[mono] lemma vsub_subset_vsub : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ -ᵥ t₁ ⊆ s₂ -ᵥ t₂ := image2_subset
+
+lemma vsub_subset_vsub_left : t₁ ⊆ t₂ → s -ᵥ t₁ ⊆ s -ᵥ t₂ := image2_subset_left
+lemma vsub_subset_vsub_right : s₁ ⊆ s₂ → s₁ -ᵥ t ⊆ s₂ -ᵥ t := image2_subset_right
+
+lemma vsub_subset_iff : s -ᵥ t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x -ᵥ y ∈ u := image2_subset_iff
+
+lemma vsub_self_mono (h : s ⊆ t) : s -ᵥ s ⊆ t -ᵥ t := vsub_subset_vsub h h
+
+lemma union_vsub : s₁ ∪ s₂ -ᵥ t = s₁ -ᵥ t ∪ (s₂ -ᵥ t) := image2_union_left
+lemma vsub_union : s -ᵥ (t₁ ∪ t₂) = s -ᵥ t₁ ∪ (s -ᵥ t₂) := image2_union_right
+
+lemma inter_vsub_subset : s₁ ∩ s₂ -ᵥ t ⊆ (s₁ -ᵥ t) ∩ (s₂ -ᵥ t) := image2_inter_subset_left
+lemma vsub_inter_subset : s -ᵥ t₁ ∩ t₂ ⊆ (s -ᵥ t₁) ∩ (s -ᵥ t₂) := image2_inter_subset_right
+
+lemma inter_vsub_union_subset_union : s₁ ∩ s₂ -ᵥ (t₁ ∪ t₂) ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) :=
+  image2_inter_union_subset_union
+
+lemma union_vsub_inter_subset_union : s₁ ∪ s₂ -ᵥ t₁ ∩ t₂ ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) :=
+  image2_union_inter_subset_union
+
+lemma iUnion_vsub_left_image : ⋃ a ∈ s, (a -ᵥ ·) '' t = s -ᵥ t := iUnion_image_left _
+lemma iUnion_vsub_right_image : ⋃ a ∈ t, (· -ᵥ a) '' s = s -ᵥ t := iUnion_image_right _
+
+lemma iUnion_vsub (s : ι → Set β) (t : Set β) : (⋃ i, s i) -ᵥ t = ⋃ i, s i -ᵥ t :=
+  image2_iUnion_left ..
+
+lemma vsub_iUnion (s : Set β) (t : ι → Set β) : (s -ᵥ ⋃ i, t i) = ⋃ i, s -ᵥ t i :=
+  image2_iUnion_right ..
+
+lemma sUnion_vsub (S : Set (Set β)) (t : Set β) : ⋃₀ S -ᵥ t = ⋃ s ∈ S, s -ᵥ t :=
+  image2_sUnion_left ..
+
+lemma vsub_sUnion (s : Set β) (T : Set (Set β)) : s -ᵥ ⋃₀ T = ⋃ t ∈ T, s -ᵥ t :=
+  image2_sUnion_right ..
+
+lemma iUnion₂_vsub (s : ∀ i, κ i → Set β) (t : Set β) :
+    (⋃ i, ⋃ j, s i j) -ᵥ t = ⋃ i, ⋃ j, s i j -ᵥ t := image2_iUnion₂_left ..
+
+lemma vsub_iUnion₂ (s : Set β) (t : ∀ i, κ i → Set β) :
+    (s -ᵥ ⋃ i, ⋃ j, t i j) = ⋃ i, ⋃ j, s -ᵥ t i j := image2_iUnion₂_right ..
+
+lemma iInter_vsub_subset (s : ι → Set β) (t : Set β) : (⋂ i, s i) -ᵥ t ⊆ ⋂ i, s i -ᵥ t :=
+  image2_iInter_subset_left ..
+
+lemma vsub_iInter_subset (s : Set β) (t : ι → Set β) : (s -ᵥ ⋂ i, t i) ⊆ ⋂ i, s -ᵥ t i :=
+  image2_iInter_subset_right ..
+
+lemma sInter_vsub_subset (S : Set (Set β)) (t : Set β) : ⋂₀ S -ᵥ t ⊆ ⋂ s ∈ S, s -ᵥ t :=
+  image2_sInter_subset_left ..
+
+lemma vsub_sInter_subset (s : Set β) (T : Set (Set β)) : s -ᵥ ⋂₀ T ⊆ ⋂ t ∈ T, s -ᵥ t :=
+  image2_sInter_subset_right ..
+
+lemma iInter₂_vsub_subset (s : ∀ i, κ i → Set β) (t : Set β) :
+    (⋂ i, ⋂ j, s i j) -ᵥ t ⊆ ⋂ i, ⋂ j, s i j -ᵥ t := image2_iInter₂_subset_left ..
+
+lemma vsub_iInter₂_subset (s : Set β) (t : ∀ i, κ i → Set β) :
+    s -ᵥ ⋂ i, ⋂ j, t i j ⊆ ⋂ i, ⋂ j, s -ᵥ t i j := image2_iInter₂_subset_right ..
+
+end VSub
+
+open Pointwise
+
+@[to_additive]
+lemma image_smul_comm [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Set β) :
+    (∀ b, f (a • b) = a • f b) → f '' (a • s) = a • f '' s := image_comm
+
 open Pointwise
 
 /-- Repeated pointwise addition (not the same as pointwise repeated addition!) of a `Set`. See
@@ -707,10 +1073,9 @@ scoped[Pointwise] attribute [instance] Set.monoid Set.addMonoid
 @[to_additive]
 theorem pow_mem_pow (ha : a ∈ s) : ∀ n : ℕ, a ^ n ∈ s ^ n
   | 0 => by
-    rw [pow_zero]
-    exact one_mem_one
+    simp only [pow_zero, mem_one]
   | n + 1 => by
-    rw [pow_succ]
+    simp only [pow_succ]
     exact mul_mem_mul (pow_mem_pow ha _) ha
 
 @[to_additive]
@@ -726,9 +1091,10 @@ theorem pow_subset_pow (hst : s ⊆ t) : ∀ n : ℕ, s ^ n ⊆ t ^ n
 theorem pow_subset_pow_of_one_mem (hs : (1 : α) ∈ s) (hn : m ≤ n) : s ^ m ⊆ s ^ n := by
   -- Porting note: `Nat.le_induction` didn't work as an induction principle in mathlib3, this was
   -- `refine Nat.le_induction ...`
-  induction' n, hn using Nat.le_induction with _ _ ih
-  · exact Subset.rfl
-  · dsimp only
+  induction n, hn using Nat.le_induction with
+  | base => exact Subset.rfl
+  | succ _ _ ih =>
+    dsimp only
     rw [pow_succ']
     exact ih.trans (subset_mul_right _ hs)
 
@@ -819,6 +1185,12 @@ theorem isUnit_iff : IsUnit s ↔ ∃ a, s = {a} ∧ IsUnit a := by
 @[to_additive (attr := simp)]
 lemma univ_div_univ : (univ / univ : Set α) = univ := by simp [div_eq_mul_inv]
 
+@[to_additive] lemma subset_div_left (ht : 1 ∈ t) : s ⊆ s / t := by
+  rw [div_eq_mul_inv]; exact subset_mul_left _ <| by simpa
+
+@[to_additive] lemma inv_subset_div_right (hs : 1 ∈ s) : t⁻¹ ⊆ s / t := by
+  rw [div_eq_mul_inv]; exact subset_mul_right _ hs
+
 end DivisionMonoid
 
 /-- `Set α` is a commutative division monoid under pointwise operations if `α` is. -/
@@ -828,55 +1200,7 @@ protected noncomputable def divisionCommMonoid [DivisionCommMonoid α] :
     DivisionCommMonoid (Set α) :=
   { Set.divisionMonoid, Set.commSemigroup with }
 
-/-- `Set α` has distributive negation if `α` has. -/
-protected noncomputable def hasDistribNeg [Mul α] [HasDistribNeg α] : HasDistribNeg (Set α) :=
-  { Set.involutiveNeg with
-    neg_mul := fun _ _ => by
-      simp_rw [← image_neg]
-      exact image2_image_left_comm neg_mul
-    mul_neg := fun _ _ => by
-      simp_rw [← image_neg]
-      exact image_image2_right_comm mul_neg }
-
-scoped[Pointwise]
-  attribute [instance] Set.divisionCommMonoid Set.subtractionCommMonoid Set.hasDistribNeg
-
-section Distrib
-
-variable [Distrib α] (s t u : Set α)
-
-/-!
-Note that `Set α` is not a `Distrib` because `s * t + s * u` has cross terms that `s * (t + u)`
-lacks.
--/
-
-
-theorem mul_add_subset : s * (t + u) ⊆ s * t + s * u :=
-  image2_distrib_subset_left mul_add
-
-theorem add_mul_subset : (s + t) * u ⊆ s * u + t * u :=
-  image2_distrib_subset_right add_mul
-
-end Distrib
-
-section MulZeroClass
-
-variable [MulZeroClass α] {s t : Set α}
-
-/-! Note that `Set` is not a `MulZeroClass` because `0 * ∅ ≠ 0`. -/
-
-
-theorem mul_zero_subset (s : Set α) : s * 0 ⊆ 0 := by simp [subset_def, mem_mul]
-
-theorem zero_mul_subset (s : Set α) : 0 * s ⊆ 0 := by simp [subset_def, mem_mul]
-
-theorem Nonempty.mul_zero (hs : s.Nonempty) : s * 0 = 0 :=
-  s.mul_zero_subset.antisymm <| by simpa [mem_mul] using hs
-
-theorem Nonempty.zero_mul (hs : s.Nonempty) : 0 * s = 0 :=
-  s.zero_mul_subset.antisymm <| by simpa [mem_mul] using hs
-
-end MulZeroClass
+scoped[Pointwise] attribute [instance] Set.divisionCommMonoid Set.subtractionCommMonoid
 
 section Group
 
@@ -908,7 +1232,7 @@ theorem isUnit_singleton (a : α) : IsUnit ({a} : Set α) :=
 
 @[to_additive (attr := simp)]
 theorem isUnit_iff_singleton : IsUnit s ↔ ∃ a, s = {a} := by
-  simp only [isUnit_iff, Group.isUnit, and_true_iff]
+  simp only [isUnit_iff, Group.isUnit, and_true]
 
 @[to_additive (attr := simp)]
 theorem image_mul_left : (a * ·) '' t = (a⁻¹ * ·) ⁻¹' t := by
@@ -949,31 +1273,15 @@ theorem preimage_mul_right_one' : (· * b⁻¹) ⁻¹' 1 = {b} := by simp
 @[to_additive (attr := simp)]
 theorem mul_univ (hs : s.Nonempty) : s * (univ : Set α) = univ :=
   let ⟨a, ha⟩ := hs
-  eq_univ_of_forall fun b => ⟨a, ha, a⁻¹ * b, trivial, mul_inv_cancel_left _ _⟩
+  eq_univ_of_forall fun b => ⟨a, ha, a⁻¹ * b, trivial, mul_inv_cancel_left ..⟩
 
 @[to_additive (attr := simp)]
 theorem univ_mul (ht : t.Nonempty) : (univ : Set α) * t = univ :=
   let ⟨a, ha⟩ := ht
-  eq_univ_of_forall fun b => ⟨b * a⁻¹, trivial, a, ha, inv_mul_cancel_right _ _⟩
+  eq_univ_of_forall fun b => ⟨b * a⁻¹, trivial, a, ha, inv_mul_cancel_right ..⟩
 
 end Group
 
-section GroupWithZero
-
-variable [GroupWithZero α] {s t : Set α}
-
-theorem div_zero_subset (s : Set α) : s / 0 ⊆ 0 := by simp [subset_def, mem_div]
-
-theorem zero_div_subset (s : Set α) : 0 / s ⊆ 0 := by simp [subset_def, mem_div]
-
-theorem Nonempty.div_zero (hs : s.Nonempty) : s / 0 = 0 :=
-  s.div_zero_subset.antisymm <| by simpa [mem_div] using hs
-
-theorem Nonempty.zero_div (hs : s.Nonempty) : 0 / s = 0 :=
-  s.zero_div_subset.antisymm <| by simpa [mem_div] using hs
-
-end GroupWithZero
-
 section Mul
 
 variable [Mul α] [Mul β] [FunLike F α β] [MulHomClass F α β] (m : F) {s t : Set α}
@@ -987,12 +1295,12 @@ lemma mul_subset_range {s t : Set β} (hs : s ⊆ range m) (ht : t ⊆ range m)
   rintro _ ⟨a, ha, b, hb, rfl⟩
   obtain ⟨a, rfl⟩ := hs ha
   obtain ⟨b, rfl⟩ := ht hb
-  exact ⟨a * b, map_mul _ _ _⟩
+  exact ⟨a * b, map_mul ..⟩
 
 @[to_additive]
 theorem preimage_mul_preimage_subset {s t : Set β} : m ⁻¹' s * m ⁻¹' t ⊆ m ⁻¹' (s * t) := by
   rintro _ ⟨_, _, _, _, rfl⟩
-  exact ⟨_, ‹_›, _, ‹_›, (map_mul m _ _).symm⟩
+  exact ⟨_, ‹_›, _, ‹_›, (map_mul m ..).symm⟩
 
 @[to_additive]
 lemma preimage_mul (hm : Injective m) {s t : Set β} (hs : s ⊆ range m) (ht : t ⊆ range m) :
@@ -1016,12 +1324,12 @@ lemma div_subset_range {s t : Set β} (hs : s ⊆ range m) (ht : t ⊆ range m)
   rintro _ ⟨a, ha, b, hb, rfl⟩
   obtain ⟨a, rfl⟩ := hs ha
   obtain ⟨b, rfl⟩ := ht hb
-  exact ⟨a / b, map_div _ _ _⟩
+  exact ⟨a / b, map_div ..⟩
 
 @[to_additive]
 theorem preimage_div_preimage_subset {s t : Set β} : m ⁻¹' s / m ⁻¹' t ⊆ m ⁻¹' (s / t) := by
   rintro _ ⟨_, _, _, _, rfl⟩
-  exact ⟨_, ‹_›, _, ‹_›, (map_div m _ _).symm⟩
+  exact ⟨_, ‹_›, _, ‹_›, (map_div m ..).symm⟩
 
 @[to_additive]
 lemma preimage_div (hm : Injective m) {s t : Set β} (hs : s ⊆ range m) (ht : t ⊆ range m) :
diff --git a/Mathlib/Data/Finset/Pointwise/Card.lean b/Mathlib/Algebra/Group/Pointwise/Set/Card.lean
similarity index 96%
rename from Mathlib/Data/Finset/Pointwise/Card.lean
rename to Mathlib/Algebra/Group/Pointwise/Set/Card.lean
index 5516539f11c50..341ed441df932 100644
--- a/Mathlib/Data/Finset/Pointwise/Card.lean
+++ b/Mathlib/Algebra/Group/Pointwise/Set/Card.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.SetTheory.Cardinal.Finite
 
 /-!
diff --git a/Mathlib/Algebra/Group/Prod.lean b/Mathlib/Algebra/Group/Prod.lean
index 50077030df988..d7cb9b0f80a16 100644
--- a/Mathlib/Algebra/Group/Prod.lean
+++ b/Mathlib/Algebra/Group/Prod.lean
@@ -34,7 +34,7 @@ assert_not_exists MonoidWithZero
 -- assert_not_exists AddMonoidWithOne
 assert_not_exists DenselyOrdered
 
-variable {A : Type*} {B : Type*} {G : Type*} {H : Type*} {M : Type*} {N : Type*} {P : Type*}
+variable {G : Type*} {H : Type*} {M : Type*} {N : Type*} {P : Type*}
 
 namespace Prod
 
@@ -218,12 +218,14 @@ instance [RightCancelSemigroup G] [RightCancelSemigroup H] : RightCancelSemigrou
 @[to_additive]
 instance [LeftCancelMonoid M] [LeftCancelMonoid N] : LeftCancelMonoid (M × N) :=
   { mul_one := by simp,
-    one_mul := by simp }
+    one_mul := by simp
+    mul_left_cancel := by simp }
 
 @[to_additive]
 instance [RightCancelMonoid M] [RightCancelMonoid N] : RightCancelMonoid (M × N) :=
   { mul_one := by simp,
-    one_mul := by simp }
+    one_mul := by simp
+    mul_right_cancel := by simp }
 
 @[to_additive]
 instance [CancelMonoid M] [CancelMonoid N] : CancelMonoid (M × N) :=
@@ -235,7 +237,7 @@ instance instCommMonoid [CommMonoid M] [CommMonoid N] : CommMonoid (M × N) :=
 
 @[to_additive]
 instance [CancelCommMonoid M] [CancelCommMonoid N] : CancelCommMonoid (M × N) :=
-  { mul_comm := fun ⟨m₁, n₁⟩ ⟨_, _⟩ => by rw [mk_mul_mk, mk_mul_mk, mul_comm m₁, mul_comm n₁] }
+  { mul_left_cancel := by simp }
 
 @[to_additive]
 instance instCommGroup [CommGroup G] [CommGroup H] : CommGroup (G × H) :=
diff --git a/Mathlib/Algebra/Group/Semiconj/Defs.lean b/Mathlib/Algebra/Group/Semiconj/Defs.lean
index 83e158027c671..cfeabeb0d12db 100644
--- a/Mathlib/Algebra/Group/Semiconj/Defs.lean
+++ b/Mathlib/Algebra/Group/Semiconj/Defs.lean
@@ -2,11 +2,11 @@
 Copyright (c) 2019 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
 -/
+-- Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
+
 import Mathlib.Algebra.Group.Defs
-import Mathlib.Init.Logic
+import Mathlib.Order.Defs
 
 /-!
 # Semiconjugate elements of a semigroup
@@ -115,7 +115,7 @@ end Monoid
 
 section Group
 
-variable [Group G] {a x y : G}
+variable [Group G]
 
 /-- `a` semiconjugates `x` to `a * x * a⁻¹`. -/
 @[to_additive "`a` semiconjugates `x` to `a + x + -a`."]
diff --git a/Mathlib/Algebra/Group/Semiconj/Units.lean b/Mathlib/Algebra/Group/Semiconj/Units.lean
index 11e0a651d129f..5a31a5e5130e9 100644
--- a/Mathlib/Algebra/Group/Semiconj/Units.lean
+++ b/Mathlib/Algebra/Group/Semiconj/Units.lean
@@ -2,9 +2,9 @@
 Copyright (c) 2019 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
-
-Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
 -/
+-- Some proofs and docs came from mathlib3 `src/algebra/commute.lean` (c) Neil Strickland
+
 import Mathlib.Algebra.Group.Semiconj.Defs
 import Mathlib.Algebra.Group.Units
 
@@ -32,7 +32,7 @@ assert_not_exists DenselyOrdered
 
 open scoped Int
 
-variable {M G : Type*}
+variable {M : Type*}
 
 namespace SemiconjBy
 
diff --git a/Mathlib/Algebra/Group/Subgroup/Basic.lean b/Mathlib/Algebra/Group/Subgroup/Basic.lean
index 01289fa8997c3..556df038ddca9 100644
--- a/Mathlib/Algebra/Group/Subgroup/Basic.lean
+++ b/Mathlib/Algebra/Group/Subgroup/Basic.lean
@@ -69,7 +69,7 @@ Definitions in the file:
 * `MonoidHom.ker f` : the kernel of a group homomorphism `f` is the subgroup of elements `x : G`
   such that `f x = 1`
 
-* `MonoidHom.eq_locus f g` : given group homomorphisms `f`, `g`, the elements of `G` such that
+* `MonoidHom.eqLocus f g` : given group homomorphisms `f`, `g`, the elements of `G` such that
   `f x = g x` form a subgroup of `G`
 
 ## Implementation notes
@@ -94,27 +94,27 @@ variable {A : Type*} [AddGroup A]
 section SubgroupClass
 
 /-- `InvMemClass S G` states `S` is a type of subsets `s ⊆ G` closed under inverses. -/
-class InvMemClass (S G : Type*) [Inv G] [SetLike S G] : Prop where
+class InvMemClass (S : Type*) (G : outParam Type*) [Inv G] [SetLike S G] : Prop where
   /-- `s` is closed under inverses -/
   inv_mem : ∀ {s : S} {x}, x ∈ s → x⁻¹ ∈ s
 
 export InvMemClass (inv_mem)
 
 /-- `NegMemClass S G` states `S` is a type of subsets `s ⊆ G` closed under negation. -/
-class NegMemClass (S G : Type*) [Neg G] [SetLike S G] : Prop where
+class NegMemClass (S : Type*) (G : outParam Type*) [Neg G] [SetLike S G] : Prop where
   /-- `s` is closed under negation -/
   neg_mem : ∀ {s : S} {x}, x ∈ s → -x ∈ s
 
 export NegMemClass (neg_mem)
 
 /-- `SubgroupClass S G` states `S` is a type of subsets `s ⊆ G` that are subgroups of `G`. -/
-class SubgroupClass (S G : Type*) [DivInvMonoid G] [SetLike S G] extends SubmonoidClass S G,
-  InvMemClass S G : Prop
+class SubgroupClass (S : Type*) (G : outParam Type*) [DivInvMonoid G] [SetLike S G]
+    extends SubmonoidClass S G, InvMemClass S G : Prop
 
 /-- `AddSubgroupClass S G` states `S` is a type of subsets `s ⊆ G` that are
 additive subgroups of `G`. -/
-class AddSubgroupClass (S G : Type*) [SubNegMonoid G] [SetLike S G] extends AddSubmonoidClass S G,
-  NegMemClass S G : Prop
+class AddSubgroupClass (S : Type*) (G : outParam Type*) [SubNegMonoid G] [SetLike S G]
+    extends AddSubmonoidClass S G, NegMemClass S G : Prop
 
 attribute [to_additive] InvMemClass SubgroupClass
 
@@ -583,8 +583,8 @@ theorem coe_pow (x : H) (n : ℕ) : ((x ^ n : H) : G) = (x : G) ^ n :=
 theorem coe_zpow (x : H) (n : ℤ) : ((x ^ n : H) : G) = (x : G) ^ n :=
   rfl
 
-@[to_additive] -- This can be proved by `Submonoid.mk_eq_one`
-theorem mk_eq_one {g : G} {h} : (⟨g, h⟩ : H) = 1 ↔ g = 1 := by simp
+@[to_additive (attr := simp)]
+theorem mk_eq_one {g : G} {h} : (⟨g, h⟩ : H) = 1 ↔ g = 1 := Submonoid.mk_eq_one ..
 
 /-- A subgroup of a group inherits a group structure. -/
 @[to_additive "An `AddSubgroup` of an `AddGroup` inherits an `AddGroup` structure."]
@@ -892,7 +892,7 @@ theorem closure_induction {p : G → Prop} {x} (h : x ∈ closure k) (mem : ∀
     (mul : ∀ x y, p x → p y → p (x * y)) (inv : ∀ x, p x → p x⁻¹) : p x :=
   (@closure_le _ _ ⟨⟨⟨setOf p, fun {x y} ↦ mul x y⟩, one⟩, fun {x} ↦ inv x⟩ k).2 mem h
 
-/-- A dependent version of `Subgroup.closure_induction`.  -/
+/-- A dependent version of `Subgroup.closure_induction`. -/
 @[to_additive (attr := elab_as_elim) "A dependent version of `AddSubgroup.closure_induction`. "]
 theorem closure_induction' {p : ∀ x, x ∈ closure k → Prop}
     (mem : ∀ (x) (h : x ∈ k), p x (subset_closure h)) (one : p 1 (one_mem _))
@@ -1020,6 +1020,10 @@ theorem mem_closure_singleton {x y : G} : y ∈ closure ({x} : Set G) ↔ ∃ n
 theorem closure_singleton_one : closure ({1} : Set G) = ⊥ := by
   simp [eq_bot_iff_forall, mem_closure_singleton]
 
+@[to_additive (attr := simp)]
+lemma mem_closure_singleton_self (x : G) : x ∈ closure ({x} : Set G) := by
+  simpa [-subset_closure] using subset_closure (k := {x})
+
 @[to_additive]
 theorem le_closure_toSubmonoid (S : Set G) : Submonoid.closure S ≤ (closure S).toSubmonoid :=
   Submonoid.closure_le.2 subset_closure
@@ -1076,7 +1080,7 @@ theorem toAddSubgroup_comap {G₂ : Type*} [Group G₂] (f : G →* G₂) (s : S
 
 @[simp]
 theorem _root_.AddSubgroup.toSubgroup_comap {A A₂ : Type*} [AddGroup A] [AddGroup A₂]
-    (f : A →+ A₂)  (s : AddSubgroup A₂) :
+    (f : A →+ A₂) (s : AddSubgroup A₂) :
     s.toSubgroup.comap (AddMonoidHom.toMultiplicative f) = AddSubgroup.toSubgroup (s.comap f) := rfl
 
 @[to_additive (attr := simp)]
@@ -1201,6 +1205,16 @@ theorem map_iSup {ι : Sort*} (f : G →* N) (s : ι → Subgroup G) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+@[to_additive]
+theorem map_inf (H K : Subgroup G) (f : G →* N) (hf : Function.Injective f) :
+    (H ⊓ K).map f = H.map f ⊓ K.map f := SetLike.coe_injective (Set.image_inter hf)
+
+@[to_additive]
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : G →* N) (hf : Function.Injective f)
+    (s : ι → Subgroup G) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 @[to_additive]
 theorem comap_sup_comap_le (H K : Subgroup N) (f : G →* N) :
     comap f H ⊔ comap f K ≤ comap f (H ⊔ K) :=
@@ -1709,11 +1723,11 @@ def _root_.NormalizerCondition :=
 variable {G}
 
 /-- Alternative phrasing of the normalizer condition: Only the full group is self-normalizing.
-This may be easier to work with, as it avoids inequalities and negations.  -/
+This may be easier to work with, as it avoids inequalities and negations. -/
 theorem _root_.normalizerCondition_iff_only_full_group_self_normalizing :
     NormalizerCondition G ↔ ∀ H : Subgroup G, H.normalizer = H → H = ⊤ := by
   apply forall_congr'; intro H
-  simp only [lt_iff_le_and_ne, le_normalizer, true_and_iff, le_top, Ne]
+  simp only [lt_iff_le_and_ne, le_normalizer, le_top, Ne]
   tauto
 
 variable (H)
@@ -1770,7 +1784,7 @@ instance subgroupOf_isCommutative [H.IsCommutative] : (H.subgroupOf K).IsCommuta
 @[to_additive]
 lemma mul_comm_of_mem_isCommutative [H.IsCommutative] {a b : G} (ha : a ∈ H) (hb : b ∈ H) :
     a * b = b * a := by
-  simpa only [Submonoid.mk_mul_mk, Subtype.mk.injEq] using mul_comm (⟨a, ha⟩ : H) (⟨b, hb⟩ : H)
+  simpa only [MulMemClass.mk_mul_mk, Subtype.mk.injEq] using mul_comm (⟨a, ha⟩ : H) (⟨b, hb⟩ : H)
 
 end Subgroup
 
@@ -2120,10 +2134,14 @@ def ker (f : G →* M) : Subgroup G :=
         f x⁻¹ = f x * f x⁻¹ := by rw [hx, one_mul]
         _ = 1 := by rw [← map_mul, mul_inv_cancel, map_one] }
 
-@[to_additive]
-theorem mem_ker (f : G →* M) {x : G} : x ∈ f.ker ↔ f x = 1 :=
+@[to_additive (attr := simp)]
+theorem mem_ker {f : G →* M} {x : G} : x ∈ f.ker ↔ f x = 1 :=
   Iff.rfl
 
+@[to_additive]
+theorem div_mem_ker_iff (f : G →* N) {x y : G} : x / y ∈ ker f ↔ f x = f y := by
+  rw [mem_ker, map_div, div_eq_one]
+
 @[to_additive]
 theorem coe_ker (f : G →* M) : (f.ker : Set G) = (f : G → M) ⁻¹' {1} :=
   rfl
@@ -2137,7 +2155,7 @@ theorem ker_toHomUnits {M} [Monoid M] (f : G →* M) : f.toHomUnits.ker = f.ker
 theorem eq_iff (f : G →* M) {x y : G} : f x = f y ↔ y⁻¹ * x ∈ f.ker := by
   constructor <;> intro h
   · rw [mem_ker, map_mul, h, ← map_mul, inv_mul_cancel, map_one]
-  · rw [← one_mul x, ← mul_inv_cancel y, mul_assoc, map_mul, f.mem_ker.1 h, mul_one]
+  · rw [← one_mul x, ← mul_inv_cancel y, mul_assoc, map_mul, mem_ker.1 h, mul_one]
 
 @[to_additive]
 instance decidableMemKer [DecidableEq M] (f : G →* M) : DecidablePred (· ∈ f.ker) := fun x =>
@@ -2208,13 +2226,13 @@ theorem range_le_ker_iff (f : G →* G') (g : G' →* G'') : f.range ≤ g.ker 
 @[to_additive]
 instance (priority := 100) normal_ker (f : G →* M) : f.ker.Normal :=
   ⟨fun x hx y => by
-    rw [mem_ker, map_mul, map_mul, f.mem_ker.1 hx, mul_one, map_mul_eq_one f (mul_inv_cancel y)]⟩
+    rw [mem_ker, map_mul, map_mul, mem_ker.1 hx, mul_one, map_mul_eq_one f (mul_inv_cancel y)]⟩
 
 @[to_additive (attr := simp)]
-lemma ker_fst : ker (fst G G') = .prod ⊥ ⊤ := SetLike.ext fun _ => (and_true_iff _).symm
+lemma ker_fst : ker (fst G G') = .prod ⊥ ⊤ := SetLike.ext fun _ => (iff_of_eq (and_true _)).symm
 
 @[to_additive (attr := simp)]
-lemma ker_snd : ker (snd G G') = .prod ⊤ ⊥ := SetLike.ext fun _ => (true_and_iff _).symm
+lemma ker_snd : ker (snd G G') = .prod ⊤ ⊥ := SetLike.ext fun _ => (iff_of_eq (true_and _)).symm
 
 @[simp]
 theorem coe_toAdditive_ker (f : G →* G') :
@@ -2376,7 +2394,7 @@ theorem map_le_map_iff {f : G →* N} {H K : Subgroup G} : H.map f ≤ K.map f 
 @[to_additive]
 theorem map_le_map_iff' {f : G →* N} {H K : Subgroup G} :
     H.map f ≤ K.map f ↔ H ⊔ f.ker ≤ K ⊔ f.ker := by
-  simp only [map_le_map_iff, sup_le_iff, le_sup_right, and_true_iff]
+  simp only [map_le_map_iff, sup_le_iff, le_sup_right, and_true]
 
 @[to_additive]
 theorem map_eq_map_iff {f : G →* N} {H K : Subgroup G} :
@@ -2581,7 +2599,7 @@ See `MonoidHom.eq_liftOfRightInverse` for the uniqueness lemma.
 def liftOfRightInverse (hf : Function.RightInverse f_inv f) :
     { g : G₁ →* G₃ // f.ker ≤ g.ker } ≃ (G₂ →* G₃) where
   toFun g := f.liftOfRightInverseAux f_inv hf g.1 g.2
-  invFun φ := ⟨φ.comp f, fun x hx => (mem_ker _).mpr <| by simp [(mem_ker _).mp hx]⟩
+  invFun φ := ⟨φ.comp f, fun x hx ↦ mem_ker.mpr <| by simp [mem_ker.mp hx]⟩
   left_inv g := by
     ext
     simp only [comp_apply, liftOfRightInverseAux_comp_apply, Subtype.coe_mk]
@@ -2912,3 +2930,5 @@ def noncenter (G : Type*) [Monoid G] : Set (ConjClasses G) :=
   g ∈ noncenter G ↔ g.carrier.Nontrivial := Iff.rfl
 
 end ConjClasses
+
+set_option linter.style.longFile 3000
diff --git a/Mathlib/Algebra/Group/Subgroup/Finite.lean b/Mathlib/Algebra/Group/Subgroup/Finite.lean
index 77f194a84dacd..6e5342a357f4e 100644
--- a/Mathlib/Algebra/Group/Subgroup/Finite.lean
+++ b/Mathlib/Algebra/Group/Subgroup/Finite.lean
@@ -165,9 +165,11 @@ theorem pi_mem_of_mulSingle_mem_aux [DecidableEq η] (I : Finset η) {H : Subgro
     x ∈ H := by
   induction I using Finset.induction_on generalizing x with
   | empty =>
-    convert one_mem H
-    ext i
-    exact h1 i (Finset.not_mem_empty i)
+    have : x = 1 := by
+      ext i
+      exact h1 i (Finset.not_mem_empty i)
+    rw [this]
+    exact one_mem H
   | insert hnmem ih =>
     rename_i i I
     have : x = Function.update x i 1 * Pi.mulSingle i (x i) := by
@@ -200,7 +202,7 @@ theorem pi_mem_of_mulSingle_mem [Finite η] [DecidableEq η] {H : Subgroup (∀
   cases nonempty_fintype η
   exact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i
 
-/-- For finite index types, the `Subgroup.pi` is generated by the embeddings of the groups.  -/
+/-- For finite index types, the `Subgroup.pi` is generated by the embeddings of the groups. -/
 @[to_additive "For finite index types, the `Subgroup.pi` is generated by the embeddings of the
  additive groups."]
 theorem pi_le_iff [DecidableEq η] [Finite η] {H : ∀ i, Subgroup (f i)} {J : Subgroup (∀ i, f i)} :
diff --git a/Mathlib/Algebra/Group/Subgroup/MulOpposite.lean b/Mathlib/Algebra/Group/Subgroup/MulOpposite.lean
index 5f6195c179e7b..296424c376749 100644
--- a/Mathlib/Algebra/Group/Subgroup/MulOpposite.lean
+++ b/Mathlib/Algebra/Group/Subgroup/MulOpposite.lean
@@ -28,6 +28,13 @@ protected def op (H : Subgroup G) : Subgroup Gᵐᵒᵖ where
   mul_mem' ha hb := H.mul_mem hb ha
   inv_mem' := H.inv_mem
 
+/- We redeclare this instance to get keys
+`SMul (@Subtype (MulOpposite _) (@Membership.mem (MulOpposite _)
+  (Subgroup (MulOpposite _) _) _ (@Subgroup.op _ _ _))) _`
+compared to the keys for `Submonoid.smul`
+`SMul (@Subtype _ (@Membership.mem _ (Submonoid _ _) _ _)) _` -/
+@[to_additive] instance instSMul (H : Subgroup G) : SMul H.op G := Submonoid.smul ..
+
 @[to_additive (attr := simp)]
 theorem mem_op {x : Gᵐᵒᵖ} {S : Subgroup G} : x ∈ S.op ↔ x.unop ∈ S := Iff.rfl
 
@@ -174,7 +181,7 @@ theorem op_closure (s : Set G) : (closure s).op = closure (MulOpposite.unop ⁻
 @[to_additive]
 theorem unop_closure (s : Set Gᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) := by
   rw [← op_inj, op_unop, op_closure]
-  rfl
+  simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
 
 /-- Bijection between a subgroup `H` and its opposite. -/
 @[to_additive (attr := simps!) "Bijection between an additive subgroup `H` and its opposite."]
diff --git a/Mathlib/Algebra/Group/Subgroup/Order.lean b/Mathlib/Algebra/Group/Subgroup/Order.lean
index 36a107b58fb58..0578e6adbde06 100644
--- a/Mathlib/Algebra/Group/Subgroup/Order.lean
+++ b/Mathlib/Algebra/Group/Subgroup/Order.lean
@@ -14,9 +14,10 @@ import Mathlib.Algebra.Order.Group.Unbundled.Abs
 
 open Subgroup
 
-@[simp] theorem abs_mem_iff {S G} [AddGroup G] [LinearOrder G] {_ : SetLike S G}
-    [NegMemClass S G] {H : S} {x : G} : |x| ∈ H ↔ x ∈ H := by
-  cases abs_choice x <;> simp [*]
+@[to_additive (attr := simp)]
+theorem mabs_mem_iff {S G} [Group G] [LinearOrder G] {_ : SetLike S G}
+    [InvMemClass S G] {H : S} {x : G} : |x|ₘ ∈ H ↔ x ∈ H := by
+  cases mabs_choice x <;> simp [*]
 
 section ModularLattice
 
@@ -110,3 +111,17 @@ instance toLinearOrderedCommGroup [LinearOrderedCommGroup G] (H : Subgroup G) :
     (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) fun _ _ => rfl
 
 end Subgroup
+
+@[to_additive]
+lemma Subsemigroup.strictMono_topEquiv {G : Type*} [OrderedCommMonoid G] :
+    StrictMono (topEquiv (M := G)) := fun _ _ ↦ id
+
+@[to_additive]
+lemma MulEquiv.strictMono_subsemigroupCongr {G : Type*} [OrderedCommMonoid G] {S T : Subsemigroup G}
+    (h : S = T) : StrictMono (subsemigroupCongr h) := fun _ _ ↦ id
+
+@[to_additive]
+lemma MulEquiv.strictMono_symm {G G' : Type*} [LinearOrderedCommMonoid G]
+    [LinearOrderedCommMonoid G'] {e : G ≃* G'} (he : StrictMono e) : StrictMono e.symm := by
+  intro
+  simp [← he.lt_iff_lt]
diff --git a/Mathlib/Algebra/Group/Subgroup/Pointwise.lean b/Mathlib/Algebra/Group/Subgroup/Pointwise.lean
index c340d58d2eccf..99bb82ddad9cc 100644
--- a/Mathlib/Algebra/Group/Subgroup/Pointwise.lean
+++ b/Mathlib/Algebra/Group/Subgroup/Pointwise.lean
@@ -72,7 +72,7 @@ theorem closure_toSubmonoid (S : Set G) :
         (fun x hx => Submonoid.closure_mono subset_union_left (Submonoid.subset_closure hx))
         (Submonoid.one_mem _) (fun x y hx hy => Submonoid.mul_mem _ hx hy) fun x hx => ?_
     rwa [← Submonoid.mem_closure_inv, Set.union_inv, inv_inv, Set.union_comm]
-  · simp only [true_and_iff, coe_toSubmonoid, union_subset_iff, subset_closure, inv_subset_closure]
+  · simp only [true_and, coe_toSubmonoid, union_subset_iff, subset_closure, inv_subset_closure]
 
 /-- For subgroups generated by a single element, see the simpler `zpow_induction_left`. -/
 @[to_additive (attr := elab_as_elim)
@@ -113,6 +113,10 @@ theorem closure_induction_right {p : (x : G) → x ∈ closure s → Prop} (one
 theorem closure_inv (s : Set G) : closure s⁻¹ = closure s := by
   simp only [← toSubmonoid_eq, closure_toSubmonoid, inv_inv, union_comm]
 
+@[to_additive (attr := simp)]
+lemma closure_singleton_inv (x : G) : closure {x⁻¹} = closure {x} := by
+  rw [← Set.inv_singleton, closure_inv]
+
 /-- An induction principle for closure membership. If `p` holds for `1` and all elements of
 `k` and their inverse, and is preserved under multiplication, then `p` holds for all elements of
 the closure of `k`. -/
@@ -388,6 +392,13 @@ theorem Normal.conjAct {G : Type*} [Group G] {H : Subgroup G} (hH : H.Normal) (g
 theorem smul_normal (g : G) (H : Subgroup G) [h : Normal H] : MulAut.conj g • H = H :=
   h.conjAct g
 
+theorem normalCore_eq_iInf_conjAct (H : Subgroup G) :
+    H.normalCore = ⨅ (g : ConjAct G), g • H := by
+  ext g
+  simp only [Subgroup.normalCore, Subgroup.mem_iInf, Subgroup.mem_pointwise_smul_iff_inv_smul_mem]
+  refine ⟨fun h x ↦ h x⁻¹, fun h x ↦ ?_⟩
+  simpa only [ConjAct.toConjAct_inv, inv_inv] using h x⁻¹
+
 end Group
 
 section GroupWithZero
diff --git a/Mathlib/Algebra/Group/Subgroup/ZPowers.lean b/Mathlib/Algebra/Group/Subgroup/ZPowers.lean
index b91454a9b1837..7d8c98bdefc66 100644
--- a/Mathlib/Algebra/Group/Subgroup/ZPowers.lean
+++ b/Mathlib/Algebra/Group/Subgroup/ZPowers.lean
@@ -227,3 +227,10 @@ theorem center_eq_infi' (S : Set G) (hS : closure S = ⊤) :
   rw [center_eq_iInf S hS, ← iInf_subtype'']
 
 end Subgroup
+
+lemma AddSubgroup.closure_singleton_int_one_eq_top : closure ({1} : Set ℤ) = ⊤ := by
+  ext
+  simp [mem_closure_singleton]
+
+lemma AddSubgroup.zmultiples_one_eq_top : zmultiples (1 : ℤ) = ⊤ := by
+  rw [zmultiples_eq_closure, closure_singleton_int_one_eq_top]
diff --git a/Mathlib/Algebra/Group/Submonoid/Basic.lean b/Mathlib/Algebra/Group/Submonoid/Basic.lean
index 35c066cced16e..623c641fbcfa6 100644
--- a/Mathlib/Algebra/Group/Submonoid/Basic.lean
+++ b/Mathlib/Algebra/Group/Submonoid/Basic.lean
@@ -65,14 +65,14 @@ variable [MulOneClass M] {s : Set M}
 variable [AddZeroClass A] {t : Set A}
 
 /-- `OneMemClass S M` says `S` is a type of subsets `s ≤ M`, such that `1 ∈ s` for all `s`. -/
-class OneMemClass (S : Type*) (M : Type*) [One M] [SetLike S M] : Prop where
+class OneMemClass (S : Type*) (M : outParam Type*) [One M] [SetLike S M] : Prop where
   /-- By definition, if we have `OneMemClass S M`, we have `1 ∈ s` for all `s : S`. -/
   one_mem : ∀ s : S, (1 : M) ∈ s
 
 export OneMemClass (one_mem)
 
 /-- `ZeroMemClass S M` says `S` is a type of subsets `s ≤ M`, such that `0 ∈ s` for all `s`. -/
-class ZeroMemClass (S : Type*) (M : Type*) [Zero M] [SetLike S M] : Prop where
+class ZeroMemClass (S : Type*) (M : outParam Type*) [Zero M] [SetLike S M] : Prop where
   /-- By definition, if we have `ZeroMemClass S M`, we have `0 ∈ s` for all `s : S`. -/
   zero_mem : ∀ s : S, (0 : M) ∈ s
 
@@ -96,7 +96,7 @@ add_decl_doc Submonoid.toSubsemigroup
 
 /-- `SubmonoidClass S M` says `S` is a type of subsets `s ≤ M` that contain `1`
 and are closed under `(*)` -/
-class SubmonoidClass (S : Type*) (M : Type*) [MulOneClass M] [SetLike S M] extends
+class SubmonoidClass (S : Type*) (M : outParam Type*) [MulOneClass M] [SetLike S M] extends
   MulMemClass S M, OneMemClass S M : Prop
 
 section
@@ -115,7 +115,7 @@ add_decl_doc AddSubmonoid.toAddSubsemigroup
 
 /-- `AddSubmonoidClass S M` says `S` is a type of subsets `s ≤ M` that contain `0`
 and are closed under `(+)` -/
-class AddSubmonoidClass (S : Type*) (M : Type*) [AddZeroClass M] [SetLike S M] extends
+class AddSubmonoidClass (S : Type*) (M : outParam Type*) [AddZeroClass M] [SetLike S M] extends
   AddMemClass S M, ZeroMemClass S M : Prop
 
 attribute [to_additive] Submonoid SubmonoidClass
@@ -376,7 +376,7 @@ theorem closure_induction {p : M → Prop} {x} (h : x ∈ closure s) (mem : ∀
     (mul : ∀ x y, p x → p y → p (x * y)) : p x :=
   (@closure_le _ _ _ ⟨⟨p, mul _ _⟩, one⟩).2 mem h
 
-/-- A dependent version of `Submonoid.closure_induction`.  -/
+/-- A dependent version of `Submonoid.closure_induction`. -/
 @[to_additive (attr := elab_as_elim) "A dependent version of `AddSubmonoid.closure_induction`. "]
 theorem closure_induction' (s : Set M) {p : ∀ x, x ∈ closure s → Prop}
     (mem : ∀ (x) (h : x ∈ s), p x (subset_closure h)) (one : p 1 (one_mem _))
@@ -387,7 +387,7 @@ theorem closure_induction' (s : Set M) {p : ∀ x, x ∈ closure s → Prop}
     closure_induction hx (fun x hx => ⟨_, mem x hx⟩) ⟨_, one⟩ fun x y ⟨hx', hx⟩ ⟨hy', hy⟩ =>
       ⟨_, mul _ _ _ _ hx hy⟩
 
-/-- An induction principle for closure membership for predicates with two arguments.  -/
+/-- An induction principle for closure membership for predicates with two arguments. -/
 @[to_additive (attr := elab_as_elim)
       "An induction principle for additive closure membership for predicates with two arguments."]
 theorem closure_induction₂ {p : M → M → Prop} {x} {y : M} (hx : x ∈ closure s) (hy : y ∈ closure s)
diff --git a/Mathlib/Algebra/Group/Submonoid/Membership.lean b/Mathlib/Algebra/Group/Submonoid/Membership.lean
index 603662f188ac8..e1ec1fb022a46 100644
--- a/Mathlib/Algebra/Group/Submonoid/Membership.lean
+++ b/Mathlib/Algebra/Group/Submonoid/Membership.lean
@@ -293,7 +293,7 @@ variable {S : Submonoid M} [Fintype S]
 open Fintype
 
 /- curly brackets `{}` are used here instead of instance brackets `[]` because
-  the instance in a goal is often not the same as the one inferred by type class inference.  -/
+  the instance in a goal is often not the same as the one inferred by type class inference. -/
 @[to_additive]
 theorem card_bot {_ : Fintype (⊥ : Submonoid M)} : card (⊥ : Submonoid M) = 1 :=
   card_eq_one_iff.2
diff --git a/Mathlib/Algebra/Group/Submonoid/MulOpposite.lean b/Mathlib/Algebra/Group/Submonoid/MulOpposite.lean
index ce69dc1d1759d..9157b526b564c 100644
--- a/Mathlib/Algebra/Group/Submonoid/MulOpposite.lean
+++ b/Mathlib/Algebra/Group/Submonoid/MulOpposite.lean
@@ -164,7 +164,7 @@ theorem op_closure (s : Set M) : (closure s).op = closure (MulOpposite.unop ⁻
 @[to_additive]
 theorem unop_closure (s : Set Mᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) := by
   rw [← op_inj, op_unop, op_closure]
-  rfl
+  simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
 
 /-- Bijection between a submonoid `H` and its opposite. -/
 @[to_additive (attr := simps!) "Bijection between an additive submonoid `H` and its opposite."]
diff --git a/Mathlib/Algebra/Group/Submonoid/Operations.lean b/Mathlib/Algebra/Group/Submonoid/Operations.lean
index a56159be66a0d..f84160423a5b5 100644
--- a/Mathlib/Algebra/Group/Submonoid/Operations.lean
+++ b/Mathlib/Algebra/Group/Submonoid/Operations.lean
@@ -274,6 +274,16 @@ theorem map_sup (S T : Submonoid M) (f : F) : (S ⊔ T).map f = S.map f ⊔ T.ma
 theorem map_iSup {ι : Sort*} (f : F) (s : ι → Submonoid M) : (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f : GaloisConnection (map f) (comap f)).l_iSup
 
+@[to_additive]
+theorem map_inf (S T : Submonoid M) (f : F) (hf : Function.Injective f) :
+    (S ⊓ T).map f = S.map f ⊓ T.map f := SetLike.coe_injective (Set.image_inter hf)
+
+@[to_additive]
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
+    (s : ι → Submonoid M) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 @[to_additive]
 theorem comap_inf (S T : Submonoid N) (f : F) : (S ⊓ T).comap f = S.comap f ⊓ T.comap f :=
   (gc_map_comap f : GaloisConnection (map f) (comap f)).u_inf
@@ -579,8 +589,9 @@ theorem coe_equivMapOfInjective_apply (f : M →* N) (hf : Function.Injective f)
 theorem closure_closure_coe_preimage {s : Set M} : closure (((↑) : closure s → M) ⁻¹' s) = ⊤ :=
   eq_top_iff.2 fun x =>
     Subtype.recOn x fun x hx _ => by
-      refine closure_induction' (p := fun y hy ↦ ⟨y, hy⟩ ∈ closure (((↑) : closure s → M) ⁻¹' s))
-        (fun g hg => subset_closure hg) ?_ (fun g₁ g₂ hg₁ hg₂ => ?_) hx
+      refine closure_induction'
+        (p := fun y hy ↦ (⟨y, hy⟩ : closure s) ∈ closure (((↑) : closure s → M) ⁻¹' s))
+          _ (fun g hg => subset_closure hg) ?_ (fun g₁ g₂ hg₁ hg₂ => ?_) hx
       · exact Submonoid.one_mem _
       · exact Submonoid.mul_mem _
 
@@ -827,6 +838,11 @@ def codRestrict {S} [SetLike S N] [SubmonoidClass S N] (f : M →* N) (s : S) (h
   map_one' := Subtype.eq f.map_one
   map_mul' x y := Subtype.eq (f.map_mul x y)
 
+@[to_additive (attr := simp)]
+lemma injective_codRestrict {S} [SetLike S N] [SubmonoidClass S N] (f : M →* N) (s : S)
+    (h : ∀ x, f x ∈ s) : Function.Injective (f.codRestrict s h) ↔ Function.Injective f :=
+  ⟨fun H _ _ hxy ↦ H <| Subtype.eq hxy, fun H _ _ hxy ↦ H (congr_arg Subtype.val hxy)⟩
+
 /-- Restriction of a monoid hom to its range interpreted as a submonoid. -/
 @[to_additive "Restriction of an `AddMonoid` hom to its range interpreted as a submonoid."]
 def mrangeRestrict {N} [MulOneClass N] (f : M →* N) : M →* (mrange f) :=
@@ -849,8 +865,8 @@ that `f x = 1` -/
 def mker (f : F) : Submonoid M :=
   (⊥ : Submonoid N).comap f
 
-@[to_additive]
-theorem mem_mker (f : F) {x : M} : x ∈ mker f ↔ f x = 1 :=
+@[to_additive (attr := simp)]
+theorem mem_mker {f : F} {x : M} : x ∈ mker f ↔ f x = 1 :=
   Iff.rfl
 
 @[to_additive]
@@ -859,7 +875,7 @@ theorem coe_mker (f : F) : (mker f : Set M) = (f : M → N) ⁻¹' {1} :=
 
 @[to_additive]
 instance decidableMemMker [DecidableEq N] (f : F) : DecidablePred (· ∈ mker f) := fun x =>
-  decidable_of_iff (f x = 1) (mem_mker f)
+  decidable_of_iff (f x = 1) mem_mker
 
 @[to_additive]
 theorem comap_mker (g : N →* P) (f : M →* N) : g.mker.comap f = mker (comp g f) :=
@@ -908,10 +924,10 @@ theorem mker_inr : mker (inr M N) = ⊥ := by
   simp [mem_mker]
 
 @[to_additive (attr := simp)]
-lemma mker_fst : mker (fst M N) = .prod ⊥ ⊤ := SetLike.ext fun _ => (and_true_iff _).symm
+lemma mker_fst : mker (fst M N) = .prod ⊥ ⊤ := SetLike.ext fun _ => (iff_of_eq (and_true _)).symm
 
 @[to_additive (attr := simp)]
-lemma mker_snd : mker (snd M N) = .prod ⊤ ⊥ := SetLike.ext fun _ => (true_and_iff _).symm
+lemma mker_snd : mker (snd M N) = .prod ⊤ ⊥ := SetLike.ext fun _ => (iff_of_eq (true_and _)).symm
 
 /-- The `MonoidHom` from the preimage of a submonoid to itself. -/
 @[to_additive (attr := simps)
diff --git a/Mathlib/Algebra/Group/Submonoid/Units.lean b/Mathlib/Algebra/Group/Submonoid/Units.lean
index 2754d556e4588..96ce6d155d1e4 100644
--- a/Mathlib/Algebra/Group/Submonoid/Units.lean
+++ b/Mathlib/Algebra/Group/Submonoid/Units.lean
@@ -40,13 +40,13 @@ variable {M : Type*} [Monoid M]
 open Units
 
 open Pointwise in
-/-- The units of `S`, packaged as a subgroup of `Mˣ`.  -/
+/-- The units of `S`, packaged as a subgroup of `Mˣ`. -/
 @[to_additive " The additive units of `S`, packaged as an additive subgroup of `AddUnits M`. "]
 def Submonoid.units (S : Submonoid M) : Subgroup Mˣ where
   toSubmonoid := S.comap (coeHom M) ⊓ (S.comap (coeHom M))⁻¹
   inv_mem' ha := ⟨ha.2, ha.1⟩
 
-/-- A subgroup of units represented as a submonoid of `M`.  -/
+/-- A subgroup of units represented as a submonoid of `M`. -/
 @[to_additive " A additive subgroup of additive units represented as a additive submonoid of `M`. "]
 def Subgroup.ofUnits (S : Subgroup Mˣ) : Submonoid M := S.toSubmonoid.map (coeHom M)
 
@@ -327,6 +327,6 @@ lemma val_mem_ofUnits_iff_mem (H : Subgroup Gˣ) (x : Gˣ) : (x : G) ∈ H.ofUni
 @[to_additive " The equivalence between the greatest subgroup of additive units
 contained within `T` and `T` itself. "]
 def unitsEquivSelf (H : Subgroup G) : H.units ≃* H :=
-  H.unitsEquivUnitsType.trans toUnits.symm
+  H.unitsEquivUnitsType.trans (toUnits (G := H)).symm
 
 end Subgroup
diff --git a/Mathlib/Algebra/Group/Subsemigroup/Basic.lean b/Mathlib/Algebra/Group/Subsemigroup/Basic.lean
index eb038acc2f5ba..c5801a38f6bdb 100644
--- a/Mathlib/Algebra/Group/Subsemigroup/Basic.lean
+++ b/Mathlib/Algebra/Group/Subsemigroup/Basic.lean
@@ -57,14 +57,14 @@ variable [Mul M] {s : Set M}
 variable [Add A] {t : Set A}
 
 /-- `MulMemClass S M` says `S` is a type of sets `s : Set M` that are closed under `(*)` -/
-class MulMemClass (S : Type*) (M : Type*) [Mul M] [SetLike S M] : Prop where
+class MulMemClass (S : Type*) (M : outParam Type*) [Mul M] [SetLike S M] : Prop where
   /-- A substructure satisfying `MulMemClass` is closed under multiplication. -/
   mul_mem : ∀ {s : S} {a b : M}, a ∈ s → b ∈ s → a * b ∈ s
 
 export MulMemClass (mul_mem)
 
 /-- `AddMemClass S M` says `S` is a type of sets `s : Set M` that are closed under `(+)` -/
-class AddMemClass (S : Type*) (M : Type*) [Add M] [SetLike S M] : Prop where
+class AddMemClass (S : Type*) (M : outParam Type*) [Add M] [SetLike S M] : Prop where
   /-- A substructure satisfying `AddMemClass` is closed under addition. -/
   add_mem : ∀ {s : S} {a b : M}, a ∈ s → b ∈ s → a + b ∈ s
 
@@ -111,7 +111,7 @@ theorem mem_mk {s : Set M} {x : M} (h_mul) : x ∈ mk s h_mul ↔ x ∈ s :=
   Iff.rfl
 
 @[to_additive (attr := simp, norm_cast)]
-theorem coe_set_mk {s : Set M} (h_mul) : (mk s h_mul : Set M) = s :=
+theorem coe_set_mk (s : Set M) (h_mul) : (mk s h_mul : Set M) = s :=
   rfl
 
 @[to_additive (attr := simp)]
@@ -298,7 +298,7 @@ theorem closure_induction {p : M → Prop} {x} (h : x ∈ closure s) (mem : ∀
     (mul : ∀ x y, p x → p y → p (x * y)) : p x :=
   (@closure_le _ _ _ ⟨p, mul _ _⟩).2 mem h
 
-/-- A dependent version of `Subsemigroup.closure_induction`.  -/
+/-- A dependent version of `Subsemigroup.closure_induction`. -/
 @[to_additive (attr := elab_as_elim) "A dependent version of `AddSubsemigroup.closure_induction`. "]
 theorem closure_induction' (s : Set M) {p : ∀ x, x ∈ closure s → Prop}
     (mem : ∀ (x) (h : x ∈ s), p x (subset_closure h))
@@ -309,7 +309,7 @@ theorem closure_induction' (s : Set M) {p : ∀ x, x ∈ closure s → Prop}
     closure_induction hx (fun x hx => ⟨_, mem x hx⟩) fun x y ⟨hx', hx⟩ ⟨hy', hy⟩ =>
       ⟨_, mul _ _ _ _ hx hy⟩
 
-/-- An induction principle for closure membership for predicates with two arguments.  -/
+/-- An induction principle for closure membership for predicates with two arguments. -/
 @[to_additive (attr := elab_as_elim) "An induction principle for additive closure membership for
   predicates with two arguments."]
 theorem closure_induction₂ {p : M → M → Prop} {x} {y : M} (hx : x ∈ closure s) (hy : y ∈ closure s)
diff --git a/Mathlib/Algebra/Group/Subsemigroup/Operations.lean b/Mathlib/Algebra/Group/Subsemigroup/Operations.lean
index 306960fc30fff..85f523619d7ca 100644
--- a/Mathlib/Algebra/Group/Subsemigroup/Operations.lean
+++ b/Mathlib/Algebra/Group/Subsemigroup/Operations.lean
@@ -278,6 +278,16 @@ theorem map_iSup {ι : Sort*} (f : M →ₙ* N) (s : ι → Subsemigroup M) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+@[to_additive]
+theorem map_inf (S T : Subsemigroup M) (f : M →ₙ* N) (hf : Function.Injective f) :
+    (S ⊓ T).map f = S.map f ⊓ T.map f := SetLike.coe_injective (Set.image_inter hf)
+
+@[to_additive]
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : M →ₙ* N) (hf : Function.Injective f)
+    (s : ι → Subsemigroup M) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 @[to_additive]
 theorem comap_inf (S T : Subsemigroup N) (f : M →ₙ* N) : (S ⊓ T).comap f = S.comap f ⊓ T.comap f :=
   (gc_map_comap f).u_inf
@@ -489,9 +499,9 @@ theorem coe_equivMapOfInjective_apply (f : M →ₙ* N) (hf : Function.Injective
 theorem closure_closure_coe_preimage {s : Set M} :
     closure ((Subtype.val : closure s → M) ⁻¹' s) = ⊤ :=
   eq_top_iff.2 fun x =>
-    Subtype.recOn x fun _ hx _ =>
-      closure_induction' (p := fun y hy ↦ ⟨y, hy⟩ ∈ closure (((↑) : closure s → M) ⁻¹' s))
-        (fun _ hg => subset_closure hg) (fun _ _ _ _ => Subsemigroup.mul_mem _) hx
+    Subtype.recOn x fun _ hx' _ => closure_induction'
+      (p := fun y hy ↦ (⟨y, hy⟩ : closure s) ∈ closure (((↑) : closure s → M) ⁻¹' s))
+        _ (fun _ hg => subset_closure hg) (fun _ _ _ _ => Subsemigroup.mul_mem _) hx'
 
 /-- Given `Subsemigroup`s `s`, `t` of semigroups `M`, `N` respectively, `s × t` as a subsemigroup
 of `M × N`. -/
@@ -595,6 +605,15 @@ theorem coe_srange (f : M →ₙ* N) : (f.srange : Set N) = Set.range f :=
 theorem mem_srange {f : M →ₙ* N} {y : N} : y ∈ f.srange ↔ ∃ x, f x = y :=
   Iff.rfl
 
+@[to_additive]
+private theorem srange_mk_aux_mul {f : M → N} (hf : ∀ (x y : M), f (x * y) = f x * f y)
+    {x y : N} (hx : x ∈ Set.range f) (hy : y ∈ Set.range f) :
+    x * y ∈ Set.range f :=
+  (srange ⟨f, hf⟩).mul_mem hx hy
+
+@[to_additive (attr := simp)] theorem srange_mk (f : M → N) (hf) :
+    srange ⟨f, hf⟩ = ⟨Set.range f, srange_mk_aux_mul hf⟩ := rfl
+
 @[to_additive]
 theorem srange_eq_map (f : M →ₙ* N) : f.srange = (⊤ : Subsemigroup M).map f :=
   copy_eq _
diff --git a/Mathlib/Algebra/Group/ULift.lean b/Mathlib/Algebra/Group/ULift.lean
index 423ee6578ba6b..ce331c7da8288 100644
--- a/Mathlib/Algebra/Group/ULift.lean
+++ b/Mathlib/Algebra/Group/ULift.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Equiv.Basic
 import Mathlib.Algebra.Group.InjSurj
diff --git a/Mathlib/Algebra/Group/UniqueProds.lean b/Mathlib/Algebra/Group/UniqueProds/Basic.lean
similarity index 98%
rename from Mathlib/Algebra/Group/UniqueProds.lean
rename to Mathlib/Algebra/Group/UniqueProds/Basic.lean
index 82f5f510f979c..99d252eee8364 100644
--- a/Mathlib/Algebra/Group/UniqueProds.lean
+++ b/Mathlib/Algebra/Group/UniqueProds/Basic.lean
@@ -3,9 +3,10 @@ Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Data.DFinsupp.Basic
-import Mathlib.Data.Finset.Pointwise.Basic
-import Mathlib.LinearAlgebra.Basis.VectorSpace
+import Mathlib.Algebra.Group.ULift
+import Mathlib.Data.Finsupp.Defs
 
 /-!
 # Unique products and related notions
@@ -27,7 +28,7 @@ Here you can see several examples of Types that have `UniqueSums/Prods`
 ```lean
 import Mathlib.Data.Real.Basic
 import Mathlib.Data.PNat.Basic
-import Mathlib.Algebra.Group.UniqueProds
+import Mathlib.Algebra.Group.UniqueProds.Basic
 
 example : UniqueSums ℕ   := inferInstance
 example : UniqueSums ℕ+  := inferInstance
@@ -45,6 +46,12 @@ about the grading type and then a generic statement of the form "look at the coe
 The file `Algebra/MonoidAlgebra/NoZeroDivisors` contains several examples of this use.
 -/
 
+assert_not_exists Cardinal
+assert_not_exists Subsemiring
+assert_not_exists Algebra
+assert_not_exists Submodule
+assert_not_exists StarModule
+
 /-- Let `G` be a Type with multiplication, let `A B : Finset G` be finite subsets and
 let `a0 b0 : G` be two elements.  `UniqueMul A B a0 b0` asserts `a0 * b0` can be written in at
 most one way as a product of an element of `A` and an element of `B`. -/
@@ -63,13 +70,16 @@ variable {G H : Type*} [Mul G] [Mul H] {A B : Finset G} {a0 b0 : G}
 theorem of_subsingleton [Subsingleton G] : UniqueMul A B a0 b0 := by
   simp [UniqueMul, eq_iff_true_of_subsingleton]
 
-@[to_additive]
+@[to_additive of_card_le_one]
 theorem of_card_le_one (hA : A.Nonempty) (hB : B.Nonempty) (hA1 : A.card ≤ 1) (hB1 : B.card ≤ 1) :
     ∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b := by
   rw [Finset.card_le_one_iff] at hA1 hB1
   obtain ⟨a, ha⟩ := hA; obtain ⟨b, hb⟩ := hB
   exact ⟨a, ha, b, hb, fun _ _ ha' hb' _ ↦ ⟨hA1 ha' ha, hB1 hb' hb⟩⟩
 
+@[deprecated (since := "2024-09-23")]
+alias _root_.UniqueAdd.of_card_nonpos := UniqueAdd.of_card_le_one
+
 @[to_additive]
 theorem mt (h : UniqueMul A B a0 b0) :
     ∀ ⦃a b⦄, a ∈ A → b ∈ B → a ≠ a0 ∨ b ≠ b0 → a * b ≠ a0 * b0 := fun _ _ ha hb k ↦ by
@@ -106,7 +116,7 @@ theorem iff_existsUnique (aA : a0 ∈ A) (bB : b0 ∈ B) :
         exact Prod.mk.inj_iff.mp (J (x, y) ⟨Finset.mk_mem_product hx hy, l⟩))⟩
 
 open Finset in
-@[to_additive]
+@[to_additive iff_card_le_one]
 theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
     UniqueMul A B a0 b0 ↔ ((A ×ˢ B).filter (fun p ↦ p.1 * p.2 = a0 * b0)).card ≤ 1 := by
   simp_rw [card_le_one_iff, mem_filter, mem_product]
@@ -117,6 +127,9 @@ theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
     · rw [h1.2, h2.2]
   · exact Prod.ext_iff.1 (@h (a, b) (a0, b0) ⟨⟨ha, hb⟩, he⟩ ⟨⟨ha0, hb0⟩, rfl⟩)
 
+@[deprecated (since := "2024-09-23")]
+alias _root_.UniqueAdd.iff_card_nonpos := UniqueAdd.iff_card_le_one
+
 -- Porting note: mathport warning: expanding binder collection
 --  (ab «expr ∈ » [finset.product/multiset.product/set.prod/list.product](A, B)) -/
 @[to_additive]
@@ -635,12 +648,6 @@ instance {ι G} [AddZeroClass G] [TwoUniqueSums G] : TwoUniqueSums (ι →₀ G)
   TwoUniqueSums.of_injective_addHom
     Finsupp.coeFnAddHom.toAddHom DFunLike.coe_injective inferInstance
 
-/-- Any `ℚ`-vector space has `TwoUniqueSums`, because it is isomorphic to some
-  `(Basis.ofVectorSpaceIndex ℚ G) →₀ ℚ` by choosing a basis, and `ℚ` already has
-  `TwoUniqueSums` because it's ordered. -/
-instance [AddCommGroup G] [Module ℚ G] : TwoUniqueSums G :=
-  TwoUniqueSums.of_injective_addHom _ (Basis.ofVectorSpace ℚ G).repr.injective inferInstance
-
 /-- Any `FreeMonoid` has the `TwoUniqueProds` property. -/
 instance FreeMonoid.instTwoUniqueProds {κ : Type*} : TwoUniqueProds (FreeMonoid κ) :=
   .of_mulHom ⟨Multiplicative.ofAdd ∘ List.length, fun _ _ ↦ congr_arg _ (List.length_append _ _)⟩
diff --git a/Mathlib/Algebra/Group/UniqueProds/VectorSpace.lean b/Mathlib/Algebra/Group/UniqueProds/VectorSpace.lean
new file mode 100644
index 0000000000000..3b1ba113d4f00
--- /dev/null
+++ b/Mathlib/Algebra/Group/UniqueProds/VectorSpace.lean
@@ -0,0 +1,19 @@
+/-
+Copyright (c) 2022 Damiano Testa. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Damiano Testa
+-/
+import Mathlib.Algebra.Group.UniqueProds.Basic
+import Mathlib.LinearAlgebra.Basis.VectorSpace
+
+/-!
+# A `ℚ`-vector space has `TwoUniqueSums`.
+-/
+
+variable {G : Type*}
+
+/-- Any `ℚ`-vector space has `TwoUniqueSums`, because it is isomorphic to some
+  `(Basis.ofVectorSpaceIndex ℚ G) →₀ ℚ` by choosing a basis, and `ℚ` already has
+  `TwoUniqueSums` because it's ordered. -/
+instance [AddCommGroup G] [Module ℚ G] : TwoUniqueSums G :=
+  TwoUniqueSums.of_injective_addHom _ (Basis.ofVectorSpace ℚ G).repr.injective inferInstance
diff --git a/Mathlib/Algebra/Group/Units.lean b/Mathlib/Algebra/Group/Units.lean
index a42617a760e41..1c93daf4e3123 100644
--- a/Mathlib/Algebra/Group/Units.lean
+++ b/Mathlib/Algebra/Group/Units.lean
@@ -379,7 +379,7 @@ theorem Units.val_mkOfMulEqOne [CommMonoid α] {a b : α} (h : a * b = 1) :
 
 section Monoid
 
-variable [Monoid α] {a b c : α}
+variable [Monoid α] {a : α}
 
 /-- Partial division. It is defined when the
   second argument is invertible, and unlike the division operator
@@ -651,9 +651,9 @@ lemma IsUnit.exists_left_inv {a : M} (h : IsUnit a) : ∃ b, b * a = 1 := by
 @[to_additive] lemma IsUnit.pow (n : ℕ) : IsUnit a → IsUnit (a ^ n) := by
   rintro ⟨u, rfl⟩; exact ⟨u ^ n, rfl⟩
 
-theorem units_eq_one [Unique Mˣ] (u : Mˣ) : u = 1 := by subsingleton
+theorem units_eq_one [Subsingleton Mˣ] (u : Mˣ) : u = 1 := by subsingleton
 
-@[to_additive] lemma isUnit_iff_eq_one [Unique Mˣ] {x : M} : IsUnit x ↔ x = 1 :=
+@[to_additive] lemma isUnit_iff_eq_one [Subsingleton Mˣ] {x : M} : IsUnit x ↔ x = 1 :=
   ⟨fun ⟨u, hu⟩ ↦ by rw [← hu, Subsingleton.elim u 1, Units.val_one], fun h ↦ h ▸ isUnit_one⟩
 
 end Monoid
diff --git a/Mathlib/Algebra/Group/Units/Equiv.lean b/Mathlib/Algebra/Group/Units/Equiv.lean
index a3ae5f7904381..12d6fb5b2285c 100644
--- a/Mathlib/Algebra/Group/Units/Equiv.lean
+++ b/Mathlib/Algebra/Group/Units/Equiv.lean
@@ -13,7 +13,7 @@ import Mathlib.Algebra.Group.Units.Hom
 assert_not_exists MonoidWithZero
 assert_not_exists DenselyOrdered
 
-variable {F α β A B M N P Q G H : Type*}
+variable {F α M N G : Type*}
 
 /-- A group is isomorphic to its group of units. -/
 @[to_additive (attr := simps apply_val symm_apply)
@@ -31,7 +31,7 @@ lemma toUnits_val_apply {G : Type*} [Group G] (x : Gˣ) : toUnits (x : G) = x :=
 
 namespace Units
 
-variable [Monoid M] [Monoid N] [Monoid P]
+variable [Monoid M] [Monoid N]
 
 /-- A multiplicative equivalence of monoids defines a multiplicative equivalence
 of their groups of units. -/
@@ -192,11 +192,10 @@ def MulEquiv.inv (G : Type*) [DivisionCommMonoid G] : G ≃* G :=
 theorem MulEquiv.inv_symm (G : Type*) [DivisionCommMonoid G] :
     (MulEquiv.inv G).symm = MulEquiv.inv G :=
   rfl
--- Porting note: no `add_equiv.neg_symm` in `mathlib3`
 
-@[to_additive]
-protected
-theorem MulEquiv.map_isUnit_iff {M N} [Monoid M] [Monoid N] [EquivLike F M N] [MonoidHomClass F M N]
-    (f : F) {m : M} : IsUnit (f m) ↔ IsUnit m :=
-  isUnit_map_of_leftInverse (MonoidHom.inverse (f : M →* N) (EquivLike.inv f)
-    (EquivLike.left_inv f) <| EquivLike.right_inv f) (EquivLike.left_inv f)
+instance isLocalRingHom_equiv [Monoid M] [Monoid N] [EquivLike F M N]
+    [MulEquivClass F M N] (f : F) : IsLocalRingHom f where
+  map_nonunit a ha := by
+    convert ha.map (f : M ≃* N).symm
+    rw [MulEquiv.eq_symm_apply]
+    rfl -- note to reviewers: ugly `rfl`
diff --git a/Mathlib/Algebra/Group/Units/Hom.lean b/Mathlib/Algebra/Group/Units/Hom.lean
index 1a84f0441934c..09f626ae87904 100644
--- a/Mathlib/Algebra/Group/Units/Hom.lean
+++ b/Mathlib/Algebra/Group/Units/Hom.lean
@@ -17,6 +17,18 @@ also contains unrelated results about `Units` that depend on `MonoidHom`.
 * `Units.map`: Turn a homomorphism from `α` to `β` monoids into a homomorphism from `αˣ` to `βˣ`.
 * `MonoidHom.toHomUnits`: Turn a homomorphism from a group `α` to `β` into a homomorphism from
   `α` to `βˣ`.
+* `IsLocalRingHom`: A predicate on monoid maps, requiring that it maps nonunits
+  to nonunits. For local rings, this means that the image of the unique maximal ideal is again
+  contained in the unique maximal ideal. This is developed earlier, and in the generality of
+  monoids, as it allows its use in non-local-ring related contexts, but it does have the
+  strange consequence that it does not require local rings, or even rings.
+
+## TODO
+
+The results that don't mention homomorphisms should be proved (earlier?) in a different file and be
+used to golf the basic `Group` lemmas.
+
+Add a `@[to_additive]` version of `IsLocalRingHom`.
 -/
 
 assert_not_exists MonoidWithZero
@@ -152,7 +164,7 @@ end MonoidHom
 
 namespace IsUnit
 
-variable {F G α M N : Type*} [FunLike F M N] [FunLike G N M]
+variable {F G M N : Type*} [FunLike F M N] [FunLike G N M]
 
 section Monoid
 
@@ -167,6 +179,7 @@ theorem of_leftInverse [MonoidHomClass G N M] {f : F} {x : M} (g : G)
     (hfg : Function.LeftInverse g f) (h : IsUnit (f x)) : IsUnit x := by
   simpa only [hfg x] using h.map g
 
+/-- Prefer `IsLocalRingHom.of_leftInverse`, but we can't get rid of this because of `ToAdditive`. -/
 @[to_additive]
 theorem _root_.isUnit_map_of_leftInverse [MonoidHomClass F M N] [MonoidHomClass G N M]
     {f : F} {x : M} (g : G) (hfg : Function.LeftInverse g f) :
@@ -194,3 +207,49 @@ theorem liftRight_inv_mul (f : M →* N) (h : ∀ x, IsUnit (f x)) (x) :
 
 end Monoid
 end IsUnit
+
+section IsLocalRingHom
+
+variable {G R S T F : Type*}
+
+variable [Monoid R] [Monoid S] [Monoid T] [FunLike F R S]
+
+/-- A local ring homomorphism is a map `f` between monoids such that `a` in the domain
+  is a unit if `f a` is a unit for any `a`. See `LocalRing.local_hom_TFAE` for other equivalent
+  definitions in the local ring case - from where this concept originates, but it is useful in
+  other contexts, so we allow this generalisation in mathlib. -/
+class IsLocalRingHom (f : F) : Prop where
+  /-- A local ring homomorphism `f : R ⟶ S` will send nonunits of `R` to nonunits of `S`. -/
+  map_nonunit : ∀ a, IsUnit (f a) → IsUnit a
+
+@[simp]
+theorem IsUnit.of_map (f : F) [IsLocalRingHom f] (a : R) (h : IsUnit (f a)) : IsUnit a :=
+  IsLocalRingHom.map_nonunit a h
+
+-- TODO : remove alias, change the parenthesis of `f` and `a`
+alias isUnit_of_map_unit := IsUnit.of_map
+
+variable [MonoidHomClass F R S]
+
+@[simp]
+theorem isUnit_map_iff (f : F) [IsLocalRingHom f] (a : R) : IsUnit (f a) ↔ IsUnit a :=
+  ⟨IsLocalRingHom.map_nonunit a, IsUnit.map f⟩
+
+theorem isLocalRingHom_of_leftInverse [FunLike G S R] [MonoidHomClass G S R]
+    {f : F} (g : G) (hfg : Function.LeftInverse g f) : IsLocalRingHom f where
+  map_nonunit a ha := by rwa [isUnit_map_of_leftInverse g hfg] at ha
+
+instance MonoidHom.isLocalRingHom_comp (g : S →* T) (f : R →* S) [IsLocalRingHom g]
+    [IsLocalRingHom f] : IsLocalRingHom (g.comp f) where
+  map_nonunit a := IsLocalRingHom.map_nonunit a ∘ IsLocalRingHom.map_nonunit (f := g) (f a)
+
+-- see note [lower instance priority]
+instance (priority := 100) isLocalRingHom_toMonoidHom (f : F) [IsLocalRingHom f] :
+    IsLocalRingHom (f : R →* S) :=
+  ⟨IsLocalRingHom.map_nonunit (f := f)⟩
+
+theorem MonoidHom.isLocalRingHom_of_comp (f : R →* S) (g : S →* T) [IsLocalRingHom (g.comp f)] :
+    IsLocalRingHom f :=
+  ⟨fun _ ha => (isUnit_map_iff (g.comp f) _).mp (ha.map g)⟩
+
+end IsLocalRingHom
diff --git a/Mathlib/Algebra/Group/WithOne/Defs.lean b/Mathlib/Algebra/Group/WithOne/Defs.lean
index 353b3a6e27864..39ed722267c5d 100644
--- a/Mathlib/Algebra/Group/WithOne/Defs.lean
+++ b/Mathlib/Algebra/Group/WithOne/Defs.lean
@@ -43,7 +43,7 @@ assert_not_exists DenselyOrdered
 
 universe u v w
 
-variable {α : Type u} {β : Type v} {γ : Type w}
+variable {α : Type u}
 
 /-- Add an extra element `1` to a type -/
 @[to_additive "Add an extra element `0` to a type"]
@@ -112,6 +112,16 @@ def recOneCoe {C : WithOne α → Sort*} (h₁ : C 1) (h₂ : ∀ a : α, C a) :
   | Option.none => h₁
   | Option.some x => h₂ x
 
+@[to_additive (attr := simp)]
+lemma recOneCoe_one {C : WithOne α → Sort*} (h₁ h₂) :
+    recOneCoe h₁ h₂ (1 : WithOne α) = (h₁ : C 1) :=
+  rfl
+
+@[to_additive (attr := simp)]
+lemma recOneCoe_coe {C : WithOne α → Sort*} (h₁ h₂) (a : α) :
+    recOneCoe h₁ h₂ (a : WithOne α) = (h₂ : ∀ a : α, C a) a :=
+  rfl
+
 /-- Deconstruct an `x : WithOne α` to the underlying value in `α`, given a proof that `x ≠ 1`. -/
 @[to_additive unzero
       "Deconstruct an `x : WithZero α` to the underlying value in `α`, given a proof that `x ≠ 0`."]
diff --git a/Mathlib/Algebra/Group/ZeroOne.lean b/Mathlib/Algebra/Group/ZeroOne.lean
index 3e72968095621..8822f97d1d17e 100644
--- a/Mathlib/Algebra/Group/ZeroOne.lean
+++ b/Mathlib/Algebra/Group/ZeroOne.lean
@@ -6,17 +6,10 @@ Authors: Gabriel Ebner, Mario Carneiro
 import Mathlib.Tactic.ToAdditive
 
 /-!
-## Classes for `Zero` and `One`
--/
-
-class Zero.{u} (α : Type u) where
-  zero : α
+## Typeclass `One`
 
-instance (priority := 300) Zero.toOfNat0 {α} [Zero α] : OfNat α (nat_lit 0) where
-  ofNat := ‹Zero α›.1
-
-instance (priority := 200) Zero.ofOfNat0 {α} [OfNat α (nat_lit 0)] : Zero α where
-  zero := 0
+`Zero` has already been defined in Lean.
+-/
 
 universe u
 
diff --git a/Mathlib/Algebra/GroupWithZero/Action/Defs.lean b/Mathlib/Algebra/GroupWithZero/Action/Defs.lean
index 28a7f1306d857..d30f29f25d6f4 100644
--- a/Mathlib/Algebra/GroupWithZero/Action/Defs.lean
+++ b/Mathlib/Algebra/GroupWithZero/Action/Defs.lean
@@ -49,7 +49,7 @@ assert_not_exists Ring
 
 open Function
 
-variable {R R' M M' N G A B α β : Type*}
+variable {M N A B α β : Type*}
 
 /-- `Monoid.toMulAction` is faithful on nontrivial cancellative monoids with zero. -/
 instance CancelMonoidWithZero.faithfulSMul [CancelMonoidWithZero α] [Nontrivial α] :
diff --git a/Mathlib/Algebra/GroupWithZero/Action/Opposite.lean b/Mathlib/Algebra/GroupWithZero/Action/Opposite.lean
index e78af0c456d38..27bab44a1aaed 100644
--- a/Mathlib/Algebra/GroupWithZero/Action/Opposite.lean
+++ b/Mathlib/Algebra/GroupWithZero/Action/Opposite.lean
@@ -27,7 +27,7 @@ With `open scoped RightActions`, this provides:
 * `p <+ᵥ v` as an alias for `AddOpposite.op v +ᵥ p`
 -/
 
-variable {R M N α : Type*}
+variable {M α : Type*}
 
 /-! ### Actions _on_ the opposite type
 
diff --git a/Mathlib/Algebra/GroupWithZero/Action/Prod.lean b/Mathlib/Algebra/GroupWithZero/Action/Prod.lean
index d523dffe4f80d..b825703247594 100644
--- a/Mathlib/Algebra/GroupWithZero/Action/Prod.lean
+++ b/Mathlib/Algebra/GroupWithZero/Action/Prod.lean
@@ -26,7 +26,7 @@ namespace Prod
 
 section
 
-variable [SMul M α] [SMul M β] [SMul N α] [SMul N β] (a : M) (x : α × β)
+variable [SMul M α] [SMul M β]
 
 theorem smul_zero_mk {α : Type*} [Monoid M] [AddMonoid α] [DistribMulAction M α] (a : M) (c : β) :
     a • ((0 : α), c) = (0, a • c) := by rw [Prod.smul_mk, smul_zero]
@@ -89,6 +89,6 @@ def DistribMulAction.prodEquiv : DistribMulAction (M × N) α ≃
     congr 1
     · funext i; congr; ext m a; clear i; (conv_rhs => rw [← one_smul N a]); rfl
     · ext n a; (conv_rhs => rw [← one_smul M (SMul.smul n a)]); rfl
-    · apply heq_prop
+    · exact proof_irrel_heq ..
 
 end Action_by_Prod
diff --git a/Mathlib/Algebra/GroupWithZero/Basic.lean b/Mathlib/Algebra/GroupWithZero/Basic.lean
index 78525d46a0706..b6f9678b599f4 100644
--- a/Mathlib/Algebra/GroupWithZero/Basic.lean
+++ b/Mathlib/Algebra/GroupWithZero/Basic.lean
@@ -37,7 +37,7 @@ assert_not_exists DenselyOrdered
 
 open Function
 
-variable {α M₀ G₀ M₀' G₀' F F' : Type*}
+variable {M₀ G₀ : Type*}
 
 section
 
@@ -136,7 +136,7 @@ theorem right_ne_zero_of_mul_eq_one (h : a * b = 1) : b ≠ 0 :=
 end
 
 section MonoidWithZero
-variable [MonoidWithZero M₀] {a : M₀} {m n : ℕ}
+variable [MonoidWithZero M₀] {a : M₀} {n : ℕ}
 
 @[simp] lemma zero_pow : ∀ {n : ℕ}, n ≠ 0 → (0 : M₀) ^ n = 0
   | n + 1, _ => by rw [pow_succ, mul_zero]
@@ -234,7 +234,7 @@ end CancelMonoidWithZero
 
 section GroupWithZero
 
-variable [GroupWithZero G₀] {a b c g h x : G₀}
+variable [GroupWithZero G₀] {a b x : G₀}
 
 theorem GroupWithZero.mul_left_injective (h : x ≠ 0) :
     Function.Injective fun y => x * y := fun y y' w => by
@@ -291,7 +291,7 @@ end GroupWithZero
 
 section GroupWithZero
 
-variable [GroupWithZero G₀] {a b c : G₀}
+variable [GroupWithZero G₀] {a : G₀}
 
 @[simp]
 theorem zero_div (a : G₀) : 0 / a = 0 := by rw [div_eq_mul_inv, zero_mul]
@@ -411,8 +411,7 @@ lemma zpow_add' {m n : ℤ} (h : a ≠ 0 ∨ m + n ≠ 0 ∨ m = 0 ∧ n = 0) :
   · simp [hn]
   by_cases ha : a = 0
   · subst a
-    simp only [false_or_iff, eq_self_iff_true, not_true, Ne, hm, hn, false_and_iff,
-      or_false_iff] at h
+    simp only [false_or, eq_self_iff_true, not_true, Ne, hm, hn, false_and, or_false] at h
     rw [zero_zpow _ h, zero_zpow _ hm, zero_mul]
   · exact zpow_add₀ ha m n
 
@@ -422,7 +421,7 @@ end GroupWithZero
 
 section CommGroupWithZero
 
-variable [CommGroupWithZero G₀] {a b c d : G₀}
+variable [CommGroupWithZero G₀]
 
 theorem div_mul_eq_mul_div₀ (a b c : G₀) : a / c * b = a * b / c := by
   simp_rw [div_eq_mul_inv, mul_assoc, mul_comm c⁻¹]
diff --git a/Mathlib/Algebra/GroupWithZero/Commute.lean b/Mathlib/Algebra/GroupWithZero/Commute.lean
index 59423cb09016f..24a2a0afe674b 100644
--- a/Mathlib/Algebra/GroupWithZero/Commute.lean
+++ b/Mathlib/Algebra/GroupWithZero/Commute.lean
@@ -14,7 +14,7 @@ import Mathlib.Tactic.Nontriviality
 
 assert_not_exists DenselyOrdered
 
-variable {α M₀ G₀ M₀' G₀' F F' : Type*}
+variable {M₀ G₀ : Type*}
 variable [MonoidWithZero M₀]
 
 namespace Ring
@@ -83,7 +83,7 @@ theorem div_left (hac : Commute a c) (hbc : Commute b c) : Commute (a / b) c :=
 end Commute
 
 section GroupWithZero
-variable {G₀ : Type*} [GroupWithZero G₀] {a : G₀} {m n : ℕ}
+variable {G₀ : Type*} [GroupWithZero G₀]
 
 theorem pow_inv_comm₀ (a : G₀) (m n : ℕ) : a⁻¹ ^ m * a ^ n = a ^ n * a⁻¹ ^ m :=
   (Commute.refl a).inv_left₀.pow_pow m n
diff --git a/Mathlib/Algebra/GroupWithZero/Defs.lean b/Mathlib/Algebra/GroupWithZero/Defs.lean
index b3d73d7ba0a35..14ace092e132d 100644
--- a/Mathlib/Algebra/GroupWithZero/Defs.lean
+++ b/Mathlib/Algebra/GroupWithZero/Defs.lean
@@ -26,7 +26,7 @@ universe u
 
 -- We have to fix the universe of `G₀` here, since the default argument to
 -- `GroupWithZero.div'` cannot contain a universe metavariable.
-variable {G₀ : Type u} {M₀ M₀' G₀' : Type*}
+variable {G₀ : Type u} {M₀ : Type*}
 
 /-- Typeclass for expressing that a type `M₀` with multiplication and a zero satisfies
 `0 * a = 0` and `a * 0 = 0` for all `a : M₀`. -/
@@ -159,17 +159,17 @@ class MulDivCancelClass (M₀ : Type*) [MonoidWithZero M₀] [Div M₀] : Prop w
   protected mul_div_cancel (a b : M₀) : b ≠ 0 → a * b / b = a
 
 section MulDivCancelClass
-variable [MonoidWithZero M₀] [Div M₀] [MulDivCancelClass M₀] {a b : M₀}
+variable [MonoidWithZero M₀] [Div M₀] [MulDivCancelClass M₀]
 
-@[simp] lemma mul_div_cancel_right₀ (a : M₀) (hb : b ≠ 0) : a * b / b = a :=
+@[simp] lemma mul_div_cancel_right₀ (a : M₀) {b : M₀} (hb : b ≠ 0) : a * b / b = a :=
   MulDivCancelClass.mul_div_cancel _ _ hb
 
 end MulDivCancelClass
 
 section MulDivCancelClass
-variable [CommMonoidWithZero M₀] [Div M₀] [MulDivCancelClass M₀] {a b : M₀}
+variable [CommMonoidWithZero M₀] [Div M₀] [MulDivCancelClass M₀]
 
-@[simp] lemma mul_div_cancel_left₀ (b : M₀) (ha : a ≠ 0) : a * b / a = b := by
+@[simp] lemma mul_div_cancel_left₀ (b : M₀) {a : M₀} (ha : a ≠ 0) : a * b / a = b := by
   rw [mul_comm, mul_div_cancel_right₀ _ ha]
 
 end MulDivCancelClass
@@ -216,7 +216,7 @@ end
 
 section GroupWithZero
 
-variable [GroupWithZero G₀] {a b c g h x : G₀}
+variable [GroupWithZero G₀] {a b : G₀}
 
 @[simp]
 theorem mul_inv_cancel_right₀ (h : b ≠ 0) (a : G₀) : a * b * b⁻¹ = a :=
diff --git a/Mathlib/Algebra/GroupWithZero/Indicator.lean b/Mathlib/Algebra/GroupWithZero/Indicator.lean
index 2b8dcaded4211..7ad962522c252 100644
--- a/Mathlib/Algebra/GroupWithZero/Indicator.lean
+++ b/Mathlib/Algebra/GroupWithZero/Indicator.lean
@@ -40,6 +40,12 @@ lemma indicator_mul_right (s : Set ι) (f g : ι → M₀) :
   · rfl
   · rw [mul_zero]
 
+lemma indicator_mul_const (s : Set ι) (f : ι → M₀) (a : M₀) (i : ι) :
+    s.indicator (f · * a) i = s.indicator f i * a := by rw [indicator_mul_left]
+
+lemma indicator_const_mul (s : Set ι) (f : ι → M₀) (a : M₀) (i : ι) :
+    s.indicator (a * f ·) i = a * s.indicator f i := by rw [indicator_mul_right]
+
 lemma inter_indicator_mul (f g : ι → M₀) (i : ι) :
     (s ∩ t).indicator (fun j ↦ f j * g j) i = s.indicator f i * t.indicator g i := by
   rw [← Set.indicator_indicator]
diff --git a/Mathlib/Algebra/GroupWithZero/InjSurj.lean b/Mathlib/Algebra/GroupWithZero/InjSurj.lean
index d7a7ff367118a..5372c67cb3f27 100644
--- a/Mathlib/Algebra/GroupWithZero/InjSurj.lean
+++ b/Mathlib/Algebra/GroupWithZero/InjSurj.lean
@@ -19,7 +19,7 @@ variable {M₀ G₀ M₀' G₀' : Type*}
 
 section MulZeroClass
 
-variable [MulZeroClass M₀] {a b : M₀}
+variable [MulZeroClass M₀]
 
 /-- Pull back a `MulZeroClass` instance along an injective function.
 See note [reducible non-instances]. -/
@@ -149,7 +149,7 @@ end MonoidWithZero
 
 section CancelMonoidWithZero
 
-variable [CancelMonoidWithZero M₀] {a b c : M₀}
+variable [CancelMonoidWithZero M₀]
 
 /-- Pull back a `CancelMonoidWithZero` along an injective function.
 See note [reducible non-instances]. -/
@@ -167,7 +167,7 @@ end CancelMonoidWithZero
 
 section CancelCommMonoidWithZero
 
-variable [CancelCommMonoidWithZero M₀] {a b c : M₀}
+variable [CancelCommMonoidWithZero M₀]
 
 /-- Pull back a `CancelCommMonoidWithZero` along an injective function.
 See note [reducible non-instances]. -/
@@ -181,7 +181,7 @@ end CancelCommMonoidWithZero
 
 section GroupWithZero
 
-variable [GroupWithZero G₀] {a b c g h x : G₀}
+variable [GroupWithZero G₀]
 
 /-- Pull back a `GroupWithZero` along an injective function.
 See note [reducible non-instances]. -/
@@ -215,7 +215,7 @@ end GroupWithZero
 
 section CommGroupWithZero
 
-variable [CommGroupWithZero G₀] {a b c d : G₀}
+variable [CommGroupWithZero G₀]
 
 /-- Pull back a `CommGroupWithZero` along an injective function.
 See note [reducible non-instances]. -/
diff --git a/Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean b/Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean
index 3c09328f414ca..2c4e1832cb5a3 100644
--- a/Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean
+++ b/Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean
@@ -304,7 +304,7 @@ theorem mk_mem_nonZeroDivisors_associates : Associates.mk a ∈ (Associates M₀
 /-- The non-zero divisors of associates of a monoid with zero `M₀` are isomorphic to the associates
 of the non-zero divisors of `M₀` under the map `⟨⟦a⟧, _⟩ ↦ ⟦⟨a, _⟩⟧`. -/
 def associatesNonZeroDivisorsEquiv : (Associates M₀)⁰ ≃* Associates M₀⁰ where
-  toEquiv := .subtypeQuotientEquivQuotientSubtype (s₂ := Associated.setoid _)
+  toEquiv := .subtypeQuotientEquivQuotientSubtype _ (s₂ := Associated.setoid _)
     (· ∈ nonZeroDivisors _)
     (by simp [mem_nonZeroDivisors_iff, Quotient.forall, Associates.mk_mul_mk])
     (by simp [Associated.setoid])
diff --git a/Mathlib/Algebra/GroupWithZero/Pi.lean b/Mathlib/Algebra/GroupWithZero/Pi.lean
index 547f6a5013092..f11c9daf124b3 100644
--- a/Mathlib/Algebra/GroupWithZero/Pi.lean
+++ b/Mathlib/Algebra/GroupWithZero/Pi.lean
@@ -22,7 +22,7 @@ variable {ι : Type*} {α : ι → Type*}
 namespace Pi
 
 section MulZeroClass
-variable [∀ i, MulZeroClass (α i)] [DecidableEq ι] {i j : ι} {f : ∀ i, α i}
+variable [∀ i, MulZeroClass (α i)] [DecidableEq ι] {i : ι} {f : ∀ i, α i}
 
 instance mulZeroClass : MulZeroClass (∀ i, α i) where
   zero_mul := by intros; ext; exact zero_mul _
diff --git a/Mathlib/Algebra/GroupWithZero/Pointwise/Set/Basic.lean b/Mathlib/Algebra/GroupWithZero/Pointwise/Set/Basic.lean
new file mode 100644
index 0000000000000..ec178273360b2
--- /dev/null
+++ b/Mathlib/Algebra/GroupWithZero/Pointwise/Set/Basic.lean
@@ -0,0 +1,59 @@
+/-
+Copyright (c) 2019 Johan Commelin. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johan Commelin, Floris van Doorn
+-/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.GroupWithZero.Basic
+
+/-!
+# Pointwise operations of sets in a group with zero
+
+This file proves properties of pointwise operations of sets in a group with zero.
+
+## Tags
+
+set multiplication, set addition, pointwise addition, pointwise multiplication,
+pointwise subtraction
+-/
+
+assert_not_exists OrderedAddCommMonoid
+assert_not_exists Ring
+
+open Function
+open scoped Pointwise
+
+variable {F α β γ : Type*}
+
+namespace Set
+
+section MulZeroClass
+variable [MulZeroClass α] {s t : Set α}
+
+/-! Note that `Set` is not a `MulZeroClass` because `0 * ∅ ≠ 0`. -/
+
+lemma mul_zero_subset (s : Set α) : s * 0 ⊆ 0 := by simp [subset_def, mem_mul]
+lemma zero_mul_subset (s : Set α) : 0 * s ⊆ 0 := by simp [subset_def, mem_mul]
+
+lemma Nonempty.mul_zero (hs : s.Nonempty) : s * 0 = 0 :=
+  s.mul_zero_subset.antisymm <| by simpa [mem_mul] using hs
+
+lemma Nonempty.zero_mul (hs : s.Nonempty) : 0 * s = 0 :=
+  s.zero_mul_subset.antisymm <| by simpa [mem_mul] using hs
+
+end MulZeroClass
+
+section GroupWithZero
+variable [GroupWithZero α] {s t : Set α}
+
+lemma div_zero_subset (s : Set α) : s / 0 ⊆ 0 := by simp [subset_def, mem_div]
+lemma zero_div_subset (s : Set α) : 0 / s ⊆ 0 := by simp [subset_def, mem_div]
+
+lemma Nonempty.div_zero (hs : s.Nonempty) : s / 0 = 0 :=
+  s.div_zero_subset.antisymm <| by simpa [mem_div] using hs
+
+lemma Nonempty.zero_div (hs : s.Nonempty) : 0 / s = 0 :=
+  s.zero_div_subset.antisymm <| by simpa [mem_div] using hs
+
+end GroupWithZero
+end Set
diff --git a/Mathlib/Algebra/GroupWithZero/Semiconj.lean b/Mathlib/Algebra/GroupWithZero/Semiconj.lean
index f3798456188e4..0dfc56212073d 100644
--- a/Mathlib/Algebra/GroupWithZero/Semiconj.lean
+++ b/Mathlib/Algebra/GroupWithZero/Semiconj.lean
@@ -13,7 +13,7 @@ import Mathlib.Algebra.Group.Semiconj.Units
 
 assert_not_exists DenselyOrdered
 
-variable {α M₀ G₀ M₀' G₀' F F' : Type*}
+variable {G₀ : Type*}
 
 namespace SemiconjBy
 
diff --git a/Mathlib/Algebra/GroupWithZero/ULift.lean b/Mathlib/Algebra/GroupWithZero/ULift.lean
index 2e370b30faa0a..f178a5b637230 100644
--- a/Mathlib/Algebra/GroupWithZero/ULift.lean
+++ b/Mathlib/Algebra/GroupWithZero/ULift.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.ULift
 import Mathlib.Algebra.GroupWithZero.InjSurj
diff --git a/Mathlib/Algebra/GroupWithZero/Units/Basic.lean b/Mathlib/Algebra/GroupWithZero/Units/Basic.lean
index a32ad81fb22e8..4daea455977cf 100644
--- a/Mathlib/Algebra/GroupWithZero/Units/Basic.lean
+++ b/Mathlib/Algebra/GroupWithZero/Units/Basic.lean
@@ -23,7 +23,7 @@ We also define `Ring.inverse`, a globally defined function on any ring
 assert_not_exists Multiplicative
 assert_not_exists DenselyOrdered
 
-variable {α M₀ G₀ M₀' G₀' F F' : Type*}
+variable {α M₀ G₀ : Type*}
 variable [MonoidWithZero M₀]
 
 namespace Units
@@ -87,6 +87,12 @@ noncomputable def inverse : M₀ → M₀ := fun x => if h : IsUnit x then ((h.u
 theorem inverse_unit (u : M₀ˣ) : inverse (u : M₀) = (u⁻¹ : M₀ˣ) := by
   rw [inverse, dif_pos u.isUnit, IsUnit.unit_of_val_units]
 
+theorem IsUnit.ringInverse {x : M₀} (h : IsUnit x) : IsUnit (inverse x) :=
+  match h with
+  | ⟨u, hu⟩ => hu ▸ ⟨u⁻¹, (inverse_unit u).symm⟩
+
+theorem inverse_of_isUnit {x : M₀} (h : IsUnit x) : inverse x = ((h.unit⁻¹ : M₀ˣ) : M₀) := dif_pos h
+
 /-- By definition, if `x` is not invertible then `inverse x = 0`. -/
 @[simp]
 theorem inverse_non_unit (x : M₀) (h : ¬IsUnit x) : inverse x = 0 :=
@@ -152,7 +158,6 @@ theorem isUnit_ring_inverse {a : M₀} : IsUnit (Ring.inverse a) ↔ IsUnit a :=
 namespace Units
 
 variable [GroupWithZero G₀]
-variable {a b : G₀}
 
 /-- Embed a non-zero element of a `GroupWithZero` into the unit group.
   By combining this function with the operations on units,
@@ -208,7 +213,7 @@ theorem _root_.GroupWithZero.eq_zero_or_unit (a : G₀) : a = 0 ∨ ∃ u : G₀
 end Units
 
 section GroupWithZero
-variable [GroupWithZero G₀] {a b c d : G₀} {m n : ℕ}
+variable [GroupWithZero G₀] {a b c : G₀} {m n : ℕ}
 
 theorem IsUnit.mk0 (x : G₀) (hx : x ≠ 0) : IsUnit x :=
   (Units.mk0 x hx).isUnit
diff --git a/Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean b/Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
index c413ab0a165b8..407a0fb1b6c75 100644
--- a/Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
+++ b/Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
@@ -15,10 +15,34 @@ import Mathlib.Algebra.GroupWithZero.Hom
 
 assert_not_exists DenselyOrdered
 
-variable {α M₀ G₀ M₀' G₀' F F' : Type*}
+variable {M M₀ G₀ M₀' G₀' F F' : Type*}
 variable [MonoidWithZero M₀]
 
+section Monoid
+
+variable [Monoid M] [GroupWithZero G₀]
+
+lemma isLocalRingHom_of_exists_map_ne_one [FunLike F G₀ M] [MonoidHomClass F G₀ M] {f : F}
+    (hf : ∃ x : G₀, f x ≠ 1) : IsLocalRingHom f where
+  map_nonunit a h := by
+    rcases eq_or_ne a 0 with (rfl | h)
+    · obtain ⟨t, ht⟩ := hf
+      refine (ht ?_).elim
+      have := map_mul f t 0
+      rw [← one_mul (f (t * 0)), mul_zero] at this
+      exact (h.mul_right_cancel this).symm
+    · exact ⟨⟨a, a⁻¹, mul_inv_cancel₀ h, inv_mul_cancel₀ h⟩, rfl⟩
+
+instance [GroupWithZero G₀] [FunLike F G₀ M₀] [MonoidWithZeroHomClass F G₀ M₀] [Nontrivial M₀]
+    (f : F) : IsLocalRingHom f :=
+  isLocalRingHom_of_exists_map_ne_one ⟨0, by simp⟩
+
+end Monoid
+
+section GroupWithZero
+
 namespace Commute
+
 variable [GroupWithZero G₀] {a b c d : G₀}
 
 /-- The `MonoidWithZero` version of `div_eq_div_iff_mul_eq_mul`. -/
@@ -94,7 +118,6 @@ def invMonoidWithZeroHom {G₀ : Type*} [CommGroupWithZero G₀] : G₀ →*₀
 namespace Units
 
 variable [GroupWithZero G₀]
-variable {a b : G₀}
 
 @[simp]
 theorem smul_mk0 {α : Type*} [SMul G₀ α] {g : G₀} (hg : g ≠ 0) (a : α) : mk0 g hg • a = g • a :=
@@ -108,3 +131,5 @@ end Units
 theorem map_zpow₀ {F G₀ G₀' : Type*} [GroupWithZero G₀] [GroupWithZero G₀'] [FunLike F G₀ G₀']
     [MonoidWithZeroHomClass F G₀ G₀'] (f : F) (x : G₀) (n : ℤ) : f (x ^ n) = f x ^ n :=
   map_zpow' f (map_inv₀ f) x n
+
+end GroupWithZero
diff --git a/Mathlib/Algebra/GroupWithZero/WithZero.lean b/Mathlib/Algebra/GroupWithZero/WithZero.lean
index 1a8e376d48a1d..9790623ef1ff9 100644
--- a/Mathlib/Algebra/GroupWithZero/WithZero.lean
+++ b/Mathlib/Algebra/GroupWithZero/WithZero.lean
@@ -266,6 +266,43 @@ def unitsWithZeroEquiv : (WithZero α)ˣ ≃* α where
   right_inv _ := rfl
   map_mul' _ _ := coe_inj.mp <| by simp only [Units.val_mul, coe_unzero, coe_mul]
 
+/-- Any group with zero is isomorphic to adjoining `0` to the units of itself. -/
+def withZeroUnitsEquiv {G : Type*} [GroupWithZero G]
+    [DecidablePred (fun a : G ↦ a = 0)] :
+    WithZero Gˣ ≃* G where
+  toFun := WithZero.recZeroCoe 0 Units.val
+  invFun a := if h : a = 0 then 0 else (Units.mk0 a h : Gˣ)
+  left_inv := (by induction · <;> simp)
+  right_inv _ := by simp only; split <;> simp_all
+  map_mul' x y := by
+    induction x <;> induction y <;>
+    simp [← WithZero.coe_mul, ← Units.val_mul]
+
+/-- A version of `Equiv.optionCongr` for `WithZero`. -/
+noncomputable def _root_.MulEquiv.withZero [Group β] (e : α ≃* β) :
+    WithZero α ≃* WithZero β where
+  toFun := map' e.toMonoidHom
+  invFun := map' e.symm.toMonoidHom
+  left_inv := (by induction · <;> simp)
+  right_inv := (by induction · <;> simp)
+  map_mul' x y := by
+    induction x <;> induction y <;>
+    simp
+
+/-- The inverse of `MulEquiv.withZero`. -/
+protected noncomputable def _root_.MulEquiv.unzero [Group β] (e : WithZero α ≃* WithZero β) :
+    α ≃* β where
+  toFun x := unzero (x := e x) (by simp [ne_eq, ← e.eq_symm_apply])
+  invFun x := unzero (x := e.symm x) (by simp [e.symm_apply_eq])
+  left_inv _ := by simp
+  right_inv _ := by simp
+  map_mul' _ _ := by
+    simp only [coe_mul, map_mul]
+    generalize_proofs A B C
+    suffices ((unzero A : β) : WithZero β) = (unzero B) * (unzero C) by
+      rwa [← WithZero.coe_mul, WithZero.coe_inj] at this
+    simp
+
 end Group
 
 instance commGroupWithZero [CommGroup α] : CommGroupWithZero (WithZero α) :=
diff --git a/Mathlib/Algebra/HierarchyDesign.lean b/Mathlib/Algebra/HierarchyDesign.lean
index 862788ae48368..b2fe9b97fe510 100644
--- a/Mathlib/Algebra/HierarchyDesign.lean
+++ b/Mathlib/Algebra/HierarchyDesign.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Eric Wieser
+Authors: Kim Morrison, Eric Wieser
 -/
 import Mathlib.Init
 import Batteries.Util.LibraryNote
diff --git a/Mathlib/Algebra/Homology/Additive.lean b/Mathlib/Algebra/Homology/Additive.lean
index b2988fbd4aba9..85979eabec627 100644
--- a/Mathlib/Algebra/Homology/Additive.lean
+++ b/Mathlib/Algebra/Homology/Additive.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.Single
 import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
diff --git a/Mathlib/Algebra/Homology/Augment.lean b/Mathlib/Algebra/Homology/Augment.lean
index 0b02a3d3140cd..c72e003fc7557 100644
--- a/Mathlib/Algebra/Homology/Augment.lean
+++ b/Mathlib/Algebra/Homology/Augment.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.Single
 
diff --git a/Mathlib/Algebra/Homology/BifunctorAssociator.lean b/Mathlib/Algebra/Homology/BifunctorAssociator.lean
index e13e10c8a242e..77c349d2de113 100644
--- a/Mathlib/Algebra/Homology/BifunctorAssociator.lean
+++ b/Mathlib/Algebra/Homology/BifunctorAssociator.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joël Riou
 -/
 import Mathlib.CategoryTheory.GradedObject.Associator
-import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
+import Mathlib.CategoryTheory.Linear.LinearFunctor
 import Mathlib.Algebra.Homology.Bifunctor
 
 /-!
@@ -65,7 +65,7 @@ variable (F₁₂ G) in
 /-- Given bifunctors `F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂`, `G : C₁₂ ⥤ C₃ ⥤ C₄`, homological complexes
 `K₁ : HomologicalComplex C₁ c₁`, `K₂ : HomologicalComplex C₂ c₂` and
 `K₃ : HomologicalComplex C₃ c₃`, and complexes shapes `c₁₂`, `c₄`, this asserts
-that for all `i₁₂ : ι₁₂` and `i₃ : ι₃`, the functor `G(-, K₃.X i₃)` commutes wich
+that for all `i₁₂ : ι₁₂` and `i₃ : ι₃`, the functor `G(-, K₃.X i₃)` commutes with
 the coproducts of the `F₁₂(X₁ i₁, X₂ i₂)` such that `π c₁ c₂ c₁₂ ⟨i₁, i₂⟩ = i₁₂`. -/
 abbrev HasGoodTrifunctor₁₂Obj :=
   GradedObject.HasGoodTrifunctor₁₂Obj F₁₂ G
@@ -76,7 +76,7 @@ variable (F G₂₃) in
 `K₁ : HomologicalComplex C₁ c₁`, `K₂ : HomologicalComplex C₂ c₂` and
 `K₃ : HomologicalComplex C₃ c₃`, and complexes shapes `c₁₂`, `c₂₃`, `c₄`
 with `ComplexShape.Associative c₁ c₂ c₃ c₁₂ c₂₃ c₄`, this asserts that for
-all `i₁ : ι₁` and `i₂₃ : ι₂₃`, the functor `F(K₁.X i₁, _)` commutes wich
+all `i₁ : ι₁` and `i₂₃ : ι₂₃`, the functor `F(K₁.X i₁, _)` commutes with
 the coproducts of the `G₂₃(K₂.X i₂, K₃.X i₃)`
 such that `π c₂ c₃ c₂₃ ⟨i₂, i₃⟩ = i₂₃`. -/
 abbrev HasGoodTrifunctor₂₃Obj :=
@@ -187,6 +187,212 @@ lemma ι_mapBifunctor₁₂Desc (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃)
 
 end
 
+variable (F₁₂ G)
+
+/-- The first differential on a summand
+of `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def d₁ (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) :
+    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j :=
+  (ComplexShape.ε₁ c₁₂ c₃ c₄ (ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩, i₃) *
+    ComplexShape.ε₁ c₁ c₂ c₁₂ (i₁, i₂)) •
+  (G.map ((F₁₂.map (K₁.d i₁ (c₁.next i₁))).app (K₂.X i₂))).app (K₃.X i₃) ≫
+    ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ _ i₂ i₃ j
+
+lemma d₁_eq_zero (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) (h : ¬ c₁.Rel i₁ (c₁.next i₁)) :
+    d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j = 0 := by
+  dsimp [d₁]
+  rw [shape _ _ _ h, Functor.map_zero, zero_app, Functor.map_zero, zero_app, zero_comp, smul_zero]
+
+lemma d₁_eq {i₁ i₁' : ι₁} (h₁ : c₁.Rel i₁ i₁') (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) :
+    d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j =
+    (ComplexShape.ε₁ c₁₂ c₃ c₄ (ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩, i₃) *
+      ComplexShape.ε₁ c₁ c₂ c₁₂ (i₁, i₂) ) •
+    (G.map ((F₁₂.map (K₁.d i₁ i₁')).app (K₂.X i₂))).app (K₃.X i₃) ≫
+      ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁' i₂ i₃ j := by
+  obtain rfl := c₁.next_eq' h₁
+  rfl
+
+/-- The second differential on a summand
+of `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def d₂ (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) :
+    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j :=
+  (c₁₂.ε₁ c₃ c₄ (ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩, i₃) * c₁.ε₂ c₂ c₁₂ (i₁, i₂)) •
+  (G.map ((F₁₂.obj (K₁.X i₁)).map (K₂.d i₂ (c₂.next i₂)))).app (K₃.X i₃) ≫
+    ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ _ i₃ j
+
+lemma d₂_eq_zero (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) (h : ¬ c₂.Rel i₂ (c₂.next i₂)) :
+    d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j = 0 := by
+  dsimp [d₂]
+  rw [shape _ _ _ h, Functor.map_zero, Functor.map_zero, zero_app, zero_comp, smul_zero]
+
+lemma d₂_eq (i₁ : ι₁) {i₂ i₂' : ι₂} (h₂ : c₂.Rel i₂ i₂') (i₃ : ι₃) (j : ι₄) :
+    d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j =
+  (c₁₂.ε₁ c₃ c₄ (ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩, i₃) * c₁.ε₂ c₂ c₁₂ (i₁, i₂)) •
+    (G.map ((F₁₂.obj (K₁.X i₁)).map (K₂.d i₂ i₂'))).app (K₃.X i₃) ≫
+      ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ _ i₃ j := by
+  obtain rfl := c₂.next_eq' h₂
+  rfl
+
+/-- The third differential on a summand
+of `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def d₃ (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) :
+    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j :=
+  (ComplexShape.ε₂ c₁₂ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃)) •
+    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).map (K₃.d i₃ (c₃.next i₃)) ≫
+      ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ _ j
+
+lemma d₃_eq_zero (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j : ι₄) (h : ¬ c₃.Rel i₃ (c₃.next i₃)) :
+    d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j = 0 := by
+  dsimp [d₃]
+  rw [shape _ _ _ h, Functor.map_zero, zero_comp, smul_zero]
+
+lemma d₃_eq (i₁ : ι₁) (i₂ : ι₂) {i₃ i₃' : ι₃} (h₃ : c₃.Rel i₃ i₃') (j : ι₄) :
+    d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j =
+  (ComplexShape.ε₂ c₁₂ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), i₃)) •
+    (G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).map (K₃.d i₃ i₃') ≫
+      ιOrZero F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ _ j := by
+  obtain rfl := c₃.next_eq' h₃
+  rfl
+
+
+section
+
+variable [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄]
+variable (j j' : ι₄)
+
+/-- The first differential on `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def D₁ :
+    (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j' :=
+  mapBifunctor₁₂Desc (fun i₁ i₂ i₃ _ ↦ d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j')
+
+/-- The second differential on `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def D₂ :
+    (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j' :=
+  mapBifunctor₁₂Desc (fun i₁ i₂ i₃ _ ↦ d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j')
+
+/-- The third differential on `mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄`. -/
+noncomputable def D₃ :
+    (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j ⟶
+      (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).X j' :=
+  mapBifunctor.D₂ _ _ _ _ _ _
+
+end
+
+section
+
+variable (i₁ : ι₁) (i₂ : ι₂) (i₃ : ι₃) (j j' : ι₄)
+    (h : ComplexShape.r c₁ c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j)
+
+@[reassoc (attr := simp)]
+lemma ι_D₁ [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄] :
+    ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' =
+      d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' := by
+  simp [D₁]
+
+@[reassoc (attr := simp)]
+lemma ι_D₂ [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄] :
+    ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' =
+      d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' := by
+  simp [D₂]
+
+@[reassoc (attr := simp)]
+lemma ι_D₃  :
+    ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' =
+      d₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' := by
+  simp only [ι_eq _ _ _ _ _ _ _ _ _ _ _ _ rfl h, D₃, assoc, mapBifunctor.ι_D₂]
+  by_cases h₁ : c₃.Rel i₃ (c₃.next i₃)
+  · rw [d₃_eq _ _ _ _ _ _ _ _ _ h₁]
+    by_cases h₂ : ComplexShape.π c₁₂ c₃ c₄ (c₁.π c₂ c₁₂ (i₁, i₂), c₃.next i₃) = j'
+    · rw [mapBifunctor.d₂_eq _ _ _ _ _ h₁ _ h₂,
+        ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ h₂,
+        Linear.comp_units_smul, smul_left_cancel_iff,
+        ι_eq _ _ _ _ _ _ _ _ _ _ _ _ rfl h₂,
+        NatTrans.naturality_assoc]
+    · rw [mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂, comp_zero,
+        ιOrZero_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₂, comp_zero, smul_zero]
+  · rw [mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₁, comp_zero,
+      d₃_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₁]
+
+end
+
+lemma d_eq (j j' : ι₄) [HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄] :
+    (mapBifunctor (mapBifunctor K₁ K₂ F₁₂ c₁₂) K₃ G c₄).d j j' =
+      D₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' + D₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' +
+        D₃ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ j j' := by
+  rw [mapBifunctor.d_eq]
+  congr 1
+  ext i₁ i₂ i₃ h
+  simp only [Preadditive.comp_add, ι_D₁, ι_D₂]
+  rw [ι_eq _ _ _ _ _ _ _ _ _ _ _ _ rfl h, assoc, mapBifunctor.ι_D₁]
+  set i₁₂ := ComplexShape.π c₁ c₂ c₁₂ ⟨i₁, i₂⟩
+  by_cases h₁ : c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
+  · by_cases h₂ : ComplexShape.π c₁₂ c₃ c₄ (c₁₂.next i₁₂, i₃) = j'
+    · rw [mapBifunctor.d₁_eq _ _ _ _ h₁ _ _ h₂]
+      simp only [mapBifunctor.d_eq, Functor.map_add, NatTrans.app_add, Preadditive.add_comp,
+        smul_add, Preadditive.comp_add, Linear.comp_units_smul]
+      congr 1
+      · rw [← NatTrans.comp_app_assoc, ← Functor.map_comp,
+          mapBifunctor.ι_D₁]
+        by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
+        · have h₄ := (ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂).symm
+          rw [mapBifunctor.d₁_eq _ _ _ _ h₃ _ _ h₄,
+            d₁_eq _ _ _ _ _ _ _ h₃,
+            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
+            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
+            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
+            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
+        · rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
+            mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₃,
+            Functor.map_zero, zero_app, zero_comp, smul_zero]
+      · rw [← NatTrans.comp_app_assoc, ← Functor.map_comp,
+          mapBifunctor.ι_D₂]
+        by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
+        · have h₄ := (ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃).symm
+          rw [mapBifunctor.d₂_eq _ _ _ _ _ h₃ _ h₄,
+            d₂_eq _ _ _ _ _ _ _ _ h₃,
+            ιOrZero_eq _ _ _ _ _ _ _ _ _ _ _ (by rw [← h₂, ← h₄]; rfl),
+            ι_eq _ _ _ _ _ _ _ _ _ _ (c₁₂.next i₁₂) _ h₄ h₂,
+            Functor.map_units_smul, Functor.map_comp, NatTrans.app_units_zsmul,
+            NatTrans.comp_app, Linear.units_smul_comp, assoc, smul_smul]
+        · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃,
+            mapBifunctor.d₂_eq_zero _ _ _ _ _ _ _ h₃,
+            Functor.map_zero, zero_app, zero_comp, smul_zero]
+    · rw [mapBifunctor.d₁_eq_zero' _ _ _ _ h₁ _ _ h₂, comp_zero]
+      trans 0 + 0
+      · simp
+      · congr 1
+        · by_cases h₃ : c₁.Rel i₁ (c₁.next i₁)
+          · rw [d₁_eq _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
+            dsimp [ComplexShape.r]
+            intro h₄
+            apply h₂
+            rw [← h₄, ComplexShape.next_π₁ c₂ c₁₂ h₃ i₂]
+          · rw [d₁_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]
+        · by_cases h₃ : c₂.Rel i₂ (c₂.next i₂)
+          · rw [d₂_eq _ _ _ _ _ _ _ _ h₃, ιOrZero_eq_zero, comp_zero, smul_zero]
+            dsimp [ComplexShape.r]
+            intro h₄
+            apply h₂
+            rw [← h₄, ComplexShape.next_π₂ c₁ c₁₂ i₁ h₃]
+          · rw [d₂_eq_zero _ _ _ _ _ _ _ _ _ _ _ h₃]
+  · rw [mapBifunctor.d₁_eq_zero _ _ _ _ _ _ _ h₁, comp_zero,
+      d₁_eq_zero, d₂_eq_zero, zero_add]
+    · intro h₂
+      apply h₁
+      have := ComplexShape.rel_π₂ c₁ c₁₂ i₁ h₂
+      rw [c₁₂.next_eq' this]
+      exact this
+    · intro h₂
+      apply h₁
+      have := ComplexShape.rel_π₁ c₂ c₁₂ h₂ i₂
+      rw [c₁₂.next_eq' this]
+      exact this
+
 end mapBifunctor₁₂
 
 end HomologicalComplex
diff --git a/Mathlib/Algebra/Homology/ComplexShape.lean b/Mathlib/Algebra/Homology/ComplexShape.lean
index e4da8ba92cfdf..2b46e9d743bc7 100644
--- a/Mathlib/Algebra/Homology/ComplexShape.lean
+++ b/Mathlib/Algebra/Homology/ComplexShape.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison
+Authors: Johan Commelin, Kim Morrison
 -/
 import Mathlib.Algebra.Group.Defs
 import Mathlib.Logic.Relation
diff --git a/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean b/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
index 781e2de9cf31f..774df032bb48c 100644
--- a/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
+++ b/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
@@ -36,7 +36,6 @@ sheaves over `X` shall be in `Type u`.
 
 ## TODO
 * compute `Ext X Y 0`
-* construct the covariant long exact sequences of `Ext`.
 * construct the contravariant long exact sequences of `Ext`.
 
 -/
diff --git a/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean b/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean
new file mode 100644
index 0000000000000..427dd7ae80159
--- /dev/null
+++ b/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExactSequences.lean
@@ -0,0 +1,164 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Exact
+import Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass
+import Mathlib.Algebra.Homology.ShortComplex.Ab
+import Mathlib.CategoryTheory.Triangulated.Yoneda
+
+/-!
+# Long exact sequences of `Ext`-groups
+
+In this file, we obtain the covariant long exact sequence of `Ext`:
+`Ext X S.X₁ n₀ → Ext X S.X₂ n₀ → Ext X S.X₃ n₀ → Ext X S.X₁ n₁ → Ext X S.X₂ n₁ → Ext X S.X₃ n₁`
+when `S` is a short exact short complex in an abelian category `C`, `n₀ + 1 = n₁` and `X : C`.
+
+-/
+
+universe w' w v u
+
+namespace CategoryTheory
+
+open Opposite DerivedCategory
+
+variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{w} C]
+
+namespace Abelian
+
+namespace Ext
+
+section CovariantSequence
+
+lemma hom_comp_singleFunctor_map_shift [HasDerivedCategory.{w'} C]
+    {X Y Z : C} {n : ℕ} (x : Ext X Y n) (f : Y ⟶ Z) :
+    x.hom ≫ ((DerivedCategory.singleFunctor C 0).map f)⟦(n : ℤ)⟧' =
+      (x.comp (mk₀ f) (add_zero n)).hom := by
+  simp only [comp_hom, mk₀_hom, ShiftedHom.comp_mk₀]
+
+variable {X : C} {S : ShortComplex C} (hS : S.ShortExact)
+
+lemma preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply
+    [HasDerivedCategory.{w'} C] {X : C} {n₀ : ℕ} (x : Ext X S.X₃ n₀)
+    {n₁ : ℕ} (h : n₀ + 1 = n₁) :
+    (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequenceδ
+      hS.singleTriangle n₀ n₁ (by omega) x.hom =
+        (x.comp hS.extClass h).hom := by
+  rw [Pretriangulated.preadditiveCoyoneda_homologySequenceδ_apply,
+    comp_hom, hS.extClass_hom, ShiftedHom.comp]
+  rfl
+
+variable (X)
+
+include hS in
+/-- Alternative formulation of `covariant_sequence_exact₂` -/
+lemma covariant_sequence_exact₂' (n : ℕ) :
+    (ShortComplex.mk (AddCommGrp.ofHom ((mk₀ S.f).postcomp X (add_zero n)))
+      (AddCommGrp.ofHom ((mk₀ S.g).postcomp X (add_zero n))) (by
+        ext x
+        dsimp [AddCommGrp.ofHom]
+        simp only [comp_assoc_of_third_deg_zero, mk₀_comp_mk₀, ShortComplex.zero, mk₀_zero,
+          comp_zero]
+        rfl)).Exact := by
+  letI := HasDerivedCategory.standard C
+  have := (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequence_exact₂ _
+    (hS.singleTriangle_distinguished) n
+  rw [ShortComplex.ab_exact_iff_function_exact] at this ⊢
+  apply Function.Exact.of_ladder_addEquiv_of_exact' (e₁ := Ext.homAddEquiv)
+    (e₂ := Ext.homAddEquiv) (e₃ := Ext.homAddEquiv) (H := this)
+  all_goals ext x; apply hom_comp_singleFunctor_map_shift (C := C)
+
+section
+
+variable (n₀ n₁ : ℕ) (h : n₀ + 1 = n₁)
+
+/-- Alternative formulation of `covariant_sequence_exact₃` -/
+lemma covariant_sequence_exact₃' :
+    (ShortComplex.mk (AddCommGrp.ofHom ((mk₀ S.g).postcomp X (add_zero n₀)))
+      (AddCommGrp.ofHom (hS.extClass.postcomp X h)) (by
+        ext x
+        dsimp [AddCommGrp.ofHom]
+        simp only [comp_assoc_of_second_deg_zero, ShortComplex.ShortExact.comp_extClass,
+          comp_zero]
+        rfl)).Exact := by
+  letI := HasDerivedCategory.standard C
+  have := (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequence_exact₃ _
+    (hS.singleTriangle_distinguished) n₀ n₁ (by omega)
+  rw [ShortComplex.ab_exact_iff_function_exact] at this ⊢
+  apply Function.Exact.of_ladder_addEquiv_of_exact' (e₁ := Ext.homAddEquiv)
+    (e₂ := Ext.homAddEquiv) (e₃ := Ext.homAddEquiv) (H := this)
+  · ext x; apply hom_comp_singleFunctor_map_shift (C := C)
+  · ext x
+    exact preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply hS x h
+
+/-- Alternative formulation of `covariant_sequence_exact₁` -/
+lemma covariant_sequence_exact₁' :
+    (ShortComplex.mk
+      (AddCommGrp.ofHom (hS.extClass.postcomp X h))
+      (AddCommGrp.ofHom ((mk₀ S.f).postcomp X (add_zero n₁))) (by
+        ext x
+        dsimp [AddCommGrp.ofHom]
+        simp only [comp_assoc_of_third_deg_zero, ShortComplex.ShortExact.extClass_comp, comp_zero]
+        rfl)).Exact := by
+  letI := HasDerivedCategory.standard C
+  have := (preadditiveCoyoneda.obj (op ((singleFunctor C 0).obj X))).homologySequence_exact₁ _
+    (hS.singleTriangle_distinguished) n₀ n₁ (by omega)
+  rw [ShortComplex.ab_exact_iff_function_exact] at this ⊢
+  apply Function.Exact.of_ladder_addEquiv_of_exact' (e₁ := Ext.homAddEquiv)
+    (e₂ := Ext.homAddEquiv) (e₃ := Ext.homAddEquiv) (H := this)
+  · ext x
+    exact preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply hS x h
+  · ext x; apply hom_comp_singleFunctor_map_shift (C := C)
+
+open ComposableArrows
+
+/-- Given a short exact short complex `S` in an abelian category `C` and an object `X : C`,
+this is the long exact sequence
+`Ext X S.X₁ n₀ → Ext X S.X₂ n₀ → Ext X S.X₃ n₀ → Ext X S.X₁ n₁ → Ext X S.X₂ n₁ → Ext X S.X₃ n₁`
+when `n₀ + 1 = n₁` -/
+noncomputable def covariantSequence : ComposableArrows AddCommGrp.{w} 5 :=
+  mk₅ (AddCommGrp.ofHom ((mk₀ S.f).postcomp X (add_zero n₀)))
+    (AddCommGrp.ofHom ((mk₀ S.g).postcomp X (add_zero n₀)))
+    (AddCommGrp.ofHom (hS.extClass.postcomp X h))
+    (AddCommGrp.ofHom ((mk₀ S.f).postcomp X (add_zero n₁)))
+    (AddCommGrp.ofHom ((mk₀ S.g).postcomp X (add_zero n₁)))
+
+lemma covariantSequence_exact :
+    (covariantSequence X hS n₀ n₁ h).Exact :=
+  exact_of_δ₀ (covariant_sequence_exact₂' X hS n₀).exact_toComposableArrows
+    (exact_of_δ₀ (covariant_sequence_exact₃' X hS n₀ n₁ h).exact_toComposableArrows
+      (exact_of_δ₀ (covariant_sequence_exact₁' X hS n₀ n₁ h).exact_toComposableArrows
+        (covariant_sequence_exact₂' X hS n₁).exact_toComposableArrows))
+
+end
+
+lemma covariant_sequence_exact₁ {n₁ : ℕ} (x₁ : Ext X S.X₁ n₁)
+    (hx₁ : x₁.comp (mk₀ S.f) (add_zero n₁) = 0) {n₀ : ℕ} (hn₀ : n₀ + 1 = n₁) :
+    ∃ (x₃ : Ext X S.X₃ n₀), x₃.comp hS.extClass hn₀ = x₁ := by
+  have := covariant_sequence_exact₁' X hS n₀ n₁ hn₀
+  rw [ShortComplex.ab_exact_iff] at this
+  exact this x₁ hx₁
+
+include hS in
+lemma covariant_sequence_exact₂ {n : ℕ} (x₂ : Ext X S.X₂ n)
+    (hx₂ : x₂.comp (mk₀ S.g) (add_zero n) = 0) :
+    ∃ (x₁ : Ext X S.X₁ n), x₁.comp (mk₀ S.f) (add_zero n) = x₂ := by
+  have := covariant_sequence_exact₂' X hS n
+  rw [ShortComplex.ab_exact_iff] at this
+  exact this x₂ hx₂
+
+lemma covariant_sequence_exact₃ {n₀ : ℕ} (x₃ : Ext X S.X₃ n₀) {n₁ : ℕ} (hn₁ : n₀ + 1 = n₁)
+    (hx₃ : x₃.comp hS.extClass hn₁ = 0) :
+    ∃ (x₂ : Ext X S.X₂ n₀), x₂.comp (mk₀ S.g) (add_zero n₀) = x₃ := by
+  have := covariant_sequence_exact₃' X hS n₀ n₁ hn₁
+  rw [ShortComplex.ab_exact_iff] at this
+  exact this x₃ hx₃
+
+end CovariantSequence
+
+end Ext
+
+end Abelian
+
+end CategoryTheory
diff --git a/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.lean b/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.lean
index 9a636124ff714..2869c7955e0c0 100644
--- a/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.lean
+++ b/Mathlib/Algebra/Homology/DerivedCategory/Ext/ExtClass.lean
@@ -29,6 +29,8 @@ namespace ShortComplex
 
 variable (S : ShortComplex C)
 
+lemma ext_mk₀_f_comp_ext_mk₀_g : (Ext.mk₀ S.f).comp (Ext.mk₀ S.g) (zero_add 0) = 0 := by simp
+
 namespace ShortExact
 
 variable {S}
diff --git a/Mathlib/Algebra/Homology/DifferentialObject.lean b/Mathlib/Algebra/Homology/DifferentialObject.lean
index cd85f3f0e9e6c..d1b9484894419 100644
--- a/Mathlib/Algebra/Homology/DifferentialObject.lean
+++ b/Mathlib/Algebra/Homology/DifferentialObject.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.HomologicalComplex
 import Mathlib.CategoryTheory.DifferentialObject
diff --git a/Mathlib/Algebra/Homology/Embedding/HomEquiv.lean b/Mathlib/Algebra/Homology/Embedding/HomEquiv.lean
new file mode 100644
index 0000000000000..ae8a61b3f4536
--- /dev/null
+++ b/Mathlib/Algebra/Homology/Embedding/HomEquiv.lean
@@ -0,0 +1,211 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Homology.Embedding.Restriction
+import Mathlib.Algebra.Homology.Embedding.Extend
+import Mathlib.Algebra.Homology.Embedding.Boundary
+import Mathlib.CategoryTheory.MorphismProperty.Basic
+
+/-!
+# Relations between `extend` and `restriction`
+
+Given an embedding `e : Embedding c c'` of complex shapes satisfying `e.IsRelIff`,
+we obtain a bijection `e.homEquiv` between the type of morphisms
+`K ⟶ L.extend e` (with `K : HomologicalComplex C c'` and `L : HomologicalComplex C c`)
+and the subtype of morphisms `φ : K.restriction e ⟶ L` which satisfy a certain
+condition `e.HasLift φ`.
+
+## TODO
+* obtain dual results for morphisms `L.extend e ⟶ K`.
+
+-/
+
+open CategoryTheory Category Limits
+
+namespace ComplexShape
+
+variable {ι ι' : Type*} {c : ComplexShape ι} {c' : ComplexShape ι'} (e : Embedding c c')
+  {C : Type*} [Category C] [HasZeroMorphisms C] [HasZeroObject C]
+
+namespace Embedding
+
+open HomologicalComplex
+
+variable {K K' : HomologicalComplex C c'} {L L' : HomologicalComplex C c}
+  [e.IsRelIff]
+
+section
+
+/-- The condition on a morphism `K.restriction e ⟶ L` which allows to
+extend it as a morphism `K ⟶ L.extend e`, see `Embedding.homEquiv`. -/
+def HasLift (φ : K.restriction e ⟶ L) : Prop :=
+  ∀ (j : ι) (_ : e.BoundaryGE j) (i' : ι')
+    (_ : c'.Rel i' (e.f j)), K.d i' _ ≫ φ.f j = 0
+
+namespace liftExtend
+
+variable (φ : K.restriction e ⟶ L)
+
+variable {e}
+
+open Classical in
+/-- Auxiliary definition for `liftExtend`. -/
+noncomputable def f (i' : ι') : K.X i' ⟶ (L.extend e).X i' :=
+  if hi' : ∃ i, e.f i = i' then
+    (K.restrictionXIso e hi'.choose_spec).inv ≫ φ.f hi'.choose ≫
+      (L.extendXIso e hi'.choose_spec).inv
+  else 0
+
+lemma f_eq {i' : ι'} {i : ι} (hi : e.f i = i') :
+    f φ i' = (K.restrictionXIso e hi).inv ≫ φ.f i ≫ (L.extendXIso e hi).inv := by
+  have hi' : ∃ k, e.f k = i' := ⟨i, hi⟩
+  have : hi'.choose = i := e.injective_f (by rw [hi'.choose_spec, hi])
+  dsimp [f]
+  rw [dif_pos ⟨i, hi⟩]
+  subst this
+  rfl
+
+@[reassoc (attr := simp)]
+lemma comm (hφ : e.HasLift φ) (i' j' : ι') :
+    f φ i' ≫ (L.extend e).d i' j' = K.d i' j' ≫ f φ j' := by
+  by_cases hij' : c'.Rel i' j'
+  · by_cases hi' : ∃ i, e.f i = i'
+    · obtain ⟨i, hi⟩ := hi'
+      rw [f_eq φ hi]
+      by_cases hj' : ∃ j, e.f j = j'
+      · obtain ⟨j, hj⟩ := hj'
+        rw [f_eq φ hj, L.extend_d_eq e hi hj]
+        subst hi hj
+        simp [HomologicalComplex.restrictionXIso]
+      · apply (L.isZero_extend_X e j' (by simpa using hj')).eq_of_tgt
+    · have : (L.extend e).d i' j' = 0 := by
+        apply (L.isZero_extend_X e i' (by simpa using hi')).eq_of_src
+      rw [this, comp_zero]
+      by_cases hj' : ∃ j, e.f j = j'
+      · obtain ⟨j, rfl⟩ := hj'
+        rw [f_eq φ rfl]
+        dsimp [restrictionXIso]
+        rw [id_comp, reassoc_of% (hφ j (e.boundaryGE hij'
+          (by simpa using hi')) i' hij'), zero_comp]
+      · have : f φ j' = 0 := by
+          apply (L.isZero_extend_X e j' (by simpa using hj')).eq_of_tgt
+        rw [this, comp_zero]
+  · simp [HomologicalComplex.shape _ _ _ hij']
+
+end liftExtend
+
+variable (φ : K.restriction e ⟶ L) (hφ : e.HasLift φ)
+
+/-- The morphism  `K ⟶ L.extend e` given by a morphism `K.restriction e ⟶ L`
+which satisfy `e.HasLift φ`. -/
+noncomputable def liftExtend :
+    K ⟶ L.extend e where
+  f i' := liftExtend.f φ i'
+  comm' _ _ _ := liftExtend.comm φ hφ _ _
+
+variable {i' : ι'} {i : ι} (hi : e.f i = i')
+
+lemma liftExtend_f :
+    (e.liftExtend φ hφ).f i' = (K.restrictionXIso e hi).inv ≫ φ.f i ≫
+      (L.extendXIso e hi).inv := by
+  apply liftExtend.f_eq
+
+/-- Given `φ : K.restriction e ⟶ L` such that `hφ : e.HasLift φ`, this is
+the isomorphisms in the category of arrows between the maps
+`(e.liftExtend φ hφ).f i'` and `φ.f i` when `e.f i = i'`. -/
+noncomputable def liftExtendfArrowIso :
+    Arrow.mk ((e.liftExtend φ hφ).f i') ≅ Arrow.mk (φ.f i) :=
+  Arrow.isoMk (K.restrictionXIso e hi).symm (L.extendXIso e hi)
+    (by simp [e.liftExtend_f φ hφ hi])
+
+lemma isIso_liftExtend_f_iff (hi : e.f i = i') :
+    IsIso ((e.liftExtend φ hφ).f i') ↔ IsIso (φ.f i) :=
+  (MorphismProperty.isomorphisms C).arrow_mk_iso_iff (e.liftExtendfArrowIso φ hφ hi)
+
+lemma mono_liftExtend_f_iff (hi : e.f i = i') :
+    Mono ((e.liftExtend φ hφ).f i') ↔ Mono (φ.f i) :=
+  (MorphismProperty.monomorphisms C).arrow_mk_iso_iff (e.liftExtendfArrowIso φ hφ hi)
+
+lemma epi_liftExtend_f_iff (hi : e.f i = i') :
+    Epi ((e.liftExtend φ hφ).f i') ↔ Epi (φ.f i) :=
+  (MorphismProperty.epimorphisms C).arrow_mk_iso_iff (e.liftExtendfArrowIso φ hφ hi)
+
+end
+
+namespace homRestrict
+
+variable {e}
+variable (ψ : K ⟶ L.extend e)
+
+/-- Auxiliary definition for `Embedding.homRestrict`. -/
+noncomputable def f (i : ι) : (K.restriction e).X i ⟶ L.X i :=
+  ψ.f (e.f i) ≫ (L.extendXIso e rfl).hom
+
+lemma f_eq {i : ι} {i' : ι'} (h : e.f i = i') :
+    f ψ i = (K.restrictionXIso e h).hom ≫ ψ.f i' ≫ (L.extendXIso e h).hom := by
+  subst h
+  simp [f, restrictionXIso]
+
+@[reassoc (attr := simp)]
+lemma comm (i j : ι) :
+    f ψ i ≫ L.d i j = K.d (e.f i) (e.f j) ≫ f ψ j := by
+  dsimp [f]
+  simp only [assoc, ← ψ.comm_assoc, L.extend_d_eq e rfl rfl, Iso.inv_hom_id, comp_id]
+
+end homRestrict
+
+/-- The morphism `K.restriction e ⟶ L` induced by a morphism `K ⟶ L.extend e`. -/
+noncomputable def homRestrict (ψ : K ⟶ L.extend e) : K.restriction e ⟶ L where
+  f i := homRestrict.f ψ i
+
+lemma homRestrict_f (ψ : K ⟶ L.extend e) {i : ι} {i' : ι'} (h : e.f i = i') :
+    (e.homRestrict ψ).f i = (K.restrictionXIso e h).hom ≫ ψ.f i' ≫ (L.extendXIso e h).hom :=
+  homRestrict.f_eq ψ h
+
+lemma homRestrict_hasLift (ψ : K ⟶ L.extend e) :
+    e.HasLift (e.homRestrict ψ) := by
+  intro j hj i' hij'
+  have : (L.extend e).d i' (e.f j) = 0 := by
+    apply (L.isZero_extend_X e i' (hj.not_mem hij')).eq_of_src
+  dsimp [homRestrict]
+  rw [homRestrict.f_eq ψ rfl, restrictionXIso, eqToIso_refl, Iso.refl_hom, id_comp,
+    ← ψ.comm_assoc, this, zero_comp, comp_zero]
+
+@[simp]
+lemma liftExtend_homRestrict (ψ : K ⟶ L.extend e) :
+    e.liftExtend (e.homRestrict ψ) (e.homRestrict_hasLift ψ) = ψ := by
+  ext i'
+  by_cases hi' : ∃ i, e.f i = i'
+  · obtain ⟨i, rfl⟩ := hi'
+    simp [e.homRestrict_f _ rfl, e.liftExtend_f _ _ rfl]
+  · apply (L.isZero_extend_X e i' (by simpa using hi')).eq_of_tgt
+
+@[simp]
+lemma homRestrict_liftExtend (φ : K.restriction e ⟶ L) (hφ : e.HasLift φ) :
+    e.homRestrict (e.liftExtend φ hφ) = φ := by
+  ext i
+  simp [e.homRestrict_f _ rfl, e.liftExtend_f _ _ rfl]
+
+@[reassoc]
+lemma homRestrict_precomp (α : K' ⟶ K) (ψ : K ⟶ L.extend e) :
+    e.homRestrict (α ≫ ψ) = restrictionMap α e ≫ e.homRestrict ψ := by
+  ext i
+  simp [homRestrict_f _ _ rfl, restrictionXIso]
+
+variable (K L)
+
+/-- The bijection between `K ⟶ L.extend e` and the subtype of `K.restriction e ⟶ L`
+consisting of morphisms `φ` such that `e.HasLift φ`. -/
+@[simps]
+noncomputable def homEquiv :
+    (K ⟶ L.extend e) ≃ { φ : K.restriction e ⟶ L // e.HasLift φ } where
+  toFun ψ := ⟨e.homRestrict ψ, e.homRestrict_hasLift ψ⟩
+  invFun φ := e.liftExtend φ.1 φ.2
+  left_inv ψ := by simp
+  right_inv φ := by simp
+
+end Embedding
+
+end ComplexShape
diff --git a/Mathlib/Algebra/Homology/Embedding/TruncGE.lean b/Mathlib/Algebra/Homology/Embedding/TruncGE.lean
index 3cfeee56bada8..c0c9a8b3cbd18 100644
--- a/Mathlib/Algebra/Homology/Embedding/TruncGE.lean
+++ b/Mathlib/Algebra/Homology/Embedding/TruncGE.lean
@@ -110,8 +110,8 @@ noncomputable def truncGE'XIsoOpcycles {i : ι} {i' : ι'} (hi' : e.f i = i') (h
     (K.truncGE' e).X i ≅ K.opcycles i' :=
   (truncGE'.XIsoOpcycles K e hi) ≪≫ eqToIso (by subst hi'; rfl)
 
-lemma truncGE'_d_eq {i j : ι} (hij : c.Rel i j)  {i' j' : ι'}
-    (hi' : e.f i = i') (hj' : e.f j = j')  (hi : ¬ e.BoundaryGE i) :
+lemma truncGE'_d_eq {i j : ι} (hij : c.Rel i j) {i' j' : ι'}
+    (hi' : e.f i = i') (hj' : e.f j = j') (hi : ¬ e.BoundaryGE i) :
     (K.truncGE' e).d i j = (K.truncGE'XIso e hi' hi).hom ≫ K.d i' j' ≫
       (K.truncGE'XIso e hj' (e.not_boundaryGE_next hij)).inv := by
   dsimp [truncGE', truncGE'.d]
diff --git a/Mathlib/Algebra/Homology/ExactSequence.lean b/Mathlib/Algebra/Homology/ExactSequence.lean
index 0afb27611c949..bf8559ea76f1a 100644
--- a/Mathlib/Algebra/Homology/ExactSequence.lean
+++ b/Mathlib/Algebra/Homology/ExactSequence.lean
@@ -159,7 +159,7 @@ lemma exact_iff_of_iso {S₁ S₂ : ComposableArrows C n} (e : S₁ ≅ S₂) :
 lemma exact₀ (S : ComposableArrows C 0) : S.Exact where
   toIsComplex := S.isComplex₀
   -- See https://github.com/leanprover/lean4/issues/2862
-  exact i hi := by simp [autoParam] at hi
+  exact i hi := by simp at hi
 
 lemma exact₁ (S : ComposableArrows C 1) : S.Exact where
   toIsComplex := S.isComplex₁
diff --git a/Mathlib/Algebra/Homology/HomologicalBicomplex.lean b/Mathlib/Algebra/Homology/HomologicalBicomplex.lean
index c693b7880842a..dbd85f9d8e710 100644
--- a/Mathlib/Algebra/Homology/HomologicalBicomplex.lean
+++ b/Mathlib/Algebra/Homology/HomologicalBicomplex.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Homology.HomologicalComplex
 
@@ -205,7 +205,7 @@ def XXIsoOfEq {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ :
 
 @[simp]
 lemma XXIsoOfEq_rfl (i₁ : I₁) (i₂ : I₂) :
-    K.XXIsoOfEq (rfl : i₁ = i₁) (rfl : i₂ = i₂) = Iso.refl _ := rfl
+    K.XXIsoOfEq _ _ _ (rfl : i₁ = i₁) (rfl : i₂ = i₂) = Iso.refl _ := rfl
 
 
 end HomologicalComplex₂
diff --git a/Mathlib/Algebra/Homology/HomologicalComplex.lean b/Mathlib/Algebra/Homology/HomologicalComplex.lean
index 59f2a43f6c5a1..2494b21c2eb58 100644
--- a/Mathlib/Algebra/Homology/HomologicalComplex.lean
+++ b/Mathlib/Algebra/Homology/HomologicalComplex.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison
+Authors: Johan Commelin, Kim Morrison
 -/
 import Mathlib.Algebra.Homology.ComplexShape
 import Mathlib.CategoryTheory.Subobject.Limits
diff --git a/Mathlib/Algebra/Homology/HomologySequence.lean b/Mathlib/Algebra/Homology/HomologySequence.lean
index 3c219686c7084..6a325db2f53a9 100644
--- a/Mathlib/Algebra/Homology/HomologySequence.lean
+++ b/Mathlib/Algebra/Homology/HomologySequence.lean
@@ -283,7 +283,7 @@ namespace ShortComplex
 namespace ShortExact
 
 /-- The connecting homoomorphism `S.X₃.homology i ⟶ S.X₁.homology j` for a short exact
-short complex `S`.  -/
+short complex `S`. -/
 noncomputable def δ : S.X₃.homology i ⟶ S.X₁.homology j := (snakeInput hS i j hij).δ
 
 @[reassoc (attr := simp)]
diff --git a/Mathlib/Algebra/Homology/HomologySequenceLemmas.lean b/Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
index acf08cce2512f..26d1411fed80d 100644
--- a/Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
+++ b/Mathlib/Algebra/Homology/HomologySequenceLemmas.lean
@@ -103,8 +103,6 @@ noncomputable def mapComposableArrows₅ (i j : ι) (hij : c.Rel i j) :
     (naturality' (mapComposableArrows₂ φ j) 0 1)
     (naturality' (mapComposableArrows₂ φ j) 1 2)
 
-attribute [local instance] epi_comp
-
 include hS₁ hS₂
 
 lemma mono_homologyMap_τ₃ (i : ι)
diff --git a/Mathlib/Algebra/Homology/Homotopy.lean b/Mathlib/Algebra/Homology/Homotopy.lean
index bc2fd8fdf31ef..f8c43c0d2b844 100644
--- a/Mathlib/Algebra/Homology/Homotopy.lean
+++ b/Mathlib/Algebra/Homology/Homotopy.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.Linear
 import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
@@ -101,7 +101,7 @@ theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D
   dsimp [dNext]
   cases i
   · simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
-      not_false_iff, zero_comp]
+      not_false_iff, zero_comp, reduceCtorEq]
   · congr <;> simp
 
 theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
@@ -109,7 +109,7 @@ theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶
   dsimp [prevD]
   cases i
   · simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
-      not_false_iff, comp_zero]
+      not_false_iff, comp_zero, reduceCtorEq]
   · congr <;> simp
 
 -- Porting note(#5171): removed @[has_nonempty_instance]
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory.lean b/Mathlib/Algebra/Homology/HomotopyCategory.lean
index a938f482ae5ea..93400ddd35d94 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.Homotopy
 import Mathlib.Algebra.Homology.Linear
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean b/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
index 957e7c2a38b45..42704741a95dc 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
@@ -51,7 +51,7 @@ open HomComplex
 
 /-- The left inclusion in the mapping cone, as a cochain of degree `-1`. -/
 noncomputable def inl : Cochain F (mappingCone φ) (-1) :=
-  Cochain.mk (fun p q hpq => homotopyCofiber.inlX φ p q  (by dsimp; omega))
+  Cochain.mk (fun p q hpq => homotopyCofiber.inlX φ p q (by dsimp; omega))
 
 /-- The right inclusion in the mapping cone. -/
 noncomputable def inr : G ⟶ mappingCone φ := homotopyCofiber.inr φ
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean b/Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean
index b49421f3f25b8..85cb5a4752ba7 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean
@@ -61,7 +61,7 @@ noncomputable def hom :
     (descCochain _ 0 (Cochain.ofHom (inr (f ≫ g))) (neg_add_cancel 1)) (by
       ext p _ rfl
       simp [mappingConeCompTriangle, map, ext_from_iff _ _ _ rfl,
-        inl_v_d_assoc _ (p+1) p (p+2) (by linarith) (by linarith)])
+        inl_v_d_assoc _ (p+1) p (p+2) (by omega) (by omega)])
 
 /-- Given two composable morphisms `f` and `g` in the category of cochain complexes, this
 is the canonical morphism (which is an homotopy equivalence) from the mapping cone of
@@ -72,7 +72,7 @@ noncomputable def inv : mappingCone (mappingConeCompTriangle f g).mor₁ ⟶ map
       ext p
       rw [ext_from_iff _ (p + 1) _ rfl, ext_to_iff _ _ (p + 1) rfl]
       simp [map, δ_zero_cochain_comp,
-        Cochain.comp_v _ _ (add_neg_cancel 1) p (p+1) p (by linarith) (by linarith)])
+        Cochain.comp_v _ _ (add_neg_cancel 1) p (p+1) p (by omega) (by omega)])
 
 @[reassoc (attr := simp)]
 lemma hom_inv_id : hom f g ≫ inv f g = 𝟙 _ := by
@@ -86,44 +86,25 @@ this is the `homotopyInvHomId` field of the homotopy equivalence
 the morphism `mappingCone f ⟶ mappingCone (f ≫ g)`. -/
 noncomputable def homotopyInvHomId : Homotopy (inv f g ≫ hom f g) (𝟙 _) :=
   (Cochain.equivHomotopy _ _).symm ⟨-((snd _).comp ((fst (f ≫ g)).1.comp
-    ((inl f).comp (inl _) (by linarith)) (show 1 + (-2) = -1 by linarith)) (zero_add (-1))), by
+    ((inl f).comp (inl _) (by omega)) (show 1 + (-2) = -1 by omega)) (zero_add (-1))), by
       rw [δ_neg, δ_zero_cochain_comp _ _ _ (neg_add_cancel 1),
         Int.negOnePow_neg, Int.negOnePow_one, Units.neg_smul, one_smul,
-        δ_comp _ _ (show 1 + (-2) = -1 by linarith) 2 (-1) 0 (by linarith)
-          (by linarith) (by linarith),
-        δ_comp _ _ (show (-1) + (-1) = -2 by linarith) 0 0 (-1) (by linarith)
-          (by linarith) (by linarith), Int.negOnePow_neg, Int.negOnePow_neg,
-        Int.negOnePow_even 2 ⟨1, by linarith⟩, Int.negOnePow_one, Units.neg_smul,
+        δ_comp _ _ (show 1 + (-2) = -1 by omega) 2 (-1) 0 (by omega)
+          (by omega) (by omega),
+        δ_comp _ _ (show (-1) + (-1) = -2 by omega) 0 0 (-1) (by omega)
+          (by omega) (by omega), Int.negOnePow_neg, Int.negOnePow_neg,
+        Int.negOnePow_even 2 ⟨1, by omega⟩, Int.negOnePow_one, Units.neg_smul,
         one_smul, one_smul, δ_inl, δ_inl, δ_snd, Cocycle.δ_eq_zero, Cochain.zero_comp, add_zero,
         Cochain.neg_comp, neg_neg]
       ext n
       rw [ext_from_iff _ (n + 1) n rfl, ext_from_iff _ (n + 1) n rfl,
-        ext_from_iff _ (n + 2) (n + 1) (by linarith)]
-      simp? [hom, inv, ext_to_iff _ n (n + 1) rfl, map, Cochain.comp_v _ _
-          (add_neg_cancel 1) n (n + 1) n (by linarith) (by linarith),
-        Cochain.comp_v _ _ (show 1 + -2 = -1 by linarith) (n + 1) (n + 2) n
-          (by linarith) (by linarith),
-        Cochain.comp_v _ _ (show (-1) + -1 = -2 by linarith) (n + 2) (n + 1) n
-          (by linarith) (by linarith)] says
-        simp only [mappingConeCompTriangle_obj₁, mappingConeCompTriangle_obj₂,
-          mappingConeCompTriangle_mor₁, map, Int.reduceNeg, inv, hom, Cochain.ofHom_comp,
-          ofHom_desc, ofHom_lift, descCocycle_coe, AddSubmonoid.coe_zero,
-          Cochain.comp_zero_cochain_v, inl_v_descCochain_v_assoc, Cochain.zero_cochain_comp_v,
-          assoc, inl_v_snd_v_assoc, zero_comp, Cochain.id_comp,
-          Cochain.comp_assoc_of_first_is_zero_cochain, Cochain.comp_add, Cochain.comp_neg,
-          Cochain.comp_assoc_of_second_is_zero_cochain, neg_add_rev, neg_neg, Cochain.add_v,
-          Cochain.neg_v,
-          Cochain.comp_v _ _ (add_neg_cancel 1) n (n + 1) n (by linarith) (by linarith),
-          Cochain.comp_v _ _ (show 1 + -2 = -1 by linarith) (n + 1) (n + 2) n (by linarith)
-            (by linarith),
-          Cochain.comp_v _ _ (show (-1) + -1 = -2 by linarith) (n + 2) (n + 1) n (by linarith)
-            (by linarith),
-          Cochain.ofHom_v, HomologicalComplex.id_f, Preadditive.comp_add, Preadditive.comp_neg,
-          inl_v_fst_v_assoc, neg_zero, add_zero, comp_id, neg_add_cancel, inr_f_snd_v_assoc,
-          inr_f_descCochain_v_assoc, inr_f_fst_v_assoc, comp_zero, zero_add,
-          ext_to_iff _ n (n + 1) rfl, liftCochain_v_fst_v, inl_v_descCochain_v, inl_v_fst_v,
-          liftCochain_v_snd_v, Cochain.zero_v, inl_v_snd_v, and_self, neg_add_cancel_right,
-          inr_f_descCochain_v, inr_f_fst_v, inr_f_snd_v]⟩
+        ext_from_iff _ (n + 2) (n + 1) (by omega)]
+      simp [hom, inv, ext_to_iff _ n (n + 1) rfl, map, Cochain.comp_v _ _
+          (add_neg_cancel 1) n (n + 1) n (by omega) (by omega),
+        Cochain.comp_v _ _ (show 1 + -2 = -1 by omega) (n + 1) (n + 2) n
+          (by omega) (by omega),
+        Cochain.comp_v _ _ (show (-1) + -1 = -2 by omega) (n + 2) (n + 1) n
+          (by omega) (by omega)]⟩
 
 end MappingConeCompHomotopyEquiv
 
diff --git a/Mathlib/Algebra/Homology/ImageToKernel.lean b/Mathlib/Algebra/Homology/ImageToKernel.lean
index 7ab7f5ff1347d..16f064b89a76f 100644
--- a/Mathlib/Algebra/Homology/ImageToKernel.lean
+++ b/Mathlib/Algebra/Homology/ImageToKernel.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.Limits
 
@@ -147,7 +147,7 @@ instance imageToKernel_epi_of_epi_of_zero [HasImages V] [Epi f] :
   simp only [imageToKernel_zero_right]
   haveI := epi_image_of_epi f
   rw [← imageSubobject_arrow]
-  exact @epi_comp _ _ _ _ _ _ (epi_comp _ _) _ _
+  infer_instance
 
 end
 
diff --git a/Mathlib/Algebra/Homology/LocalCohomology.lean b/Mathlib/Algebra/Homology/LocalCohomology.lean
index e6bcfa9d338a5..4ea0bfce99d36 100644
--- a/Mathlib/Algebra/Homology/LocalCohomology.lean
+++ b/Mathlib/Algebra/Homology/LocalCohomology.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Emily Witt. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Emily Witt, Scott Morrison, Jake Levinson, Sam van Gool
+Authors: Emily Witt, Kim Morrison, Jake Levinson, Sam van Gool
 -/
 import Mathlib.RingTheory.Ideal.Basic
 import Mathlib.Algebra.Category.ModuleCat.Colimits
@@ -232,8 +232,8 @@ instance ideal_powers_initial [hR : IsNoetherian R R] :
       -- The inclusions `J^n1 ≤ J'` and `J^n2 ≤ J'` always form a triangle, based on
       -- which exponent is larger.
       rcases le_total (unop j1.left) (unop j2.left) with h | h
-      · right; exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
-      · left; exact ⟨CostructuredArrow.homMk (homOfLE h).op (AsTrue.get trivial)⟩
+      · right; exact ⟨CostructuredArrow.homMk (homOfLE h).op rfl⟩
+      · left; exact ⟨CostructuredArrow.homMk (homOfLE h).op rfl⟩
 
 example : HasColimitsOfSize.{0, 0, u, u + 1} (ModuleCat.{u, u} R) := inferInstance
 /-- Local cohomology (defined in terms of powers of `J`) agrees with local
diff --git a/Mathlib/Algebra/Homology/QuasiIso.lean b/Mathlib/Algebra/Homology/QuasiIso.lean
index dbb3e8cbd051b..47a65f02f7de3 100644
--- a/Mathlib/Algebra/Homology/QuasiIso.lean
+++ b/Mathlib/Algebra/Homology/QuasiIso.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Homology.Homotopy
 
diff --git a/Mathlib/Algebra/Homology/ShortComplex/Ab.lean b/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
index 8b76e44a2ca8c..474f801a48272 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
@@ -109,6 +109,18 @@ lemma ab_exact_iff :
     obtain ⟨x₁, rfl⟩ := h x₂ hx₂
     exact ⟨x₁, rfl⟩
 
+lemma ab_exact_iff_function_exact :
+    S.Exact ↔ Function.Exact S.f S.g := by
+  rw [S.ab_exact_iff]
+  apply forall_congr'
+  intro x₂
+  constructor
+  · intro h
+    refine ⟨h, ?_⟩
+    rintro ⟨x₁, rfl⟩
+    simp only [ab_zero_apply]
+  · tauto
+
 lemma ab_exact_iff_ker_le_range : S.Exact ↔ S.g.ker ≤ S.f.range := S.ab_exact_iff
 
 lemma ab_exact_iff_range_eq_ker : S.Exact ↔ S.f.range = S.g.ker := by
diff --git a/Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean b/Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean
index 974a1c8685585..d61079ee62930 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/ExactFunctor.lean
@@ -116,7 +116,7 @@ lemma preservesFiniteLimits_tfae : List.TFAE
       Nonempty <| PreservesFiniteLimits F
     ] := by
   tfae_have 1 → 2
-  · rintro hF S ⟨hS, hf⟩
+  | hF, S, ⟨hS, hf⟩ => by
     have := preservesMonomorphisms_of_preserves_shortExact_left F hF
     refine ⟨?_, inferInstance⟩
     let T := ShortComplex.mk S.f (Abelian.coimage.π S.g) (Abelian.comp_coimage_π_eq_zero S.zero)
@@ -129,7 +129,7 @@ lemma preservesFiniteLimits_tfae : List.TFAE
     exact (exact_iff_of_epi_of_isIso_of_mono φ).1 (hF T ⟨(S.exact_iff_exact_coimage_π).1 hS⟩).1
 
   tfae_have 2 → 3
-  · intro hF X Y f
+  | hF, X, Y, f => by
     refine ⟨preservesLimitOfPreservesLimitCone (kernelIsKernel f) ?_⟩
     apply (KernelFork.isLimitMapConeEquiv _ F).2
     let S := ShortComplex.mk _ _ (kernel.condition f)
@@ -138,13 +138,13 @@ lemma preservesFiniteLimits_tfae : List.TFAE
     exact hS.1.fIsKernel
 
   tfae_have 3 → 4
-  · intro hF
+  | hF => by
     have := fun X Y (f : X ⟶ Y) ↦ (hF f).some
     exact ⟨preservesFiniteLimitsOfPreservesKernels F⟩
 
   tfae_have 4 → 1
-  · rintro ⟨_⟩ S hS
-    exact (S.map F).exact_and_mono_f_iff_f_is_kernel |>.2 ⟨KernelFork.mapIsLimit _ hS.fIsKernel F⟩
+  | ⟨_⟩, S, hS =>
+    (S.map F).exact_and_mono_f_iff_f_is_kernel |>.2 ⟨KernelFork.mapIsLimit _ hS.fIsKernel F⟩
 
   tfae_finish
 
@@ -175,7 +175,7 @@ lemma preservesFiniteColimits_tfae : List.TFAE
       Nonempty <| PreservesFiniteColimits F
     ] := by
   tfae_have 1 → 2
-  · rintro hF S ⟨hS, hf⟩
+  | hF, S, ⟨hS, hf⟩ => by
     have := preservesEpimorphisms_of_preserves_shortExact_right F hF
     refine ⟨?_, inferInstance⟩
     let T := ShortComplex.mk (Abelian.image.ι S.f) S.g (Abelian.image_ι_comp_eq_zero S.zero)
@@ -188,7 +188,7 @@ lemma preservesFiniteColimits_tfae : List.TFAE
     exact (exact_iff_of_epi_of_isIso_of_mono φ).2 (hF T ⟨(S.exact_iff_exact_image_ι).1 hS⟩).1
 
   tfae_have 2 → 3
-  · intro hF X Y f
+  | hF, X, Y, f => by
     refine ⟨preservesColimitOfPreservesColimitCocone (cokernelIsCokernel f) ?_⟩
     apply (CokernelCofork.isColimitMapCoconeEquiv _ F).2
     let S := ShortComplex.mk _ _ (cokernel.condition f)
@@ -197,14 +197,13 @@ lemma preservesFiniteColimits_tfae : List.TFAE
     exact hS.1.gIsCokernel
 
   tfae_have 3 → 4
-  · intro hF
+  | hF => by
     have := fun X Y (f : X ⟶ Y) ↦ (hF f).some
     exact ⟨preservesFiniteColimitsOfPreservesCokernels F⟩
 
   tfae_have 4 → 1
-  · rintro ⟨_⟩ S hS
-    exact (S.map F).exact_and_epi_g_iff_g_is_cokernel |>.2
-      ⟨CokernelCofork.mapIsColimit _ hS.gIsCokernel F⟩
+  | ⟨_⟩, S, hS => (S.map F).exact_and_epi_g_iff_g_is_cokernel |>.2
+    ⟨CokernelCofork.mapIsColimit _ hS.gIsCokernel F⟩
 
   tfae_finish
 
@@ -224,7 +223,7 @@ lemma exact_tfae : List.TFAE
       Nonempty (PreservesFiniteLimits F) ∧ Nonempty (PreservesFiniteColimits F)
     ] := by
   tfae_have 1 → 3
-  · intro hF
+  | hF => by
     refine ⟨fun {X Y} f ↦ ?_, fun {X Y} f ↦ ?_⟩
     · have h := (preservesFiniteLimits_tfae F |>.out 0 2 |>.1 fun S hS ↦
         And.intro (hF S hS).exact (hF S hS).mono_f)
@@ -234,7 +233,7 @@ lemma exact_tfae : List.TFAE
       exact h f |>.some
 
   tfae_have 2 → 1
-  · intro hF S hS
+  | hF, S, hS => by
     have : Mono (S.map F).f := exact_iff_mono _ (by simp) |>.1 <|
       hF (.mk (0 : 0 ⟶ S.X₁) S.f <| by simp) (exact_iff_mono _ (by simp) |>.2 hS.mono_f)
     have : Epi (S.map F).g := exact_iff_epi _ (by simp) |>.1 <|
@@ -242,13 +241,11 @@ lemma exact_tfae : List.TFAE
     exact ⟨hF S hS.exact⟩
 
   tfae_have 3 → 4
-  · rintro ⟨h⟩
-    exact ⟨⟨preservesFiniteLimitsOfPreservesHomology F⟩,
-      ⟨preservesFiniteColimitsOfPreservesHomology F⟩⟩
+  | ⟨h⟩ => ⟨⟨preservesFiniteLimitsOfPreservesHomology F⟩,
+    ⟨preservesFiniteColimitsOfPreservesHomology F⟩⟩
 
   tfae_have 4 → 2
-  · rintro ⟨⟨h1⟩, ⟨h2⟩⟩
-    exact fun _ h ↦ h.map F
+  | ⟨⟨h1⟩, ⟨h2⟩⟩, _, h => h.map F
 
   tfae_finish
 
diff --git a/Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean b/Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
index c14da91511cae..fb3e8617e352d 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean
@@ -318,8 +318,6 @@ lemma p_opcyclesMap : K.pOpcycles i ≫ opcyclesMap φ i = φ.f i ≫ L.pOpcycle
 
 instance [Mono (φ.f i)] : Mono (cyclesMap φ i) := mono_of_mono_fac (cyclesMap_i φ i)
 
-attribute [local instance] epi_comp
-
 instance [Epi (φ.f i)] : Epi (opcyclesMap φ i) := epi_of_epi_fac (p_opcyclesMap φ i)
 
 variable (K)
diff --git a/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean b/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
index 1d91f59bef1b9..1022a52c7ab90 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
@@ -388,10 +388,10 @@ section
 variable (S)
 variable [S.HasLeftHomology]
 
-/-- The left homology of a short complex, given by the `H` field of a chosen left homology data.  -/
+/-- The left homology of a short complex, given by the `H` field of a chosen left homology data. -/
 noncomputable def leftHomology : C := S.leftHomologyData.H
 
-/-- The cycles of a short complex, given by the `K` field of a chosen left homology data.  -/
+/-- The cycles of a short complex, given by the `K` field of a chosen left homology data. -/
 noncomputable def cycles : C := S.leftHomologyData.K
 
 /-- The "homology class" map `S.cycles ⟶ S.leftHomology`. -/
diff --git a/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean b/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
index 7523eadd214dd..59034f288455b 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean
@@ -33,7 +33,7 @@ linear maps `f` and `g` and the vanishing of their composition. -/
 def moduleCatMk {X₁ X₂ X₃ : Type v} [AddCommGroup X₁] [AddCommGroup X₂] [AddCommGroup X₃]
     [Module R X₁] [Module R X₂] [Module R X₃] (f : X₁ →ₗ[R] X₂) (g : X₂ →ₗ[R] X₃)
     (hfg : g.comp f = 0) : ShortComplex (ModuleCat.{v} R) :=
-  ShortComplex.mk (ModuleCat.ofHom f) (ModuleCat.ofHom g) hfg
+  ShortComplex.mk (ModuleCat.asHom f) (ModuleCat.asHom g) hfg
 
 variable (S : ShortComplex (ModuleCat.{v} R))
 
@@ -138,7 +138,7 @@ def moduleCatLeftHomologyData : S.LeftHomologyData where
     erw [Submodule.Quotient.mk_eq_zero]
     rw [LinearMap.mem_range]
     apply exists_apply_eq_apply
-  hπ := ModuleCat.cokernelIsColimit (ModuleCat.ofHom S.moduleCatToCycles)
+  hπ := ModuleCat.cokernelIsColimit (ModuleCat.asHom S.moduleCatToCycles)
 
 @[simp]
 lemma moduleCatLeftHomologyData_f' :
diff --git a/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean b/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
index d1364bc4bca38..8d23c46c9a207 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean
@@ -508,7 +508,7 @@ def op (h : Homotopy φ₁ φ₂) : Homotopy (opMap φ₁) (opMap φ₂) where
 /-- The homotopy between morphisms in `ShortComplex C` that is induced by a homotopy
 between morphisms in `ShortComplex Cᵒᵖ`. -/
 @[simps]
-def unop {S₁ S₂ : ShortComplex Cᵒᵖ} {φ₁ φ₂ : S₁ ⟶ S₂}  (h : Homotopy φ₁ φ₂) :
+def unop {S₁ S₂ : ShortComplex Cᵒᵖ} {φ₁ φ₂ : S₁ ⟶ S₂} (h : Homotopy φ₁ φ₂) :
     Homotopy (unopMap φ₁) (unopMap φ₂) where
   h₀ := h.h₃.unop
   h₁ := h.h₂.unop
diff --git a/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean b/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
index 5a62395fcb632..865ffa00b94fb 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
@@ -1004,7 +1004,7 @@ variable (φ : S₁ ⟶ S₂) (h : RightHomologyData S₁) [Epi φ.τ₁] [IsIso
 /-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
 and `φ.τ₃` is mono, then a right homology data for `S₁` induces a right homology data for `S₂` with
 the same `Q` and `H` fields. This is obtained by dualising `LeftHomologyData.ofEpiOfIsIsoOfMono'`.
-The inverse construction is `ofEpiOfIsIsoOfMono'`.  -/
+The inverse construction is `ofEpiOfIsIsoOfMono'`. -/
 noncomputable def ofEpiOfIsIsoOfMono : RightHomologyData S₂ := by
   haveI : Epi (opMap φ).τ₁ := by dsimp; infer_instance
   haveI : IsIso (opMap φ).τ₂ := by dsimp; infer_instance
@@ -1032,7 +1032,7 @@ variable (φ : S₁ ⟶ S₂) (h : RightHomologyData S₂) [Epi φ.τ₁] [IsIso
 /-- If `φ : S₁ ⟶ S₂` is a morphism of short complexes such that `φ.τ₁` is epi, `φ.τ₂` is an iso
 and `φ.τ₃` is mono, then a right homology data for `S₂` induces a right homology data for `S₁` with
 the same `Q` and `H` fields. This is obtained by dualising `LeftHomologyData.ofEpiOfIsIsoOfMono`.
-The inverse construction is `ofEpiOfIsIsoOfMono`.  -/
+The inverse construction is `ofEpiOfIsIsoOfMono`. -/
 noncomputable def ofEpiOfIsIsoOfMono' : RightHomologyData S₁ := by
   haveI : Epi (opMap φ).τ₁ := by dsimp; infer_instance
   haveI : IsIso (opMap φ).τ₂ := by dsimp; infer_instance
diff --git a/Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean b/Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
index 77769d32f3b80..8fe3dbac06932 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
@@ -381,7 +381,7 @@ variable (S₁ S₂ S₃ : SnakeInput C)
 /-- A morphism of snake inputs involve four morphisms of short complexes
 which make the obvious diagram commute. -/
 @[ext]
-structure Hom :=
+structure Hom where
   /-- a morphism between the zeroth lines -/
   f₀ : S₁.L₀ ⟶ S₂.L₀
   /-- a morphism between the first lines -/
diff --git a/Mathlib/Algebra/Homology/Single.lean b/Mathlib/Algebra/Homology/Single.lean
index 833e8089b75d2..6adb409ebcd24 100644
--- a/Mathlib/Algebra/Homology/Single.lean
+++ b/Mathlib/Algebra/Homology/Single.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.HomologicalComplex
 
diff --git a/Mathlib/Algebra/Homology/TotalComplex.lean b/Mathlib/Algebra/Homology/TotalComplex.lean
index 73fb3959181c0..645d747de064f 100644
--- a/Mathlib/Algebra/Homology/TotalComplex.lean
+++ b/Mathlib/Algebra/Homology/TotalComplex.lean
@@ -260,7 +260,7 @@ noncomputable def ιTotal (i₁ : I₁) (i₂ : I₂) (i₁₂ : I₁₂)
 @[reassoc (attr := simp)]
 lemma XXIsoOfEq_hom_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂)
     (i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (y₁, y₂) = i₁₂) :
-    (K.XXIsoOfEq h₁ h₂).hom ≫ K.ιTotal c₁₂ y₁ y₂ i₁₂ h =
+    (K.XXIsoOfEq _ _ _ h₁ h₂).hom ≫ K.ιTotal c₁₂ y₁ y₂ i₁₂ h =
       K.ιTotal c₁₂ x₁ x₂ i₁₂ (by rw [h₁, h₂, h]) := by
   subst h₁ h₂
   simp
@@ -268,7 +268,7 @@ lemma XXIsoOfEq_hom_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ :
 @[reassoc (attr := simp)]
 lemma XXIsoOfEq_inv_ιTotal {x₁ y₁ : I₁} (h₁ : x₁ = y₁) {x₂ y₂ : I₂} (h₂ : x₂ = y₂)
     (i₁₂ : I₁₂) (h : ComplexShape.π c₁ c₂ c₁₂ (x₁, x₂) = i₁₂) :
-    (K.XXIsoOfEq h₁ h₂).inv ≫ K.ιTotal c₁₂ x₁ x₂ i₁₂ h =
+    (K.XXIsoOfEq _ _ _ h₁ h₂).inv ≫ K.ιTotal c₁₂ x₁ x₂ i₁₂ h =
       K.ιTotal c₁₂ y₁ y₂ i₁₂ (by rw [← h, h₁, h₂]) := by
   subst h₁ h₂
   simp
diff --git a/Mathlib/Algebra/Homology/TotalComplexShift.lean b/Mathlib/Algebra/Homology/TotalComplexShift.lean
index 8d32945310df2..345677aa57934 100644
--- a/Mathlib/Algebra/Homology/TotalComplexShift.lean
+++ b/Mathlib/Algebra/Homology/TotalComplexShift.lean
@@ -129,7 +129,7 @@ noncomputable def totalShift₁XIso (n n' : ℤ) (h : n + x = n') :
     (((shiftFunctor₁ C x).obj K).total (up ℤ)).X n ≅ (K.total (up ℤ)).X n' where
   hom := totalDesc _ (fun p q hpq => K.ιTotal (up ℤ) (p + x) q n' (by dsimp at hpq ⊢; omega))
   inv := totalDesc _ (fun p q hpq =>
-    (K.XXIsoOfEq (Int.sub_add_cancel p x) rfl).inv ≫
+    (K.XXIsoOfEq _ _ _ (Int.sub_add_cancel p x) rfl).inv ≫
       ((shiftFunctor₁ C x).obj K).ιTotal (up ℤ) (p - x) q n
         (by dsimp at hpq ⊢; omega))
   hom_inv_id := by
@@ -235,7 +235,7 @@ noncomputable def totalShift₂XIso (n n' : ℤ) (h : n + y = n') :
   hom := totalDesc _ (fun p q hpq => (p * y).negOnePow • K.ιTotal (up ℤ) p (q + y) n'
     (by dsimp at hpq ⊢; omega))
   inv := totalDesc _ (fun p q hpq => (p * y).negOnePow •
-    (K.XXIsoOfEq rfl (Int.sub_add_cancel q y)).inv ≫
+    (K.XXIsoOfEq _ _ _ rfl (Int.sub_add_cancel q y)).inv ≫
       ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) p (q - y) n (by dsimp at hpq ⊢; omega))
   hom_inv_id := by
     ext p q h
diff --git a/Mathlib/Algebra/Jordan/Basic.lean b/Mathlib/Algebra/Jordan/Basic.lean
index 4098f22e03cf1..b6e455cab6209 100644
--- a/Mathlib/Algebra/Jordan/Basic.lean
+++ b/Mathlib/Algebra/Jordan/Basic.lean
@@ -101,7 +101,7 @@ instance (priority := 100) IsCommJordan.toIsJordan [CommMagma A] [IsCommJordan A
     rw [mul_comm b a, IsCommJordan.lmul_comm_rmul_rmul, mul_comm]
 
 -- see Note [lower instance priority]
-/-- Semigroup multiplication satisfies the (non-commutative) Jordan axioms-/
+/-- Semigroup multiplication satisfies the (non-commutative) Jordan axioms -/
 instance (priority := 100) Semigroup.isJordan [Semigroup A] : IsJordan A where
   lmul_comm_rmul a b := by rw [mul_assoc]
   lmul_lmul_comm_lmul a b := by rw [mul_assoc, mul_assoc]
diff --git a/Mathlib/Algebra/Lie/Abelian.lean b/Mathlib/Algebra/Lie/Abelian.lean
index 9d26a486b09c5..1c88f3135e544 100644
--- a/Mathlib/Algebra/Lie/Abelian.lean
+++ b/Mathlib/Algebra/Lie/Abelian.lean
@@ -201,20 +201,21 @@ def maxTrivLinearMapEquivLieModuleHom : maxTrivSubmodule R L (M →ₗ[R] N) ≃
 
 @[simp]
 theorem coe_maxTrivLinearMapEquivLieModuleHom (f : maxTrivSubmodule R L (M →ₗ[R] N)) :
-    (maxTrivLinearMapEquivLieModuleHom f : M → N) = f := by ext; rfl
+    (maxTrivLinearMapEquivLieModuleHom (M := M) (N := N) f : M → N) = f := by ext; rfl
 
 @[simp]
 theorem coe_maxTrivLinearMapEquivLieModuleHom_symm (f : M →ₗ⁅R,L⁆ N) :
-    (maxTrivLinearMapEquivLieModuleHom.symm f : M → N) = f :=
+    (maxTrivLinearMapEquivLieModuleHom (M := M) (N := N) |>.symm f : M → N) = f :=
   rfl
 
 @[simp]
 theorem coe_linearMap_maxTrivLinearMapEquivLieModuleHom (f : maxTrivSubmodule R L (M →ₗ[R] N)) :
-    (maxTrivLinearMapEquivLieModuleHom f : M →ₗ[R] N) = (f : M →ₗ[R] N) := by ext; rfl
+    (maxTrivLinearMapEquivLieModuleHom (M := M) (N := N) f : M →ₗ[R] N) = (f : M →ₗ[R] N) := by
+  ext; rfl
 
 @[simp]
 theorem coe_linearMap_maxTrivLinearMapEquivLieModuleHom_symm (f : M →ₗ⁅R,L⁆ N) :
-    (maxTrivLinearMapEquivLieModuleHom.symm f : M →ₗ[R] N) = (f : M →ₗ[R] N) :=
+    (maxTrivLinearMapEquivLieModuleHom (M := M) (N := N) |>.symm f : M →ₗ[R] N) = (f : M →ₗ[R] N) :=
   rfl
 
 end LieModule
diff --git a/Mathlib/Algebra/Lie/BaseChange.lean b/Mathlib/Algebra/Lie/BaseChange.lean
index 6ad77f8f9d079..232b3327a85d3 100644
--- a/Mathlib/Algebra/Lie/BaseChange.lean
+++ b/Mathlib/Algebra/Lie/BaseChange.lean
@@ -169,10 +169,10 @@ def baseChange : LieSubmodule A (A ⊗[R] L) (A ⊗[R] M) :=
       intro x m hm
       simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
         Submodule.mem_toAddSubmonoid] at hm ⊢
-      obtain ⟨c, rfl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp hm
+      obtain ⟨c, rfl⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp hm
       refine x.induction_on (by simp) (fun a y ↦ ?_) (fun y z hy hz ↦ ?_)
       · change toEnd A (A ⊗[R] L) (A ⊗[R] M) _ _ ∈ _
-        simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum, map_smul, toEnd_apply_apply]
+        simp_rw [Finsupp.linearCombination_apply, Finsupp.sum, map_sum, map_smul, toEnd_apply_apply]
         suffices ∀ n : (N : Submodule R M).map (TensorProduct.mk R A M 1),
             ⁅a ⊗ₜ[R] y, (n : A ⊗[R] M)⁆ ∈ (N : Submodule R M).baseChange A by
           exact Submodule.sum_mem _ fun n _ ↦ Submodule.smul_mem _ _ (this n)
diff --git a/Mathlib/Algebra/Lie/Basic.lean b/Mathlib/Algebra/Lie/Basic.lean
index 40207bffb861a..cb37eac2841ca 100644
--- a/Mathlib/Algebra/Lie/Basic.lean
+++ b/Mathlib/Algebra/Lie/Basic.lean
@@ -269,10 +269,10 @@ attribute [coe] LieHom.toLinearMap
 instance : Coe (L₁ →ₗ⁅R⁆ L₂) (L₁ →ₗ[R] L₂) :=
   ⟨LieHom.toLinearMap⟩
 
-instance : FunLike (L₁ →ₗ⁅R⁆ L₂) L₁ L₂ :=
-  { coe := fun f => f.toFun,
-    coe_injective' := fun x y h =>
-      by cases x; cases y; simp at h; simp [h] }
+instance : FunLike (L₁ →ₗ⁅R⁆ L₂) L₁ L₂ where
+  coe f := f.toFun
+  coe_injective' x y h := by
+    cases x; cases y; simp at h; simp [h]
 
 initialize_simps_projections LieHom (toFun → apply)
 
@@ -471,13 +471,12 @@ instance hasCoeToLieHom : Coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ →ₗ⁅R⁆ L₂
 instance hasCoeToLinearEquiv : Coe (L₁ ≃ₗ⁅R⁆ L₂) (L₁ ≃ₗ[R] L₂) :=
   ⟨toLinearEquiv⟩
 
-instance : EquivLike (L₁ ≃ₗ⁅R⁆ L₂) L₁ L₂ :=
-  { coe := fun f => f.toFun,
-    inv := fun f => f.invFun,
-    left_inv := fun f => f.left_inv,
-    right_inv := fun f => f.right_inv,
-    coe_injective' := fun f g h₁ h₂ =>
-      by cases f; cases g; simp at h₁ h₂; simp [*] }
+instance : EquivLike (L₁ ≃ₗ⁅R⁆ L₂) L₁ L₂ where
+  coe f := f.toFun
+  inv f := f.invFun
+  left_inv f := f.left_inv
+  right_inv f := f.right_inv
+  coe_injective' f g h₁ h₂ := by cases f; cases g; simp at h₁ h₂; simp [*]
 
 theorem coe_to_lieHom (e : L₁ ≃ₗ⁅R⁆ L₂) : ⇑(e : L₁ →ₗ⁅R⁆ L₂) = e :=
   rfl
@@ -622,10 +621,9 @@ attribute [coe] LieModuleHom.toLinearMap
 instance : CoeOut (M →ₗ⁅R,L⁆ N) (M →ₗ[R] N) :=
   ⟨LieModuleHom.toLinearMap⟩
 
-instance : FunLike (M →ₗ⁅R, L⁆ N) M N :=
-  { coe := fun f => f.toFun,
-    coe_injective' := fun x y h =>
-      by cases x; cases y; simp at h; simp [h] }
+instance : FunLike (M →ₗ⁅R, L⁆ N) M N where
+  coe f := f.toFun
+  coe_injective' x y h := by cases x; cases y; simp at h; simp [h]
 
 initialize_simps_projections LieModuleHom (toFun → apply)
 
@@ -855,13 +853,12 @@ instance hasCoeToLieModuleHom : Coe (M ≃ₗ⁅R,L⁆ N) (M →ₗ⁅R,L⁆ N)
 instance hasCoeToLinearEquiv : CoeOut (M ≃ₗ⁅R,L⁆ N) (M ≃ₗ[R] N) :=
   ⟨toLinearEquiv⟩
 
-instance : EquivLike (M ≃ₗ⁅R,L⁆ N) M N :=
-  { coe := fun f => f.toFun,
-    inv := fun f => f.invFun,
-    left_inv := fun f => f.left_inv,
-    right_inv := fun f => f.right_inv,
-    coe_injective' := fun f g h₁ h₂ =>
-      by cases f; cases g; simp at h₁ h₂; simp [*] }
+instance : EquivLike (M ≃ₗ⁅R,L⁆ N) M N where
+  coe f := f.toFun
+  inv f := f.invFun
+  left_inv f := f.left_inv
+  right_inv f := f.right_inv
+  coe_injective' f g h₁ h₂ := by cases f; cases g; simp at h₁ h₂; simp [*]
 
 @[simp] lemma coe_coe (e : M ≃ₗ⁅R,L⁆ N) : ⇑(e : M →ₗ⁅R,L⁆ N) = e := rfl
 
diff --git a/Mathlib/Algebra/Lie/CartanExists.lean b/Mathlib/Algebra/Lie/CartanExists.lean
index 2a91aa8017dfa..5ee149734e3b5 100644
--- a/Mathlib/Algebra/Lie/CartanExists.lean
+++ b/Mathlib/Algebra/Lie/CartanExists.lean
@@ -39,7 +39,7 @@ variable [Module.Finite K L]
 variable [Module.Finite R L] [Module.Free R L]
 variable [Module.Finite R M] [Module.Free R M]
 
-open FiniteDimensional LieSubalgebra Module.Free Polynomial
+open Module LieSubalgebra Module.Free Polynomial
 
 variable (K)
 
@@ -117,7 +117,7 @@ section Field
 
 variable {K L : Type*} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L]
 
-open FiniteDimensional LieSubalgebra LieSubmodule Polynomial Cardinal LieModule engel_isBot_of_isMin
+open Module LieSubalgebra LieSubmodule Polynomial Cardinal LieModule engel_isBot_of_isMin
 
 #adaptation_note /-- otherwise there is a spurious warning on `contrapose!` below. -/
 set_option linter.unusedVariables false in
@@ -210,7 +210,7 @@ lemma engel_isBot_of_isMin (hLK : finrank K L ≤ #K) (U : LieSubalgebra K L)
     obtain hz₀|hz₀ := eq_or_ne z 0
     · -- If `z = 0`, then `⁅α • u + x, x⁆` vanishes and we use our assumption `x ≠ 0`.
       refine ⟨⟨x, self_mem_engel K x⟩, ?_, ?_⟩
-      · simpa [coe_bracket_of_module, ne_eq, Submodule.mk_eq_zero] using hx₀
+      · exact Subtype.coe_ne_coe.mp hx₀
       · dsimp only [z] at hz₀
         simp only [coe_bracket_of_module, hz₀, LieHom.map_zero, LinearMap.zero_apply]
     -- If `z ≠ 0`, then `⁅α • u + x, z⁆` vanishes per axiom of Lie algebras
@@ -360,7 +360,7 @@ lemma exists_isCartanSubalgebra_engel_of_finrank_le_card (h : finrank K L ≤ #K
   suffices finrank K (engel K x) ≤ finrank K (engel K y) by
     suffices engel K y = engel K x from this.ge
     apply LieSubalgebra.to_submodule_injective
-    exact eq_of_le_of_finrank_le hyx this
+    exact Submodule.eq_of_le_of_finrank_le hyx this
   rw [(isRegular_iff_finrank_engel_eq_rank K x).mp hx]
   apply rank_le_finrank_engel
 
diff --git a/Mathlib/Algebra/Lie/CartanSubalgebra.lean b/Mathlib/Algebra/Lie/CartanSubalgebra.lean
index f5d08788b2163..c904d1921c59f 100644
--- a/Mathlib/Algebra/Lie/CartanSubalgebra.lean
+++ b/Mathlib/Algebra/Lie/CartanSubalgebra.lean
@@ -107,7 +107,7 @@ end LieSubalgebra
 theorem LieIdeal.normalizer_eq_top {R : Type u} {L : Type v} [CommRing R] [LieRing L]
     [LieAlgebra R L] (I : LieIdeal R L) : (I : LieSubalgebra R L).normalizer = ⊤ := by
   ext x
-  simpa only [LieSubalgebra.mem_normalizer_iff, LieSubalgebra.mem_top, iff_true_iff] using
+  simpa only [LieSubalgebra.mem_normalizer_iff, LieSubalgebra.mem_top, iff_true] using
     fun y hy => I.lie_mem hy
 
 open LieIdeal
diff --git a/Mathlib/Algebra/Lie/Classical.lean b/Mathlib/Algebra/Lie/Classical.lean
index 23b88270b6e03..cc6d16181296e 100644
--- a/Mathlib/Algebra/Lie/Classical.lean
+++ b/Mathlib/Algebra/Lie/Classical.lean
@@ -329,7 +329,7 @@ theorem indefiniteDiagonal_assoc :
       Sum.elim_inl, if_true, eq_self_iff_true, Matrix.one_apply_eq, Matrix.fromBlocks_apply₁₁,
       DMatrix.zero_apply, Equiv.sumAssoc_apply_inl_inr, if_false, Matrix.fromBlocks_apply₁₂,
       Matrix.fromBlocks_apply₂₁, Matrix.fromBlocks_apply₂₂, Equiv.sumAssoc_apply_inr,
-      Sum.elim_inr, Sum.inl_injective.eq_iff, Sum.inr_injective.eq_iff] <;>
+      Sum.elim_inr, Sum.inl_injective.eq_iff, Sum.inr_injective.eq_iff, reduceCtorEq] <;>
     congr 1
 
 /-- An equivalence between two possible definitions of the classical Lie algebra of type B. -/
diff --git a/Mathlib/Algebra/Lie/Derivation/Basic.lean b/Mathlib/Algebra/Lie/Derivation/Basic.lean
index 0d155c7a5b6fe..00c92eef36252 100644
--- a/Mathlib/Algebra/Lie/Derivation/Basic.lean
+++ b/Mathlib/Algebra/Lie/Derivation/Basic.lean
@@ -120,6 +120,30 @@ theorem ext_of_lieSpan_eq_top (s : Set L) (hs : LieSubalgebra.lieSpan R L s = 
     (h : Set.EqOn D1 D2 s) : D1 = D2 :=
   ext fun _ => eqOn_lieSpan h <| hs.symm ▸ trivial
 
+section
+
+open Finset Nat
+
+/-- The general Leibniz rule for Lie derivatives. -/
+theorem iterate_apply_lie (D : LieDerivation R L L) (n : ℕ) (a b : L) :
+    D^[n] ⁅a, b⁆ = ∑ ij in antidiagonal n, choose n ij.1 • ⁅D^[ij.1] a, D^[ij.2] b⁆ := by
+  induction n with
+  | zero => simp
+  | succ n ih =>
+    rw [sum_antidiagonal_choose_succ_nsmul (M := L) (fun i j => ⁅D^[i] a, D^[j] b⁆) n]
+    simp only [Function.iterate_succ_apply', ih, map_sum, map_nsmul, apply_lie_eq_add, smul_add,
+      sum_add_distrib, add_right_inj]
+    refine sum_congr rfl fun ⟨i, j⟩ hij ↦ ?_
+    rw [n.choose_symm_of_eq_add (mem_antidiagonal.1 hij).symm]
+
+/-- Alternate version of the general Leibniz rule for Lie derivatives. -/
+theorem iterate_apply_lie' (D : LieDerivation R L L) (n : ℕ) (a b : L) :
+    D^[n] ⁅a, b⁆ = ∑ i in range (n + 1), n.choose i • ⁅D^[i] a, D^[n - i] b⁆ := by
+  rw [iterate_apply_lie D n a b]
+  exact sum_antidiagonal_eq_sum_range_succ (fun i j ↦ n.choose i • ⁅D^[i] a, D^[j] b⁆) n
+
+end
+
 instance instZero : Zero (LieDerivation R L M) where
   zero :=
     { toLinearMap := 0
diff --git a/Mathlib/Algebra/Lie/Derivation/Killing.lean b/Mathlib/Algebra/Lie/Derivation/Killing.lean
index e6c9325939168..65e346eb954d5 100644
--- a/Mathlib/Algebra/Lie/Derivation/Killing.lean
+++ b/Mathlib/Algebra/Lie/Derivation/Killing.lean
@@ -86,8 +86,10 @@ instance instIsKilling_range_ad : LieAlgebra.IsKilling R 𝕀 :=
 
 /-- The restriction of the Killing form of a finite-dimensional Killing Lie algebra to the range of
 the adjoint action is nondegenerate. -/
-lemma killingForm_restrict_range_ad_nondegenerate : ((killingForm R 𝔻).restrict 𝕀).Nondegenerate :=
-  killingForm_restrict_range_ad R L ▸ LieAlgebra.IsKilling.killingForm_nondegenerate R _
+lemma killingForm_restrict_range_ad_nondegenerate :
+    ((killingForm R 𝔻).restrict 𝕀).Nondegenerate := by
+  convert LieAlgebra.IsKilling.killingForm_nondegenerate R 𝕀
+  exact killingForm_restrict_range_ad R L
 
 /-- The range of the adjoint action on a finite-dimensional Killing Lie algebra is full. -/
 @[simp]
diff --git a/Mathlib/Algebra/Lie/Engel.lean b/Mathlib/Algebra/Lie/Engel.lean
index ecf69a863dfff..4b96b9ada30d7 100644
--- a/Mathlib/Algebra/Lie/Engel.lean
+++ b/Mathlib/Algebra/Lie/Engel.lean
@@ -200,7 +200,9 @@ theorem LieAlgebra.exists_engelian_lieSubalgebra_of_lt_normalizer {K : LieSubalg
   have hI₂ : (R ∙ (⟨x, hxK'⟩ : K')) ⊔ (LieSubmodule.toSubmodule I) = ⊤ := by
     rw [← LieIdeal.coe_to_lieSubalgebra_to_submodule R K' I, hI₁]
     apply Submodule.map_injective_of_injective (K' : Submodule R L).injective_subtype
-    simp
+    simp only [LieSubalgebra.coe_ofLe, Submodule.map_sup, Submodule.map_subtype_range_inclusion,
+      Submodule.map_top, Submodule.range_subtype]
+    rw [Submodule.map_subtype_span_singleton]
   have e : K ≃ₗ⁅R⁆ I :=
     (LieSubalgebra.equivOfLe hKK').trans
       (LieEquiv.ofEq _ _ ((LieSubalgebra.coe_set_eq _ _).mpr hI₁.symm))
diff --git a/Mathlib/Algebra/Lie/EngelSubalgebra.lean b/Mathlib/Algebra/Lie/EngelSubalgebra.lean
index 73ab3ad4f7c2b..6720922dae1f2 100644
--- a/Mathlib/Algebra/Lie/EngelSubalgebra.lean
+++ b/Mathlib/Algebra/Lie/EngelSubalgebra.lean
@@ -127,7 +127,9 @@ lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
     clear hk; revert hy
     generalize k+1 = k
     induction k generalizing y with
-    | zero => cases y; intro hy; simpa using hy
+    | zero =>
+      cases y; intro hy; simp only [pow_zero, LinearMap.one_apply]
+      exact (AddSubmonoid.mk_eq_zero N.toAddSubmonoid).mp hy
     | succ k ih => simp only [pow_succ, LinearMap.mem_ker, LinearMap.mul_apply] at ih ⊢; apply ih
   · rw [← Submodule.map_le_iff_le_comap]
     apply le_sup_of_le_right
@@ -135,7 +137,7 @@ lemma normalizer_eq_self_of_engel_le [IsArtinian R L]
     rintro _ ⟨y, rfl⟩
     simp only [pow_succ', LinearMap.mul_apply, Submodule.mem_comap, mem_coe_submodule]
     apply aux₁
-    simp only [Submodule.coeSubtype, SetLike.coe_mem]
+    simp only [Submodule.coe_subtype, SetLike.coe_mem]
 
 /-- A Lie subalgebra of a Noetherian Lie algebra is nilpotent
 if it is contained in the Engel subalgebra of all its elements. -/
diff --git a/Mathlib/Algebra/Lie/InvariantForm.lean b/Mathlib/Algebra/Lie/InvariantForm.lean
index dea6e6da8c49d..e95fdc8b0f1bf 100644
--- a/Mathlib/Algebra/Lie/InvariantForm.lean
+++ b/Mathlib/Algebra/Lie/InvariantForm.lean
@@ -124,14 +124,14 @@ variable (hΦ_inv : Φ.lieInvariant L) (hΦ_refl : Φ.IsRefl)
 variable (hL : ∀ I : LieIdeal K L, IsAtom I → ¬IsLieAbelian I)
 include hΦ_nondeg hΦ_refl hL
 
-open FiniteDimensional Submodule in
+open Module Submodule in
 lemma orthogonal_isCompl_coe_submodule (I : LieIdeal K L) (hI : IsAtom I) :
     IsCompl I.toSubmodule (orthogonal Φ hΦ_inv I).toSubmodule := by
   rw [orthogonal_toSubmodule, LinearMap.BilinForm.isCompl_orthogonal_iff_disjoint hΦ_refl,
       ← orthogonal_toSubmodule _ hΦ_inv, ← LieSubmodule.disjoint_iff_coe_toSubmodule]
   exact orthogonal_disjoint Φ hΦ_nondeg hΦ_inv hL I hI
 
-open FiniteDimensional Submodule in
+open Module Submodule in
 lemma orthogonal_isCompl (I : LieIdeal K L) (hI : IsAtom I) :
     IsCompl I (orthogonal Φ hΦ_inv I) := by
   rw [LieSubmodule.isCompl_iff_coe_toSubmodule]
@@ -151,7 +151,7 @@ lemma restrict_orthogonal_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
     LinearMap.BilinForm.orthogonal_orthogonal hΦ_nondeg hΦ_refl]
   exact (orthogonal_isCompl_coe_submodule Φ hΦ_nondeg hΦ_inv hΦ_refl hL I hI).symm
 
-open FiniteDimensional Submodule in
+open Module Submodule in
 lemma atomistic : ∀ I : LieIdeal K L, sSup {J : LieIdeal K L | IsAtom J ∧ J ≤ I} = I := by
   intro I
   apply le_antisymm
diff --git a/Mathlib/Algebra/Lie/Killing.lean b/Mathlib/Algebra/Lie/Killing.lean
index 45f1f3365045c..4efe977e0587c 100644
--- a/Mathlib/Algebra/Lie/Killing.lean
+++ b/Mathlib/Algebra/Lie/Killing.lean
@@ -46,7 +46,7 @@ namespace LieAlgebra
 
 NB: This is not standard terminology (the literature does not seem to name Lie algebras with this
 property). -/
-class IsKilling : Prop :=
+class IsKilling : Prop where
   /-- We say a Lie algebra is Killing if its Killing form is non-singular. -/
   killingCompl_top_eq_bot : LieIdeal.killingCompl R L ⊤ = ⊥
 
@@ -111,7 +111,7 @@ lemma isKilling_of_equiv [IsKilling R L] (e : L ≃ₗ⁅R⁆ L') : IsKilling R
   refine ⟨fun hx' ↦ ?_, fun hx y _ ↦ hx ▸ LinearMap.map_zero₂ (killingForm R L') y⟩
   suffices e.symm x' ∈ LinearMap.ker (killingForm R L) by
     rw [IsKilling.ker_killingForm_eq_bot] at this
-    simpa using (e : L ≃ₗ[R] L').congr_arg this
+    simpa [map_zero] using (e : L ≃ₗ[R] L').congr_arg this
   ext y
   replace hx' : ∀ y', killingForm R L' x' y' = 0 := by simpa using hx'
   specialize hx' (e y)
diff --git a/Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.lean b/Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.lean
index 18edee2b23811..124c5f21cd8a8 100644
--- a/Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.lean
+++ b/Mathlib/Algebra/Lie/NonUnitalNonAssocAlgebra.lean
@@ -14,8 +14,8 @@ separate `Mul` typeclass used for general algebras.
 
 It is useful to have a special typeclass for Lie algebras because:
  * it enables us to use the traditional notation `⁅x, y⁆` for the Lie multiplication,
- * associative algebras carry a natural Lie algebra structure via the ring commutator and so we need
-   them to carry both `Mul` and `Bracket` simultaneously,
+ * associative algebras carry a natural Lie algebra structure via the ring commutator and so we
+   need them to carry both `Mul` and `Bracket` simultaneously,
  * more generally, Poisson algebras (not yet defined) need both typeclasses.
 
 However there are times when it is convenient to be able to regard a Lie algebra as a general
@@ -23,8 +23,8 @@ algebra and we provide some basic definitions for doing so here.
 
 ## Main definitions
 
-  * `CommutatorRing` turns a Lie ring into a `NonUnitalNonAssocSemiring` by turning its
-    `Bracket` (denoted `⁅, ⁆`) into a `Mul` (denoted `*`).
+  * `CommutatorRing` turns a Lie ring into a `NonUnitalNonAssocRing` by turning its
+    `Bracket` (denoted `⁅ , ⁆`) into a `Mul` (denoted `*`).
   * `LieHom.toNonUnitalAlgHom`
 
 ## Tags
@@ -37,17 +37,17 @@ universe u v w
 
 variable (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L]
 
-/-- Type synonym for turning a `LieRing` into a `NonUnitalNonAssocSemiring`.
+/-- Type synonym for turning a `LieRing` into a `NonUnitalNonAssocRing`.
 
-A `LieRing` can be regarded as a `NonUnitalNonAssocSemiring` by turning its
+A `LieRing` can be regarded as a `NonUnitalNonAssocRing` by turning its
 `Bracket` (denoted `⁅, ⁆`) into a `Mul` (denoted `*`). -/
 def CommutatorRing (L : Type v) : Type v := L
 
-/-- A `LieRing` can be regarded as a `NonUnitalNonAssocSemiring` by turning its
+/-- A `LieRing` can be regarded as a `NonUnitalNonAssocRing` by turning its
 `Bracket` (denoted `⁅, ⁆`) into a `Mul` (denoted `*`). -/
-instance : NonUnitalNonAssocSemiring (CommutatorRing L) :=
-  show NonUnitalNonAssocSemiring L from
-    { (inferInstance : AddCommMonoid L) with
+instance : NonUnitalNonAssocRing (CommutatorRing L) :=
+  show NonUnitalNonAssocRing L from
+    { (inferInstance : AddCommGroup L) with
       mul := Bracket.bracket
       left_distrib := lie_add
       right_distrib := add_lie
@@ -64,11 +64,11 @@ instance : LieRing (CommutatorRing L) := show LieRing L by infer_instance
 
 instance : LieAlgebra R (CommutatorRing L) := show LieAlgebra R L by infer_instance
 
-/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocSemiring`, we can
+/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocRing`, we can
 reinterpret the `smul_lie` law as an `IsScalarTower`. -/
 instance isScalarTower : IsScalarTower R (CommutatorRing L) (CommutatorRing L) := ⟨smul_lie⟩
 
-/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocSemiring`, we can
+/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocRing`, we can
 reinterpret the `lie_smul` law as an `SMulCommClass`. -/
 instance smulCommClass : SMulCommClass R (CommutatorRing L) (CommutatorRing L) :=
   ⟨fun t x y => (lie_smul t x y).symm⟩
@@ -80,7 +80,7 @@ namespace LieHom
 variable {R L}
 variable {L₂ : Type w} [LieRing L₂] [LieAlgebra R L₂]
 
-/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocSemiring`, we can
+/-- Regarding the `LieRing` of a `LieAlgebra` as a `NonUnitalNonAssocRing`, we can
 regard a `LieHom` as a `NonUnitalAlgHom`. -/
 @[simps]
 def toNonUnitalAlgHom (f : L →ₗ⁅R⁆ L₂) : CommutatorRing L →ₙₐ[R] CommutatorRing L₂ :=
diff --git a/Mathlib/Algebra/Lie/OfAssociative.lean b/Mathlib/Algebra/Lie/OfAssociative.lean
index 363b9a3da015a..10d1f4dcca458 100644
--- a/Mathlib/Algebra/Lie/OfAssociative.lean
+++ b/Mathlib/Algebra/Lie/OfAssociative.lean
@@ -300,7 +300,10 @@ theorem toEnd_comp_subtype_mem (m : M) (hm : m ∈ (N : Submodule R M)) :
 @[simp]
 theorem toEnd_restrict_eq_toEnd (h := N.toEnd_comp_subtype_mem x) :
     (toEnd R L M x).restrict h = toEnd R L N x := by
-  ext; simp [LinearMap.restrict_apply]
+  ext
+  simp only [LinearMap.restrict_coe_apply, toEnd_apply_apply, ← coe_bracket,
+    SetLike.coe_eq_coe]
+  rfl
 
 lemma mapsTo_pow_toEnd_sub_algebraMap {φ : R} {k : ℕ} {x : L} :
     MapsTo ((toEnd R L M x - algebraMap R (Module.End R M) φ) ^ k) N N := by
diff --git a/Mathlib/Algebra/Lie/Rank.lean b/Mathlib/Algebra/Lie/Rank.lean
index 3fabb5568a84c..c9c5877344913 100644
--- a/Mathlib/Algebra/Lie/Rank.lean
+++ b/Mathlib/Algebra/Lie/Rank.lean
@@ -65,13 +65,13 @@ lemma rank_eq_natTrailingDegree [Nontrivial R] [DecidableEq ι] :
     rank R L M = (polyCharpoly φ b).natTrailingDegree := by
   apply nilRank_eq_polyCharpoly_natTrailingDegree
 
-open FiniteDimensional
+open Module
 
 include bₘ in
 lemma rank_le_card [Nontrivial R] : rank R L M ≤ Fintype.card ιₘ :=
   nilRank_le_card _ bₘ
 
-open FiniteDimensional
+open Module
 lemma rank_le_finrank [Nontrivial R] : rank R L M ≤ finrank R M :=
   nilRank_le_finrank _
 
@@ -103,7 +103,7 @@ section IsDomain
 variable (L)
 variable [IsDomain R]
 
-open Cardinal FiniteDimensional MvPolynomial in
+open Cardinal Module MvPolynomial in
 lemma exists_isRegular_of_finrank_le_card (h : finrank R M ≤ #R) :
     ∃ x : L, IsRegular R M x :=
   LinearMap.exists_isNilRegular_of_finrank_le_card _ h
@@ -138,7 +138,7 @@ lemma rank_eq_natTrailingDegree [Nontrivial R] [DecidableEq ι] :
     rank R L = (polyCharpoly (ad R L).toLinearMap b).natTrailingDegree := by
   apply nilRank_eq_polyCharpoly_natTrailingDegree
 
-open FiniteDimensional
+open Module
 
 include b in
 lemma rank_le_card [Nontrivial R] : rank R L ≤ Fintype.card ι :=
@@ -175,7 +175,7 @@ section IsDomain
 variable (L)
 variable [IsDomain R]
 
-open Cardinal FiniteDimensional MvPolynomial in
+open Cardinal Module MvPolynomial in
 lemma exists_isRegular_of_finrank_le_card (h : finrank R L ≤ #R) :
     ∃ x : L, IsRegular R x :=
   LinearMap.exists_isNilRegular_of_finrank_le_card _ h
@@ -191,7 +191,7 @@ namespace LieAlgebra
 
 variable (K : Type*) {L : Type*} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L]
 
-open FiniteDimensional LieSubalgebra
+open Module LieSubalgebra
 
 lemma finrank_engel (x : L) :
     finrank K (engel K x) = (ad K L x).charpoly.natTrailingDegree :=
diff --git a/Mathlib/Algebra/Lie/Semisimple/Basic.lean b/Mathlib/Algebra/Lie/Semisimple/Basic.lean
index 3ff85afe45a6f..ada8114afb6da 100644
--- a/Mathlib/Algebra/Lie/Semisimple/Basic.lean
+++ b/Mathlib/Algebra/Lie/Semisimple/Basic.lean
@@ -140,11 +140,12 @@ lemma isSimple_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : IsSimple R I where
           Submodule.mem_toAddSubmonoid]
         apply add_mem
         -- Now `⁅a, y⁆ ∈ J` since `a ∈ I`, `y ∈ J`, and `J` is an ideal of `I`.
-        · simp only [Submodule.mem_map, LieSubmodule.mem_coeSubmodule, Submodule.coeSubtype,
-            Subtype.exists, exists_and_right, exists_eq_right, ha, lie_mem_left, exists_true_left]
+        · simp only [Submodule.mem_map, LieSubmodule.mem_coeSubmodule, Subtype.exists]
+          erw [Submodule.coe_subtype]
+          simp only [exists_and_right, exists_eq_right, ha, lie_mem_left, exists_true_left]
           exact lie_mem_right R I J ⟨a, ha⟩ y hy
         -- Finally `⁅b, y⁆ = 0`, by the independence of the atoms.
-        · suffices ⁅b, y.val⁆ = 0 by simp only [this, zero_mem]
+        · suffices ⁅b, y.val⁆ = 0 by erw [this]; simp only [zero_mem]
           rw [← LieSubmodule.mem_bot (R := R) (L := L),
               ← (IsSemisimple.setIndependent_isAtom hI).eq_bot]
           exact ⟨lie_mem_right R L I b y y.2, lie_mem_left _ _ _ _ _ hb⟩ }
@@ -157,7 +158,11 @@ lemma isSimple_of_isAtom (I : LieIdeal R L) (hI : IsAtom I) : IsSimple R I where
       rw [eq_bot_iff] at this ⊢
       intro x hx
       suffices x ∈ J → x = 0 from this hx
-      simpa [J'] using @this x.1
+      have := @this x.1
+      simp only [LieIdeal.incl_coe, LieIdeal.coe_to_lieSubalgebra_to_submodule,
+        LieSubmodule.mem_mk_iff', Submodule.mem_map, LieSubmodule.mem_coeSubmodule, Subtype.exists,
+        LieSubmodule.mem_bot, ZeroMemClass.coe_eq_zero, forall_exists_index, and_imp, J'] at this
+      exact fun _ ↦ this (↑x) x.property hx rfl
     -- We need to show that `J = ⊥`.
     -- Since `J` is an ideal of `L`, and `I` is an atom,
     -- it suffices to show that `J < I`.
@@ -301,7 +306,7 @@ theorem subsingleton_of_hasTrivialRadical_lie_abelian [HasTrivialRadical R L] [h
 
 theorem abelian_radical_of_hasTrivialRadical [HasTrivialRadical R L] :
     IsLieAbelian (radical R L) := by
-  rw [HasTrivialRadical.radical_eq_bot]; infer_instance
+  rw [HasTrivialRadical.radical_eq_bot]; exact LieIdeal.isLieAbelian_of_trivial ..
 
 /-- The two properties shown to be equivalent here are possible definitions for a Lie algebra
 to be reductive.
diff --git a/Mathlib/Algebra/Lie/SkewAdjoint.lean b/Mathlib/Algebra/Lie/SkewAdjoint.lean
index bf72da94b1715..4ba8facc11fda 100644
--- a/Mathlib/Algebra/Lie/SkewAdjoint.lean
+++ b/Mathlib/Algebra/Lie/SkewAdjoint.lean
@@ -126,10 +126,9 @@ def skewAdjointMatricesLieSubalgebraEquiv (P : Matrix n n R) (h : Invertible P)
       exact this
     simp [Matrix.IsSkewAdjoint, J.isAdjointPair_equiv _ _ P (isUnit_of_invertible P)]
 
--- TODO(mathlib4#6607): fix elaboration so annotation on `A` isn't needed
 theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
     (A : skewAdjointMatricesLieSubalgebra J) :
-    ↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * (A : Matrix n n R) * P := by
+    ↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * A * P := by
   simp [skewAdjointMatricesLieSubalgebraEquiv]
 
 /-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
diff --git a/Mathlib/Algebra/Lie/Sl2.lean b/Mathlib/Algebra/Lie/Sl2.lean
index 04d470605c27a..0fb0a5807b82b 100644
--- a/Mathlib/Algebra/Lie/Sl2.lean
+++ b/Mathlib/Algebra/Lie/Sl2.lean
@@ -148,7 +148,7 @@ lemma exists_nat [IsNoetherian R M] [NoZeroSMulDivisors R M] [IsDomain R] [CharZ
     {μ - 2 * n | n : ℕ}
     (fun ⟨s, hs⟩ ↦ ψ Classical.choose hs)
     (fun ⟨r, hr⟩ ↦ by simp [lie_h_pow_toEnd_f P, Classical.choose_spec hr, contra,
-      Module.End.HasEigenvector, Module.End.mem_eigenspace_iff])).finite
+      Module.End.hasEigenvector_iff, Module.End.mem_eigenspace_iff])).finite
 
 lemma pow_toEnd_f_ne_zero_of_eq_nat
     [CharZero R] [NoZeroSMulDivisors R M]
diff --git a/Mathlib/Algebra/Lie/Solvable.lean b/Mathlib/Algebra/Lie/Solvable.lean
index 8cbeec856550a..1ce5fd8fdcf58 100644
--- a/Mathlib/Algebra/Lie/Solvable.lean
+++ b/Mathlib/Algebra/Lie/Solvable.lean
@@ -198,11 +198,11 @@ namespace LieAlgebra
 class IsSolvable : Prop where
   solvable : ∃ k, derivedSeries R L k = ⊥
 
-instance isSolvableBot : IsSolvable R (↥(⊥ : LieIdeal R L)) :=
+instance isSolvableBot : IsSolvable R (⊥ : LieIdeal R L) :=
   ⟨⟨0, Subsingleton.elim _ ⊥⟩⟩
 
 instance isSolvableAdd {I J : LieIdeal R L} [hI : IsSolvable R I] [hJ : IsSolvable R J] :
-    IsSolvable R (↥(I + J)) := by
+    IsSolvable R (I + J) := by
   obtain ⟨k, hk⟩ := id hI; obtain ⟨l, hl⟩ := id hJ
   exact ⟨⟨k + l, LieIdeal.derivedSeries_add_eq_bot hk hl⟩⟩
 
@@ -266,8 +266,8 @@ def radical :=
 
 /-- The radical of a Noetherian Lie algebra is solvable. -/
 instance radicalIsSolvable [IsNoetherian R L] : IsSolvable R (radical R L) := by
-  have hwf := (LieSubmodule.wellFoundedGT_of_noetherian R L L).wf
-  rw [← CompleteLattice.isSupClosedCompact_iff_wellFounded] at hwf
+  have hwf := LieSubmodule.wellFoundedGT_of_noetherian R L L
+  rw [← CompleteLattice.isSupClosedCompact_iff_wellFoundedGT] at hwf
   refine hwf { I : LieIdeal R L | IsSolvable R I } ⟨⊥, ?_⟩ fun I hI J hJ => ?_
   · exact LieAlgebra.isSolvableBot R L
   · rw [Set.mem_setOf_eq] at hI hJ ⊢
@@ -287,7 +287,9 @@ instance [IsSolvable R L] : IsSolvable R (⊤ : LieSubalgebra R L) := by
 
 @[simp] lemma radical_eq_top_of_isSolvable [IsSolvable R L] :
     radical R L = ⊤ := by
-  rw [eq_top_iff]; exact le_sSup <| inferInstanceAs (IsSolvable R (⊤ : LieIdeal R L))
+  rw [eq_top_iff]
+  have h : IsSolvable R (⊤ : LieSubalgebra R L) := inferInstance
+  exact le_sSup h
 
 /-- Given a solvable Lie ideal `I` with derived series `I = D₀ ≥ D₁ ≥ ⋯ ≥ Dₖ = ⊥`, this is the
 natural number `k` (the number of inclusions).
@@ -333,11 +335,14 @@ noncomputable def derivedAbelianOfIdeal (I : LieIdeal R L) : LieIdeal R L :=
   | 0 => ⊥
   | k + 1 => derivedSeriesOfIdeal R L k I
 
+instance : Unique {x // x ∈ (⊥ : LieIdeal R L)} :=
+  inferInstanceAs <| Unique {x // x ∈ (⊥ : Submodule R L)}
+
 theorem abelian_derivedAbelianOfIdeal (I : LieIdeal R L) :
     IsLieAbelian (derivedAbelianOfIdeal I) := by
   dsimp only [derivedAbelianOfIdeal]
   cases' h : derivedLengthOfIdeal R L I with k
-  · infer_instance
+  · dsimp; infer_instance
   · rw [derivedSeries_of_derivedLength_succ] at h; exact h.1
 
 theorem derivedLength_zero (I : LieIdeal R L) [hI : IsSolvable R I] :
diff --git a/Mathlib/Algebra/Lie/Subalgebra.lean b/Mathlib/Algebra/Lie/Subalgebra.lean
index 5301d1d9b0348..a1d215735203f 100644
--- a/Mathlib/Algebra/Lie/Subalgebra.lean
+++ b/Mathlib/Algebra/Lie/Subalgebra.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
 import Mathlib.Algebra.Lie.Basic
-import Mathlib.RingTheory.Noetherian
+import Mathlib.RingTheory.Artinian
 
 /-!
 # Lie subalgebras
@@ -107,6 +107,9 @@ instance [SMul R₁ R] [Module R₁ L] [IsScalarTower R₁ R L] (L' : LieSubalge
 instance (L' : LieSubalgebra R L) [IsNoetherian R L] : IsNoetherian R L' :=
   isNoetherian_submodule' _
 
+instance (L' : LieSubalgebra R L) [IsArtinian R L] : IsArtinian R L' :=
+  isArtinian_submodule' _
+
 end
 
 /-- A Lie subalgebra forms a new Lie algebra. -/
@@ -220,7 +223,8 @@ variable [Module R M]
 /-- Given a Lie algebra `L` containing a Lie subalgebra `L' ⊆ L`, together with a Lie module `M` of
 `L`, we may regard `M` as a Lie module of `L'` by restriction. -/
 instance lieModule [LieModule R L M] : LieModule R L' M where
-  smul_lie t x m := by simp only [coe_bracket_of_module, smul_lie, Submodule.coe_smul_of_tower]
+  smul_lie t x m := by
+    rw [coe_bracket_of_module, Submodule.coe_smul_of_tower, smul_lie, coe_bracket_of_module]
   lie_smul t x m := by simp only [coe_bracket_of_module, lie_smul]
 
 /-- An `L`-equivariant map of Lie modules `M → N` is `L'`-equivariant for any Lie subalgebra
@@ -500,7 +504,8 @@ theorem eq_bot_iff : K = ⊥ ↔ ∀ x : L, x ∈ K → x = 0 := by
 instance subsingleton_of_bot : Subsingleton (LieSubalgebra R (⊥ : LieSubalgebra R L)) := by
   apply subsingleton_of_bot_eq_top
   ext ⟨x, hx⟩; change x ∈ ⊥ at hx; rw [LieSubalgebra.mem_bot] at hx; subst hx
-  simp only [true_iff_iff, eq_self_iff_true, Submodule.mk_eq_zero, mem_bot, mem_top]
+  simp only [mem_bot, mem_top, iff_true]
+  rfl
 
 theorem subsingleton_bot : Subsingleton (⊥ : LieSubalgebra R L) :=
   show Subsingleton ((⊥ : LieSubalgebra R L) : Set L) by simp
@@ -555,7 +560,7 @@ theorem coe_ofLe : (ofLe h : Submodule R K') = LinearMap.range (Submodule.inclus
   rfl
 
 /-- Given nested Lie subalgebras `K ⊆ K'`, there is a natural equivalence from `K` to its image in
-`K'`.  -/
+`K'`. -/
 noncomputable def equivOfLe : K ≃ₗ⁅R⁆ ofLe h :=
   (inclusion h).equivRangeOfInjective (inclusion_injective h)
 
diff --git a/Mathlib/Algebra/Lie/Submodule.lean b/Mathlib/Algebra/Lie/Submodule.lean
index 80e77f5112f35..1ce9ee98b69dc 100644
--- a/Mathlib/Algebra/Lie/Submodule.lean
+++ b/Mathlib/Algebra/Lie/Submodule.lean
@@ -72,12 +72,12 @@ instance : Zero (LieSubmodule R L M) :=
 instance : Inhabited (LieSubmodule R L M) :=
   ⟨0⟩
 
-instance coeSubmodule : CoeOut (LieSubmodule R L M) (Submodule R M) :=
+instance (priority := high) coeSort : CoeSort (LieSubmodule R L M) (Type w) where
+  coe N := { x : M // x ∈ N }
+
+instance (priority := mid) coeSubmodule : CoeOut (LieSubmodule R L M) (Submodule R M) :=
   ⟨toSubmodule⟩
 
-instance instCanLiftSubmoduleLieSubmodule : CanLift (Submodule R M) (LieSubmodule R L M) (·)
-    (fun N ↦ ∀ {x : L} {m : M}, m ∈ N → ⁅x, m⁆ ∈ N) where
-  prf N hN := ⟨⟨N, hN⟩, rfl⟩
 
 @[norm_cast]
 theorem coe_toSubmodule : ((N : Submodule R M) : Set M) = N :=
@@ -109,7 +109,7 @@ theorem mem_coe {x : M} : x ∈ (N : Set M) ↔ x ∈ N :=
 protected theorem zero_mem : (0 : M) ∈ N :=
   zero_mem N
 
--- Porting note (#10618): @[simp] can prove this
+@[simp]
 theorem mk_eq_zero {x} (h : x ∈ N) : (⟨x, h⟩ : N) = 0 ↔ x = 0 :=
   Subtype.ext_iff_val
 
@@ -156,17 +156,6 @@ instance : LieRingModule L N where
   lie_add := by intro x m n; apply SetCoe.ext; apply lie_add
   leibniz_lie := by intro x y m; apply SetCoe.ext; apply leibniz_lie
 
-instance module' {S : Type*} [Semiring S] [SMul S R] [Module S M] [IsScalarTower S R M] :
-    Module S N :=
-  N.toSubmodule.module'
-
-instance : Module R N :=
-  N.toSubmodule.module
-
-instance {S : Type*} [Semiring S] [SMul S R] [SMul Sᵐᵒᵖ R] [Module S M] [Module Sᵐᵒᵖ M]
-    [IsScalarTower S R M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] : IsCentralScalar S N :=
-  N.toSubmodule.isCentralScalar
-
 @[simp, norm_cast]
 theorem coe_zero : ((0 : N) : M) = (0 : M) :=
   rfl
@@ -188,9 +177,20 @@ theorem coe_smul (t : R) (m : N) : (↑(t • m) : M) = t • (m : M) :=
   rfl
 
 @[simp, norm_cast]
-theorem coe_bracket (x : L) (m : N) : (↑⁅x, m⁆ : M) = ⁅x, ↑m⁆ :=
+theorem coe_bracket (x : L) (m : N) :
+    (↑⁅x, m⁆ : M) = ⁅x, ↑m⁆ :=
   rfl
 
+-- Copying instances from `Submodule` for correct discrimination keys
+instance [IsNoetherian R M] (N : LieSubmodule R L M) : IsNoetherian R N :=
+  inferInstanceAs <| IsNoetherian R N.toSubmodule
+
+instance [IsArtinian R M] (N : LieSubmodule R L M) : IsArtinian R N :=
+  inferInstanceAs <| IsArtinian R N.toSubmodule
+
+instance [NoZeroSMulDivisors R M] : NoZeroSMulDivisors R N :=
+  inferInstanceAs <| NoZeroSMulDivisors R N.toSubmodule
+
 variable [LieAlgebra R L] [LieModule R L M]
 
 instance instLieModule : LieModule R L N where
@@ -246,7 +246,8 @@ instance LieIdeal.lieRingModule {R L : Type*} [CommRing R] [LieRing L] [LieAlgeb
 
 @[simp]
 theorem LieIdeal.coe_bracket_of_module {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
-    (I : LieIdeal R L) [LieRingModule L M] (x : I) (m : M) : ⁅x, m⁆ = ⁅(↑x : L), m⁆ :=
+    (I : LieIdeal R L) [LieRingModule L M] (x : I) (m : M) :
+    ⁅x, m⁆ = ⁅(↑x : L), m⁆ :=
   LieSubalgebra.coe_bracket_of_module (I : LieSubalgebra R L) x m
 
 /-- Transfer the `LieModule` instance from the coercion `LieIdeal → LieSubalgebra`. -/
@@ -321,6 +322,9 @@ theorem coeSubmodule_le_coeSubmodule : (N : Submodule R M) ≤ N' ↔ N ≤ N' :
 instance : Bot (LieSubmodule R L M) :=
   ⟨0⟩
 
+instance instUniqueBot : Unique (⊥ : LieSubmodule R L M) :=
+  inferInstanceAs <| Unique (⊥ : Submodule R M)
+
 @[simp]
 theorem bot_coe : ((⊥ : LieSubmodule R L M) : Set M) = {0} :=
   rfl
@@ -543,10 +547,11 @@ theorem mem_sup (x : M) : x ∈ N ⊔ N' ↔ ∃ y ∈ N, ∃ z ∈ N', y + z =
 
 nonrec theorem eq_bot_iff : N = ⊥ ↔ ∀ m : M, m ∈ N → m = 0 := by rw [eq_bot_iff]; exact Iff.rfl
 
-instance subsingleton_of_bot : Subsingleton (LieSubmodule R L ↑(⊥ : LieSubmodule R L M)) := by
+instance subsingleton_of_bot : Subsingleton (LieSubmodule R L (⊥ : LieSubmodule R L M)) := by
   apply subsingleton_of_bot_eq_top
-  ext ⟨x, hx⟩; change x ∈ ⊥ at hx; rw [Submodule.mem_bot] at hx; subst hx
-  simp only [true_iff_iff, eq_self_iff_true, Submodule.mk_eq_zero, LieSubmodule.mem_bot, mem_top]
+  ext ⟨_, hx⟩
+  simp only [mem_bot, mk_eq_zero, mem_top, iff_true]
+  exact hx
 
 instance : IsModularLattice (LieSubmodule R L M) where
   sup_inf_le_assoc_of_le _ _ := by
@@ -826,9 +831,9 @@ theorem comap_incl_eq_top : N₂.comap N.incl = ⊤ ↔ N ≤ N₂ := by
     LieSubmodule.top_coeSubmodule, Submodule.comap_subtype_eq_top, coeSubmodule_le_coeSubmodule]
 
 theorem comap_incl_eq_bot : N₂.comap N.incl = ⊥ ↔ N ⊓ N₂ = ⊥ := by
-  simp only [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.coeSubmodule_comap,
-    LieSubmodule.incl_coe, LieSubmodule.bot_coeSubmodule, ← Submodule.disjoint_iff_comap_eq_bot,
-    disjoint_iff, inf_coe_toSubmodule]
+  simp only [← coe_toSubmodule_eq_iff, coeSubmodule_comap, incl_coe, bot_coeSubmodule,
+    inf_coe_toSubmodule]
+  rw [← Submodule.disjoint_iff_comap_eq_bot, disjoint_iff]
 
 @[mono]
 theorem map_mono (h : N ≤ N₂) : N.map f ≤ N₂.map f :=
@@ -983,9 +988,9 @@ same as ideals of `L` contained in `I`. -/
 instance subsingleton_of_bot : Subsingleton (LieIdeal R (⊥ : LieIdeal R L)) := by
   apply subsingleton_of_bot_eq_top
   ext ⟨x, hx⟩
-  rw [LieSubmodule.bot_coeSubmodule, Submodule.mem_bot] at hx
+  rw [LieSubmodule.mem_bot] at hx
   subst hx
-  simp only [Submodule.mk_eq_zero, LieSubmodule.mem_bot, LieSubmodule.mem_top]
+  simp only [LieSubmodule.mk_eq_zero, LieSubmodule.mem_bot, LieSubmodule.mem_top]
 
 end LieIdeal
 
@@ -1044,6 +1049,7 @@ theorem ker_le_comap : f.ker ≤ J.comap f :=
 theorem ker_coeSubmodule : LieSubmodule.toSubmodule (ker f) = LinearMap.ker (f : L →ₗ[R] L') :=
   rfl
 
+variable {f} in
 @[simp]
 theorem mem_ker {x : L} : x ∈ ker f ↔ f x = 0 :=
   show x ∈ LieSubmodule.toSubmodule (f.ker) ↔ _ by
@@ -1151,9 +1157,12 @@ theorem map_sup_ker_eq_map : LieIdeal.map f (I ⊔ f.ker) = LieIdeal.map f I :=
   suffices LieIdeal.map f (I ⊔ f.ker) ≤ LieIdeal.map f I by
     exact le_antisymm this (LieIdeal.map_mono le_sup_left)
   apply LieSubmodule.lieSpan_mono
-  rintro x ⟨y, hy₁, hy₂⟩; rw [← hy₂]
-  erw [LieSubmodule.mem_sup] at hy₁;obtain ⟨z₁, hz₁, z₂, hz₂, hy⟩ := hy₁; rw [← hy]
-  rw [f.coe_toLinearMap, f.map_add, f.mem_ker.mp hz₂, add_zero]; exact ⟨z₁, hz₁, rfl⟩
+  rintro x ⟨y, hy₁, hy₂⟩
+  rw [← hy₂]
+  erw [LieSubmodule.mem_sup] at hy₁
+  obtain ⟨z₁, hz₁, z₂, hz₂, hy⟩ := hy₁
+  rw [← hy]
+  rw [f.coe_toLinearMap, f.map_add, LieHom.mem_ker.mp hz₂, add_zero]; exact ⟨z₁, hz₁, rfl⟩
 
 @[simp]
 theorem map_sup_ker_eq_map' :
@@ -1245,7 +1254,7 @@ theorem ker_eq_bot : f.ker = ⊥ ↔ Function.Injective f := by
 variable {f}
 
 @[simp]
-theorem mem_ker (m : M) : m ∈ f.ker ↔ f m = 0 :=
+theorem mem_ker {m : M} : m ∈ f.ker ↔ f m = 0 :=
   Iff.rfl
 
 @[simp]
@@ -1253,7 +1262,7 @@ theorem ker_id : (LieModuleHom.id : M →ₗ⁅R,L⁆ M).ker = ⊥ :=
   rfl
 
 @[simp]
-theorem comp_ker_incl : f.comp f.ker.incl = 0 := by ext ⟨m, hm⟩; exact (mem_ker m).mp hm
+theorem comp_ker_incl : f.comp f.ker.incl = 0 := by ext ⟨m, hm⟩; exact mem_ker.mp hm
 
 theorem le_ker_iff_map (M' : LieSubmodule R L M) : M' ≤ f.ker ↔ LieSubmodule.map f M' = ⊥ := by
   rw [ker, eq_bot_iff, LieSubmodule.map_le_iff_le_comap]
@@ -1266,11 +1275,11 @@ def range : LieSubmodule R L N :=
   (LieSubmodule.map f ⊤).copy (Set.range f) Set.image_univ.symm
 
 @[simp]
-theorem coe_range : (f.range : Set N) = Set.range f :=
+theorem coe_range : f.range = Set.range f :=
   rfl
 
 @[simp]
-theorem coeSubmodule_range : (f.range : Submodule R N) = LinearMap.range (f : M →ₗ[R] N) :=
+theorem coeSubmodule_range : f.range = LinearMap.range (f : M →ₗ[R] N) :=
   rfl
 
 @[simp]
@@ -1306,13 +1315,17 @@ variable [AddCommGroup M] [Module R M] [LieRingModule L M]
 variable (N : LieSubmodule R L M)
 
 @[simp]
-theorem ker_incl : N.incl.ker = ⊥ := by simp [← LieSubmodule.coe_toSubmodule_eq_iff]
+theorem ker_incl : N.incl.ker = ⊥ := (LieModuleHom.ker_eq_bot N.incl).mpr <| injective_incl N
 
 @[simp]
-theorem range_incl : N.incl.range = N := by simp [← LieSubmodule.coe_toSubmodule_eq_iff]
+theorem range_incl : N.incl.range = N := by
+  simp only [← coe_toSubmodule_eq_iff, LieModuleHom.coeSubmodule_range, incl_coe]
+  rw [Submodule.range_subtype]
 
 @[simp]
-theorem comap_incl_self : comap N.incl N = ⊤ := by simp [← LieSubmodule.coe_toSubmodule_eq_iff]
+theorem comap_incl_self : comap N.incl N = ⊤ := by
+  simp only [← coe_toSubmodule_eq_iff, coeSubmodule_comap, incl_coe, top_coeSubmodule]
+  rw [Submodule.comap_subtype_self]
 
 theorem map_incl_top : (⊤ : LieSubmodule R L N).map N.incl = N := by simp
 
diff --git a/Mathlib/Algebra/Lie/TensorProduct.lean b/Mathlib/Algebra/Lie/TensorProduct.lean
index 9d5d16473476a..0fe6b3f7fb9e8 100644
--- a/Mathlib/Algebra/Lie/TensorProduct.lean
+++ b/Mathlib/Algebra/Lie/TensorProduct.lean
@@ -195,14 +195,14 @@ applying the action of `L` on `M`, we obtain morphism of Lie modules `f : I ⊗
 
 This lemma states that `⁅I, N⁆ = range f`. -/
 theorem lieIdeal_oper_eq_tensor_map_range :
-    ⁅I, N⁆ = ((toModuleHom R L M).comp (mapIncl I N : (↥I) ⊗[R] (↥N) →ₗ⁅R,L⁆ L ⊗[R] M)).range := by
+    ⁅I, N⁆ = ((toModuleHom R L M).comp (mapIncl I N : I ⊗[R] N →ₗ⁅R,L⁆ L ⊗[R] M)).range := by
   rw [← coe_toSubmodule_eq_iff, lieIdeal_oper_eq_linear_span, LieModuleHom.coeSubmodule_range,
     LieModuleHom.coe_linearMap_comp, LinearMap.range_comp, mapIncl_def, coe_linearMap_map,
     TensorProduct.map_range_eq_span_tmul, Submodule.map_span]
   congr; ext m; constructor
   · rintro ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩; use x ⊗ₜ n; constructor
-    · use ⟨x, hx⟩, ⟨n, hn⟩; simp
+    · use ⟨x, hx⟩, ⟨n, hn⟩; rfl
     · simp
-  · rintro ⟨t, ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩, h⟩; rw [← h]; use ⟨x, hx⟩, ⟨n, hn⟩; simp
+  · rintro ⟨t, ⟨⟨x, hx⟩, ⟨n, hn⟩, rfl⟩, h⟩; rw [← h]; use ⟨x, hx⟩, ⟨n, hn⟩; rfl
 
 end LieSubmodule
diff --git a/Mathlib/Algebra/Lie/TraceForm.lean b/Mathlib/Algebra/Lie/TraceForm.lean
index 1d580721035c7..bddebc215744c 100644
--- a/Mathlib/Algebra/Lie/TraceForm.lean
+++ b/Mathlib/Algebra/Lie/TraceForm.lean
@@ -38,7 +38,7 @@ variable (R K L M : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
 local notation "φ" => LieModule.toEnd R L M
 
 open LinearMap (trace)
-open Set FiniteDimensional
+open Set Module
 
 namespace LieModule
 
@@ -226,9 +226,9 @@ lemma traceForm_eq_sum_genWeightSpaceOf
     convert finite_genWeightSpaceOf_ne_bot R L M z
     exact LieSubmodule.coeSubmodule_eq_bot_iff (genWeightSpaceOf M _ _)
   classical
-  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
-    (LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_genWeightSpaceOf R L M z)
-    (IsTriangularizable.iSup_eq_top z)
+  have h := LieSubmodule.independent_iff_coe_toSubmodule.mp <| independent_genWeightSpaceOf R L M z
+  have hds := DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top h <| by
+    simp [← LieSubmodule.iSup_coe_toSubmodule]
   simp only [LinearMap.coeFn_sum, Finset.sum_apply, traceForm_apply_apply,
     LinearMap.trace_eq_sum_trace_restrict' hds hfin hxy]
   exact Finset.sum_congr (by simp) (fun χ _ ↦ rfl)
@@ -276,9 +276,9 @@ lemma lowerCentralSeries_one_inf_center_le_ker_traceForm [Module.Free R M] [Modu
     intro y
     exact y.induction_on rfl (fun a u ↦ by simp [hzc u]) (fun u v hu hv ↦ by simp [hu, hv])
   apply LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero
-  · exact IsTriangularizable.iSup_eq_top (1 ⊗ₜ[R] x)
+  · simpa only [Module.End.maxGenEigenspace_def] using IsTriangularizable.iSup_eq_top (1 ⊗ₜ[R] x)
   · exact fun μ ↦ trace_toEnd_eq_zero_of_mem_lcs A (A ⊗[R] L)
-      (genWeightSpaceOf (A ⊗[R] M) μ (1 ⊗ₜ x)) (le_refl 1) hz
+      (genWeightSpaceOf (A ⊗[R] M) μ ((1:A) ⊗ₜ[R] x)) (le_refl 1) hz
   · exact commute_toEnd_of_mem_center_right (A ⊗[R] M) hzc (1 ⊗ₜ x)
 
 /-- A nilpotent Lie algebra with a representation whose trace form is non-singular is Abelian. -/
@@ -302,7 +302,10 @@ variable [IsDomain R] [IsPrincipalIdealRing R]
 lemma trace_eq_trace_restrict_of_le_idealizer
     (hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := fun m _ ↦ N.lie_mem (N.mem_idealizer.mp (h hy) m)) :
     trace R M (φ x ∘ₗ φ y) = trace R N ((φ x ∘ₗ φ y).restrict hy') := by
-  suffices ∀ m, ⁅x, ⁅y, m⁆⁆ ∈ N by simp [(φ x ∘ₗ φ y).trace_restrict_eq_of_forall_mem _ this]
+  suffices ∀ m, ⁅x, ⁅y, m⁆⁆ ∈ N by
+    have : (trace R { x // x ∈ N }) ((φ x ∘ₗ φ y).restrict _) = (trace R M) (φ x ∘ₗ φ y) :=
+      (φ x ∘ₗ φ y).trace_restrict_eq_of_forall_mem _ this
+    simp [this]
   exact fun m ↦ N.lie_mem (h hy m)
 
 include h in
@@ -322,7 +325,7 @@ lemma traceForm_eq_zero_of_isTrivial [LieModule.IsTrivial I N] :
   let hy' : ∀ m ∈ N, (φ x ∘ₗ φ y) m ∈ N := fun m _ ↦ N.lie_mem (N.mem_idealizer.mp (h hy) m)
   suffices (φ x ∘ₗ φ y).restrict hy' = 0 by
     simp [this, N.trace_eq_trace_restrict_of_le_idealizer I h x hy]
-  ext n
+  ext (n : N)
   suffices ⁅y, (n : M)⁆ = 0 by simp [this]
   exact Submodule.coe_eq_zero.mpr (LieModule.IsTrivial.trivial (⟨y, hy⟩ : I) n)
 
@@ -389,7 +392,7 @@ lemma killingForm_eq :
 
 end LieIdeal
 
-open LieModule FiniteDimensional
+open LieModule Module
 open Submodule (span subset_span)
 
 namespace LieModule
@@ -410,6 +413,8 @@ lemma traceForm_eq_sum_finrank_nsmul_mul (x y : L) :
     ← traceForm_genWeightSpace_eq K L M _ x y]
   rfl
 
+/-- See also `LieModule.traceForm_eq_sum_finrank_nsmul'` for an expression omitting the zero
+weights. -/
 lemma traceForm_eq_sum_finrank_nsmul :
     traceForm K L M = ∑ χ : Weight K L M, finrank K (genWeightSpace M χ) •
       (χ : L →ₗ[K] K).smulRight (χ : L →ₗ[K] K) := by
@@ -417,6 +422,21 @@ lemma traceForm_eq_sum_finrank_nsmul :
   rw [traceForm_eq_sum_finrank_nsmul_mul, ← Finset.sum_attach]
   simp
 
+/-- A variant of `LieModule.traceForm_eq_sum_finrank_nsmul` in which the sum is taken only over the
+non-zero weights. -/
+lemma traceForm_eq_sum_finrank_nsmul' :
+    traceForm K L M = ∑ χ in {χ : Weight K L M | χ.IsNonZero}, finrank K (genWeightSpace M χ) •
+      (χ : L →ₗ[K] K).smulRight (χ : L →ₗ[K] K) := by
+  classical
+  suffices ∑ χ in {χ : Weight K L M | χ.IsZero}, finrank K (genWeightSpace M χ) •
+      (χ : L →ₗ[K] K).smulRight (χ : L →ₗ[K] K) = 0 by
+    rw [traceForm_eq_sum_finrank_nsmul,
+      ← Finset.sum_filter_add_sum_filter_not (p := fun χ : Weight K L M ↦ χ.IsNonZero)]
+    simp [this]
+  refine Finset.sum_eq_zero fun χ hχ ↦ ?_
+  replace hχ : (χ : L →ₗ[K] K) = 0 := by simpa [← Weight.coe_toLinear_eq_zero_iff] using hχ
+  simp [hχ]
+
 -- The reverse inclusion should also hold: TODO prove this!
 lemma range_traceForm_le_span_weight :
     LinearMap.range (traceForm K L M) ≤ span K (range (Weight.toLinear K L M)) := by
diff --git a/Mathlib/Algebra/Lie/Weights/Basic.lean b/Mathlib/Algebra/Lie/Weights/Basic.lean
index ca39dad256a93..e0f030da4cacc 100644
--- a/Mathlib/Algebra/Lie/Weights/Basic.lean
+++ b/Mathlib/Algebra/Lie/Weights/Basic.lean
@@ -165,7 +165,7 @@ theorem mem_genWeightSpaceOf (χ : R) (x : L) (m : M) :
 
 theorem coe_genWeightSpaceOf_zero (x : L) :
     ↑(genWeightSpaceOf M (0 : R) x) = ⨆ k, LinearMap.ker (toEnd R L M x ^ k) := by
-  simp [genWeightSpaceOf, Module.End.maxGenEigenspace]
+  simp [genWeightSpaceOf, Module.End.maxGenEigenspace_def]
 
 /-- If `M` is a representation of a nilpotent Lie algebra `L`
 and `χ : L → R` is a family of scalars,
@@ -247,6 +247,8 @@ abbrev IsNonZero (χ : Weight R L M) := ¬ IsZero (χ : Weight R L M)
 lemma isNonZero_iff_ne_zero [Nontrivial (genWeightSpace M (0 : L → R))] {χ : Weight R L M} :
     χ.IsNonZero ↔ χ ≠ 0 := isZero_iff_eq_zero.not
 
+noncomputable instance : DecidablePred (IsNonZero (R := R) (L := L) (M := M)) := Classical.decPred _
+
 variable (R L M) in
 /-- The set of weights is equivalent to a subtype. -/
 def equivSetOf : Weight R L M ≃ {χ : L → R | genWeightSpace M χ ≠ ⊥} where
@@ -265,7 +267,7 @@ lemma genWeightSpaceOf_ne_bot (χ : Weight R L M) (x : L) :
 lemma hasEigenvalueAt (χ : Weight R L M) (x : L) :
     (toEnd R L M x).HasEigenvalue (χ x) := by
   obtain ⟨k : ℕ, hk : (toEnd R L M x).genEigenspace (χ x) k ≠ ⊥⟩ := by
-    simpa [Module.End.maxGenEigenspace, genWeightSpaceOf] using χ.genWeightSpaceOf_ne_bot x
+    simpa [genWeightSpaceOf, Module.End.maxGenEigenspace_def] using χ.genWeightSpaceOf_ne_bot x
   exact Module.End.hasEigenvalue_of_hasGenEigenvalue hk
 
 lemma apply_eq_zero_of_isNilpotent [NoZeroSMulDivisors R M] [IsReduced R]
@@ -306,7 +308,7 @@ theorem exists_genWeightSpace_le_ker_of_isNoetherian [IsNoetherian R M] (χ : L
   intro m hm
   replace hm : m ∈ (toEnd R L M x).maxGenEigenspace (χ x) :=
     genWeightSpace_le_genWeightSpaceOf M x χ hm
-  rwa [Module.End.maxGenEigenspace_eq] at hm
+  rwa [Module.End.maxGenEigenspace_eq, Module.End.genEigenspace_def] at hm
 
 variable (R) in
 theorem exists_genWeightSpace_zero_le_ker_of_isNoetherian
@@ -323,7 +325,9 @@ lemma isNilpotent_toEnd_sub_algebraMap [IsNoetherian R M] (χ : L → R) (x : L)
   obtain ⟨k, hk⟩ := exists_genWeightSpace_le_ker_of_isNoetherian M χ x
   use k
   ext ⟨m, hm⟩
-  simpa [this, LinearMap.pow_restrict _, LinearMap.restrict_apply] using hk hm
+  simp only [this, LinearMap.pow_restrict _, LinearMap.zero_apply, ZeroMemClass.coe_zero,
+    ZeroMemClass.coe_eq_zero]
+  exact ZeroMemClass.coe_eq_zero.mp (hk hm)
 
 /-- A (nilpotent) Lie algebra acts nilpotently on the zero weight space of a Noetherian Lie
 module. -/
@@ -626,6 +630,8 @@ end fitting_decomposition
 lemma disjoint_genWeightSpaceOf [NoZeroSMulDivisors R M] {x : L} {φ₁ φ₂ : R} (h : φ₁ ≠ φ₂) :
     Disjoint (genWeightSpaceOf M φ₁ x) (genWeightSpaceOf M φ₂ x) := by
   rw [LieSubmodule.disjoint_iff_coe_toSubmodule]
+  dsimp [genWeightSpaceOf]
+  simp_rw [Module.End.maxGenEigenspace_def]
   exact Module.End.disjoint_iSup_genEigenspace _ h
 
 lemma disjoint_genWeightSpace [NoZeroSMulDivisors R M] {χ₁ χ₂ : L → R} (h : χ₁ ≠ χ₂) :
@@ -695,17 +701,17 @@ lemma independent_genWeightSpace' [NoZeroSMulDivisors R M] :
 lemma independent_genWeightSpaceOf [NoZeroSMulDivisors R M] (x : L) :
     CompleteLattice.Independent fun (χ : R) ↦ genWeightSpaceOf M χ x := by
   rw [LieSubmodule.independent_iff_coe_toSubmodule]
+  dsimp [genWeightSpaceOf]
+  simp_rw [Module.End.maxGenEigenspace_def]
   exact (toEnd R L M x).independent_genEigenspace
 
 lemma finite_genWeightSpaceOf_ne_bot [NoZeroSMulDivisors R M] [IsNoetherian R M] (x : L) :
     {χ : R | genWeightSpaceOf M χ x ≠ ⊥}.Finite :=
-  CompleteLattice.WellFounded.finite_ne_bot_of_independent
-    IsWellFounded.wf (independent_genWeightSpaceOf R L M x)
+  CompleteLattice.WellFoundedGT.finite_ne_bot_of_independent (independent_genWeightSpaceOf R L M x)
 
 lemma finite_genWeightSpace_ne_bot [NoZeroSMulDivisors R M] [IsNoetherian R M] :
     {χ : L → R | genWeightSpace M χ ≠ ⊥}.Finite :=
-  CompleteLattice.WellFounded.finite_ne_bot_of_independent
-    IsWellFounded.wf (independent_genWeightSpace R L M)
+  CompleteLattice.WellFoundedGT.finite_ne_bot_of_independent (independent_genWeightSpace R L M)
 
 instance Weight.instFinite [NoZeroSMulDivisors R M] [IsNoetherian R M] :
     Finite (Weight R L M) := by
@@ -718,7 +724,7 @@ noncomputable instance Weight.instFintype [NoZeroSMulDivisors R M] [IsNoetherian
 
 /-- A Lie module `M` of a Lie algebra `L` is triangularizable if the endomorhpism of `M` defined by
 any `x : L` is triangularizable. -/
-class IsTriangularizable : Prop :=
+class IsTriangularizable : Prop where
   iSup_eq_top : ∀ x, ⨆ φ, ⨆ k, (toEnd R L M x).genEigenspace φ k = ⊤
 
 instance (L' : LieSubalgebra R L) [IsTriangularizable R L M] : IsTriangularizable R L' M where
@@ -735,9 +741,11 @@ lemma iSup_genWeightSpaceOf_eq_top [IsTriangularizable R L M] (x : L) :
     ⨆ (φ : R), genWeightSpaceOf M φ x = ⊤ := by
   rw [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.iSup_coe_toSubmodule,
     LieSubmodule.top_coeSubmodule]
+  dsimp [genWeightSpaceOf]
+  simp_rw [Module.End.maxGenEigenspace_def]
   exact IsTriangularizable.iSup_eq_top x
 
-open LinearMap FiniteDimensional in
+open LinearMap Module in
 @[simp]
 lemma trace_toEnd_genWeightSpace [IsDomain R] [IsPrincipalIdealRing R]
     [Module.Free R M] [Module.Finite R M] (χ : L → R) (x : L) :
@@ -751,7 +759,7 @@ lemma trace_toEnd_genWeightSpace [IsDomain R] [IsPrincipalIdealRing R]
 
 section field
 
-open FiniteDimensional
+open Module
 
 variable (K)
 variable [Field K] [LieAlgebra K L] [Module K M] [LieModule K L M] [LieAlgebra.IsNilpotent K L]
@@ -770,37 +778,12 @@ instance (N : LieSubmodule K L M) [IsTriangularizable K L M] : IsTriangularizabl
 See also `LieModule.iSup_genWeightSpace_eq_top'`. -/
 lemma iSup_genWeightSpace_eq_top [IsTriangularizable K L M] :
     ⨆ χ : L → K, genWeightSpace M χ = ⊤ := by
-  generalize h_dim : finrank K M = n
-  induction n using Nat.strongInductionOn generalizing M with | ind n ih => ?_
-  obtain h' | ⟨y : L, hy : ¬ ∃ φ, genWeightSpaceOf M φ y = ⊤⟩ :=
-    forall_or_exists_not (fun (x : L) ↦ ∃ (φ : K), genWeightSpaceOf M φ x = ⊤)
-  · choose χ hχ using h'
-    replace hχ : genWeightSpace M χ = ⊤ := by simpa only [genWeightSpace, hχ] using iInf_top
-    exact eq_top_iff.mpr <| hχ ▸ le_iSup (genWeightSpace M) χ
-  · replace hy : ∀ φ, finrank K (genWeightSpaceOf M φ y) < n := fun φ ↦ by
-      simp_rw [not_exists, ← lt_top_iff_ne_top] at hy; exact h_dim ▸ Submodule.finrank_lt (hy φ)
-    replace ih : ∀ φ, ⨆ χ : L → K, genWeightSpace (genWeightSpaceOf M φ y) χ = ⊤ :=
-      fun φ ↦ ih _ (hy φ) (genWeightSpaceOf M φ y) rfl
-    replace ih : ∀ φ, ⨆ (χ : L → K) (_ : χ y = φ),
-        genWeightSpace (genWeightSpaceOf M φ y) χ = ⊤ := by
-      intro φ
-      suffices ∀ χ : L → K, χ y ≠ φ → genWeightSpace (genWeightSpaceOf M φ y) χ = ⊥ by
-        specialize ih φ; rw [iSup_split, biSup_congr this] at ih; simpa using ih
-      intro χ hχ
-      rw [eq_bot_iff, ← (genWeightSpaceOf M φ y).ker_incl, LieModuleHom.ker,
-        ← LieSubmodule.map_le_iff_le_comap, map_genWeightSpace_eq_of_injective
-        (genWeightSpaceOf M φ y).injective_incl, LieSubmodule.range_incl, ← disjoint_iff_inf_le]
-      exact (disjoint_genWeightSpaceOf K L M hχ).mono_left
-        (genWeightSpace_le_genWeightSpaceOf M y χ)
-    replace ih : ∀ φ, ⨆ (χ : L → K) (_ : χ y = φ), genWeightSpace M χ = genWeightSpaceOf M φ y := by
-      intro φ
-      have (χ : L → K) (hχ : χ y = φ) : genWeightSpace M χ =
-          (genWeightSpace (genWeightSpaceOf M φ y) χ).map (genWeightSpaceOf M φ y).incl := by
-        rw [← hχ, genWeightSpace_genWeightSpaceOf_map_incl]
-      simp_rw [biSup_congr this, ← LieSubmodule.map_iSup, ih, LieModuleHom.map_top,
-        LieSubmodule.range_incl]
-    simpa only [← ih, iSup_comm (ι := K), iSup_iSup_eq_right] using
-      iSup_genWeightSpaceOf_eq_top K L M y
+  simp only [← LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.iSup_coe_toSubmodule,
+    LieSubmodule.iInf_coe_toSubmodule, LieSubmodule.top_coeSubmodule, genWeightSpace]
+  refine Module.End.iSup_iInf_maxGenEigenspace_eq_top_of_forall_mapsTo (toEnd K L M)
+    (fun x y φ z ↦ (genWeightSpaceOf M φ y).lie_mem) ?_
+  simp_rw [Module.End.maxGenEigenspace_def]
+  apply IsTriangularizable.iSup_eq_top
 
 lemma iSup_genWeightSpace_eq_top' [IsTriangularizable K L M] :
     ⨆ χ : Weight K L M, genWeightSpace M χ = ⊤ := by
diff --git a/Mathlib/Algebra/Lie/Weights/Cartan.lean b/Mathlib/Algebra/Lie/Weights/Cartan.lean
index 6dbc65cb2e75f..0ef812f460f9e 100644
--- a/Mathlib/Algebra/Lie/Weights/Cartan.lean
+++ b/Mathlib/Algebra/Lie/Weights/Cartan.lean
@@ -88,7 +88,7 @@ def rootSpaceWeightSpaceProductAux {χ₁ χ₂ χ₃ : H → R} (hχ : χ₁ +
     { toFun := fun m =>
         ⟨⁅(x : L), (m : M)⁆,
           hχ ▸ lie_mem_genWeightSpace_of_mem_genWeightSpace x.property m.property⟩
-      map_add' := fun m n => by simp only [LieSubmodule.coe_add, lie_add]; rfl
+      map_add' := fun m n => by simp only [LieSubmodule.coe_add, lie_add, AddMemClass.mk_add_mk]
       map_smul' := fun t m => by
         dsimp only
         conv_lhs =>
@@ -97,8 +97,8 @@ def rootSpaceWeightSpaceProductAux {χ₁ χ₂ χ₃ : H → R} (hχ : χ₁ +
         rfl }
   map_add' x y := by
     ext m
-    simp only [AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, add_lie, LinearMap.coe_mk,
-      AddHom.coe_mk, LinearMap.add_apply, AddSubmonoid.mk_add_mk]
+    simp only [LieSubmodule.coe_add, add_lie, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply,
+      AddMemClass.mk_add_mk]
   map_smul' t x := by
     simp only [RingHom.id_apply]
     ext m
@@ -115,9 +115,9 @@ def rootSpaceWeightSpaceProduct (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ
     { toLinearMap := rootSpaceWeightSpaceProductAux R L H M hχ
       map_lie' := fun {x y} => by
         ext m
-        simp only [rootSpaceWeightSpaceProductAux, LieSubmodule.coe_bracket,
-          LieSubalgebra.coe_bracket_of_module, lie_lie, LinearMap.coe_mk, AddHom.coe_mk,
-          Subtype.coe_mk, LieHom.lie_apply, LieSubmodule.coe_sub] }
+        simp only [rootSpaceWeightSpaceProductAux]
+        dsimp
+        simp only [LieSubalgebra.coe_bracket_of_module, lie_lie] }
 
 @[simp]
 theorem coe_rootSpaceWeightSpaceProduct_tmul (χ₁ χ₂ χ₃ : H → R) (hχ : χ₁ + χ₂ = χ₃)
@@ -270,7 +270,9 @@ lemma mem_corootSpace {x : H} :
   have : x ∈ corootSpace α ↔
       (x : L) ∈ LieSubmodule.map H.toLieSubmodule.incl (corootSpace α) := by
     rw [corootSpace]
-    simpa using exists_congr fun _ ↦ H.toLieSubmodule.injective_incl.eq_iff.symm
+    simp only [rootSpaceProduct_def, LieModuleHom.mem_range, LieSubmodule.mem_map,
+      LieSubmodule.incl_apply, SetLike.coe_eq_coe, exists_eq_right]
+    rfl
   simp_rw [this, corootSpace, ← LieModuleHom.map_top, ← LieSubmodule.mem_coeSubmodule,
     LieSubmodule.coeSubmodule_map, LieSubmodule.top_coeSubmodule, ← TensorProduct.span_tmul_eq_top,
     LinearMap.map_span, Set.image, Set.mem_setOf_eq, exists_exists_exists_and_eq]
@@ -287,13 +289,14 @@ lemma mem_corootSpace' {x : H} :
     erw [← (H : Submodule R L).injective_subtype.mem_set_image (s := Submodule.span R s)]
     rw [mem_image]
     simp_rw [SetLike.mem_coe]
-    rw [← Submodule.mem_map, Submodule.coeSubtype, Submodule.map_span, mem_corootSpace, ← this]
+    rw [← Submodule.mem_map, Submodule.coe_subtype, Submodule.map_span, mem_corootSpace, ← this]
   ext u
-  simp only [Submodule.coeSubtype, mem_image, Subtype.exists, LieSubalgebra.mem_coe_submodule,
+  simp only [Submodule.coe_subtype, mem_image, Subtype.exists, LieSubalgebra.mem_coe_submodule,
     exists_and_right, exists_eq_right, mem_setOf_eq, s]
   refine ⟨fun ⟨_, y, hy, z, hz, hyz⟩ ↦ ⟨y, hy, z, hz, hyz⟩,
     fun ⟨y, hy, z, hz, hyz⟩ ↦ ⟨?_, y, hy, z, hz, hyz⟩⟩
-  convert (rootSpaceProduct R L H α (-α) 0 (add_neg_cancel α) (⟨y, hy⟩ ⊗ₜ[R] ⟨z, hz⟩)).property
+  convert
+    (rootSpaceProduct R L H α (-α) 0 (add_neg_cancel α) (⟨y, hy⟩ ⊗ₜ[R] ⟨z, hz⟩)).property using 0
   simp [hyz]
 
 end LieAlgebra
diff --git a/Mathlib/Algebra/Lie/Weights/Chain.lean b/Mathlib/Algebra/Lie/Weights/Chain.lean
index e810a89aeeb6d..428b6ff73884c 100644
--- a/Mathlib/Algebra/Lie/Weights/Chain.lean
+++ b/Mathlib/Algebra/Lie/Weights/Chain.lean
@@ -41,7 +41,7 @@ We provide basic definitions and results to support `α`-chain techniques in thi
 
 -/
 
-open FiniteDimensional Function Set
+open Module Function Set
 
 variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
   (M : Type*) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
@@ -173,7 +173,11 @@ lemma trace_toEnd_genWeightSpaceChain_eq_zero
           lie_mem_genWeightSpaceChain_of_genWeightSpace_eq_bot_left M α χ p q hp hz hm⟩
         map_add' := fun _ _ ↦ by simp
         map_smul' := fun t m ↦ by simp }
-    have hfg : toEnd R H _ u = ⁅f, g⁆ := by ext; simp [f, g, ← hyz]
+    have hfg : toEnd R H _ u = ⁅f, g⁆ := by
+      ext
+      rw [toEnd_apply_apply, LieSubmodule.coe_bracket, LieSubalgebra.coe_bracket_of_module, ← hyz]
+      simp only [lie_lie, LieHom.lie_apply, LinearMap.coe_mk, AddHom.coe_mk, Module.End.lie_apply,
+      AddSubgroupClass.coe_sub, f, g]
     simp [hfg]
   · simp
   · simp_all
@@ -208,10 +212,16 @@ lemma exists_forall_mem_corootSpace_smul_add_eq_zero
   have h₃ : genWeightSpaceChain M α χ p q = ⨆ i ∈ Finset.Ioo p q, N i := by
     simp_rw [genWeightSpaceChain_def', LieSubmodule.iSup_coe_toSubmodule]
   rw [← trace_toEnd_genWeightSpaceChain_eq_zero M α χ p q hp hq hx,
-    ← LieSubmodule.toEnd_restrict_eq_toEnd,
-    LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup _ h₁ h₂ (genWeightSpaceChain M α χ p q) h₃]
-  simp_rw [LieSubmodule.toEnd_restrict_eq_toEnd,
-    trace_toEnd_genWeightSpace, Pi.add_apply, Pi.smul_apply, smul_add, ← smul_assoc,
+    ← LieSubmodule.toEnd_restrict_eq_toEnd]
+  -- The lines below illustrate the cost of treating `LieSubmodule` as both a
+  -- `Submodule` and a `LieSubmodule` simultaneously.
+  erw [LinearMap.trace_eq_sum_trace_restrict_of_eq_biSup _ h₁ h₂ (genWeightSpaceChain M α χ p q) h₃]
+  simp_rw [LieSubmodule.toEnd_restrict_eq_toEnd]
+  dsimp [N]
+  convert_to _ =
+    ∑ k ∈ Finset.Ioo p q, (LinearMap.trace R { x // x ∈ (genWeightSpace M (k • α + χ)) })
+      ((toEnd R { x // x ∈ H } { x // x ∈ genWeightSpace M (k • α + χ) }) x)
+  simp_rw [trace_toEnd_genWeightSpace, Pi.add_apply, Pi.smul_apply, smul_add, ← smul_assoc,
     Finset.sum_add_distrib, ← Finset.sum_smul, natCast_zsmul]
 
 end IsCartanSubalgebra
diff --git a/Mathlib/Algebra/Lie/Weights/Killing.lean b/Mathlib/Algebra/Lie/Weights/Killing.lean
index 03e0421bfd2dc..80e8ca2ed6bf3 100644
--- a/Mathlib/Algebra/Lie/Weights/Killing.lean
+++ b/Mathlib/Algebra/Lie/Weights/Killing.lean
@@ -85,7 +85,7 @@ end IsKilling
 
 section Field
 
-open FiniteDimensional LieModule Set
+open Module LieModule Set
 open Submodule (span subset_span)
 
 variable [FiniteDimensional K L] (H : LieSubalgebra K L) [H.IsCartanSubalgebra]
@@ -146,7 +146,9 @@ Over a perfect field a much stronger result is true, see
 `LieAlgebra.IsKilling.isSemisimple_ad_of_mem_isCartanSubalgebra`. -/
 lemma eq_zero_of_isNilpotent_ad_of_mem_isCartanSubalgebra {x : L} (hx : x ∈ H)
     (hx' : _root_.IsNilpotent (ad K L x)) : x = 0 := by
-  suffices ⟨x, hx⟩ ∈ LinearMap.ker (traceForm K H L) by simpa using this
+  suffices ⟨x, hx⟩ ∈ LinearMap.ker (traceForm K H L) by
+    simp at this
+    exact (AddSubmonoid.mk_eq_zero H.toAddSubmonoid).mp this
   simp only [LinearMap.mem_ker]
   ext y
   have comm : Commute (toEnd K H L ⟨x, hx⟩) (toEnd K H L y) := by
@@ -340,7 +342,8 @@ lemma coe_corootSpace_eq_span_singleton' (α : Weight K H L) :
     rw [Submodule.mem_span_singleton] at this ⊢
     obtain ⟨t, rfl⟩ := this
     use t
-    simp [Subtype.ext_iff]
+    simp only [Subtype.ext_iff]
+    rw [Submodule.coe_smul_of_tower]
   · simp only [Submodule.span_singleton_le_iff_mem, LieSubmodule.mem_coeSubmodule]
     exact cartanEquivDual_symm_apply_mem_corootSpace α
 
@@ -484,7 +487,9 @@ lemma exists_isSl2Triple_of_weight_isNonZero {α : Weight K H L} (hα : α.IsNon
   have hef := lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg heα hfα
   let h : H := ⟨⁅e, f'⁆, hef ▸ Submodule.smul_mem _ _ (Submodule.coe_mem _)⟩
   have hh : α h ≠ 0 := by
-    have : h = killingForm K L e f' • (cartanEquivDual H).symm α := by simp [Subtype.ext_iff, hef]
+    have : h = killingForm K L e f' • (cartanEquivDual H).symm α := by
+      simp only [Subtype.ext_iff, hef]
+      rw [Submodule.coe_smul_of_tower]
     rw [this, map_smul, smul_eq_mul, ne_eq, mul_eq_zero, not_or]
     exact ⟨hf, root_apply_cartanEquivDual_symm_ne_zero hα⟩
   let f := (2 * (α h)⁻¹) • f'
@@ -517,7 +522,8 @@ lemma _root_.IsSl2Triple.h_eq_coroot {α : Weight K H L} (hα : α.IsNonZero)
     rwa [this, one_smul] at hs
   set α' := (cartanEquivDual H).symm α with hα'
   have h_eq : h = killingForm K L e f • α' := by
-    simp only [hα', Subtype.ext_iff, Submodule.coe_smul_of_tower, ← ht.lie_e_f, hef]
+    simp only [hα', Subtype.ext_iff, ← ht.lie_e_f, hef]
+    rw [Submodule.coe_smul_of_tower]
   use (2 • (α α')⁻¹) * (killingForm K L e f)⁻¹
   have hef₀ : killingForm K L e f ≠ 0 := by
     have := ht.h_ne_zero
@@ -528,7 +534,9 @@ lemma _root_.IsSl2Triple.h_eq_coroot {α : Weight K H L} (hα : α.IsNonZero)
 lemma finrank_rootSpace_eq_one (α : Weight K H L) (hα : α.IsNonZero) :
     finrank K (rootSpace H α) = 1 := by
   suffices ¬ 1 < finrank K (rootSpace H α) by
-    have h₀ : finrank K (rootSpace H α) ≠ 0 := by simpa using α.genWeightSpace_ne_bot
+    have h₀ : finrank K (rootSpace H α) ≠ 0 := by
+      convert_to finrank K (rootSpace H α).toSubmodule ≠ 0
+      simpa using α.genWeightSpace_ne_bot
     omega
   intro contra
   obtain ⟨h, e, f, ht, heα, hfα⟩ := exists_isSl2Triple_of_weight_isNonZero hα
@@ -539,7 +547,7 @@ lemma finrank_rootSpace_eq_one (α : Weight K H L) (hα : α.IsNonZero) :
     have : killingForm K L y f = 0 := by simpa [F, traceForm_comm] using hy
     simpa [this] using lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg hyα hfα
   have P : ht.symm.HasPrimitiveVectorWith y (-2 : K) :=
-    { ne_zero := by simpa using hy₀
+    { ne_zero := by simpa [LieSubmodule.mk_eq_zero] using hy₀
       lie_h := by simp only [neg_smul, neg_lie, neg_inj, ht.h_eq_coroot hα heα hfα,
         ← H.coe_bracket_of_module, lie_eq_smul_of_mem_rootSpace hyα (coroot α),
         root_apply_coroot hα]
@@ -548,6 +556,16 @@ lemma finrank_rootSpace_eq_one (α : Weight K H L) (hα : α.IsNonZero) :
   replace hn : -2 = (n : ℤ) := by norm_cast at hn
   omega
 
+/-- The collection of roots as a `Finset`. -/
+noncomputable abbrev _root_.LieSubalgebra.root : Finset (Weight K H L) := {α | α.IsNonZero}
+
+lemma restrict_killingForm_eq_sum :
+    (killingForm K L).restrict H = ∑ α in H.root, (α : H →ₗ[K] K).smulRight (α : H →ₗ[K] K) := by
+  rw [restrict_killingForm, traceForm_eq_sum_finrank_nsmul' K H L]
+  refine Finset.sum_congr rfl fun χ hχ ↦ ?_
+  replace hχ : χ.IsNonZero := by simpa [LieSubalgebra.root] using hχ
+  simp [finrank_rootSpace_eq_one _ hχ]
+
 end CharZero
 
 end IsKilling
diff --git a/Mathlib/Algebra/Lie/Weights/Linear.lean b/Mathlib/Algebra/Lie/Weights/Linear.lean
index dc963d3db3a9c..741b68fdabb92 100644
--- a/Mathlib/Algebra/Lie/Weights/Linear.lean
+++ b/Mathlib/Algebra/Lie/Weights/Linear.lean
@@ -48,7 +48,7 @@ namespace LieModule
 
 /-- A typeclass encoding the fact that a given Lie module has linear weights, vanishing on the
 derived ideal. -/
-class LinearWeights [LieAlgebra.IsNilpotent R L] : Prop :=
+class LinearWeights [LieAlgebra.IsNilpotent R L] : Prop where
   map_add : ∀ χ : L → R, genWeightSpace M χ ≠ ⊥ → ∀ x y, χ (x + y) = χ x + χ y
   map_smul : ∀ χ : L → R, genWeightSpace M χ ≠ ⊥ → ∀ (t : R) x, χ (t • x) = t • χ x
   map_lie : ∀ χ : L → R, genWeightSpace M χ ≠ ⊥ → ∀ x y : L, χ ⁅x, y⁆ = 0
@@ -98,13 +98,15 @@ instance instLinearWeightsOfIsLieAbelian [IsLieAbelian L] [NoZeroSMulDivisors R
       rw [commute_iff_lie_eq, ← LieHom.map_lie, trivial_lie_zero, LieHom.map_zero]
     intro χ hχ x y
     simp_rw [Ne, ← LieSubmodule.coe_toSubmodule_eq_iff, genWeightSpace, genWeightSpaceOf,
-      LieSubmodule.iInf_coe_toSubmodule, LieSubmodule.bot_coeSubmodule] at hχ
+      LieSubmodule.iInf_coe_toSubmodule, LieSubmodule.bot_coeSubmodule,
+      Module.End.maxGenEigenspace_def] at hχ
     exact Module.End.map_add_of_iInf_genEigenspace_ne_bot_of_commute
       (toEnd R L M).toLinearMap χ hχ h x y
   { map_add := aux
     map_smul := fun χ hχ t x ↦ by
       simp_rw [Ne, ← LieSubmodule.coe_toSubmodule_eq_iff, genWeightSpace, genWeightSpaceOf,
-        LieSubmodule.iInf_coe_toSubmodule, LieSubmodule.bot_coeSubmodule] at hχ
+        LieSubmodule.iInf_coe_toSubmodule, LieSubmodule.bot_coeSubmodule,
+        Module.End.maxGenEigenspace_def] at hχ
       exact Module.End.map_smul_of_iInf_genEigenspace_ne_bot
         (toEnd R L M).toLinearMap χ hχ t x
     map_lie := fun χ hχ t x ↦ by
@@ -112,7 +114,7 @@ instance instLinearWeightsOfIsLieAbelian [IsLieAbelian L] [NoZeroSMulDivisors R
 
 section FiniteDimensional
 
-open FiniteDimensional
+open Module
 
 variable [IsDomain R] [IsPrincipalIdealRing R] [Module.Free R M] [Module.Finite R M]
   [LieAlgebra.IsNilpotent R L]
@@ -184,6 +186,7 @@ instance : LieRingModule L (shiftedGenWeightSpace R L M χ) where
     abel
 
 @[simp] lemma coe_lie_shiftedGenWeightSpace_apply (x : L) (m : shiftedGenWeightSpace R L M χ) :
+    letI : Bracket L (shiftedGenWeightSpace R L M χ) := LieRingModule.toBracket
     ⁅x, m⁆ = ⁅x, (m : M)⁆ - χ x • m :=
   rfl
 
@@ -191,13 +194,17 @@ instance : LieModule R L (shiftedGenWeightSpace R L M χ) where
   smul_lie t x m := by
     nontriviality shiftedGenWeightSpace R L M χ
     apply Subtype.ext
-    simp only [coe_lie_shiftedGenWeightSpace_apply, smul_lie,
-      LinearWeights.map_smul χ (aux R L M χ), smul_assoc t, SetLike.val_smul, smul_sub]
+    rw [coe_lie_shiftedGenWeightSpace_apply]
+    simp only [smul_lie, LinearWeights.map_smul χ (aux R L M χ), smul_assoc t, SetLike.val_smul]
+    rw [← smul_sub]
+    congr
   lie_smul t x m := by
     nontriviality shiftedGenWeightSpace R L M χ
     apply Subtype.ext
-    simp only [coe_lie_shiftedGenWeightSpace_apply, SetLike.val_smul, lie_smul, smul_sub,
-      smul_comm t]
+    rw [coe_lie_shiftedGenWeightSpace_apply]
+    simp only [SetLike.val_smul, lie_smul]
+    rw [smul_comm (χ x), ← smul_sub]
+    congr
 
 /-- Forgetting the action of `L`,
 the spaces `genWeightSpace M χ` and `shiftedGenWeightSpace R L M χ` are equivalent. -/
@@ -206,7 +213,14 @@ the spaces `genWeightSpace M χ` and `shiftedGenWeightSpace R L M χ` are equiva
 lemma toEnd_eq (x : L) :
     toEnd R L (shiftedGenWeightSpace R L M χ) x =
     (shift R L M χ).conj (toEnd R L (genWeightSpace M χ) x - χ x • LinearMap.id) := by
-  ext; simp [LinearEquiv.conj_apply]
+  ext
+  simp only [toEnd_apply_apply, map_sub, LinearEquiv.conj_apply, map_smul, LinearMap.comp_id,
+    LinearEquiv.comp_coe, LinearEquiv.symm_trans_self, LinearEquiv.refl_toLinearMap,
+    LinearMap.sub_apply, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
+    shift_symm_apply, shift_apply, LinearMap.smul_apply, LinearMap.id_coe, id_eq,
+    AddSubgroupClass.coe_sub, SetLike.val_smul]
+  rw [LieSubmodule.coe_bracket]
+  rfl
 
 /-- By Engel's theorem, if `M` is Noetherian, the shifted action `⁅x, m⁆ - χ x • m` makes the
 `χ`-weight space into a nilpotent Lie module. -/
@@ -215,6 +229,7 @@ instance [IsNoetherian R M] : IsNilpotent R L (shiftedGenWeightSpace R L M χ) :
 
 end shiftedGenWeightSpace
 
+open shiftedGenWeightSpace in
 /-- Given a Lie module `M` of a Lie algebra `L` with coefficients in `R`, if a function `χ : L → R`
 has a simultaneous generalized eigenvector for the action of `L` then it has a simultaneous true
 eigenvector, provided `M` is Noetherian and has linear weights. -/
@@ -224,9 +239,11 @@ lemma exists_forall_lie_eq_smul [LinearWeights R L M] [IsNoetherian R M] (χ : W
     (LieSubmodule.nontrivial_iff_ne_bot R L M).mpr χ.genWeightSpace_ne_bot
   obtain ⟨⟨⟨m, _⟩, hm₁⟩, hm₂⟩ :=
     @exists_ne _ (nontrivial_max_triv_of_isNilpotent R L (shiftedGenWeightSpace R L M χ)) 0
-  simp_rw [LieSubmodule.mem_coeSubmodule, mem_maxTrivSubmodule, Subtype.ext_iff,
-    shiftedGenWeightSpace.coe_lie_shiftedGenWeightSpace_apply,
-    ZeroMemClass.coe_zero, sub_eq_zero] at hm₁
-  exact ⟨m, by simpa using hm₂, hm₁⟩
+  simp_rw [mem_maxTrivSubmodule, Subtype.ext_iff,
+    ZeroMemClass.coe_zero] at hm₁
+  refine ⟨m, by simpa [LieSubmodule.mk_eq_zero] using hm₂, ?_⟩
+  intro x
+  have := hm₁ x
+  rwa [coe_lie_shiftedGenWeightSpace_apply, sub_eq_zero] at this
 
 end LieModule
diff --git a/Mathlib/Algebra/Lie/Weights/RootSystem.lean b/Mathlib/Algebra/Lie/Weights/RootSystem.lean
index 7f73bf98624fa..d2a8a94210cd9 100644
--- a/Mathlib/Algebra/Lie/Weights/RootSystem.lean
+++ b/Mathlib/Algebra/Lie/Weights/RootSystem.lean
@@ -5,6 +5,7 @@ Authors: Andrew Yang
 -/
 import Mathlib.Algebra.Lie.Weights.Killing
 import Mathlib.LinearAlgebra.RootSystem.Basic
+import Mathlib.LinearAlgebra.RootSystem.Finite.CanonicalBilinear
 import Mathlib.Algebra.Algebra.Rat
 
 /-!
@@ -376,7 +377,7 @@ variable (H)
 /-- The root system of a finite-dimensional Lie algebra with non-degenerate Killing form over a
 field of characteristic zero, relative to a splitting Cartan subalgebra. -/
 def rootSystem :
-    RootSystem {α : Weight K H L // α.IsNonZero} K (Dual K H) H :=
+    RootSystem H.root K (Dual K H) H :=
   RootSystem.mk'
     IsReflexive.toPerfectPairingDual
     { toFun := (↑)
@@ -384,15 +385,24 @@ def rootSystem :
         intro α β h; ext x; simpa using LinearMap.congr_fun h x  }
     { toFun := coroot ∘ (↑)
       inj' := by rintro ⟨α, hα⟩ ⟨β, hβ⟩ h; simpa using h }
-    (fun α ↦ by simpa using root_apply_coroot α.property)
+    (fun ⟨α, hα⟩ ↦ by simpa using root_apply_coroot <| by simpa using hα)
     (by
       rintro ⟨α, hα⟩ - ⟨⟨β, hβ⟩, rfl⟩
       simp only [Function.Embedding.coeFn_mk, IsReflexive.toPerfectPairingDual_toLin,
         Function.comp_apply, Set.mem_range, Subtype.exists, exists_prop]
-      exact ⟨reflectRoot α β, reflectRoot_isNonZero α β hβ, rfl⟩)
+      exact ⟨reflectRoot α β, (by simpa using reflectRoot_isNonZero α β <| by simpa using hβ), rfl⟩)
     (by convert span_weight_isNonZero_eq_top K L H; ext; simp)
 
-@[simp] lemma rootSystem_toLin_apply (f x) : (rootSystem H).toLin f x = f x := rfl
+@[simp]
+lemma corootForm_rootSystem_eq_killing :
+    (rootSystem H).CorootForm = (killingForm K L).restrict H := by
+  rw [restrict_killingForm_eq_sum, RootPairing.CorootForm, ← Finset.sum_coe_sort (s := H.root)]
+  rfl
+
+@[simp] lemma rootSystem_toPerfectPairing_apply (f x) : (rootSystem H).toPerfectPairing f x = f x :=
+  rfl
+@[deprecated (since := "2024-09-09")]
+alias rootSystem_toLin_apply := rootSystem_toPerfectPairing_apply
 @[simp] lemma rootSystem_pairing_apply (α β) : (rootSystem H).pairing β α = β.1 (coroot α.1) := rfl
 @[simp] lemma rootSystem_root_apply (α) : (rootSystem H).root α = α := rfl
 @[simp] lemma rootSystem_coroot_apply (α) : (rootSystem H).coroot α = coroot α := rfl
@@ -402,12 +412,12 @@ theorem isCrystallographic_rootSystem : (rootSystem H).IsCrystallographic := by
   exact ⟨chainBotCoeff β.1 α.1 - chainTopCoeff β.1 α.1, by simp [apply_coroot_eq_cast β.1 α.1]⟩
 
 theorem isReduced_rootSystem : (rootSystem H).IsReduced := by
-  intro α β e
+  intro ⟨α, hα⟩ ⟨β, hβ⟩ e
   rw [LinearIndependent.pair_iff' ((rootSystem H).ne_zero _), not_forall] at e
   simp only [Nat.succ_eq_add_one, Nat.reduceAdd, rootSystem_root_apply, ne_eq, not_not] at e
   obtain ⟨u, hu⟩ := e
   obtain (h | h) :=
-    eq_neg_or_eq_of_eq_smul α.1 β.1 β.2 u (by ext x; exact DFunLike.congr_fun hu.symm x)
+    eq_neg_or_eq_of_eq_smul α β (by simpa using hβ) u (by ext x; exact DFunLike.congr_fun hu.symm x)
   · right; ext x; simpa [neg_eq_iff_eq_neg] using DFunLike.congr_fun h.symm x
   · left; ext x; simpa using DFunLike.congr_fun h.symm x
 
diff --git a/Mathlib/Algebra/ModEq.lean b/Mathlib/Algebra/ModEq.lean
index 8c4bcc0fd4def..a275aad2c7254 100644
--- a/Mathlib/Algebra/ModEq.lean
+++ b/Mathlib/Algebra/ModEq.lean
@@ -5,6 +5,7 @@ Authors: Yaël Dillies
 -/
 import Mathlib.Data.Int.ModEq
 import Mathlib.Algebra.Field.Basic
+import Mathlib.Algebra.Order.Ring.Int
 import Mathlib.GroupTheory.QuotientGroup.Basic
 
 /-!
diff --git a/Mathlib/Algebra/Module/Algebra.lean b/Mathlib/Algebra/Module/Algebra.lean
index d2cee9bce327b..6fb1b53849c2b 100644
--- a/Mathlib/Algebra/Module/Algebra.lean
+++ b/Mathlib/Algebra/Module/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Module.Defs
 import Mathlib.Algebra.Algebra.Basic
diff --git a/Mathlib/Algebra/Module/Basic.lean b/Mathlib/Algebra/Module/Basic.lean
index 2e3fb20b33d35..8656e47667f24 100644
--- a/Mathlib/Algebra/Module/Basic.lean
+++ b/Mathlib/Algebra/Module/Basic.lean
@@ -101,9 +101,7 @@ section NoZeroSMulDivisors
 
 section Module
 
-variable [Ring R] [AddCommGroup M] [Module R M] [NoZeroSMulDivisors R M]
-
-instance [NoZeroSMulDivisors ℤ M] : NoZeroSMulDivisors ℕ M :=
+instance [AddCommGroup M] [NoZeroSMulDivisors ℤ M] : NoZeroSMulDivisors ℕ M :=
   ⟨fun {c x} hcx ↦ by rwa [← Nat.cast_smul_eq_nsmul ℤ c x, smul_eq_zero, Nat.cast_eq_zero] at hcx⟩
 
 end Module
@@ -135,7 +133,7 @@ lemma support_smul_subset_right [Zero M] [SMulZeroClass R M] (f : α → R) (g :
 
 lemma support_const_smul_of_ne_zero [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M]
     (c : R) (g : α → M) (hc : c ≠ 0) : support (c • g) = support g :=
-  ext fun x ↦ by simp only [hc, mem_support, Pi.smul_apply, Ne, smul_eq_zero, false_or_iff]
+  ext fun x ↦ by simp only [hc, mem_support, Pi.smul_apply, Ne, smul_eq_zero, false_or]
 
 lemma support_smul [Zero R] [Zero M] [SMulWithZero R M] [NoZeroSMulDivisors R M] (f : α → R)
     (g : α → M) : support (f • g) = support f ∩ support g :=
diff --git a/Mathlib/Algebra/Module/BigOperators.lean b/Mathlib/Algebra/Module/BigOperators.lean
index 0799ce0c60156..d15e7274ec57a 100644
--- a/Mathlib/Algebra/Module/BigOperators.lean
+++ b/Mathlib/Algebra/Module/BigOperators.lean
@@ -3,9 +3,9 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Yury Kudryashov, Yaël Dillies
 -/
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.Algebra.Module.Defs
 import Mathlib.Data.Fintype.BigOperators
-import Mathlib.GroupTheory.GroupAction.BigOperators
 
 /-!
 # Finite sums over modules over a ring
@@ -15,7 +15,7 @@ variable {ι κ α β R M : Type*}
 
 section AddCommMonoid
 
-variable [Semiring R] [AddCommMonoid M] [Module R M] (r s : R) (x y : M)
+variable [Semiring R] [AddCommMonoid M] [Module R M]
 
 theorem List.sum_smul {l : List R} {x : M} : l.sum • x = (l.map fun r ↦ r • x).sum :=
   map_list_sum ((smulAddHom R M).flip x) l
@@ -54,6 +54,6 @@ lemma sum_piFinset_apply (f : κ → α) (s : Finset κ) (i : ι) :
   classical
   rw [Finset.sum_comp]
   simp only [eval_image_piFinset_const, card_filter_piFinset_const s, ite_smul, zero_smul, smul_sum,
-    sum_ite_mem, inter_self]
+    Finset.sum_ite_mem, inter_self]
 
 end Fintype
diff --git a/Mathlib/Algebra/Module/Equiv/Basic.lean b/Mathlib/Algebra/Module/Equiv/Basic.lean
index 3d210451cac46..502f5ddd8ebdf 100644
--- a/Mathlib/Algebra/Module/Equiv/Basic.lean
+++ b/Mathlib/Algebra/Module/Equiv/Basic.lean
@@ -122,7 +122,7 @@ protected theorem smul_def (f : M ≃ₗ[R] M) (a : M) : f • a = f a :=
 
 /-- `LinearEquiv.applyDistribMulAction` is faithful. -/
 instance apply_faithfulSMul : FaithfulSMul (M ≃ₗ[R] M) M :=
-  ⟨@fun _ _ ↦ LinearEquiv.ext⟩
+  ⟨LinearEquiv.ext⟩
 
 instance apply_smulCommClass [SMul S R] [SMul S M] [IsScalarTower S R M] :
     SMulCommClass S (M ≃ₗ[R] M) M where
diff --git a/Mathlib/Algebra/Module/FinitePresentation.lean b/Mathlib/Algebra/Module/FinitePresentation.lean
index 400521dbf951a..024751ec3f4b5 100644
--- a/Mathlib/Algebra/Module/FinitePresentation.lean
+++ b/Mathlib/Algebra/Module/FinitePresentation.lean
@@ -48,6 +48,8 @@ For finitely presented algebras, see `Algebra.FinitePresentation`
 in file `Mathlib.RingTheory.FinitePresentation`.
 -/
 
+open Finsupp
+
 section Semiring
 variable (R M) [Semiring R] [AddCommMonoid M] [Module R M]
 
@@ -57,7 +59,7 @@ and the kernel of the presentation `Rˢ → M` is also finitely generated.
 -/
 class Module.FinitePresentation : Prop where
   out : ∃ (s : Finset M), Submodule.span R (s : Set M) = ⊤ ∧
-    (LinearMap.ker (Finsupp.total R ((↑) : s → M))).FG
+    (LinearMap.ker (Finsupp.linearCombination R ((↑) : s → M))).FG
 
 instance (priority := 100) [h : Module.FinitePresentation R M] : Module.Finite R M := by
   obtain ⟨s, hs₁, _⟩ := h
@@ -78,10 +80,10 @@ theorem Module.FinitePresentation.equiv_quotient [fp : Module.FinitePresentation
       Module.Free R L ∧ Module.Finite R L ∧ K.FG := by
   obtain ⟨ι, ⟨hι₁, hι₂⟩⟩ := fp
   use ι →₀ R, inferInstance, inferInstance
-  use LinearMap.ker (Finsupp.total R Subtype.val)
+  use LinearMap.ker (Finsupp.linearCombination R Subtype.val)
   refine ⟨(LinearMap.quotKerEquivOfSurjective _ ?_).symm, inferInstance, inferInstance, hι₂⟩
   apply LinearMap.range_eq_top.mp
-  simpa only [Finsupp.range_total, Subtype.range_coe_subtype, Finset.setOf_mem]
+  simpa only [Finsupp.range_linearCombination, Subtype.range_coe_subtype, Finset.setOf_mem]
 
 -- Ideally this should be an instance but it makes mathlib much slower.
 lemma Module.finitePresentation_of_finite [IsNoetherianRing R] [h : Module.Finite R M] :
@@ -119,12 +121,12 @@ lemma Module.finitePresentation_of_free_of_surjective [Module.Free R M] [Module.
   constructor
   · intro hx
     refine ⟨b.repr.symm (x.mapDomain σ), ?_, ?_⟩
-    · simp [Finsupp.apply_total, hσ₂, hx]
+    · simp [Finsupp.apply_linearCombination, hσ₂, hx]
     · simp only [f, LinearMap.comp_apply, b.repr.apply_symm_apply,
         LinearEquiv.coe_toLinearMap, Finsupp.lmapDomain_apply]
       rw [← Finsupp.mapDomain_comp, hσ₁, Finsupp.mapDomain_id]
   · rintro ⟨y, hy, rfl⟩
-    simp [f, hπ, ← Finsupp.apply_total, hy]
+    simp [f, hπ, ← Finsupp.apply_linearCombination, hy]
 
 -- Ideally this should be an instance but it makes mathlib much slower.
 variable (R M) in
@@ -145,12 +147,12 @@ lemma Module.finitePresentation_of_surjective [h : Module.FinitePresentation R M
   classical
   obtain ⟨s, hs, hs'⟩ := h
   obtain ⟨t, ht⟩ := hl'
-  have H : Function.Surjective (Finsupp.total R ((↑) : s → M)) :=
-    LinearMap.range_eq_top.mp (by rw [Finsupp.range_total, Subtype.range_val, ← hs]; rfl)
-  apply Module.finitePresentation_of_free_of_surjective (l ∘ₗ Finsupp.total R Subtype.val)
+  have H : Function.Surjective (Finsupp.linearCombination R ((↑) : s → M)) :=
+    LinearMap.range_eq_top.mp (by rw [range_linearCombination, Subtype.range_val, ← hs]; rfl)
+  apply Module.finitePresentation_of_free_of_surjective (l ∘ₗ linearCombination R Subtype.val)
     (hl.comp H)
   choose σ hσ using (show _ from H)
-  have : Finsupp.total R Subtype.val '' (σ '' t) = t := by
+  have : Finsupp.linearCombination R Subtype.val '' (σ '' t) = t := by
     simp only [Set.image_image, hσ, Set.image_id']
   rw [LinearMap.ker_comp, ← ht, ← this, ← Submodule.map_span, Submodule.comap_map_eq,
     ← Finset.coe_image]
@@ -161,11 +163,11 @@ lemma Module.FinitePresentation.fg_ker [Module.Finite R M]
     (LinearMap.ker l).FG := by
   classical
   obtain ⟨s, hs, hs'⟩ := h
-  have H : Function.Surjective (Finsupp.total R ((↑) : s → N)) :=
-    LinearMap.range_eq_top.mp (by rw [Finsupp.range_total, Subtype.range_val, ← hs]; rfl)
-  obtain ⟨f, hf⟩ : ∃ f : (s →₀ R) →ₗ[R] M, l ∘ₗ f = (Finsupp.total R Subtype.val) := by
+  have H : Function.Surjective (Finsupp.linearCombination R ((↑) : s → N)) :=
+    LinearMap.range_eq_top.mp (by rw [range_linearCombination, Subtype.range_val, ← hs]; rfl)
+  obtain ⟨f, hf⟩ : ∃ f : (s →₀ R) →ₗ[R] M, l ∘ₗ f = (Finsupp.linearCombination R Subtype.val) := by
     choose f hf using show _ from hl
-    exact ⟨Finsupp.total R (fun i ↦ f i), by ext; simp [hf]⟩
+    exact ⟨Finsupp.linearCombination R (fun i ↦ f i), by ext; simp [hf]⟩
   have : (LinearMap.ker l).map (LinearMap.range f).mkQ = ⊤ := by
     rw [← top_le_iff]
     rintro x -
@@ -192,9 +194,9 @@ lemma Module.finitePresentation_of_ker [Module.FinitePresentation R N]
     · rw [Submodule.map_top, LinearMap.range_eq_top.mpr hl]; exact Module.Finite.out
     · rw [top_inf_eq, ← Submodule.fg_top]; exact Module.Finite.out
   refine ⟨s, hs, ?_⟩
-  let π := Finsupp.total R ((↑) : s → M)
+  let π := Finsupp.linearCombination R ((↑) : s → M)
   have H : Function.Surjective π :=
-    LinearMap.range_eq_top.mp (by rw [Finsupp.range_total, Subtype.range_val, ← hs]; rfl)
+    LinearMap.range_eq_top.mp (by rw [range_linearCombination, Subtype.range_val, ← hs]; rfl)
   have inst : Module.Finite R (LinearMap.ker (l ∘ₗ π)) := by
     constructor
     rw [Submodule.fg_top]; exact Module.FinitePresentation.fg_ker _ (hl.comp H)
@@ -227,23 +229,24 @@ lemma Module.FinitePresentation.exists_lift_of_isLocalizedModule
     [h : Module.FinitePresentation R M] (g : M →ₗ[R] N') :
     ∃ (h : M →ₗ[R] N) (s : S), f ∘ₗ h = s • g := by
   obtain ⟨σ, hσ, τ, hτ⟩ := h
-  let π := Finsupp.total R ((↑) : σ → M)
+  let π := Finsupp.linearCombination R ((↑) : σ → M)
   have hπ : Function.Surjective π :=
-    LinearMap.range_eq_top.mp (by rw [Finsupp.range_total, Subtype.range_val, ← hσ]; rfl)
+    LinearMap.range_eq_top.mp (by rw [range_linearCombination, Subtype.range_val, ← hσ]; rfl)
   classical
   choose s hs using IsLocalizedModule.surj S f
   let i : σ → N :=
     fun x ↦ (∏ j ∈ σ.erase x.1, (s (g j)).2) • (s (g x)).1
   let s₀ := ∏ j ∈ σ, (s (g j)).2
-  have hi : f ∘ₗ Finsupp.total R i = (s₀ • g) ∘ₗ π := by
+  have hi : f ∘ₗ Finsupp.linearCombination R i = (s₀ • g) ∘ₗ π := by
     ext j
-    simp only [LinearMap.coe_comp, Function.comp_apply, Finsupp.lsingle_apply, Finsupp.total_single,
-      one_smul, LinearMap.map_smul_of_tower, ← hs, LinearMap.smul_apply, i, s₀, π]
+    simp only [LinearMap.coe_comp, Function.comp_apply, Finsupp.lsingle_apply,
+      linearCombination_single, one_smul, LinearMap.map_smul_of_tower, ← hs, LinearMap.smul_apply,
+      i, s₀, π]
     rw [← mul_smul, Finset.prod_erase_mul]
     exact j.prop
-  have : ∀ x : τ, ∃ s : S, s • (Finsupp.total R i x) = 0 := by
+  have : ∀ x : τ, ∃ s : S, s • (Finsupp.linearCombination R i x) = 0 := by
     intros x
-    convert_to ∃ s : S, s • (Finsupp.total R i x) = s • 0
+    convert_to ∃ s : S, s • (Finsupp.linearCombination R i x) = s • 0
     · simp only [smul_zero]
     apply IsLocalizedModule.exists_of_eq (S := S) (f := f)
     rw [← LinearMap.comp_apply, map_zero, hi, LinearMap.comp_apply]
@@ -252,7 +255,7 @@ lemma Module.FinitePresentation.exists_lift_of_isLocalizedModule
     exact Submodule.subset_span x.prop
   choose s' hs' using this
   let s₁ := ∏ i : τ, s' i
-  have : LinearMap.ker π ≤ LinearMap.ker (s₁ • Finsupp.total R i) := by
+  have : LinearMap.ker π ≤ LinearMap.ker (s₁ • Finsupp.linearCombination R i) := by
     rw [← hτ, Submodule.span_le]
     intro x hxσ
     simp only [s₁]
diff --git a/Mathlib/Algebra/Module/Injective.lean b/Mathlib/Algebra/Module/Injective.lean
index 536a41a974e42..a2e077efec5f7 100644
--- a/Mathlib/Algebra/Module/Injective.lean
+++ b/Mathlib/Algebra/Module/Injective.lean
@@ -67,8 +67,8 @@ theorem Module.injective_module_of_injective_object
     [inj : CategoryTheory.Injective <| ModuleCat.of R Q] :
     Module.Injective R Q where
   out X Y _ _ _ _ f hf g := by
-    have : CategoryTheory.Mono (ModuleCat.ofHom f) := (ModuleCat.mono_iff_injective _).mpr hf
-    obtain ⟨l, rfl⟩ := inj.factors (ModuleCat.ofHom g) (ModuleCat.ofHom f)
+    have : CategoryTheory.Mono (ModuleCat.asHom f) := (ModuleCat.mono_iff_injective _).mpr hf
+    obtain ⟨l, rfl⟩ := inj.factors (ModuleCat.asHom g) (ModuleCat.asHom f)
     exact ⟨l, fun _ ↦ rfl⟩
 
 theorem Module.injective_iff_injective_object :
@@ -249,7 +249,7 @@ variable (f)
 def ExtensionOfMaxAdjoin.ideal (y : N) : Ideal R :=
   (extensionOfMax i f).domain.comap ((LinearMap.id : R →ₗ[R] R).smulRight y)
 
-/-- A linear map `I ⟶ Q` by `x ↦ f' (x • y)` where `f'` is the maximal extension-/
+/-- A linear map `I ⟶ Q` by `x ↦ f' (x • y)` where `f'` is the maximal extension -/
 def ExtensionOfMaxAdjoin.idealTo (y : N) : ExtensionOfMaxAdjoin.ideal i f y →ₗ[R] Q where
   toFun (z : { x // x ∈ ideal i f y }) := (extensionOfMax i f).toLinearPMap ⟨(↑z : R) • y, z.prop⟩
   map_add' (z1 z2 : { x // x ∈ ideal i f y }) := by
diff --git a/Mathlib/Algebra/Module/LinearMap/Basic.lean b/Mathlib/Algebra/Module/LinearMap/Basic.lean
index f50429fa64814..1a68ecbd89506 100644
--- a/Mathlib/Algebra/Module/LinearMap/Basic.lean
+++ b/Mathlib/Algebra/Module/LinearMap/Basic.lean
@@ -21,41 +21,41 @@ open Function
 
 universe u u' v w x y z
 
-variable {R R₁ R₂ R₃ k S S₃ T M M₁ M₂ M₃ N₁ N₂ N₃ ι : Type*}
+variable {R R' S M M' : Type*}
 
 namespace LinearMap
 
 section SMul
 
-variable [Semiring R] [Semiring R₂]
-variable [AddCommMonoid M] [AddCommMonoid M₂]
-variable [Module R M] [Module R₂ M₂]
-variable {σ₁₂ : R →+* R₂}
+variable [Semiring R] [Semiring R']
+variable [AddCommMonoid M] [AddCommMonoid M']
+variable [Module R M] [Module R' M']
+variable {σ₁₂ : R →+* R'}
 
 variable {S' T' : Type*}
 variable [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M]
 variable [Monoid T'] [DistribMulAction T' M] [SMulCommClass R T' M]
 
-instance : SMul S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M₂) where
+instance : SMul S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M') where
   smul a f :=
-    { toFun := a • (f : M → M₂)
+    { toFun := a • (f : M → M')
       map_add' := fun x y ↦ by simp only [DomMulAct.smul_apply, f.map_add, smul_add]
       map_smul' := fun c x ↦ by simp_rw [DomMulAct.smul_apply, ← smul_comm, f.map_smulₛₗ] }
 
-theorem _root_.DomMulAct.smul_linearMap_apply (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂) (x : M) :
+theorem _root_.DomMulAct.smul_linearMap_apply (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') (x : M) :
     (a • f) x = f (DomMulAct.mk.symm a • x) :=
   rfl
 
 @[simp]
-theorem _root_.DomMulAct.mk_smul_linearMap_apply (a : S') (f : M →ₛₗ[σ₁₂] M₂) (x : M) :
+theorem _root_.DomMulAct.mk_smul_linearMap_apply (a : S') (f : M →ₛₗ[σ₁₂] M') (x : M) :
     (DomMulAct.mk a • f) x = f (a • x) :=
   rfl
 
-theorem  _root_.DomMulAct.coe_smul_linearMap (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M₂) :
-    (a • f : M →ₛₗ[σ₁₂] M₂) = a • (f : M → M₂) :=
+theorem  _root_.DomMulAct.coe_smul_linearMap (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') :
+    (a • f : M →ₛₗ[σ₁₂] M') = a • (f : M → M') :=
   rfl
 
-instance [SMulCommClass S' T' M] : SMulCommClass S'ᵈᵐᵃ T'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M₂) :=
+instance [SMulCommClass S' T' M] : SMulCommClass S'ᵈᵐᵃ T'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M') :=
   ⟨fun s t f ↦ ext fun m ↦ by simp_rw [DomMulAct.smul_linearMap_apply, smul_comm]⟩
 
 end SMul
@@ -63,19 +63,15 @@ end SMul
 
 section Actions
 
-variable [Semiring R] [Semiring R₂] [Semiring R₃]
-variable [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃]
-variable [Module R M] [Module R₂ M₂] [Module R₃ M₃]
-variable {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃]
+variable [Semiring R] [Semiring R']
+variable [AddCommMonoid M] [AddCommMonoid M']
+variable [Module R M] [Module R' M']
+variable {σ₁₂ : R →+* R'}
 
 section SMul
 
-variable [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂]
-variable [Monoid S₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃]
-variable [Monoid T] [DistribMulAction T M₂] [SMulCommClass R₂ T M₂]
-
 instance {S'} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] :
-    DistribMulAction S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M₂) where
+    DistribMulAction S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M') where
   one_smul _ := ext fun _ ↦ congr_arg _ (one_smul _ _)
   mul_smul _ _ _ := ext fun _ ↦ congr_arg _ (mul_smul _ _ _)
   smul_add _ _ _ := ext fun _ ↦ rfl
@@ -85,12 +81,12 @@ end SMul
 
 section Module
 
-variable [Semiring S] [Module S M] [Module S M₂] [SMulCommClass R₂ S M₂]
+variable [Semiring S] [Module S M] [Module S M'] [SMulCommClass R' S M']
 
-instance [NoZeroSMulDivisors S M₂] : NoZeroSMulDivisors S (M →ₛₗ[σ₁₂] M₂) :=
+instance [NoZeroSMulDivisors S M'] : NoZeroSMulDivisors S (M →ₛₗ[σ₁₂] M') :=
   coe_injective.noZeroSMulDivisors _ rfl coe_smul
 
-instance [SMulCommClass R S M] : Module Sᵈᵐᵃ (M →ₛₗ[σ₁₂] M₂) where
+instance [SMulCommClass R S M] : Module Sᵈᵐᵃ (M →ₛₗ[σ₁₂] M') where
   add_smul _ _ _ := ext fun _ ↦ by
     simp_rw [add_apply, DomMulAct.smul_linearMap_apply, ← map_add, ← add_smul]; rfl
   zero_smul _ := ext fun _ ↦ by erw [DomMulAct.smul_linearMap_apply, zero_smul, map_zero]; rfl
diff --git a/Mathlib/Algebra/Module/LinearMap/Defs.lean b/Mathlib/Algebra/Module/LinearMap/Defs.lean
index e442944e3cb7b..e2c0d89223846 100644
--- a/Mathlib/Algebra/Module/LinearMap/Defs.lean
+++ b/Mathlib/Algebra/Module/LinearMap/Defs.lean
@@ -933,9 +933,8 @@ end Actions
 
 section RestrictScalarsAsLinearMap
 
-variable {R S M N : Type*} [Semiring R] [Semiring S] [AddCommGroup M] [AddCommGroup N] [Module R M]
-   [Module R N] [Module S M] [Module S N]
-  [LinearMap.CompatibleSMul M N R S]
+variable {R S M N P : Type*} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N]
+  [Module R M] [Module R N] [Module S M] [Module S N] [CompatibleSMul M N R S]
 
 variable (R S M N) in
 @[simp]
@@ -948,7 +947,9 @@ theorem restrictScalars_add (f g : M →ₗ[S] N) :
   rfl
 
 @[simp]
-theorem restrictScalars_neg (f : M →ₗ[S] N) : (-f).restrictScalars R = -f.restrictScalars R :=
+theorem restrictScalars_neg {M N : Type*} [AddCommGroup M] [AddCommGroup N]
+    [Module R M] [Module R N] [Module S M] [Module S N] [CompatibleSMul M N R S]
+    (f : M →ₗ[S] N) : (-f).restrictScalars R = -f.restrictScalars R :=
   rfl
 
 variable {R₁ : Type*} [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N]
@@ -958,6 +959,18 @@ theorem restrictScalars_smul (c : R₁) (f : M →ₗ[S] N) :
     (c • f).restrictScalars R = c • f.restrictScalars R :=
   rfl
 
+@[simp]
+lemma restrictScalars_comp [AddCommMonoid P] [Module S P] [Module R P]
+    [CompatibleSMul N P R S] [CompatibleSMul M P R S] (f : N →ₗ[S] P) (g : M →ₗ[S] N) :
+    (f ∘ₗ g).restrictScalars R = f.restrictScalars R ∘ₗ g.restrictScalars R := by
+  rfl
+
+@[simp]
+lemma restrictScalars_trans {T : Type*} [CommSemiring T] [Module T M] [Module T N]
+    [CompatibleSMul M N S T] [CompatibleSMul M N R T] (f : M →ₗ[T] N) :
+    (f.restrictScalars S).restrictScalars R = f.restrictScalars R :=
+  rfl
+
 variable (S M N R R₁)
 
 /-- `LinearMap.restrictScalars` as a `LinearMap`. -/
diff --git a/Mathlib/Algebra/Module/LinearMap/Polynomial.lean b/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
index 6f0b08d7782c3..823b874234f59 100644
--- a/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
+++ b/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
@@ -351,7 +351,7 @@ lemma polyCharpolyAux_basisIndep {ιM' : Type*} [Fintype ιM'] [DecidableEq ιM'
 
 end aux
 
-open FiniteDimensional Matrix
+open Module Matrix
 
 variable [Module.Free R M] [Module.Finite R M] (b : Basis ι R L)
 
@@ -479,11 +479,11 @@ lemma polyCharpoly_coeff_nilRank_ne_zero :
   rw [nilRank_eq_polyCharpoly_natTrailingDegree _ b]
   apply polyCharpoly_coeff_nilRankAux_ne_zero
 
-open FiniteDimensional Module.Free
+open Module Module.Free
 
 lemma nilRank_le_card {ι : Type*} [Fintype ι] (b : Basis ι R M) : nilRank φ ≤ Fintype.card ι := by
   apply Polynomial.natTrailingDegree_le_of_ne_zero
-  rw [← FiniteDimensional.finrank_eq_card_basis b, ← polyCharpoly_natDegree φ (chooseBasis R L),
+  rw [← Module.finrank_eq_card_basis b, ← polyCharpoly_natDegree φ (chooseBasis R L),
     Polynomial.coeff_natDegree, (polyCharpoly_monic _ _).leadingCoeff]
   apply one_ne_zero
 
@@ -538,7 +538,7 @@ section IsDomain
 
 variable [IsDomain R]
 
-open Cardinal FiniteDimensional MvPolynomial Module.Free in
+open Cardinal Module MvPolynomial Module.Free in
 lemma exists_isNilRegular_of_finrank_le_card (h : finrank R M ≤ #R) :
     ∃ x : L, IsNilRegular φ x := by
   let b := chooseBasis R L
diff --git a/Mathlib/Algebra/Module/LocalizedModule.lean b/Mathlib/Algebra/Module/LocalizedModule.lean
index 45ebbceedbb8a..5edf2a007db89 100644
--- a/Mathlib/Algebra/Module/LocalizedModule.lean
+++ b/Mathlib/Algebra/Module/LocalizedModule.lean
@@ -402,6 +402,29 @@ noncomputable instance isModule' : Module R (LocalizedModule S M) :=
 theorem smul'_mk (r : R) (s : S) (m : M) : r • mk m s = mk (r • m) s := by
   erw [mk_smul_mk r m 1 s, one_mul]
 
+lemma smul_eq_iff_of_mem
+    (r : R) (hr : r ∈ S) (x y : LocalizedModule S M) :
+    r • x = y ↔ x = Localization.mk 1 ⟨r, hr⟩ • y := by
+  induction x using induction_on with
+  | h m s =>
+    induction y using induction_on with
+    | h n t =>
+      rw [smul'_mk, mk_smul_mk, one_smul, mk_eq, mk_eq]
+      simp only [Subtype.exists, Submonoid.mk_smul, exists_prop]
+      fconstructor
+      · rintro ⟨a, ha, eq1⟩
+        refine ⟨a, ha, ?_⟩
+        rw [mul_smul, ← eq1, Submonoid.mk_smul, smul_comm r t]
+      · rintro ⟨a, ha, eq1⟩
+        refine ⟨a, ha, ?_⟩
+        rw [← eq1, mul_comm, mul_smul, Submonoid.mk_smul]
+        rfl
+
+lemma eq_zero_of_smul_eq_zero
+    (r : R) (hr : r ∈ S) (x : LocalizedModule S M) (hx : r • x = 0) : x = 0 := by
+  rw [smul_eq_iff_of_mem (hr := hr)] at hx
+  rw [hx, smul_zero]
+
 theorem smul'_mul {A : Type*} [Semiring A] [Algebra R A] (x : T) (p₁ p₂ : LocalizedModule S A) :
     x • p₁ * p₂ = x • (p₁ * p₂) := by
   induction p₁, p₂ using induction_on₂ with | _ a₁ s₁ a₂ s₂ => _
diff --git a/Mathlib/Algebra/Module/PID.lean b/Mathlib/Algebra/Module/PID.lean
index 65924b9cdde6d..a00bd3ef0ee93 100644
--- a/Mathlib/Algebra/Module/PID.lean
+++ b/Mathlib/Algebra/Module/PID.lean
@@ -225,7 +225,7 @@ theorem torsion_by_prime_power_decomposition (hN : Module.IsTorsion' N (Submonoi
         ⟨(@hN x).choose, by rw [← Quotient.mk_smul, (@hN x).choose_spec, Quotient.mk_zero]⟩
     · have hs' := congr_arg (Submodule.map <| mkQ <| R ∙ s j) hs
       rw [Submodule.map_span, Submodule.map_top, range_mkQ] at hs'; simp only [mkQ_apply] at hs'
-      simp only [s']; rw [← Function.comp.assoc, Set.range_comp (_ ∘ s), Fin.range_succAbove]
+      simp only [s']; rw [← Function.comp_assoc, Set.range_comp (_ ∘ s), Fin.range_succAbove]
       rw [← Set.range_comp, ← Set.insert_image_compl_eq_range _ j, Function.comp_apply,
         (Quotient.mk_eq_zero _).mpr (Submodule.mem_span_singleton_self _), span_insert_zero] at hs'
       exact hs'
diff --git a/Mathlib/Algebra/Module/Projective.lean b/Mathlib/Algebra/Module/Projective.lean
index 5432bd8a44e48..b4387f01f63c9 100644
--- a/Mathlib/Algebra/Module/Projective.lean
+++ b/Mathlib/Algebra/Module/Projective.lean
@@ -72,7 +72,7 @@ open Finsupp
   definitions. -/
 class Module.Projective (R : Type*) [Semiring R] (P : Type*) [AddCommMonoid P] [Module R P] :
     Prop where
-  out : ∃ s : P →ₗ[R] P →₀ R, Function.LeftInverse (Finsupp.total R id) s
+  out : ∃ s : P →ₗ[R] P →₀ R, Function.LeftInverse (Finsupp.linearCombination R id) s
 
 namespace Module
 
@@ -82,11 +82,11 @@ variable {R : Type*} [Semiring R] {P : Type*} [AddCommMonoid P] [Module R P] {M
   [AddCommMonoid M] [Module R M] {N : Type*} [AddCommMonoid N] [Module R N]
 
 theorem projective_def :
-    Projective R P ↔ ∃ s : P →ₗ[R] P →₀ R, Function.LeftInverse (Finsupp.total R id) s :=
+    Projective R P ↔ ∃ s : P →ₗ[R] P →₀ R, Function.LeftInverse (linearCombination R id) s :=
   ⟨fun h => h.1, fun h => ⟨h⟩⟩
 
 theorem projective_def' :
-    Projective R P ↔ ∃ s : P →ₗ[R] P →₀ R, Finsupp.total R id ∘ₗ s = .id := by
+    Projective R P ↔ ∃ s : P →ₗ[R] P →₀ R, Finsupp.linearCombination R id ∘ₗ s = .id := by
   simp_rw [projective_def, DFunLike.ext_iff, Function.LeftInverse, comp_apply, id_apply]
 
 /-- A projective R-module has the property that maps from it lift along surjections. -/
@@ -95,20 +95,20 @@ theorem projective_lifting_property [h : Projective R P] (f : M →ₗ[R] N) (g
   /-
     Here's the first step of the proof.
     Recall that `X →₀ R` is Lean's way of talking about the free `R`-module
-    on a type `X`. The universal property `Finsupp.total` says that to a map
+    on a type `X`. The universal property `Finsupp.linearCombination` says that to a map
     `X → N` from a type to an `R`-module, we get an associated R-module map
     `(X →₀ R) →ₗ N`. Apply this to a (noncomputable) map `P → M` coming from the map
     `P →ₗ N` and a random splitting of the surjection `M →ₗ N`, and we get
     a map `φ : (P →₀ R) →ₗ M`.
     -/
-  let φ : (P →₀ R) →ₗ[R] M := Finsupp.total _ fun p => Function.surjInv hf (g p)
+  let φ : (P →₀ R) →ₗ[R] M := Finsupp.linearCombination _ fun p => Function.surjInv hf (g p)
   -- By projectivity we have a map `P →ₗ (P →₀ R)`;
   cases' h.out with s hs
   -- Compose to get `P →ₗ M`. This works.
   use φ.comp s
   ext p
   conv_rhs => rw [← hs p]
-  simp [φ, Finsupp.total_apply, Function.surjInv_eq hf, map_finsupp_sum]
+  simp [φ, Finsupp.linearCombination_apply, Function.surjInv_eq hf, map_finsupp_sum]
 
 /-- A module which satisfies the universal property is projective: If all surjections of
 `R`-modules `(P →₀ R) →ₗ[R] P` have `R`-linear left inverse maps, then `P` is
@@ -117,8 +117,8 @@ theorem Projective.of_lifting_property'' {R : Type u} [Semiring R] {P : Type v}
     [Module R P] (huniv : ∀ (f : (P →₀ R) →ₗ[R] P), Function.Surjective f →
       ∃ h : P →ₗ[R] (P →₀ R), f.comp h = .id) :
     Projective R P :=
-  projective_def'.2 <| huniv (Finsupp.total R (id : P → P))
-    (total_surjective _ Function.surjective_id)
+  projective_def'.2 <| huniv (Finsupp.linearCombination R (id : P → P))
+    (linearCombination_surjective _ Function.surjective_id)
 
 variable {Q : Type*} [AddCommMonoid Q] [Module R Q]
 
@@ -140,31 +140,22 @@ instance [h : ∀ i : ι, Projective R (A i)] : Projective R (Π₀ i, A i) :=
       ext i x j
       simp only [comp_apply, id_apply, DFinsupp.lsingle_apply, DFinsupp.coprodMap_apply_single, hg]
 
-end Semiring
-
-section Ring
-
-variable {R : Type u} [Ring R] {P : Type v} [AddCommGroup P] [Module R P]
-
 /-- Free modules are projective. -/
 theorem Projective.of_basis {ι : Type*} (b : Basis ι R P) : Projective R P := by
   -- need P →ₗ (P →₀ R) for definition of projective.
   -- get it from `ι → (P →₀ R)` coming from `b`.
   use b.constr ℕ fun i => Finsupp.single (b i) (1 : R)
   intro m
-  simp only [b.constr_apply, mul_one, id, Finsupp.smul_single', Finsupp.total_single,
+  simp only [b.constr_apply, mul_one, id, Finsupp.smul_single', Finsupp.linearCombination_single,
     map_finsupp_sum]
-  exact b.total_repr m
+  exact b.linearCombination_repr m
 
 instance (priority := 100) Projective.of_free [Module.Free R P] : Module.Projective R P :=
   .of_basis <| Module.Free.chooseBasis R P
 
-variable {R₀ M N} [CommRing R₀] [Algebra R₀ R] [AddCommGroup M] [Module R₀ M] [Module R M]
-variable [IsScalarTower R₀ R M] [AddCommGroup N] [Module R₀ N]
-
 theorem Projective.of_split [Module.Projective R M]
     (i : P →ₗ[R] M) (s : M →ₗ[R] P) (H : s.comp i = LinearMap.id) : Module.Projective R P := by
-  obtain ⟨g, hg⟩ := projective_lifting_property (Finsupp.total R id) s
+  obtain ⟨g, hg⟩ := projective_lifting_property (Finsupp.linearCombination R id) s
     (fun x ↦ ⟨Finsupp.single x 1, by simp⟩)
   refine ⟨g.comp i, fun x ↦ ?_⟩
   rw [LinearMap.comp_apply, ← LinearMap.comp_apply, hg,
@@ -174,19 +165,27 @@ theorem Projective.of_equiv [Module.Projective R M]
     (e : M ≃ₗ[R] P) : Module.Projective R P :=
   Projective.of_split e.symm e.toLinearMap (by ext; simp)
 
-/-- A module is projective iff it is the direct summand of a free module. -/
-theorem Projective.iff_split : Module.Projective R P ↔
-    ∃ (M : Type max u v) (_ : AddCommGroup M) (_ : Module R M) (_ : Module.Free R M)
-      (i : P →ₗ[R] M) (s : M →ₗ[R] P), s.comp i = LinearMap.id :=
-  ⟨fun ⟨i, hi⟩ ↦ ⟨P →₀ R, _, _, inferInstance, i, Finsupp.total R id, LinearMap.ext hi⟩,
-    fun ⟨_, _, _, _, i, s, H⟩ ↦ Projective.of_split i s H⟩
-
 /-- A quotient of a projective module is projective iff it is a direct summand. -/
 theorem Projective.iff_split_of_projective [Module.Projective R M] (s : M →ₗ[R] P)
     (hs : Function.Surjective s) :
     Module.Projective R P ↔ ∃ i, s ∘ₗ i = LinearMap.id :=
   ⟨fun _ ↦ projective_lifting_property _ _ hs, fun ⟨i, H⟩ ↦ Projective.of_split i s H⟩
 
+end Semiring
+
+section Ring
+
+variable {R : Type u} [Ring R] {P : Type v} [AddCommMonoid P] [Module R P]
+variable {R₀ M N} [CommRing R₀] [Algebra R₀ R] [AddCommGroup M] [Module R₀ M] [Module R M]
+variable [IsScalarTower R₀ R M] [AddCommGroup N] [Module R₀ N]
+
+/-- A module is projective iff it is the direct summand of a free module. -/
+theorem Projective.iff_split : Module.Projective R P ↔
+    ∃ (M : Type max u v) (_ : AddCommGroup M) (_ : Module R M) (_ : Module.Free R M)
+      (i : P →ₗ[R] M) (s : M →ₗ[R] P), s.comp i = LinearMap.id :=
+  ⟨fun ⟨i, hi⟩ ↦ ⟨P →₀ R, _, _, inferInstance, i, Finsupp.linearCombination R id, LinearMap.ext hi⟩,
+    fun ⟨_, _, _, _, i, s, H⟩ ↦ Projective.of_split i s H⟩
+
 set_option maxSynthPendingDepth 2 in
 open TensorProduct in
 instance Projective.tensorProduct [hM : Module.Projective R M] [hN : Module.Projective R₀ N] :
@@ -197,11 +196,11 @@ instance Projective.tensorProduct [hM : Module.Projective R M] [hN : Module.Proj
     fapply Projective.of_split (R := R) (M := ((M →₀ R) ⊗[R₀] (N →₀ R₀)))
     · exact (AlgebraTensorModule.map sM (LinearMap.id (R := R₀) (M := N →₀ R₀)))
     · exact (AlgebraTensorModule.map
-        (Finsupp.total R id) (LinearMap.id (R := R₀) (M := N →₀ R₀)))
+        (Finsupp.linearCombination R id) (LinearMap.id (R := R₀) (M := N →₀ R₀)))
     · ext; simp [hsM _]
   fapply Projective.of_split (R := R) (M := (M ⊗[R₀] (N →₀ R₀)))
   · exact (AlgebraTensorModule.map (LinearMap.id (R := R) (M := M)) sN)
-  · exact (AlgebraTensorModule.map (LinearMap.id (R := R) (M := M)) (Finsupp.total R₀ id))
+  · exact (AlgebraTensorModule.map (LinearMap.id (R := R) (M := M)) (linearCombination R₀ id))
   · ext; simp [hsN _]
 
 end Ring
diff --git a/Mathlib/Algebra/Module/Rat.lean b/Mathlib/Algebra/Module/Rat.lean
index fa255570c0b86..c0ed84a4366ed 100644
--- a/Mathlib/Algebra/Module/Rat.lean
+++ b/Mathlib/Algebra/Module/Rat.lean
@@ -5,6 +5,7 @@ Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
 -/
 import Mathlib.Algebra.Module.Basic
 import Mathlib.Algebra.Field.Rat
+import Mathlib.Algebra.Order.Field.Rat
 
 /-!
 # Basic results about modules over the rationals.
@@ -14,6 +15,13 @@ universe u v
 
 variable {M M₂ : Type*}
 
+theorem map_nnratCast_smul [AddCommMonoid M] [AddCommMonoid M₂] {F : Type*} [FunLike F M M₂]
+    [AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionSemiring R] [DivisionSemiring S]
+    [Module R M] [Module S M₂] (c : ℚ≥0) (x : M) :
+    f ((c : R) • x) = (c : S) • f x := by
+  rw [NNRat.cast_def, NNRat.cast_def, div_eq_mul_inv, div_eq_mul_inv, mul_smul, mul_smul,
+    map_natCast_smul f R S, map_inv_natCast_smul f R S]
+
 theorem map_ratCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
     [AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
     [Module S M₂] (c : ℚ) (x : M) :
@@ -24,17 +32,34 @@ theorem map_ratCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLi
 @[deprecated (since := "2024-04-17")]
 alias map_rat_cast_smul := map_ratCast_smul
 
+theorem map_nnrat_smul [AddCommMonoid M] [AddCommMonoid M₂]
+    [_instM : Module ℚ≥0 M] [_instM₂ : Module ℚ≥0 M₂]
+    {F : Type*} [FunLike F M M₂] [AddMonoidHomClass F M M₂]
+    (f : F) (c : ℚ≥0) (x : M) : f (c • x) = c • f x :=
+  map_nnratCast_smul f ℚ≥0 ℚ≥0 c x
+
 theorem map_rat_smul [AddCommGroup M] [AddCommGroup M₂]
     [_instM : Module ℚ M] [_instM₂ : Module ℚ M₂]
     {F : Type*} [FunLike F M M₂] [AddMonoidHomClass F M M₂]
     (f : F) (c : ℚ) (x : M) : f (c • x) = c • f x :=
   map_ratCast_smul f ℚ ℚ c x
 
+/-- There can be at most one `Module ℚ≥0 E` structure on an additive commutative monoid. -/
+instance subsingleton_nnrat_module (E : Type*) [AddCommMonoid E] : Subsingleton (Module ℚ≥0 E) :=
+  ⟨fun P Q => (Module.ext' P Q) fun r x =>
+    map_nnrat_smul (_instM := P) (_instM₂ := Q) (AddMonoidHom.id E) r x⟩
+
 /-- There can be at most one `Module ℚ E` structure on an additive commutative group. -/
 instance subsingleton_rat_module (E : Type*) [AddCommGroup E] : Subsingleton (Module ℚ E) :=
   ⟨fun P Q => (Module.ext' P Q) fun r x =>
     map_rat_smul (_instM := P) (_instM₂ := Q) (AddMonoidHom.id E) r x⟩
 
+/-- If `E` is a vector space over two division semirings `R` and `S`, then scalar multiplications
+agree on non-negative rational numbers in `R` and `S`. -/
+theorem nnratCast_smul_eq {E : Type*} (R S : Type*) [AddCommMonoid E] [DivisionSemiring R]
+    [DivisionSemiring S] [Module R E] [Module S E] (r : ℚ≥0) (x : E) : (r : R) • x = (r : S) • x :=
+  map_nnratCast_smul (AddMonoidHom.id E) R S r x
+
 /-- If `E` is a vector space over two division rings `R` and `S`, then scalar multiplications
 agree on rational numbers in `R` and `S`. -/
 theorem ratCast_smul_eq {E : Type*} (R S : Type*) [AddCommGroup E] [DivisionRing R]
@@ -44,22 +69,41 @@ theorem ratCast_smul_eq {E : Type*} (R S : Type*) [AddCommGroup E] [DivisionRing
 @[deprecated (since := "2024-04-17")]
 alias rat_cast_smul_eq := ratCast_smul_eq
 
+instance IsScalarTower.nnrat {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] [Module R M]
+    [Module ℚ≥0 R] [Module ℚ≥0 M] : IsScalarTower ℚ≥0 R M where
+  smul_assoc r x y := map_nnrat_smul ((smulAddHom R M).flip y) r x
+
 instance IsScalarTower.rat {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
     [Module ℚ R] [Module ℚ M] : IsScalarTower ℚ R M where
   smul_assoc r x y := map_rat_smul ((smulAddHom R M).flip y) r x
 
-instance SMulCommClass.rat {R : Type u} {M : Type v} [Semiring R] [AddCommGroup M] [Module R M]
-    [Module ℚ M] : SMulCommClass ℚ R M where
-  smul_comm r x y := (map_rat_smul (smulAddHom R M x) r y).symm
+section
+variable {α : Type u} {M : Type v}
 
-instance SMulCommClass.rat' {R : Type u} {M : Type v} [Semiring R] [AddCommGroup M] [Module R M]
-    [Module ℚ M] : SMulCommClass R ℚ M :=
+instance SMulCommClass.nnrat [Monoid α] [AddCommMonoid M] [DistribMulAction α M] [Module ℚ≥0 M] :
+    SMulCommClass ℚ≥0 α M where
+  smul_comm r x y := (map_nnrat_smul (DistribMulAction.toAddMonoidHom M x) r y).symm
+
+instance SMulCommClass.rat [Monoid α] [AddCommGroup M] [DistribMulAction α M] [Module ℚ M] :
+    SMulCommClass ℚ α M where
+  smul_comm r x y := (map_rat_smul (DistribMulAction.toAddMonoidHom M x) r y).symm
+
+instance SMulCommClass.nnrat' [Monoid α] [AddCommMonoid M] [DistribMulAction α M] [Module ℚ≥0 M] :
+    SMulCommClass α ℚ≥0 M :=
+  SMulCommClass.symm _ _ _
+
+instance SMulCommClass.rat' [Monoid α] [AddCommGroup M] [DistribMulAction α M] [Module ℚ M] :
+    SMulCommClass α ℚ M :=
   SMulCommClass.symm _ _ _
 
+end
+
+-- see note [lower instance priority]
+instance (priority := 100) NNRatModule.noZeroSMulDivisors [AddCommMonoid M] [Module ℚ≥0 M] :
+    NoZeroSMulDivisors ℕ M :=
+  ⟨fun {k} {x : M} h => by simpa [← Nat.cast_smul_eq_nsmul ℚ≥0 k x] using h⟩
+
 -- see note [lower instance priority]
 instance (priority := 100) RatModule.noZeroSMulDivisors [AddCommGroup M] [Module ℚ M] :
     NoZeroSMulDivisors ℤ M :=
-  ⟨fun {k} {x : M} h => by
-    simpa only [← Int.cast_smul_eq_zsmul ℚ k x, smul_eq_zero, Rat.zero_iff_num_zero] using h⟩
-  -- Porting note: old proof was:
-  --⟨fun {k x} h => by simpa [zsmul_eq_smul_cast ℚ k x] using h⟩
+  ⟨fun {k} {x : M} h => by simpa [← Int.cast_smul_eq_zsmul ℚ k x] using h⟩
diff --git a/Mathlib/Algebra/Module/Submodule/Basic.lean b/Mathlib/Algebra/Module/Submodule/Basic.lean
index 6c89784c098f8..3958ed075e9fb 100644
--- a/Mathlib/Algebra/Module/Submodule/Basic.lean
+++ b/Mathlib/Algebra/Module/Submodule/Basic.lean
@@ -152,7 +152,7 @@ variable [Semiring R] [AddCommMonoid M] [Module R M] {A : Type*} [SetLike A M]
   [AddSubmonoidClass A M] [SMulMemClass A R M] (S' : A)
 
 -- Prefer subclasses of `Module` over `SMulMemClass`.
-/-- A submodule of a `Module` is a `Module`.  -/
+/-- A submodule of a `Module` is a `Module`. -/
 instance (priority := 75) toModule : Module R S' :=
   Subtype.coe_injective.module R (AddSubmonoidClass.subtype S') (SetLike.val_smul S')
 
@@ -236,6 +236,7 @@ instance isCentralScalar [SMul S R] [SMul S M] [IsScalarTower S R M] [SMul Sᵐ
 protected theorem nonempty : (p : Set M).Nonempty :=
   ⟨0, p.zero_mem⟩
 
+@[simp]
 theorem mk_eq_zero {x} (h : x ∈ p) : (⟨x, h⟩ : p) = 0 ↔ x = 0 :=
   Subtype.ext_iff_val
 
diff --git a/Mathlib/Algebra/Module/Submodule/Bilinear.lean b/Mathlib/Algebra/Module/Submodule/Bilinear.lean
index 8f1e0b93e8437..d6c065576cbbf 100644
--- a/Mathlib/Algebra/Module/Submodule/Bilinear.lean
+++ b/Mathlib/Algebra/Module/Submodule/Bilinear.lean
@@ -37,7 +37,7 @@ variable [Module R M] [Module R N] [Module R P]
 
 /-- Map a pair of submodules under a bilinear map.
 
-This is the submodule version of `Set.image2`.  -/
+This is the submodule version of `Set.image2`. -/
 def map₂ (f : M →ₗ[R] N →ₗ[R] P) (p : Submodule R M) (q : Submodule R N) : Submodule R P :=
   ⨆ s : p, q.map (f s)
 
@@ -56,13 +56,15 @@ theorem map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : Set M) (t : Set N)
   apply le_antisymm
   · rw [map₂_le]
     apply @span_induction' R M _ _ _ s
-    intro a ha
-    apply @span_induction' R N _ _ _ t
-    intro b hb
-    exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
-    all_goals intros; simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add,
-                                 LinearMap.zero_apply, LinearMap.add_apply, LinearMap.smul_apply,
-                                 map_smul]
+    on_goal 1 =>
+      intro a ha
+      apply @span_induction' R N _ _ _ t
+      · intro b hb
+        exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
+    all_goals
+      intros
+      simp only [*, add_mem, smul_mem, zero_mem, _root_.map_zero, map_add,
+        LinearMap.zero_apply, LinearMap.add_apply, LinearMap.smul_apply, map_smul]
   · rw [span_le, image2_subset_iff]
     intro a ha b hb
     exact apply_mem_map₂ _ (subset_span ha) (subset_span hb)
diff --git a/Mathlib/Algebra/Module/Submodule/Equiv.lean b/Mathlib/Algebra/Module/Submodule/Equiv.lean
index df3250cb70e1e..f5de6d41357b8 100644
--- a/Mathlib/Algebra/Module/Submodule/Equiv.lean
+++ b/Mathlib/Algebra/Module/Submodule/Equiv.lean
@@ -186,6 +186,11 @@ theorem ofBijective_symm_apply_apply [RingHomInvPair σ₁₂ σ₂₁] [RingHom
     (ofBijective f h).symm (f x) = x := by
   simp [LinearEquiv.symm_apply_eq]
 
+@[simp]
+theorem apply_ofBijective_symm_apply [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] {h}
+    (x : M₂) : f ((ofBijective f h).symm x) = x := by
+  rw [← ofBijective_apply f ((ofBijective f h).symm x), apply_symm_apply]
+
 end
 
 end AddCommMonoid
diff --git a/Mathlib/Algebra/Module/Submodule/Ker.lean b/Mathlib/Algebra/Module/Submodule/Ker.lean
index 6eefd47994bea..e329b35726b80 100644
--- a/Mathlib/Algebra/Module/Submodule/Ker.lean
+++ b/Mathlib/Algebra/Module/Submodule/Ker.lean
@@ -192,11 +192,11 @@ theorem ker_eq_bot {f : M →ₛₗ[τ₁₂] M₂} : ker f = ⊥ ↔ Injective
   · intro x ⟨hx, h'x⟩
     have : ⟨x, hx⟩ ∈ LinearMap.ker (LinearMap.domRestrict f S) := by simpa using h'x
     rw [h] at this
-    simpa using this
+    simpa [mk_eq_zero] using this
   · rintro ⟨x, hx⟩ h'x
     have : x ∈ S ⊓ LinearMap.ker f := ⟨hx, h'x⟩
     rw [h] at this
-    simpa using this
+    simpa [mk_eq_zero] using this
 
 @[simp] theorem injective_restrict_iff_disjoint {p : Submodule R M} {f : M →ₗ[R] M}
     (hf : ∀ x ∈ p, f x ∈ p) :
diff --git a/Mathlib/Algebra/Module/Submodule/Lattice.lean b/Mathlib/Algebra/Module/Submodule/Lattice.lean
index 1b9c97d71bbe6..3e15b4721bc18 100644
--- a/Mathlib/Algebra/Module/Submodule/Lattice.lean
+++ b/Mathlib/Algebra/Module/Submodule/Lattice.lean
@@ -241,6 +241,10 @@ theorem mem_finset_inf {ι} {s : Finset ι} {p : ι → Submodule R M} {x : M} :
     x ∈ s.inf p ↔ ∀ i ∈ s, x ∈ p i := by
   simp only [← SetLike.mem_coe, finset_inf_coe, Set.mem_iInter]
 
+lemma inf_iInf {ι : Type*} [Nonempty ι] {p : ι → Submodule R M} (q : Submodule R M) :
+    q ⊓ ⨅ i, p i = ⨅ i, q ⊓ p i :=
+  SetLike.coe_injective <| by simpa only [inf_coe, iInf_coe] using Set.inter_iInter _ _
+
 theorem mem_sup_left {S T : Submodule R M} : ∀ {x : M}, x ∈ S → x ∈ S ⊔ T := by
   have : S ≤ S ⊔ T := le_sup_left
   rw [LE.le] at this
@@ -287,7 +291,7 @@ theorem toAddSubmonoid_sSup (s : Set (Submodule R M)) :
     { toAddSubmonoid := sSup (toAddSubmonoid '' s)
       smul_mem' := fun t {m} h ↦ by
         simp_rw [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup, sSup_eq_iSup'] at h ⊢
-        refine AddSubmonoid.iSup_induction'
+        refine AddSubmonoid.iSup_induction' _
           (C := fun x _ ↦ t • x ∈ ⨆ p : toAddSubmonoid '' s, (p : AddSubmonoid M)) ?_ ?_
           (fun x y _ _ ↦ ?_) h
         · rintro ⟨-, ⟨p : Submodule R M, hp : p ∈ s, rfl⟩⟩ x (hx : x ∈ p)
diff --git a/Mathlib/Algebra/Module/Submodule/LinearMap.lean b/Mathlib/Algebra/Module/Submodule/LinearMap.lean
index 4ff64ac81d3b1..36b04e8dc9c3d 100644
--- a/Mathlib/Algebra/Module/Submodule/LinearMap.lean
+++ b/Mathlib/Algebra/Module/Submodule/LinearMap.lean
@@ -75,9 +75,11 @@ theorem subtype_apply (x : p) : p.subtype x = x :=
   rfl
 
 @[simp]
-theorem coeSubtype : (Submodule.subtype p : p → M) = Subtype.val :=
+theorem coe_subtype : (Submodule.subtype p : p → M) = Subtype.val :=
   rfl
 
+@[deprecated (since := "2024-09-27")] alias coeSubtype := coe_subtype
+
 theorem injective_subtype : Injective p.subtype :=
   Subtype.coe_injective
 
@@ -181,6 +183,13 @@ lemma restrict_comp
     (g ∘ₗ f).restrict hfg = (g.restrict hg) ∘ₗ (f.restrict hf) :=
   rfl
 
+-- TODO Consider defining `Algebra R (p.compatibleMaps p)`, `AlgHom` version of `LinearMap.restrict`
+lemma restrict_smul_one
+    {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] {p : Submodule R M}
+    (μ : R) (h : ∀ x ∈ p, (μ • (1 : Module.End R M)) x ∈ p := fun _ ↦ p.smul_mem μ) :
+    (μ • 1 : Module.End R M).restrict h = μ • (1 : Module.End R p) :=
+  rfl
+
 lemma restrict_commute {f g : M →ₗ[R] M} (h : Commute f g) {p : Submodule R M}
     (hf : MapsTo f p p) (hg : MapsTo g p p) :
     Commute (f.restrict hf) (g.restrict hg) := by
diff --git a/Mathlib/Algebra/Module/Submodule/Map.lean b/Mathlib/Algebra/Module/Submodule/Map.lean
index 89fd9e06bb747..0e08a969e1c07 100644
--- a/Mathlib/Algebra/Module/Submodule/Map.lean
+++ b/Mathlib/Algebra/Module/Submodule/Map.lean
@@ -99,11 +99,12 @@ theorem map_comp [RingHomSurjective σ₂₃] [RingHomSurjective σ₁₃] (f :
     (g : M₂ →ₛₗ[σ₂₃] M₃) (p : Submodule R M) : map (g.comp f : M →ₛₗ[σ₁₃] M₃) p = map g (map f p) :=
   SetLike.coe_injective <| by simp only [← image_comp, map_coe, LinearMap.coe_comp, comp_apply]
 
+@[gcongr]
 theorem map_mono {f : F} {p p' : Submodule R M} : p ≤ p' → map f p ≤ map f p' :=
   image_subset _
 
 @[simp]
-theorem map_zero : map (0 : M →ₛₗ[σ₁₂] M₂) p = ⊥ :=
+protected theorem map_zero : map (0 : M →ₛₗ[σ₁₂] M₂) p = ⊥ :=
   have : ∃ x : M, x ∈ p := ⟨0, p.zero_mem⟩
   ext <| by simp [this, eq_comm]
 
@@ -119,6 +120,10 @@ theorem map_inf (f : F) {p q : Submodule R M} (hf : Injective f) :
     (p ⊓ q).map f = p.map f ⊓ q.map f :=
   SetLike.coe_injective <| Set.image_inter hf
 
+lemma map_iInf {ι : Type*} [Nonempty ι] {p : ι → Submodule R M} (f : F) (hf : Injective f) :
+    (⨅ i, p i).map f = ⨅ i, (p i).map f :=
+  SetLike.coe_injective <| by simpa only [map_coe, iInf_coe] using hf.injOn.image_iInter_eq
+
 theorem range_map_nonempty (N : Submodule R M) :
     (Set.range (fun ϕ => Submodule.map ϕ N : (M →ₛₗ[σ₁₂] M₂) → Submodule R₂ M₂)).Nonempty :=
   ⟨_, Set.mem_range.mpr ⟨0, rfl⟩⟩
@@ -186,6 +191,7 @@ theorem comap_comp (f : M →ₛₗ[σ₁₂] M₂) (g : M₂ →ₛₗ[σ₂₃
     comap (g.comp f : M →ₛₗ[σ₁₃] M₃) p = comap f (comap g p) :=
   rfl
 
+@[gcongr]
 theorem comap_mono {f : F} {q q' : Submodule R₂ M₂} : q ≤ q' → comap f q ≤ comap f q' :=
   preimage_mono
 
@@ -409,6 +415,10 @@ lemma comap_neg {f : M →ₗ[R] M₂} {p : Submodule R M₂} :
     p.comap (-f) = p.comap f := by
   ext; simp
 
+lemma map_toAddSubgroup (f : M →ₗ[R] M₂) (p : Submodule R M) :
+    (p.map f).toAddSubgroup = p.toAddSubgroup.map (f : M →+ M₂) :=
+  rfl
+
 end AddCommGroup
 
 end Submodule
@@ -620,7 +630,7 @@ This is `LinearEquiv.ofSubmodule'` but with `map` on the right instead of `comap
 def submoduleMap (p : Submodule R M) : p ≃ₛₗ[σ₁₂] ↥(p.map (e : M →ₛₗ[σ₁₂] M₂) : Submodule R₂ M₂) :=
   { ((e : M →ₛₗ[σ₁₂] M₂).domRestrict p).codRestrict (p.map (e : M →ₛₗ[σ₁₂] M₂)) fun x =>
       ⟨x, by
-        simp only [LinearMap.domRestrict_apply, eq_self_iff_true, and_true_iff, SetLike.coe_mem,
+        simp only [LinearMap.domRestrict_apply, eq_self_iff_true, and_true, SetLike.coe_mem,
           SetLike.mem_coe]⟩ with
     invFun := fun y =>
       ⟨(e.symm : M₂ →ₛₗ[σ₂₁] M) y, by
diff --git a/Mathlib/Algebra/Module/Submodule/Pointwise.lean b/Mathlib/Algebra/Module/Submodule/Pointwise.lean
index 99e50eaa44e68..c8d8a18665f39 100644
--- a/Mathlib/Algebra/Module/Submodule/Pointwise.lean
+++ b/Mathlib/Algebra/Module/Submodule/Pointwise.lean
@@ -416,12 +416,10 @@ lemma set_smul_eq_map [SMulCommClass R R N] :
   apply set_smul_eq_of_le
   · intro r n hr hn
     exact ⟨Finsupp.single r ⟨n, hn⟩, Finsupp.single_mem_supported _ _ hr, by simp⟩
-
   · intro x hx
     obtain ⟨c, hc, rfl⟩ := hx
-    simp only [LinearMap.coe_comp, coeSubtype, Finsupp.coe_lsum, Finsupp.sum, LinearMap.coe_mk,
-      AddHom.coe_mk, Function.comp_apply, AddSubmonoid.coe_finset_sum, coe_toAddSubmonoid,
-      SetLike.val_smul]
+    simp only [LinearMap.coe_comp, coe_subtype, Finsupp.coe_lsum, Finsupp.sum, Function.comp_apply]
+    rw [AddSubmonoid.coe_finset_sum]
     refine Submodule.sum_mem (p := sR • N) (t := c.support) ?_ _ ⟨sR • N, ?_⟩
     · rintro r hr
       rw [mem_set_smul_def, Submodule.mem_sInf]
@@ -442,10 +440,9 @@ lemma mem_set_smul (x : M) [SMulCommClass R R N] :
     rw [set_smul_eq_map] at h
     obtain ⟨c, hc, rfl⟩ := h
     exact ⟨c, hc, rfl⟩
-
   · rw [mem_set_smul_def, Submodule.mem_sInf]
     rintro ⟨c, hc1, rfl⟩ p hp
-    simp only [Finsupp.sum, AddSubmonoid.coe_finset_sum, coe_toAddSubmonoid, SetLike.val_smul]
+    rw [Finsupp.sum, AddSubmonoid.coe_finset_sum]
     exact Submodule.sum_mem _ fun r hr ↦ hp (hc1 hr) (c _).2
 
 @[simp] lemma empty_set_smul : (∅ : Set S) • N = ⊥ := by
@@ -496,8 +493,8 @@ protected def pointwiseSetMulAction [SMulCommClass R R M] :
     (set_smul_le _ _ _ fun r m hr hm ↦ by
       have : SMulCommClass R R x := ⟨fun r s m => Subtype.ext <| smul_comm _ _ _⟩
       obtain ⟨c, hc1, rfl⟩ := mem_set_smul _ _ _ |>.mp hm
-      simp only [Finsupp.sum, AddSubmonoid.coe_finset_sum, coe_toAddSubmonoid, SetLike.val_smul,
-        Finset.smul_sum, smul_smul]
+      rw [Finsupp.sum, AddSubmonoid.coe_finset_sum]
+      simp only [SetLike.val_smul, Finset.smul_sum, smul_smul]
       exact Submodule.sum_mem _ fun r' hr' ↦
         mem_set_smul_of_mem_mem (Set.mul_mem_mul hr (hc1 hr')) (c _).2)
 
@@ -533,11 +530,12 @@ lemma coe_span_smul {R' M' : Type*} [CommSemiring R'] [AddCommMonoid M'] [Module
     (Ideal.span s : Set R') • N = s • N :=
   set_smul_eq_of_le _ _ _
     (by rintro r n hr hn
-        induction' hr using Submodule.span_induction' with r h _ _ _ _ ihr ihs r r' hr hr'
-        · exact mem_set_smul_of_mem_mem h hn
-        · rw [zero_smul]; exact Submodule.zero_mem _
-        · rw [add_smul]; exact Submodule.add_mem _ ihr ihs
-        · rw [mem_span_set] at hr
+        induction hr using Submodule.span_induction' with
+        | mem _ h => exact mem_set_smul_of_mem_mem h hn
+        | zero => rw [zero_smul]; exact Submodule.zero_mem _
+        | add _ _ _ _ ihr ihs => rw [add_smul]; exact Submodule.add_mem _ ihr ihs
+        | smul _ _ hr =>
+          rw [mem_span_set] at hr
           obtain ⟨c, hc, rfl⟩ := hr
           rw [Finsupp.sum, Finset.smul_sum, Finset.sum_smul]
           refine Submodule.sum_mem _ fun i hi => ?_
diff --git a/Mathlib/Algebra/Module/Torsion.lean b/Mathlib/Algebra/Module/Torsion.lean
index 492b9633c7d35..5be854af327cb 100644
--- a/Mathlib/Algebra/Module/Torsion.lean
+++ b/Mathlib/Algebra/Module/Torsion.lean
@@ -3,10 +3,11 @@ Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
 -/
-import Mathlib.LinearAlgebra.Isomorphisms
+import Mathlib.Algebra.DirectSum.Module
+import Mathlib.Algebra.Module.ZMod
 import Mathlib.GroupTheory.Torsion
+import Mathlib.LinearAlgebra.Isomorphisms
 import Mathlib.RingTheory.Coprime.Ideal
-import Mathlib.Data.ZMod.Module
 
 /-!
 # Torsion submodules
@@ -360,9 +361,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf
     (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤) :
     ⨆ i ∈ S, torsionBySet R M (p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) := by
   rcases S.eq_empty_or_nonempty with h | h
-  · simp only [h]
-    -- Porting note: converts were not cooperating
-    convert iSup_emptyset (f := fun i => torsionBySet R M (p i)) <;> simp
+  · simp [h]
   apply le_antisymm
   · apply iSup_le _
     intro i
diff --git a/Mathlib/Algebra/Module/ULift.lean b/Mathlib/Algebra/Module/ULift.lean
index 730083e6948c9..1517ff6089728 100644
--- a/Mathlib/Algebra/Module/ULift.lean
+++ b/Mathlib/Algebra/Module/ULift.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.GroupWithZero.ULift
 import Mathlib.Algebra.Ring.ULift
diff --git a/Mathlib/Algebra/Module/Zlattice/Basic.lean b/Mathlib/Algebra/Module/ZLattice/Basic.lean
similarity index 77%
rename from Mathlib/Algebra/Module/Zlattice/Basic.lean
rename to Mathlib/Algebra/Module/ZLattice/Basic.lean
index c3614a576ee4a..a436852b2ab88 100644
--- a/Mathlib/Algebra/Module/Zlattice/Basic.lean
+++ b/Mathlib/Algebra/Module/ZLattice/Basic.lean
@@ -17,27 +17,34 @@ subgroup of `E` such that `L` spans `E` over `K`.
 
 A `ℤ`-lattice `L` can be defined in two ways:
 * For `b` a basis of `E`, then `L = Submodule.span ℤ (Set.range b)` is a ℤ-lattice of `E`
-* As an `AddSubgroup E` with the additional properties:
+* As an`ℤ-submodule` of `E` with the additional properties:
   * `DiscreteTopology L`, that is `L` is discrete
   * `Submodule.span ℝ (L : Set E) = ⊤`, that is `L` spans `E` over `K`.
 
-Results about the first point of view are in the `Zspan` namespace and results about the second
-point of view are in the `Zlattice` namespace.
+Results about the first point of view are in the `ZSpan` namespace and results about the second
+point of view are in the `ZLattice` namespace.
 
 ## Main results
 
-* `Zspan.isAddFundamentalDomain`: for a ℤ-lattice `Submodule.span ℤ (Set.range b)`, proves that
-the set defined by `Zspan.fundamentalDomain` is a fundamental domain.
-* `Zlattice.module_free`: an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free
+* `ZSpan.isAddFundamentalDomain`: for a ℤ-lattice `Submodule.span ℤ (Set.range b)`, proves that
+the set defined by `ZSpan.fundamentalDomain` is a fundamental domain.
+* `ZLattice.module_free`: a `ℤ`-submodule of `E` that is discrete and spans `E` over `K` is a free
 `ℤ`-module
-* `Zlattice.rank`:  an AddSubgroup of `E` that is discrete and spans `E` over `K` is a free
-`ℤ`-module of `ℤ`-rank equal to the `K`-rank of `E`
+* `ZLattice.rank`: a `ℤ`-submodule of `E` that is discrete and spans `E` over `K` is free
+of `ℤ`-rank equal to the `K`-rank of `E`
+
+## Implementation Notes
+
+A `ZLattice` could be defined either as a `AddSubgroup E` or a `Submodule ℤ E`. However, the module
+aspect appears to be the more useful one (especially in computations involving basis) and is also
+consistent with the `ZSpan` construction of `ℤ`-lattices.
+
 -/
 
 
 noncomputable section
 
-namespace Zspan
+namespace ZSpan
 
 open MeasureTheory MeasurableSet Submodule Bornology
 
@@ -51,7 +58,7 @@ variable (b : Basis ι K E)
 
 theorem span_top : span K (span ℤ (Set.range b) : Set E) = ⊤ := by simp [span_span_of_tower]
 
-/-- The fundamental domain of the ℤ-lattice spanned by `b`. See `Zspan.isAddFundamentalDomain`
+/-- The fundamental domain of the ℤ-lattice spanned by `b`. See `ZSpan.isAddFundamentalDomain`
 for the proof that it is a fundamental domain. -/
 def fundamentalDomain : Set E := {m | ∀ i, b.repr m i ∈ Set.Ico (0 : K) 1}
 
@@ -123,7 +130,7 @@ theorem ceil_eq_self_of_mem (m : E) (h : m ∈ span ℤ (Set.range b)) : (ceil b
   exact congr_arg (Int.cast : ℤ → K) (Int.ceil_intCast z)
 
 /-- The map that sends a vector `E` to the `fundamentalDomain` of the lattice,
-see `Zspan.fract_mem_fundamentalDomain`, and `fractRestrict` for the map with the codomain
+see `ZSpan.fract_mem_fundamentalDomain`, and `fractRestrict` for the map with the codomain
 restricted to `fundamentalDomain`. -/
 def fract (m : E) : E := m - floor b m
 
@@ -138,7 +145,7 @@ theorem fract_fract (m : E) : fract b (fract b m) = fract b m :=
   Basis.ext_elem b fun _ => by classical simp only [repr_fract_apply, Int.fract_fract]
 
 @[simp]
-theorem fract_zspan_add (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
+theorem fract_zSpan_add (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
     fract b (v + m) = fract b m := by
   classical
   refine (Basis.ext_elem_iff b).mpr fun i => ?_
@@ -148,8 +155,8 @@ theorem fract_zspan_add (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
     ← eq_intCast (algebraMap ℤ K) _, Basis.restrictScalars_repr_apply, coe_mk]
 
 @[simp]
-theorem fract_add_zspan (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
-    fract b (m + v) = fract b m := by rw [add_comm, fract_zspan_add b m h]
+theorem fract_add_ZSpan (m : E) {v : E} (h : v ∈ span ℤ (Set.range b)) :
+    fract b (m + v) = fract b m := by rw [add_comm, fract_zSpan_add b m h]
 
 variable {b}
 
@@ -222,7 +229,7 @@ theorem fundamentalDomain_isBounded [Finite ι] [HasSolidNorm K] :
 theorem vadd_mem_fundamentalDomain [Fintype ι] (y : span ℤ (Set.range b)) (x : E) :
     y +ᵥ x ∈ fundamentalDomain b ↔ y = -floor b x := by
   rw [Subtype.ext_iff, ← add_right_inj x, NegMemClass.coe_neg, ← sub_eq_add_neg, ← fract_apply,
-    ← fract_zspan_add b _ (Subtype.mem y), add_comm, ← vadd_eq_add, ← vadd_def, eq_comm, ←
+    ← fract_zSpan_add b _ (Subtype.mem y), add_comm, ← vadd_eq_add, ← vadd_def, eq_comm, ←
     fract_eq_self]
 
 theorem exist_unique_vadd_mem_fundamentalDomain [Finite ι] (x : E) :
@@ -232,8 +239,8 @@ theorem exist_unique_vadd_mem_fundamentalDomain [Finite ι] (x : E) :
   · exact (vadd_mem_fundamentalDomain b (-floor b x) x).mpr rfl
   · exact (vadd_mem_fundamentalDomain b y x).mp h
 
-/-- The map `Zspan.fractRestrict` defines an equiv map between `E ⧸ span ℤ (Set.range b)`
-and `Zspan.fundamentalDomain b`. -/
+/-- The map `ZSpan.fractRestrict` defines an equiv map between `E ⧸ span ℤ (Set.range b)`
+and `ZSpan.fundamentalDomain b`. -/
 def quotientEquiv [Fintype ι] :
     E ⧸ span ℤ (Set.range b) ≃ (fundamentalDomain b) := by
   refine Equiv.ofBijective ?_ ⟨fun x y => ?_, fun x => ?_⟩
@@ -307,20 +314,25 @@ theorem fundamentalDomain_measurableSet [MeasurableSpace E] [OpensMeasurableSpac
       Set.mem_preimage, Basis.equivFun_apply, Set.mem_pi, Set.mem_univ, forall_true_left]
 
 /-- For a ℤ-lattice `Submodule.span ℤ (Set.range b)`, proves that the set defined
-by `Zspan.fundamentalDomain` is a fundamental domain. -/
+by `ZSpan.fundamentalDomain` is a fundamental domain. -/
 protected theorem isAddFundamentalDomain [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E]
     (μ : Measure E) :
-    IsAddFundamentalDomain (span ℤ (Set.range b)).toAddSubgroup (fundamentalDomain b) μ := by
+    IsAddFundamentalDomain (span ℤ (Set.range b)) (fundamentalDomain b) μ := by
   cases nonempty_fintype ι
   exact IsAddFundamentalDomain.mk' (nullMeasurableSet (fundamentalDomain_measurableSet b))
     fun x => exist_unique_vadd_mem_fundamentalDomain b x
 
+/-- A version of `ZSpan.isAddFundamentalDomain` for `AddSubgroup`. -/
+protected theorem isAddFundamentalDomain' [Finite ι] [MeasurableSpace E] [OpensMeasurableSpace E]
+    (μ : Measure E) :
+    IsAddFundamentalDomain (span ℤ (Set.range b)).toAddSubgroup (fundamentalDomain b) μ :=
+  ZSpan.isAddFundamentalDomain b μ
+
 theorem measure_fundamentalDomain_ne_zero [Finite ι] [MeasurableSpace E] [BorelSpace E]
     {μ : Measure E} [Measure.IsAddHaarMeasure μ] :
     μ (fundamentalDomain b) ≠ 0 := by
-  convert (Zspan.isAddFundamentalDomain b μ).measure_ne_zero (NeZero.ne μ)
-  simp only [mem_toAddSubgroup]
-  infer_instance
+  convert (ZSpan.isAddFundamentalDomain b μ).measure_ne_zero (NeZero.ne μ)
+  exact (inferInstance : VAddInvariantMeasure (span ℤ (Set.range b)).toAddSubgroup E μ)
 
 theorem measure_fundamentalDomain [Fintype ι] [DecidableEq ι] [MeasurableSpace E] (μ : Measure E)
     [BorelSpace E] [Measure.IsAddHaarMeasure μ] (b₀ : Basis ι ℝ E) :
@@ -368,31 +380,30 @@ theorem fundamentalDomain_ae_parallelepiped [Fintype ι] [MeasurableSpace E] (μ
 
 end Real
 
-end Zspan
+end ZSpan
 
-section Zlattice
+section ZLattice
 
-open Submodule FiniteDimensional
+open Submodule Module ZSpan
 
 -- TODO: generalize this class to other rings than `ℤ`
-/-- An `L : Addsubgroup E` where `E` is a vector space over a normed field `K` is a `ℤ`-lattice if
+/-- `L : Submodule ℤ E` where `E` is a vector space over a normed field `K` is a `ℤ`-lattice if
 it is discrete and spans `E` over `K`. -/
-class IsZlattice (K : Type*) [NormedField K] {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
-    (L : AddSubgroup E) [DiscreteTopology L] : Prop where
+class IsZLattice (K : Type*) [NormedField K] {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
+    (L : Submodule ℤ E) [DiscreteTopology L] : Prop where
   /-- `L` spans the full space `E` over `K`. -/
   span_top : span K (L : Set E) = ⊤
 
-theorem _root_.Zspan.isZlattice {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+theorem _root_.ZSpan.isZLattice {E ι : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
     [Finite ι] (b : Basis ι ℝ E) :
-    IsZlattice ℝ (span ℤ (Set.range b)).toAddSubgroup where
-  span_top := Zspan.span_top b
+    IsZLattice ℝ (span ℤ (Set.range b)) where
+  span_top := ZSpan.span_top b
 
 variable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]
-variable [ProperSpace E] (L : AddSubgroup E) [DiscreteTopology L]
+variable [ProperSpace E] (L : Submodule ℤ E) [DiscreteTopology L]
 
-theorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L := by
-  suffices (AddSubgroup.toIntSubmodule L).FG by exact (fg_iff_add_subgroup_fg _).mp this
+theorem Zlattice.FG [hs : IsZLattice K L] : L.FG := by
   obtain ⟨s, ⟨h_incl, ⟨h_span, h_lind⟩⟩⟩ := exists_linearIndependent K (L : Set E)
   -- Let `s` be a maximal `K`-linear independent family of elements of `L`. We show that
   -- `L` is finitely generated (as a ℤ-module) because it fits in the exact sequence
@@ -401,75 +412,68 @@ theorem Zlattice.FG [hs : IsZlattice K L] : AddSubgroup.FG L := by
   · -- Let `b` be the `K`-basis of `E` formed by the vectors in `s`. The elements of
     -- `L ⧸ span ℤ s = L ⧸ span ℤ b` are in bijection with elements of `L ∩ fundamentalDomain b`
     -- so there are finitely many since `fundamentalDomain b` is bounded.
-    refine fg_def.mpr ⟨map (span ℤ s).mkQ (AddSubgroup.toIntSubmodule L), ?_, span_eq _⟩
+    refine fg_def.mpr ⟨map (span ℤ s).mkQ L, ?_, span_eq _⟩
     let b := Basis.mk h_lind (by
       rw [← hs.span_top, ← h_span]
       exact span_mono (by simp only [Subtype.range_coe_subtype, Set.setOf_mem_eq, subset_rfl]))
     rw [show span ℤ s = span ℤ (Set.range b) by simp [b, Basis.coe_mk, Subtype.range_coe_subtype]]
     have : Fintype s := h_lind.setFinite.fintype
-    refine Set.Finite.of_finite_image (f := ((↑) : _ →  E) ∘ Zspan.quotientEquiv b) ?_
-      (Function.Injective.injOn (Subtype.coe_injective.comp (Zspan.quotientEquiv b).injective))
-    have : Set.Finite ((Zspan.fundamentalDomain b) ∩ L) :=
-      Metric.finite_isBounded_inter_isClosed (Zspan.fundamentalDomain_isBounded b) inferInstance
+    refine Set.Finite.of_finite_image (f := ((↑) : _ →  E) ∘ quotientEquiv b) ?_
+      (Function.Injective.injOn (Subtype.coe_injective.comp (quotientEquiv b).injective))
+    have : ((fundamentalDomain b) ∩ L).Finite := by
+      change ((fundamentalDomain b) ∩ L.toAddSubgroup).Finite
+      have : DiscreteTopology L.toAddSubgroup := (inferInstance : DiscreteTopology L)
+      exact Metric.finite_isBounded_inter_isClosed (fundamentalDomain_isBounded b) inferInstance
     refine Set.Finite.subset this ?_
     rintro _ ⟨_, ⟨⟨x, ⟨h_mem, rfl⟩⟩, rfl⟩⟩
-    rw [Function.comp_apply, mkQ_apply, Zspan.quotientEquiv_apply_mk, Zspan.fractRestrict_apply]
+    rw [Function.comp_apply, mkQ_apply, quotientEquiv_apply_mk, fractRestrict_apply]
     refine ⟨?_, ?_⟩
-    · exact Zspan.fract_mem_fundamentalDomain b x
-    · rw [Zspan.fract, SetLike.mem_coe, sub_eq_add_neg]
-      refine AddSubgroup.add_mem _ h_mem
-        (neg_mem (Set.mem_of_subset_of_mem ?_ (Subtype.mem (Zspan.floor b x))))
-      rw [show (L : Set E) = AddSubgroup.toIntSubmodule L by rfl]
+    · exact fract_mem_fundamentalDomain b x
+    · rw [fract, SetLike.mem_coe, sub_eq_add_neg]
+      refine Submodule.add_mem _ h_mem
+        (neg_mem (Set.mem_of_subset_of_mem ?_ (Subtype.mem (floor b x))))
       rw [SetLike.coe_subset_coe, Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq]
       exact span_le.mpr h_incl
   · -- `span ℤ s` is finitely generated because `s` is finite
     rw [ker_mkQ, inf_of_le_right (span_le.mpr h_incl)]
     exact fg_span (LinearIndependent.setFinite h_lind)
 
-theorem Zlattice.module_finite [IsZlattice K L] : Module.Finite ℤ L :=
-  Module.Finite.iff_addGroup_fg.mpr ((AddGroup.fg_iff_addSubgroup_fg L).mpr (FG K L))
+theorem ZLattice.module_finite [IsZLattice K L] : Module.Finite ℤ L :=
+  Module.Finite.iff_fg.mpr (Zlattice.FG K L)
 
-instance instModuleFinite_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]
-    [NormedSpace ℝ E] [FiniteDimensional ℝ E] (L : AddSubgroup E) [DiscreteTopology L] :
+instance instModuleFinite_of_discrete_submodule {E : Type*} [NormedAddCommGroup E]
+    [NormedSpace ℝ E] [FiniteDimensional ℝ E] (L : Submodule ℤ E) [DiscreteTopology L] :
     Module.Finite ℤ L := by
   let f := (span ℝ (L : Set E)).subtype
-  let L₀ := (AddSubgroup.toIntSubmodule L).comap (f.restrictScalars ℤ)
+  let L₀ := L.comap (f.restrictScalars ℤ)
   have h_img : f '' L₀ = L := by
     rw [← LinearMap.coe_restrictScalars ℤ f, ← Submodule.map_coe (f.restrictScalars ℤ),
-      Submodule.map_comap_eq_self, AddSubgroup.coe_toIntSubmodule]
+      Submodule.map_comap_eq_self]
     exact fun x hx ↦ LinearMap.mem_range.mpr ⟨⟨x, Submodule.subset_span hx⟩, rfl⟩
   suffices Module.Finite ℤ L₀ by
-    have : L₀.map (f.restrictScalars ℤ) = (AddSubgroup.toIntSubmodule L) :=
+    have : L₀.map (f.restrictScalars ℤ) = L :=
       SetLike.ext'_iff.mpr h_img
     convert this ▸ Module.Finite.map L₀ (f.restrictScalars ℤ)
-  have : DiscreteTopology L₀.toAddSubgroup := by
+  have : DiscreteTopology L₀ := by
     refine DiscreteTopology.preimage_of_continuous_injective (L : Set E) ?_ (injective_subtype _)
     exact LinearMap.continuous_of_finiteDimensional f
-  have : IsZlattice ℝ L₀.toAddSubgroup := ⟨by
+  have : IsZLattice ℝ L₀ := ⟨by
     rw [← (Submodule.map_injective_of_injective (injective_subtype _)).eq_iff, Submodule.map_span,
-      Submodule.map_top, range_subtype, coe_toAddSubgroup, h_img]⟩
-  exact Zlattice.module_finite ℝ L₀.toAddSubgroup
+      Submodule.map_top, range_subtype, h_img]⟩
+  exact ZLattice.module_finite ℝ L₀
 
-theorem Zlattice.module_free [IsZlattice K L] : Module.Free ℤ L := by
+theorem ZLattice.module_free [IsZLattice K L] : Module.Free ℤ L := by
   have : Module.Finite ℤ L := module_finite K L
   have : Module ℚ E := Module.compHom E (algebraMap ℚ K)
-  have : NoZeroSMulDivisors ℤ E := RatModule.noZeroSMulDivisors
-  have : NoZeroSMulDivisors ℤ L := by
-    change NoZeroSMulDivisors ℤ (AddSubgroup.toIntSubmodule L)
-    exact noZeroSMulDivisors _
   infer_instance
 
-instance instModuleFree_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E]
-    [NormedSpace ℝ E] [FiniteDimensional ℝ E] (L : AddSubgroup E) [DiscreteTopology L] :
+instance instModuleFree_of_discrete_submodule {E : Type*} [NormedAddCommGroup E]
+    [NormedSpace ℝ E] [FiniteDimensional ℝ E] (L : Submodule ℤ E) [DiscreteTopology L] :
     Module.Free ℤ L := by
   have : Module ℚ E := Module.compHom E (algebraMap ℚ ℝ)
-  have : NoZeroSMulDivisors ℤ E := RatModule.noZeroSMulDivisors
-  have : NoZeroSMulDivisors ℤ L := by
-    change NoZeroSMulDivisors ℤ (AddSubgroup.toIntSubmodule L)
-    exact noZeroSMulDivisors _
   infer_instance
 
-theorem Zlattice.rank [hs : IsZlattice K L] : finrank ℤ L = finrank K E := by
+theorem ZLattice.rank [hs : IsZLattice K L] : finrank ℤ L = finrank K E := by
   classical
   have : Module.Finite ℤ L := module_finite K L
   have : Module.Free ℤ L := module_free K L
@@ -478,10 +482,10 @@ theorem Zlattice.rank [hs : IsZlattice K L] : finrank ℤ L = finrank K E := by
   -- Let `b` be a `ℤ`-basis of `L` formed of vectors of `E`
   let b := Subtype.val ∘ b₀
   have : LinearIndependent ℤ b :=
-    LinearIndependent.map' b₀.linearIndependent (L.toIntSubmodule.subtype) (ker_subtype _)
+    LinearIndependent.map' b₀.linearIndependent (L.subtype) (ker_subtype _)
   -- We prove some assertions that will be useful later on
-  have h_spanL : span ℤ (Set.range b) = AddSubgroup.toIntSubmodule L := by
-    convert congrArg (map (Submodule.subtype (AddSubgroup.toIntSubmodule L))) b₀.span_eq
+  have h_spanL : span ℤ (Set.range b) = L := by
+    convert congrArg (map (Submodule.subtype L)) b₀.span_eq
     · rw [map_span, Set.range_comp]
       rfl
     · exact (map_subtype_top _).symm
@@ -525,26 +529,27 @@ theorem Zlattice.rank [hs : IsZlattice K L] : finrank ℤ L = finrank K E := by
       linearIndependent_insert (Set.not_mem_of_mem_diff hv),  not_and, not_not]
     intro _
     -- But that follows from the fact that there exist `n, m : ℕ`, `n ≠ m`
-    -- such that `(n - m) • v ∈ span ℤ e` which is true since `n ↦ Zspan.fract e (n • v)`
+    -- such that `(n - m) • v ∈ span ℤ e` which is true since `n ↦ ZSpan.fract e (n • v)`
     -- takes value into the finite set `fundamentalDomain e ∩ L`
-    have h_mapsto : Set.MapsTo (fun n : ℤ => Zspan.fract e (n • v)) Set.univ
+    have h_mapsto : Set.MapsTo (fun n : ℤ => fract e (n • v)) Set.univ
         (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ (L : Set E)) := by
       rw [Set.mapsTo_inter, Set.mapsTo_univ_iff, Set.mapsTo_univ_iff]
-      refine ⟨fun _ ↦ mem_closedBall_zero_iff.mpr (Zspan.norm_fract_le e _), fun _ => ?_⟩
-      · change _ ∈ AddSubgroup.toIntSubmodule L
-        rw [← h_spanL]
+      refine ⟨fun _ ↦ mem_closedBall_zero_iff.mpr (norm_fract_le e _), fun _ => ?_⟩
+      · rw [← h_spanL]
         refine sub_mem ?_ ?_
         · exact zsmul_mem (subset_span (Set.diff_subset hv)) _
         · exact span_mono (by simp [e, ht_inc]) (coe_mem _)
-    have h_finite : Set.Finite (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ (L : Set E)) :=
-      Metric.finite_isBounded_inter_isClosed Metric.isBounded_closedBall inferInstance
+    have h_finite : Set.Finite (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ (L : Set E)) := by
+      change ((_ : Set E) ∩ L.toAddSubgroup).Finite
+      have : DiscreteTopology L.toAddSubgroup := (inferInstance : DiscreteTopology L)
+      exact Metric.finite_isBounded_inter_isClosed  Metric.isBounded_closedBall inferInstance
     obtain ⟨n, -, m, -, h_neq, h_eq⟩ := Set.Infinite.exists_ne_map_eq_of_mapsTo
       Set.infinite_univ h_mapsto h_finite
     have h_nz : (-n + m : ℚ) ≠ 0 := by
       rwa [Ne, add_eq_zero_iff_eq_neg.not, neg_inj, Rat.coe_int_inj, ← Ne]
     apply (smul_mem_iff _ h_nz).mp
     refine span_subset_span ℤ ℚ _ ?_
-    rwa [add_smul, neg_smul, SetLike.mem_coe, ← Zspan.fract_eq_fract, Int.cast_smul_eq_zsmul ℚ,
+    rwa [add_smul, neg_smul, SetLike.mem_coe, ← fract_eq_fract, Int.cast_smul_eq_zsmul ℚ,
       Int.cast_smul_eq_zsmul ℚ]
   · -- To prove that `finrank K E ≤ finrank ℤ L`, we use the fact `b` generates `E` over `K`
     -- and thus `finrank K E ≤ card b = finrank ℤ L`
@@ -553,55 +558,53 @@ theorem Zlattice.rank [hs : IsZlattice K L] : finrank ℤ L = finrank K E := by
 
 open Module
 
-variable {ι : Type*} [hs : IsZlattice K L] (b : Basis ι ℤ L)
+variable {ι : Type*} [hs : IsZLattice K L] (b : Basis ι ℤ L)
 /-- Any `ℤ`-basis of `L` is also a `K`-basis of `E`. -/
-def Basis.ofZlatticeBasis :
+def Basis.ofZLatticeBasis :
     Basis ι K E := by
-  have : Finite ℤ L := Zlattice.module_finite K L
-  have : Free ℤ L := Zlattice.module_free K L
+  have : Module.Finite ℤ L := ZLattice.module_finite K L
+  have : Free ℤ L := ZLattice.module_free K L
   let e :=  Basis.indexEquiv (Free.chooseBasis ℤ L) b
   have : Fintype ι := Fintype.ofEquiv _ e
-  refine basisOfTopLeSpanOfCardEqFinrank (L.subtype.toIntLinearMap ∘ b) ?_ ?_
+  refine basisOfTopLeSpanOfCardEqFinrank (L.subtype ∘ b) ?_ ?_
   · rw [← span_span_of_tower ℤ, Set.range_comp, ← map_span, Basis.span_eq, Submodule.map_top,
-      top_le_iff, AddMonoidHom.coe_toIntLinearMap_range, AddSubgroup.subtype_range,
-      AddSubgroup.coe_toIntSubmodule, hs.span_top]
-  · rw [← Fintype.card_congr e, ← finrank_eq_card_chooseBasisIndex, Zlattice.rank K L]
+      range_subtype, top_le_iff, hs.span_top]
+  · rw [← Fintype.card_congr e, ← finrank_eq_card_chooseBasisIndex, ZLattice.rank K L]
 
 @[simp]
-theorem Basis.ofZlatticeBasis_apply (i : ι) :
-    b.ofZlatticeBasis K L i =  b i := by simp [Basis.ofZlatticeBasis]
+theorem Basis.ofZLatticeBasis_apply (i : ι) :
+    b.ofZLatticeBasis K L i = b i := by simp [Basis.ofZLatticeBasis]
 
 @[simp]
-theorem Basis.ofZlatticeBasis_repr_apply (x : L) (i : ι) :
-    (b.ofZlatticeBasis K L).repr x i = b.repr x i := by
-  suffices ((b.ofZlatticeBasis K L).repr.toLinearMap.restrictScalars ℤ) ∘ₗ L.subtype.toIntLinearMap
+theorem Basis.ofZLatticeBasis_repr_apply (x : L) (i : ι) :
+    (b.ofZLatticeBasis K L).repr x i = b.repr x i := by
+  suffices ((b.ofZLatticeBasis K L).repr.toLinearMap.restrictScalars ℤ) ∘ₗ L.subtype
       = Finsupp.mapRange.linearMap (Algebra.linearMap ℤ K) ∘ₗ b.repr.toLinearMap by
     exact DFunLike.congr_fun (LinearMap.congr_fun this x) i
   refine Basis.ext b fun i ↦ ?_
   simp_rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.coe_restrictScalars,
-    LinearEquiv.coe_coe, AddMonoidHom.coe_toIntLinearMap, AddSubgroup.coeSubtype,
-    ← b.ofZlatticeBasis_apply K, repr_self, Finsupp.mapRange.linearMap_apply,
-    Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]
-
-theorem Basis.ofZlatticeBasis_span :
-    (span ℤ (Set.range (b.ofZlatticeBasis K))).toAddSubgroup = L := by
-  calc (span ℤ (Set.range (b.ofZlatticeBasis K))).toAddSubgroup
-    _ = (span ℤ (L.subtype.toIntLinearMap '' (Set.range b))).toAddSubgroup := by congr; ext; simp
-    _ = (map L.subtype.toIntLinearMap (span ℤ (Set.range b))).toAddSubgroup := by
-        rw [Submodule.map_span]
+    LinearEquiv.coe_coe, coe_subtype, ← b.ofZLatticeBasis_apply K, repr_self,
+    Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]
+
+theorem Basis.ofZLatticeBasis_span :
+    (span ℤ (Set.range (b.ofZLatticeBasis K))) = L := by
+  calc (span ℤ (Set.range (b.ofZLatticeBasis K)))
+    _ = (span ℤ (L.subtype '' (Set.range b))) := by congr; ext; simp
+    _ = (map L.subtype (span ℤ (Set.range b))) := by rw [Submodule.map_span]
     _ = L := by simp [b.span_eq]
 
-theorem Zlattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [FiniteDimensional ℝ E] {L : AddSubgroup E} [DiscreteTopology L] [IsZlattice ℝ L] [Finite ι]
-    (b : Basis ι ℤ L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : MeasureTheory.Measure E) :
-    MeasureTheory.IsAddFundamentalDomain L (Zspan.fundamentalDomain (b.ofZlatticeBasis ℝ)) μ := by
-  convert Zspan.isAddFundamentalDomain (b.ofZlatticeBasis ℝ) μ
-  all_goals exact (b.ofZlatticeBasis_span ℝ).symm
+open MeasureTheory in
+theorem ZLattice.isAddFundamentalDomain {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] {L : Submodule ℤ E} [DiscreteTopology L] [IsZLattice ℝ L] [Finite ι]
+    (b : Basis ι ℤ L) [MeasurableSpace E] [OpensMeasurableSpace E] (μ : Measure E) :
+    IsAddFundamentalDomain L (fundamentalDomain (b.ofZLatticeBasis ℝ)) μ := by
+  convert ZSpan.isAddFundamentalDomain (b.ofZLatticeBasis ℝ) μ
+  all_goals exact (b.ofZLatticeBasis_span ℝ).symm
 
-instance instCountable_of_discrete_addSubgroup {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [FiniteDimensional ℝ E] (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice ℝ L] :
+instance instCountable_of_discrete_submodule {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+    [FiniteDimensional ℝ E] (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice ℝ L] :
     Countable L := by
-  simp_rw [← (Module.Free.chooseBasis ℤ L).ofZlatticeBasis_span ℝ, mem_toAddSubgroup]
+  simp_rw [← (Module.Free.chooseBasis ℤ L).ofZLatticeBasis_span ℝ]
   infer_instance
 
-end Zlattice
+end ZLattice
diff --git a/Mathlib/Algebra/Module/Zlattice/Covolume.lean b/Mathlib/Algebra/Module/ZLattice/Covolume.lean
similarity index 60%
rename from Mathlib/Algebra/Module/Zlattice/Covolume.lean
rename to Mathlib/Algebra/Module/ZLattice/Covolume.lean
index 8a3b7297b4d77..04e559d80a384 100644
--- a/Mathlib/Algebra/Module/Zlattice/Covolume.lean
+++ b/Mathlib/Algebra/Module/ZLattice/Covolume.lean
@@ -3,43 +3,43 @@ Copyright (c) 2024 Xavier Roblot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Xavier Roblot
 -/
-import Mathlib.Algebra.Module.Zlattice.Basic
+import Mathlib.Algebra.Module.ZLattice.Basic
 
 /-!
 # Covolume of ℤ-lattices
 
 Let `E` be a finite dimensional real vector space with an inner product.
 
-Let `L` be a `ℤ`-lattice `L` defined as a discrete `AddSubgroup E` that spans `E` over `ℝ`.
+Let `L` be a `ℤ`-lattice `L` defined as a discrete `ℤ`-submodule of `E` that spans `E` over `ℝ`.
 
 ## Main definitions and results
 
-* `Zlattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental
+* `ZLattice.covolume`: the covolume of `L` defined as the volume of an arbitrary fundamental
 domain of `L`.
 
-* `Zlattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the
+* `ZLattice.covolume_eq_measure_fundamentalDomain`: the covolume of `L` does not depend on the
 choice of the fundamental domain of `L`.
 
-* `Zlattice.covolume_eq_det`: if `L` is a lattice in `ℝ^n`, then its covolume is the absolute
+* `ZLattice.covolume_eq_det`: if `L` is a lattice in `ℝ^n`, then its covolume is the absolute
 value of the determinant of any `ℤ`-basis of `L`.
 
 -/
 
 noncomputable section
 
-namespace Zlattice
+namespace ZLattice
 
-open Submodule MeasureTheory FiniteDimensional MeasureTheory Module
+open Submodule MeasureTheory Module MeasureTheory Module
 
 section General
 
 variable (K : Type*) [NormedLinearOrderedField K] [HasSolidNorm K] [FloorRing K]
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E] [FiniteDimensional K E]
 variable [ProperSpace E] [MeasurableSpace E]
-variable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice K L]
+variable (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice K L]
 
 /-- The covolume of a `ℤ`-lattice is the volume of some fundamental domain; see
-`Zlattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of
+`ZLattice.covolume_eq_volume` for the proof that the volume does not depend on the choice of
 the fundamental domain. -/
 def covolume (μ : Measure E := by volume_tac) : ℝ := (addCovolume L E μ).toReal
 
@@ -49,41 +49,44 @@ section Real
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
 variable [MeasurableSpace E] [BorelSpace E]
-variable (L : AddSubgroup E) [DiscreteTopology L] [IsZlattice ℝ L]
+variable (L : Submodule ℤ E) [DiscreteTopology L] [IsZLattice ℝ L]
 variable (μ : Measure E := by volume_tac) [Measure.IsAddHaarMeasure μ]
 
 theorem covolume_eq_measure_fundamentalDomain {F : Set E} (h : IsAddFundamentalDomain L F μ) :
-    covolume L μ = (μ F).toReal := congr_arg ENNReal.toReal (h.covolume_eq_volume μ)
+    covolume L μ = (μ F).toReal := by
+  have : MeasurableVAdd L E := (inferInstance : MeasurableVAdd L.toAddSubgroup E)
+  have : VAddInvariantMeasure L E μ := (inferInstance : VAddInvariantMeasure L.toAddSubgroup E μ)
+  exact congr_arg ENNReal.toReal (h.covolume_eq_volume μ)
 
 theorem covolume_ne_zero : covolume L μ ≠ 0 := by
   rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
     ENNReal.toReal_ne_zero]
-  refine ⟨Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_⟩
-  exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)
+  refine ⟨ZSpan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_⟩
+  exact Bornology.IsBounded.measure_lt_top (ZSpan.fundamentalDomain_isBounded _)
 
 theorem covolume_pos : 0 < covolume L μ :=
   lt_of_le_of_ne ENNReal.toReal_nonneg (covolume_ne_zero L μ).symm
 
 theorem covolume_eq_det_mul_measure {ι : Type*} [Fintype ι] [DecidableEq ι] (b : Basis ι ℤ L)
     (b₀ : Basis ι ℝ E) :
-    covolume L μ = |b₀.det ((↑) ∘ b)| * (μ (Zspan.fundamentalDomain b₀)).toReal := by
+    covolume L μ = |b₀.det ((↑) ∘ b)| * (μ (ZSpan.fundamentalDomain b₀)).toReal := by
   rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain b μ),
-    Zspan.measure_fundamentalDomain _ _ b₀,
-    measure_congr (Zspan.fundamentalDomain_ae_parallelepiped b₀ μ), ENNReal.toReal_mul,
+    ZSpan.measure_fundamentalDomain _ _ b₀,
+    measure_congr (ZSpan.fundamentalDomain_ae_parallelepiped b₀ μ), ENNReal.toReal_mul,
     ENNReal.toReal_ofReal (by positivity)]
   congr
   ext
-  exact b.ofZlatticeBasis_apply ℝ L _
+  exact b.ofZLatticeBasis_apply ℝ L _
 
-theorem covolume_eq_det {ι : Type*} [Fintype ι] [DecidableEq ι] (L : AddSubgroup (ι → ℝ))
-    [DiscreteTopology L] [IsZlattice ℝ L] (b : Basis ι ℤ L) :
+theorem covolume_eq_det {ι : Type*} [Fintype ι] [DecidableEq ι] (L : Submodule ℤ (ι → ℝ))
+    [DiscreteTopology L] [IsZLattice ℝ L] (b : Basis ι ℤ L) :
     covolume L = |(Matrix.of ((↑) ∘ b)).det| := by
   rw [covolume_eq_measure_fundamentalDomain L volume (isAddFundamentalDomain b volume),
-    Zspan.volume_fundamentalDomain, ENNReal.toReal_ofReal (by positivity)]
+    ZSpan.volume_fundamentalDomain, ENNReal.toReal_ofReal (by positivity)]
   congr
   ext1
-  exact b.ofZlatticeBasis_apply ℝ L _
+  exact b.ofZLatticeBasis_apply ℝ L _
 
 end Real
 
-end Zlattice
+end ZLattice
diff --git a/Mathlib/Data/ZMod/Module.lean b/Mathlib/Algebra/Module/ZMod.lean
similarity index 94%
rename from Mathlib/Data/ZMod/Module.lean
rename to Mathlib/Algebra/Module/ZMod.lean
index 64cb1d86b6160..961af9fe2c048 100644
--- a/Mathlib/Data/ZMod/Module.lean
+++ b/Mathlib/Algebra/Module/ZMod.lean
@@ -5,7 +5,6 @@ Authors: Lawrence Wu
 -/
 import Mathlib.Algebra.Module.Submodule.Lattice
 import Mathlib.Data.ZMod.Basic
-import Mathlib.Order.OmegaCompletePartialOrder
 
 /-!
 # The `ZMod n`-module structure on Abelian groups whose elements have order dividing `n`
@@ -92,9 +91,8 @@ theorem toZModSubmodule_symm :
     ⇑((toZModSubmodule n).symm : _ ≃o AddSubgroup M) = Submodule.toAddSubgroup :=
   rfl
 
-@[simp]
-theorem coe_toZModSubmodule (S : AddSubgroup M) : (toZModSubmodule n S : Set M) = S :=
-  rfl
+@[simp] lemma coe_toZModSubmodule (S : AddSubgroup M) : (toZModSubmodule n S : Set M) = S := rfl
+@[simp] lemma mem_toZModSubmodule {S : AddSubgroup M} : x ∈ toZModSubmodule n S ↔ x ∈ S := .rfl
 
 @[simp]
 theorem toZModSubmodule_toAddSubgroup (S : AddSubgroup M) :
diff --git a/Mathlib/Algebra/MonoidAlgebra/Basic.lean b/Mathlib/Algebra/MonoidAlgebra/Basic.lean
index 4022f37abb36e..c859345b543ca 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Basic.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Basic.lean
@@ -1,10 +1,11 @@
 /-
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Yury Kudryashov, Scott Morrison
+Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
 -/
-import Mathlib.Algebra.Algebra.Equiv
+import Mathlib.Algebra.MonoidAlgebra.Defs
 import Mathlib.Algebra.Algebra.NonUnitalHom
+import Mathlib.Algebra.Algebra.Equiv
 import Mathlib.Algebra.BigOperators.Finsupp
 import Mathlib.Algebra.Module.BigOperators
 import Mathlib.Data.Finsupp.Basic
@@ -13,43 +14,8 @@ import Mathlib.LinearAlgebra.Finsupp
 /-!
 # Monoid algebras
 
-When the domain of a `Finsupp` has a multiplicative or additive structure, we can define
-a convolution product. To mathematicians this structure is known as the "monoid algebra",
-i.e. the finite formal linear combinations over a given semiring of elements of the monoid.
-The "group ring" ℤ[G] or the "group algebra" k[G] are typical uses.
-
-In fact the construction of the "monoid algebra" makes sense when `G` is not even a monoid, but
-merely a magma, i.e., when `G` carries a multiplication which is not required to satisfy any
-conditions at all. In this case the construction yields a not-necessarily-unital,
-not-necessarily-associative algebra but it is still adjoint to the forgetful functor from such
-algebras to magmas, and we prove this as `MonoidAlgebra.liftMagma`.
-
-In this file we define `MonoidAlgebra k G := G →₀ k`, and `AddMonoidAlgebra k G`
-in the same way, and then define the convolution product on these.
-
-When the domain is additive, this is used to define polynomials:
-```
-Polynomial R := AddMonoidAlgebra R ℕ
-MvPolynomial σ α := AddMonoidAlgebra R (σ →₀ ℕ)
-```
-
-When the domain is multiplicative, e.g. a group, this will be used to define the group ring.
-
-## Notation
-
-We introduce the notation `R[A]` for `AddMonoidAlgebra R A`.
-
-## Implementation note
-Unfortunately because additive and multiplicative structures both appear in both cases,
-it doesn't appear to be possible to make much use of `to_additive`, and we just settle for
-saying everything twice.
-
-Similarly, I attempted to just define
-`k[G] := MonoidAlgebra k (Multiplicative G)`, but the definitional equality
-`Multiplicative G = G` leaks through everywhere, and seems impossible to use.
 -/
 
-
 noncomputable section
 
 open Finset
@@ -62,518 +28,10 @@ variable (k : Type u₁) (G : Type u₂) (H : Type*) {R : Type*}
 
 /-! ### Multiplicative monoids -/
 
-
-section
-
-variable [Semiring k]
-
-/-- The monoid algebra over a semiring `k` generated by the monoid `G`.
-It is the type of finite formal `k`-linear combinations of terms of `G`,
-endowed with the convolution product.
--/
-def MonoidAlgebra : Type max u₁ u₂ :=
-  G →₀ k
-
--- Porting note: The compiler couldn't derive this.
-instance MonoidAlgebra.inhabited : Inhabited (MonoidAlgebra k G) :=
-  inferInstanceAs (Inhabited (G →₀ k))
-
--- Porting note: The compiler couldn't derive this.
-instance MonoidAlgebra.addCommMonoid : AddCommMonoid (MonoidAlgebra k G) :=
-  inferInstanceAs (AddCommMonoid (G →₀ k))
-
-instance MonoidAlgebra.instIsCancelAdd [IsCancelAdd k] : IsCancelAdd (MonoidAlgebra k G) :=
-  inferInstanceAs (IsCancelAdd (G →₀ k))
-
-instance MonoidAlgebra.coeFun : CoeFun (MonoidAlgebra k G) fun _ => G → k :=
-  Finsupp.instCoeFun
-
-end
-
 namespace MonoidAlgebra
 
 variable {k G}
 
-section
-
-variable [Semiring k] [NonUnitalNonAssocSemiring R]
-
--- Porting note: `reducible` cannot be `local`, so we replace some definitions and theorems with
---               new ones which have new types.
-
-abbrev single (a : G) (b : k) : MonoidAlgebra k G := Finsupp.single a b
-
-theorem single_zero (a : G) : (single a 0 : MonoidAlgebra k G) = 0 := Finsupp.single_zero a
-
-theorem single_add (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
-  Finsupp.single_add a b₁ b₂
-
-@[simp]
-theorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N}
-    (h_zero : h a 0 = 0) :
-    (single a b).sum h = h a b := Finsupp.sum_single_index h_zero
-
-@[simp]
-theorem sum_single (f : MonoidAlgebra k G) : f.sum single = f :=
-  Finsupp.sum_single f
-
-theorem single_apply {a a' : G} {b : k} [Decidable (a = a')] :
-    single a b a' = if a = a' then b else 0 :=
-  Finsupp.single_apply
-
-@[simp]
-theorem single_eq_zero {a : G} {b : k} : single a b = 0 ↔ b = 0 := Finsupp.single_eq_zero
-
-abbrev mapDomain {G' : Type*} (f : G → G') (v : MonoidAlgebra k G) : MonoidAlgebra k G' :=
-  Finsupp.mapDomain f v
-
-theorem mapDomain_sum {k' G' : Type*} [Semiring k'] {f : G → G'} {s : MonoidAlgebra k' G}
-    {v : G → k' → MonoidAlgebra k G} :
-    mapDomain f (s.sum v) = s.sum fun a b => mapDomain f (v a b) :=
-  Finsupp.mapDomain_sum
-
-/-- A non-commutative version of `MonoidAlgebra.lift`: given an additive homomorphism `f : k →+ R`
-and a homomorphism `g : G → R`, returns the additive homomorphism from
-`MonoidAlgebra k G` such that `liftNC f g (single a b) = f b * g a`. If `f` is a ring homomorphism
-and the range of either `f` or `g` is in center of `R`, then the result is a ring homomorphism.  If
-`R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra homomorphism called
-`MonoidAlgebra.lift`. -/
-def liftNC (f : k →+ R) (g : G → R) : MonoidAlgebra k G →+ R :=
-  liftAddHom fun x : G => (AddMonoidHom.mulRight (g x)).comp f
-
-@[simp]
-theorem liftNC_single (f : k →+ R) (g : G → R) (a : G) (b : k) :
-    liftNC f g (single a b) = f b * g a :=
-  liftAddHom_apply_single _ _ _
-
-end
-
-section Mul
-
-variable [Semiring k] [Mul G]
-
-/-- The multiplication in a monoid algebra. We make it irreducible so that Lean doesn't unfold
-it trying to unify two things that are different. -/
-@[irreducible] def mul' (f g : MonoidAlgebra k G) : MonoidAlgebra k G :=
-  f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)
-
-/-- The product of `f g : MonoidAlgebra k G` is the finitely supported function
-  whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
-  such that `x * y = a`. (Think of the group ring of a group.) -/
-instance instMul : Mul (MonoidAlgebra k G) := ⟨MonoidAlgebra.mul'⟩
-
-theorem mul_def {f g : MonoidAlgebra k G} :
-    f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂) := by
-  with_unfolding_all rfl
-
-instance nonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring (MonoidAlgebra k G) :=
-  { Finsupp.instAddCommMonoid with
-    -- Porting note: `refine` & `exact` are required because `simp` behaves differently.
-    left_distrib := fun f g h => by
-      haveI := Classical.decEq G
-      simp only [mul_def]
-      refine Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_add_index ?_ ?_)) ?_ <;>
-        simp only [mul_add, mul_zero, single_zero, single_add, forall_true_iff, sum_add]
-    right_distrib := fun f g h => by
-      haveI := Classical.decEq G
-      simp only [mul_def]
-      refine Eq.trans (sum_add_index ?_ ?_) ?_ <;>
-        simp only [add_mul, zero_mul, single_zero, single_add, forall_true_iff, sum_zero, sum_add]
-    zero_mul := fun f => by
-      simp only [mul_def]
-      exact sum_zero_index
-    mul_zero := fun f => by
-      simp only [mul_def]
-      exact Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_zero_index)) sum_zero }
-
-variable [Semiring R]
-
-theorem liftNC_mul {g_hom : Type*} [FunLike g_hom G R] [MulHomClass g_hom G R]
-    (f : k →+* R) (g : g_hom) (a b : MonoidAlgebra k G)
-    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g y)) :
-    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b := by
-  conv_rhs => rw [← sum_single a, ← sum_single b]
-  -- Porting note: `(liftNC _ g).map_finsupp_sum` → `map_finsupp_sum`
-  simp_rw [mul_def, map_finsupp_sum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]
-  refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_
-  simp [mul_assoc, (h_comm hy).left_comm]
-
-end Mul
-
-section Semigroup
-
-variable [Semiring k] [Semigroup G] [Semiring R]
-
-instance nonUnitalSemiring : NonUnitalSemiring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalNonAssocSemiring with
-    mul_assoc := fun f g h => by
-      -- Porting note: `reducible` cannot be `local` so proof gets long.
-      simp only [mul_def]
-      rw [sum_sum_index]; congr; ext a₁ b₁
-      rw [sum_sum_index, sum_sum_index]; congr; ext a₂ b₂
-      rw [sum_sum_index, sum_single_index]; congr; ext a₃ b₃
-      rw [sum_single_index, mul_assoc, mul_assoc]
-      all_goals simp only [single_zero, single_add, forall_true_iff, add_mul,
-        mul_add, zero_mul, mul_zero, sum_zero, sum_add] }
-
-end Semigroup
-
-section One
-
-variable [NonAssocSemiring R] [Semiring k] [One G]
-
-/-- The unit of the multiplication is `single 1 1`, i.e. the function
-  that is `1` at `1` and zero elsewhere. -/
-instance one : One (MonoidAlgebra k G) :=
-  ⟨single 1 1⟩
-
-theorem one_def : (1 : MonoidAlgebra k G) = single 1 1 :=
-  rfl
-
-@[simp]
-theorem liftNC_one {g_hom : Type*} [FunLike g_hom G R] [OneHomClass g_hom G R]
-    (f : k →+* R) (g : g_hom) :
-    liftNC (f : k →+ R) g 1 = 1 := by simp [one_def]
-
-end One
-
-section MulOneClass
-
-variable [Semiring k] [MulOneClass G]
-
-instance nonAssocSemiring : NonAssocSemiring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalNonAssocSemiring with
-    natCast := fun n => single 1 n
-    natCast_zero := by simp
-    natCast_succ := fun _ => by simp; rfl
-    one_mul := fun f => by
-      simp only [mul_def, one_def, sum_single_index, zero_mul, single_zero, sum_zero, zero_add,
-        one_mul, sum_single]
-    mul_one := fun f => by
-      simp only [mul_def, one_def, sum_single_index, mul_zero, single_zero, sum_zero, add_zero,
-        mul_one, sum_single] }
-
-theorem natCast_def (n : ℕ) : (n : MonoidAlgebra k G) = single (1 : G) (n : k) :=
-  rfl
-
-@[deprecated (since := "2024-04-17")]
-alias nat_cast_def := natCast_def
-
-end MulOneClass
-
-/-! #### Semiring structure -/
-
-
-section Semiring
-
-variable [Semiring k] [Monoid G]
-
-instance semiring : Semiring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalSemiring,
-    MonoidAlgebra.nonAssocSemiring with }
-
-variable [Semiring R]
-
-/-- `liftNC` as a `RingHom`, for when `f x` and `g y` commute -/
-def liftNCRingHom (f : k →+* R) (g : G →* R) (h_comm : ∀ x y, Commute (f x) (g y)) :
-    MonoidAlgebra k G →+* R :=
-  { liftNC (f : k →+ R) g with
-    map_one' := liftNC_one _ _
-    map_mul' := fun _a _b => liftNC_mul _ _ _ _ fun {_ _} _ => h_comm _ _ }
-
-end Semiring
-
-instance nonUnitalCommSemiring [CommSemiring k] [CommSemigroup G] :
-    NonUnitalCommSemiring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalSemiring with
-    mul_comm := fun f g => by
-      simp only [mul_def, Finsupp.sum, mul_comm]
-      rw [Finset.sum_comm]
-      simp only [mul_comm] }
-
-instance nontrivial [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial (MonoidAlgebra k G) :=
-  Finsupp.instNontrivial
-
-/-! #### Derived instances -/
-
-
-section DerivedInstances
-
-instance commSemiring [CommSemiring k] [CommMonoid G] : CommSemiring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalCommSemiring, MonoidAlgebra.semiring with }
-
-instance unique [Semiring k] [Subsingleton k] : Unique (MonoidAlgebra k G) :=
-  Finsupp.uniqueOfRight
-
-instance addCommGroup [Ring k] : AddCommGroup (MonoidAlgebra k G) :=
-  Finsupp.instAddCommGroup
-
-instance nonUnitalNonAssocRing [Ring k] [Mul G] : NonUnitalNonAssocRing (MonoidAlgebra k G) :=
-  { MonoidAlgebra.addCommGroup, MonoidAlgebra.nonUnitalNonAssocSemiring with }
-
-instance nonUnitalRing [Ring k] [Semigroup G] : NonUnitalRing (MonoidAlgebra k G) :=
-  { MonoidAlgebra.addCommGroup, MonoidAlgebra.nonUnitalSemiring with }
-
-instance nonAssocRing [Ring k] [MulOneClass G] : NonAssocRing (MonoidAlgebra k G) :=
-  { MonoidAlgebra.addCommGroup,
-    MonoidAlgebra.nonAssocSemiring with
-    intCast := fun z => single 1 (z : k)
-    -- Porting note: Both were `simpa`.
-    intCast_ofNat := fun n => by simp; rfl
-    intCast_negSucc := fun n => by simp; rfl }
-
-theorem intCast_def [Ring k] [MulOneClass G] (z : ℤ) :
-    (z : MonoidAlgebra k G) = single (1 : G) (z : k) :=
-  rfl
-
-@[deprecated (since := "2024-04-17")]
-alias int_cast_def := intCast_def
-
-instance ring [Ring k] [Monoid G] : Ring (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonAssocRing, MonoidAlgebra.semiring with }
-
-instance nonUnitalCommRing [CommRing k] [CommSemigroup G] :
-    NonUnitalCommRing (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalCommSemiring, MonoidAlgebra.nonUnitalRing with }
-
-instance commRing [CommRing k] [CommMonoid G] : CommRing (MonoidAlgebra k G) :=
-  { MonoidAlgebra.nonUnitalCommRing, MonoidAlgebra.ring with }
-
-variable {S : Type*}
-
-instance smulZeroClass [Semiring k] [SMulZeroClass R k] : SMulZeroClass R (MonoidAlgebra k G) :=
-  Finsupp.smulZeroClass
-
-instance distribSMul [Semiring k] [DistribSMul R k] : DistribSMul R (MonoidAlgebra k G) :=
-  Finsupp.distribSMul _ _
-
-instance distribMulAction [Monoid R] [Semiring k] [DistribMulAction R k] :
-    DistribMulAction R (MonoidAlgebra k G) :=
-  Finsupp.distribMulAction G k
-
-instance module [Semiring R] [Semiring k] [Module R k] : Module R (MonoidAlgebra k G) :=
-  Finsupp.module G k
-
-instance faithfulSMul [Semiring k] [SMulZeroClass R k] [FaithfulSMul R k] [Nonempty G] :
-    FaithfulSMul R (MonoidAlgebra k G) :=
-  Finsupp.faithfulSMul
-
-instance isScalarTower [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S]
-    [IsScalarTower R S k] : IsScalarTower R S (MonoidAlgebra k G) :=
-  Finsupp.isScalarTower G k
-
-instance smulCommClass [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMulCommClass R S k] :
-    SMulCommClass R S (MonoidAlgebra k G) :=
-  Finsupp.smulCommClass G k
-
-instance isCentralScalar [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k]
-    [IsCentralScalar R k] : IsCentralScalar R (MonoidAlgebra k G) :=
-  Finsupp.isCentralScalar G k
-
-/-- This is not an instance as it conflicts with `MonoidAlgebra.distribMulAction` when `G = kˣ`.
--/
-def comapDistribMulActionSelf [Group G] [Semiring k] : DistribMulAction G (MonoidAlgebra k G) :=
-  Finsupp.comapDistribMulAction
-
-end DerivedInstances
-
-section MiscTheorems
-
-variable [Semiring k]
-
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-theorem mul_apply [DecidableEq G] [Mul G] (f g : MonoidAlgebra k G) (x : G) :
-    (f * g) x = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => if a₁ * a₂ = x then b₁ * b₂ else 0 := by
-  -- Porting note: `reducible` cannot be `local` so proof gets long.
-  rw [mul_def, Finsupp.sum_apply]; congr; ext
-  rw [Finsupp.sum_apply]; congr; ext
-  apply single_apply
-
-theorem mul_apply_antidiagonal [Mul G] (f g : MonoidAlgebra k G) (x : G) (s : Finset (G × G))
-    (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2 := by
-  classical exact
-      let F : G × G → k := fun p => if p.1 * p.2 = x then f p.1 * g p.2 else 0
-      calc
-        (f * g) x = ∑ a₁ ∈ f.support, ∑ a₂ ∈ g.support, F (a₁, a₂) := mul_apply f g x
-        _ = ∑ p ∈ f.support ×ˢ g.support, F p := Finset.sum_product.symm
-        _ = ∑ p ∈ (f.support ×ˢ g.support).filter fun p : G × G => p.1 * p.2 = x, f p.1 * g p.2 :=
-          (Finset.sum_filter _ _).symm
-        _ = ∑ p ∈ s.filter fun p : G × G => p.1 ∈ f.support ∧ p.2 ∈ g.support, f p.1 * g p.2 :=
-          (sum_congr
-            (by
-              ext
-              simp only [mem_filter, mem_product, hs, and_comm])
-            fun _ _ => rfl)
-        _ = ∑ p ∈ s, f p.1 * g p.2 :=
-          sum_subset (filter_subset _ _) fun p hps hp => by
-            simp only [mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢
-            by_cases h1 : f p.1 = 0
-            · rw [h1, zero_mul]
-            · rw [hp hps h1, mul_zero]
-
-@[simp]
-theorem single_mul_single [Mul G] {a₁ a₂ : G} {b₁ b₂ : k} :
-    single a₁ b₁ * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂) := by
-  rw [mul_def]
-  exact (sum_single_index (by simp only [zero_mul, single_zero, sum_zero])).trans
-    (sum_single_index (by rw [mul_zero, single_zero]))
-
-theorem single_commute_single [Mul G] {a₁ a₂ : G} {b₁ b₂ : k}
-    (ha : Commute a₁ a₂) (hb : Commute b₁ b₂) :
-    Commute (single a₁ b₁) (single a₂ b₂) :=
-  single_mul_single.trans <| congr_arg₂ single ha hb |>.trans single_mul_single.symm
-
-theorem single_commute [Mul G] {a : G} {b : k} (ha : ∀ a', Commute a a') (hb : ∀ b', Commute b b') :
-    ∀ f : MonoidAlgebra k G, Commute (single a b) f :=
-  suffices AddMonoidHom.mulLeft (single a b) = AddMonoidHom.mulRight (single a b) from
-    DFunLike.congr_fun this
-  addHom_ext' fun a' => AddMonoidHom.ext fun b' => single_commute_single (ha a') (hb b')
-
-@[simp]
-theorem single_pow [Monoid G] {a : G} {b : k} : ∀ n : ℕ, single a b ^ n = single (a ^ n) (b ^ n)
-  | 0 => by
-    simp only [pow_zero]
-    rfl
-  | n + 1 => by simp only [pow_succ, single_pow n, single_mul_single]
-
-section
-
-/-- Like `Finsupp.mapDomain_zero`, but for the `1` we define in this file -/
-@[simp]
-theorem mapDomain_one {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [One α] [One α₂]
-    {F : Type*} [FunLike F α α₂] [OneHomClass F α α₂] (f : F) :
-    (mapDomain f (1 : MonoidAlgebra β α) : MonoidAlgebra β α₂) = (1 : MonoidAlgebra β α₂) := by
-  simp_rw [one_def, mapDomain_single, map_one]
-
-/-- Like `Finsupp.mapDomain_add`, but for the convolutive multiplication we define in this file -/
-theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Mul α] [Mul α₂]
-    {F : Type*} [FunLike F α α₂] [MulHomClass F α α₂] (f : F) (x y : MonoidAlgebra β α) :
-    mapDomain f (x * y) = mapDomain f x * mapDomain f y := by
-  simp_rw [mul_def, mapDomain_sum, mapDomain_single, map_mul]
-  rw [Finsupp.sum_mapDomain_index]
-  · congr
-    ext a b
-    rw [Finsupp.sum_mapDomain_index]
-    · simp
-    · simp [mul_add]
-  · simp
-  · simp [add_mul]
-
-variable (k G)
-
-/-- The embedding of a magma into its magma algebra. -/
-@[simps]
-def ofMagma [Mul G] : G →ₙ* MonoidAlgebra k G where
-  toFun a := single a 1
-  map_mul' a b := by simp only [mul_def, mul_one, sum_single_index, single_eq_zero, mul_zero]
-
-/-- The embedding of a unital magma into its magma algebra. -/
-@[simps]
-def of [MulOneClass G] : G →* MonoidAlgebra k G :=
-  { ofMagma k G with
-    toFun := fun a => single a 1
-    map_one' := rfl }
-
-end
-
-theorem smul_of [MulOneClass G] (g : G) (r : k) : r • of k G g = single g r := by
-  -- porting note (#10745): was `simp`.
-  rw [of_apply, smul_single', mul_one]
-
-theorem of_injective [MulOneClass G] [Nontrivial k] :
-    Function.Injective (of k G) := fun a b h => by
-  simpa using (single_eq_single_iff _ _ _ _).mp h
-
-theorem of_commute [MulOneClass G] {a : G} (h : ∀ a', Commute a a') (f : MonoidAlgebra k G) :
-    Commute (of k G a) f :=
-  single_commute h Commute.one_left f
-
-/-- `Finsupp.single` as a `MonoidHom` from the product type into the monoid algebra.
-
-Note the order of the elements of the product are reversed compared to the arguments of
-`Finsupp.single`.
--/
-@[simps]
-def singleHom [MulOneClass G] : k × G →* MonoidAlgebra k G where
-  toFun a := single a.2 a.1
-  map_one' := rfl
-  map_mul' _a _b := single_mul_single.symm
-
-theorem mul_single_apply_aux [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G}
-    (H : ∀ a, a * x = z ↔ a = y) : (f * single x r) z = f y * r := by
-  classical exact
-      have A :
-        ∀ a₁ b₁,
-          ((single x r).sum fun a₂ b₂ => ite (a₁ * a₂ = z) (b₁ * b₂) 0) =
-            ite (a₁ * x = z) (b₁ * r) 0 :=
-        fun a₁ b₁ => sum_single_index <| by simp
-      calc
-        (HMul.hMul (β := MonoidAlgebra k G) f (single x r)) z =
-            sum f fun a b => if a = y then b * r else 0 := by simp only [mul_apply, A, H]
-        _ = if y ∈ f.support then f y * r else 0 := f.support.sum_ite_eq' _ _
-        _ = f y * r := by split_ifs with h <;> simp at h <;> simp [h]
-
-theorem mul_single_one_apply [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) :
-    (HMul.hMul (β := MonoidAlgebra k G) f (single 1 r)) x = f x * r :=
-  f.mul_single_apply_aux fun a => by rw [mul_one]
-
-theorem mul_single_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
-    (h : ¬∃ d, g' = d * g) : (x * single g r) g' = 0 := by
-  classical
-    rw [mul_apply, Finsupp.sum_comm, Finsupp.sum_single_index]
-    swap
-    · simp_rw [Finsupp.sum, mul_zero, ite_self, Finset.sum_const_zero]
-    · apply Finset.sum_eq_zero
-      simp_rw [ite_eq_right_iff]
-      rintro g'' _hg'' rfl
-      exfalso
-      exact h ⟨_, rfl⟩
-
-theorem single_mul_apply_aux [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G}
-    (H : ∀ a, x * a = y ↔ a = z) : (single x r * f) y = r * f z := by
-  classical exact
-      have : (f.sum fun a b => ite (x * a = y) (0 * b) 0) = 0 := by simp
-      calc
-        (HMul.hMul (α := MonoidAlgebra k G) (single x r) f) y =
-            sum f fun a b => ite (x * a = y) (r * b) 0 :=
-          (mul_apply _ _ _).trans <| sum_single_index this
-        _ = f.sum fun a b => ite (a = z) (r * b) 0 := by simp only [H]
-        _ = if z ∈ f.support then r * f z else 0 := f.support.sum_ite_eq' _ _
-        _ = _ := by split_ifs with h <;> simp at h <;> simp [h]
-
-theorem single_one_mul_apply [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) :
-    (single (1 : G) r * f) x = r * f x :=
-  f.single_mul_apply_aux fun a => by rw [one_mul]
-
-theorem single_mul_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
-    (h : ¬∃ d, g' = g * d) : (single g r * x) g' = 0 := by
-  classical
-    rw [mul_apply, Finsupp.sum_single_index]
-    swap
-    · simp_rw [Finsupp.sum, zero_mul, ite_self, Finset.sum_const_zero]
-    · apply Finset.sum_eq_zero
-      simp_rw [ite_eq_right_iff]
-      rintro g'' _hg'' rfl
-      exfalso
-      exact h ⟨_, rfl⟩
-
-theorem liftNC_smul [MulOneClass G] {R : Type*} [Semiring R] (f : k →+* R) (g : G →* R) (c : k)
-    (φ : MonoidAlgebra k G) : liftNC (f : k →+ R) g (c • φ) = f c * liftNC (f : k →+ R) g φ := by
-  suffices (liftNC (↑f) g).comp (smulAddHom k (MonoidAlgebra k G) c) =
-      (AddMonoidHom.mulLeft (f c)).comp (liftNC (↑f) g) from
-    DFunLike.congr_fun this φ
-  -- Porting note: `ext` couldn't a find appropriate theorem.
-  refine addHom_ext' fun a => AddMonoidHom.ext fun b => ?_
-  -- Porting note: `reducible` cannot be `local` so the proof gets more complex.
-  unfold MonoidAlgebra
-  simp only [AddMonoidHom.coe_comp, Function.comp_apply, singleAddHom_apply, smulAddHom_apply,
-    smul_single, smul_eq_mul, AddMonoidHom.coe_mulLeft]
-  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-  erw [liftNC_single, liftNC_single]; rw [AddMonoidHom.coe_coe, map_mul, mul_assoc]
-
-end MiscTheorems
-
 /-! #### Non-unital, non-associative algebra structure -/
 
 
@@ -581,42 +39,6 @@ section NonUnitalNonAssocAlgebra
 
 variable (k) [Semiring k] [DistribSMul R k] [Mul G]
 
-instance isScalarTower_self [IsScalarTower R k k] :
-    IsScalarTower R (MonoidAlgebra k G) (MonoidAlgebra k G) :=
-  ⟨fun t a b => by
-    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
-    refine Finsupp.ext fun m => ?_
-    -- Porting note: `refine` & `rw` are required because `simp` behaves differently.
-    classical
-      simp only [smul_eq_mul, mul_apply]
-      rw [coe_smul]
-      refine Eq.trans (sum_smul_index' (g := a) (b := t) ?_) ?_ <;>
-        simp only [mul_apply, Finsupp.smul_sum, smul_ite, smul_mul_assoc,
-          zero_mul, ite_self, imp_true_iff, sum_zero, Pi.smul_apply, smul_zero]⟩
-
-/-- Note that if `k` is a `CommSemiring` then we have `SMulCommClass k k k` and so we can take
-`R = k` in the below. In other words, if the coefficients are commutative amongst themselves, they
-also commute with the algebra multiplication. -/
-instance smulCommClass_self [SMulCommClass R k k] :
-    SMulCommClass R (MonoidAlgebra k G) (MonoidAlgebra k G) :=
-  ⟨fun t a b => by
-    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
-    refine Finsupp.ext fun m => ?_
-    -- Porting note: `refine` & `rw` are required because `simp` behaves differently.
-    classical
-      simp only [smul_eq_mul, mul_apply]
-      rw [coe_smul]
-      refine Eq.symm (Eq.trans (congr_arg (sum a)
-        (funext₂ fun a₁ b₁ => sum_smul_index' (g := b) (b := t) ?_)) ?_) <;>
-      simp only [mul_apply, Finsupp.sum, Finset.smul_sum, smul_ite, mul_smul_comm,
-        imp_true_iff, ite_eq_right_iff, Pi.smul_apply, mul_zero, smul_zero]⟩
-
-instance smulCommClass_symm_self [SMulCommClass k R k] :
-    SMulCommClass (MonoidAlgebra k G) R (MonoidAlgebra k G) :=
-  ⟨fun t a b => by
-    haveI := SMulCommClass.symm k R k
-    rw [← smul_comm]⟩
-
 variable {A : Type u₃} [NonUnitalNonAssocSemiring A]
 
 /-- A non_unital `k`-algebra homomorphism from `MonoidAlgebra k G` is uniquely defined by its
@@ -656,7 +78,7 @@ def liftMagma [Module k A] [IsScalarTower k A A] [SMulCommClass k A A] :
           intros
           rw [← add_smul]
         -- Porting note: `reducible` cannot be `local` so proof gets long.
-        simp_rw [Finsupp.mul_sum, Finsupp.sum_mul, smul_mul_smul, ← f.map_mul, mul_def,
+        simp_rw [Finsupp.mul_sum, Finsupp.sum_mul, smul_mul_smul_comm, ← f.map_mul, mul_def,
           sum_comm a₂ a₁]
         rw [sum_sum_index h₁ h₂]; congr; ext
         rw [sum_sum_index h₁ h₂]; congr; ext
@@ -678,57 +100,8 @@ end NonUnitalNonAssocAlgebra
 
 /-! #### Algebra structure -/
 
-
 section Algebra
 
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-theorem single_one_comm [CommSemiring k] [MulOneClass G] (r : k) (f : MonoidAlgebra k G) :
-    single (1 : G) r * f = f * single (1 : G) r :=
-  single_commute Commute.one_left (Commute.all _) f
-
-/-- `Finsupp.single 1` as a `RingHom` -/
-@[simps]
-def singleOneRingHom [Semiring k] [MulOneClass G] : k →+* MonoidAlgebra k G :=
-  { Finsupp.singleAddHom 1 with
-    map_one' := rfl
-    map_mul' := fun x y => by
-      -- Porting note (#10691): Was `rw`.
-      simp only [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe, singleAddHom_apply,
-        single_mul_single, mul_one] }
-
-/-- If `f : G → H` is a multiplicative homomorphism between two monoids, then
-`Finsupp.mapDomain f` is a ring homomorphism between their monoid algebras. -/
-@[simps]
-def mapDomainRingHom (k : Type*) {H F : Type*} [Semiring k] [Monoid G] [Monoid H]
-    [FunLike F G H] [MonoidHomClass F G H] (f : F) : MonoidAlgebra k G →+* MonoidAlgebra k H :=
-  { (Finsupp.mapDomain.addMonoidHom f : MonoidAlgebra k G →+ MonoidAlgebra k H) with
-    map_one' := mapDomain_one f
-    map_mul' := fun x y => mapDomain_mul f x y }
-
-/-- If two ring homomorphisms from `MonoidAlgebra k G` are equal on all `single a 1`
-and `single 1 b`, then they are equal. -/
-theorem ringHom_ext {R} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R}
-    (h₁ : ∀ b, f (single 1 b) = g (single 1 b)) (h_of : ∀ a, f (single a 1) = g (single a 1)) :
-    f = g :=
-  RingHom.coe_addMonoidHom_injective <|
-    addHom_ext fun a b => by
-      rw [← single, ← one_mul a, ← mul_one b, ← single_mul_single]
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      erw [AddMonoidHom.coe_coe f, AddMonoidHom.coe_coe g]; rw [f.map_mul, g.map_mul, h₁, h_of]
-
-/-- If two ring homomorphisms from `MonoidAlgebra k G` are equal on all `single a 1`
-and `single 1 b`, then they are equal.
-
-See note [partially-applied ext lemmas]. -/
-@[ext high]
-theorem ringHom_ext' {R} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R}
-    (h₁ : f.comp singleOneRingHom = g.comp singleOneRingHom)
-    (h_of :
-      (f : MonoidAlgebra k G →* R).comp (of k G) = (g : MonoidAlgebra k G →* R).comp (of k G)) :
-    f = g :=
-  ringHom_ext (RingHom.congr_fun h₁) (DFunLike.congr_fun h_of)
-
 /-- The instance `Algebra k (MonoidAlgebra A G)` whenever we have `Algebra k A`.
 
 In particular this provides the instance `Algebra k (MonoidAlgebra k G)`.
@@ -769,15 +142,6 @@ theorem single_algebraMap_eq_algebraMap_mul_of {A : Type*} [CommSemiring k] [Sem
     [Algebra k A] [Monoid G] (a : G) (b : k) :
     single a (algebraMap k A b) = algebraMap k (MonoidAlgebra A G) b * of A G a := by simp
 
-theorem induction_on [Semiring k] [Monoid G] {p : MonoidAlgebra k G → Prop} (f : MonoidAlgebra k G)
-    (hM : ∀ g, p (of k G g)) (hadd : ∀ f g : MonoidAlgebra k G, p f → p g → p (f + g))
-    (hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by
-  refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
-  · simpa using hsmul 0 (of k G 1) (hM 1)
-  · convert hsmul r (of k G g) (hM g)
-    -- Porting note: Was `simp only`.
-    rw [of_apply, smul_single', mul_one]
-
 end Algebra
 
 section lift
@@ -975,678 +339,17 @@ end
 
 end
 
-section
-
-universe ui
-
-variable {ι : Type ui}
-
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-theorem prod_single [CommSemiring k] [CommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} :
-    (∏ i ∈ s, single (a i) (b i)) = single (∏ i ∈ s, a i) (∏ i ∈ s, b i) :=
-  Finset.cons_induction_on s rfl fun a s has ih => by
-    rw [prod_cons has, ih, single_mul_single, prod_cons has, prod_cons has]
-
-end
-
-section
-
--- We now prove some additional statements that hold for group algebras.
-variable [Semiring k] [Group G]
-
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-@[simp]
-theorem mul_single_apply (f : MonoidAlgebra k G) (r : k) (x y : G) :
-    (f * single x r) y = f (y * x⁻¹) * r :=
-  f.mul_single_apply_aux fun _a => eq_mul_inv_iff_mul_eq.symm
-
-@[simp]
-theorem single_mul_apply (r : k) (x : G) (f : MonoidAlgebra k G) (y : G) :
-    (single x r * f) y = r * f (x⁻¹ * y) :=
-  f.single_mul_apply_aux fun _z => eq_inv_mul_iff_mul_eq.symm
-
-theorem mul_apply_left (f g : MonoidAlgebra k G) (x : G) :
-    (f * g) x = f.sum fun a b => b * g (a⁻¹ * x) :=
-  calc
-    (f * g) x = sum f fun a b => (single a b * g) x := by
-      rw [← Finsupp.sum_apply, ← Finsupp.sum_mul g f, f.sum_single]
-    _ = _ := by simp only [single_mul_apply, Finsupp.sum]
-
--- If we'd assumed `CommSemiring`, we could deduce this from `mul_apply_left`.
-theorem mul_apply_right (f g : MonoidAlgebra k G) (x : G) :
-    (f * g) x = g.sum fun a b => f (x * a⁻¹) * b :=
-  calc
-    (f * g) x = sum g fun a b => (f * single a b) x := by
-      rw [← Finsupp.sum_apply, ← Finsupp.mul_sum f g, g.sum_single]
-    _ = _ := by simp only [mul_single_apply, Finsupp.sum]
-
-end
-
-section Opposite
-
-open Finsupp MulOpposite
-
-variable [Semiring k]
-
-/-- The opposite of a `MonoidAlgebra R I` equivalent as a ring to
-the `MonoidAlgebra Rᵐᵒᵖ Iᵐᵒᵖ` over the opposite ring, taking elements to their opposite. -/
-@[simps! (config := { simpRhs := true }) apply symm_apply]
-protected noncomputable def opRingEquiv [Monoid G] :
-    (MonoidAlgebra k G)ᵐᵒᵖ ≃+* MonoidAlgebra kᵐᵒᵖ Gᵐᵒᵖ :=
-  { opAddEquiv.symm.trans <|
-      (Finsupp.mapRange.addEquiv (opAddEquiv : k ≃+ kᵐᵒᵖ)).trans <| Finsupp.domCongr opEquiv with
-    map_mul' := by
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      rw [Equiv.toFun_as_coe, AddEquiv.toEquiv_eq_coe]; erw [AddEquiv.coe_toEquiv]
-      rw [← AddEquiv.coe_toAddMonoidHom]
-      refine Iff.mpr (AddMonoidHom.map_mul_iff (R := (MonoidAlgebra k G)ᵐᵒᵖ)
-        (S := MonoidAlgebra kᵐᵒᵖ Gᵐᵒᵖ) _) ?_
-      -- Porting note: Was `ext`.
-      refine AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₁ => AddMonoidHom.ext fun r₁ =>
-        AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₂ => AddMonoidHom.ext fun r₂ => ?_
-      -- Porting note: `reducible` cannot be `local` so proof gets long.
-      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, opAddEquiv_apply,
-        Function.comp_apply, singleAddHom_apply, AddMonoidHom.compr₂_apply, AddMonoidHom.coe_mul,
-        AddMonoidHom.coe_mulLeft, AddMonoidHom.compl₂_apply]
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      erw [AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply,
-        AddEquiv.trans_apply, AddEquiv.trans_apply, MulOpposite.opAddEquiv_symm_apply]
-      rw [MulOpposite.unop_mul (α := MonoidAlgebra k G)]
-      -- This was not needed before leanprover/lean4#2644
-      erw [unop_op, unop_op, single_mul_single]
-      simp }
-
--- @[simp] -- Porting note (#10618): simp can prove this
-theorem opRingEquiv_single [Monoid G] (r : k) (x : G) :
-    MonoidAlgebra.opRingEquiv (op (single x r)) = single (op x) (op r) := by simp
-
--- @[simp] -- Porting note (#10618): simp can prove this
-theorem opRingEquiv_symm_single [Monoid G] (r : kᵐᵒᵖ) (x : Gᵐᵒᵖ) :
-    MonoidAlgebra.opRingEquiv.symm (single x r) = op (single x.unop r.unop) := by simp
-
-end Opposite
-
-section Submodule
-
-variable [CommSemiring k] [Monoid G]
-variable {V : Type*} [AddCommMonoid V]
-variable [Module k V] [Module (MonoidAlgebra k G) V] [IsScalarTower k (MonoidAlgebra k G) V]
-
-/-- A submodule over `k` which is stable under scalar multiplication by elements of `G` is a
-submodule over `MonoidAlgebra k G`  -/
-def submoduleOfSMulMem (W : Submodule k V) (h : ∀ (g : G) (v : V), v ∈ W → of k G g • v ∈ W) :
-    Submodule (MonoidAlgebra k G) V where
-  carrier := W
-  zero_mem' := W.zero_mem'
-  add_mem' := W.add_mem'
-  smul_mem' := by
-    intro f v hv
-    rw [← Finsupp.sum_single f, Finsupp.sum, Finset.sum_smul]
-    simp_rw [← smul_of, smul_assoc]
-    exact Submodule.sum_smul_mem W _ fun g _ => h g v hv
-
-end Submodule
-
 end MonoidAlgebra
 
-/-! ### Additive monoids -/
-
-
-section
-
-variable [Semiring k]
-
-/-- The monoid algebra over a semiring `k` generated by the additive monoid `G`.
-It is the type of finite formal `k`-linear combinations of terms of `G`,
-endowed with the convolution product.
--/
-def AddMonoidAlgebra :=
-  G →₀ k
-
-@[inherit_doc]
-scoped[AddMonoidAlgebra] notation:9000 R:max "[" A "]" => AddMonoidAlgebra R A
-
-namespace AddMonoidAlgebra
-
--- Porting note: The compiler couldn't derive this.
-instance inhabited : Inhabited k[G] :=
-  inferInstanceAs (Inhabited (G →₀ k))
-
--- Porting note: The compiler couldn't derive this.
-instance addCommMonoid : AddCommMonoid k[G] :=
-  inferInstanceAs (AddCommMonoid (G →₀ k))
-
-instance instIsCancelAdd [IsCancelAdd k] : IsCancelAdd (AddMonoidAlgebra k G) :=
-  inferInstanceAs (IsCancelAdd (G →₀ k))
-
-instance coeFun : CoeFun k[G] fun _ => G → k :=
-  Finsupp.instCoeFun
-
-end AddMonoidAlgebra
-
-end
-
-namespace AddMonoidAlgebra
-
-variable {k G}
-
-section
-
-variable [Semiring k] [NonUnitalNonAssocSemiring R]
-
--- Porting note: `reducible` cannot be `local`, so we replace some definitions and theorems with
---               new ones which have new types.
-
-abbrev single (a : G) (b : k) : k[G] := Finsupp.single a b
-
-theorem single_zero (a : G) : (single a 0 : k[G]) = 0 := Finsupp.single_zero a
-
-theorem single_add (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
-  Finsupp.single_add a b₁ b₂
-
-@[simp]
-theorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N}
-    (h_zero : h a 0 = 0) :
-    (single a b).sum h = h a b := Finsupp.sum_single_index h_zero
-
-@[simp]
-theorem sum_single (f : k[G]) : f.sum single = f :=
-  Finsupp.sum_single f
-
-theorem single_apply {a a' : G} {b : k} [Decidable (a = a')] :
-    single a b a' = if a = a' then b else 0 :=
-  Finsupp.single_apply
-
-@[simp]
-theorem single_eq_zero {a : G} {b : k} : single a b = 0 ↔ b = 0 := Finsupp.single_eq_zero
-
-abbrev mapDomain {G' : Type*} (f : G → G') (v : k[G]) : k[G'] :=
-  Finsupp.mapDomain f v
-
-theorem mapDomain_sum {k' G' : Type*} [Semiring k'] {f : G → G'} {s : AddMonoidAlgebra k' G}
-    {v : G → k' → k[G]} :
-    mapDomain f (s.sum v) = s.sum fun a b => mapDomain f (v a b) :=
-  Finsupp.mapDomain_sum
-
-theorem mapDomain_single {G' : Type*} {f : G → G'} {a : G} {b : k} :
-    mapDomain f (single a b) = single (f a) b :=
-  Finsupp.mapDomain_single
-
-/-- A non-commutative version of `AddMonoidAlgebra.lift`: given an additive homomorphism
-`f : k →+ R` and a map `g : Multiplicative G → R`, returns the additive
-homomorphism from `k[G]` such that `liftNC f g (single a b) = f b * g a`. If `f`
-is a ring homomorphism and the range of either `f` or `g` is in center of `R`, then the result is a
-ring homomorphism.  If `R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra
-homomorphism called `AddMonoidAlgebra.lift`. -/
-def liftNC (f : k →+ R) (g : Multiplicative G → R) : k[G] →+ R :=
-  liftAddHom fun x : G => (AddMonoidHom.mulRight (g <| Multiplicative.ofAdd x)).comp f
-
-@[simp]
-theorem liftNC_single (f : k →+ R) (g : Multiplicative G → R) (a : G) (b : k) :
-    liftNC f g (single a b) = f b * g (Multiplicative.ofAdd a) :=
-  liftAddHom_apply_single _ _ _
-
-end
-
-section Mul
-
-variable [Semiring k] [Add G]
-
-/-- The product of `f g : k[G]` is the finitely supported function
-  whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
-  such that `x + y = a`. (Think of the product of multivariate
-  polynomials where `α` is the additive monoid of monomial exponents.) -/
-instance hasMul : Mul k[G] :=
-  ⟨fun f g => MonoidAlgebra.mul' (G := Multiplicative G) f g⟩
-
-theorem mul_def {f g : k[G]} :
-    f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ + a₂) (b₁ * b₂) :=
-  MonoidAlgebra.mul_def (G := Multiplicative G)
-
-instance nonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring k[G] :=
-  { Finsupp.instAddCommMonoid with
-    -- Porting note: `refine` & `exact` are required because `simp` behaves differently.
-    left_distrib := fun f g h => by
-      haveI := Classical.decEq G
-      simp only [mul_def]
-      refine Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_add_index ?_ ?_)) ?_ <;>
-        simp only [mul_add, mul_zero, single_zero, single_add, forall_true_iff, sum_add]
-    right_distrib := fun f g h => by
-      haveI := Classical.decEq G
-      simp only [mul_def]
-      refine Eq.trans (sum_add_index ?_ ?_) ?_ <;>
-        simp only [add_mul, zero_mul, single_zero, single_add, forall_true_iff, sum_zero, sum_add]
-    zero_mul := fun f => by
-      simp only [mul_def]
-      exact sum_zero_index
-    mul_zero := fun f => by
-      simp only [mul_def]
-      exact Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_zero_index)) sum_zero
-    nsmul := fun n f => n • f
-    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
-    nsmul_zero := by
-      intros
-      refine Finsupp.ext fun _ => ?_
-      simp [-nsmul_eq_mul, add_smul]
-    nsmul_succ := by
-      intros
-      refine Finsupp.ext fun _ => ?_
-      simp [-nsmul_eq_mul, add_smul] }
-
-variable [Semiring R]
-
-theorem liftNC_mul {g_hom : Type*}
-    [FunLike g_hom (Multiplicative G) R] [MulHomClass g_hom (Multiplicative G) R]
-    (f : k →+* R) (g : g_hom) (a b : k[G])
-    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g <| Multiplicative.ofAdd y)) :
-    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b :=
-  (MonoidAlgebra.liftNC_mul f g _ _ @h_comm : _)
-
-end Mul
-
-section One
-
-variable [Semiring k] [Zero G] [NonAssocSemiring R]
-
-/-- The unit of the multiplication is `single 0 1`, i.e. the function
-  that is `1` at `0` and zero elsewhere. -/
-instance one : One k[G] :=
-  ⟨single 0 1⟩
-
-theorem one_def : (1 : k[G]) = single 0 1 :=
-  rfl
-
-@[simp]
-theorem liftNC_one {g_hom : Type*}
-    [FunLike g_hom (Multiplicative G) R] [OneHomClass g_hom (Multiplicative G) R]
-    (f : k →+* R) (g : g_hom) : liftNC (f : k →+ R) g 1 = 1 :=
-  (MonoidAlgebra.liftNC_one f g : _)
-
-end One
-
-section Semigroup
-
-variable [Semiring k] [AddSemigroup G]
-
-instance nonUnitalSemiring : NonUnitalSemiring k[G] :=
-  { AddMonoidAlgebra.nonUnitalNonAssocSemiring with
-    mul_assoc := fun f g h => by
-      -- Porting note: `reducible` cannot be `local` so proof gets long.
-      simp only [mul_def]
-      rw [sum_sum_index]; congr; ext a₁ b₁
-      rw [sum_sum_index, sum_sum_index]; congr; ext a₂ b₂
-      rw [sum_sum_index, sum_single_index]; congr; ext a₃ b₃
-      rw [sum_single_index, mul_assoc, add_assoc]
-      all_goals simp only [single_zero, single_add, forall_true_iff, add_mul,
-        mul_add, zero_mul, mul_zero, sum_zero, sum_add] }
-
-end Semigroup
-
-section MulOneClass
-
-variable [Semiring k] [AddZeroClass G]
-
-instance nonAssocSemiring : NonAssocSemiring k[G] :=
-  { AddMonoidAlgebra.nonUnitalNonAssocSemiring with
-    natCast := fun n => single 0 n
-    natCast_zero := by simp
-    natCast_succ := fun _ => by simp; rfl
-    one_mul := fun f => by
-      simp only [mul_def, one_def, sum_single_index, zero_mul, single_zero, sum_zero, zero_add,
-        one_mul, sum_single]
-    mul_one := fun f => by
-      simp only [mul_def, one_def, sum_single_index, mul_zero, single_zero, sum_zero, add_zero,
-        mul_one, sum_single] }
-
-theorem natCast_def (n : ℕ) : (n : k[G]) = single (0 : G) (n : k) :=
-  rfl
-
-@[deprecated (since := "2024-04-17")]
-alias nat_cast_def := natCast_def
-
-end MulOneClass
-
-/-! #### Semiring structure -/
-
-
-section Semiring
-
-instance smulZeroClass [Semiring k] [SMulZeroClass R k] : SMulZeroClass R k[G] :=
-  Finsupp.smulZeroClass
-
-variable [Semiring k] [AddMonoid G]
-
-instance semiring : Semiring k[G] :=
-  { AddMonoidAlgebra.nonUnitalSemiring,
-    AddMonoidAlgebra.nonAssocSemiring with }
-
-variable [Semiring R]
-
-/-- `liftNC` as a `RingHom`, for when `f` and `g` commute -/
-def liftNCRingHom (f : k →+* R) (g : Multiplicative G →* R) (h_comm : ∀ x y, Commute (f x) (g y)) :
-    k[G] →+* R :=
-  { liftNC (f : k →+ R) g with
-    map_one' := liftNC_one _ _
-    map_mul' := fun _a _b => liftNC_mul _ _ _ _ fun {_ _} _ => h_comm _ _ }
-
-end Semiring
-
-instance nonUnitalCommSemiring [CommSemiring k] [AddCommSemigroup G] :
-    NonUnitalCommSemiring k[G] :=
-  { AddMonoidAlgebra.nonUnitalSemiring with
-    mul_comm := @mul_comm (MonoidAlgebra k <| Multiplicative G) _ }
-
-instance nontrivial [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial k[G] :=
-  Finsupp.instNontrivial
-
-/-! #### Derived instances -/
-
-
-section DerivedInstances
-
-instance commSemiring [CommSemiring k] [AddCommMonoid G] : CommSemiring k[G] :=
-  { AddMonoidAlgebra.nonUnitalCommSemiring, AddMonoidAlgebra.semiring with }
-
-instance unique [Semiring k] [Subsingleton k] : Unique k[G] :=
-  Finsupp.uniqueOfRight
-
-instance addCommGroup [Ring k] : AddCommGroup k[G] :=
-  Finsupp.instAddCommGroup
-
-instance nonUnitalNonAssocRing [Ring k] [Add G] : NonUnitalNonAssocRing k[G] :=
-  { AddMonoidAlgebra.addCommGroup, AddMonoidAlgebra.nonUnitalNonAssocSemiring with }
-
-instance nonUnitalRing [Ring k] [AddSemigroup G] : NonUnitalRing k[G] :=
-  { AddMonoidAlgebra.addCommGroup, AddMonoidAlgebra.nonUnitalSemiring with }
-
-instance nonAssocRing [Ring k] [AddZeroClass G] : NonAssocRing k[G] :=
-  { AddMonoidAlgebra.addCommGroup,
-    AddMonoidAlgebra.nonAssocSemiring with
-    intCast := fun z => single 0 (z : k)
-    -- Porting note: Both were `simpa`.
-    intCast_ofNat := fun n => by simp; rfl
-    intCast_negSucc := fun n => by simp; rfl }
-
-theorem intCast_def [Ring k] [AddZeroClass G] (z : ℤ) :
-    (z : k[G]) = single (0 : G) (z : k) :=
-  rfl
-
-@[deprecated (since := "2024-04-17")]
-alias int_cast_def := intCast_def
-
-instance ring [Ring k] [AddMonoid G] : Ring k[G] :=
-  { AddMonoidAlgebra.nonAssocRing, AddMonoidAlgebra.semiring with }
-
-instance nonUnitalCommRing [CommRing k] [AddCommSemigroup G] :
-    NonUnitalCommRing k[G] :=
-  { AddMonoidAlgebra.nonUnitalCommSemiring, AddMonoidAlgebra.nonUnitalRing with }
-
-instance commRing [CommRing k] [AddCommMonoid G] : CommRing k[G] :=
-  { AddMonoidAlgebra.nonUnitalCommRing, AddMonoidAlgebra.ring with }
-
-variable {S : Type*}
-
-instance distribSMul [Semiring k] [DistribSMul R k] : DistribSMul R k[G] :=
-  Finsupp.distribSMul G k
-
-instance distribMulAction [Monoid R] [Semiring k] [DistribMulAction R k] :
-    DistribMulAction R k[G] :=
-  Finsupp.distribMulAction G k
-
-instance faithfulSMul [Semiring k] [SMulZeroClass R k] [FaithfulSMul R k] [Nonempty G] :
-    FaithfulSMul R k[G] :=
-  Finsupp.faithfulSMul
-
-instance module [Semiring R] [Semiring k] [Module R k] : Module R k[G] :=
-  Finsupp.module G k
-
-instance isScalarTower [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S]
-    [IsScalarTower R S k] : IsScalarTower R S k[G] :=
-  Finsupp.isScalarTower G k
-
-instance smulCommClass [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMulCommClass R S k] :
-    SMulCommClass R S k[G] :=
-  Finsupp.smulCommClass G k
-
-instance isCentralScalar [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k]
-    [IsCentralScalar R k] : IsCentralScalar R k[G] :=
-  Finsupp.isCentralScalar G k
-
-/-! It is hard to state the equivalent of `DistribMulAction G k[G]`
-because we've never discussed actions of additive groups. -/
-
-
-end DerivedInstances
-
-section MiscTheorems
-
-variable [Semiring k]
-
-theorem mul_apply [DecidableEq G] [Add G] (f g : k[G]) (x : G) :
-    (f * g) x = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => if a₁ + a₂ = x then b₁ * b₂ else 0 :=
-  @MonoidAlgebra.mul_apply k (Multiplicative G) _ _ _ _ _ _
-
-theorem mul_apply_antidiagonal [Add G] (f g : k[G]) (x : G) (s : Finset (G × G))
-    (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 + p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2 :=
-  @MonoidAlgebra.mul_apply_antidiagonal k (Multiplicative G) _ _ _ _ _ s @hs
-
-theorem single_mul_single [Add G] {a₁ a₂ : G} {b₁ b₂ : k} :
-    single a₁ b₁ * single a₂ b₂ = single (a₁ + a₂) (b₁ * b₂) :=
-  @MonoidAlgebra.single_mul_single k (Multiplicative G) _ _ _ _ _ _
-
-theorem single_commute_single [Add G] {a₁ a₂ : G} {b₁ b₂ : k}
-    (ha : AddCommute a₁ a₂) (hb : Commute b₁ b₂) :
-    Commute (single a₁ b₁) (single a₂ b₂) :=
-  @MonoidAlgebra.single_commute_single k (Multiplicative G) _ _ _ _ _ _ ha hb
-
--- This should be a `@[simp]` lemma, but the simp_nf linter times out if we add this.
--- Probably the correct fix is to make a `[Add]MonoidAlgebra.single` with the correct type,
--- instead of relying on `Finsupp.single`.
-theorem single_pow [AddMonoid G] {a : G} {b : k} : ∀ n : ℕ, single a b ^ n = single (n • a) (b ^ n)
-  | 0 => by
-    simp only [pow_zero, zero_nsmul]
-    rfl
-  | n + 1 => by
-    rw [pow_succ, pow_succ, single_pow n, single_mul_single, add_nsmul, one_nsmul]
-
-/-- Like `Finsupp.mapDomain_zero`, but for the `1` we define in this file -/
-@[simp]
-theorem mapDomain_one {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Zero α] [Zero α₂]
-    {F : Type*} [FunLike F α α₂] [ZeroHomClass F α α₂] (f : F) :
-    (mapDomain f (1 : AddMonoidAlgebra β α) : AddMonoidAlgebra β α₂) =
-      (1 : AddMonoidAlgebra β α₂) := by
-  simp_rw [one_def, mapDomain_single, map_zero]
-
-/-- Like `Finsupp.mapDomain_add`, but for the convolutive multiplication we define in this file -/
-theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Add α] [Add α₂]
-    {F : Type*} [FunLike F α α₂] [AddHomClass F α α₂] (f : F) (x y : AddMonoidAlgebra β α) :
-    mapDomain f (x * y) = mapDomain f x * mapDomain f y := by
-  simp_rw [mul_def, mapDomain_sum, mapDomain_single, map_add]
-  rw [Finsupp.sum_mapDomain_index]
-  · congr
-    ext a b
-    rw [Finsupp.sum_mapDomain_index]
-    · simp
-    · simp [mul_add]
-  · simp
-  · simp [add_mul]
-
-section
-
-variable (k G)
-
-/-- The embedding of an additive magma into its additive magma algebra. -/
-@[simps]
-def ofMagma [Add G] : Multiplicative G →ₙ* k[G] where
-  toFun a := single a 1
-  map_mul' a b := by simp only [mul_def, mul_one, sum_single_index, single_eq_zero, mul_zero]; rfl
-
-/-- Embedding of a magma with zero into its magma algebra. -/
-def of [AddZeroClass G] : Multiplicative G →* k[G] :=
-  { ofMagma k G with
-    toFun := fun a => single a 1
-    map_one' := rfl }
-
-/-- Embedding of a magma with zero `G`, into its magma algebra, having `G` as source. -/
-def of' : G → k[G] := fun a => single a 1
-
-end
-
-@[simp]
-theorem of_apply [AddZeroClass G] (a : Multiplicative G) :
-    of k G a = single (Multiplicative.toAdd a) 1 :=
-  rfl
-
-@[simp]
-theorem of'_apply (a : G) : of' k G a = single a 1 :=
-  rfl
-
-theorem of'_eq_of [AddZeroClass G] (a : G) : of' k G a = of k G (.ofAdd a) := rfl
-
-theorem of_injective [Nontrivial k] [AddZeroClass G] : Function.Injective (of k G) :=
-  MonoidAlgebra.of_injective
-
-theorem of'_commute [AddZeroClass G] {a : G} (h : ∀ a', AddCommute a a')
-    (f : AddMonoidAlgebra k G) :
-    Commute (of' k G a) f :=
-  MonoidAlgebra.of_commute (G := Multiplicative G) h f
-
-/-- `Finsupp.single` as a `MonoidHom` from the product type into the additive monoid algebra.
-
-Note the order of the elements of the product are reversed compared to the arguments of
-`Finsupp.single`.
--/
-@[simps]
-def singleHom [AddZeroClass G] : k × Multiplicative G →* k[G] where
-  toFun a := single (Multiplicative.toAdd a.2) a.1
-  map_one' := rfl
-  map_mul' _a _b := single_mul_single.symm
-
-theorem mul_single_apply_aux [Add G] (f : k[G]) (r : k) (x y z : G)
-    (H : ∀ a, a + x = z ↔ a = y) : (f * single x r) z = f y * r :=
-  @MonoidAlgebra.mul_single_apply_aux k (Multiplicative G) _ _ _ _ _ _ _ H
-
-theorem mul_single_zero_apply [AddZeroClass G] (f : k[G]) (r : k) (x : G) :
-    (f * single (0 : G) r) x = f x * r :=
-  f.mul_single_apply_aux r _ _ _ fun a => by rw [add_zero]
-
-theorem mul_single_apply_of_not_exists_add [Add G] (r : k) {g g' : G} (x : k[G])
-    (h : ¬∃ d, g' = d + g) : (x * single g r) g' = 0 :=
-  @MonoidAlgebra.mul_single_apply_of_not_exists_mul k (Multiplicative G) _ _ _ _ _ _ h
-
-theorem single_mul_apply_aux [Add G] (f : k[G]) (r : k) (x y z : G)
-    (H : ∀ a, x + a = y ↔ a = z) : (single x r * f) y = r * f z :=
-  @MonoidAlgebra.single_mul_apply_aux k (Multiplicative G) _ _ _ _ _ _ _ H
-
-theorem single_zero_mul_apply [AddZeroClass G] (f : k[G]) (r : k) (x : G) :
-    (single (0 : G) r * f) x = r * f x :=
-  f.single_mul_apply_aux r _ _ _ fun a => by rw [zero_add]
-
-theorem single_mul_apply_of_not_exists_add [Add G] (r : k) {g g' : G} (x : k[G])
-    (h : ¬∃ d, g' = g + d) : (single g r * x) g' = 0 :=
-  @MonoidAlgebra.single_mul_apply_of_not_exists_mul k (Multiplicative G) _ _ _ _ _ _ h
-
-theorem mul_single_apply [AddGroup G] (f : k[G]) (r : k) (x y : G) :
-    (f * single x r) y = f (y - x) * r :=
-  (sub_eq_add_neg y x).symm ▸ @MonoidAlgebra.mul_single_apply k (Multiplicative G) _ _ _ _ _ _
-
-theorem single_mul_apply [AddGroup G] (r : k) (x : G) (f : k[G]) (y : G) :
-    (single x r * f) y = r * f (-x + y) :=
-  @MonoidAlgebra.single_mul_apply k (Multiplicative G) _ _ _ _ _ _
-
-theorem liftNC_smul {R : Type*} [AddZeroClass G] [Semiring R] (f : k →+* R)
-    (g : Multiplicative G →* R) (c : k) (φ : MonoidAlgebra k G) :
-    liftNC (f : k →+ R) g (c • φ) = f c * liftNC (f : k →+ R) g φ :=
-  @MonoidAlgebra.liftNC_smul k (Multiplicative G) _ _ _ _ f g c φ
-
-theorem induction_on [AddMonoid G] {p : k[G] → Prop} (f : k[G])
-    (hM : ∀ g, p (of k G (Multiplicative.ofAdd g)))
-    (hadd : ∀ f g : k[G], p f → p g → p (f + g))
-    (hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by
-  refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
-  · simpa using hsmul 0 (of k G (Multiplicative.ofAdd 0)) (hM 0)
-  · convert hsmul r (of k G (Multiplicative.ofAdd g)) (hM g)
-    -- Porting note: Was `simp only`.
-    rw [of_apply, toAdd_ofAdd, smul_single', mul_one]
-
-/-- If `f : G → H` is an additive homomorphism between two additive monoids, then
-`Finsupp.mapDomain f` is a ring homomorphism between their add monoid algebras. -/
-@[simps]
-def mapDomainRingHom (k : Type*) [Semiring k] {H F : Type*} [AddMonoid G] [AddMonoid H]
-    [FunLike F G H] [AddMonoidHomClass F G H] (f : F) : k[G] →+* k[H] :=
-  { (Finsupp.mapDomain.addMonoidHom f : MonoidAlgebra k G →+ MonoidAlgebra k H) with
-    map_one' := mapDomain_one f
-    map_mul' := fun x y => mapDomain_mul f x y }
-
-end MiscTheorems
-
-end AddMonoidAlgebra
-
-/-!
-#### Conversions between `AddMonoidAlgebra` and `MonoidAlgebra`
-
-We have not defined `k[G] = MonoidAlgebra k (Multiplicative G)`
-because historically this caused problems;
-since the changes that have made `nsmul` definitional, this would be possible,
-but for now we just construct the ring isomorphisms using `RingEquiv.refl _`.
--/
-
-
-/-- The equivalence between `AddMonoidAlgebra` and `MonoidAlgebra` in terms of
-`Multiplicative` -/
-protected def AddMonoidAlgebra.toMultiplicative [Semiring k] [Add G] :
-    AddMonoidAlgebra k G ≃+* MonoidAlgebra k (Multiplicative G) :=
-  { Finsupp.domCongr
-      Multiplicative.ofAdd with
-    toFun := equivMapDomain Multiplicative.ofAdd
-    map_mul' := fun x y => by
-      -- Porting note: added `dsimp only`; `beta_reduce` alone is not sufficient
-      dsimp only
-      repeat' rw [equivMapDomain_eq_mapDomain (M := k)]
-      dsimp [Multiplicative.ofAdd]
-      exact MonoidAlgebra.mapDomain_mul (α := Multiplicative G) (β := k)
-        (MulHom.id (Multiplicative G)) x y }
-
-/-- The equivalence between `MonoidAlgebra` and `AddMonoidAlgebra` in terms of `Additive` -/
-protected def MonoidAlgebra.toAdditive [Semiring k] [Mul G] :
-    MonoidAlgebra k G ≃+* AddMonoidAlgebra k (Additive G) :=
-  { Finsupp.domCongr Additive.ofMul with
-    toFun := equivMapDomain Additive.ofMul
-    map_mul' := fun x y => by
-      -- Porting note: added `dsimp only`; `beta_reduce` alone is not sufficient
-      dsimp only
-      repeat' rw [equivMapDomain_eq_mapDomain (M := k)]
-      dsimp [Additive.ofMul]
-      convert MonoidAlgebra.mapDomain_mul (β := k) (MulHom.id G) x y }
-
 namespace AddMonoidAlgebra
 
 variable {k G H}
 
 /-! #### Non-unital, non-associative algebra structure -/
 
-
 section NonUnitalNonAssocAlgebra
 
 variable (k) [Semiring k] [DistribSMul R k] [Add G]
-
-instance isScalarTower_self [IsScalarTower R k k] :
-    IsScalarTower R k[G] k[G] :=
-  @MonoidAlgebra.isScalarTower_self k (Multiplicative G) R _ _ _ _
-
-/-- Note that if `k` is a `CommSemiring` then we have `SMulCommClass k k k` and so we can take
-`R = k` in the below. In other words, if the coefficients are commutative amongst themselves, they
-also commute with the algebra multiplication. -/
-instance smulCommClass_self [SMulCommClass R k k] :
-    SMulCommClass R k[G] k[G] :=
-  @MonoidAlgebra.smulCommClass_self k (Multiplicative G) R _ _ _ _
-
-instance smulCommClass_symm_self [SMulCommClass k R k] :
-    SMulCommClass k[G] R k[G] :=
-  @MonoidAlgebra.smulCommClass_symm_self k (Multiplicative G) R _ _ _ _
-
 variable {A : Type u₃} [NonUnitalNonAssocSemiring A]
 
 /-- A non_unital `k`-algebra homomorphism from `k[G]` is uniquely defined by its
@@ -1680,75 +383,6 @@ end NonUnitalNonAssocAlgebra
 
 section Algebra
 
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-/-- `Finsupp.single 0` as a `RingHom` -/
-@[simps]
-def singleZeroRingHom [Semiring k] [AddMonoid G] : k →+* k[G] :=
-  { Finsupp.singleAddHom 0 with
-    map_one' := rfl
-    -- Porting note (#10691): Was `rw`.
-    map_mul' := fun x y => by simp only [singleAddHom, single_mul_single, zero_add] }
-
-/-- If two ring homomorphisms from `k[G]` are equal on all `single a 1`
-and `single 0 b`, then they are equal. -/
-theorem ringHom_ext {R} [Semiring k] [AddMonoid G] [Semiring R] {f g : k[G] →+* R}
-    (h₀ : ∀ b, f (single 0 b) = g (single 0 b)) (h_of : ∀ a, f (single a 1) = g (single a 1)) :
-    f = g :=
-  @MonoidAlgebra.ringHom_ext k (Multiplicative G) R _ _ _ _ _ h₀ h_of
-
-/-- If two ring homomorphisms from `k[G]` are equal on all `single a 1`
-and `single 0 b`, then they are equal.
-
-See note [partially-applied ext lemmas]. -/
-@[ext high]
-theorem ringHom_ext' {R} [Semiring k] [AddMonoid G] [Semiring R] {f g : k[G] →+* R}
-    (h₁ : f.comp singleZeroRingHom = g.comp singleZeroRingHom)
-    (h_of : (f : k[G] →* R).comp (of k G) = (g : k[G] →* R).comp (of k G)) :
-    f = g :=
-  ringHom_ext (RingHom.congr_fun h₁) (DFunLike.congr_fun h_of)
-
-section Opposite
-
-open Finsupp MulOpposite
-
-variable [Semiring k]
-
-/-- The opposite of an `R[I]` is ring equivalent to
-the `AddMonoidAlgebra Rᵐᵒᵖ I` over the opposite ring, taking elements to their opposite. -/
-@[simps! (config := { simpRhs := true }) apply symm_apply]
-protected noncomputable def opRingEquiv [AddCommMonoid G] :
-    k[G]ᵐᵒᵖ ≃+* kᵐᵒᵖ[G] :=
-  { MulOpposite.opAddEquiv.symm.trans
-      (Finsupp.mapRange.addEquiv (MulOpposite.opAddEquiv : k ≃+ kᵐᵒᵖ)) with
-    map_mul' := by
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      rw [Equiv.toFun_as_coe, AddEquiv.toEquiv_eq_coe]; erw [AddEquiv.coe_toEquiv]
-      rw [← AddEquiv.coe_toAddMonoidHom]
-      refine Iff.mpr (AddMonoidHom.map_mul_iff (R := k[G]ᵐᵒᵖ) (S := kᵐᵒᵖ[G]) _) ?_
-      -- Porting note: Was `ext`.
-      refine AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₁ => AddMonoidHom.ext fun r₁ =>
-        AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₂ => AddMonoidHom.ext fun r₂ => ?_
-      -- Porting note: `reducible` cannot be `local` so proof gets long.
-      dsimp
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      erw [AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply,
-        MulOpposite.opAddEquiv_symm_apply]; rw [MulOpposite.unop_mul (α := k[G])]
-      dsimp
-      -- This was not needed before leanprover/lean4#2644
-      erw [mapRange_single, single_mul_single, mapRange_single, mapRange_single]
-      simp only [mapRange_single, single_mul_single, ← op_mul, add_comm] }
-
--- @[simp] -- Porting note (#10618): simp can prove this
-theorem opRingEquiv_single [AddCommMonoid G] (r : k) (x : G) :
-    AddMonoidAlgebra.opRingEquiv (op (single x r)) = single x (op r) := by simp
-
--- @[simp] -- Porting note (#10618): simp can prove this
-theorem opRingEquiv_symm_single [AddCommMonoid G] (r : kᵐᵒᵖ) (x : Gᵐᵒᵖ) :
-    AddMonoidAlgebra.opRingEquiv.symm (single x r) = op (single x r.unop) := by simp
-
-end Opposite
-
 /-- The instance `Algebra R k[G]` whenever we have `Algebra R k`.
 
 In particular this provides the instance `Algebra k k[G]`.
@@ -1867,21 +501,6 @@ theorem algHom_ext_iff {φ₁ φ₂ : k[G] →ₐ[k] A} :
 
 end lift
 
-section
-
--- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
-
-universe ui
-
-variable {ι : Type ui}
-
-theorem prod_single [CommSemiring k] [AddCommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} :
-    (∏ i ∈ s, single (a i) (b i)) = single (∑ i ∈ s, a i) (∏ i ∈ s, b i) :=
-  Finset.cons_induction_on s rfl fun a s has ih => by
-    rw [prod_cons has, ih, single_mul_single, sum_cons has, prod_cons has]
-
-end
-
 theorem mapDomain_algebraMap (A : Type*) {H F : Type*} [CommSemiring k] [Semiring A] [Algebra k A]
     [AddMonoid G] [AddMonoid H] [FunLike F G H] [AddMonoidHomClass F G H]
     (f : F) (r : k) :
diff --git a/Mathlib/Algebra/MonoidAlgebra/Defs.lean b/Mathlib/Algebra/MonoidAlgebra/Defs.lean
new file mode 100644
index 0000000000000..a2574ffe76219
--- /dev/null
+++ b/Mathlib/Algebra/MonoidAlgebra/Defs.lean
@@ -0,0 +1,1442 @@
+/-
+Copyright (c) 2017 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
+-/
+import Mathlib.Algebra.BigOperators.Finsupp
+import Mathlib.Algebra.Module.BigOperators
+import Mathlib.Data.Finsupp.Basic
+import Mathlib.LinearAlgebra.Finsupp
+
+/-!
+# Monoid algebras
+
+When the domain of a `Finsupp` has a multiplicative or additive structure, we can define
+a convolution product. To mathematicians this structure is known as the "monoid algebra",
+i.e. the finite formal linear combinations over a given semiring of elements of the monoid.
+The "group ring" ℤ[G] or the "group algebra" k[G] are typical uses.
+
+In fact the construction of the "monoid algebra" makes sense when `G` is not even a monoid, but
+merely a magma, i.e., when `G` carries a multiplication which is not required to satisfy any
+conditions at all. In this case the construction yields a not-necessarily-unital,
+not-necessarily-associative algebra but it is still adjoint to the forgetful functor from such
+algebras to magmas, and we prove this as `MonoidAlgebra.liftMagma`.
+
+In this file we define `MonoidAlgebra k G := G →₀ k`, and `AddMonoidAlgebra k G`
+in the same way, and then define the convolution product on these.
+
+When the domain is additive, this is used to define polynomials:
+```
+Polynomial R := AddMonoidAlgebra R ℕ
+MvPolynomial σ α := AddMonoidAlgebra R (σ →₀ ℕ)
+```
+
+When the domain is multiplicative, e.g. a group, this will be used to define the group ring.
+
+## Notation
+
+We introduce the notation `R[A]` for `AddMonoidAlgebra R A`.
+
+## Implementation note
+Unfortunately because additive and multiplicative structures both appear in both cases,
+it doesn't appear to be possible to make much use of `to_additive`, and we just settle for
+saying everything twice.
+
+Similarly, I attempted to just define
+`k[G] := MonoidAlgebra k (Multiplicative G)`, but the definitional equality
+`Multiplicative G = G` leaks through everywhere, and seems impossible to use.
+-/
+
+assert_not_exists NonUnitalAlgHom
+assert_not_exists AlgEquiv
+
+noncomputable section
+
+open Finset
+
+open Finsupp hiding single mapDomain
+
+universe u₁ u₂ u₃ u₄
+
+variable (k : Type u₁) (G : Type u₂) (H : Type*) {R : Type*}
+
+/-! ### Multiplicative monoids -/
+
+
+section
+
+variable [Semiring k]
+
+/-- The monoid algebra over a semiring `k` generated by the monoid `G`.
+It is the type of finite formal `k`-linear combinations of terms of `G`,
+endowed with the convolution product.
+-/
+def MonoidAlgebra : Type max u₁ u₂ :=
+  G →₀ k
+
+-- Porting note: The compiler couldn't derive this.
+instance MonoidAlgebra.inhabited : Inhabited (MonoidAlgebra k G) :=
+  inferInstanceAs (Inhabited (G →₀ k))
+
+-- Porting note: The compiler couldn't derive this.
+instance MonoidAlgebra.addCommMonoid : AddCommMonoid (MonoidAlgebra k G) :=
+  inferInstanceAs (AddCommMonoid (G →₀ k))
+
+instance MonoidAlgebra.instIsCancelAdd [IsCancelAdd k] : IsCancelAdd (MonoidAlgebra k G) :=
+  inferInstanceAs (IsCancelAdd (G →₀ k))
+
+instance MonoidAlgebra.coeFun : CoeFun (MonoidAlgebra k G) fun _ => G → k :=
+  Finsupp.instCoeFun
+
+end
+
+namespace MonoidAlgebra
+
+variable {k G}
+
+section
+
+variable [Semiring k] [NonUnitalNonAssocSemiring R]
+
+-- Porting note: `reducible` cannot be `local`, so we replace some definitions and theorems with
+--               new ones which have new types.
+
+abbrev single (a : G) (b : k) : MonoidAlgebra k G := Finsupp.single a b
+
+theorem single_zero (a : G) : (single a 0 : MonoidAlgebra k G) = 0 := Finsupp.single_zero a
+
+theorem single_add (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
+  Finsupp.single_add a b₁ b₂
+
+@[simp]
+theorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N}
+    (h_zero : h a 0 = 0) :
+    (single a b).sum h = h a b := Finsupp.sum_single_index h_zero
+
+@[simp]
+theorem sum_single (f : MonoidAlgebra k G) : f.sum single = f :=
+  Finsupp.sum_single f
+
+theorem single_apply {a a' : G} {b : k} [Decidable (a = a')] :
+    single a b a' = if a = a' then b else 0 :=
+  Finsupp.single_apply
+
+@[simp]
+theorem single_eq_zero {a : G} {b : k} : single a b = 0 ↔ b = 0 := Finsupp.single_eq_zero
+
+abbrev mapDomain {G' : Type*} (f : G → G') (v : MonoidAlgebra k G) : MonoidAlgebra k G' :=
+  Finsupp.mapDomain f v
+
+theorem mapDomain_sum {k' G' : Type*} [Semiring k'] {f : G → G'} {s : MonoidAlgebra k' G}
+    {v : G → k' → MonoidAlgebra k G} :
+    mapDomain f (s.sum v) = s.sum fun a b => mapDomain f (v a b) :=
+  Finsupp.mapDomain_sum
+
+/-- A non-commutative version of `MonoidAlgebra.lift`: given an additive homomorphism `f : k →+ R`
+and a homomorphism `g : G → R`, returns the additive homomorphism from
+`MonoidAlgebra k G` such that `liftNC f g (single a b) = f b * g a`. If `f` is a ring homomorphism
+and the range of either `f` or `g` is in center of `R`, then the result is a ring homomorphism.  If
+`R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra homomorphism called
+`MonoidAlgebra.lift`. -/
+def liftNC (f : k →+ R) (g : G → R) : MonoidAlgebra k G →+ R :=
+  liftAddHom fun x : G => (AddMonoidHom.mulRight (g x)).comp f
+
+@[simp]
+theorem liftNC_single (f : k →+ R) (g : G → R) (a : G) (b : k) :
+    liftNC f g (single a b) = f b * g a :=
+  liftAddHom_apply_single _ _ _
+
+end
+
+section Mul
+
+variable [Semiring k] [Mul G]
+
+/-- The multiplication in a monoid algebra. We make it irreducible so that Lean doesn't unfold
+it trying to unify two things that are different. -/
+@[irreducible] def mul' (f g : MonoidAlgebra k G) : MonoidAlgebra k G :=
+  f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)
+
+/-- The product of `f g : MonoidAlgebra k G` is the finitely supported function
+  whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
+  such that `x * y = a`. (Think of the group ring of a group.) -/
+instance instMul : Mul (MonoidAlgebra k G) := ⟨MonoidAlgebra.mul'⟩
+
+theorem mul_def {f g : MonoidAlgebra k G} :
+    f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂) := by
+  with_unfolding_all rfl
+
+instance nonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring (MonoidAlgebra k G) :=
+  { Finsupp.instAddCommMonoid with
+    -- Porting note: `refine` & `exact` are required because `simp` behaves differently.
+    left_distrib := fun f g h => by
+      haveI := Classical.decEq G
+      simp only [mul_def]
+      refine Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_add_index ?_ ?_)) ?_ <;>
+        simp only [mul_add, mul_zero, single_zero, single_add, forall_true_iff, sum_add]
+    right_distrib := fun f g h => by
+      haveI := Classical.decEq G
+      simp only [mul_def]
+      refine Eq.trans (sum_add_index ?_ ?_) ?_ <;>
+        simp only [add_mul, zero_mul, single_zero, single_add, forall_true_iff, sum_zero, sum_add]
+    zero_mul := fun f => by
+      simp only [mul_def]
+      exact sum_zero_index
+    mul_zero := fun f => by
+      simp only [mul_def]
+      exact Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_zero_index)) sum_zero }
+
+variable [Semiring R]
+
+theorem liftNC_mul {g_hom : Type*} [FunLike g_hom G R] [MulHomClass g_hom G R]
+    (f : k →+* R) (g : g_hom) (a b : MonoidAlgebra k G)
+    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g y)) :
+    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b := by
+  conv_rhs => rw [← sum_single a, ← sum_single b]
+  -- Porting note: `(liftNC _ g).map_finsupp_sum` → `map_finsupp_sum`
+  simp_rw [mul_def, map_finsupp_sum, liftNC_single, Finsupp.sum_mul, Finsupp.mul_sum]
+  refine Finset.sum_congr rfl fun y hy => Finset.sum_congr rfl fun x _hx => ?_
+  simp [mul_assoc, (h_comm hy).left_comm]
+
+end Mul
+
+section Semigroup
+
+variable [Semiring k] [Semigroup G] [Semiring R]
+
+instance nonUnitalSemiring : NonUnitalSemiring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalNonAssocSemiring with
+    mul_assoc := fun f g h => by
+      -- Porting note: `reducible` cannot be `local` so proof gets long.
+      simp only [mul_def]
+      rw [sum_sum_index] <;> congr; on_goal 1 => ext a₁ b₁
+      rw [sum_sum_index, sum_sum_index] <;> congr; on_goal 1 => ext a₂ b₂
+      rw [sum_sum_index, sum_single_index] <;> congr; on_goal 1 => ext a₃ b₃
+      on_goal 1 => rw [sum_single_index, mul_assoc, mul_assoc]
+      all_goals simp only [single_zero, single_add, forall_true_iff, add_mul,
+        mul_add, zero_mul, mul_zero, sum_zero, sum_add] }
+
+end Semigroup
+
+section One
+
+variable [NonAssocSemiring R] [Semiring k] [One G]
+
+/-- The unit of the multiplication is `single 1 1`, i.e. the function
+  that is `1` at `1` and zero elsewhere. -/
+instance one : One (MonoidAlgebra k G) :=
+  ⟨single 1 1⟩
+
+theorem one_def : (1 : MonoidAlgebra k G) = single 1 1 :=
+  rfl
+
+@[simp]
+theorem liftNC_one {g_hom : Type*} [FunLike g_hom G R] [OneHomClass g_hom G R]
+    (f : k →+* R) (g : g_hom) :
+    liftNC (f : k →+ R) g 1 = 1 := by simp [one_def]
+
+end One
+
+section MulOneClass
+
+variable [Semiring k] [MulOneClass G]
+
+instance nonAssocSemiring : NonAssocSemiring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalNonAssocSemiring with
+    natCast := fun n => single 1 n
+    natCast_zero := by simp
+    natCast_succ := fun _ => by simp; rfl
+    one_mul := fun f => by
+      simp only [mul_def, one_def, sum_single_index, zero_mul, single_zero, sum_zero, zero_add,
+        one_mul, sum_single]
+    mul_one := fun f => by
+      simp only [mul_def, one_def, sum_single_index, mul_zero, single_zero, sum_zero, add_zero,
+        mul_one, sum_single] }
+
+theorem natCast_def (n : ℕ) : (n : MonoidAlgebra k G) = single (1 : G) (n : k) :=
+  rfl
+
+@[deprecated (since := "2024-04-17")]
+alias nat_cast_def := natCast_def
+
+end MulOneClass
+
+/-! #### Semiring structure -/
+
+
+section Semiring
+
+variable [Semiring k] [Monoid G]
+
+instance semiring : Semiring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalSemiring,
+    MonoidAlgebra.nonAssocSemiring with }
+
+variable [Semiring R]
+
+/-- `liftNC` as a `RingHom`, for when `f x` and `g y` commute -/
+def liftNCRingHom (f : k →+* R) (g : G →* R) (h_comm : ∀ x y, Commute (f x) (g y)) :
+    MonoidAlgebra k G →+* R :=
+  { liftNC (f : k →+ R) g with
+    map_one' := liftNC_one _ _
+    map_mul' := fun _a _b => liftNC_mul _ _ _ _ fun {_ _} _ => h_comm _ _ }
+
+end Semiring
+
+instance nonUnitalCommSemiring [CommSemiring k] [CommSemigroup G] :
+    NonUnitalCommSemiring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalSemiring with
+    mul_comm := fun f g => by
+      simp only [mul_def, Finsupp.sum, mul_comm]
+      rw [Finset.sum_comm]
+      simp only [mul_comm] }
+
+instance nontrivial [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial (MonoidAlgebra k G) :=
+  Finsupp.instNontrivial
+
+/-! #### Derived instances -/
+
+
+section DerivedInstances
+
+instance commSemiring [CommSemiring k] [CommMonoid G] : CommSemiring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalCommSemiring, MonoidAlgebra.semiring with }
+
+instance unique [Semiring k] [Subsingleton k] : Unique (MonoidAlgebra k G) :=
+  Finsupp.uniqueOfRight
+
+instance addCommGroup [Ring k] : AddCommGroup (MonoidAlgebra k G) :=
+  Finsupp.instAddCommGroup
+
+instance nonUnitalNonAssocRing [Ring k] [Mul G] : NonUnitalNonAssocRing (MonoidAlgebra k G) :=
+  { MonoidAlgebra.addCommGroup, MonoidAlgebra.nonUnitalNonAssocSemiring with }
+
+instance nonUnitalRing [Ring k] [Semigroup G] : NonUnitalRing (MonoidAlgebra k G) :=
+  { MonoidAlgebra.addCommGroup, MonoidAlgebra.nonUnitalSemiring with }
+
+instance nonAssocRing [Ring k] [MulOneClass G] : NonAssocRing (MonoidAlgebra k G) :=
+  { MonoidAlgebra.addCommGroup,
+    MonoidAlgebra.nonAssocSemiring with
+    intCast := fun z => single 1 (z : k)
+    -- Porting note: Both were `simpa`.
+    intCast_ofNat := fun n => by simp; rfl
+    intCast_negSucc := fun n => by simp; rfl }
+
+theorem intCast_def [Ring k] [MulOneClass G] (z : ℤ) :
+    (z : MonoidAlgebra k G) = single (1 : G) (z : k) :=
+  rfl
+
+@[deprecated (since := "2024-04-17")]
+alias int_cast_def := intCast_def
+
+instance ring [Ring k] [Monoid G] : Ring (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonAssocRing, MonoidAlgebra.semiring with }
+
+instance nonUnitalCommRing [CommRing k] [CommSemigroup G] :
+    NonUnitalCommRing (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalCommSemiring, MonoidAlgebra.nonUnitalRing with }
+
+instance commRing [CommRing k] [CommMonoid G] : CommRing (MonoidAlgebra k G) :=
+  { MonoidAlgebra.nonUnitalCommRing, MonoidAlgebra.ring with }
+
+variable {S : Type*}
+
+instance smulZeroClass [Semiring k] [SMulZeroClass R k] : SMulZeroClass R (MonoidAlgebra k G) :=
+  Finsupp.smulZeroClass
+
+instance distribSMul [Semiring k] [DistribSMul R k] : DistribSMul R (MonoidAlgebra k G) :=
+  Finsupp.distribSMul _ _
+
+instance distribMulAction [Monoid R] [Semiring k] [DistribMulAction R k] :
+    DistribMulAction R (MonoidAlgebra k G) :=
+  Finsupp.distribMulAction G k
+
+instance module [Semiring R] [Semiring k] [Module R k] : Module R (MonoidAlgebra k G) :=
+  Finsupp.module G k
+
+instance faithfulSMul [Semiring k] [SMulZeroClass R k] [FaithfulSMul R k] [Nonempty G] :
+    FaithfulSMul R (MonoidAlgebra k G) :=
+  Finsupp.faithfulSMul
+
+instance isScalarTower [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S]
+    [IsScalarTower R S k] : IsScalarTower R S (MonoidAlgebra k G) :=
+  Finsupp.isScalarTower G k
+
+instance smulCommClass [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMulCommClass R S k] :
+    SMulCommClass R S (MonoidAlgebra k G) :=
+  Finsupp.smulCommClass G k
+
+instance isCentralScalar [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k]
+    [IsCentralScalar R k] : IsCentralScalar R (MonoidAlgebra k G) :=
+  Finsupp.isCentralScalar G k
+
+/-- This is not an instance as it conflicts with `MonoidAlgebra.distribMulAction` when `G = kˣ`.
+-/
+def comapDistribMulActionSelf [Group G] [Semiring k] : DistribMulAction G (MonoidAlgebra k G) :=
+  Finsupp.comapDistribMulAction
+
+end DerivedInstances
+
+section MiscTheorems
+
+variable [Semiring k]
+
+-- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
+
+theorem mul_apply [DecidableEq G] [Mul G] (f g : MonoidAlgebra k G) (x : G) :
+    (f * g) x = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => if a₁ * a₂ = x then b₁ * b₂ else 0 := by
+  -- Porting note: `reducible` cannot be `local` so proof gets long.
+  rw [mul_def, Finsupp.sum_apply]; congr; ext
+  rw [Finsupp.sum_apply]; congr; ext
+  apply single_apply
+
+theorem mul_apply_antidiagonal [Mul G] (f g : MonoidAlgebra k G) (x : G) (s : Finset (G × G))
+    (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2 := by
+  classical exact
+      let F : G × G → k := fun p => if p.1 * p.2 = x then f p.1 * g p.2 else 0
+      calc
+        (f * g) x = ∑ a₁ ∈ f.support, ∑ a₂ ∈ g.support, F (a₁, a₂) := mul_apply f g x
+        _ = ∑ p ∈ f.support ×ˢ g.support, F p := by rw [Finset.sum_product]
+        _ = ∑ p ∈ (f.support ×ˢ g.support).filter fun p : G × G => p.1 * p.2 = x, f p.1 * g p.2 :=
+          (Finset.sum_filter _ _).symm
+        _ = ∑ p ∈ s.filter fun p : G × G => p.1 ∈ f.support ∧ p.2 ∈ g.support, f p.1 * g p.2 :=
+          (sum_congr
+            (by
+              ext
+              simp only [mem_filter, mem_product, hs, and_comm])
+            fun _ _ => rfl)
+        _ = ∑ p ∈ s, f p.1 * g p.2 :=
+          sum_subset (filter_subset _ _) fun p hps hp => by
+            simp only [mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢
+            by_cases h1 : f p.1 = 0
+            · rw [h1, zero_mul]
+            · rw [hp hps h1, mul_zero]
+
+@[simp]
+theorem single_mul_single [Mul G] {a₁ a₂ : G} {b₁ b₂ : k} :
+    single a₁ b₁ * single a₂ b₂ = single (a₁ * a₂) (b₁ * b₂) := by
+  rw [mul_def]
+  exact (sum_single_index (by simp only [zero_mul, single_zero, sum_zero])).trans
+    (sum_single_index (by rw [mul_zero, single_zero]))
+
+theorem single_commute_single [Mul G] {a₁ a₂ : G} {b₁ b₂ : k}
+    (ha : Commute a₁ a₂) (hb : Commute b₁ b₂) :
+    Commute (single a₁ b₁) (single a₂ b₂) :=
+  single_mul_single.trans <| congr_arg₂ single ha hb |>.trans single_mul_single.symm
+
+theorem single_commute [Mul G] {a : G} {b : k} (ha : ∀ a', Commute a a') (hb : ∀ b', Commute b b') :
+    ∀ f : MonoidAlgebra k G, Commute (single a b) f :=
+  suffices AddMonoidHom.mulLeft (single a b) = AddMonoidHom.mulRight (single a b) from
+    DFunLike.congr_fun this
+  addHom_ext' fun a' => AddMonoidHom.ext fun b' => single_commute_single (ha a') (hb b')
+
+@[simp]
+theorem single_pow [Monoid G] {a : G} {b : k} : ∀ n : ℕ, single a b ^ n = single (a ^ n) (b ^ n)
+  | 0 => by
+    simp only [pow_zero]
+    rfl
+  | n + 1 => by simp only [pow_succ, single_pow n, single_mul_single]
+
+section
+
+/-- Like `Finsupp.mapDomain_zero`, but for the `1` we define in this file -/
+@[simp]
+theorem mapDomain_one {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [One α] [One α₂]
+    {F : Type*} [FunLike F α α₂] [OneHomClass F α α₂] (f : F) :
+    (mapDomain f (1 : MonoidAlgebra β α) : MonoidAlgebra β α₂) = (1 : MonoidAlgebra β α₂) := by
+  simp_rw [one_def, mapDomain_single, map_one]
+
+/-- Like `Finsupp.mapDomain_add`, but for the convolutive multiplication we define in this file -/
+theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Mul α] [Mul α₂]
+    {F : Type*} [FunLike F α α₂] [MulHomClass F α α₂] (f : F) (x y : MonoidAlgebra β α) :
+    mapDomain f (x * y) = mapDomain f x * mapDomain f y := by
+  simp_rw [mul_def, mapDomain_sum, mapDomain_single, map_mul]
+  rw [Finsupp.sum_mapDomain_index]
+  · congr
+    ext a b
+    rw [Finsupp.sum_mapDomain_index]
+    · simp
+    · simp [mul_add]
+  · simp
+  · simp [add_mul]
+
+variable (k G)
+
+/-- The embedding of a magma into its magma algebra. -/
+@[simps]
+def ofMagma [Mul G] : G →ₙ* MonoidAlgebra k G where
+  toFun a := single a 1
+  map_mul' a b := by simp only [mul_def, mul_one, sum_single_index, single_eq_zero, mul_zero]
+
+/-- The embedding of a unital magma into its magma algebra. -/
+@[simps]
+def of [MulOneClass G] : G →* MonoidAlgebra k G :=
+  { ofMagma k G with
+    toFun := fun a => single a 1
+    map_one' := rfl }
+
+end
+
+theorem smul_of [MulOneClass G] (g : G) (r : k) : r • of k G g = single g r := by
+  -- porting note (#10745): was `simp`.
+  rw [of_apply, smul_single', mul_one]
+
+theorem of_injective [MulOneClass G] [Nontrivial k] :
+    Function.Injective (of k G) := fun a b h => by
+  simpa using (single_eq_single_iff _ _ _ _).mp h
+
+theorem of_commute [MulOneClass G] {a : G} (h : ∀ a', Commute a a') (f : MonoidAlgebra k G) :
+    Commute (of k G a) f :=
+  single_commute h Commute.one_left f
+
+/-- `Finsupp.single` as a `MonoidHom` from the product type into the monoid algebra.
+
+Note the order of the elements of the product are reversed compared to the arguments of
+`Finsupp.single`.
+-/
+@[simps]
+def singleHom [MulOneClass G] : k × G →* MonoidAlgebra k G where
+  toFun a := single a.2 a.1
+  map_one' := rfl
+  map_mul' _a _b := single_mul_single.symm
+
+theorem mul_single_apply_aux [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G}
+    (H : ∀ a, a * x = z ↔ a = y) : (f * single x r) z = f y * r := by
+  classical exact
+      have A :
+        ∀ a₁ b₁,
+          ((single x r).sum fun a₂ b₂ => ite (a₁ * a₂ = z) (b₁ * b₂) 0) =
+            ite (a₁ * x = z) (b₁ * r) 0 :=
+        fun a₁ b₁ => sum_single_index <| by simp
+      calc
+        (HMul.hMul (β := MonoidAlgebra k G) f (single x r)) z =
+            sum f fun a b => if a = y then b * r else 0 := by simp only [mul_apply, A, H]
+        _ = if y ∈ f.support then f y * r else 0 := f.support.sum_ite_eq' _ _
+        _ = f y * r := by split_ifs with h <;> simp at h <;> simp [h]
+
+theorem mul_single_one_apply [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) :
+    (HMul.hMul (β := MonoidAlgebra k G) f (single 1 r)) x = f x * r :=
+  f.mul_single_apply_aux fun a => by rw [mul_one]
+
+theorem mul_single_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
+    (h : ¬∃ d, g' = d * g) : (x * single g r) g' = 0 := by
+  classical
+    rw [mul_apply, Finsupp.sum_comm, Finsupp.sum_single_index]
+    swap
+    · simp_rw [Finsupp.sum, mul_zero, ite_self, Finset.sum_const_zero]
+    · apply Finset.sum_eq_zero
+      simp_rw [ite_eq_right_iff]
+      rintro g'' _hg'' rfl
+      exfalso
+      exact h ⟨_, rfl⟩
+
+theorem single_mul_apply_aux [Mul G] (f : MonoidAlgebra k G) {r : k} {x y z : G}
+    (H : ∀ a, x * a = y ↔ a = z) : (single x r * f) y = r * f z := by
+  classical exact
+      have : (f.sum fun a b => ite (x * a = y) (0 * b) 0) = 0 := by simp
+      calc
+        (HMul.hMul (α := MonoidAlgebra k G) (single x r) f) y =
+            sum f fun a b => ite (x * a = y) (r * b) 0 :=
+          (mul_apply _ _ _).trans <| sum_single_index this
+        _ = f.sum fun a b => ite (a = z) (r * b) 0 := by simp only [H]
+        _ = if z ∈ f.support then r * f z else 0 := f.support.sum_ite_eq' _ _
+        _ = _ := by split_ifs with h <;> simp at h <;> simp [h]
+
+theorem single_one_mul_apply [MulOneClass G] (f : MonoidAlgebra k G) (r : k) (x : G) :
+    (single (1 : G) r * f) x = r * f x :=
+  f.single_mul_apply_aux fun a => by rw [one_mul]
+
+theorem single_mul_apply_of_not_exists_mul [Mul G] (r : k) {g g' : G} (x : MonoidAlgebra k G)
+    (h : ¬∃ d, g' = g * d) : (single g r * x) g' = 0 := by
+  classical
+    rw [mul_apply, Finsupp.sum_single_index]
+    swap
+    · simp_rw [Finsupp.sum, zero_mul, ite_self, Finset.sum_const_zero]
+    · apply Finset.sum_eq_zero
+      simp_rw [ite_eq_right_iff]
+      rintro g'' _hg'' rfl
+      exfalso
+      exact h ⟨_, rfl⟩
+
+theorem liftNC_smul [MulOneClass G] {R : Type*} [Semiring R] (f : k →+* R) (g : G →* R) (c : k)
+    (φ : MonoidAlgebra k G) : liftNC (f : k →+ R) g (c • φ) = f c * liftNC (f : k →+ R) g φ := by
+  suffices (liftNC (↑f) g).comp (smulAddHom k (MonoidAlgebra k G) c) =
+      (AddMonoidHom.mulLeft (f c)).comp (liftNC (↑f) g) from
+    DFunLike.congr_fun this φ
+  -- Porting note: `ext` couldn't a find appropriate theorem.
+  refine addHom_ext' fun a => AddMonoidHom.ext fun b => ?_
+  -- Porting note: `reducible` cannot be `local` so the proof gets more complex.
+  unfold MonoidAlgebra
+  simp only [AddMonoidHom.coe_comp, Function.comp_apply, singleAddHom_apply, smulAddHom_apply,
+    smul_single, smul_eq_mul, AddMonoidHom.coe_mulLeft]
+  -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+  erw [liftNC_single, liftNC_single]; rw [AddMonoidHom.coe_coe, map_mul, mul_assoc]
+
+end MiscTheorems
+
+/-! #### Non-unital, non-associative algebra structure -/
+
+
+section NonUnitalNonAssocAlgebra
+
+variable (k) [Semiring k] [DistribSMul R k] [Mul G]
+
+instance isScalarTower_self [IsScalarTower R k k] :
+    IsScalarTower R (MonoidAlgebra k G) (MonoidAlgebra k G) :=
+  ⟨fun t a b => by
+    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
+    refine Finsupp.ext fun m => ?_
+    -- Porting note: `refine` & `rw` are required because `simp` behaves differently.
+    classical
+      simp only [smul_eq_mul, mul_apply]
+      rw [coe_smul]
+      refine Eq.trans (sum_smul_index' (g := a) (b := t) ?_) ?_ <;>
+        simp only [mul_apply, Finsupp.smul_sum, smul_ite, smul_mul_assoc,
+          zero_mul, ite_self, imp_true_iff, sum_zero, Pi.smul_apply, smul_zero]⟩
+
+/-- Note that if `k` is a `CommSemiring` then we have `SMulCommClass k k k` and so we can take
+`R = k` in the below. In other words, if the coefficients are commutative amongst themselves, they
+also commute with the algebra multiplication. -/
+instance smulCommClass_self [SMulCommClass R k k] :
+    SMulCommClass R (MonoidAlgebra k G) (MonoidAlgebra k G) :=
+  ⟨fun t a b => by
+    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
+    refine Finsupp.ext fun m => ?_
+    -- Porting note: `refine` & `rw` are required because `simp` behaves differently.
+    classical
+      simp only [smul_eq_mul, mul_apply]
+      rw [coe_smul]
+      refine Eq.symm (Eq.trans (congr_arg (sum a)
+        (funext₂ fun a₁ b₁ => sum_smul_index' (g := b) (b := t) ?_)) ?_) <;>
+      simp only [mul_apply, Finsupp.sum, Finset.smul_sum, smul_ite, mul_smul_comm,
+        imp_true_iff, ite_eq_right_iff, Pi.smul_apply, mul_zero, smul_zero]⟩
+
+instance smulCommClass_symm_self [SMulCommClass k R k] :
+    SMulCommClass (MonoidAlgebra k G) R (MonoidAlgebra k G) :=
+  ⟨fun t a b => by
+    haveI := SMulCommClass.symm k R k
+    rw [← smul_comm]⟩
+
+end NonUnitalNonAssocAlgebra
+
+theorem single_one_comm [CommSemiring k] [MulOneClass G] (r : k) (f : MonoidAlgebra k G) :
+    single (1 : G) r * f = f * single (1 : G) r :=
+  single_commute Commute.one_left (Commute.all _) f
+
+/-- `Finsupp.single 1` as a `RingHom` -/
+@[simps]
+def singleOneRingHom [Semiring k] [MulOneClass G] : k →+* MonoidAlgebra k G :=
+  { Finsupp.singleAddHom 1 with
+    map_one' := rfl
+    map_mul' := fun x y => by
+      -- Porting note (#10691): Was `rw`.
+      simp only [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe, singleAddHom_apply,
+        single_mul_single, mul_one] }
+
+/-- If `f : G → H` is a multiplicative homomorphism between two monoids, then
+`Finsupp.mapDomain f` is a ring homomorphism between their monoid algebras. -/
+@[simps]
+def mapDomainRingHom (k : Type*) {H F : Type*} [Semiring k] [Monoid G] [Monoid H]
+    [FunLike F G H] [MonoidHomClass F G H] (f : F) : MonoidAlgebra k G →+* MonoidAlgebra k H :=
+  { (Finsupp.mapDomain.addMonoidHom f : MonoidAlgebra k G →+ MonoidAlgebra k H) with
+    map_one' := mapDomain_one f
+    map_mul' := fun x y => mapDomain_mul f x y }
+
+/-- If two ring homomorphisms from `MonoidAlgebra k G` are equal on all `single a 1`
+and `single 1 b`, then they are equal. -/
+theorem ringHom_ext {R} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R}
+    (h₁ : ∀ b, f (single 1 b) = g (single 1 b)) (h_of : ∀ a, f (single a 1) = g (single a 1)) :
+    f = g :=
+  RingHom.coe_addMonoidHom_injective <|
+    addHom_ext fun a b => by
+      rw [← single, ← one_mul a, ← mul_one b, ← single_mul_single]
+      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+      erw [AddMonoidHom.coe_coe f, AddMonoidHom.coe_coe g]; rw [f.map_mul, g.map_mul, h₁, h_of]
+
+/-- If two ring homomorphisms from `MonoidAlgebra k G` are equal on all `single a 1`
+and `single 1 b`, then they are equal.
+
+See note [partially-applied ext lemmas]. -/
+@[ext high]
+theorem ringHom_ext' {R} [Semiring k] [MulOneClass G] [Semiring R] {f g : MonoidAlgebra k G →+* R}
+    (h₁ : f.comp singleOneRingHom = g.comp singleOneRingHom)
+    (h_of :
+      (f : MonoidAlgebra k G →* R).comp (of k G) = (g : MonoidAlgebra k G →* R).comp (of k G)) :
+    f = g :=
+  ringHom_ext (RingHom.congr_fun h₁) (DFunLike.congr_fun h_of)
+
+theorem induction_on [Semiring k] [Monoid G] {p : MonoidAlgebra k G → Prop} (f : MonoidAlgebra k G)
+    (hM : ∀ g, p (of k G g)) (hadd : ∀ f g : MonoidAlgebra k G, p f → p g → p (f + g))
+    (hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by
+  refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
+  · simpa using hsmul 0 (of k G 1) (hM 1)
+  · convert hsmul r (of k G g) (hM g)
+    -- Porting note: Was `simp only`.
+    rw [of_apply, smul_single', mul_one]
+
+section
+
+universe ui
+
+variable {ι : Type ui}
+
+-- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
+
+theorem prod_single [CommSemiring k] [CommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} :
+    (∏ i ∈ s, single (a i) (b i)) = single (∏ i ∈ s, a i) (∏ i ∈ s, b i) :=
+  Finset.cons_induction_on s rfl fun a s has ih => by
+    rw [prod_cons has, ih, single_mul_single, prod_cons has, prod_cons has]
+
+end
+
+section
+
+-- We now prove some additional statements that hold for group algebras.
+variable [Semiring k] [Group G]
+
+-- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
+
+@[simp]
+theorem mul_single_apply (f : MonoidAlgebra k G) (r : k) (x y : G) :
+    (f * single x r) y = f (y * x⁻¹) * r :=
+  f.mul_single_apply_aux fun _a => eq_mul_inv_iff_mul_eq.symm
+
+@[simp]
+theorem single_mul_apply (r : k) (x : G) (f : MonoidAlgebra k G) (y : G) :
+    (single x r * f) y = r * f (x⁻¹ * y) :=
+  f.single_mul_apply_aux fun _z => eq_inv_mul_iff_mul_eq.symm
+
+theorem mul_apply_left (f g : MonoidAlgebra k G) (x : G) :
+    (f * g) x = f.sum fun a b => b * g (a⁻¹ * x) :=
+  calc
+    (f * g) x = sum f fun a b => (single a b * g) x := by
+      rw [← Finsupp.sum_apply, ← Finsupp.sum_mul g f, f.sum_single]
+    _ = _ := by simp only [single_mul_apply, Finsupp.sum]
+
+-- If we'd assumed `CommSemiring`, we could deduce this from `mul_apply_left`.
+theorem mul_apply_right (f g : MonoidAlgebra k G) (x : G) :
+    (f * g) x = g.sum fun a b => f (x * a⁻¹) * b :=
+  calc
+    (f * g) x = sum g fun a b => (f * single a b) x := by
+      rw [← Finsupp.sum_apply, ← Finsupp.mul_sum f g, g.sum_single]
+    _ = _ := by simp only [mul_single_apply, Finsupp.sum]
+
+end
+
+section Opposite
+
+open Finsupp MulOpposite
+
+variable [Semiring k]
+
+/-- The opposite of a `MonoidAlgebra R I` equivalent as a ring to
+the `MonoidAlgebra Rᵐᵒᵖ Iᵐᵒᵖ` over the opposite ring, taking elements to their opposite. -/
+@[simps! (config := { simpRhs := true }) apply symm_apply]
+protected noncomputable def opRingEquiv [Monoid G] :
+    (MonoidAlgebra k G)ᵐᵒᵖ ≃+* MonoidAlgebra kᵐᵒᵖ Gᵐᵒᵖ :=
+  { opAddEquiv.symm.trans <|
+      (Finsupp.mapRange.addEquiv (opAddEquiv : k ≃+ kᵐᵒᵖ)).trans <| Finsupp.domCongr opEquiv with
+    map_mul' := by
+      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+      rw [Equiv.toFun_as_coe, AddEquiv.toEquiv_eq_coe]; erw [AddEquiv.coe_toEquiv]
+      rw [← AddEquiv.coe_toAddMonoidHom]
+      refine Iff.mpr (AddMonoidHom.map_mul_iff (R := (MonoidAlgebra k G)ᵐᵒᵖ)
+        (S := MonoidAlgebra kᵐᵒᵖ Gᵐᵒᵖ) _) ?_
+      -- Porting note: Was `ext`.
+      refine AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₁ => AddMonoidHom.ext fun r₁ =>
+        AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₂ => AddMonoidHom.ext fun r₂ => ?_
+      -- Porting note: `reducible` cannot be `local` so proof gets long.
+      simp only [AddMonoidHom.coe_comp, AddEquiv.coe_toAddMonoidHom, opAddEquiv_apply,
+        Function.comp_apply, singleAddHom_apply, AddMonoidHom.compr₂_apply, AddMonoidHom.coe_mul,
+        AddMonoidHom.coe_mulLeft, AddMonoidHom.compl₂_apply]
+      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+      erw [AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply,
+        AddEquiv.trans_apply, AddEquiv.trans_apply, MulOpposite.opAddEquiv_symm_apply]
+      rw [MulOpposite.unop_mul (α := MonoidAlgebra k G)]
+      -- This was not needed before leanprover/lean4#2644
+      erw [unop_op, unop_op, single_mul_single]
+      simp }
+
+-- @[simp] -- Porting note (#10618): simp can prove this
+theorem opRingEquiv_single [Monoid G] (r : k) (x : G) :
+    MonoidAlgebra.opRingEquiv (op (single x r)) = single (op x) (op r) := by simp
+
+-- @[simp] -- Porting note (#10618): simp can prove this
+theorem opRingEquiv_symm_single [Monoid G] (r : kᵐᵒᵖ) (x : Gᵐᵒᵖ) :
+    MonoidAlgebra.opRingEquiv.symm (single x r) = op (single x.unop r.unop) := by simp
+
+end Opposite
+
+section Submodule
+
+variable [CommSemiring k] [Monoid G]
+variable {V : Type*} [AddCommMonoid V]
+variable [Module k V] [Module (MonoidAlgebra k G) V] [IsScalarTower k (MonoidAlgebra k G) V]
+
+/-- A submodule over `k` which is stable under scalar multiplication by elements of `G` is a
+submodule over `MonoidAlgebra k G`  -/
+def submoduleOfSMulMem (W : Submodule k V) (h : ∀ (g : G) (v : V), v ∈ W → of k G g • v ∈ W) :
+    Submodule (MonoidAlgebra k G) V where
+  carrier := W
+  zero_mem' := W.zero_mem'
+  add_mem' := W.add_mem'
+  smul_mem' := by
+    intro f v hv
+    rw [← Finsupp.sum_single f, Finsupp.sum, Finset.sum_smul]
+    simp_rw [← smul_of, smul_assoc]
+    exact Submodule.sum_smul_mem W _ fun g _ => h g v hv
+
+end Submodule
+
+end MonoidAlgebra
+
+/-! ### Additive monoids -/
+
+
+section
+
+variable [Semiring k]
+
+/-- The monoid algebra over a semiring `k` generated by the additive monoid `G`.
+It is the type of finite formal `k`-linear combinations of terms of `G`,
+endowed with the convolution product.
+-/
+def AddMonoidAlgebra :=
+  G →₀ k
+
+@[inherit_doc]
+scoped[AddMonoidAlgebra] notation:9000 R:max "[" A "]" => AddMonoidAlgebra R A
+
+namespace AddMonoidAlgebra
+
+-- Porting note: The compiler couldn't derive this.
+instance inhabited : Inhabited k[G] :=
+  inferInstanceAs (Inhabited (G →₀ k))
+
+-- Porting note: The compiler couldn't derive this.
+instance addCommMonoid : AddCommMonoid k[G] :=
+  inferInstanceAs (AddCommMonoid (G →₀ k))
+
+instance instIsCancelAdd [IsCancelAdd k] : IsCancelAdd (AddMonoidAlgebra k G) :=
+  inferInstanceAs (IsCancelAdd (G →₀ k))
+
+instance coeFun : CoeFun k[G] fun _ => G → k :=
+  Finsupp.instCoeFun
+
+end AddMonoidAlgebra
+
+end
+
+namespace AddMonoidAlgebra
+
+variable {k G}
+
+section
+
+variable [Semiring k] [NonUnitalNonAssocSemiring R]
+
+-- Porting note: `reducible` cannot be `local`, so we replace some definitions and theorems with
+--               new ones which have new types.
+
+abbrev single (a : G) (b : k) : k[G] := Finsupp.single a b
+
+theorem single_zero (a : G) : (single a 0 : k[G]) = 0 := Finsupp.single_zero a
+
+theorem single_add (a : G) (b₁ b₂ : k) : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
+  Finsupp.single_add a b₁ b₂
+
+@[simp]
+theorem sum_single_index {N} [AddCommMonoid N] {a : G} {b : k} {h : G → k → N}
+    (h_zero : h a 0 = 0) :
+    (single a b).sum h = h a b := Finsupp.sum_single_index h_zero
+
+@[simp]
+theorem sum_single (f : k[G]) : f.sum single = f :=
+  Finsupp.sum_single f
+
+theorem single_apply {a a' : G} {b : k} [Decidable (a = a')] :
+    single a b a' = if a = a' then b else 0 :=
+  Finsupp.single_apply
+
+@[simp]
+theorem single_eq_zero {a : G} {b : k} : single a b = 0 ↔ b = 0 := Finsupp.single_eq_zero
+
+abbrev mapDomain {G' : Type*} (f : G → G') (v : k[G]) : k[G'] :=
+  Finsupp.mapDomain f v
+
+theorem mapDomain_sum {k' G' : Type*} [Semiring k'] {f : G → G'} {s : AddMonoidAlgebra k' G}
+    {v : G → k' → k[G]} :
+    mapDomain f (s.sum v) = s.sum fun a b => mapDomain f (v a b) :=
+  Finsupp.mapDomain_sum
+
+theorem mapDomain_single {G' : Type*} {f : G → G'} {a : G} {b : k} :
+    mapDomain f (single a b) = single (f a) b :=
+  Finsupp.mapDomain_single
+
+/-- A non-commutative version of `AddMonoidAlgebra.lift`: given an additive homomorphism
+`f : k →+ R` and a map `g : Multiplicative G → R`, returns the additive
+homomorphism from `k[G]` such that `liftNC f g (single a b) = f b * g a`. If `f`
+is a ring homomorphism and the range of either `f` or `g` is in center of `R`, then the result is a
+ring homomorphism.  If `R` is a `k`-algebra and `f = algebraMap k R`, then the result is an algebra
+homomorphism called `AddMonoidAlgebra.lift`. -/
+def liftNC (f : k →+ R) (g : Multiplicative G → R) : k[G] →+ R :=
+  liftAddHom fun x : G => (AddMonoidHom.mulRight (g <| Multiplicative.ofAdd x)).comp f
+
+@[simp]
+theorem liftNC_single (f : k →+ R) (g : Multiplicative G → R) (a : G) (b : k) :
+    liftNC f g (single a b) = f b * g (Multiplicative.ofAdd a) :=
+  liftAddHom_apply_single _ _ _
+
+end
+
+section Mul
+
+variable [Semiring k] [Add G]
+
+/-- The product of `f g : k[G]` is the finitely supported function
+  whose value at `a` is the sum of `f x * g y` over all pairs `x, y`
+  such that `x + y = a`. (Think of the product of multivariate
+  polynomials where `α` is the additive monoid of monomial exponents.) -/
+instance hasMul : Mul k[G] :=
+  ⟨fun f g => MonoidAlgebra.mul' (G := Multiplicative G) f g⟩
+
+theorem mul_def {f g : k[G]} :
+    f * g = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => single (a₁ + a₂) (b₁ * b₂) :=
+  MonoidAlgebra.mul_def (G := Multiplicative G)
+
+instance nonUnitalNonAssocSemiring : NonUnitalNonAssocSemiring k[G] :=
+  { Finsupp.instAddCommMonoid with
+    -- Porting note: `refine` & `exact` are required because `simp` behaves differently.
+    left_distrib := fun f g h => by
+      haveI := Classical.decEq G
+      simp only [mul_def]
+      refine Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_add_index ?_ ?_)) ?_ <;>
+        simp only [mul_add, mul_zero, single_zero, single_add, forall_true_iff, sum_add]
+    right_distrib := fun f g h => by
+      haveI := Classical.decEq G
+      simp only [mul_def]
+      refine Eq.trans (sum_add_index ?_ ?_) ?_ <;>
+        simp only [add_mul, zero_mul, single_zero, single_add, forall_true_iff, sum_zero, sum_add]
+    zero_mul := fun f => by
+      simp only [mul_def]
+      exact sum_zero_index
+    mul_zero := fun f => by
+      simp only [mul_def]
+      exact Eq.trans (congr_arg (sum f) (funext₂ fun a₁ b₁ => sum_zero_index)) sum_zero
+    nsmul := fun n f => n • f
+    -- Porting note: `ext` → `refine Finsupp.ext fun _ => ?_`
+    nsmul_zero := by
+      intros
+      refine Finsupp.ext fun _ => ?_
+      simp [-nsmul_eq_mul, add_smul]
+    nsmul_succ := by
+      intros
+      refine Finsupp.ext fun _ => ?_
+      simp [-nsmul_eq_mul, add_smul] }
+
+variable [Semiring R]
+
+theorem liftNC_mul {g_hom : Type*}
+    [FunLike g_hom (Multiplicative G) R] [MulHomClass g_hom (Multiplicative G) R]
+    (f : k →+* R) (g : g_hom) (a b : k[G])
+    (h_comm : ∀ {x y}, y ∈ a.support → Commute (f (b x)) (g <| Multiplicative.ofAdd y)) :
+    liftNC (f : k →+ R) g (a * b) = liftNC (f : k →+ R) g a * liftNC (f : k →+ R) g b :=
+  (MonoidAlgebra.liftNC_mul f g _ _ @h_comm : _)
+
+end Mul
+
+section One
+
+variable [Semiring k] [Zero G] [NonAssocSemiring R]
+
+/-- The unit of the multiplication is `single 0 1`, i.e. the function
+  that is `1` at `0` and zero elsewhere. -/
+instance one : One k[G] :=
+  ⟨single 0 1⟩
+
+theorem one_def : (1 : k[G]) = single 0 1 :=
+  rfl
+
+@[simp]
+theorem liftNC_one {g_hom : Type*}
+    [FunLike g_hom (Multiplicative G) R] [OneHomClass g_hom (Multiplicative G) R]
+    (f : k →+* R) (g : g_hom) : liftNC (f : k →+ R) g 1 = 1 :=
+  (MonoidAlgebra.liftNC_one f g : _)
+
+end One
+
+section Semigroup
+
+variable [Semiring k] [AddSemigroup G]
+
+instance nonUnitalSemiring : NonUnitalSemiring k[G] :=
+  { AddMonoidAlgebra.nonUnitalNonAssocSemiring with
+    mul_assoc := fun f g h => by
+      -- Porting note: `reducible` cannot be `local` so proof gets long.
+      simp only [mul_def]
+      rw [sum_sum_index] <;> congr; on_goal 1 => ext a₁ b₁
+      rw [sum_sum_index, sum_sum_index] <;> congr; on_goal 1 => ext a₂ b₂
+      rw [sum_sum_index, sum_single_index] <;> congr; on_goal 1 => ext a₃ b₃
+      on_goal 1 => rw [sum_single_index, mul_assoc, add_assoc]
+      all_goals simp only [single_zero, single_add, forall_true_iff, add_mul,
+        mul_add, zero_mul, mul_zero, sum_zero, sum_add] }
+
+end Semigroup
+
+section MulOneClass
+
+variable [Semiring k] [AddZeroClass G]
+
+instance nonAssocSemiring : NonAssocSemiring k[G] :=
+  { AddMonoidAlgebra.nonUnitalNonAssocSemiring with
+    natCast := fun n => single 0 n
+    natCast_zero := by simp
+    natCast_succ := fun _ => by simp; rfl
+    one_mul := fun f => by
+      simp only [mul_def, one_def, sum_single_index, zero_mul, single_zero, sum_zero, zero_add,
+        one_mul, sum_single]
+    mul_one := fun f => by
+      simp only [mul_def, one_def, sum_single_index, mul_zero, single_zero, sum_zero, add_zero,
+        mul_one, sum_single] }
+
+theorem natCast_def (n : ℕ) : (n : k[G]) = single (0 : G) (n : k) :=
+  rfl
+
+@[deprecated (since := "2024-04-17")]
+alias nat_cast_def := natCast_def
+
+end MulOneClass
+
+/-! #### Semiring structure -/
+
+
+section Semiring
+
+instance smulZeroClass [Semiring k] [SMulZeroClass R k] : SMulZeroClass R k[G] :=
+  Finsupp.smulZeroClass
+
+variable [Semiring k] [AddMonoid G]
+
+instance semiring : Semiring k[G] :=
+  { AddMonoidAlgebra.nonUnitalSemiring,
+    AddMonoidAlgebra.nonAssocSemiring with }
+
+variable [Semiring R]
+
+/-- `liftNC` as a `RingHom`, for when `f` and `g` commute -/
+def liftNCRingHom (f : k →+* R) (g : Multiplicative G →* R) (h_comm : ∀ x y, Commute (f x) (g y)) :
+    k[G] →+* R :=
+  { liftNC (f : k →+ R) g with
+    map_one' := liftNC_one _ _
+    map_mul' := fun _a _b => liftNC_mul _ _ _ _ fun {_ _} _ => h_comm _ _ }
+
+end Semiring
+
+instance nonUnitalCommSemiring [CommSemiring k] [AddCommSemigroup G] :
+    NonUnitalCommSemiring k[G] :=
+  { AddMonoidAlgebra.nonUnitalSemiring with
+    mul_comm := @mul_comm (MonoidAlgebra k <| Multiplicative G) _ }
+
+instance nontrivial [Semiring k] [Nontrivial k] [Nonempty G] : Nontrivial k[G] :=
+  Finsupp.instNontrivial
+
+/-! #### Derived instances -/
+
+
+section DerivedInstances
+
+instance commSemiring [CommSemiring k] [AddCommMonoid G] : CommSemiring k[G] :=
+  { AddMonoidAlgebra.nonUnitalCommSemiring, AddMonoidAlgebra.semiring with }
+
+instance unique [Semiring k] [Subsingleton k] : Unique k[G] :=
+  Finsupp.uniqueOfRight
+
+instance addCommGroup [Ring k] : AddCommGroup k[G] :=
+  Finsupp.instAddCommGroup
+
+instance nonUnitalNonAssocRing [Ring k] [Add G] : NonUnitalNonAssocRing k[G] :=
+  { AddMonoidAlgebra.addCommGroup, AddMonoidAlgebra.nonUnitalNonAssocSemiring with }
+
+instance nonUnitalRing [Ring k] [AddSemigroup G] : NonUnitalRing k[G] :=
+  { AddMonoidAlgebra.addCommGroup, AddMonoidAlgebra.nonUnitalSemiring with }
+
+instance nonAssocRing [Ring k] [AddZeroClass G] : NonAssocRing k[G] :=
+  { AddMonoidAlgebra.addCommGroup,
+    AddMonoidAlgebra.nonAssocSemiring with
+    intCast := fun z => single 0 (z : k)
+    -- Porting note: Both were `simpa`.
+    intCast_ofNat := fun n => by simp; rfl
+    intCast_negSucc := fun n => by simp; rfl }
+
+theorem intCast_def [Ring k] [AddZeroClass G] (z : ℤ) :
+    (z : k[G]) = single (0 : G) (z : k) :=
+  rfl
+
+@[deprecated (since := "2024-04-17")]
+alias int_cast_def := intCast_def
+
+instance ring [Ring k] [AddMonoid G] : Ring k[G] :=
+  { AddMonoidAlgebra.nonAssocRing, AddMonoidAlgebra.semiring with }
+
+instance nonUnitalCommRing [CommRing k] [AddCommSemigroup G] :
+    NonUnitalCommRing k[G] :=
+  { AddMonoidAlgebra.nonUnitalCommSemiring, AddMonoidAlgebra.nonUnitalRing with }
+
+instance commRing [CommRing k] [AddCommMonoid G] : CommRing k[G] :=
+  { AddMonoidAlgebra.nonUnitalCommRing, AddMonoidAlgebra.ring with }
+
+variable {S : Type*}
+
+instance distribSMul [Semiring k] [DistribSMul R k] : DistribSMul R k[G] :=
+  Finsupp.distribSMul G k
+
+instance distribMulAction [Monoid R] [Semiring k] [DistribMulAction R k] :
+    DistribMulAction R k[G] :=
+  Finsupp.distribMulAction G k
+
+instance faithfulSMul [Semiring k] [SMulZeroClass R k] [FaithfulSMul R k] [Nonempty G] :
+    FaithfulSMul R k[G] :=
+  Finsupp.faithfulSMul
+
+instance module [Semiring R] [Semiring k] [Module R k] : Module R k[G] :=
+  Finsupp.module G k
+
+instance isScalarTower [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMul R S]
+    [IsScalarTower R S k] : IsScalarTower R S k[G] :=
+  Finsupp.isScalarTower G k
+
+instance smulCommClass [Semiring k] [SMulZeroClass R k] [SMulZeroClass S k] [SMulCommClass R S k] :
+    SMulCommClass R S k[G] :=
+  Finsupp.smulCommClass G k
+
+instance isCentralScalar [Semiring k] [SMulZeroClass R k] [SMulZeroClass Rᵐᵒᵖ k]
+    [IsCentralScalar R k] : IsCentralScalar R k[G] :=
+  Finsupp.isCentralScalar G k
+
+/-! It is hard to state the equivalent of `DistribMulAction G k[G]`
+because we've never discussed actions of additive groups. -/
+
+
+end DerivedInstances
+
+section MiscTheorems
+
+variable [Semiring k]
+
+theorem mul_apply [DecidableEq G] [Add G] (f g : k[G]) (x : G) :
+    (f * g) x = f.sum fun a₁ b₁ => g.sum fun a₂ b₂ => if a₁ + a₂ = x then b₁ * b₂ else 0 :=
+  @MonoidAlgebra.mul_apply k (Multiplicative G) _ _ _ _ _ _
+
+theorem mul_apply_antidiagonal [Add G] (f g : k[G]) (x : G) (s : Finset (G × G))
+    (hs : ∀ {p : G × G}, p ∈ s ↔ p.1 + p.2 = x) : (f * g) x = ∑ p ∈ s, f p.1 * g p.2 :=
+  @MonoidAlgebra.mul_apply_antidiagonal k (Multiplicative G) _ _ _ _ _ s @hs
+
+theorem single_mul_single [Add G] {a₁ a₂ : G} {b₁ b₂ : k} :
+    single a₁ b₁ * single a₂ b₂ = single (a₁ + a₂) (b₁ * b₂) :=
+  @MonoidAlgebra.single_mul_single k (Multiplicative G) _ _ _ _ _ _
+
+theorem single_commute_single [Add G] {a₁ a₂ : G} {b₁ b₂ : k}
+    (ha : AddCommute a₁ a₂) (hb : Commute b₁ b₂) :
+    Commute (single a₁ b₁) (single a₂ b₂) :=
+  @MonoidAlgebra.single_commute_single k (Multiplicative G) _ _ _ _ _ _ ha hb
+
+-- This should be a `@[simp]` lemma, but the simp_nf linter times out if we add this.
+-- Probably the correct fix is to make a `[Add]MonoidAlgebra.single` with the correct type,
+-- instead of relying on `Finsupp.single`.
+theorem single_pow [AddMonoid G] {a : G} {b : k} : ∀ n : ℕ, single a b ^ n = single (n • a) (b ^ n)
+  | 0 => by
+    simp only [pow_zero, zero_nsmul]
+    rfl
+  | n + 1 => by
+    rw [pow_succ, pow_succ, single_pow n, single_mul_single, add_nsmul, one_nsmul]
+
+/-- Like `Finsupp.mapDomain_zero`, but for the `1` we define in this file -/
+@[simp]
+theorem mapDomain_one {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Zero α] [Zero α₂]
+    {F : Type*} [FunLike F α α₂] [ZeroHomClass F α α₂] (f : F) :
+    (mapDomain f (1 : AddMonoidAlgebra β α) : AddMonoidAlgebra β α₂) =
+      (1 : AddMonoidAlgebra β α₂) := by
+  simp_rw [one_def, mapDomain_single, map_zero]
+
+/-- Like `Finsupp.mapDomain_add`, but for the convolutive multiplication we define in this file -/
+theorem mapDomain_mul {α : Type*} {β : Type*} {α₂ : Type*} [Semiring β] [Add α] [Add α₂]
+    {F : Type*} [FunLike F α α₂] [AddHomClass F α α₂] (f : F) (x y : AddMonoidAlgebra β α) :
+    mapDomain f (x * y) = mapDomain f x * mapDomain f y := by
+  simp_rw [mul_def, mapDomain_sum, mapDomain_single, map_add]
+  rw [Finsupp.sum_mapDomain_index]
+  · congr
+    ext a b
+    rw [Finsupp.sum_mapDomain_index]
+    · simp
+    · simp [mul_add]
+  · simp
+  · simp [add_mul]
+
+section
+
+variable (k G)
+
+/-- The embedding of an additive magma into its additive magma algebra. -/
+@[simps]
+def ofMagma [Add G] : Multiplicative G →ₙ* k[G] where
+  toFun a := single a 1
+  map_mul' a b := by simp only [mul_def, mul_one, sum_single_index, single_eq_zero, mul_zero]; rfl
+
+/-- Embedding of a magma with zero into its magma algebra. -/
+def of [AddZeroClass G] : Multiplicative G →* k[G] :=
+  { ofMagma k G with
+    toFun := fun a => single a 1
+    map_one' := rfl }
+
+/-- Embedding of a magma with zero `G`, into its magma algebra, having `G` as source. -/
+def of' : G → k[G] := fun a => single a 1
+
+end
+
+@[simp]
+theorem of_apply [AddZeroClass G] (a : Multiplicative G) :
+    of k G a = single (Multiplicative.toAdd a) 1 :=
+  rfl
+
+@[simp]
+theorem of'_apply (a : G) : of' k G a = single a 1 :=
+  rfl
+
+theorem of'_eq_of [AddZeroClass G] (a : G) : of' k G a = of k G (.ofAdd a) := rfl
+
+theorem of_injective [Nontrivial k] [AddZeroClass G] : Function.Injective (of k G) :=
+  MonoidAlgebra.of_injective
+
+theorem of'_commute [AddZeroClass G] {a : G} (h : ∀ a', AddCommute a a')
+    (f : AddMonoidAlgebra k G) :
+    Commute (of' k G a) f :=
+  MonoidAlgebra.of_commute (G := Multiplicative G) h f
+
+/-- `Finsupp.single` as a `MonoidHom` from the product type into the additive monoid algebra.
+
+Note the order of the elements of the product are reversed compared to the arguments of
+`Finsupp.single`.
+-/
+@[simps]
+def singleHom [AddZeroClass G] : k × Multiplicative G →* k[G] where
+  toFun a := single (Multiplicative.toAdd a.2) a.1
+  map_one' := rfl
+  map_mul' _a _b := single_mul_single.symm
+
+theorem mul_single_apply_aux [Add G] (f : k[G]) (r : k) (x y z : G)
+    (H : ∀ a, a + x = z ↔ a = y) : (f * single x r) z = f y * r :=
+  @MonoidAlgebra.mul_single_apply_aux k (Multiplicative G) _ _ _ _ _ _ _ H
+
+theorem mul_single_zero_apply [AddZeroClass G] (f : k[G]) (r : k) (x : G) :
+    (f * single (0 : G) r) x = f x * r :=
+  f.mul_single_apply_aux r _ _ _ fun a => by rw [add_zero]
+
+theorem mul_single_apply_of_not_exists_add [Add G] (r : k) {g g' : G} (x : k[G])
+    (h : ¬∃ d, g' = d + g) : (x * single g r) g' = 0 :=
+  @MonoidAlgebra.mul_single_apply_of_not_exists_mul k (Multiplicative G) _ _ _ _ _ _ h
+
+theorem single_mul_apply_aux [Add G] (f : k[G]) (r : k) (x y z : G)
+    (H : ∀ a, x + a = y ↔ a = z) : (single x r * f) y = r * f z :=
+  @MonoidAlgebra.single_mul_apply_aux k (Multiplicative G) _ _ _ _ _ _ _ H
+
+theorem single_zero_mul_apply [AddZeroClass G] (f : k[G]) (r : k) (x : G) :
+    (single (0 : G) r * f) x = r * f x :=
+  f.single_mul_apply_aux r _ _ _ fun a => by rw [zero_add]
+
+theorem single_mul_apply_of_not_exists_add [Add G] (r : k) {g g' : G} (x : k[G])
+    (h : ¬∃ d, g' = g + d) : (single g r * x) g' = 0 :=
+  @MonoidAlgebra.single_mul_apply_of_not_exists_mul k (Multiplicative G) _ _ _ _ _ _ h
+
+theorem mul_single_apply [AddGroup G] (f : k[G]) (r : k) (x y : G) :
+    (f * single x r) y = f (y - x) * r :=
+  (sub_eq_add_neg y x).symm ▸ @MonoidAlgebra.mul_single_apply k (Multiplicative G) _ _ _ _ _ _
+
+theorem single_mul_apply [AddGroup G] (r : k) (x : G) (f : k[G]) (y : G) :
+    (single x r * f) y = r * f (-x + y) :=
+  @MonoidAlgebra.single_mul_apply k (Multiplicative G) _ _ _ _ _ _
+
+theorem liftNC_smul {R : Type*} [AddZeroClass G] [Semiring R] (f : k →+* R)
+    (g : Multiplicative G →* R) (c : k) (φ : MonoidAlgebra k G) :
+    liftNC (f : k →+ R) g (c • φ) = f c * liftNC (f : k →+ R) g φ :=
+  @MonoidAlgebra.liftNC_smul k (Multiplicative G) _ _ _ _ f g c φ
+
+theorem induction_on [AddMonoid G] {p : k[G] → Prop} (f : k[G])
+    (hM : ∀ g, p (of k G (Multiplicative.ofAdd g)))
+    (hadd : ∀ f g : k[G], p f → p g → p (f + g))
+    (hsmul : ∀ (r : k) (f), p f → p (r • f)) : p f := by
+  refine Finsupp.induction_linear f ?_ (fun f g hf hg => hadd f g hf hg) fun g r => ?_
+  · simpa using hsmul 0 (of k G (Multiplicative.ofAdd 0)) (hM 0)
+  · convert hsmul r (of k G (Multiplicative.ofAdd g)) (hM g)
+    -- Porting note: Was `simp only`.
+    rw [of_apply, toAdd_ofAdd, smul_single', mul_one]
+
+/-- If `f : G → H` is an additive homomorphism between two additive monoids, then
+`Finsupp.mapDomain f` is a ring homomorphism between their add monoid algebras. -/
+@[simps]
+def mapDomainRingHom (k : Type*) [Semiring k] {H F : Type*} [AddMonoid G] [AddMonoid H]
+    [FunLike F G H] [AddMonoidHomClass F G H] (f : F) : k[G] →+* k[H] :=
+  { (Finsupp.mapDomain.addMonoidHom f : MonoidAlgebra k G →+ MonoidAlgebra k H) with
+    map_one' := mapDomain_one f
+    map_mul' := fun x y => mapDomain_mul f x y }
+
+end MiscTheorems
+
+end AddMonoidAlgebra
+
+/-!
+#### Conversions between `AddMonoidAlgebra` and `MonoidAlgebra`
+
+We have not defined `k[G] = MonoidAlgebra k (Multiplicative G)`
+because historically this caused problems;
+since the changes that have made `nsmul` definitional, this would be possible,
+but for now we just construct the ring isomorphisms using `RingEquiv.refl _`.
+-/
+
+
+/-- The equivalence between `AddMonoidAlgebra` and `MonoidAlgebra` in terms of
+`Multiplicative` -/
+protected def AddMonoidAlgebra.toMultiplicative [Semiring k] [Add G] :
+    AddMonoidAlgebra k G ≃+* MonoidAlgebra k (Multiplicative G) :=
+  { Finsupp.domCongr
+      Multiplicative.ofAdd with
+    toFun := equivMapDomain Multiplicative.ofAdd
+    map_mul' := fun x y => by
+      -- Porting note: added `dsimp only`; `beta_reduce` alone is not sufficient
+      dsimp only
+      repeat' rw [equivMapDomain_eq_mapDomain (M := k)]
+      dsimp [Multiplicative.ofAdd]
+      exact MonoidAlgebra.mapDomain_mul (α := Multiplicative G) (β := k)
+        (MulHom.id (Multiplicative G)) x y }
+
+/-- The equivalence between `MonoidAlgebra` and `AddMonoidAlgebra` in terms of `Additive` -/
+protected def MonoidAlgebra.toAdditive [Semiring k] [Mul G] :
+    MonoidAlgebra k G ≃+* AddMonoidAlgebra k (Additive G) :=
+  { Finsupp.domCongr Additive.ofMul with
+    toFun := equivMapDomain Additive.ofMul
+    map_mul' := fun x y => by
+      -- Porting note: added `dsimp only`; `beta_reduce` alone is not sufficient
+      dsimp only
+      repeat' rw [equivMapDomain_eq_mapDomain (M := k)]
+      dsimp [Additive.ofMul]
+      convert MonoidAlgebra.mapDomain_mul (β := k) (MulHom.id G) x y }
+
+namespace AddMonoidAlgebra
+
+variable {k G H}
+
+/-! #### Non-unital, non-associative algebra structure -/
+
+
+section NonUnitalNonAssocAlgebra
+
+variable (k) [Semiring k] [DistribSMul R k] [Add G]
+
+instance isScalarTower_self [IsScalarTower R k k] :
+    IsScalarTower R k[G] k[G] :=
+  @MonoidAlgebra.isScalarTower_self k (Multiplicative G) R _ _ _ _
+
+/-- Note that if `k` is a `CommSemiring` then we have `SMulCommClass k k k` and so we can take
+`R = k` in the below. In other words, if the coefficients are commutative amongst themselves, they
+also commute with the algebra multiplication. -/
+instance smulCommClass_self [SMulCommClass R k k] :
+    SMulCommClass R k[G] k[G] :=
+  @MonoidAlgebra.smulCommClass_self k (Multiplicative G) R _ _ _ _
+
+instance smulCommClass_symm_self [SMulCommClass k R k] :
+    SMulCommClass k[G] R k[G] :=
+  @MonoidAlgebra.smulCommClass_symm_self k (Multiplicative G) R _ _ _ _
+
+end NonUnitalNonAssocAlgebra
+
+/-! #### Algebra structure -/
+
+
+section Algebra
+
+-- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
+
+/-- `Finsupp.single 0` as a `RingHom` -/
+@[simps]
+def singleZeroRingHom [Semiring k] [AddMonoid G] : k →+* k[G] :=
+  { Finsupp.singleAddHom 0 with
+    map_one' := rfl
+    -- Porting note (#10691): Was `rw`.
+    map_mul' := fun x y => by simp only [singleAddHom, single_mul_single, zero_add] }
+
+/-- If two ring homomorphisms from `k[G]` are equal on all `single a 1`
+and `single 0 b`, then they are equal. -/
+theorem ringHom_ext {R} [Semiring k] [AddMonoid G] [Semiring R] {f g : k[G] →+* R}
+    (h₀ : ∀ b, f (single 0 b) = g (single 0 b)) (h_of : ∀ a, f (single a 1) = g (single a 1)) :
+    f = g :=
+  @MonoidAlgebra.ringHom_ext k (Multiplicative G) R _ _ _ _ _ h₀ h_of
+
+/-- If two ring homomorphisms from `k[G]` are equal on all `single a 1`
+and `single 0 b`, then they are equal.
+
+See note [partially-applied ext lemmas]. -/
+@[ext high]
+theorem ringHom_ext' {R} [Semiring k] [AddMonoid G] [Semiring R] {f g : k[G] →+* R}
+    (h₁ : f.comp singleZeroRingHom = g.comp singleZeroRingHom)
+    (h_of : (f : k[G] →* R).comp (of k G) = (g : k[G] →* R).comp (of k G)) :
+    f = g :=
+  ringHom_ext (RingHom.congr_fun h₁) (DFunLike.congr_fun h_of)
+
+section Opposite
+
+open Finsupp MulOpposite
+
+variable [Semiring k]
+
+/-- The opposite of an `R[I]` is ring equivalent to
+the `AddMonoidAlgebra Rᵐᵒᵖ I` over the opposite ring, taking elements to their opposite. -/
+@[simps! (config := { simpRhs := true }) apply symm_apply]
+protected noncomputable def opRingEquiv [AddCommMonoid G] :
+    k[G]ᵐᵒᵖ ≃+* kᵐᵒᵖ[G] :=
+  { MulOpposite.opAddEquiv.symm.trans
+      (Finsupp.mapRange.addEquiv (MulOpposite.opAddEquiv : k ≃+ kᵐᵒᵖ)) with
+    map_mul' := by
+      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+      rw [Equiv.toFun_as_coe, AddEquiv.toEquiv_eq_coe]; erw [AddEquiv.coe_toEquiv]
+      rw [← AddEquiv.coe_toAddMonoidHom]
+      refine Iff.mpr (AddMonoidHom.map_mul_iff (R := k[G]ᵐᵒᵖ) (S := kᵐᵒᵖ[G]) _) ?_
+      -- Porting note: Was `ext`.
+      refine AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₁ => AddMonoidHom.ext fun r₁ =>
+        AddMonoidHom.mul_op_ext _ _ <| addHom_ext' fun i₂ => AddMonoidHom.ext fun r₂ => ?_
+      -- Porting note: `reducible` cannot be `local` so proof gets long.
+      dsimp
+      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
+      erw [AddEquiv.trans_apply, AddEquiv.trans_apply, AddEquiv.trans_apply,
+        MulOpposite.opAddEquiv_symm_apply]; rw [MulOpposite.unop_mul (α := k[G])]
+      dsimp
+      -- This was not needed before leanprover/lean4#2644
+      erw [mapRange_single, single_mul_single, mapRange_single, mapRange_single]
+      simp only [mapRange_single, single_mul_single, ← op_mul, add_comm] }
+
+-- @[simp] -- Porting note (#10618): simp can prove this
+theorem opRingEquiv_single [AddCommMonoid G] (r : k) (x : G) :
+    AddMonoidAlgebra.opRingEquiv (op (single x r)) = single x (op r) := by simp
+
+-- @[simp] -- Porting note (#10618): simp can prove this
+theorem opRingEquiv_symm_single [AddCommMonoid G] (r : kᵐᵒᵖ) (x : Gᵐᵒᵖ) :
+    AddMonoidAlgebra.opRingEquiv.symm (single x r) = op (single x r.unop) := by simp
+
+end Opposite
+
+end Algebra
+
+section
+
+-- attribute [local reducible] MonoidAlgebra -- Porting note: `reducible` cannot be `local`.
+
+universe ui
+
+variable {ι : Type ui}
+
+theorem prod_single [CommSemiring k] [AddCommMonoid G] {s : Finset ι} {a : ι → G} {b : ι → k} :
+    (∏ i ∈ s, single (a i) (b i)) = single (∑ i ∈ s, a i) (∏ i ∈ s, b i) :=
+  Finset.cons_induction_on s rfl fun a s has ih => by
+    rw [prod_cons has, ih, single_mul_single, sum_cons has, prod_cons has]
+
+end
+
+end AddMonoidAlgebra
diff --git a/Mathlib/Algebra/MonoidAlgebra/Degree.lean b/Mathlib/Algebra/MonoidAlgebra/Degree.lean
index b5fb1511da9ed..4546eb21d1d13 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Degree.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Degree.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
 import Mathlib.Algebra.MonoidAlgebra.Support
+import Mathlib.Order.Filter.Extr
 
 /-!
 # Lemmas about the `sup` and `inf` of the support of `AddMonoidAlgebra`
@@ -213,7 +214,14 @@ For an element `f : R[A]`, the element `supDegree f : B` is the supremum of all
 support of `f`, or `⊥` if `f` is zero.
 Often, the Type `B` is `WithBot A`,
 If, further, `A` has a linear order, then this notion coincides with the usual one,
-using the maximum of the exponents. -/
+using the maximum of the exponents.
+
+If `A := σ →₀ ℕ` then `R[A] = MvPolynomial σ R`, and if we equip `σ` with a linear order then
+the induced linear order on `Lex A` equips `MvPolynomial` ring with a
+[monomial order](https://en.wikipedia.org/wiki/Monomial_order) (i.e. a linear order on `A`, the
+type of (monic) monomials in `R[A]`, that respects addition). We make use of this monomial order
+by taking `D := toLex`, and different monomial orders could be accessed via different type
+synonyms once they are added. -/
 abbrev supDegree (f : R[A]) : B :=
   f.support.sup D
 
@@ -256,6 +264,10 @@ theorem supDegree_withBot_some_comp {s : AddMonoidAlgebra R A} (hs : s.support.N
   unfold AddMonoidAlgebra.supDegree
   rw [← Finset.coe_sup' hs, Finset.sup'_eq_sup]
 
+theorem supDegree_eq_of_isMaxOn {p : R[A]} {a : A} (hmem : a ∈ p.support)
+    (hmax : IsMaxOn D p.support a) : p.supDegree D = D a :=
+  sup_eq_of_isMaxOn hmem hmax
+
 variable [AddZeroClass A] {p q : R[A]}
 
 @[simp]
@@ -266,6 +278,10 @@ theorem ne_zero_of_supDegree_ne_bot : p.supDegree D ≠ ⊥ → p ≠ 0 := mt (f
 theorem ne_zero_of_not_supDegree_le {b : B} (h : ¬ p.supDegree D ≤ b) : p ≠ 0 :=
   ne_zero_of_supDegree_ne_bot (fun he => h <| he ▸ bot_le)
 
+theorem supDegree_eq_of_max {b : B} (hb : b ∈ Set.range D) (hmem : D.invFun b ∈ p.support)
+    (hmax : ∀ a ∈ p.support, D a ≤ b) : p.supDegree D = b :=
+  sup_eq_of_max hb hmem hmax
+
 variable [Add B]
 
 theorem supDegree_mul_le (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
@@ -309,6 +325,249 @@ theorem apply_add_of_supDegree_le (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
 
 end SupDegree
 
+section LinearOrder
+
+variable [LinearOrder B] [OrderBot B] {p q : R[A]} (D : A → B)
+
+/-- If `D` is an injection into a linear order `B`, the leading coefficient of `f : R[A]` is the
+  nonzero coefficient of highest degree according to `D`, or 0 if `f = 0`. In general, it is defined
+  to be the coefficient at an inverse image of `supDegree f` (if such exists). -/
+noncomputable def leadingCoeff [Nonempty A] (f : R[A]) : R :=
+  f (D.invFun <| f.supDegree D)
+
+/-- An element `f : R[A]` is monic if its leading coefficient is one. -/
+@[reducible] def Monic [Nonempty A] (f : R[A]) : Prop :=
+  f.leadingCoeff D = 1
+
+variable {D}
+
+@[simp]
+theorem leadingCoeff_single [Nonempty A] (hD : D.Injective) (a : A) (r : R) :
+    (single a r).leadingCoeff D = r := by
+  classical
+  rw [leadingCoeff, supDegree_single]
+  split_ifs with hr
+  · simp [hr]
+  · rw [Function.leftInverse_invFun hD, single_apply, if_pos rfl]
+
+@[simp]
+theorem leadingCoeff_zero [Nonempty A] : (0 : R[A]).leadingCoeff D = 0 := rfl
+
+lemma Monic.ne_zero [Nonempty A] [Nontrivial R] (hp : p.Monic D) : p ≠ 0 := fun h => by
+  simp_rw [Monic, h, leadingCoeff_zero, zero_ne_one] at hp
+
+@[simp]
+theorem monic_one [AddZeroClass A] (hD : D.Injective) : (1 : R[A]).Monic D := by
+  rw [Monic, one_def, leadingCoeff_single hD]
+
+variable (D) in
+lemma exists_supDegree_mem_support (hp : p ≠ 0) : ∃ a ∈ p.support, p.supDegree D = D a :=
+  Finset.exists_mem_eq_sup _ (Finsupp.support_nonempty_iff.mpr hp) D
+
+variable (D) in
+lemma supDegree_mem_range (hp : p ≠ 0) : p.supDegree D ∈ Set.range D := by
+  obtain ⟨a, -, he⟩ := exists_supDegree_mem_support D hp; exact ⟨a, he.symm⟩
+
+variable {ι : Type*} {s : Finset ι} {i : ι} (hi : i ∈ s) {f : ι → R[A]}
+
+lemma supDegree_sum_lt (hs : s.Nonempty) {b : B}
+    (h : ∀ i ∈ s, (f i).supDegree D < b) : (∑ i ∈ s, f i).supDegree D < b := by
+  refine supDegree_sum_le.trans_lt ((Finset.sup_lt_iff ?_).mpr h)
+  obtain ⟨i, hi⟩ := hs; exact bot_le.trans_lt (h i hi)
+
+variable [AddZeroClass A]
+
+open Finsupp in
+lemma supDegree_add_eq_left (h : q.supDegree D < p.supDegree D) :
+    (p + q).supDegree D = p.supDegree D := by
+  apply (supDegree_add_le.trans <| sup_le le_rfl h.le).antisymm
+  obtain ⟨a, ha, he⟩ := exists_supDegree_mem_support D (ne_zero_of_not_supDegree_le h.not_le)
+  rw [he] at h ⊢
+  apply Finset.le_sup
+  rw [mem_support_iff, add_apply, apply_eq_zero_of_not_le_supDegree h.not_le, add_zero]
+  exact mem_support_iff.mp ha
+
+lemma supDegree_add_eq_right (h : p.supDegree D < q.supDegree D) :
+    (p + q).supDegree D = q.supDegree D := by
+  rw [add_comm, supDegree_add_eq_left h]
+
+lemma leadingCoeff_add_eq_left (h : q.supDegree D < p.supDegree D) :
+    (p + q).leadingCoeff D = p.leadingCoeff D := by
+  obtain ⟨a, he⟩ := supDegree_mem_range D (ne_zero_of_not_supDegree_le h.not_le)
+  rw [leadingCoeff, supDegree_add_eq_left h, Finsupp.add_apply, ← leadingCoeff,
+    apply_eq_zero_of_not_le_supDegree (D := D), add_zero]
+  rw [← he, Function.apply_invFun_apply (f := D), he]; exact h.not_le
+
+lemma leadingCoeff_add_eq_right (h : p.supDegree D < q.supDegree D) :
+    (p + q).leadingCoeff D = q.leadingCoeff D := by
+  rw [add_comm, leadingCoeff_add_eq_left h]
+
+lemma supDegree_mem_support (hD : D.Injective) (hp : p ≠ 0) :
+    D.invFun (p.supDegree D) ∈ p.support := by
+  obtain ⟨a, ha, he⟩ := exists_supDegree_mem_support D hp
+  rwa [he, Function.leftInverse_invFun hD]
+
+@[simp]
+lemma leadingCoeff_eq_zero (hD : D.Injective) : p.leadingCoeff D = 0 ↔ p = 0 := by
+  refine ⟨(fun h => ?_).mtr, fun h => h ▸ leadingCoeff_zero⟩
+  rw [leadingCoeff, ← Ne, ← Finsupp.mem_support_iff]
+  exact supDegree_mem_support hD h
+
+lemma supDegree_sub_lt_of_leadingCoeff_eq (hD : D.Injective) {R} [CommRing R] {p q : R[A]}
+    (hd : p.supDegree D = q.supDegree D) (hc : p.leadingCoeff D = q.leadingCoeff D) :
+    (p - q).supDegree D < p.supDegree D ∨ p = q := by
+  rw [or_iff_not_imp_right]
+  refine fun he => (supDegree_sub_le.trans ?_).lt_of_ne ?_
+  · rw [hd, sup_idem]
+  · rw [← sub_eq_zero, ← leadingCoeff_eq_zero hD, leadingCoeff] at he
+    refine fun h => he ?_
+    rwa [h, Finsupp.sub_apply, ← leadingCoeff, hd, ← leadingCoeff, sub_eq_zero]
+
+lemma supDegree_leadingCoeff_sum_eq
+    (hi : i ∈ s) (hmax : ∀ j ∈ s, j ≠ i → (f j).supDegree D < (f i).supDegree D) :
+    (∑ j ∈ s, f j).supDegree D = (f i).supDegree D ∧
+    (∑ j ∈ s, f j).leadingCoeff D = (f i).leadingCoeff D := by
+  classical
+  rw [← s.add_sum_erase _ hi]
+  by_cases hs : s.erase i = ∅
+  · rw [hs, Finset.sum_empty, add_zero]; exact ⟨rfl, rfl⟩
+  suffices _ from ⟨supDegree_add_eq_left this, leadingCoeff_add_eq_left this⟩
+  refine supDegree_sum_lt ?_ (fun j hj => ?_)
+  · rw [Finset.nonempty_iff_ne_empty]; exact hs
+  · rw [Finset.mem_erase] at hj; exact hmax j hj.2 hj.1
+
+open Finset in
+lemma sum_ne_zero_of_injOn_supDegree' (hs : ∃ i ∈ s, f i ≠ 0)
+    (hd : (s : Set ι).InjOn (supDegree D ∘ f)) :
+    ∑ i ∈ s, f i ≠ 0 := by
+  obtain ⟨j, hj, hne⟩ := hs
+  obtain ⟨i, hi, he⟩ := exists_mem_eq_sup _ ⟨j, hj⟩ (supDegree D ∘ f)
+  by_cases h : ∀ k ∈ s, k = i
+  · refine (sum_eq_single_of_mem j hj (fun k hk hne => ?_)).trans_ne hne
+    rw [h k hk, h j hj] at hne; exact hne.irrefl.elim
+  push_neg at h; obtain ⟨j, hj, hne⟩ := h
+  apply ne_zero_of_supDegree_ne_bot (D := D)
+  have (k) (hk : k ∈ s) (hne : k ≠ i) : supDegree D (f k) < supDegree D (f i) :=
+    ((le_sup hk).trans_eq he).lt_of_ne (hd.ne hk hi hne)
+  rw [(supDegree_leadingCoeff_sum_eq hi this).1]
+  exact (this j hj hne).ne_bot
+
+lemma sum_ne_zero_of_injOn_supDegree (hs : s ≠ ∅)
+    (hf : ∀ i ∈ s, f i ≠ 0) (hd : (s : Set ι).InjOn (supDegree D ∘ f)) :
+    ∑ i ∈ s, f i ≠ 0 :=
+  let ⟨i, hi⟩ := Finset.nonempty_iff_ne_empty.2 hs
+  sum_ne_zero_of_injOn_supDegree' ⟨i, hi, hf i hi⟩ hd
+
+variable [Add B]
+variable [CovariantClass B B (· + ·) (· < ·)] [CovariantClass B B (Function.swap (· + ·)) (· < ·)]
+
+lemma apply_supDegree_add_supDegree (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) :
+    (p * q) (D.invFun (p.supDegree D + q.supDegree D)) = p.leadingCoeff D * q.leadingCoeff D := by
+  obtain rfl | hp := eq_or_ne p 0
+  · simp_rw [leadingCoeff_zero, zero_mul, Finsupp.coe_zero, Pi.zero_apply]
+  obtain rfl | hq := eq_or_ne q 0
+  · simp_rw [leadingCoeff_zero, mul_zero, Finsupp.coe_zero, Pi.zero_apply]
+  obtain ⟨ap, -, hp⟩ := exists_supDegree_mem_support D hp
+  obtain ⟨aq, -, hq⟩ := exists_supDegree_mem_support D hq
+  simp_rw [leadingCoeff, hp, hq, ← hadd, Function.leftInverse_invFun hD _]
+  exact apply_add_of_supDegree_le hadd hD hp.le hq.le
+
+lemma supDegree_mul
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
+    (hpq : leadingCoeff D p * leadingCoeff D q ≠ 0)
+    (hp : p ≠ 0) (hq : q ≠ 0) :
+    (p * q).supDegree D = p.supDegree D + q.supDegree D := by
+  cases subsingleton_or_nontrivial R; · exact (hp (Subsingleton.elim _ _)).elim
+  apply supDegree_eq_of_max
+  · rw [← AddSubsemigroup.coe_set_mk (Set.range D), ← AddHom.srange_mk _ hadd, SetLike.mem_coe]
+    exact add_mem (supDegree_mem_range D hp) (supDegree_mem_range D hq)
+  · simp_rw [Finsupp.mem_support_iff, apply_supDegree_add_supDegree hD hadd]
+    exact hpq
+  · have := covariantClass_le_of_lt B B (· + ·)
+    have := covariantClass_le_of_lt B B (Function.swap (· + ·))
+    exact fun a ha => (Finset.le_sup ha).trans (supDegree_mul_le hadd)
+
+lemma Monic.supDegree_mul_of_ne_zero_left
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
+    (hq : q.Monic D) (hp : p ≠ 0) :
+    (p * q).supDegree D = p.supDegree D + q.supDegree D := by
+  cases subsingleton_or_nontrivial R; · exact (hp (Subsingleton.elim _ _)).elim
+  apply supDegree_mul hD hadd ?_ hp hq.ne_zero
+  simp_rw [hq, mul_one, Ne, leadingCoeff_eq_zero hD, hp, not_false_eq_true]
+
+lemma Monic.supDegree_mul_of_ne_zero_right
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
+    (hp : p.Monic D) (hq : q ≠ 0) :
+    (p * q).supDegree D = p.supDegree D + q.supDegree D := by
+  cases subsingleton_or_nontrivial R; · exact (hq (Subsingleton.elim _ _)).elim
+  apply supDegree_mul hD hadd ?_ hp.ne_zero hq
+  simp_rw [hp, one_mul, Ne, leadingCoeff_eq_zero hD, hq, not_false_eq_true]
+
+lemma Monic.supDegree_mul
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
+    (hbot : (⊥ : B) + ⊥ = ⊥) (hp : p.Monic D) (hq : q.Monic D) :
+    (p * q).supDegree D = p.supDegree D + q.supDegree D := by
+  cases subsingleton_or_nontrivial R
+  · simp_rw [Subsingleton.eq_zero p, Subsingleton.eq_zero q, mul_zero, supDegree_zero, hbot]
+  exact hq.supDegree_mul_of_ne_zero_left hD hadd hp.ne_zero
+
+lemma leadingCoeff_mul [NoZeroDivisors R]
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) :
+    (p * q).leadingCoeff D = p.leadingCoeff D * q.leadingCoeff D := by
+  obtain rfl | hp := eq_or_ne p 0
+  · simp_rw [leadingCoeff_zero, zero_mul, leadingCoeff_zero]
+  obtain rfl | hq := eq_or_ne q 0
+  · simp_rw [leadingCoeff_zero, mul_zero, leadingCoeff_zero]
+  rw [← apply_supDegree_add_supDegree hD hadd, ← supDegree_mul hD hadd ?_ hp hq, leadingCoeff]
+  apply mul_ne_zero <;> rwa [Ne, leadingCoeff_eq_zero hD]
+
+lemma Monic.leadingCoeff_mul_eq_left
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) (hq : q.Monic D) :
+    (p * q).leadingCoeff D = p.leadingCoeff D := by
+  obtain rfl | hp := eq_or_ne p 0
+  · rw [zero_mul]
+  rw [leadingCoeff, hq.supDegree_mul_of_ne_zero_left hD hadd hp,
+    apply_supDegree_add_supDegree hD hadd, hq, mul_one]
+
+lemma Monic.leadingCoeff_mul_eq_right
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) (hp : p.Monic D) :
+    (p * q).leadingCoeff D = q.leadingCoeff D := by
+  obtain rfl | hq := eq_or_ne q 0
+  · rw [mul_zero]
+  rw [leadingCoeff, hp.supDegree_mul_of_ne_zero_right hD hadd hq,
+    apply_supDegree_add_supDegree hD hadd, hp, one_mul]
+
+lemma Monic.mul
+    (hD : D.Injective) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2)
+    (hp : p.Monic D) (hq : q.Monic D) : (p * q).Monic D := by
+  rw [Monic, hq.leadingCoeff_mul_eq_left hD hadd]; exact hp
+
+section AddMonoid
+
+variable {A B : Type*} [AddMonoid A] [AddMonoid B] [LinearOrder B] [OrderBot B]
+  [CovariantClass B B (· + ·) (· < ·)] [CovariantClass B B (Function.swap (· + ·)) (· < ·)]
+  {D : A → B} {p : R[A]} {n : ℕ}
+
+lemma Monic.pow
+    (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) (hD : D.Injective)
+    (hp : p.Monic D) : (p ^ n).Monic D := by
+  induction n with
+  | zero => rw [pow_zero]; exact monic_one hD
+  | succ n ih => rw [pow_succ']; exact hp.mul hD hadd ih
+
+lemma Monic.supDegree_pow
+    (hzero : D 0 = 0) (hadd : ∀ a1 a2, D (a1 + a2) = D a1 + D a2) (hD : D.Injective)
+    [Nontrivial R] (hp : p.Monic D) :
+    (p ^ n).supDegree D = n • p.supDegree D := by
+  induction n with
+  | zero => rw [pow_zero, zero_nsmul, one_def, supDegree_single 0 1, if_neg one_ne_zero, hzero]
+  | succ n ih => rw [pow_succ', (hp.pow hadd hD).supDegree_mul_of_ne_zero_left hD hadd hp.ne_zero,
+      ih, succ_nsmul']
+
+end AddMonoid
+
+end LinearOrder
+
 section InfDegree
 
 variable [SemilatticeInf T] [OrderTop T] (D : A → T)
diff --git a/Mathlib/Algebra/MonoidAlgebra/Division.lean b/Mathlib/Algebra/MonoidAlgebra/Division.lean
index b9e13bd2bac92..3eb1337dc2017 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Division.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Division.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Algebra.MonoidAlgebra.Basic
+import Mathlib.Algebra.MonoidAlgebra.Defs
 
 /-!
 # Division of `AddMonoidAlgebra` by monomials
diff --git a/Mathlib/Algebra/MonoidAlgebra/Grading.lean b/Mathlib/Algebra/MonoidAlgebra/Grading.lean
index 6c9381050ab49..e13d03a50b709 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Grading.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Grading.lean
@@ -7,6 +7,7 @@ import Mathlib.LinearAlgebra.Finsupp
 import Mathlib.Algebra.MonoidAlgebra.Support
 import Mathlib.Algebra.DirectSum.Internal
 import Mathlib.RingTheory.GradedAlgebra.Basic
+import Mathlib.Algebra.MonoidAlgebra.Basic
 
 /-!
 # Internal grading of an `AddMonoidAlgebra`
diff --git a/Mathlib/Algebra/MonoidAlgebra/Ideal.lean b/Mathlib/Algebra/MonoidAlgebra/Ideal.lean
index 17cab16c6913a..65393744fc7ce 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Ideal.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Ideal.lean
@@ -3,8 +3,8 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Algebra.MonoidAlgebra.Basic
 import Mathlib.RingTheory.Ideal.Basic
+import Mathlib.Algebra.MonoidAlgebra.Defs
 
 /-!
 # Lemmas about ideals of `MonoidAlgebra` and `AddMonoidAlgebra`
diff --git a/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean b/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean
index 3b849d0527597..897769a72110a 100644
--- a/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathlib.Algebra.MonoidAlgebra.Basic
-import Mathlib.Algebra.Group.UniqueProds
+import Mathlib.Algebra.Group.UniqueProds.Basic
+import Mathlib.Algebra.MonoidAlgebra.Defs
 
 /-!
 # Variations on non-zero divisors in `AddMonoidAlgebra`s
diff --git a/Mathlib/Algebra/MonoidAlgebra/Support.lean b/Mathlib/Algebra/MonoidAlgebra/Support.lean
index 18069fab871f8..cfd6eaba70024 100644
--- a/Mathlib/Algebra/MonoidAlgebra/Support.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/Support.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathlib.Algebra.MonoidAlgebra.Basic
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.MonoidAlgebra.Defs
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 
 /-!
 #  Lemmas about the support of a finitely supported function
diff --git a/Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean b/Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
index 81bb2a27474e6..111fe27da2293 100644
--- a/Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
+++ b/Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
@@ -191,7 +191,7 @@ def addMonoidAlgebraEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Dec
     toFun := AddMonoidAlgebra.toDirectSum
     invFun := DirectSum.toAddMonoidAlgebra }
 
-/-- The additive version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
+/-- The additive version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`. -/
 @[simps (config := .asFn)]
 def addMonoidAlgebraAddEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M, Decidable (m ≠ 0)] :
     AddMonoidAlgebra M ι ≃+ ⨁ _ : ι, M :=
@@ -200,7 +200,7 @@ def addMonoidAlgebraAddEquivDirectSum [DecidableEq ι] [Semiring M] [∀ m : M,
     invFun := DirectSum.toAddMonoidAlgebra
     map_add' := AddMonoidAlgebra.toDirectSum_add }
 
-/-- The ring version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`.  -/
+/-- The ring version of `AddMonoidAlgebra.addMonoidAlgebraEquivDirectSum`. -/
 @[simps (config := .asFn)]
 def addMonoidAlgebraRingEquivDirectSum [DecidableEq ι] [AddMonoid ι] [Semiring M]
     [∀ m : M, Decidable (m ≠ 0)] : AddMonoidAlgebra M ι ≃+* ⨁ _ : ι, M :=
diff --git a/Mathlib/Algebra/MvPolynomial/Basic.lean b/Mathlib/Algebra/MvPolynomial/Basic.lean
index 90a2b3d0d0db8..722c1bdf7b69a 100644
--- a/Mathlib/Algebra/MvPolynomial/Basic.lean
+++ b/Mathlib/Algebra/MvPolynomial/Basic.lean
@@ -10,6 +10,7 @@ import Mathlib.Algebra.MonoidAlgebra.Support
 import Mathlib.Data.Finsupp.Antidiagonal
 import Mathlib.Order.SymmDiff
 import Mathlib.RingTheory.Adjoin.Basic
+import Mathlib.Algebra.MonoidAlgebra.Basic
 
 /-!
 # Multivariate polynomials
@@ -461,7 +462,7 @@ theorem linearMap_ext {M : Type*} [AddCommMonoid M] [Module R M] {f g : MvPolyno
 
 section Support
 
-/-- The finite set of all `m : σ →₀ ℕ` such that `X^m` has a non-zero coefficient.  -/
+/-- The finite set of all `m : σ →₀ ℕ` such that `X^m` has a non-zero coefficient. -/
 def support (p : MvPolynomial σ R) : Finset (σ →₀ ℕ) :=
   Finsupp.support p
 
@@ -1356,6 +1357,11 @@ theorem comp_aeval {B : Type*} [CommSemiring B] [Algebra R B] (φ : S₁ →ₐ[
   ext i
   simp
 
+lemma comp_aeval_apply {B : Type*} [CommSemiring B] [Algebra R B] (φ : S₁ →ₐ[R] B)
+    (p : MvPolynomial σ R) :
+    φ (aeval f p) = aeval (fun i ↦ φ (f i)) p := by
+  rw [← comp_aeval, AlgHom.coe_comp, comp_apply]
+
 @[simp]
 theorem map_aeval {B : Type*} [CommSemiring B] (g : σ → S₁) (φ : S₁ →+* B) (p : MvPolynomial σ R) :
     φ (aeval g p) = eval₂Hom (φ.comp (algebraMap R S₁)) (fun i => φ (g i)) p := by
@@ -1437,7 +1443,7 @@ variable (R)
 theorem _root_.Algebra.adjoin_range_eq_range_aeval :
     Algebra.adjoin R (Set.range f) = (MvPolynomial.aeval f).range := by
   simp only [← Algebra.map_top, ← MvPolynomial.adjoin_range_X, AlgHom.map_adjoin, ← Set.range_comp,
-    (· ∘ ·), MvPolynomial.aeval_X]
+    Function.comp_def, MvPolynomial.aeval_X]
 
 theorem _root_.Algebra.adjoin_eq_range (s : Set S₁) :
     Algebra.adjoin R s = (MvPolynomial.aeval ((↑) : s → S₁)).range := by
@@ -1536,6 +1542,19 @@ theorem eval_mem {p : MvPolynomial σ S} {s : subS} (hs : ∀ i ∈ p.support, p
 
 end EvalMem
 
+variable {S T : Type*} [CommSemiring S] [Algebra R S] [CommSemiring T] [Algebra R T] [Algebra S T]
+  [IsScalarTower R S T]
+
+lemma aeval_sum_elim {σ τ : Type*} (p : MvPolynomial (σ ⊕ τ) R) (f : τ → S) (g : σ → T) :
+    (aeval (Sum.elim g (algebraMap S T ∘ f))) p =
+      (aeval g) ((aeval (Sum.elim X (C ∘ f))) p) := by
+  induction' p using MvPolynomial.induction_on with r p q hp hq p i h
+  · simp [← IsScalarTower.algebraMap_apply]
+  · simp [hp, hq]
+  · cases i <;> simp [h]
+
 end CommSemiring
 
 end MvPolynomial
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Algebra/MvPolynomial/Division.lean b/Mathlib/Algebra/MvPolynomial/Division.lean
index eafbfef36254a..7d505e900d7d2 100644
--- a/Mathlib/Algebra/MvPolynomial/Division.lean
+++ b/Mathlib/Algebra/MvPolynomial/Division.lean
@@ -206,19 +206,19 @@ theorem monomial_dvd_monomial {r s : R} {i j : σ →₀ ℕ} :
 theorem monomial_one_dvd_monomial_one [Nontrivial R] {i j : σ →₀ ℕ} :
     monomial i (1 : R) ∣ monomial j 1 ↔ i ≤ j := by
   rw [monomial_dvd_monomial]
-  simp_rw [one_ne_zero, false_or_iff, dvd_rfl, and_true_iff]
+  simp_rw [one_ne_zero, false_or, dvd_rfl, and_true]
 
 @[simp]
 theorem X_dvd_X [Nontrivial R] {i j : σ} :
     (X i : MvPolynomial σ R) ∣ (X j : MvPolynomial σ R) ↔ i = j := by
   refine monomial_one_dvd_monomial_one.trans ?_
   simp_rw [Finsupp.single_le_iff, Nat.one_le_iff_ne_zero, Finsupp.single_apply_ne_zero,
-    ne_eq, not_false_eq_true, and_true]
+    ne_eq, reduceCtorEq,not_false_eq_true, and_true]
 
 @[simp]
 theorem X_dvd_monomial {i : σ} {j : σ →₀ ℕ} {r : R} :
     (X i : MvPolynomial σ R) ∣ monomial j r ↔ r = 0 ∨ j i ≠ 0 := by
   refine monomial_dvd_monomial.trans ?_
-  simp_rw [one_dvd, and_true_iff, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
+  simp_rw [one_dvd, and_true, Finsupp.single_le_iff, Nat.one_le_iff_ne_zero]
 
 end MvPolynomial
diff --git a/Mathlib/Algebra/MvPolynomial/Equiv.lean b/Mathlib/Algebra/MvPolynomial/Equiv.lean
index a6350e2e62a69..6b59003105ae6 100644
--- a/Mathlib/Algebra/MvPolynomial/Equiv.lean
+++ b/Mathlib/Algebra/MvPolynomial/Equiv.lean
@@ -196,6 +196,17 @@ def isEmptyAlgEquiv [he : IsEmpty σ] : MvPolynomial σ R ≃ₐ[R] R :=
       ext i m
       exact IsEmpty.elim' he i)
 
+variable {R S₁ σ} in
+@[simp]
+lemma aeval_injective_iff_of_isEmpty [IsEmpty σ] [CommSemiring S₁] [Algebra R S₁] {f : σ → S₁} :
+    Function.Injective (aeval f : MvPolynomial σ R →ₐ[R] S₁) ↔
+      Function.Injective (algebraMap R S₁) := by
+  have : aeval f = (Algebra.ofId R S₁).comp (@isEmptyAlgEquiv R σ _ _).toAlgHom := by
+    ext i
+    exact IsEmpty.elim' ‹IsEmpty σ› i
+  rw [this, ← Injective.of_comp_iff' _ (@isEmptyAlgEquiv R σ _ _).bijective]
+  rfl
+
 /-- The ring isomorphism between multivariable polynomials in no variables
 and the ground ring. -/
 @[simps!]
@@ -244,6 +255,18 @@ def sumAlgEquiv : MvPolynomial (S₁ ⊕ S₂) R ≃ₐ[R] MvPolynomial S₁ (Mv
       simp only [sumRingEquiv, mvPolynomialEquivMvPolynomial, Equiv.toFun_as_coe,
         Equiv.coe_fn_mk, B, sumToIter_C, A] }
 
+lemma sumAlgEquiv_comp_rename_inr :
+    (sumAlgEquiv R S₁ S₂).toAlgHom.comp (rename Sum.inr) = IsScalarTower.toAlgHom R
+        (MvPolynomial S₂ R) (MvPolynomial S₁ (MvPolynomial S₂ R)) := by
+  ext i
+  simp
+
+lemma sumAlgEquiv_comp_rename_inl :
+    (sumAlgEquiv R S₁ S₂).toAlgHom.comp (rename Sum.inl) =
+      MvPolynomial.mapAlgHom (Algebra.ofId _ _) := by
+  ext i
+  simp
+
 section
 
 -- this speeds up typeclass search in the lemma below
diff --git a/Mathlib/Algebra/MvPolynomial/Expand.lean b/Mathlib/Algebra/MvPolynomial/Expand.lean
index 142256c94fc0d..8b06a208a6ea6 100644
--- a/Mathlib/Algebra/MvPolynomial/Expand.lean
+++ b/Mathlib/Algebra/MvPolynomial/Expand.lean
@@ -68,7 +68,7 @@ theorem map_expand (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) :
 @[simp]
 theorem rename_expand (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) :
     rename f (expand p φ) = expand p (rename f φ) := by
-  simp [expand, bind₁_rename, rename_bind₁, Function.comp]
+  simp [expand, bind₁_rename, rename_bind₁, Function.comp_def]
 
 @[simp]
 theorem rename_comp_expand (f : σ → τ) (p : ℕ) :
diff --git a/Mathlib/Algebra/MvPolynomial/PDeriv.lean b/Mathlib/Algebra/MvPolynomial/PDeriv.lean
index a4755788c2c2a..2f36da703dccd 100644
--- a/Mathlib/Algebra/MvPolynomial/PDeriv.lean
+++ b/Mathlib/Algebra/MvPolynomial/PDeriv.lean
@@ -65,12 +65,12 @@ theorem pderiv_def [DecidableEq σ] (i : σ) : pderiv i = mkDerivation R (Pi.sin
 theorem pderiv_monomial {i : σ} :
     pderiv i (monomial s a) = monomial (s - single i 1) (a * s i) := by
   classical
-    simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, ← smul_mul_assoc,
-      ← (monomial _).map_smul]
-    refine (Finset.sum_eq_single i (fun j _ hne => ?_) fun hi => ?_).trans ?_
-    · simp [Pi.single_eq_of_ne hne]
-    · rw [Finsupp.not_mem_support_iff] at hi; simp [hi]
-    · simp
+  simp only [pderiv_def, mkDerivation_monomial, Finsupp.smul_sum, smul_eq_mul, ← smul_mul_assoc,
+    ← (monomial _).map_smul]
+  refine (Finset.sum_eq_single i (fun j _ hne => ?_) fun hi => ?_).trans ?_
+  · simp [Pi.single_eq_of_ne hne]
+  · rw [Finsupp.not_mem_support_iff] at hi; simp [hi]
+  · simp
 
 theorem pderiv_C {i : σ} : pderiv i (C a) = 0 :=
   derivation_C _ _
@@ -115,6 +115,28 @@ theorem pderiv_map {S} [CommSemiring S] {φ : R →+* S} {f : MvPolynomial σ R}
   · simp [eq]
   · simp [eq, h]
 
+lemma pderiv_rename {τ : Type*} {f : σ → τ} (hf : Function.Injective f)
+    (x : σ) (p : MvPolynomial σ R) :
+    pderiv (f x) (rename f p) = rename f (pderiv x p) := by
+  classical
+  induction' p using MvPolynomial.induction_on with a p q hp hq p a h
+  · simp
+  · simp [hp, hq]
+  · simp only [map_mul, MvPolynomial.rename_X, Derivation.leibniz, MvPolynomial.pderiv_X,
+      Pi.single_apply, hf.eq_iff, smul_eq_mul, mul_ite, mul_one, mul_zero, h, map_add, add_left_inj]
+    split_ifs <;> simp
+
+lemma aeval_sum_elim_pderiv_inl {S τ : Type*} [CommRing S] [Algebra R S]
+    (p : MvPolynomial (σ ⊕ τ) R) (f : τ → S) (j : σ) :
+    aeval (Sum.elim X (C ∘ f)) ((pderiv (Sum.inl j)) p) =
+      (pderiv j) ((aeval (Sum.elim X (C ∘ f))) p) := by
+  classical
+  induction' p using MvPolynomial.induction_on with a p q hp hq p q h
+  · simp
+  · simp [hp, hq]
+  · simp only [Derivation.leibniz, pderiv_X, smul_eq_mul, map_add, map_mul, aeval_X, h]
+    cases q <;> simp [Pi.single_apply]
+
 end PDeriv
 
 end MvPolynomial
diff --git a/Mathlib/Algebra/MvPolynomial/Polynomial.lean b/Mathlib/Algebra/MvPolynomial/Polynomial.lean
index f04fac634e9f1..529c8a050c772 100644
--- a/Mathlib/Algebra/MvPolynomial/Polynomial.lean
+++ b/Mathlib/Algebra/MvPolynomial/Polynomial.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.MvPolynomial.Equiv
 import Mathlib.Algebra.Polynomial.Eval
diff --git a/Mathlib/Algebra/MvPolynomial/Rename.lean b/Mathlib/Algebra/MvPolynomial/Rename.lean
index 3043309529923..48bfa47e642c2 100644
--- a/Mathlib/Algebra/MvPolynomial/Rename.lean
+++ b/Mathlib/Algebra/MvPolynomial/Rename.lean
@@ -72,7 +72,7 @@ theorem rename_rename (f : σ → τ) (g : τ → α) (p : MvPolynomial σ R) :
     -- Porting note: the Lean 3 proof of this was very fragile and included a nonterminal `simp`.
     -- Hopefully this is less prone to breaking
     rw [eval₂_comp_left (eval₂Hom (algebraMap R (MvPolynomial α R)) (X ∘ g)) C (X ∘ f) p]
-    simp only [(· ∘ ·), eval₂Hom_X']
+    simp only [comp_def, eval₂Hom_X']
     refine eval₂Hom_congr ?_ rfl rfl
     ext1; simp only [comp_apply, RingHom.coe_comp, eval₂Hom_C]
 
@@ -206,15 +206,14 @@ theorem exists_finset_rename (p : MvPolynomial σ R) :
   · rintro p q ⟨s, p, rfl⟩ ⟨t, q, rfl⟩
     refine ⟨s ∪ t, ⟨?_, ?_⟩⟩
     · refine rename (Subtype.map id ?_) p + rename (Subtype.map id ?_) q <;>
-        simp (config := { contextual := true }) only [id, true_or_iff, or_true_iff,
+        simp (config := { contextual := true }) only [id, true_or, or_true,
           Finset.mem_union, forall_true_iff]
     · simp only [rename_rename, map_add]
       rfl
   · rintro p n ⟨s, p, rfl⟩
     refine ⟨insert n s, ⟨?_, ?_⟩⟩
     · refine rename (Subtype.map id ?_) p * X ⟨n, s.mem_insert_self n⟩
-      simp (config := { contextual := true }) only [id, or_true_iff, Finset.mem_insert,
-        forall_true_iff]
+      simp (config := { contextual := true }) only [id, or_true, Finset.mem_insert, forall_true_iff]
     · simp only [rename_rename, rename_X, Subtype.coe_mk, map_mul]
       rfl
 
@@ -246,7 +245,7 @@ theorem exists_fin_rename (p : MvPolynomial σ R) :
   let e := Fintype.equivFin { x // x ∈ s }
   refine ⟨n, (↑) ∘ e.symm, Subtype.val_injective.comp e.symm.injective, rename e q, ?_⟩
   rw [← rename_rename, rename_rename e]
-  simp only [Function.comp, Equiv.symm_apply_apply, rename_rename]
+  simp only [Function.comp_def, Equiv.symm_apply_apply, rename_rename]
 
 end Rename
 
diff --git a/Mathlib/Algebra/MvPolynomial/Variables.lean b/Mathlib/Algebra/MvPolynomial/Variables.lean
index f431fe8b5f796..730a6443dcfec 100644
--- a/Mathlib/Algebra/MvPolynomial/Variables.lean
+++ b/Mathlib/Algebra/MvPolynomial/Variables.lean
@@ -3,6 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
 -/
+import Mathlib.Data.Finsupp.Lex
 import Mathlib.Algebra.MvPolynomial.Degrees
 
 /-!
@@ -139,7 +140,7 @@ theorem vars_prod {ι : Type*} [DecidableEq σ] {s : Finset ι} (f : ι → MvPo
 
 section IsDomain
 
-variable {A : Type*} [CommRing A] [IsDomain A]
+variable {A : Type*} [CommRing A] [NoZeroDivisors A]
 
 theorem vars_C_mul (a : A) (ha : a ≠ 0) (φ : MvPolynomial σ A) :
     (C a * φ : MvPolynomial σ A).vars = φ.vars := by
@@ -148,7 +149,7 @@ theorem vars_C_mul (a : A) (ha : a ≠ 0) (φ : MvPolynomial σ A) :
   apply exists_congr
   intro d
   apply and_congr _ Iff.rfl
-  rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and_iff]
+  rw [coeff_C_mul, mul_ne_zero_iff, eq_true ha, true_and]
 
 end IsDomain
 
@@ -272,7 +273,7 @@ theorem eval₂Hom_congr' {f₁ f₂ : R →+* S} {g₁ g₂ : σ → S} {p₁ p
 
 /-- If `f₁` and `f₂` are ring homs out of the polynomial ring and `p₁` and `p₂` are polynomials,
   then `f₁ p₁ = f₂ p₂` if `p₁ = p₂` and `f₁` and `f₂` are equal on `R` and on the variables
-  of `p₁`.  -/
+  of `p₁`. -/
 theorem hom_congr_vars {f₁ f₂ : MvPolynomial σ R →+* S} {p₁ p₂ : MvPolynomial σ R}
     (hC : f₁.comp C = f₂.comp C) (hv : ∀ i, i ∈ p₁.vars → i ∈ p₂.vars → f₁ (X i) = f₂ (X i))
     (hp : p₁ = p₂) : f₁ p₁ = f₂ p₂ :=
@@ -308,6 +309,25 @@ theorem mem_vars_rename (f : σ → τ) (φ : MvPolynomial σ R) {j : τ} (h : j
 
 end EvalVars
 
+section Lex
+
+variable [LinearOrder σ]
+
+lemma leadingCoeff_toLex : p.leadingCoeff toLex = p.coeff (ofLex <| p.supDegree toLex) := by
+  rw [leadingCoeff]
+  apply congr_arg p.coeff
+  apply toLex.injective
+  rw [Function.rightInverse_invFun toLex.surjective, toLex_ofLex]
+
+lemma supDegree_toLex_C (r : R) : supDegree toLex (C (σ := σ) r) = 0 := by
+  classical
+    exact (supDegree_single _ r).trans (ite_eq_iff'.mpr ⟨fun _ => rfl, fun _ => rfl⟩)
+
+lemma leadingCoeff_toLex_C (r : R) : leadingCoeff toLex (C (σ := σ) r) = r :=
+  leadingCoeff_single toLex.injective _ r
+
+end Lex
+
 end CommSemiring
 
 end MvPolynomial
diff --git a/Mathlib/Algebra/NeZero.lean b/Mathlib/Algebra/NeZero.lean
index 5851e7c4e2cef..3fdf3e370cb38 100644
--- a/Mathlib/Algebra/NeZero.lean
+++ b/Mathlib/Algebra/NeZero.lean
@@ -10,32 +10,12 @@ import Mathlib.Order.Defs
 /-!
 # `NeZero` typeclass
 
-We create a typeclass `NeZero n` which carries around the fact that `(n : R) ≠ 0`.
+We give basic facts about the `NeZero n` typeclass.
 
-## Main declarations
-
-* `NeZero`: `n ≠ 0` as a typeclass.
 -/
 
 variable {R : Type*} [Zero R]
 
-/-- A type-class version of `n ≠ 0`.  -/
-class NeZero (n : R) : Prop where
-  /-- The proposition that `n` is not zero. -/
-  out : n ≠ 0
-
-theorem NeZero.ne (n : R) [h : NeZero n] : n ≠ 0 :=
-  h.out
-
-theorem NeZero.ne' (n : R) [h : NeZero n] : 0 ≠ n :=
-  h.out.symm
-
-theorem neZero_iff {n : R} : NeZero n ↔ n ≠ 0 :=
-  ⟨fun h ↦ h.out, NeZero.mk⟩
-
-@[simp] lemma neZero_zero_iff_false {α : Type*} [Zero α] : NeZero (0 : α) ↔ False :=
-  ⟨fun h ↦ h.ne rfl, fun h ↦ h.elim⟩
-
 theorem not_neZero {n : R} : ¬NeZero n ↔ n = 0 := by simp [neZero_iff]
 
 theorem eq_zero_or_neZero (a : R) : a = 0 ∨ NeZero a :=
@@ -77,10 +57,6 @@ namespace NeZero
 
 variable {M : Type*} {x : M}
 
-instance succ {n : ℕ} : NeZero (n + 1) := ⟨n.succ_ne_zero⟩
-
 theorem of_pos [Preorder M] [Zero M] (h : 0 < x) : NeZero x := ⟨ne_of_gt h⟩
 
 end NeZero
-
-lemma Nat.pos_of_neZero (n : ℕ) [NeZero n] : 0 < n := Nat.pos_of_ne_zero (NeZero.ne _)
diff --git a/Mathlib/Algebra/Order/Algebra.lean b/Mathlib/Algebra/Order/Algebra.lean
index 675c50fd0e04d..dc583079f431f 100644
--- a/Mathlib/Algebra/Order/Algebra.lean
+++ b/Mathlib/Algebra/Order/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.Defs
 import Mathlib.Algebra.Order.Module.OrderedSMul
diff --git a/Mathlib/Algebra/Order/Antidiag/Pi.lean b/Mathlib/Algebra/Order/Antidiag/Pi.lean
index 2573fd6a28d1f..fea1cd40e9b83 100644
--- a/Mathlib/Algebra/Order/Antidiag/Pi.lean
+++ b/Mathlib/Algebra/Order/Antidiag/Pi.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Eric Wieser, Bhavik Mehta,
   Yaël Dillies
 -/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Algebra.Order.BigOperators.Group.Finset
-import Mathlib.Data.Finset.Pointwise.Basic
 import Mathlib.Data.Fin.Tuple.NatAntidiagonal
 
 /-!
@@ -164,7 +164,7 @@ lemma pairwiseDisjoint_piAntidiag_map_addRightEmbedding (hi : i ∉ s) (n : μ)
   rintro hfg _ f rfl - rfl g rfl - hgf
   exact hfg <| by simpa [sum_add_distrib, hi] using congr_arg (∑ j ∈ s, · j) hgf.symm
 
-lemma piAntidiag_cons (hi : i ∉ s)  (n : μ) :
+lemma piAntidiag_cons (hi : i ∉ s) (n : μ) :
     piAntidiag (cons i s hi) n = (antidiagonal n).disjiUnion (fun p : μ × μ ↦
       (piAntidiag s p.snd).map (addRightEmbedding fun t ↦ if t = i then p.fst else 0))
         (pairwiseDisjoint_piAntidiag_map_addRightEmbedding hi _) := by
diff --git a/Mathlib/Algebra/Order/Archimedean/Basic.lean b/Mathlib/Algebra/Order/Archimedean/Basic.lean
index c0968b4ac7e37..576accb749d8c 100644
--- a/Mathlib/Algebra/Order/Archimedean/Basic.lean
+++ b/Mathlib/Algebra/Order/Archimedean/Basic.lean
@@ -223,7 +223,7 @@ variable [LinearOrderedSemifield α] [Archimedean α] {x y ε : α}
 lemma exists_nat_one_div_lt (hε : 0 < ε) : ∃ n : ℕ, 1 / (n + 1 : α) < ε := by
   cases' exists_nat_gt (1 / ε) with n hn
   use n
-  rw [div_lt_iff, ← div_lt_iff' hε]
+  rw [div_lt_iff₀, ← div_lt_iff₀' hε]
   · apply hn.trans
     simp [zero_lt_one]
   · exact n.cast_add_one_pos
@@ -241,7 +241,7 @@ theorem exists_mem_Ico_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ico (
           le_of_lt
             (by
               rw [zpow_neg y ↑N, zpow_natCast]
-              exact (inv_lt hx (lt_trans (inv_pos.2 hx) hN)).1 hN)⟩
+              exact (inv_lt_comm₀ hx (lt_trans (inv_pos.2 hx) hN)).1 hN)⟩
       let ⟨M, hM⟩ := pow_unbounded_of_one_lt x hy
       have hb : ∃ b : ℤ, ∀ m, y ^ m ≤ x → m ≤ b :=
         ⟨M, fun m hm =>
@@ -257,8 +257,8 @@ but with ≤ and < the other way around. -/
 theorem exists_mem_Ioc_zpow (hx : 0 < x) (hy : 1 < y) : ∃ n : ℤ, x ∈ Ioc (y ^ n) (y ^ (n + 1)) :=
   let ⟨m, hle, hlt⟩ := exists_mem_Ico_zpow (inv_pos.2 hx) hy
   have hyp : 0 < y := lt_trans zero_lt_one hy
-  ⟨-(m + 1), by rwa [zpow_neg, inv_lt (zpow_pos_of_pos hyp _) hx], by
-    rwa [neg_add, neg_add_cancel_right, zpow_neg, le_inv hx (zpow_pos_of_pos hyp _)]⟩
+  ⟨-(m + 1), by rwa [zpow_neg, inv_lt_comm₀ (zpow_pos_of_pos hyp _) hx], by
+    rwa [neg_add, neg_add_cancel_right, zpow_neg, le_inv_comm₀ hx (zpow_pos_of_pos hyp _)]⟩
 
 /-- For any `y < 1` and any positive `x`, there exists `n : ℕ` with `y ^ n < x`. -/
 theorem exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n < x := by
@@ -267,18 +267,18 @@ theorem exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n <
     simp only [pow_one]
     exact y_pos.trans_lt hx
   rw [not_le] at y_pos
-  rcases pow_unbounded_of_one_lt x⁻¹ (one_lt_inv y_pos hy) with ⟨q, hq⟩
-  exact ⟨q, by rwa [inv_pow, inv_lt_inv hx (pow_pos y_pos _)] at hq⟩
+  rcases pow_unbounded_of_one_lt x⁻¹ ((one_lt_inv₀ y_pos).2 hy) with ⟨q, hq⟩
+  exact ⟨q, by rwa [inv_pow, inv_lt_inv₀ hx (pow_pos y_pos _)] at hq⟩
 
 /-- Given `x` and `y` between `0` and `1`, `x` is between two successive powers of `y`.
 This is the same as `exists_nat_pow_near`, but for elements between `0` and `1` -/
 theorem exists_nat_pow_near_of_lt_one (xpos : 0 < x) (hx : x ≤ 1) (ypos : 0 < y) (hy : y < 1) :
     ∃ n : ℕ, y ^ (n + 1) < x ∧ x ≤ y ^ n := by
-  rcases exists_nat_pow_near (one_le_inv_iff.2 ⟨xpos, hx⟩) (one_lt_inv_iff.2 ⟨ypos, hy⟩) with
+  rcases exists_nat_pow_near (one_le_inv_iff₀.2 ⟨xpos, hx⟩) (one_lt_inv_iff₀.2 ⟨ypos, hy⟩) with
     ⟨n, hn, h'n⟩
   refine ⟨n, ?_, ?_⟩
-  · rwa [inv_pow, inv_lt_inv xpos (pow_pos ypos _)] at h'n
-  · rwa [inv_pow, inv_le_inv (pow_pos ypos _) xpos] at hn
+  · rwa [inv_pow, inv_lt_inv₀ xpos (pow_pos ypos _)] at h'n
+  · rwa [inv_pow, inv_le_inv₀ (pow_pos ypos _) xpos] at hn
 
 end LinearOrderedSemifield
 
@@ -299,11 +299,11 @@ theorem exists_rat_btwn {x y : α} (h : x < y) : ∃ q : ℚ, x < q ∧ (q : α)
   refine ⟨(z + 1 : ℤ) / n, ?_⟩
   have n0' := (inv_pos.2 (sub_pos.2 h)).trans nh
   have n0 := Nat.cast_pos.1 n0'
-  rw [Rat.cast_div_of_ne_zero, Rat.cast_natCast, Rat.cast_intCast, div_lt_iff n0']
-  · refine ⟨(lt_div_iff n0').2 <| (lt_iff_lt_of_le_iff_le (zh _)).1 (lt_add_one _), ?_⟩
+  rw [Rat.cast_div_of_ne_zero, Rat.cast_natCast, Rat.cast_intCast, div_lt_iff₀ n0']
+  · refine ⟨(lt_div_iff₀ n0').2 <| (lt_iff_lt_of_le_iff_le (zh _)).1 (lt_add_one _), ?_⟩
     rw [Int.cast_add, Int.cast_one]
     refine lt_of_le_of_lt (add_le_add_right ((zh _).1 le_rfl) _) ?_
-    rwa [← lt_sub_iff_add_lt', ← sub_mul, ← div_lt_iff' (sub_pos.2 h), one_div]
+    rwa [← lt_sub_iff_add_lt', ← sub_mul, ← div_lt_iff₀' (sub_pos.2 h), one_div]
   · rw [Rat.den_intCast, Nat.cast_one]
     exact one_ne_zero
   · intro H
@@ -352,7 +352,7 @@ variable [LinearOrderedField α]
 theorem archimedean_iff_nat_lt : Archimedean α ↔ ∀ x : α, ∃ n : ℕ, x < n :=
   ⟨@exists_nat_gt α _, fun H =>
     ⟨fun x y y0 =>
-      (H (x / y)).imp fun n h => le_of_lt <| by rwa [div_lt_iff y0, ← nsmul_eq_mul] at h⟩⟩
+      (H (x / y)).imp fun n h => le_of_lt <| by rwa [div_lt_iff₀ y0, ← nsmul_eq_mul] at h⟩⟩
 
 theorem archimedean_iff_nat_le : Archimedean α ↔ ∀ x : α, ∃ n : ℕ, x ≤ n :=
   archimedean_iff_nat_lt.trans
@@ -427,3 +427,8 @@ instance (priority := 100) FloorRing.archimedean (α) [LinearOrderedField α] [F
     Archimedean α := by
   rw [archimedean_iff_int_le]
   exact fun x => ⟨⌈x⌉, Int.le_ceil x⟩
+
+@[to_additive]
+instance Units.instMulArchimedean (α) [OrderedCommMonoid α] [MulArchimedean α] :
+    MulArchimedean αˣ :=
+  ⟨fun x {_} h ↦ MulArchimedean.arch x.val h⟩
diff --git a/Mathlib/Algebra/Order/Archimedean/Submonoid.lean b/Mathlib/Algebra/Order/Archimedean/Submonoid.lean
new file mode 100644
index 0000000000000..fccc02aa01ab0
--- /dev/null
+++ b/Mathlib/Algebra/Order/Archimedean/Submonoid.lean
@@ -0,0 +1,31 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+
+import Mathlib.Algebra.Order.Archimedean.Basic
+import Mathlib.Algebra.Order.Monoid.Submonoid
+
+/-!
+# Submonoids of archimedean monoids
+
+This file defines the instances that show that the (mul)archimedean property is retained in a
+submonoid of the ambient group.
+
+## Main statements
+
+* `SubmonoidClass.instMulArchimedean`: the submonoid (and similar subobjects) of a mul-archimedean
+  group retains the mul-archimedean property when restricted to the submonoid.
+* `AddSubmonoidClass.instArchimedean`: the additive submonoid (and similar subobjects) of an
+  archimedean additive group retains the archimedean property when restricted to the additive
+  submonoid.
+-/
+
+@[to_additive]
+instance SubmonoidClass.instMulArchimedean {M S : Type*} [SetLike S M] [OrderedCommMonoid M]
+    [SubmonoidClass S M] [MulArchimedean M] (H : S) : MulArchimedean H := by
+  constructor
+  rintro x _
+  simp only [← Subtype.coe_lt_coe, OneMemClass.coe_one, SubmonoidClass.mk_pow, Subtype.mk_le_mk]
+  exact MulArchimedean.arch x.val
diff --git a/Mathlib/Algebra/Order/BigOperators/Expect.lean b/Mathlib/Algebra/Order/BigOperators/Expect.lean
new file mode 100644
index 0000000000000..a449b9ce36435
--- /dev/null
+++ b/Mathlib/Algebra/Order/BigOperators/Expect.lean
@@ -0,0 +1,210 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.BigOperators.Expect
+import Mathlib.Algebra.Module.Rat
+import Mathlib.Algebra.Order.BigOperators.Ring.Finset
+import Mathlib.Algebra.Order.Module.Rat
+
+/-!
+# Order properties of the average over a finset
+-/
+
+open Function
+open Fintype (card)
+open scoped BigOperators Pointwise NNRat
+
+variable {ι κ α β R : Type*}
+
+local notation a " /ℚ " q => (q : ℚ≥0)⁻¹ • a
+
+namespace Finset
+section OrderedAddCommMonoid
+variable [OrderedAddCommMonoid α] [Module ℚ≥0 α] [OrderedAddCommMonoid β] [Module ℚ≥0 β]
+  {s : Finset ι} {f g : ι → α}
+
+lemma expect_eq_zero_iff_of_nonneg (hs : s.Nonempty) (hf : ∀ i ∈ s, 0 ≤ f i) :
+    𝔼 i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0 := by
+  simp [expect, sum_eq_zero_iff_of_nonneg hf, hs.ne_empty]
+
+lemma expect_eq_zero_iff_of_nonpos (hs : s.Nonempty) (hf : ∀ i ∈ s, f i ≤ 0) :
+    𝔼 i ∈ s, f i = 0 ↔ ∀ i ∈ s, f i = 0 := by
+  simp [expect, sum_eq_zero_iff_of_nonpos hf, hs.ne_empty]
+
+section PosSMulMono
+variable [PosSMulMono ℚ≥0 α] {a : α}
+
+lemma expect_le_expect (hfg : ∀ i ∈ s, f i ≤ g i) : 𝔼 i ∈ s, f i ≤ 𝔼 i ∈ s, g i :=
+  smul_le_smul_of_nonneg_left (sum_le_sum hfg) <| by positivity
+
+/-- This is a (beta-reduced) version of the standard lemma `Finset.expect_le_expect`,
+convenient for the `gcongr` tactic. -/
+@[gcongr]
+lemma _root_.GCongr.expect_le_expect (h : ∀ i ∈ s, f i ≤ g i) : s.expect f ≤ s.expect g :=
+  Finset.expect_le_expect h
+
+lemma expect_le (hs : s.Nonempty) (h : ∀ x ∈ s, f x ≤ a) : 𝔼 i ∈ s, f i ≤ a :=
+  (inv_smul_le_iff_of_pos <| mod_cast hs.card_pos).2 <| by
+    rw [Nat.cast_smul_eq_nsmul]; exact sum_le_card_nsmul _ _ _ h
+
+lemma le_expect (hs : s.Nonempty) (h : ∀ x ∈ s, a ≤ f x) : a ≤ 𝔼 i ∈ s, f i :=
+  (le_inv_smul_iff_of_pos <| mod_cast hs.card_pos).2 <| by
+    rw [Nat.cast_smul_eq_nsmul]; exact card_nsmul_le_sum _ _ _ h
+
+lemma expect_nonneg (hf : ∀ i ∈ s, 0 ≤ f i) : 0 ≤ 𝔼 i ∈ s, f i :=
+  smul_nonneg (by positivity) <| sum_nonneg hf
+
+end PosSMulMono
+
+section PosSMulMono
+variable {M N : Type*} [AddCommMonoid M] [Module ℚ≥0 M] [OrderedAddCommMonoid N] [Module ℚ≥0 N]
+  [PosSMulMono ℚ≥0 N] {m : M → N} {p : M → Prop} {f : ι → M} {s : Finset ι}
+
+/-- Let `{a | p a}` be an additive subsemigroup of an additive commutative monoid `M`. If `m` is a
+subadditive function (`m (a + b) ≤ m a + m b`) preserved under division by a natural, `f` is a
+function valued in that subsemigroup and `s` is a nonempty set, then
+`m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)`. -/
+lemma le_expect_nonempty_of_subadditive_on_pred (h_add : ∀ a b, p a → p b → m (a + b) ≤ m a + m b)
+    (hp_add : ∀ a b, p a → p b → p (a + b)) (h_div : ∀ (n : ℕ) a, p a → m (a /ℚ n) = m a /ℚ n)
+    (hs_nonempty : s.Nonempty) (hs : ∀ i ∈ s, p (f i)) :
+    m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i) := by
+  simp only [expect, h_div _ _ (sum_induction_nonempty _ _ hp_add hs_nonempty hs)]
+  exact smul_le_smul_of_nonneg_left
+    (le_sum_nonempty_of_subadditive_on_pred _ _ h_add hp_add _ _ hs_nonempty hs) <| by positivity
+
+/-- If `m : M → N` is a subadditive function (`m (a + b) ≤ m a + m b`) and `s` is a nonempty set,
+then `m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)`. -/
+lemma le_expect_nonempty_of_subadditive (m : M → N) (h_mul : ∀ a b, m (a + b) ≤ m a + m b)
+    (h_div : ∀ (n : ℕ) a, m (a /ℚ n) = m a /ℚ n) (hs : s.Nonempty) :
+    m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i) :=
+  le_expect_nonempty_of_subadditive_on_pred (p := fun _ ↦ True) (by simpa) (by simp) (by simpa) hs
+    (by simp)
+
+/-- Let `{a | p a}` be a subsemigroup of a commutative monoid `M`. If `m` is a subadditive function
+(`m (x + y) ≤ m x + m y`, `m 0 = 0`) preserved under division by a natural and `f` is a function
+valued in that subsemigroup, then `m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)`. -/
+lemma le_expect_of_subadditive_on_pred (h_zero : m 0 = 0)
+    (h_add : ∀ a b, p a → p b → m (a + b) ≤ m a + m b) (hp_add : ∀ a b, p a → p b → p (a + b))
+    (h_div : ∀ (n : ℕ) a, p a → m (a /ℚ n) = m a /ℚ n)
+    (hs : ∀ i ∈ s, p (f i)) : m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i) := by
+  obtain rfl | hs_nonempty := s.eq_empty_or_nonempty
+  · simp [h_zero]
+  · exact le_expect_nonempty_of_subadditive_on_pred h_add hp_add h_div hs_nonempty hs
+
+-- TODO: Contribute back better docstring to `le_prod_of_submultiplicative`
+/-- If `m` is a subadditive function (`m (x + y) ≤ m x + m y`, `m 0 = 0`) preserved under division
+by a natural, then `m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)`. -/
+lemma le_expect_of_subadditive (h_zero : m 0 = 0) (h_add : ∀ a b, m (a + b) ≤ m a + m b)
+    (h_div : ∀ (n : ℕ) a, m (a /ℚ n) = m a /ℚ n) : m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i) :=
+  le_expect_of_subadditive_on_pred (p := fun _ ↦ True) h_zero (by simpa) (by simp) (by simpa)
+    (by simp)
+
+end PosSMulMono
+end OrderedAddCommMonoid
+
+section OrderedCancelAddCommMonoid
+variable [OrderedCancelAddCommMonoid α] [Module ℚ≥0 α] {s : Finset ι} {f g : ι → α}
+section PosSMulStrictMono
+variable [PosSMulStrictMono ℚ≥0 α]
+
+lemma expect_pos (hf : ∀ i ∈ s, 0 < f i) (hs : s.Nonempty) : 0 < 𝔼 i ∈ s, f i :=
+  smul_pos (inv_pos.2 <| mod_cast hs.card_pos) <| sum_pos hf hs
+
+end PosSMulStrictMono
+end OrderedCancelAddCommMonoid
+
+section LinearOrderedAddCommMonoid
+variable [LinearOrderedAddCommMonoid α] [Module ℚ≥0 α] [PosSMulMono ℚ≥0 α] {s : Finset ι}
+  {f : ι → α} {a : α}
+
+lemma exists_lt_of_lt_expect (hs : s.Nonempty) (h : a < 𝔼 i ∈ s, f i) : ∃ x ∈ s, a < f x := by
+  contrapose! h; exact expect_le hs h
+
+lemma exists_lt_of_expect_lt (hs : s.Nonempty) (h : 𝔼 i ∈ s, f i < a) : ∃ x ∈ s, f x < a := by
+  contrapose! h; exact le_expect hs h
+
+end LinearOrderedAddCommMonoid
+
+section LinearOrderedAddCommGroup
+variable [LinearOrderedAddCommGroup α] [Module ℚ≥0 α] [PosSMulMono ℚ≥0 α]
+
+lemma abs_expect_le (s : Finset ι) (f : ι → α) : |𝔼 i ∈ s, f i| ≤ 𝔼 i ∈ s, |f i| :=
+  le_expect_of_subadditive abs_zero abs_add (fun _ ↦ abs_nnqsmul _)
+
+end LinearOrderedAddCommGroup
+
+section LinearOrderedCommSemiring
+variable [LinearOrderedCommSemiring R] [ExistsAddOfLE R] [Module ℚ≥0 R] [PosSMulMono ℚ≥0 R]
+
+/-- **Cauchy-Schwarz inequality** in terms of `Finset.expect`. -/
+lemma expect_mul_sq_le_sq_mul_sq (s : Finset ι) (f g : ι → R) :
+    (𝔼 i ∈ s, f i * g i) ^ 2 ≤ (𝔼 i ∈ s, f i ^ 2) * 𝔼 i ∈ s, g i ^ 2 := by
+  simp only [expect, smul_pow, inv_pow, smul_mul_smul_comm, ← sq]
+  gcongr
+  exact sum_mul_sq_le_sq_mul_sq ..
+
+end LinearOrderedCommSemiring
+end Finset
+
+open Finset
+
+namespace Fintype
+variable [Fintype ι] [Fintype κ]
+
+section OrderedAddCommMonoid
+variable [OrderedAddCommMonoid α] [Module ℚ≥0 α] {f : ι → α}
+
+lemma expect_eq_zero_iff_of_nonneg [Nonempty ι] (hf : 0 ≤ f) : 𝔼 i, f i = 0 ↔ f = 0 := by
+  simp [expect, sum_eq_zero_iff_of_nonneg hf, univ_nonempty.ne_empty]
+
+lemma expect_eq_zero_iff_of_nonpos [Nonempty ι] (hf : f ≤ 0) : 𝔼 i, f i = 0 ↔ f = 0 := by
+  simp [expect, sum_eq_zero_iff_of_nonpos hf, univ_nonempty.ne_empty]
+
+end OrderedAddCommMonoid
+end Fintype
+
+open Finset
+
+namespace Mathlib.Meta.Positivity
+open Qq Lean Meta Finset
+open scoped BigOperators
+
+/-- Positivity extension for `Finset.expect`. -/
+@[positivity Finset.expect _ _]
+def evalFinsetExpect : PositivityExt where eval {u α} zα pα e := do
+  match e with
+  | ~q(@Finset.expect $ι _ $instα $instmod $s $f) =>
+    let i : Q($ι) ← mkFreshExprMVarQ q($ι) .syntheticOpaque
+    have body : Q($α) := .betaRev f #[i]
+    let rbody ← core zα pα body
+    let p_pos : Option Q(0 < $e) := ← (do
+      let .positive pbody := rbody | pure none -- Fail if the body is not provably positive
+      let .some ps ← proveFinsetNonempty s | pure none
+      let .some pα' ← trySynthInstanceQ q(OrderedCancelAddCommMonoid $α) | pure none
+      let .some instαordsmul ← trySynthInstanceQ q(PosSMulStrictMono ℚ≥0 $α) | pure none
+      assumeInstancesCommute
+      let pr : Q(∀ i, 0 < $f i) ← mkLambdaFVars #[i] pbody
+      return some q(@expect_pos $ι $α $pα' $instmod $s $f $instαordsmul (fun i _ ↦ $pr i) $ps))
+    -- Try to show that the sum is positive
+    if let some p_pos := p_pos then
+      return .positive p_pos
+    -- Fall back to showing that the sum is nonnegative
+    else
+      let pbody ← rbody.toNonneg
+      let pr : Q(∀ i, 0 ≤ $f i) ← mkLambdaFVars #[i] pbody
+      let instαordmon ← synthInstanceQ q(OrderedAddCommMonoid $α)
+      let instαordsmul ← synthInstanceQ q(PosSMulMono ℚ≥0 $α)
+      assumeInstancesCommute
+      return .nonnegative q(@expect_nonneg $ι $α $instαordmon $instmod $s $f $instαordsmul
+        fun i _ ↦ $pr i)
+  | _ => throwError "not Finset.expect"
+
+example (n : ℕ) (a : ℕ → ℚ) : 0 ≤ 𝔼 j ∈ range n, a j^2 := by positivity
+example (a : ULift.{2} ℕ → ℚ) (s : Finset (ULift.{2} ℕ)) : 0 ≤ 𝔼 j ∈ s, a j^2 := by positivity
+example (n : ℕ) (a : ℕ → ℚ) : 0 ≤ 𝔼 j : Fin 8, 𝔼 i ∈ range n, (a j^2 + i ^ 2) := by positivity
+example (n : ℕ) (a : ℕ → ℚ) : 0 < 𝔼 j : Fin (n + 1), (a j^2 + 1) := by positivity
+example (a : ℕ → ℚ) : 0 < 𝔼 j ∈ ({1} : Finset ℕ), (a j^2 + 1) := by positivity
+
+end Mathlib.Meta.Positivity
diff --git a/Mathlib/Algebra/Order/BigOperators/Group/Finset.lean b/Mathlib/Algebra/Order/BigOperators/Group/Finset.lean
index 93535400087e6..d5cdc6b8f86e7 100644
--- a/Mathlib/Algebra/Order/BigOperators/Group/Finset.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Group/Finset.lean
@@ -121,7 +121,7 @@ theorem one_le_prod'' (h : ∀ i : ι, 1 ≤ f i) : 1 ≤ ∏ i ∈ s, f i :=
 theorem prod_le_one' (h : ∀ i ∈ s, f i ≤ 1) : ∏ i ∈ s, f i ≤ 1 :=
   (prod_le_prod' h).trans_eq (by rw [prod_const_one])
 
-@[to_additive sum_le_sum_of_subset_of_nonneg]
+@[to_additive (attr := gcongr) sum_le_sum_of_subset_of_nonneg]
 theorem prod_le_prod_of_subset_of_one_le' (h : s ⊆ t) (hf : ∀ i ∈ t, i ∉ s → 1 ≤ f i) :
     ∏ i ∈ s, f i ≤ ∏ i ∈ t, f i := by
   classical calc
@@ -210,6 +210,16 @@ theorem prod_le_prod_fiberwise_of_prod_fiber_le_one' {t : Finset ι'} {g : ι 
 
 end OrderedCommMonoid
 
+@[to_additive]
+lemma max_prod_le [LinearOrderedCommMonoid M] {f g : ι → M} {s : Finset ι} :
+    max (s.prod f) (s.prod g) ≤ s.prod (fun i ↦ max (f i) (g i)) :=
+  Multiset.max_prod_le
+
+@[to_additive]
+lemma prod_min_le [LinearOrderedCommMonoid M] {f g : ι → M} {s : Finset ι} :
+    s.prod (fun i ↦ min (f i) (g i)) ≤ min (s.prod f) (s.prod g) :=
+  Multiset.prod_min_le
+
 theorem abs_sum_le_sum_abs {G : Type*} [LinearOrderedAddCommGroup G] (f : ι → G) (s : Finset ι) :
     |∑ i ∈ s, f i| ≤ ∑ i ∈ s, |f i| := le_sum_of_subadditive _ abs_zero abs_add s f
 
@@ -221,6 +231,16 @@ theorem abs_sum_of_nonneg' {G : Type*} [LinearOrderedAddCommGroup G] {f : ι →
     (hf : ∀ i, 0 ≤ f i) : |∑ i ∈ s, f i| = ∑ i ∈ s, f i := by
   rw [abs_of_nonneg (Finset.sum_nonneg' hf)]
 
+section CommMonoid
+variable [CommMonoid α] [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {s : Finset ι} {f : ι → α}
+
+@[to_additive (attr := simp)]
+lemma mulLECancellable_prod :
+    MulLECancellable (∏ i ∈ s, f i) ↔ ∀ ⦃i⦄, i ∈ s → MulLECancellable (f i) := by
+  induction' s using Finset.cons_induction with i s hi ih <;> simp [*]
+
+end CommMonoid
+
 section Pigeonhole
 
 variable [DecidableEq β]
@@ -545,7 +565,7 @@ theorem finset_sum_eq_sup_iff_disjoint [DecidableEq α] {β : Type*} {i : Finset
   · simp only [Finset.not_mem_empty, IsEmpty.forall_iff, imp_true_iff, Finset.sum_empty,
       Finset.sup_empty, bot_eq_zero, eq_self_iff_true]
   · simp_rw [Finset.sum_cons hz, Finset.sup_cons, Finset.mem_cons, Multiset.sup_eq_union,
-      forall_eq_or_imp, Ne, not_true_eq_false, IsEmpty.forall_iff, true_and_iff,
+      forall_eq_or_imp, Ne, not_true_eq_false, IsEmpty.forall_iff, true_and,
       imp_and, forall_and, ← hr, @eq_comm _ z]
     have := fun x (H : x ∈ i) => ne_of_mem_of_not_mem H hz
     simp (config := { contextual := true }) only [this, not_false_iff, true_imp_iff]
diff --git a/Mathlib/Algebra/Order/BigOperators/Group/List.lean b/Mathlib/Algebra/Order/BigOperators/Group/List.lean
index 7872ed429ca63..94dd5a115870c 100644
--- a/Mathlib/Algebra/Order/BigOperators/Group/List.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Group/List.lean
@@ -119,6 +119,24 @@ lemma one_le_prod_of_one_le [Preorder M] [CovariantClass M M (· * ·) (· ≤ 
   rw [prod_cons]
   exact one_le_mul (hl₁ hd (mem_cons_self hd tl)) (ih fun x h => hl₁ x (mem_cons_of_mem hd h))
 
+@[to_additive]
+lemma max_prod_le (l : List α) (f g : α → M) [LinearOrder M]
+    [CovariantClass M M (· * ·) (· ≤ ·)] [CovariantClass M M (Function.swap (· * ·)) (· ≤ ·)] :
+    max (l.map f).prod (l.map g).prod ≤ (l.map fun i ↦ max (f i) (g i)).prod := by
+  rw [max_le_iff]
+  constructor <;> apply List.prod_le_prod' <;> intros
+  · apply le_max_left
+  · apply le_max_right
+
+@[to_additive]
+lemma prod_min_le [LinearOrder M] [CovariantClass M M (· * ·) (· ≤ ·)]
+    [CovariantClass M M (Function.swap (· * ·)) (· ≤ ·)] (l : List α) (f g : α → M) :
+    (l.map fun i ↦ min (f i) (g i)).prod ≤ min (l.map f).prod (l.map g).prod := by
+  rw [le_min_iff]
+  constructor <;> apply List.prod_le_prod' <;> intros
+  · apply min_le_left
+  · apply min_le_right
+
 end Monoid
 
 -- TODO: develop theory of tropical rings
@@ -143,7 +161,8 @@ lemma one_lt_prod_of_one_lt [OrderedCommMonoid M] :
     · exact hl₁.2.1
     · exact hl₁.2.2 _ ‹_›
 
-@[to_additive]
+/-- See also `List.le_prod_of_mem`. -/
+@[to_additive "See also `List.le_sum_of_mem`."]
 lemma single_le_prod [OrderedCommMonoid M] {l : List M} (hl₁ : ∀ x ∈ l, (1 : M) ≤ x) :
     ∀ x ∈ l, x ≤ l.prod := by
   induction l
@@ -163,7 +182,7 @@ variable [CanonicallyOrderedCommMonoid M] {l : List M}
 
 @[to_additive] lemma prod_eq_one_iff : l.prod = 1 ↔ ∀ x ∈ l, x = (1 : M) :=
   ⟨all_one_of_le_one_le_of_prod_eq_one fun _ _ => one_le _, fun h => by
-    rw [List.eq_replicate.2 ⟨_, h⟩, prod_replicate, one_pow]
+    rw [List.eq_replicate_iff.2 ⟨_, h⟩, prod_replicate, one_pow]
     · exact (length l)
     · rfl⟩
 
@@ -174,5 +193,18 @@ variable [CanonicallyOrderedCommMonoid M] {l : List M}
     exact le_self_mul
   · simp [take_of_length_le h, take_of_length_le (le_trans h (Nat.le_succ _))]
 
+/-- See also `List.single_le_prod`. -/
+@[to_additive "See also `List.single_le_sum`."]
+theorem le_prod_of_mem {xs : List M} {x : M} (h₁ : x ∈ xs) : x ≤ xs.prod := by
+  induction xs with
+  | nil => simp at h₁
+  | cons y ys ih =>
+    simp only [mem_cons] at h₁
+    rcases h₁ with (rfl | h₁)
+    · simp
+    · specialize ih h₁
+      simp only [List.prod_cons]
+      exact le_mul_left ih
+
 end CanonicallyOrderedCommMonoid
 end List
diff --git a/Mathlib/Algebra/Order/BigOperators/Group/LocallyFinite.lean b/Mathlib/Algebra/Order/BigOperators/Group/LocallyFinite.lean
new file mode 100644
index 0000000000000..f43f25244887d
--- /dev/null
+++ b/Mathlib/Algebra/Order/BigOperators/Group/LocallyFinite.lean
@@ -0,0 +1,80 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Order.Interval.Finset.Basic
+
+/-!
+# Big operators indexed by intervals
+
+This file proves lemmas about `∏ x ∈ Ixx a b, f x` and `∑ x ∈ Ixx a b, f x`.
+-/
+
+variable {α β : Type*} [PartialOrder α] [CommMonoid β] {f : α → β} {a b : α}
+
+namespace Finset
+section LocallyFiniteOrder
+variable [LocallyFiniteOrder α]
+
+@[to_additive]
+lemma left_mul_prod_Ioc (h : a ≤ b) : f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x := by
+  rw [Icc_eq_cons_Ioc h, prod_cons]
+
+@[to_additive]
+lemma prod_Ioc_mul_left (h : a ≤ b) : (∏ x ∈ Ioc a b, f x) * f a = ∏ x ∈ Icc a b, f x := by
+  rw [mul_comm, left_mul_prod_Ioc h]
+
+@[to_additive]
+lemma right_mul_prod_Ico (h : a ≤ b) : f b * ∏ x ∈ Ico a b, f x = ∏ x ∈ Icc a b, f x := by
+  rw [Icc_eq_cons_Ico h, prod_cons]
+
+@[to_additive]
+lemma prod_Ico_mul_right (h : a ≤ b) : (∏ x ∈ Ico a b, f x) * f b = ∏ x ∈ Icc a b, f x := by
+  rw [mul_comm, right_mul_prod_Ico h]
+
+@[to_additive]
+lemma left_mul_prod_Ioo (h : a < b) : f a * ∏ x ∈ Ioo a b, f x = ∏ x ∈ Ico a b, f x := by
+  rw [Ico_eq_cons_Ioo h, prod_cons]
+
+@[to_additive]
+lemma prod_Ioo_mul_left (h : a < b) : (∏ x ∈ Ioo a b, f x) * f a = ∏ x ∈ Ico a b, f x := by
+  rw [mul_comm, left_mul_prod_Ioo h]
+
+@[to_additive]
+lemma right_mul_prod_Ioo (h : a < b) : f b * ∏ x ∈ Ioo a b, f x = ∏ x ∈ Ioc a b, f x := by
+  rw [Ioc_eq_cons_Ioo h, prod_cons]
+
+@[to_additive]
+lemma prod_Ioo_mul_right (h : a < b) : (∏ x ∈ Ioo a b, f x) * f b = ∏ x ∈ Ioc a b, f x := by
+  rw [mul_comm, right_mul_prod_Ioo h]
+
+end LocallyFiniteOrder
+
+section LocallyFiniteOrderTop
+variable [LocallyFiniteOrderTop α]
+
+@[to_additive]
+lemma left_mul_prod_Ioi (a : α) : f a * ∏ x ∈ Ioi a, f x = ∏ x ∈ Ici a, f x := by
+  rw [Ici_eq_cons_Ioi, prod_cons]
+
+@[to_additive]
+lemma prod_Ioi_mul_left (a : α) : (∏ x ∈ Ioi a, f x) * f a = ∏ x ∈ Ici a, f x := by
+  rw [mul_comm, left_mul_prod_Ioi]
+
+end LocallyFiniteOrderTop
+
+section LocallyFiniteOrderBot
+variable [LocallyFiniteOrderBot α]
+
+@[to_additive]
+lemma right_mul_prod_Iio (a : α) : f a * ∏ x ∈ Iio a, f x = ∏ x ∈ Iic a, f x := by
+  rw [Iic_eq_cons_Iio, prod_cons]
+
+@[to_additive]
+lemma prod_Iio_mul_right (a : α) : (∏ x ∈ Iio a, f x) * f a = ∏ x ∈ Iic a, f x := by
+  rw [mul_comm, right_mul_prod_Iio]
+
+end LocallyFiniteOrderBot
+end Finset
diff --git a/Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean b/Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean
index 19dc75bdf2115..ad405fe75fb40 100644
--- a/Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean
@@ -47,9 +47,10 @@ lemma all_one_of_le_one_le_of_prod_eq_one :
 
 @[to_additive]
 lemma prod_le_prod_of_rel_le (h : s.Rel (· ≤ ·) t) : s.prod ≤ t.prod := by
-  induction' h with _ _ _ _ rh _ rt
-  · rfl
-  · rw [prod_cons, prod_cons]
+  induction h with
+  | zero => rfl
+  | cons rh _ rt =>
+    rw [prod_cons, prod_cons]
     exact mul_le_mul' rh rt
 
 @[to_additive]
@@ -155,6 +156,20 @@ lemma max_le_of_forall_le {α : Type*} [LinearOrder α] [OrderBot α] (l : Multi
   induction l using Quotient.inductionOn
   simpa using List.max_le_of_forall_le _ _ h
 
+@[to_additive]
+lemma max_prod_le [LinearOrderedCommMonoid α] {s : Multiset ι} {f g : ι → α} :
+    max (s.map f).prod (s.map g).prod ≤ (s.map fun i ↦ max (f i) (g i)).prod := by
+  obtain ⟨l⟩ := s
+  simp_rw [Multiset.quot_mk_to_coe'', Multiset.map_coe, Multiset.prod_coe]
+  apply List.max_prod_le
+
+@[to_additive]
+lemma prod_min_le [LinearOrderedCommMonoid α] {s : Multiset ι} {f g : ι → α} :
+    (s.map fun i ↦ min (f i) (g i)).prod ≤ min (s.map f).prod (s.map g).prod := by
+  obtain ⟨l⟩ := s
+  simp_rw [Multiset.quot_mk_to_coe'', Multiset.map_coe, Multiset.prod_coe]
+  apply List.prod_min_le
+
 lemma abs_sum_le_sum_abs [LinearOrderedAddCommGroup α] {s : Multiset α} :
     |s.sum| ≤ (s.map abs).sum :=
   le_sum_of_subadditive _ abs_zero abs_add s
diff --git a/Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.lean b/Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.lean
new file mode 100644
index 0000000000000..e38120c46d7ac
--- /dev/null
+++ b/Mathlib/Algebra/Order/BigOperators/GroupWithZero/List.lean
@@ -0,0 +1,99 @@
+/-
+Copyright (c) 2021 Stuart Presnell. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Stuart Presnell, Daniel Weber
+-/
+import Mathlib.Algebra.BigOperators.Group.List
+import Mathlib.Algebra.Order.GroupWithZero.Unbundled
+
+/-!
+# Big operators on a list in ordered groups with zeros
+
+This file contains the results concerning the interaction of list big operators with ordered
+groups with zeros.
+-/
+
+namespace List
+variable {R : Type*} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R]
+
+lemma prod_nonneg {s : List R} (h : ∀ a ∈ s, 0 ≤ a) : 0 ≤ s.prod := by
+  induction s with
+  | nil => simp
+  | cons head tail hind =>
+    simp only [prod_cons]
+    simp only [mem_cons, forall_eq_or_imp] at h
+    exact mul_nonneg h.1 (hind h.2)
+
+
+lemma one_le_prod {s : List R} (h : ∀ a ∈ s, 1 ≤ a) : 1 ≤ s.prod := by
+  induction s with
+  | nil => simp
+  | cons head tail hind =>
+    simp only [prod_cons]
+    simp only [mem_cons, forall_eq_or_imp] at h
+    exact one_le_mul_of_one_le_of_one_le h.1 (hind h.2)
+
+theorem prod_map_le_prod_map₀ {ι : Type*} {s : List ι} (f : ι → R) (g : ι → R)
+    (h0 : ∀ i ∈ s, 0 ≤ f i) (h : ∀ i ∈ s, f i ≤ g i) :
+    (map f s).prod ≤ (map g s).prod := by
+  induction s with
+  | nil => simp
+  | cons a s hind =>
+    simp only [map_cons, prod_cons]
+    have := posMulMono_iff_mulPosMono.1 ‹PosMulMono R›
+    apply mul_le_mul
+    · apply h
+      simp
+    · apply hind
+      · intro i hi
+        apply h0
+        simp [hi]
+      · intro i hi
+        apply h
+        simp [hi]
+    · apply prod_nonneg
+      simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+      intro a ha
+      apply h0
+      simp [ha]
+    · apply (h0 _ _).trans (h _ _) <;> simp
+
+omit [PosMulMono R]
+variable [PosMulStrictMono R] [NeZero (1 : R)]
+
+lemma prod_pos {s : List R} (h : ∀ a ∈ s, 0 < a) : 0 < s.prod := by
+  induction s with
+  | nil => simp
+  | cons a s hind =>
+    simp only [prod_cons]
+    simp only [mem_cons, forall_eq_or_imp] at h
+    exact mul_pos h.1 (hind h.2)
+
+theorem prod_map_lt_prod_map {ι : Type*} {s : List ι} (hs : s ≠ [])
+    (f : ι → R) (g : ι → R) (h0 : ∀ i ∈ s, 0 < f i) (h : ∀ i ∈ s, f i < g i) :
+    (map f s).prod < (map g s).prod := by
+  match s with
+  | [] => contradiction
+  | a :: s =>
+    simp only [map_cons, prod_cons]
+    have := posMulStrictMono_iff_mulPosStrictMono.1 ‹PosMulStrictMono R›
+    apply mul_lt_mul
+    · apply h
+      simp
+    · apply prod_map_le_prod_map₀
+      · intro i hi
+        apply le_of_lt
+        apply h0
+        simp [hi]
+      · intro i hi
+        apply le_of_lt
+        apply h
+        simp [hi]
+    · apply prod_pos
+      simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+      intro a ha
+      apply h0
+      simp [ha]
+    · apply le_of_lt ((h0 _ _).trans (h _ _)) <;> simp
+
+end List
diff --git a/Mathlib/Algebra/Order/BigOperators/GroupWithZero/Multiset.lean b/Mathlib/Algebra/Order/BigOperators/GroupWithZero/Multiset.lean
new file mode 100644
index 0000000000000..7115296a5745e
--- /dev/null
+++ b/Mathlib/Algebra/Order/BigOperators/GroupWithZero/Multiset.lean
@@ -0,0 +1,51 @@
+/-
+Copyright (c) 2021 Ruben Van de Velde. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Ruben Van de Velde, Daniel Weber
+-/
+import Mathlib.Algebra.Order.BigOperators.GroupWithZero.List
+import Mathlib.Algebra.BigOperators.Group.Multiset
+
+/-!
+# Big operators on a multiset in ordered groups with zeros
+
+This file contains the results concerning the interaction of multiset big operators with ordered
+groups with zeros.
+-/
+
+namespace Multiset
+variable {R : Type*} [CommMonoidWithZero R] [PartialOrder R] [ZeroLEOneClass R] [PosMulMono R]
+
+lemma prod_nonneg {s : Multiset R} (h : ∀ a ∈ s, 0 ≤ a) : 0 ≤ s.prod := by
+  cases s using Quotient.ind
+  simp only [quot_mk_to_coe, mem_coe, prod_coe] at *
+  apply List.prod_nonneg h
+
+lemma one_le_prod {s : Multiset R} (h : ∀ a ∈ s, 1 ≤ a) : 1 ≤ s.prod := by
+  cases s using Quotient.ind
+  simp only [quot_mk_to_coe, mem_coe, prod_coe] at *
+  apply List.one_le_prod h
+
+theorem prod_map_le_prod_map₀ {ι : Type*} {s : Multiset ι} (f : ι → R) (g : ι → R)
+    (h0 : ∀ i ∈ s, 0 ≤ f i) (h : ∀ i ∈ s, f i ≤ g i) :
+    (map f s).prod ≤ (map g s).prod := by
+  cases s using Quotient.ind
+  simp only [quot_mk_to_coe, mem_coe, map_coe, prod_coe] at *
+  apply List.prod_map_le_prod_map₀ f g h0 h
+
+omit [PosMulMono R]
+variable [PosMulStrictMono R] [NeZero (1 : R)]
+
+lemma prod_pos {s : Multiset R} (h : ∀ a ∈ s, 0 < a) : 0 < s.prod := by
+  cases s using Quotient.ind
+  simp only [quot_mk_to_coe, mem_coe, map_coe, prod_coe] at *
+  apply List.prod_pos h
+
+theorem prod_map_lt_prod_map {ι : Type*} {s : Multiset ι} (hs : s ≠ 0)
+    (f : ι → R) (g : ι → R) (h0 : ∀ i ∈ s, 0 < f i) (h : ∀ i ∈ s, f i < g i) :
+    (map f s).prod < (map g s).prod := by
+  cases s using Quotient.ind
+  simp only [quot_mk_to_coe, mem_coe, map_coe, prod_coe, ne_eq, coe_eq_zero] at *
+  apply List.prod_map_lt_prod_map hs f g h0 h
+
+end Multiset
diff --git a/Mathlib/Algebra/Order/BigOperators/Ring/List.lean b/Mathlib/Algebra/Order/BigOperators/Ring/List.lean
index 2a108cb55955b..d728f2a6a84bf 100644
--- a/Mathlib/Algebra/Order/BigOperators/Ring/List.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Ring/List.lean
@@ -14,23 +14,10 @@ This file contains the results concerning the interaction of list big operators
 
 variable {R : Type*}
 
-namespace List
-
-/-- The product of a list of positive natural numbers is positive,
-and likewise for any nontrivial ordered semiring. -/
-lemma prod_pos [StrictOrderedSemiring R] (l : List R) (h : ∀ a ∈ l, (0 : R) < a) :
-    0 < l.prod := by
-  induction' l with a l ih
-  · simp
-  · rw [prod_cons]
-    exact mul_pos (h _ <| mem_cons_self _ _) (ih fun a ha => h a <| mem_cons_of_mem _ ha)
-
 /-- A variant of `List.prod_pos` for `CanonicallyOrderedCommSemiring`. -/
-@[simp] lemma _root_.CanonicallyOrderedCommSemiring.list_prod_pos
+@[simp] lemma CanonicallyOrderedCommSemiring.list_prod_pos
     {α : Type*} [CanonicallyOrderedCommSemiring α] [Nontrivial α] :
     ∀ {l : List α}, 0 < l.prod ↔ (∀ x ∈ l, (0 : α) < x)
   | [] => by simp
-  | (x :: xs) => by simp_rw [prod_cons, forall_mem_cons, CanonicallyOrderedCommSemiring.mul_pos,
-    list_prod_pos]
-
-end List
+  | (x :: xs) => by simp_rw [List.prod_cons, List.forall_mem_cons,
+      CanonicallyOrderedCommSemiring.mul_pos, list_prod_pos]
diff --git a/Mathlib/Algebra/Order/BigOperators/Ring/Multiset.lean b/Mathlib/Algebra/Order/BigOperators/Ring/Multiset.lean
index 3ca2b1b7c918c..0b91673c65582 100644
--- a/Mathlib/Algebra/Order/BigOperators/Ring/Multiset.lean
+++ b/Mathlib/Algebra/Order/BigOperators/Ring/Multiset.lean
@@ -13,26 +13,10 @@ This file contains the results concerning the interaction of multiset big operat
 rings.
 -/
 
-namespace Multiset
-variable {R : Type*}
-
-section OrderedCommSemiring
-variable [OrderedCommSemiring R] {s : Multiset R}
-
-lemma prod_nonneg (h : ∀ a ∈ s, 0 ≤ a) : 0 ≤ s.prod := by
-  revert h
-  refine s.induction_on ?_ fun a s hs ih ↦ ?_
-  · simp
-  · rw [prod_cons]
-    exact mul_nonneg (ih _ <| mem_cons_self _ _) (hs fun a ha ↦ ih _ <| mem_cons_of_mem ha)
-
-end OrderedCommSemiring
-
 @[simp]
-lemma _root_.CanonicallyOrderedCommSemiring.multiset_prod_pos [CanonicallyOrderedCommSemiring R]
-    [Nontrivial R] {m : Multiset R} : 0 < m.prod ↔ ∀ x ∈ m, 0 < x := by
+lemma CanonicallyOrderedCommSemiring.multiset_prod_pos {R : Type*}
+    [CanonicallyOrderedCommSemiring R] [Nontrivial R] {m : Multiset R} :
+    0 < m.prod ↔ ∀ x ∈ m, 0 < x := by
   rcases m with ⟨l⟩
   rw [Multiset.quot_mk_to_coe'', Multiset.prod_coe]
   exact CanonicallyOrderedCommSemiring.list_prod_pos
-
-end Multiset
diff --git a/Mathlib/Algebra/Order/CauSeq/Basic.lean b/Mathlib/Algebra/Order/CauSeq/Basic.lean
index 461ef07e12d62..b390f7b8dceaf 100644
--- a/Mathlib/Algebra/Order/CauSeq/Basic.lean
+++ b/Mathlib/Algebra/Order/CauSeq/Basic.lean
@@ -678,7 +678,7 @@ instance : Preorder (CauSeq α abs) where
     | Or.inr fg, Or.inl gh => Or.inl <| lt_of_eq_of_lt fg gh
     | Or.inr fg, Or.inr gh => Or.inr <| Setoid.trans fg gh
   lt_iff_le_not_le _ _ :=
-    ⟨fun h => ⟨Or.inl h, not_or_of_not (mt (lt_trans h) lt_irrefl) (not_limZero_of_pos h)⟩,
+    ⟨fun h => ⟨Or.inl h, not_or_intro (mt (lt_trans h) lt_irrefl) (not_limZero_of_pos h)⟩,
       fun ⟨h₁, h₂⟩ => h₁.resolve_right (mt (fun h => Or.inr (Setoid.symm h)) h₂)⟩
 
 theorem le_antisymm {f g : CauSeq α abs} (fg : f ≤ g) (gf : g ≤ f) : f ≈ g :=
diff --git a/Mathlib/Algebra/Order/CauSeq/BigOperators.lean b/Mathlib/Algebra/Order/CauSeq/BigOperators.lean
index 2ac291cf0c7a6..96a11d63d193a 100644
--- a/Mathlib/Algebra/Order/CauSeq/BigOperators.lean
+++ b/Mathlib/Algebra/Order/CauSeq/BigOperators.lean
@@ -188,7 +188,7 @@ lemma geo_series [Nontrivial β] (x : β) (hx1 : abv x < 1) :
     · gcongr
       exact sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)
     refine div_nonneg (sub_nonneg.2 ?_) (sub_nonneg.2 <| le_of_lt hx1)
-    exact pow_le_one _ (by positivity) hx1.le
+    exact pow_le_one₀ (by positivity) hx1.le
   · intro n _
     rw [← one_mul (abv x ^ n), pow_succ']
     gcongr
diff --git a/Mathlib/Algebra/Order/Chebyshev.lean b/Mathlib/Algebra/Order/Chebyshev.lean
index ebbe5643ee00c..439d79b0c4e2b 100644
--- a/Mathlib/Algebra/Order/Chebyshev.lean
+++ b/Mathlib/Algebra/Order/Chebyshev.lean
@@ -4,9 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mantas Bakšys, Yaël Dillies
 -/
 import Mathlib.Algebra.Order.BigOperators.Group.Finset
+import Mathlib.Algebra.Order.Monovary
 import Mathlib.Algebra.Order.Rearrangement
-import Mathlib.Algebra.Order.Ring.Basic
 import Mathlib.GroupTheory.Perm.Cycle.Basic
+import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Positivity.Basic
+import Mathlib.Tactic.Positivity.Finset
 
 /-!
 # Chebyshev's sum inequality
@@ -44,28 +47,26 @@ variable {ι α β : Type*}
 
 
 section SMul
-
-variable [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β]
-  {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
+variable [LinearOrderedSemiring α] [ExistsAddOfLE α] [LinearOrderedCancelAddCommMonoid β]
+  [Module α β] [OrderedSMul α β] {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
 
 /-- **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
 monotone/antitone), the scalar product of their sum is less than the size of the set times their
 scalar product. -/
 theorem MonovaryOn.sum_smul_sum_le_card_smul_sum (hfg : MonovaryOn f g s) :
-    ((∑ i ∈ s, f i) • ∑ i ∈ s, g i) ≤ s.card • ∑ i ∈ s, f i • g i := by
+    (∑ i ∈ s, f i) • ∑ i ∈ s, g i ≤ s.card • ∑ i ∈ s, f i • g i := by
   classical
-    obtain ⟨σ, hσ, hs⟩ := s.countable_toSet.exists_cycleOn
-    rw [← card_range s.card, sum_smul_sum_eq_sum_perm hσ]
-    exact
-      sum_le_card_nsmul _ _ _ fun n _ =>
-        hfg.sum_smul_comp_perm_le_sum_smul fun x hx => hs fun h => hx <| IsFixedPt.perm_pow h _
+  obtain ⟨σ, hσ, hs⟩ := s.countable_toSet.exists_cycleOn
+  rw [← card_range s.card, sum_smul_sum_eq_sum_perm hσ]
+  exact sum_le_card_nsmul _ _ _ fun n _ ↦
+    hfg.sum_smul_comp_perm_le_sum_smul fun x hx ↦ hs fun h ↦ hx <| IsFixedPt.perm_pow h _
 
 /-- **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
 other is antitone), the scalar product of their sum is less than the size of the set times their
 scalar product. -/
 theorem AntivaryOn.card_smul_sum_le_sum_smul_sum (hfg : AntivaryOn f g s) :
-    (s.card • ∑ i ∈ s, f i • g i) ≤ (∑ i ∈ s, f i) • ∑ i ∈ s, g i := by
-  exact hfg.dual_right.sum_smul_sum_le_card_smul_sum
+    s.card • ∑ i ∈ s, f i • g i ≤ (∑ i ∈ s, f i) • ∑ i ∈ s, g i :=
+  hfg.dual_right.sum_smul_sum_le_card_smul_sum
 
 variable [Fintype ι]
 
@@ -73,15 +74,15 @@ variable [Fintype ι]
 monotone/antitone), the scalar product of their sum is less than the size of the set times their
 scalar product. -/
 theorem Monovary.sum_smul_sum_le_card_smul_sum (hfg : Monovary f g) :
-    ((∑ i, f i) • ∑ i, g i) ≤ Fintype.card ι • ∑ i, f i • g i :=
+    (∑ i, f i) • ∑ i, g i ≤ Fintype.card ι • ∑ i, f i • g i :=
   (hfg.monovaryOn _).sum_smul_sum_le_card_smul_sum
 
 /-- **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
 other is antitone), the scalar product of their sum is less than the size of the set times their
 scalar product. -/
 theorem Antivary.card_smul_sum_le_sum_smul_sum (hfg : Antivary f g) :
-    (Fintype.card ι • ∑ i, f i • g i) ≤ (∑ i, f i) • ∑ i, g i := by
-  exact (hfg.dual_right.monovaryOn _).sum_smul_sum_le_card_smul_sum
+    Fintype.card ι • ∑ i, f i • g i ≤ (∑ i, f i) • ∑ i, g i :=
+  (hfg.dual_right.monovaryOn _).sum_smul_sum_le_card_smul_sum
 
 end SMul
 
@@ -93,14 +94,13 @@ Special cases of the above when scalar multiplication is actually multiplication
 
 
 section Mul
-
-variable [LinearOrderedRing α] {s : Finset ι} {σ : Perm ι} {f g : ι → α}
+variable [LinearOrderedSemiring α] [ExistsAddOfLE α] {s : Finset ι} {σ : Perm ι} {f g : ι → α}
 
 /-- **Chebyshev's Sum Inequality**: When `f` and `g` monovary together (eg they are both
 monotone/antitone), the product of their sum is less than the size of the set times their scalar
 product. -/
 theorem MonovaryOn.sum_mul_sum_le_card_mul_sum (hfg : MonovaryOn f g s) :
-    ((∑ i ∈ s, f i) * ∑ i ∈ s, g i) ≤ s.card * ∑ i ∈ s, f i * g i := by
+    (∑ i ∈ s, f i) * ∑ i ∈ s, g i ≤ s.card * ∑ i ∈ s, f i * g i := by
   rw [← nsmul_eq_mul]
   exact hfg.sum_smul_sum_le_card_smul_sum
 
@@ -108,10 +108,26 @@ theorem MonovaryOn.sum_mul_sum_le_card_mul_sum (hfg : MonovaryOn f g s) :
 other is antitone), the product of their sum is greater than the size of the set times their scalar
 product. -/
 theorem AntivaryOn.card_mul_sum_le_sum_mul_sum (hfg : AntivaryOn f g s) :
-    ((s.card : α) * ∑ i ∈ s, f i * g i) ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i := by
+    (s.card : α) * ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i) * ∑ i ∈ s, g i := by
   rw [← nsmul_eq_mul]
   exact hfg.card_smul_sum_le_sum_smul_sum
 
+/-- Special case of **Jensen's inequality** for sums of powers. -/
+lemma pow_sum_le_card_mul_sum_pow (hf : ∀ i ∈ s, 0 ≤ f i) :
+    ∀ n, (∑ i ∈ s, f i) ^ (n + 1) ≤ (s.card : α) ^ n * ∑ i ∈ s, f i ^ (n + 1)
+  | 0 => by simp
+  | n + 1 =>
+    calc
+      _ = (∑ i ∈ s, f i) ^ (n + 1) * ∑ i ∈ s, f i := by rw [pow_succ]
+      _ ≤ (s.card ^ n * ∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i := by
+        gcongr
+        exacts [sum_nonneg hf, pow_sum_le_card_mul_sum_pow hf _]
+      _ = s.card ^ n * ((∑ i ∈ s, f i ^ (n + 1)) * ∑ i ∈ s, f i) := by rw [mul_assoc]
+      _ ≤ s.card ^ n * (s.card * ∑ i ∈ s, f i ^ (n + 1) * f i) := by
+        gcongr _ * ?_
+        exact ((monovaryOn_self ..).pow_left₀ hf _).sum_mul_sum_le_card_mul_sum
+      _ = _ := by simp_rw [← mul_assoc, ← pow_succ]
+
 /-- Special case of **Chebyshev's Sum Inequality** or the **Cauchy-Schwarz Inequality**: The square
 of the sum is less than the size of the set times the sum of the squares. -/
 theorem sq_sum_le_card_mul_sum_sq : (∑ i ∈ s, f i) ^ 2 ≤ s.card * ∑ i ∈ s, f i ^ 2 := by
@@ -124,25 +140,32 @@ variable [Fintype ι]
 monotone/antitone), the product of their sum is less than the size of the set times their scalar
 product. -/
 theorem Monovary.sum_mul_sum_le_card_mul_sum (hfg : Monovary f g) :
-    ((∑ i, f i) * ∑ i, g i) ≤ Fintype.card ι * ∑ i, f i * g i :=
+    (∑ i, f i) * ∑ i, g i ≤ Fintype.card ι * ∑ i, f i * g i :=
   (hfg.monovaryOn _).sum_mul_sum_le_card_mul_sum
 
 /-- **Chebyshev's Sum Inequality**: When `f` and `g` antivary together (eg one is monotone, the
 other is antitone), the product of their sum is less than the size of the set times their scalar
 product. -/
 theorem Antivary.card_mul_sum_le_sum_mul_sum (hfg : Antivary f g) :
-    ((Fintype.card ι : α) * ∑ i, f i * g i) ≤ (∑ i, f i) * ∑ i, g i :=
+    Fintype.card ι * ∑ i, f i * g i ≤ (∑ i, f i) * ∑ i, g i :=
   (hfg.antivaryOn _).card_mul_sum_le_sum_mul_sum
 
 end Mul
 
-variable [LinearOrderedField α] {s : Finset ι} {f : ι → α}
+variable [LinearOrderedSemifield α] [ExistsAddOfLE α] {s : Finset ι} {f : ι → α}
+
+/-- Special case of **Jensen's inequality** for sums of powers. -/
+lemma pow_sum_div_card_le_sum_pow (hf : ∀ i ∈ s, 0 ≤ f i) (n : ℕ) :
+    (∑ i ∈ s, f i) ^ (n + 1) / s.card ^ n ≤ ∑ i ∈ s, f i ^ (n + 1) := by
+  obtain rfl | hs := s.eq_empty_or_nonempty
+  · simp
+  rw [div_le_iff₀' (by positivity)]
+  exact pow_sum_le_card_mul_sum_pow hf _
 
 theorem sum_div_card_sq_le_sum_sq_div_card :
     ((∑ i ∈ s, f i) / s.card) ^ 2 ≤ (∑ i ∈ s, f i ^ 2) / s.card := by
   obtain rfl | hs := s.eq_empty_or_nonempty
   · simp
-  rw [← card_pos, ← @Nat.cast_pos α] at hs
-  rw [div_pow, div_le_div_iff (sq_pos_of_ne_zero hs.ne') hs, sq (s.card : α), mul_left_comm, ←
-    mul_assoc]
-  exact mul_le_mul_of_nonneg_right sq_sum_le_card_mul_sum_sq hs.le
+  rw [div_pow, div_le_div_iff (by positivity) (by positivity), sq (s.card : α), mul_left_comm,
+    ← mul_assoc]
+  exact mul_le_mul_of_nonneg_right sq_sum_le_card_mul_sum_sq (by positivity)
diff --git a/Mathlib/Algebra/Order/CompleteField.lean b/Mathlib/Algebra/Order/CompleteField.lean
index 2c8942e53c15c..680ebceb3f631 100644
--- a/Mathlib/Algebra/Order/CompleteField.lean
+++ b/Mathlib/Algebra/Order/CompleteField.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alex J. Best, Yaël Dillies
 -/
 import Mathlib.Algebra.Order.Archimedean.Hom
-import Mathlib.Algebra.Order.Pointwise
+import Mathlib.Algebra.Order.Group.Pointwise.CompleteLattice
 import Mathlib.Analysis.SpecialFunctions.Pow.Real
 
 /-!
diff --git a/Mathlib/Algebra/Order/Field/Basic.lean b/Mathlib/Algebra/Order/Field/Basic.lean
index 10cb8d1525691..4e76076e79d17 100644
--- a/Mathlib/Algebra/Order/Field/Basic.lean
+++ b/Mathlib/Algebra/Order/Field/Basic.lean
@@ -5,9 +5,9 @@ Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
 -/
 import Mathlib.Algebra.CharZero.Lemmas
 import Mathlib.Algebra.Order.Field.Defs
+import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Lemmas
 import Mathlib.Algebra.Order.Ring.Abs
 import Mathlib.Order.Bounds.OrderIso
-import Mathlib.Tactic.Bound.Attribute
 import Mathlib.Tactic.Positivity.Core
 
 /-!
@@ -23,171 +23,143 @@ section LinearOrderedSemifield
 
 variable [LinearOrderedSemifield α] {a b c d e : α} {m n : ℤ}
 
-/-- `Equiv.mulLeft₀` as an order_iso. -/
-@[simps! (config := { simpRhs := true })]
-def OrderIso.mulLeft₀ (a : α) (ha : 0 < a) : α ≃o α :=
-  { Equiv.mulLeft₀ a ha.ne' with map_rel_iff' := @fun _ _ => mul_le_mul_left ha }
-
-/-- `Equiv.mulRight₀` as an order_iso. -/
-@[simps! (config := { simpRhs := true })]
-def OrderIso.mulRight₀ (a : α) (ha : 0 < a) : α ≃o α :=
-  { Equiv.mulRight₀ a ha.ne' with map_rel_iff' := @fun _ _ => mul_le_mul_right ha }
-
 /-!
 ### Relating one division with another term.
 -/
 
-theorem lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b :=
-  lt_iff_lt_of_le_iff_le <| div_le_iff₀ hc
-
-theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b := by rw [mul_comm, lt_div_iff hc]
+@[deprecated lt_div_iff₀ (since := "2024-10-02")]
+theorem lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b := lt_div_iff₀ hc
 
-theorem div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c :=
-  lt_iff_lt_of_le_iff_le (le_div_iff₀ hc)
+@[deprecated lt_div_iff₀' (since := "2024-10-02")]
+theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b := lt_div_iff₀' hc
 
-theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a := by rw [mul_comm, div_lt_iff hc]
+@[deprecated div_lt_iff₀ (since := "2024-10-02")]
+theorem div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c := div_lt_iff₀ hc
 
-lemma div_lt_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b < c ↔ a / c < b := by
-  rw [div_lt_iff hb, div_lt_iff' hc]
+@[deprecated div_lt_iff₀' (since := "2024-10-02")]
+theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a := div_lt_iff₀' hc
 
-theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c := by
-  rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
-  exact div_le_iff₀' h
+@[deprecated inv_mul_le_iff₀ (since := "2024-10-02")]
+theorem inv_mul_le_iff (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ b * c := inv_mul_le_iff₀ h
 
-theorem inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b := by rw [inv_mul_le_iff h, mul_comm]
+@[deprecated inv_mul_le_iff₀' (since := "2024-10-02")]
+theorem inv_mul_le_iff' (h : 0 < b) : b⁻¹ * a ≤ c ↔ a ≤ c * b := inv_mul_le_iff₀' h
 
-theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c := by rw [mul_comm, inv_mul_le_iff h]
+@[deprecated mul_inv_le_iff₀' (since := "2024-10-02")]
+theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c := mul_inv_le_iff₀' h
 
-theorem mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b := by rw [mul_comm, inv_mul_le_iff' h]
+@[deprecated mul_inv_le_iff₀ (since := "2024-10-02")]
+theorem mul_inv_le_iff' (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ c * b := mul_inv_le_iff₀ h
 
-theorem div_self_le_one (a : α) : a / a ≤ 1 :=
-  if h : a = 0 then by simp [h] else by simp [h]
+@[deprecated inv_mul_lt_iff₀ (since := "2024-10-02")]
+theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c := inv_mul_lt_iff₀ h
 
-theorem inv_mul_lt_iff (h : 0 < b) : b⁻¹ * a < c ↔ a < b * c := by
-  rw [inv_eq_one_div, mul_comm, ← div_eq_mul_one_div]
-  exact div_lt_iff' h
+@[deprecated inv_mul_lt_iff₀' (since := "2024-10-02")]
+theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b := inv_mul_lt_iff₀' h
 
-theorem inv_mul_lt_iff' (h : 0 < b) : b⁻¹ * a < c ↔ a < c * b := by rw [inv_mul_lt_iff h, mul_comm]
+@[deprecated mul_inv_lt_iff₀' (since := "2024-10-02")]
+theorem mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c := mul_inv_lt_iff₀' h
 
-theorem mul_inv_lt_iff (h : 0 < b) : a * b⁻¹ < c ↔ a < b * c := by rw [mul_comm, inv_mul_lt_iff h]
+@[deprecated mul_inv_lt_iff₀ (since := "2024-10-02")]
+theorem mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b := mul_inv_lt_iff₀ h
 
-theorem mul_inv_lt_iff' (h : 0 < b) : a * b⁻¹ < c ↔ a < c * b := by rw [mul_comm, inv_mul_lt_iff' h]
+@[deprecated inv_le_iff_one_le_mul₀ (since := "2024-10-03")]
+theorem inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a := inv_le_iff_one_le_mul₀ ha
 
-theorem inv_pos_le_iff_one_le_mul (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a := by
-  rw [inv_eq_one_div]
-  exact div_le_iff₀ ha
+@[deprecated inv_le_iff_one_le_mul₀' (since := "2024-10-03")]
+theorem inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b := inv_le_iff_one_le_mul₀' ha
 
-theorem inv_pos_le_iff_one_le_mul' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by
-  rw [inv_eq_one_div]
-  exact div_le_iff₀' ha
+@[deprecated inv_lt_iff_one_lt_mul₀ (since := "2024-10-03")]
+theorem inv_pos_lt_iff_one_lt_mul (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a := inv_lt_iff_one_lt_mul₀ ha
 
-theorem inv_pos_lt_iff_one_lt_mul (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a := by
-  rw [inv_eq_one_div]
-  exact div_lt_iff ha
-
-theorem inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b := by
-  rw [inv_eq_one_div]
-  exact div_lt_iff' ha
+@[deprecated inv_lt_iff_one_lt_mul₀' (since := "2024-10-03")]
+theorem inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b := inv_lt_iff_one_lt_mul₀' ha
 
 /-- One direction of `div_le_iff` where `b` is allowed to be `0` (but `c` must be nonnegative) -/
-theorem div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c := by
-  rcases eq_or_lt_of_le hb with (rfl | hb')
-  · simp only [div_zero, hc]
-  · rwa [div_le_iff₀ hb']
+@[deprecated div_le_of_le_mul₀ (since := "2024-10-03")]
+theorem div_le_of_nonneg_of_le_mul (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c :=
+  div_le_of_le_mul₀ hb hc h
 
 /-- One direction of `div_le_iff` where `c` is allowed to be `0` (but `b` must be nonnegative) -/
-lemma mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b := by
-  obtain rfl | hc := hc.eq_or_lt
-  · simpa using hb
-  · rwa [le_div_iff₀ hc] at h
+@[deprecated mul_le_of_le_div₀ (since := "2024-10-03")]
+lemma mul_le_of_nonneg_of_le_div (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b :=
+  mul_le_of_le_div₀ hb hc h
 
-@[bound]
-theorem div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 :=
-  div_le_of_nonneg_of_le_mul hb zero_le_one <| by rwa [one_mul]
+@[deprecated div_le_one_of_le₀ (since := "2024-10-03")]
+theorem div_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 := div_le_one_of_le₀ h hb
 
-@[bound]
-lemma mul_inv_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 := by
-  simpa only [← div_eq_mul_inv] using div_le_one_of_le h hb
+@[deprecated mul_inv_le_one_of_le₀ (since := "2024-10-03")]
+lemma mul_inv_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 := mul_inv_le_one_of_le₀ h hb
 
-@[bound]
-lemma inv_mul_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 := by
-  simpa only [← div_eq_inv_mul] using div_le_one_of_le h hb
+@[deprecated inv_mul_le_one_of_le₀ (since := "2024-10-03")]
+lemma inv_mul_le_one_of_le (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 := inv_mul_le_one_of_le₀ h hb
 
 /-!
 ### Bi-implications of inequalities using inversions
 -/
 
-@[gcongr, bound]
-theorem inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ := by
-  rwa [← one_div a, le_div_iff₀' ha, ← div_eq_mul_inv, div_le_iff₀ (ha.trans_le h), one_mul]
+@[deprecated inv_anti₀ (since := "2024-10-05")]
+theorem inv_le_inv_of_le (ha : 0 < a) (h : a ≤ b) : b⁻¹ ≤ a⁻¹ := inv_anti₀ ha h
 
 /-- See `inv_le_inv_of_le` for the implication from right-to-left with one fewer assumption. -/
-theorem inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by
-  rw [← one_div, div_le_iff₀ ha, ← div_eq_inv_mul, le_div_iff₀ hb, one_mul]
+@[deprecated inv_le_inv₀ (since := "2024-10-05")]
+theorem inv_le_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := inv_le_inv₀ ha hb
 
 /-- In a linear ordered field, for positive `a` and `b` we have `a⁻¹ ≤ b ↔ b⁻¹ ≤ a`.
 See also `inv_le_of_inv_le` for a one-sided implication with one fewer assumption. -/
-theorem inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by
-  rw [← inv_le_inv hb (inv_pos.2 ha), inv_inv]
+@[deprecated inv_le_comm₀ (since := "2024-10-05")]
+theorem inv_le (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := inv_le_comm₀ ha hb
 
-theorem inv_le_of_inv_le (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a :=
-  (inv_le ha ((inv_pos.2 ha).trans_le h)).1 h
+@[deprecated inv_le_of_inv_le₀ (since := "2024-10-05")]
+theorem inv_le_of_inv_le (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a := inv_le_of_inv_le₀ ha h
 
-theorem le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := by
-  rw [← inv_le_inv (inv_pos.2 hb) ha, inv_inv]
+@[deprecated le_inv_comm₀ (since := "2024-10-05")]
+theorem le_inv (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := le_inv_comm₀ ha hb
 
 /-- See `inv_lt_inv_of_lt` for the implication from right-to-left with one fewer assumption. -/
-theorem inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a :=
-  lt_iff_lt_of_le_iff_le (inv_le_inv hb ha)
+@[deprecated inv_lt_inv₀ (since := "2024-10-05")]
+theorem inv_lt_inv (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a := inv_lt_inv₀ ha hb
 
-@[gcongr]
-theorem inv_lt_inv_of_lt (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹ :=
-  (inv_lt_inv (hb.trans h) hb).2 h
+@[deprecated inv_strictAnti₀ (since := "2024-10-05")]
+theorem inv_lt_inv_of_lt (hb : 0 < b) (h : b < a) : a⁻¹ < b⁻¹ := inv_strictAnti₀ hb h
 
 /-- In a linear ordered field, for positive `a` and `b` we have `a⁻¹ < b ↔ b⁻¹ < a`.
 See also `inv_lt_of_inv_lt` for a one-sided implication with one fewer assumption. -/
-theorem inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a :=
-  lt_iff_lt_of_le_iff_le (le_inv hb ha)
+@[deprecated inv_lt_comm₀ (since := "2024-10-05")]
+theorem inv_lt (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a := inv_lt_comm₀ ha hb
 
-theorem inv_lt_of_inv_lt (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a :=
-  (inv_lt ha ((inv_pos.2 ha).trans h)).1 h
+@[deprecated inv_lt_of_inv_lt₀ (since := "2024-10-05")]
+theorem inv_lt_of_inv_lt (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a := inv_lt_of_inv_lt₀ ha h
 
-theorem lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ :=
-  lt_iff_lt_of_le_iff_le (inv_le hb ha)
+@[deprecated lt_inv_comm₀ (since := "2024-10-05")]
+theorem lt_inv (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ := lt_inv_comm₀ ha hb
 
-theorem inv_lt_one (ha : 1 < a) : a⁻¹ < 1 := by
-  rwa [inv_lt (zero_lt_one.trans ha) zero_lt_one, inv_one]
+@[deprecated inv_lt_one_of_one_lt₀ (since := "2024-10-05")]
+theorem inv_lt_one (ha : 1 < a) : a⁻¹ < 1 := inv_lt_one_of_one_lt₀ ha
 
-theorem one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ := by
-  rwa [lt_inv (@zero_lt_one α _ _ _ _ _) h₁, inv_one]
+@[deprecated one_lt_inv₀ (since := "2024-10-05")]
+theorem one_lt_inv (h₁ : 0 < a) (h₂ : a < 1) : 1 < a⁻¹ := (one_lt_inv₀ h₁).2 h₂
 
-@[bound]
-theorem inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 := by
-  rwa [inv_le (zero_lt_one.trans_le ha) zero_lt_one, inv_one]
+@[deprecated inv_le_one_of_one_le₀ (since := "2024-10-05")]
+theorem inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 := inv_le_one_of_one_le₀ ha
 
-theorem one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ := by
-  rwa [le_inv (@zero_lt_one α _ _ _ _ _) h₁, inv_one]
+@[deprecated one_le_inv₀ (since := "2024-10-05")]
+theorem one_le_inv (h₁ : 0 < a) (h₂ : a ≤ 1) : 1 ≤ a⁻¹ := (one_le_inv₀ h₁).2 h₂
 
-theorem inv_lt_one_iff_of_pos (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a :=
-  ⟨fun h₁ => inv_inv a ▸ one_lt_inv (inv_pos.2 h₀) h₁, inv_lt_one⟩
+@[deprecated inv_lt_one₀ (since := "2024-10-05")]
+theorem inv_lt_one_iff_of_pos (h₀ : 0 < a) : a⁻¹ < 1 ↔ 1 < a := inv_lt_one₀ h₀
 
-theorem inv_lt_one_iff : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a := by
-  rcases le_or_lt a 0 with ha | ha
-  · simp [ha, (inv_nonpos.2 ha).trans_lt zero_lt_one]
-  · simp only [ha.not_le, false_or_iff, inv_lt_one_iff_of_pos ha]
+@[deprecated inv_lt_one_iff₀ (since := "2024-10-05")]
+theorem inv_lt_one_iff : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a := inv_lt_one_iff₀
 
-theorem one_lt_inv_iff : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 :=
-  ⟨fun h => ⟨inv_pos.1 (zero_lt_one.trans h),
-    inv_inv a ▸ inv_lt_one h⟩, and_imp.2 one_lt_inv⟩
+@[deprecated one_lt_inv_iff₀ (since := "2024-10-05")]
+theorem one_lt_inv_iff : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 := one_lt_inv_iff₀
 
-theorem inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a := by
-  rcases em (a = 1) with (rfl | ha)
-  · simp [le_rfl]
-  · simp only [Ne.le_iff_lt (Ne.symm ha), Ne.le_iff_lt (mt inv_eq_one.1 ha), inv_lt_one_iff]
+@[deprecated inv_le_one_iff₀ (since := "2024-10-05")]
+theorem inv_le_one_iff : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a := inv_le_one_iff₀
 
-theorem one_le_inv_iff : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 :=
-  ⟨fun h => ⟨inv_pos.1 (zero_lt_one.trans_le h),
-    inv_inv a ▸ inv_le_one h⟩, and_imp.2 one_le_inv⟩
+@[deprecated one_le_inv_iff₀ (since := "2024-10-05")]
+theorem one_le_inv_iff : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 := one_le_inv_iff₀
 
 /-!
 ### Relating two divisions.
@@ -208,11 +180,11 @@ lemma div_lt_div_of_pos_right (h : a < b) (hc : 0 < c) : a / c < b / c := by
 @[gcongr]
 lemma div_le_div_of_nonneg_left (ha : 0 ≤ a) (hc : 0 < c) (h : c ≤ b) : a / b ≤ a / c := by
   rw [div_eq_mul_inv, div_eq_mul_inv]
-  exact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha
+  exact mul_le_mul_of_nonneg_left ((inv_le_inv₀ (hc.trans_le h) hc).mpr h) ha
 
 @[gcongr, bound]
 lemma div_lt_div_of_pos_left (ha : 0 < a) (hc : 0 < c) (h : c < b) : a / b < a / c := by
-  simpa only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv (hc.trans h) hc]
+  simpa only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv₀ (hc.trans h) hc]
 
 @[deprecated (since := "2024-02-16")] alias div_le_div_of_le_of_nonneg := div_le_div_of_nonneg_right
 @[deprecated (since := "2024-02-16")] alias div_lt_div_of_lt := div_lt_div_of_pos_right
@@ -231,13 +203,13 @@ theorem div_lt_div_right (hc : 0 < c) : a / c < b / c ↔ a < b :=
   lt_iff_lt_of_le_iff_le <| div_le_div_right hc
 
 theorem div_lt_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b < a / c ↔ c < b := by
-  simp only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv hb hc]
+  simp only [div_eq_mul_inv, mul_lt_mul_left ha, inv_lt_inv₀ hb hc]
 
 theorem div_le_div_left (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a / b ≤ a / c ↔ c ≤ b :=
   le_iff_le_iff_lt_iff_lt.2 (div_lt_div_left ha hc hb)
 
 theorem div_lt_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b < c / d ↔ a * d < c * b := by
-  rw [lt_div_iff d0, div_mul_eq_mul_div, div_lt_iff b0]
+  rw [lt_div_iff₀ d0, div_mul_eq_mul_div, div_lt_iff₀ b0]
 
 theorem div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b := by
   rw [le_div_iff₀ d0, div_mul_eq_mul_div, div_le_iff₀ b0]
@@ -275,17 +247,21 @@ theorem one_le_div (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a := by rw [le_div_iff
 
 theorem div_le_one (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b := by rw [div_le_iff₀ hb, one_mul]
 
-theorem one_lt_div (hb : 0 < b) : 1 < a / b ↔ b < a := by rw [lt_div_iff hb, one_mul]
+theorem one_lt_div (hb : 0 < b) : 1 < a / b ↔ b < a := by rw [lt_div_iff₀ hb, one_mul]
 
-theorem div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b := by rw [div_lt_iff hb, one_mul]
+theorem div_lt_one (hb : 0 < b) : a / b < 1 ↔ a < b := by rw [div_lt_iff₀ hb, one_mul]
 
-theorem one_div_le (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a := by simpa using inv_le ha hb
+theorem one_div_le (ha : 0 < a) (hb : 0 < b) : 1 / a ≤ b ↔ 1 / b ≤ a := by
+  simpa using inv_le_comm₀ ha hb
 
-theorem one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a := by simpa using inv_lt ha hb
+theorem one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a := by
+  simpa using inv_lt_comm₀ ha hb
 
-theorem le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a := by simpa using le_inv ha hb
+theorem le_one_div (ha : 0 < a) (hb : 0 < b) : a ≤ 1 / b ↔ b ≤ 1 / a := by
+  simpa using le_inv_comm₀ ha hb
 
-theorem lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a := by simpa using lt_inv ha hb
+theorem lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a := by
+  simpa using lt_inv_comm₀ ha hb
 
 @[bound] lemma Bound.one_lt_div_of_pos_of_lt (b0 : 0 < b) : b < a → 1 < a / b := (one_lt_div b0).mpr
 
@@ -297,10 +273,10 @@ theorem lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a := by sim
 
 
 theorem one_div_le_one_div_of_le (ha : 0 < a) (h : a ≤ b) : 1 / b ≤ 1 / a := by
-  simpa using inv_le_inv_of_le ha h
+  simpa using inv_anti₀ ha h
 
 theorem one_div_lt_one_div_of_lt (ha : 0 < a) (h : a < b) : 1 / b < 1 / a := by
-  rwa [lt_div_iff' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]
+  rwa [lt_div_iff₀' ha, ← div_eq_mul_one_div, div_lt_one (ha.trans h)]
 
 theorem le_of_one_div_le_one_div (ha : 0 < a) (h : 1 / a ≤ 1 / b) : b ≤ a :=
   le_imp_le_of_lt_imp_lt (one_div_lt_one_div_of_lt ha) h
@@ -341,7 +317,7 @@ theorem half_le_self_iff : a / 2 ≤ a ↔ 0 ≤ a := by
 
 @[simp]
 theorem half_lt_self_iff : a / 2 < a ↔ 0 < a := by
-  rw [div_lt_iff (zero_lt_two' α), mul_two, lt_add_iff_pos_left]
+  rw [div_lt_iff₀ (zero_lt_two' α), mul_two, lt_add_iff_pos_left]
 
 alias ⟨_, half_le_self⟩ := half_le_self_iff
 
@@ -355,9 +331,9 @@ theorem one_half_lt_one : (1 / 2 : α) < 1 :=
 theorem two_inv_lt_one : (2⁻¹ : α) < 1 :=
   (one_div _).symm.trans_lt one_half_lt_one
 
-theorem left_lt_add_div_two : a < (a + b) / 2 ↔ a < b := by simp [lt_div_iff, mul_two]
+theorem left_lt_add_div_two : a < (a + b) / 2 ↔ a < b := by simp [lt_div_iff₀, mul_two]
 
-theorem add_div_two_lt_right : (a + b) / 2 < b ↔ a < b := by simp [div_lt_iff, mul_two]
+theorem add_div_two_lt_right : (a + b) / 2 < b ↔ a < b := by simp [div_lt_iff₀, mul_two]
 
 theorem add_thirds (a : α) : a / 3 + a / 3 + a / 3 = a := by
   rw [div_add_div_same, div_add_div_same, ← two_mul, ← add_one_mul 2 a, two_add_one_eq_three,
@@ -385,12 +361,12 @@ theorem div_mul_le_div_mul_of_div_le_div (h : a / b ≤ c / d) (he : 0 ≤ e) :
 theorem exists_pos_mul_lt {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b * c < a := by
   have : 0 < a / max (b + 1) 1 := div_pos h (lt_max_iff.2 (Or.inr zero_lt_one))
   refine ⟨a / max (b + 1) 1, this, ?_⟩
-  rw [← lt_div_iff this, div_div_cancel' h.ne']
+  rw [← lt_div_iff₀ this, div_div_cancel' h.ne']
   exact lt_max_iff.2 (Or.inl <| lt_add_one _)
 
 theorem exists_pos_lt_mul {a : α} (h : 0 < a) (b : α) : ∃ c : α, 0 < c ∧ b < c * a :=
   let ⟨c, hc₀, hc⟩ := exists_pos_mul_lt h b;
-  ⟨c⁻¹, inv_pos.2 hc₀, by rwa [← div_eq_inv_mul, lt_div_iff hc₀]⟩
+  ⟨c⁻¹, inv_pos.2 hc₀, by rwa [← div_eq_inv_mul, lt_div_iff₀ hc₀]⟩
 
 lemma monotone_div_right_of_nonneg (ha : 0 ≤ a) : Monotone (· / a) :=
   fun _b _c hbc ↦ div_le_div_of_nonneg_right hbc ha
@@ -429,7 +405,7 @@ theorem one_div_strictAntiOn : StrictAntiOn (fun x : α => 1 / x) (Set.Ioi 0) :=
 
 theorem one_div_pow_le_one_div_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) :
     1 / a ^ n ≤ 1 / a ^ m := by
-  refine (one_div_le_one_div ?_ ?_).mpr (pow_le_pow_right a1 mn) <;>
+  refine (one_div_le_one_div ?_ ?_).mpr (pow_right_mono₀ a1 mn) <;>
     exact pow_pos (zero_lt_one.trans_le a1) _
 
 theorem one_div_pow_lt_one_div_pow_of_lt (a1 : 1 < a) {m n : ℕ} (mn : m < n) :
@@ -444,7 +420,7 @@ theorem one_div_pow_strictAnti (a1 : 1 < a) : StrictAnti fun n : ℕ => 1 / a ^
   one_div_pow_lt_one_div_pow_of_lt a1
 
 theorem inv_strictAntiOn : StrictAntiOn (fun x : α => x⁻¹) (Set.Ioi 0) := fun _ hx _ hy xy =>
-  (inv_lt_inv hy hx).2 xy
+  (inv_lt_inv₀ hy hx).2 xy
 
 theorem inv_pow_le_inv_pow_of_le (a1 : 1 ≤ a) {m n : ℕ} (mn : m ≤ n) : (a ^ n)⁻¹ ≤ (a ^ m)⁻¹ := by
   convert one_div_pow_le_one_div_pow_of_le a1 mn using 1 <;> simp
@@ -538,7 +514,7 @@ theorem lt_div_iff_of_neg' (hc : c < 0) : a < b / c ↔ b < c * a := by
   rw [mul_comm, lt_div_iff_of_neg hc]
 
 theorem div_le_one_of_ge (h : b ≤ a) (hb : b ≤ 0) : a / b ≤ 1 := by
-  simpa only [neg_div_neg_eq] using div_le_one_of_le (neg_le_neg h) (neg_nonneg_of_nonpos hb)
+  simpa only [neg_div_neg_eq] using div_le_one_of_le₀ (neg_le_neg h) (neg_nonneg_of_nonpos hb)
 
 /-! ### Bi-implications of inequalities using inversions -/
 
@@ -569,7 +545,7 @@ theorem lt_inv_of_neg (ha : a < 0) (hb : b < 0) : a < b⁻¹ ↔ b < a⁻¹ :=
 theorem sub_inv_antitoneOn_Ioi :
     AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Ioi c) :=
   antitoneOn_iff_forall_lt.mpr fun _ ha _ hb hab ↦
-    inv_le_inv (sub_pos.mpr hb) (sub_pos.mpr ha) |>.mpr <| sub_le_sub (le_of_lt hab) le_rfl
+    inv_le_inv₀ (sub_pos.mpr hb) (sub_pos.mpr ha) |>.mpr <| sub_le_sub (le_of_lt hab) le_rfl
 
 theorem sub_inv_antitoneOn_Iio :
     AntitoneOn (fun x ↦ (x-c)⁻¹) (Set.Iio c) :=
@@ -723,11 +699,11 @@ theorem add_sub_div_two_lt (h : a < b) : a + (b - a) / 2 < b := by
 /-- An inequality involving `2`. -/
 theorem sub_one_div_inv_le_two (a2 : 2 ≤ a) : (1 - 1 / a)⁻¹ ≤ 2 := by
   -- Take inverses on both sides to obtain `2⁻¹ ≤ 1 - 1 / a`
-  refine (inv_le_inv_of_le (inv_pos.2 <| zero_lt_two' α) ?_).trans_eq (inv_inv (2 : α))
+  refine (inv_anti₀ (inv_pos.2 <| zero_lt_two' α) ?_).trans_eq (inv_inv (2 : α))
   -- move `1 / a` to the left and `2⁻¹` to the right.
   rw [le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le]
   -- take inverses on both sides and use the assumption `2 ≤ a`.
-  convert (one_div a).le.trans (inv_le_inv_of_le zero_lt_two a2) using 1
+  convert (one_div a).le.trans (inv_anti₀ zero_lt_two a2) using 1
   -- show `1 - 1 / 2 = 1 / 2`.
   rw [sub_eq_iff_eq_add, ← two_mul, mul_inv_cancel₀ two_ne_zero]
 
diff --git a/Mathlib/Algebra/Order/Field/Defs.lean b/Mathlib/Algebra/Order/Field/Defs.lean
index 5e36efa7b7f81..86284a304dde6 100644
--- a/Mathlib/Algebra/Order/Field/Defs.lean
+++ b/Mathlib/Algebra/Order/Field/Defs.lean
@@ -35,3 +35,67 @@ class LinearOrderedField (α : Type*) extends LinearOrderedCommRing α, Field α
 instance (priority := 100) LinearOrderedField.toLinearOrderedSemifield [LinearOrderedField α] :
     LinearOrderedSemifield α :=
   { LinearOrderedRing.toLinearOrderedSemiring, ‹LinearOrderedField α› with }
+
+variable [LinearOrderedSemifield α] {a b c : α}
+
+/-- Equality holds when `a ≠ 0`. See `mul_inv_cancel`. -/
+lemma mul_inv_le_one : a * a⁻¹ ≤ 1 := by obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `a ≠ 0`. See `inv_mul_cancel`. -/
+lemma inv_mul_le_one : a⁻¹ * a ≤ 1 := by obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `a ≠ 0`. See `mul_inv_cancel_left`. -/
+lemma mul_inv_left_le (hb : 0 ≤ b) : a * (a⁻¹ * b) ≤ b := by
+  obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `a ≠ 0`. See `mul_inv_cancel_left`. -/
+lemma le_mul_inv_left (hb : b ≤ 0) : b ≤ a * (a⁻¹ * b) := by
+  obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `a ≠ 0`. See `inv_mul_cancel_left`. -/
+lemma inv_mul_left_le (hb : 0 ≤ b) : a⁻¹ * (a * b) ≤ b := by
+  obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `a ≠ 0`. See `inv_mul_cancel_left`. -/
+lemma le_inv_mul_left (hb : b ≤ 0) : b ≤ a⁻¹ * (a * b) := by
+  obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+/-- Equality holds when `b ≠ 0`. See `mul_inv_cancel_right`. -/
+lemma mul_inv_right_le (ha : 0 ≤ a) : a * b * b⁻¹ ≤ a := by
+  obtain rfl | hb := eq_or_ne b 0 <;> simp [*]
+
+/-- Equality holds when `b ≠ 0`. See `mul_inv_cancel_right`. -/
+lemma le_mul_inv_right (ha : a ≤ 0) : a ≤ a * b * b⁻¹ := by
+  obtain rfl | hb := eq_or_ne b 0 <;> simp [*]
+
+/-- Equality holds when `b ≠ 0`. See `inv_mul_cancel_right`. -/
+lemma inv_mul_right_le (ha : 0 ≤ a) : a * b⁻¹ * b ≤ a := by
+  obtain rfl | hb := eq_or_ne b 0 <;> simp [*]
+
+/-- Equality holds when `b ≠ 0`. See `inv_mul_cancel_right`. -/
+lemma le_inv_mul_right (ha : a ≤ 0) : a ≤ a * b⁻¹ * b := by
+  obtain rfl | hb := eq_or_ne b 0 <;> simp [*]
+
+/-- Equality holds when `c ≠ 0`. See `mul_div_mul_left`. -/
+lemma mul_div_mul_left_le (h : 0 ≤ a / b) : c * a / (c * b) ≤ a / b := by
+  obtain rfl | hc := eq_or_ne c 0
+  · simpa
+  · rw [mul_div_mul_left _ _ hc]
+
+/-- Equality holds when `c ≠ 0`. See `mul_div_mul_left`. -/
+lemma le_mul_div_mul_left (h : a / b ≤ 0) : a / b ≤ c * a / (c * b) := by
+  obtain rfl | hc := eq_or_ne c 0
+  · simpa
+  · rw [mul_div_mul_left _ _ hc]
+
+/-- Equality holds when `c ≠ 0`. See `mul_div_mul_right`. -/
+lemma mul_div_mul_right_le (h : 0 ≤ a / b) : a * c / (b * c) ≤ a / b := by
+  obtain rfl | hc := eq_or_ne c 0
+  · simpa
+  · rw [mul_div_mul_right _ _ hc]
+
+/-- Equality holds when `c ≠ 0`. See `mul_div_mul_right`. -/
+lemma le_mul_div_mul_right (h : a / b ≤ 0) : a / b ≤ a * c / (b * c) := by
+  obtain rfl | hc := eq_or_ne c 0
+  · simpa
+  · rw [mul_div_mul_right _ _ hc]
diff --git a/Mathlib/Algebra/Order/Field/Pointwise.lean b/Mathlib/Algebra/Order/Field/Pointwise.lean
new file mode 100644
index 0000000000000..01d4453c22523
--- /dev/null
+++ b/Mathlib/Algebra/Order/Field/Pointwise.lean
@@ -0,0 +1,126 @@
+/-
+Copyright (c) 2021 Alex J. Best. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Alex J. Best, Yaël Dillies
+-/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Order.Field.Defs
+import Mathlib.Algebra.SMulWithZero
+
+/-!
+# Pointwise operations on ordered algebraic objects
+
+This file contains lemmas about the effect of pointwise operations on sets with an order structure.
+-/
+
+open Function Set
+open scoped Pointwise
+
+variable {α : Type*}
+
+namespace LinearOrderedField
+
+variable {K : Type*} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
+include hr
+
+theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
+  constructor
+  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
+    constructor
+    · exact (mul_lt_mul_left hr).mpr a_h_left_left
+    · exact (mul_lt_mul_left hr).mpr a_h_left_right
+  · rintro ⟨a_left, a_right⟩
+    use x / r
+    refine ⟨⟨(lt_div_iff₀' hr).mpr a_left, (div_lt_iff₀' hr).mpr a_right⟩, ?_⟩
+    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
+
+theorem smul_Icc : r • Icc a b = Icc (r • a) (r • b) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Icc]
+  constructor
+  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
+    constructor
+    · exact (mul_le_mul_left hr).mpr a_h_left_left
+    · exact (mul_le_mul_left hr).mpr a_h_left_right
+  · rintro ⟨a_left, a_right⟩
+    use x / r
+    refine ⟨⟨(le_div_iff₀' hr).mpr a_left, (div_le_iff₀' hr).mpr a_right⟩, ?_⟩
+    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
+
+theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Ico]
+  constructor
+  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
+    constructor
+    · exact (mul_le_mul_left hr).mpr a_h_left_left
+    · exact (mul_lt_mul_left hr).mpr a_h_left_right
+  · rintro ⟨a_left, a_right⟩
+    use x / r
+    refine ⟨⟨(le_div_iff₀' hr).mpr a_left, (div_lt_iff₀' hr).mpr a_right⟩, ?_⟩
+    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
+
+theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Ioc]
+  constructor
+  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
+    constructor
+    · exact (mul_lt_mul_left hr).mpr a_h_left_left
+    · exact (mul_le_mul_left hr).mpr a_h_left_right
+  · rintro ⟨a_left, a_right⟩
+    use x / r
+    refine ⟨⟨(lt_div_iff₀' hr).mpr a_left, (div_le_iff₀' hr).mpr a_right⟩, ?_⟩
+    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
+
+theorem smul_Ioi : r • Ioi a = Ioi (r • a) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
+  constructor
+  · rintro ⟨a_w, a_h_left, rfl⟩
+    exact (mul_lt_mul_left hr).mpr a_h_left
+  · rintro h
+    use x / r
+    constructor
+    · exact (lt_div_iff₀' hr).mpr h
+    · exact mul_div_cancel₀ _ (ne_of_gt hr)
+
+theorem smul_Iio : r • Iio a = Iio (r • a) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Iio]
+  constructor
+  · rintro ⟨a_w, a_h_left, rfl⟩
+    exact (mul_lt_mul_left hr).mpr a_h_left
+  · rintro h
+    use x / r
+    constructor
+    · exact (div_lt_iff₀' hr).mpr h
+    · exact mul_div_cancel₀ _ (ne_of_gt hr)
+
+theorem smul_Ici : r • Ici a = Ici (r • a) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
+  constructor
+  · rintro ⟨a_w, a_h_left, rfl⟩
+    exact (mul_le_mul_left hr).mpr a_h_left
+  · rintro h
+    use x / r
+    constructor
+    · exact (le_div_iff₀' hr).mpr h
+    · exact mul_div_cancel₀ _ (ne_of_gt hr)
+
+theorem smul_Iic : r • Iic a = Iic (r • a) := by
+  ext x
+  simp only [mem_smul_set, smul_eq_mul, mem_Iio]
+  constructor
+  · rintro ⟨a_w, a_h_left, rfl⟩
+    exact (mul_le_mul_left hr).mpr a_h_left
+  · rintro h
+    use x / r
+    constructor
+    · exact (div_le_iff₀' hr).mpr h
+    · exact mul_div_cancel₀ _ (ne_of_gt hr)
+
+end LinearOrderedField
diff --git a/Mathlib/Algebra/Order/Field/Power.lean b/Mathlib/Algebra/Order/Field/Power.lean
index 8dfd3676717c3..70c83a1869e07 100644
--- a/Mathlib/Algebra/Order/Field/Power.lean
+++ b/Mathlib/Algebra/Order/Field/Power.lean
@@ -31,7 +31,7 @@ theorem zpow_le_of_le (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n := by
   calc
     a ^ m = a ^ m * 1 := (mul_one _).symm
     _ ≤ a ^ m * a ^ k :=
-      mul_le_mul_of_nonneg_left (one_le_pow_of_one_le ha _) (zpow_nonneg ha₀.le _)
+      mul_le_mul_of_nonneg_left (one_le_pow₀ ha) (zpow_nonneg ha₀.le _)
     _ = a ^ n := by rw [← zpow_natCast, ← zpow_add₀ ha₀.ne', hk, add_sub_cancel]
 
 theorem zpow_le_one_of_nonpos (ha : 1 ≤ a) (hn : n ≤ 0) : a ^ n ≤ 1 :=
@@ -48,7 +48,7 @@ theorem Nat.zpow_ne_zero_of_pos {a : ℕ} (h : 0 < a) (n : ℤ) : (a : α) ^ n 
   (Nat.zpow_pos_of_pos h n).ne'
 
 theorem one_lt_zpow (ha : 1 < a) : ∀ n : ℤ, 0 < n → 1 < a ^ n
-  | (n : ℕ), h => (zpow_natCast _ _).symm.subst (one_lt_pow ha <| Int.natCast_ne_zero.mp h.ne')
+  | (n : ℕ), h => (zpow_natCast _ _).symm.subst (one_lt_pow₀ ha <| Int.natCast_ne_zero.mp h.ne')
   | -[_+1], h => ((Int.negSucc_not_pos _).mp h).elim
 
 theorem zpow_strictMono (hx : 1 < a) : StrictMono (a ^ · : ℤ → α) :=
@@ -79,7 +79,7 @@ theorem zpow_le_iff_le (hx : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n :=
 
 @[simp]
 theorem div_pow_le (ha : 0 ≤ a) (hb : 1 ≤ b) (k : ℕ) : a / b ^ k ≤ a :=
-  div_le_self ha <| one_le_pow_of_one_le hb _
+  div_le_self ha <| one_le_pow₀ hb
 
 theorem zpow_injective (h₀ : 0 < a) (h₁ : a ≠ 1) : Injective (a ^ · : ℤ → α) := by
   rcases h₁.lt_or_lt with (H | H)
diff --git a/Mathlib/Algebra/Order/Floor.lean b/Mathlib/Algebra/Order/Floor.lean
index 590bbabf6d112..09bdb68063e38 100644
--- a/Mathlib/Algebra/Order/Floor.lean
+++ b/Mathlib/Algebra/Order/Floor.lean
@@ -9,9 +9,9 @@ import Mathlib.Algebra.Group.Int
 import Mathlib.Data.Int.Lemmas
 import Mathlib.Data.Nat.Cast.Order.Field
 import Mathlib.Data.Set.Subsingleton
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Order.GaloisConnection
 import Mathlib.Tactic.Abel
+import Mathlib.Tactic.FieldSimp
 import Mathlib.Tactic.Linarith
 import Mathlib.Tactic.Positivity
 
@@ -151,6 +151,7 @@ theorem floor_le (ha : 0 ≤ a) : (⌊a⌋₊ : α) ≤ a :=
 theorem lt_succ_floor (a : α) : a < ⌊a⌋₊.succ :=
   lt_of_floor_lt <| Nat.lt_succ_self _
 
+@[bound]
 theorem lt_floor_add_one (a : α) : a < ⌊a⌋₊ + 1 := by simpa using lt_succ_floor a
 
 @[simp]
@@ -183,7 +184,7 @@ theorem floor_mono : Monotone (floor : α → ℕ) := fun a b h => by
     exact Nat.zero_le _
   · exact le_floor ((floor_le ha).trans h)
 
-@[gcongr] lemma floor_le_floor (hab : a ≤ b) : ⌊a⌋₊ ≤ ⌊b⌋₊ := floor_mono hab
+@[gcongr, bound] lemma floor_le_floor (hab : a ≤ b) : ⌊a⌋₊ ≤ ⌊b⌋₊ := floor_mono hab
 
 theorem le_floor_iff' (hn : n ≠ 0) : n ≤ ⌊a⌋₊ ↔ (n : α) ≤ a := by
   obtain ha | ha := le_total a 0
@@ -259,10 +260,12 @@ theorem add_one_le_ceil_iff : n + 1 ≤ ⌈a⌉₊ ↔ (n : α) < a := by
 theorem one_le_ceil_iff : 1 ≤ ⌈a⌉₊ ↔ 0 < a := by
   rw [← zero_add 1, Nat.add_one_le_ceil_iff, Nat.cast_zero]
 
+@[bound]
 theorem ceil_le_floor_add_one (a : α) : ⌈a⌉₊ ≤ ⌊a⌋₊ + 1 := by
   rw [ceil_le, Nat.cast_add, Nat.cast_one]
   exact (lt_floor_add_one a).le
 
+@[bound]
 theorem le_ceil (a : α) : a ≤ ⌈a⌉₊ :=
   ceil_le.1 le_rfl
 
@@ -280,7 +283,7 @@ theorem ceil_natCast (n : ℕ) : ⌈(n : α)⌉₊ = n :=
 theorem ceil_mono : Monotone (ceil : α → ℕ) :=
   gc_ceil_coe.monotone_l
 
-@[gcongr] lemma ceil_le_ceil (hab : a ≤ b) : ⌈a⌉₊ ≤ ⌈b⌉₊ := ceil_mono hab
+@[gcongr, bound] lemma ceil_le_ceil (hab : a ≤ b) : ⌈a⌉₊ ≤ ⌈b⌉₊ := ceil_mono hab
 
 @[simp]
 theorem ceil_zero : ⌈(0 : α)⌉₊ = 0 := by rw [← Nat.cast_zero, ceil_natCast]
@@ -304,6 +307,7 @@ theorem lt_of_ceil_lt (h : ⌈a⌉₊ < n) : a < n :=
 theorem le_of_ceil_le (h : ⌈a⌉₊ ≤ n) : a ≤ n :=
   (le_ceil a).trans (Nat.cast_le.2 h)
 
+@[bound]
 theorem floor_le_ceil (a : α) : ⌊a⌋₊ ≤ ⌈a⌉₊ := by
   obtain ha | ha := le_total a 0
   · rw [floor_of_nonpos ha]
@@ -446,6 +450,7 @@ theorem ceil_add_ofNat (ha : 0 ≤ a) (n : ℕ) [n.AtLeastTwo] :
 theorem ceil_lt_add_one (ha : 0 ≤ a) : (⌈a⌉₊ : α) < a + 1 :=
   lt_ceil.1 <| (Nat.lt_succ_self _).trans_le (ceil_add_one ha).ge
 
+@[bound]
 theorem ceil_add_le (a b : α) : ⌈a + b⌉₊ ≤ ⌈a⌉₊ + ⌈b⌉₊ := by
   rw [ceil_le, Nat.cast_add]
   exact _root_.add_le_add (le_ceil _) (le_ceil _)
@@ -456,6 +461,7 @@ section LinearOrderedRing
 
 variable [LinearOrderedRing α] [FloorSemiring α]
 
+@[bound]
 theorem sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋₊ :=
   sub_lt_iff_lt_add.2 <| lt_floor_add_one a
 
@@ -478,7 +484,7 @@ theorem floor_div_nat (a : α) (n : ℕ) : ⌊a / n⌋₊ = ⌊a⌋₊ / n := by
   · exact div_nonneg ha n.cast_nonneg
   constructor
   · exact cast_div_le.trans (div_le_div_of_nonneg_right (floor_le ha) n.cast_nonneg)
-  rw [div_lt_iff, add_mul, one_mul, ← cast_mul, ← cast_add, ← floor_lt ha]
+  rw [div_lt_iff₀, add_mul, one_mul, ← cast_mul, ← cast_add, ← floor_lt ha]
   · exact lt_div_mul_add hn
   · exact cast_pos.2 hn
 
@@ -494,6 +500,46 @@ theorem floor_div_eq_div (m n : ℕ) : ⌊(m : α) / n⌋₊ = m / n := by
 
 end LinearOrderedSemifield
 
+section LinearOrderedField
+variable [LinearOrderedField α] [FloorSemiring α] {a b : α}
+
+lemma mul_lt_floor (hb₀ : 0 < b) (hb : b < 1) (hba : ⌈b / (1 - b)⌉₊ ≤ a) : b * a < ⌊a⌋₊ := by
+  calc
+    b * a < b * (⌊a⌋₊ + 1) := by gcongr; exacts [hb₀, lt_floor_add_one _]
+    _ ≤ ⌊a⌋₊ := by
+      rw [_root_.mul_add_one, ← le_sub_iff_add_le', ← one_sub_mul, ← div_le_iff₀' (by linarith),
+        ← ceil_le]
+      exact le_floor hba
+
+lemma ceil_lt_mul (hb : 1 < b) (hba : ⌈(b - 1)⁻¹⌉₊ / b < a) : ⌈a⌉₊ < b * a := by
+  obtain hab | hba := le_total a (b - 1)⁻¹
+  · calc
+      ⌈a⌉₊ ≤ (⌈(b - 1)⁻¹⌉₊ : α) := by gcongr
+      _ < b * a := by rwa [← div_lt_iff₀']; positivity
+  · rw [← sub_pos] at hb
+    calc
+      ⌈a⌉₊ < a + 1 := ceil_lt_add_one <| hba.trans' <| by positivity
+      _ = a + (b - 1) * (b - 1)⁻¹ := by rw [mul_inv_cancel₀]; positivity
+      _ ≤ a + (b - 1) * a := by gcongr; positivity
+      _ = b * a := by rw [sub_one_mul, add_sub_cancel]
+
+lemma ceil_le_mul (hb : 1 < b) (hba : ⌈(b - 1)⁻¹⌉₊ / b ≤ a) : ⌈a⌉₊ ≤ b * a := by
+  obtain rfl | hba := hba.eq_or_lt
+  · rw [mul_div_cancel₀, cast_le, ceil_le]
+    · exact _root_.div_le_self (by positivity) hb.le
+    · positivity
+  · exact (ceil_lt_mul hb hba).le
+
+lemma div_two_lt_floor (ha : 1 ≤ a) : a / 2 < ⌊a⌋₊ := by
+  rw [div_eq_inv_mul]; refine mul_lt_floor ?_ ?_ ?_ <;> norm_num; assumption
+
+lemma ceil_lt_two_mul (ha : 2⁻¹ < a) : ⌈a⌉₊ < 2 * a :=
+  ceil_lt_mul one_lt_two (by norm_num at ha ⊢; exact ha)
+
+lemma ceil_le_two_mul (ha : 2⁻¹ ≤ a) : ⌈a⌉₊ ≤ 2 * a :=
+  ceil_le_mul one_lt_two (by norm_num at ha ⊢; exact ha)
+
+end LinearOrderedField
 end Nat
 
 /-- There exists at most one `FloorSemiring` structure on a linear ordered semiring. -/
@@ -607,6 +653,7 @@ theorem le_floor : z ≤ ⌊a⌋ ↔ (z : α) ≤ a :=
 theorem floor_lt : ⌊a⌋ < z ↔ a < z :=
   lt_iff_lt_of_le_iff_le le_floor
 
+@[bound]
 theorem floor_le (a : α) : (⌊a⌋ : α) ≤ a :=
   gc_coe_floor.l_u_le a
 
@@ -619,6 +666,7 @@ theorem floor_le_sub_one_iff : ⌊a⌋ ≤ z - 1 ↔ a < z := by rw [← floor_l
 theorem floor_le_neg_one_iff : ⌊a⌋ ≤ -1 ↔ a < 0 := by
   rw [← zero_sub (1 : ℤ), floor_le_sub_one_iff, cast_zero]
 
+@[bound]
 theorem floor_nonpos (ha : a ≤ 0) : ⌊a⌋ ≤ 0 := by
   rw [← @cast_le α, Int.cast_zero]
   exact (floor_le a).trans ha
@@ -626,11 +674,11 @@ theorem floor_nonpos (ha : a ≤ 0) : ⌊a⌋ ≤ 0 := by
 theorem lt_succ_floor (a : α) : a < ⌊a⌋.succ :=
   floor_lt.1 <| Int.lt_succ_self _
 
-@[simp]
+@[simp, bound]
 theorem lt_floor_add_one (a : α) : a < ⌊a⌋ + 1 := by
   simpa only [Int.succ, Int.cast_add, Int.cast_one] using lt_succ_floor a
 
-@[simp]
+@[simp, bound]
 theorem sub_one_lt_floor (a : α) : a - 1 < ⌊a⌋ :=
   sub_lt_iff_lt_add.2 (lt_floor_add_one a)
 
@@ -656,7 +704,7 @@ theorem floor_one : ⌊(1 : α)⌋ = 1 := by rw [← cast_one, floor_intCast]
 theorem floor_mono : Monotone (floor : α → ℤ) :=
   gc_coe_floor.monotone_u
 
-@[gcongr] lemma floor_le_floor (hab : a ≤ b) : ⌊a⌋ ≤ ⌊b⌋ := floor_mono hab
+@[gcongr, bound] lemma floor_le_floor (hab : a ≤ b) : ⌊a⌋ ≤ ⌊b⌋ := floor_mono hab
 
 theorem floor_pos : 0 < ⌊a⌋ ↔ 1 ≤ a := by
   -- Porting note: broken `convert le_floor`
@@ -672,10 +720,12 @@ theorem floor_add_one (a : α) : ⌊a + 1⌋ = ⌊a⌋ + 1 := by
   -- Porting note: broken `convert floor_add_int a 1`
   rw [← cast_one, floor_add_int]
 
+@[bound]
 theorem le_floor_add (a b : α) : ⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋ := by
   rw [le_floor, Int.cast_add]
   exact add_le_add (floor_le _) (floor_le _)
 
+@[bound]
 theorem le_floor_add_floor (a b : α) : ⌊a + b⌋ - 1 ≤ ⌊a⌋ + ⌊b⌋ := by
   rw [← sub_le_iff_le_add, le_floor, Int.cast_sub, sub_le_comm, Int.cast_sub, Int.cast_one]
   refine le_trans ?_ (sub_one_lt_floor _).le
@@ -927,7 +977,7 @@ theorem fract_neg {x : α} (hx : fract x ≠ 0) : fract (-x) = 1 - fract x := by
 
 @[simp]
 theorem fract_neg_eq_zero {x : α} : fract (-x) = 0 ↔ fract x = 0 := by
-  simp only [fract_eq_iff, le_refl, zero_lt_one, tsub_zero, true_and_iff]
+  simp only [fract_eq_iff, le_refl, zero_lt_one, tsub_zero, true_and]
   constructor <;> rintro ⟨z, hz⟩ <;> use -z <;> simp [← hz]
 
 theorem fract_mul_nat (a : α) (b : ℕ) : ∃ z : ℤ, fract a * b - fract (a * b) = z := by
@@ -977,7 +1027,7 @@ theorem sub_floor_div_mul_nonneg (a : k) (hb : 0 < b) : 0 ≤ a - ⌊a / b⌋ *
 theorem sub_floor_div_mul_lt (a : k) (hb : 0 < b) : a - ⌊a / b⌋ * b < b :=
   sub_lt_iff_lt_add.2 <| by
     -- Porting note: `← one_add_mul` worked in mathlib3 without the argument
-    rw [← one_add_mul _ b, ← div_lt_iff hb, add_comm]
+    rw [← one_add_mul _ b, ← div_lt_iff₀ hb, add_comm]
     exact lt_floor_add_one _
 
 theorem fract_div_natCast_eq_div_natCast_mod {m n : ℕ} : fract ((m : k) / n) = ↑(m % n) / n := by
@@ -1004,7 +1054,7 @@ theorem fract_div_intCast_eq_div_intCast_mod {m : ℤ} {n : ℕ} :
     obtain ⟨l₀, rfl | rfl⟩ := l.eq_nat_or_neg
     · rw [cast_natCast, ← natCast_mod, cast_natCast, fract_div_natCast_eq_div_natCast_mod]
     · rw [Right.nonneg_neg_iff, natCast_nonpos_iff] at hl
-      simp [hl, zero_mod]
+      simp [hl]
   obtain ⟨m₀, rfl | rfl⟩ := m.eq_nat_or_neg
   · exact this (ofNat_nonneg m₀)
   let q := ⌈↑m₀ / (n : k)⌉
@@ -1052,6 +1102,7 @@ theorem add_one_le_ceil_iff : z + 1 ≤ ⌈a⌉ ↔ (z : α) < a := by rw [← l
 theorem one_le_ceil_iff : 1 ≤ ⌈a⌉ ↔ 0 < a := by
   rw [← zero_add (1 : ℤ), add_one_le_ceil_iff, cast_zero]
 
+@[bound]
 theorem ceil_le_floor_add_one (a : α) : ⌈a⌉ ≤ ⌊a⌋ + 1 := by
   rw [ceil_le, Int.cast_add, Int.cast_one]
   exact (lt_floor_add_one a).le
@@ -1060,6 +1111,9 @@ theorem ceil_le_floor_add_one (a : α) : ⌈a⌉ ≤ ⌊a⌋ + 1 := by
 theorem le_ceil (a : α) : a ≤ ⌈a⌉ :=
   gc_ceil_coe.le_u_l a
 
+lemma le_ceil_iff : z ≤ ⌈a⌉ ↔ z - 1 < a := by rw [← sub_one_lt_iff, lt_ceil]; norm_cast
+lemma ceil_lt_iff : ⌈a⌉ < z ↔ a ≤ z - 1 := by rw [← le_sub_one_iff, ceil_le]; norm_cast
+
 @[simp]
 theorem ceil_intCast (z : ℤ) : ⌈(z : α)⌉ = z :=
   eq_of_forall_ge_iff fun a => by rw [ceil_le, Int.cast_le]
@@ -1075,7 +1129,7 @@ theorem ceil_ofNat (n : ℕ) [n.AtLeastTwo] : ⌈(no_index (OfNat.ofNat n : α))
 theorem ceil_mono : Monotone (ceil : α → ℤ) :=
   gc_ceil_coe.monotone_l
 
-@[gcongr] lemma ceil_le_ceil (hab : a ≤ b) : ⌈a⌉ ≤ ⌈b⌉ := ceil_mono hab
+@[gcongr, bound] lemma ceil_le_ceil (hab : a ≤ b) : ⌈a⌉ ≤ ⌈b⌉ := ceil_mono hab
 
 @[simp]
 theorem ceil_add_int (a : α) (z : ℤ) : ⌈a + z⌉ = ⌈a⌉ + z := by
@@ -1114,14 +1168,17 @@ theorem ceil_sub_ofNat (a : α) (n : ℕ) [n.AtLeastTwo] :
     ⌈a - (no_index (OfNat.ofNat n))⌉ = ⌈a⌉ - OfNat.ofNat n :=
   ceil_sub_nat a n
 
+@[bound]
 theorem ceil_lt_add_one (a : α) : (⌈a⌉ : α) < a + 1 := by
   rw [← lt_ceil, ← Int.cast_one, ceil_add_int]
   apply lt_add_one
 
+@[bound]
 theorem ceil_add_le (a b : α) : ⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉ := by
   rw [ceil_le, Int.cast_add]
   exact add_le_add (le_ceil _) (le_ceil _)
 
+@[bound]
 theorem ceil_add_ceil_le (a b : α) : ⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1 := by
   rw [← le_sub_iff_add_le, ceil_le, Int.cast_sub, Int.cast_add, Int.cast_one, le_sub_comm]
   refine (ceil_lt_add_one _).le.trans ?_
@@ -1137,6 +1194,7 @@ theorem ceil_zero : ⌈(0 : α)⌉ = 0 := by rw [← cast_zero, ceil_intCast]
 @[simp]
 theorem ceil_one : ⌈(1 : α)⌉ = 1 := by rw [← cast_one, ceil_intCast]
 
+@[bound]
 theorem ceil_nonneg (ha : 0 ≤ a) : 0 ≤ ⌈a⌉ := mod_cast ha.trans (le_ceil a)
 
 theorem ceil_eq_iff : ⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ z := by
@@ -1152,9 +1210,11 @@ theorem ceil_eq_on_Ioc (z : ℤ) : ∀ a ∈ Set.Ioc (z - 1 : α) z, ⌈a⌉ = z
 theorem ceil_eq_on_Ioc' (z : ℤ) : ∀ a ∈ Set.Ioc (z - 1 : α) z, (⌈a⌉ : α) = z := fun a ha =>
   mod_cast ceil_eq_on_Ioc z a ha
 
+@[bound]
 theorem floor_le_ceil (a : α) : ⌊a⌋ ≤ ⌈a⌉ :=
   cast_le.1 <| (floor_le _).trans <| le_ceil _
 
+@[bound]
 theorem floor_lt_ceil_of_lt {a b : α} (h : a < b) : ⌊a⌋ < ⌈b⌉ :=
   cast_lt.1 <| (floor_le a).trans_lt <| h.trans_le <| le_ceil b
 
@@ -1184,6 +1244,56 @@ theorem ceil_sub_self_eq (ha : fract a ≠ 0) : (⌈a⌉ : α) - a = 1 - fract a
   rw [(or_iff_right ha).mp (fract_eq_zero_or_add_one_sub_ceil a)]
   abel
 
+section LinearOrderedField
+variable {k : Type*} [LinearOrderedField k] [FloorRing k] {a b : k}
+
+lemma mul_lt_floor (hb₀ : 0 < b) (hb : b < 1) (hba : ⌈b / (1 - b)⌉ ≤ a) : b * a < ⌊a⌋ := by
+  calc
+    b * a < b * (⌊a⌋ + 1) := by gcongr; exacts [hb₀, lt_floor_add_one _]
+    _ ≤ ⌊a⌋ := by
+      rwa [_root_.mul_add_one, ← le_sub_iff_add_le', ← one_sub_mul, ← div_le_iff₀' (by linarith),
+        ← ceil_le, le_floor]
+
+lemma ceil_div_ceil_inv_sub_one (ha : 1 ≤ a) : ⌈⌈(a - 1)⁻¹⌉ / a⌉ = ⌈(a - 1)⁻¹⌉ := by
+  obtain rfl | ha := ha.eq_or_lt
+  · simp
+  have : 0 < a - 1 := by linarith
+  have : 0 < ⌈(a - 1)⁻¹⌉ := ceil_pos.2 <| by positivity
+  refine le_antisymm (ceil_le.2 <| div_le_self (by positivity) ha.le) <| ?_
+  rw [le_ceil_iff, sub_lt_comm, div_eq_mul_inv, ← mul_one_sub,
+    ← lt_div_iff₀ (sub_pos.2 <| inv_lt_one_of_one_lt₀ ha)]
+  convert ceil_lt_add_one _ using 1
+  field_simp
+
+lemma ceil_lt_mul (hb : 1 < b) (hba : ⌈(b - 1)⁻¹⌉ / b < a) : ⌈a⌉ < b * a := by
+  obtain hab | hba := le_total a (b - 1)⁻¹
+  · calc
+      ⌈a⌉ ≤ (⌈(b - 1)⁻¹⌉ : k) := by gcongr
+      _ < b * a := by rwa [← div_lt_iff₀']; positivity
+  · rw [← sub_pos] at hb
+    calc
+      ⌈a⌉ < a + 1 := ceil_lt_add_one _
+      _ = a + (b - 1) * (b - 1)⁻¹ := by rw [mul_inv_cancel₀]; positivity
+      _ ≤ a + (b - 1) * a := by gcongr; positivity
+      _ = b * a := by rw [sub_one_mul, add_sub_cancel]
+
+lemma ceil_le_mul (hb : 1 < b) (hba : ⌈(b - 1)⁻¹⌉ / b ≤ a) : ⌈a⌉ ≤ b * a := by
+  obtain rfl | hba := hba.eq_or_lt
+  · rw [ceil_div_ceil_inv_sub_one hb.le, mul_div_cancel₀]
+    positivity
+  · exact (ceil_lt_mul hb hba).le
+
+lemma div_two_lt_floor (ha : 1 ≤ a) : a / 2 < ⌊a⌋ := by
+  rw [div_eq_inv_mul]; refine mul_lt_floor ?_ ?_ ?_ <;> norm_num; assumption
+
+lemma ceil_lt_two_mul (ha : 2⁻¹ < a) : ⌈a⌉ < 2 * a :=
+  ceil_lt_mul one_lt_two (by norm_num at ha ⊢; exact ha)
+
+lemma ceil_le_two_mul (ha : 2⁻¹ ≤ a) : ⌈a⌉ ≤ 2 * a :=
+  ceil_le_mul one_lt_two (by norm_num at ha ⊢; exact ha)
+
+end LinearOrderedField
+
 /-! #### Intervals -/
 
 @[simp]
@@ -1343,7 +1453,7 @@ section LinearOrderedField
 variable [LinearOrderedField α] [FloorRing α]
 
 theorem round_eq (x : α) : round x = ⌊x + 1 / 2⌋ := by
-  simp_rw [round, (by simp only [lt_div_iff', two_pos] : 2 * fract x < 1 ↔ fract x < 1 / 2)]
+  simp_rw [round, (by simp only [lt_div_iff₀', two_pos] : 2 * fract x < 1 ↔ fract x < 1 / 2)]
   cases' lt_or_le (fract x) (1 / 2) with hx hx
   · conv_rhs => rw [← fract_add_floor x, add_assoc, add_left_comm, floor_int_add]
     rw [if_pos hx, self_eq_add_right, floor_eq_iff, cast_zero, zero_add]
@@ -1391,10 +1501,12 @@ theorem abs_sub_round_div_natCast_eq {m n : ℕ} :
   rw [abs_sub_round_eq_min, Nat.cast_min, ← min_div_div_right hn'.le,
     fract_div_natCast_eq_div_natCast_mod, Nat.cast_sub (m.mod_lt hn).le, sub_div, div_self hn'.ne']
 
+@[bound]
 theorem sub_half_lt_round (x : α) : x - 1 / 2 < round x := by
   rw [round_eq x, show x - 1 / 2 = x + 1 / 2 - 1 by nlinarith]
   exact Int.sub_one_lt_floor (x + 1 / 2)
 
+@[bound]
 theorem round_le_add_half (x : α) : round x ≤ x + 1 / 2 := by
   rw [round_eq x]
   exact Int.floor_le (x + 1 / 2)
@@ -1501,7 +1613,7 @@ theorem Int.natCast_ceil_eq_ceil (ha : 0 ≤ a) : (⌈a⌉₊ : ℤ) = ⌈a⌉ :
 theorem natCast_floor_eq_intCast_floor (ha : 0 ≤ a) : (⌊a⌋₊ : α) = ⌊a⌋ := by
   rw [← Int.natCast_floor_eq_floor ha, Int.cast_natCast]
 
-theorem natCast_ceil_eq_intCast_ceil  (ha : 0 ≤ a) : (⌈a⌉₊ : α) = ⌈a⌉ := by
+theorem natCast_ceil_eq_intCast_ceil (ha : 0 ≤ a) : (⌈a⌉₊ : α) = ⌈a⌉ := by
   rw [← Int.natCast_ceil_eq_ceil ha, Int.cast_natCast]
 
 @[deprecated (since := "2024-08-20")] alias Int.ofNat_floor_eq_floor := natCast_floor_eq_floor
@@ -1581,3 +1693,5 @@ def evalIntCeil : PositivityExt where eval {u α} _zα _pα e := do
   | _, _, _ => throwError "failed to match on Int.ceil application"
 
 end Mathlib.Meta.Positivity
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Algebra/Order/Floor/Div.lean b/Mathlib/Algebra/Order/Floor/Div.lean
index 35d76dbf57197..20b853e6e5f94 100644
--- a/Mathlib/Algebra/Order/Floor/Div.lean
+++ b/Mathlib/Algebra/Order/Floor/Div.lean
@@ -119,7 +119,7 @@ end OrderedAddCommMonoid
 
 section LinearOrderedAddCommMonoid
 variable [LinearOrderedAddCommMonoid α] [OrderedAddCommMonoid β] [SMulZeroClass α β]
-  [PosSMulReflectLE α β] [FloorDiv α β] [CeilDiv α β] {a : α} {b c : β}
+  [PosSMulReflectLE α β] [FloorDiv α β] [CeilDiv α β] {a : α} {b : β}
 
 lemma floorDiv_le_ceilDiv : b ⌊/⌋ a ≤ b ⌈/⌉ a := by
   obtain ha | ha := le_or_lt a 0
diff --git a/Mathlib/Algebra/Order/Floor/Prime.lean b/Mathlib/Algebra/Order/Floor/Prime.lean
index 6c418d3d3d34b..f3545648ae511 100644
--- a/Mathlib/Algebra/Order/Floor/Prime.lean
+++ b/Mathlib/Algebra/Order/Floor/Prime.lean
@@ -3,40 +3,44 @@ Copyright (c) 2022 Yuyang Zhao. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yuyang Zhao
 -/
-
-import Mathlib.Algebra.Order.Floor
 import Mathlib.Data.Nat.Prime.Basic
+import Mathlib.Topology.Algebra.Order.Floor
 
 /-!
 # Existence of a sufficiently large prime for which `a * c ^ p / (p - 1)! < 1`
 
 This is a technical result used in the proof of the Lindemann-Weierstrass theorem.
--/
 
-namespace FloorRing
+TODO: delete this file, as all its lemmas have been deprecated.
+-/
 
 open scoped Nat
 
+@[deprecated eventually_mul_pow_lt_factorial_sub (since := "2024-09-25")]
+theorem Nat.exists_prime_mul_pow_lt_factorial (n a c : ℕ) :
+    ∃ p > n, p.Prime ∧ a * c ^ p < (p - 1)! :=
+  ((Filter.frequently_atTop.mpr Nat.exists_infinite_primes).and_eventually
+    (eventually_mul_pow_lt_factorial_sub a c 1)).forall_exists_of_atTop (n + 1)
+
+namespace FloorRing
+
 variable {K : Type*}
 
+@[deprecated FloorSemiring.eventually_mul_pow_lt_factorial_sub (since := "2024-09-25")]
 theorem exists_prime_mul_pow_lt_factorial [LinearOrderedRing K] [FloorRing K] (n : ℕ) (a c : K) :
-    ∃ p > n, p.Prime ∧ a * c ^ p < (p - 1)! := by
-  obtain ⟨p, pn, pp, h⟩ := n.exists_prime_mul_pow_lt_factorial ⌈|a|⌉.natAbs ⌈|c|⌉.natAbs
-  use p, pn, pp
-  calc a * c ^ p
-    _ ≤ |a * c ^ p| := le_abs_self _
-    _ ≤ ⌈|a|⌉ * (⌈|c|⌉ : K) ^ p := ?_
-    _ = ↑(Int.natAbs ⌈|a|⌉ * Int.natAbs ⌈|c|⌉ ^ p) := ?_
-    _ < ↑(p - 1)! := Nat.cast_lt.mpr h
-  · rw [abs_mul, abs_pow]
-    gcongr <;> try first | positivity | apply Int.le_ceil
-  · simp_rw [Nat.cast_mul, Nat.cast_pow, Int.cast_natAbs,
-      abs_eq_self.mpr (Int.ceil_nonneg (abs_nonneg (_ : K)))]
+    ∃ p > n, p.Prime ∧ a * c ^ p < (p - 1)! :=
+  ((Filter.frequently_atTop.mpr Nat.exists_infinite_primes).and_eventually
+    (FloorSemiring.eventually_mul_pow_lt_factorial_sub a c 1)).forall_exists_of_atTop (n + 1)
 
+@[deprecated FloorSemiring.tendsto_mul_pow_div_factorial_sub_atTop (since := "2024-09-25")]
 theorem exists_prime_mul_pow_div_factorial_lt_one [LinearOrderedField K] [FloorRing K]
     (n : ℕ) (a c : K) :
-    ∃ p > n, p.Prime ∧ a * c ^ p / (p - 1)! < 1 := by
-  simp_rw [div_lt_one (α := K) (Nat.cast_pos.mpr (Nat.factorial_pos _))]
-  exact exists_prime_mul_pow_lt_factorial ..
+    ∃ p > n, p.Prime ∧ a * c ^ p / (p - 1)! < 1 :=
+  letI := Preorder.topology K
+  haveI : OrderTopology K := ⟨rfl⟩
+  ((Filter.frequently_atTop.mpr Nat.exists_infinite_primes).and_eventually
+    (eventually_lt_of_tendsto_lt zero_lt_one
+      (FloorSemiring.tendsto_mul_pow_div_factorial_sub_atTop a c 1))).forall_exists_of_atTop
+    (n + 1)
 
 end FloorRing
diff --git a/Mathlib/Algebra/Order/Group/Abs.lean b/Mathlib/Algebra/Order/Group/Abs.lean
index 3d6763c037cfa..9693c189cfb6c 100644
--- a/Mathlib/Algebra/Order/Group/Abs.lean
+++ b/Mathlib/Algebra/Order/Group/Abs.lean
@@ -94,7 +94,7 @@ theorem apply_abs_le_mul_of_one_le {β : Type*} [MulOneClass β] [Preorder β]
 theorem abs_add (a b : α) : |a + b| ≤ |a| + |b| :=
   abs_le.2
     ⟨(neg_add |a| |b|).symm ▸
-        add_le_add ((@neg_le α ..).2 <| neg_le_abs _) ((@neg_le α ..).2 <| neg_le_abs _),
+        add_le_add (neg_le.2 <| neg_le_abs _) (neg_le.2 <| neg_le_abs _),
       add_le_add (le_abs_self _) (le_abs_self _)⟩
 
 theorem abs_add' (a b : α) : |a| ≤ |b| + |b + a| := by simpa using abs_add (-b) (b + a)
@@ -122,7 +122,7 @@ theorem sub_lt_of_abs_sub_lt_right (h : |a - b| < c) : a - c < b :=
   sub_lt_of_abs_sub_lt_left (abs_sub_comm a b ▸ h)
 
 theorem abs_sub_abs_le_abs_sub (a b : α) : |a| - |b| ≤ |a - b| :=
-  (@sub_le_iff_le_add α ..).2 <|
+  sub_le_iff_le_add.2 <|
     calc
       |a| = |a - b + b| := by rw [sub_add_cancel]
       _ ≤ |a - b| + |b| := abs_add _ _
diff --git a/Mathlib/Algebra/Order/Group/Basic.lean b/Mathlib/Algebra/Order/Group/Basic.lean
index da2cd7d995442..bc99b34b1b895 100644
--- a/Mathlib/Algebra/Order/Group/Basic.lean
+++ b/Mathlib/Algebra/Order/Group/Basic.lean
@@ -20,26 +20,32 @@ variable {α M R : Type*}
 section OrderedCommGroup
 variable [OrderedCommGroup α] {m n : ℤ} {a b : α}
 
+@[to_additive zsmul_left_strictMono]
+lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n := by
+  refine strictMono_int_of_lt_succ fun n ↦ ?_
+  rw [zpow_add_one]
+  exact lt_mul_of_one_lt_right' (a ^ n) ha
+
+@[deprecated (since := "2024-09-19")] alias zsmul_strictMono_left := zsmul_left_strictMono
+
 @[to_additive zsmul_pos] lemma one_lt_zpow' (ha : 1 < a) (hn : 0 < n) : 1 < a ^ n := by
-  obtain ⟨n, rfl⟩ := Int.eq_ofNat_of_zero_le hn.le
-  rw [zpow_natCast]
-  refine one_lt_pow' ha ?_
-  rintro rfl
-  simp at hn
-
-@[to_additive zsmul_strictMono_left]
-lemma zpow_right_strictMono (ha : 1 < a) : StrictMono fun n : ℤ ↦ a ^ n := fun m n h ↦
-  calc
-    a ^ m = a ^ m * 1 := (mul_one _).symm
-    _ < a ^ m * a ^ (n - m) := mul_lt_mul_left' (one_lt_zpow' ha <| Int.sub_pos_of_lt h) _
-    _ = a ^ n := by simp [← zpow_add, m.add_comm]
+  simpa using zpow_right_strictMono ha hn
+
+@[to_additive zsmul_left_strictAnti]
+lemma zpow_right_strictAnti (ha : a < 1) : StrictAnti fun n : ℤ ↦ a ^ n := by
+  refine strictAnti_int_of_succ_lt fun n ↦ ?_
+  rw [zpow_add_one]
+  exact mul_lt_of_lt_one_right' (a ^ n) ha
+
+@[to_additive zsmul_left_inj]
+lemma zpow_right_inj (ha : 1 < a) {m n : ℤ} : a ^ m = a ^ n ↔ m = n :=
+  (zpow_right_strictMono ha).injective.eq_iff
 
 @[to_additive zsmul_mono_left]
-lemma zpow_mono_right (ha : 1 ≤ a) : Monotone fun n : ℤ ↦ a ^ n := fun m n h ↦
-  calc
-    a ^ m = a ^ m * 1 := (mul_one _).symm
-    _ ≤ a ^ m * a ^ (n - m) := mul_le_mul_left' (one_le_zpow ha <| Int.sub_nonneg_of_le h) _
-    _ = a ^ n := by simp [← zpow_add, m.add_comm]
+lemma zpow_mono_right (ha : 1 ≤ a) : Monotone fun n : ℤ ↦ a ^ n := by
+  refine monotone_int_of_le_succ fun n ↦ ?_
+  rw [zpow_add_one]
+  exact le_mul_of_one_le_right' ha
 
 @[to_additive (attr := gcongr)]
 lemma zpow_le_zpow (ha : 1 ≤ a) (h : m ≤ n) : a ^ m ≤ a ^ n := zpow_mono_right ha h
@@ -125,4 +131,3 @@ theorem not_isCyclic_of_denselyOrdered [DenselyOrdered α] [Nontrivial α] : ¬I
     simp_all
 
 end LinearOrderedCommGroup
-
diff --git a/Mathlib/Algebra/Order/Group/CompleteLattice.lean b/Mathlib/Algebra/Order/Group/CompleteLattice.lean
new file mode 100644
index 0000000000000..19c8f5fda8e68
--- /dev/null
+++ b/Mathlib/Algebra/Order/Group/CompleteLattice.lean
@@ -0,0 +1,49 @@
+/-
+Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury G. Kudryashov
+-/
+import Mathlib.Algebra.Order.Group.OrderIso
+import Mathlib.Order.ConditionallyCompleteLattice.Basic
+
+/-!
+# Distributivity of group operations over supremum/infimum
+-/
+
+open Function Set
+
+variable {ι G : Type*} [Group G] [ConditionallyCompleteLattice G] [Nonempty ι] {f : ι → G}
+
+section Right
+variable [CovariantClass G G (swap (· * ·)) (· ≤ ·)]
+
+@[to_additive]
+lemma ciSup_mul (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) * a = ⨆ i, f i * a :=
+  (OrderIso.mulRight a).map_ciSup hf
+
+@[to_additive]
+lemma ciSup_div (hf : BddAbove (range f)) (a : G) : (⨆ i, f i) / a = ⨆ i, f i / a := by
+  simp only [div_eq_mul_inv, ciSup_mul hf]
+
+@[to_additive]
+lemma ciInf_mul (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) * a = ⨅ i, f i * a :=
+  (OrderIso.mulRight a).map_ciInf hf
+
+@[to_additive]
+lemma ciInf_div (hf : BddBelow (range f)) (a : G) : (⨅ i, f i) / a = ⨅ i, f i / a := by
+  simp only [div_eq_mul_inv, ciInf_mul hf]
+
+end Right
+
+section Left
+variable [CovariantClass G G (· * ·) (· ≤ ·)]
+
+@[to_additive]
+lemma mul_ciSup (hf : BddAbove (range f)) (a : G) : (a * ⨆ i, f i) = ⨆ i, a * f i :=
+  (OrderIso.mulLeft a).map_ciSup hf
+
+@[to_additive]
+lemma mul_ciInf (hf : BddBelow (range f)) (a : G) : (a * ⨅ i, f i) = ⨅ i, a * f i :=
+  (OrderIso.mulLeft a).map_ciInf hf
+
+end Left
diff --git a/Mathlib/Algebra/Order/Group/Cone.lean b/Mathlib/Algebra/Order/Group/Cone.lean
index fd0738f924ca8..6d794594f08df 100644
--- a/Mathlib/Algebra/Order/Group/Cone.lean
+++ b/Mathlib/Algebra/Order/Group/Cone.lean
@@ -1,85 +1,120 @@
 /-
 Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Scott Morrison
+Authors: Mario Carneiro, Kim Morrison, Artie Khovanov
 -/
 import Mathlib.Algebra.Order.Group.Defs
+import Mathlib.Algebra.Order.Monoid.Submonoid
 
 /-!
-# Construct ordered groups from positive cones
+# Construct ordered groups from groups with a specified positive cone.
 
-In this file we provide structures `PositiveCone` and `TotalPositiveCone`
-that encode axioms of `OrderedAddCommGroup` and `LinearOrderedAddCommGroup`
-in terms of the `(0 ≤ ·)` predicate.
+In this file we provide the structure `GroupCone` and the predicate `IsMaxCone`
+that encode axioms of `OrderedCommGroup` and `LinearOrderedCommGroup`
+in terms of the subset of non-negative elements.
 
-We also provide two constructors,
-`OrderedAddCommGroup.mkOfPositiveCone` and `LinearOrderedAddCommGroup.mkOfPositiveCone`,
-that turn these structures into instances of the corresponding typeclasses.
+We also provide constructors that convert between
+cones in groups and the corresponding ordered groups.
 -/
 
-namespace AddCommGroup
-
-/-- A collection of elements in an `AddCommGroup` designated as "non-negative".
-This is useful for constructing an `OrderedAddCommGroup`
-by choosing a positive cone in an existing `AddCommGroup`. -/
--- Porting note(#5171): @[nolint has_nonempty_instance]
-structure PositiveCone (α : Type*) [AddCommGroup α] where
-  /-- The characteristic predicate of a positive cone. `nonneg a` means that `0 ≤ a` according to
-  the cone. -/
-  nonneg : α → Prop
-  /-- The characteristic predicate of a positive cone. `pos a` means that `0 < a` according to
-  the cone. -/
-  pos : α → Prop := fun a => nonneg a ∧ ¬nonneg (-a)
-  pos_iff : ∀ a, pos a ↔ nonneg a ∧ ¬nonneg (-a) := by intros; rfl
-  zero_nonneg : nonneg 0
-  add_nonneg : ∀ {a b}, nonneg a → nonneg b → nonneg (a + b)
-  nonneg_antisymm : ∀ {a}, nonneg a → nonneg (-a) → a = 0
-
-/-- A positive cone in an `AddCommGroup` induces a linear order if
-for every `a`, either `a` or `-a` is non-negative. -/
--- Porting note(#5171): @[nolint has_nonempty_instance]
-structure TotalPositiveCone (α : Type*) [AddCommGroup α] extends PositiveCone α where
-  /-- For any `a` the proposition `nonneg a` is decidable -/
-  nonnegDecidable : DecidablePred nonneg
-  /-- Either `a` or `-a` is `nonneg` -/
-  nonneg_total : ∀ a : α, nonneg a ∨ nonneg (-a)
-
-/-- Forget that a `TotalPositiveCone` is total. -/
-add_decl_doc TotalPositiveCone.toPositiveCone
-
-end AddCommGroup
-
-namespace OrderedAddCommGroup
-
-open AddCommGroup
-
-/-- Construct an `OrderedAddCommGroup` by
-designating a positive cone in an existing `AddCommGroup`. -/
-def mkOfPositiveCone {α : Type*} [AddCommGroup α] (C : PositiveCone α) : OrderedAddCommGroup α :=
-  { ‹AddCommGroup α› with
-    le := fun a b => C.nonneg (b - a),
-    lt := fun a b => C.pos (b - a),
-    lt_iff_le_not_le := fun a b => by simp [C.pos_iff],
-    le_refl := fun a => by simp [C.zero_nonneg],
-    le_trans := fun a b c nab nbc => by simpa using C.add_nonneg nbc nab,
-    le_antisymm := fun a b nab nba =>
-      eq_of_sub_eq_zero <| C.nonneg_antisymm nba (by rwa [neg_sub]),
-    add_le_add_left := fun a b nab c => by simpa using nab }
-
-end OrderedAddCommGroup
-
-namespace LinearOrderedAddCommGroup
-
-open AddCommGroup
-
-/-- Construct a `LinearOrderedAddCommGroup` by
-designating a positive cone in an existing `AddCommGroup`
-such that for every `a`, either `a` or `-a` is non-negative. -/
-def mkOfPositiveCone {α : Type*} [AddCommGroup α] (C : TotalPositiveCone α) :
-    LinearOrderedAddCommGroup α :=
-  { OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone with
-    -- Porting note: was `C.nonneg_total (b - a)`
-    le_total := fun a b => by simpa [neg_sub] using C.nonneg_total (b - a)
-    decidableLE := fun a b => C.nonnegDecidable _ }
-
-end LinearOrderedAddCommGroup
+/-- `AddGroupConeClass S G` says that `S` is a type of cones in `G`. -/
+class AddGroupConeClass (S : Type*) (G : outParam Type*) [AddCommGroup G] [SetLike S G]
+    extends AddSubmonoidClass S G : Prop where
+  eq_zero_of_mem_of_neg_mem {C : S} {a : G} : a ∈ C → -a ∈ C → a = 0
+
+/-- `GroupConeClass S G` says that `S` is a type of cones in `G`. -/
+@[to_additive]
+class GroupConeClass (S : Type*) (G : outParam Type*) [CommGroup G] [SetLike S G] extends
+    SubmonoidClass S G : Prop where
+  eq_one_of_mem_of_inv_mem {C : S} {a : G} : a ∈ C → a⁻¹ ∈ C → a = 1
+
+export GroupConeClass (eq_one_of_mem_of_inv_mem)
+export AddGroupConeClass (eq_zero_of_mem_of_neg_mem)
+
+/-- A (positive) cone in an abelian group is a submonoid that
+does not contain both `a` and `-a` for any nonzero `a`.
+This is equivalent to being the set of non-negative elements of
+some order making the group into a partially ordered group. -/
+structure AddGroupCone (G : Type*) [AddCommGroup G] extends AddSubmonoid G where
+  eq_zero_of_mem_of_neg_mem' {a} : a ∈ carrier → -a ∈ carrier → a = 0
+
+/-- A (positive) cone in an abelian group is a submonoid that
+does not contain both `a` and `a⁻¹` for any non-identity `a`.
+This is equivalent to being the set of elements that are at least 1 in
+some order making the group into a partially ordered group. -/
+@[to_additive]
+structure GroupCone (G : Type*) [CommGroup G] extends Submonoid G where
+  eq_one_of_mem_of_inv_mem' {a} : a ∈ carrier → a⁻¹ ∈ carrier → a = 1
+
+@[to_additive]
+instance GroupCone.instSetLike (G : Type*) [CommGroup G] : SetLike (GroupCone G) G where
+  coe C := C.carrier
+  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.ext' h
+
+@[to_additive]
+instance GroupCone.instGroupConeClass (G : Type*) [CommGroup G] :
+    GroupConeClass (GroupCone G) G where
+  mul_mem {C} := C.mul_mem'
+  one_mem {C} := C.one_mem'
+  eq_one_of_mem_of_inv_mem {C} := C.eq_one_of_mem_of_inv_mem'
+
+/-- Typeclass for maximal additive cones. -/
+class IsMaxCone {S G : Type*} [AddCommGroup G] [SetLike S G] (C : S) : Prop where
+  mem_or_neg_mem (a : G) : a ∈ C ∨ -a ∈ C
+
+/-- Typeclass for maximal multiplicative cones. -/
+@[to_additive IsMaxCone]
+class IsMaxMulCone {S G : Type*} [CommGroup G] [SetLike S G] (C : S) : Prop where
+  mem_or_inv_mem (a : G) : a ∈ C ∨ a⁻¹ ∈ C
+
+export IsMaxCone (mem_or_neg_mem)
+export IsMaxMulCone (mem_or_inv_mem)
+
+namespace GroupCone
+variable {H : Type*} [OrderedCommGroup H] {a : H}
+
+variable (H) in
+/-- Construct a cone from the set of elements of
+a partially ordered abelian group that are at least 1. -/
+@[to_additive nonneg
+"Construct a cone from the set of non-negative elements of a partially ordered abelian group."]
+def oneLE : GroupCone H where
+  __ := Submonoid.oneLE H
+  eq_one_of_mem_of_inv_mem' {a} := by simpa using ge_antisymm
+
+@[to_additive (attr := simp) nonneg_toAddSubmonoid]
+lemma oneLE_toSubmonoid : (oneLE H).toSubmonoid = .oneLE H := rfl
+@[to_additive (attr := simp) mem_nonneg]
+lemma mem_oneLE : a ∈ oneLE H ↔ 1 ≤ a := Iff.rfl
+@[to_additive (attr := simp, norm_cast) coe_nonneg]
+lemma coe_oneLE : oneLE H = {x : H | 1 ≤ x} := rfl
+
+@[to_additive nonneg.isMaxCone]
+instance oneLE.isMaxMulCone {H : Type*} [LinearOrderedCommGroup H] : IsMaxMulCone (oneLE H) where
+  mem_or_inv_mem := by simpa using le_total 1
+
+end GroupCone
+
+variable {S G : Type*} [CommGroup G] [SetLike S G] (C : S)
+
+/-- Construct a partially ordered abelian group by designating a cone in an abelian group. -/
+@[to_additive (attr := reducible)
+"Construct a partially ordered abelian group by designating a cone in an abelian group."]
+def OrderedCommGroup.mkOfCone [GroupConeClass S G] :
+    OrderedCommGroup G where
+  le a b := b / a ∈ C
+  le_refl a := by simp [one_mem]
+  le_trans a b c nab nbc := by simpa using mul_mem nbc nab
+  le_antisymm a b nab nba := by
+    simpa [div_eq_one, eq_comm] using eq_one_of_mem_of_inv_mem nab (by simpa using nba)
+  mul_le_mul_left a b nab c := by simpa using nab
+
+/-- Construct a linearly ordered abelian group by designating a maximal cone in an abelian group. -/
+@[to_additive (attr := reducible)
+"Construct a linearly ordered abelian group by designating a maximal cone in an abelian group."]
+def LinearOrderedCommGroup.mkOfCone
+    [GroupConeClass S G] [IsMaxMulCone C] (dec : DecidablePred (· ∈ C)) :
+    LinearOrderedCommGroup G where
+  __ := OrderedCommGroup.mkOfCone C
+  le_total a b := by simpa using mem_or_inv_mem (b / a)
+  decidableLE a b := dec _
diff --git a/Mathlib/Algebra/Order/Group/Defs.lean b/Mathlib/Algebra/Order/Group/Defs.lean
index c486a4984cd8a..53b53337c45d3 100644
--- a/Mathlib/Algebra/Order/Group/Defs.lean
+++ b/Mathlib/Algebra/Order/Group/Defs.lean
@@ -21,9 +21,9 @@ The reason is that we did not want to change existing names in the library.
 -/
 
 /-
-`NeZero` should not be needed at this point in the ordered algebraic hierarchy.
+`NeZero` theory should not be needed at this point in the ordered algebraic hierarchy.
 -/
-assert_not_exists NeZero
+assert_not_imported Mathlib.Algebra.NeZero
 
 open Function
 
@@ -170,18 +170,16 @@ end LinearOrderedCommGroup
 section NormNumLemmas
 
 /- The following lemmas are stated so that the `norm_num` tactic can use them with the
-expected signatures.  -/
+expected signatures. -/
 variable [OrderedCommGroup α] {a b : α}
 
 @[to_additive (attr := gcongr) neg_le_neg]
 theorem inv_le_inv' : a ≤ b → b⁻¹ ≤ a⁻¹ :=
-  -- Porting note: explicit type annotation was not needed before.
-  (@inv_le_inv_iff α ..).mpr
+  inv_le_inv_iff.mpr
 
 @[to_additive (attr := gcongr) neg_lt_neg]
 theorem inv_lt_inv' : a < b → b⁻¹ < a⁻¹ :=
-  -- Porting note: explicit type annotation was not needed before.
-  (@inv_lt_inv_iff α ..).mpr
+  inv_lt_inv_iff.mpr
 
 --  The additive version is also a `linarith` lemma.
 @[to_additive]
diff --git a/Mathlib/Algebra/Order/Group/Indicator.lean b/Mathlib/Algebra/Order/Group/Indicator.lean
index 6ae4c61cca65e..a8e229c800374 100644
--- a/Mathlib/Algebra/Order/Group/Indicator.lean
+++ b/Mathlib/Algebra/Order/Group/Indicator.lean
@@ -106,18 +106,25 @@ lemma mulIndicator_le_mulIndicator' (h : a ∈ s → f a ≤ g a) :
     mulIndicator s f a ≤ mulIndicator s g a :=
   mulIndicator_rel_mulIndicator le_rfl h
 
-@[to_additive]
+@[to_additive (attr := mono, gcongr)]
 lemma mulIndicator_le_mulIndicator (h : f a ≤ g a) : mulIndicator s f a ≤ mulIndicator s g a :=
   mulIndicator_rel_mulIndicator le_rfl fun _ ↦ h
 
-attribute [mono] mulIndicator_le_mulIndicator indicator_le_indicator
+@[to_additive (attr := gcongr)]
+lemma mulIndicator_mono (h : f ≤ g) : s.mulIndicator f ≤ s.mulIndicator g :=
+  fun _ ↦ mulIndicator_le_mulIndicator (h _)
 
 @[to_additive]
-lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : ∀ a, 1 ≤ f a) (a : α) :
+lemma mulIndicator_le_mulIndicator_apply_of_subset (h : s ⊆ t) (hf : 1 ≤ f a) :
     mulIndicator s f a ≤ mulIndicator t f a :=
   mulIndicator_apply_le'
     (fun ha ↦ le_mulIndicator_apply (fun _ ↦ le_rfl) fun hat ↦ (hat <| h ha).elim) fun _ ↦
-    one_le_mulIndicator_apply fun _ ↦ hf _
+    one_le_mulIndicator_apply fun _ ↦ hf
+
+@[to_additive]
+lemma mulIndicator_le_mulIndicator_of_subset (h : s ⊆ t) (hf : 1 ≤ f) :
+    mulIndicator s f ≤ mulIndicator t f :=
+  fun _ ↦ mulIndicator_le_mulIndicator_apply_of_subset h (hf _)
 
 @[to_additive]
 lemma mulIndicator_le_self' (hf : ∀ x ∉ s, 1 ≤ f x) : mulIndicator s f ≤ f :=
@@ -174,6 +181,23 @@ lemma mulIndicator_iInter_apply (h1 : (⊥ : M) = 1) (s : ι → Set α) (f : α
     refine le_antisymm (by simp only [← h1, le_iInf_iff, bot_le, forall_const]) ?_
     simpa [mulIndicator_of_not_mem hj] using (iInf_le (fun i ↦ (s i).mulIndicator f) j) x
 
+@[to_additive]
+lemma iSup_mulIndicator {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → α → M}
+    {s : ι → Set α} (h1 : (⊥ : M) = 1) (hf : Monotone f) (hs : Monotone s) :
+    ⨆ i, (s i).mulIndicator (f i) = (⋃ i, s i).mulIndicator (⨆ i, f i) := by
+  simp only [le_antisymm_iff, iSup_le_iff]
+  refine ⟨fun i ↦ (mulIndicator_mono (le_iSup _ _)).trans (mulIndicator_le_mulIndicator_of_subset
+    (subset_iUnion _ _) (fun _ ↦ by simp [← h1])), fun a ↦ ?_⟩
+  by_cases ha : a ∈ ⋃ i, s i
+  · obtain ⟨i, hi⟩ : ∃ i, a ∈ s i := by simpa using ha
+    rw [mulIndicator_of_mem ha, iSup_apply, iSup_apply]
+    refine iSup_le fun j ↦ ?_
+    obtain ⟨k, hik, hjk⟩ := exists_ge_ge i j
+    refine le_iSup_of_le k <| (hf hjk _).trans_eq ?_
+    rw [mulIndicator_of_mem (hs hik hi)]
+  · rw [mulIndicator_of_not_mem ha, ← h1]
+    exact bot_le
+
 end CompleteLattice
 
 section CanonicallyOrderedCommMonoid
diff --git a/Mathlib/Algebra/Order/Group/MinMax.lean b/Mathlib/Algebra/Order/Group/MinMax.lean
index 4e53be43c3ca4..adf6af1c4b904 100644
--- a/Mathlib/Algebra/Order/Group/MinMax.lean
+++ b/Mathlib/Algebra/Order/Group/MinMax.lean
@@ -30,7 +30,7 @@ end
 
 section LinearOrderedCommGroup
 
-variable {α : Type*} [LinearOrderedCommGroup α] {a b c : α}
+variable {α : Type*} [LinearOrderedCommGroup α]
 
 @[to_additive min_neg_neg]
 theorem min_inv_inv' (a b : α) : min a⁻¹ b⁻¹ = (max a b)⁻¹ :=
@@ -64,7 +64,7 @@ end LinearOrderedCommGroup
 
 section LinearOrderedAddCommGroup
 
-variable {α : Type*} [LinearOrderedAddCommGroup α] {a b c : α}
+variable {α : Type*} [LinearOrderedAddCommGroup α]
 
 theorem max_sub_max_le_max (a b c d : α) : max a b - max c d ≤ max (a - c) (b - d) := by
   simp only [sub_le_iff_le_add, max_le_iff]; constructor
diff --git a/Mathlib/Algebra/Order/Group/Opposite.lean b/Mathlib/Algebra/Order/Group/Opposite.lean
new file mode 100644
index 0000000000000..e145f616c4c11
--- /dev/null
+++ b/Mathlib/Algebra/Order/Group/Opposite.lean
@@ -0,0 +1,85 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.Order.Group.Defs
+import Mathlib.Algebra.Group.Opposite
+
+/-!
+# Order instances for `MulOpposite`/`AddOpposite`
+
+This files transfers order instances and ordered monoid/group instances from `α` to `αᵐᵒᵖ` and
+`αᵃᵒᵖ`.
+-/
+
+variable {α : Type*}
+
+namespace MulOpposite
+section Preorder
+variable [Preorder α]
+
+@[to_additive] instance : Preorder αᵐᵒᵖ := Preorder.lift unop
+
+@[to_additive (attr := simp)] lemma unop_le_unop {a b : αᵐᵒᵖ} : a.unop ≤ b.unop ↔ a ≤ b := .rfl
+@[to_additive (attr := simp)] lemma op_le_op {a b : α} : op a ≤ op b ↔ a ≤ b := .rfl
+
+end Preorder
+
+@[to_additive] instance [PartialOrder α] : PartialOrder αᵐᵒᵖ := PartialOrder.lift _ unop_injective
+
+section OrderedCommMonoid
+variable [OrderedCommMonoid α]
+
+@[to_additive] instance : OrderedCommMonoid αᵐᵒᵖ where
+  mul_le_mul_left a b hab c := mul_le_mul_right' (by simpa) c.unop
+
+@[to_additive (attr := simp)] lemma unop_le_one {a : αᵐᵒᵖ} : unop a ≤ 1 ↔ a ≤ 1 := .rfl
+@[to_additive (attr := simp)] lemma one_le_unop {a : αᵐᵒᵖ} : 1 ≤ unop a ↔ 1 ≤ a := .rfl
+@[to_additive (attr := simp)] lemma op_le_one {a : α} : op a ≤ 1 ↔ a ≤ 1 := .rfl
+@[to_additive (attr := simp)] lemma one_le_op {a : α} : 1 ≤ op a ↔ 1 ≤ a := .rfl
+
+end OrderedCommMonoid
+
+@[to_additive] instance [OrderedCommGroup α] : OrderedCommGroup αᵐᵒᵖ where
+  __ := instCommGroup
+  __ := instOrderedCommMonoid
+
+section OrderedAddCommMonoid
+variable [OrderedAddCommMonoid α]
+
+instance : OrderedAddCommMonoid αᵐᵒᵖ where
+  add_le_add_left a b hab c := add_le_add_left (by simpa) c.unop
+
+@[simp] lemma unop_nonneg {a : αᵐᵒᵖ} : unop a ≤ 0 ↔ a ≤ 0 := .rfl
+@[simp] lemma unop_nonpos {a : αᵐᵒᵖ} : 0 ≤ unop a ↔ 0 ≤ a := .rfl
+@[simp] lemma op_nonneg {a : α} : op a ≤ 0 ↔ a ≤ 0 := .rfl
+@[simp] lemma op_nonpos {a : α} : 0 ≤ op a ↔ 0 ≤ a := .rfl
+
+end OrderedAddCommMonoid
+
+instance [OrderedAddCommGroup α] : OrderedAddCommGroup αᵐᵒᵖ where
+  __ := instAddCommGroup
+  __ := instOrderedAddCommMonoid
+
+end MulOpposite
+
+namespace AddOpposite
+section OrderedCommMonoid
+variable [OrderedCommMonoid α]
+
+instance : OrderedCommMonoid αᵃᵒᵖ where
+  mul_le_mul_left a b hab c := mul_le_mul_left' (by simpa) c.unop
+
+@[simp] lemma unop_le_one {a : αᵃᵒᵖ} : unop a ≤ 1 ↔ a ≤ 1 := .rfl
+@[simp] lemma one_le_unop {a : αᵃᵒᵖ} : 1 ≤ unop a ↔ 1 ≤ a := .rfl
+@[simp] lemma op_le_one {a : α} : op a ≤ 1 ↔ a ≤ 1 := .rfl
+@[simp] lemma one_le_op {a : α} : 1 ≤ op a ↔ 1 ≤ a := .rfl
+
+end OrderedCommMonoid
+
+instance [OrderedCommGroup α] : OrderedCommGroup αᵃᵒᵖ where
+  __ := instCommGroup
+  __ := instOrderedCommMonoid
+
+end AddOpposite
diff --git a/Mathlib/Algebra/Order/Group/Pointwise/Bounds.lean b/Mathlib/Algebra/Order/Group/Pointwise/Bounds.lean
new file mode 100644
index 0000000000000..1b1e0d3b70882
--- /dev/null
+++ b/Mathlib/Algebra/Order/Group/Pointwise/Bounds.lean
@@ -0,0 +1,117 @@
+/-
+Copyright (c) 2021 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Order.Group.OrderIso
+import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
+import Mathlib.Order.Bounds.OrderIso
+
+/-!
+# Upper/lower bounds in ordered monoids and groups
+
+In this file we prove a few facts like “`-s` is bounded above iff `s` is bounded below”
+(`bddAbove_neg`).
+-/
+
+open Function Set
+open scoped Pointwise
+
+variable {ι G M : Type*}
+
+section Mul
+variable [Mul M] [Preorder M] [CovariantClass M M (· * ·) (· ≤ ·)]
+  [CovariantClass M M (swap (· * ·)) (· ≤ ·)] {f g : ι → M} {s t : Set M} {a b : M}
+
+@[to_additive]
+lemma mul_mem_upperBounds_mul (ha : a ∈ upperBounds s) (hb : b ∈ upperBounds t) :
+    a * b ∈ upperBounds (s * t) := forall_image2_iff.2 fun _ hx _ hy => mul_le_mul' (ha hx) (hb hy)
+
+@[to_additive]
+lemma subset_upperBounds_mul (s t : Set M) : upperBounds s * upperBounds t ⊆ upperBounds (s * t) :=
+  image2_subset_iff.2 fun _ hx _ hy => mul_mem_upperBounds_mul hx hy
+
+@[to_additive]
+lemma mul_mem_lowerBounds_mul (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t) :
+    a * b ∈ lowerBounds (s * t) := mul_mem_upperBounds_mul (M := Mᵒᵈ) ha hb
+
+@[to_additive]
+lemma subset_lowerBounds_mul (s t : Set M) : lowerBounds s * lowerBounds t ⊆ lowerBounds (s * t) :=
+  subset_upperBounds_mul (M := Mᵒᵈ) _ _
+
+@[to_additive]
+lemma BddAbove.mul (hs : BddAbove s) (ht : BddAbove t) : BddAbove (s * t) :=
+  (Nonempty.mul hs ht).mono (subset_upperBounds_mul s t)
+
+@[to_additive]
+lemma BddBelow.mul (hs : BddBelow s) (ht : BddBelow t) : BddBelow (s * t) :=
+  (Nonempty.mul hs ht).mono (subset_lowerBounds_mul s t)
+
+@[to_additive]
+lemma BddAbove.range_mul (hf : BddAbove (range f)) (hg : BddAbove (range g)) :
+    BddAbove (range fun i ↦ f i * g i) :=
+  .range_comp (f := fun i ↦ (f i, g i)) (bddAbove_range_prod.2 ⟨hf, hg⟩)
+    (monotone_fst.mul' monotone_snd)
+
+@[to_additive]
+lemma BddBelow.range_mul (hf : BddBelow (range f)) (hg : BddBelow (range g)) :
+    BddBelow (range fun i ↦ f i * g i) := BddAbove.range_mul (M := Mᵒᵈ) hf hg
+
+end Mul
+
+section InvNeg
+variable [Group G] [Preorder G] [CovariantClass G G (· * ·) (· ≤ ·)]
+  [CovariantClass G G (swap (· * ·)) (· ≤ ·)] {s : Set G} {a : G}
+
+@[to_additive (attr := simp)]
+theorem bddAbove_inv : BddAbove s⁻¹ ↔ BddBelow s :=
+  (OrderIso.inv G).bddAbove_preimage
+
+@[to_additive (attr := simp)]
+theorem bddBelow_inv : BddBelow s⁻¹ ↔ BddAbove s :=
+  (OrderIso.inv G).bddBelow_preimage
+
+@[to_additive]
+theorem BddAbove.inv (h : BddAbove s) : BddBelow s⁻¹ :=
+  bddBelow_inv.2 h
+
+@[to_additive]
+theorem BddBelow.inv (h : BddBelow s) : BddAbove s⁻¹ :=
+  bddAbove_inv.2 h
+
+@[to_additive (attr := simp)]
+theorem isLUB_inv : IsLUB s⁻¹ a ↔ IsGLB s a⁻¹ :=
+  (OrderIso.inv G).isLUB_preimage
+
+@[to_additive]
+theorem isLUB_inv' : IsLUB s⁻¹ a⁻¹ ↔ IsGLB s a :=
+  (OrderIso.inv G).isLUB_preimage'
+
+@[to_additive]
+theorem IsGLB.inv (h : IsGLB s a) : IsLUB s⁻¹ a⁻¹ :=
+  isLUB_inv'.2 h
+
+@[to_additive (attr := simp)]
+theorem isGLB_inv : IsGLB s⁻¹ a ↔ IsLUB s a⁻¹ :=
+  (OrderIso.inv G).isGLB_preimage
+
+@[to_additive]
+theorem isGLB_inv' : IsGLB s⁻¹ a⁻¹ ↔ IsLUB s a :=
+  (OrderIso.inv G).isGLB_preimage'
+
+@[to_additive]
+theorem IsLUB.inv (h : IsLUB s a) : IsGLB s⁻¹ a⁻¹ :=
+  isGLB_inv'.2 h
+
+@[to_additive]
+lemma BddBelow.range_inv {α : Type*} {f : α → G} (hf : BddBelow (range f)) :
+    BddAbove (range (fun x => (f x)⁻¹)) :=
+  hf.range_comp (OrderIso.inv G).monotone
+
+@[to_additive]
+lemma BddAbove.range_inv {α : Type*} {f : α → G} (hf : BddAbove (range f)) :
+    BddBelow (range (fun x => (f x)⁻¹)) :=
+  BddBelow.range_inv (G := Gᵒᵈ) hf
+
+end InvNeg
diff --git a/Mathlib/Algebra/Order/Group/Pointwise/CompleteLattice.lean b/Mathlib/Algebra/Order/Group/Pointwise/CompleteLattice.lean
new file mode 100644
index 0000000000000..6f09d373cffd4
--- /dev/null
+++ b/Mathlib/Algebra/Order/Group/Pointwise/CompleteLattice.lean
@@ -0,0 +1,117 @@
+/-
+Copyright (c) 2021 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Order.Group.Pointwise.Bounds
+import Mathlib.Order.ConditionallyCompleteLattice.Basic
+
+/-!
+# Infima/suprema in ordered monoids and groups
+
+In this file we prove a few facts like “The infimum of `-s` is `-` the supremum of `s`”.
+
+## TODO
+
+`sSup (s • t) = sSup s • sSup t` and `sInf (s • t) = sInf s • sInf t` hold as well but
+`CovariantClass` is currently not polymorphic enough to state it.
+-/
+
+open Function Set
+open scoped Pointwise
+
+variable {ι G M : Type*}
+
+section ConditionallyCompleteLattice
+variable [ConditionallyCompleteLattice M]
+
+section One
+variable [One M]
+
+@[to_additive (attr := simp)] lemma csSup_one : sSup (1 : Set M) = 1 := csSup_singleton _
+@[to_additive (attr := simp)] lemma csInf_one : sInf (1 : Set M) = 1 := csInf_singleton _
+
+end One
+
+section Group
+variable [Group M] [CovariantClass M M (· * ·) (· ≤ ·)] [CovariantClass M M (swap (· * ·)) (· ≤ ·)]
+  {s t : Set M}
+
+@[to_additive]
+lemma csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ := by
+  rw [← image_inv]
+  exact ((OrderIso.inv _).map_csInf' hs₀ hs₁).symm
+
+@[to_additive]
+lemma csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by
+  rw [← image_inv]
+  exact ((OrderIso.inv _).map_csSup' hs₀ hs₁).symm
+
+@[to_additive]
+lemma csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
+    sSup (s * t) = sSup s * sSup t :=
+  csSup_image2_eq_csSup_csSup (fun _ => (OrderIso.mulRight _).to_galoisConnection)
+    (fun _ => (OrderIso.mulLeft _).to_galoisConnection) hs₀ hs₁ ht₀ ht₁
+
+@[to_additive]
+lemma csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
+    sInf (s * t) = sInf s * sInf t :=
+  csInf_image2_eq_csInf_csInf (fun _ => (OrderIso.mulRight _).symm.to_galoisConnection)
+    (fun _ => (OrderIso.mulLeft _).symm.to_galoisConnection) hs₀ hs₁ ht₀ ht₁
+
+@[to_additive]
+lemma csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
+    sSup (s / t) = sSup s / sInf t := by
+  rw [div_eq_mul_inv, csSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, csSup_inv ht₀ ht₁, div_eq_mul_inv]
+
+@[to_additive]
+lemma csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
+    sInf (s / t) = sInf s / sSup t := by
+  rw [div_eq_mul_inv, csInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, csInf_inv ht₀ ht₁, div_eq_mul_inv]
+
+end Group
+end ConditionallyCompleteLattice
+
+section CompleteLattice
+variable [CompleteLattice M]
+
+section One
+variable [One M]
+
+@[to_additive] lemma sSup_one : sSup (1 : Set M) = 1 := sSup_singleton
+@[to_additive] lemma sInf_one : sInf (1 : Set M) = 1 := sInf_singleton
+
+end One
+
+section Group
+variable [Group M] [CovariantClass M M (· * ·) (· ≤ ·)] [CovariantClass M M (swap (· * ·)) (· ≤ ·)]
+  (s t : Set M)
+
+@[to_additive]
+lemma sSup_inv (s : Set M) : sSup s⁻¹ = (sInf s)⁻¹ := by
+  rw [← image_inv, sSup_image]
+  exact ((OrderIso.inv M).map_sInf _).symm
+
+@[to_additive]
+lemma sInf_inv (s : Set M) : sInf s⁻¹ = (sSup s)⁻¹ := by
+  rw [← image_inv, sInf_image]
+  exact ((OrderIso.inv M).map_sSup _).symm
+
+@[to_additive]
+lemma sSup_mul : sSup (s * t) = sSup s * sSup t :=
+  (sSup_image2_eq_sSup_sSup fun _ => (OrderIso.mulRight _).to_galoisConnection) fun _ =>
+    (OrderIso.mulLeft _).to_galoisConnection
+
+@[to_additive]
+lemma sInf_mul : sInf (s * t) = sInf s * sInf t :=
+  (sInf_image2_eq_sInf_sInf fun _ => (OrderIso.mulRight _).symm.to_galoisConnection) fun _ =>
+    (OrderIso.mulLeft _).symm.to_galoisConnection
+
+@[to_additive]
+lemma sSup_div : sSup (s / t) = sSup s / sInf t := by simp_rw [div_eq_mul_inv, sSup_mul, sSup_inv]
+
+@[to_additive]
+lemma sInf_div : sInf (s / t) = sInf s / sSup t := by simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
+
+end Group
+end CompleteLattice
diff --git a/Mathlib/Algebra/Order/Group/PosPart.lean b/Mathlib/Algebra/Order/Group/PosPart.lean
index 078e76ef92cb4..8374c0a8ed405 100644
--- a/Mathlib/Algebra/Order/Group/PosPart.lean
+++ b/Mathlib/Algebra/Order/Group/PosPart.lean
@@ -49,7 +49,7 @@ section Lattice
 variable [Lattice α]
 
 section Group
-variable [Group α] {a : α}
+variable [Group α] {a b : α}
 
 /-- The *positive part* of an element `a` in a lattice ordered group is `a ⊔ 1`, denoted `a⁺ᵐ`. -/
 @[to_additive
@@ -130,6 +130,16 @@ lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a := by simp [leOnePart_eq_one']
 @[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ := by
   rw [← inv_div, oneLePart_div_leOnePart]
 
+@[to_additive]
+lemma oneLePart_leOnePart_injective : Injective fun a : α ↦ (a⁺ᵐ, a⁻ᵐ) := by
+  simp only [Injective, Prod.mk.injEq, and_imp]
+  rintro a b hpos hneg
+  rw [← oneLePart_div_leOnePart a, ← oneLePart_div_leOnePart b, hpos, hneg]
+
+@[to_additive]
+lemma oneLePart_leOnePart_inj : a⁺ᵐ = b⁺ᵐ ∧ a⁻ᵐ = b⁻ᵐ ↔ a = b :=
+  Prod.mk.inj_iff.symm.trans oneLePart_leOnePart_injective.eq_iff
+
 section covariantmulop
 variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
 
@@ -200,7 +210,7 @@ end CommGroup
 end Lattice
 
 section LinearOrder
-variable [LinearOrder α] [Group α] {a : α}
+variable [LinearOrder α] [Group α] {a b : α}
 
 @[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 := by
   rw [oneLePart, ← maxDefault, ← sup_eq_maxDefault]; simp_rw [sup_comm]
@@ -212,6 +222,8 @@ variable [LinearOrder α] [Group α] {a : α}
 lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a := by
   rw [oneLePart, right_lt_sup, not_le] at ha; exact oneLePart_eq_self.2 ha.le
 
+@[to_additive (attr := simp)] lemma oneLePart_lt : a⁺ᵐ < b ↔ a < b ∧ 1 < b := sup_lt_iff
+
 section covariantmul
 variable [CovariantClass α α (· * ·) (· ≤ ·)]
 
@@ -221,16 +233,21 @@ variable [CovariantClass α α (· * ·) (· ≤ ·)]
 @[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 :=
   lt_iff_lt_of_le_iff_le <| (one_le_leOnePart _).le_iff_eq.trans leOnePart_eq_one
 
+variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
+
+@[to_additive (attr := simp)] lemma leOnePart_lt : a⁻ᵐ < b ↔ b⁻¹ < a ∧ 1 < b :=
+  sup_lt_iff.trans <| by rw [inv_lt']
+
 end covariantmul
 end LinearOrder
 
 namespace Pi
-variable {ι : Type*} {α : ι → Type*} [∀ i, Lattice (α i)] [∀ i, AddCommGroup (α i)]
+variable {ι : Type*} {α : ι → Type*} [∀ i, Lattice (α i)] [∀ i, Group (α i)]
 
-@[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ i = (f i)⁺ := rfl
-@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ i = (f i)⁻ := rfl
+@[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ᵐ i = (f i)⁺ᵐ := rfl
+@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ᵐ i = (f i)⁻ᵐ := rfl
 
-@[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ = fun i ↦ (f i)⁺ := rfl
-@[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ = fun i ↦ (f i)⁻ := rfl
+@[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ᵐ = fun i ↦ (f i)⁺ᵐ := rfl
+@[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ᵐ = fun i ↦ (f i)⁻ᵐ := rfl
 
 end Pi
diff --git a/Mathlib/Algebra/Order/Group/Unbundled/Basic.lean b/Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
index f0e95176dbb3d..4d59951456a6e 100644
--- a/Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
+++ b/Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
@@ -33,7 +33,7 @@ variable [Group α]
 
 section TypeclassesLeftLE
 
-variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c d : α}
+variable [LE α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α}
 
 /-- Uses `left` co(ntra)variant. -/
 @[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
@@ -471,7 +471,7 @@ variable [Group α] [LE α]
 
 section Right
 
-variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c d : α}
+variable [CovariantClass α α (swap (· * ·)) (· ≤ ·)] {a b c : α}
 
 @[to_additive]
 theorem div_le_div_iff_right (c : α) : a / c ≤ b / c ↔ a ≤ b := by
@@ -594,7 +594,7 @@ variable [Group α] [LT α]
 
 section Right
 
-variable [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c d : α}
+variable [CovariantClass α α (swap (· * ·)) (· < ·)] {a b c : α}
 
 @[to_additive (attr := simp)]
 theorem div_lt_div_iff_right (c : α) : a / c < b / c ↔ a < b := by
@@ -722,7 +722,7 @@ variable [CovariantClass α α (· * ·) (· ≤ ·)]
 
 section VariableNames
 
-variable {a b c : α}
+variable {a b : α}
 
 @[to_additive]
 theorem le_of_forall_one_lt_lt_mul (h : ∀ ε : α, 1 < ε → a < b * ε) : a ≤ b :=
diff --git a/Mathlib/Algebra/Order/Group/Unbundled/Int.lean b/Mathlib/Algebra/Order/Group/Unbundled/Int.lean
index 773e4ce5f46eb..96c6eed502379 100644
--- a/Mathlib/Algebra/Order/Group/Unbundled/Int.lean
+++ b/Mathlib/Algebra/Order/Group/Unbundled/Int.lean
@@ -82,6 +82,15 @@ theorem abs_le_one_iff {a : ℤ} : |a| ≤ 1 ↔ a = 0 ∨ a = 1 ∨ a = -1 := b
 theorem one_le_abs {z : ℤ} (h₀ : z ≠ 0) : 1 ≤ |z| :=
   add_one_le_iff.mpr (abs_pos.mpr h₀)
 
+lemma eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by
+  by_contra h
+  have := Int.natAbs_le_of_dvd_ne_zero h1 h
+  rw [Int.abs_eq_natAbs] at h2
+  omega
+
+lemma abs_sub_lt_of_lt_lt {m a b : ℕ} (ha : a < m) (hb : b < m) : |(b : ℤ) - a| < m := by
+  rw [abs_lt]; omega
+
 /-! #### `/`  -/
 
 theorem ediv_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < |b|) : a / b = 0 :=
diff --git a/Mathlib/Algebra/Order/Group/Units.lean b/Mathlib/Algebra/Order/Group/Units.lean
index eaf081e0b36ab..af674406ea62a 100644
--- a/Mathlib/Algebra/Order/Group/Units.lean
+++ b/Mathlib/Algebra/Order/Group/Units.lean
@@ -25,3 +25,11 @@ instance Units.orderedCommGroup [OrderedCommMonoid α] : OrderedCommGroup αˣ :
 -- Porting note: the mathlib3 proof was
 -- mul_le_mul_left := fun a b h c => (mul_le_mul_left' (h : (a : α) ≤ b) _ : (c : α) * a ≤ c * b) }
 -- see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/elaboration.20failure.20in.20algebra.2Eorder.2Egroup.2Eunits
+
+/-- The units of a linearly ordered commutative monoid form a linearly ordered commutative group. -/
+@[to_additive "The units of a linearly ordered commutative additive monoid form a
+linearly ordered commutative additive group."]
+instance Units.instLinearOrderedCommGroup [LinearOrderedCommMonoid α] :
+    LinearOrderedCommGroup αˣ where
+  __ := Units.instLinearOrder
+  __ := Units.orderedCommGroup
diff --git a/Mathlib/Algebra/Order/GroupWithZero/Canonical.lean b/Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
index d7eab9e094013..1808690367f6d 100644
--- a/Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
+++ b/Mathlib/Algebra/Order/GroupWithZero/Canonical.lean
@@ -130,11 +130,6 @@ instance (priority := 100) LinearOrderedCommGroupWithZero.toMulPosStrictMono :
     MulPosStrictMono α where
   elim a b c hbc := by by_contra! h; exact hbc.not_le <| (mul_le_mul_right a.2).1 h
 
-/-- Alias of `mul_le_one'` for unification. -/
-@[deprecated mul_le_one' (since := "2024-08-21")]
-theorem mul_le_one₀ (ha : a ≤ 1) (hb : b ≤ 1) : a * b ≤ 1 :=
-  mul_le_one' ha hb
-
 /-- Alias of `one_le_mul'` for unification. -/
 @[deprecated one_le_mul (since := "2024-08-21")]
 theorem one_le_mul₀ (ha : 1 ≤ a) (hb : 1 ≤ b) : 1 ≤ a * b :=
@@ -185,25 +180,9 @@ theorem inv_mul_lt_of_lt_mul₀ (h : a < b * c) : b⁻¹ * a < c := by
 theorem mul_lt_right₀ (c : α) (h : a < b) (hc : c ≠ 0) : a * c < b * c :=
   mul_lt_mul_of_pos_right h (zero_lt_iff.2 hc)
 
-theorem inv_lt_one₀ (ha : a ≠ 0) : a⁻¹ < 1 ↔ 1 < a :=
-  inv_lt_one' (a := Units.mk0 a ha)
-
-theorem one_lt_inv₀ (ha : a ≠ 0) : 1 < a⁻¹ ↔ a < 1 :=
-  one_lt_inv' (a := Units.mk0 a ha)
-
-theorem inv_lt_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ < b⁻¹ ↔ b < a :=
-  show (Units.mk0 a ha)⁻¹ < (Units.mk0 b hb)⁻¹ ↔ Units.mk0 b hb < Units.mk0 a ha from
-    have : CovariantClass αˣ αˣ (· * ·) (· < ·) :=
-      IsLeftCancelMul.covariant_mul_lt_of_covariant_mul_le αˣ
-    inv_lt_inv_iff
-
-theorem inv_le_inv₀ (ha : a ≠ 0) (hb : b ≠ 0) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a :=
-  show (Units.mk0 a ha)⁻¹ ≤ (Units.mk0 b hb)⁻¹ ↔ Units.mk0 b hb ≤ Units.mk0 a ha from
-    inv_le_inv_iff
-
 theorem lt_of_mul_lt_mul_of_le₀ (h : a * b < c * d) (hc : 0 < c) (hh : c ≤ a) : b < d := by
   have ha : a ≠ 0 := ne_of_gt (lt_of_lt_of_le hc hh)
-  simp_rw [← inv_le_inv₀ ha (ne_of_gt hc)] at hh
+  rw [← inv_le_inv₀ (zero_lt_iff.2 ha) hc] at hh
   have := mul_lt_mul_of_lt_of_le₀ hh (inv_ne_zero (ne_of_gt hc)) h
   simpa [inv_mul_cancel_left₀ ha, inv_mul_cancel_left₀ (ne_of_gt hc)] using this
 
@@ -219,7 +198,8 @@ theorem div_le_div_right₀ (hc : c ≠ 0) : a / c ≤ b / c ↔ a ≤ b := by
   rw [div_eq_mul_inv, div_eq_mul_inv, mul_le_mul_right (zero_lt_iff.2 (inv_ne_zero hc))]
 
 theorem div_le_div_left₀ (ha : a ≠ 0) (hb : b ≠ 0) (hc : c ≠ 0) : a / b ≤ a / c ↔ c ≤ b := by
-  simp only [div_eq_mul_inv, mul_le_mul_left (zero_lt_iff.2 ha), inv_le_inv₀ hb hc]
+  simp only [div_eq_mul_inv, mul_le_mul_left (zero_lt_iff.2 ha),
+    inv_le_inv₀ (zero_lt_iff.2 hb) (zero_lt_iff.2 hc)]
 
 /-- `Equiv.mulLeft₀` as an `OrderIso` on a `LinearOrderedCommGroupWithZero.`.
 
@@ -263,8 +243,6 @@ lemma pow_lt_pow_succ (ha : 1 < a) : a ^ n < a ^ n.succ := by
 lemma pow_lt_pow_right₀ (ha : 1 < a) (hmn : m < n) : a ^ m < a ^ n := by
   induction' hmn with n _ ih; exacts [pow_lt_pow_succ ha, lt_trans ih (pow_lt_pow_succ ha)]
 
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow₀ := pow_lt_pow_right₀
-
 end LinearOrderedCommGroupWithZero
 
 instance instLinearOrderedCommMonoidWithZeroMultiplicativeOrderDual
@@ -431,3 +409,13 @@ instance instLinearOrderedCommGroupWithZero [LinearOrderedCommGroup α] :
   __ := commGroupWithZero
 
 end WithZero
+
+section MultiplicativeNotation
+
+/-- Notation for `WithZero (Multiplicative ℕ)` -/
+scoped[Multiplicative] notation "ℕₘ₀" => WithZero (Multiplicative ℕ)
+
+/-- Notation for `WithZero (Multiplicative ℤ)` -/
+scoped[Multiplicative] notation "ℤₘ₀" => WithZero (Multiplicative ℤ)
+
+end MultiplicativeNotation
diff --git a/Mathlib/Algebra/Order/GroupWithZero/Submonoid.lean b/Mathlib/Algebra/Order/GroupWithZero/Submonoid.lean
new file mode 100644
index 0000000000000..c3b553bb0ca38
--- /dev/null
+++ b/Mathlib/Algebra/Order/GroupWithZero/Submonoid.lean
@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2021 Chris Birkbeck. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Birkbeck
+-/
+import Mathlib.Algebra.Group.Submonoid.Basic
+import Mathlib.Algebra.Order.GroupWithZero.Unbundled
+
+/-!
+# The submonoid of positive elements
+-/
+
+namespace Submonoid
+variable (α) [MulZeroOneClass α] [PartialOrder α] [PosMulStrictMono α] [ZeroLEOneClass α]
+  [NeZero (1 : α)] {a : α}
+
+/-- The submonoid of positive elements. -/
+@[simps] def pos : Submonoid α where
+  carrier := Set.Ioi 0
+  one_mem' := zero_lt_one
+  mul_mem' := mul_pos
+
+variable {α}
+
+@[simp] lemma mem_pos : a ∈ pos α ↔ 0 < a := Iff.rfl
+
+end Submonoid
diff --git a/Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean b/Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
index 29e087b5c4728..d89b2b8a4fc2b 100644
--- a/Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
+++ b/Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
@@ -7,7 +7,8 @@ import Mathlib.Algebra.Group.Pi.Basic
 import Mathlib.Algebra.GroupWithZero.Units.Basic
 import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
 import Mathlib.Algebra.Order.ZeroLEOne
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.Bound.Attribute
+import Mathlib.Tactic.GCongr.CoreAttrs
 import Mathlib.Tactic.Nontriviality
 
 /-!
@@ -935,12 +936,16 @@ section MonoidWithZero
 variable [MonoidWithZero M₀]
 
 section Preorder
-variable [Preorder M₀] {a b c d : M₀} {n : ℕ}
+variable [Preorder M₀] {a b : M₀} {m n : ℕ}
 
 @[simp] lemma pow_nonneg [ZeroLEOneClass M₀] [PosMulMono M₀] (ha : 0 ≤ a) : ∀ n, 0 ≤ a ^ n
   | 0 => pow_zero a ▸ zero_le_one
   | n + 1 => pow_succ a n ▸ mul_nonneg (pow_nonneg ha _) ha
 
+lemma zero_pow_le_one [ZeroLEOneClass M₀] : ∀ n : ℕ, (0 : M₀) ^ n ≤ 1
+  | 0 => (pow_zero _).le
+  | n + 1 => by rw [zero_pow n.succ_ne_zero]; exact zero_le_one
+
 lemma pow_le_pow_of_le_one [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha₀ : 0 ≤ a)
     (ha₁ : a ≤ 1) : ∀ {m n : ℕ}, m ≤ n → a ^ n ≤ a ^ m
   | _, _, Nat.le.refl => le_rfl
@@ -958,9 +963,6 @@ lemma sq_le [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (h₀ : 0
 lemma one_le_mul_of_one_le_of_one_le [ZeroLEOneClass M₀] [PosMulMono M₀] (ha : 1 ≤ a) (hb : 1 ≤ b) :
     (1 : M₀) ≤ a * b := Left.one_le_mul_of_le_of_le ha hb <| zero_le_one.trans ha
 
-lemma mul_le_one [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀] (ha : a ≤ 1) (hb₀ : 0 ≤ b)
-    (hb : b ≤ 1) : a * b ≤ 1 := one_mul (1 : M₀) ▸ mul_le_mul ha hb hb₀ zero_le_one
-
 lemma one_lt_mul_of_le_of_lt [ZeroLEOneClass M₀] [MulPosMono M₀] (ha : 1 ≤ a) (hb : 1 < b) :
     1 < a * b := hb.trans_le <| le_mul_of_one_le_left (zero_le_one.trans hb.le) ha
 
@@ -975,6 +977,43 @@ lemma mul_lt_one_of_nonneg_of_lt_one_left [PosMulMono M₀] (ha₀ : 0 ≤ a) (h
 lemma mul_lt_one_of_nonneg_of_lt_one_right [MulPosMono M₀] (ha : a ≤ 1) (hb₀ : 0 ≤ b) (hb : b < 1) :
     a * b < 1 := (mul_le_of_le_one_left hb₀ ha).trans_lt hb
 
+section
+variable [ZeroLEOneClass M₀] [PosMulMono M₀] [MulPosMono M₀]
+
+lemma mul_le_one₀ (ha : a ≤ 1) (hb₀ : 0 ≤ b) (hb : b ≤ 1) : a * b ≤ 1 :=
+  one_mul (1 : M₀) ▸ mul_le_mul ha hb hb₀ zero_le_one
+
+lemma pow_le_one₀ : ∀ {n : ℕ}, 0 ≤ a → a ≤ 1 → a ^ n ≤ 1
+  | 0, _, _ => (pow_zero a).le
+  | n + 1, h₀, h₁ => (pow_succ a n).le.trans (mul_le_one₀ (pow_le_one₀ h₀ h₁) h₀ h₁)
+
+lemma pow_lt_one₀ (h₀ : 0 ≤ a) (h₁ : a < 1) : ∀ {n : ℕ}, n ≠ 0 → a ^ n < 1
+  | 0, h => (h rfl).elim
+  | n + 1, _ => by
+    rw [pow_succ']; exact mul_lt_one_of_nonneg_of_lt_one_left h₀ h₁ (pow_le_one₀ h₀ h₁.le)
+
+lemma one_le_pow₀ (ha : 1 ≤ a) : ∀ {n : ℕ}, 1 ≤ a ^ n
+  | 0 => by rw [pow_zero]
+  | n + 1 => by
+    simpa only [pow_succ', mul_one]
+      using mul_le_mul ha (one_le_pow₀ ha) zero_le_one (zero_le_one.trans ha)
+
+lemma one_lt_pow₀ (ha : 1 < a) : ∀ {n : ℕ}, n ≠ 0 → 1 < a ^ n
+  | 0, h => (h rfl).elim
+  | n + 1, _ => by rw [pow_succ']; exact one_lt_mul_of_lt_of_le ha (one_le_pow₀ ha.le)
+
+lemma pow_right_mono₀ (h : 1 ≤ a) : Monotone (a ^ ·) :=
+  monotone_nat_of_le_succ fun n => by
+    rw [pow_succ']; exact le_mul_of_one_le_left (pow_nonneg (zero_le_one.trans h) _) h
+
+@[gcongr]
+lemma pow_le_pow_right₀ (ha : 1 ≤ a) (hmn : m ≤ n) : a ^ m ≤ a ^ n := pow_right_mono₀ ha hmn
+
+lemma le_self_pow₀ (ha : 1 ≤ a) (hn : n ≠ 0) : a ≤ a ^ n := by
+  simpa only [pow_one] using pow_le_pow_right₀ ha <| Nat.pos_iff_ne_zero.2 hn
+
+end
+
 variable [Preorder α] {f g : α → M₀}
 
 lemma monotone_mul_left_of_nonneg [PosMulMono M₀] (ha : 0 ≤ a) : Monotone fun x ↦ a * x :=
@@ -1113,8 +1152,16 @@ end CancelMonoidWithZero
 section GroupWithZero
 variable [GroupWithZero G₀]
 
+section Preorder
+variable [Preorder G₀] [ZeroLEOneClass G₀]
+
+/-- See `div_self` for the version with equality when `a ≠ 0`. -/
+lemma div_self_le_one (a : G₀) : a / a ≤ 1 := by obtain rfl | ha := eq_or_ne a 0 <;> simp [*]
+
+end Preorder
+
 section PartialOrder
-variable [PartialOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀] {a b c d : G₀}
+variable [PartialOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀] {a b c : G₀}
 
 @[simp] lemma inv_pos : 0 < a⁻¹ ↔ 0 < a :=
   suffices ∀ a : G₀, 0 < a → 0 < a⁻¹ from ⟨fun h ↦ inv_inv a ▸ this _ h, this a⟩
@@ -1149,10 +1196,12 @@ lemma zpow_pos_of_pos [PosMulStrictMono G₀] (ha : 0 < a) : ∀ n : ℤ, 0 < a
 section PosMulMono
 variable [PosMulMono G₀]
 
+/-- See `le_inv_mul_iff₀'` for a version with multiplication on the other side. -/
 lemma le_inv_mul_iff₀ (hc : 0 < c) : a ≤ c⁻¹ * b ↔ c * a ≤ b where
   mp h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_left h hc.le
   mpr h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_left h (inv_nonneg.2 hc.le)
 
+/-- See `inv_mul_le_iff₀'` for a version with multiplication on the other side. -/
 lemma inv_mul_le_iff₀ (hc : 0 < c) : c⁻¹ * b ≤ a ↔ b ≤ c * a where
   mp h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_left h hc.le
   mpr h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_left h (inv_nonneg.2 hc.le)
@@ -1160,39 +1209,213 @@ lemma inv_mul_le_iff₀ (hc : 0 < c) : c⁻¹ * b ≤ a ↔ b ≤ c * a where
 lemma one_le_inv_mul₀ (ha : 0 < a) : 1 ≤ a⁻¹ * b ↔ a ≤ b := by rw [le_inv_mul_iff₀ ha, mul_one]
 lemma inv_mul_le_one₀ (ha : 0 < a) : a⁻¹ * b ≤ 1 ↔ b ≤ a := by rw [inv_mul_le_iff₀ ha, mul_one]
 
+/-- See `inv_le_iff_one_le_mul₀` for a version with multiplication on the other side. -/
+lemma inv_le_iff_one_le_mul₀' (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ a * b := by
+  rw [← inv_mul_le_iff₀ ha, mul_one]
+
 lemma one_le_inv₀ (ha : 0 < a) : 1 ≤ a⁻¹ ↔ a ≤ 1 := by simpa using one_le_inv_mul₀ ha (b := 1)
 lemma inv_le_one₀ (ha : 0 < a) : a⁻¹ ≤ 1 ↔ 1 ≤ a := by simpa using inv_mul_le_one₀ ha (b := 1)
 
+@[bound]
+lemma inv_le_one_of_one_le₀ (ha : 1 ≤ a) : a⁻¹ ≤ 1 := (inv_le_one₀ <| zero_lt_one.trans_le ha).2 ha
+
+lemma one_le_inv_iff₀ : 1 ≤ a⁻¹ ↔ 0 < a ∧ a ≤ 1 where
+  mp h := ⟨inv_pos.1 (zero_lt_one.trans_le h),
+    inv_inv a ▸ (inv_le_one₀ <| zero_lt_one.trans_le h).2 h⟩
+  mpr h := (one_le_inv₀ h.1).2 h.2
+
+/-- One direction of `le_inv_mul_iff₀` where `c` is allowed to be `0` (but `b` must be nonnegative).
+-/
+lemma mul_le_of_le_inv_mul₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c⁻¹ * b) : c * a ≤ b := by
+  obtain rfl | hc := hc.eq_or_lt
+  · simpa using hb
+  · rwa [le_inv_mul_iff₀ hc] at h
+
+/-- One direction of `inv_mul_le_iff₀` where `b` is allowed to be `0` (but `c` must be nonnegative).
+-/
+lemma inv_mul_le_of_le_mul₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b * c) : b⁻¹ * a ≤ c := by
+  obtain rfl | hb := hb.eq_or_lt
+  · simp [hc]
+  · rwa [inv_mul_le_iff₀ hb]
+
+@[bound]
+lemma inv_mul_le_one_of_le₀ (h : a ≤ b) (hb : 0 ≤ b) : b⁻¹ * a ≤ 1 :=
+  inv_mul_le_of_le_mul₀ hb zero_le_one <| by rwa [mul_one]
+
 end PosMulMono
 
 section MulPosMono
 variable [MulPosMono G₀]
 
+/-- See `le_mul_inv_iff₀'` for a version with multiplication on the other side. -/
 lemma le_mul_inv_iff₀ (hc : 0 < c) : a ≤ b * c⁻¹ ↔ a * c ≤ b where
   mp h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_right h hc.le
   mpr h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_right h (inv_nonneg.2 hc.le)
 
+/-- See `mul_inv_le_iff₀'` for a version with multiplication on the other side. -/
 lemma mul_inv_le_iff₀ (hc : 0 < c) : b * c⁻¹ ≤ a ↔ b ≤ a * c where
   mp h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_right h hc.le
   mpr h := by simpa [hc.ne'] using mul_le_mul_of_nonneg_right h (inv_nonneg.2 hc.le)
 
+/-- See `le_div_iff₀'` for a version with multiplication on the other side. -/
 lemma le_div_iff₀ (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b := by
   rw [div_eq_mul_inv, le_mul_inv_iff₀ hc]
 
+/-- See `div_le_iff₀'` for a version with multiplication on the other side. -/
 lemma div_le_iff₀ (hc : 0 < c) : b / c ≤ a ↔ b ≤ a * c := by
   rw [div_eq_mul_inv, mul_inv_le_iff₀ hc]
 
+/-- See `inv_le_iff_one_le_mul₀'` for a version with multiplication on the other side. -/
+lemma inv_le_iff_one_le_mul₀ (ha : 0 < a) : a⁻¹ ≤ b ↔ 1 ≤ b * a := by
+  rw [← mul_inv_le_iff₀ ha, one_mul]
+
 lemma one_le_div₀ (hb : 0 < b) : 1 ≤ a / b ↔ b ≤ a := by rw [le_div_iff₀ hb, one_mul]
 lemma div_le_one₀ (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b := by rw [div_le_iff₀ hb, one_mul]
 
+/-- One direction of `le_mul_inv_iff₀` where `c` is allowed to be `0` (but `b` must be nonnegative).
+-/
+lemma mul_le_of_le_mul_inv₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b * c⁻¹) : a * c ≤ b := by
+  obtain rfl | hc := hc.eq_or_lt
+  · simpa using hb
+  · rwa [le_mul_inv_iff₀ hc] at h
+
+/-- One direction of `mul_inv_le_iff₀` where `b` is allowed to be `0` (but `c` must be nonnegative).
+-/
+lemma mul_inv_le_of_le_mul₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a * b⁻¹ ≤ c := by
+  obtain rfl | hb := hb.eq_or_lt
+  · simp [hc]
+  · rwa [mul_inv_le_iff₀ hb]
+
+/-- One direction of `le_div_iff₀` where `c` is allowed to be `0` (but `b` must be nonnegative). -/
+lemma mul_le_of_le_div₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ b / c) : a * c ≤ b :=
+  mul_le_of_le_mul_inv₀ hb hc (div_eq_mul_inv b _ ▸ h)
+
+/-- One direction of `div_le_iff₀` where `b` is allowed to be `0` (but `c` must be nonnegative). -/
+lemma div_le_of_le_mul₀ (hb : 0 ≤ b) (hc : 0 ≤ c) (h : a ≤ c * b) : a / b ≤ c :=
+  div_eq_mul_inv a _ ▸ mul_inv_le_of_le_mul₀ hb hc h
+
+@[bound]
+lemma mul_inv_le_one_of_le₀ (h : a ≤ b) (hb : 0 ≤ b) : a * b⁻¹ ≤ 1 :=
+  mul_inv_le_of_le_mul₀ hb zero_le_one <| by rwa [one_mul]
+
+@[bound]
+lemma div_le_one_of_le₀ (h : a ≤ b) (hb : 0 ≤ b) : a / b ≤ 1 :=
+  div_le_of_le_mul₀ hb zero_le_one <| by rwa [one_mul]
+
 @[deprecated (since := "2024-08-21")] alias le_div_iff := le_div_iff₀
 @[deprecated (since := "2024-08-21")] alias div_le_iff := div_le_iff₀
 
+variable [PosMulMono G₀]
+
+/-- See `inv_anti₀` for the implication from right-to-left with one fewer assumption. -/
+lemma inv_le_inv₀ (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b⁻¹ ↔ b ≤ a := by
+  rw [inv_le_iff_one_le_mul₀' ha, le_mul_inv_iff₀ hb, one_mul]
+
+@[gcongr, bound]
+lemma inv_anti₀ (hb : 0 < b) (hba : b ≤ a) : a⁻¹ ≤ b⁻¹ := (inv_le_inv₀ (hb.trans_le hba) hb).2 hba
+
+/-- See also `inv_le_of_inv_le₀` for a one-sided implication with one fewer assumption. -/
+lemma inv_le_comm₀ (ha : 0 < a) (hb : 0 < b) : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by
+  rw [← inv_le_inv₀ hb (inv_pos.2 ha), inv_inv]
+
+lemma inv_le_of_inv_le₀ (ha : 0 < a) (h : a⁻¹ ≤ b) : b⁻¹ ≤ a :=
+  (inv_le_comm₀ ha <| (inv_pos.2 ha).trans_le h).1 h
+
+/-- See also `le_inv_of_le_inv₀` for a one-sided implication with one fewer assumption. -/
+lemma le_inv_comm₀ (ha : 0 < a) (hb : 0 < b) : a ≤ b⁻¹ ↔ b ≤ a⁻¹ := by
+  rw [← inv_le_inv₀ (inv_pos.2 hb) ha, inv_inv]
+
+lemma le_inv_of_le_inv₀ (ha : 0 < a) (h : a ≤ b⁻¹) : b ≤ a⁻¹ :=
+  (le_inv_comm₀ ha <| inv_pos.1 <| ha.trans_le h).1 h
+
 end MulPosMono
+
+section PosMulStrictMono
+variable [PosMulStrictMono G₀]
+
+/-- See `lt_inv_mul_iff₀'` for a version with multiplication on the other side. -/
+lemma lt_inv_mul_iff₀ (hc : 0 < c) : a < c⁻¹ * b ↔ c * a < b where
+  mp h := by simpa [hc.ne'] using mul_lt_mul_of_pos_left h hc
+  mpr h := by simpa [hc.ne'] using mul_lt_mul_of_pos_left h (inv_pos.2 hc)
+
+/-- See `inv_mul_lt_iff₀'` for a version with multiplication on the other side. -/
+lemma inv_mul_lt_iff₀ (hc : 0 < c) : c⁻¹ * b < a ↔ b < c * a where
+  mp h := by simpa [hc.ne'] using mul_lt_mul_of_pos_left h hc
+  mpr h := by simpa [hc.ne'] using mul_lt_mul_of_pos_left h (inv_pos.2 hc)
+
+/-- See `inv_lt_iff_one_lt_mul₀` for a version with multiplication on the other side. -/
+lemma inv_lt_iff_one_lt_mul₀' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b := by
+  rw [← inv_mul_lt_iff₀ ha, mul_one]
+
+lemma one_lt_inv_mul₀ (ha : 0 < a) : 1 < a⁻¹ * b ↔ a < b := by rw [lt_inv_mul_iff₀ ha, mul_one]
+lemma inv_mul_lt_one₀ (ha : 0 < a) : a⁻¹ * b < 1 ↔ b < a := by rw [inv_mul_lt_iff₀ ha, mul_one]
+
+lemma one_lt_inv₀ (ha : 0 < a) : 1 < a⁻¹ ↔ a < 1 := by simpa using one_lt_inv_mul₀ ha (b := 1)
+lemma inv_lt_one₀ (ha : 0 < a) : a⁻¹ < 1 ↔ 1 < a := by simpa using inv_mul_lt_one₀ ha (b := 1)
+
+@[bound]
+lemma inv_lt_one_of_one_lt₀ (ha : 1 < a) : a⁻¹ < 1 := (inv_lt_one₀ <| zero_lt_one.trans ha).2 ha
+
+lemma one_lt_inv_iff₀ : 1 < a⁻¹ ↔ 0 < a ∧ a < 1 where
+  mp h := ⟨inv_pos.1 (zero_lt_one.trans h), inv_inv a ▸ (inv_lt_one₀ <| zero_lt_one.trans h).2 h⟩
+  mpr h := (one_lt_inv₀ h.1).2 h.2
+
+end PosMulStrictMono
+
+section MulPosStrictMono
+variable [MulPosStrictMono G₀]
+
+/-- See `lt_mul_inv_iff₀'` for a version with multiplication on the other side. -/
+lemma lt_mul_inv_iff₀ (hc : 0 < c) : a < b * c⁻¹ ↔ a * c < b where
+  mp h := by simpa [hc.ne'] using mul_lt_mul_of_pos_right h hc
+  mpr h := by simpa [hc.ne'] using mul_lt_mul_of_pos_right h (inv_pos.2 hc)
+
+/-- See `mul_inv_lt_iff₀'` for a version with multiplication on the other side. -/
+lemma mul_inv_lt_iff₀ (hc : 0 < c) : b * c⁻¹ < a ↔ b < a * c where
+  mp h := by simpa [hc.ne'] using mul_lt_mul_of_pos_right h hc
+  mpr h := by simpa [hc.ne'] using mul_lt_mul_of_pos_right h (inv_pos.2 hc)
+
+/-- See `lt_div_iff₀'` for a version with multiplication on the other side. -/
+lemma lt_div_iff₀ (hc : 0 < c) : a < b / c ↔ a * c < b := by
+  rw [div_eq_mul_inv, lt_mul_inv_iff₀ hc]
+
+/-- See `div_le_iff₀'` for a version with multiplication on the other side. -/
+lemma div_lt_iff₀ (hc : 0 < c) : b / c < a ↔ b < a * c := by
+  rw [div_eq_mul_inv, mul_inv_lt_iff₀ hc]
+
+/-- See `inv_lt_iff_one_lt_mul₀'` for a version with multiplication on the other side. -/
+lemma inv_lt_iff_one_lt_mul₀ (ha : 0 < a) : a⁻¹ < b ↔ 1 < b * a := by
+  rw [← mul_inv_lt_iff₀ ha, one_mul]
+
+variable [PosMulStrictMono G₀]
+
+/-- See `inv_strictAnti₀` for the implication from right-to-left with one fewer assumption. -/
+lemma inv_lt_inv₀ (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b⁻¹ ↔ b < a := by
+  rw [inv_lt_iff_one_lt_mul₀' ha, lt_mul_inv_iff₀ hb, one_mul]
+
+@[gcongr, bound]
+lemma inv_strictAnti₀ (hb : 0 < b) (hba : b < a) : a⁻¹ < b⁻¹ :=
+  (inv_lt_inv₀ (hb.trans hba) hb).2 hba
+
+/-- See also `inv_lt_of_inv_lt₀` for a one-sided implication with one fewer assumption. -/
+lemma inv_lt_comm₀ (ha : 0 < a) (hb : 0 < b) : a⁻¹ < b ↔ b⁻¹ < a := by
+  rw [← inv_lt_inv₀ hb (inv_pos.2 ha), inv_inv]
+
+lemma inv_lt_of_inv_lt₀ (ha : 0 < a) (h : a⁻¹ < b) : b⁻¹ < a :=
+  (inv_lt_comm₀ ha <| (inv_pos.2 ha).trans h).1 h
+
+/-- See also `lt_inv_of_lt_inv₀` for a one-sided implication with one fewer assumption. -/
+lemma lt_inv_comm₀ (ha : 0 < a) (hb : 0 < b) : a < b⁻¹ ↔ b < a⁻¹ := by
+  rw [← inv_lt_inv₀ (inv_pos.2 hb) ha, inv_inv]
+
+lemma lt_inv_of_lt_inv₀ (ha : 0 < a) (h : a < b⁻¹) : b < a⁻¹ :=
+  (lt_inv_comm₀ ha <| inv_pos.1 <| ha.trans h).1 h
+
+end MulPosStrictMono
 end PartialOrder
 
 section LinearOrder
-variable [LinearOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀] {a b c : G₀}
+variable [LinearOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀] {a b : G₀}
 
 @[simp] lemma inv_neg'' : a⁻¹ < 0 ↔ a < 0 := by simp only [← not_le, inv_nonneg]
 @[simp] lemma inv_nonpos : a⁻¹ ≤ 0 ↔ a ≤ 0 := by simp only [← not_lt, inv_pos]
@@ -1205,44 +1428,125 @@ lemma one_div_nonpos : 1 / a ≤ 0 ↔ a ≤ 0 := one_div a ▸ inv_nonpos
 lemma div_nonpos_of_nonneg_of_nonpos [PosMulMono G₀] (ha : 0 ≤ a) (hb : b ≤ 0) : a / b ≤ 0 := by
   rw [div_eq_mul_inv]; exact mul_nonpos_of_nonneg_of_nonpos ha (inv_nonpos.2 hb)
 
+lemma inv_lt_one_iff₀ [PosMulStrictMono G₀] : a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a := by
+  simp_rw [← not_le, one_le_inv_iff₀, not_and_or, not_lt]
+
+lemma inv_le_one_iff₀ [PosMulStrictMono G₀] : a⁻¹ ≤ 1 ↔ a ≤ 0 ∨ 1 ≤ a := by
+  simp only [← not_lt, one_lt_inv_iff₀, not_and_or]
+
 end GroupWithZero.LinearOrder
 
 section CommSemigroupHasZero
 
-variable [Mul α] [IsSymmOp α α (· * ·)] [Zero α] [Preorder α]
+variable [Mul α] [@Std.Commutative α (· * ·)] [Zero α] [Preorder α]
 
 theorem posMulStrictMono_iff_mulPosStrictMono : PosMulStrictMono α ↔ MulPosStrictMono α := by
-  simp only [PosMulStrictMono, MulPosStrictMono, IsSymmOp.symm_op]
+  simp only [PosMulStrictMono, MulPosStrictMono, Std.Commutative.comm]
 
 theorem posMulReflectLT_iff_mulPosReflectLT : PosMulReflectLT α ↔ MulPosReflectLT α := by
-  simp only [PosMulReflectLT, MulPosReflectLT, IsSymmOp.symm_op]
+  simp only [PosMulReflectLT, MulPosReflectLT, Std.Commutative.comm]
 
 theorem posMulMono_iff_mulPosMono : PosMulMono α ↔ MulPosMono α := by
-  simp only [PosMulMono, MulPosMono, IsSymmOp.symm_op]
+  simp only [PosMulMono, MulPosMono, Std.Commutative.comm]
 
 theorem posMulReflectLE_iff_mulPosReflectLE : PosMulReflectLE α ↔ MulPosReflectLE α := by
-  simp only [PosMulReflectLE, MulPosReflectLE, IsSymmOp.symm_op]
+  simp only [PosMulReflectLE, MulPosReflectLE, Std.Commutative.comm]
 
 end CommSemigroupHasZero
 
 section CommGroupWithZero
 variable [CommGroupWithZero G₀]
-variable [PartialOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀] [MulPosMono G₀] {a b c d : G₀}
+variable [PartialOrder G₀] [ZeroLEOneClass G₀] [PosMulReflectLT G₀]
+
+section PosMulMono
+variable [PosMulMono G₀] {a b c d : G₀}
+
+/-- See `le_inv_mul_iff₀` for a version with multiplication on the other side. -/
+lemma le_inv_mul_iff₀' (hc : 0 < c) : a ≤ c⁻¹ * b ↔ c * a ≤ b := by
+  rw [le_inv_mul_iff₀ hc, mul_comm]
+
+/-- See `inv_mul_le_iff₀` for a version with multiplication on the other side. -/
+lemma inv_mul_le_iff₀' (hc : 0 < c) : c⁻¹ * b ≤ a ↔ b ≤ a * c := by
+  rw [inv_mul_le_iff₀ hc, mul_comm]
+
+/-- See `le_mul_inv_iff₀` for a version with multiplication on the other side. -/
+lemma le_mul_inv_iff₀' (hc : 0 < c) : a ≤ b * c⁻¹ ↔ c * a ≤ b := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
+  rw [le_mul_inv_iff₀ hc, mul_comm]
+
+/-- See `mul_inv_le_iff₀` for a version with multiplication on the other side. -/
+lemma mul_inv_le_iff₀' (hc : 0 < c) : b * c⁻¹ ≤ a ↔ b ≤ c * a := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
+  rw [mul_inv_le_iff₀ hc, mul_comm]
 
 lemma div_le_div₀ (hb : 0 < b) (hd : 0 < d) :
     a / b ≤ c / d ↔ a * d ≤ c * b := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
   rw [div_le_iff₀ hb, ← mul_div_right_comm, le_div_iff₀ hd]
 
-lemma le_div_iff₀' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b := by rw [le_div_iff₀ hc, mul_comm]
-lemma div_le_iff₀' (hc : 0 < c) : b / c ≤ a ↔ b ≤ c * a := by rw [div_le_iff₀ hc, mul_comm]
+/-- See `le_div_iff₀` for a version with multiplication on the other side. -/
+lemma le_div_iff₀' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
+  rw [le_div_iff₀ hc, mul_comm]
+
+/-- See `div_le_iff₀` for a version with multiplication on the other side. -/
+lemma div_le_iff₀' (hc : 0 < c) : b / c ≤ a ↔ b ≤ c * a := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
+  rw [div_le_iff₀ hc, mul_comm]
 
 lemma le_div_comm₀ (ha : 0 < a) (hc : 0 < c) : a ≤ b / c ↔ c ≤ b / a := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
   rw [le_div_iff₀ ha, le_div_iff₀' hc]
 
 lemma div_le_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b ≤ c ↔ a / c ≤ b := by
+  have := posMulMono_iff_mulPosMono.1 ‹_›
   rw [div_le_iff₀ hb, div_le_iff₀' hc]
 
 @[deprecated (since := "2024-08-21")] alias le_div_iff' := le_div_iff₀'
 @[deprecated (since := "2024-08-21")] alias div_le_iff' := div_le_iff₀'
 
+end PosMulMono
+
+section PosMulStrictMono
+variable [PosMulStrictMono G₀] {a b c : G₀}
+
+/-- See `lt_inv_mul_iff₀` for a version with multiplication on the other side. -/
+lemma lt_inv_mul_iff₀' (hc : 0 < c) : a < c⁻¹ * b ↔ a * c < b := by
+  rw [lt_inv_mul_iff₀ hc, mul_comm]
+
+/-- See `inv_mul_lt_iff₀` for a version with multiplication on the other side. -/
+lemma inv_mul_lt_iff₀' (hc : 0 < c) : c⁻¹ * b < a ↔ b < a * c := by
+  rw [inv_mul_lt_iff₀ hc, mul_comm]
+
+/-- See `lt_mul_inv_iff₀` for a version with multiplication on the other side. -/
+lemma lt_mul_inv_iff₀' (hc : 0 < c) : a < b * c⁻¹ ↔ c * a < b := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [lt_mul_inv_iff₀ hc, mul_comm]
+
+/-- See `mul_inv_lt_iff₀` for a version with multiplication on the other side. -/
+lemma mul_inv_lt_iff₀' (hc : 0 < c) : b * c⁻¹ < a ↔ b < c * a := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [mul_inv_lt_iff₀ hc, mul_comm]
+
+/-- See `lt_div_iff₀` for a version with multiplication on the other side. -/
+lemma lt_div_iff₀' (hc : 0 < c) : a < b / c ↔ c * a < b := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [lt_div_iff₀ hc, mul_comm]
+
+/-- See `div_lt_iff₀` for a version with multiplication on the other side. -/
+lemma div_lt_iff₀' (hc : 0 < c) : b / c < a ↔ b < c * a := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [div_lt_iff₀ hc, mul_comm]
+
+lemma lt_div_comm₀ (ha : 0 < a) (hc : 0 < c) : a < b / c ↔ c < b / a := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [lt_div_iff₀ ha, lt_div_iff₀' hc]
+
+lemma div_lt_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b < c ↔ a / c < b := by
+  have := posMulStrictMono_iff_mulPosStrictMono.1 ‹_›
+  rw [div_lt_iff₀ hb, div_lt_iff₀' hc]
+
+end PosMulStrictMono
 end CommGroupWithZero
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Algebra/Order/GroupWithZero/Unbundled/Lemmas.lean b/Mathlib/Algebra/Order/GroupWithZero/Unbundled/Lemmas.lean
new file mode 100644
index 0000000000000..edc0cb47b953d
--- /dev/null
+++ b/Mathlib/Algebra/Order/GroupWithZero/Unbundled/Lemmas.lean
@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2021 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+-/
+import Mathlib.Algebra.Group.Pi.Basic
+import Mathlib.Algebra.Order.GroupWithZero.Unbundled
+import Mathlib.Algebra.GroupWithZero.Units.Equiv
+import Mathlib.Order.Hom.Basic
+
+/-!
+# Multiplication by a positive element as an order isomorphism
+-/
+
+variable {G₀} [GroupWithZero G₀] [Preorder G₀] [ZeroLEOneClass G₀] {a b c d : G₀}
+
+/-- `Equiv.mulLeft₀` as an order isomorphism. -/
+@[simps! (config := { simpRhs := true })]
+def OrderIso.mulLeft₀ [PosMulMono G₀] [PosMulReflectLE G₀] (a : G₀) (ha : 0 < a) : G₀ ≃o G₀ where
+  toEquiv := .mulLeft₀ a ha.ne'
+  map_rel_iff' := mul_le_mul_left ha
+
+/-- `Equiv.mulRight₀` as an order isomorphism. -/
+@[simps! (config := { simpRhs := true })]
+def OrderIso.mulRight₀ [MulPosMono G₀] [MulPosReflectLE G₀] (a : G₀) (ha : 0 < a) : G₀ ≃o G₀ where
+  toEquiv := .mulRight₀ a ha.ne'
+  map_rel_iff' := mul_le_mul_right ha
diff --git a/Mathlib/Algebra/Order/Hom/Basic.lean b/Mathlib/Algebra/Order/Hom/Basic.lean
index dbcb880887ff3..b97b19b03d22d 100644
--- a/Mathlib/Algebra/Order/Hom/Basic.lean
+++ b/Mathlib/Algebra/Order/Hom/Basic.lean
@@ -73,29 +73,33 @@ variable {ι F α β γ δ : Type*}
 /-! ### Basics -/
 
 /-- `NonnegHomClass F α β` states that `F` is a type of nonnegative morphisms. -/
-class NonnegHomClass (F α β : Type*) [Zero β] [LE β] [FunLike F α β] : Prop where
+class NonnegHomClass (F : Type*) (α β : outParam Type*) [Zero β] [LE β] [FunLike F α β] : Prop where
   /-- the image of any element is non negative. -/
   apply_nonneg (f : F) : ∀ a, 0 ≤ f a
 
 /-- `SubadditiveHomClass F α β` states that `F` is a type of subadditive morphisms. -/
-class SubadditiveHomClass (F α β : Type*) [Add α] [Add β] [LE β] [FunLike F α β] : Prop where
+class SubadditiveHomClass (F : Type*) (α β : outParam Type*)
+    [Add α] [Add β] [LE β] [FunLike F α β] : Prop where
   /-- the image of a sum is less or equal than the sum of the images. -/
   map_add_le_add (f : F) : ∀ a b, f (a + b) ≤ f a + f b
 
 /-- `SubmultiplicativeHomClass F α β` states that `F` is a type of submultiplicative morphisms. -/
 @[to_additive SubadditiveHomClass]
-class SubmultiplicativeHomClass (F α β : Type*) [Mul α] [Mul β] [LE β] [FunLike F α β] : Prop where
+class SubmultiplicativeHomClass (F : Type*) (α β : outParam (Type*)) [Mul α] [Mul β] [LE β]
+    [FunLike F α β] : Prop where
   /-- the image of a product is less or equal than the product of the images. -/
   map_mul_le_mul (f : F) : ∀ a b, f (a * b) ≤ f a * f b
 
 /-- `MulLEAddHomClass F α β` states that `F` is a type of subadditive morphisms. -/
 @[to_additive SubadditiveHomClass]
-class MulLEAddHomClass (F α β : Type*) [Mul α] [Add β] [LE β] [FunLike F α β] : Prop where
+class MulLEAddHomClass (F : Type*) (α β : outParam Type*) [Mul α] [Add β] [LE β] [FunLike F α β] :
+    Prop where
   /-- the image of a product is less or equal than the sum of the images. -/
   map_mul_le_add (f : F) : ∀ a b, f (a * b) ≤ f a + f b
 
 /-- `NonarchimedeanHomClass F α β` states that `F` is a type of non-archimedean morphisms. -/
-class NonarchimedeanHomClass (F α β : Type*) [Add α] [LinearOrder β] [FunLike F α β] : Prop where
+class NonarchimedeanHomClass (F : Type*) (α β : outParam Type*)
+    [Add α] [LinearOrder β] [FunLike F α β] : Prop where
   /-- the image of a sum is less or equal than the maximum of the images. -/
   map_add_le_max (f : F) : ∀ a b, f (a + b) ≤ max (f a) (f b)
 
@@ -154,7 +158,8 @@ end Mathlib.Meta.Positivity
 group `α`.
 
 You should extend this class when you extend `AddGroupSeminorm`. -/
-class AddGroupSeminormClass (F α β : Type*) [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β]
+class AddGroupSeminormClass (F : Type*) (α β : outParam Type*)
+    [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β]
   extends SubadditiveHomClass F α β : Prop where
   /-- The image of zero is zero. -/
   map_zero (f : F) : f 0 = 0
@@ -165,7 +170,8 @@ class AddGroupSeminormClass (F α β : Type*) [AddGroup α] [OrderedAddCommMonoi
 
 You should extend this class when you extend `GroupSeminorm`. -/
 @[to_additive]
-class GroupSeminormClass (F α β : Type*) [Group α] [OrderedAddCommMonoid β] [FunLike F α β]
+class GroupSeminormClass (F : Type*) (α β : outParam Type*)
+    [Group α] [OrderedAddCommMonoid β] [FunLike F α β]
   extends MulLEAddHomClass F α β : Prop where
   /-- The image of one is zero. -/
   map_one_eq_zero (f : F) : f 1 = 0
@@ -176,7 +182,8 @@ class GroupSeminormClass (F α β : Type*) [Group α] [OrderedAddCommMonoid β]
 `α`.
 
 You should extend this class when you extend `AddGroupNorm`. -/
-class AddGroupNormClass (F α β : Type*) [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β]
+class AddGroupNormClass (F : Type*) (α β : outParam Type*)
+    [AddGroup α] [OrderedAddCommMonoid β] [FunLike F α β]
   extends AddGroupSeminormClass F α β : Prop where
   /-- The argument is zero if its image under the map is zero. -/
   eq_zero_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 0
@@ -185,7 +192,8 @@ class AddGroupNormClass (F α β : Type*) [AddGroup α] [OrderedAddCommMonoid β
 
 You should extend this class when you extend `GroupNorm`. -/
 @[to_additive]
-class GroupNormClass (F α β : Type*) [Group α] [OrderedAddCommMonoid β] [FunLike F α β]
+class GroupNormClass (F : Type*) (α β : outParam Type*)
+    [Group α] [OrderedAddCommMonoid β] [FunLike F α β]
   extends GroupSeminormClass F α β : Prop where
   /-- The argument is one if its image under the map is zero. -/
   eq_one_of_map_eq_zero (f : F) {a : α} : f a = 0 → a = 1
@@ -275,20 +283,23 @@ theorem map_pos_of_ne_one [Group α] [LinearOrderedAddCommMonoid β] [GroupNormC
 /-- `RingSeminormClass F α` states that `F` is a type of `β`-valued seminorms on the ring `α`.
 
 You should extend this class when you extend `RingSeminorm`. -/
-class RingSeminormClass (F α β : Type*) [NonUnitalNonAssocRing α] [OrderedSemiring β]
-  [FunLike F α β] extends AddGroupSeminormClass F α β, SubmultiplicativeHomClass F α β : Prop
+class RingSeminormClass (F : Type*) (α β : outParam Type*)
+    [NonUnitalNonAssocRing α] [OrderedSemiring β] [FunLike F α β]
+  extends AddGroupSeminormClass F α β, SubmultiplicativeHomClass F α β : Prop
 
 /-- `RingNormClass F α` states that `F` is a type of `β`-valued norms on the ring `α`.
 
 You should extend this class when you extend `RingNorm`. -/
-class RingNormClass (F α β : Type*) [NonUnitalNonAssocRing α] [OrderedSemiring β] [FunLike F α β]
+class RingNormClass (F : Type*) (α β : outParam Type*)
+    [NonUnitalNonAssocRing α] [OrderedSemiring β] [FunLike F α β]
   extends RingSeminormClass F α β, AddGroupNormClass F α β : Prop
 
 /-- `MulRingSeminormClass F α` states that `F` is a type of `β`-valued multiplicative seminorms
 on the ring `α`.
 
 You should extend this class when you extend `MulRingSeminorm`. -/
-class MulRingSeminormClass (F α β : Type*) [NonAssocRing α] [OrderedSemiring β] [FunLike F α β]
+class MulRingSeminormClass (F : Type*) (α β : outParam Type*)
+    [NonAssocRing α] [OrderedSemiring β] [FunLike F α β]
   extends AddGroupSeminormClass F α β, MonoidWithZeroHomClass F α β : Prop
 
 -- Lower the priority of these instances since they require synthesizing an order structure.
@@ -299,7 +310,8 @@ attribute [instance 50]
 ring `α`.
 
 You should extend this class when you extend `MulRingNorm`. -/
-class MulRingNormClass (F α β : Type*) [NonAssocRing α] [OrderedSemiring β] [FunLike F α β]
+class MulRingNormClass (F : Type*) (α β : outParam Type*)
+    [NonAssocRing α] [OrderedSemiring β] [FunLike F α β]
   extends MulRingSeminormClass F α β, AddGroupNormClass F α β : Prop
 
 -- See note [out-param inheritance]
diff --git a/Mathlib/Algebra/Order/Hom/Monoid.lean b/Mathlib/Algebra/Order/Hom/Monoid.lean
index d212b514c3cc4..89356b58c0ecf 100644
--- a/Mathlib/Algebra/Order/Hom/Monoid.lean
+++ b/Mathlib/Algebra/Order/Hom/Monoid.lean
@@ -18,12 +18,17 @@ This file defines morphisms between (additive) ordered monoids.
 * `OrderAddMonoidHom`: Ordered additive monoid homomorphisms.
 * `OrderMonoidHom`: Ordered monoid homomorphisms.
 * `OrderMonoidWithZeroHom`: Ordered monoid with zero homomorphisms.
+* `OrderAddMonoidIso`: Ordered additive monoid isomorphisms.
+* `OrderMonoidIso`: Ordered monoid isomorphisms.
 
 ## Notation
 
-* `→+o`: Bundled ordered additive monoid homs. Also use for additive groups homs.
-* `→*o`: Bundled ordered monoid homs. Also use for groups homs.
-* `→*₀o`: Bundled ordered monoid with zero homs. Also use for groups with zero homs.
+* `→+o`: Bundled ordered additive monoid homs. Also use for additive group homs.
+* `→*o`: Bundled ordered monoid homs. Also use for group homs.
+* `→*₀o`: Bundled ordered monoid with zero homs. Also use for group with zero homs.
+* `≃+o`: Bundled ordered additive monoid isos. Also use for additive group isos.
+* `≃*o`: Bundled ordered monoid isos. Also use for group isos.
+* `≃*₀o`: Bundled ordered monoid with zero isos. Also use for group with zero isos.
 
 ## Implementation notes
 
@@ -66,9 +71,8 @@ structure.
 `OrderAddMonoidHom` is also used for ordered group homomorphisms.
 
 When possible, instead of parametrizing results over `(f : α →+o β)`,
-you should parametrize over `(F : Type*) [OrderAddMonoidHomClass F α β] (f : F)`.
-
-When you extend this structure, make sure to extend `OrderAddMonoidHomClass`. -/
+you should parametrize over
+`(F : Type*) [FunLike F M N] [MonoidHomClass F M N] [OrderHomClass F M N] (f : F)`. -/
 structure OrderAddMonoidHom (α β : Type*) [Preorder α] [Preorder β] [AddZeroClass α]
   [AddZeroClass β] extends α →+ β where
   /-- An `OrderAddMonoidHom` is a monotone function. -/
@@ -77,6 +81,22 @@ structure OrderAddMonoidHom (α β : Type*) [Preorder α] [Preorder β] [AddZero
 /-- Infix notation for `OrderAddMonoidHom`. -/
 infixr:25 " →+o " => OrderAddMonoidHom
 
+/-- `α ≃+o β` is the type of monotone isomorphisms `α ≃ β` that preserve the `OrderedAddCommMonoid`
+structure.
+
+`OrderAddMonoidIso` is also used for ordered group isomorphisms.
+
+When possible, instead of parametrizing results over `(f : α ≃+o β)`,
+you should parametrize over
+`(F : Type*) [FunLike F M N] [AddEquivClass F M N] [OrderIsoClass F M N] (f : F)`. -/
+structure OrderAddMonoidIso (α β : Type*) [Preorder α] [Preorder β] [AddZeroClass α]
+  [AddZeroClass β] extends α ≃+ β where
+  /-- An `OrderAddMonoidIso` respects `≤`. -/
+  map_le_map_iff' {a b : α} : toFun a ≤ toFun b ↔ a ≤ b
+
+/-- Infix notation for `OrderAddMonoidIso`. -/
+infixr:25 " ≃+o " => OrderAddMonoidIso
+
 -- Instances and lemmas are defined below through `@[to_additive]`.
 end AddMonoid
 
@@ -87,9 +107,8 @@ section Monoid
 `OrderMonoidHom` is also used for ordered group homomorphisms.
 
 When possible, instead of parametrizing results over `(f : α →*o β)`,
-you should parametrize over `(F : Type*) [OrderMonoidHomClass F α β] (f : F)`.
-
-When you extend this structure, make sure to extend `OrderMonoidHomClass`. -/
+you should parametrize over
+`(F : Type*) [FunLike F M N] [MonoidHomClass F M N] [OrderHomClass F M N] (f : F)`. -/
 @[to_additive]
 structure OrderMonoidHom (α β : Type*) [Preorder α] [Preorder β] [MulOneClass α]
   [MulOneClass β] extends α →* β where
@@ -104,8 +123,9 @@ variable [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] [FunLike
 /-- Turn an element of a type `F` satisfying `OrderHomClass F α β` and `MonoidHomClass F α β`
 into an actual `OrderMonoidHom`. This is declared as the default coercion from `F` to `α →*o β`. -/
 @[to_additive (attr := coe)
-  "Turn an element of a type `F` satisfying `OrderAddMonoidHomClass F α β` into an actual
-  `OrderAddMonoidHom`. This is declared as the default coercion from `F` to `α →+o β`."]
+  "Turn an element of a type `F` satisfying `OrderHomClass F α β` and `AddMonoidHomClass F α β`
+  into an actual `OrderAddMonoidHom`.
+  This is declared as the default coercion from `F` to `α →+o β`."]
 def OrderMonoidHomClass.toOrderMonoidHom [OrderHomClass F α β] [MonoidHomClass F α β] (f : F) :
     α →*o β :=
   { (f : α →* β) with monotone' := OrderHomClass.monotone f }
@@ -117,6 +137,49 @@ def OrderMonoidHomClass.toOrderMonoidHom [OrderHomClass F α β] [MonoidHomClass
 instance [OrderHomClass F α β] [MonoidHomClass F α β] : CoeTC F (α →*o β) :=
   ⟨OrderMonoidHomClass.toOrderMonoidHom⟩
 
+/-- `α ≃*o β` is the type of isomorphisms `α ≃ β` that preserve the `OrderedCommMonoid` structure.
+
+`OrderMonoidIso` is also used for ordered group isomorphisms.
+
+When possible, instead of parametrizing results over `(f : α ≃*o β)`,
+you should parametrize over
+`(F : Type*) [FunLike F M N] [MulEquivClass F M N] [OrderIsoClass F M N] (f : F)`. -/
+@[to_additive]
+structure OrderMonoidIso (α β : Type*) [Preorder α] [Preorder β] [MulOneClass α]
+  [MulOneClass β] extends α ≃* β where
+  /-- An `OrderMonoidIso` respects `≤`. -/
+  map_le_map_iff' {a b : α} : toFun a ≤ toFun b ↔ a ≤ b
+
+/-- Infix notation for `OrderMonoidIso`. -/
+infixr:25 " ≃*o " => OrderMonoidIso
+
+variable [Preorder α] [Preorder β] [MulOneClass α] [MulOneClass β] [FunLike F α β]
+
+/-- Turn an element of a type `F` satisfying `OrderIsoClass F α β` and `MulEquivClass F α β`
+into an actual `OrderMonoidIso`. This is declared as the default coercion from `F` to `α ≃*o β`. -/
+@[to_additive (attr := coe)
+  "Turn an element of a type `F` satisfying `OrderIsoClass F α β` and `AddEquivClass F α β`
+  into an actual `OrderAddMonoidIso`.
+  This is declared as the default coercion from `F` to `α ≃+o β`."]
+def OrderMonoidIsoClass.toOrderMonoidIso [EquivLike F α β] [OrderIsoClass F α β]
+    [MulEquivClass F α β] (f : F) :
+    α ≃*o β :=
+  { (f : α ≃* β) with map_le_map_iff' := OrderIsoClass.map_le_map_iff f }
+
+/-- Any type satisfying `OrderMonoidHomClass` can be cast into `OrderMonoidHom` via
+  `OrderMonoidHomClass.toOrderMonoidHom`. -/
+@[to_additive "Any type satisfying `OrderAddMonoidHomClass` can be cast into `OrderAddMonoidHom` via
+  `OrderAddMonoidHomClass.toOrderAddMonoidHom`"]
+instance [OrderHomClass F α β] [MonoidHomClass F α β] : CoeTC F (α →*o β) :=
+  ⟨OrderMonoidHomClass.toOrderMonoidHom⟩
+
+/-- Any type satisfying `OrderMonoidIsoClass` can be cast into `OrderMonoidIso` via
+  `OrderMonoidIsoClass.toOrderMonoidIso`. -/
+@[to_additive "Any type satisfying `OrderAddMonoidIsoClass` can be cast into `OrderAddMonoidIso` via
+  `OrderAddMonoidIsoClass.toOrderAddMonoidIso`"]
+instance [EquivLike F α β] [OrderIsoClass F α β] [MulEquivClass F α β] : CoeTC F (α ≃*o β) :=
+  ⟨OrderMonoidIsoClass.toOrderMonoidIso⟩
+
 end Monoid
 
 section MonoidWithZero
@@ -129,9 +192,8 @@ the `MonoidWithZero` structure.
 `OrderMonoidWithZeroHom` is also used for group homomorphisms.
 
 When possible, instead of parametrizing results over `(f : α →+ β)`,
-you should parametrize over `(F : Type*) [OrderMonoidWithZeroHomClass F α β] (f : F)`.
-
-When you extend this structure, make sure to extend `OrderMonoidWithZeroHomClass`. -/
+you should parameterize over
+`(F : Type*) [FunLike F M N] [MonoidWithZeroHomClass F M N] [OrderHomClass F M N] (f : F)`. -/
 structure OrderMonoidWithZeroHom (α β : Type*) [Preorder α] [Preorder β] [MulZeroOneClass α]
   [MulZeroOneClass β] extends α →*₀ β where
   /-- An `OrderMonoidWithZeroHom` is a monotone function. -/
@@ -446,6 +508,172 @@ end OrderedCommGroup
 
 end OrderMonoidHom
 
+namespace OrderMonoidIso
+
+section Preorder
+
+variable [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] [MulOneClass α] [MulOneClass β]
+  [MulOneClass γ] [MulOneClass δ] {f g : α ≃*o β}
+
+@[to_additive]
+instance : EquivLike (α ≃*o β) α β where
+  coe f := f.toFun
+  inv f := f.invFun
+  left_inv f := f.left_inv
+  right_inv f := f.right_inv
+  coe_injective' f g h₁ h₂ := by
+    obtain ⟨⟨⟨_, _⟩⟩, _⟩ := f
+    obtain ⟨⟨⟨_, _⟩⟩, _⟩ := g
+    congr
+
+@[to_additive]
+instance : OrderIsoClass (α ≃*o β) α β where
+  map_le_map_iff f := f.map_le_map_iff'
+
+@[to_additive]
+instance : MulEquivClass (α ≃*o β) α β where
+  map_mul f := map_mul f.toMulEquiv
+
+-- Other lemmas should be accessed through the `FunLike` API
+@[to_additive (attr := ext)]
+theorem ext (h : ∀ a, f a = g a) : f = g :=
+  DFunLike.ext f g h
+
+@[to_additive]
+theorem toFun_eq_coe (f : α ≃*o β) : f.toFun = (f : α → β) :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem coe_mk (f : α ≃* β) (h) : (OrderMonoidIso.mk f h : α → β) = f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem mk_coe (f : α ≃*o β) (h) : OrderMonoidIso.mk (f : α ≃* β) h = f := rfl
+
+/-- Reinterpret an ordered monoid isomorphism as an order isomorphism. -/
+@[to_additive "Reinterpret an ordered additive monoid isomomorphism as an order isomomorphism."]
+def toOrderIso (f : α ≃*o β) : α ≃o β :=
+  { f with
+    map_rel_iff' := map_le_map_iff f }
+
+@[to_additive (attr := simp)]
+theorem coe_mulEquiv (f : α ≃*o β) : ((f : α ≃* β) : α → β) = f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem coe_orderIso (f : α ≃*o β) : ((f : α →o β) : α → β) = f :=
+  rfl
+
+@[to_additive]
+theorem toMulEquiv_injective : Injective (toMulEquiv : _ → α ≃* β) := fun f g h =>
+  ext <| by convert DFunLike.ext_iff.1 h using 0
+
+@[to_additive]
+theorem toOrderIso_injective : Injective (toOrderIso : _ → α ≃o β) := fun f g h =>
+  ext <| by convert DFunLike.ext_iff.1 h using 0
+
+variable (α)
+
+/-- The identity map as an ordered monoid isomorphism. -/
+@[to_additive "The identity map as an ordered additive monoid isomorphism."]
+protected def refl : α ≃*o α :=
+  { MulEquiv.refl α with map_le_map_iff' := by simp }
+
+@[to_additive (attr := simp)]
+theorem coe_refl : ⇑(OrderMonoidIso.refl α) = id :=
+  rfl
+
+@[to_additive]
+instance : Inhabited (α ≃*o α) :=
+  ⟨OrderMonoidIso.refl α⟩
+
+variable {α}
+
+/-- Transitivity of multiplication-preserving order isomorphisms -/
+@[to_additive (attr := trans) "Transitivity of addition-preserving order isomorphisms"]
+def trans (f : α ≃*o β) (g : β ≃*o γ) : α ≃*o γ :=
+  { (f : α ≃* β).trans g with map_le_map_iff' := by simp }
+
+@[to_additive (attr := simp)]
+theorem coe_trans (f : α ≃*o β) (g : β ≃*o γ) : (f.trans g : α → γ) = g ∘ f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem trans_apply (f : α ≃*o β) (g : β ≃*o γ) (a : α) : (f.trans g) a = g (f a) :=
+  rfl
+
+@[to_additive]
+theorem coe_trans_mulEquiv (f : α ≃*o β) (g : β ≃*o γ) :
+    (f.trans g : α ≃* γ) = (f : α ≃* β).trans g :=
+  rfl
+
+@[to_additive]
+theorem coe_trans_orderIso (f : α ≃*o β) (g : β ≃*o γ) :
+    (f.trans g : α ≃o γ) = (f : α ≃o β).trans g :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem trans_assoc (f : α ≃*o β) (g : β ≃*o γ) (h : γ ≃*o δ) :
+    (f.trans g).trans h = f.trans (g.trans h) :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem trans_refl (f : α ≃*o β) : f.trans (OrderMonoidIso.refl β) = f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem refl_trans (f : α ≃*o β) : (OrderMonoidIso.refl α).trans f = f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem cancel_right {g₁ g₂ : α ≃*o β} {f : β ≃*o γ} (hf : Function.Injective f) :
+    g₁.trans f = g₂.trans f ↔ g₁ = g₂ :=
+  ⟨fun h => ext fun a => hf <| by rw [← trans_apply, h, trans_apply], by rintro rfl; rfl⟩
+
+@[to_additive (attr := simp)]
+theorem cancel_left {g : α ≃*o β} {f₁ f₂ : β ≃*o γ} (hg : Function.Surjective g) :
+    g.trans f₁ = g.trans f₂ ↔ f₁ = f₂ :=
+  ⟨fun h => ext <| hg.forall.2 <| DFunLike.ext_iff.1 h, fun _ => by congr⟩
+
+@[to_additive (attr := simp)]
+theorem toMulEquiv_eq_coe (f : α ≃*o β) : f.toMulEquiv = f :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem toOrderIso_eq_coe (f : α ≃*o β) : f.toOrderIso = f :=
+  rfl
+
+variable (f)
+
+@[to_additive]
+protected lemma strictMono : StrictMono f :=
+  strictMono_of_le_iff_le fun _ _ ↦ (map_le_map_iff _).symm
+
+@[to_additive]
+protected lemma strictMono_symm : StrictMono f.symm :=
+  strictMono_of_le_iff_le <| fun a b ↦ by
+    rw [← map_le_map_iff f]
+    convert Iff.rfl <;>
+    exact f.toEquiv.apply_symm_apply _
+
+end Preorder
+
+section OrderedCommGroup
+
+variable {hα : OrderedCommGroup α} {hβ : OrderedCommGroup β}
+
+/-- Makes an ordered group isomorphism from a proof that the map preserves multiplication. -/
+@[to_additive
+      "Makes an ordered additive group isomorphism from a proof that the map preserves
+      addition."]
+def mk' (f : α ≃ β) (hf : ∀ {a b}, f a ≤ f b ↔ a ≤ b) (map_mul : ∀ a b : α, f (a * b) = f a * f b) :
+    α ≃*o β :=
+  { MulEquiv.mk' f map_mul with map_le_map_iff' := hf }
+
+end OrderedCommGroup
+
+end OrderMonoidIso
+
 namespace OrderMonoidWithZeroHom
 
 section Preorder
diff --git a/Mathlib/Algebra/Order/Interval/Basic.lean b/Mathlib/Algebra/Order/Interval/Basic.lean
index 005379a7fa950..dcd8a5d261e09 100644
--- a/Mathlib/Algebra/Order/Interval/Basic.lean
+++ b/Mathlib/Algebra/Order/Interval/Basic.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.BigOperators.Group.Finset
-import Mathlib.Data.Set.Pointwise.Basic
 import Mathlib.Order.Interval.Basic
 
 /-!
diff --git a/Mathlib/Algebra/Order/Interval/Set/Instances.lean b/Mathlib/Algebra/Order/Interval/Set/Instances.lean
index ff54fbdebf8a3..4624971afef4b 100644
--- a/Mathlib/Algebra/Order/Interval/Set/Instances.lean
+++ b/Mathlib/Algebra/Order/Interval/Set/Instances.lean
@@ -1,9 +1,10 @@
 /-
 Copyright (c) 2022 Stuart Presnell. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Scott Morrison
+Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Kim Morrison
 -/
-import Mathlib.Algebra.Order.Ring.Basic
+import Mathlib.Algebra.GroupWithZero.InjSurj
+import Mathlib.Algebra.Order.Ring.Defs
 import Mathlib.Algebra.Ring.Regular
 import Mathlib.Order.Interval.Set.Basic
 
@@ -98,10 +99,10 @@ theorem le_one {t : Icc (0 : α) 1} : t ≤ 1 :=
   t.2.2
 
 instance mul : Mul (Icc (0 : α) 1) where
-  mul p q := ⟨p * q, ⟨mul_nonneg p.2.1 q.2.1, mul_le_one p.2.2 q.2.1 q.2.2⟩⟩
+  mul p q := ⟨p * q, ⟨mul_nonneg p.2.1 q.2.1, mul_le_one₀ p.2.2 q.2.1 q.2.2⟩⟩
 
 instance pow : Pow (Icc (0 : α) 1) ℕ where
-  pow p n := ⟨p.1 ^ n, ⟨pow_nonneg p.2.1 n, pow_le_one n p.2.1 p.2.2⟩⟩
+  pow p n := ⟨p.1 ^ n, ⟨pow_nonneg p.2.1 n, pow_le_one₀ p.2.1 p.2.2⟩⟩
 
 @[simp, norm_cast]
 theorem coe_mul (x y : Icc (0 : α) 1) : ↑(x * y) = (x * y : α) :=
@@ -236,10 +237,10 @@ theorem le_one {t : Ioc (0 : α) 1} : t ≤ 1 :=
   t.2.2
 
 instance mul : Mul (Ioc (0 : α) 1) where
-  mul p q := ⟨p.1 * q.1, ⟨mul_pos p.2.1 q.2.1, mul_le_one p.2.2 (le_of_lt q.2.1) q.2.2⟩⟩
+  mul p q := ⟨p.1 * q.1, ⟨mul_pos p.2.1 q.2.1, mul_le_one₀ p.2.2 (le_of_lt q.2.1) q.2.2⟩⟩
 
 instance pow : Pow (Ioc (0 : α) 1) ℕ where
-  pow p n := ⟨p.1 ^ n, ⟨pow_pos p.2.1 n, pow_le_one n (le_of_lt p.2.1) p.2.2⟩⟩
+  pow p n := ⟨p.1 ^ n, ⟨pow_pos p.2.1 n, pow_le_one₀ (le_of_lt p.2.1) p.2.2⟩⟩
 
 @[simp, norm_cast]
 theorem coe_mul (x y : Ioc (0 : α) 1) : ↑(x * y) = (x * y : α) :=
diff --git a/Mathlib/Algebra/Order/Kleene.lean b/Mathlib/Algebra/Order/Kleene.lean
index 4a57e54379465..064a1499bf6bf 100644
--- a/Mathlib/Algebra/Order/Kleene.lean
+++ b/Mathlib/Algebra/Order/Kleene.lean
@@ -4,11 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Siddhartha Prasad, Yaël Dillies
 -/
 import Mathlib.Algebra.Order.Monoid.Canonical.Defs
-import Mathlib.Algebra.Ring.Pi
 import Mathlib.Algebra.Ring.InjSurj
-import Mathlib.Tactic.Monotonicity.Attr
+import Mathlib.Algebra.Ring.Pi
 import Mathlib.Algebra.Ring.Prod
-import Mathlib.Algebra.Order.Monoid.Canonical.Defs
+import Mathlib.Tactic.Monotonicity.Attr
 
 /-!
 # Kleene Algebras
diff --git a/Mathlib/Algebra/Order/Module/Defs.lean b/Mathlib/Algebra/Order/Module/Defs.lean
index 2d1fc4bb7948c..8323669b29434 100644
--- a/Mathlib/Algebra/Order/Module/Defs.lean
+++ b/Mathlib/Algebra/Order/Module/Defs.lean
@@ -6,6 +6,7 @@ Authors: Yaël Dillies
 import Mathlib.Algebra.Module.Defs
 import Mathlib.Algebra.Order.Field.Defs
 import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
+import Mathlib.Tactic.GCongr
 import Mathlib.Tactic.Positivity.Core
 
 /-!
@@ -769,7 +770,7 @@ instance instPosSMulReflectLE [PosSMulReflectLE α β] : PosSMulReflectLE α β
 end Left
 
 section Right
-variable [Preorder α] [Ring α] [OrderedAddCommGroup β] [Module α β]
+variable [Preorder α] [Monoid α] [OrderedAddCommGroup β] [DistribMulAction α β]
 
 instance instSMulPosMono [SMulPosMono α β] : SMulPosMono α βᵒᵈ where
   elim _b hb a₁ a₂ ha := by
@@ -794,6 +795,45 @@ instance instSMulPosReflectLE [SMulPosReflectLE α β] : SMulPosReflectLE α β
 end Right
 end OrderDual
 
+section OrderedAddCommMonoid
+variable [StrictOrderedSemiring α] [ExistsAddOfLE α] [OrderedCancelAddCommMonoid β]
+  [Module α β]
+
+section PosSMulMono
+variable [PosSMulMono α β] {a₁ a₂ : α} {b₁ b₂ : β}
+
+/-- Binary **rearrangement inequality**. -/
+lemma smul_add_smul_le_smul_add_smul (ha : a₁ ≤ a₂) (hb : b₁ ≤ b₂) :
+    a₁ • b₂ + a₂ • b₁ ≤ a₁ • b₁ + a₂ • b₂ := by
+  obtain ⟨a, ha₀, rfl⟩ := exists_nonneg_add_of_le ha
+  rw [add_smul, add_smul, add_left_comm]
+  gcongr
+
+/-- Binary **rearrangement inequality**. -/
+lemma smul_add_smul_le_smul_add_smul' (ha : a₂ ≤ a₁) (hb : b₂ ≤ b₁) :
+    a₁ • b₂ + a₂ • b₁ ≤ a₁ • b₁ + a₂ • b₂ := by
+  simp_rw [add_comm (a₁ • _)]; exact smul_add_smul_le_smul_add_smul ha hb
+
+end PosSMulMono
+
+section PosSMulStrictMono
+variable [PosSMulStrictMono α β] {a₁ a₂ : α} {b₁ b₂ : β}
+
+/-- Binary strict **rearrangement inequality**. -/
+lemma smul_add_smul_lt_smul_add_smul (ha : a₁ < a₂) (hb : b₁ < b₂) :
+    a₁ • b₂ + a₂ • b₁ < a₁ • b₁ + a₂ • b₂ := by
+  obtain ⟨a, ha₀, rfl⟩ := lt_iff_exists_pos_add.1 ha
+  rw [add_smul, add_smul, add_left_comm]
+  gcongr
+
+/-- Binary strict **rearrangement inequality**. -/
+lemma smul_add_smul_lt_smul_add_smul' (ha : a₂ < a₁) (hb : b₂ < b₁) :
+    a₁ • b₂ + a₂ • b₁ < a₁ • b₁ + a₂ • b₂ := by
+  simp_rw [add_comm (a₁ • _)]; exact smul_add_smul_lt_smul_add_smul ha hb
+
+end PosSMulStrictMono
+end OrderedAddCommMonoid
+
 section OrderedRing
 variable [OrderedRing α]
 
@@ -864,39 +904,6 @@ lemma smul_neg_iff_of_neg_left (ha : a < 0) : a • b < 0 ↔ 0 < b := by
   simpa only [smul_zero] using smul_lt_smul_iff_of_neg_left ha (b₂ := (0 : β))
 
 end PosSMulStrictMono
-
-/-- Binary **rearrangement inequality**. -/
-lemma smul_add_smul_le_smul_add_smul [PosSMulMono α β]
-    {b₁ b₂ : α} {a d : β} (hab : b₁ ≤ b₂) (hcd : a ≤ d) : b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d := by
-  obtain ⟨b₂, rfl⟩ := exists_add_of_le hab
-  obtain ⟨d, rfl⟩ := exists_add_of_le hcd
-  rw [smul_add, add_right_comm, smul_add, ← add_assoc, add_smul _ _ d]
-  rw [le_add_iff_nonneg_right] at hab hcd
-  exact add_le_add_left (le_add_of_nonneg_right <| smul_nonneg hab hcd) _
-
-/-- Binary **rearrangement inequality**. -/
-lemma smul_add_smul_le_smul_add_smul' [PosSMulMono α β]
-    {b₁ b₂ : α} {a d : β} (hba : b₂ ≤ b₁) (hdc : d ≤ a) : b₁ • d + b₂ • a ≤ b₁ • a + b₂ • d := by
-  rw [add_comm (b₁ • d), add_comm (b₁ • a)]
-  exact smul_add_smul_le_smul_add_smul hba hdc
-
-/-- Binary strict **rearrangement inequality**. -/
-lemma smul_add_smul_lt_smul_add_smul [PosSMulStrictMono α β]
-    {b₁ b₂ : α} {a d : β} (hab : b₁ < b₂) (hcd : a < d) :
-    b₁ • d + b₂ • a < b₁ • a + b₂ • d := by
-  obtain ⟨b₂, rfl⟩ := exists_add_of_le hab.le
-  obtain ⟨d, rfl⟩ := exists_add_of_le hcd.le
-  rw [smul_add, add_right_comm, smul_add, ← add_assoc, add_smul _ _ d]
-  rw [lt_add_iff_pos_right] at hab hcd
-  exact add_lt_add_left (lt_add_of_pos_right _ <| smul_pos hab hcd) _
-
-/-- Binary strict **rearrangement inequality**. -/
-lemma smul_add_smul_lt_smul_add_smul' [PosSMulStrictMono α β]
-    {b₁ b₂ : α} {a d : β} (hba : b₂ < b₁) (hdc : d < a) :
-    b₁ • d + b₂ • a < b₁ • a + b₂ • d := by
-  rw [add_comm (b₁ • d), add_comm (b₁ • a)]
-  exact smul_add_smul_lt_smul_add_smul hba hdc
-
 end OrderedAddCommGroup
 
 section LinearOrderedAddCommGroup
@@ -1163,7 +1170,7 @@ end NoZeroSMulDivisors
 
 open Lean.Meta Qq
 
-/-- Positivity extension for HSMul, i.e. (_ • _).  -/
+/-- Positivity extension for HSMul, i.e. (_ • _). -/
 @[positivity HSMul.hSMul _ _]
 def evalHSMul : PositivityExt where eval {_u α} zα pα (e : Q($α)) := do
   let .app (.app (.app (.app (.app (.app
@@ -1191,46 +1198,3 @@ def evalHSMul : PositivityExt where eval {_u α} zα pα (e : Q($α)) := do
   | _, _ => pure .none
 
 end Mathlib.Meta.Positivity
-
-/-!
-### Deprecated lemmas
-
-Those lemmas have been deprecated on 2023-12-23.
--/
-
-@[deprecated (since := "2023-12-23")] alias monotone_smul_left := monotone_smul_left_of_nonneg
-@[deprecated (since := "2023-12-23")] alias strict_mono_smul_left := strictMono_smul_left_of_pos
-@[deprecated (since := "2023-12-23")] alias smul_le_smul_of_nonneg := smul_le_smul_of_nonneg_left
-@[deprecated (since := "2023-12-23")] alias smul_lt_smul_of_pos := smul_lt_smul_of_pos_left
-
-@[deprecated (since := "2023-12-23")]
-alias lt_of_smul_lt_smul_of_nonneg := lt_of_smul_lt_smul_of_nonneg_left
-
-@[deprecated (since := "2023-12-23")] alias smul_le_smul_iff_of_pos := smul_le_smul_iff_of_pos_left
-@[deprecated (since := "2023-12-23")] alias smul_lt_smul_iff_of_pos := smul_lt_smul_iff_of_pos_left
-@[deprecated (since := "2023-12-23")] alias smul_max := smul_max_of_nonneg
-@[deprecated (since := "2023-12-23")] alias smul_min := smul_min_of_nonneg
-@[deprecated (since := "2023-12-23")] alias smul_pos_iff_of_pos := smul_pos_iff_of_pos_left
-@[deprecated (since := "2023-12-23")] alias inv_smul_le_iff := inv_smul_le_iff_of_pos
-@[deprecated (since := "2023-12-23")] alias le_inv_smul_iff := le_inv_smul_iff_of_pos
-@[deprecated (since := "2023-12-23")] alias inv_smul_lt_iff := inv_smul_lt_iff_of_pos
-@[deprecated (since := "2023-12-23")] alias lt_inv_smul_iff := lt_inv_smul_iff_of_pos
-@[deprecated (since := "2023-12-23")] alias OrderIso.smulLeft := OrderIso.smulRight
-
-@[deprecated (since := "2023-12-23")]
-alias OrderIso.smulLeft_symm_apply := OrderIso.smulRight_symm_apply
-
-@[deprecated (since := "2023-12-23")] alias OrderIso.smulLeft_apply := OrderIso.smulRight_apply
-@[deprecated (since := "2023-12-23")] alias smul_neg_iff_of_pos := smul_neg_iff_of_pos_left
-
-/-!
-Those lemmas have been deprecated on 2023-12-27.
--/
-
-@[deprecated (since := "2023-12-27")] alias strict_anti_smul_left := strictAnti_smul_left
-@[deprecated (since := "2023-12-27")] alias smul_le_smul_of_nonpos := smul_le_smul_of_nonpos_left
-@[deprecated (since := "2023-12-27")] alias smul_lt_smul_of_neg := smul_lt_smul_of_neg_left
-@[deprecated (since := "2023-12-27")] alias smul_pos_iff_of_neg := smul_pos_iff_of_neg_left
-@[deprecated (since := "2023-12-27")] alias smul_neg_iff_of_neg := smul_neg_iff_of_neg_left
-@[deprecated (since := "2023-12-27")] alias smul_le_smul_iff_of_neg := smul_le_smul_iff_of_neg_left
-@[deprecated (since := "2023-12-27")] alias smul_lt_smul_iff_of_neg := smul_lt_smul_iff_of_neg_left
diff --git a/Mathlib/Algebra/Order/Module/OrderedSMul.lean b/Mathlib/Algebra/Order/Module/OrderedSMul.lean
index bf1439508f708..ff7892c724901 100644
--- a/Mathlib/Algebra/Order/Module/OrderedSMul.lean
+++ b/Mathlib/Algebra/Order/Module/OrderedSMul.lean
@@ -9,7 +9,7 @@ import Mathlib.Algebra.Module.Prod
 import Mathlib.Algebra.Order.Module.Defs
 import Mathlib.Algebra.Order.Monoid.Prod
 import Mathlib.Algebra.Order.Pi
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 
 /-!
 # Ordered scalar product
diff --git a/Mathlib/Algebra/Order/Module/Pointwise.lean b/Mathlib/Algebra/Order/Module/Pointwise.lean
index 1214559716ed8..147291c9cdcef 100644
--- a/Mathlib/Algebra/Order/Module/Pointwise.lean
+++ b/Mathlib/Algebra/Order/Module/Pointwise.lean
@@ -3,8 +3,8 @@ Copyright (c) 2023 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.Module.Defs
-import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.Order.Bounds.OrderIso
 
 /-!
diff --git a/Mathlib/Algebra/Order/Monoid/Canonical/Basic.lean b/Mathlib/Algebra/Order/Monoid/Canonical/Basic.lean
new file mode 100644
index 0000000000000..6005ed71ee499
--- /dev/null
+++ b/Mathlib/Algebra/Order/Monoid/Canonical/Basic.lean
@@ -0,0 +1,19 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.Order.Monoid.Canonical.Defs
+import Mathlib.Data.Finset.Lattice
+
+/-!
+# Extra lemmas about canonically ordered monoids
+-/
+
+namespace Finset
+variable {ι α : Type*} [CanonicallyLinearOrderedAddCommMonoid α] {s : Finset ι} {f : ι → α}
+
+@[simp] lemma sup_eq_zero : s.sup f = 0 ↔ ∀ i ∈ s, f i = 0 := by simp [← bot_eq_zero']
+@[simp] lemma sup'_eq_zero (hs) : s.sup' hs f = 0 ↔ ∀ i ∈ s, f i = 0 := by simp [sup'_eq_sup]
+
+end Finset
diff --git a/Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean b/Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
index 87f6d53965914..7cf5e1c4143ca 100644
--- a/Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
+++ b/Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
@@ -58,7 +58,7 @@ instance (priority := 100) CanonicallyOrderedCommMonoid.existsMulOfLE (α : Type
 
 section CanonicallyOrderedCommMonoid
 
-variable [CanonicallyOrderedCommMonoid α] {a b c d : α}
+variable [CanonicallyOrderedCommMonoid α] {a b c : α}
 
 @[to_additive]
 theorem le_self_mul : a ≤ a * c :=
@@ -128,6 +128,10 @@ theorem one_lt_iff_ne_one : 1 < a ↔ a ≠ 1 :=
 @[to_additive]
 theorem eq_one_or_one_lt (a : α) : a = 1 ∨ 1 < a := (one_le a).eq_or_lt.imp_left Eq.symm
 
+@[to_additive]
+lemma one_not_mem_iff {s : Set α} : 1 ∉ s ↔ ∀ x ∈ s, 1 < x :=
+  bot_eq_one (α := α) ▸ bot_not_mem_iff
+
 @[to_additive (attr := simp) add_pos_iff]
 theorem one_lt_mul_iff : 1 < a * b ↔ 1 < a ∨ 1 < b := by
   simp only [one_lt_iff_ne_one, Ne, mul_eq_one, not_and_or]
diff --git a/Mathlib/Algebra/Order/Monoid/Defs.lean b/Mathlib/Algebra/Order/Monoid/Defs.lean
index 3e9fcaaa3a139..61e8c9e8a30d9 100644
--- a/Mathlib/Algebra/Order/Monoid/Defs.lean
+++ b/Mathlib/Algebra/Order/Monoid/Defs.lean
@@ -15,7 +15,7 @@ This file provides the definitions of ordered monoids.
 
 open Function
 
-variable {α β : Type*}
+variable {α : Type*}
 
 /-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
 addition is monotone. -/
diff --git a/Mathlib/Algebra/Order/Monoid/Prod.lean b/Mathlib/Algebra/Order/Monoid/Prod.lean
index c6998f304353c..e3d9ab5369849 100644
--- a/Mathlib/Algebra/Order/Monoid/Prod.lean
+++ b/Mathlib/Algebra/Order/Monoid/Prod.lean
@@ -40,7 +40,7 @@ instance [CanonicallyOrderedCommMonoid α] [CanonicallyOrderedCommMonoid β] :
     CanonicallyOrderedCommMonoid (α × β) :=
   { (inferInstance : OrderedCommMonoid _), (inferInstance : OrderBot _),
     (inferInstance : ExistsMulOfLE _) with
-      le_self_mul := fun _ _ ↦ ⟨le_self_mul, le_self_mul⟩ }
+      le_self_mul := fun _ _ ↦ le_def.mpr ⟨le_self_mul, le_self_mul⟩ }
 
 namespace Lex
 
diff --git a/Mathlib/Algebra/Order/Monoid/Submonoid.lean b/Mathlib/Algebra/Order/Monoid/Submonoid.lean
index 6aa9c7927f6b2..d88dd033520c1 100644
--- a/Mathlib/Algebra/Order/Monoid/Submonoid.lean
+++ b/Mathlib/Algebra/Order/Monoid/Submonoid.lean
@@ -3,16 +3,15 @@ Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-
 import Mathlib.Algebra.Group.Submonoid.Operations
-import Mathlib.Algebra.Order.GroupWithZero.Unbundled
 import Mathlib.Algebra.Order.Monoid.Basic
-import Mathlib.Algebra.Order.ZeroLEOne
 
 /-!
 # Ordered instances on submonoids
 -/
 
+assert_not_exists MonoidWithZero
+
 namespace SubmonoidClass
 variable {M S : Type*} [SetLike S M]
 
@@ -103,21 +102,4 @@ variable {M}
 @[to_additive (attr := simp) mem_nonneg] lemma mem_oneLE : a ∈ oneLE M ↔ 1 ≤ a := Iff.rfl
 
 end Preorder
-
-section MulZeroClass
-variable (α) [MulZeroOneClass α] [PartialOrder α] [PosMulStrictMono α] [ZeroLEOneClass α]
-  [NeZero (1 : α)] {a : α}
-
-/-- The submonoid of positive elements. -/
-@[simps] def pos : Submonoid α where
-  carrier := Set.Ioi 0
-  one_mem' := zero_lt_one
-  mul_mem' := mul_pos
-
-variable {α}
-
-@[simp] lemma mem_pos : a ∈ pos α ↔ 0 < a := Iff.rfl
-
-end MulZeroClass
-
 end Submonoid
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
index ec0256bb10e09..f33502550e3da 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean
@@ -1289,7 +1289,7 @@ theorem Contravariant.MulLECancellable [Mul α] [LE α] [ContravariantClass α 
     MulLECancellable a :=
   fun _ _ => le_of_mul_le_mul_left'
 
-@[to_additive]
+@[to_additive (attr := simp)]
 theorem mulLECancellable_one [Monoid α] [LE α] : MulLECancellable (1 : α) := fun a b => by
   simpa only [one_mul] using id
 
@@ -1306,12 +1306,12 @@ protected theorem inj [Mul α] [PartialOrder α] {a b c : α} (ha : MulLECancell
   ha.Injective.eq_iff
 
 @[to_additive]
-protected theorem injective_left [Mul α] [i : IsSymmOp α α (· * ·)] [PartialOrder α] {a : α}
+protected theorem injective_left [Mul α] [i : @Std.Commutative α (· * ·)] [PartialOrder α] {a : α}
     (ha : MulLECancellable a) :
-    Injective (· * a) := fun b c h => ha.Injective <| by dsimp; rwa [i.symm_op a, i.symm_op a]
+    Injective (· * a) := fun b c h => ha.Injective <| by dsimp; rwa [i.comm a, i.comm a]
 
 @[to_additive]
-protected theorem inj_left [Mul α] [IsSymmOp α α (· * ·)] [PartialOrder α] {a b c : α}
+protected theorem inj_left [Mul α] [@Std.Commutative α (· * ·)] [PartialOrder α] {a b c : α}
     (hc : MulLECancellable c) :
     a * c = b * c ↔ a = b :=
   hc.injective_left.eq_iff
@@ -1324,9 +1324,9 @@ protected theorem mul_le_mul_iff_left [Mul α] [CovariantClass α α (· * ·) (
   ⟨fun h => ha h, fun h => mul_le_mul_left' h a⟩
 
 @[to_additive]
-protected theorem mul_le_mul_iff_right [Mul α] [i : IsSymmOp α α (· * ·)]
+protected theorem mul_le_mul_iff_right [Mul α] [i : @Std.Commutative α (· * ·)]
     [CovariantClass α α (· * ·) (· ≤ ·)] {a b c : α} (ha : MulLECancellable a) :
-    b * a ≤ c * a ↔ b ≤ c := by rw [i.symm_op b, i.symm_op c, ha.mul_le_mul_iff_left]
+    b * a ≤ c * a ↔ b ≤ c := by rw [i.comm b, i.comm c, ha.mul_le_mul_iff_left]
 
 @[to_additive]
 protected theorem le_mul_iff_one_le_right [MulOneClass α] [CovariantClass α α (· * ·) (· ≤ ·)]
@@ -1341,13 +1341,29 @@ protected theorem mul_le_iff_le_one_right [MulOneClass α] [CovariantClass α α
   Iff.trans (by rw [mul_one]) ha.mul_le_mul_iff_left
 
 @[to_additive]
-protected theorem le_mul_iff_one_le_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
+protected theorem le_mul_iff_one_le_left [MulOneClass α] [i : @Std.Commutative α (· * ·)]
     [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
-    a ≤ b * a ↔ 1 ≤ b := by rw [i.symm_op, ha.le_mul_iff_one_le_right]
+    a ≤ b * a ↔ 1 ≤ b := by rw [i.comm, ha.le_mul_iff_one_le_right]
 
 @[to_additive]
-protected theorem mul_le_iff_le_one_left [MulOneClass α] [i : IsSymmOp α α (· * ·)]
+protected theorem mul_le_iff_le_one_left [MulOneClass α] [i : @Std.Commutative α (· * ·)]
     [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} (ha : MulLECancellable a) :
-    b * a ≤ a ↔ b ≤ 1 := by rw [i.symm_op, ha.mul_le_iff_le_one_right]
+    b * a ≤ a ↔ b ≤ 1 := by rw [i.comm, ha.mul_le_iff_le_one_right]
+
+@[to_additive] lemma mul [Semigroup α] {a b : α} (ha : MulLECancellable a)
+    (hb : MulLECancellable b) : MulLECancellable (a * b) :=
+  fun c d hcd ↦ hb <| ha <| by rwa [← mul_assoc, ← mul_assoc]
+
+@[to_additive] lemma of_mul_right [Semigroup α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α}
+    (h : MulLECancellable (a * b)) : MulLECancellable b :=
+  fun c d hcd ↦ h <| by rw [mul_assoc, mul_assoc]; exact mul_le_mul_left' hcd _
+
+@[to_additive] lemma of_mul_left [CommSemigroup α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α}
+    (h : MulLECancellable (a * b)) : MulLECancellable a := (mul_comm a b ▸ h).of_mul_right
 
 end MulLECancellable
+
+@[to_additive (attr := simp)]
+lemma mulLECancellable_mul [LE α] [CommSemigroup α] [CovariantClass α α (· * ·) (· ≤ ·)] {a b : α} :
+    MulLECancellable (a * b) ↔ MulLECancellable a ∧ MulLECancellable b :=
+  ⟨fun h ↦ ⟨h.of_mul_left, h.of_mul_right⟩, fun h ↦ h.1.mul h.2⟩
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean
index a86ce7532c9ee..6418be782db5c 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean
@@ -328,7 +328,7 @@ theorem Group.mulRightReflectLT_of_mulRightStrictMono [Group N] [LT N]
 
 section Trans
 
-variable [IsTrans N r] (m n : M) {a b c d : N}
+variable [IsTrans N r] (m : M) {a b c : N}
 
 --  Lemmas with 3 elements.
 theorem act_rel_of_rel_of_act_rel (ab : r a b) (rl : r (μ m b) c) : r (μ m a) c :=
@@ -361,7 +361,7 @@ theorem rel_of_act_rel_act (m : M) {a b : N} (ab : r (μ m a) (μ m b)) : r a b
 
 section Trans
 
-variable [IsTrans N r] (m n : M) {a b c d : N}
+variable [IsTrans N r] (m : M) {a b c : N}
 
 --  Lemmas with 3 elements.
 theorem act_rel_of_act_rel_of_rel_act_rel (ab : r (μ m a) b) (rl : r (μ m b) (μ m c)) :
@@ -450,11 +450,11 @@ theorem covariant_lt_iff_contravariant_le [LinearOrder N] :
 
 variable (mu : N → N → N)
 
-theorem covariant_flip_iff [IsSymmOp N N mu] :
-    Covariant N N (flip mu) r ↔ Covariant N N mu r := by rw [IsSymmOp.flip_eq]
+theorem covariant_flip_iff [h : Std.Commutative mu] :
+    Covariant N N (flip mu) r ↔ Covariant N N mu r := by unfold flip; simp_rw [h.comm]
 
-theorem contravariant_flip_iff [IsSymmOp N N mu] :
-    Contravariant N N (flip mu) r ↔ Contravariant N N mu r := by rw [IsSymmOp.flip_eq]
+theorem contravariant_flip_iff [h : Std.Commutative mu] :
+    Contravariant N N (flip mu) r ↔ Contravariant N N mu r := by unfold flip; simp_rw [h.comm]
 
 instance contravariant_lt_of_covariant_le [LinearOrder N]
     [CovariantClass N N mu (· ≤ ·)] : ContravariantClass N N mu (· < ·) where
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean
index d15dacf956b13..b509125489cc7 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean
@@ -21,7 +21,7 @@ section CommSemigroup
 variable [LinearOrder α] [CommSemigroup β]
 
 @[to_additive]
-lemma fn_min_mul_fn_max  (f : α → β) (a b : α) : f (min a b) * f (max a b) = f a * f b := by
+lemma fn_min_mul_fn_max (f : α → β) (a b : α) : f (min a b) * f (max a b) = f a * f b := by
   obtain h | h := le_total a b <;> simp [h, mul_comm]
 
 @[to_additive]
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean
index 877d5f5f47c07..eb892cfeda6b8 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean
@@ -24,96 +24,100 @@ section Preorder
 
 variable [Preorder M]
 
-section Left
-
-variable [CovariantClass M M (· * ·) (· ≤ ·)] {x : M}
+namespace Left
 
-@[to_additive (attr := mono, gcongr) nsmul_le_nsmul_right]
-theorem pow_le_pow_left' [CovariantClass M M (swap (· * ·)) (· ≤ ·)] {a b : M} (hab : a ≤ b) :
-    ∀ i : ℕ, a ^ i ≤ b ^ i
-  | 0 => by simp
-  | k + 1 => by
-    rw [pow_succ, pow_succ]
-    exact mul_le_mul' (pow_le_pow_left' hab k) hab
+variable [CovariantClass M M (· * ·) (· ≤ ·)] {a : M}
 
-@[to_additive nsmul_nonneg]
-theorem one_le_pow_of_one_le' {a : M} (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
+@[to_additive Left.nsmul_nonneg]
+theorem one_le_pow_of_le (ha : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
   | 0 => by simp
   | k + 1 => by
     rw [pow_succ]
-    exact one_le_mul (one_le_pow_of_one_le' H k) H
+    exact one_le_mul (one_le_pow_of_le ha k) ha
+
+@[deprecated (since := "2024-09-21")] alias pow_nonneg := nsmul_nonneg
 
 @[to_additive nsmul_nonpos]
-theorem pow_le_one' {a : M} (H : a ≤ 1) (n : ℕ) : a ^ n ≤ 1 :=
-  one_le_pow_of_one_le' (M := Mᵒᵈ) H n
+theorem pow_le_one_of_le (ha : a ≤ 1) (n : ℕ) : a ^ n ≤ 1 := one_le_pow_of_le (M := Mᵒᵈ) ha n
+
+@[deprecated (since := "2024-09-21")] alias pow_nonpos := nsmul_nonpos
+
+@[to_additive nsmul_neg]
+theorem pow_lt_one_of_lt {a : M} {n : ℕ} (h : a < 1) (hn : n ≠ 0) : a ^ n < 1 := by
+  rcases Nat.exists_eq_succ_of_ne_zero hn with ⟨k, rfl⟩
+  rw [pow_succ']
+  exact mul_lt_one_of_lt_of_le h (pow_le_one_of_le h.le _)
+
+@[deprecated (since := "2024-09-21")] alias pow_neg := nsmul_neg
+
+end Left
+
+@[to_additive nsmul_nonneg] alias one_le_pow_of_one_le' := Left.one_le_pow_of_le
+@[to_additive nsmul_nonpos] alias pow_le_one' := Left.pow_le_one_of_le
+@[to_additive nsmul_neg] alias pow_lt_one' := Left.pow_lt_one_of_lt
+
+section Left
+
+variable [CovariantClass M M (· * ·) (· ≤ ·)]
+
+@[to_additive nsmul_left_monotone]
+theorem pow_right_monotone {a : M} (ha : 1 ≤ a) : Monotone fun n : ℕ ↦ a ^ n :=
+  monotone_nat_of_le_succ fun n ↦ by rw [pow_succ]; exact le_mul_of_one_le_right' ha
 
 @[to_additive (attr := gcongr) nsmul_le_nsmul_left]
 theorem pow_le_pow_right' {a : M} {n m : ℕ} (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m :=
-  let ⟨k, hk⟩ := Nat.le.dest h
-  calc
-    a ^ n ≤ a ^ n * a ^ k := le_mul_of_one_le_right' (one_le_pow_of_one_le' ha _)
-    _ = a ^ m := by rw [← hk, pow_add]
+  pow_right_monotone ha h
 
 @[to_additive nsmul_le_nsmul_left_of_nonpos]
 theorem pow_le_pow_right_of_le_one' {a : M} {n m : ℕ} (ha : a ≤ 1) (h : n ≤ m) : a ^ m ≤ a ^ n :=
   pow_le_pow_right' (M := Mᵒᵈ) ha h
 
 @[to_additive nsmul_pos]
-theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k := by
-  rcases Nat.exists_eq_succ_of_ne_zero hk with ⟨l, rfl⟩
-  clear hk
-  induction l with
-  | zero => rw [pow_succ]; simpa using ha
-  | succ l IH => rw [pow_succ]; exact one_lt_mul'' IH ha
+theorem one_lt_pow' {a : M} (ha : 1 < a) {k : ℕ} (hk : k ≠ 0) : 1 < a ^ k :=
+  pow_lt_one' (M := Mᵒᵈ) ha hk
 
-@[to_additive nsmul_neg]
-theorem pow_lt_one' {a : M} (ha : a < 1) {k : ℕ} (hk : k ≠ 0) : a ^ k < 1 :=
-  one_lt_pow' (M := Mᵒᵈ) ha hk
+end Left
 
-@[to_additive (attr := gcongr) nsmul_lt_nsmul_left]
-theorem pow_lt_pow_right' [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ} (ha : 1 < a)
-    (h : n < m) : a ^ n < a ^ m := by
-  rcases Nat.le.dest h with ⟨k, rfl⟩; clear h
-  rw [pow_add, pow_succ, mul_assoc, ← pow_succ']
-  exact lt_mul_of_one_lt_right' _ (one_lt_pow' ha k.succ_ne_zero)
+section LeftLt
 
-@[to_additive nsmul_left_strictMono]
-theorem pow_right_strictMono' [CovariantClass M M (· * ·) (· < ·)] {a : M} (ha : 1 < a) :
-    StrictMono ((a ^ ·) : ℕ → M) := fun _ _ => pow_lt_pow_right' ha
+variable [CovariantClass M M (· * ·) (· < ·)] {a : M} {n m : ℕ}
 
-@[to_additive Left.pow_nonneg]
-theorem Left.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
-  | 0 => (pow_zero x).ge
-  | n + 1 => by
-    rw [pow_succ]
-    exact Left.one_le_mul (Left.one_le_pow_of_le hx) hx
+@[to_additive nsmul_left_strictMono]
+theorem pow_right_strictMono' (ha : 1 < a) : StrictMono ((a ^ ·) : ℕ → M) :=
+  strictMono_nat_of_lt_succ fun n ↦ by rw [pow_succ]; exact lt_mul_of_one_lt_right' (a ^ n) ha
 
-@[to_additive Left.pow_nonpos]
-theorem Left.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
-  | 0 => (pow_zero _).le
-  | n + 1 => by
-    rw [pow_succ]
-    exact Left.mul_le_one (Left.pow_le_one_of_le hx) hx
+@[to_additive (attr := gcongr) nsmul_lt_nsmul_left]
+theorem pow_lt_pow_right' (ha : 1 < a) (h : n < m) : a ^ n < a ^ m :=
+  pow_right_strictMono' ha h
 
-end Left
+end LeftLt
 
 section Right
 
 variable [CovariantClass M M (swap (· * ·)) (· ≤ ·)] {x : M}
 
-@[to_additive Right.pow_nonneg]
+@[to_additive Right.nsmul_nonneg]
 theorem Right.one_le_pow_of_le (hx : 1 ≤ x) : ∀ {n : ℕ}, 1 ≤ x ^ n
   | 0 => (pow_zero _).ge
   | n + 1 => by
     rw [pow_succ]
     exact Right.one_le_mul (Right.one_le_pow_of_le hx) hx
 
-@[to_additive Right.pow_nonpos]
-theorem Right.pow_le_one_of_le (hx : x ≤ 1) : ∀ {n : ℕ}, x ^ n ≤ 1
-  | 0 => (pow_zero _).le
-  | n + 1 => by
-    rw [pow_succ]
-    exact Right.mul_le_one (Right.pow_le_one_of_le hx) hx
+@[deprecated (since := "2024-09-21")] alias Right.pow_nonneg := Right.nsmul_nonneg
+
+@[to_additive Right.nsmul_nonpos]
+theorem Right.pow_le_one_of_le (hx : x ≤ 1) {n : ℕ} : x ^ n ≤ 1 :=
+  Right.one_le_pow_of_le (M := Mᵒᵈ) hx
+
+@[deprecated (since := "2024-09-21")] alias Right.pow_nonpos := Right.nsmul_nonpos
+
+@[to_additive Right.nsmul_neg]
+theorem Right.pow_lt_one_of_lt {n : ℕ} {x : M} (hn : 0 < n) (h : x < 1) : x ^ n < 1 := by
+  rcases Nat.exists_eq_succ_of_ne_zero hn.ne' with ⟨k, rfl⟩
+  rw [pow_succ]
+  exact mul_lt_one_of_le_of_lt (pow_le_one_of_le h.le) h
+
+@[deprecated (since := "2024-09-21")] alias Right.pow_neg := Right.nsmul_neg
 
 end Right
 
@@ -144,6 +148,13 @@ section CovariantLESwap
 variable [Preorder β] [CovariantClass M M (· * ·) (· ≤ ·)]
   [CovariantClass M M (swap (· * ·)) (· ≤ ·)]
 
+@[to_additive (attr := mono, gcongr) nsmul_le_nsmul_right]
+theorem pow_le_pow_left' {a b : M} (hab : a ≤ b) : ∀ i : ℕ, a ^ i ≤ b ^ i
+  | 0 => by simp
+  | k + 1 => by
+    rw [pow_succ, pow_succ]
+    exact mul_le_mul' (pow_le_pow_left' hab k) hab
+
 @[to_additive Monotone.const_nsmul]
 theorem Monotone.pow_const {f : β → M} (hf : Monotone f) : ∀ n : ℕ, Monotone fun a => f a ^ n
   | 0 => by simpa using monotone_const
@@ -156,24 +167,6 @@ theorem pow_left_mono (n : ℕ) : Monotone fun a : M => a ^ n := monotone_id.pow
 
 end CovariantLESwap
 
-@[to_additive Left.pow_neg]
-theorem Left.pow_lt_one_of_lt [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {x : M} (hn : 0 < n)
-    (h : x < 1) : x ^ n < 1 :=
-  Nat.le_induction ((pow_one _).trans_lt h)
-    (fun n _ ih => by
-      rw [pow_succ]
-      exact mul_lt_one ih h)
-    _ (Nat.succ_le_iff.2 hn)
-
-@[to_additive Right.pow_neg]
-theorem Right.pow_lt_one_of_lt [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
-    (hn : 0 < n) (h : x < 1) : x ^ n < 1 :=
-  Nat.le_induction ((pow_one _).trans_lt h)
-    (fun n _ ih => by
-      rw [pow_succ]
-      exact Right.mul_lt_one ih h)
-    _ (Nat.succ_le_iff.2 hn)
-
 end Preorder
 
 section LinearOrder
@@ -206,6 +199,10 @@ theorem pow_eq_one_iff {x : M} {n : ℕ} (hn : n ≠ 0) : x ^ n = 1 ↔ x = 1 :=
   simp only [le_antisymm_iff]
   rw [pow_le_one_iff hn, one_le_pow_iff hn]
 
+end CovariantLE
+
+section CovariantLT
+
 variable [CovariantClass M M (· * ·) (· < ·)] {a : M} {m n : ℕ}
 
 @[to_additive nsmul_le_nsmul_iff_left]
@@ -216,7 +213,7 @@ theorem pow_le_pow_iff_right' (ha : 1 < a) : a ^ m ≤ a ^ n ↔ m ≤ n :=
 theorem pow_lt_pow_iff_right' (ha : 1 < a) : a ^ m < a ^ n ↔ m < n :=
   (pow_right_strictMono' ha).lt_iff_lt
 
-end CovariantLE
+end CovariantLT
 
 section CovariantLESwap
 
@@ -266,11 +263,8 @@ theorem Left.pow_lt_one_iff [CovariantClass M M (· * ·) (· < ·)] {n : ℕ} {
 @[to_additive]
 theorem Right.pow_lt_one_iff [CovariantClass M M (swap (· * ·)) (· < ·)] {n : ℕ} {x : M}
     (hn : 0 < n) : x ^ n < 1 ↔ x < 1 :=
-  ⟨fun H =>
-    not_le.mp fun k =>
-      haveI := covariantClass_le_of_lt M M (swap (· * ·))
-      H.not_le <| Right.one_le_pow_of_le k,
-    Right.pow_lt_one_of_lt hn⟩
+  haveI := covariantClass_le_of_lt M M (swap (· * ·))
+  ⟨fun H => not_le.mp fun k => H.not_le <| Right.one_le_pow_of_le k, Right.pow_lt_one_of_lt hn⟩
 
 end LinearOrder
 
@@ -287,39 +281,3 @@ theorem one_le_zpow {x : G} (H : 1 ≤ x) {n : ℤ} (hn : 0 ≤ n) : 1 ≤ x ^ n
   apply one_le_pow_of_one_le' H
 
 end DivInvMonoid
-
-/-!
-### Deprecated lemmas
-
-Those lemmas have been deprecated on 2023-12-23.
--/
-
-@[deprecated (since := "2023-12-23")] alias pow_le_pow_of_le_left' := pow_le_pow_left'
-@[deprecated (since := "2023-12-23")] alias nsmul_le_nsmul_of_le_right := nsmul_le_nsmul_right
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow' := pow_lt_pow_right'
-@[deprecated (since := "2023-12-23")] alias nsmul_lt_nsmul := nsmul_lt_nsmul_left
-@[deprecated (since := "2023-12-23")] alias pow_strictMono_left := pow_right_strictMono'
-@[deprecated (since := "2023-12-23")] alias nsmul_strictMono_right := nsmul_left_strictMono
-@[deprecated (since := "2023-12-23")] alias StrictMono.pow_right' := StrictMono.pow_const
-@[deprecated (since := "2023-12-23")] alias StrictMono.nsmul_left := StrictMono.const_nsmul
-@[deprecated (since := "2023-12-23")] alias pow_strictMono_right' := pow_left_strictMono
-@[deprecated (since := "2023-12-23")] alias nsmul_strictMono_left := nsmul_right_strictMono
-@[deprecated (since := "2023-12-23")] alias Monotone.pow_right := Monotone.pow_const
-@[deprecated (since := "2023-12-23")] alias Monotone.nsmul_left := Monotone.const_nsmul
-@[deprecated (since := "2023-12-23")] alias lt_of_pow_lt_pow' := lt_of_pow_lt_pow_left'
-@[deprecated (since := "2023-12-23")] alias lt_of_nsmul_lt_nsmul := lt_of_nsmul_lt_nsmul_right
-@[deprecated (since := "2023-12-23")] alias pow_le_pow' := pow_le_pow_right'
-@[deprecated (since := "2023-12-23")] alias nsmul_le_nsmul := nsmul_le_nsmul_left
-@[deprecated (since := "2023-12-23")] alias pow_le_pow_of_le_one' := pow_le_pow_right_of_le_one'
-
-@[deprecated (since := "2023-12-23")]
-alias nsmul_le_nsmul_of_nonpos := nsmul_le_nsmul_left_of_nonpos
-
-@[deprecated (since := "2023-12-23")] alias le_of_pow_le_pow' := le_of_pow_le_pow_left'
-@[deprecated (since := "2023-12-23")] alias le_of_nsmul_le_nsmul := le_of_nsmul_le_nsmul_right
-@[deprecated (since := "2023-12-23")] alias pow_le_pow_iff' := pow_le_pow_iff_right'
-@[deprecated (since := "2023-12-23")] alias nsmul_le_nsmul_iff := nsmul_le_nsmul_iff_left
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow_iff' := pow_lt_pow_iff_right'
-@[deprecated (since := "2023-12-23")] alias nsmul_lt_nsmul_iff := nsmul_lt_nsmul_iff_left
-@[deprecated (since := "2023-12-23")] alias pow_mono_right := pow_left_mono
-@[deprecated (since := "2023-12-23")] alias nsmul_mono_left := nsmul_right_mono
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
index 7d7bdb4f2ff25..0b75be72f5aaf 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/WithTop.lean
@@ -80,7 +80,7 @@ end One
 
 section Add
 
-variable [Add α] {a b c d : WithTop α} {x y : α}
+variable [Add α] {a b c d : WithTop α} {x : α}
 
 instance add : Add (WithTop α) :=
   ⟨Option.map₂ (· + ·)⟩
diff --git a/Mathlib/Algebra/Order/Monovary.lean b/Mathlib/Algebra/Order/Monovary.lean
index f742a97bcf60e..e085b565b3140 100644
--- a/Mathlib/Algebra/Order/Monovary.lean
+++ b/Mathlib/Algebra/Order/Monovary.lean
@@ -238,19 +238,19 @@ variable [LinearOrderedSemifield α] [LinearOrderedSemifield β] {s : Set ι} {f
 
 @[simp]
 lemma monovaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : MonovaryOn f⁻¹ g s ↔ AntivaryOn f g s :=
-  forall₅_congr fun _i hi _j hj _ ↦ inv_le_inv (hf _ hi) (hf _ hj)
+  forall₅_congr fun _i hi _j hj _ ↦ inv_le_inv₀ (hf _ hi) (hf _ hj)
 
 @[simp]
 lemma antivaryOn_inv_left₀ (hf : ∀ i ∈ s, 0 < f i) : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s :=
-  forall₅_congr fun _i hi _j hj _ ↦ inv_le_inv (hf _ hj) (hf _ hi)
+  forall₅_congr fun _i hi _j hj _ ↦ inv_le_inv₀ (hf _ hj) (hf _ hi)
 
 @[simp]
 lemma monovaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s :=
-  forall₂_swap.trans <| forall₄_congr fun i hi j hj ↦ by erw [inv_lt_inv (hg _ hj) (hg _ hi)]
+  forall₂_swap.trans <| forall₄_congr fun i hi j hj ↦ by erw [inv_lt_inv₀ (hg _ hj) (hg _ hi)]
 
 @[simp]
 lemma antivaryOn_inv_right₀ (hg : ∀ i ∈ s, 0 < g i) : AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s :=
-  forall₂_swap.trans <| forall₄_congr fun i hi j hj ↦ by erw [inv_lt_inv (hg _ hj) (hg _ hi)]
+  forall₂_swap.trans <| forall₄_congr fun i hi j hj ↦ by erw [inv_lt_inv₀ (hg _ hj) (hg _ hi)]
 
 lemma monovaryOn_inv₀ (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i) :
     MonovaryOn f⁻¹ g⁻¹ s ↔ MonovaryOn f g s := by
@@ -260,16 +260,16 @@ lemma antivaryOn_inv₀ (hf : ∀ i ∈ s, 0 < f i) (hg : ∀ i ∈ s, 0 < g i)
   rw [antivaryOn_inv_left₀ hf, monovaryOn_inv_right₀ hg]
 
 @[simp] lemma monovary_inv_left₀ (hf : StrongLT 0 f) : Monovary f⁻¹ g ↔ Antivary f g :=
-  forall₃_congr fun _i _j _ ↦ inv_le_inv (hf _) (hf _)
+  forall₃_congr fun _i _j _ ↦ inv_le_inv₀ (hf _) (hf _)
 
 @[simp] lemma antivary_inv_left₀ (hf : StrongLT 0 f) : Antivary f⁻¹ g ↔ Monovary f g :=
-  forall₃_congr fun _i _j _ ↦ inv_le_inv (hf _) (hf _)
+  forall₃_congr fun _i _j _ ↦ inv_le_inv₀ (hf _) (hf _)
 
 @[simp] lemma monovary_inv_right₀ (hg : StrongLT 0 g) : Monovary f g⁻¹ ↔ Antivary f g :=
-  forall_swap.trans <| forall₂_congr fun i j ↦ by erw [inv_lt_inv (hg _) (hg _)]
+  forall_swap.trans <| forall₂_congr fun i j ↦ by erw [inv_lt_inv₀ (hg _) (hg _)]
 
 @[simp] lemma antivary_inv_right₀ (hg : StrongLT 0 g) : Antivary f g⁻¹ ↔ Monovary f g :=
-  forall_swap.trans <| forall₂_congr fun i j ↦ by erw [inv_lt_inv (hg _) (hg _)]
+  forall_swap.trans <| forall₂_congr fun i j ↦ by erw [inv_lt_inv₀ (hg _) (hg _)]
 
 lemma monovary_inv₀ (hf : StrongLT 0 f) (hg : StrongLT 0 g) : Monovary f⁻¹ g⁻¹ ↔ Monovary f g := by
   rw [monovary_inv_left₀ hf, antivary_inv_right₀ hg]
diff --git a/Mathlib/Algebra/Order/Pi.lean b/Mathlib/Algebra/Order/Pi.lean
index b89f80e5f895b..9710857e2382b 100644
--- a/Mathlib/Algebra/Order/Pi.lean
+++ b/Mathlib/Algebra/Order/Pi.lean
@@ -6,7 +6,6 @@ Authors: Simon Hudon, Patrick Massot
 import Mathlib.Algebra.Order.Monoid.Canonical.Defs
 import Mathlib.Algebra.Order.Ring.Defs
 import Mathlib.Algebra.Ring.Pi
-import Mathlib.Algebra.Order.Monoid.Canonical.Defs
 
 /-!
 # Pi instances for ordered groups and monoids
@@ -129,6 +128,25 @@ variable [One γ] [LE γ] {f : α → β} {g : α → γ} {e : β → γ}
 
 end extend
 end Function
+
+namespace Pi
+variable {ι : Type*} {α : ι → Type*} [DecidableEq ι] [∀ i, One (α i)] [∀ i, Preorder (α i)] {i : ι}
+  {a b : α i}
+
+@[to_additive (attr := simp)]
+lemma mulSingle_le_mulSingle : mulSingle i a ≤ mulSingle i b ↔ a ≤ b := by
+  simp [mulSingle, update_le_update_iff]
+
+@[to_additive (attr := gcongr)] alias ⟨_, GCongr.mulSingle_mono⟩ := mulSingle_le_mulSingle
+
+@[to_additive (attr := simp) single_nonneg]
+lemma one_le_mulSingle : 1 ≤ mulSingle i a ↔ 1 ≤ a := by simp [mulSingle]
+
+@[to_additive (attr := simp)]
+lemma mulSingle_le_one : mulSingle i a ≤ 1 ↔ a ≤ 1 := by simp [mulSingle]
+
+end Pi
+
 -- Porting note: Tactic code not ported yet
 -- namespace Tactic
 
diff --git a/Mathlib/Algebra/Order/Pointwise.lean b/Mathlib/Algebra/Order/Pointwise.lean
deleted file mode 100644
index 43396523eb326..0000000000000
--- a/Mathlib/Algebra/Order/Pointwise.lean
+++ /dev/null
@@ -1,250 +0,0 @@
-/-
-Copyright (c) 2021 Alex J. Best. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Alex J. Best, Yaël Dillies
--/
-import Mathlib.Algebra.Bounds
-import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
-import Mathlib.Data.Set.Pointwise.SMul
-
-/-!
-# Pointwise operations on ordered algebraic objects
-
-This file contains lemmas about the effect of pointwise operations on sets with an order structure.
-
-## TODO
-
-`sSup (s • t) = sSup s • sSup t` and `sInf (s • t) = sInf s • sInf t` hold as well but
-`CovariantClass` is currently not polymorphic enough to state it.
--/
-
-
-open Function Set
-
-open Pointwise
-
-variable {α : Type*}
-
--- Porting note: Swapped the place of `CompleteLattice` and `ConditionallyCompleteLattice`
--- due to simpNF problem between `sSup_xx` `csSup_xx`.
-
-section CompleteLattice
-
-variable [CompleteLattice α]
-
-section One
-
-variable [One α]
-
-@[to_additive (attr := simp)]
-theorem sSup_one : sSup (1 : Set α) = 1 :=
-  sSup_singleton
-
-@[to_additive (attr := simp)]
-theorem sInf_one : sInf (1 : Set α) = 1 :=
-  sInf_singleton
-
-end One
-
-section Group
-
-variable [Group α] [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
-  (s t : Set α)
-
-@[to_additive]
-theorem sSup_inv (s : Set α) : sSup s⁻¹ = (sInf s)⁻¹ := by
-  rw [← image_inv, sSup_image]
-  exact ((OrderIso.inv α).map_sInf _).symm
-
-@[to_additive]
-theorem sInf_inv (s : Set α) : sInf s⁻¹ = (sSup s)⁻¹ := by
-  rw [← image_inv, sInf_image]
-  exact ((OrderIso.inv α).map_sSup _).symm
-
-@[to_additive]
-theorem sSup_mul : sSup (s * t) = sSup s * sSup t :=
-  (sSup_image2_eq_sSup_sSup fun _ => (OrderIso.mulRight _).to_galoisConnection) fun _ =>
-    (OrderIso.mulLeft _).to_galoisConnection
-
-@[to_additive]
-theorem sInf_mul : sInf (s * t) = sInf s * sInf t :=
-  (sInf_image2_eq_sInf_sInf fun _ => (OrderIso.mulRight _).symm.to_galoisConnection) fun _ =>
-    (OrderIso.mulLeft _).symm.to_galoisConnection
-
-@[to_additive]
-theorem sSup_div : sSup (s / t) = sSup s / sInf t := by simp_rw [div_eq_mul_inv, sSup_mul, sSup_inv]
-
-@[to_additive]
-theorem sInf_div : sInf (s / t) = sInf s / sSup t := by simp_rw [div_eq_mul_inv, sInf_mul, sInf_inv]
-
-end Group
-
-end CompleteLattice
-
-section ConditionallyCompleteLattice
-
-variable [ConditionallyCompleteLattice α]
-
-section One
-
-variable [One α]
-
-@[to_additive (attr := simp)]
-theorem csSup_one : sSup (1 : Set α) = 1 :=
-  csSup_singleton _
-
-@[to_additive (attr := simp)]
-theorem csInf_one : sInf (1 : Set α) = 1 :=
-  csInf_singleton _
-
-end One
-
-section Group
-
-variable [Group α] [CovariantClass α α (· * ·) (· ≤ ·)] [CovariantClass α α (swap (· * ·)) (· ≤ ·)]
-  {s t : Set α}
-
-@[to_additive]
-theorem csSup_inv (hs₀ : s.Nonempty) (hs₁ : BddBelow s) : sSup s⁻¹ = (sInf s)⁻¹ := by
-  rw [← image_inv]
-  exact ((OrderIso.inv α).map_csInf' hs₀ hs₁).symm
-
-@[to_additive]
-theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by
-  rw [← image_inv]
-  exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm
-
-@[to_additive]
-theorem csSup_mul (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
-    sSup (s * t) = sSup s * sSup t :=
-  csSup_image2_eq_csSup_csSup (fun _ => (OrderIso.mulRight _).to_galoisConnection)
-    (fun _ => (OrderIso.mulLeft _).to_galoisConnection) hs₀ hs₁ ht₀ ht₁
-
-@[to_additive]
-theorem csInf_mul (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
-    sInf (s * t) = sInf s * sInf t :=
-  csInf_image2_eq_csInf_csInf (fun _ => (OrderIso.mulRight _).symm.to_galoisConnection)
-    (fun _ => (OrderIso.mulLeft _).symm.to_galoisConnection) hs₀ hs₁ ht₀ ht₁
-
-@[to_additive]
-theorem csSup_div (hs₀ : s.Nonempty) (hs₁ : BddAbove s) (ht₀ : t.Nonempty) (ht₁ : BddBelow t) :
-    sSup (s / t) = sSup s / sInf t := by
-  rw [div_eq_mul_inv, csSup_mul hs₀ hs₁ ht₀.inv ht₁.inv, csSup_inv ht₀ ht₁, div_eq_mul_inv]
-
-@[to_additive]
-theorem csInf_div (hs₀ : s.Nonempty) (hs₁ : BddBelow s) (ht₀ : t.Nonempty) (ht₁ : BddAbove t) :
-    sInf (s / t) = sInf s / sSup t := by
-  rw [div_eq_mul_inv, csInf_mul hs₀ hs₁ ht₀.inv ht₁.inv, csInf_inv ht₀ ht₁, div_eq_mul_inv]
-
-end Group
-
-end ConditionallyCompleteLattice
-
-namespace LinearOrderedField
-
-variable {K : Type*} [LinearOrderedField K] {a b r : K} (hr : 0 < r)
-include hr
-
-open Set
-
-theorem smul_Ioo : r • Ioo a b = Ioo (r • a) (r • b) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Ioo]
-  constructor
-  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
-    constructor
-    · exact (mul_lt_mul_left hr).mpr a_h_left_left
-    · exact (mul_lt_mul_left hr).mpr a_h_left_right
-  · rintro ⟨a_left, a_right⟩
-    use x / r
-    refine ⟨⟨(lt_div_iff' hr).mpr a_left, (div_lt_iff' hr).mpr a_right⟩, ?_⟩
-    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
-
-theorem smul_Icc : r • Icc a b = Icc (r • a) (r • b) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Icc]
-  constructor
-  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
-    constructor
-    · exact (mul_le_mul_left hr).mpr a_h_left_left
-    · exact (mul_le_mul_left hr).mpr a_h_left_right
-  · rintro ⟨a_left, a_right⟩
-    use x / r
-    refine ⟨⟨(le_div_iff₀' hr).mpr a_left, (div_le_iff₀' hr).mpr a_right⟩, ?_⟩
-    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
-
-theorem smul_Ico : r • Ico a b = Ico (r • a) (r • b) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Ico]
-  constructor
-  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
-    constructor
-    · exact (mul_le_mul_left hr).mpr a_h_left_left
-    · exact (mul_lt_mul_left hr).mpr a_h_left_right
-  · rintro ⟨a_left, a_right⟩
-    use x / r
-    refine ⟨⟨(le_div_iff₀' hr).mpr a_left, (div_lt_iff' hr).mpr a_right⟩, ?_⟩
-    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
-
-theorem smul_Ioc : r • Ioc a b = Ioc (r • a) (r • b) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Ioc]
-  constructor
-  · rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩
-    constructor
-    · exact (mul_lt_mul_left hr).mpr a_h_left_left
-    · exact (mul_le_mul_left hr).mpr a_h_left_right
-  · rintro ⟨a_left, a_right⟩
-    use x / r
-    refine ⟨⟨(lt_div_iff' hr).mpr a_left, (div_le_iff₀' hr).mpr a_right⟩, ?_⟩
-    rw [mul_div_cancel₀ _ (ne_of_gt hr)]
-
-theorem smul_Ioi : r • Ioi a = Ioi (r • a) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
-  constructor
-  · rintro ⟨a_w, a_h_left, rfl⟩
-    exact (mul_lt_mul_left hr).mpr a_h_left
-  · rintro h
-    use x / r
-    constructor
-    · exact (lt_div_iff' hr).mpr h
-    · exact mul_div_cancel₀ _ (ne_of_gt hr)
-
-theorem smul_Iio : r • Iio a = Iio (r • a) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Iio]
-  constructor
-  · rintro ⟨a_w, a_h_left, rfl⟩
-    exact (mul_lt_mul_left hr).mpr a_h_left
-  · rintro h
-    use x / r
-    constructor
-    · exact (div_lt_iff' hr).mpr h
-    · exact mul_div_cancel₀ _ (ne_of_gt hr)
-
-theorem smul_Ici : r • Ici a = Ici (r • a) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
-  constructor
-  · rintro ⟨a_w, a_h_left, rfl⟩
-    exact (mul_le_mul_left hr).mpr a_h_left
-  · rintro h
-    use x / r
-    constructor
-    · exact (le_div_iff₀' hr).mpr h
-    · exact mul_div_cancel₀ _ (ne_of_gt hr)
-
-theorem smul_Iic : r • Iic a = Iic (r • a) := by
-  ext x
-  simp only [mem_smul_set, smul_eq_mul, mem_Iio]
-  constructor
-  · rintro ⟨a_w, a_h_left, rfl⟩
-    exact (mul_le_mul_left hr).mpr a_h_left
-  · rintro h
-    use x / r
-    constructor
-    · exact (div_le_iff₀' hr).mpr h
-    · exact mul_div_cancel₀ _ (ne_of_gt hr)
-
-end LinearOrderedField
diff --git a/Mathlib/Algebra/Order/Rearrangement.lean b/Mathlib/Algebra/Order/Rearrangement.lean
index 6643fbdac7eec..f69acca4d99d3 100644
--- a/Mathlib/Algebra/Order/Rearrangement.lean
+++ b/Mathlib/Algebra/Order/Rearrangement.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mantas Bakšys
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Algebra.Order.Group.Instances
 import Mathlib.Algebra.Order.Module.OrderedSMul
 import Mathlib.Algebra.Order.Module.Synonym
 import Mathlib.Data.Prod.Lex
@@ -41,23 +40,30 @@ convenience.
 
 The case for `Monotone`/`Antitone` pairs of functions over a `LinearOrder` is not deduced in this
 file because it is easily deducible from the `Monovary` API.
+
+## TODO
+
+Add equality cases for when the permute function is injective. This comes from the following fact:
+If `Monovary f g`, `Injective g` and `σ` is a permutation, then `Monovary f (g ∘ σ) ↔ σ = 1`.
 -/
 
 
 open Equiv Equiv.Perm Finset Function OrderDual
 
-variable {ι α β : Type*}
+variable {ι α β : Type*} [LinearOrderedSemiring α] [ExistsAddOfLE α]
+  [LinearOrderedCancelAddCommMonoid β] [Module α β]
 
 /-! ### Scalar multiplication versions -/
 
-
 section SMul
 
-variable [LinearOrderedRing α] [LinearOrderedAddCommGroup β] [Module α β] [OrderedSMul α β]
-  {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
+/-! #### Weak rearrangement inequality -/
+
+section weak_inequality
+variable [PosSMulMono α β] {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
 
 /-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
-`f` and `g` monovary together. Stated by permuting the entries of `g`. -/
+`f` and `g` monovary together on `s`. Stated by permuting the entries of `g`. -/
 theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g (σ i) ≤ ∑ i ∈ s, f i • g i := by
   classical
@@ -106,9 +112,62 @@ theorem MonovaryOn.sum_smul_comp_perm_le_sum_smul (hfg : MonovaryOn f g s)
     rintro rfl
     exact has hx.2
 
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
+`f` and `g` antivary together on `s`. Stated by permuting the entries of `g`. -/
+theorem AntivaryOn.sum_smul_le_sum_smul_comp_perm (hfg : AntivaryOn f g s)
+    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f i • g (σ i) :=
+  hfg.dual_right.sum_smul_comp_perm_le_sum_smul hσ
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
+`f` and `g` monovary together on `s`. Stated by permuting the entries of `f`. -/
+theorem MonovaryOn.sum_comp_perm_smul_le_sum_smul (hfg : MonovaryOn f g s)
+    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f (σ i) • g i ≤ ∑ i ∈ s, f i • g i := by
+  convert hfg.sum_smul_comp_perm_le_sum_smul
+    (show { x | σ⁻¹ x ≠ x } ⊆ s by simp only [set_support_inv_eq, hσ]) using 1
+  exact σ.sum_comp' s (fun i j ↦ f i • g j) hσ
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
+`f` and `g` antivary together on `s`. Stated by permuting the entries of `f`. -/
+theorem AntivaryOn.sum_smul_le_sum_comp_perm_smul (hfg : AntivaryOn f g s)
+    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i :=
+  hfg.dual_right.sum_comp_perm_smul_le_sum_smul hσ
+
+variable [Fintype ι]
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
+`f` and `g` monovary together. Stated by permuting the entries of `g`. -/
+theorem Monovary.sum_smul_comp_perm_le_sum_smul (hfg : Monovary f g) :
+    ∑ i, f i • g (σ i) ≤ ∑ i, f i • g i :=
+  (hfg.monovaryOn _).sum_smul_comp_perm_le_sum_smul fun _ _ ↦ mem_univ _
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
+`f` and `g` antivary together. Stated by permuting the entries of `g`. -/
+theorem Antivary.sum_smul_le_sum_smul_comp_perm (hfg : Antivary f g) :
+    ∑ i, f i • g i ≤ ∑ i, f i • g (σ i) :=
+  (hfg.antivaryOn _).sum_smul_le_sum_smul_comp_perm fun _ _ ↦ mem_univ _
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
+`f` and `g` monovary together. Stated by permuting the entries of `f`. -/
+theorem Monovary.sum_comp_perm_smul_le_sum_smul (hfg : Monovary f g) :
+    ∑ i, f (σ i) • g i ≤ ∑ i, f i • g i :=
+  (hfg.monovaryOn _).sum_comp_perm_smul_le_sum_smul fun _ _ ↦ mem_univ _
+
+/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
+`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
+theorem Antivary.sum_smul_le_sum_comp_perm_smul (hfg : Antivary f g) :
+    ∑ i, f i • g i ≤ ∑ i, f (σ i) • g i :=
+  (hfg.antivaryOn _).sum_smul_le_sum_comp_perm_smul fun _ _ ↦ mem_univ _
+
+end weak_inequality
+
+/-! #### Equality case of the rearrangement inequality -/
+
+section equality_case
+variable [PosSMulStrictMono α β] {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
+
 /-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
-together. Stated by permuting the entries of `g`. -/
+`g`, which monovary together on `s`, is unchanged by a permutation if and only if `f` and `g ∘ σ`
+monovary together on `s`. Stated by permuting the entries of `g`. -/
 theorem MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ MonovaryOn f (g ∘ σ) s := by
@@ -134,26 +193,17 @@ theorem MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : MonovaryOn f g s)
   · convert h.sum_smul_comp_perm_le_sum_smul ((set_support_inv_eq _).subset.trans hσ) using 1
     simp_rw [Function.comp_apply, apply_inv_self]
 
-/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
-`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
-`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. -/
-theorem MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff (hfg : MonovaryOn f g s)
+/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
+`g`, which antivary together on `s`, is unchanged by a permutation if and only if `f` and `g ∘ σ`
+antivary together on `s`. Stated by permuting the entries of `g`. -/
+theorem AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
-    ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ σ) s := by
-  simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne,
-    hfg.sum_smul_comp_perm_le_sum_smul hσ]
-
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
-`f` and `g` monovary together. Stated by permuting the entries of `f`. -/
-theorem MonovaryOn.sum_comp_perm_smul_le_sum_smul (hfg : MonovaryOn f g s)
-    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f (σ i) • g i ≤ ∑ i ∈ s, f i • g i := by
-  convert hfg.sum_smul_comp_perm_le_sum_smul
-    (show { x | σ⁻¹ x ≠ x } ⊆ s by simp only [set_support_inv_eq, hσ]) using 1
-  exact σ.sum_comp' s (fun i j ↦ f i • g j) hσ
+    ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ AntivaryOn f (g ∘ σ) s :=
+  (hfg.dual_right.sum_smul_comp_perm_eq_sum_smul_iff hσ).trans monovaryOn_toDual_right
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
-together. Stated by permuting the entries of `f`. -/
+`g`, which monovary together on `s`, is unchanged by a permutation if and only if `f ∘ σ` and `g`
+monovary together on `s`. Stated by permuting the entries of `f`. -/
 theorem MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ MonovaryOn (f ∘ σ) g s := by
@@ -164,70 +214,100 @@ theorem MonovaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : MonovaryOn f g s)
     rw [σ.sum_comp' s (fun i j ↦ f i • g j) hσ]
     congr
   · convert h.comp_right σ
-    · rw [comp.assoc, inv_def, symm_comp_self, comp_id]
+    · rw [comp_assoc, inv_def, symm_comp_self, comp_id]
     · rw [σ.eq_preimage_iff_image_eq, Set.image_perm hσ]
   · convert h.comp_right σ.symm
-    · rw [comp.assoc, self_comp_symm, comp_id]
+    · rw [comp_assoc, self_comp_symm, comp_id]
     · rw [σ.symm.eq_preimage_iff_image_eq]
       exact Set.image_perm hσinv
 
-/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
-`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
-`f ∘ σ` and `g` do not monovary together. Stated by permuting the entries of `f`. -/
-theorem MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff (hfg : MonovaryOn f g s)
+/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
+`g`, which antivary together on `s`, is unchanged by a permutation if and only if `f ∘ σ` and `g`
+antivary together on `s`. Stated by permuting the entries of `f`. -/
+theorem AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
-    ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ σ) g s := by
-  simp [← hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ, lt_iff_le_and_ne,
-    hfg.sum_comp_perm_smul_le_sum_smul hσ]
+    ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ AntivaryOn (f ∘ σ) g s :=
+  (hfg.dual_right.sum_comp_perm_smul_eq_sum_smul_iff hσ).trans monovaryOn_toDual_right
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
-`f` and `g` antivary together. Stated by permuting the entries of `g`. -/
-theorem AntivaryOn.sum_smul_le_sum_smul_comp_perm (hfg : AntivaryOn f g s)
-    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f i • g (σ i) :=
-  hfg.dual_right.sum_smul_comp_perm_le_sum_smul hσ
+@[deprecated (since := "2024-06-25")]
+alias AntivaryOn.sum_smul_eq_sum_comp_perm_smul_iff := AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff
+
+variable [Fintype ι]
+
+/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
+`g`, which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
+together. Stated by permuting the entries of `g`. -/
+theorem Monovary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : Monovary f g) :
+    ∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Monovary f (g ∘ σ) := by
+  simp [(hfg.monovaryOn _).sum_smul_comp_perm_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
+
+/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
+`g`, which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
+together. Stated by permuting the entries of `g`. -/
+theorem Monovary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Monovary f g) :
+    ∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ Monovary (f ∘ σ) g := by
+  simp [(hfg.monovaryOn _).sum_comp_perm_smul_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
 `g`, which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
 together. Stated by permuting the entries of `g`. -/
-theorem AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : AntivaryOn f g s)
-    (hσ : {x | σ x ≠ x} ⊆ s) :
-    ∑ i ∈ s, f i • g (σ i) = ∑ i ∈ s, f i • g i ↔ AntivaryOn f (g ∘ σ) s :=
-  (hfg.dual_right.sum_smul_comp_perm_eq_sum_smul_iff hσ).trans monovaryOn_toDual_right
+theorem Antivary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : Antivary f g) :
+    ∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Antivary f (g ∘ σ) := by
+  simp [(hfg.antivaryOn _).sum_smul_comp_perm_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
 
 @[deprecated (since := "2024-06-25")]
-alias AntivaryOn.sum_smul_eq_sum_smul_comp_perm_iff := AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff
+alias Antivary.sum_smul_eq_sum_smul_comp_perm_iff := Antivary.sum_smul_comp_perm_eq_sum_smul_iff
+
+/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
+`g`, which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
+together. Stated by permuting the entries of `f`. -/
+theorem Antivary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Antivary f g) :
+    ∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ Antivary (f ∘ σ) g := by
+  simp [(hfg.antivaryOn _).sum_comp_perm_smul_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
+
+@[deprecated (since := "2024-06-25")]
+alias Antivary.sum_smul_eq_sum_comp_perm_smul_iff := Antivary.sum_comp_perm_smul_eq_sum_smul_iff
+
+end equality_case
+
+/-! #### Strict rearrangement inequality -/
+
+section strict_inequality
+variable [PosSMulStrictMono α β] {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
-`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
-`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. -/
+`f` and `g`, which monovary together on `s`, is strictly decreased by a permutation if and only if
+`f` and `g ∘ σ` do not monovary together on `s`. Stated by permuting the entries of `g`. -/
+theorem MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff (hfg : MonovaryOn f g s)
+    (hσ : {x | σ x ≠ x} ⊆ s) :
+    ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ σ) s := by
+  simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne,
+    hfg.sum_smul_comp_perm_le_sum_smul hσ]
+
+/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
+`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
+`f` and `g ∘ σ` do not antivary together on `s`. Stated by permuting the entries of `g`. -/
 theorem AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i • g i < ∑ i ∈ s, f i • g (σ i) ↔ ¬AntivaryOn f (g ∘ σ) s := by
   simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne, eq_comm,
     hfg.sum_smul_le_sum_smul_comp_perm hσ]
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
-`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
-theorem AntivaryOn.sum_smul_le_sum_comp_perm_smul (hfg : AntivaryOn f g s)
-    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i • g i ≤ ∑ i ∈ s, f (σ i) • g i := by
-  convert hfg.sum_smul_le_sum_smul_comp_perm
-    (show { x | σ⁻¹ x ≠ x } ⊆ s by simp only [set_support_inv_eq, hσ]) using 1
-  exact σ.sum_comp' s (fun i j ↦ f i • g j) hσ
-
-/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
-together. Stated by permuting the entries of `f`. -/
-theorem AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff (hfg : AntivaryOn f g s)
+/-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
+`f` and `g`, which monovary together on `s`, is strictly decreased by a permutation if and only if
+`f ∘ σ` and `g` do not monovary together on `s`. Stated by permuting the entries of `f`. -/
+theorem MonovaryOn.sum_comp_perm_smul_lt_sum_smul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
-    ∑ i ∈ s, f (σ i) • g i = ∑ i ∈ s, f i • g i ↔ AntivaryOn (f ∘ σ) g s :=
-  (hfg.dual_right.sum_comp_perm_smul_eq_sum_smul_iff hσ).trans monovaryOn_toDual_right
+    ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ σ) g s := by
+  simp [← hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ, lt_iff_le_and_ne,
+    hfg.sum_comp_perm_smul_le_sum_smul hσ]
 
 @[deprecated (since := "2024-06-25")]
-alias AntivaryOn.sum_smul_eq_sum_comp_perm_smul_iff := AntivaryOn.sum_comp_perm_smul_eq_sum_smul_iff
+alias AntivaryOn.sum_smul_eq_sum_smul_comp_perm_iff := AntivaryOn.sum_smul_comp_perm_eq_sum_smul_iff
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
-`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
-`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. -/
+`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
+`f ∘ σ` and `g` do not antivary together on `s`. Stated by permuting the entries of `f`. -/
 theorem AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i • g i < ∑ i ∈ s, f (σ i) • g i ↔ ¬AntivaryOn (f ∘ σ) g s := by
@@ -236,19 +316,6 @@ theorem AntivaryOn.sum_smul_lt_sum_comp_perm_smul_iff (hfg : AntivaryOn f g s)
 
 variable [Fintype ι]
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
-`f` and `g` monovary together. Stated by permuting the entries of `g`. -/
-theorem Monovary.sum_smul_comp_perm_le_sum_smul (hfg : Monovary f g) :
-    ∑ i, f i • g (σ i) ≤ ∑ i, f i • g i :=
-  (hfg.monovaryOn _).sum_smul_comp_perm_le_sum_smul fun _ _ ↦ mem_univ _
-
-/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
-together. Stated by permuting the entries of `g`. -/
-theorem Monovary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : Monovary f g) :
-    ∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Monovary f (g ∘ σ) := by
-  simp [(hfg.monovaryOn _).sum_smul_comp_perm_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
-
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
 `f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
 `f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. -/
@@ -256,19 +323,6 @@ theorem Monovary.sum_smul_comp_perm_lt_sum_smul_iff (hfg : Monovary f g) :
     ∑ i, f i • g (σ i) < ∑ i, f i • g i ↔ ¬Monovary f (g ∘ σ) := by
   simp [(hfg.monovaryOn _).sum_smul_comp_perm_lt_sum_smul_iff fun _ _ ↦ mem_univ _]
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
-`f` and `g` monovary together. Stated by permuting the entries of `f`. -/
-theorem Monovary.sum_comp_perm_smul_le_sum_smul (hfg : Monovary f g) :
-    ∑ i, f (σ i) • g i ≤ ∑ i, f i • g i :=
-  (hfg.monovaryOn _).sum_comp_perm_smul_le_sum_smul fun _ _ ↦ mem_univ _
-
-/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
-together. Stated by permuting the entries of `g`. -/
-theorem Monovary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Monovary f g) :
-    ∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ Monovary (f ∘ σ) g := by
-  simp [(hfg.monovaryOn _).sum_comp_perm_smul_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
-
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
 `f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
 `f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. -/
@@ -276,22 +330,6 @@ theorem Monovary.sum_comp_perm_smul_lt_sum_smul_iff (hfg : Monovary f g) :
     ∑ i, f (σ i) • g i < ∑ i, f i • g i ↔ ¬Monovary (f ∘ σ) g := by
   simp [(hfg.monovaryOn _).sum_comp_perm_smul_lt_sum_smul_iff fun _ _ ↦ mem_univ _]
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
-`f` and `g` antivary together. Stated by permuting the entries of `g`. -/
-theorem Antivary.sum_smul_le_sum_smul_comp_perm (hfg : Antivary f g) :
-    ∑ i, f i • g i ≤ ∑ i, f i • g (σ i) :=
-  (hfg.antivaryOn _).sum_smul_le_sum_smul_comp_perm fun _ _ ↦ mem_univ _
-
-/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
-together. Stated by permuting the entries of `g`. -/
-theorem Antivary.sum_smul_comp_perm_eq_sum_smul_iff (hfg : Antivary f g) :
-    ∑ i, f i • g (σ i) = ∑ i, f i • g i ↔ Antivary f (g ∘ σ) := by
-  simp [(hfg.antivaryOn _).sum_smul_comp_perm_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
-
-@[deprecated (since := "2024-06-25")]
-alias Antivary.sum_smul_eq_sum_smul_comp_perm_iff := Antivary.sum_smul_comp_perm_eq_sum_smul_iff
-
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
 `f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
 `f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. -/
@@ -299,22 +337,6 @@ theorem Antivary.sum_smul_lt_sum_smul_comp_perm_iff (hfg : Antivary f g) :
     ∑ i, f i • g i < ∑ i, f i • g (σ i) ↔ ¬Antivary f (g ∘ σ) := by
   simp [(hfg.antivaryOn _).sum_smul_lt_sum_smul_comp_perm_iff fun _ _ ↦ mem_univ _]
 
-/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is minimized when
-`f` and `g` antivary together. Stated by permuting the entries of `f`. -/
-theorem Antivary.sum_smul_le_sum_comp_perm_smul (hfg : Antivary f g) :
-    ∑ i, f i • g i ≤ ∑ i, f (σ i) • g i :=
-  (hfg.antivaryOn _).sum_smul_le_sum_comp_perm_smul fun _ _ ↦ mem_univ _
-
-/-- **Equality case of the Rearrangement Inequality**: Pointwise scalar multiplication of `f` and
-`g`, which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
-together. Stated by permuting the entries of `f`. -/
-theorem Antivary.sum_comp_perm_smul_eq_sum_smul_iff (hfg : Antivary f g) :
-    ∑ i, f (σ i) • g i = ∑ i, f i • g i ↔ Antivary (f ∘ σ) g := by
-  simp [(hfg.antivaryOn _).sum_comp_perm_smul_eq_sum_smul_iff fun _ _ ↦ mem_univ _]
-
-@[deprecated (since := "2024-06-25")]
-alias Antivary.sum_smul_eq_sum_comp_perm_smul_iff := Antivary.sum_comp_perm_smul_eq_sum_smul_iff
-
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
 `f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
 `f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. -/
@@ -322,6 +344,7 @@ theorem Antivary.sum_smul_lt_sum_comp_perm_smul_iff (hfg : Antivary f g) :
     ∑ i, f i • g i < ∑ i, f (σ i) • g i ↔ ¬Antivary (f ∘ σ) g := by
   simp [(hfg.antivaryOn _).sum_smul_lt_sum_comp_perm_smul_iff fun _ _ ↦ mem_univ _]
 
+end strict_inequality
 end SMul
 
 /-!
@@ -330,87 +353,84 @@ end SMul
 Special cases of the above when scalar multiplication is actually multiplication.
 -/
 
-
 section Mul
-
-
-variable [LinearOrderedRing α] {s : Finset ι} {σ : Perm ι} {f g : ι → α}
+variable {s : Finset ι} {σ : Perm ι} {f g : ι → α}
 
 /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
-`g` monovary together. Stated by permuting the entries of `g`. -/
-theorem MonovaryOn.sum_mul_comp_perm_le_sum_mul (hfg : MonovaryOn f g s)
-    (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g (σ i) ≤ ∑ i ∈ s, f i * g i :=
+`g` monovary together on `s`. Stated by permuting the entries of `g`. -/
+theorem MonovaryOn.sum_mul_comp_perm_le_sum_mul (hfg : MonovaryOn f g s) (hσ : {x | σ x ≠ x} ⊆ s) :
+    ∑ i ∈ s, f i * g (σ i) ≤ ∑ i ∈ s, f i * g i :=
   hfg.sum_smul_comp_perm_le_sum_smul hσ
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
-which monovary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` monovary
-together. Stated by permuting the entries of `g`. -/
+which monovary together on `s`, is unchanged by a permutation if and only if `f` and `g ∘ σ`
+monovary together on `s`. Stated by permuting the entries of `g`. -/
 theorem MonovaryOn.sum_mul_comp_perm_eq_sum_mul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ MonovaryOn f (g ∘ σ) s :=
   hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise scalar multiplication of
-`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
-`f` and `g ∘ σ` do not monovary together. Stated by permuting the entries of `g`. -/
+`f` and `g`, which monovary together on `s`, is strictly decreased by a permutation if and only if
+`f` and `g ∘ σ` do not monovary together on `s`. Stated by permuting the entries of `g`. -/
 theorem MonovaryOn.sum_mul_comp_perm_lt_sum_mul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ σ) s :=
   hfg.sum_smul_comp_perm_lt_sum_smul_iff hσ
 
 /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is maximized when `f` and
-`g` monovary together. Stated by permuting the entries of `f`. -/
+`g` monovary together on `s`. Stated by permuting the entries of `f`. -/
 theorem MonovaryOn.sum_comp_perm_mul_le_sum_mul (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f (σ i) * g i ≤ ∑ i ∈ s, f i * g i :=
   hfg.sum_comp_perm_smul_le_sum_smul hσ
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
-which monovary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` monovary
-together. Stated by permuting the entries of `f`. -/
+which monovary together on `s`, is unchanged by a permutation if and only if `f ∘ σ` and `g`
+monovary together on `s`. Stated by permuting the entries of `f`. -/
 theorem MonovaryOn.sum_comp_perm_mul_eq_sum_mul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ MonovaryOn (f ∘ σ) g s :=
   hfg.sum_comp_perm_smul_eq_sum_smul_iff hσ
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
-`f` and `g`, which monovary together, is strictly decreased by a permutation if and only if
-`f ∘ σ` and `g` do not monovary together. Stated by permuting the entries of `f`. -/
+`f` and `g`, which monovary together on `s`, is strictly decreased by a permutation if and only if
+`f ∘ σ` and `g` do not monovary together on `s`. Stated by permuting the entries of `f`. -/
 theorem MonovaryOn.sum_comp_perm_mul_lt_sum_mul_iff (hfg : MonovaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f (σ i) * g i < ∑ i ∈ s, f i * g i ↔ ¬MonovaryOn (f ∘ σ) g s :=
   hfg.sum_comp_perm_smul_lt_sum_smul_iff hσ
 
 /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
-`g` antivary together. Stated by permuting the entries of `g`. -/
+`g` antivary together on `s`. Stated by permuting the entries of `g`. -/
 theorem AntivaryOn.sum_mul_le_sum_mul_comp_perm (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f i * g (σ i) :=
   hfg.sum_smul_le_sum_smul_comp_perm hσ
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
-which antivary together, is unchanged by a permutation if and only if `f` and `g ∘ σ` antivary
-together. Stated by permuting the entries of `g`. -/
+which antivary together on `s`, is unchanged by a permutation if and only if `f` and `g ∘ σ`
+antivary together on `s`. Stated by permuting the entries of `g`. -/
 theorem AntivaryOn.sum_mul_eq_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i * g (σ i) = ∑ i ∈ s, f i * g i ↔ AntivaryOn f (g ∘ σ) s :=
   hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
-`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
-`f` and `g ∘ σ` do not antivary together. Stated by permuting the entries of `g`. -/
+`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
+`f` and `g ∘ σ` do not antivary together on `s`. Stated by permuting the entries of `g`. -/
 theorem AntivaryOn.sum_mul_lt_sum_mul_comp_perm_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i * g i < ∑ i ∈ s, f i * g (σ i) ↔ ¬AntivaryOn f (g ∘ σ) s :=
   hfg.sum_smul_lt_sum_smul_comp_perm_iff hσ
 
 /-- **Rearrangement Inequality**: Pointwise multiplication of `f` and `g` is minimized when `f` and
-`g` antivary together. Stated by permuting the entries of `f`. -/
+`g` antivary together on `s`. Stated by permuting the entries of `f`. -/
 theorem AntivaryOn.sum_mul_le_sum_comp_perm_mul (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) : ∑ i ∈ s, f i * g i ≤ ∑ i ∈ s, f (σ i) * g i :=
   hfg.sum_smul_le_sum_comp_perm_smul hσ
 
 /-- **Equality case of the Rearrangement Inequality**: Pointwise multiplication of `f` and `g`,
-which antivary together, is unchanged by a permutation if and only if `f ∘ σ` and `g` antivary
-together. Stated by permuting the entries of `f`. -/
+which antivary together on `s`, is unchanged by a permutation if and only if `f ∘ σ` and `g`
+antivary together on `s`. Stated by permuting the entries of `f`. -/
 theorem AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f (σ i) * g i = ∑ i ∈ s, f i * g i ↔ AntivaryOn (f ∘ σ) g s :=
@@ -420,8 +440,8 @@ theorem AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff (hfg : AntivaryOn f g s)
 alias AntivaryOn.sum_mul_eq_sum_comp_perm_mul_iff := AntivaryOn.sum_comp_perm_mul_eq_sum_mul_iff
 
 /-- **Strict inequality case of the Rearrangement Inequality**: Pointwise multiplication of
-`f` and `g`, which antivary together, is strictly decreased by a permutation if and only if
-`f ∘ σ` and `g` do not antivary together. Stated by permuting the entries of `f`. -/
+`f` and `g`, which antivary together on `s`, is strictly decreased by a permutation if and only if
+`f ∘ σ` and `g` do not antivary together on `s`. Stated by permuting the entries of `f`. -/
 theorem AntivaryOn.sum_mul_lt_sum_comp_perm_mul_iff (hfg : AntivaryOn f g s)
     (hσ : {x | σ x ≠ x} ⊆ s) :
     ∑ i ∈ s, f i * g i < ∑ i ∈ s, f (σ i) * g i ↔ ¬AntivaryOn (f ∘ σ) g s :=
diff --git a/Mathlib/Algebra/Order/Ring/Abs.lean b/Mathlib/Algebra/Order/Ring/Abs.lean
index 7d11531e4a537..e61b8d4297a31 100644
--- a/Mathlib/Algebra/Order/Ring/Abs.lean
+++ b/Mathlib/Algebra/Order/Ring/Abs.lean
@@ -43,7 +43,7 @@ lemma abs_two : |(2 : α)| = 2 := abs_of_pos zero_lt_two
 lemma abs_mul (a b : α) : |a * b| = |a| * |b| := by
   rw [abs_eq (mul_nonneg (abs_nonneg a) (abs_nonneg b))]
   rcases le_total a 0 with ha | ha <;> rcases le_total b 0 with hb | hb <;>
-    simp only [abs_of_nonpos, abs_of_nonneg, true_or_iff, or_true_iff, eq_self_iff_true, neg_mul,
+    simp only [abs_of_nonpos, abs_of_nonneg, true_or, or_true, eq_self_iff_true, neg_mul,
       mul_neg, neg_neg, *]
 
 /-- `abs` as a `MonoidWithZeroHom`. -/
diff --git a/Mathlib/Algebra/Order/Ring/Basic.lean b/Mathlib/Algebra/Order/Ring/Basic.lean
index b6456438e50c3..5e64f73cc40ea 100644
--- a/Mathlib/Algebra/Order/Ring/Basic.lean
+++ b/Mathlib/Algebra/Order/Ring/Basic.lean
@@ -37,10 +37,6 @@ section OrderedSemiring
 
 variable [OrderedSemiring R] {a b x y : R} {n m : ℕ}
 
-theorem zero_pow_le_one : ∀ n : ℕ, (0 : R) ^ n ≤ 1
-  | 0 => (pow_zero _).le
-  | n + 1 => by rw [zero_pow n.succ_ne_zero]; exact zero_le_one
-
 theorem pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y ^ n ≤ (x + y) ^ n := by
   rcases Nat.exists_eq_add_one_of_ne_zero hn with ⟨k, rfl⟩
   induction k with
@@ -60,40 +56,21 @@ theorem pow_add_pow_le (hx : 0 ≤ x) (hy : 0 ≤ y) (hn : n ≠ 0) : x ^ n + y
         rw [pow_succ' _ n]
         exact mul_le_mul_of_nonneg_left (ih (Nat.succ_ne_zero k)) h2
 
-@[bound]
-theorem pow_le_one : ∀ n : ℕ, 0 ≤ a → a ≤ 1 → a ^ n ≤ 1
-  | 0, _, _ => (pow_zero a).le
-  | n + 1, h₀, h₁ => (pow_succ a n).le.trans (mul_le_one (pow_le_one n h₀ h₁) h₀ h₁)
-
-theorem pow_lt_one (h₀ : 0 ≤ a) (h₁ : a < 1) : ∀ {n : ℕ}, n ≠ 0 → a ^ n < 1
-  | 0, h => (h rfl).elim
-  | n + 1, _ => by
-    rw [pow_succ']
-    exact mul_lt_one_of_nonneg_of_lt_one_left h₀ h₁ (pow_le_one _ h₀ h₁.le)
-
-@[bound]
-theorem one_le_pow_of_one_le (H : 1 ≤ a) : ∀ n : ℕ, 1 ≤ a ^ n
-  | 0 => by rw [pow_zero]
-  | n + 1 => by
-    rw [pow_succ']
-    simpa only [mul_one] using
-      mul_le_mul H (one_le_pow_of_one_le H n) zero_le_one (le_trans zero_le_one H)
-
-theorem pow_right_mono (h : 1 ≤ a) : Monotone (a ^ ·) :=
-  monotone_nat_of_le_succ fun n => by
-    rw [pow_succ']
-    exact le_mul_of_one_le_left (pow_nonneg (zero_le_one.trans h) _) h
-
-@[gcongr]
-theorem pow_le_pow_right (ha : 1 ≤ a) (h : n ≤ m) : a ^ n ≤ a ^ m := pow_right_mono ha h
+attribute [bound] pow_le_one₀ one_le_pow₀
 
-theorem le_self_pow (ha : 1 ≤ a) (h : m ≠ 0) : a ≤ a ^ m := by
-  simpa only [pow_one] using pow_le_pow_right ha <| Nat.pos_iff_ne_zero.2 h
+@[deprecated (since := "2024-09-28")] alias mul_le_one := mul_le_one₀
+@[deprecated (since := "2024-09-28")] alias pow_le_one := pow_le_one₀
+@[deprecated (since := "2024-09-28")] alias pow_lt_one := pow_lt_one₀
+@[deprecated (since := "2024-09-28")] alias one_le_pow_of_one_le := one_le_pow₀
+@[deprecated (since := "2024-09-28")] alias one_lt_pow := one_lt_pow₀
+@[deprecated (since := "2024-10-04")] alias pow_right_mono := pow_right_mono₀
+@[deprecated (since := "2024-10-04")] alias pow_le_pow_right := pow_le_pow_right₀
+@[deprecated (since := "2024-10-04")] alias le_self_pow := le_self_pow₀
 
 /-- The `bound` tactic can't handle `m ≠ 0` goals yet, so we express as `0 < m` -/
 @[bound]
 lemma Bound.le_self_pow_of_pos {m : ℕ} (ha : 1 ≤ a) (h : 0 < m) : a ≤ a ^ m :=
-  le_self_pow ha h.ne'
+  le_self_pow₀ ha h.ne'
 
 @[mono, gcongr, bound]
 theorem pow_le_pow_left {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) : ∀ n, a ^ n ≤ b ^ n
@@ -101,12 +78,6 @@ theorem pow_le_pow_left {a b : R} (ha : 0 ≤ a) (hab : a ≤ b) : ∀ n, a ^ n
   | n + 1 => by simpa only [pow_succ']
       using mul_le_mul hab (pow_le_pow_left ha hab _) (pow_nonneg ha _) (ha.trans hab)
 
-theorem one_lt_pow (ha : 1 < a) : ∀ {n : ℕ} (_ : n ≠ 0), 1 < a ^ n
-  | 0, h => (h rfl).elim
-  | n + 1, _ => by
-    rw [pow_succ']
-    exact one_lt_mul_of_lt_of_le ha (one_le_pow_of_one_le ha.le _)
-
 lemma pow_add_pow_le' (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ n + b ^ n ≤ 2 * (a + b) ^ n := by
   rw [two_mul]
   exact add_le_add (pow_le_pow_left ha (le_add_of_nonneg_right hb) _)
@@ -117,11 +88,19 @@ lemma pow_add_pow_le' (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ n + b ^ n ≤ 2 * (a +
 lemma Bound.pow_le_pow_right_of_le_one_or_one_le (h : 1 ≤ a ∧ n ≤ m ∨ 0 ≤ a ∧ a ≤ 1 ∧ m ≤ n) :
     a ^ n ≤ a ^ m := by
   rcases h with ⟨a1, nm⟩ | ⟨a0, a1, mn⟩
-  · exact pow_le_pow_right a1 nm
+  · exact pow_right_mono₀ a1 nm
   · exact pow_le_pow_of_le_one a0 a1 mn
 
 end OrderedSemiring
 
+-- See note [reducible non instances]
+/-- Turn an ordered domain into a strict ordered ring. -/
+abbrev OrderedRing.toStrictOrderedRing (α : Type*)
+    [OrderedRing α] [NoZeroDivisors α] [Nontrivial α] : StrictOrderedRing α where
+  __ := ‹OrderedRing α›
+  __ := ‹NoZeroDivisors α›
+  mul_pos a b ap bp := (mul_nonneg ap.le bp.le).lt_of_ne' (mul_ne_zero ap.ne' bp.ne')
+
 section StrictOrderedSemiring
 
 variable [StrictOrderedSemiring R] {a x y : R} {n m : ℕ}
@@ -353,36 +332,3 @@ lemma pow_four_le_pow_two_of_pow_two_le (h : a ^ 2 ≤ b) : a ^ 4 ≤ b ^ 2 :=
   (pow_mul a 2 2).symm ▸ pow_le_pow_left (sq_nonneg a) h 2
 
 end LinearOrderedSemiring
-
-/-!
-### Deprecated lemmas
-
-Those lemmas have been deprecated on 2023-12-23.
--/
-
-@[deprecated (since := "2023-12-23")] alias pow_mono := pow_right_mono
-@[deprecated (since := "2023-12-23")] alias pow_le_pow := pow_le_pow_right
-@[deprecated (since := "2023-12-23")] alias pow_le_pow_of_le_left := pow_le_pow_left
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow_of_lt_left := pow_lt_pow_left
-@[deprecated (since := "2023-12-23")] alias strictMonoOn_pow := pow_left_strictMonoOn
-@[deprecated (since := "2023-12-23")] alias pow_strictMono_right := pow_right_strictMono
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow := pow_lt_pow_right
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow_iff := pow_lt_pow_iff_right
-@[deprecated (since := "2023-12-23")] alias pow_le_pow_iff := pow_le_pow_iff_right
-@[deprecated (since := "2023-12-23")] alias self_lt_pow := lt_self_pow
-@[deprecated (since := "2023-12-23")] alias strictAnti_pow := pow_right_strictAnti
-
-@[deprecated (since := "2023-12-23")]
-alias pow_lt_pow_iff_of_lt_one := pow_lt_pow_iff_right_of_lt_one
-
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow_of_lt_one := pow_lt_pow_right_of_lt_one
-@[deprecated (since := "2023-12-23")] alias lt_of_pow_lt_pow := lt_of_pow_lt_pow_left
-@[deprecated (since := "2023-12-23")] alias le_of_pow_le_pow := le_of_pow_le_pow_left
-@[deprecated (since := "2023-12-23")] alias self_le_pow := le_self_pow
-@[deprecated (since := "2023-12-23")] alias Nat.pow_lt_pow_of_lt_right := pow_lt_pow_right
-
-@[deprecated (since := "2023-12-23")]
-protected alias Nat.pow_right_strictMono := pow_right_strictMono
-
-@[deprecated (since := "2023-12-23")] alias Nat.pow_le_iff_le_right := pow_le_pow_iff_right
-@[deprecated (since := "2023-12-23")] alias Nat.pow_lt_iff_lt_right := pow_lt_pow_iff_right
diff --git a/Mathlib/Algebra/Order/Ring/Cast.lean b/Mathlib/Algebra/Order/Ring/Cast.lean
index 98e9a362cfbdd..bd47e1f894415 100644
--- a/Mathlib/Algebra/Order/Ring/Cast.lean
+++ b/Mathlib/Algebra/Order/Ring/Cast.lean
@@ -37,6 +37,8 @@ lemma cast_mono : Monotone (Int.cast : ℤ → R) := by
   rw [← sub_nonneg, ← cast_sub, ← hk, cast_natCast]
   exact k.cast_nonneg'
 
+@[gcongr] protected lemma GCongr.intCast_mono {m n : ℤ} (hmn : m ≤ n) : (m : R) ≤ n := cast_mono hmn
+
 variable [NeZero (1 : R)] {m n : ℤ}
 
 @[simp] lemma cast_nonneg : ∀ {n : ℤ}, (0 : R) ≤ n ↔ 0 ≤ n
@@ -53,6 +55,8 @@ lemma cast_strictMono : StrictMono (fun x : ℤ => (x : R)) :=
 
 @[simp, norm_cast] lemma cast_lt : (m : R) < n ↔ m < n := cast_strictMono.lt_iff_lt
 
+@[gcongr] protected alias ⟨_, GCongr.intCast_strictMono⟩ := Int.cast_lt
+
 @[simp] lemma cast_nonpos : (n : R) ≤ 0 ↔ n ≤ 0 := by rw [← cast_zero, cast_le]
 
 @[simp] lemma cast_pos : (0 : R) < n ↔ 0 < n := by rw [← cast_zero, cast_lt]
diff --git a/Mathlib/Algebra/Order/Ring/Cone.lean b/Mathlib/Algebra/Order/Ring/Cone.lean
index 8789ad725e530..71c21076f0093 100644
--- a/Mathlib/Algebra/Order/Ring/Cone.lean
+++ b/Mathlib/Algebra/Order/Ring/Cone.lean
@@ -1,68 +1,86 @@
 /-
 Copyright (c) 2016 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
+Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Artie Khovanov
 -/
 import Mathlib.Algebra.Order.Group.Cone
-import Mathlib.Algebra.Order.Ring.Defs
+import Mathlib.Algebra.Order.Ring.Basic
+import Mathlib.Algebra.Ring.Subsemiring.Order
 
 /-!
-# Constructing an ordered ring from a ring with a specified positive cone.
+# Construct ordered rings from rings with a specified positive cone.
 
--/
-
-
-/-! ### Positive cones -/
-
-
-variable {α : Type*} [Ring α] [Nontrivial α]
-
-namespace Ring
-
-/-- A positive cone in a ring consists of a positive cone in underlying `AddCommGroup`,
-which contains `1` and such that the positive elements are closed under multiplication. -/
-structure PositiveCone (α : Type*) [Ring α] extends AddCommGroup.PositiveCone α where
-  /-- In a positive cone, `1` is `nonneg` -/
-  one_nonneg : nonneg 1
-  /-- In a positive cone, if `a` and `b` are `pos` then so is `a * b` -/
-  mul_pos : ∀ a b, pos a → pos b → pos (a * b)
-
-/-- Forget that a positive cone in a ring respects the multiplicative structure. -/
-add_decl_doc PositiveCone.toPositiveCone
+In this file we provide the structure `RingCone`
+that encodes axioms of `OrderedRing` and `LinearOrderedRing`
+in terms of the subset of non-negative elements.
 
-/-- A total positive cone in a nontrivial ring induces a linear order. -/
-structure TotalPositiveCone (α : Type*) [Ring α] extends PositiveCone α,
-  AddCommGroup.TotalPositiveCone α
-
-/-- Forget that a `TotalPositiveCone` in a ring is total. -/
-add_decl_doc TotalPositiveCone.toPositiveCone_1
-
-/-- Forget that a `TotalPositiveCone` in a ring respects the multiplicative structure. -/
-add_decl_doc TotalPositiveCone.toTotalPositiveCone
-
-theorem PositiveCone.one_pos (C : PositiveCone α) : C.pos 1 :=
-  (C.pos_iff _).2 ⟨C.one_nonneg, fun h => one_ne_zero <| C.nonneg_antisymm C.one_nonneg h⟩
-
-end Ring
-
-open Ring
-
-/-- Construct a `StrictOrderedRing` by designating a positive cone in an existing `Ring`. -/
-def StrictOrderedRing.mkOfPositiveCone (C : PositiveCone α) : StrictOrderedRing α :=
-  { ‹Ring α›, OrderedAddCommGroup.mkOfPositiveCone C.toPositiveCone with
-    exists_pair_ne := ⟨0, 1, fun h => by simpa [← h, C.pos_iff] using C.one_pos⟩,
-    zero_le_one := by
-      change C.nonneg (1 - 0)
-      convert C.one_nonneg
-      simp,
-    mul_pos := fun x y xp yp => by
-      change C.pos (x * y - 0)
-      -- Porting note: used to be convert, but it relied on unfolding definitions
-      rw [sub_zero]
-      exact C.mul_pos x y (by rwa [← sub_zero x]) (by rwa [← sub_zero y]) }
+We also provide constructors that convert between
+cones in rings and the corresponding ordered rings.
+-/
 
-/-- Construct a `LinearOrderedRing` by
-designating a positive cone in an existing `Ring`. -/
-def LinearOrderedRing.mkOfPositiveCone (C : TotalPositiveCone α) : LinearOrderedRing α :=
-  { LinearOrderedAddCommGroup.mkOfPositiveCone C.toTotalPositiveCone,
-    StrictOrderedRing.mkOfPositiveCone C.toPositiveCone_1 with }
+/-- `RingConeClass S R` says that `S` is a type of cones in `R`. -/
+class RingConeClass (S : Type*) (R : outParam Type*) [Ring R] [SetLike S R]
+    extends AddGroupConeClass S R, SubsemiringClass S R : Prop
+
+/-- A (positive) cone in a ring is a subsemiring that
+does not contain both `a` and `-a` for any nonzero `a`.
+This is equivalent to being the set of non-negative elements of
+some order making the ring into a partially ordered ring. -/
+structure RingCone (R : Type*) [Ring R] extends Subsemiring R, AddGroupCone R
+
+/-- Interpret a cone in a ring as a cone in the underlying additive group. -/
+add_decl_doc RingCone.toAddGroupCone
+
+instance RingCone.instSetLike (R : Type*) [Ring R] : SetLike (RingCone R) R where
+  coe C := C.carrier
+  coe_injective' p q h := by cases p; cases q; congr; exact SetLike.ext' h
+
+instance RingCone.instRingConeClass (R : Type*) [Ring R] :
+    RingConeClass (RingCone R) R where
+  add_mem {C} := C.add_mem'
+  zero_mem {C} := C.zero_mem'
+  mul_mem {C} := C.mul_mem'
+  one_mem {C} := C.one_mem'
+  eq_zero_of_mem_of_neg_mem {C} := C.eq_zero_of_mem_of_neg_mem'
+
+namespace RingCone
+
+variable {T : Type*} [OrderedRing T] {a : T}
+
+variable (T) in
+/-- Construct a cone from the set of non-negative elements of a partially ordered ring. -/
+def nonneg : RingCone T where
+  __ := Subsemiring.nonneg T
+  eq_zero_of_mem_of_neg_mem' {a} := by simpa using ge_antisymm
+
+@[simp] lemma nonneg_toSubsemiring : (nonneg T).toSubsemiring = .nonneg T := rfl
+@[simp] lemma nonneg_toAddGroupCone : (nonneg T).toAddGroupCone = .nonneg T := rfl
+@[simp] lemma mem_nonneg : a ∈ nonneg T ↔ 0 ≤ a := Iff.rfl
+@[simp, norm_cast] lemma coe_nonneg : nonneg T = {x : T | 0 ≤ x} := rfl
+
+instance nonneg.isMaxCone {T : Type*} [LinearOrderedRing T] : IsMaxCone (nonneg T) where
+  mem_or_neg_mem := mem_or_neg_mem (C := AddGroupCone.nonneg T)
+
+end RingCone
+
+variable {S R : Type*} [Ring R] [SetLike S R] (C : S)
+
+/-- Construct a partially ordered ring by designating a cone in a ring.
+Warning: using this def as a constructor in an instance can lead to diamonds
+due to non-customisable field: `lt`. -/
+@[reducible] def OrderedRing.mkOfCone [RingConeClass S R] : OrderedRing R where
+  __ := ‹Ring R›
+  __ := OrderedAddCommGroup.mkOfCone C
+  zero_le_one := show _ ∈ C by simpa using one_mem C
+  mul_nonneg x y xnn ynn := show _ ∈ C by simpa using mul_mem xnn ynn
+
+/-- Construct a linearly ordered domain by designating a maximal cone in a domain.
+Warning: using this def as a constructor in an instance can lead to diamonds
+due to non-customisable fields: `lt`, `decidableLT`, `decidableEq`, `compare`. -/
+@[reducible] def LinearOrderedRing.mkOfCone
+    [IsDomain R] [RingConeClass S R] [IsMaxCone C]
+    (dec : DecidablePred (· ∈ C)) : LinearOrderedRing R where
+  __ := OrderedRing.mkOfCone C
+  __ := OrderedRing.toStrictOrderedRing R
+  le_total a b := by simpa using mem_or_neg_mem (b - a)
+  decidableLE a b := dec _
diff --git a/Mathlib/Algebra/Order/Ring/Defs.lean b/Mathlib/Algebra/Order/Ring/Defs.lean
index 7e1dda2f4617c..2a1301cc48fc2 100644
--- a/Mathlib/Algebra/Order/Ring/Defs.lean
+++ b/Mathlib/Algebra/Order/Ring/Defs.lean
@@ -222,6 +222,22 @@ instance (priority := 100) OrderedRing.toOrderedSemiring : OrderedSemiring α :=
     mul_le_mul_of_nonneg_right := fun a b c h hc => by
       simpa only [sub_mul, sub_nonneg] using OrderedRing.mul_nonneg _ _ (sub_nonneg.2 h) hc }
 
+lemma one_add_le_one_sub_mul_one_add (h : a + b + b * c ≤ c) : 1 + a ≤ (1 - b) * (1 + c) := by
+  rw [one_sub_mul, mul_one_add, le_sub_iff_add_le, add_assoc, ← add_assoc a]
+  gcongr
+
+lemma one_add_le_one_add_mul_one_sub (h : a + c + b * c ≤ b) : 1 + a ≤ (1 + b) * (1 - c) := by
+  rw [mul_one_sub, one_add_mul, le_sub_iff_add_le, add_assoc, ← add_assoc a]
+  gcongr
+
+lemma one_sub_le_one_sub_mul_one_add (h : b + b * c ≤ a + c) : 1 - a ≤ (1 - b) * (1 + c) := by
+  rw [one_sub_mul, mul_one_add, sub_le_sub_iff, add_assoc, add_comm c]
+  gcongr
+
+lemma one_sub_le_one_add_mul_one_sub (h : c + b * c ≤ a + b) : 1 - a ≤ (1 + b) * (1 - c) := by
+  rw [mul_one_sub, one_add_mul, sub_le_sub_iff, add_assoc, add_comm b]
+  gcongr
+
 end OrderedRing
 
 section OrderedCommRing
diff --git a/Mathlib/Algebra/Order/Ring/Nat.lean b/Mathlib/Algebra/Order/Ring/Nat.lean
index 99af13373b1ec..3b3be2c922c0a 100644
--- a/Mathlib/Algebra/Order/Ring/Nat.lean
+++ b/Mathlib/Algebra/Order/Ring/Nat.lean
@@ -8,7 +8,6 @@ import Mathlib.Algebra.Order.GroupWithZero.Canonical
 import Mathlib.Algebra.Order.Ring.Canonical
 import Mathlib.Algebra.Ring.Nat
 import Mathlib.Data.Set.Basic
-import Mathlib.Init.Data.Nat.Lemmas
 
 /-!
 # The natural numbers form an ordered semiring
@@ -41,7 +40,7 @@ instance instCanonicallyOrderedCommSemiring : CanonicallyOrderedCommSemiring ℕ
   __ := instLinearOrderedCommSemiring
   exists_add_of_le h := (Nat.le.dest h).imp fun _ => Eq.symm
   le_self_add := Nat.le_add_right
-  eq_zero_or_eq_zero_of_mul_eq_zero := Nat.eq_zero_of_mul_eq_zero
+  eq_zero_or_eq_zero_of_mul_eq_zero := Nat.mul_eq_zero.mp
 
 /-!
 ### Extra instances to short-circuit type class resolution
diff --git a/Mathlib/Algebra/Order/Ring/Opposite.lean b/Mathlib/Algebra/Order/Ring/Opposite.lean
new file mode 100644
index 0000000000000..c945c5ec82242
--- /dev/null
+++ b/Mathlib/Algebra/Order/Ring/Opposite.lean
@@ -0,0 +1,50 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.Order.Group.Opposite
+import Mathlib.Algebra.Order.Ring.Defs
+import Mathlib.Algebra.Ring.Opposite
+
+/-!
+# Ordered ring instances for `MulOpposite`/`AddOpposite`
+
+This files transfers ordered (semi)ring instances from `α` to `αᵐᵒᵖ` and `αᵃᵒᵖ`.
+-/
+
+variable {α : Type*}
+
+namespace MulOpposite
+
+instance [OrderedSemiring α] : OrderedSemiring αᵐᵒᵖ where
+  __ := instSemiring
+  __ := instOrderedAddCommMonoid
+  zero_le_one := zero_le_one (α := α)
+  mul_le_mul_of_nonneg_left _ _ _ := mul_le_mul_of_nonneg_right (α := α)
+  mul_le_mul_of_nonneg_right _ _ _ := mul_le_mul_of_nonneg_left (α := α)
+
+instance [OrderedRing α] : OrderedRing αᵐᵒᵖ where
+  __ := instRing
+  __ := instOrderedAddCommGroup
+  __ := instOrderedSemiring
+  mul_nonneg _a _b ha hb := mul_nonneg (α := α) hb ha
+
+end MulOpposite
+
+namespace AddOpposite
+
+instance [OrderedSemiring α] : OrderedSemiring αᵃᵒᵖ where
+  __ := instSemiring
+  __ := instOrderedAddCommMonoid
+  zero_le_one := zero_le_one (α := α)
+  mul_le_mul_of_nonneg_left _ _ _ := mul_le_mul_of_nonneg_left (α := α)
+  mul_le_mul_of_nonneg_right _ _ _ := mul_le_mul_of_nonneg_right (α := α)
+
+instance [OrderedRing α] : OrderedRing αᵐᵒᵖ where
+  __ := instRing
+  __ := instOrderedAddCommGroup
+  __ := instOrderedSemiring
+  mul_nonneg _a _b := mul_nonneg (α := α)
+
+end AddOpposite
diff --git a/Mathlib/Algebra/Order/Ring/Star.lean b/Mathlib/Algebra/Order/Ring/Star.lean
index 5cb62bfb0e87c..06f3ffa17e0cc 100644
--- a/Mathlib/Algebra/Order/Ring/Star.lean
+++ b/Mathlib/Algebra/Order/Ring/Star.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
 import Mathlib.Algebra.Order.Ring.Defs
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 
 /-!
 # Commutative star-ordered rings are ordered rings
diff --git a/Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean b/Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
index 41655cc0c930f..ea150bfb7d6fe 100644
--- a/Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
+++ b/Mathlib/Algebra/Order/Ring/Unbundled/Basic.lean
@@ -199,7 +199,7 @@ theorem mul_le_mul_of_nonpos_of_nonpos' [ExistsAddOfLE α] [PosMulMono α] [MulP
     (hca : c ≤ a) (hdb : d ≤ b) (ha : a ≤ 0) (hd : d ≤ 0) : a * b ≤ c * d :=
   (mul_le_mul_of_nonpos_left hdb ha).trans <| mul_le_mul_of_nonpos_right hca hd
 
-/-- Variant of `mul_le_of_le_one_left` for `b` non-positive instead of non-negative.  -/
+/-- Variant of `mul_le_of_le_one_left` for `b` non-positive instead of non-negative. -/
 theorem le_mul_of_le_one_left [ExistsAddOfLE α] [MulPosMono α]
     [CovariantClass α α (swap (· + ·)) (· ≤ ·)] [ContravariantClass α α (swap (· + ·)) (· ≤ ·)]
     (hb : b ≤ 0) (h : a ≤ 1) : b ≤ a * b := by
@@ -813,10 +813,4 @@ lemma mul_self_le_mul_self_of_le_of_neg_le
       mul_le_mul h₂ h₂ (neg_nonneg.2 h) <| (neg_nonneg.2 h).trans h₂
 
 end LinearOrderedRing
-
-@[deprecated (since := "2023-12-23")] alias zero_le_mul_left := mul_nonneg_iff_of_pos_left
-@[deprecated (since := "2023-12-23")] alias zero_le_mul_right := mul_nonneg_iff_of_pos_right
-@[deprecated (since := "2023-12-23")] alias zero_lt_mul_left := mul_pos_iff_of_pos_left
-@[deprecated (since := "2023-12-23")] alias zero_lt_mul_right := mul_pos_iff_of_pos_right
-
 end OrderedCommRing
diff --git a/Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean b/Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean
index 5c59cce17cb85..b2ad2a602b376 100644
--- a/Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean
+++ b/Mathlib/Algebra/Order/Ring/Unbundled/Rat.lean
@@ -95,7 +95,7 @@ protected theorem le_iff_sub_nonneg (a b : ℚ) : a ≤ b ↔ 0 ≤ b - a :=
         decide_eq_false_iff_not, not_lt, ite_eq_left_iff, not_and, not_le, ← num_nonneg]
       split_ifs with h h'
       · rw [Rat.sub_def]
-        simp only [false_iff, not_le]
+        simp only [false_iff, not_le, reduceCtorEq]
         simp only [normalize_eq]
         apply Int.ediv_neg'
         · rw [sub_neg]
diff --git a/Mathlib/Algebra/Order/Ring/WithTop.lean b/Mathlib/Algebra/Order/Ring/WithTop.lean
index 3ee35faa46055..6ac933856d0ad 100644
--- a/Mathlib/Algebra/Order/Ring/WithTop.lean
+++ b/Mathlib/Algebra/Order/Ring/WithTop.lean
@@ -83,7 +83,7 @@ theorem mul_lt_top [LT α] {a b : WithTop α} (ha : a < ⊤) (hb : b < ⊤) : a
 @[deprecated (since := "2024-08-25")] alias mul_lt_top' := mul_lt_top
 
 instance instNoZeroDivisors [NoZeroDivisors α] : NoZeroDivisors (WithTop α) := by
-  refine ⟨fun h₁ => Decidable.by_contradiction fun h₂ => ?_⟩
+  refine ⟨fun h₁ => Decidable.byContradiction fun h₂ => ?_⟩
   rw [mul_def, if_neg h₂] at h₁
   rcases Option.mem_map₂_iff.1 h₁ with ⟨a, b, (rfl : _ = _), (rfl : _ = _), hab⟩
   exact h₂ ((eq_zero_or_eq_zero_of_mul_eq_zero hab).imp (congr_arg some) (congr_arg some))
diff --git a/Mathlib/Algebra/Star/Order.lean b/Mathlib/Algebra/Order/Star/Basic.lean
similarity index 93%
rename from Mathlib/Algebra/Star/Order.lean
rename to Mathlib/Algebra/Order/Star/Basic.lean
index 3d1200121de45..b07145ac25911 100644
--- a/Mathlib/Algebra/Star/Order.lean
+++ b/Mathlib/Algebra/Order/Star/Basic.lean
@@ -1,12 +1,13 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Submonoid.Operations
 import Mathlib.Algebra.Star.SelfAdjoint
 import Mathlib.Algebra.Star.StarRingHom
 import Mathlib.Algebra.Regular.Basic
+import Mathlib.Tactic.ContinuousFunctionalCalculus
 
 /-! # Star ordered rings
 
@@ -137,12 +138,26 @@ section NonUnitalSemiring
 
 variable [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R]
 
+lemma IsSelfAdjoint.mono {x y : R} (h : x ≤ y) (hx : IsSelfAdjoint x) : IsSelfAdjoint y := by
+  rw [StarOrderedRing.le_iff] at h
+  obtain ⟨d, hd, rfl⟩ := h
+  rw [IsSelfAdjoint, star_add, hx.star_eq]
+  congr
+  refine AddMonoidHom.eqOn_closureM (f := starAddEquiv (R := R)) (g := .id R) ?_ hd
+  rintro - ⟨s, rfl⟩
+  simp
+
+@[aesop 10% apply]
+lemma IsSelfAdjoint.of_nonneg {x : R} (hx : 0 ≤ x) : IsSelfAdjoint x :=
+  .mono hx <| .zero R
+
 theorem star_mul_self_nonneg (r : R) : 0 ≤ star r * r :=
   StarOrderedRing.nonneg_iff.mpr <| AddSubmonoid.subset_closure ⟨r, rfl⟩
 
 theorem mul_star_self_nonneg (r : R) : 0 ≤ r * star r := by
   simpa only [star_star] using star_mul_self_nonneg (star r)
 
+@[aesop safe apply]
 theorem conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c := by
   rw [StarOrderedRing.nonneg_iff] at ha
   refine AddSubmonoid.closure_induction ha (fun x hx => ?_)
@@ -155,9 +170,19 @@ theorem conjugate_nonneg {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ star c * a * c :
       _ ≤ star c * x * c + star c * y * c := add_le_add_left hy _
       _ ≤ _ := by rw [mul_add, add_mul]
 
+@[aesop safe apply]
 theorem conjugate_nonneg' {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ c * a * star c := by
   simpa only [star_star] using conjugate_nonneg ha (star c)
 
+@[aesop 90% apply (rule_sets := [CStarAlgebra])]
+protected theorem IsSelfAdjoint.conjugate_nonneg {a : R} (ha : 0 ≤ a) {c : R}
+    (hc : IsSelfAdjoint c) : 0 ≤ c * a * c := by
+  nth_rewrite 2 [← hc]; exact conjugate_nonneg' ha c
+
+theorem conjugate_nonneg_of_nonneg {a : R} (ha : 0 ≤ a) {c : R} (hc : 0 ≤ c) :
+    0 ≤ c * a * c :=
+  IsSelfAdjoint.of_nonneg hc |>.conjugate_nonneg ha
+
 theorem conjugate_le_conjugate {a b : R} (hab : a ≤ b) (c : R) :
     star c * a * c ≤ star c * b * c := by
   rw [StarOrderedRing.le_iff] at hab ⊢
@@ -169,6 +194,14 @@ theorem conjugate_le_conjugate' {a b : R} (hab : a ≤ b) (c : R) :
     c * a * star c ≤ c * b * star c := by
   simpa only [star_star] using conjugate_le_conjugate hab (star c)
 
+protected theorem IsSelfAdjoint.conjugate_le_conjugate {a b : R} (hab : a ≤ b) {c : R}
+    (hc : IsSelfAdjoint c) : c * a * c ≤ c * b * c := by
+  simpa only [hc.star_eq] using conjugate_le_conjugate hab c
+
+theorem conjugate_le_conjugate_of_nonneg {a b : R} (hab : a ≤ b) {c : R} (hc : 0 ≤ c) :
+    c * a * c ≤ c * b * c :=
+  IsSelfAdjoint.of_nonneg hc |>.conjugate_le_conjugate hab
+
 @[simp]
 lemma star_le_star_iff {x y : R} : star x ≤ star y ↔ x ≤ y := by
   suffices ∀ x y, x ≤ y → star x ≤ star y from
@@ -207,19 +240,6 @@ lemma star_pos_iff {x : R} : 0 < star x ↔ 0 < x := by
 lemma star_neg_iff {x : R} : star x < 0 ↔ x < 0 := by
   simpa using star_lt_star_iff (x := x) (y := 0)
 
-lemma IsSelfAdjoint.mono {x y : R} (h : x ≤ y) (hx : IsSelfAdjoint x) : IsSelfAdjoint y := by
-  rw [StarOrderedRing.le_iff] at h
-  obtain ⟨d, hd, rfl⟩ := h
-  rw [IsSelfAdjoint, star_add, hx.star_eq]
-  congr
-  refine AddMonoidHom.eqOn_closureM (f := starAddEquiv (R := R)) (g := .id R) ?_ hd
-  rintro - ⟨s, rfl⟩
-  simp
-
-@[aesop 10% apply]
-lemma IsSelfAdjoint.of_nonneg {x : R} (hx : 0 ≤ x) : IsSelfAdjoint x :=
-  .mono hx <| .zero R
-
 theorem conjugate_lt_conjugate {a b : R} (hab : a < b) {c : R} (hc : IsRegular c) :
     star c * a * c < star c * b * c := by
   rw [(conjugate_le_conjugate hab.le _).lt_iff_ne, hc.right.ne_iff, hc.star.left.ne_iff]
@@ -294,7 +314,7 @@ lemma StarModule.smul_lt_smul_of_pos {a b : A} {c : R} (hab : a < b) (hc : 0 < c
         obtain ⟨y, hy⟩ := hx
         apply AddSubmonoid.subset_closure
         refine ⟨z • y, ?_⟩
-        simp only [star_smul, smul_mul_smul, hz, hy]
+        simp only [star_smul, smul_mul_smul_comm, hz, hy]
       case zeroc => simpa only [zero_smul] using zero_mem _
       case addc => exact fun c' d ↦ by simpa only [add_smul] using add_mem
     case zero => simpa only [smul_zero] using zero_mem _
diff --git a/Mathlib/Algebra/Order/Star/Conjneg.lean b/Mathlib/Algebra/Order/Star/Conjneg.lean
new file mode 100644
index 0000000000000..e3a93e2c3e500
--- /dev/null
+++ b/Mathlib/Algebra/Order/Star/Conjneg.lean
@@ -0,0 +1,38 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.Order.Pi
+import Mathlib.Algebra.Order.Star.Basic
+import Mathlib.Algebra.Star.Conjneg
+
+/-!
+# Order properties of conjugation-negation
+-/
+
+open scoped ComplexConjugate
+
+variable {G R : Type*} [AddGroup G]
+
+section OrderedCommSemiring
+variable [OrderedCommSemiring R] [StarRing R] [StarOrderedRing R] {f : G → R}
+
+@[simp] lemma conjneg_nonneg : 0 ≤ conjneg f ↔ 0 ≤ f :=
+  (Equiv.neg _).forall_congr' <| by simp [starRingEnd_apply]
+
+@[simp] lemma conjneg_pos : 0 < conjneg f ↔ 0 < f := by
+  simp_rw [lt_iff_le_and_ne, ne_comm, conjneg_nonneg, conjneg_ne_zero]
+
+end OrderedCommSemiring
+
+section OrderedCommRing
+variable [OrderedCommRing R] [StarRing R] [StarOrderedRing R] {f : G → R}
+
+@[simp] lemma conjneg_nonpos : conjneg f ≤ 0 ↔ f ≤ 0 := by
+  simp_rw [← neg_nonneg, ← conjneg_neg, conjneg_nonneg]
+
+@[simp] lemma conjneg_neg' : conjneg f < 0 ↔ f < 0 := by
+  simp_rw [← neg_pos, ← conjneg_neg, conjneg_pos]
+
+end OrderedCommRing
diff --git a/Mathlib/Algebra/Order/Sub/Defs.lean b/Mathlib/Algebra/Order/Sub/Defs.lean
index eff553b812467..9966239cd270f 100644
--- a/Mathlib/Algebra/Order/Sub/Defs.lean
+++ b/Mathlib/Algebra/Order/Sub/Defs.lean
@@ -41,7 +41,7 @@ TODO: generalize `Nat.le_of_le_of_sub_le_sub_right`, `Nat.sub_le_sub_right_iff`,
 -/
 
 
-variable {α β : Type*}
+variable {α : Type*}
 
 /-- `OrderedSub α` means that `α` has a subtraction characterized by `a - b ≤ c ↔ a ≤ c + b`.
 In other words, `a - b` is the least `c` such that `a ≤ b + c`.
@@ -60,7 +60,7 @@ theorem tsub_le_iff_right [LE α] [Add α] [Sub α] [OrderedSub α] {a b c : α}
     a - b ≤ c ↔ a ≤ c + b :=
   OrderedSub.tsub_le_iff_right a b c
 
-variable [Preorder α] [Add α] [Sub α] [OrderedSub α] {a b c d : α}
+variable [Preorder α] [Add α] [Sub α] [OrderedSub α] {a b : α}
 
 /-- See `add_tsub_cancel_right` for the equality if `ContravariantClass α α (+) (≤)`. -/
 theorem add_tsub_le_right : a + b - b ≤ a :=
@@ -83,7 +83,7 @@ variable [Preorder α]
 section AddCommSemigroup
 
 variable [AddCommSemigroup α] [Sub α] [OrderedSub α] {a b c d : α}
-/- TODO: Most results can be generalized to [Add α] [IsSymmOp α α (· + ·)] -/
+/- TODO: Most results can be generalized to [Add α] [@Std.Commutative α (· + ·)] -/
 
 theorem tsub_le_iff_left : a - b ≤ c ↔ a ≤ b + c := by rw [tsub_le_iff_right, add_comm]
 
@@ -210,7 +210,7 @@ end Contra
 
 end AddCommSemigroup
 
-variable [AddCommMonoid α] [Sub α] [OrderedSub α] {a b c d : α}
+variable [AddCommMonoid α] [Sub α] [OrderedSub α] {a b : α}
 
 theorem tsub_nonpos : a - b ≤ 0 ↔ a ≤ b := by rw [tsub_le_iff_left, add_zero]
 
@@ -243,17 +243,39 @@ theorem tsub_right_comm : a - b - c = a - c - b := by
 
 namespace AddLECancellable
 
+/-- See `AddLECancellable.tsub_eq_of_eq_add'` for a version assuming that `a = c + b` itself is
+cancellable rather than `b`. -/
 protected theorem tsub_eq_of_eq_add (hb : AddLECancellable b) (h : a = c + b) : a - b = c :=
   le_antisymm (tsub_le_iff_right.mpr h.le) <| by
     rw [h]
     exact hb.le_add_tsub
 
+/-- Weaker version of `AddLECancellable.tsub_eq_of_eq_add` assuming that `a = c + b` itself is
+cancellable rather than `b`. -/
+protected lemma tsub_eq_of_eq_add' [CovariantClass α α (· + ·) (· ≤ ·)] (ha : AddLECancellable a)
+    (h : a = c + b) : a - b = c := (h ▸ ha).of_add_right.tsub_eq_of_eq_add h
+
+/-- See `AddLECancellable.eq_tsub_of_add_eq'` for a version assuming that `b = a + c` itself is
+cancellable rather than `c`. -/
 protected theorem eq_tsub_of_add_eq (hc : AddLECancellable c) (h : a + c = b) : a = b - c :=
   (hc.tsub_eq_of_eq_add h.symm).symm
 
+/-- Weaker version of `AddLECancellable.eq_tsub_of_add_eq` assuming that `b = a + c` itself is
+cancellable rather than `c`. -/
+protected lemma eq_tsub_of_add_eq' [CovariantClass α α (· + ·) (· ≤ ·)] (hb : AddLECancellable b)
+    (h : a + c = b) : a = b - c := (hb.tsub_eq_of_eq_add' h.symm).symm
+
+/-- See `AddLECancellable.tsub_eq_of_eq_add_rev'` for a version assuming that `a = b + c` itself is
+cancellable rather than `b`. -/
 protected theorem tsub_eq_of_eq_add_rev (hb : AddLECancellable b) (h : a = b + c) : a - b = c :=
   hb.tsub_eq_of_eq_add <| by rw [add_comm, h]
 
+/-- Weaker version of `AddLECancellable.tsub_eq_of_eq_add_rev` assuming that `a = b + c` itself is
+cancellable rather than `b`. -/
+protected lemma tsub_eq_of_eq_add_rev' [CovariantClass α α (· + ·) (· ≤ ·)]
+    (ha : AddLECancellable a) (h : a = b + c) : a - b = c :=
+  ha.tsub_eq_of_eq_add' <| by rw [add_comm, h]
+
 @[simp]
 protected theorem add_tsub_cancel_right (hb : AddLECancellable b) : a + b - b = a :=
   hb.tsub_eq_of_eq_add <| by rw [add_comm]
@@ -353,7 +375,7 @@ end OrderedAddCommSemigroup
 
 section LinearOrder
 
-variable {a b c d : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α]
+variable {a b c : α} [LinearOrder α] [AddCommSemigroup α] [Sub α] [OrderedSub α]
 
 /-- See `lt_of_tsub_lt_tsub_right_of_le` for a weaker statement in a partial order. -/
 theorem lt_of_tsub_lt_tsub_right (h : a - c < b - c) : a < b :=
diff --git a/Mathlib/Algebra/Order/Sub/Prod.lean b/Mathlib/Algebra/Order/Sub/Prod.lean
index de620a351ddd8..9dcb9c7e845d2 100644
--- a/Mathlib/Algebra/Order/Sub/Prod.lean
+++ b/Mathlib/Algebra/Order/Sub/Prod.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Prod
 import Mathlib.Algebra.Order.Sub.Defs
diff --git a/Mathlib/Algebra/Order/Sub/Unbundled/Hom.lean b/Mathlib/Algebra/Order/Sub/Unbundled/Hom.lean
index 2bf9585fa56a6..fedfe17b0c39c 100644
--- a/Mathlib/Algebra/Order/Sub/Unbundled/Hom.lean
+++ b/Mathlib/Algebra/Order/Sub/Unbundled/Hom.lean
@@ -16,7 +16,7 @@ variable {α β : Type*}
 
 section Add
 
-variable [Preorder α] [Add α] [Sub α] [OrderedSub α] {a b c d : α}
+variable [Preorder α] [Add α] [Sub α] [OrderedSub α]
 
 theorem AddHom.le_map_tsub [Preorder β] [Add β] [Sub β] [OrderedSub β] (f : AddHom α β)
     (hf : Monotone f) (a b : α) : f a - f b ≤ f (a - b) := by
@@ -49,7 +49,7 @@ theorem OrderIso.map_tsub {M N : Type*} [Preorder M] [Add M] [Sub M] [OrderedSub
 section Preorder
 
 variable [Preorder α]
-variable [AddCommMonoid α] [Sub α] [OrderedSub α] {a b c d : α}
+variable [AddCommMonoid α] [Sub α] [OrderedSub α]
 
 theorem AddMonoidHom.le_map_tsub [Preorder β] [AddCommMonoid β] [Sub β] [OrderedSub β] (f : α →+ β)
     (hf : Monotone f) (a b : α) : f a - f b ≤ f (a - b) :=
diff --git a/Mathlib/Algebra/Order/SuccPred.lean b/Mathlib/Algebra/Order/SuccPred.lean
new file mode 100644
index 0000000000000..9ef68870ecd16
--- /dev/null
+++ b/Mathlib/Algebra/Order/SuccPred.lean
@@ -0,0 +1,185 @@
+/-
+Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios, Yaël Dillies
+-/
+import Mathlib.Algebra.Group.Basic
+import Mathlib.Algebra.Order.ZeroLEOne
+import Mathlib.Data.Int.Cast.Defs
+import Mathlib.Order.SuccPred.Basic
+
+/-!
+# Interaction between successors and arithmetic
+
+We define the `SuccAddOrder` and `PredSubOrder` typeclasses, for orders satisfying `succ x = x + 1`
+and `pred x = x - 1` respectively. This allows us to transfer the API for successors and
+predecessors into these common arithmetical forms.
+
+## Todo
+
+In the future, we will make `x + 1` and `x - 1` the `simp`-normal forms for `succ x` and `pred x`
+respectively. This will require a refactor of `Ordinal` first, as the `simp`-normal form is
+currently set the other way around.
+-/
+
+/-- A typeclass for `succ x = x + 1`. -/
+class SuccAddOrder (α : Type*) [Preorder α] [Add α] [One α] extends SuccOrder α where
+  succ_eq_add_one (x : α) : succ x = x + 1
+
+/-- A typeclass for `pred x = x - 1`. -/
+class PredSubOrder (α : Type*) [Preorder α] [Sub α] [One α] extends PredOrder α where
+  pred_eq_sub_one (x : α) : pred x = x - 1
+
+variable {α : Type*} {x y : α}
+
+namespace Order
+
+section Preorder
+
+variable [Preorder α]
+
+section Add
+
+variable [Add α] [One α] [SuccAddOrder α]
+
+theorem succ_eq_add_one (x : α) : succ x = x + 1 :=
+  SuccAddOrder.succ_eq_add_one x
+
+theorem add_one_le_of_lt (h : x < y) : x + 1 ≤ y := by
+  rw [← succ_eq_add_one]
+  exact succ_le_of_lt h
+
+theorem add_one_le_iff_of_not_isMax (hx : ¬ IsMax x) : x + 1 ≤ y ↔ x < y := by
+  rw [← succ_eq_add_one, succ_le_iff_of_not_isMax hx]
+
+theorem add_one_le_iff [NoMaxOrder α] : x + 1 ≤ y ↔ x < y :=
+  add_one_le_iff_of_not_isMax (not_isMax x)
+
+@[simp]
+theorem wcovBy_add_one (x : α) : x ⩿ x + 1 := by
+  rw [← succ_eq_add_one]
+  exact wcovBy_succ x
+
+@[simp]
+theorem covBy_add_one [NoMaxOrder α] (x : α) : x ⋖ x + 1 := by
+  rw [← succ_eq_add_one]
+  exact covBy_succ x
+
+end Add
+
+section Sub
+
+variable [Sub α] [One α] [PredSubOrder α]
+
+theorem pred_eq_sub_one (x : α) : pred x = x - 1 :=
+  PredSubOrder.pred_eq_sub_one x
+
+theorem le_sub_one_of_lt (h : x < y) : x ≤ y - 1 := by
+  rw [← pred_eq_sub_one]
+  exact le_pred_of_lt h
+
+theorem le_sub_one_iff_of_not_isMin (hy : ¬ IsMin y) : x ≤ y - 1 ↔ x < y := by
+  rw [← pred_eq_sub_one, le_pred_iff_of_not_isMin hy]
+
+theorem le_sub_one_iff [NoMinOrder α] : x ≤ y - 1 ↔ x < y :=
+  le_sub_one_iff_of_not_isMin (not_isMin y)
+
+@[simp]
+theorem sub_one_wcovBy (x : α) : x - 1 ⩿ x := by
+  rw [← pred_eq_sub_one]
+  exact pred_wcovBy x
+
+@[simp]
+theorem sub_one_covBy [NoMinOrder α] (x : α) : x - 1 ⋖ x := by
+  rw [← pred_eq_sub_one]
+  exact pred_covBy x
+
+end Sub
+
+@[simp]
+theorem succ_iterate [AddMonoidWithOne α] [SuccAddOrder α] (x : α) (n : ℕ) :
+    succ^[n] x = x + n := by
+  induction n with
+  | zero =>
+    rw [Function.iterate_zero_apply, Nat.cast_zero, add_zero]
+  | succ n IH =>
+    rw [Function.iterate_succ_apply', IH, Nat.cast_add, succ_eq_add_one, Nat.cast_one, add_assoc]
+
+@[simp]
+theorem pred_iterate [AddCommGroupWithOne α] [PredSubOrder α] (x : α) (n : ℕ) :
+    pred^[n] x = x - n := by
+  induction n with
+  | zero =>
+    rw [Function.iterate_zero_apply, Nat.cast_zero, sub_zero]
+  | succ n IH =>
+    rw [Function.iterate_succ_apply', IH, Nat.cast_add, pred_eq_sub_one, Nat.cast_one, sub_sub]
+
+end Preorder
+
+section PartialOrder
+
+variable [PartialOrder α]
+
+theorem not_isMax_zero [Zero α] [One α] [ZeroLEOneClass α] [NeZero (1 : α)] : ¬ IsMax (0 : α) := by
+  rw [not_isMax_iff]
+  exact ⟨1, one_pos⟩
+
+theorem one_le_iff_pos [AddMonoidWithOne α] [ZeroLEOneClass α] [NeZero (1 : α)]
+    [SuccAddOrder α] : 1 ≤ x ↔ 0 < x := by
+  rw [← succ_le_iff_of_not_isMax not_isMax_zero, succ_eq_add_one, zero_add]
+
+theorem covBy_iff_add_one_eq [Add α] [One α] [SuccAddOrder α] [NoMaxOrder α] :
+    x ⋖ y ↔ x + 1 = y := by
+  rw [← succ_eq_add_one]
+  exact succ_eq_iff_covBy.symm
+
+theorem covBy_iff_sub_one_eq [Sub α] [One α] [PredSubOrder α] [NoMinOrder α] :
+    x ⋖ y ↔ y - 1 = x := by
+  rw [← pred_eq_sub_one]
+  exact pred_eq_iff_covBy.symm
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α]
+
+section Add
+
+variable [Add α] [One α] [SuccAddOrder α]
+
+theorem le_of_lt_add_one (h : x < y + 1) : x ≤ y := by
+  rw [← succ_eq_add_one] at h
+  exact le_of_lt_succ h
+
+theorem lt_add_one_iff_of_not_isMax (hy : ¬ IsMax y) : x < y + 1 ↔ x ≤ y := by
+  rw [← succ_eq_add_one, lt_succ_iff_of_not_isMax hy]
+
+theorem lt_add_one_iff [NoMaxOrder α] : x < y + 1 ↔ x ≤ y :=
+  lt_add_one_iff_of_not_isMax (not_isMax y)
+
+end Add
+
+section Sub
+
+variable [Sub α] [One α] [PredSubOrder α]
+
+theorem le_of_sub_one_lt (h : x - 1 < y) : x ≤ y := by
+  rw [← pred_eq_sub_one] at h
+  exact le_of_pred_lt h
+
+theorem sub_one_lt_iff_of_not_isMin (hx : ¬ IsMin x) : x - 1 < y ↔ x ≤ y := by
+  rw [← pred_eq_sub_one, pred_lt_iff_of_not_isMin hx]
+
+theorem sub_one_lt_iff [NoMinOrder α] : x - 1 < y ↔ x ≤ y :=
+  sub_one_lt_iff_of_not_isMin (not_isMin x)
+
+end Sub
+
+theorem lt_one_iff_nonpos [AddMonoidWithOne α] [ZeroLEOneClass α] [NeZero (1 : α)]
+    [SuccAddOrder α] : x < 1 ↔ x ≤ 0 := by
+  rw [← lt_succ_iff_of_not_isMax not_isMax_zero, succ_eq_add_one, zero_add]
+
+end LinearOrder
+
+end Order
diff --git a/Mathlib/Algebra/Order/SuccPred/TypeTags.lean b/Mathlib/Algebra/Order/SuccPred/TypeTags.lean
new file mode 100644
index 0000000000000..4293cd851694a
--- /dev/null
+++ b/Mathlib/Algebra/Order/SuccPred/TypeTags.lean
@@ -0,0 +1,50 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+import Mathlib.Algebra.Order.Monoid.TypeTags
+import Mathlib.Order.SuccPred.Archimedean
+
+/-!
+# Successor and predecessor on type tags
+
+This file declates successor and predecessor orders on type tags.
+
+-/
+
+variable {X : Type*}
+
+instance [Preorder X] [h : SuccOrder X] : SuccOrder (Multiplicative X) := h
+instance [Preorder X] [h : SuccOrder X] : SuccOrder (Additive X) := h
+
+instance [Preorder X] [h : PredOrder X] : PredOrder (Multiplicative X) := h
+instance [Preorder X] [h : PredOrder X] : PredOrder (Additive X) := h
+
+instance [Preorder X] [SuccOrder X] [h : IsSuccArchimedean X] :
+  IsSuccArchimedean (Multiplicative X) := h
+instance [Preorder X] [SuccOrder X] [h : IsSuccArchimedean X] :
+  IsSuccArchimedean (Additive X) := h
+
+instance [Preorder X] [PredOrder X] [h : IsPredArchimedean X] :
+  IsPredArchimedean (Multiplicative X) := h
+instance [Preorder X] [PredOrder X] [h : IsPredArchimedean X] :
+  IsPredArchimedean (Additive X) := h
+
+namespace Order
+
+open Additive Multiplicative
+
+@[simp] lemma succ_ofMul [Preorder X] [SuccOrder X] (x : X) : succ (ofMul x) = ofMul (succ x) := rfl
+@[simp] lemma succ_toMul [Preorder X] [SuccOrder X] (x : X) : succ (toMul x) = toMul (succ x) := rfl
+
+@[simp] lemma succ_ofAdd [Preorder X] [SuccOrder X] (x : X) : succ (ofAdd x) = ofAdd (succ x) := rfl
+@[simp] lemma succ_toAdd [Preorder X] [SuccOrder X] (x : X) : succ (toAdd x) = toAdd (succ x) := rfl
+
+@[simp] lemma pred_ofMul [Preorder X] [PredOrder X] (x : X) : pred (ofMul x) = ofMul (pred x) := rfl
+@[simp] lemma pred_toMul [Preorder X] [PredOrder X] (x : X) : pred (toMul x) = toMul (pred x) := rfl
+
+@[simp] lemma pred_ofAdd [Preorder X] [PredOrder X] (x : X) : pred (ofAdd x) = ofAdd (pred x) := rfl
+@[simp] lemma pred_toAdd [Preorder X] [PredOrder X] (x : X) : pred (toAdd x) = toAdd (pred x) := rfl
+
+end Order
diff --git a/Mathlib/Algebra/Order/ToIntervalMod.lean b/Mathlib/Algebra/Order/ToIntervalMod.lean
index 938027eaa347f..260764b5fe69c 100644
--- a/Mathlib/Algebra/Order/ToIntervalMod.lean
+++ b/Mathlib/Algebra/Order/ToIntervalMod.lean
@@ -517,23 +517,23 @@ theorem tfae_modEq :
       [a ≡ b [PMOD p], ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
         toIcoMod hp a b + p = toIocMod hp a b] := by
   rw [modEq_iff_toIcoMod_eq_left hp]
-  tfae_have 3 → 2
-  · rw [← not_exists, not_imp_not]
+  tfae_have 3 → 2 := by
+    rw [← not_exists, not_imp_not]
     exact fun ⟨i, hi⟩ =>
       ((toIcoMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ico_self hi, i, (sub_add_cancel b _).symm⟩).trans
         ((toIocMod_eq_iff hp).2 ⟨Set.Ioo_subset_Ioc_self hi, i, (sub_add_cancel b _).symm⟩).symm
   tfae_have 4 → 3
-  · intro h
+  | h => by
     rw [← h, Ne, eq_comm, add_right_eq_self]
     exact hp.ne'
   tfae_have 1 → 4
-  · intro h
+  | h => by
     rw [h, eq_comm, toIocMod_eq_iff, Set.right_mem_Ioc]
     refine ⟨lt_add_of_pos_right a hp, toIcoDiv hp a b - 1, ?_⟩
     rw [sub_one_zsmul, add_add_add_comm, add_neg_cancel, add_zero]
     conv_lhs => rw [← toIcoMod_add_toIcoDiv_zsmul hp a b, h]
-  tfae_have 2 → 1
-  · rw [← not_exists, not_imp_comm]
+  tfae_have 2 → 1 := by
+    rw [← not_exists, not_imp_comm]
     have h' := toIcoMod_mem_Ico hp a b
     exact fun h => ⟨_, h'.1.lt_of_ne' h, h'.2⟩
   tfae_finish
@@ -557,12 +557,12 @@ theorem not_modEq_iff_toIcoMod_eq_toIocMod : ¬a ≡ b [PMOD p] ↔ toIcoMod hp
 theorem not_modEq_iff_toIcoDiv_eq_toIocDiv :
     ¬a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b := by
   rw [not_modEq_iff_toIcoMod_eq_toIocMod hp, toIcoMod, toIocMod, sub_right_inj,
-    (zsmul_strictMono_left hp).injective.eq_iff]
+    zsmul_left_inj hp]
 
 theorem modEq_iff_toIcoDiv_eq_toIocDiv_add_one :
     a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1 := by
   rw [modEq_iff_toIcoMod_add_period_eq_toIocMod hp, toIcoMod, toIocMod, ← eq_sub_iff_add_eq,
-    sub_sub, sub_right_inj, ← add_one_zsmul, (zsmul_strictMono_left hp).injective.eq_iff]
+    sub_sub, sub_right_inj, ← add_one_zsmul, zsmul_left_inj hp]
 
 end AddCommGroup
 
@@ -595,7 +595,7 @@ theorem toIcoMod_le_toIocMod (a b : α) : toIcoMod hp a b ≤ toIocMod hp a b :=
 
 theorem toIocMod_le_toIcoMod_add (a b : α) : toIocMod hp a b ≤ toIcoMod hp a b + p := by
   rw [toIcoMod, toIocMod, sub_add, sub_le_sub_iff_left, sub_le_iff_le_add, ← add_one_zsmul,
-    (zsmul_strictMono_left hp).le_iff_le]
+    (zsmul_left_strictMono hp).le_iff_le]
   apply (toIocDiv_wcovBy_toIcoDiv _ _ _).le_succ
 
 end IcoIoc
diff --git a/Mathlib/Algebra/Order/ZeroLEOne.lean b/Mathlib/Algebra/Order/ZeroLEOne.lean
index ecd0e356d73ad..538c8f6985b42 100644
--- a/Mathlib/Algebra/Order/ZeroLEOne.lean
+++ b/Mathlib/Algebra/Order/ZeroLEOne.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
 -/
 import Mathlib.Order.Basic
-import Mathlib.Algebra.NeZero
 
 /-!
 # Typeclass expressing `0 ≤ 1`.
diff --git a/Mathlib/Algebra/Pointwise/Stabilizer.lean b/Mathlib/Algebra/Pointwise/Stabilizer.lean
index deab88e20079c..de0ffac8e167c 100644
--- a/Mathlib/Algebra/Pointwise/Stabilizer.lean
+++ b/Mathlib/Algebra/Pointwise/Stabilizer.lean
@@ -3,7 +3,7 @@ Copyright (c) 2023 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.GroupTheory.QuotientGroup.Basic
 
 /-!
diff --git a/Mathlib/Algebra/Polynomial/AlgebraMap.lean b/Mathlib/Algebra/Polynomial/AlgebraMap.lean
index 488fbdbe25fad..5928ff633fda6 100644
--- a/Mathlib/Algebra/Polynomial/AlgebraMap.lean
+++ b/Mathlib/Algebra/Polynomial/AlgebraMap.lean
@@ -1,11 +1,12 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Algebra.Pi
 import Mathlib.Algebra.Polynomial.Eval
 import Mathlib.RingTheory.Adjoin.Basic
+import Mathlib.Algebra.MonoidAlgebra.Basic
 
 /-!
 # Theory of univariate polynomials
@@ -102,7 +103,7 @@ instance subalgebraNontrivial [Nontrivial A] : Nontrivial (Subalgebra R A[X]) :=
   ⟨⟨⊥, ⊤, by
       rw [Ne, SetLike.ext_iff, not_forall]
       refine ⟨X, ?_⟩
-      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true_iff, Algebra.mem_top,
+      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true, Algebra.mem_top,
         algebraMap_apply, not_forall]
       intro x
       rw [ext_iff, not_forall]
@@ -384,7 +385,7 @@ instance instCommSemiringAdjoinSingleton :
   { mul_comm := fun ⟨p, hp⟩ ⟨q, hq⟩ ↦ by
       obtain ⟨p', rfl⟩ := Algebra.adjoin_singleton_eq_range_aeval R x ▸ hp
       obtain ⟨q', rfl⟩ := Algebra.adjoin_singleton_eq_range_aeval R x ▸ hq
-      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, Submonoid.mk_mul_mk, ← map_mul,
+      simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, MulMemClass.mk_mul_mk, ← map_mul,
         mul_comm p' q'] }
 
 instance instCommRingAdjoinSingleton {R A : Type*} [CommRing R] [Ring A] [Algebra R A] (x : A) :
diff --git a/Mathlib/Algebra/Polynomial/Basic.lean b/Mathlib/Algebra/Polynomial/Basic.lean
index cce2711cbb23d..b1d85a96da4bd 100644
--- a/Mathlib/Algebra/Polynomial/Basic.lean
+++ b/Mathlib/Algebra/Polynomial/Basic.lean
@@ -1,11 +1,11 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.GroupWithZero.Divisibility
-import Mathlib.Algebra.MonoidAlgebra.Basic
 import Mathlib.Data.Finset.Sort
+import Mathlib.Algebra.MonoidAlgebra.Defs
 
 /-!
 # Theory of univariate polynomials
@@ -413,6 +413,10 @@ theorem monomial_injective (n : ℕ) : Function.Injective (monomial n : R → R[
 theorem monomial_eq_zero_iff (t : R) (n : ℕ) : monomial n t = 0 ↔ t = 0 :=
   LinearMap.map_eq_zero_iff _ (Polynomial.monomial_injective n)
 
+theorem monomial_eq_monomial_iff {m n : ℕ} {a b : R} :
+    monomial m a = monomial n b ↔ m = n ∧ a = b ∨ a = 0 ∧ b = 0 := by
+  rw [← toFinsupp_inj, toFinsupp_monomial, toFinsupp_monomial, Finsupp.single_eq_single_iff]
+
 theorem support_add : (p + q).support ⊆ p.support ∪ q.support := by
   simpa [support] using Finsupp.support_add
 
@@ -482,6 +486,10 @@ theorem monomial_one_right_eq_X_pow (n : ℕ) : monomial n (1 : R) = X ^ n := by
 theorem toFinsupp_X : X.toFinsupp = Finsupp.single 1 (1 : R) :=
   rfl
 
+theorem X_ne_C [Nontrivial R] (a : R) : X ≠ C a := by
+  intro he
+  simpa using monomial_eq_monomial_iff.1 he
+
 /-- `X` commutes with everything, even when the coefficients are noncommutative. -/
 theorem X_mul : X * p = p * X := by
   rcases p with ⟨⟩
@@ -571,6 +579,13 @@ theorem toFinsupp_apply (f : R[X]) (i) : f.toFinsupp i = f.coeff i := by cases f
 theorem coeff_monomial : coeff (monomial n a) m = if n = m then a else 0 := by
   simp [coeff, Finsupp.single_apply]
 
+@[simp]
+theorem coeff_monomial_same (n : ℕ) (c : R) : (monomial n c).coeff n = c :=
+  Finsupp.single_eq_same
+
+theorem coeff_monomial_of_ne {m n : ℕ} (c : R) (h : n ≠ m) : (monomial n c).coeff m = 0 :=
+  Finsupp.single_eq_of_ne h
+
 @[simp]
 theorem coeff_zero (n : ℕ) : coeff (0 : R[X]) n = 0 :=
   rfl
@@ -930,7 +945,7 @@ section Update
 
 /-- Replace the coefficient of a `p : R[X]` at a given degree `n : ℕ`
 by a given value `a : R`. If `a = 0`, this is equal to `p.erase n`
-If `p.natDegree < n` and `a ≠ 0`, this increases the degree to `n`.  -/
+If `p.natDegree < n` and `a ≠ 0`, this increases the degree to `n`. -/
 def update (p : R[X]) (n : ℕ) (a : R) : R[X] :=
   Polynomial.ofFinsupp (p.toFinsupp.update n a)
 
diff --git a/Mathlib/Algebra/Polynomial/BigOperators.lean b/Mathlib/Algebra/Polynomial/BigOperators.lean
index ffa4e0a8e9d60..cef12c507490a 100644
--- a/Mathlib/Algebra/Polynomial/BigOperators.lean
+++ b/Mathlib/Algebra/Polynomial/BigOperators.lean
@@ -47,7 +47,7 @@ theorem natDegree_list_sum_le (l : List S[X]) : natDegree l.sum ≤ (l.map natDe
   List.sum_le_foldr_max natDegree (by simp) natDegree_add_le _
 
 theorem natDegree_multiset_sum_le (l : Multiset S[X]) :
-    natDegree l.sum ≤ (l.map natDegree).foldr max max_left_comm 0 :=
+    natDegree l.sum ≤ (l.map natDegree).foldr max 0 :=
   Quotient.inductionOn l (by simpa using natDegree_list_sum_le)
 
 theorem natDegree_sum_le (f : ι → S[X]) :
@@ -68,7 +68,7 @@ theorem degree_list_sum_le (l : List S[X]) : degree l.sum ≤ (l.map natDegree).
     rw [← List.foldr_max_of_ne_nil]
     · congr
     contrapose! h
-    rw [List.map_eq_nil] at h
+    rw [List.map_eq_nil_iff] at h
     simp [h]
 
 theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.prod ≤ (l.map natDegree).sum := by
@@ -186,7 +186,7 @@ theorem natDegree_multiset_prod_of_monic (h : ∀ f ∈ t, Monic f) :
     rw [this]
     simp
   convert prod_replicate (Multiset.card t) (1 : R)
-  · simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and_iff]
+  · simp only [eq_replicate, Multiset.card_map, eq_self_iff_true, true_and]
     rintro i hi
     obtain ⟨i, hi, rfl⟩ := Multiset.mem_map.mp hi
     apply h
diff --git a/Mathlib/Algebra/Polynomial/Coeff.lean b/Mathlib/Algebra/Polynomial/Coeff.lean
index b1cded1bdbe60..a0550adae74d1 100644
--- a/Mathlib/Algebra/Polynomial/Coeff.lean
+++ b/Mathlib/Algebra/Polynomial/Coeff.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.MonoidAlgebra.Support
 import Mathlib.Algebra.Polynomial.Basic
@@ -99,7 +99,7 @@ lemma coeff_list_sum (l : List R[X]) (n : ℕ) :
 
 lemma coeff_list_sum_map {ι : Type*} (l : List ι) (f : ι → R[X]) (n : ℕ) :
     (l.map f).sum.coeff n = (l.map (fun a => (f a).coeff n)).sum := by
-  simp_rw [coeff_list_sum, List.map_map, Function.comp, lcoeff_apply]
+  simp_rw [coeff_list_sum, List.map_map, Function.comp_def, lcoeff_apply]
 
 theorem coeff_sum [Semiring S] (n : ℕ) (f : ℕ → R → S[X]) :
     coeff (p.sum f) n = p.sum fun a b => coeff (f a b) n := by
@@ -220,7 +220,7 @@ theorem card_support_trinomial {k m n : ℕ} (hkm : k < m) (hmn : m < n) {x y z
     (hy : y ≠ 0) (hz : z ≠ 0) : card (support (C x * X ^ k + C y * X ^ m + C z * X ^ n)) = 3 := by
   rw [support_trinomial hkm hmn hx hy hz,
     card_insert_of_not_mem
-      (mt mem_insert.mp (not_or_of_not hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),
+      (mt mem_insert.mp (not_or_intro hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),
     card_insert_of_not_mem (mt mem_singleton.mp hmn.ne), card_singleton]
 
 end Fewnomials
diff --git a/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean b/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean
index 1a1dd67ce6467..8b5296df327a3 100644
--- a/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean
+++ b/Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean
@@ -50,7 +50,7 @@ noncomputable def cardPowDegree : AbsoluteValue Fq[X] ℤ :=
       · rfl
       exact pow_nonneg (Int.ofNat_zero_le _) _
     eq_zero' := fun p =>
-      ite_eq_left_iff.trans <|
+      ite_eq_left_iff.trans
         ⟨fun h => by
           contrapose! h
           exact ⟨h, (pow_pos _).ne'⟩, absurd⟩
@@ -61,7 +61,7 @@ noncomputable def cardPowDegree : AbsoluteValue Fq[X] ℤ :=
       · simp only [hpq, hp, hq, eq_self_iff_true, if_true, if_false]
         exact add_nonneg (pow_pos _).le (pow_pos _).le
       simp only [hpq, hp, hq, if_false]
-      refine le_trans (pow_le_pow_right (by omega) (Polynomial.natDegree_add_le _ _)) ?_
+      refine le_trans (pow_right_mono₀ (by omega) (Polynomial.natDegree_add_le _ _)) ?_
       refine
         le_trans (le_max_iff.mpr ?_)
           (max_le_add_of_nonneg (pow_nonneg (by omega) _) (pow_nonneg (by omega) _))
diff --git a/Mathlib/Algebra/Polynomial/Degree/Definitions.lean b/Mathlib/Algebra/Polynomial/Degree/Definitions.lean
index c1bc0023501f9..3a3fe3f22832f 100644
--- a/Mathlib/Algebra/Polynomial/Degree/Definitions.lean
+++ b/Mathlib/Algebra/Polynomial/Degree/Definitions.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.MonoidAlgebra.Degree
 import Mathlib.Algebra.Polynomial.Coeff
@@ -237,6 +237,12 @@ theorem natDegree_natCast (n : ℕ) : natDegree (n : R[X]) = 0 := by
 @[deprecated (since := "2024-04-17")]
 alias natDegree_nat_cast := natDegree_natCast
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem natDegree_ofNat (n : ℕ) [Nat.AtLeastTwo n] :
+    natDegree (no_index (OfNat.ofNat n : R[X])) = 0 :=
+  natDegree_natCast _
+
 theorem degree_natCast_le (n : ℕ) : degree (n : R[X]) ≤ 0 := degree_le_of_natDegree_le (by simp)
 
 @[deprecated (since := "2024-04-17")]
diff --git a/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean b/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
index 809a37a648c0d..649bab3a13526 100644
--- a/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
+++ b/Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Eval
 
@@ -58,6 +58,13 @@ theorem natDegree_comp_le : natDegree (p.comp q) ≤ natDegree p * natDegree q :
                   mul_le_mul_of_nonneg_right (le_natDegree_of_ne_zero (mem_support_iff.1 hn))
                     (Nat.zero_le _)
 
+theorem natDegree_comp_eq_of_mul_ne_zero (h : p.leadingCoeff * q.leadingCoeff ^ p.natDegree ≠ 0) :
+    natDegree (p.comp q) = natDegree p * natDegree q := by
+  by_cases hq : natDegree q = 0
+  · exact le_antisymm natDegree_comp_le (by simp [hq])
+  apply natDegree_eq_of_le_of_coeff_ne_zero natDegree_comp_le
+  rwa [coeff_comp_degree_mul_degree hq]
+
 theorem degree_pos_of_root {p : R[X]} (hp : p ≠ 0) (h : IsRoot p a) : 0 < degree p :=
   lt_of_not_ge fun hlt => by
     have := eq_C_of_degree_le_zero hlt
@@ -345,9 +352,8 @@ theorem natDegree_comp : natDegree (p.comp q) = natDegree p * natDegree q := by
       natDegree_C, mul_zero]
   · by_cases p0 : p = 0
     · simp only [p0, zero_comp, natDegree_zero, zero_mul]
-    refine le_antisymm natDegree_comp_le (le_natDegree_of_ne_zero ?_)
-    simp only [coeff_comp_degree_mul_degree q0, p0, mul_eq_zero, leadingCoeff_eq_zero, or_self_iff,
-      ne_zero_of_natDegree_gt (Nat.pos_of_ne_zero q0), pow_ne_zero, Ne, not_false_iff]
+    · simp only [Ne, mul_eq_zero, leadingCoeff_eq_zero, p0, natDegree_comp_eq_of_mul_ne_zero,
+        ne_zero_of_natDegree_gt (Nat.pos_of_ne_zero q0), not_false_eq_true, pow_ne_zero, or_self]
 
 @[simp]
 theorem natDegree_iterate_comp (k : ℕ) :
diff --git a/Mathlib/Algebra/Polynomial/DenomsClearable.lean b/Mathlib/Algebra/Polynomial/DenomsClearable.lean
index a9004b5762139..16046a197ca3d 100644
--- a/Mathlib/Algebra/Polynomial/DenomsClearable.lean
+++ b/Mathlib/Algebra/Polynomial/DenomsClearable.lean
@@ -5,6 +5,7 @@ Authors: Damiano Testa
 -/
 import Mathlib.Algebra.Polynomial.EraseLead
 import Mathlib.Algebra.Polynomial.Eval
+import Mathlib.Algebra.Algebra.Basic
 
 /-!
 # Denominators of evaluation of polynomials at ratios
diff --git a/Mathlib/Algebra/Polynomial/Derivative.lean b/Mathlib/Algebra/Polynomial/Derivative.lean
index e4eb94abf7e30..b2767e3897214 100644
--- a/Mathlib/Algebra/Polynomial/Derivative.lean
+++ b/Mathlib/Algebra/Polynomial/Derivative.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.GroupPower.IterateHom
 import Mathlib.Algebra.Polynomial.Eval
@@ -295,7 +295,7 @@ theorem mem_support_derivative [NoZeroSMulDivisors ℕ R] (p : R[X]) (n : ℕ) :
   suffices ¬p.coeff (n + 1) * (n + 1 : ℕ) = 0 ↔ coeff p (n + 1) ≠ 0 by
     simpa only [mem_support_iff, coeff_derivative, Ne, Nat.cast_succ]
   rw [← nsmul_eq_mul', smul_eq_zero]
-  simp only [Nat.succ_ne_zero, false_or_iff]
+  simp only [Nat.succ_ne_zero, false_or]
 
 @[simp]
 theorem degree_derivative_eq [NoZeroSMulDivisors ℕ R] (p : R[X]) (hp : 0 < natDegree p) :
diff --git a/Mathlib/Algebra/Polynomial/Div.lean b/Mathlib/Algebra/Polynomial/Div.lean
index 7533319d50143..7ac2f94a0beb8 100644
--- a/Mathlib/Algebra/Polynomial/Div.lean
+++ b/Mathlib/Algebra/Polynomial/Div.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Inductions
 import Mathlib.Algebra.Polynomial.Monic
@@ -281,7 +281,7 @@ theorem degree_divByMonic_le (p q : R[X]) : degree (p /ₘ q) ≤ degree p :=
         exact WithBot.coe_le_coe.2 (Nat.le_add_left _ _)
       else by
         unfold divByMonic divModByMonicAux
-        simp [dif_pos hq, h, false_and_iff, if_false, degree_zero, bot_le]
+        simp [dif_pos hq, h, if_false, degree_zero, bot_le]
     else (divByMonic_eq_of_not_monic p hq).symm ▸ bot_le
 
 theorem degree_divByMonic_lt (p : R[X]) {q : R[X]} (hq : Monic q) (hp0 : p ≠ 0)
@@ -486,7 +486,8 @@ def rootMultiplicity (a : R) (p : R[X]) : ℕ :=
   if h0 : p = 0 then 0
   else
     let _ : DecidablePred fun n : ℕ => ¬(X - C a) ^ (n + 1) ∣ p := fun n =>
-      @Not.decidable _ (decidableDvdMonic p ((monic_X_sub_C a).pow (n + 1)))
+      have := decidableDvdMonic p ((monic_X_sub_C a).pow (n + 1))
+      inferInstanceAs (Decidable ¬_)
     Nat.find (multiplicity_X_sub_C_finite a h0)
 
 /- Porting note: added the following due to diamond with decidableProp and
@@ -494,7 +495,8 @@ decidableDvdMonic see also [Zulip]
 (https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/non-defeq.20aliased.20instance) -/
 theorem rootMultiplicity_eq_nat_find_of_nonzero [DecidableEq R] {p : R[X]} (p0 : p ≠ 0) {a : R} :
     letI : DecidablePred fun n : ℕ => ¬(X - C a) ^ (n + 1) ∣ p := fun n =>
-      @Not.decidable _ (decidableDvdMonic p ((monic_X_sub_C a).pow (n + 1)))
+      have := decidableDvdMonic p ((monic_X_sub_C a).pow (n + 1))
+      inferInstanceAs (Decidable ¬_)
     rootMultiplicity a p = Nat.find (multiplicity_X_sub_C_finite a p0) := by
   dsimp [rootMultiplicity]
   cases Subsingleton.elim ‹DecidableEq R› (Classical.decEq R)
diff --git a/Mathlib/Algebra/Polynomial/Eval.lean b/Mathlib/Algebra/Polynomial/Eval.lean
index 13fd989d6ed3c..e6b4d34e657aa 100644
--- a/Mathlib/Algebra/Polynomial/Eval.lean
+++ b/Mathlib/Algebra/Polynomial/Eval.lean
@@ -1,10 +1,13 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Degree.Definitions
 import Mathlib.Algebra.Polynomial.Induction
+import Mathlib.Algebra.Ring.Subsemiring.Basic
+import Mathlib.Algebra.Algebra.Defs
+import Mathlib.Algebra.Ring.Subring.Basic
 
 /-!
 # Theory of univariate polynomials
diff --git a/Mathlib/Algebra/Polynomial/Expand.lean b/Mathlib/Algebra/Polynomial/Expand.lean
index e1db94a64141b..ae212d4a08496 100644
--- a/Mathlib/Algebra/Polynomial/Expand.lean
+++ b/Mathlib/Algebra/Polynomial/Expand.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
 import Mathlib.RingTheory.Polynomial.Basic
-import Mathlib.RingTheory.LocalRing.RingHom.Basic
 
 /-!
 # Expand a polynomial by a factor of p, so `∑ aₙ xⁿ` becomes `∑ aₙ xⁿᵖ`.
@@ -269,9 +268,8 @@ section IsDomain
 
 variable (R : Type u) [CommRing R] [IsDomain R]
 
-theorem isLocalRingHom_expand {p : ℕ} (hp : 0 < p) :
-    IsLocalRingHom (↑(expand R p) : R[X] →+* R[X]) := by
-  refine ⟨fun f hf1 => ?_⟩; norm_cast at hf1
+theorem isLocalRingHom_expand {p : ℕ} (hp : 0 < p) : IsLocalRingHom (expand R p) := by
+  refine ⟨fun f hf1 => ?_⟩
   have hf2 := eq_C_of_degree_eq_zero (degree_eq_zero_of_isUnit hf1)
   rw [coeff_expand hp, if_pos (dvd_zero _), p.zero_div] at hf2
   rw [hf2, isUnit_C] at hf1; rw [expand_eq_C hp] at hf2; rwa [hf2, isUnit_C]
@@ -281,7 +279,7 @@ variable {R}
 theorem of_irreducible_expand {p : ℕ} (hp : p ≠ 0) {f : R[X]} (hf : Irreducible (expand R p f)) :
     Irreducible f :=
   let _ := isLocalRingHom_expand R hp.bot_lt
-  of_irreducible_map (↑(expand R p)) hf
+  hf.of_map
 
 theorem of_irreducible_expand_pow {p : ℕ} (hp : p ≠ 0) {f : R[X]} {n : ℕ} :
     Irreducible (expand R (p ^ n) f) → Irreducible f :=
diff --git a/Mathlib/Algebra/Polynomial/FieldDivision.lean b/Mathlib/Algebra/Polynomial/FieldDivision.lean
index 0460864bfceb5..06134da2f5675 100644
--- a/Mathlib/Algebra/Polynomial/FieldDivision.lean
+++ b/Mathlib/Algebra/Polynomial/FieldDivision.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Derivative
 import Mathlib.Algebra.Polynomial.Roots
@@ -334,7 +334,7 @@ theorem mod_eq_self_iff (hq0 : q ≠ 0) : p % q = p ↔ degree p < degree q :=
     rw [mod_def, modByMonic, dif_pos (monic_mul_leadingCoeff_inv hq0)]
     unfold divModByMonicAux
     dsimp
-    simp only [this, false_and_iff, if_false]⟩
+    simp only [this, false_and, if_false]⟩
 
 theorem div_eq_zero_iff (hq0 : q ≠ 0) : p / q = 0 ↔ degree p < degree q :=
   ⟨fun h => by
@@ -382,6 +382,21 @@ theorem map_mod [Field k] (f : R →+* k) : (p % q).map f = p.map f % q.map f :=
   · rw [mod_def, mod_def, leadingCoeff_map f, ← map_inv₀ f, ← map_C f, ← Polynomial.map_mul f,
       map_modByMonic f (monic_mul_leadingCoeff_inv hq0)]
 
+lemma natDegree_mod_lt [Field k] (p : k[X]) {q : k[X]} (hq : q.natDegree ≠ 0) :
+    (p % q).natDegree < q.natDegree := by
+  have hq' : q.leadingCoeff ≠ 0 := by
+    rw [leadingCoeff_ne_zero]
+    contrapose! hq
+    simp [hq]
+  rw [mod_def]
+  refine (natDegree_modByMonic_lt p ?_ ?_).trans_le ?_
+  · refine monic_mul_C_of_leadingCoeff_mul_eq_one ?_
+    rw [mul_inv_eq_one₀ hq']
+  · contrapose! hq
+    rw [← natDegree_mul_C_eq_of_mul_eq_one ((inv_mul_eq_one₀ hq').mpr rfl)]
+    simp [hq]
+  · exact natDegree_mul_C_le q q.leadingCoeff⁻¹
+
 section
 
 open EuclideanDomain
diff --git a/Mathlib/Algebra/Polynomial/GroupRingAction.lean b/Mathlib/Algebra/Polynomial/GroupRingAction.lean
index 30813c8edd473..586acacc30c03 100644
--- a/Mathlib/Algebra/Polynomial/GroupRingAction.lean
+++ b/Mathlib/Algebra/Polynomial/GroupRingAction.lean
@@ -95,7 +95,7 @@ theorem prodXSubSMul.eval (x : R) : (prodXSubSMul G R x).eval x = 0 :=
 
 theorem prodXSubSMul.smul (x : R) (g : G) : g • prodXSubSMul G R x = prodXSubSMul G R x :=
   letI := Classical.decEq R
-  Finset.smul_prod.trans <|
+  Finset.smul_prod'.trans <|
     Fintype.prod_bijective _ (MulAction.bijective g) _ _ fun g' ↦ by
       rw [ofQuotientStabilizer_smul, smul_sub, Polynomial.smul_X, Polynomial.smul_C]
 
diff --git a/Mathlib/Algebra/Polynomial/HasseDeriv.lean b/Mathlib/Algebra/Polynomial/HasseDeriv.lean
index 526b25659bb93..7f0f31d75a921 100644
--- a/Mathlib/Algebra/Polynomial/HasseDeriv.lean
+++ b/Mathlib/Algebra/Polynomial/HasseDeriv.lean
@@ -179,7 +179,7 @@ theorem natDegree_hasseDeriv_le (p : R[X]) (n : ℕ) :
     refine (natDegree_sum_le _ _).trans ?_
     simp_rw [Function.comp, natDegree_monomial]
     rw [Finset.fold_ite, Finset.fold_const]
-    · simp only [ite_self, max_eq_right, zero_le', Finset.fold_max_le, true_and_iff, and_imp,
+    · simp only [ite_self, max_eq_right, zero_le', Finset.fold_max_le, true_and, and_imp,
         tsub_le_iff_right, mem_support_iff, Ne, Finset.mem_filter]
       intro x hx hx'
       have hxp : x ≤ p.natDegree := le_natDegree_of_ne_zero hx
diff --git a/Mathlib/Algebra/Polynomial/Identities.lean b/Mathlib/Algebra/Polynomial/Identities.lean
index 03eec7652a472..d33dfaf9c2923 100644
--- a/Mathlib/Algebra/Polynomial/Identities.lean
+++ b/Mathlib/Algebra/Polynomial/Identities.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Derivative
 import Mathlib.Tactic.LinearCombination
diff --git a/Mathlib/Algebra/Polynomial/Induction.lean b/Mathlib/Algebra/Polynomial/Induction.lean
index 3e3fdb290f72c..a4fc244b867ab 100644
--- a/Mathlib/Algebra/Polynomial/Induction.lean
+++ b/Mathlib/Algebra/Polynomial/Induction.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Basic
 import Mathlib.RingTheory.Ideal.Basic
@@ -26,12 +26,11 @@ open Polynomial
 
 universe u v w x y z
 
-variable {R : Type u} {S : Type v} {T : Type w} {ι : Type x} {k : Type y} {A : Type z} {a b : R}
-  {m n : ℕ}
+variable {R : Type u}
 
 section Semiring
 
-variable [Semiring R] {p q r : R[X]}
+variable [Semiring R]
 
 @[elab_as_elim]
 protected theorem induction_on {M : R[X] → Prop} (p : R[X]) (h_C : ∀ a, M (C a))
@@ -75,7 +74,7 @@ theorem span_le_of_C_coeff_mem (cf : ∀ i : ℕ, C (f.coeff i) ∈ I) :
 
 theorem mem_span_C_coeff : f ∈ Ideal.span { g : R[X] | ∃ i : ℕ, g = C (coeff f i) } := by
   let p := Ideal.span { g : R[X] | ∃ i : ℕ, g = C (coeff f i) }
-  nth_rw 1 [(sum_C_mul_X_pow_eq f).symm]
+  nth_rw 2 [(sum_C_mul_X_pow_eq f).symm]
   refine Submodule.sum_mem _ fun n _hn => ?_
   dsimp
   have : C (coeff f n) ∈ p := by
diff --git a/Mathlib/Algebra/Polynomial/Inductions.lean b/Mathlib/Algebra/Polynomial/Inductions.lean
index f5a98d3c5ebe0..2594bd1490eca 100644
--- a/Mathlib/Algebra/Polynomial/Inductions.lean
+++ b/Mathlib/Algebra/Polynomial/Inductions.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Damiano Testa, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Damiano Testa, Jens Wagemaker
 -/
 import Mathlib.Algebra.MonoidAlgebra.Division
 import Mathlib.Algebra.Polynomial.Degree.Definitions
diff --git a/Mathlib/Algebra/Polynomial/Laurent.lean b/Mathlib/Algebra/Polynomial/Laurent.lean
index 7368130e5cb45..c5523266588fe 100644
--- a/Mathlib/Algebra/Polynomial/Laurent.lean
+++ b/Mathlib/Algebra/Polynomial/Laurent.lean
@@ -542,10 +542,10 @@ lemma involutive_invert : Involutive (invert (R := R)) := fun _ ↦ by ext; simp
 lemma toLaurent_reverse (p : R[X]) :
     toLaurent p.reverse = invert (toLaurent p) * (T p.natDegree) := by
   nontriviality R
-  induction' p using Polynomial.recOnHorner with p t _ _ ih p hp ih
-  · simp
-  · simp [add_mul, ← ih]
-  · simpa [natDegree_mul_X hp]
+  induction p using Polynomial.recOnHorner with
+  | M0 => simp
+  | MC _ _ _ _ ih => simp [add_mul, ← ih]
+  | MX _ hp => simpa [natDegree_mul_X hp]
 
 end Inversion
 
diff --git a/Mathlib/Algebra/Polynomial/Lifts.lean b/Mathlib/Algebra/Polynomial/Lifts.lean
index d4a1f5911b893..9ec024df1967a 100644
--- a/Mathlib/Algebra/Polynomial/Lifts.lean
+++ b/Mathlib/Algebra/Polynomial/Lifts.lean
@@ -130,8 +130,7 @@ theorem monomial_mem_lifts_and_degree_eq {s : S} {n : ℕ} (hl : monomial n s 
   obtain ⟨q, hq⟩ := hl
   replace hq := (ext_iff.1 hq) n
   have hcoeff : f (q.coeff n) = s := by
-    simp? [coeff_monomial] at hq says simp only [coeff_map, coeff_monomial, ↓reduceIte] at hq
-    exact hq
+    rwa [coeff_map, coeff_monomial_same] at hq
   use monomial n (q.coeff n)
   constructor
   · simp only [hcoeff, map_monomial]
@@ -139,8 +138,7 @@ theorem monomial_mem_lifts_and_degree_eq {s : S} {n : ℕ} (hl : monomial n s 
     intro habs
     simp only [habs, RingHom.map_zero] at hcoeff
     exact hzero hcoeff.symm
-  rw [← C_mul_X_pow_eq_monomial]
-  rw [← C_mul_X_pow_eq_monomial]
+  rw [← C_mul_X_pow_eq_monomial, ← C_mul_X_pow_eq_monomial]
   simp only [hzero, hqzero, Ne, not_false_iff, degree_C_mul_X_pow]
 
 /-- A polynomial lifts if and only if it can be lifted to a polynomial of the same degree. -/
diff --git a/Mathlib/Algebra/Polynomial/Module/AEval.lean b/Mathlib/Algebra/Polynomial/Module/AEval.lean
index daa261800b354..17b934cb8564e 100644
--- a/Mathlib/Algebra/Polynomial/Module/AEval.lean
+++ b/Mathlib/Algebra/Polynomial/Module/AEval.lean
@@ -46,7 +46,8 @@ instance instAddCommMonoid : AddCommMonoid <| AEval R M a := inferInstanceAs (Ad
 
 instance instModuleOrig : Module R <| AEval R M a := inferInstanceAs (Module R M)
 
-instance instFiniteOrig [Finite R M] : Finite R <| AEval R M a := inferInstanceAs (Finite R M)
+instance instFiniteOrig [Module.Finite R M] : Module.Finite R <| AEval R M a :=
+  ‹Module.Finite R M›
 
 instance instModulePolynomial : Module R[X] <| AEval R M a := compHom M (aeval a).toRingHom
 
@@ -79,7 +80,7 @@ instance instIsScalarTowerOrigPolynomial : IsScalarTower R R[X] <| AEval R M a w
     apply (of R M a).symm.injective
     rw [of_symm_smul, map_smul, smul_assoc, map_smul, of_symm_smul]
 
-instance instFinitePolynomial [Finite R M] : Finite R[X] <| AEval R M a :=
+instance instFinitePolynomial [Module.Finite R M] : Module.Finite R[X] <| AEval R M a :=
   Finite.of_restrictScalars_finite R _ _
 
 /-- Construct an `R[X]`-linear map out of `AEval R M a` from a `R`-linear map out of `M`. -/
@@ -193,6 +194,6 @@ lemma AEval'.X_smul_of (m : M) : (X : R[X]) • AEval'.of φ m = AEval'.of φ (
 lemma AEval'.of_symm_X_smul (m : AEval' φ) :
     (AEval'.of φ).symm ((X : R[X]) • m) = φ ((AEval'.of φ).symm m) := AEval.of_symm_X_smul _ _
 
-instance [Finite R M] : Finite R[X] <| AEval' φ := inferInstance
+instance [Module.Finite R M] : Module.Finite R[X] <| AEval' φ := inferInstance
 
 end Module
diff --git a/Mathlib/Algebra/Polynomial/Module/Basic.lean b/Mathlib/Algebra/Polynomial/Module/Basic.lean
index 4beded12152cd..f21958832dbda 100644
--- a/Mathlib/Algebra/Polynomial/Module/Basic.lean
+++ b/Mathlib/Algebra/Polynomial/Module/Basic.lean
@@ -66,7 +66,7 @@ theorem add_apply (g₁ g₂ : PolynomialModule R M) (a : ℕ) : (g₁ + g₂) a
   Finsupp.add_apply g₁ g₂ a
 
 /-- The monomial `m * x ^ i`. This is defeq to `Finsupp.singleAddHom`, and is redefined here
-so that it has the desired type signature.  -/
+so that it has the desired type signature. -/
 noncomputable def single (i : ℕ) : M →+ PolynomialModule R M :=
   Finsupp.singleAddHom i
 
@@ -177,10 +177,11 @@ noncomputable def equivPolynomialSelf : PolynomialModule R R ≃ₗ[R[X]] R[X] :
     map_smul' := fun r x => by
       dsimp
       rw [← RingEquiv.coe_toEquiv_symm, RingEquiv.coe_toEquiv]
-      induction' x using induction_linear with _ _ hp hq n a
-      · rw [smul_zero, map_zero, mul_zero]
-      · rw [smul_add, map_add, map_add, mul_add, hp, hq]
-      · ext i
+      induction x using induction_linear with
+      | h0 => rw [smul_zero, map_zero, mul_zero]
+      | hadd _ _ hp hq => rw [smul_add, map_add, map_add, mul_add, hp, hq]
+      | hsingle n a =>
+        ext i
         simp only [coeff_ofFinsupp, smul_single_apply, toFinsuppIso_symm_apply, coeff_ofFinsupp,
         single_apply, smul_eq_mul, Polynomial.coeff_mul, mul_ite, mul_zero]
         split_ifs with hn
@@ -240,10 +241,10 @@ theorem map_smul (f : M →ₗ[R] M') (p : R[X]) (q : PolynomialModule R M) :
   · intro f g e₁ e₂
     rw [smul_add, map_add, e₁, e₂, map_add, smul_add]
   intro i m
-  induction' p using Polynomial.induction_on' with _ _ e₁ e₂
-  · rw [add_smul, map_add, e₁, e₂, Polynomial.map_add, add_smul]
-  · rw [monomial_smul_single, map_single, Polynomial.map_monomial, map_single, monomial_smul_single,
-      f.map_smul, algebraMap_smul]
+  induction p using Polynomial.induction_on' with
+  | h_add _ _ e₁ e₂ => rw [add_smul, map_add, e₁, e₂, Polynomial.map_add, add_smul]
+  | h_monomial => rw [monomial_smul_single, map_single, Polynomial.map_monomial, map_single,
+      monomial_smul_single, f.map_smul, algebraMap_smul]
 
 /-- Evaluate a polynomial `p : PolynomialModule R M` at `r : R`. -/
 @[simps! (config := .lemmasOnly)]
@@ -273,10 +274,9 @@ theorem eval_smul (p : R[X]) (q : PolynomialModule R M) (r : R) :
   · intro f g e₁ e₂
     rw [smul_add, map_add, e₁, e₂, map_add, smul_add]
   intro i m
-  induction' p using Polynomial.induction_on' with _ _ e₁ e₂
-  · rw [add_smul, map_add, Polynomial.eval_add, e₁, e₂, add_smul]
-  · rw [monomial_smul_single, eval_single, Polynomial.eval_monomial, eval_single, smul_comm, ←
-      smul_smul, pow_add, mul_smul]
+  induction p using Polynomial.induction_on' with
+  | h_add _ _ e₁ e₂ => rw [add_smul, map_add, Polynomial.eval_add, e₁, e₂, add_smul]
+  | h_monomial => simp only [monomial_smul_single, Polynomial.eval_monomial, eval_single]; module
 
 @[simp]
 theorem eval_map (f : M →ₗ[R] M') (q : PolynomialModule R M) (r : R) :
@@ -286,7 +286,8 @@ theorem eval_map (f : M →ₗ[R] M') (q : PolynomialModule R M) (r : R) :
   · intro f g e₁ e₂
     simp_rw [map_add, e₁, e₂]
   · intro i m
-    rw [map_single, eval_single, eval_single, f.map_smul, ← map_pow, algebraMap_smul]
+    simp only [map_single, eval_single, f.map_smul]
+    module
 
 @[simp]
 theorem eval_map' (f : M →ₗ[R] M) (q : PolynomialModule R M) (r : R) :
@@ -305,7 +306,7 @@ lemma aeval_equivPolynomial {S : Type*} [CommRing S] [Algebra S R]
     rw [equivPolynomial_single, aeval_monomial, mul_comm, map_single,
       Algebra.linearMap_apply, eval_single, smul_eq_mul]
 
-/-- `comp p q` is the composition of `p : R[X]` and `q : M[X]` as `q(p(x))`.  -/
+/-- `comp p q` is the composition of `p : R[X]` and `q : M[X]` as `q(p(x))`. -/
 @[simps!]
 noncomputable def comp (p : R[X]) : PolynomialModule R M →ₗ[R] PolynomialModule R M :=
   LinearMap.comp ((eval p).restrictScalars R) (map R[X] (lsingle R 0))
@@ -323,8 +324,8 @@ theorem comp_eval (p : R[X]) (q : PolynomialModule R M) (r : R) :
   · intro _ _ e₁ e₂
     simp_rw [map_add, e₁, e₂]
   · intro i m
-    rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, pow_zero, one_smul,
-      Polynomial.eval_pow]
+    rw [LinearMap.comp_apply, comp_single, eval_single, eval_smul, eval_single, eval_pow]
+    module
 
 theorem comp_smul (p p' : R[X]) (q : PolynomialModule R M) :
     comp p (p' • q) = p'.comp p • comp p q := by
diff --git a/Mathlib/Algebra/Polynomial/Monic.lean b/Mathlib/Algebra/Polynomial/Monic.lean
index 9ab637bcebdd6..eb0f0cf67b6b0 100644
--- a/Mathlib/Algebra/Polynomial/Monic.lean
+++ b/Mathlib/Algebra/Polynomial/Monic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Reverse
 import Mathlib.Algebra.Regular.SMul
diff --git a/Mathlib/Algebra/Polynomial/Monomial.lean b/Mathlib/Algebra/Polynomial/Monomial.lean
index e96bdc18024b0..29b15cda6f45e 100644
--- a/Mathlib/Algebra/Polynomial/Monomial.lean
+++ b/Mathlib/Algebra/Polynomial/Monomial.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.Basic
 
diff --git a/Mathlib/Algebra/Polynomial/Reverse.lean b/Mathlib/Algebra/Polynomial/Reverse.lean
index 788dcb3ce0de2..2dba679853097 100644
--- a/Mathlib/Algebra/Polynomial/Reverse.lean
+++ b/Mathlib/Algebra/Polynomial/Reverse.lean
@@ -86,7 +86,7 @@ In other words, the terms with exponent `[0, ..., N]` now have exponent `[N, ...
 
 In practice, `reflect` is only used when `N` is at least as large as the degree of `f`.
 
-Eventually, it will be used with `N` exactly equal to the degree of `f`.  -/
+Eventually, it will be used with `N` exactly equal to the degree of `f`. -/
 noncomputable def reflect (N : ℕ) : R[X] → R[X]
   | ⟨f⟩ => ⟨Finsupp.embDomain (revAt N) f⟩
 
diff --git a/Mathlib/Algebra/Polynomial/RingDivision.lean b/Mathlib/Algebra/Polynomial/RingDivision.lean
index c3fd1bd83e3f6..87a240f259415 100644
--- a/Mathlib/Algebra/Polynomial/RingDivision.lean
+++ b/Mathlib/Algebra/Polynomial/RingDivision.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
 -/
 import Mathlib.Algebra.Polynomial.AlgebraMap
 import Mathlib.Algebra.Polynomial.BigOperators
@@ -27,6 +27,26 @@ universe u v w z
 
 variable {R : Type u} {S : Type v} {T : Type w} {a b : R} {n : ℕ}
 
+section Semiring
+
+variable [Semiring R] {p q : R[X]}
+
+theorem Monic.comp (hp : p.Monic) (hq : q.Monic) (h : q.natDegree ≠ 0) : (p.comp q).Monic := by
+  nontriviality R
+  have : (p.comp q).natDegree = p.natDegree * q.natDegree := by
+    apply natDegree_comp_eq_of_mul_ne_zero
+    simp [hp.leadingCoeff, hq.leadingCoeff]
+  rw [Monic.def, Polynomial.leadingCoeff, this, coeff_comp_degree_mul_degree h, hp.leadingCoeff,
+    hq.leadingCoeff, one_pow, mul_one]
+
+theorem Monic.comp_X_add_C (hp : p.Monic) (r : R) : (p.comp (X + C r)).Monic := by
+  nontriviality R
+  refine hp.comp (monic_X_add_C _) fun ha => ?_
+  rw [natDegree_X_add_C] at ha
+  exact one_ne_zero ha
+
+end Semiring
+
 section CommRing
 
 variable [CommRing R] {p q : R[X]}
@@ -210,8 +230,6 @@ theorem not_isUnit_of_natDegree_pos (p : R[X])
     (hpl : 0 < p.natDegree) : ¬ IsUnit p :=
   not_isUnit_of_degree_pos _ (natDegree_pos_iff_degree_pos.mp hpl)
 
-variable [CharZero R]
-
 end NoZeroDivisors
 
 section NoZeroDivisors
@@ -527,6 +545,29 @@ theorem rootMultiplicity_mul' {p q : R[X]} {x : R}
   simp_rw [eval_divByMonic_eq_trailingCoeff_comp] at hpq
   simp_rw [rootMultiplicity_eq_natTrailingDegree, mul_comp, natTrailingDegree_mul' hpq]
 
+theorem Monic.comp_X_sub_C {p : R[X]} (hp : p.Monic) (r : R) : (p.comp (X - C r)).Monic := by
+  simpa using hp.comp_X_add_C (-r)
+
+@[simp]
+theorem comp_neg_X_leadingCoeff_eq (p : R[X]) :
+    (p.comp (-X)).leadingCoeff = (-1) ^ p.natDegree * p.leadingCoeff := by
+  nontriviality R
+  by_cases h : p = 0
+  · simp [h]
+  rw [Polynomial.leadingCoeff, natDegree_comp_eq_of_mul_ne_zero, coeff_comp_degree_mul_degree] <;>
+  simp [mul_comm, h]
+
+theorem Monic.neg_one_pow_natDegree_mul_comp_neg_X {p : R[X]} (hp : p.Monic) :
+    ((-1) ^ p.natDegree * p.comp (-X)).Monic := by
+  simp only [Monic]
+  calc
+    ((-1) ^ p.natDegree * p.comp (-X)).leadingCoeff =
+        (p.comp (-X) * C ((-1) ^ p.natDegree)).leadingCoeff := by
+      simp [mul_comm]
+    _ = 1 := by
+      apply monic_mul_C_of_leadingCoeff_mul_eq_one
+      simp [← pow_add, hp]
+
 variable [IsDomain R] {p q : R[X]}
 
 @[simp]
@@ -654,17 +695,6 @@ theorem natDegree_multiset_prod_X_sub_C_eq_card (s : Multiset R) :
       mul_one]
   · exact Multiset.forall_mem_map_iff.2 fun a _ => monic_X_sub_C a
 
-theorem Monic.comp (hp : p.Monic) (hq : q.Monic) (h : q.natDegree ≠ 0) : (p.comp q).Monic := by
-  rw [Monic.def, leadingCoeff_comp h, Monic.def.1 hp, Monic.def.1 hq, one_pow, one_mul]
-
-theorem Monic.comp_X_add_C (hp : p.Monic) (r : R) : (p.comp (X + C r)).Monic := by
-  refine hp.comp (monic_X_add_C _) fun ha => ?_
-  rw [natDegree_X_add_C] at ha
-  exact one_ne_zero ha
-
-theorem Monic.comp_X_sub_C (hp : p.Monic) (r : R) : (p.comp (X - C r)).Monic := by
-  simpa using hp.comp_X_add_C (-r)
-
 theorem units_coeff_zero_smul (c : R[X]ˣ) (p : R[X]) : (c : R[X]).coeff 0 • p = c * p := by
   rw [← Polynomial.C_mul', ← Polynomial.eq_C_of_degree_eq_zero (degree_coe_units c)]
 
diff --git a/Mathlib/Algebra/Polynomial/Roots.lean b/Mathlib/Algebra/Polynomial/Roots.lean
index 93797531db5dd..59f7f94c78993 100644
--- a/Mathlib/Algebra/Polynomial/Roots.lean
+++ b/Mathlib/Algebra/Polynomial/Roots.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Johan Commelin
 -/
 
 import Mathlib.Algebra.Polynomial.RingDivision
@@ -108,10 +108,13 @@ theorem ne_zero_of_mem_roots (h : a ∈ p.roots) : p ≠ 0 :=
 theorem isRoot_of_mem_roots (h : a ∈ p.roots) : IsRoot p a :=
   (mem_roots'.1 h).2
 
--- Porting note: added during port.
+theorem mem_roots_map_of_injective [Semiring S] {p : S[X]} {f : S →+* R}
+    (hf : Function.Injective f) {x : R} (hp : p ≠ 0) : x ∈ (p.map f).roots ↔ p.eval₂ f x = 0 := by
+  rw [mem_roots ((Polynomial.map_ne_zero_iff hf).mpr hp), IsRoot, eval_map]
+
 lemma mem_roots_iff_aeval_eq_zero {x : R} (w : p ≠ 0) : x ∈ roots p ↔ aeval x p = 0 := by
-  rw [mem_roots w, IsRoot.def, aeval_def, eval₂_eq_eval_map]
-  simp
+  rw [aeval_def, ← mem_roots_map_of_injective (NoZeroSMulDivisors.algebraMap_injective _ _) w,
+    Algebra.id.map_eq_id, map_id]
 
 theorem card_le_degree_of_subset_roots {p : R[X]} {Z : Finset R} (h : Z.val ⊆ p.roots) :
     Z.card ≤ p.natDegree :=
@@ -434,6 +437,13 @@ theorem aroots_monomial [CommRing S] [IsDomain S] [Algebra T S]
     (monomial n a).aroots S = n • ({0} : Multiset S) := by
   rw [← C_mul_X_pow_eq_monomial, aroots_C_mul_X_pow ha]
 
+variable (R S) in
+@[simp]
+theorem aroots_map (p : T[X]) [CommRing S] [Algebra T S] [Algebra S R] [Algebra T R]
+    [IsScalarTower T S R] :
+    (p.map (algebraMap T S)).aroots R = p.aroots R := by
+  rw [aroots_def, aroots_def, map_map, IsScalarTower.algebraMap_eq T S R]
+
 /-- The set of distinct roots of `p` in `S`.
 
 If you have a non-separable polynomial, use `Polynomial.aroots` for the multiset
@@ -525,6 +535,12 @@ theorem rootSet_mapsTo {p : T[X]} {S S'} [CommRing S] [IsDomain S] [Algebra T S]
     map_injective _ (NoZeroSMulDivisors.algebraMap_injective T S') (by rwa [Polynomial.map_zero])
   exact Polynomial.map_zero _
 
+theorem mem_rootSet_of_injective [CommRing S] {p : S[X]} [Algebra S R]
+    (h : Function.Injective (algebraMap S R)) {x : R} (hp : p ≠ 0) :
+    x ∈ p.rootSet R ↔ aeval x p = 0 := by
+  classical
+  exact Multiset.mem_toFinset.trans (mem_roots_map_of_injective h hp)
+
 end Roots
 
 lemma eq_zero_of_natDegree_lt_card_of_eval_eq_zero {R} [CommRing R] [IsDomain R]
diff --git a/Mathlib/Algebra/Polynomial/Smeval.lean b/Mathlib/Algebra/Polynomial/Smeval.lean
index 2eb2eecb587a6..d24a7089bfc93 100644
--- a/Mathlib/Algebra/Polynomial/Smeval.lean
+++ b/Mathlib/Algebra/Polynomial/Smeval.lean
@@ -180,7 +180,7 @@ the defining structures independently.  For non-associative power-associative al
 octonions), we replace the `[Semiring S]` with `[NonAssocSemiring S] [Pow S ℕ] [NatPowAssoc S]`.
 -/
 
-variable (R : Type*) [Semiring R] {p : R[X]} (r : R) (p q : R[X]) {S : Type*}
+variable (R : Type*) [Semiring R] (r : R) (p q : R[X]) {S : Type*}
   [NonAssocSemiring S] [Module R S] [Pow S ℕ] (x : S)
 
 theorem smeval_C_mul : (C r * p).smeval x = r • p.smeval x := by
diff --git a/Mathlib/Algebra/Polynomial/Splits.lean b/Mathlib/Algebra/Polynomial/Splits.lean
index c30726a113d9c..3b0f053ebb0ea 100644
--- a/Mathlib/Algebra/Polynomial/Splits.lean
+++ b/Mathlib/Algebra/Polynomial/Splits.lean
@@ -6,6 +6,7 @@ Authors: Chris Hughes
 import Mathlib.Algebra.Polynomial.FieldDivision
 import Mathlib.Algebra.Polynomial.Lifts
 import Mathlib.Data.List.Prime
+import Mathlib.RingTheory.Polynomial.Tower
 
 /-!
 # Split polynomials
@@ -300,11 +301,33 @@ theorem eq_prod_roots_of_splits_id {p : K[X]} (hsplit : Splits (RingHom.id K) p)
     p = C p.leadingCoeff * (p.roots.map fun a => X - C a).prod := by
   simpa using eq_prod_roots_of_splits hsplit
 
+theorem aeval_eq_prod_aroots_sub_of_splits [Algebra K L] {p : K[X]}
+    (hsplit : Splits (algebraMap K L) p) (v : L) :
+    aeval v p = algebraMap K L p.leadingCoeff * ((p.aroots L).map fun a ↦ v - a).prod := by
+  rw [← eval_map_algebraMap, eq_prod_roots_of_splits hsplit]
+  simp [eval_multiset_prod]
+
+theorem eval_eq_prod_roots_sub_of_splits_id {p : K[X]}
+    (hsplit : Splits (RingHom.id K) p) (v : K) :
+    eval v p = p.leadingCoeff * (p.roots.map fun a ↦ v - a).prod := by
+  convert aeval_eq_prod_aroots_sub_of_splits hsplit v
+  rw [Algebra.id.map_eq_id, map_id]
+
 theorem eq_prod_roots_of_monic_of_splits_id {p : K[X]} (m : Monic p)
     (hsplit : Splits (RingHom.id K) p) : p = (p.roots.map fun a => X - C a).prod := by
   convert eq_prod_roots_of_splits_id hsplit
   simp [m]
 
+theorem aeval_eq_prod_aroots_sub_of_monic_of_splits [Algebra K L] {p : K[X]} (m : Monic p)
+    (hsplit : Splits (algebraMap K L) p) (v : L) :
+    aeval v p = ((p.aroots L).map fun a ↦ v - a).prod := by
+  simp [aeval_eq_prod_aroots_sub_of_splits hsplit, m]
+
+theorem eval_eq_prod_roots_sub_of_monic_of_splits_id {p : K[X]} (m : Monic p)
+    (hsplit : Splits (RingHom.id K) p) (v : K) :
+    eval v p = (p.roots.map fun a ↦ v - a).prod := by
+  simp [eval_eq_prod_roots_sub_of_splits_id hsplit, m]
+
 theorem eq_X_sub_C_of_splits_of_single_root {x : K} {h : K[X]} (h_splits : Splits i h)
     (h_roots : (h.map i).roots = {i x}) : h = C h.leadingCoeff * (X - C x) := by
   apply Polynomial.map_injective _ i.injective
@@ -414,7 +437,7 @@ theorem aeval_root_derivative_of_splits [Algebra K L] [DecidableEq L] {P : K[X]}
   rw [eval_multiset_prod_X_sub_C_derivative hr]
 
 /-- If `P` is a monic polynomial that splits, then `coeff P 0` equals the product of the roots. -/
-theorem prod_roots_eq_coeff_zero_of_monic_of_split {P : K[X]} (hmo : P.Monic)
+theorem prod_roots_eq_coeff_zero_of_monic_of_splits {P : K[X]} (hmo : P.Monic)
     (hP : P.Splits (RingHom.id K)) : coeff P 0 = (-1) ^ P.natDegree * P.roots.prod := by
   nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
   rw [coeff_zero_eq_eval_zero, eval_multiset_prod, Multiset.map_map]
@@ -426,6 +449,9 @@ theorem prod_roots_eq_coeff_zero_of_monic_of_split {P : K[X]} (hmo : P.Monic)
     rw [neg_eq_neg_one_mul]
   simp only [splits_iff_card_roots.1 hP, neg_mul, one_mul, Multiset.prod_map_neg]
 
+@[deprecated (since := "2024-10-01")]
+alias prod_roots_eq_coeff_zero_of_monic_of_split := prod_roots_eq_coeff_zero_of_monic_of_splits
+
 /-- If `P` is a monic polynomial that splits, then `P.nextCoeff` equals the sum of the roots. -/
 theorem sum_roots_eq_nextCoeff_of_monic_of_split {P : K[X]} (hmo : P.Monic)
     (hP : P.Splits (RingHom.id K)) : P.nextCoeff = -P.roots.sum := by
diff --git a/Mathlib/Algebra/Polynomial/Taylor.lean b/Mathlib/Algebra/Polynomial/Taylor.lean
index 6f9f0e82544ef..4a89d16c0bea7 100644
--- a/Mathlib/Algebra/Polynomial/Taylor.lean
+++ b/Mathlib/Algebra/Polynomial/Taylor.lean
@@ -125,4 +125,11 @@ theorem sum_taylor_eq {R} [CommRing R] (f : R[X]) (r : R) :
   rw [← comp_eq_sum_left, sub_eq_add_neg, ← C_neg, ← taylor_apply, taylor_taylor, neg_add_cancel,
     taylor_zero]
 
+theorem eval_add_of_sq_eq_zero {A} [CommSemiring A] (p : Polynomial A) (x y : A) (hy : y ^ 2 = 0) :
+    p.eval (x + y) = p.eval x + p.derivative.eval x * y := by
+  rw [add_comm, ← Polynomial.taylor_eval,
+    Polynomial.eval_eq_sum_range' ((Nat.lt_succ_self _).trans (Nat.lt_succ_self _)),
+    Finset.sum_range_succ', Finset.sum_range_succ']
+  simp [pow_succ, mul_assoc, ← pow_two, hy, add_comm (eval x p)]
+
 end Polynomial
diff --git a/Mathlib/Algebra/Polynomial/UnitTrinomial.lean b/Mathlib/Algebra/Polynomial/UnitTrinomial.lean
index f22387feb2113..c21059979dbcf 100644
--- a/Mathlib/Algebra/Polynomial/UnitTrinomial.lean
+++ b/Mathlib/Algebra/Polynomial/UnitTrinomial.lean
@@ -178,7 +178,7 @@ theorem isUnitTrinomial_iff' :
   refine ⟨?_, fun hp => ?_⟩
   · rintro ⟨k, m, n, hkm, hmn, u, v, w, rfl⟩
     rw [sum_def, trinomial_support hkm hmn u.ne_zero v.ne_zero w.ne_zero,
-      sum_insert (mt mem_insert.mp (not_or_of_not hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),
+      sum_insert (mt mem_insert.mp (not_or_intro hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),
       sum_insert (mt mem_singleton.mp hmn.ne), sum_singleton, trinomial_leading_coeff' hkm hmn,
       trinomial_middle_coeff hkm hmn, trinomial_trailing_coeff' hkm hmn]
     simp_rw [← Units.val_pow_eq_pow_val, Int.units_sq, Units.val_one]
diff --git a/Mathlib/Algebra/QuadraticDiscriminant.lean b/Mathlib/Algebra/QuadraticDiscriminant.lean
index de6a5eb840dfd..157ac8292a0db 100644
--- a/Mathlib/Algebra/QuadraticDiscriminant.lean
+++ b/Mathlib/Algebra/QuadraticDiscriminant.lean
@@ -3,7 +3,7 @@ Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou
 -/
-import Mathlib.Order.Filter.AtTopBot
+import Mathlib.Order.Filter.AtTopBot.Field
 import Mathlib.Tactic.FieldSimp
 import Mathlib.Tactic.LinearCombination
 import Mathlib.Tactic.Linarith.Frontend
@@ -15,7 +15,7 @@ This file defines the discriminant of a quadratic and gives the solution to a qu
 
 ## Main definition
 
-- `discrim a b c`: the discriminant of a quadratic `a * x * x + b * x + c` is `b * b - 4 * a * c`.
+- `discrim a b c`: the discriminant of a quadratic `a * (x * x) + b * x + c` is `b * b - 4 * a * c`.
 
 ## Main statements
 
@@ -48,7 +48,7 @@ def discrim [Ring R] (a b c : R) : R :=
 
 variable [CommRing R] {a b c : R}
 
-lemma discrim_eq_sq_of_quadratic_eq_zero {x : R} (h : a * x * x + b * x + c = 0) :
+lemma discrim_eq_sq_of_quadratic_eq_zero {x : R} (h : a * (x * x) + b * x + c = 0) :
     discrim a b c = (2 * a * x + b) ^ 2 := by
   rw [discrim]
   linear_combination -4 * a * h
@@ -57,7 +57,7 @@ lemma discrim_eq_sq_of_quadratic_eq_zero {x : R} (h : a * x * x + b * x + c = 0)
 -/
 theorem quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
     (ha : a ≠ 0) (x : R) :
-    a * x * x + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2 := by
+    a * (x * x) + b * x + c = 0 ↔ discrim a b c = (2 * a * x + b) ^ 2 := by
   refine ⟨discrim_eq_sq_of_quadratic_eq_zero, fun h ↦ ?_⟩
   rw [discrim] at h
   have ha : 2 * 2 * a ≠ 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha
@@ -66,7 +66,7 @@ theorem quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
 
 /-- A quadratic has no root if its discriminant has no square root. -/
 theorem quadratic_ne_zero_of_discrim_ne_sq (h : ∀ s : R, discrim a b c ≠ s^2) (x : R) :
-    a * x * x + b * x + c ≠ 0 :=
+    a * (x * x) + b * x + c ≠ 0 :=
   mt discrim_eq_sq_of_quadratic_eq_zero (h _)
 
 end Ring
@@ -77,7 +77,7 @@ variable {K : Type*} [Field K] [NeZero (2 : K)] {a b c x : K}
 
 /-- Roots of a quadratic equation. -/
 theorem quadratic_eq_zero_iff (ha : a ≠ 0) {s : K} (h : discrim a b c = s * s) (x : K) :
-    a * x * x + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a) := by
+    a * (x * x) + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a) := by
   rw [quadratic_eq_zero_iff_discrim_eq_sq ha, h, sq, mul_self_eq_mul_self_iff]
   field_simp
   apply or_congr
@@ -86,7 +86,7 @@ theorem quadratic_eq_zero_iff (ha : a ≠ 0) {s : K} (h : discrim a b c = s * s)
 
 /-- A quadratic has roots if its discriminant has square roots -/
 theorem exists_quadratic_eq_zero (ha : a ≠ 0) (h : ∃ s, discrim a b c = s * s) :
-    ∃ x, a * x * x + b * x + c = 0 := by
+    ∃ x, a * (x * x) + b * x + c = 0 := by
   rcases h with ⟨s, hs⟩
   use (-b + s) / (2 * a)
   rw [quadratic_eq_zero_iff ha hs]
@@ -94,7 +94,7 @@ theorem exists_quadratic_eq_zero (ha : a ≠ 0) (h : ∃ s, discrim a b c = s *
 
 /-- Root of a quadratic when its discriminant equals zero -/
 theorem quadratic_eq_zero_iff_of_discrim_eq_zero (ha : a ≠ 0) (h : discrim a b c = 0) (x : K) :
-    a * x * x + b * x + c = 0 ↔ x = -b / (2 * a) := by
+    a * (x * x) + b * x + c = 0 ↔ x = -b / (2 * a) := by
   have : discrim a b c = 0 * 0 := by rw [h, mul_zero]
   rw [quadratic_eq_zero_iff ha this, add_zero, sub_zero, or_self_iff]
 
@@ -105,7 +105,7 @@ section LinearOrderedField
 variable {K : Type*} [LinearOrderedField K] {a b c : K}
 
 /-- If a polynomial of degree 2 is always nonnegative, then its discriminant is nonpositive -/
-theorem discrim_le_zero (h : ∀ x : K, 0 ≤ a * x * x + b * x + c) : discrim a b c ≤ 0 := by
+theorem discrim_le_zero (h : ∀ x : K, 0 ≤ a * (x * x) + b * x + c) : discrim a b c ≤ 0 := by
   rw [discrim, sq]
   obtain ha | rfl | ha : a < 0 ∨ a = 0 ∨ 0 < a := lt_trichotomy a 0
   -- if a < 0
@@ -114,7 +114,7 @@ theorem discrim_le_zero (h : ∀ x : K, 0 ≤ a * x * x + b * x + c) : discrim a
         ((tendsto_atBot_add_const_right _ b (tendsto_id.const_mul_atTop_of_neg ha)).atBot_mul_atTop
           tendsto_id)
     rcases (this.eventually (eventually_lt_atBot 0)).exists with ⟨x, hx⟩
-    exact False.elim ((h x).not_lt <| by rwa [← add_mul])
+    exact False.elim ((h x).not_lt <| by rwa [← mul_assoc, ← add_mul])
   -- if a = 0
   · rcases eq_or_ne b 0 with (rfl | hb)
     · simp
@@ -127,22 +127,22 @@ theorem discrim_le_zero (h : ∀ x : K, 0 ≤ a * x * x + b * x + c) : discrim a
     field_simp
     ring
 
-lemma discrim_le_zero_of_nonpos (h : ∀ x : K, a * x * x + b * x + c ≤ 0) : discrim a b c ≤ 0 :=
+lemma discrim_le_zero_of_nonpos (h : ∀ x : K, a * (x * x) + b * x + c ≤ 0) : discrim a b c ≤ 0 :=
   discrim_neg a b c ▸ discrim_le_zero <| by simpa only [neg_mul, ← neg_add, neg_nonneg]
 
 /-- If a polynomial of degree 2 is always positive, then its discriminant is negative,
 at least when the coefficient of the quadratic term is nonzero.
 -/
-theorem discrim_lt_zero (ha : a ≠ 0) (h : ∀ x : K, 0 < a * x * x + b * x + c) :
+theorem discrim_lt_zero (ha : a ≠ 0) (h : ∀ x : K, 0 < a * (x * x) + b * x + c) :
     discrim a b c < 0 := by
-  have : ∀ x : K, 0 ≤ a * x * x + b * x + c := fun x => le_of_lt (h x)
+  have : ∀ x : K, 0 ≤ a * (x * x) + b * x + c := fun x => le_of_lt (h x)
   refine lt_of_le_of_ne (discrim_le_zero this) fun h' ↦ ?_
   have := h (-b / (2 * a))
   have : a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c = 0 := by
-    rw [quadratic_eq_zero_iff_of_discrim_eq_zero ha h' (-b / (2 * a))]
+    rw [mul_assoc, quadratic_eq_zero_iff_of_discrim_eq_zero ha h' (-b / (2 * a))]
   linarith
 
-lemma discrim_lt_zero_of_neg (ha : a ≠ 0) (h : ∀ x : K, a * x * x + b * x + c < 0) :
+lemma discrim_lt_zero_of_neg (ha : a ≠ 0) (h : ∀ x : K, a * (x * x) + b * x + c < 0) :
     discrim a b c < 0 :=
   discrim_neg a b c ▸ discrim_lt_zero (neg_ne_zero.2 ha) <| by
     simpa only [neg_mul, ← neg_add, neg_pos]
diff --git a/Mathlib/Algebra/Quandle.lean b/Mathlib/Algebra/Quandle.lean
index c28296aa12d17..f037884dabe10 100644
--- a/Mathlib/Algebra/Quandle.lean
+++ b/Mathlib/Algebra/Quandle.lean
@@ -93,16 +93,16 @@ The binary operation is regarded as a left action of the type on itself.
 class Shelf (α : Type u) where
   /-- The action of the `Shelf` over `α`-/
   act : α → α → α
-  /-- A verification that `act` is self-distributive-/
+  /-- A verification that `act` is self-distributive -/
   self_distrib : ∀ {x y z : α}, act x (act y z) = act (act x y) (act x z)
 
 /--
 A *unital shelf* is a shelf equipped with an element `1` such that, for all elements `x`,
 we have both `x ◃ 1` and `1 ◃ x` equal `x`.
 -/
-class UnitalShelf (α : Type u) extends Shelf α, One α :=
-(one_act : ∀ a : α, act 1 a = a)
-(act_one : ∀ a : α, act a 1 = a)
+class UnitalShelf (α : Type u) extends Shelf α, One α where
+  one_act : ∀ a : α, act 1 a = a
+  act_one : ∀ a : α, act a 1 = a
 
 /-- The type of homomorphisms between shelves.
 This is also the notion of rack and quandle homomorphisms.
@@ -111,7 +111,7 @@ This is also the notion of rack and quandle homomorphisms.
 structure ShelfHom (S₁ : Type*) (S₂ : Type*) [Shelf S₁] [Shelf S₂] where
   /-- The function under the Shelf Homomorphism -/
   toFun : S₁ → S₂
-  /-- The homomorphism property of a Shelf Homomorphism-/
+  /-- The homomorphism property of a Shelf Homomorphism -/
   map_act' : ∀ {x y : S₁}, toFun (Shelf.act x y) = Shelf.act (toFun x) (toFun y)
 
 /-- A *rack* is an automorphic set (a set with an action on itself by
@@ -129,13 +129,13 @@ class Rack (α : Type u) extends Shelf α where
   /-- Proof of right inverse -/
   right_inv : ∀ x, Function.RightInverse (invAct x) (act x)
 
-/-- Action of a Shelf-/
+/-- Action of a Shelf -/
 scoped[Quandles] infixr:65 " ◃ " => Shelf.act
 
-/-- Inverse Action of a Rack-/
+/-- Inverse Action of a Rack -/
 scoped[Quandles] infixr:65 " ◃⁻¹ " => Rack.invAct
 
-/-- Shelf Homomorphism-/
+/-- Shelf Homomorphism -/
 scoped[Quandles] infixr:25 " →◃ " => ShelfHom
 
 open Quandles
diff --git a/Mathlib/Algebra/Quaternion.lean b/Mathlib/Algebra/Quaternion.lean
index 56d78e1cdaf61..f78db6c9f424a 100644
--- a/Mathlib/Algebra/Quaternion.lean
+++ b/Mathlib/Algebra/Quaternion.lean
@@ -296,7 +296,7 @@ variable [Ring R]
 * `i * j = k`, `j * i = -k`;
 * `k * k = -c₁ * c₂`;
 * `i * k = c₁ * j`, `k * i = -c₁ * j`;
-* `j * k = -c₂ * i`, `k * j = c₂ * i`.  -/
+* `j * k = -c₂ * i`, `k * j = c₂ * i`. -/
 instance : Mul ℍ[R,c₁,c₂] :=
   ⟨fun a b =>
     ⟨a.1 * b.1 + c₁ * a.2 * b.2 + c₂ * a.3 * b.3 - c₁ * c₂ * a.4 * b.4,
@@ -569,8 +569,8 @@ theorem rank_eq_four [StrongRankCondition R] : Module.rank R ℍ[R,c₁,c₂] =
   rw [rank_eq_card_basis (basisOneIJK c₁ c₂), Fintype.card_fin]
   norm_num
 
-theorem finrank_eq_four [StrongRankCondition R] : FiniteDimensional.finrank R ℍ[R,c₁,c₂] = 4 := by
-  rw [FiniteDimensional.finrank, rank_eq_four, Cardinal.toNat_ofNat]
+theorem finrank_eq_four [StrongRankCondition R] : Module.finrank R ℍ[R,c₁,c₂] = 4 := by
+  rw [Module.finrank, rank_eq_four, Cardinal.toNat_ofNat]
 
 /-- There is a natural equivalence when swapping the coefficients of a quaternion algebra. -/
 @[simps]
@@ -1024,7 +1024,7 @@ instance : Module.Free R ℍ[R] := inferInstanceAs <| Module.Free R ℍ[R,-1,-1]
 theorem rank_eq_four [StrongRankCondition R] : Module.rank R ℍ[R] = 4 :=
   QuaternionAlgebra.rank_eq_four _ _
 
-theorem finrank_eq_four [StrongRankCondition R] : FiniteDimensional.finrank R ℍ[R] = 4 :=
+theorem finrank_eq_four [StrongRankCondition R] : Module.finrank R ℍ[R] = 4 :=
   QuaternionAlgebra.finrank_eq_four _ _
 
 @[simp] theorem star_re : (star a).re = a.re := rfl
diff --git a/Mathlib/Algebra/Ring/Action/Basic.lean b/Mathlib/Algebra/Ring/Action/Basic.lean
index 25558ac78f579..d085e72efe2c3 100644
--- a/Mathlib/Algebra/Ring/Action/Basic.lean
+++ b/Mathlib/Algebra/Ring/Action/Basic.lean
@@ -42,8 +42,8 @@ class MulSemiringAction (M : Type u) (R : Type v) [Monoid M] [Semiring R] extend
 
 section Semiring
 
-variable (M N G : Type*) [Monoid M] [Monoid N] [Group G]
-variable (A R S F : Type v) [AddMonoid A] [Semiring R] [CommSemiring S]
+variable (M N : Type*) [Monoid M] [Monoid N]
+variable (R : Type v) [Semiring R]
 
 -- note we could not use `extends` since these typeclasses are made with `old_structure_cmd`
 instance (priority := 100) MulSemiringAction.toMulDistribMulAction [h : MulSemiringAction M R] :
@@ -92,8 +92,6 @@ end
 
 section SimpLemmas
 
-variable {M G A R F}
-
 attribute [simp] smul_one smul_mul' smul_zero smul_add
 
 end SimpLemmas
diff --git a/Mathlib/Algebra/Ring/Basic.lean b/Mathlib/Algebra/Ring/Basic.lean
index 4ae5170c85619..fd10db100aadd 100644
--- a/Mathlib/Algebra/Ring/Basic.lean
+++ b/Mathlib/Algebra/Ring/Basic.lean
@@ -40,13 +40,6 @@ def mulRight [Distrib R] (r : R) : AddHom R R where
 
 end AddHom
 
-section AddHomClass
-
-variable {α β F : Type*} [NonAssocSemiring α] [NonAssocSemiring β]
-  [FunLike F α β] [AddHomClass F α β]
-
-end AddHomClass
-
 namespace AddMonoidHom
 
 /-- Left multiplication by an element of a (semi)ring is an `AddMonoidHom` -/
@@ -105,7 +98,7 @@ end HasDistribNeg
 
 section NonUnitalCommRing
 
-variable {α : Type*} [NonUnitalCommRing α] {a b c : α}
+variable {α : Type*} [NonUnitalCommRing α]
 
 attribute [local simp] add_assoc add_comm add_left_comm mul_comm
 
diff --git a/Mathlib/Algebra/Ring/CentroidHom.lean b/Mathlib/Algebra/Ring/CentroidHom.lean
index 68eb72c1e0691..37a70431efb16 100644
--- a/Mathlib/Algebra/Ring/CentroidHom.lean
+++ b/Mathlib/Algebra/Ring/CentroidHom.lean
@@ -61,8 +61,8 @@ attribute [nolint docBlame] CentroidHom.toAddMonoidHom
 /-- `CentroidHomClass F α` states that `F` is a type of centroid homomorphisms.
 
 You should extend this class when you extend `CentroidHom`. -/
-class CentroidHomClass (F α : Type*) [NonUnitalNonAssocSemiring α] [FunLike F α α] extends
-  AddMonoidHomClass F α α : Prop where
+class CentroidHomClass (F : Type*) (α : outParam Type*)
+    [NonUnitalNonAssocSemiring α] [FunLike F α α] extends AddMonoidHomClass F α α : Prop where
   /-- Commutativity of centroid homomorphims with left multiplication. -/
   map_mul_left (f : F) (a b : α) : f (a * b) = a * f b
   /-- Commutativity of centroid homomorphims with right multiplication. -/
@@ -475,8 +475,9 @@ def centerToCentroidCenter :
     simp only [ZeroMemClass.coe_zero, map_zero]
     exact rfl
   map_add' := fun _ _ => by
-    simp only [AddSubmonoid.coe_add, NonUnitalSubsemiring.coe_toAddSubmonoid, map_add]
-    exact rfl
+    dsimp
+    simp only [map_add]
+    rfl
   map_mul' z₁ z₂ := by ext a; exact (z₁.prop.left_assoc z₂ a).symm
 
 instance : FunLike (Subsemiring.center (CentroidHom α)) α α where
diff --git a/Mathlib/Algebra/Ring/Commute.lean b/Mathlib/Algebra/Ring/Commute.lean
index 6793e2df1e30d..27887a07dc5bb 100644
--- a/Mathlib/Algebra/Ring/Commute.lean
+++ b/Mathlib/Algebra/Ring/Commute.lean
@@ -21,9 +21,9 @@ For the definitions of semirings and rings see `Mathlib.Algebra.Ring.Defs`.
 -/
 
 
-universe u v w x
+universe u
 
-variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
+variable {R : Type u}
 
 open Function
 
@@ -75,7 +75,7 @@ end
 
 section
 
-variable [MulOneClass R] [HasDistribNeg R] {a : R}
+variable [MulOneClass R] [HasDistribNeg R]
 
 -- Porting note (#10618): no longer needs to be `@[simp]` since `simp` can prove it.
 -- @[simp]
@@ -147,7 +147,7 @@ alias neg_one_pow_two := neg_one_sq
 end HasDistribNeg
 
 section Ring
-variable [Ring R] {a b : R} {n : ℕ}
+variable [Ring R] {a : R} {n : ℕ}
 
 @[simp] lemma neg_one_pow_mul_eq_zero_iff : (-1) ^ n * a = 0 ↔ a = 0 := by
   rcases neg_one_pow_eq_or R n with h | h <;> simp [h]
diff --git a/Mathlib/Algebra/Ring/Defs.lean b/Mathlib/Algebra/Ring/Defs.lean
index 18b144825141a..8ff6a13b2f8f9 100644
--- a/Mathlib/Algebra/Ring/Defs.lean
+++ b/Mathlib/Algebra/Ring/Defs.lean
@@ -24,6 +24,9 @@ the present file is about their interaction.
   addition, for example `Units`.
 * `(NonUnital)(NonAssoc)(Semi)Ring`: Typeclasses for possibly non-unital or non-associative
   rings and semirings. Some combinations are not defined yet because they haven't found use.
+  For Lie Rings, there is a type synonym `CommutatorRing` defined in
+  `Mathlib/Algebra/Algebra/NonUnitalHom.lean` turning the bracket into a multiplication so that the
+  instance `instNonUnitalNonAssocSemiringCommutatorRing` can be defined.
 
 ## Tags
 
@@ -42,9 +45,9 @@ assert_not_exists DivisionMonoid.toDivInvOneMonoid
 assert_not_exists mul_rotate
 
 
-universe u v w x
+universe u v
 
-variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
+variable {α : Type u} {R : Type v}
 
 open Function
 
@@ -109,7 +112,9 @@ that `Semiring -> NonAssocSemiring` is tried before `NonAssocRing -> NonAssocSem
 TODO: clean this once lean4#2115 is fixed
 -/
 
-/-- A not-necessarily-unital, not-necessarily-associative semiring. -/
+/-- A not-necessarily-unital, not-necessarily-associative semiring. See `CommutatorRing` and the
+  documentation thereof in case you need a `NonUnitalNonAssocSemiring` instance on a Lie ring
+  or a Lie algebra. -/
 class NonUnitalNonAssocSemiring (α : Type u) extends AddCommMonoid α, Distrib α, MulZeroClass α
 
 /-- An associative but not-necessarily unital semiring. -/
@@ -245,7 +250,7 @@ instance (priority := 100) CommSemiring.toCommMonoidWithZero [CommSemiring α] :
 
 section CommSemiring
 
-variable [CommSemiring α] {a b c : α}
+variable [CommSemiring α]
 
 theorem add_mul_self_eq (a b : α) : (a + b) * (a + b) = a * a + 2 * a * b + b * b := by
   simp only [two_mul, add_mul, mul_add, add_assoc, mul_comm b]
@@ -366,7 +371,7 @@ end NonAssocRing
 
 section Ring
 
-variable [Ring α] {a b c d e : α}
+variable [Ring α]
 
 -- A (unital, associative) ring is a not-necessarily-unital ring
 -- see Note [lower instance priority]
diff --git a/Mathlib/Algebra/Ring/Equiv.lean b/Mathlib/Algebra/Ring/Equiv.lean
index 246dcd9e49330..51c833ecb1d78 100644
--- a/Mathlib/Algebra/Ring/Equiv.lean
+++ b/Mathlib/Algebra/Ring/Equiv.lean
@@ -5,7 +5,6 @@ Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
 -/
 import Mathlib.Algebra.Group.Prod
 import Mathlib.Algebra.Group.Opposite
-import Mathlib.Algebra.Group.Units.Equiv
 import Mathlib.Algebra.GroupWithZero.InjSurj
 import Mathlib.Algebra.Ring.Hom.Defs
 import Mathlib.Logic.Equiv.Set
@@ -126,6 +125,8 @@ section Basic
 
 variable [Mul R] [Mul S] [Add R] [Add S] [Mul S'] [Add S']
 
+section coe
+
 instance : EquivLike (R ≃+* S) R S where
   coe f := f.toFun
   inv f := f.invFun
@@ -141,30 +142,18 @@ instance : RingEquivClass (R ≃+* S) R S where
   map_add f := f.map_add'
   map_mul f := f.map_mul'
 
-@[simp]
-theorem toEquiv_eq_coe (f : R ≃+* S) : f.toEquiv = f :=
-  rfl
-
--- Porting note: `toFun_eq_coe` no longer needed in Lean4
-
-@[simp]
-theorem coe_toEquiv (f : R ≃+* S) : ⇑(f : R ≃ S) = f :=
-  rfl
-
-/-- A ring isomorphism preserves multiplication. -/
-protected theorem map_mul (e : R ≃+* S) (x y : R) : e (x * y) = e x * e y :=
-  map_mul e x y
-
-/-- A ring isomorphism preserves addition. -/
-protected theorem map_add (e : R ≃+* S) (x y : R) : e (x + y) = e x + e y :=
-  map_add e x y
-
 /-- Two ring isomorphisms agree if they are defined by the
     same underlying function. -/
 @[ext]
 theorem ext {f g : R ≃+* S} (h : ∀ x, f x = g x) : f = g :=
   DFunLike.ext f g h
 
+protected theorem congr_arg {f : R ≃+* S} {x x' : R} : x = x' → f x = f x' :=
+  DFunLike.congr_arg f
+
+protected theorem congr_fun {f g : R ≃+* S} (h : f = g) (x : R) : f x = g x :=
+  DFunLike.congr_fun h x
+
 @[simp]
 theorem coe_mk (e h₃ h₄) : ⇑(⟨e, h₃, h₄⟩ : R ≃+* S) = e :=
   rfl
@@ -175,11 +164,13 @@ theorem coe_mk (e h₃ h₄) : ⇑(⟨e, h₃, h₄⟩ : R ≃+* S) = e :=
 theorem mk_coe (e : R ≃+* S) (e' h₁ h₂ h₃ h₄) : (⟨⟨e, e', h₁, h₂⟩, h₃, h₄⟩ : R ≃+* S) = e :=
   ext fun _ => rfl
 
-protected theorem congr_arg {f : R ≃+* S} {x x' : R} : x = x' → f x = f x' :=
-  DFunLike.congr_arg f
+@[simp]
+theorem toEquiv_eq_coe (f : R ≃+* S) : f.toEquiv = f :=
+  rfl
 
-protected theorem congr_fun {f g : R ≃+* S} (h : f = g) (x : R) : f x = g x :=
-  DFunLike.congr_fun h x
+@[simp]
+theorem coe_toEquiv (f : R ≃+* S) : ⇑(f : R ≃ S) = f :=
+  rfl
 
 @[simp]
 theorem toAddEquiv_eq_coe (f : R ≃+* S) : f.toAddEquiv = ↑f :=
@@ -197,22 +188,45 @@ theorem coe_toMulEquiv (f : R ≃+* S) : ⇑(f : R ≃* S) = f :=
 theorem coe_toAddEquiv (f : R ≃+* S) : ⇑(f : R ≃+ S) = f :=
   rfl
 
-/-- The `RingEquiv` between two semirings with a unique element. -/
-def ringEquivOfUnique {M N} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] : M ≃+* N :=
-  { AddEquiv.addEquivOfUnique, MulEquiv.mulEquivOfUnique with }
+end coe
 
-instance {M N} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] :
-    Unique (M ≃+* N) where
-  default := ringEquivOfUnique
-  uniq _ := ext fun _ => Subsingleton.elim _ _
+section map
+
+/-- A ring isomorphism preserves multiplication. -/
+protected theorem map_mul (e : R ≃+* S) (x y : R) : e (x * y) = e x * e y :=
+  map_mul e x y
+
+/-- A ring isomorphism preserves addition. -/
+protected theorem map_add (e : R ≃+* S) (x y : R) : e (x + y) = e x + e y :=
+  map_add e x y
+
+end map
+
+section bijective
+
+protected theorem bijective (e : R ≃+* S) : Function.Bijective e :=
+  EquivLike.bijective e
+
+protected theorem injective (e : R ≃+* S) : Function.Injective e :=
+  EquivLike.injective e
+
+protected theorem surjective (e : R ≃+* S) : Function.Surjective e :=
+  EquivLike.surjective e
+
+end bijective
 
 variable (R)
 
+section refl
+
 /-- The identity map is a ring isomorphism. -/
 @[refl]
 def refl : R ≃+* R :=
   { MulEquiv.refl R, AddEquiv.refl R with }
 
+instance : Inhabited (R ≃+* R) :=
+  ⟨RingEquiv.refl R⟩
+
 @[simp]
 theorem refl_apply (x : R) : RingEquiv.refl R x = x :=
   rfl
@@ -225,22 +239,17 @@ theorem coe_addEquiv_refl : (RingEquiv.refl R : R ≃+ R) = AddEquiv.refl R :=
 theorem coe_mulEquiv_refl : (RingEquiv.refl R : R ≃* R) = MulEquiv.refl R :=
   rfl
 
-instance : Inhabited (R ≃+* R) :=
-  ⟨RingEquiv.refl R⟩
+end refl
 
 variable {R}
 
+section symm
+
 /-- The inverse of a ring isomorphism is a ring isomorphism. -/
 @[symm]
 protected def symm (e : R ≃+* S) : S ≃+* R :=
   { e.toMulEquiv.symm, e.toAddEquiv.symm with }
 
-/-- See Note [custom simps projection] -/
-def Simps.symm_apply (e : R ≃+* S) : S → R :=
-  e.symm
-
-initialize_simps_projections RingEquiv (toFun → apply, invFun → symm_apply)
-
 @[simp]
 theorem invFun_eq_symm (f : R ≃+* S) : EquivLike.inv f = f.symm :=
   rfl
@@ -248,14 +257,6 @@ theorem invFun_eq_symm (f : R ≃+* S) : EquivLike.inv f = f.symm :=
 @[simp]
 theorem symm_symm (e : R ≃+* S) : e.symm.symm = e := rfl
 
-@[simp]
-theorem symm_refl : (RingEquiv.refl R).symm = RingEquiv.refl R :=
-  rfl
-
-@[simp]
-theorem coe_toEquiv_symm (e : R ≃+* S) : (e.symm : S ≃ R) = (e : R ≃ S).symm :=
-  rfl
-
 theorem symm_bijective : Function.Bijective (RingEquiv.symm : (R ≃+* S) → S ≃+* R) :=
   Function.bijective_iff_has_inverse.mpr ⟨_, symm_symm, symm_symm⟩
 
@@ -275,35 +276,14 @@ theorem symm_mk (f : R → S) (g h₁ h₂ h₃ h₄) :
         invFun := f } :=
   rfl
 
-/-- Transitivity of `RingEquiv`. -/
-@[trans]
-protected def trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : R ≃+* S' :=
-  { e₁.toMulEquiv.trans e₂.toMulEquiv, e₁.toAddEquiv.trans e₂.toAddEquiv with }
-
-theorem trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) : e₁.trans e₂ a = e₂ (e₁ a) :=
-  rfl
-
 @[simp]
-theorem coe_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂ : R → S') = e₂ ∘ e₁ :=
+theorem symm_refl : (RingEquiv.refl R).symm = RingEquiv.refl R :=
   rfl
 
 @[simp]
-theorem symm_trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') :
-    (e₁.trans e₂).symm a = e₁.symm (e₂.symm a) :=
-  rfl
-
-theorem symm_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm :=
+theorem coe_toEquiv_symm (e : R ≃+* S) : (e.symm : S ≃ R) = (e : R ≃ S).symm :=
   rfl
 
-protected theorem bijective (e : R ≃+* S) : Function.Bijective e :=
-  EquivLike.bijective e
-
-protected theorem injective (e : R ≃+* S) : Function.Injective e :=
-  EquivLike.injective e
-
-protected theorem surjective (e : R ≃+* S) : Function.Surjective e :=
-  EquivLike.surjective e
-
 @[simp]
 theorem apply_symm_apply (e : R ≃+* S) : ∀ x, e (e.symm x) = x :=
   e.toEquiv.apply_symm_apply
@@ -315,6 +295,40 @@ theorem symm_apply_apply (e : R ≃+* S) : ∀ x, e.symm (e x) = x :=
 theorem image_eq_preimage (e : R ≃+* S) (s : Set R) : e '' s = e.symm ⁻¹' s :=
   e.toEquiv.image_eq_preimage s
 
+end symm
+
+section simps
+
+/-- See Note [custom simps projection] -/
+def Simps.symm_apply (e : R ≃+* S) : S → R :=
+  e.symm
+
+initialize_simps_projections RingEquiv (toFun → apply, invFun → symm_apply)
+
+end simps
+
+section trans
+
+/-- Transitivity of `RingEquiv`. -/
+@[trans]
+protected def trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : R ≃+* S' :=
+  { e₁.toMulEquiv.trans e₂.toMulEquiv, e₁.toAddEquiv.trans e₂.toAddEquiv with }
+
+@[simp]
+theorem coe_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂ : R → S') = e₂ ∘ e₁ :=
+  rfl
+
+theorem trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : R) : e₁.trans e₂ a = e₂ (e₁ a) :=
+  rfl
+
+@[simp]
+theorem symm_trans_apply (e₁ : R ≃+* S) (e₂ : S ≃+* S') (a : S') :
+    (e₁.trans e₂).symm a = e₁.symm (e₂.symm a) :=
+  rfl
+
+theorem symm_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm :=
+  rfl
+
 @[simp]
 theorem coe_mulEquiv_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
     (e₁.trans e₂ : R ≃* S') = (e₁ : R ≃* S).trans ↑e₂ :=
@@ -325,6 +339,21 @@ theorem coe_addEquiv_trans (e₁ : R ≃+* S) (e₂ : S ≃+* S') :
     (e₁.trans e₂ : R ≃+ S') = (e₁ : R ≃+ S).trans ↑e₂ :=
   rfl
 
+end trans
+
+section unique
+
+/-- The `RingEquiv` between two semirings with a unique element. -/
+def ringEquivOfUnique {M N} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] : M ≃+* N :=
+  { AddEquiv.addEquivOfUnique, MulEquiv.mulEquivOfUnique with }
+
+instance {M N} [Unique M] [Unique N] [Add M] [Mul M] [Add N] [Mul N] :
+    Unique (M ≃+* N) where
+  default := ringEquivOfUnique
+  uniq _ := ext fun _ => Subsingleton.elim _ _
+
+end unique
+
 end Basic
 
 section Opposite
@@ -379,7 +408,7 @@ end Opposite
 
 section NonUnitalSemiring
 
-variable [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) (x y : R)
+variable [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] (f : R ≃+* S) (x : R)
 
 /-- A ring isomorphism sends zero to zero. -/
 protected theorem map_zero : f 0 = 0 :=
@@ -503,7 +532,7 @@ end NonUnitalSemiring
 
 section Semiring
 
-variable [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (x y : R)
+variable [NonAssocSemiring R] [NonAssocSemiring S] (f : R ≃+* S) (x : R)
 
 /-- A ring isomorphism sends one to one. -/
 protected theorem map_one : f 1 = 1 :=
@@ -574,7 +603,7 @@ end NonUnitalRing
 
 section Ring
 
-variable [NonAssocRing R] [NonAssocRing S] (f : R ≃+* S) (x y : R)
+variable [NonAssocRing R] [NonAssocRing S] (f : R ≃+* S)
 
 -- Porting note (#10618): `simp` can now prove that, so we remove the `@[simp]` tag
 theorem map_neg_one : f (-1) = -1 :=
@@ -775,9 +804,6 @@ protected theorem map_pow (f : R ≃+* S) (a) : ∀ n : ℕ, f (a ^ n) = f a ^ n
 
 end GroupPower
 
-protected theorem isUnit_iff (f : R ≃+* S) {a} : IsUnit (f a) ↔ IsUnit a :=
-  MulEquiv.map_isUnit_iff f
-
 end RingEquiv
 
 namespace MulEquiv
@@ -814,6 +840,33 @@ theorem symm_trans_self (e : R ≃+* S) : e.symm.trans e = RingEquiv.refl S :=
 
 end RingEquiv
 
+namespace RingEquiv
+
+variable [NonAssocSemiring R] [NonAssocSemiring S]
+
+/-- If a ring homomorphism has an inverse, it is a ring isomorphism. -/
+@[simps]
+def ofRingHom (f : R →+* S) (g : S →+* R) (h₁ : f.comp g = RingHom.id S)
+    (h₂ : g.comp f = RingHom.id R) : R ≃+* S :=
+  { f with
+    toFun := f
+    invFun := g
+    left_inv := RingHom.ext_iff.1 h₂
+    right_inv := RingHom.ext_iff.1 h₁ }
+
+theorem coe_ringHom_ofRingHom (f : R →+* S) (g : S →+* R) (h₁ h₂) : ofRingHom f g h₁ h₂ = f :=
+  rfl
+
+@[simp]
+theorem ofRingHom_coe_ringHom (f : R ≃+* S) (g : S →+* R) (h₁ h₂) : ofRingHom (↑f) g h₁ h₂ = f :=
+  ext fun _ ↦ rfl
+
+theorem ofRingHom_symm (f : R →+* S) (g : S →+* R) (h₁ h₂) :
+    (ofRingHom f g h₁ h₂).symm = ofRingHom g f h₂ h₁ :=
+  rfl
+
+end RingEquiv
+
 namespace MulEquiv
 
 /-- If two rings are isomorphic, and the second doesn't have zero divisors,
diff --git a/Mathlib/Algebra/Ring/Hom/Defs.lean b/Mathlib/Algebra/Ring/Hom/Defs.lean
index 6ffa90e13d5da..86672a2bb1c98 100644
--- a/Mathlib/Algebra/Ring/Hom/Defs.lean
+++ b/Mathlib/Algebra/Ring/Hom/Defs.lean
@@ -153,7 +153,6 @@ end coe
 section
 
 variable [NonUnitalNonAssocSemiring α] [NonUnitalNonAssocSemiring β]
-variable (f : α →ₙ+* β) {x y : α}
 
 @[ext]
 theorem ext ⦃f g : α →ₙ+* β⦄ : (∀ x, f x = g x) → f = g :=
@@ -225,7 +224,6 @@ theorem coe_comp (g : β →ₙ+* γ) (f : α →ₙ+* β) : ⇑(g.comp f) = g 
 @[simp]
 theorem comp_apply (g : β →ₙ+* γ) (f : α →ₙ+* β) (x : α) : g.comp f x = g (f x) :=
   rfl
-variable (g : β →ₙ+* γ) (f : α →ₙ+* β)
 
 @[simp]
 theorem coe_comp_addMonoidHom (g : β →ₙ+* γ) (f : α →ₙ+* β) :
@@ -441,7 +439,7 @@ end coe
 
 section
 
-variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β) {x y : α}
+variable {_ : NonAssocSemiring α} {_ : NonAssocSemiring β} (f : α →+* β)
 
 protected theorem congr_fun {f g : α →+* β} (h : f = g) (x : α) : f x = g x :=
   DFunLike.congr_fun h x
diff --git a/Mathlib/Algebra/Ring/Idempotents.lean b/Mathlib/Algebra/Ring/Idempotents.lean
index 415b47517507d..80a2c2e0be568 100644
--- a/Mathlib/Algebra/Ring/Idempotents.lean
+++ b/Mathlib/Algebra/Ring/Idempotents.lean
@@ -5,6 +5,7 @@ Authors: Christopher Hoskin
 -/
 import Mathlib.Algebra.Group.Basic
 import Mathlib.Algebra.Group.Commute.Defs
+import Mathlib.Algebra.Group.Hom.Defs
 import Mathlib.Algebra.Ring.Defs
 import Mathlib.Data.Subtype
 import Mathlib.Order.Notation
@@ -49,6 +50,10 @@ theorem mul_of_commute {p q : S} (h : Commute p q) (h₁ : IsIdempotentElem p)
     (h₂ : IsIdempotentElem q) : IsIdempotentElem (p * q) := by
   rw [IsIdempotentElem, mul_assoc, ← mul_assoc q, ← h.eq, mul_assoc p, h₂.eq, ← mul_assoc, h₁.eq]
 
+lemma mul {M} [CommSemigroup M] {e₁ e₂ : M}
+    (he₁ : IsIdempotentElem e₁) (he₂ : IsIdempotentElem e₂) : IsIdempotentElem (e₁ * e₂) :=
+  he₁.mul_of_commute (.all e₁ e₂) he₂
+
 theorem zero : IsIdempotentElem (0 : M₀) :=
   mul_zero _
 
@@ -83,6 +88,10 @@ theorem iff_eq_zero_or_one {p : G₀} : IsIdempotentElem p ↔ p = 0 ∨ p = 1 :
       h.elim (fun hp => hp.symm ▸ zero) fun hp => hp.symm ▸ one
   exact mul_left_cancel₀ hp (h.trans (mul_one p).symm)
 
+lemma map {M N F} [Mul M] [Mul N] [FunLike F M N] [MulHomClass F M N] {e : M}
+    (he : IsIdempotentElem e) (f : F) : IsIdempotentElem (f e) := by
+  rw [IsIdempotentElem, ← map_mul, he.eq]
+
 /-! ### Instances on `Subtype IsIdempotentElem` -/
 
 
diff --git a/Mathlib/Algebra/Ring/Int.lean b/Mathlib/Algebra/Ring/Int.lean
index 68823fef425a9..8fb0dd29430dc 100644
--- a/Mathlib/Algebra/Ring/Int.lean
+++ b/Mathlib/Algebra/Ring/Int.lean
@@ -97,6 +97,8 @@ lemma odd_iff : Odd n ↔ n % 2 = 1 where
 
 lemma not_odd_iff : ¬Odd n ↔ n % 2 = 0 := by rw [odd_iff, emod_two_ne_one]
 
+@[simp] lemma not_odd_zero : ¬Odd (0 : ℤ) := not_odd_iff.mpr rfl
+
 @[simp] lemma not_odd_iff_even : ¬Odd n ↔ Even n := by rw [not_odd_iff, even_iff]
 @[simp] lemma not_even_iff_odd : ¬Even n ↔ Odd n := by rw [not_even_iff, odd_iff]
 
@@ -118,9 +120,9 @@ lemma even_xor'_odd (n : ℤ) : Xor' (Even n) (Odd n) := by
 
 lemma even_xor'_odd' (n : ℤ) : ∃ k, Xor' (n = 2 * k) (n = 2 * k + 1) := by
   rcases even_or_odd n with (⟨k, rfl⟩ | ⟨k, rfl⟩) <;> use k
-  · simpa only [← two_mul, Xor', true_and_iff, eq_self_iff_true, not_true, or_false_iff,
-      and_false_iff] using (succ_ne_self (2 * k)).symm
-  · simp only [Xor', add_right_eq_self, false_or_iff, eq_self_iff_true, not_true, not_false_iff,
+  · simpa only [← two_mul, Xor', true_and, eq_self_iff_true, not_true, or_false,
+      and_false] using (succ_ne_self (2 * k)).symm
+  · simp only [Xor', add_right_eq_self, false_or, eq_self_iff_true, not_true, not_false_iff,
       one_ne_zero, and_self_iff]
 
 instance : DecidablePred (Odd : ℤ → Prop) := fun _ => decidable_of_iff _ not_even_iff_odd
diff --git a/Mathlib/Algebra/Ring/Parity.lean b/Mathlib/Algebra/Ring/Parity.lean
index 94a1d21b3a889..9205b9aeff65a 100644
--- a/Mathlib/Algebra/Ring/Parity.lean
+++ b/Mathlib/Algebra/Ring/Parity.lean
@@ -214,6 +214,8 @@ lemma not_odd_iff : ¬Odd n ↔ n % 2 = 0 := by rw [odd_iff, mod_two_ne_one]
 @[simp] lemma not_odd_iff_even : ¬Odd n ↔ Even n := by rw [not_odd_iff, even_iff]
 @[simp] lemma not_even_iff_odd : ¬Even n ↔ Odd n := by rw [not_even_iff, odd_iff]
 
+@[simp] lemma not_odd_zero : ¬Odd 0 := not_odd_iff.mpr rfl
+
 @[deprecated not_odd_iff_even (since := "2024-08-21")]
 lemma even_iff_not_odd : Even n ↔ ¬Odd n := by rw [not_odd_iff, even_iff]
 
@@ -324,7 +326,6 @@ namespace Involutive
 
 variable {α : Type*} {f : α → α} {n : ℕ}
 
-set_option linter.deprecated false in
 section
 
 lemma iterate_bit0 (hf : Involutive f) (n : ℕ) : f^[2 * n] = id := by
diff --git a/Mathlib/Algebra/Ring/Pointwise/Set.lean b/Mathlib/Algebra/Ring/Pointwise/Set.lean
new file mode 100644
index 0000000000000..5ec371a33579c
--- /dev/null
+++ b/Mathlib/Algebra/Ring/Pointwise/Set.lean
@@ -0,0 +1,49 @@
+/-
+Copyright (c) 2019 Johan Commelin. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johan Commelin, Floris van Doorn
+-/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Ring.Defs
+
+/-!
+# Pointwise operations of sets in a ring
+
+This file proves properties of pointwise operations of sets in a ring.
+
+## Tags
+
+set multiplication, set addition, pointwise addition, pointwise multiplication,
+pointwise subtraction
+-/
+
+assert_not_exists OrderedAddCommMonoid
+
+open Function
+open scoped Pointwise
+
+variable {F α β γ : Type*}
+
+namespace Set
+
+/-- `Set α` has distributive negation if `α` has. -/
+protected noncomputable def hasDistribNeg [Mul α] [HasDistribNeg α] : HasDistribNeg (Set α) where
+  __ := Set.involutiveNeg
+  neg_mul _ _ := by simp_rw [← image_neg]; exact image2_image_left_comm neg_mul
+  mul_neg _ _ := by simp_rw [← image_neg]; exact image_image2_right_comm mul_neg
+
+scoped[Pointwise] attribute [instance] Set.hasDistribNeg
+
+section Distrib
+variable [Distrib α] (s t u : Set α)
+
+/-!
+Note that `Set α` is not a `Distrib` because `s * t + s * u` has cross terms that `s * (t + u)`
+lacks.
+-/
+
+lemma mul_add_subset : s * (t + u) ⊆ s * t + s * u := image2_distrib_subset_left mul_add
+lemma add_mul_subset : (s + t) * u ⊆ s * u + t * u := image2_distrib_subset_right add_mul
+
+end Distrib
+end Set
diff --git a/Mathlib/Algebra/Ring/Semiconj.lean b/Mathlib/Algebra/Ring/Semiconj.lean
index 7ca35cfbf5af9..8b20e68a71a06 100644
--- a/Mathlib/Algebra/Ring/Semiconj.lean
+++ b/Mathlib/Algebra/Ring/Semiconj.lean
@@ -19,9 +19,9 @@ For the definitions of semirings and rings see `Mathlib.Algebra.Ring.Defs`.
 -/
 
 
-universe u v w x
+universe u
 
-variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
+variable {R : Type u}
 
 open Function
 
@@ -59,7 +59,7 @@ end
 
 section
 
-variable [MulOneClass R] [HasDistribNeg R] {a x y : R}
+variable [MulOneClass R] [HasDistribNeg R]
 
 -- Porting note: `simpNF` told me to remove `simp` attribute
 theorem neg_one_right (a : R) : SemiconjBy a (-1) (-1) :=
diff --git a/Mathlib/Algebra/Ring/Subring/Basic.lean b/Mathlib/Algebra/Ring/Subring/Basic.lean
index 365272a227add..4d2d0317d2fcf 100644
--- a/Mathlib/Algebra/Ring/Subring/Basic.lean
+++ b/Mathlib/Algebra/Ring/Subring/Basic.lean
@@ -72,7 +72,7 @@ section SubringClass
 
 /-- `SubringClass S R` states that `S` is a type of subsets `s ⊆ R` that
 are both a multiplicative submonoid and an additive subgroup. -/
-class SubringClass (S : Type*) (R : Type u) [Ring R] [SetLike S R] extends
+class SubringClass (S : Type*) (R : outParam (Type u)) [Ring R] [SetLike S R] extends
   SubsemiringClass S R, NegMemClass S R : Prop
 
 -- See note [lower instance priority]
@@ -433,6 +433,9 @@ theorem coe_top : ((⊤ : Subring R) : Set R) = Set.univ :=
 def topEquiv : (⊤ : Subring R) ≃+* R :=
   Subsemiring.topEquiv
 
+instance {R : Type*} [Ring R] [Fintype R] : Fintype (⊤ : Subring R) :=
+  inferInstanceAs (Fintype (⊤ : Set R))
+
 theorem card_top (R) [Ring R] [Fintype R] : Fintype.card (⊤ : Subring R) = Fintype.card R :=
   Fintype.card_congr topEquiv.toEquiv
 
@@ -819,7 +822,7 @@ theorem mem_closure_iff {s : Set R} {x} :
           mul_mem hx hy)
       (zero_mem _) (fun x y hx hy => add_mem hx hy) fun x hx => neg_mem hx⟩
 
-/-- If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring.  -/
+/-- If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring. -/
 def closureCommRingOfComm {s : Set R} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
     CommRing (closure s) :=
   { (closure s).toRing with
@@ -894,6 +897,14 @@ theorem map_iSup {ι : Sort*} (f : R →+* S) (s : ι → Subring R) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+theorem map_inf (s t : Subring R) (f : R →+* S) (hf : Function.Injective f) :
+    (s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter hf)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : R →+* S) (hf : Function.Injective f)
+    (s : ι → Subring R) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem comap_inf (s t : Subring S) (f : R →+* S) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
   (gc_map_comap f).u_inf
 
diff --git a/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean b/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean
new file mode 100644
index 0000000000000..a91e2bf11eca4
--- /dev/null
+++ b/Mathlib/Algebra/Ring/Subring/IntPolynomial.lean
@@ -0,0 +1,62 @@
+/-
+Copyright (c) 2024 María Inés de Frutos-Fernández, Filippo A. E. Nuccio. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: María Inés de Frutos-Fernández, Filippo A. E. Nuccio
+-/
+import Mathlib.Algebra.Polynomial.AlgebraMap
+
+/-!
+# Polynomials over subrings.
+
+Given a field `K` with a subring `R`, in this file we construct a map from polynomials in `K[X]`
+with coefficients in `R` to `R[X]`. We provide several lemmas to deal with
+coefficients, degree, and evaluation of `intPolynomial`.
+This is useful when dealing with integral elements in an extension of fields.
+
+# Main Definitions
+* `Polynomial.int` : given a polynomial `P`. in `K[X]` with coefficients in a field `K` with a
+  subring `R` such that all coefficients belong to `R`, `P.int R` is the corresponding polynomial
+  in `R[X]`.
+-/
+
+variable {K : Type*} [Field K] (R : Subring K)
+
+open Polynomial
+
+open scoped Polynomial
+
+/-- Given a polynomial in `K[X]` such that all coefficients belong to the subring `R`,
+  `intPolynomial` is the corresponding polynomial in `R[X]`. -/
+def Polynomial.int (P : K[X]) (hP : ∀ n : ℕ, P.coeff n ∈ R) : R[X] where
+  toFinsupp :=
+  { support := P.support
+    toFun := fun n => ⟨P.coeff n, hP n⟩
+    mem_support_toFun := fun n => by
+      rw [ne_eq, ← Subring.coe_eq_zero_iff, mem_support_iff] }
+
+namespace Polynomial
+
+variable (P : K[X]) (hP : ∀ n : ℕ, P.coeff n ∈ R)
+
+@[simp]
+theorem int_coeff_eq  (n : ℕ) : ↑((P.int R hP).coeff n) = P.coeff n := rfl
+
+@[simp]
+theorem int_leadingCoeff_eq : ↑(P.int R hP).leadingCoeff = P.leadingCoeff := rfl
+
+@[simp]
+theorem int_monic_iff : (P.int R hP).Monic ↔ P.Monic := by
+  rw [Monic, Monic, ← int_leadingCoeff_eq, OneMemClass.coe_eq_one]
+
+@[simp]
+theorem int_natDegree : (P.int R hP).natDegree = P.natDegree := rfl
+
+variable {L : Type*} [Field L] [Algebra K L]
+
+@[simp]
+theorem int_eval₂_eq (x : L) :
+    eval₂ (algebraMap R L) x (P.int R hP) = aeval x P := by
+  rw [aeval_eq_sum_range, eval₂_eq_sum_range]
+  exact Finset.sum_congr rfl (fun n _ => by rw [Algebra.smul_def]; rfl)
+
+end Polynomial
diff --git a/Mathlib/Algebra/Ring/Subring/MulOpposite.lean b/Mathlib/Algebra/Ring/Subring/MulOpposite.lean
index 77448007af326..3234c63eecfab 100644
--- a/Mathlib/Algebra/Ring/Subring/MulOpposite.lean
+++ b/Mathlib/Algebra/Ring/Subring/MulOpposite.lean
@@ -141,7 +141,7 @@ theorem op_closure (s : Set R) : (closure s).op = closure (MulOpposite.unop ⁻
 
 theorem unop_closure (s : Set Rᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) := by
   rw [← op_inj, op_unop, op_closure]
-  rfl
+  simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
 
 /-- Bijection between a subring `S` and its opposite. -/
 @[simps!]
diff --git a/Mathlib/Algebra/Ring/Subring/Pointwise.lean b/Mathlib/Algebra/Ring/Subring/Pointwise.lean
index 8dd52f6efdb1b..b472910f34900 100644
--- a/Mathlib/Algebra/Ring/Subring/Pointwise.lean
+++ b/Mathlib/Algebra/Ring/Subring/Pointwise.lean
@@ -6,7 +6,6 @@ Authors: Eric Wieser
 import Mathlib.Algebra.Group.Subgroup.Pointwise
 import Mathlib.Algebra.Ring.Subring.Basic
 import Mathlib.Algebra.Ring.Subsemiring.Pointwise
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-! # Pointwise instances on `Subring`s
 
diff --git a/Mathlib/Algebra/Ring/Subring/Units.lean b/Mathlib/Algebra/Ring/Subring/Units.lean
index 76b3f05824db9..157cdab1d1c19 100644
--- a/Mathlib/Algebra/Ring/Subring/Units.lean
+++ b/Mathlib/Algebra/Ring/Subring/Units.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Birkbeck
 -/
 import Mathlib.Algebra.Group.Subgroup.Basic
-import Mathlib.Algebra.Order.Monoid.Submonoid
+import Mathlib.Algebra.Order.GroupWithZero.Submonoid
 import Mathlib.Algebra.Order.Ring.Defs
 
 /-!
diff --git a/Mathlib/Algebra/Ring/Subsemiring/Basic.lean b/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
index 72f996c586990..3fd3c93c9cef4 100644
--- a/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
+++ b/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
@@ -27,7 +27,7 @@ section AddSubmonoidWithOneClass
 
 /-- `AddSubmonoidWithOneClass S R` says `S` is a type of subsets `s ≤ R` that contain `0`, `1`,
 and are closed under `(+)` -/
-class AddSubmonoidWithOneClass (S R : Type*) [AddMonoidWithOne R]
+class AddSubmonoidWithOneClass (S : Type*) (R : outParam Type*) [AddMonoidWithOne R]
   [SetLike S R] extends AddSubmonoidClass S R, OneMemClass S R : Prop
 
 variable {S R : Type*} [AddMonoidWithOne R] [SetLike S R] (s : S)
@@ -59,12 +59,12 @@ section SubsemiringClass
 
 /-- `SubsemiringClass S R` states that `S` is a type of subsets `s ⊆ R` that
 are both a multiplicative and an additive submonoid. -/
-class SubsemiringClass (S : Type*) (R : Type u) [NonAssocSemiring R]
+class SubsemiringClass (S : Type*) (R : outParam (Type u)) [NonAssocSemiring R]
   [SetLike S R] extends SubmonoidClass S R, AddSubmonoidClass S R : Prop
 
 -- See note [lower instance priority]
 instance (priority := 100) SubsemiringClass.addSubmonoidWithOneClass (S : Type*)
-    (R : Type u) [NonAssocSemiring R] [SetLike S R] [h : SubsemiringClass S R] :
+    (R : Type u) {_ : NonAssocSemiring R} [SetLike S R] [h : SubsemiringClass S R] :
     AddSubmonoidWithOneClass S R :=
   { h with }
 
@@ -500,6 +500,13 @@ theorem coe_sInf (S : Set (Subsemiring R)) : ((sInf S : Subsemiring R) : Set R)
 theorem mem_sInf {S : Set (Subsemiring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p :=
   Set.mem_iInter₂
 
+@[simp, norm_cast]
+theorem coe_iInf {ι : Sort*} {S : ι → Subsemiring R} : (↑(⨅ i, S i) : Set R) = ⋂ i, S i := by
+  simp only [iInf, coe_sInf, Set.biInter_range]
+
+theorem mem_iInf {ι : Sort*} {S : ι → Subsemiring R} {x : R} : (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by
+  simp only [iInf, mem_sInf, Set.forall_mem_range]
+
 @[simp]
 theorem sInf_toSubmonoid (s : Set (Subsemiring R)) :
     (sInf s).toSubmonoid = ⨅ t ∈ s, Subsemiring.toSubmonoid t :=
@@ -784,7 +791,7 @@ theorem mem_closure_iff_exists_list {R} [Semiring R] {s : Set R} {x} :
           ⟨[t], List.forall_mem_singleton.2 ht1, by
             rw [List.map_singleton, List.sum_singleton, ht2]⟩
         Submonoid.closure_induction hx
-          (fun x hx => ⟨[x], List.forall_mem_singleton.2 hx, one_mul x⟩)
+          (fun x hx => ⟨[x], List.forall_mem_singleton.2 hx, List.prod_singleton⟩)
           ⟨[], List.forall_mem_nil _, rfl⟩ fun x y ⟨t, ht1, ht2⟩ ⟨u, hu1, hu2⟩ =>
           ⟨t ++ u, List.forall_mem_append.2 ⟨ht1, hu1⟩, by rw [List.prod_append, ht2, hu2]⟩)
       ⟨[], List.forall_mem_nil _, rfl⟩ fun x y ⟨L, HL1, HL2⟩ ⟨M, HM1, HM2⟩ =>
@@ -835,6 +842,14 @@ theorem map_iSup {ι : Sort*} (f : R →+* S) (s : ι → Subsemiring R) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+theorem map_inf (s t : Subsemiring R) (f : R →+* S) (hf : Function.Injective f) :
+    (s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter hf)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : R →+* S) (hf : Function.Injective f)
+    (s : ι → Subsemiring R) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem comap_inf (s t : Subsemiring S) (f : R →+* S) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
   (gc_map_comap f).u_inf
 
diff --git a/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean b/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
index ca90d5e223c98..f709792084761 100644
--- a/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
+++ b/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean
@@ -147,7 +147,7 @@ theorem op_closure (s : Set R) : (closure s).op = closure (MulOpposite.unop ⁻
 
 theorem unop_closure (s : Set Rᵐᵒᵖ) : (closure s).unop = closure (MulOpposite.op ⁻¹' s) := by
   rw [← op_inj, op_unop, op_closure]
-  rfl
+  simp_rw [Set.preimage_preimage, MulOpposite.op_unop, Set.preimage_id']
 
 /-- Bijection between a subsemiring `S` and its opposite. -/
 @[simps!]
diff --git a/Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean b/Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean
index 842883e8a4b0b..250d388ca8b0a 100644
--- a/Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean
+++ b/Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean
@@ -6,7 +6,6 @@ Authors: Eric Wieser
 import Mathlib.Algebra.Group.Submonoid.Pointwise
 import Mathlib.Algebra.Ring.Action.Basic
 import Mathlib.Algebra.Ring.Subsemiring.Basic
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-! # Pointwise instances on `Subsemiring`s
 
diff --git a/Mathlib/Algebra/Ring/SumsOfSquares.lean b/Mathlib/Algebra/Ring/SumsOfSquares.lean
index 1eb9bb77744cf..6abec158dd334 100644
--- a/Mathlib/Algebra/Ring/SumsOfSquares.lean
+++ b/Mathlib/Algebra/Ring/SumsOfSquares.lean
@@ -3,9 +3,8 @@ Copyright (c) 2024 Florent Schaffhauser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Florent Schaffhauser
 -/
-import Mathlib.Algebra.Ring.Defs
+import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Algebra.Group.Submonoid.Basic
-import Mathlib.Algebra.Group.Even
 import Mathlib.Algebra.Order.Ring.Defs
 
 /-!
@@ -61,6 +60,15 @@ theorem IsSumSq.add [AddMonoid R] {S1 S2 : R} (p1 : IsSumSq S1)
 
 @[deprecated (since := "2024-08-09")] alias isSumSq.add := IsSumSq.add
 
+/-- A finite sum of squares is a sum of squares. -/
+theorem isSumSq_sum_mul_self {ι : Type*} [AddCommMonoid R] (s : Finset ι) (f : ι → R) :
+    IsSumSq (∑ i ∈ s, f i * f i) := by
+  induction s using Finset.cons_induction with
+  | empty =>
+    simpa only [Finset.sum_empty] using IsSumSq.zero
+  | cons i s his h =>
+    exact (Finset.sum_cons (β := R) his) ▸ IsSumSq.sq_add (f i) (∑ i ∈ s, f i * f i) h
+
 variable (R) in
 /--
 In an additive monoid with multiplication `R`, the type `sumSqIn R` is the submonoid of sums of
diff --git a/Mathlib/Algebra/Ring/ULift.lean b/Mathlib/Algebra/Ring/ULift.lean
index d22da7ec4c22e..b0d8c9a46ca7d 100644
--- a/Mathlib/Algebra/Ring/ULift.lean
+++ b/Mathlib/Algebra/Ring/ULift.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.ULift
 import Mathlib.Algebra.Ring.Equiv
diff --git a/Mathlib/Algebra/Ring/Units.lean b/Mathlib/Algebra/Ring/Units.lean
index fbcb29f67829f..619def14cb01a 100644
--- a/Mathlib/Algebra/Ring/Units.lean
+++ b/Mathlib/Algebra/Ring/Units.lean
@@ -15,7 +15,7 @@ import Mathlib.Algebra.Ring.Hom.Defs
 
 universe u v w x
 
-variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
+variable {α : Type u} {β : Type v} {R : Type x}
 
 open Function
 
@@ -23,7 +23,7 @@ namespace Units
 
 section HasDistribNeg
 
-variable [Monoid α] [HasDistribNeg α] {a b : α}
+variable [Monoid α] [HasDistribNeg α]
 
 /-- Each element of the group of units of a ring has an additive inverse. -/
 instance : Neg αˣ :=
@@ -49,7 +49,7 @@ end HasDistribNeg
 
 section Ring
 
-variable [Ring α] {a b : α}
+variable [Ring α]
 
 -- Needs to have higher simp priority than divp_add_divp. 1000 is the default priority.
 @[field_simps 1010]
diff --git a/Mathlib/Algebra/RingQuot.lean b/Mathlib/Algebra/RingQuot.lean
index f0e0f3bcd667b..1da65c2031d8e 100644
--- a/Mathlib/Algebra/RingQuot.lean
+++ b/Mathlib/Algebra/RingQuot.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.Hom
 import Mathlib.RingTheory.Ideal.Quotient
@@ -75,42 +75,42 @@ theorem Rel.smul {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : Rel r a b) :
 
 /-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
 def ringCon (r : R → R → Prop) : RingCon R where
-  r := EqvGen (Rel r)
-  iseqv := EqvGen.is_equivalence _
+  r := Relation.EqvGen (Rel r)
+  iseqv := Relation.EqvGen.is_equivalence _
   add' {a b c d} hab hcd := by
     induction hab generalizing c d with
     | rel _ _ hab =>
-      refine (EqvGen.rel _ _ hab.add_left).trans _ _ _ ?_
+      refine (Relation.EqvGen.rel _ _ hab.add_left).trans _ _ _ ?_
       induction hcd with
-      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
-      | refl => exact EqvGen.refl _
+      | rel _ _ hcd => exact Relation.EqvGen.rel _ _ hcd.add_right
+      | refl => exact Relation.EqvGen.refl _
       | symm _ _ _ h => exact h.symm _ _
       | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
     | refl => induction hcd with
-      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
-      | refl => exact EqvGen.refl _
+      | rel _ _ hcd => exact Relation.EqvGen.rel _ _ hcd.add_right
+      | refl => exact Relation.EqvGen.refl _
       | symm _ _ _ h => exact h.symm _ _
       | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
     | symm x y _ hxy => exact (hxy hcd.symm).symm
-    | trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <| EqvGen.refl _)
+    | trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <| Relation.EqvGen.refl _)
   mul' {a b c d} hab hcd := by
     induction hab generalizing c d with
     | rel _ _ hab =>
-      refine (EqvGen.rel _ _ hab.mul_left).trans _ _ _ ?_
+      refine (Relation.EqvGen.rel _ _ hab.mul_left).trans _ _ _ ?_
       induction hcd with
-      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.mul_right
-      | refl => exact EqvGen.refl _
+      | rel _ _ hcd => exact Relation.EqvGen.rel _ _ hcd.mul_right
+      | refl => exact Relation.EqvGen.refl _
       | symm _ _ _ h => exact h.symm _ _
       | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
     | refl => induction hcd with
-      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.mul_right
-      | refl => exact EqvGen.refl _
+      | rel _ _ hcd => exact Relation.EqvGen.rel _ _ hcd.mul_right
+      | refl => exact Relation.EqvGen.refl _
       | symm _ _ _ h => exact h.symm _ _
       | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
     | symm x y _ hxy => exact (hxy hcd.symm).symm
-    | trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <| EqvGen.refl _)
+    | trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <| Relation.EqvGen.refl _)
 
-theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by
+theorem eqvGen_rel_eq (r : R → R → Prop) : Relation.EqvGen (Rel r) = RingConGen.Rel r := by
   ext x₁ x₂
   constructor
   · intro h
@@ -125,7 +125,7 @@ theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r
     | trans => exact RingConGen.Rel.trans ‹_› ‹_›
   · intro h
     induction h with
-    | of => exact EqvGen.rel _ _ (Rel.of ‹_›)
+    | of => exact Relation.EqvGen.rel _ _ (Rel.of ‹_›)
     | refl => exact (RingQuot.ringCon r).refl _
     | symm => exact (RingQuot.ringCon r).symm ‹_›
     | trans => exact (RingQuot.ringCon r).trans ‹_› ‹_›
diff --git a/Mathlib/Algebra/SMulWithZero.lean b/Mathlib/Algebra/SMulWithZero.lean
index 3ef4bb96b5b9f..ae84ec92e750d 100644
--- a/Mathlib/Algebra/SMulWithZero.lean
+++ b/Mathlib/Algebra/SMulWithZero.lean
@@ -7,7 +7,6 @@ import Mathlib.Algebra.Group.Action.Opposite
 import Mathlib.Algebra.GroupWithZero.Action.Defs
 import Mathlib.Algebra.GroupWithZero.Hom
 import Mathlib.Algebra.GroupWithZero.Opposite
-import Mathlib.Algebra.Ring.Defs
 
 /-!
 # Introduce `SMulWithZero`
@@ -196,11 +195,6 @@ theorem smul_inv₀ [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (
   obtain rfl | hx := eq_or_ne x 0
   · simp only [inv_zero, smul_zero]
   · refine inv_eq_of_mul_eq_one_left ?_
-    rw [smul_mul_smul, inv_mul_cancel₀ hc, inv_mul_cancel₀ hx, one_smul]
+    rw [smul_mul_smul_comm, inv_mul_cancel₀ hc, inv_mul_cancel₀ hx, one_smul]
 
 end GroupWithZero
-
--- This instance seems a bit incongruous in this file, but `#find_home!` told me to put it here.
-instance NonUnitalNonAssocSemiring.toDistribSMul [NonUnitalNonAssocSemiring R] :
-    DistribSMul R R where
-  smul_add := mul_add
diff --git a/Mathlib/Algebra/Squarefree/Basic.lean b/Mathlib/Algebra/Squarefree/Basic.lean
index 4ea689f18ad28..2ae3fd080cb05 100644
--- a/Mathlib/Algebra/Squarefree/Basic.lean
+++ b/Mathlib/Algebra/Squarefree/Basic.lean
@@ -153,7 +153,7 @@ theorem squarefree_iff_irreducible_sq_not_dvd_of_ne_zero {r : R} (hr : r ≠ 0)
 theorem squarefree_iff_irreducible_sq_not_dvd_of_exists_irreducible {r : R}
     (hr : ∃ x : R, Irreducible x) : Squarefree r ↔ ∀ x : R, Irreducible x → ¬x * x ∣ r := by
   rw [irreducible_sq_not_dvd_iff_eq_zero_and_no_irreducibles_or_squarefree, ← not_exists]
-  simp only [hr, not_true, false_or_iff, and_false_iff]
+  simp only [hr, not_true, false_or, and_false]
 
 end Irreducible
 
diff --git a/Mathlib/Algebra/Star/Basic.lean b/Mathlib/Algebra/Star/Basic.lean
index c592e1abb4297..99fd9d0f28098 100644
--- a/Mathlib/Algebra/Star/Basic.lean
+++ b/Mathlib/Algebra/Star/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Field.Defs
 import Mathlib.Algebra.Group.Invertible.Defs
@@ -111,7 +111,7 @@ theorem star_eq_iff_star_eq [InvolutiveStar R] {r s : R} : star r = s ↔ star s
 /-- Typeclass for a trivial star operation. This is mostly meant for `ℝ`.
 -/
 class TrivialStar (R : Type u) [Star R] : Prop where
-  /-- Condition that star is trivial-/
+  /-- Condition that star is trivial -/
   star_trivial : ∀ r : R, star r = r
 
 export TrivialStar (star_trivial)
@@ -221,7 +221,7 @@ theorem star_id_of_comm {R : Type*} [CommSemiring R] {x : R} : star x = x :=
 end
 
 /-- A `*`-additive monoid `R` is an additive monoid with an involutive `star` operation which
-preserves addition.  -/
+preserves addition. -/
 class StarAddMonoid (R : Type u) [AddMonoid R] extends InvolutiveStar R where
   /-- `star` commutes with addition -/
   star_add : ∀ r s : R, star (r + s) = star r + star s
@@ -261,16 +261,16 @@ theorem star_sub [AddGroup R] [StarAddMonoid R] (r s : R) : star (r - s) = star
   (starAddEquiv : R ≃+ R).map_sub _ _
 
 @[simp]
-theorem star_nsmul [AddMonoid R] [StarAddMonoid R] (x : R) (n : ℕ) : star (n • x) = n • star x :=
+theorem star_nsmul [AddMonoid R] [StarAddMonoid R] (n : ℕ) (x : R) : star (n • x) = n • star x :=
   (starAddEquiv : R ≃+ R).toAddMonoidHom.map_nsmul _ _
 
 @[simp]
-theorem star_zsmul [AddGroup R] [StarAddMonoid R] (x : R) (n : ℤ) : star (n • x) = n • star x :=
+theorem star_zsmul [AddGroup R] [StarAddMonoid R] (n : ℤ) (x : R) : star (n • x) = n • star x :=
   (starAddEquiv : R ≃+ R).toAddMonoidHom.map_zsmul _ _
 
 /-- A `*`-ring `R` is a non-unital, non-associative (semi)ring with an involutive `star` operation
 which is additive which makes `R` with its multiplicative structure into a `*`-multiplication
-(i.e. `star (r * s) = star s * star r`).  -/
+(i.e. `star (r * s) = star s * star r`). -/
 class StarRing (R : Type u) [NonUnitalNonAssocSemiring R] extends StarMul R where
   /-- `star` commutes with addition -/
   star_add : ∀ r s : R, star (r + s) = star r + star s
@@ -415,13 +415,16 @@ attribute [simp] star_smul
 instance StarMul.toStarModule [CommMonoid R] [StarMul R] : StarModule R R :=
   ⟨star_mul'⟩
 
-instance StarAddMonoid.toStarModuleNat {α} [AddCommMonoid α] [StarAddMonoid α] : StarModule ℕ α :=
-  ⟨fun n a ↦ by rw [star_nsmul, star_trivial n]⟩
+instance StarAddMonoid.toStarModuleNat {α} [AddCommMonoid α] [StarAddMonoid α] :
+    StarModule ℕ α where star_smul := star_nsmul
+
+instance StarAddMonoid.toStarModuleInt {α} [AddCommGroup α] [StarAddMonoid α] : StarModule ℤ α where
+  star_smul := star_zsmul
 
 namespace RingHomInvPair
 
 /-- Instance needed to define star-linear maps over a commutative star ring
-(ex: conjugate-linear maps when R = ℂ).  -/
+(ex: conjugate-linear maps when R = ℂ). -/
 instance [CommSemiring R] [StarRing R] : RingHomInvPair (starRingEnd R) (starRingEnd R) :=
   ⟨RingHom.ext star_star, RingHom.ext star_star⟩
 
diff --git a/Mathlib/Algebra/Star/CHSH.lean b/Mathlib/Algebra/Star/CHSH.lean
index 72b30f5ea9329..26fc0b59288bb 100644
--- a/Mathlib/Algebra/Star/CHSH.lean
+++ b/Mathlib/Algebra/Star/CHSH.lean
@@ -1,10 +1,10 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.CharP.Invertible
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Data.Real.Sqrt
 import Mathlib.Tactic.Polyrith
 
diff --git a/Mathlib/Algebra/Star/Conjneg.lean b/Mathlib/Algebra/Star/Conjneg.lean
new file mode 100644
index 0000000000000..9b9c3b54defe4
--- /dev/null
+++ b/Mathlib/Algebra/Star/Conjneg.lean
@@ -0,0 +1,85 @@
+/-
+Copyright (c) 2024 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Algebra.BigOperators.Pi
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Star.Pi
+
+/-!
+# Conjugation-negation operator
+
+This file defines the conjugation-negation operator, useful in Fourier analysis.
+
+The way this operator enters the picture is that the adjoint of convolution with a function `f` is
+convolution with `conjneg f`.
+-/
+
+open Function
+open scoped ComplexConjugate
+
+variable {ι G R : Type*} [AddGroup G]
+
+section CommSemiring
+variable [CommSemiring R] [StarRing R] {f g : G → R}
+
+/-- Conjugation-negation. Sends `f` to `fun x ↦ conj (f (-x))`. -/
+def conjneg (f : G → R) : G → R := conj fun x ↦ f (-x)
+
+@[simp] lemma conjneg_apply (f : G → R) (x : G) : conjneg f x = conj (f (-x)) := rfl
+@[simp] lemma conjneg_conjneg (f : G → R) : conjneg (conjneg f) = f := by ext; simp
+
+lemma conjneg_involutive : Involutive (conjneg : (G → R) → G → R) := conjneg_conjneg
+lemma conjneg_bijective : Bijective (conjneg : (G → R) → G → R) := conjneg_involutive.bijective
+lemma conjneg_injective : Injective (conjneg : (G → R) → G → R) := conjneg_involutive.injective
+lemma conjneg_surjective : Surjective (conjneg : (G → R) → G → R) := conjneg_involutive.surjective
+
+@[simp] lemma conjneg_inj : conjneg f = conjneg g ↔ f = g := conjneg_injective.eq_iff
+lemma conjneg_ne_conjneg : conjneg f ≠ conjneg g ↔ f ≠ g := conjneg_injective.ne_iff
+
+@[simp] lemma conjneg_conj (f : G → R) : conjneg (conj f) = conj (conjneg f) := rfl
+
+@[simp] lemma conjneg_zero : conjneg (0 : G → R) = 0 := by ext; simp
+@[simp] lemma conjneg_one : conjneg (1 : G → R) = 1 := by ext; simp
+@[simp] lemma conjneg_add (f g : G → R) : conjneg (f + g) = conjneg f + conjneg g := by ext; simp
+@[simp] lemma conjneg_mul (f g : G → R) : conjneg (f * g) = conjneg f * conjneg g := by ext; simp
+
+@[simp] lemma conjneg_sum (s : Finset ι) (f : ι → G → R) :
+    conjneg (∑ i ∈ s, f i) = ∑ i ∈ s, conjneg (f i) := by ext; simp
+
+@[simp] lemma conjneg_prod (s : Finset ι) (f : ι → G → R) :
+    conjneg (∏ i ∈ s, f i) = ∏ i ∈ s, conjneg (f i) := by ext; simp
+
+@[simp] lemma conjneg_eq_zero : conjneg f = 0 ↔ f = 0 := by
+  rw [← conjneg_inj, conjneg_conjneg, conjneg_zero]
+
+@[simp] lemma conjneg_eq_one : conjneg f = 1 ↔ f = 1 := by
+  rw [← conjneg_inj, conjneg_conjneg, conjneg_one]
+
+lemma conjneg_ne_zero : conjneg f ≠ 0 ↔ f ≠ 0 := conjneg_eq_zero.not
+lemma conjneg_ne_one : conjneg f ≠ 1 ↔ f ≠ 1 := conjneg_eq_one.not
+
+lemma sum_conjneg [Fintype G] (f : G → R) : ∑ a, conjneg f a = ∑ a, conj (f a) :=
+  Fintype.sum_equiv (Equiv.neg _) _ _ fun _ ↦ rfl
+
+@[simp] lemma support_conjneg (f : G → R) : support (conjneg f) = -support f := by
+  ext; simp [starRingEnd_apply]
+
+/-- `conjneg` bundled as a ring homomorphism. -/
+@[simps] def conjnegRingHom : (G → R) →+* (G → R) where
+  toFun := conjneg
+  map_zero' := conjneg_zero
+  map_one' := conjneg_one
+  map_add' := conjneg_add
+  map_mul' := conjneg_mul
+
+end CommSemiring
+
+section CommRing
+variable [CommRing R] [StarRing R]
+
+@[simp] lemma conjneg_sub (f g : G → R) : conjneg (f - g) = conjneg f - conjneg g := by ext; simp
+@[simp] lemma conjneg_neg (f : G → R) : conjneg (-f) = -conjneg f := by ext; simp
+
+end CommRing
diff --git a/Mathlib/Algebra/Star/Free.lean b/Mathlib/Algebra/Star/Free.lean
index eefbd4ba0c208..b29ed2137d81e 100644
--- a/Mathlib/Algebra/Star/Free.lean
+++ b/Mathlib/Algebra/Star/Free.lean
@@ -48,7 +48,7 @@ instance : StarRing (FreeAlgebra R X) where
     unfold Star.star
     simp only [Function.comp_apply]
     let y := lift R (X := X) (MulOpposite.op ∘ ι R)
-    apply induction (C := fun x ↦ (y (y x).unop).unop = x) _ _ _ _ x
+    refine induction (C := fun x ↦ (y (y x).unop).unop = x) _ _ ?_ ?_ ?_ ?_ x
     · intros
       simp only [AlgHom.commutes, MulOpposite.algebraMap_apply, MulOpposite.unop_op]
     · intros
diff --git a/Mathlib/Algebra/Star/Module.lean b/Mathlib/Algebra/Star/Module.lean
index b062492e6acbd..214d513fed01a 100644
--- a/Mathlib/Algebra/Star/Module.lean
+++ b/Mathlib/Algebra/Star/Module.lean
@@ -75,11 +75,43 @@ theorem star_ratCast_smul [DivisionRing R] [AddCommGroup M] [Module R M] [StarAd
 @[deprecated (since := "2024-04-17")]
 alias star_rat_cast_smul := star_ratCast_smul
 
-@[simp]
-theorem star_rat_smul {R : Type*} [AddCommGroup R] [StarAddMonoid R] [Module ℚ R] (x : R) (n : ℚ) :
-    star (n • x) = n • star x :=
+/-!
+Per the naming convention, these two lemmas call `(q • ·)` `nnrat_smul` and `rat_smul` respectively,
+rather than `nnqsmul` and `qsmul` because the latter are reserved to the actions coming from
+`DivisionSemiring` and `DivisionRing`. We provide aliases with `nnqsmul` and `qsmul` for
+discoverability.
+-/
+
+/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a
+`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for
+discoverability. See `star_nnqsmul`. -/
+@[simp high]
+lemma star_nnrat_smul [AddCommMonoid R] [StarAddMonoid R] [Module ℚ≥0 R] (q : ℚ≥0) (x : R) :
+    star (q • x) = q • star x := map_nnrat_smul (starAddEquiv : R ≃+ R) _ _
+
+/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a
+`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for discoverability.
+See `star_qsmul`. -/
+@[simp high] lemma star_rat_smul [AddCommGroup R] [StarAddMonoid R] [Module ℚ R] (q : ℚ) (x : R) :
+    star (q • x) = q • star x :=
   map_rat_smul (starAddEquiv : R ≃+ R) _ _
 
+/-- Note that this lemma holds for an arbitrary `ℚ≥0`-action, rather than merely one coming from a
+`DivisionSemiring`. We keep both the `nnqsmul` and `nnrat_smul` naming conventions for
+discoverability. See `star_nnrat_smul`. -/
+alias star_nnqsmul := star_nnrat_smul
+
+/-- Note that this lemma holds for an arbitrary `ℚ`-action, rather than merely one coming from a
+`DivisionRing`. We keep both the `qsmul` and `rat_smul` naming conventions for
+discoverability. See `star_rat_smul`. -/
+alias star_qsmul := star_rat_smul
+
+instance StarAddMonoid.toStarModuleNNRat [AddCommMonoid R] [Module ℚ≥0 R] [StarAddMonoid R] :
+    StarModule ℚ≥0 R where star_smul := star_nnrat_smul
+
+instance StarAddMonoid.toStarModuleRat [AddCommGroup R] [Module ℚ R] [StarAddMonoid R] :
+    StarModule ℚ R where star_smul := star_rat_smul
+
 end SMulLemmas
 
 /-- If `A` is a module over a commutative `R` with compatible actions,
@@ -184,8 +216,8 @@ def StarModule.decomposeProdAdjoint : A ≃ₗ[R] selfAdjoint A × skewAdjoint A
   refine LinearEquiv.ofLinear ((selfAdjointPart R).prod (skewAdjointPart R))
     (LinearMap.coprod ((selfAdjoint.submodule R A).subtype) (skewAdjoint.submodule R A).subtype)
     ?_ (LinearMap.ext <| StarModule.selfAdjointPart_add_skewAdjointPart R)
-  -- Note: with #6965 `Submodule.coeSubtype` doesn't fire in `dsimp` or `simp`
-  ext x <;> dsimp <;> erw [Submodule.coeSubtype, Submodule.coeSubtype] <;> simp
+  -- Note: with #6965 `Submodule.coe_subtype` doesn't fire in `dsimp` or `simp`
+  ext x <;> dsimp <;> erw [Submodule.coe_subtype, Submodule.coe_subtype] <;> simp
 
 end SelfSkewAdjoint
 
diff --git a/Mathlib/Algebra/Star/NonUnitalSubalgebra.lean b/Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
index b0c9fd1f0330b..61e7ac629195f 100644
--- a/Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
+++ b/Mathlib/Algebra/Star/NonUnitalSubalgebra.lean
@@ -97,7 +97,7 @@ variable [CommSemiring R]
 variable [NonUnitalNonAssocSemiring A] [Module R A] [Star A]
 variable [NonUnitalNonAssocSemiring B] [Module R B] [Star B]
 variable [NonUnitalNonAssocSemiring C] [Module R C] [Star C]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
 
 instance instSetLike : SetLike (NonUnitalStarSubalgebra R A) A where
   coe {s} := s.carrier
@@ -398,7 +398,7 @@ variable [CommSemiring R]
 variable [NonUnitalNonAssocSemiring A] [Module R A] [Star A]
 variable [NonUnitalNonAssocSemiring B] [Module R B] [Star B]
 variable [NonUnitalNonAssocSemiring C] [Module R C] [Star C]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
 
 /-- Range of an `NonUnitalAlgHom` as a `NonUnitalStarSubalgebra`. -/
 protected def range (φ : F) : NonUnitalStarSubalgebra R B where
@@ -471,7 +471,7 @@ variable [CommSemiring R]
 variable [NonUnitalSemiring A] [Module R A] [Star A]
 variable [NonUnitalSemiring B] [Module R B] [Star B]
 variable [NonUnitalSemiring C] [Module R C] [Star C]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
 
 /-- Restrict a non-unital star algebra homomorphism with a left inverse to an algebra isomorphism
 to its range.
@@ -600,7 +600,7 @@ namespace NonUnitalStarAlgebra
 variable [CommSemiring R] [StarRing R]
 variable [NonUnitalSemiring A] [StarRing A] [Module R A]
 variable [NonUnitalSemiring B] [StarRing B] [Module R B]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
 
 section StarSubAlgebraA
 
@@ -733,6 +733,11 @@ theorem map_sup [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] (f
     ((S ⊔ T).map f : NonUnitalStarSubalgebra R B) = S.map f ⊔ T.map f :=
   (NonUnitalStarSubalgebra.gc_map_comap f).l_sup
 
+theorem map_inf [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] (f : F)
+    (hf : Function.Injective f) (S T : NonUnitalStarSubalgebra R A) :
+    ((S ⊓ T).map f : NonUnitalStarSubalgebra R B) = S.map f ⊓ T.map f :=
+  SetLike.coe_injective (Set.image_inter hf)
+
 @[simp, norm_cast]
 theorem coe_inf (S T : NonUnitalStarSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=
   rfl
@@ -766,6 +771,13 @@ theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} :
 theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalStarSubalgebra R A} {x : A} :
     (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range]
 
+theorem map_iInf {ι : Sort*} [Nonempty ι]
+    [IsScalarTower R B B] [SMulCommClass R B B] [StarModule R B] (f : F)
+    (hf : Function.Injective f) (S : ι → NonUnitalStarSubalgebra R A) :
+    ((⨅ i, S i).map f : NonUnitalStarSubalgebra R B) = ⨅ i, (S i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ S)
+
 @[simp]
 theorem iInf_toNonUnitalSubalgebra {ι : Sort*} (S : ι → NonUnitalStarSubalgebra R A) :
     (⨅ i, S i).toNonUnitalSubalgebra = ⨅ i, (S i).toNonUnitalSubalgebra :=
@@ -786,7 +798,7 @@ theorem toNonUnitalSubalgebra_bot :
 @[simp]
 theorem coe_bot : ((⊥ : NonUnitalStarSubalgebra R A) : Set A) = {0} := by
   simp only [Set.ext_iff, NonUnitalStarAlgebra.mem_bot, SetLike.mem_coe, Set.mem_singleton_iff,
-    iff_self_iff, forall_const]
+    forall_const]
 
 theorem eq_top_iff {S : NonUnitalStarSubalgebra R A} : S = ⊤ ↔ ∀ x : A, x ∈ S :=
   ⟨fun h x => by rw [h]; exact mem_top,
@@ -831,7 +843,7 @@ open NonUnitalStarAlgebra
 variable [CommSemiring R]
 variable [NonUnitalSemiring A] [StarRing A] [Module R A]
 variable [NonUnitalSemiring B] [StarRing B] [Module R B]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
 variable (S : NonUnitalStarSubalgebra R A)
 
 section StarSubalgebra
diff --git a/Mathlib/Algebra/Star/Pointwise.lean b/Mathlib/Algebra/Star/Pointwise.lean
index 7594743c491d5..6c37067eee6ab 100644
--- a/Mathlib/Algebra/Star/Pointwise.lean
+++ b/Mathlib/Algebra/Star/Pointwise.lean
@@ -3,9 +3,9 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Star.Basic
 import Mathlib.Data.Set.Finite
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Pointwise star operation on sets
diff --git a/Mathlib/Algebra/Star/SelfAdjoint.lean b/Mathlib/Algebra/Star/SelfAdjoint.lean
index f123298542131..32776d896fc45 100644
--- a/Mathlib/Algebra/Star/SelfAdjoint.lean
+++ b/Mathlib/Algebra/Star/SelfAdjoint.lean
@@ -91,15 +91,20 @@ lemma commute_iff {R : Type*} [Mul R] [StarMul R] {x y : R}
   · simpa only [star_mul, hx.star_eq, hy.star_eq] using h.symm
 
 /-- Functions in a `StarHomClass` preserve self-adjoint elements. -/
-theorem starHom_apply {F R S : Type*} [Star R] [Star S] [FunLike F R S] [StarHomClass F R S]
+@[aesop 10% apply]
+theorem map {F R S : Type*} [Star R] [Star S] [FunLike F R S] [StarHomClass F R S]
     {x : R} (hx : IsSelfAdjoint x) (f : F) : IsSelfAdjoint (f x) :=
   show star (f x) = f x from map_star f x ▸ congr_arg f hx
 
+@[deprecated (since := "2024-09-07")] alias starHom_apply := map
+
 /- note: this lemma is *not* marked as `simp` so that Lean doesn't look for a `[TrivialStar R]`
 instance every time it sees `⊢ IsSelfAdjoint (f x)`, which will likely occur relatively often. -/
-theorem _root_.isSelfAdjoint_starHom_apply {F R S : Type*} [Star R] [Star S] [FunLike F R S]
+theorem _root_.isSelfAdjoint_map {F R S : Type*} [Star R] [Star S] [FunLike F R S]
     [StarHomClass F R S] [TrivialStar R] (f : F) (x : R) : IsSelfAdjoint (f x) :=
-  (IsSelfAdjoint.all x).starHom_apply f
+  (IsSelfAdjoint.all x).map f
+
+@[deprecated (since := "2024-09-07")] alias _root_.isSelfAdjoint_starHom_apply := isSelfAdjoint_map
 
 section AddMonoid
 
@@ -156,6 +161,10 @@ theorem conjugate {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (z * x
 theorem conjugate' {x : R} (hx : IsSelfAdjoint x) (z : R) : IsSelfAdjoint (star z * x * z) := by
   simp only [isSelfAdjoint_iff, star_mul, star_star, mul_assoc, hx.star_eq]
 
+@[aesop 90% apply]
+theorem conjugate_self {x : R} (hx : IsSelfAdjoint x) {z : R} (hz : IsSelfAdjoint z) :
+    IsSelfAdjoint (z * x * z) := by nth_rewrite 2 [← hz]; exact conjugate hx z
+
 @[aesop 10% apply]
 theorem isStarNormal {x : R} (hx : IsSelfAdjoint x) : IsStarNormal x :=
   ⟨by simp only [Commute, SemiconjBy, hx.star_eq]⟩
diff --git a/Mathlib/Algebra/Star/StarAlgHom.lean b/Mathlib/Algebra/Star/StarAlgHom.lean
index 09ee718f7b1c2..3a2986c6d730c 100644
--- a/Mathlib/Algebra/Star/StarAlgHom.lean
+++ b/Mathlib/Algebra/Star/StarAlgHom.lean
@@ -64,6 +64,7 @@ add_decl_doc NonUnitalStarAlgHom.toNonUnitalAlgHom
 
 /-- `NonUnitalStarAlgHomClass F R A B` asserts `F` is a type of bundled non-unital ⋆-algebra
 homomorphisms from `A` to `B`. -/
+@[deprecated StarHomClass (since := "2024-09-08")]
 class NonUnitalStarAlgHomClass (F : Type*) (R A B : outParam Type*)
   [Monoid R] [Star A] [Star B] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B]
   [DistribMulAction R A] [DistribMulAction R B] [FunLike F A B] [NonUnitalAlgHomClass F R A B]
@@ -76,17 +77,18 @@ variable [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A]
 variable [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B]
 variable [FunLike F A B] [NonUnitalAlgHomClass F R A B]
 
-/-- Turn an element of a type `F` satisfying `NonUnitalStarAlgHomClass F R A B` into an actual
-`NonUnitalStarAlgHom`. This is declared as the default coercion from `F` to `A →⋆ₙₐ[R] B`. -/
+/-- Turn an element of a type `F` satisfying `NonUnitalAlgHomClass F R A B` and `StarHomClass F A B`
+into an actual `NonUnitalStarAlgHom`. This is declared as the default coercion from `F` to
+`A →⋆ₙₐ[R] B`. -/
 @[coe]
-def toNonUnitalStarAlgHom [NonUnitalStarAlgHomClass F R A B] (f : F) : A →⋆ₙₐ[R] B :=
+def toNonUnitalStarAlgHom [StarHomClass F A B] (f : F) : A →⋆ₙₐ[R] B :=
   { (f : A →ₙₐ[R] B) with
     map_star' := map_star f }
 
-instance [NonUnitalStarAlgHomClass F R A B] : CoeTC F (A →⋆ₙₐ[R] B) :=
+instance [StarHomClass F A B] : CoeTC F (A →⋆ₙₐ[R] B) :=
   ⟨toNonUnitalStarAlgHom⟩
 
-instance [NonUnitalStarAlgHomClass F R A B] : NonUnitalStarRingHomClass F A B :=
+instance [StarHomClass F A B] : NonUnitalStarRingHomClass F A B :=
   NonUnitalStarRingHomClass.mk
 
 end NonUnitalStarAlgHomClass
@@ -111,7 +113,7 @@ instance : NonUnitalAlgHomClass (A →⋆ₙₐ[R] B) R A B where
   map_zero f := f.map_zero'
   map_mul f := f.map_mul'
 
-instance : NonUnitalStarAlgHomClass (A →⋆ₙₐ[R] B) R A B where
+instance : StarHomClass (A →⋆ₙₐ[R] B) A B where
   map_star f := f.map_star'
 
 -- Porting note: in mathlib3 we didn't need the `Simps.apply` hint.
@@ -123,7 +125,7 @@ initialize_simps_projections NonUnitalStarAlgHom
 
 @[simp]
 protected theorem coe_coe {F : Type*} [FunLike F A B] [NonUnitalAlgHomClass F R A B]
-    [NonUnitalStarAlgHomClass F R A B] (f : F) :
+    [StarHomClass F A B] (f : F) :
     ⇑(f : A →⋆ₙₐ[R] B) = f := rfl
 
 @[simp]
@@ -263,7 +265,7 @@ variable (R : Type*) {S A B : Type*} [Monoid R] [Monoid S] [Star A] [Star B]
     [IsScalarTower R S A] [IsScalarTower R S B]
 
 /-- If a monoid `R` acts on another monoid `S`, then a non-unital star algebra homomorphism
-over `S` can be viewed as a non-unital star algebra homomorphism over `R`.  -/
+over `S` can be viewed as a non-unital star algebra homomorphism over `R`. -/
 def restrictScalars (f : A →⋆ₙₐ[S] B) : A →⋆ₙₐ[R] B :=
   { (f : A →ₙₐ[S] B).restrictScalars R with
     map_star' := map_star f }
@@ -304,34 +306,21 @@ by forgetting the interaction with the star operation. -/
 add_decl_doc StarAlgHom.toAlgHom
 
 /-- `StarAlgHomClass F R A B` states that `F` is a type of ⋆-algebra homomorphisms.
-
 You should also extend this typeclass when you extend `StarAlgHom`. -/
+@[deprecated StarHomClass (since := "2024-09-08")]
 class StarAlgHomClass (F : Type*) (R A B : outParam Type*)
     [CommSemiring R] [Semiring A] [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B]
     [FunLike F A B] [AlgHomClass F R A B] extends StarHomClass F A B : Prop
-
--- Porting note: no longer needed
----- `R` becomes a metavariable but that's fine because it's an `outParam`
---attribute [nolint dangerousInstance] StarAlgHomClass.toStarHomClass
-
 namespace StarAlgHomClass
 
-variable (F R A B : Type*)
-
--- See note [lower instance priority]
-instance (priority := 100) toNonUnitalStarAlgHomClass [CommSemiring R] [Semiring A] [Algebra R A]
-  [Star A] [Semiring B] [Algebra R B] [Star B] [FunLike F A B] [AlgHomClass F R A B]
-  [StarAlgHomClass F R A B] :
-  NonUnitalStarAlgHomClass F R A B :=
-  { }
+variable {F R A B : Type*}
 
 variable [CommSemiring R] [Semiring A] [Algebra R A] [Star A]
 variable [Semiring B] [Algebra R B] [Star B] [FunLike F A B] [AlgHomClass F R A B]
-variable [StarAlgHomClass F R A B]
+variable [StarHomClass F A B]
 
-variable {F R A B} in
-/-- Turn an element of a type `F` satisfying `StarAlgHomClass F R A B` into an actual
-`StarAlgHom`. This is declared as the default coercion from `F` to `A →⋆ₐ[R] B`. -/
+/-- Turn an element of a type `F` satisfying `AlgHomClass F R A B` and `StarHomClass F A B` into an
+actual `StarAlgHom`. This is declared as the default coercion from `F` to `A →⋆ₐ[R] B`. -/
 @[coe]
 def toStarAlgHom (f : F) : A →⋆ₐ[R] B :=
   { (f : A →ₐ[R] B) with
@@ -358,12 +347,12 @@ instance : AlgHomClass (A →⋆ₐ[R] B) R A B where
   map_zero f := f.map_zero'
   commutes f := f.commutes'
 
-instance : StarAlgHomClass (A →⋆ₐ[R] B) R A B where
+instance : StarHomClass (A →⋆ₐ[R] B) A B where
   map_star f := f.map_star'
 
 @[simp]
 protected theorem coe_coe {F : Type*} [FunLike F A B] [AlgHomClass F R A B]
-    [StarAlgHomClass F R A B] (f : F) :
+    [StarHomClass F A B] (f : F) :
     ⇑(f : A →⋆ₐ[R] B) = f :=
   rfl
 
@@ -660,27 +649,16 @@ class NonUnitalAlgEquivClass (F : Type*) (R A B : outParam Type*)
 
 /-- `StarAlgEquivClass F R A B` asserts `F` is a type of bundled ⋆-algebra equivalences between
 `A` and `B`.
-
 You should also extend this typeclass when you extend `StarAlgEquiv`. -/
+@[deprecated StarHomClass (since := "2024-09-08")]
 class StarAlgEquivClass (F : Type*) (R A B : outParam Type*)
   [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B]
   [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] : Prop where
   /-- By definition, a ⋆-algebra equivalence preserves the `star` operation. -/
-  map_star : ∀ (f : F) (a : A), f (star a) = star (f a)
-
--- Porting note: no longer needed
----- `R` becomes a metavariable but that's fine because it's an `outParam`
--- attribute [nolint dangerousInstance] StarAlgEquivClass.toRingEquivClass
+  protected map_star : ∀ (f : F) (a : A), f (star a) = star (f a)
 
 namespace StarAlgEquivClass
 
--- See note [lower instance priority]
-instance (priority := 50) {F R A B : Type*} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B]
-    [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B]
-    [hF : StarAlgEquivClass F R A B] :
-    StarHomClass F A B :=
-  { hF with }
-
 -- See note [lower instance priority]
 instance (priority := 100) {F R A B : Type*} [Monoid R] [NonUnitalNonAssocSemiring A]
     [DistribMulAction R A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [EquivLike F A B]
@@ -688,41 +666,27 @@ instance (priority := 100) {F R A B : Type*} [Monoid R] [NonUnitalNonAssocSemiri
     NonUnitalAlgHomClass F R A B :=
   { }
 
--- See note [lower instance priority]
-instance (priority := 100) {F R A B : Type*} [Monoid R] [NonUnitalNonAssocSemiring A]
-    [DistribMulAction R A] [Star A] [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B]
-    [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarAlgEquivClass F R A B] :
-    NonUnitalStarAlgHomClass F R A B :=
-  { }
-
 -- See note [lower instance priority]
 instance (priority := 100) instAlgHomClass (F R A B : Type*) [CommSemiring R] [Semiring A]
     [Algebra R A] [Semiring B] [Algebra R B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] :
     AlgEquivClass F R A B :=
   { commutes := fun f r => by simp only [Algebra.algebraMap_eq_smul_one, map_smul, map_one] }
 
--- See note [lower instance priority]
-instance (priority := 100) instStarAlgHomClass (F R A B : Type*) [CommSemiring R] [Semiring A]
-    [Algebra R A] [Star A] [Semiring B] [Algebra R B] [Star B] [EquivLike F A B]
-    [NonUnitalAlgEquivClass F R A B] [StarAlgEquivClass F R A B] :
-    StarAlgHomClass F R A B :=
-  { }
-
-/-- Turn an element of a type `F` satisfying `StarAlgEquivClass F R A B` into an actual
-`StarAlgEquiv`. This is declared as the default coercion from `F` to `A ≃⋆ₐ[R] B`. -/
+/-- Turn an element of a type `F` satisfying `AlgEquivClass F R A B` and `StarHomClass F A B` into
+an actual `StarAlgEquiv`. This is declared as the default coercion from `F` to `A ≃⋆ₐ[R] B`. -/
 @[coe]
 def toStarAlgEquiv {F R A B : Type*} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B] [SMul R B]
-    [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarAlgEquivClass F R A B]
+    [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarHomClass F A B]
     (f : F) : A ≃⋆ₐ[R] B :=
   { (f : A ≃+* B) with
     map_star' := map_star f
     map_smul' := map_smul f}
 
-/-- Any type satisfying `StarAlgEquivClass` can be cast into `StarAlgEquiv` via
+/-- Any type satisfying `AlgEquivClass` and `StarHomClass` can be cast into `StarAlgEquiv` via
 `StarAlgEquivClass.toStarAlgEquiv`. -/
 instance instCoeHead {F R A B : Type*} [Add A] [Mul A] [SMul R A] [Star A] [Add B] [Mul B]
-    [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B]
-    [StarAlgEquivClass F R A B] : CoeHead F (A ≃⋆ₐ[R] B) :=
+    [SMul R B] [Star B] [EquivLike F A B] [NonUnitalAlgEquivClass F R A B] [StarHomClass F A B] :
+    CoeHead F (A ≃⋆ₐ[R] B) :=
   ⟨toStarAlgEquiv⟩
 
 end StarAlgEquivClass
@@ -749,7 +713,7 @@ instance : NonUnitalAlgEquivClass (A ≃⋆ₐ[R] B) R A B where
   map_add f := f.map_add'
   map_smulₛₗ := map_smul'
 
-instance : StarAlgEquivClass (A ≃⋆ₐ[R] B) R A B where
+instance : StarHomClass (A ≃⋆ₐ[R] B) A B where
   map_star := map_star'
 
 /-- Helper instance for cases where the inference via `EquivLike` is too hard. -/
@@ -889,8 +853,8 @@ section Bijective
 variable {F G R A B : Type*} [Monoid R]
 variable [NonUnitalNonAssocSemiring A] [DistribMulAction R A] [Star A]
 variable [NonUnitalNonAssocSemiring B] [DistribMulAction R B] [Star B]
-variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [NonUnitalStarAlgHomClass F R A B]
-variable [FunLike G B A] [NonUnitalAlgHomClass G R B A] [NonUnitalStarAlgHomClass G R B A]
+variable [FunLike F A B] [NonUnitalAlgHomClass F R A B] [StarHomClass F A B]
+variable [FunLike G B A] [NonUnitalAlgHomClass G R B A] [StarHomClass G B A]
 
 /-- If a (unital or non-unital) star algebra morphism has an inverse, it is an isomorphism of
 star algebras. -/
diff --git a/Mathlib/Algebra/Star/Subalgebra.lean b/Mathlib/Algebra/Star/Subalgebra.lean
index 380f87763c95d..7d0b23cd8f043 100644
--- a/Mathlib/Algebra/Star/Subalgebra.lean
+++ b/Mathlib/Algebra/Star/Subalgebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Jireh Loreaux
+Authors: Kim Morrison, Jireh Loreaux
 -/
 import Mathlib.Algebra.Star.Center
 import Mathlib.Algebra.Star.StarAlgHom
@@ -594,6 +594,9 @@ theorem mul_mem_sup {S T : StarSubalgebra R A} {x y : A} (hx : x ∈ S) (hy : y
 theorem map_sup (f : A →⋆ₐ[R] B) (S T : StarSubalgebra R A) : map f (S ⊔ T) = map f S ⊔ map f T :=
   (StarSubalgebra.gc_map_comap f).l_sup
 
+theorem map_inf (f : A →⋆ₐ[R] B) (hf : Function.Injective f) (S T : StarSubalgebra R A) :
+    map f (S ⊓ T) = map f S ⊓ map f T := SetLike.coe_injective (Set.image_inter hf)
+
 @[simp, norm_cast]
 theorem coe_inf (S T : StarSubalgebra R A) : (↑(S ⊓ T) : Set A) = (S : Set A) ∩ T :=
   rfl
@@ -627,6 +630,11 @@ theorem coe_iInf {ι : Sort*} {S : ι → StarSubalgebra R A} : (↑(⨅ i, S i)
 theorem mem_iInf {ι : Sort*} {S : ι → StarSubalgebra R A} {x : A} :
     (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by simp only [iInf, mem_sInf, Set.forall_mem_range]
 
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : A →⋆ₐ[R] B) (hf : Function.Injective f)
+    (s : ι → StarSubalgebra R A) : map f (iInf s) = ⨅ (i : ι), map f (s i) := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 @[simp]
 theorem iInf_toSubalgebra {ι : Sort*} (S : ι → StarSubalgebra R A) :
     (⨅ i, S i).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
@@ -657,7 +665,7 @@ section
 variable [StarModule R A]
 
 theorem ext_adjoin {s : Set A} [FunLike F (adjoin R s) B]
-    [AlgHomClass F R (adjoin R s) B] [StarAlgHomClass F R (adjoin R s) B] {f g : F}
+    [AlgHomClass F R (adjoin R s) B] [StarHomClass F (adjoin R s) B] {f g : F}
     (h : ∀ x : adjoin R s, (x : A) ∈ s → f x = g x) : f = g := by
   refine DFunLike.ext f g fun a =>
     adjoin_induction' (p := fun y => f y = g y) a (fun x hx => ?_) (fun r => ?_)
@@ -669,7 +677,7 @@ theorem ext_adjoin {s : Set A} [FunLike F (adjoin R s) B]
   · simp only [map_star, hx]
 
 theorem ext_adjoin_singleton {a : A} [FunLike F (adjoin R ({a} : Set A)) B]
-    [AlgHomClass F R (adjoin R ({a} : Set A)) B] [StarAlgHomClass F R (adjoin R ({a} : Set A)) B]
+    [AlgHomClass F R (adjoin R ({a} : Set A)) B] [StarHomClass F (adjoin R ({a} : Set A)) B]
     {f g : F} (h : f ⟨a, self_mem_adjoin_singleton R a⟩ = g ⟨a, self_mem_adjoin_singleton R a⟩) :
     f = g :=
   ext_adjoin fun x hx =>
@@ -677,7 +685,7 @@ theorem ext_adjoin_singleton {a : A} [FunLike F (adjoin R ({a} : Set A)) B]
           Subtype.ext <| Set.mem_singleton_iff.mp hx).symm ▸
       h
 
-variable [FunLike F A B] [AlgHomClass F R A B] [StarAlgHomClass F R A B] (f g : F)
+variable [FunLike F A B] [AlgHomClass F R A B] [StarHomClass F A B] (f g : F)
 
 /-- The equalizer of two star `R`-algebra homomorphisms. -/
 def equalizer : StarSubalgebra R A :=
diff --git a/Mathlib/Algebra/Symmetrized.lean b/Mathlib/Algebra/Symmetrized.lean
index a99928f48d53c..f4dea0e4e8345 100644
--- a/Mathlib/Algebra/Symmetrized.lean
+++ b/Mathlib/Algebra/Symmetrized.lean
@@ -255,9 +255,9 @@ instance nonAssocSemiring [Semiring α] [Invertible (2 : α)] : NonAssocSemiring
       rw [mul_def, unsym_zero, zero_mul, mul_zero, add_zero,
         mul_zero, sym_zero]
     mul_one := fun _ => by
-      rw [mul_def, unsym_one, mul_one, one_mul, ← two_mul, invOf_mul_self_assoc, sym_unsym]
+      rw [mul_def, unsym_one, mul_one, one_mul, ← two_mul, invOf_mul_cancel_left, sym_unsym]
     one_mul := fun _ => by
-      rw [mul_def, unsym_one, mul_one, one_mul, ← two_mul, invOf_mul_self_assoc, sym_unsym]
+      rw [mul_def, unsym_one, mul_one, one_mul, ← two_mul, invOf_mul_cancel_left, sym_unsym]
     left_distrib := fun a b c => by
       -- Porting note: rewrote previous proof which used `match` in a way that seems unsupported.
       rw [mul_def, mul_def, mul_def, ← sym_add, ← mul_add, unsym_add, add_mul]
@@ -279,10 +279,11 @@ instance [Ring α] [Invertible (2 : α)] : NonAssocRing αˢʸᵐ :=
 
 
 theorem unsym_mul_self [Semiring α] [Invertible (2 : α)] (a : αˢʸᵐ) :
-    unsym (a * a) = unsym a * unsym a := by rw [mul_def, unsym_sym, ← two_mul, invOf_mul_self_assoc]
+    unsym (a * a) = unsym a * unsym a := by
+  rw [mul_def, unsym_sym, ← two_mul, invOf_mul_cancel_left]
 
 theorem sym_mul_self [Semiring α] [Invertible (2 : α)] (a : α) : sym (a * a) = sym a * sym a := by
-  rw [sym_mul_sym, ← two_mul, invOf_mul_self_assoc]
+  rw [sym_mul_sym, ← two_mul, invOf_mul_cancel_left]
 
 theorem mul_comm [Mul α] [AddCommSemigroup α] [One α] [OfNat α 2] [Invertible (2 : α)]
     (a b : αˢʸᵐ) :
diff --git a/Mathlib/Algebra/TrivSqZeroExt.lean b/Mathlib/Algebra/TrivSqZeroExt.lean
index af308e70ae3a9..c40ab975c18d7 100644
--- a/Mathlib/Algebra/TrivSqZeroExt.lean
+++ b/Mathlib/Algebra/TrivSqZeroExt.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Eric Wieser
 -/
 import Mathlib.Algebra.Algebra.Defs
-import Mathlib.GroupTheory.GroupAction.BigOperators
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.LinearAlgebra.Prod
 
 /-!
@@ -147,6 +147,12 @@ theorem snd_comp_inr [Zero R] : snd ∘ (inr : M → tsze R M) = id :=
 
 end
 
+theorem fst_surjective [Nonempty M] : Function.Surjective (fst : tsze R M → R) :=
+  Prod.fst_surjective
+
+theorem snd_surjective [Nonempty R] : Function.Surjective (snd : tsze R M → M) :=
+  Prod.snd_surjective
+
 theorem inl_injective [Zero M] : Function.Injective (inl : R → tsze R M) :=
   Function.LeftInverse.injective <| fst_inl _
 
@@ -226,6 +232,16 @@ instance module [Semiring S] [AddCommMonoid R] [AddCommMonoid M] [Module S R] [M
     Module S (tsze R M) :=
   Prod.instModule
 
+/-- The trivial square-zero extension is nontrivial if it is over a nontrivial ring. -/
+instance instNontrivial_of_left {R M : Type*} [Nontrivial R] [Nonempty M] :
+    Nontrivial (TrivSqZeroExt R M) :=
+  fst_surjective.nontrivial
+
+/-- The trivial square-zero extension is nontrivial if it is over a nontrivial module. -/
+instance instNontrivial_of_right {R M : Type*} [Nonempty R] [Nontrivial M] :
+    Nontrivial (TrivSqZeroExt R M) :=
+  snd_surjective.nontrivial
+
 @[simp]
 theorem fst_zero [Zero R] [Zero M] : (0 : tsze R M).fst = 0 :=
   rfl
@@ -624,7 +640,7 @@ instance monoid [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulActio
           · simp [List.range_succ]
           rw [List.sum_range_succ']
           simp only [pow_zero, op_one, Nat.sub_zero, one_smul, Nat.succ_sub_succ_eq_sub, fst_pow,
-            Nat.pred_succ, List.smul_sum, List.map_map, Function.comp]
+            Nat.pred_succ, List.smul_sum, List.map_map, Function.comp_def]
           simp_rw [← smul_comm (_ : R) (_ : Rᵐᵒᵖ), smul_smul, pow_succ]
           rfl) }
 
@@ -647,9 +663,9 @@ theorem snd_list_prod [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐ
   | nil => simp
   | cons x xs ih =>
     rw [List.enum_cons, ← List.map_fst_add_enum_eq_enumFrom]
-    simp_rw [List.map_cons, List.map_map, Function.comp, Prod.map_snd, Prod.map_fst, id,
-      List.take_zero, List.take_cons, List.prod_nil, List.prod_cons, snd_mul, one_smul, List.drop,
-      mul_smul, List.sum_cons, fst_list_prod, ih, List.smul_sum, List.map_map,
+    simp_rw [List.map_cons, List.map_map, Function.comp_def, Prod.map_snd, Prod.map_fst, id,
+      List.take_zero, List.take_succ_cons, List.prod_nil, List.prod_cons, snd_mul, one_smul,
+      List.drop, mul_smul, List.sum_cons, fst_list_prod, ih, List.smul_sum, List.map_map,
       ← smul_comm (_ : R) (_ : Rᵐᵒᵖ)]
     exact add_comm _ _
 
@@ -690,7 +706,9 @@ section Inv
 variable {R : Type u} {M : Type v}
 variable [Neg M] [Inv R] [SMul Rᵐᵒᵖ M] [SMul R M]
 
-/-- Inversion of the trivial-square-zero extension, sending $r + m$ to $r^{-1} - r^{-1}mr^{-1}$. -/
+/-- Inversion of the trivial-square-zero extension, sending $r + m$ to $r^{-1} - r^{-1}mr^{-1}$.
+
+Strictly this is only a _two_-sided inverse when the left and right actions associate. -/
 instance instInv : Inv (tsze R M) :=
   ⟨fun b => (b.1⁻¹, -(b.1⁻¹ •> b.2 <• b.1⁻¹))⟩
 
@@ -702,6 +720,75 @@ instance instInv : Inv (tsze R M) :=
 
 end Inv
 
+/-! This section is heavily inspired by analogous results about matrices. -/
+section Invertible
+variable {R : Type u} {M : Type v}
+variable [AddCommGroup M] [Semiring R] [Module Rᵐᵒᵖ M] [Module R M]
+
+/-- `x.fst : R` is invertible when `x : tzre R M` is. -/
+abbrev invertibleFstOfInvertible (x : tsze R M) [Invertible x] : Invertible x.fst where
+  invOf := (⅟x).fst
+  invOf_mul_self := by rw [← fst_mul, invOf_mul_self, fst_one]
+  mul_invOf_self := by rw [← fst_mul, mul_invOf_self, fst_one]
+
+theorem fst_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] : (⅟x).fst = ⅟(x.fst) := by
+  letI := invertibleFstOfInvertible x
+  convert (rfl : _ = ⅟ x.fst)
+
+theorem mul_left_eq_one (r : R) (x : tsze R M) (h : r * x.fst = 1) :
+    (inl r + inr (-((r •> x.snd) <• r))) * x = 1 := by
+  ext <;> dsimp
+  · rw [add_zero, h]
+  · rw [add_zero, zero_add, smul_neg, op_smul_op_smul, h, op_one, one_smul,
+      add_neg_cancel]
+
+theorem mul_right_eq_one (x : tsze R M) (r : R) (h : x.fst * r = 1) :
+    x * (inl r + inr (-(r •> (x.snd <• r)))) = 1 := by
+  ext <;> dsimp
+  · rw [add_zero, h]
+  · rw [add_zero, zero_add, smul_neg, smul_smul, h, one_smul, neg_add_cancel]
+
+variable [SMulCommClass R Rᵐᵒᵖ M]
+
+/-- `x : tzre R M` is invertible when `x.fst : R` is. -/
+abbrev invertibleOfInvertibleFst (x : tsze R M) [Invertible x.fst] : Invertible x where
+  invOf := (⅟x.fst, -(⅟x.fst •> x.snd <• ⅟x.fst))
+  invOf_mul_self := by
+    convert mul_left_eq_one _ _ (invOf_mul_self x.fst)
+    ext <;> simp
+  mul_invOf_self := by
+    convert mul_right_eq_one _ _ (mul_invOf_self x.fst)
+    ext <;> simp [smul_comm]
+
+theorem snd_invOf (x : tsze R M) [Invertible x] [Invertible x.fst] :
+    (⅟x).snd = -(⅟x.fst •> x.snd <• ⅟x.fst) := by
+  letI := invertibleOfInvertibleFst x
+  convert congr_arg (TrivSqZeroExt.snd (R := R) (M := M)) (_ : _ = ⅟ x)
+  convert rfl
+
+/-- Together `TrivSqZeroExt.detInvertibleOfInvertible` and `TrivSqZeroExt.invertibleOfDetInvertible`
+form an equivalence, although both sides of the equiv are subsingleton anyway. -/
+@[simps]
+def invertibleEquivInvertibleFst (x : tsze R M) : Invertible x ≃ Invertible x.fst where
+  toFun _ := invertibleFstOfInvertible x
+  invFun _ := invertibleOfInvertibleFst x
+  left_inv _ := Subsingleton.elim _ _
+  right_inv _ := Subsingleton.elim _ _
+
+/-- When lowered to a prop, `Matrix.invertibleEquivInvertibleFst` forms an `iff`. -/
+theorem isUnit_iff_isUnit_fst {x : tsze R M} : IsUnit x ↔ IsUnit x.fst := by
+  simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivInvertibleFst x).nonempty_congr]
+
+@[simp]
+theorem isUnit_inl_iff {r : R} : IsUnit (inl r : tsze R M) ↔ IsUnit r := by
+  rw [isUnit_iff_isUnit_fst, fst_inl]
+
+@[simp]
+theorem isUnit_inr_iff {m : M} : IsUnit (inr m : tsze R M) ↔ Subsingleton R := by
+  simp_rw [isUnit_iff_isUnit_fst, fst_inr, isUnit_zero_iff, subsingleton_iff_zero_eq_one]
+
+end Invertible
+
 section DivisionSemiring
 variable {R : Type u} {M : Type v}
 variable [DivisionSemiring R] [AddCommGroup M] [Module Rᵐᵒᵖ M] [Module R M]
@@ -727,18 +814,19 @@ protected theorem inv_one : (1 : tsze R M)⁻¹ = (1 : tsze R M) := by
   rw [← inl_one, TrivSqZeroExt.inv_inl, inv_one]
 
 protected theorem inv_mul_cancel {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹ * x = 1 := by
-  ext
-  · rw [fst_mul, fst_inv, inv_mul_cancel₀ hx, fst_one]
-  · rw [snd_mul, snd_inv, snd_one, smul_neg, op_smul_op_smul, inv_mul_cancel₀ hx, op_one, one_smul,
-      fst_inv, add_neg_cancel]
+  convert mul_left_eq_one _ _ (_root_.inv_mul_cancel₀ hx) using 2
+  ext <;> simp
 
 variable [SMulCommClass R Rᵐᵒᵖ M]
 
+@[simp] theorem invOf_eq_inv (x : tsze R M) [Invertible x] : ⅟x = x⁻¹ := by
+  letI := invertibleFstOfInvertible x
+  ext <;> simp [fst_invOf, snd_invOf]
+
 protected theorem mul_inv_cancel {x : tsze R M} (hx : fst x ≠ 0) : x * x⁻¹ = 1 := by
-  ext
-  · rw [fst_mul, fst_inv, fst_one, mul_inv_cancel₀ hx]
-  · rw [snd_mul, snd_inv, snd_one, smul_neg, smul_comm, smul_smul, mul_inv_cancel₀ hx, one_smul,
-      fst_inv, neg_add_cancel]
+  have : Invertible x.fst := Units.invertible (.mk0 _ hx)
+  have := invertibleOfInvertibleFst x
+  rw [← invOf_eq_inv, mul_invOf_self]
 
 protected theorem mul_inv_rev (a b : tsze R M) :
     (a * b)⁻¹ = b⁻¹ * a⁻¹ := by
@@ -763,6 +851,10 @@ protected theorem inv_inv {x : tsze R M} (hx : fst x ≠ 0) : x⁻¹⁻¹ = x :=
       rw [fst_inv]
       apply inv_ne_zero hx
 
+@[simp]
+theorem isUnit_inv_iff {x : tsze R M} : IsUnit x⁻¹ ↔ IsUnit x := by
+  simp_rw [isUnit_iff_isUnit_fst, fst_inv, isUnit_iff_ne_zero, ne_eq, inv_eq_zero]
+
 end DivisionSemiring
 
 section DivisionRing
@@ -873,7 +965,7 @@ def lift (f : R →ₐ[S] A) (g : M →ₗ[S] A)
     (TrivSqZeroExt.ind fun r₁ m₁ =>
       TrivSqZeroExt.ind fun r₂ m₂ => by
         dsimp
-        simp only [add_zero, zero_add, add_mul, mul_add, smul_mul_smul, hg, smul_zero,
+        simp only [add_zero, zero_add, add_mul, mul_add, smul_mul_smul_comm, hg, smul_zero,
           op_smul_eq_smul]
         rw [← map_mul, LinearMap.map_add, add_comm (g _), add_assoc, hfg, hgf])
 
diff --git a/Mathlib/AlgebraicGeometry/AffineScheme.lean b/Mathlib/AlgebraicGeometry/AffineScheme.lean
index eb5bc37c468a7..4b05220332013 100644
--- a/Mathlib/AlgebraicGeometry/AffineScheme.lean
+++ b/Mathlib/AlgebraicGeometry/AffineScheme.lean
@@ -52,24 +52,26 @@ deriving Category
 
 /-- A Scheme is affine if the canonical map `X ⟶ Spec Γ(X)` is an isomorphism. -/
 class IsAffine (X : Scheme) : Prop where
-  affine : IsIso (ΓSpec.adjunction.unit.app X)
+  affine : IsIso X.toSpecΓ
 
 attribute [instance] IsAffine.affine
 
+instance (X : Scheme.{u}) [IsAffine X] : IsIso (ΓSpec.adjunction.unit.app X) := @IsAffine.affine X _
+
 /-- The canonical isomorphism `X ≅ Spec Γ(X)` for an affine scheme. -/
 @[simps! (config := .lemmasOnly) hom]
 def Scheme.isoSpec (X : Scheme) [IsAffine X] : X ≅ Spec Γ(X, ⊤) :=
-  asIso (ΓSpec.adjunction.unit.app X)
+  asIso X.toSpecΓ
 
 @[reassoc]
 theorem Scheme.isoSpec_hom_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
     X.isoSpec.hom ≫ Spec.map (f.app ⊤) = f ≫ Y.isoSpec.hom := by
-  simp only [isoSpec, asIso_hom, ΓSpec.adjunction_unit_naturality]
+  simp only [isoSpec, asIso_hom, Scheme.toSpecΓ_naturality]
 
 @[reassoc]
 theorem Scheme.isoSpec_inv_naturality {X Y : Scheme} [IsAffine X] [IsAffine Y] (f : X ⟶ Y) :
     Spec.map (f.app ⊤) ≫ Y.isoSpec.inv = X.isoSpec.inv ≫ f := by
-  rw [Iso.eq_inv_comp, isoSpec, asIso_hom, ← ΓSpec.adjunction_unit_naturality_assoc, isoSpec,
+  rw [Iso.eq_inv_comp, isoSpec, asIso_hom, ← Scheme.toSpecΓ_naturality_assoc, isoSpec,
     asIso_inv, IsIso.hom_inv_id, Category.comp_id]
 
 /-- Construct an affine scheme from a scheme and the information that it is affine.
@@ -228,7 +230,7 @@ theorem iSup_affineOpens_eq_top (X : Scheme) : ⨆ i : X.affineOpens, (i : X.Ope
 theorem Scheme.map_PrimeSpectrum_basicOpen_of_affine
     (X : Scheme) [IsAffine X] (f : Scheme.Γ.obj (op X)) :
     X.isoSpec.hom ⁻¹ᵁ PrimeSpectrum.basicOpen f = X.basicOpen f :=
-  ΓSpec.adjunction_unit_map_basicOpen _ _
+  Scheme.toSpecΓ_preimage_basicOpen _ _
 
 theorem isBasis_basicOpen (X : Scheme) [IsAffine X] :
     Opens.IsBasis (Set.range (X.basicOpen : Γ(X, ⊤) → X.Opens)) := by
@@ -240,10 +242,10 @@ theorem isBasis_basicOpen (X : Scheme) [IsAffine X] :
   constructor
   · rintro ⟨_, ⟨x, rfl⟩, rfl⟩
     refine ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, ?_⟩
-    exact congr_arg Opens.carrier (ΓSpec.adjunction_unit_map_basicOpen _ _)
+    exact congr_arg Opens.carrier (Scheme.toSpecΓ_preimage_basicOpen _ _)
   · rintro ⟨_, ⟨_, ⟨x, rfl⟩, rfl⟩, rfl⟩
     refine ⟨_, ⟨x, rfl⟩, ?_⟩
-    exact congr_arg Opens.carrier (ΓSpec.adjunction_unit_map_basicOpen _ _).symm
+    exact congr_arg Opens.carrier (Scheme.toSpecΓ_preimage_basicOpen _ _).symm
 
 namespace IsAffineOpen
 
@@ -265,7 +267,7 @@ instance isOpenImmersion_fromSpec :
 theorem range_fromSpec :
     Set.range hU.fromSpec.1.base = (U : Set X) := by
   delta IsAffineOpen.fromSpec; dsimp
-  rw [Function.comp.assoc, Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ]
+  rw [Function.comp_assoc, Set.range_comp, Set.range_iff_surjective.mpr, Set.image_univ]
   · exact Subtype.range_coe
   erw [← coe_comp, ← TopCat.epi_iff_surjective] -- now `erw` after #13170
   infer_instance
@@ -312,7 +314,7 @@ theorem _root_.AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion
     (IsOpenImmersion.isoOfRangeEq (X.ofRestrict U.openEmbedding ≫ f) (Y.ofRestrict _) ?_).hom ?_ hU,
     fun hU => hU.image_of_isOpenImmersion f⟩
   · erw [Scheme.comp_val_base, coe_comp, Set.range_comp] -- now `erw` after #13170
-    dsimp [Opens.coe_inclusion, Scheme.restrict]
+    dsimp [Opens.coe_inclusion', Scheme.restrict]
     erw [Subtype.range_coe, Subtype.range_coe] -- now `erw` after #13170
     rfl
   · infer_instance
@@ -356,10 +358,9 @@ theorem SpecΓIdentity_hom_app_fromSpec :
     (Scheme.ΓSpecIso Γ(X, U)).hom ≫ hU.fromSpec.app U =
       (Spec Γ(X, U)).presheaf.map (eqToHom hU.fromSpec_preimage_self).op := by
   simp only [fromSpec, Scheme.isoSpec, asIso_inv, Scheme.comp_coeBase, Opens.map_comp_obj,
-    ΓSpecIso_obj_hom, Scheme.Opens.topIso_inv, Opens.map_top, Functor.id_obj, Functor.comp_obj,
-    Functor.rightOp_obj, Scheme.Γ_obj, unop_op, Scheme.Spec_obj, Scheme.Opens.topIso_hom,
-    Scheme.comp_app, Scheme.Opens.ι_app_self, Category.assoc, ← Functor.map_comp_assoc, ← op_comp,
-    eqToHom_trans, Scheme.Opens.eq_presheaf_map_eqToHom, Scheme.Hom.naturality_assoc,
+    ΓSpecIso_obj_hom, Scheme.Opens.topIso_inv, Opens.map_top, Scheme.Opens.topIso_hom,
+    Scheme.comp_app, Scheme.Opens.ι_app_self, unop_op, Category.assoc, ← Functor.map_comp_assoc, ←
+    op_comp, eqToHom_trans, Scheme.Opens.eq_presheaf_map_eqToHom, Scheme.Hom.naturality_assoc,
     Scheme.inv_app_top, IsIso.hom_inv_id_assoc]
   simp only [eqToHom_op, eqToHom_map, Spec.map_eqToHom, eqToHom_unop,
     Scheme.Spec_map_presheaf_map_eqToHom, eqToHom_trans]
@@ -417,8 +418,8 @@ theorem exists_basicOpen_le {V : X.Opens} (x : V) (h : ↑x ∈ U) :
     ∃ f : Γ(X, U), X.basicOpen f ≤ V ∧ ↑x ∈ X.basicOpen f := by
   have : IsAffine _ := hU
   obtain ⟨_, ⟨_, ⟨r, rfl⟩, rfl⟩, h₁, h₂⟩ :=
-    (isBasis_basicOpen U).exists_subset_of_mem_open (x.2 : ⟨x, h⟩ ∈ _)
-      ((Opens.map U.inclusion).obj V).isOpen
+    (isBasis_basicOpen U).exists_subset_of_mem_open (x.2 : (⟨x, h⟩ : U) ∈ _)
+      ((Opens.map U.inclusion').obj V).isOpen
   have :
     U.ι ''ᵁ (U.toScheme.basicOpen r) =
       X.basicOpen (X.presheaf.map (eqToHom U.openEmbedding_obj_top.symm).op r) := by
@@ -683,7 +684,7 @@ section ZeroLocus
 /-- On a locally ringed space `X`, the preimage of the zero locus of the prime spectrum
 of `Γ(X, ⊤)` under `toΓSpecFun` agrees with the associated zero locus on `X`. -/
 lemma Scheme.toΓSpec_preimage_zeroLocus_eq {X : Scheme.{u}} (s : Set Γ(X, ⊤)) :
-    (ΓSpec.adjunction.unit.app X).val.base ⁻¹' PrimeSpectrum.zeroLocus s = X.zeroLocus s :=
+    X.toSpecΓ.val.base ⁻¹' PrimeSpectrum.zeroLocus s = X.zeroLocus s :=
   LocallyRingedSpace.toΓSpec_preimage_zeroLocus_eq s
 
 open ConcreteCategory
diff --git a/Mathlib/AlgebraicGeometry/Cover/Open.lean b/Mathlib/AlgebraicGeometry/Cover/Open.lean
index 20acd2442916a..959dc19b68c73 100644
--- a/Mathlib/AlgebraicGeometry/Cover/Open.lean
+++ b/Mathlib/AlgebraicGeometry/Cover/Open.lean
@@ -92,7 +92,7 @@ theorem OpenCover.iSup_opensRange {X : Scheme.{u}} (𝒰 : X.OpenCover) :
   Opens.ext <| by rw [Opens.coe_iSup]; exact 𝒰.iUnion_range
 
 /-- Given an open cover `{ Uᵢ }` of `X`, and for each `Uᵢ` an open cover, we may combine these
-open covers to form an open cover of `X`.  -/
+open covers to form an open cover of `X`. -/
 @[simps! J obj map]
 def OpenCover.bind (f : ∀ x : 𝒰.J, OpenCover (𝒰.obj x)) : OpenCover X where
   J := Σ i : 𝒰.J, (f i).J
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
index 8e25ace403b09..d3cc6dc2264fe 100644
--- a/Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean
@@ -5,6 +5,7 @@ Authors: David Kurniadi Angdinata
 -/
 import Mathlib.Algebra.Polynomial.Bivariate
 import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
+import Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange
 
 /-!
 # Affine coordinates for Weierstrass curves
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
index cbb5edf60ed85..28d32c6a3fea5 100644
--- a/Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/Group.lean
@@ -545,7 +545,7 @@ lemma toClass_eq_zero (P : W.Point) : toClass P = 0 ↔ P = 0 := by
       rw [← finrank_quotient_span_eq_natDegree_norm (CoordinateRing.basis W) h0,
         ← (quotientEquivAlgOfEq F hp).toLinearEquiv.finrank_eq,
         (CoordinateRing.quotientXYIdealEquiv W h).toLinearEquiv.finrank_eq,
-        FiniteDimensional.finrank_self]
+        Module.finrank_self]
   · exact congr_arg toClass
 
 lemma toClass_injective : Function.Injective <| @toClass _ _ W := by
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
index 6042ae7f510d6..d0976e216dd81 100644
--- a/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean
@@ -7,6 +7,7 @@ import Mathlib.Algebra.MvPolynomial.CommRing
 import Mathlib.Algebra.MvPolynomial.PDeriv
 import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
 import Mathlib.Data.Fin.Tuple.Reflection
+import Mathlib.Tactic.LinearCombination'
 
 /-!
 # Jacobian coordinates for Weierstrass curves
@@ -359,8 +360,8 @@ variable (W') in
 /-- The proposition that a point representative $(x, y, z)$ in `W'` is nonsingular.
 In other words, either $W_X(x, y, z) \ne 0$, $W_Y(x, y, z) \ne 0$, or $W_Z(x, y, z) \ne 0$.
 
-Note that this definition is only mathematically accurate for fields.
-TODO: generalise this definition to be mathematically accurate for a larger class of rings. -/
+Note that this definition is only mathematically accurate for fields. -/
+-- TODO: generalise this definition to be mathematically accurate for a larger class of rings.
 def Nonsingular (P : Fin 3 → R) : Prop :=
   W'.Equation P ∧
     (eval P W'.polynomialX ≠ 0 ∨ eval P W'.polynomialY ≠ 0 ∨ eval P W'.polynomialZ ≠ 0)
@@ -498,7 +499,7 @@ lemma negY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
 lemma Y_sub_Y_mul_Y_sub_negY {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
     (hx : P x * Q z ^ 2 = Q x * P z ^ 2) :
     (P y * Q z ^ 3 - Q y * P z ^ 3) * (P y * Q z ^ 3 - W'.negY Q * P z ^ 3) = 0 := by
-  linear_combination (norm := (rw [negY]; ring1)) Q z ^ 6 * (equation_iff P).mp hP
+  linear_combination' (norm := (rw [negY]; ring1)) Q z ^ 6 * (equation_iff P).mp hP
     - P z ^ 6 * (equation_iff Q).mp hQ + hx * hx * hx + W'.a₂ * P z ^ 2 * Q z ^ 2 * hx * hx
     + (W'.a₄ * P z ^ 4 * Q z ^ 4 - W'.a₁ * P y * P z * Q z ^ 4) * hx
 
@@ -678,7 +679,7 @@ lemma negDblY_smul (P : Fin 3 → R) (u : R) : W'.negDblY (u • P) = (u ^ 4) ^
 
 lemma negDblY_of_Z_eq_zero {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
     W'.negDblY P = -(P x ^ 2) ^ 3 := by
-  linear_combination (norm :=
+  linear_combination' (norm :=
       (rw [negDblY, dblU_of_Z_eq_zero hPz, dblX_of_Z_eq_zero hP hPz, negY_of_Z_eq_zero hPz]; ring1))
     (8 * (equation_of_Z_eq_zero hPz).mp hP - 12 * P x ^ 3) * (equation_of_Z_eq_zero hPz).mp hP
 
@@ -1619,3 +1620,5 @@ end WeierstrassCurve.Jacobian
 abbrev WeierstrassCurve.Affine.Point.toJacobian {R : Type u} [CommRing R]
     [Nontrivial R] {W : Affine R} (P : W.Point) : W.toJacobian.Point :=
   Jacobian.Point.fromAffine P
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
new file mode 100644
index 0000000000000..565f29a05b7f1
--- /dev/null
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean
@@ -0,0 +1,706 @@
+/-
+Copyright (c) 2024 Jz Pan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jz Pan
+-/
+import Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange
+import Mathlib.Algebra.CharP.Defs
+
+/-!
+
+# Some normal forms of elliptic curves
+
+This file defines some normal forms of Weierstrass equations of elliptic curves.
+
+## Main definitions and results
+
+The following normal forms are in [silverman2009], section III.1, page 42.
+
+- `WeierstrassCurve.IsCharNeTwoNF` is a type class which asserts that a `WeierstrassCurve` is
+  of form $Y^2 = X^3 + a_2X^2 + a_4X + a_6$. It is the normal form of characteristic ≠ 2.
+
+  If 2 is invertible in the ring (for example, if it is a field of characteristic ≠ 2),
+  then for any `WeierstrassCurve` there exists a change of variables which will change
+  it into such normal form (`WeierstrassCurve.exists_variableChange_isCharNeTwoNF`).
+  See also `WeierstrassCurve.toCharNeTwoNF` and `WeierstrassCurve.toCharNeTwoNF_spec`.
+
+The following normal forms are in [silverman2009], Appendix A, Proposition 1.1.
+
+- `WeierstrassCurve.IsShortNF` is a type class which asserts that a `WeierstrassCurve` is
+  of form $Y^2 = X^3 + a_4X + a_6$. It is the normal form of characteristic ≠ 2 or 3, and
+  also the normal form of characteristic = 3 and j = 0.
+
+  If 2 and 3 are invertible in the ring (for example, if it is a field of characteristic ≠ 2 or 3),
+  then for any `WeierstrassCurve` there exists a change of variables which will change
+  it into such normal form (`WeierstrassCurve.exists_variableChange_isShortNF`).
+  See also `WeierstrassCurve.toShortNF` and `WeierstrassCurve.toShortNF_spec`.
+
+  If the ring is of characteristic = 3, then for any `WeierstrassCurve` with $b_2 = 0$ (for an
+  elliptic curve, this is equivalent to j = 0), there exists a change of variables which will
+  change it into such normal form (see `WeierstrassCurve.toShortNFOfCharThree`
+  and `WeierstrassCurve.toShortNFOfCharThree_spec`).
+
+- `WeierstrassCurve.IsCharThreeJNeZeroNF` is a type class which asserts that a `WeierstrassCurve` is
+  of form $Y^2 = X^3 + a_2X^2 + a_6$. It is the normal form of characteristic = 3 and j ≠ 0.
+
+  If the field is of characteristic = 3, then for any `WeierstrassCurve` with $b_2 \neq 0$ (for an
+  elliptic curve, this is equivalent to j ≠ 0), there exists a change of variables which will
+  change it into such normal form (see `WeierstrassCurve.toCharThreeNF`
+  and `WeierstrassCurve.toCharThreeNF_spec_of_b₂_ne_zero`).
+
+- `WeierstrassCurve.IsCharThreeNF` is the combination of the above two, that is, asserts that
+  a `WeierstrassCurve` is of form $Y^2 = X^3 + a_2X^2 + a_6$ or $Y^2 = X^3 + a_4X + a_6$.
+  It is the normal form of characteristic = 3.
+
+  If the field is of characteristic = 3, then for any `WeierstrassCurve` there exists a change of
+  variables which will change it into such normal form
+  (`WeierstrassCurve.exists_variableChange_isCharThreeNF`).
+  See also `WeierstrassCurve.toCharThreeNF` and `WeierstrassCurve.toCharThreeNF_spec`.
+
+- `WeierstrassCurve.IsCharTwoJEqZeroNF` is a type class which asserts that a `WeierstrassCurve` is
+  of form $Y^2 + a_3Y = X^3 + a_4X + a_6$. It is the normal form of characteristic = 2 and j = 0.
+
+  If the ring is of characteristic = 2, then for any `WeierstrassCurve` with $a_1 = 0$ (for an
+  elliptic curve, this is equivalent to j = 0), there exists a change of variables which will
+  change it into such normal form (see `WeierstrassCurve.toCharTwoJEqZeroNF`
+  and `WeierstrassCurve.toCharTwoJEqZeroNF_spec`).
+
+- `WeierstrassCurve.IsCharTwoJNeZeroNF` is a type class which asserts that a `WeierstrassCurve` is
+  of form $Y^2 + XY = X^3 + a_2X^2 + a_6$. It is the normal form of characteristic = 2 and j ≠ 0.
+
+  If the field is of characteristic = 2, then for any `WeierstrassCurve` with $a_1 \neq 0$ (for an
+  elliptic curve, this is equivalent to j ≠ 0), there exists a change of variables which will
+  change it into such normal form (see `WeierstrassCurve.toCharTwoJNeZeroNF`
+  and `WeierstrassCurve.toCharTwoJNeZeroNF_spec`).
+
+- `WeierstrassCurve.IsCharTwoNF` is the combination of the above two, that is, asserts that
+  a `WeierstrassCurve` is of form $Y^2 + XY = X^3 + a_2X^2 + a_6$ or
+  $Y^2 + a_3Y = X^3 + a_4X + a_6$. It is the normal form of characteristic = 2.
+
+  If the field is of characteristic = 2, then for any `WeierstrassCurve` there exists a change of
+  variables which will change it into such normal form
+  (`WeierstrassCurve.exists_variableChange_isCharTwoNF`).
+  See also `WeierstrassCurve.toCharTwoNF` and `WeierstrassCurve.toCharTwoNF_spec`.
+
+## References
+
+* [J Silverman, *The Arithmetic of Elliptic Curves*][silverman2009]
+
+## Tags
+
+elliptic curve, weierstrass equation, normal form
+
+-/
+
+variable {R : Type*} [CommRing R] (W : WeierstrassCurve R)
+variable {F : Type*} [Field F] (E : EllipticCurve F)
+
+namespace WeierstrassCurve
+
+/-! ### Normal forms of characteristic ≠ 2 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic ≠ 2, if its $a_1, a_3 = 0$.
+In other words it is $Y^2 = X^3 + a_2X^2 + a_4X + a_6$. -/
+@[mk_iff]
+class IsCharNeTwoNF : Prop where
+  a₁ : W.a₁ = 0
+  a₃ : W.a₃ = 0
+
+section Quantity
+
+variable [W.IsCharNeTwoNF]
+
+@[simp]
+theorem a₁_of_isCharNeTwoNF : W.a₁ = 0 := IsCharNeTwoNF.a₁
+
+@[simp]
+theorem a₃_of_isCharNeTwoNF : W.a₃ = 0 := IsCharNeTwoNF.a₃
+
+@[simp]
+theorem b₂_of_isCharNeTwoNF : W.b₂ = 4 * W.a₂ := by
+  rw [b₂, a₁_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem b₄_of_isCharNeTwoNF : W.b₄ = 2 * W.a₄ := by
+  rw [b₄, a₃_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem b₆_of_isCharNeTwoNF : W.b₆ = 4 * W.a₆ := by
+  rw [b₆, a₃_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem b₈_of_isCharNeTwoNF : W.b₈ = 4 * W.a₂ * W.a₆ - W.a₄ ^ 2 := by
+  rw [b₈, a₁_of_isCharNeTwoNF, a₃_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem c₄_of_isCharNeTwoNF : W.c₄ = 16 * W.a₂ ^ 2 - 48 * W.a₄ := by
+  rw [c₄, b₂_of_isCharNeTwoNF, b₄_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem c₆_of_isCharNeTwoNF : W.c₆ = -64 * W.a₂ ^ 3 + 288 * W.a₂ * W.a₄ - 864 * W.a₆ := by
+  rw [c₆, b₂_of_isCharNeTwoNF, b₄_of_isCharNeTwoNF, b₆_of_isCharNeTwoNF]
+  ring1
+
+@[simp]
+theorem Δ_of_isCharNeTwoNF : W.Δ = -64 * W.a₂ ^ 3 * W.a₆ + 16 * W.a₂ ^ 2 * W.a₄ ^ 2 - 64 * W.a₄ ^ 3
+    - 432 * W.a₆ ^ 2 + 288 * W.a₂ * W.a₄ * W.a₆ := by
+  rw [Δ, b₂_of_isCharNeTwoNF, b₄_of_isCharNeTwoNF, b₆_of_isCharNeTwoNF, b₈_of_isCharNeTwoNF]
+  ring1
+
+end Quantity
+
+section VariableChange
+
+variable [Invertible (2 : R)]
+
+/-- There is an explicit change of variables of a `WeierstrassCurve` to
+a normal form of characteristic ≠ 2, provided that 2 is invertible in the ring. -/
+@[simps]
+def toCharNeTwoNF : VariableChange R := ⟨1, 0, ⅟2 * -W.a₁, ⅟2 * -W.a₃⟩
+
+instance toCharNeTwoNF_spec : (W.variableChange W.toCharNeTwoNF).IsCharNeTwoNF := by
+  constructor <;> simp
+
+theorem exists_variableChange_isCharNeTwoNF :
+    ∃ C : VariableChange R, (W.variableChange C).IsCharNeTwoNF :=
+  ⟨_, W.toCharNeTwoNF_spec⟩
+
+end VariableChange
+
+/-! ### Short normal form -/
+
+/-- A `WeierstrassCurve` is in short normal form, if its $a_1, a_2, a_3 = 0$.
+In other words it is $Y^2 = X^3 + a_4X + a_6$.
+
+This is the normal form of characteristic ≠ 2 or 3, and
+also the normal form of characteristic = 3 and j = 0. -/
+@[mk_iff]
+class IsShortNF : Prop where
+  a₁ : W.a₁ = 0
+  a₂ : W.a₂ = 0
+  a₃ : W.a₃ = 0
+
+section Quantity
+
+variable [W.IsShortNF]
+
+instance isCharNeTwoNF_of_isShortNF : W.IsCharNeTwoNF := ⟨IsShortNF.a₁, IsShortNF.a₃⟩
+
+theorem a₁_of_isShortNF : W.a₁ = 0 := IsShortNF.a₁
+
+@[simp]
+theorem a₂_of_isShortNF : W.a₂ = 0 := IsShortNF.a₂
+
+theorem a₃_of_isShortNF : W.a₃ = 0 := IsShortNF.a₃
+
+theorem b₂_of_isShortNF : W.b₂ = 0 := by
+  simp
+
+theorem b₄_of_isShortNF : W.b₄ = 2 * W.a₄ := W.b₄_of_isCharNeTwoNF
+
+theorem b₆_of_isShortNF : W.b₆ = 4 * W.a₆ := W.b₆_of_isCharNeTwoNF
+
+theorem b₈_of_isShortNF : W.b₈ = -W.a₄ ^ 2 := by
+  simp
+
+theorem c₄_of_isShortNF : W.c₄ = -48 * W.a₄ := by
+  simp
+
+theorem c₆_of_isShortNF : W.c₆ = -864 * W.a₆ := by
+  simp
+
+theorem Δ_of_isShortNF : W.Δ = -16 * (4 * W.a₄ ^ 3 + 27 * W.a₆ ^ 2) := by
+  rw [Δ_of_isCharNeTwoNF, a₂_of_isShortNF]
+  ring1
+
+variable [CharP R 3]
+
+theorem b₄_of_isShortNF_of_char_three : W.b₄ = -W.a₄ := by
+  rw [b₄_of_isShortNF]
+  linear_combination W.a₄ * CharP.cast_eq_zero R 3
+
+theorem b₆_of_isShortNF_of_char_three : W.b₆ = W.a₆ := by
+  rw [b₆_of_isShortNF]
+  linear_combination W.a₆ * CharP.cast_eq_zero R 3
+
+theorem c₄_of_isShortNF_of_char_three : W.c₄ = 0 := by
+  rw [c₄_of_isShortNF]
+  linear_combination -16 * W.a₄ * CharP.cast_eq_zero R 3
+
+theorem c₆_of_isShortNF_of_char_three : W.c₆ = 0 := by
+  rw [c₆_of_isShortNF]
+  linear_combination -288 * W.a₆ * CharP.cast_eq_zero R 3
+
+theorem Δ_of_isShortNF_of_char_three : W.Δ = -W.a₄ ^ 3 := by
+  rw [Δ_of_isShortNF]
+  linear_combination (-21 * W.a₄ ^ 3 - 144 * W.a₆ ^ 2) * CharP.cast_eq_zero R 3
+
+variable [E.IsShortNF]
+
+theorem _root_.EllipticCurve.j_of_isShortNF :
+    E.j = 6912 * E.a₄ ^ 3 / (4 * E.a₄ ^ 3 + 27 * E.a₆ ^ 2) := by
+  have h := E.Δ'.ne_zero
+  rw [E.coe_Δ', Δ_of_isShortNF] at h
+  rw [EllipticCurve.j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, E.coe_Δ',
+    c₄_of_isShortNF, Δ_of_isShortNF, div_eq_div_iff h (right_ne_zero_of_mul h)]
+  ring1
+
+@[simp]
+theorem _root_.EllipticCurve.j_of_isShortNF_of_char_three [CharP F 3] : E.j = 0 := by
+  rw [EllipticCurve.j, c₄_of_isShortNF_of_char_three]; simp
+
+end Quantity
+
+section VariableChange
+
+variable [Invertible (2 : R)] [Invertible (3 : R)]
+
+/-- There is an explicit change of variables of a `WeierstrassCurve` to
+a short normal form, provided that 2 and 3 are invertible in the ring.
+It is the composition of an explicit change of variables with `WeierstrassCurve.toCharNeTwoNF`. -/
+def toShortNF : VariableChange R :=
+  .comp ⟨1, ⅟3 * -(W.variableChange W.toCharNeTwoNF).a₂, 0, 0⟩ W.toCharNeTwoNF
+
+instance toShortNF_spec : (W.variableChange W.toShortNF).IsShortNF := by
+  rw [toShortNF, variableChange_comp]
+  constructor <;> simp
+
+theorem exists_variableChange_isShortNF :
+    ∃ C : VariableChange R, (W.variableChange C).IsShortNF :=
+  ⟨_, W.toShortNF_spec⟩
+
+end VariableChange
+
+/-! ### Normal forms of characteristic = 3 and j ≠ 0 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic = 3 and j ≠ 0, if its
+$a_1, a_3, a_4 = 0$. In other words it is $Y^2 = X^3 + a_2X^2 + a_6$. -/
+@[mk_iff]
+class IsCharThreeJNeZeroNF : Prop where
+  a₁ : W.a₁ = 0
+  a₃ : W.a₃ = 0
+  a₄ : W.a₄ = 0
+
+section Quantity
+
+variable [W.IsCharThreeJNeZeroNF]
+
+instance isCharNeTwoNF_of_isCharThreeJNeZeroNF : W.IsCharNeTwoNF :=
+  ⟨IsCharThreeJNeZeroNF.a₁, IsCharThreeJNeZeroNF.a₃⟩
+
+theorem a₁_of_isCharThreeJNeZeroNF : W.a₁ = 0 := IsCharThreeJNeZeroNF.a₁
+
+theorem a₃_of_isCharThreeJNeZeroNF : W.a₃ = 0 := IsCharThreeJNeZeroNF.a₃
+
+@[simp]
+theorem a₄_of_isCharThreeJNeZeroNF : W.a₄ = 0 := IsCharThreeJNeZeroNF.a₄
+
+theorem b₂_of_isCharThreeJNeZeroNF : W.b₂ = 4 * W.a₂ := W.b₂_of_isCharNeTwoNF
+
+theorem b₄_of_isCharThreeJNeZeroNF : W.b₄ = 0 := by
+  simp
+
+theorem b₆_of_isCharThreeJNeZeroNF : W.b₆ = 4 * W.a₆ := W.b₆_of_isCharNeTwoNF
+
+theorem b₈_of_isCharThreeJNeZeroNF : W.b₈ = 4 * W.a₂ * W.a₆ := by
+  simp
+
+theorem c₄_of_isCharThreeJNeZeroNF : W.c₄ = 16 * W.a₂ ^ 2 := by
+  simp
+
+theorem c₆_of_isCharThreeJNeZeroNF : W.c₆ = -64 * W.a₂ ^ 3 - 864 * W.a₆ := by
+  simp
+
+theorem Δ_of_isCharThreeJNeZeroNF : W.Δ = -64 * W.a₂ ^ 3 * W.a₆ - 432 * W.a₆ ^ 2 := by
+  simp
+
+variable [CharP R 3]
+
+theorem b₂_of_isCharThreeJNeZeroNF_of_char_three : W.b₂ = W.a₂ := by
+  rw [b₂_of_isCharThreeJNeZeroNF]
+  linear_combination W.a₂ * CharP.cast_eq_zero R 3
+
+theorem b₆_of_isCharThreeJNeZeroNF_of_char_three : W.b₆ = W.a₆ := by
+  rw [b₆_of_isCharThreeJNeZeroNF]
+  linear_combination W.a₆ * CharP.cast_eq_zero R 3
+
+theorem b₈_of_isCharThreeJNeZeroNF_of_char_three : W.b₈ = W.a₂ * W.a₆ := by
+  rw [b₈_of_isCharThreeJNeZeroNF]
+  linear_combination W.a₂ * W.a₆ * CharP.cast_eq_zero R 3
+
+theorem c₄_of_isCharThreeJNeZeroNF_of_char_three : W.c₄ = W.a₂ ^ 2 := by
+  rw [c₄_of_isCharThreeJNeZeroNF]
+  linear_combination 5 * W.a₂ ^ 2 * CharP.cast_eq_zero R 3
+
+theorem c₆_of_isCharThreeJNeZeroNF_of_char_three : W.c₆ = -W.a₂ ^ 3 := by
+  rw [c₆_of_isCharThreeJNeZeroNF]
+  linear_combination (-21 * W.a₂ ^ 3 - 288 * W.a₆) * CharP.cast_eq_zero R 3
+
+theorem Δ_of_isCharThreeJNeZeroNF_of_char_three : W.Δ = -W.a₂ ^ 3 * W.a₆ := by
+  rw [Δ_of_isCharThreeJNeZeroNF]
+  linear_combination (-21 * W.a₂ ^ 3 * W.a₆ - 144 * W.a₆ ^ 2) * CharP.cast_eq_zero R 3
+
+variable [E.IsCharThreeJNeZeroNF] [CharP F 3]
+
+@[simp]
+theorem _root_.EllipticCurve.j_of_isCharThreeJNeZeroNF_of_char_three : E.j = -E.a₂ ^ 3 / E.a₆ := by
+  have h := E.Δ'.ne_zero
+  rw [E.coe_Δ', Δ_of_isCharThreeJNeZeroNF_of_char_three] at h
+  rw [EllipticCurve.j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, E.coe_Δ',
+    c₄_of_isCharThreeJNeZeroNF_of_char_three, Δ_of_isCharThreeJNeZeroNF_of_char_three,
+    div_eq_div_iff h (right_ne_zero_of_mul h)]
+  ring1
+
+theorem _root_.EllipticCurve.j_ne_zero_of_isCharThreeJNeZeroNF_of_char_three : E.j ≠ 0 := by
+  rw [E.j_of_isCharThreeJNeZeroNF_of_char_three, div_ne_zero_iff]
+  have h := E.Δ'.ne_zero
+  rwa [E.coe_Δ', Δ_of_isCharThreeJNeZeroNF_of_char_three, mul_ne_zero_iff] at h
+
+end Quantity
+
+/-! ### Normal forms of characteristic = 3 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic = 3, if it is
+$Y^2 = X^3 + a_2X^2 + a_6$ (`WeierstrassCurve.IsCharThreeJNeZeroNF`) or
+$Y^2 = X^3 + a_4X + a_6$ (`WeierstrassCurve.IsShortNF`). -/
+class inductive IsCharThreeNF : Prop
+| of_j_ne_zero [W.IsCharThreeJNeZeroNF] : IsCharThreeNF
+| of_j_eq_zero [W.IsShortNF] : IsCharThreeNF
+
+instance isCharThreeNF_of_isCharThreeJNeZeroNF [W.IsCharThreeJNeZeroNF] : W.IsCharThreeNF :=
+  IsCharThreeNF.of_j_ne_zero
+
+instance isCharThreeNF_of_isShortNF [W.IsShortNF] : W.IsCharThreeNF :=
+  IsCharThreeNF.of_j_eq_zero
+
+instance isCharNeTwoNF_of_isCharThreeNF [W.IsCharThreeNF] : W.IsCharNeTwoNF := by
+  cases ‹W.IsCharThreeNF› <;> infer_instance
+
+section VariableChange
+
+variable [CharP R 3] [CharP F 3]
+
+/-- For a `WeierstrassCurve` defined over a ring of characteristic = 3,
+there is an explicit change of variables of it to $Y^2 = X^3 + a_4X + a_6$
+(`WeierstrassCurve.IsShortNF`) if its j = 0.
+This is in fact given by `WeierstrassCurve.toCharNeTwoNF`. -/
+def toShortNFOfCharThree : VariableChange R :=
+  have h : (2 : R) * 2 = 1 := by linear_combination CharP.cast_eq_zero R 3
+  letI : Invertible (2 : R) := ⟨2, h, h⟩
+  W.toCharNeTwoNF
+
+lemma toShortNFOfCharThree_a₂ : (W.variableChange W.toShortNFOfCharThree).a₂ = W.b₂ := by
+  simp_rw [toShortNFOfCharThree, toCharNeTwoNF, variableChange_a₂, inv_one, Units.val_one, b₂]
+  linear_combination (-W.a₂ - W.a₁ ^ 2) * CharP.cast_eq_zero R 3
+
+theorem toShortNFOfCharThree_spec (hb₂ : W.b₂ = 0) :
+    (W.variableChange W.toShortNFOfCharThree).IsShortNF := by
+  have h : (2 : R) * 2 = 1 := by linear_combination CharP.cast_eq_zero R 3
+  letI : Invertible (2 : R) := ⟨2, h, h⟩
+  have H := W.toCharNeTwoNF_spec
+  exact ⟨H.a₁, hb₂ ▸ W.toShortNFOfCharThree_a₂, H.a₃⟩
+
+variable (W : WeierstrassCurve F)
+
+/-- For a `WeierstrassCurve` defined over a field of characteristic = 3,
+there is an explicit change of variables of it to `WeierstrassCurve.IsCharThreeNF`, that is,
+$Y^2 = X^3 + a_2X^2 + a_6$ (`WeierstrassCurve.IsCharThreeJNeZeroNF`) or
+$Y^2 = X^3 + a_4X + a_6$ (`WeierstrassCurve.IsShortNF`).
+It is the composition of an explicit change of variables with
+`WeierstrassCurve.toShortNFOfCharThree`. -/
+def toCharThreeNF : VariableChange F :=
+  .comp ⟨1, (W.variableChange W.toShortNFOfCharThree).a₄ /
+    (W.variableChange W.toShortNFOfCharThree).a₂, 0, 0⟩ W.toShortNFOfCharThree
+
+theorem toCharThreeNF_spec_of_b₂_ne_zero (hb₂ : W.b₂ ≠ 0) :
+    (W.variableChange W.toCharThreeNF).IsCharThreeJNeZeroNF := by
+  have h : (2 : F) * 2 = 1 := by linear_combination CharP.cast_eq_zero F 3
+  letI : Invertible (2 : F) := ⟨2, h, h⟩
+  rw [toCharThreeNF, variableChange_comp]
+  set W' := W.variableChange W.toShortNFOfCharThree
+  haveI : W'.IsCharNeTwoNF := W.toCharNeTwoNF_spec
+  constructor
+  · simp
+  · simp
+  · field_simp [W.toShortNFOfCharThree_a₂ ▸ hb₂]
+    linear_combination (W'.a₄ * W'.a₂ ^ 2 + W'.a₄ ^ 2) * CharP.cast_eq_zero F 3
+
+theorem toCharThreeNF_spec_of_b₂_eq_zero (hb₂ : W.b₂ = 0) :
+    (W.variableChange W.toCharThreeNF).IsShortNF := by
+  rw [toCharThreeNF, toShortNFOfCharThree_a₂, hb₂, div_zero, ← VariableChange.id,
+    VariableChange.id_comp]
+  exact W.toShortNFOfCharThree_spec hb₂
+
+instance toCharThreeNF_spec : (W.variableChange W.toCharThreeNF).IsCharThreeNF := by
+  by_cases hb₂ : W.b₂ = 0
+  · haveI := W.toCharThreeNF_spec_of_b₂_eq_zero hb₂
+    infer_instance
+  · haveI := W.toCharThreeNF_spec_of_b₂_ne_zero hb₂
+    infer_instance
+
+theorem exists_variableChange_isCharThreeNF :
+    ∃ C : VariableChange F, (W.variableChange C).IsCharThreeNF :=
+  ⟨_, W.toCharThreeNF_spec⟩
+
+end VariableChange
+
+/-! ### Normal forms of characteristic = 2 and j ≠ 0 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic = 2 and j ≠ 0, if its $a_1 = 1$ and
+$a_3, a_4 = 0$. In other words it is $Y^2 + XY = X^3 + a_2X^2 + a_6$. -/
+@[mk_iff]
+class IsCharTwoJNeZeroNF : Prop where
+  a₁ : W.a₁ = 1
+  a₃ : W.a₃ = 0
+  a₄ : W.a₄ = 0
+
+section Quantity
+
+variable [W.IsCharTwoJNeZeroNF]
+
+@[simp]
+theorem a₁_of_isCharTwoJNeZeroNF : W.a₁ = 1 := IsCharTwoJNeZeroNF.a₁
+
+@[simp]
+theorem a₃_of_isCharTwoJNeZeroNF : W.a₃ = 0 := IsCharTwoJNeZeroNF.a₃
+
+@[simp]
+theorem a₄_of_isCharTwoJNeZeroNF : W.a₄ = 0 := IsCharTwoJNeZeroNF.a₄
+
+@[simp]
+theorem b₂_of_isCharTwoJNeZeroNF : W.b₂ = 1 + 4 * W.a₂ := by
+  rw [b₂, a₁_of_isCharTwoJNeZeroNF]
+  ring1
+
+@[simp]
+theorem b₄_of_isCharTwoJNeZeroNF : W.b₄ = 0 := by
+  rw [b₄, a₃_of_isCharTwoJNeZeroNF, a₄_of_isCharTwoJNeZeroNF]
+  ring1
+
+@[simp]
+theorem b₆_of_isCharTwoJNeZeroNF : W.b₆ = 4 * W.a₆ := by
+  rw [b₆, a₃_of_isCharTwoJNeZeroNF]
+  ring1
+
+@[simp]
+theorem b₈_of_isCharTwoJNeZeroNF : W.b₈ = W.a₆ + 4 * W.a₂ * W.a₆ := by
+  rw [b₈, a₁_of_isCharTwoJNeZeroNF, a₃_of_isCharTwoJNeZeroNF, a₄_of_isCharTwoJNeZeroNF]
+  ring1
+
+@[simp]
+theorem c₄_of_isCharTwoJNeZeroNF : W.c₄ = W.b₂ ^ 2 := by
+  rw [c₄, b₄_of_isCharTwoJNeZeroNF]
+  ring1
+
+@[simp]
+theorem c₆_of_isCharTwoJNeZeroNF : W.c₆ = -W.b₂ ^ 3 - 864 * W.a₆ := by
+  rw [c₆, b₄_of_isCharTwoJNeZeroNF, b₆_of_isCharTwoJNeZeroNF]
+  ring1
+
+variable [CharP R 2]
+
+theorem b₂_of_isCharTwoJNeZeroNF_of_char_two : W.b₂ = 1 := by
+  rw [b₂_of_isCharTwoJNeZeroNF]
+  linear_combination 2 * W.a₂ * CharP.cast_eq_zero R 2
+
+theorem b₆_of_isCharTwoJNeZeroNF_of_char_two : W.b₆ = 0 := by
+  rw [b₆_of_isCharTwoJNeZeroNF]
+  linear_combination 2 * W.a₆ * CharP.cast_eq_zero R 2
+
+theorem b₈_of_isCharTwoJNeZeroNF_of_char_two : W.b₈ = W.a₆ := by
+  rw [b₈_of_isCharTwoJNeZeroNF]
+  linear_combination 2 * W.a₂ * W.a₆ * CharP.cast_eq_zero R 2
+
+theorem c₄_of_isCharTwoJNeZeroNF_of_char_two : W.c₄ = 1 := by
+  rw [c₄_of_isCharTwoJNeZeroNF, b₂_of_isCharTwoJNeZeroNF_of_char_two]
+  ring1
+
+theorem c₆_of_isCharTwoJNeZeroNF_of_char_two : W.c₆ = 1 := by
+  rw [c₆_of_isCharTwoJNeZeroNF, b₂_of_isCharTwoJNeZeroNF_of_char_two]
+  linear_combination (-1 - 432 * W.a₆) * CharP.cast_eq_zero R 2
+
+@[simp]
+theorem Δ_of_isCharTwoJNeZeroNF_of_char_two : W.Δ = W.a₆ := by
+  rw [Δ, b₂_of_isCharTwoJNeZeroNF_of_char_two, b₄_of_isCharTwoJNeZeroNF,
+    b₆_of_isCharTwoJNeZeroNF_of_char_two, b₈_of_isCharTwoJNeZeroNF_of_char_two]
+  linear_combination -W.a₆ * CharP.cast_eq_zero R 2
+
+variable [E.IsCharTwoJNeZeroNF] [CharP F 2]
+
+@[simp]
+theorem _root_.EllipticCurve.j_of_isCharTwoJNeZeroNF_of_char_two : E.j = 1 / E.a₆ := by
+  rw [EllipticCurve.j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, E.coe_Δ',
+    c₄_of_isCharTwoJNeZeroNF_of_char_two, Δ_of_isCharTwoJNeZeroNF_of_char_two, one_pow]
+
+theorem _root_.EllipticCurve.j_ne_zero_of_isCharTwoJNeZeroNF_of_char_two : E.j ≠ 0 := by
+  rw [E.j_of_isCharTwoJNeZeroNF_of_char_two, div_ne_zero_iff]
+  have h := E.Δ'.ne_zero
+  rw [E.coe_Δ', Δ_of_isCharTwoJNeZeroNF_of_char_two] at h
+  exact ⟨one_ne_zero, h⟩
+
+end Quantity
+
+/-! ### Normal forms of characteristic = 2 and j = 0 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic = 2 and j = 0, if its $a_1, a_2 = 0$.
+In other words it is $Y^2 + a_3Y = X^3 + a_4X + a_6$. -/
+@[mk_iff]
+class IsCharTwoJEqZeroNF : Prop where
+  a₁ : W.a₁ = 0
+  a₂ : W.a₂ = 0
+
+section Quantity
+
+variable [W.IsCharTwoJEqZeroNF]
+
+@[simp]
+theorem a₁_of_isCharTwoJEqZeroNF : W.a₁ = 0 := IsCharTwoJEqZeroNF.a₁
+
+@[simp]
+theorem a₂_of_isCharTwoJEqZeroNF : W.a₂ = 0 := IsCharTwoJEqZeroNF.a₂
+
+@[simp]
+theorem b₂_of_isCharTwoJEqZeroNF : W.b₂ = 0 := by
+  rw [b₂, a₁_of_isCharTwoJEqZeroNF, a₂_of_isCharTwoJEqZeroNF]
+  ring1
+
+@[simp]
+theorem b₄_of_isCharTwoJEqZeroNF : W.b₄ = 2 * W.a₄ := by
+  rw [b₄, a₁_of_isCharTwoJEqZeroNF]
+  ring1
+
+@[simp]
+theorem b₈_of_isCharTwoJEqZeroNF : W.b₈ = -W.a₄ ^ 2 := by
+  rw [b₈, a₁_of_isCharTwoJEqZeroNF, a₂_of_isCharTwoJEqZeroNF]
+  ring1
+
+@[simp]
+theorem c₄_of_isCharTwoJEqZeroNF : W.c₄ = -48 * W.a₄ := by
+  rw [c₄, b₂_of_isCharTwoJEqZeroNF, b₄_of_isCharTwoJEqZeroNF]
+  ring1
+
+@[simp]
+theorem c₆_of_isCharTwoJEqZeroNF : W.c₆ = -216 * W.b₆ := by
+  rw [c₆, b₂_of_isCharTwoJEqZeroNF, b₄_of_isCharTwoJEqZeroNF]
+  ring1
+
+@[simp]
+theorem Δ_of_isCharTwoJEqZeroNF : W.Δ = -(64 * W.a₄ ^ 3 + 27 * W.b₆ ^ 2) := by
+  rw [Δ, b₂_of_isCharTwoJEqZeroNF, b₄_of_isCharTwoJEqZeroNF]
+  ring1
+
+variable [CharP R 2]
+
+theorem b₄_of_isCharTwoJEqZeroNF_of_char_two : W.b₄ = 0 := by
+  rw [b₄_of_isCharTwoJEqZeroNF]
+  linear_combination W.a₄ * CharP.cast_eq_zero R 2
+
+theorem b₈_of_isCharTwoJEqZeroNF_of_char_two : W.b₈ = W.a₄ ^ 2 := by
+  rw [b₈_of_isCharTwoJEqZeroNF]
+  linear_combination -W.a₄ ^ 2 * CharP.cast_eq_zero R 2
+
+theorem c₄_of_isCharTwoJEqZeroNF_of_char_two : W.c₄ = 0 := by
+  rw [c₄_of_isCharTwoJEqZeroNF]
+  linear_combination -24 * W.a₄ * CharP.cast_eq_zero R 2
+
+theorem c₆_of_isCharTwoJEqZeroNF_of_char_two : W.c₆ = 0 := by
+  rw [c₆_of_isCharTwoJEqZeroNF]
+  linear_combination -108 * W.b₆ * CharP.cast_eq_zero R 2
+
+theorem Δ_of_isCharTwoJEqZeroNF_of_char_two : W.Δ = W.a₃ ^ 4 := by
+  rw [Δ_of_isCharTwoJEqZeroNF, b₆_of_char_two]
+  linear_combination (-32 * W.a₄ ^ 3 - 14 * W.a₃ ^ 4) * CharP.cast_eq_zero R 2
+
+variable [E.IsCharTwoJEqZeroNF]
+
+theorem _root_.EllipticCurve.j_of_isCharTwoJEqZeroNF :
+    E.j = 110592 * E.a₄ ^ 3 / (64 * E.a₄ ^ 3 + 27 * E.b₆ ^ 2) := by
+  have h := E.Δ'.ne_zero
+  rw [E.coe_Δ', Δ_of_isCharTwoJEqZeroNF] at h
+  rw [EllipticCurve.j, Units.val_inv_eq_inv_val, ← div_eq_inv_mul, E.coe_Δ',
+    c₄_of_isCharTwoJEqZeroNF, Δ_of_isCharTwoJEqZeroNF, div_eq_div_iff h (neg_ne_zero.1 h)]
+  ring1
+
+@[simp]
+theorem _root_.EllipticCurve.j_of_isCharTwoJEqZeroNF_of_char_two [CharP F 2] : E.j = 0 := by
+  rw [EllipticCurve.j, c₄_of_isCharTwoJEqZeroNF_of_char_two]; simp
+
+end Quantity
+
+/-! ### Normal forms of characteristic = 2 -/
+
+/-- A `WeierstrassCurve` is in normal form of characteristic = 2, if it is
+$Y^2 + XY = X^3 + a_2X^2 + a_6$ (`WeierstrassCurve.IsCharTwoJNeZeroNF`) or
+$Y^2 + a_3Y = X^3 + a_4X + a_6$ (`WeierstrassCurve.IsCharTwoJEqZeroNF`). -/
+class inductive IsCharTwoNF : Prop
+| of_j_ne_zero [W.IsCharTwoJNeZeroNF] : IsCharTwoNF
+| of_j_eq_zero [W.IsCharTwoJEqZeroNF] : IsCharTwoNF
+
+instance isCharTwoNF_of_isCharTwoJNeZeroNF [W.IsCharTwoJNeZeroNF] : W.IsCharTwoNF :=
+  IsCharTwoNF.of_j_ne_zero
+
+instance isCharTwoNF_of_isCharTwoJEqZeroNF [W.IsCharTwoJEqZeroNF] : W.IsCharTwoNF :=
+  IsCharTwoNF.of_j_eq_zero
+
+section VariableChange
+
+variable [CharP R 2] [CharP F 2]
+
+/-- For a `WeierstrassCurve` defined over a ring of characteristic = 2,
+there is an explicit change of variables of it to $Y^2 + a_3Y = X^3 + a_4X + a_6$
+(`WeierstrassCurve.IsCharTwoJEqZeroNF`) if its j = 0. -/
+def toCharTwoJEqZeroNF : VariableChange R := ⟨1, W.a₂, 0, 0⟩
+
+theorem toCharTwoJEqZeroNF_spec (ha₁ : W.a₁ = 0) :
+    (W.variableChange W.toCharTwoJEqZeroNF).IsCharTwoJEqZeroNF := by
+  constructor
+  · simp [toCharTwoJEqZeroNF, ha₁]
+  · simp_rw [toCharTwoJEqZeroNF, variableChange_a₂, inv_one, Units.val_one]
+    linear_combination 2 * W.a₂ * CharP.cast_eq_zero R 2
+
+variable (W : WeierstrassCurve F)
+
+/-- For a `WeierstrassCurve` defined over a field of characteristic = 2,
+there is an explicit change of variables of it to $Y^2 + XY = X^3 + a_2X^2 + a_6$
+(`WeierstrassCurve.IsCharTwoJNeZeroNF`) if its j ≠ 0. -/
+def toCharTwoJNeZeroNF (W : WeierstrassCurve F) (ha₁ : W.a₁ ≠ 0) : VariableChange F :=
+  ⟨Units.mk0 _ ha₁, W.a₃ / W.a₁, 0, (W.a₁ ^ 2 * W.a₄ + W.a₃ ^ 2) / W.a₁ ^ 3⟩
+
+theorem toCharTwoJNeZeroNF_spec (ha₁ : W.a₁ ≠ 0) :
+    (W.variableChange (W.toCharTwoJNeZeroNF ha₁)).IsCharTwoJNeZeroNF := by
+  constructor
+  · simp [toCharTwoJNeZeroNF, ha₁]
+  · field_simp [toCharTwoJNeZeroNF]
+    linear_combination (W.a₃ * W.a₁ ^ 3 + W.a₁ ^ 2 * W.a₄ + W.a₃ ^ 2) * CharP.cast_eq_zero F 2
+  · field_simp [toCharTwoJNeZeroNF]
+    linear_combination (W.a₁ ^ 4 * W.a₃ ^ 2 + W.a₁ ^ 5 * W.a₃ * W.a₂) * CharP.cast_eq_zero F 2
+
+/-- For a `WeierstrassCurve` defined over a field of characteristic = 2,
+there is an explicit change of variables of it to `WeierstrassCurve.IsCharTwoNF`, that is,
+$Y^2 + XY = X^3 + a_2X^2 + a_6$ (`WeierstrassCurve.IsCharTwoJNeZeroNF`) or
+$Y^2 + a_3Y = X^3 + a_4X + a_6$ (`WeierstrassCurve.IsCharTwoJEqZeroNF`). -/
+def toCharTwoNF [DecidableEq F] : VariableChange F :=
+  if ha₁ : W.a₁ = 0 then W.toCharTwoJEqZeroNF else W.toCharTwoJNeZeroNF ha₁
+
+instance toCharTwoNF_spec [DecidableEq F] : (W.variableChange W.toCharTwoNF).IsCharTwoNF := by
+  by_cases ha₁ : W.a₁ = 0
+  · rw [toCharTwoNF, dif_pos ha₁]
+    haveI := W.toCharTwoJEqZeroNF_spec ha₁
+    infer_instance
+  · rw [toCharTwoNF, dif_neg ha₁]
+    haveI := W.toCharTwoJNeZeroNF_spec ha₁
+    infer_instance
+
+theorem exists_variableChange_isCharTwoNF :
+    ∃ C : VariableChange F, (W.variableChange C).IsCharTwoNF := by
+  classical
+  exact ⟨_, W.toCharTwoNF_spec⟩
+
+end VariableChange
+
+end WeierstrassCurve
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
index f1314e4735540..3d4ea8337acc9 100644
--- a/Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/Projective.lean
@@ -3,9 +3,11 @@ Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Kurniadi Angdinata
 -/
-import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
 import Mathlib.Algebra.MvPolynomial.CommRing
 import Mathlib.Algebra.MvPolynomial.PDeriv
+import Mathlib.AlgebraicGeometry.EllipticCurve.Affine
+import Mathlib.Data.Fin.Tuple.Reflection
+import Mathlib.Tactic.LinearCombination'
 
 /-!
 # Projective coordinates for Weierstrass curves
@@ -18,7 +20,7 @@ condition.
 
 Let `W` be a Weierstrass curve over a field `F`. A point on the projective plane is an equivalence
 class of triples $[x:y:z]$ with coordinates in `F` such that $(x, y, z) \sim (x', y', z')$ precisely
-if there is some unit $u$ of `F` such that $(x, y, z) = (ux', uy', uz')$, with an extra condition
+if there is some unit `u` of `F` such that $(x, y, z) = (ux', uy', uz')$, with an extra condition
 that $(x, y, z) \ne (0, 0, 0)$. As described in `Mathlib.AlgebraicGeometry.EllipticCurve.Affine`, a
 rational point is a point on the projective plane satisfying a homogeneous Weierstrass equation, and
 being nonsingular means the partial derivatives $W_X(X, Y, Z)$, $W_Y(X, Y, Z)$, and $W_Z(X, Y, Z)$
@@ -27,6 +29,10 @@ derivatives are independent of the representative for $[x:y:z]$, and the nonsing
 already implies that $(x, y, z) \ne (0, 0, 0)$, so a nonsingular rational point on `W` can simply be
 given by a tuple consisting of $[x:y:z]$ and the nonsingular condition on any representative.
 
+As in `Mathlib.AlgebraicGeometry.EllipticCurve.Affine`, the set of nonsingular rational points forms
+an abelian group under the same secant-and-tangent process, but the polynomials involved are
+homogeneous, and any instances of division become multiplication in the $Z$-coordinate.
+
 ## Main definitions
 
  * `WeierstrassCurve.Projective.PointClass`: the equivalence class of a point representative.
@@ -42,11 +48,11 @@ given by a tuple consisting of $[x:y:z]$ and the nonsingular condition on any re
 
 A point representative is implemented as a term `P` of type `Fin 3 → R`, which allows for the vector
 notation `![x, y, z]`. However, `P` is not definitionally equivalent to the expanded vector
-`![P x, P y, P z]`, so the auxiliary lemma `fin3_def` can be used to convert between the two forms.
-The equivalence of two point representatives `P` and `Q` is implemented as an equivalence of orbits
-of the action of `Rˣ`, or equivalently that there is some unit `u` of `R` such that `P = u • Q`.
-However, `u • Q` is again not definitionally equal to `![u * Q x, u * Q y, u * Q z]`, so the
-auxiliary lemmas `smul_fin3` and `smul_fin3_ext` can be used to convert between the two forms.
+`![P x, P y, P z]`, so the lemmas `fin3_def` and `fin3_def_ext` can be used to convert between the
+two forms. The equivalence of two point representatives `P` and `Q` is implemented as an equivalence
+of orbits of the action of `Rˣ`, or equivalently that there is some unit `u` of `R` such that
+`P = u • Q`. However, `u • Q` is not definitionally equal to `![u * Q x, u * Q y, u * Q z]`, so the
+lemmas `smul_fin3` and `smul_fin3_ext` can be used to convert between the two forms.
 
 ## References
 
@@ -57,23 +63,27 @@ auxiliary lemmas `smul_fin3` and `smul_fin3_ext` can be used to convert between
 elliptic curve, rational point, projective coordinates
 -/
 
-local notation "x" => 0
+local notation3 "x" => (0 : Fin 3)
 
-local notation "y" => 1
+local notation3 "y" => (1 : Fin 3)
 
-local notation "z" => 2
+local notation3 "z" => (2 : Fin 3)
 
 local macro "matrix_simp" : tactic =>
   `(tactic| simp only [Matrix.head_cons, Matrix.tail_cons, Matrix.smul_empty, Matrix.smul_cons,
     Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_two])
 
-universe u
+universe u v
 
 /-! ## Weierstrass curves -/
 
 /-- An abbreviation for a Weierstrass curve in projective coordinates. -/
-abbrev WeierstrassCurve.Projective :=
-  WeierstrassCurve
+abbrev WeierstrassCurve.Projective (R : Type u) : Type u :=
+  WeierstrassCurve R
+
+/-- The coercion to a Weierstrass curve in projective coordinates. -/
+abbrev WeierstrassCurve.toProjective {R : Type u} (W : WeierstrassCurve R) : Projective R :=
+  W
 
 namespace WeierstrassCurve.Projective
 
@@ -85,231 +95,1072 @@ local macro "eval_simp" : tactic =>
 local macro "pderiv_simp" : tactic =>
   `(tactic| simp only [map_ofNat, map_neg, map_add, map_sub, map_mul, pderiv_mul, pderiv_pow,
     pderiv_C, pderiv_X_self, pderiv_X_of_ne one_ne_zero, pderiv_X_of_ne one_ne_zero.symm,
-    pderiv_X_of_ne (by decide : (2 : Fin 3) ≠ 0), pderiv_X_of_ne (by decide : (0 : Fin 3) ≠ 2),
-    pderiv_X_of_ne (by decide : (2 : Fin 3) ≠ 1), pderiv_X_of_ne (by decide : (1 : Fin 3) ≠ 2)])
+    pderiv_X_of_ne (by decide : z ≠ x), pderiv_X_of_ne (by decide : x ≠ z),
+    pderiv_X_of_ne (by decide : z ≠ y), pderiv_X_of_ne (by decide : y ≠ z)])
 
-variable {R : Type u} [CommRing R] (W : Projective R)
+variable {R : Type u} {W' : Projective R} {F : Type v} [Field F] {W : Projective F}
 
-lemma fin3_def {R : Type u} (P : Fin 3 → R) : P = ![P x, P y, P z] := by
+section Projective
+
+/-! ### Projective coordinates -/
+
+lemma fin3_def (P : Fin 3 → R) : ![P x, P y, P z] = P := by
   ext n; fin_cases n <;> rfl
 
-lemma smul_fin3 {R : Type u} [CommRing R] (P : Fin 3 → R) (u : Rˣ) :
-    u • P = ![u * P x, u * P y, u * P z] := by
-  rw [fin3_def P]
-  matrix_simp
-  simp only [Units.smul_def, smul_eq_mul]
+lemma fin3_def_ext (X Y Z : R) : ![X, Y, Z] x = X ∧ ![X, Y, Z] y = Y ∧ ![X, Y, Z] z = Z :=
+  ⟨rfl, rfl, rfl⟩
+
+lemma comp_fin3 {S : Type v} (f : R → S) (X Y Z : R) : f ∘ ![X, Y, Z] = ![f X, f Y, f Z] :=
+  (FinVec.map_eq ..).symm
+
+variable [CommRing R]
+
+lemma smul_fin3 (P : Fin 3 → R) (u : R) : u • P = ![u * P x, u * P y, u * P z] := by
+  simp [← List.ofFn_inj]
 
-lemma smul_fin3_ext {R : Type u} [CommRing R] (P : Fin 3 → R) (u : Rˣ) :
-    (u • P) x = u * P x ∧ (u • P) y = u * P y ∧ (u • P) z = u * P z := by
-  refine ⟨?_, ?_, ?_⟩ <;> simp only [Units.smul_def, Pi.smul_apply, smul_eq_mul]
+lemma smul_fin3_ext (P : Fin 3 → R) (u : R) :
+    (u • P) x = u * P x ∧ (u • P) y = u * P y ∧ (u • P) z = u * P z :=
+  ⟨rfl, rfl, rfl⟩
 
 /-- The equivalence setoid for a point representative. -/
 scoped instance instSetoidPoint : Setoid <| Fin 3 → R :=
   MulAction.orbitRel Rˣ <| Fin 3 → R
 
+variable (R) in
 /-- The equivalence class of a point representative. -/
-abbrev PointClass (R : Type u) [CommRing R] : Type u :=
+abbrev PointClass : Type u :=
   MulAction.orbitRel.Quotient Rˣ <| Fin 3 → R
 
+lemma smul_equiv (P : Fin 3 → R) {u : R} (hu : IsUnit u) : u • P ≈ P :=
+  ⟨hu.unit, rfl⟩
+
+@[simp]
+lemma smul_eq (P : Fin 3 → R) {u : R} (hu : IsUnit u) : (⟦u • P⟧ : PointClass R) = ⟦P⟧ :=
+  Quotient.eq.mpr <| smul_equiv P hu
+
+variable (W') in
 /-- The coercion to a Weierstrass curve in affine coordinates. -/
 abbrev toAffine : Affine R :=
-  W
+  W'
+
+lemma equiv_iff_eq_of_Z_eq' {P Q : Fin 3 → R} (hz : P z = Q z) (mem : Q z ∈ nonZeroDivisors R) :
+    P ≈ Q ↔ P = Q := by
+  refine ⟨?_, by rintro rfl; exact Setoid.refl _⟩
+  rintro ⟨u, rfl⟩
+  rw [← one_mul (Q z)] at hz
+  simp_rw [Units.smul_def, (mul_cancel_right_mem_nonZeroDivisors mem).mp hz, one_smul]
+
+lemma equiv_iff_eq_of_Z_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hz : P z = Q z) (hQz : Q z ≠ 0) :
+    P ≈ Q ↔ P = Q :=
+  equiv_iff_eq_of_Z_eq' hz (mem_nonZeroDivisors_of_ne_zero hQz)
+
+lemma Z_eq_zero_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : P z = 0 ↔ Q z = 0 := by
+  rcases h with ⟨_, rfl⟩
+  simp only [Units.smul_def, smul_fin3_ext, Units.mul_right_eq_zero]
+
+lemma X_eq_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : P x * Q z = Q x * P z := by
+  rcases h with ⟨u, rfl⟩
+  simp only [Units.smul_def, smul_fin3_ext]
+  ring1
+
+lemma Y_eq_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : P y * Q z = Q y * P z := by
+  rcases h with ⟨u, rfl⟩
+  simp only [Units.smul_def, smul_fin3_ext]
+  ring1
+
+lemma not_equiv_of_Z_eq_zero_left {P Q : Fin 3 → R} (hPz : P z = 0) (hQz : Q z ≠ 0) : ¬P ≈ Q :=
+  fun h => hQz <| (Z_eq_zero_of_equiv h).mp hPz
+
+lemma not_equiv_of_Z_eq_zero_right {P Q : Fin 3 → R} (hPz : P z ≠ 0) (hQz : Q z = 0) : ¬P ≈ Q :=
+  fun h => hPz <| (Z_eq_zero_of_equiv h).mpr hQz
+
+lemma not_equiv_of_X_ne {P Q : Fin 3 → R} (hx : P x * Q z ≠ Q x * P z) : ¬P ≈ Q :=
+  hx.comp X_eq_of_equiv
+
+lemma not_equiv_of_Y_ne {P Q : Fin 3 → R} (hy : P y * Q z ≠ Q y * P z) : ¬P ≈ Q :=
+  hy.comp Y_eq_of_equiv
+
+lemma equiv_of_X_eq_of_Y_eq {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) : P ≈ Q := by
+  use Units.mk0 _ hPz / Units.mk0 _ hQz
+  simp only [Units.smul_def, smul_fin3, Units.val_div_eq_div_val, Units.val_mk0, mul_comm, mul_div,
+    ← hx, ← hy, mul_div_cancel_right₀ _ hQz, fin3_def]
+
+lemma equiv_some_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) : P ≈ ![P x / P z, P y / P z, 1] :=
+  equiv_of_X_eq_of_Y_eq hPz one_ne_zero
+    (by linear_combination (norm := (matrix_simp; ring1)) -P x * div_self hPz)
+    (by linear_combination (norm := (matrix_simp; ring1)) -P y * div_self hPz)
+
+lemma X_eq_iff {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) :
+    P x * Q z = Q x * P z ↔ P x / P z = Q x / Q z :=
+  (div_eq_div_iff hPz hQz).symm
+
+lemma Y_eq_iff {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) :
+    P y * Q z = Q y * P z ↔ P y / P z = Q y / Q z :=
+  (div_eq_div_iff hPz hQz).symm
+
+end Projective
+
+variable [CommRing R]
 
 section Equation
 
-/-! ### Equations and nonsingularity -/
+/-! ### Weierstrass equations -/
 
+variable (W') in
 /-- The polynomial $W(X, Y, Z) := Y^2Z + a_1XYZ + a_3YZ^2 - (X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3)$
-associated to a Weierstrass curve `W` over `R`. This is represented as a term of type
+associated to a Weierstrass curve `W'` over `R`. This is represented as a term of type
 `MvPolynomial (Fin 3) R`, where `X 0`, `X 1`, and `X 2` represent $X$, $Y$, and $Z$ respectively. -/
 noncomputable def polynomial : MvPolynomial (Fin 3) R :=
-  X 1 ^ 2 * X 2 + C W.a₁ * X 0 * X 1 * X 2 + C W.a₃ * X 1 * X 2 ^ 2
-    - (X 0 ^ 3 + C W.a₂ * X 0 ^ 2 * X 2 + C W.a₄ * X 0 * X 2 ^ 2 + C W.a₆ * X 2 ^ 3)
+  X 1 ^ 2 * X 2 + C W'.a₁ * X 0 * X 1 * X 2 + C W'.a₃ * X 1 * X 2 ^ 2
+    - (X 0 ^ 3 + C W'.a₂ * X 0 ^ 2 * X 2 + C W'.a₄ * X 0 * X 2 ^ 2 + C W'.a₆ * X 2 ^ 3)
 
-lemma eval_polynomial (P : Fin 3 → R) : eval P W.polynomial =
-    P y ^ 2 * P z + W.a₁ * P x * P y * P z + W.a₃ * P y * P z ^ 2
-      - (P x ^ 3 + W.a₂ * P x ^ 2 * P z + W.a₄ * P x * P z ^ 2 + W.a₆ * P z ^ 3) := by
+lemma eval_polynomial (P : Fin 3 → R) : eval P W'.polynomial =
+    P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2
+      - (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3) := by
   rw [polynomial]
   eval_simp
 
-/-- The proposition that a point representative $(x, y, z)$ lies in `W`.
+lemma eval_polynomial_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) : eval P W.polynomial / P z ^ 3 =
+    W.toAffine.polynomial.evalEval (P x / P z) (P y / P z) := by
+  linear_combination (norm := (rw [eval_polynomial, Affine.evalEval_polynomial]; ring1))
+    P y ^ 2 / P z ^ 2 * div_self hPz + W.a₁ * P x * P y / P z ^ 2 * div_self hPz
+      + W.a₃ * P y / P z * div_self (pow_ne_zero 2 hPz) - W.a₂ * P x ^ 2 / P z ^ 2 * div_self hPz
+      - W.a₄ * P x / P z * div_self (pow_ne_zero 2 hPz) - W.a₆ * div_self (pow_ne_zero 3 hPz)
+
+variable (W') in
+/-- The proposition that a point representative $(x, y, z)$ lies in `W'`.
 In other words, $W(x, y, z) = 0$. -/
 def Equation (P : Fin 3 → R) : Prop :=
-  eval P W.polynomial = 0
+  eval P W'.polynomial = 0
 
-lemma equation_iff (P : Fin 3 → R) : W.Equation P ↔
-    P y ^ 2 * P z + W.a₁ * P x * P y * P z + W.a₃ * P y * P z ^ 2
-      = P x ^ 3 + W.a₂ * P x ^ 2 * P z + W.a₄ * P x * P z ^ 2 + W.a₆ * P z ^ 3 := by
+lemma equation_iff (P : Fin 3 → R) : W'.Equation P ↔
+    P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2
+      - (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3) = 0 := by
   rw [Equation, eval_polynomial, sub_eq_zero]
 
-lemma equation_zero (Y : R) : W.Equation ![0, Y, 0] :=
-  (W.equation_iff ![0, Y, 0]).mpr <| by matrix_simp; ring1
+lemma equation_smul (P : Fin 3 → R) {u : R} (hu : IsUnit u) : W'.Equation (u • P) ↔ W'.Equation P :=
+  have hP (u : R) {P : Fin 3 → R} (hP : W'.Equation P) : W'.Equation <| u • P := by
+    rw [equation_iff] at hP ⊢
+    linear_combination (norm := (simp only [smul_fin3_ext]; ring1)) u ^ 3 * hP
+  ⟨fun h => by convert hP hu.unit.inv h; erw [smul_smul, hu.val_inv_mul, one_smul], hP u⟩
+
+lemma equation_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : W'.Equation P ↔ W'.Equation Q := by
+  rcases h with ⟨u, rfl⟩
+  exact equation_smul Q u.isUnit
+
+lemma equation_of_Z_eq_zero {P : Fin 3 → R} (hPz : P z = 0) : W'.Equation P ↔ P x ^ 3 = 0 := by
+  simp only [equation_iff, hPz, add_zero, zero_sub, mul_zero, zero_pow <| OfNat.ofNat_ne_zero _,
+    neg_eq_zero]
 
-lemma equation_some (X Y : R) : W.Equation ![X, Y, 1] ↔ W.toAffine.Equation X Y := by
-  rw [equation_iff, W.toAffine.equation_iff]
-  congr! 1 <;> matrix_simp <;> ring1
+lemma equation_zero : W'.Equation ![0, 1, 0] := by
+  simp only [equation_of_Z_eq_zero, fin3_def_ext, zero_pow three_ne_zero]
 
-lemma equation_smul_iff (P : Fin 3 → R) (u : Rˣ) : W.Equation (u • P) ↔ W.Equation P :=
-  have (u : Rˣ) {P : Fin 3 → R} (h : W.Equation P) : W.Equation <| u • P := by
-    rw [equation_iff] at h ⊢
-    linear_combination (norm := (simp only [smul_fin3_ext]; ring1)) (u : R) ^ 3 * h
-  ⟨fun h => by convert this u⁻¹ h; rw [inv_smul_smul], this u⟩
+lemma equation_some (X Y : R) : W'.Equation ![X, Y, 1] ↔ W'.toAffine.Equation X Y := by
+  simp only [equation_iff, Affine.equation_iff', fin3_def_ext, one_pow, mul_one]
 
+lemma equation_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    W.Equation P ↔ W.toAffine.Equation (P x / P z) (P y / P z) :=
+  (equation_of_equiv <| equiv_some_of_Z_ne_zero hPz).trans <| equation_some ..
+
+lemma X_eq_zero_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : P x = 0 :=
+  pow_eq_zero <| (equation_of_Z_eq_zero hPz).mp hP
+
+end Equation
+
+section Nonsingular
+
+/-! ### Nonsingular Weierstrass equations -/
+
+variable (W') in
 /-- The partial derivative $W_X(X, Y, Z)$ of $W(X, Y, Z)$ with respect to $X$. -/
 noncomputable def polynomialX : MvPolynomial (Fin 3) R :=
-  pderiv x W.polynomial
+  pderiv x W'.polynomial
 
-lemma polynomialX_eq : W.polynomialX =
-    C W.a₁ * X 1 * X 2 - (C 3 * X 0 ^ 2 + C (2 * W.a₂) * X 0 * X 2 + C W.a₄ * X 2 ^ 2) := by
+lemma polynomialX_eq : W'.polynomialX =
+    C W'.a₁ * X 1 * X 2 - (C 3 * X 0 ^ 2 + C (2 * W'.a₂) * X 0 * X 2 + C W'.a₄ * X 2 ^ 2) := by
   rw [polynomialX, polynomial]
   pderiv_simp
   ring1
 
-lemma eval_polynomialX (P : Fin 3 → R) : eval P W.polynomialX =
-    W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z + W.a₄ * P z ^ 2) := by
+lemma eval_polynomialX (P : Fin 3 → R) : eval P W'.polynomialX =
+    W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z + W'.a₄ * P z ^ 2) := by
   rw [polynomialX_eq]
   eval_simp
 
+lemma eval_polynomialX_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    eval P W.polynomialX / P z ^ 2 = W.toAffine.polynomialX.evalEval (P x / P z) (P y / P z) := by
+  linear_combination (norm := (rw [eval_polynomialX, Affine.evalEval_polynomialX]; ring1))
+    W.a₁ * P y / P z * div_self hPz - 2 * W.a₂ * P x / P z * div_self hPz
+      - W.a₄ * div_self (pow_ne_zero 2 hPz)
+
+variable (W') in
 /-- The partial derivative $W_Y(X, Y, Z)$ of $W(X, Y, Z)$ with respect to $Y$. -/
 noncomputable def polynomialY : MvPolynomial (Fin 3) R :=
-  pderiv y W.polynomial
+  pderiv y W'.polynomial
 
-lemma polynomialY_eq : W.polynomialY =
-    C 2 * X 1 * X 2 + C W.a₁ * X 0 * X 2 + C W.a₃ * X 2 ^ 2 := by
+lemma polynomialY_eq : W'.polynomialY =
+    C 2 * X 1 * X 2 + C W'.a₁ * X 0 * X 2 + C W'.a₃ * X 2 ^ 2 := by
   rw [polynomialY, polynomial]
   pderiv_simp
   ring1
 
 lemma eval_polynomialY (P : Fin 3 → R) :
-    eval P W.polynomialY = 2 * P y * P z + W.a₁ * P x * P z + W.a₃ * P z ^ 2 := by
+    eval P W'.polynomialY = 2 * P y * P z + W'.a₁ * P x * P z + W'.a₃ * P z ^ 2 := by
   rw [polynomialY_eq]
   eval_simp
 
+lemma eval_polynomialY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    eval P W.polynomialY / P z ^ 2 = W.toAffine.polynomialY.evalEval (P x / P z) (P y / P z) := by
+  linear_combination (norm := (rw [eval_polynomialY, Affine.evalEval_polynomialY]; ring1))
+    2 * P y / P z * div_self hPz + W.a₁ * P x / P z * div_self hPz
+      + W.a₃ * div_self (pow_ne_zero 2 hPz)
+
+variable (W') in
 /-- The partial derivative $W_Z(X, Y, Z)$ of $W(X, Y, Z)$ with respect to $Z$. -/
 noncomputable def polynomialZ : MvPolynomial (Fin 3) R :=
-  pderiv z W.polynomial
+  pderiv z W'.polynomial
 
-lemma polynomialZ_eq : W.polynomialZ =
-    X 1 ^ 2 + C W.a₁ * X 0 * X 1 + C (2 * W.a₃) * X 1 * X 2
-      - (C W.a₂ * X 0 ^ 2 + C (2 * W.a₄) * X 0 * X 2 + C (3 * W.a₆) * X 2 ^ 2) := by
+lemma polynomialZ_eq : W'.polynomialZ =
+    X 1 ^ 2 + C W'.a₁ * X 0 * X 1 + C (2 * W'.a₃) * X 1 * X 2
+      - (C W'.a₂ * X 0 ^ 2 + C (2 * W'.a₄) * X 0 * X 2 + C (3 * W'.a₆) * X 2 ^ 2) := by
   rw [polynomialZ, polynomial]
   pderiv_simp
   ring1
 
-lemma eval_polynomialZ (P : Fin 3 → R) : eval P W.polynomialZ =
-    P y ^ 2 + W.a₁ * P x * P y + 2 * W.a₃ * P y * P z
-      - (W.a₂ * P x ^ 2 + 2 * W.a₄ * P x * P z + 3 * W.a₆ * P z ^ 2) := by
+lemma eval_polynomialZ (P : Fin 3 → R) : eval P W'.polynomialZ =
+    P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z
+      - (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2) := by
   rw [polynomialZ_eq]
   eval_simp
 
 /-- Euler's homogeneous function theorem. -/
-theorem polynomial_relation (P : Fin 3 → R) : 3 * eval P W.polynomial =
-    P x * eval P W.polynomialX + P y * eval P W.polynomialY + P z * eval P W.polynomialZ := by
+theorem polynomial_relation (P : Fin 3 → R) : 3 * eval P W'.polynomial =
+    P x * eval P W'.polynomialX + P y * eval P W'.polynomialY + P z * eval P W'.polynomialZ := by
   rw [eval_polynomial, eval_polynomialX, eval_polynomialY, eval_polynomialZ]
   ring1
 
-/-- The proposition that a point representative $(x, y, z)$ in `W` is nonsingular.
-In other words, either $W_X(x, y, z) \ne 0$, $W_Y(x, y, z) \ne 0$, or $W_Z(x, y, z) \ne 0$. -/
+variable (W') in
+/-- The proposition that a point representative $(x, y, z)$ in `W'` is nonsingular.
+In other words, either $W_X(x, y, z) \ne 0$, $W_Y(x, y, z) \ne 0$, or $W_Z(x, y, z) \ne 0$.
+
+Note that this definition is only mathematically accurate for fields. -/
+-- TODO: generalise this definition to be mathematically accurate for a larger class of rings.
 def Nonsingular (P : Fin 3 → R) : Prop :=
-  W.Equation P ∧ (eval P W.polynomialX ≠ 0 ∨ eval P W.polynomialY ≠ 0 ∨ eval P W.polynomialZ ≠ 0)
+  W'.Equation P ∧
+    (eval P W'.polynomialX ≠ 0 ∨ eval P W'.polynomialY ≠ 0 ∨ eval P W'.polynomialZ ≠ 0)
+
+lemma nonsingular_iff (P : Fin 3 → R) : W'.Nonsingular P ↔ W'.Equation P ∧
+    (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z + W'.a₄ * P z ^ 2) ≠ 0 ∨
+      2 * P y * P z + W'.a₁ * P x * P z + W'.a₃ * P z ^ 2 ≠ 0 ∨
+      P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z
+        - (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2) ≠ 0) := by
+  rw [Nonsingular, eval_polynomialX, eval_polynomialY, eval_polynomialZ]
+
+lemma nonsingular_smul (P : Fin 3 → R) {u : R} (hu : IsUnit u) :
+    W'.Nonsingular (u • P) ↔ W'.Nonsingular P :=
+  have hP {u : R} (hu : IsUnit u) {P : Fin 3 → R} (hP : W'.Nonsingular <| u • P) :
+      W'.Nonsingular P := by
+    rcases (nonsingular_iff _).mp hP with ⟨hP, hP'⟩
+    refine (nonsingular_iff P).mpr ⟨(equation_smul P hu).mp hP, ?_⟩
+    contrapose! hP'
+    simp only [smul_fin3_ext]
+    exact ⟨by linear_combination (norm := ring1) u ^ 2 * hP'.left,
+      by linear_combination (norm := ring1) u ^ 2 * hP'.right.left,
+      by linear_combination (norm := ring1) u ^ 2 * hP'.right.right⟩
+  ⟨hP hu, fun h => hP hu.unit⁻¹.isUnit <| by rwa [smul_smul, hu.val_inv_mul, one_smul]⟩
 
-lemma nonsingular_iff (P : Fin 3 → R) : W.Nonsingular P ↔ W.Equation P ∧
-    (W.a₁ * P y * P z ≠ 3 * P x ^ 2 + 2 * W.a₂ * P x * P z + W.a₄ * P z ^ 2 ∨
-      P y * P z ≠ -P y * P z - W.a₁ * P x * P z - W.a₃ * P z ^ 2 ∨
-      P y ^ 2 + W.a₁ * P x * P y + 2 * W.a₃ * P y * P z
-        ≠ W.a₂ * P x ^ 2 + 2 * W.a₄ * P x * P z + 3 * W.a₆ * P z ^ 2) := by
-  rw [Nonsingular, eval_polynomialX, eval_polynomialY, eval_polynomialZ, sub_ne_zero, sub_ne_zero,
-    ← sub_ne_zero (a := P y * P z)]
-  congr! 4
-  ring1
+lemma nonsingular_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : W'.Nonsingular P ↔ W'.Nonsingular Q := by
+  rcases h with ⟨u, rfl⟩
+  exact nonsingular_smul Q u.isUnit
+
+lemma nonsingular_of_Z_eq_zero {P : Fin 3 → R} (hPz : P z = 0) :
+    W'.Nonsingular P ↔
+      W'.Equation P ∧ (3 * P x ^ 2 ≠ 0 ∨ P y ^ 2 + W'.a₁ * P x * P y - W'.a₂ * P x ^ 2 ≠ 0) := by
+  simp only [nonsingular_iff, hPz, add_zero, sub_zero, zero_sub, mul_zero,
+    zero_pow <| OfNat.ofNat_ne_zero _, neg_ne_zero, ne_self_iff_false, false_or]
+
+lemma nonsingular_zero [Nontrivial R] : W'.Nonsingular ![0, 1, 0] := by
+  simp only [nonsingular_of_Z_eq_zero, equation_zero, true_and, fin3_def_ext, ← not_and_or]
+  exact fun h => one_ne_zero <| by linear_combination (norm := ring1) h.right
+
+lemma nonsingular_some (X Y : R) : W'.Nonsingular ![X, Y, 1] ↔ W'.toAffine.Nonsingular X Y := by
+  simp_rw [nonsingular_iff, equation_some, fin3_def_ext, Affine.nonsingular_iff',
+    Affine.equation_iff', and_congr_right_iff, ← not_and_or, not_iff_not, one_pow, mul_one,
+    and_congr_right_iff, Iff.comm, iff_self_and]
+  intro h hX hY
+  linear_combination (norm := ring1) 3 * h - X * hX - Y * hY
 
-lemma nonsingular_zero [Nontrivial R] : W.Nonsingular ![0, 1, 0] :=
-  (W.nonsingular_iff ![0, 1, 0]).mpr ⟨W.equation_zero 1, by simp⟩
+lemma nonsingular_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    W.Nonsingular P ↔ W.toAffine.Nonsingular (P x / P z) (P y / P z) :=
+  (nonsingular_of_equiv <| equiv_some_of_Z_ne_zero hPz).trans <| nonsingular_some ..
 
-lemma nonsingular_zero' [NoZeroDivisors R] {Y : R} (hy : Y ≠ 0) : W.Nonsingular ![0, Y, 0] :=
-  (W.nonsingular_iff ![0, Y, 0]).mpr ⟨W.equation_zero Y, by simpa⟩
+lemma nonsingular_iff_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    W.Nonsingular P ↔ W.Equation P ∧ (eval P W.polynomialX ≠ 0 ∨ eval P W.polynomialY ≠ 0) := by
+  rw [nonsingular_of_Z_ne_zero hPz, Affine.Nonsingular, ← equation_of_Z_ne_zero hPz,
+    ← eval_polynomialX_of_Z_ne_zero hPz, div_ne_zero_iff, and_iff_left <| pow_ne_zero 2 hPz,
+    ← eval_polynomialY_of_Z_ne_zero hPz, div_ne_zero_iff, and_iff_left <| pow_ne_zero 2 hPz]
 
-lemma nonsingular_some (X Y : R) : W.Nonsingular ![X, Y, 1] ↔ W.toAffine.Nonsingular X Y := by
-  rw [nonsingular_iff]
-  matrix_simp
-  simp only [W.toAffine.nonsingular_iff, equation_some, and_congr_right_iff,
-    W.toAffine.equation_iff, ← not_and_or, not_iff_not, one_pow, mul_one, Iff.comm, iff_self_and]
-  intro h hX hY
-  linear_combination (norm := ring1) 3 * h - X * hX - Y * hY
+lemma Y_ne_zero_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Nonsingular P)
+    (hPz : P z = 0) : P y ≠ 0 := by
+  intro hPy
+  simp only [nonsingular_of_Z_eq_zero hPz, X_eq_zero_of_Z_eq_zero hP.left hPz, hPy, add_zero,
+    sub_zero, mul_zero, zero_pow two_ne_zero, or_self, ne_self_iff_false, and_false] at hP
 
-lemma nonsingular_smul_iff (P : Fin 3 → R) (u : Rˣ) : W.Nonsingular (u • P) ↔ W.Nonsingular P :=
-  have (u : Rˣ) {P : Fin 3 → R} (h : W.Nonsingular <| u • P) : W.Nonsingular P := by
-    rcases (W.nonsingular_iff _).mp h with ⟨h, h'⟩
-    refine (W.nonsingular_iff P).mpr ⟨(W.equation_smul_iff P u).mp h, ?_⟩
-    contrapose! h'
-    simp only [smul_fin3_ext]
-    exact ⟨by linear_combination (norm := ring1) (u : R) ^ 2 * h'.left,
-      by linear_combination (norm := ring1) (u : R) ^ 2 * h'.right.left,
-      by linear_combination (norm := ring1) (u : R) ^ 2 * h'.right.right⟩
-  ⟨this u, fun h => this u⁻¹ <| by rwa [inv_smul_smul]⟩
+lemma isUnit_Y_of_Z_eq_zero {P : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P z = 0) : IsUnit (P y) :=
+  (Y_ne_zero_of_Z_eq_zero hP hPz).isUnit
 
-lemma nonsingular_of_equiv {P Q : Fin 3 → R} (h : P ≈ Q) : W.Nonsingular P ↔ W.Nonsingular Q := by
-  rcases h with ⟨u, rfl⟩
-  exact W.nonsingular_smul_iff Q u
+lemma equiv_of_Z_eq_zero {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q)
+    (hPz : P z = 0) (hQz : Q z = 0) : P ≈ Q := by
+  use (isUnit_Y_of_Z_eq_zero hP hPz).unit / (isUnit_Y_of_Z_eq_zero hQ hQz).unit
+  simp only [Units.smul_def, smul_fin3, X_eq_zero_of_Z_eq_zero hQ.left hQz, hQz, mul_zero,
+    Units.val_div_eq_div_val, IsUnit.unit_spec, (isUnit_Y_of_Z_eq_zero hQ hQz).div_mul_cancel]
+  conv_rhs => rw [← fin3_def P, X_eq_zero_of_Z_eq_zero hP.left hPz, hPz]
 
-/-- The proposition that a point class on `W` is nonsingular. If `P` is a point representative,
+lemma equiv_zero_of_Z_eq_zero {P : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P z = 0) :
+    P ≈ ![0, 1, 0] :=
+  equiv_of_Z_eq_zero hP nonsingular_zero hPz rfl
+
+variable (W') in
+/-- The proposition that a point class on `W'` is nonsingular. If `P` is a point representative,
 then `W.NonsingularLift ⟦P⟧` is definitionally equivalent to `W.Nonsingular P`. -/
 def NonsingularLift (P : PointClass R) : Prop :=
-  P.lift W.Nonsingular fun _ _ => propext ∘ W.nonsingular_of_equiv
+  P.lift W'.Nonsingular fun _ _ => propext ∘ nonsingular_of_equiv
 
-@[simp]
-lemma nonsingularLift_iff (P : Fin 3 → R) : W.NonsingularLift ⟦P⟧ ↔ W.Nonsingular P :=
+lemma nonsingularLift_iff (P : Fin 3 → R) : W'.NonsingularLift ⟦P⟧ ↔ W'.Nonsingular P :=
   Iff.rfl
 
-lemma nonsingularLift_zero [Nontrivial R] : W.NonsingularLift ⟦![0, 1, 0]⟧ :=
-  W.nonsingular_zero
-
-lemma nonsingularLift_zero' [NoZeroDivisors R] {Y : R} (hy : Y ≠ 0) :
-    W.NonsingularLift ⟦![0, Y, 0]⟧ :=
-  W.nonsingular_zero' hy
+lemma nonsingularLift_zero [Nontrivial R] : W'.NonsingularLift ⟦![0, 1, 0]⟧ :=
+  nonsingular_zero
 
 lemma nonsingularLift_some (X Y : R) :
-    W.NonsingularLift ⟦![X, Y, 1]⟧ ↔ W.toAffine.Nonsingular X Y :=
-  W.nonsingular_some X Y
+    W'.NonsingularLift ⟦![X, Y, 1]⟧ ↔ W'.toAffine.Nonsingular X Y :=
+  nonsingular_some X Y
 
-variable {F : Type u} [Field F] {W : Projective F}
+end Nonsingular
 
-lemma equiv_of_Z_eq_zero {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hQ : W.Nonsingular Q)
-    (hPz : P z = 0) (hQz : Q z = 0) : P ≈ Q := by
-  rw [fin3_def P, hPz] at hP ⊢
-  rw [fin3_def Q, hQz] at hQ ⊢
-  simp? [nonsingular_iff, equation_iff] at hP hQ says
-    simp only [Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, nonsingular_iff,
-      equation_iff, Matrix.cons_val_one, Matrix.head_cons, Matrix.cons_val_two, Matrix.tail_cons,
-      mul_zero, Matrix.cons_val_zero, add_zero, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
-      zero_pow, zero_eq_mul, pow_eq_zero_iff, not_or, sub_self, not_true_eq_false, false_or]
-    at hP hQ
-  simp? [pow_eq_zero hP.left.symm, pow_eq_zero hQ.left.symm] at * says
-    simp only [Fin.isValue, pow_eq_zero hP.left.symm, ne_eq, OfNat.ofNat_ne_zero,
-      not_false_eq_true, zero_pow, not_true_eq_false, and_false, mul_zero, zero_mul, add_zero,
-      pow_eq_zero_iff, false_or, true_and, pow_eq_zero hQ.left.symm, Nat.succ_eq_add_one,
-      Nat.reduceAdd] at *
-  exact ⟨Units.mk0 (P y / Q y) <| div_ne_zero hP hQ, by simp [div_mul_cancel₀ _ hQ]⟩
-
-lemma equiv_zero_of_Z_eq_zero {P : Fin 3 → F} (h : W.Nonsingular P) (hPz : P z = 0) :
-    P ≈ ![0, 1, 0] :=
-  equiv_of_Z_eq_zero h W.nonsingular_zero hPz rfl
+@[deprecated (since := "2024-08-27")] alias equation_smul_iff := equation_smul
+@[deprecated (since := "2024-08-27")] alias nonsingularLift_zero' := nonsingularLift_zero
+@[deprecated (since := "2024-08-27")]
+alias nonsingular_affine_of_Z_ne_zero := nonsingular_of_Z_ne_zero
+@[deprecated (since := "2024-08-27")]
+alias nonsingular_iff_affine_of_Z_ne_zero := nonsingular_of_Z_ne_zero
+@[deprecated (since := "2024-08-27")]
+alias nonsingular_of_affine_of_Z_ne_zero := nonsingular_of_Z_ne_zero
+@[deprecated (since := "2024-08-27")] alias nonsingular_smul_iff := nonsingular_smul
+@[deprecated (since := "2024-08-27")] alias nonsingular_zero' := nonsingular_zero
 
-lemma equiv_some_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) : P ≈ ![P x / P z, P y / P z, 1] :=
-  ⟨Units.mk0 _ hPz, by simp [← fin3_def P, mul_div_cancel₀ _ hPz]⟩
+section Negation
 
-lemma nonsingular_iff_affine_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
-    W.Nonsingular P ↔ W.toAffine.Nonsingular (P x / P z) (P y / P z) :=
-  (W.nonsingular_of_equiv <| equiv_some_of_Z_ne_zero hPz).trans <| W.nonsingular_some ..
+/-! ### Negation formulae -/
 
-lemma nonsingular_of_affine_of_Z_ne_zero {P : Fin 3 → F}
-    (h : W.toAffine.Nonsingular (P x / P z) (P y / P z)) (hPz : P z ≠ 0) : W.Nonsingular P :=
-  (nonsingular_iff_affine_of_Z_ne_zero hPz).mpr h
+variable (W') in
+/-- The $Y$-coordinate of a representative of `-P` for a point `P`. -/
+def negY (P : Fin 3 → R) : R :=
+  -P y - W'.a₁ * P x - W'.a₃ * P z
 
-lemma nonsingular_affine_of_Z_ne_zero {P : Fin 3 → F} (h : W.Nonsingular P) (hPz : P z ≠ 0) :
-    W.toAffine.Nonsingular (P x / P z) (P y / P z) :=
-  (nonsingular_iff_affine_of_Z_ne_zero hPz).mp h
+lemma negY_eq (X Y Z : R) : W'.negY ![X, Y, Z] = -Y - W'.a₁ * X - W'.a₃ * Z :=
+  rfl
 
-end Equation
+lemma negY_smul (P : Fin 3 → R) (u : R) : W'.negY (u • P) = u * W'.negY P := by
+  simp only [negY, smul_fin3_ext]
+  ring1
+
+lemma negY_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
+    W'.negY P = -P y := by
+  rw [negY, hPz, X_eq_zero_of_Z_eq_zero hP hPz, mul_zero, sub_zero, mul_zero, sub_zero]
+
+lemma negY_of_Z_ne_zero {P : Fin 3 → F} (hPz : P z ≠ 0) :
+    W.negY P / P z = W.toAffine.negY (P x / P z) (P y / P z) := by
+  linear_combination (norm := (rw [negY, Affine.negY]; ring1)) -W.a₃ * div_self hPz
+
+lemma Y_sub_Y_mul_Y_sub_negY {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hx : P x * Q z = Q x * P z) :
+    P z * Q z * (P y * Q z - Q y * P z) * (P y * Q z - W'.negY Q * P z) = 0 := by
+  linear_combination' (norm := (rw [negY]; ring1)) Q z ^ 3 * (equation_iff P).mp hP
+    - P z ^ 3 * (equation_iff Q).mp hQ + hx * hx * hx + W'.a₂ * P z * Q z * hx * hx
+    + (W'.a₄ * P z ^ 2 * Q z ^ 2 - W'.a₁ * P y * P z * Q z ^ 2) * hx
+
+lemma Y_eq_of_Y_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ Q y * P z) :
+    P y * Q z = W'.negY Q * P z :=
+  sub_eq_zero.mp <| (mul_eq_zero.mp <| Y_sub_Y_mul_Y_sub_negY hP hQ hx).resolve_left <|
+    mul_ne_zero (mul_ne_zero hPz hQz) <| sub_ne_zero.mpr hy
+
+lemma Y_eq_of_Y_ne' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z ≠ W'.negY Q * P z) : P y * Q z = Q y * P z :=
+  sub_eq_zero.mp <| (mul_eq_zero.mp <| (mul_eq_zero.mp <| Y_sub_Y_mul_Y_sub_negY hP hQ hx
+    ).resolve_right <| sub_ne_zero.mpr hy).resolve_left <| mul_ne_zero hPz hQz
+
+lemma Y_eq_iff' {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) :
+    P y * Q z = W.negY Q * P z ↔ P y / P z = W.toAffine.negY (Q x / Q z) (Q y / Q z) :=
+  negY_of_Z_ne_zero hQz ▸ (div_eq_div_iff hPz hQz).symm
+
+lemma Y_sub_Y_add_Y_sub_negY {P Q : Fin 3 → R} (hx : P x * Q z = Q x * P z) :
+    (P y * Q z - Q y * P z) + (P y * Q z - W'.negY Q * P z) = (P y - W'.negY P) * Q z := by
+  linear_combination (norm := (rw [negY, negY]; ring1)) -W'.a₁ * hx
+
+lemma Y_ne_negY_of_Y_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hQ : W'.Equation Q) (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z ≠ Q y * P z) : P y ≠ W'.negY P := by
+  have hy' : P y * Q z - W'.negY Q * P z = 0 := sub_eq_zero.mpr <| Y_eq_of_Y_ne hP hQ hPz hQz hx hy
+  contrapose! hy
+  linear_combination (norm := ring1) Y_sub_Y_add_Y_sub_negY hx + Q z * hy - hy'
+
+lemma Y_ne_negY_of_Y_ne' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hQ : W'.Equation Q) (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z ≠ W'.negY Q * P z) : P y ≠ W'.negY P := by
+  have hy' : P y * Q z - Q y * P z = 0 := sub_eq_zero.mpr <| Y_eq_of_Y_ne' hP hQ hPz hQz hx hy
+  contrapose! hy
+  linear_combination (norm := ring1) Y_sub_Y_add_Y_sub_negY hx + Q z * hy - hy'
+
+lemma Y_eq_negY_of_Y_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W'.negY Q * P z) :
+    P y = W'.negY P :=
+  mul_left_injective₀ hQz <| by
+    linear_combination (norm := ring1) -Y_sub_Y_add_Y_sub_negY hx + hy + hy'
+
+lemma nonsingular_iff_of_Y_eq_negY {P : Fin 3 → F} (hPz : P z ≠ 0) (hy : P y = W.negY P) :
+    W.Nonsingular P ↔ W.Equation P ∧ eval P W.polynomialX ≠ 0 := by
+  have hy' : eval P W.polynomialY = (P y - W.negY P) * P z := by rw [negY, eval_polynomialY]; ring1
+  rw [nonsingular_iff_of_Z_ne_zero hPz, hy', hy, sub_self, zero_mul, ne_self_iff_false, or_false]
+
+end Negation
+
+section Doubling
+
+/-! ### Doubling formulae -/
+
+variable (W) in
+/-- The unit associated to the doubling of a 2-torsion point `P`.
+More specifically, the unit `u` such that `W.add P P = u • ![0, 1, 0]` where `P = W.neg P`. -/
+noncomputable def dblU (P : Fin 3 → F) : F :=
+  eval P W.polynomialX ^ 3 / P z ^ 2
+
+lemma dblU_eq (P : Fin 3 → F) : W.dblU P =
+    (W.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W.a₂ * P x * P z + W.a₄ * P z ^ 2)) ^ 3 / P z ^ 2 := by
+  rw [dblU, eval_polynomialX]
+
+lemma dblU_smul {P : Fin 3 → F} (hPz : P z ≠ 0) {u : F} (hu : u ≠ 0) :
+    W.dblU (u • P) = u ^ 4 * W.dblU P := by
+  field_simp [dblU_eq, smul_fin3_ext]
+  ring1
+
+lemma dblU_of_Z_eq_zero {P : Fin 3 → F} (hPz : P z = 0) : W.dblU P = 0 := by
+  rw [dblU_eq, hPz, zero_pow two_ne_zero, div_zero]
+
+lemma dblU_ne_zero_of_Y_eq {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W.negY Q * P z) :
+    W.dblU P ≠ 0 :=
+  div_ne_zero (pow_ne_zero 3
+    ((nonsingular_iff_of_Y_eq_negY hPz <| Y_eq_negY_of_Y_eq hQz hx hy hy').mp hP).right) <|
+    pow_ne_zero 2 hPz
+
+lemma isUnit_dblU_of_Y_eq {P Q : Fin 3 → F} (hP : W.Nonsingular P) (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W.negY Q * P z) :
+    IsUnit (W.dblU P) :=
+  (dblU_ne_zero_of_Y_eq hP hPz hQz hx hy hy').isUnit
+
+variable (W') in
+/-- The $Z$-coordinate of a representative of `2 • P` for a point `P`. -/
+def dblZ (P : Fin 3 → R) : R :=
+  P z * (P y - W'.negY P) ^ 3
+
+lemma dblZ_smul (P : Fin 3 → R) (u : R) : W'.dblZ (u • P) = u ^ 4 * W'.dblZ P := by
+  simp only [dblZ, negY_smul, smul_fin3_ext]
+  ring1
+
+lemma dblZ_of_Z_eq_zero {P : Fin 3 → R} (hPz : P z = 0) : W'.dblZ P = 0 := by
+  rw [dblZ, hPz, zero_mul]
+
+lemma dblZ_of_Y_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W'.negY Q * P z) : W'.dblZ P = 0 := by
+  rw [dblZ, Y_eq_negY_of_Y_eq hQz hx hy hy', sub_self, zero_pow three_ne_zero, mul_zero]
+
+lemma dblZ_ne_zero_of_Y_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hQ : W'.Equation Q) (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z ≠ Q y * P z) : W'.dblZ P ≠ 0 :=
+  mul_ne_zero hPz <| pow_ne_zero 3 <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne hP hQ hPz hQz hx hy
+
+lemma isUnit_dblZ_of_Y_ne {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ Q y * P z) : IsUnit (W.dblZ P) :=
+  (dblZ_ne_zero_of_Y_ne hP hQ hPz hQz hx hy).isUnit
+
+lemma dblZ_ne_zero_of_Y_ne' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hQ : W'.Equation Q) (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z)
+    (hy : P y * Q z ≠ W'.negY Q * P z) : W'.dblZ P ≠ 0 :=
+  mul_ne_zero hPz <| pow_ne_zero 3 <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hPz hQz hx hy
+
+lemma isUnit_dblZ_of_Y_ne' {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    IsUnit (W.dblZ P) :=
+  (dblZ_ne_zero_of_Y_ne' hP hQ hPz hQz hx hy).isUnit
+
+private lemma toAffine_slope_of_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z) =
+      -eval P W.polynomialX / P z / (P y - W.negY P) := by
+  have hPy : P y - W.negY P ≠ 0 := sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hPz hQz hx hy
+  simp only [X_eq_iff hPz hQz, ne_eq, Y_eq_iff' hPz hQz] at hx hy
+  rw [Affine.slope_of_Y_ne hx <| negY_of_Z_ne_zero hQz ▸ hy, ← negY_of_Z_ne_zero hPz]
+  field_simp [eval_polynomialX, hPz]
+  ring1
+
+variable (W') in
+/-- The $X$-coordinate of a representative of `2 • P` for a point `P`. -/
+noncomputable def dblX (P : Fin 3 → R) : R :=
+  2 * P x * P y ^ 3 + 3 * W'.a₁ * P x ^ 2 * P y ^ 2 + 6 * W'.a₂ * P x ^ 3 * P y
+    - 8 * W'.a₂ * P y ^ 3 * P z + 9 * W'.a₃ * P x ^ 4 - 6 * W'.a₃ * P x * P y ^ 2 * P z
+    - 6 * W'.a₄ * P x ^ 2 * P y * P z - 18 * W'.a₆ * P x * P y * P z ^ 2
+    + 3 * W'.a₁ ^ 2 * P x ^ 3 * P y - 2 * W'.a₁ ^ 2 * P y ^ 3 * P z + 3 * W'.a₁ * W'.a₂ * P x ^ 4
+    - 12 * W'.a₁ * W'.a₂ * P x * P y ^ 2 * P z - 9 * W'.a₁ * W'.a₃ * P x ^ 2 * P y * P z
+    - 3 * W'.a₁ * W'.a₄ * P x ^ 3 * P z - 9 * W'.a₁ * W'.a₆ * P x ^ 2 * P z ^ 2
+    + 8 * W'.a₂ ^ 2 * P x ^ 2 * P y * P z + 12 * W'.a₂ * W'.a₃ * P x ^ 3 * P z
+    - 12 * W'.a₂ * W'.a₃ * P y ^ 2 * P z ^ 2 + 8 * W'.a₂ * W'.a₄ * P x * P y * P z ^ 2
+    - 12 * W'.a₃ ^ 2 * P x * P y * P z ^ 2 + 6 * W'.a₃ * W'.a₄ * P x ^ 2 * P z ^ 2
+    + 2 * W'.a₄ ^ 2 * P y * P z ^ 3 + W'.a₁ ^ 3 * P x ^ 4 - 3 * W'.a₁ ^ 3 * P x * P y ^ 2 * P z
+    - 2 * W'.a₁ ^ 2 * W'.a₂ * P x ^ 2 * P y * P z - 3 * W'.a₁ ^ 2 * W'.a₃ * P y ^ 2 * P z ^ 2
+    + 2 * W'.a₁ ^ 2 * W'.a₄ * P x * P y * P z ^ 2 + 4 * W'.a₁ * W'.a₂ ^ 2 * P x ^ 3 * P z
+    - 8 * W'.a₁ * W'.a₂ * W'.a₃ * P x * P y * P z ^ 2
+    + 4 * W'.a₁ * W'.a₂ * W'.a₄ * P x ^ 2 * P z ^ 2 - 3 * W'.a₁ * W'.a₃ ^ 2 * P x ^ 2 * P z ^ 2
+    + 2 * W'.a₁ * W'.a₃ * W'.a₄ * P y * P z ^ 3 + W'.a₁ * W'.a₄ ^ 2 * P x * P z ^ 3
+    + 4 * W'.a₂ ^ 2 * W'.a₃ * P x ^ 2 * P z ^ 2 - 6 * W'.a₂ * W'.a₃ ^ 2 * P y * P z ^ 3
+    + 4 * W'.a₂ * W'.a₃ * W'.a₄ * P x * P z ^ 3 - 2 * W'.a₃ ^ 3 * P x * P z ^ 3
+    + W'.a₃ * W'.a₄ ^ 2 * P z ^ 4 - W'.a₁ ^ 4 * P x ^ 2 * P y * P z
+    + W'.a₁ ^ 3 * W'.a₂ * P x ^ 3 * P z - 2 * W'.a₁ ^ 3 * W'.a₃ * P x * P y * P z ^ 2
+    + W'.a₁ ^ 3 * W'.a₄ * P x ^ 2 * P z ^ 2 + W'.a₁ ^ 2 * W'.a₂ * W'.a₃ * P x ^ 2 * P z ^ 2
+    - W'.a₁ ^ 2 * W'.a₃ ^ 2 * P y * P z ^ 3 + 2 * W'.a₁ ^ 2 * W'.a₃ * W'.a₄ * P x * P z ^ 3
+    - W'.a₁ * W'.a₂ * W'.a₃ ^ 2 * P x * P z ^ 3 - W'.a₂ * W'.a₃ ^ 3 * P z ^ 4
+    + W'.a₁ * W'.a₃ ^ 2 * W'.a₄ * P z ^ 4
+
+lemma dblX_eq' {P : Fin 3 → R} (hP : W'.Equation P) : W'.dblX P * P z =
+    (eval P W'.polynomialX ^ 2 - W'.a₁ * eval P W'.polynomialX * P z * (P y - W'.negY P)
+      - W'.a₂ * P z ^ 2 * (P y - W'.negY P) ^ 2 - 2 * P x * P z * (P y - W'.negY P) ^ 2)
+      * (P y - W'.negY P) := by
+  linear_combination (norm := (rw [dblX, eval_polynomialX, negY]; ring1))
+    9 * (W'.a₁ * P x ^ 2 + 2 * P x * P y) * (equation_iff _).mp hP
+
+lemma dblX_eq {P : Fin 3 → F} (hP : W.Equation P) (hPz : P z ≠ 0) : W.dblX P =
+    ((eval P W.polynomialX ^ 2 - W.a₁ * eval P W.polynomialX * P z * (P y - W.negY P)
+      - W.a₂ * P z ^ 2 * (P y - W.negY P) ^ 2 - 2 * P x * P z * (P y - W.negY P) ^ 2)
+      * (P y - W.negY P)) / P z := by
+  rw [← dblX_eq' hP, mul_div_cancel_right₀ _ hPz]
+
+lemma dblX_smul (P : Fin 3 → R) (u : R) : W'.dblX (u • P) = u ^ 4 * W'.dblX P := by
+  simp only [dblX, smul_fin3_ext]
+  ring1
+
+lemma dblX_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
+    W'.dblX P = 0 := by
+  rw [dblX, hPz, X_eq_zero_of_Z_eq_zero hP hPz]
+  ring1
+
+lemma dblX_of_Y_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z)
+    (hy' : P y * Q z = W'.negY Q * P z) : W'.dblX P = 0 := by
+  apply eq_zero_of_ne_zero_of_mul_right_eq_zero hPz
+  rw [dblX_eq' hP, Y_eq_negY_of_Y_eq hQz hx hy hy']
+  ring1
+
+private lemma toAffine_addX_of_eq {P : Fin 3 → F} (hPz : P z ≠ 0) {n d : F} (hd : d ≠ 0) :
+    W.toAffine.addX (P x / P z) (P x / P z) (-n / P z / d) =
+      (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2) * d / P z
+        / (P z * d ^ 3) := by
+  field_simp [mul_ne_zero hPz hd]
+  ring1
+
+lemma dblX_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    W.dblX P / W.dblZ P = W.toAffine.addX (P x / P z) (Q x / Q z)
+      (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  rw [dblX_eq hP hPz, dblZ, toAffine_slope_of_eq hP hQ hPz hQz hx hy, ← (X_eq_iff hPz hQz).mp hx,
+    toAffine_addX_of_eq hPz <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hPz hQz hx hy]
+
+variable (W') in
+/-- The $Y$-coordinate of a representative of `-(2 • P)` for a point `P`. -/
+noncomputable def negDblY (P : Fin 3 → R) : R :=
+  -P y ^ 4 - 3 * W'.a₁ * P x * P y ^ 3 - 9 * W'.a₃ * P x ^ 3 * P y + 3 * W'.a₃ * P y ^ 3 * P z
+    - 3 * W'.a₄ * P x * P y ^ 2 * P z - 27 * W'.a₆ * P x ^ 3 * P z + 9 * W'.a₆ * P y ^ 2 * P z ^ 2
+    - 3 * W'.a₁ ^ 2 * P x ^ 2 * P y ^ 2 + 4 * W'.a₁ * W'.a₂ * P y ^ 3 * P z
+    - 3 * W'.a₁ * W'.a₂ * P x ^ 3 * P y - 9 * W'.a₁ * W'.a₃ * P x ^ 4
+    + 6 * W'.a₁ * W'.a₃ * P x * P y ^ 2 * P z + 18 * W'.a₁ * W'.a₆ * P x * P y * P z ^ 2
+    + 9 * W'.a₂ ^ 2 * P x ^ 4 - 8 * W'.a₂ ^ 2 * P x * P y ^ 2 * P z
+    - 9 * W'.a₂ * W'.a₃ * P x ^ 2 * P y * P z + 9 * W'.a₂ * W'.a₄ * P x ^ 3 * P z
+    - 4 * W'.a₂ * W'.a₄ * P y ^ 2 * P z ^ 2 - 27 * W'.a₂ * W'.a₆ * P x ^ 2 * P z ^ 2
+    - 9 * W'.a₃ ^ 2 * P x ^ 3 * P z + 6 * W'.a₃ ^ 2 * P y ^ 2 * P z ^ 2
+    - 12 * W'.a₃ * W'.a₄ * P x * P y * P z ^ 2 + 9 * W'.a₄ ^ 2 * P x ^ 2 * P z ^ 2
+    - 2 * W'.a₁ ^ 3 * P x ^ 3 * P y + W'.a₁ ^ 3 * P y ^ 3 * P z + 3 * W'.a₁ ^ 2 * W'.a₂ * P x ^ 4
+    + 2 * W'.a₁ ^ 2 * W'.a₂ * P x * P y ^ 2 * P z + 3 * W'.a₁ ^ 2 * W'.a₃ * P x ^ 2 * P y * P z
+    + 3 * W'.a₁ ^ 2 * W'.a₄ * P x ^ 3 * P z - W'.a₁ ^ 2 * W'.a₄ * P y ^ 2 * P z ^ 2
+    - 12 * W'.a₁ * W'.a₂ ^ 2 * P x ^ 2 * P y * P z - 6 * W'.a₁ * W'.a₂ * W'.a₃ * P x ^ 3 * P z
+    + 4 * W'.a₁ * W'.a₂ * W'.a₃ * P y ^ 2 * P z ^ 2
+    - 8 * W'.a₁ * W'.a₂ * W'.a₄ * P x * P y * P z ^ 2 + 6 * W'.a₁ * W'.a₃ ^ 2 * P x * P y * P z ^ 2
+    - W'.a₁ * W'.a₄ ^ 2 * P y * P z ^ 3 + 8 * W'.a₂ ^ 3 * P x ^ 3 * P z
+    - 8 * W'.a₂ ^ 2 * W'.a₃ * P x * P y * P z ^ 2 + 12 * W'.a₂ ^ 2 * W'.a₄ * P x ^ 2 * P z ^ 2
+    - 9 * W'.a₂ * W'.a₃ ^ 2 * P x ^ 2 * P z ^ 2 - 4 * W'.a₂ * W'.a₃ * W'.a₄ * P y * P z ^ 3
+    + 6 * W'.a₂ * W'.a₄ ^ 2 * P x * P z ^ 3 + W'.a₃ ^ 3 * P y * P z ^ 3
+    - 3 * W'.a₃ ^ 2 * W'.a₄ * P x * P z ^ 3 + W'.a₄ ^ 3 * P z ^ 4 + W'.a₁ ^ 4 * P x * P y ^ 2 * P z
+    - 3 * W'.a₁ ^ 3 * W'.a₂ * P x ^ 2 * P y * P z + W'.a₁ ^ 3 * W'.a₃ * P y ^ 2 * P z ^ 2
+    - 2 * W'.a₁ ^ 3 * W'.a₄ * P x * P y * P z ^ 2 + 2 * W'.a₁ ^ 2 * W'.a₂ ^ 2 * P x ^ 3 * P z
+    - 2 * W'.a₁ ^ 2 * W'.a₂ * W'.a₃ * P x * P y * P z ^ 2
+    + 3 * W'.a₁ ^ 2 * W'.a₂ * W'.a₄ * P x ^ 2 * P z ^ 2
+    - 2 * W'.a₁ ^ 2 * W'.a₃ * W'.a₄ * P y * P z ^ 3 + W'.a₁ ^ 2 * W'.a₄ ^ 2 * P x * P z ^ 3
+    + W'.a₁ * W'.a₂ * W'.a₃ ^ 2 * P y * P z ^ 3 + 2 * W'.a₁ * W'.a₂ * W'.a₃ * W'.a₄ * P x * P z ^ 3
+    + W'.a₁ * W'.a₃ * W'.a₄ ^ 2 * P z ^ 4 - 2 * W'.a₂ ^ 2 * W'.a₃ ^ 2 * P x * P z ^ 3
+    - W'.a₂ * W'.a₃ ^ 2 * W'.a₄ * P z ^ 4
+
+lemma negDblY_eq' {P : Fin 3 → R} (hP : W'.Equation P) : W'.negDblY P * P z ^ 2 =
+    -eval P W'.polynomialX * (eval P W'.polynomialX ^ 2
+      - W'.a₁ * eval P W'.polynomialX * P z * (P y - W'.negY P)
+      - W'.a₂ * P z ^ 2 * (P y - W'.negY P) ^ 2 - 2 * P x * P z * (P y - W'.negY P) ^ 2
+      - P x * P z * (P y - W'.negY P) ^ 2) + P y * P z ^ 2 * (P y - W'.negY P) ^ 3 := by
+  linear_combination (norm := (rw [negDblY, eval_polynomialX, negY]; ring1))
+    -9 * (P y ^ 2 * P z + 2 * W'.a₁ * P x * P y * P z - 3 * P x ^ 3 - 3 * W'.a₂ * P x ^ 2 * P z)
+      * (equation_iff _).mp hP
+
+lemma negDblY_eq {P : Fin 3 → F} (hP : W.Equation P) (hPz : P z ≠ 0) : W.negDblY P =
+    (-eval P W.polynomialX * (eval P W.polynomialX ^ 2
+      - W.a₁ * eval P W.polynomialX * P z * (P y - W.negY P)
+      - W.a₂ * P z ^ 2 * (P y - W.negY P) ^ 2 - 2 * P x * P z * (P y - W.negY P) ^ 2
+      - P x * P z * (P y - W.negY P) ^ 2) + P y * P z ^ 2 * (P y - W.negY P) ^ 3) / P z ^ 2 := by
+  rw [← negDblY_eq' hP, mul_div_cancel_right₀ _ <| pow_ne_zero 2 hPz]
+
+lemma negDblY_smul (P : Fin 3 → R) (u : R) : W'.negDblY (u • P) = u ^ 4 * W'.negDblY P := by
+  simp only [negDblY, smul_fin3_ext]
+  ring1
+
+lemma negDblY_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
+    W'.negDblY P = -P y ^ 4 := by
+  rw [negDblY, hPz, X_eq_zero_of_Z_eq_zero hP hPz]
+  ring1
+
+lemma negDblY_of_Y_eq' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W'.negY Q * P z) :
+    W'.negDblY P * P z ^ 2 = -eval P W'.polynomialX ^ 3 := by
+  rw [negDblY_eq' hP, Y_eq_negY_of_Y_eq hQz hx hy hy']
+  ring1
+
+lemma negDblY_of_Y_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W.negY Q * P z) :
+    W.negDblY P = -W.dblU P := by
+  rw [dblU, ← neg_div, ← negDblY_of_Y_eq' hP hQz hx hy hy',
+    mul_div_cancel_right₀ _ <| pow_ne_zero 2 hPz]
+
+private lemma toAffine_negAddY_of_eq {P : Fin 3 → F} (hPz : P z ≠ 0) {n d : F} (hd : d ≠ 0) :
+    W.toAffine.negAddY (P x / P z) (P x / P z) (P y / P z) (-n / P z / d) =
+      (-n * (n ^ 2 - W.a₁ * n * P z * d - W.a₂ * P z ^ 2 * d ^ 2 - 2 * P x * P z * d ^ 2
+          - P x * P z * d ^ 2) + P y * P z ^ 2 * d ^ 3) / P z ^ 2 / (P z * d ^ 3) := by
+  rw [Affine.negAddY, toAffine_addX_of_eq hPz hd]
+  field_simp [mul_ne_zero hPz <| mul_ne_zero hPz <| pow_ne_zero 3 hd]
+  ring1
+
+lemma negDblY_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    W.negDblY P / W.dblZ P = W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)
+      (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  rw [negDblY_eq hP hPz, dblZ, toAffine_slope_of_eq hP hQ hPz hQz hx hy, ← (X_eq_iff hPz hQz).mp hx,
+    toAffine_negAddY_of_eq hPz <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hPz hQz hx hy]
+
+variable (W') in
+/-- The $Y$-coordinate of a representative of `2 • P` for a point `P`. -/
+noncomputable def dblY (P : Fin 3 → R) : R :=
+  W'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P]
+
+lemma dblY_smul (P : Fin 3 → R) (u : R) : W'.dblY (u • P) = u ^ 4 * W'.dblY P := by
+  simp only [dblY, negY_eq, negDblY_smul, dblX_smul, dblZ_smul]
+  ring1
+
+lemma dblY_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
+    W'.dblY P = P y ^ 4 := by
+  rw [dblY, negY_eq, negDblY_of_Z_eq_zero hP hPz, dblX_of_Z_eq_zero hP hPz, dblZ_of_Z_eq_zero hPz]
+  ring1
+
+lemma dblY_of_Y_eq' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z)
+    (hy' : P y * Q z = W'.negY Q * P z) : W'.dblY P * P z ^ 2 = eval P W'.polynomialX ^ 3 := by
+  linear_combination (norm := (rw [dblY, negY_eq, dblX_of_Y_eq hP hPz hQz hx hy hy',
+    dblZ_of_Y_eq hQz hx hy hy']; ring1)) -negDblY_of_Y_eq' hP hQz hx hy hy'
+
+lemma dblY_of_Y_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W.negY Q * P z) :
+    W.dblY P = W.dblU P := by
+  rw [dblU, ← dblY_of_Y_eq' hP hPz hQz hx hy hy', mul_div_cancel_right₀ _ <| pow_ne_zero 2 hPz]
+
+lemma dblY_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    W.dblY P / W.dblZ P = W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
+      (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  erw [dblY, negY_of_Z_ne_zero <| dblZ_ne_zero_of_Y_ne' hP hQ hPz hQz hx hy,
+    dblX_of_Z_ne_zero hP hQ hPz hQz hx hy, negDblY_of_Z_ne_zero hP hQ hPz hQz hx hy, Affine.addY]
+
+variable (W') in
+/-- The coordinates of a representative of `2 • P` for a point `P`. -/
+noncomputable def dblXYZ (P : Fin 3 → R) : Fin 3 → R :=
+  ![W'.dblX P, W'.dblY P, W'.dblZ P]
+
+lemma dblXYZ_X (P : Fin 3 → R) : W'.dblXYZ P x = W'.dblX P :=
+  rfl
+
+lemma dblXYZ_Y (P : Fin 3 → R) : W'.dblXYZ P y = W'.dblY P :=
+  rfl
+
+lemma dblXYZ_Z (P : Fin 3 → R) : W'.dblXYZ P z = W'.dblZ P :=
+  rfl
+
+lemma dblXYZ_smul (P : Fin 3 → R) (u : R) : W'.dblXYZ (u • P) = u ^ 4 • W'.dblXYZ P := by
+  rw [dblXYZ, dblX_smul, dblY_smul, dblZ_smul, smul_fin3, dblXYZ_X, dblXYZ_Y, dblXYZ_Z]
+
+lemma dblXYZ_of_Z_eq_zero [NoZeroDivisors R] {P : Fin 3 → R} (hP : W'.Equation P) (hPz : P z = 0) :
+    W'.dblXYZ P = P y ^ 4 • ![0, 1, 0] := by
+  erw [dblXYZ, dblX_of_Z_eq_zero hP hPz, dblY_of_Z_eq_zero hP hPz, dblZ_of_Z_eq_zero hPz, smul_fin3,
+    mul_zero, mul_one]
+
+lemma dblXYZ_of_Y_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z = Q x * P z) (hy : P y * Q z = Q y * P z) (hy' : P y * Q z = W.negY Q * P z) :
+    W.dblXYZ P = W.dblU P • ![0, 1, 0] := by
+  erw [dblXYZ, dblX_of_Y_eq hP hPz hQz hx hy hy', dblY_of_Y_eq hP hPz hQz hx hy hy',
+    dblZ_of_Y_eq hQz hx hy hy', smul_fin3, mul_zero, mul_one]
+
+lemma dblXYZ_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) (hy : P y * Q z ≠ W.negY Q * P z) :
+    W.dblXYZ P = W.dblZ P •
+      ![W.toAffine.addX (P x / P z) (Q x / Q z)
+          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
+        W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
+          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)), 1] := by
+  have hZ : IsUnit <| W.dblZ P := isUnit_dblZ_of_Y_ne' hP hQ hPz hQz hx hy
+  erw [dblXYZ, smul_fin3, ← dblX_of_Z_ne_zero hP hQ hPz hQz hx hy, hZ.mul_div_cancel,
+    ← dblY_of_Z_ne_zero hP hQ hPz hQz hx hy, hZ.mul_div_cancel, mul_one]
+
+end Doubling
+
+section Addition
+
+/-! ### Addition formulae -/
+
+/-- The unit associated to the addition of a non-2-torsion point `P` with its negation.
+More specifically, the unit `u` such that `W.add P Q = u • ![0, 1, 0]` where `P x / P z = Q x / Q z`
+but `P ≠ W.neg P`. -/
+def addU (P Q : Fin 3 → F) : F :=
+  -(P y * Q z - Q y * P z) ^ 3 / (P z * Q z)
+
+lemma addU_smul {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) {u v : F} (hu : u ≠ 0)
+    (hv : v ≠ 0) : addU (u • P) (v • Q) = (u * v) ^ 2 * addU P Q := by
+  field_simp [addU, smul_fin3_ext]
+  ring1
+
+lemma addU_of_Z_eq_zero_left {P Q : Fin 3 → F} (hPz : P z = 0) : addU P Q = 0 := by
+  rw [addU, hPz, zero_mul, div_zero]
+
+lemma addU_of_Z_eq_zero_right {P Q : Fin 3 → F} (hQz : Q z = 0) : addU P Q = 0 := by
+  rw [addU, hQz, mul_zero <| P z, div_zero]
+
+lemma addU_ne_zero_of_Y_ne {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hy : P y * Q z ≠ Q y * P z) : addU P Q ≠ 0 :=
+  div_ne_zero (neg_ne_zero.mpr <| pow_ne_zero 3 <| sub_ne_zero.mpr hy) <| mul_ne_zero hPz hQz
+
+lemma isUnit_addU_of_Y_ne {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hy : P y * Q z ≠ Q y * P z) : IsUnit (addU P Q) :=
+  (addU_ne_zero_of_Y_ne hPz hQz hy).isUnit
+
+variable (W') in
+/-- The $Z$-coordinate of a representative of `P + Q` for two distinct points `P` and `Q`.
+Note that this returns the value 0 if the representatives of `P` and `Q` are equal. -/
+def addZ (P Q : Fin 3 → R) : R :=
+  -3 * P x ^ 2 * Q x * Q z + 3 * P x * Q x ^ 2 * P z + P y ^ 2 * Q z ^ 2 - Q y ^ 2 * P z ^ 2
+    + W'.a₁ * P x * P y * Q z ^ 2 - W'.a₁ * Q x * Q y * P z ^ 2 - W'.a₂ * P x ^ 2 * Q z ^ 2
+    + W'.a₂ * Q x ^ 2 * P z ^ 2 + W'.a₃ * P y * P z * Q z ^ 2 - W'.a₃ * Q y * P z ^ 2 * Q z
+    - W'.a₄ * P x * P z * Q z ^ 2 + W'.a₄ * Q x * P z ^ 2 * Q z
+
+lemma addZ_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) :
+    W'.addZ P Q * (P z * Q z) = (P x * Q z - Q x * P z) ^ 3 := by
+  linear_combination (norm := (rw [addZ]; ring1))
+    Q z ^ 3 * (equation_iff _).mp hP - P z ^ 3 * (equation_iff _).mp hQ
+
+lemma addZ_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) : W.addZ P Q = (P x * Q z - Q x * P z) ^ 3 / (P z * Q z) := by
+  rw [← addZ_eq' hP hQ, mul_div_cancel_right₀ _ <| mul_ne_zero hPz hQz]
+
+lemma addZ_smul (P Q : Fin 3 → R) (u v : R) :
+    W'.addZ (u • P) (v • Q) = (u * v) ^ 2 * W'.addZ P Q := by
+  simp only [addZ, smul_fin3_ext]
+  ring1
+
+lemma addZ_self (P : Fin 3 → R) : W'.addZ P P = 0 := by
+  rw [addZ]
+  ring1
+
+lemma addZ_of_Z_eq_zero_left [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : W'.addZ P Q = P y ^ 2 * Q z * Q z := by
+  rw [addZ, hPz, X_eq_zero_of_Z_eq_zero hP hPz]
+  ring1
+
+lemma addZ_of_Z_eq_zero_right [NoZeroDivisors R] {P Q : Fin 3 → R} (hQ : W'.Equation Q)
+    (hQz : Q z = 0) : W'.addZ P Q = -(Q y ^ 2 * P z) * P z := by
+  rw [addZ, hQz, X_eq_zero_of_Z_eq_zero hQ hQz]
+  ring1
+
+lemma addZ_of_X_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) : W'.addZ P Q = 0 := by
+  apply eq_zero_of_ne_zero_of_mul_right_eq_zero <| mul_ne_zero hPz hQz
+  rw [addZ_eq' hP hQ, hx, sub_self, zero_pow three_ne_zero]
+
+lemma addZ_ne_zero_of_X_ne [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hQ : W'.Equation Q) (hx : P x * Q z ≠ Q x * P z) : W'.addZ P Q ≠ 0 :=
+  addZ_eq' hP hQ ▸ left_ne_zero_of_mul <| pow_ne_zero 3 <| sub_ne_zero.mpr hx
+
+lemma isUnit_addZ_of_X_ne {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q)
+    (hx : P x * Q z ≠ Q x * P z) : IsUnit <| W.addZ P Q :=
+  (addZ_ne_zero_of_X_ne hP hQ hx).isUnit
+
+private lemma toAffine_slope_of_ne {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0)
+    (hx : P x * Q z ≠ Q x * P z) :
+    W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z) =
+      (P y * Q z - Q y * P z) / (P x * Q z - Q x * P z) := by
+  field_simp [Affine.slope_of_X_ne <| by rwa [ne_eq, ← X_eq_iff hPz hQz]]
+  ring1
+
+variable (W') in
+/-- The $X$-coordinate of a representative of `P + Q` for two distinct points `P` and `Q`.
+Note that this returns the value 0 if the representatives of `P` and `Q` are equal. -/
+def addX (P Q : Fin 3 → R) : R :=
+  -P x * Q y ^ 2 * P z + Q x * P y ^ 2 * Q z - 2 * P x * P y * Q y * Q z + 2 * Q x * P y * Q y * P z
+    - W'.a₁ * P x ^ 2 * Q y * Q z + W'.a₁ * Q x ^ 2 * P y * P z + W'.a₂ * P x ^ 2 * Q x * Q z
+    - W'.a₂ * P x * Q x ^ 2 * P z - W'.a₃ * P x * P y * Q z ^ 2 + W'.a₃ * Q x * Q y * P z ^ 2
+    - 2 * W'.a₃ * P x * Q y * P z * Q z + 2 * W'.a₃ * Q x * P y * P z * Q z
+    + W'.a₄ * P x ^ 2 * Q z ^ 2 - W'.a₄ * Q x ^ 2 * P z ^ 2 + 3 * W'.a₆ * P x * P z * Q z ^ 2
+    - 3 * W'.a₆ * Q x * P z ^ 2 * Q z
+
+lemma addX_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) :
+    W'.addX P Q * (P z * Q z) ^ 2 =
+      ((P y * Q z - Q y * P z) ^ 2 * P z * Q z
+        + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z)
+        - W'.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2
+        - Q x * P z * (P x * Q z - Q x * P z) ^ 2) * (P x * Q z - Q x * P z) := by
+  linear_combination (norm := (rw [addX]; ring1))
+    (2 * Q x * P z * Q z ^ 3 - P x * Q z ^ 4) * (equation_iff _).mp hP
+      + (Q x * P z ^ 4 - 2 * P x * P z ^ 3 * Q z) * (equation_iff _).mp hQ
+
+lemma addX_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) : W.addX P Q =
+      ((P y * Q z - Q y * P z) ^ 2 * P z * Q z
+        + W.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z)
+        - W.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2
+        - Q x * P z * (P x * Q z - Q x * P z) ^ 2) * (P x * Q z - Q x * P z) / (P z * Q z) ^ 2 := by
+  rw [← addX_eq' hP hQ, mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz]
+
+lemma addX_smul (P Q : Fin 3 → R) (u v : R) :
+    W'.addX (u • P) (v • Q) = (u * v) ^ 2 * W'.addX P Q := by
+  simp only [addX, smul_fin3_ext]
+  ring1
+
+lemma addX_self (P : Fin 3 → R) : W'.addX P P = 0 := by
+  rw [addX]
+  ring1
+
+lemma addX_of_Z_eq_zero_left [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : W'.addX P Q = P y ^ 2 * Q z * Q x := by
+  rw [addX, hPz, X_eq_zero_of_Z_eq_zero hP hPz]
+  ring1
+
+lemma addX_of_Z_eq_zero_right [NoZeroDivisors R] {P Q : Fin 3 → R} (hQ : W'.Equation Q)
+    (hQz : Q z = 0) : W'.addX P Q = -(Q y ^ 2 * P z) * P x := by
+  rw [addX, hQz, X_eq_zero_of_Z_eq_zero hQ hQz]
+  ring1
+
+lemma addX_of_X_eq [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) : W'.addX P Q = 0 := by
+  apply eq_zero_of_ne_zero_of_mul_right_eq_zero <| pow_ne_zero 2 <| mul_ne_zero hPz hQz
+  rw [addX_eq' hP hQ, hx]
+  ring1
+
+private lemma toAffine_addX_of_ne {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) {n d : F}
+    (hd : d ≠ 0) : W.toAffine.addX (P x / P z) (Q x / Q z) (n / d) =
+      (n ^ 2 * P z * Q z + W.a₁ * n * P z * Q z * d - W.a₂ * P z * Q z * d ^ 2 - P x * Q z * d ^ 2
+        - Q x * P z * d ^ 2) * d / (P z * Q z) ^ 2 / (d ^ 3 / (P z * Q z)) := by
+  field_simp [hd]
+  ring1
+
+lemma addX_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z ≠ Q x * P z) : W.addX P Q / W.addZ P Q =
+    W.toAffine.addX (P x / P z) (Q x / Q z)
+      (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  rw [addX_eq hP hQ hPz hQz, addZ_eq hP hQ hPz hQz, toAffine_slope_of_ne hPz hQz hx,
+    toAffine_addX_of_ne hPz hQz <| sub_ne_zero.mpr hx]
+
+variable (W') in
+/-- The $Y$-coordinate of a representative of `-(P + Q)` for two distinct points `P` and `Q`.
+Note that this returns the value 0 if the representatives of `P` and `Q` are equal. -/
+def negAddY (P Q : Fin 3 → R) : R :=
+  -3 * P x ^ 2 * Q x * Q y + 3 * P x * Q x ^ 2 * P y - P y ^ 2 * Q y * Q z + P y * Q y ^ 2 * P z
+    + W'.a₁ * P x * Q y ^ 2 * P z - W'.a₁ * Q x * P y ^ 2 * Q z - W'.a₂ * P x ^ 2 * Q y * Q z
+    + W'.a₂ * Q x ^ 2 * P y * P z + 2 * W'.a₂ * P x * Q x * P y * Q z
+    - 2 * W'.a₂ * P x * Q x * Q y * P z - W'.a₃ * P y ^ 2 * Q z ^ 2 + W'.a₃ * Q y ^ 2 * P z ^ 2
+    + W'.a₄ * P x * P y * Q z ^ 2 - 2 * W'.a₄ * P x * Q y * P z * Q z
+    + 2 * W'.a₄ * Q x * P y * P z * Q z - W'.a₄ * Q x * Q y * P z ^ 2
+    + 3 * W'.a₆ * P y * P z * Q z ^ 2 - 3 * W'.a₆ * Q y * P z ^ 2 * Q z
+
+lemma negAddY_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q) :
+    W'.negAddY P Q * (P z * Q z) ^ 2 =
+      (P y * Q z - Q y * P z) * ((P y * Q z - Q y * P z) ^ 2 * P z * Q z
+        + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z)
+        - W'.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2
+        - Q x * P z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2)
+        + P y * Q z * (P x * Q z - Q x * P z) ^ 3 := by
+  linear_combination (norm := (rw [negAddY]; ring1))
+    (2 * Q y * P z * Q z ^ 3 - P y * Q z ^ 4) * (equation_iff _).mp hP
+      + (Q y * P z ^ 4 - 2 * P y * P z ^ 3 * Q z) * (equation_iff _).mp hQ
+
+lemma negAddY_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) : W.negAddY P Q =
+      ((P y * Q z - Q y * P z) * ((P y * Q z - Q y * P z) ^ 2 * P z * Q z
+        + W.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z)
+        - W.a₂ * P z * Q z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2
+        - Q x * P z * (P x * Q z - Q x * P z) ^ 2 - P x * Q z * (P x * Q z - Q x * P z) ^ 2)
+        + P y * Q z * (P x * Q z - Q x * P z) ^ 3) / (P z * Q z) ^ 2 := by
+  rw [← negAddY_eq' hP hQ, mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz]
+
+lemma negAddY_smul (P Q : Fin 3 → R) (u v : R) :
+    W'.negAddY (u • P) (v • Q) = (u * v) ^ 2 * W'.negAddY P Q := by
+  simp only [negAddY, smul_fin3_ext]
+  ring1
+
+lemma negAddY_self (P : Fin 3 → R) : W'.negAddY P P = 0 := by
+  rw [negAddY]
+  ring1
+
+lemma negAddY_of_Z_eq_zero_left [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : W'.negAddY P Q = P y ^ 2 * Q z * W'.negY Q := by
+  rw [negAddY, hPz, X_eq_zero_of_Z_eq_zero hP hPz, negY]
+  ring1
+
+lemma negAddY_of_Z_eq_zero_right [NoZeroDivisors R] {P Q : Fin 3 → R} (hQ : W'.Equation Q)
+    (hQz : Q z = 0) : W'.negAddY P Q = -(Q y ^ 2 * P z) * W'.negY P := by
+  rw [negAddY, hQz, X_eq_zero_of_Z_eq_zero hQ hQz, negY]
+  ring1
+
+lemma negAddY_of_X_eq' {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hx : P x * Q z = Q x * P z) :
+    W'.negAddY P Q * (P z * Q z) ^ 2 = (P y * Q z - Q y * P z) ^ 3 * (P z * Q z) := by
+  rw [negAddY_eq' hP hQ, hx]
+  ring1
+
+lemma negAddY_of_X_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) : W.negAddY P Q = -addU P Q := by
+  rw [addU, neg_div, neg_neg, ← mul_div_mul_right _ _ <| mul_ne_zero hPz hQz,
+    ← negAddY_of_X_eq' hP hQ hx, ← sq,
+    mul_div_cancel_right₀ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz]
+
+private lemma toAffine_negAddY_of_ne {P Q : Fin 3 → F} (hPz : P z ≠ 0) (hQz : Q z ≠ 0) {n d : F}
+    (hd : d ≠ 0) : W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z) (n / d) =
+      (n * (n ^ 2 * P z * Q z + W.a₁ * n * P z * Q z * d - W.a₂ * P z * Q z * d ^ 2
+        - P x * Q z * d ^ 2 - Q x * P z * d ^ 2 - P x * Q z * d ^ 2) + P y * Q z * d ^ 3)
+        / (P z * Q z) ^ 2 / (d ^ 3 / (P z * Q z)) := by
+  rw [Affine.negAddY, toAffine_addX_of_ne hPz hQz hd]
+  field_simp [mul_ne_zero (pow_ne_zero 2 <| mul_ne_zero hPz hQz) <| pow_ne_zero 3 hd]
+  ring1
+
+lemma negAddY_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z ≠ Q x * P z) : W.negAddY P Q / W.addZ P Q =
+      W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)
+        (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  rw [negAddY_eq hP hQ hPz hQz, addZ_eq hP hQ hPz hQz, toAffine_slope_of_ne hPz hQz hx,
+    toAffine_negAddY_of_ne hPz hQz <| sub_ne_zero.mpr hx]
+
+variable (W') in
+/-- The $Y$-coordinate of a representative of `P + Q` for two distinct points `P` and `Q`.
+Note that this returns the value 0 if the representatives of `P` and `Q` are equal. -/
+def addY (P Q : Fin 3 → R) : R :=
+  W'.negY ![W'.addX P Q, W'.negAddY P Q, W'.addZ P Q]
+
+lemma addY_smul (P Q : Fin 3 → R) (u v : R) :
+    W'.addY (u • P) (v • Q) = (u * v) ^ 2 * W'.addY P Q := by
+  simp only [addY, negY_eq, negAddY_smul, addX_smul, addZ_smul]
+  ring1
+
+lemma addY_self (P : Fin 3 → R) : W'.addY P P = 0 := by
+  simp only [addY, negY_eq, negAddY_self, addX_self, addZ_self, neg_zero, mul_zero, sub_zero]
+
+lemma addY_of_Z_eq_zero_left [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : W'.addY P Q = P y ^ 2 * Q z * Q y := by
+  rw [addY, negY_eq, negAddY_of_Z_eq_zero_left hP hPz, negY, addX_of_Z_eq_zero_left hP hPz,
+    addZ_of_Z_eq_zero_left hP hPz]
+  ring1
+
+lemma addY_of_Z_eq_zero_right [NoZeroDivisors R] {P Q : Fin 3 → R} (hQ : W'.Equation Q)
+    (hQz : Q z = 0) : W'.addY P Q = -(Q y ^ 2 * P z) * P y := by
+  rw [addY, negY_eq, negAddY_of_Z_eq_zero_right hQ hQz, negY, addX_of_Z_eq_zero_right hQ hQz,
+    addZ_of_Z_eq_zero_right hQ hQz]
+  ring1
+
+lemma addY_of_X_eq' [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQ : W'.Equation Q)
+    (hPz : P z ≠ 0) (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) :
+    W'.addY P Q * (P z * Q z) ^ 3 = -(P y * Q z - Q y * P z) ^ 3 * (P z * Q z) ^ 2 := by
+  linear_combination (norm := (rw [addY, negY_eq, addX_of_X_eq hP hQ hPz hQz hx,
+    addZ_of_X_eq hP hQ hPz hQz hx]; ring1)) -(P z * Q z) * negAddY_of_X_eq' hP hQ hx
+
+lemma addY_of_X_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) : W.addY P Q = addU P Q := by
+  rw [addU, ← mul_div_mul_right _ _ <| pow_ne_zero 2 <| mul_ne_zero hPz hQz,
+    ← addY_of_X_eq' hP hQ hPz hQz hx, ← pow_succ',
+    mul_div_cancel_right₀ _ <| pow_ne_zero 3 <| mul_ne_zero hPz hQz]
+
+lemma addY_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z ≠ Q x * P z) : W.addY P Q / W.addZ P Q =
+      W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
+        (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)) := by
+  erw [addY, negY_of_Z_ne_zero <| addZ_ne_zero_of_X_ne hP hQ hx, addX_of_Z_ne_zero hP hQ hPz hQz hx,
+    negAddY_of_Z_ne_zero hP hQ hPz hQz hx, Affine.addY]
+
+variable (W') in
+/-- The coordinates of a representative of `P + Q` for two distinct points `P` and `Q`.
+Note that this returns the value `![0, 0, 0]` if the representatives of `P` and `Q` are equal. -/
+noncomputable def addXYZ (P Q : Fin 3 → R) : Fin 3 → R :=
+  ![W'.addX P Q, W'.addY P Q, W'.addZ P Q]
+
+lemma addXYZ_X (P Q : Fin 3 → R) : W'.addXYZ P Q x = W'.addX P Q :=
+  rfl
+
+lemma addXYZ_Y (P Q : Fin 3 → R) : W'.addXYZ P Q y = W'.addY P Q :=
+  rfl
+
+lemma addXYZ_Z (P Q : Fin 3 → R) : W'.addXYZ P Q z = W'.addZ P Q :=
+  rfl
+
+lemma addXYZ_smul (P Q : Fin 3 → R) (u v : R) :
+    W'.addXYZ (u • P) (v • Q) = (u * v) ^ 2 • W'.addXYZ P Q := by
+  rw [addXYZ, addX_smul, addY_smul, addZ_smul, smul_fin3, addXYZ_X, addXYZ_Y, addXYZ_Z]
+
+lemma addXYZ_self (P : Fin 3 → R) : W'.addXYZ P P = ![0, 0, 0] := by
+  rw [addXYZ, addX_self, addY_self, addZ_self]
+
+lemma addXYZ_of_Z_eq_zero_left [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P)
+    (hPz : P z = 0) : W'.addXYZ P Q = (P y ^ 2 * Q z) • Q := by
+  rw [addXYZ, addX_of_Z_eq_zero_left hP hPz, addY_of_Z_eq_zero_left hP hPz,
+    addZ_of_Z_eq_zero_left hP hPz, smul_fin3]
+
+lemma addXYZ_of_Z_eq_zero_right [NoZeroDivisors R] {P Q : Fin 3 → R} (hQ : W'.Equation Q)
+    (hQz : Q z = 0) : W'.addXYZ P Q = -(Q y ^ 2 * P z) • P := by
+  rw [addXYZ, addX_of_Z_eq_zero_right hQ hQz, addY_of_Z_eq_zero_right hQ hQz,
+    addZ_of_Z_eq_zero_right hQ hQz, smul_fin3]
+
+lemma addXYZ_of_X_eq {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z = Q x * P z) : W.addXYZ P Q = addU P Q • ![0, 1, 0] := by
+  erw [addXYZ, addX_of_X_eq hP hQ hPz hQz hx, addY_of_X_eq hP hQ hPz hQz hx,
+    addZ_of_X_eq hP hQ hPz hQz hx, smul_fin3, mul_zero, mul_one]
+
+lemma addXYZ_of_Z_ne_zero {P Q : Fin 3 → F} (hP : W.Equation P) (hQ : W.Equation Q) (hPz : P z ≠ 0)
+    (hQz : Q z ≠ 0) (hx : P x * Q z ≠ Q x * P z) : W.addXYZ P Q = W.addZ P Q •
+      ![W.toAffine.addX (P x / P z) (Q x / Q z)
+          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)),
+        W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P z)
+          (W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z)), 1] := by
+  have hZ : IsUnit <| W.addZ P Q := isUnit_addZ_of_X_ne hP hQ hx
+  erw [addXYZ, smul_fin3, ← addX_of_Z_ne_zero hP hQ hPz hQz hx, hZ.mul_div_cancel,
+    ← addY_of_Z_ne_zero hP hQ hPz hQz hx, hZ.mul_div_cancel, mul_one]
+
+end Addition
 
 end WeierstrassCurve.Projective
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean
new file mode 100644
index 0000000000000..8f77cea60aaad
--- /dev/null
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/VariableChange.lean
@@ -0,0 +1,325 @@
+/-
+Copyright (c) 2024 Jz Pan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kevin Buzzard, David Kurniadi Angdinata, Jz Pan
+-/
+import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
+
+/-!
+# Change of variables of Weierstrass curves
+
+This file defines admissible linear change of variables of Weierstrass curves.
+
+## Main definitions
+
+ * `WeierstrassCurve.VariableChange`: a change of variables of Weierstrass curves.
+ * `WeierstrassCurve.variableChange`: the Weierstrass curve induced by a change of variables.
+ * `WeierstrassCurve.instMulActionVariableChange`: change of variables act on Weierstrass curves.
+ * `EllipticCurve.variableChange`: the elliptic curve induced by a change of variables.
+ * `EllipticCurve.instMulActionVariableChange`: change of variables act on elliptic curves.
+
+## Main statements
+
+ * `EllipticCurve.variableChange_j`: the j-invariant of an elliptic curve is invariant under an
+    admissible linear change of variables.
+
+## References
+
+ * [J Silverman, *The Arithmetic of Elliptic Curves*][silverman2009]
+
+## Tags
+
+elliptic curve, weierstrass equation, change of variables
+-/
+
+local macro "map_simp" : tactic =>
+  `(tactic| simp only [map_ofNat, map_neg, map_add, map_sub, map_mul, map_pow])
+
+universe s u v w
+
+namespace WeierstrassCurve
+
+variable {R : Type u} [CommRing R] (W : WeierstrassCurve R)
+
+section VariableChange
+
+/-! ### Variable changes -/
+
+/-- An admissible linear change of variables of Weierstrass curves defined over a ring `R` given by
+a tuple $(u, r, s, t)$ for some $u \in R^\times$ and some $r, s, t \in R$. As a matrix, it is
+$\begin{pmatrix} u^2 & 0 & r \cr u^2s & u^3 & t \cr 0 & 0 & 1 \end{pmatrix}$. -/
+@[ext]
+structure VariableChange (R : Type u) [CommRing R] where
+  /-- The `u` coefficient of an admissible linear change of variables, which must be a unit. -/
+  u : Rˣ
+  /-- The `r` coefficient of an admissible linear change of variables. -/
+  r : R
+  /-- The `s` coefficient of an admissible linear change of variables. -/
+  s : R
+  /-- The `t` coefficient of an admissible linear change of variables. -/
+  t : R
+
+namespace VariableChange
+
+variable (C C' C'' : VariableChange R)
+
+/-- The identity linear change of variables given by the identity matrix. -/
+def id : VariableChange R :=
+  ⟨1, 0, 0, 0⟩
+
+/-- The composition of two linear changes of variables given by matrix multiplication. -/
+def comp : VariableChange R where
+  u := C.u * C'.u
+  r := C.r * C'.u ^ 2 + C'.r
+  s := C'.u * C.s + C'.s
+  t := C.t * C'.u ^ 3 + C.r * C'.s * C'.u ^ 2 + C'.t
+
+/-- The inverse of a linear change of variables given by matrix inversion. -/
+def inv : VariableChange R where
+  u := C.u⁻¹
+  r := -C.r * C.u⁻¹ ^ 2
+  s := -C.s * C.u⁻¹
+  t := (C.r * C.s - C.t) * C.u⁻¹ ^ 3
+
+lemma id_comp (C : VariableChange R) : comp id C = C := by
+  simp only [comp, id, zero_add, zero_mul, mul_zero, one_mul]
+
+lemma comp_id (C : VariableChange R) : comp C id = C := by
+  simp only [comp, id, add_zero, mul_zero, one_mul, mul_one, one_pow, Units.val_one]
+
+lemma comp_left_inv (C : VariableChange R) : comp (inv C) C = id := by
+  rw [comp, id, inv]
+  ext <;> dsimp only
+  · exact C.u.inv_mul
+  · linear_combination (norm := ring1) -C.r * pow_mul_pow_eq_one 2 C.u.inv_mul
+  · linear_combination (norm := ring1) -C.s * C.u.inv_mul
+  · linear_combination (norm := ring1) (C.r * C.s - C.t) * pow_mul_pow_eq_one 3 C.u.inv_mul
+      + -C.r * C.s * pow_mul_pow_eq_one 2 C.u.inv_mul
+
+lemma comp_assoc (C C' C'' : VariableChange R) : comp (comp C C') C'' = comp C (comp C' C'') := by
+  ext <;> simp only [comp, Units.val_mul] <;> ring1
+
+instance instGroup : Group (VariableChange R) where
+  one := id
+  inv := inv
+  mul := comp
+  one_mul := id_comp
+  mul_one := comp_id
+  inv_mul_cancel := comp_left_inv
+  mul_assoc := comp_assoc
+
+end VariableChange
+
+variable (C : VariableChange R)
+
+/-- The Weierstrass curve over `R` induced by an admissible linear change of variables
+$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$. -/
+@[simps]
+def variableChange : WeierstrassCurve R where
+  a₁ := C.u⁻¹ * (W.a₁ + 2 * C.s)
+  a₂ := C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2)
+  a₃ := C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)
+  a₄ := C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2
+    - 2 * C.s * C.t)
+  a₆ := C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2
+    - C.r * C.t * W.a₁)
+
+lemma variableChange_id : W.variableChange VariableChange.id = W := by
+  rw [VariableChange.id, variableChange, inv_one, Units.val_one]
+  ext <;> (dsimp only; ring1)
+
+lemma variableChange_comp (C C' : VariableChange R) (W : WeierstrassCurve R) :
+    W.variableChange (C.comp C') = (W.variableChange C').variableChange C := by
+  simp only [VariableChange.comp, variableChange]
+  ext <;> simp only [mul_inv, Units.val_mul]
+  · linear_combination (norm := ring1) ↑C.u⁻¹ * C.s * 2 * C'.u.inv_mul
+  · linear_combination (norm := ring1)
+      C.s * (-C'.s * 2 - W.a₁) * C.u⁻¹ ^ 2 * ↑C'.u⁻¹ * C'.u.inv_mul
+        + (C.r * 3 - C.s ^ 2) * C.u⁻¹ ^ 2 * pow_mul_pow_eq_one 2 C'.u.inv_mul
+  · linear_combination (norm := ring1)
+      C.r * (C'.s * 2 + W.a₁) * C.u⁻¹ ^ 3 * ↑C'.u⁻¹ * pow_mul_pow_eq_one 2 C'.u.inv_mul
+        + C.t * 2 * C.u⁻¹ ^ 3 * pow_mul_pow_eq_one 3 C'.u.inv_mul
+  · linear_combination (norm := ring1)
+      C.s * (-W.a₃ - C'.r * W.a₁ - C'.t * 2) * C.u⁻¹ ^ 4 * C'.u⁻¹ ^ 3 * C'.u.inv_mul
+        + C.u⁻¹ ^ 4 * C'.u⁻¹ ^ 2 * (C.r * C'.r * 6 + C.r * W.a₂ * 2 - C'.s * C.r * W.a₁ * 2
+          - C'.s ^ 2 * C.r * 2) * pow_mul_pow_eq_one 2 C'.u.inv_mul
+        - C.u⁻¹ ^ 4 * ↑C'.u⁻¹ * (C.s * C'.s * C.r * 2 + C.s * C.r * W.a₁ + C'.s * C.t * 2
+          + C.t * W.a₁) * pow_mul_pow_eq_one 3 C'.u.inv_mul
+        + C.u⁻¹ ^ 4 * (C.r ^ 2 * 3 - C.s * C.t * 2) * pow_mul_pow_eq_one 4 C'.u.inv_mul
+  · linear_combination (norm := ring1)
+      C.r * C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 4 * (C'.r * W.a₂ * 2 - C'.r * C'.s * W.a₁ + C'.r ^ 2 * 3 + W.a₄
+          - C'.s * C'.t * 2 - C'.s * W.a₃ - C'.t * W.a₁) * pow_mul_pow_eq_one 2 C'.u.inv_mul
+        - C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 3 * C.t * (C'.r * W.a₁ + C'.t * 2 + W.a₃)
+          * pow_mul_pow_eq_one 3 C'.u.inv_mul
+        + C.r ^ 2 * C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 2 * (C'.r * 3 + W.a₂ - C'.s * W.a₁ - C'.s ^ 2)
+          * pow_mul_pow_eq_one 4 C'.u.inv_mul
+        - C.r * C.t * C.u⁻¹ ^ 6 * ↑C'.u⁻¹ * (C'.s * 2 + W.a₁) * pow_mul_pow_eq_one 5 C'.u.inv_mul
+        + C.u⁻¹ ^ 6 * (C.r ^ 3 - C.t ^ 2) * pow_mul_pow_eq_one 6 C'.u.inv_mul
+
+instance instMulActionVariableChange : MulAction (VariableChange R) (WeierstrassCurve R) where
+  smul := fun C W => W.variableChange C
+  one_smul := variableChange_id
+  mul_smul := variableChange_comp
+
+@[simp]
+lemma variableChange_b₂ : (W.variableChange C).b₂ = C.u⁻¹ ^ 2 * (W.b₂ + 12 * C.r) := by
+  simp only [b₂, variableChange_a₁, variableChange_a₂]
+  ring1
+
+@[simp]
+lemma variableChange_b₄ :
+    (W.variableChange C).b₄ = C.u⁻¹ ^ 4 * (W.b₄ + C.r * W.b₂ + 6 * C.r ^ 2) := by
+  simp only [b₂, b₄, variableChange_a₁, variableChange_a₃, variableChange_a₄]
+  ring1
+
+@[simp]
+lemma variableChange_b₆ : (W.variableChange C).b₆ =
+    C.u⁻¹ ^ 6 * (W.b₆ + 2 * C.r * W.b₄ + C.r ^ 2 * W.b₂ + 4 * C.r ^ 3) := by
+  simp only [b₂, b₄, b₆, variableChange_a₃, variableChange_a₆]
+  ring1
+
+@[simp]
+lemma variableChange_b₈ : (W.variableChange C).b₈ = C.u⁻¹ ^ 8 *
+    (W.b₈ + 3 * C.r * W.b₆ + 3 * C.r ^ 2 * W.b₄ + C.r ^ 3 * W.b₂ + 3 * C.r ^ 4) := by
+  simp only [b₂, b₄, b₆, b₈, variableChange_a₁, variableChange_a₂, variableChange_a₃,
+    variableChange_a₄, variableChange_a₆]
+  ring1
+
+@[simp]
+lemma variableChange_c₄ : (W.variableChange C).c₄ = C.u⁻¹ ^ 4 * W.c₄ := by
+  simp only [c₄, variableChange_b₂, variableChange_b₄]
+  ring1
+
+@[simp]
+lemma variableChange_c₆ : (W.variableChange C).c₆ = C.u⁻¹ ^ 6 * W.c₆ := by
+  simp only [c₆, variableChange_b₂, variableChange_b₄, variableChange_b₆]
+  ring1
+
+@[simp]
+lemma variableChange_Δ : (W.variableChange C).Δ = C.u⁻¹ ^ 12 * W.Δ := by
+  simp only [b₂, b₄, b₆, b₈, Δ, variableChange_a₁, variableChange_a₂, variableChange_a₃,
+    variableChange_a₄, variableChange_a₆]
+  ring1
+
+end VariableChange
+
+section BaseChange
+
+/-! ### Maps and base changes of variable changes -/
+
+variable {A : Type v} [CommRing A] (φ : R →+* A)
+
+namespace VariableChange
+
+variable (C : VariableChange R)
+
+/-- The change of variables mapped over a ring homomorphism `φ : R →+* A`. -/
+@[simps]
+def map : VariableChange A :=
+  ⟨Units.map φ C.u, φ C.r, φ C.s, φ C.t⟩
+
+variable (A)
+
+/-- The change of variables base changed to an algebra `A` over `R`. -/
+abbrev baseChange [Algebra R A] : VariableChange A :=
+  C.map <| algebraMap R A
+
+variable {A}
+
+@[simp]
+lemma map_id : C.map (RingHom.id R) = C :=
+  rfl
+
+lemma map_map {A : Type v} [CommRing A] (φ : R →+* A) {B : Type w} [CommRing B] (ψ : A →+* B) :
+    (C.map φ).map ψ = C.map (ψ.comp φ) :=
+  rfl
+
+@[simp]
+lemma map_baseChange {S : Type s} [CommRing S] [Algebra R S] {A : Type v} [CommRing A] [Algebra R A]
+    [Algebra S A] [IsScalarTower R S A] {B : Type w} [CommRing B] [Algebra R B] [Algebra S B]
+    [IsScalarTower R S B] (ψ : A →ₐ[S] B) : (C.baseChange A).map ψ = C.baseChange B :=
+  congr_arg C.map <| ψ.comp_algebraMap_of_tower R
+
+lemma map_injective {φ : R →+* A} (hφ : Function.Injective φ) :
+    Function.Injective <| map (φ := φ) := fun _ _ h => by
+  rcases mk.inj h with ⟨h, _, _, _⟩
+  replace h := (Units.mk.inj h).left
+  ext <;> apply_fun _ using hφ <;> assumption
+
+private lemma id_map : (id : VariableChange R).map φ = id := by
+  simp only [id, map]
+  ext <;> simp only [map_one, Units.val_one, map_zero]
+
+private lemma comp_map (C' : VariableChange R) : (C.comp C').map φ = (C.map φ).comp (C'.map φ) := by
+  simp only [comp, map]
+  ext <;> map_simp <;> simp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe]
+
+/-- The map over a ring homomorphism of a change of variables is a group homomorphism. -/
+def mapHom : VariableChange R →* VariableChange A where
+  toFun := map φ
+  map_one' := id_map φ
+  map_mul' := comp_map φ
+
+end VariableChange
+
+lemma map_variableChange (C : VariableChange R) :
+    (W.map φ).variableChange (C.map φ) = (W.variableChange C).map φ := by
+  simp only [map, variableChange, VariableChange.map]
+  ext <;> map_simp <;> simp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe]
+
+end BaseChange
+
+end WeierstrassCurve
+
+/-! ## Variable changes of elliptic curves -/
+
+namespace EllipticCurve
+
+variable {R : Type u} [CommRing R]
+
+variable (E : EllipticCurve R)
+
+section VariableChange
+
+variable (C : WeierstrassCurve.VariableChange R)
+
+-- Porting note: was just `@[simps]`
+/-- The elliptic curve over `R` induced by an admissible linear change of variables
+$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$.
+When `R` is a field, any two Weierstrass equations isomorphic to `E` are related by this. -/
+@[simps (config := { rhsMd := .default }) a₁ a₂ a₃ a₄ a₆ Δ' toWeierstrassCurve]
+def variableChange : EllipticCurve R :=
+  ⟨E.toWeierstrassCurve.variableChange C, C.u⁻¹ ^ 12 * E.Δ', by
+    rw [Units.val_mul, Units.val_pow_eq_pow_val, coe_Δ', E.variableChange_Δ]⟩
+
+lemma variableChange_id : E.variableChange WeierstrassCurve.VariableChange.id = E := by
+  simp only [variableChange, WeierstrassCurve.variableChange_id]
+  simp only [WeierstrassCurve.VariableChange.id, inv_one, one_pow, one_mul]
+
+lemma variableChange_comp (C C' : WeierstrassCurve.VariableChange R) (E : EllipticCurve R) :
+    E.variableChange (C.comp C') = (E.variableChange C').variableChange C := by
+  simp only [variableChange, WeierstrassCurve.variableChange_comp]
+  simp only [WeierstrassCurve.VariableChange.comp, mul_inv, mul_pow, ← mul_assoc]
+
+instance instMulActionVariableChange :
+    MulAction (WeierstrassCurve.VariableChange R) (EllipticCurve R) where
+  smul := fun C E => E.variableChange C
+  one_smul := variableChange_id
+  mul_smul := variableChange_comp
+
+lemma coe_variableChange_Δ' : (E.variableChange C).Δ' = C.u⁻¹ ^ 12 * E.Δ' :=
+  rfl
+
+lemma coe_inv_variableChange_Δ' : (E.variableChange C).Δ'⁻¹ = C.u ^ 12 * E.Δ'⁻¹ := by
+  rw [variableChange_Δ', mul_inv, inv_pow, inv_inv]
+
+@[simp]
+lemma variableChange_j : (E.variableChange C).j = E.j := by
+  rw [j, coe_inv_variableChange_Δ', Units.val_mul, Units.val_pow_eq_pow_val,
+    variableChange_toWeierstrassCurve, WeierstrassCurve.variableChange_c₄]
+  have hu : (C.u * C.u⁻¹ : R) ^ 12 = 1 := by rw [C.u.mul_inv, one_pow]
+  linear_combination (norm := (rw [j]; ring1)) E.j * hu
+
+end VariableChange
+
+end EllipticCurve
diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
index 5bb09227528a8..87e3695c1b4d9 100644
--- a/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
+++ b/Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
@@ -3,6 +3,7 @@ Copyright (c) 2021 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard, David Kurniadi Angdinata
 -/
+import Mathlib.Algebra.CharP.Defs
 import Mathlib.Algebra.CubicDiscriminant
 import Mathlib.Tactic.FieldSimp
 import Mathlib.Tactic.LinearCombination
@@ -33,8 +34,6 @@ splitting field of `R` are precisely the $X$-coordinates of the non-zero 2-torsi
  * `WeierstrassCurve.ofJ0`: a Weierstrass curve whose j-invariant is 0.
  * `WeierstrassCurve.ofJ1728`: a Weierstrass curve whose j-invariant is 1728.
  * `WeierstrassCurve.ofJ`: a Weierstrass curve whose j-invariant is neither 0 nor 1728.
- * `WeierstrassCurve.VariableChange`: a change of variables of Weierstrass curves.
- * `WeierstrassCurve.variableChange`: the Weierstrass curve induced by a change of variables.
  * `WeierstrassCurve.map`: the Weierstrass curve mapped over a ring homomorphism.
  * `WeierstrassCurve.twoTorsionPolynomial`: the 2-torsion polynomial of a Weierstrass curve.
  * `EllipticCurve`: an elliptic curve over a commutative ring.
@@ -48,8 +47,6 @@ splitting field of `R` are precisely the $X$-coordinates of the non-zero 2-torsi
 
  * `WeierstrassCurve.twoTorsionPolynomial_disc`: the discriminant of a Weierstrass curve is a
     constant factor of the cubic discriminant of its 2-torsion polynomial.
- * `EllipticCurve.variableChange_j`: the j-invariant of an elliptic curve is invariant under an
-    admissible linear change of variables.
  * `EllipticCurve.ofJ_j`: the j-invariant of `EllipticCurve.ofJ` is equal to j.
 
 ## Implementation notes
@@ -152,169 +149,89 @@ lemma c_relation : 1728 * W.Δ = W.c₄ ^ 3 - W.c₆ ^ 2 := by
   simp only [b₂, b₄, b₆, b₈, c₄, c₆, Δ]
   ring1
 
-end Quantity
+section CharTwo
 
-section VariableChange
+variable [CharP R 2]
 
-/-! ### Variable changes -/
+lemma b₂_of_char_two : W.b₂ = W.a₁ ^ 2 := by
+  rw [b₂]
+  linear_combination 2 * W.a₂ * CharP.cast_eq_zero R 2
 
-/-- An admissible linear change of variables of Weierstrass curves defined over a ring `R` given by
-a tuple $(u, r, s, t)$ for some $u \in R^\times$ and some $r, s, t \in R$. As a matrix, it is
-$\begin{pmatrix} u^2 & 0 & r \cr u^2s & u^3 & t \cr 0 & 0 & 1 \end{pmatrix}$. -/
-@[ext]
-structure VariableChange (R : Type u) [CommRing R] where
-  /-- The `u` coefficient of an admissible linear change of variables, which must be a unit. -/
-  u : Rˣ
-  /-- The `r` coefficient of an admissible linear change of variables. -/
-  r : R
-  /-- The `s` coefficient of an admissible linear change of variables. -/
-  s : R
-  /-- The `t` coefficient of an admissible linear change of variables. -/
-  t : R
-
-namespace VariableChange
-
-variable (C C' C'' : VariableChange R)
-
-/-- The identity linear change of variables given by the identity matrix. -/
-def id : VariableChange R :=
-  ⟨1, 0, 0, 0⟩
-
-/-- The composition of two linear changes of variables given by matrix multiplication. -/
-def comp : VariableChange R where
-  u := C.u * C'.u
-  r := C.r * C'.u ^ 2 + C'.r
-  s := C'.u * C.s + C'.s
-  t := C.t * C'.u ^ 3 + C.r * C'.s * C'.u ^ 2 + C'.t
-
-/-- The inverse of a linear change of variables given by matrix inversion. -/
-def inv : VariableChange R where
-  u := C.u⁻¹
-  r := -C.r * C.u⁻¹ ^ 2
-  s := -C.s * C.u⁻¹
-  t := (C.r * C.s - C.t) * C.u⁻¹ ^ 3
-
-lemma id_comp (C : VariableChange R) : comp id C = C := by
-  simp only [comp, id, zero_add, zero_mul, mul_zero, one_mul]
-
-lemma comp_id (C : VariableChange R) : comp C id = C := by
-  simp only [comp, id, add_zero, mul_zero, one_mul, mul_one, one_pow, Units.val_one]
-
-lemma comp_left_inv (C : VariableChange R) : comp (inv C) C = id := by
-  rw [comp, id, inv]
-  ext <;> dsimp only
-  · exact C.u.inv_mul
-  · linear_combination (norm := ring1) -C.r * pow_mul_pow_eq_one 2 C.u.inv_mul
-  · linear_combination (norm := ring1) -C.s * C.u.inv_mul
-  · linear_combination (norm := ring1) (C.r * C.s - C.t) * pow_mul_pow_eq_one 3 C.u.inv_mul
-      + -C.r * C.s * pow_mul_pow_eq_one 2 C.u.inv_mul
-
-lemma comp_assoc (C C' C'' : VariableChange R) : comp (comp C C') C'' = comp C (comp C' C'') := by
-  ext <;> simp only [comp, Units.val_mul] <;> ring1
-
-instance instGroup : Group (VariableChange R) where
-  one := id
-  inv := inv
-  mul := comp
-  one_mul := id_comp
-  mul_one := comp_id
-  inv_mul_cancel := comp_left_inv
-  mul_assoc := comp_assoc
-
-end VariableChange
-
-variable (C : VariableChange R)
-
-/-- The Weierstrass curve over `R` induced by an admissible linear change of variables
-$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$. -/
-@[simps]
-def variableChange : WeierstrassCurve R where
-  a₁ := C.u⁻¹ * (W.a₁ + 2 * C.s)
-  a₂ := C.u⁻¹ ^ 2 * (W.a₂ - C.s * W.a₁ + 3 * C.r - C.s ^ 2)
-  a₃ := C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)
-  a₄ := C.u⁻¹ ^ 4 * (W.a₄ - C.s * W.a₃ + 2 * C.r * W.a₂ - (C.t + C.r * C.s) * W.a₁ + 3 * C.r ^ 2
-    - 2 * C.s * C.t)
-  a₆ := C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2
-    - C.r * C.t * W.a₁)
-
-lemma variableChange_id : W.variableChange VariableChange.id = W := by
-  rw [VariableChange.id, variableChange, inv_one, Units.val_one]
-  ext <;> (dsimp only; ring1)
-
-lemma variableChange_comp (C C' : VariableChange R) (W : WeierstrassCurve R) :
-    W.variableChange (C.comp C') = (W.variableChange C').variableChange C := by
-  simp only [VariableChange.comp, variableChange]
-  ext <;> simp only [mul_inv, Units.val_mul]
-  · linear_combination (norm := ring1) C.u⁻¹ * C.s * 2 * C'.u.inv_mul
-  · linear_combination (norm := ring1)
-      C.s * (-C'.s * 2 - W.a₁) * C.u⁻¹ ^ 2 * C'.u⁻¹ * C'.u.inv_mul
-        + (C.r * 3 - C.s ^ 2) * C.u⁻¹ ^ 2 * pow_mul_pow_eq_one 2 C'.u.inv_mul
-  · linear_combination (norm := ring1)
-      C.r * (C'.s * 2 + W.a₁) * C.u⁻¹ ^ 3 * C'.u⁻¹ * pow_mul_pow_eq_one 2 C'.u.inv_mul
-        + C.t * 2 * C.u⁻¹ ^ 3 * pow_mul_pow_eq_one 3 C'.u.inv_mul
-  · linear_combination (norm := ring1)
-      C.s * (-W.a₃ - C'.r * W.a₁ - C'.t * 2) * C.u⁻¹ ^ 4 * C'.u⁻¹ ^ 3 * C'.u.inv_mul
-        + C.u⁻¹ ^ 4 * C'.u⁻¹ ^ 2 * (C.r * C'.r * 6 + C.r * W.a₂ * 2 - C'.s * C.r * W.a₁ * 2
-          - C'.s ^ 2 * C.r * 2) * pow_mul_pow_eq_one 2 C'.u.inv_mul
-        - C.u⁻¹ ^ 4 * C'.u⁻¹ * (C.s * C'.s * C.r * 2 + C.s * C.r * W.a₁ + C'.s * C.t * 2
-          + C.t * W.a₁) * pow_mul_pow_eq_one 3 C'.u.inv_mul
-        + C.u⁻¹ ^ 4 * (C.r ^ 2 * 3 - C.s * C.t * 2) * pow_mul_pow_eq_one 4 C'.u.inv_mul
-  · linear_combination (norm := ring1)
-      C.r * C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 4 * (C'.r * W.a₂ * 2 - C'.r * C'.s * W.a₁ + C'.r ^ 2 * 3 + W.a₄
-          - C'.s * C'.t * 2 - C'.s * W.a₃ - C'.t * W.a₁) * pow_mul_pow_eq_one 2 C'.u.inv_mul
-        - C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 3 * C.t * (C'.r * W.a₁ + C'.t * 2 + W.a₃)
-          * pow_mul_pow_eq_one 3 C'.u.inv_mul
-        + C.r ^ 2 * C.u⁻¹ ^ 6 * C'.u⁻¹ ^ 2 * (C'.r * 3 + W.a₂ - C'.s * W.a₁ - C'.s ^ 2)
-          * pow_mul_pow_eq_one 4 C'.u.inv_mul
-        - C.r * C.t * C.u⁻¹ ^ 6 * C'.u⁻¹ * (C'.s * 2 + W.a₁) * pow_mul_pow_eq_one 5 C'.u.inv_mul
-        + C.u⁻¹ ^ 6 * (C.r ^ 3 - C.t ^ 2) * pow_mul_pow_eq_one 6 C'.u.inv_mul
-
-instance instMulActionVariableChange : MulAction (VariableChange R) (WeierstrassCurve R) where
-  smul := fun C W => W.variableChange C
-  one_smul := variableChange_id
-  mul_smul := variableChange_comp
+lemma b₄_of_char_two : W.b₄ = W.a₁ * W.a₃ := by
+  rw [b₄]
+  linear_combination W.a₄ * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_b₂ : (W.variableChange C).b₂ = C.u⁻¹ ^ 2 * (W.b₂ + 12 * C.r) := by
-  simp only [b₂, variableChange_a₁, variableChange_a₂]
-  ring1
+lemma b₆_of_char_two : W.b₆ = W.a₃ ^ 2 := by
+  rw [b₆]
+  linear_combination 2 * W.a₆ * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_b₄ :
-    (W.variableChange C).b₄ = C.u⁻¹ ^ 4 * (W.b₄ + C.r * W.b₂ + 6 * C.r ^ 2) := by
-  simp only [b₂, b₄, variableChange_a₁, variableChange_a₃, variableChange_a₄]
-  ring1
+lemma b₈_of_char_two :
+    W.b₈ = W.a₁ ^ 2 * W.a₆ + W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 + W.a₄ ^ 2 := by
+  rw [b₈]
+  linear_combination (2 * W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ - W.a₄ ^ 2) * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_b₆ : (W.variableChange C).b₆ =
-    C.u⁻¹ ^ 6 * (W.b₆ + 2 * C.r * W.b₄ + C.r ^ 2 * W.b₂ + 4 * C.r ^ 3) := by
-  simp only [b₂, b₄, b₆, variableChange_a₃, variableChange_a₆]
-  ring1
+lemma c₄_of_char_two : W.c₄ = W.a₁ ^ 4 := by
+  rw [c₄, b₂_of_char_two]
+  linear_combination -12 * W.b₄ * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_b₈ : (W.variableChange C).b₈ = C.u⁻¹ ^ 8 *
-    (W.b₈ + 3 * C.r * W.b₆ + 3 * C.r ^ 2 * W.b₄ + C.r ^ 3 * W.b₂ + 3 * C.r ^ 4) := by
-  simp only [b₂, b₄, b₆, b₈, variableChange_a₁, variableChange_a₂, variableChange_a₃,
-    variableChange_a₄, variableChange_a₆]
-  ring1
+lemma c₆_of_char_two : W.c₆ = W.a₁ ^ 6 := by
+  rw [c₆, b₂_of_char_two]
+  linear_combination (18 * W.a₁ ^ 2 * W.b₄ - 108 * W.b₆ - W.a₁ ^ 6) * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_c₄ : (W.variableChange C).c₄ = C.u⁻¹ ^ 4 * W.c₄ := by
-  simp only [c₄, variableChange_b₂, variableChange_b₄]
-  ring1
+lemma Δ_of_char_two : W.Δ = W.a₁ ^ 4 * W.b₈ + W.a₃ ^ 4 + W.a₁ ^ 3 * W.a₃ ^ 3 := by
+  rw [Δ, b₂_of_char_two, b₄_of_char_two, b₆_of_char_two]
+  linear_combination (-W.a₁ ^ 4 * W.b₈ - 14 * W.a₃ ^ 4) * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_c₆ : (W.variableChange C).c₆ = C.u⁻¹ ^ 6 * W.c₆ := by
-  simp only [c₆, variableChange_b₂, variableChange_b₄, variableChange_b₆]
-  ring1
+lemma b_relation_of_char_two : W.b₂ * W.b₆ = W.b₄ ^ 2 := by
+  linear_combination -W.b_relation + 2 * W.b₈ * CharP.cast_eq_zero R 2
 
-@[simp]
-lemma variableChange_Δ : (W.variableChange C).Δ = C.u⁻¹ ^ 12 * W.Δ := by
-  simp only [b₂, b₄, b₆, b₈, Δ, variableChange_a₁, variableChange_a₂, variableChange_a₃,
-    variableChange_a₄, variableChange_a₆]
-  ring1
+lemma c_relation_of_char_two : W.c₄ ^ 3 = W.c₆ ^ 2 := by
+  linear_combination -W.c_relation + 864 * W.Δ * CharP.cast_eq_zero R 2
+
+end CharTwo
+
+section CharThree
+
+variable [CharP R 3]
+
+lemma b₂_of_char_three : W.b₂ = W.a₁ ^ 2 + W.a₂ := by
+  rw [b₂]
+  linear_combination W.a₂ * CharP.cast_eq_zero R 3
+
+lemma b₄_of_char_three : W.b₄ = -W.a₄ + W.a₁ * W.a₃ := by
+  rw [b₄]
+  linear_combination W.a₄ * CharP.cast_eq_zero R 3
+
+lemma b₆_of_char_three : W.b₆ = W.a₃ ^ 2 + W.a₆ := by
+  rw [b₆]
+  linear_combination W.a₆ * CharP.cast_eq_zero R 3
+
+lemma b₈_of_char_three :
+    W.b₈ = W.a₁ ^ 2 * W.a₆ + W.a₂ * W.a₆ - W.a₁ * W.a₃ * W.a₄ + W.a₂ * W.a₃ ^ 2 - W.a₄ ^ 2 := by
+  rw [b₈]
+  linear_combination W.a₂ * W.a₆ * CharP.cast_eq_zero R 3
 
-end VariableChange
+lemma c₄_of_char_three : W.c₄ = W.b₂ ^ 2 := by
+  rw [c₄]
+  linear_combination -8 * W.b₄ * CharP.cast_eq_zero R 3
+
+lemma c₆_of_char_three : W.c₆ = -W.b₂ ^ 3 := by
+  rw [c₆]
+  linear_combination (12 * W.b₂ * W.b₄ - 72 * W.b₆) * CharP.cast_eq_zero R 3
+
+lemma Δ_of_char_three : W.Δ = -W.b₂ ^ 2 * W.b₈ - 8 * W.b₄ ^ 3 := by
+  rw [Δ]
+  linear_combination (-9 * W.b₆ ^ 2 + 3 * W.b₂ * W.b₄ * W.b₆) * CharP.cast_eq_zero R 3
+
+lemma b_relation_of_char_three : W.b₈ = W.b₂ * W.b₆ - W.b₄ ^ 2 := by
+  linear_combination W.b_relation - W.b₈ * CharP.cast_eq_zero R 3
+
+lemma c_relation_of_char_three : W.c₄ ^ 3 = W.c₆ ^ 2 := by
+  linear_combination -W.c_relation + 576 * W.Δ * CharP.cast_eq_zero R 3
+
+end CharThree
+
+end Quantity
 
 section BaseChange
 
@@ -388,64 +305,6 @@ lemma map_injective {φ : R →+* A} (hφ : Function.Injective φ) :
   rcases mk.inj h with ⟨_, _, _, _, _⟩
   ext <;> apply_fun _ using hφ <;> assumption
 
-namespace VariableChange
-
-variable (C : VariableChange R)
-
-/-- The change of variables mapped over a ring homomorphism `φ : R →+* A`. -/
-@[simps]
-def map : VariableChange A :=
-  ⟨Units.map φ C.u, φ C.r, φ C.s, φ C.t⟩
-
-variable (A)
-
-/-- The change of variables base changed to an algebra `A` over `R`. -/
-abbrev baseChange [Algebra R A] : VariableChange A :=
-  C.map <| algebraMap R A
-
-variable {A}
-
-@[simp]
-lemma map_id : C.map (RingHom.id R) = C :=
-  rfl
-
-lemma map_map {A : Type v} [CommRing A] (φ : R →+* A) {B : Type w} [CommRing B] (ψ : A →+* B) :
-    (C.map φ).map ψ = C.map (ψ.comp φ) :=
-  rfl
-
-@[simp]
-lemma map_baseChange {S : Type s} [CommRing S] [Algebra R S] {A : Type v} [CommRing A] [Algebra R A]
-    [Algebra S A] [IsScalarTower R S A] {B : Type w} [CommRing B] [Algebra R B] [Algebra S B]
-    [IsScalarTower R S B] (ψ : A →ₐ[S] B) : (C.baseChange A).map ψ = C.baseChange B :=
-  congr_arg C.map <| ψ.comp_algebraMap_of_tower R
-
-lemma map_injective {φ : R →+* A} (hφ : Function.Injective φ) :
-    Function.Injective <| map (φ := φ) := fun _ _ h => by
-  rcases mk.inj h with ⟨h, _, _, _⟩
-  replace h := (Units.mk.inj h).left
-  ext <;> apply_fun _ using hφ <;> assumption
-
-private lemma id_map : (id : VariableChange R).map φ = id := by
-  simp only [id, map]
-  ext <;> simp only [map_one, Units.val_one, map_zero]
-
-private lemma comp_map (C' : VariableChange R) : (C.comp C').map φ = (C.map φ).comp (C'.map φ) := by
-  simp only [comp, map]
-  ext <;> map_simp <;> simp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe]
-
-/-- The map over a ring homomorphism of a change of variables is a group homomorphism. -/
-def mapHom : VariableChange R →* VariableChange A where
-  toFun := map φ
-  map_one' := id_map φ
-  map_mul' := comp_map φ
-
-end VariableChange
-
-lemma map_variableChange (C : VariableChange R) :
-    (W.map φ).variableChange (C.map φ) = (W.variableChange C).map φ := by
-  simp only [map, variableChange, VariableChange.map]
-  ext <;> map_simp <;> simp only [Units.coe_map, Units.coe_map_inv, MonoidHom.coe_coe]
-
 end BaseChange
 
 section TorsionPolynomial
@@ -462,6 +321,36 @@ lemma twoTorsionPolynomial_disc : W.twoTorsionPolynomial.disc = 16 * W.Δ := by
   simp only [b₂, b₄, b₆, b₈, Δ, twoTorsionPolynomial, Cubic.disc]
   ring1
 
+section CharTwo
+
+variable [CharP R 2]
+
+lemma twoTorsionPolynomial_of_char_two : W.twoTorsionPolynomial = ⟨0, W.b₂, 0, W.b₆⟩ := by
+  rw [twoTorsionPolynomial]
+  ext <;> dsimp
+  · linear_combination 2 * CharP.cast_eq_zero R 2
+  · linear_combination W.b₄ * CharP.cast_eq_zero R 2
+
+lemma twoTorsionPolynomial_disc_of_char_two : W.twoTorsionPolynomial.disc = 0 := by
+  linear_combination W.twoTorsionPolynomial_disc + 8 * W.Δ * CharP.cast_eq_zero R 2
+
+end CharTwo
+
+section CharThree
+
+variable [CharP R 3]
+
+lemma twoTorsionPolynomial_of_char_three : W.twoTorsionPolynomial = ⟨1, W.b₂, -W.b₄, W.b₆⟩ := by
+  rw [twoTorsionPolynomial]
+  ext <;> dsimp
+  · linear_combination CharP.cast_eq_zero R 3
+  · linear_combination W.b₄ * CharP.cast_eq_zero R 3
+
+lemma twoTorsionPolynomial_disc_of_char_three : W.twoTorsionPolynomial.disc = W.Δ := by
+  linear_combination W.twoTorsionPolynomial_disc + 5 * W.Δ * CharP.cast_eq_zero R 3
+
+end CharThree
+
 lemma twoTorsionPolynomial_disc_isUnit [Invertible (2 : R)] :
     IsUnit W.twoTorsionPolynomial.disc ↔ IsUnit W.Δ := by
   rw [twoTorsionPolynomial_disc, IsUnit.mul_iff, show (16 : R) = 2 ^ 4 by norm_num1]
@@ -526,7 +415,6 @@ end WeierstrassCurve
 
 /-- An elliptic curve over a commutative ring. Note that this definition is only mathematically
 accurate for certain rings whose Picard group has trivial 12-torsion, such as a field or a PID. -/
-@[ext]
 structure EllipticCurve (R : Type u) [CommRing R] extends WeierstrassCurve R where
   /-- The discriminant `Δ'` of an elliptic curve over `R`, which is given as a unit in `R`. -/
   Δ' : Rˣ
@@ -535,61 +423,77 @@ structure EllipticCurve (R : Type u) [CommRing R] extends WeierstrassCurve R whe
 
 namespace EllipticCurve
 
-variable {R : Type u} [CommRing R] (E : EllipticCurve R)
+variable {R : Type u} [CommRing R]
+
+theorem toWeierstrassCurve_injective : Function.Injective (toWeierstrassCurve (R := R))
+  | ⟨x1, _, x3⟩, ⟨y1, _, y3⟩, h => by
+    change x1 = y1 at h
+    congr
+    exact Units.ext (by rw [x3, y3, h])
+
+@[ext]
+theorem ext {x y : EllipticCurve R} (h₁ : x.a₁ = y.a₁) (h₂ : x.a₂ = y.a₂) (h₃ : x.a₃ = y.a₃)
+    (h₄ : x.a₄ = y.a₄) (h₆ : x.a₆ = y.a₆) : x = y :=
+  toWeierstrassCurve_injective (WeierstrassCurve.ext h₁ h₂ h₃ h₄ h₆)
+
+variable (E : EllipticCurve R)
 
 -- Porting note (#10619): removed `@[simp]` to avoid a `simpNF` linter error
 /-- The j-invariant `j` of an elliptic curve, which is invariant under isomorphisms over `R`. -/
 def j : R :=
   E.Δ'⁻¹ * E.c₄ ^ 3
 
-lemma twoTorsionPolynomial_disc_ne_zero [Nontrivial R] [Invertible (2 : R)] :
-    E.twoTorsionPolynomial.disc ≠ 0 :=
-  E.toWeierstrassCurve.twoTorsionPolynomial_disc_ne_zero <| E.coe_Δ' ▸ E.Δ'.isUnit
+/-- A variant of `EllipticCurve.j_eq_zero_iff` without assuming a reduced ring. -/
+lemma j_eq_zero_iff' : E.j = 0 ↔ E.c₄ ^ 3 = 0 := by
+  rw [j, Units.mul_right_eq_zero]
 
-section VariableChange
+lemma j_eq_zero (h : E.c₄ = 0) : E.j = 0 := by
+  rw [j_eq_zero_iff', h, zero_pow three_ne_zero]
 
-/-! ### Variable changes -/
+lemma j_eq_zero_iff [IsReduced R] : E.j = 0 ↔ E.c₄ = 0 := by
+  rw [j_eq_zero_iff', IsReduced.pow_eq_zero_iff three_ne_zero]
 
-variable (C : WeierstrassCurve.VariableChange R)
+section CharTwo
 
--- Porting note: was just `@[simps]`
-/-- The elliptic curve over `R` induced by an admissible linear change of variables
-$(X, Y) \mapsto (u^2X + r, u^3Y + u^2sX + t)$ for some $u \in R^\times$ and some $r, s, t \in R$.
-When `R` is a field, any two Weierstrass equations isomorphic to `E` are related by this. -/
-@[simps (config := { rhsMd := .default }) a₁ a₂ a₃ a₄ a₆ Δ' toWeierstrassCurve]
-def variableChange : EllipticCurve R :=
-  ⟨E.toWeierstrassCurve.variableChange C, C.u⁻¹ ^ 12 * E.Δ', by
-    rw [Units.val_mul, Units.val_pow_eq_pow_val, coe_Δ', E.variableChange_Δ]⟩
-
-lemma variableChange_id : E.variableChange WeierstrassCurve.VariableChange.id = E := by
-  simp only [variableChange, WeierstrassCurve.variableChange_id]
-  simp only [WeierstrassCurve.VariableChange.id, inv_one, one_pow, one_mul]
-
-lemma variableChange_comp (C C' : WeierstrassCurve.VariableChange R) (E : EllipticCurve R) :
-    E.variableChange (C.comp C') = (E.variableChange C').variableChange C := by
-  simp only [variableChange, WeierstrassCurve.variableChange_comp]
-  simp only [WeierstrassCurve.VariableChange.comp, mul_inv, mul_pow, ← mul_assoc]
-
-instance instMulActionVariableChange :
-    MulAction (WeierstrassCurve.VariableChange R) (EllipticCurve R) where
-  smul := fun C E => E.variableChange C
-  one_smul := variableChange_id
-  mul_smul := variableChange_comp
-
-lemma coe_variableChange_Δ' : (E.variableChange C).Δ' = C.u⁻¹ ^ 12 * E.Δ' :=
-  rfl
+variable [CharP R 2]
 
-lemma coe_inv_variableChange_Δ' : (E.variableChange C).Δ'⁻¹ = C.u ^ 12 * E.Δ'⁻¹ := by
-  rw [variableChange_Δ', mul_inv, inv_pow, inv_inv]
+lemma j_of_char_two : E.j = E.Δ'⁻¹ * E.a₁ ^ 12 := by
+  rw [j, E.c₄_of_char_two, ← pow_mul]
 
-@[simp]
-lemma variableChange_j : (E.variableChange C).j = E.j := by
-  rw [j, coe_inv_variableChange_Δ', Units.val_mul, Units.val_pow_eq_pow_val,
-    variableChange_toWeierstrassCurve, WeierstrassCurve.variableChange_c₄]
-  have hu : (C.u * C.u⁻¹ : R) ^ 12 = 1 := by rw [C.u.mul_inv, one_pow]
-  linear_combination (norm := (rw [j]; ring1)) E.j * hu
+/-- A variant of `EllipticCurve.j_eq_zero_iff_of_char_two` without assuming a reduced ring. -/
+lemma j_eq_zero_iff_of_char_two' : E.j = 0 ↔ E.a₁ ^ 12 = 0 := by
+  rw [j_of_char_two, Units.mul_right_eq_zero]
+
+lemma j_eq_zero_of_char_two (h : E.a₁ = 0) : E.j = 0 := by
+  rw [j_eq_zero_iff_of_char_two', h, zero_pow (Nat.succ_ne_zero _)]
+
+lemma j_eq_zero_iff_of_char_two [IsReduced R] : E.j = 0 ↔ E.a₁ = 0 := by
+  rw [j_eq_zero_iff_of_char_two', IsReduced.pow_eq_zero_iff (Nat.succ_ne_zero _)]
 
-end VariableChange
+end CharTwo
+
+section CharThree
+
+variable [CharP R 3]
+
+lemma j_of_char_three : E.j = E.Δ'⁻¹ * E.b₂ ^ 6 := by
+  rw [j, E.c₄_of_char_three, ← pow_mul]
+
+/-- A variant of `EllipticCurve.j_eq_zero_iff_of_char_three` without assuming a reduced ring. -/
+lemma j_eq_zero_iff_of_char_three' : E.j = 0 ↔ E.b₂ ^ 6 = 0 := by
+  rw [j_of_char_three, Units.mul_right_eq_zero]
+
+lemma j_eq_zero_of_char_three (h : E.b₂ = 0) : E.j = 0 := by
+  rw [j_eq_zero_iff_of_char_three', h, zero_pow (Nat.succ_ne_zero _)]
+
+lemma j_eq_zero_iff_of_char_three [IsReduced R] : E.j = 0 ↔ E.b₂ = 0 := by
+  rw [j_eq_zero_iff_of_char_three', IsReduced.pow_eq_zero_iff (Nat.succ_ne_zero _)]
+
+end CharThree
+
+lemma twoTorsionPolynomial_disc_ne_zero [Nontrivial R] [Invertible (2 : R)] :
+    E.twoTorsionPolynomial.disc ≠ 0 :=
+  E.toWeierstrassCurve.twoTorsionPolynomial_disc_ne_zero <| E.coe_Δ' ▸ E.Δ'.isUnit
 
 section BaseChange
 
@@ -623,8 +527,7 @@ lemma map_j : (E.map φ).j = φ E.j := by
 
 lemma map_injective {φ : R →+* A} (hφ : Function.Injective φ) :
     Function.Injective <| map (φ := φ) := fun _ _ h => by
-  rcases mk.inj h with ⟨h1, h2⟩
-  replace h2 := (Units.mk.inj h2).left
+  rcases mk.inj h with ⟨h1, _⟩
   rcases WeierstrassCurve.mk.inj h1 with ⟨_, _, _, _, _⟩
   ext <;> apply_fun _ using hφ <;> assumption
 
diff --git a/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean b/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
index 807cd5593bbde..1fe4f13ebd574 100644
--- a/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
+++ b/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
@@ -282,6 +282,8 @@ def identityToΓSpec : 𝟭 LocallyRingedSpace.{u} ⟶ Γ.rightOp ⋙ Spec.toLoc
         = toΓSpecFun X x := by rw [ContinuousMap.coe_mk]
       erw [this]
       dsimp [toΓSpecFun]
+      -- TODO: this instance was found automatically before #6045
+      have := @AlgebraicGeometry.LocallyRingedSpace.isLocalRingHomStalkMap X Y
       -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
       erw [← LocalRing.comap_closedPoint (f.stalkMap x), ←
         PrimeSpectrum.comap_comp_apply, ← PrimeSpectrum.comap_comp_apply]
@@ -314,22 +316,21 @@ theorem right_triangle (R : CommRingCat) :
 
 /-- The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `LocallyRingedSpace`. -/
 -- Porting note: `simps` cause a time out, so `Unit` and `counit` will be added manually
-def locallyRingedSpaceAdjunction : Γ.rightOp ⊣ Spec.toLocallyRingedSpace.{u} :=
-  Adjunction.mkOfUnitCounit
-    { unit := identityToΓSpec
-      counit := (NatIso.op SpecΓIdentity).inv
-      left_triangle := by
-        ext X; erw [Category.id_comp]
-        exact congr_arg Quiver.Hom.op (left_triangle X)
-      right_triangle := by
-        ext R : 2
-        -- Porting note: a little bit hand holding
-        change identityToΓSpec.app _ ≫ 𝟙 _ ≫ Spec.toLocallyRingedSpace.map _ =
-          𝟙 _
-        simp_rw [Category.id_comp, show (NatIso.op SpecΓIdentity).inv.app R =
-          (SpecΓIdentity.inv.app R.unop).op from rfl]
-        exact right_triangle R.unop
-        }
+def locallyRingedSpaceAdjunction : Γ.rightOp ⊣ Spec.toLocallyRingedSpace.{u} where
+  unit := identityToΓSpec
+  counit := (NatIso.op SpecΓIdentity).inv
+  left_triangle_components X := by
+    simp only [Functor.id_obj, Functor.rightOp_obj, Γ_obj, Functor.comp_obj,
+      Spec.toLocallyRingedSpace_obj, Spec.locallyRingedSpaceObj_toSheafedSpace,
+      Spec.sheafedSpaceObj_carrier, Spec.sheafedSpaceObj_presheaf, Functor.rightOp_map, Γ_map,
+      Quiver.Hom.unop_op, NatIso.op_inv, NatTrans.op_app, SpecΓIdentity_inv_app]
+    exact congr_arg Quiver.Hom.op (left_triangle X)
+  right_triangle_components R := by
+    simp only [Spec.toLocallyRingedSpace_obj, Functor.id_obj, Functor.comp_obj, Functor.rightOp_obj,
+      Γ_obj, Spec.locallyRingedSpaceObj_toSheafedSpace, Spec.sheafedSpaceObj_carrier,
+      Spec.sheafedSpaceObj_presheaf, NatIso.op_inv, NatTrans.op_app, op_unop, SpecΓIdentity_inv_app,
+      Spec.toLocallyRingedSpace_map, Quiver.Hom.unop_op]
+    exact right_triangle R.unop
 
 lemma locallyRingedSpaceAdjunction_unit :
     locallyRingedSpaceAdjunction.unit = identityToΓSpec := rfl
@@ -375,14 +376,15 @@ lemma toOpen_comp_locallyRingedSpaceAdjunction_homEquiv_app
   rfl
 
 /-- The adjunction `Γ ⊣ Spec` from `CommRingᵒᵖ` to `Scheme`. -/
-def adjunction : Scheme.Γ.rightOp ⊣ Scheme.Spec.{u} where
-  homEquiv X Y := locallyRingedSpaceAdjunction.{u}.homEquiv X.toLocallyRingedSpace Y
-  unit :=
-  { app := fun X ↦ locallyRingedSpaceAdjunction.{u}.unit.app X.toLocallyRingedSpace
-    naturality := fun _ _ f ↦ locallyRingedSpaceAdjunction.{u}.unit.naturality f }
-  counit := (NatIso.op Scheme.SpecΓIdentity.{u}).inv
-  homEquiv_unit := rfl
-  homEquiv_counit := rfl
+def adjunction : Scheme.Γ.rightOp ⊣ Scheme.Spec.{u} :=
+  Adjunction.mk' {
+    homEquiv := fun X Y ↦ locallyRingedSpaceAdjunction.{u}.homEquiv X.toLocallyRingedSpace Y
+    unit :=
+    { app := fun X ↦ locallyRingedSpaceAdjunction.{u}.unit.app X.toLocallyRingedSpace
+      naturality := fun _ _ f ↦ locallyRingedSpaceAdjunction.{u}.unit.naturality f }
+    counit := (NatIso.op Scheme.SpecΓIdentity.{u}).inv
+    homEquiv_unit := rfl
+    homEquiv_counit := rfl }
 
 theorem adjunction_homEquiv_apply {X : Scheme} {R : CommRingCatᵒᵖ}
     (f : (op <| Scheme.Γ.obj <| op X) ⟶ R) :
@@ -403,14 +405,13 @@ theorem adjunction_counit_app' {R : CommRingCatᵒᵖ} :
 theorem adjunction_counit_app {R : CommRingCatᵒᵖ} :
     ΓSpec.adjunction.counit.app R = (Scheme.ΓSpecIso (unop R)).inv.op := rfl
 
--- This is not a simp lemma to respect the abstraction
-theorem adjunction_unit_app {X : Scheme} :
-    ΓSpec.adjunction.unit.app X = locallyRingedSpaceAdjunction.unit.app X.1 := rfl
+/-- The canonical map `X ⟶ Spec Γ(X, ⊤)`. This is the unit of the `Γ-Spec` adjunction. -/
+def _root_.AlgebraicGeometry.Scheme.toSpecΓ (X : Scheme.{u}) : X ⟶ Spec Γ(X, ⊤) :=
+  ΓSpec.adjunction.unit.app X
 
-@[reassoc (attr := simp)]
-theorem adjunction_unit_naturality {X Y : Scheme.{u}} (f : X ⟶ Y) :
-    f ≫ ΓSpec.adjunction.unit.app Y = ΓSpec.adjunction.unit.app X ≫ Spec.map (f.app ⊤) :=
-  ΓSpec.adjunction.unit.naturality f
+@[simp]
+theorem adjunction_unit_app {X : Scheme} :
+    ΓSpec.adjunction.unit.app X = X.toSpecΓ := rfl
 
 instance isIso_locallyRingedSpaceAdjunction_counit :
     IsIso.{u + 1, u + 1} locallyRingedSpaceAdjunction.counit :=
@@ -422,56 +423,62 @@ instance isIso_adjunction_counit : IsIso ΓSpec.adjunction.counit := by
   rw [adjunction_counit_app]
   infer_instance
 
+end ΓSpec
+
+@[reassoc (attr := simp)]
+theorem Scheme.toSpecΓ_naturality {X Y : Scheme.{u}} (f : X ⟶ Y) :
+    f ≫ Y.toSpecΓ = X.toSpecΓ ≫ Spec.map (f.app ⊤) :=
+  ΓSpec.adjunction.unit.naturality f
+
 @[simp]
-theorem adjunction_unit_app_app_top (X : Scheme.{u}) :
-    (ΓSpec.adjunction.unit.app X).app ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom := by
+theorem Scheme.toSpecΓ_app_top (X : Scheme.{u}) :
+    X.toSpecΓ.app ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom := by
   have := ΓSpec.adjunction.left_triangle_components X
   dsimp at this
   rw [← IsIso.eq_comp_inv] at this
-  simp only [adjunction_counit_app, Functor.id_obj, Functor.comp_obj, Functor.rightOp_obj,
+  simp only [ΓSpec.adjunction_counit_app, Functor.id_obj, Functor.comp_obj, Functor.rightOp_obj,
     Scheme.Γ_obj, Category.id_comp] at this
   rw [← Quiver.Hom.op_inj.eq_iff, this, ← op_inv, IsIso.Iso.inv_inv]
 
 @[simp]
 theorem SpecMap_ΓSpecIso_hom (R : CommRingCat.{u}) :
-    Spec.map ((Scheme.ΓSpecIso R).hom) = adjunction.unit.app (Spec R) := by
+    Spec.map ((Scheme.ΓSpecIso R).hom) = (Spec R).toSpecΓ := by
   have := ΓSpec.adjunction.right_triangle_components (op R)
   dsimp at this
   rwa [← IsIso.eq_comp_inv, Category.id_comp, ← Spec.map_inv, IsIso.Iso.inv_inv, eq_comm] at this
 
-lemma adjunction_unit_map_basicOpen (X : Scheme.{u}) (r : Γ(X, ⊤)) :
-    (ΓSpec.adjunction.unit.app X ⁻¹ᵁ (PrimeSpectrum.basicOpen r)) = X.basicOpen r := by
-  rw [← basicOpen_eq_of_affine]
-  erw [Scheme.preimage_basicOpen]
+lemma Scheme.toSpecΓ_preimage_basicOpen (X : Scheme.{u}) (r : Γ(X, ⊤)) :
+    X.toSpecΓ ⁻¹ᵁ (PrimeSpectrum.basicOpen r) = X.basicOpen r := by
+  rw [← basicOpen_eq_of_affine, Scheme.preimage_basicOpen]
   congr
-  rw [ΓSpec.adjunction_unit_app_app_top]
+  rw [Scheme.toSpecΓ_app_top]
   exact Iso.inv_hom_id_apply _ _
 
-theorem toOpen_unit_app_val_c_app {X : Scheme.{u}} (U) :
-    StructureSheaf.toOpen _ _ ≫ (ΓSpec.adjunction.unit.app X).val.c.app U =
+-- Warning: this LHS of this lemma breaks the structure-sheaf abstraction.
+@[reassoc (attr := simp)]
+theorem toOpen_toSpecΓ_app {X : Scheme.{u}} (U) :
+    StructureSheaf.toOpen _ _ ≫ X.toSpecΓ.app U =
       X.presheaf.map (homOfLE (by exact le_top)).op := by
   rw [← StructureSheaf.toOpen_res _ _ _ (homOfLE le_top), Category.assoc,
-    NatTrans.naturality _ (homOfLE (le_top (a := U.unop))).op]
+    NatTrans.naturality _ (homOfLE (le_top (a := U))).op]
   show (ΓSpec.adjunction.counit.app (Scheme.Γ.rightOp.obj X)).unop ≫
     (Scheme.Γ.rightOp.map (ΓSpec.adjunction.unit.app X)).unop ≫ _ = _
   rw [← Category.assoc, ← unop_comp, ΓSpec.adjunction.left_triangle_components]
   dsimp
   exact Category.id_comp _
 
--- Warning: this LHS of this lemma breaks the structure-sheaf abstraction.
-@[reassoc (attr := simp)]
-theorem toOpen_unit_app_val_c_app' {X : Scheme.{u}} (U : Opens (PrimeSpectrum Γ(X, ⊤))) :
-    toOpen Γ(X, ⊤) U ≫ (adjunction.unit.app X).app U =
-      X.presheaf.map (homOfLE (by exact le_top)).op :=
-  ΓSpec.toOpen_unit_app_val_c_app (op U)
-
-end ΓSpec
-
 theorem ΓSpecIso_obj_hom {X : Scheme.{u}} (U : X.Opens) :
     (Scheme.ΓSpecIso Γ(X, U)).hom = (Spec.map U.topIso.inv).app ⊤ ≫
-      (ΓSpec.adjunction.unit.app U).app ⊤ ≫ U.topIso.hom := by
-  rw [ΓSpec.adjunction_unit_app_app_top] -- why can't simp find this
-  simp
+      U.toScheme.toSpecΓ.app ⊤ ≫ U.topIso.hom := by simp
+
+@[deprecated (since := "2024-07-24")]
+alias ΓSpec.adjunction_unit_naturality := Scheme.toSpecΓ_naturality
+@[deprecated (since := "2024-07-24")]
+alias ΓSpec.adjunction_unit_naturality_assoc := Scheme.toSpecΓ_naturality_assoc
+@[deprecated (since := "2024-07-24")]
+alias ΓSpec.adjunction_unit_app_app_top := Scheme.toSpecΓ_app_top
+@[deprecated (since := "2024-07-24")]
+alias ΓSpec.adjunction_unit_map_basicOpen := Scheme.toSpecΓ_preimage_basicOpen
 
 /-! Immediate consequences of the adjunction. -/
 
diff --git a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean
index 6233014ed00a7..b3aa7de64c852 100644
--- a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean
+++ b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean
@@ -1,13 +1,15 @@
 /-
 Copyright (c) 2024 Weihong Xu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Weihong Xu
+Authors: Kevin Buzzard, Johan Commelin, Amelia Livingston, Sophie Morel, Jujian Zhang, Weihong Xu
 -/
 
 import Mathlib.Algebra.Module.LocalizedModule
 import Mathlib.AlgebraicGeometry.StructureSheaf
 import Mathlib.AlgebraicGeometry.Modules.Sheaf
 import Mathlib.Algebra.Category.ModuleCat.Sheaf
+import Mathlib.Algebra.Category.ModuleCat.FilteredColimits
+import Mathlib.CategoryTheory.Limits.ConcreteCategory.WithAlgebraicStructures
 
 /-!
 
@@ -18,9 +20,15 @@ such that `M^~(U)` is the set of dependent functions that are locally fractions.
 
 ## Main definitions
 
-* `ModuleCat.tildeInAddCommGrp` : `M^~` as a sheaf of abelian groups.
+* `ModuleCat.tildeInType` : `M^~` as a sheaf of types groups.
 * `ModuleCat.tilde` : `M^~` as a sheaf of `𝒪_{Spec R}`-modules.
 
+## Technical note
+
+To get the `R`-module structure on the stalks on `M^~`, we had to define
+`ModuleCat.tildeInModuleCat`, which is `M^~` seen as sheaf of `R`-modules. We get it by
+applying a forgetful functor to `ModuleCat.tilde M`.
+
 -/
 
 universe u
@@ -138,37 +146,112 @@ instance (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
     AddCommGroup (M.tildeInType.1.obj U) :=
   inferInstanceAs <| AddCommGroup (Tilde.sectionsSubmodule M U)
 
+noncomputable instance (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
+    Module ((Spec (.of R)).ringCatSheaf.1.obj U) (M.tildeInType.1.obj U) :=
+  inferInstanceAs <| Module _ (Tilde.sectionsSubmodule M U)
+
 /--
-`M^~` as a presheaf of abelian groups over `Spec R`
+`M^~` as a sheaf of `𝒪_{Spec R}`-modules
 -/
-def preTildeInAddCommGrp : Presheaf AddCommGrp (PrimeSpectrum.Top R) where
-  obj U := .of ((M.tildeInType).1.obj U)
-  map {U V} i :=
-    { toFun := M.tildeInType.1.map i
-      map_zero' := rfl
-      map_add' := fun x y => rfl}
+noncomputable def tilde : (Spec (CommRingCat.of R)).Modules where
+  val :=
+    { obj := fun U ↦ ModuleCat.of _ (M.tildeInType.val.obj U)
+      map := fun {U V} i ↦
+        { toFun := M.tildeInType.val.map i
+          map_smul' := by intros; rfl
+          map_add' := by intros; rfl } }
+  isSheaf := (TopCat.Presheaf.isSheaf_iff_isSheaf_comp (forget AddCommGrp) _ ).2
+    M.tildeInType.2
 
 /--
-`M^~` as a sheaf of abelian groups over `Spec R`
+This is `M^~` as a sheaf of `R`-modules.
 -/
-def tildeInAddCommGrp : Sheaf AddCommGrp (PrimeSpectrum.Top R) :=
-  ⟨M.preTildeInAddCommGrp,
-    TopCat.Presheaf.isSheaf_iff_isSheaf_comp (forget AddCommGrp) _ |>.mpr
-      (TopCat.Presheaf.isSheaf_of_iso (NatIso.ofComponents (fun _ => Iso.refl _) fun _ => rfl)
-        M.tildeInType.2)⟩
+noncomputable def tildeInModuleCat :
+    TopCat.Presheaf (ModuleCat R) (PrimeSpectrum.Top R) :=
+  (PresheafOfModules.forgetToPresheafModuleCat (op ⊤) <|
+    Limits.initialOpOfTerminal Limits.isTerminalTop).obj (tilde M).1 ⋙
+  ModuleCat.restrictScalars (StructureSheaf.globalSectionsIso R).hom
 
-noncomputable instance (U : (Opens (PrimeSpectrum.Top R))ᵒᵖ) :
-    Module ((Spec (.of R)).ringCatSheaf.1.obj U) (M.tildeInAddCommGrp.1.obj U) :=
-  inferInstanceAs <| Module _ (Tilde.sectionsSubmodule M U)
+namespace Tilde
+
+@[simp]
+theorem res_apply (U V : Opens (PrimeSpectrum.Top R)) (i : V ⟶ U)
+    (s : (tildeInModuleCat M).obj (op U)) (x : V) :
+    ((tildeInModuleCat M).map i.op s).1 x = (s.1 (i x) : _) :=
+  rfl
+
+lemma smul_section_apply (r : R) (U : Opens (PrimeSpectrum.Top R))
+    (s : (tildeInModuleCat M).1.obj (op U)) (x : U) :
+    (r • s).1 x = r • (s.1 x) := rfl
+
+lemma smul_stalk_no_nonzero_divisor {x : PrimeSpectrum R}
+    (r : x.asIdeal.primeCompl) (st : (tildeInModuleCat M).stalk x) (hst : r.1 • st = 0) :
+    st = 0 := by
+  refine Limits.Concrete.colimit_no_zero_smul_divisor
+    _ _ _ ⟨op ⟨PrimeSpectrum.basicOpen r.1, r.2⟩, fun U i s hs ↦ Subtype.eq <| funext fun pt ↦ ?_⟩
+    _ hst
+  apply LocalizedModule.eq_zero_of_smul_eq_zero _ (i.unop pt).2 _
+    (congr_fun (Subtype.ext_iff.1 hs) pt)
 
 /--
-`M^~` as a sheaf of `𝒪_{Spec R}`-modules
+If `U` is an open subset of `Spec R`, this is the morphism of `R`-modules from `M` to
+`M^~(U)`.
 -/
-noncomputable def tilde : (Spec (CommRingCat.of R)).Modules where
-  val :=
-  { presheaf := M.tildeInAddCommGrp.1
-    module := inferInstance
-    map_smul := fun _ _ _ => rfl }
-  isSheaf := M.tildeInAddCommGrp.2
+def toOpen (U : Opens (PrimeSpectrum.Top R)) :
+    ModuleCat.of R M ⟶ (tildeInModuleCat M).1.obj (op U) where
+  toFun f :=
+  ⟨fun x ↦ LocalizedModule.mkLinearMap _ _ f, fun x ↦
+    ⟨U, x.2, 𝟙 _, f, 1, fun y ↦ ⟨(Ideal.ne_top_iff_one _).1 y.1.2.1, by simp⟩⟩⟩
+  map_add' f g := Subtype.eq <| funext fun x ↦ LinearMap.map_add _ _ _
+  map_smul' r m := by
+    simp only [isLocallyFraction_pred, LocalizedModule.mkLinearMap_apply, LinearMapClass.map_smul,
+      RingHom.id_apply]
+    rfl
+
+@[simp]
+theorem toOpen_res (U V : Opens (PrimeSpectrum.Top R)) (i : V ⟶ U) :
+    toOpen M U ≫ (tildeInModuleCat M).map i.op = toOpen M V :=
+  rfl
+
+/--
+If `x` is a point of `Spec R`, this is the morphism of `R`-modules from `M` to the stalk of
+`M^~` at `x`.
+-/
+noncomputable def toStalk (x : PrimeSpectrum.Top R) :
+    ModuleCat.of R M ⟶ TopCat.Presheaf.stalk (tildeInModuleCat M) x :=
+  (toOpen M ⊤ ≫ TopCat.Presheaf.germ (tildeInModuleCat M) ⟨x, by trivial⟩)
+
+open LocalizedModule TopCat.Presheaf in
+lemma isUnit_toStalk (x : PrimeSpectrum.Top R) (r : x.asIdeal.primeCompl) :
+    IsUnit ((algebraMap R (Module.End R ((tildeInModuleCat M).stalk x))) r) := by
+  rw [Module.End_isUnit_iff]
+  refine ⟨LinearMap.ker_eq_bot.1 <| eq_bot_iff.2 fun st (h : r.1 • st = 0) ↦
+    smul_stalk_no_nonzero_divisor M r st h, fun st ↦ ?_⟩
+  obtain ⟨U, mem, s, rfl⟩ := germ_exist (F := M.tildeInModuleCat) x st
+  let O := U ⊓ (PrimeSpectrum.basicOpen r)
+  refine ⟨germ M.tildeInModuleCat (⟨x, ⟨mem, r.2⟩⟩ : O)
+    ⟨fun q ↦ (Localization.mk 1 ⟨r, q.2.2⟩ : Localization.AtPrime q.1.asIdeal) • s.1
+      ⟨q.1, q.2.1⟩, fun q ↦ ?_⟩, by
+        simpa only [Module.algebraMap_end_apply, ← map_smul] using
+          germ_ext (W := O) (hxW := ⟨mem, r.2⟩) (iWU := 𝟙 _) (iWV := homOfLE inf_le_left) _ <|
+          Subtype.eq <| funext fun y ↦ smul_eq_iff_of_mem (S := y.1.1.primeCompl) r _ _ _ |>.2 rfl⟩
+  obtain ⟨V, mem_V, iV, num, den, hV⟩ := s.2 ⟨q.1, q.2.1⟩
+  refine ⟨V ⊓ O, ⟨mem_V, q.2⟩, homOfLE inf_le_right, num, r * den, fun y ↦ ?_⟩
+  obtain ⟨h1, h2⟩ := hV ⟨y, y.2.1⟩
+  refine ⟨y.1.asIdeal.primeCompl.mul_mem y.2.2.2 h1, ?_⟩
+  simp only [Opens.coe_inf, isLocallyFraction_pred, mkLinearMap_apply,
+    smul_eq_iff_of_mem (S := y.1.1.primeCompl) (hr := h1), mk_smul_mk, one_smul, mul_one] at h2 ⊢
+  simpa only [h2, mk_smul_mk, one_smul, smul'_mk, mk_eq] using ⟨1, by simp only [one_smul]; rfl⟩
+
+/--
+The morphism of `R`-modules from the localization of `M` at the prime ideal corresponding to `x`
+to the stalk of `M^~` at `x`.
+-/
+noncomputable def localizationToStalk (x : PrimeSpectrum.Top R) :
+    ModuleCat.of R (LocalizedModule x.asIdeal.primeCompl M) ⟶
+    (TopCat.Presheaf.stalk (tildeInModuleCat M) x) :=
+  LocalizedModule.lift _ (toStalk M x) <| isUnit_toStalk M x
+
+end Tilde
 
 end ModuleCat
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
index 9cf21c35955ab..c990b0ee818d7 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean
@@ -122,8 +122,7 @@ lemma isAffine_of_isAffineOpen_basicOpen (s : Set Γ(X, ⊤))
     simp only [← basicOpen_eq_of_affine]
     exact (isAffineOpen_top (Scheme.Spec.obj (op _))).basicOpen _
   · rw [PrimeSpectrum.iSup_basicOpen_eq_top_iff, Subtype.range_coe_subtype, Set.setOf_mem_eq, hs]
-  · show IsAffineOpen (ΓSpec.adjunction.unit.app X ⁻¹ᵁ PrimeSpectrum.basicOpen i.1)
-    rw [ΓSpec.adjunction_unit_map_basicOpen]
+  · rw [Scheme.toSpecΓ_preimage_basicOpen]
     exact hs₂ _ i.2
   · simp only [Functor.comp_obj, Functor.rightOp_obj, Scheme.Γ_obj, Scheme.Spec_obj, id_eq,
       eq_mpr_eq_cast, Functor.id_obj, Opens.map_top, morphismRestrict_app]
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean b/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
index e92624a6197d0..9b5b16f7d5f16 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/Basic.lean
@@ -108,7 +108,7 @@ Also see `IsLocalAtTarget.mk'` for a convenient constructor.
 class IsLocalAtTarget (P : MorphismProperty Scheme) : Prop where
   /-- `P` respects isomorphisms. -/
   respectsIso : P.RespectsIso := by infer_instance
-  /-- `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`.  -/
+  /-- `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`. -/
   iff_of_openCover' :
     ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y),
       P f ↔ ∀ i, P (𝒰.pullbackHom f i)
@@ -186,7 +186,7 @@ Also see `IsLocalAtSource.mk'` for a convenient constructor.
 class IsLocalAtSource (P : MorphismProperty Scheme) : Prop where
   /-- `P` respects isomorphisms. -/
   respectsIso : P.RespectsIso := by infer_instance
-  /-- `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`.  -/
+  /-- `P` holds for `f ∣_ U` for an open cover `U` of `Y` if and only if `P` holds for `f`. -/
   iff_of_openCover' :
     ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} X),
       P f ↔ ∀ i, P (𝒰.map i ≫ f)
@@ -326,7 +326,7 @@ theorem respectsIso_mk {P : AffineTargetMorphismProperty}
     (h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y],
       P f → @P _ _ (f ≫ e.hom) (isAffine_of_isIso e.inv)) :
     P.toProperty.RespectsIso := by
-  constructor
+  apply MorphismProperty.RespectsIso.mk
   · rintro X Y Z e f ⟨a, h⟩; exact ⟨a, h₁ e f h⟩
   · rintro X Y Z e f ⟨a, h⟩; exact ⟨isAffine_of_isIso e.inv, h₂ e f h⟩
 
@@ -398,7 +398,7 @@ theorem of_targetAffineLocally_of_isPullback
 
 instance (P : AffineTargetMorphismProperty) [P.toProperty.RespectsIso] :
     (targetAffineLocally P).RespectsIso := by
-  constructor
+  apply MorphismProperty.RespectsIso.mk
   · introv H U
     rw [morphismRestrict_comp, P.cancel_left_of_respectsIso]
     exact H U
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean
index 4affb2c6194ba..e0f201805fa3b 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean
@@ -84,7 +84,7 @@ instance comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsClosedImmersion f]
 
 /-- Composition with an isomorphism preserves closed immersions. -/
 instance respectsIso : MorphismProperty.RespectsIso @IsClosedImmersion := by
-  constructor <;> intro X Y Z e f hf <;> infer_instance
+  apply MorphismProperty.RespectsIso.mk <;> intro X Y Z e f hf <;> infer_instance
 
 /-- Given two commutative rings `R S : CommRingCat` and a surjective morphism
 `f : R ⟶ S`, the induced scheme morphism `specObj S ⟶ specObj R` is a
@@ -95,11 +95,11 @@ theorem spec_of_surjective {R S : CommRingCat} (f : R ⟶ S) (h : Function.Surje
   surj_on_stalks x := by
     haveI : (RingHom.toMorphismProperty (fun f ↦ Function.Surjective f)).RespectsIso := by
       rw [← RingHom.toMorphismProperty_respectsIso_iff]
-      exact surjective_respectsIso
+      exact RingHom.surjective_respectsIso
     apply (MorphismProperty.arrow_mk_iso_iff
       (RingHom.toMorphismProperty (fun f ↦ Function.Surjective f))
       (Scheme.arrowStalkMapSpecIso f x)).mpr
-    exact surjective_localRingHom_of_surjective f h x.asIdeal
+    exact RingHom.surjective_localRingHom_of_surjective f h x.asIdeal
 
 /-- For any ideal `I` in a commutative ring `R`, the quotient map `specObj R ⟶ specObj (R ⧸ I)`
 is a closed immersion. -/
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean b/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
index 91b9cb9c05db1..878d9b451bd27 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean
@@ -251,12 +251,12 @@ variable {P : ∀ {R S : Type u} [CommRing R] [CommRing S], (R →+* S) → Prop
 /-- If `P` respects isos, then `stalkwise P` respects isos. -/
 lemma stalkwise_respectsIso (hP : RingHom.RespectsIso P) :
     (stalkwise P).RespectsIso where
-  precomp {X Y Z} e f hf := by
+  precomp {X Y Z} e (he : IsIso e) f hf := by
     simp only [stalkwise, Scheme.comp_coeBase, TopCat.coe_comp, Function.comp_apply]
     intro x
     rw [Scheme.stalkMap_comp]
-    exact (RingHom.RespectsIso.cancel_right_isIso hP _ _).mpr <| hf (e.hom.val.base x)
-  postcomp {X Y Z} e f hf := by
+    exact (RingHom.RespectsIso.cancel_right_isIso hP _ _).mpr <| hf (e.val.base x)
+  postcomp {X Y Z} e (he : IsIso e) f hf := by
     simp only [stalkwise, Scheme.comp_coeBase, TopCat.coe_comp, Function.comp_apply]
     intro x
     rw [Scheme.stalkMap_comp]
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean b/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean
new file mode 100644
index 0000000000000..d6dee82946a83
--- /dev/null
+++ b/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean
@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
+import Mathlib.RingTheory.RingHom.FinitePresentation
+
+/-!
+
+# Morphisms of finite presentation
+
+A morphism of schemes `f : X ⟶ Y` is locally of finite presentation if for each affine `U ⊆ Y` and
+`V ⊆ f ⁻¹' U`, The induced map `Γ(Y, U) ⟶ Γ(X, V)` is of finite presentation.
+
+A morphism of schemes is of finite presentation if it is both locally of finite presentation and
+quasi-compact. We do not provide a separate declaration for this, instead simply assume both
+conditions.
+
+We show that these properties are local, and are stable under compositions.
+
+-/
+
+
+noncomputable section
+
+open CategoryTheory
+
+universe v u
+
+namespace AlgebraicGeometry
+
+variable {X Y : Scheme.{u}} (f : X ⟶ Y)
+
+/-- A morphism of schemes `f : X ⟶ Y` is locally of finite presentation if for each affine `U ⊆ Y`
+and `V ⊆ f ⁻¹' U`, The induced map `Γ(Y, U) ⟶ Γ(X, V)` is of finite presentation. -/
+@[mk_iff]
+class LocallyOfFinitePresentation : Prop where
+  finitePresentation_of_affine_subset :
+    ∀ (U : Y.affineOpens) (V : X.affineOpens) (e : V.1 ≤ f ⁻¹ᵁ U.1),
+      (f.appLE U V e).FinitePresentation
+
+instance : HasRingHomProperty @LocallyOfFinitePresentation RingHom.FinitePresentation where
+  isLocal_ringHomProperty := RingHom.finitePresentation_isLocal
+  eq_affineLocally' := by
+    ext X Y f
+    rw [locallyOfFinitePresentation_iff, affineLocally_iff_affineOpens_le]
+
+instance (priority := 900) locallyOfFinitePresentation_of_isOpenImmersion [IsOpenImmersion f] :
+    LocallyOfFinitePresentation f :=
+  HasRingHomProperty.of_isOpenImmersion
+    RingHom.finitePresentation_holdsForLocalizationAway.containsIdentities
+
+instance : MorphismProperty.IsStableUnderComposition @LocallyOfFinitePresentation :=
+  HasRingHomProperty.stableUnderComposition RingHom.finitePresentation_stableUnderComposition
+
+instance locallyOfFinitePresentation_comp {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z)
+    [hf : LocallyOfFinitePresentation f] [hg : LocallyOfFinitePresentation g] :
+    LocallyOfFinitePresentation (f ≫ g) :=
+  MorphismProperty.comp_mem _ f g hf hg
+
+lemma locallyOfFinitePresentation_stableUnderBaseChange :
+    MorphismProperty.StableUnderBaseChange @LocallyOfFinitePresentation :=
+  HasRingHomProperty.stableUnderBaseChange RingHom.finitePresentation_stableUnderBaseChange
+
+end AlgebraicGeometry
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean b/Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean
index ffc351acd412a..4424a0eed0994 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean
@@ -46,6 +46,10 @@ instance : HasRingHomProperty @LocallyOfFiniteType RingHom.FiniteType where
 instance (priority := 900) locallyOfFiniteType_of_isOpenImmersion [IsOpenImmersion f] :
     LocallyOfFiniteType f :=
   HasRingHomProperty.of_isOpenImmersion
+    RingHom.finiteType_holdsForLocalizationAway.containsIdentities
+
+instance : MorphismProperty.IsStableUnderComposition @LocallyOfFiniteType :=
+  HasRingHomProperty.stableUnderComposition RingHom.finiteType_stableUnderComposition
 
 instance locallyOfFiniteType_comp {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z)
     [hf : LocallyOfFiniteType f] [hg : LocallyOfFiniteType g] : LocallyOfFiniteType (f ≫ g) :=
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean
index 25d3e57f11b98..b9472fc3ea550 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.RingHom.Surjective
 
 /-!
 
@@ -52,7 +52,7 @@ lemma isPreimmersion_eq_inf :
 /-- Being surjective on stalks is local at the target. -/
 instance isSurjectiveOnStalks_isLocalAtTarget : IsLocalAtTarget
     (stalkwise (Function.Surjective ·)) :=
-  stalkwiseIsLocalAtTarget_of_respectsIso surjective_respectsIso
+  stalkwiseIsLocalAtTarget_of_respectsIso RingHom.surjective_respectsIso
 
 namespace IsPreimmersion
 
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean b/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
index 211a07cc5edf0..cf3a6124cc51f 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
@@ -198,7 +198,15 @@ theorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen (X : Sch
     {U : X.Opens} (hU : IsAffineOpen U) (x f : Γ(X, U))
     (H : x |_ X.basicOpen f = 0) : ∃ n : ℕ, f ^ n * x = 0 := by
   rw [← map_zero (X.presheaf.map (homOfLE <| X.basicOpen_le f : X.basicOpen f ⟶ U).op)] at H
-  obtain ⟨⟨_, n, rfl⟩, e⟩ := (hU.isLocalization_basicOpen f).exists_of_eq H
+  #adaptation_note
+  /--
+  Prior to nightly-2024-09-29, we could use dot notation here:
+  `(hU.isLocalization_basicOpen f).exists_of_eq H`
+  This is no longer possible;
+  likely changing the signature of `IsLocalization.Away.exists_of_eq` is in order.
+  -/
+  obtain ⟨n, e⟩ :=
+    @IsLocalization.Away.exists_of_eq _ _ _ _ _ _ (hU.isLocalization_basicOpen f) _ _ H
   exact ⟨n, by simpa [mul_comm x] using e⟩
 
 /-- If `x : Γ(X, U)` is zero on `D(f)` for some `f : Γ(X, U)`, and `U` is quasi-compact, then
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean b/Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
index 83373737cbce2..fc114dd60f6a8 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean
@@ -353,7 +353,7 @@ theorem isIso_ΓSpec_adjunction_unit_app_basicOpen {X : Scheme} [CompactSpace X]
     [QuasiSeparatedSpace X] (f : X.presheaf.obj (op ⊤)) :
     IsIso ((ΓSpec.adjunction.unit.app X).val.c.app (op (PrimeSpectrum.basicOpen f))) := by
   refine @IsIso.of_isIso_comp_right _ _ _ _ _ _ (X.presheaf.map
-    (eqToHom (ΓSpec.adjunction_unit_map_basicOpen _ _).symm).op) _ ?_
+    (eqToHom (Scheme.toSpecΓ_preimage_basicOpen _ _).symm).op) _ ?_
   rw [ConcreteCategory.isIso_iff_bijective, CommRingCat.forget_map]
   apply (config := { allowSynthFailures := true }) IsLocalization.bijective
   · exact StructureSheaf.IsLocalization.to_basicOpen _ _
@@ -361,8 +361,6 @@ theorem isIso_ΓSpec_adjunction_unit_app_basicOpen {X : Scheme} [CompactSpace X]
     · exact isCompact_univ
     · exact isQuasiSeparated_univ
   · rw [← CommRingCat.comp_eq_ring_hom_comp]
-    simp [RingHom.algebraMap_toAlgebra]
-    rw [ΓSpec.toOpen_unit_app_val_c_app'_assoc, ← Functor.map_comp]
-    rfl
+    simp [RingHom.algebraMap_toAlgebra, ← Functor.map_comp]
 
 end AlgebraicGeometry
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
index 3ed0b2cf6b132..1a867be2b4d2a 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.AlgebraicGeometry.Morphisms.Basic
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.LocalProperties.Basic
 
 /-!
 
@@ -228,9 +228,8 @@ theorem of_source_openCover [IsAffine Y]
   | basicOpen U r H =>
     simp_rw [Scheme.affineBasicOpen_coe,
       ← f.appLE_map (U := ⊤) le_top (homOfLE (X.basicOpen_le r)).op]
-    apply (isLocal_ringHomProperty P).StableUnderComposition _ _ H
     have := U.2.isLocalization_basicOpen r
-    apply (isLocal_ringHomProperty P).HoldsForLocalizationAway _ r
+    exact (isLocal_ringHomProperty P).StableUnderCompositionWithLocalizationAway.left _ r _ H
   | openCover U s hs H =>
     apply (isLocal_ringHomProperty P).OfLocalizationSpanTarget.ofIsLocalization
       (isLocal_ringHomProperty P).respectsIso _ _ hs
@@ -283,16 +282,16 @@ instance : IsLocalAtSource P := by
     fun i ↦ iff_of_source_openCover (P := P) (f := 𝒰.map i ≫ f) (𝒰.obj i).affineCover]
   simp [Scheme.OpenCover.affineRefinement, Sigma.forall]
 
-instance : P.ContainsIdentities where
+lemma containsIdentities (hP : RingHom.ContainsIdentities Q) : P.ContainsIdentities where
   id_mem X := by
     rw [IsLocalAtTarget.iff_of_iSup_eq_top (P := P) _ (iSup_affineOpens_eq_top _)]
     intro U
     have : IsAffine (𝟙 X ⁻¹ᵁ U.1) := U.2
     rw [morphismRestrict_id, iff_of_isAffine (P := P), Scheme.id_app]
-    exact (isLocal_ringHomProperty P).HoldsForLocalizationAway.of_bijective _ _
-      Function.bijective_id
+    apply hP
 
-instance : P.IsStableUnderComposition where
+lemma stableUnderComposition (hP : RingHom.StableUnderComposition Q) :
+    P.IsStableUnderComposition where
   comp_mem {X Y Z} f g hf hg := by
     wlog hZ : IsAffine Z generalizing X Y Z
     · rw [IsLocalAtTarget.iff_of_iSup_eq_top (P := P) _ (iSup_affineOpens_eq_top _)]
@@ -311,7 +310,7 @@ instance : P.IsStableUnderComposition where
       rw [← Category.assoc]
       exact this _ (comp_of_isOpenImmersion _ _ _ hf) inferInstance
     rw [iff_of_isAffine (P := P)] at hf hg ⊢
-    exact (isLocal_ringHomProperty P).StableUnderComposition _ _ hg hf
+    exact hP _ _ hg hf
 
 theorem of_comp
     (H : ∀ {R S T : Type u} [CommRing R] [CommRing S] [CommRing T],
@@ -339,10 +338,16 @@ theorem of_comp
   rw [iff_of_isAffine (P := P)] at h ⊢
   exact H _ _ h
 
-instance : P.IsMultiplicative where
+lemma isMultiplicative (hPc : RingHom.StableUnderComposition Q)
+    (hPi : RingHom.ContainsIdentities Q) :
+    P.IsMultiplicative where
+  comp_mem := (stableUnderComposition hPc).comp_mem
+  id_mem := (containsIdentities hPi).id_mem
 
 include Q in
-lemma of_isOpenImmersion [IsOpenImmersion f] : P f := IsLocalAtSource.of_isOpenImmersion f
+lemma of_isOpenImmersion (hP : RingHom.ContainsIdentities Q) [IsOpenImmersion f] : P f :=
+  haveI : P.ContainsIdentities := containsIdentities hP
+  IsLocalAtSource.of_isOpenImmersion f
 
 lemma stableUnderBaseChange (hP : RingHom.StableUnderBaseChange Q) : P.StableUnderBaseChange := by
   apply HasAffineProperty.stableUnderBaseChange
diff --git a/Mathlib/AlgebraicGeometry/OpenImmersion.lean b/Mathlib/AlgebraicGeometry/OpenImmersion.lean
index 0a6b512618e52..70d24f7442e9b 100644
--- a/Mathlib/AlgebraicGeometry/OpenImmersion.lean
+++ b/Mathlib/AlgebraicGeometry/OpenImmersion.lean
@@ -105,6 +105,12 @@ lemma preimage_image_eq (U : X.Opens) : f ⁻¹ᵁ f ''ᵁ U = U := by
   apply Opens.ext
   simp [Set.preimage_image_eq _ f.openEmbedding.inj]
 
+lemma image_le_image_iff (f : X ⟶ Y) [IsOpenImmersion f] (U U' : X.Opens) :
+    f ''ᵁ U ≤ f ''ᵁ U' ↔ U ≤ U' := by
+  refine ⟨fun h ↦ ?_, image_le_image_of_le f⟩
+  rw [← preimage_image_eq f U, ← preimage_image_eq f U']
+  apply preimage_le_preimage_of_le f h
+
 lemma image_preimage_eq_opensRange_inter (U : Y.Opens) : f ''ᵁ f ⁻¹ᵁ U = f.opensRange ⊓ U := by
   apply Opens.ext
   simp [Set.image_preimage_eq_range_inter]
@@ -209,7 +215,7 @@ theorem exists_affine_mem_range_and_range_subset
     ∃ (R : CommRingCat) (f : Spec R ⟶ X),
       IsOpenImmersion f ∧ x ∈ Set.range f.1.base ∧ Set.range f.1.base ⊆ U := by
   obtain ⟨⟨V, hxV⟩, R, ⟨e⟩⟩ := X.2 x
-  have : e.hom.1.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.1.base ≫ V.inclusion)).obj U :=
+  have : e.hom.1.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.1.base ≫ V.inclusion')).obj U :=
     show ((e.hom ≫ e.inv).1.base ⟨x, hxV⟩).1 ∈ U from e.hom_inv_id ▸ hxU
   obtain ⟨_, ⟨_, ⟨r : R, rfl⟩, rfl⟩, hr, hr'⟩ :=
     PrimeSpectrum.isBasis_basic_opens.exists_subset_of_mem_open this (Opens.is_open' _)
@@ -431,7 +437,7 @@ instance pullback_fst_of_right : IsOpenImmersion (pullback.fst g f) := by
   rw [← pullbackSymmetry_hom_comp_snd]
   -- Porting note: was just `infer_instance`, it is a bit weird that no explicit class instance is
   -- provided but still class inference fail to find this
-  exact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _
+  exact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _ _
 
 instance pullback_to_base [IsOpenImmersion g] :
     IsOpenImmersion (limit.π (cospan f g) WalkingCospan.one) := by
@@ -439,14 +445,14 @@ instance pullback_to_base [IsOpenImmersion g] :
   change IsOpenImmersion (_ ≫ f)
   -- Porting note: was just `infer_instance`, it is a bit weird that no explicit class instance is
   -- provided but still class inference fail to find this
-  exact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _
+  exact LocallyRingedSpace.IsOpenImmersion.comp (H := inferInstance) _ _
 
 instance forgetToTopPreservesOfLeft : PreservesLimit (cospan f g) Scheme.forgetToTop := by
   delta Scheme.forgetToTop
-  apply @Limits.compPreservesLimit (K := cospan f g) (F := forget)
+  refine @Limits.compPreservesLimit _ _ _ _ _ _ (K := cospan f g) _ _ (F := forget)
     (G := LocallyRingedSpace.forgetToTop) ?_ ?_
   · infer_instance
-  apply @preservesLimitOfIsoDiagram (F := _) _ _ _ _ _ _ (diagramIsoCospan.{u} _).symm ?_
+  refine @preservesLimitOfIsoDiagram _ _ _ _ _ _ _ _ _ (diagramIsoCospan.{u} _).symm ?_
   dsimp [LocallyRingedSpace.forgetToTop]
   infer_instance
 
diff --git a/Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean b/Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
index fbd5e0acc99f5..f3fa7a4fbdf4a 100644
--- a/Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
+++ b/Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
@@ -3,14 +3,14 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.RingTheory.PrimeSpectrum
-import Mathlib.Topology.Irreducible
-import Mathlib.Topology.Sets.Closeds
+import Mathlib.RingTheory.KrullDimension.Basic
+import Mathlib.Topology.KrullDimension
 import Mathlib.Topology.Sober
 import Mathlib.RingTheory.Ideal.MinimalPrime
 import Mathlib.RingTheory.Ideal.Over
 import Mathlib.RingTheory.Localization.Away.Basic
 import Mathlib.RingTheory.LocalRing.ResidueField.Defs
+import Mathlib.RingTheory.LocalRing.RingHom.Basic
 
 /-!
 # The Zariski topology on the prime spectrum of a commutative (semi)ring
@@ -207,21 +207,14 @@ section Comap
 
 variable {S' : Type*} [CommSemiring S']
 
-theorem preimage_comap_zeroLocus_aux (f : R →+* S) (s : Set R) :
-    (fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩ : PrimeSpectrum S → PrimeSpectrum R) ⁻¹'
-        zeroLocus s =
-      zeroLocus (f '' s) := by
-  ext x
-  simp only [mem_zeroLocus, Set.image_subset_iff, Set.mem_preimage, mem_zeroLocus, Ideal.coe_comap]
-
-/-- The function between prime spectra of commutative (semi)rings induced by a ring homomorphism.
-This function is continuous. -/
+/-- The continuous function between prime spectra of commutative (semi)rings induced by a ring
+homomorphism. -/
 def comap (f : R →+* S) : C(PrimeSpectrum S, PrimeSpectrum R) where
-  toFun y := ⟨Ideal.comap f y.asIdeal, inferInstance⟩
+  toFun := f.specComap
   continuous_toFun := by
     simp only [continuous_iff_isClosed, isClosed_iff_zeroLocus]
     rintro _ ⟨s, rfl⟩
-    exact ⟨_, preimage_comap_zeroLocus_aux f s⟩
+    exact ⟨_, preimage_specComap_zeroLocus_aux f s⟩
 
 variable (f : R →+* S)
 
@@ -244,12 +237,10 @@ theorem comap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S')
 
 @[simp]
 theorem preimage_comap_zeroLocus (s : Set R) : comap f ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
-  preimage_comap_zeroLocus_aux f s
+  preimage_specComap_zeroLocus_aux f s
 
 theorem comap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
-    Function.Injective (comap f) := fun x y h =>
-  PrimeSpectrum.ext (Ideal.comap_injective_of_surjective f hf
-    (congr_arg PrimeSpectrum.asIdeal h : (comap f x).asIdeal = (comap f y).asIdeal))
+    Function.Injective (comap f) := fun _ _ h => specComap_injective_of_surjective _ hf h
 
 variable (S)
 
@@ -266,29 +257,16 @@ theorem localization_comap_inducing [Algebra R S] (M : Submonoid R) [IsLocalizat
     exact ⟨_, rfl⟩
 
 theorem localization_comap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
-    Function.Injective (comap (algebraMap R S)) := by
-  intro p q h
-  replace h := congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
-  dsimp only [comap, ContinuousMap.coe_mk] at h
-  rw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
-  ext1
-  exact h
+    Function.Injective (comap (algebraMap R S)) :=
+  fun _ _ h => localization_specComap_injective S M h
 
 theorem localization_comap_embedding [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
     Embedding (comap (algebraMap R S)) :=
   ⟨localization_comap_inducing S M, localization_comap_injective S M⟩
 
 theorem localization_comap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
-    Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } := by
-  ext x
-  constructor
-  · simp_rw [disjoint_iff_inf_le]
-    rintro ⟨p, rfl⟩ x ⟨hx₁, hx₂⟩
-    exact (p.2.1 : ¬_) (p.asIdeal.eq_top_of_isUnit_mem hx₂ (IsLocalization.map_units S ⟨x, hx₁⟩))
-  · intro h
-    use ⟨x.asIdeal.map (algebraMap R S), IsLocalization.isPrime_of_isPrime_disjoint M S _ x.2 h⟩
-    ext1
-    exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h
+    Set.range (comap (algebraMap R S)) = { p | Disjoint (M : Set R) p.asIdeal } :=
+  localization_specComap_range ..
 
 open Function RingHom
 
@@ -336,28 +314,12 @@ theorem comap_singleton_isClosed_of_isIntegral (f : R →+* S) (hf : f.IsIntegra
     (Ideal.isMaximal_comap_of_isIntegral_of_isMaximal' f hf x.asIdeal)
 
 theorem image_comap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
-    comap f '' zeroLocus I = zeroLocus (I.comap f) := by
-  simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]
-  refine fun p => ⟨?_, fun h_I_p => ?_⟩
-  · rintro ⟨p, hp, rfl⟩ a ha
-    exact hp ha
-  · have hp : ker f ≤ p.asIdeal := (Ideal.comap_mono bot_le).trans h_I_p
-    refine ⟨⟨p.asIdeal.map f, Ideal.map_isPrime_of_surjective hf hp⟩, fun x hx => ?_, ?_⟩
-    · obtain ⟨x', rfl⟩ := hf x
-      exact Ideal.mem_map_of_mem f (h_I_p hx)
-    · ext x
-      rw [comap_asIdeal, Ideal.mem_comap, Ideal.mem_map_iff_of_surjective f hf]
-      refine ⟨?_, fun hx => ⟨x, hx, rfl⟩⟩
-      rintro ⟨x', hx', heq⟩
-      rw [← sub_sub_cancel x' x]
-      refine p.asIdeal.sub_mem hx' (hp ?_)
-      rwa [mem_ker, map_sub, sub_eq_zero]
+    comap f '' zeroLocus I = zeroLocus (I.comap f) :=
+  image_specComap_zeroLocus_eq_zeroLocus_comap _ f hf I
 
 theorem range_comap_of_surjective (hf : Surjective f) :
-    Set.range (comap f) = zeroLocus (ker f) := by
-  rw [← Set.image_univ]
-  convert image_comap_zeroLocus_eq_zeroLocus_comap _ _ hf _
-  rw [zeroLocus_bot]
+    Set.range (comap f) = zeroLocus (ker f) :=
+  range_specComap_of_surjective _ f hf
 
 theorem isClosed_range_comap_of_surjective (hf : Surjective f) :
     IsClosed (Set.range (comap f)) := by
@@ -391,7 +353,7 @@ def primeSpectrumProdHomeo :
   refine ((primeSpectrumProd R S).symm.toHomeomorphOfInducing ?_).symm
   refine (closedEmbedding_of_continuous_injective_closed ?_ (Equiv.injective _) ?_).toInducing
   · rw [continuous_sum_dom]
-    simp only [Function.comp, primeSpectrumProd_symm_inl, primeSpectrumProd_symm_inr]
+    simp only [Function.comp_def, primeSpectrumProd_symm_inl, primeSpectrumProd_symm_inr]
     exact ⟨(comap _).2, (comap _).2⟩
   · rw [isClosedMap_sum]
     constructor
@@ -622,7 +584,7 @@ lemma vanishingIdeal_irreducibleComponents :
   rw [irreducibleComponents_eq_maximals_closed, minimalPrimes_eq_minimals,
     image_antitone_setOf_maximal (fun s t hs _ ↦ (vanishingIdeal_anti_mono_iff hs.1).symm),
     ← funext (@Set.mem_setOf_eq _ · Ideal.IsPrime), ← vanishingIdeal_isClosed_isIrreducible]
-  simp only [Set.mem_image, Set.mem_setOf]
+  rfl
 
 lemma zeroLocus_minimalPrimes :
     zeroLocus ∘ (↑) '' minimalPrimes R =
@@ -693,3 +655,12 @@ theorem PrimeSpectrum.comap_residue (T : Type u) [CommRing T] [LocalRing T]
   exact Ideal.mk_ker
 
 end LocalRing
+
+section KrullDimension
+
+theorem PrimeSpectrum.topologicalKrullDim_eq_ringKrullDim [CommRing R] :
+    topologicalKrullDim (PrimeSpectrum R) = ringKrullDim R :=
+  Order.krullDim_orderDual.symm.trans <| Order.krullDim_eq_of_orderIso
+  (PrimeSpectrum.pointsEquivIrreducibleCloseds R).symm
+
+end KrullDimension
diff --git a/Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean b/Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean
index 1e92a49a98670..7522a16f45dbe 100644
--- a/Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean
+++ b/Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean
@@ -19,9 +19,8 @@ open TopologicalSpace
 
 variable (R : Type u) [CommRing R] [IsNoetherianRing R]
 
-instance : NoetherianSpace (PrimeSpectrum R) := by
-  apply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr
-  exact (closedsEmbedding R).dual.wellFounded IsWellFounded.wf
+instance : NoetherianSpace (PrimeSpectrum R) :=
+  ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr (closedsEmbedding R).dual.wellFoundedLT
 
 lemma _root_.minimalPrimes.finite_of_isNoetherianRing : (minimalPrimes R).Finite :=
   minimalPrimes.equivIrreducibleComponents R
diff --git a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
index 79096073211a1..133dcd36da1a9 100644
--- a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
+++ b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
@@ -264,11 +264,12 @@ The set `{a | aᵢᵐ/fⁱ ∈ q}`
 * is relevant, as proved in `carrier.relevant`.
 -/
 def carrier (f_deg : f ∈ 𝒜 m) (q : Spec.T A⁰_ f) : Set A :=
-  {a | ∀ i, (HomogeneousLocalization.mk ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
+  {a | ∀ i, (HomogeneousLocalization.mk ⟨m * i, ⟨proj 𝒜 i a ^ m, by rw [← smul_eq_mul]; mem_tac⟩,
               ⟨f ^ i, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.1}
 
 theorem mem_carrier_iff (q : Spec.T A⁰_ f) (a : A) :
-    a ∈ carrier f_deg q ↔ ∀ i, (HomogeneousLocalization.mk ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
+    a ∈ carrier f_deg q ↔ ∀ i, (HomogeneousLocalization.mk ⟨m * i, ⟨proj 𝒜 i a ^ m, by
+      rw [← smul_eq_mul]; mem_tac⟩,
       ⟨f ^ i, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.1 :=
   Iff.rfl
 
@@ -290,7 +291,7 @@ theorem mem_carrier_iff_of_mem (hm : 0 < m) (q : Spec.T A⁰_ f) (a : A) {n} (hn
     a ∈ carrier f_deg q ↔
       (HomogeneousLocalization.mk ⟨m * n, ⟨a ^ m, pow_mem_graded m hn⟩,
         ⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal := by
-  trans (HomogeneousLocalization.mk ⟨m * n, ⟨proj 𝒜 n a ^ m, by mem_tac⟩,
+  trans (HomogeneousLocalization.mk ⟨m * n, ⟨proj 𝒜 n a ^ m, by rw [← smul_eq_mul]; mem_tac⟩,
     ⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal
   · refine ⟨fun h ↦ h n, fun h i ↦ if hi : i = n then hi ▸ h else ?_⟩
     convert zero_mem q.asIdeal
@@ -337,12 +338,12 @@ theorem carrier.add_mem (q : Spec.T A⁰_ f) {a b : A} (ha : a ∈ carrier f_deg
             ⟨m * i, ⟨proj 𝒜 i a ^ j * proj 𝒜 i b ^ (m - j), ?_⟩,
               ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
           letI r : A⁰_ f := HomogeneousLocalization.mk
-            ⟨m * i, ⟨proj 𝒜 i b ^ m, by mem_tac⟩,
+            ⟨m * i, ⟨proj 𝒜 i b ^ m, by rw [← smul_eq_mul]; mem_tac⟩,
               ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
           l * r
         else
           letI l : A⁰_ f := HomogeneousLocalization.mk
-            ⟨m * i, ⟨proj 𝒜 i a ^ m, by mem_tac⟩,
+            ⟨m * i, ⟨proj 𝒜 i a ^ m, by rw [← smul_eq_mul]; mem_tac⟩,
               ⟨_, by rw [mul_comm]; mem_tac⟩, ⟨i, rfl⟩⟩
           letI r : A⁰_ f := HomogeneousLocalization.mk
             ⟨m * i, ⟨proj 𝒜 i a ^ (j - m) * proj 𝒜 i b ^ (m + m - j), ?_⟩,
@@ -529,7 +530,8 @@ lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
     toSpec 𝒜 f '' (Subtype.val ⁻¹' (pbo (decompose 𝒜 a i) : Set (ProjectiveSpectrum 𝒜))) =
     (PrimeSpectrum.basicOpen (R := A⁰_ f) <|
       HomogeneousLocalization.mk
-        ⟨m * i, ⟨decompose 𝒜 a i ^ m, SetLike.pow_mem_graded _ (Submodule.coe_mem _)⟩,
+        ⟨m * i, ⟨decompose 𝒜 a i ^ m,
+          (smul_eq_mul ℕ) ▸ SetLike.pow_mem_graded _ (Submodule.coe_mem _)⟩,
           ⟨f^i, by rw [mul_comm]; exact SetLike.pow_mem_graded _ f_deg⟩, ⟨i, rfl⟩⟩).1 :=
   Set.preimage_injective.mpr (toSpec_surjective 𝒜 f_deg hm) <|
     Set.preimage_image_eq _ (toSpec_injective 𝒜 f_deg hm) ▸ by
@@ -804,7 +806,7 @@ If `f ∈ A` is a homogeneous element of positive degree, then the projective sp
 -/
 def projIsoSpec (f) {m} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) :
     (Proj| pbo f) ≅ (Spec (A⁰_ f)) :=
-  @asIso (f := toSpec 𝒜 f) (isIso_toSpec 𝒜 f f_deg hm)
+  @asIso _ _ _ _ (f := toSpec 𝒜 f) (isIso_toSpec 𝒜 f f_deg hm)
 
 /--
 This is the scheme `Proj(A)` for any `ℕ`-graded ring `A`.
diff --git a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
index 3b17cd563ee74..634828aaafdef 100644
--- a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
+++ b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
@@ -126,7 +126,7 @@ theorem gc_set :
     @GaloisConnection (Set A) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
       (fun s => zeroLocus 𝒜 s) fun t => vanishingIdeal t := by
   have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi A _).gc
-  simpa [zeroLocus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
+  simpa [zeroLocus_span, Function.comp_def] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)
 
 theorem gc_homogeneousIdeal :
     @GaloisConnection (HomogeneousIdeal 𝒜) (Set (ProjectiveSpectrum 𝒜))ᵒᵈ _ _
diff --git a/Mathlib/AlgebraicGeometry/Properties.lean b/Mathlib/AlgebraicGeometry/Properties.lean
index d4cb9d574a2c2..70e632782f02a 100644
--- a/Mathlib/AlgebraicGeometry/Properties.lean
+++ b/Mathlib/AlgebraicGeometry/Properties.lean
@@ -4,10 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.AlgebraicGeometry.AffineScheme
-import Mathlib.RingTheory.Nilpotent.Lemmas
-import Mathlib.Topology.Sheaves.SheafCondition.Sites
-import Mathlib.Algebra.Category.Ring.Constructions
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.LocalProperties.Reduced
 
 /-!
 # Basic properties of schemes
diff --git a/Mathlib/AlgebraicGeometry/Pullbacks.lean b/Mathlib/AlgebraicGeometry/Pullbacks.lean
index 36ae5c042ab8c..1b43cf9c18bfd 100644
--- a/Mathlib/AlgebraicGeometry/Pullbacks.lean
+++ b/Mathlib/AlgebraicGeometry/Pullbacks.lean
@@ -453,8 +453,8 @@ instance isAffine_of_isAffine_isAffine_isAffine {X Y Z : Scheme}
     IsAffine (pullback f g) :=
   isAffine_of_isIso
     (pullback.map f g (Spec.map (Γ.map f.op)) (Spec.map (Γ.map g.op))
-        (ΓSpec.adjunction.unit.app X) (ΓSpec.adjunction.unit.app Y) (ΓSpec.adjunction.unit.app Z)
-        (ΓSpec.adjunction.unit.naturality f) (ΓSpec.adjunction.unit.naturality g) ≫
+        X.toSpecΓ Y.toSpecΓ Z.toSpecΓ
+        (Scheme.toSpecΓ_naturality f) (Scheme.toSpecΓ_naturality g) ≫
       (PreservesPullback.iso Scheme.Spec _ _).inv)
 
 /-- Given an open cover `{ Xᵢ }` of `X`, then `X ×[Z] Y` is covered by `Xᵢ ×[Z] Y`. -/
@@ -507,7 +507,7 @@ def openCoverOfBase' (𝒰 : OpenCover Z) (f : X ⟶ Z) (g : Y ⟶ Z) : OpenCove
     pasteVertIsPullback rfl (pullbackIsPullback g (𝒰.map i))
       (pullbackIsPullback (pullback.snd g (𝒰.map i)) (pullback.snd f (𝒰.map i)))
   refine
-    @openCoverOfIsIso
+    @openCoverOfIsIso _ _
       (f := (pullbackSymmetry _ _).hom ≫ (limit.isoLimitCone ⟨_, this⟩).inv ≫
         pullback.map _ _ _ _ (𝟙 _) (𝟙 _) (𝟙 _) ?_ ?_) inferInstance
   · simp [← pullback.condition]
@@ -583,7 +583,8 @@ the morphism `Spec (S ⊗[R] T) ⟶ Spec T` obtained by applying `Spec.map` to t
 -/
 @[reassoc (attr := simp)]
 lemma pullbackSpecIso_inv_snd :
-    (pullbackSpecIso R S T).inv ≫ pullback.snd _ _ = Spec.map (ofHom (toRingHom includeRight)) :=
+    (pullbackSpecIso R S T).inv ≫ pullback.snd _ _ =
+      Spec.map (ofHom (R := T) (S := S ⊗[R] T) (toRingHom includeRight)) :=
   limit.isoLimitCone_inv_π _ _
 /--
 The composition of the isomorphism `pullbackSepcIso R S T` (from the pullback of
diff --git a/Mathlib/AlgebraicGeometry/ResidueField.lean b/Mathlib/AlgebraicGeometry/ResidueField.lean
new file mode 100644
index 0000000000000..e127f7a5bccab
--- /dev/null
+++ b/Mathlib/AlgebraicGeometry/ResidueField.lean
@@ -0,0 +1,187 @@
+/-
+Copyright (c) 2024 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+-/
+import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField
+import Mathlib.AlgebraicGeometry.Stalk
+
+/-!
+
+# Residue fields of points
+
+## Main definitions
+
+The following are in the `AlgebraicGeometry.Scheme` namespace:
+
+- `AlgebraicGeometry.Scheme.residueField`: The residue field of the stalk at `x`.
+- `AlgebraicGeometry.Scheme.evaluation`: For open subsets `U` of `X` containing `x`,
+  the evaluation map from sections over `U` to the residue field at `x`.
+- `AlgebraicGeometry.Scheme.Hom.residueFieldMap`: A morphism of schemes induce a homomorphism of
+  residue fields.
+- `AlgebraicGeometry.Scheme.fromSpecResidueField`: The canonical map `Spec κ(x) ⟶ X`.
+
+-/
+
+universe u
+
+open CategoryTheory TopologicalSpace Opposite
+
+noncomputable section
+
+namespace AlgebraicGeometry.Scheme
+
+variable (X : Scheme.{u}) {U : X.Opens}
+
+/-- The residue field of `X` at a point `x` is the residue field of the stalk of `X`
+at `x`. -/
+def residueField (x : X) : CommRingCat :=
+  CommRingCat.of <| LocalRing.ResidueField (X.presheaf.stalk x)
+
+instance (x : X) : Field (X.residueField x) :=
+  inferInstanceAs <| Field (LocalRing.ResidueField (X.presheaf.stalk x))
+
+/-- The residue map from the stalk to the residue field. -/
+def residue (X : Scheme.{u}) (x) : X.presheaf.stalk x ⟶ X.residueField x :=
+  LocalRing.residue _
+
+lemma residue_surjective (X : Scheme.{u}) (x) : Function.Surjective (X.residue x) :=
+  Ideal.Quotient.mk_surjective
+
+instance (X : Scheme.{u}) (x) : Epi (X.residue x) :=
+  ConcreteCategory.epi_of_surjective _ (X.residue_surjective x)
+
+/--
+If `U` is an open of `X` containing `x`, we have a canonical ring map from the sections
+over `U` to the residue field of `x`.
+
+If we interpret sections over `U` as functions of `X` defined on `U`, then this ring map
+corresponds to evaluation at `x`.
+-/
+def evaluation (U : X.Opens) (x : X) (hx : x ∈ U) : Γ(X, U) ⟶ X.residueField x :=
+  X.presheaf.germ ⟨x, hx⟩ ≫ X.residue _
+
+@[reassoc]
+lemma germ_residue (x : U) : X.presheaf.germ x ≫ X.residue x.1 = X.evaluation U x x.2 := rfl
+
+/-- The global evaluation map from `Γ(X, ⊤)` to the residue field at `x`. -/
+abbrev Γevaluation (x : X) : Γ(X, ⊤) ⟶ X.residueField x :=
+  X.evaluation ⊤ x trivial
+
+@[simp]
+lemma evaluation_eq_zero_iff_not_mem_basicOpen (x : X) (hx : x ∈ U) (f : Γ(X, U)) :
+    X.evaluation U x hx f = 0 ↔ x ∉ X.basicOpen f :=
+  X.toLocallyRingedSpace.evaluation_eq_zero_iff_not_mem_basicOpen ⟨x, hx⟩ f
+
+lemma evaluation_ne_zero_iff_mem_basicOpen (x : X) (hx : x ∈ U) (f : Γ(X, U)) :
+    X.evaluation U x hx f ≠ 0 ↔ x ∈ X.basicOpen f := by
+  simp
+
+variable {X Y : Scheme.{u}} (f : X ⟶ Y)
+
+
+-- TODO: This instance is found before #6045.
+-- We need this strange instance for `residueFieldMap`, the type of `F` must be fixed
+-- like this. The instance `IsLocalRingHom (f.stalkMap x)` already exists, but does not work for
+-- `residueFieldMap`.
+instance (x): IsLocalRingHom (F := Y.presheaf.stalk (f.val.base x) →+* X.presheaf.stalk x)
+    (f.stalkMap x) :=
+  f.2 x
+
+/-- If `X ⟶ Y` is a morphism of locally ringed spaces and `x` a point of `X`, we obtain
+a morphism of residue fields in the other direction. -/
+def Hom.residueFieldMap (f : X.Hom Y) (x : X) :
+    Y.residueField (f.val.base x) ⟶ X.residueField x :=
+  LocalRing.ResidueField.map (f.stalkMap x)
+
+@[reassoc]
+lemma residue_residueFieldMap (x : X) :
+    Y.residue (f.val.base x) ≫ f.residueFieldMap x = f.stalkMap x ≫ X.residue x := by
+  simp [Hom.residueFieldMap]
+  rfl
+
+@[simp]
+lemma residueFieldMap_id (x : X) :
+    Hom.residueFieldMap (𝟙 X) x = 𝟙 (X.residueField x) :=
+  LocallyRingedSpace.residueFieldMap_id _
+
+@[simp]
+lemma residueFieldMap_comp {Z : Scheme.{u}} (g : Y ⟶ Z) (x : X) :
+    (f ≫ g).residueFieldMap x = g.residueFieldMap (f.val.base x) ≫ f.residueFieldMap x :=
+  LocallyRingedSpace.residueFieldMap_comp _ _ _
+
+@[reassoc]
+lemma evaluation_naturality {V : Opens Y} (x : X) (hx : f.val.base x ∈ V) :
+    Y.evaluation V (f.val.base x) hx ≫ f.residueFieldMap x =
+      f.app V ≫ X.evaluation (f ⁻¹ᵁ V) x hx :=
+  LocallyRingedSpace.evaluation_naturality f ⟨x, hx⟩
+
+lemma evaluation_naturality_apply {V : Opens Y} (x : X) (hx : f.val.base x ∈ V) (s) :
+    f.residueFieldMap x (Y.evaluation V (f.val.base x) hx s) =
+      X.evaluation (f ⁻¹ᵁ V) x hx (f.app V s) :=
+  LocallyRingedSpace.evaluation_naturality_apply f ⟨x, hx⟩ s
+
+instance [IsOpenImmersion f] (x) : IsIso (f.residueFieldMap x) :=
+  (LocalRing.ResidueField.mapEquiv
+    (asIso (f.stalkMap x)).commRingCatIsoToRingEquiv).toCommRingCatIso.isIso_hom
+
+section congr
+
+-- replace this def if hard to work with
+/-- The isomorphism between residue fields of equal points. -/
+def residueFieldCongr {x y : X} (h : x = y) :
+    X.residueField x ≅ X.residueField y :=
+  eqToIso (by subst h; rfl)
+
+@[simp]
+lemma residueFieldCongr_refl {x : X} :
+    X.residueFieldCongr (refl x) = Iso.refl _ := rfl
+
+@[simp]
+lemma residueFieldCongr_symm {x y : X} (e : x = y) :
+    (X.residueFieldCongr e).symm = X.residueFieldCongr e.symm := rfl
+
+@[simp]
+lemma residueFieldCongr_inv {x y : X} (e : x = y) :
+    (X.residueFieldCongr e).inv = (X.residueFieldCongr e.symm).hom := rfl
+
+@[simp]
+lemma residueFieldCongr_trans {x y z : X} (e : x = y) (e' : y = z) :
+    X.residueFieldCongr e ≪≫ X.residueFieldCongr e' = X.residueFieldCongr (e.trans e') := by
+  subst e e'
+  rfl
+
+@[reassoc (attr := simp)]
+lemma residueFieldCongr_trans_hom (X : Scheme) {x y z : X} (e : x = y) (e' : y = z) :
+    (X.residueFieldCongr e).hom ≫ (X.residueFieldCongr e').hom =
+      (X.residueFieldCongr (e.trans e')).hom := by
+  subst e e'
+  rfl
+
+@[reassoc]
+lemma residue_residueFieldCongr (X : Scheme) {x y : X} (h : x = y) :
+    X.residue x ≫ (X.residueFieldCongr h).hom =
+      (X.presheaf.stalkCongr (.of_eq h)).hom ≫ X.residue y := by
+  subst h
+  simp
+
+end congr
+
+section fromResidueField
+
+/-- The canonical map `Spec κ(x) ⟶ X`. -/
+def fromSpecResidueField (X : Scheme) (x : X) :
+    Spec (X.residueField x) ⟶ X :=
+  Spec.map (CommRingCat.ofHom (X.residue x)) ≫ X.fromSpecStalk x
+
+@[reassoc (attr := simp)]
+lemma residueFieldCongr_fromSpecResidueField {x y : X} (h : x = y) :
+    Spec.map (X.residueFieldCongr h).hom ≫ X.fromSpecResidueField _ =
+      X.fromSpecResidueField _ := by
+  subst h; simp
+
+end fromResidueField
+
+end Scheme
+
+end AlgebraicGeometry
diff --git a/Mathlib/AlgebraicGeometry/Restrict.lean b/Mathlib/AlgebraicGeometry/Restrict.lean
index aa6bf66cace12..489e0dc07ca6a 100644
--- a/Mathlib/AlgebraicGeometry/Restrict.lean
+++ b/Mathlib/AlgebraicGeometry/Restrict.lean
@@ -69,7 +69,7 @@ lemma ι_appLE (V W e) :
     U.ι.appLE V W e =
       X.presheaf.map (homOfLE (x := U.ι ''ᵁ W) (Set.image_subset_iff.mpr ‹_›)).op := by
   simp only [Hom.appLE, ι_app, Functor.op_obj, Opens.carrier_eq_coe, toScheme_presheaf_map,
-    Quiver.Hom.unop_op, Hom.opensFunctor_map_homOfLE, Opens.coe_inclusion, ← Functor.map_comp]
+    Quiver.Hom.unop_op, Hom.opensFunctor_map_homOfLE, Opens.coe_inclusion', ← Functor.map_comp]
   rfl
 
 @[simp]
@@ -87,9 +87,13 @@ lemma range_ι : Set.range U.ι.val.base = U :=
 lemma ι_image_top : U.ι ''ᵁ ⊤ = U :=
   U.openEmbedding_obj_top
 
+lemma ι_image_le (W : U.toScheme.Opens) : U.ι ''ᵁ W ≤ U := by
+  simp_rw [← U.ι_image_top]
+  exact U.ι.image_le_image_of_le le_top
+
 @[simp]
 lemma ι_preimage_self : U.ι ⁻¹ᵁ U = ⊤ :=
-  Opens.inclusion_map_eq_top _
+  Opens.inclusion'_map_eq_top _
 
 instance ι_appLE_isIso :
     IsIso (U.ι.appLE U ⊤ U.ι_preimage_self.ge) := by
@@ -216,8 +220,10 @@ def Scheme.restrictFunctor : X.Opens ⥤ Over X where
 @[simp] lemma Scheme.restrictFunctor_obj_hom (U : X.Opens) :
   (X.restrictFunctor.obj U).hom = U.ι := rfl
 
-@[simp] lemma Scheme.restrictFunctor_map_left {U V : X.Opens} (i : U ⟶ V) :
-  (X.restrictFunctor.map i).left = IsOpenImmersion.lift (V.ι) U.ι (by simpa using i.le) := rfl
+/-- This is not a `simp` lemma, as `(X.restricFunctor.map i).left` is used as the `simp`
+normal-form for the induced morphism `U.toScheme ⟶ V.toScheme`. -/
+lemma Scheme.restrictFunctor_map_left {U V : X.Opens} (i : U ⟶ V) :
+    (X.restrictFunctor.map i).left = IsOpenImmersion.lift (V.ι) U.ι (by simpa using i.le) := rfl
 
 -- Porting note: the `by ...` used to be automatically done by unification magic
 @[reassoc]
@@ -234,7 +240,7 @@ theorem Scheme.restrictFunctor_map_base {U V : X.Opens} (i : U ⟶ V) :
 theorem Scheme.restrictFunctor_map_app_aux {U V : X.Opens} (i : U ⟶ V) (W : Opens V) :
     U.ι ''ᵁ ((X.restrictFunctor.map i).1 ⁻¹ᵁ W) ≤ V.ι ''ᵁ W := by
   simp only [← SetLike.coe_subset_coe, IsOpenMap.functor_obj_coe, Set.image_subset_iff,
-    Scheme.restrictFunctor_map_base, Opens.map_coe, Opens.inclusion_apply]
+    Scheme.restrictFunctor_map_base, Opens.map_coe, Opens.inclusion'_apply]
   rintro _ h
   exact ⟨_, h, rfl⟩
 
@@ -258,7 +264,7 @@ def Scheme.restrictFunctorΓ : X.restrictFunctor.op ⋙ (Over.forget X).op ⋙ S
     (fun U => X.presheaf.mapIso ((eqToIso (unop U).openEmbedding_obj_top).symm.op : _))
     (by
       intro U V i
-      dsimp [-Scheme.restrictFunctor_map_left]
+      dsimp
       rw [X.restrictFunctor_map_app, ← Functor.map_comp, ← Functor.map_comp]
       congr 1)
 
@@ -466,7 +472,7 @@ def morphismRestrictRestrict {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V :
   refine Arrow.isoMk' _ _ (Scheme.restrictRestrict _ _ _ ≪≫ Scheme.restrictIsoOfEq _ ?_)
     (Scheme.restrictRestrict _ _ _) ?_
   · ext x
-    simp only [IsOpenMap.functor_obj_coe, Opens.coe_inclusion,
+    simp only [IsOpenMap.functor_obj_coe, Opens.coe_inclusion',
       Opens.map_coe, Set.mem_image, Set.mem_preimage, SetLike.mem_coe, morphismRestrict_val_base]
     constructor
     · rintro ⟨⟨a, h₁⟩, h₂, rfl⟩
@@ -480,7 +486,7 @@ def morphismRestrictRestrict {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (V :
       Scheme.restrictRestrict_hom_restrict,
       morphismRestrict_ι_assoc, morphismRestrict_ι]
 
-/-- Restricting a morphism twice onto a basic open set is isomorphic to one restriction.  -/
+/-- Restricting a morphism twice onto a basic open set is isomorphic to one restriction. -/
 def morphismRestrictRestrictBasicOpen {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) (r : Γ(Y, U)) :
     Arrow.mk (f ∣_ U ∣_
           U.toScheme.basicOpen (Y.presheaf.map (eqToHom U.openEmbedding_obj_top).op r)) ≅
@@ -488,7 +494,7 @@ def morphismRestrictRestrictBasicOpen {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Op
   refine morphismRestrictRestrict _ _ _ ≪≫ morphismRestrictEq _ ?_
   have e := Scheme.preimage_basicOpen U.ι r
   rw [Scheme.Opens.ι_app] at e
-  rw [← U.toScheme.basicOpen_res_eq _ (eqToHom U.inclusion_map_eq_top).op]
+  rw [← U.toScheme.basicOpen_res_eq _ (eqToHom U.inclusion'_map_eq_top).op]
   erw [← comp_apply]
   erw [← Y.presheaf.map_comp]
   rw [eqToHom_op, eqToHom_op, eqToHom_map, eqToHom_trans]
@@ -515,6 +521,80 @@ instance {X Y : Scheme.{u}} (f : X ⟶ Y) (U : Y.Opens) [IsOpenImmersion f] :
   delta morphismRestrict
   exact PresheafedSpace.IsOpenImmersion.comp _ _
 
+variable {X Y : Scheme.{u}}
+
+namespace Scheme.Hom
+
+/-- The restriction of a morphism `f : X ⟶ Y` to open sets on the source and target. -/
+def resLE (f : Hom X Y) (U : Y.Opens) (V : X.Opens) (e : V ≤ f ⁻¹ᵁ U) : V.toScheme ⟶ U.toScheme :=
+  (X.restrictFunctor.map (homOfLE e)).left ≫ f ∣_ U
+
+variable (f : X ⟶ Y) {U U' : Y.Opens} {V V' : X.Opens} (e : V ≤ f ⁻¹ᵁ U)
+
+lemma resLE_eq_morphismRestrict : f.resLE U (f ⁻¹ᵁ U) le_rfl = f ∣_ U := by
+  simp [Scheme.Hom.resLE]
+
+lemma resLE_id (i : V ⟶ V') : resLE (𝟙 X) V' V i.le = (X.restrictFunctor.map i).left := by
+  simp only [resLE, id_val_base, morphismRestrict_id, Category.comp_id]
+  rfl
+
+@[reassoc (attr := simp)]
+lemma resLE_comp_ι : f.resLE U V e ≫ U.ι = V.ι ≫ f := by
+  simp [resLE, restrictFunctor_map_ofRestrict_assoc]
+
+@[reassoc]
+lemma resLE_comp_resLE {Z : Scheme.{u}} (g : Y ⟶ Z) {W : Z.Opens} (e') :
+    f.resLE U V e ≫ g.resLE W U e' = (f ≫ g).resLE W V
+      (e.trans ((Opens.map f.val.base).map (homOfLE e')).le) := by
+  simp [← cancel_mono W.ι]
+
+@[reassoc (attr := simp)]
+lemma map_resLE (i : V' ⟶ V) :
+    (X.restrictFunctor.map i).left ≫ f.resLE U V e = f.resLE U V' (i.le.trans e) := by
+  simp_rw [← resLE_id, resLE_comp_resLE, Category.id_comp]
+
+@[reassoc (attr := simp)]
+lemma resLE_map (i : U ⟶ U') :
+    f.resLE U V e ≫ (Y.restrictFunctor.map i).left =
+      f.resLE U' V (e.trans ((Opens.map f.1.base).map i).le) := by
+  simp_rw [← resLE_id, resLE_comp_resLE, Category.comp_id]
+
+lemma resLE_congr (e₁ : U = U') (e₂ : V = V') (P : MorphismProperty Scheme.{u}) :
+    P (f.resLE U V e) ↔ P (f.resLE U' V' (e₁ ▸ e₂ ▸ e)) := by
+  subst e₁; subst e₂; rfl
+
+lemma resLE_preimage (f : X ⟶ Y) {U : Y.Opens} {V : X.Opens} (e : V ≤ f ⁻¹ᵁ U)
+    (O : U.toScheme.Opens) :
+    f.resLE U V e ⁻¹ᵁ O = V.ι ⁻¹ᵁ (f ⁻¹ᵁ U.ι ''ᵁ O) := by
+  rw [← preimage_comp, ← resLE_comp_ι f e, preimage_comp, preimage_image_eq]
+
+lemma le_preimage_resLE_iff {U : Y.Opens} {V : X.Opens} (e : V ≤ f ⁻¹ᵁ U)
+    (O : U.toScheme.Opens) (W : V.toScheme.Opens) :
+    W ≤ (f.resLE U V e) ⁻¹ᵁ O ↔ V.ι ''ᵁ W ≤ f ⁻¹ᵁ U.ι ''ᵁ O := by
+  simp [resLE_preimage, ← image_le_image_iff V.ι, image_preimage_eq_opensRange_inter, V.ι_image_le]
+
+lemma resLE_appLE {U : Y.Opens} {V : X.Opens} (e : V ≤ f ⁻¹ᵁ U)
+    (O : U.toScheme.Opens) (W : V.toScheme.Opens) (e' : W ≤ resLE f U V e ⁻¹ᵁ O) :
+    (f.resLE U V e).appLE O W e' =
+      f.appLE (U.ι ''ᵁ O) (V.ι ''ᵁ W) ((le_preimage_resLE_iff f e O W).mp e') := by
+  simp only [Scheme.Hom.appLE, Scheme.Hom.resLE, Scheme.restrictFunctor_map_left, Opens.map_coe,
+    id_eq, Scheme.comp_app, morphismRestrict_app', Category.assoc, IsOpenImmersion.lift_app,
+    Scheme.Opens.ι_appIso, Scheme.Opens.ι_app, Scheme.Opens.toScheme_presheaf_map, Category.assoc]
+  rw [← X.presheaf.map_comp, ← X.presheaf.map_comp]
+  erw [Category.id_comp]
+  rw [← X.presheaf.map_comp]
+  rfl
+
+end Scheme.Hom
+
+/-- `f.resLE U V` induces `f.appLE U V` on global sections. -/
+noncomputable def arrowResLEAppIso (f : X ⟶ Y) (U : Y.Opens) (V : X.Opens) (e : V ≤ f ⁻¹ᵁ U) :
+    Arrow.mk ((f.resLE U V e).app ⊤) ≅ Arrow.mk (f.appLE U V e) :=
+  Arrow.isoMk U.topIso V.topIso <| by
+  simp only [Opens.map_top, Arrow.mk_left, Arrow.mk_right, Functor.id_obj, Scheme.Opens.topIso_hom,
+    eqToHom_op, Arrow.mk_hom, Scheme.Hom.map_appLE]
+  rw [← Scheme.Hom.appLE_eq_app, Scheme.Hom.resLE_appLE, Scheme.Hom.appLE_map]
+
 end MorphismRestrict
 
 /-- The restriction of an open cover to an open subset. -/
diff --git a/Mathlib/AlgebraicGeometry/Scheme.lean b/Mathlib/AlgebraicGeometry/Scheme.lean
index fab17a4f32795..97a64d277db0a 100644
--- a/Mathlib/AlgebraicGeometry/Scheme.lean
+++ b/Mathlib/AlgebraicGeometry/Scheme.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.AlgebraicGeometry.Spec
 import Mathlib.Algebra.Category.Ring.Constructions
@@ -165,6 +165,10 @@ lemma preimage_iSup {ι} (U : ι → Opens Y) : f ⁻¹ᵁ iSup U = ⨆ i, f ⁻
 lemma preimage_iSup_eq_top {ι} {U : ι → Opens Y} (hU : iSup U = ⊤) :
     ⨆ i, f ⁻¹ᵁ U i = ⊤ := f.preimage_iSup U ▸ hU ▸ rfl
 
+lemma preimage_le_preimage_of_le {U U' : Y.Opens} (hUU' : U ≤ U') :
+    f ⁻¹ᵁ U ≤ f ⁻¹ᵁ U' :=
+  fun _ ha ↦ hUU' ha
+
 end Hom
 
 @[simp]
diff --git a/Mathlib/AlgebraicGeometry/Spec.lean b/Mathlib/AlgebraicGeometry/Spec.lean
index ccb5927d0074e..78b6f5fd8c9f9 100644
--- a/Mathlib/AlgebraicGeometry/Spec.lean
+++ b/Mathlib/AlgebraicGeometry/Spec.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Justus Springer
+Authors: Kim Morrison, Justus Springer
 -/
 import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
 import Mathlib.AlgebraicGeometry.StructureSheaf
@@ -236,6 +236,8 @@ def Spec.locallyRingedSpaceMap {R S : CommRingCat.{u}} (f : R ⟶ S) :
       #adaptation_note /-- nightly-2024-04-01
       It's this `erw` that is blowing up. The implicit arguments differ significantly. -/
       erw [← localRingHom_comp_stalkIso_apply] at ha
+      -- TODO: this instance was found automatically before #6045
+      haveI : IsLocalRingHom (stalkIso (↑S) p).inv := isLocalRingHom_of_isIso _
       replace ha := (isUnit_map_iff (stalkIso S p).inv _).mp ha
       -- Porting note: `f` had to be made explicit
       replace ha := IsLocalRingHom.map_nonunit
@@ -269,7 +271,7 @@ section SpecΓ
 
 open AlgebraicGeometry.LocallyRingedSpace
 
-/-- The counit morphism `R ⟶ Γ(Spec R)` given by `AlgebraicGeometry.StructureSheaf.toOpen`.  -/
+/-- The counit morphism `R ⟶ Γ(Spec R)` given by `AlgebraicGeometry.StructureSheaf.toOpen`. -/
 @[simps!]
 def toSpecΓ (R : CommRingCat.{u}) : R ⟶ Γ.obj (op (Spec.toLocallyRingedSpace.obj (op R))) :=
   StructureSheaf.toOpen R ⊤
diff --git a/Mathlib/AlgebraicGeometry/StructureSheaf.lean b/Mathlib/AlgebraicGeometry/StructureSheaf.lean
index a145948515d7c..8cfb094cb6c53 100644
--- a/Mathlib/AlgebraicGeometry/StructureSheaf.lean
+++ b/Mathlib/AlgebraicGeometry/StructureSheaf.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison
+Authors: Johan Commelin, Kim Morrison
 -/
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
 import Mathlib.Algebra.Category.Ring.Colimits
@@ -543,9 +543,15 @@ def stalkIso (x : PrimeSpectrum.Top R) :
 instance (x : PrimeSpectrum R) : IsIso (stalkToFiberRingHom R x) :=
   (stalkIso R x).isIso_hom
 
+instance (x : PrimeSpectrum R) : IsLocalRingHom (stalkToFiberRingHom R x) :=
+  isLocalRingHom_of_isIso _
+
 instance (x : PrimeSpectrum R) : IsIso (localizationToStalk R x) :=
   (stalkIso R x).isIso_inv
 
+instance (x : PrimeSpectrum R) : IsLocalRingHom (localizationToStalk R x) :=
+  isLocalRingHom_of_isIso _
+
 @[simp, reassoc]
 theorem stalkToFiberRingHom_localizationToStalk (x : PrimeSpectrum.Top R) :
     stalkToFiberRingHom R x ≫ localizationToStalk R x = 𝟙 _ :=
@@ -799,9 +805,9 @@ theorem toBasicOpen_surjective (f : R) : Function.Surjective (toBasicOpen R f) :
     apply PrimeSpectrum.vanishingIdeal_anti_mono ht_cover
     exact PrimeSpectrum.subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)
   replace hn := Ideal.mul_mem_right f _ hn
-  erw [← pow_succ, Finsupp.mem_span_image_iff_total] at hn
+  erw [← pow_succ, Finsupp.mem_span_image_iff_linearCombination] at hn
   rcases hn with ⟨b, b_supp, hb⟩
-  rw [Finsupp.total_apply_of_mem_supported R b_supp] at hb
+  rw [Finsupp.linearCombination_apply_of_mem_supported R b_supp] at hb
   dsimp at hb
   -- Finally, we have all the ingredients.
   -- We claim that our preimage is given by `(∑ (i : ι) ∈ t, b i * a i) / f ^ (n+1)`
diff --git a/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean b/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
index 4a3ffcf186da8..de7b73ca2ffc3 100644
--- a/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
+++ b/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
@@ -80,7 +80,7 @@ theorem d_squared (n : ℕ) : objD X (n + 1) ≫ objD X n = 0 := by
   apply Finset.sum_bij φ
   · -- φ(S) is contained in Sᶜ
     intro ij hij
-    simp only [S, Finset.mem_univ, Finset.compl_filter, Finset.mem_filter, true_and_iff,
+    simp only [S, Finset.mem_univ, Finset.compl_filter, Finset.mem_filter, true_and,
       Fin.val_succ, Fin.coe_castLT] at hij ⊢
     linarith
   · -- φ : S → Sᶜ is injective
diff --git a/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean b/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean
index dd949cbd7cb71..dde1b3dea55c9 100644
--- a/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean
+++ b/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean
@@ -49,7 +49,7 @@ theorem HigherFacesVanish.comp_σ {Y : C} {X : SimplicialObject C} {n b q : ℕ}
     linarith
   · intro hj'
     simp only [hnbq, add_comm b, add_assoc, hj', Fin.val_zero, zero_add, add_le_iff_nonpos_right,
-      nonpos_iff_eq_zero, add_eq_zero, false_and] at hj
+      nonpos_iff_eq_zero, add_eq_zero, false_and, reduceCtorEq] at hj
 
 theorem σ_comp_P_eq_zero (X : SimplicialObject C) {n q : ℕ} (i : Fin (n + 1)) (hi : n + 1 ≤ i + q) :
     X.σ i ≫ (P q).f (n + 1) = 0 := by
@@ -95,7 +95,7 @@ theorem σ_comp_P_eq_zero (X : SimplicialObject C) {n q : ℕ} (i : Fin (n + 1))
           comp_id, v.comp_Hσ_eq hi, assoc, SimplicialObject.δ_comp_σ_succ_assoc, Fin.eta,
           decomposition_Q n q, sum_comp, sum_comp, Finset.sum_eq_zero, add_zero, add_neg_eq_zero]
         intro j hj
-        simp only [true_and_iff, Finset.mem_univ, Finset.mem_filter] at hj
+        simp only [Finset.mem_univ, Finset.mem_filter] at hj
         obtain ⟨k, hk⟩ := Nat.le.dest (Nat.lt_succ_iff.mp (Fin.is_lt j))
         rw [add_comm] at hk
         have hi' : i = Fin.castSucc ⟨i, by omega⟩ := by
diff --git a/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean b/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean
index 59cf4f6cf84aa..72a3b619b766a 100644
--- a/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean
+++ b/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean
@@ -62,7 +62,7 @@ we define the functor in the other direction
 extension of a functor `Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C` which is
 defined similarly as in [*Simplicial Homotopy Theory* by Goerss-Jardine][goerss-jardine-2009].
 In `Degeneracies.lean`, we show that `PInfty` vanishes on the image of degeneracy
-operators, which is one of the key properties that makes it possible to contruct
+operators, which is one of the key properties that makes it possible to construct
 the isomorphism `N₂Γ₂ : Γ₂ ⋙ N₂ ≅ 𝟭 (Karoubi (ChainComplex C ℕ))`.
 
 The rest of the proof follows the strategy in the [original paper by Dold][dold1958]. We show
diff --git a/Mathlib/AlgebraicTopology/DoldKan/Faces.lean b/Mathlib/AlgebraicTopology/DoldKan/Faces.lean
index a232a8b12caaf..845dd32f779d1 100644
--- a/Mathlib/AlgebraicTopology/DoldKan/Faces.lean
+++ b/Mathlib/AlgebraicTopology/DoldKan/Faces.lean
@@ -155,7 +155,7 @@ theorem comp_Hσ_eq_zero {Y : C} {n q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : Higher
         omega
       · intro h
         simp only [Fin.pred, Fin.subNat, Fin.ext_iff, Nat.succ_add_sub_one,
-          Fin.val_zero, add_eq_zero, false_and] at h
+          Fin.val_zero, add_eq_zero, false_and, reduceCtorEq] at h
       · simp only [Fin.pred, Fin.subNat, Nat.pred_eq_sub_one, Nat.succ_add_sub_one]
         omega
 
diff --git a/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean b/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
index 72511e140c0f6..67d99a2628ff5 100644
--- a/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
+++ b/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean
@@ -148,7 +148,7 @@ theorem mapMono_comp (i' : Δ'' ⟶ Δ') (i : Δ' ⟶ Δ) [Mono i'] [Mono i] :
   have eq : Δ.len = Δ''.len + (k + k' + 2) := by omega
   rw [mapMono_eq_zero K (i' ≫ i) _ _]; rotate_left
   · by_contra h
-    simp only [self_eq_add_right, h, add_eq_zero, and_false] at eq
+    simp only [self_eq_add_right, h, add_eq_zero, and_false, reduceCtorEq] at eq
   · by_contra h
     simp only [h.1, add_right_inj] at eq
     omega
diff --git a/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean b/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean
index d458777633750..43aa24e304f15 100644
--- a/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean
+++ b/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean
@@ -124,10 +124,13 @@ def N₁_iso_normalizedMooreComplex_comp_toKaroubi : N₁ ≅ normalizedMooreCom
           comm := by erw [inclusionOfMooreComplexMap_comp_PInfty, id_comp] }
       naturality := fun X Y f => by
         ext
-        simp only [Functor.comp_map, normalizedMooreComplex_map, Karoubi.comp_f, toKaroubi_map_f,
-          HomologicalComplex.comp_f, NormalizedMooreComplex.map_f,
-          inclusionOfMooreComplexMap_f, factorThru_arrow, N₁_map_f,
-          inclusionOfMooreComplexMap_comp_PInfty_assoc, AlternatingFaceMapComplex.map_f] }
+        simp only [Functor.comp_obj, normalizedMooreComplex_obj, toKaroubi_obj_X,
+          NormalizedMooreComplex.obj_X, N₁_obj_X, AlternatingFaceMapComplex.obj_X, Functor.comp_map,
+          normalizedMooreComplex_map, Karoubi.comp_f, toKaroubi_map_f, HomologicalComplex.comp_f,
+          NormalizedMooreComplex.map_f, inclusionOfMooreComplexMap_f, NormalizedMooreComplex.objX,
+          factorThru_arrow, N₁_map_f, inclusionOfMooreComplexMap_comp_PInfty_assoc,
+          AlternatingFaceMapComplex.map_f]
+         }
   hom_inv_id := by
     ext X : 3
     simp only [PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap,
diff --git a/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean b/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean
index 4c1705a39ed6d..4610e95420a71 100644
--- a/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean
+++ b/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean
@@ -153,7 +153,7 @@ theorem shiftFun_succ {n : ℕ} {X : Type*} [Zero X] (f : Fin n → X) (i : Fin
   dsimp [shiftFun]
   split_ifs with h
   · exfalso
-    simp only [Fin.ext_iff, Fin.val_succ, Fin.val_zero, add_eq_zero, and_false] at h
+    simp only [Fin.ext_iff, Fin.val_succ, Fin.val_zero, add_eq_zero, and_false, reduceCtorEq] at h
   · simp only [Fin.pred_succ]
 
 /-- The shift of a morphism `f : [n] → Δ` in `SimplexCategory` corresponds to
@@ -282,7 +282,7 @@ theorem ExtraDegeneracy.s_comp_π_succ (n : ℕ) (i : Fin (n + 1)) :
   dsimp [ExtraDegeneracy.s]
   simp only [WidePullback.lift_π]
   split_ifs with h
-  · simp only [Fin.ext_iff, Fin.val_succ, Fin.val_zero, add_eq_zero, and_false] at h
+  · simp only [Fin.ext_iff, Fin.val_succ, Fin.val_zero, add_eq_zero, and_false, reduceCtorEq] at h
   · simp only [Fin.pred_succ]
 
 -- Porting note (#11119): @[simp] removed as the linter complains the LHS is not in normal form
diff --git a/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean b/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
index 84980eaca3178..2a8bb0c418f7e 100644
--- a/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
+++ b/Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean
@@ -59,7 +59,7 @@ theorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by
         · norm_num
       · unit_interval
     · rw [mul_assoc]
-      apply mul_le_one
+      apply mul_le_one₀
       · unit_interval
       · apply mul_nonneg
         · norm_num
@@ -69,7 +69,7 @@ theorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by
     · apply mul_nonneg
       · unit_interval
       linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]
-    · apply mul_le_one
+    · apply mul_le_one₀
       · unit_interval
       · linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]
       · linarith [unitInterval.nonneg x.2, unitInterval.le_one x.2]
diff --git a/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean b/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean
index c47f77c73c988..d00c54679772a 100644
--- a/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean
+++ b/Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean
@@ -96,7 +96,7 @@ private theorem end_path : f x₁ = g x₃ := by convert hfg 1 <;> simp only [Pa
 
 theorem eq_path_of_eq_image :
     (πₘ f).map ⟦p⟧ = hcast (start_path hfg) ≫ (πₘ g).map ⟦q⟧ ≫ hcast (end_path hfg).symm := by
-  rw [Functor.conj_eqToHom_iff_heq
+  rw [conj_eqToHom_iff_heq
     ((πₘ f).map ⟦p⟧) ((πₘ g).map ⟦q⟧)
     (FundamentalGroupoid.ext <| start_path hfg)
     (FundamentalGroupoid.ext <| end_path hfg)]
@@ -178,7 +178,7 @@ theorem evalAt_eq (x : X) : ⟦H.evalAt x⟧ = hcast (H.apply_zero x).symm ≫
     (πₘ H.uliftMap).map (prodToProdTopI uhpath01 (𝟙 (fromTop x))) ≫
       hcast (H.apply_one x).symm.symm := by
   dsimp only [prodToProdTopI, uhpath01, hcast]
-  refine (@Functor.conj_eqToHom_iff_heq (πₓ Y) _ _ _ _ _ _ _ _
+  refine (@conj_eqToHom_iff_heq (πₓ Y) _ _ _ _ _ _ _ _
     (FundamentalGroupoid.ext <| H.apply_one x).symm).mpr ?_
   simp only [id_eq_path_refl, prodToProdTop_map, Path.Homotopic.prod_lift, map_eq, ←
     Path.Homotopic.map_lift]
diff --git a/Mathlib/AlgebraicTopology/MooreComplex.lean b/Mathlib/AlgebraicTopology/MooreComplex.lean
index 62e4f9ca124ea..5ab0ded88a8b6 100644
--- a/Mathlib/AlgebraicTopology/MooreComplex.lean
+++ b/Mathlib/AlgebraicTopology/MooreComplex.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Homology.HomologicalComplex
 import Mathlib.AlgebraicTopology.SimplicialObject
diff --git a/Mathlib/AlgebraicTopology/SimplexCategory.lean b/Mathlib/AlgebraicTopology/SimplexCategory.lean
index 3003a0457ca57..e72f2e8cd7bec 100644
--- a/Mathlib/AlgebraicTopology/SimplexCategory.lean
+++ b/Mathlib/AlgebraicTopology/SimplexCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison, Adam Topaz
+Authors: Johan Commelin, Kim Morrison, Adam Topaz
 -/
 import Mathlib.Tactic.Linarith
 import Mathlib.CategoryTheory.Skeletal
@@ -173,9 +173,11 @@ without identifying `n` with `[n].len`.
 def mkHom {n m : ℕ} (f : Fin (n + 1) →o Fin (m + 1)) : ([n] : SimplexCategory) ⟶ [m] :=
   SimplexCategory.Hom.mk f
 
+instance (Δ : SimplexCategory) : Subsingleton (Δ ⟶ [0]) where
+  allEq f g := by ext : 3; apply Subsingleton.elim (α := Fin 1)
+
 theorem hom_zero_zero (f : ([0] : SimplexCategory) ⟶ [0]) : f = 𝟙 _ := by
-  ext : 3
-  apply @Subsingleton.elim (Fin 1)
+  apply Subsingleton.elim
 
 end
 
@@ -520,7 +522,7 @@ theorem epi_iff_surjective {n m : SimplexCategory} {f : n ⟶ m} :
   simp only [skeletalFunctor_obj, skeletalFunctor_map,
     NonemptyFinLinOrd.epi_iff_surjective, NonemptyFinLinOrd.coe_of]
 
-/-- A monomorphism in `SimplexCategory` must increase lengths-/
+/-- A monomorphism in `SimplexCategory` must increase lengths -/
 theorem len_le_of_mono {x y : SimplexCategory} {f : x ⟶ y} : Mono f → x.len ≤ y.len := by
   intro hyp_f_mono
   have f_inj : Function.Injective f.toOrderHom.toFun := mono_iff_injective.1 hyp_f_mono
@@ -529,7 +531,7 @@ theorem len_le_of_mono {x y : SimplexCategory} {f : x ⟶ y} : Mono f → x.len
 theorem le_of_mono {n m : ℕ} {f : ([n] : SimplexCategory) ⟶ [m]} : CategoryTheory.Mono f → n ≤ m :=
   len_le_of_mono
 
-/-- An epimorphism in `SimplexCategory` must decrease lengths-/
+/-- An epimorphism in `SimplexCategory` must decrease lengths -/
 theorem len_le_of_epi {x y : SimplexCategory} {f : x ⟶ y} : Epi f → y.len ≤ x.len := by
   intro hyp_f_epi
   have f_surj : Function.Surjective f.toOrderHom.toFun := epi_iff_surjective.1 hyp_f_epi
@@ -725,7 +727,7 @@ theorem eq_id_of_mono {x : SimplexCategory} (i : x ⟶ x) [Mono i] : i = 𝟙 _
   apply isIso_of_bijective
   dsimp
   rw [Fintype.bijective_iff_injective_and_card i.toOrderHom, ← mono_iff_injective,
-    eq_self_iff_true, and_true_iff]
+    eq_self_iff_true, and_true]
   infer_instance
 
 theorem eq_id_of_epi {x : SimplexCategory} (i : x ⟶ x) [Epi i] : i = 𝟙 _ := by
@@ -735,7 +737,7 @@ theorem eq_id_of_epi {x : SimplexCategory} (i : x ⟶ x) [Epi i] : i = 𝟙 _ :=
   apply isIso_of_bijective
   dsimp
   rw [Fintype.bijective_iff_surjective_and_card i.toOrderHom, ← epi_iff_surjective,
-    eq_self_iff_true, and_true_iff]
+    eq_self_iff_true, and_true]
   infer_instance
 
 theorem eq_σ_of_epi {n : ℕ} (θ : mk (n + 1) ⟶ mk n) [Epi θ] : ∃ i : Fin (n + 1), θ = σ i := by
diff --git a/Mathlib/AlgebraicTopology/SimplicialCategory/SimplicialObject.lean b/Mathlib/AlgebraicTopology/SimplicialCategory/SimplicialObject.lean
new file mode 100644
index 0000000000000..966e00873375e
--- /dev/null
+++ b/Mathlib/AlgebraicTopology/SimplicialCategory/SimplicialObject.lean
@@ -0,0 +1,41 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.AlgebraicTopology.SimplicialCategory.Basic
+import Mathlib.CategoryTheory.Functor.FunctorHom
+
+/-!
+# The category of simplicial objects is simplicial
+
+In `CategoryTheory.Functor.FunctorHom`, it was shown that a category of functors
+`C ⥤ D` is enriched over a suitable category `C ⥤ Type _` of functors to types.
+
+In this file, we deduce that `SimplicialObject D` is enriched over `SSet.{v} D`
+(when `D : Type u` and `[Category.{v} D]`) and that `SimplicialObject D`
+is actually a simplicial category. In particular, the category of simplicial
+sets is a simplicial category.
+
+-/
+
+universe v u
+
+namespace CategoryTheory
+
+variable {D : Type u} [Category.{v} D]
+
+namespace SimplicialObject
+
+noncomputable instance : EnrichedCategory SSet.{v} (SimplicialObject D)  :=
+  inferInstanceAs (EnrichedCategory (_ ⥤ Type v) (_ ⥤ D))
+
+noncomputable instance : SimplicialCategory (SimplicialObject D) where
+  homEquiv K L := Functor.natTransEquiv.symm
+
+noncomputable instance : SimplicialCategory SSet.{v} :=
+  inferInstanceAs (SimplicialCategory (SimplicialObject (Type v)))
+
+end SimplicialObject
+
+end CategoryTheory
diff --git a/Mathlib/AlgebraicTopology/SimplicialObject.lean b/Mathlib/AlgebraicTopology/SimplicialObject.lean
index 4d65c1b5fcf57..ea02f63a11c95 100644
--- a/Mathlib/AlgebraicTopology/SimplicialObject.lean
+++ b/Mathlib/AlgebraicTopology/SimplicialObject.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison, Adam Topaz
+Authors: Johan Commelin, Kim Morrison, Adam Topaz
 -/
 import Mathlib.AlgebraicTopology.SimplexCategory
 import Mathlib.CategoryTheory.Comma.Arrow
diff --git a/Mathlib/AlgebraicTopology/SimplicialSet.lean b/Mathlib/AlgebraicTopology/SimplicialSet.lean
index e20ccf4e006ff..a8198f848a6c7 100644
--- a/Mathlib/AlgebraicTopology/SimplicialSet.lean
+++ b/Mathlib/AlgebraicTopology/SimplicialSet.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison, Adam Topaz
+Authors: Johan Commelin, Kim Morrison, Adam Topaz
 -/
 import Mathlib.AlgebraicTopology.SimplicialObject
 import Mathlib.CategoryTheory.Limits.Shapes.Types
diff --git a/Mathlib/AlgebraicTopology/SingularSet.lean b/Mathlib/AlgebraicTopology/SingularSet.lean
index 21364f30fc245..bc3d246833408 100644
--- a/Mathlib/AlgebraicTopology/SingularSet.lean
+++ b/Mathlib/AlgebraicTopology/SingularSet.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison, Adam Topaz
+Authors: Johan Commelin, Kim Morrison, Adam Topaz
 -/
 import Mathlib.AlgebraicTopology.SimplicialSet
 import Mathlib.AlgebraicTopology.TopologicalSimplex
diff --git a/Mathlib/AlgebraicTopology/SplitSimplicialObject.lean b/Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
index c3a4875c7d176..447b1a9770800 100644
--- a/Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
+++ b/Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
@@ -208,14 +208,14 @@ structure Splitting (X : SimplicialObject C) where
   /-- The "inclusion" `N n ⟶ X _[n]` for all `n : ℕ`. -/
   ι : ∀ n, N n ⟶ X _[n]
   /-- For each `Δ`, `X.obj Δ` identifies to the coproduct of the objects `N A.1.unop.len`
-  for all `A : IndexSet Δ`.  -/
+  for all `A : IndexSet Δ`. -/
   isColimit' : ∀ Δ : SimplexCategoryᵒᵖ, IsColimit (Splitting.cofan' N X ι Δ)
 
 namespace Splitting
 
 variable {X Y : SimplicialObject C} (s : Splitting X)
 
-/-- The cofan for `summand s.N Δ` induced by a splitting of a simplicial object.  -/
+/-- The cofan for `summand s.N Δ` induced by a splitting of a simplicial object. -/
 def cofan (Δ : SimplexCategoryᵒᵖ) : Cofan (summand s.N Δ) :=
   Cofan.mk (X.obj Δ) (fun A => s.ι A.1.unop.len ≫ X.map A.e.op)
 
diff --git a/Mathlib/Analysis/Analytic/Basic.lean b/Mathlib/Analysis/Analytic/Basic.lean
index 8ef5a0aed701c..cd994fb7364b8 100644
--- a/Mathlib/Analysis/Analytic/Basic.lean
+++ b/Mathlib/Analysis/Analytic/Basic.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Analysis.Calculus.FormalMultilinearSeries
 import Mathlib.Analysis.SpecificLimits.Normed
 import Mathlib.Logic.Equiv.Fin
@@ -49,14 +49,13 @@ Additionally, let `f` be a function from `E` to `F`.
 * `HasFPowerSeriesAt f p x`: on some ball of center `x` with positive radius, holds
   `HasFPowerSeriesOnBall f p x r`.
 * `AnalyticAt 𝕜 f x`: there exists a power series `p` such that holds `HasFPowerSeriesAt f p x`.
-* `AnalyticOn 𝕜 f s`: the function `f` is analytic at every point of `s`.
+* `AnalyticOnNhd 𝕜 f s`: the function `f` is analytic at every point of `s`.
 
-We also define versions of `HasFPowerSeriesOnBall`, `AnalyticAt`, and `AnalyticOn` restricted to a
-set, similar to `ContinuousWithinAt`. See `Mathlib.Analysis.Analytic.Within` for basic properties.
+We also define versions of `HasFPowerSeriesOnBall`, `AnalyticAt`, and `AnalyticOnNhd` restricted to
+a set, similar to `ContinuousWithinAt`. See `Mathlib.Analysis.Analytic.Within` for basic properties.
 
-* `AnalyticWithinAt 𝕜 f s x` means a power series at `x` converges to `f` on `𝓝[s] x`, and
-  `f` is continuous within `s` at `x`.
-* `AnalyticWithinOn 𝕜 f s t` means `∀ x ∈ t, AnalyticWithinAt 𝕜 f s x`.
+* `AnalyticWithinAt 𝕜 f s x` means a power series at `x` converges to `f` on `𝓝[s ∪ {x}] x`.
+* `AnalyticOn 𝕜 f s t` means `∀ x ∈ t, AnalyticWithinAt 𝕜 f s x`.
 
 We develop the basic properties of these notions, notably:
 * If a function admits a power series, it is continuous (see
@@ -64,10 +63,6 @@ We develop the basic properties of these notions, notably:
   `AnalyticAt.continuousAt`).
 * In a complete space, the sum of a formal power series with positive radius is well defined on the
   disk of convergence, see `FormalMultilinearSeries.hasFPowerSeriesOnBall`.
-* If a function admits a power series in a ball, then it is analytic at any point `y` of this ball,
-  and the power series there can be expressed in terms of the initial power series `p` as
-  `p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular that
-  the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
 
 ## Implementation details
 
@@ -230,7 +225,7 @@ theorem lt_radius_of_isBigO (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ
 theorem norm_mul_pow_le_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
     (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * (r : ℝ) ^ n ≤ C :=
   let ⟨_, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
-  ⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩
+  ⟨C, hC, fun n => (h n).trans <| mul_le_of_le_one_right hC.lt.le (pow_le_one₀ ha.1.le ha.2.le)⟩
 
 /-- For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. -/
 theorem norm_le_div_pow_of_pos_of_lt_radius (p : FormalMultilinearSeries 𝕜 E F) {r : ℝ≥0}
@@ -306,6 +301,18 @@ theorem le_mul_pow_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (h : 0 <
   rw [inv_pow, ← div_eq_mul_inv]
   exact hCp n
 
+lemma radius_le_of_le {𝕜' E' F' : Type*}
+    [NontriviallyNormedField 𝕜'] [NormedAddCommGroup E'] [NormedSpace 𝕜' E']
+    [NormedAddCommGroup F'] [NormedSpace 𝕜' F']
+    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜' E' F'}
+    (h : ∀ n, ‖p n‖ ≤ ‖q n‖) : q.radius ≤ p.radius := by
+  apply le_of_forall_nnreal_lt (fun r hr ↦ ?_)
+  rcases norm_mul_pow_le_of_lt_radius _ hr with ⟨C, -, hC⟩
+  apply le_radius_of_bound _ C (fun n ↦ ?_)
+  apply le_trans _ (hC n)
+  gcongr
+  exact h n
+
 /-- The radius of the sum of two formal series is at least the minimum of their two radii. -/
 theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
     min p.radius q.radius ≤ (p + q).radius := by
@@ -340,7 +347,7 @@ end FormalMultilinearSeries
 
 section
 
-variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {x : E} {r r' : ℝ≥0∞}
+variable {f g : E → F} {p pf pg : FormalMultilinearSeries 𝕜 E F} {s t : Set E} {x : E} {r r' : ℝ≥0∞}
 
 /-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
 a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`.
@@ -352,7 +359,8 @@ structure HasFPowerSeriesOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜
   hasSum :
     ∀ {y}, y ∈ EMetric.ball (0 : E) r → HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
 
-/-- Analogue of `HasFPowerSeriesOnBall` where convergence is required only on a set `s`. -/
+/-- Analogue of `HasFPowerSeriesOnBall` where convergence is required only on a set `s`. We also
+require convergence at `x` as the behavior of this notion is very bad otherwise. -/
 structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries 𝕜 E F) (s : Set E)
     (x : E) (r : ℝ≥0∞) : Prop where
   /-- `p` converges on `ball 0 r` -/
@@ -360,10 +368,8 @@ structure HasFPowerSeriesWithinOnBall (f : E → F) (p : FormalMultilinearSeries
   /-- The radius of convergence is positive -/
   r_pos : 0 < r
   /-- `p converges to f` within `s` -/
-  hasSum : ∀ {y}, x + y ∈ s → y ∈ EMetric.ball (0 : E) r →
+  hasSum : ∀ {y}, x + y ∈ insert x s → y ∈ EMetric.ball (0 : E) r →
     HasSum (fun n : ℕ => p n fun _ : Fin n => y) (f (x + y))
-  /-- We require `ContinuousWithinAt f s x` to ensure `f x` is nice -/
-  continuousWithinAt : ContinuousWithinAt f s x
 
 /-- Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
 a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. -/
@@ -390,14 +396,21 @@ def AnalyticWithinAt (f : E → F) (s : Set E) (x : E) : Prop :=
 
 /-- Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
 every point of `s`. -/
-def AnalyticOn (f : E → F) (s : Set E) :=
+def AnalyticOnNhd (f : E → F) (s : Set E) :=
   ∀ x, x ∈ s → AnalyticAt 𝕜 f x
 
-/-- `f` is analytic within `s` if it is analytic within `s` at each point of `t`.  Note that
-this is weaker than `AnalyticOn 𝕜 f s`, as `f` is allowed to be arbitrary outside `s`. -/
-def AnalyticWithinOn (f : E → F) (s : Set E) : Prop :=
+/-- `f` is analytic within `s` if it is analytic within `s` at each point of `s`.  Note that
+this is weaker than `AnalyticOnNhd 𝕜 f s`, as `f` is allowed to be arbitrary outside `s`. -/
+def AnalyticOn (f : E → F) (s : Set E) : Prop :=
   ∀ x ∈ s, AnalyticWithinAt 𝕜 f s x
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn := AnalyticOn
+
+/-!
+### `HasFPowerSeriesOnBall` and `HasFPowerSeriesWithinOnBall`
+-/
+
 variable {𝕜}
 
 theorem HasFPowerSeriesOnBall.hasFPowerSeriesAt (hf : HasFPowerSeriesOnBall f p x r) :
@@ -410,14 +423,16 @@ theorem HasFPowerSeriesAt.analyticAt (hf : HasFPowerSeriesAt f p x) : AnalyticAt
 theorem HasFPowerSeriesOnBall.analyticAt (hf : HasFPowerSeriesOnBall f p x r) : AnalyticAt 𝕜 f x :=
   hf.hasFPowerSeriesAt.analyticAt
 
-theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
-    (hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
-  { r_le := hf.r_le
-    r_pos := hf.r_pos
-    hasSum := fun {y} hy => by
-      convert hf.hasSum hy using 1
-      apply hg.symm
-      simpa [edist_eq_coe_nnnorm_sub] using hy }
+theorem HasFPowerSeriesWithinOnBall.hasFPowerSeriesWithinAt
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) : HasFPowerSeriesWithinAt f p s x :=
+  ⟨r, hf⟩
+
+theorem HasFPowerSeriesWithinAt.analyticWithinAt (hf : HasFPowerSeriesWithinAt f p s x) :
+    AnalyticWithinAt 𝕜 f s x := ⟨p, hf⟩
+
+theorem HasFPowerSeriesWithinOnBall.analyticWithinAt (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    AnalyticWithinAt 𝕜 f s x :=
+  hf.hasFPowerSeriesWithinAt.analyticWithinAt
 
 /-- If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
 same power series around `x + y`. -/
@@ -429,6 +444,13 @@ theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y :
       convert hf.hasSum hz using 2
       abel }
 
+theorem HasFPowerSeriesWithinOnBall.hasSum_sub (hf : HasFPowerSeriesWithinOnBall f p s x r) {y : E}
+    (hy : y ∈ (insert x s) ∩ EMetric.ball x r) :
+    HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
+  have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy.2
+  have := hf.hasSum (by simpa only [add_sub_cancel] using hy.1) this
+  simpa only [add_sub_cancel]
+
 theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
     (hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
   have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
@@ -437,14 +459,69 @@ theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y
 theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
   lt_of_lt_of_le hf.r_pos hf.r_le
 
+theorem HasFPowerSeriesWithinOnBall.radius_pos (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    0 < p.radius :=
+  lt_of_lt_of_le hf.r_pos hf.r_le
+
 theorem HasFPowerSeriesAt.radius_pos (hf : HasFPowerSeriesAt f p x) : 0 < p.radius :=
   let ⟨_, hr⟩ := hf
   hr.radius_pos
 
+theorem HasFPowerSeriesWithinOnBall.of_le
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (r'_pos : 0 < r') (hr : r' ≤ r) :
+    HasFPowerSeriesWithinOnBall f p s x r' :=
+  ⟨le_trans hr hf.1, r'_pos, fun hy h'y => hf.hasSum hy (EMetric.ball_subset_ball hr h'y)⟩
+
 theorem HasFPowerSeriesOnBall.mono (hf : HasFPowerSeriesOnBall f p x r) (r'_pos : 0 < r')
     (hr : r' ≤ r) : HasFPowerSeriesOnBall f p x r' :=
   ⟨le_trans hr hf.1, r'_pos, fun hy => hf.hasSum (EMetric.ball_subset_ball hr hy)⟩
 
+lemma HasFPowerSeriesWithinOnBall.congr {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F}
+    {s : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p s x r)
+    (h' : EqOn g f (s ∩ EMetric.ball x r)) (h'' : g x = f x) :
+    HasFPowerSeriesWithinOnBall g p s x r := by
+  refine ⟨h.r_le, h.r_pos, ?_⟩
+  intro y hy h'y
+  convert h.hasSum hy h'y using 1
+  simp only [mem_insert_iff, add_right_eq_self] at hy
+  rcases hy with rfl | hy
+  · simpa using h''
+  · apply h'
+    refine ⟨hy, ?_⟩
+    simpa [edist_eq_coe_nnnorm_sub] using h'y
+
+/-- Variant of `HasFPowerSeriesWithinOnBall.congr` in which one requests equality on `insert x s`
+instead of separating `x` and `s`. -/
+lemma HasFPowerSeriesWithinOnBall.congr' {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F}
+    {s : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p s x r)
+    (h' : EqOn g f (insert x s ∩ EMetric.ball x r)) :
+    HasFPowerSeriesWithinOnBall g p s x r := by
+  refine ⟨h.r_le, h.r_pos, fun {y} hy h'y ↦ ?_⟩
+  convert h.hasSum hy h'y using 1
+  exact h' ⟨hy, by simpa [edist_eq_coe_nnnorm_sub] using h'y⟩
+
+lemma HasFPowerSeriesWithinAt.congr {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E}
+    {x : E} (h : HasFPowerSeriesWithinAt f p s x) (h' : g =ᶠ[𝓝[s] x] f) (h'' : g x = f x) :
+    HasFPowerSeriesWithinAt g p s x := by
+  rcases h with ⟨r, hr⟩
+  obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, EMetric.ball x ε ∩ s ⊆ {y | g y = f y} :=
+    EMetric.mem_nhdsWithin_iff.1 h'
+  let r' := min r ε
+  refine ⟨r', ?_⟩
+  have := hr.of_le (r' := r') (by simp [r', εpos, hr.r_pos]) (min_le_left _ _)
+  apply this.congr _ h''
+  intro z hz
+  exact hε ⟨EMetric.ball_subset_ball (min_le_right _ _) hz.2, hz.1⟩
+
+theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
+    (hg : EqOn f g (EMetric.ball x r)) : HasFPowerSeriesOnBall g p x r :=
+  { r_le := hf.r_le
+    r_pos := hf.r_pos
+    hasSum := fun {y} hy => by
+      convert hf.hasSum hy using 1
+      apply hg.symm
+      simpa [edist_eq_coe_nnnorm_sub] using hy }
+
 theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
     HasFPowerSeriesAt g p x := by
   rcases hf with ⟨r₁, h₁⟩
@@ -453,6 +530,12 @@ theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝
     (h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
       fun y hy => h₂ (EMetric.ball_subset_ball inf_le_right hy)⟩
 
+protected theorem HasFPowerSeriesWithinAt.eventually (hf : HasFPowerSeriesWithinAt f p s x) :
+    ∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesWithinOnBall f p s x r :=
+  let ⟨_, hr⟩ := hf
+  mem_of_superset (Ioo_mem_nhdsWithin_Ioi (left_mem_Ico.2 hr.r_pos)) fun _ hr' =>
+    hr.of_le hr'.1 hr'.2.le
+
 protected theorem HasFPowerSeriesAt.eventually (hf : HasFPowerSeriesAt f p x) :
     ∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, HasFPowerSeriesOnBall f p x r :=
   let ⟨_, hr⟩ := hf
@@ -487,149 +570,352 @@ theorem HasFPowerSeriesAt.eventually_eq_zero
   let ⟨_, hr⟩ := hf
   hr.eventually_eq_zero
 
-theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
-    HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
-  refine ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => ?_⟩
-  simp [constFormalMultilinearSeries_apply hn]
+@[simp] lemma hasFPowerSeriesWithinOnBall_univ :
+    HasFPowerSeriesWithinOnBall f p univ x r ↔ HasFPowerSeriesOnBall f p x r := by
+  constructor
+  · intro h
+    refine ⟨h.r_le, h.r_pos, fun {y} m ↦ h.hasSum (by simp) m⟩
+  · intro h
+    exact ⟨h.r_le, h.r_pos, fun {y} _ m => h.hasSum m⟩
+
+@[simp] lemma hasFPowerSeriesWithinAt_univ :
+    HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x := by
+  simp only [HasFPowerSeriesWithinAt, hasFPowerSeriesWithinOnBall_univ, HasFPowerSeriesAt]
+
+lemma HasFPowerSeriesWithinOnBall.mono (hf : HasFPowerSeriesWithinOnBall f p s x r) (h : t ⊆ s) :
+    HasFPowerSeriesWithinOnBall f p t x r where
+  r_le := hf.r_le
+  r_pos := hf.r_pos
+  hasSum hy h'y := hf.hasSum (insert_subset_insert h hy) h'y
+
+lemma HasFPowerSeriesOnBall.hasFPowerSeriesWithinOnBall (hf : HasFPowerSeriesOnBall f p x r) :
+    HasFPowerSeriesWithinOnBall f p s x r := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  exact hf.mono (subset_univ _)
+
+lemma HasFPowerSeriesWithinAt.mono (hf : HasFPowerSeriesWithinAt f p s x) (h : t ⊆ s) :
+    HasFPowerSeriesWithinAt f p t x := by
+  obtain ⟨r, hp⟩ := hf
+  exact ⟨r, hp.mono h⟩
+
+lemma HasFPowerSeriesAt.hasFPowerSeriesWithinAt (hf : HasFPowerSeriesAt f p x) :
+    HasFPowerSeriesWithinAt f p s x := by
+  rw [← hasFPowerSeriesWithinAt_univ] at hf
+  apply hf.mono (subset_univ _)
+
+theorem HasFPowerSeriesWithinAt.mono_of_mem
+    (h : HasFPowerSeriesWithinAt f p s x) (hst : s ∈ 𝓝[t] x) :
+    HasFPowerSeriesWithinAt f p t x := by
+  rcases h with ⟨r, hr⟩
+  rcases EMetric.mem_nhdsWithin_iff.1 hst with ⟨r', r'_pos, hr'⟩
+  refine ⟨min r r', ?_⟩
+  have Z := hr.of_le (by simp [r'_pos, hr.r_pos]) (min_le_left r r')
+  refine ⟨Z.r_le, Z.r_pos, fun {y} hy h'y ↦ ?_⟩
+  apply Z.hasSum ?_ h'y
+  simp only [mem_insert_iff, add_right_eq_self] at hy
+  rcases hy with rfl | hy
+  · simp
+  apply mem_insert_of_mem _ (hr' ?_)
+  simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, sub_zero, lt_min_iff, mem_inter_iff,
+    add_sub_cancel_left, hy, and_true] at h'y ⊢
+  exact h'y.2
+
+@[simp] lemma hasFPowerSeriesWithinOnBall_insert_self :
+    HasFPowerSeriesWithinOnBall f p (insert x s) x r ↔ HasFPowerSeriesWithinOnBall f p s x r := by
+  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩  <;>
+  exact ⟨h.r_le, h.r_pos, fun {y} ↦ by simpa only [insert_idem] using h.hasSum (y := y)⟩
+
+@[simp] theorem hasFPowerSeriesWithinAt_insert {y : E} :
+    HasFPowerSeriesWithinAt f p (insert y s) x ↔ HasFPowerSeriesWithinAt f p s x := by
+  rcases eq_or_ne x y with rfl | hy
+  · simp [HasFPowerSeriesWithinAt]
+  · refine ⟨fun h ↦ h.mono (subset_insert _ _), fun h ↦ ?_⟩
+    apply HasFPowerSeriesWithinAt.mono_of_mem h
+    rw [nhdsWithin_insert_of_ne hy]
+    exact self_mem_nhdsWithin
+
+theorem HasFPowerSeriesWithinOnBall.coeff_zero (hf : HasFPowerSeriesWithinOnBall f pf s x r)
+    (v : Fin 0 → E) : pf 0 v = f x := by
+  have v_eq : v = fun i => 0 := Subsingleton.elim _ _
+  have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
+  have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
+    intro i hi
+    have : 0 < i := pos_iff_ne_zero.2 hi
+    exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
+  have A := (hf.hasSum (by simp) zero_mem).unique (hasSum_single _ this)
+  simpa [v_eq] using A.symm
 
-theorem hasFPowerSeriesAt_const {c : F} {e : E} :
-    HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
-  ⟨⊤, hasFPowerSeriesOnBall_const⟩
+theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r)
+    (v : Fin 0 → E) : pf 0 v = f x := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  exact hf.coeff_zero v
 
-theorem analyticAt_const {v : F} : AnalyticAt 𝕜 (fun _ => v) x :=
-  ⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
+theorem HasFPowerSeriesWithinAt.coeff_zero (hf : HasFPowerSeriesWithinAt f pf s x) (v : Fin 0 → E) :
+    pf 0 v = f x :=
+  let ⟨_, hrf⟩ := hf
+  hrf.coeff_zero v
 
-theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
-  fun _ _ => analyticAt_const
+theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
+    pf 0 v = f x :=
+  let ⟨_, hrf⟩ := hf
+  hrf.coeff_zero v
 
-theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
-    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
-  { r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
-    r_pos := hf.r_pos
-    hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
+/-!
+### Analytic functions
+-/
 
-theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
-    HasFPowerSeriesAt (f + g) (pf + pg) x := by
-  rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
-  exact ⟨r, hr.1.add hr.2⟩
+@[simp] lemma analyticWithinAt_univ :
+    AnalyticWithinAt 𝕜 f univ x ↔ AnalyticAt 𝕜 f x := by
+  simp [AnalyticWithinAt, AnalyticAt]
 
-theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
-  let ⟨_, hpf⟩ := hf
-  (hpf.congr hg).analyticAt
+@[simp] lemma analyticOn_univ {f : E → F} :
+    AnalyticOn 𝕜 f univ ↔ AnalyticOnNhd 𝕜 f univ := by
+  simp only [AnalyticOn, analyticWithinAt_univ, AnalyticOnNhd]
 
-theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
-  ⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_univ := analyticOn_univ
 
-theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
-  let ⟨_, hpf⟩ := hf
-  let ⟨_, hqf⟩ := hg
-  (hpf.add hqf).analyticAt
-
-theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
-    HasFPowerSeriesOnBall (-f) (-pf) x r :=
-  { r_le := by
-      rw [pf.radius_neg]
-      exact hf.r_le
-    r_pos := hf.r_pos
-    hasSum := fun hy => (hf.hasSum hy).neg }
+lemma AnalyticWithinAt.mono (hf : AnalyticWithinAt 𝕜 f s x) (h : t ⊆ s) :
+    AnalyticWithinAt 𝕜 f t x := by
+  obtain ⟨p, hp⟩ := hf
+  exact ⟨p, hp.mono h⟩
 
-theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
-  let ⟨_, hrf⟩ := hf
-  hrf.neg.hasFPowerSeriesAt
+lemma AnalyticAt.analyticWithinAt (hf : AnalyticAt 𝕜 f x) : AnalyticWithinAt 𝕜 f s x := by
+  rw [← analyticWithinAt_univ] at hf
+  apply hf.mono (subset_univ _)
 
-theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
-  let ⟨_, hpf⟩ := hf
-  hpf.neg.analyticAt
+lemma AnalyticOnNhd.analyticOn (hf : AnalyticOnNhd 𝕜 f s) : AnalyticOn 𝕜 f s :=
+  fun x hx ↦ (hf x hx).analyticWithinAt
 
-theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
-    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
-  simpa only [sub_eq_add_neg] using hf.add hg.neg
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.analyticWithinOn := AnalyticOnNhd.analyticOn
 
-theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
-    HasFPowerSeriesAt (f - g) (pf - pg) x := by
-  simpa only [sub_eq_add_neg] using hf.add hg.neg
+lemma AnalyticWithinAt.congr_of_eventuallyEq {f g : E → F} {s : Set E} {x : E}
+    (hf : AnalyticWithinAt 𝕜 f s x) (hs : g =ᶠ[𝓝[s] x] f) (hx : g x = f x) :
+    AnalyticWithinAt 𝕜 g s x := by
+  rcases hf with ⟨p, hp⟩
+  exact ⟨p, hp.congr hs hx⟩
 
-theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
-    AnalyticAt 𝕜 (f - g) x := by
-  simpa only [sub_eq_add_neg] using hf.add hg.neg
+lemma AnalyticWithinAt.congr_of_eventuallyEq_insert {f g : E → F} {s : Set E} {x : E}
+    (hf : AnalyticWithinAt 𝕜 f s x) (hs : g =ᶠ[𝓝[insert x s] x] f) :
+    AnalyticWithinAt 𝕜 g s x := by
+  apply hf.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) hs)
+  apply mem_of_mem_nhdsWithin (mem_insert x s) hs
 
-theorem AnalyticOn.mono {s t : Set E} (hf : AnalyticOn 𝕜 f t) (hst : s ⊆ t) : AnalyticOn 𝕜 f s :=
-  fun z hz => hf z (hst hz)
+lemma AnalyticWithinAt.congr {f g : E → F} {s : Set E} {x : E}
+    (hf : AnalyticWithinAt 𝕜 f s x) (hs : EqOn g f s) (hx : g x = f x) :
+    AnalyticWithinAt 𝕜 g s x :=
+  hf.congr_of_eventuallyEq hs.eventuallyEq_nhdsWithin hx
 
-theorem AnalyticOn.congr' {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
+lemma AnalyticOn.congr {f g : E → F} {s : Set E}
+    (hf : AnalyticOn 𝕜 f s) (hs : EqOn g f s) :
     AnalyticOn 𝕜 g s :=
+  fun x m ↦ (hf x m).congr hs (hs m)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.congr := AnalyticOn.congr
+
+theorem AnalyticAt.congr (hf : AnalyticAt 𝕜 f x) (hg : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 g x :=
+  let ⟨_, hpf⟩ := hf
+  (hpf.congr hg).analyticAt
+
+theorem analyticAt_congr (h : f =ᶠ[𝓝 x] g) : AnalyticAt 𝕜 f x ↔ AnalyticAt 𝕜 g x :=
+  ⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
+
+theorem AnalyticOnNhd.mono {s t : Set E} (hf : AnalyticOnNhd 𝕜 f t) (hst : s ⊆ t) :
+    AnalyticOnNhd 𝕜 f s :=
+  fun z hz => hf z (hst hz)
+
+theorem AnalyticOnNhd.congr' (hf : AnalyticOnNhd 𝕜 f s) (hg : f =ᶠ[𝓝ˢ s] g) :
+    AnalyticOnNhd 𝕜 g s :=
   fun z hz => (hf z hz).congr (mem_nhdsSet_iff_forall.mp hg z hz)
 
-theorem analyticOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOn 𝕜 f s ↔ AnalyticOn 𝕜 g s :=
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.congr' := AnalyticOnNhd.congr'
+
+theorem analyticOnNhd_congr' (h : f =ᶠ[𝓝ˢ s] g) : AnalyticOnNhd 𝕜 f s ↔ AnalyticOnNhd 𝕜 g s :=
   ⟨fun hf => hf.congr' h, fun hg => hg.congr' h.symm⟩
 
-theorem AnalyticOn.congr {s : Set E} (hs : IsOpen s) (hf : AnalyticOn 𝕜 f s) (hg : s.EqOn f g) :
-    AnalyticOn 𝕜 g s :=
+@[deprecated (since := "2024-09-26")]
+alias analyticOn_congr' := analyticOnNhd_congr'
+
+theorem AnalyticOnNhd.congr (hs : IsOpen s) (hf : AnalyticOnNhd 𝕜 f s) (hg : s.EqOn f g) :
+    AnalyticOnNhd 𝕜 g s :=
   hf.congr' <| mem_nhdsSet_iff_forall.mpr
     (fun _ hz => eventuallyEq_iff_exists_mem.mpr ⟨s, hs.mem_nhds hz, hg⟩)
 
-theorem analyticOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOn 𝕜 f s ↔
-    AnalyticOn 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
+theorem analyticOnNhd_congr (hs : IsOpen s) (h : s.EqOn f g) : AnalyticOnNhd 𝕜 f s ↔
+    AnalyticOnNhd 𝕜 g s := ⟨fun hf => hf.congr hs h, fun hg => hg.congr hs h.symm⟩
 
-theorem AnalyticOn.add {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
-    AnalyticOn 𝕜 (f + g) s :=
-  fun z hz => (hf z hz).add (hg z hz)
+@[deprecated (since := "2024-09-26")]
+alias analyticOn_congr := analyticOnNhd_congr
 
-theorem AnalyticOn.neg {s : Set E} (hf : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (-f) s :=
-  fun z hz ↦ (hf z hz).neg
+theorem AnalyticWithinAt.mono_of_mem
+    (h : AnalyticWithinAt 𝕜 f s x) (hst : s ∈ 𝓝[t] x) : AnalyticWithinAt 𝕜 f t x := by
+  rcases h with ⟨p, hp⟩
+  exact ⟨p, hp.mono_of_mem hst⟩
 
-theorem AnalyticOn.sub {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
-    AnalyticOn 𝕜 (f - g) s :=
-  fun z hz => (hf z hz).sub (hg z hz)
+lemma AnalyticOn.mono {f : E → F} {s t : Set E} (h : AnalyticOn 𝕜 f t)
+    (hs : s ⊆ t) : AnalyticOn 𝕜 f s :=
+  fun _ m ↦ (h _ (hs m)).mono hs
 
-theorem HasFPowerSeriesOnBall.coeff_zero (hf : HasFPowerSeriesOnBall f pf x r) (v : Fin 0 → E) :
-    pf 0 v = f x := by
-  have v_eq : v = fun i => 0 := Subsingleton.elim _ _
-  have zero_mem : (0 : E) ∈ EMetric.ball (0 : E) r := by simp [hf.r_pos]
-  have : ∀ i, i ≠ 0 → (pf i fun j => 0) = 0 := by
-    intro i hi
-    have : 0 < i := pos_iff_ne_zero.2 hi
-    exact ContinuousMultilinearMap.map_coord_zero _ (⟨0, this⟩ : Fin i) rfl
-  have A := (hf.hasSum zero_mem).unique (hasSum_single _ this)
-  simpa [v_eq] using A.symm
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.mono := AnalyticOn.mono
 
-theorem HasFPowerSeriesAt.coeff_zero (hf : HasFPowerSeriesAt f pf x) (v : Fin 0 → E) :
-    pf 0 v = f x :=
-  let ⟨_, hrf⟩ := hf
-  hrf.coeff_zero v
+@[simp] theorem analyticWithinAt_insert {f : E → F} {s : Set E} {x y : E} :
+    AnalyticWithinAt 𝕜 f (insert y s) x ↔ AnalyticWithinAt 𝕜 f s x := by
+  simp [AnalyticWithinAt]
+
+/-!
+### Composition with linear maps
+-/
+
+/-- If a function `f` has a power series `p` on a ball within a set and `g` is linear,
+then `g ∘ f` has the power series `g ∘ p` on the same ball. -/
+theorem ContinuousLinearMap.comp_hasFPowerSeriesWithinOnBall (g : F →L[𝕜] G)
+    (h : HasFPowerSeriesWithinOnBall f p s x r) :
+    HasFPowerSeriesWithinOnBall (g ∘ f) (g.compFormalMultilinearSeries p) s x r where
+  r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
+  r_pos := h.r_pos
+  hasSum hy h'y := by
+    simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
+      ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
+      g.hasSum (h.hasSum hy h'y)
 
 /-- If a function `f` has a power series `p` on a ball and `g` is linear, then `g ∘ f` has the
 power series `g ∘ p` on the same ball. -/
 theorem ContinuousLinearMap.comp_hasFPowerSeriesOnBall (g : F →L[𝕜] G)
     (h : HasFPowerSeriesOnBall f p x r) :
-    HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r :=
-  { r_le := h.r_le.trans (p.radius_le_radius_continuousLinearMap_comp _)
-    r_pos := h.r_pos
-    hasSum := fun hy => by
-      simpa only [ContinuousLinearMap.compFormalMultilinearSeries_apply,
-        ContinuousLinearMap.compContinuousMultilinearMap_coe, Function.comp_apply] using
-        g.hasSum (h.hasSum hy) }
+    HasFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x r := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at h ⊢
+  exact g.comp_hasFPowerSeriesWithinOnBall h
 
 /-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
 on `s`. -/
-theorem ContinuousLinearMap.comp_analyticOn {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
+theorem ContinuousLinearMap.comp_analyticOn (g : F →L[𝕜] G) (h : AnalyticOn 𝕜 f s) :
     AnalyticOn 𝕜 (g ∘ f) s := by
   rintro x hx
   rcases h x hx with ⟨p, r, hp⟩
+  exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesWithinOnBall hp⟩
+
+/-- If a function `f` is analytic on a set `s` and `g` is linear, then `g ∘ f` is analytic
+on `s`. -/
+theorem ContinuousLinearMap.comp_analyticOnNhd
+    {s : Set E} (g : F →L[𝕜] G) (h : AnalyticOnNhd 𝕜 f s) :
+    AnalyticOnNhd 𝕜 (g ∘ f) s := by
+  rintro x hx
+  rcases h x hx with ⟨p, r, hp⟩
   exact ⟨g.compFormalMultilinearSeries p, r, g.comp_hasFPowerSeriesOnBall hp⟩
 
+/-!
+### Relation between analytic function and the partial sums of its power series
+-/
+
+theorem HasFPowerSeriesWithinOnBall.tendsto_partialSum
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) {y : E} (hy : y ∈ EMetric.ball (0 : E) r)
+    (h'y : x + y ∈ insert x s) :
+    Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y))) :=
+  (hf.hasSum h'y hy).tendsto_sum_nat
+
+theorem HasFPowerSeriesOnBall.tendsto_partialSum
+    (hf : HasFPowerSeriesOnBall f p x r) {y : E} (hy : y ∈ EMetric.ball (0 : E) r) :
+    Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y))) :=
+  (hf.hasSum hy).tendsto_sum_nat
+
+theorem HasFPowerSeriesAt.tendsto_partialSum
+    (hf : HasFPowerSeriesAt f p x) :
+    ∀ᶠ y in 𝓝 0, Tendsto (fun n => p.partialSum n y) atTop (𝓝 (f (x + y))) := by
+  rcases hf with ⟨r, hr⟩
+  filter_upwards [EMetric.ball_mem_nhds (0 : E) hr.r_pos] with y hy
+  exact hr.tendsto_partialSum hy
+
+open Finset in
+/-- If a function admits a power series expansion within a ball, then the partial sums
+`p.partialSum n z` converge to `f (x + y)` as `n → ∞` and `z → y`. Note that `x + z` doesn't need
+to belong to the set where the power series expansion holds. -/
+theorem HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod {y : E}
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (hy : y ∈ EMetric.ball (0 : E) r)
+    (h'y : x + y ∈ insert x s) :
+    Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) := by
+  have A : Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 y) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) := by
+    apply (hf.tendsto_partialSum hy h'y).comp tendsto_fst
+  suffices Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2 - p.partialSum z.1 y)
+    (atTop ×ˢ 𝓝 y) (𝓝 0) by simpa using A.add this
+  apply Metric.tendsto_nhds.2 (fun ε εpos ↦ ?_)
+  obtain ⟨r', yr', r'r⟩ : ∃ (r' : ℝ≥0), ‖y‖₊ < r' ∧ r' < r := by
+    simp [edist_eq_coe_nnnorm] at hy
+    simpa using ENNReal.lt_iff_exists_nnreal_btwn.1 hy
+  have yr'_2 : ‖y‖ < r' := by simpa [← coe_nnnorm] using yr'
+  have S : Summable fun n ↦ ‖p n‖ * ↑r' ^ n := p.summable_norm_mul_pow (r'r.trans_le hf.r_le)
+  obtain ⟨k, hk⟩ : ∃ k, ∑' (n : ℕ), ‖p (n + k)‖ * ↑r' ^ (n + k) < ε / 4 := by
+    have : Tendsto (fun k ↦ ∑' n, ‖p (n + k)‖ * ↑r' ^ (n + k)) atTop (𝓝 0) := by
+      apply _root_.tendsto_sum_nat_add (f := fun n ↦ ‖p n‖ * ↑r' ^ n)
+    exact ((tendsto_order.1 this).2 _ (by linarith)).exists
+  have A : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y,
+      dist (p.partialSum k z.2) (p.partialSum k y) < ε / 4 := by
+    have : ContinuousAt (fun z ↦ p.partialSum k z) y := (p.partialSum_continuous k).continuousAt
+    exact tendsto_snd (Metric.tendsto_nhds.1 this.tendsto (ε / 4) (by linarith))
+  have B : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, ‖z.2‖₊ < r' := by
+    suffices ∀ᶠ (z : E) in 𝓝 y, ‖z‖₊ < r' from tendsto_snd this
+    have : Metric.ball 0 r' ∈ 𝓝 y := Metric.isOpen_ball.mem_nhds (by simpa using yr'_2)
+    filter_upwards [this] with a ha using by simpa [← coe_nnnorm] using ha
+  have C : ∀ᶠ (z : ℕ × E) in atTop ×ˢ 𝓝 y, k ≤ z.1 := tendsto_fst (Ici_mem_atTop _)
+  filter_upwards [A, B, C]
+  rintro ⟨n, z⟩ hz h'z hkn
+  simp only [dist_eq_norm, sub_zero] at hz ⊢
+  have I (w : E) (hw : ‖w‖₊ < r') : ‖∑ i ∈ Ico k n, p i (fun _ ↦ w)‖ ≤ ε / 4 := calc
+    ‖∑ i ∈ Ico k n, p i (fun _ ↦ w)‖
+    _ = ‖∑ i ∈ range (n - k), p (i + k) (fun _ ↦ w)‖ := by
+        rw [sum_Ico_eq_sum_range]
+        congr with i
+        rw [add_comm k]
+    _ ≤ ∑ i ∈ range (n - k), ‖p (i + k) (fun _ ↦ w)‖ := norm_sum_le _ _
+    _ ≤ ∑ i ∈ range (n - k), ‖p (i + k)‖ * ‖w‖ ^ (i + k) := by
+        gcongr with i _hi; exact ((p (i + k)).le_opNorm _).trans_eq (by simp)
+    _ ≤ ∑ i ∈ range (n - k), ‖p (i + k)‖ * ↑r' ^ (i + k) := by
+        gcongr with i _hi; simpa [← coe_nnnorm] using hw.le
+    _ ≤ ∑' i, ‖p (i + k)‖ * ↑r' ^ (i + k) := by
+        apply sum_le_tsum _ (fun i _hi ↦ by positivity)
+        apply ((_root_.summable_nat_add_iff k).2 S)
+    _ ≤ ε / 4 := hk.le
+  calc
+  ‖p.partialSum n z - p.partialSum n y‖
+  _ = ‖∑ i ∈ range n, p i (fun _ ↦ z) - ∑ i ∈ range n, p i (fun _ ↦ y)‖ := rfl
+  _ = ‖(∑ i ∈ range k, p i (fun _ ↦ z) + ∑ i ∈ Ico k n, p i (fun _ ↦ z))
+        - (∑ i ∈ range k, p i (fun _ ↦ y) + ∑ i ∈ Ico k n, p i (fun _ ↦ y))‖ := by
+    simp [sum_range_add_sum_Ico _ hkn]
+  _ = ‖(p.partialSum k z - p.partialSum k y) + (∑ i ∈ Ico k n, p i (fun _ ↦ z))
+        + (- ∑ i ∈ Ico k n, p i (fun _ ↦ y))‖ := by
+    congr 1
+    simp only [FormalMultilinearSeries.partialSum]
+    abel
+  _ ≤ ‖p.partialSum k z - p.partialSum k y‖ + ‖∑ i ∈ Ico k n, p i (fun _ ↦ z)‖
+      + ‖- ∑ i ∈ Ico k n, p i (fun _ ↦ y)‖ := norm_add₃_le _ _ _
+  _ ≤ ε / 4 + ε / 4 + ε / 4 := by
+    gcongr
+    · exact I _ h'z
+    · simp only [norm_neg]; exact I _ yr'
+  _ < ε := by linarith
+
+/-- If a function admits a power series on a ball, then the partial sums
+`p.partialSum n z` converges to `f (x + y)` as `n → ∞` and `z → y`. -/
+theorem HasFPowerSeriesOnBall.tendsto_partialSum_prod {y : E}
+    (hf : HasFPowerSeriesOnBall f p x r) (hy : y ∈ EMetric.ball (0 : E) r) :
+    Tendsto (fun (z : ℕ × E) ↦ p.partialSum z.1 z.2) (atTop ×ˢ 𝓝 y) (𝓝 (f (x + y))) :=
+  (hf.hasFPowerSeriesWithinOnBall (s := univ)).tendsto_partialSum_prod hy (by simp)
+
 /-- If a function admits a power series expansion, then it is exponentially close to the partial
 sums of this power series on strict subdisks of the disk of convergence.
 
 This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
-`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
-theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
-    (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
-    ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
+`HasFPowerSeriesWithinOnBall.uniform_geometric_approx` for a weaker version. -/
+theorem HasFPowerSeriesWithinOnBall.uniform_geometric_approx' {r' : ℝ≥0}
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
+    ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, x + y ∈ insert x s →
       ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
   obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
     p.norm_mul_pow_le_mul_pow_of_lt_radius (h.trans_le hf.r_le)
-  refine ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n => ?_⟩
+  refine ⟨a, ha, C / (1 - a), div_pos hC (sub_pos.2 ha.2), fun y hy n ys => ?_⟩
   have yr' : ‖y‖ < r' := by
     rw [ball_zero_eq] at hy
     exact hy
@@ -639,13 +925,13 @@ theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
     exact mod_cast yr'
   rw [norm_sub_rev, ← mul_div_right_comm]
   have ya : a * (‖y‖ / ↑r') ≤ a :=
-    mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
+    mul_le_of_le_one_right ha.1.le (div_le_one_of_le₀ yr'.le r'.coe_nonneg)
   suffices ‖p.partialSum n y - f (x + y)‖ ≤ C * (a * (‖y‖ / r')) ^ n / (1 - a * (‖y‖ / r')) by
     refine this.trans ?_
     have : 0 < a := ha.1
     gcongr
     apply_rules [sub_pos.2, ha.2]
-  apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum this)
+  apply norm_sub_le_of_geometric_bound_of_hasSum (ya.trans_lt ha.2) _ (hf.hasSum ys this)
   intro n
   calc
     ‖(p n) fun _ : Fin n => y‖
@@ -655,58 +941,94 @@ theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
     _ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc]
 
 /-- If a function admits a power series expansion, then it is exponentially close to the partial
-sums of this power series on strict subdisks of the disk of convergence. -/
-theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
+sums of this power series on strict subdisks of the disk of convergence.
+
+This version provides an upper estimate that decreases both in `‖y‖` and `n`. See also
+`HasFPowerSeriesOnBall.uniform_geometric_approx` for a weaker version. -/
+theorem HasFPowerSeriesOnBall.uniform_geometric_approx' {r' : ℝ≥0}
     (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
+    ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
+      ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n := by
+   rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+   simpa using hf.uniform_geometric_approx' h
+
+/-- If a function admits a power series expansion within a set in a ball, then it is exponentially
+close to the partial sums of this power series on strict subdisks of the disk of convergence. -/
+theorem HasFPowerSeriesWithinOnBall.uniform_geometric_approx {r' : ℝ≥0}
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
     ∃ a ∈ Ioo (0 : ℝ) 1,
-      ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
+      ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n, x + y ∈ insert x s →
+      ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
   obtain ⟨a, ha, C, hC, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
-      ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
+      x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
     hf.uniform_geometric_approx' h
-  refine ⟨a, ha, C, hC, fun y hy n => (hp y hy n).trans ?_⟩
+  refine ⟨a, ha, C, hC, fun y hy n ys => (hp y hy n ys).trans ?_⟩
   have yr' : ‖y‖ < r' := by rwa [ball_zero_eq] at hy
   have := ha.1.le -- needed to discharge a side goal on the next line
   gcongr
-  exact mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)
+  exact mul_le_of_le_one_right ha.1.le (div_le_one_of_le₀ yr'.le r'.coe_nonneg)
 
-/-- Taylor formula for an analytic function, `IsBigO` version. -/
-theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
-    (fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
+/-- If a function admits a power series expansion, then it is exponentially close to the partial
+sums of this power series on strict subdisks of the disk of convergence. -/
+theorem HasFPowerSeriesOnBall.uniform_geometric_approx {r' : ℝ≥0}
+    (hf : HasFPowerSeriesOnBall f p x r) (h : (r' : ℝ≥0∞) < r) :
+    ∃ a ∈ Ioo (0 : ℝ) 1,
+      ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
+      ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.uniform_geometric_approx h
+
+/-- Taylor formula for an analytic function within a set, `IsBigO` version. -/
+theorem HasFPowerSeriesWithinAt.isBigO_sub_partialSum_pow
+    (hf : HasFPowerSeriesWithinAt f p s x) (n : ℕ) :
+    (fun y : E => f (x + y) - p.partialSum n y)
+      =O[𝓝[(x + ·)⁻¹' insert x s] 0] fun y => ‖y‖ ^ n := by
   rcases hf with ⟨r, hf⟩
   rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
   obtain ⟨a, -, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
-      ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
+      x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * (a * (‖y‖ / r')) ^ n :=
     hf.uniform_geometric_approx' h
   refine isBigO_iff.2 ⟨C * (a / r') ^ n, ?_⟩
   replace r'0 : 0 < (r' : ℝ) := mod_cast r'0
-  filter_upwards [Metric.ball_mem_nhds (0 : E) r'0] with y hy
-  simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div] using hp y hy n
+  filter_upwards [inter_mem_nhdsWithin _ (Metric.ball_mem_nhds (0 : E) r'0)] with y hy
+  simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div, div_pow]
+    using hp y hy.2 n (by simpa using hy.1)
 
-/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
-ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
-`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
-`Filter.principal` on `E × E`. -/
-theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
-    (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
-    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓟 (EMetric.ball (x, x) r')]
+/-- Taylor formula for an analytic function, `IsBigO` version. -/
+theorem HasFPowerSeriesAt.isBigO_sub_partialSum_pow
+    (hf : HasFPowerSeriesAt f p x) (n : ℕ) :
+    (fun y : E => f (x + y) - p.partialSum n y) =O[𝓝 0] fun y => ‖y‖ ^ n := by
+  rw [← hasFPowerSeriesWithinAt_univ] at hf
+  simpa using hf.isBigO_sub_partialSum_pow n
+
+/-- If `f` has formal power series `∑ n, pₙ` in a set, within a ball of radius `r`, then
+for `y, z` in any smaller ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is
+bounded above by `C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property
+using `IsBigO` and `Filter.principal` on `E × E`. -/
+theorem HasFPowerSeriesWithinOnBall.isBigO_image_sub_image_sub_deriv_principal
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (hr : r' < r) :
+    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2)
+      =O[𝓟 (EMetric.ball (x, x) r' ∩ ((insert x s) ×ˢ (insert x s)))]
       fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
   lift r' to ℝ≥0 using ne_top_of_lt hr
   rcases (zero_le r').eq_or_lt with (rfl | hr'0)
-  · simp only [isBigO_bot, EMetric.ball_zero, principal_empty, ENNReal.coe_zero]
+  · simp only [ENNReal.coe_zero, EMetric.ball_zero, empty_inter, principal_empty, isBigO_bot]
   obtain ⟨a, ha, C, hC : 0 < C, hp⟩ :
       ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ n : ℕ, ‖p n‖ * (r' : ℝ) ^ n ≤ C * a ^ n :=
     p.norm_mul_pow_le_mul_pow_of_lt_radius (hr.trans_le hf.r_le)
   simp only [← le_div_iff₀ (pow_pos (NNReal.coe_pos.2 hr'0) _)] at hp
   set L : E × E → ℝ := fun y =>
     C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * (a / (1 - a) ^ 2 + 2 / (1 - a))
-  have hL : ∀ y ∈ EMetric.ball (x, x) r', ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
-    intro y hy'
+  have hL : ∀ y ∈ EMetric.ball (x, x) r' ∩ ((insert x s) ×ˢ (insert x s)),
+      ‖f y.1 - f y.2 - p 1 fun _ => y.1 - y.2‖ ≤ L y := by
+    intro y ⟨hy', ys⟩
     have hy : y ∈ EMetric.ball x r ×ˢ EMetric.ball x r := by
       rw [EMetric.ball_prod_same]
       exact EMetric.ball_subset_ball hr.le hy'
     set A : ℕ → F := fun n => (p n fun _ => y.1 - x) - p n fun _ => y.2 - x
     have hA : HasSum (fun n => A (n + 2)) (f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) := by
-      convert (hasSum_nat_add_iff' 2).2 ((hf.hasSum_sub hy.1).sub (hf.hasSum_sub hy.2)) using 1
+      convert (hasSum_nat_add_iff' 2).2
+        ((hf.hasSum_sub ⟨ys.1, hy.1⟩).sub (hf.hasSum_sub ⟨ys.2, hy.2⟩)) using 1
       rw [Finset.sum_range_succ, Finset.sum_range_one, hf.coeff_zero, hf.coeff_zero, sub_self,
         zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
         ← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_sub, ← Pi.single,
@@ -742,23 +1064,60 @@ theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
       exact (hasSum_coe_mul_geometric_of_norm_lt_one this).add  -- Porting note: was `convert`!
           ((hasSum_geometric_of_norm_lt_one this).mul_left 2)
     exact hA.norm_le_of_bounded hBL hAB
-  suffices L =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
+  suffices L =O[𝓟 (EMetric.ball (x, x) r' ∩ ((insert x s) ×ˢ (insert x s)))]
+      fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ by
     refine (IsBigO.of_bound 1 (eventually_principal.2 fun y hy => ?_)).trans this
     rw [one_mul]
     exact (hL y hy).trans (le_abs_self _)
   simp_rw [L, mul_right_comm _ (_ * _)]
   exact (isBigO_refl _ _).const_mul_left _
 
+/-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
+ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
+`C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. This lemma formulates this property using `IsBigO` and
+`Filter.principal` on `E × E`. -/
+theorem HasFPowerSeriesOnBall.isBigO_image_sub_image_sub_deriv_principal
+    (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
+    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2)
+      =O[𝓟 (EMetric.ball (x, x) r')] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.isBigO_image_sub_image_sub_deriv_principal hr
+
+/-- If `f` has formal power series `∑ n, pₙ` within a set, on a ball of radius `r`, then for `y, z`
+in any smaller ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above
+by `C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
+theorem HasFPowerSeriesWithinOnBall.image_sub_sub_deriv_le
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (hr : r' < r) :
+    ∃ C, ∀ᵉ (y ∈ insert x s ∩ EMetric.ball x r') (z ∈ insert x s ∩ EMetric.ball x r'),
+      ‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
+  have := hf.isBigO_image_sub_image_sub_deriv_principal hr
+  simp only [isBigO_principal, mem_inter_iff, EMetric.mem_ball, Prod.edist_eq, max_lt_iff, mem_prod,
+    norm_mul, Real.norm_eq_abs, abs_norm, and_imp, Prod.forall, mul_assoc] at this ⊢
+  rcases this with ⟨C, hC⟩
+  exact ⟨C, fun y ys hy z zs hz ↦ hC y z hy hz ys zs⟩
+
 /-- If `f` has formal power series `∑ n, pₙ` on a ball of radius `r`, then for `y, z` in any smaller
 ball, the norm of the difference `f y - f z - p 1 (fun _ ↦ y - z)` is bounded above by
 `C * (max ‖y - x‖ ‖z - x‖) * ‖y - z‖`. -/
-theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le (hf : HasFPowerSeriesOnBall f p x r)
-    (hr : r' < r) :
+theorem HasFPowerSeriesOnBall.image_sub_sub_deriv_le
+    (hf : HasFPowerSeriesOnBall f p x r) (hr : r' < r) :
     ∃ C, ∀ᵉ (y ∈ EMetric.ball x r') (z ∈ EMetric.ball x r'),
       ‖f y - f z - p 1 fun _ => y - z‖ ≤ C * max ‖y - x‖ ‖z - x‖ * ‖y - z‖ := by
-  simpa only [isBigO_principal, mul_assoc, norm_mul, norm_norm, Prod.forall, EMetric.mem_ball,
-    Prod.edist_eq, max_lt_iff, and_imp, @forall_swap (_ < _) E] using
-    hf.isBigO_image_sub_image_sub_deriv_principal hr
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa only [mem_univ, insert_eq_of_mem, univ_inter] using hf.image_sub_sub_deriv_le hr
+
+/-- If `f` has formal power series `∑ n, pₙ` at `x` within a set `s`, then
+`f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`
+within `s × s`. -/
+theorem HasFPowerSeriesWithinAt.isBigO_image_sub_norm_mul_norm_sub
+    (hf : HasFPowerSeriesWithinAt f p s x) :
+    (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2)
+      =O[𝓝[(insert x s) ×ˢ (insert x s)] (x, x)] fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
+  rcases hf with ⟨r, hf⟩
+  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
+  refine (hf.isBigO_image_sub_image_sub_deriv_principal h).mono ?_
+  rw [inter_comm]
+  exact le_principal_iff.2 (inter_mem_nhdsWithin _ (EMetric.ball_mem_nhds _ r'0))
 
 /-- If `f` has formal power series `∑ n, pₙ` at `x`, then
 `f y - f z - p 1 (fun _ ↦ y - z) = O(‖(y, z) - (x, x)‖ * ‖y - z‖)` as `(y, z) → (x, x)`.
@@ -766,27 +1125,52 @@ In particular, `f` is strictly differentiable at `x`. -/
 theorem HasFPowerSeriesAt.isBigO_image_sub_norm_mul_norm_sub (hf : HasFPowerSeriesAt f p x) :
     (fun y : E × E => f y.1 - f y.2 - p 1 fun _ => y.1 - y.2) =O[𝓝 (x, x)] fun y =>
       ‖y - (x, x)‖ * ‖y.1 - y.2‖ := by
-  rcases hf with ⟨r, hf⟩
-  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hf.r_pos with ⟨r', r'0, h⟩
-  refine (hf.isBigO_image_sub_image_sub_deriv_principal h).mono ?_
-  exact le_principal_iff.2 (EMetric.ball_mem_nhds _ r'0)
-
-/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
-partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
-is the uniform limit of `p.partialSum n y` there. -/
-theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
-    (h : (r' : ℝ≥0∞) < r) :
+  rw [← hasFPowerSeriesWithinAt_univ] at hf
+  simpa using hf.isBigO_image_sub_norm_mul_norm_sub
+
+/-- If a function admits a power series expansion within a set at `x`, then it is the uniform limit
+of the partial sums of this power series on strict subdisks of the disk of convergence, i.e.,
+`f (x + y)` is the uniform limit of `p.partialSum n y` there. -/
+theorem HasFPowerSeriesWithinOnBall.tendstoUniformlyOn {r' : ℝ≥0}
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
     TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
-      (Metric.ball (0 : E) r') := by
+      ((x + ·)⁻¹' (insert x s) ∩ Metric.ball (0 : E) r') := by
   obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
-    ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := hf.uniform_geometric_approx h
+    x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := hf.uniform_geometric_approx h
   refine Metric.tendstoUniformlyOn_iff.2 fun ε εpos => ?_
   have L : Tendsto (fun n => (C : ℝ) * a ^ n) atTop (𝓝 ((C : ℝ) * 0)) :=
     tendsto_const_nhds.mul (tendsto_pow_atTop_nhds_zero_of_lt_one ha.1.le ha.2)
   rw [mul_zero] at L
   refine (L.eventually (gt_mem_nhds εpos)).mono fun n hn y hy => ?_
   rw [dist_eq_norm]
-  exact (hp y hy n).trans_lt hn
+  exact (hp y hy.2 n hy.1).trans_lt hn
+
+/-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
+partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f (x + y)`
+is the uniform limit of `p.partialSum n y` there. -/
+theorem HasFPowerSeriesOnBall.tendstoUniformlyOn {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
+    (h : (r' : ℝ≥0∞) < r) :
+    TendstoUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
+      (Metric.ball (0 : E) r') := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.tendstoUniformlyOn h
+
+/-- If a function admits a power series expansion within a set at `x`, then it is the locally
+uniform limit of the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
+is the locally uniform limit of `p.partialSum n y` there. -/
+theorem HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
+      ((x + ·)⁻¹' (insert x s) ∩ EMetric.ball (0 : E) r) := by
+  intro u hu y hy
+  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hy.2 with ⟨r', yr', hr'⟩
+  have : EMetric.ball (0 : E) r' ∈ 𝓝 y := IsOpen.mem_nhds EMetric.isOpen_ball yr'
+  refine ⟨(x + ·)⁻¹' (insert x s) ∩ EMetric.ball (0 : E) r', ?_, ?_⟩
+  · rw [nhdsWithin_inter_of_mem']
+    · exact inter_mem_nhdsWithin _ this
+    · apply mem_nhdsWithin_of_mem_nhds
+      apply Filter.mem_of_superset this (EMetric.ball_subset_ball hr'.le)
+  · simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
 
 /-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
 the partial sums of this power series on the disk of convergence, i.e., `f (x + y)`
@@ -794,11 +1178,20 @@ is the locally uniform limit of `p.partialSum n y` there. -/
 theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn (hf : HasFPowerSeriesOnBall f p x r) :
     TendstoLocallyUniformlyOn (fun n y => p.partialSum n y) (fun y => f (x + y)) atTop
       (EMetric.ball (0 : E) r) := by
-  intro u hu x hx
-  rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hx with ⟨r', xr', hr'⟩
-  have : EMetric.ball (0 : E) r' ∈ 𝓝 x := IsOpen.mem_nhds EMetric.isOpen_ball xr'
-  refine ⟨EMetric.ball (0 : E) r', mem_nhdsWithin_of_mem_nhds this, ?_⟩
-  simpa [Metric.emetric_ball_nnreal] using hf.tendstoUniformlyOn hr' u hu
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.tendstoLocallyUniformlyOn
+
+/-- If a function admits a power series expansion within a set at `x`, then it is the uniform limit
+of the partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
+is the uniform limit of `p.partialSum n (y - x)` there. -/
+theorem HasFPowerSeriesWithinOnBall.tendstoUniformlyOn' {r' : ℝ≥0}
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) (h : (r' : ℝ≥0∞) < r) :
+    TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
+      (insert x s ∩ Metric.ball (x : E) r') := by
+  convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
+  · simp [Function.comp_def]
+  · ext z
+    simp [dist_eq_norm]
 
 /-- If a function admits a power series expansion at `x`, then it is the uniform limit of the
 partial sums of this power series on strict subdisks of the disk of convergence, i.e., `f y`
@@ -806,18 +1199,17 @@ is the uniform limit of `p.partialSum n (y - x)` there. -/
 theorem HasFPowerSeriesOnBall.tendstoUniformlyOn' {r' : ℝ≥0} (hf : HasFPowerSeriesOnBall f p x r)
     (h : (r' : ℝ≥0∞) < r) :
     TendstoUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop (Metric.ball (x : E) r') := by
-  convert (hf.tendstoUniformlyOn h).comp fun y => y - x using 1
-  · simp [(· ∘ ·)]
-  · ext z
-    simp [dist_eq_norm]
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.tendstoUniformlyOn' h
 
-/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
-the partial sums of this power series on the disk of convergence, i.e., `f y`
+/-- If a function admits a power series expansion within a set at `x`, then it is the locally
+uniform limit of the partial sums of this power series on the disk of convergence, i.e., `f y`
 is the locally uniform limit of `p.partialSum n (y - x)` there. -/
-theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
+theorem HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn'
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) :
     TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
-      (EMetric.ball (x : E) r) := by
-  have A : ContinuousOn (fun y : E => y - x) (EMetric.ball (x : E) r) :=
+      (insert x s ∩ EMetric.ball (x : E) r) := by
+  have A : ContinuousOn (fun y : E => y - x) (insert x s ∩ EMetric.ball (x : E) r) :=
     (continuous_id.sub continuous_const).continuousOn
   convert hf.tendstoLocallyUniformlyOn.comp (fun y : E => y - x) _ A using 1
   · ext z
@@ -825,29 +1217,88 @@ theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOn
   · intro z
     simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
 
-/-- If a function admits a power series expansion on a disk, then it is continuous there. -/
-protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
-    ContinuousOn f (EMetric.ball x r) :=
+/-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
+the partial sums of this power series on the disk of convergence, i.e., `f y`
+is the locally uniform limit of `p.partialSum n (y - x)` there. -/
+theorem HasFPowerSeriesOnBall.tendstoLocallyUniformlyOn' (hf : HasFPowerSeriesOnBall f p x r) :
+    TendstoLocallyUniformlyOn (fun n y => p.partialSum n (y - x)) f atTop
+      (EMetric.ball (x : E) r) := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.tendstoLocallyUniformlyOn'
+
+/-- If a function admits a power series expansion within a set on a ball, then it is
+continuous there. -/
+protected theorem HasFPowerSeriesWithinOnBall.continuousOn
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    ContinuousOn f (insert x s ∩ EMetric.ball x r) :=
   hf.tendstoLocallyUniformlyOn'.continuousOn <|
     Eventually.of_forall fun n =>
       ((p.partialSum_continuous n).comp (continuous_id.sub continuous_const)).continuousOn
 
+/-- If a function admits a power series expansion on a ball, then it is continuous there. -/
+protected theorem HasFPowerSeriesOnBall.continuousOn (hf : HasFPowerSeriesOnBall f p x r) :
+    ContinuousOn f (EMetric.ball x r) := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  simpa using hf.continuousOn
+
+protected theorem HasFPowerSeriesWithinOnBall.continuousWithinAt_insert
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    ContinuousWithinAt f (insert x s) x := by
+  apply (hf.continuousOn.continuousWithinAt (x := x) (by simp [hf.r_pos])).mono_of_mem
+  exact inter_mem_nhdsWithin _ (EMetric.ball_mem_nhds x hf.r_pos)
+
+protected theorem HasFPowerSeriesWithinOnBall.continuousWithinAt
+    (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    ContinuousWithinAt f s x :=
+  hf.continuousWithinAt_insert.mono (subset_insert x s)
+
+protected theorem HasFPowerSeriesWithinAt.continuousWithinAt_insert
+    (hf : HasFPowerSeriesWithinAt f p s x) :
+    ContinuousWithinAt f (insert x s) x := by
+  rcases hf with ⟨r, hr⟩
+  apply hr.continuousWithinAt_insert
+
+protected theorem HasFPowerSeriesWithinAt.continuousWithinAt
+    (hf : HasFPowerSeriesWithinAt f p s x) :
+    ContinuousWithinAt f s x :=
+  hf.continuousWithinAt_insert.mono (subset_insert x s)
+
 protected theorem HasFPowerSeriesAt.continuousAt (hf : HasFPowerSeriesAt f p x) :
     ContinuousAt f x :=
   let ⟨_, hr⟩ := hf
   hr.continuousOn.continuousAt (EMetric.ball_mem_nhds x hr.r_pos)
 
+protected theorem AnalyticWithinAt.continuousWithinAt_insert (hf : AnalyticWithinAt 𝕜 f s x) :
+    ContinuousWithinAt f (insert x s) x :=
+  let ⟨_, hp⟩ := hf
+  hp.continuousWithinAt_insert
+
+protected theorem AnalyticWithinAt.continuousWithinAt (hf : AnalyticWithinAt 𝕜 f s x) :
+    ContinuousWithinAt f s x :=
+  hf.continuousWithinAt_insert.mono (subset_insert x s)
+
 protected theorem AnalyticAt.continuousAt (hf : AnalyticAt 𝕜 f x) : ContinuousAt f x :=
   let ⟨_, hp⟩ := hf
   hp.continuousAt
 
-protected theorem AnalyticOn.continuousOn {s : Set E} (hf : AnalyticOn 𝕜 f s) : ContinuousOn f s :=
+protected theorem AnalyticOnNhd.continuousOn {s : Set E} (hf : AnalyticOnNhd 𝕜 f s) :
+    ContinuousOn f s :=
   fun x hx => (hf x hx).continuousAt.continuousWithinAt
 
+protected lemma AnalyticOn.continuousOn {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s) :
+    ContinuousOn f s :=
+  fun x m ↦ (h x m).continuousWithinAt
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.continuousOn := AnalyticOn.continuousOn
+
 /-- Analytic everywhere implies continuous -/
-theorem AnalyticOn.continuous {f : E → F} (fa : AnalyticOn 𝕜 f univ) : Continuous f := by
+theorem AnalyticOnNhd.continuous {f : E → F} (fa : AnalyticOnNhd 𝕜 f univ) : Continuous f := by
   rw [continuous_iff_continuousOn_univ]; exact fa.continuousOn
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.continuous := AnalyticOnNhd.continuous
+
 /-- In a complete space, the sum of a converging power series `p` admits `p` as a power series.
 This is not totally obvious as we need to check the convergence of the series. -/
 protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F]
@@ -859,6 +1310,10 @@ protected theorem FormalMultilinearSeries.hasFPowerSeriesOnBall [CompleteSpace F
       rw [zero_add]
       exact p.hasSum hy }
 
+theorem HasFPowerSeriesWithinOnBall.sum (h : HasFPowerSeriesWithinOnBall f p s x r) {y : E}
+    (h'y : x + y ∈ insert x s) (hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
+  (h.hasSum h'y hy).tsum_eq.symm
+
 theorem HasFPowerSeriesOnBall.sum (h : HasFPowerSeriesOnBall f p x r) {y : E}
     (hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y :=
   (h.hasSum hy).tsum_eq.symm
@@ -872,465 +1327,6 @@ protected theorem FormalMultilinearSeries.continuousOn [CompleteSpace F] :
 
 end
 
-/-!
-### Uniqueness of power series
-If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
-to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
-for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
-when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
-`ContinuousMultilinearMap.mkPiRing`, and hence are determined completely by the value of
-`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
-transferred to the other.
--/
-
-
-section Uniqueness
-
-open ContinuousMultilinearMap
-
-theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
-    (h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
-  obtain ⟨c, c_pos, hc⟩ := h.exists_pos
-  obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
-  obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
-  clear h hc z_mem
-  cases' n with n
-  · exact norm_eq_zero.mp (by
-      -- Porting note: the symmetric difference of the `simpa only` sets:
-      -- added `zero_add, pow_one`
-      -- removed `zero_pow, Ne.def, Nat.one_ne_zero, not_false_iff`
-      simpa only [fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
-        mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
-  · refine Or.elim (Classical.em (y = 0))
-      (fun hy => by simpa only [hy] using p.map_zero) fun hy => ?_
-    replace hy := norm_pos_iff.mpr hy
-    refine norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => ?_) (norm_nonneg _))
-    have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
-    obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
-      (lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
-    have h₁ : ‖k • y‖ < δ := by
-      rw [norm_smul]
-      exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
-        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
-    have h₂ :=
-      calc
-        ‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
-          -- Porting note: now Lean wants `_root_.`
-          simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
-          --simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
-        _ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-          -- Porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
-          simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
-          -- Porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
-          ring
-    have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
-      inv_mul_cancel_right₀ h₀.ne.symm ε ▸
-        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
-    calc
-      ‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
-        simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
-          Finset.card_fin] using
-          congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
-      _ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
-      _ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
-        rw [← mul_assoc]
-        simp [norm_mul, mul_pow]
-      _ ≤ 0 + ε := by
-        rw [inv_mul_cancel₀ (norm_pos_iff.mp k_pos)]
-        simpa using h₃.le
-
-/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
-terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
-theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
-    (h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
-  refine Nat.strong_induction_on n fun k hk => ?_
-  have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
-    funext z
-    refine Finset.sum_eq_single _ (fun b hb hnb => ?_) fun hn => ?_
-    · have := Finset.mem_range_succ_iff.mp hb
-      simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
-    · exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
-  replace h := h.isBigO_sub_partialSum_pow k.succ
-  simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
-  exact h.continuousMultilinearMap_apply_eq_zero
-
-/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
-theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
-    (h : HasFPowerSeriesAt 0 p x) : p = 0 := by
-  ext n x
-  rw [← mkPiRing_apply_one_eq_self (p n)]
-  simp [h.apply_eq_zero n 1]
-
-/-- One-dimensional formal multilinear series representing the same function are equal. -/
-theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
-    {f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
-  sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (x := x) (by simpa only [sub_self] using h₁.sub h₂))
-
-theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
-    {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
-    (hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
-  (hp.congr heq).eq_formalMultilinearSeries hq
-
-/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
-theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
-    {x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
-  (hp.congr hf).eq_zero
-
-/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
-which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
-series in one representation has a particularly nice form, but the other has a larger radius. -/
-theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
-    {r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
-    (h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
-  h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
-
-/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
-each positive radius it has some power series representation, then `p` converges to `f` on the whole
-`𝕜`. -/
-theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
-    {p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
-    (h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
-      HasFPowerSeriesOnBall f p' x r') :
-    HasFPowerSeriesOnBall f p x ∞ :=
-  { r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
-      let ⟨_, hp'⟩ := h' r hr
-      (h.exchange_radius hp').r_le
-    r_pos := ENNReal.coe_lt_top
-    hasSum := fun {y} _ =>
-      let ⟨r', hr'⟩ := exists_gt ‖y‖₊
-      let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
-      (h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
-
-end Uniqueness
-
-/-!
-### Changing origin in a power series
-
-If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
-one. Indeed, one can write
-$$
-f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
-= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
-$$
-The corresponding power series has thus a `k`-th coefficient equal to
-$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
-to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
-possible subsets `s` of `Fin n` of cardinality `k`, and attribute `z` to the indices in `s` and
-`y` to the indices outside of `s`.
-
-In this paragraph, we implement this. The new power series is called `p.changeOrigin y`. Then, we
-check its convergence and the fact that its sum coincides with the original sum. The outcome of this
-discussion is that the set of points where a function is analytic is open.
--/
-
-
-namespace FormalMultilinearSeries
-
-section
-
-variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
-
-/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
-
-Given a formal multilinear series `p` and a point `x` in its ball of convergence,
-`p.changeOrigin x` is a formal multilinear series such that
-`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
-is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
-`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
-The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
-p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
--/
-def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
-    E[×l]→L[𝕜] E[×k]→L[𝕜] F :=
-  let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
-    (by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
-  a (p (k + l))
-
-theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
-    (x y : E) :
-    (p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
-      p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
-  ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
-
-@[simp]
-theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
-    ‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
-  simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
-
-@[simp]
-theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
-    ‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
-  simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
-
-theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
-    (hs : s.card = l) (x y : E) :
-    ‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
-      ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
-  rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
-  apply ContinuousMultilinearMap.le_of_opNNNorm_le
-  apply ContinuousMultilinearMap.le_opNNNorm
-
-/-- The power series for `f.changeOrigin k`.
-
-Given a formal multilinear series `p` and a point `x` in its ball of convergence,
-`p.changeOrigin x` is a formal multilinear series such that
-`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
-the series `p.changeOriginSeries k`. -/
-def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
-  ∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
-
-theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
-    ‖p.changeOriginSeries k l‖₊ ≤
-      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
-  (nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
-    simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
-
-theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
-    ‖p.changeOriginSeries k l fun _ => x‖₊ ≤
-      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
-  rw [NNReal.tsum_mul_right, ← Fin.prod_const]
-  exact (p.changeOriginSeries k l).le_of_opNNNorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
-
-/-- Changing the origin of a formal multilinear series `p`, so that
-`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
--/
-def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
-  fun k => (p.changeOriginSeries k).sum x
-
-/-- An auxiliary equivalence useful in the proofs about
-`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
-`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
-`Finset (Fin n)`.
-
-The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
-`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
-with non-definitional equalities. -/
-@[simps]
-def changeOriginIndexEquiv :
-    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) where
-  toFun s := ⟨s.1 + s.2.1, s.2.2⟩
-  invFun s :=
-    ⟨s.1 - s.2.card, s.2.card,
-      ⟨s.2.map
-        (finCongr <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEmbedding,
-        Finset.card_map _⟩⟩
-  left_inv := by
-    rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
-    dsimp only [Subtype.coe_mk]
-    -- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
-    -- formulate the generalized goal
-    suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
-        (⟨k', l', ⟨s.map (finCongr hkl).toEmbedding, hs'⟩⟩ :
-          Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
-      apply this <;> simp only [hs, add_tsub_cancel_right]
-    rintro _ _ rfl rfl hkl hs'
-    simp only [Equiv.refl_toEmbedding, finCongr_refl, Finset.map_refl, eq_self_iff_true,
-      OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
-  right_inv := by
-    rintro ⟨n, s⟩
-    simp [tsub_add_cancel_of_le (card_finset_fin_le s), finCongr_eq_equivCast]
-
-lemma changeOriginSeriesTerm_changeOriginIndexEquiv_symm (n t) :
-    let s := changeOriginIndexEquiv.symm ⟨n, t⟩
-    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ ↦ x) (fun _ ↦ y) =
-    p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) := by
-  have : ∀ (m) (hm : n = m), p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) =
-      p m ((t.map (finCongr hm).toEmbedding).piecewise (fun _ ↦ x) fun _ ↦ y) := by
-    rintro m rfl
-    simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
-  simp_rw [changeOriginSeriesTerm_apply, eq_comm]; apply this
-
-theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
-    Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
-      ‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
-  rw [← changeOriginIndexEquiv.symm.summable_iff]
-  dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
-    changeOriginIndexEquiv_symm_apply_snd_fst]
-  have : ∀ n : ℕ,
-      HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
-        (‖p n‖₊ * (r + r') ^ n) := by
-    intro n
-    -- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
-    convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
-    · ext1 s
-      rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
-    rw [← Fin.sum_pow_mul_eq_add_pow]
-    exact (hasSum_fintype _).mul_left _
-  refine NNReal.summable_sigma.2 ⟨fun n => (this n).summable, ?_⟩
-  simp only [(this _).tsum_eq]
-  exact p.summable_nnnorm_mul_pow hr
-
-theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
-    Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
-      ‖p (k + s.1)‖₊ * r ^ s.1 := by
-  rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
-  simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
-    ((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
-
-theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
-    Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
-  refine NNReal.summable_of_le
-    (fun n => ?_) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
-  simp only [NNReal.tsum_mul_right]
-  exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
-
-theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
-  ENNReal.le_of_forall_nnreal_lt fun _r hr =>
-    le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
-
-theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
-    ‖p.changeOrigin x k‖₊ ≤
-      ∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
-  refine tsum_of_nnnorm_bounded ?_ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
-  have := p.changeOriginSeries_summable_aux₂ h k
-  refine HasSum.sigma this.hasSum fun l => ?_
-  exact ((NNReal.summable_sigma.1 this).1 l).hasSum
-
-/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
-`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
-convergence. -/
-theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
-  refine ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => ?_
-  rw [lt_tsub_iff_right, add_comm] at hr
-  have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
-  apply le_radius_of_summable_nnnorm
-  have : ∀ k : ℕ,
-      ‖p.changeOrigin x k‖₊ * r ^ k ≤
-        (∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
-          r ^ k :=
-    fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
-  refine NNReal.summable_of_le this ?_
-  simpa only [← NNReal.tsum_mul_right] using
-    (NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
-
-/-- `derivSeries p` is a power series for `fderiv 𝕜 f` if `p` is a power series for `f`,
-see `HasFPowerSeriesOnBall.fderiv`. -/
-def derivSeries : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F) :=
-  (continuousMultilinearCurryFin1 𝕜 E F : (E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F)
-    |>.compFormalMultilinearSeries (p.changeOriginSeries 1)
-
-end
-
--- From this point on, assume that the space is complete, to make sure that series that converge
--- in norm also converge in `F`.
-variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
-
-theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
-    HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
-  have := p.le_changeOriginSeries_radius k
-  ((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
-
-/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
-theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
-    (p.changeOrigin x).sum y = p.sum (x + y) := by
-  have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
-  have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
-    mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
-  have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
-    refine mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le ?_ p.changeOrigin_radius)
-    rwa [lt_tsub_iff_right, add_comm]
-  have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
-    refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ h)
-    exact mod_cast nnnorm_add_le x y
-  set f : (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
-    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
-  have hsf : Summable f := by
-    refine .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) ?_
-    rintro ⟨k, l, s, hs⟩
-    dsimp only [Subtype.coe_mk]
-    exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
-  have hf : HasSum f ((p.changeOrigin x).sum y) := by
-    refine HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => ?_) hsf
-    · dsimp only [f]
-      refine ContinuousMultilinearMap.hasSum_eval ?_ _
-      have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
-      rw [zero_add] at this
-      refine HasSum.sigma_of_hasSum this (fun l => ?_) ?_
-      · simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
-        apply hasSum_fintype
-      · refine .of_nnnorm_bounded _
-          (p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k)
-            fun s => ?_
-        refine (ContinuousMultilinearMap.le_opNNNorm _ _).trans_eq ?_
-        simp
-  refine hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 ?_)
-  refine HasSum.sigma_of_hasSum
-    (p.hasSum x_add_y_mem_ball) (fun n => ?_) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
-  erw [(p n).map_add_univ (fun _ => x) fun _ => y]
-  simp_rw [← changeOriginSeriesTerm_changeOriginIndexEquiv_symm]
-  exact hasSum_fintype (fun c => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
-
-/-- Power series terms are analytic as we vary the origin -/
-theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
-    AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
-  (FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
-
-end FormalMultilinearSeries
-
-section
-
-variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x y : E} {r : ℝ≥0∞}
-
-/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
-power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
--/
-theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
-    (h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) :=
-  { r_le := by
-      apply le_trans _ p.changeOrigin_radius
-      exact tsub_le_tsub hf.r_le le_rfl
-    r_pos := by simp [h]
-    hasSum := fun {z} hz => by
-      have : f (x + y + z) =
-          FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
-        rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
-        rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
-        refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)
-        exact mod_cast nnnorm_add_le y z
-      rw [this]
-      apply (p.changeOrigin y).hasSum
-      refine EMetric.ball_subset_ball (le_trans ?_ p.changeOrigin_radius) hz
-      exact tsub_le_tsub hf.r_le le_rfl }
-
-/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
-it is analytic at every point of this ball. -/
-theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
-    (h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
-  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
-  have := hf.changeOrigin this
-  rw [add_sub_cancel] at this
-  exact this.analyticAt
-
-theorem HasFPowerSeriesOnBall.analyticOn (hf : HasFPowerSeriesOnBall f p x r) :
-    AnalyticOn 𝕜 f (EMetric.ball x r) :=
-  fun _y hy => hf.analyticAt_of_mem hy
-
-variable (𝕜 f)
-
-/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
-that `f` is analytic at `x` is open. -/
-theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
-  rw [isOpen_iff_mem_nhds]
-  rintro x ⟨p, r, hr⟩
-  exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
-
-variable {𝕜}
-
-theorem AnalyticAt.eventually_analyticAt {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
-    ∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
-(isOpen_analyticAt 𝕜 f).mem_nhds h
-
-theorem AnalyticAt.exists_mem_nhds_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
-    ∃ s ∈ 𝓝 x, AnalyticOn 𝕜 f s :=
-h.eventually_analyticAt.exists_mem
-
-/-- If we're analytic at a point, we're analytic in a nonempty ball -/
-theorem AnalyticAt.exists_ball_analyticOn {f : E → F} {x : E} (h : AnalyticAt 𝕜 f x) :
-    ∃ r : ℝ, 0 < r ∧ AnalyticOn 𝕜 f (Metric.ball x r) :=
-  Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
-
-end
-
 section
 
 open FormalMultilinearSeries
@@ -1347,7 +1343,7 @@ theorem hasFPowerSeriesAt_iff :
   simp only [Metric.eventually_nhds_iff]
   rintro ⟨r, r_pos, h⟩
   refine ⟨p.radius ⊓ r.toNNReal, by simp, ?_, ?_⟩
-  · simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true_iff]
+  · simp only [r_pos.lt, lt_inf_iff, ENNReal.coe_pos, Real.toNNReal_pos, and_true]
     obtain ⟨z, z_pos, le_z⟩ := NormedField.exists_norm_lt 𝕜 r_pos.lt
     have : (‖z‖₊ : ENNReal) ≤ p.radius := by
       simp only [dist_zero_right] at h
diff --git a/Mathlib/Analysis/Analytic/CPolynomial.lean b/Mathlib/Analysis/Analytic/CPolynomial.lean
index e217c3fc8e1c6..9576f0b22c613 100644
--- a/Mathlib/Analysis/Analytic/CPolynomial.lean
+++ b/Mathlib/Analysis/Analytic/CPolynomial.lean
@@ -3,7 +3,8 @@ Copyright (c) 2023 Sophie Morel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sophie Morel
 -/
-import Mathlib.Analysis.Analytic.Basic
+import Mathlib.Analysis.Analytic.ChangeOrigin
+import Mathlib.Analysis.Analytic.Constructions
 
 /-! We specialize the theory fo analytic functions to the case of functions that admit a
 development given by a *finite* formal multilinear series. We call them "continuously polynomial",
@@ -28,15 +29,19 @@ for `n : ℕ`, and let `f` be a function from `E` to `F`.
 We develop the basic properties of these notions, notably:
 * If a function is continuously polynomial, then it is analytic, see
   `HasFiniteFPowerSeriesOnBall.hasFPowerSeriesOnBall`, `HasFiniteFPowerSeriesAt.hasFPowerSeriesAt`,
-  `CPolynomialAt.analyticAt` and `CPolynomialOn.analyticOn`.
+  `CPolynomialAt.analyticAt` and `CPolynomialOn.analyticOnNhd`.
 * The sum of a finite formal power series with positive radius is well defined on the whole space,
   see `FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_of_finite`.
 * If a function admits a finite power series in a ball, then it is continuously polynomial at
   any point `y` of this ball, and the power series there can be expressed in terms of the initial
   power series `p` as `p.changeOrigin y`, which is finite (with the same bound as `p`) by
-  `changeOrigin_finite_of_finite`. See `HasFiniteFPowerSeriesOnBall.changeOrigin `. It follows in
+  `changeOrigin_finite_of_finite`. See `HasFiniteFPowerSeriesOnBall.changeOrigin`. It follows in
   particular that the set of points at which a given function is continuously polynomial is open,
   see `isOpen_cPolynomialAt`.
+
+We prove in particular that continuous multilinear maps are continuously polynomial, and so
+are continuous linear maps into continuous multilinear maps. In particular, such maps are
+analytic.
 -/
 
 variable {𝕜 E F G : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
@@ -111,9 +116,16 @@ theorem CPolynomialAt.analyticAt (hf : CPolynomialAt 𝕜 f x) : AnalyticAt 𝕜
   let ⟨p, _, hp⟩ := hf
   ⟨p, hp.toHasFPowerSeriesAt⟩
 
-theorem CPolynomialOn.analyticOn {s : Set E} (hf : CPolynomialOn 𝕜 f s) : AnalyticOn 𝕜 f s :=
+theorem CPolynomialAt.analyticWithinAt {s : Set E} (hf : CPolynomialAt 𝕜 f x) :
+    AnalyticWithinAt 𝕜 f s x :=
+  hf.analyticAt.analyticWithinAt
+
+theorem CPolynomialOn.analyticOnNhd {s : Set E} (hf : CPolynomialOn 𝕜 f s) : AnalyticOnNhd 𝕜 f s :=
   fun x hx ↦ (hf x hx).analyticAt
 
+theorem CPolynomialOn.analyticOn {s : Set E} (hf : CPolynomialOn 𝕜 f s) : AnalyticOn 𝕜 f s :=
+  hf.analyticOnNhd.analyticOn
+
 theorem HasFiniteFPowerSeriesOnBall.congr (hf : HasFiniteFPowerSeriesOnBall f p x n r)
     (hg : EqOn f g (EMetric.ball x r)) : HasFiniteFPowerSeriesOnBall g p x n r :=
   ⟨hf.1.congr hg, hf.finite⟩
@@ -330,7 +342,7 @@ protected theorem CPolynomialAt.continuousAt (hf : CPolynomialAt 𝕜 f x) : Con
 
 protected theorem CPolynomialOn.continuousOn {s : Set E} (hf : CPolynomialOn 𝕜 f s) :
     ContinuousOn f s :=
-  hf.analyticOn.continuousOn
+  hf.analyticOnNhd.continuousOn
 
 /-- Continuously polynomial everywhere implies continuous -/
 theorem CPolynomialOn.continuous {f : E → F} (fa : CPolynomialOn 𝕜 f univ) : Continuous f := by
@@ -381,7 +393,7 @@ main point is that the new series `p.changeOrigin x` is still finite, with the s
 variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
 
 /-- If `p` is a formal multilinear series such that `p m = 0` for `n ≤ m`, then
-`p.changeOriginSeriesTerm k l = 0` for `n ≤ k + l`.  -/
+`p.changeOriginSeriesTerm k l = 0` for `n ≤ k + l`. -/
 lemma changeOriginSeriesTerm_bound (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
     (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k l : ℕ) {s : Finset (Fin (k + l))}
     (hs : s.card = l) (hkl : n ≤ k + l) :
@@ -393,7 +405,7 @@ lemma changeOriginSeriesTerm_bound (p : FormalMultilinearSeries 𝕜 E F) {n : 
 
 /-- If `p` is a finite formal multilinear series, then so is `p.changeOriginSeries k` for every
 `k` in `ℕ`. More precisely, if `p m = 0` for `n ≤ m`, then `p.changeOriginSeries k m = 0` for
-`n ≤ k + m`.  -/
+`n ≤ k + m`. -/
 lemma changeOriginSeries_finite_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
     (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) : ∀ {m : ℕ}, n ≤ k + m →
     p.changeOriginSeries k m = 0 := by
@@ -413,7 +425,7 @@ lemma changeOriginSeries_sum_eq_partialSum_of_finite (p : FormalMultilinearSerie
     ContinuousMultilinearMap.zero_apply]
 
 /-- If `p` is a formal multilinear series such that `p m = 0` for `n ≤ m`, then
-`p.changeOrigin x k = 0` for `n ≤ k`.  -/
+`p.changeOrigin x k = 0` for `n ≤ k`. -/
 lemma changeOrigin_finite_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
     (hn : ∀ (m : ℕ), n ≤ m → p m = 0) {k : ℕ} (hk : n ≤ k) :
     p.changeOrigin x k = 0 := by
@@ -539,3 +551,104 @@ theorem CPolynomialAt.exists_ball_cPolynomialOn {f : E → F} {x : E} (h : CPoly
   Metric.isOpen_iff.mp (isOpen_cPolynomialAt _ _) _ h
 
 end
+
+/-!
+### Continuous multilinear maps
+
+We show that continuous multilinear maps are continuously polynomial, and therefore analytic.
+-/
+
+namespace ContinuousMultilinearMap
+
+variable {ι : Type*} {Em : ι → Type*} [∀ i, NormedAddCommGroup (Em i)] [∀ i, NormedSpace 𝕜 (Em i)]
+  [Fintype ι] (f : ContinuousMultilinearMap 𝕜 Em F) {x : Π i, Em i} {s : Set (Π i, Em i)}
+
+open FormalMultilinearSeries
+
+protected theorem hasFiniteFPowerSeriesOnBall :
+    HasFiniteFPowerSeriesOnBall f f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤ :=
+  .mk' (fun m hm ↦ dif_neg (Nat.succ_le_iff.mp hm).ne) ENNReal.zero_lt_top fun y _ ↦ by
+    rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]
+    · rw [toFormalMultilinearSeries, dif_pos rfl]; rfl
+    · intro m _ ne; rw [toFormalMultilinearSeries, dif_neg ne.symm]; rfl
+
+lemma cpolynomialAt  : CPolynomialAt 𝕜 f x :=
+  f.hasFiniteFPowerSeriesOnBall.cPolynomialAt_of_mem
+    (by simp only [Metric.emetric_ball_top, Set.mem_univ])
+
+lemma cpolyomialOn : CPolynomialOn 𝕜 f s := fun _ _ ↦ f.cpolynomialAt
+
+lemma analyticOnNhd : AnalyticOnNhd 𝕜 f s := f.cpolyomialOn.analyticOnNhd
+
+lemma analyticOn : AnalyticOn 𝕜 f s := f.analyticOnNhd.analyticOn
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn := analyticOn
+
+lemma analyticAt : AnalyticAt 𝕜 f x := f.cpolynomialAt.analyticAt
+
+lemma analyticWithinAt : AnalyticWithinAt 𝕜 f s x := f.analyticAt.analyticWithinAt
+
+end ContinuousMultilinearMap
+
+
+/-!
+### Continuous linear maps into continuous multilinear maps
+
+We show that a continuous linear map into continuous multilinear maps is continuously polynomial
+(as a function of two variables, i.e., uncurried). Therefore, it is also analytic.
+-/
+
+namespace ContinuousLinearMap
+
+variable {ι : Type*} {Em : ι → Type*} [∀ i, NormedAddCommGroup (Em i)] [∀ i, NormedSpace 𝕜 (Em i)]
+  [Fintype ι] (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 Em F)
+  {s : Set (G × (Π i, Em i))} {x : G × (Π i, Em i)}
+
+/-- Formal multilinear series associated to a linear map into multilinear maps. -/
+noncomputable def toFormalMultilinearSeriesOfMultilinear :
+    FormalMultilinearSeries 𝕜 (G × (Π i, Em i)) F :=
+  fun n ↦ if h : Fintype.card (Option ι) = n then
+    (f.continuousMultilinearMapOption).domDomCongr (Fintype.equivFinOfCardEq h)
+  else 0
+
+protected theorem hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear :
+    HasFiniteFPowerSeriesOnBall (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2)
+      f.toFormalMultilinearSeriesOfMultilinear 0 (Fintype.card (Option ι) + 1) ⊤ := by
+  apply HasFiniteFPowerSeriesOnBall.mk' ?_ ENNReal.zero_lt_top  ?_
+  · intro m hm
+    apply dif_neg
+    exact Nat.ne_of_lt hm
+  · intro y _
+    rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]
+    · rw [toFormalMultilinearSeriesOfMultilinear, dif_pos rfl]; rfl
+    · intro m _ ne; rw [toFormalMultilinearSeriesOfMultilinear, dif_neg ne.symm]; rfl
+
+lemma cpolynomialAt_uncurry_of_multilinear :
+    CPolynomialAt 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) x :=
+  f.hasFiniteFPowerSeriesOnBall_uncurry_of_multilinear.cPolynomialAt_of_mem
+    (by simp only [Metric.emetric_ball_top, Set.mem_univ])
+
+lemma cpolyomialOn_uncurry_of_multilinear :
+    CPolynomialOn 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) s :=
+  fun _ _ ↦ f.cpolynomialAt_uncurry_of_multilinear
+
+lemma analyticOnNhd_uncurry_of_multilinear :
+    AnalyticOnNhd 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) s :=
+  f.cpolyomialOn_uncurry_of_multilinear.analyticOnNhd
+
+lemma analyticOn_uncurry_of_multilinear :
+    AnalyticOn 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) s :=
+  f.analyticOnNhd_uncurry_of_multilinear.analyticOn
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_uncurry_of_multilinear := analyticOn_uncurry_of_multilinear
+
+lemma analyticAt_uncurry_of_multilinear : AnalyticAt 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) x :=
+  f.cpolynomialAt_uncurry_of_multilinear.analyticAt
+
+lemma analyticWithinAt_uncurry_of_multilinear :
+    AnalyticWithinAt 𝕜 (fun (p : G × (Π i, Em i)) ↦ f p.1 p.2) s x :=
+  f.analyticAt_uncurry_of_multilinear.analyticWithinAt
+
+end ContinuousLinearMap
diff --git a/Mathlib/Analysis/Analytic/ChangeOrigin.lean b/Mathlib/Analysis/Analytic/ChangeOrigin.lean
new file mode 100644
index 0000000000000..1658276bf5b4a
--- /dev/null
+++ b/Mathlib/Analysis/Analytic/ChangeOrigin.lean
@@ -0,0 +1,391 @@
+/-
+Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sébastien Gouëzel, Yury Kudryashov
+-/
+import Mathlib.Analysis.Analytic.Basic
+
+/-!
+# Changing origin in a power series
+
+If a function is analytic in a disk `D(x, R)`, then it is analytic in any disk contained in that
+one. Indeed, one can write
+$$
+f (x + y + z) = \sum_{n} p_n (y + z)^n = \sum_{n, k} \binom{n}{k} p_n y^{n-k} z^k
+= \sum_{k} \Bigl(\sum_{n} \binom{n}{k} p_n y^{n-k}\Bigr) z^k.
+$$
+The corresponding power series has thus a `k`-th coefficient equal to
+$\sum_{n} \binom{n}{k} p_n y^{n-k}$. In the general case where `pₙ` is a multilinear map, this has
+to be interpreted suitably: instead of having a binomial coefficient, one should sum over all
+possible subsets `s` of `Fin n` of cardinality `k`, and attribute `z` to the indices in `s` and
+`y` to the indices outside of `s`.
+
+In this file, we implement this. The new power series is called `p.changeOrigin y`. Then, we
+check its convergence and the fact that its sum coincides with the original sum. The outcome of this
+discussion is that the set of points where a function is analytic is open. All these arguments
+require the target space to be complete, as otherwise the series might not converge.
+
+### Main results
+
+In a complete space, if a function admits a power series in a ball, then it is analytic at any
+point `y` of this ball, and the power series there can be expressed in terms of the initial power
+series `p` as `p.changeOrigin y`. See `HasFPowerSeriesOnBall.changeOrigin`. It follows in particular
+that the set of points at which a given function is analytic is open, see `isOpen_analyticAt`.
+-/
+
+noncomputable section
+
+open scoped NNReal ENNReal Topology
+open Filter Set
+
+variable {𝕜 E F : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+[NormedAddCommGroup F] [NormedSpace 𝕜 F]
+
+namespace FormalMultilinearSeries
+
+section
+
+variable (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
+
+/-- A term of `FormalMultilinearSeries.changeOriginSeries`.
+
+Given a formal multilinear series `p` and a point `x` in its ball of convergence,
+`p.changeOrigin x` is a formal multilinear series such that
+`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Each term of `p.changeOrigin x`
+is itself an analytic function of `x` given by the series `p.changeOriginSeries`. Each term in
+`changeOriginSeries` is the sum of `changeOriginSeriesTerm`'s over all `s` of cardinality `l`.
+The definition is such that `p.changeOriginSeriesTerm k l s hs (fun _ ↦ x) (fun _ ↦ y) =
+p (k + l) (s.piecewise (fun _ ↦ x) (fun _ ↦ y))`
+-/
+def changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
+    E[×l]→L[𝕜] E[×k]→L[𝕜] F :=
+  let a := ContinuousMultilinearMap.curryFinFinset 𝕜 E F hs
+    (by erw [Finset.card_compl, Fintype.card_fin, hs, add_tsub_cancel_right])
+  a (p (k + l))
+
+theorem changeOriginSeriesTerm_apply (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l)
+    (x y : E) :
+    (p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y) =
+      p (k + l) (s.piecewise (fun _ => x) fun _ => y) :=
+  ContinuousMultilinearMap.curryFinFinset_apply_const _ _ _ _ _
+
+@[simp]
+theorem norm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
+    ‖p.changeOriginSeriesTerm k l s hs‖ = ‖p (k + l)‖ := by
+  simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]
+
+@[simp]
+theorem nnnorm_changeOriginSeriesTerm (k l : ℕ) (s : Finset (Fin (k + l))) (hs : s.card = l) :
+    ‖p.changeOriginSeriesTerm k l s hs‖₊ = ‖p (k + l)‖₊ := by
+  simp only [changeOriginSeriesTerm, LinearIsometryEquiv.nnnorm_map]
+
+theorem nnnorm_changeOriginSeriesTerm_apply_le (k l : ℕ) (s : Finset (Fin (k + l)))
+    (hs : s.card = l) (x y : E) :
+    ‖p.changeOriginSeriesTerm k l s hs (fun _ => x) fun _ => y‖₊ ≤
+      ‖p (k + l)‖₊ * ‖x‖₊ ^ l * ‖y‖₊ ^ k := by
+  rw [← p.nnnorm_changeOriginSeriesTerm k l s hs, ← Fin.prod_const, ← Fin.prod_const]
+  apply ContinuousMultilinearMap.le_of_opNNNorm_le
+  apply ContinuousMultilinearMap.le_opNNNorm
+
+/-- The power series for `f.changeOrigin k`.
+
+Given a formal multilinear series `p` and a point `x` in its ball of convergence,
+`p.changeOrigin x` is a formal multilinear series such that
+`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense. Its `k`-th term is the sum of
+the series `p.changeOriginSeries k`. -/
+def changeOriginSeries (k : ℕ) : FormalMultilinearSeries 𝕜 E (E[×k]→L[𝕜] F) := fun l =>
+  ∑ s : { s : Finset (Fin (k + l)) // Finset.card s = l }, p.changeOriginSeriesTerm k l s s.2
+
+theorem nnnorm_changeOriginSeries_le_tsum (k l : ℕ) :
+    ‖p.changeOriginSeries k l‖₊ ≤
+      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ :=
+  (nnnorm_sum_le _ (fun t => changeOriginSeriesTerm p k l (Subtype.val t) t.prop)).trans_eq <| by
+    simp_rw [tsum_fintype, nnnorm_changeOriginSeriesTerm (p := p) (k := k) (l := l)]
+
+theorem nnnorm_changeOriginSeries_apply_le_tsum (k l : ℕ) (x : E) :
+    ‖p.changeOriginSeries k l fun _ => x‖₊ ≤
+      ∑' _ : { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + l)‖₊ * ‖x‖₊ ^ l := by
+  rw [NNReal.tsum_mul_right, ← Fin.prod_const]
+  exact (p.changeOriginSeries k l).le_of_opNNNorm_le _ (p.nnnorm_changeOriginSeries_le_tsum _ _)
+
+/-- Changing the origin of a formal multilinear series `p`, so that
+`p.sum (x+y) = (p.changeOrigin x).sum y` when this makes sense.
+-/
+def changeOrigin (x : E) : FormalMultilinearSeries 𝕜 E F :=
+  fun k => (p.changeOriginSeries k).sum x
+
+/-- An auxiliary equivalence useful in the proofs about
+`FormalMultilinearSeries.changeOriginSeries`: the set of triples `(k, l, s)`, where `s` is a
+`Finset (Fin (k + l))` of cardinality `l` is equivalent to the set of pairs `(n, s)`, where `s` is a
+`Finset (Fin n)`.
+
+The forward map sends `(k, l, s)` to `(k + l, s)` and the inverse map sends `(n, s)` to
+`(n - Finset.card s, Finset.card s, s)`. The actual definition is less readable because of problems
+with non-definitional equalities. -/
+@[simps]
+def changeOriginIndexEquiv :
+    (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) ≃ Σ n : ℕ, Finset (Fin n) where
+  toFun s := ⟨s.1 + s.2.1, s.2.2⟩
+  invFun s :=
+    ⟨s.1 - s.2.card, s.2.card,
+      ⟨s.2.map
+        (finCongr <| (tsub_add_cancel_of_le <| card_finset_fin_le s.2).symm).toEmbedding,
+        Finset.card_map _⟩⟩
+  left_inv := by
+    rintro ⟨k, l, ⟨s : Finset (Fin <| k + l), hs : s.card = l⟩⟩
+    dsimp only [Subtype.coe_mk]
+    -- Lean can't automatically generalize `k' = k + l - s.card`, `l' = s.card`, so we explicitly
+    -- formulate the generalized goal
+    suffices ∀ k' l', k' = k → l' = l → ∀ (hkl : k + l = k' + l') (hs'),
+        (⟨k', l', ⟨s.map (finCongr hkl).toEmbedding, hs'⟩⟩ :
+          Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) = ⟨k, l, ⟨s, hs⟩⟩ by
+      apply this <;> simp only [hs, add_tsub_cancel_right]
+    rintro _ _ rfl rfl hkl hs'
+    simp only [Equiv.refl_toEmbedding, finCongr_refl, Finset.map_refl, eq_self_iff_true,
+      OrderIso.refl_toEquiv, and_self_iff, heq_iff_eq]
+  right_inv := by
+    rintro ⟨n, s⟩
+    simp [tsub_add_cancel_of_le (card_finset_fin_le s), finCongr_eq_equivCast]
+
+lemma changeOriginSeriesTerm_changeOriginIndexEquiv_symm (n t) :
+    let s := changeOriginIndexEquiv.symm ⟨n, t⟩
+    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ ↦ x) (fun _ ↦ y) =
+    p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) := by
+  have : ∀ (m) (hm : n = m), p n (t.piecewise (fun _ ↦ x) fun _ ↦ y) =
+      p m ((t.map (finCongr hm).toEmbedding).piecewise (fun _ ↦ x) fun _ ↦ y) := by
+    rintro m rfl
+    simp (config := { unfoldPartialApp := true }) [Finset.piecewise]
+  simp_rw [changeOriginSeriesTerm_apply, eq_comm]; apply this
+
+theorem changeOriginSeries_summable_aux₁ {r r' : ℝ≥0} (hr : (r + r' : ℝ≥0∞) < p.radius) :
+    Summable fun s : Σk l : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
+      ‖p (s.1 + s.2.1)‖₊ * r ^ s.2.1 * r' ^ s.1 := by
+  rw [← changeOriginIndexEquiv.symm.summable_iff]
+  dsimp only [Function.comp_def, changeOriginIndexEquiv_symm_apply_fst,
+    changeOriginIndexEquiv_symm_apply_snd_fst]
+  have : ∀ n : ℕ,
+      HasSum (fun s : Finset (Fin n) => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card))
+        (‖p n‖₊ * (r + r') ^ n) := by
+    intro n
+    -- TODO: why `simp only [tsub_add_cancel_of_le (card_finset_fin_le _)]` fails?
+    convert_to HasSum (fun s : Finset (Fin n) => ‖p n‖₊ * (r ^ s.card * r' ^ (n - s.card))) _
+    · ext1 s
+      rw [tsub_add_cancel_of_le (card_finset_fin_le _), mul_assoc]
+    rw [← Fin.sum_pow_mul_eq_add_pow]
+    exact (hasSum_fintype _).mul_left _
+  refine NNReal.summable_sigma.2 ⟨fun n => (this n).summable, ?_⟩
+  simp only [(this _).tsum_eq]
+  exact p.summable_nnnorm_mul_pow hr
+
+theorem changeOriginSeries_summable_aux₂ (hr : (r : ℝ≥0∞) < p.radius) (k : ℕ) :
+    Summable fun s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l } =>
+      ‖p (k + s.1)‖₊ * r ^ s.1 := by
+  rcases ENNReal.lt_iff_exists_add_pos_lt.1 hr with ⟨r', h0, hr'⟩
+  simpa only [mul_inv_cancel_right₀ (pow_pos h0 _).ne'] using
+    ((NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr')).1 k).mul_right (r' ^ k)⁻¹
+
+theorem changeOriginSeries_summable_aux₃ {r : ℝ≥0} (hr : ↑r < p.radius) (k : ℕ) :
+    Summable fun l : ℕ => ‖p.changeOriginSeries k l‖₊ * r ^ l := by
+  refine NNReal.summable_of_le
+    (fun n => ?_) (NNReal.summable_sigma.1 <| p.changeOriginSeries_summable_aux₂ hr k).2
+  simp only [NNReal.tsum_mul_right]
+  exact mul_le_mul' (p.nnnorm_changeOriginSeries_le_tsum _ _) le_rfl
+
+theorem le_changeOriginSeries_radius (k : ℕ) : p.radius ≤ (p.changeOriginSeries k).radius :=
+  ENNReal.le_of_forall_nnreal_lt fun _r hr =>
+    le_radius_of_summable_nnnorm _ (p.changeOriginSeries_summable_aux₃ hr k)
+
+theorem nnnorm_changeOrigin_le (k : ℕ) (h : (‖x‖₊ : ℝ≥0∞) < p.radius) :
+    ‖p.changeOrigin x k‖₊ ≤
+      ∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1 := by
+  refine tsum_of_nnnorm_bounded ?_ fun l => p.nnnorm_changeOriginSeries_apply_le_tsum k l x
+  have := p.changeOriginSeries_summable_aux₂ h k
+  refine HasSum.sigma this.hasSum fun l => ?_
+  exact ((NNReal.summable_sigma.1 this).1 l).hasSum
+
+/-- The radius of convergence of `p.changeOrigin x` is at least `p.radius - ‖x‖`. In other words,
+`p.changeOrigin x` is well defined on the largest ball contained in the original ball of
+convergence. -/
+theorem changeOrigin_radius : p.radius - ‖x‖₊ ≤ (p.changeOrigin x).radius := by
+  refine ENNReal.le_of_forall_pos_nnreal_lt fun r _h0 hr => ?_
+  rw [lt_tsub_iff_right, add_comm] at hr
+  have hr' : (‖x‖₊ : ℝ≥0∞) < p.radius := (le_add_right le_rfl).trans_lt hr
+  apply le_radius_of_summable_nnnorm
+  have : ∀ k : ℕ,
+      ‖p.changeOrigin x k‖₊ * r ^ k ≤
+        (∑' s : Σl : ℕ, { s : Finset (Fin (k + l)) // s.card = l }, ‖p (k + s.1)‖₊ * ‖x‖₊ ^ s.1) *
+          r ^ k :=
+    fun k => mul_le_mul_right' (p.nnnorm_changeOrigin_le k hr') (r ^ k)
+  refine NNReal.summable_of_le this ?_
+  simpa only [← NNReal.tsum_mul_right] using
+    (NNReal.summable_sigma.1 (p.changeOriginSeries_summable_aux₁ hr)).2
+
+/-- `derivSeries p` is a power series for `fderiv 𝕜 f` if `p` is a power series for `f`,
+see `HasFPowerSeriesOnBall.fderiv`. -/
+def derivSeries : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F) :=
+  (continuousMultilinearCurryFin1 𝕜 E F : (E[×1]→L[𝕜] F) →L[𝕜] E →L[𝕜] F)
+    |>.compFormalMultilinearSeries (p.changeOriginSeries 1)
+
+theorem radius_le_radius_derivSeries : p.radius ≤ p.derivSeries.radius := by
+  apply (p.le_changeOriginSeries_radius 1).trans (radius_le_of_le (fun n ↦ ?_))
+  apply (ContinuousLinearMap.norm_compContinuousMultilinearMap_le _ _).trans
+  apply mul_le_of_le_one_left (norm_nonneg  _)
+  exact ContinuousLinearMap.opNorm_le_bound _ zero_le_one (by simp)
+
+end
+
+-- From this point on, assume that the space is complete, to make sure that series that converge
+-- in norm also converge in `F`.
+variable [CompleteSpace F] (p : FormalMultilinearSeries 𝕜 E F) {x y : E} {r R : ℝ≥0}
+
+theorem hasFPowerSeriesOnBall_changeOrigin (k : ℕ) (hr : 0 < p.radius) :
+    HasFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 p.radius :=
+  have := p.le_changeOriginSeries_radius k
+  ((p.changeOriginSeries k).hasFPowerSeriesOnBall (hr.trans_le this)).mono hr this
+
+/-- Summing the series `p.changeOrigin x` at a point `y` gives back `p (x + y)`. -/
+theorem changeOrigin_eval (h : (‖x‖₊ + ‖y‖₊ : ℝ≥0∞) < p.radius) :
+    (p.changeOrigin x).sum y = p.sum (x + y) := by
+  have radius_pos : 0 < p.radius := lt_of_le_of_lt (zero_le _) h
+  have x_mem_ball : x ∈ EMetric.ball (0 : E) p.radius :=
+    mem_emetric_ball_zero_iff.2 ((le_add_right le_rfl).trans_lt h)
+  have y_mem_ball : y ∈ EMetric.ball (0 : E) (p.changeOrigin x).radius := by
+    refine mem_emetric_ball_zero_iff.2 (lt_of_lt_of_le ?_ p.changeOrigin_radius)
+    rwa [lt_tsub_iff_right, add_comm]
+  have x_add_y_mem_ball : x + y ∈ EMetric.ball (0 : E) p.radius := by
+    refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ h)
+    exact mod_cast nnnorm_add_le x y
+  set f : (Σ k l : ℕ, { s : Finset (Fin (k + l)) // s.card = l }) → F := fun s =>
+    p.changeOriginSeriesTerm s.1 s.2.1 s.2.2 s.2.2.2 (fun _ => x) fun _ => y
+  have hsf : Summable f := by
+    refine .of_nnnorm_bounded _ (p.changeOriginSeries_summable_aux₁ h) ?_
+    rintro ⟨k, l, s, hs⟩
+    dsimp only [Subtype.coe_mk]
+    exact p.nnnorm_changeOriginSeriesTerm_apply_le _ _ _ _ _ _
+  have hf : HasSum f ((p.changeOrigin x).sum y) := by
+    refine HasSum.sigma_of_hasSum ((p.changeOrigin x).summable y_mem_ball).hasSum (fun k => ?_) hsf
+    · dsimp only [f]
+      refine ContinuousMultilinearMap.hasSum_eval ?_ _
+      have := (p.hasFPowerSeriesOnBall_changeOrigin k radius_pos).hasSum x_mem_ball
+      rw [zero_add] at this
+      refine HasSum.sigma_of_hasSum this (fun l => ?_) ?_
+      · simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply]
+        apply hasSum_fintype
+      · refine .of_nnnorm_bounded _
+          (p.changeOriginSeries_summable_aux₂ (mem_emetric_ball_zero_iff.1 x_mem_ball) k)
+            fun s => ?_
+        refine (ContinuousMultilinearMap.le_opNNNorm _ _).trans_eq ?_
+        simp
+  refine hf.unique (changeOriginIndexEquiv.symm.hasSum_iff.1 ?_)
+  refine HasSum.sigma_of_hasSum
+    (p.hasSum x_add_y_mem_ball) (fun n => ?_) (changeOriginIndexEquiv.symm.summable_iff.2 hsf)
+  erw [(p n).map_add_univ (fun _ => x) fun _ => y]
+  simp_rw [← changeOriginSeriesTerm_changeOriginIndexEquiv_symm]
+  exact hasSum_fintype (fun c => f (changeOriginIndexEquiv.symm ⟨n, c⟩))
+
+/-- Power series terms are analytic as we vary the origin -/
+theorem analyticAt_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) (rp : p.radius > 0) (n : ℕ) :
+    AnalyticAt 𝕜 (fun x ↦ p.changeOrigin x n) 0 :=
+  (FormalMultilinearSeries.hasFPowerSeriesOnBall_changeOrigin p n rp).analyticAt
+
+end FormalMultilinearSeries
+
+
+section
+
+variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E}
+  {x y : E} {r : ℝ≥0∞}
+
+/-- If a function admits a power series expansion `p` within a set `s` on a ball `B (x, r)`, then
+it also admits a power series on any subball of this ball (even with a different center provided
+it belongs to `s`), given by `p.changeOrigin`. -/
+theorem HasFPowerSeriesWithinOnBall.changeOrigin (hf : HasFPowerSeriesWithinOnBall f p s x r)
+    (h : (‖y‖₊ : ℝ≥0∞) < r) (hy : x + y ∈ insert x s) :
+    HasFPowerSeriesWithinOnBall f (p.changeOrigin y) s (x + y) (r - ‖y‖₊) where
+  r_le := by
+    apply le_trans _ p.changeOrigin_radius
+    exact tsub_le_tsub hf.r_le le_rfl
+  r_pos := by simp [h]
+  hasSum {z} h'z hz := by
+    have : f (x + y + z) =
+        FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
+      rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
+      rw [p.changeOrigin_eval (hz.trans_le hf.r_le), add_assoc, hf.sum]
+      · have : insert (x + y) s ⊆ insert (x + y) (insert x s) := by
+          apply insert_subset_insert (subset_insert _ _)
+        rw [insert_eq_of_mem hy] at this
+        apply this
+        simpa [add_assoc] using h'z
+      refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)
+      exact mod_cast nnnorm_add_le y z
+    rw [this]
+    apply (p.changeOrigin y).hasSum
+    refine EMetric.ball_subset_ball (le_trans ?_ p.changeOrigin_radius) hz
+    exact tsub_le_tsub hf.r_le le_rfl
+
+/-- If a function admits a power series expansion `p` on a ball `B (x, r)`, then it also admits a
+power series on any subball of this ball (even with a different center), given by `p.changeOrigin`.
+-/
+theorem HasFPowerSeriesOnBall.changeOrigin (hf : HasFPowerSeriesOnBall f p x r)
+    (h : (‖y‖₊ : ℝ≥0∞) < r) : HasFPowerSeriesOnBall f (p.changeOrigin y) (x + y) (r - ‖y‖₊) := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf ⊢
+  exact hf.changeOrigin h (by simp)
+
+/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
+it is analytic at every point of this ball. -/
+theorem HasFPowerSeriesWithinOnBall.analyticWithinAt_of_mem
+    (hf : HasFPowerSeriesWithinOnBall f p s x r)
+    (h : y ∈ insert x s ∩ EMetric.ball x r) : AnalyticWithinAt 𝕜 f s y := by
+  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h.2
+  have := hf.changeOrigin this (by simpa using h.1)
+  rw [add_sub_cancel] at this
+  exact this.analyticWithinAt
+
+/-- If a function admits a power series expansion `p` on an open ball `B (x, r)`, then
+it is analytic at every point of this ball. -/
+theorem HasFPowerSeriesOnBall.analyticAt_of_mem (hf : HasFPowerSeriesOnBall f p x r)
+    (h : y ∈ EMetric.ball x r) : AnalyticAt 𝕜 f y := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf
+  rw [← analyticWithinAt_univ]
+  exact hf.analyticWithinAt_of_mem (by simpa using h)
+
+theorem HasFPowerSeriesWithinOnBall.analyticOn (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    AnalyticOn 𝕜 f (insert x s ∩ EMetric.ball x r) :=
+  fun _ hy ↦ ((analyticWithinAt_insert (y := x)).2 (hf.analyticWithinAt_of_mem hy)).mono
+    inter_subset_left
+
+theorem HasFPowerSeriesOnBall.analyticOnNhd (hf : HasFPowerSeriesOnBall f p x r) :
+    AnalyticOnNhd 𝕜 f (EMetric.ball x r) :=
+  fun _y hy => hf.analyticAt_of_mem hy
+
+@[deprecated (since := "2024-09-26")]
+alias HasFPowerSeriesOnBall.analyticOn := HasFPowerSeriesOnBall.analyticOnNhd
+
+variable (𝕜 f) in
+/-- For any function `f` from a normed vector space to a Banach space, the set of points `x` such
+that `f` is analytic at `x` is open. -/
+theorem isOpen_analyticAt : IsOpen { x | AnalyticAt 𝕜 f x } := by
+  rw [isOpen_iff_mem_nhds]
+  rintro x ⟨p, r, hr⟩
+  exact mem_of_superset (EMetric.ball_mem_nhds _ hr.r_pos) fun y hy => hr.analyticAt_of_mem hy
+
+theorem AnalyticAt.eventually_analyticAt (h : AnalyticAt 𝕜 f x) :
+    ∀ᶠ y in 𝓝 x, AnalyticAt 𝕜 f y :=
+  (isOpen_analyticAt 𝕜 f).mem_nhds h
+
+theorem AnalyticAt.exists_mem_nhds_analyticOnNhd (h : AnalyticAt 𝕜 f x) :
+    ∃ s ∈ 𝓝 x, AnalyticOnNhd 𝕜 f s :=
+  h.eventually_analyticAt.exists_mem
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticAt.exists_mem_nhds_analyticOn := AnalyticAt.exists_mem_nhds_analyticOnNhd
+
+/-- If we're analytic at a point, we're analytic in a nonempty ball -/
+theorem AnalyticAt.exists_ball_analyticOnNhd (h : AnalyticAt 𝕜 f x) :
+    ∃ r : ℝ, 0 < r ∧ AnalyticOnNhd 𝕜 f (Metric.ball x r) :=
+  Metric.isOpen_iff.mp (isOpen_analyticAt _ _) _ h
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticAt.exists_ball_analyticOn := AnalyticAt.exists_ball_analyticOnNhd
+
+end
diff --git a/Mathlib/Analysis/Analytic/Composition.lean b/Mathlib/Analysis/Analytic/Composition.lean
index 52ae7340407a9..c73176c9a5ec5 100644
--- a/Mathlib/Analysis/Analytic/Composition.lean
+++ b/Mathlib/Analysis/Analytic/Composition.lean
@@ -331,29 +331,34 @@ section
 variable (𝕜 E)
 
 /-- The identity formal multilinear series, with all coefficients equal to `0` except for `n = 1`
-where it is (the continuous multilinear version of) the identity. -/
-def id : FormalMultilinearSeries 𝕜 E E
-  | 0 => 0
+where it is (the continuous multilinear version of) the identity. We allow an arbitrary
+constant coefficient `x`. -/
+def id (x : E) : FormalMultilinearSeries 𝕜 E E
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ x
   | 1 => (continuousMultilinearCurryFin1 𝕜 E E).symm (ContinuousLinearMap.id 𝕜 E)
   | _ => 0
 
+@[simp] theorem id_apply_zero (x : E) (v : Fin 0 → E) :
+    (FormalMultilinearSeries.id 𝕜 E x) 0 v = x := rfl
+
 /-- The first coefficient of `id 𝕜 E` is the identity. -/
 @[simp]
-theorem id_apply_one (v : Fin 1 → E) : (FormalMultilinearSeries.id 𝕜 E) 1 v = v 0 :=
+theorem id_apply_one (x : E) (v : Fin 1 → E) : (FormalMultilinearSeries.id 𝕜 E x) 1 v = v 0 :=
   rfl
 
 /-- The `n`th coefficient of `id 𝕜 E` is the identity when `n = 1`. We state this in a dependent
 way, as it will often appear in this form. -/
-theorem id_apply_one' {n : ℕ} (h : n = 1) (v : Fin n → E) :
-    (id 𝕜 E) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ := by
+theorem id_apply_one' (x : E) {n : ℕ} (h : n = 1) (v : Fin n → E) :
+    (id 𝕜 E x) n v = v ⟨0, h.symm ▸ zero_lt_one⟩ := by
   subst n
   apply id_apply_one
 
 /-- For `n ≠ 1`, the `n`-th coefficient of `id 𝕜 E` is zero, by definition. -/
 @[simp]
-theorem id_apply_ne_one {n : ℕ} (h : n ≠ 1) : (FormalMultilinearSeries.id 𝕜 E) n = 0 := by
+theorem id_apply_of_one_lt (x : E) {n : ℕ} (h : 1 < n) :
+    (FormalMultilinearSeries.id 𝕜 E x) n = 0 := by
   cases' n with n
-  · rfl
+  · contradiction
   · cases n
     · contradiction
     · rfl
@@ -361,11 +366,11 @@ theorem id_apply_ne_one {n : ℕ} (h : n ≠ 1) : (FormalMultilinearSeries.id 
 end
 
 @[simp]
-theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) : p.comp (id 𝕜 E) = p := by
+theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) (x : E) : p.comp (id 𝕜 E x) = p := by
   ext1 n
   dsimp [FormalMultilinearSeries.comp]
   rw [Finset.sum_eq_single (Composition.ones n)]
-  · show compAlongComposition p (id 𝕜 E) (Composition.ones n) = p n
+  · show compAlongComposition p (id 𝕜 E x) (Composition.ones n) = p n
     ext v
     rw [compAlongComposition_apply]
     apply p.congr (Composition.ones_length n)
@@ -375,51 +380,61 @@ theorem comp_id (p : FormalMultilinearSeries 𝕜 E F) : p.comp (id 𝕜 E) = p
     rw [Fin.ext_iff, Fin.coe_castLE, Fin.val_mk]
   · show
     ∀ b : Composition n,
-      b ∈ Finset.univ → b ≠ Composition.ones n → compAlongComposition p (id 𝕜 E) b = 0
+      b ∈ Finset.univ → b ≠ Composition.ones n → compAlongComposition p (id 𝕜 E x) b = 0
     intro b _ hb
     obtain ⟨k, hk, lt_k⟩ : ∃ (k : ℕ), k ∈ Composition.blocks b ∧ 1 < k :=
       Composition.ne_ones_iff.1 hb
     obtain ⟨i, hi⟩ : ∃ (i : Fin b.blocks.length), b.blocks[i] = k :=
       List.get_of_mem hk
-
     let j : Fin b.length := ⟨i.val, b.blocks_length ▸ i.prop⟩
     have A : 1 < b.blocksFun j := by convert lt_k
     ext v
     rw [compAlongComposition_apply, ContinuousMultilinearMap.zero_apply]
     apply ContinuousMultilinearMap.map_coord_zero _ j
     dsimp [applyComposition]
-    rw [id_apply_ne_one _ _ (ne_of_gt A)]
+    rw [id_apply_of_one_lt _ _ _ A]
     rfl
   · simp
 
 @[simp]
-theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (h : p 0 = 0) : (id 𝕜 F).comp p = p := by
+theorem id_comp (p : FormalMultilinearSeries 𝕜 E F) (v0 : Fin 0 → E) :
+    (id 𝕜 F (p 0 v0)).comp p = p := by
   ext1 n
   by_cases hn : n = 0
-  · rw [hn, h]
+  · rw [hn]
     ext v
-    rw [comp_coeff_zero', id_apply_ne_one _ _ zero_ne_one]
-    rfl
+    simp only [comp_coeff_zero', id_apply_zero]
+    congr with i
+    exact i.elim0
   · dsimp [FormalMultilinearSeries.comp]
     have n_pos : 0 < n := bot_lt_iff_ne_bot.mpr hn
     rw [Finset.sum_eq_single (Composition.single n n_pos)]
-    · show compAlongComposition (id 𝕜 F) p (Composition.single n n_pos) = p n
+    · show compAlongComposition (id 𝕜 F (p 0 v0)) p (Composition.single n n_pos) = p n
       ext v
-      rw [compAlongComposition_apply, id_apply_one' _ _ (Composition.single_length n_pos)]
+      rw [compAlongComposition_apply, id_apply_one' _ _ _ (Composition.single_length n_pos)]
       dsimp [applyComposition]
       refine p.congr rfl fun i him hin => congr_arg v <| ?_
       ext; simp
     · show
-      ∀ b : Composition n,
-        b ∈ Finset.univ → b ≠ Composition.single n n_pos → compAlongComposition (id 𝕜 F) p b = 0
+      ∀ b : Composition n, b ∈ Finset.univ → b ≠ Composition.single n n_pos →
+        compAlongComposition (id 𝕜 F (p 0 v0)) p b = 0
       intro b _ hb
-      have A : b.length ≠ 1 := by simpa [Composition.eq_single_iff_length] using hb
+      have A : 1 < b.length := by
+        have : b.length ≠ 1 := by simpa [Composition.eq_single_iff_length] using hb
+        have : 0 < b.length := Composition.length_pos_of_pos b n_pos
+        omega
       ext v
-      rw [compAlongComposition_apply, id_apply_ne_one _ _ A]
+      rw [compAlongComposition_apply, id_apply_of_one_lt _ _ _ A]
       rfl
     · simp
 
-/-! ### Summability properties of the composition of formal power series-/
+/-- Variant of `id_comp` in which the zero coefficient is given by an equality hypothesis instead
+of a definitional equality. Useful for rewriting or simplifying out in some situations. -/
+theorem id_comp' (p : FormalMultilinearSeries 𝕜 E F) (x : F) (v0 : Fin 0 → E) (h : x = p 0 v0) :
+    (id 𝕜 F x).comp p = p := by
+  simp [h]
+
+/-! ### Summability properties of the composition of formal power series -/
 
 
 section
@@ -458,7 +473,7 @@ theorem comp_summable_nnreal (q : FormalMultilinearSeries 𝕜 F G) (p : FormalM
         simp only [Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, c.sum_blocksFun]
       _ ≤ ∏ _i : Fin c.length, Cp := Finset.prod_le_prod' fun i _ => hCp _
       _ = Cp ^ c.length := by simp
-      _ ≤ Cp ^ n := pow_le_pow_right hCp1 c.length_le
+      _ ≤ Cp ^ n := pow_right_mono₀ hCp1 c.length_le
     calc
       ‖q.compAlongComposition p c‖₊ * r ^ n ≤
           (‖q c.length‖₊ * ∏ i, ‖p (c.blocksFun i)‖₊) * r ^ n :=
@@ -527,8 +542,7 @@ def compPartialSumSource (m M N : ℕ) : Finset (Σ n, Fin n → ℕ) :=
 theorem mem_compPartialSumSource_iff (m M N : ℕ) (i : Σ n, Fin n → ℕ) :
     i ∈ compPartialSumSource m M N ↔
       (m ≤ i.1 ∧ i.1 < M) ∧ ∀ a : Fin i.1, 1 ≤ i.2 a ∧ i.2 a < N := by
-  simp only [compPartialSumSource, Finset.mem_Ico, Fintype.mem_piFinset, Finset.mem_sigma,
-    iff_self_iff]
+  simp only [compPartialSumSource, Finset.mem_Ico, Fintype.mem_piFinset, Finset.mem_sigma]
 
 /-- Change of variables appearing to compute the composition of partial sums of formal
 power series -/
@@ -569,7 +583,7 @@ theorem compPartialSumTargetSet_image_compPartialSumSource (m M N : ℕ)
   rcases i with ⟨n, c⟩
   refine ⟨⟨c.length, c.blocksFun⟩, ?_, ?_⟩
   · simp only [compPartialSumTargetSet, Set.mem_setOf_eq] at hi
-    simp only [mem_compPartialSumSource_iff, hi.left, hi.right, true_and_iff, and_true_iff]
+    simp only [mem_compPartialSumSource_iff, hi.left, hi.right, true_and, and_true]
     exact fun a => c.one_le_blocks' _
   · dsimp [compChangeOfVariables]
     rw [Composition.sigma_eq_iff_blocks_eq]
@@ -608,7 +622,7 @@ theorem compChangeOfVariables_sum {α : Type*} [AddCommMonoid α] (m M N : ℕ)
     -- Porting note: added
     simp only at H
     simp only [mem_compPartialSumTarget_iff, Composition.length, Composition.blocks, H.left,
-      map_ofFn, length_ofFn, true_and_iff, compChangeOfVariables]
+      map_ofFn, length_ofFn, true_and, compChangeOfVariables]
     intro j
     simp only [Composition.blocksFun, (H.right _).right, List.get_ofFn]
   -- 2 - show that the map is injective
@@ -635,11 +649,12 @@ theorem compChangeOfVariables_sum {α : Type*} [AddCommMonoid α] (m M N : ℕ)
 
 /-- The auxiliary set corresponding to the composition of partial sums asymptotically contains
 all possible compositions. -/
-theorem compPartialSumTarget_tendsto_atTop :
-    Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop := by
+theorem compPartialSumTarget_tendsto_prod_atTop :
+    Tendsto (fun (p : ℕ × ℕ) => compPartialSumTarget 0 p.1 p.2) atTop atTop := by
   apply Monotone.tendsto_atTop_finset
   · intro m n hmn a ha
-    have : ∀ i, i < m → i < n := fun i hi => lt_of_lt_of_le hi hmn
+    have : ∀ i, i < m.1 → i < n.1 := fun i hi => lt_of_lt_of_le hi hmn.1
+    have : ∀ i, i < m.2 → i < n.2 := fun i hi => lt_of_lt_of_le hi hmn.2
     aesop
   · rintro ⟨n, c⟩
     simp only [mem_compPartialSumTarget_iff]
@@ -651,40 +666,48 @@ theorem compPartialSumTarget_tendsto_atTop :
     apply hn
     simp only [Finset.mem_image_of_mem, Finset.mem_coe, Finset.mem_univ]
 
+/-- The auxiliary set corresponding to the composition of partial sums asymptotically contains
+all possible compositions. -/
+theorem compPartialSumTarget_tendsto_atTop :
+    Tendsto (fun N => compPartialSumTarget 0 N N) atTop atTop := by
+  apply Tendsto.comp compPartialSumTarget_tendsto_prod_atTop tendsto_atTop_diagonal
+
 /-- Composing the partial sums of two multilinear series coincides with the sum over all
 compositions in `compPartialSumTarget 0 N N`. This is precisely the motivation for the
 definition of `compPartialSumTarget`. -/
 theorem comp_partialSum (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F)
-    (N : ℕ) (z : E) :
-    q.partialSum N (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
-      ∑ i ∈ compPartialSumTarget 0 N N, q.compAlongComposition p i.2 fun _j => z := by
+    (M N : ℕ) (z : E) :
+    q.partialSum M (∑ i ∈ Finset.Ico 1 N, p i fun _j => z) =
+      ∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z := by
   -- we expand the composition, using the multilinearity of `q` to expand along each coordinate.
   suffices H :
-    (∑ n ∈ Finset.range N,
+    (∑ n ∈ Finset.range M,
         ∑ r ∈ Fintype.piFinset fun i : Fin n => Finset.Ico 1 N,
           q n fun i : Fin n => p (r i) fun _j => z) =
-      ∑ i ∈ compPartialSumTarget 0 N N, q.compAlongComposition p i.2 fun _j => z by
+      ∑ i ∈ compPartialSumTarget 0 M N, q.compAlongComposition p i.2 fun _j => z by
     simpa only [FormalMultilinearSeries.partialSum, ContinuousMultilinearMap.map_sum_finset] using H
   -- rewrite the first sum as a big sum over a sigma type, in the finset
   -- `compPartialSumTarget 0 N N`
   rw [Finset.range_eq_Ico, Finset.sum_sigma']
   -- use `compChangeOfVariables_sum`, saying that this change of variables respects sums
-  apply compChangeOfVariables_sum 0 N N
+  apply compChangeOfVariables_sum 0 M N
   rintro ⟨k, blocks_fun⟩ H
-  apply congr _ (compChangeOfVariables_length 0 N N H).symm
+  apply congr _ (compChangeOfVariables_length 0 M N H).symm
   intros
-  rw [← compChangeOfVariables_blocksFun 0 N N H]
+  rw [← compChangeOfVariables_blocksFun 0 M N H]
   rfl
 
 end FormalMultilinearSeries
 
 open FormalMultilinearSeries
 
-/-- If two functions `g` and `f` have power series `q` and `p` respectively at `f x` and `x`, then
-`g ∘ f` admits the power series `q.comp p` at `x`. -/
-theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
-    {p : FormalMultilinearSeries 𝕜 E F} {x : E} (hg : HasFPowerSeriesAt g q (f x))
-    (hf : HasFPowerSeriesAt f p x) : HasFPowerSeriesAt (g ∘ f) (q.comp p) x := by
+/-- If two functions `g` and `f` have power series `q` and `p` respectively at `f x` and `x`, within
+two sets `s` and `t` such that `f` maps `s` to `t`, then `g ∘ f` admits the power
+series `q.comp p` at `x` within `s`. -/
+theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
+    {p : FormalMultilinearSeries 𝕜 E F} {x : E} {t : Set F} {s : Set E}
+    (hg : HasFPowerSeriesWithinAt g q t (f x)) (hf : HasFPowerSeriesWithinAt f p s x)
+    (hs : Set.MapsTo f s t) : HasFPowerSeriesWithinAt (g ∘ f) (q.comp p) s x := by
   /- Consider `rf` and `rg` such that `f` and `g` have power series expansion on the disks
     of radius `rf` and `rg`. -/
   rcases hg with ⟨rg, Hg⟩
@@ -695,10 +718,14 @@ theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinea
     `f (x + y)` is close enough to `f x` to be in the disk where `g` is well behaved. Let
     `min (r, rf, δ)` be this new radius. -/
   obtain ⟨δ, δpos, hδ⟩ :
-    ∃ δ : ℝ≥0∞, 0 < δ ∧ ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg := by
-    have : EMetric.ball (f x) rg ∈ 𝓝 (f x) := EMetric.ball_mem_nhds _ Hg.r_pos
-    rcases EMetric.mem_nhds_iff.1 (Hf.analyticAt.continuousAt this) with ⟨δ, δpos, Hδ⟩
-    exact ⟨δ, δpos, fun hz => Hδ hz⟩
+    ∃ δ : ℝ≥0∞, 0 < δ ∧ ∀ {z : E}, z ∈ insert x s ∩ EMetric.ball x δ
+      → f z ∈ insert (f x) t ∩ EMetric.ball (f x) rg := by
+    have : insert (f x) t ∩ EMetric.ball (f x) rg ∈ 𝓝[insert (f x) t] (f x) := by
+      apply inter_mem_nhdsWithin
+      exact EMetric.ball_mem_nhds _ Hg.r_pos
+    have := Hf.analyticWithinAt.continuousWithinAt_insert.tendsto_nhdsWithin (hs.insert x) this
+    rcases EMetric.mem_nhdsWithin_iff.1 this with ⟨δ, δpos, Hδ⟩
+    exact ⟨δ, δpos, fun {z} hz => Hδ (by rwa [Set.inter_comm])⟩
   let rf' := min rf δ
   have min_pos : 0 < min rf' r := by
     simp only [rf', r_pos, Hf.r_pos, δpos, lt_min_iff, ENNReal.coe_pos, and_self_iff]
@@ -707,17 +734,17 @@ theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinea
   refine ⟨min rf' r, ?_⟩
   refine
     ⟨le_trans (min_le_right rf' r) (FormalMultilinearSeries.le_comp_radius_of_summable q p r hr),
-      min_pos, @fun y hy => ?_⟩
+      min_pos, fun {y} h'y hy ↦ ?_⟩
   /- Let `y` satisfy `‖y‖ < min (r, rf', δ)`. We want to show that `g (f (x + y))` is the sum of
     `q.comp p` applied to `y`. -/
   -- First, check that `y` is small enough so that estimates for `f` and `g` apply.
   have y_mem : y ∈ EMetric.ball (0 : E) rf :=
     (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_left _ _))) hy
-  have fy_mem : f (x + y) ∈ EMetric.ball (f x) rg := by
+  have fy_mem : f (x + y) ∈ insert (f x) t ∩ EMetric.ball (f x) rg := by
     apply hδ
     have : y ∈ EMetric.ball (0 : E) δ :=
       (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_right _ _))) hy
-    simpa [edist_eq_coe_nnnorm_sub, edist_eq_coe_nnnorm]
+    simpa [-Set.mem_insert_iff, edist_eq_coe_nnnorm_sub, h'y]
   /- Now the proof starts. To show that the sum of `q.comp p` at `y` is `g (f (x + y))`,
     we will write `q.comp p` applied to `y` as a big sum over all compositions.
     Since the sum is summable, to get its convergence it suffices to get
@@ -727,11 +754,11 @@ theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinea
     To show that it converges to `g (f (x + y))`, pointwise convergence would not be enough,
     but we have uniform convergence to save the day. -/
   -- First step: the partial sum of `p` converges to `f (x + y)`.
-  have A : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, p a fun _b => y)
-      atTop (𝓝 (f (x + y) - f x)) := by
-    have L :
-      ∀ᶠ n in atTop, (∑ a ∈ Finset.range n, p a fun _b => y) - f x
-        = ∑ a ∈ Finset.Ico 1 n, p a fun _b => y := by
+  have A : Tendsto (fun n ↦ (n, ∑ a ∈ Finset.Ico 1 n, p a fun _ ↦ y))
+      atTop (atTop ×ˢ 𝓝 (f (x + y) - f x)) := by
+    apply Tendsto.prod_mk tendsto_id
+    have L : ∀ᶠ n in atTop, (∑ a ∈ Finset.range n, p a fun _b ↦ y) - f x
+        = ∑ a ∈ Finset.Ico 1 n, p a fun _b ↦ y := by
       rw [eventually_atTop]
       refine ⟨1, fun n hn => ?_⟩
       symm
@@ -740,23 +767,19 @@ theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinea
     have :
       Tendsto (fun n => (∑ a ∈ Finset.range n, p a fun _b => y) - f x) atTop
         (𝓝 (f (x + y) - f x)) :=
-      (Hf.hasSum y_mem).tendsto_sum_nat.sub tendsto_const_nhds
+      (Hf.hasSum h'y y_mem).tendsto_sum_nat.sub tendsto_const_nhds
     exact Tendsto.congr' L this
   -- Second step: the composition of the partial sums of `q` and `p` converges to `g (f (x + y))`.
-  have B :
-    Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, p a fun _b => y)) atTop
+  have B : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, p a fun _b ↦ y)) atTop
       (𝓝 (g (f (x + y)))) := by
-    -- we use the fact that the partial sums of `q` converge locally uniformly to `g`, and that
-    -- composition passes to the limit under locally uniform convergence.
-    have B₁ : ContinuousAt (fun z : F => g (f x + z)) (f (x + y) - f x) := by
-      refine ContinuousAt.comp ?_ (continuous_const.add continuous_id).continuousAt
-      simp only [add_sub_cancel, _root_.id]
-      exact Hg.continuousOn.continuousAt (IsOpen.mem_nhds EMetric.isOpen_ball fy_mem)
-    have B₂ : f (x + y) - f x ∈ EMetric.ball (0 : F) rg := by
-      simpa [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub] using fy_mem
-    rw [← EMetric.isOpen_ball.nhdsWithin_eq B₂] at A
-    convert Hg.tendstoLocallyUniformlyOn.tendsto_comp B₁.continuousWithinAt B₂ A
-    simp only [add_sub_cancel]
+    -- we use the fact that the partial sums of `q` converge to `g (f (x + y))`, uniformly on a
+    -- neighborhood of `f (x + y)`.
+    have : Tendsto (fun (z : ℕ × F) ↦ q.partialSum z.1 z.2)
+        (atTop ×ˢ 𝓝 (f (x + y) - f x)) (𝓝 (g (f x + (f (x + y) - f x)))) := by
+      apply Hg.tendsto_partialSum_prod (y := f (x + y) - f x)
+      · simpa [edist_eq_coe_nnnorm_sub] using fy_mem.2
+      · simpa using fy_mem.1
+    simpa using this.comp A
   -- Third step: the sum over all compositions in `compPartialSumTarget 0 n n` converges to
   -- `g (f (x + y))`. As this sum is exactly the composition of the partial sum, this is a direct
   -- consequence of the second step
@@ -803,13 +826,46 @@ theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinea
   rw [Function.comp_apply]
   exact E
 
+/-- If two functions `g` and `f` have power series `q` and `p` respectively at `f x` and `x`,
+then `g ∘ f` admits the power  series `q.comp p` at `x` within `s`. -/
+theorem HasFPowerSeriesAt.comp {g : F → G} {f : E → F} {q : FormalMultilinearSeries 𝕜 F G}
+    {p : FormalMultilinearSeries 𝕜 E F} {x : E}
+    (hg : HasFPowerSeriesAt g q (f x)) (hf : HasFPowerSeriesAt f p x) :
+    HasFPowerSeriesAt (g ∘ f) (q.comp p) x := by
+  rw [← hasFPowerSeriesWithinAt_univ] at hf hg ⊢
+  apply hg.comp hf (by simp)
+
+/-- If two functions `g` and `f` are analytic respectively at `f x` and `x`, within
+two sets `s` and `t` such that `f` maps `s` to `t`, then `g ∘ f` is analytic at `x` within `s`. -/
+theorem AnalyticWithinAt.comp {g : F → G} {f : E → F} {x : E} {t : Set F} {s : Set E}
+    (hg : AnalyticWithinAt 𝕜 g t (f x)) (hf : AnalyticWithinAt 𝕜 f s x) (h : Set.MapsTo f s t) :
+    AnalyticWithinAt 𝕜 (g ∘ f) s x := by
+  let ⟨_q, hq⟩ := hg
+  let ⟨_p, hp⟩ := hf
+  exact (hq.comp hp h).analyticWithinAt
+
+/-- Version of `AnalyticWithinAt.comp` where point equality is a separate hypothesis. -/
+theorem AnalyticWithinAt.comp_of_eq {g : F → G} {f : E → F} {y : F} {x : E} {t : Set F} {s : Set E}
+    (hg : AnalyticWithinAt 𝕜 g t y) (hf : AnalyticWithinAt 𝕜 f s x) (h : Set.MapsTo f s t)
+    (hy : f x = y) :
+    AnalyticWithinAt 𝕜 (g ∘ f) s x := by
+  rw [← hy] at hg
+  exact hg.comp hf h
+
+lemma AnalyticOn.comp {f : F → G} {g : E → F} {s : Set F}
+    {t : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g t) (h : Set.MapsTo g t s) :
+    AnalyticOn 𝕜 (f ∘ g) t :=
+  fun x m ↦ (hf _ (h m)).comp (hg x m) h
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.comp := AnalyticOn.comp
+
 /-- If two functions `g` and `f` are analytic respectively at `f x` and `x`, then `g ∘ f` is
 analytic at `x`. -/
 theorem AnalyticAt.comp {g : F → G} {f : E → F} {x : E} (hg : AnalyticAt 𝕜 g (f x))
-    (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (g ∘ f) x :=
-  let ⟨_q, hq⟩ := hg
-  let ⟨_p, hp⟩ := hf
-  (hq.comp hp).analyticAt
+    (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (g ∘ f) x := by
+  rw [← analyticWithinAt_univ] at hg hf ⊢
+  apply hg.comp hf (by simp)
 
 /-- Version of `AnalyticAt.comp` where point equality is a separate hypothesis. -/
 theorem AnalyticAt.comp_of_eq {g : F → G} {f : E → F} {y : F} {x : E} (hg : AnalyticAt 𝕜 g y)
@@ -817,16 +873,40 @@ theorem AnalyticAt.comp_of_eq {g : F → G} {f : E → F} {y : F} {x : E} (hg :
   rw [← hy] at hg
   exact hg.comp hf
 
+theorem AnalyticAt.comp_analyticWithinAt {g : F → G} {f : E → F} {x : E} {s : Set E}
+    (hg : AnalyticAt 𝕜 g (f x)) (hf : AnalyticWithinAt 𝕜 f s x) :
+    AnalyticWithinAt 𝕜 (g ∘ f) s x := by
+  rw [← analyticWithinAt_univ] at hg
+  exact hg.comp hf (Set.mapsTo_univ _ _)
+
+theorem AnalyticAt.comp_analyticWithinAt_of_eq {g : F → G} {f : E → F} {x : E} {y : F} {s : Set E}
+    (hg : AnalyticAt 𝕜 g y) (hf : AnalyticWithinAt 𝕜 f s x) (h : f x = y) :
+    AnalyticWithinAt 𝕜 (g ∘ f) s x := by
+  rw [← h] at hg
+  exact hg.comp_analyticWithinAt hf
+
 /-- If two functions `g` and `f` are analytic respectively on `s.image f` and `s`, then `g ∘ f` is
 analytic on `s`. -/
-theorem AnalyticOn.comp' {s : Set E} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g (s.image f))
-    (hf : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (g ∘ f) s :=
+theorem AnalyticOnNhd.comp' {s : Set E} {g : F → G} {f : E → F} (hg : AnalyticOnNhd 𝕜 g (s.image f))
+    (hf : AnalyticOnNhd 𝕜 f s) : AnalyticOnNhd 𝕜 (g ∘ f) s :=
   fun z hz => (hg (f z) (Set.mem_image_of_mem f hz)).comp (hf z hz)
 
-theorem AnalyticOn.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F} (hg : AnalyticOn 𝕜 g t)
-    (hf : AnalyticOn 𝕜 f s) (st : Set.MapsTo f s t) : AnalyticOn 𝕜 (g ∘ f) s :=
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.comp' := AnalyticOnNhd.comp'
+
+theorem AnalyticOnNhd.comp {s : Set E} {t : Set F} {g : F → G} {f : E → F}
+    (hg : AnalyticOnNhd 𝕜 g t) (hf : AnalyticOnNhd 𝕜 f s) (st : Set.MapsTo f s t) :
+    AnalyticOnNhd 𝕜 (g ∘ f) s :=
   comp' (mono hg (Set.mapsTo'.mp st)) hf
 
+lemma AnalyticOnNhd.comp_analyticOn {f : F → G} {g : E → F} {s : Set F}
+    {t : Set E} (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOn 𝕜 g t) (h : Set.MapsTo g t s) :
+    AnalyticOn 𝕜 (f ∘ g) t :=
+  fun x m ↦ (hf _ (h m)).comp_analyticWithinAt (hg x m)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.comp_analyticWithinOn := AnalyticOnNhd.comp_analyticOn
+
 /-!
 ### Associativity of the composition of formal multilinear series
 
@@ -963,7 +1043,7 @@ def sigmaCompositionAux (a : Composition n) (b : Composition a.length)
     a.blocks_pos
       (by
         rw [← a.blocks.join_splitWrtComposition b]
-        exact mem_join_of_mem (List.getElem_mem _ _ _) hi)
+        exact mem_join_of_mem (List.getElem_mem _) hi)
   blocks_sum := by simp [Composition.blocksFun, getElem_map, Composition.gather]
 
 theorem length_sigmaCompositionAux (a : Composition n) (b : Composition a.length)
@@ -1029,7 +1109,7 @@ theorem sizeUpTo_sizeUpTo_add (a : Composition n) (b : Composition a.length) {i
     have : sizeUpTo b i + Nat.succ j = (sizeUpTo b i + j).succ := rfl
     rw [this, sizeUpTo_succ _ D, IHj A, sizeUpTo_succ _ B]
     simp only [sigmaCompositionAux, add_assoc, add_left_inj, Fin.val_mk]
-    rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop', getElem_take _ _ C]
+    rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take' _ _ C]
 
 /-- Natural equivalence between `(Σ (a : Composition n), Composition a.length)` and
 `(Σ (c : Composition n), Π (i : Fin c.length), Composition (c.blocksFun i))`, that shows up as a
@@ -1086,7 +1166,7 @@ def sigmaEquivSigmaPi (n : ℕ) :
     -- but we need to massage it to take care of the dependent setting.
     rintro ⟨c, d⟩
     have : map List.sum (ofFn fun i : Fin (Composition.length c) => (d i).blocks) = c.blocks := by
-      simp [map_ofFn, (· ∘ ·), Composition.blocks_sum, Composition.ofFn_blocksFun]
+      simp [map_ofFn, Function.comp_def, Composition.blocks_sum, Composition.ofFn_blocksFun]
     rw [sigma_pi_composition_eq_iff]
     dsimp
     congr! 1
@@ -1100,7 +1180,7 @@ def sigmaEquivSigmaPi (n : ℕ) :
       · intro i
         dsimp [Composition.sigmaCompositionAux]
         rw [getElem_of_eq (splitWrtComposition_join _ _ _)]
-        · simp only [getElem_ofFn]
+        · simp only [List.getElem_ofFn]
         · simp only [map_ofFn]
           rfl
       · congr
diff --git a/Mathlib/Analysis/Analytic/Constructions.lean b/Mathlib/Analysis/Analytic/Constructions.lean
index b66134ef40759..42de038a5780f 100644
--- a/Mathlib/Analysis/Analytic/Constructions.lean
+++ b/Mathlib/Analysis/Analytic/Constructions.lean
@@ -6,6 +6,7 @@ Authors: David Loeffler, Geoffrey Irving
 import Mathlib.Analysis.Analytic.Composition
 import Mathlib.Analysis.Analytic.Linear
 import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
+import Mathlib.Analysis.Normed.Ring.Units
 
 /-!
 # Various ways to combine analytic functions
@@ -31,6 +32,169 @@ variable {E F G H : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAd
 variable {𝕝 : Type*} [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝]
 variable {A : Type*} [NormedRing A] [NormedAlgebra 𝕜 A]
 
+/-!
+### Constants are analytic
+-/
+
+theorem hasFPowerSeriesOnBall_const {c : F} {e : E} :
+    HasFPowerSeriesOnBall (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e ⊤ := by
+  refine ⟨by simp, WithTop.zero_lt_top, fun _ => hasSum_single 0 fun n hn => ?_⟩
+  simp [constFormalMultilinearSeries_apply hn]
+
+theorem hasFPowerSeriesAt_const {c : F} {e : E} :
+    HasFPowerSeriesAt (fun _ => c) (constFormalMultilinearSeries 𝕜 E c) e :=
+  ⟨⊤, hasFPowerSeriesOnBall_const⟩
+
+theorem analyticAt_const {v : F} {x : E} : AnalyticAt 𝕜 (fun _ => v) x :=
+  ⟨constFormalMultilinearSeries 𝕜 E v, hasFPowerSeriesAt_const⟩
+
+theorem analyticOnNhd_const {v : F} {s : Set E} : AnalyticOnNhd 𝕜 (fun _ => v) s :=
+  fun _ _ => analyticAt_const
+
+theorem analyticWithinAt_const {v : F} {s : Set E} {x : E} : AnalyticWithinAt 𝕜 (fun _ => v) s x :=
+  analyticAt_const.analyticWithinAt
+
+theorem analyticOn_const {v : F} {s : Set E} : AnalyticOn 𝕜 (fun _ => v) s :=
+  analyticOnNhd_const.analyticOn
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_const := analyticOn_const
+
+/-!
+### Addition, negation, subtraction
+-/
+
+section
+
+variable {f g : E → F} {pf pg : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞}
+
+theorem HasFPowerSeriesWithinOnBall.add (hf : HasFPowerSeriesWithinOnBall f pf s x r)
+    (hg : HasFPowerSeriesWithinOnBall g pg s x r) :
+    HasFPowerSeriesWithinOnBall (f + g) (pf + pg) s x r :=
+  { r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
+    r_pos := hf.r_pos
+    hasSum := fun hy h'y => (hf.hasSum hy h'y).add (hg.hasSum hy h'y) }
+
+theorem HasFPowerSeriesOnBall.add (hf : HasFPowerSeriesOnBall f pf x r)
+    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f + g) (pf + pg) x r :=
+  { r_le := le_trans (le_min_iff.2 ⟨hf.r_le, hg.r_le⟩) (pf.min_radius_le_radius_add pg)
+    r_pos := hf.r_pos
+    hasSum := fun hy => (hf.hasSum hy).add (hg.hasSum hy) }
+
+theorem HasFPowerSeriesWithinAt.add
+    (hf : HasFPowerSeriesWithinAt f pf s x) (hg : HasFPowerSeriesWithinAt g pg s x) :
+    HasFPowerSeriesWithinAt (f + g) (pf + pg) s x := by
+  rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
+  exact ⟨r, hr.1.add hr.2⟩
+
+theorem HasFPowerSeriesAt.add (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
+    HasFPowerSeriesAt (f + g) (pf + pg) x := by
+  rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
+  exact ⟨r, hr.1.add hr.2⟩
+
+theorem AnalyticWithinAt.add (hf : AnalyticWithinAt 𝕜 f s x) (hg : AnalyticWithinAt 𝕜 g s x) :
+    AnalyticWithinAt 𝕜 (f + g) s x :=
+  let ⟨_, hpf⟩ := hf
+  let ⟨_, hqf⟩ := hg
+  (hpf.add hqf).analyticWithinAt
+
+theorem AnalyticAt.add (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) : AnalyticAt 𝕜 (f + g) x :=
+  let ⟨_, hpf⟩ := hf
+  let ⟨_, hqf⟩ := hg
+  (hpf.add hqf).analyticAt
+
+theorem HasFPowerSeriesWithinOnBall.neg (hf : HasFPowerSeriesWithinOnBall f pf s x r) :
+    HasFPowerSeriesWithinOnBall (-f) (-pf) s x r :=
+  { r_le := by
+      rw [pf.radius_neg]
+      exact hf.r_le
+    r_pos := hf.r_pos
+    hasSum := fun hy h'y => (hf.hasSum hy h'y).neg }
+
+theorem HasFPowerSeriesOnBall.neg (hf : HasFPowerSeriesOnBall f pf x r) :
+    HasFPowerSeriesOnBall (-f) (-pf) x r :=
+  { r_le := by
+      rw [pf.radius_neg]
+      exact hf.r_le
+    r_pos := hf.r_pos
+    hasSum := fun hy => (hf.hasSum hy).neg }
+
+theorem HasFPowerSeriesWithinAt.neg (hf : HasFPowerSeriesWithinAt f pf s x) :
+    HasFPowerSeriesWithinAt (-f) (-pf) s x :=
+  let ⟨_, hrf⟩ := hf
+  hrf.neg.hasFPowerSeriesWithinAt
+
+theorem HasFPowerSeriesAt.neg (hf : HasFPowerSeriesAt f pf x) : HasFPowerSeriesAt (-f) (-pf) x :=
+  let ⟨_, hrf⟩ := hf
+  hrf.neg.hasFPowerSeriesAt
+
+theorem AnalyticWithinAt.neg (hf : AnalyticWithinAt 𝕜 f s x) : AnalyticWithinAt 𝕜 (-f) s x :=
+  let ⟨_, hpf⟩ := hf
+  hpf.neg.analyticWithinAt
+
+theorem AnalyticAt.neg (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (-f) x :=
+  let ⟨_, hpf⟩ := hf
+  hpf.neg.analyticAt
+
+theorem HasFPowerSeriesWithinOnBall.sub (hf : HasFPowerSeriesWithinOnBall f pf s x r)
+    (hg : HasFPowerSeriesWithinOnBall g pg s x r) :
+    HasFPowerSeriesWithinOnBall (f - g) (pf - pg) s x r := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem HasFPowerSeriesOnBall.sub (hf : HasFPowerSeriesOnBall f pf x r)
+    (hg : HasFPowerSeriesOnBall g pg x r) : HasFPowerSeriesOnBall (f - g) (pf - pg) x r := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem HasFPowerSeriesWithinAt.sub
+    (hf : HasFPowerSeriesWithinAt f pf s x) (hg : HasFPowerSeriesWithinAt g pg s x) :
+    HasFPowerSeriesWithinAt (f - g) (pf - pg) s x := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem HasFPowerSeriesAt.sub (hf : HasFPowerSeriesAt f pf x) (hg : HasFPowerSeriesAt g pg x) :
+    HasFPowerSeriesAt (f - g) (pf - pg) x := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem AnalyticWithinAt.sub (hf : AnalyticWithinAt 𝕜 f s x) (hg : AnalyticWithinAt 𝕜 g s x) :
+    AnalyticWithinAt 𝕜 (f - g) s x := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem AnalyticAt.sub (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) :
+    AnalyticAt 𝕜 (f - g) x := by
+  simpa only [sub_eq_add_neg] using hf.add hg.neg
+
+theorem AnalyticOn.add (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
+    AnalyticOn 𝕜 (f + g) s :=
+  fun z hz => (hf z hz).add (hg z hz)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.add := AnalyticOn.add
+
+theorem AnalyticOnNhd.add (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
+    AnalyticOnNhd 𝕜 (f + g) s :=
+  fun z hz => (hf z hz).add (hg z hz)
+
+theorem AnalyticOn.neg (hf : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (-f) s :=
+  fun z hz ↦ (hf z hz).neg
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.neg := AnalyticOn.neg
+
+theorem AnalyticOnNhd.neg (hf : AnalyticOnNhd 𝕜 f s) : AnalyticOnNhd 𝕜 (-f) s :=
+  fun z hz ↦ (hf z hz).neg
+
+theorem AnalyticOn.sub (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
+    AnalyticOn 𝕜 (f - g) s :=
+  fun z hz => (hf z hz).sub (hg z hz)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.sub := AnalyticOn.sub
+
+theorem AnalyticOnNhd.sub (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
+    AnalyticOnNhd 𝕜 (f - g) s :=
+  fun z hz => (hf z hz).sub (hg z hz)
+
+end
+
 /-!
 ### Cartesian products are analytic
 -/
@@ -63,21 +227,36 @@ lemma FormalMultilinearSeries.radius_prod_eq_min
     refine (max_le_add_of_nonneg (norm_nonneg _) (norm_nonneg _)).trans ?_
     apply Real.le_norm_self
 
-lemma HasFPowerSeriesOnBall.prod {e : E} {f : E → F} {g : E → G} {r s : ℝ≥0∞}
+lemma HasFPowerSeriesWithinOnBall.prod {e : E} {f : E → F} {g : E → G} {r s : ℝ≥0∞} {t : Set E}
     {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
-    (hf : HasFPowerSeriesOnBall f p e r) (hg : HasFPowerSeriesOnBall g q e s) :
-    HasFPowerSeriesOnBall (fun x ↦ (f x, g x)) (p.prod q) e (min r s) where
+    (hf : HasFPowerSeriesWithinOnBall f p t e r) (hg : HasFPowerSeriesWithinOnBall g q t e s) :
+    HasFPowerSeriesWithinOnBall (fun x ↦ (f x, g x)) (p.prod q) t e (min r s) where
   r_le := by
     rw [p.radius_prod_eq_min]
     exact min_le_min hf.r_le hg.r_le
   r_pos := lt_min hf.r_pos hg.r_pos
   hasSum := by
-    intro y hy
+    intro y h'y hy
     simp_rw [FormalMultilinearSeries.prod, ContinuousMultilinearMap.prod_apply]
-    refine (hf.hasSum ?_).prod_mk (hg.hasSum ?_)
+    refine (hf.hasSum h'y ?_).prod_mk (hg.hasSum h'y ?_)
     · exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_left _ _))
     · exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_right _ _))
 
+lemma HasFPowerSeriesOnBall.prod {e : E} {f : E → F} {g : E → G} {r s : ℝ≥0∞}
+    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
+    (hf : HasFPowerSeriesOnBall f p e r) (hg : HasFPowerSeriesOnBall g q e s) :
+    HasFPowerSeriesOnBall (fun x ↦ (f x, g x)) (p.prod q) e (min r s) := by
+  rw [← hasFPowerSeriesWithinOnBall_univ] at hf hg ⊢
+  exact hf.prod hg
+
+lemma HasFPowerSeriesWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
+    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
+    (hf : HasFPowerSeriesWithinAt f p s e) (hg : HasFPowerSeriesWithinAt g q s e) :
+    HasFPowerSeriesWithinAt (fun x ↦ (f x, g x)) (p.prod q) s e := by
+  rcases hf with ⟨_, hf⟩
+  rcases hg with ⟨_, hg⟩
+  exact ⟨_, hf.prod hg⟩
+
 lemma HasFPowerSeriesAt.prod {e : E} {f : E → F} {g : E → G}
     {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
     (hf : HasFPowerSeriesAt f p e) (hg : HasFPowerSeriesAt g q e) :
@@ -86,6 +265,14 @@ lemma HasFPowerSeriesAt.prod {e : E} {f : E → F} {g : E → G}
   rcases hg with ⟨_, hg⟩
   exact ⟨_, hf.prod hg⟩
 
+/-- The Cartesian product of analytic functions is analytic. -/
+lemma AnalyticWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
+    (hf : AnalyticWithinAt 𝕜 f s e) (hg : AnalyticWithinAt 𝕜 g s e) :
+    AnalyticWithinAt 𝕜 (fun x ↦ (f x, g x)) s e := by
+  rcases hf with ⟨_, hf⟩
+  rcases hg with ⟨_, hg⟩
+  exact ⟨_, hf.prod hg⟩
+
 /-- The Cartesian product of analytic functions is analytic. -/
 lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
     (hf : AnalyticAt 𝕜 f e) (hg : AnalyticAt 𝕜 g e) :
@@ -94,12 +281,21 @@ lemma AnalyticAt.prod {e : E} {f : E → F} {g : E → G}
   rcases hg with ⟨_, hg⟩
   exact ⟨_, hf.prod hg⟩
 
-/-- The Cartesian product of analytic functions is analytic. -/
+/-- The Cartesian product of analytic functions within a set is analytic. -/
 lemma AnalyticOn.prod {f : E → F} {g : E → G} {s : Set E}
     (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
     AnalyticOn 𝕜 (fun x ↦ (f x, g x)) s :=
   fun x hx ↦ (hf x hx).prod (hg x hx)
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.prod := AnalyticOn.prod
+
+/-- The Cartesian product of analytic functions is analytic. -/
+lemma AnalyticOnNhd.prod {f : E → F} {g : E → G} {s : Set E}
+    (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
+    AnalyticOnNhd 𝕜 (fun x ↦ (f x, g x)) s :=
+  fun x hx ↦ (hf x hx).prod (hg x hx)
+
 /-- `AnalyticAt.comp` for functions on product spaces -/
 theorem AnalyticAt.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {x : E}
     (ha : AnalyticAt 𝕜 h (f x, g x)) (fa : AnalyticAt 𝕜 f x)
@@ -107,35 +303,249 @@ theorem AnalyticAt.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {x : E
     AnalyticAt 𝕜 (fun x ↦ h (f x, g x)) x :=
   AnalyticAt.comp ha (fa.prod ga)
 
-/-- `AnalyticOn.comp` for functions on product spaces -/
-theorem AnalyticOn.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)} {t : Set E}
-    (ha : AnalyticOn 𝕜 h s) (fa : AnalyticOn 𝕜 f t) (ga : AnalyticOn 𝕜 g t)
-    (m : ∀ x, x ∈ t → (f x, g x) ∈ s) : AnalyticOn 𝕜 (fun x ↦ h (f x, g x)) t :=
+/-- `AnalyticWithinAt.comp` for functions on product spaces -/
+theorem AnalyticWithinAt.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)}
+    {t : Set E} {x : E}
+    (ha : AnalyticWithinAt 𝕜 h s (f x, g x)) (fa : AnalyticWithinAt 𝕜 f t x)
+    (ga : AnalyticWithinAt 𝕜 g t x) (hf : Set.MapsTo (fun y ↦ (f y, g y)) t s) :
+    AnalyticWithinAt 𝕜 (fun x ↦ h (f x, g x)) t x :=
+  AnalyticWithinAt.comp ha (fa.prod ga) hf
+
+/-- `AnalyticAt.comp_analyticWithinAt` for functions on product spaces -/
+theorem AnalyticAt.comp₂_analyticWithinAt
+    {h : F × G → H} {f : E → F} {g : E → G} {x : E} {s : Set E}
+    (ha : AnalyticAt 𝕜 h (f x, g x)) (fa : AnalyticWithinAt 𝕜 f s x)
+    (ga : AnalyticWithinAt 𝕜 g s x) :
+    AnalyticWithinAt 𝕜 (fun x ↦ h (f x, g x)) s x :=
+  AnalyticAt.comp_analyticWithinAt ha (fa.prod ga)
+
+/-- `AnalyticOnNhd.comp` for functions on product spaces -/
+theorem AnalyticOnNhd.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)} {t : Set E}
+    (ha : AnalyticOnNhd 𝕜 h s) (fa : AnalyticOnNhd 𝕜 f t) (ga : AnalyticOnNhd 𝕜 g t)
+    (m : ∀ x, x ∈ t → (f x, g x) ∈ s) : AnalyticOnNhd 𝕜 (fun x ↦ h (f x, g x)) t :=
   fun _ xt ↦ (ha _ (m _ xt)).comp₂ (fa _ xt) (ga _ xt)
 
+/-- `AnalyticOn.comp` for functions on product spaces -/
+theorem AnalyticOn.comp₂ {h : F × G → H} {f : E → F} {g : E → G} {s : Set (F × G)}
+    {t : Set E}
+    (ha : AnalyticOn 𝕜 h s) (fa : AnalyticOn 𝕜 f t)
+    (ga : AnalyticOn 𝕜 g t) (m : Set.MapsTo (fun y ↦ (f y, g y)) t s) :
+    AnalyticOn 𝕜 (fun x ↦ h (f x, g x)) t :=
+  fun x hx ↦ (ha _ (m hx)).comp₂ (fa x hx) (ga x hx) m
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.comp₂ := AnalyticOn.comp₂
+
 /-- Analytic functions on products are analytic in the first coordinate -/
 theorem AnalyticAt.curry_left {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
     AnalyticAt 𝕜 (fun x ↦ f (x, p.2)) p.1 :=
-  AnalyticAt.comp₂ fa (analyticAt_id _ _) analyticAt_const
+  AnalyticAt.comp₂ fa analyticAt_id analyticAt_const
 alias AnalyticAt.along_fst := AnalyticAt.curry_left
 
+theorem AnalyticWithinAt.curry_left
+    {f : E × F → G} {s : Set (E × F)} {p : E × F} (fa : AnalyticWithinAt 𝕜 f s p) :
+    AnalyticWithinAt 𝕜 (fun x ↦ f (x, p.2)) {x | (x, p.2) ∈ s} p.1 :=
+  AnalyticWithinAt.comp₂ fa analyticWithinAt_id analyticWithinAt_const (fun _ hx ↦ hx)
+
 /-- Analytic functions on products are analytic in the second coordinate -/
 theorem AnalyticAt.curry_right {f : E × F → G} {p : E × F} (fa : AnalyticAt 𝕜 f p) :
     AnalyticAt 𝕜 (fun y ↦ f (p.1, y)) p.2 :=
-  AnalyticAt.comp₂ fa analyticAt_const (analyticAt_id _ _)
+  AnalyticAt.comp₂ fa analyticAt_const analyticAt_id
 alias AnalyticAt.along_snd := AnalyticAt.curry_right
 
+theorem AnalyticWithinAt.curry_right
+    {f : E × F → G} {s : Set (E × F)} {p : E × F} (fa : AnalyticWithinAt 𝕜 f s p) :
+    AnalyticWithinAt 𝕜 (fun y ↦ f (p.1, y)) {y | (p.1, y) ∈ s} p.2 :=
+  AnalyticWithinAt.comp₂ fa  analyticWithinAt_const analyticWithinAt_id (fun _ hx ↦ hx)
+
 /-- Analytic functions on products are analytic in the first coordinate -/
-theorem AnalyticOn.curry_left {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s) :
+theorem AnalyticOnNhd.curry_left {f : E × F → G} {s : Set (E × F)} {y : F}
+    (fa : AnalyticOnNhd 𝕜 f s) :
+    AnalyticOnNhd 𝕜 (fun x ↦ f (x, y)) {x | (x, y) ∈ s} :=
+  fun x m ↦ (fa (x, y) m).curry_left
+alias AnalyticOnNhd.along_fst := AnalyticOnNhd.curry_left
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.along_fst := AnalyticOnNhd.curry_left
+
+theorem AnalyticOn.curry_left
+    {f : E × F → G} {s : Set (E × F)} {y : F} (fa : AnalyticOn 𝕜 f s) :
     AnalyticOn 𝕜 (fun x ↦ f (x, y)) {x | (x, y) ∈ s} :=
-  fun x m ↦ (fa (x, y) m).along_fst
-alias AnalyticOn.along_fst := AnalyticOn.curry_left
+  fun x m ↦ (fa (x, y) m).curry_left
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.curry_left := AnalyticOn.curry_left
 
 /-- Analytic functions on products are analytic in the second coordinate -/
-theorem AnalyticOn.curry_right {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s) :
+theorem AnalyticOnNhd.curry_right {f : E × F → G} {x : E} {s : Set (E × F)}
+    (fa : AnalyticOnNhd 𝕜 f s) :
+    AnalyticOnNhd 𝕜 (fun y ↦ f (x, y)) {y | (x, y) ∈ s} :=
+  fun y m ↦ (fa (x, y) m).curry_right
+alias AnalyticOnNhd.along_snd := AnalyticOnNhd.curry_right
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.along_snd := AnalyticOnNhd.curry_right
+
+theorem AnalyticOn.curry_right
+    {f : E × F → G} {x : E} {s : Set (E × F)} (fa : AnalyticOn 𝕜 f s) :
     AnalyticOn 𝕜 (fun y ↦ f (x, y)) {y | (x, y) ∈ s} :=
-  fun y m ↦ (fa (x, y) m).along_snd
-alias AnalyticOn.along_snd := AnalyticOn.curry_right
+  fun y m ↦ (fa (x, y) m).curry_right
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.curry_right := AnalyticOn.curry_right
+
+/-!
+### Analyticity in Pi spaces
+
+In this section, `f : Π i, E → Fm i` is a family of functions, i.e., each `f i` is a function,
+from `E` to a space `Fm i`. We discuss whether the family as a whole is analytic as a function
+of `x : E`, i.e., whether `x ↦ (f 1 x, ..., f n x)` is analytic from `E` to the product space
+`Π i, Fm i`. This function is denoted either by `fun x ↦ (fun i ↦ f i x)`, or `fun x i ↦ f i x`,
+or `fun x ↦ (f ⬝ x)`. We use the latter spelling in the statements, for readability purposes.
+-/
+
+section
+
+variable {ι : Type*} [Fintype ι] {e : E} {Fm : ι → Type*}
+    [∀ i, NormedAddCommGroup (Fm i)] [∀ i, NormedSpace 𝕜 (Fm i)]
+    {f : Π i, E → Fm i} {s : Set E} {r : ℝ≥0∞}
+    {p : Π i, FormalMultilinearSeries 𝕜 E (Fm i)}
+
+lemma FormalMultilinearSeries.radius_pi_le (p : Π i, FormalMultilinearSeries 𝕜 E (Fm i)) (i : ι) :
+    (FormalMultilinearSeries.pi p).radius ≤ (p i).radius := by
+  apply le_of_forall_nnreal_lt (fun r' hr' ↦ ?_)
+  obtain ⟨C, -, hC⟩ :  ∃ C > 0, ∀ (n : ℕ),
+    ‖pi p n‖ * ↑r' ^ n ≤ C := norm_mul_pow_le_of_lt_radius _ hr'
+  apply le_radius_of_bound _ C (fun n ↦ ?_)
+  apply le_trans _ (hC n)
+  gcongr
+  rw [pi, ContinuousMultilinearMap.opNorm_pi]
+  exact norm_le_pi_norm (fun i ↦ p i n) i
+
+lemma FormalMultilinearSeries.le_radius_pi (h : ∀ i, r ≤ (p i).radius) :
+    r ≤ (FormalMultilinearSeries.pi p).radius := by
+  apply le_of_forall_nnreal_lt (fun r' hr' ↦ ?_)
+  have I i : ∃ C > 0, ∀ n, ‖p i n‖ * (r' : ℝ) ^ n ≤ C :=
+    norm_mul_pow_le_of_lt_radius _ (hr'.trans_le (h i))
+  choose C C_pos hC using I
+  obtain ⟨D, D_nonneg, hD⟩ : ∃ D ≥ 0, ∀ i, C i ≤ D :=
+    ⟨∑ i, C i, Finset.sum_nonneg (fun i _ ↦ (C_pos i).le),
+      fun i ↦ Finset.single_le_sum (fun j _ ↦ (C_pos j).le) (Finset.mem_univ _)⟩
+  apply le_radius_of_bound _ D (fun n ↦ ?_)
+  rcases le_or_lt ((r' : ℝ)^n) 0 with hr' | hr'
+  · exact le_trans (mul_nonpos_of_nonneg_of_nonpos (by positivity) hr') D_nonneg
+  · simp only [pi]
+    rw [← le_div_iff₀ hr', ContinuousMultilinearMap.opNorm_pi,
+      pi_norm_le_iff_of_nonneg (by positivity)]
+    intro i
+    exact (le_div_iff₀ hr').2 ((hC i n).trans (hD i))
+
+lemma FormalMultilinearSeries.radius_pi_eq_iInf :
+    (FormalMultilinearSeries.pi p).radius = ⨅ i, (p i).radius := by
+  refine le_antisymm (by simp [radius_pi_le]) ?_
+  apply le_of_forall_nnreal_lt (fun r' hr' ↦ ?_)
+  exact le_radius_pi (fun i ↦ le_iInf_iff.1 hr'.le i)
+
+/-- If each function in a finite family has a power series within a ball, then so does the
+family as a whole. Note that the positivity assumption on the radius is only needed when
+the family is empty. -/
+lemma HasFPowerSeriesWithinOnBall.pi
+    (hf : ∀ i, HasFPowerSeriesWithinOnBall (f i) (p i) s e r) (hr : 0 < r) :
+    HasFPowerSeriesWithinOnBall (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) s e r where
+  r_le := by
+    apply FormalMultilinearSeries.le_radius_pi (fun i ↦ ?_)
+    exact (hf i).r_le
+  r_pos := hr
+  hasSum {y} m hy := Pi.hasSum.2 (fun i ↦ (hf i).hasSum m hy)
+
+lemma hasFPowerSeriesWithinOnBall_pi_iff (hr : 0 < r) :
+    HasFPowerSeriesWithinOnBall (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) s e r
+      ↔ ∀ i, HasFPowerSeriesWithinOnBall (f i) (p i) s e r :=
+  ⟨fun h i ↦ ⟨h.r_le.trans (FormalMultilinearSeries.radius_pi_le _ _), hr,
+    fun m hy ↦ Pi.hasSum.1 (h.hasSum m hy) i⟩, fun h ↦ .pi h hr⟩
+
+lemma HasFPowerSeriesOnBall.pi
+    (hf : ∀ i, HasFPowerSeriesOnBall (f i) (p i) e r) (hr : 0 < r) :
+    HasFPowerSeriesOnBall (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) e r := by
+  simp_rw [← hasFPowerSeriesWithinOnBall_univ] at hf ⊢
+  exact HasFPowerSeriesWithinOnBall.pi hf hr
+
+lemma hasFPowerSeriesOnBall_pi_iff (hr : 0 < r) :
+    HasFPowerSeriesOnBall (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) e r
+      ↔ ∀ i, HasFPowerSeriesOnBall (f i) (p i) e r := by
+  simp_rw [← hasFPowerSeriesWithinOnBall_univ]
+  exact hasFPowerSeriesWithinOnBall_pi_iff hr
+
+lemma HasFPowerSeriesWithinAt.pi
+    (hf : ∀ i, HasFPowerSeriesWithinAt (f i) (p i) s e) :
+    HasFPowerSeriesWithinAt (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) s e := by
+  have : ∀ᶠ r in 𝓝[>] 0, ∀ i, HasFPowerSeriesWithinOnBall (f i) (p i) s e r :=
+    eventually_all.mpr (fun i ↦ (hf i).eventually)
+  obtain ⟨r, hr, r_pos⟩ := (this.and self_mem_nhdsWithin).exists
+  exact ⟨r, HasFPowerSeriesWithinOnBall.pi hr r_pos⟩
+
+lemma hasFPowerSeriesWithinAt_pi_iff :
+    HasFPowerSeriesWithinAt (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) s e
+      ↔ ∀ i, HasFPowerSeriesWithinAt (f i) (p i) s e := by
+  refine ⟨fun h i ↦ ?_, fun h ↦ .pi h⟩
+  obtain ⟨r, hr⟩ := h
+  exact ⟨r, (hasFPowerSeriesWithinOnBall_pi_iff hr.r_pos).1 hr i⟩
+
+lemma HasFPowerSeriesAt.pi
+    (hf : ∀ i, HasFPowerSeriesAt (f i) (p i) e) :
+    HasFPowerSeriesAt (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) e := by
+  simp_rw [← hasFPowerSeriesWithinAt_univ] at hf ⊢
+  exact HasFPowerSeriesWithinAt.pi hf
+
+lemma hasFPowerSeriesAt_pi_iff :
+    HasFPowerSeriesAt (fun x ↦ (f · x)) (FormalMultilinearSeries.pi p) e
+      ↔ ∀ i, HasFPowerSeriesAt (f i) (p i) e := by
+  simp_rw [← hasFPowerSeriesWithinAt_univ]
+  exact hasFPowerSeriesWithinAt_pi_iff
+
+lemma AnalyticWithinAt.pi (hf : ∀ i, AnalyticWithinAt 𝕜 (f i) s e) :
+    AnalyticWithinAt 𝕜 (fun x ↦ (f · x)) s e := by
+  choose p hp using hf
+  exact ⟨FormalMultilinearSeries.pi p, HasFPowerSeriesWithinAt.pi hp⟩
+
+lemma analyticWithinAt_pi_iff :
+    AnalyticWithinAt 𝕜 (fun x ↦ (f · x)) s e ↔ ∀ i, AnalyticWithinAt 𝕜 (f i) s e := by
+  refine ⟨fun h i ↦ ?_, fun h ↦ .pi h⟩
+  exact ((ContinuousLinearMap.proj (R := 𝕜) i).analyticAt _).comp_analyticWithinAt h
+
+lemma AnalyticAt.pi (hf : ∀ i, AnalyticAt 𝕜 (f i) e) :
+    AnalyticAt 𝕜 (fun x ↦ (f · x)) e := by
+  simp_rw [← analyticWithinAt_univ] at hf ⊢
+  exact AnalyticWithinAt.pi hf
+
+lemma analyticAt_pi_iff :
+    AnalyticAt 𝕜 (fun x ↦ (f · x)) e ↔ ∀ i, AnalyticAt 𝕜 (f i) e := by
+  simp_rw [← analyticWithinAt_univ]
+  exact analyticWithinAt_pi_iff
+
+lemma AnalyticOn.pi (hf : ∀ i, AnalyticOn 𝕜 (f i) s) :
+    AnalyticOn 𝕜 (fun x ↦ (f · x)) s :=
+  fun x hx ↦ AnalyticWithinAt.pi (fun i ↦ hf i x hx)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.pi := AnalyticOn.pi
+
+lemma analyticOn_pi_iff :
+    AnalyticOn 𝕜 (fun x ↦ (f · x)) s ↔ ∀ i, AnalyticOn 𝕜 (f i) s :=
+  ⟨fun h i x hx ↦ analyticWithinAt_pi_iff.1 (h x hx) i, fun h ↦ .pi h⟩
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_pi_iff := analyticOn_pi_iff
+
+lemma AnalyticOnNhd.pi (hf : ∀ i, AnalyticOnNhd 𝕜 (f i) s) :
+    AnalyticOnNhd 𝕜 (fun x ↦ (f · x)) s :=
+  fun x hx ↦ AnalyticAt.pi (fun i ↦ hf i x hx)
+
+lemma analyticOnNhd_pi_iff :
+    AnalyticOnNhd 𝕜 (fun x ↦ (f · x)) s ↔ ∀ i, AnalyticOnNhd 𝕜 (f i) s :=
+  ⟨fun h i x hx ↦ analyticAt_pi_iff.1 (h x hx) i, fun h ↦ .pi h⟩
+
+end
 
 /-!
 ### Arithmetic on analytic functions
@@ -154,6 +564,13 @@ lemma analyticAt_smul [NormedSpace 𝕝 E] [IsScalarTower 𝕜 𝕝 E] (z : 𝕝
 lemma analyticAt_mul (z : A × A) : AnalyticAt 𝕜 (fun x : A × A ↦ x.1 * x.2) z :=
   (ContinuousLinearMap.mul 𝕜 A).analyticAt_bilinear z
 
+/-- Scalar multiplication of one analytic function by another. -/
+lemma AnalyticWithinAt.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F]
+    {f : E → 𝕝} {g : E → F} {s : Set E} {z : E}
+    (hf : AnalyticWithinAt 𝕜 f s z) (hg : AnalyticWithinAt 𝕜 g s z) :
+    AnalyticWithinAt 𝕜 (fun x ↦ f x • g x) s z :=
+  (analyticAt_smul _).comp₂_analyticWithinAt hf hg
+
 /-- Scalar multiplication of one analytic function by another. -/
 lemma AnalyticAt.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {z : E}
     (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z) :
@@ -161,53 +578,165 @@ lemma AnalyticAt.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E 
   (analyticAt_smul _).comp₂ hf hg
 
 /-- Scalar multiplication of one analytic function by another. -/
-lemma AnalyticOn.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {s : Set E}
+lemma AnalyticOn.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F]
+    {f : E → 𝕝} {g : E → F} {s : Set E}
     (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
     AnalyticOn 𝕜 (fun x ↦ f x • g x) s :=
   fun _ m ↦ (hf _ m).smul (hg _ m)
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.smul := AnalyticOn.smul
+
+/-- Scalar multiplication of one analytic function by another. -/
+lemma AnalyticOnNhd.smul [NormedSpace 𝕝 F] [IsScalarTower 𝕜 𝕝 F] {f : E → 𝕝} {g : E → F} {s : Set E}
+    (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
+    AnalyticOnNhd 𝕜 (fun x ↦ f x • g x) s :=
+  fun _ m ↦ (hf _ m).smul (hg _ m)
+
+/-- Multiplication of analytic functions (valued in a normed `𝕜`-algebra) is analytic. -/
+lemma AnalyticWithinAt.mul {f g : E → A} {s : Set E} {z : E}
+    (hf : AnalyticWithinAt 𝕜 f s z) (hg : AnalyticWithinAt 𝕜 g s z) :
+    AnalyticWithinAt 𝕜 (fun x ↦ f x * g x) s z :=
+  (analyticAt_mul _).comp₂_analyticWithinAt hf hg
+
 /-- Multiplication of analytic functions (valued in a normed `𝕜`-algebra) is analytic. -/
 lemma AnalyticAt.mul {f g : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (hg : AnalyticAt 𝕜 g z) :
     AnalyticAt 𝕜 (fun x ↦ f x * g x) z :=
   (analyticAt_mul _).comp₂ hf hg
 
 /-- Multiplication of analytic functions (valued in a normed `𝕜`-algebra) is analytic. -/
-lemma AnalyticOn.mul {f g : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
+lemma AnalyticOn.mul {f g : E → A} {s : Set E}
+    (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
     AnalyticOn 𝕜 (fun x ↦ f x * g x) s :=
   fun _ m ↦ (hf _ m).mul (hg _ m)
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.mul := AnalyticOn.mul
+
+/-- Multiplication of analytic functions (valued in a normed `𝕜`-algebra) is analytic. -/
+lemma AnalyticOnNhd.mul {f g : E → A} {s : Set E}
+    (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
+    AnalyticOnNhd 𝕜 (fun x ↦ f x * g x) s :=
+  fun _ m ↦ (hf _ m).mul (hg _ m)
+
 /-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
-lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
-    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
+lemma AnalyticWithinAt.pow {f : E → A} {z : E} {s : Set E} (hf : AnalyticWithinAt 𝕜 f s z) (n : ℕ) :
+    AnalyticWithinAt 𝕜 (fun x ↦ f x ^ n) s z := by
   induction n with
   | zero =>
     simp only [pow_zero]
-    apply analyticAt_const
+    apply analyticWithinAt_const
   | succ m hm =>
     simp only [pow_succ]
     exact hm.mul hf
 
+/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
+lemma AnalyticAt.pow {f : E → A} {z : E} (hf : AnalyticAt 𝕜 f z) (n : ℕ) :
+    AnalyticAt 𝕜 (fun x ↦ f x ^ n) z := by
+  rw [← analyticWithinAt_univ] at hf ⊢
+  exact hf.pow n
+
 /-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
 lemma AnalyticOn.pow {f : E → A} {s : Set E} (hf : AnalyticOn 𝕜 f s) (n : ℕ) :
     AnalyticOn 𝕜 (fun x ↦ f x ^ n) s :=
   fun _ m ↦ (hf _ m).pow n
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.pow := AnalyticOn.pow
+
+/-- Powers of analytic functions (into a normed `𝕜`-algebra) are analytic. -/
+lemma AnalyticOnNhd.pow {f : E → A} {s : Set E} (hf : AnalyticOnNhd 𝕜 f s) (n : ℕ) :
+    AnalyticOnNhd 𝕜 (fun x ↦ f x ^ n) s :=
+  fun _ m ↦ (hf _ m).pow n
+
+
+/-!
+### Restriction of scalars
+-/
+
+section
+
+variable {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜 𝕜']
+  [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E]
+  [NormedSpace 𝕜' F] [IsScalarTower 𝕜 𝕜' F]
+  {f : E → F} {p : FormalMultilinearSeries 𝕜' E F} {x : E} {s : Set E} {r : ℝ≥0∞}
+
+lemma HasFPowerSeriesWithinOnBall.restrictScalars (hf : HasFPowerSeriesWithinOnBall f p s x r) :
+    HasFPowerSeriesWithinOnBall f (p.restrictScalars 𝕜) s x r :=
+  ⟨hf.r_le.trans (FormalMultilinearSeries.radius_le_of_le (fun n ↦ by simp)), hf.r_pos, hf.hasSum⟩
+
+lemma HasFPowerSeriesOnBall.restrictScalars (hf : HasFPowerSeriesOnBall f p x r) :
+    HasFPowerSeriesOnBall f (p.restrictScalars 𝕜) x r :=
+  ⟨hf.r_le.trans (FormalMultilinearSeries.radius_le_of_le (fun n ↦ by simp)), hf.r_pos, hf.hasSum⟩
+
+lemma HasFPowerSeriesWithinAt.restrictScalars (hf : HasFPowerSeriesWithinAt f p s x) :
+    HasFPowerSeriesWithinAt f (p.restrictScalars 𝕜) s x := by
+  rcases hf with ⟨r, hr⟩
+  exact ⟨r, hr.restrictScalars⟩
+
+lemma HasFPowerSeriesAt.restrictScalars (hf : HasFPowerSeriesAt f p x) :
+    HasFPowerSeriesAt f (p.restrictScalars 𝕜) x := by
+  rcases hf with ⟨r, hr⟩
+  exact ⟨r, hr.restrictScalars⟩
+
+lemma AnalyticWithinAt.restrictScalars (hf : AnalyticWithinAt 𝕜' f s x) :
+    AnalyticWithinAt 𝕜 f s x := by
+  rcases hf with ⟨p, hp⟩
+  exact ⟨p.restrictScalars 𝕜, hp.restrictScalars⟩
+
+lemma AnalyticAt.restrictScalars (hf : AnalyticAt 𝕜' f x) :
+    AnalyticAt 𝕜 f x := by
+  rcases hf with ⟨p, hp⟩
+  exact ⟨p.restrictScalars 𝕜, hp.restrictScalars⟩
+
+lemma AnalyticOn.restrictScalars (hf : AnalyticOn 𝕜' f s) :
+    AnalyticOn 𝕜 f s :=
+  fun x hx ↦ (hf x hx).restrictScalars
+
+lemma AnalyticOnNhd.restrictScalars (hf : AnalyticOnNhd 𝕜' f s) :
+    AnalyticOnNhd 𝕜 f s :=
+  fun x hx ↦ (hf x hx).restrictScalars
+
+end
+
+
+/-!
+### Inversion is analytic
+-/
+
 section Geometric
 
 variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NormedRing A] [NormedAlgebra 𝕜 A]
-  [NormOneClass A]
 
 /-- The geometric series `1 + x + x ^ 2 + ...` as a `FormalMultilinearSeries`. -/
 def formalMultilinearSeries_geometric : FormalMultilinearSeries 𝕜 A A :=
   fun n ↦ ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A
 
-lemma formalMultilinearSeries_geometric_apply_norm (n : ℕ) :
+lemma formalMultilinearSeries_geometric_apply_norm_le (n : ℕ) :
+    ‖formalMultilinearSeries_geometric 𝕜 A n‖ ≤ max 1 ‖(1 : A)‖ :=
+  ContinuousMultilinearMap.norm_mkPiAlgebraFin_le
+
+lemma formalMultilinearSeries_geometric_apply_norm [NormOneClass A] (n : ℕ) :
     ‖formalMultilinearSeries_geometric 𝕜 A n‖ = 1 :=
   ContinuousMultilinearMap.norm_mkPiAlgebraFin
 
 end Geometric
 
-lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 𝕜]
+lemma one_le_formalMultilinearSeries_geometric_radius (𝕜 : Type*) [NontriviallyNormedField 𝕜]
+    (A : Type*) [NormedRing A] [NormedAlgebra 𝕜 A] :
+    1 ≤ (formalMultilinearSeries_geometric 𝕜 A).radius := by
+  refine le_of_forall_nnreal_lt (fun r hr ↦ ?_)
+  rw [← Nat.cast_one, ENNReal.coe_lt_natCast, Nat.cast_one] at hr
+  apply FormalMultilinearSeries.le_radius_of_isBigO
+  apply isBigO_of_le' (c := max 1 ‖(1 : A)‖) atTop (fun n ↦ ?_)
+  simp only [norm_mul, norm_norm, norm_pow, Real.norm_eq_abs, NNReal.abs_eq, norm_one, mul_one,
+    abs_norm]
+  apply le_trans ?_ (formalMultilinearSeries_geometric_apply_norm_le 𝕜 A n)
+  conv_rhs => rw [← mul_one (‖formalMultilinearSeries_geometric 𝕜 A n‖)]
+  gcongr
+  exact pow_le_one₀ (coe_nonneg r) hr.le
+
+lemma formalMultilinearSeries_geometric_radius (𝕜 : Type*) [NontriviallyNormedField 𝕜]
     (A : Type*) [NormedRing A] [NormOneClass A] [NormedAlgebra 𝕜 A] :
     (formalMultilinearSeries_geometric 𝕜 A).radius = 1 := by
   apply le_antisymm
@@ -231,22 +760,68 @@ lemma formalMultilinearSeries_geometric_radius (𝕜) [NontriviallyNormedField 
     simp_rw [formalMultilinearSeries_geometric_apply_norm, one_mul]
     refine isBigO_of_le atTop (fun n ↦ ?_)
     rw [norm_one, Real.norm_of_nonneg (pow_nonneg (coe_nonneg r) _)]
-    exact pow_le_one _ (coe_nonneg r) hr.le
+    exact pow_le_one₀ (coe_nonneg r) hr.le
 
-lemma hasFPowerSeriesOnBall_inv_one_sub
-    (𝕜 𝕝 : Type*) [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] :
-    HasFPowerSeriesOnBall (fun x : 𝕝 ↦ (1 - x)⁻¹) (formalMultilinearSeries_geometric 𝕜 𝕝) 0 1 := by
+lemma hasFPowerSeriesOnBall_inverse_one_sub
+    (𝕜 : Type*) [NontriviallyNormedField 𝕜]
+    (A : Type*) [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] :
+    HasFPowerSeriesOnBall (fun x : A ↦ Ring.inverse (1 - x))
+      (formalMultilinearSeries_geometric 𝕜 A) 0 1 := by
   constructor
-  · exact le_of_eq (formalMultilinearSeries_geometric_radius 𝕜 𝕝).symm
+  · exact one_le_formalMultilinearSeries_geometric_radius 𝕜 A
   · exact one_pos
   · intro y hy
-    simp_rw [zero_add, formalMultilinearSeries_geometric,
-        ContinuousMultilinearMap.mkPiAlgebraFin_apply,
-        List.prod_ofFn, Finset.prod_const,
-        Finset.card_univ, Fintype.card_fin]
-    apply hasSum_geometric_of_norm_lt_one
-    simpa only [← ofReal_one, Metric.emetric_ball, Metric.ball,
-      dist_eq_norm, sub_zero] using hy
+    simp only [EMetric.mem_ball, edist_dist, dist_zero_right, ofReal_lt_one] at hy
+    simp only [zero_add, NormedRing.inverse_one_sub _ hy, Units.oneSub, Units.inv_mk,
+      formalMultilinearSeries_geometric, ContinuousMultilinearMap.mkPiAlgebraFin_apply,
+      List.ofFn_const, List.prod_replicate]
+    exact (summable_geometric_of_norm_lt_one hy).hasSum
+
+lemma analyticAt_inverse_one_sub (𝕜 : Type*) [NontriviallyNormedField 𝕜]
+    (A : Type*) [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] :
+    AnalyticAt 𝕜 (fun x : A ↦ Ring.inverse (1 - x)) 0 :=
+  ⟨_, ⟨_, hasFPowerSeriesOnBall_inverse_one_sub 𝕜 A⟩⟩
+
+/-- If `A` is a normed algebra over `𝕜` with summable geometric series, then inversion on `A` is
+analytic at any unit. -/
+lemma analyticAt_inverse {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+    {A : Type*} [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] (z : Aˣ) :
+    AnalyticAt 𝕜 Ring.inverse (z : A) := by
+  rcases subsingleton_or_nontrivial A with hA|hA
+  · convert analyticAt_const (v := (0 : A))
+  · let f1 : A → A := fun a ↦ a * z.inv
+    let f2 : A → A := fun b ↦ Ring.inverse (1 - b)
+    let f3 : A → A := fun c ↦ 1 - z.inv * c
+    have feq : ∀ᶠ y in 𝓝 (z : A), (f1 ∘ f2 ∘ f3) y = Ring.inverse y := by
+      have : Metric.ball (z : A) (‖(↑z⁻¹ : A)‖⁻¹) ∈ 𝓝 (z : A) := by
+        apply Metric.ball_mem_nhds
+        simp
+      filter_upwards [this] with y hy
+      simp only [Metric.mem_ball, dist_eq_norm] at hy
+      have : y = Units.ofNearby z y hy := rfl
+      rw [this, Eq.comm]
+      simp only [Ring.inverse_unit, Function.comp_apply]
+      simp [Units.ofNearby, f1, f2, f3, Units.add, _root_.mul_sub]
+      rw [← Ring.inverse_unit]
+      congr
+      simp
+    apply AnalyticAt.congr _ feq
+    apply (analyticAt_id.mul analyticAt_const).comp
+    apply AnalyticAt.comp
+    · simp only [Units.inv_eq_val_inv, Units.inv_mul, sub_self, f2, f3]
+      exact analyticAt_inverse_one_sub 𝕜 A
+    · exact analyticAt_const.sub (analyticAt_const.mul analyticAt_id)
+
+lemma analyticOnNhd_inverse {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+    {A : Type*} [NormedRing A] [NormedAlgebra 𝕜 A] [HasSummableGeomSeries A] :
+    AnalyticOnNhd 𝕜 Ring.inverse {x : A | IsUnit x} :=
+  fun _ hx ↦ analyticAt_inverse (IsUnit.unit hx)
+
+lemma hasFPowerSeriesOnBall_inv_one_sub
+    (𝕜 𝕝 : Type*) [NontriviallyNormedField 𝕜] [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] :
+    HasFPowerSeriesOnBall (fun x : 𝕝 ↦ (1 - x)⁻¹) (formalMultilinearSeries_geometric 𝕜 𝕝) 0 1 := by
+  convert hasFPowerSeriesOnBall_inverse_one_sub 𝕜 𝕝
+  exact Ring.inverse_eq_inv'.symm
 
 lemma analyticAt_inv_one_sub (𝕝 : Type*) [NontriviallyNormedField 𝕝] [NormedAlgebra 𝕜 𝕝] :
     AnalyticAt 𝕜 (fun x : 𝕝 ↦ (1 - x)⁻¹) 0 :=
@@ -255,34 +830,48 @@ lemma analyticAt_inv_one_sub (𝕝 : Type*) [NontriviallyNormedField 𝕝] [Norm
 /-- If `𝕝` is a normed field extension of `𝕜`, then the inverse map `𝕝 → 𝕝` is `𝕜`-analytic
 away from 0. -/
 lemma analyticAt_inv {z : 𝕝} (hz : z ≠ 0) : AnalyticAt 𝕜 Inv.inv z := by
-  let f1 : 𝕝 → 𝕝 := fun a ↦ 1 / z * a
-  let f2 : 𝕝 → 𝕝 := fun b ↦ (1 - b)⁻¹
-  let f3 : 𝕝 → 𝕝 := fun c ↦ 1 - c / z
-  have feq : f1 ∘ f2 ∘ f3 = Inv.inv := by
-    ext1 x
-    dsimp only [f1, f2, f3, Function.comp_apply]
-    field_simp
-  have f3val : f3 z = 0 := by simp only [f3, div_self hz, sub_self]
-  have f3an : AnalyticAt 𝕜 f3 z := by
-    apply analyticAt_const.sub
-    simpa only [div_eq_inv_mul] using analyticAt_const.mul (analyticAt_id 𝕜 z)
-  exact feq ▸ (analyticAt_const.mul (analyticAt_id _ _)).comp
-    ((f3val.symm ▸ analyticAt_inv_one_sub 𝕝).comp f3an)
+  convert analyticAt_inverse (𝕜 := 𝕜) (Units.mk0 _ hz)
+  exact Ring.inverse_eq_inv'.symm
 
 /-- `x⁻¹` is analytic away from zero -/
-lemma analyticOn_inv : AnalyticOn 𝕜 (fun z ↦ z⁻¹) {z : 𝕝 | z ≠ 0} := by
+lemma analyticOnNhd_inv : AnalyticOnNhd 𝕜 (fun z ↦ z⁻¹) {z : 𝕝 | z ≠ 0} := by
   intro z m; exact analyticAt_inv m
 
+lemma analyticOn_inv : AnalyticOn 𝕜 (fun z ↦ z⁻¹) {z : 𝕝 | z ≠ 0} :=
+  analyticOnNhd_inv.analyticOn
+
+/-- `(f x)⁻¹` is analytic away from `f x = 0` -/
+theorem AnalyticWithinAt.inv {f : E → 𝕝} {x : E} {s : Set E}
+    (fa : AnalyticWithinAt 𝕜 f s x) (f0 : f x ≠ 0) :
+    AnalyticWithinAt 𝕜 (fun x ↦ (f x)⁻¹) s x :=
+  (analyticAt_inv f0).comp_analyticWithinAt fa
+
 /-- `(f x)⁻¹` is analytic away from `f x = 0` -/
 theorem AnalyticAt.inv {f : E → 𝕝} {x : E} (fa : AnalyticAt 𝕜 f x) (f0 : f x ≠ 0) :
     AnalyticAt 𝕜 (fun x ↦ (f x)⁻¹) x :=
   (analyticAt_inv f0).comp fa
 
-/-- `x⁻¹` is analytic away from zero -/
-theorem AnalyticOn.inv {f : E → 𝕝} {s : Set E} (fa : AnalyticOn 𝕜 f s) (f0 : ∀ x ∈ s, f x ≠ 0) :
+/-- `(f x)⁻¹` is analytic away from `f x = 0` -/
+theorem AnalyticOn.inv {f : E → 𝕝} {s : Set E}
+    (fa : AnalyticOn 𝕜 f s) (f0 : ∀ x ∈ s, f x ≠ 0) :
     AnalyticOn 𝕜 (fun x ↦ (f x)⁻¹) s :=
   fun x m ↦ (fa x m).inv (f0 x m)
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.inv := AnalyticOn.inv
+
+/-- `(f x)⁻¹` is analytic away from `f x = 0` -/
+theorem AnalyticOnNhd.inv {f : E → 𝕝} {s : Set E}
+    (fa : AnalyticOnNhd 𝕜 f s) (f0 : ∀ x ∈ s, f x ≠ 0) :
+    AnalyticOnNhd 𝕜 (fun x ↦ (f x)⁻¹) s :=
+  fun x m ↦ (fa x m).inv (f0 x m)
+
+/-- `f x / g x` is analytic away from `g x = 0` -/
+theorem AnalyticWithinAt.div {f g : E → 𝕝} {s : Set E} {x : E}
+    (fa : AnalyticWithinAt 𝕜 f s x) (ga : AnalyticWithinAt 𝕜 g s x) (g0 : g x ≠ 0) :
+    AnalyticWithinAt 𝕜 (fun x ↦ f x / g x) s x := by
+  simp_rw [div_eq_mul_inv]; exact fa.mul (ga.inv g0)
+
 /-- `f x / g x` is analytic away from `g x = 0` -/
 theorem AnalyticAt.div {f g : E → 𝕝} {x : E}
     (fa : AnalyticAt 𝕜 f x) (ga : AnalyticAt 𝕜 g x) (g0 : g x ≠ 0) :
@@ -295,40 +884,81 @@ theorem AnalyticOn.div {f g : E → 𝕝} {s : Set E}
     AnalyticOn 𝕜 (fun x ↦ f x / g x) s := fun x m ↦
   (fa x m).div (ga x m) (g0 x m)
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.div := AnalyticOn.div
+
+/-- `f x / g x` is analytic away from `g x = 0` -/
+theorem AnalyticOnNhd.div {f g : E → 𝕝} {s : Set E}
+    (fa : AnalyticOnNhd 𝕜 f s) (ga : AnalyticOnNhd 𝕜 g s) (g0 : ∀ x ∈ s, g x ≠ 0) :
+    AnalyticOnNhd 𝕜 (fun x ↦ f x / g x) s := fun x m ↦
+  (fa x m).div (ga x m) (g0 x m)
+
 /-!
 ### Finite sums and products of analytic functions
 -/
 
 /-- Finite sums of analytic functions are analytic -/
-theorem Finset.analyticAt_sum {f : α → E → F} {c : E}
-    (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
-    AnalyticAt 𝕜 (fun z ↦ ∑ n ∈ N, f n z) c := by
+theorem Finset.analyticWithinAt_sum {f : α → E → F} {c : E} {s : Set E}
+    (N : Finset α) (h : ∀ n ∈ N, AnalyticWithinAt 𝕜 (f n) s c) :
+    AnalyticWithinAt 𝕜 (fun z ↦ ∑ n ∈ N, f n z) s c := by
   induction' N using Finset.induction with a B aB hB
   · simp only [Finset.sum_empty]
-    exact analyticAt_const
+    exact analyticWithinAt_const
   · simp_rw [Finset.sum_insert aB]
     simp only [Finset.mem_insert] at h
     exact (h a (Or.inl rfl)).add (hB fun b m ↦ h b (Or.inr m))
 
+/-- Finite sums of analytic functions are analytic -/
+theorem Finset.analyticAt_sum {f : α → E → F} {c : E}
+    (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
+    AnalyticAt 𝕜 (fun z ↦ ∑ n ∈ N, f n z) c := by
+  simp_rw [← analyticWithinAt_univ] at h ⊢
+  exact N.analyticWithinAt_sum h
+
 /-- Finite sums of analytic functions are analytic -/
 theorem Finset.analyticOn_sum {f : α → E → F} {s : Set E}
     (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s) :
     AnalyticOn 𝕜 (fun z ↦ ∑ n ∈ N, f n z) s :=
+  fun z zs ↦ N.analyticWithinAt_sum (fun n m ↦ h n m z zs)
+
+@[deprecated (since := "2024-09-26")]
+alias Finset.analyticWithinOn_sum := Finset.analyticOn_sum
+
+/-- Finite sums of analytic functions are analytic -/
+theorem Finset.analyticOnNhd_sum {f : α → E → F} {s : Set E}
+    (N : Finset α) (h : ∀ n ∈ N, AnalyticOnNhd 𝕜 (f n) s) :
+    AnalyticOnNhd 𝕜 (fun z ↦ ∑ n ∈ N, f n z) s :=
   fun z zs ↦ N.analyticAt_sum (fun n m ↦ h n m z zs)
 
 /-- Finite products of analytic functions are analytic -/
-theorem Finset.analyticAt_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
-    {f : α → E → A} {c : E} (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
-    AnalyticAt 𝕜 (fun z ↦ ∏ n ∈ N, f n z) c := by
+theorem Finset.analyticWithinAt_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
+    {f : α → E → A} {c : E} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticWithinAt 𝕜 (f n) s c) :
+    AnalyticWithinAt 𝕜 (fun z ↦ ∏ n ∈ N, f n z) s c := by
   induction' N using Finset.induction with a B aB hB
   · simp only [Finset.prod_empty]
-    exact analyticAt_const
+    exact analyticWithinAt_const
   · simp_rw [Finset.prod_insert aB]
     simp only [Finset.mem_insert] at h
     exact (h a (Or.inl rfl)).mul (hB fun b m ↦ h b (Or.inr m))
 
+/-- Finite products of analytic functions are analytic -/
+theorem Finset.analyticAt_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
+    {f : α → E → A} {c : E} (N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
+    AnalyticAt 𝕜 (fun z ↦ ∏ n ∈ N, f n z) c := by
+  simp_rw [← analyticWithinAt_univ] at h ⊢
+  exact N.analyticWithinAt_prod h
+
 /-- Finite products of analytic functions are analytic -/
 theorem Finset.analyticOn_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
     {f : α → E → A} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticOn 𝕜 (f n) s) :
     AnalyticOn 𝕜 (fun z ↦ ∏ n ∈ N, f n z) s :=
+  fun z zs ↦ N.analyticWithinAt_prod (fun n m ↦ h n m z zs)
+
+@[deprecated (since := "2024-09-26")]
+alias Finset.analyticWithinOn_prod := Finset.analyticOn_prod
+
+/-- Finite products of analytic functions are analytic -/
+theorem Finset.analyticOnNhd_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A]
+    {f : α → E → A} {s : Set E} (N : Finset α) (h : ∀ n ∈ N, AnalyticOnNhd 𝕜 (f n) s) :
+    AnalyticOnNhd 𝕜 (fun z ↦ ∏ n ∈ N, f n z) s :=
   fun z zs ↦ N.analyticAt_prod (fun n m ↦ h n m z zs)
diff --git a/Mathlib/Analysis/Analytic/Inverse.lean b/Mathlib/Analysis/Analytic/Inverse.lean
index 704910afccb1c..eadc577c91124 100644
--- a/Mathlib/Analysis/Analytic/Inverse.lean
+++ b/Mathlib/Analysis/Analytic/Inverse.lean
@@ -4,36 +4,43 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
 import Mathlib.Analysis.Analytic.Composition
+import Mathlib.Analysis.Analytic.Linear
 
 /-!
 
 # Inverse of analytic functions
 
 We construct the left and right inverse of a formal multilinear series with invertible linear term,
-we prove that they coincide and study their properties (notably convergence).
+we prove that they coincide and study their properties (notably convergence). We deduce that the
+inverse of an analytic partial homeomorphism is analytic.
 
 ## Main statements
 
-* `p.leftInv i`: the formal left inverse of the formal multilinear series `p`,
-  for `i : E ≃L[𝕜] F` which coincides with `p₁`.
-* `p.rightInv i`: the formal right inverse of the formal multilinear series `p`,
-  for `i : E ≃L[𝕜] F` which coincides with `p₁`.
-* `p.leftInv_comp` says that `p.leftInv i` is indeed a left inverse to `p` when `p₁ = i`.
-* `p.rightInv_comp` says that `p.rightInv i` is indeed a right inverse to `p` when `p₁ = i`.
+* `p.leftInv i x`: the formal left inverse of the formal multilinear series `p`, with constant
+  coefficient `x`, for `i : E ≃L[𝕜] F` which coincides with `p₁`.
+* `p.rightInv i x`: the formal right inverse of the formal multilinear series `p`, with constant
+  coefficient `x`, for `i : E ≃L[𝕜] F` which coincides with `p₁`.
+* `p.leftInv_comp` says that `p.leftInv i x` is indeed a left inverse to `p` when `p₁ = i`.
+* `p.rightInv_comp` says that `p.rightInv i x` is indeed a right inverse to `p` when `p₁ = i`.
 * `p.leftInv_eq_rightInv`: the two inverses coincide.
 * `p.radius_rightInv_pos_of_radius_pos`: if a power series has a positive radius of convergence,
   then so does its inverse.
 
+* `PartialHomeomorph.hasFPowerSeriesAt_symm` shows that, if a partial homeomorph has a power series
+  `p` at a point, with invertible linear part, then the inverse also has a power series at the
+  image point, given by `p.leftInv`.
 -/
 
-open scoped Topology
+open scoped Topology ENNReal
 
 open Finset Filter
 
-namespace FormalMultilinearSeries
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+  {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+  {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+  {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
 
-variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
-  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+namespace FormalMultilinearSeries
 
 /-! ### The left inverse of a formal multilinear series -/
 
@@ -51,27 +58,27 @@ term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`.
 These formulas only make sense when the constant term `p₀` vanishes. The definition we give is
 general, but it ignores the value of `p₀`.
 -/
-noncomputable def leftInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
+noncomputable def leftInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
     FormalMultilinearSeries 𝕜 F E
-  | 0 => 0
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ x
   | 1 => (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm
   | n + 2 =>
     -∑ c : { c : Composition (n + 2) // c.length < n + 2 },
-        (leftInv p i (c : Composition (n + 2)).length).compAlongComposition
+        (leftInv p i x (c : Composition (n + 2)).length).compAlongComposition
           (p.compContinuousLinearMap i.symm) c
 
 @[simp]
-theorem leftInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.leftInv i 0 = 0 := by rw [leftInv]
+theorem leftInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.leftInv i x 0 = ContinuousMultilinearMap.uncurry0 𝕜 _ x := by rw [leftInv]
 
 @[simp]
-theorem leftInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.leftInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [leftInv]
+theorem leftInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.leftInv i x 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [leftInv]
 
 /-- The left inverse does not depend on the zeroth coefficient of a formal multilinear
 series. -/
-theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.removeZero.leftInv i = p.leftInv i := by
+theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.removeZero.leftInv i x = p.leftInv i x := by
   ext1 n
   induction' n using Nat.strongRec' with n IH
   match n with
@@ -87,14 +94,15 @@ theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[
 
 /-- The left inverse to a formal multilinear series is indeed a left inverse, provided its linear
 term is invertible. -/
-theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
-    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : (leftInv p i).comp p = id 𝕜 E := by
-  ext (n v)
+theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
+    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
+    (leftInv p i x).comp p = id 𝕜 E x := by
+  ext n v
   classical
   match n with
   | 0 =>
-    simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne,
-      not_false_iff, zero_ne_one, comp_coeff_zero']
+    simp only [comp_coeff_zero', leftInv_coeff_zero, ContinuousMultilinearMap.uncurry0_apply,
+      id_apply_zero]
   | 1 =>
     simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
       ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply]
@@ -111,16 +119,16 @@ theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
         {Composition.ones (n + 2)} := by
       simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})]
     have C :
-      ((p.leftInv i (Composition.ones (n + 2)).length)
+      ((p.leftInv i x (Composition.ones (n + 2)).length)
           fun j : Fin (Composition.ones n.succ.succ).length =>
           p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) =
-        p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j := by
+        p.leftInv i x (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j := by
       apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_
       exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr
     have D :
-      (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =
+      (p.leftInv i x (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =
         -∑ c ∈ {c : Composition (n + 2) | c.length < n + 2}.toFinset,
-            (p.leftInv i c.length) (p.applyComposition c v) := by
+            (p.leftInv i x c.length) (p.applyComposition c v) := by
       simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj,
         ContinuousMultilinearMap.sum_apply]
       convert
@@ -128,7 +136,7 @@ theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
           (fun c : Composition (n + 2) => c.length < n + 2)
           (fun c : Composition (n + 2) =>
           (ContinuousMultilinearMap.compAlongComposition
-            (p.compContinuousLinearMap (i.symm : F →L[𝕜] E)) c (p.leftInv i c.length))
+            (p.compContinuousLinearMap (i.symm : F →L[𝕜] E)) c (p.leftInv i x c.length))
             fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans
           _
       simp only [compContinuousLinearMap_applyComposition,
@@ -137,7 +145,7 @@ theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
       ext c
       congr
       ext k
-      simp [h, Function.comp]
+      simp [h, Function.comp_def]
     simp [FormalMultilinearSeries.comp, show n + 2 ≠ 1 by omega, A, Finset.sum_union B,
       applyComposition_ones, C, D, -Set.toFinset_setOf]
 
@@ -157,26 +165,26 @@ term compensates the rest of the sum, using `i⁻¹` as an inverse to `p₁`.
 These formulas only make sense when the constant term `p₀` vanishes. The definition we give is
 general, but it ignores the value of `p₀`.
 -/
-noncomputable def rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
+noncomputable def rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
     FormalMultilinearSeries 𝕜 F E
-  | 0 => 0
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ x
   | 1 => (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm
   | n + 2 =>
-    let q : FormalMultilinearSeries 𝕜 F E := fun k => if k < n + 2 then rightInv p i k else 0;
+    let q : FormalMultilinearSeries 𝕜 F E := fun k => if k < n + 2 then rightInv p i x k else 0;
     -(i.symm : F →L[𝕜] E).compContinuousMultilinearMap ((p.comp q) (n + 2))
 
 @[simp]
-theorem rightInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.rightInv i 0 = 0 := by rw [rightInv]
+theorem rightInv_coeff_zero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.rightInv i x 0 = ContinuousMultilinearMap.uncurry0 𝕜 _ x := by rw [rightInv]
 
 @[simp]
-theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.rightInv i 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [rightInv]
+theorem rightInv_coeff_one (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.rightInv i x 1 = (continuousMultilinearCurryFin1 𝕜 F E).symm i.symm := by rw [rightInv]
 
 /-- The right inverse does not depend on the zeroth coefficient of a formal multilinear
 series. -/
-theorem rightInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
-    p.removeZero.rightInv i = p.rightInv i := by
+theorem rightInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) :
+    p.removeZero.rightInv i x = p.rightInv i x := by
   ext1 n
   induction' n using Nat.strongRec' with n IH
   match n with
@@ -216,12 +224,12 @@ theorem comp_rightInv_aux1 {n : ℕ} (hn : 0 < n) (p : FormalMultilinearSeries 
   simp [FormalMultilinearSeries.comp, A, Finset.sum_union B, C, -Set.toFinset_setOf,
     -add_right_inj, -Composition.single_length]
 
-theorem comp_rightInv_aux2 (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ)
+theorem comp_rightInv_aux2 (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E) (n : ℕ)
     (v : Fin (n + 2) → F) :
     ∑ c ∈ {c : Composition (n + 2) | 1 < c.length}.toFinset,
-        p c.length (applyComposition (fun k : ℕ => ite (k < n + 2) (p.rightInv i k) 0) c v) =
+        p c.length (applyComposition (fun k : ℕ => ite (k < n + 2) (p.rightInv i x k) 0) c v) =
       ∑ c ∈ {c : Composition (n + 2) | 1 < c.length}.toFinset,
-        p c.length ((p.rightInv i).applyComposition c v) := by
+        p c.length ((p.rightInv i x).applyComposition c v) := by
   have N : 0 < n + 2 := by norm_num
   refine sum_congr rfl fun c hc => p.congr rfl fun j hj1 hj2 => ?_
   have : ∀ k, c.blocksFun k < n + 2 := by
@@ -232,14 +240,16 @@ theorem comp_rightInv_aux2 (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[
 
 /-- The right inverse to a formal multilinear series is indeed a right inverse, provided its linear
 term is invertible and its constant term vanishes. -/
-theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
-    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
-    p.comp (rightInv p i) = id 𝕜 F := by
+theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
+    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
+    p.comp (rightInv p i x) = id 𝕜 F (p 0 0) := by
   ext (n v)
   match n with
   | 0 =>
-    simp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne, not_false_iff,
-      zero_ne_one, comp_coeff_zero']
+    simp only [comp_coeff_zero', Matrix.zero_empty, id_apply_zero]
+    congr
+    ext i
+    exact i.elim0
   | 1 =>
     simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
       id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinearCurryFin1_symm_apply]
@@ -248,11 +258,12 @@ theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F
     simp [comp_rightInv_aux1 N, h, rightInv, lt_irrefl n, show n + 2 ≠ 1 by omega,
       ← sub_eq_add_neg, sub_eq_zero, comp_rightInv_aux2, -Set.toFinset_setOf]
 
-theorem rightInv_coeff (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (n : ℕ) (hn : 2 ≤ n) :
-    p.rightInv i n =
+theorem rightInv_coeff (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
+    (n : ℕ) (hn : 2 ≤ n) :
+    p.rightInv i x n =
       -(i.symm : F →L[𝕜] E).compContinuousMultilinearMap
           (∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition n)),
-            p.compAlongComposition (p.rightInv i) c) := by
+            p.compAlongComposition (p.rightInv i x) c) := by
   match n with
   | 0 => exact False.elim (zero_lt_two.not_le hn)
   | 1 => exact False.elim (one_lt_two.not_le hn)
@@ -267,26 +278,15 @@ theorem rightInv_coeff (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜]
 /-! ### Coincidence of the left and the right inverse -/
 
 
-private theorem leftInv_eq_rightInv_aux (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
-    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
-    leftInv p i = rightInv p i :=
-  calc
-    leftInv p i = (leftInv p i).comp (id 𝕜 F) := by simp
-    _ = (leftInv p i).comp (p.comp (rightInv p i)) := by rw [comp_rightInv p i h h0]
-    _ = ((leftInv p i).comp p).comp (rightInv p i) := by rw [comp_assoc]
-    _ = (id 𝕜 E).comp (rightInv p i) := by rw [leftInv_comp p i h]
-    _ = rightInv p i := by simp
-
-/-- The left inverse and the right inverse of a formal multilinear series coincide. This is not at
-all obvious from their definition, but it follows from uniqueness of inverses (which comes from the
-fact that composition is associative on formal multilinear series). -/
-theorem leftInv_eq_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
-    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : leftInv p i = rightInv p i :=
+theorem leftInv_eq_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) (x : E)
+    (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
+    leftInv p i x = rightInv p i x :=
   calc
-    leftInv p i = leftInv p.removeZero i := by rw [leftInv_removeZero]
-    _ = rightInv p.removeZero i := by
-      apply leftInv_eq_rightInv_aux _ _ (by simpa using h) (by simp)
-    _ = rightInv p i := by rw [rightInv_removeZero]
+    leftInv p i x = (leftInv p i x).comp (id 𝕜 F (p 0 0)) := by simp
+    _ = (leftInv p i x).comp (p.comp (rightInv p i x)) := by rw [comp_rightInv p i _ h]
+    _ = ((leftInv p i x).comp p).comp (rightInv p i x) := by rw [comp_assoc]
+    _ = (id 𝕜 E x).comp (rightInv p i x) := by rw [leftInv_comp p i _ h]
+    _ = rightInv p i x := by simp [id_comp' _ _ 0]
 
 /-!
 ### Convergence of the inverse of a power series
@@ -423,17 +423,17 @@ theorem radius_right_inv_pos_of_radius_pos_aux1 (n : ℕ) (p : ℕ → ℝ) (hp
 expression for `∑_{k<n+1} aᵏ Qₖ` in terms of a sum of powers of the same sum one step before,
 in the specific setup we are interesting in, by reducing to the general bound in
 `radius_rightInv_pos_of_radius_pos_aux1`. -/
-theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
+theorem radius_rightInv_pos_of_radius_pos_aux2 {x : E} {n : ℕ} (hn : 2 ≤ n + 1)
     (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) {r a C : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a)
     (hC : 0 ≤ C) (hp : ∀ n, ‖p n‖ ≤ C * r ^ n) :
-    ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ ≤
+    ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i x k‖ ≤
       ‖(i.symm : F →L[𝕜] E)‖ * a +
         ‖(i.symm : F →L[𝕜] E)‖ * C *
-          ∑ k ∈ Ico 2 (n + 1), (r * ∑ j ∈ Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k :=
+          ∑ k ∈ Ico 2 (n + 1), (r * ∑ j ∈ Ico 1 n, a ^ j * ‖p.rightInv i x j‖) ^ k :=
   let I := ‖(i.symm : F →L[𝕜] E)‖
   calc
-    ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ =
-        a * I + ∑ k ∈ Ico 2 (n + 1), a ^ k * ‖p.rightInv i k‖ := by
+    ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i x k‖ =
+        a * I + ∑ k ∈ Ico 2 (n + 1), a ^ k * ‖p.rightInv i x k‖ := by
       simp only [LinearIsometryEquiv.norm_map, pow_one, rightInv_coeff_one,
         show Ico (1 : ℕ) 2 = {1} from Nat.Ico_succ_singleton 1,
         sum_singleton, ← sum_Ico_consecutive _ one_le_two hn]
@@ -443,16 +443,16 @@ theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
             a ^ k *
               ‖(i.symm : F →L[𝕜] E).compContinuousMultilinearMap
                   (∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
-                    p.compAlongComposition (p.rightInv i) c)‖ := by
+                    p.compAlongComposition (p.rightInv i x) c)‖ := by
       congr! 2 with j hj
-      rw [rightInv_coeff _ _ _ (mem_Ico.1 hj).1, norm_neg]
+      rw [rightInv_coeff _ _ _ _ (mem_Ico.1 hj).1, norm_neg]
     _ ≤
         a * ‖(i.symm : F →L[𝕜] E)‖ +
           ∑ k ∈ Ico 2 (n + 1),
             a ^ k *
               (I *
                 ∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
-                  C * r ^ c.length * ∏ j, ‖p.rightInv i (c.blocksFun j)‖) := by
+                  C * r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖) := by
       gcongr with j
       apply (ContinuousLinearMap.norm_compContinuousMultilinearMap_le _ _).trans
       gcongr
@@ -461,28 +461,25 @@ theorem radius_rightInv_pos_of_radius_pos_aux2 {n : ℕ} (hn : 2 ≤ n + 1)
       apply (compAlongComposition_norm _ _ _).trans
       gcongr
       apply hp
-    _ =
-        I * a +
-          I * C *
-            ∑ k ∈ Ico 2 (n + 1),
-              a ^ k *
-                ∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
-                  r ^ c.length * ∏ j, ‖p.rightInv i (c.blocksFun j)‖ := by
+    _ = I * a + I * C * ∑ k ∈ Ico 2 (n + 1), a ^ k *
+          ∑ c ∈ ({c | 1 < Composition.length c}.toFinset : Finset (Composition k)),
+            r ^ c.length * ∏ j, ‖p.rightInv i x (c.blocksFun j)‖ := by
       simp_rw [mul_assoc C, ← mul_sum, ← mul_assoc, mul_comm _ ‖(i.symm : F →L[𝕜] E)‖, mul_assoc,
         ← mul_sum, ← mul_assoc, mul_comm _ C, mul_assoc, ← mul_sum]
       ring
     _ ≤ I * a + I * C *
-        ∑ k ∈ Ico 2 (n + 1), (r * ∑ j ∈ Ico 1 n, a ^ j * ‖p.rightInv i j‖) ^ k := by
+        ∑ k ∈ Ico 2 (n + 1), (r * ∑ j ∈ Ico 1 n, a ^ j * ‖p.rightInv i x j‖) ^ k := by
       gcongr _ + _ * _ * ?_
       simp_rw [mul_pow]
       apply
-        radius_right_inv_pos_of_radius_pos_aux1 n (fun k => ‖p.rightInv i k‖)
+        radius_right_inv_pos_of_radius_pos_aux1 n (fun k => ‖p.rightInv i x k‖)
           (fun k => norm_nonneg _) hr ha
 
 /-- If a a formal multilinear series has a positive radius of convergence, then its right inverse
 also has a positive radius of convergence. -/
-theorem radius_rightInv_pos_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
-    (hp : 0 < p.radius) : 0 < (p.rightInv i).radius := by
+theorem radius_rightInv_pos_of_radius_pos
+    {p : FormalMultilinearSeries 𝕜 E F} {i : E ≃L[𝕜] F} {x : E}
+    (hp : 0 < p.radius) : 0 < (p.rightInv i x).radius := by
   obtain ⟨C, r, Cpos, rpos, ple⟩ :
     ∃ (C r : _) (_ : 0 < C) (_ : 0 < r), ∀ n : ℕ, ‖p n‖ ≤ C * r ^ n :=
     le_mul_pow_of_radius_pos p hp
@@ -508,7 +505,7 @@ theorem radius_rightInv_pos_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F)
     exact ⟨a, ha.1, ha.2.1.le, ha.2.2.le⟩
   -- check by induction that the partial sums are suitably bounded, using the choice of `a` and the
   -- inductive control from Lemma `radius_rightInv_pos_of_radius_pos_aux2`.
-  let S n := ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i k‖
+  let S n := ∑ k ∈ Ico 1 n, a ^ k * ‖p.rightInv i x k‖
   have IRec : ∀ n, 1 ≤ n → S n ≤ (I + 1) * a := by
     apply Nat.le_induction
     · simp only [S]
@@ -536,21 +533,159 @@ theorem radius_rightInv_pos_of_radius_pos (p : FormalMultilinearSeries 𝕜 E F)
         _ ≤ (I + 1) * a := by gcongr
   -- conclude that all coefficients satisfy `aⁿ Qₙ ≤ (I + 1) a`.
   let a' : NNReal := ⟨a, apos.le⟩
-  suffices H : (a' : ENNReal) ≤ (p.rightInv i).radius by
+  suffices H : (a' : ENNReal) ≤ (p.rightInv i x).radius by
     apply lt_of_lt_of_le _ H
     -- Prior to leanprover/lean4#2734, this was `exact_mod_cast apos`.
     simpa only [ENNReal.coe_pos]
-  apply le_radius_of_bound _ ((I + 1) * a) fun n => ?_
-  by_cases hn : n = 0
-  · have : ‖p.rightInv i n‖ = ‖p.rightInv i 0‖ := by congr <;> try rw [hn]
-    simp only [this, norm_zero, zero_mul, rightInv_coeff_zero]
-    positivity
-  · have one_le_n : 1 ≤ n := bot_lt_iff_ne_bot.2 hn
-    calc
-      ‖p.rightInv i n‖ * (a' : ℝ) ^ n = a ^ n * ‖p.rightInv i n‖ := mul_comm _ _
-      _ ≤ ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i k‖ :=
-        (haveI : ∀ k ∈ Ico 1 (n + 1), 0 ≤ a ^ k * ‖p.rightInv i k‖ := fun k _ => by positivity
-        single_le_sum this (by simp [one_le_n]))
-      _ ≤ (I + 1) * a := IRec (n + 1) (by norm_num)
+  apply le_radius_of_eventually_le _ ((I + 1) * a)
+  filter_upwards [Ici_mem_atTop 1] with n (hn : 1 ≤ n)
+  calc
+    ‖p.rightInv i x n‖ * (a' : ℝ) ^ n = a ^ n * ‖p.rightInv i x n‖ := mul_comm _ _
+    _ ≤ ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i x k‖ :=
+      (haveI : ∀ k ∈ Ico 1 (n + 1), 0 ≤ a ^ k * ‖p.rightInv i x k‖ := fun k _ => by positivity
+      single_le_sum this (by simp [hn]))
+    _ ≤ (I + 1) * a := IRec (n + 1) (by norm_num)
+
+/-- If a a formal multilinear series has a positive radius of convergence, then its left inverse
+also has a positive radius of convergence. -/
+theorem radius_leftInv_pos_of_radius_pos
+    {p : FormalMultilinearSeries 𝕜 E F} {i : E ≃L[𝕜] F} {x : E}
+    (hp : 0 < p.radius) (h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
+    0 < (p.leftInv i x).radius := by
+  rw [leftInv_eq_rightInv _ _ _ h]
+  exact radius_rightInv_pos_of_radius_pos hp
 
 end FormalMultilinearSeries
+
+/-!
+### The inverse of an analytic partial homeomorphism is analytic
+-/
+
+open FormalMultilinearSeries List
+
+lemma HasFPowerSeriesAt.tendsto_partialSum_prod_of_comp
+    {f : E → G} {q : FormalMultilinearSeries 𝕜 F G}
+    {p : FormalMultilinearSeries 𝕜 E F} {x : E}
+    (hf : HasFPowerSeriesAt f (q.comp p) x) (hq : 0 < q.radius) (hp : 0 < p.radius) :
+    ∀ᶠ y in 𝓝 0, Tendsto (fun (a : ℕ × ℕ) ↦ q.partialSum a.1 (p.partialSum a.2 y
+      - p 0 (fun _ ↦ 0))) atTop (𝓝 (f (x + y))) := by
+  rcases hf with ⟨r0, h0⟩
+  rcases q.comp_summable_nnreal p hq hp with ⟨r1, r1_pos : 0 < r1, hr1⟩
+  let r : ℝ≥0∞ := min r0 r1
+  have : EMetric.ball (0 : E) r ∈ 𝓝 0 :=
+    EMetric.ball_mem_nhds 0 (lt_min h0.r_pos (by exact_mod_cast r1_pos))
+  filter_upwards [this] with y hy
+  have hy0 : y ∈ EMetric.ball 0 r0 := EMetric.ball_subset_ball (min_le_left _ _) hy
+  have A : HasSum (fun i : Σ n, Composition n => q.compAlongComposition p i.2 fun _j => y)
+      (f (x + y)) := by
+    have cau : CauchySeq fun s : Finset (Σ n, Composition n) =>
+        ∑ i ∈ s, q.compAlongComposition p i.2 fun _j => y := by
+      apply cauchySeq_finset_of_norm_bounded _ (NNReal.summable_coe.2 hr1) _
+      simp only [coe_nnnorm, NNReal.coe_mul, NNReal.coe_pow]
+      rintro ⟨n, c⟩
+      calc
+        ‖(compAlongComposition q p c) fun _j : Fin n => y‖ ≤
+            ‖compAlongComposition q p c‖ * ∏ _j : Fin n, ‖y‖ := by
+          apply ContinuousMultilinearMap.le_opNorm
+        _ ≤ ‖compAlongComposition q p c‖ * (r1 : ℝ) ^ n := by
+          apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
+          rw [Finset.prod_const, Finset.card_fin]
+          apply pow_le_pow_left (norm_nonneg _)
+          rw [EMetric.mem_ball, edist_eq_coe_nnnorm] at hy
+          have := le_trans (le_of_lt hy) (min_le_right _ _)
+          rwa [ENNReal.coe_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this
+    apply HasSum.of_sigma (fun b ↦ hasSum_fintype _) ?_ cau
+    simpa [FormalMultilinearSeries.comp] using h0.hasSum hy0
+  have B : Tendsto (fun (n : ℕ × ℕ) => ∑ i ∈ compPartialSumTarget 0 n.1 n.2,
+      q.compAlongComposition p i.2 fun _j => y) atTop (𝓝 (f (x + y))) := by
+    apply Tendsto.comp A compPartialSumTarget_tendsto_prod_atTop
+  have C : Tendsto (fun (n : ℕ × ℕ) => q.partialSum n.1 (∑ a ∈ Finset.Ico 1 n.2, p a fun _b ↦ y))
+      atTop (𝓝 (f (x + y))) := by simpa [comp_partialSum] using B
+  apply C.congr'
+  filter_upwards [Ici_mem_atTop (0, 1)]
+  rintro ⟨-, n⟩ ⟨-, (hn : 1 ≤ n)⟩
+  congr
+  rw [partialSum, eq_sub_iff_add_eq', Finset.range_eq_Ico,
+        Finset.sum_eq_sum_Ico_succ_bot hn]
+  congr with i
+  exact i.elim0
+
+lemma HasFPowerSeriesAt.eventually_hasSum_of_comp  {f : E → F} {g : F → G}
+    {q : FormalMultilinearSeries 𝕜 F G} {p : FormalMultilinearSeries 𝕜 E F} {x : E}
+    (hgf : HasFPowerSeriesAt (g ∘ f) (q.comp p) x) (hf : HasFPowerSeriesAt f p x)
+    (hq : 0 < q.radius) :
+    ∀ᶠ y in 𝓝 0, HasSum (fun n : ℕ => q n fun _ : Fin n => (f (x + y) - f x)) (g (f (x + y))) := by
+  have : ∀ᶠ y in 𝓝 (0 : E), f (x + y) - f x ∈ EMetric.ball 0 q.radius := by
+    have A : ContinuousAt (fun y ↦ f (x + y) - f x) 0 := by
+      apply ContinuousAt.sub _ continuousAt_const
+      exact hf.continuousAt.comp_of_eq (continuous_add_left x).continuousAt (by simp)
+    have B : EMetric.ball 0 q.radius ∈ 𝓝 (f (x + 0) - f x) := by
+      simpa using EMetric.ball_mem_nhds _ hq
+    exact A.preimage_mem_nhds B
+  filter_upwards [hgf.tendsto_partialSum_prod_of_comp hq (hf.radius_pos),
+    hf.tendsto_partialSum, this] with y hy h'y h''y
+  have L : Tendsto (fun n ↦ q.partialSum n (f (x + y) - f x)) atTop (𝓝 (g (f (x + y)))) := by
+    apply (closed_nhds_basis (g (f (x + y)))).tendsto_right_iff.2
+    rintro u ⟨hu, u_closed⟩
+    simp only [id_eq, eventually_atTop, ge_iff_le]
+    rcases mem_nhds_iff.1 hu with ⟨v, vu, v_open, hv⟩
+    obtain ⟨a₀, b₀, hab⟩ : ∃ a₀ b₀, ∀ (a b : ℕ), a₀ ≤ a → b₀ ≤ b →
+        q.partialSum a (p.partialSum b y - (p 0) fun x ↦ 0) ∈ v := by
+      simpa using hy (v_open.mem_nhds hv)
+    refine ⟨a₀, fun a ha ↦ ?_⟩
+    have : Tendsto (fun b ↦ q.partialSum a (p.partialSum b y - (p 0) fun x ↦ 0)) atTop
+        (𝓝 (q.partialSum a (f (x + y) - f x))) := by
+      have : ContinuousAt (q.partialSum a) (f (x + y) - f x) :=
+        (partialSum_continuous q a).continuousAt
+      apply this.tendsto.comp
+      apply Tendsto.sub h'y
+      convert tendsto_const_nhds
+      exact (HasFPowerSeriesAt.coeff_zero hf fun _ ↦ 0).symm
+    apply u_closed.mem_of_tendsto this
+    filter_upwards [Ici_mem_atTop b₀] with b hb using vu (hab _ _ ha hb)
+  have C : CauchySeq (fun (s : Finset ℕ) ↦ ∑ n ∈ s, q n fun _ : Fin n => (f (x + y) - f x)) := by
+    have Z := q.summable_norm_apply (x := f (x + y) - f x) h''y
+    exact cauchySeq_finset_of_norm_bounded _ Z (fun i ↦ le_rfl)
+  exact tendsto_nhds_of_cauchySeq_of_subseq C tendsto_finset_range L
+
+/-- If a partial homeomorphism `f` is defined at `a` and has a power series expansion there with
+invertible linear term, then `f.symm` has a power series expansion at `f a`, given by the inverse
+of the initial power series. -/
+theorem PartialHomeomorph.hasFPowerSeriesAt_symm (f : PartialHomeomorph E F) {a : E}
+    {i : E ≃L[𝕜] F} (h0 : a ∈ f.source) {p : FormalMultilinearSeries 𝕜 E F}
+    (h : HasFPowerSeriesAt f p a) (hp : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) :
+    HasFPowerSeriesAt f.symm (p.leftInv i a) (f a) := by
+  have A : HasFPowerSeriesAt (f.symm ∘ f) ((p.leftInv i a).comp p) a := by
+    have : HasFPowerSeriesAt (ContinuousLinearMap.id 𝕜 E) ((p.leftInv i a).comp p) a := by
+      rw [leftInv_comp _ _ _ hp]
+      exact (ContinuousLinearMap.id 𝕜 E).hasFPowerSeriesAt a
+    apply this.congr
+    filter_upwards [f.open_source.mem_nhds h0] with x hx using by simp [hx]
+  have B : ∀ᶠ (y : E) in 𝓝 0, HasSum (fun n ↦ (p.leftInv i a n) fun _ ↦ f (a + y) - f a)
+      (f.symm (f (a + y))) := by
+    simpa using A.eventually_hasSum_of_comp h (radius_leftInv_pos_of_radius_pos h.radius_pos hp)
+  have C : ∀ᶠ (y : E) in 𝓝 a, HasSum (fun n ↦ (p.leftInv i a n) fun _ ↦ f y - f a)
+      (f.symm (f y)) := by
+    rw [← sub_eq_zero_of_eq (a := a) rfl] at B
+    have : ContinuousAt (fun x ↦ x - a) a := by fun_prop
+    simpa using this.preimage_mem_nhds B
+  have D : ∀ᶠ (y : E) in 𝓝 (f.symm (f a)),
+      HasSum (fun n ↦ (p.leftInv i a n) fun _ ↦ f y - f a) y := by
+    simp only [h0, PartialHomeomorph.left_inv]
+    filter_upwards [C, f.open_source.mem_nhds h0] with x hx h'x
+    simpa [h'x] using hx
+  have E : ∀ᶠ z in 𝓝 (f a), HasSum (fun n ↦ (p.leftInv i a n) fun _ ↦ f (f.symm z) - f a)
+      (f.symm z) := by
+    have : ContinuousAt f.symm (f a) := f.continuousAt_symm (f.map_source h0)
+    exact this D
+  have F : ∀ᶠ z in 𝓝 (f a), HasSum (fun n ↦ (p.leftInv i a n) fun _ ↦ z - f a) (f.symm z) := by
+    filter_upwards [f.open_target.mem_nhds (f.map_source h0), E] with z hz h'z
+    simpa [hz] using h'z
+  rcases EMetric.mem_nhds_iff.1 F with ⟨r, r_pos, hr⟩
+  refine ⟨min r (p.leftInv i a).radius, min_le_right _ _,
+    lt_min r_pos (radius_leftInv_pos_of_radius_pos h.radius_pos hp), fun {y} hy ↦ ?_⟩
+  have : y + f a ∈ EMetric.ball (f a) r := by
+    simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, sub_zero, lt_min_iff,
+      add_sub_cancel_right] at hy ⊢
+    exact hy.1
+  simpa [add_comm] using hr this
diff --git a/Mathlib/Analysis/Analytic/IsolatedZeros.lean b/Mathlib/Analysis/Analytic/IsolatedZeros.lean
index 5646e3721fa72..18aad9fb34b3e 100644
--- a/Mathlib/Analysis/Analytic/IsolatedZeros.lean
+++ b/Mathlib/Analysis/Analytic/IsolatedZeros.lean
@@ -20,7 +20,7 @@ useful in this setup.
 * `AnalyticAt.eventually_eq_zero_or_eventually_ne_zero` is the main statement that if a function is
   analytic at `z₀`, then either it is identically zero in a neighborhood of `z₀`, or it does not
   vanish in a punctured neighborhood of `z₀`.
-* `AnalyticOn.eqOn_of_preconnected_of_frequently_eq` is the identity theorem for analytic
+* `AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq` is the identity theorem for analytic
   functions: if a function `f` is analytic on a connected set `U` and is zero on a set with an
   accumulation point in `U` then `f` is identically `0` on `U`.
 -/
@@ -157,7 +157,7 @@ lemma unique_eventuallyEq_zpow_smul_nonzero {m n : ℤ}
   rw [frequently_eq_iff_eventually_eq hj_an] at this
   · rw [EventuallyEq.eq_of_nhds this, sub_self, zero_zpow _ (sub_ne_zero.mpr hj_ne), zero_smul]
   conv => enter [2, z, 1]; rw [← Int.toNat_sub_of_le h_le, zpow_natCast]
-  exact (((analyticAt_id _ _).sub analyticAt_const).pow _).smul hg_an
+  exact ((analyticAt_id.sub analyticAt_const).pow _).smul hg_an
 
 /-- For a function `f` on `𝕜`, and `z₀ ∈ 𝕜`, there exists at most one `n` such that on a
 neighbourhood of `z₀` we have `f z = (z - z₀) ^ n • g z`, with `g` analytic and nonvanishing at
@@ -221,7 +221,7 @@ lemma order_eq_nat_iff (hf : AnalyticAt 𝕜 f z₀) (n : ℕ) : hf.order = ↑n
 
 end AnalyticAt
 
-namespace AnalyticOn
+namespace AnalyticOnNhd
 
 variable {U : Set 𝕜}
 
@@ -229,13 +229,22 @@ variable {U : Set 𝕜}
 analytic on a connected set `U` and vanishes in arbitrary neighborhoods of a point `z₀ ∈ U`, then
 it is identically zero in `U`.
 For higher-dimensional versions requiring that the function vanishes in a neighborhood of `z₀`,
-see `AnalyticOn.eqOn_zero_of_preconnected_of_eventuallyEq_zero`. -/
-theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOn 𝕜 f U)
+see `AnalyticOnNhd.eqOn_zero_of_preconnected_of_eventuallyEq_zero`. -/
+theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOnNhd 𝕜 f U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfw : ∃ᶠ z in 𝓝[≠] z₀, f z = 0) : EqOn f 0 U :=
   hf.eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h₀
     ((hf z₀ h₀).frequently_zero_iff_eventually_zero.1 hfw)
 
-theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U)
+theorem eqOn_zero_or_eventually_ne_zero_of_preconnected (hf : AnalyticOnNhd 𝕜 f U)
+    (hU : IsPreconnected U) : EqOn f 0 U ∨ ∀ᶠ x in codiscreteWithin U, f x ≠ 0 := by
+  simp only [or_iff_not_imp_right, ne_eq, eventually_iff, mem_codiscreteWithin,
+    disjoint_principal_right, not_forall]
+  rintro ⟨x, hx, hx2⟩
+  refine hf.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hx fun nh ↦ hx2 ?_
+  filter_upwards [nh] with a ha
+  simp_all
+
+theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U)
     (h₀ : z₀ ∈ U) (hfz₀ : z₀ ∈ closure ({z | f z = 0} \ {z₀})) : EqOn f 0 U :=
   hf.eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀
     (mem_closure_ne_iff_frequently_within.mp hfz₀)
@@ -244,15 +253,21 @@ theorem eqOn_zero_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hU
 analytic on a connected set `U` and coincide at points which accumulate to a point `z₀ ∈ U`, then
 they coincide globally in `U`.
 For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
-see `AnalyticOn.eqOn_of_preconnected_of_eventuallyEq`. -/
-theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
+see `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`. -/
+theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOnNhd 𝕜 f U) (hg : AnalyticOnNhd 𝕜 g U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : EqOn f g U := by
   have hfg' : ∃ᶠ z in 𝓝[≠] z₀, (f - g) z = 0 :=
     hfg.mono fun z h => by rw [Pi.sub_apply, h, sub_self]
   simpa [sub_eq_zero] using fun z hz =>
     (hf.sub hg).eqOn_zero_of_preconnected_of_frequently_eq_zero hU h₀ hfg' hz
 
-theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : AnalyticOn 𝕜 g U)
+theorem eqOn_or_eventually_ne_of_preconnected (hf : AnalyticOnNhd 𝕜 f U) (hg : AnalyticOnNhd 𝕜 g U)
+    (hU : IsPreconnected U) : EqOn f g U ∨ ∀ᶠ x in codiscreteWithin U, f x ≠ g x :=
+  (eqOn_zero_or_eventually_ne_zero_of_preconnected (hf.sub hg) hU).imp
+    (fun h _ hx ↦ eq_of_sub_eq_zero (h hx))
+    (by simp only [Pi.sub_apply, ne_eq, sub_eq_zero, imp_self])
+
+theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOnNhd 𝕜 f U) (hg : AnalyticOnNhd 𝕜 g U)
     (hU : IsPreconnected U) (h₀ : z₀ ∈ U) (hfg : z₀ ∈ closure ({z | f z = g z} \ {z₀})) :
     EqOn f g U :=
   hf.eqOn_of_preconnected_of_frequently_eq hg hU h₀ (mem_closure_ne_iff_frequently_within.mp hfg)
@@ -261,10 +276,13 @@ theorem eqOn_of_preconnected_of_mem_closure (hf : AnalyticOn 𝕜 f U) (hg : Ana
 field `𝕜` are analytic everywhere and coincide at points which accumulate to a point `z₀`, then
 they coincide globally.
 For higher-dimensional versions requiring that the functions coincide in a neighborhood of `z₀`,
-see `AnalyticOn.eq_of_eventuallyEq`. -/
-theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOn 𝕜 f univ) (hg : AnalyticOn 𝕜 g univ)
-    (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g :=
+see `AnalyticOnNhd.eq_of_eventuallyEq`. -/
+theorem eq_of_frequently_eq [ConnectedSpace 𝕜] (hf : AnalyticOnNhd 𝕜 f univ)
+    (hg : AnalyticOnNhd 𝕜 g univ) (hfg : ∃ᶠ z in 𝓝[≠] z₀, f z = g z) : f = g :=
   funext fun x =>
     eqOn_of_preconnected_of_frequently_eq hf hg isPreconnected_univ (mem_univ z₀) hfg (mem_univ x)
 
-end AnalyticOn
+@[deprecated (since := "2024-09-26")]
+alias _root_.AnalyticOn.eq_of_frequently_eq := eq_of_frequently_eq
+
+end AnalyticOnNhd
diff --git a/Mathlib/Analysis/Analytic/Linear.lean b/Mathlib/Analysis/Analytic/Linear.lean
index 6bd87d9e23035..e527cda45057c 100644
--- a/Mathlib/Analysis/Analytic/Linear.lean
+++ b/Mathlib/Analysis/Analytic/Linear.lean
@@ -9,7 +9,10 @@ import Mathlib.Analysis.Analytic.Basic
 # Linear functions are analytic
 
 In this file we prove that a `ContinuousLinearMap` defines an analytic function with
-the formal power series `f x = f a + f (x - a)`. We also prove similar results for multilinear maps.
+the formal power series `f x = f a + f (x - a)`. We also prove similar results for bilinear maps.
+
+TODO: port to use `CPolynomial`, and prove the stronger result that continuous linear maps are
+continuously polynomial
 -/
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
@@ -41,6 +44,18 @@ protected theorem hasFPowerSeriesAt (f : E →L[𝕜] F) (x : E) :
 protected theorem analyticAt (f : E →L[𝕜] F) (x : E) : AnalyticAt 𝕜 f x :=
   (f.hasFPowerSeriesAt x).analyticAt
 
+protected theorem analyticOnNhd (f : E →L[𝕜] F) (s : Set E) : AnalyticOnNhd 𝕜 f s :=
+  fun x _ ↦ f.analyticAt x
+
+protected theorem analyticWithinAt (f : E →L[𝕜] F) (s : Set E) (x : E) : AnalyticWithinAt 𝕜 f s x :=
+  (f.analyticAt x).analyticWithinAt
+
+protected theorem analyticOn (f : E →L[𝕜] F) (s : Set E) : AnalyticOn 𝕜 f s :=
+  fun x _ ↦ f.analyticWithinAt _ x
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn := ContinuousLinearMap.analyticOn
+
 /-- Reinterpret a bilinear map `f : E →L[𝕜] F →L[𝕜] G` as a multilinear map
 `(E × F) [×2]→L[𝕜] G`. This multilinear map is the second term in the formal
 multilinear series expansion of `uncurry f`. It is given by
@@ -57,13 +72,13 @@ theorem uncurryBilinear_apply (f : E →L[𝕜] F →L[𝕜] G) (m : Fin 2 → E
 
 /-- Formal multilinear series expansion of a bilinear function `f : E →L[𝕜] F →L[𝕜] G`. -/
 def fpowerSeriesBilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) : FormalMultilinearSeries 𝕜 (E × F) G
-  | 0 => ContinuousMultilinearMap.curry0 𝕜 _ (f x.1 x.2)
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ (f x.1 x.2)
   | 1 => (continuousMultilinearCurryFin1 𝕜 (E × F) G).symm (f.deriv₂ x)
   | 2 => f.uncurryBilinear
   | _ => 0
 
 theorem fpowerSeriesBilinear_apply_zero (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
-    fpowerSeriesBilinear f x 0 = ContinuousMultilinearMap.curry0 𝕜 _ (f x.1 x.2) :=
+    fpowerSeriesBilinear f x 0 = ContinuousMultilinearMap.uncurry0 𝕜 _ (f x.1 x.2) :=
   rfl
 
 theorem fpowerSeriesBilinear_apply_one (f : E →L[𝕜] F →L[𝕜] G) (x : E × F) :
@@ -109,29 +124,110 @@ protected theorem analyticAt_bilinear (f : E →L[𝕜] F →L[𝕜] G) (x : E 
     AnalyticAt 𝕜 (fun x : E × F => f x.1 x.2) x :=
   (f.hasFPowerSeriesAt_bilinear x).analyticAt
 
+protected theorem analyticWithinAt_bilinear (f : E →L[𝕜] F →L[𝕜] G) (s : Set (E × F)) (x : E × F) :
+    AnalyticWithinAt 𝕜 (fun x : E × F => f x.1 x.2) s x :=
+  (f.analyticAt_bilinear x).analyticWithinAt
+
+protected theorem analyticOnNhd_bilinear (f : E →L[𝕜] F →L[𝕜] G) (s : Set (E × F)) :
+    AnalyticOnNhd 𝕜 (fun x : E × F => f x.1 x.2) s :=
+  fun x _ ↦ f.analyticAt_bilinear x
+
+protected theorem analyticOn_bilinear (f : E →L[𝕜] F →L[𝕜] G) (s : Set (E × F)) :
+    AnalyticOn 𝕜 (fun x : E × F => f x.1 x.2) s :=
+  (f.analyticOnNhd_bilinear s).analyticOn
+
 end ContinuousLinearMap
 
-variable (𝕜)
+variable {s : Set E} {z : E} {t : Set (E × F)} {p : E × F}
 
-lemma analyticAt_id (z : E) : AnalyticAt 𝕜 (id : E → E) z :=
+lemma analyticAt_id : AnalyticAt 𝕜 (id : E → E) z :=
   (ContinuousLinearMap.id 𝕜 E).analyticAt z
 
+lemma analyticWithinAt_id : AnalyticWithinAt 𝕜 (id : E → E) s z :=
+  analyticAt_id.analyticWithinAt
+
 /-- `id` is entire -/
-theorem analyticOn_id {s : Set E} : AnalyticOn 𝕜 (fun x : E ↦ x) s :=
-  fun _ _ ↦ analyticAt_id _ _
+theorem analyticOnNhd_id : AnalyticOnNhd 𝕜 (fun x : E ↦ x) s :=
+  fun _ _ ↦ analyticAt_id
+
+theorem analyticOn_id : AnalyticOn 𝕜 (fun x : E ↦ x) s :=
+  fun _ _ ↦ analyticWithinAt_id
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_id := analyticOn_id
 
 /-- `fst` is analytic -/
-theorem analyticAt_fst {p : E × F} : AnalyticAt 𝕜 (fun p : E × F ↦ p.fst) p :=
+theorem analyticAt_fst  : AnalyticAt 𝕜 (fun p : E × F ↦ p.fst) p :=
   (ContinuousLinearMap.fst 𝕜 E F).analyticAt p
 
+theorem analyticWithinAt_fst  : AnalyticWithinAt 𝕜 (fun p : E × F ↦ p.fst) t p :=
+  analyticAt_fst.analyticWithinAt
+
 /-- `snd` is analytic -/
-theorem analyticAt_snd {p : E × F} : AnalyticAt 𝕜 (fun p : E × F ↦ p.snd) p :=
+theorem analyticAt_snd : AnalyticAt 𝕜 (fun p : E × F ↦ p.snd) p :=
   (ContinuousLinearMap.snd 𝕜 E F).analyticAt p
 
+theorem analyticWithinAt_snd : AnalyticWithinAt 𝕜 (fun p : E × F ↦ p.snd) t p :=
+  analyticAt_snd.analyticWithinAt
+
 /-- `fst` is entire -/
-theorem analyticOn_fst {s : Set (E × F)} : AnalyticOn 𝕜 (fun p : E × F ↦ p.fst) s :=
-  fun _ _ ↦ analyticAt_fst _
+theorem analyticOnNhd_fst : AnalyticOnNhd 𝕜 (fun p : E × F ↦ p.fst) t :=
+  fun _ _ ↦ analyticAt_fst
+
+theorem analyticOn_fst : AnalyticOn 𝕜 (fun p : E × F ↦ p.fst) t :=
+  fun _ _ ↦ analyticWithinAt_fst
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_fst := analyticOn_fst
 
 /-- `snd` is entire -/
-theorem analyticOn_snd {s : Set (E × F)} : AnalyticOn 𝕜 (fun p : E × F ↦ p.snd) s :=
-  fun _ _ ↦ analyticAt_snd _
+theorem analyticOnNhd_snd : AnalyticOnNhd 𝕜 (fun p : E × F ↦ p.snd) t :=
+  fun _ _ ↦ analyticAt_snd
+
+theorem analyticOn_snd : AnalyticOn 𝕜 (fun p : E × F ↦ p.snd) t :=
+  fun _ _ ↦ analyticWithinAt_snd
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_snd := analyticOn_snd
+
+namespace ContinuousLinearEquiv
+
+variable (f : E ≃L[𝕜] F) (s : Set E) (x : E)
+
+protected theorem analyticAt : AnalyticAt 𝕜 f x :=
+  ((f : E →L[𝕜] F).hasFPowerSeriesAt x).analyticAt
+
+protected theorem analyticOnNhd : AnalyticOnNhd 𝕜 f s :=
+  fun x _ ↦ f.analyticAt x
+
+protected theorem analyticWithinAt (f : E →L[𝕜] F) (s : Set E) (x : E) : AnalyticWithinAt 𝕜 f s x :=
+  (f.analyticAt x).analyticWithinAt
+
+protected theorem analyticOn (f : E →L[𝕜] F) (s : Set E) : AnalyticOn 𝕜 f s :=
+  fun x _ ↦ f.analyticWithinAt _ x
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn := ContinuousLinearEquiv.analyticOn
+
+end ContinuousLinearEquiv
+
+namespace LinearIsometryEquiv
+
+variable (f : E ≃ₗᵢ[𝕜] F) (s : Set E) (x : E)
+
+protected theorem analyticAt : AnalyticAt 𝕜 f x :=
+  ((f : E →L[𝕜] F).hasFPowerSeriesAt x).analyticAt
+
+protected theorem analyticOnNhd : AnalyticOnNhd 𝕜 f s :=
+  fun x _ ↦ f.analyticAt x
+
+protected theorem analyticWithinAt (f : E →L[𝕜] F) (s : Set E) (x : E) : AnalyticWithinAt 𝕜 f s x :=
+  (f.analyticAt x).analyticWithinAt
+
+protected theorem analyticOn (f : E →L[𝕜] F) (s : Set E) : AnalyticOn 𝕜 f s :=
+  fun x _ ↦ f.analyticWithinAt _ x
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn := LinearIsometryEquiv.analyticOn
+
+end LinearIsometryEquiv
diff --git a/Mathlib/Analysis/Analytic/Meromorphic.lean b/Mathlib/Analysis/Analytic/Meromorphic.lean
index 12ec5f6379e2c..be73e28bbbfe1 100644
--- a/Mathlib/Analysis/Analytic/Meromorphic.lean
+++ b/Mathlib/Analysis/Analytic/Meromorphic.lean
@@ -35,7 +35,7 @@ lemma AnalyticAt.meromorphicAt {f : 𝕜 → E} {x : 𝕜} (hf : AnalyticAt 𝕜
 
 namespace MeromorphicAt
 
-lemma id (x : 𝕜) : MeromorphicAt id x := (analyticAt_id 𝕜 x).meromorphicAt
+lemma id (x : 𝕜) : MeromorphicAt id x := analyticAt_id.meromorphicAt
 
 lemma const (e : E) (x : 𝕜) : MeromorphicAt (fun _ ↦ e) x :=
   analyticAt_const.meromorphicAt
@@ -50,8 +50,8 @@ lemma add {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : Meromorph
     simp_rw [← mul_smul, ← pow_add, Nat.sub_add_cancel (Nat.le_max_left _ _),
       Nat.sub_add_cancel (Nat.le_max_right _ _), Pi.add_apply, smul_add]
   rw [this]
-  exact ((((analyticAt_id 𝕜 x).sub analyticAt_const).pow _).smul hf).add
-   ((((analyticAt_id 𝕜 x).sub analyticAt_const).pow _).smul hg)
+  exact (((analyticAt_id.sub analyticAt_const).pow _).smul hf).add
+   (((analyticAt_id.sub analyticAt_const).pow _).smul hg)
 
 lemma smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
     MeromorphicAt (f • g) x := by
@@ -59,8 +59,8 @@ lemma smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f
   rcases hg with ⟨n, hg⟩
   refine ⟨m + n, ?_⟩
   convert hf.smul hg using 2 with z
-  rw [smul_eq_mul, ← mul_smul, mul_assoc, mul_comm (f z), ← mul_assoc, pow_add,
-    ← smul_eq_mul (a' := f z), smul_assoc, Pi.smul_apply']
+  rw [Pi.smul_apply', smul_eq_mul]
+  module
 
 lemma mul {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
     MeromorphicAt (f * g) x :=
@@ -88,7 +88,7 @@ lemma congr {f g : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) (hfg : f =ᶠ
     MeromorphicAt g x := by
   rcases hf with ⟨m, hf⟩
   refine ⟨m + 1, ?_⟩
-  have : AnalyticAt 𝕜 (fun z ↦ z - x) x := (analyticAt_id 𝕜 x).sub analyticAt_const
+  have : AnalyticAt 𝕜 (fun z ↦ z - x) x := analyticAt_id.sub analyticAt_const
   refine (this.smul hf).congr ?_
   rw [eventuallyEq_nhdsWithin_iff] at hfg
   filter_upwards [hfg] with z hz
@@ -108,7 +108,7 @@ lemma inv {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) : MeromorphicA
   · -- interesting case: use local formula for `f`
     obtain ⟨n, g, hg_an, hg_ne, hg_eq⟩ := hf.exists_eventuallyEq_pow_smul_nonzero_iff.mpr h_eq
     have : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ (m + 1)) x :=
-      ((analyticAt_id 𝕜 x).sub analyticAt_const).pow _
+      (analyticAt_id.sub analyticAt_const).pow _
     -- use `m + 1` rather than `m` to damp out any silly issues with the value at `z = x`
     refine ⟨n + 1, (this.smul <| hg_an.inv hg_ne).congr ?_⟩
     filter_upwards [hg_eq, hg_an.continuousAt.eventually_ne hg_ne] with z hfg hg_ne'
@@ -142,6 +142,23 @@ lemma zpow {f : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (n : ℤ) : M
   | ofNat m => simpa only [Int.ofNat_eq_coe, zpow_natCast] using hf.pow m
   | negSucc m => simpa only [zpow_negSucc, inv_iff] using hf.pow (m + 1)
 
+theorem eventually_analyticAt [CompleteSpace E] {f : 𝕜 → E} {x : 𝕜}
+    (h : MeromorphicAt f x) : ∀ᶠ y in 𝓝[≠] x, AnalyticAt 𝕜 f y := by
+  rw [MeromorphicAt] at h
+  obtain ⟨n, h⟩ := h
+  apply AnalyticAt.eventually_analyticAt at h
+  refine (h.filter_mono ?_).mp ?_
+  · simp [nhdsWithin]
+  · rw [eventually_nhdsWithin_iff]
+    apply Filter.Eventually.of_forall
+    intro y hy hf
+    rw [Set.mem_compl_iff, Set.mem_singleton_iff] at hy
+    have := ((analyticAt_id (𝕜 := 𝕜).sub analyticAt_const).pow n).inv
+      (pow_ne_zero _ (sub_ne_zero_of_ne hy))
+    apply (this.smul hf).congr ∘ (eventually_ne_nhds hy).mono
+    intro z hz
+    simp [smul_smul, hz, sub_eq_zero]
+
 /-- The order of vanishing of a meromorphic function, as an element of `ℤ ∪ ∞` (to include the
 case of functions identically 0 near `x`). -/
 noncomputable def order {f : 𝕜 → E} {x : 𝕜} (hf : MeromorphicAt f x) : WithTop ℤ :=
@@ -210,8 +227,8 @@ lemma iff_eventuallyEq_zpow_smul_analyticAt {f : 𝕜 → E} {x : 𝕜} : Meromo
     ∃ (n : ℤ) (g : 𝕜 → E), AnalyticAt 𝕜 g x ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z := by
   refine ⟨fun ⟨n, hn⟩ ↦ ⟨-n, _, ⟨hn, eventually_nhdsWithin_iff.mpr ?_⟩⟩, ?_⟩
   · filter_upwards with z hz
-    rw [← mul_smul, ← zpow_natCast, ← zpow_add₀ (sub_ne_zero.mpr hz), neg_add_cancel,
-      zpow_zero, one_smul]
+    match_scalars
+    field_simp [sub_ne_zero.mpr hz]
   · refine fun ⟨n, g, hg_an, hg_eq⟩ ↦ MeromorphicAt.congr ?_ (EventuallyEq.symm hg_eq)
     exact (((MeromorphicAt.id x).sub (.const _ x)).zpow _).smul hg_an.meromorphicAt
 
@@ -220,10 +237,12 @@ end MeromorphicAt
 /-- Meromorphy of a function on a set. -/
 def MeromorphicOn (f : 𝕜 → E) (U : Set 𝕜) : Prop := ∀ x ∈ U, MeromorphicAt f x
 
-lemma AnalyticOn.meromorphicOn {f : 𝕜 → E} {U : Set 𝕜} (hf : AnalyticOn 𝕜 f U) :
+lemma AnalyticOnNhd.meromorphicOn {f : 𝕜 → E} {U : Set 𝕜} (hf : AnalyticOnNhd 𝕜 f U) :
     MeromorphicOn f U :=
   fun x hx ↦ (hf x hx).meromorphicAt
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.meromorphicOn := AnalyticOnNhd.meromorphicOn
 
 namespace MeromorphicOn
 
@@ -281,4 +300,14 @@ lemma congr (h_eq : Set.EqOn f g U) (hu : IsOpen U) : MeromorphicOn g U := by
   refine fun x hx ↦ (hf x hx).congr (EventuallyEq.filter_mono ?_ nhdsWithin_le_nhds)
   exact eventually_of_mem (hu.mem_nhds hx) h_eq
 
+theorem eventually_codiscreteWithin_analyticAt
+    [CompleteSpace E] (f : 𝕜 → E) (h : MeromorphicOn f U) :
+    ∀ᶠ (y : 𝕜) in codiscreteWithin U, AnalyticAt 𝕜 f y := by
+  rw [eventually_iff, mem_codiscreteWithin]
+  intro x hx
+  rw [disjoint_principal_right]
+  apply Filter.mem_of_superset ((h x hx).eventually_analyticAt)
+  intro x hx
+  simp [hx]
+
 end MeromorphicOn
diff --git a/Mathlib/Analysis/Analytic/Polynomial.lean b/Mathlib/Analysis/Analytic/Polynomial.lean
index e32bceb99f93f..77e62fd5b542b 100644
--- a/Mathlib/Analysis/Analytic/Polynomial.lean
+++ b/Mathlib/Analysis/Analytic/Polynomial.lean
@@ -23,19 +23,30 @@ open Polynomial
 
 variable [NormedRing B] [NormedAlgebra 𝕜 B] [Algebra A B] {f : E → B}
 
-theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
-    AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by
+theorem AnalyticWithinAt.aeval_polynomial (hf : AnalyticWithinAt 𝕜 f s z) (p : A[X]) :
+    AnalyticWithinAt 𝕜 (fun x ↦ aeval (f x) p) s z := by
   refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_
-  · simp_rw [aeval_C]; apply analyticAt_const
+  · simp_rw [aeval_C]; apply analyticWithinAt_const
   · simp_rw [aeval_add]; exact hp.add hq
   · convert hp.mul hf
     simp_rw [pow_succ, aeval_mul, ← mul_assoc, aeval_X]
 
+theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
+    AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by
+  rw [← analyticWithinAt_univ] at hf ⊢
+  exact hf.aeval_polynomial p
+
+theorem AnalyticOnNhd.aeval_polynomial (hf : AnalyticOnNhd 𝕜 f s) (p : A[X]) :
+    AnalyticOnNhd 𝕜 (fun x ↦ aeval (f x) p) s := fun x hx ↦ (hf x hx).aeval_polynomial p
+
 theorem AnalyticOn.aeval_polynomial (hf : AnalyticOn 𝕜 f s) (p : A[X]) :
     AnalyticOn 𝕜 (fun x ↦ aeval (f x) p) s := fun x hx ↦ (hf x hx).aeval_polynomial p
 
+theorem AnalyticOnNhd.eval_polynomial {A} [NormedCommRing A] [NormedAlgebra 𝕜 A] (p : A[X]) :
+    AnalyticOnNhd 𝕜 (eval · p) Set.univ := analyticOnNhd_id.aeval_polynomial p
+
 theorem AnalyticOn.eval_polynomial {A} [NormedCommRing A] [NormedAlgebra 𝕜 A] (p : A[X]) :
-    AnalyticOn 𝕜 (eval · p) Set.univ := (analyticOn_id 𝕜).aeval_polynomial p
+    AnalyticOn 𝕜 (eval · p) Set.univ := analyticOn_id.aeval_polynomial p
 
 end Polynomial
 
@@ -51,27 +62,47 @@ theorem AnalyticAt.aeval_mvPolynomial (hf : ∀ i, AnalyticAt 𝕜 (f · i) z) (
   · simp_rw [map_add]; exact hp.add hq
   · simp_rw [map_mul, aeval_X]; exact hp.mul (hf i)
 
-theorem AnalyticOn.aeval_mvPolynomial (hf : ∀ i, AnalyticOn 𝕜 (f · i) s) (p : MvPolynomial σ A) :
-    AnalyticOn 𝕜 (fun x ↦ aeval (f x) p) s := fun x hx ↦ .aeval_mvPolynomial (hf · x hx) p
+theorem AnalyticOnNhd.aeval_mvPolynomial
+    (hf : ∀ i, AnalyticOnNhd 𝕜 (f · i) s) (p : MvPolynomial σ A) :
+    AnalyticOnNhd 𝕜 (fun x ↦ aeval (f x) p) s := fun x hx ↦ .aeval_mvPolynomial (hf · x hx) p
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.aeval_mvPolynomial := AnalyticOnNhd.aeval_mvPolynomial
 
-theorem AnalyticOn.eval_continuousLinearMap (f : E →L[𝕜] σ → B) (p : MvPolynomial σ B) :
-    AnalyticOn 𝕜 (fun x ↦ eval (f x) p) Set.univ :=
+theorem AnalyticOnNhd.eval_continuousLinearMap (f : E →L[𝕜] σ → B) (p : MvPolynomial σ B) :
+    AnalyticOnNhd 𝕜 (fun x ↦ eval (f x) p) Set.univ :=
   fun x _ ↦ .aeval_mvPolynomial (fun i ↦ ((ContinuousLinearMap.proj i).comp f).analyticAt x) p
 
-theorem AnalyticOn.eval_continuousLinearMap' (f : σ → E →L[𝕜] B) (p : MvPolynomial σ B) :
-    AnalyticOn 𝕜 (fun x ↦ eval (f · x) p) Set.univ :=
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.eval_continuousLinearMap := AnalyticOnNhd.eval_continuousLinearMap
+
+theorem AnalyticOnNhd.eval_continuousLinearMap' (f : σ → E →L[𝕜] B) (p : MvPolynomial σ B) :
+    AnalyticOnNhd 𝕜 (fun x ↦ eval (f · x) p) Set.univ :=
   fun x _ ↦ .aeval_mvPolynomial (fun i ↦ (f i).analyticAt x) p
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.eval_continuousLinearMap' := AnalyticOnNhd.eval_continuousLinearMap'
+
 variable [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E]
 
-theorem AnalyticOn.eval_linearMap (f : E →ₗ[𝕜] σ → B) (p : MvPolynomial σ B) :
-    AnalyticOn 𝕜 (fun x ↦ eval (f x) p) Set.univ :=
-  AnalyticOn.eval_continuousLinearMap { f with cont := f.continuous_of_finiteDimensional } p
+theorem AnalyticOnNhd.eval_linearMap (f : E →ₗ[𝕜] σ → B) (p : MvPolynomial σ B) :
+    AnalyticOnNhd 𝕜 (fun x ↦ eval (f x) p) Set.univ :=
+  AnalyticOnNhd.eval_continuousLinearMap { f with cont := f.continuous_of_finiteDimensional } p
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.eval_linearMap := AnalyticOnNhd.eval_linearMap
+
+theorem AnalyticOnNhd.eval_linearMap' (f : σ → E →ₗ[𝕜] B) (p : MvPolynomial σ B) :
+    AnalyticOnNhd 𝕜 (fun x ↦ eval (f · x) p) Set.univ := AnalyticOnNhd.eval_linearMap (.pi f) p
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.eval_linearMap' := AnalyticOnNhd.eval_linearMap'
 
-theorem AnalyticOn.eval_linearMap' (f : σ → E →ₗ[𝕜] B) (p : MvPolynomial σ B) :
-    AnalyticOn 𝕜 (fun x ↦ eval (f · x) p) Set.univ := AnalyticOn.eval_linearMap (.pi f) p
+theorem AnalyticOnNhd.eval_mvPolynomial [Fintype σ] (p : MvPolynomial σ 𝕜) :
+    AnalyticOnNhd 𝕜 (eval · p) Set.univ :=
+  AnalyticOnNhd.eval_linearMap (.id (R := 𝕜) (M := σ → 𝕜)) p
 
-theorem AnalyticOn.eval_mvPolynomial [Fintype σ] (p : MvPolynomial σ 𝕜) :
-    AnalyticOn 𝕜 (eval · p) Set.univ := AnalyticOn.eval_linearMap (.id (R := 𝕜) (M := σ → 𝕜)) p
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.eval_mvPolynomial := AnalyticOnNhd.eval_mvPolynomial
 
 end MvPolynomial
diff --git a/Mathlib/Analysis/Analytic/RadiusLiminf.lean b/Mathlib/Analysis/Analytic/RadiusLiminf.lean
index 175d6acca7461..6aefcab0e900e 100644
--- a/Mathlib/Analysis/Analytic/RadiusLiminf.lean
+++ b/Mathlib/Analysis/Analytic/RadiusLiminf.lean
@@ -53,7 +53,7 @@ theorem radius_eq_liminf :
       refine
         H.mp ((eventually_gt_atTop 0).mono fun n hn₀ hn => (this _ hn₀).2 (NNReal.coe_le_coe.1 ?_))
       push_cast
-      exact (le_abs_self _).trans (hn.trans (pow_le_one _ ha.1.le ha.2.le))
+      exact (le_abs_self _).trans (hn.trans (pow_le_one₀ ha.1.le ha.2.le))
   · refine p.le_radius_of_isBigO (IsBigO.of_bound 1 ?_)
     refine (eventually_lt_of_lt_liminf hr).mp ((eventually_gt_atTop 0).mono fun n hn₀ hn => ?_)
     simpa using NNReal.coe_le_coe.2 ((this _ hn₀).1 hn.le)
diff --git a/Mathlib/Analysis/Analytic/Uniqueness.lean b/Mathlib/Analysis/Analytic/Uniqueness.lean
index b18db8f4e34b7..6c836c8afcab6 100644
--- a/Mathlib/Analysis/Analytic/Uniqueness.lean
+++ b/Mathlib/Analysis/Analytic/Uniqueness.lean
@@ -5,13 +5,15 @@ Authors: Sébastien Gouëzel
 -/
 import Mathlib.Analysis.Analytic.Linear
 import Mathlib.Analysis.Analytic.Composition
+import Mathlib.Analysis.Analytic.Constructions
 import Mathlib.Analysis.Normed.Module.Completion
+import Mathlib.Analysis.Analytic.ChangeOrigin
 
 /-!
 # Uniqueness principle for analytic functions
 
 We show that two analytic functions which coincide around a point coincide on whole connected sets,
-in `AnalyticOn.eqOn_of_preconnected_of_eventuallyEq`.
+in `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`.
 -/
 
 
@@ -20,15 +22,146 @@ variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCom
 
 open Set
 
-open scoped Topology ENNReal
+open scoped Topology ENNReal NNReal
 
-namespace AnalyticOn
+/-!
+### Uniqueness of power series
+If a function `f : E → F` has two representations as power series at a point `x : E`, corresponding
+to formal multilinear series `p₁` and `p₂`, then these representations agree term-by-term. That is,
+for any `n : ℕ` and `y : E`, `p₁ n (fun i ↦ y) = p₂ n (fun i ↦ y)`. In the one-dimensional case,
+when `f : 𝕜 → E`, the continuous multilinear maps `p₁ n` and `p₂ n` are given by
+`ContinuousMultilinearMap.mkPiRing`, and hence are determined completely by the value of
+`p₁ n (fun i ↦ 1)`, so `p₁ = p₂`. Consequently, the radius of convergence for one series can be
+transferred to the other.
+-/
+
+section Uniqueness
+
+open ContinuousMultilinearMap
+
+theorem Asymptotics.IsBigO.continuousMultilinearMap_apply_eq_zero {n : ℕ} {p : E[×n]→L[𝕜] F}
+    (h : (fun y => p fun _ => y) =O[𝓝 0] fun y => ‖y‖ ^ (n + 1)) (y : E) : (p fun _ => y) = 0 := by
+  obtain ⟨c, c_pos, hc⟩ := h.exists_pos
+  obtain ⟨t, ht, t_open, z_mem⟩ := eventually_nhds_iff.mp (isBigOWith_iff.mp hc)
+  obtain ⟨δ, δ_pos, δε⟩ := (Metric.isOpen_iff.mp t_open) 0 z_mem
+  clear h hc z_mem
+  cases' n with n
+  · exact norm_eq_zero.mp (by
+      -- Porting note: the symmetric difference of the `simpa only` sets:
+      -- added `zero_add, pow_one`
+      -- removed `zero_pow, Ne.def, Nat.one_ne_zero, not_false_iff`
+      simpa only [fin0_apply_norm, norm_eq_zero, norm_zero, zero_add, pow_one,
+        mul_zero, norm_le_zero_iff] using ht 0 (δε (Metric.mem_ball_self δ_pos)))
+  · refine Or.elim (Classical.em (y = 0))
+      (fun hy => by simpa only [hy] using p.map_zero) fun hy => ?_
+    replace hy := norm_pos_iff.mpr hy
+    refine norm_eq_zero.mp (le_antisymm (le_of_forall_pos_le_add fun ε ε_pos => ?_) (norm_nonneg _))
+    have h₀ := _root_.mul_pos c_pos (pow_pos hy (n.succ + 1))
+    obtain ⟨k, k_pos, k_norm⟩ := NormedField.exists_norm_lt 𝕜
+      (lt_min (mul_pos δ_pos (inv_pos.mpr hy)) (mul_pos ε_pos (inv_pos.mpr h₀)))
+    have h₁ : ‖k • y‖ < δ := by
+      rw [norm_smul]
+      exact inv_mul_cancel_right₀ hy.ne.symm δ ▸
+        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_left _ _)) hy
+    have h₂ :=
+      calc
+        ‖p fun _ => k • y‖ ≤ c * ‖k • y‖ ^ (n.succ + 1) := by
+          -- Porting note: now Lean wants `_root_.`
+          simpa only [norm_pow, _root_.norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
+          --simpa only [norm_pow, norm_norm] using ht (k • y) (δε (mem_ball_zero_iff.mpr h₁))
+        _ = ‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
+          -- Porting note: added `Nat.succ_eq_add_one` since otherwise `ring` does not conclude.
+          simp only [norm_smul, mul_pow, Nat.succ_eq_add_one]
+          -- Porting note: removed `rw [pow_succ]`, since it now becomes superfluous.
+          ring
+    have h₃ : ‖k‖ * (c * ‖y‖ ^ (n.succ + 1)) < ε :=
+      inv_mul_cancel_right₀ h₀.ne.symm ε ▸
+        mul_lt_mul_of_pos_right (lt_of_lt_of_le k_norm (min_le_right _ _)) h₀
+    calc
+      ‖p fun _ => y‖ = ‖k⁻¹ ^ n.succ‖ * ‖p fun _ => k • y‖ := by
+        simpa only [inv_smul_smul₀ (norm_pos_iff.mp k_pos), norm_smul, Finset.prod_const,
+          Finset.card_fin] using
+          congr_arg norm (p.map_smul_univ (fun _ : Fin n.succ => k⁻¹) fun _ : Fin n.succ => k • y)
+      _ ≤ ‖k⁻¹ ^ n.succ‖ * (‖k‖ ^ n.succ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1)))) := by gcongr
+      _ = ‖(k⁻¹ * k) ^ n.succ‖ * (‖k‖ * (c * ‖y‖ ^ (n.succ + 1))) := by
+        rw [← mul_assoc]
+        simp [norm_mul, mul_pow]
+      _ ≤ 0 + ε := by
+        rw [inv_mul_cancel₀ (norm_pos_iff.mp k_pos)]
+        simpa using h₃.le
+
+/-- If a formal multilinear series `p` represents the zero function at `x : E`, then the
+terms `p n (fun i ↦ y)` appearing in the sum are zero for any `n : ℕ`, `y : E`. -/
+theorem HasFPowerSeriesAt.apply_eq_zero {p : FormalMultilinearSeries 𝕜 E F} {x : E}
+    (h : HasFPowerSeriesAt 0 p x) (n : ℕ) : ∀ y : E, (p n fun _ => y) = 0 := by
+  refine Nat.strong_induction_on n fun k hk => ?_
+  have psum_eq : p.partialSum (k + 1) = fun y => p k fun _ => y := by
+    funext z
+    refine Finset.sum_eq_single _ (fun b hb hnb => ?_) fun hn => ?_
+    · have := Finset.mem_range_succ_iff.mp hb
+      simp only [hk b (this.lt_of_ne hnb), Pi.zero_apply]
+    · exact False.elim (hn (Finset.mem_range.mpr (lt_add_one k)))
+  replace h := h.isBigO_sub_partialSum_pow k.succ
+  simp only [psum_eq, zero_sub, Pi.zero_apply, Asymptotics.isBigO_neg_left] at h
+  exact h.continuousMultilinearMap_apply_eq_zero
+
+/-- A one-dimensional formal multilinear series representing the zero function is zero. -/
+theorem HasFPowerSeriesAt.eq_zero {p : FormalMultilinearSeries 𝕜 𝕜 E} {x : 𝕜}
+    (h : HasFPowerSeriesAt 0 p x) : p = 0 := by
+  ext n x
+  rw [← mkPiRing_apply_one_eq_self (p n)]
+  simp [h.apply_eq_zero n 1]
+
+/-- One-dimensional formal multilinear series representing the same function are equal. -/
+theorem HasFPowerSeriesAt.eq_formalMultilinearSeries {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E}
+    {f : 𝕜 → E} {x : 𝕜} (h₁ : HasFPowerSeriesAt f p₁ x) (h₂ : HasFPowerSeriesAt f p₂ x) : p₁ = p₂ :=
+  sub_eq_zero.mp (HasFPowerSeriesAt.eq_zero (x := x) (by simpa only [sub_self] using h₁.sub h₂))
+
+theorem HasFPowerSeriesAt.eq_formalMultilinearSeries_of_eventually
+    {p q : FormalMultilinearSeries 𝕜 𝕜 E} {f g : 𝕜 → E} {x : 𝕜} (hp : HasFPowerSeriesAt f p x)
+    (hq : HasFPowerSeriesAt g q x) (heq : ∀ᶠ z in 𝓝 x, f z = g z) : p = q :=
+  (hp.congr heq).eq_formalMultilinearSeries hq
+
+/-- A one-dimensional formal multilinear series representing a locally zero function is zero. -/
+theorem HasFPowerSeriesAt.eq_zero_of_eventually {p : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
+    {x : 𝕜} (hp : HasFPowerSeriesAt f p x) (hf : f =ᶠ[𝓝 x] 0) : p = 0 :=
+  (hp.congr hf).eq_zero
+
+/-- If a function `f : 𝕜 → E` has two power series representations at `x`, then the given radii in
+which convergence is guaranteed may be interchanged. This can be useful when the formal multilinear
+series in one representation has a particularly nice form, but the other has a larger radius. -/
+theorem HasFPowerSeriesOnBall.exchange_radius {p₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E} {f : 𝕜 → E}
+    {r₁ r₂ : ℝ≥0∞} {x : 𝕜} (h₁ : HasFPowerSeriesOnBall f p₁ x r₁)
+    (h₂ : HasFPowerSeriesOnBall f p₂ x r₂) : HasFPowerSeriesOnBall f p₁ x r₂ :=
+  h₂.hasFPowerSeriesAt.eq_formalMultilinearSeries h₁.hasFPowerSeriesAt ▸ h₂
+
+/-- If a function `f : 𝕜 → E` has power series representation `p` on a ball of some radius and for
+each positive radius it has some power series representation, then `p` converges to `f` on the whole
+`𝕜`. -/
+theorem HasFPowerSeriesOnBall.r_eq_top_of_exists {f : 𝕜 → E} {r : ℝ≥0∞} {x : 𝕜}
+    {p : FormalMultilinearSeries 𝕜 𝕜 E} (h : HasFPowerSeriesOnBall f p x r)
+    (h' : ∀ (r' : ℝ≥0) (_ : 0 < r'), ∃ p' : FormalMultilinearSeries 𝕜 𝕜 E,
+      HasFPowerSeriesOnBall f p' x r') :
+    HasFPowerSeriesOnBall f p x ∞ :=
+  { r_le := ENNReal.le_of_forall_pos_nnreal_lt fun r hr _ =>
+      let ⟨_, hp'⟩ := h' r hr
+      (h.exchange_radius hp').r_le
+    r_pos := ENNReal.coe_lt_top
+    hasSum := fun {y} _ =>
+      let ⟨r', hr'⟩ := exists_gt ‖y‖₊
+      let ⟨_, hp'⟩ := h' r' hr'.ne_bot.bot_lt
+      (h.exchange_radius hp').hasSum <| mem_emetric_ball_zero_iff.mpr (ENNReal.coe_lt_coe.2 hr') }
+
+end Uniqueness
+
+namespace AnalyticOnNhd
 
 /-- If an analytic function vanishes around a point, then it is uniformly zero along
 a connected set. Superseded by `eqOn_zero_of_preconnected_of_locally_zero` which does not assume
 completeness of the target space. -/
 theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E → F} {U : Set E}
-    (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
+    (hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U)
+    {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
     EqOn f 0 U := by
   /- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
     that its limit points in `U` still belong to it, from which the inclusion `U ⊆ u` will follow
@@ -72,11 +205,12 @@ neighborhood of a point `z₀`, then it is uniformly zero along a connected set.
 version assuming only that the function vanishes at some points arbitrarily close to `z₀`, see
 `eqOn_zero_of_preconnected_of_frequently_eq_zero`. -/
 theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero {f : E → F} {U : Set E}
-    (hf : AnalyticOn 𝕜 f U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
+    (hf : AnalyticOnNhd 𝕜 f U) (hU : IsPreconnected U)
+    {z₀ : E} (h₀ : z₀ ∈ U) (hfz₀ : f =ᶠ[𝓝 z₀] 0) :
     EqOn f 0 U := by
   let F' := UniformSpace.Completion F
   set e : F →L[𝕜] F' := UniformSpace.Completion.toComplL
-  have : AnalyticOn 𝕜 (e ∘ f) U := fun x hx => (e.analyticAt _).comp (hf x hx)
+  have : AnalyticOnNhd 𝕜 (e ∘ f) U := fun x hx => (e.analyticAt _).comp (hf x hx)
   have A : EqOn (e ∘ f) 0 U := by
     apply eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux this hU h₀
     filter_upwards [hfz₀] with x hx
@@ -89,8 +223,8 @@ theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero {f : E → F} {U : Set E}
 neighborhood of a point `z₀`, then they coincide globally along a connected set.
 For a one-dimensional version assuming only that the functions coincide at some points
 arbitrarily close to `z₀`, see `eqOn_of_preconnected_of_frequently_eq`. -/
-theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : AnalyticOn 𝕜 f U)
-    (hg : AnalyticOn 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
+theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : AnalyticOnNhd 𝕜 f U)
+    (hg : AnalyticOnNhd 𝕜 g U) (hU : IsPreconnected U) {z₀ : E} (h₀ : z₀ ∈ U) (hfg : f =ᶠ[𝓝 z₀] g) :
     EqOn f g U := by
   have hfg' : f - g =ᶠ[𝓝 z₀] 0 := hfg.mono fun z h => by simp [h]
   simpa [sub_eq_zero] using fun z hz =>
@@ -100,9 +234,9 @@ theorem eqOn_of_preconnected_of_eventuallyEq {f g : E → F} {U : Set E} (hf : A
 coincide in a neighborhood of a point `z₀`, then they coincide everywhere.
 For a one-dimensional version assuming only that the functions coincide at some points
 arbitrarily close to `z₀`, see `eq_of_frequently_eq`. -/
-theorem eq_of_eventuallyEq {f g : E → F} [PreconnectedSpace E] (hf : AnalyticOn 𝕜 f univ)
-    (hg : AnalyticOn 𝕜 g univ) {z₀ : E} (hfg : f =ᶠ[𝓝 z₀] g) : f = g :=
+theorem eq_of_eventuallyEq {f g : E → F} [PreconnectedSpace E] (hf : AnalyticOnNhd 𝕜 f univ)
+    (hg : AnalyticOnNhd 𝕜 g univ) {z₀ : E} (hfg : f =ᶠ[𝓝 z₀] g) : f = g :=
   funext fun x =>
     eqOn_of_preconnected_of_eventuallyEq hf hg isPreconnected_univ (mem_univ z₀) hfg (mem_univ x)
 
-end AnalyticOn
+end AnalyticOnNhd
diff --git a/Mathlib/Analysis/Analytic/Within.lean b/Mathlib/Analysis/Analytic/Within.lean
index a35cbce9538cf..7703eb9524ef0 100644
--- a/Mathlib/Analysis/Analytic/Within.lean
+++ b/Mathlib/Analysis/Analytic/Within.lean
@@ -4,24 +4,20 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Geoffrey Irving
 -/
 import Mathlib.Analysis.Analytic.Constructions
-import Mathlib.Analysis.Calculus.FDeriv.Analytic
 
 /-!
 # Properties of analyticity restricted to a set
 
-From `Mathlib.Analysis.Analytic.Basic`, we have the definitons
+From `Mathlib.Analysis.Analytic.Basic`, we have the definitions
 
-1. `AnalyticWithinAt 𝕜 f s x` means a power series at `x` converges to `f` on `𝓝[s] x`, and
-    `f` is continuous within `s` at `x`.
-2. `AnalyticWithinOn 𝕜 f s t` means `∀ x ∈ t, AnalyticWithinAt 𝕜 f s x`.
+1. `AnalyticWithinAt 𝕜 f s x` means a power series at `x` converges to `f` on `𝓝[insert x s] x`.
+2. `AnalyticOn 𝕜 f s t` means `∀ x ∈ t, AnalyticWithinAt 𝕜 f s x`.
 
 This means there exists an extension of `f` which is analytic and agrees with `f` on `s ∪ {x}`, but
-`f` is allowed to be arbitrary elsewhere.  Requiring `ContinuousWithinAt` is essential if `x ∉ s`:
-it is required for composition and smoothness to follow without extra hypotheses (we could
-alternately require convergence at `x` even if `x ∉ s`).
+`f` is allowed to be arbitrary elsewhere.
 
 Here we prove basic properties of these definitions. Where convenient we assume completeness of the
-ambient space, which allows us to related `AnalyticWithinAt` to analyticity of a local extension.
+ambient space, which allows us to relate `AnalyticWithinAt` to analyticity of a local extension.
 -/
 
 noncomputable section
@@ -40,43 +36,9 @@ variable {E F G H : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAd
 ### Basic properties
 -/
 
-@[simp] lemma hasFPowerSeriesWithinOnBall_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
-    {x : E} {r : ℝ≥0∞} :
-    HasFPowerSeriesWithinOnBall f p univ x r ↔ HasFPowerSeriesOnBall f p x r := by
-  constructor
-  · intro h
-    exact ⟨h.r_le, h.r_pos, fun {y} m ↦ h.hasSum (mem_univ _) m⟩
-  · intro h
-    refine ⟨h.r_le, h.r_pos, fun {y} _ m => h.hasSum m, ?_⟩
-    exact (h.continuousOn.continuousAt (EMetric.ball_mem_nhds x h.r_pos)).continuousWithinAt
-
-@[simp] lemma hasFPowerSeriesWithinAt_univ {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} :
-    HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x := by
-  simp only [HasFPowerSeriesWithinAt, hasFPowerSeriesWithinOnBall_univ, HasFPowerSeriesAt]
-
-@[simp] lemma analyticWithinAt_univ {f : E → F} {x : E} :
-    AnalyticWithinAt 𝕜 f univ x ↔ AnalyticAt 𝕜 f x := by
-  simp only [AnalyticWithinAt, hasFPowerSeriesWithinAt_univ, AnalyticAt]
-
-lemma analyticWithinOn_univ {f : E → F} :
-    AnalyticWithinOn 𝕜 f univ ↔ AnalyticOn 𝕜 f univ := by
-  simp only [AnalyticWithinOn, analyticWithinAt_univ, AnalyticOn]
-
-lemma HasFPowerSeriesWithinAt.continuousWithinAt {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
-    {s : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p s x) : ContinuousWithinAt f s x := by
-  rcases h with ⟨r, h⟩
-  exact h.continuousWithinAt
-
-lemma AnalyticWithinAt.continuousWithinAt {f : E → F} {s : Set E} {x : E}
-    (h : AnalyticWithinAt 𝕜 f s x) : ContinuousWithinAt f s x := by
-  rcases h with ⟨p, h⟩
-  exact h.continuousWithinAt
-
 /-- `AnalyticWithinAt` is trivial if `{x} ∈ 𝓝[s] x` -/
 lemma analyticWithinAt_of_singleton_mem {f : E → F} {s : Set E} {x : E} (h : {x} ∈ 𝓝[s] x) :
     AnalyticWithinAt 𝕜 f s x := by
-  have fc : ContinuousWithinAt f s x :=
-    Filter.Tendsto.mono_left (tendsto_pure_nhds _ _) (Filter.le_pure_iff.mpr h)
   rcases mem_nhdsWithin.mp h with ⟨t, ot, xt, st⟩
   rcases Metric.mem_nhds_iff.mp (ot.mem_nhds xt) with ⟨r, r0, rt⟩
   exact ⟨constFormalMultilinearSeries 𝕜 E (f x), .ofReal r, {
@@ -85,38 +47,19 @@ lemma analyticWithinAt_of_singleton_mem {f : E → F} {s : Set E} {x : E} (h : {
     hasSum := by
       intro y ys yr
       simp only [subset_singleton_iff, mem_inter_iff, and_imp] at st
-      specialize st (x + y) (rt (by simpa using yr)) ys
-      simp only [st]
+      simp only [mem_insert_iff, add_right_eq_self] at ys
+      have : x + y = x := by
+        rcases ys with rfl | ys
+        · simp
+        · exact st (x + y) (rt (by simpa using yr)) ys
+      simp only [this]
       apply (hasFPowerSeriesOnBall_const (e := 0)).hasSum
-      simp only [Metric.emetric_ball_top, mem_univ]
-    continuousWithinAt := fc
-  }⟩
-
-/-- Analyticity implies analyticity within any `s` -/
-lemma AnalyticAt.analyticWithinAt {f : E → F} {s : Set E} {x : E} (h : AnalyticAt 𝕜 f x) :
-    AnalyticWithinAt 𝕜 f s x := by
-  rcases h with ⟨p, r, hp⟩
-  exact ⟨p, r, {
-    r_le := hp.r_le
-    r_pos := hp.r_pos
-    hasSum := fun {y} _ yr ↦ hp.hasSum yr
-    continuousWithinAt :=
-      (hp.continuousOn.continuousAt (EMetric.ball_mem_nhds x hp.r_pos)).continuousWithinAt
-  }⟩
-
-/-- Analyticity on `s` implies analyticity within `s` -/
-lemma AnalyticOn.analyticWithinOn {f : E → F} {s : Set E} (h : AnalyticOn 𝕜 f s) :
-    AnalyticWithinOn 𝕜 f s :=
-  fun x m ↦ (h x m).analyticWithinAt
-
-lemma AnalyticWithinOn.continuousOn {f : E → F} {s : Set E} (h : AnalyticWithinOn 𝕜 f s) :
-    ContinuousOn f s :=
-  fun x m ↦ (h x m).continuousWithinAt
+      simp only [Metric.emetric_ball_top, mem_univ] }⟩
 
-/-- If `f` is `AnalyticWithinOn` near each point in a set, it is `AnalyticWithinOn` the set -/
-lemma analyticWithinOn_of_locally_analyticWithinOn {f : E → F} {s : Set E}
-    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ AnalyticWithinOn 𝕜 f (s ∩ u)) :
-    AnalyticWithinOn 𝕜 f s := by
+/-- If `f` is `AnalyticOn` near each point in a set, it is `AnalyticOn` the set -/
+lemma analyticOn_of_locally_analyticOn {f : E → F} {s : Set E}
+    (h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ AnalyticOn 𝕜 f (s ∩ u)) :
+    AnalyticOn 𝕜 f s := by
   intro x m
   rcases h x m with ⟨u, ou, xu, fu⟩
   rcases Metric.mem_nhds_iff.mp (ou.mem_nhds xu) with ⟨r, r0, ru⟩
@@ -125,20 +68,24 @@ lemma analyticWithinOn_of_locally_analyticWithinOn {f : E → F} {s : Set E}
     r_pos := lt_min (by positivity) fp.r_pos
     r_le := min_le_of_right_le fp.r_le
     hasSum := by
-      intro y ys yr
-      simp only [EMetric.mem_ball, lt_min_iff, edist_lt_ofReal, dist_zero_right] at yr
-      apply fp.hasSum ⟨ys, ru ?_⟩
-      · simp only [EMetric.mem_ball, yr]
-      · simp only [Metric.mem_ball, dist_self_add_left, yr]
-    continuousWithinAt := by
-      refine (fu.continuousOn x ⟨m, xu⟩).mono_left (le_of_eq ?_)
-      exact nhdsWithin_eq_nhdsWithin xu ou (by simp only [inter_assoc, inter_self])
-  }⟩
-
-/-- On open sets, `AnalyticOn` and `AnalyticWithinOn` coincide -/
-@[simp] lemma IsOpen.analyticWithinOn_iff_analyticOn {f : E → F} {s : Set E} (hs : IsOpen s) :
-    AnalyticWithinOn 𝕜 f s ↔ AnalyticOn 𝕜 f s := by
-  refine ⟨?_, AnalyticOn.analyticWithinOn⟩
+        intro y ys yr
+        simp only [EMetric.mem_ball, lt_min_iff, edist_lt_ofReal, dist_zero_right] at yr
+        apply fp.hasSum
+        · simp only [mem_insert_iff, add_right_eq_self] at ys
+          rcases ys with rfl | ys
+          · simp
+          · simp only [mem_insert_iff, add_right_eq_self, mem_inter_iff, ys, true_and]
+            apply Or.inr (ru ?_)
+            simp only [Metric.mem_ball, dist_self_add_left, yr]
+        · simp only [EMetric.mem_ball, yr] }⟩
+
+@[deprecated (since := "2024-09-26")]
+alias analyticWithinOn_of_locally_analyticWithinOn := analyticOn_of_locally_analyticOn
+
+/-- On open sets, `AnalyticOnNhd` and `AnalyticOn` coincide -/
+lemma IsOpen.analyticOn_iff_analyticOnNhd {f : E → F} {s : Set E} (hs : IsOpen s) :
+    AnalyticOn 𝕜 f s ↔ AnalyticOnNhd 𝕜 f s := by
+  refine ⟨?_, AnalyticOnNhd.analyticOn⟩
   intro hf x m
   rcases Metric.mem_nhds_iff.mp (hs.mem_nhds m) with ⟨r, r0, rs⟩
   rcases hf x m with ⟨p, t, fp⟩
@@ -148,16 +95,21 @@ lemma analyticWithinOn_of_locally_analyticWithinOn {f : E → F} {s : Set E}
     hasSum := by
       intro y ym
       simp only [EMetric.mem_ball, lt_min_iff, edist_lt_ofReal, dist_zero_right] at ym
-      refine fp.hasSum (rs ?_) ym.2
-      simp only [Metric.mem_ball, dist_self_add_left, ym.1]
-  }⟩
+      refine fp.hasSum ?_ ym.2
+      apply mem_insert_of_mem
+      apply rs
+      simp only [Metric.mem_ball, dist_self_add_left, ym.1] }⟩
+
+@[deprecated (since := "2024-09-26")]
+alias IsOpen.analyticWithinOn_iff_analyticOn := IsOpen.analyticOn_iff_analyticOnNhd
+
 
 /-!
 ### Equivalence to analyticity of a local extension
 
 We show that `HasFPowerSeriesWithinOnBall`, `HasFPowerSeriesWithinAt`, and `AnalyticWithinAt` are
 equivalent to the existence of a local extension with full analyticity.  We do not yet show a
-result for `AnalyticWithinOn`, as this requires a bit more work to show that local extensions can
+result for `AnalyticOn`, as this requires a bit more work to show that local extensions can
 be stitched together.
 -/
 
@@ -165,11 +117,11 @@ be stitched together.
 lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpace F] {f : E → F}
     {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} {r : ℝ≥0∞} :
     HasFPowerSeriesWithinOnBall f p s x r ↔
-      ContinuousWithinAt f s x ∧ ∃ g, EqOn f g (s ∩ EMetric.ball x r) ∧
+      ∃ g, EqOn f g (insert x s ∩ EMetric.ball x r) ∧
         HasFPowerSeriesOnBall g p x r := by
   constructor
   · intro h
-    refine ⟨h.continuousWithinAt, fun y ↦ p.sum (y - x), ?_, ?_⟩
+    refine ⟨fun y ↦ p.sum (y - x), ?_, ?_⟩
     · intro y ⟨ys,yb⟩
       simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub] at yb
       have e0 := p.hasSum (x := y - x) ?_
@@ -186,8 +138,8 @@ lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpac
       apply p.hasSum
       simp only [EMetric.mem_ball] at lt ⊢
       exact lt_of_lt_of_le lt h.r_le
-  · intro ⟨mem, g, hfg, hg⟩
-    refine ⟨hg.r_le, hg.r_pos, ?_, mem⟩
+  · intro ⟨g, hfg, hg⟩
+    refine ⟨hg.r_le, hg.r_pos, ?_⟩
     intro y ys lt
     rw [hfg]
     · exact hg.hasSum lt
@@ -198,18 +150,18 @@ lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpac
 lemma hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt [CompleteSpace F] {f : E → F}
     {p : FormalMultilinearSeries 𝕜 E F} {s : Set E} {x : E} :
     HasFPowerSeriesWithinAt f p s x ↔
-      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ HasFPowerSeriesAt g p x := by
+      ∃ g, f =ᶠ[𝓝[insert x s] x] g ∧ HasFPowerSeriesAt g p x := by
   constructor
   · intro ⟨r, h⟩
-    rcases hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall.mp h with ⟨fc, g, e, h⟩
-    refine ⟨fc, g, ?_, ⟨r, h⟩⟩
+    rcases hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall.mp h with ⟨g, e, h⟩
+    refine ⟨g, ?_, ⟨r, h⟩⟩
     refine Filter.eventuallyEq_iff_exists_mem.mpr ⟨_, ?_, e⟩
     exact inter_mem_nhdsWithin _ (EMetric.ball_mem_nhds _ h.r_pos)
-  · intro ⟨mem, g, hfg, ⟨r, hg⟩⟩
+  · intro ⟨g, hfg, ⟨r, hg⟩⟩
     simp only [eventuallyEq_nhdsWithin_iff, Metric.eventually_nhds_iff] at hfg
     rcases hfg with ⟨e, e0, hfg⟩
     refine ⟨min r (.ofReal e), ?_⟩
-    refine hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall.mpr ⟨mem, g, ?_, ?_⟩
+    refine hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall.mpr ⟨g, ?_, ?_⟩
     · intro y ⟨ys, xy⟩
       refine hfg ?_ ys
       simp only [EMetric.mem_ball, lt_min_iff, edist_lt_ofReal] at xy
@@ -219,147 +171,32 @@ lemma hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt [CompleteSpace F] {f
 /-- `f` is analytic within `s` at `x` iff some local extension of `f` is analytic at `x` -/
 lemma analyticWithinAt_iff_exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E} :
     AnalyticWithinAt 𝕜 f s x ↔
-      ContinuousWithinAt f s x ∧ ∃ g, f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x := by
+      ∃ g, f =ᶠ[𝓝[insert x s] x] g ∧ AnalyticAt 𝕜 g x := by
   simp only [AnalyticWithinAt, AnalyticAt, hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt]
   tauto
 
-/-- If `f` is analytic within `s` at `x`, some local extension of `f` is analytic at `x` -/
-lemma AnalyticWithinAt.exists_analyticAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
-    (h : AnalyticWithinAt 𝕜 f s x) : ∃ g, f x = g x ∧ f =ᶠ[𝓝[s] x] g ∧ AnalyticAt 𝕜 g x := by
-  by_cases s0 : 𝓝[s] x = ⊥
-  · refine ⟨fun _ ↦ f x, rfl, ?_, analyticAt_const⟩
-    simp only [EventuallyEq, s0, eventually_bot]
-  · rcases analyticWithinAt_iff_exists_analyticAt.mp h with ⟨_, g, fg, hg⟩
-    refine ⟨g, ?_, fg, hg⟩
-    exact tendsto_nhds_unique' ⟨s0⟩ h.continuousWithinAt
-      (hg.continuousAt.continuousWithinAt.congr' fg.symm)
-
-/-!
-### Congruence
-
-We require completeness to use equivalence to locally extensions, but this is nonessential.
--/
-
-lemma AnalyticWithinAt.congr_of_eventuallyEq [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
-    (hf : AnalyticWithinAt 𝕜 f s x) (hs : f =ᶠ[𝓝[s] x] g) (hx : f x = g x) :
-    AnalyticWithinAt 𝕜 g s x := by
-  rcases hf.exists_analyticAt with ⟨f', fx, ef, hf'⟩
-  rw [analyticWithinAt_iff_exists_analyticAt]
-  have eg := hs.symm.trans ef
-  refine ⟨?_, f', eg, hf'⟩
-  exact hf'.continuousAt.continuousWithinAt.congr_of_eventuallyEq eg (hx.symm.trans fx)
-
-lemma AnalyticWithinAt.congr [CompleteSpace F] {f g : E → F} {s : Set E} {x : E}
-    (hf : AnalyticWithinAt 𝕜 f s x) (hs : EqOn f g s) (hx : f x = g x) :
-    AnalyticWithinAt 𝕜 g s x :=
-  hf.congr_of_eventuallyEq hs.eventuallyEq_nhdsWithin hx
-
-lemma AnalyticWithinOn.congr [CompleteSpace F] {f g : E → F} {s : Set E}
-    (hf : AnalyticWithinOn 𝕜 f s) (hs : EqOn f g s) :
-    AnalyticWithinOn 𝕜 g s :=
-  fun x m ↦ (hf x m).congr hs (hs m)
-
-/-!
-### Monotonicity w.r.t. the set we're analytic within
--/
-
-lemma HasFPowerSeriesWithinOnBall.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
-    {s t : Set E} {x : E} {r : ℝ≥0∞} (h : HasFPowerSeriesWithinOnBall f p t x r)
-    (hs : s ⊆ t) : HasFPowerSeriesWithinOnBall f p s x r where
-  r_le := h.r_le
-  r_pos := h.r_pos
-  hasSum {_} ys yb := h.hasSum (hs ys) yb
-  continuousWithinAt := h.continuousWithinAt.mono hs
-
-lemma HasFPowerSeriesWithinAt.mono {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
-    {s t : Set E} {x : E} (h : HasFPowerSeriesWithinAt f p t x)
-    (hs : s ⊆ t) : HasFPowerSeriesWithinAt f p s x := by
-  rcases h with ⟨r, hr⟩
-  exact ⟨r, hr.mono hs⟩
-
-lemma AnalyticWithinAt.mono {f : E → F} {s t : Set E} {x : E} (h : AnalyticWithinAt 𝕜 f t x)
-    (hs : s ⊆ t) : AnalyticWithinAt 𝕜 f s x := by
-  rcases h with ⟨p, hp⟩
-  exact ⟨p, hp.mono hs⟩
-
-lemma AnalyticWithinOn.mono {f : E → F} {s t : Set E} (h : AnalyticWithinOn 𝕜 f t)
-    (hs : s ⊆ t) : AnalyticWithinOn 𝕜 f s :=
-  fun _ m ↦ (h _ (hs m)).mono hs
-
-/-!
-### Analyticity within respects composition
-
-Currently we require `CompleteSpace`s to use equivalence to local extensions, but this is not
-essential.
--/
-
-lemma AnalyticWithinAt.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
-    {t : Set E} {x : E} (hf : AnalyticWithinAt 𝕜 f s (g x)) (hg : AnalyticWithinAt 𝕜 g t x)
-    (h : MapsTo g t s) : AnalyticWithinAt 𝕜 (f ∘ g) t x := by
-  rcases hf.exists_analyticAt with ⟨f', _, ef, hf'⟩
-  rcases hg.exists_analyticAt with ⟨g', gx, eg, hg'⟩
-  refine analyticWithinAt_iff_exists_analyticAt.mpr ⟨?_, f' ∘ g', ?_, ?_⟩
-  · exact hf.continuousWithinAt.comp hg.continuousWithinAt h
-  · have gt := hg.continuousWithinAt.tendsto_nhdsWithin h
-    filter_upwards [eg, gt.eventually ef]
-    intro y gy fgy
-    simp only [Function.comp_apply, fgy, ← gy]
-  · exact hf'.comp_of_eq hg' gx.symm
-
-lemma AnalyticWithinOn.comp [CompleteSpace F] [CompleteSpace G] {f : F → G} {g : E → F} {s : Set F}
-    {t : Set E} (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g t) (h : MapsTo g t s) :
-    AnalyticWithinOn 𝕜 (f ∘ g) t :=
-  fun x m ↦ (hf _ (h m)).comp (hg x m) h
-
-/-!
-### Analyticity within implies smoothness
--/
-
-lemma AnalyticWithinAt.contDiffWithinAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
-    (h : AnalyticWithinAt 𝕜 f s x) {n : ℕ∞} : ContDiffWithinAt 𝕜 n f s x := by
-  rcases h.exists_analyticAt with ⟨g, fx, fg, hg⟩
-  exact hg.contDiffAt.contDiffWithinAt.congr_of_eventuallyEq fg fx
-
-lemma AnalyticWithinOn.contDiffOn [CompleteSpace F] {f : E → F} {s : Set E}
-    (h : AnalyticWithinOn 𝕜 f s) {n : ℕ∞} : ContDiffOn 𝕜 n f s :=
-  fun x m ↦ (h x m).contDiffWithinAt
-
-/-!
-### Analyticity within respects products
--/
-
-lemma HasFPowerSeriesWithinOnBall.prod {e : E} {f : E → F} {g : E → G} {s : Set E} {r t : ℝ≥0∞}
-    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
-    (hf : HasFPowerSeriesWithinOnBall f p s e r) (hg : HasFPowerSeriesWithinOnBall g q s e t) :
-    HasFPowerSeriesWithinOnBall (fun x ↦ (f x, g x)) (p.prod q) s e (min r t) where
-  r_le := by
-    rw [p.radius_prod_eq_min]
-    exact min_le_min hf.r_le hg.r_le
-  r_pos := lt_min hf.r_pos hg.r_pos
-  hasSum := by
-    intro y m hy
-    simp_rw [FormalMultilinearSeries.prod, ContinuousMultilinearMap.prod_apply]
-    refine (hf.hasSum m ?_).prod_mk (hg.hasSum m ?_)
-    · exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_left _ _))
-    · exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_right _ _))
-  continuousWithinAt := hf.continuousWithinAt.prod hg.continuousWithinAt
-
-lemma HasFPowerSeriesWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
-    {p : FormalMultilinearSeries 𝕜 E F} {q : FormalMultilinearSeries 𝕜 E G}
-    (hf : HasFPowerSeriesWithinAt f p s e) (hg : HasFPowerSeriesWithinAt g q s e) :
-    HasFPowerSeriesWithinAt (fun x ↦ (f x, g x)) (p.prod q) s e := by
-  rcases hf with ⟨_, hf⟩
-  rcases hg with ⟨_, hg⟩
-  exact ⟨_, hf.prod hg⟩
-
-lemma AnalyticWithinAt.prod {e : E} {f : E → F} {g : E → G} {s : Set E}
-    (hf : AnalyticWithinAt 𝕜 f s e) (hg : AnalyticWithinAt 𝕜 g s e) :
-    AnalyticWithinAt 𝕜 (fun x ↦ (f x, g x)) s e := by
-  rcases hf with ⟨_, hf⟩
-  rcases hg with ⟨_, hg⟩
-  exact ⟨_, hf.prod hg⟩
-
-lemma AnalyticWithinOn.prod {f : E → F} {g : E → G} {s : Set E}
-    (hf : AnalyticWithinOn 𝕜 f s) (hg : AnalyticWithinOn 𝕜 g s) :
-    AnalyticWithinOn 𝕜 (fun x ↦ (f x, g x)) s :=
-  fun x hx ↦ (hf x hx).prod (hg x hx)
+/-- `f` is analytic within `s` at `x` iff some local extension of `f` is analytic at `x`. In this
+version, we make sure that the extension coincides with `f` on all of `insert x s`. -/
+lemma analyticWithinAt_iff_exists_analyticAt' [CompleteSpace F] {f : E → F} {s : Set E} {x : E} :
+    AnalyticWithinAt 𝕜 f s x ↔
+      ∃ g, f x = g x ∧ EqOn f g (insert x s) ∧ AnalyticAt 𝕜 g x := by
+  classical
+  simp only [analyticWithinAt_iff_exists_analyticAt]
+  refine ⟨?_, ?_⟩
+  · rintro ⟨g, hf, hg⟩
+    rcases mem_nhdsWithin.1 hf with ⟨u, u_open, xu, hu⟩
+    let g' := Set.piecewise u g f
+    refine ⟨g', ?_, ?_, ?_⟩
+    · have : x ∈ u ∩ insert x s := ⟨xu, by simp⟩
+      simpa [g', xu, this] using hu this
+    · intro y hy
+      by_cases h'y : y ∈ u
+      · have : y ∈ u ∩ insert x s := ⟨h'y, hy⟩
+        simpa [g', h'y, this] using hu this
+      · simp [g', h'y]
+    · apply hg.congr
+      filter_upwards [u_open.mem_nhds xu] with y hy using by simp [g', hy]
+  · rintro ⟨g, -, hf, hg⟩
+    exact ⟨g, by filter_upwards [self_mem_nhdsWithin] using hf, hg⟩
+
+alias ⟨AnalyticWithinAt.exists_analyticAt, _⟩ := analyticWithinAt_iff_exists_analyticAt'
diff --git a/Mathlib/Analysis/Asymptotics/Asymptotics.lean b/Mathlib/Analysis/Asymptotics/Asymptotics.lean
index f9a9eb6bce641..15ae960d1d41e 100644
--- a/Mathlib/Analysis/Asymptotics/Asymptotics.lean
+++ b/Mathlib/Analysis/Asymptotics/Asymptotics.lean
@@ -3,6 +3,7 @@ Copyright (c) 2019 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Yury Kudryashov
 -/
+import Mathlib.Analysis.Normed.Group.Bounded
 import Mathlib.Analysis.Normed.Group.InfiniteSum
 import Mathlib.Analysis.Normed.MulAction
 import Mathlib.Topology.Algebra.Order.LiminfLimsup
@@ -127,13 +128,13 @@ theorem isBigO_iff'' {g : α → E'''} :
     obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
     refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
     filter_upwards [hc] with x hx
-    rwa [inv_mul_le_iff (by positivity)]
+    rwa [inv_mul_le_iff₀ (by positivity)]
   case mpr =>
     rw [isBigO_iff']
     obtain ⟨c, ⟨hc_pos, hc⟩⟩ := h
     refine ⟨c⁻¹, ⟨by positivity, ?_⟩⟩
     filter_upwards [hc] with x hx
-    rwa [← inv_inv c, inv_mul_le_iff (by positivity)] at hx
+    rwa [← inv_inv c, inv_mul_le_iff₀ (by positivity)] at hx
 
 theorem IsBigO.of_bound (c : ℝ) (h : ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g :=
   isBigO_iff.2 ⟨c, h⟩
@@ -250,7 +251,7 @@ theorem isLittleO_iff_nat_mul_le_aux (h₀ : (∀ x, 0 ≤ ‖f x‖) ∨ ∀ x,
     rcases exists_nat_gt ε⁻¹ with ⟨n, hn⟩
     have hn₀ : (0 : ℝ) < n := (inv_pos.2 ε0).trans hn
     refine ((isBigOWith_inv hn₀).2 (H n)).bound.mono fun x hfg => ?_
-    refine hfg.trans (mul_le_mul_of_nonneg_right (inv_le_of_inv_le ε0 hn.le) ?_)
+    refine hfg.trans (mul_le_mul_of_nonneg_right (inv_le_of_inv_le₀ ε0 hn.le) ?_)
     refine h₀.elim (fun hf => nonneg_of_mul_nonneg_right ((hf x).trans hfg) ?_) fun h => h x
     exact inv_pos.2 hn₀
 
@@ -592,7 +593,7 @@ theorem isLittleO_sup : f =o[l ⊔ l'] g ↔ f =o[l] g ∧ f =o[l'] g :=
 theorem isBigOWith_insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E} {g' : α → F}
     (h : ‖g x‖ ≤ C * ‖g' x‖) : IsBigOWith C (𝓝[insert x s] x) g g' ↔
     IsBigOWith C (𝓝[s] x) g g' := by
-  simp_rw [IsBigOWith_def, nhdsWithin_insert, eventually_sup, eventually_pure, h, true_and_iff]
+  simp_rw [IsBigOWith_def, nhdsWithin_insert, eventually_sup, eventually_pure, h, true_and]
 
 protected theorem IsBigOWith.insert [TopologicalSpace α] {x : α} {s : Set α} {C : ℝ} {g : α → E}
     {g' : α → F} (h1 : IsBigOWith C (𝓝[s] x) g g') (h2 : ‖g x‖ ≤ C * ‖g' x‖) :
@@ -820,6 +821,46 @@ theorem IsLittleO.prod_rightr (h : f =o[l] g') : f =o[l] fun x => (f' x, g' x) :
 
 end
 
+section
+
+variable {f : α × β → E} {g : α × β → F} {l' : Filter β}
+
+protected theorem IsBigO.fiberwise_right :
+    f =O[l ×ˢ l'] g → ∀ᶠ a in l, (f ⟨a, ·⟩) =O[l'] (g ⟨a, ·⟩) := by
+  simp only [isBigO_iff, eventually_iff, mem_prod_iff]
+  rintro ⟨c, t₁, ht₁, t₂, ht₂, ht⟩
+  exact mem_of_superset ht₁ fun _ ha ↦ ⟨c, mem_of_superset ht₂ fun _ hb ↦ ht ⟨ha, hb⟩⟩
+
+protected theorem IsBigO.fiberwise_left :
+    f =O[l ×ˢ l'] g → ∀ᶠ b in l', (f ⟨·, b⟩) =O[l] (g ⟨·, b⟩) := by
+  simp only [isBigO_iff, eventually_iff, mem_prod_iff]
+  rintro ⟨c, t₁, ht₁, t₂, ht₂, ht⟩
+  exact mem_of_superset ht₂ fun _ hb ↦ ⟨c, mem_of_superset ht₁ fun _ ha ↦ ht ⟨ha, hb⟩⟩
+
+end
+
+section
+
+variable (l' : Filter β)
+
+protected theorem IsBigO.comp_fst : f =O[l] g → (f ∘ Prod.fst) =O[l ×ˢ l'] (g ∘ Prod.fst) := by
+  simp only [isBigO_iff, eventually_prod_iff]
+  exact fun ⟨c, hc⟩ ↦ ⟨c, _, hc, fun _ ↦ True, eventually_true l', fun {_} h {_} _ ↦ h⟩
+
+protected theorem IsBigO.comp_snd : f =O[l] g → (f ∘ Prod.snd) =O[l' ×ˢ l] (g ∘ Prod.snd) := by
+  simp only [isBigO_iff, eventually_prod_iff]
+  exact fun ⟨c, hc⟩ ↦ ⟨c, fun _ ↦ True, eventually_true l', _, hc, fun _ ↦ id⟩
+
+protected theorem IsLittleO.comp_fst : f =o[l] g → (f ∘ Prod.fst) =o[l ×ˢ l'] (g ∘ Prod.fst) := by
+  simp only [isLittleO_iff, eventually_prod_iff]
+  exact fun h _ hc ↦ ⟨_, h hc, fun _ ↦ True, eventually_true l', fun {_} h {_} _ ↦ h⟩
+
+protected theorem IsLittleO.comp_snd : f =o[l] g → (f ∘ Prod.snd) =o[l' ×ˢ l] (g ∘ Prod.snd) := by
+  simp only [isLittleO_iff, eventually_prod_iff]
+  exact fun h _ hc ↦ ⟨fun _ ↦ True, eventually_true l', _, h hc, fun _ ↦ id⟩
+
+end
+
 theorem IsBigOWith.prod_left_same (hf : IsBigOWith c l f' k') (hg : IsBigOWith c l g' k') :
     IsBigOWith c l (fun x => (f' x, g' x)) k' := by
   rw [isBigOWith_iff] at *; filter_upwards [hf, hg] with x using max_le
@@ -1018,7 +1059,7 @@ variable {g g' l}
 
 @[simp]
 theorem isBigOWith_zero_right_iff : (IsBigOWith c l f'' fun _x => (0 : F')) ↔ f'' =ᶠ[l] 0 := by
-  simp only [IsBigOWith_def, exists_prop, true_and_iff, norm_zero, mul_zero,
+  simp only [IsBigOWith_def, exists_prop, norm_zero, mul_zero,
     norm_le_zero_iff, EventuallyEq, Pi.zero_apply]
 
 @[simp]
@@ -1210,6 +1251,9 @@ theorem IsLittleO.trans_tendsto (hfg : f'' =o[l] g'') (hg : Tendsto g'' l (𝓝
     Tendsto f'' l (𝓝 0) :=
   hfg.isBigO.trans_tendsto hg
 
+lemma isLittleO_id_one [One F''] [NeZero (1 : F'')] : (fun x : E'' => x) =o[𝓝 0] (1 : E'' → F'') :=
+  isLittleO_id_const one_ne_zero
+
 /-! ### Multiplication by a constant -/
 
 
@@ -1406,7 +1450,7 @@ theorem IsBigOWith.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : IsBigOWith
   rcases eq_or_ne (f x) 0 with hx | hx
   · simp only [hx, h₀ hx, inv_zero, norm_zero, mul_zero, le_rfl]
   · have hc : 0 < c := pos_of_mul_pos_left ((norm_pos_iff.2 hx).trans_le hle) (norm_nonneg _)
-    replace hle := inv_le_inv_of_le (norm_pos_iff.2 hx) hle
+    replace hle := inv_anti₀ (norm_pos_iff.2 hx) hle
     simpa only [norm_inv, mul_inv, ← div_eq_inv_mul, div_le_iff₀ hc] using hle
 
 theorem IsBigO.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : f =O[l] g)
@@ -1581,8 +1625,8 @@ theorem isLittleO_const_left_of_ne {c : E''} (hc : c ≠ 0) :
 theorem isLittleO_const_left {c : E''} :
     (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop := by
   rcases eq_or_ne c 0 with (rfl | hc)
-  · simp only [isLittleO_zero, eq_self_iff_true, true_or_iff]
-  · simp only [hc, false_or_iff, isLittleO_const_left_of_ne hc]; rfl
+  · simp only [isLittleO_zero, eq_self_iff_true, true_or]
+  · simp only [hc, false_or, isLittleO_const_left_of_ne hc]; rfl
 
 @[simp 1001] -- Porting note: increase priority so that this triggers before `isLittleO_const_left`
 theorem isLittleO_const_const_iff [NeBot l] {d : E''} {c : F''} :
@@ -1659,7 +1703,7 @@ theorem isBigOWith_iff_exists_eq_mul (hc : 0 ≤ c) :
   · intro h
     use fun x => u x / v x
     refine ⟨Eventually.mono h.bound fun y hy => ?_, h.eventually_mul_div_cancel.symm⟩
-    simpa using div_le_of_nonneg_of_le_mul (norm_nonneg _) hc hy
+    simpa using div_le_of_le_mul₀ (norm_nonneg _) hc hy
   · rintro ⟨φ, hφ, h⟩
     exact isBigOWith_of_eq_mul φ hφ h
 
@@ -1700,7 +1744,7 @@ theorem div_isBoundedUnder_of_isBigO {α : Type*} {l : Filter α} {f g : α →
   obtain ⟨c, h₀, hc⟩ := h.exists_nonneg
   refine ⟨c, eventually_map.2 (hc.bound.mono fun x hx => ?_)⟩
   rw [norm_div]
-  exact div_le_of_nonneg_of_le_mul (norm_nonneg _) h₀ hx
+  exact div_le_of_le_mul₀ (norm_nonneg _) h₀ hx
 
 theorem isBigO_iff_div_isBoundedUnder {α : Type*} {l : Filter α} {f g : α → 𝕜}
     (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) :
@@ -1874,6 +1918,13 @@ theorem isBigO_atTop_iff_eventually_exists_pos {α : Type*}
     f =O[atTop] g ↔ ∀ᶠ n₀ in atTop, ∃ c > 0, ∀ n ≥ n₀, c * ‖f n‖ ≤ ‖g n‖ := by
   simp_rw [isBigO_iff'', ← exists_prop, Subtype.exists', exists_eventually_atTop]
 
+lemma isBigO_mul_iff_isBigO_div {f g h : α → 𝕜} (hf : ∀ᶠ x in l, f x ≠ 0) :
+    (fun x ↦ f x * g x) =O[l] h ↔ g =O[l] (fun x ↦ h x / f x) := by
+  rw [isBigO_iff', isBigO_iff']
+  refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;>
+  · refine H.congr <| Eventually.mp hf <| Eventually.of_forall fun x hx ↦ ?_
+    rw [norm_mul, norm_div, ← mul_div_assoc, le_div_iff₀' (norm_pos_iff.mpr hx)]
+
 end Asymptotics
 
 open Asymptotics
@@ -1949,3 +2000,52 @@ theorem isLittleO_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F
   exact forall₂_congr fun c _hc => e.isBigOWith_congr
 
 end Homeomorph
+
+namespace ContinuousOn
+
+variable {α E F : Type*} [TopologicalSpace α] {s : Set α} {f : α → E} {c : F}
+
+section IsBigO
+
+variable [SeminormedAddGroup E] [Norm F]
+
+protected theorem isBigOWith_principal
+    (hf : ContinuousOn f s) (hs : IsCompact s) (hc : ‖c‖ ≠ 0) :
+    IsBigOWith (sSup (Norm.norm '' (f '' s)) / ‖c‖) (𝓟 s) f fun _ => c := by
+  rw [isBigOWith_principal, div_mul_cancel₀ _ hc]
+  exact fun x hx ↦ hs.image_of_continuousOn hf |>.image continuous_norm
+   |>.isLUB_sSup (Set.image_nonempty.mpr <| Set.image_nonempty.mpr ⟨x, hx⟩)
+   |>.left <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx
+
+protected theorem isBigO_principal (hf : ContinuousOn f s) (hs : IsCompact s)
+    (hc : ‖c‖ ≠ 0) : f =O[𝓟 s] fun _ => c :=
+  (hf.isBigOWith_principal hs hc).isBigO
+
+end IsBigO
+
+section IsBigORev
+
+variable [NormedAddGroup E] [SeminormedAddGroup F]
+
+protected theorem isBigOWith_rev_principal
+    (hf : ContinuousOn f s) (hs : IsCompact s) (hC : ∀ i ∈ s, f i ≠ 0) (c : F) :
+    IsBigOWith (‖c‖ / sInf (Norm.norm '' (f '' s))) (𝓟 s) (fun _ => c) f := by
+  refine isBigOWith_principal.mpr fun x hx ↦ ?_
+  rw [mul_comm_div]
+  replace hs := hs.image_of_continuousOn hf |>.image continuous_norm
+  have h_sInf := hs.isGLB_sInf <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr ⟨x, hx⟩
+  refine le_mul_of_one_le_right (norm_nonneg c) <| (one_le_div ?_).mpr <|
+    h_sInf.1 <| Set.mem_image_of_mem _ <| Set.mem_image_of_mem _ hx
+  obtain ⟨_, ⟨x, hx, hCx⟩, hnormCx⟩ := hs.sInf_mem h_sInf.nonempty
+  rw [← hnormCx, ← hCx]
+  exact (norm_ne_zero_iff.mpr (hC x hx)).symm.lt_of_le (norm_nonneg _)
+
+protected theorem isBigO_rev_principal (hf : ContinuousOn f s)
+    (hs : IsCompact s) (hC : ∀ i ∈ s, f i ≠ 0) (c : F) : (fun _ => c) =O[𝓟 s] f :=
+  (hf.isBigOWith_rev_principal hs hC c).isBigO
+
+end IsBigORev
+
+end ContinuousOn
+
+set_option linter.style.longFile 2200
diff --git a/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean b/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
index a0281a8808ecb..72724bc1976ef 100644
--- a/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
+++ b/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
@@ -27,7 +27,7 @@ theorem Filter.IsBoundedUnder.isLittleO_sub_self_inv {𝕜 E : Type*} [NormedFie
     {f : 𝕜 → E} (h : IsBoundedUnder (· ≤ ·) (𝓝[≠] a) (norm ∘ f)) :
     f =o[𝓝[≠] a] fun x => (x - a)⁻¹ := by
   refine (h.isBigO_const (one_ne_zero' ℝ)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr ?_)
-  simp only [(· ∘ ·), norm_inv]
+  simp only [Function.comp_def, norm_inv]
   exact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero
 
 end NormedField
@@ -107,11 +107,9 @@ theorem Asymptotics.IsLittleO.sum_range {α : Type*} [NormedAddCommGroup α] {f
       (add_le_add le_rfl (norm_sum_le_of_le _ fun i hi => hN _ (mem_Ico.1 hi).1))
     _ ≤ ‖∑ i ∈ range N, f i‖ + ∑ i ∈ range n, ε / 2 * g i := by
       gcongr
-      apply sum_le_sum_of_subset_of_nonneg
+      · exact fun i _ _ ↦ mul_nonneg (half_pos εpos).le (hg i)
       · rw [range_eq_Ico]
         exact Ico_subset_Ico (zero_le _) le_rfl
-      · intro i _ _
-        exact mul_nonneg (half_pos εpos).le (hg i)
     _ ≤ ε / 2 * ‖∑ i ∈ range n, g i‖ + ε / 2 * ∑ i ∈ range n, g i := by rw [← mul_sum]; gcongr
     _ = ε * ‖∑ i ∈ range n, g i‖ := by
       simp only [B]
diff --git a/Mathlib/Analysis/Asymptotics/Theta.lean b/Mathlib/Analysis/Asymptotics/Theta.lean
index ab9b0e2af090e..bf570aafef4e4 100644
--- a/Mathlib/Analysis/Asymptotics/Theta.lean
+++ b/Mathlib/Analysis/Asymptotics/Theta.lean
@@ -131,6 +131,9 @@ instance : Trans (α := α → E) (β := α → E) (γ := α → F) (EventuallyE
 lemma _root_.Filter.EventuallyEq.isTheta {f g : α → E} (h : f =ᶠ[l] g) : f =Θ[l] g :=
   h.trans_isTheta isTheta_rfl
 
+@[simp]
+theorem isTheta_bot : f =Θ[⊥] g := by simp [IsTheta]
+
 @[simp]
 theorem isTheta_norm_left : (fun x ↦ ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g := by simp [IsTheta]
 
@@ -185,7 +188,8 @@ theorem IsTheta.tendsto_zero_iff (h : f'' =Θ[l] g'') :
 
 theorem IsTheta.tendsto_norm_atTop_iff (h : f' =Θ[l] g') :
     Tendsto (norm ∘ f') l atTop ↔ Tendsto (norm ∘ g') l atTop := by
-  simp only [Function.comp, ← isLittleO_const_left_of_ne (one_ne_zero' ℝ), h.isLittleO_congr_right]
+  simp only [Function.comp_def, ← isLittleO_const_left_of_ne (one_ne_zero' ℝ),
+    h.isLittleO_congr_right]
 
 theorem IsTheta.isBoundedUnder_le_iff (h : f' =Θ[l] g') :
     IsBoundedUnder (· ≤ ·) l (norm ∘ f') ↔ IsBoundedUnder (· ≤ ·) l (norm ∘ g') := by
@@ -234,7 +238,7 @@ theorem isTheta_const_const_iff [NeBot l] {c₁ : E''} {c₂ : F''} :
 
 @[simp]
 theorem isTheta_zero_left : (fun _ ↦ (0 : E')) =Θ[l] g'' ↔ g'' =ᶠ[l] 0 := by
-  simp only [IsTheta, isBigO_zero, isBigO_zero_right_iff, true_and_iff]
+  simp only [IsTheta, isBigO_zero, isBigO_zero_right_iff, true_and]
 
 @[simp]
 theorem isTheta_zero_right : (f'' =Θ[l] fun _ ↦ (0 : F')) ↔ f'' =ᶠ[l] 0 :=
@@ -280,4 +284,46 @@ lemma IsLittleO.add_isTheta {f₁ f₂ : α → E'} {g : α → F}
     (ho : f₁ =o[l] g) (hΘ : f₂ =Θ[l] g) : (f₁ + f₂) =Θ[l] g :=
   add_comm f₁ f₂ ▸ hΘ.add_isLittleO ho
 
+section
+
+variable {f : α × β → E} {g : α × β → F} {l' : Filter β}
+
+protected theorem IsTheta.fiberwise_right :
+    f =Θ[l ×ˢ l'] g → ∀ᶠ x in l, (f ⟨x, ·⟩) =Θ[l'] (g ⟨x, ·⟩) := by
+  simp only [IsTheta, eventually_and]
+  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.fiberwise_right, h₂.fiberwise_right⟩
+
+protected theorem IsTheta.fiberwise_left :
+    f =Θ[l ×ˢ l'] g → ∀ᶠ y in l', (f ⟨·, y⟩) =Θ[l] (g ⟨·, y⟩) := by
+  simp only [IsTheta, eventually_and]
+  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.fiberwise_left, h₂.fiberwise_left⟩
+
+end
+
+section
+
+variable (l' : Filter β)
+
+protected theorem IsTheta.comp_fst : f =Θ[l] g → (f ∘ Prod.fst) =Θ[l ×ˢ l'] (g ∘ Prod.fst) := by
+  simp only [IsTheta, eventually_and]
+  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.comp_fst l', h₂.comp_fst l'⟩
+
+protected theorem IsTheta.comp_snd : f =Θ[l] g → (f ∘ Prod.snd) =Θ[l' ×ˢ l] (g ∘ Prod.snd) := by
+  simp only [IsTheta, eventually_and]
+  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.comp_snd l', h₂.comp_snd l'⟩
+
+end
+
 end Asymptotics
+
+namespace ContinuousOn
+
+variable {α E F : Type*} [NormedAddGroup E] [SeminormedAddGroup F] [TopologicalSpace α]
+  {s : Set α} {f : α → E} {c : F}
+
+protected theorem isTheta_principal
+    (hf : ContinuousOn f s) (hs : IsCompact s) (hc : ‖c‖ ≠ 0) (hC : ∀ i ∈ s, f i ≠ 0) :
+    f =Θ[𝓟 s] fun _ => c :=
+  ⟨hf.isBigO_principal hs hc, hf.isBigO_rev_principal hs hC c⟩
+
+end ContinuousOn
diff --git a/Mathlib/Analysis/BoundedVariation.lean b/Mathlib/Analysis/BoundedVariation.lean
index 0d69ebc8834bd..4f281d9bfca6d 100644
--- a/Mathlib/Analysis/BoundedVariation.lean
+++ b/Mathlib/Analysis/BoundedVariation.lean
@@ -192,7 +192,7 @@ protected theorem lowerSemicontinuous (s : Set α) :
   simpa only [UniformOnFun.tendsto_iff_tendstoUniformlyOn, mem_image, forall_exists_index, and_imp,
     forall_apply_eq_imp_iff₂, tendstoUniformlyOn_singleton_iff_tendsto] using @tendsto_id _ (𝓝 f)
 
-/-- The map `(eVariationOn · s)` is lower semicontinuous for uniform convergence on `s`.  -/
+/-- The map `(eVariationOn · s)` is lower semicontinuous for uniform convergence on `s`. -/
 theorem lowerSemicontinuous_uniformOn (s : Set α) :
     LowerSemicontinuous fun f : α →ᵤ[{s}] E => eVariationOn f s := fun f ↦ by
   apply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun α E {s}) id (𝓝 f) f s _
@@ -292,7 +292,7 @@ theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ 
         apply Finset.sum_congr rfl fun i _hi => ?_
         dsimp only [w]
         simp only [← Npos, Nat.not_lt_zero, Nat.add_succ_sub_one, add_zero, if_false,
-          add_eq_zero, Nat.one_ne_zero, false_and_iff, Nat.succ_add_sub_one, zero_add]
+          add_eq_zero, Nat.one_ne_zero, false_and, Nat.succ_add_sub_one, zero_add]
         rw [add_comm 1 i]
       _ = ∑ i ∈ Finset.Ico 1 (n + 1), edist (f (w (i + 1))) (f (w i)) := by
         rw [Finset.range_eq_Ico]
@@ -317,7 +317,7 @@ theorem add_point (f : α → E) {s : Set α} {x : α} (hx : x ∈ s) (u : ℕ 
         congr 1
         · congr 1
           · apply Finset.sum_congr rfl fun i hi => ?_
-            simp only [Finset.mem_Ico, zero_le', true_and_iff] at hi
+            simp only [Finset.mem_Ico, zero_le', true_and] at hi
             dsimp only [w]
             have A : i + 1 < N := Nat.lt_pred_iff.1 hi
             have B : i < N := Nat.lt_of_succ_lt A
diff --git a/Mathlib/Analysis/BoxIntegral/Basic.lean b/Mathlib/Analysis/BoxIntegral/Basic.lean
index b2c6da7cff285..e9d54c402baef 100644
--- a/Mathlib/Analysis/BoxIntegral/Basic.lean
+++ b/Mathlib/Analysis/BoxIntegral/Basic.lean
@@ -184,7 +184,7 @@ theorem hasIntegral_iff : HasIntegral I l f vol y ↔
     simp [@forall_swap ℝ≥0 (TaggedPrepartition I)]
 
 /-- Quite often it is more natural to prove an estimate of the form `a * ε`, not `ε` in the RHS of
-`BoxIntegral.hasIntegral_iff`, so we provide this auxiliary lemma.  -/
+`BoxIntegral.hasIntegral_iff`, so we provide this auxiliary lemma. -/
 theorem HasIntegral.of_mul (a : ℝ)
     (h : ∀ ε : ℝ, 0 < ε → ∃ r : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ), (∀ c, l.RCond (r c)) ∧ ∀ c π,
       l.MemBaseSet I c (r c) π → IsPartition π → dist (integralSum f vol π) y ≤ a * ε) :
@@ -391,7 +391,7 @@ additional distortion estimates if `BoxIntegral.IntegrationParams.bDistortion l
 corresponding integral sum is `ε`-close to the integral.
 
 If `BoxIntegral.IntegrationParams.bRiemann = true`, then `r c x` does not depend on `x`. If
-`ε ≤ 0`, then we use `r c x = 1`.  -/
+`ε ≤ 0`, then we use `r c x = 1`. -/
 def convergenceR (h : Integrable I l f vol) (ε : ℝ) : ℝ≥0 → ℝⁿ → Ioi (0 : ℝ) :=
   if hε : 0 < ε then (hasIntegral_iff.1 h.hasIntegral ε hε).choose
   else fun _ _ => ⟨1, Set.mem_Ioi.2 zero_lt_one⟩
@@ -447,7 +447,7 @@ theorem dist_integralSum_le_of_memBaseSet (h : Integrable I l f vol) (hpos₁ :
 
 /-- If `f` is integrable on `I` along `l`, then for two sufficiently fine tagged prepartitions
 (in the sense of the filter `BoxIntegral.IntegrationParams.toFilter l I`) such that they cover
-the same part of `I`, the integral sums of `f` over `π₁` and `π₂` are very close to each other.  -/
+the same part of `I`, the integral sums of `f` over `π₁` and `π₂` are very close to each other. -/
 theorem tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity (h : Integrable I l f vol) :
     Tendsto (fun π : TaggedPrepartition I × TaggedPrepartition I =>
       (integralSum f vol π.1, integralSum f vol π.2))
diff --git a/Mathlib/Analysis/BoxIntegral/Box/Basic.lean b/Mathlib/Analysis/BoxIntegral/Box/Basic.lean
index 9017646668d13..551320c2187d8 100644
--- a/Mathlib/Analysis/BoxIntegral/Box/Basic.lean
+++ b/Mathlib/Analysis/BoxIntegral/Box/Basic.lean
@@ -90,7 +90,7 @@ theorem lower_ne_upper (i) : I.lower i ≠ I.upper i :=
   (I.lower_lt_upper i).ne
 
 instance : Membership (ι → ℝ) (Box ι) :=
-  ⟨fun x I ↦ ∀ i, x i ∈ Ioc (I.lower i) (I.upper i)⟩
+  ⟨fun I x ↦ ∀ i, x i ∈ Ioc (I.lower i) (I.upper i)⟩
 
 -- Porting note: added
 /-- The set of points in this box: this is the product of half-open intervals `(lower i, upper i]`,
@@ -140,15 +140,12 @@ theorem le_def : I ≤ J ↔ ∀ x ∈ I, x ∈ J := Iff.rfl
 
 theorem le_TFAE : List.TFAE [I ≤ J, (I : Set (ι → ℝ)) ⊆ J,
     Icc I.lower I.upper ⊆ Icc J.lower J.upper, J.lower ≤ I.lower ∧ I.upper ≤ J.upper] := by
-  tfae_have 1 ↔ 2
-  · exact Iff.rfl
+  tfae_have 1 ↔ 2 := Iff.rfl
   tfae_have 2 → 3
-  · intro h
-    simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
-  tfae_have 3 ↔ 4
-  · exact Icc_subset_Icc_iff I.lower_le_upper
+  | h => by simpa [coe_eq_pi, closure_pi_set, lower_ne_upper] using closure_mono h
+  tfae_have 3 ↔ 4 := Icc_subset_Icc_iff I.lower_le_upper
   tfae_have 4 → 2
-  · exact fun h x hx i ↦ Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)
+  | h, x, hx, i => Ioc_subset_Ioc (h.1 i) (h.2 i) (hx i)
   tfae_finish
 
 variable {I J}
@@ -275,7 +272,7 @@ theorem withBotCoe_inj {I J : WithBot (Box ι)} : (I : Set (ι → ℝ)) = J ↔
 
 /-- Make a `WithBot (Box ι)` from a pair of corners `l u : ι → ℝ`. If `l i < u i` for all `i`,
 then the result is `⟨l, u, _⟩ : Box ι`, otherwise it is `⊥`. In any case, the result interpreted
-as a set in `ι → ℝ` is the set `{x : ι → ℝ | ∀ i, x i ∈ Ioc (l i) (u i)}`.  -/
+as a set in `ι → ℝ` is the set `{x : ι → ℝ | ∀ i, x i ∈ Ioc (l i) (u i)}`. -/
 def mk' (l u : ι → ℝ) : WithBot (Box ι) :=
   if h : ∀ i, l i < u i then ↑(⟨l, u, h⟩ : Box ι) else ⊥
 
diff --git a/Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean b/Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
index c516b55504b50..f70f3e2cb58e1 100644
--- a/Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
+++ b/Mathlib/Analysis/BoxIntegral/Box/SubboxInduction.lean
@@ -52,7 +52,7 @@ theorem mem_splitCenterBox {s : Set ι} {y : ι → ℝ} :
   simp only [splitCenterBox, mem_def, ← forall_and]
   refine forall_congr' fun i ↦ ?_
   dsimp only [Set.piecewise]
-  split_ifs with hs <;> simp only [hs, iff_true_iff, iff_false_iff, not_lt]
+  split_ifs with hs <;> simp only [hs, iff_true, iff_false, not_lt]
   exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
       fun H ↦ ⟨H.2, H.1.2⟩⟩,
     ⟨fun H ↦ ⟨⟨H.1, H.2.trans (add_div_two_lt_right.2 (I.lower_lt_upper i)).le⟩, H.2⟩,
diff --git a/Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean b/Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
index 79429fdb79725..9c979a7202bd1 100644
--- a/Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
+++ b/Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean
@@ -77,7 +77,7 @@ theorem norm_volume_sub_integral_face_upper_sub_lower_smul_le {f : (Fin (n + 1)
     `f y - a - f' (y - x)` over each of these faces is less than or equal to `ε * c * vol I`. We
     integrate a function of the norm `≤ ε * diam I.Icc` over a box of volume
     `∏ j ≠ i, (I.upper j - I.lower j)`. Since `diam I.Icc ≤ c * (I.upper i - I.lower i)`, we get the
-    required estimate.  -/
+    required estimate. -/
   have Hl : I.lower i ∈ Icc (I.lower i) (I.upper i) := Set.left_mem_Icc.2 (I.lower_le_upper i)
   have Hu : I.upper i ∈ Icc (I.lower i) (I.upper i) := Set.right_mem_Icc.2 (I.lower_le_upper i)
   have Hi : ∀ x ∈ Icc (I.lower i) (I.upper i),
diff --git a/Mathlib/Analysis/BoxIntegral/Integrability.lean b/Mathlib/Analysis/BoxIntegral/Integrability.lean
index 12a2a1c87c00d..eaf2f17700f98 100644
--- a/Mathlib/Analysis/BoxIntegral/Integrability.lean
+++ b/Mathlib/Analysis/BoxIntegral/Integrability.lean
@@ -113,7 +113,7 @@ theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι 
   have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
     refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
     cases' n with n
-    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
+    · simp [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne']
     · refine (measure_mono_null ?_ hf).le.trans_lt ?_
       · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
       · simp [(δ0 _).ne']
@@ -152,7 +152,7 @@ theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι 
   exact (mul_le_mul_left' this.le _).trans ENNReal.mul_div_le
 
 /-- If `f` has integral `y` on a box `I` with respect to a locally finite measure `μ` and `g` is
-a.e. equal to `f` on `I`, then `g` has the same integral on `I`.  -/
+a.e. equal to `f` on `I`, then `g` has the same integral on `I`. -/
 theorem HasIntegral.congr_ae {l : IntegrationParams} {I : Box ι} {y : E} {f g : (ι → ℝ) → E}
     {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ]
     (hf : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul y) (hfg : f =ᵐ[μ.restrict I] g)
@@ -191,7 +191,7 @@ end SimpleFunc
 open TopologicalSpace
 
 /-- If `f : ℝⁿ → E` is Bochner integrable w.r.t. a locally finite measure `μ` on a rectangular box
-`I`, then it is McShane integrable on `I` with the same integral.  -/
+`I`, then it is McShane integrable on `I` with the same integral. -/
 theorem IntegrableOn.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} {μ : Measure (ι → ℝ)}
     [IsLocallyFiniteMeasure μ] {I : Box ι} (hf : IntegrableOn f I μ) (l : IntegrationParams)
     (hl : l.bRiemann = false) :
diff --git a/Mathlib/Analysis/BoxIntegral/Partition/Additive.lean b/Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
index dfa3b333012c5..a82e54fd54bb5 100644
--- a/Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
+++ b/Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
@@ -190,7 +190,7 @@ def upperSubLower.{u} {G : Type u} [AddCommGroup G] (I₀ : Box (Fin (n + 1))) (
       rw [WithTop.coe_le_coe] at hJ
       refine i.succAboveCases (fun hx => ?_) (fun j hx => ?_) j
       · simp only [Box.splitLower_def hx, Box.splitUpper_def hx, update_same, ← WithBot.some_eq_coe,
-          Option.elim', Box.face, (· ∘ ·), update_noteq (Fin.succAbove_ne _ _)]
+          Option.elim', Box.face, Function.comp_def, update_noteq (Fin.succAbove_ne _ _)]
         abel
       · have : (J.face i : WithTop (Box (Fin n))) ≤ I₀.face i :=
           WithTop.coe_le_coe.2 (face_mono hJ i)
diff --git a/Mathlib/Analysis/BoxIntegral/Partition/Basic.lean b/Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
index ea45c95bef466..788c55499703d 100644
--- a/Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
+++ b/Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
@@ -60,7 +60,7 @@ namespace Prepartition
 variable {I J J₁ J₂ : Box ι} (π : Prepartition I) {π₁ π₂ : Prepartition I} {x : ι → ℝ}
 
 instance : Membership (Box ι) (Prepartition I) :=
-  ⟨fun J π => J ∈ π.boxes⟩
+  ⟨fun π J => J ∈ π.boxes⟩
 
 @[simp]
 theorem mem_boxes : J ∈ π.boxes ↔ J ∈ π := Iff.rfl
@@ -541,7 +541,8 @@ theorem filter_true : (π.filter fun _ => True) = π :=
 theorem iUnion_filter_not (π : Prepartition I) (p : Box ι → Prop) :
     (π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion := by
   simp only [Prepartition.iUnion]
-  convert (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm
+  convert
+    (@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm using 4
   · simp (config := { contextual := true })
   · rw [Set.PairwiseDisjoint]
     convert π.pairwiseDisjoint
@@ -622,7 +623,7 @@ def IsPartition (π : Prepartition I) :=
   ∀ x ∈ I, ∃ J ∈ π, x ∈ J
 
 theorem isPartition_iff_iUnion_eq {π : Prepartition I} : π.IsPartition ↔ π.iUnion = I := by
-  simp_rw [IsPartition, Set.Subset.antisymm_iff, π.iUnion_subset, true_and_iff, Set.subset_def,
+  simp_rw [IsPartition, Set.Subset.antisymm_iff, π.iUnion_subset, true_and, Set.subset_def,
     mem_iUnion, Box.mem_coe]
 
 @[simp]
diff --git a/Mathlib/Analysis/BoxIntegral/Partition/Filter.lean b/Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
index 3a2ae15d2bd93..c1b1d7ea49571 100644
--- a/Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
+++ b/Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
@@ -223,7 +223,7 @@ instance : Inhabited IntegrationParams :=
   ⟨⊥⟩
 
 instance : DecidableRel ((· ≤ ·) : IntegrationParams → IntegrationParams → Prop) :=
-  fun _ _ => And.decidable
+  fun _ _ => inferInstanceAs (Decidable (_ ∧ _))
 
 instance : DecidableEq IntegrationParams :=
   fun _ _ => decidable_of_iff _ IntegrationParams.ext_iff.symm
diff --git a/Mathlib/Analysis/BoxIntegral/Partition/Split.lean b/Mathlib/Analysis/BoxIntegral/Partition/Split.lean
index 5d3d61fca5246..06363982a7538 100644
--- a/Mathlib/Analysis/BoxIntegral/Partition/Split.lean
+++ b/Mathlib/Analysis/BoxIntegral/Partition/Split.lean
@@ -149,7 +149,7 @@ def split (I : Box ι) (i : ι) (x : ℝ) : Prepartition I :=
       rintro J (rfl | rfl)
       exacts [Box.splitLower_le, Box.splitUpper_le])
     (by
-      simp only [Finset.coe_insert, Finset.coe_singleton, true_and_iff, Set.mem_singleton_iff,
+      simp only [Finset.coe_insert, Finset.coe_singleton, true_and, Set.mem_singleton_iff,
         pairwise_insert_of_symmetric symmetric_disjoint, pairwise_singleton]
       rintro J rfl -
       exact I.disjoint_splitLower_splitUpper i x)
@@ -244,7 +244,7 @@ theorem inf_splitMany {I : Box ι} (π : Prepartition I) (s : Finset (ι × ℝ)
 /-- Let `s : Finset (ι × ℝ)` be a set of hyperplanes `{x : ι → ℝ | x i = r}` in `ι → ℝ` encoded as
 pairs `(i, r)`. Suppose that this set contains all faces of a box `J`. The hyperplanes of `s` split
 a box `I` into subboxes. Let `Js` be one of them. If `J` and `Js` have nonempty intersection, then
-`Js` is a subbox of `J`.  -/
+`Js` is a subbox of `J`. -/
 theorem not_disjoint_imp_le_of_subset_of_mem_splitMany {I J Js : Box ι} {s : Finset (ι × ℝ)}
     (H : ∀ i, {(i, J.lower i), (i, J.upper i)} ⊆ s) (HJs : Js ∈ splitMany I s)
     (Hn : ¬Disjoint (J : WithBot (Box ι)) Js) : Js ≤ J := by
diff --git a/Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean b/Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
index 71fea77c5b71a..967500dc84169 100644
--- a/Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
+++ b/Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
@@ -49,7 +49,7 @@ namespace TaggedPrepartition
 variable {I J J₁ J₂ : Box ι} (π : TaggedPrepartition I) {x : ι → ℝ}
 
 instance : Membership (Box ι) (TaggedPrepartition I) :=
-  ⟨fun J π => J ∈ π.boxes⟩
+  ⟨fun π J => J ∈ π.boxes⟩
 
 @[simp]
 theorem mem_toPrepartition {π : TaggedPrepartition I} : J ∈ π.toPrepartition ↔ J ∈ π := Iff.rfl
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
index a475c96675c2d..073d532f7a479 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Basic.lean
@@ -90,7 +90,7 @@ theorem elementalStarAlgebra.bijective_characterSpaceToSpectrum :
       ContinuousMap.coe_mk] using h
   · rintro ⟨z, hz⟩
     have hz' := (StarSubalgebra.spectrum_eq (hS := elementalStarAlgebra.isClosed ℂ a)
-      (a := ⟨a, self_mem ℂ a⟩).symm.subst hz)
+      (a := ⟨a, self_mem ℂ a⟩) ▸ hz)
     rw [CharacterSpace.mem_spectrum_iff_exists] at hz'
     obtain ⟨φ, rfl⟩ := hz'
     exact ⟨φ, rfl⟩
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
index eaa3333737924..180ca90ca5288 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Instances.lean
@@ -23,7 +23,7 @@ import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
   elements in an `ℝ`-algebra with a continuous functional calculus for selfadjoint elements,
   where every element has compact spectrum, and where nonnegative elements have nonnegative
   spectrum. In particular, this includes unital C⋆-algebras over `ℝ`.
-* `CStarRing.instNonnegSpectrumClass`: In a unital C⋆-algebra over `ℂ` which is also a
+* `CStarAlgebra.instNonnegSpectrumClass`: In a unital C⋆-algebra over `ℂ` which is also a
   `StarOrderedRing`, the spectrum of a nonnegative element is nonnegative.
 
 ## Tags
@@ -218,7 +218,7 @@ lemma QuasispectrumRestricts.isSelfAdjoint (a : A) (ha : QuasispectrumRestricts
 instance IsSelfAdjoint.instNonUnitalContinuousFunctionalCalculus :
     NonUnitalContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop) :=
   QuasispectrumRestricts.cfc (q := IsStarNormal) (p := IsSelfAdjoint) Complex.reCLM
-    Complex.isometry_ofReal.uniformEmbedding (.zero _)
+    Complex.isometry_ofReal.isUniformEmbedding (.zero _)
     (fun _ ↦ isSelfAdjoint_iff_isStarNormal_and_quasispectrumRestricts)
 
 end SelfAdjointNonUnital
@@ -264,9 +264,14 @@ lemma SpectrumRestricts.isSelfAdjoint (a : A) (ha : SpectrumRestricts a Complex.
 instance IsSelfAdjoint.instContinuousFunctionalCalculus :
     ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop) :=
   SpectrumRestricts.cfc (q := IsStarNormal) (p := IsSelfAdjoint) Complex.reCLM
-    Complex.isometry_ofReal.uniformEmbedding (.zero _)
+    Complex.isometry_ofReal.isUniformEmbedding (.zero _)
     (fun _ ↦ isSelfAdjoint_iff_isStarNormal_and_spectrumRestricts)
 
+lemma IsSelfAdjoint.spectrum_nonempty {A : Type*} [Ring A] [StarRing A]
+    [TopologicalSpace A] [Algebra ℝ A] [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
+    [Nontrivial A] {a : A} (ha : IsSelfAdjoint a) : (σ ℝ a).Nonempty :=
+  CFC.spectrum_nonempty ℝ a ha
+
 end SelfAdjointUnital
 
 /-!
@@ -308,9 +313,15 @@ open NNReal in
 instance Nonneg.instNonUnitalContinuousFunctionalCalculus :
     NonUnitalContinuousFunctionalCalculus ℝ≥0 (fun x : A ↦ 0 ≤ x) :=
   QuasispectrumRestricts.cfc (q := IsSelfAdjoint) ContinuousMap.realToNNReal
-    uniformEmbedding_subtype_val le_rfl
+    isUniformEmbedding_subtype_val le_rfl
     (fun _ ↦ nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts)
 
+open NNReal in
+lemma NNReal.spectrum_nonempty {A : Type*} [Ring A] [StarRing A] [PartialOrder A]
+    [TopologicalSpace A] [Algebra ℝ≥0 A] [ContinuousFunctionalCalculus ℝ≥0 (fun x : A ↦ 0 ≤ x)]
+    [Nontrivial A] {a : A} (ha : 0 ≤ a) : (spectrum ℝ≥0 a).Nonempty :=
+  CFC.spectrum_nonempty ℝ≥0 a ha
+
 end Nonneg
 
 
@@ -348,7 +359,7 @@ open NNReal in
 instance Nonneg.instContinuousFunctionalCalculus :
     ContinuousFunctionalCalculus ℝ≥0 (fun x : A ↦ 0 ≤ x) :=
   SpectrumRestricts.cfc (q := IsSelfAdjoint) ContinuousMap.realToNNReal
-    uniformEmbedding_subtype_val le_rfl (fun _ ↦ nonneg_iff_isSelfAdjoint_and_spectrumRestricts)
+    isUniformEmbedding_subtype_val le_rfl (fun _ ↦ nonneg_iff_isSelfAdjoint_and_spectrumRestricts)
 
 end Nonneg
 
@@ -484,7 +495,7 @@ variable {A : Type*} [NormedRing A] [CompleteSpace A]
 variable [PartialOrder A] [StarRing A] [StarOrderedRing A] [CStarRing A]
 variable [NormedAlgebra ℂ A] [StarModule ℂ A]
 
-instance CStarRing.instNonnegSpectrumClass : NonnegSpectrumClass ℝ A :=
+instance CStarAlgebra.instNonnegSpectrumClass : NonnegSpectrumClass ℝ A :=
   .of_spectrum_nonneg fun a ha ↦ by
     rw [StarOrderedRing.nonneg_iff] at ha
     induction ha using AddSubmonoid.closure_induction' with
@@ -500,7 +511,7 @@ instance CStarRing.instNonnegSpectrumClass : NonnegSpectrumClass ℝ A :=
       exact hx.nnreal_add (.of_nonneg x_mem) (.of_nonneg y_mem) hy
 
 open ComplexOrder in
-instance CStarRing.instNonnegSpectrumClassComplexUnital : NonnegSpectrumClass ℂ A where
+instance CStarAlgebra.instNonnegSpectrumClassComplexUnital : NonnegSpectrumClass ℂ A where
   quasispectrum_nonneg_of_nonneg a ha x := by
     rw [mem_quasispectrum_iff]
     refine (Or.elim · ge_of_eq fun hx ↦ ?_)
@@ -520,7 +531,7 @@ selfadjoint and has nonnegative spectrum.
 This is not declared as an instance because one may already have a partial order with better
 definitional properties. However, it can be useful to invoke this as an instance in proofs. -/
 @[reducible]
-def CStarRing.spectralOrder : PartialOrder A where
+def CStarAlgebra.spectralOrder : PartialOrder A where
   le x y := IsSelfAdjoint (y - x) ∧ SpectrumRestricts (y - x) ContinuousMap.realToNNReal
   le_refl := by
     simp only [sub_self, IsSelfAdjoint.zero, true_and, forall_const]
@@ -533,9 +544,9 @@ def CStarRing.spectralOrder : PartialOrder A where
   le_trans x y z hxy hyz :=
     ⟨by simpa using hyz.1.add hxy.1, by simpa using hyz.2.nnreal_add hyz.1 hxy.1 hxy.2⟩
 
-/-- The `CStarRing.spectralOrder` on a unital C⋆-algebra is a `StarOrderedRing`. -/
-lemma CStarRing.spectralOrderedRing : @StarOrderedRing A _ (CStarRing.spectralOrder A) _ :=
-  let _ := CStarRing.spectralOrder A
+/-- The `CStarAlgebra.spectralOrder` on a unital C⋆-algebra is a `StarOrderedRing`. -/
+lemma CStarAlgebra.spectralOrderedRing : @StarOrderedRing A _ (CStarAlgebra.spectralOrder A) _ :=
+  let _ := CStarAlgebra.spectralOrder A
   { le_iff := by
       intro x y
       constructor
@@ -567,12 +578,12 @@ variable {A : Type*} [NonUnitalNormedRing A] [CompleteSpace A]
 variable [PartialOrder A] [StarRing A] [StarOrderedRing A] [CStarRing A]
 variable [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [StarModule ℂ A]
 
-instance CStarRing.instNonnegSpectrumClass' : NonnegSpectrumClass ℝ A where
+instance CStarAlgebra.instNonnegSpectrumClass' : NonnegSpectrumClass ℝ A where
   quasispectrum_nonneg_of_nonneg a ha := by
     rw [Unitization.quasispectrum_eq_spectrum_inr' _ ℂ]
     -- should this actually be an instance on the `Unitization`? (probably scoped)
-    let _ := CStarRing.spectralOrder (Unitization ℂ A)
-    have := CStarRing.spectralOrderedRing (Unitization ℂ A)
+    let _ := CStarAlgebra.spectralOrder (Unitization ℂ A)
+    have := CStarAlgebra.spectralOrderedRing (Unitization ℂ A)
     apply spectrum_nonneg_of_nonneg
     rw [StarOrderedRing.nonneg_iff] at ha ⊢
     have := AddSubmonoid.mem_map_of_mem (Unitization.inrNonUnitalStarAlgHom ℂ A) ha
@@ -594,17 +605,38 @@ variable {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A] [Algebra ℂ A]
 
 lemma cfcHom_real_eq_restrict {a : A} (ha : IsSelfAdjoint a) :
     cfcHom ha = ha.spectrumRestricts.starAlgHom (cfcHom ha.isStarNormal) (f := Complex.reCLM) :=
-  ha.spectrumRestricts.cfcHom_eq_restrict Complex.isometry_ofReal.uniformEmbedding
+  ha.spectrumRestricts.cfcHom_eq_restrict _ Complex.isometry_ofReal.isUniformEmbedding
     ha ha.isStarNormal
 
 lemma cfc_real_eq_complex {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac)  :
     cfc f a = cfc (fun x ↦ f x.re : ℂ → ℂ) a := by
   replace ha : IsSelfAdjoint a := ha -- hack to avoid issues caused by autoParam
   exact ha.spectrumRestricts.cfc_eq_restrict (f := Complex.reCLM)
-    Complex.isometry_ofReal.uniformEmbedding ha ha.isStarNormal f
+    Complex.isometry_ofReal.isUniformEmbedding ha ha.isStarNormal f
 
 end RealEqComplex
 
+section RealEqComplexNonUnital
+
+variable {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [StarRing A] [Module ℂ A]
+  [IsScalarTower ℂ A A] [SMulCommClass ℂ A A]
+  [NonUnitalContinuousFunctionalCalculus ℂ (IsStarNormal : A → Prop)]
+  [UniqueNonUnitalContinuousFunctionalCalculus ℝ A]
+
+lemma cfcₙHom_real_eq_restrict {a : A} (ha : IsSelfAdjoint a) :
+    cfcₙHom ha = (ha.quasispectrumRestricts.2).nonUnitalStarAlgHom (cfcₙHom ha.isStarNormal)
+      (f := Complex.reCLM) :=
+  ha.quasispectrumRestricts.2.cfcₙHom_eq_restrict _ Complex.isometry_ofReal.isUniformEmbedding
+    ha ha.isStarNormal
+
+lemma cfcₙ_real_eq_complex {a : A} (f : ℝ → ℝ) (ha : IsSelfAdjoint a := by cfc_tac)  :
+    cfcₙ f a = cfcₙ (fun x ↦ f x.re : ℂ → ℂ) a := by
+  replace ha : IsSelfAdjoint a := ha -- hack to avoid issues caused by autoParam
+  exact ha.quasispectrumRestricts.2.cfcₙ_eq_restrict (f := Complex.reCLM)
+    Complex.isometry_ofReal.isUniformEmbedding ha ha.isStarNormal f
+
+end RealEqComplexNonUnital
+
 section NNRealEqReal
 
 open NNReal
@@ -618,14 +650,39 @@ variable {A : Type*} [TopologicalSpace A] [Ring A] [PartialOrder A] [StarRing A]
 lemma cfcHom_nnreal_eq_restrict {a : A} (ha : 0 ≤ a) :
     cfcHom ha = (SpectrumRestricts.nnreal_of_nonneg ha).starAlgHom
       (cfcHom (IsSelfAdjoint.of_nonneg ha)) := by
-  apply (SpectrumRestricts.nnreal_of_nonneg ha).cfcHom_eq_restrict uniformEmbedding_subtype_val
+  apply (SpectrumRestricts.nnreal_of_nonneg ha).cfcHom_eq_restrict _ isUniformEmbedding_subtype_val
 
 lemma cfc_nnreal_eq_real {a : A} (f : ℝ≥0 → ℝ≥0) (ha : 0 ≤ a := by cfc_tac)  :
     cfc f a = cfc (fun x ↦ f x.toNNReal : ℝ → ℝ) a := by
   replace ha : 0 ≤ a := ha -- hack to avoid issues caused by autoParam
-  apply (SpectrumRestricts.nnreal_of_nonneg ha).cfc_eq_restrict
-    uniformEmbedding_subtype_val ha (.of_nonneg ha)
+  apply (SpectrumRestricts.nnreal_of_nonneg ha).cfc_eq_restrict _
+    isUniformEmbedding_subtype_val ha (.of_nonneg ha)
 
 end NNRealEqReal
 
+section NNRealEqRealNonUnital
+
+open NNReal
+
+variable {A : Type*} [TopologicalSpace A] [NonUnitalRing A] [PartialOrder A] [StarRing A]
+  [StarOrderedRing A] [Module ℝ A] [TopologicalRing A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A]
+  [NonUnitalContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
+  [NonUnitalContinuousFunctionalCalculus ℝ≥0 ((0 : A) ≤ ·)]
+  [UniqueNonUnitalContinuousFunctionalCalculus ℝ A]
+  [NonnegSpectrumClass ℝ A]
+
+lemma cfcₙHom_nnreal_eq_restrict {a : A} (ha : 0 ≤ a) :
+    cfcₙHom ha = (QuasispectrumRestricts.nnreal_of_nonneg ha).nonUnitalStarAlgHom
+      (cfcₙHom (IsSelfAdjoint.of_nonneg ha)) := by
+  apply (QuasispectrumRestricts.nnreal_of_nonneg ha).cfcₙHom_eq_restrict _
+    isUniformEmbedding_subtype_val
+
+lemma cfcₙ_nnreal_eq_real {a : A} (f : ℝ≥0 → ℝ≥0) (ha : 0 ≤ a := by cfc_tac)  :
+    cfcₙ f a = cfcₙ (fun x ↦ f x.toNNReal : ℝ → ℝ) a := by
+  replace ha : 0 ≤ a := ha -- hack to avoid issues caused by autoParam
+  apply (QuasispectrumRestricts.nnreal_of_nonneg ha).cfcₙ_eq_restrict _
+    isUniformEmbedding_subtype_val ha (.of_nonneg ha)
+
+end NNRealEqRealNonUnital
+
 end
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean
index f2aaa20d55f86..f56a3cf3d1927 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Integral.lean
@@ -6,6 +6,7 @@ Authors: Frédéric Dupuis
 
 import Mathlib.Analysis.Normed.Algebra.Spectrum
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
+import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
 import Mathlib.MeasureTheory.Integral.SetIntegral
 
 /-!
@@ -19,15 +20,18 @@ that the integral commutes with the continuous functional calculus under appropr
 + `cfc_integral`: given a function `f : X → 𝕜 → 𝕜`, we have that
   `cfc (fun r => ∫ x, f x r ∂μ) a = ∫ x, cfc (f x) a ∂μ`
   under appropriate conditions
++ `cfcₙ_integral`: given a function `f : X → 𝕜 → 𝕜`, we have that
+  `cfcₙ (fun r => ∫ x, f x r ∂μ) a = ∫ x, cfcₙ (f x) a ∂μ`
+  under appropriate conditions
 
 ## TODO
 
-+ Prove a similar result for the non-unital case
 + Lift this to the case where the CFC is over `ℝ≥0`
 + Use this to prove operator monotonicity and concavity/convexity of `rpow` and `log`
 -/
 
 open MeasureTheory
+open scoped ContinuousMapZero
 
 section unital
 
@@ -36,12 +40,14 @@ variable {X : Type*} {𝕜 : Type*} {A : Type*} {p : A → Prop} [RCLike 𝕜]
   [NormedRing A] [StarRing A] [NormedAlgebra 𝕜 A] [NormedAlgebra ℝ A] [CompleteSpace A]
   [ContinuousFunctionalCalculus 𝕜 p]
 
-lemma cfcL_integral (a : A) (f : X → C(spectrum 𝕜 a, 𝕜)) (hf₁ : Integrable f μ) (ha : p a) :
-    ∫ x, cfcL ha (f x) ∂μ = cfcL ha (∫ x, f x ∂μ) := by
+lemma cfcL_integral (a : A) (f : X → C(spectrum 𝕜 a, 𝕜)) (hf₁ : Integrable f μ)
+    (ha : p a := by cfc_tac) :
+    ∫ x, cfcL (a := a) ha (f x) ∂μ = cfcL (a := a) ha (∫ x, f x ∂μ) := by
   rw [ContinuousLinearMap.integral_comp_comm _ hf₁]
 
-lemma cfcHom_integral (a : A) (f : X → C(spectrum 𝕜 a, 𝕜)) (hf₁ : Integrable f μ) (ha : p a) :
-    ∫ x, cfcHom ha (f x) ∂μ = cfcHom ha (∫ x, f x ∂μ) :=
+lemma cfcHom_integral (a : A) (f : X → C(spectrum 𝕜 a, 𝕜)) (hf₁ : Integrable f μ)
+    (ha : p a := by cfc_tac) :
+    ∫ x, cfcHom (a := a) ha (f x) ∂μ = cfcHom (a := a) ha (∫ x, f x ∂μ) :=
   cfcL_integral a f hf₁ ha
 
 open ContinuousMap in
@@ -80,3 +86,62 @@ lemma cfc_integral' [TopologicalSpace X] [OpensMeasurableSpace X] (f : X → 
   · exact ContinuousMap.curry ⟨_, hf⟩ |>.continuous
 
 end unital
+
+section nonunital
+
+variable {X : Type*} {𝕜 : Type*} {A : Type*} {p : A → Prop} [RCLike 𝕜]
+  [MeasurableSpace X] {μ : Measure X} [NonUnitalNormedRing A] [StarRing A] [CompleteSpace A]
+  [NormedSpace 𝕜 A] [NormedSpace ℝ A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
+  [NonUnitalContinuousFunctionalCalculus 𝕜 p]
+
+lemma cfcₙL_integral (a : A) (f : X → C(quasispectrum 𝕜 a, 𝕜)₀) (hf₁ : Integrable f μ)
+    (ha : p a := by cfc_tac) :
+    ∫ x, cfcₙL (a := a) ha (f x) ∂μ = cfcₙL (a := a) ha (∫ x, f x ∂μ) := by
+  rw [ContinuousLinearMap.integral_comp_comm _ hf₁]
+
+lemma cfcₙHom_integral (a : A) (f : X → C(quasispectrum 𝕜 a, 𝕜)₀) (hf₁ : Integrable f μ)
+    (ha : p a := by cfc_tac) :
+    ∫ x, cfcₙHom (a := a) ha (f x) ∂μ = cfcₙHom (a := a) ha (∫ x, f x ∂μ) :=
+  cfcₙL_integral a f hf₁ ha
+
+open ContinuousMapZero in
+/-- The non-unital continuous functional calculus commutes with integration. -/
+lemma cfcₙ_integral [TopologicalSpace X] [OpensMeasurableSpace X] (f : X → 𝕜 → 𝕜)
+    (bound : X → ℝ) (a : A) [SecondCountableTopologyEither X C(quasispectrum 𝕜 a, 𝕜)₀]
+    (hf₁ : ∀ x, ContinuousOn (f x) (quasispectrum 𝕜 a))
+    (hf₂ : ∀ x, f x 0 = 0)
+    (hf₃ : Continuous (fun x ↦ (⟨⟨_, hf₁ x |>.restrict⟩, hf₂ x⟩ : C(quasispectrum 𝕜 a, 𝕜)₀)))
+    (hbound : ∀ x, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ ‖bound x‖)
+    (hbound_finite_integral : HasFiniteIntegral bound μ) (ha : p a := by cfc_tac) :
+    cfcₙ (fun r => ∫ x, f x r ∂μ) a = ∫ x, cfcₙ (f x) a ∂μ := by
+  let fc : X → C(quasispectrum 𝕜 a, 𝕜)₀ := fun x => ⟨⟨_, (hf₁ x).restrict⟩, hf₂ x⟩
+  have fc_integrable : Integrable fc μ := by
+    refine ⟨hf₃.aestronglyMeasurable, ?_⟩
+    refine hbound_finite_integral.mono <| .of_forall fun x ↦ ?_
+    change ‖(fc x : C(quasispectrum  𝕜 a, 𝕜))‖ ≤ ‖bound x‖
+    rw [ContinuousMap.norm_le _ (norm_nonneg (bound x))]
+    exact fun z ↦ hbound x z.1 z.2
+  have h_int_fc : (quasispectrum 𝕜 a).restrict (∫ x, f x · ∂μ) = ∫ x, fc x ∂μ := by
+    ext; simp [integral_apply fc_integrable, fc]
+  have hcont₂ : ContinuousOn (fun r => ∫ x, f x r ∂μ) (quasispectrum 𝕜 a) := by
+    rw [continuousOn_iff_continuous_restrict]
+    convert map_continuous (∫ x, fc x ∂μ)
+  rw [integral_congr_ae (.of_forall fun _ ↦ cfcₙ_apply ..), cfcₙ_apply ..,
+    cfcₙHom_integral _ _ fc_integrable]
+  congr
+
+/-- The non-unital continuous functional calculus commutes with integration. -/
+lemma cfcₙ_integral' [TopologicalSpace X] [OpensMeasurableSpace X] (f : X → 𝕜 → 𝕜)
+    (bound : X → ℝ) (a : A) [SecondCountableTopologyEither X C(quasispectrum 𝕜 a, 𝕜)₀]
+    (hf : Continuous (fun x => (quasispectrum 𝕜 a).restrict (f x)).uncurry)
+    (hf₂ : ∀ x, f x 0 = 0)
+    (hbound : ∀ x, ∀ z ∈ quasispectrum 𝕜 a, ‖f x z‖ ≤ ‖bound x‖)
+    (hbound_finite_integral : HasFiniteIntegral bound μ) (ha : p a := by cfc_tac) :
+    cfcₙ (fun r => ∫ x, f x r ∂μ) a = ∫ x, cfcₙ (f x) a ∂μ := by
+  refine cfcₙ_integral f bound a ?_ hf₂ ?_ hbound hbound_finite_integral
+  · exact (continuousOn_iff_continuous_restrict.mpr <| hf.uncurry_left ·)
+  · let g := ((↑) : C(quasispectrum 𝕜 a, 𝕜)₀ → C(quasispectrum 𝕜 a, 𝕜))
+    refine (Inducing.continuous_iff (g := g) ((inducing_iff g).mpr rfl)).mpr ?_
+    exact ContinuousMap.curry ⟨_, hf⟩ |>.continuous
+
+end nonunital
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
index 816e5257d8754..361cdc8aefebc 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/NonUnital.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
 import Mathlib.Algebra.Algebra.Quasispectrum
-import Mathlib.Topology.ContinuousFunction.Compact
-import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
+import Mathlib.Topology.ContinuousMap.Compact
+import Mathlib.Topology.ContinuousMap.ContinuousMapZero
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
 import Mathlib.Topology.UniformSpace.CompactConvergence
 
@@ -127,6 +127,10 @@ lemma cfcₙHom_closedEmbedding :
     ClosedEmbedding <| (cfcₙHom ha : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) :=
   (NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.1
 
+@[fun_prop]
+lemma cfcₙHom_continuous : Continuous (cfcₙHom ha : C(σₙ R a, R)₀ →⋆ₙₐ[R] A) :=
+  cfcₙHom_closedEmbedding ha |>.continuous
+
 lemma cfcₙHom_id :
     cfcₙHom ha ⟨(ContinuousMap.id R).restrict <| σₙ R a, rfl⟩ = a :=
   (NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.2.1
@@ -170,6 +174,17 @@ theorem cfcₙHom_comp [UniqueNonUnitalContinuousFunctionalCalculus R A] (f : C(
 
 end cfcₙHom
 
+section cfcₙL
+
+/-- `cfcₙHom` bundled as a continuous linear map. -/
+@[simps apply]
+noncomputable def cfcₙL {a : A} (ha : p a) : C(σₙ R a, R)₀ →L[R] A :=
+  { cfcₙHom ha with
+    toFun := cfcₙHom ha
+    map_smul' := map_smul _
+    cont := (cfcₙHom_closedEmbedding ha).continuous }
+
+end cfcₙL
 
 section CFCn
 
@@ -406,7 +421,7 @@ lemma cfcₙ_comp (g f : R → R) (a : A)
     ext
     simp
   rw [cfcₙ_apply .., cfcₙ_apply f a,
-    cfcₙ_apply _ (by convert hg) (ha := cfcₙHom_predicate (show p a from ha) _) ,
+    cfcₙ_apply _ _ (by convert hg) (ha := cfcₙHom_predicate (show p a from ha) _),
     ← cfcₙHom_comp _ _]
   swap
   · exact ⟨.mk _ <| hf.restrict.codRestrict fun x ↦ by rw [sp_eq]; use x.1; simp, Subtype.ext hf0⟩
@@ -458,6 +473,7 @@ lemma CFC.quasispectrum_zero_eq : σₙ R (0 : A) = {0} := by
     simpa [CFC.quasispectrum_zero_eq]
   · exact cfcₙ_apply_of_not_map_zero _ hf0
 
+@[simp]
 instance IsStarNormal.cfcₙ_map (f : R → R) (a : A) : IsStarNormal (cfcₙ f a) where
   star_comm_self := by
     refine cfcₙ_cases (fun x ↦ Commute (star x) x) _ _ (Commute.zero_right _) fun _ _ _ ↦ ?_
@@ -466,6 +482,19 @@ instance IsStarNormal.cfcₙ_map (f : R → R) (a : A) : IsStarNormal (cfcₙ f
     congr! 2
     exact mul_comm _ _
 
+-- The following two lemmas are just `cfcₙ_predicate`, but specific enough for the `@[simp]` tag.
+@[simp]
+protected lemma IsSelfAdjoint.cfcₙ
+    [NonUnitalContinuousFunctionalCalculus R (IsSelfAdjoint : A → Prop)] {f : R → R} {a : A} :
+    IsSelfAdjoint (cfcₙ f a) :=
+  cfcₙ_predicate _ _
+
+@[simp]
+lemma cfcₙ_nonneg_of_predicate [PartialOrder A]
+    [NonUnitalContinuousFunctionalCalculus R (fun (a : A) => 0 ≤ a)] {f : R → R} {a : A} :
+    0 ≤ cfcₙ f a :=
+  cfcₙ_predicate _ _
+
 end CFCn
 
 end Main
@@ -663,7 +692,7 @@ lemma closedEmbedding_cfcₙHom_of_cfcHom {a : A} (ha : p a) :
   let f : C(spectrum R a, σₙ R a) :=
     ⟨_, continuous_inclusion <| spectrum_subset_quasispectrum R a⟩
   refine (cfcHom_closedEmbedding ha).comp <|
-    (UniformInducing.uniformEmbedding ⟨?_⟩).toClosedEmbedding
+    (IsUniformInducing.isUniformEmbedding ⟨?_⟩).toClosedEmbedding
   have := uniformSpace_eq_inf_precomp_of_cover (β := R) f (0 : C(Unit, σₙ R a))
     (map_continuous f).isProperMap (map_continuous 0).isProperMap <| by
       simp only [← Subtype.val_injective.image_injective.eq_iff, f, ContinuousMap.coe_mk,
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
index 2bebd79a58986..8536353d9475b 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
@@ -5,8 +5,9 @@ Authors: Frédéric Dupuis
 -/
 
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
-import Mathlib.Topology.ContinuousFunction.StarOrdered
 import Mathlib.Analysis.CStarAlgebra.Unitization
+import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow
+import Mathlib.Topology.ContinuousMap.StarOrdered
 
 /-! # Facts about star-ordered rings that depend on the continuous functional calculus
 
@@ -23,10 +24,11 @@ the spectral order.
   C⋆-algebra.
 * `mul_star_le_algebraMap_norm_sq` and `star_mul_le_algebraMap_norm_sq`, which give similar
   statements for `a * star a` and `star a * a`.
-* `CStarRing.norm_le_norm_of_nonneg_of_le`: in a non-unital C⋆-algebra, if `0 ≤ a ≤ b`, then
+* `CStarAlgebra.norm_le_norm_of_nonneg_of_le`: in a non-unital C⋆-algebra, if `0 ≤ a ≤ b`, then
   `‖a‖ ≤ ‖b‖`.
-* `CStarRing.conjugate_le_norm_smul`: in a non-unital C⋆-algebra, we have that
+* `CStarAlgebra.conjugate_le_norm_smul`: in a non-unital C⋆-algebra, we have that
   `star a * b * a ≤ ‖b‖ • (star a * a)` (and a primed version for the `a * b * star a` case).
+* `CStarAlgebra.inv_le_inv_iff`: in a unital C⋆-algebra, `b⁻¹ ≤ a⁻¹` iff `a ≤ b`.
 
 ## Tags
 
@@ -41,9 +43,11 @@ variable {A : Type*} [NonUnitalNormedRing A] [CompleteSpace A]
   [PartialOrder A] [StarRing A] [StarOrderedRing A] [CStarRing A] [NormedSpace ℂ A] [StarModule ℂ A]
   [SMulCommClass ℂ A A] [IsScalarTower ℂ A A]
 
-instance instPartialOrder : PartialOrder (Unitization ℂ A) := CStarRing.spectralOrder _
+instance instPartialOrder : PartialOrder (Unitization ℂ A) :=
+    CStarAlgebra.spectralOrder _
 
-instance instStarOrderedRing : StarOrderedRing (Unitization ℂ A) := CStarRing.spectralOrderedRing _
+instance instStarOrderedRing : StarOrderedRing (Unitization ℂ A) :=
+    CStarAlgebra.spectralOrderedRing _
 
 lemma inr_le_iff (a b : A) (ha : IsSelfAdjoint a := by cfc_tac)
     (hb : IsSelfAdjoint b := by cfc_tac) :
@@ -64,6 +68,39 @@ lemma inr_nonneg_iff {a : A} : 0 ≤ (a : Unitization ℂ A) ↔ 0 ≤ a := by
 
 end Unitization
 
+/-- `cfc_le_iff` only applies to a scalar ring where `R` is an actual `Ring`, and not a `Semiring`.
+However, this theorem still holds for `ℝ≥0` as long as the algebra `A` itself is an `ℝ`-algebra. -/
+lemma cfc_nnreal_le_iff {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A]
+    [StarOrderedRing A] [Algebra ℝ A] [TopologicalRing A] [NonnegSpectrumClass ℝ A]
+    [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
+    [UniqueContinuousFunctionalCalculus ℝ A]
+    (f : ℝ≥0 → ℝ≥0) (g : ℝ≥0 → ℝ≥0) (a : A)
+    (ha_spec : SpectrumRestricts a ContinuousMap.realToNNReal)
+    (hf : ContinuousOn f (spectrum ℝ≥0 a) := by cfc_cont_tac)
+    (hg : ContinuousOn g (spectrum ℝ≥0 a) := by cfc_cont_tac)
+    (ha : 0 ≤ a := by cfc_tac) :
+    cfc f a ≤ cfc g a ↔ ∀ x ∈ spectrum ℝ≥0 a, f x ≤ g x := by
+  have hf' := hf.ofReal_map_toNNReal <| ha_spec.image ▸ Set.mapsTo_image ..
+  have hg' := hg.ofReal_map_toNNReal <| ha_spec.image ▸ Set.mapsTo_image ..
+  rw [cfc_nnreal_eq_real, cfc_nnreal_eq_real, cfc_le_iff ..]
+  simp [NNReal.coe_le_coe, ← ha_spec.image]
+
+/-- In a unital `ℝ`-algebra `A` with a continuous functional calculus, an element `a : A` is larger
+than some `algebraMap ℝ A r` if and only if every element of the `ℝ`-spectrum is nonnegative. -/
+lemma CFC.exists_pos_algebraMap_le_iff {A : Type*} [TopologicalSpace A] [Ring A] [StarRing A]
+    [PartialOrder A] [StarOrderedRing A] [Algebra ℝ A] [NonnegSpectrumClass ℝ A]
+    [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
+    {a : A} [CompactSpace (spectrum ℝ a)]
+    (h_non : (spectrum ℝ a).Nonempty) (ha : IsSelfAdjoint a := by cfc_tac) :
+    (∃ r > 0, algebraMap ℝ A r ≤ a) ↔ (∀ x ∈ spectrum ℝ a, 0 < x) := by
+  have h_cpct : IsCompact (spectrum ℝ a) := isCompact_iff_compactSpace.mpr inferInstance
+  simp_rw [algebraMap_le_iff_le_spectrum (a := a)]
+  refine ⟨?_, fun h ↦ ?_⟩
+  · rintro ⟨r, hr, hr_le⟩
+    exact (hr.trans_le <| hr_le · ·)
+  · obtain ⟨r, hr, hr_min⟩ := h_cpct.exists_isMinOn h_non continuousOn_id
+    exact ⟨r, h _ hr, hr_min⟩
+
 section CStar_unital
 
 variable {A : Type*} [NormedRing A] [StarRing A] [CStarRing A] [CompleteSpace A]
@@ -89,11 +126,13 @@ lemma IsSelfAdjoint.neg_algebraMap_norm_le_self {a : A} (ha : IsSelfAdjoint a :=
     exact IsSelfAdjoint.le_algebraMap_norm_self (neg ha)
   exact neg_le.mp this
 
-lemma CStarRing.mul_star_le_algebraMap_norm_sq {a : A} : a * star a ≤ algebraMap ℝ A (‖a‖ ^ 2) := by
+lemma CStarAlgebra.mul_star_le_algebraMap_norm_sq {a : A} :
+    a * star a ≤ algebraMap ℝ A (‖a‖ ^ 2) := by
   have : a * star a ≤ algebraMap ℝ A ‖a * star a‖ := IsSelfAdjoint.le_algebraMap_norm_self
   rwa [CStarRing.norm_self_mul_star, ← pow_two] at this
 
-lemma CStarRing.star_mul_le_algebraMap_norm_sq {a : A} : star a * a ≤ algebraMap ℝ A (‖a‖ ^ 2) := by
+lemma CStarAlgebra.star_mul_le_algebraMap_norm_sq {a : A} :
+    star a * a ≤ algebraMap ℝ A (‖a‖ ^ 2) := by
   have : star a * a ≤ algebraMap ℝ A ‖star a * a‖ := IsSelfAdjoint.le_algebraMap_norm_self
   rwa [CStarRing.norm_star_mul_self, ← pow_two] at this
 
@@ -103,7 +142,9 @@ lemma IsSelfAdjoint.toReal_spectralRadius_eq_norm {a : A} (ha : IsSelfAdjoint a)
     (spectralRadius ℝ a).toReal = ‖a‖ := by
   simp [ha.spectrumRestricts.spectralRadius_eq, ha.spectralRadius_eq_nnnorm]
 
-lemma CStarRing.norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
+namespace CStarAlgebra
+
+lemma norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
     (ha : IsSelfAdjoint a := by cfc_tac) : ‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a := by
   have ha' : SpectrumRestricts a Complex.reCLM := ha.spectrumRestricts
   rw [← ha.toReal_spectralRadius_eq_norm]
@@ -111,16 +152,168 @@ lemma CStarRing.norm_or_neg_norm_mem_spectrum [Nontrivial A] {a : A}
 
 variable [PartialOrder A] [StarOrderedRing A]
 
-lemma CStarRing.nnnorm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
+lemma nnnorm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
     ‖a‖₊ ∈ spectrum ℝ≥0 a := by
   have : IsSelfAdjoint a := .of_nonneg ha
   convert NNReal.spectralRadius_mem_spectrum (a := a) ?_ (.nnreal_of_nonneg ha)
   · simp [this.spectrumRestricts.spectralRadius_eq, this.spectralRadius_eq_nnnorm]
   · exact this.spectrumRestricts.image ▸ (spectrum.nonempty a).image _
 
-lemma CStarRing.norm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
+lemma norm_mem_spectrum_of_nonneg [Nontrivial A] {a : A} (ha : 0 ≤ a := by cfc_tac) :
     ‖a‖ ∈ spectrum ℝ a := by
-  simpa using spectrum.algebraMap_mem ℝ <| CStarRing.nnnorm_mem_spectrum_of_nonneg ha
+  simpa using spectrum.algebraMap_mem ℝ <| nnnorm_mem_spectrum_of_nonneg ha
+
+lemma norm_le_iff_le_algebraMap (a : A) {r : ℝ} (hr : 0 ≤ r) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖ ≤ r ↔ a ≤ algebraMap ℝ A r := by
+  rw [le_algebraMap_iff_spectrum_le]
+  obtain (h | _) := subsingleton_or_nontrivial A
+  · simp [Subsingleton.elim a 0, hr]
+  · exact ⟨fun h x hx ↦ Real.le_norm_self x |>.trans (spectrum.norm_le_norm_of_mem hx) |>.trans h,
+      fun h ↦ h ‖a‖ <| norm_mem_spectrum_of_nonneg⟩
+
+lemma nnnorm_le_iff_of_nonneg (a : A) (r : ℝ≥0) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖₊ ≤ r ↔ a ≤ algebraMap ℝ≥0 A r := by
+  rw [← NNReal.coe_le_coe]
+  exact norm_le_iff_le_algebraMap a r.2
+
+lemma norm_le_one_iff_of_nonneg (a : A) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖ ≤ 1 ↔ a ≤ 1 := by
+  simpa using norm_le_iff_le_algebraMap a zero_le_one
+
+lemma nnnorm_le_one_iff_of_nonneg (a : A) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖₊ ≤ 1 ↔ a ≤ 1 := by
+  rw [← NNReal.coe_le_coe]
+  exact norm_le_one_iff_of_nonneg a
+
+lemma norm_le_natCast_iff_of_nonneg (a : A) (n : ℕ) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖ ≤ n ↔ a ≤ n := by
+  simpa using norm_le_iff_le_algebraMap a n.cast_nonneg
+
+lemma nnnorm_le_natCast_iff_of_nonneg (a : A) (n : ℕ) (ha : 0 ≤ a := by cfc_tac) :
+    ‖a‖₊ ≤ n ↔ a ≤ n := by
+  simpa using nnnorm_le_iff_of_nonneg a n
+
+end CStarAlgebra
+
+section Inv
+
+open CFC
+
+variable [PartialOrder A] [StarOrderedRing A]
+
+-- TODO : relate everything in this section to strict positivity
+
+lemma CFC.conjugate_rpow_neg_one_half {a : A} (h₀ : IsUnit a) (ha : 0 ≤ a := by cfc_tac) :
+    a ^ (-(1 / 2) : ℝ) * a * a ^ (-(1 / 2) : ℝ) = 1 := by
+  lift a to Aˣ using h₀
+  nth_rw 2 [← rpow_one (a : A)]
+  simp only [← rpow_add (a.zero_not_mem_spectrum ℝ≥0)]
+  norm_num
+  exact rpow_zero _
+
+/-- In a unital C⋆-algebra, if `a` is nonnegative and invertible, and `a ≤ b`, then `b` is
+invertible. -/
+lemma CStarAlgebra.isUnit_of_le {a b : A} (h₀ : IsUnit a) (ha : 0 ≤ a := by cfc_tac)
+    (hab : a ≤ b) : IsUnit b := by
+  rw [← spectrum.zero_not_mem_iff ℝ≥0] at h₀ ⊢
+  nontriviality A
+  have hb := (show 0 ≤ a from ha).trans hab
+  have ha' := IsSelfAdjoint.of_nonneg ha |>.spectrum_nonempty
+  have hb' := IsSelfAdjoint.of_nonneg hb |>.spectrum_nonempty
+  rw [zero_not_mem_iff, SpectrumRestricts.nnreal_lt_iff (.nnreal_of_nonneg ‹_›),
+    NNReal.coe_zero, ← CFC.exists_pos_algebraMap_le_iff ‹_›] at h₀ ⊢
+  peel h₀ with r hr _
+  exact this.trans hab
+
+lemma le_iff_norm_sqrt_mul_rpow {a b : A} (hbu : IsUnit b) (ha : 0 ≤ a) (hb : 0 ≤ (b : A)) :
+    a ≤ b ↔ ‖sqrt a * (b : A) ^ (-(1 / 2) : ℝ)‖ ≤ 1 := by
+  lift b to Aˣ using hbu
+  have hbab : 0 ≤ (b : A) ^ (-(1 / 2) : ℝ) * a * (b : A) ^ (-(1 / 2) : ℝ) :=
+    conjugate_nonneg_of_nonneg ha rpow_nonneg
+  conv_rhs =>
+    rw [← sq_le_one_iff (norm_nonneg _), sq, ← CStarRing.norm_star_mul_self, star_mul,
+      IsSelfAdjoint.of_nonneg sqrt_nonneg, IsSelfAdjoint.of_nonneg rpow_nonneg,
+      ← mul_assoc, mul_assoc _ _ (sqrt a), sqrt_mul_sqrt_self a,
+      CStarAlgebra.norm_le_one_iff_of_nonneg _ hbab]
+  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
+  · calc
+      _ ≤ ↑b ^ (-(1 / 2) : ℝ) * (b : A) * ↑b ^ (-(1 / 2) : ℝ) :=
+        IsSelfAdjoint.of_nonneg rpow_nonneg |>.conjugate_le_conjugate h
+      _ = 1 := conjugate_rpow_neg_one_half b.isUnit
+  · calc
+      a = (sqrt ↑b * ↑b ^ (-(1 / 2) : ℝ)) * a * (↑b ^ (-(1 / 2) : ℝ) * sqrt ↑b) := by
+        simp only [CFC.sqrt_eq_rpow .., ← CFC.rpow_add (b.zero_not_mem_spectrum ℝ≥0)]
+        norm_num
+        simp [CFC.rpow_zero (b : A)]
+      _ = sqrt ↑b * (↑b ^ (-(1 / 2) : ℝ) * a * ↑b ^ (-(1 / 2) : ℝ)) * sqrt ↑b := by
+        simp only [mul_assoc]
+      _ ≤ b := conjugate_le_conjugate_of_nonneg h sqrt_nonneg |>.trans <| by
+        simp [CFC.sqrt_mul_sqrt_self (b : A)]
+
+lemma le_iff_norm_sqrt_mul_sqrt_inv {a : A} {b : Aˣ} (ha : 0 ≤ a) (hb : 0 ≤ (b : A)) :
+    a ≤ b ↔ ‖sqrt a * sqrt (↑b⁻¹ : A)‖ ≤ 1 := by
+  rw [CFC.sqrt_eq_rpow (a := (↑b⁻¹ : A)), ← CFC.rpow_neg_one_eq_inv b,
+    CFC.rpow_rpow (b : A) _ _ (by simp) (by norm_num), le_iff_norm_sqrt_mul_rpow b.isUnit ha hb]
+  norm_num
+
+namespace CStarAlgebra
+
+/-- In a unital C⋆-algebra, if `0 ≤ a ≤ b` and `a` and `b` are units, then `b⁻¹ ≤ a⁻¹`. -/
+protected lemma inv_le_inv {a b : Aˣ} (ha : 0 ≤ (a : A))
+    (hab : (a : A) ≤ b) : (↑b⁻¹ : A) ≤ a⁻¹ := by
+  have hb := ha.trans hab
+  have hb_inv : (0 : A) ≤ b⁻¹ := inv_nonneg_of_nonneg b hb
+  have ha_inv : (0 : A) ≤ a⁻¹ := inv_nonneg_of_nonneg a ha
+  rw [le_iff_norm_sqrt_mul_sqrt_inv ha hb, ← sq_le_one_iff (norm_nonneg _), sq,
+    ← CStarRing.norm_star_mul_self] at hab
+  rw [le_iff_norm_sqrt_mul_sqrt_inv hb_inv ha_inv, inv_inv, ← sq_le_one_iff (norm_nonneg _), sq,
+    ← CStarRing.norm_self_mul_star]
+  rwa [star_mul, IsSelfAdjoint.of_nonneg sqrt_nonneg,
+    IsSelfAdjoint.of_nonneg sqrt_nonneg] at hab ⊢
+
+/-- In a unital C⋆-algebra, if `0 ≤ a` and `0 ≤ b` and `a` and `b` are units, then `a⁻¹ ≤ b⁻¹`
+if and only if `b ≤ a`. -/
+protected lemma inv_le_inv_iff {a b : Aˣ} (ha : 0 ≤ (a : A)) (hb : 0 ≤ (b : A)) :
+    (↑a⁻¹ : A) ≤ b⁻¹ ↔ (b : A) ≤ a :=
+  ⟨CStarAlgebra.inv_le_inv (inv_nonneg_of_nonneg a ha), CStarAlgebra.inv_le_inv hb⟩
+
+lemma inv_le_iff {a b : Aˣ} (ha : 0 ≤ (a : A)) (hb : 0 ≤ (↑b : A)) :
+    (↑a⁻¹ : A) ≤ b ↔ (↑b⁻¹ : A) ≤ a := by
+  simpa using CStarAlgebra.inv_le_inv_iff ha (inv_nonneg_of_nonneg b hb)
+
+lemma le_inv_iff {a b : Aˣ} (ha : 0 ≤ (a : A)) (hb : 0 ≤ (↑b : A)) :
+    a ≤ (↑b⁻¹ : A) ↔ b ≤ (↑a⁻¹ : A) := by
+  simpa using CStarAlgebra.inv_le_inv_iff (inv_nonneg_of_nonneg a ha) hb
+
+lemma one_le_inv_iff_le_one {a : Aˣ} (ha : 0 ≤ (a : A)) :
+    1 ≤ (↑a⁻¹ : A) ↔ a ≤ 1 := by
+  simpa using CStarAlgebra.le_inv_iff (a := 1) (by simp) ha
+
+lemma inv_le_one_iff_one_le {a : Aˣ} (ha : 0 ≤ (a : A)) :
+    (↑a⁻¹ : A) ≤ 1 ↔ 1 ≤ a := by
+  simpa using CStarAlgebra.inv_le_iff ha (b := 1) (by simp)
+
+lemma inv_le_one {a : Aˣ} (ha : 1 ≤ a) : (↑a⁻¹ : A) ≤ 1 :=
+  CStarAlgebra.inv_le_one_iff_one_le (zero_le_one.trans ha) |>.mpr ha
+
+lemma le_one_of_one_le_inv {a : Aˣ} (ha : 1 ≤ (↑a⁻¹ : A)) : (a : A) ≤ 1 := by
+  simpa using CStarAlgebra.inv_le_one ha
+
+lemma rpow_neg_one_le_rpow_neg_one {a b : A} (ha : 0 ≤ a) (hab : a ≤ b) (hau : IsUnit a) :
+    b ^ (-1 : ℝ) ≤ a ^ (-1 : ℝ) := by
+  lift b to Aˣ using isUnit_of_le hau ha hab
+  lift a to Aˣ using hau
+  rw [rpow_neg_one_eq_inv a ha, rpow_neg_one_eq_inv b (ha.trans hab)]
+  exact CStarAlgebra.inv_le_inv ha hab
+
+lemma rpow_neg_one_le_one {a : A} (ha : 1 ≤ a) : a ^ (-1 : ℝ) ≤ 1 := by
+  lift a to Aˣ using isUnit_of_le isUnit_one zero_le_one ha
+  rw [rpow_neg_one_eq_inv a (zero_le_one.trans ha)]
+  exact inv_le_one ha
+
+end CStarAlgebra
+
+end Inv
 
 end CStar_unital
 
@@ -130,13 +323,15 @@ variable {A : Type*} [NonUnitalNormedRing A] [CompleteSpace A] [PartialOrder A]
   [StarOrderedRing A] [CStarRing A] [NormedSpace ℂ A] [StarModule ℂ A]
   [SMulCommClass ℂ A A] [IsScalarTower ℂ A A]
 
+namespace CStarAlgebra
+
 open ComplexOrder in
-instance CStarRing.instNonnegSpectrumClassComplexNonUnital : NonnegSpectrumClass ℂ A where
+instance instNonnegSpectrumClassComplexNonUnital : NonnegSpectrumClass ℂ A where
   quasispectrum_nonneg_of_nonneg a ha x hx := by
     rw [Unitization.quasispectrum_eq_spectrum_inr' ℂ ℂ a] at hx
     exact spectrum_nonneg_of_nonneg (Unitization.inr_nonneg_iff.mpr ha) hx
 
-lemma CStarRing.norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b) :
+lemma norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_tac) (hab : a ≤ b) :
     ‖a‖ ≤ ‖b‖ := by
   suffices ∀ a b : Unitization ℂ A, 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖ by
     have hb := ha.trans hab
@@ -156,7 +351,7 @@ lemma CStarRing.norm_le_norm_of_nonneg_of_le {a b : A} (ha : 0 ≤ a := by cfc_t
   rw [cfc_le_iff id (fun _ => ‖b‖) a] at h₂
   exact h₂ ‖a‖ <| norm_mem_spectrum_of_nonneg ha
 
-lemma CStarRing.conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
+lemma conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
     star a * b * a ≤ ‖b‖ • (star a * a) := by
   suffices ∀ a b : Unitization ℂ A, IsSelfAdjoint b → star a * b * a ≤ ‖b‖ • (star a * a) by
     rw [← Unitization.inr_le_iff _ _ (by aesop) ((IsSelfAdjoint.all _).smul (.star_mul_self a))]
@@ -167,11 +362,27 @@ lemma CStarRing.conjugate_le_norm_smul {a b : A} (hb : IsSelfAdjoint b := by cfc
       conjugate_le_conjugate hb.le_algebraMap_norm_self _
     _ = ‖b‖ • (star a * a) := by simp [Algebra.algebraMap_eq_smul_one]
 
-lemma CStarRing.conjugate_le_norm_smul' {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
+lemma conjugate_le_norm_smul' {a b : A} (hb : IsSelfAdjoint b := by cfc_tac) :
     a * b * star a ≤ ‖b‖ • (a * star a) := by
   have h₁ : a * b * star a = star (star a) * b * star a := by simp
   have h₂ : a * star a = star (star a) * star a := by simp
   simp only [h₁, h₂]
   exact conjugate_le_norm_smul
 
+/-- The set of nonnegative elements in a C⋆-algebra is closed. -/
+lemma isClosed_nonneg : IsClosed {a : A | 0 ≤ a} := by
+  suffices IsClosed {a : Unitization ℂ A | 0 ≤ a} by
+    rw [Unitization.isometry_inr (𝕜 := ℂ) |>.closedEmbedding.closed_iff_image_closed]
+    convert this.inter <| (Unitization.isometry_inr (𝕜 := ℂ)).closedEmbedding.isClosed_range
+    ext a
+    simp only [Set.mem_image, Set.mem_setOf_eq, Set.mem_inter_iff, Set.mem_range, ← exists_and_left]
+    congr! 2 with x
+    exact and_congr_left fun h ↦ by simp [← h]
+  simp only [nonneg_iff_isSelfAdjoint_and_spectrumRestricts,
+    and_congr_right (SpectrumRestricts.nnreal_iff_nnnorm · le_rfl), Set.setOf_and]
+  refine isClosed_eq ?_ ?_ |>.inter <| isClosed_le ?_ ?_
+  all_goals fun_prop
+
+end CStarAlgebra
+
 end CStar_nonunital
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
index 2f9adf1d2187e..cf36028a7922e 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Restrict.lean
@@ -84,17 +84,17 @@ variable [CompleteSpace R]
 
 lemma closedEmbedding_starAlgHom {a : A} {φ : C(spectrum S a, S) →⋆ₐ[S] A}
     (hφ : ClosedEmbedding φ) {f : C(S, R)} (h : SpectrumRestricts a f)
-    (halg : UniformEmbedding (algebraMap R S)) :
+    (halg : IsUniformEmbedding (algebraMap R S)) :
     ClosedEmbedding (h.starAlgHom φ) :=
-  hφ.comp <| UniformEmbedding.toClosedEmbedding <| .comp
-    (ContinuousMap.uniformEmbedding_comp _ halg)
-    (UniformEquiv.arrowCongr h.homeomorph.symm (.refl _) |>.uniformEmbedding)
+  hφ.comp <| IsUniformEmbedding.toClosedEmbedding <| .comp
+    (ContinuousMap.isUniformEmbedding_comp _ halg)
+    (UniformEquiv.arrowCongr h.homeomorph.symm (.refl _) |>.isUniformEmbedding)
 
 /-- Given a `ContinuousFunctionalCalculus S q`. If we form the predicate `p` for `a : A`
 characterized by: `q a` and the spectrum of `a` restricts to the scalar subring `R` via
 `f : C(S, R)`, then we can get a restricted functional calculus
 `ContinuousFunctionalCalculus R p`. -/
-protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (h0 : p 0)
+protected theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
     (h : ∀ a, p a ↔ q a ∧ SpectrumRestricts a f) :
     ContinuousFunctionalCalculus R p where
   predicate_zero := h0
@@ -133,14 +133,14 @@ protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (
 
 variable [ContinuousFunctionalCalculus R p] [UniqueContinuousFunctionalCalculus R A]
 
-lemma cfcHom_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S))
+lemma cfcHom_eq_restrict (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S))
     {a : A} (hpa : p a) (hqa : q a) (h : SpectrumRestricts a f) :
     cfcHom hpa = h.starAlgHom (cfcHom hqa) := by
   apply cfcHom_eq_of_continuous_of_map_id
   · exact h.closedEmbedding_starAlgHom (cfcHom_closedEmbedding hqa) halg |>.continuous
   · exact h.starAlgHom_id (cfcHom_id hqa)
 
-lemma cfc_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A} (hpa : p a)
+lemma cfc_eq_restrict (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) {a : A} (hpa : p a)
     (hqa : q a) (h : SpectrumRestricts a f) (g : R → R) :
     cfc g a = cfc (fun x ↦ algebraMap R S (g (f x))) a := by
   by_cases hg : ContinuousOn g (spectrum R a)
@@ -152,7 +152,7 @@ lemma cfc_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {
   · have : ¬ ContinuousOn (fun x ↦ algebraMap R S (g (f x)) : S → S) (spectrum S a) := by
       refine fun hg' ↦ hg ?_
       rw [halg.embedding.continuousOn_iff]
-      simpa [halg.embedding.continuousOn_iff, Function.comp, h.left_inv _] using
+      simpa [halg.embedding.continuousOn_iff, Function.comp_def, h.left_inv _] using
         hg'.comp halg.embedding.continuous.continuousOn (fun _ : R ↦ spectrum.algebraMap_mem S)
     rw [cfc_apply_of_not_continuousOn a hg, cfc_apply_of_not_continuousOn a this]
 
@@ -218,12 +218,12 @@ variable [CompleteSpace R]
 
 lemma closedEmbedding_nonUnitalStarAlgHom {a : A} {φ : C(σₙ S a, S)₀ →⋆ₙₐ[S] A}
     (hφ : ClosedEmbedding φ) {f : C(S, R)} (h : QuasispectrumRestricts a f)
-    (halg : UniformEmbedding (algebraMap R S)) :
+    (halg : IsUniformEmbedding (algebraMap R S)) :
     ClosedEmbedding (h.nonUnitalStarAlgHom φ) := by
   have : h.homeomorph.symm 0 = 0 := Subtype.ext (map_zero <| algebraMap _ _)
-  refine hφ.comp <| UniformEmbedding.toClosedEmbedding <| .comp
-    (ContinuousMapZero.uniformEmbedding_comp _ halg)
-    (UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.uniformEmbedding)
+  refine hφ.comp <| IsUniformEmbedding.toClosedEmbedding <| .comp
+    (ContinuousMapZero.isUniformEmbedding_comp _ halg)
+    (UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.isUniformEmbedding)
 
 variable [IsScalarTower R A A] [SMulCommClass R A A]
 
@@ -231,7 +231,7 @@ variable [IsScalarTower R A A] [SMulCommClass R A A]
 characterized by: `q a` and the quasispectrum of `a` restricts to the scalar subring `R` via
 `f : C(S, R)`, then we can get a restricted functional calculus
 `NonUnitalContinuousFunctionalCalculus R p`. -/
-protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (h0 : p 0)
+protected theorem cfc (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) (h0 : p 0)
     (h : ∀ a, p a ↔ q a ∧ QuasispectrumRestricts a f) :
     NonUnitalContinuousFunctionalCalculus R p where
   predicate_zero := h0
@@ -275,15 +275,15 @@ protected theorem cfc (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) (
 variable [NonUnitalContinuousFunctionalCalculus R p]
 variable [UniqueNonUnitalContinuousFunctionalCalculus R A]
 
-lemma cfcₙHom_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A}
+lemma cfcₙHom_eq_restrict (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) {a : A}
     (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a f) :
     cfcₙHom hpa = h.nonUnitalStarAlgHom (cfcₙHom hqa) := by
   apply cfcₙHom_eq_of_continuous_of_map_id
   · exact h.closedEmbedding_nonUnitalStarAlgHom (cfcₙHom_closedEmbedding hqa) halg |>.continuous
   · exact h.nonUnitalStarAlgHom_id (cfcₙHom_id hqa)
 
-lemma cfcₙ_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)) {a : A} (hpa : p a)
-    (hqa : q a) (h : QuasispectrumRestricts a f) (g : R → R) :
+lemma cfcₙ_eq_restrict (f : C(S, R)) (halg : IsUniformEmbedding (algebraMap R S)) {a : A}
+    (hpa : p a) (hqa : q a) (h : QuasispectrumRestricts a f) (g : R → R) :
     cfcₙ g a = cfcₙ (fun x ↦ algebraMap R S (g (f x))) a := by
   by_cases hg : ContinuousOn g (σₙ R a) ∧ g 0 = 0
   · obtain ⟨hg, hg0⟩ := hg
@@ -297,7 +297,7 @@ lemma cfcₙ_eq_restrict (f : C(S, R)) (halg : UniformEmbedding (algebraMap R S)
     · have : ¬ ContinuousOn (fun x ↦ algebraMap R S (g (f x)) : S → S) (σₙ S a) := by
         refine fun hg' ↦ hg ?_
         rw [halg.embedding.continuousOn_iff]
-        simpa [halg.embedding.continuousOn_iff, Function.comp, h.left_inv _] using
+        simpa [halg.embedding.continuousOn_iff, Function.comp_def, h.left_inv _] using
           hg'.comp halg.embedding.continuous.continuousOn
           (fun _ : R ↦ quasispectrum.algebraMap_mem S)
       rw [cfcₙ_apply_of_not_continuousOn a hg, cfcₙ_apply_of_not_continuousOn a this]
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.lean
index 830edf4006647..155a1b6dd519c 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unique.lean
@@ -5,7 +5,7 @@ Authors: Jireh Loreaux
 -/
 import Mathlib.Analysis.Normed.Algebra.Spectrum
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital
-import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
+import Mathlib.Topology.ContinuousMap.StoneWeierstrass
 
 /-!
 # Uniqueness of the continuous functional calculus
@@ -420,3 +420,103 @@ instance NNReal.instUniqueNonUnitalContinuousFunctionalCalculus
 end NNReal
 
 end UniqueNonUnital
+
+section NonUnitalStarAlgHom
+
+open scoped ContinuousMapZero
+
+variable {F R S A B : Type*} {p : A → Prop} {q : B → Prop}
+  [CommSemiring R] [Nontrivial R] [StarRing R] [MetricSpace R] [TopologicalSemiring R]
+  [ContinuousStar R] [CommRing S] [Algebra R S]
+  [NonUnitalRing A] [StarRing A] [TopologicalSpace A] [Module R A]
+  [IsScalarTower R A A] [SMulCommClass R A A]
+  [NonUnitalRing B] [StarRing B] [TopologicalSpace B] [Module R B]
+  [IsScalarTower R B B] [SMulCommClass R B B]
+  [Module S A] [Module S B] [IsScalarTower R S A] [IsScalarTower R S B]
+  [NonUnitalContinuousFunctionalCalculus R p] [NonUnitalContinuousFunctionalCalculus R q]
+  [UniqueNonUnitalContinuousFunctionalCalculus R B] [FunLike F A B] [NonUnitalAlgHomClass F S A B]
+  [StarHomClass F A B]
+
+include S in
+/-- Non-unital star algebra homomorphisms commute with the non-unital continuous functional
+calculus. -/
+lemma NonUnitalStarAlgHomClass.map_cfcₙ (φ : F) (f : R → R) (a : A)
+    [CompactSpace (quasispectrum R a)] (hf : ContinuousOn f (quasispectrum R a) := by cfc_cont_tac)
+    (hf₀ : f 0 = 0 := by cfc_zero_tac) (hφ : Continuous φ := by fun_prop) (ha : p a := by cfc_tac)
+    (hφa : q (φ a) := by cfc_tac) : φ (cfcₙ f a) = cfcₙ f (φ a) := by
+  let ψ : A →⋆ₙₐ[R] B := (φ : A →⋆ₙₐ[S] B).restrictScalars R
+  have : Continuous ψ := hφ
+  have h_spec := NonUnitalAlgHom.quasispectrum_apply_subset' (R := R) S φ a
+  have hψa : q (ψ a) := hφa
+  let ι : C(quasispectrum R (ψ a), quasispectrum R a)₀ :=
+    ⟨⟨Set.inclusion h_spec, continuous_id.subtype_map h_spec⟩, rfl⟩
+  suffices ψ.comp (cfcₙHom ha) =
+      (cfcₙHom hψa).comp (ContinuousMapZero.nonUnitalStarAlgHom_precomp R ι) by
+    have hf' : ContinuousOn f (quasispectrum R (ψ a)) := hf.mono h_spec
+    rw [cfcₙ_apply .., cfcₙ_apply ..]
+    exact DFunLike.congr_fun this _
+  refine UniqueNonUnitalContinuousFunctionalCalculus.eq_of_continuous_of_map_id _ rfl _ _
+    ?_ ?_ ?apply_id
+  case apply_id =>
+    trans cfcₙHom hψa ⟨.restrict (quasispectrum R (ψ a)) (.id R), rfl⟩
+    · simp [cfcₙHom_id]
+    · congr
+  all_goals
+    simp [ContinuousMapZero.nonUnitalStarAlgHom_precomp]
+    fun_prop
+
+/-- Non-unital star algebra homomorphisms commute with the non-unital continuous functional
+calculus.  This version is specialized to `A →⋆ₙₐ[S] B` to allow for dot notation. -/
+lemma NonUnitalStarAlgHom.map_cfcₙ (φ : A →⋆ₙₐ[S] B) (f : R → R) (a : A)
+    [CompactSpace (quasispectrum R a)] (hf : ContinuousOn f (quasispectrum R a) := by cfc_cont_tac)
+    (hf₀ : f 0 = 0 := by cfc_zero_tac) (hφ : Continuous φ := by fun_prop) (ha : p a := by cfc_tac)
+    (hφa : q (φ a) := by cfc_tac) : φ (cfcₙ f a) = cfcₙ f (φ a) :=
+  NonUnitalStarAlgHomClass.map_cfcₙ φ f a
+
+end NonUnitalStarAlgHom
+
+section StarAlgHom
+
+variable {F R S A B : Type*} {p : A → Prop} {q : B → Prop}
+  [CommSemiring R] [StarRing R] [MetricSpace R] [TopologicalSemiring R] [ContinuousStar R]
+  [Ring A] [StarRing A] [TopologicalSpace A] [Algebra R A]
+  [Ring B] [StarRing B] [TopologicalSpace B] [Algebra R B]
+  [CommSemiring S] [Algebra R S] [Algebra S A] [Algebra S B] [IsScalarTower R S A]
+  [IsScalarTower R S B] [ContinuousFunctionalCalculus R p] [ContinuousFunctionalCalculus R q]
+  [UniqueContinuousFunctionalCalculus R B] [FunLike F A B] [AlgHomClass F S A B]
+  [StarHomClass F A B]
+
+include S in
+/-- Star algebra homomorphisms commute with the continuous functional calculus. -/
+lemma StarAlgHomClass.map_cfc (φ : F) (f : R → R) (a : A)
+    [CompactSpace (spectrum R a)] (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac)
+    (hφ : Continuous φ := by fun_prop) (ha : p a := by cfc_tac) (hφa : q (φ a) := by cfc_tac) :
+    φ (cfc f a) = cfc f (φ a) := by
+  let ψ : A →⋆ₐ[R] B := (φ : A →⋆ₐ[S] B).restrictScalars R
+  have : Continuous ψ := hφ
+  have h_spec := AlgHom.spectrum_apply_subset ψ a
+  have hψa : q (ψ a) := hφa
+  let ι : C(spectrum R (ψ a), spectrum R a) :=
+    ⟨Set.inclusion h_spec, continuous_id.subtype_map h_spec⟩
+  suffices ψ.comp (cfcHom ha) = (cfcHom hψa).comp (ContinuousMap.compStarAlgHom' R R ι) by
+    have hf' : ContinuousOn f (spectrum R (ψ a)) := hf.mono h_spec
+    rw [cfc_apply .., cfc_apply ..]
+    congrm($(this) ⟨_, hf.restrict⟩)
+  refine UniqueContinuousFunctionalCalculus.eq_of_continuous_of_map_id _ _ _ ?_ ?_ ?apply_id
+  case apply_id =>
+    trans cfcHom hψa (.restrict (spectrum R (ψ a)) (.id R))
+    · simp [cfcHom_id]
+    · congr
+  all_goals
+    simp [ContinuousMap.compStarAlgHom']
+    fun_prop
+
+/-- Star algebra homomorphisms commute with the continuous functional calculus.
+This version is specialized to `A →⋆ₐ[S] B` to allow for dot notation. -/
+lemma StarAlgHom.map_cfc (φ : A →⋆ₐ[S] B) (f : R → R) (a : A) [CompactSpace (spectrum R a)]
+    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (hφ : Continuous φ := by fun_prop)
+    (ha : p a := by cfc_tac) (hφa : q (φ a) := by cfc_tac) :
+    φ (cfc f a) = cfc f (φ a) :=
+  StarAlgHomClass.map_cfc φ f a
+
+end StarAlgHom
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean
index 804929b679d15..8b6473cba0de3 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unital.lean
@@ -5,9 +5,9 @@ Authors: Jireh Loreaux
 -/
 import Mathlib.Algebra.Algebra.Quasispectrum
 import Mathlib.Algebra.Algebra.Spectrum
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Topology.Algebra.Polynomial
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Tactic.ContinuousFunctionalCalculus
 
 /-!
@@ -107,7 +107,7 @@ b = cfc id b = cfc (NNReal.sqrt ∘ (· ^ 2)) b =
 + `cfc : (R → R) → A → A`: an unbundled version of `cfcHom` which takes the junk value `0` when
   `cfcHom` is not defined.
 + `cfcUnits`: builds a unit from `cfc f a` when `f` is nonzero and continuous on the
-  specturm of `a`.
+  spectrum of `a`.
 
 ## Main theorems
 
@@ -225,6 +225,10 @@ lemma cfcHom_closedEmbedding :
     ClosedEmbedding <| (cfcHom ha : C(spectrum R a, R) →⋆ₐ[R] A) :=
   (ContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.1
 
+@[fun_prop]
+lemma cfcHom_continuous : Continuous (cfcHom ha : C(spectrum R a, R) →⋆ₐ[R] A) :=
+  cfcHom_closedEmbedding ha |>.continuous
+
 lemma cfcHom_id :
     cfcHom ha ((ContinuousMap.id R).restrict <| spectrum R a) = a :=
   (ContinuousFunctionalCalculus.exists_cfc_of_predicate a ha).choose_spec.2.1
@@ -580,7 +584,7 @@ lemma cfc_comp_smul {S : Type*} [SMul S R] [ContinuousConstSMul S R] [SMulZeroCl
   rw [cfc_comp' .., cfc_smul_id ..]
 
 lemma cfc_comp_const_mul (r : R) (f : R → R) (a : A)
-    (hf : ContinuousOn f ((r * ·) '' (spectrum R a)) := by cfc_cont_tac)  (ha : p a := by cfc_tac) :
+    (hf : ContinuousOn f ((r * ·) '' (spectrum R a)) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
     cfc (f <| r * ·) a = cfc f (r • a) := by
   rw [cfc_comp' .., cfc_const_mul_id ..]
 
@@ -653,6 +657,7 @@ lemma cfc_algebraMap (r : R) (f : R → R) : cfc f (algebraMap R A r) = algebraM
 @[simp] lemma cfc_apply_one {f : R → R} : cfc f (1 : A) = algebraMap R A (f 1) := by
   simpa using cfc_algebraMap (A := A) 1 f
 
+@[simp]
 instance IsStarNormal.cfc_map (f : R → R) (a : A) : IsStarNormal (cfc f a) where
   star_comm_self := by
     rw [Commute, SemiconjBy]
@@ -662,6 +667,27 @@ instance IsStarNormal.cfc_map (f : R → R) (a : A) : IsStarNormal (cfc f a) whe
       exact mul_comm _ _
     · simp [cfc_apply_of_not_continuousOn a h]
 
+-- The following two lemmas are just `cfc_predicate`, but specific enough for the `@[simp]` tag.
+@[simp]
+protected lemma IsSelfAdjoint.cfc [ContinuousFunctionalCalculus R (IsSelfAdjoint : A → Prop)]
+    {f : R → R} {a : A} : IsSelfAdjoint (cfc f a) :=
+  cfc_predicate _ _
+
+@[simp]
+lemma cfc_nonneg_of_predicate [PartialOrder A]
+    [ContinuousFunctionalCalculus R (fun (a : A) => 0 ≤ a)] {f : R → R} {a : A} : 0 ≤ cfc f a :=
+  cfc_predicate _ _
+
+variable (R) in
+/-- In an `R`-algebra with a continuous functional calculus, every element satisfying the predicate
+has nonempty `R`-spectrum. -/
+lemma CFC.spectrum_nonempty [Nontrivial A] (a : A) (ha : p a := by cfc_tac) :
+    (spectrum R a).Nonempty := by
+  by_contra! h
+  apply one_ne_zero (α := A)
+  rw [← cfc_one R a, ← cfc_zero R a]
+  exact cfc_congr fun x hx ↦ by simp_all
+
 end CFC
 
 end Basic
@@ -697,7 +723,7 @@ noncomputable def cfcUnits (hf' : ∀ x ∈ spectrum R a, f x ≠ 0)
 lemma cfcUnits_pow (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℕ)
     (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
     (cfcUnits f a hf') ^ n =
-      cfcUnits (forall₂_imp (fun _ _ ↦ pow_ne_zero n) hf') (hf := hf.pow n) := by
+      cfcUnits _ _ (forall₂_imp (fun _ _ ↦ pow_ne_zero n) hf') (hf := hf.pow n) := by
   ext
   cases n with
   | zero => simp [cfc_const_one R a]
@@ -752,7 +778,7 @@ lemma cfcUnits_zpow (hf' : ∀ x ∈ spectrum R a, f x ≠ 0) (n : ℤ)
   | negSucc n =>
     simp only [zpow_negSucc, ← inv_pow]
     ext
-    exact cfc_pow (hf := hf.inv₀ hf') _ |>.symm
+    exact cfc_pow (hf := hf.inv₀ hf') .. |>.symm
 
 lemma cfc_zpow (a : Aˣ) (n : ℤ) (ha : p a := by cfc_tac) :
     cfc (fun x : R ↦ x ^ n) (a : A) = ↑(a ^ n) := by
@@ -898,6 +924,14 @@ lemma one_le_cfc (f : R → R) (a : A) (h : ∀ x ∈ spectrum R a, 1 ≤ f x)
     1 ≤ cfc f a := by
   simpa using algebraMap_le_cfc f 1 a h
 
+lemma CFC.le_one {a : A} (h : ∀ x ∈ spectrum R a, x ≤ 1) (ha : p a := by cfc_tac) :
+    a ≤ 1 := by
+  simpa using le_algebraMap_of_spectrum_le h
+
+lemma CFC.one_le {a : A} (h : ∀ x ∈ spectrum R a, 1 ≤ x) (ha : p a := by cfc_tac) :
+    1 ≤ a := by
+  simpa using algebraMap_le_of_le_spectrum h
+
 end Semiring
 
 section NNReal
@@ -921,7 +955,7 @@ variable {R A : Type*} {p : A → Prop} [OrderedCommRing R] [StarRing R]
 variable [MetricSpace R] [TopologicalRing R] [ContinuousStar R]
 variable [∀ (α) [TopologicalSpace α], StarOrderedRing C(α, R)]
 variable [TopologicalSpace A] [Ring A] [StarRing A] [PartialOrder A] [StarOrderedRing A]
-variable [Algebra R A] [ContinuousFunctionalCalculus R p]
+variable [Algebra R A] [instCFC : ContinuousFunctionalCalculus R p]
 variable [NonnegSpectrumClass R A]
 
 lemma cfcHom_le_iff {a : A} (ha : p a) {f g : C(spectrum R a, R)} :
@@ -939,6 +973,44 @@ lemma cfc_nonpos_iff (f : R → R) (a : A) (hf : ContinuousOn f (spectrum R a) :
   simp_rw [← neg_nonneg, ← cfc_neg]
   exact cfc_nonneg_iff (fun x ↦ -f x) a
 
+lemma cfc_le_algebraMap_iff (f : R → R) (r : R) (a : A)
+    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
+    cfc f a ≤ algebraMap R A r ↔ ∀ x ∈ spectrum R a, f x ≤ r := by
+  rw [← cfc_const r a, cfc_le_iff ..]
+
+lemma algebraMap_le_cfc_iff (f : R → R) (r : R) (a : A)
+    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
+    algebraMap R A r ≤ cfc f a ↔ ∀ x ∈ spectrum R a, r ≤ f x := by
+  rw [← cfc_const r a, cfc_le_iff ..]
+
+lemma le_algebraMap_iff_spectrum_le {r : R} {a : A} (ha : p a := by cfc_tac) :
+    a ≤ algebraMap R A r ↔ ∀ x ∈ spectrum R a, x ≤ r := by
+  nth_rw 1 [← cfc_id R a]
+  exact cfc_le_algebraMap_iff id r a
+
+lemma algebraMap_le_iff_le_spectrum {r : R} {a : A} (ha : p a := by cfc_tac) :
+    algebraMap R A r ≤ a ↔ ∀ x ∈ spectrum R a, r ≤ x:= by
+  nth_rw 1 [← cfc_id R a]
+  exact algebraMap_le_cfc_iff id r a
+
+lemma cfc_le_one_iff (f : R → R) (a : A)
+    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
+    cfc f a ≤ 1 ↔ ∀ x ∈ spectrum R a, f x ≤ 1 := by
+  simpa using cfc_le_algebraMap_iff f 1 a
+
+lemma one_le_cfc_iff (f : R → R) (a : A)
+    (hf : ContinuousOn f (spectrum R a) := by cfc_cont_tac) (ha : p a := by cfc_tac) :
+    1 ≤ cfc f a ↔ ∀ x ∈ spectrum R a, 1 ≤ f x := by
+  simpa using algebraMap_le_cfc_iff f 1 a
+
+lemma CFC.le_one_iff (a : A) (ha : p a := by cfc_tac) :
+    a ≤ 1 ↔ ∀ x ∈ spectrum R a, x ≤ 1 := by
+  simpa using le_algebraMap_iff_spectrum_le (r := (1 : R)) (a := a)
+
+lemma CFC.one_le_iff (a : A) (ha : p a := by cfc_tac) :
+    1 ≤ a ↔ ∀ x ∈ spectrum R a, 1 ≤ x := by
+  simpa using algebraMap_le_iff_le_spectrum (r := (1 : R)) (a := a)
+
 end Ring
 
 end Order
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unitary.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unitary.lean
index d39bee3d59145..3ff6426b45606 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unitary.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Unitary.lean
@@ -43,8 +43,7 @@ lemma unitary_iff_isStarNormal_and_spectrum_subset_unitary {u : A} :
   refine and_congr_right fun hu ↦ ?_
   nth_rw 1 [← cfc_id ℂ u]
   rw [cfc_unitary_iff id u, Set.subset_def]
-  congr! with x -
-  simp [unitary.mem_iff_star_mul_self]
+  simp only [id_eq, RCLike.star_def, SetLike.mem_coe, unitary.mem_iff_star_mul_self]
 
 lemma mem_unitary_of_spectrum_subset_unitary {u : A}
     [IsStarNormal u] (hu : spectrum ℂ u ⊆ unitary ℂ) : u ∈ unitary A :=
diff --git a/Mathlib/Analysis/CStarAlgebra/GelfandDuality.lean b/Mathlib/Analysis/CStarAlgebra/GelfandDuality.lean
index c3e2791a52a09..af17ea013e3fc 100644
--- a/Mathlib/Analysis/CStarAlgebra/GelfandDuality.lean
+++ b/Mathlib/Analysis/CStarAlgebra/GelfandDuality.lean
@@ -6,11 +6,11 @@ Authors: Jireh Loreaux
 import Mathlib.Analysis.CStarAlgebra.Spectrum
 import Mathlib.Analysis.Normed.Group.Quotient
 import Mathlib.Analysis.Normed.Algebra.Basic
-import Mathlib.Topology.ContinuousFunction.Units
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.Units
+import Mathlib.Topology.ContinuousMap.Compact
 import Mathlib.Topology.Algebra.Algebra
-import Mathlib.Topology.ContinuousFunction.Ideals
-import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
+import Mathlib.Topology.ContinuousMap.Ideals
+import Mathlib.Topology.ContinuousMap.StoneWeierstrass
 
 /-!
 # Gelfand Duality
diff --git a/Mathlib/Analysis/CStarAlgebra/Hom.lean b/Mathlib/Analysis/CStarAlgebra/Hom.lean
new file mode 100644
index 0000000000000..1f062615cf409
--- /dev/null
+++ b/Mathlib/Analysis/CStarAlgebra/Hom.lean
@@ -0,0 +1,86 @@
+/-
+Copyright (c) 2024 Jireh Loreaux. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jireh Loreaux
+-/
+
+import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order
+
+/-! # Properties of C⋆-algebra homomorphisms
+
+Here we collect properties of C⋆-algebra homomorphisms.
+
+## Main declarations
+
++ `NonUnitalStarAlgHom.norm_map`: A non-unital star algebra monomorphism of complex C⋆-algebras
+  is isometric.
+-/
+
+open CStarAlgebra in
+lemma IsSelfAdjoint.map_spectrum_real {F A B : Type*}
+    [NormedRing A] [CompleteSpace A] [StarRing A] [CStarRing A]
+    [NormedAlgebra ℂ A] [StarModule ℂ A]
+    [NormedRing B] [CompleteSpace B] [StarRing B] [CStarRing B]
+    [NormedAlgebra ℂ B] [StarModule ℂ B]
+    [FunLike F A B] [AlgHomClass F ℂ A B] [StarHomClass F A B]
+    {a : A} (ha : IsSelfAdjoint a) (φ : F) (hφ : Function.Injective φ) :
+    spectrum ℝ (φ a) = spectrum ℝ a := by
+  have h_spec := AlgHom.spectrum_apply_subset ((φ : A →⋆ₐ[ℂ] B).restrictScalars ℝ) a
+  refine Set.eq_of_subset_of_subset h_spec fun x hx ↦ ?_
+  /- we prove the reverse inclusion by contradiction, so assume that `x ∈ spectrum ℝ a`, but
+  `x ∉ spectrum ℝ (φ a)`. Then by Urysohn's lemma we can get a function for which `f x = 1`, but
+  `f = 0` on `spectrum ℝ a`. -/
+  by_contra hx'
+  obtain ⟨f, h_eqOn, h_eqOn_x, -⟩ := exists_continuous_zero_one_of_isClosed
+    (spectrum.isClosed (𝕜 := ℝ) (φ a)) (isClosed_singleton (x := x)) <| by simpa
+  /- it suffices to show that `φ (f a) = 0`, for if so, then `f a = 0` by injectivity of `φ`, and
+  hence `f = 0` on `spectrum ℝ a`, contradicting the fact that `f x = 1`. -/
+  suffices φ (cfc f a) = 0 by
+    rw [map_eq_zero_iff φ hφ, ← cfc_zero ℝ a, cfc_eq_cfc_iff_eqOn] at this
+    exact zero_ne_one <| calc
+      0 = f x := (this hx).symm
+      _ = 1 := h_eqOn_x <| Set.mem_singleton x
+  /- Finally, `φ (f a) = f (φ a) = 0`, where the last equality follows since `f = 0` on
+  `spectrum ℝ (φ a)`. -/
+  calc φ (cfc f a) = cfc f (φ a) := StarAlgHomClass.map_cfc φ f a
+    _ = cfc (0 : ℝ → ℝ) (φ a) := cfc_congr h_eqOn
+    _ = 0 := by simp
+
+namespace NonUnitalStarAlgHom
+
+variable {F A B : Type*}
+variable [NonUnitalNormedRing A] [CompleteSpace A] [StarRing A] [CStarRing A]
+variable [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [StarModule ℂ A]
+variable [NonUnitalNormedRing B] [CompleteSpace B] [StarRing B] [CStarRing B]
+variable [NormedSpace ℂ B] [IsScalarTower ℂ B B] [SMulCommClass ℂ B B] [StarModule ℂ B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F ℂ A B] [StarHomClass F A B]
+
+open CStarAlgebra Unitization in
+/-- A non-unital star algebra monomorphism of complex C⋆-algebras is isometric. -/
+lemma norm_map (φ : F) (hφ : Function.Injective φ) (a : A) : ‖φ a‖ = ‖a‖ := by
+  /- Since passing to the unitization is functorial, and it is an isometric embedding, we may assume
+  that `φ` is a unital star algebra monomorphism and that `A` and `B` are unital C⋆-algebras. -/
+  suffices ∀ {ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B} (_ : Function.Injective ψ)
+      (a : Unitization ℂ A), ‖ψ a‖ = ‖a‖ by
+    simpa [norm_inr] using this (starMap_injective (φ := (φ : A →⋆ₙₐ[ℂ] B)) hφ) a
+  intro ψ hψ a
+  -- to show `‖ψ a‖ = ‖a‖`, by the C⋆-property it suffices to show `‖ψ (star a * a)‖ = ‖star a * a‖`
+  rw [← sq_eq_sq (by positivity) (by positivity)]
+  simp only [sq, ← CStarRing.norm_star_mul_self, ← map_star, ← map_mul]
+  /- since `star a * a` is selfadjoint, it has the same `ℝ`-spectrum as `ψ (star a * a)`.
+  Since the spectral radius over `ℝ` coincides with the norm, `‖ψ (star a * a)‖ = ‖star a * a‖`. -/
+  have ha : IsSelfAdjoint (star a * a) := .star_mul_self a
+  calc ‖ψ (star a * a)‖ = (spectralRadius ℝ (ψ (star a * a))).toReal :=
+      ha.map ψ |>.toReal_spectralRadius_eq_norm.symm
+    _ = (spectralRadius ℝ (star a * a)).toReal := by
+      simp only [spectralRadius, ha.map_spectrum_real ψ hψ]
+    _ = ‖star a * a‖ := ha.toReal_spectralRadius_eq_norm
+
+/-- A non-unital star algebra monomorphism of complex C⋆-algebras is isometric. -/
+lemma nnnorm_map (φ : F) (hφ : Function.Injective φ) (a : A) : ‖φ a‖₊ = ‖a‖₊ :=
+  Subtype.ext <| norm_map φ hφ a
+
+lemma isometry (φ : F) (hφ : Function.Injective φ) : Isometry φ :=
+  AddMonoidHomClass.isometry_of_norm φ (norm_map φ hφ)
+
+end NonUnitalStarAlgHom
diff --git a/Mathlib/Analysis/CStarAlgebra/Matrix.lean b/Mathlib/Analysis/CStarAlgebra/Matrix.lean
index 2a7915775d319..49e853c8a1438 100644
--- a/Mathlib/Analysis/CStarAlgebra/Matrix.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Matrix.lean
@@ -125,14 +125,14 @@ lemma piLp_equiv_toEuclideanCLM (A : Matrix n n 𝕜) (x : EuclideanSpace 𝕜 n
   rfl
 
 /-- An auxiliary definition used only to construct the true `NormedAddCommGroup` (and `Metric`)
-structure provided by `Matrix.instMetricSpaceL2Op` and `Matrix.instNormedAddCommGroupL2Op`.  -/
+structure provided by `Matrix.instMetricSpaceL2Op` and `Matrix.instNormedAddCommGroupL2Op`. -/
 def l2OpNormedAddCommGroupAux : NormedAddCommGroup (Matrix m n 𝕜) :=
   @NormedAddCommGroup.induced ((Matrix m n 𝕜) ≃ₗ[𝕜] (EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 m)) _
     _ _ _ ContinuousLinearMap.toNormedAddCommGroup.toNormedAddGroup _ _ <|
     (toEuclideanLin.trans toContinuousLinearMap).injective
 
 /-- An auxiliary definition used only to construct the true `NormedRing` (and `Metric`) structure
-provided by `Matrix.instMetricSpaceL2Op` and `Matrix.instNormedRingL2Op`.  -/
+provided by `Matrix.instMetricSpaceL2Op` and `Matrix.instNormedRingL2Op`. -/
 def l2OpNormedRingAux : NormedRing (Matrix n n 𝕜) :=
   @NormedRing.induced ((Matrix n n 𝕜) ≃⋆ₐ[𝕜] (EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n)) _
     _ _ _ ContinuousLinearMap.toNormedRing _ _ toEuclideanCLM.injective
diff --git a/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean b/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
index 7fb42a09b19aa..a2bc174a1c07d 100644
--- a/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
@@ -81,6 +81,7 @@ instance : CStarModule A A where
     rw [← sq_eq_sq (norm_nonneg _) (by positivity)]
     simpa [sq] using Eq.symm <| CStarRing.norm_star_mul_self
 
+open scoped InnerProductSpace in
 lemma inner_def (x y : A) : ⟪x, y⟫_A = star x * y := rfl
 
 end Self
@@ -89,6 +90,8 @@ end Self
 
 section Prod
 
+open scoped InnerProductSpace
+
 variable {E F : Type*}
 variable [NormedAddCommGroup E] [Module ℂ E] [SMul Aᵐᵒᵖ E]
 variable [NormedAddCommGroup F] [Module ℂ F] [SMul Aᵐᵒᵖ F]
@@ -139,7 +142,7 @@ lemma max_le_prod_norm (x : C⋆ᵐᵒᵈ (E × F)) : max ‖x.1‖ ‖x.2‖ 
     Real.sqrt_le_sqrt_iff]
   constructor
   all_goals
-    apply norm_le_norm_of_nonneg_of_le
+    apply CStarAlgebra.norm_le_norm_of_nonneg_of_le
     all_goals
       aesop (add safe apply CStarModule.inner_self_nonneg)
 
@@ -189,6 +192,8 @@ end Prod
 
 section Pi
 
+open scoped InnerProductSpace
+
 variable {ι : Type*} {E : ι → Type*} [Fintype ι]
 variable [∀ i, NormedAddCommGroup (E i)] [∀ i, Module ℂ (E i)] [∀ i, SMul Aᵐᵒᵖ (E i)]
 variable [∀ i, CStarModule A (E i)]
@@ -255,7 +260,7 @@ lemma norm_apply_le_norm (x : C⋆ᵐᵒᵈ (Π i, E i)) (i : ι) : ‖x i‖ 
   let _ : NormedAddCommGroup (C⋆ᵐᵒᵈ (Π i, E i)) := normedAddCommGroup
   refine abs_le_of_sq_le_sq' ?_ (by positivity) |>.2
   rw [pi_norm_sq, norm_sq_eq]
-  refine norm_le_norm_of_nonneg_of_le inner_self_nonneg ?_
+  refine CStarAlgebra.norm_le_norm_of_nonneg_of_le inner_self_nonneg ?_
   exact Finset.single_le_sum (fun j _ ↦ inner_self_nonneg (x := x j)) (Finset.mem_univ i)
 
 open Finset in
@@ -312,6 +317,7 @@ variable {E : Type*}
 variable [NormedAddCommGroup E] [InnerProductSpace ℂ E]
 variable [instSMulOp : SMul ℂᵐᵒᵖ E] [instCentral : IsCentralScalar ℂ E]
 
+open scoped InnerProductSpace in
 /-- Reinterpret an inner product space `E` over `ℂ` as a `CStarModule` over `ℂ`.
 
 Note: this instance requires `SMul ℂᵐᵒᵖ E` and `IsCentralScalar ℂ E` instances to exist on `E`,
diff --git a/Mathlib/Analysis/CStarAlgebra/Module/Defs.lean b/Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
index daf2909a3ed1a..a94e81711816a 100644
--- a/Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Module/Defs.lean
@@ -158,6 +158,7 @@ variable {A E : Type*} [NonUnitalNormedRing A] [StarRing A] [PartialOrder A]
 
 local notation "⟪" x ", " y "⟫" => inner (𝕜 := A) x y
 
+open scoped InnerProductSpace in
 /-- The norm associated with a Hilbert C⋆-module. It is not registered as a norm, since a type
 might already have a norm defined on it. -/
 noncomputable def norm (A : Type*) {E : Type*} [Norm A] [Inner A E] : Norm E where
@@ -168,7 +169,7 @@ lemma norm_sq_eq {x : E} : ‖x‖ ^ 2 = ‖⟪x, x⟫‖ := by simp [norm_eq_sq
 section
 include A
 
-protected lemma norm_nonneg {x : E} : 0 ≤ ‖x‖ := by simp [norm_eq_sqrt_norm_inner_self]; positivity
+protected lemma norm_nonneg {x : E} : 0 ≤ ‖x‖ := by simp [norm_eq_sqrt_norm_inner_self]
 
 protected lemma norm_pos {x : E} (hx : x ≠ 0) : 0 < ‖x‖ := by
   simp only [norm_eq_sqrt_norm_inner_self, Real.sqrt_pos, norm_pos_iff]
@@ -189,6 +190,7 @@ end
 variable [CStarRing A] [StarOrderedRing A] [StarModule ℂ A]
   [IsScalarTower ℂ A A] [SMulCommClass ℂ A A]
 
+open scoped InnerProductSpace in
 /-- The C⋆-algebra-valued Cauchy-Schwarz inequality for Hilbert C⋆-modules. -/
 lemma inner_mul_inner_swap_le [CompleteSpace A] {x y : E} : ⟪y, x⟫ * ⟪x, y⟫ ≤ ‖x‖ ^ 2 • ⟪y, y⟫ := by
   rcases eq_or_ne x 0 with h|h
@@ -207,7 +209,7 @@ lemma inner_mul_inner_swap_le [CompleteSpace A] {x y : E} : ⟪y, x⟫ * ⟪x, y
             _ ≤ ‖x‖ ^ 2 • (star a * a) - ‖x‖ ^ 2 • (⟪y, x⟫ * a)
                   - ‖x‖ ^ 2 • (star a * ⟪x, y⟫) + ‖x‖ ^ 2 • (‖x‖ ^ 2 • ⟪y, y⟫) := by
                       gcongr
-                      calc _ ≤ ‖⟪x, x⟫_A‖ • (star a * a) := CStarRing.conjugate_le_norm_smul
+                      calc _ ≤ ‖⟪x, x⟫_A‖ • (star a * a) := CStarAlgebra.conjugate_le_norm_smul
                         _ = (Real.sqrt ‖⟪x, x⟫_A‖) ^ 2 • (star a * a) := by
                                   congr
                                   have : 0 ≤ ‖⟪x, x⟫_A‖ := by positivity
@@ -217,13 +219,14 @@ lemma inner_mul_inner_swap_le [CompleteSpace A] {x y : E} : ⟪y, x⟫ * ⟪x, y
     simp only [star_inner, sub_self, zero_sub, le_neg_add_iff_add_le, add_zero] at h₁
     rwa [smul_le_smul_iff_of_pos_left (pow_pos (CStarModule.norm_pos h) _)] at h₁
 
+open scoped InnerProductSpace in
 variable (E) in
 /-- The Cauchy-Schwarz inequality for Hilbert C⋆-modules. -/
 lemma norm_inner_le [CompleteSpace A] {x y : E} : ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖ := by
   have := calc ‖⟪x, y⟫‖ ^ 2 = ‖⟪y, x⟫ * ⟪x, y⟫‖ := by
                 rw [← star_inner x, CStarRing.norm_star_mul_self, pow_two]
     _ ≤ ‖‖x‖^ 2 • ⟪y, y⟫‖ := by
-                refine CStarRing.norm_le_norm_of_nonneg_of_le ?_ inner_mul_inner_swap_le
+                refine CStarAlgebra.norm_le_norm_of_nonneg_of_le ?_ inner_mul_inner_swap_le
                 rw [← star_inner x]
                 exact star_mul_self_nonneg ⟪x, y⟫_A
     _ = ‖x‖ ^ 2 * ‖⟪y, y⟫‖ := by simp [norm_smul]
@@ -260,12 +263,13 @@ and bornology instead of inheriting them from the norm. -/
 abbrev normedAddCommGroup [CompleteSpace A] : NormedAddCommGroup E :=
   NormedAddCommGroup.ofCore CStarModule.normedSpaceCore
 
+open scoped InnerProductSpace in
 lemma norm_eq_csSup [CompleteSpace A] (v : E) :
     ‖v‖ = sSup { ‖⟪w, v⟫_A‖ | (w : E) (_ : ‖w‖ ≤ 1) } := by
   let instNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore normedSpaceCore
   let instNS : NormedSpace ℂ E := .ofCore normedSpaceCore
   refine Eq.symm <| IsGreatest.csSup_eq ⟨⟨‖v‖⁻¹ • v, ?_, ?_⟩, ?_⟩
-  · simpa only [norm_smul, norm_inv, norm_norm] using inv_mul_le_one_of_le le_rfl (by positivity)
+  · simpa only [norm_smul, norm_inv, norm_norm] using inv_mul_le_one_of_le₀ le_rfl (by positivity)
   · simp [norm_smul, ← norm_sq_eq, pow_two, ← mul_assoc]
   · rintro - ⟨w, hw, rfl⟩
     calc _ ≤ ‖w‖ * ‖v‖ := norm_inner_le E
@@ -276,6 +280,8 @@ end norm
 
 section NormedAddCommGroup
 
+open scoped InnerProductSpace
+
 /- Note: one generally creates a `CStarModule` instance for a type `E` first before getting the
 `NormedAddCommGroup` and `NormedSpace` instances via `CStarModule.normedSpaceCore`, especially by
 using `NormedAddCommGroup.ofCoreReplaceAll` and `NormedSpace.ofCore`. See
diff --git a/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean b/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
index afee7eea2b49e..82554bb415dee 100644
--- a/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
@@ -167,7 +167,8 @@ instance [u : UniformSpace E] : UniformSpace (C⋆ᵐᵒᵈ E) := u.comap <| equ
 instance [Bornology E] : Bornology (C⋆ᵐᵒᵈ E) := Bornology.induced <| equiv E
 
 /-- `WithCStarModule.equiv` as a uniform equivalence between `C⋆ᵐᵒᵈ E` and `E`. -/
-def uniformEquiv [UniformSpace E] : C⋆ᵐᵒᵈ E ≃ᵤ E := equiv E |>.toUniformEquivOfUniformInducing ⟨rfl⟩
+def uniformEquiv [UniformSpace E] : C⋆ᵐᵒᵈ E ≃ᵤ E :=
+  equiv E |>.toUniformEquivOfIsUniformInducing ⟨rfl⟩
 
 instance [UniformSpace E] [CompleteSpace E] : CompleteSpace (C⋆ᵐᵒᵈ E) :=
   uniformEquiv.completeSpace_iff.mpr inferInstance
diff --git a/Mathlib/Analysis/CStarAlgebra/Multiplier.lean b/Mathlib/Analysis/CStarAlgebra/Multiplier.lean
index 248d3fa9e31ac..3c84852d4f405 100644
--- a/Mathlib/Analysis/CStarAlgebra/Multiplier.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Multiplier.lean
@@ -534,11 +534,15 @@ instance instNormedSpace : NormedSpace 𝕜 𝓜(𝕜, A) :=
 instance instNormedAlgebra : NormedAlgebra 𝕜 𝓜(𝕜, A) :=
   { DoubleCentralizer.instAlgebra, DoubleCentralizer.instNormedSpace with }
 
-theorem uniformEmbedding_toProdMulOpposite : UniformEmbedding (@toProdMulOpposite 𝕜 A _ _ _ _ _) :=
-  uniformEmbedding_comap toProdMulOpposite_injective
+theorem isUniformEmbedding_toProdMulOpposite :
+    IsUniformEmbedding (toProdMulOpposite (𝕜 := 𝕜) (A := A)) :=
+  isUniformEmbedding_comap toProdMulOpposite_injective
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toProdMulOpposite := isUniformEmbedding_toProdMulOpposite
 
 instance [CompleteSpace A] : CompleteSpace 𝓜(𝕜, A) := by
-  rw [completeSpace_iff_isComplete_range uniformEmbedding_toProdMulOpposite.toUniformInducing]
+  rw [completeSpace_iff_isComplete_range isUniformEmbedding_toProdMulOpposite.isUniformInducing]
   apply IsClosed.isComplete
   simp only [range_toProdMulOpposite, Set.setOf_forall]
   refine isClosed_iInter fun x => isClosed_iInter fun y => isClosed_eq ?_ ?_
diff --git a/Mathlib/Analysis/CStarAlgebra/Spectrum.lean b/Mathlib/Analysis/CStarAlgebra/Spectrum.lean
index 80057039f51d6..6d91e465acfeb 100644
--- a/Mathlib/Analysis/CStarAlgebra/Spectrum.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Spectrum.lean
@@ -78,7 +78,7 @@ theorem unitary.spectrum_subset_circle (u : unitary E) :
     rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_inv] at hk
     have : ‖k‖⁻¹ ≤ ‖(↑(unitary.toUnits u)⁻¹ : E)‖ := by
       simpa only [norm_inv] using norm_le_norm_of_mem hk
-    simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this
+    simpa using inv_le_of_inv_le₀ (norm_pos_iff.mpr hnk) this
 
 theorem spectrum.subset_circle_of_unitary {u : E} (h : u ∈ unitary E) :
     spectrum 𝕜 u ⊆ Metric.sphere 0 1 :=
@@ -198,7 +198,7 @@ lemma coe_isUnit {a : S} : IsUnit (a : A) ↔ IsUnit a := by
   have ha₂ := ha.mul ha.star
   have spec_eq {x : S} (hx : IsSelfAdjoint x) : spectrum ℂ x = spectrum ℂ (x : A) :=
     Subalgebra.spectrum_eq_of_isPreconnected_compl S _ <|
-      (hx.starHom_apply S.subtype).isConnected_spectrum_compl.isPreconnected
+      (hx.map S.subtype).isConnected_spectrum_compl.isPreconnected
   rw [← StarMemClass.coe_star, ← MulMemClass.coe_mul, ← spectrum.zero_not_mem_iff ℂ, ← spec_eq,
     spectrum.zero_not_mem_iff] at ha₁ ha₂
   · have h₁ : ha₁.unit⁻¹ * star a * a = 1 := mul_assoc _ _ a ▸ ha₁.val_inv_mul
@@ -226,7 +226,7 @@ variable [NonUnitalNormedRing A] [CompleteSpace A] [StarRing A] [CStarRing A]
 variable [NormedSpace ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] [StarModule ℂ A]
 variable [NonUnitalNormedRing B] [CompleteSpace B] [StarRing B] [CStarRing B]
 variable [NormedSpace ℂ B] [IsScalarTower ℂ B B] [SMulCommClass ℂ B B] [StarModule ℂ B]
-variable [FunLike F A B] [NonUnitalAlgHomClass F ℂ A B] [NonUnitalStarAlgHomClass F ℂ A B]
+variable [FunLike F A B] [NonUnitalAlgHomClass F ℂ A B] [StarHomClass F A B]
 
 open Unitization
 
@@ -241,7 +241,7 @@ lemma nnnorm_apply_le (φ : F) (a : A) : ‖φ a‖₊ ≤ ‖a‖₊ := by
     intro s hs
     suffices this : spectralRadius ℂ (ψ s) ≤ spectralRadius ℂ s by
       -- changing the order of `rw`s below runs into https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/weird.20type.20class.20synthesis.20error/near/421224482
-      rwa [(hs.starHom_apply ψ).spectralRadius_eq_nnnorm, hs.spectralRadius_eq_nnnorm, coe_le_coe]
+      rwa [(hs.map ψ).spectralRadius_eq_nnnorm, hs.spectralRadius_eq_nnnorm, coe_le_coe]
         at this
     exact iSup_le_iSup_of_subset (AlgHom.spectrum_apply_subset ψ s)
   simpa [nnnorm_inr] using h (starLift (inrNonUnitalStarAlgHom ℂ B |>.comp (φ : A →⋆ₙₐ[ℂ] B))) a
@@ -263,11 +263,11 @@ end NonUnitalStarAlgHom
 
 namespace StarAlgEquiv
 
-variable {F A B : Type*} [NormedRing A] [NormedSpace ℂ A] [SMulCommClass ℂ A A]
+variable {F A B : Type*} [NonUnitalNormedRing A] [NormedSpace ℂ A] [SMulCommClass ℂ A A]
 variable [IsScalarTower ℂ A A] [CompleteSpace A] [StarRing A] [CStarRing A] [StarModule ℂ A]
-variable [NormedRing B] [NormedSpace ℂ B] [SMulCommClass ℂ B B] [IsScalarTower ℂ B B]
+variable [NonUnitalNormedRing B] [NormedSpace ℂ B] [SMulCommClass ℂ B B] [IsScalarTower ℂ B B]
 variable [CompleteSpace B] [StarRing B] [CStarRing B] [StarModule ℂ B] [EquivLike F A B]
-variable [NonUnitalAlgEquivClass F ℂ A B] [StarAlgEquivClass F ℂ A B]
+variable [NonUnitalAlgEquivClass F ℂ A B] [StarHomClass F A B]
 
 lemma nnnorm_map (φ : F) (a : A) : ‖φ a‖₊ = ‖a‖₊ :=
   le_antisymm (NonUnitalStarAlgHom.nnnorm_apply_le φ a) <| by
@@ -292,7 +292,7 @@ open scoped ComplexStarModule
 variable {F A : Type*} [NormedRing A] [NormedAlgebra ℂ A] [CompleteSpace A] [StarRing A]
   [CStarRing A] [StarModule ℂ A] [FunLike F A ℂ] [hF : AlgHomClass F ℂ A ℂ]
 
-/-- This instance is provided instead of `StarAlgHomClass` to avoid type class inference loops.
+/-- This instance is provided instead of `StarHomClass` to avoid type class inference loops.
 See note [lower instance priority] -/
 noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass F A ℂ where
   map_star φ a := by
@@ -309,13 +309,13 @@ noncomputable instance (priority := 100) Complex.instStarHomClass : StarHomClass
 
 /-- This is not an instance to avoid type class inference loops. See
 `WeakDual.Complex.instStarHomClass`. -/
-lemma _root_.AlgHomClass.instStarAlgHomClass : StarAlgHomClass F ℂ A ℂ :=
+lemma _root_.AlgHomClass.instStarHomClass : StarHomClass F A ℂ :=
   { WeakDual.Complex.instStarHomClass, hF with }
 
 namespace CharacterSpace
 
-noncomputable instance instStarAlgHomClass : StarAlgHomClass (characterSpace ℂ A) ℂ A ℂ :=
-  { AlgHomClass.instStarAlgHomClass with }
+noncomputable instance instStarHomClass : StarHomClass (characterSpace ℂ A) A ℂ :=
+  { AlgHomClass.instStarHomClass with }
 
 end CharacterSpace
 
diff --git a/Mathlib/Analysis/CStarAlgebra/Unitization.lean b/Mathlib/Analysis/CStarAlgebra/Unitization.lean
index 0ccc38b7262f0..7a5bdee2eef6e 100644
--- a/Mathlib/Analysis/CStarAlgebra/Unitization.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Unitization.lean
@@ -72,7 +72,7 @@ instance CStarRing.instRegularNormedAlgebra : RegularNormedAlgebra 𝕜 E where
       · simpa only [mem_closedBall_zero_iff, norm_smul, one_mul, norm_star] using
           (NNReal.le_inv_iff_mul_le ha.ne').1 (one_mul ‖a‖₊⁻¹ ▸ hk₂.le : ‖k‖₊ ≤ ‖a‖₊⁻¹)
       · simp only [map_smul, nnnorm_smul, mul_apply', mul_smul_comm, CStarRing.nnnorm_self_mul_star]
-        rwa [← NNReal.div_lt_iff (mul_pos ha ha).ne', div_eq_mul_inv, mul_inv, ← mul_assoc]
+        rwa [← div_lt_iff₀ (mul_pos ha ha), div_eq_mul_inv, mul_inv, ← mul_assoc]
 
 section CStarProperty
 
@@ -129,7 +129,7 @@ instance Unitization.instCStarRing : CStarRing (Unitization 𝕜 E) where
   norm_mul_self_le x := by
     -- rewrite both sides as a `⊔`
     simp only [Unitization.norm_def, Prod.norm_def, ← sup_eq_max]
-    -- Show that `(Unitization.splitMul 𝕜 E x).snd` satisifes the C⋆-property, in two stages:
+    -- Show that `(Unitization.splitMul 𝕜 E x).snd` satisfies the C⋆-property, in two stages:
     have h₁ : ∀ x : Unitization 𝕜 E,
         ‖(Unitization.splitMul 𝕜 E x).snd‖ ≤ ‖(Unitization.splitMul 𝕜 E (star x)).snd‖ := by
       simp only [add_zero, Unitization.splitMul_apply, Unitization.snd_star, Unitization.fst_star]
diff --git a/Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean b/Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean
index 6640b13a2e378..4b70bd94ec863 100644
--- a/Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean
+++ b/Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean
@@ -115,7 +115,7 @@ theorem ae_convolution_tendsto_right_of_locallyIntegrable
     tendsto_nhdsWithin_iff.2 ⟨hφ, Eventually.of_forall (fun i ↦ (φ i).rOut_pos)⟩
   have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
   simp only [Function.comp] at this
-  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
+  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (Module.finrank ℝ G)) this
   · filter_upwards with i using
       hg.integrableOn_isCompact (isCompact_closedBall _ _)
   · apply tendsto_const_nhds.congr (fun i ↦ ?_)
diff --git a/Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean b/Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
index 314fbbe3fa861..d0f390973978c 100644
--- a/Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
+++ b/Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean
@@ -25,7 +25,7 @@ the indicator function of `closedBall 0 1` with a function as above with `s = ba
 
 noncomputable section
 
-open Set Metric TopologicalSpace Function Asymptotics MeasureTheory FiniteDimensional
+open Set Metric TopologicalSpace Function Asymptotics MeasureTheory Module
   ContinuousLinearMap Filter MeasureTheory.Measure Bornology
 
 open scoped Pointwise Topology NNReal Convolution
@@ -407,7 +407,7 @@ theorem y_pos_of_mem_ball {D : ℝ} {x : E} (Dpos : 0 < D) (D_lt_one : D < 1)
       intro y hy
       simp only [support_mul, w_support E Dpos]
       simp only [φ, mem_inter_iff, mem_support, Ne, indicator_apply_eq_zero,
-        mem_closedBall_zero_iff, one_ne_zero, not_forall, not_false_iff, exists_prop, and_true_iff]
+        mem_closedBall_zero_iff, one_ne_zero, not_forall, not_false_iff, exists_prop, and_true]
       constructor
       · apply ball_subset_ball' _ hy
         simp only [hz, norm_smul, abs_of_nonneg Dpos.le, abs_of_nonneg B.le, dist_zero_right,
@@ -512,7 +512,7 @@ instance (priority := 100) {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E
         · rintro ⟨R, x⟩ ⟨hR : 1 < R, _⟩
           have A : 0 < (R - 1) / (R + 1) := by apply div_pos <;> linarith
           have B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith
-          simp only [mem_preimage, prod_mk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true_iff, A, B]
+          simp only [mem_preimage, prod_mk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true, A, B]
       eq_one := fun R hR x hx => by
         have A : 0 < R + 1 := by linarith
         simp only [hR, if_true]
diff --git a/Mathlib/Analysis/Calculus/BumpFunction/Normed.lean b/Mathlib/Analysis/Calculus/BumpFunction/Normed.lean
index 1846ced215ae9..a61f635da6766 100644
--- a/Mathlib/Analysis/Calculus/BumpFunction/Normed.lean
+++ b/Mathlib/Analysis/Calculus/BumpFunction/Normed.lean
@@ -16,7 +16,7 @@ In this file we define `ContDiffBump.normed f μ` to be the bump function `f` no
 
 noncomputable section
 
-open Function Filter Set Metric MeasureTheory FiniteDimensional Measure
+open Function Filter Set Metric MeasureTheory Module Measure
 open scoped Topology
 
 namespace ContDiffBump
diff --git a/Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean b/Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean
index b64a5bf2e5cc8..5e7a4a62baea7 100644
--- a/Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean
+++ b/Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean
@@ -63,7 +63,7 @@ theorem conformalAt_const_smul {c : ℝ} (h : c ≠ 0) (x : X) : ConformalAt (fu
 theorem Subsingleton.conformalAt [Subsingleton X] (f : X → Y) (x : X) : ConformalAt f x :=
   ⟨0, hasFDerivAt_of_subsingleton _ _, isConformalMap_of_subsingleton _⟩
 
-/-- A function is a conformal map if and only if its differential is a conformal linear map-/
+/-- A function is a conformal map if and only if its differential is a conformal linear map -/
 theorem conformalAt_iff_isConformalMap_fderiv {f : X → Y} {x : X} :
     ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x) := by
   constructor
diff --git a/Mathlib/Analysis/Calculus/ContDiff/Basic.lean b/Mathlib/Analysis/Calculus/ContDiff/Basic.lean
index 3dc61bad5603b..f9cc1aee3a4f1 100644
--- a/Mathlib/Analysis/Calculus/ContDiff/Basic.lean
+++ b/Mathlib/Analysis/Calculus/ContDiff/Basic.lean
@@ -299,7 +299,7 @@ point in a domain. -/
 theorem ContinuousLinearEquiv.comp_contDiffWithinAt_iff (e : F ≃L[𝕜] G) :
     ContDiffWithinAt 𝕜 n (e ∘ f) s x ↔ ContDiffWithinAt 𝕜 n f s x :=
   ⟨fun H => by
-    simpa only [(· ∘ ·), e.symm.coe_coe, e.symm_apply_apply] using
+    simpa only [Function.comp_def, e.symm.coe_coe, e.symm_apply_apply] using
       H.continuousLinearMap_comp (e.symm : G →L[𝕜] F),
     fun H => H.continuousLinearMap_comp (e : F →L[𝕜] G)⟩
 
@@ -432,7 +432,7 @@ theorem ContinuousLinearEquiv.contDiffWithinAt_comp_iff (e : G ≃L[𝕜] E) :
     ContDiffWithinAt 𝕜 n (f ∘ e) (e ⁻¹' s) (e.symm x) ↔ ContDiffWithinAt 𝕜 n f s x := by
   constructor
   · intro H
-    simpa [← preimage_comp, (· ∘ ·)] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G)
+    simpa [← preimage_comp, Function.comp_def] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G)
   · intro H
     rw [← e.apply_symm_apply x, ← e.coe_coe] at H
     exact H.comp_continuousLinearMap _
@@ -448,8 +448,8 @@ theorem ContinuousLinearEquiv.contDiffAt_comp_iff (e : G ≃L[𝕜] E) :
 domains. -/
 theorem ContinuousLinearEquiv.contDiffOn_comp_iff (e : G ≃L[𝕜] E) :
     ContDiffOn 𝕜 n (f ∘ e) (e ⁻¹' s) ↔ ContDiffOn 𝕜 n f s :=
-  ⟨fun H => by simpa [(· ∘ ·)] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G), fun H =>
-    H.comp_continuousLinearMap (e : G →L[𝕜] E)⟩
+  ⟨fun H => by simpa [Function.comp_def] using H.comp_continuousLinearMap (e.symm : E →L[𝕜] G),
+    fun H => H.comp_continuousLinearMap (e : G →L[𝕜] E)⟩
 
 /-- Composition by continuous linear equivs on the right respects higher differentiability. -/
 theorem ContinuousLinearEquiv.contDiff_comp_iff (e : G ≃L[𝕜] E) :
@@ -802,7 +802,7 @@ theorem ContDiff.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F}
 theorem ContDiffOn.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F} {s : Set X}
     (hg : ContDiffOn 𝕜 n g s) (hf : ContDiffOn 𝕜 n f s) :
     ContDiffOn 𝕜 n (fun x => (g x).comp (f x)) s :=
-  isBoundedBilinearMap_comp.contDiff.comp_contDiff_on₂ hg hf
+  (isBoundedBilinearMap_comp (𝕜 := 𝕜) (E := E) (F := F) (G := G)).contDiff.comp_contDiff_on₂ hg hf
 
 theorem ContDiff.clm_apply {f : E → F →L[𝕜] G} {g : E → F} {n : ℕ∞} (hf : ContDiff 𝕜 n f)
     (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n fun x => (f x) (g x) :=
@@ -882,7 +882,7 @@ then there is a function `f' : E → F →L[𝕜] G` that is `C^n` at `x₀` wit
 sufficiently close to `x₀` within `s ∪ {x₀}` the function `y ↦ f x y` has derivative `f' x` at `g x`
 within `t ⊆ F`.  For convenience, we return an explicit set of `x`'s where this holds that is a
 subset of `s ∪ {x₀}`.  We need one additional condition, namely that `t` is a neighborhood of
-`g(x₀)` within `g '' s`.  -/
+`g(x₀)` within `g '' s`. -/
 theorem ContDiffWithinAt.hasFDerivWithinAt_nhds {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ}
     {x₀ : E} (hf : ContDiffWithinAt 𝕜 (n + 1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
     (hg : ContDiffWithinAt 𝕜 n g s x₀) (hgt : t ∈ 𝓝[g '' s] g x₀) :
@@ -892,7 +892,7 @@ theorem ContDiffWithinAt.hasFDerivWithinAt_nhds {f : E → F → G} {g : E → F
   have hst : insert x₀ s ×ˢ t ∈ 𝓝[(fun x => (x, g x)) '' s] (x₀, g x₀) := by
     refine nhdsWithin_mono _ ?_ (nhdsWithin_prod self_mem_nhdsWithin hgt)
     simp_rw [image_subset_iff, mk_preimage_prod, preimage_id', subset_inter_iff, subset_insert,
-      true_and_iff, subset_preimage_image]
+      true_and, subset_preimage_image]
   obtain ⟨v, hv, hvs, f', hvf', hf'⟩ := contDiffWithinAt_succ_iff_hasFDerivWithinAt'.mp hf
   refine
     ⟨(fun z => (z, g z)) ⁻¹' v ∩ insert x₀ s, ?_, inter_subset_right, fun z =>
@@ -982,7 +982,8 @@ theorem ContDiffWithinAt.iteratedFderivWithin_right {i : ℕ} (hf : ContDiffWith
       ((continuousMultilinearCurryFin0 𝕜 E F).symm : _ →L[𝕜] E [×0]→L[𝕜] F)
   · rw [Nat.cast_succ, add_comm _ 1, ← add_assoc] at hmn
     exact ((hi hmn).fderivWithin_right hs le_rfl hx₀s).continuousLinearMap_comp
-      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i+1) ↦ E) F : _ →L[𝕜] E [×(i+1)]→L[𝕜] F)
+      ((continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i+1) ↦ E) F).symm :
+        _ →L[𝕜] E [×(i+1)]→L[𝕜] F)
 
 /-- `x ↦ fderiv 𝕜 (f x) (g x)` is smooth at `x₀`. -/
 protected theorem ContDiffAt.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞}
@@ -1070,15 +1071,14 @@ theorem hasFTaylorSeriesUpToOn_pi :
   set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
     ContinuousMultilinearMap.piₗᵢ _ _
   refine ⟨fun h i => ?_, fun h => ⟨fun x hx => ?_, ?_, ?_⟩⟩
-  · convert h.continuousLinearMap_comp (pr i)
+  · exact h.continuousLinearMap_comp (pr i)
   · ext1 i
     exact (h i).zero_eq x hx
   · intro m hm x hx
-    have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
-    convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this
+    exact (L m).hasFDerivAt.comp_hasFDerivWithinAt x <|
+      hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
   · intro m hm
-    have := continuousOn_pi.2 fun i => (h i).cont m hm
-    convert (L m).continuous.comp_continuousOn this
+    exact (L m).continuous.comp_continuousOn <| continuousOn_pi.2 fun i => (h i).cont m hm
 
 @[simp]
 theorem hasFTaylorSeriesUpToOn_pi' :
@@ -1948,3 +1948,5 @@ theorem ContDiff.restrict_scalars (h : ContDiff 𝕜' n f) : ContDiff 𝕜 n f :
   contDiff_iff_contDiffAt.2 fun _ => h.contDiffAt.restrict_scalars _
 
 end RestrictScalars
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/Analysis/Calculus/ContDiff/Bounds.lean b/Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
index d5cce4ded4d53..cc97d67c10ce6 100644
--- a/Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
+++ b/Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
@@ -304,7 +304,7 @@ theorem norm_iteratedFDerivWithin_prod_le [DecidableEq ι] [NormOneClass A'] {u
       (g := (fun v ↦ v.multinomial *
           ∏ j ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (v.count j) (f j) s x‖) ∘
         Sym.toMultiset ∘ Subtype.val ∘ (Finset.symInsertEquiv hi).symm)
-      (by simp) (by simp only [← comp_apply (g := Finset.symInsertEquiv hi), comp.assoc]; simp)]
+      (by simp) (by simp only [← comp_apply (g := Finset.symInsertEquiv hi), comp_assoc]; simp)]
     rw [← Finset.univ_sigma_univ, Finset.sum_sigma, Finset.sum_range]
     simp only [comp_apply, Finset.symInsertEquiv_symm_apply_coe]
     refine Finset.sum_le_sum ?_
@@ -449,7 +449,7 @@ theorem norm_iteratedFDerivWithin_comp_le_aux {Fu Gu : Type u} [NormedAddCommGro
         exact Nat.add_sub_of_le (Finset.mem_range_succ_iff.1 hi)
     _ ≤ ∑ i ∈ Finset.range (n + 1), (n ! : ℝ) * 1 * C * D ^ (n + 1) * 1 := by
       gcongr with i
-      apply inv_le_one
+      apply inv_le_one_of_one_le₀
       simpa only [Nat.one_le_cast] using (n - i).factorial_pos
     _ = (n + 1)! * C * D ^ (n + 1) := by
       simp only [mul_assoc, mul_one, Finset.sum_const, Finset.card_range, nsmul_eq_mul,
diff --git a/Mathlib/Analysis/Calculus/ContDiff/Defs.lean b/Mathlib/Analysis/Calculus/ContDiff/Defs.lean
index 1bde1bf2752a2..3f934a6deec9b 100644
--- a/Mathlib/Analysis/Calculus/ContDiff/Defs.lean
+++ b/Mathlib/Analysis/Calculus/ContDiff/Defs.lean
@@ -3,8 +3,7 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Analysis.Calculus.FDeriv.Equiv
-import Mathlib.Analysis.Calculus.FormalMultilinearSeries
+import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
 
 /-!
 # Higher differentiability
@@ -14,11 +13,7 @@ By induction, it is `C^n` if it is `C^{n-1}` and its (n-1)-th derivative is `C^1
 equivalently, if it is `C^1` and its derivative is `C^{n-1}`.
 Finally, it is `C^∞` if it is `C^n` for all n.
 
-We formalize these notions by defining iteratively the `n+1`-th derivative of a function as the
-derivative of the `n`-th derivative. It is called `iteratedFDeriv 𝕜 n f x` where `𝕜` is the
-field, `n` is the number of iterations, `f` is the function and `x` is the point, and it is given
-as an `n`-multilinear map. We also define a version `iteratedFDerivWithin` relative to a domain,
-as well as predicates `ContDiffWithinAt`, `ContDiffAt`, `ContDiffOn` and
+We formalize these notions with predicates `ContDiffWithinAt`, `ContDiffAt`, `ContDiffOn` and
 `ContDiff` saying that the function is `C^n` within a set at a point, at a point, on a set
 and on the whole space respectively.
 
@@ -26,28 +21,18 @@ To avoid the issue of choice when choosing a derivative in sets where the deriva
 necessarily unique, `ContDiffOn` is not defined directly in terms of the
 regularity of the specific choice `iteratedFDerivWithin 𝕜 n f s` inside `s`, but in terms of the
 existence of a nice sequence of derivatives, expressed with a predicate
-`HasFTaylorSeriesUpToOn`.
+`HasFTaylorSeriesUpToOn` defined in the file `FTaylorSeries`.
 
 We prove basic properties of these notions.
 
 ## Main definitions and results
 Let `f : E → F` be a map between normed vector spaces over a nontrivially normed field `𝕜`.
 
-* `HasFTaylorSeriesUpTo n f p`: expresses that the formal multilinear series `p` is a sequence
-  of iterated derivatives of `f`, up to the `n`-th term (where `n` is a natural number or `∞`).
-* `HasFTaylorSeriesUpToOn n f p s`: same thing, but inside a set `s`. The notion of derivative
-  is now taken inside `s`. In particular, derivatives don't have to be unique.
 * `ContDiff 𝕜 n f`: expresses that `f` is `C^n`, i.e., it admits a Taylor series up to
   rank `n`.
 * `ContDiffOn 𝕜 n f s`: expresses that `f` is `C^n` in `s`.
 * `ContDiffAt 𝕜 n f x`: expresses that `f` is `C^n` around `x`.
 * `ContDiffWithinAt 𝕜 n f s x`: expresses that `f` is `C^n` around `x` within the set `s`.
-* `iteratedFDerivWithin 𝕜 n f s x` is an `n`-th derivative of `f` over the field `𝕜` on the
-  set `s` at the point `x`. It is a continuous multilinear map from `E^n` to `F`, defined as a
-  derivative within `s` of `iteratedFDerivWithin 𝕜 (n-1) f s` if one exists, and `0` otherwise.
-* `iteratedFDeriv 𝕜 n f x` is the `n`-th derivative of `f` over the field `𝕜` at the point `x`.
-  It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of
-  `iteratedFDeriv 𝕜 (n-1) f` if one exists, and `0` otherwise.
 
 In sets of unique differentiability, `ContDiffOn 𝕜 n f s` can be expressed in terms of the
 properties of `iteratedFDerivWithin 𝕜 m f s` for `m ≤ n`. In the whole space,
@@ -91,55 +76,6 @@ within `s`. However, this does not imply continuity or differentiability within
 at `x` when `x` does not belong to `s`. Therefore, we require such existence and good behavior on
 a neighborhood of `x` within `s ∪ {x}` (which appears as `insert x s` in this file).
 
-### Side of the composition, and universe issues
-
-With a naïve direct definition, the `n`-th derivative of a function belongs to the space
-`E →L[𝕜] (E →L[𝕜] (E ... F)...)))` where there are n iterations of `E →L[𝕜]`. This space
-may also be seen as the space of continuous multilinear functions on `n` copies of `E` with
-values in `F`, by uncurrying. This is the point of view that is usually adopted in textbooks,
-and that we also use. This means that the definition and the first proofs are slightly involved,
-as one has to keep track of the uncurrying operation. The uncurrying can be done from the
-left or from the right, amounting to defining the `n+1`-th derivative either as the derivative of
-the `n`-th derivative, or as the `n`-th derivative of the derivative.
-For proofs, it would be more convenient to use the latter approach (from the right),
-as it means to prove things at the `n+1`-th step we only need to understand well enough the
-derivative in `E →L[𝕜] F` (contrary to the approach from the left, where one would need to know
-enough on the `n`-th derivative to deduce things on the `n+1`-th derivative).
-
-However, the definition from the right leads to a universe polymorphism problem: if we define
-`iteratedFDeriv 𝕜 (n + 1) f x = iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x` by induction, we need to
-generalize over all spaces (as `f` and `fderiv 𝕜 f` don't take values in the same space). It is
-only possible to generalize over all spaces in some fixed universe in an inductive definition.
-For `f : E → F`, then `fderiv 𝕜 f` is a map `E → (E →L[𝕜] F)`. Therefore, the definition will only
-work if `F` and `E →L[𝕜] F` are in the same universe.
-
-This issue does not appear with the definition from the left, where one does not need to generalize
-over all spaces. Therefore, we use the definition from the left. This means some proofs later on
-become a little bit more complicated: to prove that a function is `C^n`, the most efficient approach
-is to exhibit a formula for its `n`-th derivative and prove it is continuous (contrary to the
-inductive approach where one would prove smoothness statements without giving a formula for the
-derivative). In the end, this approach is still satisfactory as it is good to have formulas for the
-iterated derivatives in various constructions.
-
-One point where we depart from this explicit approach is in the proof of smoothness of a
-composition: there is a formula for the `n`-th derivative of a composition (Faà di Bruno's formula),
-but it is very complicated and barely usable, while the inductive proof is very simple. Thus, we
-give the inductive proof. As explained above, it works by generalizing over the target space, hence
-it only works well if all spaces belong to the same universe. To get the general version, we lift
-things to a common universe using a trick.
-
-### Variables management
-
-The textbook definitions and proofs use various identifications and abuse of notations, for instance
-when saying that the natural space in which the derivative lives, i.e.,
-`E →L[𝕜] (E →L[𝕜] ( ... →L[𝕜] F))`, is the same as a space of multilinear maps. When doing things
-formally, we need to provide explicit maps for these identifications, and chase some diagrams to see
-everything is compatible with the identifications. In particular, one needs to check that taking the
-derivative and then doing the identification, or first doing the identification and then taking the
-derivative, gives the same result. The key point for this is that taking the derivative commutes
-with continuous linear equivalences. Therefore, we need to implement all our identifications with
-continuous linear equivs.
-
 ## Notations
 
 We use the notation `E [×n]→L[𝕜] F` for the space of continuous multilinear maps on `E^n` with
@@ -175,221 +111,6 @@ variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAdd
   {s s₁ t u : Set E} {f f₁ : E → F} {g : F → G} {x x₀ : E} {c : F} {m n : ℕ∞}
   {p : E → FormalMultilinearSeries 𝕜 E F}
 
-/-! ### Functions with a Taylor series on a domain -/
-
-/-- `HasFTaylorSeriesUpToOn n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a
-derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
-`HasFDerivWithinAt` but for higher order derivatives.
-
-Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
-`f` is analytic and `n = ∞`: an additional `1/m!` factor on the `m`th term is necessary for that. -/
-structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
-  (s : Set E) : Prop where
-  zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x
-  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
-    HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
-  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
-
-theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
-    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by
-  rw [← h.zero_eq x hx]
-  exact (p x 0).uncurry0_curry0.symm
-
-/-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a
-Taylor series for the second one. -/
-theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
-    (h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s := by
-  refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩
-  rw [h₁ x hx]
-  exact h.zero_eq x hx
-
-theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
-    HasFTaylorSeriesUpToOn n f p t :=
-  ⟨fun x hx => h.zero_eq x (hst hx), fun m hm x hx => (h.fderivWithin m hm x (hst hx)).mono hst,
-    fun m hm => (h.cont m hm).mono hst⟩
-
-theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
-    HasFTaylorSeriesUpToOn m f p s :=
-  ⟨h.zero_eq, fun k hk x hx => h.fderivWithin k (lt_of_lt_of_le hk hmn) x hx, fun k hk =>
-    h.cont k (le_trans hk hmn)⟩
-
-theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
-    ContinuousOn f s := by
-  have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
-  rwa [← (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_continuousOn_iff]
-
-theorem hasFTaylorSeriesUpToOn_zero_iff :
-    HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).uncurry0 = f x := by
-  refine ⟨fun H => ⟨H.continuousOn, H.zero_eq⟩, fun H =>
-      ⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm ↦ ?_⟩⟩
-  obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
-  have : EqOn (p · 0) ((continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s := fun x hx ↦
-    (continuousMultilinearCurryFin0 𝕜 E F).eq_symm_apply.2 (H.2 x hx)
-  rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]
-  exact H.1
-
-theorem hasFTaylorSeriesUpToOn_top_iff :
-    HasFTaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFTaylorSeriesUpToOn n f p s := by
-  constructor
-  · intro H n; exact H.of_le le_top
-  · intro H
-    constructor
-    · exact (H 0).zero_eq
-    · intro m _
-      apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
-    · intro m _
-      apply (H m).cont m le_rfl
-
-/-- In the case that `n = ∞` we don't need the continuity assumption in
-`HasFTaylorSeriesUpToOn`. -/
-theorem hasFTaylorSeriesUpToOn_top_iff' :
-    HasFTaylorSeriesUpToOn ∞ f p s ↔
-      (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
-        ∀ m : ℕ, ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
-  -- Everything except for the continuity is trivial:
-  ⟨fun h => ⟨h.1, fun m => h.2 m (WithTop.coe_lt_top m)⟩, fun h =>
-    ⟨h.1, fun m _ => h.2 m, fun m _ x hx =>
-      -- The continuity follows from the existence of a derivative:
-      (h.2 m x hx).continuousWithinAt⟩⟩
-
-/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
-series is a derivative of `f`. -/
-theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
-    (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x := by
-  have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) := fun y hy ↦
-    (h.zero_eq y hy).symm
-  suffices H : HasFDerivWithinAt (continuousMultilinearCurryFin0 𝕜 E F ∘ (p · 0))
-    (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x from H.congr A (A x hx)
-  rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
-  have : ((0 : ℕ) : ℕ∞) < n := zero_lt_one.trans_le hn
-  convert h.fderivWithin _ this x hx
-  ext y v
-  change (p x 1) (snoc 0 y) = (p x 1) (cons y v)
-  congr with i
-  rw [Unique.eq_default (α := Fin 1) i]
-  rfl
-
-theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
-    DifferentiableOn 𝕜 f s := fun _x hx => (h.hasFDerivWithinAt hn hx).differentiableWithinAt
-
-/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
-of order `1` of this series is a derivative of `f` at `x`. -/
-theorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
-    (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
-  (h.hasFDerivWithinAt hn (mem_of_mem_nhds hx)).hasFDerivAt hx
-
-/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
-in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/
-theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)
-    (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
-    ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
-  (eventually_eventually_nhds.2 hx).mono fun _y hy => h.hasFDerivAt hn hy
-
-/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
-it is differentiable at `x`. -/
-theorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
-    (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x :=
-  (h.hasFDerivAt hn hx).differentiableAt
-
-/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
-`p (n + 1)` is a derivative of `p n`. -/
-theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
-    HasFTaylorSeriesUpToOn (n + 1) f p s ↔
-      HasFTaylorSeriesUpToOn n f p s ∧
-        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
-          ContinuousOn (fun x => p x (n + 1)) s := by
-  constructor
-  · exact fun h ↦ ⟨h.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n)),
-      h.fderivWithin _ (WithTop.coe_lt_coe.2 (lt_add_one n)), h.cont (n + 1) le_rfl⟩
-  · intro h
-    constructor
-    · exact h.1.zero_eq
-    · intro m hm
-      by_cases h' : m < n
-      · exact h.1.fderivWithin m (WithTop.coe_lt_coe.2 h')
-      · have : m = n := Nat.eq_of_lt_succ_of_not_lt (WithTop.coe_lt_coe.1 hm) h'
-        rw [this]
-        exact h.2.1
-    · intro m hm
-      by_cases h' : m ≤ n
-      · apply h.1.cont m (WithTop.coe_le_coe.2 h')
-      · have : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')
-        rw [this]
-        exact h.2.2
-
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4119,
-without `set_option maxSynthPendingDepth 2` this proof needs substantial repair.
--/
-set_option maxSynthPendingDepth 2 in
--- Porting note: this was split out from `hasFTaylorSeriesUpToOn_succ_iff_right` to avoid a timeout.
-theorem HasFTaylorSeriesUpToOn.shift_of_succ
-    {n : ℕ} (H : HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s) :
-    (HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
-      (fun x => (p x).shift)) s := by
-  constructor
-  · intro x _
-    rfl
-  · intro m (hm : (m : ℕ∞) < n) x (hx : x ∈ s)
-    have A : (m.succ : ℕ∞) < n.succ := by
-      rw [Nat.cast_lt] at hm ⊢
-      exact Nat.succ_lt_succ hm
-    change HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ (p · m.succ))
-      (p x m.succ.succ).curryRight.curryLeft s x
-    rw [((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm).comp_hasFDerivWithinAt_iff']
-    convert H.fderivWithin _ A x hx
-    ext y v
-    change p x (m + 2) (snoc (cons y (init v)) (v (last _))) = p x (m + 2) (cons y v)
-    rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
-  · intro m (hm : (m : ℕ∞) ≤ n)
-    suffices A : ContinuousOn (p · (m + 1)) s from
-      ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm).continuous.comp_continuousOn A
-    refine H.cont _ ?_
-    rw [Nat.cast_le] at hm ⊢
-    exact Nat.succ_le_succ hm
-
-/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
-for `p 1`, which is a derivative of `f`. -/
-theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
-    HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
-      (∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
-        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
-          HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
-            (fun x => (p x).shift) s := by
-  constructor
-  · intro H
-    refine ⟨H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), ?_⟩
-    exact H.shift_of_succ
-  · rintro ⟨Hzero_eq, Hfderiv_zero, Htaylor⟩
-    constructor
-    · exact Hzero_eq
-    · intro m (hm : (m : ℕ∞) < n.succ) x (hx : x ∈ s)
-      cases' m with m
-      · exact Hfderiv_zero x hx
-      · have A : (m : ℕ∞) < n := by
-          rw [Nat.cast_lt] at hm ⊢
-          exact Nat.lt_of_succ_lt_succ hm
-        have :
-          HasFDerivWithinAt ((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm ∘ (p · m.succ))
-            ((p x).shift m.succ).curryLeft s x := Htaylor.fderivWithin _ A x hx
-        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
-        convert this
-        ext y v
-        change
-          (p x (Nat.succ (Nat.succ m))) (cons y v) =
-            (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
-        rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
-    · intro m (hm : (m : ℕ∞) ≤ n.succ)
-      cases' m with m
-      · have : DifferentiableOn 𝕜 (fun x => p x 0) s := fun x hx =>
-          (Hfderiv_zero x hx).differentiableWithinAt
-        exact this.continuousOn
-      · refine (continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm.comp_continuousOn_iff.mp ?_
-        refine Htaylor.cont _ ?_
-        rw [Nat.cast_le] at hm ⊢
-        exact Nat.lt_succ_iff.mp hm
-
 /-! ### Smooth functions within a set around a point -/
 
 variable (𝕜)
@@ -506,8 +227,8 @@ theorem ContDiffWithinAt.differentiable_within_at' (h : ContDiffWithinAt 𝕜 n
   rcases h 1 hn with ⟨u, hu, p, H⟩
   rcases mem_nhdsWithin.1 hu with ⟨t, t_open, xt, tu⟩
   rw [inter_comm] at tu
-  have := ((H.mono tu).differentiableOn le_rfl) x ⟨mem_insert x s, xt⟩
-  exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 this
+  exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 <|
+    ((H.mono tu).differentiableOn le_rfl) x ⟨mem_insert x s, xt⟩
 
 theorem ContDiffWithinAt.differentiableWithinAt (h : ContDiffWithinAt 𝕜 n f s x) (hn : 1 ≤ n) :
     DifferentiableWithinAt 𝕜 f s x :=
@@ -602,7 +323,7 @@ theorem HasFTaylorSeriesUpToOn.contDiffOn {f' : E → FormalMultilinearSeries 
     (hf : HasFTaylorSeriesUpToOn n f f' s) : ContDiffOn 𝕜 n f s := by
   intro x hx m hm
   use s
-  simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and_iff]
+  simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and]
   exact ⟨f', hf.of_le hm⟩
 
 theorem ContDiffOn.contDiffWithinAt (h : ContDiffOn 𝕜 n f s) (hx : x ∈ s) :
@@ -626,7 +347,7 @@ protected theorem ContDiffWithinAt.eventually {n : ℕ} (h : ContDiffWithinAt 
     ∀ᶠ y in 𝓝[insert x s] x, ContDiffWithinAt 𝕜 n f s y := by
   rcases h.contDiffOn le_rfl with ⟨u, hu, _, hd⟩
   have : ∀ᶠ y : E in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u :=
-    (eventually_nhdsWithin_nhdsWithin.2 hu).and hu
+    (eventually_eventually_nhdsWithin.2 hu).and hu
   refine this.mono fun y hy => (hd y hy.2).mono_of_mem ?_
   exact nhdsWithin_mono y (subset_insert _ _) hy.1
 
@@ -701,211 +422,6 @@ theorem contDiffOn_succ_iff_hasFDerivWithinAt {n : ℕ} :
     have : x ∈ u := mem_of_mem_nhdsWithin (mem_insert _ _) u_nhbd
     exact ⟨u, u_nhbd, f', hu, hf' x this⟩
 
-/-! ### Iterated derivative within a set -/
-
-
-variable (𝕜)
-
-/-- The `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th
-derivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with
-an uncurrying step to see it as a multilinear map in `n+1` variables..
--/
-noncomputable def iteratedFDerivWithin (n : ℕ) (f : E → F) (s : Set E) : E → E[×n]→L[𝕜] F :=
-  Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 𝕜 E (f x)) fun _ rec x =>
-    ContinuousLinearMap.uncurryLeft (fderivWithin 𝕜 rec s x)
-
-/-- Formal Taylor series associated to a function within a set. -/
-def ftaylorSeriesWithin (f : E → F) (s : Set E) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
-  iteratedFDerivWithin 𝕜 n f s x
-
-variable {𝕜}
-
-@[simp]
-theorem iteratedFDerivWithin_zero_apply (m : Fin 0 → E) :
-    (iteratedFDerivWithin 𝕜 0 f s x : (Fin 0 → E) → F) m = f x :=
-  rfl
-
-theorem iteratedFDerivWithin_zero_eq_comp :
-    iteratedFDerivWithin 𝕜 0 f s = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
-  rfl
-
-@[simp]
-theorem norm_iteratedFDerivWithin_zero : ‖iteratedFDerivWithin 𝕜 0 f s x‖ = ‖f x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDerivWithin_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]
-
-theorem iteratedFDerivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
-      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
-  rfl
-
-/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
-and the derivative of the `n`-th derivative. -/
-theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
-    iteratedFDerivWithin 𝕜 (n + 1) f s =
-      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F :
-          (E →L[𝕜] (E [×n]→L[𝕜] F)) → (E [×n.succ]→L[𝕜] F)) ∘
-        fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s :=
-  rfl
-
-theorem fderivWithin_iteratedFDerivWithin {s : Set E} {n : ℕ} :
-    fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s =
-      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
-        iteratedFDerivWithin 𝕜 (n + 1) f s := by
-  rw [iteratedFDerivWithin_succ_eq_comp_left]
-  ext1 x
-  simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
-
-theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
-    ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
-      ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDerivWithin_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]
-
-theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
-    (m : Fin (n + 1) → E) :
-    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
-      iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x (init m) (m (last n)) := by
-  induction' n with n IH generalizing x
-  · rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,
-      iteratedFDerivWithin_zero_apply, Function.comp_apply,
-      LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
-    rfl
-  · let I := continuousMultilinearCurryRightEquiv' 𝕜 n E F
-    have A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y =
-        (I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y := fun y hy ↦ by
-      ext m
-      rw [@IH y hy m]
-      rfl
-    calc
-      (iteratedFDerivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =
-          (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)
-            (tail m) :=
-        rfl
-      _ = (fderivWithin 𝕜 (I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
-              E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by
-        rw [fderivWithin_congr A (A x hx)]
-      _ = (I ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
-              E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by
-        #adaptation_note
-        /--
-        After https://github.com/leanprover/lean4/pull/4119 we need to either use
-        `set_option maxSynthPendingDepth 2 in`
-        or fill in an explicit argument as
-        ```
-        simp only [LinearIsometryEquiv.comp_fderivWithin _
-          (f := iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) (hs x hx)]
-        ```
-        -/
-        set_option maxSynthPendingDepth 2 in
-          simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
-        rfl
-      _ = (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :
-              E → E[×n]→L[𝕜] E →L[𝕜] F) (m 0) (init (tail m)) ((tail m) (last n)) := rfl
-      _ = iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
-            (m (last (n + 1))) := by
-        rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]
-        rfl
-
-/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
-and the `n`-th derivative of the derivative. -/
-theorem iteratedFDerivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
-    iteratedFDerivWithin 𝕜 (n + 1) f s x =
-      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘
-          iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s)
-        x := by
-  ext m; rw [iteratedFDerivWithin_succ_apply_right hs hx]; rfl
-
-theorem norm_iteratedFDerivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
-    ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ =
-      ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDerivWithin_succ_eq_comp_right hs hx, comp_apply, LinearIsometryEquiv.norm_map]
-
-@[simp]
-theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
-    iteratedFDerivWithin 𝕜 1 f s x m = fderivWithin 𝕜 f s x (m 0) := by
-  simp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,
-    (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_fderivWithin h]
-  rfl
-
-/-- On a set of unique differentiability, the second derivative is obtained by taking the
-derivative of the derivative. -/
-lemma iteratedFDerivWithin_two_apply (f : E → F) {z : E} (hs : UniqueDiffOn 𝕜 s) (hz : z ∈ s)
-    (m : Fin 2 → E) :
-    iteratedFDerivWithin 𝕜 2 f s z m = fderivWithin 𝕜 (fderivWithin 𝕜 f s) s z (m 0) (m 1) := by
-  simp only [iteratedFDerivWithin_succ_apply_right hs hz]
-  rfl
-
-theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t := by
-  induction n with
-  | zero => exact h.mono fun y hy => DFunLike.ext _ _ fun _ => hy
-  | succ n ihn =>
-    have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderivWithin' ht
-    refine this.mono fun y hy => ?_
-    simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
-
-protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s :=
-  h.iteratedFDerivWithin' Subset.rfl n
-
-/-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
-iterated differentials within this set at `x` coincide. -/
-theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
-    (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
-  have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [EventuallyEq, hx]
-  (this.iteratedFDerivWithin' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
-
-/-- If two functions coincide on a set `s`, then their iterated differentials within this set
-coincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and
-`Filter.EventuallyEq.iteratedFDerivWithin`. -/
-theorem iteratedFDerivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
-  (hs.eventuallyEq.filter_mono inf_le_right).iteratedFDerivWithin_eq (hs hx) _
-
-/-- If two functions coincide on a set `s`, then their iterated differentials within this set
-coincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and
-`Filter.EventuallyEq.iteratedFDerivWithin`. -/
-protected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f₁ f s) (n : ℕ) :
-    EqOn (iteratedFDerivWithin 𝕜 n f₁ s) (iteratedFDerivWithin 𝕜 n f s) s := fun _x hx =>
-  iteratedFDerivWithin_congr hs hx n
-
-theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t := by
-  induction n generalizing x with
-  | zero => rfl
-  | succ n ihn =>
-    refine (eventually_nhds_nhdsWithin.2 h).mono fun y hy => ?_
-    simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
-    rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
-
-theorem iteratedFDerivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t :=
-  iteratedFDerivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n
-
-theorem iteratedFDerivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
-    iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f t x :=
-  (iteratedFDerivWithin_eventually_congr_set h n).self_of_nhds
-
-/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
-`s` with a neighborhood of `x` within `s`. -/
-theorem iteratedFDerivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
-    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
-  iteratedFDerivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _
-
-/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
-`s` with a neighborhood of `x`. -/
-theorem iteratedFDerivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
-    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
-  iteratedFDerivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)
-
-/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
-`s` with an open set containing `x`. -/
-theorem iteratedFDerivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
-    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
-  iteratedFDerivWithin_inter (hu.mem_nhds hx)
-
 @[simp]
 theorem contDiffOn_zero : ContDiffOn 𝕜 0 f s ↔ ContinuousOn f s := by
   refine ⟨fun H => H.continuousOn, fun H => fun x hx m hm ↦ ?_⟩
@@ -929,36 +445,16 @@ theorem contDiffWithinAt_zero (hx : x ∈ s) :
     have h' : x ∈ s ∩ u := ⟨hx, mem_of_mem_nhdsWithin hx H⟩
     exact (contDiffOn_zero.mpr hu).contDiffWithinAt h'
 
-/-- On a set with unique differentiability, any choice of iterated differential has to coincide
-with the one we have chosen in `iteratedFDerivWithin 𝕜 m f s`. -/
-theorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
-    (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
-    (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x := by
-  induction' m with m IH generalizing x
-  · rw [h.zero_eq' hx, iteratedFDerivWithin_zero_eq_comp]; rfl
-  · have A : (m : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn
-    have :
-      HasFDerivWithinAt (fun y : E => iteratedFDerivWithin 𝕜 m f s y)
-        (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=
-      (h.fderivWithin m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm)
-        (IH (le_of_lt A) hx).symm
-    rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderivWithin (hs x hx)]
-    exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
-
-@[deprecated (since := "2024-03-28")]
-alias HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn :=
-  HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
 `ftaylorSeriesWithin 𝕜 f s` as a Taylor series up to order `n` in `s`. -/
 protected theorem ContDiffOn.ftaylorSeriesWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s) :
     HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s := by
   constructor
   · intro x _
-    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
+    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.curry0_apply,
       iteratedFDerivWithin_zero_apply]
   · intro m hm x hx
-    rcases (h x hx) m.succ (ENat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
+    rcases (h x hx) m.succ (Order.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩
     rw [insert_eq_of_mem hx] at hu
     rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
     rw [inter_comm] at ho
@@ -1002,7 +498,7 @@ theorem contDiffOn_of_continuousOn_differentiableOn
   refine ⟨s, self_mem_nhdsWithin, ftaylorSeriesWithin 𝕜 f s, ?_⟩
   constructor
   · intro y _
-    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.uncurry0_apply,
+    simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.curry0_apply,
       iteratedFDerivWithin_zero_apply]
   · intro k hk y hy
     convert (Hdiff k (lt_of_lt_of_le hk hm) y hy).hasFDerivWithinAt
@@ -1027,7 +523,7 @@ theorem ContDiffOn.differentiableOn_iteratedFDerivWithin {m : ℕ} (h : ContDiff
 theorem ContDiffWithinAt.differentiableWithinAt_iteratedFDerivWithin {m : ℕ}
     (h : ContDiffWithinAt 𝕜 n f s x) (hmn : (m : ℕ∞) < n) (hs : UniqueDiffOn 𝕜 (insert x s)) :
     DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x := by
-  rcases h.contDiffOn' (ENat.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩
+  rcases h.contDiffOn' (Order.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩
   set t := insert x s ∩ u
   have A : t =ᶠ[𝓝[≠] x] s := by
     simp only [set_eventuallyEq_iff_inf_principal, ← nhdsWithin_inter']
@@ -1139,99 +635,6 @@ theorem ContDiffOn.continuousOn_fderiv_of_isOpen (h : ContDiffOn 𝕜 n f s) (hs
     (hn : 1 ≤ n) : ContinuousOn (fun x => fderiv 𝕜 f x) s :=
   ((contDiffOn_succ_iff_fderiv_of_isOpen hs).1 (h.of_le hn)).2.continuousOn
 
-/-! ### Functions with a Taylor series on the whole space -/
-
-/-- `HasFTaylorSeriesUpTo n f p` registers the fact that `p 0 = f` and `p (m+1)` is a
-derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
-`HasFDerivAt` but for higher order derivatives.
-
-Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
-`f` is analytic and `n = ∞`: an addition `1/m!` factor on the `m`th term is necessary for that. -/
-structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
-  Prop where
-  zero_eq : ∀ x, (p x 0).uncurry0 = f x
-  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
-  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
-
-theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :
-    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by
-  rw [← h.zero_eq x]
-  exact (p x 0).uncurry0_curry0.symm
-
-theorem hasFTaylorSeriesUpToOn_univ_iff :
-    HasFTaylorSeriesUpToOn n f p univ ↔ HasFTaylorSeriesUpTo n f p := by
-  constructor
-  · intro H
-    constructor
-    · exact fun x => H.zero_eq x (mem_univ x)
-    · intro m hm x
-      rw [← hasFDerivWithinAt_univ]
-      exact H.fderivWithin m hm x (mem_univ x)
-    · intro m hm
-      rw [continuous_iff_continuousOn_univ]
-      exact H.cont m hm
-  · intro H
-    constructor
-    · exact fun x _ => H.zero_eq x
-    · intro m hm x _
-      rw [hasFDerivWithinAt_univ]
-      exact H.fderiv m hm x
-    · intro m hm
-      rw [← continuous_iff_continuousOn_univ]
-      exact H.cont m hm
-
-theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :
-    HasFTaylorSeriesUpToOn n f p s :=
-  (hasFTaylorSeriesUpToOn_univ_iff.2 h).mono (subset_univ _)
-
-theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
-    HasFTaylorSeriesUpTo m f p := by
-  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn
-
-theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f := by
-  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
-  rw [continuous_iff_continuousOn_univ]
-  exact h.continuousOn
-
-theorem hasFTaylorSeriesUpTo_zero_iff :
-    HasFTaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ x, (p x 0).uncurry0 = f x := by
-  simp [hasFTaylorSeriesUpToOn_univ_iff.symm, continuous_iff_continuousOn_univ,
-    hasFTaylorSeriesUpToOn_zero_iff]
-
-theorem hasFTaylorSeriesUpTo_top_iff :
-    HasFTaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFTaylorSeriesUpTo n f p := by
-  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]
-
-/-- In the case that `n = ∞` we don't need the continuity assumption in
-`HasFTaylorSeriesUpTo`. -/
-theorem hasFTaylorSeriesUpTo_top_iff' :
-    HasFTaylorSeriesUpTo ∞ f p ↔
-      (∀ x, (p x 0).uncurry0 = f x) ∧
-        ∀ (m : ℕ) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x := by
-  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff', mem_univ,
-    forall_true_left, hasFDerivWithinAt_univ]
-
-/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
-series is a derivative of `f`. -/
-theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
-    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x := by
-  rw [← hasFDerivWithinAt_univ]
-  exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _)
-
-theorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
-    Differentiable 𝕜 f := fun x => (h.hasFDerivAt hn x).differentiableAt
-
-/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
-for `p 1`, which is a derivative of `f`. -/
-theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
-    HasFTaylorSeriesUpTo (n + 1 : ℕ) f p ↔
-      (∀ x, (p x 0).uncurry0 = f x) ∧
-        (∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
-          HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
-            (p x).shift := by
-  simp only [hasFTaylorSeriesUpToOn_succ_iff_right, ← hasFTaylorSeriesUpToOn_univ_iff, mem_univ,
-    forall_true_left, hasFDerivWithinAt_univ]
-
 /-! ### Smooth functions at a point -/
 
 variable (𝕜)
@@ -1392,152 +795,6 @@ theorem contDiff_one_iff_hasFDerivAt : ContDiff 𝕜 1 f ↔
     ∃ f' : E → E →L[𝕜] F, Continuous f' ∧ ∀ x, HasFDerivAt f (f' x) x := by
   convert contDiff_succ_iff_hasFDerivAt using 4; simp
 
-/-! ### Iterated derivative -/
-
-
-variable (𝕜)
-
-/-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -/
-noncomputable def iteratedFDeriv (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F :=
-  Nat.recOn n (fun x => ContinuousMultilinearMap.curry0 𝕜 E (f x)) fun _ rec x =>
-    ContinuousLinearMap.uncurryLeft (fderiv 𝕜 rec x)
-
-/-- Formal Taylor series associated to a function. -/
-def ftaylorSeries (f : E → F) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
-  iteratedFDeriv 𝕜 n f x
-
-variable {𝕜}
-
-@[simp]
-theorem iteratedFDeriv_zero_apply (m : Fin 0 → E) :
-    (iteratedFDeriv 𝕜 0 f x : (Fin 0 → E) → F) m = f x :=
-  rfl
-
-theorem iteratedFDeriv_zero_eq_comp :
-    iteratedFDeriv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
-  rfl
-
-@[simp]
-theorem norm_iteratedFDeriv_zero : ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]
-
-theorem iteratedFDerivWithin_zero_eq : iteratedFDerivWithin 𝕜 0 f s = iteratedFDeriv 𝕜 0 f := rfl
-
-theorem iteratedFDeriv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
-      (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
-  rfl
-
-/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
-and the derivative of the `n`-th derivative. -/
-theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
-    iteratedFDeriv 𝕜 (n + 1) f =
-      continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F ∘
-        fderiv 𝕜 (iteratedFDeriv 𝕜 n f) :=
-  rfl
-
-/-- Writing explicitly the derivative of the `n`-th derivative as the composition of a currying
-linear equiv, and the `n + 1`-th derivative. -/
-theorem fderiv_iteratedFDeriv {n : ℕ} :
-    fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
-      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
-        iteratedFDeriv 𝕜 (n + 1) f := by
-  rw [iteratedFDeriv_succ_eq_comp_left]
-  ext1 x
-  simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
-
-theorem tsupport_iteratedFDeriv_subset (n : ℕ) : tsupport (iteratedFDeriv 𝕜 n f) ⊆ tsupport f := by
-  induction n with
-  | zero =>
-    rw [iteratedFDeriv_zero_eq_comp]
-    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
-      subset_closure) isClosed_closure
-  | succ n IH =>
-    rw [iteratedFDeriv_succ_eq_comp_left]
-    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
-      ((support_fderiv_subset 𝕜).trans IH)) isClosed_closure
-
-theorem support_iteratedFDeriv_subset (n : ℕ) : support (iteratedFDeriv 𝕜 n f) ⊆ tsupport f :=
-  subset_closure.trans (tsupport_iteratedFDeriv_subset n)
-
-theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
-    HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
-  hf.of_isClosed_subset isClosed_closure (tsupport_iteratedFDeriv_subset n)
-
-theorem norm_fderiv_iteratedFDeriv {n : ℕ} :
-    ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]
-
-theorem iteratedFDerivWithin_univ {n : ℕ} :
-    iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f := by
-  induction n with
-  | zero => ext x; simp
-  | succ n IH =>
-    ext x m
-    rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
-
-theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv
-    (h : HasFTaylorSeriesUpTo n f p) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (x : E) :
-    p x m = iteratedFDeriv 𝕜 m f x := by
-  rw [← iteratedFDerivWithin_univ]
-  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
-  exact h.eq_iteratedFDerivWithin_of_uniqueDiffOn hmn uniqueDiffOn_univ (mem_univ _)
-
-/-- In an open set, the iterated derivative within this set coincides with the global iterated
-derivative. -/
-theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
-    EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s := by
-  induction n with
-  | zero =>
-    intro x _
-    ext1
-    simp only [iteratedFDerivWithin_zero_apply, iteratedFDeriv_zero_apply]
-  | succ n IH =>
-    intro x hx
-    rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
-    dsimp
-    congr 1
-    rw [fderivWithin_of_isOpen hs hx]
-    apply Filter.EventuallyEq.fderiv_eq
-    filter_upwards [hs.mem_nhds hx]
-    exact IH
-
-theorem ftaylorSeriesWithin_univ : ftaylorSeriesWithin 𝕜 f univ = ftaylorSeries 𝕜 f := by
-  ext1 x; ext1 n
-  change iteratedFDerivWithin 𝕜 n f univ x = iteratedFDeriv 𝕜 n f x
-  rw [iteratedFDerivWithin_univ]
-
-theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
-    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
-      iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) := by
-  rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ, ← fderivWithin_univ]
-  exact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _
-
-/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
-and the `n`-th derivative of the derivative. -/
-theorem iteratedFDeriv_succ_eq_comp_right {n : ℕ} :
-    iteratedFDeriv 𝕜 (n + 1) f x =
-      (continuousMultilinearCurryRightEquiv' 𝕜 n E F ∘ iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y)
-        x := by
-  ext m; rw [iteratedFDeriv_succ_apply_right]; rfl
-
-theorem norm_iteratedFDeriv_fderiv {n : ℕ} :
-    ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
-  -- Porting note: added `comp_apply`.
-  rw [iteratedFDeriv_succ_eq_comp_right, comp_apply, LinearIsometryEquiv.norm_map]
-
-@[simp]
-theorem iteratedFDeriv_one_apply (m : Fin 1 → E) :
-    iteratedFDeriv 𝕜 1 f x m = fderiv 𝕜 f x (m 0) := by
-  rw [iteratedFDeriv_succ_apply_right, iteratedFDeriv_zero_apply]; rfl
-
-lemma iteratedFDeriv_two_apply (f : E → F) (z : E) (m : Fin 2 → E) :
-    iteratedFDeriv 𝕜 2 f z m = fderiv 𝕜 (fderiv 𝕜 f) z (m 0) (m 1) := by
-  simp only [iteratedFDeriv_succ_apply_right]
-  rfl
-
 /-- When a function is `C^n` in a set `s` of unique differentiability, it admits
 `ftaylorSeriesWithin 𝕜 f s` as a Taylor series up to order `n` in `s`. -/
 theorem contDiff_iff_ftaylorSeries :
diff --git a/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
new file mode 100644
index 0000000000000..bce65e8bbdf7d
--- /dev/null
+++ b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
@@ -0,0 +1,796 @@
+/-
+Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sébastien Gouëzel
+-/
+import Mathlib.Analysis.Calculus.FDeriv.Equiv
+import Mathlib.Analysis.Calculus.FormalMultilinearSeries
+
+/-!
+# Iterated derivatives of a function
+
+In this file, we define iteratively the `n+1`-th derivative of a function as the
+derivative of the `n`-th derivative. It is called `iteratedFDeriv 𝕜 n f x` where `𝕜` is the
+field, `n` is the number of iterations, `f` is the function and `x` is the point, and it is given
+as an `n`-multilinear map. We also define a version `iteratedFDerivWithin` relative to a domain.
+Note that, in domains, there may be several choices of possible derivative, so we make some
+arbitrary choice in the definition.
+
+We also define a predicate `HasFTaylorSeriesUpTo` (and its localized version
+`HasFTaylorSeriesUpToOn`), saying that a sequence of multilinear maps is *a* sequence of
+derivatives of `f`. Contrary to `iteratedFDerivWithin`, it accomodates well the
+non-uniqueness of derivatives.
+
+## Main definitions and results
+
+Let `f : E → F` be a map between normed vector spaces over a nontrivially normed field `𝕜`.
+
+* `HasFTaylorSeriesUpTo n f p`: expresses that the formal multilinear series `p` is a sequence
+  of iterated derivatives of `f`, up to the `n`-th term (where `n` is a natural number or `∞`).
+* `HasFTaylorSeriesUpToOn n f p s`: same thing, but inside a set `s`. The notion of derivative
+  is now taken inside `s`. In particular, derivatives don't have to be unique.
+
+* `iteratedFDerivWithin 𝕜 n f s x` is an `n`-th derivative of `f` over the field `𝕜` on the
+  set `s` at the point `x`. It is a continuous multilinear map from `E^n` to `F`, defined as a
+  derivative within `s` of `iteratedFDerivWithin 𝕜 (n-1) f s` if one exists, and `0` otherwise.
+* `iteratedFDeriv 𝕜 n f x` is the `n`-th derivative of `f` over the field `𝕜` at the point `x`.
+  It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of
+  `iteratedFDeriv 𝕜 (n-1) f` if one exists, and `0` otherwise.
+
+
+### Side of the composition, and universe issues
+
+With a naïve direct definition, the `n`-th derivative of a function belongs to the space
+`E →L[𝕜] (E →L[𝕜] (E ... F)...)))` where there are n iterations of `E →L[𝕜]`. This space
+may also be seen as the space of continuous multilinear functions on `n` copies of `E` with
+values in `F`, by uncurrying. This is the point of view that is usually adopted in textbooks,
+and that we also use. This means that the definition and the first proofs are slightly involved,
+as one has to keep track of the uncurrying operation. The uncurrying can be done from the
+left or from the right, amounting to defining the `n+1`-th derivative either as the derivative of
+the `n`-th derivative, or as the `n`-th derivative of the derivative.
+For proofs, it would be more convenient to use the latter approach (from the right),
+as it means to prove things at the `n+1`-th step we only need to understand well enough the
+derivative in `E →L[𝕜] F` (contrary to the approach from the left, where one would need to know
+enough on the `n`-th derivative to deduce things on the `n+1`-th derivative).
+
+However, the definition from the right leads to a universe polymorphism problem: if we define
+`iteratedFDeriv 𝕜 (n + 1) f x = iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x` by induction, we need to
+generalize over all spaces (as `f` and `fderiv 𝕜 f` don't take values in the same space). It is
+only possible to generalize over all spaces in some fixed universe in an inductive definition.
+For `f : E → F`, then `fderiv 𝕜 f` is a map `E → (E →L[𝕜] F)`. Therefore, the definition will only
+work if `F` and `E →L[𝕜] F` are in the same universe.
+
+This issue does not appear with the definition from the left, where one does not need to generalize
+over all spaces. Therefore, we use the definition from the left. This means some proofs later on
+become a little bit more complicated: to prove that a function is `C^n`, the most efficient approach
+is to exhibit a formula for its `n`-th derivative and prove it is continuous (contrary to the
+inductive approach where one would prove smoothness statements without giving a formula for the
+derivative). In the end, this approach is still satisfactory as it is good to have formulas for the
+iterated derivatives in various constructions.
+
+One point where we depart from this explicit approach is in the proof of smoothness of a
+composition: there is a formula for the `n`-th derivative of a composition (Faà di Bruno's formula),
+but it is very complicated and barely usable, while the inductive proof is very simple. Thus, we
+give the inductive proof. As explained above, it works by generalizing over the target space, hence
+it only works well if all spaces belong to the same universe. To get the general version, we lift
+things to a common universe using a trick.
+
+### Variables management
+
+The textbook definitions and proofs use various identifications and abuse of notations, for instance
+when saying that the natural space in which the derivative lives, i.e.,
+`E →L[𝕜] (E →L[𝕜] ( ... →L[𝕜] F))`, is the same as a space of multilinear maps. When doing things
+formally, we need to provide explicit maps for these identifications, and chase some diagrams to see
+everything is compatible with the identifications. In particular, one needs to check that taking the
+derivative and then doing the identification, or first doing the identification and then taking the
+derivative, gives the same result. The key point for this is that taking the derivative commutes
+with continuous linear equivalences. Therefore, we need to implement all our identifications with
+continuous linear equivs.
+
+## Notations
+
+We use the notation `E [×n]→L[𝕜] F` for the space of continuous multilinear maps on `E^n` with
+values in `F`. This is the space in which the `n`-th derivative of a function from `E` to `F` lives.
+
+In this file, we denote `⊤ : ℕ∞` with `∞`.
+-/
+
+
+noncomputable section
+
+open scoped Classical
+open NNReal Topology Filter
+
+local notation "∞" => (⊤ : ℕ∞)
+
+/-
+Porting note: These lines are not required in Mathlib4.
+attribute [local instance 1001]
+  NormedAddCommGroup.toAddCommGroup NormedSpace.toModule' AddCommGroup.toAddCommMonoid
+-/
+
+open Set Fin Filter Function
+
+universe u uE uF uG uX
+
+variable {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Type uE} [NormedAddCommGroup E]
+  [NormedSpace 𝕜 E] {F : Type uF} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type uG}
+  [NormedAddCommGroup G] [NormedSpace 𝕜 G] {X : Type uX} [NormedAddCommGroup X] [NormedSpace 𝕜 X]
+  {s s₁ t u : Set E} {f f₁ : E → F} {g : F → G} {x x₀ : E} {c : F} {m n : ℕ∞}
+  {p : E → FormalMultilinearSeries 𝕜 E F}
+
+/-! ### Functions with a Taylor series on a domain -/
+
+/-- `HasFTaylorSeriesUpToOn n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a
+derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
+`HasFDerivWithinAt` but for higher order derivatives.
+
+Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
+`f` is analytic and `n = ∞`: an additional `1/m!` factor on the `m`th term is necessary for that. -/
+structure HasFTaylorSeriesUpToOn (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F)
+  (s : Set E) : Prop where
+  zero_eq : ∀ x ∈ s, (p x 0).curry0 = f x
+  protected fderivWithin : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x ∈ s,
+    HasFDerivWithinAt (p · m) (p x m.succ).curryLeft s x
+  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → ContinuousOn (p · m) s
+
+theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x ∈ s) :
+    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by
+  rw [← h.zero_eq x hx]
+  exact (p x 0).uncurry0_curry0.symm
+
+/-- If two functions coincide on a set `s`, then a Taylor series for the first one is as well a
+Taylor series for the second one. -/
+theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
+    (h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s := by
+  refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩
+  rw [h₁ x hx]
+  exact h.zero_eq x hx
+
+theorem HasFTaylorSeriesUpToOn.mono (h : HasFTaylorSeriesUpToOn n f p s) {t : Set E} (hst : t ⊆ s) :
+    HasFTaylorSeriesUpToOn n f p t :=
+  ⟨fun x hx => h.zero_eq x (hst hx), fun m hm x hx => (h.fderivWithin m hm x (hst hx)).mono hst,
+    fun m hm => (h.cont m hm).mono hst⟩
+
+theorem HasFTaylorSeriesUpToOn.of_le (h : HasFTaylorSeriesUpToOn n f p s) (hmn : m ≤ n) :
+    HasFTaylorSeriesUpToOn m f p s :=
+  ⟨h.zero_eq, fun k hk x hx => h.fderivWithin k (lt_of_lt_of_le hk hmn) x hx, fun k hk =>
+    h.cont k (le_trans hk hmn)⟩
+
+theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
+    ContinuousOn f s := by
+  have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
+  rwa [← (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_continuousOn_iff]
+
+theorem hasFTaylorSeriesUpToOn_zero_iff :
+    HasFTaylorSeriesUpToOn 0 f p s ↔ ContinuousOn f s ∧ ∀ x ∈ s, (p x 0).curry0 = f x := by
+  refine ⟨fun H => ⟨H.continuousOn, H.zero_eq⟩, fun H =>
+      ⟨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm ↦ ?_⟩⟩
+  obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
+  have : EqOn (p · 0) ((continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f) s := fun x hx ↦
+    (continuousMultilinearCurryFin0 𝕜 E F).eq_symm_apply.2 (H.2 x hx)
+  rw [continuousOn_congr this, LinearIsometryEquiv.comp_continuousOn_iff]
+  exact H.1
+
+theorem hasFTaylorSeriesUpToOn_top_iff :
+    HasFTaylorSeriesUpToOn ∞ f p s ↔ ∀ n : ℕ, HasFTaylorSeriesUpToOn n f p s := by
+  constructor
+  · intro H n; exact H.of_le le_top
+  · intro H
+    constructor
+    · exact (H 0).zero_eq
+    · intro m _
+      apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
+    · intro m _
+      apply (H m).cont m le_rfl
+
+/-- In the case that `n = ∞` we don't need the continuity assumption in
+`HasFTaylorSeriesUpToOn`. -/
+theorem hasFTaylorSeriesUpToOn_top_iff' :
+    HasFTaylorSeriesUpToOn ∞ f p s ↔
+      (∀ x ∈ s, (p x 0).curry0 = f x) ∧
+        ∀ m : ℕ, ∀ x ∈ s, HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft s x :=
+  -- Everything except for the continuity is trivial:
+  ⟨fun h => ⟨h.1, fun m => h.2 m (WithTop.coe_lt_top m)⟩, fun h =>
+    ⟨h.1, fun m _ => h.2 m, fun m _ x hx =>
+      -- The continuity follows from the existence of a derivative:
+      (h.2 m x hx).continuousWithinAt⟩⟩
+
+/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
+series is a derivative of `f`. -/
+theorem HasFTaylorSeriesUpToOn.hasFDerivWithinAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+    (hx : x ∈ s) : HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x := by
+  have A : ∀ y ∈ s, f y = (continuousMultilinearCurryFin0 𝕜 E F) (p y 0) := fun y hy ↦
+    (h.zero_eq y hy).symm
+  suffices H : HasFDerivWithinAt (continuousMultilinearCurryFin0 𝕜 E F ∘ (p · 0))
+    (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) s x from H.congr A (A x hx)
+  rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff']
+  have : ((0 : ℕ) : ℕ∞) < n := zero_lt_one.trans_le hn
+  convert h.fderivWithin _ this x hx
+  ext y v
+  change (p x 1) (snoc 0 y) = (p x 1) (cons y v)
+  congr with i
+  rw [Unique.eq_default (α := Fin 1) i]
+  rfl
+
+theorem HasFTaylorSeriesUpToOn.differentiableOn (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n) :
+    DifferentiableOn 𝕜 f s := fun _x hx => (h.hasFDerivWithinAt hn hx).differentiableWithinAt
+
+/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
+of order `1` of this series is a derivative of `f` at `x`. -/
+theorem HasFTaylorSeriesUpToOn.hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+    (hx : s ∈ 𝓝 x) : HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x :=
+  (h.hasFDerivWithinAt hn (mem_of_mem_nhds hx)).hasFDerivAt hx
+
+/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
+in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. -/
+theorem HasFTaylorSeriesUpToOn.eventually_hasFDerivAt (h : HasFTaylorSeriesUpToOn n f p s)
+    (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
+    ∀ᶠ y in 𝓝 x, HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p y 1)) y :=
+  (eventually_eventually_nhds.2 hx).mono fun _y hy => h.hasFDerivAt hn hy
+
+/-- If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
+it is differentiable at `x`. -/
+theorem HasFTaylorSeriesUpToOn.differentiableAt (h : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
+    (hx : s ∈ 𝓝 x) : DifferentiableAt 𝕜 f x :=
+  (h.hasFDerivAt hn hx).differentiableAt
+
+/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
+`p (n + 1)` is a derivative of `p n`. -/
+theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
+    HasFTaylorSeriesUpToOn (n + 1) f p s ↔
+      HasFTaylorSeriesUpToOn n f p s ∧
+        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y n) (p x n.succ).curryLeft s x) ∧
+          ContinuousOn (fun x => p x (n + 1)) s := by
+  constructor
+  · exact fun h ↦ ⟨h.of_le (WithTop.coe_le_coe.2 (Nat.le_succ n)),
+      h.fderivWithin _ (WithTop.coe_lt_coe.2 (lt_add_one n)), h.cont (n + 1) le_rfl⟩
+  · intro h
+    constructor
+    · exact h.1.zero_eq
+    · intro m hm
+      by_cases h' : m < n
+      · exact h.1.fderivWithin m (WithTop.coe_lt_coe.2 h')
+      · have : m = n := Nat.eq_of_lt_succ_of_not_lt (WithTop.coe_lt_coe.1 hm) h'
+        rw [this]
+        exact h.2.1
+    · intro m hm
+      by_cases h' : m ≤ n
+      · apply h.1.cont m (WithTop.coe_le_coe.2 h')
+      · have : m = n + 1 := le_antisymm (WithTop.coe_le_coe.1 hm) (not_le.1 h')
+        rw [this]
+        exact h.2.2
+
+#adaptation_note
+/--
+After https://github.com/leanprover/lean4/pull/4119,
+without `set_option maxSynthPendingDepth 2` this proof needs substantial repair.
+-/
+set_option maxSynthPendingDepth 2 in
+-- Porting note: this was split out from `hasFTaylorSeriesUpToOn_succ_iff_right` to avoid a timeout.
+theorem HasFTaylorSeriesUpToOn.shift_of_succ
+    {n : ℕ} (H : HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s) :
+    (HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
+      (fun x => (p x).shift)) s := by
+  constructor
+  · intro x _
+    rfl
+  · intro m (hm : (m : ℕ∞) < n) x (hx : x ∈ s)
+    have A : (m.succ : ℕ∞) < n.succ := by
+      rw [Nat.cast_lt] at hm ⊢
+      exact Nat.succ_lt_succ hm
+    change HasFDerivWithinAt (continuousMultilinearCurryRightEquiv' 𝕜 m E F ∘ (p · m.succ))
+      (p x m.succ.succ).curryRight.curryLeft s x
+    rw [(continuousMultilinearCurryRightEquiv' 𝕜 m E F).comp_hasFDerivWithinAt_iff']
+    convert H.fderivWithin _ A x hx
+    ext y v
+    change p x (m + 2) (snoc (cons y (init v)) (v (last _))) = p x (m + 2) (cons y v)
+    rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
+  · intro m (hm : (m : ℕ∞) ≤ n)
+    suffices A : ContinuousOn (p · (m + 1)) s from
+      (continuousMultilinearCurryRightEquiv' 𝕜 m E F).continuous.comp_continuousOn A
+    refine H.cont _ ?_
+    rw [Nat.cast_le] at hm ⊢
+    exact Nat.succ_le_succ hm
+
+/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
+for `p 1`, which is a derivative of `f`. -/
+theorem hasFTaylorSeriesUpToOn_succ_iff_right {n : ℕ} :
+    HasFTaylorSeriesUpToOn (n + 1 : ℕ) f p s ↔
+      (∀ x ∈ s, (p x 0).curry0 = f x) ∧
+        (∀ x ∈ s, HasFDerivWithinAt (fun y => p y 0) (p x 1).curryLeft s x) ∧
+          HasFTaylorSeriesUpToOn n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1))
+            (fun x => (p x).shift) s := by
+  constructor
+  · intro H
+    refine ⟨H.zero_eq, H.fderivWithin 0 (Nat.cast_lt.2 (Nat.succ_pos n)), ?_⟩
+    exact H.shift_of_succ
+  · rintro ⟨Hzero_eq, Hfderiv_zero, Htaylor⟩
+    constructor
+    · exact Hzero_eq
+    · intro m (hm : (m : ℕ∞) < n.succ) x (hx : x ∈ s)
+      cases' m with m
+      · exact Hfderiv_zero x hx
+      · have A : (m : ℕ∞) < n := by
+          rw [Nat.cast_lt] at hm ⊢
+          exact Nat.lt_of_succ_lt_succ hm
+        have :
+          HasFDerivWithinAt (continuousMultilinearCurryRightEquiv' 𝕜 m E F ∘ (p · m.succ))
+            ((p x).shift m.succ).curryLeft s x := Htaylor.fderivWithin _ A x hx
+        rw [LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'] at this
+        convert this
+        ext y v
+        change
+          (p x (Nat.succ (Nat.succ m))) (cons y v) =
+            (p x m.succ.succ) (snoc (cons y (init v)) (v (last _)))
+        rw [← cons_snoc_eq_snoc_cons, snoc_init_self]
+    · intro m (hm : (m : ℕ∞) ≤ n.succ)
+      cases' m with m
+      · have : DifferentiableOn 𝕜 (fun x => p x 0) s := fun x hx =>
+          (Hfderiv_zero x hx).differentiableWithinAt
+        exact this.continuousOn
+      · refine (continuousMultilinearCurryRightEquiv' 𝕜 m E F).comp_continuousOn_iff.mp ?_
+        refine Htaylor.cont _ ?_
+        rw [Nat.cast_le] at hm ⊢
+        exact Nat.lt_succ_iff.mp hm
+
+/-! ### Iterated derivative within a set -/
+
+
+variable (𝕜)
+
+/-- The `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th
+derivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with
+an uncurrying step to see it as a multilinear map in `n+1` variables..
+-/
+noncomputable def iteratedFDerivWithin (n : ℕ) (f : E → F) (s : Set E) : E → E[×n]→L[𝕜] F :=
+  Nat.recOn n (fun x => ContinuousMultilinearMap.uncurry0 𝕜 E (f x)) fun _ rec x =>
+    ContinuousLinearMap.uncurryLeft (fderivWithin 𝕜 rec s x)
+
+/-- Formal Taylor series associated to a function within a set. -/
+def ftaylorSeriesWithin (f : E → F) (s : Set E) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
+  iteratedFDerivWithin 𝕜 n f s x
+
+variable {𝕜}
+
+@[simp]
+theorem iteratedFDerivWithin_zero_apply (m : Fin 0 → E) :
+    (iteratedFDerivWithin 𝕜 0 f s x : (Fin 0 → E) → F) m = f x :=
+  rfl
+
+theorem iteratedFDerivWithin_zero_eq_comp :
+    iteratedFDerivWithin 𝕜 0 f s = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
+  rfl
+
+@[simp]
+theorem norm_iteratedFDerivWithin_zero : ‖iteratedFDerivWithin 𝕜 0 f s x‖ = ‖f x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDerivWithin_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]
+
+theorem iteratedFDerivWithin_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
+      (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
+  rfl
+
+/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
+and the derivative of the `n`-th derivative. -/
+theorem iteratedFDerivWithin_succ_eq_comp_left {n : ℕ} :
+    iteratedFDerivWithin 𝕜 (n + 1) f s =
+      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
+        fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s :=
+  rfl
+
+theorem fderivWithin_iteratedFDerivWithin {s : Set E} {n : ℕ} :
+    fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s =
+      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F) ∘
+        iteratedFDerivWithin 𝕜 (n + 1) f s :=
+  rfl
+
+theorem norm_fderivWithin_iteratedFDerivWithin {n : ℕ} :
+    ‖fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n f s) s x‖ =
+      ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDerivWithin_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]
+
+theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s)
+    (m : Fin (n + 1) → E) :
+    (iteratedFDerivWithin 𝕜 (n + 1) f s x : (Fin (n + 1) → E) → F) m =
+      iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s x (init m) (m (last n)) := by
+  induction' n with n IH generalizing x
+  · rw [iteratedFDerivWithin_succ_eq_comp_left, iteratedFDerivWithin_zero_eq_comp,
+      iteratedFDerivWithin_zero_apply, Function.comp_apply,
+      LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
+    rfl
+  · let I := (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm
+    have A : ∀ y ∈ s, iteratedFDerivWithin 𝕜 n.succ f s y =
+        (I ∘ iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) y := fun y hy ↦ by
+      ext m
+      rw [@IH y hy m]
+      rfl
+    calc
+      (iteratedFDerivWithin 𝕜 (n + 2) f s x : (Fin (n + 2) → E) → F) m =
+          (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n.succ f s) s x : E → E[×n + 1]→L[𝕜] F) (m 0)
+            (tail m) :=
+        rfl
+      _ = (fderivWithin 𝕜 (I ∘ iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
+              E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by
+        rw [fderivWithin_congr A (A x hx)]
+      _ = (I ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
+              E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by
+        #adaptation_note
+        /--
+        After https://github.com/leanprover/lean4/pull/4119 we need to either use
+        `set_option maxSynthPendingDepth 2 in`
+        or fill in an explicit argument as
+        ```
+        simp only [LinearIsometryEquiv.comp_fderivWithin _
+          (f := iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) (hs x hx)]
+        ```
+        -/
+        set_option maxSynthPendingDepth 2 in
+          simp only [LinearIsometryEquiv.comp_fderivWithin _ (hs x hx)]
+        rfl
+      _ = (fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s) s x :
+              E → E[×n]→L[𝕜] E →L[𝕜] F) (m 0) (init (tail m)) ((tail m) (last n)) := rfl
+      _ = iteratedFDerivWithin 𝕜 (Nat.succ n) (fun y => fderivWithin 𝕜 f s y) s x (init m)
+            (m (last (n + 1))) := by
+        rw [iteratedFDerivWithin_succ_apply_left, tail_init_eq_init_tail]
+        rfl
+
+/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
+and the `n`-th derivative of the derivative. -/
+theorem iteratedFDerivWithin_succ_eq_comp_right {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
+    iteratedFDerivWithin 𝕜 (n + 1) f s x =
+      ((continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm ∘
+          iteratedFDerivWithin 𝕜 n (fun y => fderivWithin 𝕜 f s y) s)
+        x := by
+  ext m; rw [iteratedFDerivWithin_succ_apply_right hs hx]; rfl
+
+theorem norm_iteratedFDerivWithin_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
+    ‖iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s x‖ =
+      ‖iteratedFDerivWithin 𝕜 (n + 1) f s x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDerivWithin_succ_eq_comp_right hs hx, comp_apply, LinearIsometryEquiv.norm_map]
+
+@[simp]
+theorem iteratedFDerivWithin_one_apply (h : UniqueDiffWithinAt 𝕜 s x) (m : Fin 1 → E) :
+    iteratedFDerivWithin 𝕜 1 f s x m = fderivWithin 𝕜 f s x (m 0) := by
+  simp only [iteratedFDerivWithin_succ_apply_left, iteratedFDerivWithin_zero_eq_comp,
+    (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_fderivWithin h]
+  rfl
+
+/-- On a set of unique differentiability, the second derivative is obtained by taking the
+derivative of the derivative. -/
+lemma iteratedFDerivWithin_two_apply (f : E → F) {z : E} (hs : UniqueDiffOn 𝕜 s) (hz : z ∈ s)
+    (m : Fin 2 → E) :
+    iteratedFDerivWithin 𝕜 2 f s z m = fderivWithin 𝕜 (fderivWithin 𝕜 f s) s z (m 0) (m 1) := by
+  simp only [iteratedFDerivWithin_succ_apply_right hs hz]
+  rfl
+
+theorem Filter.EventuallyEq.iteratedFDerivWithin' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f₁ t =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f t := by
+  induction n with
+  | zero => exact h.mono fun y hy => DFunLike.ext _ _ fun _ => hy
+  | succ n ihn =>
+    have : fderivWithin 𝕜 _ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 _ t := ihn.fderivWithin' ht
+    refine this.mono fun y hy => ?_
+    simp only [iteratedFDerivWithin_succ_eq_comp_left, hy, (· ∘ ·)]
+
+protected theorem Filter.EventuallyEq.iteratedFDerivWithin (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f₁ s =ᶠ[𝓝[s] x] iteratedFDerivWithin 𝕜 n f s :=
+  h.iteratedFDerivWithin' Subset.rfl n
+
+/-- If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
+iterated differentials within this set at `x` coincide. -/
+theorem Filter.EventuallyEq.iteratedFDerivWithin_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
+    (n : ℕ) : iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
+  have : f₁ =ᶠ[𝓝[insert x s] x] f := by simpa [EventuallyEq, hx]
+  (this.iteratedFDerivWithin' (subset_insert _ _) n).self_of_nhdsWithin (mem_insert _ _)
+
+/-- If two functions coincide on a set `s`, then their iterated differentials within this set
+coincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and
+`Filter.EventuallyEq.iteratedFDerivWithin`. -/
+theorem iteratedFDerivWithin_congr (hs : EqOn f₁ f s) (hx : x ∈ s) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f₁ s x = iteratedFDerivWithin 𝕜 n f s x :=
+  (hs.eventuallyEq.filter_mono inf_le_right).iteratedFDerivWithin_eq (hs hx) _
+
+/-- If two functions coincide on a set `s`, then their iterated differentials within this set
+coincide. See also `Filter.EventuallyEq.iteratedFDerivWithin_eq` and
+`Filter.EventuallyEq.iteratedFDerivWithin`. -/
+protected theorem Set.EqOn.iteratedFDerivWithin (hs : EqOn f₁ f s) (n : ℕ) :
+    EqOn (iteratedFDerivWithin 𝕜 n f₁ s) (iteratedFDerivWithin 𝕜 n f s) s := fun _x hx =>
+  iteratedFDerivWithin_congr hs hx n
+
+theorem iteratedFDerivWithin_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t := by
+  induction n generalizing x with
+  | zero => rfl
+  | succ n ihn =>
+    refine (eventually_nhds_nhdsWithin.2 h).mono fun y hy => ?_
+    simp only [iteratedFDerivWithin_succ_eq_comp_left, (· ∘ ·)]
+    rw [(ihn hy).fderivWithin_eq_nhds, fderivWithin_congr_set' _ hy]
+
+theorem iteratedFDerivWithin_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t :=
+  iteratedFDerivWithin_eventually_congr_set' x (h.filter_mono inf_le_left) n
+
+theorem iteratedFDerivWithin_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
+    iteratedFDerivWithin 𝕜 n f s x = iteratedFDerivWithin 𝕜 n f t x :=
+  (iteratedFDerivWithin_eventually_congr_set h n).self_of_nhds
+
+/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
+`s` with a neighborhood of `x` within `s`. -/
+theorem iteratedFDerivWithin_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_congr_set (nhdsWithin_eq_iff_eventuallyEq.1 <| nhdsWithin_inter_of_mem' hu) _
+
+/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
+`s` with a neighborhood of `x`. -/
+theorem iteratedFDerivWithin_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_inter' (mem_nhdsWithin_of_mem_nhds hu)
+
+/-- The iterated differential within a set `s` at a point `x` is not modified if one intersects
+`s` with an open set containing `x`. -/
+theorem iteratedFDerivWithin_inter_open {n : ℕ} (hu : IsOpen u) (hx : x ∈ u) :
+    iteratedFDerivWithin 𝕜 n f (s ∩ u) x = iteratedFDerivWithin 𝕜 n f s x :=
+  iteratedFDerivWithin_inter (hu.mem_nhds hx)
+
+/-- On a set with unique differentiability, any choice of iterated differential has to coincide
+with the one we have chosen in `iteratedFDerivWithin 𝕜 m f s`. -/
+theorem HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
+    (h : HasFTaylorSeriesUpToOn n f p s) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : UniqueDiffOn 𝕜 s)
+    (hx : x ∈ s) : p x m = iteratedFDerivWithin 𝕜 m f s x := by
+  induction' m with m IH generalizing x
+  · rw [h.zero_eq' hx, iteratedFDerivWithin_zero_eq_comp]; rfl
+  · have A : (m : ℕ∞) < n := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (lt_add_one m)) hmn
+    have :
+      HasFDerivWithinAt (fun y : E => iteratedFDerivWithin 𝕜 m f s y)
+        (ContinuousMultilinearMap.curryLeft (p x (Nat.succ m))) s x :=
+      (h.fderivWithin m A x hx).congr (fun y hy => (IH (le_of_lt A) hy).symm)
+        (IH (le_of_lt A) hx).symm
+    rw [iteratedFDerivWithin_succ_eq_comp_left, Function.comp_apply, this.fderivWithin (hs x hx)]
+    exact (ContinuousMultilinearMap.uncurry_curryLeft _).symm
+
+@[deprecated (since := "2024-03-28")]
+alias HasFTaylorSeriesUpToOn.eq_ftaylor_series_of_uniqueDiffOn :=
+  HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn
+
+
+/-! ### Functions with a Taylor series on the whole space -/
+
+/-- `HasFTaylorSeriesUpTo n f p` registers the fact that `p 0 = f` and `p (m+1)` is a
+derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
+`HasFDerivAt` but for higher order derivatives.
+
+Notice that `p` does not sum up to `f` on the diagonal (`FormalMultilinearSeries.sum`), even if
+`f` is analytic and `n = ∞`: an addition `1/m!` factor on the `m`th term is necessary for that. -/
+structure HasFTaylorSeriesUpTo (n : ℕ∞) (f : E → F) (p : E → FormalMultilinearSeries 𝕜 E F) :
+  Prop where
+  zero_eq : ∀ x, (p x 0).curry0 = f x
+  fderiv : ∀ m : ℕ, (m : ℕ∞) < n → ∀ x, HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x
+  cont : ∀ m : ℕ, (m : ℕ∞) ≤ n → Continuous fun x => p x m
+
+theorem HasFTaylorSeriesUpTo.zero_eq' (h : HasFTaylorSeriesUpTo n f p) (x : E) :
+    p x 0 = (continuousMultilinearCurryFin0 𝕜 E F).symm (f x) := by
+  rw [← h.zero_eq x]
+  exact (p x 0).uncurry0_curry0.symm
+
+theorem hasFTaylorSeriesUpToOn_univ_iff :
+    HasFTaylorSeriesUpToOn n f p univ ↔ HasFTaylorSeriesUpTo n f p := by
+  constructor
+  · intro H
+    constructor
+    · exact fun x => H.zero_eq x (mem_univ x)
+    · intro m hm x
+      rw [← hasFDerivWithinAt_univ]
+      exact H.fderivWithin m hm x (mem_univ x)
+    · intro m hm
+      rw [continuous_iff_continuousOn_univ]
+      exact H.cont m hm
+  · intro H
+    constructor
+    · exact fun x _ => H.zero_eq x
+    · intro m hm x _
+      rw [hasFDerivWithinAt_univ]
+      exact H.fderiv m hm x
+    · intro m hm
+      rw [← continuous_iff_continuousOn_univ]
+      exact H.cont m hm
+
+theorem HasFTaylorSeriesUpTo.hasFTaylorSeriesUpToOn (h : HasFTaylorSeriesUpTo n f p) (s : Set E) :
+    HasFTaylorSeriesUpToOn n f p s :=
+  (hasFTaylorSeriesUpToOn_univ_iff.2 h).mono (subset_univ _)
+
+theorem HasFTaylorSeriesUpTo.ofLe (h : HasFTaylorSeriesUpTo n f p) (hmn : m ≤ n) :
+    HasFTaylorSeriesUpTo m f p := by
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn
+
+theorem HasFTaylorSeriesUpTo.continuous (h : HasFTaylorSeriesUpTo n f p) : Continuous f := by
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
+  rw [continuous_iff_continuousOn_univ]
+  exact h.continuousOn
+
+theorem hasFTaylorSeriesUpTo_zero_iff :
+    HasFTaylorSeriesUpTo 0 f p ↔ Continuous f ∧ ∀ x, (p x 0).curry0 = f x := by
+  simp [hasFTaylorSeriesUpToOn_univ_iff.symm, continuous_iff_continuousOn_univ,
+    hasFTaylorSeriesUpToOn_zero_iff]
+
+theorem hasFTaylorSeriesUpTo_top_iff :
+    HasFTaylorSeriesUpTo ∞ f p ↔ ∀ n : ℕ, HasFTaylorSeriesUpTo n f p := by
+  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff]
+
+/-- In the case that `n = ∞` we don't need the continuity assumption in
+`HasFTaylorSeriesUpTo`. -/
+theorem hasFTaylorSeriesUpTo_top_iff' :
+    HasFTaylorSeriesUpTo ∞ f p ↔
+      (∀ x, (p x 0).curry0 = f x) ∧
+        ∀ (m : ℕ) (x), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x := by
+  simp only [← hasFTaylorSeriesUpToOn_univ_iff, hasFTaylorSeriesUpToOn_top_iff', mem_univ,
+    forall_true_left, hasFDerivWithinAt_univ]
+
+/-- If a function has a Taylor series at order at least `1`, then the term of order `1` of this
+series is a derivative of `f`. -/
+theorem HasFTaylorSeriesUpTo.hasFDerivAt (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) (x : E) :
+    HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p x 1)) x := by
+  rw [← hasFDerivWithinAt_univ]
+  exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _)
+
+theorem HasFTaylorSeriesUpTo.differentiable (h : HasFTaylorSeriesUpTo n f p) (hn : 1 ≤ n) :
+    Differentiable 𝕜 f := fun x => (h.hasFDerivAt hn x).differentiableAt
+
+/-- `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
+for `p 1`, which is a derivative of `f`. -/
+theorem hasFTaylorSeriesUpTo_succ_iff_right {n : ℕ} :
+    HasFTaylorSeriesUpTo (n + 1 : ℕ) f p ↔
+      (∀ x, (p x 0).curry0 = f x) ∧
+        (∀ x, HasFDerivAt (fun y => p y 0) (p x 1).curryLeft x) ∧
+          HasFTaylorSeriesUpTo n (fun x => continuousMultilinearCurryFin1 𝕜 E F (p x 1)) fun x =>
+            (p x).shift := by
+  simp only [hasFTaylorSeriesUpToOn_succ_iff_right, ← hasFTaylorSeriesUpToOn_univ_iff, mem_univ,
+    forall_true_left, hasFDerivWithinAt_univ]
+
+
+/-! ### Iterated derivative -/
+
+
+variable (𝕜)
+
+/-- The `n`-th derivative of a function, as a multilinear map, defined inductively. -/
+noncomputable def iteratedFDeriv (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F :=
+  Nat.recOn n (fun x => ContinuousMultilinearMap.uncurry0 𝕜 E (f x)) fun _ rec x =>
+    ContinuousLinearMap.uncurryLeft (fderiv 𝕜 rec x)
+
+/-- Formal Taylor series associated to a function. -/
+def ftaylorSeries (f : E → F) (x : E) : FormalMultilinearSeries 𝕜 E F := fun n =>
+  iteratedFDeriv 𝕜 n f x
+
+variable {𝕜}
+
+@[simp]
+theorem iteratedFDeriv_zero_apply (m : Fin 0 → E) :
+    (iteratedFDeriv 𝕜 0 f x : (Fin 0 → E) → F) m = f x :=
+  rfl
+
+theorem iteratedFDeriv_zero_eq_comp :
+    iteratedFDeriv 𝕜 0 f = (continuousMultilinearCurryFin0 𝕜 E F).symm ∘ f :=
+  rfl
+
+@[simp]
+theorem norm_iteratedFDeriv_zero : ‖iteratedFDeriv 𝕜 0 f x‖ = ‖f x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDeriv_zero_eq_comp, comp_apply, LinearIsometryEquiv.norm_map]
+
+theorem iteratedFDerivWithin_zero_eq : iteratedFDerivWithin 𝕜 0 f s = iteratedFDeriv 𝕜 0 f := rfl
+
+theorem iteratedFDeriv_succ_apply_left {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
+      (fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x : E → E[×n]→L[𝕜] F) (m 0) (tail m) :=
+  rfl
+
+/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
+and the derivative of the `n`-th derivative. -/
+theorem iteratedFDeriv_succ_eq_comp_left {n : ℕ} :
+    iteratedFDeriv 𝕜 (n + 1) f =
+      (continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
+        fderiv 𝕜 (iteratedFDeriv 𝕜 n f) :=
+  rfl
+
+/-- Writing explicitly the derivative of the `n`-th derivative as the composition of a currying
+linear equiv, and the `n + 1`-th derivative. -/
+theorem fderiv_iteratedFDeriv {n : ℕ} :
+    fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
+      continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F ∘
+        iteratedFDeriv 𝕜 (n + 1) f :=
+  rfl
+
+theorem tsupport_iteratedFDeriv_subset (n : ℕ) : tsupport (iteratedFDeriv 𝕜 n f) ⊆ tsupport f := by
+  induction n with
+  | zero =>
+    rw [iteratedFDeriv_zero_eq_comp]
+    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
+      subset_closure) isClosed_closure
+  | succ n IH =>
+    rw [iteratedFDeriv_succ_eq_comp_left]
+    exact closure_minimal ((support_comp_subset (LinearIsometryEquiv.map_zero _) _).trans
+      ((support_fderiv_subset 𝕜).trans IH)) isClosed_closure
+
+theorem support_iteratedFDeriv_subset (n : ℕ) : support (iteratedFDeriv 𝕜 n f) ⊆ tsupport f :=
+  subset_closure.trans (tsupport_iteratedFDeriv_subset n)
+
+theorem HasCompactSupport.iteratedFDeriv (hf : HasCompactSupport f) (n : ℕ) :
+    HasCompactSupport (iteratedFDeriv 𝕜 n f) :=
+  hf.of_isClosed_subset isClosed_closure (tsupport_iteratedFDeriv_subset n)
+
+theorem norm_fderiv_iteratedFDeriv {n : ℕ} :
+    ‖fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDeriv_succ_eq_comp_left, comp_apply, LinearIsometryEquiv.norm_map]
+
+theorem iteratedFDerivWithin_univ {n : ℕ} :
+    iteratedFDerivWithin 𝕜 n f univ = iteratedFDeriv 𝕜 n f := by
+  induction n with
+  | zero => ext x; simp
+  | succ n IH =>
+    ext x m
+    rw [iteratedFDeriv_succ_apply_left, iteratedFDerivWithin_succ_apply_left, IH, fderivWithin_univ]
+
+theorem HasFTaylorSeriesUpTo.eq_iteratedFDeriv
+    (h : HasFTaylorSeriesUpTo n f p) {m : ℕ} (hmn : (m : ℕ∞) ≤ n) (x : E) :
+    p x m = iteratedFDeriv 𝕜 m f x := by
+  rw [← iteratedFDerivWithin_univ]
+  rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
+  exact h.eq_iteratedFDerivWithin_of_uniqueDiffOn hmn uniqueDiffOn_univ (mem_univ _)
+
+/-- In an open set, the iterated derivative within this set coincides with the global iterated
+derivative. -/
+theorem iteratedFDerivWithin_of_isOpen (n : ℕ) (hs : IsOpen s) :
+    EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s := by
+  induction n with
+  | zero =>
+    intro x _
+    ext1
+    simp only [iteratedFDerivWithin_zero_apply, iteratedFDeriv_zero_apply]
+  | succ n IH =>
+    intro x hx
+    rw [iteratedFDeriv_succ_eq_comp_left, iteratedFDerivWithin_succ_eq_comp_left]
+    dsimp
+    congr 1
+    rw [fderivWithin_of_isOpen hs hx]
+    apply Filter.EventuallyEq.fderiv_eq
+    filter_upwards [hs.mem_nhds hx]
+    exact IH
+
+theorem ftaylorSeriesWithin_univ : ftaylorSeriesWithin 𝕜 f univ = ftaylorSeries 𝕜 f := by
+  ext1 x; ext1 n
+  change iteratedFDerivWithin 𝕜 n f univ x = iteratedFDeriv 𝕜 n f x
+  rw [iteratedFDerivWithin_univ]
+
+theorem iteratedFDeriv_succ_apply_right {n : ℕ} (m : Fin (n + 1) → E) :
+    (iteratedFDeriv 𝕜 (n + 1) f x : (Fin (n + 1) → E) → F) m =
+      iteratedFDeriv 𝕜 n (fun y => fderiv 𝕜 f y) x (init m) (m (last n)) := by
+  rw [← iteratedFDerivWithin_univ, ← iteratedFDerivWithin_univ, ← fderivWithin_univ]
+  exact iteratedFDerivWithin_succ_apply_right uniqueDiffOn_univ (mem_univ _) _
+
+/-- Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
+and the `n`-th derivative of the derivative. -/
+theorem iteratedFDeriv_succ_eq_comp_right {n : ℕ} :
+    iteratedFDeriv 𝕜 (n + 1) f x =
+      ((continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm ∘
+          iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y) x := by
+  ext m; rw [iteratedFDeriv_succ_apply_right]; rfl
+
+theorem norm_iteratedFDeriv_fderiv {n : ℕ} :
+    ‖iteratedFDeriv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iteratedFDeriv 𝕜 (n + 1) f x‖ := by
+  -- Porting note: added `comp_apply`.
+  rw [iteratedFDeriv_succ_eq_comp_right, comp_apply, LinearIsometryEquiv.norm_map]
+
+@[simp]
+theorem iteratedFDeriv_one_apply (m : Fin 1 → E) :
+    iteratedFDeriv 𝕜 1 f x m = fderiv 𝕜 f x (m 0) := by
+  rw [iteratedFDeriv_succ_apply_right, iteratedFDeriv_zero_apply]; rfl
+
+lemma iteratedFDeriv_two_apply (f : E → F) (z : E) (m : Fin 2 → E) :
+    iteratedFDeriv 𝕜 2 f z m = fderiv 𝕜 (fderiv 𝕜 f) z (m 0) (m 1) := by
+  simp only [iteratedFDeriv_succ_apply_right]
+  rfl
diff --git a/Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean b/Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean
index 7b0d3e391e4fd..5b3ed3d9a9e4f 100644
--- a/Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean
+++ b/Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean
@@ -24,7 +24,7 @@ variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {D : Type uD} [NormedAddC
 
 section FiniteDimensional
 
-open Function FiniteDimensional
+open Function Module
 
 variable [CompleteSpace 𝕜]
 
diff --git a/Mathlib/Analysis/Calculus/Deriv/Abs.lean b/Mathlib/Analysis/Calculus/Deriv/Abs.lean
new file mode 100644
index 0000000000000..77194eadb8363
--- /dev/null
+++ b/Mathlib/Analysis/Calculus/Deriv/Abs.lean
@@ -0,0 +1,200 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.Analysis.Calculus.Deriv.Add
+import Mathlib.Analysis.InnerProductSpace.Calculus
+
+/-!
+# Derivative of the absolute value
+
+This file compiles basic derivability properties of the absolute value, and is largely inspired
+from `Mathlib.Analysis.InnerProductSpace.Calculus`, which is the analoguous file for norms derived
+from an inner product space.
+
+## Tags
+
+absolute value, derivative
+-/
+
+open Filter Real Set
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable {n : ℕ∞} {f : E → ℝ} {f' : E →L[ℝ] ℝ} {s : Set E} {x : E}
+
+theorem contDiffAt_abs {x : ℝ} (hx : x ≠ 0) : ContDiffAt ℝ n (|·|) x := contDiffAt_norm ℝ hx
+
+theorem ContDiffAt.abs (hf : ContDiffAt ℝ n f x) (h₀ : f x ≠ 0) :
+    ContDiffAt ℝ n (fun x ↦ |f x|) x := hf.norm ℝ h₀
+
+theorem contDiffWithinAt_abs {x : ℝ} (hx : x ≠ 0) (s : Set ℝ) :
+    ContDiffWithinAt ℝ n (|·|) s x := (contDiffAt_abs hx).contDiffWithinAt
+
+theorem ContDiffWithinAt.abs (hf : ContDiffWithinAt ℝ n f s x) (h₀ : f x ≠ 0) :
+    ContDiffWithinAt ℝ n (fun y ↦ |f y|) s x :=
+  (contDiffAt_abs h₀).comp_contDiffWithinAt x hf
+
+theorem contDiffOn_abs {s : Set ℝ} (hs : ∀ x ∈ s, x ≠ 0) :
+    ContDiffOn ℝ n (|·|) s := fun x hx ↦ contDiffWithinAt_abs (hs x hx) s
+
+theorem ContDiffOn.abs (hf : ContDiffOn ℝ n f s) (h₀ : ∀ x ∈ s, f x ≠ 0) :
+    ContDiffOn ℝ n (fun y ↦ |f y|) s := fun x hx ↦ (hf x hx).abs (h₀ x hx)
+
+theorem ContDiff.abs (hf : ContDiff ℝ n f) (h₀ : ∀ x, f x ≠ 0) : ContDiff ℝ n fun y ↦ |f y| :=
+  contDiff_iff_contDiffAt.2 fun x ↦ hf.contDiffAt.abs (h₀ x)
+
+theorem hasStrictDerivAt_abs_neg {x : ℝ} (hx : x < 0) :
+    HasStrictDerivAt (|·|) (-1) x :=
+  (hasStrictDerivAt_neg x).congr_of_eventuallyEq <|
+    EqOn.eventuallyEq_of_mem (fun _ hy ↦ (abs_of_neg (mem_Iio.1 hy)).symm) (Iio_mem_nhds hx)
+
+theorem hasDerivAt_abs_neg {x : ℝ} (hx : x < 0) :
+    HasDerivAt (|·|) (-1) x := (hasStrictDerivAt_abs_neg hx).hasDerivAt
+
+theorem hasStrictDerivAt_abs_pos {x : ℝ} (hx : 0 < x) :
+    HasStrictDerivAt (|·|) 1 x :=
+  (hasStrictDerivAt_id x).congr_of_eventuallyEq <|
+    EqOn.eventuallyEq_of_mem (fun _ hy ↦ (abs_of_pos (mem_Iio.1 hy)).symm) (Ioi_mem_nhds hx)
+
+theorem hasDerivAt_abs_pos {x : ℝ} (hx : 0 < x) :
+    HasDerivAt (|·|) 1 x := (hasStrictDerivAt_abs_pos hx).hasDerivAt
+
+theorem hasStrictDerivAt_abs {x : ℝ} (hx : x ≠ 0) :
+    HasStrictDerivAt (|·|) (SignType.sign x : ℝ) x := by
+  obtain hx | hx := hx.lt_or_lt
+  · simpa [hx] using hasStrictDerivAt_abs_neg hx
+  · simpa [hx] using hasStrictDerivAt_abs_pos hx
+
+theorem hasDerivAt_abs {x : ℝ} (hx : x ≠ 0) :
+    HasDerivAt (|·|) (SignType.sign x : ℝ) x := (hasStrictDerivAt_abs hx).hasDerivAt
+
+theorem HasStrictFDerivAt.abs_of_neg (hf : HasStrictFDerivAt f f' x)
+    (h₀ : f x < 0) : HasStrictFDerivAt (fun x ↦ |f x|) (-f') x := by
+  convert (hasStrictDerivAt_abs_neg h₀).hasStrictFDerivAt.comp x hf using 1
+  ext y
+  simp
+
+theorem HasFDerivAt.abs_of_neg (hf : HasFDerivAt f f' x)
+    (h₀ : f x < 0) : HasFDerivAt (fun x ↦ |f x|) (-f') x := by
+  convert (hasDerivAt_abs_neg h₀).hasFDerivAt.comp x hf using 1
+  ext y
+  simp
+
+theorem HasStrictFDerivAt.abs_of_pos (hf : HasStrictFDerivAt f f' x)
+    (h₀ : 0 < f x) : HasStrictFDerivAt (fun x ↦ |f x|) f' x := by
+  convert (hasStrictDerivAt_abs_pos h₀).hasStrictFDerivAt.comp x hf using 1
+  ext y
+  simp
+
+theorem HasFDerivAt.abs_of_pos (hf : HasFDerivAt f f' x)
+    (h₀ : 0 < f x) : HasFDerivAt (fun x ↦ |f x|) f' x := by
+  convert (hasDerivAt_abs_pos h₀).hasFDerivAt.comp x hf using 1
+  ext y
+  simp
+
+theorem HasStrictFDerivAt.abs (hf : HasStrictFDerivAt f f' x)
+    (h₀ : f x ≠ 0) : HasStrictFDerivAt (fun x ↦ |f x|) ((SignType.sign (f x) : ℝ) • f') x := by
+  convert (hasStrictDerivAt_abs h₀).hasStrictFDerivAt.comp x hf using 1
+  ext y
+  simp [mul_comm]
+
+theorem HasFDerivAt.abs (hf : HasFDerivAt f f' x)
+    (h₀ : f x ≠ 0) : HasFDerivAt (fun x ↦ |f x|) ((SignType.sign (f x) : ℝ) • f') x := by
+  convert (hasDerivAt_abs h₀).hasFDerivAt.comp x hf using 1
+  ext y
+  simp [mul_comm]
+
+theorem hasDerivWithinAt_abs_neg (s : Set ℝ) {x : ℝ} (hx : x < 0) :
+    HasDerivWithinAt (|·|) (-1) s x := (hasDerivAt_abs_neg hx).hasDerivWithinAt
+
+theorem hasDerivWithinAt_abs_pos (s : Set ℝ) {x : ℝ} (hx : 0 < x) :
+    HasDerivWithinAt (|·|) 1 s x := (hasDerivAt_abs_pos hx).hasDerivWithinAt
+
+theorem hasDerivWithinAt_abs (s : Set ℝ) {x : ℝ} (hx : x ≠ 0) :
+    HasDerivWithinAt (|·|) (SignType.sign x : ℝ) s x := (hasDerivAt_abs hx).hasDerivWithinAt
+
+theorem HasFDerivWithinAt.abs_of_neg (hf : HasFDerivWithinAt f f' s x)
+    (h₀ : f x < 0) : HasFDerivWithinAt (fun x ↦ |f x|) (-f') s x := by
+  convert (hasDerivAt_abs_neg h₀).comp_hasFDerivWithinAt x hf using 1
+  simp
+
+theorem HasFDerivWithinAt.abs_of_pos (hf : HasFDerivWithinAt f f' s x)
+    (h₀ : 0 < f x) : HasFDerivWithinAt (fun x ↦ |f x|) f' s x := by
+  convert (hasDerivAt_abs_pos h₀).comp_hasFDerivWithinAt x hf using 1
+  simp
+
+theorem HasFDerivWithinAt.abs (hf : HasFDerivWithinAt f f' s x)
+    (h₀ : f x ≠ 0) : HasFDerivWithinAt (fun x ↦ |f x|) ((SignType.sign (f x) : ℝ) • f') s x :=
+  (hasDerivAt_abs h₀).comp_hasFDerivWithinAt x hf
+
+theorem differentiableAt_abs_neg {x : ℝ} (hx : x < 0) :
+    DifferentiableAt ℝ (|·|) x := (hasDerivAt_abs_neg hx).differentiableAt
+
+theorem differentiableAt_abs_pos {x : ℝ} (hx : 0 < x) :
+    DifferentiableAt ℝ (|·|) x := (hasDerivAt_abs_pos hx).differentiableAt
+
+theorem differentiableAt_abs {x : ℝ} (hx : x ≠ 0) :
+    DifferentiableAt ℝ (|·|) x := (hasDerivAt_abs hx).differentiableAt
+
+theorem DifferentiableAt.abs_of_neg (hf : DifferentiableAt ℝ f x) (h₀ : f x < 0) :
+    DifferentiableAt ℝ (fun x ↦ |f x|) x := (differentiableAt_abs_neg h₀).comp x hf
+
+theorem DifferentiableAt.abs_of_pos (hf : DifferentiableAt ℝ f x) (h₀ : 0 < f x) :
+    DifferentiableAt ℝ (fun x ↦ |f x|) x := (differentiableAt_abs_pos h₀).comp x hf
+
+theorem DifferentiableAt.abs (hf : DifferentiableAt ℝ f x) (h₀ : f x ≠ 0) :
+    DifferentiableAt ℝ (fun x ↦ |f x|) x := (differentiableAt_abs h₀).comp x hf
+
+theorem differentiableWithinAt_abs_neg (s : Set ℝ) {x : ℝ} (hx : x < 0) :
+    DifferentiableWithinAt ℝ (|·|) s x := (differentiableAt_abs_neg hx).differentiableWithinAt
+
+theorem differentiableWithinAt_abs_pos (s : Set ℝ) {x : ℝ} (hx : 0 < x) :
+    DifferentiableWithinAt ℝ (|·|) s x := (differentiableAt_abs_pos hx).differentiableWithinAt
+
+theorem differentiableWithinAt_abs (s : Set ℝ) {x : ℝ} (hx : x ≠ 0) :
+    DifferentiableWithinAt ℝ (|·|) s x := (differentiableAt_abs hx).differentiableWithinAt
+
+theorem DifferentiableWithinAt.abs_of_neg (hf : DifferentiableWithinAt ℝ f s x) (h₀ : f x < 0) :
+    DifferentiableWithinAt ℝ (fun x ↦ |f x|) s x :=
+  (differentiableAt_abs_neg h₀).comp_differentiableWithinAt x hf
+
+theorem DifferentiableWithinAt.abs_of_pos (hf : DifferentiableWithinAt ℝ f s x) (h₀ : 0 < f x) :
+    DifferentiableWithinAt ℝ (fun x ↦ |f x|) s x :=
+  (differentiableAt_abs_pos h₀).comp_differentiableWithinAt x hf
+
+theorem DifferentiableWithinAt.abs (hf : DifferentiableWithinAt ℝ f s x) (h₀ : f x ≠ 0) :
+    DifferentiableWithinAt ℝ (fun x ↦ |f x|) s x :=
+  (differentiableAt_abs h₀).comp_differentiableWithinAt x hf
+
+theorem differentiableOn_abs {s : Set ℝ} (hs : ∀ x ∈ s, x ≠ 0) : DifferentiableOn ℝ (|·|) s :=
+  fun x hx ↦ differentiableWithinAt_abs s (hs x hx)
+
+theorem DifferentiableOn.abs (hf : DifferentiableOn ℝ f s) (h₀ : ∀ x ∈ s, f x ≠ 0) :
+    DifferentiableOn ℝ (fun x ↦ |f x|) s :=
+  fun x hx ↦ (hf x hx).abs (h₀ x hx)
+
+theorem Differentiable.abs (hf : Differentiable ℝ f) (h₀ : ∀ x, f x ≠ 0) :
+    Differentiable ℝ (fun x ↦ |f x|) := fun x ↦ (hf x).abs (h₀ x)
+
+theorem not_differentiableAt_abs_zero : ¬ DifferentiableAt ℝ (abs : ℝ → ℝ) 0 := by
+  intro h
+  have h₁ : deriv abs (0 : ℝ) = 1 :=
+    (uniqueDiffOn_Ici _ _ Set.left_mem_Ici).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
+      (hasDerivWithinAt_id _ _).congr_of_mem (fun _ h ↦ abs_of_nonneg h) Set.left_mem_Ici
+  have h₂ : deriv abs (0 : ℝ) = -1 :=
+    (uniqueDiffOn_Iic _ _ Set.right_mem_Iic).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
+      (hasDerivWithinAt_neg _ _).congr_of_mem (fun _ h ↦ abs_of_nonpos h) Set.right_mem_Iic
+  linarith
+
+theorem deriv_abs_neg {x : ℝ} (hx : x < 0) : deriv (|·|) x = -1 := (hasDerivAt_abs_neg hx).deriv
+
+theorem deriv_abs_pos {x : ℝ} (hx : 0 < x) : deriv (|·|) x = 1 := (hasDerivAt_abs_pos hx).deriv
+
+theorem deriv_abs_zero : deriv (|·|) (0 : ℝ) = 0 :=
+  deriv_zero_of_not_differentiableAt not_differentiableAt_abs_zero
+
+theorem deriv_abs (x : ℝ) : deriv (|·|) x = SignType.sign x := by
+  obtain rfl | hx := eq_or_ne x 0
+  · simpa using deriv_abs_zero
+  · simpa [hx] using (hasDerivAt_abs hx).deriv
diff --git a/Mathlib/Analysis/Calculus/Deriv/Add.lean b/Mathlib/Analysis/Calculus/Deriv/Add.lean
index 6f51814995811..169722bc11409 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Add.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Add.lean
@@ -247,16 +247,6 @@ theorem differentiable_neg : Differentiable 𝕜 (Neg.neg : 𝕜 → 𝕜) :=
 theorem differentiableOn_neg : DifferentiableOn 𝕜 (Neg.neg : 𝕜 → 𝕜) s :=
   DifferentiableOn.neg differentiableOn_id
 
-theorem not_differentiableAt_abs_zero : ¬ DifferentiableAt ℝ (abs : ℝ → ℝ) 0 := by
-  intro h
-  have h₁ : deriv abs (0 : ℝ) = 1 :=
-    (uniqueDiffOn_Ici _ _ Set.left_mem_Ici).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
-      (hasDerivWithinAt_id _ _).congr_of_mem (fun _ h ↦ abs_of_nonneg h) Set.left_mem_Ici
-  have h₂ : deriv abs (0 : ℝ) = -1 :=
-    (uniqueDiffOn_Iic _ _ Set.right_mem_Iic).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
-      (hasDerivWithinAt_neg _ _).congr_of_mem (fun _ h ↦ abs_of_nonpos h) Set.right_mem_Iic
-  linarith
-
 lemma differentiableAt_comp_neg {a : 𝕜} :
     DifferentiableAt 𝕜 (fun x ↦ f (-x)) a ↔ DifferentiableAt 𝕜 f (-a) := by
   refine ⟨fun H ↦ ?_, fun H ↦ H.comp a differentiable_neg.differentiableAt⟩
diff --git a/Mathlib/Analysis/Calculus/Deriv/Basic.lean b/Mathlib/Analysis/Calculus/Deriv/Basic.lean
index edda887e12373..5ed888191d0e0 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Basic.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Basic.lean
@@ -416,7 +416,7 @@ theorem norm_deriv_eq_norm_fderiv : ‖deriv f x‖ = ‖fderiv 𝕜 f x‖ := b
 
 theorem DifferentiableAt.derivWithin (h : DifferentiableAt 𝕜 f x) (hxs : UniqueDiffWithinAt 𝕜 s x) :
     derivWithin f s x = deriv f x := by
-  unfold derivWithin deriv
+  unfold _root_.derivWithin deriv
   rw [h.fderivWithin hxs]
 
 theorem HasDerivWithinAt.deriv_eq_zero (hd : HasDerivWithinAt f 0 s x)
diff --git a/Mathlib/Analysis/Calculus/Deriv/Inv.lean b/Mathlib/Analysis/Calculus/Deriv/Inv.lean
index ffa6762826f8e..8687b3af9373d 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Inv.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Inv.lean
@@ -63,20 +63,13 @@ theorem hasDerivWithinAt_inv (x_ne_zero : x ≠ 0) (s : Set 𝕜) :
     HasDerivWithinAt (fun x => x⁻¹) (-(x ^ 2)⁻¹) s x :=
   (hasDerivAt_inv x_ne_zero).hasDerivWithinAt
 
-theorem differentiableAt_inv : DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0 :=
+theorem differentiableAt_inv_iff : DifferentiableAt 𝕜 (fun x => x⁻¹) x ↔ x ≠ 0 :=
   ⟨fun H => NormedField.continuousAt_inv.1 H.continuousAt, fun H =>
     (hasDerivAt_inv H).differentiableAt⟩
 
-theorem differentiableWithinAt_inv (x_ne_zero : x ≠ 0) :
-    DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x :=
-  (differentiableAt_inv.2 x_ne_zero).differentiableWithinAt
-
-theorem differentiableOn_inv : DifferentiableOn 𝕜 (fun x : 𝕜 => x⁻¹) { x | x ≠ 0 } := fun _x hx =>
-  differentiableWithinAt_inv hx
-
 theorem deriv_inv : deriv (fun x => x⁻¹) x = -(x ^ 2)⁻¹ := by
   rcases eq_or_ne x 0 with (rfl | hne)
-  · simp [deriv_zero_of_not_differentiableAt (mt differentiableAt_inv.1 (not_not.2 rfl))]
+  · simp [deriv_zero_of_not_differentiableAt (mt differentiableAt_inv_iff.1 (not_not.2 rfl))]
   · exact (hasDerivAt_inv hne).deriv
 
 @[simp]
@@ -85,13 +78,17 @@ theorem deriv_inv' : (deriv fun x : 𝕜 => x⁻¹) = fun x => -(x ^ 2)⁻¹ :=
 
 theorem derivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
     derivWithin (fun x => x⁻¹) s x = -(x ^ 2)⁻¹ := by
-  rw [DifferentiableAt.derivWithin (differentiableAt_inv.2 x_ne_zero) hxs]
+  rw [DifferentiableAt.derivWithin (differentiableAt_inv x_ne_zero) hxs]
   exact deriv_inv
 
 theorem hasFDerivAt_inv (x_ne_zero : x ≠ 0) :
     HasFDerivAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) x :=
   hasDerivAt_inv x_ne_zero
 
+theorem hasStrictFDerivAt_inv (x_ne_zero : x ≠ 0) :
+    HasStrictFDerivAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) x :=
+  hasStrictDerivAt_inv x_ne_zero
+
 theorem hasFDerivWithinAt_inv (x_ne_zero : x ≠ 0) :
     HasFDerivWithinAt (fun x => x⁻¹) (smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) : 𝕜 →L[𝕜] 𝕜) s x :=
   (hasFDerivAt_inv x_ne_zero).hasFDerivWithinAt
@@ -101,7 +98,7 @@ theorem fderiv_inv : fderiv 𝕜 (fun x => x⁻¹) x = smulRight (1 : 𝕜 →L[
 
 theorem fderivWithin_inv (x_ne_zero : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
     fderivWithin 𝕜 (fun x => x⁻¹) s x = smulRight (1 : 𝕜 →L[𝕜] 𝕜) (-(x ^ 2)⁻¹) := by
-  rw [DifferentiableAt.fderivWithin (differentiableAt_inv.2 x_ne_zero) hxs]
+  rw [DifferentiableAt.fderivWithin (differentiableAt_inv x_ne_zero) hxs]
   exact fderiv_inv
 
 variable {c : 𝕜 → 𝕜} {h : E → 𝕜} {c' : 𝕜} {z : E} {S : Set E}
@@ -116,22 +113,6 @@ theorem HasDerivAt.inv (hc : HasDerivAt c c' x) (hx : c x ≠ 0) :
   rw [← hasDerivWithinAt_univ] at *
   exact hc.inv hx
 
-theorem DifferentiableWithinAt.inv (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
-    DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z :=
-  (differentiableAt_inv.mpr hz).comp_differentiableWithinAt z hf
-
-@[simp]
-theorem DifferentiableAt.inv (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
-    DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z :=
-  (differentiableAt_inv.mpr hz).comp z hf
-
-theorem DifferentiableOn.inv (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
-    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S := fun x h => (hf x h).inv (hz x h)
-
-@[simp]
-theorem Differentiable.inv (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
-    Differentiable 𝕜 fun x => (h x)⁻¹ := fun x => (hf x).inv (hz x)
-
 theorem derivWithin_inv' (hc : DifferentiableWithinAt 𝕜 c s x) (hx : c x ≠ 0)
     (hxs : UniqueDiffWithinAt 𝕜 s x) :
     derivWithin (fun x => (c x)⁻¹) s x = -derivWithin c s x / c x ^ 2 :=
diff --git a/Mathlib/Analysis/Calculus/Deriv/Mul.lean b/Mathlib/Analysis/Calculus/Deriv/Mul.lean
index 64628ceb448d0..c89e9ca6d7c04 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Mul.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Mul.lean
@@ -324,22 +324,26 @@ end HasDeriv
 variable {ι : Type*} {𝔸' : Type*} [NormedCommRing 𝔸'] [NormedAlgebra 𝕜 𝔸']
   {u : Finset ι} {f : ι → 𝕜 → 𝔸'} {f' : ι → 𝔸'}
 
+@[fun_prop]
 theorem DifferentiableAt.finset_prod (hd : ∀ i ∈ u, DifferentiableAt 𝕜 (f i) x) :
     DifferentiableAt 𝕜 (∏ i ∈ u, f i ·) x := by
   classical
   exact
     (HasDerivAt.finset_prod (fun i hi ↦ DifferentiableAt.hasDerivAt (hd i hi))).differentiableAt
 
+@[fun_prop]
 theorem DifferentiableWithinAt.finset_prod (hd : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (f i) s x) :
     DifferentiableWithinAt 𝕜 (∏ i ∈ u, f i ·) s x := by
   classical
   exact (HasDerivWithinAt.finset_prod (fun i hi ↦
     DifferentiableWithinAt.hasDerivWithinAt (hd i hi))).differentiableWithinAt
 
+@[fun_prop]
 theorem DifferentiableOn.finset_prod (hd : ∀ i ∈ u, DifferentiableOn 𝕜 (f i) s) :
     DifferentiableOn 𝕜 (∏ i ∈ u, f i ·) s :=
   fun x hx ↦ .finset_prod (fun i hi ↦ hd i hi x hx)
 
+@[fun_prop]
 theorem Differentiable.finset_prod (hd : ∀ i ∈ u, Differentiable 𝕜 (f i)) :
     Differentiable 𝕜 (∏ i ∈ u, f i ·) :=
   fun x ↦ .finset_prod (fun i hi ↦ hd i hi x)
@@ -362,19 +366,21 @@ theorem HasStrictDerivAt.div_const (hc : HasStrictDerivAt c c' x) (d : 𝕜') :
     HasStrictDerivAt (fun x => c x / d) (c' / d) x := by
   simpa only [div_eq_mul_inv] using hc.mul_const d⁻¹
 
+@[fun_prop]
 theorem DifferentiableWithinAt.div_const (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝕜') :
     DifferentiableWithinAt 𝕜 (fun x => c x / d) s x :=
   (hc.hasDerivWithinAt.div_const _).differentiableWithinAt
 
-@[simp]
+@[simp, fun_prop]
 theorem DifferentiableAt.div_const (hc : DifferentiableAt 𝕜 c x) (d : 𝕜') :
     DifferentiableAt 𝕜 (fun x => c x / d) x :=
   (hc.hasDerivAt.div_const _).differentiableAt
 
+@[fun_prop]
 theorem DifferentiableOn.div_const (hc : DifferentiableOn 𝕜 c s) (d : 𝕜') :
     DifferentiableOn 𝕜 (fun x => c x / d) s := fun x hx => (hc x hx).div_const d
 
-@[simp]
+@[simp, fun_prop]
 theorem Differentiable.div_const (hc : Differentiable 𝕜 c) (d : 𝕜') :
     Differentiable 𝕜 fun x => c x / d := fun x => (hc x).div_const d
 
diff --git a/Mathlib/Analysis/Calculus/Deriv/Slope.lean b/Mathlib/Analysis/Calculus/Deriv/Slope.lean
index 3b9f51dde30cf..6ffd40c200519 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Slope.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Slope.lean
@@ -76,7 +76,7 @@ theorem hasDerivAt_iff_tendsto_slope_zero :
     HasDerivAt f f' x ↔ Tendsto (fun t ↦ t⁻¹ • (f (x + t) - f x)) (𝓝[≠] 0) (𝓝 f') := by
   have : 𝓝[≠] x = Filter.map (fun t ↦ x + t) (𝓝[≠] 0) := by
     simp [nhdsWithin, map_add_left_nhds_zero x, Filter.map_inf, add_right_injective x]
-  simp [hasDerivAt_iff_tendsto_slope, this, slope, Function.comp]
+  simp [hasDerivAt_iff_tendsto_slope, this, slope, Function.comp_def]
 
 alias ⟨HasDerivAt.tendsto_slope_zero, _⟩ := hasDerivAt_iff_tendsto_slope_zero
 
diff --git a/Mathlib/Analysis/Calculus/Deriv/ZPow.lean b/Mathlib/Analysis/Calculus/Deriv/ZPow.lean
index c8b4e4b0dd3f7..73cd119155a71 100644
--- a/Mathlib/Analysis/Calculus/Deriv/ZPow.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/ZPow.lean
@@ -46,7 +46,7 @@ theorem hasStrictDerivAt_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
   · have hx : x ≠ 0 := h.resolve_right hm.not_le
     have := (hasStrictDerivAt_inv ?_).scomp _ (this (-m) (neg_pos.2 hm)) <;>
       [skip; exact zpow_ne_zero _ hx]
-    simp only [(· ∘ ·), zpow_neg, one_div, inv_inv, smul_eq_mul] at this
+    simp only [Function.comp_def, zpow_neg, one_div, inv_inv, smul_eq_mul] at this
     convert this using 1
     rw [sq, mul_inv, inv_inv, Int.cast_neg, neg_mul, neg_mul_neg, ← zpow_add₀ hx, mul_assoc, ←
       zpow_add₀ hx]
diff --git a/Mathlib/Analysis/Calculus/DiffContOnCl.lean b/Mathlib/Analysis/Calculus/DiffContOnCl.lean
index 10a6fd3b0a289..03f7ad0b9609c 100644
--- a/Mathlib/Analysis/Calculus/DiffContOnCl.lean
+++ b/Mathlib/Analysis/Calculus/DiffContOnCl.lean
@@ -3,8 +3,9 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Analysis.Calculus.Deriv.Inv
 import Mathlib.Analysis.NormedSpace.Real
+import Mathlib.Analysis.Calculus.FDeriv.Add
+import Mathlib.Analysis.Calculus.FDeriv.Mul
 
 /-!
 # Functions differentiable on a domain and continuous on its closure
diff --git a/Mathlib/Analysis/Calculus/Dslope.lean b/Mathlib/Analysis/Calculus/Dslope.lean
index a35fb76d7fa45..df304274fc2ac 100644
--- a/Mathlib/Analysis/Calculus/Dslope.lean
+++ b/Mathlib/Analysis/Calculus/Dslope.lean
@@ -4,7 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.Analysis.Calculus.Deriv.Slope
-import Mathlib.Analysis.Calculus.Deriv.Inv
+import Mathlib.Analysis.Calculus.Deriv.Comp
+import Mathlib.Analysis.Calculus.FDeriv.Add
+import Mathlib.Analysis.Calculus.FDeriv.Mul
 
 /-!
 # Slope of a differentiable function
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean b/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
index 982fb6e51c423..80735fc0538cf 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
@@ -3,11 +3,14 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Analysis.Analytic.Basic
 import Mathlib.Analysis.Analytic.CPolynomial
+import Mathlib.Analysis.Analytic.Inverse
+import Mathlib.Analysis.Analytic.Within
 import Mathlib.Analysis.Calculus.Deriv.Basic
 import Mathlib.Analysis.Calculus.ContDiff.Defs
 import Mathlib.Analysis.Calculus.FDeriv.Add
+import Mathlib.Analysis.Calculus.FDeriv.Prod
+import Mathlib.Analysis.Normed.Module.Completion
 
 /-!
 # Frechet derivatives of analytic functions.
@@ -17,11 +20,50 @@ Also the special case in terms of `deriv` when the domain is 1-dimensional.
 
 As an application, we show that continuous multilinear maps are smooth. We also compute their
 iterated derivatives, in `ContinuousMultilinearMap.iteratedFDeriv_eq`.
+
+## Main definitions and results
+
+* `AnalyticAt.differentiableAt` : an analytic function at a point is differentiable there.
+* `AnalyticOnNhd.fderiv` : in a complete space, if a function is analytic on a
+  neighborhood of a set `s`, so is its derivative.
+* `AnalyticOnNhd.fderiv_of_isOpen` : if a function is analytic on a neighborhood of an
+  open set `s`, so is its derivative.
+* `AnalyticOn.fderivWithin` : if a function is analytic on a set of unique differentiability,
+  so is its derivative within this set.
+* `PartialHomeomorph.analyticAt_symm` : if a partial homeomorphism `f` is analytic at a
+  point `f.symm a`, with invertible derivative, then its inverse is analytic at `a`.
+
+## Comments on completeness
+
+Some theorems need a complete space, some don't, for the following reason.
+
+(1) If a function is analytic at a point `x`, then it is differentiable there (with derivative given
+by the first term in the power series). There is no issue of convergence here.
+
+(2) If a function has a power series on a ball `B (x, r)`, there is no guarantee that the power
+series for the derivative will converge at `y ≠ x`, if the space is not complete. So, to deduce
+that `f` is differentiable at `y`, one needs completeness in general.
+
+(3) However, if a function `f` has a power series on a ball `B (x, r)`, and is a priori known to be
+differentiable at some point `y ≠ x`, then the power series for the derivative of `f` will
+automatically converge at `y`, towards the given derivative: this follows from the facts that this
+is true in the completion (thanks to the previous point) and that the map to the completion is
+an embedding.
+
+(4) Therefore, if one assumes `AnalyticOn 𝕜 f s` where `s` is an open set, then `f` is analytic
+therefore differentiable at every point of `s`, by (1), so by (3) the power series for its
+derivative converges on whole balls. Therefore, the derivative of `f` is also analytic on `s`. The
+same holds if `s` is merely a set with unique differentials.
+
+(5) However, this does not work for `AnalyticOnNhd 𝕜 f s`, as we don't get for free
+differentiability at points in a neighborhood of `s`. Therefore, the theorem that deduces
+`AnalyticOnNhd 𝕜 (fderiv 𝕜 f) s` from `AnalyticOnNhd 𝕜 f s` requires completeness of the space.
+
 -/
 
-open Filter Asymptotics
+open Filter Asymptotics Set
 
-open scoped ENNReal
+open scoped ENNReal Topology
 
 universe u v
 
@@ -34,49 +76,123 @@ section fderiv
 variable {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞}
 variable {f : E → F} {x : E} {s : Set E}
 
-theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
-    HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
+/-- A function which is analytic within a set is strictly differentiable there. Since we
+don't have a predicate `HasStrictFDerivWithinAt`, we spell out what it would mean. -/
+theorem HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt (h : HasFPowerSeriesWithinAt f p s x) :
+    (fun y ↦ f y.1 - f y.2 - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (y.1 - y.2))
+      =o[𝓝[insert x s ×ˢ insert x s] (x, x)] fun y ↦ y.1 - y.2 := by
   refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)
   refine isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, ?_, EventuallyEq.rfl⟩
+  apply Tendsto.mono_left _ nhdsWithin_le_nhds
   refine (continuous_id.sub continuous_const).norm.tendsto' _ _ ?_
   rw [_root_.id, sub_self, norm_zero]
 
+theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
+    HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by
+  simpa only [Set.insert_eq_of_mem, Set.mem_univ, Set.univ_prod_univ, nhdsWithin_univ]
+    using (h.hasFPowerSeriesWithinAt (s := Set.univ)).hasStrictFDerivWithinAt
+
+theorem HasFPowerSeriesWithinAt.hasFDerivWithinAt (h : HasFPowerSeriesWithinAt f p s x) :
+    HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) (insert x s) x := by
+  rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO, isLittleO_iff]
+  intro c hc
+  have : Tendsto (fun y ↦ (y, x)) (𝓝[insert x s] x) (𝓝[insert x s ×ˢ insert x s] (x, x)) := by
+    rw [nhdsWithin_prod_eq]
+    exact Tendsto.prod_mk tendsto_id (tendsto_const_nhdsWithin (by simp))
+  exact this (isLittleO_iff.1 h.hasStrictFDerivWithinAt hc)
+
 theorem HasFPowerSeriesAt.hasFDerivAt (h : HasFPowerSeriesAt f p x) :
     HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x :=
   h.hasStrictFDerivAt.hasFDerivAt
 
+theorem HasFPowerSeriesWithinAt.differentiableWithinAt (h : HasFPowerSeriesWithinAt f p s x) :
+    DifferentiableWithinAt 𝕜 f (insert x s) x :=
+  h.hasFDerivWithinAt.differentiableWithinAt
+
 theorem HasFPowerSeriesAt.differentiableAt (h : HasFPowerSeriesAt f p x) : DifferentiableAt 𝕜 f x :=
   h.hasFDerivAt.differentiableAt
 
+theorem AnalyticWithinAt.differentiableWithinAt (h : AnalyticWithinAt 𝕜 f s x) :
+    DifferentiableWithinAt 𝕜 f (insert x s) x := by
+  obtain ⟨p, hp⟩ := h
+  exact hp.differentiableWithinAt
+
 theorem AnalyticAt.differentiableAt : AnalyticAt 𝕜 f x → DifferentiableAt 𝕜 f x
   | ⟨_, hp⟩ => hp.differentiableAt
 
 theorem AnalyticAt.differentiableWithinAt (h : AnalyticAt 𝕜 f x) : DifferentiableWithinAt 𝕜 f s x :=
   h.differentiableAt.differentiableWithinAt
 
+theorem HasFPowerSeriesWithinAt.fderivWithin_eq
+    (h : HasFPowerSeriesWithinAt f p s x) (hu : UniqueDiffWithinAt 𝕜 (insert x s) x) :
+    fderivWithin 𝕜 f (insert x s) x = continuousMultilinearCurryFin1 𝕜 E F (p 1) :=
+  h.hasFDerivWithinAt.fderivWithin hu
+
 theorem HasFPowerSeriesAt.fderiv_eq (h : HasFPowerSeriesAt f p x) :
     fderiv 𝕜 f x = continuousMultilinearCurryFin1 𝕜 E F (p 1) :=
   h.hasFDerivAt.fderiv
 
+theorem AnalyticAt.hasStrictFDerivAt (h : AnalyticAt 𝕜 f x) :
+    HasStrictFDerivAt f (fderiv 𝕜 f x) x := by
+  rcases h with ⟨p, hp⟩
+  rw [hp.fderiv_eq]
+  exact hp.hasStrictFDerivAt
+
+theorem HasFPowerSeriesWithinOnBall.differentiableOn [CompleteSpace F]
+    (h : HasFPowerSeriesWithinOnBall f p s x r) :
+    DifferentiableOn 𝕜 f (insert x s ∩ EMetric.ball x r) := by
+  intro y hy
+  have Z := (h.analyticWithinAt_of_mem hy).differentiableWithinAt
+  rcases eq_or_ne y x with rfl | hy
+  · exact Z.mono inter_subset_left
+  · apply (Z.mono (subset_insert _ _)).mono_of_mem
+    suffices s ∈ 𝓝[insert x s] y from nhdsWithin_mono _ inter_subset_left this
+    rw [nhdsWithin_insert_of_ne hy]
+    exact self_mem_nhdsWithin
+
 theorem HasFPowerSeriesOnBall.differentiableOn [CompleteSpace F]
     (h : HasFPowerSeriesOnBall f p x r) : DifferentiableOn 𝕜 f (EMetric.ball x r) := fun _ hy =>
   (h.analyticAt_of_mem hy).differentiableWithinAt
 
-theorem AnalyticOn.differentiableOn (h : AnalyticOn 𝕜 f s) : DifferentiableOn 𝕜 f s := fun y hy =>
-  (h y hy).differentiableWithinAt
+theorem AnalyticOn.differentiableOn (h : AnalyticOn 𝕜 f s) : DifferentiableOn 𝕜 f s :=
+  fun y hy ↦  (h y hy).differentiableWithinAt.mono (by simp)
+
+theorem AnalyticOnNhd.differentiableOn (h : AnalyticOnNhd 𝕜 f s) : DifferentiableOn 𝕜 f s :=
+  fun y hy ↦ (h y hy).differentiableWithinAt
+
+theorem HasFPowerSeriesWithinOnBall.hasFDerivWithinAt [CompleteSpace F]
+    (h : HasFPowerSeriesWithinOnBall f p s x r)
+    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) (h'y : x + y ∈ insert x s) :
+    HasFDerivWithinAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1))
+      (insert x s) (x + y) := by
+  rcases eq_or_ne y 0 with rfl | h''y
+  · convert (h.changeOrigin hy h'y).hasFPowerSeriesWithinAt.hasFDerivWithinAt
+    simp
+  · have Z := (h.changeOrigin hy h'y).hasFPowerSeriesWithinAt.hasFDerivWithinAt
+    apply (Z.mono (subset_insert _ _)).mono_of_mem
+    rw [nhdsWithin_insert_of_ne]
+    · exact self_mem_nhdsWithin
+    · simpa using h''y
 
 theorem HasFPowerSeriesOnBall.hasFDerivAt [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
     {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
     HasFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1)) (x + y) :=
   (h.changeOrigin hy).hasFPowerSeriesAt.hasFDerivAt
 
+theorem HasFPowerSeriesWithinOnBall.fderivWithin_eq [CompleteSpace F]
+    (h : HasFPowerSeriesWithinOnBall f p s x r)
+    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) (h'y : x + y ∈ insert x s) (hu : UniqueDiffOn 𝕜 (insert x s)) :
+    fderivWithin 𝕜 f (insert x s) (x + y) =
+      continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1) :=
+  (h.hasFDerivWithinAt hy h'y).fderivWithin (hu _ h'y)
+
 theorem HasFPowerSeriesOnBall.fderiv_eq [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r)
     {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
     fderiv 𝕜 f (x + y) = continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin y 1) :=
   (h.hasFDerivAt hy).fderiv
 
-/-- If a function has a power series on a ball, then so does its derivative. -/
-theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBall f p x r) :
+protected theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F]
+    (h : HasFPowerSeriesOnBall f p x r) :
     HasFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x r := by
   refine .congr (f := fun z ↦ continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1)) ?_
     fun z hz ↦ ?_
@@ -88,33 +204,76 @@ theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F] (h : HasFPowerSeriesOnBal
   rw [← h.fderiv_eq, add_sub_cancel]
   simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
 
+/-- If a function has a power series within a set on a ball, then so does its derivative. -/
+protected theorem HasFPowerSeriesWithinOnBall.fderivWithin [CompleteSpace F]
+    (h : HasFPowerSeriesWithinOnBall f p s x r) (hu : UniqueDiffOn 𝕜 (insert x s)) :
+    HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f (insert x s)) p.derivSeries s x r := by
+  refine .congr' (f := fun z ↦ continuousMultilinearCurryFin1 𝕜 E F (p.changeOrigin (z - x) 1)) ?_
+    (fun z hz ↦ ?_)
+  · refine continuousMultilinearCurryFin1 𝕜 E F
+      |>.toContinuousLinearEquiv.toContinuousLinearMap.comp_hasFPowerSeriesWithinOnBall ?_
+    apply HasFPowerSeriesOnBall.hasFPowerSeriesWithinOnBall
+    simpa using ((p.hasFPowerSeriesOnBall_changeOrigin 1
+      (h.r_pos.trans_le h.r_le)).mono h.r_pos h.r_le).comp_sub x
+  · dsimp only
+    rw [← h.fderivWithin_eq _ _ hu, add_sub_cancel]
+    · simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz.2
+    · simpa using hz.1
+
 /-- If a function is analytic on a set `s`, so is its Fréchet derivative. -/
-theorem AnalyticOn.fderiv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) :
-    AnalyticOn 𝕜 (fderiv 𝕜 f) s := by
-  intro y hy
-  rcases h y hy with ⟨p, r, hp⟩
+protected theorem AnalyticAt.fderiv [CompleteSpace F] (h : AnalyticAt 𝕜 f x) :
+    AnalyticAt 𝕜 (fderiv 𝕜 f) x := by
+  rcases h with ⟨p, r, hp⟩
   exact hp.fderiv.analyticAt
 
-/-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. -/
-theorem AnalyticOn.iteratedFDeriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
-    AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s := by
+/-- If a function is analytic on a set `s`, so is its Fréchet derivative. See also
+`AnalyticOnNhd.fderiv_of_isOpen`, removing the completeness assumption but requiring the set
+to be open. -/
+protected theorem AnalyticOnNhd.fderiv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) :
+    AnalyticOnNhd 𝕜 (fderiv 𝕜 f) s :=
+  fun y hy ↦ AnalyticAt.fderiv (h y hy)
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.fderiv := AnalyticOnNhd.fderiv
+
+/-- If a function is analytic on a set `s`, so are its successive Fréchet derivative. See also
+`AnalyticOnNhd.iteratedFDeruv_of_isOpen`, removing the completeness assumption but requiring the set
+to be open.-/
+protected theorem AnalyticOnNhd.iteratedFDeriv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) (n : ℕ) :
+    AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s := by
   induction n with
   | zero =>
     rw [iteratedFDeriv_zero_eq_comp]
-    exact ((continuousMultilinearCurryFin0 𝕜 E F).symm : F →L[𝕜] E[×0]→L[𝕜] F).comp_analyticOn h
+    exact ((continuousMultilinearCurryFin0 𝕜 E F).symm : F →L[𝕜] E[×0]→L[𝕜] F).comp_analyticOnNhd h
   | succ n IH =>
     rw [iteratedFDeriv_succ_eq_comp_left]
     -- Porting note: for reasons that I do not understand at all, `?g` cannot be inlined.
-    convert ContinuousLinearMap.comp_analyticOn ?g IH.fderiv
-    case g => exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F)
+    convert ContinuousLinearMap.comp_analyticOnNhd ?g IH.fderiv
+    case g => exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F).symm
     simp
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.iteratedFDeriv := AnalyticOnNhd.iteratedFDeriv
+
+/-- If a function is analytic on a neighborhood of a set `s`, then it has a Taylor series given
+by the sequence of its derivatives. Note that, if the function were just analytic on `s`, then
+one would have to use instead the sequence of derivatives inside the set, as in
+`AnalyticOn.hasFTaylorSeriesUpToOn`. -/
+lemma AnalyticOnNhd.hasFTaylorSeriesUpToOn [CompleteSpace F]
+    (n : ℕ∞) (h : AnalyticOnNhd 𝕜 f s) :
+    HasFTaylorSeriesUpToOn n f (ftaylorSeries 𝕜 f) s := by
+  refine ⟨fun x _hx ↦ rfl, fun m _hm x hx ↦ ?_, fun m _hm x hx ↦ ?_⟩
+  · apply HasFDerivAt.hasFDerivWithinAt
+    exact ((h.iteratedFDeriv m x hx).differentiableAt).hasFDerivAt
+  · apply (DifferentiableAt.continuousAt (𝕜 := 𝕜) ?_).continuousWithinAt
+    exact (h.iteratedFDeriv m x hx).differentiableAt
+
 /-- An analytic function is infinitely differentiable. -/
-theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : ℕ∞} :
+protected theorem AnalyticOnNhd.contDiffOn [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) {n : ℕ∞} :
     ContDiffOn 𝕜 n f s :=
   let t := { x | AnalyticAt 𝕜 f x }
   suffices ContDiffOn 𝕜 n f t from this.mono h
-  have H : AnalyticOn 𝕜 f t := fun _x hx ↦ hx
+  have H : AnalyticOnNhd 𝕜 f t := fun _x hx ↦ hx
   have t_open : IsOpen t := isOpen_analyticAt 𝕜 f
   contDiffOn_of_continuousOn_differentiableOn
     (fun m _ ↦ (H.iteratedFDeriv m).continuousOn.congr
@@ -122,11 +281,150 @@ theorem AnalyticOn.contDiffOn [CompleteSpace F] (h : AnalyticOn 𝕜 f s) {n : 
     (fun m _ ↦ (H.iteratedFDeriv m).differentiableOn.congr
       fun _ hx ↦ iteratedFDerivWithin_of_isOpen _ t_open hx)
 
+/-- An analytic function on the whole space is infinitely differentiable there. -/
+theorem AnalyticOnNhd.contDiff [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f univ) {n : ℕ∞} :
+    ContDiff 𝕜 n f := by
+  rw [← contDiffOn_univ]
+  exact h.contDiffOn
+
 theorem AnalyticAt.contDiffAt [CompleteSpace F] (h : AnalyticAt 𝕜 f x) {n : ℕ∞} :
     ContDiffAt 𝕜 n f x := by
-  obtain ⟨s, hs, hf⟩ := h.exists_mem_nhds_analyticOn
+  obtain ⟨s, hs, hf⟩ := h.exists_mem_nhds_analyticOnNhd
   exact hf.contDiffOn.contDiffAt hs
 
+protected lemma AnalyticWithinAt.contDiffWithinAt [CompleteSpace F] {f : E → F} {s : Set E} {x : E}
+    (h : AnalyticWithinAt 𝕜 f s x) {n : ℕ∞} : ContDiffWithinAt 𝕜 n f s x := by
+  rcases h.exists_analyticAt with ⟨g, fx, fg, hg⟩
+  exact hg.contDiffAt.contDiffWithinAt.congr (fg.mono (subset_insert _ _)) fx
+
+protected lemma AnalyticOn.contDiffOn [CompleteSpace F] {f : E → F} {s : Set E}
+    (h : AnalyticOn 𝕜 f s) {n : ℕ∞} : ContDiffOn 𝕜 n f s :=
+  fun x m ↦ (h x m).contDiffWithinAt
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticWithinOn.contDiffOn := AnalyticOn.contDiffOn
+
+lemma AnalyticWithinAt.exists_hasFTaylorSeriesUpToOn [CompleteSpace F]
+    (n : ℕ∞) (h : AnalyticWithinAt 𝕜 f s x) :
+    ∃ u ∈ 𝓝[insert x s] x, ∃ (p : E → FormalMultilinearSeries 𝕜 E F),
+    HasFTaylorSeriesUpToOn n f p u ∧ ∀ i, AnalyticOn 𝕜 (fun x ↦ p x i) u := by
+  rcases h.exists_analyticAt with ⟨g, -, fg, hg⟩
+  rcases hg.exists_mem_nhds_analyticOnNhd with ⟨v, vx, hv⟩
+  refine ⟨insert x s ∩ v, inter_mem_nhdsWithin _ vx, ftaylorSeries 𝕜 g, ?_, fun i ↦ ?_⟩
+  · suffices HasFTaylorSeriesUpToOn n g (ftaylorSeries 𝕜 g) (insert x s ∩ v) from
+      this.congr (fun y hy ↦ fg hy.1)
+    exact AnalyticOnNhd.hasFTaylorSeriesUpToOn _ (hv.mono Set.inter_subset_right)
+  · exact (hv.iteratedFDeriv i).analyticOn.mono Set.inter_subset_right
+
+/-- If a function has a power series `p` within a set of unique differentiability, inside a ball,
+and is differentiable at a point, then the derivative series of `p` is summable at a point, with
+sum the given differential. Note that this theorem does not require completeness of the space.-/
+theorem HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt
+    (h : HasFPowerSeriesWithinOnBall f p s x r)
+    {f' : E →L[𝕜] F}
+    {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) (h'y : x + y ∈ insert x s)
+    (hf' : HasFDerivWithinAt f f' (insert x s) (x + y))
+    (hu : UniqueDiffOn 𝕜 (insert x s)) :
+    HasSum (fun n ↦ p.derivSeries n (fun _ ↦ y)) f' := by
+  /- In the completion of the space, the derivative series is summable, and its sum is a derivative
+  of the function. Therefore, by uniqueness of derivatives, its sum is the image of `f'` under
+  the canonical embedding. As this is an embedding, it means that there was also convergence in
+  the original space, to `f'`. -/
+  let F' := UniformSpace.Completion F
+  let a : F →L[𝕜] F' := UniformSpace.Completion.toComplL
+  let b : (E →L[𝕜] F) →ₗᵢ[𝕜] (E →L[𝕜] F') := UniformSpace.Completion.toComplₗᵢ.postcomp
+  rw [← b.embedding.hasSum_iff]
+  have : HasFPowerSeriesWithinOnBall (a ∘ f) (a.compFormalMultilinearSeries p) s x r :=
+    a.comp_hasFPowerSeriesWithinOnBall h
+  have Z := (this.fderivWithin hu).hasSum h'y (by simpa [edist_eq_coe_nnnorm] using hy)
+  have : fderivWithin 𝕜 (a ∘ f) (insert x s) (x + y) = a ∘L f' := by
+    apply HasFDerivWithinAt.fderivWithin _ (hu _ h'y)
+    exact a.hasFDerivAt.comp_hasFDerivWithinAt (x + y) hf'
+  rw [this] at Z
+  convert Z with n
+  ext v
+  simp only [FormalMultilinearSeries.derivSeries,
+    ContinuousLinearMap.compFormalMultilinearSeries_apply,
+    FormalMultilinearSeries.changeOriginSeries,
+    ContinuousLinearMap.compContinuousMultilinearMap_coe, ContinuousLinearEquiv.coe_coe,
+    LinearIsometryEquiv.coe_coe, Function.comp_apply, ContinuousMultilinearMap.sum_apply, map_sum,
+    ContinuousLinearMap.coe_sum', Finset.sum_apply,
+    Matrix.zero_empty]
+  rfl
+
+/-- If a function is analytic within a set with unique differentials, then so is its derivative.
+Note that this theorem does not require completeness of the space. -/
+protected theorem AnalyticOn.fderivWithin (h : AnalyticOn 𝕜 f s) (hu : UniqueDiffOn 𝕜 s) :
+    AnalyticOn 𝕜 (fderivWithin 𝕜 f s) s := by
+  intro x hx
+  rcases h x hx with ⟨p, r, hr⟩
+  refine ⟨p.derivSeries, r, ?_⟩
+  refine ⟨hr.r_le.trans p.radius_le_radius_derivSeries, hr.r_pos, fun {y} hy h'y ↦ ?_⟩
+  apply hr.hasSum_derivSeries_of_hasFDerivWithinAt (by simpa [edist_eq_coe_nnnorm] using h'y) hy
+  · rw [insert_eq_of_mem hx] at hy ⊢
+    apply DifferentiableWithinAt.hasFDerivWithinAt
+    exact h.differentiableOn _ hy
+  · rwa [insert_eq_of_mem hx]
+
+/-- If a function is analytic on a set `s`, so are its successive Fréchet derivative within this
+set. Note that this theorem does not require completeness of the space. -/
+protected theorem AnalyticOn.iteratedFDerivWithin (h : AnalyticOn 𝕜 f s)
+    (hu : UniqueDiffOn 𝕜 s) (n : ℕ) :
+    AnalyticOn 𝕜 (iteratedFDerivWithin 𝕜 n f s) s := by
+  induction n with
+  | zero =>
+    rw [iteratedFDerivWithin_zero_eq_comp]
+    exact ((continuousMultilinearCurryFin0 𝕜 E F).symm : F →L[𝕜] E[×0]→L[𝕜] F)
+      |>.comp_analyticOn h
+  | succ n IH =>
+    rw [iteratedFDerivWithin_succ_eq_comp_left]
+    apply AnalyticOnNhd.comp_analyticOn _ (IH.fderivWithin hu) (mapsTo_univ _ _)
+    apply LinearIsometryEquiv.analyticOnNhd
+
+lemma AnalyticOn.hasFTaylorSeriesUpToOn {n : ℕ∞}
+    (h : AnalyticOn 𝕜 f s) (hu : UniqueDiffOn 𝕜 s) :
+    HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s := by
+  refine ⟨fun x _hx ↦ rfl, fun m _hm x hx ↦ ?_, fun m _hm x hx ↦ ?_⟩
+  · have := (h.iteratedFDerivWithin hu m x hx).differentiableWithinAt.hasFDerivWithinAt
+    rwa [insert_eq_of_mem hx] at this
+  · exact (h.iteratedFDerivWithin hu m x hx).continuousWithinAt
+
+lemma AnalyticOn.exists_hasFTaylorSeriesUpToOn
+    (h : AnalyticOn 𝕜 f s) (hu : UniqueDiffOn 𝕜 s) :
+    ∃ (p : E → FormalMultilinearSeries 𝕜 E F),
+      HasFTaylorSeriesUpToOn ⊤ f p s ∧ ∀ i, AnalyticOn 𝕜 (fun x ↦ p x i) s :=
+  ⟨ftaylorSeriesWithin 𝕜 f s, h.hasFTaylorSeriesUpToOn hu, h.iteratedFDerivWithin hu⟩
+
+theorem AnalyticOnNhd.fderiv_of_isOpen (h : AnalyticOnNhd 𝕜 f s) (hs : IsOpen s) :
+    AnalyticOnNhd 𝕜 (fderiv 𝕜 f) s := by
+  rw [← hs.analyticOn_iff_analyticOnNhd] at h ⊢
+  exact (h.fderivWithin hs.uniqueDiffOn).congr (fun x hx ↦ (fderivWithin_of_isOpen hs hx).symm)
+
+theorem AnalyticOnNhd.iteratedFDeriv_of_isOpen (h : AnalyticOnNhd 𝕜 f s) (hs : IsOpen s) (n : ℕ) :
+    AnalyticOnNhd 𝕜 (iteratedFDeriv 𝕜 n f) s := by
+  rw [← hs.analyticOn_iff_analyticOnNhd] at h ⊢
+  exact (h.iteratedFDerivWithin hs.uniqueDiffOn n).congr
+    (fun x hx ↦ (iteratedFDerivWithin_of_isOpen n hs hx).symm)
+
+/-- If a partial homeomorphism `f` is analytic at a point `a`, with invertible derivative, then
+its inverse is analytic at `f a`. -/
+theorem PartialHomeomorph.analyticAt_symm' (f : PartialHomeomorph E F) {a : E}
+    {i : E ≃L[𝕜] F} (h0 : a ∈ f.source) (h : AnalyticAt 𝕜 f a) (h' : fderiv 𝕜 f a = i) :
+    AnalyticAt 𝕜 f.symm (f a) := by
+  rcases h with ⟨p, hp⟩
+  have : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i := by simp [← h', hp.fderiv_eq]
+  exact (f.hasFPowerSeriesAt_symm h0 hp this).analyticAt
+
+/-- If a partial homeomorphism `f` is analytic at a point `f.symm a`, with invertible derivative,
+then its inverse is analytic at `a`. -/
+theorem PartialHomeomorph.analyticAt_symm (f : PartialHomeomorph E F) {a : F}
+    {i : E ≃L[𝕜] F} (h0 : a ∈ f.target) (h : AnalyticAt 𝕜 f (f.symm a))
+    (h' : fderiv 𝕜 f (f.symm a) = i) :
+    AnalyticAt 𝕜 f.symm a := by
+  have : a = f (f.symm a) := by simp [h0]
+  rw [this]
+  exact f.analyticAt_symm' (by simp [h0]) h h'
+
 end fderiv
 
 section deriv
@@ -146,17 +444,29 @@ protected theorem HasFPowerSeriesAt.deriv (h : HasFPowerSeriesAt f p x) :
     deriv f x = p 1 fun _ => 1 :=
   h.hasDerivAt.deriv
 
-/-- If a function is analytic on a set `s`, so is its derivative. -/
-theorem AnalyticOn.deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) : AnalyticOn 𝕜 (deriv f) s :=
-  (ContinuousLinearMap.apply 𝕜 F (1 : 𝕜)).comp_analyticOn h.fderiv
+/-- If a function is analytic on a set `s` in a complete space, so is its derivative. -/
+protected theorem AnalyticOnNhd.deriv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) :
+    AnalyticOnNhd 𝕜 (deriv f) s :=
+  (ContinuousLinearMap.apply 𝕜 F (1 : 𝕜)).comp_analyticOnNhd h.fderiv
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.deriv := AnalyticOnNhd.deriv
+
+/-- If a function is analytic on an open set `s`, so is its derivative. -/
+theorem AnalyticOnNhd.deriv_of_isOpen (h : AnalyticOnNhd 𝕜 f s) (hs : IsOpen s) :
+    AnalyticOnNhd 𝕜 (deriv f) s :=
+  (ContinuousLinearMap.apply 𝕜 F (1 : 𝕜)).comp_analyticOnNhd (h.fderiv_of_isOpen hs)
 
 /-- If a function is analytic on a set `s`, so are its successive derivatives. -/
-theorem AnalyticOn.iterated_deriv [CompleteSpace F] (h : AnalyticOn 𝕜 f s) (n : ℕ) :
-    AnalyticOn 𝕜 (_root_.deriv^[n] f) s := by
+theorem AnalyticOnNhd.iterated_deriv [CompleteSpace F] (h : AnalyticOnNhd 𝕜 f s) (n : ℕ) :
+    AnalyticOnNhd 𝕜 (_root_.deriv^[n] f) s := by
   induction n with
   | zero => exact h
   | succ n IH => simpa only [Function.iterate_succ', Function.comp_apply] using IH.deriv
 
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.iterated_deriv := AnalyticOnNhd.iterated_deriv
+
 end deriv
 section fderiv
 
@@ -230,7 +540,7 @@ theorem CPolynomialOn.iteratedFDeriv (h : CPolynomialOn 𝕜 f s) (n : ℕ) :
   | succ n IH =>
     rw [iteratedFDeriv_succ_eq_comp_left]
     convert ContinuousLinearMap.comp_cPolynomialOn ?g IH.fderiv
-    case g => exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F)
+    case g => exact ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) ↦ E) F).symm
     simp
 
 /-- A polynomial function is infinitely differentiable. -/
@@ -243,7 +553,7 @@ theorem CPolynomialOn.contDiffOn (h : CPolynomialOn 𝕜 f s) {n : ℕ∞} :
   contDiffOn_of_continuousOn_differentiableOn
     (fun m _ ↦ (H.iteratedFDeriv m).continuousOn.congr
       fun  _ hx ↦ iteratedFDerivWithin_of_isOpen _ t_open hx)
-    (fun m _ ↦ (H.iteratedFDeriv m).analyticOn.differentiableOn.congr
+    (fun m _ ↦ (H.iteratedFDeriv m).analyticOnNhd.differentiableOn.congr
       fun _ hx ↦ iteratedFDerivWithin_of_isOpen _ t_open hx)
 
 theorem CPolynomialAt.contDiffAt (h : CPolynomialAt 𝕜 f x) {n : ℕ∞} :
@@ -278,13 +588,6 @@ variable {ι : Type*} {E : ι → Type*} [∀ i, NormedAddCommGroup (E i)] [∀
 
 open FormalMultilinearSeries
 
-protected theorem hasFiniteFPowerSeriesOnBall :
-    HasFiniteFPowerSeriesOnBall f f.toFormalMultilinearSeries 0 (Fintype.card ι + 1) ⊤ :=
-  .mk' (fun m hm ↦ dif_neg (Nat.succ_le_iff.mp hm).ne) ENNReal.zero_lt_top fun y _ ↦ by
-    rw [Finset.sum_eq_single_of_mem _ (Finset.self_mem_range_succ _), zero_add]
-    · rw [toFormalMultilinearSeries, dif_pos rfl]; rfl
-    · intro m _ ne; rw [toFormalMultilinearSeries, dif_neg ne.symm]; rfl
-
 theorem changeOriginSeries_support {k l : ℕ} (h : k + l ≠ Fintype.card ι) :
     f.toFormalMultilinearSeries.changeOriginSeries k l = 0 :=
   Finset.sum_eq_zero fun _ _ ↦ by
@@ -313,7 +616,7 @@ theorem changeOrigin_toFormalMultilinearSeries [DecidableEq ι] :
   refine (Fintype.sum_bijective (?_ ∘ Fintype.equivFinOfCardEq (Nat.add_sub_of_le
     Fintype.card_pos).symm) (.comp ?_ <| Equiv.bijective _) _ _ fun i ↦ ?_).symm
   · exact (⟨{·}ᶜ, by
-      rw [card_compl, Fintype.card_fin, card_singleton, Nat.add_sub_cancel_left]⟩)
+      rw [card_compl, Fintype.card_fin, Finset.card_singleton, Nat.add_sub_cancel_left]⟩)
   · use fun _ _ ↦ (singleton_injective <| compl_injective <| Subtype.ext_iff.mp ·)
     intro ⟨s, hs⟩
     have h : sᶜ.card = 1 := by rw [card_compl, hs, Fintype.card_fin, Nat.add_sub_cancel]
@@ -333,6 +636,30 @@ protected theorem hasFDerivAt [DecidableEq ι] : HasFDerivAt f (f.linearDeriv x)
   convert f.hasFiniteFPowerSeriesOnBall.hasFDerivAt (y := x) ENNReal.coe_lt_top
   rw [zero_add]
 
+/-- Given `f` a multilinear map, then the derivative of `x ↦ f (g₁ x, ..., gₙ x)` at `x` applied
+to a vector `v` is given by `∑ i, f (g₁ x, ..., g'ᵢ v, ..., gₙ x)`. Version inside a set. -/
+theorem _root_.HasFDerivWithinAt.multilinear_comp
+    [DecidableEq ι] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+    {g : ∀ i, G → E i} {g' : ∀ i, G →L[𝕜] E i} {s : Set G} {x : G}
+    (hg : ∀ i, HasFDerivWithinAt (g i) (g' i) s x) :
+    HasFDerivWithinAt (fun x ↦ f (fun i ↦ g i x))
+      ((∑ i : ι, (f.toContinuousLinearMap (fun j ↦ g j x) i) ∘L (g' i))) s x := by
+  convert (f.hasFDerivAt (fun j ↦ g j x)).comp_hasFDerivWithinAt x (hasFDerivWithinAt_pi.2 hg)
+  ext v
+  simp [linearDeriv]
+
+/-- Given `f` a multilinear map, then the derivative of `x ↦ f (g₁ x, ..., gₙ x)` at `x` applied
+to a vector `v` is given by `∑ i, f (g₁ x, ..., g'ᵢ v, ..., gₙ x)`. -/
+theorem _root_.HasFDerivAt.multilinear_comp
+    [DecidableEq ι] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+    {g : ∀ i, G → E i} {g' : ∀ i, G →L[𝕜] E i} {x : G}
+    (hg : ∀ i, HasFDerivAt (g i) (g' i) x) :
+    HasFDerivAt (fun x ↦ f (fun i ↦ g i x))
+      ((∑ i : ι, (f.toContinuousLinearMap (fun j ↦ g j x) i) ∘L (g' i))) x := by
+  convert (f.hasFDerivAt (fun j ↦ g j x)).comp x (hasFDerivAt_pi.2 hg)
+  ext v
+  simp [linearDeriv]
+
 /-- Technical lemma used in the proof of `hasFTaylorSeriesUpTo_iteratedFDeriv`, to compare sums
 over embedding of `Fin k` and `Fin (k + 1)`. -/
 private lemma _root_.Equiv.succ_embeddingFinSucc_fst_symm_apply {ι : Type*} [DecidableEq ι]
@@ -463,7 +790,7 @@ private theorem factorial_smul' {n : ℕ} : ∀ {F : Type max u v} [NormedAddCom
     [NormedSpace 𝕜 F] [CompleteSpace F] {p : FormalMultilinearSeries 𝕜 E F}
     {f : E → F}, HasFPowerSeriesOnBall f p x r →
     n ! • p n (fun _ ↦ y) = iteratedFDeriv 𝕜 n f x (fun _ ↦ y) := by
-  induction' n with n ih <;> intro F _ _ _ p f h
+  induction n with | zero => _ | succ n ih => _ <;> intro F _ _ _ p f h
   · rw [factorial_zero, one_smul, h.iteratedFDeriv_zero_apply_diag]
   · rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag, ← smul_apply,
       ih h.fderiv, iteratedFDeriv_succ_apply_right]
@@ -487,3 +814,73 @@ theorem hasSum_iteratedFDeriv [CharZero 𝕜] {y : E} (hy : y ∈ EMetric.ball 0
     mul_inv_cancel₀ <| cast_ne_zero.mpr n.factorial_ne_zero, one_smul]
 
 end HasFPowerSeriesOnBall
+
+/-!
+### Derivative of a linear map into multilinear maps
+-/
+
+namespace ContinuousLinearMap
+
+variable {ι : Type*} {G : ι → Type*} [∀ i, NormedAddCommGroup (G i)] [∀ i, NormedSpace 𝕜 (G i)]
+  [Fintype ι]  {H : Type*} [NormedAddCommGroup H]
+  [NormedSpace 𝕜 H]
+
+theorem hasFDerivAt_uncurry_of_multilinear [DecidableEq ι]
+    (f : E →L[𝕜] ContinuousMultilinearMap 𝕜 G F) (v : E × Π i, G i) :
+    HasFDerivAt (fun (p : E × Π i, G i) ↦ f p.1 p.2)
+      ((f.flipMultilinear v.2) ∘L (.fst _ _ _) +
+        ∑ i : ι, ((f v.1).toContinuousLinearMap v.2 i) ∘L (.proj _) ∘L (.snd _ _ _)) v := by
+  convert HasFDerivAt.multilinear_comp (f.continuousMultilinearMapOption)
+    (g := fun (_ : Option ι) p ↦ p) (g' := fun _ ↦ ContinuousLinearMap.id _ _) (x := v)
+    (fun _ ↦ hasFDerivAt_id _)
+  have I : f.continuousMultilinearMapOption.toContinuousLinearMap (fun _ ↦ v) none =
+      (f.flipMultilinear v.2) ∘L (.fst _ _ _) := by
+    simp [ContinuousMultilinearMap.toContinuousLinearMap, continuousMultilinearMapOption]
+    apply ContinuousLinearMap.ext (fun w ↦ ?_)
+    simp
+  have J : ∀ (i : ι), f.continuousMultilinearMapOption.toContinuousLinearMap (fun _ ↦ v) (some i)
+      = ((f v.1).toContinuousLinearMap v.2 i) ∘L (.proj _) ∘L (.snd _ _ _) := by
+    intro i
+    apply ContinuousLinearMap.ext (fun w ↦ ?_)
+    simp only [ContinuousMultilinearMap.toContinuousLinearMap, continuousMultilinearMapOption,
+      coe_mk', MultilinearMap.toLinearMap_apply, ContinuousMultilinearMap.coe_coe,
+      MultilinearMap.coe_mkContinuous, MultilinearMap.coe_mk, ne_eq, reduceCtorEq,
+      not_false_eq_true, Function.update_noteq, coe_comp', coe_snd', Function.comp_apply,
+      proj_apply]
+    congr
+    ext j
+    rcases eq_or_ne j i with rfl | hij
+    · simp
+    · simp [hij]
+  simp [I, J]
+
+/-- Given `f` a linear map into multilinear maps, then the derivative
+of `x ↦ f (a x) (b₁ x, ..., bₙ x)` at `x` applied to a vector `v` is given by
+`f (a' v) (b₁ x, ...., bₙ x) + ∑ i, f a (b₁ x, ..., b'ᵢ v, ..., bₙ x)`. Version inside a set. -/
+theorem _root_.HasFDerivWithinAt.linear_multilinear_comp
+    [DecidableEq ι] {a : H → E} {a' : H →L[𝕜] E}
+    {b : ∀ i, H → G i} {b' : ∀ i, H →L[𝕜] G i} {s : Set H} {x : H}
+    (ha : HasFDerivWithinAt a a' s x) (hb : ∀ i, HasFDerivWithinAt (b i) (b' i) s x)
+    (f : E →L[𝕜] ContinuousMultilinearMap 𝕜 G F) :
+    HasFDerivWithinAt (fun y ↦ f (a y) (fun i ↦ b i y))
+      ((f.flipMultilinear (fun i ↦ b i x)) ∘L a' +
+        ∑ i, ((f (a x)).toContinuousLinearMap (fun j ↦ b j x) i) ∘L (b' i)) s x := by
+  convert (hasFDerivAt_uncurry_of_multilinear f (a x, fun i ↦ b i x)).comp_hasFDerivWithinAt x
+    (ha.prod (hasFDerivWithinAt_pi.mpr hb))
+  ext v
+  simp
+
+/-- Given `f` a linear map into multilinear maps, then the derivative
+of `x ↦ f (a x) (b₁ x, ..., bₙ x)` at `x` applied to a vector `v` is given by
+`f (a' v) (b₁ x, ...., bₙ x) + ∑ i, f a (b₁ x, ..., b'ᵢ v, ..., bₙ x)`. -/
+theorem _root_.HasFDerivAt.linear_multilinear_comp [DecidableEq ι] {a : H → E} {a' : H →L[𝕜] E}
+    {b : ∀ i, H → G i} {b' : ∀ i, H →L[𝕜] G i} {x : H}
+    (ha : HasFDerivAt a a' x) (hb : ∀ i, HasFDerivAt (b i) (b' i) x)
+    (f : E →L[𝕜] ContinuousMultilinearMap 𝕜 G F) :
+    HasFDerivAt (fun y ↦ f (a y) (fun i ↦ b i y))
+      ((f.flipMultilinear (fun i ↦ b i x)) ∘L a' +
+        ∑ i, ((f (a x)).toContinuousLinearMap (fun j ↦ b j x) i) ∘L (b' i)) x := by
+  simp_rw [← hasFDerivWithinAt_univ] at ha hb ⊢
+  exact HasFDerivWithinAt.linear_multilinear_comp ha hb f
+
+end ContinuousLinearMap
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Basic.lean b/Mathlib/Analysis/Calculus/FDeriv/Basic.lean
index 7da598a12bd5b..88e55627bf87e 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Basic.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Basic.lean
@@ -306,7 +306,7 @@ theorem hasFDerivAt_iff_tendsto :
 theorem hasFDerivAt_iff_isLittleO_nhds_zero :
     HasFDerivAt f f' x ↔ (fun h : E => f (x + h) - f x - f' h) =o[𝓝 0] fun h => h := by
   rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, ← map_add_left_nhds_zero x, isLittleO_map]
-  simp [(· ∘ ·)]
+  simp [Function.comp_def]
 
 /-- Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
 on a neighborhood of `x₀` then its derivative at `x₀` has norm bounded by `C`. This version
@@ -895,7 +895,7 @@ theorem Filter.EventuallyEq.fderivWithin_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx :
 
 theorem Filter.EventuallyEq.fderivWithin' (hs : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) :
     fderivWithin 𝕜 f₁ t =ᶠ[𝓝[s] x] fderivWithin 𝕜 f t :=
-  (eventually_nhdsWithin_nhdsWithin.2 hs).mp <|
+  (eventually_eventually_nhdsWithin.2 hs).mp <|
     eventually_mem_nhdsWithin.mono fun _y hys hs =>
       EventuallyEq.fderivWithin_eq (hs.filter_mono <| nhdsWithin_mono _ ht)
         (hs.self_of_nhdsWithin hys)
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean b/Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
index 9505419593904..b5afecc0aa536 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean
@@ -69,7 +69,7 @@ theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap 𝕜 b)
       refine (isBigO_refl _ _).mul_isLittleO ((isLittleO_one_iff _).2 ?_)
       -- TODO: `continuity` fails
       exact (continuous_snd.fst.prod_mk continuous_fst.snd).norm.tendsto' _ _ (by simp)
-    _ = _ := by simp [(· ∘ ·)]
+    _ = _ := by simp [Function.comp_def]
 
 @[fun_prop]
 theorem IsBoundedBilinearMap.hasFDerivAt (h : IsBoundedBilinearMap 𝕜 b) (p : E × F) :
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Comp.lean b/Mathlib/Analysis/Calculus/FDeriv/Comp.lean
index beaf6fad6b9fe..410d6d1cf3e69 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Comp.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Comp.lean
@@ -172,7 +172,7 @@ theorem Differentiable.comp_differentiableOn {g : F → G} (hg : Differentiable
 protected theorem HasStrictFDerivAt.comp {g : F → G} {g' : F →L[𝕜] G}
     (hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
     HasStrictFDerivAt (fun x => g (f x)) (g'.comp f') x :=
-  ((hg.comp_tendsto (hf.continuousAt.prod_map' hf.continuousAt)).trans_isBigO
+  ((hg.comp_tendsto (hf.continuousAt.prodMap' hf.continuousAt)).trans_isBigO
       hf.isBigO_sub).triangle <| by
     simpa only [g'.map_sub, f'.coe_comp'] using (g'.isBigO_comp _ _).trans_isLittleO hf
 
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Equiv.lean b/Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
index e2f34a6adc71a..ac47ac267e108 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Equiv.lean
@@ -87,7 +87,7 @@ theorem comp_differentiableWithinAt_iff {f : G → E} {s : Set G} {x : G} :
     ⟨fun H => ?_, fun H => iso.differentiable.differentiableAt.comp_differentiableWithinAt x H⟩
   have : DifferentiableWithinAt 𝕜 (iso.symm ∘ iso ∘ f) s x :=
     iso.symm.differentiable.differentiableAt.comp_differentiableWithinAt x H
-  rwa [← Function.comp.assoc iso.symm iso f, iso.symm_comp_self] at this
+  rwa [← Function.comp_assoc iso.symm iso f, iso.symm_comp_self] at this
 
 theorem comp_differentiableAt_iff {f : G → E} {x : G} :
     DifferentiableAt 𝕜 (iso ∘ f) x ↔ DifferentiableAt 𝕜 f x := by
@@ -107,7 +107,7 @@ theorem comp_hasFDerivWithinAt_iff {f : G → E} {s : Set G} {x : G} {f' : G →
     HasFDerivWithinAt (iso ∘ f) ((iso : E →L[𝕜] F).comp f') s x ↔ HasFDerivWithinAt f f' s x := by
   refine ⟨fun H => ?_, fun H => iso.hasFDerivAt.comp_hasFDerivWithinAt x H⟩
   have A : f = iso.symm ∘ iso ∘ f := by
-    rw [← Function.comp.assoc, iso.symm_comp_self]
+    rw [← Function.comp_assoc, iso.symm_comp_self]
     rfl
   have B : f' = (iso.symm : F →L[𝕜] E).comp ((iso : E →L[𝕜] F).comp f') := by
     rw [← ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe, ContinuousLinearMap.id_comp]
@@ -174,7 +174,7 @@ theorem comp_right_differentiableWithinAt_iff {f : F → G} {s : Set F} {x : E}
     apply H.comp (iso x) iso.symm.differentiableWithinAt
     intro y hy
     simpa only [mem_preimage, apply_symm_apply] using hy
-  rwa [Function.comp.assoc, iso.self_comp_symm] at this
+  rwa [Function.comp_assoc, iso.self_comp_symm] at this
 
 theorem comp_right_differentiableAt_iff {f : F → G} {x : E} :
     DifferentiableAt 𝕜 (f ∘ iso) x ↔ DifferentiableAt 𝕜 f (iso x) := by
@@ -198,7 +198,7 @@ theorem comp_right_hasFDerivWithinAt_iff {f : F → G} {s : Set F} {x : E} {f' :
   refine ⟨fun H => ?_, fun H => H.comp x iso.hasFDerivWithinAt (mapsTo_preimage _ s)⟩
   rw [← iso.symm_apply_apply x] at H
   have A : f = (f ∘ iso) ∘ iso.symm := by
-    rw [Function.comp.assoc, iso.self_comp_symm]
+    rw [Function.comp_assoc, iso.self_comp_symm]
     rfl
   have B : f' = (f'.comp (iso : E →L[𝕜] F)).comp (iso.symm : F →L[𝕜] E) := by
     rw [ContinuousLinearMap.comp_assoc, iso.coe_comp_coe_symm, ContinuousLinearMap.comp_id]
@@ -339,7 +339,7 @@ inverse function. -/
 theorem HasStrictFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
     (hg : ContinuousAt g a) (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) (g a))
     (hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) a := by
-  replace hg := hg.prod_map' hg
+  replace hg := hg.prodMap' hg
   replace hfg := hfg.prod_mk_nhds hfg
   have :
     (fun p : F × F => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)] fun p : F × F =>
@@ -355,7 +355,7 @@ theorem HasStrictFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜]
   · refine (hf.isBigO_sub_rev.comp_tendsto hg).congr' (Eventually.of_forall fun _ => rfl)
       (hfg.mono ?_)
     rintro p ⟨hp1, hp2⟩
-    simp only [(· ∘ ·), hp1, hp2]
+    simp only [(· ∘ ·), hp1, hp2, Prod.map]
 
 /-- If `f (g y) = y` for `y` in some neighborhood of `a`, `g` is continuous at `a`, and `f` has an
 invertible derivative `f'` at `g a`, then `g` has the derivative `f'⁻¹` at `a`.
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean b/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
index 5623962121150..28e1539f0e516 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
@@ -303,7 +303,7 @@ theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete
     · simp [y_pos]
     have yzero : 0 < ‖y‖ := norm_pos_iff.mpr y_pos
     have y_lt : ‖y‖ < (1 / 2) ^ (n e + 1) := by simpa using mem_ball_iff_norm.1 hy
-    have yone : ‖y‖ ≤ 1 := le_trans y_lt.le (pow_le_one _ (by norm_num) (by norm_num))
+    have yone : ‖y‖ ≤ 1 := le_trans y_lt.le (pow_le_one₀ (by norm_num) (by norm_num))
     -- define the scale `k`.
     obtain ⟨k, hk, h'k⟩ : ∃ k : ℕ, (1 / 2) ^ (k + 1) < ‖y‖ ∧ ‖y‖ ≤ (1 / 2) ^ k :=
       exists_nat_pow_near_of_lt_one yzero yone (by norm_num : (0 : ℝ) < 1 / 2)
@@ -638,7 +638,7 @@ theorem D_subset_differentiable_set {K : Set F} (hK : IsComplete K) :
     · simp only [sub_self, zero_smul, norm_zero, mul_zero, le_rfl]
     have yzero : 0 < y - x := sub_pos.2 xy
     have y_le : y - x ≤ (1 / 2) ^ (n e + 1) := by linarith [hy.2]
-    have yone : y - x ≤ 1 := le_trans y_le (pow_le_one _ (by norm_num) (by norm_num))
+    have yone : y - x ≤ 1 := le_trans y_le (pow_le_one₀ (by norm_num) (by norm_num))
     -- define the scale `k`.
     obtain ⟨k, hk, h'k⟩ : ∃ k : ℕ, (1 / 2) ^ (k + 1) < y - x ∧ y - x ≤ (1 / 2) ^ k :=
       exists_nat_pow_near_of_lt_one yzero yone (by norm_num : (0 : ℝ) < 1 / 2)
@@ -661,7 +661,7 @@ theorem D_subset_differentiable_set {K : Set F} (hK : IsComplete K) :
           · simp only [one_div, inv_pow, left_mem_Icc, le_add_iff_nonneg_right]
             positivity
           · simp only [pow_add, tsub_le_iff_left] at h'k
-            simpa only [hy.1, mem_Icc, true_and_iff, one_div, pow_one] using h'k
+            simpa only [hy.1, mem_Icc, true_and, one_div, pow_one] using h'k
         _ = 4 * (1 / 2) ^ e * (1 / 2) ^ (m + 2) := by field_simp; ring
         _ ≤ 4 * (1 / 2) ^ e * (y - x) := by gcongr
         _ = 4 * (1 / 2) ^ e * ‖y - x‖ := by rw [Real.norm_of_nonneg yzero.le]
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Mul.lean b/Mathlib/Analysis/Calculus/FDeriv/Mul.lean
index 9c79b4bf2279c..990bb02bd9599 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Mul.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Mul.lean
@@ -3,6 +3,8 @@ Copyright (c) 2019 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
 -/
+import Mathlib.Analysis.Analytic.Constructions
+import Mathlib.Analysis.Calculus.FDeriv.Analytic
 import Mathlib.Analysis.Calculus.FDeriv.Bilinear
 
 /-!
@@ -50,20 +52,20 @@ variable {H : Type*} [NormedAddCommGroup H] [NormedSpace 𝕜 H] {c : E → G 
 theorem HasStrictFDerivAt.clm_comp (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x) :
     HasStrictFDerivAt (fun y => (c y).comp (d y))
       ((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x :=
-  (isBoundedBilinearMap_comp.hasStrictFDerivAt (c x, d x)).comp x <| hc.prod hd
+  (isBoundedBilinearMap_comp.hasStrictFDerivAt (c x, d x) :).comp x <| hc.prod hd
 
 @[fun_prop]
 theorem HasFDerivWithinAt.clm_comp (hc : HasFDerivWithinAt c c' s x)
     (hd : HasFDerivWithinAt d d' s x) :
     HasFDerivWithinAt (fun y => (c y).comp (d y))
       ((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') s x :=
-  (isBoundedBilinearMap_comp.hasFDerivAt (c x, d x)).comp_hasFDerivWithinAt x <| hc.prod hd
+  (isBoundedBilinearMap_comp.hasFDerivAt (c x, d x) :).comp_hasFDerivWithinAt x <| hc.prod hd
 
 @[fun_prop]
 theorem HasFDerivAt.clm_comp (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :
     HasFDerivAt (fun y => (c y).comp (d y))
       ((compL 𝕜 F G H (c x)).comp d' + ((compL 𝕜 F G H).flip (d x)).comp c') x :=
-  (isBoundedBilinearMap_comp.hasFDerivAt (c x, d x)).comp x <| hc.prod hd
+  (isBoundedBilinearMap_comp.hasFDerivAt (c x, d x) :).comp x <| hc.prod hd
 
 @[fun_prop]
 theorem DifferentiableWithinAt.clm_comp (hc : DifferentiableWithinAt 𝕜 c s x)
@@ -107,12 +109,12 @@ theorem HasStrictFDerivAt.clm_apply (hc : HasStrictFDerivAt c c' x)
 theorem HasFDerivWithinAt.clm_apply (hc : HasFDerivWithinAt c c' s x)
     (hu : HasFDerivWithinAt u u' s x) :
     HasFDerivWithinAt (fun y => (c y) (u y)) ((c x).comp u' + c'.flip (u x)) s x :=
-  (isBoundedBilinearMap_apply.hasFDerivAt (c x, u x)).comp_hasFDerivWithinAt x (hc.prod hu)
+  (isBoundedBilinearMap_apply.hasFDerivAt (c x, u x) :).comp_hasFDerivWithinAt x (hc.prod hu)
 
 @[fun_prop]
 theorem HasFDerivAt.clm_apply (hc : HasFDerivAt c c' x) (hu : HasFDerivAt u u' x) :
     HasFDerivAt (fun y => (c y) (u y)) ((c x).comp u' + c'.flip (u x)) x :=
-  (isBoundedBilinearMap_apply.hasFDerivAt (c x, u x)).comp x (hc.prod hu)
+  (isBoundedBilinearMap_apply.hasFDerivAt (c x, u x) :).comp x (hc.prod hu)
 
 @[fun_prop]
 theorem DifferentiableWithinAt.clm_apply (hc : DifferentiableWithinAt 𝕜 c s x)
@@ -239,12 +241,13 @@ theorem HasStrictFDerivAt.smul (hc : HasStrictFDerivAt c c' x) (hf : HasStrictFD
 @[fun_prop]
 theorem HasFDerivWithinAt.smul (hc : HasFDerivWithinAt c c' s x) (hf : HasFDerivWithinAt f f' s x) :
     HasFDerivWithinAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) s x :=
-  (isBoundedBilinearMap_smul.hasFDerivAt (c x, f x)).comp_hasFDerivWithinAt x <| hc.prod hf
+  (isBoundedBilinearMap_smul.hasFDerivAt (𝕜 := 𝕜) (c x, f x) :).comp_hasFDerivWithinAt x <|
+    hc.prod hf
 
 @[fun_prop]
 theorem HasFDerivAt.smul (hc : HasFDerivAt c c' x) (hf : HasFDerivAt f f' x) :
     HasFDerivAt (fun y => c y • f y) (c x • f' + c'.smulRight (f x)) x :=
-  (isBoundedBilinearMap_smul.hasFDerivAt (c x, f x)).comp x <| hc.prod hf
+  (isBoundedBilinearMap_smul.hasFDerivAt (𝕜 := 𝕜) (c x, f x) :).comp x <| hc.prod hf
 
 @[fun_prop]
 theorem DifferentiableWithinAt.smul (hc : DifferentiableWithinAt 𝕜 c s x)
@@ -570,10 +573,10 @@ theorem hasStrictFDerivAt_list_prod_finRange' {n : ℕ} {x : Fin n → 𝔸} :
 theorem hasStrictFDerivAt_list_prod_attach' [DecidableEq ι] {l : List ι} {x : {i // i ∈ l} → 𝔸} :
     HasStrictFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.attach.map x).prod)
       (∑ i : Fin l.length, ((l.attach.take i).map x).prod •
-        smulRight (proj l.attach[i.cast l.length_attach.symm])
+        smulRight (proj l.attach[i.cast List.length_attach.symm])
           ((l.attach.drop (.succ i)).map x).prod) x :=
   hasStrictFDerivAt_list_prod'.congr_fderiv <| Eq.symm <|
-    Finset.sum_equiv (finCongr l.length_attach.symm) (by simp) (by simp)
+    Finset.sum_equiv (finCongr List.length_attach.symm) (by simp) (by simp)
 
 @[fun_prop]
 theorem hasFDerivAt_list_prod' [Fintype ι] {l : List ι} {x : ι → 𝔸'} :
@@ -593,7 +596,7 @@ theorem hasFDerivAt_list_prod_finRange' {n : ℕ} {x : Fin n → 𝔸} :
 theorem hasFDerivAt_list_prod_attach' [DecidableEq ι] {l : List ι} {x : {i // i ∈ l} → 𝔸} :
     HasFDerivAt (𝕜 := 𝕜) (fun x ↦ (l.attach.map x).prod)
       (∑ i : Fin l.length, ((l.attach.take i).map x).prod •
-        smulRight (proj l.attach[i.cast l.length_attach.symm])
+        smulRight (proj l.attach[i.cast List.length_attach.symm])
           ((l.attach.drop (.succ i)).map x).prod) x :=
   hasStrictFDerivAt_list_prod_attach'.hasFDerivAt
 
@@ -645,7 +648,7 @@ theorem HasStrictFDerivAt.list_prod' {l : List ι} {x : E}
         smulRight (f' l[i]) ((l.drop (.succ i)).map (f · x)).prod) x := by
   simp only [← List.finRange_map_get l, List.map_map]
   refine .congr_fderiv (hasStrictFDerivAt_list_prod_finRange'.comp x
-    (hasStrictFDerivAt_pi.mpr fun i ↦ h l[i] (l.getElem_mem ..))) ?_
+    (hasStrictFDerivAt_pi.mpr fun i ↦ h l[i] (List.getElem_mem ..))) ?_
   ext m
   simp [← List.map_map]
 
@@ -660,7 +663,7 @@ theorem HasFDerivAt.list_prod' {l : List ι} {x : E}
         smulRight (f' l[i]) ((l.drop (.succ i)).map (f · x)).prod) x := by
   simp only [← List.finRange_map_get l, List.map_map]
   refine .congr_fderiv (hasFDerivAt_list_prod_finRange'.comp x
-    (hasFDerivAt_pi.mpr fun i ↦ h l[i] (l.getElem_mem i i.isLt))) ?_
+    (hasFDerivAt_pi.mpr fun i ↦ h l[i] (List.getElem_mem i.isLt))) ?_
   ext m
   simp [← List.map_map]
 
@@ -774,16 +777,15 @@ end Prod
 
 section AlgebraInverse
 
-variable {R : Type*} [NormedRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R]
+variable {R : Type*} [NormedRing R] [HasSummableGeomSeries R] [NormedAlgebra 𝕜 R]
 
 open NormedRing ContinuousLinearMap Ring
 
 /-- At an invertible element `x` of a normed algebra `R`, the Fréchet derivative of the inversion
 operation is the linear map `fun t ↦ - x⁻¹ * t * x⁻¹`.
 
-TODO: prove that `Ring.inverse` is analytic and use it to prove a `HasStrictFDerivAt` lemma.
-TODO (low prio): prove a version without assumption `[CompleteSpace R]` but within the set of
-units. -/
+TODO (low prio): prove a version without assumption `[HasSummableGeomSeries R]` but within the set
+of units. -/
 @[fun_prop]
 theorem hasFDerivAt_ring_inverse (x : Rˣ) :
     HasFDerivAt Ring.inverse (-mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹) x :=
@@ -808,6 +810,11 @@ theorem differentiableOn_inverse : DifferentiableOn 𝕜 (@Ring.inverse R _) {x
 theorem fderiv_inverse (x : Rˣ) : fderiv 𝕜 (@Ring.inverse R _) x = -mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹ :=
   (hasFDerivAt_ring_inverse x).fderiv
 
+theorem hasStrictFDerivAt_ring_inverse (x : Rˣ) :
+    HasStrictFDerivAt Ring.inverse (-mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹) x := by
+  convert (analyticAt_inverse (𝕜 := 𝕜) x).hasStrictFDerivAt
+  exact (fderiv_inverse x).symm
+
 variable {h : E → R} {z : E} {S : Set E}
 
 @[fun_prop]
@@ -832,35 +839,51 @@ end AlgebraInverse
 
 /-! ### Derivative of the inverse in a division ring
 
-Note these lemmas are primed as they need `CompleteSpace R`, whereas the other lemmas in
-`Mathlib/Analysis/Calculus/Deriv/Inv.lean` do not, but instead need `NontriviallyNormedField R`.
+Note that some lemmas are primed as they are expressed without commutativity, whereas their
+counterparts in commutative fields involve simpler expressions, and are given in
+`Mathlib/Analysis/Calculus/Deriv/Inv.lean`.
 -/
 
 section DivisionRingInverse
 
-variable {R : Type*} [NormedDivisionRing R] [NormedAlgebra 𝕜 R] [CompleteSpace R]
+variable {R : Type*} [NormedDivisionRing R] [NormedAlgebra 𝕜 R]
 
 open NormedRing ContinuousLinearMap Ring
 
+/-- At an invertible element `x` of a normed division algebra `R`, the inversion is strictly
+differentiable, with derivative the linear map `fun t ↦ - x⁻¹ * t * x⁻¹`. For a nicer formula in
+the commutative case, see `hasStrictFDerivAt_inv`. -/
+theorem hasStrictFDerivAt_inv' {x : R} (hx : x ≠ 0) :
+    HasStrictFDerivAt Inv.inv (-mulLeftRight 𝕜 R x⁻¹ x⁻¹) x := by
+  simpa using hasStrictFDerivAt_ring_inverse (Units.mk0 _ hx)
+
 /-- At an invertible element `x` of a normed division algebra `R`, the Fréchet derivative of the
-inversion operation is the linear map `fun t ↦ - x⁻¹ * t * x⁻¹`. -/
+inversion operation is the linear map `fun t ↦ - x⁻¹ * t * x⁻¹`. For a nicer formula in the
+commutative case, see `hasFDerivAt_inv`. -/
 @[fun_prop]
 theorem hasFDerivAt_inv' {x : R} (hx : x ≠ 0) :
     HasFDerivAt Inv.inv (-mulLeftRight 𝕜 R x⁻¹ x⁻¹) x := by
   simpa using hasFDerivAt_ring_inverse (Units.mk0 _ hx)
 
 @[fun_prop]
-theorem differentiableAt_inv' {x : R} (hx : x ≠ 0) : DifferentiableAt 𝕜 Inv.inv x :=
+theorem differentiableAt_inv {x : R} (hx : x ≠ 0) : DifferentiableAt 𝕜 Inv.inv x :=
   (hasFDerivAt_inv' hx).differentiableAt
 
+@[deprecated (since := "2024-09-21")] alias differentiableAt_inv' := differentiableAt_inv
+
 @[fun_prop]
-theorem differentiableWithinAt_inv' {x : R} (hx : x ≠ 0) (s : Set R) :
+theorem differentiableWithinAt_inv {x : R} (hx : x ≠ 0) (s : Set R) :
     DifferentiableWithinAt 𝕜 (fun x => x⁻¹) s x :=
-  (differentiableAt_inv' hx).differentiableWithinAt
+  (differentiableAt_inv hx).differentiableWithinAt
+
+@[deprecated (since := "2024-09-21")]
+alias differentiableWithinAt_inv' := differentiableWithinAt_inv
 
 @[fun_prop]
-theorem differentiableOn_inv' : DifferentiableOn 𝕜 (fun x : R => x⁻¹) {x | x ≠ 0} := fun _x hx =>
-  differentiableWithinAt_inv' hx _
+theorem differentiableOn_inv : DifferentiableOn 𝕜 (fun x : R => x⁻¹) {x | x ≠ 0} := fun _x hx =>
+  differentiableWithinAt_inv hx _
+
+@[deprecated (since := "2024-09-21")] alias differentiableOn_inv' := differentiableOn_inv
 
 /-- Non-commutative version of `fderiv_inv` -/
 theorem fderiv_inv' {x : R} (hx : x ≠ 0) : fderiv 𝕜 Inv.inv x = -mulLeftRight 𝕜 R x⁻¹ x⁻¹ :=
@@ -869,28 +892,37 @@ theorem fderiv_inv' {x : R} (hx : x ≠ 0) : fderiv 𝕜 Inv.inv x = -mulLeftRig
 /-- Non-commutative version of `fderivWithin_inv` -/
 theorem fderivWithin_inv' {s : Set R} {x : R} (hx : x ≠ 0) (hxs : UniqueDiffWithinAt 𝕜 s x) :
     fderivWithin 𝕜 (fun x => x⁻¹) s x = -mulLeftRight 𝕜 R x⁻¹ x⁻¹ := by
-  rw [DifferentiableAt.fderivWithin (differentiableAt_inv' hx) hxs]
+  rw [DifferentiableAt.fderivWithin (differentiableAt_inv hx) hxs]
   exact fderiv_inv' hx
 
 variable {h : E → R} {z : E} {S : Set E}
 
 @[fun_prop]
-theorem DifferentiableWithinAt.inv' (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
+theorem DifferentiableWithinAt.inv (hf : DifferentiableWithinAt 𝕜 h S z) (hz : h z ≠ 0) :
     DifferentiableWithinAt 𝕜 (fun x => (h x)⁻¹) S z :=
-  (differentiableAt_inv' hz).comp_differentiableWithinAt z hf
+  (differentiableAt_inv hz).comp_differentiableWithinAt z hf
+
+@[deprecated (since := "2024-09-21")]
+alias DifferentiableWithinAt.inv' := DifferentiableWithinAt.inv
 
 @[simp, fun_prop]
-theorem DifferentiableAt.inv' (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
+theorem DifferentiableAt.inv (hf : DifferentiableAt 𝕜 h z) (hz : h z ≠ 0) :
     DifferentiableAt 𝕜 (fun x => (h x)⁻¹) z :=
-  (differentiableAt_inv' hz).comp z hf
+  (differentiableAt_inv hz).comp z hf
+
+@[deprecated (since := "2024-09-21")] alias DifferentiableAt.inv' := DifferentiableAt.inv
 
 @[fun_prop]
-theorem DifferentiableOn.inv' (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
-    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S := fun x h => (hf x h).inv' (hz x h)
+theorem DifferentiableOn.inv (hf : DifferentiableOn 𝕜 h S) (hz : ∀ x ∈ S, h x ≠ 0) :
+    DifferentiableOn 𝕜 (fun x => (h x)⁻¹) S := fun x h => (hf x h).inv (hz x h)
+
+@[deprecated (since := "2024-09-21")] alias DifferentiableOn.inv' := DifferentiableOn.inv
 
 @[simp, fun_prop]
-theorem Differentiable.inv' (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
-    Differentiable 𝕜 fun x => (h x)⁻¹ := fun x => (hf x).inv' (hz x)
+theorem Differentiable.inv (hf : Differentiable 𝕜 h) (hz : ∀ x, h x ≠ 0) :
+    Differentiable 𝕜 fun x => (h x)⁻¹ := fun x => (hf x).inv (hz x)
+
+@[deprecated (since := "2024-09-21")] alias Differentiable.inv' := Differentiable.inv
 
 end DivisionRingInverse
 
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Norm.lean b/Mathlib/Analysis/Calculus/FDeriv/Norm.lean
new file mode 100644
index 0000000000000..b43a36ea81d3a
--- /dev/null
+++ b/Mathlib/Analysis/Calculus/FDeriv/Norm.lean
@@ -0,0 +1,200 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.Analysis.Calculus.Deriv.Abs
+import Mathlib.Analysis.Calculus.LineDeriv.Basic
+
+/-!
+# Differentiability of the norm in a real normed vector space
+
+This file provides basic results about the differentiability of the norm in a real vector space.
+Most are of the following kind: if the norm has some differentiability property
+(`DifferentiableAt`, `ContDiffAt`, `HasStrictFDerivAt`, `HasFDerivAt`) at `x`, then so it has
+at `t • x` when `t ≠ 0`.
+
+## Main statements
+
+* `ContDiffAt.contDiffAt_norm_smul`: If the norm is continuously differentiable up to order `n`
+  at `x`, then so it is at `t • x` when `t ≠ 0`.
+* `differentiableAt_norm_smul`: If `t ≠ 0`, the norm is differentiable at `x` if and only if
+  it is at `t • x`.
+* `HasFDerivAt.hasFDerivAt_norm_smul`: If the norm has a Fréchet derivative `f` at `x` and `t ≠ 0`,
+  then it has `(SignType t) • f` as a Fréchet derivative at `t · x`.
+* `fderiv_norm_smul` : `fderiv ℝ (‖·‖) (t • x) = (SignType.sign t : ℝ) • (fderiv ℝ (‖·‖) x)`,
+  this holds without any differentiability assumptions.
+* `DifferentiableAt.fderiv_norm_self`: if the norm is differentiable at `x`,
+  then `fderiv ℝ (‖·‖) x x = ‖x‖`.
+* `norm_fderiv_norm`: if the norm is differentiable at `x` then the operator norm of its derivative
+  is `1` (on a non trivial space).
+
+## Tags
+
+differentiability, norm
+
+-/
+
+open ContinuousLinearMap Filter NNReal Real Set
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+variable {n : ℕ∞} {f : E →L[ℝ] ℝ} {x : E} {t : ℝ}
+
+variable (E) in
+theorem not_differentiableAt_norm_zero [Nontrivial E] :
+    ¬DifferentiableAt ℝ (‖·‖) (0 : E) := by
+  obtain ⟨x, hx⟩ := NormedSpace.exists_lt_norm ℝ E 0
+  intro h
+  have : DifferentiableAt ℝ (fun t : ℝ ↦ ‖t • x‖) 0 := DifferentiableAt.comp _ (by simpa) (by simp)
+  have : DifferentiableAt ℝ (|·|) (0 : ℝ) := by
+    simp_rw [norm_smul, norm_eq_abs] at this
+    have aux : abs = fun t ↦ (1 / ‖x‖) * (|t| * ‖x‖) := by field_simp
+    rw [aux]
+    exact this.const_mul _
+  exact not_differentiableAt_abs_zero this
+
+theorem ContDiffAt.contDiffAt_norm_smul (ht : t ≠ 0) (h : ContDiffAt ℝ n (‖·‖) x) :
+    ContDiffAt ℝ n (‖·‖) (t • x) := by
+  have h1 : ContDiffAt ℝ n (fun y ↦ t⁻¹ • y) (t • x) := (contDiff_const_smul t⁻¹).contDiffAt
+  have h2 : ContDiffAt ℝ n (fun y ↦ |t| * ‖y‖) x := h.const_smul |t|
+  conv at h2 => enter [4]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul]
+  convert h2.comp (t • x) h1 using 1
+  ext y
+  simp only [Function.comp_apply]
+  rw [norm_smul, ← mul_assoc, norm_eq_abs, ← abs_mul, mul_inv_cancel₀ ht, abs_one, one_mul]
+
+theorem contDiffAt_norm_smul_iff (ht : t ≠ 0) :
+    ContDiffAt ℝ n (‖·‖) x ↔ ContDiffAt ℝ n (‖·‖) (t • x) where
+  mp h := h.contDiffAt_norm_smul ht
+  mpr hd := by
+    convert hd.contDiffAt_norm_smul (inv_ne_zero ht)
+    rw [smul_smul, inv_mul_cancel₀ ht, one_smul]
+
+theorem ContDiffAt.contDiffAt_norm_of_smul (h : ContDiffAt ℝ n (‖·‖) (t • x)) :
+    ContDiffAt ℝ n (‖·‖) x := by
+  obtain rfl | hn : n = 0 ∨ 1 ≤ n := by
+    rw [← ENat.lt_one_iff_eq_zero]
+    exact lt_or_le ..
+  · rw [contDiffAt_zero]
+    exact ⟨univ, univ_mem, continuous_norm.continuousOn⟩
+  obtain rfl | ht := eq_or_ne t 0
+  · by_cases hE : Nontrivial E
+    · rw [zero_smul] at h
+      exact (mt (ContDiffAt.differentiableAt · hn)) (not_differentiableAt_norm_zero E) h |>.elim
+    · rw [not_nontrivial_iff_subsingleton] at hE
+      rw [eq_const_of_subsingleton (‖·‖) 0]
+      exact contDiffAt_const
+  · exact contDiffAt_norm_smul_iff ht |>.2 h
+
+theorem HasStrictFDerivAt.hasStrictFDerivAt_norm_smul
+    (ht : t ≠ 0) (h : HasStrictFDerivAt (‖·‖) f x) :
+    HasStrictFDerivAt (‖·‖) ((SignType.sign t : ℝ) • f) (t • x) := by
+  have h1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x) :=
+    hasStrictFDerivAt_id (t • x) |>.const_smul t⁻¹
+  have h2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x := h.const_smul |t|
+  conv at h2 => enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul]
+  convert h2.comp (t • x) h1 with y
+  · rw [norm_smul, ← mul_assoc, norm_eq_abs, ← abs_mul, mul_inv_cancel₀ ht, abs_one, one_mul]
+  ext y
+  simp only [coe_smul', Pi.smul_apply, smul_eq_mul, comp_smulₛₗ, map_inv₀, RingHom.id_apply,
+    comp_id]
+  rw [eq_inv_mul_iff_mul_eq₀ ht, ← mul_assoc, self_mul_sign]
+
+theorem HasStrictFDerivAt.hasStrictDerivAt_norm_smul_neg
+    (ht : t < 0) (h : HasStrictFDerivAt (‖·‖) f x) :
+    HasStrictFDerivAt (‖·‖) (-f) (t • x) := by
+  simpa [ht] using h.hasStrictFDerivAt_norm_smul ht.ne
+
+theorem HasStrictFDerivAt.hasStrictDerivAt_norm_smul_pos
+    (ht : 0 < t) (h : HasStrictFDerivAt (‖·‖) f x) :
+    HasStrictFDerivAt (‖·‖) f (t • x) := by
+  simpa [ht] using h.hasStrictFDerivAt_norm_smul ht.ne'
+
+theorem HasFDerivAt.hasFDerivAt_norm_smul
+    (ht : t ≠ 0) (h : HasFDerivAt (‖·‖) f x) :
+    HasFDerivAt (‖·‖) ((SignType.sign t : ℝ) • f) (t • x) := by
+  have h1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x) :=
+    hasFDerivAt_id (t • x) |>.const_smul t⁻¹
+  have h2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x := h.const_smul |t|
+  conv at h2 => enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul]
+  convert h2.comp (t • x) h1 using 2 with y
+  · simp only [Function.comp_apply]
+    rw [norm_smul, ← mul_assoc, norm_eq_abs, ← abs_mul, mul_inv_cancel₀ ht, abs_one, one_mul]
+  · ext y
+    simp only [coe_smul', Pi.smul_apply, smul_eq_mul, comp_smulₛₗ, map_inv₀, RingHom.id_apply,
+      comp_id]
+    rw [eq_inv_mul_iff_mul_eq₀ ht, ← mul_assoc, self_mul_sign]
+
+theorem HasFDerivAt.hasFDerivAt_norm_smul_neg
+    (ht : t < 0) (h : HasFDerivAt (‖·‖) f x) :
+    HasFDerivAt (‖·‖) (-f) (t • x) := by
+  simpa [ht] using h.hasFDerivAt_norm_smul ht.ne
+
+theorem HasFDerivAt.hasFDerivAt_norm_smul_pos
+    (ht : 0 < t) (h : HasFDerivAt (‖·‖) f x) :
+    HasFDerivAt (‖·‖) f (t • x) := by
+  simpa [ht] using h.hasFDerivAt_norm_smul ht.ne'
+
+theorem differentiableAt_norm_smul (ht : t ≠ 0) :
+    DifferentiableAt ℝ (‖·‖) x ↔ DifferentiableAt ℝ (‖·‖) (t • x) where
+  mp hd := (hd.hasFDerivAt.hasFDerivAt_norm_smul ht).differentiableAt
+  mpr hd := by
+    convert (hd.hasFDerivAt.hasFDerivAt_norm_smul (inv_ne_zero ht)).differentiableAt
+    rw [smul_smul, inv_mul_cancel₀ ht, one_smul]
+
+theorem DifferentiableAt.differentiableAt_norm_of_smul (h : DifferentiableAt ℝ (‖·‖) (t • x)) :
+    DifferentiableAt ℝ (‖·‖) x := by
+  obtain rfl | ht := eq_or_ne t 0
+  · by_cases hE : Nontrivial E
+    · rw [zero_smul] at h
+      exact not_differentiableAt_norm_zero E h |>.elim
+    · rw [not_nontrivial_iff_subsingleton] at hE
+      exact (hasFDerivAt_of_subsingleton _ _).differentiableAt
+  · exact differentiableAt_norm_smul ht |>.2 h
+
+theorem DifferentiableAt.fderiv_norm_self {x : E} (h : DifferentiableAt ℝ (‖·‖) x) :
+    fderiv ℝ (‖·‖) x x = ‖x‖ := by
+  rw [← h.lineDeriv_eq_fderiv, lineDeriv]
+  have this (t : ℝ) : ‖x + t • x‖ = |1 + t| * ‖x‖ := by
+    rw [← norm_eq_abs, ← norm_smul, add_smul, one_smul]
+  simp_rw [this]
+  rw [deriv_mul_const]
+  · conv_lhs => enter [1, 1]; change _root_.abs ∘ (fun t ↦ 1 + t)
+    rw [deriv.comp, deriv_abs, deriv_const_add]
+    · simp
+    · exact differentiableAt_abs (by norm_num)
+    · exact differentiableAt_id.const_add _
+  · exact (differentiableAt_abs (by norm_num)).comp _ (differentiableAt_id.const_add _)
+
+variable (x t) in
+theorem fderiv_norm_smul :
+    fderiv ℝ (‖·‖) (t • x) = (SignType.sign t : ℝ) • (fderiv ℝ (‖·‖) x) := by
+  by_cases hE : Nontrivial E
+  · by_cases hd : DifferentiableAt ℝ (‖·‖) x
+    · obtain rfl | ht := eq_or_ne t 0
+      · simp only [zero_smul, _root_.sign_zero, SignType.coe_zero]
+        exact fderiv_zero_of_not_differentiableAt <| not_differentiableAt_norm_zero E
+      · rw [(hd.hasFDerivAt.hasFDerivAt_norm_smul ht).fderiv]
+    · rw [fderiv_zero_of_not_differentiableAt hd, fderiv_zero_of_not_differentiableAt]
+      · simp
+      · exact mt DifferentiableAt.differentiableAt_norm_of_smul hd
+  · rw [not_nontrivial_iff_subsingleton] at hE
+    simp_rw [(hasFDerivAt_of_subsingleton _ _).fderiv, smul_zero]
+
+theorem fderiv_norm_smul_pos (ht : 0 < t) :
+    fderiv ℝ (‖·‖) (t • x) = fderiv ℝ (‖·‖) x := by
+  simp [fderiv_norm_smul, ht]
+
+theorem fderiv_norm_smul_neg (ht : t < 0) :
+    fderiv ℝ (‖·‖) (t • x) = -fderiv ℝ (‖·‖) x := by
+  simp [fderiv_norm_smul, ht]
+
+theorem norm_fderiv_norm [Nontrivial E] (h : DifferentiableAt ℝ (‖·‖) x) :
+    ‖fderiv ℝ (‖·‖) x‖ = 1 := by
+  have : x ≠ 0 := fun hx ↦ not_differentiableAt_norm_zero E (hx ▸ h)
+  refine le_antisymm (NNReal.coe_one ▸ norm_fderiv_le_of_lipschitz ℝ lipschitzWith_one_norm) ?_
+  apply le_of_mul_le_mul_right _ (norm_pos_iff.2 this)
+  calc
+    1 * ‖x‖ = fderiv ℝ (‖·‖) x x := by rw [one_mul, h.fderiv_norm_self]
+    _ ≤ ‖fderiv ℝ (‖·‖) x x‖ := le_norm_self _
+    _ ≤ ‖fderiv ℝ (‖·‖) x‖ * ‖x‖ := le_opNorm _ _
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean b/Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean
index b24d6c6e85fe1..399dada358cca 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean
@@ -97,8 +97,8 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     rw [← smul_smul]
     apply s_conv.interior.add_smul_mem this _ ht
     rw [add_assoc] at hw
-    rw [add_assoc, ← smul_add]
-    exact s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩
+    convert s_conv.add_smul_mem_interior xs hw ⟨hpos, h_lt_1.le⟩ using 1
+    module
   -- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the
   -- quantity to be estimated. We will check that its derivative is given by an explicit
   -- expression `g'`, that we can bound. Then the desired bound for `g 1 - g 0` follows from the
@@ -139,14 +139,14 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
     calc
       ‖g' t‖ = ‖(f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)) (h • w)‖ := by
         rw [hg']
-        have : h * (t * h) = t * (h * h) := by ring
-        simp only [ContinuousLinearMap.coe_sub', ContinuousLinearMap.map_add, pow_two,
-          ContinuousLinearMap.add_apply, Pi.smul_apply, smul_sub, smul_add, smul_smul, ← sub_sub,
-          ContinuousLinearMap.coe_smul', Pi.sub_apply, ContinuousLinearMap.map_smul, this]
+        congrm ‖?_‖
+        simp only [ContinuousLinearMap.sub_apply, ContinuousLinearMap.add_apply,
+          ContinuousLinearMap.smul_apply, map_add, map_smul]
+        module
       _ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
         (ContinuousLinearMap.le_opNorm _ _)
       _ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ := by
-        apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
+        gcongr
         have H : x + h • v + (t * h) • w ∈ Metric.ball x δ ∩ interior s := by
           refine ⟨?_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
           rw [add_assoc, add_mem_ball_iff_norm]
@@ -157,7 +157,7 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
         apply (norm_add_le _ _).trans
         gcongr
         simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]
-        exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le
+        exact mul_le_of_le_one_left (by positivity) ht.2.le
       _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
         simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]; ring
   -- conclude using the mean value inequality
@@ -167,10 +167,10 @@ theorem Convex.taylor_approx_two_segment {v w : E} (hv : x + v ∈ interior s)
   convert I using 1
   · congr 1
     simp only [g, Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero,
-      zero_smul, Ne, not_false_iff, zero_pow]
+      zero_smul, Ne, not_false_iff, zero_pow, reduceCtorEq]
     abel
-  · simp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg,
-      hpos.le, mul_assoc, norm_nonneg, abs_pow]
+  · simp (discharger := positivity) only [Real.norm_eq_abs, abs_mul, abs_of_nonneg, abs_pow]
+    ring
 
 /-- One can get `f'' v w` as the limit of `h ^ (-2)` times the alternate sum of the values of `f`
 along the vertices of a quadrilateral with sides `h v` and `h w` based at `x`.
@@ -183,40 +183,27 @@ theorem Convex.isLittleO_alternate_sum_square {v w : E} (h4v : x + (4 : ℝ) •
       fun h => h ^ 2 := by
   have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
   have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩
-  have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_smul]
   have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s := by
     convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1
-    simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]
-    norm_num
-    simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]
-    abel
+    module
   have h2vww : x + (2 • v + w) + w ∈ interior s := by
     convert h2v2w using 1
-    simp only [two_smul]
-    abel
+    module
   have h2v : x + (2 : ℝ) • v ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h4v A using 1
-    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
-    norm_num
+    module
   have h2w : x + (2 : ℝ) • w ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h4w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj]
-    norm_num
+    module
   have hvw : x + (v + w) ∈ interior s := by
     convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1
-    simp only [smul_smul, one_div, add_sub_cancel_left, add_right_inj, smul_add, smul_sub]
-    norm_num
-    abel
+    module
   have h2vw : x + (2 • v + w) ∈ interior s := by
     convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1
-    simp only [smul_add, smul_sub, smul_smul, ← C]
-    norm_num
-    abel
+    module
   have hvww : x + (v + w) + w ∈ interior s := by
     convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1
-    rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel_left, inv_smul_smul₀ two_ne_zero,
-      two_smul]
-    abel
+    module
   have TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww
   have TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww
   convert TA1.sub TA2 using 1
@@ -245,11 +232,9 @@ theorem Convex.second_derivative_within_at_symmetric_of_mem_interior {v w : E}
     apply C.congr' _ _
     · filter_upwards [self_mem_nhdsWithin]
       intro h (hpos : 0 < h)
-      rw [← one_smul ℝ (f'' w v - f'' v w), smul_smul, smul_smul]
-      congr 1
-      field_simp [LT.lt.ne' hpos]
+      match_scalars <;> field_simp
     · filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)
-      field_simp [LT.lt.ne' hpos, SMul.smul]
+      field_simp
   simpa only [sub_eq_zero] using isLittleO_const_const_iff.1 B
 
 end
@@ -298,8 +283,8 @@ theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Conve
     s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)
   simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,
     ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this
-  rw [add_assoc, add_assoc, add_right_inj, add_left_comm, add_right_inj, add_right_inj, mul_comm]
-    at this
+  have : (t v * t w) • (f'' v) w = (t v * t w) • (f'' w) v := by
+    linear_combination (norm := module) this
   apply smul_right_injective F _ this
   simp [(tpos v).ne', (tpos w).ne']
 
diff --git a/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean b/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
index 2a99f4113e978..06d42a093e031 100644
--- a/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
+++ b/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
@@ -88,6 +88,15 @@ def prod (p : FormalMultilinearSeries 𝕜 E F) (q : FormalMultilinearSeries 
     FormalMultilinearSeries 𝕜 E (F × G)
   | n => (p n).prod (q n)
 
+/-- Product of formal multilinear series (with the same field `𝕜` and the same source
+space, but possibly different target spaces). -/
+@[simp] def pi {ι : Type*} {F : ι → Type*}
+    [∀ i, AddCommGroup (F i)] [∀ i, Module 𝕜 (F i)] [∀ i, TopologicalSpace (F i)]
+    [∀ i, TopologicalAddGroup (F i)] [∀ i, ContinuousConstSMul 𝕜 (F i)]
+    (p : Π i, FormalMultilinearSeries 𝕜 E (F i)) :
+    FormalMultilinearSeries 𝕜 E (Π i, F i)
+  | n => ContinuousMultilinearMap.pi (fun i ↦ p i n)
+
 /-- Killing the zeroth coefficient in a formal multilinear series -/
 def removeZero (p : FormalMultilinearSeries 𝕜 E F) : FormalMultilinearSeries 𝕜 E F
   | 0 => 0
@@ -158,9 +167,7 @@ corresponds to starting from a Taylor series (`HasFTaylorSeriesUpTo`) for the de
 function, and building a Taylor series for the function itself. -/
 def unshift (q : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)) (z : F) : FormalMultilinearSeries 𝕜 E F
   | 0 => (continuousMultilinearCurryFin0 𝕜 E F).symm z
-  | n + 1 => -- Porting note: added type hint here and explicit universes to fix compile
-    (continuousMultilinearCurryRightEquiv' 𝕜 n E F :
-      (E [×n]→L[𝕜] E →L[𝕜] F) → (E [×n.succ]→L[𝕜] F)) (q n)
+  | n + 1 => (continuousMultilinearCurryRightEquiv' 𝕜 n E F).symm (q n)
 
 end FormalMultilinearSeries
 
@@ -295,7 +302,7 @@ noncomputable def fslope (p : FormalMultilinearSeries 𝕜 𝕜 E) : FormalMulti
 theorem coeff_fslope : p.fslope.coeff n = p.coeff (n + 1) := by
   simp only [fslope, coeff, ContinuousMultilinearMap.curryLeft_apply]
   congr 1
-  exact Fin.cons_self_tail 1
+  exact Fin.cons_self_tail (fun _ => (1 : 𝕜))
 
 @[simp]
 theorem coeff_iterate_fslope (k n : ℕ) : (fslope^[k] p).coeff n = p.coeff (n + k) := by
@@ -316,7 +323,7 @@ def constFormalMultilinearSeries (𝕜 : Type*) [NontriviallyNormedField 𝕜] (
     [NormedAddCommGroup E] [NormedSpace 𝕜 E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup E]
     {F : Type*} [NormedAddCommGroup F] [TopologicalAddGroup F] [NormedSpace 𝕜 F]
     [ContinuousConstSMul 𝕜 F] (c : F) : FormalMultilinearSeries 𝕜 E F
-  | 0 => ContinuousMultilinearMap.curry0 _ _ c
+  | 0 => ContinuousMultilinearMap.uncurry0 _ _ c
   | _ => 0
 
 @[simp]
@@ -333,7 +340,7 @@ lemma constFormalMultilinearSeries_zero [NontriviallyNormedField 𝕜] [NormedAd
   simp only [FormalMultilinearSeries.zero_apply, ContinuousMultilinearMap.zero_apply,
     constFormalMultilinearSeries]
   induction n
-  · simp only [ContinuousMultilinearMap.curry0_apply]
+  · simp only [ContinuousMultilinearMap.uncurry0_apply]
   · simp only [constFormalMultilinearSeries.match_1.eq_2, ContinuousMultilinearMap.zero_apply]
 
 end Const
@@ -349,12 +356,12 @@ namespace ContinuousLinearMap
 /-- Formal power series of a continuous linear map `f : E →L[𝕜] F` at `x : E`:
 `f y = f x + f (y - x)`. -/
 def fpowerSeries (f : E →L[𝕜] F) (x : E) : FormalMultilinearSeries 𝕜 E F
-  | 0 => ContinuousMultilinearMap.curry0 𝕜 _ (f x)
+  | 0 => ContinuousMultilinearMap.uncurry0 𝕜 _ (f x)
   | 1 => (continuousMultilinearCurryFin1 𝕜 E F).symm f
   | _ => 0
 
 theorem fpowerSeries_apply_zero (f : E →L[𝕜] F) (x : E) :
-    f.fpowerSeries x 0 = ContinuousMultilinearMap.curry0 𝕜 _ (f x) :=
+    f.fpowerSeries x 0 = ContinuousMultilinearMap.uncurry0 𝕜 _ (f x) :=
   rfl
 
 theorem fpowerSeries_apply_one (f : E →L[𝕜] F) (x : E) :
diff --git a/Mathlib/Analysis/Calculus/Implicit.lean b/Mathlib/Analysis/Calculus/Implicit.lean
index 72620c2d28d46..2e323d8872a82 100644
--- a/Mathlib/Analysis/Calculus/Implicit.lean
+++ b/Mathlib/Analysis/Calculus/Implicit.lean
@@ -326,12 +326,12 @@ theorem to_implicitFunctionOfComplemented (hf : HasStrictFDerivAt f f' a) (hf' :
   · ext
     -- Porting note: added parentheses to help `simp`
     simp only [Classical.choose_spec hker, implicitFunctionDataOfComplemented,
-      ContinuousLinearMap.comp_apply, Submodule.coe_subtypeL', Submodule.coeSubtype,
+      ContinuousLinearMap.comp_apply, Submodule.coe_subtypeL', Submodule.coe_subtype,
       ContinuousLinearMap.id_apply]
   swap
   · ext
     -- Porting note: added parentheses to help `simp`
-    simp only [(ContinuousLinearMap.comp_apply), Submodule.coe_subtypeL', Submodule.coeSubtype,
+    simp only [(ContinuousLinearMap.comp_apply), Submodule.coe_subtypeL', Submodule.coe_subtype,
       LinearMap.map_coe_ker, (ContinuousLinearMap.zero_apply)]
   simp only [implicitFunctionDataOfComplemented, map_sub, sub_self]
 
diff --git a/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean b/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean
index 56145cc2bf428..67b3db1e4cd80 100644
--- a/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean
+++ b/Mathlib/Analysis/Calculus/InverseFunctionTheorem/ApproximatesLinearOn.lean
@@ -144,7 +144,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse
     simp only [dist_le_zero] at this
     rw [this]
   have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc
-  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff If'] at hc
+  have Icf' : (c : ℝ) * f'symm.nnnorm < 1 := by rwa [inv_eq_one_div, lt_div_iff₀ If'] at hc
   have Jf' : (f'symm.nnnorm : ℝ) ≠ 0 := ne_of_gt If'
   have Jcf' : (1 : ℝ) - c * f'symm.nnnorm ≠ 0 := by apply ne_of_gt; linarith
   /- We have to show that `y` can be written as `f x` for some `x ∈ closedBall b ε`.
@@ -257,7 +257,7 @@ theorem surjOn_closedBall_of_nonlinearRightInverse
   -- It remains to check that `f x = y`. This follows from continuity of `f` on `closedBall b ε`
   -- and from the fact that `f uₙ` is converging to `y` by construction.
   have hx' : Tendsto u atTop (𝓝[closedBall b ε] x) := by
-    simp only [nhdsWithin, tendsto_inf, hx, true_and_iff, tendsto_principal]
+    simp only [nhdsWithin, tendsto_inf, hx, true_and, tendsto_principal]
     exact Eventually.of_forall fun n => C n _ (D n).2
   have T1 : Tendsto (f ∘ u) atTop (𝓝 (f x)) :=
     (hf.continuousOn.mono hε x xmem).tendsto.comp hx'
diff --git a/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean b/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean
index 82d6364c31eb6..43a83ae9b58a1 100644
--- a/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean
+++ b/Mathlib/Analysis/Calculus/InverseFunctionTheorem/FDeriv.lean
@@ -186,7 +186,7 @@ theorem to_localInverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :
 and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
 
 For a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[CompleteSpace E]`
-see `of_local_left_inverse`.  -/
+see `of_local_left_inverse`. -/
 theorem to_local_left_inverse (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) {g : F → E}
     (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) (f a) :=
   hf.to_localInverse.congr_of_eventuallyEq <| (hf.localInverse_unique hg).mono fun _ => Eq.symm
diff --git a/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean b/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
index b662ec13e36fc..f5790c73a631f 100644
--- a/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
+++ b/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
@@ -82,7 +82,7 @@ iterated derivative. -/
 theorem iteratedFDerivWithin_eq_equiv_comp :
     iteratedFDerivWithin 𝕜 n f s =
       ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDerivWithin n f s := by
-  rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
+  rw [iteratedDerivWithin_eq_equiv_comp, ← Function.comp_assoc, LinearIsometryEquiv.self_comp_symm,
     Function.id_comp]
 
 /-- The `n`-th Fréchet derivative applied to a vector `(m 0, ..., m (n-1))` is the derivative
@@ -203,7 +203,7 @@ theorem iteratedDeriv_eq_equiv_comp : iteratedDeriv n f =
 iterated derivative. -/
 theorem iteratedFDeriv_eq_equiv_comp : iteratedFDeriv 𝕜 n f =
     ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F ∘ iteratedDeriv n f := by
-  rw [iteratedDeriv_eq_equiv_comp, ← Function.comp.assoc, LinearIsometryEquiv.self_comp_symm,
+  rw [iteratedDeriv_eq_equiv_comp, ← Function.comp_assoc, LinearIsometryEquiv.self_comp_symm,
     Function.id_comp]
 
 /-- The `n`-th Fréchet derivative applied to a vector `(m 0, ..., m (n-1))` is the derivative
diff --git a/Mathlib/Analysis/Calculus/LHopital.lean b/Mathlib/Analysis/Calculus/LHopital.lean
index 653749f47e528..610346d0119ac 100644
--- a/Mathlib/Analysis/Calculus/LHopital.lean
+++ b/Mathlib/Analysis/Calculus/LHopital.lean
@@ -143,7 +143,7 @@ theorem lhopital_zero_atTop_on_Ioi (hff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x)
     ⟨lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),
       lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)⟩⟩
   have fact1 : ∀ x : ℝ, x ∈ Ioo 0 a'⁻¹ → x ≠ 0 := fun _ hx => (ne_of_lt hx.1).symm
-  have fact2 : ∀ x ∈ Ioo 0 a'⁻¹, a < x⁻¹ := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)
+  have fact2 (x) (hx : x ∈ Ioo 0 a'⁻¹) : a < x⁻¹ := lt_trans haa' ((lt_inv_comm₀ ha' hx.1).mpr hx.2)
   have hdnf : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (f ∘ Inv.inv) (f' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
     comp x (hff' x⁻¹ <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)
   have hdng : ∀ x ∈ Ioo 0 a'⁻¹, HasDerivAt (g ∘ Inv.inv) (g' x⁻¹ * -(x ^ 2)⁻¹) x := fun x hx =>
diff --git a/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean b/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean
index eb08a5c169706..c3b79965c35ff 100644
--- a/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean
+++ b/Mathlib/Analysis/Calculus/LineDeriv/Basic.lean
@@ -499,7 +499,7 @@ theorem HasLineDerivWithinAt.smul (h : HasLineDerivWithinAt 𝕜 f f' s x v) (c
   have A : HasDerivAt g c 0 := by simpa using (hasDerivAt_id (0 : 𝕜)).const_smul c
   have B : HasDerivWithinAt (fun t ↦ f (x + t • v)) f' s' (g 0) := by simpa [g] using h
   have Z := B.scomp (0 : 𝕜) A.hasDerivWithinAt (mapsTo_preimage g s')
-  simp only [g, s', Function.comp, smul_eq_mul, mul_comm c, ← smul_smul] at Z
+  simp only [g, s', Function.comp_def, smul_eq_mul, mul_comm c, ← smul_smul] at Z
   convert Z
   ext t
   simp [← smul_smul]
diff --git a/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean b/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean
index e46ccfaeb5601..81863eb235bbc 100644
--- a/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean
+++ b/Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean
@@ -44,7 +44,7 @@ TODO: prove similar theorems assuming that the functions tend to zero at infinit
 integrable derivatives.
 -/
 
-open MeasureTheory Measure FiniteDimensional
+open MeasureTheory Measure Module
 
 variable {E F G W : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F]
   [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] [NormedAddCommGroup W]
@@ -134,9 +134,9 @@ theorem integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable
     simpa [this, hL.integral_map] using H
   have L_emb : MeasurableEmbedding L := L.toHomeomorph.measurableEmbedding
   apply integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
-  · simpa [L_emb.integrable_map_iff, Function.comp] using hf'g
-  · simpa [L_emb.integrable_map_iff, Function.comp] using hfg'
-  · simpa [L_emb.integrable_map_iff, Function.comp] using hfg
+  · simpa [L_emb.integrable_map_iff, Function.comp_def] using hf'g
+  · simpa [L_emb.integrable_map_iff, Function.comp_def] using hfg'
+  · simpa [L_emb.integrable_map_iff, Function.comp_def] using hfg
   · intro x
     have : f = (f ∘ L.symm) ∘ (L : E →ₗ[ℝ] (E' × ℝ)) := by ext y; simp
     specialize hf (L.symm x)
diff --git a/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean b/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean
index 1aa1c46474630..062cf3b3b7880 100644
--- a/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean
+++ b/Mathlib/Analysis/Calculus/LocalExtr/Polynomial.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Benjamin Davidson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Benjamin Davidson, Yury Kudryashov
 -/
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Analysis.Calculus.LocalExtr.Rolle
 import Mathlib.Analysis.Calculus.Deriv.Polynomial
 import Mathlib.Topology.Algebra.Polynomial
diff --git a/Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean b/Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean
index dbaffb564b895..bd4dace2f8094 100644
--- a/Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean
+++ b/Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean
@@ -58,7 +58,7 @@ theorem exists_deriv_eq_zero (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI
 
 /-- **Rolle's Theorem**, a version for a function on an open interval: if `f` has derivative `f'`
 on `(a, b)` and has the same limit `l` at `𝓝[>] a` and `𝓝[<] b`, then `f' c = 0`
-for some `c ∈ (a, b)`.  -/
+for some `c ∈ (a, b)`. -/
 theorem exists_hasDerivAt_eq_zero' (hab : a < b) (hfa : Tendsto f (𝓝[>] a) (𝓝 l))
     (hfb : Tendsto f (𝓝[<] b) (𝓝 l)) (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) :
     ∃ c ∈ Ioo a b, f' c = 0 :=
diff --git a/Mathlib/Analysis/Calculus/MeanValue.lean b/Mathlib/Analysis/Calculus/MeanValue.lean
index 707218f441f7c..703da3c9d5c78 100644
--- a/Mathlib/Analysis/Calculus/MeanValue.lean
+++ b/Mathlib/Analysis/Calculus/MeanValue.lean
@@ -416,22 +416,28 @@ end
 ### Vector-valued functions `f : E → G`
 
 Theorems in this section work both for real and complex differentiable functions. We use assumptions
-`[RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to achieve this result. For the domain `E` we
-also assume `[NormedSpace ℝ E]` to have a notion of a `Convex` set. -/
+`[NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 G]` to
+achieve this result. For the domain `E` we also assume `[NormedSpace ℝ E]` to have a notion
+of a `Convex` set. -/
 
 section
 
-variable {𝕜 G : Type*} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
-
 namespace Convex
 
-variable {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G}
+variable {𝕜 G : Type*} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]
+  [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+  {f g : E → G} {C : ℝ} {s : Set E} {x y : E} {f' g' : E → E →L[𝕜] G} {φ : E →L[𝕜] G}
+
+instance (priority := 100) : PathConnectedSpace 𝕜 := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
+  infer_instance
 
 /-- The mean value theorem on a convex set: if the derivative of a function is bounded by `C`, then
 the function is `C`-Lipschitz. Version with `HasFDerivWithinAt`. -/
 theorem norm_image_sub_le_of_norm_hasFDerivWithin_le
     (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
     (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   letI : NormedSpace ℝ G := RestrictScalars.normedSpace ℝ 𝕜 G
   /- By composition with `AffineMap.lineMap x y`, we reduce to a statement for functions defined
     on `[0,1]`, for which it is proved in `norm_image_sub_le_of_norm_deriv_le_segment`.
@@ -524,6 +530,7 @@ theorem _root_.lipschitzWith_of_nnnorm_fderiv_le
     {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
     {C : ℝ≥0} (hf : Differentiable 𝕜 f)
     (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   rw [← lipschitzOnWith_univ]
   exact lipschitzOnWith_of_nnnorm_fderiv_le (fun x _ ↦ hf x) (fun x _ ↦ bound x) convex_univ
@@ -573,6 +580,7 @@ theorem _root_.is_const_of_fderiv_eq_zero
     {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
     (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
     (x y : E) : f x = f y := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   exact convex_univ.is_const_of_fderivWithin_eq_zero hf.differentiableOn
     (fun x _ => by rw [fderivWithin_univ]; exact hf' x) trivial trivial
@@ -591,6 +599,7 @@ theorem _root_.eq_of_fderiv_eq
     {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G}
     (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
     (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   let A : NormedSpace ℝ E := RestrictScalars.normedSpace ℝ 𝕜 E
   suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x
   exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn
@@ -600,7 +609,8 @@ end Convex
 
 namespace Convex
 
-variable {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜}
+variable {𝕜 G : Type*} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+  {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜}
 
 /-- The mean value theorem on a convex set in dimension 1: if the derivative of a function is
 bounded by `C`, then the function is `C`-Lipschitz. Version with `HasDerivWithinAt`. -/
@@ -891,7 +901,7 @@ theorem Convex.mul_sub_lt_image_sub_of_lt_deriv {D : Set ℝ} (hD : Convex ℝ D
   obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
     exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')
   have : C < (f y - f x) / (y - x) := ha ▸ hf'_gt _ (hxyD' a_mem)
-  exact (lt_div_iff (sub_pos.2 hxy)).1 this
+  exact (lt_div_iff₀ (sub_pos.2 hxy)).1 this
 
 /-- Let `f : ℝ → ℝ` be a differentiable function. If `C < f'`, then `f` grows faster than
 `C * x`, i.e., `C * (y - x) < f y - f x` whenever `x < y`. -/
diff --git a/Mathlib/Analysis/Calculus/ParametricIntegral.lean b/Mathlib/Analysis/Calculus/ParametricIntegral.lean
index e6088a80f8437..6bafbed0ec063 100644
--- a/Mathlib/Analysis/Calculus/ParametricIntegral.lean
+++ b/Mathlib/Analysis/Calculus/ParametricIntegral.lean
@@ -140,7 +140,7 @@ theorem hasFDerivAt_integral_of_dominated_loc_of_lip' {F' : α → H →L[𝕜]
         gcongr; exact (F' a).le_opNorm _
       _ ≤ b a + ‖F' a‖ := ?_
     simp only [← div_eq_inv_mul]
-    apply_rules [add_le_add, div_le_of_nonneg_of_le_mul] <;> first | rfl | positivity
+    apply_rules [add_le_add, div_le_of_le_mul₀] <;> first | rfl | positivity
   · exact b_int.add hF'_int.norm
   · apply h_diff.mono
     intro a ha
diff --git a/Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean b/Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
index a6d43de3221c4..fa25dfa3f5ee9 100644
--- a/Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
+++ b/Mathlib/Analysis/Calculus/ParametricIntervalIntegral.lean
@@ -10,7 +10,7 @@ import Mathlib.MeasureTheory.Integral.IntervalIntegral
 # Derivatives of interval integrals depending on parameters
 
 In this file we restate theorems about derivatives of integrals depending on parameters for interval
-integrals.  -/
+integrals. -/
 
 
 open TopologicalSpace MeasureTheory Filter Metric
diff --git a/Mathlib/Analysis/Calculus/Rademacher.lean b/Mathlib/Analysis/Calculus/Rademacher.lean
index b1924f51dde25..67771abec0dce 100644
--- a/Mathlib/Analysis/Calculus/Rademacher.lean
+++ b/Mathlib/Analysis/Calculus/Rademacher.lean
@@ -42,7 +42,7 @@ See `LipschitzWith.hasFderivAt_of_hasLineDerivAt_of_closure`.
 * [Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Theorem 7.3][Federer1996]
 -/
 
-open Filter MeasureTheory Measure FiniteDimensional Metric Set Asymptotics
+open Filter MeasureTheory Measure Module Metric Set Asymptotics
 
 open scoped NNReal ENNReal Topology
 
@@ -386,3 +386,18 @@ theorem LipschitzWith.ae_differentiableAt {f : E → F} (h : LipschitzWith C f)
     ∀ᵐ x ∂μ, DifferentiableAt ℝ f x := by
   rw [← lipschitzOnWith_univ] at h
   simpa [differentiableWithinAt_univ] using h.ae_differentiableWithinAt_of_mem
+
+/-- In a real finite-dimensional normed vector space,
+  the norm is almost everywhere differentiable. -/
+theorem ae_differentiableAt_norm :
+    ∀ᵐ x ∂μ, DifferentiableAt ℝ (‖·‖) x := lipschitzWith_one_norm.ae_differentiableAt
+
+omit [MeasurableSpace E] in
+/-- In a real finite-dimensional normed vector space,
+  the set of points where the norm is differentiable at is dense. -/
+theorem dense_differentiableAt_norm :
+    Dense {x : E | DifferentiableAt ℝ (‖·‖) x} :=
+  let _ : MeasurableSpace E := borel E
+  have _ : BorelSpace E := ⟨rfl⟩
+  let w := Basis.ofVectorSpace ℝ E
+  MeasureTheory.Measure.dense_of_ae (ae_differentiableAt_norm (μ := w.addHaar))
diff --git a/Mathlib/Analysis/Calculus/SmoothSeries.lean b/Mathlib/Analysis/Calculus/SmoothSeries.lean
index 803f02e84a103..58d0f39d86c54 100644
--- a/Mathlib/Analysis/Calculus/SmoothSeries.lean
+++ b/Mathlib/Analysis/Calculus/SmoothSeries.lean
@@ -26,8 +26,8 @@ open Set Metric TopologicalSpace Function Asymptotics Filter
 
 open scoped Topology NNReal
 
-variable {α β 𝕜 E F : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-  [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
+variable {α β 𝕜 E F : Type*} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]
+  [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [CompleteSpace F] {u : α → ℝ}
 
 /-! ### Differentiability -/
 
@@ -77,7 +77,7 @@ theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
       apply Summable.hasSum
       exact summable_of_summable_hasFDerivAt_of_isPreconnected hu hs h's hf hf' hx₀ hf0 hy
     refine hasFDerivAt_of_tendstoUniformlyOn hs (tendstoUniformlyOn_tsum hu hf')
-      (fun t y hy => ?_) A _ hx
+      (fun t y hy => ?_) A hx
     exact HasFDerivAt.sum fun n _ => hf n y hy
 
 /-- Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
@@ -100,6 +100,7 @@ then the series converges everywhere. -/
 theorem summable_of_summable_hasFDerivAt (hu : Summable u)
     (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
     (hf0 : Summable fun n => f n x₀) (x : E) : Summable fun n => f n x := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
   exact summable_of_summable_hasFDerivAt_of_isPreconnected hu isOpen_univ isPreconnected_univ
     (fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)
@@ -119,6 +120,7 @@ then the series is differentiable and its derivative is the sum of the derivativ
 theorem hasFDerivAt_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
     (hf' : ∀ n x, ‖f' n x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E) :
     HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   let A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
   exact hasFDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ
     (fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)
@@ -199,7 +201,7 @@ theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
     rw [fderiv_tsum (hv _ hk) (fun n => (hf n).differentiable_iteratedFDeriv h'k) _ A]
     · ext1 x
       exact (continuousMultilinearCurryLeftEquiv 𝕜
-        (fun _ : Fin (k + 1) => E) F).toContinuousLinearEquiv.map_tsum
+        (fun _ : Fin (k + 1) => E) F).symm.toContinuousLinearEquiv.map_tsum
     · intro n x
       simpa only [iteratedFDeriv_succ_eq_comp_left, LinearIsometryEquiv.norm_map, comp_apply]
         using h'f k.succ n x hk
@@ -230,7 +232,7 @@ theorem contDiff_tsum (hf : ∀ i, ContDiff 𝕜 N (f i)) (hv : ∀ k : ℕ, (k
       exact h'f _ _ _ hm
   · intro m hm
     have h'm : ((m + 1 : ℕ) : ℕ∞) ≤ N := by
-      simpa only [ENat.coe_add, ENat.coe_one] using ENat.add_one_le_of_lt hm
+      simpa only [ENat.coe_add, ENat.coe_one] using Order.add_one_le_of_lt hm
     rw [iteratedFDeriv_tsum hf hv h'f hm.le]
     have A :
       ∀ n x, HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x :=
diff --git a/Mathlib/Analysis/Calculus/TangentCone.lean b/Mathlib/Analysis/Calculus/TangentCone.lean
index 280e5356cb943..f3ac765b70a30 100644
--- a/Mathlib/Analysis/Calculus/TangentCone.lean
+++ b/Mathlib/Analysis/Calculus/TangentCone.lean
@@ -81,7 +81,7 @@ theorem mem_tangentConeAt_of_pow_smul {r : 𝕜} (hr₀ : r ≠ 0) (hr : ‖r‖
     (hs : ∀ᶠ n : ℕ in atTop, x + r ^ n • y ∈ s) : y ∈ tangentConeAt 𝕜 s x := by
   refine ⟨fun n ↦ (r ^ n)⁻¹, fun n ↦ r ^ n • y, hs, ?_, ?_⟩
   · simp only [norm_inv, norm_pow, ← inv_pow]
-    exact tendsto_pow_atTop_atTop_of_one_lt <| one_lt_inv (norm_pos_iff.2 hr₀) hr
+    exact tendsto_pow_atTop_atTop_of_one_lt <| (one_lt_inv₀ (norm_pos_iff.2 hr₀)).2 hr
   · simp only [inv_smul_smul₀ (pow_ne_zero _ hr₀), tendsto_const_nhds]
 
 theorem tangentCone_univ : tangentConeAt 𝕜 univ x = univ :=
@@ -196,7 +196,7 @@ theorem mem_tangentCone_of_openSegment_subset {s : Set G} {x y : G} (h : openSeg
   rw [openSegment_eq_image]
   refine ⟨(1 / 2) ^ n, ⟨?_, ?_⟩, ?_⟩
   · exact pow_pos one_half_pos _
-  · exact pow_lt_one one_half_pos.le one_half_lt_one hn
+  · exact pow_lt_one₀ one_half_pos.le one_half_lt_one hn
   · simp only [sub_smul, one_smul, smul_sub]; abel
 
 /-- If a subset of a real vector space contains a segment, then the direction of this
diff --git a/Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean b/Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
index 75ca96be5b582..434945a20c58c 100644
--- a/Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
+++ b/Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
@@ -99,7 +99,8 @@ open scoped uniformity Filter Topology
 
 section LimitsOfDerivatives
 
-variable {ι : Type*} {l : Filter ι} {E : Type*} [NormedAddCommGroup E] {𝕜 : Type*} [RCLike 𝕜]
+variable {ι : Type*} {l : Filter ι} {E : Type*} [NormedAddCommGroup E] {𝕜 : Type*}
+  [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜]
   [NormedSpace 𝕜 E] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G}
   {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E}
 
@@ -110,6 +111,7 @@ sequence in a neighborhood of `x`. -/
 theorem uniformCauchySeqOnFilter_of_fderiv (hf' : UniformCauchySeqOnFilter f' l (𝓝 x))
     (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
     (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
   rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢
   suffices
@@ -173,6 +175,7 @@ convergence. See `cauchy_map_of_uniformCauchySeqOn_fderiv`.
 theorem uniformCauchySeqOn_ball_of_fderiv {r : ℝ} (hf' : UniformCauchySeqOn f' l (Metric.ball x r))
     (hf : ∀ n : ι, ∀ y : E, y ∈ Metric.ball x r → HasFDerivAt (f n) (f' n y) y)
     (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOn f l (Metric.ball x r) := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
   have : NeBot l := (cauchy_map_iff.1 hfg).1
   rcases le_or_lt r 0 with (hr | hr)
@@ -249,15 +252,16 @@ theorem cauchy_map_of_uniformCauchySeqOn_fderiv {s : Set E} (hs : IsOpen s) (h's
 /-- If `f_n → g` pointwise and the derivatives `(f_n)' → h` _uniformly_ converge, then
 in fact for a fixed `y`, the difference quotients `‖z - y‖⁻¹ • (f_n z - f_n y)` converge
 _uniformly_ to `‖z - y‖⁻¹ • (g z - g y)` -/
-theorem difference_quotients_converge_uniformly (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
+theorem difference_quotients_converge_uniformly
+    {E : Type*} [NormedAddCommGroup E] {𝕜 : Type*} [RCLike 𝕜]
+    [NormedSpace 𝕜 E] {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ι → E → G}
+    {g : E → G} {f' : ι → E → E →L[𝕜] G} {g' : E → E →L[𝕜] G} {x : E}
+    (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
     (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
     (hfg : ∀ᶠ y : E in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) :
     TendstoUniformlyOnFilter (fun n : ι => fun y : E => (‖y - x‖⁻¹ : 𝕜) • (f n y - f n x))
       (fun y : E => (‖y - x‖⁻¹ : 𝕜) • (g y - g x)) l (𝓝 x) := by
   let A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
-  rcases eq_or_ne l ⊥ with (hl | hl)
-  · simp only [hl, TendstoUniformlyOnFilter, bot_prod, eventually_bot, imp_true_iff]
-  haveI : NeBot l := ⟨hl⟩
   refine
     UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto ?_
       ((hfg.and (eventually_const.mpr hfg.self_of_nhds)).mono fun y hy =>
@@ -282,7 +286,7 @@ theorem difference_quotients_converge_uniformly (hf' : TendstoUniformlyOnFilter
   refine lt_of_le_of_lt ?_ hqε
   by_cases hyz' : x = y; · simp [hyz', hqpos.le]
   have hyz : 0 < ‖y - x‖ := by rw [norm_pos_iff]; intro hy'; exact hyz' (eq_of_sub_eq_zero hy').symm
-  rw [inv_mul_le_iff hyz, mul_comm, sub_sub_sub_comm]
+  rw [inv_mul_le_iff₀ hyz, mul_comm, sub_sub_sub_comm]
   simp only [Pi.zero_apply, dist_zero_left] at e
   refine
     Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
@@ -302,6 +306,7 @@ theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
     (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
     (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
     (hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasFDerivAt g (g' x) x := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   -- The proof strategy follows several steps:
   --   1. The quantifiers in the definition of the derivative are
   --      `∀ ε > 0, ∃δ > 0, ∀y ∈ B_δ(x)`. We will introduce a quantifier in the middle:
@@ -351,8 +356,8 @@ theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
     apply ((this ε hε).filter_mono curry_le_prod).mono
     intro n hn
     rw [dist_eq_norm] at hn ⊢
-    rw [← smul_sub] at hn
-    rwa [sub_zero]
+    convert hn using 2
+    module
   · -- (Almost) the definition of the derivatives
     rw [Metric.tendsto_nhds]
     intro ε hε
@@ -380,7 +385,7 @@ theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
     by_cases hx : x = n.2; · simp [hx]
     have hnx : 0 < ‖n.2 - x‖ := by
       rw [norm_pos_iff]; intro hx'; exact hx (eq_of_sub_eq_zero hx').symm
-    rw [inv_mul_le_iff hnx, mul_comm]
+    rw [inv_mul_le_iff₀ hnx, mul_comm]
     simp only [Function.comp_apply, Prod.map_apply']
     rw [norm_sub_rev]
     exact (f' n.1 x - g' x).le_opNorm (n.2 - x)
@@ -411,20 +416,19 @@ _uniformly_ to their limit on an open set containing `x`. -/
 theorem hasFDerivAt_of_tendstoUniformlyOn [NeBot l] {s : Set E} (hs : IsOpen s)
     (hf' : TendstoUniformlyOn f' g' l s)
     (hf : ∀ n : ι, ∀ x : E, x ∈ s → HasFDerivAt (f n) (f' n x) x)
-    (hfg : ∀ x : E, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) :
-    ∀ x : E, x ∈ s → HasFDerivAt g (g' x) x := fun _ =>
-  hasFDerivAt_of_tendstoLocallyUniformlyOn hs hf'.tendstoLocallyUniformlyOn hf hfg
+    (hfg : ∀ x : E, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
+    HasFDerivAt g (g' x) x :=
+  hasFDerivAt_of_tendstoLocallyUniformlyOn hs hf'.tendstoLocallyUniformlyOn hf hfg hx
 
 /-- `(d/dx) lim_{n → ∞} f n x = lim_{n → ∞} f' n x` when the `f' n` converge
 _uniformly_ to their limit. -/
 theorem hasFDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
     (hf : ∀ n : ι, ∀ x : E, HasFDerivAt (f n) (f' n x) x)
-    (hfg : ∀ x : E, Tendsto (fun n => f n x) l (𝓝 (g x))) : ∀ x : E, HasFDerivAt g (g' x) x := by
-  intro x
+    (hfg : ∀ x : E, Tendsto (fun n => f n x) l (𝓝 (g x))) (x : E) : HasFDerivAt g (g' x) x := by
   have hf : ∀ n : ι, ∀ x : E, x ∈ Set.univ → HasFDerivAt (f n) (f' n x) x := by simp [hf]
   have hfg : ∀ x : E, x ∈ Set.univ → Tendsto (fun n => f n x) l (𝓝 (g x)) := by simp [hfg]
   have hf' : TendstoUniformlyOn f' g' l Set.univ := by rwa [tendstoUniformlyOn_univ]
-  exact hasFDerivAt_of_tendstoUniformlyOn isOpen_univ hf' hf hfg x (Set.mem_univ x)
+  exact hasFDerivAt_of_tendstoUniformlyOn isOpen_univ hf' hf hfg (Set.mem_univ x)
 
 end LimitsOfDerivatives
 
@@ -436,7 +440,8 @@ In this section, we provide `deriv` equivalents of the `fderiv` lemmas in the pr
 -/
 
 
-variable {ι : Type*} {l : Filter ι} {𝕜 : Type*} [RCLike 𝕜] {G : Type*} [NormedAddCommGroup G]
+variable {ι : Type*} {l : Filter ι} {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+  {G : Type*} [NormedAddCommGroup G]
   [NormedSpace 𝕜 G] {f : ι → 𝕜 → G} {g : 𝕜 → G} {f' : ι → 𝕜 → G} {g' : 𝕜 → G} {x : 𝕜}
 
 /-- If our derivatives converge uniformly, then the Fréchet derivatives converge uniformly -/
@@ -461,6 +466,8 @@ theorem UniformCauchySeqOnFilter.one_smulRight {l' : Filter 𝕜}
   rw [← smul_sub, norm_smul, mul_comm]
   gcongr
 
+variable [IsRCLikeNormedField 𝕜]
+
 theorem uniformCauchySeqOnFilter_of_deriv (hf' : UniformCauchySeqOnFilter f' l (𝓝 x))
     (hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2)
     (hfg : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x) := by
@@ -530,18 +537,17 @@ theorem hasDerivAt_of_tendsto_locally_uniformly_on' [NeBot l] {s : Set 𝕜} (hs
 theorem hasDerivAt_of_tendstoUniformlyOn [NeBot l] {s : Set 𝕜} (hs : IsOpen s)
     (hf' : TendstoUniformlyOn f' g' l s)
     (hf : ∀ᶠ n in l, ∀ x : 𝕜, x ∈ s → HasDerivAt (f n) (f' n x) x)
-    (hfg : ∀ x : 𝕜, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) :
-    ∀ x : 𝕜, x ∈ s → HasDerivAt g (g' x) x := fun _ =>
-  hasDerivAt_of_tendstoLocallyUniformlyOn hs hf'.tendstoLocallyUniformlyOn hf hfg
+    (hfg : ∀ x : 𝕜, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
+    HasDerivAt g (g' x) x :=
+  hasDerivAt_of_tendstoLocallyUniformlyOn hs hf'.tendstoLocallyUniformlyOn hf hfg hx
 
 theorem hasDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
     (hf : ∀ᶠ n in l, ∀ x : 𝕜, HasDerivAt (f n) (f' n x) x)
-    (hfg : ∀ x : 𝕜, Tendsto (fun n => f n x) l (𝓝 (g x))) : ∀ x : 𝕜, HasDerivAt g (g' x) x := by
-  intro x
+    (hfg : ∀ x : 𝕜, Tendsto (fun n => f n x) l (𝓝 (g x))) (x : 𝕜) : HasDerivAt g (g' x) x := by
   have hf : ∀ᶠ n in l, ∀ x : 𝕜, x ∈ Set.univ → HasDerivAt (f n) (f' n x) x := by
     filter_upwards [hf] with n h x _ using h x
   have hfg : ∀ x : 𝕜, x ∈ Set.univ → Tendsto (fun n => f n x) l (𝓝 (g x)) := by simp [hfg]
   have hf' : TendstoUniformlyOn f' g' l Set.univ := by rwa [tendstoUniformlyOn_univ]
-  exact hasDerivAt_of_tendstoUniformlyOn isOpen_univ hf' hf hfg x (Set.mem_univ x)
+  exact hasDerivAt_of_tendstoUniformlyOn isOpen_univ hf' hf hfg (Set.mem_univ x)
 
 end deriv
diff --git a/Mathlib/Analysis/Complex/AbelLimit.lean b/Mathlib/Analysis/Complex/AbelLimit.lean
index 0e14c1272df17..167aa0434d806 100644
--- a/Mathlib/Analysis/Complex/AbelLimit.lean
+++ b/Mathlib/Analysis/Complex/AbelLimit.lean
@@ -206,9 +206,9 @@ theorem tendsto_tsum_powerSeries_nhdsWithin_stolzSet
     calc
       _ ≤ ‖1 - z‖ * ∑ i ∈ range B₁, ‖l - s (i + 1)‖ := by
         gcongr; nth_rw 3 [← mul_one ‖_‖]
-        gcongr; exact pow_le_one _ (norm_nonneg _) zn.le
+        gcongr; exact pow_le_one₀ (norm_nonneg _) zn.le
       _ ≤ ‖1 - z‖ * (F + 1) := by gcongr; linarith only
-      _ < _ := by rwa [norm_sub_rev, lt_div_iff (by positivity)] at zd
+      _ < _ := by rwa [norm_sub_rev, lt_div_iff₀ (by positivity)] at zd
   have S₂ : ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ‖l - s (i + 1)‖ * ‖z‖ ^ i < ε / 4 :=
     calc
       _ ≤ ‖1 - z‖ * ∑ i ∈ Ico B₁ (max B₁ B₂), ε / 4 / M * ‖z‖ ^ i := by
@@ -268,6 +268,5 @@ theorem tendsto_tsum_powerSeries_nhdsWithin_lt
   convert h
   simp_rw [Function.comp_apply, dist_eq_norm]
   norm_cast
-  rw [norm_real]
 
 end Real
diff --git a/Mathlib/Analysis/Complex/AbsMax.lean b/Mathlib/Analysis/Complex/AbsMax.lean
index e7257a7aab9fb..0d367d1d608d3 100644
--- a/Mathlib/Analysis/Complex/AbsMax.lean
+++ b/Mathlib/Analysis/Complex/AbsMax.lean
@@ -143,8 +143,8 @@ theorem norm_max_aux₂ {f : ℂ → F} {z w : ℂ} (hd : DiffContOnCl ℂ f (ba
   set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
   have he : ∀ x, ‖e x‖ = ‖x‖ := UniformSpace.Completion.norm_coe
   replace hz : IsMaxOn (norm ∘ e ∘ f) (closedBall z (dist w z)) z := by
-    simpa only [IsMaxOn, (· ∘ ·), he] using hz
-  simpa only [he, (· ∘ ·)]
+    simpa only [IsMaxOn, Function.comp_def, he] using hz
+  simpa only [he, Function.comp_def]
     using norm_max_aux₁ (e.differentiable.comp_diffContOnCl hd) hz
 
 /-!
@@ -172,7 +172,7 @@ Finally, we generalize the theorem from a disk in `ℂ` to a closed ball in any
 
 /-- **Maximum modulus principle** on a closed ball: if `f : E → F` is continuous on a closed ball,
 is complex differentiable on the corresponding open ball, and the norm `‖f w‖` takes its maximum
-value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball.  -/
+value on the open ball at its center, then the norm `‖f w‖` is constant on the closed ball. -/
 theorem norm_eqOn_closedBall_of_isMaxOn {f : E → F} {z : E} {r : ℝ}
     (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
     EqOn (norm ∘ f) (const E ‖f z‖) (closedBall z r) := by
@@ -306,7 +306,7 @@ normed complex space to a strictly convex normed complex space has the following
 - it is complex differentiable on the corresponding open ball;
 - the norm `‖f w‖` takes its maximum value on the open ball at its center.
 
-Then `f` is a constant on the closed ball.  -/
+Then `f` is a constant on the closed ball. -/
 theorem eqOn_closedBall_of_isMaxOn_norm {f : E → F} {z : E} {r : ℝ}
     (hd : DiffContOnCl ℂ f (ball z r)) (hz : IsMaxOn (norm ∘ f) (ball z r) z) :
     EqOn f (const E (f z)) (closedBall z r) := fun _x hx =>
diff --git a/Mathlib/Analysis/Complex/Angle.lean b/Mathlib/Analysis/Complex/Angle.lean
index 81cc8f077e489..e7031e3748729 100644
--- a/Mathlib/Analysis/Complex/Angle.lean
+++ b/Mathlib/Analysis/Complex/Angle.lean
@@ -96,7 +96,7 @@ lemma norm_sub_mem_Icc_angle (hx : ‖x‖ = 1) (hy : ‖y‖ = 1) :
       calc
         _ = 2 * (1 - (1 - 2 / π ^ 2 * θ ^ 2)) := by ring
         _ ≤ 2 * (1 - θ.cos) := by
-            gcongr; exact Real.cos_quadratic_upper_bound <| abs_le.2 <| Ioc_subset_Icc_self hθ
+            gcongr; exact Real.cos_le_one_sub_mul_cos_sq <| abs_le.2 <| Ioc_subset_Icc_self hθ
         _  = _ := by linear_combination -θ.cos_sq_add_sin_sq
     · rw [Real.sqrt_le_left (by positivity), ← _root_.abs_pow, abs_sq]
       calc
diff --git a/Mathlib/Analysis/Complex/Arg.lean b/Mathlib/Analysis/Complex/Arg.lean
index ec78377f02673..13b4096755966 100644
--- a/Mathlib/Analysis/Complex/Arg.lean
+++ b/Mathlib/Analysis/Complex/Arg.lean
@@ -31,7 +31,7 @@ theorem sameRay_iff : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg := b
   · simp
   rcases eq_or_ne y 0 with (rfl | hy)
   · simp
-  simp only [hx, hy, false_or_iff, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
+  simp only [hx, hy, sameRay_iff_norm_smul_eq, arg_eq_arg_iff hx hy]
   field_simp [hx, hy]
   rw [mul_comm, eq_comm]
 
diff --git a/Mathlib/Analysis/Complex/Basic.lean b/Mathlib/Analysis/Complex/Basic.lean
index 5cce5c3e7eeb1..b2b77580329f5 100644
--- a/Mathlib/Analysis/Complex/Basic.lean
+++ b/Mathlib/Analysis/Complex/Basic.lean
@@ -92,6 +92,8 @@ instance (priority := 900) _root_.NormedAlgebra.complexToReal {A : Type*} [Semin
     [NormedAlgebra ℂ A] : NormedAlgebra ℝ A :=
   NormedAlgebra.restrictScalars ℝ ℂ A
 
+@[simp] lemma nnnorm_I : ‖I‖₊ = 1 := by simp [nnnorm]
+
 theorem dist_eq (z w : ℂ) : dist z w = abs (z - w) :=
   rfl
 
@@ -138,23 +140,38 @@ theorem nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * Real.nnabs z.im :=
 theorem comap_abs_nhds_zero : comap abs (𝓝 0) = 𝓝 0 :=
   comap_norm_nhds_zero
 
-theorem norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖ :=
-  abs_ofReal _
+@[simp 1100, norm_cast] lemma norm_real (r : ℝ) : ‖(r : ℂ)‖ = ‖r‖ := abs_ofReal _
+@[simp, norm_cast] lemma nnnorm_real (r : ℝ) : ‖(r : ℂ)‖₊ = ‖r‖₊ := by ext; exact norm_real _
 
-@[simp 1100]
-theorem norm_rat (r : ℚ) : ‖(r : ℂ)‖ = |(r : ℝ)| := by
-  rw [← ofReal_ratCast]
-  exact norm_real _
+@[simp 1100, norm_cast] lemma norm_natCast (n : ℕ) : ‖(n : ℂ)‖ = n := abs_natCast _
+@[simp 1100, norm_cast] lemma norm_intCast (n : ℤ) : ‖(n : ℂ)‖ = |(n : ℝ)| := abs_intCast n
+@[simp 1100, norm_cast] lemma norm_ratCast (q : ℚ) : ‖(q : ℂ)‖ = |(q : ℝ)| := norm_real _
 
-@[simp 1100]
-theorem norm_nat (n : ℕ) : ‖(n : ℂ)‖ = n :=
-  abs_natCast _
+@[simp 1100, norm_cast] lemma nnnorm_natCast (n : ℕ) : ‖(n : ℂ)‖₊ = n := Subtype.ext <| by simp
+@[simp 1100, norm_cast] lemma nnnorm_intCast (n : ℤ) : ‖(n : ℂ)‖₊ = ‖n‖₊ := by
+  ext; exact norm_intCast n
+@[simp 1100, norm_cast] lemma nnnorm_ratCast (q : ℚ) : ‖(q : ℂ)‖₊ = ‖(q : ℝ)‖₊ := nnnorm_real q
 
-@[simp 1100]
-lemma norm_int {n : ℤ} : ‖(n : ℂ)‖ = |(n : ℝ)| := abs_intCast n
+@[simp 1100] lemma norm_ofNat (n : ℕ) [n.AtLeastTwo] :
+    ‖(no_index (OfNat.ofNat n) : ℂ)‖ = OfNat.ofNat n := norm_natCast n
+
+@[simp 1100] lemma nnnorm_ofNat (n : ℕ) [n.AtLeastTwo] :
+    ‖(no_index (OfNat.ofNat n) : ℂ)‖₊ = OfNat.ofNat n := nnnorm_natCast n
+
+@[deprecated (since := "2024-08-25")] alias norm_nat := norm_natCast
+@[deprecated (since := "2024-08-25")] alias norm_int := norm_intCast
+@[deprecated (since := "2024-08-25")] alias norm_rat := norm_ratCast
+@[deprecated (since := "2024-08-25")] alias nnnorm_nat := nnnorm_natCast
+@[deprecated (since := "2024-08-25")] alias nnnorm_int := nnnorm_intCast
+
+@[simp 1100, norm_cast]
+lemma norm_nnratCast (q : ℚ≥0) : ‖(q : ℂ)‖ = q := abs_of_nonneg q.cast_nonneg
+
+@[simp 1100, norm_cast]
+lemma nnnorm_nnratCast (q : ℚ≥0) : ‖(q : ℂ)‖₊ = q := by simp [nnnorm, -norm_eq_abs]
 
 theorem norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖(n : ℂ)‖ = n := by
-  rw [norm_int, ← Int.cast_abs, _root_.abs_of_nonneg hn]
+  rw [norm_intCast, ← Int.cast_abs, _root_.abs_of_nonneg hn]
 
 lemma normSq_eq_norm_sq (z : ℂ) : normSq z = ‖z‖ ^ 2 := by
   rw [normSq_eq_abs, norm_eq_abs]
@@ -167,17 +184,6 @@ theorem continuous_abs : Continuous abs :=
 theorem continuous_normSq : Continuous normSq := by
   simpa [← normSq_eq_abs] using continuous_abs.pow 2
 
-@[simp, norm_cast]
-theorem nnnorm_real (r : ℝ) : ‖(r : ℂ)‖₊ = ‖r‖₊ :=
-  Subtype.ext <| norm_real r
-
-@[simp, norm_cast]
-theorem nnnorm_nat (n : ℕ) : ‖(n : ℂ)‖₊ = n :=
-  Subtype.ext <| by simp
-
-@[simp, norm_cast]
-theorem nnnorm_int (n : ℤ) : ‖(n : ℂ)‖₊ = ‖n‖₊ :=
-  Subtype.ext norm_int
 
 theorem nnnorm_eq_one_of_pow_eq_one {ζ : ℂ} {n : ℕ} (h : ζ ^ n = 1) (hn : n ≠ 0) : ‖ζ‖₊ = 1 :=
   (pow_left_inj zero_le' zero_le' hn).1 <| by rw [← nnnorm_pow, h, nnnorm_one, one_pow]
@@ -202,11 +208,14 @@ theorem antilipschitz_equivRealProd : AntilipschitzWith (NNReal.sqrt 2) equivRea
   AddMonoidHomClass.antilipschitz_of_bound equivRealProdLm fun z ↦ by
     simpa only [Real.coe_sqrt, NNReal.coe_ofNat] using abs_le_sqrt_two_mul_max z
 
-theorem uniformEmbedding_equivRealProd : UniformEmbedding equivRealProd :=
-  antilipschitz_equivRealProd.uniformEmbedding lipschitz_equivRealProd.uniformContinuous
+theorem isUniformEmbedding_equivRealProd : IsUniformEmbedding equivRealProd :=
+  antilipschitz_equivRealProd.isUniformEmbedding lipschitz_equivRealProd.uniformContinuous
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_equivRealProd := isUniformEmbedding_equivRealProd
 
 instance : CompleteSpace ℂ :=
-  (completeSpace_congr uniformEmbedding_equivRealProd).mpr inferInstance
+  (completeSpace_congr isUniformEmbedding_equivRealProd).mpr inferInstance
 
 instance instT2Space : T2Space ℂ := TopologicalSpace.t2Space_of_metrizableSpace
 
@@ -275,7 +284,7 @@ theorem restrictScalars_one_smulRight' (x : E) :
     ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] E) =
       reCLM.smulRight x + I • imCLM.smulRight x := by
   ext ⟨a, b⟩
-  simp [mk_eq_add_mul_I, mul_smul, smul_comm I b x]
+  simp [map_add, mk_eq_add_mul_I, mul_smul, smul_comm I b x]
 
 theorem restrictScalars_one_smulRight (x : ℂ) :
     ContinuousLinearMap.restrictScalars ℝ ((1 : ℂ →L[ℂ] ℂ).smulRight x : ℂ →L[ℂ] ℂ) =
diff --git a/Mathlib/Analysis/Complex/CauchyIntegral.lean b/Mathlib/Analysis/Complex/CauchyIntegral.lean
index adda35554977c..ebc991f8801e8 100644
--- a/Mathlib/Analysis/Complex/CauchyIntegral.lean
+++ b/Mathlib/Analysis/Complex/CauchyIntegral.lean
@@ -3,15 +3,14 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
-import Mathlib.MeasureTheory.Integral.DivergenceTheorem
-import Mathlib.MeasureTheory.Integral.CircleIntegral
-import Mathlib.Analysis.Calculus.Dslope
-import Mathlib.Analysis.Analytic.Basic
-import Mathlib.Analysis.Complex.ReImTopology
+import Mathlib.Analysis.Analytic.Uniqueness
 import Mathlib.Analysis.Calculus.DiffContOnCl
+import Mathlib.Analysis.Calculus.Dslope
 import Mathlib.Analysis.Calculus.FDeriv.Analytic
-import Mathlib.Data.Real.Cardinality
+import Mathlib.Analysis.Complex.ReImTopology
+import Mathlib.MeasureTheory.Integral.CircleIntegral
+import Mathlib.MeasureTheory.Integral.DivergenceTheorem
+import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
 
 /-!
 # Cauchy integral formula
@@ -569,14 +568,18 @@ protected theorem _root_.DifferentiableOn.analyticAt {s : Set ℂ} {f : ℂ →
   lift R to ℝ≥0 using hR0.le
   exact ((hd.mono hRs).hasFPowerSeriesOnBall hR0).analyticAt
 
+theorem _root_.DifferentiableOn.analyticOnNhd {s : Set ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f s)
+    (hs : IsOpen s) : AnalyticOnNhd ℂ f s := fun _z hz => hd.analyticAt (hs.mem_nhds hz)
+
 theorem _root_.DifferentiableOn.analyticOn {s : Set ℂ} {f : ℂ → E} (hd : DifferentiableOn ℂ f s)
-    (hs : IsOpen s) : AnalyticOn ℂ f s := fun _z hz => hd.analyticAt (hs.mem_nhds hz)
+    (hs : IsOpen s) : AnalyticOn ℂ f s :=
+  (hd.analyticOnNhd hs).analyticOn
 
 /-- If `f : ℂ → E` is complex differentiable on some open set `s`, then it is continuously
 differentiable on `s`. -/
 protected theorem _root_.DifferentiableOn.contDiffOn {s : Set ℂ} {f : ℂ → E} {n : ℕ}
     (hd : DifferentiableOn ℂ f s) (hs : IsOpen s) : ContDiffOn ℂ n f s :=
-  (hd.analyticOn hs).contDiffOn
+  (hd.analyticOnNhd hs).contDiffOn
 
 /-- A complex differentiable function `f : ℂ → E` is analytic at every point. -/
 protected theorem _root_.Differentiable.analyticAt {f : ℂ → E} (hf : Differentiable ℂ f) (z : ℂ) :
@@ -595,16 +598,27 @@ protected theorem _root_.Differentiable.hasFPowerSeriesOnBall {f : ℂ → E} (h
   (h.differentiableOn.hasFPowerSeriesOnBall hR).r_eq_top_of_exists fun _r hr =>
     ⟨_, h.differentiableOn.hasFPowerSeriesOnBall hr⟩
 
+/-- On an open set, `f : ℂ → E` is analytic iff it is differentiable -/
+theorem analyticOnNhd_iff_differentiableOn {f : ℂ → E} {s : Set ℂ} (o : IsOpen s) :
+    AnalyticOnNhd ℂ f s ↔ DifferentiableOn ℂ f s :=
+  ⟨AnalyticOnNhd.differentiableOn, fun d _ zs ↦ d.analyticAt (o.mem_nhds zs)⟩
+
 /-- On an open set, `f : ℂ → E` is analytic iff it is differentiable -/
 theorem analyticOn_iff_differentiableOn {f : ℂ → E} {s : Set ℂ} (o : IsOpen s) :
-    AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s :=
-  ⟨AnalyticOn.differentiableOn, fun d _ zs ↦ d.analyticAt (o.mem_nhds zs)⟩
+    AnalyticOn ℂ f s ↔ DifferentiableOn ℂ f s := by
+  rw [o.analyticOn_iff_analyticOnNhd]
+  exact analyticOnNhd_iff_differentiableOn o
 
 /-- `f : ℂ → E` is entire iff it's differentiable -/
+theorem analyticOnNhd_univ_iff_differentiable {f : ℂ → E} :
+    AnalyticOnNhd ℂ f univ ↔ Differentiable ℂ f := by
+  simp only [← differentiableOn_univ]
+  exact analyticOnNhd_iff_differentiableOn isOpen_univ
+
 theorem analyticOn_univ_iff_differentiable {f : ℂ → E} :
     AnalyticOn ℂ f univ ↔ Differentiable ℂ f := by
-  simp only [← differentiableOn_univ]
-  exact analyticOn_iff_differentiableOn isOpen_univ
+  rw [analyticOn_univ]
+  exact analyticOnNhd_univ_iff_differentiable
 
 /-- `f : ℂ → E` is analytic at `z` iff it's differentiable near `z` -/
 theorem analyticAt_iff_eventually_differentiableAt {f : ℂ → E} {c : ℂ} :
@@ -615,8 +629,8 @@ theorem analyticAt_iff_eventually_differentiableAt {f : ℂ → E} {c : ℂ} :
     apply AnalyticAt.differentiableAt
   · intro d
     rcases _root_.eventually_nhds_iff.mp d with ⟨s, d, o, m⟩
-    have h : AnalyticOn ℂ f s := by
-      refine DifferentiableOn.analyticOn ?_ o
+    have h : AnalyticOnNhd ℂ f s := by
+      refine DifferentiableOn.analyticOnNhd ?_ o
       intro z m
       exact (d z m).differentiableWithinAt
     exact h _ m
diff --git a/Mathlib/Analysis/Complex/Circle.lean b/Mathlib/Analysis/Complex/Circle.lean
index bda86fd651ad1..72e66ae46c1b7 100644
--- a/Mathlib/Analysis/Complex/Circle.lean
+++ b/Mathlib/Analysis/Complex/Circle.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
 import Mathlib.Analysis.SpecialFunctions.Exp
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Analysis.Normed.Field.UnitBall
 
 /-!
diff --git a/Mathlib/Analysis/Complex/Hadamard.lean b/Mathlib/Analysis/Complex/Hadamard.lean
index 88d6c801d518c..37d20a8d07540 100644
--- a/Mathlib/Analysis/Complex/Hadamard.lean
+++ b/Mathlib/Analysis/Complex/Hadamard.lean
@@ -189,14 +189,14 @@ lemma F_edge_le_one (f : ℂ → E) (ε : ℝ) (hε : ε > 0) (z : ℂ)
   rcases hz with hz0 | hz1
   -- `z.re = 0`
   · simp only [hz0, zero_sub, Real.rpow_neg_one, neg_zero, Real.rpow_zero, mul_one,
-      inv_mul_le_iff (sSupNormIm_eps_pos f hε 0)]
+      inv_mul_le_iff₀ (sSupNormIm_eps_pos f hε 0)]
     rw [← hz0]
     apply le_of_lt (norm_lt_sSupNormIm_eps f ε hε _ _ hB)
     simp only [verticalClosedStrip, mem_preimage, zero_le_one, left_mem_Icc, hz0]
   -- `z.re = 1`
   · rw [mem_singleton_iff] at hz1
     simp only [hz1, one_mul, Real.rpow_zero, sub_self, Real.rpow_neg_one,
-      inv_mul_le_iff (sSupNormIm_eps_pos f hε 1), mul_one]
+      inv_mul_le_iff₀ (sSupNormIm_eps_pos f hε 1), mul_one]
     rw [← hz1]
     apply le_of_lt (norm_lt_sSupNormIm_eps f ε hε _ _ hB)
     simp only [verticalClosedStrip, mem_preimage, zero_le_one, hz1, right_mem_Icc]
@@ -304,7 +304,7 @@ lemma norm_le_interpStrip_of_mem_verticalClosedStrip_eps (ε : ℝ) (hε : ε >
     ‖f z‖ ≤  ‖((ε + sSupNormIm f 0) ^ (1-z) * (ε + sSupNormIm f 1) ^ z : ℂ)‖ := by
   simp only [F, abs_invInterpStrip _ _ hε, norm_smul, norm_mul, norm_eq_abs,
     ← ofReal_add, abs_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _, sub_re, one_re]
-  rw [← mul_inv_le_iff, ← one_mul (((ε + sSupNormIm f 1) ^ z.re)), ← mul_inv_le_iff',
+  rw [← mul_inv_le_iff₀', ← one_mul (((ε + sSupNormIm f 1) ^ z.re)), ← mul_inv_le_iff₀,
     ← Real.rpow_neg_one, ← Real.rpow_neg_one]
   · simp only [← Real.rpow_mul (le_of_lt (sSupNormIm_eps_pos f hε _)),
     mul_neg, mul_one, neg_sub, mul_assoc]
diff --git a/Mathlib/Analysis/Complex/IsIntegral.lean b/Mathlib/Analysis/Complex/IsIntegral.lean
new file mode 100644
index 0000000000000..939434a7a666d
--- /dev/null
+++ b/Mathlib/Analysis/Complex/IsIntegral.lean
@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2022 Yuyang Zhao. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuyang Zhao
+-/
+
+import Mathlib.Analysis.Complex.Basic
+import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
+
+/-!
+# Integral elements of ℂ
+
+This file proves that `Complex.I` is integral over ℤ and ℚ.
+-/
+
+open Polynomial
+
+namespace Complex
+
+theorem isIntegral_int_I : IsIntegral ℤ I := by
+  refine ⟨X ^ 2 + C 1, monic_X_pow_add_C _ two_ne_zero, ?_⟩
+  rw [eval₂_add, eval₂_X_pow, eval₂_C, I_sq, eq_intCast, Int.cast_one, neg_add_cancel]
+
+theorem isIntegral_rat_I : IsIntegral ℚ I :=
+  isIntegral_int_I.tower_top
+
+end Complex
diff --git a/Mathlib/Analysis/Complex/OpenMapping.lean b/Mathlib/Analysis/Complex/OpenMapping.lean
index 9ce4d01945319..c37d82d1058c9 100644
--- a/Mathlib/Analysis/Complex/OpenMapping.lean
+++ b/Mathlib/Analysis/Complex/OpenMapping.lean
@@ -27,7 +27,7 @@ That second step is implemented in `DiffContOnCl.ball_subset_image_closedBall`.
 
 * `AnalyticAt.eventually_constant_or_nhds_le_map_nhds` is the local version of the open mapping
   theorem around a point;
-* `AnalyticOn.is_constant_or_isOpen` is the open mapping theorem on a connected open set.
+* `AnalyticOnNhd.is_constant_or_isOpen` is the open mapping theorem on a connected open set.
 -/
 
 
@@ -51,7 +51,7 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
   have h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r) := h.sub_const v
   have h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r) :=
     continuous_norm.comp_continuousOn (closure_ball z₀ hr.ne.symm ▸ h1.continuousOn)
-  have h3 : AnalyticOn ℂ f (ball z₀ r) := h.differentiableOn.analyticOn isOpen_ball
+  have h3 : AnalyticOnNhd ℂ f (ball z₀ r) := h.differentiableOn.analyticOnNhd isOpen_ball
   have h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖ := fun z hz => by
     linarith [hf z hz, show ‖v - f z₀‖ < ε / 2 from mem_ball.mp hv,
       norm_sub_sub_norm_sub_le_norm_sub (f z) v (f z₀)]
@@ -64,7 +64,7 @@ theorem DiffContOnCl.ball_subset_image_closedBall (h : DiffContOnCl ℂ f (ball
   have h7 : ∀ᶠ w in 𝓝 z, f w = f z := by filter_upwards [key] with h; field_simp
   replace h7 : ∃ᶠ w in 𝓝[≠] z, f w = f z := (h7.filter_mono nhdsWithin_le_nhds).frequently
   have h8 : IsPreconnected (ball z₀ r) := (convex_ball z₀ r).isPreconnected
-  have h9 := h3.eqOn_of_preconnected_of_frequently_eq analyticOn_const h8 hz1 h7
+  have h9 := h3.eqOn_of_preconnected_of_frequently_eq analyticOnNhd_const h8 hz1 h7
   have h10 : f z = f z₀ := (h9 (mem_ball_self hr)).symm
   exact not_eventually.mpr hz₀ (mem_of_superset (ball_mem_nhds z₀ hr) (h10 ▸ h9))
 
@@ -83,7 +83,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds_aux (hf : AnalyticAt
   have h1 := (hf.eventually_eq_or_eventually_ne analyticAt_const).resolve_left h
   have h2 : ∀ᶠ z in 𝓝 z₀, AnalyticAt ℂ f z := (isOpen_analyticAt ℂ f).eventually_mem hf
   obtain ⟨ρ, hρ, h3, h4⟩ :
-    ∃ ρ > 0, AnalyticOn ℂ f (closedBall z₀ ρ) ∧ ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀ := by
+    ∃ ρ > 0, AnalyticOnNhd ℂ f (closedBall z₀ ρ) ∧ ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀ := by
     simpa only [setOf_and, subset_inter_iff] using
       nhds_basis_closedBall.mem_iff.mp (h2.and (eventually_nhdsWithin_iff.mp h1))
   replace h3 : DiffContOnCl ℂ f (ball z₀ ρ) :=
@@ -118,7 +118,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
   let ray : E → ℂ → E := fun z t => z₀ + t • z
   let gray : E → ℂ → ℂ := fun z => g ∘ ray z
   obtain ⟨r, hr, hgr⟩ := isOpen_iff.mp (isOpen_analyticAt ℂ g) z₀ hg
-  have h1 : ∀ z ∈ sphere (0 : E) 1, AnalyticOn ℂ (gray z) (ball 0 r) := by
+  have h1 : ∀ z ∈ sphere (0 : E) 1, AnalyticOnNhd ℂ (gray z) (ball 0 r) := by
     refine fun z hz t ht => AnalyticAt.comp ?_ ?_
     · exact hgr (by simpa [ray, norm_smul, mem_sphere_zero_iff_norm.mp hz] using ht)
     · exact analyticAt_const.add
@@ -134,7 +134,7 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
       have e1 : IsPreconnected (ball (0 : ℂ) r) := (convex_ball 0 r).isPreconnected
       have e2 : w ∈ sphere (0 : E) 1 := by simp [w, norm_smul, inv_mul_cancel₀ h']
       specialize h1 w e2
-      apply h1.eqOn_of_preconnected_of_eventuallyEq analyticOn_const e1 (mem_ball_self hr)
+      apply h1.eqOn_of_preconnected_of_eventuallyEq analyticOnNhd_const e1 (mem_ball_self hr)
       simpa [ray, gray] using h w e2
     have h4 : ‖z - z₀‖ < r := by simpa [dist_eq_norm] using mem_ball.mp hz
     replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by
@@ -156,13 +156,16 @@ theorem AnalyticAt.eventually_constant_or_nhds_le_map_nhds {z₀ : E} (hg : Anal
 /-- The *open mapping theorem* for holomorphic functions, global version: if a function `g : E → ℂ`
 is analytic on a connected set `U`, then either it is constant on `U`, or it is open on `U` (in the
 sense that it maps any open set contained in `U` to an open set in `ℂ`). -/
-theorem AnalyticOn.is_constant_or_isOpen (hg : AnalyticOn ℂ g U) (hU : IsPreconnected U) :
+theorem AnalyticOnNhd.is_constant_or_isOpen (hg : AnalyticOnNhd ℂ g U) (hU : IsPreconnected U) :
     (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen s → IsOpen (g '' s) := by
   by_cases h : ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀
   · obtain ⟨z₀, hz₀, h⟩ := h
-    exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOn_const hU hz₀ h⟩
+    exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOnNhd_const hU hz₀ h⟩
   · push_neg at h
     refine Or.inr fun s hs1 hs2 => isOpen_iff_mem_nhds.mpr ?_
     rintro z ⟨w, hw1, rfl⟩
     exact (hg w (hs1 hw1)).eventually_constant_or_nhds_le_map_nhds.resolve_left (h w (hs1 hw1))
         (image_mem_map (hs2.mem_nhds hw1))
+
+@[deprecated (since := "2024-09-26")]
+alias AnalyticOn.is_constant_or_isOpen := AnalyticOnNhd.is_constant_or_isOpen
diff --git a/Mathlib/Analysis/Complex/PhragmenLindelof.lean b/Mathlib/Analysis/Complex/PhragmenLindelof.lean
index c08ca1fd6b87e..a457d182cf842 100644
--- a/Mathlib/Analysis/Complex/PhragmenLindelof.lean
+++ b/Mathlib/Analysis/Complex/PhragmenLindelof.lean
@@ -133,7 +133,7 @@ theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
   rcases hB with ⟨c, hc, B, hO⟩
   obtain ⟨d, ⟨hcd, hd₀⟩, hd⟩ : ∃ d, (c < d ∧ 0 < d) ∧ d < π / 2 / b := by
     simpa only [max_lt_iff] using exists_between (max_lt hc hπb)
-  have hb' : d * b < π / 2 := (lt_div_iff hb).1 hd
+  have hb' : d * b < π / 2 := (lt_div_iff₀ hb).1 hd
   set aff := (fun w => d * (w - a * I) : ℂ → ℂ)
   set g := fun (ε : ℝ) (w : ℂ) => exp (ε * (exp (aff w) + exp (-aff w)))
   /- Since `g ε z → 1` as `ε → 0⁻`, it suffices to prove that `‖g ε z • f z‖ ≤ C`
@@ -282,8 +282,8 @@ theorem vertical_strip (hfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b))
     have : Tendsto (· * -I) (comap (|re ·|) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b))
         (comap (|im ·|) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)) := by
       refine (tendsto_comap_iff.2 ?_).inf H.tendsto
-      simpa [(· ∘ ·)] using tendsto_comap
-    simpa [(· ∘ ·)] using hO.comp_tendsto this
+      simpa [Function.comp_def] using tendsto_comap
+    simpa [Function.comp_def] using hO.comp_tendsto this
   all_goals simpa
 
 /-- **Phragmen-Lindelöf principle** in a strip `U = {z : ℂ | a < re z < b}`.
@@ -453,7 +453,7 @@ theorem quadrant_II (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0))
   rcases hB with ⟨c, hc, B, hO⟩
   refine quadrant_I (hd.comp (differentiable_id.mul_const _).diffContOnCl H) ⟨c, hc, B, ?_⟩ him
     (fun x hx => ?_) hz_im hz_re
-  · simpa only [(· ∘ ·), map_mul, abs_I, mul_one]
+  · simpa only [Function.comp_def, map_mul, abs_I, mul_one]
       using hO.comp_tendsto ((tendsto_mul_right_cobounded I_ne_zero).inf H.tendsto)
   · rw [comp_apply, mul_assoc, I_mul_I, mul_neg_one, ← ofReal_neg]
     exact hre _ (neg_nonpos.2 hx)
@@ -518,7 +518,7 @@ theorem quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
       hz_re hz_im
   · rcases hB with ⟨c, hc, B, hO⟩
     refine ⟨c, hc, B, ?_⟩
-    simpa only [(· ∘ ·), Complex.abs.map_neg]
+    simpa only [Function.comp_def, Complex.abs.map_neg]
       using hO.comp_tendsto (tendsto_neg_cobounded.inf H.tendsto)
   · rw [comp_apply, ← ofReal_neg]
     exact hre (-x) (neg_nonpos.2 hx)
@@ -582,7 +582,7 @@ theorem quadrant_IV (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
     (hd.comp differentiable_neg.diffContOnCl H) ?_ (fun x hx => ?_) (fun x hx => ?_) hz_re hz_im
   · rcases hB with ⟨c, hc, B, hO⟩
     refine ⟨c, hc, B, ?_⟩
-    simpa only [(· ∘ ·), Complex.abs.map_neg]
+    simpa only [Function.comp_def, Complex.abs.map_neg]
       using hO.comp_tendsto (tendsto_neg_cobounded.inf H.tendsto)
   · rw [comp_apply, ← ofReal_neg]
     exact hre (-x) (neg_nonneg.2 hx)
diff --git a/Mathlib/Analysis/Complex/Polynomial/Basic.lean b/Mathlib/Analysis/Complex/Polynomial/Basic.lean
index 145fa099c8971..82e38a8adc578 100644
--- a/Mathlib/Analysis/Complex/Polynomial/Basic.lean
+++ b/Mathlib/Analysis/Complex/Polynomial/Basic.lean
@@ -128,7 +128,7 @@ theorem galActionHom_bijective_of_prime_degree {p : ℚ[X]} (p_irr : Irreducible
   let conj' := restrict p ℂ (Complex.conjAe.restrictScalars ℚ)
   refine
     ⟨galActionHom_injective p ℂ, fun x =>
-      (congr_arg (Membership.mem x) (show (galActionHom p ℂ).range = ⊤ from ?_)).mpr
+      (congr_arg (x ∈ ·) (show (galActionHom p ℂ).range = ⊤ from ?_)).mpr
         (Subgroup.mem_top x)⟩
   apply Equiv.Perm.subgroup_eq_top_of_swap_mem
   · rwa [h1]
diff --git a/Mathlib/Analysis/Complex/RemovableSingularity.lean b/Mathlib/Analysis/Complex/RemovableSingularity.lean
index 9702422a17c49..b6d73c9132378 100644
--- a/Mathlib/Analysis/Complex/RemovableSingularity.lean
+++ b/Mathlib/Analysis/Complex/RemovableSingularity.lean
@@ -48,7 +48,7 @@ theorem differentiableOn_compl_singleton_and_continuousAt_iff {f : ℂ → E} {s
   rcases eq_or_ne x c with (rfl | hne)
   · refine (analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt
       ?_ hc).differentiableAt.differentiableWithinAt
-    refine eventually_nhdsWithin_iff.2 ((eventually_mem_nhds.2 hs).mono fun z hz hzx => ?_)
+    refine eventually_nhdsWithin_iff.2 ((eventually_mem_nhds_iff.2 hs).mono fun z hz hzx => ?_)
     exact hd.differentiableAt (inter_mem hz (isOpen_ne.mem_nhds hzx))
   · simpa only [DifferentiableWithinAt, HasFDerivWithinAt, hne.nhdsWithin_diff_singleton] using
       hd x ⟨hx, hne⟩
diff --git a/Mathlib/Analysis/Complex/TaylorSeries.lean b/Mathlib/Analysis/Complex/TaylorSeries.lean
index 90a8b74ad8c80..dbd3037467b3a 100644
--- a/Mathlib/Analysis/Complex/TaylorSeries.lean
+++ b/Mathlib/Analysis/Complex/TaylorSeries.lean
@@ -11,7 +11,7 @@ import Mathlib.Analysis.Complex.CauchyIntegral
 We show that the Taylor series around some point `c : ℂ` of a function `f` that is complex
 differentiable on the open ball of radius `r` around `c` converges to `f` on that open ball;
 see `Complex.hasSum_taylorSeries_on_ball` and `Complex.taylorSeries_eq_on_ball` for versions
-(in terms of `HasSum` and `tsum`, repsectively) for functions to a complete normed
+(in terms of `HasSum` and `tsum`, respectively) for functions to a complete normed
 space over `ℂ`, and `Complex.taylorSeries_eq_on_ball'` for a variant when `f : ℂ → ℂ`.
 
 There are corresponding statements for `EMEtric.ball`s; see
@@ -24,16 +24,24 @@ see `Complex.hasSum_taylorSeries_of_entire`, `Complex.taylorSeries_eq_of_entire`
 `Complex.taylorSeries_eq_of_entire'`.
 -/
 
+#adaptation_note
+/--
+Due to https://github.com/leanprover/lean4/issues/5126, we've had to change all
+the `⦃⦄` stricit implicit variable statements in this file to normal `{}` implicit variables.
+
+Once this issue is fixed, we should change them back. For now it doesn't break anything downstream.
+-/
+
 namespace Complex
 
 open Nat
 
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] ⦃f : ℂ → E⦄
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] [CompleteSpace E] {f : ℂ → E}
 
 section ball
 
-variable ⦃c : ℂ⦄ ⦃r : ℝ⦄ (hf : DifferentiableOn ℂ f (Metric.ball c r))
-variable ⦃z : ℂ⦄ (hz : z ∈ Metric.ball c r)
+variable {c : ℂ} {r : ℝ} (hf : DifferentiableOn ℂ f (Metric.ball c r))
+variable {z : ℂ} (hz : z ∈ Metric.ball c r)
 
 include hf hz in
 /-- A function that is complex differentiable on the open ball of radius `r` around `c`
@@ -74,8 +82,8 @@ end ball
 
 section emetric
 
-variable ⦃c : ℂ⦄ ⦃r : ENNReal⦄ (hf : DifferentiableOn ℂ f (EMetric.ball c r))
-variable ⦃z : ℂ⦄ (hz : z ∈ EMetric.ball c r)
+variable {c : ℂ} {r : ENNReal} (hf : DifferentiableOn ℂ f (EMetric.ball c r))
+variable {z : ℂ} (hz : z ∈ EMetric.ball c r)
 
 include hf hz in
 /-- A function that is complex differentiable on the open ball of radius `r ≤ ∞` around `c`
@@ -108,7 +116,7 @@ end emetric
 
 section entire
 
-variable ⦃f : ℂ → E⦄ (hf : Differentiable ℂ f) (c z : ℂ)
+variable {f : ℂ → E} (hf : Differentiable ℂ f) (c z : ℂ)
 
 include hf in
 /-- A function that is complex differentiable on the complex plane is given by evaluating
diff --git a/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean b/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
index 6afa53eec6ef1..f9c084804d861 100644
--- a/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
+++ b/Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
@@ -3,13 +3,9 @@ Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rig
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
 -/
-import Mathlib.Algebra.GroupWithZero.Action.Defs
 import Mathlib.Analysis.Complex.Basic
 import Mathlib.Data.Fintype.Parity
 import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
-import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
-import Mathlib.Tactic.AdaptationNote
-import Mathlib.Tactic.LinearCombination
 
 /-!
 # The upper half plane and its automorphisms
@@ -182,7 +178,7 @@ theorem linear_ne_zero (cd : Fin 2 → ℝ) (z : ℍ) (h : cd ≠ 0) : (cd 0 : 
   have : cd 0 = 0 := by
     -- we will need this twice
     apply_fun Complex.im at h
-    simpa only [z.im_ne_zero, Complex.add_im, add_zero, coe_im, zero_mul, or_false_iff,
+    simpa only [z.im_ne_zero, Complex.add_im, add_zero, coe_im, zero_mul, or_false,
       Complex.ofReal_im, Complex.zero_im, Complex.mul_im, mul_eq_zero] using h
   simp only [this, zero_mul, Complex.ofReal_zero, zero_add, Complex.ofReal_eq_zero]
     at h
diff --git a/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean b/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
index dec00ac0e5457..256120e306349 100644
--- a/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
+++ b/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
@@ -261,7 +261,7 @@ instance : MetricSpace ℍ :=
       have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
       refine ⟨log (R / im z + 1), Real.log_pos h₁, ?_⟩
       refine fun w hw => (dist_coe_le w z).trans_lt ?_
-      rwa [← lt_div_iff' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
+      rwa [← lt_div_iff₀' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
 
 theorem im_pos_of_dist_center_le {z : ℍ} {r : ℝ} {w : ℂ}
     (h : dist w (center z r) ≤ z.im * Real.sinh r) : 0 < w.im :=
@@ -327,7 +327,7 @@ instance : IsometricSMul SL(2, ℝ) ℍ :=
         have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
         simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im,
           div_one, Complex.inv_im, Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg,
-          mul_eq_mul_left_iff, Real.arsinh_inj, one_ne_zero, or_false_iff,
+          mul_eq_mul_left_iff, Real.arsinh_inj, one_ne_zero,
           dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ←
           AbsoluteValue.map_mul, ← Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply,
           mul_div (2 : ℝ), div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
diff --git a/Mathlib/Analysis/Convex/Basic.lean b/Mathlib/Analysis/Convex/Basic.lean
index f07e88d57f245..a06894150ee98 100644
--- a/Mathlib/Analysis/Convex/Basic.lean
+++ b/Mathlib/Analysis/Convex/Basic.lean
@@ -8,6 +8,8 @@ import Mathlib.Algebra.Order.Module.OrderedSMul
 import Mathlib.Algebra.Order.Module.Synonym
 import Mathlib.Analysis.Convex.Star
 import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.NoncommRing
 
 /-!
 # Convex sets and functions in vector spaces
@@ -158,8 +160,7 @@ theorem convex_segment (x y : E) : Convex 𝕜 [x -[𝕜] y] := by
     ⟨a * ap + b * aq, a * bp + b * bq, add_nonneg (mul_nonneg ha hap) (mul_nonneg hb haq),
       add_nonneg (mul_nonneg ha hbp) (mul_nonneg hb hbq), ?_, ?_⟩
   · rw [add_add_add_comm, ← mul_add, ← mul_add, habp, habq, mul_one, mul_one, hab]
-  · simp_rw [add_smul, mul_smul, smul_add]
-    exact add_add_add_comm _ _ _ _
+  · match_scalars <;> noncomm_ring
 
 theorem Convex.linear_image (hs : Convex 𝕜 s) (f : E →ₗ[𝕜] F) : Convex 𝕜 (f '' s) := by
   rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩ a b ha hb hab
@@ -406,8 +407,8 @@ theorem convex_openSegment (a b : E) : Convex 𝕜 (openSegment 𝕜 a b) := by
   rw [convex_iff_openSegment_subset]
   rintro p ⟨ap, bp, hap, hbp, habp, rfl⟩ q ⟨aq, bq, haq, hbq, habq, rfl⟩ z ⟨a, b, ha, hb, hab, rfl⟩
   refine ⟨a * ap + b * aq, a * bp + b * bq, by positivity, by positivity, ?_, ?_⟩
-  · rw [add_add_add_comm, ← mul_add, ← mul_add, habp, habq, mul_one, mul_one, hab]
-  · simp_rw [add_smul, mul_smul, smul_add, add_add_add_comm]
+  · linear_combination (norm := noncomm_ring) a * habp + b * habq + hab
+  · module
 
 end StrictOrderedCommSemiring
 
@@ -425,8 +426,7 @@ theorem convex_vadd (a : E) : Convex 𝕜 (a +ᵥ s) ↔ Convex 𝕜 s :=
 
 theorem Convex.add_smul_mem (hs : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ s) {t : 𝕜}
     (ht : t ∈ Icc (0 : 𝕜) 1) : x + t • y ∈ s := by
-  have h : x + t • y = (1 - t) • x + t • (x + y) := by
-    rw [smul_add, ← add_assoc, ← add_smul, sub_add_cancel, one_smul]
+  have h : x + t • y = (1 - t) • x + t • (x + y) := by match_scalars <;> noncomm_ring
   rw [h]
   exact hs hx hy (sub_nonneg_of_le ht.2) ht.1 (sub_add_cancel _ _)
 
@@ -505,7 +505,8 @@ theorem convex_iff_div :
 theorem Convex.mem_smul_of_zero_mem (h : Convex 𝕜 s) {x : E} (zero_mem : (0 : E) ∈ s) (hx : x ∈ s)
     {t : 𝕜} (ht : 1 ≤ t) : x ∈ t • s := by
   rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']
-  exact h.smul_mem_of_zero_mem zero_mem hx ⟨inv_nonneg.2 (zero_le_one.trans ht), inv_le_one ht⟩
+  exact h.smul_mem_of_zero_mem zero_mem hx
+    ⟨inv_nonneg.2 (zero_le_one.trans ht), inv_le_one_of_one_le₀ ht⟩
 
 theorem Convex.exists_mem_add_smul_eq (h : Convex 𝕜 s) {x y : E} {p q : 𝕜} (hx : x ∈ s) (hy : y ∈ s)
     (hp : 0 ≤ p) (hq : 0 ≤ q) : ∃ z ∈ s, (p + q) • z = p • x + q • y := by
@@ -515,7 +516,7 @@ theorem Convex.exists_mem_add_smul_eq (h : Convex 𝕜 s) {x y : E} {p q : 𝕜}
   · replace hpq : 0 < p + q :=
       (add_nonneg hp hq).lt_of_ne' (mt (add_eq_zero_iff_of_nonneg hp hq).1 hpq)
     refine ⟨_, convex_iff_div.1 h hx hy hp hq hpq, ?_⟩
-    simp only [smul_add, smul_smul, mul_div_cancel₀ _ hpq.ne']
+    match_scalars <;> field_simp
 
 theorem Convex.add_smul (h_conv : Convex 𝕜 s) {p q : 𝕜} (hp : 0 ≤ p) (hq : 0 ≤ q) :
     (p + q) • s = p • s + q • s := (add_smul_subset _ _ _).antisymm <| by
diff --git a/Mathlib/Analysis/Convex/Between.lean b/Mathlib/Analysis/Convex/Between.lean
index ccbae3b13d6c8..d57a70c2bf9d9 100644
--- a/Mathlib/Analysis/Convex/Between.lean
+++ b/Mathlib/Analysis/Convex/Between.lean
@@ -392,7 +392,7 @@ theorem sbtw_one_zero_iff {x : R} : Sbtw R 1 x 0 ↔ x ∈ Set.Ioo (0 : R) 1 :=
 theorem Wbtw.trans_left {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R w x y) : Wbtw R w x z := by
   rcases h₁ with ⟨t₁, ht₁, rfl⟩
   rcases h₂ with ⟨t₂, ht₂, rfl⟩
-  refine ⟨t₂ * t₁, ⟨mul_nonneg ht₂.1 ht₁.1, mul_le_one ht₂.2 ht₁.1 ht₁.2⟩, ?_⟩
+  refine ⟨t₂ * t₁, ⟨mul_nonneg ht₂.1 ht₁.1, mul_le_one₀ ht₂.2 ht₁.1 ht₁.2⟩, ?_⟩
   rw [lineMap_apply, lineMap_apply, lineMap_vsub_left, smul_smul]
 
 theorem Wbtw.trans_right {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z) : Wbtw R w y z := by
@@ -583,14 +583,14 @@ theorem wbtw_iff_left_eq_or_right_mem_image_Ici {x y z : P} :
   · rcases h with ⟨r, ⟨hr0, hr1⟩, rfl⟩
     rcases hr0.lt_or_eq with (hr0' | rfl)
     · rw [Set.mem_image]
-      refine Or.inr ⟨r⁻¹, one_le_inv hr0' hr1, ?_⟩
+      refine .inr ⟨r⁻¹, (one_le_inv₀ hr0').2 hr1, ?_⟩
       simp only [lineMap_apply, smul_smul, vadd_vsub]
       rw [inv_mul_cancel₀ hr0'.ne', one_smul, vsub_vadd]
     · simp
   · rcases h with (rfl | ⟨r, ⟨hr, rfl⟩⟩)
     · exact wbtw_self_left _ _ _
     · rw [Set.mem_Ici] at hr
-      refine ⟨r⁻¹, ⟨inv_nonneg.2 (zero_le_one.trans hr), inv_le_one hr⟩, ?_⟩
+      refine ⟨r⁻¹, ⟨inv_nonneg.2 (zero_le_one.trans hr), inv_le_one_of_one_le₀ hr⟩, ?_⟩
       simp only [lineMap_apply, smul_smul, vadd_vsub]
       rw [inv_mul_cancel₀ (one_pos.trans_le hr).ne', one_smul, vsub_vadd]
 
@@ -655,7 +655,7 @@ theorem Sbtw.left_mem_affineSpan {x y z : P} (h : Sbtw R x y z) : x ∈ line[R,
 
 theorem wbtw_smul_vadd_smul_vadd_of_nonneg_of_le (x : P) (v : V) {r₁ r₂ : R} (hr₁ : 0 ≤ r₁)
     (hr₂ : r₁ ≤ r₂) : Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x) := by
-  refine ⟨r₁ / r₂, ⟨div_nonneg hr₁ (hr₁.trans hr₂), div_le_one_of_le hr₂ (hr₁.trans hr₂)⟩, ?_⟩
+  refine ⟨r₁ / r₂, ⟨div_nonneg hr₁ (hr₁.trans hr₂), div_le_one_of_le₀ hr₂ (hr₁.trans hr₂)⟩, ?_⟩
   by_cases h : r₁ = 0; · simp [h]
   simp [lineMap_apply, smul_smul, ((hr₁.lt_of_ne' h).trans_le hr₂).ne.symm]
 
@@ -695,8 +695,8 @@ theorem Wbtw.trans_left_right {w x y z : P} (h₁ : Wbtw R w y z) (h₂ : Wbtw R
   refine
     ⟨(t₁ - t₂ * t₁) / (1 - t₂ * t₁),
       ⟨div_nonneg (sub_nonneg.2 (mul_le_of_le_one_left ht₁.1 ht₂.2))
-          (sub_nonneg.2 (mul_le_one ht₂.2 ht₁.1 ht₁.2)),
-        div_le_one_of_le (sub_le_sub_right ht₁.2 _) (sub_nonneg.2 (mul_le_one ht₂.2 ht₁.1 ht₁.2))⟩,
+          (sub_nonneg.2 (mul_le_one₀ ht₂.2 ht₁.1 ht₁.2)), div_le_one_of_le₀
+            (sub_le_sub_right ht₁.2 _) (sub_nonneg.2 (mul_le_one₀ ht₂.2 ht₁.1 ht₁.2))⟩,
       ?_⟩
   simp only [lineMap_apply, smul_smul, ← add_vadd, vsub_vadd_eq_vsub_sub, smul_sub, ← sub_smul,
     ← add_smul, vadd_vsub, vadd_right_cancel_iff, div_mul_eq_mul_div, div_sub_div_same]
@@ -772,7 +772,7 @@ theorem wbtw_iff_sameRay_vsub {x y z : P} : Wbtw R x y z ↔ SameRay R (y -ᵥ x
   · refine
       ⟨r₂ / (r₁ + r₂),
         ⟨div_nonneg hr₂.le (add_nonneg hr₁.le hr₂.le),
-          div_le_one_of_le (le_add_of_nonneg_left hr₁.le) (add_nonneg hr₁.le hr₂.le)⟩,
+          div_le_one_of_le₀ (le_add_of_nonneg_left hr₁.le) (add_nonneg hr₁.le hr₂.le)⟩,
         ?_⟩
     have h' : z = r₂⁻¹ • r₁ • (y -ᵥ x) +ᵥ y := by simp [h, hr₂.ne']
     rw [eq_comm]
diff --git a/Mathlib/Analysis/Convex/Birkhoff.lean b/Mathlib/Analysis/Convex/Birkhoff.lean
new file mode 100644
index 0000000000000..a754f5db135c9
--- /dev/null
+++ b/Mathlib/Analysis/Convex/Birkhoff.lean
@@ -0,0 +1,172 @@
+/-
+Copyright (c) 2024 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+-/
+
+import Mathlib.Analysis.Convex.Combination
+import Mathlib.Combinatorics.Hall.Basic
+import Mathlib.Data.Matrix.DoublyStochastic
+import Mathlib.Tactic.Linarith
+
+/-!
+# Birkhoff's theorem
+
+## Main statements
+
+* `doublyStochastic_eq_sum_perm`: If `M` is a doubly stochastic matrix, then it is a convex
+  combination of permutation matrices.
+* `doublyStochastic_eq_convexHull_perm`: The set of doubly stochastic matrices is the convex hull
+  of the permutation matrices.
+
+## TODO
+
+* Show that the extreme points of doubly stochastic matrices are the permutation matrices.
+* Show that for `x y : n → R`, `x` is majorized by `y` if and only if there is a doubly stochastic
+  matrix `M` such that `M *ᵥ y = x`.
+
+## Tags
+
+Doubly stochastic, Birkhoff's theorem, Birkhoff-von Neumann theorem
+-/
+
+open Finset Function Matrix
+
+variable {R n : Type*} [Fintype n] [DecidableEq n]
+
+section LinearOrderedSemifield
+
+variable [LinearOrderedSemifield R] {M : Matrix n n R}
+
+/--
+If M is a positive scalar multiple of a doubly stochastic matrix, then there is a permutation matrix
+whose support is contained in the support of M.
+-/
+private lemma exists_perm_eq_zero_implies_eq_zero [Nonempty n] {s : R} (hs : 0 < s)
+    (hM : ∃ M' ∈ doublyStochastic R n, M = s • M') :
+    ∃ σ : Equiv.Perm n, ∀ i j, M i j = 0 → σ.permMatrix R i j = 0 := by
+  rw [exists_mem_doublyStochastic_eq_smul_iff hs.le] at hM
+  let f (i : n) : Finset n := univ.filter (M i · ≠ 0)
+  have hf (A : Finset n) : A.card ≤ (A.biUnion f).card := by
+    have (i) : ∑ j ∈ f i, M i j = s := by simp [sum_subset (filter_subset _ _), hM.2.1]
+    have h₁ : ∑ i ∈ A, ∑ j ∈ f i, M i j = A.card * s := by simp [this]
+    have h₂ : ∑ i, ∑ j ∈ A.biUnion f, M i j = (A.biUnion f).card * s := by
+      simp [sum_comm (t := A.biUnion f), hM.2.2, mul_comm s]
+    suffices A.card * s ≤ (A.biUnion f).card * s by exact_mod_cast le_of_mul_le_mul_right this hs
+    rw [← h₁, ← h₂]
+    trans ∑ i ∈ A, ∑ j ∈ A.biUnion f, M i j
+    · refine sum_le_sum fun i hi => ?_
+      exact sum_le_sum_of_subset_of_nonneg (subset_biUnion_of_mem f hi) (by simp [*])
+    · exact sum_le_sum_of_subset_of_nonneg (by simp) fun _ _ _ => sum_nonneg fun j _ => hM.1 _ _
+  obtain ⟨g, hg, hg'⟩ := (all_card_le_biUnion_card_iff_exists_injective f).1 hf
+  rw [Finite.injective_iff_bijective] at hg
+  refine ⟨Equiv.ofBijective g hg, fun i j hij => ?_⟩
+  simp only [PEquiv.toMatrix_apply, Option.mem_def, ite_eq_right_iff, one_ne_zero, imp_false,
+    Equiv.toPEquiv_apply, Equiv.ofBijective_apply, Option.some.injEq]
+  rintro rfl
+  simpa [f, hij] using hg' i
+
+end LinearOrderedSemifield
+
+section LinearOrderedField
+
+variable [LinearOrderedField R] {M : Matrix n n R}
+
+/--
+If M is a scalar multiple of a doubly stochastic matrix, then it is a conical combination of
+permutation matrices. This is most useful when M is a doubly stochastic matrix, in which case
+the combination is convex.
+
+This particular formulation is chosen to make the inductive step easier: we no longer need to
+rescale each time a permutation matrix is subtracted.
+-/
+private lemma doublyStochastic_sum_perm_aux (M : Matrix n n R)
+    (s : R) (hs : 0 ≤ s)
+    (hM : ∃ M' ∈ doublyStochastic R n, M = s • M') :
+    ∃ w : Equiv.Perm n → R, (∀ σ, 0 ≤ w σ) ∧ ∑ σ, w σ • σ.permMatrix R = M := by
+  rcases isEmpty_or_nonempty n
+  case inl => exact ⟨1, by simp, Subsingleton.elim _ _⟩
+  set d : ℕ := (Finset.univ.filter fun i : n × n => M i.1 i.2 ≠ 0).card with ← hd
+  clear_value d
+  induction d using Nat.strongRecOn generalizing M s
+  case ind d ih =>
+  rcases eq_or_lt_of_le hs with rfl | hs'
+  case inl =>
+    use 0
+    simp only [zero_smul, exists_and_right] at hM
+    simp [hM]
+  obtain ⟨σ, hσ⟩ := exists_perm_eq_zero_implies_eq_zero hs' hM
+  obtain ⟨i, hi, hi'⟩ := exists_min_image _ (fun i => M i (σ i)) univ_nonempty
+  rw [exists_mem_doublyStochastic_eq_smul_iff hs] at hM
+  let N : Matrix n n R := M - M i (σ i) • σ.permMatrix R
+  have hMi' : 0 < M i (σ i) := (hM.1 _ _).lt_of_ne' fun h => by
+    simpa [Equiv.toPEquiv_apply] using hσ _ _ h
+  let s' : R := s - M i (σ i)
+  have hs' : 0 ≤ s' := by
+    simp only [s', sub_nonneg, ← hM.2.1 i]
+    exact single_le_sum (fun j _ => hM.1 i j) (by simp)
+  have : ∃ M' ∈ doublyStochastic R n, N = s' • M' := by
+    rw [exists_mem_doublyStochastic_eq_smul_iff hs']
+    simp only [sub_apply, smul_apply, PEquiv.toMatrix_apply, Equiv.toPEquiv_apply, Option.mem_def,
+      Option.some.injEq, smul_eq_mul, mul_ite, mul_one, mul_zero, sub_nonneg,
+      sum_sub_distrib, sum_ite_eq, mem_univ, ↓reduceIte, N]
+    refine ⟨fun i' j => ?_, by simp [hM.2.1], by simp [← σ.eq_symm_apply, hM]⟩
+    split
+    case isTrue h => exact (hi' i' (by simp)).trans_eq (by rw [h])
+    case isFalse h => exact hM.1 _ _
+  have hd' : (univ.filter fun i : n × n => N i.1 i.2 ≠ 0).card < d := by
+    rw [← hd]
+    refine card_lt_card ?_
+    rw [ssubset_iff_of_subset (monotone_filter_right _ _)]
+    · simp only [ne_eq, mem_filter, mem_univ, true_and, Decidable.not_not, Prod.exists]
+      refine ⟨i, σ i, hMi'.ne', ?_⟩
+      simp [N, Equiv.toPEquiv_apply]
+    · rintro ⟨i', j'⟩ hN' hM'
+      dsimp at hN' hM'
+      simp only [sub_apply, hM', smul_apply, PEquiv.toMatrix_apply, Equiv.toPEquiv_apply,
+        Option.mem_def, Option.some.injEq, smul_eq_mul, mul_ite, mul_one, mul_zero, zero_sub,
+        neg_eq_zero, ite_eq_right_iff, Classical.not_imp, N] at hN'
+      obtain ⟨rfl, _⟩ := hN'
+      linarith [hi' i' (by simp)]
+  obtain ⟨w, hw, hw'⟩ := ih _ hd' _ s' hs' this rfl
+  refine ⟨w + fun σ' => if σ' = σ then M i (σ i) else 0, ?_⟩
+  simp only [Pi.add_apply, add_smul, sum_add_distrib, hw', ite_smul, zero_smul,
+    sum_ite_eq', mem_univ, ↓reduceIte, N, sub_add_cancel, and_true]
+  intro σ'
+  split <;> simp [add_nonneg, hw, hM.1]
+
+/--
+If M is a doubly stochastic matrix, then it is an convex combination of permutation matrices. Note
+`doublyStochastic_eq_convexHull_permMatrix` shows `doublyStochastic n` is exactly the convex hull of
+the permutation matrices, and this lemma is instead most useful for accessing the coefficients of
+each permutation matrices directly.
+-/
+lemma exists_eq_sum_perm_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) :
+    ∃ w : Equiv.Perm n → R, (∀ σ, 0 ≤ w σ) ∧ ∑ σ, w σ = 1 ∧ ∑ σ, w σ • σ.permMatrix R = M := by
+  rcases isEmpty_or_nonempty n
+  case inl => exact ⟨fun _ => 1, by simp, by simp, Subsingleton.elim _ _⟩
+  obtain ⟨w, hw1, hw3⟩ := doublyStochastic_sum_perm_aux M 1 (by simp) ⟨M, hM, by simp⟩
+  refine ⟨w, hw1, ?_, hw3⟩
+  inhabit n
+  have : ∑ j, ∑ σ : Equiv.Perm n, w σ • σ.permMatrix R default j = 1 := by
+    simp only [← smul_apply (m := n), ← Finset.sum_apply, hw3]
+    rw [sum_row_of_mem_doublyStochastic hM]
+  simpa [sum_comm (γ := n), Equiv.toPEquiv_apply] using this
+
+/--
+**Birkhoff's theorem**
+The set of doubly stochastic matrices is the convex hull of the permutation matrices.  Note
+`exists_eq_sum_perm_of_mem_doublyStochastic` gives a convex weighting of each permutation matrix
+directly.  To show `doublyStochastic n` is convex, use `convex_doublyStochastic`.
+-/
+theorem doublyStochastic_eq_convexHull_permMatrix :
+    doublyStochastic R n = convexHull R {σ.permMatrix R | σ : Equiv.Perm n} := by
+  refine (convexHull_min ?g1 convex_doublyStochastic).antisymm' fun M hM => ?g2
+  case g1 =>
+    rintro x ⟨h, rfl⟩
+    exact permMatrix_mem_doublyStochastic
+  case g2 =>
+    obtain ⟨w, hw1, hw2, hw3⟩ := exists_eq_sum_perm_of_mem_doublyStochastic hM
+    exact mem_convexHull_of_exists_fintype w (·.permMatrix R) hw1 hw2 (by simp) hw3
+
+end LinearOrderedField
diff --git a/Mathlib/Analysis/Convex/Body.lean b/Mathlib/Analysis/Convex/Body.lean
index ab8c493ff10c3..219c7807cb05d 100644
--- a/Mathlib/Analysis/Convex/Body.lean
+++ b/Mathlib/Analysis/Convex/Body.lean
@@ -212,7 +212,7 @@ theorem iInter_smul_eq_self [T2Space V] {u : ℕ → ℝ≥0} (K : ConvexBody V)
     rw [show (1 + u n : ℝ) • y - y = (u n : ℝ) • y by rw [add_smul, one_smul, add_sub_cancel_left],
       norm_smul, Real.norm_eq_abs]
     specialize hn n le_rfl
-    rw [_root_.lt_div_iff' hC_pos, mul_comm, NNReal.coe_zero, sub_zero, Real.norm_eq_abs] at hn
+    rw [lt_div_iff₀' hC_pos, mul_comm, NNReal.coe_zero, sub_zero, Real.norm_eq_abs] at hn
     refine lt_of_le_of_lt ?_ hn
     exact mul_le_mul_of_nonneg_left (hC_bdd _ hyK) (abs_nonneg _)
   · refine Set.mem_iInter.mpr (fun n => Convex.mem_smul_of_zero_mem K.convex h_zero h ?_)
diff --git a/Mathlib/Analysis/Convex/Caratheodory.lean b/Mathlib/Analysis/Convex/Caratheodory.lean
index 75f0b5f68dfeb..5d81223dc517b 100644
--- a/Mathlib/Analysis/Convex/Caratheodory.lean
+++ b/Mathlib/Analysis/Convex/Caratheodory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison
+Authors: Johan Commelin, Kim Morrison
 -/
 import Mathlib.Analysis.Convex.Combination
 import Mathlib.LinearAlgebra.AffineSpace.Independent
@@ -85,7 +85,7 @@ theorem mem_convexHull_erase [DecidableEq E] {t : Finset E} (h : ¬AffineIndepen
         _ ≤ 0 := by
           apply mul_nonpos_of_nonneg_of_nonpos
           · apply div_nonneg (fpos i₀ (mem_of_subset (filter_subset _ t) mem)) (le_of_lt hg)
-          · simpa only [s, mem_filter, het, true_and_iff, not_lt] using hes
+          · simpa only [s, mem_filter, het, true_and, not_lt] using hes
         _ ≤ f e := fpos e het
   · rw [Subtype.coe_mk, centerMass_eq_of_sum_1 _ id ksum]
     calc
diff --git a/Mathlib/Analysis/Convex/Combination.lean b/Mathlib/Analysis/Convex/Combination.lean
index 20fcc69a29fb2..399464ada7406 100644
--- a/Mathlib/Analysis/Convex/Combination.lean
+++ b/Mathlib/Analysis/Convex/Combination.lean
@@ -50,7 +50,8 @@ theorem Finset.centerMass_empty : (∅ : Finset ι).centerMass w z = 0 := by
 
 theorem Finset.centerMass_pair (hne : i ≠ j) :
     ({i, j} : Finset ι).centerMass w z = (w i / (w i + w j)) • z i + (w j / (w i + w j)) • z j := by
-  simp only [centerMass, sum_pair hne, smul_add, (mul_smul _ _ _).symm, div_eq_inv_mul]
+  simp only [centerMass, sum_pair hne]
+  module
 
 variable {w}
 
@@ -63,7 +64,9 @@ theorem Finset.centerMass_insert (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0) :
   rw [div_mul_eq_mul_div, mul_inv_cancel₀ hw, one_div]
 
 theorem Finset.centerMass_singleton (hw : w i ≠ 0) : ({i} : Finset ι).centerMass w z = z i := by
-  rw [centerMass, sum_singleton, sum_singleton, ← mul_smul, inv_mul_cancel₀ hw, one_smul]
+  rw [centerMass, sum_singleton, sum_singleton]
+  match_scalars
+  field_simp
 
 @[simp] lemma Finset.centerMass_neg_left : t.centerMass (-w) z = t.centerMass w z := by
   simp [centerMass, inv_neg]
@@ -124,7 +127,7 @@ theorem Finset.centerMass_subset {t' : Finset ι} (ht : t ⊆ t') (h : ∀ i ∈
 theorem Finset.centerMass_filter_ne_zero :
     (t.filter fun i => w i ≠ 0).centerMass w z = t.centerMass w z :=
   Finset.centerMass_subset z (filter_subset _ _) fun i hit hit' => by
-    simpa only [hit, mem_filter, true_and_iff, Ne, Classical.not_not] using hit'
+    simpa only [hit, mem_filter, true_and, Ne, Classical.not_not] using hit'
 
 namespace Finset
 
@@ -481,7 +484,7 @@ theorem Set.Finite.convexHull_eq_image {s : Set E} (hs : s.Finite) : convexHull
   letI := hs.fintype
   rw [← convexHull_basis_eq_stdSimplex, LinearMap.image_convexHull, ← Set.range_comp]
   apply congr_arg
-  simp_rw [Function.comp]
+  simp_rw [Function.comp_def]
   convert Subtype.range_coe.symm
   simp [LinearMap.sum_apply, ite_smul, Finset.filter_eq, Finset.mem_univ]
 
diff --git a/Mathlib/Analysis/Convex/Cone/InnerDual.lean b/Mathlib/Analysis/Convex/Cone/InnerDual.lean
index 555882971d866..8bf4feeae7d46 100644
--- a/Mathlib/Analysis/Convex/Cone/InnerDual.lean
+++ b/Mathlib/Analysis/Convex/Cone/InnerDual.lean
@@ -147,6 +147,7 @@ section CompleteSpace
 
 variable [CompleteSpace H]
 
+open scoped InnerProductSpace in
 /-- This is a stronger version of the Hahn-Banach separation theorem for closed convex cones. This
 is also the geometric interpretation of Farkas' lemma. -/
 theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem (K : ConvexCone ℝ H)
@@ -181,7 +182,7 @@ theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem (K : Co
       _ = ⟪b - z, b - z + z⟫_ℝ := (inner_add_right _ _ _).symm
       _ = ⟪b - z, b⟫_ℝ := by rw [sub_add_cancel]
 
-/-- The inner dual of inner dual of a non-empty, closed convex cone is itself.  -/
+/-- The inner dual of inner dual of a non-empty, closed convex cone is itself. -/
 theorem ConvexCone.innerDualCone_of_innerDualCone_eq_self (K : ConvexCone ℝ H)
     (ne : (K : Set H).Nonempty) (hc : IsClosed (K : Set H)) :
     ((K : Set H).innerDualCone : Set H).innerDualCone = K := by
diff --git a/Mathlib/Analysis/Convex/Cone/Pointed.lean b/Mathlib/Analysis/Convex/Cone/Pointed.lean
index 80484baf0da84..b8353729d6675 100644
--- a/Mathlib/Analysis/Convex/Cone/Pointed.lean
+++ b/Mathlib/Analysis/Convex/Cone/Pointed.lean
@@ -186,6 +186,7 @@ def dual (S : PointedCone ℝ E) : PointedCone ℝ E :=
 theorem toConvexCone_dual (S : PointedCone ℝ E) : ↑(dual S) = (S : Set E).innerDualCone :=
   rfl
 
+open scoped InnerProductSpace in
 @[simp]
 theorem mem_dual {S : PointedCone ℝ E} {y : E} : y ∈ dual S ↔ ∀ ⦃x⦄, x ∈ S → 0 ≤ ⟪x, y⟫_ℝ := by
   rfl
diff --git a/Mathlib/Analysis/Convex/Cone/Proper.lean b/Mathlib/Analysis/Convex/Cone/Proper.lean
index fe4f8e9acd54a..a2ebcdc0f1ef2 100644
--- a/Mathlib/Analysis/Convex/Cone/Proper.lean
+++ b/Mathlib/Analysis/Convex/Cone/Proper.lean
@@ -170,6 +170,7 @@ def dual (K : ProperCone ℝ E) : ProperCone ℝ E where
 theorem coe_dual (K : ProperCone ℝ E) : K.dual = (K : Set E).innerDualCone :=
   rfl
 
+open scoped InnerProductSpace in
 @[simp]
 theorem mem_dual {K : ProperCone ℝ E} {y : E} : y ∈ dual K ↔ ∀ ⦃x⦄, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ := by
   aesop
@@ -201,6 +202,8 @@ end InnerProductSpace
 
 section CompleteSpace
 
+open scoped InnerProductSpace
+
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
 variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F]
 
diff --git a/Mathlib/Analysis/Convex/Continuous.lean b/Mathlib/Analysis/Convex/Continuous.lean
new file mode 100644
index 0000000000000..4b2ebd632b395
--- /dev/null
+++ b/Mathlib/Analysis/Convex/Continuous.lean
@@ -0,0 +1,232 @@
+/-
+Copyright (c) 2023 Yaël Dillies, Zichen Wang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies, Zichen Wang
+-/
+import Mathlib.Analysis.Convex.Normed
+
+/-!
+# Convex functions are continuous
+
+This file proves that a convex function from a finite dimensional real normed space to `ℝ` is
+continuous.
+-/
+
+open FiniteDimensional Metric Set List Bornology
+open scoped Topology
+
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
+  {C : Set E} {f : E → ℝ} {x₀ : E} {ε r r' M : ℝ}
+
+lemma ConvexOn.lipschitzOnWith_of_abs_le (hf : ConvexOn ℝ (ball x₀ r) f) (hε : 0 < ε)
+    (hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
+    LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε)) := by
+  set K := 2 * M / ε with hK
+  have oneside {x y : E} (hx : x ∈ ball x₀ (r - ε)) (hy : y ∈ ball x₀ (r - ε)) :
+      f x - f y ≤ K * ‖x - y‖ := by
+    obtain rfl | hxy := eq_or_ne x y
+    · simp
+    have hx₀r : ball x₀ (r - ε) ⊆ ball x₀ r := ball_subset_ball <| by linarith
+    have hx' : x ∈ ball x₀ r := hx₀r hx
+    have hy' : y ∈ ball x₀ r := hx₀r hy
+    let z := x + (ε / ‖x - y‖) • (x - y)
+    replace hxy : 0 < ‖x - y‖ := by rwa [norm_sub_pos_iff]
+    have hz : z ∈ ball x₀ r := mem_ball_iff_norm.2 <| by
+      calc
+        _ = ‖(x - x₀) + (ε / ‖x - y‖) • (x - y)‖ := by simp only [z, add_sub_right_comm]
+        _ ≤ ‖x - x₀‖ + ‖(ε / ‖x - y‖) • (x - y)‖ := norm_add_le ..
+        _ < r - ε + ε :=
+          add_lt_add_of_lt_of_le (mem_ball_iff_norm.1 hx) <| by
+            simp [norm_smul, abs_of_nonneg, hε.le, hxy.ne']
+        _ = r := by simp
+    let a := ε / (ε + ‖x - y‖)
+    let b := ‖x - y‖ / (ε + ‖x - y‖)
+    have hab : a + b = 1 := by field_simp [a, b]
+    have hxyz : x = a • y + b • z := by
+      calc
+        x = a • x + b • x := by rw [Convex.combo_self hab]
+        _ = a • y + b • z := by simp [z, a, b, smul_smul, hxy.ne', smul_sub]; abel
+    rw [hK, mul_comm, ← mul_div_assoc, le_div_iff₀' hε]
+    calc
+      ε * (f x - f y) ≤ ‖x - y‖ * (f z - f x) := by
+        rw [mul_sub, mul_sub, sub_le_sub_iff, ← add_mul]
+        have h := hf.2 hy' hz (by positivity) (by positivity) hab
+        field_simp [← hxyz, a, b, ← mul_div_right_comm] at h
+        rwa [← le_div_iff₀' (by positivity), add_comm (_ * _)]
+      _ ≤ _ := by
+        rw [sub_eq_add_neg (f _), two_mul]
+        gcongr
+        · exact (le_abs_self _).trans <| hM _ hz
+        · exact (neg_le_abs _).trans <| hM _ hx'
+  refine .of_dist_le' fun x hx y hy ↦ ?_
+  simp_rw [dist_eq_norm_sub, Real.norm_eq_abs, abs_sub_le_iff]
+  exact ⟨oneside hx hy, norm_sub_rev x _ ▸ oneside hy hx⟩
+
+lemma ConcaveOn.lipschitzOnWith_of_abs_le (hf : ConcaveOn ℝ (ball x₀ r) f) (hε : 0 < ε)
+    (hM : ∀ a, dist a x₀ < r → |f a| ≤ M) :
+    LipschitzOnWith (2 * M / ε).toNNReal f (ball x₀ (r - ε)) := by
+  simpa using hf.neg.lipschitzOnWith_of_abs_le hε <| by simpa using hM
+
+lemma ConvexOn.exists_lipschitzOnWith_of_isBounded (hf : ConvexOn ℝ (ball x₀ r) f) (hr : r' < r)
+    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r') := by
+  rw [isBounded_iff_subset_ball 0] at hf'
+  simp only [Set.subset_def, mem_image, mem_ball, dist_zero_right, Real.norm_eq_abs,
+    forall_exists_index, and_imp, forall_apply_eq_imp_iff₂] at hf'
+  obtain ⟨M, hM⟩ := hf'
+  rw [← sub_sub_cancel r r']
+  exact ⟨_, hf.lipschitzOnWith_of_abs_le (sub_pos.2 hr) fun a ha ↦ (hM a ha).le⟩
+
+lemma ConcaveOn.exists_lipschitzOnWith_of_isBounded (hf : ConcaveOn ℝ (ball x₀ r) f) (hr : r' < r)
+    (hf' : IsBounded (f '' ball x₀ r)) : ∃ K, LipschitzOnWith K f (ball x₀ r') := by
+  replace hf' : IsBounded ((-f) '' ball x₀ r) := by convert hf'.neg; ext; simp [neg_eq_iff_eq_neg]
+  simpa using hf.neg.exists_lipschitzOnWith_of_isBounded hr hf'
+
+lemma ConvexOn.isBoundedUnder_abs (hf : ConvexOn ℝ C f) {x₀ : E} (hC : C ∈ 𝓝 x₀) :
+    (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f| ↔ (𝓝 x₀).IsBoundedUnder (· ≤ ·) f := by
+  refine ⟨fun h ↦ h.mono_le <| .of_forall fun x ↦ le_abs_self _, ?_⟩
+  rintro ⟨r, hr⟩
+  refine ⟨|r| + 2 * |f x₀|, ?_⟩
+  have : (𝓝 x₀).Tendsto (fun y => 2 • x₀ - y) (𝓝 x₀) :=
+    tendsto_nhds_nhds.2 (⟨·, ·, by simp [two_nsmul, dist_comm]⟩)
+  simp only [Filter.eventually_map, Pi.abs_apply, abs_le'] at hr ⊢
+  filter_upwards [this.eventually_mem hC, hC, hr, this.eventually hr] with y hx hx' hfr hfr'
+  refine ⟨hfr.trans <| (le_abs_self _).trans <| by simp, ?_⟩
+  rw [← sub_le_iff_le_add, neg_sub_comm, sub_le_iff_le_add', ← abs_two, ← abs_mul]
+  calc
+    -|2 * f x₀| ≤ 2 * f x₀ := neg_abs_le _
+    _ ≤ f y + f (2 • x₀ - y) := by
+      have := hf.2 hx' hx (by positivity) (by positivity) (add_halves _)
+      simp only [one_div, ← Nat.cast_smul_eq_nsmul ℝ, Nat.cast_ofNat, smul_sub, ne_eq,
+        OfNat.ofNat_ne_zero, not_false_eq_true, inv_smul_smul₀, add_sub_cancel, smul_eq_mul] at this
+      cancel_denoms at this
+      rwa [← Nat.cast_two, Nat.cast_smul_eq_nsmul] at this
+    _ ≤ f y + |r| := by gcongr; exact hfr'.trans (le_abs_self _)
+
+lemma ConcaveOn.isBoundedUnder_abs (hf : ConcaveOn ℝ C f) {x₀ : E} (hC : C ∈ 𝓝 x₀) :
+    (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f| ↔ (𝓝 x₀).IsBoundedUnder (· ≥ ·) f := by
+  simpa [Pi.neg_def, Pi.abs_def] using hf.neg.isBoundedUnder_abs hC
+
+lemma ConvexOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConvexOn ℝ C f) : TFAE [
+    LocallyLipschitzOn C f,
+    ContinuousOn f C,
+    ∃ x₀ ∈ C, ContinuousAt f x₀,
+    ∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
+    ∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) f,
+    ∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f|] := by
+  tfae_have 1 → 2
+  · exact LocallyLipschitzOn.continuousOn
+  tfae_have 2 → 3
+  · obtain ⟨x₀, hx₀⟩ := hC'
+    exact fun h ↦ ⟨x₀, hx₀, h.continuousAt <| hC.mem_nhds hx₀⟩
+  tfae_have 3 → 4
+  · rintro ⟨x₀, hx₀, h⟩
+    exact ⟨x₀, hx₀, f x₀ + 1, by simpa using h.eventually (eventually_le_nhds (by simp))⟩
+  tfae_have 4 → 5
+  · rintro ⟨x₀, hx₀, r, hr⟩ x hx
+    have : ∀ᶠ δ in 𝓝 (0 : ℝ), (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀ ∈ C := by
+      have h : ContinuousAt (fun δ : ℝ ↦ (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀) 0 := by
+        fun_prop (disch := norm_num)
+      exact h (by simpa using hC.mem_nhds hx)
+    obtain ⟨δ, hδ₀, hy, hδ₁⟩ := (this.and <| eventually_lt_nhds zero_lt_one).exists_gt
+    set y := (1 - δ)⁻¹ • x - (δ / (1 - δ)) • x₀
+    refine ⟨max r (f y), ?_⟩
+    simp only [Filter.eventually_map, Pi.abs_apply] at hr ⊢
+    obtain ⟨ε, hε, hr⟩ := Metric.eventually_nhds_iff.1 <| hr.and (hC.eventually_mem hx₀)
+    refine Metric.eventually_nhds_iff.2 ⟨ε * δ, by positivity, fun z hz ↦ ?_⟩
+    have hx₀' : δ⁻¹ • (x - y) + y = x₀ := MulAction.injective₀ (sub_ne_zero.2 hδ₁.ne') <| by
+      simp [y, smul_sub, smul_smul, hδ₀.ne', div_eq_mul_inv, sub_ne_zero.2 hδ₁.ne', mul_left_comm,
+        sub_mul, sub_smul]
+    let w := δ⁻¹ • (z - y) + y
+    have hwyz : δ • w + (1 - δ) • y = z := by simp [w, hδ₀.ne', sub_smul]
+    have hw : dist w x₀ < ε := by
+      simpa [w, ← hx₀', dist_smul₀, abs_of_nonneg, hδ₀.le, inv_mul_lt_iff₀', hδ₀]
+    calc
+      f z ≤ max (f w) (f y) :=
+        hf.le_max_of_mem_segment (hr hw).2 hy ⟨_, _, hδ₀.le, sub_nonneg.2 hδ₁.le, by simp, hwyz⟩
+      _ ≤ max r (f y) := by gcongr; exact (hr hw).1
+  tfae_have 6 ↔ 5
+  · exact forall₂_congr fun x₀ hx₀ ↦ hf.isBoundedUnder_abs (hC.mem_nhds hx₀)
+  tfae_have 6 → 1
+  · rintro h x hx
+    obtain ⟨r, hr⟩ := h hx
+    obtain ⟨ε, hε, hεD⟩ := Metric.mem_nhds_iff.1 <| Filter.inter_mem (hC.mem_nhds hx) hr
+    simp only [preimage_setOf_eq, Pi.abs_apply, subset_inter_iff, hC.nhdsWithin_eq hx] at hεD ⊢
+    obtain ⟨K, hK⟩ := exists_lipschitzOnWith_of_isBounded (hf.subset hεD.1 (convex_ball ..))
+      (half_lt_self hε) <| isBounded_iff_forall_norm_le.2 ⟨r, by simpa using hεD.2⟩
+    exact ⟨K, _, ball_mem_nhds _ (by simpa), hK⟩
+  tfae_finish
+
+lemma ConcaveOn.continuousOn_tfae (hC : IsOpen C) (hC' : C.Nonempty) (hf : ConcaveOn ℝ C f) : TFAE [
+    LocallyLipschitzOn C f,
+    ContinuousOn f C,
+    ∃ x₀ ∈ C, ContinuousAt f x₀,
+    ∃ x₀ ∈ C, (𝓝 x₀).IsBoundedUnder (· ≥ ·) f,
+    ∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≥ ·) f,
+    ∀ ⦃x₀⦄, x₀ ∈ C → (𝓝 x₀).IsBoundedUnder (· ≤ ·) |f|] := by
+  have := hf.neg.continuousOn_tfae hC hC'
+  simp at this
+  convert this using 8 <;> exact (Equiv.neg ℝ).exists_congr (by simp)
+
+lemma ConvexOn.locallyLipschitzOn_iff_continuousOn (hC : IsOpen C) (hf : ConvexOn ℝ C f) :
+    LocallyLipschitzOn C f ↔ ContinuousOn f C := by
+  obtain rfl | hC' := C.eq_empty_or_nonempty
+  · simp
+  · exact (hf.continuousOn_tfae hC hC').out 0 1
+
+lemma ConcaveOn.locallyLipschitzOn_iff_continuousOn (hC : IsOpen C) (hf : ConcaveOn ℝ C f) :
+    LocallyLipschitzOn C f ↔ ContinuousOn f C := by
+  simpa using hf.neg.locallyLipschitzOn_iff_continuousOn hC
+
+variable [FiniteDimensional ℝ E]
+
+protected lemma ConvexOn.locallyLipschitzOn (hC : IsOpen C) (hf : ConvexOn ℝ C f) :
+    LocallyLipschitzOn C f := by
+  obtain rfl | ⟨x₀, hx₀⟩ := C.eq_empty_or_nonempty
+  · simp
+  · obtain ⟨b, hx₀b, hbC⟩ := exists_mem_interior_convexHull_affineBasis (hC.mem_nhds hx₀)
+    refine ((hf.continuousOn_tfae hC ⟨x₀, hx₀⟩).out 3 0).mp ?_
+    refine ⟨x₀, hx₀, BddAbove.isBoundedUnder (IsOpen.mem_nhds isOpen_interior hx₀b) ?_⟩
+    exact (hf.bddAbove_convexHull ((subset_convexHull ..).trans hbC)
+      ((finite_range _).image _).bddAbove).mono (by gcongr; exact interior_subset)
+
+protected lemma ConcaveOn.locallyLipschitzOn (hC : IsOpen C) (hf : ConcaveOn ℝ C f) :
+    LocallyLipschitzOn C f := by simpa using hf.neg.locallyLipschitzOn hC
+
+protected lemma ConvexOn.continuousOn (hC : IsOpen C) (hf : ConvexOn ℝ C f) :
+    ContinuousOn f C := (hf.locallyLipschitzOn hC).continuousOn
+
+protected lemma ConcaveOn.continuousOn (hC : IsOpen C) (hf : ConcaveOn ℝ C f) :
+    ContinuousOn f C := (hf.locallyLipschitzOn hC).continuousOn
+
+lemma ConvexOn.locallyLipschitzOn_interior (hf : ConvexOn ℝ C f) :
+    LocallyLipschitzOn (interior C) f :=
+  (hf.subset interior_subset hf.1.interior).locallyLipschitzOn isOpen_interior
+
+lemma ConcaveOn.locallyLipschitzOn_interior (hf : ConcaveOn ℝ C f) :
+    LocallyLipschitzOn (interior C) f :=
+  (hf.subset interior_subset hf.1.interior).locallyLipschitzOn isOpen_interior
+
+lemma ConvexOn.continuousOn_interior (hf : ConvexOn ℝ C f) : ContinuousOn f (interior C) :=
+  hf.locallyLipschitzOn_interior.continuousOn
+
+lemma ConcaveOn.continuousOn_interior (hf : ConcaveOn ℝ C f) : ContinuousOn f (interior C) :=
+  hf.locallyLipschitzOn_interior.continuousOn
+
+protected lemma ConvexOn.locallyLipschitz (hf : ConvexOn ℝ univ f) : LocallyLipschitz f := by
+  simpa using hf.locallyLipschitzOn_interior
+
+protected lemma ConcaveOn.locallyLipschitz (hf : ConcaveOn ℝ univ f) : LocallyLipschitz f := by
+  simpa using hf.locallyLipschitzOn_interior
+
+-- Commented out since `intrinsicInterior` is not imported (but should be once these are proved)
+-- proof_wanted ConvexOn.locallyLipschitzOn_intrinsicInterior (hf : ConvexOn ℝ C f) :
+--     ContinuousOn f (intrinsicInterior ℝ C)
+
+-- proof_wanted ConcaveOn.locallyLipschitzOn_intrinsicInterior (hf : ConcaveOn ℝ C f) :
+--     ContinuousOn f (intrinsicInterior ℝ C)
+
+-- proof_wanted ConvexOn.continuousOn_intrinsicInterior (hf : ConvexOn ℝ C f) :
+--     ContinuousOn f (intrinsicInterior ℝ C)
+
+-- proof_wanted ConcaveOn.continuousOn_intrinsicInterior (hf : ConcaveOn ℝ C f) :
+--     ContinuousOn f (intrinsicInterior ℝ C)
diff --git a/Mathlib/Analysis/Convex/Deriv.lean b/Mathlib/Analysis/Convex/Deriv.lean
index 6954ba6c48f2b..4d71b529373a4 100644
--- a/Mathlib/Analysis/Convex/Deriv.lean
+++ b/Mathlib/Analysis/Convex/Deriv.lean
@@ -95,7 +95,7 @@ theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine ⟨b, ⟨hxw.trans hwb, hby⟩, ?_⟩
-    simp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
+    simp only [div_lt_iff₀, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (w - x) < deriv f b * (w - x) := by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
       · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
@@ -139,7 +139,7 @@ theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : C
         apply ne_of_gt
         exact hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hz.1, hz.2⟩ hz.1
     refine ⟨a, ⟨hxa, haw.trans hwy⟩, ?_⟩
-    simp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb ⊢
+    simp only [lt_div_iff₀, hxy, hxw, hwy, sub_pos] at ha hb ⊢
     have : deriv f a * (y - w) < deriv f b * (y - w) := by
       apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _
       · exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
diff --git a/Mathlib/Analysis/Convex/Gauge.lean b/Mathlib/Analysis/Convex/Gauge.lean
index 4083b51c775b0..df43239a123de 100644
--- a/Mathlib/Analysis/Convex/Gauge.lean
+++ b/Mathlib/Analysis/Convex/Gauge.lean
@@ -67,7 +67,7 @@ private theorem gauge_set_bddBelow : BddBelow { r : ℝ | 0 < r ∧ x ∈ r •
   ⟨0, fun _ hr => hr.1.le⟩
 
 /-- If the given subset is `Absorbent` then the set we take an infimum over in `gauge` is nonempty,
-which is useful for proving many properties about the gauge.  -/
+which is useful for proving many properties about the gauge. -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
     { r : ℝ | 0 < r ∧ x ∈ r • s }.Nonempty :=
   let ⟨r, hr₁, hr₂⟩ := (absorbs x).exists_pos
@@ -111,7 +111,7 @@ theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
 
 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
-  Real.sInf_nonneg _ fun _ hx => hx.1.le
+  Real.sInf_nonneg fun _ hx => hx.1.le
 
 theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x := by
   have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using symmetric _ h, symmetric x⟩
@@ -139,7 +139,7 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
     suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
     rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
     refine hs₁.smul_mem_of_zero_mem hs₀ hδ ⟨by positivity, ?_⟩
-    rw [inv_mul_le_iff hr', mul_one]
+    rw [inv_mul_le_iff₀ hr', mul_one]
     exact hδr.le
   · have hε' := (lt_add_iff_pos_right a).2 (half_pos hε)
     exact
@@ -214,7 +214,7 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
   have ha := hb.trans hba
   refine ⟨(a⁻¹ * b) • x, hs₀ hx' (by positivity) ?_, ?_⟩
   · rw [← div_eq_inv_mul]
-    exact div_le_one_of_le hba.le ha.le
+    exact div_le_one_of_le₀ hba.le ha.le
   · dsimp only
     rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']
 
@@ -316,7 +316,7 @@ theorem comap_gauge_nhds_zero_le (ha : Absorbent ℝ s) (hb : Bornology.IsVonNBo
   rcases (hb hu).exists_pos with ⟨r, hr₀, hr⟩
   filter_upwards [preimage_mem_comap (gt_mem_nhds (inv_pos.2 hr₀))] with x (hx : gauge s x < r⁻¹)
   rcases exists_lt_of_gauge_lt ha hx with ⟨c, hc₀, hcr, y, hy, rfl⟩
-  have hrc := (lt_inv hr₀ hc₀).2 hcr
+  have hrc := (lt_inv_comm₀ hr₀ hc₀).2 hcr
   rcases hr c⁻¹ (hrc.le.trans (le_abs_self _)) hy with ⟨z, hz, rfl⟩
   simpa only [smul_inv_smul₀ hc₀.ne']
 
@@ -359,17 +359,17 @@ theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) 
   convert interior_subset_gauge_lt_one s
   exact hs₂.interior_eq.symm
 
--- Porting note: droped unneeded assumptions
+-- Porting note: dropped unneeded assumptions
 theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
     gauge s x < 1 :=
   interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
 
--- Porting note: droped unneeded assumptions
+-- Porting note: dropped unneeded assumptions
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
     gauge s x < ε := by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
   have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₂ this
-  rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one]
+  rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff₀ hε, mul_one]
     at h_gauge_lt
 
 theorem mem_closure_of_gauge_le_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
@@ -500,7 +500,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
     refine hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, ?_, smul_inv_smul₀ hpx₂.ne' _⟩
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
-      inv_mul_lt_iff hpx₂, mul_one]
+      inv_mul_lt_iff₀ hpx₂, mul_one]
     exact lt_mul_of_one_lt_left hpx one_lt_two
   refine IsGLB.csInf_eq ⟨fun r => ?_, fun r hr => le_of_forall_pos_le_add fun ε hε => ?_⟩ hp
   · rintro ⟨hr, y, hy, rfl⟩
@@ -512,7 +512,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
       add_pos_of_nonneg_of_pos (apply_nonneg _ _) hε
     refine hr ⟨hpε, (p x + ε)⁻¹ • x, ?_, smul_inv_smul₀ hpε.ne' _⟩
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpε),
-      inv_mul_lt_iff hpε, mul_one]
+      inv_mul_lt_iff₀ hpε, mul_one]
     exact lt_add_of_pos_right _ hε
 
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
diff --git a/Mathlib/Analysis/Convex/GaugeRescale.lean b/Mathlib/Analysis/Convex/GaugeRescale.lean
index c16d558914d31..5355e87f1273a 100644
--- a/Mathlib/Analysis/Convex/GaugeRescale.lean
+++ b/Mathlib/Analysis/Convex/GaugeRescale.lean
@@ -115,7 +115,7 @@ theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈
   rcases eq_or_ne x 0 with rfl | hx
   · rw [ContinuousAt, gaugeRescale_zero]
     nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀]
-    simp only [tendsto_comap_iff, (· ∘ ·), gauge_gaugeRescale _ hta htb]
+    simp only [tendsto_comap_iff, Function.comp_def, gauge_gaugeRescale _ hta htb]
     exact tendsto_gauge_nhds_zero hs₀
   · exact ((continuousAt_gauge hs hs₀).div (continuousAt_gauge ht ht₀)
       ((gauge_pos hta htb).2 hx).ne').smul continuousAt_id
diff --git a/Mathlib/Analysis/Convex/Integral.lean b/Mathlib/Analysis/Convex/Integral.lean
index 12b11b68057db..dd75c412a4ffc 100644
--- a/Mathlib/Analysis/Convex/Integral.lean
+++ b/Mathlib/Analysis/Convex/Integral.lean
@@ -328,7 +328,7 @@ theorem ae_eq_const_or_norm_integral_lt_of_norm_le_const [StrictConvexSpace ℝ
     simp [ENNReal.toReal_pos_iff, pos_iff_ne_zero, h₀, measure_lt_top]
   refine (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => ?_
   rwa [average_eq, norm_smul, norm_inv, Real.norm_eq_abs, abs_of_pos hμ, ← div_eq_inv_mul,
-    div_lt_iff' hμ] at H
+    div_lt_iff₀' hμ] at H
 
 /-- If `E` is a strictly convex normed space and `f : α → E` is a function such that `‖f x‖ ≤ C`
 a.e. on a set `t` of finite measure, then either this function is a.e. equal to its average value on
diff --git a/Mathlib/Analysis/Convex/Intrinsic.lean b/Mathlib/Analysis/Convex/Intrinsic.lean
index 7e5b5e4ee2e13..82cc1b1b0da2d 100644
--- a/Mathlib/Analysis/Convex/Intrinsic.lean
+++ b/Mathlib/Analysis/Convex/Intrinsic.lean
@@ -116,8 +116,8 @@ alias ⟨Set.Nonempty.ofIntrinsicClosure, Set.Nonempty.intrinsicClosure⟩ := in
 
 @[simp]
 theorem intrinsicInterior_singleton (x : P) : intrinsicInterior 𝕜 ({x} : Set P) = {x} := by
-  simpa only [intrinsicInterior, preimage_coe_affineSpan_singleton, interior_univ, image_univ,
-    Subtype.range_coe] using coe_affineSpan_singleton _ _ _
+  simp only [intrinsicInterior, preimage_coe_affineSpan_singleton, interior_univ, image_univ,
+    Subtype.range_coe_subtype, mem_affineSpan_singleton, setOf_eq_eq_singleton]
 
 @[simp]
 theorem intrinsicFrontier_singleton (x : P) : intrinsicFrontier 𝕜 ({x} : Set P) = ∅ := by
@@ -125,8 +125,8 @@ theorem intrinsicFrontier_singleton (x : P) : intrinsicFrontier 𝕜 ({x} : Set
 
 @[simp]
 theorem intrinsicClosure_singleton (x : P) : intrinsicClosure 𝕜 ({x} : Set P) = {x} := by
-  simpa only [intrinsicClosure, preimage_coe_affineSpan_singleton, closure_univ, image_univ,
-    Subtype.range_coe] using coe_affineSpan_singleton _ _ _
+  simp only [intrinsicClosure, preimage_coe_affineSpan_singleton, closure_univ, image_univ,
+    Subtype.range_coe_subtype, mem_affineSpan_singleton, setOf_eq_eq_singleton]
 
 /-!
 Note that neither `intrinsicInterior` nor `intrinsicFrontier` is monotone.
@@ -223,8 +223,8 @@ theorem image_intrinsicInterior (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
   let f := ((affineSpan 𝕜 s).isometryEquivMap φ).toHomeomorph
   have : φ.toAffineMap ∘ (↑) ∘ f.symm = (↑) := funext isometryEquivMap.apply_symm_apply
   rw [intrinsicInterior, intrinsicInterior, ← φ.coe_toAffineMap, ← map_span φ.toAffineMap s, ← this,
-    ← Function.comp.assoc, image_comp, image_comp, f.symm.image_interior, f.image_symm,
-    ← preimage_comp, Function.comp.assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
+    ← Function.comp_assoc, image_comp, image_comp, f.symm.image_interior, f.image_symm,
+    ← preimage_comp, Function.comp_assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
     Function.comp_id, preimage_comp, φ.injective.preimage_image]
 
 @[simp]
@@ -236,8 +236,8 @@ theorem image_intrinsicFrontier (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
   let f := ((affineSpan 𝕜 s).isometryEquivMap φ).toHomeomorph
   have : φ.toAffineMap ∘ (↑) ∘ f.symm = (↑) := funext isometryEquivMap.apply_symm_apply
   rw [intrinsicFrontier, intrinsicFrontier, ← φ.coe_toAffineMap, ← map_span φ.toAffineMap s, ← this,
-    ← Function.comp.assoc, image_comp, image_comp, f.symm.image_frontier, f.image_symm,
-    ← preimage_comp, Function.comp.assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
+    ← Function.comp_assoc, image_comp, image_comp, f.symm.image_frontier, f.image_symm,
+    ← preimage_comp, Function.comp_assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
     Function.comp_id, preimage_comp, φ.injective.preimage_image]
 
 @[simp]
@@ -249,8 +249,8 @@ theorem image_intrinsicClosure (φ : P →ᵃⁱ[𝕜] Q) (s : Set P) :
   let f := ((affineSpan 𝕜 s).isometryEquivMap φ).toHomeomorph
   have : φ.toAffineMap ∘ (↑) ∘ f.symm = (↑) := funext isometryEquivMap.apply_symm_apply
   rw [intrinsicClosure, intrinsicClosure, ← φ.coe_toAffineMap, ← map_span φ.toAffineMap s, ← this,
-    ← Function.comp.assoc, image_comp, image_comp, f.symm.image_closure, f.image_symm,
-    ← preimage_comp, Function.comp.assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
+    ← Function.comp_assoc, image_comp, image_comp, f.symm.image_closure, f.image_symm,
+    ← preimage_comp, Function.comp_assoc, f.symm_comp_self, AffineIsometry.coe_toAffineMap,
     Function.comp_id, preimage_comp, φ.injective.preimage_image]
 
 end AffineIsometry
diff --git a/Mathlib/Analysis/Convex/Jensen.lean b/Mathlib/Analysis/Convex/Jensen.lean
index 20e5b017a492d..93f195e014393 100644
--- a/Mathlib/Analysis/Convex/Jensen.lean
+++ b/Mathlib/Analysis/Convex/Jensen.lean
@@ -118,11 +118,10 @@ lemma StrictConvexOn.map_sum_lt (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈
   have := h₀ k <| by simp
   let c := w j + w k
   have hc : w j / c + w k / c = 1 := by field_simp
-  have hcj : c * (w j / c) = w j := by field_simp
-  have hck : c * (w k / c) = w k := by field_simp
   calc f (w j • p j + (w k • p k + ∑ x ∈ u, w x • p x))
     _ = f (c • ((w j / c) • p j + (w k / c) • p k) + ∑ x ∈ u, w x • p x) := by
-      rw [smul_add, ← mul_smul, ← mul_smul, hcj, hck, add_assoc]
+      congrm f ?_
+      match_scalars <;> field_simp
     _ ≤ c • f ((w j / c) • p j + (w k / c) • p k) + ∑ x ∈ u, w x • f (p x) :=
       -- apply the usual Jensen's inequality wrt the weighted average of the two distinguished
       -- points and all the other points
@@ -134,7 +133,7 @@ lemma StrictConvexOn.map_sum_lt (hf : StrictConvexOn 𝕜 s f) (h₀ : ∀ i ∈
       -- then apply the definition of strict convexity for the two distinguished points
       gcongr; refine hf.2 (hmem _ <| by simp) (hmem _ <| by simp) hjk ?_ ?_ hc <;> positivity
     _ = (w j • f (p j) + w k • f (p k)) + ∑ x ∈ u, w x • f (p x) := by
-      rw [smul_add, ← mul_smul, ← mul_smul, hcj, hck]
+      match_scalars <;> field_simp
     _ = w j • f (p j) + (w k • f (p k) + ∑ x ∈ u, w x • f (p x)) := by abel_nf
 
 /-- Concave **strict Jensen inequality**.
diff --git a/Mathlib/Analysis/Convex/Join.lean b/Mathlib/Analysis/Convex/Join.lean
index a8641487cfccd..590ebaa4aa614 100644
--- a/Mathlib/Analysis/Convex/Join.lean
+++ b/Mathlib/Analysis/Convex/Join.lean
@@ -113,19 +113,13 @@ theorem convexJoin_assoc_aux (s t u : Set E) :
   rintro _ ⟨z, ⟨x, hx, y, hy, a₁, b₁, ha₁, hb₁, hab₁, rfl⟩, z, hz, a₂, b₂, ha₂, hb₂, hab₂, rfl⟩
   obtain rfl | hb₂ := hb₂.eq_or_lt
   · refine ⟨x, hx, y, ⟨y, hy, z, hz, left_mem_segment 𝕜 _ _⟩, a₁, b₁, ha₁, hb₁, hab₁, ?_⟩
-    rw [add_zero] at hab₂
-    rw [hab₂, one_smul, zero_smul, add_zero]
-  have ha₂b₁ : 0 ≤ a₂ * b₁ := mul_nonneg ha₂ hb₁
-  have hab : 0 < a₂ * b₁ + b₂ := add_pos_of_nonneg_of_pos ha₂b₁ hb₂
+    linear_combination (norm := module) congr(-$hab₂ • (a₁ • x + b₁ • y))
   refine
     ⟨x, hx, (a₂ * b₁ / (a₂ * b₁ + b₂)) • y + (b₂ / (a₂ * b₁ + b₂)) • z,
-      ⟨y, hy, z, hz, _, _, ?_, ?_, ?_, rfl⟩,
-      a₂ * a₁, a₂ * b₁ + b₂, mul_nonneg ha₂ ha₁, hab.le, ?_, ?_⟩
-  · exact div_nonneg ha₂b₁ hab.le
-  · exact div_nonneg hb₂.le hab.le
-  · rw [← add_div, div_self hab.ne']
-  · rw [← add_assoc, ← mul_add, hab₁, mul_one, hab₂]
-  · simp_rw [smul_add, ← mul_smul, mul_div_cancel₀ _ hab.ne', add_assoc]
+      ⟨y, hy, z, hz, _, _, by positivity, by positivity, by field_simp, rfl⟩,
+      a₂ * a₁, a₂ * b₁ + b₂, by positivity, by positivity, ?_, ?_⟩
+  · linear_combination a₂ * hab₁ + hab₂
+  · match_scalars <;> field_simp
 
 theorem convexJoin_assoc (s t u : Set E) :
     convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 s (convexJoin 𝕜 t u) := by
@@ -155,9 +149,9 @@ protected theorem Convex.convexJoin (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) :
   rcases hs.exists_mem_add_smul_eq hx₁ hx₂ (mul_nonneg hp ha₁) (mul_nonneg hq ha₂) with ⟨x, hxs, hx⟩
   rcases ht.exists_mem_add_smul_eq hy₁ hy₂ (mul_nonneg hp hb₁) (mul_nonneg hq hb₂) with ⟨y, hyt, hy⟩
   refine ⟨_, hxs, _, hyt, p * a₁ + q * a₂, p * b₁ + q * b₂, ?_, ?_, ?_, ?_⟩ <;> try positivity
-  · rwa [add_add_add_comm, ← mul_add, ← mul_add, hab₁, hab₂, mul_one, mul_one]
-  · rw [hx, hy, add_add_add_comm]
-    simp only [smul_add, smul_smul]
+  · linear_combination p * hab₁ + q * hab₂ + hpq
+  · rw [hx, hy]
+    module
 
 protected theorem Convex.convexHull_union (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hs₀ : s.Nonempty)
     (ht₀ : t.Nonempty) : convexHull 𝕜 (s ∪ t) = convexJoin 𝕜 s t :=
diff --git a/Mathlib/Analysis/Convex/Measure.lean b/Mathlib/Analysis/Convex/Measure.lean
index 6dec1b9d305d3..cf276ed3f8958 100644
--- a/Mathlib/Analysis/Convex/Measure.lean
+++ b/Mathlib/Analysis/Convex/Measure.lean
@@ -18,7 +18,7 @@ convex set in `E`. Then the frontier of `s` has measure zero (see `Convex.addHaa
 
 open MeasureTheory MeasureTheory.Measure Set Metric Filter Bornology
 
-open FiniteDimensional (finrank)
+open Module (finrank)
 
 open scoped Topology NNReal ENNReal
 
@@ -59,12 +59,12 @@ theorem addHaar_frontier (hs : Convex ℝ s) : μ (frontier s) = 0 := by
     `μ (closure s) ≤ μ (interior s)`. -/
   replace hb : μ (interior s) ≠ ∞ := (hb.subset interior_subset).measure_lt_top.ne
   suffices μ (closure s) ≤ μ (interior s) by
-    rwa [frontier, measure_diff interior_subset_closure isOpen_interior.measurableSet hb,
+    rwa [frontier, measure_diff interior_subset_closure isOpen_interior.nullMeasurableSet hb,
       tsub_eq_zero_iff_le]
   /- Due to `Convex.closure_subset_image_homothety_interior_of_one_lt`, for any `r > 1` we have
     `closure s ⊆ homothety x r '' interior s`, hence `μ (closure s) ≤ r ^ d * μ (interior s)`,
     where `d = finrank ℝ E`. -/
-  set d : ℕ := FiniteDimensional.finrank ℝ E
+  set d : ℕ := Module.finrank ℝ E
   have : ∀ r : ℝ≥0, 1 < r → μ (closure s) ≤ ↑(r ^ d) * μ (interior s) := fun r hr ↦ by
     refine (measure_mono <|
       hs.closure_subset_image_homothety_interior_of_one_lt hx r hr).trans_eq ?_
diff --git a/Mathlib/Analysis/Convex/Normed.lean b/Mathlib/Analysis/Convex/Normed.lean
index 2cb4a8b2db0d7..bfb9ac2e29cd3 100644
--- a/Mathlib/Analysis/Convex/Normed.lean
+++ b/Mathlib/Analysis/Convex/Normed.lean
@@ -27,7 +27,7 @@ We prove the following facts:
 
 variable {ι : Type*} {E P : Type*}
 
-open AffineBasis FiniteDimensional Metric Set
+open AffineBasis Module Metric Set
 open scoped Convex Pointwise Topology
 
 section SeminormedAddCommGroup
diff --git a/Mathlib/Analysis/Convex/Radon.lean b/Mathlib/Analysis/Convex/Radon.lean
index ee505ba41b251..f297649b6f470 100644
--- a/Mathlib/Analysis/Convex/Radon.lean
+++ b/Mathlib/Analysis/Convex/Radon.lean
@@ -62,7 +62,7 @@ theorem radon_partition {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) :
   · linarith only [hI, hJI]
   · exact (mem_filter.mp hi').2.not_lt (mem_filter.mp hi).2
 
-open FiniteDimensional
+open Module
 
 /-- Corner case for `helly_theorem'`. -/
 private lemma helly_theorem_corner {F : ι → Set E} {s : Finset ι}
diff --git a/Mathlib/Analysis/Convex/Segment.lean b/Mathlib/Analysis/Convex/Segment.lean
index 87adc0a70ac2e..9f5026dbeb8cc 100644
--- a/Mathlib/Analysis/Convex/Segment.lean
+++ b/Mathlib/Analysis/Convex/Segment.lean
@@ -122,7 +122,7 @@ theorem segment_same (x : E) : [x -[𝕜] x] = {x} :=
 theorem insert_endpoints_openSegment (x y : E) :
     insert x (insert y (openSegment 𝕜 x y)) = [x -[𝕜] y] := by
   simp only [subset_antisymm_iff, insert_subset_iff, left_mem_segment, right_mem_segment,
-    openSegment_subset_segment, true_and_iff]
+    openSegment_subset_segment, true_and]
   rintro z ⟨a, b, ha, hb, hab, rfl⟩
   refine hb.eq_or_gt.imp ?_ fun hb' => ha.eq_or_gt.imp ?_ fun ha' => ?_
   · rintro rfl
@@ -140,7 +140,7 @@ theorem mem_openSegment_of_ne_left_right (hx : x ≠ z) (hy : y ≠ z) (hz : z 
 
 theorem openSegment_subset_iff_segment_subset (hx : x ∈ s) (hy : y ∈ s) :
     openSegment 𝕜 x y ⊆ s ↔ [x -[𝕜] y] ⊆ s := by
-  simp only [← insert_endpoints_openSegment, insert_subset_iff, *, true_and_iff]
+  simp only [← insert_endpoints_openSegment, insert_subset_iff, *, true_and]
 
 end Module
 
diff --git a/Mathlib/Analysis/Convex/Side.lean b/Mathlib/Analysis/Convex/Side.lean
index 13933585a56e0..45bc7d0036eb8 100644
--- a/Mathlib/Analysis/Convex/Side.lean
+++ b/Mathlib/Analysis/Convex/Side.lean
@@ -333,12 +333,8 @@ theorem _root_.Wbtw.wOppSide₁₃ {s : AffineSubspace R P} {x y z : P} (h : Wbt
   rcases ht0.lt_or_eq with (ht0' | rfl); swap
   · rw [lineMap_apply_zero]; simp
   refine Or.inr (Or.inr ⟨1 - t, t, sub_pos.2 ht1', ht0', ?_⟩)
-  -- TODO: after lean4#2336 "simp made no progress feature"
-  -- had to add `_` to several lemmas here. Not sure why!
-  simp_rw [lineMap_apply _, vadd_vsub_assoc _, vsub_vadd_eq_vsub_sub _,
-    ← neg_vsub_eq_vsub_rev z x, vsub_self _, zero_sub, ← neg_one_smul R (z -ᵥ x),
-    ← add_smul, smul_neg, ← neg_smul, smul_smul]
-  ring_nf
+  rw [lineMap_apply, vadd_vsub_assoc, vsub_vadd_eq_vsub_sub, ← neg_vsub_eq_vsub_rev z, vsub_self]
+  module
 
 theorem _root_.Wbtw.wOppSide₃₁ {s : AffineSubspace R P} {x y z : P} (h : Wbtw R x y z)
     (hy : y ∈ s) : s.WOppSide z x :=
@@ -411,9 +407,9 @@ theorem wOppSide_iff_exists_left {s : AffineSubspace R P} {x y p₁ : P} (h : p
       exact SameRay.zero_right _
     · refine Or.inr ⟨(-r₁ / r₂) • (p₁ -ᵥ p₁') +ᵥ p₂', s.smul_vsub_vadd_mem _ h hp₁' hp₂',
         Or.inr (Or.inr ⟨r₁, r₂, hr₁, hr₂, ?_⟩)⟩
-      rw [vadd_vsub_assoc, smul_add, ← hr, smul_smul, neg_div, mul_neg,
-        mul_div_cancel₀ _ hr₂.ne.symm, neg_smul, neg_add_eq_sub, ← smul_sub,
-        vsub_sub_vsub_cancel_right]
+      rw [vadd_vsub_assoc, ← vsub_sub_vsub_cancel_right x p₁ p₁']
+      linear_combination (norm := match_scalars <;> field_simp) hr
+      ring
   · rintro (h' | ⟨h₁, h₂, h₃⟩)
     · exact wOppSide_of_left_mem y h'
     · exact ⟨p₁, h, h₁, h₂, h₃⟩
@@ -584,16 +580,15 @@ theorem wOppSide_iff_exists_wbtw {s : AffineSubspace R P} {x y : P} :
   · refine ⟨lineMap x y (r₂ / (r₁ + r₂)), ?_, ?_⟩
     · have : (r₂ / (r₁ + r₂)) • (y -ᵥ p₂ + (p₂ -ᵥ p₁) - (x -ᵥ p₁)) + (x -ᵥ p₁) =
           (r₂ / (r₁ + r₂)) • (p₂ -ᵥ p₁) := by
-        rw [add_comm (y -ᵥ p₂), smul_sub, smul_add, add_sub_assoc, add_assoc, add_right_eq_self,
-          div_eq_inv_mul, ← neg_vsub_eq_vsub_rev, smul_neg, ← smul_smul, ← h, smul_smul, ← neg_smul,
-          ← sub_smul, ← div_eq_inv_mul, ← div_eq_inv_mul, ← neg_div, ← sub_div, sub_eq_add_neg,
-          ← neg_add, neg_div, div_self (Left.add_pos hr₁ hr₂).ne.symm, neg_one_smul, neg_add_cancel]
+        rw [← neg_vsub_eq_vsub_rev p₂ y]
+        linear_combination (norm := match_scalars <;> field_simp) congr((r₁ + r₂)⁻¹ • $h)
+        ring
       rw [lineMap_apply, ← vsub_vadd x p₁, ← vsub_vadd y p₂, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc,
         ← vadd_assoc, vadd_eq_add, this]
       exact s.smul_vsub_vadd_mem (r₂ / (r₁ + r₂)) hp₂ hp₁ hp₁
     · exact Set.mem_image_of_mem _
-        ⟨div_nonneg hr₂.le (Left.add_pos hr₁ hr₂).le,
-          div_le_one_of_le (le_add_of_nonneg_left hr₁.le) (Left.add_pos hr₁ hr₂).le⟩
+        ⟨by positivity,
+          div_le_one_of_le₀ (le_add_of_nonneg_left hr₁.le) (Left.add_pos hr₁ hr₂).le⟩
 
 theorem SOppSide.exists_sbtw {s : AffineSubspace R P} {x y : P} (h : s.SOppSide x y) :
     ∃ p ∈ s, Sbtw R x p y := by
@@ -776,7 +771,7 @@ theorem isPreconnected_setOf_sSameSide (s : AffineSubspace ℝ P) (x : P) :
     simp only [h, not_sSameSide_bot]
     exact isPreconnected_empty
   · by_cases hx : x ∈ s
-    · simp only [hx, SSameSide, not_true, false_and_iff, and_false_iff]
+    · simp only [hx, SSameSide, not_true, false_and, and_false]
       exact isPreconnected_empty
     · exact (isConnected_setOf_sSameSide hx h).isPreconnected
 
@@ -817,7 +812,7 @@ theorem isPreconnected_setOf_sOppSide (s : AffineSubspace ℝ P) (x : P) :
     simp only [h, not_sOppSide_bot]
     exact isPreconnected_empty
   · by_cases hx : x ∈ s
-    · simp only [hx, SOppSide, not_true, false_and_iff, and_false_iff]
+    · simp only [hx, SOppSide, not_true, false_and, and_false]
       exact isPreconnected_empty
     · exact (isConnected_setOf_sOppSide hx h).isPreconnected
 
diff --git a/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean b/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
index b1f12fdb8b073..41111c8c7ea85 100644
--- a/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
+++ b/Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
@@ -72,7 +72,7 @@ variable {K : SimplicialComplex 𝕜 E} {s t : Finset E} {x : E}
 
 /-- A `Finset` belongs to a `SimplicialComplex` if it's a face of it. -/
 instance : Membership (Finset E) (SimplicialComplex 𝕜 E) :=
-  ⟨fun s K => s ∈ K.faces⟩
+  ⟨fun K s => s ∈ K.faces⟩
 
 /-- The underlying space of a simplicial complex is the union of its faces. -/
 def space (K : SimplicialComplex 𝕜 E) : Set E :=
@@ -166,7 +166,7 @@ theorem face_subset_face_iff (hs : s ∈ K.faces) (ht : t ∈ K.faces) :
   ⟨fun h _ hxs =>
     (vertex_mem_convexHull_iff
           (K.down_closed hs (Finset.singleton_subset_iff.2 hxs) <| singleton_ne_empty _) ht).1
-      (h (subset_convexHull 𝕜 (↑s) hxs)),
+      (h (subset_convexHull 𝕜 (E := E) s hxs)),
     convexHull_mono⟩
 
 /-! ### Facets -/
diff --git a/Mathlib/Analysis/Convex/Slope.lean b/Mathlib/Analysis/Convex/Slope.lean
index ea7ca88a2e853..38072902d3e91 100644
--- a/Mathlib/Analysis/Convex/Slope.lean
+++ b/Mathlib/Analysis/Convex/Slope.lean
@@ -158,7 +158,7 @@ theorem strictConvexOn_of_slope_strict_mono_adjacent (hs : Convex 𝕜 s)
       simp_rw [div_eq_iff hxz.ne', ← hab]
       ring
     rwa [sub_mul, sub_mul, sub_lt_iff_lt_add', ← add_sub_assoc, lt_sub_iff_add_lt, ← mul_add,
-      sub_add_sub_cancel, ← lt_div_iff hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x),
+      sub_add_sub_cancel, ← lt_div_iff₀ hxz, add_div, mul_div_assoc, mul_div_assoc, mul_comm (f x),
       mul_comm (f z), ha, hb] at this
 
 /-- If for any three points `x < y < z`, the slope of the secant line of `f : 𝕜 → 𝕜` on `[x, y]` is
@@ -267,7 +267,7 @@ theorem StrictConvexOn.secant_strict_mono_aux1 (hf : StrictConvexOn 𝕜 s f) {x
   have hxy' : 0 < y - x := by linarith
   have hyz' : 0 < z - y := by linarith
   have hxz' : 0 < z - x := by linarith
-  rw [← lt_div_iff' hxz']
+  rw [← lt_div_iff₀' hxz']
   have ha : 0 < (z - y) / (z - x) := by positivity
   have hb : 0 < (y - x) / (z - x) := by positivity
   calc
diff --git a/Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean b/Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
index 922b18543ffe0..b293f4c6a7eb3 100644
--- a/Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
+++ b/Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
@@ -40,14 +40,14 @@ theorem strictConvexOn_exp : StrictConvexOn ℝ univ exp := by
   trans exp y
   · have h1 : 0 < y - x := by linarith
     have h2 : x - y < 0 := by linarith
-    rw [div_lt_iff h1]
+    rw [div_lt_iff₀ h1]
     calc
       exp y - exp x = exp y - exp y * exp (x - y) := by rw [← exp_add]; ring_nf
       _ = exp y * (1 - exp (x - y)) := by ring
       _ < exp y * -(x - y) := by gcongr; linarith [add_one_lt_exp h2.ne]
       _ = exp y * (y - x) := by ring
   · have h1 : 0 < z - y := by linarith
-    rw [lt_div_iff h1]
+    rw [lt_div_iff₀ h1]
     calc
       exp y * (z - y) < exp y * (exp (z - y) - 1) := by
         gcongr _ * ?_
@@ -66,7 +66,7 @@ theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log := by
   have hy : 0 < y := hx.trans hxy
   trans y⁻¹
   · have h : 0 < z - y := by linarith
-    rw [div_lt_iff h]
+    rw [div_lt_iff₀ h]
     have hyz' : 0 < z / y := by positivity
     have hyz'' : z / y ≠ 1 := by
       contrapose! h
@@ -77,7 +77,7 @@ theorem strictConcaveOn_log_Ioi : StrictConcaveOn ℝ (Ioi 0) log := by
       _ < z / y - 1 := log_lt_sub_one_of_pos hyz' hyz''
       _ = y⁻¹ * (z - y) := by field_simp
   · have h : 0 < y - x := by linarith
-    rw [lt_div_iff h]
+    rw [lt_div_iff₀ h]
     have hxy' : 0 < x / y := by positivity
     have hxy'' : x / y ≠ 1 := by
       contrapose! h
@@ -105,12 +105,12 @@ theorem one_add_mul_self_lt_rpow_one_add {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠
   rw [rpow_def_of_pos hs1, ← exp_log hs2]
   apply exp_strictMono
   cases' lt_or_gt_of_ne hs' with hs' hs'
-  · rw [← div_lt_iff hp', ← div_lt_div_right_of_neg hs']
+  · rw [← div_lt_iff₀ hp', ← div_lt_div_right_of_neg hs']
     convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs2 hs1 hs4 hs3 _ using 1
     · rw [add_sub_cancel_left, log_one, sub_zero]
     · rw [add_sub_cancel_left, div_div, log_one, sub_zero]
     · apply add_lt_add_left (mul_lt_of_one_lt_left hs' hp)
-  · rw [← div_lt_iff hp', ← div_lt_div_right hs']
+  · rw [← div_lt_iff₀ hp', ← div_lt_div_right hs']
     convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs1 hs2 hs3 hs4 _ using 1
     · rw [add_sub_cancel_left, div_div, log_one, sub_zero]
     · rw [add_sub_cancel_left, log_one, sub_zero]
@@ -144,12 +144,12 @@ theorem rpow_one_add_lt_one_add_mul_self {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠
   rw [rpow_def_of_pos hs1, ← exp_log hs2]
   apply exp_strictMono
   cases' lt_or_gt_of_ne hs' with hs' hs'
-  · rw [← lt_div_iff hp1, ← div_lt_div_right_of_neg hs']
+  · rw [← lt_div_iff₀ hp1, ← div_lt_div_right_of_neg hs']
     convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs1 hs2 hs3 hs4 _ using 1
     · rw [add_sub_cancel_left, div_div, log_one, sub_zero]
     · rw [add_sub_cancel_left, log_one, sub_zero]
     · apply add_lt_add_left (lt_mul_of_lt_one_left hs' hp2)
-  · rw [← lt_div_iff hp1, ← div_lt_div_right hs']
+  · rw [← lt_div_iff₀ hp1, ← div_lt_div_right hs']
     convert strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs2 hs1 hs4 hs3 _ using 1
     · rw [add_sub_cancel_left, log_one, sub_zero]
     · rw [add_sub_cancel_left, div_div, log_one, sub_zero]
@@ -175,20 +175,21 @@ theorem strictConvexOn_rpow {p : ℝ} (hp : 1 < p) : StrictConvexOn ℝ (Ici 0)
   have hy' : 0 < y ^ p := rpow_pos_of_pos hy _
   trans p * y ^ (p - 1)
   · have q : 0 < y - x := by rwa [sub_pos]
-    rw [div_lt_iff q, ← div_lt_div_right hy', _root_.sub_div, div_self hy'.ne', ← div_rpow hx hy.le,
-      sub_lt_comm, ← add_sub_cancel_right (x / y) 1, add_comm, add_sub_assoc, ← div_mul_eq_mul_div,
-      mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left, rpow_neg_one, mul_assoc, ← div_eq_inv_mul,
-      sub_eq_add_neg, ← mul_neg, ← neg_div, neg_sub, _root_.sub_div, div_self hy.ne']
+    rw [div_lt_iff₀ q, ← div_lt_div_right hy', _root_.sub_div, div_self hy'.ne',
+      ← div_rpow hx hy.le, sub_lt_comm, ← add_sub_cancel_right (x / y) 1, add_comm, add_sub_assoc,
+      ← div_mul_eq_mul_div, mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left, rpow_neg_one,
+      mul_assoc, ← div_eq_inv_mul, sub_eq_add_neg, ← mul_neg, ← neg_div, neg_sub, _root_.sub_div,
+      div_self hy.ne']
     apply one_add_mul_self_lt_rpow_one_add _ _ hp
     · rw [le_sub_iff_add_le, neg_add_cancel, div_nonneg_iff]
       exact Or.inl ⟨hx, hy.le⟩
     · rw [sub_ne_zero]
       exact ((div_lt_one hy).mpr hxy).ne
   · have q : 0 < z - y := by rwa [sub_pos]
-    rw [lt_div_iff q, ← div_lt_div_right hy', _root_.sub_div, div_self hy'.ne', ← div_rpow hz hy.le,
-      lt_sub_iff_add_lt', ← add_sub_cancel_right (z / y) 1, add_comm _ 1, add_sub_assoc,
-      ← div_mul_eq_mul_div, mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left, rpow_neg_one,
-      mul_assoc, ← div_eq_inv_mul, _root_.sub_div, div_self hy.ne']
+    rw [lt_div_iff₀ q, ← div_lt_div_right hy', _root_.sub_div, div_self hy'.ne',
+      ← div_rpow hz hy.le, lt_sub_iff_add_lt', ← add_sub_cancel_right (z / y) 1, add_comm _ 1,
+      add_sub_assoc, ← div_mul_eq_mul_div, mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left,
+      rpow_neg_one, mul_assoc, ← div_eq_inv_mul, _root_.sub_div, div_self hy.ne']
     apply one_add_mul_self_lt_rpow_one_add _ _ hp
     · rw [le_sub_iff_add_le, neg_add_cancel, div_nonneg_iff]
       exact Or.inl ⟨hz, hy.le⟩
diff --git a/Mathlib/Analysis/Convex/Star.lean b/Mathlib/Analysis/Convex/Star.lean
index 44f1a60ccabfb..ad0ee8acf1ca5 100644
--- a/Mathlib/Analysis/Convex/Star.lean
+++ b/Mathlib/Analysis/Convex/Star.lean
@@ -8,6 +8,7 @@ import Mathlib.Algebra.Order.Module.Synonym
 import Mathlib.Algebra.Order.Group.Instances
 import Mathlib.Analysis.Convex.Segment
 import Mathlib.Tactic.GCongr
+import Mathlib.Tactic.Module
 
 /-!
 # Star-convex sets
@@ -210,14 +211,14 @@ theorem StarConvex.add_left (hs : StarConvex 𝕜 x s) (z : E) :
   intro y hy a b ha hb hab
   obtain ⟨y', hy', rfl⟩ := hy
   refine ⟨a • x + b • y', hs hy' ha hb hab, ?_⟩
-  rw [smul_add, smul_add, add_add_add_comm, ← add_smul, hab, one_smul]
+  match_scalars <;> simp [hab]
 
 theorem StarConvex.add_right (hs : StarConvex 𝕜 x s) (z : E) :
     StarConvex 𝕜 (x + z) ((fun x => x + z) '' s) := by
   intro y hy a b ha hb hab
   obtain ⟨y', hy', rfl⟩ := hy
   refine ⟨a • x + b • y', hs hy' ha hb hab, ?_⟩
-  rw [smul_add, smul_add, add_add_add_comm, ← add_smul, hab, one_smul]
+  match_scalars <;> simp [hab]
 
 /-- The translation of a star-convex set is also star-convex. -/
 theorem StarConvex.preimage_add_right (hs : StarConvex 𝕜 (z + x) s) :
@@ -385,7 +386,7 @@ theorem starConvex_iff_div : StarConvex 𝕜 x s ↔ ∀ ⦃y⦄, y ∈ s →
 theorem StarConvex.mem_smul (hs : StarConvex 𝕜 0 s) (hx : x ∈ s) {t : 𝕜} (ht : 1 ≤ t) :
     x ∈ t • s := by
   rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans_le ht).ne']
-  exact hs.smul_mem hx (by positivity) (inv_le_one ht)
+  exact hs.smul_mem hx (by positivity) (inv_le_one_of_one_le₀ ht)
 
 end AddCommGroup
 
diff --git a/Mathlib/Analysis/Convex/StoneSeparation.lean b/Mathlib/Analysis/Convex/StoneSeparation.lean
index 3a0fdf9701a5f..1027690d2b881 100644
--- a/Mathlib/Analysis/Convex/StoneSeparation.lean
+++ b/Mathlib/Analysis/Convex/StoneSeparation.lean
@@ -46,7 +46,6 @@ theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ s
   · positivity
   · positivity
   · rw [← add_div, div_self]; positivity
-  rw [smul_add, smul_add, add_add_add_comm, add_comm, ← mul_smul, ← mul_smul]
   classical
     let w : Fin 3 → 𝕜 := ![az * av * bu, bz * au * bv, au * av]
     let z : Fin 3 → E := ![p, q, az • x + bz • y]
@@ -59,18 +58,15 @@ theorem not_disjoint_segment_convexHull_triple {p q u v x y z : E} (hz : z ∈ s
     have hw : ∑ i, w i = az * av + bz * au := by
       trans az * av * bu + (bz * au * bv + au * av)
       · simp [w, Fin.sum_univ_succ, Fin.sum_univ_zero]
-      rw [← one_mul (au * av), ← habz, add_mul, ← add_assoc, add_add_add_comm, mul_assoc, ← mul_add,
-        mul_assoc, ← mul_add, mul_comm av, ← add_mul, ← mul_add, add_comm bu, add_comm bv, habu,
-        habv, one_mul, mul_one]
+      linear_combination (au * bv - 1 * au) * habz + (-(1 * az * au) + au) * habv + az * av * habu
     have hz : ∀ i, z i ∈ ({p, q, az • x + bz • y} : Set E) := fun i => by fin_cases i <;> simp [z]
-    convert Finset.centerMass_mem_convexHull (Finset.univ : Finset (Fin 3)) (fun i _ => hw₀ i)
-        (by rwa [hw]) fun i _ => hz i
-    rw [Finset.centerMass]
-    simp_rw [div_eq_inv_mul, hw, mul_assoc, mul_smul (az * av + bz * au)⁻¹, ← smul_add, add_assoc, ←
-      mul_assoc]
+    convert (Finset.centerMass_mem_convexHull (Finset.univ : Finset (Fin 3)) (fun i _ => hw₀ i)
+        (by rwa [hw]) fun i _ => hz i : Finset.univ.centerMass w z ∈ _)
+    rw [Finset.centerMass, hw]
+    trans (az * av + bz * au)⁻¹ •
+      ((az * av * bu) • p + ((bz * au * bv) • q + (au * av) • (az • x + bz • y)))
+    · module
     congr 3
-    rw [← mul_smul, ← mul_rotate, mul_right_comm, mul_smul, ← mul_smul _ av, mul_rotate,
-      mul_smul _ bz, ← smul_add]
     simp only [w, z, smul_add, List.foldr, Matrix.cons_val_succ', Fin.mk_one,
       Matrix.cons_val_one, Matrix.head_cons, add_zero]
 
diff --git a/Mathlib/Analysis/Convex/Strict.lean b/Mathlib/Analysis/Convex/Strict.lean
index da771a82ffc55..1eabcd86ec380 100644
--- a/Mathlib/Analysis/Convex/Strict.lean
+++ b/Mathlib/Analysis/Convex/Strict.lean
@@ -302,8 +302,7 @@ theorem StrictConvex.eq_of_openSegment_subset_frontier [Nontrivial 𝕜] [Densel
 
 theorem StrictConvex.add_smul_mem (hs : StrictConvex 𝕜 s) (hx : x ∈ s) (hxy : x + y ∈ s)
     (hy : y ≠ 0) {t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • y ∈ interior s := by
-  have h : x + t • y = (1 - t) • x + t • (x + y) := by
-    rw [smul_add, ← add_assoc, ← _root_.add_smul, sub_add_cancel, one_smul]
+  have h : x + t • y = (1 - t) • x + t • (x + y) := by match_scalars <;> field_simp
   rw [h]
   exact hs hx hxy (fun h => hy <| add_left_cancel (a := x) (by rw [← h, add_zero]))
     (sub_pos_of_lt ht₁) ht₀ (sub_add_cancel 1 t)
@@ -359,16 +358,14 @@ theorem strictConvex_iff_div :
     StrictConvex 𝕜 s ↔
       s.Pairwise fun x y =>
         ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → (a / (a + b)) • x + (b / (a + b)) • y ∈ interior s :=
-  ⟨fun h x hx y hy hxy a b ha hb => by
-    apply h hx hy hxy (div_pos ha <| add_pos ha hb) (div_pos hb <| add_pos ha hb)
-    rw [← add_div]
-    exact div_self (add_pos ha hb).ne', fun h x hx y hy hxy a b ha hb hab => by
+  ⟨fun h x hx y hy hxy a b ha hb ↦ h hx hy hxy (by positivity) (by positivity) (by field_simp),
+    fun h x hx y hy hxy a b ha hb hab ↦ by
     convert h hx hy hxy ha hb <;> rw [hab, div_one]⟩
 
 theorem StrictConvex.mem_smul_of_zero_mem (hs : StrictConvex 𝕜 s) (zero_mem : (0 : E) ∈ s)
     (hx : x ∈ s) (hx₀ : x ≠ 0) {t : 𝕜} (ht : 1 < t) : x ∈ t • interior s := by
-  rw [mem_smul_set_iff_inv_smul_mem₀ (zero_lt_one.trans ht).ne']
-  exact hs.smul_mem_of_zero_mem zero_mem hx hx₀ (inv_pos.2 <| zero_lt_one.trans ht) (inv_lt_one ht)
+  rw [mem_smul_set_iff_inv_smul_mem₀ (by positivity)]
+  exact hs.smul_mem_of_zero_mem zero_mem hx hx₀ (by positivity) (inv_lt_one_of_one_lt₀ ht)
 
 end AddCommGroup
 
diff --git a/Mathlib/Analysis/Convex/StrictConvexBetween.lean b/Mathlib/Analysis/Convex/StrictConvexBetween.lean
index c3bb34bd7fb30..08a6eed3ee484 100644
--- a/Mathlib/Analysis/Convex/StrictConvexBetween.lean
+++ b/Mathlib/Analysis/Convex/StrictConvexBetween.lean
@@ -58,13 +58,13 @@ theorem Collinear.wbtw_of_dist_eq_of_dist_le {p p₁ p₂ p₃ : P} {r : ℝ}
     · simp [hp₃p₂]
     have hs : Sbtw ℝ p₂ p₃ p₁ := ⟨hw, hp₃p₂, hp₁p₃.symm⟩
     have hs' := hs.dist_lt_max_dist p
-    rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, or_false_iff] at hs'
+    rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, or_false] at hs'
     exact False.elim (hp₂.not_lt hs')
   · by_cases hp₁p₂ : p₁ = p₂
     · simp [hp₁p₂]
     have hs : Sbtw ℝ p₃ p₁ p₂ := ⟨hw, hp₁p₃, hp₁p₂⟩
     have hs' := hs.dist_lt_max_dist p
-    rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, false_or_iff] at hs'
+    rw [hp₁, hp₃, lt_max_iff, lt_self_iff_false, false_or] at hs'
     exact False.elim (hp₂.not_lt hs')
 
 /-- Given three collinear points, two (not equal) with distance `r` from `p` and one with
diff --git a/Mathlib/Analysis/Convex/StrictConvexSpace.lean b/Mathlib/Analysis/Convex/StrictConvexSpace.lean
index 80a4039dba35a..32a1cef9cd3a9 100644
--- a/Mathlib/Analysis/Convex/StrictConvexSpace.lean
+++ b/Mathlib/Analysis/Convex/StrictConvexSpace.lean
@@ -4,10 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Yury Kudryashov
 -/
 import Mathlib.Analysis.Convex.Normed
-import Mathlib.Analysis.Convex.Strict
-import Mathlib.Analysis.Normed.Order.Basic
-import Mathlib.Analysis.Normed.Affine.AddTorsor
-import Mathlib.Analysis.NormedSpace.Pointwise
 import Mathlib.Analysis.Normed.Module.Ray
 
 /-!
@@ -208,5 +204,5 @@ theorem not_sameRay_iff_abs_lt_norm_sub : ¬SameRay ℝ x y ↔ |‖x‖ - ‖y
 
 theorem norm_midpoint_lt_iff (h : ‖x‖ = ‖y‖) : ‖(1 / 2 : ℝ) • (x + y)‖ < ‖x‖ ↔ x ≠ y := by
   rw [norm_smul, Real.norm_of_nonneg (one_div_nonneg.2 zero_le_two), ← inv_eq_one_div, ←
-    div_eq_inv_mul, div_lt_iff (zero_lt_two' ℝ), mul_two, ← not_sameRay_iff_of_norm_eq h,
+    div_eq_inv_mul, div_lt_iff₀ (zero_lt_two' ℝ), mul_two, ← not_sameRay_iff_of_norm_eq h,
     not_sameRay_iff_norm_add_lt, h]
diff --git a/Mathlib/Analysis/Convex/Topology.lean b/Mathlib/Analysis/Convex/Topology.lean
index 312786bb4c364..c6d92df1bdcf2 100644
--- a/Mathlib/Analysis/Convex/Topology.lean
+++ b/Mathlib/Analysis/Convex/Topology.lean
@@ -26,9 +26,20 @@ open Metric Bornology Set Pointwise Convex
 
 variable {ι 𝕜 E : Type*}
 
-theorem Real.convex_iff_isPreconnected {s : Set ℝ} : Convex ℝ s ↔ IsPreconnected s :=
+namespace Real
+variable {s : Set ℝ} {r ε : ℝ}
+
+lemma closedBall_eq_segment (hε : 0 ≤ ε) : closedBall r ε = segment ℝ (r - ε) (r + ε) := by
+  rw [closedBall_eq_Icc, segment_eq_Icc ((sub_le_self _ hε).trans <| le_add_of_nonneg_right hε)]
+
+lemma ball_eq_openSegment (hε : 0 < ε) : ball r ε = openSegment ℝ (r - ε) (r + ε) := by
+  rw [ball_eq_Ioo, openSegment_eq_Ioo ((sub_lt_self _ hε).trans <| lt_add_of_pos_right _ hε)]
+
+theorem convex_iff_isPreconnected : Convex ℝ s ↔ IsPreconnected s :=
   convex_iff_ordConnected.trans isPreconnected_iff_ordConnected.symm
 
+end Real
+
 alias ⟨_, IsPreconnected.convex⟩ := Real.convex_iff_isPreconnected
 
 /-! ### Standard simplex -/
diff --git a/Mathlib/Analysis/Convex/Uniform.lean b/Mathlib/Analysis/Convex/Uniform.lean
index c57da9a0c876f..9e22c41ca4d16 100644
--- a/Mathlib/Analysis/Convex/Uniform.lean
+++ b/Mathlib/Analysis/Convex/Uniform.lean
@@ -72,7 +72,8 @@ theorem exists_forall_closed_ball_dist_add_le_two_sub (hε : 0 < ε) :
   have h₂ : ∀ z : E, ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ' := by
     rintro z hz hδz
     nth_rw 3 [← one_smul ℝ z]
-    rwa [← sub_smul, norm_smul_of_nonneg (sub_nonneg_of_le <| one_le_inv (hδ'.trans_le hδz) hz),
+    rwa [← sub_smul,
+      norm_smul_of_nonneg (sub_nonneg_of_le <| (one_le_inv₀ (hδ'.trans_le hδz)).2 hz),
       sub_mul, inv_mul_cancel₀ (hδ'.trans_le hδz).ne', one_mul, sub_le_comm]
   set x' := ‖x‖⁻¹ • x
   set y' := ‖y‖⁻¹ • y
diff --git a/Mathlib/Analysis/Convolution.lean b/Mathlib/Analysis/Convolution.lean
index 4b2168a084049..55806cb2f24fa 100644
--- a/Mathlib/Analysis/Convolution.lean
+++ b/Mathlib/Analysis/Convolution.lean
@@ -1031,10 +1031,10 @@ theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P
   let g' := fderiv 𝕜 ↿g
   have A : ∀ p ∈ s, Continuous (g p) := fun p hp ↦ by
     refine hg.continuousOn.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_
-    simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hp
+    simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp
   have A' : ∀ q : P × G, q.1 ∈ s → s ×ˢ univ ∈ 𝓝 q := fun q hq ↦ by
     apply (hs.prod isOpen_univ).mem_nhds
-    simpa only [mem_prod, mem_univ, and_true_iff] using hq
+    simpa only [mem_prod, mem_univ, and_true] using hq
   -- The derivative of `g` vanishes away from `k`.
   have g'_zero : ∀ p x, p ∈ s → x ∉ k → g' (p, x) = 0 := by
     intro p x hp hx
@@ -1055,7 +1055,7 @@ theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P
         hg.continuousOn_fderiv_of_isOpen (hs.prod isOpen_univ) le_rfl
       apply exists_isOpen_isBounded_image_of_isCompact_of_continuousOn A (hs.prod isOpen_univ) _ B
       simp only [prod_subset_prod_iff, hq₀, singleton_subset_iff, subset_univ, and_self_iff,
-        true_or_iff]
+        true_or]
     obtain ⟨ε, εpos, hε, h'ε⟩ :
       ∃ ε : ℝ, 0 < ε ∧ thickening ε ({q₀.fst} ×ˢ k) ⊆ t ∧ ball q₀.1 ε ⊆ s := by
       obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ thickening ε (({q₀.fst} : Set P) ×ˢ k) ⊆ t :=
@@ -1071,10 +1071,10 @@ theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P
     · have H : (p, x) ∈ t := by
         apply hε
         refine mem_thickening_iff.2 ⟨(q₀.1, x), ?_, ?_⟩
-        · simp only [hx, singleton_prod, mem_image, Prod.mk.inj_iff, eq_self_iff_true, true_and_iff,
+        · simp only [hx, singleton_prod, mem_image, Prod.mk.inj_iff, eq_self_iff_true, true_and,
             exists_eq_right]
         · rw [← dist_eq_norm] at hp
-          simpa only [Prod.dist_eq, εpos, dist_self, max_lt_iff, and_true_iff] using hp
+          simpa only [Prod.dist_eq, εpos, dist_self, max_lt_iff, and_true] using hp
       have : g' (p, x) ∈ closedBall (0 : P × G →L[𝕜] E') C := hC (mem_image_of_mem _ H)
       rwa [mem_closedBall_zero_iff] at this
     · have : g' (p, x) = 0 := g'_zero _ _ hps hx
@@ -1102,7 +1102,7 @@ theorem hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P
       hg.continuousOn_fderiv_of_isOpen (hs.prod isOpen_univ) le_rfl
     apply this.comp_continuous (continuous_const.prod_mk continuous_id')
     intro x
-    simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hq₀
+    simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hq₀
   set K' := (-k + {q₀.2} : Set G) with K'_def
   have hK' : IsCompact K' := hk.neg.add isCompact_singleton
   obtain ⟨U, U_open, K'U, hU⟩ : ∃ U, IsOpen U ∧ K' ⊆ U ∧ IntegrableOn f U μ :=
@@ -1251,12 +1251,12 @@ theorem contDiffOn_convolution_right_with_param {f : G → E} {n : ℕ∞} (L :
       apply hg.comp (isoP.prod isoG).contDiff.contDiffOn
       rintro ⟨p, x⟩ ⟨hp, -⟩
       simpa only [mem_preimage, ContinuousLinearEquiv.prod_apply, prod_mk_mem_set_prod_eq, mem_univ,
-        and_true_iff] using hp
+        and_true] using hp
   have A : ContDiffOn 𝕜 n (isoF ∘ R ∘ (isoP.prod isoG).symm) (s ×ˢ univ) := by
     apply isoF.contDiff.comp_contDiffOn
     apply R_contdiff.comp (ContinuousLinearEquiv.contDiff _).contDiffOn
     rintro ⟨p, x⟩ ⟨hp, -⟩
-    simpa only [mem_preimage, mem_prod, mem_univ, and_true_iff, ContinuousLinearEquiv.prod_symm,
+    simpa only [mem_preimage, mem_prod, mem_univ, and_true, ContinuousLinearEquiv.prod_symm,
       ContinuousLinearEquiv.prod_apply, ContinuousLinearEquiv.apply_symm_apply] using hp
   have : isoF ∘ R ∘ (isoP.prod isoG).symm = fun q : P × G => (f ⋆[L, μ] g q.1) q.2 := by
     apply funext
diff --git a/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean b/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean
index 8919c573b7372..530b964dfa9a1 100644
--- a/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean
+++ b/Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean
@@ -112,6 +112,8 @@ theorem ae_eq_zero_of_integral_smooth_smul_eq_zero [SigmaCompactSpace M]
     simpa [g_supp] using vK n
   simpa [this] using L
 
+-- An instance with keys containing `Opens`
+instance (U : Opens M) : BorelSpace U := inferInstanceAs (BorelSpace (U : Set M))
 
 /-- If a function `f` locally integrable on an open subset `U` of a finite-dimensional real
   manifold has zero integral when multiplied by any smooth function compactly supported
diff --git a/Mathlib/Analysis/Distribution/SchwartzSpace.lean b/Mathlib/Analysis/Distribution/SchwartzSpace.lean
index cb092fd0a5552..63f73727e30da 100644
--- a/Mathlib/Analysis/Distribution/SchwartzSpace.lean
+++ b/Mathlib/Analysis/Distribution/SchwartzSpace.lean
@@ -380,7 +380,7 @@ end Module
 
 section Seminorms
 
-/-! ### Seminorms on Schwartz space-/
+/-! ### Seminorms on Schwartz space -/
 
 
 variable [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F]
@@ -480,7 +480,7 @@ end Seminorms
 
 section Topology
 
-/-! ### The topology on the Schwartz space-/
+/-! ### The topology on the Schwartz space -/
 
 
 variable [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F]
@@ -533,7 +533,7 @@ theorem _root_.Function.HasTemperateGrowth.norm_iteratedFDeriv_le_uniform_aux {f
   choose k C f using hf_temperate.2
   use (Finset.range (n + 1)).sup k
   let C' := max (0 : ℝ) ((Finset.range (n + 1)).sup' (by simp) C)
-  have hC' : 0 ≤ C' := by simp only [C', le_refl, Finset.le_sup'_iff, true_or_iff, le_max_iff]
+  have hC' : 0 ≤ C' := by simp only [C', le_refl, Finset.le_sup'_iff, true_or, le_max_iff]
   use C', hC'
   intro N hN x
   rw [← Finset.mem_range_succ_iff] at hN
@@ -575,11 +575,11 @@ lemma _root_.ContinuousLinearMap.hasTemperateGrowth (f : E →L[ℝ] F) :
 
 variable [NormedAddCommGroup D] [MeasurableSpace D]
 
-open MeasureTheory FiniteDimensional
+open MeasureTheory Module
 
 /-- A measure `μ` has temperate growth if there is an `n : ℕ` such that `(1 + ‖x‖) ^ (- n)` is
 `μ`-integrable. -/
-class _root_.MeasureTheory.Measure.HasTemperateGrowth (μ : Measure D) : Prop :=
+class _root_.MeasureTheory.Measure.HasTemperateGrowth (μ : Measure D) : Prop where
   exists_integrable : ∃ (n : ℕ), Integrable (fun x ↦ (1 + ‖x‖) ^ (- (n : ℝ))) μ
 
 open Classical in
@@ -616,7 +616,7 @@ lemma pow_mul_le_of_le_of_pow_mul_le {C₁ C₂ : ℝ} {k l : ℕ} {x f : ℝ} (
   rw [this]
   rcases le_total x 1 with h'x|h'x
   · gcongr
-    · apply (pow_le_one k hx h'x).trans
+    · apply (pow_le_one₀ hx h'x).trans
       apply Real.one_le_rpow_of_pos_of_le_one_of_nonpos
       · linarith
       · linarith
@@ -854,7 +854,7 @@ def compCLM {g : D → E} (hg : g.HasTemperateGrowth)
         refine add_le_add ?_ (hg_upper' x)
         nth_rw 1 [← one_mul (1 : ℝ)]
         gcongr
-        apply one_le_pow_of_one_le
+        apply one_le_pow₀
         simp only [le_add_iff_nonneg_right, norm_nonneg]
       have hbound :
         ∀ i, i ≤ n → ‖iteratedFDeriv ℝ i f (g x)‖ ≤ 2 ^ k' * seminorm_f / (1 + ‖g x‖) ^ k' := by
@@ -870,8 +870,8 @@ def compCLM {g : D → E} (hg : g.HasTemperateGrowth)
         rw [mul_pow]
         have hN₁' := (lt_of_lt_of_le zero_lt_one hN₁).ne'
         gcongr
-        · exact le_trans (by simp [hC]) (le_self_pow (by simp [hC]) hN₁')
-        · refine le_self_pow (one_le_pow_of_one_le ?_ l) hN₁'
+        · exact le_trans (by simp [hC]) (le_self_pow₀ (by simp [hC]) hN₁')
+        · refine le_self_pow₀ (one_le_pow₀ ?_) hN₁'
           simp only [le_add_iff_nonneg_right, norm_nonneg]
       have := norm_iteratedFDeriv_comp_le f.smooth' hg.1 le_top x hbound hgrowth'
       have hxk : ‖x‖ ^ k ≤ (1 + ‖x‖) ^ k :=
@@ -1036,7 +1036,7 @@ section Integration
 /-! ### Integration -/
 
 
-open Real Complex Filter MeasureTheory MeasureTheory.Measure FiniteDimensional
+open Real Complex Filter MeasureTheory MeasureTheory.Measure Module
 
 variable [RCLike 𝕜]
 variable [NormedAddCommGroup D] [NormedSpace ℝ D]
@@ -1182,7 +1182,7 @@ instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass 𝓢(
     intro ε hε
     use (SchwartzMap.seminorm ℝ 1 0) f / ε
     intro x hx
-    rw [div_lt_iff hε] at hx
+    rw [div_lt_iff₀ hε] at hx
     have hxpos : 0 < ‖x‖ := by
       rw [norm_pos_iff']
       intro hxzero
@@ -1191,7 +1191,7 @@ instance instZeroAtInftyContinuousMapClass : ZeroAtInftyContinuousMapClass 𝓢(
     have := norm_pow_mul_le_seminorm ℝ f 1 x
     rw [pow_one, ← le_div_iff₀' hxpos] at this
     apply lt_of_le_of_lt this
-    rwa [div_lt_iff' hxpos]
+    rwa [div_lt_iff₀' hxpos]
 
 /-- Schwartz functions as continuous functions vanishing at infinity. -/
 def toZeroAtInfty (f : 𝓢(E, F)) : C₀(E, F) where
diff --git a/Mathlib/Analysis/Fourier/AddCircle.lean b/Mathlib/Analysis/Fourier/AddCircle.lean
index 86ac394b06e8d..7ca64c6a78543 100644
--- a/Mathlib/Analysis/Fourier/AddCircle.lean
+++ b/Mathlib/Analysis/Fourier/AddCircle.lean
@@ -10,7 +10,7 @@ import Mathlib.MeasureTheory.Function.ContinuousMapDense
 import Mathlib.MeasureTheory.Function.L2Space
 import Mathlib.MeasureTheory.Group.Integral
 import Mathlib.MeasureTheory.Integral.Periodic
-import Mathlib.Topology.ContinuousFunction.StoneWeierstrass
+import Mathlib.Topology.ContinuousMap.StoneWeierstrass
 import Mathlib.MeasureTheory.Integral.FundThmCalculus
 
 /-!
diff --git a/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean b/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
index 588af56dc65eb..0fbe303edbe46 100644
--- a/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
+++ b/Mathlib/Analysis/Fourier/FourierTransformDeriv.lean
@@ -327,8 +327,7 @@ lemma norm_fourierPowSMulRight_le (f : V → E) (v : V) (n : ℕ) :
       simp [_root_.abs_of_nonneg pi_nonneg, norm_smul]
   _ ≤ (2 * π) ^ n * ((∏ x : Fin n, ‖L‖ * ‖v‖ * ‖m x‖) * ‖f v‖) := by
       gcongr with i _hi
-      · exact fun i _hi ↦ abs_nonneg _
-      · exact L.le_opNorm₂ v (m i)
+      exact L.le_opNorm₂ v (m i)
   _ = (2 * π * ‖L‖) ^ n * ‖v‖ ^ n * ‖f v‖ * ∏ i : Fin n, ‖m i‖ := by
       simp [Finset.prod_mul_distrib, mul_pow]; ring
 
@@ -414,7 +413,7 @@ lemma norm_iteratedFDeriv_fourierPowSMulRight
     · rw [← Nat.cast_pow, Nat.cast_le]
       calc n.descFactorial i ≤ n ^ i := Nat.descFactorial_le_pow _ _
       _ ≤ (n + 1) ^ i := pow_le_pow_left (by omega) (by omega) i
-      _ ≤ (n + 1) ^ k := pow_le_pow_right (by omega) (Finset.mem_range_succ_iff.mp hi)
+      _ ≤ (n + 1) ^ k := pow_right_mono₀ (by omega) (Finset.mem_range_succ_iff.mp hi)
     · exact hv _ (by omega) _ (by omega)
   _ = (2 * n + 2) ^ k * (‖L‖^n * C) := by
     simp only [← Finset.sum_mul, ← Nat.cast_sum, Nat.sum_range_choose, mul_one, ← mul_assoc,
@@ -448,7 +447,7 @@ lemma hasFTaylorSeriesUpTo_fourierIntegral {N : ℕ∞}
       (fun w n ↦ fourierIntegral 𝐞 μ L.toLinearMap₂ (fun v ↦ fourierPowSMulRight L f v n) w) := by
   constructor
   · intro w
-    rw [uncurry0_apply, Matrix.zero_empty, fourierIntegral_continuousMultilinearMap_apply'
+    rw [curry0_apply, Matrix.zero_empty, fourierIntegral_continuousMultilinearMap_apply'
       (integrable_fourierPowSMulRight L (hf 0 bot_le) h'f)]
     simp only [fourierPowSMulRight_apply, pow_zero, Finset.univ_eq_empty, Finset.prod_empty,
       one_smul]
@@ -456,7 +455,7 @@ lemma hasFTaylorSeriesUpTo_fourierIntegral {N : ℕ∞}
     have I₁ : Integrable (fun v ↦ fourierPowSMulRight L f v n) μ :=
       integrable_fourierPowSMulRight L (hf n hn.le) h'f
     have I₂ : Integrable (fun v ↦ ‖v‖ * ‖fourierPowSMulRight L f v n‖) μ := by
-      apply ((hf (n+1) (ENat.add_one_le_of_lt hn)).const_mul ((2 * π * ‖L‖) ^ n)).mono'
+      apply ((hf (n+1) (Order.add_one_le_of_lt hn)).const_mul ((2 * π * ‖L‖) ^ n)).mono'
         (continuous_norm.aestronglyMeasurable.mul (h'f.fourierPowSMulRight L n).norm)
       filter_upwards with v
       simp only [Pi.mul_apply, norm_mul, norm_norm]
@@ -466,7 +465,7 @@ lemma hasFTaylorSeriesUpTo_fourierIntegral {N : ℕ∞}
           gcongr; apply norm_fourierPowSMulRight_le
       _ = (2 * π * ‖L‖) ^ n * (‖v‖ ^ (n + 1) * ‖f v‖) := by rw [pow_succ]; ring
     have I₃ : Integrable (fun v ↦ fourierPowSMulRight L f v (n + 1)) μ :=
-      integrable_fourierPowSMulRight L (hf (n + 1) (ENat.add_one_le_of_lt hn)) h'f
+      integrable_fourierPowSMulRight L (hf (n + 1) (Order.add_one_le_of_lt hn)) h'f
     have I₄ : Integrable
         (fun v ↦ fourierSMulRight L (fun v ↦ fourierPowSMulRight L f v n) v) μ := by
       apply (I₂.const_mul ((2 * π * ‖L‖))).mono' (h'f.fourierPowSMulRight L n).fourierSMulRight
@@ -628,7 +627,7 @@ lemma pow_mul_norm_iteratedFDeriv_fourierIntegral_le [FiniteDimensional ℝ V]
   _ ≤ (2 * π) ^ n
       * (|L v w| ^ n * ‖iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f) w‖) := by
     apply le_mul_of_one_le_left (by positivity)
-    apply one_le_pow_of_one_le
+    apply one_le_pow₀
     linarith [one_le_pi_div_two]
   _ = ‖fourierPowSMulRight (-L.flip)
         (iteratedFDeriv ℝ k (fourierIntegral 𝐞 μ L.toLinearMap₂ f)) w n (fun _ ↦ v)‖ := by
diff --git a/Mathlib/Analysis/Fourier/Inversion.lean b/Mathlib/Analysis/Fourier/Inversion.lean
index cea8c2f5ade12..158aae8391b20 100644
--- a/Mathlib/Analysis/Fourier/Inversion.lean
+++ b/Mathlib/Analysis/Fourier/Inversion.lean
@@ -37,7 +37,7 @@ To check the concentration property of the middle factor and the fact that it ha
 rely on the explicit computation of the Fourier transform of Gaussians.
 -/
 
-open Filter MeasureTheory Complex FiniteDimensional Metric Real Bornology
+open Filter MeasureTheory Complex Module Metric Real Bornology
 
 open scoped Topology FourierTransform RealInnerProductSpace Complex
 
diff --git a/Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean b/Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean
index 9d6a346e16240..1531edb4f4be0 100644
--- a/Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean
+++ b/Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean
@@ -9,9 +9,9 @@ import Mathlib.Analysis.InnerProductSpace.EuclideanDist
 import Mathlib.MeasureTheory.Function.ContinuousMapDense
 import Mathlib.MeasureTheory.Group.Integral
 import Mathlib.MeasureTheory.Integral.SetIntegral
-import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
 import Mathlib.Topology.EMetricSpace.Paracompact
 import Mathlib.MeasureTheory.Measure.Haar.Unique
+import Mathlib.Topology.Algebra.Module.WeakDual
 
 /-!
 # The Riemann-Lebesgue Lemma
@@ -45,7 +45,7 @@ equivalence to an inner-product space.
 
 noncomputable section
 
-open MeasureTheory Filter Complex Set FiniteDimensional
+open MeasureTheory Filter Complex Set Module
 
 open scoped Filter Topology Real ENNReal FourierTransform RealInnerProductSpace NNReal
 
@@ -140,7 +140,7 @@ theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
   have : ‖(1 / 2 : ℂ)‖ = 2⁻¹ := by norm_num
   rw [fourierIntegral_eq_half_sub_half_period_translate hw_ne
       (hf1.integrable_of_hasCompactSupport hf2),
-    norm_smul, this, inv_mul_eq_div, div_lt_iff' two_pos]
+    norm_smul, this, inv_mul_eq_div, div_lt_iff₀' two_pos]
   refine lt_of_le_of_lt (norm_integral_le_integral_norm _) ?_
   simp_rw [Circle.norm_smul]
   --* Show integral can be taken over A only.
@@ -164,8 +164,8 @@ theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
     simp_rw [norm_norm]
     simp_rw [dist_eq_norm] at hδ2
     refine fun x _ => (hδ2 ?_).le
-    rw [sub_add_cancel_left, norm_neg, hw'_nm, ← div_div, div_lt_iff (norm_pos_iff.mpr hw_ne), ←
-      div_lt_iff' hδ1, div_div]
+    rw [sub_add_cancel_left, norm_neg, hw'_nm, ← div_div, div_lt_iff₀ (norm_pos_iff.mpr hw_ne), ←
+      div_lt_iff₀' hδ1, div_div]
     exact (lt_add_of_pos_left _ one_half_pos).trans_le hw_bd
   have bdA2 := norm_setIntegral_le_of_norm_le_const (hB_vol.trans_lt ENNReal.coe_lt_top) bdA ?_
   swap
@@ -177,7 +177,7 @@ theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
     Real.norm_of_nonneg (setIntegral_nonneg mA fun x _ => norm_nonneg _)
   rw [this] at bdA2
   refine bdA2.trans_lt ?_
-  rw [div_mul_eq_mul_div, div_lt_iff (NNReal.coe_pos.mpr hB_pos), mul_comm (2 : ℝ), mul_assoc,
+  rw [div_mul_eq_mul_div, div_lt_iff₀ (NNReal.coe_pos.mpr hB_pos), mul_comm (2 : ℝ), mul_assoc,
     mul_lt_mul_left hε]
   rw [← ENNReal.toReal_le_toReal] at hB_vol
   · refine hB_vol.trans_lt ?_
@@ -262,41 +262,11 @@ theorem tendsto_integral_exp_smul_cocompact (μ : Measure V) [μ.IsAddHaarMeasur
   borelize V'
   -- various equivs derived from A
   let Aₘ : MeasurableEquiv V V' := A.toHomeomorph.toMeasurableEquiv
-  -- isomorphism between duals derived from A -- need to do continuity as a separate step in order
-  -- to apply `LinearMap.continuous_of_finiteDimensional`.
-  let Adualₗ : (V →L[ℝ] ℝ) ≃ₗ[ℝ] V' →L[ℝ] ℝ :=
-    { toFun := fun t => t.comp A.symm.toContinuousLinearMap
-      invFun := fun t => t.comp A.toContinuousLinearMap
-      map_add' := by
-        intro t s
-        ext1 v
-        simp only [ContinuousLinearMap.coe_comp', Function.comp_apply,
-          ContinuousLinearMap.add_apply]
-      map_smul' := by
-        intro x f
-        ext1 v
-        simp only [RingHom.id_apply, ContinuousLinearMap.coe_comp', Function.comp_apply,
-          ContinuousLinearMap.smul_apply]
-      left_inv := by
-        intro w
-        ext1 v
-        simp only [ContinuousLinearMap.coe_comp',
-          ContinuousLinearEquiv.coe_coe, Function.comp_apply,
-          ContinuousLinearEquiv.symm_apply_apply]
-      right_inv := by
-        intro w
-        ext1 v
-        simp only [ContinuousLinearMap.coe_comp',
-          ContinuousLinearEquiv.coe_coe, Function.comp_apply,
-          ContinuousLinearEquiv.apply_symm_apply] }
-  let Adual : (V →L[ℝ] ℝ) ≃L[ℝ] V' →L[ℝ] ℝ :=
-    { Adualₗ with
-      continuous_toFun := Adualₗ.toLinearMap.continuous_of_finiteDimensional
-      continuous_invFun := Adualₗ.symm.toLinearMap.continuous_of_finiteDimensional }
-  have : (μ.map Aₘ).IsAddHaarMeasure := Measure.MapContinuousLinearEquiv.isAddHaarMeasure _ A
-  convert
-    (tendsto_integral_exp_smul_cocompact_of_inner_product (f ∘ A.symm) (μ.map Aₘ)).comp
-      Adual.toHomeomorph.toCocompactMap.cocompact_tendsto' with w
+  -- isomorphism between duals derived from A
+  let Adual : (V →L[ℝ] ℝ) ≃L[ℝ] V' →L[ℝ] ℝ := A.arrowCongrSL (.refl _ _)
+  have : (μ.map Aₘ).IsAddHaarMeasure := A.isAddHaarMeasure_map _
+  convert (tendsto_integral_exp_smul_cocompact_of_inner_product (f ∘ A.symm) (μ.map Aₘ)).comp
+    Adual.toHomeomorph.toCocompactMap.cocompact_tendsto' with w
   rw [Function.comp_apply, integral_map_equiv]
   congr 1 with v : 1
   congr
diff --git a/Mathlib/Analysis/Fourier/ZMod.lean b/Mathlib/Analysis/Fourier/ZMod.lean
index e817057d937c2..4fad478f0bc2c 100644
--- a/Mathlib/Analysis/Fourier/ZMod.lean
+++ b/Mathlib/Analysis/Fourier/ZMod.lean
@@ -3,6 +3,7 @@ Copyright (c) 2024 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
+import Mathlib.Algebra.Group.EvenFunction
 import Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar
 import Mathlib.Analysis.Fourier.FourierTransform
 import Mathlib.NumberTheory.DirichletCharacter.GaussSum
@@ -177,6 +178,25 @@ lemma dft_comp_unitMul (Φ : ZMod N → E) (u : (ZMod N)ˣ) (k : ZMod N) :
   refine Fintype.sum_equiv u.mulLeft _ _ fun x ↦ ?_
   simp only [mul_comm u.val, u.mulLeft_apply, ← mul_assoc, u.mul_inv_cancel_right]
 
+section signs
+
+/-- The discrete Fourier transform of `Φ` is even if and only if `Φ` itself is. -/
+lemma dft_even_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Even ↔ Φ.Even := by
+  have h {f : ZMod N → ℂ} (hf : f.Even) : (𝓕 f).Even := by
+    simp only [Function.Even, ← congr_fun (dft_comp_neg f), funext hf, implies_true]
+  refine ⟨fun hΦ x ↦ ?_, h⟩
+  simpa only [neg_neg, smul_right_inj (NeZero.ne (N : ℂ)), dft_dft] using h hΦ (-x)
+
+/-- The discrete Fourier transform of `Φ` is odd if and only if `Φ` itself is. -/
+lemma dft_odd_iff {Φ : ZMod N → ℂ} : (𝓕 Φ).Odd ↔ Φ.Odd := by
+  have h {f : ZMod N → ℂ} (hf : f.Odd) : (𝓕 f).Odd := by
+    simp only [Function.Odd, ← congr_fun (dft_comp_neg f), funext hf, ← Pi.neg_apply, map_neg,
+      implies_true]
+  refine ⟨fun hΦ x ↦ ?_, h⟩
+  simpa only [neg_neg, dft_dft, ← smul_neg, smul_right_inj (NeZero.ne (N : ℂ))] using h hΦ (-x)
+
+end signs
+
 end ZMod
 
 namespace DirichletCharacter
diff --git a/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean b/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
index 786527166c2b3..8164bbd406fee 100644
--- a/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
+++ b/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
@@ -349,7 +349,7 @@ theorem lintegral_pow_le_pow_lintegral_fderiv_aux [Fintype ι]
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]
   [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
 
-open FiniteDimensional
+open Module
 
 /-- The constant factor occurring in the conclusion of `lintegral_pow_le_pow_lintegral_fderiv`.
 It only depends on `E`, `μ` and `p`.
@@ -421,7 +421,7 @@ theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
         refine (Continuous.nnnorm ?_).measurable.coe_nnreal_ennreal
         exact (hu.continuous_fderiv le_rfl).comp e.symm.continuous
     _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p : ℝ≥0) * (∫⁻ y, ‖fderiv ℝ u (e.symm y)‖₊) ^ p := by
-        rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ENNReal.coe_rpow_of_nonneg _ h0p]
+        rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← ENNReal.coe_rpow_of_nonneg _ h0p]
     _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p : ℝ≥0)
         * (∫⁻ x, ‖fderiv ℝ u x‖₊ ∂(volume : Measure (ι → ℝ)).map e.symm) ^ p := by
         congr
@@ -430,7 +430,7 @@ theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
         fun_prop
   rw [← ENNReal.mul_le_mul_left h3c ENNReal.coe_ne_top, ← mul_assoc, ← ENNReal.coe_mul, ← hC,
     ENNReal.coe_mul] at this
-  rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← mul_assoc, ENNReal.coe_rpow_of_ne_zero hc.ne']
+  rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← mul_assoc, ← ENNReal.coe_rpow_of_ne_zero hc.ne']
   exact this
 
 /-- The constant factor occurring in the conclusion of `eLpNorm_le_eLpNorm_fderiv_one`.
@@ -448,7 +448,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_one  {u : E → F} (hu : ContDiff ℝ 1 u) (h2
   have h0p : 0 < (p : ℝ) := hp.coe.symm.pos
   rw [eLpNorm_one_eq_lintegral_nnnorm,
     ← ENNReal.rpow_le_rpow_iff h0p, ENNReal.mul_rpow_of_nonneg _ _ h0p.le,
-    ENNReal.coe_rpow_of_nonneg _ h0p.le, eLpNormLESNormFDerivOneConst, ← NNReal.rpow_mul,
+    ← ENNReal.coe_rpow_of_nonneg _ h0p.le, eLpNormLESNormFDerivOneConst, ← NNReal.rpow_mul,
     eLpNorm_nnreal_pow_eq_lintegral hp.symm.pos.ne',
     inv_mul_cancel₀ h0p.ne', NNReal.rpow_one]
   exact lintegral_pow_le_pow_lintegral_fderiv μ hu h2u hp.coe
@@ -489,7 +489,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
     have : 0 < p⁻¹ - (n : ℝ)⁻¹ :=
       NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (inv_ne_zero hp'0)) |>.trans_eq hp'
     rwa [NNReal.coe_inv, sub_pos,
-      inv_lt_inv _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this
+      inv_lt_inv₀ _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this
     exact_mod_cast hn
   have h0n : 2 ≤ n := Nat.succ_le_of_lt <| Nat.one_lt_cast.mp <| hp.trans_lt h2p
   have hn : NNReal.IsConjExponent n n' := .conjExponent (by norm_cast)
@@ -514,7 +514,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
   have h0p' : p' ≠ 0 := by
     suffices 0 < (p' : ℝ) from (show 0 < p' from this) |>.ne'
     rw [← inv_pos, hp', sub_pos]
-    exact inv_lt_inv_of_lt hq.pos h2p
+    exact inv_strictAnti₀ hq.pos h2p
   have h2q : 1 / n' - 1 / q = 1 / p' := by
     simp_rw (config := {zeta := false}) [one_div, hp']
     rw [← hq.one_sub_inv, ← hn.coe.one_sub_inv, sub_sub_sub_cancel_left]
@@ -553,13 +553,13 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
   calc (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / (n' : ℝ)) = eLpNorm v n' μ := by
         rw [← h2γ, eLpNorm_nnreal_eq_lintegral hn.symm.pos.ne']
         simp (discharger := positivity) [v, Real.nnnorm_rpow_of_nonneg, ENNReal.rpow_mul,
-          ENNReal.coe_rpow_of_nonneg]
+          ← ENNReal.coe_rpow_of_nonneg]
     _ ≤ C * eLpNorm (fderiv ℝ v) 1 μ := eLpNorm_le_eLpNorm_fderiv_one μ hv h2v hn
     _ = C * ∫⁻ x, ‖fderiv ℝ v x‖₊ ∂μ := by rw [eLpNorm_one_eq_lintegral_nnnorm]
     _ ≤ C * γ * ∫⁻ x, ‖u x‖₊ ^ ((γ : ℝ) - 1) * ‖fderiv ℝ u x‖₊ ∂μ := by
       rw [mul_assoc, ← lintegral_const_mul γ]
       gcongr
-      simp_rw [← mul_assoc, ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.mpr h1γ.le)]
+      simp_rw [← mul_assoc, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.mpr h1γ.le)]
       exact ENNReal.coe_le_coe.mpr <| nnnorm_fderiv_norm_rpow_le (hu.differentiable le_rfl) h1γ
       fun_prop
     _ ≤ C * γ * ((∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) *
@@ -634,7 +634,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq [FiniteDimensional ℝ F]
       (fderiv ℝ e (u x)).opNorm_comp_le (fderiv ℝ u x)
     _ = C₂ * ‖fderiv ℝ u x‖ := by simp_rw [e.fderiv, C₂, coe_nnnorm]
   calc eLpNorm u p' μ
-      = eLpNorm (e.symm ∘ v) p' μ := by simp_rw [v, Function.comp, e.symm_apply_apply]
+      = eLpNorm (e.symm ∘ v) p' μ := by simp_rw [v, Function.comp_def, e.symm_apply_apply]
     _ ≤ C₁ • eLpNorm v p' μ := by
       apply eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul
       exact Eventually.of_forall (fun x ↦ (e.symm : F' →L[ℝ] F).le_opNNNorm _)
@@ -684,7 +684,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
   have : (q : ℝ≥0∞) ≤ p' := by
     have H : (p' : ℝ)⁻¹ ≤ (↑q)⁻¹ := trans hp' hpq
     norm_cast at H ⊢
-    rwa [inv_le_inv] at H
+    rwa [inv_le_inv₀] at H
     · dsimp
       have : 0 < p⁻¹ - (finrank ℝ E : ℝ≥0)⁻¹ := by
         simp only [tsub_pos_iff_lt]
@@ -697,7 +697,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
       = eLpNorm u q (μ.restrict s) := by rw [eLpNorm_restrict_eq_of_support_subset h2u]
     _ ≤ eLpNorm u p' (μ.restrict s) * t := by
         convert eLpNorm_le_eLpNorm_mul_rpow_measure_univ this hu.continuous.aestronglyMeasurable
-        rw [← ENNReal.coe_rpow_of_nonneg]
+        rw [ENNReal.coe_rpow_of_nonneg]
         · simp [ENNReal.coe_toNNReal hs.measure_lt_top.ne]
         · rw [one_div, one_div]
           norm_cast
diff --git a/Mathlib/Analysis/InnerProductSpace/Adjoint.lean b/Mathlib/Analysis/InnerProductSpace/Adjoint.lean
index c4a1bcd72f1b5..d70316a5027b3 100644
--- a/Mathlib/Analysis/InnerProductSpace/Adjoint.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Adjoint.lean
@@ -564,7 +564,7 @@ lemma Matrix.toLin_conjTranspose (A : Matrix m n 𝕜) :
     orthonormal_iff_ite.mp v₁.orthonormal, orthonormal_iff_ite.mp v₂.orthonormal]
 
 /-- The matrix associated to the adjoint of a linear map corresponding to two orthonormal bases
-is the conjugate tranpose of the matrix associated to the linear map. -/
+is the conjugate transpose of the matrix associated to the linear map. -/
 lemma LinearMap.toMatrix_adjoint (f : E →ₗ[𝕜] F) :
     toMatrix v₂.toBasis v₁.toBasis (adjoint f) = (toMatrix v₁.toBasis v₂.toBasis f)ᴴ :=
   toLin v₂.toBasis v₁.toBasis |>.injective <| by simp [toLin_conjTranspose]
diff --git a/Mathlib/Analysis/InnerProductSpace/Basic.lean b/Mathlib/Analysis/InnerProductSpace/Basic.lean
index f1bde23312b08..d4672630df1ed 100644
--- a/Mathlib/Analysis/InnerProductSpace/Basic.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Basic.lean
@@ -68,7 +68,7 @@ noncomputable section
 
 open RCLike Real Filter
 
-open Topology ComplexConjugate
+open Topology ComplexConjugate Finsupp
 
 open LinearMap (BilinForm)
 
@@ -82,7 +82,7 @@ class Inner (𝕜 E : Type*) where
 export Inner (inner)
 
 /-- The inner product with values in `𝕜`. -/
-notation3:max "⟪" x ", " y "⟫_" 𝕜:max => @inner 𝕜 _ _ x y
+scoped[InnerProductSpace] notation3:max "⟪" x ", " y "⟫_" 𝕜:max => @inner 𝕜 _ _ x y
 
 section Notations
 
@@ -447,7 +447,7 @@ theorem inner_self_eq_zero {x : F} : ⟪x, x⟫ = 0 ↔ x = 0 :=
 
 theorem normSq_eq_zero {x : F} : normSqF x = 0 ↔ x = 0 :=
   Iff.trans
-    (by simp only [normSq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true_iff])
+    (by simp only [normSq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true])
     (@inner_self_eq_zero 𝕜 _ _ _ _ _ x)
 
 theorem inner_self_ne_zero {x : F} : ⟪x, x⟫ ≠ 0 ↔ x ≠ 0 :=
@@ -508,9 +508,10 @@ end
 
 /-! ### Properties of inner product spaces -/
 
-
 section BasicProperties_Seminormed
 
+open scoped InnerProductSpace
+
 variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 variable [SeminormedAddCommGroup F] [InnerProductSpace ℝ F]
 
@@ -547,8 +548,30 @@ theorem inner_re_symm (x y : E) : re ⟪x, y⟫ = re ⟪y, x⟫ := by rw [← in
 
 theorem inner_im_symm (x y : E) : im ⟪x, y⟫ = -im ⟪y, x⟫ := by rw [← inner_conj_symm, conj_im]
 
+section Algebra
+variable {𝕝 : Type*} [CommSemiring 𝕝] [StarRing 𝕝] [Algebra 𝕝 𝕜] [Module 𝕝 E]
+  [IsScalarTower 𝕝 𝕜 E] [StarModule 𝕝 𝕜]
+
+/-- See `inner_smul_left` for the common special when `𝕜 = 𝕝`. -/
+lemma inner_smul_left_eq_star_smul (x y : E) (r : 𝕝) : ⟪r • x, y⟫ = r† • ⟪x, y⟫ := by
+  rw [← algebraMap_smul 𝕜 r, InnerProductSpace.smul_left, starRingEnd_apply, starRingEnd_apply,
+    ← algebraMap_star_comm, ← smul_eq_mul, algebraMap_smul]
+
+/-- Special case of `inner_smul_left_eq_star_smul` when the acting ring has a trivial star
+(eg `ℕ`, `ℤ`, `ℚ≥0`, `ℚ`, `ℝ`). -/
+lemma inner_smul_left_eq_smul [TrivialStar 𝕝] (x y : E) (r : 𝕝) : ⟪r • x, y⟫ = r • ⟪x, y⟫ := by
+  rw [inner_smul_left_eq_star_smul, starRingEnd_apply, star_trivial]
+
+/-- See `inner_smul_right` for the common special when `𝕜 = 𝕝`. -/
+lemma inner_smul_right_eq_smul (x y : E) (r : 𝕝) : ⟪x, r • y⟫ = r • ⟪x, y⟫ := by
+  rw [← inner_conj_symm, inner_smul_left_eq_star_smul, starRingEnd_apply, starRingEnd_apply,
+    star_smul, star_star, ← starRingEnd_apply, inner_conj_symm]
+
+end Algebra
+
+/-- See `inner_smul_left_eq_star_smul` for the case of a general algebra action. -/
 theorem inner_smul_left (x y : E) (r : 𝕜) : ⟪r • x, y⟫ = r† * ⟪x, y⟫ :=
-  InnerProductSpace.smul_left _ _ _
+  inner_smul_left_eq_star_smul ..
 
 theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
   inner_smul_left _ _ _
@@ -556,8 +579,9 @@ theorem real_inner_smul_left (x y : F) (r : ℝ) : ⟪r • x, y⟫_ℝ = r * 
 theorem inner_smul_real_left (x y : E) (r : ℝ) : ⟪(r : 𝕜) • x, y⟫ = r • ⟪x, y⟫ := by
   rw [inner_smul_left, conj_ofReal, Algebra.smul_def]
 
-theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by
-  rw [← inner_conj_symm, inner_smul_left, RingHom.map_mul, conj_conj, inner_conj_symm]
+/-- See `inner_smul_right_eq_smul` for the case of a general algebra action. -/
+theorem inner_smul_right (x y : E) (r : 𝕜) : ⟪x, r • y⟫ = r * ⟪x, y⟫ :=
+  inner_smul_right_eq_smul ..
 
 theorem real_inner_smul_right (x y : F) (r : ℝ) : ⟪x, r • y⟫_ℝ = r * ⟪x, y⟫_ℝ :=
   inner_smul_right _ _ _
@@ -591,26 +615,26 @@ theorem inner_sum {ι : Type*} (s : Finset ι) (f : ι → E) (x : E) :
   map_sum (LinearMap.flip sesqFormOfInner x) _ _
 
 /-- An inner product with a sum on the left, `Finsupp` version. -/
-theorem Finsupp.sum_inner {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
+protected theorem Finsupp.sum_inner {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
     ⟪l.sum fun (i : ι) (a : 𝕜) => a • v i, x⟫ = l.sum fun (i : ι) (a : 𝕜) => conj a • ⟪v i, x⟫ := by
-  convert _root_.sum_inner (𝕜 := 𝕜) l.support (fun a => l a • v a) x
+  convert sum_inner (𝕜 := 𝕜) l.support (fun a => l a • v a) x
   simp only [inner_smul_left, Finsupp.sum, smul_eq_mul]
 
 /-- An inner product with a sum on the right, `Finsupp` version. -/
-theorem Finsupp.inner_sum {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
+protected theorem Finsupp.inner_sum {ι : Type*} (l : ι →₀ 𝕜) (v : ι → E) (x : E) :
     ⟪x, l.sum fun (i : ι) (a : 𝕜) => a • v i⟫ = l.sum fun (i : ι) (a : 𝕜) => a • ⟪x, v i⟫ := by
-  convert _root_.inner_sum (𝕜 := 𝕜) l.support (fun a => l a • v a) x
+  convert inner_sum (𝕜 := 𝕜) l.support (fun a => l a • v a) x
   simp only [inner_smul_right, Finsupp.sum, smul_eq_mul]
 
-theorem DFinsupp.sum_inner {ι : Type*} [DecidableEq ι] {α : ι → Type*}
+protected theorem DFinsupp.sum_inner {ι : Type*} [DecidableEq ι] {α : ι → Type*}
     [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E)
     (l : Π₀ i, α i) (x : E) : ⟪l.sum f, x⟫ = l.sum fun i a => ⟪f i a, x⟫ := by
-  simp (config := { contextual := true }) only [DFinsupp.sum, _root_.sum_inner, smul_eq_mul]
+  simp (config := { contextual := true }) only [DFinsupp.sum, sum_inner, smul_eq_mul]
 
-theorem DFinsupp.inner_sum {ι : Type*} [DecidableEq ι] {α : ι → Type*}
+protected theorem DFinsupp.inner_sum {ι : Type*} [DecidableEq ι] {α : ι → Type*}
     [∀ i, AddZeroClass (α i)] [∀ (i) (x : α i), Decidable (x ≠ 0)] (f : ∀ i, α i → E)
     (l : Π₀ i, α i) (x : E) : ⟪x, l.sum f⟫ = l.sum fun i a => ⟪x, f i a⟫ := by
-  simp (config := { contextual := true }) only [DFinsupp.sum, _root_.inner_sum, smul_eq_mul]
+  simp (config := { contextual := true }) only [DFinsupp.sum, inner_sum, smul_eq_mul]
 
 @[simp]
 theorem inner_zero_left (x : E) : ⟪0, x⟫ = 0 := by
@@ -754,6 +778,7 @@ variable {𝕜}
 theorem inner_self_nonpos {x : E} : re ⟪x, x⟫ ≤ 0 ↔ x = 0 := by
   rw [← norm_sq_eq_inner, (sq_nonneg _).le_iff_eq, sq_eq_zero_iff, norm_eq_zero]
 
+open scoped InnerProductSpace in
 theorem real_inner_self_nonpos {x : F} : ⟪x, x⟫_ℝ ≤ 0 ↔ x = 0 :=
   @inner_self_nonpos ℝ F _ _ _ x
 
@@ -829,9 +854,9 @@ theorem orthonormal_subtype_iff_ite [DecidableEq E] {s : Set E} :
 /-- The inner product of a linear combination of a set of orthonormal vectors with one of those
 vectors picks out the coefficient of that vector. -/
 theorem Orthonormal.inner_right_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
-    ⟪v i, Finsupp.total 𝕜 v l⟫ = l i := by
+    ⟪v i, linearCombination 𝕜 v l⟫ = l i := by
   classical
-  simpa [Finsupp.total_apply, Finsupp.inner_sum, orthonormal_iff_ite.mp hv] using Eq.symm
+  simpa [linearCombination_apply, Finsupp.inner_sum, orthonormal_iff_ite.mp hv] using Eq.symm
 
 /-- The inner product of a linear combination of a set of orthonormal vectors with one of those
 vectors picks out the coefficient of that vector. -/
@@ -849,7 +874,7 @@ theorem Orthonormal.inner_right_fintype [Fintype ι] {v : ι → E} (hv : Orthon
 /-- The inner product of a linear combination of a set of orthonormal vectors with one of those
 vectors picks out the coefficient of that vector. -/
 theorem Orthonormal.inner_left_finsupp {v : ι → E} (hv : Orthonormal 𝕜 v) (l : ι →₀ 𝕜) (i : ι) :
-    ⟪Finsupp.total 𝕜 v l, v i⟫ = conj (l i) := by rw [← inner_conj_symm, hv.inner_right_finsupp]
+    ⟪linearCombination 𝕜 v l, v i⟫ = conj (l i) := by rw [← inner_conj_symm, hv.inner_right_finsupp]
 
 /-- The inner product of a linear combination of a set of orthonormal vectors with one of those
 vectors picks out the coefficient of that vector. -/
@@ -868,19 +893,20 @@ theorem Orthonormal.inner_left_fintype [Fintype ι] {v : ι → E} (hv : Orthono
 /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as
 a sum over the first `Finsupp`. -/
 theorem Orthonormal.inner_finsupp_eq_sum_left {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
-    ⟪Finsupp.total 𝕜 v l₁, Finsupp.total 𝕜 v l₂⟫ = l₁.sum fun i y => conj y * l₂ i := by
-  simp only [l₁.total_apply _, Finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul]
+    ⟪linearCombination 𝕜 v l₁, linearCombination 𝕜 v l₂⟫ = l₁.sum fun i y => conj y * l₂ i := by
+  simp only [l₁.linearCombination_apply _, Finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul]
 
 /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as
 a sum over the second `Finsupp`. -/
 theorem Orthonormal.inner_finsupp_eq_sum_right {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι →₀ 𝕜) :
-    ⟪Finsupp.total 𝕜 v l₁, Finsupp.total 𝕜 v l₂⟫ = l₂.sum fun i y => conj (l₁ i) * y := by
-  simp only [l₂.total_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul]
+    ⟪linearCombination 𝕜 v l₁, linearCombination 𝕜 v l₂⟫ = l₂.sum fun i y => conj (l₁ i) * y := by
+  simp only [l₂.linearCombination_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm,
+             smul_eq_mul]
 
 /-- The inner product of two linear combinations of a set of orthonormal vectors, expressed as
 a sum. -/
-theorem Orthonormal.inner_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜) (s : Finset ι) :
-    ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, conj (l₁ i) * l₂ i := by
+protected theorem Orthonormal.inner_sum {v : ι → E} (hv : Orthonormal 𝕜 v) (l₁ l₂ : ι → 𝕜)
+    (s : Finset ι) : ⟪∑ i ∈ s, l₁ i • v i, ∑ i ∈ s, l₂ i • v i⟫ = ∑ i ∈ s, conj (l₁ i) * l₂ i := by
   simp_rw [sum_inner, inner_smul_left]
   refine Finset.sum_congr rfl fun i hi => ?_
   rw [hv.inner_right_sum l₂ hi]
@@ -900,7 +926,7 @@ theorem Orthonormal.linearIndependent {v : ι → E} (hv : Orthonormal 𝕜 v) :
   rw [linearIndependent_iff]
   intro l hl
   ext i
-  have key : ⟪v i, Finsupp.total 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw [hl]
+  have key : ⟪v i, Finsupp.linearCombination 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw [hl]
   simpa only [hv.inner_right_finsupp, inner_zero_right] using key
 
 /-- A subfamily of an orthonormal family (i.e., a composition with an injective map) is an
@@ -932,7 +958,7 @@ theorem Orthonormal.toSubtypeRange {v : ι → E} (hv : Orthonormal 𝕜 v) :
 set. -/
 theorem Orthonormal.inner_finsupp_eq_zero {v : ι → E} (hv : Orthonormal 𝕜 v) {s : Set ι} {i : ι}
     (hi : i ∉ s) {l : ι →₀ 𝕜} (hl : l ∈ Finsupp.supported 𝕜 𝕜 s) :
-    ⟪Finsupp.total 𝕜 v l, v i⟫ = 0 := by
+    ⟪Finsupp.linearCombination 𝕜 v l, v i⟫ = 0 := by
   rw [Finsupp.mem_supported'] at hl
   simp only [hv.inner_left_finsupp, hl i hi, map_zero]
 
@@ -987,7 +1013,7 @@ theorem exists_maximal_orthonormal {s : Set E} (hs : Orthonormal 𝕜 (Subtype.v
     · exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc
     · exact fun _ => Set.subset_sUnion_of_mem
 
-open FiniteDimensional
+open Module
 
 /-- A family of orthonormal vectors with the correct cardinality forms a basis. -/
 def basisOfOrthonormalOfCardEqFinrank [Fintype ι] [Nonempty ι] {v : ι → E} (hv : Orthonormal 𝕜 v)
@@ -1000,30 +1026,17 @@ theorem coe_basisOfOrthonormalOfCardEqFinrank [Fintype ι] [Nonempty ι] {v : ι
     (basisOfOrthonormalOfCardEqFinrank hv card_eq : ι → E) = v :=
   coe_basisOfLinearIndependentOfCardEqFinrank _ _
 
-end OrthonormalSets_Seminormed
-
-section OrthonormalSets
-
-variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
-variable [NormedAddCommGroup F] [InnerProductSpace ℝ F]
-variable {ι : Type*}
-
-local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
-
-local notation "IK" => @RCLike.I 𝕜 _
-
-local postfix:90 "†" => starRingEnd _
-
 theorem Orthonormal.ne_zero {v : ι → E} (hv : Orthonormal 𝕜 v) (i : ι) : v i ≠ 0 := by
-  have : ‖v i‖ ≠ 0 := by
-    rw [hv.1 i]
-    norm_num
-  simpa using this
+  refine ne_of_apply_ne norm ?_
+  rw [hv.1 i, norm_zero]
+  norm_num
 
-end OrthonormalSets
+end OrthonormalSets_Seminormed
 
 section Norm_Seminormed
 
+open scoped InnerProductSpace
+
 variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 variable [SeminormedAddCommGroup F] [InnerProductSpace ℝ F]
 
@@ -1193,9 +1206,9 @@ instance (priority := 100) InnerProductSpace.toUniformConvexSpace : UniformConve
     ring_nf
     exact sub_le_sub_left (pow_le_pow_left hε.le hxy _) 4⟩
 
-section Complex
+section Complex_Seminormed
 
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+variable {V : Type*} [SeminormedAddCommGroup V] [InnerProductSpace ℂ V]
 
 /-- A complex polarization identity, with a linear map
 -/
@@ -1221,6 +1234,12 @@ theorem inner_map_polarization' (T : V →ₗ[ℂ] V) (x y : V) :
     mul_add, ← mul_assoc, mul_neg, neg_neg, sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub]
   ring
 
+end Complex_Seminormed
+
+section Complex
+
+variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+
 /-- A linear map `T` is zero, if and only if the identity `⟪T x, x⟫_ℂ = 0` holds for all `x`.
 -/
 theorem inner_map_self_eq_zero (T : V →ₗ[ℂ] V) : (∀ x : V, ⟪T x, x⟫_ℂ = 0) ↔ T = 0 := by
@@ -1321,7 +1340,8 @@ theorem Orthonormal.mapLinearIsometryEquiv {v : Basis ι 𝕜 E} (hv : Orthonorm
 def LinearMap.isometryOfOrthonormal (f : E →ₗ[𝕜] E') {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v)
     (hf : Orthonormal 𝕜 (f ∘ v)) : E →ₗᵢ[𝕜] E' :=
   f.isometryOfInner fun x y => by
-    classical rw [← v.total_repr x, ← v.total_repr y, Finsupp.apply_total, Finsupp.apply_total,
+    classical rw [← v.linearCombination_repr x, ← v.linearCombination_repr y,
+      Finsupp.apply_linearCombination, Finsupp.apply_linearCombination,
       hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]
 
 @[simp]
@@ -1341,8 +1361,9 @@ def LinearEquiv.isometryOfOrthonormal (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜
     (hf : Orthonormal 𝕜 (f ∘ v)) : E ≃ₗᵢ[𝕜] E' :=
   f.isometryOfInner fun x y => by
     rw [← LinearEquiv.coe_coe] at hf
-    classical rw [← v.total_repr x, ← v.total_repr y, ← LinearEquiv.coe_coe f, Finsupp.apply_total,
-      Finsupp.apply_total, hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]
+    classical rw [← v.linearCombination_repr x, ← v.linearCombination_repr y,
+      ← LinearEquiv.coe_coe f, Finsupp.apply_linearCombination,
+      Finsupp.apply_linearCombination, hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]
 
 @[simp]
 theorem LinearEquiv.coe_isometryOfOrthonormal (f : E ≃ₗ[𝕜] E') {v : Basis ι 𝕜 E}
@@ -1474,7 +1495,7 @@ theorem norm_sub_eq_norm_add {v w : E} (h : ⟪v, w⟫ = 0) : ‖w - v‖ = ‖w
 norms, has absolute value at most 1. -/
 theorem abs_real_inner_div_norm_mul_norm_le_one (x y : F) : |⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)| ≤ 1 := by
   rw [abs_div, abs_mul, abs_norm, abs_norm]
-  exact div_le_one_of_le (abs_real_inner_le_norm x y) (by positivity)
+  exact div_le_one_of_le₀ (abs_real_inner_le_norm x y) (by positivity)
 
 /-- The inner product of a vector with a multiple of itself. -/
 theorem real_inner_smul_self_left (x : F) (r : ℝ) : ⟪r • x, x⟫_ℝ = r * (‖x‖ * ‖x‖) := by
@@ -1544,7 +1565,7 @@ variable {𝕜}
 
 namespace ContinuousLinearMap
 
-variable {E' : Type*} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E']
+variable {E' : Type*} [SeminormedAddCommGroup E'] [InnerProductSpace 𝕜 E']
 
 -- Note: odd and expensive build behavior is explicitly turned off using `noncomputable`
 /-- Given `f : E →L[𝕜] E'`, construct the continuous sesquilinear form `fun x y ↦ ⟪x, A y⟫`, given
@@ -1569,10 +1590,89 @@ theorem toSesqForm_apply_norm_le {f : E →L[𝕜] E'} {v : E'} : ‖toSesqForm
 
 end ContinuousLinearMap
 
+section
+
+variable {ι : Type*} {ι' : Type*} {ι'' : Type*}
+variable {E' : Type*} [SeminormedAddCommGroup E'] [InnerProductSpace 𝕜 E']
+variable {E'' : Type*} [SeminormedAddCommGroup E''] [InnerProductSpace 𝕜 E'']
+
+@[simp]
+theorem Orthonormal.equiv_refl {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) :
+    hv.equiv hv (Equiv.refl ι) = LinearIsometryEquiv.refl 𝕜 E :=
+  v.ext_linearIsometryEquiv fun i => by
+    simp only [Orthonormal.equiv_apply, Equiv.coe_refl, id, LinearIsometryEquiv.coe_refl]
+
+@[simp]
+theorem Orthonormal.equiv_symm {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
+    (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') : (hv.equiv hv' e).symm = hv'.equiv hv e.symm :=
+  v'.ext_linearIsometryEquiv fun i =>
+    (hv.equiv hv' e).injective <| by
+      simp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]
+
+end
+
+variable (𝕜)
+
+/-- `innerSL` is an isometry. Note that the associated `LinearIsometry` is defined in
+`InnerProductSpace.Dual` as `toDualMap`. -/
+@[simp]
+theorem innerSL_apply_norm (x : E) : ‖innerSL 𝕜 x‖ = ‖x‖ := by
+  refine
+    le_antisymm ((innerSL 𝕜 x).opNorm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) ?_
+  rcases (norm_nonneg x).eq_or_gt with (h | h)
+  · simp [h]
+  · refine (mul_le_mul_right h).mp ?_
+    calc
+      ‖x‖ * ‖x‖ = ‖(⟪x, x⟫ : 𝕜)‖ := by
+        rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]
+      _ ≤ ‖innerSL 𝕜 x‖ * ‖x‖ := (innerSL 𝕜 x).le_opNorm _
+
+lemma norm_innerSL_le : ‖innerSL 𝕜 (E := E)‖ ≤ 1 :=
+  ContinuousLinearMap.opNorm_le_bound _ zero_le_one (by simp)
+
+variable {𝕜}
+
+/-- When an inner product space `E` over `𝕜` is considered as a real normed space, its inner
+product satisfies `IsBoundedBilinearMap`.
+
+In order to state these results, we need a `NormedSpace ℝ E` instance. We will later establish
+such an instance by restriction-of-scalars, `InnerProductSpace.rclikeToReal 𝕜 E`, but this
+instance may be not definitionally equal to some other “natural” instance. So, we assume
+`[NormedSpace ℝ E]`.
+-/
+theorem _root_.isBoundedBilinearMap_inner [NormedSpace ℝ E] [IsScalarTower ℝ 𝕜 E] :
+    IsBoundedBilinearMap ℝ fun p : E × E => ⟪p.1, p.2⟫ :=
+  { add_left := inner_add_left
+    smul_left := fun r x y => by
+      simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]
+    add_right := inner_add_right
+    smul_right := fun r x y => by
+      simp only [← algebraMap_smul 𝕜 r y, algebraMap_eq_ofReal, inner_smul_real_right]
+    bound :=
+      ⟨1, zero_lt_one, fun x y => by
+        rw [one_mul]
+        exact norm_inner_le_norm x y⟩ }
+
+/-- The inner product of two weighted sums, where the weights in each
+sum add to 0, in terms of the norms of pairwise differences. -/
+theorem inner_sum_smul_sum_smul_of_sum_eq_zero {ι₁ : Type*} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ}
+    (v₁ : ι₁ → F) (h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ}
+    (v₂ : ι₂ → F) (h₂ : ∑ i ∈ s₂, w₂ i = 0) :
+    ⟪∑ i₁ ∈ s₁, w₁ i₁ • v₁ i₁, ∑ i₂ ∈ s₂, w₂ i₂ • v₂ i₂⟫_ℝ =
+      (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2 := by
+  simp_rw [sum_inner, inner_sum, real_inner_smul_left, real_inner_smul_right,
+    real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two, ← div_sub_div_same,
+    ← div_add_div_same, mul_sub_left_distrib, left_distrib, Finset.sum_sub_distrib,
+    Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.sum_mul, h₁, h₂, zero_mul,
+    mul_zero, Finset.sum_const_zero, zero_add, zero_sub, Finset.mul_sum, neg_div,
+    Finset.sum_div, mul_div_assoc, mul_assoc]
+
 end Norm_Seminormed
 
 section Norm
 
+open scoped InnerProductSpace
+
 variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 variable [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 variable {ι : Type*} {ι' : Type*} {ι'' : Type*}
@@ -1602,27 +1702,6 @@ theorem dist_div_norm_sq_smul {x y : F} (hx : x ≠ 0) (hy : y ≠ 0) (R : ℝ)
     _ = R ^ 2 / (‖x‖ * ‖y‖) * dist x y := by
       rw [sqrt_mul, sqrt_sq, sqrt_sq, dist_eq_norm] <;> positivity
 
-section
-
-variable {ι : Type*} {ι' : Type*} {ι'' : Type*}
-variable {E' : Type*} [NormedAddCommGroup E'] [InnerProductSpace 𝕜 E']
-variable {E'' : Type*} [NormedAddCommGroup E''] [InnerProductSpace 𝕜 E'']
-
-@[simp]
-theorem Orthonormal.equiv_refl {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) :
-    hv.equiv hv (Equiv.refl ι) = LinearIsometryEquiv.refl 𝕜 E :=
-  v.ext_linearIsometryEquiv fun i => by
-    simp only [Orthonormal.equiv_apply, Equiv.coe_refl, id, LinearIsometryEquiv.coe_refl]
-
-@[simp]
-theorem Orthonormal.equiv_symm {v : Basis ι 𝕜 E} (hv : Orthonormal 𝕜 v) {v' : Basis ι' 𝕜 E'}
-    (hv' : Orthonormal 𝕜 v') (e : ι ≃ ι') : (hv.equiv hv' e).symm = hv'.equiv hv e.symm :=
-  v'.ext_linearIsometryEquiv fun i =>
-    (hv.equiv hv' e).injective <| by
-      simp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]
-
-end
-
 /-- The inner product of a nonzero vector with a nonzero multiple of
 itself, divided by the product of their norms, has absolute value
 1. -/
@@ -1662,8 +1741,8 @@ theorem norm_inner_eq_norm_tfae (x y : E) :
       x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫) • x,
       x = 0 ∨ ∃ r : 𝕜, y = r • x,
       x = 0 ∨ y ∈ 𝕜 ∙ x] := by
-  tfae_have 1 → 2
-  · refine fun h => or_iff_not_imp_left.2 fun hx₀ => ?_
+  tfae_have 1 → 2 := by
+    refine fun h => or_iff_not_imp_left.2 fun hx₀ => ?_
     have : ‖x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx₀)
     rw [← sq_eq_sq, mul_pow, ← mul_right_inj' this, eq_comm, ← sub_eq_zero, ← mul_sub] at h <;>
       try positivity
@@ -1673,13 +1752,12 @@ theorem norm_inner_eq_norm_tfae (x y : E) :
       sub_eq_zero] at h
     rw [div_eq_inv_mul, mul_smul, h, inv_smul_smul₀]
     rwa [inner_self_ne_zero]
-  tfae_have 2 → 3
-  · exact fun h => h.imp_right fun h' => ⟨_, h'⟩
-  tfae_have 3 → 1
-  · rintro (rfl | ⟨r, rfl⟩) <;>
+  tfae_have 2 → 3 := fun h => h.imp_right fun h' => ⟨_, h'⟩
+  tfae_have 3 → 1 := by
+    rintro (rfl | ⟨r, rfl⟩) <;>
     simp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K, inner_self_eq_norm_mul_norm,
       sq, mul_left_comm]
-  tfae_have 3 ↔ 4; · simp only [Submodule.mem_span_singleton, eq_comm]
+  tfae_have 3 ↔ 4 := by simp only [Submodule.mem_span_singleton, eq_comm]
   tfae_finish
 
 /-- If the inner product of two vectors is equal to the product of their norms, then the two vectors
@@ -1796,62 +1874,6 @@ theorem eq_of_norm_le_re_inner_eq_norm_sq {x y : E} (hle : ‖x‖ ≤ ‖y‖)
   have H₂ : re ⟪y, x⟫ = ‖y‖ ^ 2 := by rwa [← inner_conj_symm, conj_re]
   simpa [inner_sub_left, inner_sub_right, ← norm_sq_eq_inner, h, H₂] using H₁
 
-/-- The inner product of two weighted sums, where the weights in each
-sum add to 0, in terms of the norms of pairwise differences. -/
-theorem inner_sum_smul_sum_smul_of_sum_eq_zero {ι₁ : Type*} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ}
-    (v₁ : ι₁ → F) (h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ}
-    (v₂ : ι₂ → F) (h₂ : ∑ i ∈ s₂, w₂ i = 0) :
-    ⟪∑ i₁ ∈ s₁, w₁ i₁ • v₁ i₁, ∑ i₂ ∈ s₂, w₂ i₂ • v₂ i₂⟫_ℝ =
-      (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (‖v₁ i₁ - v₂ i₂‖ * ‖v₁ i₁ - v₂ i₂‖)) / 2 := by
-  simp_rw [sum_inner, inner_sum, real_inner_smul_left, real_inner_smul_right,
-    real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two, ← div_sub_div_same,
-    ← div_add_div_same, mul_sub_left_distrib, left_distrib, Finset.sum_sub_distrib,
-    Finset.sum_add_distrib, ← Finset.mul_sum, ← Finset.sum_mul, h₁, h₂, zero_mul,
-    mul_zero, Finset.sum_const_zero, zero_add, zero_sub, Finset.mul_sum, neg_div,
-    Finset.sum_div, mul_div_assoc, mul_assoc]
-
-variable (𝕜)
-
-/-- `innerSL` is an isometry. Note that the associated `LinearIsometry` is defined in
-`InnerProductSpace.Dual` as `toDualMap`.  -/
-@[simp]
-theorem innerSL_apply_norm (x : E) : ‖innerSL 𝕜 x‖ = ‖x‖ := by
-  refine
-    le_antisymm ((innerSL 𝕜 x).opNorm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) ?_
-  rcases eq_or_ne x 0 with (rfl | h)
-  · simp
-  · refine (mul_le_mul_right (norm_pos_iff.2 h)).mp ?_
-    calc
-      ‖x‖ * ‖x‖ = ‖(⟪x, x⟫ : 𝕜)‖ := by
-        rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]
-      _ ≤ ‖innerSL 𝕜 x‖ * ‖x‖ := (innerSL 𝕜 x).le_opNorm _
-
-lemma norm_innerSL_le : ‖innerSL 𝕜 (E := E)‖ ≤ 1 :=
-  ContinuousLinearMap.opNorm_le_bound _ zero_le_one (by simp)
-
-variable {𝕜}
-
-/-- When an inner product space `E` over `𝕜` is considered as a real normed space, its inner
-product satisfies `IsBoundedBilinearMap`.
-
-In order to state these results, we need a `NormedSpace ℝ E` instance. We will later establish
-such an instance by restriction-of-scalars, `InnerProductSpace.rclikeToReal 𝕜 E`, but this
-instance may be not definitionally equal to some other “natural” instance. So, we assume
-`[NormedSpace ℝ E]`.
--/
-theorem _root_.isBoundedBilinearMap_inner [NormedSpace ℝ E] :
-    IsBoundedBilinearMap ℝ fun p : E × E => ⟪p.1, p.2⟫ :=
-  { add_left := inner_add_left
-    smul_left := fun r x y => by
-      simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]
-    add_right := inner_add_right
-    smul_right := fun r x y => by
-      simp only [← algebraMap_smul 𝕜 r y, algebraMap_eq_ofReal, inner_smul_real_right]
-    bound :=
-      ⟨1, zero_lt_one, fun x y => by
-        rw [one_mul]
-        exact norm_inner_le_norm x y⟩ }
-
 end Norm
 
 section BesselsInequality
@@ -1880,7 +1902,7 @@ theorem Orthonormal.sum_inner_products_le {s : Finset ι} (hv : Orthonormal 𝕜
     rw [← sub_nonneg, ← hbf]
     simp only [norm_nonneg, pow_nonneg]
   rw [@norm_sub_sq 𝕜, sub_add]
-  simp only [@InnerProductSpace.norm_sq_eq_inner 𝕜, _root_.inner_sum, _root_.sum_inner]
+  simp only [@InnerProductSpace.norm_sq_eq_inner 𝕜, inner_sum, sum_inner]
   simp only [inner_smul_right, two_mul, inner_smul_left, inner_conj_symm, ← mul_assoc, h₂,
     add_sub_cancel_right, sub_right_inj]
   simp only [map_sum, ← inner_conj_symm x, ← h₃]
@@ -2036,12 +2058,12 @@ theorem OrthogonalFamily.inner_right_fintype [Fintype ι] (l : ∀ i, G i) (i :
     _ = ⟪v, l i⟫ := by
       simp only [Finset.sum_ite_eq, Finset.mem_univ, (V i).inner_map_map, if_true]
 
-theorem OrthogonalFamily.inner_sum (l₁ l₂ : ∀ i, G i) (s : Finset ι) :
+nonrec theorem OrthogonalFamily.inner_sum (l₁ l₂ : ∀ i, G i) (s : Finset ι) :
     ⟪∑ i ∈ s, V i (l₁ i), ∑ j ∈ s, V j (l₂ j)⟫ = ∑ i ∈ s, ⟪l₁ i, l₂ i⟫ := by
   classical
   calc
     ⟪∑ i ∈ s, V i (l₁ i), ∑ j ∈ s, V j (l₂ j)⟫ = ∑ j ∈ s, ∑ i ∈ s, ⟪V i (l₁ i), V j (l₂ j)⟫ := by
-      simp only [_root_.sum_inner, _root_.inner_sum]
+      simp only [sum_inner, inner_sum]
     _ = ∑ j ∈ s, ∑ i ∈ s, ite (i = j) ⟪V i (l₁ i), V j (l₂ j)⟫ 0 := by
       congr with i
       congr with j
@@ -2193,6 +2215,8 @@ end OrthogonalFamily
 
 section RCLikeToReal
 
+open scoped InnerProductSpace
+
 variable {G : Type*}
 variable (𝕜 E)
 variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
@@ -2258,7 +2282,7 @@ end RCLikeToReal
 
 section Continuous
 
-variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
@@ -2273,6 +2297,7 @@ local postfix:90 "†" => starRingEnd _
 
 theorem continuous_inner : Continuous fun p : E × E => ⟪p.1, p.2⟫ :=
   letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
+  letI : IsScalarTower ℝ 𝕜 E := RestrictScalars.isScalarTower _ _ _
   isBoundedBilinearMap_inner.continuous
 
 variable {α : Type*}
@@ -2323,7 +2348,7 @@ end ReApplyInnerSelf
 
 section ReApplyInnerSelf_Seminormed
 
-variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
@@ -2345,7 +2370,7 @@ end ReApplyInnerSelf_Seminormed
 
 section UniformSpace.Completion
 
-variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
@@ -2359,11 +2384,11 @@ open UniformSpace Function
 
 instance toInner {𝕜' E' : Type*} [TopologicalSpace 𝕜'] [UniformSpace E'] [Inner 𝕜' E'] :
     Inner 𝕜' (Completion E') where
-  inner := curry <| (denseInducing_coe.prod denseInducing_coe).extend (uncurry inner)
+  inner := curry <| (isDenseInducing_coe.prodMap isDenseInducing_coe).extend (uncurry inner)
 
 @[simp]
 theorem inner_coe (a b : E) : inner (a : Completion E) (b : Completion E) = (inner a b : 𝕜) :=
-  (denseInducing_coe.prod denseInducing_coe).extend_eq
+  (isDenseInducing_coe.prodMap isDenseInducing_coe).extend_eq
     (continuous_inner : Continuous (uncurry inner : E × E → 𝕜)) (a, b)
 
 protected theorem continuous_inner :
@@ -2376,9 +2401,9 @@ protected theorem continuous_inner :
   rw [Completion.toInner, inner, uncurry_curry _]
   change
     Continuous
-      (((denseInducing_toCompl E).prod (denseInducing_toCompl E)).extend fun p : E × E =>
+      (((isDenseInducing_toCompl E).prodMap (isDenseInducing_toCompl E)).extend fun p : E × E =>
         inner' p.1 p.2)
-  exact (denseInducing_toCompl E).extend_Z_bilin (denseInducing_toCompl E) this
+  exact (isDenseInducing_toCompl E).extend_Z_bilin (isDenseInducing_toCompl E) this
 
 protected theorem Continuous.inner {α : Type*} [TopologicalSpace α] {f g : α → Completion E}
     (hf : Continuous f) (hg : Continuous g) : Continuous (fun x : α => inner (f x) (g x) : α → 𝕜) :=
@@ -2413,3 +2438,5 @@ instance innerProductSpace : InnerProductSpace 𝕜 (Completion E) where
 end UniformSpace.Completion
 
 end UniformSpace.Completion
+
+set_option linter.style.longFile 2500
diff --git a/Mathlib/Analysis/InnerProductSpace/Calculus.lean b/Mathlib/Analysis/InnerProductSpace/Calculus.lean
index 031592910f445..44808e4d72423 100644
--- a/Mathlib/Analysis/InnerProductSpace/Calculus.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Calculus.lean
@@ -80,15 +80,18 @@ theorem HasFDerivWithinAt.inner (hf : HasFDerivWithinAt f f' s x)
     (hg : HasFDerivWithinAt g g' s x) :
     HasFDerivWithinAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') s
       x :=
-  (isBoundedBilinearMap_inner.hasFDerivAt (f x, g x)).comp_hasFDerivWithinAt x (hf.prod hg)
+  isBoundedBilinearMap_inner (𝕜 := 𝕜) (E := E)
+    |>.hasFDerivAt (f x, g x) |>.comp_hasFDerivWithinAt x (hf.prod hg)
 
 theorem HasStrictFDerivAt.inner (hf : HasStrictFDerivAt f f' x) (hg : HasStrictFDerivAt g g' x) :
     HasStrictFDerivAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') x :=
-  (isBoundedBilinearMap_inner.hasStrictFDerivAt (f x, g x)).comp x (hf.prod hg)
+  isBoundedBilinearMap_inner (𝕜 := 𝕜) (E := E)
+    |>.hasStrictFDerivAt (f x, g x) |>.comp x (hf.prod hg)
 
 theorem HasFDerivAt.inner (hf : HasFDerivAt f f' x) (hg : HasFDerivAt g g' x) :
     HasFDerivAt (fun t => ⟪f t, g t⟫) ((fderivInnerCLM 𝕜 (f x, g x)).comp <| f'.prod g') x :=
-  (isBoundedBilinearMap_inner.hasFDerivAt (f x, g x)).comp x (hf.prod hg)
+  isBoundedBilinearMap_inner (𝕜 := 𝕜) (E := E)
+    |>.hasFDerivAt (f x, g x) |>.comp x (hf.prod hg)
 
 theorem HasDerivWithinAt.inner {f g : ℝ → E} {f' g' : E} {s : Set ℝ} {x : ℝ}
     (hf : HasDerivWithinAt f f' s x) (hg : HasDerivWithinAt g g' s x) :
diff --git a/Mathlib/Analysis/InnerProductSpace/Dual.lean b/Mathlib/Analysis/InnerProductSpace/Dual.lean
index e6af204a8c575..58aba1f6281ae 100644
--- a/Mathlib/Analysis/InnerProductSpace/Dual.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Dual.lean
@@ -45,8 +45,10 @@ namespace InnerProductSpace
 
 open RCLike ContinuousLinearMap
 
-variable (𝕜 : Type*)
-variable (E : Type*) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable (𝕜 E : Type*)
+
+section Seminormed
+variable [RCLike 𝕜] [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
 
@@ -67,10 +69,19 @@ variable {E}
 theorem toDualMap_apply {x y : E} : toDualMap 𝕜 E x y = ⟪x, y⟫ :=
   rfl
 
+end Seminormed
+
+section Normed
+variable [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+
+local notation "⟪" x ", " y "⟫" => @inner 𝕜 E _ x y
+
+local postfix:90 "†" => starRingEnd _
+
 theorem innerSL_norm [Nontrivial E] : ‖(innerSL 𝕜 : E →L⋆[𝕜] E →L[𝕜] 𝕜)‖ = 1 :=
   show ‖(toDualMap 𝕜 E).toContinuousLinearMap‖ = 1 from LinearIsometry.norm_toContinuousLinearMap _
 
-variable {𝕜}
+variable {E 𝕜}
 
 theorem ext_inner_left_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
     (h : ∀ i : ι, ⟪b i, x⟫ = ⟪b i, y⟫) : x = y := by
@@ -170,4 +181,6 @@ theorem unique_continuousLinearMapOfBilin {v f : E} (is_lax_milgram : ∀ w, ⟪
   rw [continuousLinearMapOfBilin_apply]
   exact is_lax_milgram w
 
+end Normed
+
 end InnerProductSpace
diff --git a/Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean b/Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
index 69889f1ed8e41..50173dd6a64c2 100644
--- a/Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
+++ b/Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
@@ -29,7 +29,7 @@ variable {E : Type*} [AddCommGroup E] [TopologicalSpace E] [TopologicalAddGroup
 
 noncomputable section
 
-open FiniteDimensional
+open Module
 
 /-- If `E` is a finite dimensional space over `ℝ`, then `toEuclidean` is a continuous `ℝ`-linear
 equivalence between `E` and the Euclidean space of the same dimension. -/
diff --git a/Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean b/Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
index dc8c71696a394..b0351cdc69d46 100644
--- a/Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
+++ b/Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
@@ -35,7 +35,7 @@ and outputs a set of orthogonal vectors which have the same span.
 -/
 
 
-open Finset Submodule FiniteDimensional
+open Finset Submodule Module
 
 variable (𝕜 : Type*) {E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
 variable {ι : Type*} [LinearOrder ι] [LocallyFiniteOrderBot ι] [IsWellOrder ι (· < ·)]
@@ -137,7 +137,7 @@ theorem gramSchmidt_mem_span (f : ι → E) :
     (Submodule.sum_mem _ fun k hk => ?_)
   let hkj : k < j := (Finset.mem_Iio.1 hk).trans_le hij
   exact smul_mem _ _
-    (span_mono (image_subset f <| Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)
+    (span_mono (image_subset f <| Set.Iic_subset_Iic.2 hkj.le) <| gramSchmidt_mem_span _ le_rfl)
 termination_by j => j
 
 theorem span_gramSchmidt_Iic (f : ι → E) (c : ι) :
diff --git a/Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean b/Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean
new file mode 100644
index 0000000000000..049ba1532a238
--- /dev/null
+++ b/Mathlib/Analysis/InnerProductSpace/JointEigenspace.lean
@@ -0,0 +1,113 @@
+/-
+Copyright (c) 2024 Jon Bannon. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jon Bannon, Jack Cheverton, Samyak Dhar Tuladhar
+-/
+
+import Mathlib.Analysis.InnerProductSpace.Spectrum
+import Mathlib.Analysis.InnerProductSpace.Projection
+import Mathlib.Order.CompleteLattice
+import Mathlib.LinearAlgebra.Eigenspace.Basic
+
+/-! # Joint eigenspaces of a commuting pair of symmetric operators
+
+This file collects various decomposition results for joint eigenspaces of a commuting pair
+of symmetric operators on a finite-dimensional inner product space.
+
+# Main Result
+
+* `LinearMap.IsSymmetric.directSum_isInternal_of_commute` establishes that
+   if `{A B : E →ₗ[𝕜] E}`, then `IsSymmetric A`, `IsSymmetric B` and `A ∘ₗ B = B ∘ₗ A` imply that
+   `E` decomposes as an internal direct sum of the pairwise orthogonal spaces
+   `eigenspace B μ ⊓ eigenspace A ν`
+
+## TODO
+
+Develop a `Diagonalization` structure for linear maps and / or matrices which consists of a basis,
+and a proof obligation that the basis vectors are eigenvectors.
+
+## Tags
+
+self-adjoint operator, simultaneous eigenspaces, joint eigenspaces
+
+-/
+
+variable {𝕜 E : Type*} [RCLike 𝕜]
+variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+
+open Module.End
+
+namespace LinearMap
+
+namespace IsSymmetric
+
+section Pair
+
+variable {α : 𝕜} {A B : E →ₗ[𝕜] E}
+
+/--If a pair of operators commute, then the eigenspaces of one are invariant under the other.-/
+theorem eigenspace_invariant_of_commute
+    (hAB : A ∘ₗ B = B ∘ₗ A) (α : 𝕜) : ∀ v ∈ (eigenspace A α), (B v ∈ eigenspace A α) := by
+  intro v hv
+  rw [eigenspace_def, mem_ker, sub_apply, smul_apply, one_apply, ← comp_apply A B v, hAB,
+    comp_apply B A v, ← map_smul, ← map_sub, hv, map_zero] at *
+
+/--The simultaneous eigenspaces of a pair of commuting symmetric operators form an
+`OrthogonalFamily`.-/
+theorem orthogonalFamily_eigenspace_inf_eigenspace (hA : A.IsSymmetric) (hB : B.IsSymmetric) :
+    OrthogonalFamily 𝕜 (fun (i : 𝕜 × 𝕜) => (eigenspace A i.2 ⊓ eigenspace B i.1 : Submodule 𝕜 E))
+    (fun i => (eigenspace A i.2 ⊓ eigenspace B i.1).subtypeₗᵢ) :=
+     OrthogonalFamily.of_pairwise fun i j hij v ⟨hv1 , hv2⟩ ↦ by
+    obtain (h₁ | h₂) : i.1 ≠ j.1 ∨ i.2 ≠ j.2 := by rwa [Ne.eq_def, Prod.ext_iff, not_and_or] at hij
+    all_goals intro w ⟨hw1, hw2⟩
+    · exact hB.orthogonalFamily_eigenspaces.pairwise h₁ hv2 w hw2
+    · exact hA.orthogonalFamily_eigenspaces.pairwise h₂ hv1 w hw1
+
+open Submodule in
+
+/-- The intersection of eigenspaces of commuting selfadjoint operators is equal to the eigenspace of
+one operator restricted to the eigenspace of the other, which is an invariant subspace because the
+operators commute. -/
+theorem eigenspace_inf_eigenspace
+    (hAB : A ∘ₗ B = B ∘ₗ A) (γ : 𝕜) :
+    eigenspace A α ⊓ eigenspace B γ = map (Submodule.subtype (eigenspace A α))
+      (eigenspace (B.restrict (eigenspace_invariant_of_commute hAB α)) γ) :=
+  (eigenspace A α).inf_genEigenspace _ _ (k := 1)
+
+variable [FiniteDimensional 𝕜 E]
+
+/-- If A and B are commuting symmetric operators on a finite dimensional inner product space
+then the eigenspaces of the restriction of B to any eigenspace of A exhaust that eigenspace.-/
+theorem iSup_eigenspace_inf_eigenspace (hB : B.IsSymmetric)
+    (hAB : A ∘ₗ B = B ∘ₗ A):
+    (⨆ γ, eigenspace A α ⊓ eigenspace B γ) = eigenspace A α := by
+  conv_rhs => rw [← (eigenspace A α).map_subtype_top]
+  simp only [eigenspace_inf_eigenspace hAB, ← Submodule.map_iSup]
+  congr 1
+  rw [← Submodule.orthogonal_eq_bot_iff]
+  exact orthogonalComplement_iSup_eigenspaces_eq_bot <|
+    hB.restrict_invariant <| eigenspace_invariant_of_commute hAB α
+
+/-- If A and B are commuting symmetric operators acting on a finite dimensional inner product space,
+then the simultaneous eigenspaces of A and B exhaust the space. -/
+theorem iSup_iSup_eigenspace_inf_eigenspace_eq_top (hA : A.IsSymmetric) (hB : B.IsSymmetric)
+    (hAB : A ∘ₗ B = B ∘ₗ A) :
+    (⨆ α, ⨆ γ, eigenspace A α ⊓ eigenspace B γ) = ⊤ := by
+  simpa [iSup_eigenspace_inf_eigenspace hB hAB] using
+    Submodule.orthogonal_eq_bot_iff.mp <| hA.orthogonalComplement_iSup_eigenspaces_eq_bot
+
+/-- Given a commuting pair of symmetric linear operators on a finite dimensional inner product
+space, the space decomposes as an internal direct sum of simultaneous eigenspaces of these
+operators. -/
+theorem directSum_isInteral_of_commute (hA : A.IsSymmetric) (hB : B.IsSymmetric)
+    (hAB : A ∘ₗ B = B ∘ₗ A) :
+    DirectSum.IsInternal (fun (i : 𝕜 × 𝕜) ↦ (eigenspace A i.2 ⊓ eigenspace B i.1)):= by
+  apply (orthogonalFamily_eigenspace_inf_eigenspace hA hB).isInternal_iff.mpr
+  rw [Submodule.orthogonal_eq_bot_iff, iSup_prod, iSup_comm]
+  exact iSup_iSup_eigenspace_inf_eigenspace_eq_top hA hB hAB
+
+end Pair
+
+end IsSymmetric
+
+end LinearMap
diff --git a/Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean b/Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
index 7ac440395d2e8..9aa623e738fae 100644
--- a/Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
+++ b/Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean
@@ -64,7 +64,7 @@ theorem antilipschitz (coercive : IsCoercive B) : ∃ C : ℝ≥0, 0 < C ∧ Ant
   refine ⟨C⁻¹.toNNReal, Real.toNNReal_pos.mpr (inv_pos.mpr C_pos), ?_⟩
   refine ContinuousLinearMap.antilipschitz_of_bound B♯ ?_
   simp_rw [Real.coe_toNNReal', max_eq_left_of_lt (inv_pos.mpr C_pos), ←
-    inv_mul_le_iff (inv_pos.mpr C_pos)]
+    inv_mul_le_iff₀ (inv_pos.mpr C_pos)]
   simpa using below_bound
 
 theorem ker_eq_bot (coercive : IsCoercive B) : ker B♯ = ⊥ := by
diff --git a/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean b/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean
index 9019b62655b0c..13396afdb97d0 100644
--- a/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean
+++ b/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean
@@ -103,7 +103,7 @@ variable (hT : Dense (T.domain : Set E))
 /-- The unique continuous extension of the operator `adjointDomainMkCLM` to `E`. -/
 def adjointDomainMkCLMExtend (y : T.adjointDomain) : E →L[𝕜] 𝕜 :=
   (T.adjointDomainMkCLM y).extend (Submodule.subtypeL T.domain) hT.denseRange_val
-    uniformEmbedding_subtype_val.toUniformInducing
+    isUniformEmbedding_subtype_val.isUniformInducing
 
 @[simp]
 theorem adjointDomainMkCLMExtend_apply (y : T.adjointDomain) (x : T.domain) :
@@ -202,7 +202,7 @@ as taking the `ContinuousLinearMap.adjoint` interpreted as a `LinearPMap`. -/
 theorem toPMap_adjoint_eq_adjoint_toPMap_of_dense (hp : Dense (p : Set E)) :
     (A.toPMap p).adjoint = A.adjoint.toPMap ⊤ := by
   ext x y hxy
-  · simp only [LinearMap.toPMap_domain, Submodule.mem_top, iff_true_iff,
+  · simp only [LinearMap.toPMap_domain, Submodule.mem_top, iff_true,
       LinearPMap.mem_adjoint_domain_iff, LinearMap.coe_comp, innerₛₗ_apply_coe]
     exact ((innerSL 𝕜 x).comp <| A.comp <| Submodule.subtypeL _).cont
   refine LinearPMap.adjoint_apply_eq ?_ _ fun v => ?_
diff --git a/Mathlib/Analysis/InnerProductSpace/OfNorm.lean b/Mathlib/Analysis/InnerProductSpace/OfNorm.lean
index ad51b9d339160..e2e23c0b46a9d 100644
--- a/Mathlib/Analysis/InnerProductSpace/OfNorm.lean
+++ b/Mathlib/Analysis/InnerProductSpace/OfNorm.lean
@@ -5,6 +5,8 @@ Authors: Heather Macbeth
 -/
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Algebra.Module.LinearMap.Rat
+import Mathlib.Tactic.Module
 
 /-!
 # Inner product space derived from a norm
@@ -97,199 +99,118 @@ private def innerProp' (r : 𝕜) : Prop :=
 
 variable {E}
 
-theorem innerProp_neg_one : innerProp' E ((-1 : ℤ) : 𝕜) := by
-  intro x y
-  simp only [inner_, neg_mul_eq_neg_mul, one_mul, Int.cast_one, one_smul, RingHom.map_one, map_neg,
-    Int.cast_neg, neg_smul, neg_one_mul]
-  rw [neg_mul_comm]
-  congr 1
-  have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
-  have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
-  have h₃ : ‖(I : 𝕜) • -x + y‖ = ‖(I : 𝕜) • x - y‖ := by
-    rw [← neg_sub, norm_neg, sub_eq_neg_add, ← smul_neg]
-  have h₄ : ‖(I : 𝕜) • -x - y‖ = ‖(I : 𝕜) • x + y‖ := by rw [smul_neg, ← neg_add', norm_neg]
-  rw [h₁, h₂, h₃, h₄]
-  ring
-
 theorem _root_.Continuous.inner_ {f g : ℝ → E} (hf : Continuous f) (hg : Continuous g) :
     Continuous fun x => inner_ 𝕜 (f x) (g x) := by
-  unfold inner_
+  unfold _root_.inner_
   fun_prop
 
 theorem inner_.norm_sq (x : E) : ‖x‖ ^ 2 = re (inner_ 𝕜 x x) := by
-  simp only [inner_]
-  have h₁ : RCLike.normSq (4 : 𝕜) = 16 := by
-    have : ((4 : ℝ) : 𝕜) = (4 : 𝕜) := by norm_cast
-    rw [← this, normSq_eq_def', RCLike.norm_of_nonneg (by norm_num : (0 : ℝ) ≤ 4)]
-    norm_num
-  have h₂ : ‖x + x‖ = 2 * ‖x‖ := by rw [← two_smul 𝕜, norm_smul, RCLike.norm_two]
-  simp only [h₁, h₂, algebraMap_eq_ofReal, sub_self, norm_zero, mul_re, inv_re, ofNat_re, map_sub,
-    map_add, ofReal_re, ofNat_im, ofReal_im, mul_im, I_re, inv_im]
+  simp only [inner_, normSq_apply, ofNat_re, ofNat_im, map_sub, map_add, map_zero, map_mul,
+    ofReal_re, ofReal_im, mul_re, inv_re, mul_im, I_re, inv_im]
+  have h₁ : ‖x - x‖ = 0 := by simp
+  have h₂ : ‖x + x‖ = 2 • ‖x‖ := by convert norm_nsmul 𝕜 2 x using 2; module
+  rw [h₁, h₂]
   ring
 
-attribute [local simp] map_ofNat in -- use `ofNat` simp theorem with bad keys
 theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by
-  simp only [inner_]
-  have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
-  rw [map_mul, h4]
-  congr 1
-  simp only [map_sub, map_add, conj_ofReal, map_mul, conj_I]
+  simp only [inner_, map_sub, map_add, map_mul, map_inv₀, map_ofNat, conj_ofReal, conj_I]
   rw [add_comm y x, norm_sub_rev]
   by_cases hI : (I : 𝕜) = 0
   · simp only [hI, neg_zero, zero_mul]
-  -- Porting note: this replaces `norm_I_of_ne_zero` which does not exist in Lean 4
-  have : ‖(I : 𝕜)‖ = 1 := by
-    rw [← mul_self_inj_of_nonneg (norm_nonneg I) zero_le_one, one_mul, ← norm_mul,
-      I_mul_I_of_nonzero hI, norm_neg, norm_one]
+  have hI' := I_mul_I_of_nonzero hI
+  have I_smul (v : E) : ‖(I : 𝕜) • v‖ = ‖v‖ := by rw [norm_smul, norm_I_of_ne_zero hI, one_mul]
   have h₁ : ‖(I : 𝕜) • y - x‖ = ‖(I : 𝕜) • x + y‖ := by
-    trans ‖(I : 𝕜) • ((I : 𝕜) • y - x)‖
-    · rw [norm_smul, this, one_mul]
-    · rw [smul_sub, smul_smul, I_mul_I_of_nonzero hI, neg_one_smul, ← neg_add', add_comm, norm_neg]
+    convert I_smul ((I : 𝕜) • x + y) using 2
+    linear_combination (norm := module) congr(-$hI' • x)
   have h₂ : ‖(I : 𝕜) • y + x‖ = ‖(I : 𝕜) • x - y‖ := by
-    trans ‖(I : 𝕜) • ((I : 𝕜) • y + x)‖
-    · rw [norm_smul, this, one_mul]
-    · rw [smul_add, smul_smul, I_mul_I_of_nonzero hI, neg_one_smul, ← neg_add_eq_sub]
-  rw [h₁, h₂, ← sub_add_eq_add_sub]
-  simp only [neg_mul, sub_eq_add_neg, neg_neg]
+    convert (I_smul ((I : 𝕜) • y + x)).symm using 2
+    linear_combination (norm := module) congr(-$hI' • y)
+  rw [h₁, h₂]
+  ring
 
 variable [InnerProductSpaceable E]
 
-private theorem add_left_aux1 (x y z : E) : ‖x + y + z‖ * ‖x + y + z‖ =
-    (‖2 • x + y‖ * ‖2 • x + y‖ + ‖2 • z + y‖ * ‖2 • z + y‖) / 2 - ‖x - z‖ * ‖x - z‖ := by
-  rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm]
-  convert parallelogram_identity (x + y + z) (x - z) using 4 <;> · rw [two_smul]; abel
-
-private theorem add_left_aux2 (x y z : E) : ‖x + y - z‖ * ‖x + y - z‖ =
-    (‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 - ‖x + z‖ * ‖x + z‖ := by
-  rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm]
-  have h₀ := parallelogram_identity (x + y - z) (x + z)
-  convert h₀ using 4 <;> · rw [two_smul]; abel
+private theorem add_left_aux1 (x y z : E) :
+    ‖2 • x + y‖ * ‖2 • x + y‖ + ‖2 • z + y‖ * ‖2 • z + y‖
+    = 2 * (‖x + y + z‖ * ‖x + y + z‖ + ‖x - z‖ * ‖x - z‖) := by
+  convert parallelogram_identity (x + y + z) (x - z) using 4 <;> abel
 
-private theorem add_left_aux2' (x y z : E) :
-    ‖x + y + z‖ * ‖x + y + z‖ - ‖x + y - z‖ * ‖x + y - z‖ =
-    ‖x + z‖ * ‖x + z‖ - ‖x - z‖ * ‖x - z‖ +
-    (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 := by
-  rw [add_left_aux1, add_left_aux2]; ring
+private theorem add_left_aux2 (x y z : E) : ‖2 • x + y‖ * ‖2 • x + y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖
+    = 2 * (‖x + y - z‖ * ‖x + y - z‖ + ‖x + z‖ * ‖x + z‖) := by
+  convert parallelogram_identity (x + y - z) (x + z) using 4 <;> abel
 
 private theorem add_left_aux3 (y z : E) :
-    ‖2 • z + y‖ * ‖2 • z + y‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ := by
-  apply eq_sub_of_add_eq
-  convert parallelogram_identity (y + z) z using 4 <;> (try rw [two_smul]) <;> abel
+    ‖2 • z + y‖ * ‖2 • z + y‖ + ‖y‖ * ‖y‖ = 2 * (‖y + z‖ * ‖y + z‖ + ‖z‖ * ‖z‖) := by
+  convert parallelogram_identity (y + z) z using 4 <;> abel
 
 private theorem add_left_aux4 (y z : E) :
-    ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) - ‖y‖ * ‖y‖ := by
-  apply eq_sub_of_add_eq'
-  have h₀ := parallelogram_identity (y - z) z
-  convert h₀ using 4 <;> (try rw [two_smul]) <;> abel
+    ‖y‖ * ‖y‖ + ‖y - 2 • z‖ * ‖y - 2 • z‖ = 2 * (‖y - z‖ * ‖y - z‖ + ‖z‖ * ‖z‖) := by
+  convert parallelogram_identity (y - z) z using 4 <;> abel
 
-private theorem add_left_aux4' (y z : E) :
-    (‖2 • z + y‖ * ‖2 • z + y‖ - ‖y - 2 • z‖ * ‖y - 2 • z‖) / 2 =
-    ‖y + z‖ * ‖y + z‖ - ‖y - z‖ * ‖y - z‖ := by
-  rw [add_left_aux3, add_left_aux4]; ring
+variable (𝕜)
 
 private theorem add_left_aux5 (x y z : E) :
-    ‖(I : 𝕜) • (x + y) + z‖ * ‖(I : 𝕜) • (x + y) + z‖ =
-    (‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
-    ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖) / 2 -
-    ‖(I : 𝕜) • x - z‖ * ‖(I : 𝕜) • x - z‖ := by
-  rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm]
-  have h₀ := parallelogram_identity ((I : 𝕜) • (x + y) + z) ((I : 𝕜) • x - z)
-  convert h₀ using 4 <;> · try simp only [two_smul, smul_add]; abel
+    ‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖
+    + ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖
+    = 2 * (‖(I : 𝕜) • (x + y) + z‖ * ‖(I : 𝕜) • (x + y) + z‖
+    + ‖(I : 𝕜) • x - z‖ * ‖(I : 𝕜) • x - z‖) := by
+  convert parallelogram_identity ((I : 𝕜) • (x + y) + z) ((I : 𝕜) • x - z) using 4 <;> module
 
 private theorem add_left_aux6 (x y z : E) :
-    ‖(I : 𝕜) • (x + y) - z‖ * ‖(I : 𝕜) • (x + y) - z‖ =
     (‖(I : 𝕜) • (2 • x + y)‖ * ‖(I : 𝕜) • (2 • x + y)‖ +
-    ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖) / 2 -
-    ‖(I : 𝕜) • x + z‖ * ‖(I : 𝕜) • x + z‖ := by
-  rw [eq_sub_iff_add_eq, eq_div_iff (two_ne_zero' ℝ), mul_comm _ (2 : ℝ), eq_comm]
-  have h₀ := parallelogram_identity ((I : 𝕜) • (x + y) - z) ((I : 𝕜) • x + z)
-  convert h₀ using 4 <;> · try simp only [two_smul, smul_add]; abel
+    ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖)
+    = 2 * (‖(I : 𝕜) • (x + y) - z‖ * ‖(I : 𝕜) • (x + y) - z‖ +
+    ‖(I : 𝕜) • x + z‖ * ‖(I : 𝕜) • x + z‖) := by
+  convert parallelogram_identity ((I : 𝕜) • (x + y) - z) ((I : 𝕜) • x + z) using 4 <;> module
 
 private theorem add_left_aux7 (y z : E) :
-    ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖ =
-    2 * (‖(I : 𝕜) • y + z‖ * ‖(I : 𝕜) • y + z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ := by
-  apply eq_sub_of_add_eq
-  have h₀ := parallelogram_identity ((I : 𝕜) • y + z) z
-  convert h₀ using 4 <;> · (try simp only [two_smul, smul_add]); abel
+    ‖(I : 𝕜) • y + 2 • z‖ * ‖(I : 𝕜) • y + 2 • z‖ + ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ =
+    2 * (‖(I : 𝕜) • y + z‖ * ‖(I : 𝕜) • y + z‖ + ‖z‖ * ‖z‖) := by
+  convert parallelogram_identity ((I : 𝕜) • y + z) z using 4 <;> module
 
 private theorem add_left_aux8 (y z : E) :
-    ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖ =
-    2 * (‖(I : 𝕜) • y - z‖ * ‖(I : 𝕜) • y - z‖ + ‖z‖ * ‖z‖) - ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ := by
-  apply eq_sub_of_add_eq'
-  have h₀ := parallelogram_identity ((I : 𝕜) • y - z) z
-  convert h₀ using 4 <;> · (try simp only [two_smul, smul_add]); abel
+    ‖(I : 𝕜) • y‖ * ‖(I : 𝕜) • y‖ + ‖(I : 𝕜) • y - 2 • z‖ * ‖(I : 𝕜) • y - 2 • z‖ =
+    2 * (‖(I : 𝕜) • y - z‖ * ‖(I : 𝕜) • y - z‖ + ‖z‖ * ‖z‖) := by
+  convert parallelogram_identity ((I : 𝕜) • y - z) z using 4 <;> module
+
+variable {𝕜}
 
 theorem add_left (x y z : E) : inner_ 𝕜 (x + y) z = inner_ 𝕜 x z + inner_ 𝕜 y z := by
-  simp only [inner_, ← mul_add]
-  congr
-  simp only [mul_assoc, ← map_mul, add_sub_assoc, ← mul_sub, ← map_sub]
-  rw [add_add_add_comm]
-  simp only [← map_add, ← mul_add]
-  congr
-  · rw [← add_sub_assoc, add_left_aux2', add_left_aux4']
-  · rw [add_left_aux5, add_left_aux6, add_left_aux7, add_left_aux8]
-    simp only [map_sub, map_mul, map_add, div_eq_mul_inv]
-    ring
-
-theorem nat (n : ℕ) (x y : E) : inner_ 𝕜 ((n : 𝕜) • x) y = (n : 𝕜) * inner_ 𝕜 x y := by
-  induction' n with n ih
-  · simp only [inner_, zero_sub, Nat.cast_zero, zero_mul,
-      eq_self_iff_true, zero_smul, zero_add, mul_zero, sub_self, norm_neg, smul_zero]
-  · simp only [Nat.cast_succ, add_smul, one_smul]
-    rw [add_left, ih, add_mul, one_mul]
-
-private theorem nat_prop (r : ℕ) : innerProp' E (r : 𝕜) := fun x y => by
-  simp only [map_natCast]; exact nat r x y
-
-private theorem int_prop (n : ℤ) : innerProp' E (n : 𝕜) := by
-  intro x y
-  rw [← n.sign_mul_natAbs]
-  simp only [Int.cast_natCast, map_natCast, map_intCast, Int.cast_mul, map_mul, mul_smul]
-  obtain hn | rfl | hn := lt_trichotomy n 0
-  · rw [Int.sign_eq_neg_one_of_neg hn, innerProp_neg_one ((n.natAbs : 𝕜) • x), nat]
-    simp only [map_neg, neg_mul, one_mul, mul_eq_mul_left_iff, true_or_iff, Int.natAbs_eq_zero,
-      eq_self_iff_true, Int.cast_one, map_one, neg_inj, Nat.cast_eq_zero, Int.cast_neg]
-  · simp only [inner_, Int.cast_zero, zero_sub, Nat.cast_zero, zero_mul,
-      eq_self_iff_true, Int.sign_zero, zero_smul, zero_add, mul_zero, smul_zero,
-      sub_self, norm_neg, Int.natAbs_zero]
-  · rw [Int.sign_eq_one_of_pos hn]
-    simp only [one_mul, mul_eq_mul_left_iff, true_or_iff, Int.natAbs_eq_zero, eq_self_iff_true,
-      Int.cast_one, one_smul, Nat.cast_eq_zero, nat]
+  have H_re := congr(- $(add_left_aux1 x y z) + $(add_left_aux2 x y z)
+    + $(add_left_aux3 y z) - $(add_left_aux4 y z))
+  have H_im := congr(- $(add_left_aux5 𝕜 x y z) + $(add_left_aux6 𝕜 x y z)
+      + $(add_left_aux7 𝕜 y z) - $(add_left_aux8 𝕜 y z))
+  have H := congr(𝓚 $H_re + I * 𝓚 $H_im)
+  simp only [inner_, map_add, map_sub, map_neg, map_mul, map_ofNat] at H ⊢
+  linear_combination H / 8
 
 private theorem rat_prop (r : ℚ) : innerProp' E (r : 𝕜) := by
   intro x y
-  have : (r.den : 𝕜) ≠ 0 := by
-    haveI : CharZero 𝕜 := RCLike.charZero_rclike
-    exact mod_cast r.pos.ne'
-  rw [← r.num_div_den, ← mul_right_inj' this, ← nat r.den _ y, smul_smul, Rat.cast_div]
-  simp only [map_natCast, Rat.cast_natCast, map_intCast, Rat.cast_intCast, map_div₀]
-  rw [← mul_assoc, mul_div_cancel₀ _ this, int_prop _ x, map_intCast]
+  let hom : 𝕜 →ₗ[ℚ] 𝕜 := AddMonoidHom.toRatLinearMap <|
+    AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) <| fun a b ↦ by
+      simpa [add_smul] using add_left (a • x) (b • x) y
+  simpa [hom, Rat.smul_def] using map_smul hom r 1
 
 private theorem real_prop (r : ℝ) : innerProp' E (r : 𝕜) := by
   intro x y
   revert r
   rw [← Function.funext_iff]
-  refine Rat.denseEmbedding_coe_real.dense.equalizer ?_ ?_ (funext fun X => ?_)
+  refine Rat.isDenseEmbedding_coe_real.dense.equalizer ?_ ?_ (funext fun X => ?_)
   · exact (continuous_ofReal.smul continuous_const).inner_ continuous_const
   · exact (continuous_conj.comp continuous_ofReal).mul continuous_const
   · simp only [Function.comp_apply, RCLike.ofReal_ratCast, rat_prop _ _]
 
 private theorem I_prop : innerProp' E (I : 𝕜) := by
   by_cases hI : (I : 𝕜) = 0
-  · rw [hI, ← Nat.cast_zero]; exact nat_prop _
+  · rw [hI]
+    simpa using real_prop (𝕜 := 𝕜) 0
   intro x y
-  have hI' : (-I : 𝕜) * I = 1 := by rw [← inv_I, inv_mul_cancel₀ hI]
-  rw [conj_I, inner_, inner_, mul_left_comm]
-  congr 1
-  rw [smul_smul, I_mul_I_of_nonzero hI, neg_one_smul]
-  rw [mul_sub, mul_add, mul_sub, mul_assoc I (𝓚 ‖I • x - y‖), ← mul_assoc (-I) I, hI', one_mul,
-    mul_assoc I (𝓚 ‖I • x + y‖), ← mul_assoc (-I) I, hI', one_mul]
+  have hI' := I_mul_I_of_nonzero hI
+  rw [conj_I, inner_, inner_, mul_left_comm, smul_smul, hI', neg_one_smul]
   have h₁ : ‖-x - y‖ = ‖x + y‖ := by rw [← neg_add', norm_neg]
   have h₂ : ‖-x + y‖ = ‖x - y‖ := by rw [← neg_sub, norm_neg, sub_eq_neg_add]
   rw [h₁, h₂]
-  simp only [sub_eq_add_neg, mul_assoc]
-  rw [← neg_mul_eq_neg_mul, ← neg_mul_eq_neg_mul]
-  abel
+  linear_combination (- 𝓚 ‖(I : 𝕜) • x - y‖ ^ 2 + 𝓚 ‖(I : 𝕜) • x + y‖ ^ 2) * hI' / 4
 
 theorem innerProp (r : 𝕜) : innerProp' E r := by
   intro x y
diff --git a/Mathlib/Analysis/InnerProductSpace/Orientation.lean b/Mathlib/Analysis/InnerProductSpace/Orientation.lean
index 60b27da658fe3..fb73168b5bab2 100644
--- a/Mathlib/Analysis/InnerProductSpace/Orientation.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Orientation.lean
@@ -38,7 +38,7 @@ noncomputable section
 
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
-open FiniteDimensional
+open Module
 
 open scoped RealInnerProductSpace
 
diff --git a/Mathlib/Analysis/InnerProductSpace/Orthogonal.lean b/Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
index c2b5db639db34..7b857c4ae5eba 100644
--- a/Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
@@ -112,8 +112,7 @@ theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v :
 /-- The orthogonal complement of any submodule `K` is closed. -/
 theorem isClosed_orthogonal : IsClosed (Kᗮ : Set E) := by
   rw [orthogonal_eq_inter K]
-  have := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
-  convert isClosed_iInter this
+  convert isClosed_iInter <| fun v : K => ContinuousLinearMap.isClosed_ker (innerSL 𝕜 (v : E))
   simp only [iInf_coe]
 
 /-- In a complete space, the orthogonal complement of any submodule `K` is complete. -/
diff --git a/Mathlib/Analysis/InnerProductSpace/PiL2.lean b/Mathlib/Analysis/InnerProductSpace/PiL2.lean
index b157563aab735..5fb4aa1c20293 100644
--- a/Mathlib/Analysis/InnerProductSpace/PiL2.lean
+++ b/Mathlib/Analysis/InnerProductSpace/PiL2.lean
@@ -144,11 +144,11 @@ variable [Fintype ι]
 
 @[simp]
 theorem finrank_euclideanSpace :
-    FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι := by
+    Module.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι := by
   simp [EuclideanSpace, PiLp, WithLp]
 
 theorem finrank_euclideanSpace_fin {n : ℕ} :
-    FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 (Fin n)) = n := by simp
+    Module.finrank 𝕜 (EuclideanSpace 𝕜 (Fin n)) = n := by simp
 
 theorem EuclideanSpace.inner_eq_star_dotProduct (x y : EuclideanSpace 𝕜 ι) :
     ⟪x, y⟫ = Matrix.dotProduct (star <| WithLp.equiv _ _ x) (WithLp.equiv _ _ y) :=
@@ -386,6 +386,7 @@ protected theorem sum_repr (b : OrthonormalBasis ι 𝕜 E) (x : E) : ∑ i, b.r
   simp_rw [← b.coe_toBasis_repr_apply, ← b.coe_toBasis]
   exact b.toBasis.sum_repr x
 
+open scoped InnerProductSpace in
 protected theorem sum_repr' (b : OrthonormalBasis ι 𝕜 E) (x : E) : ∑ i, ⟪b i, x⟫_𝕜 • b i = x := by
   nth_rw 2 [← (b.sum_repr x)]
   simp_rw [b.repr_apply_apply x]
@@ -668,7 +669,7 @@ theorem Complex.isometryOfOrthonormal_apply (v : OrthonormalBasis (Fin 2) ℝ F)
 
 end Complex
 
-open FiniteDimensional
+open Module
 
 /-! ### Matrix representation of an orthonormal basis with respect to another -/
 
@@ -678,6 +679,28 @@ section ToMatrix
 variable [DecidableEq ι]
 
 section
+open scoped Matrix
+
+/-- A version of `OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary` that works for bases with
+different index types. -/
+@[simp]
+theorem OrthonormalBasis.toMatrix_orthonormalBasis_conjTranspose_mul_self [Fintype ι']
+    (a : OrthonormalBasis ι' 𝕜 E) (b : OrthonormalBasis ι 𝕜 E) :
+    (a.toBasis.toMatrix b)ᴴ * a.toBasis.toMatrix b = 1 := by
+  ext i j
+  convert a.repr.inner_map_map (b i) (b j)
+  rw [orthonormal_iff_ite.mp b.orthonormal i j]
+  rfl
+
+/-- A version of `OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary` that works for bases with
+different index types. -/
+@[simp]
+theorem OrthonormalBasis.toMatrix_orthonormalBasis_self_mul_conjTranspose [Fintype ι']
+    (a : OrthonormalBasis ι 𝕜 E) (b : OrthonormalBasis ι' 𝕜 E) :
+    a.toBasis.toMatrix b * (a.toBasis.toMatrix b)ᴴ = 1 := by
+  classical
+  rw [Matrix.mul_eq_one_comm_of_equiv (a.toBasis.indexEquiv b.toBasis),
+    a.toMatrix_orthonormalBasis_conjTranspose_mul_self b]
 
 variable (a b : OrthonormalBasis ι 𝕜 E)
 
@@ -685,10 +708,7 @@ variable (a b : OrthonormalBasis ι 𝕜 E)
 theorem OrthonormalBasis.toMatrix_orthonormalBasis_mem_unitary :
     a.toBasis.toMatrix b ∈ Matrix.unitaryGroup ι 𝕜 := by
   rw [Matrix.mem_unitaryGroup_iff']
-  ext i j
-  convert a.repr.inner_map_map (b i) (b j)
-  rw [orthonormal_iff_ite.mp b.orthonormal i j]
-  rfl
+  exact a.toMatrix_orthonormalBasis_conjTranspose_mul_self b
 
 /-- The determinant of the change-of-basis matrix between two orthonormal bases `a`, `b` has
 unit length. -/
@@ -772,7 +792,7 @@ theorem Orthonormal.exists_orthonormalBasis_extension_of_card_eq {ι : Type*} [F
   obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormalBasis_extension
   have hιY : Fintype.card ι = Y.card := by
     refine card_ι.symm.trans ?_
-    exact FiniteDimensional.finrank_eq_card_finset_basis b₀.toBasis
+    exact Module.finrank_eq_card_finset_basis b₀.toBasis
   have hvsY : s.MapsTo v Y := (s.mapsTo_image v).mono_right (by rwa [← range_restrict])
   have hsv' : Set.InjOn v s := by
     rw [Set.injOn_iff_injective]
@@ -820,7 +840,7 @@ irreducible_def DirectSum.IsInternal.sigmaOrthonormalBasisIndexEquiv
     (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
     (Σi, Fin (finrank 𝕜 (V i))) ≃ Fin n :=
   let b := hV.collectedOrthonormalBasis hV' fun i => stdOrthonormalBasis 𝕜 (V i)
-  Fintype.equivFinOfCardEq <| (FiniteDimensional.finrank_eq_card_basis b.toBasis).symm.trans hn
+  Fintype.equivFinOfCardEq <| (Module.finrank_eq_card_basis b.toBasis).symm.trans hn
 
 /-- An `n`-dimensional `InnerProductSpace` equipped with a decomposition as an internal direct
 sum has an orthonormal basis indexed by `Fin n` and subordinate to that direct sum. -/
@@ -865,7 +885,7 @@ section LinearIsometry
 variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace 𝕜 V] [FiniteDimensional 𝕜 V]
 variable {S : Submodule 𝕜 V} {L : S →ₗᵢ[𝕜] V}
 
-open FiniteDimensional
+open Module
 
 /-- Let `S` be a subspace of a finite-dimensional complex inner product space `V`.  A linear
 isometry mapping `S` into `V` can be extended to a full isometry of `V`.
@@ -915,7 +935,7 @@ noncomputable def LinearIsometry.extend (L : S →ₗᵢ[𝕜] V) : V →ₗᵢ[
     simp only [sq, Mx_decomp]
     rw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth]
     simp only [p1, p2, LinearIsometry.norm_map, _root_.add_left_inj, mul_eq_mul_left_iff,
-      norm_eq_zero, true_or_iff, eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm,
+      norm_eq_zero, eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm,
       Submodule.coe_eq_zero]
   exact
     { toLinearMap := M
@@ -927,7 +947,7 @@ theorem LinearIsometry.extend_apply (L : S →ₗᵢ[𝕜] V) (s : S) : L.extend
   simp only [add_right_eq_self, LinearIsometry.coe_toLinearMap,
     LinearIsometryEquiv.coe_toLinearIsometry, LinearIsometry.coe_comp, Function.comp_apply,
     orthogonalProjection_mem_subspace_eq_self, LinearMap.coe_comp, ContinuousLinearMap.coe_coe,
-    Submodule.coeSubtype, LinearMap.add_apply, Submodule.coe_eq_zero,
+    Submodule.coe_subtype, LinearMap.add_apply, Submodule.coe_eq_zero,
     LinearIsometryEquiv.map_eq_zero_iff, Submodule.coe_subtypeₗᵢ,
     orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.orthogonal_orthogonal,
     Submodule.coe_mem]
diff --git a/Mathlib/Analysis/InnerProductSpace/Positive.lean b/Mathlib/Analysis/InnerProductSpace/Positive.lean
index c874c7b049ac5..d3c5fef5c6c27 100644
--- a/Mathlib/Analysis/InnerProductSpace/Positive.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Positive.lean
@@ -99,6 +99,31 @@ theorem IsPositive.orthogonalProjection_comp {T : E →L[𝕜] E} (hT : T.IsPosi
   have := hT.conj_adjoint (orthogonalProjection U : E →L[𝕜] U)
   rwa [U.adjoint_orthogonalProjection] at this
 
+open scoped NNReal
+
+lemma antilipschitz_of_forall_le_inner_map {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace 𝕜 H] (f : H →L[𝕜] H) {c : ℝ≥0} (hc : 0 < c)
+    (h : ∀ x, ‖x‖ ^ 2 * c ≤ ‖⟪f x, x⟫_𝕜‖) : AntilipschitzWith c⁻¹ f := by
+  refine f.antilipschitz_of_bound (K := c⁻¹) fun x ↦ ?_
+  rw [NNReal.coe_inv, inv_mul_eq_div, le_div_iff₀ (by exact_mod_cast hc)]
+  simp_rw [sq, mul_assoc] at h
+  by_cases hx0 : x = 0
+  · simp [hx0]
+  · apply (map_le_map_iff <| OrderIso.mulLeft₀ ‖x‖ (norm_pos_iff'.mpr hx0)).mp
+    exact (h x).trans <| (norm_inner_le_norm _ _).trans <| (mul_comm _ _).le
+
+lemma isUnit_of_forall_le_norm_inner_map (f : E →L[𝕜] E) {c : ℝ≥0} (hc : 0 < c)
+    (h : ∀ x, ‖x‖ ^ 2 * c ≤ ‖⟪f x, x⟫_𝕜‖) : IsUnit f := by
+  rw [isUnit_iff_bijective, bijective_iff_dense_range_and_antilipschitz]
+  have h_anti : AntilipschitzWith c⁻¹ f := antilipschitz_of_forall_le_inner_map f hc h
+  refine ⟨?_, ⟨_, h_anti⟩⟩
+  have _inst := h_anti.completeSpace_range_clm
+  rw [Submodule.topologicalClosure_eq_top_iff, Submodule.eq_bot_iff]
+  intro x hx
+  have : ‖x‖ ^ 2 * c = 0 := le_antisymm (by simpa only [hx (f x) ⟨x, rfl⟩, norm_zero] using h x)
+    (by positivity)
+  aesop
+
 section Complex
 
 variable {E' : Type*} [NormedAddCommGroup E'] [InnerProductSpace ℂ E'] [CompleteSpace E']
diff --git a/Mathlib/Analysis/InnerProductSpace/Projection.lean b/Mathlib/Analysis/InnerProductSpace/Projection.lean
index 707697625b2cc..28ef290eefc60 100644
--- a/Mathlib/Analysis/InnerProductSpace/Projection.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Projection.lean
@@ -42,11 +42,13 @@ The Coq code is available at the following address: <http://www.lri.fr/~sboldo/e
 
 noncomputable section
 
+open InnerProductSpace
+
 open RCLike Real Filter
 
 open LinearMap (ker range)
 
-open Topology
+open Topology Finsupp
 
 variable {𝕜 E F : Type*} [RCLike 𝕜]
 variable [NormedAddCommGroup E] [NormedAddCommGroup F]
@@ -216,8 +218,7 @@ theorem norm_eq_iInf_iff_real_inner_le_zero {K : Set F} (h : Convex ℝ K) {u :
         θ * θ * ‖w - v‖ ^ 2 - 2 * θ * inner (u - v) (w - v) =
           θ * (θ * ‖w - v‖ ^ 2 - 2 * inner (u - v) (w - v)) := by ring
       rw [eq₂] at this
-      have := le_of_sub_nonneg (nonneg_of_mul_nonneg_right this hθ₁)
-      exact this
+      exact le_of_sub_nonneg (nonneg_of_mul_nonneg_right this hθ₁)
     by_cases hq : q = 0
     · rw [hq] at this
       have : p ≤ 0 := by
@@ -278,7 +279,7 @@ theorem exists_norm_eq_iInf_of_complete_subspace (h : IsComplete (↑K : Set E))
 Let `u` be a point in a real inner product space, and let `K` be a nonempty subspace.
 Then point `v` minimizes the distance `‖u - v‖` over points in `K` if and only if
 for all `w ∈ K`, `⟪u - v, w⟫ = 0` (i.e., `u - v` is orthogonal to the subspace `K`).
-This is superceded by `norm_eq_iInf_iff_inner_eq_zero` that gives the same conclusion over
+This is superseded by `norm_eq_iInf_iff_inner_eq_zero` that gives the same conclusion over
 any `RCLike` field.
 -/
 theorem norm_eq_iInf_iff_real_inner_eq_zero (K : Submodule ℝ F) {u : F} {v : F} (hv : v ∈ K) :
@@ -332,7 +333,7 @@ theorem norm_eq_iInf_iff_inner_eq_zero {u : E} {v : E} (hv : v ∈ K) :
   · intro H
     have A : ∀ w ∈ K, re ⟪u - v, w⟫ = 0 := (norm_eq_iInf_iff_real_inner_eq_zero K' hv).1 H
     intro w hw
-    apply ext
+    apply RCLike.ext
     · simp [A w hw]
     · symm
       calc
@@ -347,7 +348,7 @@ theorem norm_eq_iInf_iff_inner_eq_zero {u : E} {v : E} (hv : v ∈ K) :
       exact zero_re'
     exact (norm_eq_iInf_iff_real_inner_eq_zero K' hv).2 this
 
-/-- A subspace `K : Submodule 𝕜 E` has an orthogonal projection if evey vector `v : E` admits an
+/-- A subspace `K : Submodule 𝕜 E` has an orthogonal projection if every vector `v : E` admits an
 orthogonal projection to `K`. -/
 class HasOrthogonalProjection (K : Submodule 𝕜 E) : Prop where
   exists_orthogonal (v : E) : ∃ w ∈ K, v - w ∈ Kᗮ
@@ -471,13 +472,13 @@ theorem orthogonalProjectionFn_eq (v : E) :
     orthogonalProjectionFn K v = (orthogonalProjection K v : E) :=
   rfl
 
-/-- The characterization of the orthogonal projection.  -/
+/-- The characterization of the orthogonal projection. -/
 @[simp]
 theorem orthogonalProjection_inner_eq_zero (v : E) :
     ∀ w ∈ K, ⟪v - orthogonalProjection K v, w⟫ = 0 :=
   orthogonalProjectionFn_inner_eq_zero v
 
-/-- The difference of `v` from its orthogonal projection onto `K` is in `Kᗮ`.  -/
+/-- The difference of `v` from its orthogonal projection onto `K` is in `Kᗮ`. -/
 @[simp]
 theorem sub_orthogonalProjection_mem_orthogonal (v : E) : v - orthogonalProjection K v ∈ Kᗮ := by
   intro w hw
@@ -780,7 +781,7 @@ is the topological closure of `K`.
 
 Note that the completeness assumption is necessary. Let `E` be the space `ℕ →₀ ℝ` with inner space
 structure inherited from `PiLp 2 (fun _ : ℕ ↦ ℝ)`. Let `K` be the subspace of sequences with the sum
-of all elements equal to zero. Then `Kᗮ = ⊥`, `Kᗮᗮ = ⊤`.  -/
+of all elements equal to zero. Then `Kᗮ = ⊥`, `Kᗮᗮ = ⊤`. -/
 theorem Submodule.orthogonal_orthogonal_eq_closure [CompleteSpace E] :
     Kᗮᗮ = K.topologicalClosure := by
   refine le_antisymm ?_ ?_
@@ -862,12 +863,13 @@ theorem orthogonalProjection_orthogonalProjection_of_le {U V : Submodule 𝕜 E}
 /-- Given a monotone family `U` of complete submodules of `E` and a fixed `x : E`,
 the orthogonal projection of `x` on `U i` tends to the orthogonal projection of `x` on
 `(⨆ i, U i).topologicalClosure` along `atTop`. -/
-theorem orthogonalProjection_tendsto_closure_iSup [CompleteSpace E] {ι : Type*} [SemilatticeSup ι]
-    (U : ι → Submodule 𝕜 E) [∀ i, CompleteSpace (U i)] (hU : Monotone U) (x : E) :
+theorem orthogonalProjection_tendsto_closure_iSup {ι : Type*} [Preorder ι]
+    (U : ι → Submodule 𝕜 E) [∀ i, HasOrthogonalProjection (U i)]
+    [HasOrthogonalProjection (⨆ i, U i).topologicalClosure] (hU : Monotone U) (x : E) :
     Filter.Tendsto (fun i => (orthogonalProjection (U i) x : E)) atTop
       (𝓝 (orthogonalProjection (⨆ i, U i).topologicalClosure x : E)) := by
-  cases isEmpty_or_nonempty ι
-  · exact tendsto_of_isEmpty
+  refine .of_neBot_imp fun h ↦ ?_
+  cases atTop_neBot_iff.mp h
   let y := (orthogonalProjection (⨆ i, U i).topologicalClosure x : E)
   have proj_x : ∀ i, orthogonalProjection (U i) x = orthogonalProjection (U i) y := fun i =>
     (orthogonalProjection_orthogonalProjection_of_le
@@ -889,14 +891,15 @@ theorem orthogonalProjection_tendsto_closure_iSup [CompleteSpace E] {ι : Type*}
 
 /-- Given a monotone family `U` of complete submodules of `E` with dense span supremum,
 and a fixed `x : E`, the orthogonal projection of `x` on `U i` tends to `x` along `at_top`. -/
-theorem orthogonalProjection_tendsto_self [CompleteSpace E] {ι : Type*} [SemilatticeSup ι]
-    (U : ι → Submodule 𝕜 E) [∀ t, CompleteSpace (U t)] (hU : Monotone U) (x : E)
+theorem orthogonalProjection_tendsto_self {ι : Type*} [Preorder ι]
+    (U : ι → Submodule 𝕜 E) [∀ t, HasOrthogonalProjection (U t)] (hU : Monotone U) (x : E)
     (hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure) :
     Filter.Tendsto (fun t => (orthogonalProjection (U t) x : E)) atTop (𝓝 x) := by
-  rw [← eq_top_iff] at hU'
+  have : HasOrthogonalProjection (⨆ i, U i).topologicalClosure := by
+    rw [top_unique hU']
+    infer_instance
   convert orthogonalProjection_tendsto_closure_iSup U hU x
-  rw [orthogonalProjection_eq_self_iff.mpr _]
-  rw [hU']
+  rw [eq_comm, orthogonalProjection_eq_self_iff, top_unique hU']
   trivial
 
 /-- The orthogonal complement satisfies `Kᗮᗮᗮ = Kᗮ`. -/
@@ -1028,7 +1031,7 @@ theorem orthogonalProjection_isSymmetric [HasOrthogonalProjection K] :
     (K.subtypeL ∘L orthogonalProjection K : E →ₗ[𝕜] E).IsSymmetric :=
   inner_orthogonalProjection_left_eq_right K
 
-open FiniteDimensional
+open Module
 
 /-- Given a finite-dimensional subspace `K₂`, and a subspace `K₁`
 contained in it, the dimensions of `K₁` and the intersection of its
@@ -1037,7 +1040,7 @@ theorem Submodule.finrank_add_inf_finrank_orthogonal {K₁ K₂ : Submodule 𝕜
     [FiniteDimensional 𝕜 K₂] (h : K₁ ≤ K₂) :
     finrank 𝕜 K₁ + finrank 𝕜 (K₁ᗮ ⊓ K₂ : Submodule 𝕜 E) = finrank 𝕜 K₂ := by
   haveI : FiniteDimensional 𝕜 K₁ := Submodule.finiteDimensional_of_le h
-  haveI := proper_rclike 𝕜 K₁
+  haveI := FiniteDimensional.proper_rclike 𝕜 K₁
   have hd := Submodule.finrank_sup_add_finrank_inf_eq K₁ (K₁ᗮ ⊓ K₂)
   rw [← inf_assoc, (Submodule.orthogonal_disjoint K₁).eq_bot, bot_inf_eq, finrank_bot,
     Submodule.sup_orthogonal_inf_of_completeSpace h] at hd
@@ -1187,7 +1190,7 @@ theorem OrthogonalFamily.isInternal_iff_of_isComplete [DecidableEq ι] {V : ι 
     (hc : IsComplete (↑(iSup V) : Set E)) : DirectSum.IsInternal V ↔ (iSup V)ᗮ = ⊥ := by
   haveI : CompleteSpace (↥(iSup V)) := hc.completeSpace_coe
   simp only [DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top, hV.independent,
-    true_and_iff, Submodule.orthogonal_eq_bot_iff]
+    true_and, Submodule.orthogonal_eq_bot_iff]
 
 /-- An orthogonal family of subspaces of `E` satisfies `DirectSum.IsInternal` (that is,
 they provide an internal direct sum decomposition of `E`) if and only if their span has trivial
@@ -1269,7 +1272,7 @@ section OrthonormalBasis
 
 variable {v : Set E}
 
-open FiniteDimensional Submodule Set
+open Module Submodule Set
 
 /-- An orthonormal set in an `InnerProductSpace` is maximal, if and only if the orthogonal
 complement of its span is empty. -/
@@ -1329,8 +1332,8 @@ theorem maximal_orthonormal_iff_orthogonalComplement_eq_bot (hv : Orthonormal 
     intro hxv y hy
     have hxv' : (⟨x, hxu⟩ : u) ∉ ((↑) ⁻¹' v : Set u) := by simp [huv, hxv]
     obtain ⟨l, hl, rfl⟩ :
-      ∃ l ∈ Finsupp.supported 𝕜 𝕜 ((↑) ⁻¹' v : Set u), (Finsupp.total 𝕜 ((↑) : u → E)) l = y := by
-      rw [← Finsupp.mem_span_image_iff_total]
+      ∃ l ∈ supported 𝕜 𝕜 ((↑) ⁻¹' v : Set u), (linearCombination 𝕜 ((↑) : u → E)) l = y := by
+      rw [← Finsupp.mem_span_image_iff_linearCombination]
       simp [huv, inter_eq_self_of_subset_right, hy]
     exact hu.inner_finsupp_eq_zero hxv' hl
 
@@ -1340,7 +1343,7 @@ variable [FiniteDimensional 𝕜 E]
 is a basis. -/
 theorem maximal_orthonormal_iff_basis_of_finiteDimensional (hv : Orthonormal 𝕜 ((↑) : v → E)) :
     (∀ u ⊇ v, Orthonormal 𝕜 ((↑) : u → E) → u = v) ↔ ∃ b : Basis v 𝕜 E, ⇑b = ((↑) : v → E) := by
-  haveI := proper_rclike 𝕜 (span 𝕜 v)
+  haveI := FiniteDimensional.proper_rclike 𝕜 (span 𝕜 v)
   rw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hv]
   rw [Submodule.orthogonal_eq_bot_iff]
   have hv_coe : range ((↑) : v → E) = v := by simp
diff --git a/Mathlib/Analysis/InnerProductSpace/Rayleigh.lean b/Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
index 1758106c5c5f0..8d29cca646b3d 100644
--- a/Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Rayleigh.lean
@@ -130,6 +130,7 @@ theorem linearly_dependent_of_isLocalExtrOn (hT : IsSelfAdjoint T) {x₀ : F}
   apply smul_right_injective (F →L[ℝ] ℝ) (two_ne_zero : (2 : ℝ) ≠ 0)
   simpa only [two_smul, smul_add, add_smul, add_zero] using h₂
 
+open scoped InnerProductSpace in
 theorem eq_smul_self_of_isLocalExtrOn_real (hT : IsSelfAdjoint T) {x₀ : F}
     (hextr : IsLocalExtrOn T.reApplyInnerSelf (sphere (0 : F) ‖x₀‖) x₀) :
     T x₀ = T.rayleighQuotient x₀ • x₀ := by
diff --git a/Mathlib/Analysis/InnerProductSpace/Spectrum.lean b/Mathlib/Analysis/InnerProductSpace/Spectrum.lean
index 68d1c7ecfc1b4..0f4bd55b7bae8 100644
--- a/Mathlib/Analysis/InnerProductSpace/Spectrum.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Spectrum.lean
@@ -184,7 +184,7 @@ end Version1
 
 section Version2
 
-variable {n : ℕ} (hn : FiniteDimensional.finrank 𝕜 E = n)
+variable {n : ℕ} (hn : Module.finrank 𝕜 E = n)
 
 /-- A choice of orthonormal basis of eigenvectors for self-adjoint operator `T` on a
 finite-dimensional inner product space `E`.
diff --git a/Mathlib/Analysis/InnerProductSpace/StarOrder.lean b/Mathlib/Analysis/InnerProductSpace/StarOrder.lean
new file mode 100644
index 0000000000000..7ed9ab6de0e4b
--- /dev/null
+++ b/Mathlib/Analysis/InnerProductSpace/StarOrder.lean
@@ -0,0 +1,77 @@
+/-
+Copyright (c) 2024 Jireh Loreaux. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jireh Loreaux
+-/
+import Mathlib.Analysis.InnerProductSpace.Positive
+import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
+
+/-!
+# Continuous linear maps on a Hilbert space are a `StarOrderedRing`
+
+In this file we show that the continuous linear maps on a complex Hilbert space form a
+`StarOrderedRing`.  Note that they are already equipped with the Loewner partial order. We also
+prove that, with respect to this partial order, a map is positive if every element of the
+real spectrum is nonnegative. Consequently, when `H` is a Hilbert space, then `H →L[ℂ] H` is
+equipped with all the usual instances of the continuous functional calculus.
+
+ -/
+
+namespace ContinuousLinearMap
+
+open RCLike
+open scoped NNReal
+
+variable {𝕜 H : Type*} [RCLike 𝕜] [NormedAddCommGroup H] [InnerProductSpace 𝕜 H] [CompleteSpace H]
+variable [Algebra ℝ (H →L[𝕜] H)] [IsScalarTower ℝ 𝕜 (H →L[𝕜] H)]
+
+open scoped InnerProductSpace in
+lemma IsPositive.spectrumRestricts {f : H →L[𝕜] H} (hf : f.IsPositive) :
+    SpectrumRestricts f ContinuousMap.realToNNReal := by
+  rw [SpectrumRestricts.nnreal_iff]
+  intro c hc
+  contrapose! hc
+  rw [spectrum.not_mem_iff, IsUnit.sub_iff, sub_eq_add_neg, ← map_neg]
+  rw [← neg_pos] at hc
+  set c := -c
+  exact isUnit_of_forall_le_norm_inner_map _ (c := ⟨c, hc.le⟩) hc fun x ↦ calc
+    ‖x‖ ^ 2 * c = re ⟪algebraMap ℝ (H →L[𝕜] H) c x, x⟫_𝕜 := by
+      rw [Algebra.algebraMap_eq_smul_one, ← algebraMap_smul 𝕜 c (1 : (H →L[𝕜] H)), coe_smul',
+        Pi.smul_apply, one_apply, inner_smul_left, RCLike.algebraMap_eq_ofReal, conj_ofReal,
+        re_ofReal_mul, inner_self_eq_norm_sq, mul_comm]
+    _ ≤ re ⟪(f + (algebraMap ℝ (H →L[𝕜] H)) c) x, x⟫_𝕜 := by
+      simpa only [add_apply, inner_add_left, map_add, le_add_iff_nonneg_left]
+        using hf.inner_nonneg_left x
+    _ ≤ ‖⟪(f + (algebraMap ℝ (H →L[𝕜] H)) c) x, x⟫_𝕜‖ := RCLike.re_le_norm _
+
+instance : NonnegSpectrumClass ℝ (H →L[𝕜] H) where
+  quasispectrum_nonneg_of_nonneg f hf :=
+    QuasispectrumRestricts.nnreal_iff.mp <| sub_zero f ▸ hf.spectrumRestricts
+
+/-- Because this takes `ContinuousFunctionalCalculus ℝ IsSelfAdjoint` as an argument, and for
+the moment we only have this for `𝕜 := ℂ`, this is not registered as an instance. -/
+lemma instStarOrderedRingRCLike
+    [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : (H →L[𝕜] H) → Prop)] :
+    StarOrderedRing (H →L[𝕜] H) where
+  le_iff f g := by
+    constructor
+    · intro h
+      rw [le_def] at h
+      obtain ⟨p, hp₁, -, hp₃⟩ :=
+        CFC.exists_sqrt_of_isSelfAdjoint_of_spectrumRestricts h.1 h.spectrumRestricts
+      refine ⟨p ^ 2, ?_, by symm; rwa [add_comm, ← eq_sub_iff_add_eq]⟩
+      exact AddSubmonoid.subset_closure ⟨p, by simp only [hp₁.star_eq, sq]⟩
+    · rintro ⟨p, hp, rfl⟩
+      rw [le_def, add_sub_cancel_left]
+      induction hp using AddSubmonoid.closure_induction' with
+      | mem _ hf =>
+        obtain ⟨f, rfl⟩ := hf
+        simpa using ContinuousLinearMap.IsPositive.adjoint_conj isPositive_one f
+      | one => exact isPositive_zero
+      | mul f _ g _ hf hg => exact hf.add hg
+
+instance instStarOrderedRing {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℂ H] [CompleteSpace H] : StarOrderedRing (H →L[ℂ] H) :=
+  instStarOrderedRingRCLike
+
+end ContinuousLinearMap
diff --git a/Mathlib/Analysis/InnerProductSpace/Symmetric.lean b/Mathlib/Analysis/InnerProductSpace/Symmetric.lean
index cc99f2f374367..9acf7b33a0635 100644
--- a/Mathlib/Analysis/InnerProductSpace/Symmetric.lean
+++ b/Mathlib/Analysis/InnerProductSpace/Symmetric.lean
@@ -36,11 +36,13 @@ open RCLike
 
 open ComplexConjugate
 
+section Seminormed
+
 variable {𝕜 E E' F G : Type*} [RCLike 𝕜]
-variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
-variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F]
-variable [NormedAddCommGroup G] [InnerProductSpace 𝕜 G]
-variable [NormedAddCommGroup E'] [InnerProductSpace ℝ E']
+variable [SeminormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable [SeminormedAddCommGroup F] [InnerProductSpace 𝕜 F]
+variable [SeminormedAddCommGroup G] [InnerProductSpace 𝕜 G]
+variable [SeminormedAddCommGroup E'] [InnerProductSpace ℝ E']
 
 local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
 
@@ -72,33 +74,47 @@ theorem IsSymmetric.apply_clm {T : E →L[𝕜] E} (hT : IsSymmetric (T : E →
     ⟪T x, y⟫ = ⟪x, T y⟫ :=
   hT x y
 
-theorem isSymmetric_zero : (0 : E →ₗ[𝕜] E).IsSymmetric := fun x y =>
+@[simp]
+protected theorem IsSymmetric.zero : (0 : E →ₗ[𝕜] E).IsSymmetric := fun x y =>
   (inner_zero_right x : ⟪x, 0⟫ = 0).symm ▸ (inner_zero_left y : ⟪0, y⟫ = 0)
 
-theorem isSymmetric_id : (LinearMap.id : E →ₗ[𝕜] E).IsSymmetric := fun _ _ => rfl
+@[deprecated (since := "2024-09-30")] alias isSymmetric_zero := IsSymmetric.zero
+
+@[simp]
+protected theorem IsSymmetric.id : (LinearMap.id : E →ₗ[𝕜] E).IsSymmetric := fun _ _ => rfl
+
+@[deprecated (since := "2024-09-30")] alias isSymmetric_id := IsSymmetric.id
 
+@[aesop safe apply]
 theorem IsSymmetric.add {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
     (T + S).IsSymmetric := by
   intro x y
   rw [LinearMap.add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right]
   rfl
 
-/-- The **Hellinger--Toeplitz theorem**: if a symmetric operator is defined on a complete space,
-  then it is automatically continuous. -/
-theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
-    Continuous T := by
-  -- We prove it by using the closed graph theorem
-  refine T.continuous_of_seq_closed_graph fun u x y hu hTu => ?_
-  rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜]
-  have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
-    intro k
-    rw [← T.map_sub, hT]
-  refine tendsto_nhds_unique ((hTu.sub_const _).inner tendsto_const_nhds) ?_
-  simp_rw [Function.comp_apply, hlhs]
-  rw [← inner_zero_left (T (y - T x))]
-  refine Filter.Tendsto.inner ?_ tendsto_const_nhds
-  rw [← sub_self x]
-  exact hu.sub_const _
+@[aesop safe apply]
+theorem IsSymmetric.sub {T S : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (hS : S.IsSymmetric) :
+    (T - S).IsSymmetric := by
+  intro x y
+  rw [LinearMap.sub_apply, inner_sub_left, hT x y, hS x y, ← inner_sub_right]
+  rfl
+
+@[aesop safe apply]
+theorem IsSymmetric.smul {c : 𝕜} (hc : conj c = c) {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
+    c • T |>.IsSymmetric := by
+  intro x y
+  simp only [smul_apply, inner_smul_left, hc, hT x y, inner_smul_right]
+
+@[aesop 30% apply]
+lemma IsSymmetric.mul_of_commute {S T : E →ₗ[𝕜] E} (hS : S.IsSymmetric) (hT : T.IsSymmetric)
+    (hST : Commute S T) : (S * T).IsSymmetric :=
+  fun _ _ ↦ by rw [mul_apply, hS, hT, hST, mul_apply]
+
+@[aesop safe apply]
+lemma IsSymmetric.pow {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) (n : ℕ) : (T ^ n).IsSymmetric := by
+  refine Nat.le_induction (by simp [one_eq_id]) (fun k _ ih ↦ ?_) n n.zero_le
+  rw [iterate_succ, ← mul_eq_comp]
+  exact ih.mul_of_commute hT <| .pow_left rfl k
 
 /-- For a symmetric operator `T`, the function `fun x ↦ ⟪T x, x⟫` is real-valued. -/
 @[simp]
@@ -115,16 +131,17 @@ theorem IsSymmetric.restrict_invariant {T : E →ₗ[𝕜] E} (hT : IsSymmetric
     (hV : ∀ v ∈ V, T v ∈ V) : IsSymmetric (T.restrict hV) := fun v w => hT v w
 
 theorem IsSymmetric.restrictScalars {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
-    @LinearMap.IsSymmetric ℝ E _ _ (InnerProductSpace.rclikeToReal 𝕜 E)
-      (@LinearMap.restrictScalars ℝ 𝕜 _ _ _ _ _ _ (InnerProductSpace.rclikeToReal 𝕜 E).toModule
-        (InnerProductSpace.rclikeToReal 𝕜 E).toModule _ _ _ T) :=
+    letI := InnerProductSpace.rclikeToReal 𝕜 E
+    letI : IsScalarTower ℝ 𝕜 E := RestrictScalars.isScalarTower _ _ _
+    (T.restrictScalars ℝ).IsSymmetric :=
   fun x y => by simp [hT x y, real_inner_eq_re_inner, LinearMap.coe_restrictScalars ℝ]
 
 section Complex
 
-variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V]
+variable {V : Type*} [SeminormedAddCommGroup V] [InnerProductSpace ℂ V]
 
 attribute [local simp] map_ofNat in -- use `ofNat` simp theorem with bad keys
+open scoped InnerProductSpace in
 /-- A linear operator on a complex inner product space is symmetric precisely when
 `⟪T v, v⟫_ℂ` is real for all v. -/
 theorem isSymmetric_iff_inner_map_self_real (T : V →ₗ[ℂ] V) :
@@ -166,6 +183,39 @@ theorem IsSymmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.IsSymm
       sub_sub, ← mul_assoc, mul_neg, h, neg_neg, one_mul, neg_one_mul]
     ring
 
+end LinearMap
+
+end Seminormed
+
+section Normed
+
+variable {𝕜 E E' F G : Type*} [RCLike 𝕜]
+variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
+variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F]
+variable [NormedAddCommGroup G] [InnerProductSpace 𝕜 G]
+variable [NormedAddCommGroup E'] [InnerProductSpace ℝ E']
+
+local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
+
+namespace LinearMap
+
+/-- The **Hellinger--Toeplitz theorem**: if a symmetric operator is defined on a complete space,
+  then it is automatically continuous. -/
+theorem IsSymmetric.continuous [CompleteSpace E] {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) :
+    Continuous T := by
+  -- We prove it by using the closed graph theorem
+  refine T.continuous_of_seq_closed_graph fun u x y hu hTu => ?_
+  rw [← sub_eq_zero, ← @inner_self_eq_zero 𝕜]
+  have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ := by
+    intro k
+    rw [← T.map_sub, hT]
+  refine tendsto_nhds_unique ((hTu.sub_const _).inner tendsto_const_nhds) ?_
+  simp_rw [Function.comp_apply, hlhs]
+  rw [← inner_zero_left (T (y - T x))]
+  refine Filter.Tendsto.inner ?_ tendsto_const_nhds
+  rw [← sub_self x]
+  exact hu.sub_const _
+
 /-- A symmetric linear map `T` is zero if and only if `⟪T x, x⟫_ℝ = 0` for all `x`.
 See `inner_map_self_eq_zero` for the complex version without the symmetric assumption. -/
 theorem IsSymmetric.inner_map_self_eq_zero {T : E →ₗ[𝕜] E} (hT : T.IsSymmetric) :
@@ -177,3 +227,5 @@ theorem IsSymmetric.inner_map_self_eq_zero {T : E →ₗ[𝕜] E} (hT : T.IsSymm
   ring
 
 end LinearMap
+
+end Normed
diff --git a/Mathlib/Analysis/InnerProductSpace/TwoDim.lean b/Mathlib/Analysis/InnerProductSpace/TwoDim.lean
index 5f43d060381e7..329dd441e2adc 100644
--- a/Mathlib/Analysis/InnerProductSpace/TwoDim.lean
+++ b/Mathlib/Analysis/InnerProductSpace/TwoDim.lean
@@ -71,7 +71,7 @@ noncomputable section
 
 open scoped RealInnerProductSpace ComplexConjugate
 
-open FiniteDimensional
+open Module
 
 lemma FiniteDimensional.of_fact_finrank_eq_two {K V : Type*} [DivisionRing K]
     [AddCommGroup V] [Module K V] [Fact (finrank K V = 2)] : FiniteDimensional K V :=
@@ -204,7 +204,7 @@ def rightAngleRotationAux₂ : E →ₗᵢ[ℝ] E :=
         exact o.areaForm_le x (o.rightAngleRotationAux₁ x)
       · let K : Submodule ℝ E := ℝ ∙ x
         have : Nontrivial Kᗮ := by
-          apply @FiniteDimensional.nontrivial_of_finrank_pos ℝ
+          apply nontrivial_of_finrank_pos (R := ℝ)
           have : finrank ℝ K ≤ Finset.card {x} := by
             rw [← Set.toFinset_singleton]
             exact finrank_span_le_card ({x} : Set E)
@@ -422,8 +422,7 @@ theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
   · intro h
     obtain ⟨r, hr, rfl⟩ := h.exists_nonneg_left hx
     simp only [inner_smul_right, real_inner_self_eq_norm_sq, LinearMap.map_smulₛₗ,
-      areaForm_apply_self, Algebra.id.smul_eq_mul, mul_zero, eq_self_iff_true,
-      and_true_iff]
+      areaForm_apply_self, Algebra.id.smul_eq_mul, mul_zero, eq_self_iff_true, and_true]
     positivity
 
 /-- A complex-valued real-bilinear map on an oriented real inner product space of dimension 2. Its
diff --git a/Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean b/Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean
index 34f154ad37514..52173af64e2c0 100644
--- a/Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean
+++ b/Mathlib/Analysis/InnerProductSpace/WeakOperatorTopology.lean
@@ -15,7 +15,7 @@ Hilbert spaces. This mostly involves using the Fréchet-Riesz representation to
 applications of elements of the dual and inner products with vectors in the space.
 -/
 
-open scoped Topology
+open scoped Topology InnerProductSpace
 
 namespace ContinuousLinearMapWOT
 
diff --git a/Mathlib/Analysis/InnerProductSpace/l2Space.lean b/Mathlib/Analysis/InnerProductSpace/l2Space.lean
index 9eb925f593b03..54a08fc3d190b 100644
--- a/Mathlib/Analysis/InnerProductSpace/l2Space.lean
+++ b/Mathlib/Analysis/InnerProductSpace/l2Space.lean
@@ -249,7 +249,7 @@ protected theorem range_linearIsometry [∀ i, CompleteSpace (G i)] :
       rintro i x ⟨x, rfl⟩
       use lp.single 2 i x
       exact hV.linearIsometry_apply_single x
-    exact hV.linearIsometry.isometry.uniformInducing.isComplete_range.isClosed
+    exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed
 
 end OrthogonalFamily
 
diff --git a/Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean b/Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
index 5667478982a21..08b82218840f4 100644
--- a/Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
+++ b/Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
@@ -131,7 +131,7 @@ theorem balancedHull.balanced (s : Set E) : Balanced 𝕜 (balancedHull 𝕜 s)
   simp_rw [balancedHull, smul_set_iUnion₂, subset_def, mem_iUnion₂]
   rintro x ⟨r, hr, hx⟩
   rw [← smul_assoc] at hx
-  exact ⟨a • r, (SeminormedRing.norm_mul _ _).trans (mul_le_one ha (norm_nonneg r) hr), hx⟩
+  exact ⟨a • r, (SeminormedRing.norm_mul _ _).trans (mul_le_one₀ ha (norm_nonneg r) hr), hx⟩
 
 end Module
 
@@ -158,7 +158,7 @@ theorem balancedCoreAux_balanced (h0 : (0 : E) ∈ balancedCoreAux 𝕜 s) :
   intro r hr
   have h'' : 1 ≤ ‖a⁻¹ • r‖ := by
     rw [norm_smul, norm_inv]
-    exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr
+    exact one_le_mul_of_one_le_of_one_le ((one_le_inv₀ (norm_pos_iff.mpr h)).2 ha) hr
   have h' := hy (a⁻¹ • r) h''
   rwa [smul_assoc, mem_inv_smul_set_iff₀ h] at h'
 
@@ -167,7 +167,7 @@ theorem balancedCoreAux_maximal (h : t ⊆ s) (ht : Balanced 𝕜 t) : t ⊆ bal
   rw [mem_smul_set_iff_inv_smul_mem₀ (norm_pos_iff.mp <| zero_lt_one.trans_le hr)]
   refine h (ht.smul_mem ?_ hx)
   rw [norm_inv]
-  exact inv_le_one hr
+  exact inv_le_one_of_one_le₀ hr
 
 theorem balancedCore_subset_balancedCoreAux : balancedCore 𝕜 s ⊆ balancedCoreAux 𝕜 s :=
   balancedCoreAux_maximal (balancedCore_subset s) (balancedCore_balanced s)
@@ -185,7 +185,7 @@ theorem subset_balancedCore (ht : (0 : E) ∈ t) (hst : ∀ a : 𝕜, ‖a‖ 
   rw [subset_set_smul_iff₀ (norm_pos_iff.mp <| zero_lt_one.trans_le ha)]
   apply hst
   rw [norm_inv]
-  exact inv_le_one ha
+  exact inv_le_one_of_one_le₀ ha
 
 end NormedField
 
diff --git a/Mathlib/Analysis/LocallyConvex/Basic.lean b/Mathlib/Analysis/LocallyConvex/Basic.lean
index c5bf0dfa6a4e3..4b242d6809928 100644
--- a/Mathlib/Analysis/LocallyConvex/Basic.lean
+++ b/Mathlib/Analysis/LocallyConvex/Basic.lean
@@ -183,18 +183,18 @@ theorem Balanced.smul_mono (hs : Balanced 𝕝 s) {a : 𝕝} {b : 𝕜} (h : ‖
       a • s = b • (b⁻¹ • a) • s := by rw [smul_assoc, smul_inv_smul₀ hb]
       _ ⊆ b • s := smul_set_mono <| hs _ <| by
         rw [norm_smul, norm_inv, ← div_eq_inv_mul]
-        exact div_le_one_of_le h (norm_nonneg _)
+        exact div_le_one_of_le₀ h (norm_nonneg _)
 
 theorem Balanced.smul_mem_mono [SMulCommClass 𝕝 𝕜 E] (hs : Balanced 𝕝 s) {a : 𝕜} {b : 𝕝}
     (ha : a • x ∈ s) (hba : ‖b‖ ≤ ‖a‖) : b • x ∈ s := by
   rcases eq_or_ne a 0 with rfl | ha₀
   · simp_all
   · calc
-      b • x = (a⁻¹ • b) • a • x := by rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀]
-      _ ∈ s := by
+      (a⁻¹ • b) • a • x ∈ s := by
         refine hs.smul_mem ?_ ha
         rw [norm_smul, norm_inv, ← div_eq_inv_mul]
-        exact div_le_one_of_le hba (norm_nonneg _)
+        exact div_le_one_of_le₀ hba (norm_nonneg _)
+      (a⁻¹ • b) • a • x = b • x := by rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀]
 
 theorem Balanced.subset_smul (hA : Balanced 𝕜 A) (ha : 1 ≤ ‖a‖) : A ⊆ a • A := by
   rw [← @norm_one 𝕜] at ha; simpa using hA.smul_mono ha
diff --git a/Mathlib/Analysis/LocallyConvex/Bounded.lean b/Mathlib/Analysis/LocallyConvex/Bounded.lean
index 66eed51d7bf1e..9463b8d6760d7 100644
--- a/Mathlib/Analysis/LocallyConvex/Bounded.lean
+++ b/Mathlib/Analysis/LocallyConvex/Bounded.lean
@@ -176,11 +176,16 @@ lemma isVonNBounded_iff_tendsto_smallSets_nhds {𝕜 E : Type*} [NormedDivisionR
 
 alias ⟨IsVonNBounded.tendsto_smallSets_nhds, _⟩ := isVonNBounded_iff_tendsto_smallSets_nhds
 
+lemma isVonNBounded_iff_absorbing_le {𝕜 E : Type*} [NormedDivisionRing 𝕜]
+    [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] {S : Set E} :
+    IsVonNBounded 𝕜 S ↔ Filter.absorbing 𝕜 S ≤ 𝓝 0 :=
+  .rfl
+
 lemma isVonNBounded_pi_iff {𝕜 ι : Type*} {E : ι → Type*} [NormedDivisionRing 𝕜]
     [∀ i, AddCommGroup (E i)] [∀ i, Module 𝕜 (E i)] [∀ i, TopologicalSpace (E i)]
     {S : Set (∀ i, E i)} : IsVonNBounded 𝕜 S ↔ ∀ i, IsVonNBounded 𝕜 (eval i '' S) := by
   simp_rw [isVonNBounded_iff_tendsto_smallSets_nhds, nhds_pi, Filter.pi, smallSets_iInf,
-    smallSets_comap_eq_comap_image, tendsto_iInf, tendsto_comap_iff, Function.comp,
+    smallSets_comap_eq_comap_image, tendsto_iInf, tendsto_comap_iff, Function.comp_def,
     ← image_smul, image_image, eval, Pi.smul_apply, Pi.zero_apply]
 
 section Image
diff --git a/Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean b/Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean
index a58db852ed74d..d1b8d23fb2ca3 100644
--- a/Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean
+++ b/Mathlib/Analysis/LocallyConvex/ContinuousOfBounded.lean
@@ -99,7 +99,7 @@ theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
     refine bE.1.to_hasBasis ?_ ?_
     · intro n _
       use n + 1
-      simp only [Ne, Nat.succ_ne_zero, not_false_iff, Nat.cast_add, Nat.cast_one, true_and_iff]
+      simp only [Ne, Nat.succ_ne_zero, not_false_iff, Nat.cast_add, Nat.cast_one, true_and]
       -- `b (n + 1) ⊆ b n` follows from `Antitone`.
       have h : b (n + 1) ⊆ b n := bE.2 (by simp)
       refine _root_.trans ?_ h
@@ -109,7 +109,7 @@ theorem LinearMap.continuousAt_zero_of_locally_bounded (f : E →ₛₗ[σ] F)
       refine (bE1 (n + 1)).2.smul_mem ?_ hx
       have h' : 0 < (n : ℝ) + 1 := n.cast_add_one_pos
       rw [norm_inv, ← Nat.cast_one, ← Nat.cast_add, RCLike.norm_natCast, Nat.cast_add,
-        Nat.cast_one, inv_le h' zero_lt_one]
+        Nat.cast_one, inv_le_comm₀ h' zero_lt_one]
       simp
     intro n hn
     -- The converse direction follows from continuity of the scalar multiplication
diff --git a/Mathlib/Analysis/LocallyConvex/Polar.lean b/Mathlib/Analysis/LocallyConvex/Polar.lean
index e0235090926dc..dba80d95bd857 100644
--- a/Mathlib/Analysis/LocallyConvex/Polar.lean
+++ b/Mathlib/Analysis/LocallyConvex/Polar.lean
@@ -5,7 +5,7 @@ Authors: Moritz Doll, Kalle Kytölä
 -/
 import Mathlib.Analysis.Normed.Field.Basic
 import Mathlib.LinearAlgebra.SesquilinearForm
-import Mathlib.Topology.Algebra.Module.WeakDual
+import Mathlib.Topology.Algebra.Module.WeakBilin
 
 /-!
 # Polar set
@@ -64,6 +64,10 @@ theorem polar_mem (s : Set E) (y : F) (hy : y ∈ B.polar s) : ∀ x ∈ s, ‖B
 theorem zero_mem_polar (s : Set E) : (0 : F) ∈ B.polar s := fun _ _ => by
   simp only [map_zero, norm_zero, zero_le_one]
 
+theorem polar_nonempty (s : Set E) : Set.Nonempty (B.polar s) := by
+  use 0
+  exact zero_mem_polar B s
+
 theorem polar_eq_iInter {s : Set E} : B.polar s = ⋂ x ∈ s, { y : F | ‖B x y‖ ≤ 1 } := by
   ext
   simp only [polar_mem_iff, Set.mem_iInter, Set.mem_setOf_eq]
diff --git a/Mathlib/Analysis/LocallyConvex/WeakDual.lean b/Mathlib/Analysis/LocallyConvex/WeakDual.lean
index ff5b801a83d03..13d4746c6c707 100644
--- a/Mathlib/Analysis/LocallyConvex/WeakDual.lean
+++ b/Mathlib/Analysis/LocallyConvex/WeakDual.lean
@@ -3,9 +3,9 @@ Copyright (c) 2022 Moritz Doll. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Moritz Doll
 -/
-import Mathlib.Topology.Algebra.Module.WeakDual
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.LocallyConvex.WithSeminorms
+import Mathlib.Topology.Algebra.Module.WeakBilin
 
 /-!
 # Weak Dual in Topological Vector Spaces
@@ -112,7 +112,7 @@ theorem LinearMap.hasBasis_weakBilin (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
       refine lt_of_le_of_lt (hp x) (lt_of_lt_of_le hx ?_)
       exact Finset.inf'_le _ hyU'
     rw [Set.not_nonempty_iff_eq_empty.mp hU₃]
-    simp only [Set.empty_pi, Set.preimage_univ, Set.subset_univ, and_true_iff]
+    simp only [Set.empty_pi, Set.preimage_univ, Set.subset_univ, and_true]
     exact Exists.intro ((p 0).ball 0 1) (p.basisSets_singleton_mem 0 one_pos)
   rintro U (hU : U ∈ p.basisSets)
   rw [SeminormFamily.basisSets_iff] at hU
diff --git a/Mathlib/Analysis/LocallyConvex/WeakSpace.lean b/Mathlib/Analysis/LocallyConvex/WeakSpace.lean
new file mode 100644
index 0000000000000..ff4823a65b3fb
--- /dev/null
+++ b/Mathlib/Analysis/LocallyConvex/WeakSpace.lean
@@ -0,0 +1,100 @@
+/-
+Copyright (c) 2024 Jireh Loreaux. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jireh Loreaux
+-/
+import Mathlib.LinearAlgebra.Dual
+import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
+import Mathlib.Topology.Algebra.Module.WeakDual
+
+/-! # Closures of convex sets in locally convex spaces
+
+This file contains the standard result that if `E` is a vector space with two locally convex
+topologies, then the closure of a convex set is the same in either topology, provided they have the
+same collection of continuous linear functionals. In particular, the weak closure of a convex set
+in a locally convex space coincides with the closure in the original topology.
+Of course, we phrase this in terms of linear maps between locally convex spaces, rather than
+creating two separate topologies on the same space.
+-/
+
+variable {𝕜 E F : Type*}
+variable [RCLike 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F]
+variable [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [Module ℝ F] [IsScalarTower ℝ 𝕜 F]
+variable [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [LocallyConvexSpace ℝ E]
+variable [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜 F] [LocallyConvexSpace ℝ F]
+
+variable (𝕜) in
+/-- If `E` is a locally convex space over `𝕜` (with `RCLike 𝕜`), and `s : Set E` is `ℝ`-convex, then
+the closure of `s` and the weak closure of `s` coincide. More precisely, the topological closure
+commutes with `toWeakSpace 𝕜 E`.
+
+This holds more generally for any linear equivalence `e : E ≃ₗ[𝕜] F` between locally convex spaces
+such that precomposition with `e` and `e.symm` preserves continuity of linear functionals. See
+`LinearEquiv.image_closure_of_convex`. -/
+theorem Convex.toWeakSpace_closure {s : Set E} (hs : Convex ℝ s) :
+    (toWeakSpace 𝕜 E) '' (closure s) = closure (toWeakSpace 𝕜 E '' s) := by
+  refine le_antisymm (map_continuous <| toWeakSpaceCLM 𝕜 E).continuousOn.image_closure
+    (Set.compl_subset_compl.mp fun x hx ↦ ?_)
+  obtain ⟨x, -, rfl⟩ := (toWeakSpace 𝕜 E).toEquiv.image_compl (closure s) |>.symm.subset hx
+  have : ContinuousSMul ℝ E := IsScalarTower.continuousSMul 𝕜
+  obtain ⟨f, u, hus, hux⟩ := RCLike.geometric_hahn_banach_closed_point (𝕜 := 𝕜)
+    hs.closure isClosed_closure (by simpa using hx)
+  let f' : WeakSpace 𝕜 E →L[𝕜] 𝕜 :=
+    { toLinearMap := (f : E →ₗ[𝕜] 𝕜).comp ((toWeakSpace 𝕜 E).symm : WeakSpace 𝕜 E →ₗ[𝕜] E)
+      cont := WeakBilin.eval_continuous (topDualPairing 𝕜 E).flip _ }
+  have hux' : u < RCLike.reCLM.comp (f'.restrictScalars ℝ) (toWeakSpace 𝕜 E x) := by simpa [f']
+  have hus' : closure (toWeakSpace 𝕜 E '' s) ⊆
+      {y | RCLike.reCLM.comp (f'.restrictScalars ℝ) y ≤ u} := by
+    refine closure_minimal ?_ <| isClosed_le (by fun_prop) (by fun_prop)
+    rintro - ⟨y, hy, rfl⟩
+    simpa [f'] using (hus y <| subset_closure hy).le
+  exact (hux'.not_le <| hus' ·)
+
+/-- If `e : E →ₗ[𝕜] F` is a linear map between locally convex spaces, and `f ∘ e` is continuous
+for every continuous linear functional `f : F →L[𝕜] 𝕜`, then `e` commutes with the closure on
+convex sets. -/
+theorem LinearMap.image_closure_of_convex {s : Set E} (hs : Convex ℝ s) (e : E →ₗ[𝕜] F)
+    (he : ∀ f : F →L[𝕜] 𝕜, Continuous (e.dualMap f)) :
+    e '' (closure s) ⊆ closure (e '' s) := by
+  suffices he' : Continuous (toWeakSpace 𝕜 F <| e <| (toWeakSpace 𝕜 E).symm ·) by
+    have h_convex : Convex ℝ (e '' s) := hs.linear_image e
+    rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
+    simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEquiv.symm_apply_apply]
+      using he'.continuousOn.image_closure (s := toWeakSpace 𝕜 E '' s)
+  exact WeakBilin.continuous_of_continuous_eval _ fun f ↦
+    WeakBilin.eval_continuous _ { toLinearMap := e.dualMap f : E →L[𝕜] 𝕜 }
+
+/-- If `e` is a linear isomorphism between two locally convex spaces, and `e` induces (via
+precomposition) an isomorphism between their continuous duals, then `e` commutes with the closure
+on convex sets.
+
+The hypotheses hold automatically for `e := toWeakSpace 𝕜 E`, see `Convex.toWeakSpace_closure`. -/
+theorem LinearEquiv.image_closure_of_convex {s : Set E} (hs : Convex ℝ s) (e : E ≃ₗ[𝕜] F)
+    (he₁ : ∀ f : F →L[𝕜] 𝕜, Continuous (e.dualMap f))
+    (he₂ : ∀ f : E →L[𝕜] 𝕜, Continuous (e.symm.dualMap f)) :
+    e '' (closure s) = closure (e '' s) := by
+  refine le_antisymm ((e : E →ₗ[𝕜] F).image_closure_of_convex hs he₁) ?_
+  simp only [Set.le_eq_subset, ← Set.image_subset_image_iff e.symm.injective]
+  simpa [Set.image_image]
+    using (e.symm : F →ₗ[𝕜] E).image_closure_of_convex (hs.linear_image (e : E →ₗ[𝕜] F)) he₂
+
+/-- If `e` is a linear isomorphism between two locally convex spaces, and `e` induces (via
+precomposition) an isomorphism between their continuous duals, then `e` commutes with the closure
+on convex sets.
+
+The hypotheses hold automatically for `e := toWeakSpace 𝕜 E`, see `Convex.toWeakSpace_closure`. -/
+theorem LinearEquiv.image_closure_of_convex' {s : Set E} (hs : Convex ℝ s) (e : E ≃ₗ[𝕜] F)
+    (e_dual : (F →L[𝕜] 𝕜) ≃ (E →L[𝕜] 𝕜))
+    (he : ∀ f : F →L[𝕜] 𝕜, (e_dual f : E →ₗ[𝕜] 𝕜) = e.dualMap f) :
+    e '' (closure s) = closure (e '' s) := by
+  have he' (f : E →L[𝕜] 𝕜) : (e_dual.symm f : F →ₗ[𝕜] 𝕜) = e.symm.dualMap f := by
+    simp only [DFunLike.ext'_iff, ContinuousLinearMap.coe_coe] at he ⊢
+    have (g : E →L[𝕜] 𝕜) : ⇑g = e_dual.symm g ∘ e := by
+      have := he _ ▸ congr(⇑$(e_dual.apply_symm_apply g)).symm
+      simpa
+    ext x
+    conv_rhs => rw [LinearEquiv.dualMap_apply, ContinuousLinearMap.coe_coe, this]
+    simp
+  refine e.image_closure_of_convex hs ?_ ?_
+  · simpa [← he] using fun f ↦ map_continuous (e_dual f)
+  · simpa [← he'] using fun f ↦ map_continuous (e_dual.symm f)
diff --git a/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean b/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
index c562e61f2eae3..b4165a1a77053 100644
--- a/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
+++ b/Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
@@ -135,7 +135,7 @@ theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
   rw [hU, Filter.eventually_iff]
   simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]
   by_cases h : 0 < (s.sup p) v
-  · simp_rw [(lt_div_iff h).symm]
+  · simp_rw [(lt_div_iff₀ h).symm]
     rw [← _root_.ball_zero_eq]
     exact Metric.ball_mem_nhds 0 (div_pos hr h)
   simp_rw [le_antisymm (not_lt.mp h) (apply_nonneg _ v), mul_zero, hr]
@@ -623,14 +623,11 @@ protected theorem _root_.WithSeminorms.equicontinuous_TFAE {κ : Type*}
   clear u hu hq
   -- Now we can prove the equivalence in this setting
   simp only [List.map]
-  tfae_have 1 → 3
-  · exact uniformEquicontinuous_of_equicontinuousAt_zero f
-  tfae_have 3 → 2
-  · exact UniformEquicontinuous.equicontinuous
-  tfae_have 2 → 1
-  · exact fun H ↦ H 0
+  tfae_have 1 → 3 := uniformEquicontinuous_of_equicontinuousAt_zero f
+  tfae_have 3 → 2 := UniformEquicontinuous.equicontinuous
+  tfae_have 2 → 1 := fun H ↦ H 0
   tfae_have 3 → 5
-  · intro H
+  | H => by
     have : ∀ᶠ x in 𝓝 0, ∀ k, q i (f k x) ≤ 1 := by
       filter_upwards [Metric.equicontinuousAt_iff_right.mp (H.equicontinuous 0) 1 one_pos]
         with x hx k
@@ -642,11 +639,10 @@ protected theorem _root_.WithSeminorms.equicontinuous_TFAE {κ : Type*}
     refine ⟨bdd, Seminorm.continuous' (r := 1) ?_⟩
     filter_upwards [this] with x hx
     simpa only [closedBall_iSup bdd _ one_pos, mem_iInter, mem_closedBall_zero] using hx
-  tfae_have 5 → 4
-  · exact fun H ↦ ⟨⨆ k, (q i).comp (f k), Seminorm.coe_iSup_eq H.1 ▸ H.2, le_ciSup H.1⟩
+  tfae_have 5 → 4 := fun H ↦ ⟨⨆ k, (q i).comp (f k), Seminorm.coe_iSup_eq H.1 ▸ H.2, le_ciSup H.1⟩
   tfae_have 4 → 1 -- This would work over any `NormedField`
-  · intro ⟨p, hp, hfp⟩
-    exact Metric.equicontinuousAt_of_continuity_modulus p (map_zero p ▸ hp.tendsto 0) _ <|
+  | ⟨p, hp, hfp⟩ =>
+    Metric.equicontinuousAt_of_continuity_modulus p (map_zero p ▸ hp.tendsto 0) _ <|
       Eventually.of_forall fun x k ↦ by simpa using hfp k x
   tfae_finish
 
diff --git a/Mathlib/Analysis/MeanInequalities.lean b/Mathlib/Analysis/MeanInequalities.lean
index b35c1fbdcb152..ebc7bb9056c0f 100644
--- a/Mathlib/Analysis/MeanInequalities.lean
+++ b/Mathlib/Analysis/MeanInequalities.lean
@@ -3,6 +3,7 @@ Copyright (c) 2019 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
 -/
+import Mathlib.Algebra.BigOperators.Expect
 import Mathlib.Analysis.Convex.Jensen
 import Mathlib.Analysis.Convex.SpecificFunctions.Basic
 import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
@@ -105,6 +106,7 @@ less than or equal to the sum of the maximum values of the summands.
 universe u v
 
 open Finset NNReal ENNReal
+open scoped BigOperators
 
 noncomputable section
 
@@ -270,11 +272,11 @@ theorem harm_mean_le_geom_mean_weighted (w z : ι → ℝ) (hs : s.Nonempty) (hw
     have s_pos : 0 < ∑ i in s, w i * (z i)⁻¹ :=
       sum_pos (fun i hi => mul_pos (hw i hi) (inv_pos.2 (hz i hi))) hs
     norm_num at this
-    rw [← inv_le_inv s_pos p_pos] at this
+    rw [← inv_le_inv₀ s_pos p_pos] at this
     apply le_trans this
     have p_pos₂ : 0 < (∏ i in s, (z i) ^ w i)⁻¹ :=
       inv_pos.2 (prod_pos fun i hi => rpow_pos_of_pos ((hz i hi)) _ )
-    rw [← inv_inv (∏ i in s, z i ^ w i), inv_le_inv p_pos p_pos₂, ← Finset.prod_inv_distrib]
+    rw [← inv_inv (∏ i in s, z i ^ w i), inv_le_inv₀ p_pos p_pos₂, ← Finset.prod_inv_distrib]
     gcongr
     · exact fun i hi ↦ inv_nonneg.mpr (Real.rpow_nonneg (le_of_lt (hz i hi)) _)
     · rw [Real.inv_rpow]; apply fun i hi ↦ le_of_lt (hz i hi); assumption
@@ -330,7 +332,7 @@ end Real
 namespace NNReal
 
 /-- **Young's inequality**, `ℝ≥0` version. We use `{p q : ℝ≥0}` in order to avoid constructing
-witnesses of `0 ≤ p` and `0 ≤ q` for the denominators.  -/
+witnesses of `0 ≤ p` and `0 ≤ q` for the denominators. -/
 theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hpq : p.IsConjExponent q) :
     a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q :=
   Real.young_inequality_of_nonneg a.coe_nonneg b.coe_nonneg hpq.coe
@@ -353,8 +355,8 @@ theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjExponent
     cases' h with h h <;> rw [h] <;> simp [h, hpq.pos, hpq.symm.pos]
   push_neg at h
   -- if a ≠ ⊤ and b ≠ ⊤, use the nnreal version: nnreal.young_inequality_real
-  rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,
-    coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
+  rw [← coe_toNNReal h.left, ← coe_toNNReal h.right, ← coe_mul, ← coe_rpow_of_nonneg _ hpq.nonneg,
+    ← coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, ←
     @coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), ←
     @coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), ← coe_add, coe_le_coe]
   exact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq
@@ -476,7 +478,7 @@ theorem inner_le_Lp_mul_Lq_tsum' {f g : ι → ℝ≥0} {p q : ℝ} (hpq : p.IsC
 /-- **Hölder inequality**: the scalar product of two functions is bounded by the product of their
 `L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
 functions. For an alternative version, convenient if the infinite sums are not already expressed as
-`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`.  -/
+`p`-th powers, see `inner_le_Lp_mul_Lq_tsum`. -/
 theorem inner_le_Lp_mul_Lq_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p q : ℝ}
     (hpq : p.IsConjExponent q) (hf : HasSum (fun i => f i ^ p) (A ^ p))
     (hg : HasSum (fun i => g i ^ q) (B ^ q)) : ∃ C, C ≤ A * B ∧ HasSum (fun i => f i * g i) C := by
@@ -520,7 +522,7 @@ theorem isGreatest_Lp (f : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q
         simp [h, hpq.ne_zero]
       simp only [Set.mem_setOf_eq, div_rpow, ← sum_div, ← rpow_mul,
         div_mul_cancel₀ _ hpq.symm.ne_zero, rpow_one, div_le_iff₀ hf, one_mul, hpq.mul_eq_add, ←
-        rpow_sub' _ A, add_sub_cancel_right, le_refl, true_and_iff, ← mul_div_assoc, B]
+        rpow_sub' A, add_sub_cancel_right, le_refl, true_and, ← mul_div_assoc, B]
       rw [div_eq_iff, ← rpow_add hf.ne', one_div, one_div, hpq.inv_add_inv_conj, rpow_one]
       simpa [hpq.symm.ne_zero] using hf.ne'
   · rintro _ ⟨g, hg, rfl⟩
@@ -584,7 +586,7 @@ theorem Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Su
 /-- **Minkowski inequality**: the `L_p` seminorm of the infinite sum of two vectors is less than or
 equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
 exist. A version for `NNReal`-valued functions. For an alternative version, convenient if the
-infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`.  -/
+infinite sums are not already expressed as `p`-th powers, see `Lp_add_le_tsum_of_nonneg`. -/
 theorem Lp_add_le_hasSum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
     (hf : HasSum (fun i => f i ^ p) (A ^ p)) (hg : HasSum (fun i => g i ^ p) (B ^ p)) :
     ∃ C, C ≤ A + B ∧ HasSum (fun i => (f i + g i) ^ p) (C ^ p) := by
@@ -660,6 +662,18 @@ lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p)
   norm_cast at *
   exact NNReal.inner_le_weight_mul_Lp _ hp _ _
 
+/-- **Weighted Hölder inequality** in terms of `Finset.expect`. -/
+lemma compact_inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) {w f : ι → ℝ}
+    (hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 ≤ f i) :
+    𝔼 i ∈ s, w i * f i ≤ (𝔼 i ∈ s, w i) ^ (1 - p⁻¹) * (𝔼 i ∈ s, w i * f i ^ p) ^ p⁻¹ := by
+  simp_rw [expect_eq_sum_div_card]
+  rw [div_rpow, div_rpow, div_mul_div_comm, ← rpow_add', sub_add_cancel, rpow_one]
+  · gcongr
+    exact inner_le_weight_mul_Lp_of_nonneg s hp _ _ hw hf
+  any_goals simp
+  · exact sum_nonneg fun i _ ↦ by have := hw i; have := hf i; positivity
+  · exact sum_nonneg fun i _ ↦ by have := hw i; positivity
+
 /-- **Hölder inequality**: the scalar product of two functions is bounded by the product of their
 `L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
 For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
@@ -688,7 +702,7 @@ theorem inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.IsConjExponent q) (hf : ∀
 /-- **Hölder inequality**: the scalar product of two functions is bounded by the product of their
 `L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `NNReal`-valued
 functions. For an alternative version, convenient if the infinite sums are not already expressed as
-`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`.  -/
+`p`-th powers, see `inner_le_Lp_mul_Lq_tsum_of_nonneg`. -/
 theorem inner_le_Lp_mul_Lq_hasSum_of_nonneg (hpq : p.IsConjExponent q) {A B : ℝ} (hA : 0 ≤ A)
     (hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
     (hf_sum : HasSum (fun i => f i ^ p) (A ^ p)) (hg_sum : HasSum (fun i => g i ^ q) (B ^ q)) :
@@ -795,8 +809,8 @@ theorem inner_le_Lp_mul_Lq (hpq : p.IsConjExponent q) :
       ENNReal.sum_eq_top, not_or] using H'
   have := ENNReal.coe_le_coe.2 (@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i))
     (fun i => ENNReal.toNNReal (g i)) _ _ hpq)
-  simp [← ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos,
-    le_of_lt hpq.symm.pos, le_of_lt hpq.symm.one_div_pos] at this
+  simp [ENNReal.coe_rpow_of_nonneg, hpq.pos.le, hpq.one_div_pos.le, hpq.symm.pos.le,
+    hpq.symm.one_div_pos.le] at this
   convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
     · refine Finset.sum_congr rfl fun i hi => ?_
       simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul]
@@ -807,7 +821,7 @@ lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p)
   obtain rfl | hp := hp.eq_or_lt
   · simp
   have hp₀ : 0 < p := by positivity
-  have hp₁ : p⁻¹ < 1 := inv_lt_one hp
+  have hp₁ : p⁻¹ < 1 := inv_lt_one_of_one_lt₀ hp
   by_cases H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) = 0 ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = 0
   · replace H : (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0 := by
       simpa [hp₀, hp₁, hp₀.not_lt, hp₁.not_lt, sum_eq_zero_iff_of_nonneg] using H
@@ -821,9 +835,9 @@ lemma inner_le_weight_mul_Lp_of_nonneg (s : Finset ι) {p : ℝ} (hp : 1 ≤ p)
   have := coe_le_coe.2 <| NNReal.inner_le_weight_mul_Lp s hp.le (fun i ↦ ENNReal.toNNReal (w i))
     fun i ↦ ENNReal.toNNReal (f i)
   rw [coe_mul] at this
-  simp_rw [← coe_rpow_of_nonneg _ <| inv_nonneg.2 hp₀.le, coe_finset_sum, ENNReal.toNNReal_rpow,
+  simp_rw [coe_rpow_of_nonneg _ <| inv_nonneg.2 hp₀.le, coe_finset_sum, ← ENNReal.toNNReal_rpow,
     ← ENNReal.toNNReal_mul, sum_congr rfl fun i hi ↦ coe_toNNReal (H'.2 i hi)] at this
-  simp [← ENNReal.coe_rpow_of_nonneg, hp₀.le, hp₁.le] at this
+  simp [ENNReal.coe_rpow_of_nonneg, hp₀.le, hp₁.le] at this
   convert this using 2 with i hi
   · obtain hw | hw := eq_or_ne (w i) 0
     · simp [hw]
@@ -864,7 +878,7 @@ theorem Lp_add_le (hp : 1 ≤ p) :
     ENNReal.coe_le_coe.2
       (@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _
         hp)
-  push_cast [← ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
+  push_cast [ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this
   convert this using 2 <;> [skip; congr 1; congr 1] <;>
     · refine Finset.sum_congr rfl fun i hi => ?_
       simp [H'.1 i hi, H'.2 i hi]
diff --git a/Mathlib/Analysis/MeanInequalitiesPow.lean b/Mathlib/Analysis/MeanInequalitiesPow.lean
index 7fcc0ddceed9a..fdee6d8fdbe46 100644
--- a/Mathlib/Analysis/MeanInequalitiesPow.lean
+++ b/Mathlib/Analysis/MeanInequalitiesPow.lean
@@ -61,23 +61,6 @@ theorem pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i
     (∑ i ∈ s, w i * z i) ^ n ≤ ∑ i ∈ s, w i * z i ^ n :=
   hn.convexOn_pow.map_sum_le hw hw' fun _ _ => Set.mem_univ _
 
-/-- Specific case of Jensen's inequality for sums of powers -/
-theorem pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
-    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by
-  rcases s.eq_empty_or_nonempty with (rfl | hs)
-  · simp_rw [Finset.sum_empty, zero_pow n.succ_ne_zero, zero_div]; rfl
-  · have hs0 : 0 < (s.card : ℝ) := Nat.cast_pos.2 hs.card_pos
-    suffices (∑ x ∈ s, f x / s.card) ^ (n + 1) ≤ ∑ x ∈ s, f x ^ (n + 1) / s.card by
-      rwa [← Finset.sum_div, ← Finset.sum_div, div_pow, pow_succ (s.card : ℝ), ← div_div,
-        div_le_iff₀ hs0, div_mul, div_self hs0.ne', div_one] at this
-    have :=
-      @ConvexOn.map_sum_le ℝ ℝ ℝ ι _ _ _ _ _ _ (Set.Ici 0) (fun x => x ^ (n + 1)) s
-        (fun _ => 1 / s.card) ((↑) ∘ f) (convexOn_pow (n + 1)) ?_ ?_ fun i hi =>
-        Set.mem_Ici.2 (hf i hi)
-    · simpa only [inv_mul_eq_div, one_div, Algebra.id.smul_eq_mul] using this
-    · simp only [one_div, inv_nonneg, Nat.cast_nonneg, imp_true_iff]
-    · simpa only [one_div, Finset.sum_const, nsmul_eq_mul] using mul_inv_cancel₀ hs0.ne'
-
 theorem zpow_arith_mean_le_arith_mean_zpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
     (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) :
     (∑ i ∈ s, w i * z i) ^ m ≤ ∑ i ∈ s, w i * z i ^ m :=
@@ -111,11 +94,6 @@ theorem pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ≥0) (hw' : ∑ i ∈
     Real.pow_arith_mean_le_arith_mean_pow s _ _ (fun i _ => (w i).coe_nonneg)
       (mod_cast hw') (fun i _ => (z i).coe_nonneg) n
 
-theorem pow_sum_div_card_le_sum_pow (f : ι → ℝ≥0) (n : ℕ) :
-    (∑ x ∈ s, f x) ^ (n + 1) / (s.card : ℝ) ^ n ≤ ∑ x ∈ s, f x ^ (n + 1) := by
-  simpa only [← NNReal.coe_le_coe, NNReal.coe_sum, Nonneg.coe_div, NNReal.coe_pow] using
-    @Real.pow_sum_div_card_le_sum_pow ι s (((↑) : ℝ≥0 → ℝ) ∘ f) n fun _ _ => NNReal.coe_nonneg _
-
 /-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
 functions and real exponents. -/
 theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) {p : ℝ}
@@ -141,7 +119,7 @@ theorem rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 
   · simp only [one_div, inv_mul_cancel_left₀, Ne, mul_eq_zero, two_ne_zero, one_ne_zero,
       not_false_iff]
   · have A : p - 1 ≠ 0 := ne_of_gt (sub_pos.2 h'p)
-    simp only [mul_rpow, rpow_sub' _ A, div_eq_inv_mul, rpow_one, mul_one]
+    simp only [mul_rpow, rpow_sub' A, div_eq_inv_mul, rpow_one, mul_one]
     ring
 
 /-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
@@ -221,7 +199,7 @@ theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑
   · -- first, prove `(∑ i ∈ s, w i * z i) ^ p = ⊤ → ∑ i ∈ s, (w i * z i ^ p) = ⊤`
     rw [rpow_eq_top_iff, sum_eq_top, sum_eq_top]
     intro h
-    simp only [and_false_iff, hp_not_neg, false_or_iff] at h
+    simp only [and_false, hp_not_neg, false_or] at h
     rcases h.left with ⟨a, H, ha⟩
     use a, H
     rwa [← h_top_iff_rpow_top a H]
@@ -241,7 +219,7 @@ theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑
       specialize h_top i hi
       rwa [Ne, ← h_top_iff_rpow_top i hi]
     -- put the `.toNNReal` inside the sums.
-    simp_rw [toNNReal_sum h_top_rpow, ← toNNReal_rpow, toNNReal_sum h_top, toNNReal_mul, ←
+    simp_rw [toNNReal_sum h_top_rpow, toNNReal_rpow, toNNReal_sum h_top, toNNReal_mul,
       toNNReal_rpow]
     -- use corresponding nnreal result
     refine
@@ -282,7 +260,7 @@ theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) : a ^
   obtain ⟨ha_top, hb_top⟩ := add_ne_top.mp h_top
   lift a to ℝ≥0 using ha_top
   lift b to ℝ≥0 using hb_top
-  simpa [← ENNReal.coe_rpow_of_nonneg _ hp_pos.le] using
+  simpa [ENNReal.coe_rpow_of_nonneg _ hp_pos.le] using
     ENNReal.coe_le_coe.2 (NNReal.add_rpow_le_rpow_add a b hp1)
 
 theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) :
diff --git a/Mathlib/Analysis/MellinTransform.lean b/Mathlib/Analysis/MellinTransform.lean
index 95ff3017bcc71..6712829f036dc 100644
--- a/Mathlib/Analysis/MellinTransform.lean
+++ b/Mathlib/Analysis/MellinTransform.lean
@@ -256,7 +256,7 @@ theorem mellin_convergent_of_isBigO_scalar {a b : ℝ} {f : ℝ → ℝ} {s : 
   rw [this, integrableOn_union, integrableOn_union]
   refine ⟨⟨hc2', integrableOn_Icc_iff_integrableOn_Ioc.mp ?_⟩, hc1'⟩
   refine
-    (hfc.continuousOn_mul ?_ isOpen_Ioi).integrableOn_compact_subset
+    (hfc.continuousOn_mul ?_ isOpen_Ioi.isLocallyClosed).integrableOn_compact_subset
       (fun t ht => (hc2.trans_le ht.1 : 0 < t)) isCompact_Icc
   exact ContinuousAt.continuousOn fun t ht => continuousAt_rpow_const _ _ <| Or.inl <| ne_of_gt ht
 
@@ -327,7 +327,8 @@ theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
     exact mellinConvergent_of_isBigO_rpow hfc hf_top hs_top hf_bot hs_bot
   have h3 : AEStronglyMeasurable (F' s) (volume.restrict <| Ioi 0) := by
     apply LocallyIntegrableOn.aestronglyMeasurable
-    refine hfc.continuousOn_smul isOpen_Ioi ((ContinuousAt.continuousOn fun t ht => ?_).mul ?_)
+    refine hfc.continuousOn_smul isOpen_Ioi.isLocallyClosed
+      ((ContinuousAt.continuousOn fun t ht => ?_).mul ?_)
     · exact continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt ht)
     · refine continuous_ofReal.comp_continuousOn ?_
       exact continuousOn_log.mono (subset_compl_singleton_iff.mpr not_mem_Ioi_self)
@@ -362,7 +363,7 @@ theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ}
       obtain ⟨w', hw1', hw2'⟩ := exists_between hj'
       refine mellin_convergent_of_isBigO_scalar ?_ ?_ hw1' ?_ hw2
       · simp_rw [mul_comm]
-        refine hfc.norm.mul_continuousOn ?_ isOpen_Ioi
+        refine hfc.norm.mul_continuousOn ?_ isOpen_Ioi.isLocallyClosed
         refine Continuous.comp_continuousOn _root_.continuous_abs (continuousOn_log.mono ?_)
         exact subset_compl_singleton_iff.mpr not_mem_Ioi_self
       · refine (isBigO_rpow_top_log_smul hw2' hf_top).norm_left.congr_left fun t ↦ ?_
diff --git a/Mathlib/Analysis/Normed/Affine/AddTorsor.lean b/Mathlib/Analysis/Normed/Affine/AddTorsor.lean
index caa4e080fffdf..2e2257c96f378 100644
--- a/Mathlib/Analysis/Normed/Affine/AddTorsor.lean
+++ b/Mathlib/Analysis/Normed/Affine/AddTorsor.lean
@@ -220,7 +220,7 @@ theorem eventually_homothety_mem_of_mem_interior (x : Q) {s : Set Q} {y : Q} (hy
   obtain ⟨u, hu₁, hu₂, hu₃⟩ := mem_interior.mp hy
   obtain ⟨ε, hε, hyε⟩ := Metric.isOpen_iff.mp hu₂ y hu₃
   refine ⟨ε / ‖y -ᵥ x‖, div_pos hε hxy, fun δ (hδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖) => hu₁ (hyε ?_)⟩
-  rw [lt_div_iff hxy, ← norm_smul, sub_smul, one_smul] at hδ
+  rw [lt_div_iff₀ hxy, ← norm_smul, sub_smul, one_smul] at hδ
   rwa [homothety_apply, Metric.mem_ball, dist_eq_norm_vsub W, vadd_vsub_eq_sub_vsub]
 
 theorem eventually_homothety_image_subset_of_finite_subset_interior (x : Q) {s : Set Q} {t : Set Q}
diff --git a/Mathlib/Analysis/Normed/Affine/Isometry.lean b/Mathlib/Analysis/Normed/Affine/Isometry.lean
index f0545ea11cc87..67f6a72b44673 100644
--- a/Mathlib/Analysis/Normed/Affine/Isometry.lean
+++ b/Mathlib/Analysis/Normed/Affine/Isometry.lean
@@ -69,9 +69,9 @@ theorem linear_eq_linearIsometry : f.linear = f.linearIsometry.toLinearMap := by
   ext
   rfl
 
-instance : FunLike (P →ᵃⁱ[𝕜] P₂) P P₂ :=
-  { coe := fun f => f.toFun,
-    coe_injective' := fun f g => by cases f; cases g; simp }
+instance : FunLike (P →ᵃⁱ[𝕜] P₂) P P₂ where
+  coe f := f.toFun
+  coe_injective' f g := by cases f; cases g; simp
 
 @[simp]
 theorem coe_toAffineMap : ⇑f.toAffineMap = f := by
@@ -282,16 +282,16 @@ theorem linear_eq_linear_isometry : e.linear = e.linearIsometryEquiv.toLinearEqu
   ext
   rfl
 
-instance : EquivLike (P ≃ᵃⁱ[𝕜] P₂) P P₂ :=
-  { coe := fun f => f.toFun
-    inv := fun f => f.invFun
-    left_inv := fun f => f.left_inv
-    right_inv := fun f => f.right_inv,
-    coe_injective' := fun f g h _ => by
-      cases f
-      cases g
-      congr
-      simpa [DFunLike.coe_injective.eq_iff] using h }
+instance : EquivLike (P ≃ᵃⁱ[𝕜] P₂) P P₂ where
+  coe f := f.toFun
+  inv f := f.invFun
+  left_inv f := f.left_inv
+  right_inv f := f.right_inv
+  coe_injective' f g h _ := by
+    cases f
+    cases g
+    congr
+    simpa [DFunLike.coe_injective.eq_iff] using h
 
 @[simp]
 theorem coe_mk (e : P ≃ᵃ[𝕜] P₂) (he : ∀ x, ‖e.linear x‖ = ‖x‖) : ⇑(mk e he) = e :=
diff --git a/Mathlib/Analysis/Normed/Algebra/Exponential.lean b/Mathlib/Analysis/Normed/Algebra/Exponential.lean
index 5f50363a1084b..e211c1cfd4a8d 100644
--- a/Mathlib/Analysis/Normed/Algebra/Exponential.lean
+++ b/Mathlib/Analysis/Normed/Algebra/Exponential.lean
@@ -3,12 +3,9 @@ Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Eric Wieser
 -/
-import Mathlib.Analysis.Analytic.Basic
+import Mathlib.Analysis.Analytic.ChangeOrigin
 import Mathlib.Analysis.Complex.Basic
-import Mathlib.Analysis.Normed.Field.InfiniteSum
 import Mathlib.Data.Nat.Choose.Cast
-import Mathlib.Data.Finset.NoncommProd
-import Mathlib.Topology.Algebra.Algebra
 
 /-!
 # Exponential in a Banach algebra
@@ -154,6 +151,7 @@ theorem star_exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸)
 
 variable (𝕂)
 
+@[aesop safe apply]
 theorem _root_.IsSelfAdjoint.exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {x : 𝔸}
     (h : IsSelfAdjoint x) : IsSelfAdjoint (exp 𝕂 x) :=
   (star_exp x).trans <| h.symm ▸ rfl
@@ -269,7 +267,7 @@ theorem exp_add_of_commute_of_mem_ball [CharZero 𝕂] {x y : 𝔸} (hxy : Commu
     ext
     rw [hxy.add_pow' _, Finset.smul_sum]
   refine tsum_congr fun n => Finset.sum_congr rfl fun kl hkl => ?_
-  rw [← Nat.cast_smul_eq_nsmul 𝕂, smul_smul, smul_mul_smul, ← Finset.mem_antidiagonal.mp hkl,
+  rw [← Nat.cast_smul_eq_nsmul 𝕂, smul_smul, smul_mul_smul_comm, ← Finset.mem_antidiagonal.mp hkl,
     Nat.cast_add_choose, Finset.mem_antidiagonal.mp hkl]
   congr 1
   have : (n ! : 𝕂) ≠ 0 := Nat.cast_ne_zero.mpr n.factorial_ne_zero
diff --git a/Mathlib/Analysis/Normed/Algebra/MatrixExponential.lean b/Mathlib/Analysis/Normed/Algebra/MatrixExponential.lean
index 2f0463ab0aa36..a481e6f7c4823 100644
--- a/Mathlib/Analysis/Normed/Algebra/MatrixExponential.lean
+++ b/Mathlib/Analysis/Normed/Algebra/MatrixExponential.lean
@@ -145,17 +145,15 @@ nonrec theorem isUnit_exp (A : Matrix m m 𝔸) : IsUnit (exp 𝕂 A) := by
   letI : NormedAlgebra 𝕂 (Matrix m m 𝔸) := Matrix.linftyOpNormedAlgebra
   exact isUnit_exp _ A
 
--- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
 nonrec theorem exp_units_conj (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :
-    exp 𝕂 (U.val * A * (U⁻¹).val) = U.val * exp 𝕂 A * (U⁻¹).val := by
+    exp 𝕂 (U * A * U⁻¹) = U * exp 𝕂 A * U⁻¹ := by
   letI : SeminormedRing (Matrix m m 𝔸) := Matrix.linftyOpSemiNormedRing
   letI : NormedRing (Matrix m m 𝔸) := Matrix.linftyOpNormedRing
   letI : NormedAlgebra 𝕂 (Matrix m m 𝔸) := Matrix.linftyOpNormedAlgebra
   exact exp_units_conj _ U A
 
--- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
 theorem exp_units_conj' (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :
-    exp 𝕂 ((U⁻¹).val * A * U.val) = (U⁻¹).val * exp 𝕂 A * U.val :=
+    exp 𝕂 (U⁻¹ * A * U) = U⁻¹ * exp 𝕂 A * U :=
   exp_units_conj 𝕂 U⁻¹ A
 
 end Normed
diff --git a/Mathlib/Analysis/Normed/Algebra/Norm.lean b/Mathlib/Analysis/Normed/Algebra/Norm.lean
new file mode 100644
index 0000000000000..eb815ac91298e
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Algebra/Norm.lean
@@ -0,0 +1,202 @@
+/-
+Copyright (c) 2024 María Inés de Frutos-Fernández. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: María Inés de Frutos-Fernández
+-/
+import Mathlib.Analysis.Normed.Ring.Seminorm
+import Mathlib.Analysis.Seminorm
+
+/-!
+# Algebra norms
+
+We define algebra norms and multiplicative algebra norms.
+
+## Main Definitions
+* `AlgebraNorm` : an algebra norm on an `R`-algebra `S` is a ring norm on `S` compatible with
+  the action of `R`.
+* `MulAlgebraNorm` : a multiplicative algebra norm on an `R`-algebra `S` is a multiplicative
+  ring norm on `S` compatible with the action of `R`.
+
+## Tags
+
+norm, algebra norm
+-/
+
+/-- An algebra norm on an `R`-algebra `S` is a ring norm on `S` compatible with the
+action of `R`. -/
+structure AlgebraNorm (R : Type*) [SeminormedCommRing R] (S : Type*) [Ring S] [Algebra R S] extends
+  RingNorm S, Seminorm R S
+
+attribute [nolint docBlame] AlgebraNorm.toSeminorm AlgebraNorm.toRingNorm
+
+instance (K : Type*) [NormedField K] : Inhabited (AlgebraNorm K K) :=
+  ⟨{  toFun     := norm
+      map_zero' := norm_zero
+      add_le'   := norm_add_le
+      neg'      := norm_neg
+      smul'     := norm_mul
+      mul_le'   := norm_mul_le
+      eq_zero_of_map_eq_zero' := fun _ => norm_eq_zero.mp }⟩
+
+/-- `AlgebraNormClass F R S` states that `F` is a type of `R`-algebra norms on the ring `S`.
+You should extend this class when you extend `AlgebraNorm`. -/
+class AlgebraNormClass (F : Type*) (R : outParam <| Type*) [SeminormedCommRing R]
+    (S : outParam <| Type*) [Ring S] [Algebra R S] [FunLike F S ℝ] extends RingNormClass F S ℝ,
+    SeminormClass F R S : Prop
+
+namespace AlgebraNorm
+
+variable {R : Type*} [SeminormedCommRing R] {S : Type*} [Ring S] [Algebra R S] {f : AlgebraNorm R S}
+
+/-- The ring seminorm underlying an algebra norm. -/
+def toRingSeminorm' (f : AlgebraNorm R S) : RingSeminorm S :=
+  f.toRingNorm.toRingSeminorm
+
+instance : FunLike (AlgebraNorm R S) S ℝ where
+  coe f := f.toFun
+  coe_injective' f f' h := by
+    simp only [AddGroupSeminorm.toFun_eq_coe, RingSeminorm.toFun_eq_coe] at h
+    cases f; cases f'; congr;
+    simp only at h
+    ext s
+    erw [h]
+    rfl
+
+instance algebraNormClass : AlgebraNormClass (AlgebraNorm R S) R S where
+  map_zero f        := f.map_zero'
+  map_add_le_add f  := f.add_le'
+  map_mul_le_mul f  := f.mul_le'
+  map_neg_eq_map f  := f.neg'
+  eq_zero_of_map_eq_zero f := f.eq_zero_of_map_eq_zero' _
+  map_smul_eq_mul f := f.smul'
+
+/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`. -/
+instance : CoeFun (AlgebraNorm R S) fun _ => S → ℝ :=
+  DFunLike.hasCoeToFun
+
+theorem toFun_eq_coe (p : AlgebraNorm R S) : p.toFun = p := rfl
+
+@[ext]
+theorem ext {p q : AlgebraNorm R S} : (∀ x, p x = q x) → p = q :=
+  DFunLike.ext p q
+
+/-- An `R`-algebra norm such that `f 1 = 1` extends the norm on `R`. -/
+theorem extends_norm' (hf1 : f 1 = 1) (a : R) : f (a • (1 : S)) = ‖a‖ := by
+  rw [← mul_one ‖a‖, ← hf1]; exact f.smul' _ _
+
+/-- An `R`-algebra norm such that `f 1 = 1` extends the norm on `R`. -/
+theorem extends_norm (hf1 : f 1 = 1) (a : R) : f (algebraMap R S a) = ‖a‖ := by
+  rw [Algebra.algebraMap_eq_smul_one]; exact extends_norm' hf1 _
+
+/-- The restriction of an algebra norm to a subalgebra. -/
+def restriction (A : Subalgebra R S) (f : AlgebraNorm R S) : AlgebraNorm R A where
+  toFun       := fun x : A => f x.val
+  map_zero'   := map_zero f
+  add_le' x y := map_add_le_add _ _ _
+  neg' x      := map_neg_eq_map _ _
+  mul_le' x y := map_mul_le_mul _ _ _
+  eq_zero_of_map_eq_zero' x hx := by
+    rw [← ZeroMemClass.coe_eq_zero]; exact eq_zero_of_map_eq_zero f hx
+  smul' r x := map_smul_eq_mul _ _ _
+
+/-- The restriction of an algebra norm in a scalar tower. -/
+def isScalarTower_restriction {A : Type*} [CommRing A] [Algebra R A] [Algebra A S]
+    [IsScalarTower R A S] (hinj : Function.Injective (algebraMap A S)) (f : AlgebraNorm R S) :
+    AlgebraNorm R A where
+  toFun       := fun x : A => f (algebraMap A S x)
+  map_zero'   := by simp only [map_zero]
+  add_le' x y := by simp only [map_add, map_add_le_add]
+  neg' x      := by simp only [map_neg, map_neg_eq_map]
+  mul_le' x y := by simp only [map_mul, map_mul_le_mul]
+  eq_zero_of_map_eq_zero' x hx := by
+    rw [← map_eq_zero_iff (algebraMap A S) hinj]
+    exact eq_zero_of_map_eq_zero f hx
+  smul' r x := by
+    simp only [Algebra.smul_def, map_mul, ← IsScalarTower.algebraMap_apply]
+    simp only [← smul_eq_mul, algebraMap_smul, map_smul_eq_mul]
+
+end AlgebraNorm
+
+/-- A multiplicative algebra norm on an `R`-algebra norm `S` is a multiplicative ring norm on `S`
+  compatible with the action of `R`. -/
+structure MulAlgebraNorm (R : Type*) [SeminormedCommRing R] (S : Type*) [Ring S] [Algebra R S]
+  extends MulRingNorm S, Seminorm R S
+
+attribute [nolint docBlame] MulAlgebraNorm.toSeminorm MulAlgebraNorm.toMulRingNorm
+
+instance (K : Type*) [NormedField K] : Inhabited (MulAlgebraNorm K K) :=
+  ⟨{  toFun     := norm
+      map_zero' := norm_zero
+      add_le'   := norm_add_le
+      neg'      := norm_neg
+      smul'     := norm_mul
+      map_one'  := norm_one
+      map_mul'  := norm_mul
+      eq_zero_of_map_eq_zero' := fun _ => norm_eq_zero.mp }⟩
+
+/-- `MulAlgebraNormClass F R S` states that `F` is a type of multiplicative `R`-algebra norms on
+the ring `S`. You should extend this class when you extend `MulAlgebraNorm`. -/
+class MulAlgebraNormClass (F : Type*) (R : outParam <| Type*) [SeminormedCommRing R]
+    (S : outParam <| Type*) [Ring S] [Algebra R S] [FunLike F S ℝ] extends MulRingNormClass F S ℝ,
+    SeminormClass F R S : Prop
+
+namespace MulAlgebraNorm
+
+variable {R S : outParam <| Type*} [SeminormedCommRing R] [Ring S] [Algebra R S]
+  {f : AlgebraNorm R S}
+
+instance : FunLike (MulAlgebraNorm R S) S ℝ where
+  coe f := f.toFun
+  coe_injective' f f' h:= by
+    simp only [AddGroupSeminorm.toFun_eq_coe, MulRingSeminorm.toFun_eq_coe, DFunLike.coe_fn_eq] at h
+    obtain ⟨⟨_, _⟩, _⟩ := f; obtain ⟨⟨_, _⟩, _⟩ := f'; congr;
+
+instance mulAlgebraNormClass : MulAlgebraNormClass (MulAlgebraNorm R S) R S where
+  map_zero f        := f.map_zero'
+  map_add_le_add f  := f.add_le'
+  map_one f         := f.map_one'
+  map_mul f         := f.map_mul'
+  map_neg_eq_map f  := f.neg'
+  eq_zero_of_map_eq_zero f := f.eq_zero_of_map_eq_zero' _
+  map_smul_eq_mul f := f.smul'
+
+/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`. -/
+instance : CoeFun (MulAlgebraNorm R S) fun _ => S → ℝ :=
+  DFunLike.hasCoeToFun
+
+theorem toFun_eq_coe (p : MulAlgebraNorm R S) : p.toFun = p := rfl
+
+@[ext]
+theorem ext {p q : MulAlgebraNorm R S} : (∀ x, p x = q x) → p = q :=
+  DFunLike.ext p q
+
+/-- A multiplicative `R`-algebra norm extends the norm on `R`. -/
+theorem extends_norm' (f : MulAlgebraNorm R S) (a : R) : f (a • (1 : S)) = ‖a‖ := by
+  rw [← mul_one ‖a‖, ← f.map_one', ← f.smul']; rfl
+
+/-- A multiplicative `R`-algebra norm extends the norm on `R`. -/
+theorem extends_norm (f : MulAlgebraNorm R S) (a : R) : f (algebraMap R S a) = ‖a‖ := by
+  rw [Algebra.algebraMap_eq_smul_one]; exact extends_norm' _ _
+
+end MulAlgebraNorm
+
+namespace MulRingNorm
+
+variable {R : Type*} [NonAssocRing R]
+
+/-- The ring norm underlying a multiplicative ring norm. -/
+def toRingNorm (f : MulRingNorm R) : RingNorm R where
+  toFun       := f
+  map_zero'   := f.map_zero'
+  add_le'     := f.add_le'
+  neg'        := f.neg'
+  mul_le' x y := le_of_eq (f.map_mul' x y)
+  eq_zero_of_map_eq_zero' := f.eq_zero_of_map_eq_zero'
+
+/-- A multiplicative ring norm is power-multiplicative. -/
+theorem isPowMul {A : Type*} [Ring A] (f : MulRingNorm A) : IsPowMul f := fun x n hn => by
+  cases n
+  · exfalso; linarith
+  · rw [map_pow]
+
+end MulRingNorm
diff --git a/Mathlib/Analysis/Normed/Algebra/Spectrum.lean b/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
index d74a1d6c50562..fa49b3701161c 100644
--- a/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
+++ b/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
@@ -101,7 +101,7 @@ theorem mem_resolventSet_of_norm_lt_mul {a : A} {k : 𝕜} (h : ‖a‖ * ‖(1
     ne_zero_of_norm_ne_zero ((mul_nonneg (norm_nonneg _) (norm_nonneg _)).trans_lt h).ne'
   letI ku := Units.map ↑ₐ.toMonoidHom (Units.mk0 k hk)
   rw [← inv_inv ‖(1 : A)‖,
-    mul_inv_lt_iff (inv_pos.2 <| norm_pos_iff.2 (one_ne_zero : (1 : A) ≠ 0))] at h
+    mul_inv_lt_iff₀' (inv_pos.2 <| norm_pos_iff.2 (one_ne_zero : (1 : A) ≠ 0))] at h
   have hku : ‖-a‖ < ‖(↑ku⁻¹ : A)‖⁻¹ := by simpa [ku, norm_algebraMap] using h
   simpa [ku, sub_eq_add_neg, Algebra.algebraMap_eq_smul_one] using (ku.add (-a) hku).isUnit
 
@@ -203,11 +203,9 @@ theorem spectralRadius_le_liminf_pow_nnnorm_pow_one_div (a : A) :
   refine ENNReal.le_of_forall_lt_one_mul_le fun ε hε => ?_
   by_cases h : ε = 0
   · simp only [h, zero_mul, zero_le']
-  have hε' : ε⁻¹ ≠ ∞ := fun h' =>
-    h (by simpa only [inv_inv, inv_top] using congr_arg (fun x : ℝ≥0∞ => x⁻¹) h')
   simp only [ENNReal.mul_le_iff_le_inv h (hε.trans_le le_top).ne, mul_comm ε⁻¹,
     liminf_eq_iSup_iInf_of_nat', ENNReal.iSup_mul]
-  conv_rhs => arg 1; intro i; rw [ENNReal.iInf_mul hε']
+  conv_rhs => arg 1; intro i; rw [ENNReal.iInf_mul (by simp [h])]
   rw [← ENNReal.inv_lt_inv, inv_one] at hε
   obtain ⟨N, hN⟩ := eventually_atTop.mp
     (ENNReal.eventually_pow_one_div_le (ENNReal.coe_ne_top : ↑‖(1 : A)‖₊ ≠ ∞) hε)
@@ -272,7 +270,7 @@ variable (𝕜)
 /-- In a Banach algebra `A` over a nontrivially normed field `𝕜`, for any `a : A` the
 power series with coefficients `a ^ n` represents the function `(1 - z • a)⁻¹` in a disk of
 radius `‖a‖₊⁻¹`. -/
-theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
+theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [HasSummableGeomSeries A] (a : A) :
     HasFPowerSeriesOnBall (fun z : 𝕜 => Ring.inverse (1 - z • a))
       (fun n => ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)) 0 ‖a‖₊⁻¹ :=
   { r_le := by
@@ -280,7 +278,7 @@ theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
         le_radius_of_bound_nnreal _ (max 1 ‖(1 : A)‖₊) fun n => ?_
       rw [← norm_toNNReal, norm_mkPiRing, norm_toNNReal]
       cases' n with n
-      · simp only [le_refl, mul_one, or_true_iff, le_max_iff, pow_zero]
+      · simp only [le_refl, mul_one, or_true, le_max_iff, pow_zero]
       · refine
           le_trans (le_trans (mul_le_mul_right' (nnnorm_pow_le' a n.succ_pos) (r ^ n.succ)) ?_)
             (le_max_left _ _)
@@ -297,7 +295,7 @@ theorem hasFPowerSeriesOnBall_inverse_one_sub_smul [CompleteSpace A] (a : A) :
             simpa only [← coe_inv h, mem_ball_zero_iff, Metric.emetric_ball_nnreal] using hy
           rwa [← coe_nnnorm, ← Real.lt_toNNReal_iff_coe_lt, Real.toNNReal_one, nnnorm_smul,
             ← NNReal.lt_inv_iff_mul_lt h]
-      simpa [← smul_pow, (NormedRing.summable_geometric_of_norm_lt_one _ norm_lt).hasSum_iff] using
+      simpa [← smul_pow, (summable_geometric_of_norm_lt_one norm_lt).hasSum_iff] using
         (NormedRing.inverse_one_sub _ norm_lt).symm }
 
 variable {𝕜}
@@ -740,6 +738,26 @@ lemma real_iff [Algebra ℂ A] {a : A} :
     simp
   · exact .of_subset_range_algebraMap Complex.ofReal_re fun x hx ↦ ⟨x.re, (h x hx).symm⟩
 
+lemma nnreal_le_iff [Algebra ℝ A] {a : A}
+    (ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ spectrum ℝ≥0 a, r ≤ x) ↔ ∀ x ∈ spectrum ℝ a, r ≤ x := by
+  simp [← ha.algebraMap_image]
+
+lemma nnreal_lt_iff [Algebra ℝ A] {a : A}
+    (ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ spectrum ℝ≥0 a, r < x) ↔ ∀ x ∈ spectrum ℝ a, r < x := by
+  simp [← ha.algebraMap_image]
+
+lemma le_nnreal_iff [Algebra ℝ A] {a : A}
+    (ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ spectrum ℝ≥0 a, x ≤ r) ↔ ∀ x ∈ spectrum ℝ a, x ≤ r := by
+  simp [← ha.algebraMap_image]
+
+lemma lt_nnreal_iff [Algebra ℝ A] {a : A}
+    (ha : SpectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ spectrum ℝ≥0 a, x < r) ↔ ∀ x ∈ spectrum ℝ a, x < r := by
+  simp [← ha.algebraMap_image]
+
 lemma nnreal_iff_spectralRadius_le [Algebra ℝ A] {a : A} {t : ℝ≥0} (ht : spectralRadius ℝ a ≤ t) :
     SpectrumRestricts a ContinuousMap.realToNNReal ↔
       spectralRadius ℝ (algebraMap ℝ A t - a) ≤ t := by
@@ -817,6 +835,16 @@ lemma real_iff [Module ℂ A] [IsScalarTower ℂ A A] [SMulCommClass ℂ A A] {a
   rw [quasispectrumRestricts_iff_spectrumRestricts_inr,
     Unitization.quasispectrum_eq_spectrum_inr' _ ℂ, SpectrumRestricts.real_iff]
 
+lemma le_nnreal_iff [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] {a : A}
+    (ha : QuasispectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ quasispectrum ℝ≥0 a, x ≤ r) ↔ ∀ x ∈ quasispectrum ℝ a, x ≤ r := by
+  simp [← ha.algebraMap_image]
+
+lemma lt_nnreal_iff [Module ℝ A] [IsScalarTower ℝ A A] [SMulCommClass ℝ A A] {a : A}
+    (ha : QuasispectrumRestricts a ContinuousMap.realToNNReal) {r : ℝ≥0} :
+    (∀ x ∈ quasispectrum ℝ≥0 a, x < r) ↔ ∀ x ∈ quasispectrum ℝ a, x < r := by
+  simp [← ha.algebraMap_image]
+
 end QuasispectrumRestricts
 
 variable {A : Type*} [Ring A] [PartialOrder A]
diff --git a/Mathlib/Analysis/Normed/Algebra/Unitization.lean b/Mathlib/Analysis/Normed/Algebra/Unitization.lean
index 789e764e8c4e0..de8b39cdbddce 100644
--- a/Mathlib/Analysis/Normed/Algebra/Unitization.lean
+++ b/Mathlib/Analysis/Normed/Algebra/Unitization.lean
@@ -184,8 +184,8 @@ open scoped Uniformity Topology
 
 theorem uniformity_eq_aux :
     𝓤[instUniformSpaceProd.comap <| addEquiv 𝕜 A] = 𝓤 (Unitization 𝕜 A) := by
-  have key : UniformInducing (addEquiv 𝕜 A) :=
-    antilipschitzWith_addEquiv.uniformInducing lipschitzWith_addEquiv.uniformContinuous
+  have key : IsUniformInducing (addEquiv 𝕜 A) :=
+    antilipschitzWith_addEquiv.isUniformInducing lipschitzWith_addEquiv.uniformContinuous
   rw [← key.comap_uniformity]
   rfl
 
@@ -202,21 +202,24 @@ instance instUniformSpace : UniformSpace (Unitization 𝕜 A) :=
 
 /-- The natural equivalence between `Unitization 𝕜 A` and `𝕜 × A` as a uniform equivalence. -/
 def uniformEquivProd : (Unitization 𝕜 A) ≃ᵤ (𝕜 × A) :=
-  Equiv.toUniformEquivOfUniformInducing (addEquiv 𝕜 A) ⟨rfl⟩
+  Equiv.toUniformEquivOfIsUniformInducing (addEquiv 𝕜 A) ⟨rfl⟩
 
 /-- The bornology on `Unitization 𝕜 A` is inherited from `𝕜 × A`. -/
 instance instBornology : Bornology (Unitization 𝕜 A) :=
   Bornology.induced <| addEquiv 𝕜 A
 
-theorem uniformEmbedding_addEquiv {𝕜} [NontriviallyNormedField 𝕜] :
-    UniformEmbedding (addEquiv 𝕜 A) where
+theorem isUniformEmbedding_addEquiv {𝕜} [NontriviallyNormedField 𝕜] :
+    IsUniformEmbedding (addEquiv 𝕜 A) where
   comap_uniformity := rfl
   inj := (addEquiv 𝕜 A).injective
 
-/-- `Unitization 𝕜 A` is complete whenever `𝕜` and `A` are also.  -/
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_addEquiv := isUniformEmbedding_addEquiv
+
+/-- `Unitization 𝕜 A` is complete whenever `𝕜` and `A` are also. -/
 instance instCompleteSpace [CompleteSpace 𝕜] [CompleteSpace A] :
     CompleteSpace (Unitization 𝕜 A) :=
-  (completeSpace_congr uniformEmbedding_addEquiv).mpr CompleteSpace.prod
+  uniformEquivProd.completeSpace_iff.2 .prod
 
 /-- Pull back the metric structure from `𝕜 × (A →L[𝕜] A)` to `Unitization 𝕜 A` using the
 algebra homomorphism `Unitization.splitMul 𝕜 A`, but replace the bornology and the uniformity so
diff --git a/Mathlib/Analysis/Normed/Algebra/UnitizationL1.lean b/Mathlib/Analysis/Normed/Algebra/UnitizationL1.lean
index 4a0ab143ce2a1..bb8f42626297c 100644
--- a/Mathlib/Analysis/Normed/Algebra/UnitizationL1.lean
+++ b/Mathlib/Analysis/Normed/Algebra/UnitizationL1.lean
@@ -50,7 +50,7 @@ noncomputable def uniformEquiv_unitization_addEquiv_prod :
 
 instance instCompleteSpace [CompleteSpace 𝕜] [CompleteSpace A] :
     CompleteSpace (WithLp 1 (Unitization 𝕜 A)) :=
-  completeSpace_congr (uniformEquiv_unitization_addEquiv_prod 𝕜 A).uniformEmbedding |>.mpr
+  completeSpace_congr (uniformEquiv_unitization_addEquiv_prod 𝕜 A).isUniformEmbedding |>.mpr
     CompleteSpace.prod
 
 variable {𝕜 A}
diff --git a/Mathlib/Analysis/Normed/Field/Basic.lean b/Mathlib/Analysis/Normed/Field/Basic.lean
index c9a3e6d3393c9..405b9ae12e4fa 100644
--- a/Mathlib/Analysis/Normed/Field/Basic.lean
+++ b/Mathlib/Analysis/Normed/Field/Basic.lean
@@ -5,31 +5,35 @@ Authors: Patrick Massot, Johannes Hölzl
 -/
 import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
-import Mathlib.Algebra.Group.AddChar
-import Mathlib.Algebra.Order.Ring.Finset
-import Mathlib.Analysis.Normed.Group.Bounded
 import Mathlib.Analysis.Normed.Group.Constructions
-import Mathlib.Analysis.Normed.Group.Rat
 import Mathlib.Analysis.Normed.Group.Submodule
-import Mathlib.Analysis.Normed.Group.Uniform
-import Mathlib.GroupTheory.OrderOfElement
-import Mathlib.Topology.Instances.NNReal
-import Mathlib.Topology.MetricSpace.DilationEquiv
+import Mathlib.Data.Set.Pointwise.Interval
 
 /-!
 # Normed fields
 
 In this file we define (semi)normed rings and fields. We also prove some theorems about these
 definitions.
+
+Some useful results that relate the topology of the normed field to the discrete topology include:
+* `norm_eq_one_iff_ne_zero_of_discrete`
 -/
 
 -- Guard against import creep.
+assert_not_exists AddChar
+assert_not_exists comap_norm_atTop
+assert_not_exists DilationEquiv
+assert_not_exists Finset.sup_mul_le_mul_sup_of_nonneg
+assert_not_exists IsOfFinOrder
+assert_not_exists Isometry.norm_map_of_map_one
+assert_not_exists NNReal.isOpen_Ico_zero
+assert_not_exists Rat.norm_cast_real
 assert_not_exists RestrictScalars
 
-variable {α : Type*} {β : Type*} {γ : Type*} {ι : Type*}
+variable {α : Type*} {β : Type*} {ι : Type*}
 
-open Filter Metric Bornology
-open scoped Topology NNReal ENNReal uniformity Pointwise
+open Filter
+open scoped Topology NNReal
 
 /-- A non-unital seminormed ring is a not-necessarily-unital ring
 endowed with a seminorm which satisfies the inequality `‖x y‖ ≤ ‖x‖ ‖y‖`. -/
@@ -224,17 +228,6 @@ theorem one_le_norm_one (β) [NormedRing β] [Nontrivial β] : 1 ≤ ‖(1 : β)
 theorem one_le_nnnorm_one (β) [NormedRing β] [Nontrivial β] : 1 ≤ ‖(1 : β)‖₊ :=
   one_le_norm_one β
 
-theorem Filter.Tendsto.zero_mul_isBoundedUnder_le {f g : ι → α} {l : Filter ι}
-    (hf : Tendsto f l (𝓝 0)) (hg : IsBoundedUnder (· ≤ ·) l ((‖·‖) ∘ g)) :
-    Tendsto (fun x => f x * g x) l (𝓝 0) :=
-  hf.op_zero_isBoundedUnder_le hg (· * ·) norm_mul_le
-
-theorem Filter.isBoundedUnder_le_mul_tendsto_zero {f g : ι → α} {l : Filter ι}
-    (hf : IsBoundedUnder (· ≤ ·) l (norm ∘ f)) (hg : Tendsto g l (𝓝 0)) :
-    Tendsto (fun x => f x * g x) l (𝓝 0) :=
-  hg.op_zero_isBoundedUnder_le hf (flip (· * ·)) fun x y =>
-    (norm_mul_le y x).trans_eq (mul_comm _ _)
-
 /-- In a seminormed ring, the left-multiplication `AddMonoidHom` is bounded. -/
 theorem mulLeft_bound (x : α) : ∀ y : α, ‖AddMonoidHom.mulLeft x y‖ ≤ ‖x‖ * ‖y‖ :=
   norm_mul_le x
@@ -245,7 +238,7 @@ theorem mulRight_bound (x : α) : ∀ y : α, ‖AddMonoidHom.mulRight x y‖ 
   exact norm_mul_le y x
 
 /-- A non-unital subalgebra of a non-unital seminormed ring is also a non-unital seminormed ring,
-with the restriction of the norm.  -/
+with the restriction of the norm. -/
 instance NonUnitalSubalgebra.nonUnitalSeminormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
     [NonUnitalSeminormedRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
     NonUnitalSeminormedRing s :=
@@ -253,7 +246,7 @@ instance NonUnitalSubalgebra.nonUnitalSeminormedRing {𝕜 : Type*} [CommRing 
     norm_mul := fun a b => norm_mul_le a.1 b.1 }
 
 /-- A non-unital subalgebra of a non-unital seminormed ring is also a non-unital seminormed ring,
-with the restriction of the norm.  -/
+with the restriction of the norm. -/
 -- necessary to require `SMulMemClass S 𝕜 E` so that `𝕜` can be determined as an `outParam`
 @[nolint unusedArguments]
 instance (priority := 75) NonUnitalSubalgebraClass.nonUnitalSeminormedRing {S 𝕜 E : Type*}
@@ -264,14 +257,14 @@ instance (priority := 75) NonUnitalSubalgebraClass.nonUnitalSeminormedRing {S 
     norm_mul := fun a b => norm_mul_le a.1 b.1 }
 
 /-- A non-unital subalgebra of a non-unital normed ring is also a non-unital normed ring, with the
-restriction of the norm.  -/
+restriction of the norm. -/
 instance NonUnitalSubalgebra.nonUnitalNormedRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
     [NonUnitalNormedRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) : NonUnitalNormedRing s :=
   { s.nonUnitalSeminormedRing with
     eq_of_dist_eq_zero := eq_of_dist_eq_zero }
 
 /-- A non-unital subalgebra of a non-unital normed ring is also a non-unital normed ring,
-with the restriction of the norm.  -/
+with the restriction of the norm. -/
 instance (priority := 75) NonUnitalSubalgebraClass.nonUnitalNormedRing {S 𝕜 E : Type*}
     [CommRing 𝕜] [NonUnitalNormedRing E] [Module 𝕜 E] [SetLike S E] [NonUnitalSubringClass S E]
     [SMulMemClass S 𝕜 E] (s : S) :
@@ -301,21 +294,6 @@ instance Prod.nonUnitalSeminormedRing [NonUnitalSeminormedRing β] :
         _ = ‖x‖ * ‖y‖ := rfl
          }
 
-/-- Non-unital seminormed ring structure on the product of finitely many non-unital seminormed
-rings, using the sup norm. -/
-instance Pi.nonUnitalSeminormedRing {π : ι → Type*} [Fintype ι]
-    [∀ i, NonUnitalSeminormedRing (π i)] : NonUnitalSeminormedRing (∀ i, π i) :=
-  { Pi.seminormedAddCommGroup, Pi.nonUnitalRing with
-    norm_mul := fun x y =>
-      NNReal.coe_mono <|
-        calc
-          (Finset.univ.sup fun i => ‖x i * y i‖₊) ≤
-              Finset.univ.sup ((fun i => ‖x i‖₊) * fun i => ‖y i‖₊) :=
-            Finset.sup_mono_fun fun _ _ => norm_mul_le _ _
-          _ ≤ (Finset.univ.sup fun i => ‖x i‖₊) * Finset.univ.sup fun i => ‖y i‖₊ :=
-            Finset.sup_mul_le_mul_sup_of_nonneg _ (fun _ _ => zero_le _) fun _ _ => zero_le _
-           }
-
 instance MulOpposite.instNonUnitalSeminormedRing : NonUnitalSeminormedRing αᵐᵒᵖ where
   __ := instNonUnitalRing
   __ := instSeminormedAddCommGroup
@@ -326,7 +304,7 @@ end NonUnitalSeminormedRing
 
 section SeminormedRing
 
-variable [SeminormedRing α]
+variable [SeminormedRing α] {a b c : α}
 
 /-- A subalgebra of a seminormed ring is also a seminormed ring, with the restriction of the
 norm. -/
@@ -359,6 +337,7 @@ instance (priority := 75) SubalgebraClass.normedRing {S 𝕜 E : Type*} [CommRin
   { seminormedRing s with
     eq_of_dist_eq_zero := eq_of_dist_eq_zero }
 
+
 theorem Nat.norm_cast_le : ∀ n : ℕ, ‖(n : α)‖ ≤ n * ‖(1 : α)‖
   | 0 => by simp
   | n + 1 => by
@@ -369,7 +348,7 @@ theorem List.norm_prod_le' : ∀ {l : List α}, l ≠ [] → ‖l.prod‖ ≤ (l
   | [], h => (h rfl).elim
   | [a], _ => by simp
   | a::b::l, _ => by
-    rw [List.map_cons, List.prod_cons, @List.prod_cons _ _ _ ‖a‖]
+    rw [List.map_cons, List.prod_cons, List.prod_cons (a := ‖a‖)]
     refine le_trans (norm_mul_le _ _) (mul_le_mul_of_nonneg_left ?_ (norm_nonneg _))
     exact List.norm_prod_le' (List.cons_ne_nil b l)
 
@@ -437,16 +416,39 @@ instance ULift.seminormedRing : SeminormedRing (ULift α) :=
 instance Prod.seminormedRing [SeminormedRing β] : SeminormedRing (α × β) :=
   { nonUnitalSeminormedRing, instRing with }
 
-/-- Seminormed ring structure on the product of finitely many seminormed rings,
-  using the sup norm. -/
-instance Pi.seminormedRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedRing (π i)] :
-    SeminormedRing (∀ i, π i) :=
-  { Pi.nonUnitalSeminormedRing, Pi.ring with }
-
 instance MulOpposite.instSeminormedRing : SeminormedRing αᵐᵒᵖ where
   __ := instRing
   __ := instNonUnitalSeminormedRing
 
+/-- This inequality is particularly useful when `c = 1` and `‖a‖ = ‖b‖ = 1` as it then shows that
+chord length is a metric on the unit complex numbers. -/
+lemma norm_sub_mul_le (ha : ‖a‖ ≤ 1) : ‖c - a * b‖ ≤ ‖c - a‖ + ‖1 - b‖ :=
+  calc
+    _ ≤ ‖c - a‖ + ‖a * (1 - b)‖ := by
+        simpa [mul_one_sub] using norm_sub_le_norm_sub_add_norm_sub c a (a * b)
+    _ ≤ ‖c - a‖ + ‖a‖ * ‖1 - b‖ := by gcongr; exact norm_mul_le ..
+    _ ≤ ‖c - a‖ + 1 * ‖1 - b‖ := by gcongr
+    _ = ‖c - a‖ + ‖1 - b‖ := by simp
+
+/-- This inequality is particularly useful when `c = 1` and `‖a‖ = ‖b‖ = 1` as it then shows that
+chord length is a metric on the unit complex numbers. -/
+lemma norm_sub_mul_le' (hb : ‖b‖ ≤ 1) : ‖c - a * b‖ ≤ ‖1 - a‖ + ‖c - b‖ := by
+  rw [add_comm]; exact norm_sub_mul_le (α := αᵐᵒᵖ) hb
+
+/-- This inequality is particularly useful when `c = 1` and `‖a‖ = ‖b‖ = 1` as it then shows that
+chord length is a metric on the unit complex numbers. -/
+lemma nnnorm_sub_mul_le (ha : ‖a‖₊ ≤ 1) : ‖c - a * b‖₊ ≤ ‖c - a‖₊ + ‖1 - b‖₊ := norm_sub_mul_le ha
+
+/-- This inequality is particularly useful when `c = 1` and `‖a‖ = ‖b‖ = 1` as it then shows that
+chord length is a metric on the unit complex numbers. -/
+lemma nnnorm_sub_mul_le' (hb : ‖b‖₊ ≤ 1) : ‖c - a * b‖₊ ≤ ‖1 - a‖₊ + ‖c - b‖₊ := norm_sub_mul_le' hb
+
+/-- A homomorphism `f` between semi_normed_rings is bounded if there exists a positive
+  constant `C` such that for all `x` in `α`, `norm (f x) ≤ C * norm x`. -/
+def RingHom.IsBounded {α : Type*} [SeminormedRing α] {β : Type*} [SeminormedRing β]
+    (f : α →+* β) : Prop :=
+  ∃ C : ℝ, 0 < C ∧ ∀ x : α, norm (f x) ≤ C * norm x
+
 end SeminormedRing
 
 section NonUnitalNormedRing
@@ -461,12 +463,6 @@ using the sup norm. -/
 instance Prod.nonUnitalNormedRing [NonUnitalNormedRing β] : NonUnitalNormedRing (α × β) :=
   { Prod.nonUnitalSeminormedRing, Prod.normedAddCommGroup with }
 
-/-- Normed ring structure on the product of finitely many non-unital normed rings, using the sup
-norm. -/
-instance Pi.nonUnitalNormedRing {π : ι → Type*} [Fintype ι] [∀ i, NonUnitalNormedRing (π i)] :
-    NonUnitalNormedRing (∀ i, π i) :=
-  { Pi.nonUnitalSeminormedRing, Pi.normedAddCommGroup with }
-
 instance MulOpposite.instNonUnitalNormedRing : NonUnitalNormedRing αᵐᵒᵖ where
   __ := instNonUnitalRing
   __ := instNonUnitalSeminormedRing
@@ -491,11 +487,6 @@ instance ULift.normedRing : NormedRing (ULift α) :=
 instance Prod.normedRing [NormedRing β] : NormedRing (α × β) :=
   { nonUnitalNormedRing, instRing with }
 
-/-- Normed ring structure on the product of finitely many normed rings, using the sup norm. -/
-instance Pi.normedRing {π : ι → Type*} [Fintype ι] [∀ i, NormedRing (π i)] :
-    NormedRing (∀ i, π i) :=
-  { Pi.seminormedRing, Pi.normedAddCommGroup with }
-
 instance MulOpposite.instNormedRing : NormedRing αᵐᵒᵖ where
   __ := instRing
   __ := instSeminormedRing
@@ -516,12 +507,6 @@ instance Prod.nonUnitalSeminormedCommRing [NonUnitalSeminormedCommRing β] :
     NonUnitalSeminormedCommRing (α × β) :=
   { nonUnitalSeminormedRing, instNonUnitalCommRing with }
 
-/-- Non-unital seminormed commutative ring structure on the product of finitely many non-unital
-seminormed commutative rings, using the sup norm. -/
-instance Pi.nonUnitalSeminormedCommRing {π : ι → Type*} [Fintype ι]
-    [∀ i, NonUnitalSeminormedCommRing (π i)] : NonUnitalSeminormedCommRing (∀ i, π i) :=
-  { Pi.nonUnitalSeminormedRing, Pi.nonUnitalCommRing with }
-
 instance MulOpposite.instNonUnitalSeminormedCommRing : NonUnitalSeminormedCommRing αᵐᵒᵖ where
   __ := instNonUnitalSeminormedRing
   __ := instNonUnitalCommRing
@@ -533,14 +518,14 @@ section NonUnitalNormedCommRing
 variable [NonUnitalNormedCommRing α]
 
 /-- A non-unital subalgebra of a non-unital seminormed commutative ring is also a non-unital
-seminormed commutative ring, with the restriction of the norm.  -/
+seminormed commutative ring, with the restriction of the norm. -/
 instance NonUnitalSubalgebra.nonUnitalSeminormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
     [NonUnitalSeminormedCommRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
     NonUnitalSeminormedCommRing s :=
   { s.nonUnitalSeminormedRing, s.toNonUnitalCommRing with }
 
 /-- A non-unital subalgebra of a non-unital normed commutative ring is also a non-unital normed
-commutative ring, with the restriction of the norm.  -/
+commutative ring, with the restriction of the norm. -/
 instance NonUnitalSubalgebra.nonUnitalNormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*}
     [NonUnitalNormedCommRing E] [Module 𝕜 E] (s : NonUnitalSubalgebra 𝕜 E) :
     NonUnitalNormedCommRing s :=
@@ -555,12 +540,6 @@ instance Prod.nonUnitalNormedCommRing [NonUnitalNormedCommRing β] :
     NonUnitalNormedCommRing (α × β) :=
   { Prod.nonUnitalSeminormedCommRing, Prod.normedAddCommGroup with }
 
-/-- Normed commutative ring structure on the product of finitely many non-unital normed
-commutative rings, using the sup norm. -/
-instance Pi.nonUnitalNormedCommRing {π : ι → Type*} [Fintype ι]
-    [∀ i, NonUnitalNormedCommRing (π i)] : NonUnitalNormedCommRing (∀ i, π i) :=
-  { Pi.nonUnitalSeminormedCommRing, Pi.normedAddCommGroup with }
-
 instance MulOpposite.instNonUnitalNormedCommRing : NonUnitalNormedCommRing αᵐᵒᵖ where
   __ := instNonUnitalNormedRing
   __ := instNonUnitalSeminormedCommRing
@@ -579,12 +558,6 @@ instance ULift.seminormedCommRing : SeminormedCommRing (ULift α) :=
 instance Prod.seminormedCommRing [SeminormedCommRing β] : SeminormedCommRing (α × β) :=
   { Prod.nonUnitalSeminormedCommRing, instCommRing with }
 
-/-- Seminormed commutative ring structure on the product of finitely many seminormed commutative
-rings, using the sup norm. -/
-instance Pi.seminormedCommRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedCommRing (π i)] :
-    SeminormedCommRing (∀ i, π i) :=
-  { Pi.nonUnitalSeminormedCommRing, Pi.ring with }
-
 instance MulOpposite.instSeminormedCommRing : SeminormedCommRing αᵐᵒᵖ where
   __ := instSeminormedRing
   __ := instNonUnitalSeminormedCommRing
@@ -594,13 +567,13 @@ end SeminormedCommRing
 section NormedCommRing
 
 /-- A subalgebra of a seminormed commutative ring is also a seminormed commutative ring, with the
-restriction of the norm.  -/
+restriction of the norm. -/
 instance Subalgebra.seminormedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [SeminormedCommRing E]
     [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : SeminormedCommRing s :=
   { s.seminormedRing, s.toCommRing with }
 
 /-- A subalgebra of a normed commutative ring is also a normed commutative ring, with the
-restriction of the norm.  -/
+restriction of the norm. -/
 instance Subalgebra.normedCommRing {𝕜 : Type*} [CommRing 𝕜] {E : Type*} [NormedCommRing E]
     [Algebra 𝕜 E] (s : Subalgebra 𝕜 E) : NormedCommRing s :=
   { s.seminormedCommRing, s.normedRing with }
@@ -615,47 +588,17 @@ norm. -/
 instance Prod.normedCommRing [NormedCommRing β] : NormedCommRing (α × β) :=
   { nonUnitalNormedRing, instCommRing with }
 
-/-- Normed commutative ring structure on the product of finitely many normed commutative rings,
-using the sup norm. -/
-instance Pi.normedCommutativeRing {π : ι → Type*} [Fintype ι] [∀ i, NormedCommRing (π i)] :
-    NormedCommRing (∀ i, π i) :=
-  { Pi.seminormedCommRing, Pi.normedAddCommGroup with }
-
 instance MulOpposite.instNormedCommRing : NormedCommRing αᵐᵒᵖ where
   __ := instNormedRing
   __ := instSeminormedCommRing
 
-end NormedCommRing
-
--- see Note [lower instance priority]
-instance (priority := 100) semi_normed_ring_top_monoid [NonUnitalSeminormedRing α] :
-    ContinuousMul α :=
-  ⟨continuous_iff_continuousAt.2 fun x =>
-      tendsto_iff_norm_sub_tendsto_zero.2 <| by
-        have : ∀ e : α × α,
-            ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1‖ * ‖e.2 - x.2‖ + ‖e.1 - x.1‖ * ‖x.2‖ := by
-          intro e
-          calc
-            ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1 * (e.2 - x.2) + (e.1 - x.1) * x.2‖ := by
-              rw [_root_.mul_sub, _root_.sub_mul, sub_add_sub_cancel]
-            -- Porting note: `ENNReal.{mul_sub, sub_mul}` should be protected
-            _ ≤ ‖e.1‖ * ‖e.2 - x.2‖ + ‖e.1 - x.1‖ * ‖x.2‖ :=
-              norm_add_le_of_le (norm_mul_le _ _) (norm_mul_le _ _)
-        refine squeeze_zero (fun e => norm_nonneg _) this ?_
-        convert
-          ((continuous_fst.tendsto x).norm.mul
-                ((continuous_snd.tendsto x).sub tendsto_const_nhds).norm).add
-            (((continuous_fst.tendsto x).sub tendsto_const_nhds).norm.mul _)
-        -- Porting note: `show` used to select a goal to work on
-        rotate_right
-        · show Tendsto _ _ _
-          exact tendsto_const_nhds
-        · simp⟩
+/-- The restriction of a power-multiplicative function to a subalgebra is power-multiplicative. -/
+theorem IsPowMul.restriction {R S : Type*} [NormedCommRing R] [CommRing S] [Algebra R S]
+    (A : Subalgebra R S) {f : S → ℝ} (hf_pm : IsPowMul f) :
+    IsPowMul fun x : A => f x.val := fun x n hn => by
+  simpa [SubsemiringClass.coe_pow] using hf_pm (↑x) hn
 
--- see Note [lower instance priority]
-/-- A seminormed ring is a topological ring. -/
-instance (priority := 100) semi_normed_top_ring [NonUnitalSeminormedRing α] :
-    TopologicalRing α where
+end NormedCommRing
 
 section NormedDivisionRing
 
@@ -741,116 +684,45 @@ theorem nndist_inv_inv₀ {z w : α} (hz : z ≠ 0) (hw : w ≠ 0) :
     nndist z⁻¹ w⁻¹ = nndist z w / (‖z‖₊ * ‖w‖₊) :=
   NNReal.eq <| dist_inv_inv₀ hz hw
 
-lemma antilipschitzWith_mul_left {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (a * ·) :=
-  AntilipschitzWith.of_le_mul_dist fun _ _ ↦ by simp [dist_eq_norm, ← _root_.mul_sub, ha]
-
-lemma antilipschitzWith_mul_right {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (· * a) :=
-  AntilipschitzWith.of_le_mul_dist fun _ _ ↦ by
-    simp [dist_eq_norm, ← _root_.sub_mul, ← mul_comm (‖a‖), ha]
-
-/-- Multiplication by a nonzero element `a` on the left
-as a `DilationEquiv` of a normed division ring. -/
-@[simps!]
-def DilationEquiv.mulLeft (a : α) (ha : a ≠ 0) : α ≃ᵈ α where
-  toEquiv := Equiv.mulLeft₀ a ha
-  edist_eq' := ⟨‖a‖₊, nnnorm_ne_zero_iff.2 ha, fun x y ↦ by
-    simp [edist_nndist, nndist_eq_nnnorm, ← mul_sub]⟩
-
-/-- Multiplication by a nonzero element `a` on the right
-as a `DilationEquiv` of a normed division ring. -/
-@[simps!]
-def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α where
-  toEquiv := Equiv.mulRight₀ a ha
-  edist_eq' := ⟨‖a‖₊, nnnorm_ne_zero_iff.2 ha, fun x y ↦ by
-    simp [edist_nndist, nndist_eq_nnnorm, ← sub_mul, ← mul_comm (‖a‖₊)]⟩
-
-namespace Filter
-
-@[simp]
-lemma comap_mul_left_cobounded {a : α} (ha : a ≠ 0) :
-    comap (a * ·) (cobounded α) = cobounded α :=
-  Dilation.comap_cobounded (DilationEquiv.mulLeft a ha)
-
-@[simp]
-lemma map_mul_left_cobounded {a : α} (ha : a ≠ 0) :
-    map (a * ·) (cobounded α) = cobounded α :=
-  DilationEquiv.map_cobounded (DilationEquiv.mulLeft a ha)
-
-@[simp]
-lemma comap_mul_right_cobounded {a : α} (ha : a ≠ 0) :
-    comap (· * a) (cobounded α) = cobounded α :=
-  Dilation.comap_cobounded (DilationEquiv.mulRight a ha)
-
-@[simp]
-lemma map_mul_right_cobounded {a : α} (ha : a ≠ 0) :
-    map (· * a) (cobounded α) = cobounded α :=
-  DilationEquiv.map_cobounded (DilationEquiv.mulRight a ha)
-
-/-- Multiplication on the left by a nonzero element of a normed division ring tends to infinity at
-infinity. -/
-theorem tendsto_mul_left_cobounded {a : α} (ha : a ≠ 0) :
-    Tendsto (a * ·) (cobounded α) (cobounded α) :=
-  (map_mul_left_cobounded ha).le
-
-/-- Multiplication on the right by a nonzero element of a normed division ring tends to infinity at
-infinity. -/
-theorem tendsto_mul_right_cobounded {a : α} (ha : a ≠ 0) :
-    Tendsto (· * a) (cobounded α) (cobounded α) :=
-  (map_mul_right_cobounded ha).le
+namespace NormedDivisionRing
+
+section Discrete
+
+variable {𝕜 : Type*} [NormedDivisionRing 𝕜] [DiscreteTopology 𝕜]
+
+lemma norm_eq_one_iff_ne_zero_of_discrete {x : 𝕜} : ‖x‖ = 1 ↔ x ≠ 0 := by
+  constructor <;> intro hx
+  · contrapose! hx
+    simp [hx]
+  · have : IsOpen {(0 : 𝕜)} := isOpen_discrete {0}
+    simp_rw [Metric.isOpen_singleton_iff, dist_eq_norm, sub_zero] at this
+    obtain ⟨ε, εpos, h'⟩ := this
+    wlog h : ‖x‖ < 1 generalizing 𝕜 with H
+    · push_neg at h
+      rcases h.eq_or_lt with h|h
+      · rw [h]
+      replace h := norm_inv x ▸ inv_lt_one_of_one_lt₀ h
+      rw [← inv_inj, inv_one, ← norm_inv]
+      exact H (by simpa) h' h
+    obtain ⟨k, hk⟩ : ∃ k : ℕ, ‖x‖ ^ k < ε := exists_pow_lt_of_lt_one εpos h
+    rw [← norm_pow] at hk
+    specialize h' _ hk
+    simp [hx] at h'
 
 @[simp]
-lemma inv_cobounded₀ : (cobounded α)⁻¹ = 𝓝[≠] 0 := by
-  rw [← comap_norm_atTop, ← Filter.comap_inv, ← comap_norm_nhdsWithin_Ioi_zero,
-    ← inv_atTop₀, ← Filter.comap_inv]
-  simp only [comap_comap, (· ∘ ·), norm_inv]
+lemma norm_le_one_of_discrete
+    (x : 𝕜) : ‖x‖ ≤ 1 := by
+  rcases eq_or_ne x 0 with rfl|hx
+  · simp
+  · simp [norm_eq_one_iff_ne_zero_of_discrete.mpr hx]
 
-@[simp]
-lemma inv_nhdsWithin_ne_zero : (𝓝[≠] (0 : α))⁻¹ = cobounded α := by
-  rw [← inv_cobounded₀, inv_inv]
-
-lemma tendsto_inv₀_cobounded' : Tendsto Inv.inv (cobounded α) (𝓝[≠] 0) :=
-  inv_cobounded₀.le
-
-theorem tendsto_inv₀_cobounded : Tendsto Inv.inv (cobounded α) (𝓝 0) :=
-  tendsto_inv₀_cobounded'.mono_right inf_le_left
-
-lemma tendsto_inv₀_nhdsWithin_ne_zero : Tendsto Inv.inv (𝓝[≠] 0) (cobounded α) :=
-  inv_nhdsWithin_ne_zero.le
-
-end Filter
-
--- see Note [lower instance priority]
-instance (priority := 100) NormedDivisionRing.to_hasContinuousInv₀ : HasContinuousInv₀ α := by
-  refine ⟨fun r r0 => tendsto_iff_norm_sub_tendsto_zero.2 ?_⟩
-  have r0' : 0 < ‖r‖ := norm_pos_iff.2 r0
-  rcases exists_between r0' with ⟨ε, ε0, εr⟩
-  have : ∀ᶠ e in 𝓝 r, ‖e⁻¹ - r⁻¹‖ ≤ ‖r - e‖ / ‖r‖ / ε := by
-    filter_upwards [(isOpen_lt continuous_const continuous_norm).eventually_mem εr] with e he
-    have e0 : e ≠ 0 := norm_pos_iff.1 (ε0.trans he)
-    calc
-      ‖e⁻¹ - r⁻¹‖ = ‖r‖⁻¹ * ‖r - e‖ * ‖e‖⁻¹ := by
-        rw [← norm_inv, ← norm_inv, ← norm_mul, ← norm_mul, _root_.mul_sub, _root_.sub_mul,
-          mul_assoc _ e, inv_mul_cancel₀ r0, mul_inv_cancel₀ e0, one_mul, mul_one]
-      -- Porting note: `ENNReal.{mul_sub, sub_mul}` should be `protected`
-      _ = ‖r - e‖ / ‖r‖ / ‖e‖ := by field_simp [mul_comm]
-      _ ≤ ‖r - e‖ / ‖r‖ / ε := by gcongr
-  refine squeeze_zero' (Eventually.of_forall fun _ => norm_nonneg _) this ?_
-  refine (((continuous_const.sub continuous_id).norm.div_const _).div_const _).tendsto' _ _ ?_
+lemma discreteTopology_unit_closedBall_eq_univ : (Metric.closedBall 0 1 : Set 𝕜) = Set.univ := by
+  ext
   simp
 
--- see Note [lower instance priority]
-/-- A normed division ring is a topological division ring. -/
-instance (priority := 100) NormedDivisionRing.to_topologicalDivisionRing :
-    TopologicalDivisionRing α where
-
-protected lemma IsOfFinOrder.norm_eq_one (ha : IsOfFinOrder a) : ‖a‖ = 1 :=
-  ((normHom : α →*₀ ℝ).toMonoidHom.isOfFinOrder ha).eq_one <| norm_nonneg _
+end Discrete
 
-example [Monoid β] (φ : β →* α) {x : β} {k : ℕ+} (h : x ^ (k : ℕ) = 1) :
-    ‖φ x‖ = 1 := (φ.isOfFinOrder <| isOfFinOrder_iff_pow_eq_one.2 ⟨_, k.2, h⟩).norm_eq_one
-
-@[simp] lemma AddChar.norm_apply {G : Type*} [AddLeftCancelMonoid G] [Finite G] (ψ : AddChar G α)
-    (x : G) : ‖ψ x‖ = 1 := (ψ.toMonoidHom.isOfFinOrder <| isOfFinOrder_of_finite _).norm_eq_one
+end NormedDivisionRing
 
 end NormedDivisionRing
 
@@ -964,11 +836,6 @@ instance denselyOrdered_range_nnnorm : DenselyOrdered (Set.range (nnnorm : α 
     let ⟨z, h⟩ := exists_lt_nnnorm_lt α hxy
     exact ⟨⟨‖z‖₊, z, rfl⟩, h⟩
 
-theorem denseRange_nnnorm : DenseRange (nnnorm : α → ℝ≥0) :=
-  dense_of_exists_between fun _ _ hr =>
-    let ⟨x, h⟩ := exists_lt_nnnorm_lt α hr
-    ⟨‖x‖₊, ⟨x, rfl⟩, h⟩
-
 end Densely
 
 end NormedField
@@ -983,7 +850,7 @@ def NontriviallyNormedField.ofNormNeOne {𝕜 : Type*} [h' : NormedField 𝕜]
     rcases hx1.lt_or_lt with hlt | hlt
     · use x⁻¹
       rw [norm_inv]
-      exact one_lt_inv (norm_pos_iff.2 hx) hlt
+      exact (one_lt_inv₀ (norm_pos_iff.2 hx)).2 hlt
     · exact ⟨x, hlt⟩
 
 instance Real.normedCommRing : NormedCommRing ℝ :=
@@ -1021,14 +888,6 @@ theorem norm_eq (x : ℝ≥0) : ‖(x : ℝ)‖ = x := by rw [Real.norm_eq_abs,
 theorem nnnorm_eq (x : ℝ≥0) : ‖(x : ℝ)‖₊ = x :=
   NNReal.eq <| Real.norm_of_nonneg x.2
 
-lemma lipschitzWith_sub : LipschitzWith 2 (fun (p : ℝ≥0 × ℝ≥0) ↦ p.1 - p.2) := by
-  rw [← isometry_subtype_coe.lipschitzWith_iff]
-  have : Isometry (Prod.map ((↑) : ℝ≥0 → ℝ) ((↑) : ℝ≥0 → ℝ)) :=
-    isometry_subtype_coe.prod_map isometry_subtype_coe
-  convert (((LipschitzWith.prod_fst.comp this.lipschitz).sub
-    (LipschitzWith.prod_snd.comp this.lipschitz)).max_const 0)
-  norm_num
-
 end NNReal
 
 @[simp 1001] -- Porting note: increase priority so that the LHS doesn't simplify
@@ -1040,37 +899,19 @@ theorem nnnorm_norm [SeminormedAddCommGroup α] (a : α) : ‖‖a‖‖₊ = 
   rw [Real.nnnorm_of_nonneg (norm_nonneg a)]; rfl
 
 /-- A restatement of `MetricSpace.tendsto_atTop` in terms of the norm. -/
-theorem NormedAddCommGroup.tendsto_atTop [Nonempty α] [SemilatticeSup α] {β : Type*}
-    [SeminormedAddCommGroup β] {f : α → β} {b : β} :
+theorem NormedAddCommGroup.tendsto_atTop [Nonempty α] [Preorder α] [IsDirected α (· ≤ ·)]
+    {β : Type*} [SeminormedAddCommGroup β] {f : α → β} {b : β} :
     Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N ≤ n → ‖f n - b‖ < ε :=
   (atTop_basis.tendsto_iff Metric.nhds_basis_ball).trans (by simp [dist_eq_norm])
 
 /-- A variant of `NormedAddCommGroup.tendsto_atTop` that
 uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
 -/
-theorem NormedAddCommGroup.tendsto_atTop' [Nonempty α] [SemilatticeSup α] [NoMaxOrder α]
-    {β : Type*} [SeminormedAddCommGroup β] {f : α → β} {b : β} :
+theorem NormedAddCommGroup.tendsto_atTop' [Nonempty α] [Preorder α] [IsDirected α (· ≤ ·)]
+    [NoMaxOrder α] {β : Type*} [SeminormedAddCommGroup β] {f : α → β} {b : β} :
     Tendsto f atTop (𝓝 b) ↔ ∀ ε, 0 < ε → ∃ N, ∀ n, N < n → ‖f n - b‖ < ε :=
   (atTop_basis_Ioi.tendsto_iff Metric.nhds_basis_ball).trans (by simp [dist_eq_norm])
 
-instance Int.instNormedCommRing : NormedCommRing ℤ where
-  __ := instCommRing
-  __ := instNormedAddCommGroup
-  norm_mul m n := by simp only [norm, Int.cast_mul, abs_mul, le_rfl]
-
-instance Int.instNormOneClass : NormOneClass ℤ :=
-  ⟨by simp [← Int.norm_cast_real]⟩
-
-instance Rat.instNormedField : NormedField ℚ where
-  __ := instField
-  __ := instNormedAddCommGroup
-  norm_mul' a b := by simp only [norm, Rat.cast_mul, abs_mul]
-
-instance Rat.instDenselyNormedField : DenselyNormedField ℚ where
-  lt_norm_lt r₁ r₂ h₀ hr :=
-    let ⟨q, h⟩ := exists_rat_btwn hr
-    ⟨q, by rwa [← Rat.norm_cast_real, Real.norm_eq_abs, abs_of_pos (h₀.trans_lt h.1)]⟩
-
 section RingHomIsometric
 
 variable {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
diff --git a/Mathlib/Analysis/Normed/Field/InfiniteSum.lean b/Mathlib/Analysis/Normed/Field/InfiniteSum.lean
index 4defadc79c84a..720c5451551ec 100644
--- a/Mathlib/Analysis/Normed/Field/InfiniteSum.lean
+++ b/Mathlib/Analysis/Normed/Field/InfiniteSum.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 -/
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.Normed.Group.InfiniteSum
 import Mathlib.Topology.Algebra.InfiniteSum.Real
 
@@ -21,9 +21,9 @@ We first establish results about arbitrary index types, `ι` and `ι'`, and then
 
 variable {R : Type*} {ι : Type*} {ι' : Type*} [NormedRing R]
 
-open scoped Classical
+open scoped Topology
 
-open Finset
+open Finset Filter
 
 /-! ### Arbitrary index types -/
 
@@ -43,13 +43,35 @@ theorem summable_mul_of_summable_norm [CompleteSpace R] {f : ι → R} {g : ι'
     Summable fun x : ι × ι' => f x.1 * g x.2 :=
   (hf.mul_norm hg).of_norm
 
-/-- Product of two infinites sums indexed by arbitrary types.
-    See also `tsum_mul_tsum` if `f` and `g` are *not* absolutely summable. -/
+theorem summable_mul_of_summable_norm' {f : ι → R} {g : ι' → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    Summable fun x : ι × ι' => f x.1 * g x.2 := by
+  classical
+  suffices HasSum (fun x : ι × ι' => f x.1 * g x.2) ((∑' i, f i) * (∑' j, g j)) from this.summable
+  let s : Finset ι × Finset ι' → Finset (ι × ι') := fun p ↦ p.1 ×ˢ p.2
+  apply hasSum_of_subseq_of_summable (hf.mul_norm hg) tendsto_finset_prod_atTop
+  rw [← prod_atTop_atTop_eq]
+  have := Tendsto.prod_map h'f.hasSum h'g.hasSum
+  rw [← nhds_prod_eq] at this
+  convert ((continuous_mul (M := R)).continuousAt
+      (x := (∑' (i : ι), f i, ∑' (j : ι'), g j))).tendsto.comp this with p
+  simp [s, sum_product, ← mul_sum, ← sum_mul]
+
+/-- Product of two infinite sums indexed by arbitrary types.
+    See also `tsum_mul_tsum` if `f` and `g` are *not* absolutely summable, and
+    `tsum_mul_tsum_of_summable_norm'` when the space is not complete. -/
 theorem tsum_mul_tsum_of_summable_norm [CompleteSpace R] {f : ι → R} {g : ι' → R}
     (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
     ((∑' x, f x) * ∑' y, g y) = ∑' z : ι × ι', f z.1 * g z.2 :=
   tsum_mul_tsum hf.of_norm hg.of_norm (summable_mul_of_summable_norm hf hg)
 
+theorem tsum_mul_tsum_of_summable_norm' {f : ι → R} {g : ι' → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    ((∑' x, f x) * ∑' y, g y) = ∑' z : ι × ι', f z.1 * g z.2 :=
+  tsum_mul_tsum h'f h'g (summable_mul_of_summable_norm' hf h'f hg h'g)
+
 /-! ### `ℕ`-indexed families (Cauchy product)
 
 We prove two versions of the Cauchy product formula. The first one is
@@ -76,28 +98,69 @@ theorem summable_norm_sum_mul_antidiagonal_of_summable_norm {f g : ℕ → R}
       norm_sum_le _ _
     _ ≤ ∑ kl ∈ antidiagonal n, ‖f kl.1‖ * ‖g kl.2‖ := by gcongr; apply norm_mul_le
 
+theorem summable_sum_mul_antidiagonal_of_summable_norm' {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    Summable fun n => ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 :=
+  summable_sum_mul_antidiagonal_of_summable_mul (summable_mul_of_summable_norm' hf h'f hg h'g)
+
 /-- The Cauchy product formula for the product of two infinite sums indexed by `ℕ`,
     expressed by summing on `Finset.antidiagonal`.
     See also `tsum_mul_tsum_eq_tsum_sum_antidiagonal` if `f` and `g` are
-    *not* absolutely summable. -/
+    *not* absolutely summable, and `tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm'`
+    when the space is not complete. -/
 theorem tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm [CompleteSpace R] {f g : ℕ → R}
     (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
     ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 :=
   tsum_mul_tsum_eq_tsum_sum_antidiagonal hf.of_norm hg.of_norm (summable_mul_of_summable_norm hf hg)
 
+theorem tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm' {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ kl ∈ antidiagonal n, f kl.1 * g kl.2 :=
+  tsum_mul_tsum_eq_tsum_sum_antidiagonal h'f h'g (summable_mul_of_summable_norm' hf h'f hg h'g)
+
 theorem summable_norm_sum_mul_range_of_summable_norm {f g : ℕ → R} (hf : Summable fun x => ‖f x‖)
     (hg : Summable fun x => ‖g x‖) : Summable fun n => ‖∑ k ∈ range (n + 1), f k * g (n - k)‖ := by
   simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
   exact summable_norm_sum_mul_antidiagonal_of_summable_norm hf hg
 
+theorem summable_sum_mul_range_of_summable_norm' {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    Summable fun n => ∑ k ∈ range (n + 1), f k * g (n - k) := by
+  simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
+  exact summable_sum_mul_antidiagonal_of_summable_norm' hf h'f hg h'g
+
 /-- The Cauchy product formula for the product of two infinite sums indexed by `ℕ`,
     expressed by summing on `Finset.range`.
     See also `tsum_mul_tsum_eq_tsum_sum_range` if `f` and `g` are
-    *not* absolutely summable. -/
+    *not* absolutely summable, and `tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm'` when the
+    space is not complete. -/
 theorem tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm [CompleteSpace R] {f g : ℕ → R}
     (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
     ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ k ∈ range (n + 1), f k * g (n - k) := by
   simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
   exact tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm hf hg
 
+theorem hasSum_sum_range_mul_of_summable_norm [CompleteSpace R] {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (hg : Summable fun x => ‖g x‖) :
+    HasSum (fun n ↦ ∑ k ∈ range (n + 1), f k * g (n - k)) ((∑' n, f n) * ∑' n, g n) := by
+  convert (summable_norm_sum_mul_range_of_summable_norm hf hg).of_norm.hasSum
+  exact tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm hf hg
+
+theorem tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm' {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    ((∑' n, f n) * ∑' n, g n) = ∑' n, ∑ k ∈ range (n + 1), f k * g (n - k) := by
+  simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
+  exact tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm' hf h'f hg h'g
+
+theorem hasSum_sum_range_mul_of_summable_norm' {f g : ℕ → R}
+    (hf : Summable fun x => ‖f x‖) (h'f : Summable f)
+    (hg : Summable fun x => ‖g x‖) (h'g : Summable g) :
+    HasSum (fun n ↦ ∑ k ∈ range (n + 1), f k * g (n - k)) ((∑' n, f n) * ∑' n, g n) := by
+  convert (summable_sum_mul_range_of_summable_norm' hf h'f hg h'g).hasSum
+  exact tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm' hf h'f hg h'g
+
 end Nat
diff --git a/Mathlib/Analysis/Normed/Field/Lemmas.lean b/Mathlib/Analysis/Normed/Field/Lemmas.lean
new file mode 100644
index 0000000000000..a2236fabdefec
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Field/Lemmas.lean
@@ -0,0 +1,422 @@
+/-
+Copyright (c) 2018 Patrick Massot. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Patrick Massot, Johannes Hölzl
+-/
+
+import Mathlib.Algebra.Group.AddChar
+import Mathlib.Algebra.Order.Ring.Finset
+import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Group.Bounded
+import Mathlib.Analysis.Normed.Group.Rat
+import Mathlib.Analysis.Normed.Group.Uniform
+import Mathlib.Topology.Instances.NNReal
+import Mathlib.Topology.MetricSpace.DilationEquiv
+
+/-!
+# Normed fields
+
+In this file we continue building the theory of (semi)normed rings and fields.
+
+Some useful results that relate the topology of the normed field to the discrete topology include:
+* `discreteTopology_or_nontriviallyNormedField`
+* `discreteTopology_of_bddAbove_range_norm`
+
+-/
+
+-- Guard against import creep.
+assert_not_exists RestrictScalars
+
+variable {α : Type*} {β : Type*} {ι : Type*}
+
+open Filter Bornology
+open scoped Topology NNReal Pointwise
+
+section NonUnitalSeminormedRing
+
+variable [NonUnitalSeminormedRing α]
+
+theorem Filter.Tendsto.zero_mul_isBoundedUnder_le {f g : ι → α} {l : Filter ι}
+    (hf : Tendsto f l (𝓝 0)) (hg : IsBoundedUnder (· ≤ ·) l ((‖·‖) ∘ g)) :
+    Tendsto (fun x => f x * g x) l (𝓝 0) :=
+  hf.op_zero_isBoundedUnder_le hg (· * ·) norm_mul_le
+
+theorem Filter.isBoundedUnder_le_mul_tendsto_zero {f g : ι → α} {l : Filter ι}
+    (hf : IsBoundedUnder (· ≤ ·) l (norm ∘ f)) (hg : Tendsto g l (𝓝 0)) :
+    Tendsto (fun x => f x * g x) l (𝓝 0) :=
+  hg.op_zero_isBoundedUnder_le hf (flip (· * ·)) fun x y =>
+    (norm_mul_le y x).trans_eq (mul_comm _ _)
+
+/-- Non-unital seminormed ring structure on the product of finitely many non-unital seminormed
+rings, using the sup norm. -/
+instance Pi.nonUnitalSeminormedRing {π : ι → Type*} [Fintype ι]
+    [∀ i, NonUnitalSeminormedRing (π i)] : NonUnitalSeminormedRing (∀ i, π i) :=
+  { Pi.seminormedAddCommGroup, Pi.nonUnitalRing with
+    norm_mul := fun x y =>
+      NNReal.coe_mono <|
+        calc
+          (Finset.univ.sup fun i => ‖x i * y i‖₊) ≤
+              Finset.univ.sup ((fun i => ‖x i‖₊) * fun i => ‖y i‖₊) :=
+            Finset.sup_mono_fun fun _ _ => norm_mul_le _ _
+          _ ≤ (Finset.univ.sup fun i => ‖x i‖₊) * Finset.univ.sup fun i => ‖y i‖₊ :=
+            Finset.sup_mul_le_mul_sup_of_nonneg _ (fun _ _ => zero_le _) fun _ _ => zero_le _
+           }
+
+end NonUnitalSeminormedRing
+
+section SeminormedRing
+
+variable [SeminormedRing α]
+
+/-- Seminormed ring structure on the product of finitely many seminormed rings,
+  using the sup norm. -/
+instance Pi.seminormedRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedRing (π i)] :
+    SeminormedRing (∀ i, π i) :=
+  { Pi.nonUnitalSeminormedRing, Pi.ring with }
+
+end SeminormedRing
+
+section NonUnitalNormedRing
+
+variable [NonUnitalNormedRing α]
+
+/-- Normed ring structure on the product of finitely many non-unital normed rings, using the sup
+norm. -/
+instance Pi.nonUnitalNormedRing {π : ι → Type*} [Fintype ι] [∀ i, NonUnitalNormedRing (π i)] :
+    NonUnitalNormedRing (∀ i, π i) :=
+  { Pi.nonUnitalSeminormedRing, Pi.normedAddCommGroup with }
+
+end NonUnitalNormedRing
+
+section NormedRing
+
+variable [NormedRing α]
+
+/-- Normed ring structure on the product of finitely many normed rings, using the sup norm. -/
+instance Pi.normedRing {π : ι → Type*} [Fintype ι] [∀ i, NormedRing (π i)] :
+    NormedRing (∀ i, π i) :=
+  { Pi.seminormedRing, Pi.normedAddCommGroup with }
+
+end NormedRing
+
+section NonUnitalSeminormedCommRing
+
+variable [NonUnitalSeminormedCommRing α]
+
+/-- Non-unital seminormed commutative ring structure on the product of finitely many non-unital
+seminormed commutative rings, using the sup norm. -/
+instance Pi.nonUnitalSeminormedCommRing {π : ι → Type*} [Fintype ι]
+    [∀ i, NonUnitalSeminormedCommRing (π i)] : NonUnitalSeminormedCommRing (∀ i, π i) :=
+  { Pi.nonUnitalSeminormedRing, Pi.nonUnitalCommRing with }
+
+end NonUnitalSeminormedCommRing
+
+section NonUnitalNormedCommRing
+
+variable [NonUnitalNormedCommRing α]
+
+/-- Normed commutative ring structure on the product of finitely many non-unital normed
+commutative rings, using the sup norm. -/
+instance Pi.nonUnitalNormedCommRing {π : ι → Type*} [Fintype ι]
+    [∀ i, NonUnitalNormedCommRing (π i)] : NonUnitalNormedCommRing (∀ i, π i) :=
+  { Pi.nonUnitalSeminormedCommRing, Pi.normedAddCommGroup with }
+
+end NonUnitalNormedCommRing
+
+section SeminormedCommRing
+
+variable [SeminormedCommRing α]
+
+/-- Seminormed commutative ring structure on the product of finitely many seminormed commutative
+rings, using the sup norm. -/
+instance Pi.seminormedCommRing {π : ι → Type*} [Fintype ι] [∀ i, SeminormedCommRing (π i)] :
+    SeminormedCommRing (∀ i, π i) :=
+  { Pi.nonUnitalSeminormedCommRing, Pi.ring with }
+
+end SeminormedCommRing
+
+section NormedCommRing
+
+variable [NormedCommRing α]
+
+/-- Normed commutative ring structure on the product of finitely many normed commutative rings,
+using the sup norm. -/
+instance Pi.normedCommutativeRing {π : ι → Type*} [Fintype ι] [∀ i, NormedCommRing (π i)] :
+    NormedCommRing (∀ i, π i) :=
+  { Pi.seminormedCommRing, Pi.normedAddCommGroup with }
+
+end NormedCommRing
+
+-- see Note [lower instance priority]
+instance (priority := 100) NonUnitalSeminormedRing.toContinuousMul [NonUnitalSeminormedRing α] :
+    ContinuousMul α :=
+  ⟨continuous_iff_continuousAt.2 fun x =>
+      tendsto_iff_norm_sub_tendsto_zero.2 <| by
+        have : ∀ e : α × α,
+            ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1‖ * ‖e.2 - x.2‖ + ‖e.1 - x.1‖ * ‖x.2‖ := by
+          intro e
+          calc
+            ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1 * (e.2 - x.2) + (e.1 - x.1) * x.2‖ := by
+              rw [_root_.mul_sub, _root_.sub_mul, sub_add_sub_cancel]
+            -- Porting note: `ENNReal.{mul_sub, sub_mul}` should be protected
+            _ ≤ ‖e.1‖ * ‖e.2 - x.2‖ + ‖e.1 - x.1‖ * ‖x.2‖ :=
+              norm_add_le_of_le (norm_mul_le _ _) (norm_mul_le _ _)
+        refine squeeze_zero (fun e => norm_nonneg _) this ?_
+        convert
+          ((continuous_fst.tendsto x).norm.mul
+                ((continuous_snd.tendsto x).sub tendsto_const_nhds).norm).add
+            (((continuous_fst.tendsto x).sub tendsto_const_nhds).norm.mul _)
+        -- Porting note: `show` used to select a goal to work on
+        rotate_right
+        · show Tendsto _ _ _
+          exact tendsto_const_nhds
+        · simp⟩
+
+-- see Note [lower instance priority]
+/-- A seminormed ring is a topological ring. -/
+instance (priority := 100) NonUnitalSeminormedRing.toTopologicalRing [NonUnitalSeminormedRing α] :
+    TopologicalRing α where
+
+namespace SeparationQuotient
+
+instance [NonUnitalSeminormedRing α] : NonUnitalNormedRing (SeparationQuotient α) where
+  __ : NonUnitalRing (SeparationQuotient α) := inferInstance
+  __ : NormedAddCommGroup (SeparationQuotient α) := inferInstance
+  norm_mul := Quotient.ind₂ norm_mul_le
+
+instance [NonUnitalSeminormedCommRing α] : NonUnitalNormedCommRing (SeparationQuotient α) where
+  __ : NonUnitalCommRing (SeparationQuotient α) := inferInstance
+  __ : NormedAddCommGroup (SeparationQuotient α) := inferInstance
+  norm_mul := Quotient.ind₂ norm_mul_le
+
+instance [SeminormedRing α] : NormedRing (SeparationQuotient α) where
+  __ : Ring (SeparationQuotient α) := inferInstance
+  __ : NormedAddCommGroup (SeparationQuotient α) := inferInstance
+  norm_mul := Quotient.ind₂ norm_mul_le
+
+instance [SeminormedCommRing α] : NormedCommRing (SeparationQuotient α) where
+  __ : CommRing (SeparationQuotient α) := inferInstance
+  __ : NormedAddCommGroup (SeparationQuotient α) := inferInstance
+  norm_mul := Quotient.ind₂ norm_mul_le
+
+instance [SeminormedAddCommGroup α] [One α] [NormOneClass α] :
+    NormOneClass (SeparationQuotient α) where
+  norm_one := norm_one (α := α)
+
+end SeparationQuotient
+
+section NormedDivisionRing
+
+variable [NormedDivisionRing α] {a : α}
+
+lemma antilipschitzWith_mul_left {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (a * ·) :=
+  AntilipschitzWith.of_le_mul_dist fun _ _ ↦ by simp [dist_eq_norm, ← _root_.mul_sub, ha]
+
+lemma antilipschitzWith_mul_right {a : α} (ha : a ≠ 0) : AntilipschitzWith (‖a‖₊⁻¹) (· * a) :=
+  AntilipschitzWith.of_le_mul_dist fun _ _ ↦ by
+    simp [dist_eq_norm, ← _root_.sub_mul, ← mul_comm (‖a‖), ha]
+
+/-- Multiplication by a nonzero element `a` on the left
+as a `DilationEquiv` of a normed division ring. -/
+@[simps!]
+def DilationEquiv.mulLeft (a : α) (ha : a ≠ 0) : α ≃ᵈ α where
+  toEquiv := Equiv.mulLeft₀ a ha
+  edist_eq' := ⟨‖a‖₊, nnnorm_ne_zero_iff.2 ha, fun x y ↦ by
+    simp [edist_nndist, nndist_eq_nnnorm, ← mul_sub]⟩
+
+/-- Multiplication by a nonzero element `a` on the right
+as a `DilationEquiv` of a normed division ring. -/
+@[simps!]
+def DilationEquiv.mulRight (a : α) (ha : a ≠ 0) : α ≃ᵈ α where
+  toEquiv := Equiv.mulRight₀ a ha
+  edist_eq' := ⟨‖a‖₊, nnnorm_ne_zero_iff.2 ha, fun x y ↦ by
+    simp [edist_nndist, nndist_eq_nnnorm, ← sub_mul, ← mul_comm (‖a‖₊)]⟩
+
+namespace Filter
+
+@[simp]
+lemma comap_mul_left_cobounded {a : α} (ha : a ≠ 0) :
+    comap (a * ·) (cobounded α) = cobounded α :=
+  Dilation.comap_cobounded (DilationEquiv.mulLeft a ha)
+
+@[simp]
+lemma map_mul_left_cobounded {a : α} (ha : a ≠ 0) :
+    map (a * ·) (cobounded α) = cobounded α :=
+  DilationEquiv.map_cobounded (DilationEquiv.mulLeft a ha)
+
+@[simp]
+lemma comap_mul_right_cobounded {a : α} (ha : a ≠ 0) :
+    comap (· * a) (cobounded α) = cobounded α :=
+  Dilation.comap_cobounded (DilationEquiv.mulRight a ha)
+
+@[simp]
+lemma map_mul_right_cobounded {a : α} (ha : a ≠ 0) :
+    map (· * a) (cobounded α) = cobounded α :=
+  DilationEquiv.map_cobounded (DilationEquiv.mulRight a ha)
+
+/-- Multiplication on the left by a nonzero element of a normed division ring tends to infinity at
+infinity. -/
+theorem tendsto_mul_left_cobounded {a : α} (ha : a ≠ 0) :
+    Tendsto (a * ·) (cobounded α) (cobounded α) :=
+  (map_mul_left_cobounded ha).le
+
+/-- Multiplication on the right by a nonzero element of a normed division ring tends to infinity at
+infinity. -/
+theorem tendsto_mul_right_cobounded {a : α} (ha : a ≠ 0) :
+    Tendsto (· * a) (cobounded α) (cobounded α) :=
+  (map_mul_right_cobounded ha).le
+
+@[simp]
+lemma inv_cobounded₀ : (cobounded α)⁻¹ = 𝓝[≠] 0 := by
+  rw [← comap_norm_atTop, ← Filter.comap_inv, ← comap_norm_nhdsWithin_Ioi_zero,
+    ← inv_atTop₀, ← Filter.comap_inv]
+  simp only [comap_comap, Function.comp_def, norm_inv]
+
+@[simp]
+lemma inv_nhdsWithin_ne_zero : (𝓝[≠] (0 : α))⁻¹ = cobounded α := by
+  rw [← inv_cobounded₀, inv_inv]
+
+lemma tendsto_inv₀_cobounded' : Tendsto Inv.inv (cobounded α) (𝓝[≠] 0) :=
+  inv_cobounded₀.le
+
+theorem tendsto_inv₀_cobounded : Tendsto Inv.inv (cobounded α) (𝓝 0) :=
+  tendsto_inv₀_cobounded'.mono_right inf_le_left
+
+lemma tendsto_inv₀_nhdsWithin_ne_zero : Tendsto Inv.inv (𝓝[≠] 0) (cobounded α) :=
+  inv_nhdsWithin_ne_zero.le
+
+end Filter
+
+-- see Note [lower instance priority]
+instance (priority := 100) NormedDivisionRing.to_hasContinuousInv₀ : HasContinuousInv₀ α := by
+  refine ⟨fun r r0 => tendsto_iff_norm_sub_tendsto_zero.2 ?_⟩
+  have r0' : 0 < ‖r‖ := norm_pos_iff.2 r0
+  rcases exists_between r0' with ⟨ε, ε0, εr⟩
+  have : ∀ᶠ e in 𝓝 r, ‖e⁻¹ - r⁻¹‖ ≤ ‖r - e‖ / ‖r‖ / ε := by
+    filter_upwards [(isOpen_lt continuous_const continuous_norm).eventually_mem εr] with e he
+    have e0 : e ≠ 0 := norm_pos_iff.1 (ε0.trans he)
+    calc
+      ‖e⁻¹ - r⁻¹‖ = ‖r‖⁻¹ * ‖r - e‖ * ‖e‖⁻¹ := by
+        rw [← norm_inv, ← norm_inv, ← norm_mul, ← norm_mul, _root_.mul_sub, _root_.sub_mul,
+          mul_assoc _ e, inv_mul_cancel₀ r0, mul_inv_cancel₀ e0, one_mul, mul_one]
+      -- Porting note: `ENNReal.{mul_sub, sub_mul}` should be `protected`
+      _ = ‖r - e‖ / ‖r‖ / ‖e‖ := by field_simp [mul_comm]
+      _ ≤ ‖r - e‖ / ‖r‖ / ε := by gcongr
+  refine squeeze_zero' (Eventually.of_forall fun _ => norm_nonneg _) this ?_
+  refine (((continuous_const.sub continuous_id).norm.div_const _).div_const _).tendsto' _ _ ?_
+  simp
+
+-- see Note [lower instance priority]
+/-- A normed division ring is a topological division ring. -/
+instance (priority := 100) NormedDivisionRing.to_topologicalDivisionRing :
+    TopologicalDivisionRing α where
+
+protected lemma IsOfFinOrder.norm_eq_one (ha : IsOfFinOrder a) : ‖a‖ = 1 :=
+  ((normHom : α →*₀ ℝ).toMonoidHom.isOfFinOrder ha).eq_one <| norm_nonneg _
+
+example [Monoid β] (φ : β →* α) {x : β} {k : ℕ+} (h : x ^ (k : ℕ) = 1) :
+    ‖φ x‖ = 1 := (φ.isOfFinOrder <| isOfFinOrder_iff_pow_eq_one.2 ⟨_, k.2, h⟩).norm_eq_one
+
+@[simp] lemma AddChar.norm_apply {G : Type*} [AddLeftCancelMonoid G] [Finite G] (ψ : AddChar G α)
+    (x : G) : ‖ψ x‖ = 1 := (ψ.toMonoidHom.isOfFinOrder <| isOfFinOrder_of_finite _).norm_eq_one
+
+end NormedDivisionRing
+
+namespace NormedField
+
+/-- A normed field is either nontrivially normed or has a discrete topology.
+In the discrete topology case, all the norms are 1, by `norm_eq_one_iff_ne_zero_of_discrete`.
+The nontrivially normed field instance is provided by a subtype with a proof that the
+forgetful inheritance to the existing `NormedField` instance is definitionally true.
+This allows one to have the new `NontriviallyNormedField` instance without data clashes. -/
+lemma discreteTopology_or_nontriviallyNormedField (𝕜 : Type*) [h : NormedField 𝕜] :
+    DiscreteTopology 𝕜 ∨ Nonempty ({h' : NontriviallyNormedField 𝕜 // h'.toNormedField = h}) := by
+  by_cases H : ∃ x : 𝕜, x ≠ 0 ∧ ‖x‖ ≠ 1
+  · exact Or.inr ⟨(⟨NontriviallyNormedField.ofNormNeOne H, rfl⟩)⟩
+  · simp_rw [discreteTopology_iff_isOpen_singleton_zero, Metric.isOpen_singleton_iff, dist_eq_norm,
+             sub_zero]
+    refine Or.inl ⟨1, zero_lt_one, ?_⟩
+    contrapose! H
+    refine H.imp ?_
+    -- contextual to reuse the `a ≠ 0` hypothesis in the proof of `a ≠ 0 ∧ ‖a‖ ≠ 1`
+    simp (config := {contextual := true}) [add_comm, ne_of_lt]
+
+lemma discreteTopology_of_bddAbove_range_norm {𝕜 : Type*} [NormedField 𝕜]
+    (h : BddAbove (Set.range fun k : 𝕜 ↦ ‖k‖)) :
+    DiscreteTopology 𝕜 := by
+  refine (NormedField.discreteTopology_or_nontriviallyNormedField _).resolve_right ?_
+  rintro ⟨_, rfl⟩
+  obtain ⟨x, h⟩ := h
+  obtain ⟨k, hk⟩ := NormedField.exists_lt_norm 𝕜 x
+  exact hk.not_le (h (Set.mem_range_self k))
+
+section Densely
+
+variable (α) [DenselyNormedField α]
+
+theorem denseRange_nnnorm : DenseRange (nnnorm : α → ℝ≥0) :=
+  dense_of_exists_between fun _ _ hr =>
+    let ⟨x, h⟩ := exists_lt_nnnorm_lt α hr
+    ⟨‖x‖₊, ⟨x, rfl⟩, h⟩
+
+end Densely
+
+end NormedField
+
+namespace NNReal
+
+lemma lipschitzWith_sub : LipschitzWith 2 (fun (p : ℝ≥0 × ℝ≥0) ↦ p.1 - p.2) := by
+  rw [← isometry_subtype_coe.lipschitzWith_iff]
+  have : Isometry (Prod.map ((↑) : ℝ≥0 → ℝ) ((↑) : ℝ≥0 → ℝ)) :=
+    isometry_subtype_coe.prod_map isometry_subtype_coe
+  convert (((LipschitzWith.prod_fst.comp this.lipschitz).sub
+    (LipschitzWith.prod_snd.comp this.lipschitz)).max_const 0)
+  norm_num
+
+end NNReal
+
+instance Int.instNormedCommRing : NormedCommRing ℤ where
+  __ := instCommRing
+  __ := instNormedAddCommGroup
+  norm_mul m n := by simp only [norm, Int.cast_mul, abs_mul, le_rfl]
+
+instance Int.instNormOneClass : NormOneClass ℤ :=
+  ⟨by simp [← Int.norm_cast_real]⟩
+
+instance Rat.instNormedField : NormedField ℚ where
+  __ := instField
+  __ := instNormedAddCommGroup
+  norm_mul' a b := by simp only [norm, Rat.cast_mul, abs_mul]
+
+instance Rat.instDenselyNormedField : DenselyNormedField ℚ where
+  lt_norm_lt r₁ r₂ h₀ hr :=
+    let ⟨q, h⟩ := exists_rat_btwn hr
+    ⟨q, by rwa [← Rat.norm_cast_real, Real.norm_eq_abs, abs_of_pos (h₀.trans_lt h.1)]⟩
+
+section Complete
+
+lemma NormedField.completeSpace_iff_isComplete_closedBall {K : Type*} [NormedField K] :
+    CompleteSpace K ↔ IsComplete (Metric.closedBall 0 1 : Set K) := by
+  constructor <;> intro h
+  · exact Metric.isClosed_ball.isComplete
+  rcases NormedField.discreteTopology_or_nontriviallyNormedField K with _|⟨_, rfl⟩
+  · rwa [completeSpace_iff_isComplete_univ,
+         ← NormedDivisionRing.discreteTopology_unit_closedBall_eq_univ]
+  refine Metric.complete_of_cauchySeq_tendsto fun u hu ↦ ?_
+  obtain ⟨k, hk⟩ := hu.norm_bddAbove
+  have kpos : 0 ≤ k := (_root_.norm_nonneg (u 0)).trans (hk (by simp))
+  obtain ⟨x, hx⟩ := NormedField.exists_lt_norm K k
+  have hu' : CauchySeq ((· / x) ∘ u) := (uniformContinuous_div_const' x).comp_cauchySeq hu
+  have hb : ∀ n, ((· / x) ∘ u) n ∈ Metric.closedBall 0 1 := by
+    intro
+    simp only [Function.comp_apply, Metric.mem_closedBall, dist_zero_right, norm_div]
+    rw [div_le_one (kpos.trans_lt hx)]
+    exact hx.le.trans' (hk (by simp))
+  obtain ⟨a, -, ha'⟩ := cauchySeq_tendsto_of_isComplete h hb hu'
+  refine ⟨a * x, (((continuous_mul_right x).tendsto a).comp ha').congr ?_⟩
+  have hx' : x ≠ 0 := by
+    contrapose! hx
+    simp [hx, kpos]
+  simp [div_mul_cancel₀ _ hx']
+
+end Complete
diff --git a/Mathlib/Analysis/Normed/Field/ProperSpace.lean b/Mathlib/Analysis/Normed/Field/ProperSpace.lean
new file mode 100644
index 0000000000000..58a51d47a39f0
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Field/ProperSpace.lean
@@ -0,0 +1,48 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+
+import Mathlib.Analysis.Normed.Field.Lemmas
+import Mathlib.Analysis.SpecificLimits.Basic
+import Mathlib.Topology.MetricSpace.ProperSpace
+
+/-!
+# Proper nontrivally normed fields
+
+Nontrivially normed fields are `ProperSpaces` when they are `WeaklyLocallyCompact`.
+
+## Main results
+
+* `ProperSpace.of_nontriviallyNormedField_of_weaklyLocallyCompactSpace`
+
+## Implementation details
+
+This is a special case of `ProperSpace.of_locallyCompactSpace` from
+`Mathlib.Analysis.Normed.Module.FiniteDimension`, specialized to be on the field itself
+with a proof that requires fewer imports.
+-/
+
+assert_not_exists FiniteDimensional
+
+open Metric Filter
+
+/-- A weakly locally compact normed field is proper.
+This is a specialization of `ProperSpace.of_locallyCompactSpace`
+which holds for `NormedSpace`s but requires more imports. -/
+lemma ProperSpace.of_nontriviallyNormedField_of_weaklyLocallyCompactSpace
+    (𝕜 : Type*) [NontriviallyNormedField 𝕜] [WeaklyLocallyCompactSpace 𝕜] :
+    ProperSpace 𝕜 := by
+  rcases exists_isCompact_closedBall (0 : 𝕜) with ⟨r, rpos, hr⟩
+  rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
+  have hC n : IsCompact (closedBall (0 : 𝕜) (‖c‖^n * r)) := by
+    have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_lt] at hc
+    convert hr.smul (c ^ n)
+    ext
+    simp only [mem_closedBall, dist_zero_right, Set.mem_smul_set_iff_inv_smul_mem₀ this,
+      smul_eq_mul, norm_mul, norm_inv, norm_pow,
+      inv_mul_le_iff₀ (by simpa only [norm_pow] using norm_pos_iff.mpr this)]
+  have hTop : Tendsto (fun n ↦ ‖c‖^n * r) atTop atTop :=
+    Tendsto.atTop_mul_const rpos (tendsto_pow_atTop_atTop_of_one_lt hc)
+  exact .of_seq_closedBall hTop (Eventually.of_forall hC)
diff --git a/Mathlib/Analysis/Normed/Field/UnitBall.lean b/Mathlib/Analysis/Normed/Field/UnitBall.lean
index 8652d79164427..146e37b0e5af2 100644
--- a/Mathlib/Analysis/Normed/Field/UnitBall.lean
+++ b/Mathlib/Analysis/Normed/Field/UnitBall.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Heather Macbeth
 -/
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.Normed.Group.BallSphere
 
 /-!
@@ -51,7 +51,7 @@ def Subsemigroup.unitClosedBall (𝕜 : Type*) [NonUnitalSeminormedRing 𝕜] :
   carrier := closedBall 0 1
   mul_mem' hx hy := by
     rw [mem_closedBall_zero_iff] at *
-    exact (norm_mul_le _ _).trans (mul_le_one hx (norm_nonneg _) hy)
+    exact (norm_mul_le _ _).trans (mul_le_one₀ hx (norm_nonneg _) hy)
 
 instance Metric.unitClosedBall.semigroup [NonUnitalSeminormedRing 𝕜] :
     Semigroup (closedBall (0 : 𝕜) 1) :=
diff --git a/Mathlib/Analysis/Normed/Group/Basic.lean b/Mathlib/Analysis/Normed/Group/Basic.lean
index f1f7f33fd7d48..04e1413e4bea8 100644
--- a/Mathlib/Analysis/Normed/Group/Basic.lean
+++ b/Mathlib/Analysis/Normed/Group/Basic.lean
@@ -770,6 +770,16 @@ theorem continuous_norm' : Continuous fun a : E => ‖a‖ := by
 theorem continuous_nnnorm' : Continuous fun a : E => ‖a‖₊ :=
   continuous_norm'.subtype_mk _
 
+set_option linter.docPrime false in
+@[to_additive Inseparable.norm_eq_norm]
+theorem Inseparable.norm_eq_norm' {u v : E} (h : Inseparable u v) : ‖u‖ = ‖v‖ :=
+  h.map continuous_norm' |>.eq
+
+set_option linter.docPrime false in
+@[to_additive Inseparable.nnnorm_eq_nnnorm]
+theorem Inseparable.nnnorm_eq_nnnorm' {u v : E} (h : Inseparable u v) : ‖u‖₊ = ‖v‖₊ :=
+  h.map continuous_nnnorm' |>.eq
+
 @[to_additive]
 theorem mem_closure_one_iff_norm {x : E} : x ∈ closure ({1} : Set E) ↔ ‖x‖ = 0 := by
   rw [← closedBall_zero', mem_closedBall_one_iff, (norm_nonneg' x).le_iff_eq]
@@ -1129,7 +1139,7 @@ theorem nnnorm_prod_le (s : Finset ι) (f : ι → E) : ‖∏ a ∈ s, f a‖
 @[to_additive]
 theorem nnnorm_prod_le_of_le (s : Finset ι) {f : ι → E} {n : ι → ℝ≥0} (h : ∀ b ∈ s, ‖f b‖₊ ≤ n b) :
     ‖∏ b ∈ s, f b‖₊ ≤ ∑ b ∈ s, n b :=
-  (norm_prod_le_of_le s h).trans_eq NNReal.coe_sum.symm
+  (norm_prod_le_of_le s h).trans_eq (NNReal.coe_sum ..).symm
 
 namespace Real
 
@@ -1152,24 +1162,26 @@ theorem norm_of_nonpos (hr : r ≤ 0) : ‖r‖ = -r :=
 theorem le_norm_self (r : ℝ) : r ≤ ‖r‖ :=
   le_abs_self r
 
--- Porting note (#10618): `simp` can prove this
-theorem norm_natCast (n : ℕ) : ‖(n : ℝ)‖ = n :=
-  abs_of_nonneg n.cast_nonneg
-
-@[simp]
-theorem nnnorm_natCast (n : ℕ) : ‖(n : ℝ)‖₊ = n :=
-  NNReal.eq <| norm_natCast _
+@[simp 1100] lemma norm_natCast (n : ℕ) : ‖(n : ℝ)‖ = n := abs_of_nonneg n.cast_nonneg
+@[simp 1100] lemma nnnorm_natCast (n : ℕ) : ‖(n : ℝ)‖₊ = n := NNReal.eq <| norm_natCast _
 
 @[deprecated (since := "2024-04-05")] alias norm_coe_nat := norm_natCast
 @[deprecated (since := "2024-04-05")] alias nnnorm_coe_nat := nnnorm_natCast
 
--- Porting note (#10618): `simp` can prove this
-theorem norm_two : ‖(2 : ℝ)‖ = 2 :=
-  abs_of_pos zero_lt_two
+@[simp 1100] lemma norm_ofNat (n : ℕ) [n.AtLeastTwo] :
+    ‖(no_index (OfNat.ofNat n) : ℝ)‖ = OfNat.ofNat n := norm_natCast n
 
-@[simp]
-theorem nnnorm_two : ‖(2 : ℝ)‖₊ = 2 :=
-  NNReal.eq <| by simp
+@[simp 1100] lemma nnnorm_ofNat (n : ℕ) [n.AtLeastTwo] :
+    ‖(no_index (OfNat.ofNat n) : ℝ)‖₊ = OfNat.ofNat n := nnnorm_natCast n
+
+lemma norm_two : ‖(2 : ℝ)‖ = 2 := abs_of_pos zero_lt_two
+lemma nnnorm_two : ‖(2 : ℝ)‖₊ = 2 := NNReal.eq <| by simp
+
+@[simp 1100, norm_cast]
+lemma norm_nnratCast (q : ℚ≥0) : ‖(q : ℝ)‖ = q := norm_of_nonneg q.cast_nonneg
+
+@[simp 1100, norm_cast]
+lemma nnnorm_nnratCast (q : ℚ≥0) : ‖(q : ℝ)‖₊ = q := by simp [nnnorm, -norm_eq_abs]
 
 theorem nnnorm_of_nonneg (hr : 0 ≤ r) : ‖r‖₊ = ⟨r, hr⟩ :=
   NNReal.eq <| norm_of_nonneg hr
diff --git a/Mathlib/Analysis/Normed/Group/Bounded.lean b/Mathlib/Analysis/Normed/Group/Bounded.lean
index 9a4c7dac9e078..329d2bfa734ce 100644
--- a/Mathlib/Analysis/Normed/Group/Bounded.lean
+++ b/Mathlib/Analysis/Normed/Group/Bounded.lean
@@ -59,7 +59,7 @@ lemma tendsto_norm_cocompact_atTop' [ProperSpace E] : Tendsto norm (cocompact E)
 
 @[to_additive (attr := simp)]
 lemma Filter.inv_cobounded : (cobounded E)⁻¹ = cobounded E := by
-  simp only [← comap_norm_atTop', ← Filter.comap_inv, comap_comap, (· ∘ ·), norm_inv']
+  simp only [← comap_norm_atTop', ← Filter.comap_inv, comap_comap, Function.comp_def, norm_inv']
 
 /-- In a (semi)normed group, inversion `x ↦ x⁻¹` tends to infinity at infinity. -/
 @[to_additive "In a (semi)normed group, negation `x ↦ -x` tends to infinity at infinity."]
diff --git a/Mathlib/Analysis/Normed/Group/CocompactMap.lean b/Mathlib/Analysis/Normed/Group/CocompactMap.lean
index ff947f47393ef..40889eae41c45 100644
--- a/Mathlib/Analysis/Normed/Group/CocompactMap.lean
+++ b/Mathlib/Analysis/Normed/Group/CocompactMap.lean
@@ -5,7 +5,7 @@ Authors: Moritz Doll
 -/
 
 import Mathlib.Analysis.Normed.Group.Basic
-import Mathlib.Topology.ContinuousFunction.CocompactMap
+import Mathlib.Topology.ContinuousMap.CocompactMap
 import Mathlib.Topology.MetricSpace.Bounded
 
 /-!
diff --git a/Mathlib/Analysis/Normed/Group/Completeness.lean b/Mathlib/Analysis/Normed/Group/Completeness.lean
index 6390ce5cd3cc9..8fee862ad3d6d 100644
--- a/Mathlib/Analysis/Normed/Group/Completeness.lean
+++ b/Mathlib/Analysis/Normed/Group/Completeness.lean
@@ -51,7 +51,7 @@ section Normed
 variable {E : Type*} [NormedAddCommGroup E]
 
 /-- A normed additive group is complete if any absolutely convergent series converges in the
-space.  -/
+space. -/
 lemma NormedAddCommGroup.completeSpace_of_summable_imp_tendsto
     (h : ∀ u : ℕ → E,
       Summable (‖u ·‖) → ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a)) :
@@ -73,14 +73,14 @@ lemma NormedAddCommGroup.completeSpace_of_summable_imp_tendsto
   simpa only [sub_add_cancel] using h₁
 
 /-- In a complete normed additive group, every absolutely convergent series converges in the
-space.  -/
+space. -/
 lemma NormedAddCommGroup.summable_imp_tendsto_of_complete [CompleteSpace E] (u : ℕ → E)
     (hu : Summable (‖u ·‖)) : ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a) := by
   refine cauchySeq_tendsto_of_complete <| cauchySeq_of_summable_dist ?_
   simp [dist_eq_norm, sum_range_succ, hu]
 
 /-- In a normed additive group, every absolutely convergent series converges in the
-space iff the space is complete.  -/
+space iff the space is complete. -/
 lemma NormedAddCommGroup.summable_imp_tendsto_iff_completeSpace :
     (∀ u : ℕ → E, Summable (‖u ·‖) → ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a))
      ↔ CompleteSpace E :=
diff --git a/Mathlib/Analysis/Normed/Group/Constructions.lean b/Mathlib/Analysis/Normed/Group/Constructions.lean
index a43e9e963c9ee..c9ee9cc5d178f 100644
--- a/Mathlib/Analysis/Normed/Group/Constructions.lean
+++ b/Mathlib/Analysis/Normed/Group/Constructions.lean
@@ -362,7 +362,7 @@ lemma Pi.sum_norm_apply_le_norm' : ∑ i, ‖f i‖ ≤ Fintype.card ι • ‖f
 @[to_additive Pi.sum_nnnorm_apply_le_nnnorm "The $L^1$ norm is less than the $L^\\infty$ norm
 scaled by the cardinality."]
 lemma Pi.sum_nnnorm_apply_le_nnnorm' : ∑ i, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊ :=
-  NNReal.coe_sum.trans_le <| Pi.sum_norm_apply_le_norm' _
+  (NNReal.coe_sum ..).trans_le <| Pi.sum_norm_apply_le_norm' _
 
 end SeminormedGroup
 
diff --git a/Mathlib/Analysis/Normed/Group/Hom.lean b/Mathlib/Analysis/Normed/Group/Hom.lean
index 0b30d2fdc3c07..36439765ebe21 100644
--- a/Mathlib/Analysis/Normed/Group/Hom.lean
+++ b/Mathlib/Analysis/Normed/Group/Hom.lean
@@ -82,7 +82,7 @@ def ofLipschitz (f : V₁ →+ V₂) {K : ℝ≥0} (h : LipschitzWith K f) : Nor
 
 instance funLike : FunLike (NormedAddGroupHom V₁ V₂) V₁ V₂ where
   coe := toFun
-  coe_injective' := fun f g h => by cases f; cases g; congr
+  coe_injective' f g h := by cases f; cases g; congr
 
 -- Porting note: moved this declaration up so we could get a `FunLike` instance sooner.
 instance toAddMonoidHomClass : AddMonoidHomClass (NormedAddGroupHom V₁ V₂) V₁ V₂ where
@@ -231,7 +231,7 @@ protected theorem continuous (f : NormedAddGroupHom V₁ V₂) : Continuous f :=
   f.uniformContinuous.continuous
 
 theorem ratio_le_opNorm (x : V₁) : ‖f x‖ / ‖x‖ ≤ ‖f‖ :=
-  div_le_of_nonneg_of_le_mul (norm_nonneg _) f.opNorm_nonneg (le_opNorm _ _)
+  div_le_of_le_mul₀ (norm_nonneg _) f.opNorm_nonneg (le_opNorm _ _)
 
 /-- If one controls the norm of every `f x`, then one controls the norm of `f`. -/
 theorem opNorm_le_bound {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ x, ‖f x‖ ≤ M * ‖x‖) : ‖f‖ ≤ M :=
@@ -532,7 +532,7 @@ instance toNormedAddCommGroup {V₁ V₂ : Type*} [NormedAddCommGroup V₁] [Nor
       add_le' := opNorm_add_le
       eq_zero_of_map_eq_zero' := fun _f => opNorm_zero_iff.1 }
 
-/-- Coercion of a `NormedAddGroupHom` is an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`.  -/
+/-- Coercion of a `NormedAddGroupHom` is an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`. -/
 @[simps]
 def coeAddHom : NormedAddGroupHom V₁ V₂ →+ V₁ → V₂ where
   toFun := DFunLike.coe
@@ -656,7 +656,7 @@ theorem mem_ker (v : V₁) : v ∈ f.ker ↔ f v = 0 := by
 @[simps]
 def ker.lift (h : g.comp f = 0) : NormedAddGroupHom V₁ g.ker where
   toFun v := ⟨f v, by rw [g.mem_ker, ← comp_apply g f, h, zero_apply]⟩
-  map_add' v w := by simp only [map_add, AddSubmonoid.mk_add_mk]
+  map_add' v w := by simp only [map_add, AddMemClass.mk_add_mk]
   bound' := f.bound'
 
 @[simp]
diff --git a/Mathlib/Analysis/Normed/Group/HomCompletion.lean b/Mathlib/Analysis/Normed/Group/HomCompletion.lean
index ca7368753ff56..3ef2814acd733 100644
--- a/Mathlib/Analysis/Normed/Group/HomCompletion.lean
+++ b/Mathlib/Analysis/Normed/Group/HomCompletion.lean
@@ -132,7 +132,7 @@ theorem NormedAddCommGroup.norm_toCompl (x : G) : ‖toCompl x‖ = ‖x‖ :=
   Completion.norm_coe x
 
 theorem NormedAddCommGroup.denseRange_toCompl : DenseRange (toCompl : G → Completion G) :=
-  Completion.denseInducing_coe.dense
+  Completion.isDenseInducing_coe.dense
 
 @[simp]
 theorem NormedAddGroupHom.completion_toCompl (f : NormedAddGroupHom G H) :
@@ -158,12 +158,13 @@ theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : ℝ}
   rcases h.exists_pos with ⟨C', C'_pos, hC'⟩
   rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩
   obtain ⟨_, ⟨g : G, rfl⟩, hg : ‖hatg - g‖ < δ⟩ :=
-    SeminormedAddCommGroup.mem_closure_iff.mp (Completion.denseInducing_coe.dense hatg) δ δ_pos
+    SeminormedAddCommGroup.mem_closure_iff.mp (Completion.isDenseInducing_coe.dense hatg) δ δ_pos
   obtain ⟨g' : G, hgg' : f g' = f g, hfg : ‖g'‖ ≤ C' * ‖f g‖⟩ := hC' (f g) (mem_range_self _ g)
   have mem_ker : g - g' ∈ f.ker := by rw [f.mem_ker, map_sub, sub_eq_zero.mpr hgg'.symm]
   refine ⟨_, ⟨⟨g - g', mem_ker⟩, rfl⟩, ?_⟩
   have : ‖f g‖ ≤ ‖f‖ * δ := calc
-    ‖f g‖ ≤ ‖f‖ * ‖hatg - g‖ := by simpa [hatg_in] using f.completion.le_opNorm (hatg - g)
+    ‖f g‖ ≤ ‖f‖ * ‖hatg - g‖ := by
+      simpa [map_sub, hatg_in] using f.completion.le_opNorm (hatg - g)
     _ ≤ ‖f‖ * δ := by gcongr
   calc ‖hatg - ↑(g - g')‖ = ‖hatg - g + g'‖ := by rw [Completion.coe_sub, sub_add]
     _ ≤ ‖hatg - g‖ + ‖(g' : Completion G)‖ := norm_add_le _ _
diff --git a/Mathlib/Analysis/Normed/Group/Quotient.lean b/Mathlib/Analysis/Normed/Group/Quotient.lean
index 7406b8e2cf248..9acc022a5fcfb 100644
--- a/Mathlib/Analysis/Normed/Group/Quotient.lean
+++ b/Mathlib/Analysis/Normed/Group/Quotient.lean
@@ -156,7 +156,7 @@ theorem quotient_norm_mk_eq (S : AddSubgroup M) (m : M) :
 
 /-- The quotient norm is nonnegative. -/
 theorem quotient_norm_nonneg (S : AddSubgroup M) (x : M ⧸ S) : 0 ≤ ‖x‖ :=
-  Real.sInf_nonneg _ <| forall_mem_image.2 fun _ _ ↦ norm_nonneg _
+  Real.sInf_nonneg <| forall_mem_image.2 fun _ _ ↦ norm_nonneg _
 
 /-- The quotient norm is nonnegative. -/
 theorem norm_mk_nonneg (S : AddSubgroup M) (m : M) : 0 ≤ ‖mk' S m‖ :=
@@ -213,7 +213,7 @@ theorem quotient_nhd_basis (S : AddSubgroup M) :
     refine fun ε ↦ Set.ext <| forall_mk.2 fun x ↦ ?_
     rw [ball_zero_eq, mem_setOf_eq, norm_lt_iff, mem_image]
     exact exists_congr fun _ ↦ and_comm
-  rw [← mk_zero, nhds_eq, ← funext this]
+  rw [← QuotientAddGroup.mk_zero, nhds_eq, ← funext this]
   exact .map _ Metric.nhds_basis_ball
 
 /-- The seminormed group structure on the quotient by an additive subgroup. -/
@@ -282,9 +282,9 @@ theorem _root_.QuotientAddGroup.norm_lift_apply_le {S : AddSubgroup M} (f : Norm
     rcases mk_surjective x with ⟨x, rfl⟩
     simpa [h] using le_opNorm f x
   | inr h =>
-    rw [← not_lt, ← _root_.lt_div_iff' h, norm_lt_iff]
+    rw [← not_lt, ← lt_div_iff₀' h, norm_lt_iff]
     rintro ⟨x, rfl, hx⟩
-    exact ((lt_div_iff' h).1 hx).not_le (le_opNorm f x)
+    exact ((lt_div_iff₀' h).1 hx).not_le (le_opNorm f x)
 
 /-- The operator norm of the projection is `1` if the subspace is not dense. -/
 theorem norm_normedMk (S : AddSubgroup M) (h : (S.topologicalClosure : Set M) ≠ univ) :
diff --git a/Mathlib/Analysis/Normed/Group/SemiNormedGrp.lean b/Mathlib/Analysis/Normed/Group/SemiNormedGrp.lean
index 2e56d519dc3b2..f63a09356056d 100644
--- a/Mathlib/Analysis/Normed/Group/SemiNormedGrp.lean
+++ b/Mathlib/Analysis/Normed/Group/SemiNormedGrp.lean
@@ -56,7 +56,7 @@ instance (M : SemiNormedGrp) : SeminormedAddCommGroup M :=
 -- Porting note (#10754): added instance
 instance funLike {V W : SemiNormedGrp} : FunLike (V ⟶ W) V W where
   coe := (forget SemiNormedGrp).map
-  coe_injective' := fun f g h => by cases f; cases g; congr
+  coe_injective' f g h := by cases f; cases g; congr
 
 instance toAddMonoidHomClass {V W : SemiNormedGrp} : AddMonoidHomClass (V ⟶ W) V W where
   map_add f := f.map_add'
diff --git a/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean b/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
index 1f32d6b23b009..e5aff9ade9489 100644
--- a/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
+++ b/Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Riccardo Brasca, Johan Commelin, Scott Morrison
+Authors: Riccardo Brasca, Johan Commelin, Kim Morrison
 -/
 import Mathlib.Analysis.Normed.Group.SemiNormedGrp
 import Mathlib.Analysis.Normed.Group.Quotient
diff --git a/Mathlib/Analysis/Normed/Group/Seminorm.lean b/Mathlib/Analysis/Normed/Group/Seminorm.lean
index 39f1d2b490a43..06d80361d16b3 100644
--- a/Mathlib/Analysis/Normed/Group/Seminorm.lean
+++ b/Mathlib/Analysis/Normed/Group/Seminorm.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: María Inés de Frutos-Fernández, Yaël Dillies
 -/
 import Mathlib.Data.NNReal.Basic
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 
 /-!
 # Group seminorms
@@ -48,7 +48,7 @@ open Set
 
 open NNReal
 
-variable {ι R R' E F G : Type*}
+variable {R R' E F G : Type*}
 
 /-- A seminorm on an additive group `G` is a function `f : G → ℝ` that preserves zero, is
 subadditive and such that `f (-x) = f x` for all `x`. -/
@@ -325,7 +325,7 @@ end Group
 
 section CommGroup
 
-variable [CommGroup E] [CommGroup F] (p q : GroupSeminorm E) (x y : E)
+variable [CommGroup E] [CommGroup F] (p q : GroupSeminorm E) (x : E)
 
 @[to_additive]
 theorem comp_mul_le (f g : F →* E) : p.comp (f * g) ≤ p.comp f + p.comp g := fun _ =>
@@ -381,7 +381,7 @@ end GroupSeminorm
 see that `SMul R ℝ` should be fixed because `ℝ` is fixed. -/
 namespace AddGroupSeminorm
 
-variable [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ] (p : AddGroupSeminorm E)
+variable [AddGroup E] [SMul R ℝ] [SMul R ℝ≥0] [IsScalarTower R ℝ≥0 ℝ]
 
 instance toOne [DecidableEq E] : One (AddGroupSeminorm E) :=
   ⟨{  toFun := fun x => if x = 0 then 0 else 1
@@ -435,7 +435,7 @@ namespace NonarchAddGroupSeminorm
 
 section AddGroup
 
-variable [AddGroup E] [AddGroup F] [AddGroup G] {p q : NonarchAddGroupSeminorm E}
+variable [AddGroup E] {p q : NonarchAddGroupSeminorm E}
 
 instance funLike : FunLike (NonarchAddGroupSeminorm E) E ℝ where
   coe f := f.toFun
@@ -477,7 +477,7 @@ theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q :=
 theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q :=
   Iff.rfl
 
-variable (p q) (f : F →+ E)
+variable (p q)
 
 instance : Zero (NonarchAddGroupSeminorm E) :=
   ⟨{  toFun := 0
@@ -522,7 +522,7 @@ end AddGroup
 
 section AddCommGroup
 
-variable [AddCommGroup E] [AddCommGroup F] (p q : NonarchAddGroupSeminorm E) (x y : E)
+variable [AddCommGroup E]
 
 theorem add_bddBelow_range_add {p q : NonarchAddGroupSeminorm E} {x : E} :
     BddBelow (range fun y => p y + q (x - y)) :=
@@ -653,7 +653,7 @@ namespace GroupNorm
 
 section Group
 
-variable [Group E] [Group F] [Group G] {p q : GroupNorm E}
+variable [Group E] {p q : GroupNorm E}
 
 @[to_additive]
 instance funLike : FunLike (GroupNorm E) E ℝ where
@@ -703,7 +703,7 @@ theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q :=
 theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q :=
   Iff.rfl
 
-variable (p q) (f : F →* E)
+variable (p q)
 
 @[to_additive]
 instance : Add (GroupNorm E) :=
@@ -787,7 +787,7 @@ namespace NonarchAddGroupNorm
 
 section AddGroup
 
-variable [AddGroup E] [AddGroup F] {p q : NonarchAddGroupNorm E}
+variable [AddGroup E] {p q : NonarchAddGroupNorm E}
 
 instance funLike : FunLike (NonarchAddGroupNorm E) E ℝ where
   coe f := f.toFun
@@ -829,7 +829,7 @@ theorem coe_le_coe : (p : E → ℝ) ≤ q ↔ p ≤ q :=
 theorem coe_lt_coe : (p : E → ℝ) < q ↔ p < q :=
   Iff.rfl
 
-variable (p q) (f : F →+ E)
+variable (p q)
 
 instance : Sup (NonarchAddGroupNorm E) :=
   ⟨fun p q =>
diff --git a/Mathlib/Analysis/Normed/Group/Ultra.lean b/Mathlib/Analysis/Normed/Group/Ultra.lean
new file mode 100644
index 0000000000000..8e83b3535e430
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Group/Ultra.lean
@@ -0,0 +1,240 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky, David Loeffler
+-/
+import Mathlib.Analysis.Normed.Group.Uniform
+import Mathlib.Topology.Algebra.Nonarchimedean.Basic
+import Mathlib.Topology.MetricSpace.Ultra.Basic
+
+/-!
+# Ultrametric norms
+
+This file contains results on the behavior of norms in ultrametric groups.
+
+## Main results
+
+* `IsUltrametricDist.isUltrametricDist_of_isNonarchimedean_norm`:
+  a normed additive group has an ultrametric iff the norm is nonarchimedean
+* `IsUltrametricDist.nonarchimedeanGroup` and its additive version: instance showing that a
+  commutative group with a nonarchimedean seminorm is a nonarchimedean topological group (i.e.
+  there is a neighbourhood basis of the identity consisting of open subgroups).
+
+## Implementation details
+
+Some results are proved first about `nnnorm : X → ℝ≥0` because the bottom element
+in `NNReal` is 0, so easier to make statements about maxima of empty sets.
+
+## Tags
+
+ultrametric, nonarchimedean
+-/
+open Metric NNReal
+
+namespace IsUltrametricDist
+
+section Group
+
+variable {S S' ι : Type*} [SeminormedGroup S] [SeminormedGroup S'] [IsUltrametricDist S]
+
+@[to_additive]
+lemma norm_mul_le_max (x y : S) :
+    ‖x * y‖ ≤ max ‖x‖ ‖y‖ := by
+  simpa only [le_max_iff, dist_eq_norm_div, div_inv_eq_mul, div_one, one_mul] using
+    dist_triangle_max x 1 y⁻¹
+
+@[to_additive]
+lemma isUltrametricDist_of_forall_norm_mul_le_max_norm
+    (h : ∀ x y : S', ‖x * y‖ ≤ max ‖x‖ ‖y‖) : IsUltrametricDist S' where
+  dist_triangle_max x y z := by
+    simpa only [dist_eq_norm_div, le_max_iff, div_mul_div_cancel] using h (x / y) (y / z)
+
+lemma isUltrametricDist_of_isNonarchimedean_norm {S' : Type*} [SeminormedAddGroup S']
+    (h : IsNonarchimedean (norm : S' → ℝ)) : IsUltrametricDist S' :=
+  isUltrametricDist_of_forall_norm_add_le_max_norm h
+
+@[to_additive]
+lemma nnnorm_mul_le_max (x y : S) :
+    ‖x * y‖₊ ≤ max ‖x‖₊ ‖y‖₊ :=
+  norm_mul_le_max _ _
+
+@[to_additive]
+lemma isUltrametricDist_of_forall_nnnorm_mul_le_max_nnnorm
+    (h : ∀ x y : S', ‖x * y‖₊ ≤ max ‖x‖₊ ‖y‖₊) : IsUltrametricDist S' :=
+  isUltrametricDist_of_forall_norm_mul_le_max_norm h
+
+lemma isUltrametricDist_of_isNonarchimedean_nnnorm {S' : Type*} [SeminormedAddGroup S']
+    (h : IsNonarchimedean ((↑) ∘ (nnnorm : S' → ℝ≥0))) : IsUltrametricDist S' :=
+  isUltrametricDist_of_forall_nnnorm_add_le_max_nnnorm h
+
+/-- All triangles are isosceles in an ultrametric normed group. -/
+@[to_additive "All triangles are isosceles in an ultrametric normed additive group."]
+lemma norm_mul_eq_max_of_norm_ne_norm
+    {x y : S} (h : ‖x‖ ≠ ‖y‖) : ‖x * y‖ = max ‖x‖ ‖y‖ := by
+  rw [← div_inv_eq_mul, ← dist_eq_norm_div, dist_eq_max_of_dist_ne_dist _ 1 _ (by simp [h])]
+  simp only [dist_one_right, dist_one_left, norm_inv']
+
+@[to_additive]
+lemma norm_eq_of_mul_norm_lt_max {x y : S} (h : ‖x * y‖ < max ‖x‖ ‖y‖) :
+    ‖x‖ = ‖y‖ :=
+  not_ne_iff.mp (h.ne ∘ norm_mul_eq_max_of_norm_ne_norm)
+
+/-- All triangles are isosceles in an ultrametric normed group. -/
+@[to_additive "All triangles are isosceles in an ultrametric normed additive group."]
+lemma nnnorm_mul_eq_max_of_nnnorm_ne_nnnorm
+    {x y : S} (h : ‖x‖₊ ≠ ‖y‖₊) : ‖x * y‖₊ = max ‖x‖₊ ‖y‖₊ := by
+  simpa only [← NNReal.coe_inj, NNReal.coe_max] using
+    norm_mul_eq_max_of_norm_ne_norm (NNReal.coe_injective.ne h)
+
+@[to_additive]
+lemma nnnorm_eq_of_mul_nnnorm_lt_max {x y : S} (h : ‖x * y‖₊ < max ‖x‖₊ ‖y‖₊) :
+    ‖x‖₊ = ‖y‖₊ :=
+  not_ne_iff.mp (h.ne ∘ nnnorm_mul_eq_max_of_nnnorm_ne_nnnorm)
+
+/-- All triangles are isosceles in an ultrametric normed group. -/
+@[to_additive "All triangles are isosceles in an ultrametric normed additive group."]
+lemma norm_div_eq_max_of_norm_div_ne_norm_div (x y z : S) (h : ‖x / y‖ ≠ ‖y / z‖) :
+    ‖x / z‖ = max ‖x / y‖ ‖y / z‖ := by
+  simpa only [div_mul_div_cancel] using norm_mul_eq_max_of_norm_ne_norm h
+
+/-- All triangles are isosceles in an ultrametric normed group. -/
+@[to_additive "All triangles are isosceles in an ultrametric normed additive group."]
+lemma nnnorm_div_eq_max_of_nnnorm_div_ne_nnnorm_div (x y z : S) (h : ‖x / y‖₊ ≠ ‖y / z‖₊) :
+    ‖x / z‖₊ = max ‖x / y‖₊ ‖y / z‖₊ := by
+  simpa only [← NNReal.coe_inj, NNReal.coe_max] using
+    norm_div_eq_max_of_norm_div_ne_norm_div _ _ _ (NNReal.coe_injective.ne h)
+
+@[to_additive]
+lemma nnnorm_pow_le (x : S) (n : ℕ) :
+    ‖x ^ n‖₊ ≤ ‖x‖₊ := by
+  induction n with
+  | zero => simp
+  | succ n hn => simpa [pow_add, hn] using nnnorm_mul_le_max (x ^ n) x
+
+@[to_additive]
+lemma norm_pow_le (x : S) (n : ℕ) :
+    ‖x ^ n‖ ≤ ‖x‖ :=
+  nnnorm_pow_le x n
+
+@[to_additive]
+lemma nnnorm_zpow_le (x : S) (z : ℤ) :
+    ‖x ^ z‖₊ ≤ ‖x‖₊ := by
+  cases z <;>
+  simpa using nnnorm_pow_le _ _
+
+@[to_additive]
+lemma norm_zpow_le (x : S) (z : ℤ) :
+    ‖x ^ z‖ ≤ ‖x‖ :=
+  nnnorm_zpow_le x z
+
+section nonarch
+
+variable (S)
+/--
+In a group with an ultrametric norm, open balls around 1 of positive radius are open subgroups.
+-/
+@[to_additive "In an additive group with an ultrametric norm, open balls around 0 of
+positive radius are open subgroups."]
+def ball_openSubgroup {r : ℝ} (hr : 0 < r) : OpenSubgroup S where
+  carrier := Metric.ball (1 : S) r
+  mul_mem' {x} {y} hx hy := by
+    simp only [Metric.mem_ball, dist_eq_norm_div, div_one] at hx hy ⊢
+    exact (norm_mul_le_max x y).trans_lt (max_lt hx hy)
+  one_mem' := Metric.mem_ball_self hr
+  inv_mem' := by simp only [Metric.mem_ball, dist_one_right, norm_inv', imp_self, implies_true]
+  isOpen' := Metric.isOpen_ball
+
+/--
+In a group with an ultrametric norm, closed balls around 1 of positive radius are open subgroups.
+-/
+@[to_additive "In an additive group with an ultrametric norm, closed balls around 0 of positive
+radius are open subgroups."]
+def closedBall_openSubgroup {r : ℝ} (hr : 0 < r) : OpenSubgroup S where
+  carrier := Metric.closedBall (1 : S) r
+  mul_mem' {x} {y} hx hy := by
+    simp only [Metric.mem_closedBall, dist_eq_norm_div, div_one] at hx hy ⊢
+    exact (norm_mul_le_max x y).trans (max_le hx hy)
+  one_mem' := Metric.mem_closedBall_self hr.le
+  inv_mem' := by simp only [mem_closedBall, dist_one_right, norm_inv', imp_self, implies_true]
+  isOpen' := IsUltrametricDist.isOpen_closedBall _ hr.ne'
+
+end nonarch
+
+end Group
+
+section CommGroup
+
+variable {M ι : Type*} [SeminormedCommGroup M] [IsUltrametricDist M]
+
+/-- A commutative group with an ultrametric group seminorm is nonarchimedean (as a topological
+group, i.e. every neighborhood of 1 contains an open subgroup). -/
+@[to_additive "A commutative additive group with an ultrametric group seminorm is nonarchimedean
+(as a topological group, i.e. every neighborhood of 0 contains an open subgroup)."]
+instance nonarchimedeanGroup : NonarchimedeanGroup M where
+  is_nonarchimedean := by simpa only [Metric.mem_nhds_iff]
+    using fun U ⟨ε, hεp, hεU⟩ ↦ ⟨ball_openSubgroup M hεp, hεU⟩
+
+/-- Nonarchimedean norm of a product is less than or equal the norm of any term in the product.
+This version is phrased using `Finset.sup'` and `Finset.Nonempty` due to `Finset.sup`
+operating over an `OrderBot`, which `ℝ` is not.
+-/
+@[to_additive "Nonarchimedean norm of a sum is less than or equal the norm of any term in the sum.
+This version is phrased using `Finset.sup'` and `Finset.Nonempty` due to `Finset.sup`
+operating over an `OrderBot`, which `ℝ` is not. "]
+lemma _root_.Finset.Nonempty.norm_prod_le_sup'_norm {s : Finset ι} (hs : s.Nonempty) (f : ι → M) :
+    ‖∏ i ∈ s, f i‖ ≤ s.sup' hs (‖f ·‖) := by
+  simp only [Finset.le_sup'_iff]
+  induction hs using Finset.Nonempty.cons_induction with
+  | singleton j => simp only [Finset.mem_singleton, Finset.prod_singleton, exists_eq_left, le_refl]
+  | cons j t hj _ IH =>
+      simp only [Finset.prod_cons, Finset.mem_cons, exists_eq_or_imp]
+      refine (le_total ‖∏ i ∈ t, f i‖ ‖f j‖).imp ?_ ?_ <;> intro h
+      · exact (norm_mul_le_max _ _).trans (max_eq_left h).le
+      · exact ⟨_, IH.choose_spec.left, (norm_mul_le_max _ _).trans <|
+          ((max_eq_right h).le.trans IH.choose_spec.right)⟩
+
+/-- Nonarchimedean norm of a product is less than or equal to the largest norm of a term in the
+product. -/
+@[to_additive "Nonarchimedean norm of a sum is less than or equal to the largest norm of a term in
+the sum."]
+lemma _root_.Finset.nnnorm_prod_le_sup_nnnorm (s : Finset ι) (f : ι → M) :
+    ‖∏ i ∈ s, f i‖₊ ≤ s.sup (‖f ·‖₊) := by
+  rcases s.eq_empty_or_nonempty with rfl|hs
+  · simp only [Finset.prod_empty, nnnorm_one', Finset.sup_empty, bot_eq_zero', le_refl]
+  · simpa only [← Finset.sup'_eq_sup hs, Finset.le_sup'_iff, coe_le_coe, coe_nnnorm']
+      using hs.norm_prod_le_sup'_norm f
+
+/--
+Generalised ultrametric triangle inequality for finite products in commutative groups with
+an ultrametric norm.
+-/
+@[to_additive "Generalised ultrametric triangle inequality for finite sums in additive commutative
+groups with an ultrametric norm."]
+lemma nnnorm_prod_le_of_forall_le {s : Finset ι} {f : ι → M} {C : ℝ≥0}
+    (hC : ∀ i ∈ s, ‖f i‖₊ ≤ C) : ‖∏ i ∈ s, f i‖₊ ≤ C :=
+  (s.nnnorm_prod_le_sup_nnnorm f).trans <| Finset.sup_le hC
+
+/--
+Generalised ultrametric triangle inequality for nonempty finite products in commutative groups with
+an ultrametric norm.
+-/
+@[to_additive "Generalised ultrametric triangle inequality for nonempty finite sums in additive
+commutative groups with an ultrametric norm."]
+lemma norm_prod_le_of_forall_le_of_nonempty {s : Finset ι} (hs : s.Nonempty) {f : ι → M} {C : ℝ}
+    (hC : ∀ i ∈ s, ‖f i‖ ≤ C) : ‖∏ i ∈ s, f i‖ ≤ C :=
+  (hs.norm_prod_le_sup'_norm f).trans (Finset.sup'_le hs _ hC)
+
+/--
+Generalised ultrametric triangle inequality for finite products in commutative groups with
+an ultrametric norm.
+-/
+@[to_additive "Generalised ultrametric triangle inequality for finite sums in additive commutative
+groups with an ultrametric norm."]
+lemma norm_prod_le_of_forall_le_of_nonneg {s : Finset ι} {f : ι → M} {C : ℝ}
+    (h_nonneg : 0 ≤ C) (hC : ∀ i ∈ s, ‖f i‖ ≤ C) : ‖∏ i ∈ s, f i‖ ≤ C := by
+  lift C to NNReal using h_nonneg
+  exact nnnorm_prod_le_of_forall_le hC
+
+end CommGroup
+
+end IsUltrametricDist
diff --git a/Mathlib/Analysis/Normed/Group/Uniform.lean b/Mathlib/Analysis/Normed/Group/Uniform.lean
index 42621d3976ca7..f8e72a4535b45 100644
--- a/Mathlib/Analysis/Normed/Group/Uniform.lean
+++ b/Mathlib/Analysis/Normed/Group/Uniform.lean
@@ -365,6 +365,29 @@ instance (priority := 100) SeminormedCommGroup.to_uniformGroup : UniformGroup E
 instance (priority := 100) SeminormedCommGroup.toTopologicalGroup : TopologicalGroup E :=
   inferInstance
 
+/-! ### SeparationQuotient -/
+
+namespace SeparationQuotient
+
+@[to_additive instNorm]
+instance instMulNorm : Norm (SeparationQuotient E) where
+  norm := lift Norm.norm fun _ _ h => h.norm_eq_norm'
+
+set_option linter.docPrime false in
+@[to_additive (attr := simp) norm_mk]
+theorem norm_mk' (p : E) : ‖mk p‖ = ‖p‖ := rfl
+
+@[to_additive]
+instance : NormedCommGroup (SeparationQuotient E) where
+  __ : CommGroup (SeparationQuotient E) := instCommGroup
+  dist_eq := Quotient.ind₂ dist_eq_norm_div
+
+set_option linter.docPrime false in
+@[to_additive (attr := simp) nnnorm_mk]
+theorem nnnorm_mk' (p : E) : ‖mk p‖₊ = ‖p‖₊ := rfl
+
+end SeparationQuotient
+
 @[to_additive]
 theorem cauchySeq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n ≥ N, u n = v n)
     (hv : CauchySeq fun n => ∏ k ∈ range (n + 1), v k) :
@@ -379,4 +402,17 @@ theorem cauchySeq_prod_of_eventually_eq {u v : ℕ → E} {N : ℕ} (huv : ∀ n
   intro m hm
   simp [huv m (le_of_lt hm)]
 
+@[to_additive CauchySeq.norm_bddAbove]
+lemma CauchySeq.mul_norm_bddAbove {G : Type*} [SeminormedGroup G] {u : ℕ → G}
+    (hu : CauchySeq u) : BddAbove (Set.range (fun n ↦ ‖u n‖)) := by
+  obtain ⟨C, -, hC⟩ := cauchySeq_bdd hu
+  simp_rw [SeminormedGroup.dist_eq] at hC
+  have : ∀ n, ‖u n‖ ≤ C + ‖u 0‖ := by
+    intro n
+    rw [add_comm]
+    refine (norm_le_norm_add_norm_div' (u n) (u 0)).trans ?_
+    simp [(hC _ _).le]
+  rw [bddAbove_def]
+  exact ⟨C + ‖u 0‖, by simpa using this⟩
+
 end SeminormedCommGroup
diff --git a/Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean b/Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean
index a2cabd37f31d1..825bb7c7941a6 100644
--- a/Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean
+++ b/Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Moritz Doll
 -/
 
-import Mathlib.Topology.ContinuousFunction.ZeroAtInfty
+import Mathlib.Topology.ContinuousMap.ZeroAtInfty
 
 /-!
 # ZeroAtInftyContinuousMapClass in normed additive groups
diff --git a/Mathlib/Analysis/Normed/Lp/LpEquiv.lean b/Mathlib/Analysis/Normed/Lp/LpEquiv.lean
index d43a285b86665..144ad83e536db 100644
--- a/Mathlib/Analysis/Normed/Lp/LpEquiv.lean
+++ b/Mathlib/Analysis/Normed/Lp/LpEquiv.lean
@@ -5,7 +5,7 @@ Authors: Jireh Loreaux
 -/
 import Mathlib.Analysis.Normed.Lp.lpSpace
 import Mathlib.Analysis.Normed.Lp.PiLp
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 
 /-!
 # Equivalences among $L^p$ spaces
@@ -72,7 +72,7 @@ theorem coe_equiv_lpPiLp_symm (f : PiLp p E) : (Equiv.lpPiLp.symm f : ∀ i, E i
 def AddEquiv.lpPiLp : lp E p ≃+ PiLp p E :=
   { Equiv.lpPiLp with map_add' := fun _f _g ↦ rfl }
 
-theorem coe_addEquiv_lpPiLp  (f : lp E p) : AddEquiv.lpPiLp f = ⇑f :=
+theorem coe_addEquiv_lpPiLp (f : lp E p) : AddEquiv.lpPiLp f = ⇑f :=
   rfl
 
 theorem coe_addEquiv_lpPiLp_symm (f : PiLp p E) :
diff --git a/Mathlib/Analysis/Normed/Lp/PiLp.lean b/Mathlib/Analysis/Normed/Lp/PiLp.lean
index 45556b51f15c5..decc915289980 100644
--- a/Mathlib/Analysis/Normed/Lp/PiLp.lean
+++ b/Mathlib/Analysis/Normed/Lp/PiLp.lean
@@ -354,7 +354,7 @@ abbrev pseudoMetricAux : PseudoMetricSpace (PiLp p α) :=
             PseudoMetricSpace.edist_dist]
           -- Porting note: `le_iSup` needed some help
           exact le_iSup (fun k => edist (f k) (g k)) i
-        · refine ENNReal.toReal_le_of_le_ofReal (Real.sSup_nonneg _ ?_) (iSup_le fun i => ?_)
+        · refine ENNReal.toReal_le_of_le_ofReal (Real.sSup_nonneg ?_) (iSup_le fun i => ?_)
           · rintro - ⟨i, rfl⟩
             exact dist_nonneg
           · change PseudoMetricSpace.edist _ _ ≤ _
@@ -413,8 +413,8 @@ theorem antilipschitzWith_equiv_aux :
         rw [this, ENNReal.coe_rpow_of_nonneg _ nonneg]
 
 theorem aux_uniformity_eq : 𝓤 (PiLp p β) = 𝓤[Pi.uniformSpace _] := by
-  have A : UniformInducing (WithLp.equiv p (∀ i, β i)) :=
-    (antilipschitzWith_equiv_aux p β).uniformInducing
+  have A : IsUniformInducing (WithLp.equiv p (∀ i, β i)) :=
+    (antilipschitzWith_equiv_aux p β).isUniformInducing
       (lipschitzWith_equiv_aux p β).uniformContinuous
   have : (fun x : PiLp p β × PiLp p β => (WithLp.equiv p _ x.fst, WithLp.equiv p _ x.snd)) = id :=
     by ext i <;> rfl
@@ -640,7 +640,7 @@ variable (p 𝕜)
 variable (E : Type*) [SeminormedAddCommGroup E] [Module 𝕜 E]
 
 /-- An equivalence of finite domains induces a linearly isometric equivalence of finitely supported
-functions-/
+functions -/
 def _root_.LinearIsometryEquiv.piLpCongrLeft (e : ι ≃ ι') :
     (PiLp p fun _ : ι => E) ≃ₗᵢ[𝕜] PiLp p fun _ : ι' => E where
   toLinearEquiv := LinearEquiv.piCongrLeft' 𝕜 (fun _ : ι => E) e
@@ -936,7 +936,7 @@ nonrec theorem basis_toMatrix_basisFun_mul [Fintype ι]
       Matrix.of fun i j => b.repr ((WithLp.equiv _ _).symm (Aᵀ j)) i := by
   have := basis_toMatrix_basisFun_mul (b.map (WithLp.linearEquiv _ 𝕜 _)) A
   simp_rw [← PiLp.basisFun_map p, Basis.map_repr, LinearEquiv.trans_apply,
-    WithLp.linearEquiv_symm_apply, Basis.toMatrix_map, Function.comp, Basis.map_apply,
+    WithLp.linearEquiv_symm_apply, Basis.toMatrix_map, Function.comp_def, Basis.map_apply,
     LinearEquiv.symm_apply_apply] at this
   exact this
 
diff --git a/Mathlib/Analysis/Normed/Lp/ProdLp.lean b/Mathlib/Analysis/Normed/Lp/ProdLp.lean
index 6d23a952e2391..23d86ced09013 100644
--- a/Mathlib/Analysis/Normed/Lp/ProdLp.lean
+++ b/Mathlib/Analysis/Normed/Lp/ProdLp.lean
@@ -325,7 +325,7 @@ def prodPseudoEMetricAux [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
             (if · = 0 then edist f.fst g.fst else edist f.snd g.snd)
             (if · = 0 then edist g.fst h.fst else edist g.snd h.snd) hp
           simp only [Finset.mem_singleton, not_false_eq_true, Finset.sum_insert,
-            Finset.sum_singleton] at this
+            Finset.sum_singleton, reduceCtorEq] at this
           exact this
 
 attribute [local instance] WithLp.prodPseudoEMetricAux
@@ -420,12 +420,12 @@ theorem prod_antilipschitzWith_equiv_aux [PseudoEMetricSpace α] [PseudoEMetricS
         gcongr <;> simp [edist]
       _ = (2 ^ (1 / p.toReal) : ℝ≥0) * edist (WithLp.equiv p _ x) (WithLp.equiv p _ y) := by
         simp only [← two_mul, ENNReal.mul_rpow_of_nonneg _ _ nonneg, ← ENNReal.rpow_mul, cancel,
-          ENNReal.rpow_one, ← ENNReal.coe_rpow_of_nonneg _ nonneg, coe_ofNat]
+          ENNReal.rpow_one, ENNReal.coe_rpow_of_nonneg _ nonneg, coe_ofNat]
 
 theorem prod_aux_uniformity_eq [PseudoEMetricSpace α] [PseudoEMetricSpace β] :
     𝓤 (WithLp p (α × β)) = 𝓤[instUniformSpaceProd] := by
-  have A : UniformInducing (WithLp.equiv p (α × β)) :=
-    (prod_antilipschitzWith_equiv_aux p α β).uniformInducing
+  have A : IsUniformInducing (WithLp.equiv p (α × β)) :=
+    (prod_antilipschitzWith_equiv_aux p α β).isUniformInducing
       (prod_lipschitzWith_equiv_aux p α β).uniformContinuous
   have : (fun x : WithLp p (α × β) × WithLp p (α × β) =>
     ((WithLp.equiv p (α × β)) x.fst, (WithLp.equiv p (α × β)) x.snd)) = id := by
diff --git a/Mathlib/Analysis/Normed/Lp/WithLp.lean b/Mathlib/Analysis/Normed/Lp/WithLp.lean
index b636322d0d7e8..d921359dcb5a5 100644
--- a/Mathlib/Analysis/Normed/Lp/WithLp.lean
+++ b/Mathlib/Analysis/Normed/Lp/WithLp.lean
@@ -46,7 +46,7 @@ instance, on `Prod` and finite `Pi` types). -/
 @[nolint unusedArguments]
 def WithLp (_p : ℝ≥0∞) (V : Type uV) : Type uV := V
 
-variable (p : ℝ≥0∞) (K : Type uK) (K' : Type uK')  (V : Type uV)
+variable (p : ℝ≥0∞) (K : Type uK) (K' : Type uK') (V : Type uV)
 
 namespace WithLp
 
diff --git a/Mathlib/Analysis/Normed/Lp/lpSpace.lean b/Mathlib/Analysis/Normed/Lp/lpSpace.lean
index 8bfe697508229..dfe8bef80fdec 100644
--- a/Mathlib/Analysis/Normed/Lp/lpSpace.lean
+++ b/Mathlib/Analysis/Normed/Lp/lpSpace.lean
@@ -595,7 +595,7 @@ theorem norm_const_smul_le (hp : p ≠ 0) (c : 𝕜) (f : lp E p) : ‖c • f
     · simp [lp.eq_zero' f]
     have hcf := lp.isLUB_norm (c • f)
     have hfc := (lp.isLUB_norm f).mul_left (norm_nonneg c)
-    simp_rw [← Set.range_comp, Function.comp] at hfc
+    simp_rw [← Set.range_comp, Function.comp_def] at hfc
     -- TODO: some `IsLUB` API should make it a one-liner from here.
     refine hcf.right ?_
     have := hfc.left
diff --git a/Mathlib/Analysis/Normed/Module/Basic.lean b/Mathlib/Analysis/Normed/Module/Basic.lean
index e010869a152e5..d1bb26f18316d 100644
--- a/Mathlib/Analysis/Normed/Module/Basic.lean
+++ b/Mathlib/Analysis/Normed/Module/Basic.lean
@@ -8,7 +8,7 @@ import Mathlib.Algebra.Algebra.Prod
 import Mathlib.Algebra.Algebra.Rat
 import Mathlib.Algebra.Algebra.RestrictScalars
 import Mathlib.Algebra.Module.Rat
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.Normed.MulAction
 
 /-!
@@ -119,6 +119,9 @@ instance Pi.normedSpace {ι : Type*} {E : ι → Type*} [Fintype ι] [∀ i, Sem
       NNReal.mul_finset_sup]
     exact Finset.sup_mono_fun fun _ _ => norm_smul_le a _
 
+instance SeparationQuotient.instNormedSpace : NormedSpace 𝕜 (SeparationQuotient E) where
+  norm_smul_le := norm_smul_le
+
 instance MulOpposite.instNormedSpace : NormedSpace 𝕜 Eᵐᵒᵖ where
   norm_smul_le _ x := norm_smul_le _ x.unop
 
@@ -186,7 +189,7 @@ theorem NormedSpace.exists_lt_norm (c : ℝ) : ∃ x : E, c < ‖x‖ := by
   rcases exists_ne (0 : E) with ⟨x, hx⟩
   rcases NormedField.exists_lt_norm 𝕜 (c / ‖x‖) with ⟨r, hr⟩
   use r • x
-  rwa [norm_smul, ← _root_.div_lt_iff]
+  rwa [norm_smul, ← div_lt_iff₀]
   rwa [norm_pos_iff]
 
 protected theorem NormedSpace.unbounded_univ : ¬Bornology.IsBounded (univ : Set E) := fun h =>
@@ -348,6 +351,10 @@ instance Pi.normedAlgebra {ι : Type*} {E : ι → Type*} [Fintype ι] [∀ i, S
 
 variable [SeminormedRing E] [NormedAlgebra 𝕜 E]
 
+instance SeparationQuotient.instNormedAlgebra : NormedAlgebra 𝕜 (SeparationQuotient E) where
+  __ : NormedSpace 𝕜 (SeparationQuotient E) := inferInstance
+  __ : Algebra 𝕜 (SeparationQuotient E) := inferInstance
+
 instance MulOpposite.instNormedAlgebra {E : Type*} [SeminormedRing E] [NormedAlgebra 𝕜 E] :
     NormedAlgebra 𝕜 Eᵐᵒᵖ where
   __ := instAlgebra
diff --git a/Mathlib/Analysis/Normed/Module/Completion.lean b/Mathlib/Analysis/Normed/Module/Completion.lean
index 4e50f3b1130b0..f0a8fdeed667d 100644
--- a/Mathlib/Analysis/Normed/Module/Completion.lean
+++ b/Mathlib/Analysis/Normed/Module/Completion.lean
@@ -6,6 +6,7 @@ Authors: Yury Kudryashov
 import Mathlib.Analysis.Normed.Group.Completion
 import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
 import Mathlib.Topology.Algebra.UniformRing
+import Mathlib.Topology.Algebra.UniformField
 
 /-!
 # Normed space structure on the completion of a normed space
@@ -26,20 +27,16 @@ namespace UniformSpace
 
 namespace Completion
 
-variable (𝕜 E : Type*) [NormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable (𝕜 E : Type*)
 
-instance (priority := 100) NormedSpace.to_uniformContinuousConstSMul :
-    UniformContinuousConstSMul 𝕜 E :=
-  ⟨fun c => (lipschitzWith_smul c).uniformContinuous⟩
+instance [NormedField 𝕜] [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] :
+    NormedSpace 𝕜 (Completion E) where
+  norm_smul_le := norm_smul_le
 
-instance : NormedSpace 𝕜 (Completion E) :=
-  { Completion.instModule with
-    norm_smul_le := fun c x =>
-      induction_on x
-        (isClosed_le (continuous_const_smul _).norm (continuous_const.mul continuous_norm)) fun y =>
-        by simp only [← coe_smul, norm_coe, norm_smul, le_rfl] }
+section Module
 
 variable {𝕜 E}
+variable [Semiring 𝕜] [SeminormedAddCommGroup E] [Module 𝕜 E] [UniformContinuousConstSMul 𝕜 E]
 
 /-- Embedding of a normed space to its completion as a linear isometry. -/
 def toComplₗᵢ : E →ₗᵢ[𝕜] Completion E :=
@@ -65,40 +62,36 @@ theorem norm_toComplL {𝕜 E : Type*} [NontriviallyNormedField 𝕜] [NormedAdd
     [NormedSpace 𝕜 E] [Nontrivial E] : ‖(toComplL : E →L[𝕜] Completion E)‖ = 1 :=
   (toComplₗᵢ : E →ₗᵢ[𝕜] Completion E).norm_toContinuousLinearMap
 
+end Module
+
 section Algebra
 
-variable (𝕜) (A : Type*)
-
-instance [SeminormedRing A] : NormedRing (Completion A) :=
-  { Completion.ring,
-    Completion.instMetricSpace with
-    dist_eq := fun x y => by
-      refine Completion.induction_on₂ x y ?_ ?_ <;> clear x y
-      · refine isClosed_eq (Completion.uniformContinuous_extension₂ _).continuous ?_
-        exact Continuous.comp Completion.continuous_extension continuous_sub
-      · intro x y
-        rw [← Completion.coe_sub, norm_coe, Completion.dist_eq, dist_eq_norm]
-    norm_mul := fun x y => by
-      refine Completion.induction_on₂ x y ?_ ?_ <;> clear x y
-      · exact
-          isClosed_le (Continuous.comp continuous_norm continuous_mul)
-            (Continuous.comp _root_.continuous_mul
-              (Continuous.prod_map continuous_norm continuous_norm))
-      · intro x y
-        simp only [← coe_mul, norm_coe]
-        exact norm_mul_le x y }
-
-instance [SeminormedCommRing A] [NormedAlgebra 𝕜 A] [UniformContinuousConstSMul 𝕜 A] :
-    NormedAlgebra 𝕜 (Completion A) :=
-  { Completion.algebra A 𝕜 with
-    norm_smul_le := fun r x => by
-      refine Completion.induction_on x ?_ ?_ <;> clear x
-      · exact
-          isClosed_le (Continuous.comp continuous_norm (continuous_const_smul r))
-            (Continuous.comp (continuous_mul_left _) continuous_norm)
-      · intro x
-        simp only [← coe_smul, norm_coe]
-        exact norm_smul_le r x }
+variable (A : Type*)
+
+instance [SeminormedRing A] : NormedRing (Completion A) where
+  __ : NormedAddCommGroup (Completion A) := inferInstance
+  __ : Ring (Completion A) := inferInstance
+  norm_mul x y := by
+    induction x, y using induction_on₂ with
+    | hp =>
+      apply isClosed_le <;> fun_prop
+    | ih x y =>
+      simp only [← coe_mul, norm_coe]
+      exact norm_mul_le x y
+
+instance [SeminormedCommRing A] : NormedCommRing (Completion A) where
+  __ : CommRing (Completion A) := inferInstance
+  __ : NormedRing (Completion A) := inferInstance
+
+instance [NormedField 𝕜] [SeminormedCommRing A] [NormedAlgebra 𝕜 A] :
+    NormedAlgebra 𝕜 (Completion A) where
+  norm_smul_le := norm_smul_le
+
+instance [NormedField A] [CompletableTopField A] :
+    NormedField (UniformSpace.Completion A) where
+  __ : NormedCommRing (Completion A) := inferInstance
+  __ : Field (Completion A) := inferInstance
+  norm_mul' x y := induction_on₂ x y (isClosed_eq (by fun_prop) (by fun_prop)) (by simp [← coe_mul])
 
 end Algebra
 
diff --git a/Mathlib/Analysis/Normed/Module/Dual.lean b/Mathlib/Analysis/Normed/Module/Dual.lean
index 0c1b5814b92f1..f8115711046c5 100644
--- a/Mathlib/Analysis/Normed/Module/Dual.lean
+++ b/Mathlib/Analysis/Normed/Module/Dual.lean
@@ -159,6 +159,13 @@ variable {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 theorem mem_polar_iff {x' : Dual 𝕜 E} (s : Set E) : x' ∈ polar 𝕜 s ↔ ∀ z ∈ s, ‖x' z‖ ≤ 1 :=
   Iff.rfl
 
+@[simp]
+theorem zero_mem_polar (s : Set E) : (0 : Dual 𝕜 E) ∈ polar 𝕜 s :=
+  LinearMap.zero_mem_polar _ s
+
+theorem polar_nonempty (s : Set E) : Set.Nonempty (polar 𝕜 s) :=
+  LinearMap.polar_nonempty _ _
+
 @[simp]
 theorem polar_univ : polar 𝕜 (univ : Set E) = {(0 : Dual 𝕜 E)} :=
   (dualPairing 𝕜 E).flip.polar_univ
@@ -205,7 +212,7 @@ theorem polar_ball_subset_closedBall_div {c : 𝕜} (hc : 1 < ‖c‖) {r : ℝ}
   refine ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => ?_
   calc
     ‖x' x‖ ≤ 1 := hx' _ h₂
-    _ ≤ ‖c‖ / r * ‖x‖ := (inv_pos_le_iff_one_le_mul' hcr).1 (by rwa [inv_div])
+    _ ≤ ‖c‖ / r * ‖x‖ := (inv_le_iff_one_le_mul₀' hcr).1 (by rwa [inv_div])
 
 variable (𝕜)
 
@@ -229,6 +236,23 @@ theorem polar_closedBall {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [
   refine ContinuousLinearMap.opNorm_le_of_ball hr (inv_nonneg.mpr hr.le) fun z _ => ?_
   simpa only [one_div] using LinearMap.bound_of_ball_bound' hr 1 x'.toLinearMap h z
 
+theorem polar_ball {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] {r : ℝ}
+    (hr : 0 < r) : polar 𝕜 (ball (0 : E) r) = closedBall (0 : Dual 𝕜 E) r⁻¹ := by
+  apply le_antisymm
+  · intro x hx
+    rw [mem_closedBall_zero_iff]
+    apply le_of_forall_le_of_dense
+    intro a ha
+    rw [← mem_closedBall_zero_iff, ← (mul_div_cancel_left₀ a (Ne.symm (ne_of_lt hr)))]
+    rw [← RCLike.norm_of_nonneg (K := 𝕜) (le_trans zero_le_one
+      (le_of_lt ((inv_lt_iff_one_lt_mul₀' hr).mp ha)))]
+    apply polar_ball_subset_closedBall_div _ hr hx
+    rw [RCLike.norm_of_nonneg (K := 𝕜) (le_trans zero_le_one
+      (le_of_lt ((inv_lt_iff_one_lt_mul₀' hr).mp ha)))]
+    exact (inv_lt_iff_one_lt_mul₀' hr).mp ha
+  · rw [← polar_closedBall hr]
+    exact LinearMap.polar_antitone _ ball_subset_closedBall
+
 /-- Given a neighborhood `s` of the origin in a normed space `E`, the dual norms
 of all elements of the polar `polar 𝕜 s` are bounded by a constant. -/
 theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)) :
@@ -239,6 +263,10 @@ theorem isBounded_polar_of_mem_nhds_zero {s : Set E} (s_nhd : s ∈ 𝓝 (0 : E)
     (((dualPairing 𝕜 E).flip.polar_antitone r_ball).trans <|
       polar_ball_subset_closedBall_div ha r_pos)
 
+@[simp]
+theorem polar_empty : polar 𝕜 (∅ : Set E) = Set.univ :=
+  LinearMap.polar_empty _
+
 @[simp]
 theorem polar_singleton {a : E} : polar 𝕜 {a} = { x | ‖x a‖ ≤ 1 } := by
   simp only [polar, LinearMap.polar_singleton, LinearMap.flip_apply, dualPairing_apply]
@@ -246,6 +274,9 @@ theorem polar_singleton {a : E} : polar 𝕜 {a} = { x | ‖x a‖ ≤ 1 } := by
 theorem mem_polar_singleton {a : E} (y : Dual 𝕜 E) : y ∈ polar 𝕜 {a} ↔ ‖y a‖ ≤ 1 := by
   simp only [polar_singleton, mem_setOf_eq]
 
+theorem polar_zero : polar 𝕜 ({0} : Set E) = Set.univ :=
+  LinearMap.polar_zero _
+
 theorem sInter_polar_eq_closedBall {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
     {r : ℝ} (hr : 0 < r) :
     ⋂₀ (polar 𝕜 '' { F | F.Finite ∧ F ⊆ closedBall (0 : E) r⁻¹ }) = closedBall 0 r := by
diff --git a/Mathlib/Analysis/Normed/Module/FiniteDimension.lean b/Mathlib/Analysis/Normed/Module/FiniteDimension.lean
index 4c55f7d09dafa..4735888f210dd 100644
--- a/Mathlib/Analysis/Normed/Module/FiniteDimension.lean
+++ b/Mathlib/Analysis/Normed/Module/FiniteDimension.lean
@@ -48,7 +48,7 @@ universe u v w x
 
 noncomputable section
 
-open Set FiniteDimensional TopologicalSpace Filter Asymptotics Topology NNReal Metric
+open Asymptotics Filter Module Metric Module NNReal Set TopologicalSpace Topology
 
 namespace LinearIsometry
 
@@ -323,14 +323,14 @@ theorem Basis.exists_opNorm_le {ι : Type*} [Finite ι] (v : Basis ι 𝕜 E) :
 
 instance [FiniteDimensional 𝕜 E] [SecondCountableTopology F] :
     SecondCountableTopology (E →L[𝕜] F) := by
-  set d := FiniteDimensional.finrank 𝕜 E
+  set d := Module.finrank 𝕜 E
   suffices
     ∀ ε > (0 : ℝ), ∃ n : (E →L[𝕜] F) → Fin d → ℕ, ∀ f g : E →L[𝕜] F, n f = n g → dist f g ≤ ε from
     Metric.secondCountable_of_countable_discretization fun ε ε_pos =>
       ⟨Fin d → ℕ, by infer_instance, this ε ε_pos⟩
   intro ε ε_pos
   obtain ⟨u : ℕ → F, hu : DenseRange u⟩ := exists_dense_seq F
-  let v := FiniteDimensional.finBasis 𝕜 E
+  let v := Module.finBasis 𝕜 E
   obtain
     ⟨C : ℝ, C_pos : 0 < C, hC :
       ∀ {φ : E →L[𝕜] F} {M : ℝ}, 0 ≤ M → (∀ i, ‖φ (v i)‖ ≤ M) → ‖φ‖ ≤ C * M⟩ :=
@@ -647,7 +647,7 @@ theorem summable_norm_iff {α E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ
   refine ⟨Summable.of_norm, fun hf ↦ ?_⟩
   -- First we use a finite basis to reduce the problem to the case `E = Fin N → ℝ`
   suffices ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x => ‖g x‖ by
-    obtain v := finBasis ℝ E
+    obtain v := Module.finBasis ℝ E
     set e := v.equivFunL
     have H : Summable fun x => ‖e (f x)‖ := this (e.summable.2 hf)
     refine .of_norm_bounded _ (H.mul_left ↑‖(e.symm : (Fin (finrank ℝ E) → ℝ) →L[ℝ] E)‖₊) fun i ↦ ?_
diff --git a/Mathlib/Analysis/Normed/Module/Ray.lean b/Mathlib/Analysis/Normed/Module/Ray.lean
index 93b860a1c1924..f01da59497153 100644
--- a/Mathlib/Analysis/Normed/Module/Ray.lean
+++ b/Mathlib/Analysis/Normed/Module/Ray.lean
@@ -81,7 +81,7 @@ the unit vectors `‖x‖⁻¹ • x` and `‖y‖⁻¹ • y` are equal. -/
 theorem sameRay_iff_inv_norm_smul_eq : SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y := by
   rcases eq_or_ne x 0 with (rfl | hx); · simp [SameRay.zero_left]
   rcases eq_or_ne y 0 with (rfl | hy); · simp [SameRay.zero_right]
-  simp only [sameRay_iff_inv_norm_smul_eq_of_ne hx hy, *, false_or_iff]
+  simp only [sameRay_iff_inv_norm_smul_eq_of_ne hx hy, *, false_or]
 
 /-- Two vectors of the same norm are on the same ray if and only if they are equal. -/
 theorem sameRay_iff_of_norm_eq (h : ‖x‖ = ‖y‖) : SameRay ℝ x y ↔ x = y := by
diff --git a/Mathlib/Analysis/Normed/Module/WeakDual.lean b/Mathlib/Analysis/Normed/Module/WeakDual.lean
index 71dc2da624481..9eceea12586f7 100644
--- a/Mathlib/Analysis/Normed/Module/WeakDual.lean
+++ b/Mathlib/Analysis/Normed/Module/WeakDual.lean
@@ -5,6 +5,7 @@ Authors: Kalle Kytölä, Yury Kudryashov
 -/
 import Mathlib.Analysis.Normed.Module.Dual
 import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
+import Mathlib.Topology.Algebra.Module.WeakDual
 
 /-!
 # Weak dual of normed space
diff --git a/Mathlib/Analysis/Normed/MulAction.lean b/Mathlib/Analysis/Normed/MulAction.lean
index 618fab80dff3f..2c9c706fdb6c1 100644
--- a/Mathlib/Analysis/Normed/MulAction.lean
+++ b/Mathlib/Analysis/Normed/MulAction.lean
@@ -3,8 +3,9 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Topology.MetricSpace.Algebra
 import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Topology.MetricSpace.Algebra
+import Mathlib.Topology.Algebra.UniformMulAction
 
 /-!
 # Lemmas for `BoundedSMul` over normed additive groups
diff --git a/Mathlib/Analysis/Normed/Operator/Banach.lean b/Mathlib/Analysis/Normed/Operator/Banach.lean
index 5d6eca063bbe1..8561d451ba771 100644
--- a/Mathlib/Analysis/Normed/Operator/Banach.lean
+++ b/Mathlib/Analysis/Normed/Operator/Banach.lean
@@ -531,7 +531,7 @@ section BijectivityCriteria
 
 namespace ContinuousLinearMap
 
-variable {σ : 𝕜 →+* 𝕜'}  {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {f : E →SL[σ] F}
+variable {σ : 𝕜 →+* 𝕜'} {σ' : 𝕜' →+* 𝕜} [RingHomInvPair σ σ'] {f : E →SL[σ] F}
 variable {F : Type u_4} [NormedAddCommGroup F] [NormedSpace 𝕜' F]
 variable [CompleteSpace E]
 
diff --git a/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean b/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
index 114bd3205cea4..4d0fea4e147c4 100644
--- a/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
+++ b/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
@@ -59,14 +59,14 @@ open Filter (Tendsto)
 
 open Metric ContinuousLinearMap
 
-variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
-  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type*}
-  [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E]
+  [NormedSpace 𝕜 E] {F : Type*} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] {G : Type*}
+  [SeminormedAddCommGroup G] [NormedSpace 𝕜 G]
 
 /-- A function `f` satisfies `IsBoundedLinearMap 𝕜 f` if it is linear and satisfies the
 inequality `‖f x‖ ≤ M * ‖x‖` for some positive constant `M`. -/
-structure IsBoundedLinearMap (𝕜 : Type*) [NormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
-  [NormedSpace 𝕜 E] {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) extends
+structure IsBoundedLinearMap (𝕜 : Type*) [NormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E]
+  [NormedSpace 𝕜 E] {F : Type*} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F) extends
   IsLinearMap 𝕜 f : Prop where
   bound : ∃ M, 0 < M ∧ ∀ x : E, ‖f x‖ ≤ M * ‖x‖
 
@@ -186,7 +186,7 @@ variable {ι : Type*} [Fintype ι]
 
 /-- Taking the cartesian product of two continuous multilinear maps is a bounded linear
 operation. -/
-theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, NormedAddCommGroup (E i)]
+theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, SeminormedAddCommGroup (E i)]
     [∀ i, NormedSpace 𝕜 (E i)] :
     IsBoundedLinearMap 𝕜 fun p : ContinuousMultilinearMap 𝕜 E F × ContinuousMultilinearMap 𝕜 E G =>
       p.1.prod p.2 where
@@ -244,7 +244,7 @@ variable {R : Type*}
 variable {𝕜₂ 𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NontriviallyNormedField 𝕜₂]
 variable {M : Type*} [TopologicalSpace M]
 variable {σ₁₂ : 𝕜 →+* 𝕜₂}
-variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G']
+variable {G' : Type*} [SeminormedAddCommGroup G'] [NormedSpace 𝕜₂ G'] [NormedSpace 𝕜' G']
 variable [SMulCommClass 𝕜₂ 𝕜' G']
 
 section Semiring
@@ -374,7 +374,7 @@ theorem IsBoundedBilinearMap.isBoundedLinearMap_right (h : IsBoundedBilinearMap
   (h.toContinuousLinearMap x).isBoundedLinearMap
 
 theorem isBoundedBilinearMap_smul {𝕜' : Type*} [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] {E : Type*}
-    [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] :
+    [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedSpace 𝕜' E] [IsScalarTower 𝕜 𝕜' E] :
     IsBoundedBilinearMap 𝕜 fun p : 𝕜' × E => p.1 • p.2 :=
   (lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E).isBoundedBilinearMap
 
@@ -436,7 +436,7 @@ variable (𝕜)
 
 /-- The function `ContinuousLinearMap.mulLeftRight : 𝕜' × 𝕜' → (𝕜' →L[𝕜] 𝕜')` is a bounded
 bilinear map. -/
-theorem ContinuousLinearMap.mulLeftRight_isBoundedBilinear (𝕜' : Type*) [NormedRing 𝕜']
+theorem ContinuousLinearMap.mulLeftRight_isBoundedBilinear (𝕜' : Type*) [SeminormedRing 𝕜']
     [NormedAlgebra 𝕜 𝕜'] :
     IsBoundedBilinearMap 𝕜 fun p : 𝕜' × 𝕜' => ContinuousLinearMap.mulLeftRight 𝕜 𝕜' p.1 p.2 :=
   (ContinuousLinearMap.mulLeftRight 𝕜 𝕜').isBoundedBilinearMap
@@ -471,9 +471,16 @@ theorem ContinuousOn.clm_apply {X} [TopologicalSpace X] {f : X → (E →L[𝕜]
     ContinuousOn (fun x ↦ f x (g x)) s :=
   isBoundedBilinearMap_apply.continuous.comp_continuousOn (hf.prod hg)
 
+end
+
 namespace ContinuousLinearEquiv
 
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable {F : Type*} [SeminormedAddCommGroup F] [NormedSpace 𝕜 F]
+
 open Set
+open scoped Topology
 
 /-!
 ### The set of continuous linear equivalences between two Banach spaces is open
diff --git a/Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.lean b/Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.lean
index da7b7a93c50c0..64c5ba0eb60b5 100644
--- a/Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.lean
+++ b/Mathlib/Analysis/Normed/Operator/ContinuousLinearMap.lean
@@ -3,8 +3,9 @@ Copyright (c) 2019 Jan-David Salchow. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
 -/
-import Mathlib.Topology.Algebra.Module.Basic
+import Mathlib.Analysis.Normed.Group.Uniform
 import Mathlib.Analysis.Normed.MulAction
+import Mathlib.Topology.Algebra.Module.Basic
 
 /-! # Constructions of continuous linear maps between (semi-)normed spaces
 
@@ -152,9 +153,13 @@ variable [Ring 𝕜] [Ring 𝕜₂]
 variable [NormedAddCommGroup E] [NormedAddCommGroup F] [Module 𝕜 E] [Module 𝕜₂ F]
 variable {σ : 𝕜 →+* 𝕜₂} (f g : E →SL[σ] F) (x y z : E)
 
-theorem ContinuousLinearMap.uniformEmbedding_of_bound {K : ℝ≥0} (hf : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
-    UniformEmbedding f :=
-  (AddMonoidHomClass.antilipschitz_of_bound f hf).uniformEmbedding f.uniformContinuous
+theorem ContinuousLinearMap.isUniformEmbedding_of_bound {K : ℝ≥0} (hf : ∀ x, ‖x‖ ≤ K * ‖f x‖) :
+    IsUniformEmbedding f :=
+  (AddMonoidHomClass.antilipschitz_of_bound f hf).isUniformEmbedding f.uniformContinuous
+
+@[deprecated (since := "2024-10-01")]
+alias ContinuousLinearMap.uniformEmbedding_of_bound :=
+  ContinuousLinearMap.isUniformEmbedding_of_bound
 
 end Normed
 
diff --git a/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean b/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
index 578fd81859e6d..92b76035981f0 100644
--- a/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
+++ b/Mathlib/Analysis/Normed/Operator/LinearIsometry.lean
@@ -205,10 +205,12 @@ theorem nnnorm_map (x : E) : ‖f x‖₊ = ‖x‖₊ :=
 protected theorem isometry : Isometry f :=
   AddMonoidHomClass.isometry_of_norm f.toLinearMap (norm_map _)
 
+protected lemma embedding (f : F →ₛₗᵢ[σ₁₂] E₂) : Embedding f := f.isometry.embedding
+
 -- Should be `@[simp]` but it doesn't fire due to `lean4#3107`.
 theorem isComplete_image_iff [SemilinearIsometryClass 𝓕 σ₁₂ E E₂] (f : 𝓕) {s : Set E} :
     IsComplete (f '' s) ↔ IsComplete s :=
-  _root_.isComplete_image_iff (SemilinearIsometryClass.isometry f).uniformInducing
+  _root_.isComplete_image_iff (SemilinearIsometryClass.isometry f).isUniformInducing
 
 @[simp] -- Should be replaced with `LinearIsometry.isComplete_image_iff` when `lean4#3107` is fixed.
 theorem isComplete_image_iff' (f : LinearIsometry σ₁₂ E E₂) {s : Set E} :
diff --git a/Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean b/Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean
index b7a5ea7136815..45dfa1892cbcc 100644
--- a/Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean
+++ b/Mathlib/Analysis/Normed/Operator/WeakOperatorTopology.lean
@@ -85,7 +85,7 @@ variable (𝕜) (E) (F)
 
 unseal ContinuousLinearMapWOT in
 /-- The linear equivalence that sends a continuous linear map to the type copy endowed with the
-weak operator topology.  -/
+weak operator topology. -/
 def _root_.ContinuousLinearMap.toWOT : (E →L[𝕜] F) ≃ₗ[𝕜] (E →WOT[𝕜] F) := LinearEquiv.refl 𝕜 _
 
 variable {𝕜} {E} {F}
@@ -211,7 +211,7 @@ end Topology
 section Seminorms
 
 /-- The family of seminorms that induce the weak operator topology, namely `‖y (A x)‖` for
-all `x` and `y`.  -/
+all `x` and `y`. -/
 def seminorm (x : E) (y : F⋆) : Seminorm 𝕜 (E →WOT[𝕜] F) where
   toFun A := ‖y (A x)‖
   map_zero' := by simp
@@ -221,7 +221,7 @@ def seminorm (x : E) (y : F⋆) : Seminorm 𝕜 (E →WOT[𝕜] F) where
 
 variable (𝕜) (E) (F) in
 /-- The family of seminorms that induce the weak operator topology, namely `‖y (A x)‖` for
-all `x` and `y`.  -/
+all `x` and `y`. -/
 def seminormFamily : SeminormFamily 𝕜 (E →WOT[𝕜] F) (E × F⋆) :=
   fun ⟨x, y⟩ => seminorm x y
 
diff --git a/Mathlib/Analysis/Normed/Order/Basic.lean b/Mathlib/Analysis/Normed/Order/Basic.lean
index d78be19a7ae1f..e3f1e8a5b1e77 100644
--- a/Mathlib/Analysis/Normed/Order/Basic.lean
+++ b/Mathlib/Analysis/Normed/Order/Basic.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker, Yaël Dillies
 -/
 import Mathlib.Algebra.Order.Group.TypeTags
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 
 /-!
 # Ordered normed spaces
diff --git a/Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean b/Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean
new file mode 100644
index 0000000000000..615d91284df1a
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Ring/IsPowMulFaithful.lean
@@ -0,0 +1,96 @@
+/-
+Copyright (c) 2024 María Inés de Frutos-Fernández. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: María Inés de Frutos-Fernández
+-/
+import Mathlib.Analysis.Normed.Algebra.Norm
+import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
+
+/-!
+# Equivalent power-multiplicative norms
+
+In this file, we prove [BGR, Proposition 3.1.5/1][bosch-guntzer-remmert]: if `R` is a normed
+commutative ring and `f₁` and `f₂` are two power-multiplicative `R`-algebra norms on `S`, then if
+`f₁` and `f₂` are equivalent on every subring `R[y]` for `y : S`, it follows that `f₁ = f₂`.
+
+## Main Results
+* `eq_of_powMul_faithful` : the proof of [BGR, Proposition 3.1.5/1][bosch-guntzer-remmert].
+
+## References
+* [S. Bosch, U. Güntzer, R. Remmert, *Non-Archimedean Analysis*][bosch-guntzer-remmert]
+
+## Tags
+
+norm, equivalent, power-multiplicative
+-/
+
+open Filter Real
+open scoped Topology
+
+/-- If `f : α →+* β` is bounded with respect to a ring seminorm `nα` on `α` and a
+  power-multiplicative function `nβ : β → ℝ`, then `∀ x : α, nβ (f x) ≤ nα x`. -/
+theorem contraction_of_isPowMul_of_boundedWrt {F : Type*} {α : outParam (Type*)} [Ring α]
+    [FunLike F α ℝ] [RingSeminormClass F α ℝ] {β : Type*} [Ring β] (nα : F) {nβ : β → ℝ}
+    (hβ : IsPowMul nβ) {f : α →+* β} (hf : f.IsBoundedWrt nα nβ) (x : α) : nβ (f x) ≤ nα x := by
+  obtain ⟨C, hC0, hC⟩ := hf
+  have hlim : Tendsto (fun n : ℕ => C ^ (1 / (n : ℝ)) * nα x) atTop (𝓝 (nα x)) := by
+    nth_rewrite 2 [← one_mul (nα x)]
+    exact ((rpow_zero C ▸ ContinuousAt.tendsto (continuousAt_const_rpow (ne_of_gt hC0))).comp
+      (tendsto_const_div_atTop_nhds_zero_nat 1)).mul tendsto_const_nhds
+  apply ge_of_tendsto hlim
+  simp only [eventually_atTop, ge_iff_le]
+  use 1
+  intro n hn
+  have h : (C ^ (1 / n : ℝ)) ^ n = C := by
+    have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
+    rw [← rpow_natCast, ← rpow_mul (le_of_lt hC0), one_div, inv_mul_cancel₀ hn0, rpow_one]
+  apply le_of_pow_le_pow_left (ne_of_gt hn)
+    (mul_nonneg (rpow_nonneg (le_of_lt hC0) _) (apply_nonneg _ _))
+  · rw [mul_pow, h, ← hβ _ hn, ← RingHom.map_pow]
+    apply le_trans (hC (x ^ n))
+    rw [mul_le_mul_left hC0]
+    exact map_pow_le_pow _ _ (Nat.one_le_iff_ne_zero.mp hn)
+
+/-- Given a bounded `f : α →+* β` between seminormed rings, is the seminorm on `β` is
+  power-multiplicative, then `f` is a contraction. -/
+theorem contraction_of_isPowMul {α β : Type*} [SeminormedRing α] [SeminormedRing β]
+    (hβ : IsPowMul (norm : β → ℝ)) {f : α →+* β} (hf : f.IsBounded) (x : α) : norm (f x) ≤ norm x :=
+  contraction_of_isPowMul_of_boundedWrt (SeminormedRing.toRingSeminorm α) hβ hf x
+
+/-- Given two power-multiplicative ring seminorms `f, g` on `α`, if `f` is bounded by a positive
+  multiple of `g` and viceversa, then `f = g`. -/
+theorem eq_seminorms {F : Type*} {α : outParam (Type*)} [Ring α] [FunLike F α ℝ]
+    [RingSeminormClass F α ℝ] {f g : F} (hfpm : IsPowMul f) (hgpm : IsPowMul g)
+    (hfg : ∃ (r : ℝ) (_ : 0 < r), ∀ a : α, f a ≤ r * g a)
+    (hgf : ∃ (r : ℝ) (_ : 0 < r), ∀ a : α, g a ≤ r * f a) : f = g := by
+  obtain ⟨r, hr0, hr⟩ := hfg
+  obtain ⟨s, hs0, hs⟩ := hgf
+  have hle : RingHom.IsBoundedWrt f g (RingHom.id _) := ⟨s, hs0, hs⟩
+  have hge : RingHom.IsBoundedWrt g f (RingHom.id _) := ⟨r, hr0, hr⟩
+  rw [← Function.Injective.eq_iff DFunLike.coe_injective']
+  ext x
+  exact le_antisymm (contraction_of_isPowMul_of_boundedWrt g hfpm hge x)
+    (contraction_of_isPowMul_of_boundedWrt f hgpm hle x)
+
+variable {R S : Type*} [NormedCommRing R] [CommRing S] [Algebra R S]
+
+/-- If `R` is a normed commutative ring and `f₁` and `f₂` are two power-multiplicative `R`-algebra
+  norms on `S`, then if `f₁` and `f₂` are equivalent on every  subring `R[y]` for `y : S`, it
+  follows that `f₁ = f₂` [BGR, Proposition 3.1.5/1][bosch-guntzer-remmert]. -/
+theorem eq_of_powMul_faithful (f₁ : AlgebraNorm R S) (hf₁_pm : IsPowMul f₁) (f₂ : AlgebraNorm R S)
+    (hf₂_pm : IsPowMul f₂)
+    (h_eq : ∀ y : S, ∃ (C₁ C₂ : ℝ) (_ : 0 < C₁) (_ : 0 < C₂),
+      ∀ x : Algebra.adjoin R {y}, f₁ x.val ≤ C₁ * f₂ x.val ∧ f₂ x.val ≤ C₂ * f₁ x.val) :
+    f₁ = f₂ := by
+  ext x
+  set g₁ : AlgebraNorm R (Algebra.adjoin R ({x} : Set S)) := AlgebraNorm.restriction _ f₁
+  set g₂ : AlgebraNorm R (Algebra.adjoin R ({x} : Set S)) := AlgebraNorm.restriction _ f₂
+  have hg₁_pm : IsPowMul g₁ := IsPowMul.restriction _ hf₁_pm
+  have hg₂_pm : IsPowMul g₂ := IsPowMul.restriction _ hf₂_pm
+  let y : Algebra.adjoin R ({x} : Set S) := ⟨x, Algebra.self_mem_adjoin_singleton R x⟩
+  have hy : x = y.val := rfl
+  have h1 : f₁ y.val = g₁ y := rfl
+  have h2 : f₂ y.val = g₂ y := rfl
+  obtain ⟨C₁, C₂, hC₁_pos, hC₂_pos, hC⟩ := h_eq x
+  obtain ⟨hC₁, hC₂⟩ := forall_and.mp hC
+  rw [hy, h1, h2, eq_seminorms hg₁_pm hg₂_pm ⟨C₁, hC₁_pos, hC₁⟩ ⟨C₂, hC₂_pos, hC₂⟩]
diff --git a/Mathlib/Analysis/Normed/Ring/Seminorm.lean b/Mathlib/Analysis/Normed/Ring/Seminorm.lean
index 9d88d4bb64e68..650ace74daac1 100644
--- a/Mathlib/Analysis/Normed/Ring/Seminorm.lean
+++ b/Mathlib/Analysis/Normed/Ring/Seminorm.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: María Inés de Frutos-Fernández, Yaël Dillies
 -/
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.SpecialFunctions.Pow.Real
 
 /-!
@@ -151,6 +151,26 @@ end Ring
 
 end RingSeminorm
 
+/-- If `f` is a ring seminorm on `a`, then `∀ {n : ℕ}, n ≠ 0 → f (a ^ n) ≤ f a ^ n`. -/
+theorem map_pow_le_pow {F α : Type*} [Ring α] [FunLike F α ℝ] [RingSeminormClass F α ℝ] (f : F)
+    (a : α) : ∀ {n : ℕ}, n ≠ 0 → f (a ^ n) ≤ f a ^ n
+  | 0, h => absurd rfl h
+  | 1, _ => by simp only [pow_one, le_refl]
+  | n + 2, _ => by
+    simp only [pow_succ _ (n + 1)];
+      exact
+        le_trans (map_mul_le_mul f _ a)
+          (mul_le_mul_of_nonneg_right (map_pow_le_pow _ _ n.succ_ne_zero) (apply_nonneg f a))
+
+/-- If `f` is a ring seminorm on `a` with `f 1 ≤ 1`, then `∀ (n : ℕ), f (a ^ n) ≤ f a ^ n`. -/
+theorem map_pow_le_pow' {F α : Type*} [Ring α] [FunLike F α ℝ] [RingSeminormClass F α ℝ] {f : F}
+    (hf1 : f 1 ≤ 1) (a : α) : ∀ n : ℕ, f (a ^ n) ≤ f a ^ n
+  | 0 => by simp only [pow_zero, hf1]
+  | n + 1 => by
+    simp only [pow_succ _ n];
+      exact le_trans (map_mul_le_mul f _ a)
+        (mul_le_mul_of_nonneg_right (map_pow_le_pow' hf1 _ n) (apply_nonneg f a))
+
 /-- The norm of a `NonUnitalSeminormedRing` as a `RingSeminorm`. -/
 def normRingSeminorm (R : Type*) [NonUnitalSeminormedRing R] : RingSeminorm R :=
   { normAddGroupSeminorm R with
@@ -298,7 +318,7 @@ instance : Inhabited (MulRingNorm R) :=
 variable {R : Type*} [Ring R]
 
 /-- Two multiplicative ring norms `f, g` on `R` are equivalent if there exists a positive constant
-  `c` such that for all `x ∈ R`, `(f x)^c = g x`.  -/
+  `c` such that for all `x ∈ R`, `(f x)^c = g x`. -/
 
 def equiv (f : MulRingNorm R) (g : MulRingNorm R) :=
   ∃ c : ℝ, 0 < c ∧ (fun x => (f x) ^ c) = g
@@ -368,3 +388,36 @@ lemma MulRingNorm.apply_natAbs_eq {R : Type*} [Ring R] (x : ℤ) (f : MulRingNor
     f x := by
   obtain ⟨n, rfl | rfl⟩ := eq_nat_or_neg x <;>
   simp only [natAbs_neg, natAbs_ofNat, cast_neg, cast_natCast, map_neg_eq_map]
+
+/-- The seminorm on a `SeminormedRing`, as a `RingSeminorm`. -/
+def SeminormedRing.toRingSeminorm (R : Type*) [SeminormedRing R] : RingSeminorm R where
+  toFun     := norm
+  map_zero' := norm_zero
+  add_le'   := norm_add_le
+  mul_le'   := norm_mul_le
+  neg'      := norm_neg
+
+/-- The norm on a `NormedRing`, as a `RingNorm`. -/
+@[simps]
+def NormedRing.toRingNorm (R : Type*) [NormedRing R] : RingNorm R where
+  toFun     := norm
+  map_zero' := norm_zero
+  add_le'   := norm_add_le
+  mul_le'   := norm_mul_le
+  neg'      := norm_neg
+  eq_zero_of_map_eq_zero' x hx := by rw [← norm_eq_zero]; exact hx
+
+@[simp]
+theorem NormedRing.toRingNorm_apply (R : Type*) [NormedRing R] (x : R) :
+    (NormedRing.toRingNorm R) x = ‖x‖ :=
+  rfl
+
+/-- The norm on a `NormedField`, as a `MulRingNorm`. -/
+def NormedField.toMulRingNorm (R : Type*) [NormedField R] : MulRingNorm R where
+  toFun     := norm
+  map_zero' := norm_zero
+  map_one'  := norm_one
+  add_le'   := norm_add_le
+  map_mul'  := norm_mul
+  neg'      := norm_neg
+  eq_zero_of_map_eq_zero' x hx := by rw [← norm_eq_zero]; exact hx
diff --git a/Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean b/Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean
index 2855ee36ba27b..a98f45db8cc03 100644
--- a/Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean
+++ b/Mathlib/Analysis/Normed/Ring/SeminormFromBounded.lean
@@ -153,11 +153,11 @@ theorem seminormFromBounded_eq_zero_iff (f_nonneg : 0 ≤ f)
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · have hf := seminormFromBounded_ge f_nonneg f_mul x
     rw [h, mul_zero] at hf
-    exact hf.antisymm  (f_nonneg _)
+    exact hf.antisymm (f_nonneg _)
   · have hf : seminormFromBounded' f x ≤ c * f x :=
       seminormFromBounded_le f_nonneg f_mul x
     rw [h, mul_zero] at hf
-    exact hf.antisymm  (seminormFromBounded_nonneg f_nonneg f_mul x)
+    exact hf.antisymm (seminormFromBounded_nonneg f_nonneg f_mul x)
 
 /-- If `f` is invariant under negation of `x`, then so is `seminormFromBounded'`.-/
 theorem seminormFromBounded_neg (f_neg : ∀ x : R, f (-x) = f x) (x : R) :
diff --git a/Mathlib/Analysis/Normed/Ring/Ultra.lean b/Mathlib/Analysis/Normed/Ring/Ultra.lean
new file mode 100644
index 0000000000000..1e8016ac72edb
--- /dev/null
+++ b/Mathlib/Analysis/Normed/Ring/Ultra.lean
@@ -0,0 +1,78 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Group.Ultra
+
+/-!
+# Ultrametric norms on rings where the norm of one is one
+
+This file contains results on the behavior of norms in ultrametric normed rings.
+The norm must send one to one.
+
+## Main results
+
+* `norm_intCast_le_one`:
+  the norm of the image of an integer in the ring is always less than or equal to one
+
+## Implementation details
+
+A `[NormedRing R]` only assumes a submultiplicative norm and does not have `[NormOneClass R]`.
+The weakest ring-like structure that has a bundled norm such that `‖1‖ = 1` is
+`[NormedDivisionRing K]`.
+Since the statements below hold in any context, we can state them
+in an unbundled fashion using `[NormOneClass R]`.
+In fact one can actually prove all these lemmas only assuming
+`{R : Type*} [SeminormedAddGroup R] [One R] [NormOneClass R] [IsUltrametricDist R]`.
+But one has to give the typeclass machinery a little help in order to get it to recognise that there
+is a coercion from `ℕ` or `ℤ` to `R`.
+Instead, we use weakest pre-existing typeclass that implies both
+`[SeminormedAddGroup R]` and `[AddGroupWithOne R]`, which is `[SeminormedRing R]`.
+
+## Tags
+
+ultrametric, nonarchimedean
+-/
+open Metric NNReal
+
+namespace IsUltrametricDist
+
+section NormOneClass
+
+variable {R : Type*} [SeminormedRing R] [NormOneClass R] [IsUltrametricDist R]
+
+lemma norm_add_one_le_max_norm_one (x : R) :
+    ‖x + 1‖ ≤ max ‖x‖ 1 := by
+  simpa only [le_max_iff, norm_one] using norm_add_le_max x 1
+
+lemma nnnorm_add_one_le_max_nnnorm_one (x : R) :
+    ‖x + 1‖₊ ≤ max ‖x‖₊ 1 :=
+  norm_add_one_le_max_norm_one _
+
+variable (R)
+lemma nnnorm_natCast_le_one (n : ℕ) :
+    ‖(n : R)‖₊ ≤ 1 := by
+  induction n with
+  | zero => simp only [Nat.cast_zero, nnnorm_zero, zero_le]
+  | succ n hn => simpa only [Nat.cast_add, Nat.cast_one, hn, max_eq_right] using
+    nnnorm_add_one_le_max_nnnorm_one (n : R)
+
+lemma norm_natCast_le_one (n : ℕ) :
+    ‖(n : R)‖ ≤ 1 :=
+  nnnorm_natCast_le_one R n
+
+lemma nnnorm_intCast_le_one (z : ℤ) :
+    ‖(z : R)‖₊ ≤ 1 := by
+  induction z <;>
+  simpa only [Int.ofNat_eq_coe, Int.cast_natCast, Int.cast_negSucc, Nat.cast_one, nnnorm_neg]
+    using nnnorm_natCast_le_one _ _
+
+lemma norm_intCast_le_one (z : ℤ) :
+    ‖(z : R)‖ ≤ 1 :=
+  nnnorm_intCast_le_one _ z
+
+end NormOneClass
+
+end IsUltrametricDist
diff --git a/Mathlib/Analysis/Normed/Ring/Units.lean b/Mathlib/Analysis/Normed/Ring/Units.lean
index c46b35e61a4d4..86883d4c5d017 100644
--- a/Mathlib/Analysis/Normed/Ring/Units.lean
+++ b/Mathlib/Analysis/Normed/Ring/Units.lean
@@ -14,12 +14,12 @@ normed ring (Banach algebras being a notable special case).
 
 ## Main results
 
-The constructions `Units.oneSub`, `Units.add`, and `Units.ofNearby` state, in varying forms, that
-perturbations of a unit are units. The latter two are not stated in their optimal form; more precise
-versions would use the spectral radius.
+The constructions `Units.add` and `Units.ofNearby` (based on `Units.oneSub` defined elsewhere)
+state, in varying forms, that perturbations of a unit are units. They are not stated
+in their optimal form; more precise versions would use the spectral radius.
 
-The first main result is `Units.isOpen`: the group of units of a complete normed ring is an open
-subset of the ring.
+The first main result is `Units.isOpen`: the group of units of a normed ring with summable
+geometric series is an open subset of the ring.
 
 The function `Ring.inverse` (defined elsewhere), for a ring `R`, sends `a : R` to `a⁻¹` if `a` is a
 unit and `0` if not.  The other major results of this file (notably `NormedRing.inverse_add`,
@@ -31,21 +31,13 @@ noncomputable section
 
 open Topology
 
-variable {R : Type*} [NormedRing R] [CompleteSpace R]
+variable {R : Type*} [NormedRing R] [HasSummableGeomSeries R]
 
 namespace Units
 
-/-- In a complete normed ring, a perturbation of `1` by an element `t` of distance less than `1`
-from `1` is a unit.  Here we construct its `Units` structure.  -/
-@[simps val]
-def oneSub (t : R) (h : ‖t‖ < 1) : Rˣ where
-  val := 1 - t
-  inv := ∑' n : ℕ, t ^ n
-  val_inv := mul_neg_geom_series t h
-  inv_val := geom_series_mul_neg t h
-
-/-- In a complete normed ring, a perturbation of a unit `x` by an element `t` of distance less than
-`‖x⁻¹‖⁻¹` from `x` is a unit.  Here we construct its `Units` structure. -/
+/-- In a normed ring with summable geometric series, a perturbation of a unit `x` by an
+element `t` of distance less than `‖x⁻¹‖⁻¹` from `x` is a unit.
+Here we construct its `Units` structure. -/
 @[simps! val]
 def add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ :=
   Units.copy -- to make `add_val` true definitionally, for convenience
@@ -59,13 +51,14 @@ def add (x : Rˣ) (t : R) (h : ‖t‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ :=
         _ = 1 := mul_inv_cancel₀ (ne_of_gt hpos)))
     (x + t) (by simp [mul_add]) _ rfl
 
-/-- In a complete normed ring, an element `y` of distance less than `‖x⁻¹‖⁻¹` from `x` is a unit.
-Here we construct its `Units` structure. -/
+/-- In a normed ring with summable geometric series, an element `y` of distance less
+than `‖x⁻¹‖⁻¹` from `x` is a unit. Here we construct its `Units` structure. -/
 @[simps! val]
 def ofNearby (x : Rˣ) (y : R) (h : ‖y - x‖ < ‖(↑x⁻¹ : R)‖⁻¹) : Rˣ :=
   (x.add (y - x : R) h).copy y (by simp) _ rfl
 
-/-- The group of units of a complete normed ring is an open subset of the ring. -/
+/-- The group of units of a normed ring with summable geometric series is an open subset
+of the ring. -/
 protected theorem isOpen : IsOpen { x : R | IsUnit x } := by
   nontriviality R
   rw [Metric.isOpen_iff]
@@ -81,12 +74,12 @@ end Units
 
 namespace nonunits
 
-/-- The `nonunits` in a complete normed ring are contained in the complement of the ball of radius
-`1` centered at `1 : R`. -/
+/-- The `nonunits` in a normed ring with summable geometric series are contained in the
+complement of the ball of radius `1` centered at `1 : R`. -/
 theorem subset_compl_ball : nonunits R ⊆ (Metric.ball (1 : R) 1)ᶜ := fun x hx h₁ ↦ hx <|
   sub_sub_self 1 x ▸ (Units.oneSub (1 - x) (by rwa [mem_ball_iff_norm'] at h₁)).isUnit
 
--- The `nonunits` in a complete normed ring are a closed set
+-- The `nonunits` in a normed ring with summable geometric series are a closed set
 protected theorem isClosed : IsClosed (nonunits R) :=
   Units.isOpen.isClosed_compl
 
@@ -114,7 +107,7 @@ theorem inverse_add (x : Rˣ) :
 
 theorem inverse_one_sub_nth_order' (n : ℕ) {t : R} (ht : ‖t‖ < 1) :
     inverse ((1 : R) - t) = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) :=
-  have := NormedRing.summable_geometric_of_norm_lt_one t ht
+  have := _root_.summable_geometric_of_norm_lt_one ht
   calc inverse (1 - t) = ∑' i : ℕ, t ^ i := inverse_one_sub t ht
     _ = ∑ i ∈ range n, t ^ i + ∑' i : ℕ, t ^ (i + n) := (sum_add_tsum_nat_add _ this).symm
     _ = (∑ i ∈ range n, t ^ i) + t ^ n * inverse (1 - t) := by
@@ -149,10 +142,10 @@ theorem inverse_one_sub_norm : (fun t : R => inverse (1 - t)) =O[𝓝 0] (fun _t
     linarith
   simp only [inverse_one_sub t ht', norm_one, mul_one, Set.mem_setOf_eq]
   change ‖∑' n : ℕ, t ^ n‖ ≤ _
-  have := NormedRing.tsum_geometric_of_norm_lt_one t ht'
+  have := tsum_geometric_le_of_norm_lt_one t ht'
   have : (1 - ‖t‖)⁻¹ ≤ 2 := by
     rw [← inv_inv (2 : ℝ)]
-    refine inv_le_inv_of_le (by norm_num) ?_
+    refine inv_anti₀ (by norm_num) ?_
     have : (2 : ℝ)⁻¹ + (2 : ℝ)⁻¹ = 1 := by ring
     linarith
   linarith
@@ -197,7 +190,7 @@ theorem inverse_continuousAt (x : Rˣ) : ContinuousAt inverse (x : R) := by
     refine tendsto_zero_iff_norm_tendsto_zero.mpr ?_
     exact tendsto_iff_norm_sub_tendsto_zero.mp tendsto_id
   rw [ContinuousAt, tendsto_iff_norm_sub_tendsto_zero, inverse_unit]
-  simpa [(· ∘ ·)] using h_is_o.norm_left.tendsto_div_nhds_zero.comp h_lim
+  simpa [Function.comp_def] using h_is_o.norm_left.tendsto_div_nhds_zero.comp h_lim
 
 end NormedRing
 
@@ -205,15 +198,15 @@ namespace Units
 
 open MulOpposite Filter NormedRing
 
-/-- In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
-embedding in `R × R`) to `R` is an open embedding. -/
+/-- In a normed ring with summable geometric series, the coercion from `Rˣ` (equipped with the
+induced topology from the embedding in `R × R`) to `R` is an open embedding. -/
 theorem openEmbedding_val : OpenEmbedding (val : Rˣ → R) where
   toEmbedding := embedding_val_mk'
     (fun _ ⟨u, hu⟩ ↦ hu ▸ (inverse_continuousAt u).continuousWithinAt) Ring.inverse_unit
   isOpen_range := Units.isOpen
 
-/-- In a normed ring, the coercion from `Rˣ` (equipped with the induced topology from the
-embedding in `R × R`) to `R` is an open map. -/
+/-- In a normed ring with summable geometric series, the coercion from `Rˣ` (equipped with the
+induced topology from the embedding in `R × R`) to `R` is an open map. -/
 theorem isOpenMap_val : IsOpenMap (val : Rˣ → R) :=
   openEmbedding_val.isOpenMap
 
@@ -227,16 +220,18 @@ theorem eq_top_of_norm_lt_one (I : Ideal R) {x : R} (hxI : x ∈ I) (hx : ‖1 -
   I.eq_top_iff_one.mpr <| by
     simpa only [show u.inv * x = 1 by simp [u]] using I.mul_mem_left u.inv hxI
 
-/-- The `Ideal.closure` of a proper ideal in a complete normed ring is proper. -/
+/-- The `Ideal.closure` of a proper ideal in a normed ring with summable
+geometric series is proper. -/
 theorem closure_ne_top (I : Ideal R) (hI : I ≠ ⊤) : I.closure ≠ ⊤ := by
   have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed
   simpa only [I.closure.eq_top_iff_one, Ne] using mt (@h 1) one_not_mem_nonunits
 
-/-- The `Ideal.closure` of a maximal ideal in a complete normed ring is the ideal itself. -/
+/-- The `Ideal.closure` of a maximal ideal in a normed ring with summable
+geometric series is the ideal itself. -/
 theorem IsMaximal.closure_eq {I : Ideal R} (hI : I.IsMaximal) : I.closure = I :=
   (hI.eq_of_le (I.closure_ne_top hI.ne_top) subset_closure).symm
 
-/-- Maximal ideals in complete normed rings are closed. -/
+/-- Maximal ideals in normed rings with summable geometric series are closed. -/
 instance IsMaximal.isClosed {I : Ideal R} [hI : I.IsMaximal] : IsClosed (I : Set R) :=
   isClosed_of_closure_subset <| Eq.subset <| congr_arg ((↑) : Ideal R → Set R) hI.closure_eq
 
diff --git a/Mathlib/Analysis/NormedSpace/Connected.lean b/Mathlib/Analysis/NormedSpace/Connected.lean
index 64d98223dda5e..e14389a41de74 100644
--- a/Mathlib/Analysis/NormedSpace/Connected.lean
+++ b/Mathlib/Analysis/NormedSpace/Connected.lean
@@ -52,13 +52,11 @@ theorem Set.Countable.isPathConnected_compl_of_one_lt_rank
   let c := (2 : ℝ)⁻¹ • (a + b)
   let x := (2 : ℝ)⁻¹ • (b - a)
   have Ia : c - x = a := by
-    simp only [c, x, smul_add, smul_sub]
-    abel_nf
-    simp [← Int.cast_smul_eq_zsmul ℝ 2]
+    simp only [c, x]
+    module
   have Ib : c + x = b := by
-    simp only [c, x, smul_add, smul_sub]
-    abel_nf
-    simp [← Int.cast_smul_eq_zsmul ℝ 2]
+    simp only [c, x]
+    module
   have x_ne_zero : x ≠ 0 := by simpa [x] using sub_ne_zero.2 hab.symm
   obtain ⟨y, hy⟩ : ∃ y, LinearIndependent ℝ ![x, y] :=
     exists_linearIndependent_pair_of_one_lt_rank h x_ne_zero
diff --git a/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean b/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
index b759259be866a..b158eee9c860d 100644
--- a/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
+++ b/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
@@ -62,13 +62,15 @@ section RCLike
 
 open RCLike
 
-variable {𝕜 : Type*} [RCLike 𝕜] {E F : Type*}
+variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [IsRCLikeNormedField 𝕜] {E F : Type*}
   [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
   [NormedAddCommGroup F] [NormedSpace 𝕜 F]
 
-/-- **Hahn-Banach theorem** for continuous linear functions over `𝕜` satisfying `RCLike 𝕜`. -/
+/-- **Hahn-Banach theorem** for continuous linear functions over `𝕜`
+satisfying `IsRCLikeNormedField 𝕜`. -/
 theorem exists_extension_norm_eq (p : Subspace 𝕜 E) (f : p →L[𝕜] 𝕜) :
     ∃ g : E →L[𝕜] 𝕜, (∀ x : p, g x = f x) ∧ ‖g‖ = ‖f‖ := by
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
   letI : Module ℝ E := RestrictScalars.module ℝ 𝕜 E
   letI : IsScalarTower ℝ 𝕜 E := RestrictScalars.isScalarTower _ _ _
   letI : NormedSpace ℝ E := NormedSpace.restrictScalars _ 𝕜 _
@@ -105,12 +107,11 @@ theorem exists_extension_norm_eq (p : Subspace 𝕜 E) (f : p →L[𝕜] 𝕜) :
       _ = ‖f‖ := by rw [reCLM_norm, one_mul]
   · exact f.opNorm_le_bound g.extendTo𝕜.opNorm_nonneg fun x => h x ▸ g.extendTo𝕜.le_opNorm x
 
-open FiniteDimensional
+open Module
 
 /-- Corollary of the **Hahn-Banach theorem**: if `f : p → F` is a continuous linear map
 from a submodule of a normed space `E` over `𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
-with a finite dimensional range,
-then `f` admits an extension to a continuous linear map `E → F`.
+with a finite dimensional range, then `f` admits an extension to a continuous linear map `E → F`.
 
 Note that contrary to the case `F = 𝕜`, see `exists_extension_norm_eq`,
 we provide no estimates on the norm of the extension.
@@ -118,7 +119,8 @@ we provide no estimates on the norm of the extension.
 lemma ContinuousLinearMap.exist_extension_of_finiteDimensional_range {p : Submodule 𝕜 E}
     (f : p →L[𝕜] F) [FiniteDimensional 𝕜 (LinearMap.range f)] :
     ∃ g : E →L[𝕜] F, f = g.comp p.subtypeL := by
-  set b := finBasis 𝕜 (LinearMap.range f)
+  letI : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
+  set b := Module.finBasis 𝕜 (LinearMap.range f)
   set e := b.equivFunL
   set fi := fun i ↦ (LinearMap.toContinuousLinearMap (b.coord i)).comp
     (f.codRestrict _ <| LinearMap.mem_range_self _)
@@ -142,8 +144,6 @@ variable {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 
 open ContinuousLinearEquiv Submodule
 
-open scoped Classical
-
 theorem coord_norm' {x : E} (h : x ≠ 0) : ‖(‖x‖ : 𝕜) • coord 𝕜 x h‖ = 1 := by
   #adaptation_note
   /--
diff --git a/Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean b/Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
index a054142703272..0cb8fc167da35 100644
--- a/Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
+++ b/Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
@@ -7,6 +7,7 @@ import Mathlib.Analysis.Convex.Cone.Extension
 import Mathlib.Analysis.Convex.Gauge
 import Mathlib.Topology.Algebra.Module.FiniteDimension
 import Mathlib.Topology.Algebra.Module.LocallyConvex
+import Mathlib.Topology.Algebra.MulAction
 import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Analysis.NormedSpace.Extend
 
@@ -210,10 +211,10 @@ end
 
 namespace RCLike
 
-variable [RCLike 𝕜] [Module 𝕜 E] [ContinuousSMul 𝕜 E] [IsScalarTower ℝ 𝕜 E]
+variable [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 /--Real linear extension of continuous extension of `LinearMap.extendTo𝕜'` -/
-noncomputable def extendTo𝕜'ₗ : (E →L[ℝ] ℝ) →ₗ[ℝ] (E →L[𝕜] 𝕜) :=
+noncomputable def extendTo𝕜'ₗ [ContinuousConstSMul 𝕜 E]: (E →L[ℝ] ℝ) →ₗ[ℝ] (E →L[𝕜] 𝕜) :=
   letI to𝕜 (fr : (E →L[ℝ] ℝ)) : (E →L[𝕜] 𝕜) :=
     { toLinearMap := LinearMap.extendTo𝕜' fr
       cont := show Continuous fun x ↦ (fr x : 𝕜) - (I : 𝕜) * (fr ((I : 𝕜) • x) : 𝕜) by fun_prop }
@@ -223,16 +224,18 @@ noncomputable def extendTo𝕜'ₗ : (E →L[ℝ] ℝ) →ₗ[ℝ] (E →L[𝕜]
     map_smul' := by intros; ext; simp [h, real_smul_eq_coe_mul]; ring }
 
 @[simp]
-lemma re_extendTo𝕜'ₗ (g : E →L[ℝ] ℝ) (x : E) : re ((extendTo𝕜'ₗ g) x : 𝕜) = g x := by
+lemma re_extendTo𝕜'ₗ [ContinuousConstSMul 𝕜 E] (g : E →L[ℝ] ℝ) (x : E) : re ((extendTo𝕜'ₗ g) x : 𝕜)
+    = g x := by
   have h g (x : E) : extendTo𝕜'ₗ g x = ((g x : 𝕜) - (I : 𝕜) * (g ((I : 𝕜) • x) : 𝕜)) := rfl
   simp only [h , map_sub, ofReal_re, mul_re, I_re, zero_mul, ofReal_im, mul_zero,
     sub_self, sub_zero]
 
-variable [TopologicalAddGroup E] [ContinuousSMul ℝ E]
+variable [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
 
 theorem separate_convex_open_set {s : Set E}
     (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) {x₀ : E} (hx₀ : x₀ ∉ s) :
     ∃ f : E →L[𝕜] 𝕜, re (f x₀) = 1 ∧ ∀ x ∈ s, re (f x) < 1 := by
+  have := IsScalarTower.continuousSMul (M := ℝ) (α := E) 𝕜
   obtain ⟨g, hg⟩ := _root_.separate_convex_open_set hs₀ hs₁ hs₂ hx₀
   use extendTo𝕜'ₗ g
   simp only [re_extendTo𝕜'ₗ]
@@ -241,6 +244,7 @@ theorem separate_convex_open_set {s : Set E}
 theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht : Convex ℝ t)
     (disj : Disjoint s t) : ∃ (f : E →L[𝕜] 𝕜) (u : ℝ), (∀ a ∈ s, re (f a) < u) ∧
     ∀ b ∈ t, u ≤ re (f b) := by
+  have := IsScalarTower.continuousSMul (M := ℝ) (α := E) 𝕜
   obtain ⟨f, u, h⟩ := _root_.geometric_hahn_banach_open hs₁ hs₂ ht disj
   use extendTo𝕜'ₗ f
   simp only [re_extendTo𝕜'ₗ]
@@ -248,6 +252,7 @@ theorem geometric_hahn_banach_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (ht
 
 theorem geometric_hahn_banach_open_point (hs₁ : Convex ℝ s) (hs₂ : IsOpen s) (disj : x ∉ s) :
     ∃ f : E →L[𝕜] 𝕜, ∀ a ∈ s, re (f a) < re (f x) := by
+  have := IsScalarTower.continuousSMul (M := ℝ) (α := E) 𝕜
   obtain ⟨f, h⟩ := _root_.geometric_hahn_banach_open_point hs₁ hs₂ disj
   use extendTo𝕜'ₗ f
   simp only [re_extendTo𝕜'ₗ]
@@ -261,6 +266,7 @@ theorem geometric_hahn_banach_point_open (ht₁ : Convex ℝ t) (ht₂ : IsOpen
 theorem geometric_hahn_banach_open_open (hs₁ : Convex ℝ s) (hs₂ : IsOpen s)
     (ht₁ : Convex ℝ t) (ht₃ : IsOpen t) (disj : Disjoint s t) :
     ∃ (f : E →L[𝕜] 𝕜) (u : ℝ), (∀ a ∈ s, re (f a) < u) ∧ ∀ b ∈ t, u < re (f b) := by
+  have := IsScalarTower.continuousSMul (M := ℝ) (α := E) 𝕜
   obtain ⟨f, u, h⟩ := _root_.geometric_hahn_banach_open_open hs₁ hs₂ ht₁ ht₃ disj
   use extendTo𝕜'ₗ f
   simp only [re_extendTo𝕜'ₗ]
@@ -271,6 +277,7 @@ variable [LocallyConvexSpace ℝ E]
 theorem geometric_hahn_banach_compact_closed (hs₁ : Convex ℝ s) (hs₂ : IsCompact s)
     (ht₁ : Convex ℝ t) (ht₂ : IsClosed t) (disj : Disjoint s t) :
     ∃ (f : E →L[𝕜] 𝕜) (u v : ℝ), (∀ a ∈ s, re (f a) < u) ∧ u < v ∧ ∀ b ∈ t, v < re (f b) := by
+  have := IsScalarTower.continuousSMul (M := ℝ) (α := E) 𝕜
   obtain ⟨g, u, v, h1⟩ := _root_.geometric_hahn_banach_compact_closed hs₁ hs₂ ht₁ ht₂ disj
   use extendTo𝕜'ₗ g
   simp only [re_extendTo𝕜'ₗ, exists_and_left]
diff --git a/Mathlib/Analysis/NormedSpace/Int.lean b/Mathlib/Analysis/NormedSpace/Int.lean
index fe43e855c5bea..b08f55039887a 100644
--- a/Mathlib/Analysis/NormedSpace/Int.lean
+++ b/Mathlib/Analysis/NormedSpace/Int.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 
 /-!
 # The integers as normed ring
diff --git a/Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean b/Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
index e7d06672107a6..4b87da345fe31 100644
--- a/Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
+++ b/Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean
@@ -74,19 +74,48 @@ We use the following type variables in this file:
 
 universe u v v' wE wE₁ wE' wG wG'
 
-/-- Applying a multilinear map to a vector is continuous in both coordinates. -/
-theorem ContinuousMultilinearMap.continuous_eval {𝕜 ι : Type*} {E : ι → Type*} {F : Type*}
+section continuous_eval
+
+variable {𝕜 ι : Type*} {E : ι → Type*} {F : Type*}
     [NormedField 𝕜] [Finite ι] [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
-    [TopologicalSpace F] [AddCommGroup F] [TopologicalAddGroup F] [Module 𝕜 F] :
+    [TopologicalSpace F] [AddCommGroup F] [TopologicalAddGroup F] [Module 𝕜 F]
+
+/-- Applying a multilinear map to a vector is continuous in both coordinates. -/
+theorem ContinuousMultilinearMap.continuous_eval :
     Continuous fun p : ContinuousMultilinearMap 𝕜 E F × ∀ i, E i => p.1 p.2 := by
   cases nonempty_fintype ι
   let _ := TopologicalAddGroup.toUniformSpace F
   have := comm_topologicalAddGroup_is_uniform (G := F)
   refine (UniformOnFun.continuousOn_eval₂ fun m ↦ ?_).comp_continuous
-    (embedding_toUniformOnFun.continuous.prod_map continuous_id) fun (f, x) ↦ f.cont.continuousAt
+    (embedding_toUniformOnFun.continuous.prodMap continuous_id) fun (f, x) ↦ f.cont.continuousAt
   exact ⟨ball m 1, NormedSpace.isVonNBounded_of_isBounded _ isBounded_ball,
     ball_mem_nhds _ one_pos⟩
 
+namespace ContinuousLinearMap
+
+variable {G : Type*} [AddCommGroup G] [TopologicalSpace G] [Module 𝕜 G] [ContinuousConstSMul 𝕜 F]
+  (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F)
+
+lemma continuous_uncurry_of_multilinear :
+    Continuous (fun (p : G × (Π i, E i)) ↦ f p.1 p.2) :=
+  ContinuousMultilinearMap.continuous_eval.comp <| .prodMap (map_continuous f) continuous_id
+
+lemma continuousOn_uncurry_of_multilinear {s} :
+    ContinuousOn (fun (p : G × (Π i, E i)) ↦ f p.1 p.2) s :=
+  f.continuous_uncurry_of_multilinear.continuousOn
+
+lemma continuousAt_uncurry_of_multilinear {x} :
+    ContinuousAt (fun (p : G × (Π i, E i)) ↦ f p.1 p.2) x :=
+  f.continuous_uncurry_of_multilinear.continuousAt
+
+lemma continuousWithinAt_uncurry_of_multilinear {s x} :
+    ContinuousWithinAt (fun (p : G × (Π i, E i)) ↦ f p.1 p.2) s x :=
+  f.continuous_uncurry_of_multilinear.continuousWithinAt
+
+end ContinuousLinearMap
+
+end continuous_eval
+
 section Seminorm
 
 variable {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E₁ : ι → Type wE₁}
@@ -189,7 +218,6 @@ theorem norm_image_sub_le_of_bound' [DecidableEq ι] {C : ℝ} (hC : 0 ≤ C)
       rw [B, A, ← f.map_sub]
       apply le_trans (H _)
       gcongr with j
-      · exact fun j _ => norm_nonneg _
       by_cases h : j = i
       · rw [h]
         simp
@@ -340,7 +368,7 @@ theorem isLeast_opNorm : IsLeast {c : ℝ | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c *
 @[deprecated (since := "2024-02-02")] alias isLeast_op_norm := isLeast_opNorm
 
 theorem opNorm_nonneg : 0 ≤ ‖f‖ :=
-  Real.sInf_nonneg _ fun _ ⟨hx, _⟩ => hx
+  Real.sInf_nonneg fun _ ⟨hx, _⟩ => hx
 
 @[deprecated (since := "2024-02-02")] alias op_norm_nonneg := opNorm_nonneg
 
@@ -391,7 +419,7 @@ theorem le_of_opNorm_le {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i,
 variable (f)
 
 theorem ratio_le_opNorm : (‖f m‖ / ∏ i, ‖m i‖) ≤ ‖f‖ :=
-  div_le_of_nonneg_of_le_mul (by positivity) (opNorm_nonneg _) (f.le_opNorm m)
+  div_le_of_le_mul₀ (by positivity) (opNorm_nonneg _) (f.le_opNorm m)
 
 @[deprecated (since := "2024-02-02")] alias ratio_le_op_norm := ratio_le_opNorm
 
@@ -657,17 +685,6 @@ def restrictScalarsₗᵢ : ContinuousMultilinearMap 𝕜 E G →ₗᵢ[𝕜'] C
   map_smul' _ _ := rfl
   norm_map' _ := rfl
 
-/-- `ContinuousMultilinearMap.restrictScalars` as a `ContinuousLinearMap`. -/
-def restrictScalarsLinear : ContinuousMultilinearMap 𝕜 E G →L[𝕜'] ContinuousMultilinearMap 𝕜' E G :=
-  (restrictScalarsₗᵢ 𝕜').toContinuousLinearMap
-
-variable {𝕜'}
-
-theorem continuous_restrictScalars :
-    Continuous
-      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E G → ContinuousMultilinearMap 𝕜' E G) :=
-  (restrictScalarsLinear 𝕜').continuous
-
 end RestrictScalars
 
 /-- The difference `f m₁ - f m₂` is controlled in terms of `‖f‖` and `‖m₁ - m₂‖`, precise version.
@@ -759,7 +776,7 @@ theorem norm_mkPiAlgebraFin_succ_le : ‖ContinuousMultilinearMap.mkPiAlgebraFin
   simp only [ContinuousMultilinearMap.mkPiAlgebraFin_apply, one_mul, List.ofFn_eq_map,
     Fin.prod_univ_def, Multiset.map_coe, Multiset.prod_coe]
   refine (List.norm_prod_le' ?_).trans_eq ?_
-  · rw [Ne, List.map_eq_nil, List.finRange_eq_nil]
+  · rw [Ne, List.map_eq_nil_iff, List.finRange_eq_nil]
     exact Nat.succ_ne_zero _
   rw [List.map_map, Function.comp_def]
 
@@ -775,6 +792,12 @@ theorem norm_mkPiAlgebraFin_zero : ‖ContinuousMultilinearMap.mkPiAlgebraFin 
   · convert ratio_le_opNorm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun _ => (1 : A)
     simp
 
+theorem norm_mkPiAlgebraFin_le :
+    ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ max 1 ‖(1 : A)‖ := by
+  cases n
+  · exact norm_mkPiAlgebraFin_zero.le.trans (le_max_right _ _)
+  · exact (norm_mkPiAlgebraFin_le_of_pos (Nat.zero_lt_succ _)).trans (le_max_left _ _)
+
 @[simp]
 theorem norm_mkPiAlgebraFin [NormOneClass A] :
     ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1 := by
@@ -1252,7 +1275,6 @@ lemma norm_iteratedFDerivComponent_le {α : Type*} [Fintype α]
   _ ≤ ‖f‖ * ∏ _i : {a : ι // a ∉ s}, ‖x‖ := by
       gcongr
       · exact MultilinearMap.mkContinuousMultilinear_norm_le _ (norm_nonneg _) _
-      · exact fun _ _ ↦ norm_nonneg _
       · exact norm_le_pi_norm _ _
   _ = ‖f‖ * ‖x‖ ^ (Fintype.card {a : ι // a ∉ s}) := by rw [prod_const, card_univ]
   _ = ‖f‖ * ‖x‖ ^ (Fintype.card ι - Fintype.card α) := by simp [Fintype.card_congr e]
diff --git a/Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean b/Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean
index 888c0aecdb488..433b3db5d9d54 100644
--- a/Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean
+++ b/Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean
@@ -162,53 +162,49 @@ the space of continuous linear maps from `E 0` to the space of continuous multil
 `continuousMultilinearCurryLeftEquiv 𝕜 E E₂`. The algebraic version (without topology) is given
 in `multilinearCurryLeftEquiv 𝕜 E E₂`.
 
-The direct and inverse maps are given by `f.uncurryLeft` and `f.curryLeft`. Use these
+The direct and inverse maps are given by `f.curryLeft` and `f.uncurryLeft`. Use these
 unless you need the full framework of linear isometric equivs. -/
 def continuousMultilinearCurryLeftEquiv :
-    (Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) ≃ₗᵢ[𝕜]
-      ContinuousMultilinearMap 𝕜 Ei G :=
+    ContinuousMultilinearMap 𝕜 Ei G ≃ₗᵢ[𝕜]
+      Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G :=
   LinearIsometryEquiv.ofBounds
-    { toFun := ContinuousLinearMap.uncurryLeft
-      map_add' := fun f₁ f₂ => by
-        ext m
-        rfl
-      map_smul' := fun c f => by
-        ext m
-        rfl
-      invFun := ContinuousMultilinearMap.curryLeft
-      left_inv := ContinuousLinearMap.curry_uncurryLeft
-      right_inv := ContinuousMultilinearMap.uncurry_curryLeft }
+    { toFun := ContinuousMultilinearMap.curryLeft
+      map_add' := fun f₁ f₂ => rfl
+      map_smul' := fun c f => rfl
+      invFun := ContinuousLinearMap.uncurryLeft
+      left_inv := ContinuousMultilinearMap.uncurry_curryLeft
+      right_inv := ContinuousLinearMap.curry_uncurryLeft }
     (fun f => by
       simp only [LinearEquiv.coe_mk]
-      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
+      exact LinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
     (fun f => by
       simp only [LinearEquiv.coe_symm_mk]
-      exact LinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
+      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
 
 variable {𝕜 Ei G}
 
 @[simp]
 theorem continuousMultilinearCurryLeftEquiv_apply
-    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (v : ∀ i, Ei i) :
-    continuousMultilinearCurryLeftEquiv 𝕜 Ei G f v = f (v 0) (tail v) :=
+    (f : ContinuousMultilinearMap 𝕜 Ei G) (x : Ei 0) (v : Π i : Fin n, Ei i.succ) :
+    continuousMultilinearCurryLeftEquiv 𝕜 Ei G f x v = f (cons x v) :=
   rfl
 
 @[simp]
-theorem continuousMultilinearCurryLeftEquiv_symm_apply (f : ContinuousMultilinearMap 𝕜 Ei G)
-    (x : Ei 0) (v : ∀ i : Fin n, Ei i.succ) :
-    (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm f x v = f (cons x v) :=
+theorem continuousMultilinearCurryLeftEquiv_symm_apply
+    (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) (v : Π i, Ei i) :
+    (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm f v = f (v 0) (tail v) :=
   rfl
 
 @[simp]
 theorem ContinuousMultilinearMap.curryLeft_norm (f : ContinuousMultilinearMap 𝕜 Ei G) :
     ‖f.curryLeft‖ = ‖f‖ :=
-  (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm.norm_map f
+  (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).norm_map f
 
 @[simp]
 theorem ContinuousLinearMap.uncurryLeft_norm
     (f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) :
     ‖f.uncurryLeft‖ = ‖f‖ :=
-  (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).norm_map f
+  (continuousMultilinearCurryLeftEquiv 𝕜 Ei G).symm.norm_map f
 
 /-! #### Right currying -/
 
@@ -279,27 +275,25 @@ space of continuous linear maps on `Ei (last n)`, by separating the last variabl
 isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
 The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.
 
-The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
+The direct and inverse maps are given by `f.curryRight` and `f.uncurryRight`. Use these
 unless you need the full framework of linear isometric equivs.
 -/
 def continuousMultilinearCurryRightEquiv :
-    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
-      ContinuousMultilinearMap 𝕜 Ei G :=
+    ContinuousMultilinearMap 𝕜 Ei G ≃ₗᵢ[𝕜]
+      ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) :=
   LinearIsometryEquiv.ofBounds
-    { toFun := ContinuousMultilinearMap.uncurryRight
-      map_add' := fun f₁ f₂ => by
-        ext m
-        rfl
-      map_smul' := fun c f => by
-        ext m
-        rfl
-      invFun := ContinuousMultilinearMap.curryRight
-      left_inv := ContinuousMultilinearMap.curry_uncurryRight
-      right_inv := ContinuousMultilinearMap.uncurry_curryRight } (fun f => by
-        simp only [uncurryRight, LinearEquiv.coe_mk]
-        exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _) fun f => by
-          simp only [curryRight, LinearEquiv.coe_symm_mk]
-          exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _
+    { toFun := ContinuousMultilinearMap.curryRight
+      map_add' := fun f₁ f₂ => rfl
+      map_smul' := fun c f => rfl
+      invFun := ContinuousMultilinearMap.uncurryRight
+      left_inv := ContinuousMultilinearMap.uncurry_curryRight
+      right_inv := ContinuousMultilinearMap.curry_uncurryRight }
+    (fun f => by
+      simp only [curryRight, LinearEquiv.coe_mk]
+      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
+    (fun f => by
+      simp only [uncurryRight, LinearEquiv.coe_symm_mk]
+      exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _)
 
 variable (n G')
 
@@ -310,47 +304,48 @@ isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEqui
 For a version allowing dependent types, see `continuousMultilinearCurryRightEquiv`. When there
 are no dependent types, use the primed version as it helps Lean a lot for unification.
 
-The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
+The direct and inverse maps are given by `f.curryRight` and `f.uncurryRight`. Use these
 unless you need the full framework of linear isometric equivs. -/
-def continuousMultilinearCurryRightEquiv' : (G[×n]→L[𝕜] G →L[𝕜] G') ≃ₗᵢ[𝕜] G[×n.succ]→L[𝕜] G' :=
+def continuousMultilinearCurryRightEquiv' : (G[×n.succ]→L[𝕜] G') ≃ₗᵢ[𝕜] G[×n]→L[𝕜] G →L[𝕜] G' :=
   continuousMultilinearCurryRightEquiv 𝕜 (fun _ => G) G'
 
 variable {n 𝕜 G Ei G'}
 
 @[simp]
 theorem continuousMultilinearCurryRightEquiv_apply
-    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G))
-    (v : ∀ i, Ei i) : (continuousMultilinearCurryRightEquiv 𝕜 Ei G) f v = f (init v) (v (last n)) :=
+    (f : ContinuousMultilinearMap 𝕜 Ei G) (v : Π i : Fin n, Ei <| castSucc i) (x : Ei (last n)) :
+    continuousMultilinearCurryRightEquiv 𝕜 Ei G f v x = f (snoc v x) :=
   rfl
 
 @[simp]
-theorem continuousMultilinearCurryRightEquiv_symm_apply (f : ContinuousMultilinearMap 𝕜 Ei G)
-    (v : ∀ i : Fin n, Ei <| castSucc i) (x : Ei (last n)) :
-    (continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm f v x = f (snoc v x) :=
+theorem continuousMultilinearCurryRightEquiv_symm_apply
+    (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G))
+    (v : Π i, Ei i) :
+    (continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm f v = f (init v) (v (last n)) :=
   rfl
 
 @[simp]
-theorem continuousMultilinearCurryRightEquiv_apply' (f : G[×n]→L[𝕜] G →L[𝕜] G')
-    (v : Fin (n + 1) → G) :
-    continuousMultilinearCurryRightEquiv' 𝕜 n G G' f v = f (init v) (v (last n)) :=
+theorem continuousMultilinearCurryRightEquiv_apply'
+    (f : G[×n.succ]→L[𝕜] G') (v : Fin n → G) (x : G) :
+    continuousMultilinearCurryRightEquiv' 𝕜 n G G' f v x = f (snoc v x) :=
   rfl
 
 @[simp]
-theorem continuousMultilinearCurryRightEquiv_symm_apply' (f : G[×n.succ]→L[𝕜] G')
-    (v : Fin n → G) (x : G) :
-    (continuousMultilinearCurryRightEquiv' 𝕜 n G G').symm f v x = f (snoc v x) :=
+theorem continuousMultilinearCurryRightEquiv_symm_apply'
+    (f : G[×n]→L[𝕜] G →L[𝕜] G') (v : Fin (n + 1) → G) :
+    (continuousMultilinearCurryRightEquiv' 𝕜 n G G').symm f v = f (init v) (v (last n)) :=
   rfl
 
 @[simp]
 theorem ContinuousMultilinearMap.curryRight_norm (f : ContinuousMultilinearMap 𝕜 Ei G) :
     ‖f.curryRight‖ = ‖f‖ :=
-  (continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm.norm_map f
+  (continuousMultilinearCurryRightEquiv 𝕜 Ei G).norm_map f
 
 @[simp]
 theorem ContinuousMultilinearMap.uncurryRight_norm
     (f : ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G)) :
     ‖f.uncurryRight‖ = ‖f‖ :=
-  (continuousMultilinearCurryRightEquiv 𝕜 Ei G).norm_map f
+  (continuousMultilinearCurryRightEquiv 𝕜 Ei G).symm.norm_map f
 
 /-!
 #### Currying with `0` variables
@@ -365,63 +360,57 @@ derivatives, we register this isomorphism. -/
 section
 
 /-- Associating to a continuous multilinear map in `0` variables the unique value it takes. -/
-def ContinuousMultilinearMap.uncurry0 (f : ContinuousMultilinearMap 𝕜 (fun _ : Fin 0 => G) G') :
+def ContinuousMultilinearMap.curry0 (f : ContinuousMultilinearMap 𝕜 (fun _ : Fin 0 => G) G') :
     G' :=
   f 0
 
-variable (𝕜 G)
-
+variable (𝕜 G) in
 /-- Associating to an element `x` of a vector space `E₂` the continuous multilinear map in `0`
 variables taking the (unique) value `x` -/
-def ContinuousMultilinearMap.curry0 (x : G') : G[×0]→L[𝕜] G' :=
+def ContinuousMultilinearMap.uncurry0 (x : G') : G[×0]→L[𝕜] G' :=
   ContinuousMultilinearMap.constOfIsEmpty 𝕜 _ x
 
-variable {G}
-
+variable (𝕜) in
 @[simp]
-theorem ContinuousMultilinearMap.curry0_apply (x : G') (m : Fin 0 → G) :
-    ContinuousMultilinearMap.curry0 𝕜 G x m = x :=
+theorem ContinuousMultilinearMap.uncurry0_apply (x : G') (m : Fin 0 → G) :
+    ContinuousMultilinearMap.uncurry0 𝕜 G x m = x :=
   rfl
 
-variable {𝕜}
-
 @[simp]
-theorem ContinuousMultilinearMap.uncurry0_apply (f : G[×0]→L[𝕜] G') : f.uncurry0 = f 0 :=
+theorem ContinuousMultilinearMap.curry0_apply (f : G[×0]→L[𝕜] G') : f.curry0 = f 0 :=
   rfl
 
 @[simp]
-theorem ContinuousMultilinearMap.apply_zero_curry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
-    ContinuousMultilinearMap.curry0 𝕜 G (f x) = f := by
+theorem ContinuousMultilinearMap.apply_zero_uncurry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
+    ContinuousMultilinearMap.uncurry0 𝕜 G (f x) = f := by
   ext m
   simp [Subsingleton.elim x m]
 
 theorem ContinuousMultilinearMap.uncurry0_curry0 (f : G[×0]→L[𝕜] G') :
-    ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0 = f := by simp
-
-variable (𝕜 G)
+    ContinuousMultilinearMap.uncurry0 𝕜 G f.curry0 = f := by simp
 
+variable (𝕜 G) in
 theorem ContinuousMultilinearMap.curry0_uncurry0 (x : G') :
-    (ContinuousMultilinearMap.curry0 𝕜 G x).uncurry0 = x :=
+    (ContinuousMultilinearMap.uncurry0 𝕜 G x).curry0 = x :=
   rfl
 
+variable (𝕜 G) in
 @[simp]
-theorem ContinuousMultilinearMap.curry0_norm (x : G') :
-    ‖ContinuousMultilinearMap.curry0 𝕜 G x‖ = ‖x‖ :=
+theorem ContinuousMultilinearMap.uncurry0_norm (x : G') :
+    ‖ContinuousMultilinearMap.uncurry0 𝕜 G x‖ = ‖x‖ :=
   norm_constOfIsEmpty _ _ _
 
-variable {𝕜 G}
-
 @[simp]
 theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
     ‖f x‖ = ‖f‖ := by
   obtain rfl : x = 0 := Subsingleton.elim _ _
   refine le_antisymm (by simpa using f.le_opNorm 0) ?_
-  have : ‖ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0‖ ≤ ‖f.uncurry0‖ :=
+  have : ‖ContinuousMultilinearMap.uncurry0 𝕜 G f.curry0‖ ≤ ‖f.curry0‖ :=
     ContinuousMultilinearMap.opNorm_le_bound _ (norm_nonneg _) fun m => by
-      simp [-ContinuousMultilinearMap.apply_zero_curry0]
+      simp [-ContinuousMultilinearMap.apply_zero_uncurry0]
   simpa [-Matrix.zero_empty] using this
 
-theorem ContinuousMultilinearMap.uncurry0_norm (f : G[×0]→L[𝕜] G') : ‖f.uncurry0‖ = ‖f‖ := by simp
+theorem ContinuousMultilinearMap.curry0_norm (f : G[×0]→L[𝕜] G') : ‖f.curry0‖ = ‖f‖ := by simp
 
 variable (𝕜 G G')
 
@@ -431,13 +420,13 @@ maps in `0` variables with values in this normed space.
 The direct and inverse maps are `uncurry0` and `curry0`. Use these unless you need the full
 framework of linear isometric equivs. -/
 def continuousMultilinearCurryFin0 : (G[×0]→L[𝕜] G') ≃ₗᵢ[𝕜] G' where
-  toFun f := ContinuousMultilinearMap.uncurry0 f
-  invFun f := ContinuousMultilinearMap.curry0 𝕜 G f
+  toFun f := ContinuousMultilinearMap.curry0 f
+  invFun f := ContinuousMultilinearMap.uncurry0 𝕜 G f
   map_add' _ _ := rfl
   map_smul' _ _ := rfl
   left_inv := ContinuousMultilinearMap.uncurry0_curry0
   right_inv := ContinuousMultilinearMap.curry0_uncurry0 𝕜 G
-  norm_map' := ContinuousMultilinearMap.uncurry0_norm
+  norm_map' := ContinuousMultilinearMap.curry0_norm
 
 variable {𝕜 G G'}
 
@@ -461,7 +450,7 @@ variable (𝕜 G G')
 /-- Continuous multilinear maps from `G^1` to `G'` are isomorphic with continuous linear maps from
 `G` to `G'`. -/
 def continuousMultilinearCurryFin1 : (G[×1]→L[𝕜] G') ≃ₗᵢ[𝕜] G →L[𝕜] G' :=
-  (continuousMultilinearCurryRightEquiv 𝕜 (fun _ : Fin 1 => G) G').symm.trans
+  (continuousMultilinearCurryRightEquiv 𝕜 (fun _ : Fin 1 => G) G').trans
     (continuousMultilinearCurryFin0 𝕜 G (G →L[𝕜] G'))
 
 variable {𝕜 G G'}
@@ -614,3 +603,58 @@ theorem curryFinFinset_apply_const (hk : s.card = k) (hl : sᶜ.card = l) (f : G
 end
 
 end ContinuousMultilinearMap
+
+namespace ContinuousLinearMap
+
+variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+  [NormedAddCommGroup G] [NormedSpace 𝕜 G]
+
+/-- Given a linear map into continuous multilinear maps
+`B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F`, one can not always uncurry it as `G` and `E` might
+live in a different universe. However, one can always lift it to a continuous multilinear map
+on `(G × (Π i, E i)) ^ (1 + n)`, which maps `(v_0, ..., v_n)` to `B (g_0) (u_1, ..., u_n)` where
+`g_0` is the `G`-coordinate of `v_0` and `u_i` is the `E_i` coordinate of `v_i`. -/
+noncomputable def continuousMultilinearMapOption (B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F) :
+    ContinuousMultilinearMap 𝕜 (fun (_ : Option ι) ↦ (G × (Π i, E i))) F :=
+  MultilinearMap.mkContinuous
+  { toFun := fun p ↦ B (p none).1 (fun i ↦ (p i).2 i)
+    map_add' := by
+      intro inst v j x y
+      match j with
+      | none => simp
+      | some j =>
+        classical
+        have B z : (fun i ↦ (Function.update v (some j) z (some i)).2 i) =
+            Function.update (fun (i : ι) ↦ (v i).2 i) j (z.2 j) := by
+          ext i
+          rcases eq_or_ne i j with rfl | hij
+          · simp
+          · simp [hij]
+        simp [B]
+    map_smul' := by
+      intro inst v j c x
+      match j with
+      | none => simp
+      | some j =>
+        classical
+        have B z : (fun i ↦ (Function.update v (some j) z (some i)).2 i) =
+            Function.update (fun (i : ι) ↦ (v i).2 i) j (z.2 j) := by
+          ext i
+          rcases eq_or_ne i j with rfl | hij
+          · simp
+          · simp [hij]
+        simp [B] } (‖B‖) <| by
+  intro b
+  simp only [MultilinearMap.coe_mk, Fintype.prod_option]
+  apply (ContinuousMultilinearMap.le_opNorm _ _).trans
+  rw [← mul_assoc]
+  gcongr with i _
+  · apply (B.le_opNorm _).trans
+    gcongr
+    exact norm_fst_le _
+  · exact (norm_le_pi_norm _ _).trans (norm_snd_le _)
+
+lemma continuousMultilinearMapOption_apply_eq_self (B : G →L[𝕜] ContinuousMultilinearMap 𝕜 E F)
+    (a : G) (v : Π i, E i) : B.continuousMultilinearMapOption (fun _ ↦ (a, v)) = B a v := rfl
+
+end ContinuousLinearMap
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean
index 7c3c4398246cc..5d787cb4490a6 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean
@@ -174,7 +174,7 @@ theorem opNorm_neg (f : E →SL[σ₁₂] F) : ‖-f‖ = ‖f‖ := by simp onl
 @[deprecated (since := "2024-02-02")] alias op_norm_neg := opNorm_neg
 
 theorem opNorm_nonneg (f : E →SL[σ₁₂] F) : 0 ≤ ‖f‖ :=
-  Real.sInf_nonneg _ fun _ ↦ And.left
+  Real.sInf_nonneg fun _ ↦ And.left
 
 @[deprecated (since := "2024-02-02")] alias op_norm_nonneg := opNorm_nonneg
 
@@ -227,7 +227,7 @@ theorem opNorm_le_iff {f : E →SL[σ₁₂] F} {M : ℝ} (hMp : 0 ≤ M) :
 @[deprecated (since := "2024-02-02")] alias op_norm_le_iff := opNorm_le_iff
 
 theorem ratio_le_opNorm : ‖f x‖ / ‖x‖ ≤ ‖f‖ :=
-  div_le_of_nonneg_of_le_mul (norm_nonneg _) f.opNorm_nonneg (le_opNorm _ _)
+  div_le_of_le_mul₀ (norm_nonneg _) f.opNorm_nonneg (le_opNorm _ _)
 
 @[deprecated (since := "2024-02-02")] alias ratio_le_op_norm := ratio_le_opNorm
 
@@ -264,7 +264,7 @@ theorem opNorm_le_of_shell' {f : E →SL[σ₁₂] F} {ε C : ℝ} (ε_pos : 0 <
   · refine opNorm_le_of_ball ε_pos hC fun x hx => hf x ?_ ?_
     · simp [h0]
     · rwa [ball_zero_eq] at hx
-  · rw [← inv_inv c, norm_inv, inv_lt_one_iff_of_pos (norm_pos_iff.2 <| inv_ne_zero h0)] at hc
+  · rw [← inv_inv c, norm_inv, inv_lt_one₀ (norm_pos_iff.2 <| inv_ne_zero h0)] at hc
     refine opNorm_le_of_shell ε_pos hC hc ?_
     rwa [norm_inv, div_eq_mul_inv, inv_inv]
 
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
index 07c05d0a62e73..61f5199b50531 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
@@ -382,7 +382,7 @@ variable {𝕜₁' : Type*} {𝕜₂' : Type*} [NontriviallyNormedField 𝕜₁'
   [RingHomIsometric σ₂₃] [RingHomIsometric σ₁₃'] [RingHomIsometric σ₂₃']
 
 /-- Compose a bilinear map `E →SL[σ₁₃] F →SL[σ₂₃] G` with two linear maps
-`E' →SL[σ₁'] E` and `F' →SL[σ₂'] F`.  -/
+`E' →SL[σ₁'] E` and `F' →SL[σ₂'] F`. -/
 def bilinearComp (f : E →SL[σ₁₃] F →SL[σ₂₃] G) (gE : E' →SL[σ₁'] E) (gF : F' →SL[σ₂'] F) :
     E' →SL[σ₁₃'] F' →SL[σ₂₃'] G :=
   ((f.comp gE).flip.comp gF).flip
@@ -409,6 +409,54 @@ theorem map_add_add (f : E →L[𝕜] Fₗ →L[𝕜] Gₗ) (x x' : E) (y y' : F
   simp only [map_add, add_apply, coe_deriv₂, add_assoc]
   abel
 
+/-- The norm of the tensor product of a scalar linear map and of an element of a normed space
+is the product of the norms. -/
+@[simp]
+theorem norm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖ = ‖c‖ * ‖f‖ := by
+  refine le_antisymm ?_ ?_
+  · refine opNorm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) fun x => ?_
+    calc
+      ‖c x • f‖ = ‖c x‖ * ‖f‖ := norm_smul _ _
+      _ ≤ ‖c‖ * ‖x‖ * ‖f‖ := mul_le_mul_of_nonneg_right (le_opNorm _ _) (norm_nonneg _)
+      _ = ‖c‖ * ‖f‖ * ‖x‖ := by ring
+  · obtain hf | hf := (norm_nonneg f).eq_or_gt
+    · simp [hf]
+    · rw [← le_div_iff₀ hf]
+      refine opNorm_le_bound _ (div_nonneg (norm_nonneg _) (norm_nonneg f)) fun x => ?_
+      rw [div_mul_eq_mul_div, le_div_iff₀ hf]
+      calc
+        ‖c x‖ * ‖f‖ = ‖c x • f‖ := (norm_smul _ _).symm
+        _ = ‖smulRight c f x‖ := rfl
+        _ ≤ ‖smulRight c f‖ * ‖x‖ := le_opNorm _ _
+
+/-- The non-negative norm of the tensor product of a scalar linear map and of an element of a normed
+space is the product of the non-negative norms. -/
+@[simp]
+theorem nnnorm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖₊ = ‖c‖₊ * ‖f‖₊ :=
+  NNReal.eq <| c.norm_smulRight_apply f
+
+variable (𝕜 E Fₗ) in
+/-- `ContinuousLinearMap.smulRight` as a continuous trilinear map:
+`smulRightL (c : E →L[𝕜] 𝕜) (f : F) (x : E) = c x • f`. -/
+def smulRightL : (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ :=
+  LinearMap.mkContinuous₂
+    { toFun := smulRightₗ
+      map_add' := fun c₁ c₂ => by
+        ext x
+        simp only [add_smul, coe_smulRightₗ, add_apply, smulRight_apply, LinearMap.add_apply]
+      map_smul' := fun m c => by
+        ext x
+        dsimp
+        rw [smul_smul] }
+    1 fun c x => by
+      simp only [coe_smulRightₗ, one_mul, norm_smulRight_apply, LinearMap.coe_mk, AddHom.coe_mk,
+        le_refl]
+
+
+@[simp]
+theorem norm_smulRightL_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRightL 𝕜 E Fₗ c f‖ = ‖c‖ * ‖f‖ :=
+  norm_smulRight_apply c f
+
 end ContinuousLinearMap
 
 end SemiNormed
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean
index 03c7738155ba1..783b55750d8fb 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean
@@ -181,16 +181,16 @@ variable [CompleteSpace F] (e : E →L[𝕜] Fₗ) (h_dense : DenseRange e)
 
 section
 
-variable (h_e : UniformInducing e)
+variable (h_e : IsUniformInducing e)
 
 /-- Extension of a continuous linear map `f : E →SL[σ₁₂] F`, with `E` a normed space and `F` a
-complete normed space, along a uniform and dense embedding `e : E →L[𝕜] Fₗ`.  -/
+complete normed space, along a uniform and dense embedding `e : E →L[𝕜] Fₗ`. -/
 def extend : Fₗ →SL[σ₁₂] F :=
   -- extension of `f` is continuous
   have cont := (uniformContinuous_uniformly_extend h_e h_dense f.uniformContinuous).continuous
   -- extension of `f` agrees with `f` on the domain of the embedding `e`
   have eq := uniformly_extend_of_ind h_e h_dense f.uniformContinuous
-  { toFun := (h_e.denseInducing h_dense).extend f
+  { toFun := (h_e.isDenseInducing h_dense).extend f
     map_add' := by
       refine h_dense.induction_on₂ ?_ ?_
       · exact isClosed_eq (cont.comp continuous_add)
@@ -208,10 +208,10 @@ def extend : Fₗ →SL[σ₁₂] F :=
         exact ContinuousLinearMap.map_smulₛₗ _ _ _
     cont }
 
--- Porting note: previously `(h_e.denseInducing h_dense)` was inferred.
+-- Porting note: previously `(h_e.isDenseInducing h_dense)` was inferred.
 @[simp]
 theorem extend_eq (x : E) : extend f e h_dense h_e (e x) = f x :=
-  DenseInducing.extend_eq (h_e.denseInducing h_dense) f.cont _
+  IsDenseInducing.extend_eq (h_e.isDenseInducing h_dense) f.cont _
 
 theorem extend_unique (g : Fₗ →SL[σ₁₂] F) (H : g.comp e = f) : extend f e h_dense h_e = g :=
   ContinuousLinearMap.coeFn_injective <|
@@ -230,7 +230,7 @@ variable {N : ℝ≥0} (h_e : ∀ x, ‖x‖ ≤ N * ‖e x‖) [RingHomIsometri
 /-- If a dense embedding `e : E →L[𝕜] G` expands the norm by a constant factor `N⁻¹`, then the
 norm of the extension of `f` along `e` is bounded by `N * ‖f‖`. -/
 theorem opNorm_extend_le :
-    ‖f.extend e h_dense (uniformEmbedding_of_bound _ h_e).toUniformInducing‖ ≤ N * ‖f‖ := by
+    ‖f.extend e h_dense (isUniformEmbedding_of_bound _ h_e).isUniformInducing‖ ≤ N * ‖f‖ := by
   -- Add `opNorm_le_of_dense`?
   refine opNorm_le_bound _ ?_ (isClosed_property h_dense (isClosed_le ?_ ?_) fun x ↦ ?_)
   · cases le_total 0 N with
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean
index 9eae72e5fb45b..a42244f918fa7 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean
@@ -121,7 +121,7 @@ examples. Any algebra with an approximate identity (e.g., $$L^1$$) is also regul
 
 This is a useful class because it gives rise to a nice norm on the unitization; in particular it is
 a C⋆-norm when the norm on `A` is a C⋆-norm. -/
-class _root_.RegularNormedAlgebra : Prop :=
+class _root_.RegularNormedAlgebra : Prop where
   /-- The left regular representation of the algebra on itself is an isometry. -/
   isometry_mul' : Isometry (mul 𝕜 𝕜')
 
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean
index 7461f70c18dfc..020de596c8f44 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean
@@ -156,7 +156,7 @@ theorem exists_lt_apply_of_lt_opNNNorm {𝕜 𝕜₂ E F : Type*} [NormedAddComm
   obtain ⟨k, hk₁, hk₂⟩ := NormedField.exists_lt_nnnorm_lt 𝕜 hy
   refine ⟨k • y, (nnnorm_smul k y).symm ▸ (NNReal.lt_inv_iff_mul_lt hy').1 hk₂, ?_⟩
   have : ‖σ₁₂ k‖₊ = ‖k‖₊ := Subtype.ext RingHomIsometric.is_iso
-  rwa [map_smulₛₗ f, nnnorm_smul, ← NNReal.div_lt_iff hfy, div_eq_mul_inv, this]
+  rwa [map_smulₛₗ f, nnnorm_smul, ← div_lt_iff₀ hfy.bot_lt, div_eq_mul_inv, this]
 
 @[deprecated (since := "2024-02-02")]
 alias exists_lt_apply_of_lt_op_nnnorm := exists_lt_apply_of_lt_opNNNorm
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean
index d07b5ac33cae4..253805834044d 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean
@@ -165,6 +165,15 @@ theorem norm_toContinuousLinearMap_comp [RingHomIsometric σ₁₂] (f : F →
   opNorm_ext (f.toContinuousLinearMap.comp g) g fun x => by
     simp only [norm_map, coe_toContinuousLinearMap, coe_comp', Function.comp_apply]
 
+/-- Composing on the left with a linear isometry gives a linear isometry between spaces of
+continuous linear maps. -/
+def postcomp [RingHomIsometric σ₁₂] [RingHomIsometric σ₁₃] (a : F →ₛₗᵢ[σ₂₃] G) :
+    (E →SL[σ₁₂] F) →ₛₗᵢ[σ₂₃] (E →SL[σ₁₃] G) where
+  toFun f := a.toContinuousLinearMap.comp f
+  map_add' f g := by simp
+  map_smul' c f := by simp
+  norm_map' f := by simp [a.norm_toContinuousLinearMap_comp]
+
 end LinearIsometry
 
 end
@@ -200,58 +209,6 @@ theorem opNorm_comp_linearIsometryEquiv (f : F →SL[σ₂₃] G) (g : F' ≃ₛ
 @[deprecated (since := "2024-02-02")]
 alias op_norm_comp_linearIsometryEquiv := opNorm_comp_linearIsometryEquiv
 
-/-- The norm of the tensor product of a scalar linear map and of an element of a normed space
-is the product of the norms. -/
-@[simp]
-theorem norm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖ = ‖c‖ * ‖f‖ := by
-  refine le_antisymm ?_ ?_
-  · refine opNorm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) fun x => ?_
-    calc
-      ‖c x • f‖ = ‖c x‖ * ‖f‖ := norm_smul _ _
-      _ ≤ ‖c‖ * ‖x‖ * ‖f‖ := mul_le_mul_of_nonneg_right (le_opNorm _ _) (norm_nonneg _)
-      _ = ‖c‖ * ‖f‖ * ‖x‖ := by ring
-  · by_cases h : f = 0
-    · simp [h]
-    · have : 0 < ‖f‖ := norm_pos_iff.2 h
-      rw [← le_div_iff₀ this]
-      refine opNorm_le_bound _ (div_nonneg (norm_nonneg _) (norm_nonneg f)) fun x => ?_
-      rw [div_mul_eq_mul_div, le_div_iff₀ this]
-      calc
-        ‖c x‖ * ‖f‖ = ‖c x • f‖ := (norm_smul _ _).symm
-        _ = ‖smulRight c f x‖ := rfl
-        _ ≤ ‖smulRight c f‖ * ‖x‖ := le_opNorm _ _
-
-/-- The non-negative norm of the tensor product of a scalar linear map and of an element of a normed
-space is the product of the non-negative norms. -/
-@[simp]
-theorem nnnorm_smulRight_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRight c f‖₊ = ‖c‖₊ * ‖f‖₊ :=
-  NNReal.eq <| c.norm_smulRight_apply f
-
-variable (𝕜 E Fₗ)
-
-
-/-- `ContinuousLinearMap.smulRight` as a continuous trilinear map:
-`smulRightL (c : E →L[𝕜] 𝕜) (f : F) (x : E) = c x • f`. -/
-def smulRightL : (E →L[𝕜] 𝕜) →L[𝕜] Fₗ →L[𝕜] E →L[𝕜] Fₗ :=
-  LinearMap.mkContinuous₂
-    { toFun := smulRightₗ
-      map_add' := fun c₁ c₂ => by
-        ext x
-        simp only [add_smul, coe_smulRightₗ, add_apply, smulRight_apply, LinearMap.add_apply]
-      map_smul' := fun m c => by
-        ext x
-        dsimp
-        rw [smul_smul] }
-    1 fun c x => by
-      simp only [coe_smulRightₗ, one_mul, norm_smulRight_apply, LinearMap.coe_mk, AddHom.coe_mk,
-        le_refl]
-
-variable {𝕜 E Fₗ}
-
-@[simp]
-theorem norm_smulRightL_apply (c : E →L[𝕜] 𝕜) (f : Fₗ) : ‖smulRightL 𝕜 E Fₗ c f‖ = ‖c‖ * ‖f‖ :=
-  norm_smulRight_apply c f
-
 @[simp]
 theorem norm_smulRightL (c : E →L[𝕜] 𝕜) [Nontrivial Fₗ] : ‖smulRightL 𝕜 E Fₗ c‖ = ‖c‖ :=
   ContinuousLinearMap.homothety_norm _ c.norm_smulRight_apply
@@ -365,33 +322,30 @@ protected theorem NormedSpace.equicontinuous_TFAE : List.TFAE
       BddAbove (Set.range (‖f ·‖)),
       (⨆ i, (‖f i‖₊ : ENNReal)) < ⊤ ] := by
   -- `1 ↔ 2 ↔ 3` follows from `uniformEquicontinuous_of_equicontinuousAt_zero`
-  tfae_have 1 → 3
-  · exact uniformEquicontinuous_of_equicontinuousAt_zero f
-  tfae_have 3 → 2
-  · exact UniformEquicontinuous.equicontinuous
-  tfae_have 2 → 1
-  · exact fun H ↦ H 0
+  tfae_have 1 → 3 := uniformEquicontinuous_of_equicontinuousAt_zero f
+  tfae_have 3 → 2 := UniformEquicontinuous.equicontinuous
+  tfae_have 2 → 1 := fun H ↦ H 0
   -- `4 ↔ 5 ↔ 6 ↔ 7 ↔ 8 ↔ 9` is morally trivial, we just have to use a lot of rewriting
   -- and `congr` lemmas
-  tfae_have 4 ↔ 5
-  · rw [exists_ge_and_iff_exists]
+  tfae_have 4 ↔ 5 := by
+    rw [exists_ge_and_iff_exists]
     exact fun C₁ C₂ hC ↦ forall₂_imp fun i x ↦ le_trans' <| by gcongr
-  tfae_have 5 ↔ 7
-  · refine exists_congr (fun C ↦ and_congr_right fun hC ↦ forall_congr' fun i ↦ ?_)
+  tfae_have 5 ↔ 7 := by
+    refine exists_congr (fun C ↦ and_congr_right fun hC ↦ forall_congr' fun i ↦ ?_)
     rw [ContinuousLinearMap.opNorm_le_iff hC]
-  tfae_have 7 ↔ 8
-  · simp_rw [bddAbove_iff_exists_ge (0 : ℝ), Set.forall_mem_range]
-  tfae_have 6 ↔ 8
-  · simp_rw [bddAbove_def, Set.forall_mem_range]
-  tfae_have 8 ↔ 9
-  · rw [ENNReal.iSup_coe_lt_top, ← NNReal.bddAbove_coe, ← Set.range_comp]
+  tfae_have 7 ↔ 8 := by
+    simp_rw [bddAbove_iff_exists_ge (0 : ℝ), Set.forall_mem_range]
+  tfae_have 6 ↔ 8 := by
+    simp_rw [bddAbove_def, Set.forall_mem_range]
+  tfae_have 8 ↔ 9 := by
+    rw [ENNReal.iSup_coe_lt_top, ← NNReal.bddAbove_coe, ← Set.range_comp]
     rfl
   -- `3 ↔ 4` is the interesting part of the result. It is essentially a combination of
   -- `WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm` which turns
   -- equicontinuity into existence of some continuous seminorm and
   -- `Seminorm.bound_of_continuous_normedSpace` which characterize such seminorms.
-  tfae_have 3 ↔ 4
-  · refine ((norm_withSeminorms 𝕜₂ F).uniformEquicontinuous_iff_exists_continuous_seminorm _).trans
+  tfae_have 3 ↔ 4 := by
+    refine ((norm_withSeminorms 𝕜₂ F).uniformEquicontinuous_iff_exists_continuous_seminorm _).trans
       ?_
     rw [forall_const]
     constructor
diff --git a/Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean b/Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
index a31afd3501b4f..299585582989c 100644
--- a/Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
+++ b/Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean
@@ -179,12 +179,13 @@ theorem norm_eval_le_injectiveSeminorm (f : ContinuousMultilinearMap 𝕜 E F) (
   set f' := MultilinearMap.mkContinuous f'₀ ‖f‖ hf'₀
   have hnorm : ‖f'‖ ≤ ‖f‖ := (f'.opNorm_le_iff (norm_nonneg f)).mpr hf'₀
   have heq : e (lift f'.toMultilinearMap x) = lift f.toMultilinearMap x := by
-    induction' x using PiTensorProduct.induction_on with a m _ _ hx hy
-    · simp only [lift_symm, map_smul, lift.tprod, ContinuousMultilinearMap.coe_coe,
+    induction x using PiTensorProduct.induction_on with
+    | smul_tprod =>
+      simp only [lift_symm, map_smul, lift.tprod, ContinuousMultilinearMap.coe_coe,
       MultilinearMap.coe_mkContinuous, LinearMap.compMultilinearMap_apply, LinearMap.coe_comp,
       LinearEquiv.coe_coe, Function.comp_apply, LinearEquiv.apply_symm_apply, SetLike.val_smul,
       LinearMap.codRestrict_apply, f', f'₀]
-    · simp only [map_add, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, hx, hy]
+    | add _ _ hx hy => simp only [map_add, Submodule.coe_add, hx, hy]
   suffices h : ‖lift f'.toMultilinearMap x‖ ≤ ‖f'‖ * injectiveSeminorm x by
     change ‖(e (lift f'.toMultilinearMap x)).1‖ ≤ _ at h
     rw [heq] at h
@@ -210,8 +211,7 @@ theorem injectiveSeminorm_le_projectiveSeminorm :
     existsi PUnit, inferInstance, inferInstance
     ext x
     simp only [Seminorm.zero_apply, Seminorm.comp_apply, coe_normSeminorm]
-    have heq : toDualContinuousMultilinearMap PUnit x = 0 := by ext _
-    rw [heq, norm_zero]
+    rw [Subsingleton.elim (toDualContinuousMultilinearMap PUnit x) 0, norm_zero]
   · intro p hp
     simp only [Set.mem_setOf_eq] at hp
     obtain ⟨G, _, _, h⟩ := hp
@@ -343,11 +343,12 @@ theorem mapL_coe : (mapL f).toLinearMap = map (fun i ↦ (f i).toLinearMap) := b
 
 @[simp]
 theorem mapL_apply (x : ⨂[𝕜] i, E i) : mapL f x = map (fun i ↦ (f i).toLinearMap) x := by
-  induction' x using PiTensorProduct.induction_on with _ _ _ _ hx hy
-  · simp only [mapL, map_smul, liftIsometry_apply_apply, lift.tprod,
+  induction x using PiTensorProduct.induction_on with
+  | smul_tprod =>
+    simp only [mapL, map_smul, liftIsometry_apply_apply, lift.tprod,
     ContinuousMultilinearMap.coe_coe, ContinuousMultilinearMap.compContinuousLinearMap_apply,
     tprodL_toFun, map_tprod, ContinuousLinearMap.coe_coe]
-  · simp only [map_add, hx, hy]
+  | add _ _ hx hy => simp only [map_add, hx, hy]
 
 /-- Given submodules `pᵢ ⊆ Eᵢ`, this is the natural map: `⨂[𝕜] i, pᵢ → ⨂[𝕜] i, Eᵢ`.
 This is the continuous version of `PiTensorProduct.mapIncl`.
diff --git a/Mathlib/Analysis/NormedSpace/Pointwise.lean b/Mathlib/Analysis/NormedSpace/Pointwise.lean
index bb061dfe9e186..866ca33561073 100644
--- a/Mathlib/Analysis/NormedSpace/Pointwise.lean
+++ b/Mathlib/Analysis/NormedSpace/Pointwise.lean
@@ -78,7 +78,7 @@ theorem smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • ball x r =
   ext y
   rw [mem_smul_set_iff_inv_smul_mem₀ hc]
   conv_lhs => rw [← inv_smul_smul₀ hc x]
-  simp [← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀]
+  simp [← div_eq_inv_mul, div_lt_iff₀ (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀]
 
 theorem smul_unitBall {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) ‖c‖ := by
   rw [_root_.smul_ball hc, smul_zero, mul_one]
diff --git a/Mathlib/Analysis/NormedSpace/RieszLemma.lean b/Mathlib/Analysis/NormedSpace/RieszLemma.lean
index 415a55f95a3f4..553d89feaaa42 100644
--- a/Mathlib/Analysis/NormedSpace/RieszLemma.lean
+++ b/Mathlib/Analysis/NormedSpace/RieszLemma.lean
@@ -50,7 +50,7 @@ theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃
       simp only [r', max_lt_iff, hr, true_and]
       norm_num
     have hlt : 0 < r' := lt_of_lt_of_le (by norm_num) (le_max_right r 2⁻¹)
-    have hdlt : d < d / r' := (lt_div_iff hlt).mpr ((mul_lt_iff_lt_one_right hdp).2 hr')
+    have hdlt : d < d / r' := (lt_div_iff₀ hlt).mpr ((mul_lt_iff_lt_one_right hdp).2 hr')
     obtain ⟨y₀, hy₀F, hxy₀⟩ : ∃ y ∈ F, dist x y < d / r' := (Metric.infDist_lt_iff hFn).mp hdlt
     have x_ne_y₀ : x - y₀ ∉ F := by
       by_contra h
@@ -63,7 +63,7 @@ theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃
       r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖ := by gcongr; apply le_max_left
       _ < d := by
         rw [← dist_eq_norm]
-        exact (lt_div_iff' hlt).1 hxy₀
+        exact (lt_div_iff₀' hlt).1 hxy₀
       _ ≤ dist x (y₀ + y) := Metric.infDist_le_dist_of_mem hy₀y
       _ = ‖x - y₀ - y‖ := by rw [sub_sub, dist_eq_norm]
 
@@ -82,7 +82,7 @@ theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖
     ∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ := by
   have Rpos : 0 < R := (norm_nonneg _).trans_lt hR
   have : ‖c‖ / R < 1 := by
-    rw [div_lt_iff Rpos]
+    rw [div_lt_iff₀ Rpos]
     simpa using hR
   rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩
   have x0 : x ≠ 0 := fun H => by simp [H] at xF
diff --git a/Mathlib/Analysis/ODE/PicardLindelof.lean b/Mathlib/Analysis/ODE/PicardLindelof.lean
index ad6c05fedff2b..9c46abf777613 100644
--- a/Mathlib/Analysis/ODE/PicardLindelof.lean
+++ b/Mathlib/Analysis/ODE/PicardLindelof.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Winston Yin
 -/
 import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.Topology.Algebra.Order.Floor
 import Mathlib.Topology.MetricSpace.Contracting
 
 /-!
@@ -169,9 +170,12 @@ def toContinuousMap : v.FunSpace ↪ C(Icc v.tMin v.tMax, E) :=
 instance : MetricSpace v.FunSpace :=
   MetricSpace.induced toContinuousMap toContinuousMap.injective inferInstance
 
-theorem uniformInducing_toContinuousMap : UniformInducing (@toContinuousMap _ _ _ v) :=
+theorem isUniformInducing_toContinuousMap : IsUniformInducing (@toContinuousMap _ _ _ v) :=
   ⟨rfl⟩
 
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_toContinuousMap := isUniformInducing_toContinuousMap
+
 theorem range_toContinuousMap :
     range toContinuousMap =
       {f : C(Icc v.tMin v.tMax, E) | f v.t₀ = v.x₀ ∧ LipschitzWith v.C f} := by
@@ -216,7 +220,7 @@ theorem dist_le_of_forall {f₁ f₂ : FunSpace v} {d : ℝ} (h : ∀ t, dist (f
     v.nonempty_Icc.to_subtype).2 h
 
 instance [CompleteSpace E] : CompleteSpace v.FunSpace := by
-  refine (completeSpace_iff_isComplete_range uniformInducing_toContinuousMap).2
+  refine (completeSpace_iff_isComplete_range isUniformInducing_toContinuousMap).2
       (IsClosed.isComplete ?_)
   rw [range_toContinuousMap, setOf_and]
   refine (isClosed_eq (ContinuousMap.continuous_eval_const _) continuous_const).inter ?_
@@ -284,7 +288,7 @@ variable [CompleteSpace E]
 theorem hasDerivWithinAt_next (t : Icc v.tMin v.tMax) :
     HasDerivWithinAt (f.next ∘ v.proj) (v t (f t)) (Icc v.tMin v.tMax) t := by
   haveI : Fact ((t : ℝ) ∈ Icc v.tMin v.tMax) := ⟨t.2⟩
-  simp only [(· ∘ ·), next_apply]
+  simp only [Function.comp_def, next_apply]
   refine HasDerivWithinAt.const_add _ ?_
   have : HasDerivWithinAt (∫ τ in v.t₀..·, f.vComp τ) (f.vComp t) (Icc v.tMin v.tMax) t :=
     integral_hasDerivWithinAt_right (f.intervalIntegrable_vComp _ _)
@@ -301,7 +305,7 @@ section
 
 theorem exists_contracting_iterate :
     ∃ (N : ℕ) (K : _), ContractingWith K (FunSpace.next : v.FunSpace → v.FunSpace)^[N] := by
-  rcases ((Real.tendsto_pow_div_factorial_atTop (v.L * v.tDist)).eventually
+  rcases ((FloorSemiring.tendsto_pow_div_factorial_atTop (v.L * v.tDist)).eventually
     (gt_mem_nhds zero_lt_one)).exists with ⟨N, hN⟩
   have : (0 : ℝ) ≤ (v.L * v.tDist) ^ N / N ! :=
     div_nonneg (pow_nonneg (mul_nonneg v.L.2 v.tDist_nonneg) _) (Nat.cast_nonneg _)
diff --git a/Mathlib/Analysis/Oscillation.lean b/Mathlib/Analysis/Oscillation.lean
index 16bc8089b08b1..c6327e3b9a059 100644
--- a/Mathlib/Analysis/Oscillation.lean
+++ b/Mathlib/Analysis/Oscillation.lean
@@ -3,8 +3,9 @@ Copyright (c) 2024 James Sundstrom. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: James Sundstrom
 -/
-import Mathlib.Topology.EMetricSpace.Diam
+import Mathlib.Data.ENNReal.Real
 import Mathlib.Order.WellFoundedSet
+import Mathlib.Topology.EMetricSpace.Diam
 
 /-!
 # Oscillation
diff --git a/Mathlib/Analysis/PSeries.lean b/Mathlib/Analysis/PSeries.lean
index 6b1fde6df52ee..a0365a83cfc7b 100644
--- a/Mathlib/Analysis/PSeries.lean
+++ b/Mathlib/Analysis/PSeries.lean
@@ -63,7 +63,7 @@ theorem le_sum_schlomilch' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f
 theorem le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
     (∑ k ∈ Ico 1 (2 ^ n), f k) ≤ ∑ k ∈ range n, 2 ^ k • f (2 ^ k) := by
   convert le_sum_schlomilch' hf (fun n => pow_pos zero_lt_two n)
-    (fun m n hm => pow_le_pow_right one_le_two hm) n using 2
+    (fun m n hm => pow_right_mono₀ one_le_two hm) n using 2
   simp [pow_succ, mul_two, two_mul]
 
 theorem le_sum_schlomilch (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
@@ -98,7 +98,7 @@ theorem sum_schlomilch_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f
 theorem sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
     (∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by
   convert sum_schlomilch_le' hf (fun n => pow_pos zero_lt_two n)
-    (fun m n hm => pow_le_pow_right one_le_two hm) n using 2
+    (fun m n hm => pow_right_mono₀ one_le_two hm) n using 2
   simp [pow_succ, mul_two, two_mul]
 
 theorem sum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (h_pos : ∀ n, 0 < u n)
@@ -162,8 +162,8 @@ theorem tsum_schlomilch_le {C : ℕ} (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n →
       le_trans ?_
         (add_le_add_left
           (mul_le_mul_of_nonneg_left (ENNReal.sum_le_tsum <| Finset.Ico (u 0 + 1) (u n + 1)) ?_) _)
-  simpa using Finset.sum_schlomilch_le hf h_pos h_nonneg hu h_succ_diff n
-  exact zero_le _
+  · simpa using Finset.sum_schlomilch_le hf h_pos h_nonneg hu h_succ_diff n
+  · exact zero_le _
 
 theorem tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
     (∑' k : ℕ, 2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k := by
@@ -283,7 +283,7 @@ theorem summable_nat_rpow_inv {p : ℝ} :
           (eventually_cofinite_ne 0)).exists
     apply hk₀
     rw [← pos_iff_ne_zero, ← @Nat.cast_pos ℝ] at hk₀
-    simpa [inv_lt_one_iff_of_pos (rpow_pos_of_pos hk₀ _), one_lt_rpow_iff_of_pos hk₀, hp,
+    simpa [inv_lt_one₀ (rpow_pos_of_pos hk₀ _), one_lt_rpow_iff_of_pos hk₀, hp,
       hp.not_lt, hk₀] using hk₁
 
 @[simp]
@@ -401,7 +401,7 @@ theorem sum_Ioo_inv_sq_le (k n : ℕ) : (∑ i ∈ Ioo k n, (i ^ 2 : α)⁻¹) 
       apply sum_le_sum_of_subset_of_nonneg
       · intro x hx
         simp only [mem_Ioo] at hx
-        simp only [hx, hx.2.le, mem_Ioc, le_max_iff, or_true_iff, and_self_iff]
+        simp only [hx, hx.2.le, mem_Ioc, le_max_iff, or_true, and_self_iff]
       · intro i _hi _hident
         positivity
     _ ≤ ((k + 1 : α) ^ 2)⁻¹ + ∑ i ∈ Ioc k.succ (max (k + 1) n), ((i : α) ^ 2)⁻¹ := by
@@ -416,7 +416,7 @@ theorem sum_Ioo_inv_sq_le (k n : ℕ) : (∑ i ∈ Ioo k n, (i ^ 2 : α)⁻¹) 
       have A : (1 : α) ≤ k + 1 := by simp only [le_add_iff_nonneg_left, Nat.cast_nonneg]
       simp_rw [← one_div]
       gcongr
-      simpa using pow_le_pow_right A one_le_two
+      simpa using pow_right_mono₀ A one_le_two
     _ = 2 / (k + 1) := by ring
 
 end
diff --git a/Mathlib/Analysis/Quaternion.lean b/Mathlib/Analysis/Quaternion.lean
index 0991d3a5949c9..3466df2251849 100644
--- a/Mathlib/Analysis/Quaternion.lean
+++ b/Mathlib/Analysis/Quaternion.lean
@@ -195,9 +195,9 @@ theorem continuous_im : Continuous fun q : ℍ => q.im := by
   simpa only [← sub_self_re] using continuous_id.sub (continuous_coe.comp continuous_re)
 
 instance : CompleteSpace ℍ :=
-  haveI : UniformEmbedding linearIsometryEquivTuple.toLinearEquiv.toEquiv.symm :=
-    linearIsometryEquivTuple.toContinuousLinearEquiv.symm.uniformEmbedding
-  (completeSpace_congr this).1 (by infer_instance)
+  haveI : IsUniformEmbedding linearIsometryEquivTuple.toLinearEquiv.toEquiv.symm :=
+    linearIsometryEquivTuple.toContinuousLinearEquiv.symm.isUniformEmbedding
+  (completeSpace_congr this).1 inferInstance
 
 section infinite_sum
 
diff --git a/Mathlib/Analysis/RCLike/Basic.lean b/Mathlib/Analysis/RCLike/Basic.lean
index 3b63c5f4bbcbc..69097ee6513a0 100644
--- a/Mathlib/Analysis/RCLike/Basic.lean
+++ b/Mathlib/Analysis/RCLike/Basic.lean
@@ -3,12 +3,13 @@ Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Frédéric Dupuis
 -/
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Algebra.Field
+import Mathlib.Algebra.BigOperators.Balance
+import Mathlib.Algebra.Order.BigOperators.Expect
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Analysis.CStarAlgebra.Basic
 import Mathlib.Analysis.Normed.Operator.ContinuousLinearMap
-import Mathlib.Analysis.Normed.Module.Basic
 import Mathlib.Data.Real.Sqrt
-import Mathlib.Algebra.Algebra.Field
 
 /-!
 # `RCLike`: a typeclass for ℝ or ℂ
@@ -40,7 +41,8 @@ their counterparts in `Mathlib/Analysis/Complex/Basic.lean` (which causes linter
 A few lemmas requiring heavier imports are in `Mathlib/Data/RCLike/Lemmas.lean`.
 -/
 
-open scoped ComplexConjugate
+open Fintype
+open scoped BigOperators ComplexConjugate
 
 section
 
@@ -231,10 +233,21 @@ theorem norm_ofReal (r : ℝ) : ‖(r : K)‖ = |r| :=
 /-! ### Characteristic zero -/
 
 -- see Note [lower instance priority]
-/-- ℝ and ℂ are both of characteristic zero.  -/
+/-- ℝ and ℂ are both of characteristic zero. -/
 instance (priority := 100) charZero_rclike : CharZero K :=
   (RingHom.charZero_iff (algebraMap ℝ K).injective).1 inferInstance
 
+@[rclike_simps, norm_cast]
+lemma ofReal_expect {α : Type*} (s : Finset α) (f : α → ℝ) : 𝔼 i ∈ s, f i = 𝔼 i ∈ s, (f i : K) :=
+  map_expect (algebraMap ..) ..
+
+@[norm_cast]
+lemma ofReal_balance {ι : Type*} [Fintype ι] (f : ι → ℝ) (i : ι) :
+    ((balance f i : ℝ) : K) = balance ((↑) ∘ f) i := map_balance (algebraMap ..) ..
+
+@[simp] lemma ofReal_comp_balance {ι : Type*} [Fintype ι] (f : ι → ℝ) :
+    ofReal ∘ balance f = balance (ofReal ∘ f : ι → K) := funext <| ofReal_balance _
+
 /-! ### The imaginary unit, `I` -/
 
 /-- The imaginary unit. -/
@@ -318,16 +331,14 @@ open List in
 /-- There are several equivalent ways to say that a number `z` is in fact a real number. -/
 theorem is_real_TFAE (z : K) : TFAE [conj z = z, ∃ r : ℝ, (r : K) = z, ↑(re z) = z, im z = 0] := by
   tfae_have 1 → 4
-  · intro h
+  | h => by
     rw [← @ofReal_inj K, im_eq_conj_sub, h, sub_self, mul_zero, zero_div,
       ofReal_zero]
   tfae_have 4 → 3
-  · intro h
+  | h => by
     conv_rhs => rw [← re_add_im z, h, ofReal_zero, zero_mul, add_zero]
-  tfae_have 3 → 2
-  · exact fun h => ⟨_, h⟩
-  tfae_have 2 → 1
-  · exact fun ⟨r, hr⟩ => hr ▸ conj_ofReal _
+  tfae_have 3 → 2 := fun h => ⟨_, h⟩
+  tfae_have 2 → 1 := fun ⟨r, hr⟩ => hr ▸ conj_ofReal _
   tfae_finish
 
 theorem conj_eq_iff_real {z : K} : conj z = z ↔ ∃ r : ℝ, z = (r : K) :=
@@ -503,11 +514,13 @@ theorem normSq_inv (z : K) : normSq z⁻¹ = (normSq z)⁻¹ :=
 theorem normSq_div (z w : K) : normSq (z / w) = normSq z / normSq w :=
   map_div₀ normSq z w
 
-@[rclike_simps] -- porting note (#10618): was `simp`
-theorem norm_conj {z : K} : ‖conj z‖ = ‖z‖ := by simp only [← sqrt_normSq_eq_norm, normSq_conj]
+@[simp 1100, rclike_simps]
+theorem norm_conj (z : K) : ‖conj z‖ = ‖z‖ := by simp only [← sqrt_normSq_eq_norm, normSq_conj]
+
+@[simp, rclike_simps] lemma nnnorm_conj (z : K) : ‖conj z‖₊ = ‖z‖₊ := by simp [nnnorm]
 
 instance (priority := 100) : CStarRing K where
-  norm_mul_self_le x := le_of_eq <| ((norm_mul _ _).trans <| congr_arg (· * ‖x‖) norm_conj).symm
+  norm_mul_self_le x := le_of_eq <| ((norm_mul _ _).trans <| congr_arg (· * ‖x‖) (norm_conj _)).symm
 
 /-! ### Cast lemmas -/
 
@@ -515,6 +528,9 @@ instance (priority := 100) : CStarRing K where
 theorem ofReal_natCast (n : ℕ) : ((n : ℝ) : K) = n :=
   map_natCast (algebraMap ℝ K) n
 
+@[rclike_simps, norm_cast]
+lemma ofReal_nnratCast (q : ℚ≥0) : ((q : ℝ) : K) = q := map_nnratCast (algebraMap ℝ K) _
+
 @[simp, rclike_simps] -- Porting note: removed `norm_cast`
 theorem natCast_re (n : ℕ) : re (n : K) = n := by rw [← ofReal_natCast, ofReal_re]
 
@@ -575,21 +591,56 @@ theorem norm_natCast (n : ℕ) : ‖(n : K)‖ = n := by
   rw [← ofReal_natCast]
   exact norm_of_nonneg (Nat.cast_nonneg n)
 
+@[simp, rclike_simps, norm_cast] lemma nnnorm_natCast (n : ℕ) : ‖(n : K)‖₊ = n := by simp [nnnorm]
+
 @[simp, rclike_simps]
 theorem norm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖(no_index (OfNat.ofNat n) : K)‖ = OfNat.ofNat n :=
   norm_natCast n
 
+@[simp, rclike_simps]
+lemma nnnorm_ofNat (n : ℕ) [n.AtLeastTwo] : ‖(no_index (OfNat.ofNat n) : K)‖₊ = OfNat.ofNat n :=
+  nnnorm_natCast n
+
+lemma norm_two : ‖(2 : K)‖ = 2 := norm_ofNat 2
+lemma nnnorm_two : ‖(2 : K)‖₊ = 2 := nnnorm_ofNat 2
+
+@[simp, rclike_simps, norm_cast]
+lemma norm_nnratCast (q : ℚ≥0) : ‖(q : K)‖ = q := by
+  rw [← ofReal_nnratCast]; exact norm_of_nonneg q.cast_nonneg
+
+@[simp, rclike_simps, norm_cast]
+lemma nnnorm_nnratCast (q : ℚ≥0) : ‖(q : K)‖₊ = q := by simp [nnnorm]
+
 variable (K) in
 lemma norm_nsmul [NormedAddCommGroup E] [NormedSpace K E] (n : ℕ) (x : E) : ‖n • x‖ = n • ‖x‖ := by
-  rw [← Nat.cast_smul_eq_nsmul K, norm_smul, RCLike.norm_natCast, nsmul_eq_mul]
+  simpa [Nat.cast_smul_eq_nsmul] using norm_smul (n : K) x
+
+variable (K) in
+lemma nnnorm_nsmul [NormedAddCommGroup E] [NormedSpace K E] (n : ℕ) (x : E) :
+    ‖n • x‖₊ = n • ‖x‖₊ := by simpa [Nat.cast_smul_eq_nsmul] using nnnorm_smul (n : K) x
+
+section NormedField
+variable [NormedField E] [CharZero E] [NormedSpace K E]
+include K
+
+variable (K) in
+lemma norm_nnqsmul (q : ℚ≥0) (x : E) : ‖q • x‖ = q • ‖x‖ := by
+  simpa [NNRat.cast_smul_eq_nnqsmul] using norm_smul (q : K) x
+
+variable (K) in
+lemma nnnorm_nnqsmul (q : ℚ≥0) (x : E) : ‖q • x‖₊ = q • ‖x‖₊ := by
+  simpa [NNRat.cast_smul_eq_nnqsmul] using nnnorm_smul (q : K) x
+
+@[bound]
+lemma norm_expect_le {ι : Type*} {s : Finset ι} {f : ι → E} : ‖𝔼 i ∈ s, f i‖ ≤ 𝔼 i ∈ s, ‖f i‖ :=
+  Finset.le_expect_of_subadditive norm_zero norm_add_le fun _ _ ↦ by rw [norm_nnqsmul K]
+
+end NormedField
 
 theorem mul_self_norm (z : K) : ‖z‖ * ‖z‖ = normSq z := by rw [normSq_eq_def', sq]
 
 attribute [rclike_simps] norm_zero norm_one norm_eq_zero abs_norm norm_inv norm_div
 
--- Porting note: removed @[simp, rclike_simps], b/c generalized to `norm_ofNat`
-theorem norm_two : ‖(2 : K)‖ = 2 := norm_ofNat 2
-
 theorem abs_re_le_norm (z : K) : |re z| ≤ ‖z‖ := by
   rw [mul_self_le_mul_self_iff (abs_nonneg _) (norm_nonneg _), abs_mul_abs_self, mul_self_norm]
   apply re_sq_le_normSq
@@ -621,11 +672,11 @@ open IsAbsoluteValue
 
 theorem abs_re_div_norm_le_one (z : K) : |re z / ‖z‖| ≤ 1 := by
   rw [abs_div, abs_norm]
-  exact div_le_one_of_le (abs_re_le_norm _) (norm_nonneg _)
+  exact div_le_one_of_le₀ (abs_re_le_norm _) (norm_nonneg _)
 
 theorem abs_im_div_norm_le_one (z : K) : |im z / ‖z‖| ≤ 1 := by
   rw [abs_div, abs_norm]
-  exact div_le_one_of_le (abs_im_le_norm _) (norm_nonneg _)
+  exact div_le_one_of_le₀ (abs_im_le_norm _) (norm_nonneg _)
 
 theorem norm_I_of_ne_zero (hI : (I : K) ≠ 0) : ‖(I : K)‖ = 1 := by
   rw [← mul_self_inj_of_nonneg (norm_nonneg I) zero_le_one, one_mul, ← norm_mul,
@@ -803,7 +854,7 @@ theorem toOrderedSMul : OrderedSMul ℝ K :=
 
 scoped[ComplexOrder] attribute [instance] RCLike.toOrderedSMul
 
-/-- A star algebra over `K` has a scalar multiplication that respects the order.  -/
+/-- A star algebra over `K` has a scalar multiplication that respects the order. -/
 lemma _root_.StarModule.instOrderedSMul {A : Type*} [NonUnitalRing A] [StarRing A] [PartialOrder A]
     [StarOrderedRing A] [Module K A] [StarModule K A] [IsScalarTower K A A] [SMulCommClass K A A] :
     OrderedSMul K A where
@@ -922,7 +973,7 @@ theorem conjAe_coe : (conjAe : K → K) = conj :=
 
 /-- Conjugate as a linear isometry -/
 noncomputable def conjLIE : K ≃ₗᵢ[ℝ] K :=
-  ⟨conjAe.toLinearEquiv, fun _ => norm_conj⟩
+  ⟨conjAe.toLinearEquiv, norm_conj⟩
 
 @[simp, rclike_simps]
 theorem conjLIE_apply : (conjLIE : K → K) = conj :=
@@ -1029,3 +1080,64 @@ lemma map_neg_eq_conj [AddCommGroup G] (ψ : AddChar G K) (x : G) : ψ (-x) = co
   rw [map_neg_eq_inv, inv_apply_eq_conj]
 
 end AddChar
+
+section
+
+/-- A mixin over a normed field, saying that the norm field structure is the same as `ℝ` or `ℂ`.
+To endow such a field with a compatible `RCLike` structure in a proof, use
+`letI := IsRCLikeNormedField.rclike 𝕜`.-/
+class IsRCLikeNormedField (𝕜 : Type*) [hk : NormedField 𝕜] : Prop where
+  out : ∃ h : RCLike 𝕜, hk = h.toNormedField
+
+instance (priority := 100) (𝕜 : Type*) [h : RCLike 𝕜] : IsRCLikeNormedField 𝕜 := ⟨⟨h, rfl⟩⟩
+
+/-- A copy of an `RCLike` field in which the `NormedField` field is adjusted to be become defeq
+to a propeq one. -/
+noncomputable def RCLike.copy_of_normedField {𝕜 : Type*} (h : RCLike 𝕜) (hk : NormedField 𝕜)
+    (h'' : hk = h.toNormedField) : RCLike 𝕜 where
+  __ := hk
+  toPartialOrder := h.toPartialOrder
+  toDecidableEq := h.toDecidableEq
+  complete := by subst h''; exact h.complete
+  lt_norm_lt := by subst h''; exact h.lt_norm_lt
+  -- star fields
+  star := (@StarMul.toInvolutiveStar _ (_) (@StarRing.toStarMul _ (_) h.toStarRing)).star
+  star_involutive := by subst h''; exact h.star_involutive
+  star_mul := by subst h''; exact h.star_mul
+  star_add := by subst h''; exact h.star_add
+  -- algebra fields
+  smul := (@Algebra.toSMul _ _ _ (_) (@NormedAlgebra.toAlgebra _ _ _ (_) h.toNormedAlgebra)).smul
+  toFun := @Algebra.toRingHom _ _ _ (_) (@NormedAlgebra.toAlgebra _ _ _ (_) h.toNormedAlgebra)
+  map_one' := by subst h''; exact h.map_one'
+  map_mul' := by subst h''; exact h.map_mul'
+  map_zero' := by subst h''; exact h.map_zero'
+  map_add' := by subst h''; exact h.map_add'
+  commutes' := by subst h''; exact h.commutes'
+  smul_def' := by subst h''; exact h.smul_def'
+  norm_smul_le := by subst h''; exact h.norm_smul_le
+  -- RCLike fields
+  re := by subst h''; exact h.re
+  im := by subst h''; exact h.im
+  I := h.I
+  I_re_ax := by subst h''; exact h.I_re_ax
+  I_mul_I_ax := by subst h''; exact h.I_mul_I_ax
+  re_add_im_ax := by subst h''; exact h.re_add_im_ax
+  ofReal_re_ax := by subst h''; exact h.ofReal_re_ax
+  ofReal_im_ax := by subst h''; exact h.ofReal_im_ax
+  mul_re_ax := by subst h''; exact h.mul_re_ax
+  mul_im_ax := by subst h''; exact h.mul_im_ax
+  conj_re_ax := by subst h''; exact h.conj_re_ax
+  conj_im_ax := by subst h''; exact h.conj_im_ax
+  conj_I_ax := by subst h''; exact h.conj_I_ax
+  norm_sq_eq_def_ax := by subst h''; exact h.norm_sq_eq_def_ax
+  mul_im_I_ax := by subst h''; exact h.mul_im_I_ax
+  le_iff_re_im := by subst h''; exact h.le_iff_re_im
+
+/-- Given a normed field `𝕜` satisfying `IsRCLikeNormedField 𝕜`, build an associated `RCLike 𝕜`
+structure on `𝕜` which is definitionally compatible with the given normed field structure. -/
+noncomputable def IsRCLikeNormedField.rclike (𝕜 : Type*)
+    [hk : NormedField 𝕜] [h : IsRCLikeNormedField 𝕜] : RCLike 𝕜 := by
+  choose p hp using h.out
+  exact p.copy_of_normedField hk hp
+
+end
diff --git a/Mathlib/Analysis/Seminorm.lean b/Mathlib/Analysis/Seminorm.lean
index 46659a1b853cf..c59be01c21689 100644
--- a/Mathlib/Analysis/Seminorm.lean
+++ b/Mathlib/Analysis/Seminorm.lean
@@ -495,7 +495,7 @@ noncomputable instance instSupSet : SupSet (Seminorm 𝕜 E) where
     if h : BddAbove ((↑) '' s : Set (E → ℝ)) then
       { toFun := ⨆ p : s, ((p : Seminorm 𝕜 E) : E → ℝ)
         map_zero' := by
-          rw [iSup_apply, ← @Real.ciSup_const_zero s]
+          rw [iSup_apply, ← @Real.iSup_const_zero s]
           congr!
           rename_i _ _ _ i
           exact map_zero i.1
@@ -658,7 +658,7 @@ theorem ball_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x :
     (c • p).ball x r = p.ball x (r / c) := by
   ext
   rw [mem_ball, mem_ball, smul_apply, NNReal.smul_def, smul_eq_mul, mul_comm,
-    lt_div_iff (NNReal.coe_pos.mpr hc)]
+    lt_div_iff₀ (NNReal.coe_pos.mpr hc)]
 
 theorem closedBall_smul (p : Seminorm 𝕜 E) {c : NNReal} (hc : 0 < c) (r : ℝ) (x : E) :
     (c • p).closedBall x r = p.closedBall x (r / c) := by
@@ -815,14 +815,14 @@ theorem closedBall_finset_sup (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x :
 @[simp]
 theorem ball_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r ≤ 0) : p.ball x r = ∅ := by
   ext
-  rw [Seminorm.mem_ball, Set.mem_empty_iff_false, iff_false_iff, not_lt]
+  rw [Seminorm.mem_ball, Set.mem_empty_iff_false, iff_false, not_lt]
   exact hr.trans (apply_nonneg p _)
 
 @[simp]
 theorem closedBall_eq_emptyset (p : Seminorm 𝕜 E) {x : E} {r : ℝ} (hr : r < 0) :
     p.closedBall x r = ∅ := by
   ext
-  rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false_iff, not_le]
+  rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false, not_le]
   exact hr.trans_le (apply_nonneg _ _)
 
 theorem closedBall_smul_ball (p : Seminorm 𝕜 E) {r₁ : ℝ} (hr₁ : r₁ ≠ 0) (r₂ : ℝ) :
@@ -899,7 +899,7 @@ theorem smul_ball_zero {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0)
     k • p.ball 0 r = p.ball 0 (‖k‖ * r) := by
   ext
   rw [mem_smul_set_iff_inv_smul_mem₀ hk, p.mem_ball_zero, p.mem_ball_zero, map_smul_eq_mul,
-    norm_inv, ← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hk), mul_comm]
+    norm_inv, ← div_eq_inv_mul, div_lt_iff₀ (norm_pos_iff.2 hk), mul_comm]
 
 theorem smul_closedBall_subset {p : Seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
     k • p.closedBall 0 r ⊆ p.closedBall 0 (‖k‖ * r) := by
@@ -952,7 +952,7 @@ protected theorem absorbent_closedBall (hpr : p x < r) : Absorbent 𝕜 (closedB
 theorem smul_ball_preimage (p : Seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0) :
     (a • ·) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ‖a‖) :=
   Set.ext fun _ => by
-    rw [mem_preimage, mem_ball, mem_ball, lt_div_iff (norm_pos_iff.mpr ha), mul_comm, ←
+    rw [mem_preimage, mem_ball, mem_ball, lt_div_iff₀ (norm_pos_iff.mpr ha), mul_comm, ←
       map_smul_eq_mul p, smul_sub, smul_inv_smul₀ ha]
 
 end NormedField
@@ -1047,7 +1047,7 @@ theorem continuousAt_zero' [TopologicalSpace E] [ContinuousConstSMul 𝕜 E] {p
   obtain ⟨k, hk₀, hk⟩ : ∃ k : 𝕜, 0 < ‖k‖ ∧ ‖k‖ * r < ε := by
     rcases le_or_lt r 0 with hr | hr
     · use 1; simpa using hr.trans_lt hε
-    · simpa [lt_div_iff hr] using exists_norm_lt 𝕜 (div_pos hε hr)
+    · simpa [lt_div_iff₀ hr] using exists_norm_lt 𝕜 (div_pos hε hr)
   rw [← set_smul_mem_nhds_zero_iff (norm_pos_iff.1 hk₀), smul_closedBall_zero hk₀] at hp
   exact mem_of_superset hp <| p.closedBall_mono hk.le
 
@@ -1184,9 +1184,9 @@ lemma rescale_to_shell_zpow (p : Seminorm 𝕜 E) {c : 𝕜} (hc : 1 < ‖c‖)
   refine ⟨-(n+1), ?_, ?_, ?_, ?_⟩
   · show c ^ (-(n + 1)) ≠ 0; exact zpow_ne_zero _ (norm_pos_iff.1 cpos)
   · show p ((c ^ (-(n + 1))) • x) < ε
-    rw [map_smul_eq_mul, zpow_neg, norm_inv, ← div_eq_inv_mul, div_lt_iff cnpos, mul_comm,
+    rw [map_smul_eq_mul, zpow_neg, norm_inv, ← div_eq_inv_mul, div_lt_iff₀ cnpos, mul_comm,
         norm_zpow]
-    exact (div_lt_iff εpos).1 (hn.2)
+    exact (div_lt_iff₀ εpos).1 (hn.2)
   · show ε / ‖c‖ ≤ p (c ^ (-(n + 1)) • x)
     rw [zpow_neg, div_le_iff₀ cpos, map_smul_eq_mul, norm_inv, norm_zpow, zpow_add₀ (ne_of_gt cpos),
         zpow_one, mul_inv_rev, mul_comm, ← mul_assoc, ← mul_assoc, mul_inv_cancel₀ (ne_of_gt cpos),
diff --git a/Mathlib/Analysis/SpecialFunctions/Bernstein.lean b/Mathlib/Analysis/SpecialFunctions/Bernstein.lean
index 66d2388a494f1..50a7c0ab65155 100644
--- a/Mathlib/Analysis/SpecialFunctions/Bernstein.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Bernstein.lean
@@ -1,12 +1,12 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.RingTheory.Polynomial.Bernstein
-import Mathlib.Topology.ContinuousFunction.Polynomial
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.Polynomial
+import Mathlib.Topology.ContinuousMap.Compact
 
 /-!
 # Bernstein approximations and Weierstrass' theorem
@@ -101,8 +101,8 @@ local postfix:90 "/ₙ" => z
 theorem probability (n : ℕ) (x : I) : (∑ k : Fin (n + 1), bernstein n k x) = 1 := by
   have := bernsteinPolynomial.sum ℝ n
   apply_fun fun p => Polynomial.aeval (x : ℝ) p at this
-  simp? [AlgHom.map_sum, Finset.sum_range] at this says
-    simp only [Finset.sum_range, map_sum, Polynomial.coe_aeval_eq_eval, map_one] at this
+  simp? [map_sum, Finset.sum_range] at this says
+    simp only [Finset.sum_range, map_sum, Polynomial.coe_aeval_eq_eval, Polynomial.eval_one] at this
   exact this
 
 theorem variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
@@ -115,9 +115,10 @@ theorem variance {n : ℕ} (h : 0 < (n : ℝ)) (x : I) :
   conv_rhs => rw [div_mul_cancel₀ _ h']
   have := bernsteinPolynomial.variance ℝ n
   apply_fun fun p => Polynomial.aeval (x : ℝ) p at this
-  simp? [AlgHom.map_sum, Finset.sum_range, ← Polynomial.natCast_mul] at this says
-    simp only [nsmul_eq_mul, Finset.sum_range, map_sum, map_mul, map_pow, map_sub, map_natCast,
-      Polynomial.aeval_X, Polynomial.coe_aeval_eq_eval, map_one] at this
+  simp? [map_sum, Finset.sum_range, ← Polynomial.natCast_mul] at this says
+    simp only [nsmul_eq_mul, Finset.sum_range, map_sum, Polynomial.coe_aeval_eq_eval,
+      Polynomial.eval_mul, Polynomial.eval_pow, Polynomial.eval_sub, Polynomial.eval_natCast,
+      Polynomial.eval_X, Polynomial.eval_one] at this
   convert this using 1
   · congr 1; funext k
     rw [mul_comm _ (n : ℝ), mul_comm _ (n : ℝ), ← mul_assoc, ← mul_assoc]
@@ -251,9 +252,7 @@ theorem bernsteinApproximation_uniform (f : C(I, ℝ)) :
       _ = ε / 2 * ∑ k ∈ S, bernstein n k x := by rw [Finset.mul_sum]
       -- In this step we increase the sum over `S` back to a sum over all of `Fin (n+1)`,
       -- so that we can use `bernstein.probability`.
-      _ ≤ ε / 2 * ∑ k : Fin (n + 1), bernstein n k x := by
-        gcongr
-        exact Finset.sum_le_univ_sum_of_nonneg fun k => bernstein_nonneg
+      _ ≤ ε / 2 * ∑ k : Fin (n + 1), bernstein n k x := by gcongr; exact S.subset_univ
       _ = ε / 2 := by rw [bernstein.probability, mul_one]
   · -- We now turn to working on `Sᶜ`: we control the difference term just using `‖f‖`,
     -- and then insert a `δ^(-2) * (x - k/n)^2` factor
@@ -270,9 +269,7 @@ theorem bernsteinApproximation_uniform (f : C(I, ℝ)) :
         exact le_of_mem_S_compl m
       -- Again enlarging the sum from `Sᶜ` to all of `Fin (n+1)`
       _ ≤ 2 * ‖f‖ * ∑ k : Fin (n + 1), δ ^ (-2 : ℤ) * ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
-        gcongr
-        refine Finset.sum_le_univ_sum_of_nonneg fun k => ?_
-        positivity
+        gcongr; exact Sᶜ.subset_univ
       _ = 2 * ‖f‖ * δ ^ (-2 : ℤ) * ∑ k : Fin (n + 1), ((x : ℝ) - k/ₙ) ^ 2 * bernstein n k x := by
         conv_rhs =>
           rw [mul_assoc, Finset.mul_sum]
diff --git a/Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean b/Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
index c76c7577fe79d..c74cbff25c9e5 100644
--- a/Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
+++ b/Mathlib/Analysis/SpecialFunctions/BinaryEntropy.lean
@@ -80,8 +80,8 @@ lemma binEntropy_two_inv_add (p : ℝ) : binEntropy (2⁻¹ + p) = binEntropy (2
 lemma binEntropy_pos (hp₀ : 0 < p) (hp₁ : p < 1) : 0 < binEntropy p := by
   unfold binEntropy
   have : 0 < 1 - p := sub_pos.2 hp₁
-  have : 0 < log p⁻¹ := log_pos <| one_lt_inv hp₀ hp₁
-  have : 0 < log (1 - p)⁻¹ := log_pos <| one_lt_inv ‹_› (sub_lt_self _ hp₀)
+  have : 0 < log p⁻¹ := log_pos <| (one_lt_inv₀ hp₀).2 hp₁
+  have : 0 < log (1 - p)⁻¹ := log_pos <| (one_lt_inv₀ ‹_›).2 (sub_lt_self _ hp₀)
   positivity
 
 lemma binEntropy_nonneg (hp₀ : 0 ≤ p) (hp₁ : p ≤ 1) : 0 ≤ binEntropy p := by
@@ -174,11 +174,11 @@ lemma differentiableAt_binEntropy_iff_ne_zero_one :
   refine ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ differentiableAt_binEntropy h.1 h.2⟩
     <;> rintro rfl <;> unfold binEntropy at h
   · rw [DifferentiableAt.add_iff_left] at h
-    simp [log_inv, mul_neg, ← neg_mul, ← negMulLog_def, differentiableAt_negMulLog_iff] at h
-    fun_prop (disch := simp)
+    · simp [log_inv, mul_neg, ← neg_mul, ← negMulLog_def, differentiableAt_negMulLog_iff] at h
+    · fun_prop (disch := simp)
   · rw [DifferentiableAt.add_iff_right, differentiableAt_iff_comp_const_sub (b := 1)] at h
-    simp [log_inv, mul_neg, ← neg_mul, ← negMulLog_def, differentiableAt_negMulLog_iff] at h
-    fun_prop (disch := simp)
+    · simp [log_inv, mul_neg, ← neg_mul, ← negMulLog_def, differentiableAt_negMulLog_iff] at h
+    · fun_prop (disch := simp)
 
 set_option push_neg.use_distrib true in
 /-- Binary entropy has derivative `log (1 - p) - log p`.
@@ -189,7 +189,7 @@ lemma deriv_binEntropy (p : ℝ) : deriv binEntropy p = log (1 - p) - log p := b
     rw [ne_comm, ← sub_ne_zero] at hp₁
     rw [binEntropy_eq_negMulLog_add_negMulLog_one_sub', deriv_add, deriv_comp_const_sub,
       deriv_negMulLog hp₀, deriv_negMulLog hp₁]
-    ring
+    · ring
     all_goals fun_prop (disch := assumption)
   -- pathological case where `deriv = 0` since `binEntropy` is not differentiable there
   · rw [deriv_zero_of_not_differentiableAt (differentiableAt_binEntropy_iff_ne_zero_one.not.2 hp)]
@@ -246,8 +246,8 @@ lemma deriv_qaryEntropy (hp₀ : p ≠ 0) (hp₁ : p ≠ 1) :
     deriv (qaryEntropy q) p = log (q - 1) + log (1 - p) - log p := by
   unfold qaryEntropy
   rw [deriv_add]
-  simp only [Int.cast_sub, Int.cast_natCast, Int.cast_one, differentiableAt_id', deriv_mul_const,
-    deriv_id'', one_mul, deriv_binEntropy, add_sub_assoc]
+  · simp only [Int.cast_sub, Int.cast_natCast, Int.cast_one, differentiableAt_id', deriv_mul_const,
+      deriv_id'', one_mul, deriv_binEntropy, add_sub_assoc]
   all_goals fun_prop (disch := assumption)
 
 /-- Binary entropy has derivative `log (1 - p) - log p`. -/
@@ -276,7 +276,7 @@ private lemma tendsto_log_one_sub_sub_log_nhdsWithin_one_atBot :
   · have : Tendsto log (𝓝[>] 0) atBot := Real.tendsto_log_nhdsWithin_zero_right
     apply Tendsto.comp (f := (1 - ·)) (g := log) this
     have contF : Continuous ((1 : ℝ) - ·) := continuous_sub_left 1
-    have : MapsTo ((1 : ℝ) - ·)  (Iio 1) (Ioi 0) := by
+    have : MapsTo ((1 : ℝ) - ·) (Iio 1) (Ioi 0) := by
       intro p hx
       simp_all only [mem_Iio, mem_Ioi, sub_pos]
     convert ContinuousWithinAt.tendsto_nhdsWithin (x :=(1 : ℝ)) contF.continuousWithinAt this
@@ -297,12 +297,12 @@ lemma not_continuousAt_deriv_qaryEntropy_one :
     apply tendsto_atBot_add_const_left
     exact tendsto_log_one_sub_sub_log_nhdsWithin_one_atBot
   apply not_continuousAt_of_tendsto (Filter.Tendsto.congr' _ tendstoBot) nhdsWithin_le_nhds
-  simp only [disjoint_nhds_atBot_iff, not_isBot, not_false_eq_true]
+  · simp only [disjoint_nhds_atBot_iff, not_isBot, not_false_eq_true]
   filter_upwards [Ioo_mem_nhdsWithin_Iio' (show 1 - 2⁻¹ < (1 : ℝ) by norm_num)]
   intros
   apply (deriv_qaryEntropy _ _).symm
-  simp_all only [mem_Ioo, ne_eq]
-  · linarith [show (1 : ℝ) = 2⁻¹ + 2⁻¹ by norm_num]
+  · simp_all only [mem_Ioo, ne_eq]
+    linarith [show (1 : ℝ) = 2⁻¹ + 2⁻¹ by norm_num]
   · simp_all only [mem_Ioo, ne_eq]
     linarith [two_inv_lt_one (α := ℝ)]
 
@@ -314,12 +314,12 @@ lemma not_continuousAt_deriv_qaryEntropy_zero :
     rw [this]
     exact tendsto_atTop_add_const_left _ _ tendsto_log_one_sub_sub_log_nhdsWithin_atAtop
   apply not_continuousAt_of_tendsto (Filter.Tendsto.congr' _ tendstoTop) nhdsWithin_le_nhds
-  simp only [disjoint_nhds_atTop_iff, not_isTop, not_false_eq_true]
+  · simp only [disjoint_nhds_atTop_iff, not_isTop, not_false_eq_true]
   filter_upwards [Ioo_mem_nhdsWithin_Ioi' (show (0 : ℝ) < 2⁻¹ by norm_num)]
   intros
   apply (deriv_qaryEntropy _ _).symm
-  simp_all only [zero_add, mem_Ioo, ne_eq]
-  · linarith
+  · simp_all only [zero_add, mem_Ioo, ne_eq]
+    linarith
   · simp_all only [zero_add, mem_Ioo, ne_eq]
     linarith [two_inv_lt_one (α := ℝ)]
 
@@ -397,7 +397,7 @@ lemma qaryEntropy_strictAntiOn (qLe2 : 2 ≤ q) :
   · exact qaryEntropy_continuous.continuousOn
   · intro p hp
     have : 2 ≤ (q : ℝ) := Nat.ofNat_le_cast.mpr qLe2
-    have qinv_lt_1 : (q : ℝ)⁻¹ < 1 := inv_lt_one (by linarith)
+    have qinv_lt_1 : (q : ℝ)⁻¹ < 1 := inv_lt_one_of_one_lt₀ (by linarith)
     have zero_lt_1_sub_p : 0 < 1 - p := by simp_all only [sub_pos, hp.2, interior_Icc, mem_Ioo]
     simp only [one_div, interior_Icc, mem_Ioo] at hp
     rw [deriv_qaryEntropy (by linarith)]
@@ -440,8 +440,8 @@ lemma strictConcaveOn_qaryEntropy : StrictConcaveOn ℝ (Icc 0 1) (qaryEntropy q
   rw [deriv2_qaryEntropy]
   · simp_all only [interior_Icc, mem_Ioo]
     apply div_neg_of_neg_of_pos
-    norm_num [show 0 < log 2 by positivity]
-    simp_all only [gt_iff_lt, mul_pos_iff_of_pos_left, sub_pos, hp]
+    · norm_num [show 0 < log 2 by positivity]
+    · simp_all only [gt_iff_lt, mul_pos_iff_of_pos_left, sub_pos, hp]
 
 lemma strictConcave_binEntropy : StrictConcaveOn ℝ (Icc 0 1) binEntropy :=
   qaryEntropy_two ▸ strictConcaveOn_qaryEntropy
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean b/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
index 18ea3db3671d5..d717114a27b82 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
@@ -11,7 +11,7 @@ import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
 /-!
 # Various complex special functions are analytic
 
-`exp`, `log`, and `cpow` are analytic, since they are differentiable.
+`log`, and `cpow` are analytic, since they are differentiable.
 -/
 
 open Complex Set
@@ -20,22 +20,6 @@ open scoped Topology
 variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
 variable {f g : E → ℂ} {z : ℂ} {x : E} {s : Set E}
 
-/-- `exp` is entire -/
-theorem analyticOn_cexp : AnalyticOn ℂ exp univ := by
-  rw [analyticOn_univ_iff_differentiable]; exact differentiable_exp
-
-/-- `exp` is analytic at any point -/
-theorem analyticAt_cexp : AnalyticAt ℂ exp z :=
-  analyticOn_cexp z (mem_univ _)
-
-/-- `exp ∘ f` is analytic -/
-theorem AnalyticAt.cexp (fa : AnalyticAt ℂ f x) : AnalyticAt ℂ (fun z ↦ exp (f z)) x :=
-  analyticAt_cexp.comp fa
-
-/-- `exp ∘ f` is analytic -/
-theorem AnalyticOn.cexp (fs : AnalyticOn ℂ f s) : AnalyticOn ℂ (fun z ↦ exp (f z)) s :=
-  fun z n ↦ analyticAt_cexp.comp (fs z n)
-
 /-- `log` is analytic away from nonpositive reals -/
 theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z := by
   rw [analyticAt_iff_eventually_differentiableAt]
@@ -48,21 +32,40 @@ theorem AnalyticAt.clog (fa : AnalyticAt ℂ f x) (m : f x ∈ slitPlane) :
     AnalyticAt ℂ (fun z ↦ log (f z)) x :=
   (analyticAt_clog m).comp fa
 
+theorem AnalyticWithinAt.clog (fa : AnalyticWithinAt ℂ f s x) (m : f x ∈ slitPlane) :
+    AnalyticWithinAt ℂ (fun z ↦ log (f z)) s x :=
+  (analyticAt_clog m).comp_analyticWithinAt fa
+
 /-- `log` is analytic away from nonpositive reals -/
+theorem AnalyticOnNhd.clog (fs : AnalyticOnNhd ℂ f s) (m : ∀ z ∈ s, f z ∈ slitPlane) :
+    AnalyticOnNhd ℂ (fun z ↦ log (f z)) s :=
+  fun z n ↦ (analyticAt_clog (m z n)).comp (fs z n)
+
 theorem AnalyticOn.clog (fs : AnalyticOn ℂ f s) (m : ∀ z ∈ s, f z ∈ slitPlane) :
     AnalyticOn ℂ (fun z ↦ log (f z)) s :=
-  fun z n ↦ (analyticAt_clog (m z n)).comp (fs z n)
+  fun z n ↦ (analyticAt_clog (m z n)).analyticWithinAt.comp (fs z n) m
 
 /-- `f z ^ g z` is analytic if `f z` is not a nonpositive real -/
-theorem AnalyticAt.cpow (fa : AnalyticAt ℂ f x) (ga : AnalyticAt ℂ g x)
-    (m : f x ∈ slitPlane) : AnalyticAt ℂ (fun z ↦ f z ^ g z) x := by
-  have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝 x] fun z ↦ exp (log (f z) * g z) := by
-    filter_upwards [(fa.continuousAt.eventually_ne (slitPlane_ne_zero m))]
+theorem AnalyticWithinAt.cpow (fa : AnalyticWithinAt ℂ f s x) (ga : AnalyticWithinAt ℂ g s x)
+    (m : f x ∈ slitPlane) : AnalyticWithinAt ℂ (fun z ↦ f z ^ g z) s x := by
+  have e : (fun z ↦ f z ^ g z) =ᶠ[𝓝[insert x s] x] fun z ↦ exp (log (f z) * g z) := by
+    filter_upwards [(fa.continuousWithinAt_insert.eventually_ne (slitPlane_ne_zero m))]
     intro z fz
     simp only [fz, cpow_def, if_false]
-  rw [analyticAt_congr e]
+  apply AnalyticWithinAt.congr_of_eventuallyEq_insert _ e
   exact ((fa.clog m).mul ga).cexp
 
+/-- `f z ^ g z` is analytic if `f z` is not a nonpositive real -/
+theorem AnalyticAt.cpow (fa : AnalyticAt ℂ f x) (ga : AnalyticAt ℂ g x)
+    (m : f x ∈ slitPlane) : AnalyticAt ℂ (fun z ↦ f z ^ g z) x := by
+  rw [← analyticWithinAt_univ] at fa ga ⊢
+  exact fa.cpow ga m
+
+/-- `f z ^ g z` is analytic if `f z` avoids nonpositive reals -/
+theorem AnalyticOnNhd.cpow (fs : AnalyticOnNhd ℂ f s) (gs : AnalyticOnNhd ℂ g s)
+    (m : ∀ z ∈ s, f z ∈ slitPlane) : AnalyticOnNhd ℂ (fun z ↦ f z ^ g z) s :=
+  fun z n ↦ (fs z n).cpow (gs z n) (m z n)
+
 /-- `f z ^ g z` is analytic if `f z` avoids nonpositive reals -/
 theorem AnalyticOn.cpow (fs : AnalyticOn ℂ f s) (gs : AnalyticOn ℂ g s)
     (m : ∀ z ∈ s, f z ∈ slitPlane) : AnalyticOn ℂ (fun z ↦ f z ^ g z) s :=
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean b/Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean
index d9d625ead931a..d5bc0c1f1b5ba 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean
@@ -73,7 +73,7 @@ theorem arctan_tan {z : ℂ} (h₀ : z ≠ π / 2) (h₁ : -(π / 2) < z.re) (h
   rw [← exp_mul_I, ← exp_mul_I, ← exp_sub, show z * I - -z * I = 2 * (I * z) by ring, log_exp,
     show -I / 2 * (2 * (I * z)) = -(I * I) * z by ring, I_mul_I, neg_neg, one_mul]
   all_goals norm_num
-  · rwa [← div_lt_iff' two_pos, neg_div]
+  · rwa [← div_lt_iff₀' two_pos, neg_div]
   · rwa [← le_div_iff₀' two_pos]
 
 @[simp, norm_cast]
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean b/Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
index 94901e3540e4f..b7b5fa601494d 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
@@ -172,7 +172,7 @@ theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x :=
 theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
     arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by
   simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs,
-    div_mul_cancel₀ _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff]
+    div_mul_cancel₀ _ (abs.ne_zero hx), eq_self_iff_true, true_and]
   rw [← ofReal_div, arg_real_mul]
   exact div_pos (abs.pos hy) (abs.pos hx)
 
@@ -322,28 +322,28 @@ lemma image_exp_Ioc_eq_sphere : (fun θ : ℝ ↦ exp (θ * I)) '' Set.Ioc (-π)
 
 theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by
   rcases le_or_lt 0 (re z) with hre | hre
-  · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff]
-  simp only [hre.not_le, false_or_iff]
+  · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or]
+  simp only [hre.not_le, false_or]
   rcases le_or_lt 0 (im z) with him | him
   · simp only [him.not_lt]
-    rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub,
+    rw [iff_false, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub,
       Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ←
       _root_.abs_of_nonneg him, abs_im_lt_abs]
     exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne]
   · simp only [him]
-    rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him]
+    rw [iff_true, arg_of_re_neg_of_im_neg hre him]
     exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _)
 
 theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by
   rcases le_or_lt 0 (re z) with hre | hre
-  · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff]
-  simp only [hre.not_le, false_or_iff]
+  · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or]
+  simp only [hre.not_le, false_or]
   rcases le_or_lt 0 (im z) with him | him
   · simp only [him]
-    rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him]
+    rw [iff_true, arg_of_re_neg_of_im_nonneg hre him]
     exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le)
   · simp only [him.not_le]
-    rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ←
+    rw [iff_false, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ←
       sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him,
       abs_im_lt_abs]
     exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne]
@@ -367,7 +367,7 @@ lemma arg_lt_pi_div_two_iff {z : ℂ} : arg z < π / 2 ↔ 0 < re z ∨ im z < 0
 @[simp]
 theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by
   rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le,
-    and_not_self_iff, or_false_iff]
+    and_not_self_iff, or_false]
 
 @[simp]
 theorem abs_arg_lt_pi_div_two_iff {z : ℂ} : |arg z| < π / 2 ↔ 0 < re z ∨ z = 0 := by
@@ -598,7 +598,7 @@ theorem continuousAt_arg_coe_angle (h : x ≠ 0) : ContinuousAt ((↑) ∘ arg :
   · exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs)
   · rw [← Function.comp_id (((↑) : ℝ → Real.Angle) ∘ arg),
       (Function.funext_iff.2 fun _ => (neg_neg _).symm : (id : ℂ → ℂ) = Neg.neg ∘ Neg.neg), ←
-      Function.comp.assoc]
+      Function.comp_assoc]
     refine ContinuousAt.comp ?_ continuous_neg.continuousAt
     suffices ContinuousAt (Function.update (((↑) ∘ arg) ∘ Neg.neg : ℂ → Real.Angle) 0 π) (-x) by
       rwa [continuousAt_update_of_ne (neg_ne_zero.2 h)] at this
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/Log.lean b/Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
index 48b5d9c9e10b9..ad7f981e284ec 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
@@ -250,7 +250,7 @@ variable {α ι: Type*}
 
 open Real
 
-lemma Real.HasSum_rexp_HasProd  (f : ι → α → ℝ) (hfn : ∀ x n, 0 < f n x)
+lemma Real.HasSum_rexp_HasProd (f : ι → α → ℝ) (hfn : ∀ x n, 0 < f n x)
     (hf : ∀ x : α, HasSum (fun n => log (f n x)) (∑' i, log (f i x))) (a : α) :
        HasProd (fun b ↦ f b a) (∏' n : ι, (f n a)) := by
   have : HasProd (fun b ↦ f b a) ((rexp ∘ fun a ↦ ∑' (n : ι), log (f n a)) a) := by
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean b/Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
index 6bd3f9647113e..f5e05a1e202df 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean
@@ -110,7 +110,7 @@ lemma norm_one_add_mul_inv_le {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz :
     ‖(1 + t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹ := by
   rw [Set.mem_Icc] at ht
   rw [norm_inv, norm_eq_abs]
-  refine inv_le_inv_of_le (by linarith) ?_
+  refine inv_anti₀ (by linarith) ?_
   calc 1 - ‖z‖
     _ ≤ 1 - t * ‖z‖ := by
       nlinarith [norm_nonneg z]
@@ -227,7 +227,7 @@ lemma hasSum_taylorSeries_log {z : ℂ} (hz : ‖z‖ < 1) :
   refine (hasSum_iff_tendsto_nat_of_summable_norm ?_).mpr ?_
   · refine (summable_geometric_of_norm_lt_one hz).norm.of_nonneg_of_le (fun _ ↦ norm_nonneg _) ?_
     intro n
-    simp only [norm_div, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul, norm_nat]
+    simp only [norm_div, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul, norm_natCast]
     rcases n.eq_zero_or_pos with rfl | hn
     · simp
     conv => enter [2]; rw [← div_one (‖z‖ ^ n)]
diff --git a/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean b/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean
index 4879f5d427b76..8c7bd3f605070 100644
--- a/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean
+++ b/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/ExpLog.lean
@@ -7,6 +7,7 @@ Authors: Frédéric Dupuis
 import Mathlib.Analysis.Normed.Algebra.Spectrum
 import Mathlib.Analysis.SpecialFunctions.Exponential
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital
+import Mathlib.Topology.ContinuousMap.StarOrdered
 
 /-!
 # The exponential and logarithm based on the continuous functional calculus
@@ -74,14 +75,21 @@ end RCLikeNormed
 
 section RealNormed
 
-variable {A : Type*} {p : A → Prop} [NormedRing A] [StarRing A]
+variable {A : Type*} [NormedRing A] [StarRing A]
   [TopologicalRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
-  [ContinuousFunctionalCalculus ℝ p]
+  [ContinuousFunctionalCalculus ℝ (IsSelfAdjoint : A → Prop)]
 
-lemma real_exp_eq_normedSpace_exp {a : A} (ha : p a := by cfc_tac) :
+lemma real_exp_eq_normedSpace_exp {a : A} (ha : IsSelfAdjoint a := by cfc_tac) :
     cfc Real.exp a = exp ℝ a :=
   Real.exp_eq_exp_ℝ ▸ exp_eq_normedSpace_exp ha
 
+@[aesop safe apply (rule_sets := [CStarAlgebra])]
+lemma _root_.IsSelfAdjoint.exp_nonneg {𝕜 : Type*} [Field 𝕜] [Algebra 𝕜 A]
+    [PartialOrder A] [StarOrderedRing A] {a : A} (ha : IsSelfAdjoint a) :
+    0 ≤ exp 𝕜 a := by
+  rw [exp_eq_exp 𝕜 ℝ, ← real_exp_eq_normedSpace_exp]
+  exact cfc_nonneg fun x _ => Real.exp_nonneg x
+
 end RealNormed
 
 section ComplexNormed
diff --git a/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow.lean b/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow.lean
index fc706358df05d..9fcf8e77c60a6 100644
--- a/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow.lean
+++ b/Mathlib/Analysis/SpecialFunctions/ContinuousFunctionalCalculus/Rpow.lean
@@ -93,7 +93,7 @@ lemma nnrpow_add {a : A} {x y : ℝ≥0} (hx : 0 < x) (hy : 0 < y) :
   simp only [nnrpow_def]
   rw [← cfcₙ_mul _ _ a]
   congr! 2 with z
-  exact_mod_cast NNReal.rpow_add' z <| ne_of_gt (add_pos hx hy)
+  exact mod_cast z.rpow_add' <| ne_of_gt (add_pos hx hy)
 
 @[simp]
 lemma nnrpow_zero {a : A} : a ^ (0 : ℝ≥0) = 0 := by
diff --git a/Mathlib/Analysis/SpecialFunctions/Exp.lean b/Mathlib/Analysis/SpecialFunctions/Exp.lean
index e280153f8a2e3..8d018a967ecb0 100644
--- a/Mathlib/Analysis/SpecialFunctions/Exp.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Exp.lean
@@ -162,7 +162,7 @@ lemma exp_sub_sum_range_isBigO_pow (n : ℕ) :
     (fun x ↦ exp x - ∑ i ∈ Finset.range n, x ^ i / i !) =O[𝓝 0] (· ^ n) := by
   have := (Complex.exp_sub_sum_range_isBigO_pow n).comp_tendsto
     (Complex.continuous_ofReal.tendsto' 0 0 rfl)
-  simp only [(· ∘ ·)] at this
+  simp only [Function.comp_def] at this
   norm_cast at this
 
 lemma exp_sub_sum_range_succ_isLittleO_pow (n : ℕ) :
@@ -263,7 +263,7 @@ theorem tendsto_exp_div_pow_atTop (n : ℕ) : Tendsto (fun x => exp x / x ^ n) a
     eventually_atTop.1
       ((tendsto_pow_const_div_const_pow_of_one_lt n (one_lt_exp_iff.2 zero_lt_one)).eventually
         (gt_mem_nhds this))
-  simp only [← exp_nat_mul, mul_one, div_lt_iff, exp_pos, ← div_eq_inv_mul] at hN
+  simp only [← exp_nat_mul, mul_one, div_lt_iff₀, exp_pos, ← div_eq_inv_mul] at hN
   refine ⟨N, trivial, fun x hx => ?_⟩
   rw [Set.mem_Ioi] at hx
   have hx₀ : 0 < x := (Nat.cast_nonneg N).trans_lt hx
@@ -315,8 +315,8 @@ theorem tendsto_div_pow_mul_exp_add_atTop (b c : ℝ) (n : ℕ) (hb : 0 ≠ b) :
 def expOrderIso : ℝ ≃o Ioi (0 : ℝ) :=
   StrictMono.orderIsoOfSurjective _ (exp_strictMono.codRestrict exp_pos) <|
     (continuous_exp.subtype_mk _).surjective
-      (by simp only [tendsto_Ioi_atTop, Subtype.coe_mk, tendsto_exp_atTop])
-      (by simp [tendsto_exp_atBot_nhdsWithin])
+      (by rw [tendsto_Ioi_atTop]; simp only [tendsto_exp_atTop])
+      (by rw [tendsto_Ioi_atBot]; simp only [tendsto_exp_atBot_nhdsWithin])
 
 @[simp]
 theorem coe_expOrderIso_apply (x : ℝ) : (expOrderIso x : ℝ) = exp x :=
@@ -456,14 +456,14 @@ namespace Complex
 theorem comap_exp_cobounded : comap exp (cobounded ℂ) = comap re atTop :=
   calc
     comap exp (cobounded ℂ) = comap re (comap Real.exp atTop) := by
-      simp only [← comap_norm_atTop, Complex.norm_eq_abs, comap_comap, (· ∘ ·), abs_exp]
+      simp only [← comap_norm_atTop, Complex.norm_eq_abs, comap_comap, Function.comp_def, abs_exp]
     _ = comap re atTop := by rw [Real.comap_exp_atTop]
 
 @[simp]
 theorem comap_exp_nhds_zero : comap exp (𝓝 0) = comap re atBot :=
   calc
     comap exp (𝓝 0) = comap re (comap Real.exp (𝓝 0)) := by
-      simp only [comap_comap, ← comap_abs_nhds_zero, (· ∘ ·), abs_exp]
+      simp only [comap_comap, ← comap_abs_nhds_zero, Function.comp_def, abs_exp]
     _ = comap re atBot := by rw [Real.comap_exp_nhds_zero]
 
 theorem comap_exp_nhdsWithin_zero : comap exp (𝓝[≠] 0) = comap re atBot := by
diff --git a/Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean b/Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
index 31ff5558fe80a..8506d9e055ab1 100644
--- a/Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
+++ b/Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
@@ -6,6 +6,7 @@ Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
 import Mathlib.Analysis.Complex.RealDeriv
 import Mathlib.Analysis.Calculus.ContDiff.RCLike
 import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
+import Mathlib.Analysis.SpecialFunctions.Exponential
 
 /-!
 # Complex and real exponential
@@ -24,6 +25,41 @@ open scoped Topology
 
 /-! ## `Complex.exp` -/
 
+section
+
+open Complex
+
+variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
+variable {f g : E → ℂ} {z : ℂ} {x : E} {s : Set E}
+
+/-- `exp` is entire -/
+theorem analyticOnNhd_cexp : AnalyticOnNhd ℂ exp univ := by
+  rw [Complex.exp_eq_exp_ℂ]
+  exact fun x _ ↦ NormedSpace.exp_analytic x
+
+theorem analyticOn_cexp : AnalyticOn ℂ exp univ := analyticOnNhd_cexp.analyticOn
+
+/-- `exp` is analytic at any point -/
+theorem analyticAt_cexp : AnalyticAt ℂ exp z :=
+  analyticOnNhd_cexp z (mem_univ _)
+
+/-- `exp ∘ f` is analytic -/
+theorem AnalyticAt.cexp (fa : AnalyticAt ℂ f x) : AnalyticAt ℂ (fun z ↦ exp (f z)) x :=
+  analyticAt_cexp.comp fa
+
+theorem AnalyticWithinAt.cexp (fa : AnalyticWithinAt ℂ f s x) :
+    AnalyticWithinAt ℂ (fun z ↦ exp (f z)) s x :=
+  analyticAt_cexp.comp_analyticWithinAt fa
+
+/-- `exp ∘ f` is analytic -/
+theorem AnalyticOnNhd.cexp (fs : AnalyticOnNhd ℂ f s) : AnalyticOnNhd ℂ (fun z ↦ exp (f z)) s :=
+  fun z n ↦ analyticAt_cexp.comp (fs z n)
+
+theorem AnalyticOn.cexp (fs : AnalyticOn ℂ f s) : AnalyticOn ℂ (fun z ↦ exp (f z)) s :=
+  analyticOnNhd_cexp.comp_analyticOn fs (mapsTo_univ _ _)
+
+end
+
 namespace Complex
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ]
@@ -52,17 +88,8 @@ theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
   | 0 => rfl
   | n + 1 => by rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]
 
-theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp := by
-  -- Porting note: added `@` due to `∀ {n}` weirdness above
-  refine @(contDiff_all_iff_nat.2 fun n => ?_)
-  have : ContDiff ℂ (↑n) exp := by
-    induction n with
-    | zero => exact contDiff_zero.2 continuous_exp
-    | succ n ihn =>
-      rw [contDiff_succ_iff_deriv]
-      use differentiable_exp
-      rwa [deriv_exp]
-  exact this.restrict_scalars 𝕜
+theorem contDiff_exp {n : ℕ∞} : ContDiff 𝕜 n exp :=
+  analyticOnNhd_cexp.restrictScalars.contDiff
 
 theorem hasStrictDerivAt_exp (x : ℂ) : HasStrictDerivAt exp (exp x) x :=
   contDiff_exp.contDiffAt.hasStrictDerivAt' (hasDerivAt_exp x) le_rfl
@@ -156,12 +183,44 @@ theorem iteratedDeriv_cexp_const_mul (n : ℕ) (c : ℂ) :
     (iteratedDeriv n fun s : ℂ => exp (c * s)) = fun s => c ^ n * exp (c * s) := by
   rw [iteratedDeriv_const_mul contDiff_exp, iteratedDeriv_eq_iterate, iter_deriv_exp]
 
-
 /-! ## `Real.exp` -/
 
-namespace Real
+section
+
+open Real
+
+variable {x : ℝ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E}
+
+/-- `exp` is entire -/
+theorem analyticOnNhd_rexp : AnalyticOnNhd ℝ exp univ := by
+  rw [Real.exp_eq_exp_ℝ]
+  exact fun x _ ↦ NormedSpace.exp_analytic x
+
+theorem analyticOn_rexp : AnalyticOn ℝ exp univ := analyticOnNhd_rexp.analyticOn
+
+/-- `exp` is analytic at any point -/
+theorem analyticAt_rexp : AnalyticAt ℝ exp x :=
+  analyticOnNhd_rexp x (mem_univ _)
 
-variable {x y z : ℝ}
+/-- `exp ∘ f` is analytic -/
+theorem AnalyticAt.rexp {x : E} (fa : AnalyticAt ℝ f x) : AnalyticAt ℝ (fun z ↦ exp (f z)) x :=
+  analyticAt_rexp.comp fa
+
+theorem AnalyticWithinAt.rexp {x : E} (fa : AnalyticWithinAt ℝ f s x) :
+    AnalyticWithinAt ℝ (fun z ↦ exp (f z)) s x :=
+  analyticAt_rexp.comp_analyticWithinAt fa
+
+/-- `exp ∘ f` is analytic -/
+theorem AnalyticOnNhd.rexp {s : Set E} (fs : AnalyticOnNhd ℝ f s) :
+    AnalyticOnNhd ℝ (fun z ↦ exp (f z)) s :=
+  fun z n ↦ analyticAt_rexp.comp (fs z n)
+
+theorem AnalyticOn.rexp (fs : AnalyticOn ℝ f s) : AnalyticOn ℝ (fun z ↦ exp (f z)) s :=
+  analyticOnNhd_rexp.comp_analyticOn fs (mapsTo_univ _ _)
+
+end
+
+namespace Real
 
 theorem hasStrictDerivAt_exp (x : ℝ) : HasStrictDerivAt exp (exp x) x :=
   (Complex.hasStrictDerivAt_exp x).real_of_complex
@@ -169,12 +228,12 @@ theorem hasStrictDerivAt_exp (x : ℝ) : HasStrictDerivAt exp (exp x) x :=
 theorem hasDerivAt_exp (x : ℝ) : HasDerivAt exp (exp x) x :=
   (Complex.hasDerivAt_exp x).real_of_complex
 
-theorem contDiff_exp {n} : ContDiff ℝ n exp :=
+theorem contDiff_exp {n : ℕ∞} : ContDiff ℝ n exp :=
   Complex.contDiff_exp.real_of_complex
 
 theorem differentiable_exp : Differentiable ℝ exp := fun x => (hasDerivAt_exp x).differentiableAt
 
-theorem differentiableAt_exp : DifferentiableAt ℝ exp x :=
+theorem differentiableAt_exp {x : ℝ} : DifferentiableAt ℝ exp x :=
   differentiable_exp x
 
 @[simp]
diff --git a/Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean b/Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
index d07803c56bc15..86b3f777ce844 100644
--- a/Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
@@ -461,7 +461,7 @@ end GammaHasDeriv
 theorem tendsto_self_mul_Gamma_nhds_zero : Tendsto (fun z : ℂ => z * Gamma z) (𝓝[≠] 0) (𝓝 1) := by
   rw [show 𝓝 (1 : ℂ) = 𝓝 (Gamma (0 + 1)) by simp only [zero_add, Complex.Gamma_one]]
   convert (Tendsto.mono_left _ nhdsWithin_le_nhds).congr'
-    (eventuallyEq_of_mem self_mem_nhdsWithin Complex.Gamma_add_one)
+    (eventuallyEq_of_mem self_mem_nhdsWithin Complex.Gamma_add_one) using 1
   refine ContinuousAt.comp (g := Gamma) ?_ (continuous_id.add continuous_const).continuousAt
   refine (Complex.differentiableAt_Gamma _ fun m => ?_).continuousAt
   rw [zero_add, ← ofReal_natCast, ← ofReal_neg, ← ofReal_one, Ne, ofReal_inj]
@@ -553,8 +553,7 @@ lemma integral_rpow_mul_exp_neg_mul_Ioi {a r : ℝ} (ha : 0 < a) (hr : 0 < r) :
   convert integral_cpow_mul_exp_neg_mul_Ioi (by rwa [ofReal_re] : 0 < (a : ℂ).re) hr
   refine _root_.integral_ofReal.symm.trans <| setIntegral_congr measurableSet_Ioi (fun t ht ↦ ?_)
   norm_cast
-  rw [← ofReal_cpow (le_of_lt ht), RCLike.ofReal_mul]
-  rfl
+  simp_rw [← ofReal_cpow ht.le, RCLike.ofReal_mul, coe_algebraMap]
 
 open Lean.Meta Qq Mathlib.Meta.Positivity in
 /-- The `positivity` extension which identifies expressions of the form `Gamma a`. -/
diff --git a/Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean b/Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
index 674fd91181743..f449779354eda 100644
--- a/Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
@@ -123,7 +123,7 @@ theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
     push_cast
     rw [mul_sub, mul_one, mul_div_cancel₀ _ ha']
 
-/-- Relation between Beta integral and Gamma function.  -/
+/-- Relation between Beta integral and Gamma function. -/
 theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
     Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by
   -- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
@@ -183,11 +183,11 @@ theorem betaIntegral_recurrence {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
   have int_ev := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le zero_le_one hc hder h_int
   have hF0 : F 0 = 0 := by
     simp only [F, mul_eq_zero, ofReal_zero, cpow_eq_zero_iff, eq_self_iff_true, Ne,
-      true_and_iff, sub_zero, one_cpow, one_ne_zero, or_false_iff]
+      true_and, sub_zero, one_cpow, one_ne_zero, or_false]
     contrapose! hu; rw [hu, zero_re]
   have hF1 : F 1 = 0 := by
     simp only [F, mul_eq_zero, ofReal_one, one_cpow, one_ne_zero, sub_self, cpow_eq_zero_iff,
-      eq_self_iff_true, Ne, true_and_iff, false_or_iff]
+      eq_self_iff_true, Ne, true_and, false_or]
     contrapose! hv; rw [hv, zero_re]
   rw [hF0, hF1, sub_zero, intervalIntegral.integral_sub, intervalIntegral.integral_const_mul,
     intervalIntegral.integral_const_mul] at int_ev
@@ -270,7 +270,7 @@ theorem GammaSeq_eq_approx_Gamma_integral {s : ℂ} (hs : 0 < re s) {n : ℕ} (h
       mul_cpow_ofReal_nonneg hx.1.le (Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn)).le]
   rw [A, B, cpow_natCast]; ring
 
-/-- The main techical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
+/-- The main technical lemma for `GammaSeq_tendsto_Gamma`, expressing the integral defining the
 Gamma function for `0 < re s` as the limit of a sequence of integrals over finite intervals. -/
 theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
     Tendsto (fun n : ℕ => ∫ x : ℝ in (0)..n, ((1 - x / n) ^ n : ℝ) * (x : ℂ) ^ (s - 1)) atTop
@@ -328,7 +328,7 @@ theorem approx_Gamma_integral_tendsto_Gamma_integral {s : ℂ} (hs : 0 < re s) :
       exact rpow_nonneg (le_of_lt hx) _
     · rw [indicator_of_mem (mem_Ioc.mpr ⟨mem_Ioi.mp hx, hxn⟩), norm_mul, Complex.norm_eq_abs,
         Complex.abs_of_nonneg
-          (pow_nonneg (sub_nonneg.mpr <| div_le_one_of_le hxn <| by positivity) _),
+          (pow_nonneg (sub_nonneg.mpr <| div_le_one_of_le₀ hxn <| by positivity) _),
         Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos hx, sub_re, one_re,
         mul_le_mul_right (rpow_pos_of_pos hx _)]
       exact one_sub_div_pow_le_exp_neg hxn
@@ -407,7 +407,7 @@ theorem Gamma_mul_Gamma_one_sub (z : ℂ) : Gamma z * Gamma (1 - z) = π / sin (
     rw [hs, div_zero]
     rw [← neg_eq_zero, ← Complex.sin_neg, ← mul_neg, Complex.sin_eq_zero_iff, mul_comm] at hs
     obtain ⟨k, hk⟩ := hs
-    rw [mul_eq_mul_right_iff, eq_false (ofReal_ne_zero.mpr pi_pos.ne'), or_false_iff,
+    rw [mul_eq_mul_right_iff, eq_false (ofReal_ne_zero.mpr pi_pos.ne'), or_false,
       neg_eq_iff_eq_neg] at hk
     rw [hk]
     cases k
@@ -547,13 +547,13 @@ theorem Gamma_mul_Gamma_add_half (s : ℂ) :
     convert congr_arg Inv.inv (congr_fun this s) using 1
     · rw [mul_inv, inv_inv, inv_inv]
     · rw [div_eq_mul_inv, mul_inv, mul_inv, inv_inv, inv_inv, ← cpow_neg, neg_sub]
-  have h1 : AnalyticOn ℂ (fun z : ℂ => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ := by
-    refine DifferentiableOn.analyticOn ?_ isOpen_univ
+  have h1 : AnalyticOnNhd ℂ (fun z : ℂ => (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ := by
+    refine DifferentiableOn.analyticOnNhd ?_ isOpen_univ
     refine (differentiable_one_div_Gamma.mul ?_).differentiableOn
     exact differentiable_one_div_Gamma.comp (differentiable_id.add (differentiable_const _))
-  have h2 : AnalyticOn ℂ
+  have h2 : AnalyticOnNhd ℂ
       (fun z => (Gamma (2 * z))⁻¹ * (2 : ℂ) ^ (2 * z - 1) / ↑(√π)) univ := by
-    refine DifferentiableOn.analyticOn ?_ isOpen_univ
+    refine DifferentiableOn.analyticOnNhd ?_ isOpen_univ
     refine (Differentiable.mul ?_ (differentiable_const _)).differentiableOn
     apply Differentiable.mul
     · exact differentiable_one_div_Gamma.comp (differentiable_id'.const_mul _)
@@ -563,7 +563,7 @@ theorem Gamma_mul_Gamma_add_half (s : ℂ) :
     rw [tendsto_nhdsWithin_iff]; constructor
     · exact tendsto_nhdsWithin_of_tendsto_nhds continuous_ofReal.continuousAt
     · exact eventually_nhdsWithin_iff.mpr (Eventually.of_forall fun t ht => ofReal_ne_one.mpr ht)
-  refine AnalyticOn.eq_of_frequently_eq h1 h2 (h3.frequently ?_)
+  refine AnalyticOnNhd.eq_of_frequently_eq h1 h2 (h3.frequently ?_)
   refine ((Eventually.filter_mono nhdsWithin_le_nhds) ?_).frequently
   refine (eventually_gt_nhds zero_lt_one).mp (Eventually.of_forall fun t ht => ?_)
   rw [← mul_inv, Gamma_ofReal, (by norm_num : (t : ℂ) + 1 / 2 = ↑(t + 1 / 2)), Gamma_ofReal, ←
diff --git a/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean b/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean
index a60fa19f4171c..89db8290e3687 100644
--- a/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean
@@ -41,7 +41,7 @@ namespace GaussianFourier
 variable {b : ℂ}
 
 /-- The integral of the Gaussian function over the vertical edges of a rectangle
-with vertices at `(±T, 0)` and `(±T, c)`.  -/
+with vertices at `(±T, 0)` and `(±T, c)`. -/
 def verticalIntegral (b : ℂ) (c T : ℝ) : ℂ :=
   ∫ y : ℝ in (0 : ℝ)..c, I * (cexp (-b * (T + y * I) ^ 2) - cexp (-b * (T - y * I) ^ 2))
 
@@ -332,18 +332,18 @@ theorem integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
 theorem integral_cexp_neg_mul_sq_norm_add
     (hb : 0 < b.re) (c : ℂ) (w : V) :
     ∫ v : V, cexp (- b * ‖v‖^2 + c * ⟪w, v⟫) =
-      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) := by
+      (π / b) ^ (Module.finrank ℝ V / 2 : ℂ) * cexp (c ^ 2 * ‖w‖^2 / (4 * b)) := by
   let e := (stdOrthonormalBasis ℝ V).repr.symm
   rw [← e.measurePreserving.integral_comp e.toHomeomorph.measurableEmbedding]
   convert integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace
     hb c (e.symm w) <;> simp [LinearIsometryEquiv.inner_map_eq_flip]
 
 theorem integral_cexp_neg_mul_sq_norm (hb : 0 < b.re) :
-    ∫ v : V, cexp (- b * ‖v‖^2) = (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) := by
+    ∫ v : V, cexp (- b * ‖v‖^2) = (π / b) ^ (Module.finrank ℝ V / 2 : ℂ) := by
   simpa using integral_cexp_neg_mul_sq_norm_add hb 0 (0 : V)
 
 theorem integral_rexp_neg_mul_sq_norm {b : ℝ} (hb : 0 < b) :
-    ∫ v : V, rexp (- b * ‖v‖^2) = (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℝ) := by
+    ∫ v : V, rexp (- b * ‖v‖^2) = (π / b) ^ (Module.finrank ℝ V / 2 : ℝ) := by
   rw [← ofReal_inj]
   convert integral_cexp_neg_mul_sq_norm (show 0 < (b : ℂ).re from hb) (V := V)
   · change ofRealLI (∫ (v : V), rexp (-b * ‖v‖ ^ 2)) = ∫ (v : V), cexp (-↑b * ↑‖v‖ ^ 2)
@@ -354,7 +354,7 @@ theorem integral_rexp_neg_mul_sq_norm {b : ℝ} (hb : 0 < b) :
 
 theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :
     𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =
-      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) := by
+      (π / b) ^ (Module.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) := by
   simp only [neg_mul, fourierIntegral_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,
     smul_eq_mul, ← Complex.exp_add, real_inner_comm w]
   convert integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v
@@ -367,7 +367,7 @@ theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w
 
 theorem _root_.fourierIntegral_gaussian_innerProductSpace (hb : 0 < b.re) (w : V) :
     𝓕 (fun v ↦ cexp (- b * ‖v‖^2)) w =
-      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖w‖^2 / b) := by
+      (π / b) ^ (Module.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖w‖^2 / b) := by
   simpa using fourierIntegral_gaussian_innerProductSpace' hb 0 w
 
 end InnerProductSpace
diff --git a/Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean b/Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
index 121367c2358ac..163d700eb16bb 100644
--- a/Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean
@@ -109,7 +109,7 @@ theorem integrable_rpow_mul_exp_neg_mul_sq {b : ℝ} (hb : 0 < b) {s : ℝ} (hs
   refine ⟨?_, integrableOn_rpow_mul_exp_neg_mul_sq hb hs⟩
   rw [← (Measure.measurePreserving_neg (volume : Measure ℝ)).integrableOn_comp_preimage
       (Homeomorph.neg ℝ).measurableEmbedding]
-  simp only [Function.comp, neg_sq, neg_preimage, preimage_neg_Iio, neg_neg, neg_zero]
+  simp only [Function.comp_def, neg_sq, neg_preimage, preimage_neg_Iio, neg_neg, neg_zero]
   apply Integrable.mono' (integrableOn_rpow_mul_exp_neg_mul_sq hb hs)
   · apply Measurable.aestronglyMeasurable
     exact (measurable_id'.neg.pow measurable_const).mul
@@ -185,7 +185,7 @@ theorem integral_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
   convert integral_Ioi_of_hasDerivAt_of_tendsto' (fun x _ => (A ↑x).comp_ofReal)
     (integrable_mul_cexp_neg_mul_sq hb).integrableOn B using 1
   simp only [mul_zero, ofReal_zero, zero_pow, Ne, Nat.one_ne_zero,
-    not_false_iff, Complex.exp_zero, mul_one, sub_neg_eq_add, zero_add]
+    not_false_iff, Complex.exp_zero, mul_one, sub_neg_eq_add, zero_add, reduceCtorEq]
 
 /-- The *square* of the Gaussian integral `∫ x:ℝ, exp (-b * x^2)` is equal to `π / b`. -/
 theorem integral_gaussian_sq_complex {b : ℂ} (hb : 0 < b.re) :
diff --git a/Mathlib/Analysis/SpecialFunctions/Integrals.lean b/Mathlib/Analysis/SpecialFunctions/Integrals.lean
index ca08532bc9736..31f51791cdb31 100644
--- a/Mathlib/Analysis/SpecialFunctions/Integrals.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Integrals.lean
@@ -33,7 +33,7 @@ integrate, integration, integrable, integrability
 -/
 
 
-open Real Nat Set Finset
+open Real Set Finset
 
 open scoped Real Interval
 
@@ -117,7 +117,7 @@ theorem intervalIntegrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [[a,
   · -- Easy case #1: 0 ∉ [a, b] -- use continuity.
     refine (ContinuousAt.continuousOn fun x hx => ?_).intervalIntegrable
     exact Complex.continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_mem_of_not_mem hx h2)
-  rw [eq_false h2, or_false_iff] at h
+  rw [eq_false h2, or_false] at h
   rcases lt_or_eq_of_le h with (h' | h')
   · -- Easy case #2: 0 < re r -- again use continuity
     exact (Complex.continuous_ofReal_cpow_const h').intervalIntegrable _ _
@@ -580,8 +580,9 @@ theorem integral_mul_rpow_one_add_sq {t : ℝ} (ht : t ≠ -1) :
 
 end RpowCpow
 
-/-! ### Integral of `sin x ^ n` -/
+open Nat
 
+/-! ### Integral of `sin x ^ n` -/
 
 theorem integral_sin_pow_aux :
     (∫ x in a..b, sin x ^ (n + 2)) =
diff --git a/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean b/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
index 79be49beb847d..5b9cbab201ab2 100644
--- a/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
+++ b/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
@@ -27,7 +27,7 @@ noncomputable section
 
 open scoped NNReal Filter Topology ENNReal
 
-open Asymptotics Filter Set Real MeasureTheory FiniteDimensional
+open Asymptotics Filter Set Real MeasureTheory Module
 
 variable {E : Type*} [NormedAddCommGroup E]
 
@@ -85,7 +85,7 @@ theorem finite_integral_rpow_sub_one_pow_aux {r : ℝ} (n : ℕ) (hnr : (n : ℝ
   refine IntegrableOn.setLIntegral_lt_top ?_
   rw [← intervalIntegrable_iff_integrableOn_Ioc_of_le zero_le_one]
   apply intervalIntegral.intervalIntegrable_rpow'
-  rwa [neg_lt_neg_iff, inv_mul_lt_iff' hr, one_mul]
+  rwa [neg_lt_neg_iff, inv_mul_lt_iff₀' hr, one_mul]
 
 variable [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure]
 
diff --git a/Mathlib/Analysis/SpecialFunctions/Log/Base.lean b/Mathlib/Analysis/SpecialFunctions/Log/Base.lean
index e537e8132736f..f7fafb38f92f6 100644
--- a/Mathlib/Analysis/SpecialFunctions/Log/Base.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Log/Base.lean
@@ -47,6 +47,12 @@ theorem logb_zero : logb b 0 = 0 := by simp [logb]
 @[simp]
 theorem logb_one : logb b 1 = 0 := by simp [logb]
 
+@[simp]
+theorem logb_zero_left : logb 0 x = 0 := by simp only [← log_div_log, log_zero, div_zero]
+
+@[simp]
+theorem logb_one_left : logb 1 x = 0 := by simp only [← log_div_log, log_one, div_zero]
+
 @[simp]
 lemma logb_self_eq_one (hb : 1 < b) : logb b b = 1 :=
   div_self (log_pos hb).ne'
@@ -342,36 +348,50 @@ theorem tendsto_logb_atTop_of_base_lt_one : Tendsto (logb b) atTop atBot := by
 
 end BPosAndBLtOne
 
-theorem floor_logb_natCast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
+theorem floor_logb_natCast {b : ℕ} {r : ℝ} (hr : 0 ≤ r) :
     ⌊logb b r⌋ = Int.log b r := by
   obtain rfl | hr := hr.eq_or_lt
   · rw [logb_zero, Int.log_zero_right, Int.floor_zero]
-  have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
-  apply le_antisymm
-  · rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_intCast b]
-    refine le_of_le_of_eq ?_ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)
-    exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _)
-  · rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_intCast]
-    exact Int.zpow_log_le_self hb hr
+  by_cases hb : 1 < b
+  · have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
+    apply le_antisymm
+    · rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_intCast b]
+      refine le_of_le_of_eq ?_ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)
+      exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _)
+    · rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_intCast]
+      exact Int.zpow_log_le_self hb hr
+  · rw [Nat.one_lt_iff_ne_zero_and_ne_one, ← or_iff_not_and_not] at hb
+    cases hb
+    · simp_all only [CharP.cast_eq_zero, logb_zero_left, Int.floor_zero, Int.log_zero_left]
+    · simp_all only [Nat.cast_one, logb_one_left, Int.floor_zero, Int.log_one_left]
 
 @[deprecated (since := "2024-04-17")]
 alias floor_logb_nat_cast := floor_logb_natCast
 
-theorem ceil_logb_natCast {b : ℕ} {r : ℝ} (hb : 1 < b) (hr : 0 ≤ r) :
+theorem ceil_logb_natCast {b : ℕ} {r : ℝ} (hr : 0 ≤ r) :
     ⌈logb b r⌉ = Int.clog b r := by
   obtain rfl | hr := hr.eq_or_lt
   · rw [logb_zero, Int.clog_zero_right, Int.ceil_zero]
-  have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
-  apply le_antisymm
-  · rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpow_intCast]
-    exact Int.self_le_zpow_clog hb r
-  · rw [← Int.le_zpow_iff_clog_le hb hr, ← rpow_intCast b]
-    refine (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr).symm.trans_le ?_
-    exact rpow_le_rpow_of_exponent_le hb1'.le (Int.le_ceil _)
+  by_cases hb : 1 < b
+  · have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
+    apply le_antisymm
+    · rw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpow_intCast]
+      exact Int.self_le_zpow_clog hb r
+    · rw [← Int.le_zpow_iff_clog_le hb hr, ← rpow_intCast b]
+      refine (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr).symm.trans_le ?_
+      exact rpow_le_rpow_of_exponent_le hb1'.le (Int.le_ceil _)
+  · rw [Nat.one_lt_iff_ne_zero_and_ne_one, ← or_iff_not_and_not] at hb
+    cases hb
+    · simp_all only [CharP.cast_eq_zero, logb_zero_left, Int.ceil_zero, Int.clog_zero_left]
+    · simp_all only [Nat.cast_one, logb_one_left, Int.ceil_zero, Int.clog_one_left]
 
 @[deprecated (since := "2024-04-17")]
 alias ceil_logb_nat_cast := ceil_logb_natCast
 
+lemma natLog_le_logb (a b : ℕ) : Nat.log b a ≤ Real.logb b a := by
+  apply le_trans _ (Int.floor_le ((b : ℝ).logb a))
+  rw [Real.floor_logb_natCast (Nat.cast_nonneg a), Int.log_natCast, Int.cast_natCast]
+
 @[simp]
 theorem logb_eq_zero : logb b x = 0 ↔ b = 0 ∨ b = 1 ∨ b = -1 ∨ x = 0 ∨ x = 1 ∨ x = -1 := by
   simp_rw [logb, div_eq_zero_iff, log_eq_zero]
@@ -403,7 +423,7 @@ lemma Real.induction_Ico_mul {P : ℝ → Prop} (x₀ r : ℝ) (hr : 1 < r) (hx
     intro x hx
     have hx' : 0 < x / x₀ := div_pos (hx₀.trans_le hx) hx₀
     refine this ⌊logb r (x / x₀)⌋₊ x ?_
-    rw [mem_Ico, ← div_lt_iff hx₀, ← rpow_natCast, ← logb_lt_iff_lt_rpow hr hx', Nat.cast_add,
+    rw [mem_Ico, ← div_lt_iff₀ hx₀, ← rpow_natCast, ← logb_lt_iff_lt_rpow hr hx', Nat.cast_add,
       Nat.cast_one]
     exact ⟨hx, Nat.lt_floor_add_one _⟩
   intro n
diff --git a/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean b/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
index ac860ad68c9a6..ccc7141a90155 100644
--- a/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
@@ -124,7 +124,7 @@ theorem log_le_log_iff (h : 0 < x) (h₁ : 0 < y) : log x ≤ log y ↔ x ≤ y
 lemma log_le_log (hx : 0 < x) (hxy : x ≤ y) : log x ≤ log y :=
   (log_le_log_iff hx (hx.trans_le hxy)).2 hxy
 
-@[gcongr]
+@[gcongr, bound]
 theorem log_lt_log (hx : 0 < x) (h : x < y) : log x < log y := by
   rwa [← exp_lt_exp, exp_log hx, exp_log (lt_trans hx h)]
 
@@ -143,6 +143,7 @@ theorem log_pos_iff (hx : 0 < x) : 0 < log x ↔ 1 < x := by
   rw [← log_one]
   exact log_lt_log_iff zero_lt_one hx
 
+@[bound]
 theorem log_pos (hx : 1 < x) : 0 < log x :=
   (log_pos_iff (lt_trans zero_lt_one hx)).2 hx
 
@@ -155,6 +156,7 @@ theorem log_neg_iff (h : 0 < x) : log x < 0 ↔ x < 1 := by
   rw [← log_one]
   exact log_lt_log_iff h zero_lt_one
 
+@[bound]
 theorem log_neg (h0 : 0 < x) (h1 : x < 1) : log x < 0 :=
   (log_neg_iff h0).2 h1
 
@@ -177,6 +179,7 @@ theorem log_nonpos_iff' (hx : 0 ≤ x) : log x ≤ 0 ↔ x ≤ 1 := by
   · simp [le_refl, zero_le_one]
   exact log_nonpos_iff hx
 
+@[bound]
 theorem log_nonpos (hx : 0 ≤ x) (h'x : x ≤ 1) : log x ≤ 0 :=
   (log_nonpos_iff' hx).2 h'x
 
@@ -278,17 +281,27 @@ theorem log_le_sub_one_of_pos {x : ℝ} (hx : 0 < x) : log x ≤ x - 1 := by
 lemma one_sub_inv_le_log_of_pos (hx : 0 < x) : 1 - x⁻¹ ≤ log x := by
   simpa [add_comm] using log_le_sub_one_of_pos (inv_pos.2 hx)
 
+/-- See `Real.log_le_sub_one_of_pos` for the stronger version when `x ≠ 0`. -/
+lemma log_le_self (hx : 0 ≤ x) : log x ≤ x := by
+  obtain rfl | hx := hx.eq_or_lt
+  · simp
+  · exact (log_le_sub_one_of_pos hx).trans (by linarith)
+
+/-- See `Real.one_sub_inv_le_log_of_pos` for the stronger version when `x ≠ 0`. -/
+lemma neg_inv_le_log (hx : 0 ≤ x) : -x⁻¹ ≤ log x := by
+  rw [neg_le, ← log_inv]; exact log_le_self <| inv_nonneg.2 hx
+
 /-- Bound for `|log x * x|` in the interval `(0, 1]`. -/
 theorem abs_log_mul_self_lt (x : ℝ) (h1 : 0 < x) (h2 : x ≤ 1) : |log x * x| < 1 := by
   have : 0 < 1 / x := by simpa only [one_div, inv_pos] using h1
   replace := log_le_sub_one_of_pos this
   replace : log (1 / x) < 1 / x := by linarith
-  rw [log_div one_ne_zero h1.ne', log_one, zero_sub, lt_div_iff h1] at this
+  rw [log_div one_ne_zero h1.ne', log_one, zero_sub, lt_div_iff₀ h1] at this
   have aux : 0 ≤ -log x * x := by
     refine mul_nonneg ?_ h1.le
     rw [← log_inv]
     apply log_nonneg
-    rw [← le_inv h1 zero_lt_one, inv_one]
+    rw [← le_inv_comm₀ h1 zero_lt_one, inv_one]
     exact h2
   rw [← abs_of_nonneg aux, neg_mul, abs_neg] at this
   exact this
diff --git a/Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean b/Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
index 5c2378f2c0801..958e650ae3268 100644
--- a/Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
@@ -192,7 +192,7 @@ theorem tendsto_mul_log_one_plus_div_atTop (t : ℝ) :
       (((hasDerivAt_id (0 : ℝ)).const_mul t).const_add 1).log (by simp)
   have h₂ : Tendsto (fun x : ℝ => x⁻¹) atTop (𝓝[≠] 0) :=
     tendsto_inv_atTop_zero'.mono_right (nhdsWithin_mono _ fun x hx => (Set.mem_Ioi.mp hx).ne')
-  simpa only [(· ∘ ·), inv_inv] using h₁.comp h₂
+  simpa only [Function.comp_def, inv_inv] using h₁.comp h₂
 
 /-- A crude lemma estimating the difference between `log (1-x)` and its Taylor series at `0`,
 where the main point of the bound is that it tends to `0`. The goal is to deduce the series
diff --git a/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean b/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
index 2c0c4d15f5e57..68243cfc4a5e8 100644
--- a/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Log/ENNRealLog.lean
@@ -51,24 +51,20 @@ noncomputable def log (x : ℝ≥0∞) : EReal :=
 @[simp]
 lemma log_ofReal (x : ℝ) : log (ENNReal.ofReal x) = if x ≤ 0 then ⊥ else ↑(Real.log x) := by
   simp only [log, ENNReal.none_eq_top, ENNReal.ofReal_ne_top, IsEmpty.forall_iff,
-    ENNReal.ofReal_eq_zero, EReal.coe_ennreal_ofReal]
+    ENNReal.ofReal_eq_zero, EReal.coe_ennreal_ofReal, if_false]
   split_ifs with h_nonpos
   · rfl
-  · trivial
-  · rw [ENNReal.toReal_ofReal]
-    exact (not_le.mp h_nonpos).le
+  · rw [ENNReal.toReal_ofReal (not_le.mp h_nonpos).le]
 
 lemma log_ofReal_of_pos {x : ℝ} (hx : 0 < x) : log (ENNReal.ofReal x) = Real.log x := by
-  rw [log_ofReal, if_neg]
-  exact not_le.mpr hx
+  rw [log_ofReal, if_neg hx.not_le]
 
 theorem log_pos_real {x : ℝ≥0∞} (h : x ≠ 0) (h' : x ≠ ⊤) :
     log x = Real.log (ENNReal.toReal x) := by simp [log, h, h']
 
 theorem log_pos_real' {x : ℝ≥0∞} (h : 0 < x.toReal) :
     log x = Real.log (ENNReal.toReal x) := by
-  simp [log, Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 h).1),
-    ne_of_lt (ENNReal.toReal_pos_iff.1 h).2]
+  simp [log, (ENNReal.toReal_pos_iff.1 h).1.ne', (ENNReal.toReal_pos_iff.1 h).2.ne]
 
 theorem log_of_nnreal {x : ℝ≥0} (h : x ≠ 0) :
     log (x : ℝ≥0∞) = Real.log x := by simp [log, h]
@@ -85,19 +81,19 @@ theorem log_strictMono : StrictMono log := by
   · rcases ENNReal.trichotomy y with (rfl | rfl | y_real)
     · exfalso; exact lt_irrefl 0 h
     · simp
-    · simp [Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).1),
-      ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).2, EReal.bot_lt_coe]
-  · exfalso; exact (ne_top_of_lt h) (Eq.refl ⊤)
-  · simp only [Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 x_real).1),
-      ne_of_lt (ENNReal.toReal_pos_iff.1 x_real).2]
+    · simp [(ENNReal.toReal_pos_iff.1 y_real).1.ne',
+        (ENNReal.toReal_pos_iff.1 y_real).2.ne, EReal.bot_lt_coe]
+  · exfalso; exact not_top_lt h
+  · simp only [(ENNReal.toReal_pos_iff.1 x_real).1.ne',
+      (ENNReal.toReal_pos_iff.1 x_real).2.ne, if_false]
     rcases ENNReal.trichotomy y with (rfl | rfl | y_real)
     · exfalso; rw [← ENNReal.bot_eq_zero] at h; exact not_lt_bot h
     · simp
-    · simp only [Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).1), ↓reduceIte,
-        ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).2, EReal.coe_lt_coe_iff]
+    · simp only [(ENNReal.toReal_pos_iff.1 y_real).1.ne', ↓reduceIte,
+        (ENNReal.toReal_pos_iff.1 y_real).2.ne, EReal.coe_lt_coe_iff]
       apply Real.log_lt_log x_real
-      exact (ENNReal.toReal_lt_toReal (ne_of_lt (ENNReal.toReal_pos_iff.1 x_real).2)
-        (ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).2)).2 h
+      exact (ENNReal.toReal_lt_toReal (ENNReal.toReal_pos_iff.1 x_real).2.ne
+        (ENNReal.toReal_pos_iff.1 y_real).2.ne).2 h
 
 theorem log_monotone : Monotone log := log_strictMono.monotone
 
@@ -112,14 +108,14 @@ theorem log_surjective : Function.Surjective log := by
   use ENNReal.ofReal (Real.exp y.toReal)
   have exp_y_pos := not_le_of_lt (Real.exp_pos y.toReal)
   simp only [log, ofReal_eq_zero, exp_y_pos, ↓reduceIte, ofReal_ne_top,
-    ENNReal.toReal_ofReal (le_of_lt (Real.exp_pos y.toReal)), Real.log_exp y.toReal]
-  exact EReal.coe_toReal (ne_of_lt y_ntop) (Ne.symm (ne_of_lt y_nbot))
+    ENNReal.toReal_ofReal (Real.exp_pos y.toReal).le, Real.log_exp y.toReal]
+  exact EReal.coe_toReal y_ntop.ne y_nbot.ne'
 
 theorem log_bijective : Function.Bijective log := ⟨log_injective, log_surjective⟩
 
 @[simp]
 theorem log_eq_iff {x y : ℝ≥0∞} : log x = log y ↔ x = y :=
-  Iff.intro (@log_injective x y) (fun h ↦ by rw [h])
+  log_injective.eq_iff
 
 @[simp] theorem log_eq_bot_iff {x : ℝ≥0∞} : log x = ⊥ ↔ x = 0 := log_zero ▸ @log_eq_iff x 0
 
@@ -158,13 +154,13 @@ theorem log_mul_add {x y : ℝ≥0∞} : log (x * y) = log x + log y := by
     · simp
     · rw [log_pos_real' y_real, ENNReal.top_mul', EReal.top_add_coe, log_eq_top_iff]
       simp only [ite_eq_right_iff, zero_ne_top, imp_false]
-      exact Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 y_real).1)
+      exact (ENNReal.toReal_pos_iff.1 y_real).1.ne'
   · rw [log_pos_real' x_real]
     rcases ENNReal.trichotomy y with (rfl | rfl | y_real)
     · simp
-    · simp [Ne.symm (ne_of_lt (ENNReal.toReal_pos_iff.1 x_real).1)]
+    · simp [(ENNReal.toReal_pos_iff.1 x_real).1.ne']
     · rw_mod_cast [log_pos_real', log_pos_real' y_real, ENNReal.toReal_mul]
-      exact Real.log_mul (Ne.symm (ne_of_lt x_real)) (Ne.symm (ne_of_lt y_real))
+      · exact Real.log_mul x_real.ne' y_real.ne'
       rw [toReal_mul]
       positivity
 
@@ -172,24 +168,20 @@ theorem log_pow {x : ℝ≥0∞} {n : ℕ} : log (x ^ n) = n * log x := by
   cases' Nat.eq_zero_or_pos n with n_zero n_pos
   · simp [n_zero, pow_zero x]
   rcases ENNReal.trichotomy x with (rfl | rfl | x_real)
-  · rw [zero_pow (Ne.symm (ne_of_lt n_pos)), log_zero, EReal.mul_bot_of_pos (Nat.cast_pos'.2 n_pos)]
+  · rw [zero_pow n_pos.ne', log_zero, EReal.mul_bot_of_pos (Nat.cast_pos'.2 n_pos)]
   · rw [ENNReal.top_pow n_pos, log_top, EReal.mul_top_of_pos (Nat.cast_pos'.2 n_pos)]
   · replace x_real := ENNReal.toReal_pos_iff.1 x_real
-    have x_ne_zero := Ne.symm (LT.lt.ne x_real.1)
-    have x_ne_top := LT.lt.ne x_real.2
-    simp only [log, pow_eq_zero_iff', x_ne_zero, false_and, ↓reduceIte, pow_eq_top_iff, x_ne_top,
-      toReal_pow, Real.log_pow, EReal.coe_mul]
-    rfl
+    simp only [log, pow_eq_zero_iff', x_real.1.ne', false_and, ↓reduceIte, pow_eq_top_iff,
+      x_real.2.ne, toReal_pow, Real.log_pow, EReal.coe_mul, EReal.coe_coe_eq_natCast]
 
 theorem log_rpow {x : ℝ≥0∞} {y : ℝ} : log (x ^ y) = y * log x := by
   rcases lt_trichotomy y 0 with (y_neg | rfl | y_pos)
   · rcases ENNReal.trichotomy x with (rfl | rfl | x_real)
-    · simp only [ENNReal.zero_rpow_def y, not_lt_of_lt y_neg, ne_of_lt y_neg, log_top, log_zero]
-      exact Eq.symm (EReal.coe_mul_bot_of_neg y_neg)
-    · rw [ENNReal.top_rpow_of_neg y_neg, log_zero, log_top]
-      exact Eq.symm (EReal.coe_mul_top_of_neg y_neg)
-    · have x_ne_zero := Ne.symm (LT.lt.ne (ENNReal.toReal_pos_iff.1 x_real).1)
-      have x_ne_top := LT.lt.ne (ENNReal.toReal_pos_iff.1 x_real).2
+    · simp only [ENNReal.zero_rpow_def y, not_lt_of_lt y_neg, y_neg.ne, if_false, log_top,
+        log_zero, EReal.coe_mul_bot_of_neg y_neg]
+    · rw [ENNReal.top_rpow_of_neg y_neg, log_zero, log_top, EReal.coe_mul_top_of_neg y_neg]
+    · have x_ne_zero := (ENNReal.toReal_pos_iff.1 x_real).1.ne'
+      have x_ne_top := (ENNReal.toReal_pos_iff.1 x_real).2.ne
       simp only [log, rpow_eq_zero_iff, x_ne_zero, false_and, x_ne_top, or_self, ↓reduceIte,
         rpow_eq_top_iff]
       norm_cast
@@ -198,8 +190,8 @@ theorem log_rpow {x : ℝ≥0∞} {y : ℝ} : log (x ^ y) = y * log x := by
   · rcases ENNReal.trichotomy x with (rfl | rfl | x_real)
     · rw [ENNReal.zero_rpow_of_pos y_pos, log_zero, EReal.mul_bot_of_pos]; norm_cast
     · rw [ENNReal.top_rpow_of_pos y_pos, log_top, EReal.mul_top_of_pos]; norm_cast
-    · have x_ne_zero := Ne.symm (LT.lt.ne (ENNReal.toReal_pos_iff.1 x_real).1)
-      have x_ne_top := LT.lt.ne (ENNReal.toReal_pos_iff.1 x_real).2
+    · have x_ne_zero := (ENNReal.toReal_pos_iff.1 x_real).1.ne'
+      have x_ne_top := (ENNReal.toReal_pos_iff.1 x_real).2.ne
       simp only [log, rpow_eq_zero_iff, x_ne_zero, false_and, x_ne_top, or_self, ↓reduceIte,
         rpow_eq_top_iff]
       norm_cast
diff --git a/Mathlib/Analysis/SpecialFunctions/PolarCoord.lean b/Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
index e11dde25a0bad..62dafde2c1433 100644
--- a/Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
+++ b/Mathlib/Analysis/SpecialFunctions/PolarCoord.lean
@@ -39,12 +39,12 @@ def polarCoord : PartialHomeomorph (ℝ × ℝ) (ℝ × ℝ) where
     · simpa using hr
     · right
       simp at hr
-      simpa only [ne_of_gt hr, Ne, mem_setOf_eq, mul_eq_zero, false_or_iff,
+      simpa only [ne_of_gt hr, Ne, mem_setOf_eq, mul_eq_zero, false_or,
         sin_eq_zero_iff_of_lt_of_lt hθ.1 hθ.2] using h'θ
   map_source' := by
     rintro ⟨x, y⟩ hxy
     simp only [prod_mk_mem_set_prod_eq, mem_Ioi, sqrt_pos, mem_Ioo, Complex.neg_pi_lt_arg,
-      true_and_iff, Complex.arg_lt_pi_iff]
+      true_and, Complex.arg_lt_pi_iff]
     constructor
     · cases' hxy with hxy hxy
       · dsimp at hxy; linarith [sq_pos_of_ne_zero hxy.ne', sq_nonneg y]
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
index a4b120ba2ef3c..95944cd30ee97 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean
@@ -166,7 +166,7 @@ theorem ENNReal.tendsto_rpow_at_top {y : ℝ} (hy : 0 < y) :
   lift a to ℝ≥0 using ha'
   -- Porting note: reduced defeq abuse
   simp only [Set.mem_Ioi, coe_lt_coe] at ha hc
-  rw [ENNReal.coe_rpow_of_nonneg _ hy.le]
+  rw [← ENNReal.coe_rpow_of_nonneg _ hy.le]
   exact mod_cast hc a ha
 
 end Limits
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
index 9a2469906a661..c9a10865fd507 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
@@ -175,7 +175,7 @@ lemma cpow_ofNat_mul' {x : ℂ} {n : ℕ} [n.AtLeastTwo] (hlt : -π < OfNat.ofNa
 lemma pow_cpow_nat_inv {x : ℂ} {n : ℕ} (h₀ : n ≠ 0) (hlt : -(π / n) < x.arg) (hle : x.arg ≤ π / n) :
     (x ^ n) ^ (n⁻¹ : ℂ) = x := by
   rw [← cpow_nat_mul', mul_inv_cancel₀ (Nat.cast_ne_zero.2 h₀), cpow_one]
-  · rwa [← div_lt_iff' (Nat.cast_pos.2 h₀.bot_lt), neg_div]
+  · rwa [← div_lt_iff₀' (Nat.cast_pos.2 h₀.bot_lt), neg_div]
   · rwa [← le_div_iff₀' (Nat.cast_pos.2 h₀.bot_lt)]
 
 lemma pow_cpow_ofNat_inv {x : ℂ} {n : ℕ} [n.AtLeastTwo] (hlt : -(π / OfNat.ofNat n) < x.arg)
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
index d62254dfce817..841d4fc4de757 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
@@ -299,7 +299,7 @@ theorem continuousAt_cpow_zero_of_re_pos {z : ℂ} (hz : 0 < z.re) :
         (continuous_fst.norm.tendsto ((0 : ℂ), z)).rpow
           ((continuous_re.comp continuous_snd).tendsto _) _ <;>
       simp [hz, Real.zero_rpow hz.ne']
-  · simp only [Function.comp, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
+  · simp only [Function.comp_def, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
     rcases exists_gt |im z| with ⟨C, hC⟩
     refine ⟨Real.exp (π * C), eventually_map.2 ?_⟩
     refine
@@ -332,8 +332,7 @@ theorem continuousAt_ofReal_cpow (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0)
     ContinuousAt (fun p => (p.1 : ℂ) ^ p.2 : ℝ × ℂ → ℂ) (x, y) := by
   rcases lt_trichotomy (0 : ℝ) x with (hx | rfl | hx)
   · -- x > 0 : easy case
-    have : ContinuousAt (fun p => ⟨↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) :=
-      continuous_ofReal.continuousAt.prod_map continuousAt_id
+    have : ContinuousAt (fun p => ⟨↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) := by fun_prop
     refine (continuousAt_cpow (Or.inl ?_)).comp this
     rwa [ofReal_re]
   · -- x = 0 : reduce to continuousAt_cpow_zero_of_re_pos
@@ -341,15 +340,13 @@ theorem continuousAt_ofReal_cpow (x : ℝ) (y : ℂ) (h : 0 < y.re ∨ x ≠ 0)
       rw [ofReal_zero]
       apply continuousAt_cpow_zero_of_re_pos
       tauto
-    have B : ContinuousAt (fun p => ⟨↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) ⟨0, y⟩ :=
-      continuous_ofReal.continuousAt.prod_map continuousAt_id
+    have B : ContinuousAt (fun p => ⟨↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) ⟨0, y⟩ := by fun_prop
     exact A.comp_of_eq B rfl
   · -- x < 0 : difficult case
     suffices ContinuousAt (fun p => (-(p.1 : ℂ)) ^ p.2 * exp (π * I * p.2) : ℝ × ℂ → ℂ) (x, y) by
       refine this.congr (eventually_of_mem (prod_mem_nhds (Iio_mem_nhds hx) univ_mem) ?_)
       exact fun p hp => (ofReal_cpow_of_nonpos (le_of_lt hp.1) p.2).symm
-    have A : ContinuousAt (fun p => ⟨-↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) :=
-      ContinuousAt.prod_map continuous_ofReal.continuousAt.neg continuousAt_id
+    have A : ContinuousAt (fun p => ⟨-↑p.1, p.2⟩ : ℝ × ℂ → ℂ × ℂ) (x, y) := by fun_prop
     apply ContinuousAt.mul
     · refine (continuousAt_cpow (Or.inl ?_)).comp A
       rwa [neg_re, ofReal_re, neg_pos]
@@ -394,7 +391,7 @@ theorem eventually_pow_one_div_le (x : ℝ≥0) {y : ℝ≥0} (hy : 1 < y) :
   refine eventually_atTop.2 ⟨m + 1, fun n hn => ?_⟩
   simp only [one_div]
   simpa only [NNReal.rpow_inv_le_iff (Nat.cast_pos.2 <| m.succ_pos.trans_le hn),
-    NNReal.rpow_natCast] using hm.le.trans (pow_le_pow_right hy.le (m.le_succ.trans hn))
+    NNReal.rpow_natCast] using hm.le.trans (pow_right_mono₀ hy.le (m.le_succ.trans hn))
 
 end NNReal
 
@@ -445,7 +442,7 @@ theorem eventually_pow_one_div_le {x : ℝ≥0∞} (hx : x ≠ ∞) {y : ℝ≥0
   · lift y to ℝ≥0 using h
     have := NNReal.eventually_pow_one_div_le x (mod_cast hy : 1 < y)
     refine this.congr (Eventually.of_forall fun n => ?_)
-    rw [coe_rpow_of_nonneg x (by positivity : 0 ≤ (1 / n : ℝ)), coe_le_coe]
+    rw [← coe_rpow_of_nonneg x (by positivity : 0 ≤ (1 / n : ℝ)), coe_le_coe]
 
 private theorem continuousAt_rpow_const_of_pos {x : ℝ≥0∞} {y : ℝ} (h : 0 < y) :
     ContinuousAt (fun a : ℝ≥0∞ => a ^ y) x := by
@@ -457,7 +454,7 @@ private theorem continuousAt_rpow_const_of_pos {x : ℝ≥0∞} {y : ℝ} (h : 0
   rw [continuousAt_coe_iff]
   convert continuous_coe.continuousAt.comp (NNReal.continuousAt_rpow_const (Or.inr h.le)) using 1
   ext1 x
-  simp [coe_rpow_of_nonneg _ h.le]
+  simp [← coe_rpow_of_nonneg _ h.le]
 
 @[continuity, fun_prop]
 theorem continuous_rpow_const {y : ℝ} : Continuous fun a : ℝ≥0∞ => a ^ y := by
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean b/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
index e114e6562e484..d2306464d57ff 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
@@ -24,7 +24,7 @@ open Real NNReal ENNReal ComplexConjugate
 open Finset Function Set
 
 namespace NNReal
-variable {w x y z : ℝ}
+variable {x : ℝ≥0} {w y z : ℝ}
 
 /-- The nonnegative real power function `x^y`, defined for `x : ℝ≥0` and `y : ℝ` as the
 restriction of the real power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`,
@@ -52,6 +52,8 @@ theorem rpow_eq_zero_iff {x : ℝ≥0} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ y ≠
   rw [← NNReal.coe_inj, coe_rpow, ← NNReal.coe_eq_zero]
   exact Real.rpow_eq_zero_iff_of_nonneg x.2
 
+lemma rpow_eq_zero (hy : y ≠ 0) : x ^ y = 0 ↔ x = 0 := by simp [hy]
+
 @[simp]
 theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
   NNReal.eq <| Real.zero_rpow h
@@ -60,6 +62,18 @@ theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ≥0) ^ x = 0 :=
 theorem rpow_one (x : ℝ≥0) : x ^ (1 : ℝ) = x :=
   NNReal.eq <| Real.rpow_one _
 
+lemma rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
+  NNReal.eq <| Real.rpow_neg x.2 _
+
+@[simp, norm_cast]
+lemma rpow_natCast (x : ℝ≥0) (n : ℕ) : x ^ (n : ℝ) = x ^ n :=
+  NNReal.eq <| by simpa only [coe_rpow, coe_pow] using Real.rpow_natCast x n
+
+@[simp, norm_cast]
+lemma rpow_intCast (x : ℝ≥0) (n : ℤ) : x ^ (n : ℝ) = x ^ n := by
+  cases n <;> simp only [Int.ofNat_eq_coe, Int.cast_natCast, rpow_natCast, zpow_natCast,
+    Int.cast_negSucc, rpow_neg, zpow_negSucc]
+
 @[simp]
 theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
   NNReal.eq <| Real.one_rpow _
@@ -67,9 +81,45 @@ theorem one_rpow (x : ℝ) : (1 : ℝ≥0) ^ x = 1 :=
 theorem rpow_add {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y + z) = x ^ y * x ^ z :=
   NNReal.eq <| Real.rpow_add (pos_iff_ne_zero.2 hx) _ _
 
-theorem rpow_add' (x : ℝ≥0) {y z : ℝ} (h : y + z ≠ 0) : x ^ (y + z) = x ^ y * x ^ z :=
+theorem rpow_add' (h : y + z ≠ 0) (x : ℝ≥0) : x ^ (y + z) = x ^ y * x ^ z :=
   NNReal.eq <| Real.rpow_add' x.2 h
 
+lemma rpow_add_intCast (hx : x ≠ 0) (y : ℝ) (n : ℤ) : x ^ (y + n) = x ^ y * x ^ n := by
+  ext; exact Real.rpow_add_intCast (mod_cast hx) _ _
+
+lemma rpow_add_natCast (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y + n) = x ^ y * x ^ n := by
+  ext; exact Real.rpow_add_natCast (mod_cast hx) _ _
+
+lemma rpow_sub_intCast (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n := by
+  ext; exact Real.rpow_sub_intCast (mod_cast hx) _ _
+
+lemma rpow_sub_natCast (hx : x ≠ 0) (y : ℝ) (n : ℕ) : x ^ (y - n) = x ^ y / x ^ n := by
+  ext; exact Real.rpow_sub_natCast (mod_cast hx) _ _
+
+lemma rpow_add_intCast' {n : ℤ} (h : y + n ≠ 0) (x : ℝ≥0) : x ^ (y + n) = x ^ y * x ^ n := by
+  ext; exact Real.rpow_add_intCast' (mod_cast x.2) h
+
+lemma rpow_add_natCast' {n : ℕ} (h : y + n ≠ 0) (x : ℝ≥0) : x ^ (y + n) = x ^ y * x ^ n := by
+  ext; exact Real.rpow_add_natCast' (mod_cast x.2) h
+
+lemma rpow_sub_intCast' {n : ℤ} (h : y - n ≠ 0) (x : ℝ≥0) : x ^ (y - n) = x ^ y / x ^ n := by
+  ext; exact Real.rpow_sub_intCast' (mod_cast x.2) h
+
+lemma rpow_sub_natCast' {n : ℕ} (h : y - n ≠ 0) (x : ℝ≥0) : x ^ (y - n) = x ^ y / x ^ n := by
+  ext; exact Real.rpow_sub_natCast' (mod_cast x.2) h
+
+lemma rpow_add_one (hx : x ≠ 0) (y : ℝ) : x ^ (y + 1) = x ^ y * x := by
+  simpa using rpow_add_natCast hx y 1
+
+lemma rpow_sub_one (hx : x ≠ 0) (y : ℝ) : x ^ (y - 1) = x ^ y / x := by
+  simpa using rpow_sub_natCast hx y 1
+
+lemma rpow_add_one' (h : y + 1 ≠ 0) (x : ℝ≥0) : x ^ (y + 1) = x ^ y * x := by
+  rw [rpow_add' h, rpow_one]
+
+lemma rpow_one_add' (h : 1 + y ≠ 0) (x : ℝ≥0) : x ^ (1 + y) = x * x ^ y := by
+  rw [rpow_add' h, rpow_one]
+
 theorem rpow_add_of_nonneg (x : ℝ≥0) {y z : ℝ} (hy : 0 ≤ y) (hz : 0 ≤ z) :
     x ^ (y + z) = x ^ y * x ^ z := by
   ext; exact Real.rpow_add_of_nonneg x.2 hy hz
@@ -81,17 +131,32 @@ lemma rpow_of_add_eq (x : ℝ≥0) (hw : w ≠ 0) (h : y + z = w) : x ^ w = x ^
 theorem rpow_mul (x : ℝ≥0) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z :=
   NNReal.eq <| Real.rpow_mul x.2 y z
 
-theorem rpow_neg (x : ℝ≥0) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ :=
-  NNReal.eq <| Real.rpow_neg x.2 _
+lemma rpow_natCast_mul (x : ℝ≥0) (n : ℕ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by
+  rw [rpow_mul, rpow_natCast]
+
+lemma rpow_mul_natCast (x : ℝ≥0) (y : ℝ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n := by
+  rw [rpow_mul, rpow_natCast]
+
+lemma rpow_intCast_mul (x : ℝ≥0) (n : ℤ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by
+  rw [rpow_mul, rpow_intCast]
+
+lemma rpow_mul_intCast (x : ℝ≥0) (y : ℝ) (n : ℤ) : x ^ (y * n) = (x ^ y) ^ n := by
+  rw [rpow_mul, rpow_intCast]
 
 theorem rpow_neg_one (x : ℝ≥0) : x ^ (-1 : ℝ) = x⁻¹ := by simp [rpow_neg]
 
 theorem rpow_sub {x : ℝ≥0} (hx : x ≠ 0) (y z : ℝ) : x ^ (y - z) = x ^ y / x ^ z :=
   NNReal.eq <| Real.rpow_sub (pos_iff_ne_zero.2 hx) y z
 
-theorem rpow_sub' (x : ℝ≥0) {y z : ℝ} (h : y - z ≠ 0) : x ^ (y - z) = x ^ y / x ^ z :=
+theorem rpow_sub' (h : y - z ≠ 0) (x : ℝ≥0) : x ^ (y - z) = x ^ y / x ^ z :=
   NNReal.eq <| Real.rpow_sub' x.2 h
 
+lemma rpow_sub_one' (h : y - 1 ≠ 0) (x : ℝ≥0) : x ^ (y - 1) = x ^ y / x := by
+  rw [rpow_sub' h, rpow_one]
+
+lemma rpow_one_sub' (h : 1 - y ≠ 0) (x : ℝ≥0) : x ^ (1 - y) = x / x ^ y := by
+  rw [rpow_sub' h, rpow_one]
+
 theorem rpow_inv_rpow_self {y : ℝ} (hy : y ≠ 0) (x : ℝ≥0) : (x ^ y) ^ (1 / y) = x := by
   field_simp [← rpow_mul]
 
@@ -109,18 +174,9 @@ theorem sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1 / (2 : ℝ)) := by
   push_cast
   exact Real.sqrt_eq_rpow x.1
 
-@[simp, norm_cast]
-theorem rpow_natCast (x : ℝ≥0) (n : ℕ) : x ^ (n : ℝ) = x ^ n :=
-  NNReal.eq <| by simpa only [coe_rpow, coe_pow] using Real.rpow_natCast x n
-
 @[deprecated (since := "2024-04-17")]
 alias rpow_nat_cast := rpow_natCast
 
-@[simp, norm_cast]
-lemma rpow_intCast (x : ℝ≥0) (n : ℤ) : x ^ (n : ℝ) = x ^ n := by
-  cases n <;> simp only [Int.ofNat_eq_coe, Int.cast_natCast, rpow_natCast, zpow_natCast,
-    Int.cast_negSucc, rpow_neg, zpow_negSucc]
-
 @[simp]
 lemma rpow_ofNat (x : ℝ≥0) (n : ℕ) [n.AtLeastTwo] :
     x ^ (no_index (OfNat.ofNat n) : ℝ) = x ^ (OfNat.ofNat n : ℕ) :=
@@ -224,6 +280,47 @@ theorem lt_rpow_inv_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x < y ^ z⁻¹
 theorem rpow_inv_lt_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ^ z⁻¹ < y ↔ x < y ^ z := by
   simp only [← not_le, le_rpow_inv_iff hz]
 
+section
+variable {y : ℝ≥0}
+
+lemma rpow_lt_rpow_of_neg (hx : 0 < x) (hxy : x < y) (hz : z < 0) : y ^ z < x ^ z :=
+  Real.rpow_lt_rpow_of_neg hx hxy hz
+
+lemma rpow_le_rpow_of_nonpos (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0) : y ^ z ≤ x ^ z :=
+  Real.rpow_le_rpow_of_nonpos hx hxy hz
+
+lemma rpow_lt_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z < y ^ z ↔ y < x :=
+  Real.rpow_lt_rpow_iff_of_neg hx hy hz
+
+lemma rpow_le_rpow_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z ≤ y ^ z ↔ y ≤ x :=
+  Real.rpow_le_rpow_iff_of_neg hx hy hz
+
+lemma le_rpow_inv_iff_of_pos (hy : 0 ≤ y) (hz : 0 < z) (x : ℝ≥0) : x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y :=
+  Real.le_rpow_inv_iff_of_pos x.2 hy hz
+
+lemma rpow_inv_le_iff_of_pos (hy : 0 ≤ y) (hz : 0 < z) (x : ℝ≥0) : x ^ z⁻¹ ≤ y ↔ x ≤ y ^ z :=
+  Real.rpow_inv_le_iff_of_pos x.2 hy hz
+
+lemma lt_rpow_inv_iff_of_pos (hy : 0 ≤ y) (hz : 0 < z) (x : ℝ≥0) : x < y ^ z⁻¹ ↔ x ^ z < y :=
+  Real.lt_rpow_inv_iff_of_pos x.2 hy hz
+
+lemma rpow_inv_lt_iff_of_pos (hy : 0 ≤ y) (hz : 0 < z) (x : ℝ≥0) : x ^ z⁻¹ < y ↔ x < y ^ z :=
+  Real.rpow_inv_lt_iff_of_pos x.2 hy hz
+
+lemma le_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z :=
+  Real.le_rpow_inv_iff_of_neg hx hy hz
+
+lemma lt_rpow_inv_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x < y ^ z⁻¹ ↔ y < x ^ z :=
+  Real.lt_rpow_inv_iff_of_neg hx hy hz
+
+lemma rpow_inv_lt_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z⁻¹ < y ↔ y ^ z < x :=
+  Real.rpow_inv_lt_iff_of_neg hx hy hz
+
+lemma rpow_inv_le_iff_of_neg (hx : 0 < x) (hy : 0 < y) (hz : z < 0) : x ^ z⁻¹ ≤ y ↔ y ^ z ≤ x :=
+  Real.rpow_inv_le_iff_of_neg hx hy hz
+
+end
+
 @[gcongr] theorem rpow_lt_rpow_of_exponent_lt {x : ℝ≥0} {y z : ℝ} (hx : 1 < x) (hyz : y < z) :
     x ^ y < x ^ z :=
   Real.rpow_lt_rpow_of_exponent_lt hx hyz
@@ -412,13 +509,13 @@ theorem zero_rpow_mul_self (y : ℝ) : (0 : ℝ≥0∞) ^ y * (0 : ℝ≥0∞) ^
   exacts [zero_mul _, one_mul _, top_mul_top]
 
 @[norm_cast]
-theorem coe_rpow_of_ne_zero {x : ℝ≥0} (h : x ≠ 0) (y : ℝ) : (x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) := by
+theorem coe_rpow_of_ne_zero {x : ℝ≥0} (h : x ≠ 0) (y : ℝ) : (↑(x ^ y) : ℝ≥0∞) = x ^ y := by
   rw [← ENNReal.some_eq_coe]
   dsimp only [(· ^ ·), Pow.pow, rpow]
   simp [h]
 
 @[norm_cast]
-theorem coe_rpow_of_nonneg (x : ℝ≥0) {y : ℝ} (h : 0 ≤ y) : (x : ℝ≥0∞) ^ y = (x ^ y : ℝ≥0) := by
+theorem coe_rpow_of_nonneg (x : ℝ≥0) {y : ℝ} (h : 0 ≤ y) : ↑(x ^ y) = (x : ℝ≥0∞) ^ y := by
   by_cases hx : x = 0
   · rcases le_iff_eq_or_lt.1 h with (H | H)
     · simp [hx, H.symm]
@@ -439,7 +536,7 @@ theorem rpow_one (x : ℝ≥0∞) : x ^ (1 : ℝ) = x := by
 
 @[simp]
 theorem one_rpow (x : ℝ) : (1 : ℝ≥0∞) ^ x = 1 := by
-  rw [← coe_one, coe_rpow_of_ne_zero one_ne_zero]
+  rw [← coe_one, ← coe_rpow_of_ne_zero one_ne_zero]
   simp
 
 @[simp]
@@ -450,7 +547,7 @@ theorem rpow_eq_zero_iff {x : ℝ≥0∞} {y : ℝ} : x ^ y = 0 ↔ x = 0 ∧ 0
   · by_cases h : x = 0
     · rcases lt_trichotomy y 0 with (H | H | H) <;>
         simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt]
-    · simp [coe_rpow_of_ne_zero h, h]
+    · simp [← coe_rpow_of_ne_zero h, h]
 
 lemma rpow_eq_zero_iff_of_pos {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y) : x ^ y = 0 ↔ x = 0 := by
   simp [hy, hy.not_lt]
@@ -463,7 +560,7 @@ theorem rpow_eq_top_iff {x : ℝ≥0∞} {y : ℝ} : x ^ y = ⊤ ↔ x = 0 ∧ y
   · by_cases h : x = 0
     · rcases lt_trichotomy y 0 with (H | H | H) <;>
         simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt]
-    · simp [coe_rpow_of_ne_zero h, h]
+    · simp [← coe_rpow_of_ne_zero h, h]
 
 theorem rpow_eq_top_iff_of_pos {x : ℝ≥0∞} {y : ℝ} (hy : 0 < y) : x ^ y = ⊤ ↔ x = ⊤ := by
   simp [rpow_eq_top_iff, hy, asymm hy]
@@ -489,7 +586,7 @@ theorem rpow_add {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) :
   cases' x with x
   · exact (h'x rfl).elim
   have : x ≠ 0 := fun h => by simp [h] at hx
-  simp [coe_rpow_of_ne_zero this, NNReal.rpow_add this]
+  simp [← coe_rpow_of_ne_zero this, NNReal.rpow_add this]
 
 theorem rpow_add_of_nonneg {x : ℝ≥0∞} (y z : ℝ) (hy : 0 ≤ y) (hz : 0 ≤ z) :
     x ^ (y + z) = x ^ y * x ^ z := by
@@ -499,7 +596,7 @@ theorem rpow_add_of_nonneg {x : ℝ≥0∞} (y z : ℝ) (hy : 0 ≤ y) (hz : 0 
     rcases hz.eq_or_lt with rfl|hz
     · rw [rpow_zero, mul_one, add_zero]
     simp [top_rpow_of_pos, hy, hz, add_pos hy hz]
-  simp [coe_rpow_of_nonneg, hy, hz, add_nonneg hy hz, NNReal.rpow_add_of_nonneg _ hy hz]
+  simp [← coe_rpow_of_nonneg, hy, hz, add_nonneg hy hz, NNReal.rpow_add_of_nonneg _ hy hz]
 
 theorem rpow_neg (x : ℝ≥0∞) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ := by
   cases' x with x
@@ -509,7 +606,7 @@ theorem rpow_neg (x : ℝ≥0∞) (y : ℝ) : x ^ (-y) = (x ^ y)⁻¹ := by
     · rcases lt_trichotomy y 0 with (H | H | H) <;>
         simp [h, zero_rpow_of_pos, zero_rpow_of_neg, H, neg_pos.mpr]
     · have A : x ^ y ≠ 0 := by simp [h]
-      simp [coe_rpow_of_ne_zero h, ← coe_inv A, NNReal.rpow_neg]
+      simp [← coe_rpow_of_ne_zero h, ← coe_inv A, NNReal.rpow_neg]
 
 theorem rpow_sub {x : ℝ≥0∞} (y z : ℝ) (hx : x ≠ 0) (h'x : x ≠ ⊤) : x ^ (y - z) = x ^ y / x ^ z := by
   rw [sub_eq_add_neg, rpow_add _ _ hx h'x, rpow_neg, div_eq_mul_inv]
@@ -528,13 +625,13 @@ theorem rpow_mul (x : ℝ≥0∞) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z := by
         simp [h, Hy, Hz, zero_rpow_of_neg, zero_rpow_of_pos, top_rpow_of_neg, top_rpow_of_pos,
           mul_pos_of_neg_of_neg, mul_neg_of_neg_of_pos, mul_neg_of_pos_of_neg]
     · have : x ^ y ≠ 0 := by simp [h]
-      simp [coe_rpow_of_ne_zero h, coe_rpow_of_ne_zero this, NNReal.rpow_mul]
+      simp [← coe_rpow_of_ne_zero, h, this, NNReal.rpow_mul]
 
 @[simp, norm_cast]
 theorem rpow_natCast (x : ℝ≥0∞) (n : ℕ) : x ^ (n : ℝ) = x ^ n := by
   cases x
   · cases n <;> simp [top_rpow_of_pos (Nat.cast_add_one_pos _), top_pow (Nat.succ_pos _)]
-  · simp [coe_rpow_of_nonneg _ (Nat.cast_nonneg n)]
+  · simp [← coe_rpow_of_nonneg _ (Nat.cast_nonneg n)]
 
 @[deprecated (since := "2024-04-17")]
 alias rpow_nat_cast := rpow_natCast
@@ -569,9 +666,9 @@ theorem mul_rpow_eq_ite (x y : ℝ≥0∞) (z : ℝ) :
   induction y
   · rw [ne_eq, coe_eq_zero] at hx0
     cases' hz with hz hz <;> simp [*]
-  simp only [*, false_and_iff, and_false_iff, false_or_iff, if_false]
+  simp only [*, if_false]
   norm_cast at *
-  rw [coe_rpow_of_ne_zero (mul_ne_zero hx0 hy0), NNReal.mul_rpow]
+  rw [← coe_rpow_of_ne_zero (mul_ne_zero hx0 hy0), NNReal.mul_rpow]
   norm_cast
 
 theorem mul_rpow_of_ne_top {x y : ℝ≥0∞} (hx : x ≠ ⊤) (hy : y ≠ ⊤) (z : ℝ) :
@@ -624,9 +721,9 @@ theorem strictMono_rpow_of_pos {z : ℝ} (h : 0 < z) : StrictMono fun x : ℝ≥
   intro x y hxy
   lift x to ℝ≥0 using ne_top_of_lt hxy
   rcases eq_or_ne y ∞ with (rfl | hy)
-  · simp only [top_rpow_of_pos h, coe_rpow_of_nonneg _ h.le, coe_lt_top]
+  · simp only [top_rpow_of_pos h, ← coe_rpow_of_nonneg _ h.le, coe_lt_top]
   · lift y to ℝ≥0 using hy
-    simp only [coe_rpow_of_nonneg _ h.le, NNReal.rpow_lt_rpow (coe_lt_coe.1 hxy) h, coe_lt_coe]
+    simp only [← coe_rpow_of_nonneg _ h.le, NNReal.rpow_lt_rpow (coe_lt_coe.1 hxy) h, coe_lt_coe]
 
 theorem monotone_rpow_of_nonneg {z : ℝ} (h : 0 ≤ z) : Monotone fun x : ℝ≥0∞ => x ^ z :=
   h.eq_or_lt.elim (fun h0 => h0 ▸ by simp only [rpow_zero, monotone_const]) fun h0 =>
@@ -698,7 +795,7 @@ theorem rpow_lt_rpow_of_exponent_lt {x : ℝ≥0∞} {y z : ℝ} (hx : 1 < x) (h
     x ^ y < x ^ z := by
   lift x to ℝ≥0 using hx'
   rw [one_lt_coe_iff] at hx
-  simp [coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
+  simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
     NNReal.rpow_lt_rpow_of_exponent_lt hx hyz]
 
 @[gcongr] theorem rpow_le_rpow_of_exponent_le {x : ℝ≥0∞} {y z : ℝ} (hx : 1 ≤ x) (hyz : y ≤ z) :
@@ -709,14 +806,14 @@ theorem rpow_lt_rpow_of_exponent_lt {x : ℝ≥0∞} {y z : ℝ} (hx : 1 < x) (h
     simp [Hy, Hz, top_rpow_of_neg, top_rpow_of_pos, le_refl] <;>
     linarith
   · simp only [one_le_coe_iff, some_eq_coe] at hx
-    simp [coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
+    simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
       NNReal.rpow_le_rpow_of_exponent_le hx hyz]
 
 theorem rpow_lt_rpow_of_exponent_gt {x : ℝ≥0∞} {y z : ℝ} (hx0 : 0 < x) (hx1 : x < 1) (hyz : z < y) :
     x ^ y < x ^ z := by
   lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx1 le_top)
   simp only [coe_lt_one_iff, coe_pos] at hx0 hx1
-  simp [coe_rpow_of_ne_zero (ne_of_gt hx0), NNReal.rpow_lt_rpow_of_exponent_gt hx0 hx1 hyz]
+  simp [← coe_rpow_of_ne_zero (ne_of_gt hx0), NNReal.rpow_lt_rpow_of_exponent_gt hx0 hx1 hyz]
 
 theorem rpow_le_rpow_of_exponent_ge {x : ℝ≥0∞} {y z : ℝ} (hx1 : x ≤ 1) (hyz : z ≤ y) :
     x ^ y ≤ x ^ z := by
@@ -727,7 +824,7 @@ theorem rpow_le_rpow_of_exponent_ge {x : ℝ≥0∞} {y z : ℝ} (hx1 : x ≤ 1)
     simp [Hy, Hz, h, zero_rpow_of_neg, zero_rpow_of_pos, le_refl] <;>
     linarith
   · rw [coe_le_one_iff] at hx1
-    simp [coe_rpow_of_ne_zero h,
+    simp [← coe_rpow_of_ne_zero h,
       NNReal.rpow_le_rpow_of_exponent_ge (bot_lt_iff_ne_bot.mpr h) hx1 hyz]
 
 theorem rpow_le_self_of_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (h_one_le : 1 ≤ z) : x ^ z ≤ x := by
@@ -755,63 +852,63 @@ theorem rpow_pos {p : ℝ} {x : ℝ≥0∞} (hx_pos : 0 < x) (hx_ne_top : x ≠
 theorem rpow_lt_one {x : ℝ≥0∞} {z : ℝ} (hx : x < 1) (hz : 0 < z) : x ^ z < 1 := by
   lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx le_top)
   simp only [coe_lt_one_iff] at hx
-  simp [coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.rpow_lt_one hx hz]
+  simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.rpow_lt_one hx hz]
 
 theorem rpow_le_one {x : ℝ≥0∞} {z : ℝ} (hx : x ≤ 1) (hz : 0 ≤ z) : x ^ z ≤ 1 := by
   lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx coe_lt_top)
   simp only [coe_le_one_iff] at hx
-  simp [coe_rpow_of_nonneg _ hz, NNReal.rpow_le_one hx hz]
+  simp [← coe_rpow_of_nonneg _ hz, NNReal.rpow_le_one hx hz]
 
 theorem rpow_lt_one_of_one_lt_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : z < 0) : x ^ z < 1 := by
   cases x
   · simp [top_rpow_of_neg hz, zero_lt_one]
   · simp only [some_eq_coe, one_lt_coe_iff] at hx
-    simp [coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
+    simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
       NNReal.rpow_lt_one_of_one_lt_of_neg hx hz]
 
 theorem rpow_le_one_of_one_le_of_neg {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : z < 0) : x ^ z ≤ 1 := by
   cases x
   · simp [top_rpow_of_neg hz, zero_lt_one]
   · simp only [one_le_coe_iff, some_eq_coe] at hx
-    simp [coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
+    simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
       NNReal.rpow_le_one_of_one_le_of_nonpos hx (le_of_lt hz)]
 
 theorem one_lt_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x ^ z := by
   cases x
   · simp [top_rpow_of_pos hz]
   · simp only [some_eq_coe, one_lt_coe_iff] at hx
-    simp [coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_lt_rpow hx hz]
+    simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_lt_rpow hx hz]
 
 theorem one_le_rpow {x : ℝ≥0∞} {z : ℝ} (hx : 1 ≤ x) (hz : 0 < z) : 1 ≤ x ^ z := by
   cases x
   · simp [top_rpow_of_pos hz]
   · simp only [one_le_coe_iff, some_eq_coe] at hx
-    simp [coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_le_rpow hx (le_of_lt hz)]
+    simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_le_rpow hx (le_of_lt hz)]
 
 theorem one_lt_rpow_of_pos_of_lt_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x < 1)
     (hz : z < 0) : 1 < x ^ z := by
   lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx2 le_top)
   simp only [coe_lt_one_iff, coe_pos] at hx1 hx2 ⊢
-  simp [coe_rpow_of_ne_zero (ne_of_gt hx1), NNReal.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz]
+  simp [← coe_rpow_of_ne_zero (ne_of_gt hx1), NNReal.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz]
 
 theorem one_le_rpow_of_pos_of_le_one_of_neg {x : ℝ≥0∞} {z : ℝ} (hx1 : 0 < x) (hx2 : x ≤ 1)
     (hz : z < 0) : 1 ≤ x ^ z := by
   lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx2 coe_lt_top)
   simp only [coe_le_one_iff, coe_pos] at hx1 hx2 ⊢
-  simp [coe_rpow_of_ne_zero (ne_of_gt hx1),
+  simp [← coe_rpow_of_ne_zero (ne_of_gt hx1),
     NNReal.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 (le_of_lt hz)]
 
-theorem toNNReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toNNReal ^ z = (x ^ z).toNNReal := by
+@[simp] lemma toNNReal_rpow (x : ℝ≥0∞) (z : ℝ) : (x ^ z).toNNReal = x.toNNReal ^ z := by
   rcases lt_trichotomy z 0 with (H | H | H)
   · cases' x with x
     · simp [H, ne_of_lt]
     by_cases hx : x = 0
     · simp [hx, H, ne_of_lt]
-    · simp [coe_rpow_of_ne_zero hx]
+    · simp [← coe_rpow_of_ne_zero hx]
   · simp [H]
   · cases x
     · simp [H, ne_of_gt]
-    simp [coe_rpow_of_nonneg _ (le_of_lt H)]
+    simp [← coe_rpow_of_nonneg _ (le_of_lt H)]
 
 theorem toReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toReal ^ z = (x ^ z).toReal := by
   rw [ENNReal.toReal, ENNReal.toReal, ← NNReal.coe_rpow, ENNReal.toNNReal_rpow]
@@ -819,7 +916,7 @@ theorem toReal_rpow (x : ℝ≥0∞) (z : ℝ) : x.toReal ^ z = (x ^ z).toReal :
 theorem ofReal_rpow_of_pos {x p : ℝ} (hx_pos : 0 < x) :
     ENNReal.ofReal x ^ p = ENNReal.ofReal (x ^ p) := by
   simp_rw [ENNReal.ofReal]
-  rw [coe_rpow_of_ne_zero, coe_inj, Real.toNNReal_rpow_of_nonneg hx_pos.le]
+  rw [← coe_rpow_of_ne_zero, coe_inj, Real.toNNReal_rpow_of_nonneg hx_pos.le]
   simp [hx_pos]
 
 theorem ofReal_rpow_of_nonneg {x p : ℝ} (hx_nonneg : 0 ≤ x) (hp_nonneg : 0 ≤ p) :
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
index c037a96c86a69..b712f2e3cd568 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
@@ -166,6 +166,22 @@ theorem abs_rpow_le_exp_log_mul (x y : ℝ) : |x ^ y| ≤ exp (log x * y) := by
   · by_cases hy : y = 0 <;> simp [hx, hy, zero_le_one]
   · rw [rpow_def_of_pos (abs_pos.2 hx), log_abs]
 
+lemma rpow_inv_log (hx₀ : 0 < x) (hx₁ : x ≠ 1) : x ^ (log x)⁻¹ = exp 1 := by
+  rw [rpow_def_of_pos hx₀, mul_inv_cancel₀]
+  exact log_ne_zero.2 ⟨hx₀.ne', hx₁, (hx₀.trans' <| by norm_num).ne'⟩
+
+/-- See `Real.rpow_inv_log` for the equality when `x ≠ 1` is strictly positive. -/
+lemma rpow_inv_log_le_exp_one : x ^ (log x)⁻¹ ≤ exp 1 := by
+  calc
+    _ ≤ |x ^ (log x)⁻¹| := le_abs_self _
+    _ ≤ |x| ^ (log x)⁻¹ := abs_rpow_le_abs_rpow ..
+  rw [← log_abs]
+  obtain hx | hx := (abs_nonneg x).eq_or_gt
+  · simp [hx]
+  · rw [rpow_def_of_pos hx]
+    gcongr
+    exact mul_inv_le_one
+
 theorem norm_rpow_of_nonneg {x y : ℝ} (hx_nonneg : 0 ≤ x) : ‖x ^ y‖ = ‖x‖ ^ y := by
   simp_rw [Real.norm_eq_abs]
   exact abs_rpow_of_nonneg hx_nonneg
@@ -512,13 +528,13 @@ theorem monotoneOn_rpow_Ici_of_exponent_nonneg {r : ℝ} (hr : 0 ≤ r) :
 
 lemma rpow_lt_rpow_of_neg (hx : 0 < x) (hxy : x < y) (hz : z < 0) : y ^ z < x ^ z := by
   have := hx.trans hxy
-  rw [← inv_lt_inv, ← rpow_neg, ← rpow_neg]
+  rw [← inv_lt_inv₀, ← rpow_neg, ← rpow_neg]
   on_goal 1 => refine rpow_lt_rpow ?_ hxy (neg_pos.2 hz)
   all_goals positivity
 
 lemma rpow_le_rpow_of_nonpos (hx : 0 < x) (hxy : x ≤ y) (hz : z ≤ 0) : y ^ z ≤ x ^ z := by
   have := hx.trans_le hxy
-  rw [← inv_le_inv, ← rpow_neg, ← rpow_neg]
+  rw [← inv_le_inv₀, ← rpow_neg, ← rpow_neg]
   on_goal 1 => refine rpow_le_rpow ?_ hxy (neg_nonneg.2 hz)
   all_goals positivity
 
@@ -576,7 +592,7 @@ theorem rpow_lt_rpow_of_exponent_neg {x y z : ℝ} (hy : 0 < y) (hxy : y < x) (h
     x ^ z < y ^ z := by
   have hx : 0 < x := hy.trans hxy
   rw [← neg_neg z, Real.rpow_neg (le_of_lt hx) (-z), Real.rpow_neg (le_of_lt hy) (-z),
-      inv_lt_inv (rpow_pos_of_pos hx _) (rpow_pos_of_pos hy _)]
+      inv_lt_inv₀ (rpow_pos_of_pos hx _) (rpow_pos_of_pos hy _)]
   exact Real.rpow_lt_rpow (by positivity) hxy <| neg_pos_of_neg hz
 
 theorem strictAntiOn_rpow_Ioi_of_exponent_neg {r : ℝ} (hr : r < 0) :
@@ -775,7 +791,7 @@ lemma zpow_le_of_le_log {n : ℤ} (hy : 0 < y) (h : log x ≤ n * log y) : x ≤
 lemma rpow_lt_iff_lt_log (hx : 0 < x) (hy : 0 < y) : x ^ z < y ↔ z * log x < log y := by
   rw [← log_lt_log_iff (rpow_pos_of_pos hx _) hy, log_rpow hx]
 
-lemma pow_lt_iff_lt_log  (hx : 0 < x) (hy : 0 < y) : x ^ n < y ↔ n * log x < log y := by
+lemma pow_lt_iff_lt_log (hx : 0 < x) (hy : 0 < y) : x ^ n < y ↔ n * log x < log y := by
   rw [← rpow_lt_iff_lt_log hx hy, rpow_natCast]
 
 lemma zpow_lt_iff_lt_log {n : ℤ} (hx : 0 < x) (hy : 0 < y) : x ^ n < y ↔ n * log x < log y := by
@@ -807,7 +823,7 @@ theorem rpow_le_one_iff_of_pos (hx : 0 < x) : x ^ y ≤ 1 ↔ 1 ≤ x ∧ y ≤
 /-- Bound for `|log x * x ^ t|` in the interval `(0, 1]`, for positive real `t`. -/
 theorem abs_log_mul_self_rpow_lt (x t : ℝ) (h1 : 0 < x) (h2 : x ≤ 1) (ht : 0 < t) :
     |log x * x ^ t| < 1 / t := by
-  rw [lt_div_iff ht]
+  rw [lt_div_iff₀ ht]
   have := abs_log_mul_self_lt (x ^ t) (rpow_pos_of_pos h1 t) (rpow_le_one h1.le h2 ht.le)
   rwa [log_rpow h1, mul_assoc, abs_mul, abs_of_pos ht, mul_comm] at this
 
diff --git a/Mathlib/Analysis/SpecialFunctions/Stirling.lean b/Mathlib/Analysis/SpecialFunctions/Stirling.lean
index 0ff3d40698098..35d9ac4c47d5b 100644
--- a/Mathlib/Analysis/SpecialFunctions/Stirling.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Stirling.lean
@@ -102,12 +102,12 @@ theorem log_stirlingSeq_diff_le_geo_sum (n : ℕ) :
     · simp_rw [← _root_.pow_succ'] at this
       exact this
     rw [one_div, inv_pow]
-    exact inv_lt_one (one_lt_pow ((lt_add_iff_pos_left 1).mpr <| by positivity) two_ne_zero)
+    exact inv_lt_one_of_one_lt₀ (one_lt_pow₀ (lt_add_of_pos_left _ <| by positivity) two_ne_zero)
   have hab (k : ℕ) : (1 : ℝ) / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) ≤
       (((1 : ℝ) / (2 * ↑(n + 1) + 1)) ^ 2) ^ ↑(k + 1) := by
     refine mul_le_of_le_one_left (pow_nonneg h_nonneg ↑(k + 1)) ?_
     rw [one_div]
-    exact inv_le_one (le_add_of_nonneg_left <| by positivity)
+    exact inv_le_one_of_one_le₀ (le_add_of_nonneg_left <| by positivity)
   exact hasSum_le hab (log_stirlingSeq_diff_hasSum n) g
 
 /-- We have the bound `log (stirlingSeq n) - log (stirlingSeq (n+1))` ≤ 1/(4 n^2)
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
index 806f844aeaaeb..c5ed691712471 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
@@ -204,7 +204,7 @@ theorem cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
   constructor
   · intro Hcos
     rw [← sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero,
-      eq_false (two_ne_zero' ℝ), false_or_iff, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos
+      eq_false (two_ne_zero' ℝ), false_or, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos
     rcases Hcos with (⟨n, hn⟩ | ⟨n, hn⟩)
     · right
       rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), ← sub_eq_iff_eq_add] at hn
@@ -252,7 +252,7 @@ theorem cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin
   rw [eq_neg_iff_add_eq_zero, hs] at hc
   obtain ⟨n, hn⟩ : ∃ n, n • _ = _ := QuotientAddGroup.leftRel_apply.mp (Quotient.exact' hc)
   rw [← neg_one_mul, add_zero, ← sub_eq_zero, zsmul_eq_mul, ← mul_assoc, ← sub_mul, mul_eq_zero,
-    eq_false (ne_of_gt pi_pos), or_false_iff, sub_neg_eq_add, ← Int.cast_zero, ← Int.cast_one,
+    eq_false (ne_of_gt pi_pos), or_false, sub_neg_eq_add, ← Int.cast_zero, ← Int.cast_one,
     ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_add, Int.cast_inj] at hn
   have : (n * 2 + 1) % (2 : ℤ) = 0 % (2 : ℤ) := congr_arg (· % (2 : ℤ)) hn
   rw [add_comm, Int.add_mul_emod_self] at this
@@ -554,7 +554,7 @@ theorem nsmul_toReal_eq_mul {n : ℕ} (h : n ≠ 0) {θ : Angle} :
     (n • θ).toReal = n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / n) (π / n) := by
   nth_rw 1 [← coe_toReal θ]
   have h' : 0 < (n : ℝ) := mod_cast Nat.pos_of_ne_zero h
-  rw [← coe_nsmul, nsmul_eq_mul, toReal_coe_eq_self_iff, Set.mem_Ioc, div_lt_iff' h',
+  rw [← coe_nsmul, nsmul_eq_mul, toReal_coe_eq_self_iff, Set.mem_Ioc, div_lt_iff₀' h',
     le_div_iff₀' h']
 
 theorem two_nsmul_toReal_eq_two_mul {θ : Angle} :
@@ -585,7 +585,7 @@ theorem two_nsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
   rw [← coe_nsmul, two_nsmul, ← two_mul, toReal_coe_eq_self_sub_two_pi_iff, Set.mem_Ioc]
   exact
     ⟨fun h => by linarith, fun h =>
-      ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩
+      ⟨(div_lt_iff₀' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, toReal_le_pi θ]⟩⟩
 
 theorem two_zsmul_toReal_eq_two_mul_sub_two_pi {θ : Angle} :
     ((2 : ℤ) • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal := by
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
index 2ea81c6afbe95..6919f9dcbb728 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
@@ -208,7 +208,7 @@ lemma arctan_add_arctan_lt_pi_div_two {x y : ℝ} (h : x * y < 1) : arctan x + a
   cases' le_or_lt y 0 with hy hy
   · rw [← add_zero (π / 2), ← arctan_zero]
     exact add_lt_add_of_lt_of_le (arctan_lt_pi_div_two _) (tanOrderIso.symm.monotone hy)
-  · rw [← lt_div_iff hy, ← inv_eq_one_div] at h
+  · rw [← lt_div_iff₀ hy, ← inv_eq_one_div] at h
     replace h : arctan x < arctan y⁻¹ := tanOrderIso.symm.strictMono h
     rwa [arctan_inv_of_pos hy, lt_tsub_iff_right] at h
 
@@ -228,7 +228,7 @@ theorem arctan_add_eq_add_pi {x y : ℝ} (h : 1 < x * y) (hx : 0 < x) :
   have hy : 0 < y := by
     have := mul_pos_iff.mp (zero_lt_one.trans h)
     simpa [hx, hx.asymm]
-  have k := arctan_add (mul_inv x y ▸ inv_lt_one h)
+  have k := arctan_add (mul_inv x y ▸ inv_lt_one_of_one_lt₀ h)
   rw [arctan_inv_of_pos hx, arctan_inv_of_pos hy, show _ + _ = π - (arctan x + arctan y) by ring,
     sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', sub_eq_add_neg, ← arctan_neg, add_comm] at k
   convert k.symm using 3
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
index c31a03b44e31f..ff16ee4d5471a 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
@@ -3,6 +3,7 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
 -/
+import Mathlib.Algebra.QuadraticDiscriminant
 import Mathlib.Analysis.SpecialFunctions.Exp
 import Mathlib.Tactic.Positivity.Core
 import Mathlib.Algebra.Ring.NegOnePow
@@ -517,7 +518,7 @@ theorem cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * 
       (Int.emod_two_eq_zero_or_one n).elim
         (fun hn0 => by
           rwa [← mul_assoc, ← @Int.cast_two ℝ, ← Int.cast_mul,
-            Int.ediv_mul_cancel ((Int.dvd_iff_emod_eq_zero _ _).2 hn0)])
+            Int.ediv_mul_cancel (Int.dvd_iff_emod_eq_zero.2 hn0)])
         fun hn1 => by
         rw [← Int.emod_add_ediv n 2, hn1, Int.cast_add, Int.cast_one, add_mul, one_mul, add_comm,
               mul_comm (2 : ℤ), Int.cast_mul, mul_assoc, Int.cast_two] at hn
@@ -661,7 +662,7 @@ theorem sqrtTwoAddSeries_monotone_left {x y : ℝ} (h : x ≤ y) :
 theorem cos_pi_over_two_pow : ∀ n : ℕ, cos (π / 2 ^ (n + 1)) = sqrtTwoAddSeries 0 n / 2
   | 0 => by simp
   | n + 1 => by
-    have A : (1 : ℝ) < 2 ^ (n + 1) := one_lt_pow one_lt_two n.succ_ne_zero
+    have A : (1 : ℝ) < 2 ^ (n + 1) := one_lt_pow₀ one_lt_two n.succ_ne_zero
     have B : π / 2 ^ (n + 1) < π := div_lt_self pi_pos A
     have C : 0 < π / 2 ^ (n + 1) := by positivity
     rw [pow_succ, div_mul_eq_div_div, cos_half, cos_pi_over_two_pow n, sqrtTwoAddSeries,
@@ -690,7 +691,7 @@ theorem sin_pi_over_two_pow_succ (n : ℕ) :
     exact (sqrtTwoAddSeries_lt_two _).le
   refine mul_nonneg (sin_nonneg_of_nonneg_of_le_pi ?_ ?_) zero_le_two
   · positivity
-  · exact div_le_self pi_pos.le <| one_le_pow_of_one_le one_le_two _
+  · exact div_le_self pi_pos.le <| one_le_pow₀ one_le_two
 
 @[simp]
 theorem cos_pi_div_four : cos (π / 4) = √2 / 2 := by
@@ -795,6 +796,49 @@ theorem sin_pi_div_three : sin (π / 3) = √3 / 2 := by
   congr
   ring
 
+theorem quadratic_root_cos_pi_div_five :
+    letI c := cos (π / 5)
+    4 * c ^ 2 - 2 * c - 1 = 0 := by
+  set θ := π / 5 with hθ
+  set c := cos θ
+  set s := sin θ
+  suffices 2 * c = 4 * c ^ 2 - 1 by simp [this]
+  have hs : s ≠ 0 := by
+    rw [ne_eq, sin_eq_zero_iff, hθ]
+    push_neg
+    intro n hn
+    replace hn : n * 5 = 1 := by field_simp [mul_comm _ π, mul_assoc] at hn; norm_cast at hn
+    rcases Int.mul_eq_one_iff_eq_one_or_neg_one.mp hn with ⟨_, h⟩ | ⟨_, h⟩ <;> norm_num at h
+  suffices s * (2 * c) = s * (4 * c ^ 2 - 1) from mul_left_cancel₀ hs this
+  calc s * (2 * c) = 2 * s * c := by rw [← mul_assoc, mul_comm 2]
+                 _ = sin (2 * θ) := by rw [sin_two_mul]
+                 _ = sin (π - 2 * θ) := by rw [sin_pi_sub]
+                 _ = sin (2 * θ + θ) := by congr; field_simp [hθ]; linarith
+                 _ = sin (2 * θ) * c + cos (2 * θ) * s := sin_add (2 * θ) θ
+                 _ = 2 * s * c * c + cos (2 * θ) * s := by rw [sin_two_mul]
+                 _ = 2 * s * c * c + (2 * c ^ 2 - 1) * s := by rw [cos_two_mul]
+                 _ = s * (2 * c * c) + s * (2 * c ^ 2 - 1) := by linarith
+                 _ = s * (4 * c ^ 2 - 1) := by linarith
+
+open Polynomial in
+theorem Polynomial.isRoot_cos_pi_div_five :
+    (4 • X ^ 2 - 2 • X - C 1 : ℝ[X]).IsRoot (cos (π / 5)) := by
+  simpa using quadratic_root_cos_pi_div_five
+
+/-- The cosine of `π / 5` is `(1 + √5) / 4`. -/
+@[simp]
+theorem cos_pi_div_five : cos (π / 5) = (1 + √5) / 4 := by
+  set c := cos (π / 5)
+  have : 4 * (c * c) + (-2) * c + (-1) = 0 := by
+    rw [← sq, neg_mul, ← sub_eq_add_neg, ← sub_eq_add_neg]
+    exact quadratic_root_cos_pi_div_five
+  have hd : discrim 4 (-2) (-1) = (2 * √5) * (2 * √5) := by norm_num [discrim, mul_mul_mul_comm]
+  rcases (quadratic_eq_zero_iff (by norm_num) hd c).mp this with h | h
+  · field_simp [h]; linarith
+  · absurd (show 0 ≤ c from cos_nonneg_of_mem_Icc <| by constructor <;> linarith [pi_pos.le])
+    rw [not_le, h]
+    exact div_neg_of_neg_of_pos (by norm_num [lt_sqrt]) (by positivity)
+
 end CosDivSq
 
 /-- `Real.sin` as an `OrderIso` between `[-(π / 2), π / 2]` and `[-1, 1]`. -/
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
index c270fdc3cf837..f3a9858bddf37 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean
@@ -1,8 +1,9 @@
 /-
 Copyright (c) 2022 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: David Loeffler, Yaël Dillies
+Authors: David Loeffler, Yaël Dillies, Bhavik Mehta
 -/
+import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
 
 /-!
@@ -54,81 +55,94 @@ lemma sin_le (hx : 0 ≤ x) : sin x ≤ x := by
 lemma lt_sin (hx : x < 0) : x < sin x := by simpa using sin_lt <| neg_pos.2 hx
 lemma le_sin (hx : x ≤ 0) : x ≤ sin x := by simpa using sin_le <| neg_nonneg.2 hx
 
-lemma one_sub_sq_div_two_le_cos : 1 - x ^ 2 / 2 ≤ cos x := by
+theorem lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin (π / 2 * x) := by
+  simpa [mul_comm x] using
+    strictConcaveOn_sin_Icc.2 ⟨le_rfl, pi_pos.le⟩ ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩
+      pi_div_two_pos.ne (sub_pos.2 hx') hx
+
+theorem le_sin_mul {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1) : x ≤ sin (π / 2 * x) := by
+  simpa [mul_comm x] using
+    strictConcaveOn_sin_Icc.concaveOn.2 ⟨le_rfl, pi_pos.le⟩
+      ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩ (sub_nonneg.2 hx') hx
+
+theorem mul_lt_sin {x : ℝ} (hx : 0 < x) (hx' : x < π / 2) : 2 / π * x < sin x := by
+  rw [← inv_div]
+  simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @lt_sin_mul ((π / 2)⁻¹ * x)
+    (mul_pos (inv_pos.2 pi_div_two_pos) hx) (by rwa [← div_eq_inv_mul, div_lt_one pi_div_two_pos])
+
+/-- One half of **Jordan's inequality**.
+
+In the range `[0, π / 2]`, we have a linear lower bound on `sin`. The other half is given by
+`Real.sin_le`.
+-/
+theorem mul_le_sin {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ π / 2) : 2 / π * x ≤ sin x := by
+  rw [← inv_div]
+  simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @le_sin_mul ((π / 2)⁻¹ * x)
+    (mul_nonneg (inv_nonneg.2 pi_div_two_pos.le) hx)
+    (by rwa [← div_eq_inv_mul, div_le_one pi_div_two_pos])
+
+/-- Half of **Jordan's inequality** for negative values. -/
+lemma sin_le_mul (hx : -(π / 2) ≤ x) (hx₀ : x ≤ 0) : sin x ≤ 2 / π * x := by
+  simpa using mul_le_sin (neg_nonneg.2 hx₀) (neg_le.2 hx)
+
+/-- Half of **Jordan's inequality** for absolute values. -/
+lemma mul_abs_le_abs_sin (hx : |x| ≤ π / 2) : 2 / π * |x| ≤ |sin x| := by
   wlog hx₀ : 0 ≤ x
-  · simpa using this <| neg_nonneg.2 <| le_of_not_le hx₀
-  suffices MonotoneOn (fun x ↦ cos x + x ^ 2 / 2) (Ici 0) by
-    simpa using this left_mem_Ici hx₀ hx₀
-  refine monotoneOn_of_hasDerivWithinAt_nonneg
-    (convex_Ici _) (by fun_prop)
-    (fun x _ ↦ ((hasDerivAt_cos ..).add <| (hasDerivAt_pow ..).div_const _).hasDerivWithinAt)
-    fun x hx ↦ ?_
-  simpa [mul_div_cancel_left₀] using sin_le <| interior_subset hx
-
-/-- **Jordan's inequality**. -/
-lemma two_div_pi_mul_le_sin (hx₀ : 0 ≤ x) (hx : x ≤ π / 2) : 2 / π * x ≤ sin x := by
-  rw [← sub_nonneg]
-  suffices ConcaveOn ℝ (Icc 0 (π / 2)) (fun x ↦ sin x - 2 / π * x) by
-    refine (le_min ?_ ?_).trans <| this.min_le_of_mem_Icc (by simp [Real.pi_nonneg, div_nonneg])
-      (by simp [Real.pi_nonneg, div_nonneg]) ⟨hx₀, hx⟩ <;> field_simp
-  exact concaveOn_of_hasDerivWithinAt2_nonpos (convex_Icc ..)
-    (Continuous.continuousOn <| by fun_prop)
-    (fun x _ ↦ ((hasDerivAt_sin ..).sub <| (hasDerivAt_id ..).const_mul (2 / π)).hasDerivWithinAt)
-    (fun x _ ↦ (hasDerivAt_cos ..).hasDerivWithinAt.sub_const _)
-    fun x hx ↦ neg_nonpos.2 <| sin_nonneg_of_mem_Icc <| Icc_subset_Icc_right (by linarith) <|
-    interior_subset hx
-
-/-- **Jordan's inequality** for negative values. -/
-lemma sin_le_two_div_pi_mul (hx : -(π / 2) ≤ x) (hx₀ : x ≤ 0) : sin x ≤ 2 / π * x := by
-  simpa using two_div_pi_mul_le_sin (neg_nonneg.2 hx₀) (neg_le.2 hx)
-
-/-- **Jordan's inequality** for `cos`. -/
-lemma one_sub_two_div_pi_mul_le_cos (hx₀ : 0 ≤ x) (hx : x ≤ π / 2) : 1 - 2 / π * x ≤ cos x := by
+  case inr => simpa using this (by rwa [abs_neg]) <| neg_nonneg.2 <| le_of_not_le hx₀
+  rw [abs_of_nonneg hx₀] at hx ⊢
+  exact (mul_le_sin hx₀ hx).trans (le_abs_self _)
+
+lemma sin_sq_lt_sq (hx : x ≠ 0) : sin x ^ 2 < x ^ 2 := by
+  wlog hx₀ : 0 < x
+  case inr =>
+    simpa using this (neg_ne_zero.2 hx) <| neg_pos_of_neg <| hx.lt_of_le <| le_of_not_lt hx₀
+  rcases le_or_lt x 1 with hxπ | hxπ
+  case inl =>
+    exact pow_lt_pow_left (sin_lt hx₀)
+      (sin_nonneg_of_nonneg_of_le_pi hx₀.le (by linarith [two_le_pi])) (by simp)
+  case inr =>
+    exact (sin_sq_le_one _).trans_lt (by rwa [one_lt_sq_iff hx₀.le])
+
+lemma sin_sq_le_sq : sin x ^ 2 ≤ x ^ 2 := by
+  rcases eq_or_ne x 0 with rfl | hx
+  case inl => simp
+  case inr => exact (sin_sq_lt_sq hx).le
+
+lemma abs_sin_lt_abs (hx : x ≠ 0) : |sin x| < |x| := sq_lt_sq.1 (sin_sq_lt_sq hx)
+lemma abs_sin_le_abs : |sin x| ≤ |x| := sq_le_sq.1 sin_sq_le_sq
+
+lemma one_sub_sq_div_two_lt_cos (hx : x ≠ 0) : 1 - x ^ 2 / 2 < cos x := by
+  have := (sin_sq_lt_sq (by positivity)).trans_eq' (sin_sq_eq_half_sub (x / 2)).symm
+  ring_nf at this
+  linarith
+
+lemma one_sub_sq_div_two_le_cos : 1 - x ^ 2 / 2 ≤ cos x := by
+  rcases eq_or_ne x 0 with rfl | hx
+  case inl => simp
+  case inr => exact (one_sub_sq_div_two_lt_cos hx).le
+
+/-- Half of **Jordan's inequality** for `cos`. -/
+lemma one_sub_mul_le_cos (hx₀ : 0 ≤ x) (hx : x ≤ π / 2) : 1 - 2 / π * x ≤ cos x := by
   simpa [sin_pi_div_two_sub, mul_sub, div_mul_div_comm, mul_comm π, pi_pos.ne']
-    using two_div_pi_mul_le_sin (x := π / 2 - x) (by simpa) (by simpa)
+    using mul_le_sin (x := π / 2 - x) (by simpa) (by simpa)
 
-lemma cos_quadratic_upper_bound (hx : |x| ≤ π) : cos x ≤ 1 - 2 / π ^ 2 * x ^ 2 := by
+/-- Half of **Jordan's inequality** for `cos` and negative values. -/
+lemma one_add_mul_le_cos (hx₀ : -(π / 2) ≤ x) (hx : x ≤ 0) : 1 + 2 / π * x ≤ cos x := by
+  simpa using one_sub_mul_le_cos (x := -x) (by linarith) (by linarith)
+
+lemma cos_le_one_sub_mul_cos_sq (hx : |x| ≤ π) : cos x ≤ 1 - 2 / π ^ 2 * x ^ 2 := by
   wlog hx₀ : 0 ≤ x
-  · simpa using this (by rwa [abs_neg]) <| neg_nonneg.2 <| le_of_not_le hx₀
+  case inr => simpa using this (by rwa [abs_neg]) <| neg_nonneg.2 <| le_of_not_le hx₀
   rw [abs_of_nonneg hx₀] at hx
-  -- TODO: `compute_deriv` tactic?
-  have hderiv (x) : HasDerivAt (fun x ↦ 1 - 2 / π ^ 2 * x ^ 2 - cos x) _ x :=
-    (((hasDerivAt_pow ..).const_mul _).const_sub _).sub <| hasDerivAt_cos _
-  simp only [Nat.cast_ofNat, Nat.succ_sub_succ_eq_sub, tsub_zero, pow_one, ← neg_sub', neg_sub,
-    ← mul_assoc] at hderiv
-  have hmono : MonotoneOn (fun x ↦ 1 - 2 / π ^ 2 * x ^ 2 - cos x) (Icc 0 (π / 2)) := by
-    -- Compiles without this option, but somewhat slower.
-    set_option tactic.skipAssignedInstances false in
-    refine monotoneOn_of_hasDerivWithinAt_nonneg
-      (convex_Icc ..)
-      (Continuous.continuousOn <| by fun_prop)
-      (fun x _ ↦ (hderiv _).hasDerivWithinAt)
-      fun x hx ↦ sub_nonneg.2 ?_
-    have ⟨hx₀, hx⟩ := interior_subset hx
-    calc 2 / π ^ 2 * 2 * x
-        = 2 / π * (2 / π * x) := by ring
-      _ ≤ 1 * sin x := by
-          gcongr; exacts [div_le_one_of_le two_le_pi (by positivity), two_div_pi_mul_le_sin hx₀ hx]
-      _ = sin x := one_mul _
-  have hconc : ConcaveOn ℝ (Icc (π / 2) π) (fun x ↦ 1 - 2 / π ^ 2 * x ^ 2 - cos x) := by
-    -- Compiles without this option, but somewhat slower.
-    set_option tactic.skipAssignedInstances false in
-    refine concaveOn_of_hasDerivWithinAt2_nonpos (convex_Icc ..)
-      (Continuous.continuousOn <| by fun_prop) (fun x _ ↦ (hderiv _).hasDerivWithinAt)
-      (fun x _ ↦ ((hasDerivAt_sin ..).sub <| (hasDerivAt_id ..).const_mul _).hasDerivWithinAt)
-      fun x hx ↦ ?_
-    have ⟨hx, hx'⟩ := interior_subset hx
-    calc
-      _ ≤ (0 : ℝ) - 0 := by
-          gcongr
-          · exact cos_nonpos_of_pi_div_two_le_of_le hx <| hx'.trans <| by linarith
-          · positivity
-      _ = 0 := sub_zero _
-  rw [← sub_nonneg]
-  obtain hx' | hx' := le_total x (π / 2)
-  · simpa using hmono (left_mem_Icc.2 <| by positivity) ⟨hx₀, hx'⟩ hx₀
-  · refine (le_min ?_ ?_).trans <| hconc.min_le_of_mem_Icc (by simp [Real.pi_nonneg])
-      (by simp [Real.pi_nonneg]) ⟨hx', hx⟩ <;> field_simp <;> norm_num
+  have : x / π ≤ sin (x / 2) := by simpa using mul_le_sin (x := x / 2) (by positivity) (by linarith)
+  have := (pow_le_pow_left (by positivity) this 2).trans_eq (sin_sq_eq_half_sub _)
+  ring_nf at this ⊢
+  linarith
+
+@[deprecated (since := "2024-08-29")] alias two_div_pi_mul_le_sin := mul_le_sin
+@[deprecated (since := "2024-08-29")] alias sin_le_two_div_pi_mul := sin_le_mul
+@[deprecated (since := "2024-08-29")] alias one_sub_two_div_pi_mul_le_cos := one_sub_mul_le_cos
+@[deprecated (since := "2024-08-29")] alias cos_quadratic_upper_bound := cos_le_one_sub_mul_cos_sq
 
 /-- For 0 < x ≤ 1 we have x - x ^ 3 / 4 < sin x.
 
@@ -159,11 +173,9 @@ theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x := by
   let U := Ico 0 (π / 2)
   have intU : interior U = Ioo 0 (π / 2) := interior_Ico
   have half_pi_pos : 0 < π / 2 := div_pos pi_pos two_pos
-  have cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y := by
-    intro y hy
+  have cos_pos {y : ℝ} (hy : y ∈ U) : 0 < cos y := by
     exact cos_pos_of_mem_Ioo (Ico_subset_Ioo_left (neg_lt_zero.mpr half_pi_pos) hy)
-  have sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y := by
-    intro y hy
+  have sin_pos {y : ℝ} (hy : y ∈ interior U) : 0 < sin y := by
     rw [intU] at hy
     exact sin_pos_of_mem_Ioo (Ioo_subset_Ioo_right (div_le_self pi_pos.le one_le_two) hy)
   have tan_cts_U : ContinuousOn tan U := by
@@ -172,8 +184,7 @@ theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x := by
     simp only [mem_setOf_eq]
     exact (cos_pos hz).ne'
   have tan_minus_id_cts : ContinuousOn (fun y : ℝ => tan y - y) U := tan_cts_U.sub continuousOn_id
-  have deriv_pos : ∀ y : ℝ, y ∈ interior U → 0 < deriv (fun y' : ℝ => tan y' - y') y := by
-    intro y hy
+  have deriv_pos (y : ℝ) (hy : y ∈ interior U) : 0 < deriv (fun y' : ℝ => tan y' - y') y := by
     have := cos_pos (interior_subset hy)
     simp only [deriv_tan_sub_id y this.ne', one_div, gt_iff_lt, sub_pos]
     norm_cast
@@ -181,7 +192,7 @@ theorem lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x := by
       apply lt_of_le_of_ne y.cos_sq_le_one
       rw [cos_sq']
       simpa only [Ne, sub_eq_self, sq_eq_zero_iff] using (sin_pos hy).ne'
-    rwa [lt_inv, inv_one]
+    rwa [lt_inv_comm₀, inv_one]
     · exact zero_lt_one
     simpa only [sq, mul_self_pos] using this.ne'
   have mono := strictMonoOn_of_deriv_pos (convex_Ico 0 (π / 2)) tan_minus_id_cts deriv_pos
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
index 32502108c05af..34180b091bfb1 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
 -/
 import Mathlib.Algebra.QuadraticDiscriminant
-import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
 import Mathlib.Analysis.SpecialFunctions.Pow.Complex
 
 /-!
@@ -59,7 +58,7 @@ theorem sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π
 iff this number is equal to `k * π / 2` for an integer `k`.
 
 Note that this lemma takes into account that we use zero as the junk value for division by zero.
-See also `Complex.tan_eq_zero_iff'`.  -/
+See also `Complex.tan_eq_zero_iff'`. -/
 theorem tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, k * π / 2 = θ := by
   rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← mul_right_inj' two_ne_zero, mul_zero,
     ← mul_assoc, ← sin_two_mul, sin_eq_zero_iff]
@@ -173,7 +172,7 @@ theorem cos_eq_iff_quadratic {z w : ℂ} :
 
 theorem cos_surjective : Function.Surjective cos := by
   intro x
-  obtain ⟨w, w₀, hw⟩ : ∃ w ≠ 0, 1 * w * w + -2 * x * w + 1 = 0 := by
+  obtain ⟨w, w₀, hw⟩ : ∃ w ≠ 0, 1 * (w * w) + -2 * x * w + 1 = 0 := by
     rcases exists_quadratic_eq_zero one_ne_zero
         ⟨_, (cpow_nat_inv_pow _ two_ne_zero).symm.trans <| pow_two _⟩ with
       ⟨w, hw⟩
@@ -236,27 +235,4 @@ theorem tan_eq_zero_iff' {θ : ℝ} (hθ : cos θ ≠ 0) : tan θ = 0 ↔ ∃ k
 theorem tan_ne_zero_iff {θ : ℝ} : tan θ ≠ 0 ↔ ∀ k : ℤ, k * π / 2 ≠ θ :=
   mod_cast @Complex.tan_ne_zero_iff θ
 
-theorem lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin (π / 2 * x) := by
-  simpa [mul_comm x] using
-    strictConcaveOn_sin_Icc.2 ⟨le_rfl, pi_pos.le⟩ ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩
-      pi_div_two_pos.ne (sub_pos.2 hx') hx
-
-theorem le_sin_mul {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1) : x ≤ sin (π / 2 * x) := by
-  simpa [mul_comm x] using
-    strictConcaveOn_sin_Icc.concaveOn.2 ⟨le_rfl, pi_pos.le⟩
-      ⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩ (sub_nonneg.2 hx') hx
-
-theorem mul_lt_sin {x : ℝ} (hx : 0 < x) (hx' : x < π / 2) : 2 / π * x < sin x := by
-  rw [← inv_div]
-  simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @lt_sin_mul ((π / 2)⁻¹ * x)
-    (mul_pos (inv_pos.2 pi_div_two_pos) hx) (by rwa [← div_eq_inv_mul, div_lt_one pi_div_two_pos])
-
-/-- In the range `[0, π / 2]`, we have a linear lower bound on `sin`. This inequality forms one half
-of Jordan's inequality, the other half is `Real.sin_lt` -/
-theorem mul_le_sin {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ π / 2) : 2 / π * x ≤ sin x := by
-  rw [← inv_div]
-  simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @le_sin_mul ((π / 2)⁻¹ * x)
-    (mul_nonneg (inv_nonneg.2 pi_div_two_pos.le) hx)
-    (by rwa [← div_eq_inv_mul, div_le_one pi_div_two_pos])
-
 end Real
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
index 6d74a3ace85b2..e414b1421381a 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
 -/
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
 
 /-!
 # Complex trigonometric functions
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
index e11d7973f3b2c..862ffd739dff4 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
@@ -45,8 +45,10 @@ theorem contDiff_sin {n} : ContDiff ℂ n sin :=
   (((contDiff_neg.mul contDiff_const).cexp.sub (contDiff_id.mul contDiff_const).cexp).mul
     contDiff_const).div_const _
 
+@[fun_prop]
 theorem differentiable_sin : Differentiable ℂ sin := fun x => (hasDerivAt_sin x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_sin {x : ℂ} : DifferentiableAt ℂ sin x :=
   differentiable_sin x
 
@@ -70,8 +72,10 @@ theorem hasDerivAt_cos (x : ℂ) : HasDerivAt cos (-sin x) x :=
 theorem contDiff_cos {n} : ContDiff ℂ n cos :=
   ((contDiff_id.mul contDiff_const).cexp.add (contDiff_neg.mul contDiff_const).cexp).div_const _
 
+@[fun_prop]
 theorem differentiable_cos : Differentiable ℂ cos := fun x => (hasDerivAt_cos x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_cos {x : ℂ} : DifferentiableAt ℂ cos x :=
   differentiable_cos x
 
@@ -98,8 +102,10 @@ theorem hasDerivAt_sinh (x : ℂ) : HasDerivAt sinh (cosh x) x :=
 theorem contDiff_sinh {n} : ContDiff ℂ n sinh :=
   (contDiff_exp.sub contDiff_neg.cexp).div_const _
 
+@[fun_prop]
 theorem differentiable_sinh : Differentiable ℂ sinh := fun x => (hasDerivAt_sinh x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_sinh {x : ℂ} : DifferentiableAt ℂ sinh x :=
   differentiable_sinh x
 
@@ -123,8 +129,10 @@ theorem hasDerivAt_cosh (x : ℂ) : HasDerivAt cosh (sinh x) x :=
 theorem contDiff_cosh {n} : ContDiff ℂ n cosh :=
   (contDiff_exp.add contDiff_neg.cexp).div_const _
 
+@[fun_prop]
 theorem differentiable_cosh : Differentiable ℂ cosh := fun x => (hasDerivAt_cosh x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_cosh {x : ℂ} : DifferentiableAt ℂ cosh x :=
   differentiable_cosh x
 
@@ -482,8 +490,10 @@ theorem hasDerivAt_sin (x : ℝ) : HasDerivAt sin (cos x) x :=
 theorem contDiff_sin {n} : ContDiff ℝ n sin :=
   Complex.contDiff_sin.real_of_complex
 
+@[fun_prop]
 theorem differentiable_sin : Differentiable ℝ sin := fun x => (hasDerivAt_sin x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_sin : DifferentiableAt ℝ sin x :=
   differentiable_sin x
 
@@ -500,8 +510,10 @@ theorem hasDerivAt_cos (x : ℝ) : HasDerivAt cos (-sin x) x :=
 theorem contDiff_cos {n} : ContDiff ℝ n cos :=
   Complex.contDiff_cos.real_of_complex
 
+@[fun_prop]
 theorem differentiable_cos : Differentiable ℝ cos := fun x => (hasDerivAt_cos x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_cos : DifferentiableAt ℝ cos x :=
   differentiable_cos x
 
@@ -521,8 +533,10 @@ theorem hasDerivAt_sinh (x : ℝ) : HasDerivAt sinh (cosh x) x :=
 theorem contDiff_sinh {n} : ContDiff ℝ n sinh :=
   Complex.contDiff_sinh.real_of_complex
 
+@[fun_prop]
 theorem differentiable_sinh : Differentiable ℝ sinh := fun x => (hasDerivAt_sinh x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_sinh : DifferentiableAt ℝ sinh x :=
   differentiable_sinh x
 
@@ -539,8 +553,10 @@ theorem hasDerivAt_cosh (x : ℝ) : HasDerivAt cosh (sinh x) x :=
 theorem contDiff_cosh {n} : ContDiff ℝ n cosh :=
   Complex.contDiff_cosh.real_of_complex
 
+@[fun_prop]
 theorem differentiable_cosh : Differentiable ℝ cosh := fun x => (hasDerivAt_cosh x).differentiableAt
 
+@[fun_prop]
 theorem differentiableAt_cosh : DifferentiableAt ℝ cosh x :=
   differentiable_cosh x
 
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
index 7b8d099c1e98a..fc6711228b07f 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
@@ -141,7 +141,7 @@ theorem integral_sin_mul_sin_mul_cos_pow_eq (hn : 2 ≤ n) (hz : z ≠ 0) :
   · apply Continuous.intervalIntegrable
     exact Complex.continuous_sin.comp (continuous_const.mul Complex.continuous_ofReal)
 
-/-- Note this also holds for `z = 0`, but we do not need this case for `sin_pi_mul_eq`.  -/
+/-- Note this also holds for `z = 0`, but we do not need this case for `sin_pi_mul_eq`. -/
 theorem integral_cos_mul_cos_pow (hn : 2 ≤ n) (hz : z ≠ 0) :
     (((1 : ℂ) - (4 : ℂ) * z ^ 2 / (n : ℂ) ^ 2) *
       ∫ x in (0 : ℝ)..π / 2, Complex.cos (2 * z * x) * (cos x : ℂ) ^ n) =
diff --git a/Mathlib/Analysis/SpecificLimits/Basic.lean b/Mathlib/Analysis/SpecificLimits/Basic.lean
index 647d3cef3d49b..c0a2df7d23e61 100644
--- a/Mathlib/Analysis/SpecificLimits/Basic.lean
+++ b/Mathlib/Analysis/SpecificLimits/Basic.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
 -/
 import Mathlib.Algebra.GeomSum
-import Mathlib.Order.Filter.Archimedean
+import Mathlib.Order.Filter.AtTopBot.Archimedean
 import Mathlib.Order.Iterate
 import Mathlib.Topology.Algebra.Algebra
 import Mathlib.Topology.Algebra.InfiniteSum.Real
@@ -119,7 +119,7 @@ theorem tendsto_natCast_div_add_atTop {𝕜 : Type*} [DivisionRing 𝕜] [Topolo
 
 theorem tendsto_add_one_pow_atTop_atTop_of_pos [LinearOrderedSemiring α] [Archimedean α] {r : α}
     (h : 0 < r) : Tendsto (fun n : ℕ ↦ (r + 1) ^ n) atTop atTop :=
-  tendsto_atTop_atTop_of_monotone' (fun _ _ ↦ pow_le_pow_right <| le_add_of_nonneg_left h.le) <|
+  tendsto_atTop_atTop_of_monotone' (pow_right_mono₀ <| le_add_of_nonneg_left h.le) <|
     not_bddAbove_iff.2 fun _ ↦ Set.exists_range_iff.2 <| add_one_pow_unbounded_of_pos _ h
 
 theorem tendsto_pow_atTop_atTop_of_one_lt [LinearOrderedRing α] [Archimedean α] {r : α}
@@ -137,7 +137,7 @@ theorem tendsto_pow_atTop_nhds_zero_of_lt_one {𝕜 : Type*} [LinearOrderedField
     (fun hr ↦ (tendsto_add_atTop_iff_nat 1).mp <| by
       simp [_root_.pow_succ, ← hr, tendsto_const_nhds])
     (fun hr ↦
-      have := one_lt_inv hr h₂ |> tendsto_pow_atTop_atTop_of_one_lt
+      have := (one_lt_inv₀ hr).2 h₂ |> tendsto_pow_atTop_atTop_of_one_lt
       (tendsto_inv_atTop_zero.comp this).congr fun n ↦ by simp)
 @[deprecated (since := "2024-01-31")]
 alias tendsto_pow_atTop_nhds_0_of_lt_1 := tendsto_pow_atTop_nhds_zero_of_lt_one
@@ -252,13 +252,13 @@ protected theorem ENNReal.tendsto_pow_atTop_nhds_top_iff {r : ℝ≥0∞} :
     specialize h_tends (Ioi_mem_nhds one_lt_top)
     simp only [Filter.mem_map, mem_atTop_sets, ge_iff_le, Set.mem_preimage, Set.mem_Ioi] at h_tends
     obtain ⟨n, hn⟩ := h_tends
-    exact lt_irrefl _ <| lt_of_lt_of_le (hn n le_rfl) <| pow_le_one n (zero_le _) r_le_one
+    exact lt_irrefl _ <| lt_of_lt_of_le (hn n le_rfl) <| pow_le_one₀ (zero_le _) r_le_one
   · intro r_gt_one
     have obs := @Tendsto.inv ℝ≥0∞ ℕ _ _ _ (fun n ↦ (r⁻¹)^n) atTop 0
     simp only [ENNReal.tendsto_pow_atTop_nhds_zero_iff, inv_zero] at obs
     simpa [← ENNReal.inv_pow] using obs <| ENNReal.inv_lt_one.mpr r_gt_one
 
-/-! ### Geometric series-/
+/-! ### Geometric series -/
 
 
 section Geometric
@@ -360,7 +360,7 @@ theorem ENNReal.tsum_geometric (r : ℝ≥0∞) : ∑' n : ℕ, r ^ n = (1 - r)
       (ENNReal.exists_nat_gt (lt_top_iff_ne_top.1 ha)).imp fun n hn ↦ lt_of_lt_of_le hn ?_
     calc
       (n : ℝ≥0∞) = ∑ i ∈ range n, 1 := by rw [sum_const, nsmul_one, card_range]
-      _ ≤ ∑ i ∈ range n, r ^ i := by gcongr; apply one_le_pow_of_one_le' hr
+      _ ≤ ∑ i ∈ range n, r ^ i := by gcongr; apply one_le_pow₀ hr
 
 theorem ENNReal.tsum_geometric_add_one (r : ℝ≥0∞) : ∑' n : ℕ, r ^ (n + 1) = r * (1 - r)⁻¹ := by
   simp only [_root_.pow_succ', ENNReal.tsum_mul_left, ENNReal.tsum_geometric]
@@ -513,7 +513,7 @@ theorem summable_one_div_pow_of_le {m : ℝ} {f : ℕ → ℕ} (hm : 1 < m) (fi
       (summable_geometric_of_lt_one (one_div_nonneg.mpr (zero_le_one.trans hm.le))
         ((one_div_lt (zero_lt_one.trans hm) zero_lt_one).mpr (one_div_one.le.trans_lt hm)))
   rw [div_pow, one_pow]
-  refine (one_div_le_one_div ?_ ?_).mpr (pow_le_pow_right hm.le (fi a)) <;>
+  refine (one_div_le_one_div ?_ ?_).mpr (pow_right_mono₀ hm.le (fi a)) <;>
     exact pow_pos (zero_lt_one.trans hm) _
 
 /-! ### Positive sequences with small sums on countable types -/
diff --git a/Mathlib/Analysis/SpecificLimits/FloorPow.lean b/Mathlib/Analysis/SpecificLimits/FloorPow.lean
index 0916d04a949a1..0d410a3b11645 100644
--- a/Mathlib/Analysis/SpecificLimits/FloorPow.lean
+++ b/Mathlib/Analysis/SpecificLimits/FloorPow.lean
@@ -195,8 +195,7 @@ theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l :
   have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
     intro n
     refine zero_lt_one.trans_le ?_
-    simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff,
-      one_le_pow_of_one_le (cone k).le n]
+    simp only [Real.rpow_natCast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow₀ (cone k).le]
   have A :
     Tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / c k ^ (n + 1) * c k / (⌊c k ^ n⌋₊ / c k ^ n))
       atTop (𝓝 (1 * c k / 1)) := by
@@ -223,21 +222,21 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
   have B : c ^ 2 * ((1 : ℝ) - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹ := by
     rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]
     swap
-    · exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)
+    · exact sub_pos.2 (pow_lt_one₀ (inv_nonneg.2 cpos.le) (inv_lt_one_of_one_lt₀ hc) two_ne_zero)
     have : c ^ 3 = c ^ 2 * c := by ring
     simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
     rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel₀ (sq_pos_of_pos cpos).ne', one_mul]
-    simpa using pow_le_pow_right hc.le one_le_two
-  have C : c⁻¹ ^ 2 < 1 := pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero
+    simpa using pow_right_mono₀ hc.le one_le_two
+  have C : c⁻¹ ^ 2 < 1 := pow_lt_one₀ (inv_nonneg.2 cpos.le) (inv_lt_one_of_one_lt₀ hc) two_ne_zero
   calc
     (∑ i ∈ (range N).filter (j < c ^ ·), (1 : ℝ) / (c ^ i) ^ 2) ≤
         ∑ i ∈ Ico ⌊Real.log j / Real.log c⌋₊ N, (1 : ℝ) / (c ^ i) ^ 2 := by
       refine sum_le_sum_of_subset_of_nonneg (fun i hi ↦ ?_) (by intros; positivity)
       simp only [mem_filter, mem_range] at hi
-      simp only [hi.1, mem_Ico, and_true_iff]
+      simp only [hi.1, mem_Ico, and_true]
       apply Nat.floor_le_of_le
       apply le_of_lt
-      rw [div_lt_iff (Real.log_pos hc), ← Real.log_pow]
+      rw [div_lt_iff₀ (Real.log_pos hc), ← Real.log_pow]
       exact Real.log_lt_log hj hi.2
     _ = ∑ i ∈ Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i := by
       congr 1 with i
@@ -258,7 +257,7 @@ theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc
         field_simp [(Real.log_pos hc).ne']
         ring
       rw [Real.rpow_sub A, I]
-      have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'
+      have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow₀ hc two_ne_zero)).ne'
       field_simp [hj.ne', (zero_lt_one.trans hc).ne']
       ring
     _ ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 := by gcongr
@@ -271,7 +270,7 @@ theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹)
     (1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
     _ ≤ c ^ i - 1 := by
       gcongr
-      simpa only [← div_eq_mul_inv, one_le_div cpos, pow_one] using le_self_pow hc.le hi
+      simpa only [← div_eq_mul_inv, one_le_div cpos, pow_one] using le_self_pow₀ hc.le hi
     _ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
 
 /-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
@@ -280,7 +279,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
     (∑ i ∈ (range N).filter (j < ⌊c ^ ·⌋₊), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
       c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 := by
   have cpos : 0 < c := zero_lt_one.trans hc
-  have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one hc)
+  have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one_of_one_lt₀ hc)
   calc
     (∑ i ∈ (range N).filter (j < ⌊c ^ ·⌋₊), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
         ∑ i ∈ (range N).filter (j < c ^ ·), (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2 := by
@@ -292,7 +291,7 @@ theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c :
       rw [mul_div_assoc', mul_one, div_le_div_iff]; rotate_left
       · apply sq_pos_of_pos
         refine zero_lt_one.trans_le ?_
-        simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
+        simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]
       · exact sq_pos_of_pos (pow_pos cpos _)
       rw [one_mul, ← mul_pow]
       gcongr
diff --git a/Mathlib/Analysis/SpecificLimits/Normed.lean b/Mathlib/Analysis/SpecificLimits/Normed.lean
index 1e0cb4ba7e313..525b05e39a0cf 100644
--- a/Mathlib/Analysis/SpecificLimits/Normed.lean
+++ b/Mathlib/Analysis/SpecificLimits/Normed.lean
@@ -5,30 +5,28 @@ Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie
 -/
 import Mathlib.Algebra.BigOperators.Module
 import Mathlib.Algebra.Order.Field.Basic
-import Mathlib.Order.Filter.ModEq
 import Mathlib.Analysis.Asymptotics.Asymptotics
+import Mathlib.Analysis.Normed.Field.InfiniteSum
+import Mathlib.Analysis.Normed.Module.Basic
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Data.List.TFAE
-import Mathlib.Analysis.Normed.Module.Basic
+import Mathlib.Data.Nat.Choose.Bounds
+import Mathlib.Order.Filter.AtTopBot.ModEq
+import Mathlib.RingTheory.Polynomial.Pochhammer
+import Mathlib.Tactic.NoncommRing
 
 /-!
 # A collection of specific limit computations
 
 This file contains important specific limit computations in (semi-)normed groups/rings/spaces, as
 well as such computations in `ℝ` when the natural proof passes through a fact about normed spaces.
-
 -/
 
-
 noncomputable section
 
-open scoped Classical
-open Set Function Filter Finset Metric Asymptotics
+open Set Function Filter Finset Metric Asymptotics Topology Nat NNReal ENNReal
 
-open scoped Classical
-open Topology Nat uniformity NNReal ENNReal
-
-variable {α : Type*} {β : Type*} {ι : Type*}
+variable {α β ι : Type*}
 
 theorem tendsto_norm_atTop_atTop : Tendsto (norm : ℝ → ℝ) atTop atTop :=
   tendsto_abs_atTop_atTop
@@ -126,52 +124,43 @@ theorem TFAE_exists_lt_isLittleO_pow (f : ℕ → ℝ) (R : ℝ) :
     fun x hx ↦ ⟨(neg_lt_zero.2 (hx.1.trans_lt hx.2)).trans_le hx.1, hx.2⟩
   have B : Ioo 0 R ⊆ Ioo (-R) R := Subset.trans Ioo_subset_Ico_self A
   -- First we prove that 1-4 are equivalent using 2 → 3 → 4, 1 → 3, and 2 → 1
-  tfae_have 1 → 3
-  · exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
-  tfae_have 2 → 1
-  · exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
+  tfae_have 1 → 3 := fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
+  tfae_have 2 → 1 := fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
   tfae_have 3 → 2
-  · rintro ⟨a, ha, H⟩
+  | ⟨a, ha, H⟩ => by
     rcases exists_between (abs_lt.2 ha) with ⟨b, hab, hbR⟩
     exact ⟨b, ⟨(abs_nonneg a).trans_lt hab, hbR⟩,
       H.trans_isLittleO (isLittleO_pow_pow_of_abs_lt_left (hab.trans_le (le_abs_self b)))⟩
-  tfae_have 2 → 4
-  · exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
-  tfae_have 4 → 3
-  · exact fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
+  tfae_have 2 → 4 := fun ⟨a, ha, H⟩ ↦ ⟨a, ha, H.isBigO⟩
+  tfae_have 4 → 3 := fun ⟨a, ha, H⟩ ↦ ⟨a, B ha, H⟩
   -- Add 5 and 6 using 4 → 6 → 5 → 3
   tfae_have 4 → 6
-  · rintro ⟨a, ha, H⟩
+  | ⟨a, ha, H⟩ => by
     rcases bound_of_isBigO_nat_atTop H with ⟨C, hC₀, hC⟩
     refine ⟨a, ha, C, hC₀, fun n ↦ ?_⟩
     simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne')
-  tfae_have 6 → 5
-  · exact fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
+  tfae_have 6 → 5 := fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
   tfae_have 5 → 3
-  · rintro ⟨a, ha, C, h₀, H⟩
+  | ⟨a, ha, C, h₀, H⟩ => by
     rcases sign_cases_of_C_mul_pow_nonneg fun n ↦ (abs_nonneg _).trans (H n) with (rfl | ⟨hC₀, ha₀⟩)
     · obtain rfl : f = 0 := by
         ext n
         simpa using H n
-      simp only [lt_irrefl, false_or_iff] at h₀
+      simp only [lt_irrefl, false_or] at h₀
       exact ⟨0, ⟨neg_lt_zero.2 h₀, h₀⟩, isBigO_zero _ _⟩
     exact ⟨a, A ⟨ha₀, ha⟩,
       isBigO_of_le' _ fun n ↦ (H n).trans <| mul_le_mul_of_nonneg_left (le_abs_self _) hC₀.le⟩
   -- Add 7 and 8 using 2 → 8 → 7 → 3
   tfae_have 2 → 8
-  · rintro ⟨a, ha, H⟩
+  | ⟨a, ha, H⟩ => by
     refine ⟨a, ha, (H.def zero_lt_one).mono fun n hn ↦ ?_⟩
     rwa [Real.norm_eq_abs, Real.norm_eq_abs, one_mul, abs_pow, abs_of_pos ha.1] at hn
-  tfae_have 8 → 7
-  · exact fun ⟨a, ha, H⟩ ↦ ⟨a, ha.2, H⟩
+  tfae_have 8 → 7 := fun ⟨a, ha, H⟩ ↦ ⟨a, ha.2, H⟩
   tfae_have 7 → 3
-  · rintro ⟨a, ha, H⟩
+  | ⟨a, ha, H⟩ => by
     have : 0 ≤ a := nonneg_of_eventually_pow_nonneg (H.mono fun n ↦ (abs_nonneg _).trans)
     refine ⟨a, A ⟨this, ha⟩, IsBigO.of_bound 1 ?_⟩
     simpa only [Real.norm_eq_abs, one_mul, abs_pow, abs_of_nonneg this]
-  -- Porting note: used to work without explicitly having 6 → 7
-  tfae_have 6 → 7
-  · exact fun h ↦ tfae_8_to_7 <| tfae_2_to_8 <| tfae_3_to_2 <| tfae_5_to_3 <| tfae_6_to_5 h
   tfae_finish
 
 /-- For any natural `k` and a real `r > 1` we have `n ^ k = o(r ^ n)` as `n → ∞`. -/
@@ -208,7 +197,7 @@ theorem isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt {R : Type*} [Norm
   have A : (fun n ↦ (n : R) ^ k : ℕ → R) =o[atTop] fun n ↦ (r₂ / ‖r₁‖) ^ n :=
     isLittleO_pow_const_const_pow_of_one_lt k ((one_lt_div h0).2 h)
   suffices (fun n ↦ r₁ ^ n) =O[atTop] fun n ↦ ‖r₁‖ ^ n by
-    simpa [div_mul_cancel₀ _ (pow_pos h0 _).ne'] using A.mul_isBigO this
+    simpa [div_mul_cancel₀ _ (pow_pos h0 _).ne', div_pow] using A.mul_isBigO this
   exact IsBigO.of_bound 1 (by simpa using eventually_norm_pow_le r₁)
 
 theorem tendsto_pow_const_div_const_pow_of_one_lt (k : ℕ) {r : ℝ} (hr : 1 < r) :
@@ -221,7 +210,7 @@ theorem tendsto_pow_const_mul_const_pow_of_abs_lt_one (k : ℕ) {r : ℝ} (hr :
   by_cases h0 : r = 0
   · exact tendsto_const_nhds.congr'
       (mem_atTop_sets.2 ⟨1, fun n hn ↦ by simp [zero_lt_one.trans_le hn |>.ne', h0]⟩)
-  have hr' : 1 < |r|⁻¹ := one_lt_inv (abs_pos.2 h0) hr
+  have hr' : 1 < |r|⁻¹ := (one_lt_inv₀ (abs_pos.2 h0)).2 hr
   rw [tendsto_zero_iff_norm_tendsto_zero]
   simpa [div_eq_mul_inv] using tendsto_pow_const_div_const_pow_of_one_lt k hr'
 
@@ -258,8 +247,115 @@ theorem tendsto_pow_atTop_nhds_zero_of_abs_lt_one {r : ℝ} (h : |r| < 1) :
 @[deprecated (since := "2024-01-31")]
 alias tendsto_pow_atTop_nhds_0_of_abs_lt_1 := tendsto_pow_atTop_nhds_zero_of_abs_lt_one
 
-/-! ### Geometric series-/
+/-! ### Geometric series -/
+
+/-- A normed ring has summable geometric series if, for all `ξ` of norm `< 1`, the geometric series
+`∑ ξ ^ n` converges. This holds both in complete normed rings and in normed fields, providing a
+convenient abstraction of these two classes to avoid repeating the same proofs. -/
+class HasSummableGeomSeries (K : Type*) [NormedRing K] : Prop where
+  summable_geometric_of_norm_lt_one : ∀ (ξ : K), ‖ξ‖ < 1 → Summable (fun n ↦ ξ ^ n)
+
+lemma summable_geometric_of_norm_lt_one {K : Type*} [NormedRing K] [HasSummableGeomSeries K]
+    {x : K} (h : ‖x‖ < 1) : Summable (fun n ↦ x ^ n) :=
+  HasSummableGeomSeries.summable_geometric_of_norm_lt_one x h
+
+instance {R : Type*} [NormedRing R] [CompleteSpace R] : HasSummableGeomSeries R := by
+  constructor
+  intro x hx
+  have h1 : Summable fun n : ℕ ↦ ‖x‖ ^ n := summable_geometric_of_lt_one (norm_nonneg _) hx
+  exact h1.of_norm_bounded_eventually_nat _ (eventually_norm_pow_le x)
+
+section HasSummableGeometricSeries
+
+variable {R : Type*} [NormedRing R]
+
+open NormedSpace
+
+/-- Bound for the sum of a geometric series in a normed ring. This formula does not assume that the
+normed ring satisfies the axiom `‖1‖ = 1`. -/
+theorem tsum_geometric_le_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
+    ‖∑' n : ℕ, x ^ n‖ ≤ ‖(1 : R)‖ - 1 + (1 - ‖x‖)⁻¹ := by
+  by_cases hx : Summable (fun n ↦ x ^ n)
+  · rw [tsum_eq_zero_add hx]
+    simp only [_root_.pow_zero]
+    refine le_trans (norm_add_le _ _) ?_
+    have : ‖∑' b : ℕ, (fun n ↦ x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1 := by
+      refine tsum_of_norm_bounded ?_ fun b ↦ norm_pow_le' _ (Nat.succ_pos b)
+      convert (hasSum_nat_add_iff' 1).mpr (hasSum_geometric_of_lt_one (norm_nonneg x) h)
+      simp
+    linarith
+  · simp [tsum_eq_zero_of_not_summable hx]
+    nontriviality R
+    have : 1 ≤ ‖(1 : R)‖ := one_le_norm_one R
+    have : 0 ≤ (1 - ‖x‖) ⁻¹ := inv_nonneg.2 (by linarith)
+    linarith
+
+variable [HasSummableGeomSeries R]
+
+@[deprecated (since := "2024-01-31")]
+alias NormedRing.tsum_geometric_of_norm_lt_1 := tsum_geometric_le_of_norm_lt_one
+
+@[deprecated (since := "2024-07-27")]
+alias NormedRing.tsum_geometric_of_norm_lt_one := tsum_geometric_le_of_norm_lt_one
+
+theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 := by
+  have := (summable_geometric_of_norm_lt_one h).hasSum.mul_right (1 - x)
+  refine tendsto_nhds_unique this.tendsto_sum_nat ?_
+  have : Tendsto (fun n : ℕ ↦ 1 - x ^ n) atTop (𝓝 1) := by
+    simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h)
+  convert← this
+  rw [← geom_sum_mul_neg, Finset.sum_mul]
+
+theorem mul_neg_geom_series (x : R) (h : ‖x‖ < 1) : (1 - x) * ∑' i : ℕ, x ^ i = 1 := by
+  have := (summable_geometric_of_norm_lt_one h).hasSum.mul_left (1 - x)
+  refine tendsto_nhds_unique this.tendsto_sum_nat ?_
+  have : Tendsto (fun n : ℕ ↦ 1 - x ^ n) atTop (𝓝 1) := by
+    simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h)
+  convert← this
+  rw [← mul_neg_geom_sum, Finset.mul_sum]
 
+theorem geom_series_succ (x : R) (h : ‖x‖ < 1) : ∑' i : ℕ, x ^ (i + 1) = ∑' i : ℕ, x ^ i - 1 := by
+  rw [eq_sub_iff_add_eq, tsum_eq_zero_add (summable_geometric_of_norm_lt_one h),
+    pow_zero, add_comm]
+
+theorem geom_series_mul_shift (x : R) (h : ‖x‖ < 1) :
+    x * ∑' i : ℕ, x ^ i = ∑' i : ℕ, x ^ (i + 1) := by
+  simp_rw [← (summable_geometric_of_norm_lt_one h).tsum_mul_left, ← _root_.pow_succ']
+
+theorem geom_series_mul_one_add (x : R) (h : ‖x‖ < 1) :
+    (1 + x) * ∑' i : ℕ, x ^ i = 2 * ∑' i : ℕ, x ^ i - 1 := by
+  rw [add_mul, one_mul, geom_series_mul_shift x h, geom_series_succ x h, two_mul, add_sub_assoc]
+
+/-- In a normed ring with summable geometric series, a perturbation of `1` by an element `t`
+of distance less than `1` from `1` is a unit.  Here we construct its `Units` structure. -/
+@[simps val]
+def Units.oneSub (t : R) (h : ‖t‖ < 1) : Rˣ where
+  val := 1 - t
+  inv := ∑' n : ℕ, t ^ n
+  val_inv := mul_neg_geom_series t h
+  inv_val := geom_series_mul_neg t h
+
+theorem geom_series_eq_inverse (x : R) (h : ‖x‖ < 1) :
+    ∑' i, x ^ i = Ring.inverse (1 - x) := by
+  change (Units.oneSub x h) ⁻¹ = Ring.inverse (1 - x)
+  rw [← Ring.inverse_unit]
+  rfl
+
+theorem hasSum_geom_series_inverse (x : R) (h : ‖x‖ < 1) :
+    HasSum (fun i ↦ x ^ i) (Ring.inverse (1 - x)) := by
+  convert (summable_geometric_of_norm_lt_one h).hasSum
+  exact (geom_series_eq_inverse x h).symm
+
+lemma isUnit_one_sub_of_norm_lt_one {x : R} (h : ‖x‖ < 1) : IsUnit (1 - x) :=
+  ⟨Units.oneSub x h, rfl⟩
+
+end HasSummableGeometricSeries
+
+@[deprecated (since := "2024-01-31")]
+alias NormedRing.summable_geometric_of_norm_lt_1 := summable_geometric_of_norm_lt_one
+
+@[deprecated (since := "2024-07-27")]
+alias NormedRing.summable_geometric_of_norm_lt_one := summable_geometric_of_norm_lt_one
 
 section Geometric
 
@@ -278,8 +374,8 @@ theorem hasSum_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : HasSum (fun n : ℕ
 @[deprecated (since := "2024-01-31")]
 alias hasSum_geometric_of_norm_lt_1 := hasSum_geometric_of_norm_lt_one
 
-theorem summable_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : Summable fun n : ℕ ↦ ξ ^ n :=
-  ⟨_, hasSum_geometric_of_norm_lt_one h⟩
+instance : HasSummableGeomSeries K :=
+  ⟨fun _ h ↦ (hasSum_geometric_of_norm_lt_one h).summable⟩
 
 @[deprecated (since := "2024-01-31")]
 alias summable_geometric_of_norm_lt_1 := summable_geometric_of_norm_lt_one
@@ -327,55 +423,162 @@ end Geometric
 
 section MulGeometric
 
-theorem summable_norm_pow_mul_geometric_of_norm_lt_one {R : Type*} [NormedRing R] (k : ℕ) {r : R}
-    (hr : ‖r‖ < 1) : Summable fun n : ℕ ↦ ‖((n : R) ^ k * r ^ n : R)‖ := by
+variable {R : Type*} [NormedRing R] {𝕜 : Type*} [NormedDivisionRing 𝕜]
+
+theorem summable_norm_mul_geometric_of_norm_lt_one {k : ℕ} {r : R}
+    (hr : ‖r‖ < 1) {u : ℕ → ℕ} (hu : (fun n ↦ (u n : ℝ)) =O[atTop] (fun n ↦ (↑(n ^ k) : ℝ))) :
+    Summable fun n : ℕ ↦ ‖(u n * r ^ n : R)‖ := by
   rcases exists_between hr with ⟨r', hrr', h⟩
-  exact summable_of_isBigO_nat (summable_geometric_of_lt_one ((norm_nonneg _).trans hrr'.le) h)
-    (isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt _ hrr').isBigO.norm_left
+  rw [← norm_norm] at hrr'
+  apply summable_of_isBigO_nat (summable_geometric_of_lt_one ((norm_nonneg _).trans hrr'.le) h)
+  calc
+  fun n ↦ ‖↑(u n) * r ^ n‖
+  _ =O[atTop] fun n ↦ u n * ‖r‖ ^ n := by
+      apply (IsBigOWith.of_bound (c := ‖(1 : R)‖) ?_).isBigO
+      filter_upwards [eventually_norm_pow_le r] with n hn
+      simp only [norm_norm, norm_mul, Real.norm_eq_abs, abs_cast, norm_pow, abs_norm]
+      apply (norm_mul_le _ _).trans
+      have : ‖(u n : R)‖ * ‖r ^ n‖ ≤ (u n * ‖(1 : R)‖) * ‖r‖ ^ n := by
+        gcongr; exact norm_cast_le (u n)
+      exact this.trans (le_of_eq (by ring))
+  _ =O[atTop] fun n ↦ ↑(n ^ k) * ‖r‖ ^ n := hu.mul (isBigO_refl _ _)
+  _ =O[atTop] fun n ↦ r' ^ n := by
+      simp only [cast_pow]
+      exact (isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt k hrr').isBigO
+
+theorem summable_norm_pow_mul_geometric_of_norm_lt_one (k : ℕ) {r : R}
+    (hr : ‖r‖ < 1) : Summable fun n : ℕ ↦ ‖((n : R) ^ k * r ^ n : R)‖ := by
+  simp only [← cast_pow]
+  exact summable_norm_mul_geometric_of_norm_lt_one (k := k) (u := fun n ↦ n ^ k) hr
+    (isBigO_refl _ _)
+
+theorem summable_norm_geometric_of_norm_lt_one {r : R}
+    (hr : ‖r‖ < 1) : Summable fun n : ℕ ↦ ‖(r ^ n : R)‖ := by
+  simpa using summable_norm_pow_mul_geometric_of_norm_lt_one 0 hr
+
+variable [HasSummableGeomSeries R]
+
+lemma hasSum_choose_mul_geometric_of_norm_lt_one'
+    (k : ℕ) {r : R} (hr : ‖r‖ < 1) :
+    HasSum (fun n ↦ (n + k).choose k * r ^ n) (Ring.inverse (1 - r) ^ (k + 1)) := by
+  induction k with
+  | zero => simpa using hasSum_geom_series_inverse r hr
+  | succ k ih =>
+      have I1 : Summable (fun (n : ℕ) ↦ ‖(n + k).choose k * r ^ n‖) := by
+        apply summable_norm_mul_geometric_of_norm_lt_one (k := k) hr
+        apply isBigO_iff.2 ⟨2 ^ k, ?_⟩
+        filter_upwards [Ioi_mem_atTop k] with n (hn : k < n)
+        simp only [Real.norm_eq_abs, abs_cast, cast_pow, norm_pow]
+        norm_cast
+        calc (n + k).choose k
+          _ ≤ (2 * n).choose k := choose_le_choose k (by omega)
+          _ ≤ (2 * n) ^ k := Nat.choose_le_pow _ _
+          _ = 2 ^ k * n ^ k := Nat.mul_pow 2 n k
+      convert hasSum_sum_range_mul_of_summable_norm' I1 ih.summable
+        (summable_norm_geometric_of_norm_lt_one hr) (summable_geometric_of_norm_lt_one hr) with n
+      · have : ∑ i ∈ Finset.range (n + 1), ↑((i + k).choose k) * r ^ i * r ^ (n - i) =
+            ∑ i ∈ Finset.range (n + 1), ↑((i + k).choose k) * r ^ n := by
+          apply Finset.sum_congr rfl (fun i hi ↦ ?_)
+          simp only [Finset.mem_range] at hi
+          rw [mul_assoc, ← pow_add, show i + (n - i) = n by omega]
+        simp [this, ← sum_mul, ← Nat.cast_sum, sum_range_add_choose n k, add_assoc]
+      · rw [ih.tsum_eq, (hasSum_geom_series_inverse r hr).tsum_eq, pow_succ]
+
+lemma summable_choose_mul_geometric_of_norm_lt_one (k : ℕ) {r : R} (hr : ‖r‖ < 1) :
+    Summable (fun n ↦ (n + k).choose k * r ^ n) :=
+  (hasSum_choose_mul_geometric_of_norm_lt_one' k hr).summable
+
+lemma tsum_choose_mul_geometric_of_norm_lt_one' (k : ℕ) {r : R} (hr : ‖r‖ < 1) :
+    ∑' n, (n + k).choose k * r ^ n = (Ring.inverse (1 - r)) ^ (k + 1) :=
+  (hasSum_choose_mul_geometric_of_norm_lt_one' k hr).tsum_eq
+
+lemma hasSum_choose_mul_geometric_of_norm_lt_one
+    (k : ℕ) {r : 𝕜} (hr : ‖r‖ < 1) :
+    HasSum (fun n ↦ (n + k).choose k * r ^ n) (1 / (1 - r) ^ (k + 1)) := by
+  convert hasSum_choose_mul_geometric_of_norm_lt_one' k hr
+  simp
+
+lemma tsum_choose_mul_geometric_of_norm_lt_one (k : ℕ) {r : 𝕜} (hr : ‖r‖ < 1) :
+    ∑' n, (n + k).choose k * r ^ n = 1/ (1 - r) ^ (k + 1) :=
+  (hasSum_choose_mul_geometric_of_norm_lt_one k hr).tsum_eq
+
+lemma summable_descFactorial_mul_geometric_of_norm_lt_one (k : ℕ) {r : R} (hr : ‖r‖ < 1) :
+    Summable (fun n ↦ (n + k).descFactorial k * r ^ n) := by
+  convert (summable_choose_mul_geometric_of_norm_lt_one k hr).mul_left (k.factorial : R)
+    using 2 with n
+  simp [← mul_assoc, descFactorial_eq_factorial_mul_choose (n + k) k]
+
+open Polynomial in
+theorem summable_pow_mul_geometric_of_norm_lt_one (k : ℕ) {r : R} (hr : ‖r‖ < 1) :
+    Summable (fun n ↦ (n : R) ^ k * r ^ n : ℕ → R) := by
+  refine Nat.strong_induction_on k fun k hk => ?_
+  obtain ⟨a, ha⟩ : ∃ (a : ℕ → ℕ), ∀ n, (n + k).descFactorial k
+      = n ^ k + ∑ i ∈ range k, a i * n ^ i := by
+    let P : Polynomial ℕ := (ascPochhammer ℕ k).comp (Polynomial.X + C 1)
+    refine ⟨fun i ↦ P.coeff i, fun n ↦ ?_⟩
+    have mP : Monic P := Monic.comp_X_add_C (monic_ascPochhammer ℕ k) _
+    have dP : P.natDegree = k := by
+      simp only [P, natDegree_comp, ascPochhammer_natDegree, mul_one, natDegree_X_add_C]
+    have A : (n + k).descFactorial k = P.eval n := by
+      have : n + 1 + k - 1 = n + k := by omega
+      simp [P, ascPochhammer_nat_eq_descFactorial, this]
+    conv_lhs => rw [A, mP.as_sum, dP]
+    simp [eval_finset_sum]
+  have : Summable (fun n ↦ (n + k).descFactorial k * r ^ n
+      - ∑ i ∈ range k, a i * n ^ (i : ℕ) * r ^ n) := by
+    apply (summable_descFactorial_mul_geometric_of_norm_lt_one k hr).sub
+    apply summable_sum (fun i hi ↦ ?_)
+    simp_rw [mul_assoc]
+    simp only [Finset.mem_range] at hi
+    exact (hk _ hi).mul_left _
+  convert this using 1
+  ext n
+  simp [ha n, add_mul, sum_mul]
 
 @[deprecated (since := "2024-01-31")]
 alias summable_norm_pow_mul_geometric_of_norm_lt_1 := summable_norm_pow_mul_geometric_of_norm_lt_one
 
-theorem summable_pow_mul_geometric_of_norm_lt_one {R : Type*} [NormedRing R] [CompleteSpace R]
-    (k : ℕ) {r : R} (hr : ‖r‖ < 1) : Summable (fun n ↦ (n : R) ^ k * r ^ n : ℕ → R) :=
-  .of_norm <| summable_norm_pow_mul_geometric_of_norm_lt_one _ hr
-
 @[deprecated (since := "2024-01-31")]
 alias summable_pow_mul_geometric_of_norm_lt_1 := summable_pow_mul_geometric_of_norm_lt_one
 
+/-- If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`, `HasSum` version in a general ring
+with summable geometric series. For a version in a field, using division instead of `Ring.inverse`,
+see `hasSum_coe_mul_geometric_of_norm_lt_one`. -/
+theorem hasSum_coe_mul_geometric_of_norm_lt_one'
+    {x : R} (h : ‖x‖ < 1) :
+    HasSum (fun n ↦ n * x ^ n : ℕ → R) (x * (Ring.inverse (1 - x)) ^ 2) := by
+  have A : HasSum (fun (n : ℕ) ↦ (n + 1) * x ^ n) (Ring.inverse (1 - x) ^ 2) := by
+    convert hasSum_choose_mul_geometric_of_norm_lt_one' 1 h with n
+    simp
+  have B : HasSum (fun (n : ℕ) ↦ x ^ n) (Ring.inverse (1 - x)) := hasSum_geom_series_inverse x h
+  convert A.sub B using 1
+  · ext n
+    simp [add_mul]
+  · symm
+    calc Ring.inverse (1 - x) ^ 2 - Ring.inverse (1 - x)
+    _ = Ring.inverse (1 - x) ^ 2 - ((1 - x) * Ring.inverse (1 - x)) * Ring.inverse (1 - x) := by
+      simp [Ring.mul_inverse_cancel (1 - x) (isUnit_one_sub_of_norm_lt_one h)]
+    _ = x * Ring.inverse (1 - x) ^ 2 := by noncomm_ring
+
+/-- If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`, version in a general ring with
+summable geometric series. For a version in a field, using division instead of `Ring.inverse`,
+see `tsum_coe_mul_geometric_of_norm_lt_one`. -/
+theorem tsum_coe_mul_geometric_of_norm_lt_one'
+    {r : 𝕜} (hr : ‖r‖ < 1) : (∑' n : ℕ, n * r ^ n : 𝕜) = r * Ring.inverse (1 - r) ^ 2 :=
+  (hasSum_coe_mul_geometric_of_norm_lt_one' hr).tsum_eq
+
 /-- If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`, `HasSum` version. -/
-theorem hasSum_coe_mul_geometric_of_norm_lt_one {𝕜 : Type*} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜]
-    {r : 𝕜} (hr : ‖r‖ < 1) : HasSum (fun n ↦ n * r ^ n : ℕ → 𝕜) (r / (1 - r) ^ 2) := by
-  have A : Summable (fun n ↦ (n : 𝕜) * r ^ n : ℕ → 𝕜) := by
-    simpa only [pow_one] using summable_pow_mul_geometric_of_norm_lt_one 1 hr
-  have B : HasSum (r ^ · : ℕ → 𝕜) (1 - r)⁻¹ := hasSum_geometric_of_norm_lt_one hr
-  refine A.hasSum_iff.2 ?_
-  have hr' : r ≠ 1 := by
-    rintro rfl
-    simp [lt_irrefl] at hr
-  set s : 𝕜 := ∑' n : ℕ, n * r ^ n
-  have : Commute (1 - r) s :=
-    .tsum_right _ fun _ =>
-      .sub_left (.one_left _) (.mul_right (Nat.commute_cast _ _) (.pow_right (.refl _) _))
-  calc
-    s = s * (1 - r) / (1 - r) := (mul_div_cancel_right₀ _ (sub_ne_zero.2 hr'.symm)).symm
-    _ = (1 - r) * s / (1 - r) := by rw [this.eq]
-    _ = (s - r * s) / (1 - r) := by rw [_root_.sub_mul, one_mul]
-    _ = (((0 : ℕ) * r ^ 0 + ∑' n : ℕ, (n + 1 : ℕ) * r ^ (n + 1)) - r * s) / (1 - r) := by
-      rw [← tsum_eq_zero_add A]
-    _ = ((r * ∑' n : ℕ, ↑(n + 1) * r ^ n) - r * s) / (1 - r) := by
-      simp only [cast_zero, pow_zero, mul_one, _root_.pow_succ', (Nat.cast_commute _ r).left_comm,
-        _root_.tsum_mul_left, zero_add]
-    _ = r / (1 - r) ^ 2 := by
-      simp [add_mul, tsum_add A B.summable, mul_add, B.tsum_eq, ← div_eq_mul_inv, sq,
-        div_mul_eq_div_div_swap]
+theorem hasSum_coe_mul_geometric_of_norm_lt_one {r : 𝕜} (hr : ‖r‖ < 1) :
+    HasSum (fun n ↦ n * r ^ n : ℕ → 𝕜) (r / (1 - r) ^ 2) := by
+  convert hasSum_coe_mul_geometric_of_norm_lt_one' hr using 1
+  simp [div_eq_mul_inv]
 
 @[deprecated (since := "2024-01-31")]
 alias hasSum_coe_mul_geometric_of_norm_lt_1 := hasSum_coe_mul_geometric_of_norm_lt_one
 
 /-- If `‖r‖ < 1`, then `∑' n : ℕ, n * r ^ n = r / (1 - r) ^ 2`. -/
-theorem tsum_coe_mul_geometric_of_norm_lt_one {𝕜 : Type*} [NormedDivisionRing 𝕜] [CompleteSpace 𝕜]
-    {r : 𝕜} (hr : ‖r‖ < 1) : (∑' n : ℕ, n * r ^ n : 𝕜) = r / (1 - r) ^ 2 :=
+theorem tsum_coe_mul_geometric_of_norm_lt_one {r : 𝕜} (hr : ‖r‖ < 1) :
+    (∑' n : ℕ, n * r ^ n : 𝕜) = r / (1 - r) ^ 2 :=
   (hasSum_coe_mul_geometric_of_norm_lt_one hr).tsum_eq
 
 @[deprecated (since := "2024-01-31")]
@@ -459,67 +662,6 @@ lemma exists_norm_le_of_cauchySeq (h : CauchySeq fun n ↦ ∑ k ∈ range n, f
 
 end SummableLeGeometric
 
-section NormedRingGeometric
-
-variable {R : Type*} [NormedRing R] [CompleteSpace R]
-
-open NormedSpace
-
-/-- A geometric series in a complete normed ring is summable.
-Proved above (same name, different namespace) for not-necessarily-complete normed fields. -/
-theorem NormedRing.summable_geometric_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
-    Summable fun n : ℕ ↦ x ^ n :=
-  have h1 : Summable fun n : ℕ ↦ ‖x‖ ^ n := summable_geometric_of_lt_one (norm_nonneg _) h
-  h1.of_norm_bounded_eventually_nat _ (eventually_norm_pow_le x)
-@[deprecated (since := "2024-01-31")]
-alias NormedRing.summable_geometric_of_norm_lt_1 := NormedRing.summable_geometric_of_norm_lt_one
-
-/-- Bound for the sum of a geometric series in a normed ring. This formula does not assume that the
-normed ring satisfies the axiom `‖1‖ = 1`. -/
-theorem NormedRing.tsum_geometric_of_norm_lt_one (x : R) (h : ‖x‖ < 1) :
-    ‖∑' n : ℕ, x ^ n‖ ≤ ‖(1 : R)‖ - 1 + (1 - ‖x‖)⁻¹ := by
-  rw [tsum_eq_zero_add (summable_geometric_of_norm_lt_one x h)]
-  simp only [_root_.pow_zero]
-  refine le_trans (norm_add_le _ _) ?_
-  have : ‖∑' b : ℕ, (fun n ↦ x ^ (n + 1)) b‖ ≤ (1 - ‖x‖)⁻¹ - 1 := by
-    refine tsum_of_norm_bounded ?_ fun b ↦ norm_pow_le' _ (Nat.succ_pos b)
-    convert (hasSum_nat_add_iff' 1).mpr (hasSum_geometric_of_lt_one (norm_nonneg x) h)
-    simp
-  linarith
-
-@[deprecated (since := "2024-01-31")]
-alias NormedRing.tsum_geometric_of_norm_lt_1 := NormedRing.tsum_geometric_of_norm_lt_one
-
-theorem geom_series_mul_neg (x : R) (h : ‖x‖ < 1) : (∑' i : ℕ, x ^ i) * (1 - x) = 1 := by
-  have := (NormedRing.summable_geometric_of_norm_lt_one x h).hasSum.mul_right (1 - x)
-  refine tendsto_nhds_unique this.tendsto_sum_nat ?_
-  have : Tendsto (fun n : ℕ ↦ 1 - x ^ n) atTop (𝓝 1) := by
-    simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h)
-  convert← this
-  rw [← geom_sum_mul_neg, Finset.sum_mul]
-
-theorem mul_neg_geom_series (x : R) (h : ‖x‖ < 1) : ((1 - x) * ∑' i : ℕ, x ^ i) = 1 := by
-  have := (NormedRing.summable_geometric_of_norm_lt_one x h).hasSum.mul_left (1 - x)
-  refine tendsto_nhds_unique this.tendsto_sum_nat ?_
-  have : Tendsto (fun n : ℕ ↦ 1 - x ^ n) atTop (𝓝 1) := by
-    simpa using tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_norm_lt_one h)
-  convert← this
-  rw [← mul_neg_geom_sum, Finset.mul_sum]
-
-theorem geom_series_succ (x : R) (h : ‖x‖ < 1) : ∑' i : ℕ, x ^ (i + 1) = ∑' i : ℕ, x ^ i - 1 := by
-  rw [eq_sub_iff_add_eq, tsum_eq_zero_add (NormedRing.summable_geometric_of_norm_lt_one x h),
-    pow_zero, add_comm]
-
-theorem geom_series_mul_shift (x : R) (h : ‖x‖ < 1) :
-    x * ∑' i : ℕ, x ^ i = ∑' i : ℕ, x ^ (i + 1) := by
-  simp_rw [← (NormedRing.summable_geometric_of_norm_lt_one _ h).tsum_mul_left, ← _root_.pow_succ']
-
-theorem geom_series_mul_one_add (x : R) (h : ‖x‖ < 1) :
-    (1 + x) * ∑' i : ℕ, x ^ i = 2 * ∑' i : ℕ, x ^ i - 1 := by
-  rw [add_mul, one_mul, geom_series_mul_shift x h, geom_series_succ x h, two_mul, add_sub_assoc]
-
-end NormedRingGeometric
-
 /-! ### Summability tests based on comparison with geometric series -/
 
 theorem summable_of_ratio_norm_eventually_le {α : Type*} [SeminormedAddCommGroup α]
@@ -766,6 +908,8 @@ theorem Real.summable_pow_div_factorial (x : ℝ) : Summable (fun n ↦ x ^ n /
         norm_div, Real.norm_natCast, Nat.cast_succ]
     _ ≤ ‖x‖ / (⌊‖x‖⌋₊ + 1) * ‖x ^ n / (n !)‖ := by gcongr
 
+@[deprecated "`Real.tendsto_pow_div_factorial_atTop` has been deprecated, use
+`FloorSemiring.tendsto_pow_div_factorial_atTop` instead" (since := "2024-10-05")]
 theorem Real.tendsto_pow_div_factorial_atTop (x : ℝ) :
     Tendsto (fun n ↦ x ^ n / n ! : ℕ → ℝ) atTop (𝓝 0) :=
   (Real.summable_pow_div_factorial x).tendsto_atTop_zero
diff --git a/Mathlib/Analysis/Subadditive.lean b/Mathlib/Analysis/Subadditive.lean
index 10867220bdf25..8b9b5336d51ef 100644
--- a/Mathlib/Analysis/Subadditive.lean
+++ b/Mathlib/Analysis/Subadditive.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
 import Mathlib.Topology.Instances.Real
-import Mathlib.Order.Filter.Archimedean
+import Mathlib.Order.Filter.AtTopBot.Archimedean
 
 /-!
 # Convergence of subadditive sequences
diff --git a/Mathlib/Analysis/SumOverResidueClass.lean b/Mathlib/Analysis/SumOverResidueClass.lean
index 92432ccab72a0..d0109a33064e6 100644
--- a/Mathlib/Analysis/SumOverResidueClass.lean
+++ b/Mathlib/Analysis/SumOverResidueClass.lean
@@ -3,9 +3,9 @@ Copyright (c) 2024 Michael Stoll. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Stoll
 -/
+import Mathlib.Analysis.Normed.Field.Basic
 import Mathlib.Data.ZMod.Basic
 import Mathlib.Topology.Instances.ENNReal
-import Mathlib.Analysis.Normed.Field.Basic
 
 /-!
 # Sums over residue classes
@@ -92,3 +92,14 @@ lemma summable_indicator_mod_iff {m : ℕ} [NeZero m] {f : ℕ → ℝ} (hf : An
   convert summable_sum (s := Finset.univ)
     fun a _ ↦ summable_indicator_mod_iff_summable_indicator_mod hf a H
   simp only [Finset.sum_apply]
+
+open ZMod
+
+/-- If `f` is a summable function on `ℕ`, and `0 < N`, then we may compute `∑' n : ℕ, f n` by
+summing each residue class mod `N` separately. -/
+lemma Nat.sumByResidueClasses {R : Type*} [AddCommGroup R] [UniformSpace R] [UniformAddGroup R]
+    [CompleteSpace R] [T0Space R] {f : ℕ → R} (hf : Summable f) (N : ℕ) [NeZero N] :
+    ∑' n, f n = ∑ j : ZMod N, ∑' m, f (j.val + N * m) := by
+  rw [← (residueClassesEquiv N).symm.tsum_eq f, tsum_prod, tsum_fintype, residueClassesEquiv,
+    Equiv.coe_fn_symm_mk]
+  exact hf.comp_injective (residueClassesEquiv N).symm.injective
diff --git a/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean b/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean
index 6486b7bbab466..0d05481d7f037 100644
--- a/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean
+++ b/Mathlib/Analysis/VonNeumannAlgebra/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Analysis.Normed.Module.Dual
 import Mathlib.Analysis.Complex.Basic
diff --git a/Mathlib/CategoryTheory/Abelian/Basic.lean b/Mathlib/CategoryTheory/Abelian/Basic.lean
index b7e4dbcf98889..cc0db03e5a10b 100644
--- a/Mathlib/CategoryTheory/Abelian/Basic.lean
+++ b/Mathlib/CategoryTheory/Abelian/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Johan Commelin, Scott Morrison
+Authors: Markus Himmel, Johan Commelin, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
 import Mathlib.CategoryTheory.Preadditive.Biproducts
diff --git a/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean b/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
index 1919022e9a035..5812a21f56858 100644
--- a/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
+++ b/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
@@ -87,8 +87,6 @@ theorem mono_of_epi_of_mono_of_mono (hR₁ : R₁.Exact) (hR₂ : R₂.Exact)
     (by simpa only [R₁.map'_comp 0 1 2] using hR₁.toIsComplex.zero 0)
     (hR₁.exact 1).exact_toComposableArrows (hR₂.exact 0).exact_toComposableArrows h₀ h₁ h₃
 
-attribute [local instance] epi_comp
-
 theorem epi_of_epi_of_epi_of_mono'
     (hR₁ : (mk₂ (R₁.map' 1 2) (R₁.map' 2 3)).Exact)
     (hR₂ : (mk₂ (R₂.map' 0 1) (R₂.map' 1 2)).Exact) (hR₂' : R₂.map' 1 3 = 0)
diff --git a/Mathlib/CategoryTheory/Abelian/Exact.lean b/Mathlib/CategoryTheory/Abelian/Exact.lean
index cfc118215dd0f..04967ba6b7f80 100644
--- a/Mathlib/CategoryTheory/Abelian/Exact.lean
+++ b/Mathlib/CategoryTheory/Abelian/Exact.lean
@@ -169,25 +169,19 @@ section
 open List in
 theorem Abelian.tfae_mono {X Y : C} (f : X ⟶ Y) (Z : C) :
     TFAE [Mono f, kernel.ι f = 0, (ShortComplex.mk (0 : Z ⟶ X) f zero_comp).Exact] := by
-  tfae_have 2 → 1
-  · exact mono_of_kernel_ι_eq_zero _
+  tfae_have 2 → 1 := mono_of_kernel_ι_eq_zero _
   tfae_have 1 → 2
-  · intro
-    rw [← cancel_mono f, kernel.condition, zero_comp]
-  tfae_have 3 ↔ 1
-  · exact ShortComplex.exact_iff_mono _ (by simp)
+  | _ => by rw [← cancel_mono f, kernel.condition, zero_comp]
+  tfae_have 3 ↔ 1 := ShortComplex.exact_iff_mono _ (by simp)
   tfae_finish
 
 open List in
 theorem Abelian.tfae_epi {X Y : C} (f : X ⟶ Y) (Z : C ) :
     TFAE [Epi f, cokernel.π f = 0, (ShortComplex.mk f (0 : Y ⟶ Z) comp_zero).Exact] := by
-  tfae_have 2 → 1
-  · exact epi_of_cokernel_π_eq_zero _
+  tfae_have 2 → 1 := epi_of_cokernel_π_eq_zero _
   tfae_have 1 → 2
-  · intro
-    rw [← cancel_epi f, cokernel.condition, comp_zero]
-  tfae_have 3 ↔ 1
-  · exact ShortComplex.exact_iff_epi _ (by simp)
+  | _ => by rw [← cancel_epi f, cokernel.condition, comp_zero]
+  tfae_have 3 ↔ 1 := ShortComplex.exact_iff_epi _ (by simp)
   tfae_finish
 
 end
@@ -240,14 +234,14 @@ theorem preservesMonomorphisms_of_map_exact : L.PreservesMonomorphisms where
   preserves f hf := by
     apply ((Abelian.tfae_mono (L.map f) (L.obj 0)).out 2 0).mp
     refine ShortComplex.exact_of_iso ?_ (hL _ (((tfae_mono f 0).out 0 2).mp hf))
-    exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _)  (Iso.refl _)
+    exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
 
 /-- A functor which preserves exactness preserves epimorphisms. -/
 theorem preservesEpimorphisms_of_map_exact : L.PreservesEpimorphisms where
   preserves f hf := by
     apply ((Abelian.tfae_epi (L.map f) (L.obj 0)).out 2 0).mp
     refine ShortComplex.exact_of_iso ?_ (hL _ (((tfae_epi f 0).out 0 2).mp hf))
-    exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _)  (Iso.refl _)
+    exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
 
 /-- A functor which preserves the exactness of short complexes preserves homology. -/
 def preservesHomologyOfMapExact : L.PreservesHomology where
diff --git a/Mathlib/CategoryTheory/Abelian/Ext.lean b/Mathlib/CategoryTheory/Abelian/Ext.lean
index 63698a865b338..223c0b659b3b2 100644
--- a/Mathlib/CategoryTheory/Abelian/Ext.lean
+++ b/Mathlib/CategoryTheory/Abelian/Ext.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Adam Topaz
+Authors: Kim Morrison, Adam Topaz
 -/
 import Mathlib.Algebra.Category.ModuleCat.Abelian
 import Mathlib.Algebra.Homology.Opposite
diff --git a/Mathlib/CategoryTheory/Abelian/FunctorCategory.lean b/Mathlib/CategoryTheory/Abelian/FunctorCategory.lean
index b3b746622ceed..8178bfae61ecf 100644
--- a/Mathlib/CategoryTheory/Abelian/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Abelian/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Abelian.Basic
 import Mathlib.CategoryTheory.Preadditive.FunctorCategory
diff --git a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms.lean b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms.lean
new file mode 100644
index 0000000000000..ff25be4567c6f
--- /dev/null
+++ b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms.lean
@@ -0,0 +1,93 @@
+/-
+Copyright (c) 2023 Adam Topaz. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Isaac Hernando, Coleton Kotch, Adam Topaz
+-/
+
+import Mathlib.CategoryTheory.Limits.Filtered
+import Mathlib.CategoryTheory.Limits.Preserves.Finite
+
+/-!
+
+# Grothendieck Axioms
+
+This file defines some of the Grothendieck Axioms for abelian categories, and proves
+basic facts about them.
+
+## Definitions
+
+- `AB4` -- an abelian category satisfies `AB4` provided that coproducts are exact.
+- `AB5` -- an abelian category satisfies `AB5` provided that filtered colimits are exact.
+- The duals of the above definitions, called `AB4Star` and `AB5Star`.
+
+## Remarks
+
+For `AB4` and `AB5`, we only require left exactness as right exactness is automatic.
+A comparison with Grothendieck's original formulation of the properties can be found in the
+comments of the linked Stacks page.
+Exactness as the preservation of short exact sequences is introduced in
+`CategoryTheory.Abelian.Exact`.
+
+## Projects
+
+- Add additional axioms, especially define Grothendieck categories.
+- Prove that `AB5` implies `AB4`.
+
+## References
+* [Stacks: Grothendieck's AB conditions](https://stacks.math.columbia.edu/tag/079A)
+
+-/
+
+namespace CategoryTheory
+
+open Limits
+
+universe v v' u u'
+
+variable (C : Type u) [Category.{v} C]
+
+/--
+A category `C` which has coproducts is said to have `AB4` provided that
+coproducts are exact.
+-/
+class AB4 [HasCoproducts C] where
+  /-- Exactness of coproducts stated as `colim : (Discrete α ⥤ C) ⥤ C` preserving limits. -/
+  preservesFiniteLimits (α : Type v) :
+    PreservesFiniteLimits (colim (J := Discrete α) (C := C))
+
+attribute [instance] AB4.preservesFiniteLimits
+
+/-- A category `C` which has products is said to have `AB4Star` (in literature `AB4*`)
+provided that products are exact. -/
+class AB4Star [HasProducts C] where
+  /-- Exactness of products stated as `lim : (Discrete α ⥤ C) ⥤ C` preserving colimits. -/
+  preservesFiniteColimits (α : Type v) :
+    PreservesFiniteColimits (lim (J := Discrete α) (C := C))
+
+attribute [instance] AB4Star.preservesFiniteColimits
+
+/--
+A category `C` which has filtered colimits is said to have `AB5` provided that
+filtered colimits are exact.
+-/
+class AB5 [HasFilteredColimits C] where
+  /-- Exactness of filtered colimits stated as `colim : (J ⥤ C) ⥤ C` on filtered `J`
+  preserving limits. -/
+  preservesFiniteLimits (J : Type v) [SmallCategory J] [IsFiltered J] :
+    PreservesFiniteLimits (colim (J := J) (C := C))
+
+attribute [instance] AB5.preservesFiniteLimits
+
+/--
+A category `C` which has cofiltered limits is said to have `AB5Star` (in literature `AB5*`)
+provided that cofiltered limits are exact.
+-/
+class AB5Star [HasCofilteredLimits C] where
+  /-- Exactness of cofiltered limits stated as `lim : (J ⥤ C) ⥤ C` on cofiltered `J`
+  preserving colimits. -/
+  preservesFiniteColimits (J : Type v) [SmallCategory J] [IsCofiltered J] :
+    PreservesFiniteColimits (lim (J := J) (C := C))
+
+attribute [instance] AB5Star.preservesFiniteColimits
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Abelian/Images.lean b/Mathlib/CategoryTheory/Abelian/Images.lean
index 58a0dcda3e32b..c753cc0c81887 100644
--- a/Mathlib/CategoryTheory/Abelian/Images.lean
+++ b/Mathlib/CategoryTheory/Abelian/Images.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Kernels
 
diff --git a/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean b/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
index a7565550d51fd..675ab43cfcce7 100644
--- a/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
+++ b/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jujian Zhang, Scott Morrison
+Authors: Jujian Zhang, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.InjectiveResolution
 import Mathlib.Algebra.Homology.HomotopyCategory
diff --git a/Mathlib/CategoryTheory/Abelian/LeftDerived.lean b/Mathlib/CategoryTheory/Abelian/LeftDerived.lean
index be9b3ab46d0c2..f9333349b110b 100644
--- a/Mathlib/CategoryTheory/Abelian/LeftDerived.lean
+++ b/Mathlib/CategoryTheory/Abelian/LeftDerived.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Riccardo Brasca, Adam Topaz, Jujian Zhang, Joël Riou
+Authors: Kim Morrison, Riccardo Brasca, Adam Topaz, Jujian Zhang, Joël Riou
 -/
 import Mathlib.Algebra.Homology.Additive
 import Mathlib.CategoryTheory.Abelian.ProjectiveResolution
diff --git a/Mathlib/CategoryTheory/Abelian/NonPreadditive.lean b/Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
index c8c9e63e83d6f..680956963fe88 100644
--- a/Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
+++ b/Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
@@ -109,7 +109,6 @@ instance : Epi (Abelian.factorThruImage f) :=
   _ fun R (g : I ⟶ R) (hpg : p ≫ g = 0) => by
   -- Since C is abelian, u := ker g ≫ i is the kernel of some morphism h.
   let u := kernel.ι g ≫ i
-  haveI : Mono u := mono_comp _ _
   haveI hu := normalMonoOfMono u
   let h := hu.g
   -- By hypothesis, p factors through the kernel of g via some t.
@@ -146,7 +145,6 @@ instance : Mono (Abelian.factorThruCoimage f) :=
   NormalEpiCategory.mono_of_cancel_zero _ fun R (g : R ⟶ I) (hgi : g ≫ i = 0) => by
     -- Since C is abelian, u := p ≫ coker g is the cokernel of some morphism h.
     let u := p ≫ cokernel.π g
-    haveI : Epi u := epi_comp _ _
     haveI hu := normalEpiOfEpi u
     let h := hu.g
     -- By hypothesis, i factors through the cokernel of g via some t.
diff --git a/Mathlib/CategoryTheory/Abelian/Opposite.lean b/Mathlib/CategoryTheory/Abelian/Opposite.lean
index 6c2747ade568d..e862e8a81c331 100644
--- a/Mathlib/CategoryTheory/Abelian/Opposite.lean
+++ b/Mathlib/CategoryTheory/Abelian/Opposite.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Abelian.Basic
 import Mathlib.CategoryTheory.Preadditive.Opposite
diff --git a/Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean b/Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
index 854dacb44ab3e..b46ed6087d1b3 100644
--- a/Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
+++ b/Mathlib/CategoryTheory/Abelian/ProjectiveResolution.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison, Jakob von Raumer, Joël Riou
+Authors: Markus Himmel, Kim Morrison, Jakob von Raumer, Joël Riou
 -/
 import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
 import Mathlib.Algebra.Homology.HomotopyCategory
diff --git a/Mathlib/CategoryTheory/Abelian/RightDerived.lean b/Mathlib/CategoryTheory/Abelian/RightDerived.lean
index 2c12f1e56530f..8cf24315d8930 100644
--- a/Mathlib/CategoryTheory/Abelian/RightDerived.lean
+++ b/Mathlib/CategoryTheory/Abelian/RightDerived.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jujian Zhang, Scott Morrison, Joël Riou
+Authors: Jujian Zhang, Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Homology.Additive
 import Mathlib.CategoryTheory.Abelian.InjectiveResolution
diff --git a/Mathlib/CategoryTheory/Abelian/Transfer.lean b/Mathlib/CategoryTheory/Abelian/Transfer.lean
index 1948ebfb7f9a6..5a3c93413c248 100644
--- a/Mathlib/CategoryTheory/Abelian/Transfer.lean
+++ b/Mathlib/CategoryTheory/Abelian/Transfer.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Abelian.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
@@ -128,7 +128,7 @@ def coimageIsoImage (i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F) {X Y : C} (f : X 
     _ ≅ Abelian.image f := Iso.refl _
 
 -- The account of this proof in the Stacks project omits this calculation.
-theorem coimageIsoImage_hom  (i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F) {X Y : C} (f : X ⟶ Y) :
+theorem coimageIsoImage_hom (i : F ⋙ G ≅ 𝟭 C) (adj : G ⊣ F) {X Y : C} (f : X ⟶ Y) :
     (coimageIsoImage F G i adj f).hom = Abelian.coimageImageComparison f := by
   dsimp [coimageIsoImage, cokernelIso, cokernelEpiComp, cokernelCompIsIso_inv,
     coimageIsoImageAux, kernelCompMono]
diff --git a/Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean b/Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean
index fdc950a181b49..3d5e083b0ab9b 100644
--- a/Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean
+++ b/Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean
@@ -62,7 +62,7 @@ section GeneralAdjointFunctorTheorem
 variable {D : Type u} [Category.{v} D]
 variable (G : D ⥤ C)
 
-/-- If `G : D ⥤ C` is a right adjoint it satisfies the solution set condition.  -/
+/-- If `G : D ⥤ C` is a right adjoint it satisfies the solution set condition. -/
 theorem solutionSetCondition_of_isRightAdjoint [G.IsRightAdjoint] : SolutionSetCondition G := by
   intro A
   refine
diff --git a/Mathlib/CategoryTheory/Adjunction/Basic.lean b/Mathlib/CategoryTheory/Adjunction/Basic.lean
index 6bb76fb19fe02..3d9c1506b6275 100644
--- a/Mathlib/CategoryTheory/Adjunction/Basic.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Basic.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Reid Barton, Johan Commelin, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Equivalence
+import Mathlib.CategoryTheory.Yoneda
 
 /-!
 # Adjunctions between functors
@@ -13,7 +14,9 @@ import Mathlib.CategoryTheory.Equivalence
 
 We provide various useful constructors:
 * `mkOfHomEquiv`
-* `mkOfUnitCounit`
+* `mk'`: construct an adjunction from the data of a hom set equivalence, unit and counit natural
+  transformations together with proofs of the equalities `homEquiv_unit` and `homEquiv_counit`
+  relating them to each other.
 * `leftAdjointOfEquiv` / `rightAdjointOfEquiv`
   construct a left/right adjoint of a given functor given the action on objects and
   the relevant equivalence of morphism spaces.
@@ -29,6 +32,44 @@ adjoint can be obtained as `F.rightAdjoint`.
 `toEquivalence` upgrades an adjunction to an equivalence,
 given witnesses that the unit and counit are pointwise isomorphisms.
 Conversely `Equivalence.toAdjunction` recovers the underlying adjunction from an equivalence.
+
+## Overview of the directory `CategoryTheory.Adjunction`
+
+* Adjoint lifting theorems are in the directory `Lifting`.
+* The file `AdjointFunctorTheorems` proves the adjoint functor theorems.
+* The file `Comma` shows that for a functor `G : D ⥤ C` the data of an initial object in each
+  `StructuredArrow` category on `G` is equivalent to a left adjoint to `G`, as well as the dual.
+* The file `Evaluation` shows that products and coproducts are adjoint to evaluation of functors.
+* The file `FullyFaithful` characterizes when adjoints are full or faithful in terms of the unit
+  and counit.
+* The file `Limits` proves that left adjoints preserve colimits and right adjoints preserve limits.
+* The file `Mates` establishes the bijection between the 2-cells
+  ```
+          L₁                  R₁
+        C --→ D             C ←-- D
+      G ↓  ↗  ↓ H         G ↓  ↘  ↓ H
+        E --→ F             E ←-- F
+          L₂                  R₂
+  ```
+  where `L₁ ⊣ R₁` and `L₂ ⊣ R₂`. Specializing to a pair of adjoints `L₁ L₂ : C ⥤ D`,
+  `R₁ R₂ : D ⥤ C`, it provides equivalences `(L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂)` and `(L₂ ≅ L₁) ≃ (R₁ ≅ R₂)`.
+* The file `Opposites` contains constructions to relate adjunctions of functors to adjunctions of
+  their opposites.
+* The file `Reflective` defines reflective functors, i.e. fully faithful right adjoints. Note that
+  many facts about reflective functors are proved in the earlier file `FullyFaithful`.
+* The file `Restrict` defines the restriction of an adjunction along fully faithful functors.
+* The file `Triple` proves that in an adjoint triple, the left adjoint is fully faithful if and
+  only if the right adjoint is.
+* The file `Unique` proves uniqueness of adjoints.
+* The file `Whiskering` proves that functors `F : D ⥤ E` and `G : E ⥤ D` with an adjunction
+  `F ⊣ G`, induce adjunctions between the functor categories `C ⥤ D` and `C ⥤ E`,
+  and the functor categories `E ⥤ C` and `D ⥤ C`.
+
+## Other files related to adjunctions
+
+* The file `CategoryTheory.Monad.Adjunction` develops the basic relationship between adjunctions
+  and (co)monads. There it is also shown that given an adjunction `L ⊣ R` and an isomorphism
+  `L ⋙ R ≅ 𝟭 C`, the unit is an isomorphism, and similarly for the counit.
 -/
 
 
@@ -47,26 +88,30 @@ variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
 /-- `F ⊣ G` represents the data of an adjunction between two functors
 `F : C ⥤ D` and `G : D ⥤ C`. `F` is the left adjoint and `G` is the right adjoint.
 
-To construct an `adjunction` between two functors, it's often easier to instead use the
-constructors `mkOfHomEquiv` or `mkOfUnitCounit`. To construct a left adjoint,
-there are also constructors `leftAdjointOfEquiv` and `adjunctionOfEquivLeft` (as
-well as their duals) which can be simpler in practice.
+We use the unit-counit definition of an adjunction. There is a constructor `Adjunction.mk'`
+which constructs an adjunction from the data of a hom set equivalence, a unit, and a counit,
+together with proofs of the equalities `homEquiv_unit` and `homEquiv_counit` relating them to each
+other.
+
+There is also a constructor `Adjunction.mkOfHomEquiv` which constructs an adjunction from a natural
+hom set equivalence.
 
-Uniqueness of adjoints is shown in `CategoryTheory.Adjunction.Unique`.
+To construct adjoints to a given functor, there are constructors `leftAdjointOfEquiv` and
+`adjunctionOfEquivLeft` (as well as their duals).
 
 See <https://stacks.math.columbia.edu/tag/0037>.
 -/
 structure Adjunction (F : C ⥤ D) (G : D ⥤ C) where
-  /-- The equivalence between `Hom (F X) Y` and `Hom X (G Y)` coming from an adjunction -/
-  homEquiv : ∀ X Y, (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y)
   /-- The unit of an adjunction -/
   unit : 𝟭 C ⟶ F.comp G
   /-- The counit of an adjunction -/
   counit : G.comp F ⟶ 𝟭 D
-  /-- The relationship between the unit and hom set equivalence of an adjunction -/
-  homEquiv_unit : ∀ {X Y f}, (homEquiv X Y) f = unit.app X ≫ G.map f := by aesop_cat
-  /-- The relationship between the counit and hom set equivalence of an adjunction -/
-  homEquiv_counit : ∀ {X Y g}, (homEquiv X Y).symm g = F.map g ≫ counit.app Y := by aesop_cat
+  /-- Equality of the composition of the unit and counit with the identity `F ⟶ FGF ⟶ F = 𝟙` -/
+  left_triangle_components (X : C) :
+      F.map (unit.app X) ≫ counit.app (F.obj X) = 𝟙 (F.obj X) := by aesop_cat
+  /-- Equality of the composition of the unit and counit with the identity `G ⟶ GFG ⟶ G = 𝟙` -/
+  right_triangle_components (Y : D) :
+      unit.app (G.obj Y) ≫ G.map (counit.app Y) = 𝟙 (G.obj Y) := by aesop_cat
 
 /-- The notation `F ⊣ G` stands for `Adjunction F G` representing that `F` is left adjoint to `G` -/
 infixl:15 " ⊣ " => Adjunction
@@ -103,6 +148,23 @@ noncomputable def Adjunction.ofIsRightAdjoint (right : C ⥤ D) [right.IsRightAd
 
 namespace Adjunction
 
+attribute [reassoc (attr := simp)] left_triangle_components right_triangle_components
+
+/-- The hom set equivalence associated to an adjunction. -/
+@[simps]
+def homEquiv {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) (X : C) (Y : D) :
+    (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y) where
+  toFun := fun f => adj.unit.app X ≫ G.map f
+  invFun := fun g => F.map g ≫ adj.counit.app Y
+  left_inv := fun f => by
+    dsimp
+    rw [F.map_comp, assoc, ← Functor.comp_map, adj.counit.naturality, ← assoc]
+    simp
+  right_inv := fun g => by
+    simp [← assoc, ← Functor.comp_map, ← adj.unit.naturality, assoc]
+
+alias homEquiv_unit := homEquiv_apply
+alias homEquiv_counit := homEquiv_symm_apply
 attribute [simp] homEquiv_unit homEquiv_counit
 
 section
@@ -173,29 +235,11 @@ theorem homEquiv_naturality_right_square_iff (f : X' ⟶ X) (g : X ⟶ G.obj Y')
 
 @[simp]
 theorem left_triangle : whiskerRight adj.unit F ≫ whiskerLeft F adj.counit = 𝟙 _ := by
-  ext; dsimp
-  rw [← adj.homEquiv_counit, Equiv.symm_apply_eq, adj.homEquiv_unit]
-  simp
+  ext; simp
 
 @[simp]
 theorem right_triangle : whiskerLeft G adj.unit ≫ whiskerRight adj.counit G = 𝟙 _ := by
-  ext; dsimp
-  rw [← adj.homEquiv_unit, ← Equiv.eq_symm_apply, adj.homEquiv_counit]
-  simp
-
-variable (X Y)
-
-@[reassoc (attr := simp)]
-theorem left_triangle_components :
-    F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X) :=
-  congr_arg (fun t : NatTrans _ (𝟭 C ⋙ F) => t.app X) adj.left_triangle
-
-@[reassoc (attr := simp)]
-theorem right_triangle_components :
-    adj.unit.app (G.obj Y) ≫ G.map (adj.counit.app Y) = 𝟙 (G.obj Y) :=
-  congr_arg (fun t : NatTrans _ (G ⋙ 𝟭 C) => t.app Y) adj.right_triangle
-
-variable {X Y}
+  ext; simp
 
 @[reassoc (attr := simp)]
 theorem counit_naturality {X Y : D} (f : X ⟶ Y) :
@@ -207,21 +251,21 @@ theorem unit_naturality {X Y : C} (f : X ⟶ Y) :
     adj.unit.app X ≫ G.map (F.map f) = f ≫ adj.unit.app Y :=
   (adj.unit.naturality f).symm
 
+lemma unit_comp_map_eq_iff {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
+    adj.unit.app A ≫ G.map f = g ↔ f = F.map g ≫ adj.counit.app B :=
+  ⟨fun h => by simp [← h], fun h => by simp [h]⟩
+
+lemma eq_unit_comp_map_iff {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
+    g = adj.unit.app A ≫ G.map f ↔ F.map g ≫ adj.counit.app B = f :=
+  ⟨fun h => by simp [h], fun h => by simp [← h]⟩
+
 theorem homEquiv_apply_eq {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
     adj.homEquiv A B f = g ↔ f = (adj.homEquiv A B).symm g :=
-  ⟨fun h => by
-    cases h
-    simp, fun h => by
-    cases h
-    simp⟩
+  unit_comp_map_eq_iff adj f g
 
 theorem eq_homEquiv_apply {A : C} {B : D} (f : F.obj A ⟶ B) (g : A ⟶ G.obj B) :
     g = adj.homEquiv A B f ↔ (adj.homEquiv A B).symm g = f :=
-  ⟨fun h => by
-    cases h
-    simp, fun h => by
-    cases h
-    simp⟩
+  eq_unit_comp_map_iff adj f g
 
 end
 
@@ -229,6 +273,28 @@ end Adjunction
 
 namespace Adjunction
 
+/--
+This is an auxiliary data structure useful for constructing adjunctions.
+See `Adjunction.mk'`. This structure won't typically be used anywhere else.
+-/
+structure CoreHomEquivUnitCounit (F : C ⥤ D) (G : D ⥤ C) where
+  /-- The equivalence between `Hom (F X) Y` and `Hom X (G Y)` coming from an adjunction -/
+  homEquiv : ∀ X Y, (F.obj X ⟶ Y) ≃ (X ⟶ G.obj Y)
+  /-- The unit of an adjunction -/
+  unit : 𝟭 C ⟶ F ⋙ G
+  /-- The counit of an adjunction -/
+  counit : G ⋙ F ⟶ 𝟭 D
+  /-- The relationship between the unit and hom set equivalence of an adjunction -/
+  homEquiv_unit : ∀ {X Y f}, (homEquiv X Y) f = unit.app X ≫ G.map f := by aesop_cat
+  /-- The relationship between the counit and hom set equivalence of an adjunction -/
+  homEquiv_counit : ∀ {X Y g}, (homEquiv X Y).symm g = F.map g ≫ counit.app Y := by aesop_cat
+
+namespace CoreHomEquivUnitCounit
+
+attribute [simp] homEquiv_unit homEquiv_counit
+
+end CoreHomEquivUnitCounit
+
 /-- This is an auxiliary data structure useful for constructing adjunctions.
 See `Adjunction.mkOfHomEquiv`.
 This structure won't typically be used anywhere else.
@@ -297,11 +363,30 @@ end CoreUnitCounit
 
 variable {F : C ⥤ D} {G : D ⥤ C}
 
+/--
+Construct an adjunction from the data of a `CoreHomEquivUnitCounit`, i.e. a hom set
+equivalence, unit and counit natural transformations together with proofs of the equalities
+`homEquiv_unit` and `homEquiv_counit` relating them to each other.
+-/
+@[simps]
+def mk' (adj : CoreHomEquivUnitCounit F G) : F ⊣ G where
+  unit := adj.unit
+  counit := adj.counit
+  left_triangle_components X := by
+    rw [← adj.homEquiv_counit, (adj.homEquiv _ _).symm_apply_eq]
+    simp
+  right_triangle_components Y := by
+    rw [← adj.homEquiv_unit, ← (adj.homEquiv _ _).eq_symm_apply]
+    simp
+
+lemma mk'_homEquiv (adj : CoreHomEquivUnitCounit F G) : (mk' adj).homEquiv = adj.homEquiv := by
+  ext; simp
+
 /-- Construct an adjunction between `F` and `G` out of a natural bijection between each
 `F.obj X ⟶ Y` and `X ⟶ G.obj Y`. -/
-@[simps]
+@[simps!]
 def mkOfHomEquiv (adj : CoreHomEquiv F G) : F ⊣ G :=
-  { adj with
+  mk' {
     unit :=
       { app := fun X => (adj.homEquiv X (F.obj X)) (𝟙 (F.obj X))
         naturality := by
@@ -312,38 +397,32 @@ def mkOfHomEquiv (adj : CoreHomEquiv F G) : F ⊣ G :=
         naturality := by
           intros
           simp [← adj.homEquiv_naturality_left_symm, ← adj.homEquiv_naturality_right_symm] }
-    homEquiv_unit := @fun X Y f => by simp [← adj.homEquiv_naturality_right]
-    homEquiv_counit := @fun X Y f => by simp [← adj.homEquiv_naturality_left_symm] }
+    homEquiv := adj.homEquiv
+    homEquiv_unit := fun {X Y f} => by simp [← adj.homEquiv_naturality_right]
+    homEquiv_counit := fun {X Y f} => by simp [← adj.homEquiv_naturality_left_symm] }
+
+lemma mkOfHomEquiv_homEquiv (adj : CoreHomEquiv F G) :
+    (mkOfHomEquiv adj).homEquiv = adj.homEquiv := by
+  ext X Y g
+  simp [mkOfHomEquiv, ← adj.homEquiv_naturality_right (𝟙 _) g]
 
 /-- Construct an adjunction between functors `F` and `G` given a unit and counit for the adjunction
 satisfying the triangle identities. -/
-
 @[simps!]
-def mkOfUnitCounit (adj : CoreUnitCounit F G) : F ⊣ G :=
-  { adj with
-    homEquiv := fun X Y =>
-      { toFun := fun f => adj.unit.app X ≫ G.map f
-        invFun := fun g => F.map g ≫ adj.counit.app Y
-        left_inv := fun f => by
-          change F.map (_ ≫ _) ≫ _ = _
-          rw [F.map_comp, assoc, ← Functor.comp_map, adj.counit.naturality, ← assoc]
-          convert id_comp f
-          have t := congrArg (fun (s : NatTrans (𝟭 C ⋙ F) (F ⋙ 𝟭 D)) => s.app X) adj.left_triangle
-          dsimp at t
-          simp only [id_comp] at t
-          exact t
-        right_inv := fun g => by
-          change _ ≫ G.map (_ ≫ _) = _
-          rw [G.map_comp, ← assoc, ← Functor.comp_map, ← adj.unit.naturality, assoc]
-          convert comp_id g
-          have t := congrArg (fun t : NatTrans (G ⋙ 𝟭 C) (𝟭 D ⋙ G) => t.app Y) adj.right_triangle
-          dsimp at t
-          simp only [id_comp] at t
-          exact t } }
+def mkOfUnitCounit (adj : CoreUnitCounit F G) : F ⊣ G where
+  unit := adj.unit
+  counit := adj.counit
+  left_triangle_components X := by
+    have := adj.left_triangle
+    rw [NatTrans.ext_iff, funext_iff] at this
+    simpa [-CoreUnitCounit.left_triangle] using this X
+  right_triangle_components Y := by
+    have := adj.right_triangle
+    rw [NatTrans.ext_iff, funext_iff] at this
+    simpa [-CoreUnitCounit.right_triangle] using this Y
 
 /-- The adjunction between the identity functor on a category and itself. -/
 def id : 𝟭 C ⊣ 𝟭 C where
-  homEquiv X Y := Equiv.refl _
   unit := 𝟙 _
   counit := 𝟙 _
 
@@ -379,6 +458,22 @@ def ofNatIsoRight {F : C ⥤ D} {G H : D ⥤ C} (adj : F ⊣ G) (iso : G ≅ H)
   Adjunction.mkOfHomEquiv
     { homEquiv := fun X Y => (adj.homEquiv X Y).trans (equivHomsetRightOfNatIso iso) }
 
+/-- The isomorpism which an adjunction `F ⊣ G` induces on `G ⋙ yoneda`. This states that
+`Adjunction.homEquiv` is natural in both arguments. -/
+@[simps!]
+def compYonedaIso {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₁} D]
+    {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
+    G ⋙ yoneda ≅ yoneda ⋙ (whiskeringLeft _ _ _).obj F.op :=
+  NatIso.ofComponents fun X => NatIso.ofComponents fun Y => (adj.homEquiv Y.unop X).toIso.symm
+
+/-- The isomorpism which an adjunction `F ⊣ G` induces on `F.op ⋙ coyoneda`. This states that
+`Adjunction.homEquiv` is natural in both arguments. -/
+@[simps!]
+def compCoyonedaIso {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₁} D]
+    {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
+    F.op ⋙ coyoneda ≅ coyoneda ⋙ (whiskeringLeft _ _ _).obj G :=
+  NatIso.ofComponents fun X => NatIso.ofComponents fun Y => (adj.homEquiv X.unop Y).toIso
+
 section
 
 variable {E : Type u₃} [ℰ : Category.{v₃} E] {H : D ⥤ E} {I : E ⥤ D}
@@ -388,11 +483,12 @@ variable {E : Type u₃} [ℰ : Category.{v₃} E] {H : D ⥤ E} {I : E ⥤ D}
 
 See <https://stacks.math.columbia.edu/tag/0DV0>.
 -/
-def comp : F ⋙ H ⊣ I ⋙ G where
-  homEquiv X Z := Equiv.trans (adj₂.homEquiv _ _) (adj₁.homEquiv _ _)
-  unit := adj₁.unit ≫ (whiskerLeft F <| whiskerRight adj₂.unit G) ≫ (Functor.associator _ _ _).inv
-  counit :=
-    (Functor.associator _ _ _).hom ≫ (whiskerLeft I <| whiskerRight adj₁.counit H) ≫ adj₂.counit
+def comp : F ⋙ H ⊣ I ⋙ G :=
+  mk' {
+    homEquiv := fun _ _ ↦ Equiv.trans (adj₂.homEquiv _ _) (adj₁.homEquiv _ _)
+    unit := adj₁.unit ≫ (whiskerLeft F <| whiskerRight adj₂.unit G) ≫ (Functor.associator _ _ _).inv
+    counit :=
+      (Functor.associator _ _ _).hom ≫ (whiskerLeft I <| whiskerRight adj₁.counit H) ≫ adj₂.counit }
 
 @[simp, reassoc]
 lemma comp_unit_app (X : C) :
@@ -404,6 +500,10 @@ lemma comp_counit_app (X : E) :
     (adj₁.comp adj₂).counit.app X = H.map (adj₁.counit.app (I.obj X)) ≫ adj₂.counit.app X := by
   simp [Adjunction.comp]
 
+lemma comp_homEquiv :  (adj₁.comp adj₂).homEquiv =
+    fun _ _ ↦ Equiv.trans (adj₂.homEquiv _ _) (adj₁.homEquiv _ _) :=
+  mk'_homEquiv _
+
 end
 
 section ConstructLeft
@@ -518,9 +618,10 @@ variable (e : C ≌ D)
 
 /-- The adjunction given by an equivalence of categories. (To obtain the opposite adjunction,
 simply use `e.symm.toAdjunction`. -/
-@[simps! unit counit]
-def toAdjunction : e.functor ⊣ e.inverse :=
-  mkOfUnitCounit ⟨e.unit, e.counit, by ext; simp, by ext; simp⟩
+@[simps]
+def toAdjunction : e.functor ⊣ e.inverse where
+  unit := e.unit
+  counit := e.counit
 
 lemma isLeftAdjoint_functor : e.functor.IsLeftAdjoint where
   exists_rightAdjoint := ⟨_, ⟨e.toAdjunction⟩⟩
diff --git a/Mathlib/CategoryTheory/Adjunction/Comma.lean b/Mathlib/CategoryTheory/Adjunction/Comma.lean
index 07015c84df266..b74e9b326b43d 100644
--- a/Mathlib/CategoryTheory/Adjunction/Comma.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Comma.lean
@@ -132,10 +132,7 @@ def mkInitialOfLeftAdjoint (h : F ⊣ G) (A : C) :
   desc B := StructuredArrow.homMk ((h.homEquiv _ _).symm B.pt.hom)
   uniq s m _ := by
     apply StructuredArrow.ext
-    dsimp
-    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-    erw [Equiv.eq_symm_apply, Adjunction.homEquiv_unit]
-    apply StructuredArrow.w m
+    simp [← StructuredArrow.w m]
 
 /-- Given a right adjoint to `F`, we can construct a terminal object in each costructured arrow
 category on `F`. -/
@@ -144,9 +141,7 @@ def mkTerminalOfRightAdjoint (h : F ⊣ G) (A : D) :
   lift B := CostructuredArrow.homMk (h.homEquiv _ _ B.pt.hom)
   uniq s m _ := by
     apply CostructuredArrow.ext
-    dsimp
-    rw [h.eq_homEquiv_apply, Adjunction.homEquiv_counit]
-    exact CostructuredArrow.w m
+    simp [← CostructuredArrow.w m]
 
 end
 
diff --git a/Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean b/Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean
index c1e9ac499f5a3..2bdec90917b38 100644
--- a/Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean
+++ b/Mathlib/CategoryTheory/Adjunction/FullyFaithful.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Dagur Asgeirsson
+Authors: Kim Morrison, Dagur Asgeirsson
 -/
 import Mathlib.CategoryTheory.Adjunction.Basic
 import Mathlib.CategoryTheory.MorphismProperty.Basic
diff --git a/Mathlib/CategoryTheory/Adjunction/Lifting.lean b/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
similarity index 86%
rename from Mathlib/CategoryTheory/Adjunction/Lifting.lean
rename to Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
index d175dc78fa3e1..e4d139ca0cb2b 100644
--- a/Mathlib/CategoryTheory/Adjunction/Lifting.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
@@ -3,8 +3,6 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta
 -/
-import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
-import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
 import Mathlib.CategoryTheory.Monad.Adjunction
 import Mathlib.CategoryTheory.Monad.Coequalizer
 
@@ -13,23 +11,26 @@ import Mathlib.CategoryTheory.Monad.Coequalizer
 
 This file gives two constructions for building left adjoints: the adjoint triangle theorem and the
 adjoint lifting theorem.
+
 The adjoint triangle theorem concerns a functor `U : B ⥤ C` with a left adjoint `F` such
 that `ε_X : FUX ⟶ X` is a regular epi. Then for any category `A` with coequalizers of reflexive
 pairs, a functor `R : A ⥤ B` has a left adjoint if (and only if) the composite `R ⋙ U` does.
 Note that the condition on `U` regarding `ε_X` is automatically satisfied in the case when `U` is
-a monadic functor, giving the corollary: `monadicAdjointTriangleLift`, i.e. if `U` is monadic,
-`A` has reflexive coequalizers then `R : A ⥤ B` has a left adjoint provided `R ⋙ U` does.
+a monadic functor, giving the corollary: `isRightAdjoint_triangle_lift_monadic`, i.e. if `U` is
+monadic, `A` has reflexive coequalizers then `R : A ⥤ B` has a left adjoint provided `R ⋙ U` does.
 
 The adjoint lifting theorem says that given a commutative square of functors (up to isomorphism):
 
+```
       Q
     A → B
   U ↓   ↓ V
     C → D
       R
+```
 
-where `U` and `V` are monadic and `A` has reflexive coequalizers, then if `R` has a left adjoint
-then `Q` has a left adjoint.
+where `V` is monadic, `U` has a left adjoint, and `A` has reflexive coequalizers, then if `R` has a
+left adjoint then `Q` has a left adjoint.
 
 ## Implementation
 
@@ -38,10 +39,14 @@ than just a functor known to be a right adjoint. In docstrings, we write `(η, 
 and counit of the adjunction `adj₁ : F ⊣ U` and `(ι, δ)` for the unit and counit of the adjunction
 `adj₂ : F' ⊣ R ⋙ U`.
 
+This file has been adapted to `Mathlib.CategoryTheory.Adjunction.Lifting.Right`.
+Please try to keep them in sync.
+
 ## TODO
 
-Dualise to lift right adjoints through comonads (by reversing 1-cells) and dualise to lift right
-adjoints through monads (by reversing 2-cells), and the combination.
+- Dualise to lift right adjoints through monads (by reversing 2-cells).
+- Investigate whether it is possible to give a more explicit description of the lifted adjoint,
+  especially in the case when the isomorphism `comm` is `Iso.refl _`
 
 ## References
 * https://ncatlab.org/nlab/show/adjoint+triangle+theorem
@@ -61,7 +66,7 @@ variable {A : Type u₁} {B : Type u₂} {C : Type u₃}
 variable [Category.{v₁} A] [Category.{v₂} B] [Category.{v₃} C]
 
 -- Hide implementation details in this namespace
-namespace LiftAdjoint
+namespace LiftLeftAdjoint
 
 variable {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (F' : C ⥤ A)
 variable (adj₁ : F ⊣ U) (adj₂ : F' ⊣ R ⋙ U)
@@ -96,7 +101,7 @@ def otherMap (X) : F'.obj (U.obj (F.obj (U.obj X))) ⟶ F'.obj (U.obj X) :=
   F'.map (U.map (F.map (adj₂.unit.app _) ≫ adj₁.counit.app _)) ≫ adj₂.counit.app _
 
 /-- `(F'Uε_X, otherMap X)` is a reflexive pair: in particular if `A` has reflexive coequalizers then
-it has a coequalizer.
+this pair has a coequalizer.
 -/
 instance (X : B) :
     IsReflexivePair (F'.map (U.map (adj₁.counit.app X))) (otherMap _ _ adj₁ adj₂ X) :=
@@ -159,7 +164,7 @@ noncomputable def constructLeftAdjoint [∀ X : B, RegularEpi (adj₁.counit.app
   -- This used to be `simp`, but we need `aesop_cat` after leanprover/lean4#2644
   aesop_cat
 
-end LiftAdjoint
+end LiftLeftAdjoint
 
 /-- The adjoint triangle theorem: Suppose `U : B ⥤ C` has a left adjoint `F` such that each counit
 `ε_X : FUX ⟶ X` is a regular epimorphism. Then if a category `A` has coequalizers of reflexive
@@ -168,18 +173,18 @@ pairs, then a functor `R : A ⥤ B` has a left adjoint if the composite `R ⋙ U
 Note the converse is true (with weaker assumptions), by `Adjunction.comp`.
 See https://ncatlab.org/nlab/show/adjoint+triangle+theorem
 -/
-lemma adjointTriangleLift {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (adj₁ : F ⊣ U)
+lemma isRightAdjoint_triangle_lift {U : B ⥤ C} {F : C ⥤ B} (R : A ⥤ B) (adj₁ : F ⊣ U)
     [∀ X : B, RegularEpi (adj₁.counit.app X)] [HasReflexiveCoequalizers A]
     [(R ⋙ U).IsRightAdjoint ] : R.IsRightAdjoint where
   exists_leftAdjoint :=
-    ⟨LiftAdjoint.constructLeftAdjoint R _ adj₁ (Adjunction.ofIsRightAdjoint _),
+    ⟨LiftLeftAdjoint.constructLeftAdjoint R _ adj₁ (Adjunction.ofIsRightAdjoint _),
       ⟨Adjunction.adjunctionOfEquivLeft _ _⟩⟩
 
 /-- If `R ⋙ U` has a left adjoint, the domain of `R` has reflexive coequalizers and `U` is a monadic
 functor, then `R` has a left adjoint.
-This is a special case of `adjointTriangleLift` which is often more useful in practice.
+This is a special case of `isRightAdjoint_triangle_lift` which is often more useful in practice.
 -/
-lemma monadicAdjointTriangleLift (U : B ⥤ C) [MonadicRightAdjoint U] {R : A ⥤ B}
+lemma isRightAdjoint_triangle_lift_monadic (U : B ⥤ C) [MonadicRightAdjoint U] {R : A ⥤ B}
     [HasReflexiveCoequalizers A] [(R ⋙ U).IsRightAdjoint] : R.IsRightAdjoint := by
   let R' : A ⥤ _ := R ⋙ Monad.comparison (monadicAdjunction U)
   rsuffices : R'.IsRightAdjoint
@@ -195,18 +200,20 @@ lemma monadicAdjointTriangleLift (U : B ⥤ C) [MonadicRightAdjoint U] {R : A 
     intro X
     simp only [Monad.adj_counit]
     exact ⟨_, _, _, _, Monad.beckAlgebraCoequalizer X⟩
-  exact adjointTriangleLift R' (Monad.adj _)
+  exact isRightAdjoint_triangle_lift R' (Monad.adj _)
 
 variable {D : Type u₄}
 variable [Category.{v₄} D]
 
 /-- Suppose we have a commutative square of functors
 
+```
       Q
     A → B
   U ↓   ↓ V
     C → D
       R
+```
 
 where `U` has a left adjoint, `A` has reflexive coequalizers and `V` has a left adjoint such that
 each component of the counit is a regular epi.
@@ -214,30 +221,32 @@ Then `Q` has a left adjoint if `R` has a left adjoint.
 
 See https://ncatlab.org/nlab/show/adjoint+lifting+theorem
 -/
-lemma adjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
+lemma isRightAdjoint_square_lift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
     (comm : U ⋙ R ≅ Q ⋙ V) [U.IsRightAdjoint] [V.IsRightAdjoint] [R.IsRightAdjoint]
     [∀ X, RegularEpi ((Adjunction.ofIsRightAdjoint V).counit.app X)] [HasReflexiveCoequalizers A] :
     Q.IsRightAdjoint :=
   have := ((Adjunction.ofIsRightAdjoint (U ⋙ R)).ofNatIsoRight comm).isRightAdjoint
-  adjointTriangleLift Q (Adjunction.ofIsRightAdjoint V)
+  isRightAdjoint_triangle_lift Q (Adjunction.ofIsRightAdjoint V)
 
 /-- Suppose we have a commutative square of functors
 
+```
       Q
     A → B
   U ↓   ↓ V
     C → D
       R
+```
 
 where `U` has a left adjoint, `A` has reflexive coequalizers and `V` is monadic.
 Then `Q` has a left adjoint if `R` has a left adjoint.
 
 See https://ncatlab.org/nlab/show/adjoint+lifting+theorem
 -/
-lemma monadicAdjointSquareLift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
+lemma isRightAdjoint_square_lift_monadic (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (R : C ⥤ D)
     (comm : U ⋙ R ≅ Q ⋙ V) [U.IsRightAdjoint] [MonadicRightAdjoint V] [R.IsRightAdjoint]
     [HasReflexiveCoequalizers A] : Q.IsRightAdjoint :=
   have := ((Adjunction.ofIsRightAdjoint (U ⋙ R)).ofNatIsoRight comm).isRightAdjoint
-  monadicAdjointTriangleLift V
+  isRightAdjoint_triangle_lift_monadic V
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Adjunction/Lifting/Right.lean b/Mathlib/CategoryTheory/Adjunction/Lifting/Right.lean
new file mode 100644
index 0000000000000..3d3893abeac50
--- /dev/null
+++ b/Mathlib/CategoryTheory/Adjunction/Lifting/Right.lean
@@ -0,0 +1,245 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.CategoryTheory.Monad.Adjunction
+import Mathlib.CategoryTheory.Monad.Equalizer
+
+/-!
+# Adjoint lifting
+
+This file gives two constructions for building right adjoints: the adjoint triangle theorem and the
+adjoint lifting theorem.
+
+The adjoint triangle theorem concerns a functor `F : B ⥤ A` with a right adjoint `U` such
+that `η_X : X ⟶ UFX` is a regular mono. Then for any category `C` with equalizers of coreflexive
+pairs, a functor `L : C ⥤ B` has a right adjoint if (and only if) the composite `L ⋙ F` does.
+Note that the condition on `F` regarding `η_X` is automatically satisfied in the case when `F` is
+a comonadic functor, giving the corollary: `isLeftAdjoint_triangle_lift_comonadic`, i.e. if `F` is
+comonadic, `C` has coreflexive equalizers then `L : C ⥤ B` has a right adjoint provided `L ⋙ F`
+does.
+
+The adjoint lifting theorem says that given a commutative square of functors (up to isomorphism):
+
+```
+      Q
+    A → B
+  U ↓   ↓ V
+    C → D
+      L
+```
+
+where `V` is comonadic, `U` has a right adjoint, and `A` has coreflexive equalizers, then if `L` has
+a right adjoint then `Q` has a right adjoint.
+
+## Implementation
+
+It is more convenient to prove this theorem by assuming we are given the explicit adjunction rather
+than just a functor known to be a right adjoint. In docstrings, we write `(η, ε)` for the unit
+and counit of the adjunction `adj₁ : F ⊣ U` and `(ι, δ)` for the unit and counit of the adjunction
+`adj₂ : L ⋙ F ⊣ U'`.
+
+This file has been adapted from `Mathlib.CategoryTheory.Adjunction.Lifting.Left`.
+Please try to keep them in sync.
+
+## TODO
+
+- Dualise to lift left adjoints through comonads (by reversing 2-cells).
+- Investigate whether it is possible to give a more explicit description of the lifted adjoint,
+  especially in the case when the isomorphism `comm` is `Iso.refl _`
+
+## References
+* https://ncatlab.org/nlab/show/adjoint+triangle+theorem
+* https://ncatlab.org/nlab/show/adjoint+lifting+theorem
+* Adjoint Lifting Theorems for Categories of Algebras (PT Johnstone, 1975)
+* A unified approach to the lifting of adjoints (AJ Power, 1988)
+-/
+
+
+namespace CategoryTheory
+
+open Category Limits
+
+universe v₁ v₂ v₃ v₄ u₁ u₂ u₃ u₄
+
+variable {A : Type u₁} {B : Type u₂} {C : Type u₃}
+variable [Category.{v₁} A] [Category.{v₂} B] [Category.{v₃} C]
+
+-- Hide implementation details in this namespace
+namespace LiftRightAdjoint
+
+variable {U : A ⥤ B} {F : B ⥤ A} (L : C ⥤ B) (U' : A ⥤ C)
+variable (adj₁ : F ⊣ U) (adj₂ : L ⋙ F ⊣ U')
+
+/-- To show that `η_X` is an equalizer for `(UFη_X, η_UFX)`, it suffices to assume it's always an
+equalizer of something (i.e. a regular mono).
+-/
+def unitEqualises [∀ X : B, RegularMono (adj₁.unit.app X)] (X : B) :
+    IsLimit (Fork.ofι (adj₁.unit.app X) (adj₁.unit_naturality _)) :=
+  Fork.IsLimit.mk' _ fun s => by
+    refine ⟨(RegularMono.lift' (adj₁.unit.app X) s.ι ?_).1, ?_, ?_⟩
+    · rw [← cancel_mono (adj₁.unit.app (RegularMono.Z (adj₁.unit.app X)))]
+      rw [assoc, ← adj₁.unit_naturality RegularMono.left]
+      dsimp only [Functor.comp_obj]
+      erw [← assoc, ← s.condition, assoc, ← U.map_comp, ← F.map_comp, RegularMono.w, F.map_comp,
+        U.map_comp, s.condition_assoc, assoc, ← adj₁.unit_naturality RegularMono.right]
+      rfl
+    · apply (RegularMono.lift' (adj₁.unit.app X) s.ι _).2
+    · intro m hm
+      rw [← cancel_mono (adj₁.unit.app X)]
+      apply hm.trans (RegularMono.lift' (adj₁.unit.app X) s.ι _).2.symm
+
+/-- (Implementation)
+To construct the right adjoint, we use the equalizer of `U' F η_X` with the composite
+
+`U' F X ⟶ U' F L U' F X ⟶ U' F U F L U' F X ⟶ U' F U F X`
+
+where the first morphism is `ι_U'FX`, the second is `U' F η_LU'FX` and the third is `U' F U δ_FX`.
+We will show that this equalizer exists and that it forms the object map for a right adjoint to `L`.
+-/
+def otherMap (X : B) : U'.obj (F.obj X) ⟶  U'.obj (F.obj (U.obj (F.obj X))) :=
+  adj₂.unit.app _ ≫ U'.map (F.map (adj₁.unit.app _ ≫ (U.map (adj₂.counit.app _))))
+
+/-- `(U'Fη_X, otherMap X)` is a coreflexive pair: in particular if `C` has coreflexive equalizers
+then this pair has an equalizer.
+-/
+instance (X : B) :
+    IsCoreflexivePair (U'.map (F.map (adj₁.unit.app X))) (otherMap _ _ adj₁ adj₂ X) :=
+  IsCoreflexivePair.mk' (U'.map (adj₁.counit.app (F.obj X)))
+    (by simp [← Functor.map_comp])
+    (by simp only [otherMap, assoc, ← Functor.map_comp]; simp)
+
+variable [HasCoreflexiveEqualizers C]
+
+/-- Construct the object part of the desired right adjoint as the equalizer of `U'Fη_Y` with
+`otherMap`.
+-/
+noncomputable def constructRightAdjointObj (Y : B) : C :=
+  equalizer (U'.map (F.map (adj₁.unit.app Y))) (otherMap _ _ adj₁ adj₂ Y)
+
+/-- The homset equivalence which helps show that `L` is a left adjoint. -/
+@[simps!]
+noncomputable def constructRightAdjointEquiv [∀ X : B, RegularMono (adj₁.unit.app X)] (Y : C)
+    (X : B) : (Y ⟶ constructRightAdjointObj _ _ adj₁ adj₂ X) ≃ (L.obj Y ⟶ X) :=
+  calc
+    (Y ⟶ constructRightAdjointObj _ _ adj₁ adj₂ X) ≃
+        { f : Y ⟶ U'.obj (F.obj X) //
+          f ≫ U'.map (F.map (adj₁.unit.app X)) = f ≫ (otherMap _ _ adj₁ adj₂ X) } :=
+      Fork.IsLimit.homIso (limit.isLimit _) _
+    _ ≃ { g : F.obj (L.obj Y) ⟶ F.obj X // F.map (adj₁.unit.app _≫ U.map g) =
+        g ≫ F.map (adj₁.unit.app _) } := by
+      apply (adj₂.homEquiv _ _).symm.subtypeEquiv _
+      intro f
+      rw [← (adj₂.homEquiv _ _).injective.eq_iff, eq_comm, otherMap,
+        ← adj₂.homEquiv_naturality_right_symm, adj₂.homEquiv_unit, ← adj₂.unit_naturality_assoc,
+        adj₂.homEquiv_counit]
+      simp
+    _ ≃ { z : L.obj Y ⟶ U.obj (F.obj X) //
+        z ≫ U.map (F.map (adj₁.unit.app X)) = z ≫ adj₁.unit.app (U.obj (F.obj X)) } := by
+      apply (adj₁.homEquiv _ _).subtypeEquiv
+      intro g
+      rw [← (adj₁.homEquiv _ _).injective.eq_iff, adj₁.homEquiv_unit,
+        adj₁.homEquiv_unit, adj₁.homEquiv_unit, eq_comm]
+      simp
+    _ ≃ (L.obj Y ⟶ X) := (Fork.IsLimit.homIso (unitEqualises adj₁ X) _).symm
+
+/-- Construct the right adjoint to `L`, with object map `constructRightAdjointObj`. -/
+noncomputable def constructRightAdjoint [∀ X : B, RegularMono (adj₁.unit.app X)] : B ⥤ C := by
+  refine Adjunction.rightAdjointOfEquiv
+    (fun X Y => (constructRightAdjointEquiv L _ adj₁ adj₂ X Y).symm) ?_
+  intro X Y Y' g h
+  rw [constructRightAdjointEquiv_symm_apply, constructRightAdjointEquiv_symm_apply,
+    Equiv.symm_apply_eq, Subtype.ext_iff]
+  dsimp
+  erw [Fork.IsLimit.homIso_natural, Fork.IsLimit.homIso_natural]
+  simp only [Fork.ofι_pt, Functor.map_comp, assoc, limit.cone_x]
+  erw [adj₂.homEquiv_naturality_left, Equiv.rightInverse_symm]
+  simp
+
+end LiftRightAdjoint
+
+/-- The adjoint triangle theorem: Suppose `U : A ⥤ B` has a left adjoint `F` such that each unit
+`η_X : X ⟶ UFX` is a regular monomorphism. Then if a category `C` has equalizers of coreflexive
+pairs, then a functor `L : C ⥤ B` has a right adjoint if the composite `L ⋙ F` does.
+
+Note the converse is true (with weaker assumptions), by `Adjunction.comp`.
+See https://ncatlab.org/nlab/show/adjoint+triangle+theorem
+-/
+lemma isLeftAdjoint_triangle_lift {U : A ⥤ B} {F : B ⥤ A} (L : C ⥤ B) (adj₁ : F ⊣ U)
+    [∀ X, RegularMono (adj₁.unit.app X)] [HasCoreflexiveEqualizers C]
+    [(L ⋙ F).IsLeftAdjoint ] : L.IsLeftAdjoint where
+  exists_rightAdjoint :=
+    ⟨LiftRightAdjoint.constructRightAdjoint L _ adj₁ (Adjunction.ofIsLeftAdjoint _),
+      ⟨Adjunction.adjunctionOfEquivRight _ _⟩⟩
+
+/-- If `L ⋙ F` has a right adjoint, the domain of `L` has coreflexive equalizers and `F` is a
+comonadic functor, then `L` has a right adjoint.
+This is a special case of `isLeftAdjoint_triangle_lift` which is often more useful in practice.
+-/
+lemma isLeftAdjoint_triangle_lift_comonadic (F : B ⥤ A) [ComonadicLeftAdjoint F] {L : C ⥤ B}
+    [HasCoreflexiveEqualizers C] [(L ⋙ F).IsLeftAdjoint] : L.IsLeftAdjoint := by
+  let L' : _ ⥤ _ := L ⋙ Comonad.comparison (comonadicAdjunction F)
+  rsuffices : L'.IsLeftAdjoint
+  · let this : (L' ⋙ (Comonad.comparison (comonadicAdjunction F)).inv).IsLeftAdjoint := by
+      infer_instance
+    refine ((Adjunction.ofIsLeftAdjoint
+      (L' ⋙ (Comonad.comparison (comonadicAdjunction F)).inv)).ofNatIsoLeft ?_).isLeftAdjoint
+    exact isoWhiskerLeft L (Comonad.comparison _).asEquivalence.unitIso.symm ≪≫ L.leftUnitor
+  let this : (L' ⋙ Comonad.forget (comonadicAdjunction F).toComonad).IsLeftAdjoint := by
+    refine ((Adjunction.ofIsLeftAdjoint (L ⋙ F)).ofNatIsoLeft ?_).isLeftAdjoint
+    exact isoWhiskerLeft L (Comonad.comparisonForget (comonadicAdjunction F)).symm
+  let this : ∀ X, RegularMono ((Comonad.adj (comonadicAdjunction F).toComonad).unit.app X) := by
+    intro X
+    simp only [Comonad.adj_unit]
+    exact ⟨_, _, _, _, Comonad.beckCoalgebraEqualizer X⟩
+  exact isLeftAdjoint_triangle_lift L' (Comonad.adj _)
+
+variable {D : Type u₄}
+variable [Category.{v₄} D]
+
+/-- Suppose we have a commutative square of functors
+
+```
+      Q
+    A → B
+  U ↓   ↓ V
+    C → D
+      R
+```
+
+where `U` has a right adjoint, `A` has coreflexive equalizers and `V` has a right adjoint such that
+each component of the counit is a regular mono.
+Then `Q` has a right adjoint if `L` has a right adjoint.
+
+See https://ncatlab.org/nlab/show/adjoint+lifting+theorem
+-/
+lemma isLeftAdjoint_square_lift (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (L : C ⥤ D)
+    (comm : U ⋙ L ≅ Q ⋙ V) [U.IsLeftAdjoint] [V.IsLeftAdjoint] [L.IsLeftAdjoint]
+    [∀ X, RegularMono ((Adjunction.ofIsLeftAdjoint V).unit.app X)] [HasCoreflexiveEqualizers A] :
+    Q.IsLeftAdjoint :=
+  have := ((Adjunction.ofIsLeftAdjoint (U ⋙ L)).ofNatIsoLeft comm).isLeftAdjoint
+  isLeftAdjoint_triangle_lift Q (Adjunction.ofIsLeftAdjoint V)
+
+/-- Suppose we have a commutative square of functors
+
+```
+      Q
+    A → B
+  U ↓   ↓ V
+    C → D
+      R
+```
+
+where `U` has a right adjoint, `A` has reflexive equalizers and `V` is comonadic.
+Then `Q` has a right adjoint if `L` has a right adjoint.
+
+See https://ncatlab.org/nlab/show/adjoint+lifting+theorem
+-/
+lemma isLeftAdjoint_square_lift_comonadic (Q : A ⥤ B) (V : B ⥤ D) (U : A ⥤ C) (L : C ⥤ D)
+    (comm : U ⋙ L ≅ Q ⋙ V) [U.IsLeftAdjoint] [ComonadicLeftAdjoint V] [L.IsLeftAdjoint]
+    [HasCoreflexiveEqualizers A] : Q.IsLeftAdjoint :=
+  have := ((Adjunction.ofIsLeftAdjoint (U ⋙ L)).ofNatIsoLeft comm).isLeftAdjoint
+  isLeftAdjoint_triangle_lift_comonadic V
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Adjunction/Limits.lean b/Mathlib/CategoryTheory/Adjunction/Limits.lean
index e66f18e83e81d..6a2f33e8ce6ad 100644
--- a/Mathlib/CategoryTheory/Adjunction/Limits.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Limits.lean
@@ -74,10 +74,9 @@ def functorialityCounit :
   app c := { hom := adj.counit.app c.pt }
 
 /-- The functor `Cocones.functoriality K F : Cocone K ⥤ Cocone (K ⋙ F)` is a left adjoint. -/
-def functorialityAdjunction : Cocones.functoriality K F ⊣ functorialityRightAdjoint adj K :=
-  mkOfUnitCounit
-    { unit := functorialityUnit adj K
-      counit := functorialityCounit adj K}
+def functorialityAdjunction : Cocones.functoriality K F ⊣ functorialityRightAdjoint adj K where
+  unit := functorialityUnit adj K
+  counit := functorialityCounit adj K
 
 /-- A left adjoint preserves colimits.
 
@@ -91,6 +90,11 @@ def leftAdjointPreservesColimits : PreservesColimitsOfSize.{v, u} F where
               @Equiv.unique _ _ (IsColimit.isoUniqueCoconeMorphism.hom hc _)
                 ((adj.functorialityAdjunction _).homEquiv _ _) } }
 
+noncomputable
+instance colimPreservesColimits [HasColimitsOfShape J C] :
+    PreservesColimits (colim (J := J) (C := C)) :=
+  colimConstAdj.leftAdjointPreservesColimits
+
 -- see Note [lower instance priority]
 noncomputable instance (priority := 100) isEquivalencePreservesColimits
     (E : C ⥤ D) [E.IsEquivalence] :
@@ -178,10 +182,9 @@ def functorialityCounit' :
   app c := { hom := adj.counit.app c.pt }
 
 /-- The functor `Cones.functoriality K G : Cone K ⥤ Cone (K ⋙ G)` is a right adjoint. -/
-def functorialityAdjunction' : functorialityLeftAdjoint adj K ⊣ Cones.functoriality K G :=
-  mkOfUnitCounit
-    { unit := functorialityUnit' adj K
-      counit := functorialityCounit' adj K }
+def functorialityAdjunction' : functorialityLeftAdjoint adj K ⊣ Cones.functoriality K G where
+  unit := functorialityUnit' adj K
+  counit := functorialityCounit' adj K
 
 /-- A right adjoint preserves limits.
 
@@ -195,6 +198,11 @@ def rightAdjointPreservesLimits : PreservesLimitsOfSize.{v, u} G where
               @Equiv.unique _ _ (IsLimit.isoUniqueConeMorphism.hom hc _)
                 ((adj.functorialityAdjunction' _).homEquiv _ _).symm } }
 
+noncomputable
+instance limPreservesLimits [HasLimitsOfShape J C] :
+    PreservesLimits (lim (J := J) (C := C)) :=
+  constLimAdj.rightAdjointPreservesLimits
+
 -- see Note [lower instance priority]
 noncomputable instance (priority := 100) isEquivalencePreservesLimits
     (E : D ⥤ C) [E.IsEquivalence] :
diff --git a/Mathlib/CategoryTheory/Adjunction/Mates.lean b/Mathlib/CategoryTheory/Adjunction/Mates.lean
index 2e15c2012bd9e..ca344b3aa3796 100644
--- a/Mathlib/CategoryTheory/Adjunction/Mates.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Mates.lean
@@ -13,11 +13,13 @@ import Mathlib.Tactic.ApplyFun
 
 This file establishes the bijection between the 2-cells
 
+```
          L₁                  R₁
       C --→ D             C ←-- D
     G ↓  ↗  ↓ H         G ↓  ↘  ↓ H
       E --→ F             E ←-- F
          L₂                  R₂
+```
 
 where `L₁ ⊣ R₁` and `L₂ ⊣ R₂`. The corresponding natural transformations are called mates.
 
@@ -206,8 +208,8 @@ theorem mateEquiv_hcomp
       rightAdjointSquare.hcomp (mateEquiv adj₁ adj₂ α) (mateEquiv adj₃ adj₄ β) := by
   unfold leftAdjointSquare.hcomp rightAdjointSquare.hcomp mateEquiv Adjunction.comp
   ext c
-  simp only [comp_obj, whiskerLeft_comp, whiskerLeft_twice, whiskerRight_comp, assoc,
-    Equiv.coe_fn_mk, comp_app, whiskerLeft_app, whiskerRight_app, id_obj, associator_inv_app,
+  simp only [comp_obj, mk'_unit, whiskerLeft_comp, whiskerLeft_twice, mk'_counit, whiskerRight_comp,
+    assoc, Equiv.coe_fn_mk, comp_app, whiskerLeft_app, whiskerRight_app, id_obj, associator_inv_app,
     Functor.comp_map, associator_hom_app, map_id, id_comp, whiskerRight_twice]
   slice_rhs 2 4 =>
     rw [← R₂.map_comp, ← R₂.map_comp, ← assoc, ← unit_naturality (adj₄)]
@@ -328,6 +330,7 @@ Furthermore, this bijection preserves (and reflects) isomorphisms, i.e. a transf
 iff its image under the bijection is an iso, see eg `CategoryTheory.conjugateIsoEquiv`.
 This is in contrast to the general case `mateEquiv` which does not in general have this property.
 -/
+@[simps!]
 def conjugateEquiv : (L₂ ⟶ L₁) ≃ (R₁ ⟶ R₂) :=
   calc
     (L₂ ⟶ L₁) ≃ _ := (Iso.homCongr L₂.leftUnitor L₁.rightUnitor).symm
@@ -401,6 +404,7 @@ variable [Category.{v₁} C] [Category.{v₂} D]
 variable {L₁ L₂ L₃ : C ⥤ D} {R₁ R₂ R₃ : D ⥤ C}
 variable (adj₁ : L₁ ⊣ R₁) (adj₂ : L₂ ⊣ R₂) (adj₃ : L₃ ⊣ R₃)
 
+@[simp]
 theorem conjugateEquiv_comp (α : L₂ ⟶ L₁) (β : L₃ ⟶ L₂) :
     conjugateEquiv adj₁ adj₂ α ≫ conjugateEquiv adj₂ adj₃ β =
       conjugateEquiv adj₁ adj₃ (β ≫ α) := by
@@ -414,6 +418,7 @@ theorem conjugateEquiv_comp (α : L₂ ⟶ L₁) (β : L₃ ⟶ L₂) :
   simp only [comp_id, id_comp, assoc, map_comp] at vcompd ⊢
   rw [vcompd]
 
+@[simp]
 theorem conjugateEquiv_symm_comp (α : R₁ ⟶ R₂) (β : R₂ ⟶ R₃) :
     (conjugateEquiv adj₂ adj₃).symm β ≫ (conjugateEquiv adj₁ adj₂).symm α =
       (conjugateEquiv adj₁ adj₃).symm (α ≫ β) := by
@@ -473,9 +478,16 @@ theorem conjugateEquiv_symm_of_iso (α : R₁ ⟶ R₂)
   infer_instance
 
 /-- Thus conjugation defines an equivalence between natural isomorphisms. -/
-noncomputable def conjugateIsoEquiv : (L₂ ≅ L₁) ≃ (R₁ ≅ R₂) where
-  toFun α := asIso (conjugateEquiv adj₁ adj₂ α.hom)
-  invFun β := asIso ((conjugateEquiv adj₁ adj₂).symm β.hom)
+@[simps]
+def conjugateIsoEquiv : (L₂ ≅ L₁) ≃ (R₁ ≅ R₂) where
+  toFun α := {
+    hom := conjugateEquiv adj₁ adj₂ α.hom
+    inv := conjugateEquiv adj₂ adj₁ α.inv
+  }
+  invFun β := {
+    hom := (conjugateEquiv adj₁ adj₂).symm β.hom
+    inv := (conjugateEquiv adj₂ adj₁).symm β.inv
+  }
   left_inv := by aesop_cat
   right_inv := by aesop_cat
 
@@ -505,11 +517,7 @@ theorem iterated_mateEquiv_conjugateEquiv (α : F₁ ⋙ L₂ ⟶ L₁ ⋙ F₂)
       conjugateEquiv (adj₁.comp adj₄) (adj₃.comp adj₂) α := by
   ext d
   unfold conjugateEquiv mateEquiv Adjunction.comp
-  simp only [comp_obj, Equiv.coe_fn_mk, whiskerLeft_comp, whiskerLeft_twice, whiskerRight_comp,
-    assoc, comp_app, whiskerLeft_app, whiskerRight_app, id_obj, Functor.comp_map, Iso.homCongr_symm,
-    Equiv.instTrans_trans, Equiv.trans_apply, Iso.homCongr_apply, Iso.symm_inv, Iso.symm_hom,
-    rightUnitor_inv_app, associator_inv_app, leftUnitor_hom_app, map_id, associator_hom_app,
-    Functor.id_map, comp_id, id_comp]
+  simp
 
 theorem iterated_mateEquiv_conjugateEquiv_symm (α : U₂ ⋙ R₁ ⟶ R₂ ⋙ U₁) :
     (mateEquiv adj₁ adj₂).symm ((mateEquiv adj₄ adj₃).symm α) =
diff --git a/Mathlib/CategoryTheory/Adjunction/Opposites.lean b/Mathlib/CategoryTheory/Adjunction/Opposites.lean
index ea57e651e6b63..f4362585c27a4 100644
--- a/Mathlib/CategoryTheory/Adjunction/Opposites.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Opposites.lean
@@ -35,21 +35,17 @@ def adjointOfOpAdjointOp (F : C ⥤ D) (G : D ⥤ C) (h : G.op ⊣ F.op) : F ⊣
       ((h.homEquiv (Opposite.op Y) (Opposite.op X)).trans (opEquiv _ _)).symm.trans
         (opEquiv _ _)
     homEquiv_naturality_left_symm := by
-      -- Porting note: This proof was handled by `obviously` in mathlib3.
-      intros X' X Y f g
-      dsimp [opEquiv]
-      -- Porting note: Why is `erw` needed here?
-      -- https://github.com/leanprover-community/mathlib4/issues/5164
-      erw [homEquiv_unit, homEquiv_unit]
-      simp
+      -- Porting note: This proof was handled by `obviously` in mathlib3. The only obstruction to
+      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` and
+      -- `homEquiv` aren't firing.
+      intros
+      simp [opEquiv, homEquiv]
     homEquiv_naturality_right := by
-      -- Porting note: This proof was handled by `obviously` in mathlib3.
-      intros X Y Y' f g
-      dsimp [opEquiv]
-      -- Porting note: Why is `erw` needed here?
-      -- https://github.com/leanprover-community/mathlib4/issues/5164
-      erw [homEquiv_counit, homEquiv_counit]
-      simp }
+      -- Porting note: This proof was handled by `obviously` in mathlib3. The only obstruction to
+      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` and
+      -- `homEquiv` aren't firing.
+      intros
+      simp [opEquiv, homEquiv] }
 
 /-- If `G` is adjoint to `F.op` then `F` is adjoint to `G.unop`. -/
 def adjointUnopOfAdjointOp (F : C ⥤ D) (G : Dᵒᵖ ⥤ Cᵒᵖ) (h : G ⊣ F.op) : F ⊣ G.unop :=
@@ -71,12 +67,12 @@ def opAdjointOpOfAdjoint (F : C ⥤ D) (G : D ⥤ C) (h : G ⊣ F) : F.op ⊣ G.
       (opEquiv _ Y).trans ((h.homEquiv _ _).symm.trans (opEquiv X (Opposite.op _)).symm)
     homEquiv_naturality_left_symm := by
       -- Porting note: This proof was handled by `obviously` in mathlib3. The only obstruction to
-      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` aren't firing.
+      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` aren't firing.
       intros
       simp [opEquiv]
     homEquiv_naturality_right := by
       -- Porting note: This proof was handled by `obviously` in mathlib3. The only obstruction to
-      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` aren't firing.
+      -- automation fully kicking in here is that the `@[simps]` lemmas of `opEquiv` aren't firing.
       intros
       simp [opEquiv] }
 
diff --git a/Mathlib/CategoryTheory/Adjunction/Reflective.lean b/Mathlib/CategoryTheory/Adjunction/Reflective.lean
index fd2b0d3f2dcf4..2ddd0414b0f2f 100644
--- a/Mathlib/CategoryTheory/Adjunction/Reflective.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Reflective.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
-import Mathlib.CategoryTheory.Functor.ReflectsIso
+import Mathlib.CategoryTheory.Functor.EpiMono
 import Mathlib.CategoryTheory.HomCongr
 
 /-!
@@ -153,60 +153,20 @@ instance [Reflective i] (X : Functor.EssImageSubcategory i) :
   IsIso (NatTrans.app (reflectorAdjunction i).unit X.obj) :=
 Functor.essImage.unit_isIso X.property
 
--- Porting note: the following auxiliary definition and the next two lemmas were
--- introduced in order to ease the port
-/-- The counit isomorphism of the equivalence `D ≌ i.EssImageSubcategory` given
-by `equivEssImageOfReflective` when the functor `i` is reflective. -/
-def equivEssImageOfReflective_counitIso_app [Reflective i] (X : Functor.EssImageSubcategory i) :
-    ((Functor.essImageInclusion i ⋙ reflector i) ⋙ Functor.toEssImage i).obj X ≅ X := by
-  refine Iso.symm (@asIso _ _ X _ ((reflectorAdjunction i).unit.app X.obj) ?_)
-  refine @isIso_of_reflects_iso _ _ _ _ _ _ _ i.essImageInclusion ?_ _
-  dsimp
-  exact inferInstance
-
-lemma equivEssImageOfReflective_map_counitIso_app_hom [Reflective i]
-    (X : Functor.EssImageSubcategory i) :
-  (Functor.essImageInclusion i).map (equivEssImageOfReflective_counitIso_app X).hom =
-    inv (NatTrans.app (reflectorAdjunction i).unit X.obj) := by
-    simp only [Functor.comp_obj, Functor.essImageInclusion_obj, Functor.toEssImage_obj_obj,
-      equivEssImageOfReflective_counitIso_app, asIso, Iso.symm_mk, Functor.essImageInclusion_map,
-      Functor.id_obj]
-    rfl
-
-lemma equivEssImageOfReflective_map_counitIso_app_inv [Reflective i]
-    (X : Functor.EssImageSubcategory i) :
-  (Functor.essImageInclusion i).map (equivEssImageOfReflective_counitIso_app X).inv =
-    (NatTrans.app (reflectorAdjunction i).unit X.obj) := rfl
-
+-- These attributes are necessary to make automation work in `equivEssImageOfReflective`.
+-- Making them global doesn't break anything elsewhere, but this is enough for now.
+-- TODO: investigate further.
+attribute [local simp 900] Functor.essImageInclusion_map in
+attribute [local ext] Functor.essImage_ext in
 /-- If `i : D ⥤ C` is reflective, the inverse functor of `i ≌ F.essImage` can be explicitly
 defined by the reflector. -/
 @[simps]
 def equivEssImageOfReflective [Reflective i] : D ≌ i.EssImageSubcategory where
   functor := i.toEssImage
   inverse := i.essImageInclusion ⋙ reflector i
-  unitIso :=
-    NatIso.ofComponents (fun X => (asIso <| (reflectorAdjunction i).counit.app X).symm)
-      (by
-        intro X Y f
-        dsimp
-        rw [IsIso.comp_inv_eq, Category.assoc, IsIso.eq_inv_comp]
-        exact ((reflectorAdjunction i).counit.naturality f).symm)
-  counitIso :=
-    NatIso.ofComponents equivEssImageOfReflective_counitIso_app
-      (by
-        intro X Y f
-        apply (Functor.essImageInclusion i).map_injective
-        have h := ((reflectorAdjunction i).unit.naturality f).symm
-        rw [Functor.id_map] at h
-        erw [Functor.map_comp, Functor.map_comp,
-          equivEssImageOfReflective_map_counitIso_app_hom,
-          equivEssImageOfReflective_map_counitIso_app_hom,
-          IsIso.comp_inv_eq, assoc, ← h, IsIso.inv_hom_id_assoc, Functor.comp_map])
-  functor_unitIso_comp := fun X => by
-    -- Porting note: this proof was automatically handled by the automation in mathlib
-    apply (Functor.essImageInclusion i).map_injective
-    erw [Functor.map_comp, equivEssImageOfReflective_map_counitIso_app_hom]
-    aesop_cat
+  unitIso := (asIso <| (reflectorAdjunction i).counit).symm
+  counitIso := Functor.fullyFaithfulCancelRight i.essImageInclusion <|
+    NatIso.ofComponents (fun X ↦ (asIso ((reflectorAdjunction i).unit.app X.obj)).symm)
 
 /--
 A functor is *coreflective*, or *a coreflective inclusion*, if it is fully faithful and left
diff --git a/Mathlib/CategoryTheory/Adjunction/Restrict.lean b/Mathlib/CategoryTheory/Adjunction/Restrict.lean
index e433f14102099..d3e8258ab5062 100644
--- a/Mathlib/CategoryTheory/Adjunction/Restrict.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Restrict.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.Basic
 import Mathlib.CategoryTheory.HomCongr
@@ -69,6 +69,11 @@ lemma restrictFullyFaithful_homEquiv_apply {X : C} {Y : D} (f : L.obj X ⟶ Y) :
     (adj.restrictFullyFaithful hiC hiD comm1 comm2).homEquiv X Y f =
       hiC.preimage (adj.unit.app (iC.obj X) ≫ R'.map (comm1.hom.app X) ≫
         R'.map (iD.map f) ≫ comm2.hom.app Y) := by
-  simp [restrictFullyFaithful]
+  -- This proof was just `simp [restrictFullyFaithful]` before #16317
+  apply hiC.map_injective
+  simp only [homEquiv_apply, Functor.comp_obj, Functor.map_comp, map_restrictFullyFaithful_unit_app,
+    Functor.id_obj, assoc, Functor.FullyFaithful.map_preimage]
+  congr 2
+  exact (comm2.hom.naturality _).symm
 
 end CategoryTheory.Adjunction
diff --git a/Mathlib/CategoryTheory/Adjunction/Triple.lean b/Mathlib/CategoryTheory/Adjunction/Triple.lean
new file mode 100644
index 0000000000000..d1ee7a410b19f
--- /dev/null
+++ b/Mathlib/CategoryTheory/Adjunction/Triple.lean
@@ -0,0 +1,55 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.CategoryTheory.Adjunction.Unique
+import Mathlib.CategoryTheory.Monad.Adjunction
+/-!
+
+# Adjoint triples
+
+This file concerns adjoint triples `F ⊣ G ⊣ H` of functors `F H : C ⥤ D`, `G : D ⥤ C`.
+
+Currently, the only result is that `F` is fully faithful if and only if `H` is fully faithful.
+-/
+
+namespace CategoryTheory.Adjunction
+
+variable {C D : Type*} [Category C] [Category D]
+variable {F H : C ⥤ D} {G : D ⥤ C}
+variable (adj₁ : F ⊣ G) (adj₂ : G ⊣ H)
+
+lemma isIso_unit_iff_isIso_counit : IsIso adj₁.unit ↔ IsIso adj₂.counit := by
+  let adj : F ⋙ G ⊣ H ⋙ G := adj₁.comp adj₂
+  constructor
+  · intro h
+    let idAdj : 𝟭 C ⊣ H ⋙ G := adj.ofNatIsoLeft (asIso adj₁.unit).symm
+    exact adj₂.isIso_counit_of_iso (idAdj.rightAdjointUniq id)
+  · intro h
+    let adjId : F ⋙ G ⊣ 𝟭 C := adj.ofNatIsoRight (asIso adj₂.counit)
+    exact adj₁.isIso_unit_of_iso (adjId.leftAdjointUniq id)
+
+/--
+Given an adjoint triple `F ⊣ G ⊣ H`, the left adjoint `F` is fully faithful if and only if the
+right adjoint `H` is fully faithful.
+-/
+noncomputable def fullyFaithfulEquiv : F.FullyFaithful ≃ H.FullyFaithful where
+  toFun h :=
+    haveI := h.full
+    haveI := h.faithful
+    haveI : IsIso adj₂.counit := by
+      rw [← adj₁.isIso_unit_iff_isIso_counit adj₂]
+      infer_instance
+    adj₂.fullyFaithfulROfIsIsoCounit
+  invFun h :=
+    haveI := h.full
+    haveI := h.faithful
+    haveI : IsIso adj₁.unit := by
+      rw [adj₁.isIso_unit_iff_isIso_counit adj₂]
+      infer_instance
+    adj₁.fullyFaithfulLOfIsIsoUnit
+  left_inv _ := Subsingleton.elim _ _
+  right_inv _ := Subsingleton.elim _ _
+
+end CategoryTheory.Adjunction
diff --git a/Mathlib/CategoryTheory/Adjunction/Unique.lean b/Mathlib/CategoryTheory/Adjunction/Unique.lean
index 118731c61129b..9c4b07886fc7d 100644
--- a/Mathlib/CategoryTheory/Adjunction/Unique.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Unique.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
 -/
-import Mathlib.CategoryTheory.Adjunction.Basic
+import Mathlib.CategoryTheory.Adjunction.Mates
 /-!
 
 # Uniqueness of adjoints
@@ -11,9 +11,6 @@ import Mathlib.CategoryTheory.Adjunction.Basic
 This file shows that adjoints are unique up to natural isomorphism.
 
 ## Main results
-* `Adjunction.natTransEquiv` and `Adjunction.natIsoEquiv` If `F ⊣ G` and `F' ⊣ G'` are adjunctions,
-  then there are equivalences `(G ⟶ G') ≃ (F' ⟶ F)` and `(G ≅ G') ≃ (F' ≅ F)`.
-Everything else is deduced from this:
 
 * `Adjunction.leftAdjointUniq` : If `F` and `F'` are both left adjoint to `G`, then they are
   naturally isomorphic.
@@ -21,12 +18,6 @@ Everything else is deduced from this:
 * `Adjunction.rightAdjointUniq` : If `G` and `G'` are both right adjoint to `F`, then they are
   naturally isomorphic.
 
-## TODO
-
-There some overlap with the file `Adjunction.Mates`. In particular, `natTransEquiv` is just a
-special case of `mateEquiv`. However, before removing `natTransEquiv`, in favour of `mateEquiv`,
-the latter needs some more API lemmas such as `natTransEquiv_apply_app`, `natTransEquiv_id`, etc.
-in order to make automation work better in the rest of this file.
 -/
 
 open CategoryTheory
@@ -35,89 +26,9 @@ variable {C D : Type*} [Category C] [Category D]
 
 namespace CategoryTheory.Adjunction
 
-/--
-If `F ⊣ G` and `F' ⊣ G'` are adjunctions, then giving a natural transformation `G ⟶ G'` is the
-same as giving a natural transformation `F' ⟶ F`.
--/
-@[simps]
-def natTransEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
-    (G ⟶ G') ≃ (F' ⟶ F) where
-  toFun f := {
-    app := fun X ↦ F'.map ((adj1.unit ≫ whiskerLeft F f).app X) ≫ adj2.counit.app _
-    naturality := by
-      intro X Y g
-      simp only [← Category.assoc, ← Functor.map_comp]
-      erw [(adj1.unit ≫ (whiskerLeft F f)).naturality]
-      simp
-  }
-  invFun f := {
-    app := fun X ↦ adj2.unit.app (G.obj X) ≫ G'.map (f.app (G.obj X) ≫ adj1.counit.app X)
-    naturality := by
-      intro X Y g
-      erw [← adj2.unit_naturality_assoc]
-      simp only [← Functor.map_comp]
-      simp
-  }
-  left_inv f := by
-    ext X
-    simp only [Functor.comp_obj, NatTrans.comp_app, Functor.id_obj, whiskerLeft_app,
-      Functor.map_comp, Category.assoc, unit_naturality_assoc, right_triangle_components_assoc]
-    erw [← f.naturality (adj1.counit.app X), ← Category.assoc]
-    simp
-  right_inv f := by
-    ext
-    simp
-
-@[simp]
-lemma natTransEquiv_id {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
-    natTransEquiv adj adj (𝟙 _) = 𝟙 _ := by ext; simp
-
-@[simp]
-lemma natTransEquiv_id_symm {F : C ⥤ D} {G : D ⥤ C} (adj : F ⊣ G) :
-    (natTransEquiv adj adj).symm (𝟙 _) = 𝟙 _ := by ext; simp
-
-@[simp]
-lemma natTransEquiv_comp {F F' F'' : C ⥤ D} {G G' G'' : D ⥤ C}
-    (adj1 : F ⊣ G) (adj2 : F' ⊣ G') (adj3 : F'' ⊣ G'') (f : G ⟶ G') (g : G' ⟶ G'') :
-    natTransEquiv adj2 adj3 g ≫ natTransEquiv adj1 adj2 f = natTransEquiv adj1 adj3 (f ≫ g) := by
-  apply (natTransEquiv adj1 adj3).symm.injective
-  ext X
-  simp only [natTransEquiv_symm_apply_app, Functor.comp_obj, NatTrans.comp_app,
-    natTransEquiv_apply_app, Functor.id_obj, whiskerLeft_app, Functor.map_comp, Category.assoc,
-    unit_naturality_assoc, right_triangle_components_assoc, Equiv.symm_apply_apply,
-    ← g.naturality_assoc, ← g.naturality]
-  simp only [← Category.assoc, unit_naturality, Functor.comp_obj, right_triangle_components,
-    Category.comp_id, ← f.naturality, Category.id_comp]
-
-@[simp]
-lemma natTransEquiv_comp_symm {F F' F'' : C ⥤ D} {G G' G'' : D ⥤ C}
-    (adj1 : F ⊣ G) (adj2 : F' ⊣ G') (adj3 : F'' ⊣ G'') (f : F' ⟶ F) (g : F'' ⟶ F') :
-    (natTransEquiv adj1 adj2).symm f ≫ (natTransEquiv adj2 adj3).symm g =
-      (natTransEquiv adj1 adj3).symm (g ≫ f) := by
-  apply (natTransEquiv adj1 adj3).injective
-  ext
-  simp
-
-/--
-If `F ⊣ G` and `F' ⊣ G'` are adjunctions, then giving a natural isomorphism `G ≅ G'` is the
-same as giving a natural transformation `F' ≅ F`.
--/
-@[simps]
-def natIsoEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
-    (G ≅ G') ≃ (F' ≅ F) where
-  toFun i := {
-    hom := natTransEquiv adj1 adj2 i.hom
-    inv := natTransEquiv adj2 adj1 i.inv
-  }
-  invFun i := {
-    hom := (natTransEquiv adj1 adj2).symm i.hom
-    inv := (natTransEquiv adj2 adj1).symm i.inv }
-  left_inv i := by simp
-  right_inv i := by simp
-
 /-- If `F` and `F'` are both left adjoint to `G`, then they are naturally isomorphic. -/
 def leftAdjointUniq {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) : F ≅ F' :=
-  (natIsoEquiv adj1 adj2 (Iso.refl _)).symm
+  ((conjugateIsoEquiv adj1 adj2).symm (Iso.refl G)).symm
 
 -- Porting note (#10618): removed simp as simp can prove this
 theorem homEquiv_leftAdjointUniq_hom_app {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G)
@@ -141,9 +52,10 @@ theorem unit_leftAdjointUniq_hom_app
 theorem leftAdjointUniq_hom_counit {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
     whiskerLeft G (leftAdjointUniq adj1 adj2).hom ≫ adj2.counit = adj1.counit := by
   ext x
-  simp only [Functor.comp_obj, Functor.id_obj, leftAdjointUniq, Iso.symm_hom, natIsoEquiv_apply_inv,
-    Iso.refl_inv, NatTrans.comp_app, whiskerLeft_app, natTransEquiv_apply_app, whiskerLeft_id',
-    Category.comp_id, Category.assoc]
+  simp only [Functor.comp_obj, Functor.id_obj, leftAdjointUniq, Iso.symm_hom,
+    conjugateIsoEquiv_symm_apply_inv, Iso.refl_inv, NatTrans.comp_app, whiskerLeft_app,
+    conjugateEquiv_symm_apply_app, NatTrans.id_app, Functor.map_id, Category.id_comp,
+    Category.assoc]
   rw [← adj1.counit_naturality, ← Category.assoc, ← F.map_comp]
   simp
 
@@ -180,7 +92,7 @@ theorem leftAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
 
 /-- If `G` and `G'` are both right adjoint to `F`, then they are naturally isomorphic. -/
 def rightAdjointUniq {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G') : G ≅ G' :=
-  (natIsoEquiv adj1 adj2).symm (Iso.refl _)
+  conjugateIsoEquiv adj1 adj2 (Iso.refl _)
 
 -- Porting note (#10618): simp can prove this
 theorem homEquiv_symm_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G)
@@ -192,8 +104,8 @@ theorem homEquiv_symm_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (a
 theorem unit_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
     (x : C) : adj1.unit.app x ≫ (rightAdjointUniq adj1 adj2).hom.app (F.obj x) =
       adj2.unit.app x := by
-  simp only [Functor.id_obj, Functor.comp_obj, rightAdjointUniq, natIsoEquiv_symm_apply_hom,
-    Iso.refl_hom, natTransEquiv_symm_apply_app, NatTrans.id_app, Category.id_comp]
+  simp only [Functor.id_obj, Functor.comp_obj, rightAdjointUniq, conjugateIsoEquiv_apply_hom,
+    Iso.refl_hom, conjugateEquiv_apply_app, NatTrans.id_app, Functor.map_id, Category.id_comp]
   rw [← adj2.unit_naturality_assoc, ← G'.map_comp]
   simp
 
@@ -243,4 +155,7 @@ theorem rightAdjointUniq_refl {F : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) :
 
 end Adjunction
 
+@[deprecated (since := "2024-10-07")] alias Adjunction.natTransEquiv := conjugateEquiv
+@[deprecated (since := "2024-10-07")] alias Adjunction.natIsoEquiv := conjugateIsoEquiv
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Adjunction/Whiskering.lean b/Mathlib/CategoryTheory/Adjunction/Whiskering.lean
index 83f8ab2a242f4..0a174fd66170b 100644
--- a/Mathlib/CategoryTheory/Adjunction/Whiskering.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Whiskering.lean
@@ -25,36 +25,28 @@ variable (C : Type*) {D E : Type*} [Category C] [Category D] [Category E] {F : D
   `(whiskeringRight C _ _).obj F ⊣ (whiskeringRight C _ _).obj G`. -/
 @[simps! unit_app_app counit_app_app]
 protected def whiskerRight (adj : F ⊣ G) :
-    (whiskeringRight C D E).obj F ⊣ (whiskeringRight C E D).obj G :=
-  mkOfUnitCounit
-    { unit :=
-        { app := fun X =>
-            (Functor.rightUnitor _).inv ≫ whiskerLeft X adj.unit ≫ (Functor.associator _ _ _).inv
-          naturality := by intros; ext; dsimp; simp }
-      counit :=
-        { app := fun X =>
-            (Functor.associator _ _ _).hom ≫ whiskerLeft X adj.counit ≫ (Functor.rightUnitor _).hom
-          naturality := by intros; ext; dsimp; simp }
-      left_triangle  := by ext; dsimp; simp
-      right_triangle := by ext; dsimp; simp
-    }
+    (whiskeringRight C D E).obj F ⊣ (whiskeringRight C E D).obj G where
+  unit :=
+    { app := fun X =>
+        (Functor.rightUnitor _).inv ≫ whiskerLeft X adj.unit ≫ (Functor.associator _ _ _).inv
+      naturality := by intros; ext; dsimp; simp }
+  counit :=
+    { app := fun X =>
+        (Functor.associator _ _ _).hom ≫ whiskerLeft X adj.counit ≫ (Functor.rightUnitor _).hom
+      naturality := by intros; ext; dsimp; simp }
 
 /-- Given an adjunction `F ⊣ G`, this provides the natural adjunction
   `(whiskeringLeft _ _ C).obj G ⊣ (whiskeringLeft _ _ C).obj F`. -/
 @[simps! unit_app_app counit_app_app]
 protected def whiskerLeft (adj : F ⊣ G) :
-    (whiskeringLeft E D C).obj G ⊣ (whiskeringLeft D E C).obj F :=
-  mkOfUnitCounit
-    { unit :=
-        { app := fun X =>
-            (Functor.leftUnitor _).inv ≫ whiskerRight adj.unit X ≫ (Functor.associator _ _ _).hom
-          naturality := by intros; ext; dsimp; simp }
-      counit :=
-        { app := fun X =>
-            (Functor.associator _ _ _).inv ≫ whiskerRight adj.counit X ≫ (Functor.leftUnitor _).hom
-          naturality := by intros; ext; dsimp; simp }
-      left_triangle  := by ext x; dsimp; simp [Category.id_comp, Category.comp_id, ← x.map_comp]
-      right_triangle := by ext x; dsimp; simp [Category.id_comp, Category.comp_id, ← x.map_comp]
-    }
+    (whiskeringLeft E D C).obj G ⊣ (whiskeringLeft D E C).obj F where
+  unit :=
+    { app := fun X =>
+        (Functor.leftUnitor _).inv ≫ whiskerRight adj.unit X ≫ (Functor.associator _ _ _).hom }
+  counit :=
+    { app := fun X =>
+        (Functor.associator _ _ _).inv ≫ whiskerRight adj.counit X ≫ (Functor.leftUnitor _).hom }
+  left_triangle_components X := by ext; simp [← X.map_comp]
+  right_triangle_components X := by ext; simp [← X.map_comp]
 
 end CategoryTheory.Adjunction
diff --git a/Mathlib/CategoryTheory/Bicategory/Adjunction.lean b/Mathlib/CategoryTheory/Bicategory/Adjunction.lean
index ae25dba05c39c..1c3d1f82e0da8 100644
--- a/Mathlib/CategoryTheory/Bicategory/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Adjunction.lean
@@ -3,7 +3,7 @@ Copyright (c) 2023 Yuma Mizuno. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yuma Mizuno
 -/
-import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
 
 /-!
 # Adjunctions in bicategories
@@ -19,18 +19,6 @@ identities. The 2-morphism `η` is called the unit and `ε` is called the counit
 * `Bicategory.mkOfAdjointifyCounit`: construct an adjoint equivalence from 2-isomorphisms
   `η : 𝟙 a ≅ f ≫ g` and `ε : g ≫ f ≅ 𝟙 b`, by upgrading `ε` to a counit.
 
-## Implementation notes
-
-The computation of 2-morphisms in the proof is done using `calc` blocks. Typically,
-the LHS and the RHS in each step of `calc` are related by simple rewriting up to associators
-and unitors. So the proof for each step should be of the form `rw [...]; coherence`. In practice,
-our proofs look like `rw [...]; simp [bicategoricalComp]; coherence`. The `simp` is not strictly
-necessary, but it speeds up the proof and allow us to avoid increasing the `maxHeartbeats`.
-The speedup is probably due to reducing the length of the expression e.g. by absorbing
-identity maps or applying the pentagon relation. Such a hack may not be necessary if the
-coherence tactic is improved. One possible way would be to perform such a simplification in the
-preprocessing of the coherence tactic.
-
 ## TODO
 
 * `Bicategory.mkOfAdjointifyUnit`: construct an adjoint equivalence from 2-isomorphisms
@@ -58,7 +46,7 @@ a －－－－－－ ▸ a
         b －－－－－－ ▸ b
 ```
 -/
-def leftZigzag (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) :=
+abbrev leftZigzag (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) :=
   η ▷ f ⊗≫ f ◁ ε
 
 /-- The 2-morphism defined by the following pasting diagram:
@@ -70,7 +58,7 @@ def leftZigzag (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) :=
 b －－－－－－ ▸ b
 ```
 -/
-def rightZigzag (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) :=
+abbrev rightZigzag (η : 𝟙 a ⟶ f ≫ g) (ε : g ≫ f ⟶ 𝟙 b) :=
   g ◁ η ⊗≫ ε ▷ g
 
 theorem rightZigzag_idempotent_of_left_triangle
@@ -79,13 +67,13 @@ theorem rightZigzag_idempotent_of_left_triangle
   dsimp only [rightZigzag]
   calc
     _ = g ◁ η ⊗≫ ((ε ▷ g ▷ 𝟙 a) ≫ (𝟙 b ≫ g) ◁ η) ⊗≫ ε ▷ g := by
-      simp [bicategoricalComp]; coherence
+      bicategory
     _ = 𝟙 _ ⊗≫ g ◁ (η ▷ 𝟙 a ≫ (f ≫ g) ◁ η) ⊗≫ (ε ▷ (g ≫ f) ≫ 𝟙 b ◁ ε) ▷ g ⊗≫ 𝟙 _ := by
-      rw [← whisker_exchange]; simp [bicategoricalComp]; coherence
+      rw [← whisker_exchange]; bicategory
     _ = g ◁ η ⊗≫ g ◁ leftZigzag η ε ▷ g ⊗≫ ε ▷ g := by
-      rw [← whisker_exchange,  ← whisker_exchange]; simp [leftZigzag, bicategoricalComp]; coherence
+      rw [← whisker_exchange,  ← whisker_exchange, leftZigzag]; bicategory
     _ = g ◁ η ⊗≫ ε ▷ g := by
-      rw [h]; simp [bicategoricalComp]; coherence
+      rw [h]; bicategory
 
 /-- Adjunction between two 1-morphisms. -/
 structure Adjunction (f : a ⟶ b) (g : b ⟶ a) where
@@ -104,14 +92,14 @@ namespace Adjunction
 
 attribute [simp] left_triangle right_triangle
 
-attribute [local simp] leftZigzag rightZigzag
+-- attribute [local simp] leftZigzag rightZigzag
 
 /-- Adjunction between identities. -/
 def id (a : B) : 𝟙 a ⊣ 𝟙 a where
   unit := (ρ_ _).inv
   counit := (ρ_ _).hom
-  left_triangle := by dsimp; coherence
-  right_triangle := by dsimp; coherence
+  left_triangle := by bicategory_coherence
+  right_triangle := by bicategory_coherence
 
 instance : Inhabited (Adjunction (𝟙 a) (𝟙 a)) :=
   ⟨id a⟩
@@ -137,13 +125,13 @@ theorem comp_left_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂)
           adj₁.unit ▷ (f₁ ≫ f₂) ⊗≫
             f₁ ◁ (adj₂.unit ▷ (g₁ ≫ f₁) ≫ (f₂ ≫ g₂) ◁ adj₁.counit) ▷ f₂ ⊗≫
               (f₁ ≫ f₂) ◁ adj₂.counit ⊗≫ 𝟙 _ := by
-      simp [bicategoricalComp]; coherence
+      dsimp only [compUnit, compCounit]; bicategory
     _ = 𝟙 _ ⊗≫
           (leftZigzag adj₁.unit adj₁.counit) ▷ f₂ ⊗≫
             f₁ ◁ (leftZigzag adj₂.unit adj₂.counit) ⊗≫ 𝟙 _ := by
-      rw [← whisker_exchange]; simp [bicategoricalComp]; coherence
+      rw [← whisker_exchange]; bicategory
     _ = _ := by
-      simp_rw [left_triangle]; simp [bicategoricalComp]
+      simp_rw [left_triangle]; bicategory
 
 theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
     rightZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (ρ_ _).hom ≫ (λ_ _).inv := by
@@ -152,13 +140,13 @@ theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂
           (g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
             g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
               adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
-      simp [bicategoricalComp]; coherence
+      dsimp only [compUnit, compCounit]; bicategory
     _ = 𝟙 _ ⊗≫
           g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
             (rightZigzag adj₂.unit adj₂.counit) ▷ g₁ ⊗≫ 𝟙 _ := by
-      rw [whisker_exchange]; simp [bicategoricalComp]; coherence
+      rw [whisker_exchange]; bicategory
     _ = _ := by
-      simp_rw [right_triangle]; simp [bicategoricalComp]
+      simp_rw [right_triangle]; bicategory
 
 /-- Composition of adjunctions. -/
 @[simps]
@@ -177,15 +165,13 @@ noncomputable section
 variable (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b)
 
 /-- The isomorphism version of `leftZigzag`. -/
-def leftZigzagIso (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :=
+abbrev leftZigzagIso (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :=
   whiskerRightIso η f ≪⊗≫ whiskerLeftIso f ε
 
 /-- The isomorphism version of `rightZigzag`. -/
-def rightZigzagIso (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :=
+abbrev rightZigzagIso (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :=
   whiskerLeftIso g η ≪⊗≫ whiskerRightIso ε g
 
-attribute [local simp] leftZigzagIso rightZigzagIso leftZigzag rightZigzag
-
 @[simp]
 theorem leftZigzagIso_hom : (leftZigzagIso η ε).hom = leftZigzag η.hom ε.hom :=
   rfl
@@ -218,7 +204,7 @@ theorem right_triangle_of_left_triangle (h : leftZigzag η.hom ε.hom = (λ_ f).
     rightZigzag η.hom ε.hom = (ρ_ g).hom ≫ (λ_ g).inv := by
   rw [← cancel_epi (rightZigzag η.hom ε.hom ≫ (λ_ g).hom ≫ (ρ_ g).inv)]
   calc
-    _ = rightZigzag η.hom ε.hom ⊗≫ rightZigzag η.hom ε.hom := by coherence
+    _ = rightZigzag η.hom ε.hom ⊗≫ rightZigzag η.hom ε.hom := by bicategory
     _ = rightZigzag η.hom ε.hom := rightZigzag_idempotent_of_left_triangle _ _ h
     _ = _ := by simp
 
@@ -233,15 +219,15 @@ theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f 
   calc
     _ = 𝟙 _ ⊗≫ (η.hom ▷ (f ≫ 𝟙 b) ≫ (f ≫ g) ◁ f ◁ ε.inv) ⊗≫
           f ◁ g ◁ η.inv ▷ f ⊗≫ f ◁ ε.hom := by
-      simp [bicategoricalComp]; coherence
+      bicategory
     _ = 𝟙 _ ⊗≫ f ◁ ε.inv ⊗≫ (η.hom ▷ (f ≫ g) ≫ (f ≫ g) ◁ η.inv) ▷ f ⊗≫ f ◁ ε.hom := by
-      rw [← whisker_exchange η.hom (f ◁ ε.inv)]; simp [bicategoricalComp]; coherence
+      rw [← whisker_exchange η.hom (f ◁ ε.inv)]; bicategory
     _ = 𝟙 _ ⊗≫ f ◁ ε.inv ⊗≫ (η.inv ≫ η.hom) ▷ f ⊗≫ f ◁ ε.hom := by
-      rw [← whisker_exchange η.hom η.inv]; coherence
+      rw [← whisker_exchange η.hom η.inv]; bicategory
     _ = 𝟙 _ ⊗≫ f ◁ (ε.inv ≫ ε.hom) := by
-      rw [Iso.inv_hom_id]; simp [bicategoricalComp]
+      rw [Iso.inv_hom_id]; bicategory
     _ = _ := by
-      rw [Iso.inv_hom_id]; simp [bicategoricalComp]
+      rw [Iso.inv_hom_id]; bicategory
 
 /-- Adjoint equivalences between two objects. -/
 structure Equivalence (a b : B) where
diff --git a/Mathlib/CategoryTheory/Bicategory/Coherence.lean b/Mathlib/CategoryTheory/Bicategory/Coherence.lean
index acb01f1bd07bb..8721c45df98f3 100644
--- a/Mathlib/CategoryTheory/Bicategory/Coherence.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Coherence.lean
@@ -204,11 +204,11 @@ def normalizeUnitIso (a b : FreeBicategory B) :
 def normalizeEquiv (a b : B) : Hom a b ≌ Discrete (Path.{v + 1} a b) :=
   Equivalence.mk ((normalize _).mapFunctor a b) (inclusionPath a b) (normalizeUnitIso a b)
     (Discrete.natIso fun f => eqToIso (by
-      induction' f with f
-      induction' f with _ _ _ _ ih
-      -- Porting note: `tidy` closes the goal in mathlib3 but `aesop` doesn't here.
-      · rfl
-      · ext1
+      obtain ⟨f⟩ := f
+      induction f with
+      | nil => rfl
+      | cons _ _ ih =>
+        ext1 -- Porting note: `tidy` closes the goal in mathlib3 but `aesop` doesn't here.
         injection ih with ih
         conv_rhs => rw [← ih]
         rfl))
diff --git a/Mathlib/CategoryTheory/Bicategory/End.lean b/Mathlib/CategoryTheory/Bicategory/End.lean
index 1caa7d9c14e97..8210d2a2cdcd0 100644
--- a/Mathlib/CategoryTheory/Bicategory/End.lean
+++ b/Mathlib/CategoryTheory/Bicategory/End.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Bicategory.Basic
 import Mathlib.CategoryTheory.Monoidal.Category
diff --git a/Mathlib/CategoryTheory/Bicategory/Free.lean b/Mathlib/CategoryTheory/Bicategory/Free.lean
index 2063a717cc38f..e423dd24b31be 100644
--- a/Mathlib/CategoryTheory/Bicategory/Free.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Free.lean
@@ -334,19 +334,16 @@ def lift : Pseudofunctor (FreeBicategory B) C where
   -- in mathlib3 `tidy` did these inductions for us.
   map₂_comp := by
     intros a b f g h η θ
-    apply Quot.rec _ _ η
-    · intro η
-      apply Quot.rec _ _ θ
-      · intro θ; rfl
-      · intros; rfl
-    · intros; rfl
+    induction η using Quot.rec
+    · induction θ using Quot.rec <;> rfl
+    · rfl
   -- Porting note: still borked from here. The infoview doesn't update properly for me.
   map₂_whisker_left := by
     intro a b c f g h η
-    apply Quot.rec _ _ η
-    · intros; aesop_cat
-    · intros; rfl
-  map₂_whisker_right := by intro _ _ _ _ _ η h; dsimp; apply Quot.rec _ _ η <;> aesop_cat
+    induction η using Quot.rec
+    · aesop_cat
+    · rfl
+  map₂_whisker_right := by intro _ _ _ _ _ η h; dsimp; induction η using Quot.rec <;> aesop_cat
 
 end
 
diff --git a/Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean b/Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean
index 616625791af23..f4333e8be9f1e 100644
--- a/Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Functor/Lax.lean
@@ -6,6 +6,7 @@ Authors: Calle Sönne
 
 import Mathlib.CategoryTheory.Bicategory.Functor.Prelax
 import Mathlib.Tactic.CategoryTheory.Slice
+import Mathlib.Tactic.CategoryTheory.ToApp
 
 /-!
 # Lax functors
@@ -58,24 +59,24 @@ structure LaxFunctor (B: Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂)
   mapId (a : B) : 𝟙 (obj a) ⟶ map (𝟙 a)
   /-- The 2-morphism underlying the lax functoriality constraint. -/
   mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map f ≫ map g ⟶ map (f ≫ g)
-  /-- Naturality of the lax functoriality constraight, on the left. -/
+  /-- Naturality of the lax functoriality constraint, on the left. -/
   mapComp_naturality_left :
     ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
       mapComp f g ≫ map₂ (η ▷ g) = map₂ η ▷ map g ≫ mapComp f' g:= by aesop_cat
-  /-- Naturality of the lax functoriality constraight, on the right. -/
+  /-- Naturality of the lax functoriality constraint, on the right. -/
   mapComp_naturality_right :
     ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
      mapComp f g ≫ map₂ (f ◁ η) = map f ◁ map₂ η ≫ mapComp f g' := by aesop_cat
-  /-- Lax associativity -/
+  /-- Lax associativity. -/
   map₂_associator :
     ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
       mapComp f g ▷ map h ≫ mapComp (f ≫ g) h ≫ map₂ (α_ f g h).hom =
       (α_ (map f) (map g) (map h)).hom ≫ map f ◁ mapComp g h ≫ mapComp f (g ≫ h) := by aesop_cat
-  /-- Lax left unity -/
+  /-- Lax left unity. -/
   map₂_leftUnitor :
     ∀ {a b : B} (f : a ⟶ b),
       map₂ (λ_ f).inv = (λ_ (map f)).inv ≫ mapId a ▷ map f ≫ mapComp (𝟙 a) f := by aesop_cat
-  /-- Lax right unity -/
+  /-- Lax right unity. -/
   map₂_rightUnitor :
     ∀ {a b : B} (f : a ⟶ b),
       map₂ (ρ_ f).inv = (ρ_ (map f)).inv ≫ map f ◁ mapId b ≫ mapComp f (𝟙 b) := by aesop_cat
@@ -86,36 +87,36 @@ namespace LaxFunctor
 
 variable {B : Type u₁} [Bicategory.{w₁, v₁} B] {C : Type u₂} [Bicategory.{w₂, v₂} C]
 
-attribute [reassoc (attr := simp)]
+attribute [reassoc (attr := simp), to_app (attr := simp)]
   mapComp_naturality_left mapComp_naturality_right map₂_associator
-attribute [simp, reassoc] map₂_leftUnitor map₂_rightUnitor
+attribute [simp, reassoc, to_app] map₂_leftUnitor map₂_rightUnitor
 
 /-- The underlying prelax functor. -/
 add_decl_doc LaxFunctor.toPrelaxFunctor
 
 variable (F : LaxFunctor B C)
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_left {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     F.mapComp f g ▷ F.map h ≫ F.mapComp (f ≫ g) h = (α_ (F.map f) (F.map g) (F.map h)).hom ≫
       F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) ≫ F.map₂ (α_ f g h).inv := by
   rw [← F.map₂_associator_assoc, ← F.map₂_comp]
   simp only [Iso.hom_inv_id, PrelaxFunctor.map₂_id, comp_id]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_right {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     F.map f ◁ F.mapComp g h ≫ F.mapComp f (g ≫ h) =
       (α_ (F.map f) (F.map g) (F.map h)).inv ≫ F.mapComp f g ▷ F.map h ≫
         F.mapComp (f ≫ g) h ≫ F.map₂ (α_ f g h).hom := by
   simp only [map₂_associator, Iso.inv_hom_id_assoc]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma map₂_leftUnitor_hom {a b : B} (f : a ⟶ b) :
     (λ_ (F.map f)).hom = F.mapId a ▷ F.map f ≫ F.mapComp (𝟙 a) f ≫ F.map₂ (λ_ f).hom := by
   rw [← PrelaxFunctor.map₂Iso_hom, ← assoc, ← Iso.comp_inv_eq, ← Iso.eq_inv_comp]
   simp only [Functor.mapIso_inv, PrelaxFunctor.mapFunctor_map, map₂_leftUnitor]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma map₂_rightUnitor_hom {a b : B} (f : a ⟶ b) :
     (ρ_ (F.map f)).hom = F.map f ◁ F.mapId b ≫ F.mapComp f (𝟙 b) ≫ F.map₂ (ρ_ f).hom := by
   rw [← PrelaxFunctor.map₂Iso_hom, ← assoc, ← Iso.comp_inv_eq, ← Iso.eq_inv_comp]
@@ -147,7 +148,7 @@ def comp {D : Type u₃} [Bicategory.{w₃, v₃} D] (F : LaxFunctor B C) (G : L
   map₂_associator := fun f g h => by
     dsimp
     slice_rhs 1 3 =>
-      rw [whiskerLeft_comp, assoc, ← mapComp_naturality_right, ← map₂_associator_assoc]
+      rw [Bicategory.whiskerLeft_comp, assoc, ← mapComp_naturality_right, ← map₂_associator_assoc]
     slice_rhs 3 5 =>
       rw [← G.map₂_comp, ← G.map₂_comp, ← F.map₂_associator, G.map₂_comp, G.map₂_comp]
     slice_lhs 1 3 =>
@@ -160,23 +161,22 @@ def comp {D : Type u₃} [Bicategory.{w₃, v₃} D] (F : LaxFunctor B C) (G : L
   map₂_rightUnitor := fun f => by
     dsimp
     simp only [map₂_rightUnitor, PrelaxFunctor.map₂_comp, assoc, mapComp_naturality_right_assoc,
-      whiskerLeft_comp]
+      Bicategory.whiskerLeft_comp]
 
 /-- A structure on an Lax functor that promotes an Lax functor to a pseudofunctor.
 
 See `Pseudofunctor.mkOfLax`. -/
 structure PseudoCore (F : LaxFunctor B C) where
+  /-- The isomorphism giving rise to the lax unity constraint -/
   mapIdIso (a : B) : F.map (𝟙 a) ≅ 𝟙 (F.obj a)
+  /-- The isomorphism giving rise to the lax functoriality constraint -/
   mapCompIso {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : F.map (f ≫ g) ≅ F.map f ≫ F.map g
+  /-- `mapIdIso` gives rise to the lax unity constraint -/
   mapIdIso_inv {a : B} : (mapIdIso a).inv = F.mapId a := by aesop_cat
+  /-- `mapCompIso` gives rise to the lax functoriality constraint -/
   mapCompIso_inv {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : (mapCompIso f g).inv = F.mapComp f g := by
     aesop_cat
 
-attribute [nolint docBlame] CategoryTheory.LaxFunctor.PseudoCore.mapIdIso
-  CategoryTheory.LaxFunctor.PseudoCore.mapCompIso
-  CategoryTheory.LaxFunctor.PseudoCore.mapIdIso_inv
-  CategoryTheory.LaxFunctor.PseudoCore.mapCompIso_inv
-
 attribute [simp] PseudoCore.mapIdIso_inv PseudoCore.mapCompIso_inv
 
 end LaxFunctor
diff --git a/Mathlib/CategoryTheory/Bicategory/Functor/Oplax.lean b/Mathlib/CategoryTheory/Bicategory/Functor/Oplax.lean
index bc1c66a21dae8..9cf82c77a5732 100644
--- a/Mathlib/CategoryTheory/Bicategory/Functor/Oplax.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Functor/Oplax.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yuma Mizuno
 -/
 import Mathlib.CategoryTheory.Bicategory.Functor.Prelax
+import Mathlib.Tactic.CategoryTheory.ToApp
 
 /-!
 # Oplax functors
@@ -36,17 +37,6 @@ section
 variable {B : Type u₁} [Bicategory.{w₁, v₁} B] {C : Type u₂} [Bicategory.{w₂, v₂} C]
 variable {D : Type u₃} [Bicategory.{w₃, v₃} D]
 
--- Porting note: in Lean 3 the below auxiliary definition was only used once, in the definition
--- of oplax functor, with a comment that it had to be used to fix a timeout. The timeout is
--- not present in Lean 4, however Lean 4 is not as good at seeing through the definition,
--- meaning that `simp` wasn't functioning as well as it should. I have hence removed
--- the auxiliary definition.
---@[simp]
---def OplaxFunctor.Map₂AssociatorAux (obj : B → C) (map : ∀ {X Y : B}, (X ⟶ Y) → (obj X ⟶ obj Y))
---    (map₂ : ∀ {a b : B} {f g : a ⟶ b}, (f ⟶ g) → (map f ⟶ map g))
---    (map_comp : ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), map (f ≫ g) ⟶ map f ≫ map g) {a b c d : B}
---    (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) : Prop := ...
-
 /-- An oplax functor `F` between bicategories `B` and `C` consists of a function between objects
 `F.obj`, a function between 1-morphisms `F.map`, and a function between 2-morphisms `F.map₂`.
 
@@ -60,28 +50,32 @@ of 2-morphisms.
 -/
 structure OplaxFunctor (B : Type u₁) [Bicategory.{w₁, v₁} B] (C : Type u₂)
   [Bicategory.{w₂, v₂} C] extends PrelaxFunctor B C where
+  /-- The 2-morphism underlying the oplax unity constraint. -/
   mapId (a : B) : map (𝟙 a) ⟶ 𝟙 (obj a)
+  /-- The 2-morphism underlying the oplax functoriality constraint. -/
   mapComp {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : map (f ≫ g) ⟶ map f ≫ map g
+  /-- Naturality of the oplax functoriality constraint, on the left. -/
   mapComp_naturality_left :
     ∀ {a b c : B} {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c),
       map₂ (η ▷ g) ≫ mapComp f' g = mapComp f g ≫ map₂ η ▷ map g := by
     aesop_cat
+  /-- Naturality of the lax functoriality constraight, on the right. -/
   mapComp_naturality_right :
     ∀ {a b c : B} (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g'),
       map₂ (f ◁ η) ≫ mapComp f g' = mapComp f g ≫ map f ◁ map₂ η := by
     aesop_cat
-  -- Porting note: `map₂_associator_aux` was used here in lean 3, but this was a hack
-  -- to avoid a timeout; we revert this hack here (because it was causing other problems
-  -- and was not necessary in lean 4)
+  /-- Oplax associativity. -/
   map₂_associator :
     ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d),
       map₂ (α_ f g h).hom ≫ mapComp f (g ≫ h) ≫ map f ◁ mapComp g h =
     mapComp (f ≫ g) h ≫ mapComp f g ▷ map h ≫ (α_ (map f) (map g) (map h)).hom := by
     aesop_cat
+  /-- Oplax left unity. -/
   map₂_leftUnitor :
     ∀ {a b : B} (f : a ⟶ b),
       map₂ (λ_ f).hom = mapComp (𝟙 a) f ≫ mapId a ▷ map f ≫ (λ_ (map f)).hom := by
     aesop_cat
+  /-- Oplax right unity. -/
   map₂_rightUnitor :
     ∀ {a b : B} (f : a ⟶ b),
       map₂ (ρ_ f).hom = mapComp f (𝟙 b) ≫ map f ◁ mapId b ≫ (ρ_ (map f)).hom := by
@@ -91,49 +85,32 @@ initialize_simps_projections OplaxFunctor (+toPrelaxFunctor, -obj, -map, -map₂
 
 namespace OplaxFunctor
 
-attribute [reassoc (attr := simp)]
+attribute [reassoc (attr := simp), to_app (attr := simp)]
   mapComp_naturality_left mapComp_naturality_right map₂_associator
-attribute [simp, reassoc] map₂_leftUnitor map₂_rightUnitor
+attribute [simp, reassoc, to_app] map₂_leftUnitor map₂_rightUnitor
 
 section
 
 /-- The underlying prelax functor. -/
 add_decl_doc OplaxFunctor.toPrelaxFunctor
 
-attribute [nolint docBlame] CategoryTheory.OplaxFunctor.mapId
-  CategoryTheory.OplaxFunctor.mapComp
-  CategoryTheory.OplaxFunctor.mapComp_naturality_left
-  CategoryTheory.OplaxFunctor.mapComp_naturality_right
-  CategoryTheory.OplaxFunctor.map₂_associator
-  CategoryTheory.OplaxFunctor.map₂_leftUnitor
-  CategoryTheory.OplaxFunctor.map₂_rightUnitor
-
 variable (F : OplaxFunctor B C)
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_right {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     F.mapComp f (g ≫ h) ≫ F.map f ◁ F.mapComp g h = F.map₂ (α_ f g h).inv ≫
     F.mapComp (f ≫ g) h ≫ F.mapComp f g ▷ F.map h ≫
     (α_ (F.map f) (F.map g) (F.map h)).hom := by
-  rw [← @map₂_associator, ← F.map₂_comp_assoc]
+  rw [← F.map₂_associator, ← F.map₂_comp_assoc]
   simp
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_left {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     F.mapComp (f ≫ g) h ≫ F.mapComp f g ▷ F.map h =
     F.map₂ (α_ f g h).hom ≫ F.mapComp f (g ≫ h) ≫ F.map f ◁ F.mapComp g h
     ≫ (α_ (F.map f) (F.map g) (F.map h)).inv := by
   simp
 
--- Porting note: `to_prelax_eq_coe` and `to_prelaxFunctor_obj` are
--- syntactic tautologies in lean 4
-
--- Porting note: removed lemma `to_prelaxFunctor_map` relating the now
--- nonexistent `PrelaxFunctor.map` and `OplaxFunctor.map`
-
--- Porting note: removed lemma `to_prelaxFunctor_map₂` relating
--- `PrelaxFunctor.map₂` to nonexistent `OplaxFunctor.map₂`
-
 /-- The identity oplax functor. -/
 @[simps]
 def id (B : Type u₁) [Bicategory.{w₁, v₁} B] : OplaxFunctor B B where
@@ -161,10 +138,8 @@ def comp (F : OplaxFunctor B C) (G : OplaxFunctor C D) : OplaxFunctor B D where
       mapComp_naturality_right, assoc]
   map₂_associator := fun f g h => by
     dsimp
-    -- Porting note: if you use the `map₂_associator_aux` hack in the definition of
-    -- `map₂_associator` then the `simp only` call below does not seem to apply `map₂_associator`
     simp only [map₂_associator, ← PrelaxFunctor.map₂_comp_assoc, ← mapComp_naturality_right_assoc,
-      whiskerLeft_comp, assoc]
+      Bicategory.whiskerLeft_comp, assoc]
     simp only [map₂_associator, PrelaxFunctor.map₂_comp, mapComp_naturality_left_assoc,
       comp_whiskerRight, assoc]
   map₂_leftUnitor := fun f => by
@@ -174,27 +149,24 @@ def comp (F : OplaxFunctor B C) (G : OplaxFunctor C D) : OplaxFunctor B D where
   map₂_rightUnitor := fun f => by
     dsimp
     simp only [map₂_rightUnitor, PrelaxFunctor.map₂_comp, mapComp_naturality_right_assoc,
-      whiskerLeft_comp, assoc]
+      Bicategory.whiskerLeft_comp, assoc]
 
 /-- A structure on an oplax functor that promotes an oplax functor to a pseudofunctor.
-See `Pseudofunctor.mkOfOplax`.
--/
+
+See `Pseudofunctor.mkOfOplax`. -/
 -- Porting note(#5171): linter not ported yet
 -- @[nolint has_nonempty_instance]
--- Porting note: removing primes in structure name because
--- my understanding is that they're no longer needed
 structure PseudoCore (F : OplaxFunctor B C) where
+  /-- The isomorphism giving rise to the oplax unity constraint -/
   mapIdIso (a : B) : F.map (𝟙 a) ≅ 𝟙 (F.obj a)
+  /-- The isomorphism giving rise to the oplax functoriality constraint -/
   mapCompIso {a b c : B} (f : a ⟶ b) (g : b ⟶ c) : F.map (f ≫ g) ≅ F.map f ≫ F.map g
+  /-- `mapIdIso` gives rise to the oplax unity constraint -/
   mapIdIso_hom : ∀ {a : B}, (mapIdIso a).hom = F.mapId a := by aesop_cat
+  /-- `mapCompIso` gives rise to the oplax functoriality constraint -/
   mapCompIso_hom :
     ∀ {a b c : B} (f : a ⟶ b) (g : b ⟶ c), (mapCompIso f g).hom = F.mapComp f g := by aesop_cat
 
-attribute [nolint docBlame] CategoryTheory.OplaxFunctor.PseudoCore.mapIdIso
-  CategoryTheory.OplaxFunctor.PseudoCore.mapCompIso
-  CategoryTheory.OplaxFunctor.PseudoCore.mapIdIso_hom
-  CategoryTheory.OplaxFunctor.PseudoCore.mapCompIso_hom
-
 attribute [simp] PseudoCore.mapIdIso_hom PseudoCore.mapCompIso_hom
 
 end
diff --git a/Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean b/Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
index 20e57f3061b67..7ec881a92dc67 100644
--- a/Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Functor/Pseudofunctor.lean
@@ -94,7 +94,7 @@ initialize_simps_projections Pseudofunctor (+toPrelaxFunctor, -obj, -map, -map
 
 namespace Pseudofunctor
 
-attribute [simp, reassoc]
+attribute [simp, reassoc, to_app]
   map₂_whisker_left map₂_whisker_right map₂_associator map₂_left_unitor map₂_right_unitor
 
 section
@@ -168,35 +168,35 @@ section
 
 variable (F : Pseudofunctor B C) {a b : B}
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_right_hom {c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     (F.mapComp f (g ≫ h)).hom ≫ F.map f ◁ (F.mapComp g h).hom = F.map₂ (α_ f g h).inv ≫
     (F.mapComp (f ≫ g) h).hom ≫ (F.mapComp f g).hom ▷ F.map h ≫
     (α_ (F.map f) (F.map g) (F.map h)).hom :=
   F.toOplax.mapComp_assoc_right _ _ _
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_left_hom {c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     (F.mapComp (f ≫ g) h).hom ≫ (F.mapComp f g).hom ▷ F.map h =
     F.map₂ (α_ f g h).hom ≫ (F.mapComp f (g ≫ h)).hom ≫ F.map f ◁ (F.mapComp g h).hom
     ≫ (α_ (F.map f) (F.map g) (F.map h)).inv :=
   F.toOplax.mapComp_assoc_left _ _ _
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_right_inv {c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     F.map f ◁ (F.mapComp g h).inv ≫ (F.mapComp f (g ≫ h)).inv =
     (α_ (F.map f) (F.map g) (F.map h)).inv ≫ (F.mapComp f g).inv ▷ F.map h ≫
     (F.mapComp (f ≫ g) h).inv ≫ F.map₂ (α_ f g h).hom :=
   F.toLax.mapComp_assoc_right _ _ _
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_assoc_left_inv {c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) :
     (F.mapComp f g).inv ▷ F.map h ≫ (F.mapComp (f ≫ g) h).inv =
     (α_ (F.map f) (F.map g) (F.map h)).hom ≫ F.map f ◁ (F.mapComp g h).inv ≫
     (F.mapComp f (g ≫ h)).inv ≫ F.map₂ (α_ f g h).inv :=
   F.toLax.mapComp_assoc_left _ _ _
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_id_left_hom (f : a ⟶ b) : (F.mapComp (𝟙 a) f).hom =
     F.map₂ (λ_ f).hom ≫ (λ_ (F.map f)).inv ≫ (F.mapId a).inv ▷ F.map f := by
   simp
@@ -205,7 +205,7 @@ lemma mapComp_id_left (f : a ⟶ b) : (F.mapComp (𝟙 a) f) = F.map₂Iso (λ_
     (λ_ (F.map f)).symm ≪≫ (whiskerRightIso (F.mapId a) (F.map f)).symm :=
   Iso.ext <| F.mapComp_id_left_hom f
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_id_left_inv (f : a ⟶ b) : (F.mapComp (𝟙 a) f).inv =
     (F.mapId a).hom ▷ F.map f ≫ (λ_ (F.map f)).hom ≫ F.map₂ (λ_ f).inv := by
   simp [mapComp_id_left]
@@ -214,17 +214,17 @@ lemma whiskerRightIso_mapId (f : a ⟶ b) : whiskerRightIso (F.mapId a) (F.map f
     (F.mapComp (𝟙 a) f).symm ≪≫ F.map₂Iso (λ_ f) ≪≫ (λ_ (F.map f)).symm := by
   simp [mapComp_id_left]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma whiskerRight_mapId_hom (f : a ⟶ b) : (F.mapId a).hom ▷ F.map f =
     (F.mapComp (𝟙 a) f).inv ≫ F.map₂ (λ_ f).hom ≫ (λ_ (F.map f)).inv := by
   simp [whiskerRightIso_mapId]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma whiskerRight_mapId_inv (f : a ⟶ b) : (F.mapId a).inv ▷ F.map f =
     (λ_ (F.map f)).hom ≫ F.map₂ (λ_ f).inv ≫ (F.mapComp (𝟙 a) f).hom := by
   simpa using congrArg (·.inv) (F.whiskerRightIso_mapId f)
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_id_right_hom (f : a ⟶ b) : (F.mapComp f (𝟙 b)).hom =
     F.map₂ (ρ_ f).hom ≫ (ρ_ (F.map f)).inv ≫ F.map f ◁ (F.mapId b).inv := by
   simp
@@ -233,7 +233,7 @@ lemma mapComp_id_right (f : a ⟶ b) : (F.mapComp f (𝟙 b)) = F.map₂Iso (ρ_
     (ρ_ (F.map f)).symm ≪≫ (whiskerLeftIso (F.map f) (F.mapId b)).symm :=
   Iso.ext <| F.mapComp_id_right_hom f
 
-@[reassoc]
+@[reassoc, to_app]
 lemma mapComp_id_right_inv (f : a ⟶ b) : (F.mapComp f (𝟙 b)).inv =
     F.map f ◁ (F.mapId b).hom ≫ (ρ_ (F.map f)).hom ≫ F.map₂ (ρ_ f).inv := by
   simp [mapComp_id_right]
@@ -242,12 +242,12 @@ lemma whiskerLeftIso_mapId (f : a ⟶ b) : whiskerLeftIso (F.map f) (F.mapId b)
     (F.mapComp f (𝟙 b)).symm ≪≫ F.map₂Iso (ρ_ f) ≪≫ (ρ_ (F.map f)).symm := by
   simp [mapComp_id_right]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma whiskerLeft_mapId_hom (f : a ⟶ b) : F.map f ◁ (F.mapId b).hom =
     (F.mapComp f (𝟙 b)).inv ≫ F.map₂ (ρ_ f).hom ≫ (ρ_ (F.map f)).inv := by
   simp [whiskerLeftIso_mapId]
 
-@[reassoc]
+@[reassoc, to_app]
 lemma whiskerLeft_mapId_inv (f : a ⟶ b) : F.map f ◁ (F.mapId b).inv =
     (ρ_ (F.map f)).hom ≫ F.map₂ (ρ_ f).inv ≫ (F.mapComp f (𝟙 b)).hom := by
   simpa using congrArg (·.inv) (F.whiskerLeftIso_mapId f)
diff --git a/Mathlib/CategoryTheory/Bicategory/Kan/Adjunction.lean b/Mathlib/CategoryTheory/Bicategory/Kan/Adjunction.lean
index f35b3606bbc88..ac3e11be6fb37 100644
--- a/Mathlib/CategoryTheory/Bicategory/Kan/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Kan/Adjunction.lean
@@ -29,8 +29,6 @@ similar results for right Kan extensions and right Kan lifts.
 
 namespace CategoryTheory
 
-open Mathlib.Tactic.BicategoryCoherence bicategoricalComp
-
 namespace Bicategory
 
 universe w v u
@@ -49,11 +47,13 @@ def Adjunction.isAbsoluteLeftKan {f : a ⟶ b} {u : b ⟶ a} (adj : f ⊣ u) :
     (𝟙 _ ⊗≫ u ◁ s.unit ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ : u ≫ h ⟶ s.extension) <|
       calc _
         _ = 𝟙 _ ⊗≫ (adj.unit ▷ _ ≫ _ ◁ s.unit) ⊗≫ f ◁ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
-          simp [bicategoricalComp]
+          dsimp only [whisker_extension, StructuredArrow.mk_right, whisker_unit,
+            StructuredArrow.mk_hom_eq_self]
+          bicategory
         _ = 𝟙 _ ⊗≫ s.unit ⊗≫ leftZigzag adj.unit adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
-          rw [← whisker_exchange, leftZigzag]; simp [bicategoricalComp]
+          rw [← whisker_exchange]; bicategory
         _ = s.unit := by
-          rw [adj.left_triangle]; simp [bicategoricalComp]) <| by
+          rw [adj.left_triangle]; bicategory) <| by
     intro s τ₀
     ext
     /- We need to specify the type of `τ` to use the notation `⊗≫`. -/
@@ -62,13 +62,13 @@ def Adjunction.isAbsoluteLeftKan {f : a ⟶ b} {u : b ⟶ a} (adj : f ⊣ u) :
       simpa [bicategoricalComp] using LeftExtension.w τ₀
     calc τ
       _ = 𝟙 _ ⊗≫ rightZigzag adj.unit adj.counit ▷ h ⊗≫ τ ⊗≫ 𝟙 _ := by
-        rw [adj.right_triangle]; simp [bicategoricalComp]
+        rw [adj.right_triangle]; bicategory
       _ = 𝟙 _ ⊗≫ u ◁ adj.unit ▷ h ⊗≫ (adj.counit ▷ _ ≫ _ ◁ τ) ⊗≫ 𝟙 _ := by
-        rw [rightZigzag]; simp [bicategoricalComp]
+        rw [rightZigzag]; bicategory
       _ = 𝟙 _ ⊗≫ u ◁ (adj.unit ▷ h ⊗≫ f ◁ τ) ⊗≫ adj.counit ▷ s.extension ⊗≫ 𝟙 _ := by
-        rw [← whisker_exchange]; simp [bicategoricalComp]
+        rw [← whisker_exchange]; bicategory
       _ = _ := by
-        rw [hτ]; simp [bicategoricalComp]
+        rw [hτ]; dsimp only [StructuredArrow.homMk_right]
 
 /-- A left Kan extension of the identity along `f` such that `f` commutes with is a right adjoint
 to `f`. The unit of this adjoint is given by the unit of the Kan extension. -/
@@ -86,13 +86,13 @@ def LeftExtension.IsKan.adjunction {f : a ⟶ b} {t : LeftExtension f (𝟙 a)}
       apply H.hom_ext
       calc _
         _ = 𝟙 _ ⊗≫ t.unit ⊗≫ f ◁ rightZigzag t.unit ε ⊗≫ 𝟙 _ := by
-          simp [bicategoricalComp]
+          bicategory
         _ = 𝟙 _ ⊗≫ (t.unit ▷ _ ≫ _ ◁ t.unit) ⊗≫ f ◁ ε ▷ t.extension ⊗≫ 𝟙 _ := by
-          rw [rightZigzag]; simp [bicategoricalComp]
+          rw [rightZigzag]; bicategory
         _ = 𝟙 _ ⊗≫ t.unit ⊗≫ (t.unit ▷ f ⊗≫ f ◁ ε) ▷ t.extension ⊗≫ 𝟙 _ := by
-          rw [← whisker_exchange]; simp [bicategoricalComp]
+          rw [← whisker_exchange]; bicategory
         _ = _ := by
-          rw [← leftZigzag, Hε]; simp [bicategoricalComp] }
+          rw [← leftZigzag, Hε]; bicategory }
 
 /-- For an adjuntion `f ⊣ u`, `u` is a left Kan extension of the identity along `f`.
 The unit of this Kan extension is given by the unit of the adjunction. -/
@@ -106,14 +106,11 @@ theorem isLeftAdjoint_TFAE (f : a ⟶ b) :
       HasAbsLeftKanExtension f (𝟙 a),
       ∃ _ : HasLeftKanExtension f (𝟙 a), Lan.CommuteWith f (𝟙 a) f] := by
   tfae_have 1 → 2
-  · intro h
-    exact IsAbsKan.hasAbsLeftKanExtension (Adjunction.ofIsLeftAdjoint f).isAbsoluteLeftKan
+  | h => IsAbsKan.hasAbsLeftKanExtension (Adjunction.ofIsLeftAdjoint f).isAbsoluteLeftKan
   tfae_have 2 → 3
-  · intro h
-    exact ⟨inferInstance, inferInstance⟩
+  | h => ⟨inferInstance, inferInstance⟩
   tfae_have 3 → 1
-  · intro ⟨h, h'⟩
-    exact .mk <| (lanIsKan f (𝟙 a)).adjunction <| Lan.CommuteWith.isKan f (𝟙 a) f
+  | ⟨h, h'⟩ => .mk <| (lanIsKan f (𝟙 a)).adjunction <| Lan.CommuteWith.isKan f (𝟙 a) f
   tfae_finish
 
 end LeftExtension
@@ -130,11 +127,13 @@ def Adjunction.isAbsoluteLeftKanLift {f : a ⟶ b} {u : b ⟶ a} (adj : f ⊣ u)
     (𝟙 _ ⊗≫ s.unit ▷ f ⊗≫ s.lift ◁ adj.counit ⊗≫ 𝟙 _ : h ≫ f ⟶ s.lift) <|
       calc _
       _ = 𝟙 _ ⊗≫ (_ ◁ adj.unit ≫ s.unit ▷ _) ⊗≫ s.lift ◁ adj.counit ▷ u ⊗≫ 𝟙 _ := by
-        simp [bicategoricalComp]
+        dsimp only [whisker_lift, StructuredArrow.mk_right, whisker_unit,
+          StructuredArrow.mk_hom_eq_self]
+        bicategory
       _ = s.unit ⊗≫ s.lift ◁ (rightZigzag adj.unit adj.counit) ⊗≫ 𝟙 _ := by
-        rw [whisker_exchange, rightZigzag]; simp [bicategoricalComp]
+        rw [whisker_exchange, rightZigzag]; bicategory
       _ = s.unit := by
-        rw [adj.right_triangle]; simp [bicategoricalComp]) <| by
+        rw [adj.right_triangle]; bicategory) <| by
       intro s τ₀
       ext
       /- We need to specify the type of `τ` to use the notation `⊗≫`. -/
@@ -142,13 +141,13 @@ def Adjunction.isAbsoluteLeftKanLift {f : a ⟶ b} {u : b ⟶ a} (adj : f ⊣ u)
       have hτ : h ◁ adj.unit ⊗≫ τ ▷ u = s.unit := by simpa [bicategoricalComp] using LeftLift.w τ₀
       calc τ
         _ = 𝟙 _ ⊗≫ h ◁ leftZigzag adj.unit adj.counit ⊗≫ τ ⊗≫ 𝟙 _ := by
-          rw [adj.left_triangle]; simp [bicategoricalComp]
+          rw [adj.left_triangle]; bicategory
         _ = 𝟙 _ ⊗≫ h ◁ adj.unit ▷ f ⊗≫ (_ ◁ adj.counit ≫ τ ▷ _) ⊗≫ 𝟙 _ := by
-          rw [leftZigzag]; simp [bicategoricalComp]
+          rw [leftZigzag]; bicategory
         _ = 𝟙 _ ⊗≫ (h ◁ adj.unit ⊗≫ τ ▷ u) ▷ f ⊗≫ s.lift ◁ adj.counit ⊗≫ 𝟙 _ := by
-          rw [whisker_exchange]; simp [bicategoricalComp]
+          rw [whisker_exchange]; bicategory
         _ = _ := by
-          rw [hτ]; simp [bicategoricalComp]
+          rw [hτ]; dsimp only [StructuredArrow.homMk_right]
 
 /-- A left Kan lift of the identity along `u` such that `u` commutes with is a left adjoint
 to `u`. The unit of this adjoint is given by the unit of the Kan lift. -/
@@ -165,13 +164,13 @@ def LeftLift.IsKan.adjunction {u : b ⟶ a} {t : LeftLift u (𝟙 a)}
       apply H.hom_ext
       calc _
         _ = 𝟙 _ ⊗≫ t.unit ⊗≫ leftZigzag t.unit ε ▷ u ⊗≫ 𝟙 _ := by
-          simp [bicategoricalComp]
+          bicategory
         _ = 𝟙 _ ⊗≫ (_ ◁ t.unit ≫ t.unit ▷ _) ⊗≫ t.lift ◁ ε ▷ u ⊗≫ 𝟙 _ := by
-          rw [leftZigzag]; simp [bicategoricalComp]
+          rw [leftZigzag]; bicategory
         _ = 𝟙 _ ⊗≫ t.unit ⊗≫ t.lift ◁ (u ◁ t.unit ⊗≫ ε ▷ u) ⊗≫ 𝟙 _ := by
-          rw [whisker_exchange]; simp [bicategoricalComp]
+          rw [whisker_exchange]; bicategory
         _ = _ := by
-          rw [← rightZigzag, Hε]; simp [bicategoricalComp]
+          rw [← rightZigzag, Hε]; bicategory
     right_triangle := Hε }
 
 /-- For an adjuntion `f ⊣ u`, `f` is a left Kan lift of the identity along `u`.
@@ -186,14 +185,11 @@ theorem isRightAdjoint_TFAE (u : b ⟶ a) :
       HasAbsLeftKanLift u (𝟙 a),
       ∃ _ : HasLeftKanLift u (𝟙 a), LanLift.CommuteWith u (𝟙 a) u] := by
   tfae_have 1 → 2
-  · intro h
-    exact IsAbsKan.hasAbsLeftKanLift (Adjunction.ofIsRightAdjoint u).isAbsoluteLeftKanLift
+  | h => IsAbsKan.hasAbsLeftKanLift (Adjunction.ofIsRightAdjoint u).isAbsoluteLeftKanLift
   tfae_have 2 → 3
-  · intro h
-    exact ⟨inferInstance, inferInstance⟩
+  | h => ⟨inferInstance, inferInstance⟩
   tfae_have 3 → 1
-  · intro ⟨h, h'⟩
-    exact .mk <| (lanLiftIsKan u (𝟙 a)).adjunction <| LanLift.CommuteWith.isKan u (𝟙 a) u
+  | ⟨h, h'⟩ => .mk <| (lanLiftIsKan u (𝟙 a)).adjunction <| LanLift.CommuteWith.isKan u (𝟙 a) u
   tfae_finish
 
 end LeftLift
@@ -210,18 +206,28 @@ def isKanOfWhiskerLeftAdjoint
   .mk (fun s ↦
     let k := s.extension
     let θ := s.unit
-    let τ : t.extension ⟶ k ≫ u := H.desc (.mk _ <| 𝟙 _ ⊗≫ g ◁ η' ⊗≫ θ ▷ u ⊗≫ 𝟙 _)
-    let σ : t.extension ≫ h ⟶ k := H'.desc <| (.mk _ <| (ρ_ _).hom ≫ τ)
+    let sτ := LeftExtension.mk _ <| 𝟙 _ ⊗≫ g ◁ η' ⊗≫ θ ▷ u ⊗≫ 𝟙 _
+    let τ : t.extension ⟶ k ≫ u := H.desc sτ
+    let sσ := LeftLift.mk _ <| (ρ_ _).hom ≫ τ
+    let σ : t.extension ≫ h ⟶ k := H'.desc sσ
     LeftExtension.homMk σ <| (H' g).hom_ext <| by
       have Hσ : t.extension ◁ η' ⊗≫ σ ▷ u  = 𝟙 _ ⊗≫ τ := by
         simpa [bicategoricalComp] using (H' _).fac (.mk _ <| (ρ_ _).hom ≫ τ)
+      dsimp only [LeftLift.whisker_lift, StructuredArrow.mk_right, LeftLift.whisker_unit,
+        StructuredArrow.mk_hom_eq_self, whisker_extension, whisker_unit]
       calc _
-        _ = 𝟙 _ ⊗≫ (g ◁ η' ≫ t.unit ▷ (h ≫ u)) ⊗≫ f ◁ σ ▷ u ⊗≫ 𝟙 _ := by
-          simp [bicategoricalComp]
-        _ = 𝟙 _ ⊗≫ t.unit ▷ (𝟙 c) ⊗≫ f ◁ (t.extension ◁ η' ⊗≫ σ ▷ u) ⊗≫ 𝟙 _ := by
-          rw [whisker_exchange]; simp [bicategoricalComp]
+        _ = (g ◁ η' ≫ t.unit ▷ (h ≫ u)) ⊗≫ f ◁ σ ▷ u ⊗≫ 𝟙 _ := by
+          bicategory
+        _ = t.unit ▷ (𝟙 c) ⊗≫ f ◁ (t.extension ◁ η' ⊗≫ σ ▷ u) ⊗≫ 𝟙 _ := by
+          rw [whisker_exchange]; bicategory
+        _ = (ρ_ g).hom ≫ t.unit ≫ f ◁ H.desc sτ ≫ (α_ f s.extension u).inv := by
+          rw [Hσ]
+          dsimp only [τ]
+          bicategory
         _ = _ := by
-          rw [Hσ]; simp [τ, bicategoricalComp]) <| by
+          rw [IsKan.fac_assoc]
+          dsimp only [StructuredArrow.mk_right, StructuredArrow.mk_hom_eq_self, sτ]
+          bicategory) <| by
     intro s' τ₀'
     let τ' : t.extension ≫ h ⟶ s'.extension := τ₀'.right
     have Hτ' : t.unit ▷ h ⊗≫ f ◁ τ' = s'.unit := by simpa [bicategoricalComp] using τ₀'.w.symm
@@ -231,13 +237,20 @@ def isKanOfWhiskerLeftAdjoint
     rw [(H' _).fac]
     apply (cancel_epi (ρ_ _).inv).mp
     apply H.hom_ext
+    dsimp only [LeftLift.whisker_lift, StructuredArrow.mk_right, LeftLift.whisker_unit,
+      StructuredArrow.mk_hom_eq_self]
+    let σs' := LeftExtension.mk (s'.extension ≫ u)
+      (𝟙 g ⊗≫ g ◁ η' ⊗≫ s'.unit ▷ u ⊗≫ 𝟙 (f ≫ s'.extension ≫ u))
     calc _
       _ = 𝟙 _ ⊗≫ (t.unit ▷ (𝟙 c) ≫ (f ≫ t.extension) ◁ η') ⊗≫ f ◁ τ' ▷ u := by
-        simp [bicategoricalComp]
+        bicategory
       _ = 𝟙 g ⊗≫ g ◁ η' ⊗≫ (t.unit ▷ h ⊗≫ f ◁ τ') ▷ u ⊗≫ 𝟙 _ := by
-        rw [← whisker_exchange]; simp [bicategoricalComp]
+        rw [← whisker_exchange]; bicategory
+      _ = t.unit ≫ f ◁ H.desc σs' := by
+        rw [Hτ', IsKan.fac]
+        dsimp only [StructuredArrow.mk_hom_eq_self, σs']
       _ = _ := by
-        rw [Hτ']; simp [bicategoricalComp]
+        bicategory
 
 instance {f : a ⟶ b} {g : a ⟶ c} {x : B} {h : c ⟶ x} [IsLeftAdjoint h] [HasLeftKanExtension f g] :
     Lan.CommuteWith f g h :=
diff --git a/Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean b/Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean
index 4bce6d2322466..3ddbc22d951aa 100644
--- a/Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean
+++ b/Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean
@@ -106,6 +106,9 @@ instance locallyDiscreteBicategory.strict : Strict (LocallyDiscrete C) where
   comp_id f := Discrete.ext (Category.comp_id _)
   assoc f g h := Discrete.ext (Category.assoc _ _ _)
 
+attribute [local simp]
+  Strict.leftUnitor_eqToIso Strict.rightUnitor_eqToIso Strict.associator_eqToIso
+
 variable {I : Type u₁} [Category.{v₁} I] {B : Type u₂} [Bicategory.{w₂, v₂} B] [Strict B]
 
 /--
diff --git a/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.lean b/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.lean
index 5a1b8b335b237..d6c4159d767cf 100644
--- a/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.lean
+++ b/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Oplax.lean
@@ -92,7 +92,7 @@ variable {a b c : B} {a' : C}
 theorem whiskerLeft_naturality_naturality (f : a' ⟶ G.obj a) {g h : a ⟶ b} (β : g ⟶ h) :
     f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ θ.naturality h =
       f ◁ θ.naturality g ≫ f ◁ θ.app a ◁ H.map₂ β := by
-  simp_rw [← whiskerLeft_comp, naturality_naturality]
+  simp_rw [← Bicategory.whiskerLeft_comp, naturality_naturality]
 
 @[reassoc (attr := simp)]
 theorem whiskerRight_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :
@@ -107,7 +107,7 @@ theorem whiskerLeft_naturality_comp (f : a' ⟶ G.obj a) (g : a ⟶ b) (h : b 
         f ◁ (α_ _ _ _).hom ≫
           f ◁ G.map g ◁ θ.naturality h ≫
             f ◁ (α_ _ _ _).inv ≫ f ◁ θ.naturality g ▷ H.map h ≫ f ◁ (α_ _ _ _).hom := by
-  simp_rw [← whiskerLeft_comp, naturality_comp]
+  simp_rw [← Bicategory.whiskerLeft_comp, naturality_comp]
 
 @[reassoc (attr := simp)]
 theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c ⟶ a') :
@@ -125,7 +125,7 @@ theorem whiskerRight_naturality_comp (f : a ⟶ b) (g : b ⟶ c) (h : G.obj c 
 theorem whiskerLeft_naturality_id (f : a' ⟶ G.obj a) :
     f ◁ θ.naturality (𝟙 a) ≫ f ◁ θ.app a ◁ H.mapId a =
       f ◁ G.mapId a ▷ θ.app a ≫ f ◁ (λ_ (θ.app a)).hom ≫ f ◁ (ρ_ (θ.app a)).inv := by
-  simp_rw [← whiskerLeft_comp, naturality_id]
+  simp_rw [← Bicategory.whiskerLeft_comp, naturality_id]
 
 @[reassoc (attr := simp)]
 theorem whiskerRight_naturality_id (f : G.obj a ⟶ a') :
@@ -221,7 +221,7 @@ variable (Γ : Modification η θ) {a b c : B} {a' : C}
 @[reassoc (attr := simp)]
 theorem whiskerLeft_naturality (f : a' ⟶ F.obj b) (g : b ⟶ c) :
     f ◁ F.map g ◁ Γ.app c ≫ f ◁ θ.naturality g = f ◁ η.naturality g ≫ f ◁ Γ.app b ▷ G.map g := by
-  simp_rw [← whiskerLeft_comp, naturality]
+  simp_rw [← Bicategory.whiskerLeft_comp, naturality]
 
 @[reassoc (attr := simp)]
 theorem whiskerRight_naturality (f : a ⟶ b) (g : G.obj b ⟶ a') :
diff --git a/Mathlib/CategoryTheory/Bicategory/SingleObj.lean b/Mathlib/CategoryTheory/Bicategory/SingleObj.lean
index 7619fce58c22c..73341c73d8e16 100644
--- a/Mathlib/CategoryTheory/Bicategory/SingleObj.lean
+++ b/Mathlib/CategoryTheory/Bicategory/SingleObj.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Bicategory.End
 import Mathlib.CategoryTheory.Monoidal.Functor
diff --git a/Mathlib/CategoryTheory/Bicategory/Strict.lean b/Mathlib/CategoryTheory/Bicategory/Strict.lean
index 7e0bd006126ab..1edfaa816f646 100644
--- a/Mathlib/CategoryTheory/Bicategory/Strict.lean
+++ b/Mathlib/CategoryTheory/Bicategory/Strict.lean
@@ -50,15 +50,6 @@ class Bicategory.Strict : Prop where
     ∀ {a b c d : B} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d), α_ f g h = eqToIso (assoc f g h) := by
     aesop_cat
 
--- Porting note: not adding simp to:
--- Bicategory.Strict.id_comp
--- Bicategory.Strict.comp_id
--- Bicategory.Strict.assoc
-attribute [simp]
-   Bicategory.Strict.leftUnitor_eqToIso
-   Bicategory.Strict.rightUnitor_eqToIso
-   Bicategory.Strict.associator_eqToIso
-
 -- see Note [lower instance priority]
 /-- Category structure on a strict bicategory -/
 instance (priority := 100) StrictBicategory.category [Bicategory.Strict B] : Category B where
diff --git a/Mathlib/CategoryTheory/Category/Basic.lean b/Mathlib/CategoryTheory/Category/Basic.lean
index 7ec4a03391b45..d314808670795 100644
--- a/Mathlib/CategoryTheory/Category/Basic.lean
+++ b/Mathlib/CategoryTheory/Category/Basic.lean
@@ -1,12 +1,13 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
+Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Reid Barton
 -/
 import Mathlib.CategoryTheory.Category.Init
 import Mathlib.Combinatorics.Quiver.Basic
 import Mathlib.Tactic.PPWithUniv
 import Mathlib.Tactic.Common
+import Mathlib.Tactic.StacksAttribute
 
 /-!
 # Categories
@@ -260,10 +261,18 @@ instance (X : C) : Mono (𝟙 X) :=
 theorem cancel_epi (f : X ⟶ Y) [Epi f] {g h : Y ⟶ Z} : f ≫ g = f ≫ h ↔ g = h :=
   ⟨fun p => Epi.left_cancellation g h p, congr_arg _⟩
 
+theorem cancel_epi_assoc_iff (f : X ⟶ Y) [Epi f] {g h : Y ⟶ Z} {W : C} {k l : Z ⟶ W} :
+    (f ≫ g) ≫ k = (f ≫ h) ≫ l ↔ g ≫ k = h ≫ l :=
+  ⟨fun p => (cancel_epi f).1 <| by simpa using p, fun p => by simp only [Category.assoc, p]⟩
+
 theorem cancel_mono (f : X ⟶ Y) [Mono f] {g h : Z ⟶ X} : g ≫ f = h ≫ f ↔ g = h :=
   -- Porting note: in Lean 3 we could just write `congr_arg _` here.
   ⟨fun p => Mono.right_cancellation g h p, congr_arg (fun k => k ≫ f)⟩
 
+theorem cancel_mono_assoc_iff (f : X ⟶ Y) [Mono f] {g h : Z ⟶ X} {W : C} {k l : W ⟶ Z} :
+    k ≫ (g ≫ f) = l ≫ (h ≫ f) ↔ k ≫ g = l ≫ h :=
+  ⟨fun p => (cancel_mono f).1 <| by simpa using p, fun p => by simp only [← Category.assoc, p]⟩
+
 theorem cancel_epi_id (f : X ⟶ Y) [Epi f] {h : Y ⟶ Y} : f ≫ h = f ↔ h = 𝟙 Y := by
   convert cancel_epi f
   simp
@@ -272,40 +281,21 @@ theorem cancel_mono_id (f : X ⟶ Y) [Mono f] {g : X ⟶ X} : g ≫ f = f ↔ g
   convert cancel_mono f
   simp
 
-theorem epi_comp {X Y Z : C} (f : X ⟶ Y) [Epi f] (g : Y ⟶ Z) [Epi g] : Epi (f ≫ g) := by
-  constructor
-  intro Z a b w
-  apply (cancel_epi g).1
-  apply (cancel_epi f).1
-  simpa using w
-
-theorem mono_comp {X Y Z : C} (f : X ⟶ Y) [Mono f] (g : Y ⟶ Z) [Mono g] : Mono (f ≫ g) := by
-  constructor
-  intro Z a b w
-  apply (cancel_mono f).1
-  apply (cancel_mono g).1
-  simpa using w
-
-theorem mono_of_mono {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Mono (f ≫ g)] : Mono f := by
-  constructor
-  intro Z a b w
-  replace w := congr_arg (fun k => k ≫ g) w
-  dsimp at w
-  rw [Category.assoc, Category.assoc] at w
-  exact (cancel_mono _).1 w
+instance epi_comp {X Y Z : C} (f : X ⟶ Y) [Epi f] (g : Y ⟶ Z) [Epi g] : Epi (f ≫ g) :=
+  ⟨fun _ _ w => (cancel_epi g).1 <| (cancel_epi_assoc_iff f).1 w⟩
+
+instance mono_comp {X Y Z : C} (f : X ⟶ Y) [Mono f] (g : Y ⟶ Z) [Mono g] : Mono (f ≫ g) :=
+  ⟨fun _ _ w => (cancel_mono f).1 <| (cancel_mono_assoc_iff g).1 w⟩
+
+theorem mono_of_mono {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Mono (f ≫ g)] : Mono f :=
+  ⟨fun _ _ w => (cancel_mono (f ≫ g)).1 <| by simp only [← Category.assoc, w]⟩
 
 theorem mono_of_mono_fac {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [Mono h]
     (w : f ≫ g = h) : Mono f := by
-  subst h
-  exact mono_of_mono f g
-
-theorem epi_of_epi {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Epi (f ≫ g)] : Epi g := by
-  constructor
-  intro Z a b w
-  replace w := congr_arg (fun k => f ≫ k) w
-  dsimp at w
-  rw [← Category.assoc, ← Category.assoc] at w
-  exact (cancel_epi _).1 w
+  subst h; exact mono_of_mono f g
+
+theorem epi_of_epi {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Epi (f ≫ g)] : Epi g :=
+  ⟨fun _ _ w => (cancel_epi (f ≫ g)).1 <| by simp only [Category.assoc, w]⟩
 
 theorem epi_of_epi_fac {X Y Z : C} {f : X ⟶ Y} {g : Y ⟶ Z} {h : X ⟶ Z} [Epi h]
     (w : f ≫ g = h) : Epi g := by
diff --git a/Mathlib/CategoryTheory/Category/Cat.lean b/Mathlib/CategoryTheory/Category/Cat.lean
index 3d4788b016f0d..9838804baf16f 100644
--- a/Mathlib/CategoryTheory/Category/Cat.lean
+++ b/Mathlib/CategoryTheory/Category/Cat.lean
@@ -133,6 +133,14 @@ lemma associator_inv_app {B C D E : Cat} (F : B ⟶ C) (G : C ⟶ D) (H : D ⟶
     (α_ F G H).inv.app X = eqToHom (by simp) :=
   rfl
 
+/-- The identity in the category of categories equals the identity functor.-/
+theorem id_eq_id (X : Cat) : 𝟙 X = 𝟭 X := rfl
+
+/-- Composition in the category of categories equals functor composition.-/
+theorem comp_eq_comp {X Y Z : Cat} (F : X ⟶ Y) (G : Y ⟶ Z) : F ≫ G = F ⋙ G := rfl
+
+@[simp] theorem of_α (C) [Category C] : (of C).α = C := rfl
+
 /-- Functor that gets the set of objects of a category. It is not
 called `forget`, because it is not a faithful functor. -/
 def objects : Cat.{v, u} ⥤ Type u where
diff --git a/Mathlib/CategoryTheory/Category/Cat/Adjunction.lean b/Mathlib/CategoryTheory/Category/Cat/Adjunction.lean
index 55e17bc93bec4..229d0fbddea78 100644
--- a/Mathlib/CategoryTheory/Category/Cat/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Category/Cat/Adjunction.lean
@@ -39,15 +39,16 @@ private def typeToCatObjectsAdjCounitApp : (Cat.objects ⋙ typeToCat).obj C ⥤
   map := eqToHom ∘ Discrete.eq_of_hom
 
 /-- `typeToCat : Type ⥤ Cat` is left adjoint to `Cat.objects : Cat ⥤ Type` -/
-def typeToCatObjectsAdj : typeToCat ⊣ Cat.objects where
-  homEquiv := typeToCatObjectsAdjHomEquiv
-  unit := { app:= fun _  ↦ Discrete.mk }
-  counit := {
-    app := typeToCatObjectsAdjCounitApp
-    naturality := fun _ _ _  ↦  Functor.hext (fun _ ↦ rfl)
-      (by intro ⟨_⟩ ⟨_⟩ f
-          obtain rfl := Discrete.eq_of_hom f
-          aesop_cat ) }
+def typeToCatObjectsAdj : typeToCat ⊣ Cat.objects :=
+  Adjunction.mk' {
+    homEquiv := typeToCatObjectsAdjHomEquiv
+    unit := { app:= fun _  ↦ Discrete.mk }
+    counit := {
+      app := typeToCatObjectsAdjCounitApp
+      naturality := fun _ _ _  ↦  Functor.hext (fun _ ↦ rfl)
+        (by intro ⟨_⟩ ⟨_⟩ f
+            obtain rfl := Discrete.eq_of_hom f
+            aesop_cat ) } }
 
 /-- The connected components functor -/
 def connectedComponents : Cat.{v, u} ⥤ Type u where
@@ -57,15 +58,20 @@ def connectedComponents : Cat.{v, u} ⥤ Type u where
   map_comp _ _ := funext fun x ↦ (Quotient.exists_rep x).elim (fun _ h => by subst h; rfl)
 
 /-- `typeToCat : Type ⥤ Cat` is right adjoint to `connectedComponents : Cat ⥤ Type` -/
-def connectedComponentsTypeToCatAdj : connectedComponents ⊣ typeToCat where
-  homEquiv C X := ConnectedComponents.typeToCatHomEquiv C X
-  unit := { app:= fun C  ↦ ConnectedComponents.functorToDiscrete _ (𝟙 (connectedComponents.obj C)) }
-  counit :=  {
-      app := fun X => ConnectedComponents.liftFunctor _ (𝟙 typeToCat.obj X)
-      naturality := fun _ _ _ =>
-        funext (fun xcc => by
-          obtain ⟨x,h⟩ := Quotient.exists_rep xcc
-          aesop_cat) }
-  homEquiv_counit := fun {C X G} => by funext cc;obtain ⟨_,_⟩ := Quotient.exists_rep cc; aesop_cat
+def connectedComponentsTypeToCatAdj : connectedComponents ⊣ typeToCat :=
+  Adjunction.mk' {
+    homEquiv := fun C X ↦ ConnectedComponents.typeToCatHomEquiv C X
+    unit :=
+      { app:= fun C  ↦ ConnectedComponents.functorToDiscrete _ (𝟙 (connectedComponents.obj C)) }
+    counit :=  {
+        app := fun X => ConnectedComponents.liftFunctor _ (𝟙 typeToCat.obj X)
+        naturality := fun _ _ _ =>
+          funext (fun xcc => by
+            obtain ⟨x,h⟩ := Quotient.exists_rep xcc
+            aesop_cat) }
+    homEquiv_counit := fun {C X G} => by
+      funext cc
+      obtain ⟨_,_⟩ := Quotient.exists_rep cc
+      aesop_cat }
 
 end CategoryTheory.Cat
diff --git a/Mathlib/CategoryTheory/Category/Cat/Limit.lean b/Mathlib/CategoryTheory/Category/Cat/Limit.lean
index 6b3e9d578c572..8fb276edac4be 100644
--- a/Mathlib/CategoryTheory/Category/Cat/Limit.lean
+++ b/Mathlib/CategoryTheory/Category/Cat/Limit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Category.Cat
 import Mathlib.CategoryTheory.Limits.Types
diff --git a/Mathlib/CategoryTheory/Category/GaloisConnection.lean b/Mathlib/CategoryTheory/Category/GaloisConnection.lean
index 41362d5292353..a79e29a39e4c2 100644
--- a/Mathlib/CategoryTheory/Category/GaloisConnection.lean
+++ b/Mathlib/CategoryTheory/Category/GaloisConnection.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
+Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Reid Barton
 -/
 import Mathlib.CategoryTheory.Category.Preorder
 import Mathlib.CategoryTheory.Adjunction.Basic
diff --git a/Mathlib/CategoryTheory/Category/Pairwise.lean b/Mathlib/CategoryTheory/Category/Pairwise.lean
index a8e795bdc19ac..7b800d357beb0 100644
--- a/Mathlib/CategoryTheory/Category/Pairwise.lean
+++ b/Mathlib/CategoryTheory/Category/Pairwise.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Order.CompleteLattice
 import Mathlib.CategoryTheory.Category.Preorder
diff --git a/Mathlib/CategoryTheory/Category/PartialFun.lean b/Mathlib/CategoryTheory/Category/PartialFun.lean
index 1cb8f8be49550..0ed2b108fb523 100644
--- a/Mathlib/CategoryTheory/Category/PartialFun.lean
+++ b/Mathlib/CategoryTheory/Category/PartialFun.lean
@@ -80,6 +80,7 @@ def typeToPartialFun : Type u ⥤ PartialFun where
 instance : typeToPartialFun.Faithful where
   map_injective {_ _} := PFun.lift_injective
 
+-- b ∈ PFun.toSubtype (fun x ↦ x ≠ X.point) Subtype.val a ↔ b ∈ Part.some a
 /-- The functor which deletes the point of a pointed type. In return, this makes the maps partial.
 This is the computable part of the equivalence `PartialFunEquivPointed`. -/
 @[simps obj map]
@@ -87,7 +88,8 @@ def pointedToPartialFun : Pointed.{u} ⥤ PartialFun where
   obj X := { x : X // x ≠ X.point }
   map f := PFun.toSubtype _ f.toFun ∘ Subtype.val
   map_id X :=
-    PFun.ext fun a b => PFun.mem_toSubtype_iff.trans (Subtype.coe_inj.trans Part.mem_some_iff.symm)
+    PFun.ext fun a b =>
+      PFun.mem_toSubtype_iff (b := b).trans (Subtype.coe_inj.trans Part.mem_some_iff.symm)
   map_comp f g := by
     -- Porting note: the proof was changed because the original mathlib3 proof no longer works
     apply PFun.ext _
diff --git a/Mathlib/CategoryTheory/Category/Preorder.lean b/Mathlib/CategoryTheory/Category/Preorder.lean
index f46afa90197a5..f2935460a2529 100644
--- a/Mathlib/CategoryTheory/Category/Preorder.lean
+++ b/Mathlib/CategoryTheory/Category/Preorder.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
+Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Reid Barton
 -/
 import Mathlib.CategoryTheory.Equivalence
 import Mathlib.CategoryTheory.EqToHom
diff --git a/Mathlib/CategoryTheory/Category/Quiv.lean b/Mathlib/CategoryTheory/Category/Quiv.lean
index 40f840da97067..be83ad9b038c2 100644
--- a/Mathlib/CategoryTheory/Category/Quiv.lean
+++ b/Mathlib/CategoryTheory/Category/Quiv.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.Basic
 import Mathlib.CategoryTheory.Category.Cat
@@ -11,10 +11,8 @@ import Mathlib.CategoryTheory.PathCategory
 # The category of quivers
 
 The category of (bundled) quivers, and the free/forgetful adjunction between `Cat` and `Quiv`.
-
 -/
 
-
 universe v u
 
 namespace CategoryTheory
@@ -51,6 +49,12 @@ def forget : Cat.{v, u} ⥤ Quiv.{v, u} where
   obj C := Quiv.of C
   map F := F.toPrefunctor
 
+/-- The identity in the category of quivers equals the identity prefunctor.-/
+theorem id_eq_id (X : Quiv) : 𝟙 X = 𝟭q X := rfl
+
+/-- Composition in the category of quivers equals prefunctor composition.-/
+theorem comp_eq_comp {X Y Z : Quiv} (F : X ⟶ Y) (G : Y ⟶ Z) : F ≫ G = F ⋙q G := rfl
+
 end Quiv
 
 namespace Cat
@@ -65,14 +69,14 @@ def free : Quiv.{v, u} ⥤ Cat.{max u v, u} where
       map_comp := fun f g => F.mapPath_comp f g }
   map_id V := by
     change (show Paths V ⥤ _ from _) = _
-    ext; swap
-    · apply eq_conj_eqToHom
+    ext
     · rfl
+    · exact eq_conj_eqToHom _
   map_comp {U _ _} F G := by
     change (show Paths U ⥤ _ from _) = _
-    ext; swap
-    · apply eq_conj_eqToHom
+    ext
     · rfl
+    · exact eq_conj_eqToHom _
 
 end Cat
 
@@ -105,9 +109,9 @@ def adj : Cat.free ⊣ Quiv.forget :=
             exact Category.id_comp _ }
       homEquiv_naturality_left_symm := fun {V _ _} f g => by
         change (show Paths V ⥤ _ from _) = _
-        ext; swap
-        · apply eq_conj_eqToHom
-        · rfl }
+        ext
+        · rfl
+        · apply eq_conj_eqToHom }
 
 end Quiv
 
diff --git a/Mathlib/CategoryTheory/Category/ReflQuiv.lean b/Mathlib/CategoryTheory/Category/ReflQuiv.lean
new file mode 100644
index 0000000000000..f26afec53930b
--- /dev/null
+++ b/Mathlib/CategoryTheory/Category/ReflQuiv.lean
@@ -0,0 +1,251 @@
+/-
+Copyright (c) 2024 Mario Carneiro and Emily Riehl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro, Emily Riehl
+-/
+import Mathlib.Combinatorics.Quiver.ReflQuiver
+import Mathlib.CategoryTheory.Category.Cat
+import Mathlib.CategoryTheory.Category.Quiv
+
+/-!
+# The category of refl quivers
+
+The category `ReflQuiv` of (bundled) reflexive quivers, and the free/forgetful adjunction between
+`Cat` and `ReflQuiv`.
+-/
+
+namespace CategoryTheory
+universe v u
+
+/-- Category of refl quivers. -/
+@[nolint checkUnivs]
+def ReflQuiv :=
+  Bundled ReflQuiver.{v + 1, u}
+
+namespace ReflQuiv
+
+instance : CoeSort ReflQuiv (Type u) where coe := Bundled.α
+
+instance (C : ReflQuiv.{v, u}) : ReflQuiver.{v + 1, u} C := C.str
+
+/-- The underlying quiver of a reflexive quiver.-/
+def toQuiv (C : ReflQuiv.{v, u}) : Quiv.{v, u} := Quiv.of C.α
+
+/-- Construct a bundled `ReflQuiv` from the underlying type and the typeclass. -/
+def of (C : Type u) [ReflQuiver.{v + 1} C] : ReflQuiv.{v, u} := Bundled.of C
+
+instance : Inhabited ReflQuiv := ⟨ReflQuiv.of (Discrete default)⟩
+
+@[simp] theorem of_val (C : Type u) [ReflQuiver C] : (ReflQuiv.of C) = C := rfl
+
+/-- Category structure on `ReflQuiv` -/
+instance category : LargeCategory.{max v u} ReflQuiv.{v, u} where
+  Hom C D := ReflPrefunctor C D
+  id C := ReflPrefunctor.id C
+  comp F G := ReflPrefunctor.comp F G
+
+theorem id_eq_id (X : ReflQuiv) : 𝟙 X = 𝟭rq X := rfl
+theorem comp_eq_comp {X Y Z : ReflQuiv} (F : X ⟶ Y) (G : Y ⟶ Z) : F ≫ G = F ⋙rq G := rfl
+
+/-- The forgetful functor from categories to quivers. -/
+@[simps]
+def forget : Cat.{v, u} ⥤ ReflQuiv.{v, u} where
+  obj C := ReflQuiv.of C
+  map F := F.toReflPrefunctor
+
+theorem forget_faithful {C D : Cat.{v, u}} (F G : C ⥤ D)
+    (hyp : forget.map F = forget.map G) : F = G := by
+  cases F; cases G; cases hyp; rfl
+
+theorem forget.Faithful : Functor.Faithful (forget) where
+  map_injective := fun hyp ↦ forget_faithful _ _ hyp
+
+/-- The forgetful functor from categories to quivers. -/
+@[simps]
+def forgetToQuiv : ReflQuiv.{v, u} ⥤ Quiv.{v, u} where
+  obj V := Quiv.of V
+  map F := F.toPrefunctor
+
+theorem forgetToQuiv_faithful {V W : ReflQuiv} (F G : V ⥤rq W)
+    (hyp : forgetToQuiv.map F = forgetToQuiv.map G) : F = G := by
+  cases F; cases G; cases hyp; rfl
+
+theorem forgetToQuiv.Faithful : Functor.Faithful (forgetToQuiv) where
+  map_injective := fun hyp ↦ forgetToQuiv_faithful _ _ hyp
+
+theorem forget_forgetToQuiv : forget ⋙ forgetToQuiv = Quiv.forget := rfl
+
+end ReflQuiv
+
+namespace ReflPrefunctor
+
+/-- A refl prefunctor can be promoted to a functor if it respects composition.-/
+def toFunctor {C D : Cat} (F : (ReflQuiv.of C) ⟶ (ReflQuiv.of D))
+    (hyp : ∀ {X Y Z : ↑C} (f : X ⟶ Y) (g : Y ⟶ Z),
+      F.map (CategoryStruct.comp (obj := C) f g) =
+        CategoryStruct.comp (obj := D) (F.map f) (F.map g)) : C ⥤ D where
+  obj := F.obj
+  map := F.map
+  map_id := F.map_id
+  map_comp := hyp
+
+end ReflPrefunctor
+
+namespace Cat
+
+/-- The hom relation that identifies the specified reflexivity arrows with the nil paths.-/
+inductive FreeReflRel {V} [ReflQuiver V] : (X Y : Paths V) → (f g : X ⟶ Y) → Prop
+  | mk {X : V} : FreeReflRel X X (Quiver.Hom.toPath (𝟙rq X)) .nil
+
+/-- A reflexive quiver generates a free category, defined as as quotient of the free category
+on its underlying quiver (called the "path category") by the hom relation that uses the specified
+reflexivity arrows as the identity arrows. -/
+def FreeRefl (V) [ReflQuiver V] :=
+  Quotient (C := Cat.free.obj (Quiv.of V)) (FreeReflRel (V := V))
+
+instance (V) [ReflQuiver V] : Category (FreeRefl V) :=
+  inferInstanceAs (Category (Quotient _))
+
+/-- The quotient functor associated to a quotient category defines a natural map from the free
+category on the underlying quiver of a refl quiver to the free category on the reflexive quiver.-/
+def FreeRefl.quotientFunctor (V) [ReflQuiver V] : Cat.free.obj (Quiv.of V) ⥤ FreeRefl V :=
+  Quotient.functor (C := Cat.free.obj (Quiv.of V)) (FreeReflRel (V := V))
+
+/-- This is a specialization of `Quotient.lift_unique'` rather than `Quotient.lift_unique`, hence
+the prime in the name.-/
+theorem FreeRefl.lift_unique' {V} [ReflQuiver V] {D} [Category D] (F₁ F₂ : FreeRefl V ⥤ D)
+    (h : quotientFunctor V ⋙ F₁ = quotientFunctor V ⋙ F₂) :
+    F₁ = F₂ :=
+  Quotient.lift_unique' (C := Cat.free.obj (Quiv.of V)) (FreeReflRel (V := V)) _ _ h
+
+/-- The functor sending a reflexive quiver to the free category it generates, a quotient of
+its path category.-/
+@[simps!]
+def freeRefl : ReflQuiv.{v, u} ⥤ Cat.{max u v, u} where
+  obj V := Cat.of (FreeRefl V)
+  map f := Quotient.lift _ ((by exact Cat.free.map f.toPrefunctor) ⋙ FreeRefl.quotientFunctor _)
+    (fun X Y f g hfg => by
+      apply Quotient.sound
+      cases hfg
+      simp [ReflPrefunctor.map_id]
+      constructor)
+  map_id X := by
+    dsimp
+    refine (Quotient.lift_unique _ _ _ _ ((Functor.comp_id _).trans <|
+      (Functor.id_comp _).symm.trans ?_)).symm
+    congr 1
+    exact (free.map_id X.toQuiv).symm
+  map_comp {X Y Z} f g := by
+    dsimp
+    apply (Quotient.lift_unique _ _ _ _ _).symm
+    have : free.map (f ≫ g).toPrefunctor =
+        free.map (X := X.toQuiv) (Y := Y.toQuiv) f.toPrefunctor ⋙
+        free.map (X := Y.toQuiv) (Y := Z.toQuiv) g.toPrefunctor := by
+      show _ = _ ≫ _
+      rw [← Functor.map_comp]; rfl
+    rw [this, Functor.assoc]
+    show _ ⋙ _ ⋙ _ = _
+    rw [← Functor.assoc, Quotient.lift_spec, Functor.assoc, FreeRefl.quotientFunctor,
+      Quotient.lift_spec]
+
+theorem freeRefl_naturality {X Y} [ReflQuiver X] [ReflQuiver Y] (f : X ⥤rq Y) :
+    free.map (X := Quiv.of X) (Y := Quiv.of Y) f.toPrefunctor ⋙
+    FreeRefl.quotientFunctor ↑Y =
+    FreeRefl.quotientFunctor ↑X ⋙ freeRefl.map (X := ReflQuiv.of X) (Y := ReflQuiv.of Y) f := by
+  simp only [free_obj, FreeRefl.quotientFunctor, freeRefl, ReflQuiv.of_val]
+  rw [Quotient.lift_spec]
+
+/-- We will make use of the natural quotient map from the free category on the underlying
+quiver of a refl quiver to the free category on the reflexive quiver.-/
+def freeReflNatTrans : ReflQuiv.forgetToQuiv ⋙ Cat.free ⟶ freeRefl where
+  app V := FreeRefl.quotientFunctor V
+  naturality _ _ f := freeRefl_naturality f
+
+end Cat
+
+namespace ReflQuiv
+open Category Functor
+
+/-- The unit components are defined as the composite of the corresponding unit component for the
+adjunction between categories and quivers with the map underlying the quotient functor.-/
+@[simps! toPrefunctor obj map]
+def adj.unit.app (V : ReflQuiv.{max u v, u}) : V ⥤rq forget.obj (Cat.freeRefl.obj V) where
+  toPrefunctor := Quiv.adj.unit.app (V.toQuiv) ⋙q
+    Quiv.forget.map (Cat.FreeRefl.quotientFunctor V)
+  map_id := fun _ => Quotient.sound _ ⟨⟩
+
+/-- This is used in the proof of both triangle equalities.-/
+theorem adj.unit.component_eq (V : ReflQuiv.{max u v, u}) :
+    forgetToQuiv.map (adj.unit.app V) = Quiv.adj.unit.app (V.toQuiv) ≫
+    Quiv.forget.map (Y := Cat.of _) (Cat.FreeRefl.quotientFunctor V) := rfl
+
+/-- The counit components are defined using the universal property of the quotient
+from the corresponding counit component for the adjunction between categories and quivers.-/
+@[simps!]
+def adj.counit.app (C : Cat) : Cat.freeRefl.obj (forget.obj C) ⥤ C := by
+  fapply Quotient.lift
+  · exact Quiv.adj.counit.app C
+  · intro x y f g rel
+    cases rel
+    unfold Quiv.adj
+    simp only [Adjunction.mkOfHomEquiv_counit_app, Equiv.coe_fn_symm_mk,
+      Quiv.lift_map, Prefunctor.mapPath_toPath, composePath_toPath]
+    rfl
+
+/-- The counit of `ReflQuiv.adj` is closely related to the counit of `Quiv.adj`.-/
+@[simp]
+theorem adj.counit.component_eq (C : Cat) :
+    Cat.FreeRefl.quotientFunctor C ⋙ adj.counit.app C =
+    Quiv.adj.counit.app C := rfl
+
+/-- The counit of `ReflQuiv.adj` is closely related to the counit of `Quiv.adj`. For ease of use,
+we introduce primed version for unbundled categories.-/
+@[simp]
+theorem adj.counit.component_eq' (C) [Category C] :
+    Cat.FreeRefl.quotientFunctor C ⋙ adj.counit.app (Cat.of C) =
+    Quiv.adj.counit.app (Cat.of C) := rfl
+
+/--
+The adjunction between forming the free category on a reflexive quiver, and forgetting a category
+to a reflexive quiver.
+-/
+nonrec def adj : Cat.freeRefl.{max u v, u} ⊣ ReflQuiv.forget :=
+  Adjunction.mkOfUnitCounit {
+    unit := {
+      app := adj.unit.app
+      naturality := fun V W f ↦ by exact rfl
+    }
+    counit := {
+      app := adj.counit.app
+      naturality := fun C D F ↦ Quotient.lift_unique' _ _ _ (Quiv.adj.counit.naturality F)
+    }
+    left_triangle := by
+      ext V
+      apply Cat.FreeRefl.lift_unique'
+      simp only [id_obj, Cat.free_obj, comp_obj, Cat.freeRefl_obj_α, NatTrans.comp_app,
+        forget_obj, whiskerRight_app, associator_hom_app, whiskerLeft_app, id_comp,
+        NatTrans.id_app']
+      rw [Cat.id_eq_id, Cat.comp_eq_comp]
+      simp only [Cat.freeRefl_obj_α, Functor.comp_id]
+      rw [← Functor.assoc, ← Cat.freeRefl_naturality, Functor.assoc]
+      dsimp [Cat.freeRefl]
+      rw [adj.counit.component_eq' (Cat.FreeRefl V)]
+      conv =>
+        enter [1, 1, 2]
+        apply (Quiv.comp_eq_comp (X := Quiv.of _) (Y := Quiv.of _) (Z := Quiv.of _) ..).symm
+      rw [Cat.free.map_comp]
+      show (_ ⋙ ((Quiv.forget ⋙ Cat.free).map (X := Cat.of _) (Y := Cat.of _)
+        (Cat.FreeRefl.quotientFunctor V))) ⋙ _ = _
+      rw [Functor.assoc, ← Cat.comp_eq_comp]
+      conv => enter [1, 2]; apply Quiv.adj.counit.naturality
+      rw [Cat.comp_eq_comp, ← Functor.assoc, ← Cat.comp_eq_comp]
+      conv => enter [1, 1]; apply Quiv.adj.left_triangle_components V.toQuiv
+      exact Functor.id_comp _
+    right_triangle := by
+      ext C
+      exact forgetToQuiv_faithful _ _ (Quiv.adj.right_triangle_components C)
+  }
+
+end ReflQuiv
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Category/RelCat.lean b/Mathlib/CategoryTheory/Category/RelCat.lean
index 48f8c610da96b..7136d606db093 100644
--- a/Mathlib/CategoryTheory/Category/RelCat.lean
+++ b/Mathlib/CategoryTheory/Category/RelCat.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Uni Marx
+Authors: Kim Morrison, Uni Marx
 -/
 import Mathlib.CategoryTheory.Iso
 import Mathlib.CategoryTheory.EssentialImage
diff --git a/Mathlib/CategoryTheory/Closed/Cartesian.lean b/Mathlib/CategoryTheory/Closed/Cartesian.lean
index 7dc0f610b435c..e25efebbe63d0 100644
--- a/Mathlib/CategoryTheory/Closed/Cartesian.lean
+++ b/Mathlib/CategoryTheory/Closed/Cartesian.lean
@@ -205,10 +205,10 @@ theorem eq_curry_iff (f : A ⨯ Y ⟶ X) (g : Y ⟶ A ⟹ X) : g = curry f ↔ u
 
 -- I don't think these two should be simp.
 theorem uncurry_eq (g : Y ⟶ A ⟹ X) : uncurry g = Limits.prod.map (𝟙 A) g ≫ (exp.ev A).app X :=
-  Adjunction.homEquiv_counit _
+  rfl
 
 theorem curry_eq (g : A ⨯ Y ⟶ X) : curry g = (exp.coev A).app Y ≫ (exp A).map g :=
-  Adjunction.homEquiv_unit _
+  rfl
 
 theorem uncurry_id_eq_ev (A X : C) [Exponentiable A] : uncurry (𝟙 (A ⟹ X)) = (exp.ev A).app X := by
   rw [uncurry_eq, prod.map_id_id, id_comp]
@@ -226,21 +226,15 @@ end CartesianClosed
 
 open CartesianClosed
 
-/-- Show that the exponential of the terminal object is isomorphic to itself, i.e. `X^1 ≅ X`.
+/-- The exponential with the terminal object is naturally isomorphic to the identity. The typeclass
+argument is explicit: any instance can be used.-/
+def expTerminalNatIso [Exponentiable (⊤_ C)] : 𝟭 C ≅ exp (⊤_ C) :=
+  MonoidalClosed.unitNatIso (C := C)
 
-The typeclass argument is explicit: any instance can be used.
--/
+/-- The exponential of any object with the terminal object is isomorphic to itself, i.e. `X^1 ≅ X`.
+The typeclass argument is explicit: any instance can be used.-/
 def expTerminalIsoSelf [Exponentiable (⊤_ C)] : (⊤_ C) ⟹ X ≅ X :=
-  Yoneda.ext ((⊤_ C) ⟹ X) X
-    (fun {Y} f => (Limits.prod.leftUnitor Y).inv ≫ CartesianClosed.uncurry f)
-    (fun {Y} f => CartesianClosed.curry ((Limits.prod.leftUnitor Y).hom ≫ f))
-    (fun g => by
-      rw [curry_eq_iff, Iso.hom_inv_id_assoc])
-    (fun g => by simp)
-    (fun f g => by
-      -- Porting note: `rw` is a bit brittle here, requiring the `dsimp` rule cancellation.
-      dsimp [-prod.leftUnitor_inv]
-      rw [uncurry_natural_left, prod.leftUnitor_inv_naturality_assoc f])
+  (expTerminalNatIso.app X).symm
 
 /-- The internal element which points at the given morphism. -/
 def internalizeHom (f : A ⟶ Y) : ⊤_ C ⟶ A ⟹ Y :=
@@ -308,7 +302,7 @@ def powZero {I : C} (t : IsInitial I) [CartesianClosed C] : I ⟹ B ≅ ⊤_ C w
     rw [← curry_natural_left, curry_eq_iff, ← cancel_epi (mulZero t).inv]
     apply t.hom_ext
 
--- TODO: Generalise the below to its commutated variants.
+-- TODO: Generalise the below to its commuted variants.
 -- TODO: Define a distributive category, so that zero_mul and friends can be derived from this.
 /-- In a CCC with binary coproducts, the distribution morphism is an isomorphism. -/
 def prodCoprodDistrib [HasBinaryCoproducts C] [CartesianClosed C] (X Y Z : C) :
@@ -335,8 +329,9 @@ This actually shows a slightly stronger version: any morphism to an initial obje
 exponentiable object is an isomorphism.
 -/
 theorem strict_initial {I : C} (t : IsInitial I) (f : A ⟶ I) : IsIso f := by
-  haveI : Mono (prod.lift (𝟙 A) f ≫ (zeroMul t).hom) := mono_comp _ _
-  rw [zeroMul_hom, prod.lift_snd] at this
+  haveI : Mono f := by
+    rw [← prod.lift_snd (𝟙 A) f, ← zeroMul_hom t]
+    exact mono_comp _ _
   haveI : IsSplitEpi f := IsSplitEpi.mk' ⟨t.to _, t.hom_ext _ _⟩
   apply isIso_of_mono_of_isSplitEpi
 
diff --git a/Mathlib/CategoryTheory/Closed/FunctorCategory.lean b/Mathlib/CategoryTheory/Closed/FunctorCategory.lean
index 49176e1f107a3..a304a8071b6d1 100644
--- a/Mathlib/CategoryTheory/Closed/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Closed/FunctorCategory.lean
@@ -62,9 +62,8 @@ closed in the functor category `F : D ⥤ C` with the pointwise monoidal structu
 instance closed (F : D ⥤ C) : Closed F where
   rightAdj := closedIhom F
   adj :=
-    Adjunction.mkOfUnitCounit
-      { unit := closedUnit F
-        counit := closedCounit F }
+    { unit := closedUnit F
+      counit := closedCounit F }
 
 /-- If `C` is a monoidal closed category and `D` is groupoid, then the functor category `D ⥤ C`,
 with the pointwise monoidal structure, is monoidal closed. -/
diff --git a/Mathlib/CategoryTheory/Closed/FunctorToTypes.lean b/Mathlib/CategoryTheory/Closed/FunctorToTypes.lean
new file mode 100644
index 0000000000000..35a8c0eccbad2
--- /dev/null
+++ b/Mathlib/CategoryTheory/Closed/FunctorToTypes.lean
@@ -0,0 +1,68 @@
+/-
+Copyright (c) 2024 Jack McKoen. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jack McKoen
+-/
+import Mathlib.CategoryTheory.Functor.FunctorHom
+import Mathlib.CategoryTheory.Closed.Monoidal
+
+/-!
+# Functors to Type are closed.
+
+Show that `C ⥤ Type max w v u` is monoidal closed for `C` a category in `Type u` with morphisms in
+`Type v`, and `w` an arbitrary universe.
+
+## TODO
+It should be shown that `C ⥤ Type max w v u` is cartesian closed.
+
+-/
+
+
+universe w v' v u u'
+
+open CategoryTheory Functor MonoidalCategory
+
+namespace CategoryTheory.FunctorToTypes
+
+variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
+
+variable (F : C ⥤ Type max w v u)
+
+/-- When `F G H : C ⥤ Type max w v u`, we have `(G ⟶ F.functorHom H) ≃ (F ⊗ G ⟶ H)`. -/
+@[simps!]
+def functorHomEquiv (G H : C ⥤ Type max w v u) : (G ⟶ F.functorHom H) ≃ (F ⊗ G ⟶ H) :=
+  (Functor.functorHomEquiv F H G).trans (homObjEquiv F H G)
+
+/-- Given a morphism `f : G ⟶ H`, an object `c : C`, and an element of `(F.functorHom G).obj c`,
+construct an element of `(F.functorHom H).obj c`. -/
+@[simps]
+def rightAdj_map {F G H : C ⥤ Type max w v u} (f : G ⟶ H) (c : C) (a : (F.functorHom G).obj c) :
+    (F.functorHom H).obj c where
+  app d b := a.app d b ≫ f.app d
+  naturality g h := by
+    have := a.naturality g h
+    change (F.map g ≫ a.app _ (h ≫ g)) ≫ _ = _
+    aesop
+
+/-- A right adjoint of `tensorLeft F`. -/
+@[simps!]
+def rightAdj : (C ⥤ Type max w v u) ⥤ C ⥤ Type max w v u where
+  obj G := F.functorHom G
+  map f := { app := rightAdj_map f }
+
+/-- The adjunction `tensorLeft F ⊣ rightAdj F`. -/
+def adj : tensorLeft F ⊣ rightAdj F where
+  unit := {
+    app := fun G ↦ (functorHomEquiv F G _).2 (𝟙 _)
+    naturality := fun G H f ↦ by
+      dsimp [rightAdj]
+      ext _
+      simp [FunctorToTypes.naturality] }
+  counit := { app := fun G ↦ functorHomEquiv F _ G (𝟙 _) }
+
+instance closed : Closed F where
+  adj := adj F
+
+instance monoidalClosed : MonoidalClosed (C ⥤ Type max w v u) where
+
+end CategoryTheory.FunctorToTypes
diff --git a/Mathlib/CategoryTheory/Closed/Ideal.lean b/Mathlib/CategoryTheory/Closed/Ideal.lean
index 9fd9a5396913e..da5c0efda1a01 100644
--- a/Mathlib/CategoryTheory/Closed/Ideal.lean
+++ b/Mathlib/CategoryTheory/Closed/Ideal.lean
@@ -202,7 +202,7 @@ theorem bijection_symm_apply_id (A B : C) :
   -- Porting note: added
   dsimp only [Functor.comp_obj]
   rw [prod.comp_lift_assoc, prod.lift_snd, prod.lift_fst_assoc, prod.lift_fst_comp_snd_comp,
-    ← Adjunction.eq_homEquiv_apply, Adjunction.homEquiv_unit, Iso.comp_inv_eq, assoc]
+    ← Adjunction.eq_unit_comp_map_iff, Iso.comp_inv_eq, assoc]
   rw [PreservesLimitPair.iso_hom i ((reflector i).obj A) ((reflector i).obj B)]
   apply prod.hom_ext
   · rw [Limits.prod.map_fst, assoc, assoc, prodComparison_fst, ← i.map_comp, prodComparison_fst]
@@ -217,9 +217,9 @@ theorem bijection_natural (A B : C) (X X' : D) (f : (reflector i).obj (A ⨯ B)
   erw [homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq,
     homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq]
   apply i.map_injective
-  rw [Functor.FullyFaithful.map_preimage, i.map_comp, Functor.FullyFaithful.map_preimage,
-    comp_id, comp_id, comp_id, comp_id, comp_id,
-    comp_id, Adjunction.homEquiv_naturality_right, ← assoc, curry_natural_right _ (i.map g),
+  rw [Functor.FullyFaithful.map_preimage, i.map_comp]
+  simp only [comp_id, Functor.map_comp, Functor.FullyFaithful.map_preimage, assoc]
+  rw [← assoc, ← assoc, curry_natural_right _ (i.map g),
     unitCompPartialBijective_natural, uncurry_natural_right, ← assoc, curry_natural_right,
     unitCompPartialBijective_natural, uncurry_natural_right, assoc]
 
diff --git a/Mathlib/CategoryTheory/Closed/Monoidal.lean b/Mathlib/CategoryTheory/Closed/Monoidal.lean
index e28b77954b24f..43af4d461c0de 100644
--- a/Mathlib/CategoryTheory/Closed/Monoidal.lean
+++ b/Mathlib/CategoryTheory/Closed/Monoidal.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Monoidal.Functor
 import Mathlib.CategoryTheory.Adjunction.Limits
@@ -175,11 +175,11 @@ theorem eq_curry_iff (f : A ⊗ Y ⟶ X) (g : Y ⟶ A ⟶[C] X) : g = curry f 
   Adjunction.eq_homEquiv_apply (ihom.adjunction A) f g
 
 -- I don't think these two should be simp.
-theorem uncurry_eq (g : Y ⟶ A ⟶[C] X) : uncurry g = (A ◁ g) ≫ (ihom.ev A).app X :=
-  Adjunction.homEquiv_counit _
+theorem uncurry_eq (g : Y ⟶ A ⟶[C] X) : uncurry g = (A ◁ g) ≫ (ihom.ev A).app X := by
+  rfl
 
 theorem curry_eq (g : A ⊗ Y ⟶ X) : curry g = (ihom.coev A).app Y ≫ (ihom A).map g :=
-  Adjunction.homEquiv_unit _
+  rfl
 
 theorem curry_injective : Function.Injective (curry : (A ⊗ Y ⟶ X) → (Y ⟶ A ⟶[C] X)) :=
   (Closed.adj.homEquiv _ _).injective
@@ -196,6 +196,10 @@ theorem curry_id_eq_coev : curry (𝟙 _) = (ihom.coev A).app X := by
   rw [curry_eq, (ihom A).map_id (A ⊗ _)]
   apply comp_id
 
+/-- The internal hom out of the unit is naturally isomorphic to the identity functor.-/
+def unitNatIso [Closed (𝟙_ C)] : 𝟭 C ≅ ihom (𝟙_ C) :=
+  conjugateIsoEquiv (Adjunction.id (C := C)) (ihom.adjunction (𝟙_ C))
+    (leftUnitorNatIso C)
 section Pre
 
 variable {A B}
@@ -266,7 +270,14 @@ theorem ofEquiv_curry_def {X Y Z : C} (f : X ⊗ Y ⟶ Z) :
       adj.homEquiv Y ((ihom (F.obj X)).obj (F.obj Z))
         (MonoidalClosed.curry (adj.toEquivalence.symm.toAdjunction.homEquiv (F.obj X ⊗ F.obj Y) Z
         ((Iso.compInverseIso (H := adj.toEquivalence)
-          (MonoidalFunctor.commTensorLeft F X)).hom.app Y ≫ f))) :=
+          (MonoidalFunctor.commTensorLeft F X)).hom.app Y ≫ f))) := by
+  -- This whole proof used to be `rfl` before #16317.
+  change ((adj.comp ((ihom.adjunction (F.obj X)).comp
+      adj.toEquivalence.symm.toAdjunction)).ofNatIsoLeft _).homEquiv _ _ _ = _
+  dsimp only [Adjunction.ofNatIsoLeft]
+  rw [Adjunction.mkOfHomEquiv_homEquiv]
+  dsimp
+  rw [Adjunction.comp_homEquiv, Adjunction.comp_homEquiv]
   rfl
 
 /-- Suppose we have a monoidal equivalence `F : C ≌ D`, with `D` monoidal closed. We can pull the
@@ -280,8 +291,17 @@ theorem ofEquiv_uncurry_def {X Y Z : C} :
       ((Iso.compInverseIso (H := adj.toEquivalence)
           (MonoidalFunctor.commTensorLeft F X)).inv.app Y) ≫
             (adj.toEquivalence.symm.toAdjunction.homEquiv _ _).symm
-              (MonoidalClosed.uncurry ((adj.homEquiv _ _).symm f)) :=
-  fun _ => rfl
+              (MonoidalClosed.uncurry ((adj.homEquiv _ _).symm f)) := by
+  intro f
+  -- This whole proof used to be `rfl` before #16317.
+  change (((adj.comp ((ihom.adjunction (F.obj X)).comp
+      adj.toEquivalence.symm.toAdjunction)).ofNatIsoLeft _).homEquiv _ _).symm _ = _
+  dsimp only [Adjunction.ofNatIsoLeft]
+  rw [Adjunction.mkOfHomEquiv_homEquiv]
+  dsimp
+  rw [Adjunction.comp_homEquiv, Adjunction.comp_homEquiv]
+  rfl
+
 end OfEquiv
 
 end MonoidalClosed
diff --git a/Mathlib/CategoryTheory/Closed/Types.lean b/Mathlib/CategoryTheory/Closed/Types.lean
index 76a1e1194bd3e..503e71b9441cf 100644
--- a/Mathlib/CategoryTheory/Closed/Types.lean
+++ b/Mathlib/CategoryTheory/Closed/Types.lean
@@ -32,10 +32,9 @@ section CartesianClosed
 /-- The adjunction `Limits.Types.binaryProductFunctor.obj X ⊣ coyoneda.obj (Opposite.op X)`
 for any `X : Type v₁`. -/
 def Types.binaryProductAdjunction (X : Type v₁) :
-    Limits.Types.binaryProductFunctor.obj X ⊣ coyoneda.obj (Opposite.op X) :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun Z (z : Z) x => ⟨x, z⟩ }
-      counit := { app := fun Z xf => xf.2 xf.1 } }
+    Limits.Types.binaryProductFunctor.obj X ⊣ coyoneda.obj (Opposite.op X) where
+  unit := { app := fun Z (z : Z) x => ⟨x, z⟩ }
+  counit := { app := fun Z xf => xf.2 xf.1 }
 
 instance (X : Type v₁) : (Types.binaryProductFunctor.obj X).IsLeftAdjoint :=
   ⟨_, ⟨Types.binaryProductAdjunction X⟩⟩
@@ -51,7 +50,7 @@ instance {C : Type v₁} [SmallCategory C] : CartesianClosed (C ⥤ Type v₁) :
       have := Presheaf.isLeftAdjoint_of_preservesColimits (prod.functor.obj F)
       exact Exponentiable.mk _ _ (Adjunction.ofIsLeftAdjoint (prod.functor.obj F)))
 
--- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
+-- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
 -- closed categories, replace this with that, as it will be a more explicit construction.
 /-- This is not a good instance because of the universe levels. Below is the instance where the
 target category is `Type (max u₁ v₁)`. -/
@@ -63,12 +62,12 @@ def cartesianClosedFunctorToTypes {C : Type u₁} [Category.{v₁} C] :
         (ULift.equivalence.trans ULiftHom.equiv).functor)
   cartesianClosedOfEquiv e
 
--- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
+-- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
 -- closed categories, replace this with that, as it will be a more explicit construction.
 instance {C : Type u₁} [Category.{v₁} C] : CartesianClosed (C ⥤ Type (max u₁ v₁)) :=
   cartesianClosedFunctorToTypes
 
--- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
+-- TODO: once we have `MonoidalClosed` instances for functor categories into general monoidal
 -- closed categories, replace this with that, as it will be a more explicit construction.
 instance {C : Type u₁} [Category.{v₁} C] [EssentiallySmall.{v₁} C] :
     CartesianClosed (C ⥤ Type v₁) :=
diff --git a/Mathlib/CategoryTheory/CofilteredSystem.lean b/Mathlib/CategoryTheory/CofilteredSystem.lean
index eb69db1cc53a2..2424d1f7ae761 100644
--- a/Mathlib/CategoryTheory/CofilteredSystem.lean
+++ b/Mathlib/CategoryTheory/CofilteredSystem.lean
@@ -3,11 +3,7 @@ Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rig
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
 -/
-import Mathlib.CategoryTheory.Filtered.Basic
-import Mathlib.Data.Set.Finite
-import Mathlib.Data.Set.Subsingleton
 import Mathlib.Topology.Category.TopCat.Limits.Konig
-import Mathlib.Tactic.AdaptationNote
 
 /-!
 # Cofiltered systems
@@ -351,7 +347,7 @@ theorem eventually_injective [Nonempty J] [Finite F.sections] :
   refine ⟨fn.argmin Nat.lt_wfRel.wf,
     fun i f => ((Fintype.bijective_iff_surjective_and_card _).2
       ⟨Fsur f, le_antisymm ?_ (Fintype.card_le_of_surjective _ <| Fsur f)⟩).1⟩
-  rw [← Nat.sub_sub_self (card_le i), tsub_le_iff_tsub_le]
+  rw [← Nat.sub_le_sub_iff_left (card_le i)]
   apply fn.argmin_le
 
 end FiniteCofilteredSystem
diff --git a/Mathlib/CategoryTheory/CommSq.lean b/Mathlib/CategoryTheory/CommSq.lean
index ecb323faed1c4..f40df65859a4d 100644
--- a/Mathlib/CategoryTheory/CommSq.lean
+++ b/Mathlib/CategoryTheory/CommSq.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.CategoryTheory.Comma.Arrow
 
diff --git a/Mathlib/CategoryTheory/Comma/Arrow.lean b/Mathlib/CategoryTheory/Comma/Arrow.lean
index 1f396ae0ab27c..972c94b207848 100644
--- a/Mathlib/CategoryTheory/Comma/Arrow.lean
+++ b/Mathlib/CategoryTheory/Comma/Arrow.lean
@@ -34,7 +34,7 @@ variable (T)
 def Arrow :=
   Comma.{v, v, v} (𝟭 T) (𝟭 T)
 
-/- Porting note: could not derive `Category` above so this instance works in its place-/
+/- Porting note: could not derive `Category` above so this instance works in its place -/
 instance : Category (Arrow T) := commaCategory
 
 -- Satisfying the inhabited linter
diff --git a/Mathlib/CategoryTheory/Comma/Basic.lean b/Mathlib/CategoryTheory/Comma/Basic.lean
index 770ff3e2e5e03..c88e7997a1eeb 100644
--- a/Mathlib/CategoryTheory/Comma/Basic.lean
+++ b/Mathlib/CategoryTheory/Comma/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin, Bhavik Mehta
+Authors: Kim Morrison, Johan Commelin, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Iso
 import Mathlib.CategoryTheory.Functor.Category
@@ -60,7 +60,7 @@ variable {B' : Type u₅} [Category.{v₅} B']
 variable {T' : Type u₆} [Category.{v₆} T']
 
 /-- The objects of the comma category are triples of an object `left : A`, an object
-   `right : B` and a morphism `hom : L.obj left ⟶ R.obj right`.  -/
+   `right : B` and a morphism `hom : L.obj left ⟶ R.obj right`. -/
 structure Comma (L : A ⥤ T) (R : B ⥤ T) : Type max u₁ u₂ v₃ where
   left : A
   right : B
diff --git a/Mathlib/CategoryTheory/Comma/Over.lean b/Mathlib/CategoryTheory/Comma/Over.lean
index a1d85c11217fa..3e27eb8b7b4b1 100644
--- a/Mathlib/CategoryTheory/Comma/Over.lean
+++ b/Mathlib/CategoryTheory/Comma/Over.lean
@@ -29,6 +29,7 @@ universe v₁ v₂ u₁ u₂
 
 -- morphism levels before object levels. See note [CategoryTheory universes].
 variable {T : Type u₁} [Category.{v₁} T]
+variable {D : Type u₂} [Category.{v₂} D]
 
 /-- The over category has as objects arrows in `T` with codomain `X` and as morphisms commutative
 triangles.
@@ -65,7 +66,7 @@ theorem over_right (U : Over X) : U.right = ⟨⟨⟩⟩ := by simp only
 theorem id_left (U : Over X) : CommaMorphism.left (𝟙 U) = 𝟙 U.left :=
   rfl
 
-@[simp]
+@[simp, reassoc]
 theorem comp_left (a b c : Over X) (f : a ⟶ b) (g : b ⟶ c) : (f ≫ g).left = f.left ≫ g.left :=
   rfl
 
@@ -311,10 +312,6 @@ theorem iteratedSliceBackward_forget_forget :
 
 end IteratedSlice
 
-section
-
-variable {D : Type u₂} [Category.{v₂} D]
-
 /-- A functor `F : T ⥤ D` induces a functor `Over X ⥤ Over (F.obj X)` in the obvious way. -/
 @[simps]
 def post (F : T ⥤ D) : Over X ⥤ Over (F.obj X) where
@@ -322,14 +319,10 @@ def post (F : T ⥤ D) : Over X ⥤ Over (F.obj X) where
   map f := Over.homMk (F.map f.left)
     (by simp only [Functor.id_obj, mk_left, Functor.const_obj_obj, mk_hom, ← F.map_comp, w])
 
-end
-
 end Over
 
 namespace CostructuredArrow
 
-variable {D : Type u₂} [Category.{v₂} D]
-
 /-- Reinterpreting an `F`-costructured arrow `F.obj d ⟶ X` as an arrow over `X` induces a functor
     `CostructuredArrow F X ⥤ Over X`. -/
 @[simps!]
@@ -572,10 +565,6 @@ instance epi_right_of_epi {f g : Under X} (k : f ⟶ g) [Epi k] : Epi k.right :=
   suffices l' = (homMk m : g ⟶ mk (g.hom ≫ m)) by apply congrArg CommaMorphism.right this
   rw [← cancel_epi k]; ext; apply a
 
-section
-
-variable {D : Type u₂} [Category.{v₂} D]
-
 /-- A functor `F : T ⥤ D` induces a functor `Under X ⥤ Under (F.obj X)` in the obvious way. -/
 @[simps]
 def post {X : T} (F : T ⥤ D) : Under X ⥤ Under (F.obj X) where
@@ -583,8 +572,6 @@ def post {X : T} (F : T ⥤ D) : Under X ⥤ Under (F.obj X) where
   map f := Under.homMk (F.map f.right)
     (by simp only [Functor.id_obj, Functor.const_obj_obj, mk_right, mk_hom, ← F.map_comp, w])
 
-end
-
 end Under
 
 namespace StructuredArrow
@@ -638,7 +625,7 @@ lemma toOver_comp_forget (F : S ⥤ T) (X : T) (f : (Y : S) → F.obj Y ⟶ X)
 
 /-- Given `X : T`, to upgrade a functor `F : S ⥤ T` to a functor `S ⥤ Under X`, it suffices to
     provide maps `X ⟶ F.obj Y` for all `Y` making the obvious triangles involving all `F.map g`
-    commute.  -/
+    commute. -/
 @[simps! obj_right map_right]
 def toUnder (F : S ⥤ T) (X : T) (f : (Y : S) → X ⟶ F.obj Y)
     (h : ∀ {Y Z : S} (g : Y ⟶ Z), f Y ≫ F.map g = f Z) : S ⥤ Under X :=
@@ -657,4 +644,136 @@ lemma toUnder_comp_forget (F : S ⥤ T) (X : T) (f : (Y : S) → X ⟶ F.obj Y)
 
 end Functor
 
+namespace StructuredArrow
+
+/-- A functor from the structured arrow category on the projection functor for any structured
+arrow category. -/
+@[simps!]
+def ofStructuredArrowProjEquivalence.functor (F : D ⥤ T) (Y : T) (X : D) :
+    StructuredArrow X (StructuredArrow.proj Y F) ⥤ StructuredArrow Y (Under.forget X ⋙ F) :=
+  Functor.toStructuredArrow
+    (Functor.toUnder (StructuredArrow.proj X _ ⋙ StructuredArrow.proj Y _) _
+      (fun g => by exact g.hom) (fun m => by have := m.w; aesop_cat)) _ _
+    (fun f => f.right.hom) (by simp)
+
+/-- The inverse functor of `ofStructuredArrowProjEquivalence.functor`. -/
+@[simps!]
+def ofStructuredArrowProjEquivalence.inverse (F : D ⥤ T) (Y : T) (X : D) :
+    StructuredArrow Y (Under.forget X ⋙ F) ⥤ StructuredArrow X (StructuredArrow.proj Y F) :=
+  Functor.toStructuredArrow
+    (Functor.toStructuredArrow (StructuredArrow.proj Y _ ⋙ Under.forget X) _ _
+      (fun g => by exact g.hom) (fun m => by have := m.w; aesop_cat)) _ _
+    (fun f => f.right.hom) (by simp)
+
+/-- Characterization of the structured arrow category on the projection functor of any
+structured arrow category. -/
+def ofStructuredArrowProjEquivalence (F : D ⥤ T) (Y : T) (X : D) :
+    StructuredArrow X (StructuredArrow.proj Y F) ≌ StructuredArrow Y (Under.forget X ⋙ F) where
+  functor := ofStructuredArrowProjEquivalence.functor F Y X
+  inverse := ofStructuredArrowProjEquivalence.inverse F Y X
+  unitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by simp)
+  counitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by aesop_cat)
+
+/-- The canonical functor from the structured arrow category on the diagonal functor
+`T ⥤ T × T` to the the structured arrow category on `Under.forget`. -/
+@[simps!]
+def ofDiagEquivalence.functor (X : T × T) :
+    StructuredArrow X (Functor.diag _) ⥤ StructuredArrow X.2 (Under.forget X.1) :=
+  Functor.toStructuredArrow
+    (Functor.toUnder (StructuredArrow.proj X _) _
+      (fun f => by exact f.hom.1) (fun m => by have := m.w; aesop_cat)) _ _
+    (fun f => f.hom.2) (fun m => by have := m.w; aesop_cat)
+
+/-- The inverse functor of `ofDiagEquivalence.functor`. -/
+@[simps!]
+def ofDiagEquivalence.inverse (X : T × T) :
+    StructuredArrow X.2 (Under.forget X.1) ⥤ StructuredArrow X (Functor.diag _) :=
+  Functor.toStructuredArrow (StructuredArrow.proj _ _ ⋙ Under.forget _) _ _
+    (fun f => (f.right.hom, f.hom)) (fun m => by have := m.w; aesop_cat)
+
+/-- Characterization of the structured arrow category on the diagonal functor `T ⥤ T × T`. -/
+def ofDiagEquivalence (X : T × T) :
+    StructuredArrow X (Functor.diag _) ≌ StructuredArrow X.2 (Under.forget X.1) where
+  functor := ofDiagEquivalence.functor X
+  inverse := ofDiagEquivalence.inverse X
+  unitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by simp)
+  counitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by aesop_cat)
+
+/-- A version of `StructuredArrow.ofDiagEquivalence` with the roles of the first and second
+projection swapped. -/
+def ofDiagEquivalence' (X : T × T) :
+    StructuredArrow X (Functor.diag _) ≌ StructuredArrow X.1 (Under.forget X.2) :=
+  (ofDiagEquivalence X).trans <|
+    (ofStructuredArrowProjEquivalence (𝟭 T) X.1 X.2).trans <|
+    StructuredArrow.mapNatIso (Under.forget X.2).rightUnitor
+
+end StructuredArrow
+
+namespace CostructuredArrow
+
+/-- A functor from the costructured arrow category on the projection functor for any costructured
+arrow category. -/
+@[simps!]
+def ofCostructuredArrowProjEquivalence.functor (F : T ⥤ D) (Y : D) (X : T) :
+    CostructuredArrow (CostructuredArrow.proj F Y) X ⥤ CostructuredArrow (Over.forget X ⋙ F) Y :=
+  Functor.toCostructuredArrow
+    (Functor.toOver (CostructuredArrow.proj _ X ⋙ CostructuredArrow.proj F Y) _
+      (fun g => by exact g.hom) (fun m => by have := m.w; aesop_cat)) _ _
+    (fun f => f.left.hom) (by simp)
+
+/-- The inverse functor of `ofCostructuredArrowProjEquivalence.functor`. -/
+@[simps!]
+def ofCostructuredArrowProjEquivalence.inverse (F : T ⥤ D) (Y : D) (X : T) :
+    CostructuredArrow (Over.forget X ⋙ F) Y ⥤ CostructuredArrow (CostructuredArrow.proj F Y) X :=
+  Functor.toCostructuredArrow
+    (Functor.toCostructuredArrow (CostructuredArrow.proj _ Y ⋙ Over.forget X) _ _
+      (fun g => by exact g.hom) (fun m => by have := m.w; aesop_cat)) _ _
+    (fun f => f.left.hom) (by simp)
+
+/-- Characterization of the costructured arrow category on the projection functor of any
+costructured arrow category. -/
+def ofCostructuredArrowProjEquivalence (F : T ⥤ D) (Y : D) (X : T) :
+    CostructuredArrow (CostructuredArrow.proj F Y) X
+      ≌ CostructuredArrow (Over.forget X ⋙ F) Y where
+  functor := ofCostructuredArrowProjEquivalence.functor F Y X
+  inverse := ofCostructuredArrowProjEquivalence.inverse F Y X
+  unitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by simp)
+  counitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by aesop_cat)
+
+/-- The canonical functor from the costructured arrow category on the diagonal functor
+`T ⥤ T × T` to the the costructured arrow category on `Under.forget`. -/
+@[simps!]
+def ofDiagEquivalence.functor (X : T × T) :
+    CostructuredArrow (Functor.diag _) X ⥤ CostructuredArrow (Over.forget X.1) X.2 :=
+  Functor.toCostructuredArrow
+    (Functor.toOver (CostructuredArrow.proj _ X) _
+      (fun g => by exact g.hom.1) (fun m => by have := congrArg (·.1) m.w; aesop_cat))
+    _ _
+    (fun f => f.hom.2) (fun m => by have := congrArg (·.2) m.w; aesop_cat)
+
+/-- The inverse functor of `ofDiagEquivalence.functor`. -/
+@[simps!]
+def ofDiagEquivalence.inverse (X : T × T) :
+    CostructuredArrow (Over.forget X.1) X.2 ⥤ CostructuredArrow (Functor.diag _) X :=
+  Functor.toCostructuredArrow (CostructuredArrow.proj _ _ ⋙ Over.forget _) _ X
+    (fun f => (f.left.hom, f.hom)) (fun m => by have := m.w; aesop_cat)
+
+/-- Characterization of the costructured arrow category on the diagonal functor `T ⥤ T × T`. -/
+def ofDiagEquivalence (X : T × T) :
+    CostructuredArrow (Functor.diag _) X ≌ CostructuredArrow (Over.forget X.1) X.2 where
+  functor := ofDiagEquivalence.functor X
+  inverse := ofDiagEquivalence.inverse X
+  unitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by simp)
+  counitIso := NatIso.ofComponents (fun _ => Iso.refl _) (by aesop_cat)
+
+/-- A version of `CostructuredArrow.ofDiagEquivalence` with the roles of the first and second
+projection swapped. -/
+def ofDiagEquivalence' (X : T × T) :
+    CostructuredArrow (Functor.diag _) X ≌ CostructuredArrow (Over.forget X.2) X.1 :=
+  (ofDiagEquivalence X).trans <|
+    (ofCostructuredArrowProjEquivalence (𝟭 T) X.1 X.2).trans <|
+    CostructuredArrow.mapNatIso (Over.forget X.2).rightUnitor
+
+end CostructuredArrow
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Comma/StructuredArrow.lean b/Mathlib/CategoryTheory/Comma/StructuredArrow.lean
index 7c9348d8e83bf..3ee2b719536a8 100644
--- a/Mathlib/CategoryTheory/Comma/StructuredArrow.lean
+++ b/Mathlib/CategoryTheory/Comma/StructuredArrow.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Adam Topaz, Scott Morrison
+Authors: Adam Topaz, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Comma.Basic
 import Mathlib.CategoryTheory.PUnit
@@ -111,7 +111,7 @@ def homMk {f f' : StructuredArrow S T} (g : f.right ⟶ f'.right)
     simpa using w.symm
 
 /- Porting note: it appears the simp lemma is not getting generated but the linter
-picks up on it (seems like a bug). Either way simp solves it.  -/
+picks up on it (seems like a bug). Either way simp solves it. -/
 attribute [-simp, nolint simpNF] homMk_left
 
 theorem homMk_surjective {f f' : StructuredArrow S T} (φ : f ⟶ f') :
@@ -120,7 +120,7 @@ theorem homMk_surjective {f f' : StructuredArrow S T} (φ : f ⟶ f') :
   ⟨φ.right, StructuredArrow.w φ, rfl⟩
 
 /-- Given a structured arrow `X ⟶ T(Y)`, and an arrow `Y ⟶ Y'`, we can construct a morphism of
-    structured arrows given by `(X ⟶ T(Y)) ⟶ (X ⟶ T(Y) ⟶ T(Y'))`.  -/
+    structured arrows given by `(X ⟶ T(Y)) ⟶ (X ⟶ T(Y) ⟶ T(Y'))`. -/
 @[simps]
 def homMk' (f : StructuredArrow S T) (g : f.right ⟶ Y') : f ⟶ mk (f.hom ≫ T.map g) where
   left := 𝟙 _
@@ -889,4 +889,71 @@ def costructuredArrowOpEquivalence (F : C ⥤ D) (d : D) :
         apply CommaMorphism.ext <;>
           dsimp [StructuredArrow.isoMk, StructuredArrow.homMk, Comma.isoMk]; simp
 
+section Pre
+
+variable {E : Type u₃} [Category.{v₃} E] (F : C ⥤ D) {G : D ⥤ E} {e : E}
+
+/-- The functor establishing the equivalence `StructuredArrow.preEquivalence`. -/
+@[simps!]
+def StructuredArrow.preEquivalence.functor (f : StructuredArrow e G) :
+    StructuredArrow f (pre e F G) ⥤ StructuredArrow f.right F where
+  obj g := mk g.hom.right
+  map φ := homMk φ.right.right <| by
+    have := w φ
+    simp only [Functor.const_obj_obj] at this ⊢
+    rw [← this, comp_right]
+    simp
+
+/-- The inverse functor establishing the equivalence `StructuredArrow.preEquivalence`. -/
+@[simps!]
+def StructuredArrow.preEquivalence.inverse (f : StructuredArrow e G) :
+    StructuredArrow f.right F ⥤ StructuredArrow f (pre e F G) where
+  obj g := mk
+            (Y := mk (Y := g.right)
+              (f.hom ≫ (G.map g.hom : G.obj f.right ⟶ (F ⋙ G).obj g.right)))
+            (homMk g.hom)
+  map φ := homMk <| homMk φ.right <| by
+    simp only [Functor.const_obj_obj, Functor.comp_obj, mk_right, mk_left, mk_hom_eq_self,
+      Functor.comp_map, Category.assoc, ← w φ, Functor.map_comp]
+
+/-- A structured arrow category on a `StructuredArrow.pre e F G` functor is equivalent to the
+structured arrow category on F -/
+def StructuredArrow.preEquivalence (f : StructuredArrow e G) :
+    StructuredArrow f (pre e F G) ≌ StructuredArrow f.right F where
+  functor := StructuredArrow.preEquivalence.functor F f
+  inverse := StructuredArrow.preEquivalence.inverse F f
+  unitIso := NatIso.ofComponents (fun _ => isoMk (isoMk (Iso.refl _)))
+  counitIso := NatIso.ofComponents (fun _ => isoMk (Iso.refl _))
+
+/-- The functor establishing the equivalence `CostructuredArrow.preEquivalence`. -/
+@[simps!]
+def CostructuredArrow.preEquivalence.functor (f : CostructuredArrow G e) :
+    CostructuredArrow (pre F G e) f ⥤ CostructuredArrow F f.left where
+  obj g := mk g.hom.left
+  map φ := homMk φ.left.left <| by
+    have := w φ
+    simp only [Functor.const_obj_obj] at this ⊢
+    rw [← this, comp_left]
+    simp
+
+/-- The inverse functor establishing the equivalence `CostructuredArrow.preEquivalence`. -/
+@[simps!]
+def CostructuredArrow.preEquivalence.inverse (f : CostructuredArrow G e) :
+    CostructuredArrow F f.left ⥤ CostructuredArrow (pre F G e) f where
+  obj g := mk (Y := mk (Y := g.left) (G.map g.hom ≫ f.hom)) (homMk g.hom)
+  map φ := homMk <| homMk φ.left <| by
+    simp only [Functor.const_obj_obj, Functor.comp_obj, mk_left, Functor.comp_map, mk_hom_eq_self,
+      ← w φ, Functor.map_comp, Category.assoc]
+
+/-- A costructured arrow category on a `CostructuredArrow.pre F G e` functor is equivalent to the
+costructured arrow category on F -/
+def CostructuredArrow.preEquivalence (f : CostructuredArrow G e) :
+    CostructuredArrow (pre F G e) f ≌ CostructuredArrow F f.left where
+  functor := CostructuredArrow.preEquivalence.functor F f
+  inverse := CostructuredArrow.preEquivalence.inverse F f
+  unitIso := NatIso.ofComponents (fun _ => isoMk (isoMk (Iso.refl _)))
+  counitIso := NatIso.ofComponents (fun _ => isoMk (Iso.refl _))
+
+end Pre
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/ConcreteCategory/Basic.lean b/Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
index 8ee159d8afb77..38882d4e9f448 100644
--- a/Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
+++ b/Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johannes Hölzl, Reid Barton, Sean Leather, Yury Kudryashov
+Authors: Kim Morrison, Johannes Hölzl, Reid Barton, Sean Leather, Yury Kudryashov
 -/
 import Mathlib.CategoryTheory.Types
 
@@ -76,7 +76,7 @@ instance : HasCoeToSort X := ConcreteCategory.hasCoeToSort X
 -/
 def ConcreteCategory.hasCoeToSort (C : Type u) [Category.{v} C] [ConcreteCategory.{w} C] :
     CoeSort C (Type w) where
-  coe := fun X => (forget C).obj X
+  coe X := (forget C).obj X
 
 section
 
diff --git a/Mathlib/CategoryTheory/ConcreteCategory/Bundled.lean b/Mathlib/CategoryTheory/ConcreteCategory/Bundled.lean
index 365a1c9c6a877..1fe01ebf2d14a 100644
--- a/Mathlib/CategoryTheory/ConcreteCategory/Bundled.lean
+++ b/Mathlib/CategoryTheory/ConcreteCategory/Bundled.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johannes Hölzl, Reid Barton, Sean Leather
+Authors: Kim Morrison, Johannes Hölzl, Reid Barton, Sean Leather
 -/
 import Mathlib.Init
 import Batteries.Tactic.Lint.Misc
@@ -22,7 +22,7 @@ universe u v
 
 namespace CategoryTheory
 
-variable {c d : Type u → Type v} {α : Type u}
+variable {c d : Type u → Type v}
 
 /-- `Bundled` is a type bundled with a type class instance for that type. Only
 the type class is exposed as a parameter. -/
diff --git a/Mathlib/CategoryTheory/ConcreteCategory/BundledHom.lean b/Mathlib/CategoryTheory/ConcreteCategory/BundledHom.lean
index 201557ff6bf23..7293c742fab92 100644
--- a/Mathlib/CategoryTheory/ConcreteCategory/BundledHom.lean
+++ b/Mathlib/CategoryTheory/ConcreteCategory/BundledHom.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Yury Kudryashov
+Authors: Kim Morrison, Yury Kudryashov
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
 import Mathlib.CategoryTheory.ConcreteCategory.Bundled
diff --git a/Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean b/Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean
index 4bd471e67da9a..69e90e646f526 100644
--- a/Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean
+++ b/Mathlib/CategoryTheory/ConcreteCategory/ReflectsIso.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
 import Mathlib.CategoryTheory.Functor.ReflectsIso
diff --git a/Mathlib/CategoryTheory/ConnectedComponents.lean b/Mathlib/CategoryTheory/ConnectedComponents.lean
index b6522e987ae80..0d205fcbb0170 100644
--- a/Mathlib/CategoryTheory/ConnectedComponents.lean
+++ b/Mathlib/CategoryTheory/ConnectedComponents.lean
@@ -56,7 +56,7 @@ def ConnectedComponents.functorToDiscrete   (X : Type*)
   map g := Discrete.eqToHom (congrArg f (Quotient.sound (Zigzag.of_hom g)))
 
 /-- Every functor to a discrete category gives a function from connected components -/
-def ConnectedComponents.liftFunctor  (J) [Category J] {X : Type*} (F :J ⥤ Discrete X) :
+def ConnectedComponents.liftFunctor (J) [Category J] {X : Type*} (F :J ⥤ Discrete X) :
     (ConnectedComponents J → X) :=
   Quotient.lift (fun c => (F.obj c).as)
     (fun _ _ h => eq_of_zigzag X (zigzag_obj_of_zigzag F h))
@@ -116,7 +116,7 @@ instance (j : ConnectedComponents J) : IsConnected (Component j) := by
   -- Everything in our chosen zigzag from `j₁` to `j₂` has a zigzag to `j₂`.
   have hf : ∀ a : J, a ∈ l → Zigzag a j₂ := by
     intro i hi
-    apply List.Chain.induction (fun t => Zigzag t j₂) _ hl₁ hl₂ _ _ _ (List.mem_of_mem_tail hi)
+    apply hl₁.backwards_induction (fun t => Zigzag t j₂) _ hl₂ _ _ _ (List.mem_of_mem_tail hi)
     · intro j k
       apply Relation.ReflTransGen.head
     · apply Relation.ReflTransGen.refl
@@ -124,7 +124,7 @@ instance (j : ConnectedComponents J) : IsConnected (Component j) := by
   refine ⟨l.pmap f hf, ?_, ?_⟩
   · refine @List.chain_pmap_of_chain _ _ _ _ _ f (fun x y _ _ h => ?_) _ _ hl₁ h₁₂ _
     exact zag_of_zag_obj (Component.ι _) h
-  · erw [List.getLast_pmap _ f (j₁ :: l) (by simpa [h₁₂] using hf) (List.cons_ne_nil _ _)]
+  · erw [List.getLast_pmap f (j₁ :: l) (by simpa [h₁₂] using hf) (List.cons_ne_nil _ _)]
     exact FullSubcategory.ext hl₂
 
 /-- The disjoint union of `J`s connected components, written explicitly as a sigma-type with the
diff --git a/Mathlib/CategoryTheory/Core.lean b/Mathlib/CategoryTheory/Core.lean
index e124cbff9a5bc..e7b1e69ea7ddc 100644
--- a/Mathlib/CategoryTheory/Core.lean
+++ b/Mathlib/CategoryTheory/Core.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Control.EquivFunctor
 import Mathlib.CategoryTheory.Groupoid
diff --git a/Mathlib/CategoryTheory/DifferentialObject.lean b/Mathlib/CategoryTheory/DifferentialObject.lean
index 324755f59b43d..b5463218d54bf 100644
--- a/Mathlib/CategoryTheory/DifferentialObject.lean
+++ b/Mathlib/CategoryTheory/DifferentialObject.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Basic
 import Mathlib.Data.Int.Cast.Defs
diff --git a/Mathlib/CategoryTheory/DiscreteCategory.lean b/Mathlib/CategoryTheory/DiscreteCategory.lean
index 3c5a2e2c0acd1..59798018c9623 100644
--- a/Mathlib/CategoryTheory/DiscreteCategory.lean
+++ b/Mathlib/CategoryTheory/DiscreteCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.Pi.Basic
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean b/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean
index b835df66d8be2..4259ae6bf495b 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean
@@ -24,7 +24,7 @@ then these definitions are equivalent, see the file
 `CategoryTheory/EffectiveEpi/RegularEpi.lean`
 See [nlab: *Effective Epimorphism*](https://ncatlab.org/nlab/show/effective+epimorphism) and
 [Stacks 00WP](https://stacks.math.columbia.edu/tag/00WP) for the standard definitions. Note that
-our notion of `EffectiveEpi` is often called "strict epi" in the literature.
+our notion of `EffectiveEpi` is often called "strict epi" in the literature.
 
 ## References
 - [Elephant]: *Sketches of an Elephant*, P. T. Johnstone: C2.1, Example 2.1.12.
@@ -45,7 +45,7 @@ This structure encodes the data required for a morphism to be an effective epimo
 structure EffectiveEpiStruct {X Y : C} (f : Y ⟶ X) where
   /--
   For every `W` with a morphism `e : Y ⟶ W` that coequalizes every pair of morphisms
-  `g₁ g₂ : Z ⟶ Y` which `f` coequalizes, `desc e h` is a morphism `X ⟶ W`... 
+  `g₁ g₂ : Z ⟶ Y` which `f` coequalizes, `desc e h` is a morphism `X ⟶ W`...
   -/
   desc : ∀ {W : C} (e : Y ⟶ W),
     (∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e) → (X ⟶ W)
@@ -65,7 +65,7 @@ If `f` has a kernel pair, then this is equivalent to showing that the correspond
 a colimit.
 -/
 class EffectiveEpi {X Y : C} (f : Y ⟶ X) : Prop where
-  /-- `f` is an effective epimorphism if there exists an `EffectiveEpiStruct` for `f`. -/
+  /-- `f` is an effective epimorphism if there exists an `EffectiveEpiStruct` for `f`. -/
   effectiveEpi : Nonempty (EffectiveEpiStruct f)
 
 /-- Some chosen `EffectiveEpiStruct` associated to an effective epi. -/
@@ -107,7 +107,7 @@ structure EffectiveEpiFamilyStruct {B : C} {α : Type*}
   /--
   For every `W` with a family of morphisms `e a : Y a ⟶ W` that coequalizes every pair of morphisms
   `g₁ : Z ⟶ Y a₁`, `g₂ : Z ⟶ Y a₂` which the family `π` coequalizes, `desc e h` is a morphism
-  `X ⟶ W`... 
+  `X ⟶ W`...
   -/
   desc : ∀ {W} (e : (a : α) → (X a ⟶ W)),
           (∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),
@@ -129,7 +129,7 @@ provided that the `π a` exhibit `B` as a colimit of the diagram of all "relatio
 `R → X a₁`, `R ⟶ X a₂` for all `a₁ a₂ : α`.
 -/
 class EffectiveEpiFamily {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) : Prop where
-  /-- `π` is an effective epimorphic family if there exists an `EffectiveEpiFamilyStruct` for `π` -/
+  /-- `π` is an effective epimorphic family if there exists an `EffectiveEpiFamilyStruct` for `π` -/
   effectiveEpiFamily : Nonempty (EffectiveEpiFamilyStruct X π)
 
 /-- Some chosen `EffectiveEpiFamilyStruct` associated to an effective epi family. -/
@@ -186,7 +186,7 @@ instance {B X : C} (f : X ⟶ B) [EffectiveEpi f] : EffectiveEpiFamily (fun () 
   ⟨⟨effectiveEpiFamilyStructSingletonOfEffectiveEpi f⟩⟩
 
 /--
-A single element `EffectiveEpiFamily` consists of an `EffectiveEpi`
+A single element `EffectiveEpiFamily` consists of an `EffectiveEpi`
 -/
 noncomputable
 def effectiveEpiStructOfEffectiveEpiFamilySingleton {B X : C} (f : X ⟶ B)
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean b/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean
index 94fd3321440d7..a8b4353556eb6 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean
@@ -8,7 +8,7 @@ import Mathlib.CategoryTheory.EffectiveEpi.Basic
 
 # Composition of effective epimorphisms
 
-This file provides `EffectiveEpi` instances for certain compositions.
+This file provides `EffectiveEpi` instances for certain compositions.
 
 -/
 
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean b/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean
index 85b4dfd1e7fc3..80a53351c358d 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean
@@ -56,7 +56,7 @@ example {B : C} {α : Type*} (X : α → C) (π : (a : α) → (X a ⟶ B)) [Eff
 
 /--
 This is an auxiliary lemma used twice in the definition of  `EffectiveEpiFamilyOfEffectiveEpiDesc`.
-It is the `h` hypothesis of `EffectiveEpi.desc` and `EffectiveEpi.fac`. 
+It is the `h` hypothesis of `EffectiveEpi.desc` and `EffectiveEpi.fac`.
 -/
 theorem effectiveEpiFamilyStructOfEffectiveEpiDesc_aux {B : C} {α : Type*} {X : α → C}
     {π : (a : α) → X a ⟶ B} [HasCoproduct X]
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean b/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean
index 1c4b96149fb57..dc7f679ddd6fb 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean
@@ -8,7 +8,7 @@ import Mathlib.CategoryTheory.EffectiveEpi.Basic
 
 # Effectively enough objects in the image of a functor
 
-We define the class `F.EffectivelyEnough` on a functor `F : C ⥤ D` which says that for every object
+We define the class `F.EffectivelyEnough` on a functor `F : C ⥤ D` which says that for every object
 in `D`, there exists an effective epi to it from an object in the image of `F`.
 -/
 
@@ -21,11 +21,11 @@ variable {C D : Type*} [Category C] [Category D] (F : C ⥤ D)
 namespace Functor
 
 /--
-An effective presentation of an object `X` with respect to a functor `F` is the data of an effective
+An effective presentation of an object `X` with respect to a functor `F` is the data of an effective
 epimorphism of the form `F.obj p ⟶ X`.
 -/
 structure EffectivePresentation (X : D) where
-  /-- The object of `C` giving the source of the effective epi -/
+  /-- The object of `C` giving the source of the effective epi -/
   p : C
   /-- The morphism `F.obj p ⟶ X` -/
   f : F.obj p ⟶ X
@@ -33,11 +33,11 @@ structure EffectivePresentation (X : D) where
   effectiveEpi : EffectiveEpi f
 
 /--
-`D` has *effectively enough objects with respect to the functor `F` if every object has an
+`D` has *effectively enough objects with respect to the functor `F` if every object has an
 effective presentation.
 -/
 class EffectivelyEnough : Prop where
-  /-- For every `X : D`, there exists an object `p` of `C` with an effective epi `F.obj p ⟶ X`. -/
+  /-- For every `X : D`, there exists an object `p` of `C` with an effective epi `F.obj p ⟶ X`. -/
   presentation : ∀ (X : D), Nonempty (F.EffectivePresentation X)
 
 variable [F.EffectivelyEnough]
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean b/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean
index 433f695391d2d..00ace33a89767 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean
@@ -11,7 +11,7 @@ import Mathlib.CategoryTheory.Limits.Preserves.Finite
 
 # Preserving and reflecting effective epis on extensive categories
 
-We prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite
+We prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite
 effective epi families if it preserves (resp. reflects) effective epis.
 -/
 
diff --git a/Mathlib/CategoryTheory/EffectiveEpi/RegularEpi.lean b/Mathlib/CategoryTheory/EffectiveEpi/RegularEpi.lean
index 8f471641e5b1f..9272c878d3ca1 100644
--- a/Mathlib/CategoryTheory/EffectiveEpi/RegularEpi.lean
+++ b/Mathlib/CategoryTheory/EffectiveEpi/RegularEpi.lean
@@ -19,7 +19,7 @@ open Limits RegularEpi
 
 variable {C : Type*} [Category C]
 
-/-- The data of an `EffectiveEpi` structure on a `RegularEpi`. -/
+/-- The data of an `EffectiveEpi` structure on a `RegularEpi`. -/
 def effectiveEpiStructOfRegularEpi {B X : C} (f : X ⟶ B) [RegularEpi f] :
     EffectiveEpiStruct f where
   desc _ h := Cofork.IsColimit.desc isColimit _ (h _ _ w)
diff --git a/Mathlib/CategoryTheory/Elements.lean b/Mathlib/CategoryTheory/Elements.lean
index d5fda0bbb9e35..3269eddc6bd86 100644
--- a/Mathlib/CategoryTheory/Elements.lean
+++ b/Mathlib/CategoryTheory/Elements.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Comma.StructuredArrow
 import Mathlib.CategoryTheory.Groupoid
diff --git a/Mathlib/CategoryTheory/Elementwise.lean b/Mathlib/CategoryTheory/Elementwise.lean
index 3619c79b713f1..06fe4ef41ec18 100644
--- a/Mathlib/CategoryTheory/Elementwise.lean
+++ b/Mathlib/CategoryTheory/Elementwise.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Tactic.CategoryTheory.Elementwise
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
diff --git a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
index 0ba3c1ced4604..59d61ad43c4c1 100644
--- a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
+++ b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Joseph Hua. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua
+Authors: Kim Morrison, Bhavik Mehta, Johan Commelin, Reid Barton, Robert Y. Lewis, Joseph Hua
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Terminal
 
diff --git a/Mathlib/CategoryTheory/Endomorphism.lean b/Mathlib/CategoryTheory/Endomorphism.lean
index 30f789b4ad257..93109c730da29 100644
--- a/Mathlib/CategoryTheory/Endomorphism.lean
+++ b/Mathlib/CategoryTheory/Endomorphism.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov, Scott Morrison, Simon Hudon
+Authors: Yury Kudryashov, Kim Morrison, Simon Hudon
 -/
 import Mathlib.Algebra.Group.Action.Defs
 import Mathlib.Algebra.Group.Equiv.Basic
diff --git a/Mathlib/CategoryTheory/Enriched/Basic.lean b/Mathlib/CategoryTheory/Enriched/Basic.lean
index 325790ef8b742..f8f9518c24feb 100644
--- a/Mathlib/CategoryTheory/Enriched/Basic.lean
+++ b/Mathlib/CategoryTheory/Enriched/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Types.Symmetric
 import Mathlib.CategoryTheory.Monoidal.Types.Coyoneda
diff --git a/Mathlib/CategoryTheory/EpiMono.lean b/Mathlib/CategoryTheory/EpiMono.lean
index a9cd25cb109d7..13cfd3e4ad1b3 100644
--- a/Mathlib/CategoryTheory/EpiMono.lean
+++ b/Mathlib/CategoryTheory/EpiMono.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Scott Morrison
+Authors: Reid Barton, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Opposites
 import Mathlib.CategoryTheory.Groupoid
diff --git a/Mathlib/CategoryTheory/EqToHom.lean b/Mathlib/CategoryTheory/EqToHom.lean
index 2c70a5bfcb94d..1a0622b180b50 100644
--- a/Mathlib/CategoryTheory/EqToHom.lean
+++ b/Mathlib/CategoryTheory/EqToHom.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Scott Morrison
+Authors: Reid Barton, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Opposites
 
@@ -51,6 +51,18 @@ theorem eqToHom_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
   cases q
   simp
 
+/-- Two morphisms are conjugate via eqToHom if and only if they are heterogeneously equal.
+Note this used to be in the Functor namespace, where it doesn't belong. -/
+theorem conj_eqToHom_iff_heq {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) (h : W = Y) (h' : X = Z) :
+    f = eqToHom h ≫ g ≫ eqToHom h'.symm ↔ HEq f g := by
+  cases h
+  cases h'
+  simp
+
+theorem conj_eqToHom_iff_heq' {C} [Category C] {W X Y Z : C}
+    (f : W ⟶ X) (g : Y ⟶ Z) (h : W = Y) (h' : Z = X) :
+    f = eqToHom h ≫ g ≫ eqToHom h' ↔ HEq f g := conj_eqToHom_iff_heq _ _ _ h'.symm
+
 theorem comp_eqToHom_iff {X Y Y' : C} (p : Y = Y') (f : X ⟶ Y) (g : X ⟶ Y') :
     f ≫ eqToHom p = g ↔ f = g ≫ eqToHom p.symm :=
   { mp := fun h => h ▸ by simp
@@ -61,6 +73,41 @@ theorem eqToHom_comp_iff {X X' Y : C} (p : X = X') (f : X ⟶ Y) (g : X' ⟶ Y)
   { mp := fun h => h ▸ by simp
     mpr := fun h => h ▸ by simp [whisker_eq _ h] }
 
+theorem eqToHom_comp_heq {C} [Category C] {W X Y : C}
+    (f : Y ⟶ X) (h : W = Y) : HEq (eqToHom h ≫ f) f := by
+  rw [← conj_eqToHom_iff_heq _ _ h rfl, eqToHom_refl, Category.comp_id]
+
+@[simp] theorem eqToHom_comp_heq_iff {C} [Category C] {W X Y Z Z' : C}
+    (f : Y ⟶ X) (g : Z ⟶ Z') (h : W = Y) :
+    HEq (eqToHom h ≫ f) g ↔ HEq f g :=
+  ⟨(eqToHom_comp_heq ..).symm.trans, (eqToHom_comp_heq ..).trans⟩
+
+@[simp] theorem heq_eqToHom_comp_iff {C} [Category C] {W X Y Z Z' : C}
+    (f : Y ⟶ X) (g : Z ⟶ Z') (h : W = Y) :
+    HEq g (eqToHom h ≫ f) ↔ HEq g f :=
+  ⟨(·.trans (eqToHom_comp_heq ..)), (·.trans (eqToHom_comp_heq ..).symm)⟩
+
+theorem comp_eqToHom_heq {C} [Category C] {X Y Z : C}
+    (f : X ⟶ Y) (h : Y = Z) : HEq (f ≫ eqToHom h) f := by
+  rw [← conj_eqToHom_iff_heq' _ _ rfl h, eqToHom_refl, Category.id_comp]
+
+@[simp] theorem comp_eqToHom_heq_iff {C} [Category C] {W X Y Z Z' : C}
+    (f : X ⟶ Y) (g : Z ⟶ Z') (h : Y = W) :
+    HEq (f ≫ eqToHom h) g ↔ HEq f g :=
+  ⟨(comp_eqToHom_heq ..).symm.trans, (comp_eqToHom_heq ..).trans⟩
+
+@[simp] theorem heq_comp_eqToHom_iff {C} [Category C] {W X Y Z Z' : C}
+    (f : X ⟶ Y) (g : Z ⟶ Z') (h : Y = W) :
+    HEq g (f ≫ eqToHom h) ↔ HEq g f :=
+  ⟨(·.trans (comp_eqToHom_heq ..)), (·.trans (comp_eqToHom_heq ..).symm)⟩
+
+theorem heq_comp {C} [Category C] {X Y Z X' Y' Z' : C}
+    {f : X ⟶ Y} {g : Y ⟶ Z} {f' : X' ⟶ Y'} {g' : Y' ⟶ Z'}
+    (eq1 : X = X') (eq2 : Y = Y') (eq3 : Z = Z')
+    (H1 : HEq f f') (H2 : HEq g g') :
+    HEq (f ≫ g) (f' ≫ g') := by
+  cases eq1; cases eq2; cases eq3; cases H1; cases H2; rfl
+
 variable {β : Sort*}
 
 /-- We can push `eqToHom` to the left through families of morphisms. -/
@@ -197,13 +244,6 @@ lemma ext_of_iso {F G : C ⥤ D} (e : F ≅ G) (hobj : ∀ X, F.obj X = G.obj X)
     rw [← cancel_mono (e.hom.app Y), e.hom.naturality f, happ, happ, Category.assoc,
     Category.assoc, eqToHom_trans, eqToHom_refl, Category.comp_id])
 
-/-- Two morphisms are conjugate via eqToHom if and only if they are heterogeneously equal. -/
-theorem conj_eqToHom_iff_heq {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) (h : W = Y) (h' : X = Z) :
-    f = eqToHom h ≫ g ≫ eqToHom h'.symm ↔ HEq f g := by
-  cases h
-  cases h'
-  simp
-
 /-- Proving equality between functors using heterogeneous equality. -/
 theorem hext {F G : C ⥤ D} (h_obj : ∀ X, F.obj X = G.obj X)
     (h_map : ∀ (X Y) (f : X ⟶ Y), HEq (F.map f) (G.map f)) : F = G :=
diff --git a/Mathlib/CategoryTheory/Equivalence.lean b/Mathlib/CategoryTheory/Equivalence.lean
index 6e8a5e55a7204..0f2676be1e1e8 100644
--- a/Mathlib/CategoryTheory/Equivalence.lean
+++ b/Mathlib/CategoryTheory/Equivalence.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Functor.FullyFaithful
 import Mathlib.CategoryTheory.FullSubcategory
diff --git a/Mathlib/CategoryTheory/EssentialImage.lean b/Mathlib/CategoryTheory/EssentialImage.lean
index 083afcf335209..a9a14f0935a2b 100644
--- a/Mathlib/CategoryTheory/EssentialImage.lean
+++ b/Mathlib/CategoryTheory/EssentialImage.lean
@@ -89,6 +89,10 @@ instance : Full (essImageInclusion F) :=
 instance : Faithful (essImageInclusion F) :=
   (inferInstance : Faithful (fullSubcategoryInclusion _))
 
+lemma essImage_ext (F : C ⥤ D) {X Y : F.EssImageSubcategory} (f g : X ⟶ Y)
+    (h : F.essImageInclusion.map f = F.essImageInclusion.map g) : f = g := by
+  simpa using h
+
 /--
 Given a functor `F : C ⥤ D`, we have an (essentially surjective) functor from `C` to the essential
 image of `F`.
diff --git a/Mathlib/CategoryTheory/EssentiallySmall.lean b/Mathlib/CategoryTheory/EssentiallySmall.lean
index 88de842c58417..d3d3eb2c323fb 100644
--- a/Mathlib/CategoryTheory/EssentiallySmall.lean
+++ b/Mathlib/CategoryTheory/EssentiallySmall.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Category.ULift
 import Mathlib.CategoryTheory.Skeletal
diff --git a/Mathlib/CategoryTheory/Extensive.lean b/Mathlib/CategoryTheory/Extensive.lean
index aea742746748d..94619f08a21bb 100644
--- a/Mathlib/CategoryTheory/Extensive.lean
+++ b/Mathlib/CategoryTheory/Extensive.lean
@@ -79,7 +79,7 @@ and binary coproducts are universal. -/
 class FinitaryPreExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
   [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
-  /-- In a finitary extensive category, all coproducts are van Kampen-/
+  /-- In a finitary extensive category, all coproducts are van Kampen -/
   universal' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsUniversalColimit c
 
 attribute [instance] FinitaryPreExtensive.hasFiniteCoproducts
@@ -90,7 +90,7 @@ and binary coproducts are van Kampen. -/
 class FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where
   [hasFiniteCoproducts : HasFiniteCoproducts C]
   [hasPullbacksOfInclusions : HasPullbacksOfInclusions C]
-  /-- In a finitary extensive category, all coproducts are van Kampen-/
+  /-- In a finitary extensive category, all coproducts are van Kampen -/
   van_kampen' : ∀ {X Y : C} (c : BinaryCofan X Y), IsColimit c → IsVanKampenColimit c
 
 attribute [instance] FinitaryExtensive.hasFiniteCoproducts
@@ -527,7 +527,7 @@ instance FinitaryPreExtensive.hasPullbacks_of_inclusions [FinitaryPreExtensive C
     {α : Type*} (f : X ⟶ Z) {Y : (a : α) → C} (i : (a : α) → Y a ⟶ Z) [Finite α]
     [hi : IsIso (Sigma.desc i)] (a : α) : HasPullback f (i a) := by
   apply FinitaryPreExtensive.hasPullbacks_of_is_coproduct (c := Cofan.mk Z i)
-  exact @IsColimit.ofPointIso (t := Cofan.mk Z i) (P := _) hi
+  exact @IsColimit.ofPointIso (t := Cofan.mk Z i) (P := _) (i := hi)
 
 lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [Finite α] {X : C}
     {Z : α → C} (π : (a : α) → Z a ⟶ X) {Y : C} (f : Y ⟶ X) (hπ : IsIso (Sigma.desc π)) :
@@ -536,7 +536,7 @@ lemma FinitaryPreExtensive.sigma_desc_iso [FinitaryPreExtensive C] {α : Type} [
     change IsIso (this.coconePointUniqueUpToIso (getColimitCocone _).2).inv
     infer_instance
   let this : IsColimit (Cofan.mk X π) := by
-    refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) ?_
+    refine @IsColimit.ofPointIso (t := Cofan.mk X π) (P := coproductIsCoproduct Z) (i := ?_)
     convert hπ
     simp [coproductIsCoproduct]
   refine (FinitaryPreExtensive.isUniversal_finiteCoproducts this
diff --git a/Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean b/Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean
index c2f4a20e934c0..b6f222bcedcbc 100644
--- a/Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean
+++ b/Mathlib/CategoryTheory/FiberedCategory/Cartesian.lean
@@ -119,6 +119,7 @@ lemma map_self : IsCartesian.map p f φ φ = 𝟙 a := by
 
 /-- The canonical isomorphism between the domains of two cartesian morphisms
 lying over the same object. -/
+@[simps]
 noncomputable def domainUniqueUpToIso {a' : 𝒳} (φ' : a' ⟶ b) [IsCartesian p f φ'] : a' ≅ a where
   hom := IsCartesian.map p f φ φ'
   inv := IsCartesian.map p f φ' φ
@@ -131,6 +132,14 @@ noncomputable def domainUniqueUpToIso {a' : 𝒳} (φ' : a' ⟶ b) [IsCartesian
     apply IsCartesian.ext p (p.map φ) φ
     simp only [assoc, fac, id_comp]
 
+instance domainUniqueUpToIso_inv_isHomLift {a' : 𝒳} (φ' : a' ⟶ b) [IsCartesian p f φ'] :
+    IsHomLift p (𝟙 R) (domainUniqueUpToIso p f φ φ').hom :=
+  domainUniqueUpToIso_hom p f φ φ' ▸ IsCartesian.map_isHomLift p f φ φ'
+
+instance domainUniqueUpToIso_hom_isHomLift {a' : 𝒳} (φ' : a' ⟶ b) [IsCartesian p f φ'] :
+    IsHomLift p (𝟙 R) (domainUniqueUpToIso p f φ φ').inv :=
+  domainUniqueUpToIso_inv p f φ φ' ▸ IsCartesian.map_isHomLift p f φ' φ
+
 /-- Precomposing a cartesian morphism with an isomorphism lifting the identity is cartesian. -/
 instance of_iso_comp {a' : 𝒳} (φ' : a' ≅ a) [IsHomLift p (𝟙 R) φ'.hom] :
     IsCartesian p f (φ'.hom ≫ φ) where
@@ -351,15 +360,34 @@ lemma isIso_of_base_isIso (φ : a ⟶ b) [IsStronglyCartesian p f φ] [IsIso f]
 
 end
 
+section
+
+variable {R R' S : 𝒮} {a a' b : 𝒳} {f : R ⟶ S} {f' : R' ⟶ S} {g : R' ≅ R}
+
 /-- The canonical isomorphism between the domains of two strongly cartesian morphisms lying over
 isomorphic objects. -/
-noncomputable def domainIsoOfBaseIso {R R' S : 𝒮} {a a' b : 𝒳} {f : R ⟶ S} {f' : R' ⟶ S}
-  {g : R' ≅ R} (h : f' = g.hom ≫ f) (φ : a ⟶ b) (φ' : a' ⟶ b) [IsStronglyCartesian p f φ]
-    [IsStronglyCartesian p f' φ'] : a' ≅ a where
+@[simps]
+noncomputable def domainIsoOfBaseIso (h : f' = g.hom ≫ f) (φ : a ⟶ b) (φ' : a' ⟶ b)
+    [IsStronglyCartesian p f φ] [IsStronglyCartesian p f' φ'] : a' ≅ a where
   hom := map p f φ h φ'
-  inv := by
-    convert map p f' φ' (congrArg (g.inv ≫ ·) h.symm) φ
+  inv :=
+    haveI : p.IsHomLift ((fun x ↦ g.inv ≫ x) (g.hom ≫ f)) φ := by
+      simpa using IsCartesian.toIsHomLift
+    map p f' φ' (congrArg (g.inv ≫ ·) h.symm) φ
+
+instance domainUniqueUpToIso_inv_isHomLift (h : f' = g.hom ≫ f) (φ : a ⟶ b) (φ' : a' ⟶ b)
+    [IsStronglyCartesian p f φ] [IsStronglyCartesian p f' φ'] :
+    IsHomLift p g.hom (domainIsoOfBaseIso p h φ φ').hom :=
+  domainIsoOfBaseIso_hom p h φ φ' ▸ IsStronglyCartesian.map_isHomLift p f φ h φ'
+
+instance domainUniqueUpToIso_hom_isHomLift (h : f' = g.hom ≫ f) (φ : a ⟶ b) (φ' : a' ⟶ b)
+    [IsStronglyCartesian p f φ] [IsStronglyCartesian p f' φ'] :
+    IsHomLift p g.inv (domainIsoOfBaseIso p h φ φ').inv := by
+  haveI : p.IsHomLift ((fun x ↦ g.inv ≫ x) (g.hom ≫ f)) φ := by
     simpa using IsCartesian.toIsHomLift
+  simpa using IsStronglyCartesian.map_isHomLift p f' φ' (congrArg (g.inv ≫ ·) h.symm) φ
+
+end
 
 end IsStronglyCartesian
 
diff --git a/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean b/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean
new file mode 100644
index 0000000000000..b826fc429e660
--- /dev/null
+++ b/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean
@@ -0,0 +1,186 @@
+/-
+Copyright (c) 2024 Paul Lezeau. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Paul Lezeau, Calle Sönne
+-/
+
+import Mathlib.CategoryTheory.FiberedCategory.Cartesian
+
+/-!
+
+# Fibered categories
+
+This file defines what it means for a functor `p : 𝒳 ⥤ 𝒮` to be (pre)fibered.
+
+## Main definitions
+
+- `IsPreFibered p` expresses `𝒳` is fibered over `𝒮` via a functor `p : 𝒳 ⥤ 𝒮`, as in SGA VI.6.1.
+This means that any morphism in the base `𝒮` can be lifted to a cartesian morphism in `𝒳`.
+
+- `IsFibered p` expresses `𝒳` is fibered over `𝒮` via a functor `p : 𝒳 ⥤ 𝒮`, as in SGA VI.6.1.
+This means that it is prefibered, and that the composition of any two cartesian morphisms is
+cartesian.
+
+In the literature one often sees the notion of a fibered category defined as the existence of
+strongly cartesian morphisms lying over any given morphism in the base. This is equivalent to the
+notion above, and we give an alternate constructor `IsFibered.of_exists_isCartesian'` for
+constructing a fibered category this way.
+
+## Implementation
+
+The constructor of `IsPreFibered` is called `exists_isCartesian'`. The reason for the prime is that
+when wanting to apply this condition, it is recommended to instead use the lemma
+`exists_isCartesian` (without the prime), which is more applicable with respect to non-definitional
+equalities.
+
+## References
+* [A. Grothendieck, M. Raynaud, *SGA 1*](https://arxiv.org/abs/math/0206203)
+
+-/
+
+universe v₁ v₂ u₁ u₂
+
+open CategoryTheory Functor Category IsHomLift
+
+namespace CategoryTheory
+
+variable {𝒮 : Type u₁} {𝒳 : Type u₂} [Category.{v₁} 𝒮] [Category.{v₂} 𝒳]
+
+/-- Definition of a prefibered category.
+
+See SGA 1 VI.6.1. -/
+class Functor.IsPreFibered (p : 𝒳 ⥤ 𝒮) : Prop where
+  exists_isCartesian' {a : 𝒳} {R : 𝒮} (f : R ⟶ p.obj a) : ∃ (b : 𝒳) (φ : b ⟶ a), IsCartesian p f φ
+
+protected lemma IsPreFibered.exists_isCartesian (p : 𝒳 ⥤ 𝒮) [p.IsPreFibered] {a : 𝒳} {R S : 𝒮}
+    (ha : p.obj a = S) (f : R ⟶ S) : ∃ (b : 𝒳) (φ : b ⟶ a), IsCartesian p f φ := by
+  subst ha; exact IsPreFibered.exists_isCartesian' f
+
+/-- Definition of a fibered category.
+
+See SGA 1 VI.6.1. -/
+class Functor.IsFibered (p : 𝒳 ⥤ 𝒮) extends IsPreFibered p : Prop where
+  comp {R S T : 𝒮} (f : R ⟶ S) (g : S ⟶ T) {a b c : 𝒳} (φ : a ⟶ b) (ψ : b ⟶ c)
+    [IsCartesian p f φ] [IsCartesian p g ψ] : IsCartesian p (f ≫ g) (φ ≫ ψ)
+
+instance (p : 𝒳 ⥤ 𝒮) [p.IsFibered] {R S T : 𝒮} (f : R ⟶ S) (g : S ⟶ T) {a b c : 𝒳} (φ : a ⟶ b)
+    (ψ : b ⟶ c) [IsCartesian p f φ] [IsCartesian p g ψ] : IsCartesian p (f ≫ g) (φ ≫ ψ) :=
+  IsFibered.comp f g φ ψ
+
+namespace Functor.IsPreFibered
+
+open IsCartesian
+
+variable {p : 𝒳 ⥤ 𝒮} [IsPreFibered p] {R S : 𝒮} {a : 𝒳} (ha : p.obj a = S) (f : R ⟶ S)
+
+/-- Given a fibered category `p : 𝒳 ⥤ 𝒫`, a morphism `f : R ⟶ S` and an object `a` lying over `S`,
+then `pullbackObj` is the domain of some choice of a cartesian morphism lying over `f` with
+codomain `a`. -/
+noncomputable def pullbackObj : 𝒳 :=
+  Classical.choose (IsPreFibered.exists_isCartesian p ha f)
+
+/-- Given a fibered category `p : 𝒳 ⥤ 𝒫`, a morphism `f : R ⟶ S` and an object `a` lying over `S`,
+then `pullbackMap` is a choice of a cartesian morphism lying over `f` with codomain `a`. -/
+noncomputable def pullbackMap : pullbackObj ha f ⟶ a :=
+  Classical.choose (Classical.choose_spec (IsPreFibered.exists_isCartesian p ha f))
+
+instance pullbackMap.IsCartesian : IsCartesian p f (pullbackMap ha f) :=
+  Classical.choose_spec (Classical.choose_spec (IsPreFibered.exists_isCartesian p ha f))
+
+lemma pullbackObj_proj : p.obj (pullbackObj ha f) = R :=
+  domain_eq p f (pullbackMap ha f)
+
+end Functor.IsPreFibered
+
+namespace Functor.IsFibered
+
+open IsCartesian IsPreFibered
+
+/-- In a fibered category, any cartesian morphism is strongly cartesian. -/
+instance isStronglyCartesian_of_isCartesian (p : 𝒳 ⥤ 𝒮) [p.IsFibered] {R S : 𝒮} (f : R ⟶ S)
+    {a b : 𝒳} (φ : a ⟶ b) [p.IsCartesian f φ] : p.IsStronglyCartesian f φ where
+  universal_property' g φ' hφ' := by
+    -- Let `ψ` be a cartesian arrow lying over `g`
+    let ψ := pullbackMap (domain_eq p f φ) g
+    -- Let `τ` be the map induced by the universal property of `ψ ≫ φ`.
+    let τ := IsCartesian.map p (g ≫ f) (ψ ≫ φ) φ'
+    use τ ≫ ψ
+    -- It is easily verified that `τ ≫ ψ` lifts `g` and `τ ≫ ψ ≫ φ = φ'`
+    refine ⟨⟨inferInstance, by simp only [assoc, IsCartesian.fac, τ]⟩, ?_⟩
+    -- It remains to check that `τ ≫ ψ` is unique.
+    -- So fix another lift `π` of `g` satisfying `π ≫ φ = φ'`.
+    intro π ⟨hπ, hπ_comp⟩
+    -- Write `π` as `π = τ' ≫ ψ` for some `τ'` induced by the universal property of `ψ`.
+    rw [← fac p g ψ π]
+    -- It remains to show that `τ' = τ`. This follows again from the universal property of `ψ`.
+    congr 1
+    apply map_uniq
+    rwa [← assoc, IsCartesian.fac]
+
+/-- In a category which admits strongly cartesian pullbacks, any cartesian morphism is
+strongly cartesian. This is a helper-lemma for the fact that admitting strongly cartesian pullbacks
+implies being fibered. -/
+lemma isStronglyCartesian_of_exists_isCartesian (p : 𝒳 ⥤ 𝒮) (h : ∀ (a : 𝒳) (R : 𝒮)
+    (f : R ⟶ p.obj a), ∃ (b : 𝒳) (φ : b ⟶ a), IsStronglyCartesian p f φ) {R S : 𝒮} (f : R ⟶ S)
+      {a b : 𝒳} (φ : a ⟶ b) [p.IsCartesian f φ] : p.IsStronglyCartesian f φ := by
+  constructor
+  intro c g φ' hφ'
+  subst_hom_lift p f φ; clear a b R S
+  -- Let `ψ` be a cartesian arrow lying over `g`
+  obtain ⟨a', ψ, hψ⟩ := h _ _ (p.map φ)
+  -- Let `τ' : c ⟶ a'` be the map induced by the universal property of `ψ`
+  let τ' := IsStronglyCartesian.map p (p.map φ) ψ (f':= g ≫ p.map φ) rfl φ'
+  -- Let `Φ : a' ≅ a` be natural isomorphism induced between `φ` and `ψ`.
+  let Φ := domainUniqueUpToIso p (p.map φ) φ ψ
+  -- The map induced by `φ` will be `τ' ≫ Φ.hom`
+  use τ' ≫ Φ.hom
+  -- It is easily verified that `τ' ≫ Φ.hom` lifts `g` and `τ' ≫ Φ.hom ≫ φ = φ'`
+  refine ⟨⟨by simp only [Φ]; infer_instance, ?_⟩, ?_⟩
+  · simp [τ', Φ, IsStronglyCartesian.map_uniq p (p.map φ) ψ rfl φ']
+  -- It remains to check that it is unique. This follows from the universal property of `ψ`.
+  intro π ⟨hπ, hπ_comp⟩
+  rw [← Iso.comp_inv_eq]
+  apply IsStronglyCartesian.map_uniq p (p.map φ) ψ rfl φ'
+  simp [hπ_comp, Φ]
+
+/-- Alternate constructor for `IsFibered`, a functor `p : 𝒳 ⥤ 𝒴` is fibered if any diagram of the
+form
+```
+          a
+          -
+          |
+          v
+R --f--> p(a)
+```
+admits a strongly cartesian lift `b ⟶ a` of `f`. -/
+lemma of_exists_isStronglyCartesian {p : 𝒳 ⥤ 𝒮}
+    (h : ∀ (a : 𝒳) (R : 𝒮) (f : R ⟶ p.obj a),
+      ∃ (b : 𝒳) (φ : b ⟶ a), IsStronglyCartesian p f φ) :
+    IsFibered p where
+  exists_isCartesian' := by
+    intro a R f
+    obtain ⟨b, φ, hφ⟩ := h a R f
+    refine ⟨b, φ, inferInstance⟩
+  comp := fun R S T f g {a b c} φ ψ _ _ =>
+    have : p.IsStronglyCartesian f φ := isStronglyCartesian_of_exists_isCartesian p h _ _
+    have : p.IsStronglyCartesian g ψ := isStronglyCartesian_of_exists_isCartesian p h _ _
+    inferInstance
+
+/-- Given a diagram
+```
+                  a
+                  -
+                  |
+                  v
+T --g--> R --f--> S
+```
+we have an isomorphism `T ×_S a ≅ T ×_R (R ×_S a)` -/
+noncomputable def pullbackPullbackIso {p : 𝒳 ⥤ 𝒮} [IsFibered p]
+    {R S T : 𝒮}  {a : 𝒳} (ha : p.obj a = S) (f : R ⟶ S) (g : T ⟶ R) :
+      pullbackObj ha (g ≫ f) ≅ pullbackObj (pullbackObj_proj ha f) g :=
+  domainUniqueUpToIso p (g ≫ f) (pullbackMap (pullbackObj_proj ha f) g ≫ pullbackMap ha f)
+    (pullbackMap ha (g ≫ f))
+
+end Functor.IsFibered
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean b/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean
index f98c7a397dccd..8de8bf2c92401 100644
--- a/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean
+++ b/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean
@@ -52,7 +52,7 @@ class Functor.IsHomLift {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) :
   cond : IsHomLiftAux p f φ
 
 /-- `subst_hom_lift p f φ` tries to substitute `f` with `p(φ)` by using `p.IsHomLift f φ` -/
-macro "subst_hom_lift" p:ident f:ident φ:ident : tactic =>
+macro "subst_hom_lift" p:term:max f:term:max φ:term:max : tactic =>
   `(tactic| obtain ⟨⟩ := Functor.IsHomLift.cond (p := $p) (f := $f) (φ := $φ))
 
 /-- For any arrow `φ : a ⟶ b` in `𝒳`, `φ` lifts the arrow `p.map φ` in the base `𝒮`-/
@@ -142,7 +142,7 @@ instance comp_lift_id_left {a b c : 𝒳} {S T : 𝒮} (f : S ⟶ T) (ψ : b ⟶
 lemma comp_lift_id_left' {a b c : 𝒳} (R : 𝒮) (φ : a ⟶ b) [p.IsHomLift (𝟙 R) φ]
     {S T : 𝒮} (f : S ⟶ T) (ψ : b ⟶ c) [p.IsHomLift f ψ] : p.IsHomLift f (φ ≫ ψ) := by
   obtain rfl : R = S := by rw [← codomain_eq p (𝟙 R) φ, domain_eq p f ψ]
-  simpa using inferInstanceAs (p.IsHomLift (𝟙 R ≫ f) (φ ≫ ψ))
+  infer_instance
 
 lemma eqToHom_domain_lift_id {p : 𝒳 ⥤ 𝒮} {a b : 𝒳} (hab : a = b) {R : 𝒮} (hR : p.obj a = R) :
     p.IsHomLift (𝟙 R) (eqToHom hab) := by
diff --git a/Mathlib/CategoryTheory/Filtered/Basic.lean b/Mathlib/CategoryTheory/Filtered/Basic.lean
index f4da1d923d0e6..688a1d9ce43a8 100644
--- a/Mathlib/CategoryTheory/Filtered/Basic.lean
+++ b/Mathlib/CategoryTheory/Filtered/Basic.lean
@@ -1,15 +1,9 @@
 /-
 Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Scott Morrison
+Authors: Reid Barton, Kim Morrison
 -/
-import Mathlib.CategoryTheory.FinCategory.Basic
-import Mathlib.CategoryTheory.Limits.Cones
 import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
-import Mathlib.CategoryTheory.Adjunction.Basic
-import Mathlib.CategoryTheory.Category.Preorder
-import Mathlib.CategoryTheory.Category.ULift
-import Mathlib.CategoryTheory.PEmpty
 
 /-!
 # Filtered categories
diff --git a/Mathlib/CategoryTheory/Filtered/Final.lean b/Mathlib/CategoryTheory/Filtered/Final.lean
index 73d630e449fad..23d7d5c9ec781 100644
--- a/Mathlib/CategoryTheory/Filtered/Final.lean
+++ b/Mathlib/CategoryTheory/Filtered/Final.lean
@@ -20,10 +20,11 @@ final can be restated. We show:
   if `StructuredArrow d F` is connected for all `d : D`.
 * Under categories of objects of filtered categories are filtered and their forgetful functors
   are final.
-
-Additionally, we show that if `D` is a filtered category and `F : C ⥤ D` is fully faithful and
-satisfies the additional condition that for every `d : D` there is an object `c : D` and a morphism
-`d ⟶ F.obj c`, then `C` is filtered and `F` is final.
+* If `D` is a filtered category and `F : C ⥤ D` is fully faithful and satisfies the additional
+  condition that for every `d : D` there is an object `c : D` and a morphism `d ⟶ F.obj c`, then
+  `C` is filtered and `F` is final.
+* Finality and initiality of diagonal functors `diag : C ⥤ C × C` and of projection functors
+  of (co)structured arrow categories.
 
 ## References
 
@@ -264,11 +265,36 @@ theorem Functor.initial_iff_isCofiltered_costructuredArrow [IsCofilteredOrEmpty
   rw [initial_iff_of_isCofiltered]
   exact fun h => isCofiltered_costructuredArrow_of_isCofiltered_of_exists F h.1 h.2
 
+/-- If `C` is filtered, then the structured arrow category on the diagonal functor `C ⥤ C × C`
+is filtered as well. -/
+instance [IsFiltered C] (X : C × C) : IsFiltered (StructuredArrow X (diag C)) := by
+  haveI : ∀ Y, IsFiltered (StructuredArrow Y (Under.forget X.1)) := by
+    rw [← final_iff_isFiltered_structuredArrow (Under.forget X.1)]
+    infer_instance
+  apply IsFiltered.of_equivalence (StructuredArrow.ofDiagEquivalence X).symm
+
+/-- The diagonal functor on any filtered category is final. -/
+instance Functor.final_diag_of_isFiltered [IsFiltered C] : Final (Functor.diag C) :=
+  final_of_isFiltered_structuredArrow _
+
+/-- If `C` is cofiltered, then the costructured arrow category on the diagonal functor `C ⥤ C × C`
+is cofiltered as well. -/
+instance [IsCofiltered C] (X : C × C) : IsCofiltered (CostructuredArrow (diag C) X) := by
+  haveI : ∀ Y, IsCofiltered (CostructuredArrow (Over.forget X.1) Y) := by
+    rw [← initial_iff_isCofiltered_costructuredArrow (Over.forget X.1)]
+    infer_instance
+  apply IsCofiltered.of_equivalence (CostructuredArrow.ofDiagEquivalence X).symm
+
+/-- The diagonal functor on any cofiltered category is initial. -/
+instance Functor.initial_diag_of_isFiltered [IsCofiltered C] : Initial (Functor.diag C) :=
+  initial_of_isCofiltered_costructuredArrow _
+
 end LocallySmall
 
+variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
+
 /-- If `C` is filtered, then every functor `F : C ⥤ Discrete PUnit` is final. -/
-theorem Functor.final_of_isFiltered_of_pUnit {C : Type u₁} [Category.{v₁} C]
-    [IsFiltered C] (F : C ⥤ Discrete PUnit) :
+theorem Functor.final_of_isFiltered_of_pUnit [IsFiltered C] (F : C ⥤ Discrete PUnit) :
     Final F := by
   refine final_of_exists_of_isFiltered F (fun _ => ?_) (fun {_} {c} _ _ => ?_)
   · use Classical.choice IsFiltered.nonempty
@@ -277,8 +303,7 @@ theorem Functor.final_of_isFiltered_of_pUnit {C : Type u₁} [Category.{v₁} C]
     apply Subsingleton.elim
 
 /-- If `C` is cofiltered, then every functor `F : C ⥤ Discrete PUnit` is initial. -/
-theorem Functor.initial_of_isCofiltered_pUnit {C : Type u₁} [Category.{v₁} C]
-    [IsCofiltered C] (F : C ⥤ Discrete PUnit) :
+theorem Functor.initial_of_isCofiltered_pUnit [IsCofiltered C] (F : C ⥤ Discrete PUnit) :
     Initial F := by
   refine initial_of_exists_of_isCofiltered F (fun _ => ?_) (fun {_} {c} _ _ => ?_)
   · use Classical.choice IsCofiltered.nonempty
@@ -286,4 +311,20 @@ theorem Functor.initial_of_isCofiltered_pUnit {C : Type u₁} [Category.{v₁} C
   · use c; use 𝟙 c
     apply Subsingleton.elim
 
+/-- The functor `StructuredArrow.proj : StructuredArrow Y T ⥤ C` is final if `T : C ⥤ D` is final
+and `C` is filtered. -/
+instance StructuredArrow.final_proj_of_isFiltered [IsFilteredOrEmpty C]
+    (T : C ⥤ D) [Final T] (Y : D) : Final (StructuredArrow.proj Y T) := by
+  refine ⟨fun X => ?_⟩
+  rw [isConnected_iff_of_equivalence (ofStructuredArrowProjEquivalence T Y X)]
+  exact (final_comp (Under.forget X) T).out _
+
+/-- The functor `CostructuredArrow.proj : CostructuredArrow Y T ⥤ C` is initial if `T : C ⥤ D` is
+initial and `C` is cofiltered. -/
+instance CostructuredArrow.initial_proj_of_isCofiltered [IsCofilteredOrEmpty C]
+    (T : C ⥤ D) [Initial T] (Y : D) : Initial (CostructuredArrow.proj T Y) := by
+  refine ⟨fun X => ?_⟩
+  rw [isConnected_iff_of_equivalence (ofCostructuredArrowProjEquivalence T Y X)]
+  exact (initial_comp (Over.forget X) T).out _
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Filtered/Small.lean b/Mathlib/CategoryTheory/Filtered/Small.lean
index 7948241296129..5206fd8c67956 100644
--- a/Mathlib/CategoryTheory/Filtered/Small.lean
+++ b/Mathlib/CategoryTheory/Filtered/Small.lean
@@ -56,11 +56,11 @@ namespace FilteredClosureSmall
     in `C`. This would require some kind of inductive-recursive definition, which Lean does not
     allow. Our solution is to define a function `ℕ → Σ t : Type (max v w), t → C` by (strong)
     induction and then take the union over all natural numbers, mimicking what one would do in a
-    set-theoretic setting.  -/
+    set-theoretic setting. -/
 
 /-- One step of the inductive procedure consists of adjoining all maxima and coequalizers of all
     objects and morphisms obtained so far. This is quite redundant, picking up many objects which we
-    already hit in earlier iterations, but this is easier to work with later.  -/
+    already hit in earlier iterations, but this is easier to work with later. -/
 private inductive InductiveStep (n : ℕ) (X : ∀ (k : ℕ), k < n → Σ t : Type (max v w), t → C) :
     Type (max v w)
   | max : {k k' : ℕ} → (hk : k < n) → (hk' : k' < n) → (X _ hk).1 → (X _ hk').1 → InductiveStep n X
@@ -78,7 +78,7 @@ private noncomputable def inductiveStepRealization (n : ℕ)
     as a function of `ℕ`.
 
    The function is defined by well-founded recursion, but we really want to use its
-   definitional equalities in the proofs below, so lets make it semireducible.  -/
+   definitional equalities in the proofs below, so lets make it semireducible. -/
 @[semireducible] private noncomputable def bundledAbstractFilteredClosure :
     ℕ → Σ t : Type (max v w), t → C
   | 0 => ⟨ULift.{v} α, f ∘ ULift.down⟩
@@ -211,7 +211,7 @@ private noncomputable def inductiveStepRealization (n : ℕ)
    `EssentiallySmall.{max v w} (FullSubcategory (CofilteredClosure f))`.
 
    The function is defined by well-founded recursion, but we really want to use its
-   definitional equalities in the proofs below, so lets make it semireducible.  -/
+   definitional equalities in the proofs below, so lets make it semireducible. -/
 @[semireducible] private noncomputable def bundledAbstractCofilteredClosure :
     ℕ → Σ t : Type (max v w), t → C
   | 0 => ⟨ULift.{v} α, f ∘ ULift.down⟩
diff --git a/Mathlib/CategoryTheory/FinCategory/AsType.lean b/Mathlib/CategoryTheory/FinCategory/AsType.lean
index 9d42330a17b22..1ba7c2d3cd220 100644
--- a/Mathlib/CategoryTheory/FinCategory/AsType.lean
+++ b/Mathlib/CategoryTheory/FinCategory/AsType.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Fintype.Card
 import Mathlib.CategoryTheory.FinCategory.Basic
diff --git a/Mathlib/CategoryTheory/FinCategory/Basic.lean b/Mathlib/CategoryTheory/FinCategory/Basic.lean
index 4ae9b8ca21320..294cc5a1a352f 100644
--- a/Mathlib/CategoryTheory/FinCategory/Basic.lean
+++ b/Mathlib/CategoryTheory/FinCategory/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Fintype.Basic
 import Mathlib.CategoryTheory.DiscreteCategory
diff --git a/Mathlib/CategoryTheory/FintypeCat.lean b/Mathlib/CategoryTheory/FintypeCat.lean
index 618a4c72bae25..30efd8e330d9f 100644
--- a/Mathlib/CategoryTheory/FintypeCat.lean
+++ b/Mathlib/CategoryTheory/FintypeCat.lean
@@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta, Adam Topaz
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
-import Mathlib.CategoryTheory.FullSubcategory
+import Mathlib.CategoryTheory.Endomorphism
 import Mathlib.CategoryTheory.Skeletal
-import Mathlib.Data.Fintype.Card
+import Mathlib.Data.Finite.Basic
 
 /-!
 # The category of finite types.
@@ -99,6 +99,15 @@ def equivEquivIso {A B : FintypeCat} : A ≃ B ≃ (A ≅ B) where
   left_inv := by aesop_cat
   right_inv := by aesop_cat
 
+instance (X Y : FintypeCat) : Finite (X ⟶ Y) :=
+  inferInstanceAs <| Finite (X → Y)
+
+instance (X Y : FintypeCat) : Finite (X ≅ Y) :=
+  Finite.of_injective _ (fun _ _ h ↦ Iso.ext h)
+
+instance (X : FintypeCat) : Finite (Aut X) :=
+  inferInstanceAs <| Finite (X ≅ X)
+
 universe u
 
 /--
@@ -189,5 +198,76 @@ lemma isSkeleton : IsSkeletonOf FintypeCat Skeleton Skeleton.incl where
   skel := Skeleton.is_skeletal
   eqv := by infer_instance
 
+section Universes
+
+universe v
+
+/-- If `u` and `v` are two arbitrary universes, we may construct a functor
+`uSwitch.{u, v} : FintypeCat.{u} ⥤ FintypeCat.{v}` by sending
+`X : FintypeCat.{u}` to `ULift.{v} (Fin (Fintype.card X))`. -/
+noncomputable def uSwitch : FintypeCat.{u} ⥤ FintypeCat.{v} where
+  obj X := FintypeCat.of <| ULift.{v} (Fin (Fintype.card X))
+  map {X Y} f x := ULift.up <| (Fintype.equivFin Y) (f ((Fintype.equivFin X).symm x.down))
+  map_comp {X Y Z} f g := by ext; simp
+
+/-- Switching the universe of an object `X : FintypeCat.{u}` does not change `X` up to equivalence
+of types. This is natural in the sense that it commutes with `uSwitch.map f` for
+any `f : X ⟶ Y` in `FintypeCat.{u}`. -/
+noncomputable def uSwitchEquiv (X : FintypeCat.{u}) :
+    uSwitch.{u, v}.obj X ≃ X :=
+  Equiv.ulift.trans (Fintype.equivFin X).symm
+
+lemma uSwitchEquiv_naturality {X Y : FintypeCat.{u}} (f : X ⟶ Y)
+    (x : uSwitch.{u, v}.obj X) :
+    f (X.uSwitchEquiv x) = Y.uSwitchEquiv (uSwitch.map f x) := by
+  simp only [uSwitch, uSwitchEquiv, Equiv.trans_apply]
+  erw [Equiv.ulift_apply, Equiv.ulift_apply]
+  simp only [Equiv.symm_apply_apply]
+
+lemma uSwitchEquiv_symm_naturality {X Y : FintypeCat.{u}} (f : X ⟶ Y) (x : X) :
+    uSwitch.map f (X.uSwitchEquiv.symm x) = Y.uSwitchEquiv.symm (f x) := by
+  rw [← Equiv.apply_eq_iff_eq_symm_apply, ← uSwitchEquiv_naturality f,
+    Equiv.apply_symm_apply]
+
+lemma uSwitch_map_uSwitch_map {X Y : FintypeCat.{u}} (f : X ⟶ Y) :
+    uSwitch.map (uSwitch.map f) =
+    (equivEquivIso ((uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv)).hom ≫
+      f ≫ (equivEquivIso ((uSwitch.obj Y).uSwitchEquiv.trans
+      Y.uSwitchEquiv)).inv := by
+  ext x
+  simp only [comp_apply, equivEquivIso_apply_hom, Equiv.trans_apply]
+  rw [uSwitchEquiv_naturality f, ← uSwitchEquiv_naturality]
+  rfl
+
+/-- `uSwitch.{u, v}` is an equivalence of categories with quasi-inverse `uSwitch.{v, u}`. -/
+noncomputable def uSwitchEquivalence : FintypeCat.{u} ≌ FintypeCat.{v} where
+  functor := uSwitch
+  inverse := uSwitch
+  unitIso := NatIso.ofComponents (fun X ↦ (equivEquivIso <|
+    (uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv).symm) <| by
+    simp [uSwitch_map_uSwitch_map]
+  counitIso := NatIso.ofComponents (fun X ↦ equivEquivIso <|
+    (uSwitch.obj X).uSwitchEquiv.trans X.uSwitchEquiv) <| by
+    simp [uSwitch_map_uSwitch_map]
+  functor_unitIso_comp X := by
+    ext x
+    simp [← uSwitchEquiv_naturality]
+
+instance : uSwitch.IsEquivalence :=
+  uSwitchEquivalence.isEquivalence_functor
+
+end Universes
 
 end FintypeCat
+
+namespace FunctorToFintypeCat
+
+universe u v w
+
+variable {C : Type u} [Category.{v} C] (F G : C ⥤ FintypeCat.{w}) {X Y : C}
+
+lemma naturality (σ : F ⟶ G) (f : X ⟶ Y) (x : F.obj X) :
+    σ.app Y (F.map f x) = G.map f (σ.app X x) :=
+  congr_fun (σ.naturality f) x
+
+end FunctorToFintypeCat
diff --git a/Mathlib/CategoryTheory/FullSubcategory.lean b/Mathlib/CategoryTheory/FullSubcategory.lean
index 49bfb77d86356..bf98131940b9e 100644
--- a/Mathlib/CategoryTheory/FullSubcategory.lean
+++ b/Mathlib/CategoryTheory/FullSubcategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Reid Barton
+Authors: Kim Morrison, Reid Barton
 -/
 import Mathlib.CategoryTheory.Functor.FullyFaithful
 
@@ -97,9 +97,9 @@ See <https://stacks.math.columbia.edu/tag/001D>. We do not define 'strictly full
 -/
 @[ext]
 structure FullSubcategory where
-  /-- The category of which this is a full subcategory-/
+  /-- The category of which this is a full subcategory -/
   obj : C
-  /-- The predicate satisfied by all objects in this subcategory-/
+  /-- The predicate satisfied by all objects in this subcategory -/
   property : Z obj
 
 instance FullSubcategory.category : Category.{v} (FullSubcategory Z) :=
diff --git a/Mathlib/CategoryTheory/Functor/Basic.lean b/Mathlib/CategoryTheory/Functor/Basic.lean
index bff78cc031a50..4be276945c1e4 100644
--- a/Mathlib/CategoryTheory/Functor/Basic.lean
+++ b/Mathlib/CategoryTheory/Functor/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Category.Basic
 
diff --git a/Mathlib/CategoryTheory/Functor/Category.lean b/Mathlib/CategoryTheory/Functor/Category.lean
index 654f3f72d8875..e1f0445ab692b 100644
--- a/Mathlib/CategoryTheory/Functor/Category.lean
+++ b/Mathlib/CategoryTheory/Functor/Category.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.NatTrans
 import Mathlib.CategoryTheory.Iso
diff --git a/Mathlib/CategoryTheory/Functor/Const.lean b/Mathlib/CategoryTheory/Functor/Const.lean
index 006504deb58b6..5c783b8742f00 100644
--- a/Mathlib/CategoryTheory/Functor/Const.lean
+++ b/Mathlib/CategoryTheory/Functor/Const.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Opposites
 
diff --git a/Mathlib/CategoryTheory/Functor/Currying.lean b/Mathlib/CategoryTheory/Functor/Currying.lean
index caa5be9b7f6d8..e130aedb422c8 100644
--- a/Mathlib/CategoryTheory/Functor/Currying.lean
+++ b/Mathlib/CategoryTheory/Functor/Currying.lean
@@ -1,8 +1,9 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
+import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.Products.Basic
 
 /-!
diff --git a/Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean b/Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean
index c926a1b32013f..8acc96f8e2d2d 100644
--- a/Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean
+++ b/Mathlib/CategoryTheory/Functor/Derived/RightDerived.lean
@@ -97,7 +97,7 @@ lemma rightDerived_ext (G : D ⥤ H) (γ₁ γ₂ : RF ⟶ G)
   RF.hom_ext_of_isLeftKanExtension α γ₁ γ₂ hγ
 
 /-- The natural transformation `RF ⟶ RF'` on right derived functors that is
-induced by a natural transformation `F ⟶ F'`.  -/
+induced by a natural transformation `F ⟶ F'`. -/
 noncomputable def rightDerivedNatTrans (τ : F ⟶ F') : RF ⟶ RF' :=
   RF.rightDerivedDesc α W RF' (τ ≫ α')
 
@@ -128,7 +128,7 @@ lemma rightDerivedNatTrans_comp (τ : F ⟶ F') (τ' : F' ⟶ F'') :
   rightDerived_ext RF α W _ _ _ (by aesop_cat)
 
 /-- The natural isomorphism `RF ≅ RF'` on right derived functors that is
-induced by a natural isomorphism `F ≅ F'`.  -/
+induced by a natural isomorphism `F ≅ F'`. -/
 @[simps]
 noncomputable def rightDerivedNatIso (τ : F ≅ F') :
     RF ≅ RF' where
diff --git a/Mathlib/CategoryTheory/Functor/EpiMono.lean b/Mathlib/CategoryTheory/Functor/EpiMono.lean
index 0f25145dea713..428607bf319fa 100644
--- a/Mathlib/CategoryTheory/Functor/EpiMono.lean
+++ b/Mathlib/CategoryTheory/Functor/EpiMono.lean
@@ -103,8 +103,7 @@ theorem reflectsMonomorphisms_of_preserves_of_reflects (F : C ⥤ D) (G : D ⥤
 theorem preservesMonomorphisms.of_iso {F G : C ⥤ D} [PreservesMonomorphisms F] (α : F ≅ G) :
     PreservesMonomorphisms G :=
   { preserves := fun {X} {Y} f h => by
-      haveI : Mono (F.map f ≫ (α.app Y).hom) := mono_comp _ _
-      convert (mono_comp _ _ : Mono ((α.app X).inv ≫ F.map f ≫ (α.app Y).hom))
+      suffices G.map f = (α.app X).inv ≫ F.map f ≫ (α.app Y).hom from this ▸ mono_comp _ _
       rw [Iso.eq_inv_comp, Iso.app_hom, Iso.app_hom, NatTrans.naturality] }
 
 theorem preservesMonomorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
@@ -114,8 +113,7 @@ theorem preservesMonomorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
 theorem preservesEpimorphisms.of_iso {F G : C ⥤ D} [PreservesEpimorphisms F] (α : F ≅ G) :
     PreservesEpimorphisms G :=
   { preserves := fun {X} {Y} f h => by
-      haveI : Epi (F.map f ≫ (α.app Y).hom) := epi_comp _ _
-      convert (epi_comp _ _ : Epi ((α.app X).inv ≫ F.map f ≫ (α.app Y).hom))
+      suffices G.map f = (α.app X).inv ≫ F.map f ≫ (α.app Y).hom from this ▸ epi_comp _ _
       rw [Iso.eq_inv_comp, Iso.app_hom, Iso.app_hom, NatTrans.naturality] }
 
 theorem preservesEpimorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
@@ -126,8 +124,7 @@ theorem reflectsMonomorphisms.of_iso {F G : C ⥤ D} [ReflectsMonomorphisms F] (
     ReflectsMonomorphisms G :=
   { reflects := fun {X} {Y} f h => by
       apply F.mono_of_mono_map
-      haveI : Mono (G.map f ≫ (α.app Y).inv) := mono_comp _ _
-      convert (mono_comp _ _ : Mono ((α.app X).hom ≫ G.map f ≫ (α.app Y).inv))
+      suffices F.map f = (α.app X).hom ≫ G.map f ≫ (α.app Y).inv from this ▸ mono_comp _ _
       rw [← Category.assoc, Iso.eq_comp_inv, Iso.app_hom, Iso.app_hom, NatTrans.naturality] }
 
 theorem reflectsMonomorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
@@ -138,8 +135,7 @@ theorem reflectsEpimorphisms.of_iso {F G : C ⥤ D} [ReflectsEpimorphisms F] (α
     ReflectsEpimorphisms G :=
   { reflects := fun {X} {Y} f h => by
       apply F.epi_of_epi_map
-      haveI : Epi (G.map f ≫ (α.app Y).inv) := epi_comp _ _
-      convert (epi_comp _ _ : Epi ((α.app X).hom ≫ G.map f ≫ (α.app Y).inv))
+      suffices F.map f = (α.app X).hom ≫ G.map f ≫ (α.app Y).inv from this ▸ epi_comp _ _
       rw [← Category.assoc, Iso.eq_comp_inv, Iso.app_hom, Iso.app_hom, NatTrans.naturality] }
 
 theorem reflectsEpimorphisms.iso_iff {F G : C ⥤ D} (α : F ≅ G) :
@@ -272,7 +268,7 @@ instance (adj : F ⊣ F') {X : C} {Y : D} (f : F.obj X ⟶ Y) [hf : Mono f] [F.R
     Mono (adj.homEquiv _ _ f) :=
   F.mono_of_mono_map <| by
     rw [← (homEquiv adj X Y).symm_apply_apply f] at hf
-    exact mono_of_mono_fac adj.homEquiv_counit.symm
+    exact mono_of_mono_fac (adj.homEquiv_counit _ _ _).symm
 
 end CategoryTheory.Adjunction
 
diff --git a/Mathlib/CategoryTheory/Functor/FullyFaithful.lean b/Mathlib/CategoryTheory/Functor/FullyFaithful.lean
index 97f7070645c30..82f3de969be51 100644
--- a/Mathlib/CategoryTheory/Functor/FullyFaithful.lean
+++ b/Mathlib/CategoryTheory/Functor/FullyFaithful.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.NatIso
 import Mathlib.Logic.Equiv.Defs
@@ -171,6 +171,15 @@ lemma full : F.Full where
 lemma faithful : F.Faithful where
   map_injective := hF.map_injective
 
+instance : Subsingleton F.FullyFaithful where
+  allEq h₁ h₂ := by
+    have := h₁.faithful
+    cases h₁ with | mk f₁ hf₁ _ => cases h₂ with | mk f₂ hf₂ _ =>
+    simp only [Functor.FullyFaithful.mk.injEq]
+    ext
+    apply F.map_injective
+    rw [hf₁, hf₂]
+
 /-- The unique isomorphism `X ≅ Y` which induces an isomorphism `F.obj X ≅ F.obj Y`
 when `hF : F.FullyFaithful`. -/
 @[simps]
diff --git a/Mathlib/CategoryTheory/Functor/FunctorHom.lean b/Mathlib/CategoryTheory/Functor/FunctorHom.lean
new file mode 100644
index 0000000000000..bd8ab6a9e031e
--- /dev/null
+++ b/Mathlib/CategoryTheory/Functor/FunctorHom.lean
@@ -0,0 +1,208 @@
+/-
+Copyright (c) 2024 Jack McKoen. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jack McKoen, Joël Riou
+-/
+import Mathlib.CategoryTheory.Monoidal.FunctorCategory
+import Mathlib.CategoryTheory.Monoidal.Types.Basic
+import Mathlib.CategoryTheory.Enriched.Basic
+
+/-!
+# Internal hom in functor categories
+
+Given functors `F G : C ⥤ D`, define a functor `functorHom F G` from `C` to `Type max v' v u`,
+which is a proxy for the "internal hom" functor Hom(F ⊗ coyoneda(-), G). This is used to show
+that the functor category `C ⥤ D` is enriched over `C ⥤ Type max v' v u`. This is also useful
+for showing that `C ⥤ Type max w v u` is monoidal closed.
+
+See `Mathlib.CategoryTheory.Closed.FunctorToTypes`.
+
+-/
+
+
+universe w v' v u u'
+
+open CategoryTheory MonoidalCategory
+
+variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
+
+variable (F G : C ⥤ D)
+
+namespace CategoryTheory.Functor
+
+/-- Given functors `F G : C ⥤ D`, `HomObj F G A` is a proxy for the type
+of "morphisms" `F ⊗ A ⟶ G`, where `A : C ⥤ Type w` (`w` an arbitrary universe). -/
+@[ext]
+structure HomObj (A : C ⥤ Type w) where
+  /-- The morphism `F.obj c ⟶ G.obj c` associated with `a : A.obj c`. -/
+  app (c : C) (a : A.obj c) : F.obj c ⟶ G.obj c
+  naturality {c d : C} (f : c ⟶ d) (a : A.obj c) :
+    F.map f ≫ app d (A.map f a) = app c a ≫ G.map f := by aesop_cat
+
+/-- When `F`, `G`, and `A` are all functors `C ⥤ Type w`, then `HomObj F G A` is in
+bijection with `F ⊗ A ⟶ G`. -/
+@[simps]
+def homObjEquiv (F G A : C ⥤ Type w) : (HomObj F G A) ≃ (F ⊗ A ⟶ G) where
+  toFun a := ⟨fun X ⟨x, y⟩ ↦ a.app X y x, fun X Y f ↦ by
+    ext ⟨x, y⟩
+    erw [congr_fun (a.naturality f y) x]
+    rfl ⟩
+  invFun a := ⟨fun X y x ↦ a.app X (x, y), fun φ y ↦ by
+    ext x
+    erw [congr_fun (a.naturality φ) (x, y)]
+    rfl ⟩
+  left_inv _ := by aesop
+  right_inv _ := by aesop
+
+namespace HomObj
+
+attribute [reassoc (attr := simp)] naturality
+
+variable {F G} {A : C ⥤ Type w}
+
+lemma congr_app {f g : HomObj F G A} (h : f = g) (X : C)
+    (a : A.obj X) : f.app X a = g.app X a := by subst h; rfl
+
+/-- Given a natural transformation `F ⟶ G`, get a term of `HomObj F G A` by "ignoring" `A`. -/
+@[simps]
+def ofNatTrans (f : F ⟶ G) : HomObj F G A where
+  app X _ := f.app X
+
+/-- The identity `HomObj F F A`. -/
+@[simps!]
+def id (A : C ⥤ Type w) : HomObj F F A := ofNatTrans (𝟙 F)
+
+/-- Composition of `f : HomObj F G A` with `g : HomObj G M A`. -/
+@[simps]
+def comp {M : C ⥤ D} (f : HomObj F G A) (g : HomObj G M A) : HomObj F M A where
+  app X a := f.app X a ≫ g.app X a
+
+/-- Given a morphism `A' ⟶ A`, send a term of `HomObj F G A` to a term of `HomObj F G A'`. -/
+@[simps]
+def map {A' : C ⥤ Type w} (f : A' ⟶ A) (x : HomObj F G A) : HomObj F G A' where
+  app Δ a := x.app Δ (f.app Δ a)
+  naturality {Δ Δ'} φ a := by
+    dsimp
+    rw [← x.naturality φ (f.app Δ a), FunctorToTypes.naturality _ _ f φ a]
+
+end HomObj
+
+/-- The contravariant functor taking `A : C ⥤ Type w` to `HomObj F G A`, i.e. Hom(F ⊗ -, G). -/
+@[simps]
+def homObjFunctor : (C ⥤ Type w)ᵒᵖ ⥤ Type max w v' u where
+  obj A := HomObj F G A.unop
+  map {A A'} f x :=
+    { app := fun X a ↦ x.app X (f.unop.app _ a)
+      naturality := fun {X Y} φ a ↦ by
+        dsimp
+        rw [← HomObj.naturality]
+        congr 2
+        exact congr_fun (f.unop.naturality φ) a }
+
+/-- Composition of `homObjFunctor` with the co-Yoneda embedding, i.e. Hom(F ⊗ coyoneda(-), G).
+When `F G : C ⥤ Type max v' v u`, this is the internal hom of `F` and `G`: see
+`Mathlib.CategoryTheory.Closed.FunctorToTypes`. -/
+def functorHom (F G : C ⥤ D) : C ⥤ Type max v' v u := coyoneda.rightOp ⋙ homObjFunctor.{v} F G
+
+variable {F G} in
+@[ext]
+lemma functorHom_ext {X : C} {x y : (F.functorHom G).obj X}
+    (h : ∀ (Y : C) (f : X ⟶ Y), x.app Y f = y.app Y f) : x = y :=
+  HomObj.ext (by ext; apply h)
+
+/-- The equivalence `(A ⟶ F.functorHom G) ≃ HomObj F G A`. -/
+@[simps]
+def functorHomEquiv (A : C ⥤ Type max u v v') : (A ⟶ F.functorHom G) ≃ HomObj F G A where
+  toFun φ :=
+    { app := fun X a ↦ (φ.app X a).app X (𝟙 _)
+      naturality := fun {X Y} f a => by
+        rw [← (φ.app X a).naturality f (𝟙 _)]
+        have := HomObj.congr_app (congr_fun (φ.naturality f) a) Y (𝟙 _)
+        dsimp [functorHom, homObjFunctor] at this
+        aesop }
+  invFun x :=
+    { app := fun X a ↦ { app := fun Y f => x.app Y (A.map f a) }
+      naturality := fun X Y f => by
+        ext
+        dsimp only [types_comp_apply]
+        rw [← FunctorToTypes.map_comp_apply]
+        rfl }
+  left_inv φ := by
+    ext X a Y f
+    exact (HomObj.congr_app (congr_fun (φ.naturality f) a) Y (𝟙 _)).trans
+      (congr_arg ((φ.app X a).app Y) (by simp))
+  right_inv x := by aesop
+
+variable {F G} in
+/-- Morphisms `(𝟙_ (C ⥤ Type max v' v u) ⟶ F.functorHom G)` are in bijection with
+morphisms `F ⟶ G`. -/
+@[simps]
+def natTransEquiv : (𝟙_ (C ⥤ Type max v' v u) ⟶ F.functorHom G) ≃ (F ⟶ G) where
+  toFun f := ⟨fun X ↦ (f.app X (PUnit.unit)).app X (𝟙 _), by
+    intro X Y φ
+    rw [← (f.app X (PUnit.unit)).naturality φ]
+    congr 1
+    have := HomObj.congr_app (congr_fun (f.naturality φ) PUnit.unit) Y (𝟙 Y)
+    dsimp [functorHom, homObjFunctor] at this
+    aesop ⟩
+  invFun f := ⟨fun _ _ ↦ HomObj.ofNatTrans f, _⟩
+  left_inv f := by
+    ext X a Y φ
+    have := HomObj.congr_app (congr_fun (f.naturality φ) PUnit.unit) Y (𝟙 Y)
+    dsimp [functorHom, homObjFunctor] at this
+    aesop
+  right_inv _ := rfl
+
+end CategoryTheory.Functor
+
+open Functor
+
+namespace CategoryTheory.Enriched.Functor
+
+@[simp]
+lemma natTransEquiv_symm_app_app_apply (F G : C ⥤ D) (f : F ⟶ G)
+    {X : C} {a : (𝟙_ (C ⥤ Type (max v' v u))).obj X} (Y : C) {φ : X ⟶ Y} :
+    ((natTransEquiv.symm f).app X a).app Y φ = f.app Y := rfl
+
+@[simp]
+lemma natTransEquiv_symm_whiskerRight_functorHom_app (K L : C ⥤ D) (X : C) (f : K ⟶ K)
+    (x : 𝟙_ _ ⊗ (K.functorHom L).obj X) :
+    ((natTransEquiv.symm f ▷ K.functorHom L).app X x) =
+    (HomObj.ofNatTrans f, x.2) := rfl
+
+@[simp]
+lemma functorHom_whiskerLeft_natTransEquiv_symm_app (K L : C ⥤ D) (X : C) (f : L ⟶ L)
+    (x : (K.functorHom L).obj X ⊗ 𝟙_ _) :
+    ((K.functorHom L ◁ natTransEquiv.symm f).app X x) =
+    (x.1, HomObj.ofNatTrans f) := rfl
+
+@[simp]
+lemma whiskerLeft_app_apply (K L M N : C ⥤ D)
+    (g : L.functorHom M ⊗ M.functorHom N ⟶ L.functorHom N)
+    {X : C} (a : (K.functorHom L ⊗ L.functorHom M ⊗ M.functorHom N).obj X) :
+    (K.functorHom L ◁ g).app X a = ⟨a.1, g.app X a.2⟩ := rfl
+
+@[simp]
+lemma whiskerRight_app_apply (K L M N : C ⥤ D)
+    (f : K.functorHom L ⊗ L.functorHom M ⟶ K.functorHom M)
+    {X : C} (a : ((K.functorHom L ⊗ L.functorHom M) ⊗ M.functorHom N).obj X) :
+    (f ▷  M.functorHom N).app X a = ⟨f.app X a.1, a.2⟩ := rfl
+
+@[simp]
+lemma associator_inv_apply (K L M N : C ⥤ D) {X : C}
+    (x : ((K.functorHom L) ⊗ (L.functorHom M) ⊗ (M.functorHom N)).obj X) :
+    (α_ ((K.functorHom L).obj X) ((L.functorHom M).obj X) ((M.functorHom N).obj X)).inv x =
+    ⟨⟨x.1, x.2.1⟩, x.2.2⟩ := rfl
+
+@[simp]
+lemma associator_hom_apply (K L M N : C ⥤ D) {X : C}
+    (x : (((K.functorHom L) ⊗ (L.functorHom M)) ⊗ (M.functorHom N)).obj X) :
+    (α_ ((K.functorHom L).obj X) ((L.functorHom M).obj X) ((M.functorHom N).obj X)).hom x =
+    ⟨x.1.1, x.1.2, x.2⟩ := rfl
+
+noncomputable instance : EnrichedCategory (C ⥤ Type max v' v u) (C ⥤ D) where
+  Hom := functorHom
+  id F := natTransEquiv.symm (𝟙 F)
+  comp F G H := { app := fun X ⟨f, g⟩ => f.comp g }
+
+end CategoryTheory.Enriched.Functor
diff --git a/Mathlib/CategoryTheory/Functor/Functorial.lean b/Mathlib/CategoryTheory/Functor/Functorial.lean
index 7b9fadaad4f11..f55993471df00 100644
--- a/Mathlib/CategoryTheory/Functor/Functorial.lean
+++ b/Mathlib/CategoryTheory/Functor/Functorial.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Functor.Basic
 
diff --git a/Mathlib/CategoryTheory/Functor/Hom.lean b/Mathlib/CategoryTheory/Functor/Hom.lean
index 01f3ac9dd321f..225924cf02b07 100644
--- a/Mathlib/CategoryTheory/Functor/Hom.lean
+++ b/Mathlib/CategoryTheory/Functor/Hom.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Scott Morrison
+Authors: Reid Barton, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Products.Basic
 import Mathlib.CategoryTheory.Types
diff --git a/Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean b/Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean
index 57c9c1c9353e0..fe8433c905fb7 100644
--- a/Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Functor/KanExtension/Adjunction.lean
@@ -168,7 +168,7 @@ precomposition by `L`. -/
 noncomputable def ranAdjunction : (whiskeringLeft C D H).obj L ⊣ L.ran :=
   Adjunction.mkOfHomEquiv
     { homEquiv := fun F G =>
-        (homEquivOfIsRightKanExtension (α := L.ranCounit.app G) F).symm
+        (homEquivOfIsRightKanExtension (α := L.ranCounit.app G) _ F).symm
       homEquiv_naturality_right := fun {F G₁ G₂} β f ↦
         hom_ext_of_isRightKanExtension _ (L.ranCounit.app G₂) _ _ (by
         ext X
diff --git a/Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean b/Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
index cf9297f47d979..9d4cb7a6c2ea1 100644
--- a/Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
+++ b/Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean
@@ -343,6 +343,27 @@ instance : (pointwiseLeftKanExtension L F).IsLeftKanExtension
 instance : HasLeftKanExtension L F :=
   HasLeftKanExtension.mk _ (pointwiseLeftKanExtensionUnit L F)
 
+/-- An auxiliary cocone used in the lemma `pointwiseLeftKanExtension_desc_app` -/
+@[simps]
+def costructuredArrowMapCocone (G : D ⥤ H) (α : F ⟶ L ⋙ G) (Y : D) :
+    Cocone (CostructuredArrow.proj L Y ⋙ F) where
+  pt := G.obj Y
+  ι := {
+    app := fun f ↦ α.app f.left ≫ G.map f.hom
+    naturality := by simp [← G.map_comp] }
+
+@[simp]
+lemma pointwiseLeftKanExtension_desc_app (G : D ⥤ H) (α :  F ⟶ L ⋙ G) (Y : D) :
+    ((pointwiseLeftKanExtension L F).descOfIsLeftKanExtension (pointwiseLeftKanExtensionUnit L F)
+      G α |>.app Y) = colimit.desc _ (costructuredArrowMapCocone L F G α Y) := by
+  let β : L.pointwiseLeftKanExtension F ⟶ G :=
+    { app := fun Y ↦ colimit.desc _ (costructuredArrowMapCocone L F G α Y) }
+  have h : (pointwiseLeftKanExtension L F).descOfIsLeftKanExtension
+      (pointwiseLeftKanExtensionUnit L F) G α = β := by
+    apply hom_ext_of_isLeftKanExtension (α := pointwiseLeftKanExtensionUnit L F)
+    aesop
+  exact NatTrans.congr_app h Y
+
 variable {F L}
 
 /-- If `F` admits a pointwise left Kan extension along `L`, then any left Kan extension of `F`
@@ -421,6 +442,28 @@ instance : (pointwiseRightKanExtension L F).IsRightKanExtension
 instance : HasRightKanExtension L F :=
   HasRightKanExtension.mk _ (pointwiseRightKanExtensionCounit L F)
 
+/-- An auxiliary cocone used in the lemma `pointwiseRightKanExtension_lift_app` -/
+@[simps]
+def structuredArrowMapCone (G : D ⥤ H) (α : L ⋙ G ⟶ F) (Y : D) :
+    Cone (StructuredArrow.proj Y L ⋙ F) where
+  pt := G.obj Y
+  π := {
+    app := fun f ↦ G.map f.hom ≫ α.app f.right
+    naturality := by simp [← α.naturality, ← G.map_comp_assoc] }
+
+@[simp]
+lemma pointwiseRightKanExtension_lift_app (G : D ⥤ H) (α : L ⋙ G ⟶ F) (Y : D) :
+    ((pointwiseRightKanExtension L F).liftOfIsRightKanExtension
+      (pointwiseRightKanExtensionCounit L F) G α |>.app Y) =
+        limit.lift _ (structuredArrowMapCone L F G α Y) := by
+  let β : G ⟶ L.pointwiseRightKanExtension F :=
+    { app := fun Y ↦ limit.lift _ (structuredArrowMapCone L F G α Y) }
+  have h : (pointwiseRightKanExtension L F).liftOfIsRightKanExtension
+      (pointwiseRightKanExtensionCounit L F) G α = β := by
+    apply hom_ext_of_isRightKanExtension (α := pointwiseRightKanExtensionCounit L F)
+    aesop
+  exact NatTrans.congr_app h Y
+
 variable {F L}
 
 /-- If `F` admits a pointwise right Kan extension along `L`, then any right Kan extension of `F`
diff --git a/Mathlib/CategoryTheory/Galois/Action.lean b/Mathlib/CategoryTheory/Galois/Action.lean
new file mode 100644
index 0000000000000..fbf5fbd2f2b86
--- /dev/null
+++ b/Mathlib/CategoryTheory/Galois/Action.lean
@@ -0,0 +1,89 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.CategoryTheory.Galois.Examples
+import Mathlib.CategoryTheory.Galois.Prorepresentability
+
+/-!
+
+# Induced functor to finite `Aut F`-sets
+
+Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor
+from finite `Aut F`-sets to finite sets. In this file we collect basic properties
+of the induced functor `H : C ⥤ Action FintypeCat (MonCat.of (Aut F))`.
+
+See `Mathlib.CategoryTheory.Galois.Full` for the proof that `H` is (faithfully) full.
+
+-/
+
+universe u v
+
+namespace CategoryTheory
+
+namespace PreGaloisCategory
+
+open Limits Functor
+
+variable {C : Type u} [Category.{v} C] (F : C ⥤ FintypeCat.{u})
+
+/-- Any (fiber) functor `F : C ⥤ FintypeCat` naturally factors via
+the forgetful functor from `Action FintypeCat (MonCat.of (Aut F))` to `FintypeCat`. -/
+def functorToAction : C ⥤ Action FintypeCat.{u} (MonCat.of (Aut F)) where
+  obj X := Action.FintypeCat.ofMulAction (Aut F) (F.obj X)
+  map f := {
+    hom := F.map f
+    comm := fun g ↦ symm <| g.hom.naturality f
+  }
+
+lemma functorToAction_comp_forget₂_eq : functorToAction F ⋙ forget₂ _ FintypeCat = F := rfl
+
+@[simp]
+lemma functorToAction_map {X Y : C} (f : X ⟶ Y) : ((functorToAction F).map f).hom = F.map f :=
+  rfl
+
+instance (X : C) : MulAction (Aut X) ((functorToAction F).obj X).V :=
+  inferInstanceAs <| MulAction (Aut X) (F.obj X)
+
+variable [GaloisCategory C] [FiberFunctor F]
+
+instance (X : C) [IsGalois X] : MulAction.IsPretransitive (Aut X) ((functorToAction F).obj X).V :=
+  isPretransitive_of_isGalois F X
+
+instance : Functor.Faithful (functorToAction F) :=
+  have : Functor.Faithful (functorToAction F ⋙ forget₂ _ FintypeCat) :=
+    inferInstanceAs <| Functor.Faithful F
+  Functor.Faithful.of_comp (functorToAction F) (forget₂ _ FintypeCat)
+
+instance : PreservesMonomorphisms (functorToAction F) :=
+  have : PreservesMonomorphisms (functorToAction F ⋙ forget₂ _ FintypeCat) :=
+    inferInstanceAs <| PreservesMonomorphisms F
+  preservesMonomorphisms_of_preserves_of_reflects (functorToAction F) (forget₂ _ FintypeCat)
+
+instance : ReflectsMonomorphisms (functorToAction F) := reflectsMonomorphisms_of_faithful _
+
+instance : Functor.ReflectsIsomorphisms (functorToAction F) where
+  reflects f _ :=
+    have : IsIso (F.map f) := (forget₂ _ FintypeCat).map_isIso ((functorToAction F).map f)
+    isIso_of_reflects_iso f F
+
+noncomputable instance : PreservesFiniteCoproducts (functorToAction F) :=
+  ⟨fun J _ ↦ Action.preservesColimitsOfShapeOfPreserves (functorToAction F)
+    (inferInstanceAs <| PreservesColimitsOfShape (Discrete J) F)⟩
+
+noncomputable instance : PreservesFiniteProducts (functorToAction F) :=
+  ⟨fun J _ ↦ Action.preservesLimitsOfShapeOfPreserves (functorToAction F)
+    (inferInstanceAs <| PreservesLimitsOfShape (Discrete J) F)⟩
+
+noncomputable instance (G : Type*) [Group G] [Finite G] :
+    PreservesColimitsOfShape (SingleObj G) (functorToAction F) :=
+  Action.preservesColimitsOfShapeOfPreserves _ <|
+    inferInstanceAs <| PreservesColimitsOfShape (SingleObj G) F
+
+instance : PreservesIsConnected (functorToAction F) :=
+  ⟨fun {X} _ ↦ FintypeCat.Action.isConnected_of_transitive (Aut F) (F.obj X)⟩
+
+end PreGaloisCategory
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/Basic.lean b/Mathlib/CategoryTheory/Galois/Basic.lean
index 60874987826dd..a0cd79ac0d2fe 100644
--- a/Mathlib/CategoryTheory/Galois/Basic.lean
+++ b/Mathlib/CategoryTheory/Galois/Basic.lean
@@ -3,16 +3,14 @@ Copyright (c) 2024 Christian Merten. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christian Merten
 -/
-import Mathlib.CategoryTheory.FintypeCat
 import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
 import Mathlib.CategoryTheory.Limits.FintypeCat
 import Mathlib.CategoryTheory.Limits.MonoCoprod
-import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
-import Mathlib.CategoryTheory.Limits.Shapes.Types
 import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
 import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
 import Mathlib.CategoryTheory.SingleObj
 import Mathlib.Data.Finite.Card
+import Mathlib.Logic.Equiv.TransferInstance
 
 /-!
 # Definition and basic properties of Galois categories
@@ -41,7 +39,7 @@ as this is not needed for the proof of the fundamental theorem on Galois categor
 
 -/
 
-universe u₁ u₂ v₁ v₂ w
+universe u₁ u₂ v₁ v₂ w t
 
 namespace CategoryTheory
 
@@ -117,6 +115,11 @@ instance : HasBinaryProducts C := hasBinaryProducts_of_hasTerminal_and_pullbacks
 
 instance : HasEqualizers C := hasEqualizers_of_hasPullbacks_and_binary_products
 
+-- A `PreGaloisCategory` has quotients by finite groups in arbitrary universes. -/
+instance {G : Type*} [Group G] [Finite G] : HasColimitsOfShape (SingleObj G) C := by
+  obtain ⟨G', hg, hf, ⟨e⟩⟩ := Finite.exists_type_univ_nonempty_mulEquiv G
+  exact Limits.hasColimitsOfShape_of_equivalence e.toSingleObjEquiv.symm
+
 end
 
 namespace FiberFunctor
@@ -136,6 +139,12 @@ noncomputable instance : ReflectsColimitsOfShape (Discrete PEmpty.{1}) F :=
 noncomputable instance : PreservesFiniteLimits F :=
   preservesFiniteLimitsOfPreservesTerminalAndPullbacks F
 
+/-- Fiber functors preserve quotients by finite groups in arbitrary universes. -/
+noncomputable instance {G : Type*} [Group G] [Finite G] :
+    PreservesColimitsOfShape (SingleObj G) F := by
+  choose G' hg hf he using Finite.exists_type_univ_nonempty_mulEquiv G
+  exact Limits.preservesColimitsOfShapeOfEquiv he.some.toSingleObjEquiv.symm F
+
 /-- Fiber functors reflect monomorphisms. -/
 instance : ReflectsMonomorphisms F := ReflectsMonomorphisms.mk <| by
   intro X Y f _
@@ -157,6 +166,16 @@ instance : F.Faithful where
     haveI : IsIso (equalizer.ι f g) := isIso_of_reflects_iso _ F
     exact eq_of_epi_equalizer
 
+section
+
+/-- If `F` is a fiber functor and `E` is an equivalence between categories of finite types,
+then `F ⋙ E` is again a fiber functor. -/
+noncomputable def compRight (E : FintypeCat.{w} ⥤ FintypeCat.{t}) [E.IsEquivalence] :
+    FiberFunctor (F ⋙ E) where
+  preservesQuotientsByFiniteGroups G := compPreservesColimitsOfShape F E
+
+end
+
 end FiberFunctor
 
 variable {C : Type u₁} [Category.{u₂, u₁} C]
@@ -172,6 +191,10 @@ lemma mulAction_def {X : C} (σ : Aut F) (x : F.obj X) :
     σ • x = σ.hom.app X x :=
   rfl
 
+lemma mulAction_naturality {X Y : C} (σ : Aut F) (f : X ⟶ Y) (x : F.obj X) :
+    σ • F.map f x = F.map f (σ • x) :=
+  FunctorToFintypeCat.naturality F F σ.hom f x
+
 /-- An object that is neither initial or connected has a non-trivial subobject. -/
 lemma has_non_trivial_subobject_of_not_isConnected_of_not_initial (X : C) (hc : ¬ IsConnected X)
     (hi : IsInitial X → False) :
@@ -309,6 +332,15 @@ lemma surjective_of_nonempty_fiber_of_isConnected {X A : C} [Nonempty (F.obj X)]
   have : Epi f := epi_of_nonempty_of_isConnected F f
   exact surjective_on_fiber_of_epi F f
 
+/-- If `X : ι → C` is a finite family of objects with non-empty fiber, then
+also `∏ᶜ X` has non-empty fiber. -/
+instance nonempty_fiber_pi_of_nonempty_of_finite {ι : Type*} [Finite ι] (X : ι → C)
+    [∀ i, Nonempty (F.obj (X i))] : Nonempty (F.obj (∏ᶜ X)) := by
+  cases nonempty_fintype ι
+  let f (i : ι) : FintypeCat.{w} := F.obj (X i)
+  let i : F.obj (∏ᶜ X) ≅ ∏ᶜ f := PreservesProduct.iso F _
+  exact Nonempty.elim inferInstance fun x : (∏ᶜ f : FintypeCat.{w}) ↦ ⟨i.inv x⟩
+
 section CardFiber
 
 open ConcreteCategory
diff --git a/Mathlib/CategoryTheory/Galois/Decomposition.lean b/Mathlib/CategoryTheory/Galois/Decomposition.lean
index b9febf20bbfc8..367dfd4597ae6 100644
--- a/Mathlib/CategoryTheory/Galois/Decomposition.lean
+++ b/Mathlib/CategoryTheory/Galois/Decomposition.lean
@@ -308,6 +308,15 @@ lemma exists_hom_from_galois_of_connected (X : C) [IsConnected X] :
     ∃ (A : C) (_ : A ⟶ X), IsGalois A :=
   exists_hom_from_galois_of_fiber_nonempty F X inferInstance
 
+/-- To check equality of natural transformations `F ⟶ G`, it suffices to check it on
+Galois objects. -/
+lemma natTrans_ext_of_isGalois {G : C ⥤ FintypeCat.{w}} {t s : F ⟶ G}
+    (h : ∀ (X : C) [IsGalois X], t.app X = s.app X) :
+    t = s := by
+  ext X x
+  obtain ⟨A, f, a, _, rfl⟩ := exists_hom_from_galois_of_fiber F X x
+  rw [FunctorToFintypeCat.naturality, FunctorToFintypeCat.naturality, h A]
+
 end GaloisRep
 
 end PreGaloisCategory
diff --git a/Mathlib/CategoryTheory/Galois/EssSurj.lean b/Mathlib/CategoryTheory/Galois/EssSurj.lean
new file mode 100644
index 0000000000000..06f9db4d4af8d
--- /dev/null
+++ b/Mathlib/CategoryTheory/Galois/EssSurj.lean
@@ -0,0 +1,263 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.CategoryTheory.Galois.Full
+import Mathlib.CategoryTheory.Galois.Topology
+import Mathlib.Topology.Algebra.OpenSubgroup
+
+/-!
+
+# Essential surjectivity of fiber functors
+
+Let `F : C ⥤ FintypeCat` be a fiber functor of a Galois category `C` and denote by
+`H` the induced functor `C ⥤ Action FintypeCat (Aut F)`.
+
+In this file we show that the essential image of `H` consists of the finite `Aut F`-sets where
+the `Aut F` action is continuous.
+
+## Main results
+
+- `exists_lift_of_quotient_openSubgroup`: If `U` is an open subgroup of `Aut F`, then
+  there exists an object `X` such that `F.obj X` is isomorphic to `Aut F ⧸ U` as
+  `Aut F`-sets.
+- `exists_lift_of_continuous`: If `X` is a finite, discrete `Aut F`-set, then
+  there exists an object `A` such that `F.obj A` is isomorphic to `X` as
+  `Aut F`-sets.
+
+## Strategy
+
+We first show that every finite, discrete `Aut F`-set `Y` has a decomposition into connected
+components and each connected component is of the form `Aut F ⧸ U` for an open subgroup `U`.
+Since `H` preserves finite coproducts, it hence suffices to treat the case `Y = Aut F ⧸ U`.
+For the case `Y = Aut F ⧸ U` we closely follow the second part of Stacks Project Tag 0BN4.
+
+-/
+
+noncomputable section
+
+universe u₁ u₂
+
+namespace CategoryTheory
+
+namespace PreGaloisCategory
+
+variable {C : Type u₁} [Category.{u₂} C] {F : C ⥤ FintypeCat.{u₁}}
+
+open Limits Functor
+
+variable [GaloisCategory C] [FiberFunctor F]
+
+variable {G : Type*} [Group G] [TopologicalSpace G] [TopologicalGroup G] [CompactSpace G]
+
+private local instance fintypeQuotient (H : OpenSubgroup (G)) :
+    Fintype (G ⧸ (H : Subgroup (G))) :=
+  have : Finite (G ⧸ H.toSubgroup) := H.toSubgroup.quotient_finite_of_isOpen H.isOpen'
+  Fintype.ofFinite _
+
+private local instance fintypeQuotientStabilizer {X : Type*} [MulAction G X]
+    [TopologicalSpace X] [ContinuousSMul G X] [DiscreteTopology X] (x : X) :
+    Fintype (G ⧸ (MulAction.stabilizer (G) x)) :=
+  fintypeQuotient ⟨MulAction.stabilizer (G) x, stabilizer_isOpen (G) x⟩
+
+/-- If `X` is a finite discrete `G`-set, it can be written as the finite disjoint union
+of quotients of the form `G ⧸ Uᵢ` for open subgroups `(Uᵢ)`. Note that this
+is simply the decomposition into orbits. -/
+lemma has_decomp_quotients (X : Action FintypeCat (MonCat.of G))
+    [TopologicalSpace X.V] [DiscreteTopology X.V] [ContinuousSMul G X.V] :
+    ∃ (ι : Type) (_ : Finite ι) (f : ι → OpenSubgroup (G)),
+      Nonempty ((∐ fun i ↦ G ⧸ₐ (f i).toSubgroup) ≅ X) := by
+  obtain ⟨ι, hf, f, u, hc⟩ := has_decomp_connected_components' X
+  letI (i : ι) : TopologicalSpace (f i).V := ⊥
+  haveI (i : ι) : DiscreteTopology (f i).V := ⟨rfl⟩
+  have (i : ι) : ContinuousSMul G (f i).V := ContinuousSMul.mk <| by
+    let r : f i ⟶ X := Sigma.ι f i ≫ u.hom
+    let r'' (p : G × (f i).V) : G × X.V := (p.1, r.hom p.2)
+    let q (p : G × X.V) : X.V := X.ρ p.1 p.2
+    let q' (p : G × (f i).V) : (f i).V := (f i).ρ p.1 p.2
+    have heq : q ∘ r'' = r.hom ∘ q' := by
+      ext (p : G × (f i).V)
+      exact (congr_fun (r.comm p.1) p.2).symm
+    have hrinj : Function.Injective r.hom :=
+      (ConcreteCategory.mono_iff_injective_of_preservesPullback r).mp <| mono_comp _ _
+    let t₁ : TopologicalSpace (G × (f i).V) := inferInstance
+    show @Continuous _ _ _ ⊥ q'
+    have : TopologicalSpace.induced r.hom inferInstance = ⊥ := by
+      rw [← le_bot_iff]
+      exact fun s _ ↦ ⟨r.hom '' s, ⟨isOpen_discrete (r.hom '' s), Set.preimage_image_eq s hrinj⟩⟩
+    rw [← this, continuous_induced_rng, ← heq]
+    exact Continuous.comp continuous_smul (by fun_prop)
+  have (i : ι) : ∃ (U : OpenSubgroup (G)), (Nonempty ((f i) ≅ G ⧸ₐ U.toSubgroup)) := by
+    obtain ⟨(x : (f i).V)⟩ := nonempty_fiber_of_isConnected (forget₂ _ _) (f i)
+    let U : OpenSubgroup (G) := ⟨MulAction.stabilizer (G) x, stabilizer_isOpen (G) x⟩
+    letI : Fintype (G ⧸ MulAction.stabilizer (G) x) := fintypeQuotient U
+    exact ⟨U, ⟨FintypeCat.isoQuotientStabilizerOfIsConnected (f i) x⟩⟩
+  choose g ui using this
+  exact ⟨ι, hf, g, ⟨(Sigma.mapIso (fun i ↦ (ui i).some)).symm ≪≫ u⟩⟩
+
+/-- If `X` is connected and `x` is in the fiber of `X`, `F.obj X` is isomorphic
+to the quotient of `Aut F` by the stabilizer of `x` as `Aut F`-sets. -/
+def fiberIsoQuotientStabilizer (X : C) [IsConnected X] (x : F.obj X) :
+    (functorToAction F).obj X ≅ Aut F ⧸ₐ MulAction.stabilizer (Aut F) x :=
+  haveI : IsConnected ((functorToAction F).obj X) := PreservesIsConnected.preserves
+  letI : Fintype (Aut F ⧸ MulAction.stabilizer (Aut F) x) := fintypeQuotientStabilizer x
+  FintypeCat.isoQuotientStabilizerOfIsConnected ((functorToAction F).obj X) x
+
+section
+
+open Action.FintypeCat
+
+variable (V : OpenSubgroup (Aut F)) {U : OpenSubgroup (Aut F)}
+  (h : Subgroup.Normal U.toSubgroup) {A : C} (u : (functorToAction F).obj A ≅ Aut F ⧸ₐ U.toSubgroup)
+
+/-
+
+### Strategy outline
+
+Let `A` be an object of `C` with fiber `Aut F`-isomorphic to `Aut F ⧸ U` for an open normal
+subgroup `U`. Then for any open subgroup `V` of `Aut F`, `V ⧸ (U ⊓ V)` acts on `A`. This
+induces the diagram `quotientDiag`. Now assume `U ≤ V`. Then we can also postcompose
+the diagram `quotientDiag` with `F`. The goal of this section is to compute that the colimit
+of this composed diagram is `Aut F ⧸ V`. Finally, we obtain `F.obj (A ⧸ V) ≅ Aut F ⧸ V` as
+`Aut F`-sets.
+-/
+
+private def quotientToEndObjectHom :
+    V.toSubgroup ⧸ Subgroup.subgroupOf U.toSubgroup V.toSubgroup →* End A :=
+  let ff : (functorToAction F).FullyFaithful := FullyFaithful.ofFullyFaithful (functorToAction F)
+  let e : End A ≃* End (Aut F ⧸ₐ U.toSubgroup) := (ff.mulEquivEnd A).trans (Iso.conj u)
+  e.symm.toMonoidHom.comp (quotientToEndHom V.toSubgroup U.toSubgroup)
+
+private lemma functorToAction_map_quotientToEndObjectHom
+    (m : SingleObj.star (V ⧸ Subgroup.subgroupOf U.toSubgroup V.toSubgroup) ⟶
+      SingleObj.star (V ⧸ Subgroup.subgroupOf U.toSubgroup V.toSubgroup)) :
+    (functorToAction F).map (quotientToEndObjectHom V h u m) =
+      u.hom ≫ quotientToEndHom V.toSubgroup U.toSubgroup m ≫ u.inv := by
+  simp [← cancel_epi u.inv, ← cancel_mono u.hom, ← Iso.conj_apply, quotientToEndObjectHom]
+
+@[simps!]
+private def quotientDiag : SingleObj (V.toSubgroup ⧸ Subgroup.subgroupOf U V) ⥤ C :=
+  SingleObj.functor (quotientToEndObjectHom V h u)
+
+variable {V} (hUinV : U ≤ V)
+
+@[simps]
+private def coconeQuotientDiag :
+    Cocone (quotientDiag V h u ⋙ functorToAction F) where
+  pt := Aut F ⧸ₐ V.toSubgroup
+  ι := SingleObj.natTrans (u.hom ≫ quotientToQuotientOfLE V.toSubgroup U.toSubgroup hUinV) <| by
+    intro (m : V ⧸ Subgroup.subgroupOf U V)
+    simp only [const_obj_obj, Functor.comp_map, const_obj_map, Category.comp_id]
+    rw [← cancel_epi (u.inv), Iso.inv_hom_id_assoc]
+    apply Action.hom_ext
+    ext (x : Aut F ⧸ U.toSubgroup)
+    induction' m, x using Quotient.inductionOn₂ with σ μ
+    suffices h : ⟦μ * σ⁻¹⟧ = ⟦μ⟧ by
+      simp only [quotientToQuotientOfLE_hom_mk, quotientDiag_map,
+        functorToAction_map_quotientToEndObjectHom V _ u]
+      simpa
+    apply Quotient.sound
+    apply (QuotientGroup.leftRel_apply).mpr
+    simp
+
+@[simps]
+private def coconeQuotientDiagDesc
+    (s : Cocone (quotientDiag V h u ⋙ functorToAction F)) :
+      (coconeQuotientDiag h u hUinV).pt ⟶ s.pt where
+  hom := Quotient.lift (fun σ ↦ (u.inv ≫ s.ι.app (SingleObj.star _)).hom ⟦σ⟧) <| fun σ τ hst ↦ by
+    let J' := quotientDiag V h u ⋙ functorToAction F
+    let m : End (SingleObj.star (V.toSubgroup ⧸ Subgroup.subgroupOf U V)) :=
+      ⟦⟨σ⁻¹ * τ, (QuotientGroup.leftRel_apply).mp hst⟩⟧
+    have h1 : J'.map m ≫ s.ι.app (SingleObj.star _) = s.ι.app (SingleObj.star _) := s.ι.naturality m
+    conv_rhs => rw [← h1]
+    have h2 : (J'.map m).hom (u.inv.hom ⟦τ⟧) = u.inv.hom ⟦σ⟧ := by
+      simp only [comp_obj, quotientDiag_obj, Functor.comp_map, quotientDiag_map, J',
+        functorToAction_map_quotientToEndObjectHom V h u m]
+      show (u.inv ≫ u.hom ≫ _ ≫ u.inv).hom ⟦τ⟧ = u.inv.hom ⟦σ⟧
+      simp [m]
+    simp only [← h2, const_obj_obj, Action.comp_hom, FintypeCat.comp_apply]
+  comm g := by
+    ext (x : Aut F ⧸ V.toSubgroup)
+    induction' x using Quotient.inductionOn with σ
+    simp only [const_obj_obj]
+    show (((Aut F ⧸ₐ U.toSubgroup).ρ g ≫ u.inv.hom) ≫ (s.ι.app (SingleObj.star _)).hom) ⟦σ⟧ =
+      ((s.ι.app (SingleObj.star _)).hom ≫ s.pt.ρ g) (u.inv.hom ⟦σ⟧)
+    have : ((functorToAction F).obj A).ρ g ≫ (s.ι.app (SingleObj.star _)).hom =
+        (s.ι.app (SingleObj.star _)).hom ≫ s.pt.ρ g :=
+      (s.ι.app (SingleObj.star _)).comm g
+    rw [← this, u.inv.comm g]
+    rfl
+
+/-- The constructed cocone `coconeQuotientDiag` on the diagram `quotientDiag` is colimiting. -/
+private def coconeQuotientDiagIsColimit :
+    IsColimit (coconeQuotientDiag h u hUinV) where
+  desc := coconeQuotientDiagDesc h u hUinV
+  fac s j := by
+    apply (cancel_epi u.inv).mp
+    apply Action.hom_ext
+    ext (x : Aut F ⧸ U.toSubgroup)
+    induction' x using Quotient.inductionOn with σ
+    simp
+    rfl
+  uniq s f hf := by
+    apply Action.hom_ext
+    ext (x : Aut F ⧸ V.toSubgroup)
+    induction' x using Quotient.inductionOn with σ
+    simp [← hf (SingleObj.star _)]
+
+end
+
+/-- For every open subgroup `V` of `Aut F`, there exists an `X : C` such that
+`F.obj X ≅ Aut F ⧸ V` as `Aut F`-sets. -/
+lemma exists_lift_of_quotient_openSubgroup (V : OpenSubgroup (Aut F)) :
+    ∃ (X : C), Nonempty ((functorToAction F).obj X ≅ Aut F ⧸ₐ V.toSubgroup) := by
+  obtain ⟨I, hf, hc, hi⟩ := exists_set_ker_evaluation_subset_of_isOpen F (one_mem V) V.isOpen'
+  haveI (X : I) : IsConnected X.val := hc X X.property
+  haveI (X : I) : Nonempty (F.obj X.val) := nonempty_fiber_of_isConnected F X
+  have hn : Nonempty (F.obj <| (∏ᶜ fun X : I => X)) := nonempty_fiber_pi_of_nonempty_of_finite F _
+  obtain ⟨A, f, hgal⟩ := exists_hom_from_galois_of_fiber_nonempty F (∏ᶜ fun X : I => X) hn
+  obtain ⟨a⟩ := nonempty_fiber_of_isConnected F A
+  let U : OpenSubgroup (Aut F) := ⟨MulAction.stabilizer (Aut F) a, stabilizer_isOpen (Aut F) a⟩
+  let u := fiberIsoQuotientStabilizer A a
+  have hUnormal : U.toSubgroup.Normal := stabilizer_normal_of_isGalois F A a
+  have h1 (σ : Aut F) (σinU : σ ∈ U) : σ.hom.app A = 𝟙 (F.obj A) := by
+    have hi : (Aut F ⧸ₐ MulAction.stabilizer (Aut F) a).ρ σ = 𝟙 _ := by
+      refine FintypeCat.hom_ext _ _ (fun x ↦ ?_)
+      induction' x using Quotient.inductionOn with τ
+      show ⟦σ * τ⟧ = ⟦τ⟧
+      apply Quotient.sound
+      apply (QuotientGroup.leftRel_apply).mpr
+      simp only [mul_inv_rev]
+      exact Subgroup.Normal.conj_mem hUnormal _ (Subgroup.inv_mem U.toSubgroup σinU) _
+    simp [← cancel_mono u.hom.hom, show σ.hom.app A ≫ u.hom.hom = _ from u.hom.comm σ, hi]
+  have h2 (σ : Aut F) (σinU : σ ∈ U) : ∀ X : I, σ.hom.app X = 𝟙 (F.obj X) := by
+    intro ⟨X, hX⟩
+    ext (x : F.obj X)
+    let p : A ⟶ X := f ≫ Pi.π (fun Z : I => (Z : C)) ⟨X, hX⟩
+    have : IsConnected X := hc X hX
+    obtain ⟨a, rfl⟩ := surjective_of_nonempty_fiber_of_isConnected F p x
+    simp only [FintypeCat.id_apply, FunctorToFintypeCat.naturality, h1 σ σinU]
+  have hUinV : (U : Set (Aut F)) ≤ V := fun u uinU ↦ hi u (h2 u uinU)
+  have := V.quotient_finite_of_isOpen' (U.subgroupOf V) V.isOpen (V.subgroupOf_isOpen U U.isOpen)
+  exact ⟨colimit (quotientDiag V hUnormal u),
+    ⟨preservesColimitIso (functorToAction F) (quotientDiag V hUnormal u) ≪≫
+    colimit.isoColimitCocone ⟨coconeQuotientDiag hUnormal u hUinV,
+    coconeQuotientDiagIsColimit hUnormal u hUinV⟩⟩⟩
+
+/--
+If `X` is a finite, discrete `Aut F`-set with continuous `Aut F`-action, then
+there exists `A : C` such that `F.obj A ≅ X` as `Aut F`-sets.
+-/
+@[stacks 0BN4 "Essential surjectivity part"]
+theorem exists_lift_of_continuous (X : Action FintypeCat (MonCat.of (Aut F)))
+    [TopologicalSpace X.V] [DiscreteTopology X.V] [ContinuousSMul (Aut F) X.V] :
+    ∃ A, Nonempty ((functorToAction F).obj A ≅ X) := by
+  obtain ⟨ι, hfin, f, ⟨u⟩⟩ := has_decomp_quotients X
+  choose g gu using (fun i ↦ exists_lift_of_quotient_openSubgroup (f i))
+  exact ⟨∐ g, ⟨PreservesCoproduct.iso (functorToAction F) g ≪≫
+    Sigma.mapIso (fun i ↦ (gu i).some) ≪≫ u⟩⟩
+
+end PreGaloisCategory
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/Examples.lean b/Mathlib/CategoryTheory/Galois/Examples.lean
index f7b722ebca408..a3bbd1400e05c 100644
--- a/Mathlib/CategoryTheory/Galois/Examples.lean
+++ b/Mathlib/CategoryTheory/Galois/Examples.lean
@@ -4,12 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christian Merten
 -/
 import Mathlib.CategoryTheory.Galois.Basic
-import Mathlib.RepresentationTheory.Action.Basic
 import Mathlib.RepresentationTheory.Action.Concrete
 import Mathlib.RepresentationTheory.Action.Limits
-import Mathlib.CategoryTheory.Limits.FintypeCat
-import Mathlib.CategoryTheory.Limits.Shapes.Types
-import Mathlib.Logic.Equiv.TransferInstance
 
 /-!
 # Examples of Galois categories and fiber functors
@@ -25,10 +21,10 @@ universe u v w
 
 namespace CategoryTheory
 
-namespace FintypeCat
-
 open Limits Functor PreGaloisCategory
 
+namespace FintypeCat
+
 /-- Complement of the image of a morphism `f : X ⟶ Y` in `FintypeCat`. -/
 noncomputable def imageComplement {X Y : FintypeCat.{u}} (f : X ⟶ Y) :
     FintypeCat.{u} := by
@@ -74,7 +70,7 @@ instance {X Y : Action FintypeCat (MonCat.of G)} (f : X ⟶ Y) :
 
 /-- The category of finite sets has quotients by finite groups in arbitrary universes. -/
 instance [Finite G] : HasColimitsOfShape (SingleObj G) FintypeCat.{w} := by
-  obtain ⟨G', hg, hf, ⟨e⟩⟩ := Finite.exists_type_zero_nonempty_mulEquiv G
+  obtain ⟨G', hg, hf, ⟨e⟩⟩ := Finite.exists_type_univ_nonempty_mulEquiv G
   exact Limits.hasColimitsOfShape_of_equivalence e.toSingleObjEquiv.symm
 
 noncomputable instance : PreservesFiniteLimits (forget (Action FintypeCat (MonCat.of G))) := by
@@ -96,6 +92,10 @@ noncomputable instance : FiberFunctor (Action.forget FintypeCat (MonCat.of G)) w
   preservesQuotientsByFiniteGroups _ _ _ := inferInstance
   reflectsIsos := ⟨fun f (h : IsIso f.hom) => inferInstance⟩
 
+/-- The forgetful functor from finite `G`-sets to sets is a `FiberFunctor`. -/
+noncomputable instance : FiberFunctor (forget₂ (Action FintypeCat (MonCat.of G)) FintypeCat) :=
+  inferInstanceAs <| FiberFunctor (Action.forget FintypeCat (MonCat.of G))
+
 /-- The category of finite `G`-sets is a `GaloisCategory`. -/
 instance : GaloisCategory (Action FintypeCat (MonCat.of G)) where
   hasFiberFunctor := ⟨Action.forget FintypeCat (MonCat.of G), ⟨inferInstance⟩⟩
@@ -149,6 +149,23 @@ theorem Action.isConnected_iff_transitive (X : Action FintypeCat (MonCat.of G))
     IsConnected X ↔ MulAction.IsPretransitive G X.V :=
   ⟨fun _ ↦ pretransitive_of_isConnected G X, fun _ ↦ isConnected_of_transitive G X.V⟩
 
+variable {G}
+
+/-- If `X` is a connected `G`-set and `x` is an element of `X`, `X` is isomorphic
+to the quotient of `G` by the stabilizer of `x` as `G`-sets. -/
+noncomputable def isoQuotientStabilizerOfIsConnected (X : Action FintypeCat (MonCat.of G))
+    [IsConnected X] (x : X.V) [Fintype (G ⧸ (MulAction.stabilizer G x))] :
+    X ≅ G ⧸ₐ MulAction.stabilizer G x :=
+  haveI : MulAction.IsPretransitive G X.V := Action.pretransitive_of_isConnected G X
+  let e : X.V ≃ G ⧸ MulAction.stabilizer G x :=
+    (Equiv.Set.univ X.V).symm.trans <|
+      (Equiv.setCongr ((MulAction.orbit_eq_univ G x).symm)).trans <|
+      MulAction.orbitEquivQuotientStabilizer G x
+  Iso.symm <| Action.mkIso (FintypeCat.equivEquivIso e.symm) <| fun σ : G ↦ by
+    ext (a : G ⧸ MulAction.stabilizer G x)
+    obtain ⟨τ, rfl⟩ := Quotient.exists_rep a
+    exact mul_smul σ τ x
+
 end FintypeCat
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/Full.lean b/Mathlib/CategoryTheory/Galois/Full.lean
new file mode 100644
index 0000000000000..4c09add2e62f9
--- /dev/null
+++ b/Mathlib/CategoryTheory/Galois/Full.lean
@@ -0,0 +1,129 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.CategoryTheory.Galois.Action
+
+/-!
+
+# Fiber functors are (faithfully) full
+
+Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor
+from finite `Aut F`-sets to finite sets. The induced functor
+`H : C ⥤ Action FintypeCat (MonCat.of (Aut F))` is faithfully full. The faithfulness
+follows easily from the faithfulness of `F`. In this file we show that `H` is also full.
+
+## Main results
+
+- `PreGaloisCategory.exists_lift_of_mono`: If `Y` is a sub-`Aut F`-set of `F.obj X`, there exists
+  a sub-object `Z` of `X` such that `F.obj Z ≅ Y` as `Aut F`-sets.
+- `PreGaloisCategory.functorToAction_full`: The induced functor `H` from above is full.
+
+The main input for this is that the induced functor `H : C ⥤ Action FintypeCat (MonCat.of (Aut F))`
+preserves connectedness, which translates to the fact that `Aut F` acts transitively on
+the fibers of connected objects.
+
+## Implementation details
+
+We only show this for small categories, because the preservation of connectedness result as it
+is currently in Mathlib is only shown for
+`(C : Type u₁) [Category.{u₂} C] (F : C ⥤ FintypeCat.{u₂})` and by the definition of `Action`,
+this forces `u₁ = u₂` for the definition of `functorToAction`. Mathematically there should
+be no obstruction to generalizing the results of this file to arbitrary universes.
+
+-/
+
+universe u v
+
+namespace CategoryTheory
+
+namespace PreGaloisCategory
+
+open Limits Functor
+
+variable {C : Type u} [Category.{v} C] (F : C ⥤ FintypeCat.{u}) [GaloisCategory C] [FiberFunctor F]
+
+/--
+Let `X` be an object of a Galois category with fiber functor `F` and `Y` a sub-`Aut F`-set
+of `F.obj X`, on which `Aut F` acts transitively (i.e. which is connected in the Galois category
+of finite `Aut F`-sets). Then there exists a connected sub-object `Z` of `X` and an isomorphism
+`Y ≅ F.obj X` as `Aut F`-sets such that the obvious triangle commutes.
+
+For a version without the connectedness assumption, see `exists_lift_of_mono`.
+-/
+lemma exists_lift_of_mono_of_isConnected (X : C) (Y : Action FintypeCat.{u} (MonCat.of (Aut F)))
+    (i : Y ⟶ (functorToAction F).obj X) [Mono i] [IsConnected Y] : ∃ (Z : C) (f : Z ⟶ X)
+    (u : Y ≅ (functorToAction F).obj Z),
+    IsConnected Z ∧ Mono f ∧ i = u.hom ≫ (functorToAction F).map f := by
+  obtain ⟨y⟩ := nonempty_fiber_of_isConnected (forget₂ _ FintypeCat) Y
+  obtain ⟨Z, f, z, hz, hc, hm⟩ := fiber_in_connected_component F X (i.hom y)
+  have : IsConnected ((functorToAction F).obj Z) := PreservesIsConnected.preserves
+  obtain ⟨u, hu⟩ := connected_component_unique
+    (forget₂ (Action FintypeCat (MonCat.of (Aut F))) FintypeCat) (B := (functorToAction F).obj Z)
+    y z i ((functorToAction F).map f) hz.symm
+  refine ⟨Z, f, u, hc, hm, ?_⟩
+  apply evaluation_injective_of_isConnected
+    (forget₂ (Action FintypeCat (MonCat.of (Aut F))) FintypeCat) Y ((functorToAction F).obj X) y
+  suffices h : i.hom y = F.map f z by simpa [hu]
+  exact hz.symm
+
+/--
+Let `X` be an object of a Galois category with fiber functor `F` and `Y` a sub-`Aut F`-set
+of `F.obj X`. Then there exists a sub-object `Z` of `X` and an isomorphism
+`Y ≅ F.obj X` as `Aut F`-sets such that the obvious triangle commutes.
+-/
+lemma exists_lift_of_mono (X : C) (Y : Action FintypeCat.{u} (MonCat.of (Aut F)))
+    (i : Y ⟶ (functorToAction F).obj X) [Mono i] : ∃ (Z : C) (f : Z ⟶ X)
+    (u : Y ≅ (functorToAction F).obj Z), Mono f ∧ u.hom ≫ (functorToAction F).map f = i := by
+  obtain ⟨ι, hf, f, t, hc⟩ := has_decomp_connected_components' Y
+  let i' (j : ι) : f j ⟶ (functorToAction F).obj X := Sigma.ι f j ≫ t.hom ≫ i
+  have (j : ι) : Mono (i' j) :=
+    have : Mono (Sigma.ι f j) := MonoCoprod.mono_ι f j
+    have : Mono (t.hom ≫ i) := mono_comp _ _
+    mono_comp _ _
+  choose gZ gf gu _ _ h using fun i ↦ exists_lift_of_mono_of_isConnected F X (f i) (i' i)
+  let is2 : (functorToAction F).obj (∐ gZ) ≅ ∐ fun i => (functorToAction F).obj (gZ i) :=
+    PreservesCoproduct.iso (functorToAction F) gZ
+  let u' : ∐ f ≅ ∐ fun i => (functorToAction F).obj (gZ i) := Sigma.mapIso gu
+  have heq : (functorToAction F).map (Sigma.desc gf) = (t.symm ≪≫ u' ≪≫ is2.symm).inv ≫ i := by
+    simp only [Iso.trans_inv, Iso.symm_inv, Category.assoc]
+    rw [← Iso.inv_comp_eq]
+    refine Sigma.hom_ext _ _ (fun j ↦ ?_)
+    suffices (functorToAction F).map (gf j) = (gu j).inv ≫ i' j by
+      simpa [is2, u']
+    simp only [h, Iso.inv_hom_id_assoc]
+  refine ⟨∐ gZ, Sigma.desc gf, t.symm ≪≫ u' ≪≫ is2.symm, ?_, by simp [heq]⟩
+  · exact mono_of_mono_map (functorToAction F) (heq ▸ mono_comp _ _)
+
+/-- The by a fiber functor `F : C ⥤ FintypeCat` induced functor `functorToAction F` to
+finite `Aut F`-sets is full. -/
+instance functorToAction_full : Functor.Full (functorToAction F) where
+  map_surjective {X Y} f := by
+    let u : (functorToAction F).obj X ⟶ (functorToAction F).obj X ⨯ (functorToAction F).obj Y :=
+      prod.lift (𝟙 _) f
+    let i : (functorToAction F).obj X ⟶ (functorToAction F).obj (X ⨯ Y) :=
+      u ≫ (PreservesLimitPair.iso (functorToAction F) X Y).inv
+    have : Mono i := by
+      have : Mono (u ≫ prod.fst) := prod.lift_fst (𝟙 _) f ▸ inferInstance
+      have : Mono u := mono_of_mono u prod.fst
+      apply mono_comp u _
+    obtain ⟨Z, g, v, _, hvgi⟩ := exists_lift_of_mono F (Limits.prod X Y)
+      ((functorToAction F).obj X) i
+    let ψ : Z ⟶ X := g ≫ prod.fst
+    have hgvi : (functorToAction F).map g = v.inv ≫ i := by simp [← hvgi]
+    have : IsIso ((functorToAction F).map ψ) := by
+      simp only [map_comp, hgvi, Category.assoc, ψ]
+      have : IsIso (i ≫ (functorToAction F).map prod.fst) := by
+        suffices h : IsIso (𝟙 ((functorToAction F).obj X)) by simpa [i, u]
+        infer_instance
+      apply IsIso.comp_isIso
+    have : IsIso ψ := isIso_of_reflects_iso ψ (functorToAction F)
+    use inv ψ ≫ g ≫ prod.snd
+    rw [← cancel_epi ((functorToAction F).map ψ)]
+    ext (z : F.obj Z)
+    simp [-FintypeCat.comp_apply, -Action.comp_hom, i, u, ψ, hgvi]
+
+end PreGaloisCategory
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/GaloisObjects.lean b/Mathlib/CategoryTheory/Galois/GaloisObjects.lean
index 1ceed246f266d..25fbc42d60592 100644
--- a/Mathlib/CategoryTheory/Galois/GaloisObjects.lean
+++ b/Mathlib/CategoryTheory/Galois/GaloisObjects.lean
@@ -35,7 +35,7 @@ open Limits Functor
 
 noncomputable instance {G : Type v} [Group G] [Finite G] :
     PreservesColimitsOfShape (SingleObj G) FintypeCat.incl.{w} := by
-  choose G' hg hf e using Finite.exists_type_zero_nonempty_mulEquiv G
+  choose G' hg hf e using Finite.exists_type_univ_nonempty_mulEquiv G
   exact Limits.preservesColimitsOfShapeOfEquiv (Classical.choice e).toSingleObjEquiv.symm _
 
 /-- A connected object `X` of `C` is Galois if the quotient `X / Aut X` is terminal. -/
@@ -83,7 +83,7 @@ theorem isGalois_iff_pretransitive (X : C) [IsConnected X] :
   rw [isGalois_iff_aux, Equiv.nonempty_congr <| quotientByAutTerminalEquivUniqueQuotient F X]
   exact (MulAction.pretransitive_iff_unique_quotient_of_nonempty (Aut X) (F.obj X)).symm
 
-/-- If `X` is Galois, the quotient `X / Aut X` is terminal.  -/
+/-- If `X` is Galois, the quotient `X / Aut X` is terminal. -/
 noncomputable def isTerminalQuotientOfIsGalois (X : C) [IsGalois X] :
     IsTerminal <| colimit <| SingleObj.functor <| Aut.toEnd X :=
   Nonempty.some IsGalois.quotientByAutTerminal
@@ -95,6 +95,17 @@ instance isPretransitive_of_isGalois (X : C) [IsGalois X] :
   rw [← isGalois_iff_pretransitive]
   infer_instance
 
+lemma stabilizer_normal_of_isGalois (X : C) [IsGalois X] (x : F.obj X) :
+    Subgroup.Normal (MulAction.stabilizer (Aut F) x) where
+  conj_mem n ninstab g := by
+    rw [MulAction.mem_stabilizer_iff]
+    show g • n • (g⁻¹ • x) = x
+    have : ∃ (φ : Aut X), F.map φ.hom x = g⁻¹ • x :=
+      MulAction.IsPretransitive.exists_smul_eq x (g⁻¹ • x)
+    obtain ⟨φ, h⟩ := this
+    rw [← h, mulAction_naturality, ninstab, h]
+    simp
+
 theorem evaluation_aut_surjective_of_isGalois (A : C) [IsGalois A] (a : F.obj A) :
     Function.Surjective (fun f : Aut A ↦ F.map f.hom a) :=
   MulAction.IsPretransitive.exists_smul_eq a
diff --git a/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean b/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean
new file mode 100644
index 0000000000000..f5398eaad344f
--- /dev/null
+++ b/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean
@@ -0,0 +1,288 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.CategoryTheory.Galois.Basic
+import Mathlib.CategoryTheory.Galois.Topology
+import Mathlib.CategoryTheory.Galois.Prorepresentability
+import Mathlib.Topology.Algebra.OpenSubgroup
+
+/-!
+
+# Universal property of fundamental group
+
+Let `C` be a Galois category with fiber functor `F`. While in informal mathematics, we tend to
+identify known groups from other contexts (e.g. the absolute Galois group of a field) with
+the automorphism group `Aut F` of certain fiber functors `F`, this causes friction in formalization.
+
+Hence, in this file we develop conditions when a topological group `G` is canonically isomorphic to
+the automorphism group `Aut F` of `F`. Consequently, the API for Galois categories and their fiber
+functors should be stated in terms of an abstract topological group `G` satisfying
+`IsFundamentalGroup` in the places where `Aut F` would appear.
+
+## Main definition
+
+Given a compact, topological group `G` with an action on `F.obj X` on each `X`, we say that
+`G` is a fundamental group of `F` (`IsFundamentalGroup F G`), if
+
+- `naturality`: the `G`-action on `F.obj X` is compatible with morphisms in `C`
+- `transitive_of_isGalois`: `G` acts transitively on `F.obj X` for all Galois objects `X : C`
+- `continuous_smul`: the action of `G` on `F.obj X` is continuous if `F.obj X` is equipped with the
+  discrete topology for all `X : C`.
+- `non_trivial': if `g : G` acts trivial on all `F.obj X`, then `g = 1`.
+
+Given this data, we define `toAut F G : G →* Aut F` in the natural way.
+
+## Main results
+
+- `toAut_bijective`: `toAut F G` is a group isomorphism given `IsFundamentalGroup F G`.
+- `toAut_isHomeomorph`: `toAut F G` is a homeomorphism given `IsFundamentalGroup F G`.
+
+## TODO
+
+- Develop further equivalent conditions, in particular, relate the condition `non_trivial` with
+  `G` being a `T2Space`.
+
+-/
+universe u₁ u₂ w
+
+namespace CategoryTheory
+
+namespace PreGaloisCategory
+
+open Limits Functor
+
+variable {C : Type u₁} [Category.{u₂} C] (F : C ⥤ FintypeCat.{w})
+
+section
+
+variable (G : Type*) [Group G] [∀ X, MulAction G (F.obj X)]
+
+/-- We say `G` acts naturally on the fibers of `F` if for every `f : X ⟶ Y`, the `G`-actions
+on `F.obj X` and `F.obj Y` are compatible with `F.map f`. -/
+class IsNaturalSMul : Prop where
+  naturality (g : G) {X Y : C} (f : X ⟶ Y) (x : F.obj X) : F.map f (g • x) = g • F.map f x
+
+variable {G} in
+@[simps!]
+private def isoOnObj (g : G) (X : C) : F.obj X ≅ F.obj X :=
+  FintypeCat.equivEquivIso <| {
+    toFun := fun x ↦ g • x
+    invFun := fun x ↦ g⁻¹ • x
+    left_inv := fun _ ↦ by simp
+    right_inv := fun _ ↦ by simp
+  }
+
+variable [IsNaturalSMul F G]
+
+/-- If `G` acts naturally on `F.obj X` for each `X : C`, this is the canonical
+group homomorphism into the automorphism group of `F`. -/
+def toAut : G →* Aut F where
+  toFun g := NatIso.ofComponents (isoOnObj F g) <| by
+    intro X Y f
+    ext
+    simp [IsNaturalSMul.naturality]
+  map_one' := by
+    ext
+    simp only [NatIso.ofComponents_hom_app, isoOnObj_hom, one_smul]
+    rfl
+  map_mul' := by
+    intro g h
+    ext X x
+    simp only [NatIso.ofComponents_hom_app, isoOnObj_hom, mul_smul]
+    rfl
+
+variable {G} in
+@[simp]
+lemma toAut_hom_app_apply (g : G) {X : C} (x : F.obj X) : (toAut F G g).hom.app X x = g • x :=
+  rfl
+
+/-- `toAut` is injective, if only the identity acts trivially on every fiber. -/
+lemma toAut_injective_of_non_trivial (h : ∀ (g : G), (∀ (X : C) (x : F.obj X), g • x = x) → g = 1) :
+    Function.Injective (toAut F G) := by
+  rw [← MonoidHom.ker_eq_bot_iff, eq_bot_iff]
+  intro g (hg : toAut F G g = 1)
+  refine h g (fun X x ↦ ?_)
+  have : (toAut F G g).hom.app X = 𝟙 (F.obj X) := by
+    rw [hg]
+    rfl
+  rw [← toAut_hom_app_apply, this, FintypeCat.id_apply]
+
+variable [GaloisCategory C] [FiberFunctor F]
+
+lemma toAut_continuous [TopologicalSpace G] [TopologicalGroup G]
+    [∀ (X : C), ContinuousSMul G (F.obj X)] :
+    Continuous (toAut F G) := by
+  apply continuous_of_continuousAt_one
+  rw [continuousAt_def, map_one]
+  intro A hA
+  obtain ⟨X, _, hX⟩ := ((nhds_one_has_basis_stabilizers F).mem_iff' A).mp hA
+  rw [mem_nhds_iff]
+  exact ⟨MulAction.stabilizer G X.pt, Set.preimage_mono (f := toAut F G) hX,
+    stabilizer_isOpen G X.pt, one_mem _⟩
+
+variable {G}
+
+lemma action_ext_of_isGalois {t : F ⟶ F} {X : C} [IsGalois X] {g : G} (x : F.obj X)
+    (hg : g • x = t.app X x) (y : F.obj X) : g • y = t.app X y := by
+  obtain ⟨φ, (rfl : F.map φ.hom y = x)⟩ := MulAction.exists_smul_eq (Aut X) y x
+  have : Function.Injective (F.map φ.hom) :=
+    ConcreteCategory.injective_of_mono_of_preservesPullback (F.map φ.hom)
+  apply this
+  rw [IsNaturalSMul.naturality, hg, FunctorToFintypeCat.naturality]
+
+variable (G)
+
+lemma toAut_surjective_isGalois (t : Aut F) (X : C) [IsGalois X]
+    [MulAction.IsPretransitive G (F.obj X)] :
+    ∃ (g : G), ∀ (x : F.obj X), g • x = t.hom.app X x := by
+  obtain ⟨a⟩ := nonempty_fiber_of_isConnected F X
+  obtain ⟨g, hg⟩ := MulAction.exists_smul_eq G a (t.hom.app X a)
+  exact ⟨g, action_ext_of_isGalois F _ hg⟩
+
+lemma toAut_surjective_isGalois_finite_family (t : Aut F) {ι : Type*} [Finite ι] (X : ι → C)
+    [∀ i, IsGalois (X i)] (h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X)) :
+    ∃ (g : G), ∀ (i : ι) (x : F.obj (X i)), g • x = t.hom.app (X i) x := by
+  let x (i : ι) : F.obj (X i) := (nonempty_fiber_of_isConnected F (X i)).some
+  let P : C := ∏ᶜ X
+  letI : Fintype ι := Fintype.ofFinite ι
+  let is₁ : F.obj P ≅ ∏ᶜ fun i ↦ (F.obj (X i)) := PreservesProduct.iso F X
+  let is₂ : (∏ᶜ fun i ↦ F.obj (X i) : FintypeCat) ≃ ∀ i, F.obj (X i) :=
+    Limits.FintypeCat.productEquiv (fun i ↦ (F.obj (X i)))
+  let px : F.obj P := is₁.inv (is₂.symm x)
+  have hpx (i : ι) : F.map (Pi.π X i) px = x i := by
+    simp only [px, is₁, is₂, ← piComparison_comp_π, ← PreservesProduct.iso_hom]
+    simp only [FintypeCat.comp_apply, FintypeCat.inv_hom_id_apply,
+      FintypeCat.productEquiv_symm_comp_π_apply]
+  obtain ⟨A, f, a, _, hfa⟩ := exists_hom_from_galois_of_fiber F P px
+  obtain ⟨g, hg⟩ := toAut_surjective_isGalois F G t A
+  refine ⟨g, fun i y ↦ action_ext_of_isGalois F (x i) ?_ _⟩
+  rw [← hpx i, ← IsNaturalSMul.naturality, FunctorToFintypeCat.naturality,
+    ← hfa, FunctorToFintypeCat.naturality, ← IsNaturalSMul.naturality, hg]
+
+open Pointwise
+
+/-- If `G` is a compact, topological group that acts continuously and naturally on the
+fibers of `F`, `toAut F G` is surjective if and only if it acts transitively on the fibers
+of all Galois objects. This is the `if` direction. For the `only if` see
+`isPretransitive_of_surjective`. -/
+lemma toAut_surjective_of_isPretransitive [TopologicalSpace G] [TopologicalGroup G] [CompactSpace G]
+    [∀ (X : C), ContinuousSMul G (F.obj X)]
+    (h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X)) :
+    Function.Surjective (toAut F G) := by
+  intro t
+  choose gi hgi using (fun X : PointedGaloisObject F ↦ toAut_surjective_isGalois F G t X)
+  let cl (X : PointedGaloisObject F) : Set G := gi X • MulAction.stabilizer G X.pt
+  let c : Set G := ⋂ i, cl i
+  have hne : c.Nonempty := by
+    rw [← Set.univ_inter c]
+    apply CompactSpace.isCompact_univ.inter_iInter_nonempty
+    · intro X
+      apply IsClosed.leftCoset
+      exact Subgroup.isClosed_of_isOpen _ (stabilizer_isOpen G X.pt)
+    · intro s
+      rw [Set.univ_inter]
+      obtain ⟨gs, hgs⟩ :=
+        toAut_surjective_isGalois_finite_family F G t (fun X : s ↦ X.val.obj) h
+      use gs
+      simp only [Set.mem_iInter]
+      intro X hXmem
+      rw [mem_leftCoset_iff, SetLike.mem_coe, MulAction.mem_stabilizer_iff, mul_smul,
+        hgs ⟨X, hXmem⟩, ← hgi X, inv_smul_smul]
+  obtain ⟨g, hg⟩ := hne
+  refine ⟨g, Iso.ext <| natTrans_ext_of_isGalois _ <| fun X _ ↦ ?_⟩
+  ext x
+  simp only [toAut_hom_app_apply]
+  have : g ∈ (gi ⟨X, x, inferInstance⟩ • MulAction.stabilizer G x : Set G) := by
+    simp only [Set.mem_iInter, c] at hg
+    exact hg _
+  obtain ⟨s, (hsmem : s • x = x), (rfl : gi ⟨X, x, inferInstance⟩ • s = _)⟩ := this
+  rw [smul_eq_mul, mul_smul, hsmem]
+  exact hgi ⟨X, x, inferInstance⟩ x
+
+/-- If `toAut F G` is surjective, then `G` acts transitively on the fibers of connected objects.
+For a converse see `toAut_surjective`. -/
+lemma isPretransitive_of_surjective (h : Function.Surjective (toAut F G)) (X : C)
+    [IsConnected X] : MulAction.IsPretransitive G (F.obj X) where
+  exists_smul_eq x y := by
+    obtain ⟨t, ht⟩ := MulAction.exists_smul_eq (Aut F) x y
+    obtain ⟨g, rfl⟩ := h t
+    exact ⟨g, ht⟩
+
+end
+
+section
+
+variable [GaloisCategory C]
+variable (G : Type*) [Group G] [∀ (X : C), MulAction G (F.obj X)]
+
+/-- A compact, topological group `G` with a natural action on `F.obj X` for each `X : C`
+is a fundamental group of `F`, if `G` acts transitively on the fibers of Galois objects,
+the action on `F.obj X` is continuous for all `X : C` and the only trivally acting element of `G`
+is the identity. -/
+class IsFundamentalGroup [TopologicalSpace G] [TopologicalGroup G] [CompactSpace G]
+    extends IsNaturalSMul F G : Prop where
+  transitive_of_isGalois (X : C) [IsGalois X] : MulAction.IsPretransitive G (F.obj X)
+  continuous_smul (X : C) : ContinuousSMul G (F.obj X)
+  non_trivial' (g : G) : (∀ (X : C) (x : F.obj X), g • x = x) → g = 1
+
+namespace IsFundamentalGroup
+
+attribute [instance] continuous_smul transitive_of_isGalois
+
+variable {G} [TopologicalSpace G] [TopologicalGroup G] [CompactSpace G] [IsFundamentalGroup F G]
+
+lemma non_trivial (g : G) (h : ∀ (X : C) (x : F.obj X), g • x = x) : g = 1 :=
+  IsFundamentalGroup.non_trivial' g h
+
+end IsFundamentalGroup
+
+variable [FiberFunctor F]
+
+/-- `Aut F` is a fundamental group for `F`. -/
+instance : IsFundamentalGroup F (Aut F) where
+  naturality g X Y f x := (FunctorToFintypeCat.naturality F F g.hom f x).symm
+  transitive_of_isGalois X := FiberFunctor.isPretransitive_of_isConnected F X
+  continuous_smul X := continuousSMul_aut_fiber F X
+  non_trivial' g h := by
+    ext X x
+    exact h X x
+
+variable [TopologicalSpace G] [TopologicalGroup G] [CompactSpace G] [IsFundamentalGroup F G]
+
+lemma toAut_bijective : Function.Bijective (toAut F G) where
+  left := toAut_injective_of_non_trivial F G IsFundamentalGroup.non_trivial'
+  right := toAut_surjective_of_isPretransitive F G IsFundamentalGroup.transitive_of_isGalois
+
+instance (X : C) [IsConnected X] : MulAction.IsPretransitive G (F.obj X) :=
+  isPretransitive_of_surjective F G (toAut_bijective F G).surjective X
+
+/-- If `G` is the fundamental group for `F`, it is isomorphic to `Aut F` as groups and
+this isomorphism is also a homeomorphism (see `toAutMulEquiv_isHomeomorph`). -/
+noncomputable def toAutMulEquiv : G ≃* Aut F :=
+  MulEquiv.ofBijective (toAut F G) (toAut_bijective F G)
+
+lemma toAut_isHomeomorph : IsHomeomorph (toAut F G) := by
+  rw [isHomeomorph_iff_continuous_bijective]
+  exact ⟨toAut_continuous F G, toAut_bijective F G⟩
+
+lemma toAutMulEquiv_isHomeomorph : IsHomeomorph (toAutMulEquiv F G) :=
+  toAut_isHomeomorph F G
+
+/-- If `G` is a fundamental group for `F`, it is canonically homeomorphic to `Aut F`. -/
+noncomputable def toAutHomeo : G ≃ₜ Aut F := (toAut_isHomeomorph F G).homeomorph
+
+variable {G}
+
+@[simp]
+lemma toAutMulEquiv_apply (g : G) : toAutMulEquiv F G g = toAut F G g := rfl
+
+@[simp]
+lemma toAutHomeo_apply (g : G) : toAutHomeo F G g = toAut F G g := rfl
+
+end
+
+end PreGaloisCategory
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/Prorepresentability.lean b/Mathlib/CategoryTheory/Galois/Prorepresentability.lean
index 0edce049c34ca..fba4b027d12de 100644
--- a/Mathlib/CategoryTheory/Galois/Prorepresentability.lean
+++ b/Mathlib/CategoryTheory/Galois/Prorepresentability.lean
@@ -6,7 +6,6 @@ Authors: Christian Merten
 import Mathlib.Algebra.Category.Grp.Limits
 import Mathlib.CategoryTheory.CofilteredSystem
 import Mathlib.CategoryTheory.Galois.Decomposition
-import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
 import Mathlib.CategoryTheory.Limits.IndYoneda
 import Mathlib.CategoryTheory.Limits.Preserves.Ulift
 
@@ -34,6 +33,20 @@ groups of all Galois objects.
 - `FiberFunctor.isPretransitive_of_isConnected`: The `Aut F` action on the fiber of a connected
   object is transitive.
 
+## Implementation details
+
+The pro-representability statement and the isomorphism of `Aut F` with the limit over the
+automorphism groups of all Galois objects naturally forces `F` to take values in `FintypeCat.{u₂}`
+where `u₂` is the `Hom`-universe of `C`. Since this is used to show that `Aut F` acts
+transitively on `F.obj X` for connected `X`, we a priori only obtain this result for
+the mentioned specialized universe setup. To obtain the result for `F` taking values in an arbitrary
+`FintypeCat.{w}`, we postcompose with an equivalence `FintypeCat.{w} ≌ FintypeCat.{u₂}` and apply
+the specialized result.
+
+In the following the section `Specialized` is reserved for the setup where `F` takes values in
+`FintypeCat.{u₂}` and the section `General` contains results holding for `F` taking values in
+an arbitrary `FintypeCat.{w}`.
+
 ## References
 
 * [lenstraGSchemes]: H. W. Lenstra. Galois theory for schemes.
@@ -49,9 +62,9 @@ namespace PreGaloisCategory
 open Limits Functor
 
 variable {C : Type u₁} [Category.{u₂} C] [GaloisCategory C]
-variable (F : C ⥤ FintypeCat.{u₂})
+
 /-- A pointed Galois object is a Galois object with a fixed point of its fiber. -/
-structure PointedGaloisObject : Type (max u₁ u₂) where
+structure PointedGaloisObject (F : C ⥤ FintypeCat.{w}) : Type (max u₁ u₂ w) where
   /-- The underlying object of `C`. -/
   obj : C
   /-- An element of the fiber of `obj`. -/
@@ -61,6 +74,10 @@ structure PointedGaloisObject : Type (max u₁ u₂) where
 
 namespace PointedGaloisObject
 
+section General
+
+variable (F : C ⥤ FintypeCat.{w})
+
 attribute [instance] isGalois
 
 instance (X : PointedGaloisObject F) : CoeDep (PointedGaloisObject F) X C where
@@ -117,6 +134,12 @@ lemma incl_obj (A : PointedGaloisObject F) : (incl F).obj A = A :=
 lemma incl_map {A B : PointedGaloisObject F} (f : A ⟶ B) : (incl F).map f = f.val :=
   rfl
 
+end General
+
+section Specialized
+
+variable (F : C ⥤ FintypeCat.{u₂})
+
 /-- `F ⋙ FintypeCat.incl` as a cocone over `(can F).op ⋙ coyoneda`.
 This is a colimit cocone (see `PreGaloisCategory.isColimìt`) -/
 def cocone : Cocone ((incl F).op ⋙ coyoneda) where
@@ -172,10 +195,16 @@ noncomputable def isColimit : IsColimit (cocone F) := by
 instance : HasColimit ((incl F).op ⋙ coyoneda) where
   exists_colimit := ⟨cocone F, isColimit F⟩
 
+end Specialized
+
 end PointedGaloisObject
 
 open PointedGaloisObject
 
+section Specialized
+
+variable (F : C ⥤ FintypeCat.{u₂})
+
 /-- The diagram sending each pointed Galois object to its automorphism group
 as an object of `C`. -/
 @[simps]
@@ -189,7 +218,7 @@ noncomputable def autGaloisSystem : PointedGaloisObject F ⥤ Grp.{u₂} where
     ext (σ : Aut A.obj)
     simp
 
-/-- The limit of `autGaloisSystem`.  -/
+/-- The limit of `autGaloisSystem`. -/
 noncomputable def AutGalois : Type (max u₁ u₂) :=
   (autGaloisSystem F ⋙ forget _).sections
 
@@ -344,7 +373,7 @@ lemma endMulEquivAutGalois_pi (f : End F) (A : PointedGaloisObject F) :
 
 /-- Any endomorphism of a fiber functor is a unit. -/
 theorem FibreFunctor.end_isUnit (f : End F) : IsUnit f :=
-  (MulEquiv.map_isUnit_iff (endMulEquivAutGalois F)).mp
+  (isUnit_map_iff (endMulEquivAutGalois F) _).mp
     (Group.isUnit ((endMulEquivAutGalois F) f))
 
 /-- Any endomorphism of a fiber functor is an isomorphism. -/
@@ -365,7 +394,7 @@ noncomputable def autMulEquivAutGalois : Aut F ≃* (AutGalois F)ᵐᵒᵖ where
   right_inv t := by
     simp only [MonoidHom.coe_comp, MonoidHom.coe_coe, Function.comp_apply, Aut.toEnd_apply]
     exact (MulEquiv.eq_symm_apply (endMulEquivAutGalois F)).mp rfl
-  map_mul' := by simp
+  map_mul' := by simp [map_mul]
 
 lemma autMulEquivAutGalois_π (f : Aut F) (A : C) [IsGalois A] (a : F.obj A) :
     F.map (AutGalois.π F { obj := A, pt := a } (autMulEquivAutGalois F f).unop).hom a =
@@ -393,8 +422,9 @@ theorem FiberFunctor.isPretransitive_of_isGalois (X : C) [IsGalois X] :
   use (autMulEquivAutGalois F).symm ⟨a⟩
   simpa [mulAction_def, ha]
 
-/-- The `Aut F` action on the fiber of a connected object is transitive. -/
-instance FiberFunctor.isPretransitive_of_isConnected (X : C) [IsConnected X] :
+/-- The `Aut F` action on the fiber of a connected object is transitive. For a version
+with less restrictive universe assumptions, see `FiberFunctor.isPretransitive_of_isConnected`. -/
+private instance FiberFunctor.isPretransitive_of_isConnected' (X : C) [IsConnected X] :
     MulAction.IsPretransitive (Aut F) (F.obj X) := by
   obtain ⟨A, f, hgal⟩ := exists_hom_from_galois_of_connected F X
   have hs : Function.Surjective (F.map f) := surjective_of_nonempty_fiber_of_isConnected F f
@@ -408,6 +438,39 @@ instance FiberFunctor.isPretransitive_of_isConnected (X : C) [IsConnected X] :
   show (F.map f ≫ σ.hom.app X) a = F.map f b
   rw [σ.hom.naturality, FintypeCat.comp_apply, hσ]
 
+end Specialized
+
+section General
+
+variable (F : C ⥤ FintypeCat.{w}) [FiberFunctor F]
+
+/-- The `Aut F` action on the fiber of a connected object is transitive. -/
+instance FiberFunctor.isPretransitive_of_isConnected (X : C) [IsConnected X] :
+    MulAction.IsPretransitive (Aut F) (F.obj X) where
+  exists_smul_eq x y := by
+    let F' : C ⥤ FintypeCat.{u₂} := F ⋙ FintypeCat.uSwitch.{w, u₂}
+    letI : FiberFunctor F' := FiberFunctor.compRight _
+    let e (Y : C) : F'.obj Y ≃ F.obj Y := (F.obj Y).uSwitchEquiv
+    set x' : F'.obj X := (e X).symm x with hx'
+    set y' : F'.obj X := (e X).symm y with hy'
+    obtain ⟨g', (hg' : g'.hom.app X x' = y')⟩ := MulAction.exists_smul_eq (Aut F') x' y'
+    let gapp (Y : C) : F.obj Y ≅ F.obj Y := FintypeCat.equivEquivIso <|
+      (e Y).symm.trans <| (FintypeCat.equivEquivIso.symm (g'.app Y)).trans (e Y)
+    let g : F ≅ F := NatIso.ofComponents gapp <| fun {X Y} f ↦ by
+      ext x
+      simp only [FintypeCat.comp_apply, FintypeCat.equivEquivIso_apply_hom,
+        Equiv.trans_apply, FintypeCat.equivEquivIso_symm_apply_apply, Iso.app_hom, gapp, e]
+      erw [FintypeCat.uSwitchEquiv_naturality (F.map f)]
+      rw [← Functor.comp_map, ← FunctorToFintypeCat.naturality]
+      simp only [comp_obj, Functor.comp_map, F']
+      rw [FintypeCat.uSwitchEquiv_symm_naturality (F.map f)]
+    refine ⟨g, show (gapp X).hom x = y from ?_⟩
+    simp only [FintypeCat.equivEquivIso_apply_hom, Equiv.trans_apply,
+      FintypeCat.equivEquivIso_symm_apply_apply, Iso.app_hom, gapp]
+    rw [← hx', hg', hy', Equiv.apply_symm_apply]
+
+end General
+
 end PreGaloisCategory
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Galois/Topology.lean b/Mathlib/CategoryTheory/Galois/Topology.lean
new file mode 100644
index 0000000000000..62e4145b36bc7
--- /dev/null
+++ b/Mathlib/CategoryTheory/Galois/Topology.lean
@@ -0,0 +1,183 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.CategoryTheory.Galois.Prorepresentability
+import Mathlib.Topology.Algebra.Group.Basic
+
+/-!
+
+# Topology of fundamental group
+
+In this file we define a natural topology on the automorphism group of a functor
+`F : C ⥤ FintypeCat`: It is defined as the subspace topology induced by the natural
+embedding of `Aut F` into `∀ X, Aut (F.obj X)` where
+`Aut (F.obj X)` carries the discrete topology.
+
+## References
+
+- Stacks Project: Tag 0BMQ
+
+-/
+universe u₁ u₂ v₁ v₂ v w
+
+namespace CategoryTheory
+
+namespace PreGaloisCategory
+
+open Functor
+
+variable {C : Type u₁} [Category.{u₂} C] (F : C ⥤ FintypeCat.{w})
+
+/-- For a functor `F : C ⥤ FintypeCat`, the canonical embedding of `Aut F` into
+the product over `Aut (F.obj X)` for all objects `X`. -/
+def autEmbedding : Aut F →* ∀ X, Aut (F.obj X) :=
+  MonoidHom.mk' (fun σ X ↦ σ.app X) (fun _ _ ↦ rfl)
+
+@[simp]
+lemma autEmbedding_apply (σ : Aut F) (X : C) : autEmbedding F σ X = σ.app X :=
+  rfl
+
+lemma autEmbedding_injective : Function.Injective (autEmbedding F) := by
+  intro σ τ h
+  ext X x
+  have : σ.app X = τ.app X := congr_fun h X
+  rw [← Iso.app_hom, ← Iso.app_hom, this]
+
+/-- We put the discrete topology on `F.obj X`. -/
+scoped instance (X : C) : TopologicalSpace (F.obj X) := ⊥
+
+@[scoped instance]
+lemma obj_discreteTopology (X : C) : DiscreteTopology (F.obj X) := ⟨rfl⟩
+
+/-- We put the discrete topology on `Aut (F.obj X)`. -/
+scoped instance (X : C) : TopologicalSpace (Aut (F.obj X)) := ⊥
+
+@[scoped instance]
+lemma aut_discreteTopology (X : C) : DiscreteTopology (Aut (F.obj X)) := ⟨rfl⟩
+
+/-- `Aut F` is equipped with the by the embedding into `∀ X, Aut (F.obj X)` induced embedding. -/
+instance : TopologicalSpace (Aut F) :=
+  TopologicalSpace.induced (autEmbedding F) inferInstance
+
+/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` are precisely the compatible families of
+automorphisms. -/
+lemma autEmbedding_range :
+    Set.range (autEmbedding F) =
+      ⋂ (f : Arrow C), { a | F.map f.hom ≫ (a f.right).hom = (a f.left).hom ≫ F.map f.hom } := by
+  ext a
+  simp only [Set.mem_range, id_obj, Set.mem_iInter, Set.mem_setOf_eq]
+  refine ⟨fun ⟨σ, h⟩ i ↦ h.symm ▸ σ.hom.naturality i.hom, fun h ↦ ?_⟩
+  · use NatIso.ofComponents (fun X => a X) (fun {X Y} f ↦ h ⟨X, Y, f⟩)
+    rfl
+
+/-- The image of `Aut F` in `∀ X, Aut (F.obj X)` is closed. -/
+lemma autEmbedding_range_isClosed : IsClosed (Set.range (autEmbedding F)) := by
+  rw [autEmbedding_range]
+  refine isClosed_iInter (fun f ↦ isClosed_eq (X := F.obj f.left → F.obj f.right) ?_ ?_)
+  · fun_prop
+  · fun_prop
+
+lemma autEmbedding_closedEmbedding : ClosedEmbedding (autEmbedding F) where
+  induced := rfl
+  inj := autEmbedding_injective F
+  isClosed_range := autEmbedding_range_isClosed F
+
+instance : CompactSpace (Aut F) := ClosedEmbedding.compactSpace (autEmbedding_closedEmbedding F)
+
+instance : T2Space (Aut F) :=
+  T2Space.of_injective_continuous (autEmbedding_injective F) continuous_induced_dom
+
+instance : TotallyDisconnectedSpace (Aut F) :=
+  (Embedding.isTotallyDisconnected_range (autEmbedding_closedEmbedding F).embedding).mp
+    (isTotallyDisconnected_of_totallyDisconnectedSpace _)
+
+instance : ContinuousMul (Aut F) :=
+  Inducing.continuousMul (autEmbedding F)
+    (autEmbedding_closedEmbedding F).toInducing
+
+instance : ContinuousInv (Aut F) :=
+  Inducing.continuousInv (autEmbedding_closedEmbedding F).toInducing (fun _ ↦ rfl)
+
+instance : TopologicalGroup (Aut F) := ⟨⟩
+
+instance (X : C) : SMul (Aut (F.obj X)) (F.obj X) := ⟨fun σ a => σ.hom a⟩
+
+instance (X : C) : ContinuousSMul (Aut (F.obj X)) (F.obj X) := by
+  constructor
+  fun_prop
+
+instance continuousSMul_aut_fiber (X : C) : ContinuousSMul (Aut F) (F.obj X) where
+  continuous_smul := by
+    let g : Aut (F.obj X) × F.obj X → F.obj X := fun ⟨σ, x⟩ ↦ σ.hom x
+    let h (q : Aut F × F.obj X) : Aut (F.obj X) × F.obj X :=
+      ⟨((fun p ↦ p X) ∘ autEmbedding F) q.1, q.2⟩
+    show Continuous (g ∘ h)
+    fun_prop
+
+variable [GaloisCategory C] [FiberFunctor F]
+
+/--
+If `H` is an open subset of `Aut F` such that `1 ∈ H`, there exists a finite
+set `I` of connected objects of `C` such that every `σ : Aut F` that induces the identity
+on `F.obj X` for all `X ∈ I` is contained in `H`. In other words: The kernel
+of the evaluation map `Aut F →* ∏ X : I ↦ Aut (F.obj X)` is contained in `H`.
+-/
+lemma exists_set_ker_evaluation_subset_of_isOpen
+    {H : Set (Aut F)} (h1 : 1 ∈ H) (h : IsOpen H) :
+    ∃ (I : Set C) (_ : Fintype I), (∀ X ∈ I, IsConnected X) ∧
+      (∀ σ : Aut F, (∀ X : I, σ.hom.app X = 𝟙 (F.obj X)) → σ ∈ H) := by
+  obtain ⟨U, hUopen, rfl⟩ := isOpen_induced_iff.mp h
+  obtain ⟨I, u, ho, ha⟩ := isOpen_pi_iff.mp hUopen 1 h1
+  choose fι ff fc h4 h5 h6 using (fun X : I => has_decomp_connected_components X.val)
+  refine ⟨⋃ X, Set.range (ff X), Fintype.ofFinite _, ?_, ?_⟩
+  · rintro X ⟨A, ⟨Y, rfl⟩, hA2⟩
+    obtain ⟨i, rfl⟩ := hA2
+    exact h5 Y i
+  · refine fun σ h ↦ ha (fun X XinI ↦ ?_)
+    suffices h : autEmbedding F σ X = 1 by
+      rw [h]
+      exact (ho X XinI).right
+    have h : σ.hom.app X = 𝟙 (F.obj X) := by
+      have : Fintype (fι ⟨X, XinI⟩) := Fintype.ofFinite _
+      ext x
+      obtain ⟨⟨j⟩, a, ha : F.map _ a = x⟩ := Limits.FintypeCat.jointly_surjective
+        (Discrete.functor (ff ⟨X, XinI⟩) ⋙ F) _ (Limits.isColimitOfPreserves F (h4 ⟨X, XinI⟩)) x
+      rw [FintypeCat.id_apply, ← ha, FunctorToFintypeCat.naturality]
+      simp [h ⟨(ff _) j, ⟨Set.range (ff ⟨X, XinI⟩), ⟨⟨_, rfl⟩, ⟨j, rfl⟩⟩⟩⟩]
+    exact Iso.ext h
+
+open Limits
+
+/-- The stabilizers of points in the fibers of Galois objects form a neighbourhood basis
+of the identity in `Aut F`. -/
+lemma nhds_one_has_basis_stabilizers : (nhds (1 : Aut F)).HasBasis (fun _ ↦ True)
+    (fun X : PointedGaloisObject F ↦ MulAction.stabilizer (Aut F) X.pt) where
+  mem_iff' S := by
+    rw [mem_nhds_iff]
+    refine ⟨?_, ?_⟩
+    · intro ⟨U, hU, hUopen, hUone⟩
+      obtain ⟨I, _, hc, hmem⟩ := exists_set_ker_evaluation_subset_of_isOpen F hUone hUopen
+      let P : C := ∏ᶜ fun X : I ↦ X.val
+      obtain ⟨A, a, hgal, hbij⟩ := exists_galois_representative F P
+      refine ⟨⟨A, a, hgal⟩, trivial, ?_⟩
+      intro t (ht : t.hom.app A a = a)
+      apply hU
+      apply hmem
+      haveI (X : I) : IsConnected X.val := hc X.val X.property
+      haveI (X : I) : Nonempty (F.obj X.val) := nonempty_fiber_of_isConnected F X
+      intro X
+      ext x
+      simp only [FintypeCat.id_apply]
+      obtain ⟨z, rfl⟩ :=
+        surjective_of_nonempty_fiber_of_isConnected F (Pi.π (fun X : I ↦ X.val) X) x
+      obtain ⟨f, rfl⟩ := hbij.surjective z
+      rw [FunctorToFintypeCat.naturality, FunctorToFintypeCat.naturality, ht]
+    · intro ⟨X, _, h⟩
+      exact ⟨MulAction.stabilizer (Aut F) X.pt, h, stabilizer_isOpen (Aut F) X.pt,
+        Subgroup.one_mem _⟩
+
+end PreGaloisCategory
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Generator.lean b/Mathlib/CategoryTheory/Generator.lean
index f4689ea78b469..bd219bca91faa 100644
--- a/Mathlib/CategoryTheory/Generator.lean
+++ b/Mathlib/CategoryTheory/Generator.lean
@@ -3,14 +3,10 @@ Copyright (c) 2022 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 -/
-import Mathlib.CategoryTheory.Balanced
 import Mathlib.CategoryTheory.Limits.EssentiallySmall
 import Mathlib.CategoryTheory.Limits.Opposites
-import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
 import Mathlib.CategoryTheory.Subobject.Lattice
-import Mathlib.CategoryTheory.Subobject.WellPowered
 import Mathlib.Data.Set.Opposite
-import Mathlib.Data.Set.Subsingleton
 
 /-!
 # Separating and detecting sets
diff --git a/Mathlib/CategoryTheory/GlueData.lean b/Mathlib/CategoryTheory/GlueData.lean
index 672861742b53b..826e0567ead86 100644
--- a/Mathlib/CategoryTheory/GlueData.lean
+++ b/Mathlib/CategoryTheory/GlueData.lean
@@ -388,7 +388,7 @@ structure GlueData' where
   cocycle : ∀ i j k hij hik hjk, t' i j k hij hik hjk ≫
     t' j k i hjk hij.symm hik.symm ≫ t' k i j hik.symm hjk.symm hij = 𝟙 _
 
-attribute [local instance] GlueData'.f_mono GlueData'.f_hasPullback mono_comp
+attribute [local instance] GlueData'.f_mono GlueData'.f_hasPullback
 
 attribute [reassoc (attr := simp)] GlueData'.t_inv GlueData'.cocycle
 
diff --git a/Mathlib/CategoryTheory/GradedObject.lean b/Mathlib/CategoryTheory/GradedObject.lean
index fcee74c33d22f..c3fd1e2c4e0aa 100644
--- a/Mathlib/CategoryTheory/GradedObject.lean
+++ b/Mathlib/CategoryTheory/GradedObject.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Group.Int
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
diff --git a/Mathlib/CategoryTheory/GradedObject/Monoidal.lean b/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
index 88a32f2bf3bcd..b005a034c6b49 100644
--- a/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
+++ b/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
@@ -292,7 +292,14 @@ lemma associator_naturality (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃
     [HasGoodTensor₁₂Tensor X₁ X₂ X₃] [HasGoodTensorTensor₂₃ X₁ X₂ X₃]
     [HasGoodTensor₁₂Tensor Y₁ Y₂ Y₃] [HasGoodTensorTensor₂₃ Y₁ Y₂ Y₃] :
     tensorHom (tensorHom f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom =
-      (associator X₁ X₂ X₃).hom ≫ tensorHom f₁ (tensorHom f₂ f₃) := by aesop_cat
+      (associator X₁ X₂ X₃).hom ≫ tensorHom f₁ (tensorHom f₂ f₃) := by
+        #adaptation_note
+        /-- this used to be aesop_cat, but that broke with
+        https://github.com/leanprover/lean4/pull/4154 -/
+        ext x i₁ i₂ i₃ h : 2
+        simp only [categoryOfGradedObjects_comp, ιTensorObj₃'_tensorHom_assoc,
+          associator_conjugation, ιTensorObj₃'_associator_hom, assoc, Iso.inv_hom_id_assoc,
+          ιTensorObj₃'_associator_hom_assoc, ιTensorObj₃_tensorHom]
 
 end
 
@@ -318,7 +325,7 @@ lemma left_tensor_tensorObj₃_ext {j : I} {A : C} (Z : C)
       (_ ◁ ιTensorObj₃ X₁ X₂ X₃ i₁ i₂ i₃ j h) ≫ f =
         (_ ◁ ιTensorObj₃ X₁ X₂ X₃ i₁ i₂ i₃ j h) ≫ g) : f = g := by
     refine (@isColimitOfPreserves C _ C _ _ _ _ ((curriedTensor C).obj Z) _
-      (isColimitCofan₃MapBifunctorBifunctor₂₃MapObj (H := H) j) hZ).hom_ext ?_
+      (isColimitCofan₃MapBifunctorBifunctor₂₃MapObj (H := H) (j := j)) hZ).hom_ext ?_
     intro ⟨⟨i₁, i₂, i₃⟩, hi⟩
     exact h _ _ _ hi
 
@@ -399,10 +406,10 @@ lemma pentagon_inv :
         tensorHom (associator X₁ X₂ X₃).inv (𝟙 X₄) =
     (associator X₁ X₂ (tensorObj X₃ X₄)).inv ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv := by
   ext j i₁ i₂ i₃ i₄ h
-  dsimp
+  dsimp only [categoryOfGradedObjects_comp]
   conv_lhs =>
     rw [ιTensorObj₄_eq X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h _ rfl, assoc, ι_tensorHom_assoc]
-    dsimp
+    dsimp only [categoryOfGradedObjects_id, id_eq, eq_mpr_eq_cast, cast_eq]
     rw [id_tensorHom, ← MonoidalCategory.whiskerLeft_comp_assoc, ιTensorObj₃_associator_inv,
       ιTensorObj₃'_eq X₂ X₃ X₄ i₂ i₃ i₄ _ rfl _ rfl, MonoidalCategory.whiskerLeft_comp_assoc,
       MonoidalCategory.whiskerLeft_comp_assoc,
@@ -410,7 +417,7 @@ lemma pentagon_inv :
         (by simp only [← add_assoc, h]) _ rfl, ιTensorObj₃_associator_inv_assoc,
       ιTensorObj₃'_eq_assoc X₁ (tensorObj X₂ X₃) X₄ i₁ (i₂ + i₃) i₄ j
         (by simp only [← add_assoc, h]) (i₁ + i₂ + i₃) (by rw [add_assoc]), ι_tensorHom]
-    dsimp
+    dsimp only [id_eq, eq_mpr_eq_cast, categoryOfGradedObjects_id]
     rw [tensorHom_id, whisker_assoc_symm_assoc, Iso.hom_inv_id_assoc,
       ← MonoidalCategory.comp_whiskerRight_assoc, ← MonoidalCategory.comp_whiskerRight_assoc,
       ← ιTensorObj₃_eq X₁ X₂ X₃ i₁ i₂ i₃ _ rfl _ rfl, ιTensorObj₃_associator_inv,
diff --git a/Mathlib/CategoryTheory/GradedObject/Trifunctor.lean b/Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
index 268083af2e28c..b0d783ab327b1 100644
--- a/Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
+++ b/Mathlib/CategoryTheory/GradedObject/Trifunctor.lean
@@ -215,13 +215,17 @@ noncomputable def mapTrifunctorMapFunctorObj (X₁ : GradedObject I₁ C₁)
       NatTrans.id_app, categoryOfGradedObjects_comp, Functor.map_comp, NatTrans.comp_app,
       id_comp, assoc, ι_mapTrifunctorMapMap_assoc]
 
+#adaptation_note
+/--
+At nightly-2024-08-08 we needed to significantly increase the maxHeartbeats here.
+-/
+set_option maxHeartbeats 800000 in
 /-- Given a trifunctor `F : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄` and a map `p : I₁ × I₂ × I₃ → J`,
 this is the functor
 `GradedObject I₁ C₁ ⥤ GradedObject I₂ C₂ ⥤ GradedObject I₃ C₃ ⥤ GradedObject J C₄`
 sending `X₁ : GradedObject I₁ C₁`, `X₂ : GradedObject I₂ C₂` and `X₃ : GradedObject I₃ C₃`
 to the `J`-graded object sending `j` to the coproduct of
 `((F.obj (X₁ i₁)).obj (X₂ i₂)).obj (X₃ i₃)` for `p ⟨i₁, i₂, i₃⟩ = j`. -/
-@[simps]
 noncomputable def mapTrifunctorMap
     [∀ X₁ X₂ X₃, HasMap ((((mapTrifunctor F I₁ I₂ I₃).obj X₁).obj X₂).obj X₃) p] :
     GradedObject I₁ C₁ ⥤ GradedObject I₂ C₂ ⥤ GradedObject I₃ C₃ ⥤ GradedObject J C₄ where
@@ -243,6 +247,8 @@ noncomputable def mapTrifunctorMap
           NatTrans.id_app, ι_mapTrifunctorMapMap, id_comp,
           NatTrans.naturality_app_assoc] }
 
+attribute [simps] mapTrifunctorMap
+
 end
 
 section
@@ -253,7 +259,7 @@ variable (F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂) (G : C₁₂ ⥤ C₃ ⥤ C₄)
 /-- Given a map `r : I₁ × I₂ × I₃ → J`, a `BifunctorComp₁₂IndexData r` consists of the data
 of a type `I₁₂`, maps `p : I₁ × I₂ → I₁₂` and `q : I₁₂ × I₃ → J`, such that `r` is obtained
 by composition of `p` and `q`. -/
-structure BifunctorComp₁₂IndexData :=
+structure BifunctorComp₁₂IndexData where
   /-- an auxiliary type -/
   I₁₂ : Type*
   /-- a map `I₁ × I₂ → I₁₂` -/
@@ -361,7 +367,7 @@ section
 variable [HasMap ((((mapTrifunctor (bifunctorComp₁₂ F₁₂ G) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃) r]
 
 /-- The action on graded objects of a trifunctor obtained by composition of two
-bifunctors can be computed as a composition of the actions of these two bifunctors.  -/
+bifunctors can be computed as a composition of the actions of these two bifunctors. -/
 noncomputable def mapBifunctorComp₁₂MapObjIso :
     mapTrifunctorMapObj (bifunctorComp₁₂ F₁₂ G) r X₁ X₂ X₃ ≅
     mapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ :=
@@ -433,7 +439,7 @@ variable (F : C₁ ⥤ C₂₃ ⥤ C₄) (G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃)
 /-- Given a map `r : I₁ × I₂ × I₃ → J`, a `BifunctorComp₂₃IndexData r` consists of the data
 of a type `I₂₃`, maps `p : I₂ × I₃ → I₂₃` and `q : I₁ × I₂₃ → J`, such that `r` is obtained
 by composition of `p` and `q`. -/
-structure BifunctorComp₂₃IndexData :=
+structure BifunctorComp₂₃IndexData where
   /-- an auxiliary type -/
   I₂₃ : Type*
   /-- a map `I₂ × I₃ → I₂₃` -/
@@ -539,7 +545,7 @@ section
 variable [HasMap ((((mapTrifunctor (bifunctorComp₂₃ F G₂₃) I₁ I₂ I₃).obj X₁).obj X₂).obj X₃) r]
 
 /-- The action on graded objects of a trifunctor obtained by composition of two
-bifunctors can be computed as a composition of the actions of these two bifunctors.  -/
+bifunctors can be computed as a composition of the actions of these two bifunctors. -/
 noncomputable def mapBifunctorComp₂₃MapObjIso :
     mapTrifunctorMapObj (bifunctorComp₂₃ F G₂₃) r X₁ X₂ X₃ ≅
     mapBifunctorMapObj F ρ₂₃.q X₁ (mapBifunctorMapObj G₂₃ ρ₂₃.p X₂ X₃) :=
diff --git a/Mathlib/CategoryTheory/Grothendieck.lean b/Mathlib/CategoryTheory/Grothendieck.lean
index 60f2b56671b81..59e0d242c7d00 100644
--- a/Mathlib/CategoryTheory/Grothendieck.lean
+++ b/Mathlib/CategoryTheory/Grothendieck.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Sina Hazratpour
+Authors: Kim Morrison, Sina Hazratpour
 -/
 import Mathlib.CategoryTheory.Category.Cat
 import Mathlib.CategoryTheory.Elements
diff --git a/Mathlib/CategoryTheory/Groupoid.lean b/Mathlib/CategoryTheory/Groupoid.lean
index 7bf9c52f3bc73..4799bc3879317 100644
--- a/Mathlib/CategoryTheory/Groupoid.lean
+++ b/Mathlib/CategoryTheory/Groupoid.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Scott Morrison, David Wärn
+Authors: Reid Barton, Kim Morrison, David Wärn
 -/
 import Mathlib.CategoryTheory.FullSubcategory
 import Mathlib.CategoryTheory.Products.Basic
diff --git a/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean b/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean
index 2521a5e952967..79f597c240e2e 100644
--- a/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean
+++ b/Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean
@@ -3,13 +3,8 @@ Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémi Bottinelli
 -/
-import Mathlib.CategoryTheory.Category.Basic
-import Mathlib.CategoryTheory.Functor.Basic
 import Mathlib.CategoryTheory.Groupoid
-import Mathlib.Tactic.NthRewrite
 import Mathlib.CategoryTheory.PathCategory
-import Mathlib.CategoryTheory.Quotient
-import Mathlib.Combinatorics.Quiver.Symmetric
 
 /-!
 # Free groupoid on a quiver
@@ -80,10 +75,11 @@ theorem congr_reverse {X Y : Paths <| Quiver.Symmetrify V} (p q : X ⟶ Y) :
     Quiver.Path.reverse_comp, Quiver.reverse_reverse, Quiver.Path.reverse_toPath,
     Quiver.Path.comp_assoc] using this
 
+open Relation in
 theorem congr_comp_reverse {X Y : Paths <| Quiver.Symmetrify V} (p : X ⟶ Y) :
     Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (p ≫ p.reverse) =
       Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (𝟙 X) := by
-  apply Quot.EqvGen_sound
+  apply Quot.eqvGen_sound
   induction' p with a b q f ih
   · apply EqvGen.refl
   · simp only [Quiver.Path.reverse]
@@ -136,7 +132,7 @@ theorem of_eq :
 
 section UniversalProperty
 
-variable {V' : Type u'} [Groupoid V'] (φ : V ⥤q V')
+variable {V' : Type u'} [Groupoid V']
 
 /-- The lift of a prefunctor to a groupoid, to a functor from `FreeGroupoid V` -/
 def lift (φ : V ⥤q V') : FreeGroupoid V ⥤ V' :=
diff --git a/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean b/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
index b2efc306cfa42..54c934333769c 100644
--- a/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
+++ b/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean
@@ -139,7 +139,7 @@ theorem coe_inv_coe' {c d : S.objs} (p : c ⟶ d) :
     (CategoryTheory.inv p).val = CategoryTheory.inv p.val := by
   simp only [← inv_eq_inv, coe_inv_coe]
 
-/-- The embedding of the coerced subgroupoid to its parent-/
+/-- The embedding of the coerced subgroupoid to its parent -/
 def hom : S.objs ⥤ C where
   obj c := c.val
   map f := f.val
@@ -292,7 +292,7 @@ theorem IsWide.id_mem {S : Subgroupoid C} (Sw : S.IsWide) (c : C) : 𝟙 c ∈ S
 theorem IsWide.eqToHom_mem {S : Subgroupoid C} (Sw : S.IsWide) {c d : C} (h : c = d) :
     eqToHom h ∈ S.arrows c d := by cases h; simp only [eqToHom_refl]; apply Sw.id_mem c
 
-/-- A subgroupoid is normal if it is wide and satisfies the expected stability under conjugacy. -/
+/-- A subgroupoid is normal if it is wide and satisfies the expected stability under conjugacy. -/
 structure IsNormal extends IsWide S : Prop where
   conj : ∀ {c d} (p : c ⟶ d) {γ : c ⟶ c}, γ ∈ S.arrows c c → Groupoid.inv p ≫ γ ≫ p ∈ S.arrows d d
 
@@ -540,8 +540,7 @@ theorem isTotallyDisconnected_iff :
     S.IsTotallyDisconnected ↔ ∀ c d, (S.arrows c d).Nonempty → c = d := by
   constructor
   · rintro h c d ⟨f, fS⟩
-    have := h ⟨c, mem_objs_of_src S fS⟩ ⟨d, mem_objs_of_tgt S fS⟩ ⟨f, fS⟩
-    exact congr_arg Subtype.val this
+    exact congr_arg Subtype.val <| h ⟨c, mem_objs_of_src S fS⟩ ⟨d, mem_objs_of_tgt S fS⟩ ⟨f, fS⟩
   · rintro h ⟨c, hc⟩ ⟨d, hd⟩ ⟨f, fS⟩
     simp only [Subtype.mk_eq_mk]
     exact h c d ⟨f, fS⟩
diff --git a/Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean b/Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean
index ebcbb63e295ad..b1b1f5f30c036 100644
--- a/Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean
+++ b/Mathlib/CategoryTheory/Idempotents/FunctorCategories.lean
@@ -58,7 +58,7 @@ instance functor_category_isIdempotentComplete [IsIdempotentComplete C] :
   have hC := (isIdempotentComplete_iff_hasEqualizer_of_id_and_idempotent C).mp inferInstance
   haveI : ∀ j : J, HasEqualizer (𝟙 _) (p.app j) := fun j => hC _ _ (congr_app hp j)
   /- We construct the direct factor `Y` associated to `p : F ⟶ F` by computing
-      the equalizer of the identity and `p.app j` on each object `(j : J)`.  -/
+      the equalizer of the identity and `p.app j` on each object `(j : J)`. -/
   let Y : J ⥤ C :=
     { obj := fun j => Limits.equalizer (𝟙 _) (p.app j)
       map := fun {j j'} φ =>
diff --git a/Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean b/Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
index 0fbbb2cf5e249..4c4de5bce5d99 100644
--- a/Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
+++ b/Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
@@ -106,8 +106,12 @@ def functorExtension₁CompWhiskeringLeftToKaroubiIso :
       (fun {X Y} f => by aesop_cat))
     (by aesop_cat)
 
+#adaptation_note
+/--
+At nightly-2024-08-08 we needed to increase the maxHeartbeats here.
+-/
+set_option maxHeartbeats 400000 in
 /-- The counit isomorphism of the equivalence `(C ⥤ Karoubi D) ≌ (Karoubi C ⥤ Karoubi D)`. -/
-@[simps!]
 def KaroubiUniversal₁.counitIso :
     (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D ≅ 𝟭 _ :=
   NatIso.ofComponents
@@ -144,6 +148,8 @@ def KaroubiUniversal₁.counitIso :
       simp only [Functor.map_comp, comp_f, assoc]
       rfl)
 
+attribute [simps!] KaroubiUniversal₁.counitIso
+
 /-- The equivalence of categories `(C ⥤ Karoubi D) ≌ (Karoubi C ⥤ Karoubi D)`. -/
 @[simps]
 def karoubiUniversal₁ : C ⥤ Karoubi D ≌ Karoubi C ⥤ Karoubi D where
diff --git a/Mathlib/CategoryTheory/Iso.lean b/Mathlib/CategoryTheory/Iso.lean
index 5294a89d8becb..8d7ad6c8ea53a 100644
--- a/Mathlib/CategoryTheory/Iso.lean
+++ b/Mathlib/CategoryTheory/Iso.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.Tactic.CategoryTheory.Reassoc
 
@@ -355,7 +355,7 @@ theorem inv_id : inv (𝟙 X) = 𝟙 X := by
   apply inv_eq_of_hom_inv_id
   simp
 
-@[simp]
+@[simp, reassoc]
 theorem inv_comp [IsIso f] [IsIso h] : inv (f ≫ h) = inv h ≫ inv f := by
   apply inv_eq_of_hom_inv_id
   simp
diff --git a/Mathlib/CategoryTheory/LiftingProperties/Basic.lean b/Mathlib/CategoryTheory/LiftingProperties/Basic.lean
index 034992348ba84..6b3ab2bd33ac2 100644
--- a/Mathlib/CategoryTheory/LiftingProperties/Basic.lean
+++ b/Mathlib/CategoryTheory/LiftingProperties/Basic.lean
@@ -41,7 +41,7 @@ class HasLiftingProperty : Prop where
   sq_hasLift : ∀ {f : A ⟶ X} {g : B ⟶ Y} (sq : CommSq f i p g), sq.HasLift
 
 instance (priority := 100) sq_hasLift_of_hasLiftingProperty {f : A ⟶ X} {g : B ⟶ Y}
-    (sq : CommSq f i p g) [hip : HasLiftingProperty i p] : sq.HasLift := by apply hip.sq_hasLift
+    (sq : CommSq f i p g) [hip : HasLiftingProperty i p] : sq.HasLift := hip.sq_hasLift _
 
 namespace HasLiftingProperty
 
diff --git a/Mathlib/CategoryTheory/Limits/ColimitLimit.lean b/Mathlib/CategoryTheory/Limits/ColimitLimit.lean
index 2628945db2029..23386c1aed4c1 100644
--- a/Mathlib/CategoryTheory/Limits/ColimitLimit.lean
+++ b/Mathlib/CategoryTheory/Limits/ColimitLimit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Types
 import Mathlib.CategoryTheory.Functor.Currying
diff --git a/Mathlib/CategoryTheory/Limits/ConcreteCategory/Basic.lean b/Mathlib/CategoryTheory/Limits/ConcreteCategory/Basic.lean
index 18cd6a039d479..2ecb02a4b57c2 100644
--- a/Mathlib/CategoryTheory/Limits/ConcreteCategory/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/ConcreteCategory/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Adam Topaz
+Authors: Kim Morrison, Adam Topaz
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
@@ -27,7 +27,7 @@ section Limits
 is corepresentable, then `(G ⋙ forget C).sections` is small. -/
 lemma small_sections_of_hasLimit
     {C : Type u} [Category.{v} C] [ConcreteCategory.{v} C]
-    [(forget C).Corepresentable] {J : Type w} [Category.{t} J] (G : J ⥤ C) [HasLimit G] :
+    [(forget C).IsCorepresentable] {J : Type w} [Category.{t} J] (G : J ⥤ C) [HasLimit G] :
     Small.{v} (G ⋙ forget C).sections := by
   rw [← Types.hasLimit_iff_small_sections]
   infer_instance
diff --git a/Mathlib/CategoryTheory/Limits/Cones.lean b/Mathlib/CategoryTheory/Limits/Cones.lean
index 1d468d8221bd1..157d042f4667c 100644
--- a/Mathlib/CategoryTheory/Limits/Cones.lean
+++ b/Mathlib/CategoryTheory/Limits/Cones.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Functor.Const
 import Mathlib.CategoryTheory.DiscreteCategory
@@ -668,7 +668,7 @@ def mapCone (c : Cone F) : Cone (F ⋙ H) :=
 def mapCocone (c : Cocone F) : Cocone (F ⋙ H) :=
   (Cocones.functoriality F H).obj c
 
-/-- Given a cone morphism `c ⟶ c'`, construct a cone morphism on the mapped cones functorially.  -/
+/-- Given a cone morphism `c ⟶ c'`, construct a cone morphism on the mapped cones functorially. -/
 def mapConeMorphism {c c' : Cone F} (f : c ⟶ c') : H.mapCone c ⟶ H.mapCone c' :=
   (Cones.functoriality F H).map f
 
diff --git a/Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean b/Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean
index 109be53d93b7b..38bfd50516c2a 100644
--- a/Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean
+++ b/Mathlib/CategoryTheory/Limits/Constructions/Filtered.lean
@@ -32,10 +32,12 @@ namespace CategoryTheory.Limits
 
 namespace CoproductsFromFiniteFiltered
 
+variable [HasFiniteCoproducts C]
+
 /-- If `C` has finite coproducts, a functor `Discrete α ⥤ C` lifts to a functor
     `Finset (Discrete α) ⥤ C` by taking coproducts. -/
 @[simps!]
-def liftToFinset [HasFiniteCoproducts C] (F : Discrete α ⥤ C) : Finset (Discrete α) ⥤ C where
+def liftToFinsetObj (F : Discrete α ⥤ C) : Finset (Discrete α) ⥤ C where
   obj s := ∐ fun x : s => F.obj x
   map {_ Y} h := Sigma.desc fun y =>
     Sigma.ι (fun (x : { x // x ∈ Y }) => F.obj x) ⟨y, h.down.down y.2⟩
@@ -44,30 +46,39 @@ def liftToFinset [HasFiniteCoproducts C] (F : Discrete α ⥤ C) : Finset (Discr
     taking the colimit of the diagram formed by the coproducts of finite sets over the indexing
     type. -/
 @[simps!]
-def liftToFinsetColimitCocone [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
-    (F : Discrete α ⥤ C) : ColimitCocone F where
+def liftToFinsetColimitCocone [HasFilteredColimitsOfSize.{w, w} C] (F : Discrete α ⥤ C) :
+    ColimitCocone F where
   cocone :=
-    { pt := colimit (liftToFinset F)
+    { pt := colimit (liftToFinsetObj F)
       ι :=
         Discrete.natTrans fun j =>
           @Sigma.ι _ _ _ (fun x : ({j} : Finset (Discrete α)) => F.obj x) _ ⟨j, by simp⟩ ≫
-            colimit.ι (liftToFinset F) {j} }
+            colimit.ι (liftToFinsetObj F) {j} }
   isColimit :=
     { desc := fun s =>
-        colimit.desc (liftToFinset F)
+        colimit.desc (liftToFinsetObj F)
           { pt := s.pt
             ι := { app := fun t => Sigma.desc fun x => s.ι.app x } }
       uniq := fun s m h => by
         apply colimit.hom_ext
         rintro t
-        dsimp [liftToFinset]
+        dsimp [liftToFinsetObj]
         apply colimit.hom_ext
         rintro ⟨⟨j, hj⟩⟩
         convert h j using 1
-        · simp [← colimit.w (liftToFinset F) ⟨⟨Finset.singleton_subset_iff.2 hj⟩⟩]
+        · simp [← colimit.w (liftToFinsetObj F) ⟨⟨Finset.singleton_subset_iff.2 hj⟩⟩]
           rfl
         · aesop_cat }
 
+variable (C) (α)
+
+/-- The functor taking a functor `Discrete α ⥤ C` to a functor `Finset (Discrete α) ⥤ C` by taking
+coproducts. -/
+@[simps!]
+def liftToFinset : (Discrete α ⥤ C) ⥤ (Finset (Discrete α) ⥤ C) where
+  obj := liftToFinsetObj
+  map := fun β => { app := fun _ => Sigma.map (fun x => β.app x.val) }
+
 end CoproductsFromFiniteFiltered
 
 open CoproductsFromFiniteFiltered
@@ -91,4 +102,34 @@ theorem has_limits_of_finite_and_cofiltered [HasFiniteLimits C]
   have : HasProducts.{w} C := hasProducts_of_finite_and_cofiltered
   has_limits_of_hasEqualizers_and_products
 
+namespace CoproductsFromFiniteFiltered
+
+variable [HasFiniteCoproducts C] [HasFilteredColimitsOfSize.{w, w} C]
+
+attribute [local instance] hasCoproducts_of_finite_and_filtered
+
+/-- Helper construction for `liftToFinsetColimIso`. -/
+@[reassoc]
+theorem liftToFinsetColimIso_aux (F : Discrete α ⥤ C) {J : Finset (Discrete α)} (j : J) :
+    Sigma.ι (F.obj ·.val) j ≫ colimit.ι (liftToFinsetObj F) J ≫
+      (colimit.isoColimitCocone (liftToFinsetColimitCocone F)).inv
+    = colimit.ι F j := by
+  simp [colimit.isoColimitCocone, IsColimit.coconePointUniqueUpToIso]
+
+/-- The `liftToFinset` functor, precomposed with forming a colimit, is a coproduct on the original
+functor. -/
+def liftToFinsetColimIso : liftToFinset C α ⋙ colim ≅ colim :=
+  NatIso.ofComponents
+    (fun F => Iso.symm <| colimit.isoColimitCocone (liftToFinsetColimitCocone F))
+    (fun β => by
+      simp only [Functor.comp_obj, colim_obj, Functor.comp_map, colim_map, Iso.symm_hom]
+      ext J
+      simp only [liftToFinset_obj_obj, liftToFinset_map_app]
+      ext j
+      simp only [liftToFinset, ι_colimMap_assoc, liftToFinsetObj_obj, Discrete.functor_obj_eq_as,
+        Discrete.natTrans_app, liftToFinsetColimIso_aux, liftToFinsetColimIso_aux_assoc,
+        ι_colimMap])
+
+end CoproductsFromFiniteFiltered
+
 end CategoryTheory.Limits
diff --git a/Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean b/Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
index 824a394d211e9..5cf6eebed9923 100644
--- a/Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
+++ b/Mathlib/CategoryTheory/Limits/Constructions/LimitsOfProductsAndEqualizers.lean
@@ -1,18 +1,15 @@
 /-
 Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
+import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
+import Mathlib.CategoryTheory.Limits.Constructions.Equalizers
+import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
+import Mathlib.CategoryTheory.Limits.Preserves.Finite
+import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
 import Mathlib.Data.Fintype.Prod
 import Mathlib.Data.Fintype.Sigma
-import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
-import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
-import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
-import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
-import Mathlib.CategoryTheory.Limits.Preserves.Finite
-import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
-import Mathlib.CategoryTheory.Limits.Constructions.Equalizers
-import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
 
 /-!
 # Constructing limits from products and equalizers.
diff --git a/Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean b/Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean
index c67246dbd4d34..d978a5c54dda2 100644
--- a/Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean
+++ b/Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
 import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
diff --git a/Mathlib/CategoryTheory/Limits/Filtered.lean b/Mathlib/CategoryTheory/Limits/Filtered.lean
index 9917f097ee9bf..be5e2d3faf47f 100644
--- a/Mathlib/CategoryTheory/Limits/Filtered.lean
+++ b/Mathlib/CategoryTheory/Limits/Filtered.lean
@@ -77,6 +77,12 @@ class HasFilteredColimitsOfSize : Prop where
   /-- For all filtered types of a size `w`, we have colimits -/
   HasColimitsOfShape : ∀ (I : Type w) [Category.{w'} I] [IsFiltered I], HasColimitsOfShape I C
 
+/-- Class for having cofiltered limits. -/
+abbrev HasCofilteredLimits := HasCofilteredLimitsOfSize.{v, v} C
+
+/-- Class for having filtered colimits. -/
+abbrev HasFilteredColimits := HasFilteredColimitsOfSize.{v, v} C
+
 end
 
 instance (priority := 100) hasLimitsOfShape_of_has_cofiltered_limits
diff --git a/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean b/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
index 4d3665564222e..70ea3b65e93a5 100644
--- a/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
+++ b/Mathlib/CategoryTheory/Limits/FilteredColimitCommutesFiniteLimit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.ColimitLimit
 import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory
@@ -112,7 +112,7 @@ theorem colimitLimitToLimitColimit_injective :
       Finset.mem_union.mpr
         (Or.inl
           (by
-            simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,
+            simp only [true_and, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,
               Finset.mem_image, heq_iff_eq]
             refine ⟨j, ?_⟩
             simp only [heq_iff_eq] ))
@@ -122,7 +122,7 @@ theorem colimitLimitToLimitColimit_injective :
       Finset.mem_union.mpr
         (Or.inr
           (by
-            simp only [true_and_iff, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,
+            simp only [true_and, Finset.mem_univ, eq_self_iff_true, exists_prop_of_true,
               Finset.mem_image, heq_iff_eq]
             refine ⟨j, ?_⟩
             simp only [heq_iff_eq]))
diff --git a/Mathlib/CategoryTheory/Limits/Final.lean b/Mathlib/CategoryTheory/Limits/Final.lean
index 70bc2fb14a714..64975ca699d17 100644
--- a/Mathlib/CategoryTheory/Limits/Final.lean
+++ b/Mathlib/CategoryTheory/Limits/Final.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Comma.StructuredArrow
 import Mathlib.CategoryTheory.IsConnected
@@ -358,7 +358,7 @@ variable {C : Type v} [Category.{v} C] {D : Type u₁} [Category.{v} D] (F : C 
 namespace Final
 
 theorem zigzag_of_eqvGen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : ΣX, d ⟶ F.obj X}
-    (t : EqvGen (Types.Quot.Rel.{v, v} (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
+    (t : Relation.EqvGen (Types.Quot.Rel.{v, v} (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
     Zigzag (StructuredArrow.mk f₁.2) (StructuredArrow.mk f₂.2) := by
   induction t with
   | rel x y r =>
@@ -813,4 +813,28 @@ theorem IsCofiltered.of_initial (F : C ⥤ D) [Initial F] [IsCofiltered C] : IsC
 
 end Filtered
 
+section
+
+variable {C : Type u₁} [Category.{v₁} C]
+variable {D : Type u₂} [Category.{v₂} D]
+variable {E : Type u₃} [Category.{v₃} E]
+
+open Functor
+
+/-- The functor `StructuredArrow.pre X T S` is final if `T` is final. -/
+instance StructuredArrow.final_pre (T : C ⥤ D) [Final T] (S : D ⥤ E) (X : E) :
+    Final (pre X T S) := by
+  refine ⟨fun f => ?_⟩
+  rw [isConnected_iff_of_equivalence (StructuredArrow.preEquivalence T f)]
+  exact Final.out f.right
+
+/-- The functor `CostructuredArrow.pre X T S` is initial if `T` is initial. -/
+instance CostructuredArrow.initial_pre (T : C ⥤ D) [Initial T] (S : D ⥤ E) (X : E) :
+    Initial (CostructuredArrow.pre T S X) := by
+  refine ⟨fun f => ?_⟩
+  rw [isConnected_iff_of_equivalence (CostructuredArrow.preEquivalence T f)]
+  exact Initial.out f.left
+
+end
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean b/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean
index b75d801a67a5d..938e98c7e64e6 100644
--- a/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean
+++ b/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean
@@ -6,10 +6,10 @@ Authors: Dagur Asgeirsson, Joël Riou
 import Mathlib.CategoryTheory.Limits.Final
 /-!
 
-# Conditions for `parallelPair` to be initial
+# Conditions for `parallelPair` to be initial
 
-In this file we give sufficient conditions on a category `C` and parallel morphisms `f g : X ⟶ Y` 
-in `C` so that `parallelPair f g` becomes an initial functor.
+In this file we give sufficient conditions on a category `C` and parallel morphisms `f g : X ⟶ Y`
+in `C` so that `parallelPair f g` becomes an initial functor.
 
 The conditions are that there is a morphism out of `X` to every object of `C` and that any two
 parallel morphisms out of `X` factor through the parallel pair `f`, `g`
diff --git a/Mathlib/CategoryTheory/Limits/FintypeCat.lean b/Mathlib/CategoryTheory/Limits/FintypeCat.lean
index 34a080b4ccb92..db0a572fc3b92 100644
--- a/Mathlib/CategoryTheory/Limits/FintypeCat.lean
+++ b/Mathlib/CategoryTheory/Limits/FintypeCat.lean
@@ -4,11 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christian Merten
 -/
 import Mathlib.CategoryTheory.FintypeCat
-import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
-import Mathlib.CategoryTheory.Limits.Types
-import Mathlib.CategoryTheory.Limits.Creates
 import Mathlib.CategoryTheory.Limits.Preserves.Finite
-import Mathlib.Data.Finite.Basic
+import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
+import Mathlib.CategoryTheory.Limits.Shapes.Types
 
 /-!
 # (Co)limits in the category of finite types
@@ -60,6 +58,34 @@ noncomputable instance inclusionPreservesFiniteLimits :
 noncomputable instance : PreservesFiniteLimits (forget FintypeCat) :=
   FintypeCat.inclusionPreservesFiniteLimits
 
+/-- The categorical product of a finite family in `FintypeCat` is equivalent to the product
+as types. -/
+noncomputable def productEquiv {ι : Type*} [Finite ι] (X : ι → FintypeCat.{u}) :
+    (∏ᶜ X : FintypeCat) ≃ ∀ i, X i :=
+  letI : Fintype ι := Fintype.ofFinite _
+  haveI : Small.{u} ι :=
+    ⟨ULift (Fin (Fintype.card ι)), ⟨(Fintype.equivFin ι).trans Equiv.ulift.symm⟩⟩
+  let is₁ : FintypeCat.incl.obj (∏ᶜ fun i ↦ X i) ≅ (∏ᶜ fun i ↦ X i : Type u) :=
+    PreservesProduct.iso FintypeCat.incl (fun i ↦ X i)
+  let is₂ : (∏ᶜ fun i ↦ X i : Type u) ≅ Shrink.{u} (∀ i, X i) :=
+    Types.Small.productIso (fun i ↦ X i)
+  let e : (∀ i, X i) ≃ Shrink.{u} (∀ i, X i) := equivShrink _
+  (equivEquivIso.symm is₁).trans ((equivEquivIso.symm is₂).trans e.symm)
+
+@[simp]
+lemma productEquiv_apply {ι : Type*} [Finite ι] (X : ι → FintypeCat.{u})
+    (x : (∏ᶜ X : FintypeCat)) (i : ι) : productEquiv X x i = Pi.π X i x := by
+  simpa [productEquiv] using (elementwise_of% piComparison_comp_π FintypeCat.incl X i) x
+
+@[simp]
+lemma productEquiv_symm_comp_π_apply {ι : Type*} [Finite ι] (X : ι → FintypeCat.{u})
+    (x : ∀ i, X i) (i : ι) : Pi.π X i ((productEquiv X).symm x) = x i := by
+  rw [← productEquiv_apply, Equiv.apply_symm_apply]
+
+instance nonempty_pi_of_nonempty {ι : Type*} [Finite ι] (X : ι → FintypeCat.{u})
+    [∀ i, Nonempty (X i)] : Nonempty (∏ᶜ X : FintypeCat.{u}) :=
+  (Equiv.nonempty_congr <| productEquiv X).mpr inferInstance
+
 /-- Any functor from a finite category to Types that only involves finite objects,
 has a finite colimit. -/
 noncomputable instance finiteColimitOfFiniteDiagram {J : Type} [SmallCategory J] [FinCategory J]
@@ -93,4 +119,10 @@ noncomputable instance inclusionPreservesFiniteColimits :
 noncomputable instance : PreservesFiniteColimits (forget FintypeCat) :=
   FintypeCat.inclusionPreservesFiniteColimits
 
+lemma jointly_surjective {J : Type*} [Category J] [FinCategory J]
+    (F : J ⥤ FintypeCat.{u}) (t : Cocone F) (h : IsColimit t) (x : t.pt) :
+    ∃ j y, t.ι.app j y = x :=
+  let hs := isColimitOfPreserves FintypeCat.incl.{u} h
+  Types.jointly_surjective (F ⋙ FintypeCat.incl) hs x
+
 end CategoryTheory.Limits.FintypeCat
diff --git a/Mathlib/CategoryTheory/Limits/Fubini.lean b/Mathlib/CategoryTheory/Limits/Fubini.lean
index 3bd7d06ef9472..898d073faa5a4 100644
--- a/Mathlib/CategoryTheory/Limits/Fubini.lean
+++ b/Mathlib/CategoryTheory/Limits/Fubini.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.HasLimits
 import Mathlib.CategoryTheory.Products.Basic
diff --git a/Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean b/Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean
index b212bc76981ba..0b2e0d72e2f6e 100644
--- a/Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Functor.Currying
 import Mathlib.CategoryTheory.Limits.Preserves.Limits
@@ -146,18 +146,23 @@ def combinedIsColimit (F : J ⥤ K ⥤ C) (c : ∀ k : K, ColimitCocone (F.flip.
 
 noncomputable section
 
+instance functorCategoryHasLimit (F : J ⥤ K ⥤ C) [∀ k, HasLimit (F.flip.obj k)] : HasLimit F :=
+  HasLimit.mk
+    { cone := combineCones F fun _ => getLimitCone _
+      isLimit := combinedIsLimit _ _ }
+
 instance functorCategoryHasLimitsOfShape [HasLimitsOfShape J C] : HasLimitsOfShape J (K ⥤ C) where
-  has_limit F :=
-    HasLimit.mk
-      { cone := combineCones F fun _ => getLimitCone _
-        isLimit := combinedIsLimit _ _ }
+  has_limit _ := inferInstance
+
+instance functorCategoryHasColimit (F : J ⥤ K ⥤ C) [∀ k, HasColimit (F.flip.obj k)] :
+    HasColimit F :=
+  HasColimit.mk
+    { cocone := combineCocones F fun _ => getColimitCocone _
+      isColimit := combinedIsColimit _ _ }
 
 instance functorCategoryHasColimitsOfShape [HasColimitsOfShape J C] :
     HasColimitsOfShape J (K ⥤ C) where
-  has_colimit _ :=
-    HasColimit.mk
-      { cocone := combineCocones _ fun _ => getColimitCocone _
-        isColimit := combinedIsColimit _ _ }
+  has_colimit _ := inferInstance
 
 -- Porting note: previously Lean could see through the binders and infer_instance sufficed
 instance functorCategoryHasLimitsOfSize [HasLimitsOfSize.{v₁, u₁} C] :
@@ -169,14 +174,20 @@ instance functorCategoryHasColimitsOfSize [HasColimitsOfSize.{v₁, u₁} C] :
     HasColimitsOfSize.{v₁, u₁} (K ⥤ C) where
   has_colimits_of_shape := fun _ _ => inferInstance
 
+instance hasLimitCompEvalution (F : J ⥤ K ⥤ C) (k : K) [HasLimit (F.flip.obj k)] :
+    HasLimit (F ⋙ (evaluation _ _).obj k) :=
+  hasLimitOfIso (F := F.flip.obj k) (Iso.refl _)
+
+instance evaluationPreservesLimit (F : J ⥤ K ⥤ C) [∀ k, HasLimit (F.flip.obj k)] (k : K) :
+    PreservesLimit F ((evaluation K C).obj k) :=
+    -- Porting note: added a let because X was not inferred
+  let X : (k : K) → LimitCone (F.flip.obj k) := fun k => getLimitCone (F.flip.obj k)
+  preservesLimitOfPreservesLimitCone (combinedIsLimit _ X) <|
+    IsLimit.ofIsoLimit (limit.isLimit _) (evaluateCombinedCones F X k).symm
+
 instance evaluationPreservesLimitsOfShape [HasLimitsOfShape J C] (k : K) :
     PreservesLimitsOfShape J ((evaluation K C).obj k) where
-  preservesLimit {F} := by
-    -- Porting note: added a let because X was not inferred
-    let X : (k : K) → LimitCone (Prefunctor.obj (Functor.flip F).toPrefunctor k) :=
-      fun k => getLimitCone (Prefunctor.obj (Functor.flip F).toPrefunctor k)
-    exact preservesLimitOfPreservesLimitCone (combinedIsLimit _ _) <|
-      IsLimit.ofIsoLimit (limit.isLimit _) (evaluateCombinedCones F X k).symm
+  preservesLimit := inferInstance
 
 /-- If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a limit,
 then the evaluation of that limit at `k` is the limit of the evaluations of `F.obj j` at `k`.
@@ -225,14 +236,20 @@ theorem limit_obj_ext {H : J ⥤ K ⥤ C} [HasLimitsOfShape J C] {k : K} {W : C}
   ext j
   simpa using w j
 
+instance hasColimitCompEvaluation (F : J ⥤ K ⥤ C) (k : K) [HasColimit (F.flip.obj k)] :
+    HasColimit (F ⋙ (evaluation _ _).obj k) :=
+  hasColimitOfIso (F := F.flip.obj k) (Iso.refl _)
+
+instance evaluationPreservesColimit (F : J ⥤ K ⥤ C) [∀ k, HasColimit (F.flip.obj k)] (k : K) :
+    PreservesColimit F ((evaluation K C).obj k) :=
+  -- Porting note: added a let because X was not inferred
+  let X : (k : K) → ColimitCocone (F.flip.obj k) := fun k => getColimitCocone (F.flip.obj k)
+  preservesColimitOfPreservesColimitCocone (combinedIsColimit _ X) <|
+    IsColimit.ofIsoColimit (colimit.isColimit _) (evaluateCombinedCocones F X k).symm
+
 instance evaluationPreservesColimitsOfShape [HasColimitsOfShape J C] (k : K) :
     PreservesColimitsOfShape J ((evaluation K C).obj k) where
-  preservesColimit {F} := by
-    -- Porting note: added a let because X was not inferred
-    let X : (k : K) → ColimitCocone (Prefunctor.obj (Functor.flip F).toPrefunctor k) :=
-      fun k => getColimitCocone (Prefunctor.obj (Functor.flip F).toPrefunctor k)
-    refine preservesColimitOfPreservesColimitCocone (combinedIsColimit _ _) <|
-      IsColimit.ofIsoColimit (colimit.isColimit _) (evaluateCombinedCocones F X k).symm
+  preservesColimit := inferInstance
 
 /-- If `F : J ⥤ K ⥤ C` is a functor into a functor category which has a colimit,
 then the evaluation of that colimit at `k` is the colimit of the evaluations of `F.obj j` at `k`.
diff --git a/Mathlib/CategoryTheory/Limits/FunctorToTypes.lean b/Mathlib/CategoryTheory/Limits/FunctorToTypes.lean
index 62eb96c6be392..ba9abc78c25d6 100644
--- a/Mathlib/CategoryTheory/Limits/FunctorToTypes.lean
+++ b/Mathlib/CategoryTheory/Limits/FunctorToTypes.lean
@@ -20,15 +20,20 @@ open CategoryTheory.Limits
 universe w v₁ v₂ u₁ u₂
 
 variable {J : Type u₁} [Category.{v₁} J] {K : Type u₂} [Category.{v₂} K]
-variable (F : J ⥤ K ⥤ TypeMax.{u₁, w})
+variable (F : J ⥤ K ⥤ Type w)
 
-theorem jointly_surjective (k : K) {t : Cocone F} (h : IsColimit t) (x : t.pt.obj k) :
-    ∃ j y, x = (t.ι.app j).app k y := by
+theorem jointly_surjective (k : K) {t : Cocone F} (h : IsColimit t) (x : t.pt.obj k)
+    [∀ k, HasColimit (F.flip.obj k)] : ∃ j y, x = (t.ι.app j).app k y := by
   let hev := isColimitOfPreserves ((evaluation _ _).obj k) h
   obtain ⟨j, y, rfl⟩ := Types.jointly_surjective _ hev x
   exact ⟨j, y, by simp⟩
 
-theorem jointly_surjective' (k : K) (x : (colimit F).obj k) : ∃ j y, x = (colimit.ι F j).app k y :=
+theorem jointly_surjective' [∀ k, HasColimit (F.flip.obj k)] (k : K) (x : (colimit F).obj k) :
+    ∃ j y, x = (colimit.ι F j).app k y :=
   jointly_surjective _ _ (colimit.isColimit _) x
 
+theorem colimit.map_ι_apply [HasColimit F] (j : J) {k k' : K} {f : k ⟶ k'} {x} :
+    (colimit F).map f ((colimit.ι F j).app _ x) = (colimit.ι F j).app _ ((F.obj j).map f x) :=
+  congrFun ((colimit.ι F j).naturality _).symm _
+
 end CategoryTheory.FunctorToTypes
diff --git a/Mathlib/CategoryTheory/Limits/HasLimits.lean b/Mathlib/CategoryTheory/Limits/HasLimits.lean
index a2ac485b93211..5478f806f7e4b 100644
--- a/Mathlib/CategoryTheory/Limits/HasLimits.lean
+++ b/Mathlib/CategoryTheory/Limits/HasLimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
+Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Limits.IsLimit
 import Mathlib.CategoryTheory.Category.ULift
@@ -489,23 +489,16 @@ def limYoneda :
     lim ⋙ yoneda ⋙ (whiskeringRight _ _ _).obj uliftFunctor.{u₁} ≅ CategoryTheory.cones J C :=
   NatIso.ofComponents fun F => NatIso.ofComponents fun W => limit.homIso F (unop W)
 
-/-- The constant functor and limit functor are adjoint to each other-/
-def constLimAdj : (const J : C ⥤ J ⥤ C) ⊣ lim where
-  homEquiv c g :=
+/-- The constant functor and limit functor are adjoint to each other -/
+def constLimAdj : (const J : C ⥤ J ⥤ C) ⊣ lim := Adjunction.mk' {
+  homEquiv := fun c g ↦
     { toFun := fun f => limit.lift _ ⟨c, f⟩
       invFun := fun f =>
         { app := fun j => f ≫ limit.π _ _ }
       left_inv := by aesop_cat
       right_inv := by aesop_cat }
   unit := { app := fun c => limit.lift _ ⟨_, 𝟙 _⟩ }
-  counit := { app := fun g => { app := limit.π _ } }
-  -- This used to be automatic before leanprover/lean4#2644
-  homEquiv_unit := by
-    -- Sad that aesop can no longer do this!
-    intros
-    dsimp
-    ext
-    simp
+  counit := { app := fun g => { app := limit.π _ } } }
 
 instance : IsRightAdjoint (lim : (J ⥤ C) ⥤ C) :=
   ⟨_, ⟨constLimAdj⟩⟩
@@ -546,7 +539,7 @@ def isLimitConeOfAdj (F : J ⥤ C) :
     have eq := NatTrans.congr_app (adj.counit.naturality s.π) j
     have eq' := NatTrans.congr_app (adj.left_triangle_components s.pt) j
     dsimp at eq eq' ⊢
-    rw [Adjunction.homEquiv_unit, assoc, eq, reassoc_of% eq']
+    rw [assoc, eq, reassoc_of% eq']
   uniq s m hm := (adj.homEquiv _ _).symm.injective (by ext j; simpa using hm j)
 
 end Adjunction
@@ -1029,15 +1022,15 @@ def colimCoyoneda : colim.op ⋙ coyoneda ⋙ (whiskeringRight _ _ _).obj uliftF
 
 /-- The colimit functor and constant functor are adjoint to each other
 -/
-def colimConstAdj : (colim : (J ⥤ C) ⥤ C) ⊣ const J where
-  homEquiv f c :=
+def colimConstAdj : (colim : (J ⥤ C) ⥤ C) ⊣ const J := Adjunction.mk' {
+  homEquiv := fun f c ↦
     { toFun := fun g =>
         { app := fun _ => colimit.ι _ _ ≫ g }
       invFun := fun g => colimit.desc _ ⟨_, g⟩
       left_inv := by aesop_cat
       right_inv := by aesop_cat }
   unit := { app := fun g => { app := colimit.ι _ } }
-  counit := { app := fun c => colimit.desc _ ⟨_, 𝟙 _⟩ }
+  counit := { app := fun c => colimit.desc _ ⟨_, 𝟙 _⟩ } }
 
 instance : IsLeftAdjoint (colim : (J ⥤ C) ⥤ C) :=
   ⟨_, ⟨colimConstAdj⟩⟩
diff --git a/Mathlib/CategoryTheory/Limits/IsConnected.lean b/Mathlib/CategoryTheory/Limits/IsConnected.lean
index 7dcd61474ff13..a0e0dd26b49f2 100644
--- a/Mathlib/CategoryTheory/Limits/IsConnected.lean
+++ b/Mathlib/CategoryTheory/Limits/IsConnected.lean
@@ -85,7 +85,7 @@ noncomputable def colimitConstPUnitIsoPUnit [IsConnected C] :
 /-- Let `F` be a `Type`-valued functor. If two elements `a : F c` and `b : F d` represent the same
 element of `colimit F`, then `c` and `d` are related by a `Zigzag`. -/
 theorem zigzag_of_eqvGen_quot_rel (F : C ⥤ Type w) (c d : Σ j, F.obj j)
-    (h : EqvGen (Quot.Rel F) c d) : Zigzag c.1 d.1 := by
+    (h : Relation.EqvGen (Quot.Rel F) c d) : Zigzag c.1 d.1 := by
   induction h with
   | rel _ _ h => exact Zigzag.of_hom <| Exists.choose h
   | refl _ => exact Zigzag.refl _
diff --git a/Mathlib/CategoryTheory/Limits/IsLimit.lean b/Mathlib/CategoryTheory/Limits/IsLimit.lean
index ff385b709488f..76802822f383d 100644
--- a/Mathlib/CategoryTheory/Limits/IsLimit.lean
+++ b/Mathlib/CategoryTheory/Limits/IsLimit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
+Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Adjunction.Basic
 import Mathlib.CategoryTheory.Limits.Cones
diff --git a/Mathlib/CategoryTheory/Limits/Lattice.lean b/Mathlib/CategoryTheory/Limits/Lattice.lean
index 882daf68cd30e..ccf7e6cf11624 100644
--- a/Mathlib/CategoryTheory/Limits/Lattice.lean
+++ b/Mathlib/CategoryTheory/Limits/Lattice.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Justus Springer
+Authors: Kim Morrison, Justus Springer
 -/
 import Mathlib.Order.CompleteLattice
 import Mathlib.Data.Finset.Lattice
diff --git a/Mathlib/CategoryTheory/Limits/Opposites.lean b/Mathlib/CategoryTheory/Limits/Opposites.lean
index fab462149abac..a988e6bc1335d 100644
--- a/Mathlib/CategoryTheory/Limits/Opposites.lean
+++ b/Mathlib/CategoryTheory/Limits/Opposites.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Floris van Doorn
+Authors: Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Limits.Filtered
 import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
@@ -322,11 +322,11 @@ instance : HasProduct (op <| Z ·) := hasLimitOfIso
     (Discrete.opposite α).inverse ⋙ (Discrete.functor Z).op ≅
     Discrete.functor (op <| Z ·))
 
-/-- A `Cofan` gives a `Fan` in the opposite category.  -/
+/-- A `Cofan` gives a `Fan` in the opposite category. -/
 @[simp]
 def Cofan.op (c : Cofan Z) : Fan (op <| Z ·) := Fan.mk _ (fun a ↦ (c.inj a).op)
 
-/-- If a `Cofan` is colimit, then its opposite is limit. -/
+/-- If a `Cofan` is colimit, then its opposite is limit. -/
 def Cofan.IsColimit.op {c : Cofan Z} (hc : IsColimit c) : IsLimit c.op := by
   let e : Discrete.functor (Opposite.op <| Z ·) ≅ (Discrete.opposite α).inverse ⋙
     (Discrete.functor Z).op := Discrete.natIso (fun _ ↦ Iso.refl _)
@@ -422,7 +422,7 @@ instance : HasCoproduct (op <| Z ·) := hasColimitOfIso
 @[simp]
 def Fan.op (f : Fan Z) : Cofan (op <| Z ·) := Cofan.mk _ (fun a ↦ (f.proj a).op)
 
-/-- If a `Fan` is limit, then its opposite is colimit. -/
+/-- If a `Fan` is limit, then its opposite is colimit. -/
 def Fan.IsLimit.op {f : Fan Z} (hf : IsLimit f) : IsColimit f.op := by
   let e : Discrete.functor (Opposite.op <| Z ·) ≅ (Discrete.opposite α).inverse ⋙
     (Discrete.functor Z).op := Discrete.natIso (fun _ ↦ Iso.refl _)
diff --git a/Mathlib/CategoryTheory/Limits/Pi.lean b/Mathlib/CategoryTheory/Limits/Pi.lean
index 605261fd6d113..eb68356542cfc 100644
--- a/Mathlib/CategoryTheory/Limits/Pi.lean
+++ b/Mathlib/CategoryTheory/Limits/Pi.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Pi.Basic
 import Mathlib.CategoryTheory.Limits.HasLimits
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Basic.lean b/Mathlib/CategoryTheory/Limits/Preserves/Basic.lean
index 803fcd12b6cc8..0926bbae01d71 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Reid Barton, Bhavik Mehta, Jakob von Raumer
+Authors: Kim Morrison, Reid Barton, Bhavik Mehta, Jakob von Raumer
 -/
 import Mathlib.CategoryTheory.Limits.HasLimits
 
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Filtered.lean b/Mathlib/CategoryTheory/Limits/Preserves/Filtered.lean
index 1eb21050d5910..932826e7f60e0 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Filtered.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Filtered.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Justus Springer
+Authors: Kim Morrison, Justus Springer
 -/
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
 import Mathlib.CategoryTheory.Filtered.Basic
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Finite.lean b/Mathlib/CategoryTheory/Limits/Preserves/Finite.lean
index 87c35359a609f..eea8e1113af03 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Finite.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Finite.lean
@@ -27,7 +27,7 @@ open CategoryTheory
 namespace CategoryTheory.Limits
 
 -- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
-universe w w₂ v₁ v₂ v₃ u₁ u₂ u₃
+universe u w w₂ v₁ v₂ v₃ u₁ u₂ u₃
 
 variable {C : Type u₁} [Category.{v₁} C]
 variable {D : Type u₂} [Category.{v₂} D]
@@ -84,6 +84,11 @@ def compPreservesFiniteLimits (F : C ⥤ D) (G : D ⥤ E) [PreservesFiniteLimits
     [PreservesFiniteLimits G] : PreservesFiniteLimits (F ⋙ G) :=
   ⟨fun _ _ _ => inferInstance⟩
 
+/-- Transfer preservation of finite limits along a natural isomorphism in the functor. -/
+def preservesFiniteLimitsOfNatIso {F G : C ⥤ D} (h : F ≅ G) [PreservesFiniteLimits F] :
+    PreservesFiniteLimits G where
+  preservesFiniteLimits _ _ _ := preservesLimitsOfShapeOfNatIso h
+
 /- Porting note: adding this class because quantified classes don't behave well
 [#2764](https://github.com/leanprover-community/mathlib4/pull/2764) -/
 /-- A functor `F` preserves finite products if it preserves all from `Discrete J`
@@ -93,6 +98,12 @@ class PreservesFiniteProducts (F : C ⥤ D) where
 
 attribute [instance] PreservesFiniteProducts.preserves
 
+noncomputable instance (priority := 100) (F : C ⥤ D) (J : Type u) [Finite J]
+    [PreservesFiniteProducts F] : PreservesLimitsOfShape (Discrete J) F := by
+  apply Nonempty.some
+  obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
+  exact ⟨preservesLimitsOfShapeOfEquiv (Discrete.equivalence e.symm) F⟩
+
 instance compPreservesFiniteProducts (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteProducts F] [PreservesFiniteProducts G] :
     PreservesFiniteProducts (F ⋙ G) where
@@ -225,6 +236,11 @@ def compPreservesFiniteColimits (F : C ⥤ D) (G : D ⥤ E) [PreservesFiniteColi
     [PreservesFiniteColimits G] : PreservesFiniteColimits (F ⋙ G) :=
   ⟨fun _ _ _ => inferInstance⟩
 
+/-- Transfer preservation of finite colimits along a natural isomorphism in the functor. -/
+def preservesFiniteColimitsOfNatIso {F G : C ⥤ D} (h : F ≅ G) [PreservesFiniteColimits F] :
+    PreservesFiniteColimits G where
+  preservesFiniteColimits _ _ _ := preservesColimitsOfShapeOfNatIso h
+
 /- Porting note: adding this class because quantified classes don't behave well
 [#2764](https://github.com/leanprover-community/mathlib4/pull/2764) -/
 /-- A functor `F` preserves finite products if it preserves all from `Discrete J`
@@ -233,14 +249,15 @@ class PreservesFiniteCoproducts (F : C ⥤ D) where
   /-- preservation of colimits indexed by `Discrete J` when `[Fintype J]` -/
   preserves : ∀ (J : Type) [Fintype J], PreservesColimitsOfShape (Discrete J) F
 
-noncomputable instance (F : C ⥤ D) (J : Type*) [Finite J] [PreservesFiniteCoproducts F] :
-    PreservesColimitsOfShape (Discrete J) F := by
+attribute [instance] PreservesFiniteCoproducts.preserves
+
+noncomputable instance (priority := 100) (F : C ⥤ D) (J : Type u) [Finite J]
+    [PreservesFiniteCoproducts F] : PreservesColimitsOfShape (Discrete J) F := by
   apply Nonempty.some
   obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin J
-  have : PreservesColimitsOfShape (Discrete (Fin n)) F := PreservesFiniteCoproducts.preserves _
   exact ⟨preservesColimitsOfShapeOfEquiv (Discrete.equivalence e.symm) F⟩
 
-noncomputable instance compPreservesFiniteCoproducts (F : C ⥤ D) (G : D ⥤ E)
+instance compPreservesFiniteCoproducts (F : C ⥤ D) (G : D ⥤ E)
     [PreservesFiniteCoproducts F] [PreservesFiniteCoproducts G] :
     PreservesFiniteCoproducts (F ⋙ G) where
   preserves _ _ := inferInstance
@@ -248,7 +265,6 @@ noncomputable instance compPreservesFiniteCoproducts (F : C ⥤ D) (G : D ⥤ E)
 noncomputable instance (F : C ⥤ D) [PreservesFiniteColimits F] : PreservesFiniteCoproducts F where
   preserves _ _ := inferInstance
 
-
 /--
 A functor is said to reflect finite colimits, if it reflects all colimits of shape `J`,
 where `J : Type` is a finite category.
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean b/Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean
index 4b4cde050f773..0d33dbbf72e5a 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/FunctorCategory.lean
@@ -5,6 +5,7 @@ Authors: Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
+import Mathlib.CategoryTheory.Limits.Preserves.Finite
 import Mathlib.CategoryTheory.Limits.Yoneda
 import Mathlib.CategoryTheory.Limits.Presheaf
 
@@ -104,4 +105,16 @@ noncomputable def preservesLimitOfLanPreservesLimit {C D : Type u} [SmallCategor
   apply @preservesLimitsOfShapeOfReflectsOfPreserves _ _ _ _ _ _ _ _ F yoneda ?_
   exact preservesLimitsOfShapeOfNatIso (Presheaf.compYonedaIsoYonedaCompLan F).symm
 
+/-- `F : C ⥤ D ⥤ E` preserves finite limits if it does for each `d : D`. -/
+def preservesFiniteLimitsOfEvaluation {D : Type*} [Category D] {E : Type*} [Category E]
+    (F : C ⥤ D ⥤ E) (h : ∀ d : D, PreservesFiniteLimits (F ⋙ (evaluation D E).obj d)) :
+    PreservesFiniteLimits F :=
+  ⟨fun J _ _ => preservesLimitsOfShapeOfEvaluation F J fun k => (h k).preservesFiniteLimits _⟩
+
+/-- `F : C ⥤ D ⥤ E` preserves finite limits if it does for each `d : D`. -/
+def preservesFiniteColimitsOfEvaluation {D : Type*} [Category D] {E : Type*} [Category E]
+    (F : C ⥤ D ⥤ E) (h : ∀ d : D, PreservesFiniteColimits (F ⋙ (evaluation D E).obj d)) :
+    PreservesFiniteColimits F :=
+  ⟨fun J _ _ => preservesColimitsOfShapeOfEvaluation F J fun k => (h k).preservesFiniteColimits _⟩
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Limits.lean b/Mathlib/CategoryTheory/Limits/Preserves/Limits.lean
index ad85619badaad..37599d523f5f6 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Limits.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
 
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean
index a0006f3d434ff..b72dd4a247aae 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean
@@ -88,13 +88,13 @@ def PreservesLimitPair.iso : G.obj (X ⨯ Y) ≅ G.obj X ⨯ G.obj Y :=
 theorem PreservesLimitPair.iso_hom : (PreservesLimitPair.iso G X Y).hom = prodComparison G X Y :=
   rfl
 
-@[simp]
+@[simp, reassoc]
 theorem PreservesLimitPair.iso_inv_fst :
     (PreservesLimitPair.iso G X Y).inv ≫ G.map prod.fst = prod.fst := by
   rw [← Iso.cancel_iso_hom_left (PreservesLimitPair.iso G X Y), ← Category.assoc, Iso.hom_inv_id]
   simp
 
-@[simp]
+@[simp, reassoc]
 theorem PreservesLimitPair.iso_inv_snd :
     (PreservesLimitPair.iso G X Y).inv ≫ G.map prod.snd = prod.snd := by
   rw [← Iso.cancel_iso_hom_left (PreservesLimitPair.iso G X Y), ← Category.assoc, Iso.hom_inv_id]
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
index c5f4f3bd95437..c3d4d473c6c16 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
@@ -224,7 +224,7 @@ theorem ι_biproductComparison' (j : J) :
 variable [PreservesZeroMorphisms F]
 
 /-- The composition in the opposite direction is equal to the identity if and only if `F` preserves
-    the biproduct, see `preservesBiproduct_of_monoBiproductComparison`.  -/
+    the biproduct, see `preservesBiproduct_of_monoBiproductComparison`. -/
 @[reassoc (attr := simp)]
 theorem biproductComparison'_comp_biproductComparison :
     biproductComparison' F f ≫ biproductComparison F f = 𝟙 (⨁ F.obj ∘ f) := by
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean
index e273955285c7a..b6333a08c0d86 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Kernels
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
@@ -176,7 +176,7 @@ def isColimitMapCoconeEquiv :
 
 /-- A colimit cokernel cofork is mapped to a colimit cokernel cofork by a functor `G`
 when this functor preserves the corresponding colimit. -/
-def mapIsColimit  (hc : IsColimit c) (G : C ⥤ D)
+def mapIsColimit (hc : IsColimit c) (G : C ⥤ D)
     [Functor.PreservesZeroMorphisms G] [PreservesColimit (parallelPair f 0) G] :
     IsColimit (c.map G) :=
   c.isColimitMapCoconeEquiv G (isColimitOfPreserves G hc)
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.lean b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.lean
index 4d54f0f91c9b6..9d1105f497bd6 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Shapes/Products.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison, Bhavik Mehta. All rights reserved.
+Copyright (c) 2020 Kim Morrison, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Products
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean b/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean
index 9bab8cb39bdb4..e11c09ddc138d 100644
--- a/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean
+++ b/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean
@@ -23,8 +23,8 @@ universe v w w' u
 namespace CategoryTheory.Limits.Types
 
 /--
-The equivalence between `K.sections` and `(K ⋙ uliftFunctor.{v, u}).sections`. This is used to show
-that `uliftFunctor` preserves limits that are potentially too large to exist in the source
+The equivalence between `K.sections` and `(K ⋙ uliftFunctor.{v, u}).sections`. This is used to show
+that `uliftFunctor` preserves limits that are potentially too large to exist in the source
 category.
 -/
 def sectionsEquiv {J : Type*} [Category J] (K : J ⥤ Type u) :
diff --git a/Mathlib/CategoryTheory/Limits/Presheaf.lean b/Mathlib/CategoryTheory/Limits/Presheaf.lean
index 659bf465adea6..d7b3844ba9d24 100644
--- a/Mathlib/CategoryTheory/Limits/Presheaf.lean
+++ b/Mathlib/CategoryTheory/Limits/Presheaf.lean
@@ -6,12 +6,8 @@ Authors: Bhavik Mehta, Joël Riou
 import Mathlib.CategoryTheory.Comma.Presheaf
 import Mathlib.CategoryTheory.Elements
 import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
-import Mathlib.CategoryTheory.Limits.ConeCategory
 import Mathlib.CategoryTheory.Limits.Final
-import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
 import Mathlib.CategoryTheory.Limits.Over
-import Mathlib.CategoryTheory.Limits.Shapes.Terminal
-import Mathlib.CategoryTheory.Limits.Types
 
 /-!
 # Colimit of representables
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean b/Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
index 160d8386ff9bc..4ea865a6b1d0d 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Comma.Over
 import Mathlib.CategoryTheory.DiscreteCategory
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean b/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
index 8be6740de1be5..c5ab7690aefbc 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Jakob von Raumer
+Authors: Kim Morrison, Jakob von Raumer
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
 import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
@@ -386,7 +386,7 @@ theorem hasBiproductsOfShape_of_equiv {K : Type w'} [HasBiproductsOfShape K C] (
     let ⟨⟨h⟩⟩ := HasBiproductsOfShape.has_biproduct (F ∘ e.symm)
     let ⟨c, hc⟩ := h
     HasBiproduct.mk <| by
-      simpa only [(· ∘ ·), e.symm_apply_apply] using
+      simpa only [Function.comp_def, e.symm_apply_apply] using
         LimitBicone.mk (c.whisker e) ((c.whiskerIsBilimitIff _).2 hc)⟩
 
 instance (priority := 100) hasBiproductsOfShape_finite [HasFiniteBiproducts C] [Finite J] :
@@ -403,6 +403,14 @@ instance (priority := 100) hasFiniteCoproducts_of_hasFiniteBiproducts [HasFinite
     HasFiniteCoproducts C where
   out _ := ⟨fun _ => hasColimitOfIso Discrete.natIsoFunctor⟩
 
+instance (priority := 100) hasProductsOfShape_of_hasBiproductsOfShape [HasBiproductsOfShape J C] :
+    HasProductsOfShape J C where
+  has_limit _ := hasLimitOfIso Discrete.natIsoFunctor.symm
+
+instance (priority := 100) hasCoproductsOfShape_of_hasBiproductsOfShape [HasBiproductsOfShape J C] :
+    HasCoproductsOfShape J C where
+  has_colimit _ := hasColimitOfIso Discrete.natIsoFunctor
+
 variable {C}
 
 /-- The isomorphism between the specified limit and the specified colimit for
@@ -547,6 +555,17 @@ theorem biproduct.isoCoproduct_hom {f : J → C} [HasBiproduct f] :
     (biproduct.isoCoproduct f).hom = biproduct.desc (Sigma.ι f) :=
   biproduct.hom_ext' _ _ fun j => by simp [← Iso.eq_comp_inv]
 
+/-- If a category has biproducts of a shape `J`, its `colim` and `lim` functor on diagrams over `J`
+are isomorphic. -/
+@[simps!]
+def HasBiproductsOfShape.colimIsoLim [HasBiproductsOfShape J C] :
+    colim (J := Discrete J) (C := C) ≅ lim :=
+  NatIso.ofComponents (fun F => (Sigma.isoColimit F).symm ≪≫
+      (biproduct.isoCoproduct _).symm ≪≫ biproduct.isoProduct _ ≪≫ Pi.isoLimit F)
+    fun η => colimit.hom_ext fun ⟨i⟩ => limit.hom_ext fun ⟨j⟩ => by
+      by_cases h : i = j <;>
+       simp_all [h, Sigma.isoColimit, Pi.isoLimit, biproduct.ι_π, biproduct.ι_π_assoc]
+
 theorem biproduct.map_eq_map' {f g : J → C} [HasBiproduct f] [HasBiproduct g] (p : ∀ b, f b ⟶ g b) :
     biproduct.map p = biproduct.map' p := by
   ext
@@ -1276,7 +1295,7 @@ abbrev toBinaryBicone {X Y : C} (b : Bicone (pairFunction X Y)) : BinaryBicone X
   toBinaryBiconeFunctor.obj b
 
 /-- A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit
-    cone.  -/
+    cone. -/
 def toBinaryBiconeIsLimit {X Y : C} (b : Bicone (pairFunction X Y)) :
     IsLimit b.toBinaryBicone.toCone ≃ IsLimit b.toCone :=
   IsLimit.equivIsoLimit <| Cones.ext (Iso.refl _) fun j => by rcases j with ⟨⟨⟩⟩ <;> simp
@@ -1762,7 +1781,7 @@ def biprod.fstKernelFork : KernelFork (biprod.fst : X ⊞ Y ⟶ X) :=
 theorem biprod.fstKernelFork_ι : Fork.ι (biprod.fstKernelFork X Y) = (biprod.inr : Y ⟶ X ⊞ Y) :=
   rfl
 
-/-- The fork `biprod.fstKernelFork` is indeed a limit.  -/
+/-- The fork `biprod.fstKernelFork` is indeed a limit. -/
 def biprod.isKernelFstKernelFork : IsLimit (biprod.fstKernelFork X Y) :=
   BinaryBicone.isLimitFstKernelFork (BinaryBiproduct.isLimit _ _)
 
@@ -1775,7 +1794,7 @@ def biprod.sndKernelFork : KernelFork (biprod.snd : X ⊞ Y ⟶ Y) :=
 theorem biprod.sndKernelFork_ι : Fork.ι (biprod.sndKernelFork X Y) = (biprod.inl : X ⟶ X ⊞ Y) :=
   rfl
 
-/-- The fork `biprod.sndKernelFork` is indeed a limit.  -/
+/-- The fork `biprod.sndKernelFork` is indeed a limit. -/
 def biprod.isKernelSndKernelFork : IsLimit (biprod.sndKernelFork X Y) :=
   BinaryBicone.isLimitSndKernelFork (BinaryBiproduct.isLimit _ _)
 
@@ -1788,7 +1807,7 @@ def biprod.inlCokernelCofork : CokernelCofork (biprod.inl : X ⟶ X ⊞ Y) :=
 theorem biprod.inlCokernelCofork_π : Cofork.π (biprod.inlCokernelCofork X Y) = biprod.snd :=
   rfl
 
-/-- The cofork `biprod.inlCokernelFork` is indeed a colimit.  -/
+/-- The cofork `biprod.inlCokernelFork` is indeed a colimit. -/
 def biprod.isCokernelInlCokernelFork : IsColimit (biprod.inlCokernelCofork X Y) :=
   BinaryBicone.isColimitInlCokernelCofork (BinaryBiproduct.isColimit _ _)
 
@@ -1801,7 +1820,7 @@ def biprod.inrCokernelCofork : CokernelCofork (biprod.inr : Y ⟶ X ⊞ Y) :=
 theorem biprod.inrCokernelCofork_π : Cofork.π (biprod.inrCokernelCofork X Y) = biprod.fst :=
   rfl
 
-/-- The cofork `biprod.inrCokernelFork` is indeed a colimit.  -/
+/-- The cofork `biprod.inrCokernelFork` is indeed a colimit. -/
 def biprod.isCokernelInrCokernelFork : IsColimit (biprod.inrCokernelCofork X Y) :=
   BinaryBicone.isColimitInrCokernelCofork (BinaryBiproduct.isColimit _ _)
 
@@ -1997,3 +2016,5 @@ theorem isIso_right_of_isIso_biprod_map {W X Y Z : C} (f : W ⟶ Y) (g : X ⟶ Z
   isIso_left_of_isIso_biprod_map g f
 
 end CategoryTheory
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/CombinedProducts.lean b/Mathlib/CategoryTheory/Limits/Shapes/CombinedProducts.lean
index 1d6c74aed120e..4563605f05c24 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/CombinedProducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/CombinedProducts.lean
@@ -41,7 +41,7 @@ abbrev combPairHoms : (i : ι₁ ⊕ ι₂) → bc.pt ⟶ Sum.elim f₁ f₂ i
 variable {c₁ c₂ bc}
 
 /-- If `c₁` and `c₂` are limit fans and `bc` is a limit binary fan on their cone
-points, then the fan constructed from `combPairHoms` is a limit cone.  -/
+points, then the fan constructed from `combPairHoms` is a limit cone. -/
 def combPairIsLimit : IsLimit (Fan.mk bc.pt (combPairHoms c₁ c₂ bc)) :=
   mkFanLimit _
     (fun s ↦ Fan.IsLimit.desc h <| fun i ↦ by
@@ -76,7 +76,7 @@ abbrev combPairHoms : (i : ι₁ ⊕ ι₂) → Sum.elim f₁ f₂ i ⟶ bc.pt
 variable {c₁ c₂ bc}
 
 /-- If `c₁` and `c₂` are colimit cofans and `bc` is a colimit binary cofan on their cocone
-points, then the cofan constructed from `combPairHoms` is a colimit cocone.  -/
+points, then the cofan constructed from `combPairHoms` is a colimit cocone. -/
 def combPairIsColimit : IsColimit (Cofan.mk bc.pt (combPairHoms c₁ c₂ bc)) :=
   mkCofanColimit _
     (fun s ↦ Cofan.IsColimit.desc h <| fun i ↦ by
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean b/Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
index d1f39b75f730a..3388588964ed5 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Joël Riou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joël Riou, Scott Morrison, Adam Topaz
+Authors: Joël Riou, Kim Morrison, Adam Topaz
 -/
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean b/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
index bd6d44de0f55a..a50342b46d933 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
@@ -17,7 +17,7 @@ limits, see `sequentialFunctor_initial`.
 
 ## Projects
 
-* There is a series of `proof_wanted` at the bottom of this file, implying that all cofiltered
+* There is a series of `proof_wanted` at the bottom of this file, implying that all cofiltered
   limits over countable categories are isomorphic to sequential limits.
 
 * Prove the dual result for filtered colimits.
@@ -36,7 +36,7 @@ instance and `J : Type` has a limit.
 -/
 class HasCountableLimits : Prop where
   /-- `C` has all limits over any type `J` whose objects and morphisms lie in the same universe
-  and which has countably many objects and morphisms-/
+  and which has countably many objects and morphisms -/
   out (J : Type) [SmallCategory J] [CountableCategory J] : HasLimitsOfShape J C
 
 instance (priority := 100) hasFiniteLimits_of_hasCountableLimits [HasCountableLimits C] :
@@ -58,7 +58,7 @@ instance and `J : Type` has a colimit.
 -/
 class HasCountableColimits : Prop where
   /-- `C` has all limits over any type `J` whose objects and morphisms lie in the same universe
-  and which has countably many objects and morphisms-/
+  and which has countably many objects and morphisms -/
   out (J : Type) [SmallCategory J] [CountableCategory J] : HasColimitsOfShape J C
 
 instance (priority := 100) hasFiniteColimits_of_hasCountableColimits [HasCountableColimits C] :
@@ -102,7 +102,7 @@ noncomputable def sequentialFunctor : ℕᵒᵖ ⥤ J where
 theorem sequentialFunctor_initial_aux (j : J) : ∃ (n : ℕ), sequentialFunctor_obj J n ≤ j := by
   obtain ⟨m, h⟩ := (exists_surjective_nat _).choose_spec j
   refine ⟨m + 1, ?_⟩
-  simpa [h] using leOfHom (IsCofilteredOrEmpty.cone_objs ((exists_surjective_nat _).choose m)
+  simpa only [h] using leOfHom (IsCofilteredOrEmpty.cone_objs ((exists_surjective_nat _).choose m)
     (sequentialFunctor_obj J m)).choose_spec.choose
 
 instance sequentialFunctor_initial : (sequentialFunctor J).Initial where
@@ -113,7 +113,7 @@ instance sequentialFunctor_initial : (sequentialFunctor J).Initial where
     apply isConnected_of_zigzag
     refine fun i j ↦ ⟨[j], ?_⟩
     simp only [List.chain_cons, Zag, List.Chain.nil, and_true, ne_eq, not_false_eq_true,
-      List.getLast_cons, not_true_eq_false, List.getLast_singleton']
+      List.getLast_cons, not_true_eq_false, List.getLast_singleton', reduceCtorEq]
     clear! C
     wlog h : (unop i.left) ≤ (unop j.left)
     · exact or_comm.1 (this J d n g inferInstance j i (le_of_lt (not_le.mp h)))
@@ -127,7 +127,7 @@ proof_wanted preorder_of_cofiltered (J : Type*) [Category J] [IsCofiltered J] :
     ∃ (I : Type*) (_ : Preorder I) (_ : IsCofiltered I) (F : I ⥤ J), F.Initial
 
 /--
-The proof of `preorder_of_cofiltered` should give a countable `I` in the case that `J` is a
+The proof of `preorder_of_cofiltered` should give a countable `I` in the case that `J` is a
 countable category.
 -/
 proof_wanted preorder_of_cofiltered_countable
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean b/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
index bb067d0c9eef4..05a92677fd909 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel
+Authors: Kim Morrison, Markus Himmel
 -/
 import Mathlib.CategoryTheory.EpiMono
 import Mathlib.CategoryTheory.Limits.HasLimits
@@ -925,7 +925,7 @@ variable {f g}
 def idCofork (h : f = g) : Cofork f g :=
   Cofork.ofπ (𝟙 Y) <| h ▸ rfl
 
-/-- The identity on `Y` is a coequalizer of `(f, g)`, where `f = g`.  -/
+/-- The identity on `Y` is a coequalizer of `(f, g)`, where `f = g`. -/
 def isColimitIdCofork (h : f = g) : IsColimit (idCofork h) :=
   Cofork.IsColimit.mk _ (fun s => Cofork.π s) (fun s => Category.id_comp _) fun s m h => by
     convert h
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean b/Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean
index 154159c993a8c..812d28171a0d8 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/FiniteLimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.FinCategory.AsType
 import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
@@ -36,7 +36,7 @@ This is often called 'finitely complete'.
 -/
 class HasFiniteLimits : Prop where
   /-- `C` has all limits over any type `J` whose objects and morphisms lie in the same universe
-  and which has `FinType` objects and morphisms-/
+  and which has `FinType` objects and morphisms -/
   out (J : Type) [𝒥 : SmallCategory J] [@FinCategory J 𝒥] : @HasLimitsOfShape J 𝒥 C _
 
 instance (priority := 100) hasLimitsOfShape_of_hasFiniteLimits (J : Type w) [SmallCategory J]
@@ -85,7 +85,7 @@ This is often called 'finitely cocomplete'.
 -/
 class HasFiniteColimits : Prop where
   /-- `C` has all colimits over any type `J` whose objects and morphisms lie in the same universe
-  and which has `Fintype` objects and morphisms-/
+  and which has `Fintype` objects and morphisms -/
   out (J : Type) [𝒥 : SmallCategory J] [@FinCategory J 𝒥] : @HasColimitsOfShape J 𝒥 C _
 
 instance (priority := 100) hasColimitsOfShape_of_hasFiniteColimits (J : Type w) [SmallCategory J]
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean b/Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean
index a8fc25c47121c..76f92fb28042f 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/FiniteProducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
 import Mathlib.CategoryTheory.Limits.Shapes.Products
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/FunctorCategory.lean b/Mathlib/CategoryTheory/Limits/Shapes/FunctorCategory.lean
index 113f2c55cc2c3..0303e8d18d601 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
 import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Images.lean b/Mathlib/CategoryTheory/Limits/Shapes/Images.lean
index b403910cf1751..10e2d68009de0 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Images.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Images.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel
+Authors: Kim Morrison, Markus Himmel
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
 import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
@@ -81,8 +81,6 @@ attribute [reassoc (attr := simp)] MonoFactorisation.fac
 
 attribute [instance] MonoFactorisation.m_mono
 
-attribute [instance] MonoFactorisation.m_mono
-
 namespace MonoFactorisation
 
 /-- The obvious factorisation of a monomorphism through itself. -/
@@ -391,7 +389,7 @@ theorem image.ext [HasImage f] {W : C} {g h : image f ⟶ W} [HasLimit (parallel
   let F' : MonoFactorisation f :=
     { I := equalizer g h
       m := q ≫ image.ι f
-      m_mono := by apply mono_comp
+      m_mono := mono_comp _ _
       e := e' }
   let v := image.lift F'
   have t₀ : v ≫ q ≫ image.ι f = image.ι f := image.lift_fac F'
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean b/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
index ec4bee1404520..c05a2ec92eaab 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel
+Authors: Kim Morrison, Markus Himmel
 -/
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
 
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean b/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean
index 93dfec509e2d4..e16617bd21fec 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.RegularMono
 import Mathlib.CategoryTheory.Limits.Shapes.Kernels
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Products.lean b/Mathlib/CategoryTheory/Limits/Shapes/Products.lean
index 0821b11254338..aa816cb1ee4da 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Products.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Products.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.HasLimits
 import Mathlib.CategoryTheory.DiscreteCategory
@@ -258,7 +258,7 @@ instance {f : β → C} [HasCoproduct f] : IsIso (Sigma.desc (fun a ↦ Sigma.ι
   ext
   simp
 
-/-- A version of `Cocones.ext` for `Cofan`s. -/
+/-- A version of `Cocones.ext` for `Cofan`s. -/
 @[simps!]
 def Cofan.ext {f : β → C} {c₁ c₂ : Cofan f} (e : c₁.pt ≅ c₂.pt)
     (w : ∀ (b : β), c₁.inj b ≫ e.hom = c₂.inj b := by aesop_cat) : c₁ ≅ c₂ :=
@@ -481,6 +481,50 @@ from a family of isomorphisms between the factors.
 abbrev Sigma.mapIso {f g : β → C} [HasCoproductsOfShape β C] (p : ∀ b, f b ≅ g b) : ∐ f ≅ ∐ g :=
   colim.mapIso (Discrete.natIso fun X => p X.as)
 
+section
+
+/- In this section, we provide some API for coproducts when we are given a functor
+`Discrete α ⥤ C` instead of a map `α → C`. -/
+
+variable (X : Discrete α ⥤ C) [HasCoproduct (fun j => X.obj (Discrete.mk j))]
+
+/-- A colimit cocone for `X : Discrete α ⥤ C` that is given
+by `∐ (fun j => X.obj (Discrete.mk j))`. -/
+@[simps]
+def Sigma.cocone : Cocone X where
+  pt := ∐ (fun j => X.obj (Discrete.mk j))
+  ι := Discrete.natTrans (fun _ => Sigma.ι (fun j ↦ X.obj ⟨j⟩) _)
+
+/-- The cocone `Sigma.cocone X` is a colimit cocone. -/
+def coproductIsCoproduct' :
+    IsColimit (Sigma.cocone X) where
+  desc s := Sigma.desc (fun j => s.ι.app ⟨j⟩)
+  fac s := by simp
+  uniq s m hm := by
+    dsimp
+    ext
+    simp only [colimit.ι_desc, Cofan.mk_pt, Cofan.mk_ι_app]
+    apply hm
+
+variable [HasColimit X]
+
+/-- The isomorphism `∐ (fun j => X.obj (Discrete.mk j)) ≅ colimit X`. -/
+def Sigma.isoColimit :
+    ∐ (fun j => X.obj (Discrete.mk j)) ≅ colimit X :=
+  IsColimit.coconePointUniqueUpToIso (coproductIsCoproduct' X) (colimit.isColimit X)
+
+@[reassoc (attr := simp)]
+lemma Sigma.ι_isoColimit_hom (j : α) :
+    Sigma.ι _ j ≫ (Sigma.isoColimit X).hom = colimit.ι _ (Discrete.mk j) :=
+  IsColimit.comp_coconePointUniqueUpToIso_hom (coproductIsCoproduct' X) _ _
+
+@[reassoc (attr := simp)]
+lemma Sigma.ι_isoColimit_inv (j : α) :
+    colimit.ι _ ⟨j⟩ ≫ (Sigma.isoColimit X).inv = Sigma.ι (fun j ↦ X.obj ⟨j⟩) _ :=
+  IsColimit.comp_coconePointUniqueUpToIso_inv _ _ _
+
+end
+
 /-- Two products which differ by an equivalence in the indexing type,
 and up to isomorphism in the factors, are isomorphic.
 -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean
index b6008f36b9768..541f09bc26155 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Assoc.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
index 665866bf2883a..423fed53f4080 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou, Calle Sönne
+Authors: Kim Morrison, Joël Riou, Calle Sönne
 -/
 import Mathlib.CategoryTheory.CommSq
 import Mathlib.CategoryTheory.Limits.Opposites
@@ -280,22 +280,22 @@ noncomputable def isoIsPullback (h : IsPullback fst snd f g) (h' : IsPullback fs
 
 @[reassoc (attr := simp)]
 theorem isoIsPullback_hom_fst (h : IsPullback fst snd f g) (h' : IsPullback fst' snd' f g) :
-    (h.isoIsPullback h').hom ≫ fst' = fst :=
+    (h.isoIsPullback _ _ h').hom ≫ fst' = fst :=
   IsLimit.conePointUniqueUpToIso_hom_comp h.isLimit h'.isLimit WalkingCospan.left
 
 @[reassoc (attr := simp)]
 theorem isoIsPullback_hom_snd (h : IsPullback fst snd f g) (h' : IsPullback fst' snd' f g) :
-    (h.isoIsPullback h').hom ≫ snd' = snd :=
+    (h.isoIsPullback _ _ h').hom ≫ snd' = snd :=
   IsLimit.conePointUniqueUpToIso_hom_comp h.isLimit h'.isLimit WalkingCospan.right
 
 @[reassoc (attr := simp)]
 theorem isoIsPullback_inv_fst (h : IsPullback fst snd f g) (h' : IsPullback fst' snd' f g) :
-    (h.isoIsPullback h').inv ≫ fst = fst' := by
+    (h.isoIsPullback _ _ h').inv ≫ fst = fst' := by
   simp only [Iso.inv_comp_eq, isoIsPullback_hom_fst]
 
 @[reassoc (attr := simp)]
 theorem isoIsPullback_inv_snd (h : IsPullback fst snd f g) (h' : IsPullback fst' snd' f g) :
-    (h.isoIsPullback h').inv ≫ snd = snd' := by
+    (h.isoIsPullback _ _ h').inv ≫ snd = snd' := by
   simp only [Iso.inv_comp_eq, isoIsPullback_hom_snd]
 
 end
@@ -468,22 +468,22 @@ noncomputable def isoIsPushout (h : IsPushout f g inl inr) (h' : IsPushout f g i
 
 @[reassoc (attr := simp)]
 theorem inl_isoIsPushout_hom (h : IsPushout f g inl inr) (h' : IsPushout f g inl' inr') :
-    inl ≫ (h.isoIsPushout h').hom = inl' :=
+    inl ≫ (h.isoIsPushout _ _ h').hom = inl' :=
   IsColimit.comp_coconePointUniqueUpToIso_hom h.isColimit h'.isColimit WalkingSpan.left
 
 @[reassoc (attr := simp)]
 theorem inr_isoIsPushout_hom (h : IsPushout f g inl inr) (h' : IsPushout f g inl' inr') :
-    inr ≫ (h.isoIsPushout h').hom = inr' :=
+    inr ≫ (h.isoIsPushout _ _ h').hom = inr' :=
   IsColimit.comp_coconePointUniqueUpToIso_hom h.isColimit h'.isColimit WalkingSpan.right
 
 @[reassoc (attr := simp)]
 theorem inl_isoIsPushout_inv (h : IsPushout f g inl inr) (h' : IsPushout f g inl' inr') :
-    inl' ≫ (h.isoIsPushout h').inv = inl := by
+    inl' ≫ (h.isoIsPushout _ _ h').inv = inl := by
   simp only [Iso.comp_inv_eq, inl_isoIsPushout_hom]
 
 @[reassoc (attr := simp)]
 theorem inr_isoIsPushout_inv (h : IsPushout f g inl inr) (h' : IsPushout f g inl' inr') :
-    inr' ≫ (h.isoIsPushout h').inv = inr := by
+    inr' ≫ (h.isoIsPushout _ _ h').inv = inr := by
   simp only [Iso.comp_inv_eq, inr_isoIsPushout_hom]
 
 end
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean
index dbd52eb783c0c..03f0894ee74ee 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Cospan.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel, Bhavik Mehta
+Authors: Kim Morrison, Markus Himmel, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
 import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean
index d832ed2573d16..f6455c517bc0b 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/HasPullback.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl, Calle Sönne
+Authors: Kim Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl, Calle Sönne
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone
 
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Iso.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Iso.lean
index 0aec735dd25af..3007f243e923c 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Iso.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Iso.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
index 17d8d712cac9a..5393b03bb61a3 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta, Andrew Yang
 -/
@@ -79,7 +79,7 @@ theorem mono_of_isLimitMkIdId (f : X ⟶ Y) (t : IsLimit (mk (𝟙 X) (𝟙 X) r
 /-- Suppose `f` and `g` are two morphisms with a common codomain and `s` is a limit cone over the
     diagram formed by `f` and `g`. Suppose `f` and `g` both factor through a monomorphism `h` via
     `x` and `y`, respectively.  Then `s` is also a limit cone over the diagram formed by `x` and
-    `y`.  -/
+    `y`. -/
 def isLimitOfFactors (f : X ⟶ Z) (g : Y ⟶ Z) (h : W ⟶ Z) [Mono h] (x : X ⟶ W) (y : Y ⟶ W)
     (hxh : x ≫ h = f) (hyh : y ≫ h = g) (s : PullbackCone f g) (hs : IsLimit s) :
     IsLimit
@@ -250,7 +250,7 @@ theorem epi_of_isColimitMkIdId (f : X ⟶ Y)
 /-- Suppose `f` and `g` are two morphisms with a common domain and `s` is a colimit cocone over the
     diagram formed by `f` and `g`. Suppose `f` and `g` both factor through an epimorphism `h` via
     `x` and `y`, respectively. Then `s` is also a colimit cocone over the diagram formed by `x` and
-    `y`.  -/
+    `y`. -/
 def isColimitOfFactors (f : X ⟶ Y) (g : X ⟶ Z) (h : X ⟶ W) [Epi h] (x : W ⟶ Y) (y : W ⟶ Z)
     (hhx : h ≫ x = f) (hhy : h ≫ y = g) (s : PushoutCocone f g) (hs : IsColimit s) :
     have reassoc₁ : h ≫ x ≫ inl s = f ≫ inl s := by  -- Porting note: working around reassoc
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean
index ddf8657ecc188..6790398f308af 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Pasting.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang, Calle Sönne
 -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean
index 5c57de436f965..0c26846165918 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/PullbackCone.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl, Calle Sönne
+Authors: Kim Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl, Calle Sönne
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Cospan
 
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean b/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean
index 36ef641c0374a..c568df46e30c4 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
 import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean b/Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean
index 5168baa185fa4..e501871d0f757 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/SingleObj.lean
@@ -72,7 +72,7 @@ section Colimits
 variable {G : Type v} [Group G] (J : SingleObj G ⥤ Type u)
 
 /-- The relation used to construct colimits in types for `J : SingleObj G ⥤ Type u` is
-equivalent to the `MulAction.orbitRel` equivalence relation on `J.obj  (SingleObj.star G)`. -/
+equivalent to the `MulAction.orbitRel` equivalence relation on `J.obj (SingleObj.star G)`. -/
 lemma Types.Quot.Rel.iff_orbitRel (x y : J.obj (SingleObj.star G)) :
     Types.Quot.Rel J ⟨SingleObj.star G, x⟩ ⟨SingleObj.star G, y⟩
     ↔ Setoid.Rel (MulAction.orbitRel G (J.obj (SingleObj.star G))) x y := by
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean b/Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
index 413d4b2416447..ba11c501a2ea9 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
@@ -136,8 +136,7 @@ theorem StrongEpi.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
     (e : Arrow.mk f ≅ Arrow.mk g) [h : StrongEpi f] : StrongEpi g :=
   { epi := by
       rw [Arrow.iso_w' e]
-      haveI := epi_comp f e.hom.right
-      apply epi_comp
+      infer_instance
     llp := fun {X Y} z => by
       intro
       apply HasLiftingProperty.of_arrow_iso_left e z }
@@ -146,8 +145,7 @@ theorem StrongMono.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
     (e : Arrow.mk f ≅ Arrow.mk g) [h : StrongMono f] : StrongMono g :=
   { mono := by
       rw [Arrow.iso_w' e]
-      haveI := mono_comp f e.hom.right
-      apply mono_comp
+      infer_instance
     rlp := fun {X Y} z => by
       intro
       apply HasLiftingProperty.of_arrow_iso_right z e }
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean b/Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
index c9f8914c57de7..37a4e622da924 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.PEmpty
 import Mathlib.CategoryTheory.Limits.HasLimits
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Types.lean b/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
index ec3762f3af920..5f9021a6e2f50 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Types
 import Mathlib.CategoryTheory.Limits.Shapes.Products
@@ -553,7 +553,7 @@ theorem coequalizer_preimage_image_eq_of_preimage_eq (π : Y ⟶ Z) (e : f ≫ 
       (mono_iff_injective
             (h.coconePointUniqueUpToIso (coequalizerColimit f g).isColimit).inv).mp
         inferInstance e'
-    exact (eqv.eqvGen_iff.mp (EqvGen.mono lem (Quot.exact _ e'))).mp hy
+    exact (eqv.eqvGen_iff.mp (Relation.EqvGen.mono lem (Quot.eqvGen_exact e'))).mp hy
   · exact fun hx => ⟨_, hx, rfl⟩
 
 /-- The categorical coequalizer in `Type u` is the quotient by `f g ~ g x`. -/
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean b/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
index c39731729d5e1..6193cbb5c6587 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
@@ -138,7 +138,7 @@ def diagramIsoParallelFamily (F : WalkingParallelFamily J ⥤ C) :
     rintro _ _ (_|_) <;> aesop_cat
 
 /-- `WalkingParallelPair` as a category is equivalent to a special case of
-`WalkingParallelFamily`.  -/
+`WalkingParallelFamily`. -/
 @[simps!]
 def walkingParallelFamilyEquivWalkingParallelPair :
     WalkingParallelFamily.{w} (ULift Bool) ≌ WalkingParallelPair where
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean b/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
index 6aa46f0c5a128..259375ddca69a 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Pi.Basic
 import Mathlib.CategoryTheory.Limits.Shapes.Products
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean b/Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean
index d6dd788599bbf..54f3a9e56b996 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/ZeroObjects.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin
+Authors: Kim Morrison, Johan Commelin
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Terminal
 
diff --git a/Mathlib/CategoryTheory/Limits/Types.lean b/Mathlib/CategoryTheory/Limits/Types.lean
index 1b69256ba80da..cfa274313257e 100644
--- a/Mathlib/CategoryTheory/Limits/Types.lean
+++ b/Mathlib/CategoryTheory/Limits/Types.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Reid Barton
+Authors: Kim Morrison, Reid Barton
 -/
 import Mathlib.Data.TypeMax
 import Mathlib.Logic.UnivLE
@@ -548,7 +548,7 @@ theorem colimit_sound' {j j' : J} {x : F.obj j} {x' : F.obj j'} {j'' : J}
 variable {F} in
 theorem colimit_eq {j j' : J} {x : F.obj j} {x' : F.obj j'}
     (w : colimit.ι F j x = colimit.ι F j' x') :
-      EqvGen (Quot.Rel F) ⟨j, x⟩ ⟨j', x'⟩ := by
+      Relation.EqvGen (Quot.Rel F) ⟨j, x⟩ ⟨j', x'⟩ := by
   apply Quot.eq.1
   simpa using congr_arg (colimitEquivQuot F) w
 
diff --git a/Mathlib/CategoryTheory/Limits/TypesFiltered.lean b/Mathlib/CategoryTheory/Limits/TypesFiltered.lean
index 48b2c53fe834a..05dec6451b29c 100644
--- a/Mathlib/CategoryTheory/Limits/TypesFiltered.lean
+++ b/Mathlib/CategoryTheory/Limits/TypesFiltered.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Reid Barton
+Authors: Kim Morrison, Reid Barton
 -/
 import Mathlib.CategoryTheory.Limits.Types
 import Mathlib.CategoryTheory.Filtered.Basic
@@ -27,7 +27,7 @@ attribute [local instance] small_quot_of_hasColimit
 
 /- For filtered colimits of types, we can give an explicit description
   of the equivalence relation generated by the relation used to form
-  the colimit.  -/
+  the colimit. -/
 
 /-- An alternative relation on `Σ j, F.obj j`,
 which generates the same equivalence relation as we use to define the colimit in `Type` above,
@@ -44,10 +44,10 @@ theorem rel_of_quot_rel (x y : Σ j, F.obj j) :
   fun ⟨f, h⟩ => ⟨y.1, f, 𝟙 y.1, by rw [← h, FunctorToTypes.map_id_apply]⟩
 
 theorem eqvGen_quot_rel_of_rel (x y : Σ j, F.obj j) :
-    FilteredColimit.Rel.{v, u} F x y → EqvGen (Quot.Rel F) x y := fun ⟨k, f, g, h⟩ => by
-  refine EqvGen.trans _ ⟨k, F.map f x.2⟩ _ ?_ ?_
-  · exact (EqvGen.rel _ _ ⟨f, rfl⟩)
-  · exact (EqvGen.symm _ _ (EqvGen.rel _ _ ⟨g, h⟩))
+    FilteredColimit.Rel.{v, u} F x y → Relation.EqvGen (Quot.Rel F) x y := fun ⟨k, f, g, h⟩ => by
+  refine Relation.EqvGen.trans _ ⟨k, F.map f x.2⟩ _ ?_ ?_
+  · exact (Relation.EqvGen.rel _ _ ⟨f, rfl⟩)
+  · exact (Relation.EqvGen.symm _ _ (Relation.EqvGen.rel _ _ ⟨g, h⟩))
 
 /-- Recognizing filtered colimits of types. -/
 noncomputable def isColimitOf (t : Cocone F) (hsurj : ∀ x : t.pt, ∃ i xi, x = t.ι.app i xi)
@@ -92,12 +92,12 @@ protected theorem rel_equiv : _root_.Equivalence (FilteredColimit.Rel.{v, u} F)
         _ = F.map (g' ≫ gl ≫ n) z.2 := by simp⟩
 
 protected theorem rel_eq_eqvGen_quot_rel :
-    FilteredColimit.Rel.{v, u} F = EqvGen (Quot.Rel F) := by
+    FilteredColimit.Rel.{v, u} F = Relation.EqvGen (Quot.Rel F) := by
   ext ⟨j, x⟩ ⟨j', y⟩
   constructor
   · apply eqvGen_quot_rel_of_rel
   · rw [← (FilteredColimit.rel_equiv F).eqvGen_iff]
-    exact EqvGen.mono (rel_of_quot_rel F)
+    exact Relation.EqvGen.mono (rel_of_quot_rel F)
 
 variable [HasColimit F]
 
diff --git a/Mathlib/CategoryTheory/Limits/Unit.lean b/Mathlib/CategoryTheory/Limits/Unit.lean
index 7b572e4455d15..21e52a45493b7 100644
--- a/Mathlib/CategoryTheory/Limits/Unit.lean
+++ b/Mathlib/CategoryTheory/Limits/Unit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.PUnit
 import Mathlib.CategoryTheory.Limits.HasLimits
diff --git a/Mathlib/CategoryTheory/Limits/Yoneda.lean b/Mathlib/CategoryTheory/Limits/Yoneda.lean
index 8f2956bb7a8ef..2442748a1c2e5 100644
--- a/Mathlib/CategoryTheory/Limits/Yoneda.lean
+++ b/Mathlib/CategoryTheory/Limits/Yoneda.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
 import Mathlib.CategoryTheory.Limits.Types
@@ -193,7 +193,7 @@ namespace Functor
 
 section Representable
 
-variable (F : Cᵒᵖ ⥤ Type v) [F.Representable] {J : Type*} [Category J]
+variable (F : Cᵒᵖ ⥤ Type v) [F.IsRepresentable] {J : Type*} [Category J]
 
 noncomputable instance representablePreservesLimit (G : J ⥤ Cᵒᵖ) :
     PreservesLimit G F :=
@@ -210,7 +210,7 @@ end Representable
 
 section Corepresentable
 
-variable (F : C ⥤ Type v) [F.Corepresentable] {J : Type*} [Category J]
+variable (F : C ⥤ Type v) [F.IsCorepresentable] {J : Type*} [Category J]
 
 noncomputable instance corepresentablePreservesLimit (G : J ⥤ C) :
     PreservesLimit G F :=
diff --git a/Mathlib/CategoryTheory/Linear/Basic.lean b/Mathlib/CategoryTheory/Linear/Basic.lean
index 0a45ae4d24a02..5f724dcf36cb9 100644
--- a/Mathlib/CategoryTheory/Linear/Basic.lean
+++ b/Mathlib/CategoryTheory/Linear/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.Defs
 import Mathlib.Algebra.Module.Equiv.Defs
diff --git a/Mathlib/CategoryTheory/Linear/FunctorCategory.lean b/Mathlib/CategoryTheory/Linear/FunctorCategory.lean
index 5ecd51cc0413a..54207426b1605 100644
--- a/Mathlib/CategoryTheory/Linear/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Linear/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.FunctorCategory
 import Mathlib.CategoryTheory.Linear.Basic
@@ -21,6 +21,11 @@ open CategoryTheory.Limits Linear
 variable {R : Type*} [Semiring R]
 variable {C D : Type*} [Category C] [Category D] [Preadditive D] [Linear R D]
 
+#adaptation_note
+/--
+At nightly-2024-08-08 we needed to significantly increase the maxHeartbeats here.
+-/
+set_option maxHeartbeats 800000 in
 instance functorCategoryLinear : Linear R (C ⥤ D) where
   homModule F G :=
     { smul := fun r α =>
diff --git a/Mathlib/CategoryTheory/Linear/LinearFunctor.lean b/Mathlib/CategoryTheory/Linear/LinearFunctor.lean
index c2ef88551f038..0e58e65be5bae 100644
--- a/Mathlib/CategoryTheory/Linear/LinearFunctor.lean
+++ b/Mathlib/CategoryTheory/Linear/LinearFunctor.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
 import Mathlib.CategoryTheory.Linear.Basic
diff --git a/Mathlib/CategoryTheory/Linear/Yoneda.lean b/Mathlib/CategoryTheory/Linear/Yoneda.lean
index 7022906c6dc3a..ee468da7d7708 100644
--- a/Mathlib/CategoryTheory/Linear/Yoneda.lean
+++ b/Mathlib/CategoryTheory/Linear/Yoneda.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Basic
 import Mathlib.CategoryTheory.Linear.Basic
@@ -28,7 +28,7 @@ namespace CategoryTheory
 variable (R : Type w) [Ring R] {C : Type u} [Category.{v} C] [Preadditive C] [Linear R C]
 variable (C)
 
--- Porting note: inserted specific `ModuleCat.ofHom` in the definition of `linearYoneda`
+-- Porting note: inserted specific `ModuleCat.asHom` in the definition of `linearYoneda`
 -- and similarly in `linearCoyoneda`, otherwise many simp lemmas are not triggered automatically.
 -- Eventually, doing so allows more proofs to be automatic!
 /-- The Yoneda embedding for `R`-linear categories `C`,
@@ -38,9 +38,9 @@ with value on `Y : Cᵒᵖ` given by `ModuleCat.of R (unop Y ⟶ X)`. -/
 def linearYoneda : C ⥤ Cᵒᵖ ⥤ ModuleCat R where
   obj X :=
     { obj := fun Y => ModuleCat.of R (unop Y ⟶ X)
-      map := fun f => ModuleCat.ofHom (Linear.leftComp R _ f.unop) }
+      map := fun f => ModuleCat.asHom (Linear.leftComp R _ f.unop) }
   map {X₁ X₂} f :=
-    { app := fun Y => @ModuleCat.ofHom R _ (Y.unop ⟶ X₁) (Y.unop ⟶ X₂) _ _ _ _
+    { app := fun Y => @ModuleCat.asHom R _ (Y.unop ⟶ X₁) (Y.unop ⟶ X₂) _ _ _ _
         (Linear.rightComp R _ f) }
 
 /-- The Yoneda embedding for `R`-linear categories `C`,
@@ -50,9 +50,9 @@ with value on `X : C` given by `ModuleCat.of R (unop Y ⟶ X)`. -/
 def linearCoyoneda : Cᵒᵖ ⥤ C ⥤ ModuleCat R where
   obj Y :=
     { obj := fun X => ModuleCat.of R (unop Y ⟶ X)
-      map := fun f => ModuleCat.ofHom (Linear.rightComp R _ f) }
+      map := fun f => ModuleCat.asHom (Linear.rightComp R _ f) }
   map {Y₁ Y₂} f :=
-    { app := fun X => @ModuleCat.ofHom R _ (unop Y₁ ⟶ X) (unop Y₂ ⟶ X) _ _ _ _
+    { app := fun X => @ModuleCat.asHom R _ (unop Y₁ ⟶ X) (unop Y₂ ⟶ X) _ _ _ _
         (Linear.leftComp _ _ f.unop) }
 
 instance linearYoneda_obj_additive (X : C) : ((linearYoneda R C).obj X).Additive where
diff --git a/Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean b/Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
index b6cc4f0377426..a46ca7ab8e22f 100644
--- a/Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
+++ b/Mathlib/CategoryTheory/Localization/CalculusOfFractions.lean
@@ -226,7 +226,7 @@ lemma LeftFraction.rightFraction_fac [W.HasRightCalculusOfFractions] {X Y : C}
 
 /-- The equivalence relation on left fractions for a morphism property `W`. -/
 def LeftFractionRel {X Y : C} (z₁ z₂ : W.LeftFraction X Y) : Prop :=
-  ∃ (Z : C)  (t₁ : z₁.Y' ⟶ Z) (t₂ : z₂.Y' ⟶ Z) (_ : z₁.s ≫ t₁ = z₂.s ≫ t₂)
+  ∃ (Z : C) (t₁ : z₁.Y' ⟶ Z) (t₂ : z₂.Y' ⟶ Z) (_ : z₁.s ≫ t₁ = z₂.s ≫ t₂)
     (_ : z₁.f ≫ t₁ = z₂.f ≫ t₂), W (z₁.s ≫ t₁)
 
 namespace LeftFractionRel
@@ -877,7 +877,7 @@ instance (W : MorphismProperty Cᵒᵖ) [h : W.HasRightCalculusOfFractions] :
 
 /-- The equivalence relation on right fractions for a morphism property `W`. -/
 def RightFractionRel {X Y : C} (z₁ z₂ : W.RightFraction X Y) : Prop :=
-  ∃ (Z : C)  (t₁ : Z ⟶ z₁.X') (t₂ : Z ⟶ z₂.X') (_ : t₁ ≫ z₁.s = t₂ ≫ z₂.s)
+  ∃ (Z : C) (t₁ : Z ⟶ z₁.X') (t₂ : Z ⟶ z₂.X') (_ : t₁ ≫ z₁.s = t₂ ≫ z₂.s)
     (_ : t₁ ≫ z₁.f = t₂ ≫ z₂.f), W (t₁ ≫ z₁.s)
 
 lemma RightFractionRel.op {X Y : C} {z₁ z₂ : W.RightFraction X Y}
diff --git a/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean b/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean
index bf5ec2d0d65aa..465d603552c31 100644
--- a/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean
+++ b/Mathlib/CategoryTheory/Localization/CalculusOfFractions/Fractions.lean
@@ -87,7 +87,7 @@ variable {W}
 for a morphism property `W`. The fact it is an equivalence relation is not
 formalized, but it would follow easily from `LeftFraction₂.map_eq_iff`. -/
 def LeftFraction₂Rel {X Y : C} (z₁ z₂ : W.LeftFraction₂ X Y) : Prop :=
-  ∃ (Z : C)  (t₁ : z₁.Y' ⟶ Z) (t₂ : z₂.Y' ⟶ Z) (_ : z₁.s ≫ t₁ = z₂.s ≫ t₂)
+  ∃ (Z : C) (t₁ : z₁.Y' ⟶ Z) (t₂ : z₂.Y' ⟶ Z) (_ : z₁.s ≫ t₁ = z₂.s ≫ t₂)
     (_ : z₁.f ≫ t₁ = z₂.f ≫ t₂) (_ : z₁.f' ≫ t₁ = z₂.f' ≫ t₂), W (z₁.s ≫ t₁)
 
 namespace LeftFraction₂
diff --git a/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.lean b/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.lean
index 916c45adea7b8..2adefbca8f183 100644
--- a/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.lean
+++ b/Mathlib/CategoryTheory/Localization/DerivabilityStructure/Constructor.lean
@@ -87,7 +87,7 @@ lemma isConnected :
   refine ⟨RightResolution.mk ρ.w.left ρ.hw.1, ?_⟩
   have := zigzag_obj_of_zigzag
     (fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac)
-      (isPreconnected_zigzag  (RightResolution.mk (𝟙 _) (W₂.id_mem _))
+      (isPreconnected_zigzag (RightResolution.mk (𝟙 _) (W₂.id_mem _))
         (RightResolution.mk ρ.w.right ρ.hw.2))
   refine Zigzag.trans ?_ (Zigzag.trans this ?_)
   · exact Zigzag.of_hom (eqToHom (by aesop))
diff --git a/Mathlib/CategoryTheory/Localization/HasLocalization.lean b/Mathlib/CategoryTheory/Localization/HasLocalization.lean
index 4078cb38e28f0..76deb68f57da1 100644
--- a/Mathlib/CategoryTheory/Localization/HasLocalization.lean
+++ b/Mathlib/CategoryTheory/Localization/HasLocalization.lean
@@ -46,7 +46,7 @@ variable (W : MorphismProperty C)
 namespace MorphismProperty
 
 /-- The data of a localized category with a given universe
-for the morphisms.  -/
+for the morphisms. -/
 class HasLocalization where
   /-- the objects of the localized category. -/
   {D : Type u}
diff --git a/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean b/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean
index c0f59f13389ca..a0c71108f57aa 100644
--- a/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean
+++ b/Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean
@@ -128,7 +128,7 @@ lemma isEquivalence_iff : G.IsEquivalence ↔ G'.IsEquivalence :=
 end
 
 /-- Condition that a `LocalizerMorphism` induces an equivalence on the localized categories -/
-class IsLocalizedEquivalence : Prop :=
+class IsLocalizedEquivalence : Prop where
   /-- the induced functor on the constructed localized categories is an equivalence -/
   isEquivalence : (Φ.localizedFunctor W₁.Q W₂.Q).IsEquivalence
 
diff --git a/Mathlib/CategoryTheory/Localization/Predicate.lean b/Mathlib/CategoryTheory/Localization/Predicate.lean
index 0b955a5df9ff8..dd06454204670 100644
--- a/Mathlib/CategoryTheory/Localization/Predicate.lean
+++ b/Mathlib/CategoryTheory/Localization/Predicate.lean
@@ -369,7 +369,7 @@ lemma compLeft_iso (W) (F : D ⥤ E) : Localization.Lifting.iso L W (L ⋙ F) F
 
 /-- Given a localization functor `L : C ⥤ D` for `W : MorphismProperty C`,
 if `F₁' : D ⥤ E` lifts a functor `F₁ : C ⥤ D`, then a functor `F₂'` which
-is isomorphic to `F₁'` also lifts a functor `F₂` that is isomorphic to `F₁`.  -/
+is isomorphic to `F₁'` also lifts a functor `F₂` that is isomorphic to `F₁`. -/
 @[simps]
 def ofIsos {F₁ F₂ : C ⥤ E} {F₁' F₂' : D ⥤ E} (e : F₁ ≅ F₂) (e' : F₁' ≅ F₂') [Lifting L W F₁ F₁'] :
     Lifting L W F₂ F₂' :=
diff --git a/Mathlib/CategoryTheory/Monad/Adjunction.lean b/Mathlib/CategoryTheory/Monad/Adjunction.lean
index f84f4ac8a2247..ebd45343a99da 100644
--- a/Mathlib/CategoryTheory/Monad/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Monad/Adjunction.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Adjunction.Reflective
 import Mathlib.CategoryTheory.Monad.Algebra
@@ -73,7 +73,7 @@ def toComonad (h : L ⊣ R) : Comonad D where
     rw [← L.map_comp]
     simp
 
-/-- The monad induced by the Eilenberg-Moore adjunction is the original monad.  -/
+/-- The monad induced by the Eilenberg-Moore adjunction is the original monad. -/
 @[simps!]
 def adjToMonadIso (T : Monad C) : T.adj.toMonad ≅ T :=
   MonadIso.mk (NatIso.ofComponents fun X => Iso.refl _)
@@ -84,7 +84,7 @@ def adjToComonadIso (G : Comonad C) : G.adj.toComonad ≅ G :=
   ComonadIso.mk (NatIso.ofComponents fun X => Iso.refl _)
 
 /--
-Given an adjunction `L ⊣ R`, if `L ⋙ R` is abstractly isomorphic to the identity functor, then the
+Given an adjunction `L ⊣ R`, if `L ⋙ R` is abstractly isomorphic to the identity functor, then the
 unit is an isomorphism.
 -/
 def unitAsIsoOfIso (adj : L ⊣ R) (i : L ⋙ R ≅ 𝟭 C) : 𝟭 C ≅ L ⋙ R where
@@ -100,11 +100,11 @@ def unitAsIsoOfIso (adj : L ⊣ R) (i : L ⋙ R ≅ 𝟭 C) : 𝟭 C ≅ L ⋙ R
     ext X
     exact (adj.toMonad.transport i).right_unit X
 
-lemma isIso_unit_of_iso  (adj : L ⊣ R) (i : L ⋙ R ≅ 𝟭 C) : IsIso adj.unit :=
+lemma isIso_unit_of_iso (adj : L ⊣ R) (i : L ⋙ R ≅ 𝟭 C) : IsIso adj.unit :=
   (inferInstanceAs (IsIso (unitAsIsoOfIso adj i).hom))
 
 /--
-Given an adjunction `L ⊣ R`, if `L ⋙ R` is isomorphic to the identity functor, then `L` is
+Given an adjunction `L ⊣ R`, if `L ⋙ R` is isomorphic to the identity functor, then `L` is
 fully faithful.
 -/
 noncomputable def fullyFaithfulLOfCompIsoId (adj : L ⊣ R) (i : L ⋙ R ≅ 𝟭 C) : L.FullyFaithful :=
@@ -112,7 +112,7 @@ noncomputable def fullyFaithfulLOfCompIsoId (adj : L ⊣ R) (i : L ⋙ R ≅ 
   adj.fullyFaithfulLOfIsIsoUnit
 
 /--
-Given an adjunction `L ⊣ R`, if `R ⋙ L` is abstractly isomorphic to the identity functor, then the
+Given an adjunction `L ⊣ R`, if `R ⋙ L` is abstractly isomorphic to the identity functor, then the
 counit is an isomorphism.
 -/
 def counitAsIsoOfIso (adj : L ⊣ R) (j : R ⋙ L ≅ 𝟭 D) : R ⋙ L ≅ 𝟭 D where
@@ -132,7 +132,7 @@ lemma isIso_counit_of_iso (adj : L ⊣ R) (j : R ⋙ L ≅ 𝟭 D) : IsIso adj.c
   inferInstanceAs (IsIso (counitAsIsoOfIso adj j).hom)
 
 /--
-Given an adjunction `L ⊣ R`, if `R ⋙ L` is isomorphic to the identity functor, then `R` is
+Given an adjunction `L ⊣ R`, if `R ⋙ L` is isomorphic to the identity functor, then `R` is
 fully faithful.
 -/
 noncomputable def fullyFaithfulROfCompIsoId (adj : L ⊣ R) (j : R ⋙ L ≅ 𝟭 D) : R.FullyFaithful :=
diff --git a/Mathlib/CategoryTheory/Monad/Algebra.lean b/Mathlib/CategoryTheory/Monad/Algebra.lean
index 7997a3d8c91e9..3c17e5635d306 100644
--- a/Mathlib/CategoryTheory/Monad/Algebra.lean
+++ b/Mathlib/CategoryTheory/Monad/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Monad.Basic
 import Mathlib.CategoryTheory.Adjunction.Basic
diff --git a/Mathlib/CategoryTheory/Monad/Basic.lean b/Mathlib/CategoryTheory/Monad/Basic.lean
index 911f458280a80..6a82f3d208edd 100644
--- a/Mathlib/CategoryTheory/Monad/Basic.lean
+++ b/Mathlib/CategoryTheory/Monad/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta, Adam Topaz
+Authors: Kim Morrison, Bhavik Mehta, Adam Topaz
 -/
 import Mathlib.CategoryTheory.Functor.Category
 import Mathlib.CategoryTheory.Functor.FullyFaithful
@@ -250,7 +250,7 @@ variable {C}
 -/
 /- Porting note: removed
 `@[simps (config := { rhsMd := semireducible })]`
-and replaced with `@[simps]` in the two declarations below-/
+and replaced with `@[simps]` in the two declarations below -/
 @[simps!]
 def MonadIso.toNatIso {M N : Monad C} (h : M ≅ N) : (M : C ⥤ C) ≅ N :=
   (monadToFunctor C).mapIso h
diff --git a/Mathlib/CategoryTheory/Monad/Comonadicity.lean b/Mathlib/CategoryTheory/Monad/Comonadicity.lean
index 6097def112e98..7aeb43cea5361 100644
--- a/Mathlib/CategoryTheory/Monad/Comonadicity.lean
+++ b/Mathlib/CategoryTheory/Monad/Comonadicity.lean
@@ -175,7 +175,7 @@ def unitFork (B : C) :
   Fork.ofι (adj.unit.app B) (adj.unit_naturality _)
 
 variable {adj} in
-/-- The counit fork is a limit provided `F` preserves it.  -/
+/-- The counit fork is a limit provided `F` preserves it. -/
 def counitLimitOfPreservesEqualizer (A : adj.toComonad.Coalgebra)
     [HasEqualizer (G.map A.a) (adj.unit.app (G.obj A.A))]
     [PreservesLimit (parallelPair (G.map A.a) (adj.unit.app (G.obj A.A))) F] :
diff --git a/Mathlib/CategoryTheory/Monad/Limits.lean b/Mathlib/CategoryTheory/Monad/Limits.lean
index f447048e82490..1e1128d105453 100644
--- a/Mathlib/CategoryTheory/Monad/Limits.lean
+++ b/Mathlib/CategoryTheory/Monad/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta, Jack McKoen
+Authors: Kim Morrison, Bhavik Mehta, Jack McKoen
 -/
 import Mathlib.CategoryTheory.Monad.Adjunction
 import Mathlib.CategoryTheory.Adjunction.Limits
diff --git a/Mathlib/CategoryTheory/Monad/Monadicity.lean b/Mathlib/CategoryTheory/Monad/Monadicity.lean
index 5890e073e784f..5dbca1937d5e9 100644
--- a/Mathlib/CategoryTheory/Monad/Monadicity.lean
+++ b/Mathlib/CategoryTheory/Monad/Monadicity.lean
@@ -187,7 +187,7 @@ def counitCofork (B : D) :
   Cofork.ofπ (adj.counit.app B) (adj.counit_naturality _)
 
 variable {adj} in
-/-- The unit cofork is a colimit provided `G` preserves it.  -/
+/-- The unit cofork is a colimit provided `G` preserves it. -/
 def unitColimitOfPreservesCoequalizer (A : adj.toMonad.Algebra)
     [HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))]
     [PreservesColimit (parallelPair (F.map A.a) (adj.counit.app (F.obj A.A))) G] :
diff --git a/Mathlib/CategoryTheory/Monoidal/Bimod.lean b/Mathlib/CategoryTheory/Monoidal/Bimod.lean
index a29f338d9a4d7..b4118c7d293bf 100644
--- a/Mathlib/CategoryTheory/Monoidal/Bimod.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Bimod.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Oleksandr Manzyuk
+Authors: Kim Morrison, Oleksandr Manzyuk
 -/
 import Mathlib.CategoryTheory.Bicategory.Basic
 import Mathlib.CategoryTheory.Monoidal.Mon_
@@ -201,13 +201,13 @@ noncomputable def actLeft : R.X ⊗ X P Q ⟶ X P Q :=
           simp only [Category.assoc]
           slice_lhs 1 2 => rw [associator_inv_naturality_middle]
           slice_rhs 3 4 => rw [← comp_whiskerRight, middle_assoc, comp_whiskerRight]
-          coherence)
+          monoidal)
         (by
           dsimp
           slice_lhs 1 1 => rw [MonoidalCategory.whiskerLeft_comp]
           slice_lhs 2 3 => rw [associator_inv_naturality_right]
           slice_lhs 3 4 => rw [whisker_exchange]
-          coherence))
+          monoidal))
 
 theorem whiskerLeft_π_actLeft :
     (R.X ◁ coequalizer.π _ _) ≫ actLeft P Q =
@@ -224,7 +224,7 @@ theorem one_act_left' : (R.one ▷ _) ≫ actLeft P Q = (λ_ _).hom := by
   slice_lhs 1 2 => rw [associator_inv_naturality_left]
   slice_lhs 2 3 => rw [← comp_whiskerRight, one_actLeft]
   slice_rhs 1 2 => rw [leftUnitor_naturality]
-  coherence
+  monoidal
 
 theorem left_assoc' :
     (R.mul ▷ _) ≫ actLeft P Q = (α_ R.X R.X _).hom ≫ (R.X ◁ actLeft P Q) ≫ actLeft P Q := by
@@ -240,7 +240,7 @@ theorem left_assoc' :
       MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp]
   slice_rhs 4 5 => rw [whiskerLeft_π_actLeft]
   slice_rhs 3 4 => rw [associator_inv_naturality_middle]
-  coherence
+  monoidal
 
 end
 
@@ -484,7 +484,7 @@ theorem hom_left_act_hom' :
   slice_rhs 3 4 => erw [TensorBimod.whiskerLeft_π_actLeft P (Q.tensorBimod L)]
   slice_rhs 2 3 => erw [associator_inv_naturality_right]
   slice_rhs 3 4 => erw [whisker_exchange]
-  coherence
+  monoidal
 
 theorem hom_right_act_hom' :
     ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
@@ -510,7 +510,7 @@ theorem hom_right_act_hom' :
   slice_rhs 3 4 =>
     rw [← MonoidalCategory.whiskerLeft_comp, TensorBimod.π_tensor_id_actRight,
       MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp]
-  coherence
+  monoidal
 
 /-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
 the associator isomorphism. -/
@@ -529,7 +529,7 @@ noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).te
         slice_rhs 1 2 => rw [MonoidalCategory.whiskerLeft_comp]
         slice_rhs 2 3 => rw [associator_inv_naturality_right]
         slice_rhs 3 4 => rw [whisker_exchange]
-        coherence)
+        monoidal)
 
 /-- The underlying morphism of the inverse component of the associator isomorphism. -/
 noncomputable def inv :
@@ -550,7 +550,7 @@ noncomputable def inv :
           MonoidalCategory.whiskerLeft_comp, MonoidalCategory.whiskerLeft_comp]
       slice_rhs 4 6 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
       slice_rhs 3 4 => rw [associator_inv_naturality_middle]
-      coherence)
+      monoidal)
 
 theorem hom_inv_id : hom P Q L ≫ inv P Q L = 𝟙 _ := by
   dsimp [hom, homAux, inv, invAux]
@@ -605,7 +605,7 @@ theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
   slice_lhs 2 3 => rw [associator_inv_naturality_left]
   slice_lhs 3 4 => rw [← comp_whiskerRight, Mon_.one_mul]
   slice_rhs 1 2 => rw [Category.comp_id]
-  coherence
+  monoidal
 
 theorem inv_hom_id : inv P ≫ hom P = 𝟙 _ := by
   dsimp [hom, inv]
@@ -659,7 +659,7 @@ theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
   slice_lhs 2 3 => rw [associator_naturality_right]
   slice_lhs 3 4 => rw [← MonoidalCategory.whiskerLeft_comp, Mon_.mul_one]
   slice_rhs 1 2 => rw [Category.comp_id]
-  coherence
+  monoidal
 
 theorem inv_hom_id : inv P ≫ hom P = 𝟙 _ := by
   dsimp [hom, inv]
@@ -764,7 +764,7 @@ theorem id_whiskerLeft_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
   slice_rhs 3 4 => rw [associator_inv_naturality_left]
   slice_rhs 4 5 => rw [← comp_whiskerRight, Mon_.one_mul]
   have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ▷ N.X) = 𝟙 _ := by
-    coherence
+    monoidal
   slice_rhs 2 4 => rw [this]
   slice_rhs 1 2 => rw [Category.comp_id]
 
@@ -926,7 +926,7 @@ theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P :
     rw [← whisker_exchange]
   slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
   slice_rhs 2 3 => rw [associator_naturality_right]
-  coherence
+  monoidal
 
 theorem triangle_bimod {X Y Z : Mon_ C} (M : Bimod X Y) (N : Bimod Y Z) :
     (associatorBimod M (regular Y) N).hom ≫ whiskerLeft M (leftUnitorBimod N).hom =
diff --git a/Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
index f0ba170eb47a1..ecac5bcd3be12 100644
--- a/Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean
@@ -1,12 +1,12 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Discrete
 import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
 import Mathlib.CategoryTheory.Monoidal.Opposite
-import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
 import Mathlib.CategoryTheory.CommSq
 
 /-!
@@ -153,7 +153,7 @@ theorem yang_baxter' (X Y Z : C) :
       𝟙 _ ⊗≫ (X ◁ (β_ Y Z).hom ⊗≫ (β_ X Z).hom ▷ Y ⊗≫ Z ◁ (β_ X Y).hom) ⊗≫ 𝟙 _ := by
   rw [← cancel_epi (α_ X Y Z).inv, ← cancel_mono (α_ Z Y X).hom]
   convert yang_baxter X Y Z using 1
-  all_goals coherence
+  all_goals monoidal
 
 theorem yang_baxter_iso (X Y Z : C) :
     (α_ X Y Z).symm ≪≫ whiskerRightIso (β_ X Y) Z ≪≫ α_ Y X Z ≪≫
@@ -256,20 +256,20 @@ I couldn't find a detailed proof in print, but this is discussed in:
   "Tensor categories", vol 25, Mathematical Surveys and Monographs (2015), AMS.
 -/
 
-variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCategory C]
+variable {C : Type u₁} [Category.{v₁} C] [MonoidalCategory C] [BraidedCategory C]
 
 theorem braiding_leftUnitor_aux₁ (X : C) :
     (α_ (𝟙_ C) (𝟙_ C) X).hom ≫
         (𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
       ((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by
-  coherence
+  monoidal
 
 theorem braiding_leftUnitor_aux₂ (X : C) :
     ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) = (ρ_ X).hom ▷ 𝟙_ C :=
   calc
     ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) =
       ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) := by
-      coherence
+      monoidal
     _ = ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (_ ◁ (β_ X _).hom) ≫
           (_ ◁ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) := by
       simp
@@ -291,14 +291,14 @@ theorem braiding_rightUnitor_aux₁ (X : C) :
     (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
         ((β_ (𝟙_ C) X).inv ▷ 𝟙_ C) ≫ (α_ _ X _).hom ≫ (_ ◁ (ρ_ X).hom) =
       (X ◁ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv := by
-  coherence
+  monoidal
 
 theorem braiding_rightUnitor_aux₂ (X : C) :
     (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom :=
   calc
     (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) =
       (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) := by
-      coherence
+      monoidal
     _ = (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ ((β_ _ X).hom ▷ _) ≫
           ((β_ _ X).inv ▷ _) ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) := by
       simp
@@ -324,7 +324,7 @@ theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom 
 theorem braiding_inv_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).inv = (ρ_ X).hom ≫ (λ_ X).inv := by
   rw [Iso.inv_ext]
   rw [braiding_tensorUnit_left]
-  coherence
+  monoidal
 
 @[reassoc]
 theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv := by
@@ -343,7 +343,7 @@ theorem braiding_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).hom = (ρ_ X).hom
 theorem braiding_inv_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv := by
   rw [Iso.inv_ext]
   rw [braiding_tensorUnit_right]
-  coherence
+  monoidal
 
 end
 
@@ -455,8 +455,6 @@ def id : BraidedFunctor C C :=
 instance : Inhabited (BraidedFunctor C C) :=
   ⟨id C⟩
 
-variable {C D E}
-
 /-- The composition of braided monoidal functors. -/
 @[simps!]
 def comp (F : BraidedFunctor C D) (G : BraidedFunctor D E) : BraidedFunctor C E :=
@@ -504,18 +502,21 @@ end CommMonoid
 
 section Tensor
 
-/-- The strength of the tensor product functor from `C × C` to `C`. -/
-def tensor_μ (X Y : C × C) : (X.1 ⊗ X.2) ⊗ Y.1 ⊗ Y.2 ⟶ (X.1 ⊗ Y.1) ⊗ X.2 ⊗ Y.2 :=
-  (α_ X.1 X.2 (Y.1 ⊗ Y.2)).hom ≫
-    (X.1 ◁ (α_ X.2 Y.1 Y.2).inv) ≫
-      (X.1 ◁ (β_ X.2 Y.1).hom ▷ Y.2) ≫
-        (X.1 ◁ (α_ Y.1 X.2 Y.2).hom) ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv
+variable {C}
+
+/-- Swap the second and third objects in `(X₁ ⊗ X₂) ⊗ (Y₁ ⊗ Y₂)`. This is used to strength the
+tensor product functor from `C × C` to `C` as a monoidal functor. -/
+def tensor_μ (X₁ X₂ Y₁ Y₂ : C) : (X₁ ⊗ X₂) ⊗ Y₁ ⊗ Y₂ ⟶ (X₁ ⊗ Y₁) ⊗ X₂ ⊗ Y₂ :=
+  (α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ≫
+    (X₁ ◁ (α_ X₂ Y₁ Y₂).inv) ≫
+      (X₁ ◁ (β_ X₂ Y₁).hom ▷ Y₂) ≫
+        (X₁ ◁ (α_ Y₁ X₂ Y₂).hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv
 
 @[reassoc]
 theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁)
     (g₂ : U₂ ⟶ V₂) :
-    ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensor_μ C (Y₁, Y₂) (V₁, V₂) =
-      tensor_μ C (X₁, X₂) (U₁, U₂) ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂) := by
+    ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensor_μ Y₁ Y₂ V₁ V₂ =
+      tensor_μ X₁ X₂ U₁ U₂ ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂) := by
   dsimp only [tensor_μ]
   simp_rw [← id_tensorHom, ← tensorHom_id]
   slice_lhs 1 2 => rw [associator_naturality]
@@ -530,27 +531,27 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
 
 @[reassoc]
 theorem tensor_μ_natural_left {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (Z₁ Z₂ : C) :
-    (f₁ ⊗ f₂) ▷ (Z₁ ⊗ Z₂) ≫ tensor_μ C (Y₁, Y₂) (Z₁, Z₂) =
-      tensor_μ C (X₁, X₂) (Z₁, Z₂) ≫ (f₁ ▷ Z₁ ⊗ f₂ ▷ Z₂) := by
-  convert tensor_μ_natural C f₁ f₂ (𝟙 Z₁) (𝟙 Z₂) using 1 <;> simp
+    (f₁ ⊗ f₂) ▷ (Z₁ ⊗ Z₂) ≫ tensor_μ Y₁ Y₂ Z₁ Z₂ =
+      tensor_μ X₁ X₂ Z₁ Z₂ ≫ (f₁ ▷ Z₁ ⊗ f₂ ▷ Z₂) := by
+  convert tensor_μ_natural f₁ f₂ (𝟙 Z₁) (𝟙 Z₂) using 1 <;> simp
 
 @[reassoc]
 theorem tensor_μ_natural_right (Z₁ Z₂ : C) {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :
-    (Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) =
-      tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by
-  convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp
+    (Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ Z₁ Z₂ Y₁ Y₂ =
+      tensor_μ Z₁ Z₂ X₁ X₂ ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by
+  convert tensor_μ_natural (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp
 
 @[reassoc]
 theorem tensor_left_unitality (X₁ X₂ : C) :
     (λ_ (X₁ ⊗ X₂)).hom =
       ((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
-        tensor_μ C (𝟙_ C, 𝟙_ C) (X₁, X₂) ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) := by
+        tensor_μ (𝟙_ C) (𝟙_ C) X₁ X₂ ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) := by
   dsimp only [tensor_μ]
   have :
     ((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
         (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫ (𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).inv) =
       𝟙_ C ◁ (λ_ X₁).inv ▷ X₂ := by
-    coherence
+    monoidal
   slice_rhs 1 3 => rw [this]
   clear this
   slice_rhs 1 2 => rw [← MonoidalCategory.whiskerLeft_comp, ← comp_whiskerRight,
@@ -561,97 +562,92 @@ theorem tensor_left_unitality (X₁ X₂ : C) :
 theorem tensor_right_unitality (X₁ X₂ : C) :
     (ρ_ (X₁ ⊗ X₂)).hom =
       ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
-        tensor_μ C (X₁, X₂) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ X₁).hom ⊗ (ρ_ X₂).hom) := by
+        tensor_μ X₁ X₂ (𝟙_ C) (𝟙_ C) ≫ ((ρ_ X₁).hom ⊗ (ρ_ X₂).hom) := by
   dsimp only [tensor_μ]
   have :
     ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
         (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv) =
       (α_ X₁ X₂ (𝟙_ C)).hom ≫ (X₁ ◁ (ρ_ X₂).inv ▷ 𝟙_ C) := by
-    coherence
+    monoidal
   slice_rhs 1 3 => rw [this]
   clear this
   slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, ← comp_whiskerRight,
     rightUnitor_inv_braiding]
   simp [tensorHom_id, id_tensorHom, tensorHom_def]
 
+@[reassoc]
 theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
-    (tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (Z₁ ⊗ Z₂)) ≫
-        tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
+    (tensor_μ X₁ X₂ Y₁ Y₂ ▷ (Z₁ ⊗ Z₂)) ≫
+        tensor_μ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) Z₁ Z₂ ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
       (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫
-        ((X₁ ⊗ X₂) ◁ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) := by
+        ((X₁ ⊗ X₂) ◁ tensor_μ Y₁ Y₂ Z₁ Z₂) ≫ tensor_μ X₁ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂) := by
   dsimp only [tensor_obj, prodMonoidal_tensorObj, tensor_μ]
-  simp only [whiskerRight_tensor, comp_whiskerRight, whisker_assoc, assoc, Iso.inv_hom_id_assoc,
-    tensor_whiskerLeft, braiding_tensor_left, MonoidalCategory.whiskerLeft_comp,
-    braiding_tensor_right]
+  simp only [braiding_tensor_left, braiding_tensor_right]
   calc
     _ = 𝟙 _ ⊗≫
       X₁ ◁ ((β_ X₂ Y₁).hom ▷ (Y₂ ⊗ Z₁) ≫ (Y₁ ⊗ X₂) ◁ (β_ Y₂ Z₁).hom) ▷ Z₂ ⊗≫
-        X₁ ◁ Y₁ ◁ (β_ X₂ Z₁).hom ▷ Y₂ ▷ Z₂ ⊗≫ 𝟙 _ := by coherence
-    _ = _ := by rw [← whisker_exchange]; coherence
-
--- We got a timeout if `reassoc` was at the declaration, so we put it here instead.
-attribute [reassoc] tensor_associativity
+        X₁ ◁ Y₁ ◁ (β_ X₂ Z₁).hom ▷ Y₂ ▷ Z₂ ⊗≫ 𝟙 _ := by monoidal
+    _ = _ := by rw [← whisker_exchange]; monoidal
 
 /-- The tensor product functor from `C × C` to `C` as a monoidal functor. -/
 @[simps!]
 def tensorMonoidal : MonoidalFunctor (C × C) C :=
   { tensor C with
     ε := (λ_ (𝟙_ C)).inv
-    μ := tensor_μ C
+    μ := fun X Y ↦ tensor_μ X.1 X.2 Y.1 Y.2
     μ_natural_left := fun f Z => by
       -- `simpa` will be not needed when we define `μ_natural_left` in terms of the whiskerings.
-      simpa using tensor_μ_natural_left C f.1 f.2 Z.1 Z.2
+      simpa using tensor_μ_natural_left f.1 f.2 Z.1 Z.2
     μ_natural_right := fun Z f => by
-      simpa using tensor_μ_natural_right C Z.1 Z.2 f.1 f.2
+      simpa using tensor_μ_natural_right Z.1 Z.2 f.1 f.2
     associativity := fun X Y Z => by
-      simpa using tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
+      simpa using tensor_associativity X.1 X.2 Y.1 Y.2 Z.1 Z.2
     left_unitality := fun ⟨X₁, X₂⟩ => by
-      simpa using tensor_left_unitality C X₁ X₂
+      simpa using tensor_left_unitality X₁ X₂
     right_unitality := fun ⟨X₁, X₂⟩ => by
-      simpa using tensor_right_unitality C X₁ X₂
+      simpa using tensor_right_unitality X₁ X₂
     μ_isIso := by dsimp [tensor_μ]; infer_instance }
 
 @[reassoc]
 theorem leftUnitor_monoidal (X₁ X₂ : C) :
     (λ_ X₁).hom ⊗ (λ_ X₂).hom =
-      tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).hom := by
+      tensor_μ (𝟙_ C) X₁ (𝟙_ C) X₂ ≫ ((λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).hom := by
   dsimp only [tensor_μ]
   have :
     (λ_ X₁).hom ⊗ (λ_ X₂).hom =
       (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫
         (𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv) ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((ρ_ X₁).hom ▷ X₂) := by
-    coherence
+    monoidal
   rw [this]; clear this
   rw [← braiding_leftUnitor]
-  dsimp only [tensor_obj, prodMonoidal_tensorObj]
-  coherence
+  monoidal
 
 @[reassoc]
 theorem rightUnitor_monoidal (X₁ X₂ : C) :
     (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
-      tensor_μ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom := by
+      tensor_μ X₁ (𝟙_ C) X₂ (𝟙_ C) ≫ ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom := by
   dsimp only [tensor_μ]
   have :
     (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
       (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
         (X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv) ≫ (X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom) ≫ (X₁ ◁ (λ_ X₂).hom) := by
-    coherence
+    monoidal
   rw [this]; clear this
   rw [← braiding_rightUnitor]
-  dsimp only [tensor_obj, prodMonoidal_tensorObj]
-  coherence
+  monoidal
 
 theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
-    tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
-        (tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
+    tensor_μ (X₁ ⊗ X₂) X₃ (Y₁ ⊗ Y₂) Y₃ ≫
+        (tensor_μ X₁ X₂ Y₁ Y₂ ▷ (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
       ((α_ X₁ X₂ X₃).hom ⊗ (α_ Y₁ Y₂ Y₃).hom) ≫
-        tensor_μ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ ((X₁ ⊗ Y₁) ◁ tensor_μ C (X₂, X₃) (Y₂, Y₃)) := by
+        tensor_μ X₁ (X₂ ⊗ X₃) Y₁ (Y₂ ⊗ Y₃) ≫ ((X₁ ⊗ Y₁) ◁ tensor_μ X₂ X₃ Y₂ Y₃) := by
   dsimp only [tensor_μ]
   calc
     _ = 𝟙 _ ⊗≫ X₁ ◁ X₂ ◁ (β_ X₃ Y₁).hom ▷ Y₂ ▷ Y₃ ⊗≫
       X₁ ◁ ((X₂ ⊗ Y₁) ◁ (β_ X₃ Y₂).hom ≫
-        (β_ X₂ Y₁).hom ▷ (Y₂ ⊗ X₃)) ▷ Y₃ ⊗≫ 𝟙 _ := by simp; coherence
-    _ = _ := by rw [whisker_exchange]; simp; coherence
+        (β_ X₂ Y₁).hom ▷ (Y₂ ⊗ X₃)) ▷ Y₃ ⊗≫ 𝟙 _ := by
+          rw [braiding_tensor_right]; monoidal
+    _ = _ := by rw [whisker_exchange, braiding_tensor_left]; monoidal
 
 -- We got a timeout if `reassoc` was at the declaration, so we put it here instead.
 attribute [reassoc] associator_monoidal
diff --git a/Mathlib/CategoryTheory/Monoidal/Braided/Opposite.lean b/Mathlib/CategoryTheory/Monoidal/Braided/Opposite.lean
index db18fda7de208..a4c907e15f044 100644
--- a/Mathlib/CategoryTheory/Monoidal/Braided/Opposite.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Braided/Opposite.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Lean FRO LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Monoidal.Opposite
@@ -23,13 +23,13 @@ namespace CategoryTheory.BraidedCategory
 
 @[simp] lemma unop_tensor_μ {C : Type*} [Category C] [MonoidalCategory C]
     [BraidedCategory C] (X Y W Z : Cᵒᵖ) :
-    (tensor_μ Cᵒᵖ (X, W) (Y, Z)).unop = tensor_μ C (X.unop, Y.unop) (W.unop, Z.unop) := by
+    (tensor_μ X W Y Z).unop = tensor_μ X.unop Y.unop W.unop Z.unop := by
   simp only [unop_tensorObj, tensor_μ, unop_comp, unop_inv_associator, unop_whiskerLeft,
     unop_hom_associator, unop_whiskerRight, unop_hom_braiding, Category.assoc]
 
 @[simp] lemma op_tensor_μ {C : Type*} [Category C] [MonoidalCategory C]
     [BraidedCategory C] (X Y W Z : C) :
-    (tensor_μ C (X, W) (Y, Z)).op = tensor_μ Cᵒᵖ (op X, op Y) (op W, op Z) := by
+    (tensor_μ X W Y Z).op = tensor_μ (op X) (op Y) (op W) (op Z) := by
   simp only [op_tensorObj, tensor_μ, op_comp, op_inv_associator, op_whiskerLeft, op_hom_associator,
     op_whiskerRight, op_hom_braiding, Category.assoc]
 
diff --git a/Mathlib/CategoryTheory/Monoidal/Category.lean b/Mathlib/CategoryTheory/Monoidal/Category.lean
index 53371440fcf7e..d80764aa71802 100644
--- a/Mathlib/CategoryTheory/Monoidal/Category.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Category.lean
@@ -1,8 +1,9 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
+Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
 -/
+import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.Functor.Trifunctor
 import Mathlib.CategoryTheory.Products.Basic
 
@@ -279,15 +280,6 @@ theorem tensorHom_def' {X₁ Y₁ X₂ Y₂ : C} (f : X₁ ⟶ Y₁) (g : X₂ 
     f ⊗ g = X₁ ◁ g ≫ f ▷ Y₂ :=
   whisker_exchange f g ▸ tensorHom_def f g
 
-end MonoidalCategory
-
-open scoped MonoidalCategory
-open MonoidalCategory
-
-variable {C : Type u} [𝒞 : Category.{v} C] [MonoidalCategory C]
-
-namespace MonoidalCategory
-
 @[reassoc (attr := simp)]
 theorem whiskerLeft_hom_inv (X : C) {Y Z : C} (f : Y ≅ Z) :
     X ◁ f.hom ≫ X ◁ f.inv = 𝟙 (X ⊗ Y) := by
@@ -384,11 +376,9 @@ lemma whiskerRightIso_trans {X Y Z : C} (f : X ≅ Y) (g : Y ≅ Z) (W : C) :
 lemma whiskerRightIso_symm {X Y : C} (f : X ≅ Y) (W : C) :
     (whiskerRightIso f W).symm = whiskerRightIso f.symm W := rfl
 
-end MonoidalCategory
-
 /-- The tensor product of two isomorphisms is an isomorphism. -/
 @[simps]
-def tensorIso {C : Type u} {X Y X' Y' : C} [Category.{v} C] [MonoidalCategory.{v} C] (f : X ≅ Y)
+def tensorIso {X Y X' Y' : C} (f : X ≅ Y)
     (g : X' ≅ Y') : X ⊗ X' ≅ Y ⊗ Y' where
   hom := f.hom ⊗ g.hom
   inv := f.inv ⊗ g.inv
@@ -398,11 +388,13 @@ def tensorIso {C : Type u} {X Y X' Y' : C} [Category.{v} C] [MonoidalCategory.{v
 /-- Notation for `tensorIso`, the tensor product of isomorphisms -/
 infixr:70 " ⊗ " => tensorIso
 
-namespace MonoidalCategory
+theorem tensorIso_def {X Y X' Y' : C} (f : X ≅ Y) (g : X' ≅ Y') :
+    f ⊗ g = whiskerRightIso f X' ≪≫ whiskerLeftIso Y g :=
+  Iso.ext (tensorHom_def f.hom g.hom)
 
-section
-
-variable {C : Type u} [Category.{v} C] [MonoidalCategory.{v} C]
+theorem tensorIso_def' {X Y X' Y' : C} (f : X ≅ Y) (g : X' ≅ Y') :
+    f ⊗ g = whiskerLeftIso X g ≪≫ whiskerRightIso f Y' :=
+  Iso.ext (tensorHom_def' f.hom g.hom)
 
 instance tensor_isIso {W X Y Z : C} (f : W ⟶ X) [IsIso f] (g : Y ⟶ Z) [IsIso g] : IsIso (f ⊗ g) :=
   (asIso f ⊗ asIso g).isIso_hom
@@ -759,8 +751,6 @@ theorem tensor_left_iff {X Y : C} (f g : X ⟶ Y) : 𝟙 (𝟙_ C) ⊗ f = 𝟙
 
 theorem tensor_right_iff {X Y : C} (f g : X ⟶ Y) : f ⊗ 𝟙 (𝟙_ C) = g ⊗ 𝟙 (𝟙_ C) ↔ f = g := by simp
 
-end
-
 section
 
 variable (C)
diff --git a/Mathlib/CategoryTheory/Monoidal/Center.lean b/Mathlib/CategoryTheory/Monoidal/Center.lean
index 4d27a6af5c4be..5eab3c67802ba 100644
--- a/Mathlib/CategoryTheory/Monoidal/Center.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Center.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Functor.ReflectsIso
@@ -137,19 +137,19 @@ def tensorObj (X Y : Center C) : Center C :=
             X.1 ◁ (HalfBraiding.β Y.2 U).hom ▷ U' ⊗≫
               (_ ◁ (HalfBraiding.β Y.2 U').hom ≫
                 (HalfBraiding.β X.2 U).hom ▷ _) ⊗≫
-                  U ◁ (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by coherence
-          _ = _ := by rw [whisker_exchange]; coherence
+                  U ◁ (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by monoidal
+          _ = _ := by rw [whisker_exchange]; monoidal
       naturality := fun {U U'} f => by
         dsimp only [Iso.trans_hom, whiskerLeftIso_hom, Iso.symm_hom, whiskerRightIso_hom]
         calc
           _ = 𝟙 _ ⊗≫
             (X.1 ◁ (Y.1 ◁ f ≫ (HalfBraiding.β Y.2 U').hom)) ⊗≫
-              (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by coherence
+              (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by monoidal
           _ = 𝟙 _ ⊗≫
             X.1 ◁ (HalfBraiding.β Y.2 U).hom ⊗≫
               (X.1 ◁ f ≫ (HalfBraiding.β X.2 U').hom) ▷ Y.1 ⊗≫ 𝟙 _ := by
-            rw [HalfBraiding.naturality]; coherence
-          _ = _ := by rw [HalfBraiding.naturality]; coherence }⟩
+            rw [HalfBraiding.naturality]; monoidal
+          _ = _ := by rw [HalfBraiding.naturality]; monoidal }⟩
 
 @[reassoc]
 theorem whiskerLeft_comm (X : Center C) {Y₁ Y₂ : Center C} (f : Y₁ ⟶ Y₂) (U : C) :
@@ -160,12 +160,12 @@ theorem whiskerLeft_comm (X : Center C) {Y₁ Y₂ : Center C} (f : Y₁ ⟶ Y
   calc
     _ = 𝟙 _ ⊗≫
       X.fst ◁ (f.f ▷ U ≫ (HalfBraiding.β Y₂.snd U).hom) ⊗≫
-        (HalfBraiding.β X.snd U).hom ▷ Y₂.fst ⊗≫ 𝟙 _ := by coherence
+        (HalfBraiding.β X.snd U).hom ▷ Y₂.fst ⊗≫ 𝟙 _ := by monoidal
     _ = 𝟙 _ ⊗≫
       X.fst ◁ (HalfBraiding.β Y₁.snd U).hom ⊗≫
         ((X.fst ⊗ U) ◁ f.f ≫ (HalfBraiding.β X.snd U).hom ▷ Y₂.fst) ⊗≫ 𝟙 _ := by
-      rw [f.comm]; coherence
-    _ = _ := by rw [whisker_exchange]; coherence
+      rw [f.comm]; monoidal
+    _ = _ := by rw [whisker_exchange]; monoidal
 
 /-- Auxiliary definition for the `MonoidalCategory` instance on `Center C`. -/
 def whiskerLeft (X : Center C) {Y₁ Y₂ : Center C} (f : Y₁ ⟶ Y₂) :
@@ -182,12 +182,12 @@ theorem whiskerRight_comm {X₁ X₂ : Center C} (f : X₁ ⟶ X₂) (Y : Center
   calc
     _ = 𝟙 _ ⊗≫
       (f.f ▷ (Y.fst ⊗ U) ≫ X₂.fst ◁ (HalfBraiding.β Y.snd U).hom) ⊗≫
-        (HalfBraiding.β X₂.snd U).hom ▷ Y.fst ⊗≫ 𝟙 _ := by coherence
+        (HalfBraiding.β X₂.snd U).hom ▷ Y.fst ⊗≫ 𝟙 _ := by monoidal
     _ = 𝟙 _ ⊗≫
       X₁.fst ◁ (HalfBraiding.β Y.snd U).hom ⊗≫
         (f.f ▷ U ≫ (HalfBraiding.β X₂.snd U).hom) ▷ Y.fst ⊗≫ 𝟙 _ := by
-      rw [← whisker_exchange]; coherence
-    _ = _ := by rw [f.comm]; coherence
+      rw [← whisker_exchange]; monoidal
+    _ = _ := by rw [f.comm]; monoidal
 
 /-- Auxiliary definition for the `MonoidalCategory` instance on `Center C`. -/
 def whiskerRight {X₁ X₂ : Center C} (f : X₁ ⟶ X₂) (Y : Center C) :
diff --git a/Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean b/Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
index 430680b7fe1cf..ef4ecbd8d3584 100644
--- a/Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
+++ b/Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean
@@ -1,9 +1,9 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
+Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
 -/
-import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
 
 /-!
 # Lemmas which are consequences of monoidal coherence
@@ -26,47 +26,47 @@ variable {C : Type*} [Category C] [MonoidalCategory C]
 @[reassoc]
 theorem leftUnitor_tensor'' (X Y : C) :
     (α_ (𝟙_ C) X Y).hom ≫ (λ_ (X ⊗ Y)).hom = (λ_ X).hom ⊗ 𝟙 Y := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem leftUnitor_tensor' (X Y : C) :
     (λ_ (X ⊗ Y)).hom = (α_ (𝟙_ C) X Y).inv ≫ ((λ_ X).hom ⊗ 𝟙 Y) := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem leftUnitor_tensor_inv' (X Y : C) :
-    (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom := by coherence
+    (λ_ (X ⊗ Y)).inv = ((λ_ X).inv ⊗ 𝟙 Y) ≫ (α_ (𝟙_ C) X Y).hom := by monoidal_coherence
 
 @[reassoc]
 theorem id_tensor_rightUnitor_inv (X Y : C) : 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ _).inv ≫ (α_ _ _ _).hom := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem leftUnitor_inv_tensor_id (X Y : C) : (λ_ X).inv ⊗ 𝟙 Y = (λ_ _).inv ≫ (α_ _ _ _).inv := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem pentagon_inv_inv_hom (W X Y Z : C) :
     (α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ X) Y Z).hom =
       (𝟙 W ⊗ (α_ X Y Z).hom) ≫ (α_ W X (Y ⊗ Z)).inv := by
-  coherence
+  monoidal_coherence
 
 theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by
-  coherence
+  monoidal_coherence
 
 theorem unitors_inv_equal : (λ_ (𝟙_ C)).inv = (ρ_ (𝟙_ C)).inv := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem pentagon_hom_inv {W X Y Z : C} :
     (α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) =
       (α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) ≫ (α_ W (X ⊗ Y) Z).hom := by
-  coherence
+  monoidal_coherence
 
 @[reassoc]
 theorem pentagon_inv_hom (W X Y Z : C) :
     (α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) =
       (α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv := by
-  coherence
+  monoidal_coherence
 
 end CategoryTheory.MonoidalCategory
diff --git a/Mathlib/CategoryTheory/Monoidal/CommMon_.lean b/Mathlib/CategoryTheory/Monoidal/CommMon_.lean
index 7eaef1920b0fc..b8a3c549ea911 100644
--- a/Mathlib/CategoryTheory/Monoidal/CommMon_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/CommMon_.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Monoidal.Mon_
diff --git a/Mathlib/CategoryTheory/Monoidal/Comon_.lean b/Mathlib/CategoryTheory/Monoidal/Comon_.lean
index b58290d9ba93a..61c63c58ac902 100644
--- a/Mathlib/CategoryTheory/Monoidal/Comon_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Comon_.lean
@@ -59,9 +59,9 @@ def trivial : Comon_ C where
   X := 𝟙_ C
   counit := 𝟙 _
   comul := (λ_ _).inv
-  comul_assoc := by coherence
-  counit_comul := by coherence
-  comul_counit := by coherence
+  comul_assoc := by monoidal_coherence
+  counit_comul := by monoidal_coherence
+  comul_counit := by monoidal_coherence
 
 instance : Inhabited (Comon_ C) :=
   ⟨trivial C⟩
@@ -243,7 +243,8 @@ Comonoid objects in a braided category form a monoidal category.
 
 This definition is via transporting back and forth to monoids in the opposite category,
 -/
-instance [BraidedCategory C] : MonoidalCategory (Comon_ C) :=
+@[simps!]
+instance monoidal [BraidedCategory C] : MonoidalCategory (Comon_ C) :=
   Monoidal.transport (Comon_EquivMon_OpOp C).symm
 
 variable [BraidedCategory C]
@@ -260,7 +261,7 @@ the version provided in `tensorObj_comul` below.
 -/
 theorem tensorObj_comul' (A B : Comon_ C) :
     (A ⊗ B).comul =
-      (A.comul ⊗ B.comul) ≫ (tensor_μ Cᵒᵖ (op A.X, op B.X) (op A.X, op B.X)).unop := by
+      (A.comul ⊗ B.comul) ≫ (tensor_μ (op A.X) (op B.X) (op A.X) (op B.X)).unop := by
   rfl
 
 /--
@@ -269,7 +270,7 @@ the tensor product of the comultiplications followed by the tensor strength
 (to shuffle the factors back into order).
 -/
 theorem tensorObj_comul (A B : Comon_ C) :
-    (A ⊗ B).comul = (A.comul ⊗ B.comul) ≫ tensor_μ C (A.X, A.X) (B.X, B.X) := by
+    (A ⊗ B).comul = (A.comul ⊗ B.comul) ≫ tensor_μ A.X A.X B.X B.X := by
   rw [tensorObj_comul']
   congr
   simp only [tensor_μ, unop_tensorObj, unop_op]
diff --git a/Mathlib/CategoryTheory/Monoidal/Discrete.lean b/Mathlib/CategoryTheory/Monoidal/Discrete.lean
index dbd834083cb6e..9ff8e09d5151e 100644
--- a/Mathlib/CategoryTheory/Monoidal/Discrete.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Discrete.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Hom.Defs
 import Mathlib.CategoryTheory.DiscreteCategory
diff --git a/Mathlib/CategoryTheory/Monoidal/End.lean b/Mathlib/CategoryTheory/Monoidal/End.lean
index 52c8651d1e0ed..5a48831f3a044 100644
--- a/Mathlib/CategoryTheory/Monoidal/End.lean
+++ b/Mathlib/CategoryTheory/Monoidal/End.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Andrew Yang
+Authors: Kim Morrison, Andrew Yang
 -/
 import Mathlib.CategoryTheory.Monoidal.Functor
 
diff --git a/Mathlib/CategoryTheory/Monoidal/Functor.lean b/Mathlib/CategoryTheory/Monoidal/Functor.lean
index 8bf05788f4dbd..1cd5fbc682a6c 100644
--- a/Mathlib/CategoryTheory/Monoidal/Functor.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Functor.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta
+Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Monoidal.Category
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
@@ -365,6 +365,22 @@ theorem map_whiskerLeft (X : C) {Y Z : C} (f : Y ⟶ Z) :
 theorem map_whiskerRight {X Y : C} (f : X ⟶ Y) (Z : C) :
     F.map (f ▷ Z) = inv (F.μ X Z) ≫ F.map f ▷ F.obj Z ≫ F.μ Y Z := by simp
 
+@[reassoc]
+theorem map_associator (X Y Z : C) :
+    F.map (α_ X Y Z).hom =
+      inv (F.μ (X ⊗ Y) Z) ≫ inv (F.μ X Y) ▷ F.obj Z ≫
+        (α_ (F.obj X) (F.obj Y) (F.obj Z)).hom ≫ F.obj X ◁ F.μ Y Z ≫ F.μ X (Y ⊗ Z) := by
+  rw [← inv_whiskerRight, ← IsIso.inv_comp_assoc, IsIso.eq_inv_comp]
+  simp
+
+@[reassoc]
+theorem map_associator_inv (X Y Z : C) :
+    F.map (α_ X Y Z).inv =
+      inv (F.μ X (Y ⊗ Z)) ≫ F.obj X ◁ inv (F.μ Y Z) ≫
+        (α_ (F.obj X) (F.obj Y) (F.obj Z)).inv ≫ F.μ X Y ▷ F.obj Z ≫ F.μ (X ⊗ Y) Z := by
+  rw [← inv_whiskerLeft, ← IsIso.inv_comp_assoc, IsIso.eq_inv_comp]
+  simp
+
 @[reassoc]
 theorem map_leftUnitor (X : C) :
     F.map (λ_ X).hom = inv (F.μ (𝟙_ C) X) ≫ inv F.ε ▷ F.obj X ≫ (λ_ (F.obj X)).hom := by
@@ -699,12 +715,10 @@ noncomputable def monoidalAdjoint :
 
 instance [F.IsEquivalence] : IsIso (monoidalAdjoint F h).ε := by
   dsimp
-  rw [Adjunction.homEquiv_unit]
   infer_instance
 
 instance (X Y : D) [F.IsEquivalence] : IsIso ((monoidalAdjoint F h).μ X Y) := by
   dsimp
-  rw [Adjunction.homEquiv_unit]
   infer_instance
 
 /-- If a monoidal functor `F` is an equivalence of categories then its inverse is also monoidal. -/
diff --git a/Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean b/Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean
index 90286ae948670..5a69edb48708b 100644
--- a/Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Functor.Category
diff --git a/Mathlib/CategoryTheory/Monoidal/Functorial.lean b/Mathlib/CategoryTheory/Monoidal/Functorial.lean
index 26bd834c33628..5f9d7be6fbb24 100644
--- a/Mathlib/CategoryTheory/Monoidal/Functorial.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Functorial.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Functor
 import Mathlib.CategoryTheory.Functor.Functorial
diff --git a/Mathlib/CategoryTheory/Monoidal/Hopf_.lean b/Mathlib/CategoryTheory/Monoidal/Hopf_.lean
index 12419c19a3575..4bf50784e7cc7 100644
--- a/Mathlib/CategoryTheory/Monoidal/Hopf_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Hopf_.lean
@@ -223,7 +223,7 @@ theorem antipode_comul₂ (A : Hopf_ C) :
     MonoidalCategory.whiskerRight_id, whiskerLeft_rightUnitor, Category.assoc, Iso.hom_inv_id_assoc,
     Iso.inv_hom_id_assoc, whiskerLeft_inv_hom_assoc, antipode_right_assoc]
   rw [rightUnitor_inv_naturality_assoc, tensorHom_def]
-  coherence
+  monoidal
 
 theorem antipode_comul (A : Hopf_ C) :
     A.antipode ≫ A.X.comul.hom = A.X.comul.hom ≫ (β_ _ _).hom ≫ (A.antipode ⊗ A.antipode) := by
@@ -400,7 +400,7 @@ theorem mul_antipode₂ (A : Hopf_ C) :
   slice_lhs 2 3 =>
     rw [rightUnitor_naturality]
   simp only [Mon_.tensorUnit_X]
-  coherence
+  monoidal
 
 theorem mul_antipode (A : Hopf_ C) :
     A.X.X.mul ≫ A.antipode = (A.antipode ⊗ A.antipode) ≫ (β_ _ _).hom ≫ A.X.X.mul := by
diff --git a/Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean b/Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean
index 19c7811649daf..6d90406b86532 100644
--- a/Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Internal/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.CommMon_
 import Mathlib.CategoryTheory.Monoidal.Comon_
diff --git a/Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean b/Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean
index 8986e18168ceb..53ddce8988862 100644
--- a/Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Internal/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Internal.FunctorCategory
 import Mathlib.CategoryTheory.Monoidal.Limits
diff --git a/Mathlib/CategoryTheory/Monoidal/Internal/Module.lean b/Mathlib/CategoryTheory/Monoidal/Internal/Module.lean
index 25946bbd79708..2e6df6e377f89 100644
--- a/Mathlib/CategoryTheory/Monoidal/Internal/Module.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Internal/Module.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
 import Mathlib.Algebra.Category.AlgebraCat.Basic
@@ -45,24 +45,19 @@ instance Ring_of_Mon_ (A : Mon_ (ModuleCat.{u} R)) : Ring A.X :=
     one := A.one (1 : R)
     mul := fun x y => A.mul (x ⊗ₜ y)
     one_mul := fun x => by
-      have := LinearMap.congr_fun A.one_mul ((1 : R) ⊗ₜ x)
-      convert this
+      convert LinearMap.congr_fun A.one_mul ((1 : R) ⊗ₜ x)
       rw [MonoidalCategory.leftUnitor_hom_apply, one_smul]
     mul_one := fun x => by
-      have := LinearMap.congr_fun A.mul_one (x ⊗ₜ (1 : R))
-      convert this
+      convert LinearMap.congr_fun A.mul_one (x ⊗ₜ (1 : R))
       erw [MonoidalCategory.leftUnitor_hom_apply, one_smul]
     mul_assoc := fun x y z => by
-      have := LinearMap.congr_fun A.mul_assoc (x ⊗ₜ y ⊗ₜ z)
-      convert this
+      convert LinearMap.congr_fun A.mul_assoc (x ⊗ₜ y ⊗ₜ z)
     left_distrib := fun x y z => by
-      have := A.mul.map_add (x ⊗ₜ y) (x ⊗ₜ z)
-      convert this
+      convert A.mul.map_add (x ⊗ₜ y) (x ⊗ₜ z)
       rw [← TensorProduct.tmul_add]
       rfl
     right_distrib := fun x y z => by
-      have := A.mul.map_add (x ⊗ₜ z) (y ⊗ₜ z)
-      convert this
+      convert A.mul.map_add (x ⊗ₜ z) (y ⊗ₜ z)
       rw [← TensorProduct.add_tmul]
       rfl
     zero_mul := fun x => show A.mul _ = 0 by
diff --git a/Mathlib/CategoryTheory/Monoidal/Internal/Types.lean b/Mathlib/CategoryTheory/Monoidal/Internal/Types.lean
index 1c8d1b2ee61fc..00d58af2f33d2 100644
--- a/Mathlib/CategoryTheory/Monoidal/Internal/Types.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Internal/Types.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.MonCat.Basic
 import Mathlib.CategoryTheory.Monoidal.CommMon_
diff --git a/Mathlib/CategoryTheory/Monoidal/Limits.lean b/Mathlib/CategoryTheory/Monoidal/Limits.lean
index 4ddc2b63e9618..e13b3fa646109 100644
--- a/Mathlib/CategoryTheory/Monoidal/Limits.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Functorial
 import Mathlib.CategoryTheory.Monoidal.FunctorCategory
diff --git a/Mathlib/CategoryTheory/Monoidal/Linear.lean b/Mathlib/CategoryTheory/Monoidal/Linear.lean
index f736296f7147d..d9c475fc8dc58 100644
--- a/Mathlib/CategoryTheory/Monoidal/Linear.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Linear.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Linear.LinearFunctor
 import Mathlib.CategoryTheory.Monoidal.Preadditive
diff --git a/Mathlib/CategoryTheory/Monoidal/Mod_.lean b/Mathlib/CategoryTheory/Monoidal/Mod_.lean
index 79ddde1156f2d..bee5d1a324c2f 100644
--- a/Mathlib/CategoryTheory/Monoidal/Mod_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Mod_.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Mon_
 
diff --git a/Mathlib/CategoryTheory/Monoidal/Mon_.lean b/Mathlib/CategoryTheory/Monoidal/Mon_.lean
index 0279c6c4d67d2..2202d88535f6a 100644
--- a/Mathlib/CategoryTheory/Monoidal/Mon_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Mon_.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Monoidal.Discrete
@@ -57,8 +57,8 @@ def trivial : Mon_ C where
   X := 𝟙_ C
   one := 𝟙 _
   mul := (λ_ _).hom
-  mul_assoc := by coherence
-  mul_one := by coherence
+  mul_assoc := by monoidal_coherence
+  mul_one := by monoidal_coherence
 
 instance : Inhabited (Mon_ C) :=
   ⟨trivial C⟩
@@ -350,30 +350,30 @@ variable [BraidedCategory C]
 
 theorem Mon_tensor_one_mul (M N : Mon_ C) :
     (((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ▷ (M.X ⊗ N.X)) ≫
-        tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
+        tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) =
       (λ_ (M.X ⊗ N.X)).hom := by
   simp only [comp_whiskerRight_assoc]
   slice_lhs 2 3 => rw [tensor_μ_natural_left]
   slice_lhs 3 4 => rw [← tensor_comp, one_mul M, one_mul N]
   symm
-  exact tensor_left_unitality C M.X N.X
+  exact tensor_left_unitality M.X N.X
 
 theorem Mon_tensor_mul_one (M N : Mon_ C) :
     (M.X ⊗ N.X) ◁ ((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)) ≫
-        tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
+        tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) =
       (ρ_ (M.X ⊗ N.X)).hom := by
   simp only [MonoidalCategory.whiskerLeft_comp_assoc]
   slice_lhs 2 3 => rw [tensor_μ_natural_right]
   slice_lhs 3 4 => rw [← tensor_comp, mul_one M, mul_one N]
   symm
-  exact tensor_right_unitality C M.X N.X
+  exact tensor_right_unitality M.X N.X
 
 theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
-    ((tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ▷ (M.X ⊗ N.X)) ≫
-        tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
+    ((tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul)) ▷ (M.X ⊗ N.X)) ≫
+        tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) =
       (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
-        ((M.X ⊗ N.X) ◁ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))) ≫
-          tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) := by
+        ((M.X ⊗ N.X) ◁ (tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul))) ≫
+          tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) := by
   simp only [comp_whiskerRight_assoc, MonoidalCategory.whiskerLeft_comp_assoc]
   slice_lhs 2 3 => rw [tensor_μ_natural_left]
   slice_lhs 3 4 => rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]
@@ -382,12 +382,12 @@ theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
   simp
 
 theorem mul_associator {M N P : Mon_ C} :
-    (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫
-          (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
+    (tensor_μ (M.X ⊗ N.X) P.X (M.X ⊗ N.X) P.X ≫
+          (tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫
         (α_ M.X N.X P.X).hom =
       ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫
-        tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫
-          (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul)) := by
+        tensor_μ M.X (N.X ⊗ P.X) M.X (N.X ⊗ P.X) ≫
+          (M.mul ⊗ tensor_μ N.X P.X N.X P.X ≫ (N.mul ⊗ P.mul)) := by
   simp only [tensor_obj, prodMonoidal_tensorObj, Category.assoc]
   slice_lhs 2 3 => rw [← Category.id_comp P.mul, tensor_comp]
   slice_lhs 3 4 => rw [associator_naturality]
@@ -397,7 +397,7 @@ theorem mul_associator {M N P : Mon_ C} :
   simp only [Category.assoc]
 
 theorem mul_leftUnitor {M : Mon_ C} :
-    (tensor_μ C (𝟙_ C, M.X) (𝟙_ C, M.X) ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
+    (tensor_μ (𝟙_ C) M.X (𝟙_ C) M.X ≫ ((λ_ (𝟙_ C)).hom ⊗ M.mul)) ≫ (λ_ M.X).hom =
       ((λ_ M.X).hom ⊗ (λ_ M.X).hom) ≫ M.mul := by
   rw [← Category.comp_id (λ_ (𝟙_ C)).hom, ← Category.id_comp M.mul, tensor_comp]
   simp only [tensorHom_id, id_tensorHom]
@@ -406,7 +406,7 @@ theorem mul_leftUnitor {M : Mon_ C} :
   simp only [Category.assoc, Category.id_comp]
 
 theorem mul_rightUnitor {M : Mon_ C} :
-    (tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
+    (tensor_μ M.X (𝟙_ C) M.X (𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =
       ((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul := by
   rw [← Category.id_comp M.mul, ← Category.comp_id (λ_ (𝟙_ C)).hom, tensor_comp]
   simp only [tensorHom_id, id_tensorHom]
@@ -419,7 +419,7 @@ instance monMonoidalStruct : MonoidalCategoryStruct (Mon_ C) :=
   let tensorObj (M N : Mon_ C) : Mon_ C :=
     { X := M.X ⊗ N.X
       one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one)
-      mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)
+      mul := tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul)
       one_mul := Mon_tensor_one_mul M N
       mul_one := Mon_tensor_mul_one M N
       mul_assoc := Mon_tensor_mul_assoc M N }
@@ -457,7 +457,7 @@ theorem tensorObj_one (X Y : Mon_ C) : (X ⊗ Y).one = (λ_ (𝟙_ C)).inv ≫ (
 
 @[simp]
 theorem tensorObj_mul (X Y : Mon_ C) :
-    (X ⊗ Y).mul = tensor_μ C (X.X, Y.X) (X.X, Y.X) ≫ (X.mul ⊗ Y.mul) := rfl
+    (X ⊗ Y).mul = tensor_μ X.X Y.X X.X Y.X ≫ (X.mul ⊗ Y.mul) := rfl
 
 @[simp]
 theorem whiskerLeft_hom {X Y : Mon_ C} (f : X ⟶ Y) (Z : Mon_ C) :
@@ -492,7 +492,7 @@ theorem tensor_one (M N : Mon_ C) : (M ⊗ N).one = (λ_ (𝟙_ C)).inv ≫ (M.o
 
 @[simp]
 theorem tensor_mul (M N : Mon_ C) : (M ⊗ N).mul =
-    tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) := rfl
+    tensor_μ M.X N.X M.X N.X ≫ (M.mul ⊗ N.mul) := rfl
 
 instance monMonoidal : MonoidalCategory (Mon_ C) where
   tensorHom_def := by intros; ext; simp [tensorHom_def]
@@ -514,7 +514,7 @@ variable {C}
 theorem one_braiding {X Y : Mon_ C} : (X ⊗ Y).one ≫ (β_ X.X Y.X).hom = (Y ⊗ X).one := by
   simp only [monMonoidalStruct_tensorObj_X, tensor_one, Category.assoc,
     BraidedCategory.braiding_naturality, braiding_tensorUnit_right, Iso.cancel_iso_inv_left]
-  coherence
+  monoidal
 
 end BraidedCategory
 
diff --git a/Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean b/Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean
index c10f52bc77441..37d92020f5dfd 100644
--- a/Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean
+++ b/Mathlib/CategoryTheory/Monoidal/NaturalTransformation.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 import Mathlib.CategoryTheory.Monoidal.Functor
@@ -189,9 +189,8 @@ def monoidalCounit :
   unit := by
     have eq := h.counit.naturality F.ε
     dsimp at eq ⊢
-    rw [Adjunction.homEquiv_unit, map_inv, map_comp, assoc, assoc, map_inv,
-      ← cancel_mono F.ε, assoc, assoc, assoc, ← eq, IsIso.inv_hom_id_assoc,
-      Adjunction.left_triangle_components, comp_id, id_comp]
+    rw [map_inv, map_comp, assoc, assoc, map_inv, ← cancel_mono F.ε, assoc, assoc, assoc, ← eq,
+      IsIso.inv_hom_id_assoc, Adjunction.left_triangle_components, comp_id, id_comp]
 
 instance [F.IsEquivalence] : IsIso (monoidalUnit F h) := by
   dsimp [monoidalUnit]
diff --git a/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean b/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
index 3fc5f0c7ece26..9b23a5a2e1f1a 100644
--- a/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Simon Hudon
+Authors: Kim Morrison, Simon Hudon
 -/
 import Mathlib.CategoryTheory.Monoidal.Category
 import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
diff --git a/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean b/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean
index eeea00aadcc4d..cd1586181796e 100644
--- a/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean
+++ b/Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Simon Hudon
+Authors: Kim Morrison, Simon Hudon
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
diff --git a/Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean b/Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean
index 043177863cb9e..5395fd61f0a08 100644
--- a/Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean
+++ b/Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Simon Hudon
+Authors: Kim Morrison, Simon Hudon
 -/
 import Mathlib.CategoryTheory.Monoidal.Braided.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
diff --git a/Mathlib/CategoryTheory/Monoidal/Opposite.lean b/Mathlib/CategoryTheory/Monoidal/Opposite.lean
index cecef27b955d5..3866aa1f3735a 100644
--- a/Mathlib/CategoryTheory/Monoidal/Opposite.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Opposite.lean
@@ -1,9 +1,9 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
-import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
 
 /-!
 # Monoidal opposites
@@ -160,8 +160,8 @@ instance monoidalCategoryOp : MonoidalCategory Cᵒᵖ where
   associator_naturality f g h := Quiver.Hom.unop_inj <| by simp
   leftUnitor_naturality f := Quiver.Hom.unop_inj <| by simp
   rightUnitor_naturality f := Quiver.Hom.unop_inj <| by simp
-  triangle X Y := Quiver.Hom.unop_inj <| by dsimp; coherence
-  pentagon W X Y Z := Quiver.Hom.unop_inj <| by dsimp; coherence
+  triangle X Y := Quiver.Hom.unop_inj <| by dsimp; monoidal_coherence
+  pentagon W X Y Z := Quiver.Hom.unop_inj <| by dsimp; monoidal_coherence
 
 section OppositeLemmas
 
@@ -241,7 +241,7 @@ instance monoidalCategoryMop : MonoidalCategory Cᴹᵒᵖ where
   rightUnitor_naturality f := Quiver.Hom.unmop_inj <| by simp
   -- Porting note: Changed `by coherence` to `by simp` below
   triangle X Y := Quiver.Hom.unmop_inj <| by simp
-  pentagon W X Y Z := Quiver.Hom.unmop_inj <| by dsimp; coherence
+  pentagon W X Y Z := Quiver.Hom.unmop_inj <| by dsimp; monoidal_coherence
 
 -- it would be nice if we could autogenerate all of these somehow
 section MonoidalOppositeLemmas
diff --git a/Mathlib/CategoryTheory/Monoidal/Preadditive.lean b/Mathlib/CategoryTheory/Monoidal/Preadditive.lean
index 429a44d694e4c..7133ba4faabc0 100644
--- a/Mathlib/CategoryTheory/Monoidal/Preadditive.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Preadditive.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
 import Mathlib.CategoryTheory.Monoidal.Functor
diff --git a/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
index 9a2f2fa079c42..09afedcc3b9aa 100644
--- a/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Jakob von Raumer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jakob von Raumer
 -/
-import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
 import Mathlib.CategoryTheory.Closed.Monoidal
 import Mathlib.Tactic.ApplyFun
 
@@ -113,11 +113,11 @@ lemma evaluation_coevaluation :
   evaluation_coevaluation'
 
 lemma coevaluation_evaluation'' :
-    Y ◁ η_ X Y ⊗≫ ε_ X Y ▷ Y = ⊗𝟙 := by
+    Y ◁ η_ X Y ⊗≫ ε_ X Y ▷ Y = ⊗𝟙.hom := by
   convert coevaluation_evaluation X Y <;> simp [monoidalComp]
 
 lemma evaluation_coevaluation'' :
-    η_ X Y ▷ X ⊗≫ X ◁ ε_ X Y = ⊗𝟙 := by
+    η_ X Y ▷ X ⊗≫ X ◁ ε_ X Y = ⊗𝟙.hom := by
   convert evaluation_coevaluation X Y <;> simp [monoidalComp]
 
 end ExactPairing
@@ -128,8 +128,8 @@ attribute [reassoc (attr := simp)] ExactPairing.evaluation_coevaluation
 instance exactPairingUnit : ExactPairing (𝟙_ C) (𝟙_ C) where
   coevaluation' := (ρ_ _).inv
   evaluation' := (ρ_ _).hom
-  coevaluation_evaluation' := by rw [← id_tensorHom, ← tensorHom_id]; coherence
-  evaluation_coevaluation' := by rw [← id_tensorHom, ← tensorHom_id]; coherence
+  coevaluation_evaluation' := by monoidal_coherence
+  evaluation_coevaluation' := by monoidal_coherence
 
 /-- A class of objects which have a right dual. -/
 class HasRightDual (X : C) where
@@ -204,9 +204,9 @@ theorem rightAdjointMate_comp {X Y Z : C} [HasRightDual X] [HasRightDual Y] {f :
         _ ◁ η_ X (Xᘁ) ≫ _ ◁ (f ⊗ g) ≫ (α_ (Yᘁ) Y Z).inv ≫ ε_ Y (Yᘁ) ▷ _ ≫ (λ_ Z).hom :=
   calc
     _ = 𝟙 _ ⊗≫ (Yᘁ : C) ◁ η_ X Xᘁ ≫ Yᘁ ◁ f ▷ Xᘁ ⊗≫ (ε_ Y Yᘁ ▷ Xᘁ ≫ 𝟙_ C ◁ g) ⊗≫ 𝟙 _ := by
-      dsimp only [rightAdjointMate]; coherence
+      dsimp only [rightAdjointMate]; monoidal
     _ = _ := by
-      rw [← whisker_exchange, tensorHom_def]; coherence
+      rw [← whisker_exchange, tensorHom_def]; monoidal
 
 theorem leftAdjointMate_comp {X Y Z : C} [HasLeftDual X] [HasLeftDual Y] {f : X ⟶ Y}
     {g : (ᘁX) ⟶ Z} :
@@ -215,9 +215,9 @@ theorem leftAdjointMate_comp {X Y Z : C} [HasLeftDual X] [HasLeftDual Y] {f : X
         η_ (ᘁX : C) X ▷ _ ≫ (g ⊗ f) ▷ _ ≫ (α_ _ _ _).hom ≫ _ ◁ ε_ _ _ ≫ (ρ_ _).hom :=
   calc
     _ = 𝟙 _ ⊗≫ η_ (ᘁX : C) X ▷ (ᘁY) ⊗≫ (ᘁX) ◁ f ▷ (ᘁY) ⊗≫ ((ᘁX) ◁ ε_ (ᘁY) Y ≫ g ▷ 𝟙_ C) ⊗≫ 𝟙 _ := by
-      dsimp only [leftAdjointMate]; coherence
+      dsimp only [leftAdjointMate]; monoidal
     _ = _ := by
-      rw [whisker_exchange, tensorHom_def']; coherence
+      rw [whisker_exchange, tensorHom_def']; monoidal
 
 /-- The composition of right adjoint mates is the adjoint mate of the composition. -/
 @[reassoc]
@@ -231,14 +231,14 @@ theorem comp_rightAdjointMate {X Y Z : C} [HasRightDual X] [HasRightDual Y] [Has
   calc
     _ = 𝟙 _ ⊗≫ (η_ Y Yᘁ ▷ 𝟙_ C ≫ (Y ⊗ Yᘁ) ◁ η_ X Xᘁ) ⊗≫ Y ◁ Yᘁ ◁ f ▷ Xᘁ ⊗≫
         Y ◁ ε_ Y Yᘁ ▷ Xᘁ ⊗≫ g ▷ Xᘁ ⊗≫ 𝟙 _ := by
-      rw [tensorHom_def']; coherence
+      rw [tensorHom_def']; monoidal
     _ = η_ X Xᘁ ⊗≫ (η_ Y Yᘁ ▷ (X ⊗ Xᘁ) ≫ (Y ⊗ Yᘁ) ◁ f ▷ Xᘁ) ⊗≫
         Y ◁ ε_ Y Yᘁ ▷ Xᘁ ⊗≫ g ▷ Xᘁ ⊗≫ 𝟙 _ := by
-      rw [← whisker_exchange]; coherence
+      rw [← whisker_exchange]; monoidal
     _ = η_ X Xᘁ ⊗≫ f ▷ Xᘁ ⊗≫ (η_ Y Yᘁ ▷ Y ⊗≫ Y ◁ ε_ Y Yᘁ) ▷ Xᘁ ⊗≫ g ▷ Xᘁ ⊗≫ 𝟙 _ := by
-      rw [← whisker_exchange]; coherence
+      rw [← whisker_exchange]; monoidal
     _ = η_ X Xᘁ ≫ f ▷ Xᘁ ≫ g ▷ Xᘁ := by
-      rw [evaluation_coevaluation'']; coherence
+      rw [evaluation_coevaluation'']; monoidal
 
 /-- The composition of left adjoint mates is the adjoint mate of the composition. -/
 @[reassoc]
@@ -252,14 +252,14 @@ theorem comp_leftAdjointMate {X Y Z : C} [HasLeftDual X] [HasLeftDual Y] [HasLef
   calc
     _ = 𝟙 _ ⊗≫ ((𝟙_ C) ◁ η_ (ᘁY) Y ≫ η_ (ᘁX) X ▷ ((ᘁY) ⊗ Y)) ⊗≫ (ᘁX) ◁ f ▷ (ᘁY) ▷ Y ⊗≫
         (ᘁX) ◁ ε_ (ᘁY) Y ▷ Y ⊗≫ (ᘁX) ◁ g := by
-      rw [tensorHom_def]; coherence
+      rw [tensorHom_def]; monoidal
     _ = η_ (ᘁX) X ⊗≫ (((ᘁX) ⊗ X) ◁ η_ (ᘁY) Y ≫ ((ᘁX) ◁ f) ▷ ((ᘁY) ⊗ Y)) ⊗≫
         (ᘁX) ◁ ε_ (ᘁY) Y ▷ Y ⊗≫ (ᘁX) ◁ g := by
-      rw [whisker_exchange]; coherence
+      rw [whisker_exchange]; monoidal
     _ = η_ (ᘁX) X ⊗≫ ((ᘁX) ◁ f) ⊗≫ (ᘁX) ◁ (Y ◁ η_ (ᘁY) Y ⊗≫ ε_ (ᘁY) Y ▷ Y) ⊗≫ (ᘁX) ◁ g := by
-      rw [whisker_exchange]; coherence
+      rw [whisker_exchange]; monoidal
     _ = η_ (ᘁX) X ≫ (ᘁX) ◁ f ≫ (ᘁX) ◁ g := by
-      rw [coevaluation_evaluation'']; coherence
+      rw [coevaluation_evaluation'']; monoidal
 
 /-- Given an exact pairing on `Y Y'`,
 we get a bijection on hom-sets `(Y' ⊗ X ⟶ Z) ≃ (X ⟶ Y ⊗ Z)`
@@ -276,19 +276,19 @@ def tensorLeftHomEquiv (X Y Y' Z : C) [ExactPairing Y Y'] : (Y' ⊗ X ⟶ Z) ≃
   left_inv f := by
     calc
       _ = 𝟙 _ ⊗≫ Y' ◁ η_ Y Y' ▷ X ⊗≫ ((Y' ⊗ Y) ◁ f ≫ ε_ Y Y' ▷ Z) ⊗≫ 𝟙 _ := by
-        coherence
+        monoidal
       _ = 𝟙 _ ⊗≫ (Y' ◁ η_ Y Y' ⊗≫ ε_ Y Y' ▷ Y') ▷ X ⊗≫ f := by
-        rw [whisker_exchange]; coherence
+        rw [whisker_exchange]; monoidal
       _ = f := by
-        rw [coevaluation_evaluation'']; coherence
+        rw [coevaluation_evaluation'']; monoidal
   right_inv f := by
     calc
       _ = 𝟙 _ ⊗≫ (η_ Y Y' ▷ X ≫ (Y ⊗ Y') ◁ f) ⊗≫ Y ◁ ε_ Y Y' ▷ Z ⊗≫ 𝟙 _ := by
-        coherence
+        monoidal
       _ = f ⊗≫ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ▷ Z ⊗≫ 𝟙 _ := by
-        rw [← whisker_exchange]; coherence
+        rw [← whisker_exchange]; monoidal
       _ = f := by
-        rw [evaluation_coevaluation'']; coherence
+        rw [evaluation_coevaluation'']; monoidal
 
 /-- Given an exact pairing on `Y Y'`,
 we get a bijection on hom-sets `(X ⊗ Y ⟶ Z) ≃ (X ⟶ Z ⊗ Y')`
@@ -300,19 +300,19 @@ def tensorRightHomEquiv (X Y Y' Z : C) [ExactPairing Y Y'] : (X ⊗ Y ⟶ Z) ≃
   left_inv f := by
     calc
       _ = 𝟙 _ ⊗≫ X ◁ η_ Y Y' ▷ Y ⊗≫ (f ▷ (Y' ⊗ Y) ≫ Z ◁ ε_ Y Y') ⊗≫ 𝟙 _ := by
-        coherence
+        monoidal
       _ = 𝟙 _ ⊗≫ X ◁ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ⊗≫ f := by
-        rw [← whisker_exchange]; coherence
+        rw [← whisker_exchange]; monoidal
       _ = f := by
-        rw [evaluation_coevaluation'']; coherence
+        rw [evaluation_coevaluation'']; monoidal
   right_inv f := by
     calc
       _ = 𝟙 _ ⊗≫ (X ◁ η_ Y Y' ≫ f ▷ (Y ⊗ Y')) ⊗≫ Z ◁ ε_ Y Y' ▷ Y' ⊗≫ 𝟙 _ := by
-        coherence
+        monoidal
       _ = f ⊗≫ Z ◁ (Y' ◁ η_ Y Y' ⊗≫ ε_ Y Y' ▷ Y') ⊗≫ 𝟙 _ := by
-        rw [whisker_exchange]; coherence
+        rw [whisker_exchange]; monoidal
       _ = f := by
-        rw [coevaluation_evaluation'']; coherence
+        rw [coevaluation_evaluation'']; monoidal
 
 theorem tensorLeftHomEquiv_naturality {X Y Y' Z Z' : C} [ExactPairing Y Y'] (f : Y' ⊗ X ⟶ Z)
     (g : Z ⟶ Z') :
@@ -387,10 +387,10 @@ theorem tensorLeftHomEquiv_symm_coevaluation_comp_whiskerLeft {Y Y' Z : C} [Exac
     (f : Y' ⟶ Z) : (tensorLeftHomEquiv _ _ _ _).symm (η_ _ _ ≫ Y ◁ f) = (ρ_ _).hom ≫ f := by
   calc
     _ = Y' ◁ η_ Y Y' ⊗≫ ((Y' ⊗ Y) ◁ f ≫ ε_ Y Y' ▷ Z) ⊗≫ 𝟙 _ := by
-      dsimp [tensorLeftHomEquiv]; coherence
+      dsimp [tensorLeftHomEquiv]; monoidal
     _ = (Y' ◁ η_ Y Y' ⊗≫ ε_ Y Y' ▷ Y') ⊗≫ f := by
-      rw [whisker_exchange]; coherence
-    _ = _ := by rw [coevaluation_evaluation'']; coherence
+      rw [whisker_exchange]; monoidal
+    _ = _ := by rw [coevaluation_evaluation'']; monoidal
 
 @[simp]
 theorem tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight {X Y : C} [HasRightDual X]
@@ -411,22 +411,22 @@ theorem tensorRightHomEquiv_symm_coevaluation_comp_whiskerRight {Y Y' Z : C} [Ex
     (f : Y ⟶ Z) : (tensorRightHomEquiv _ Y _ _).symm (η_ Y Y' ≫ f ▷ Y') = (λ_ _).hom ≫ f :=
   calc
     _ = η_ Y Y' ▷ Y ⊗≫ (f ▷ (Y' ⊗ Y) ≫ Z ◁ ε_ Y Y') ⊗≫ 𝟙 _ := by
-      dsimp [tensorRightHomEquiv]; coherence
+      dsimp [tensorRightHomEquiv]; monoidal
     _ = (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ⊗≫ f := by
-      rw [← whisker_exchange]; coherence
+      rw [← whisker_exchange]; monoidal
     _ = _ := by
-      rw [evaluation_coevaluation'']; coherence
+      rw [evaluation_coevaluation'']; monoidal
 
 @[simp]
 theorem tensorLeftHomEquiv_whiskerLeft_comp_evaluation {Y Z : C} [HasLeftDual Z] (f : Y ⟶ ᘁZ) :
     (tensorLeftHomEquiv _ _ _ _) (Z ◁ f ≫ ε_ _ _) = f ≫ (ρ_ _).inv :=
   calc
     _ = 𝟙 _ ⊗≫ (η_ (ᘁZ : C) Z ▷ Y ≫ ((ᘁZ) ⊗ Z) ◁ f) ⊗≫ (ᘁZ) ◁ ε_ (ᘁZ) Z := by
-      dsimp [tensorLeftHomEquiv]; coherence
+      dsimp [tensorLeftHomEquiv]; monoidal
     _ = f ⊗≫ (η_ (ᘁZ) Z ▷ (ᘁZ) ⊗≫ (ᘁZ) ◁ ε_ (ᘁZ) Z) := by
-      rw [← whisker_exchange]; coherence
+      rw [← whisker_exchange]; monoidal
     _ = _ := by
-      rw [evaluation_coevaluation'']; coherence
+      rw [evaluation_coevaluation'']; monoidal
 
 @[simp]
 theorem tensorLeftHomEquiv_whiskerRight_comp_evaluation {X Y : C} [HasLeftDual X] [HasLeftDual Y]
@@ -445,11 +445,11 @@ theorem tensorRightHomEquiv_whiskerRight_comp_evaluation {X Y : C} [HasRightDual
     (tensorRightHomEquiv _ _ _ _) (f ▷ X ≫ ε_ X (Xᘁ)) = f ≫ (λ_ _).inv :=
   calc
     _ = 𝟙 _ ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ := by
-      dsimp [tensorRightHomEquiv]; coherence
+      dsimp [tensorRightHomEquiv]; monoidal
     _ = f ⊗≫ (Xᘁ ◁ η_ X Xᘁ ⊗≫ ε_ X Xᘁ ▷ Xᘁ) := by
-      rw [whisker_exchange]; coherence
+      rw [whisker_exchange]; monoidal
     _ = _ := by
-      rw [coevaluation_evaluation'']; coherence
+      rw [coevaluation_evaluation'']; monoidal
 
 -- Next four lemmas passing `fᘁ` or `ᘁf` through (co)evaluations.
 @[reassoc]
@@ -483,28 +483,26 @@ def exactPairingCongrLeft {X X' Y : C} [ExactPairing X' Y] (i : X ≅ X') : Exac
   evaluation_coevaluation' :=
     calc
       _ = η_ X' Y ▷ X ⊗≫ (i.inv ▷ (Y ⊗ X) ≫ X ◁ (Y ◁ i.hom)) ⊗≫ X ◁ ε_ X' Y := by
-        coherence
+        monoidal
       _ = 𝟙 _ ⊗≫ (η_ X' Y ▷ X ≫ (X' ⊗ Y) ◁ i.hom) ⊗≫
           (i.inv ▷ (Y ⊗ X') ≫ X ◁ ε_ X' Y) ⊗≫ 𝟙 _ := by
-        rw [← whisker_exchange]; coherence
+        rw [← whisker_exchange]; monoidal
       _ = 𝟙 _ ⊗≫ i.hom ⊗≫ (η_ X' Y ▷ X' ⊗≫ X' ◁ ε_ X' Y) ⊗≫ i.inv ⊗≫ 𝟙 _ := by
-        rw [← whisker_exchange, ← whisker_exchange]; coherence
+        rw [← whisker_exchange, ← whisker_exchange]; monoidal
       _ = 𝟙 _ ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 _ := by
-        rw [evaluation_coevaluation'']; coherence
+        rw [evaluation_coevaluation'']; monoidal
       _ = (λ_ X).hom ≫ (ρ_ X).inv := by
         rw [Iso.hom_inv_id]
-        -- coherence failed
-        simp [monoidalComp]
+        monoidal
   coevaluation_evaluation' := by
     calc
       _ = Y ◁ η_ X' Y ≫ Y ◁ (i.inv ≫ i.hom) ▷ Y ⊗≫ ε_ X' Y ▷ Y := by
-        coherence
+        monoidal
       _ = Y ◁ η_ X' Y ⊗≫ ε_ X' Y ▷ Y := by
-        rw [Iso.inv_hom_id]; coherence
+        rw [Iso.inv_hom_id]; monoidal
       _ = _ := by
         rw [coevaluation_evaluation'']
-        -- coherence failed
-        simp [monoidalComp]
+        monoidal
 
 /-- Transport an exact pairing across an isomorphism in the second argument. -/
 def exactPairingCongrRight {X Y Y' : C} [ExactPairing X Y'] (i : Y ≅ Y') : ExactPairing X Y where
@@ -513,28 +511,26 @@ def exactPairingCongrRight {X Y Y' : C} [ExactPairing X Y'] (i : Y ≅ Y') : Exa
   evaluation_coevaluation' := by
     calc
       _ = η_ X Y' ▷ X ⊗≫ X ◁ (i.inv ≫ i.hom) ▷ X ≫ X ◁ ε_ X Y' := by
-        coherence
+        monoidal
       _ = η_ X Y' ▷ X ⊗≫ X ◁ ε_ X Y' := by
-        rw [Iso.inv_hom_id]; coherence
+        rw [Iso.inv_hom_id]; monoidal
       _ = _ := by
         rw [evaluation_coevaluation'']
-        -- coherence failed
-        simp [monoidalComp]
+        monoidal
   coevaluation_evaluation' :=
     calc
       _ = Y ◁ η_ X Y' ⊗≫ (Y ◁ (X ◁ i.inv) ≫ i.hom ▷ (X ⊗ Y)) ⊗≫ ε_ X Y' ▷ Y := by
-        coherence
+        monoidal
       _ = 𝟙 _ ⊗≫ (Y ◁ η_ X Y' ≫ i.hom ▷ (X ⊗ Y')) ⊗≫
           ((Y' ⊗ X) ◁ i.inv ≫ ε_ X Y' ▷ Y) ⊗≫ 𝟙 _ := by
-        rw [whisker_exchange]; coherence
+        rw [whisker_exchange]; monoidal
       _ = 𝟙 _ ⊗≫ i.hom ⊗≫ (Y' ◁ η_ X Y' ⊗≫ ε_ X Y' ▷ Y') ⊗≫ i.inv ⊗≫ 𝟙 _ := by
-        rw [whisker_exchange, whisker_exchange]; coherence
+        rw [whisker_exchange, whisker_exchange]; monoidal
       _ = 𝟙 _ ⊗≫ (i.hom ≫ i.inv) ⊗≫ 𝟙 _ := by
-        rw [coevaluation_evaluation'']; coherence
+        rw [coevaluation_evaluation'']; monoidal
       _ = (ρ_ Y).hom ≫ (λ_ Y).inv := by
         rw [Iso.hom_inv_id]
-        -- coherence failed
-        simp [monoidalComp]
+        monoidal
 
 /-- Transport an exact pairing across isomorphisms. -/
 def exactPairingCongr {X X' Y Y' : C} [ExactPairing X' Y'] (i : X ≅ X') (j : Y ≅ Y') :
diff --git a/Mathlib/CategoryTheory/Monoidal/Rigid/Braided.lean b/Mathlib/CategoryTheory/Monoidal/Rigid/Braided.lean
index be429c98c1740..6e6d9d68e265d 100644
--- a/Mathlib/CategoryTheory/Monoidal/Rigid/Braided.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Rigid/Braided.lean
@@ -25,13 +25,13 @@ private theorem coevaluation_evaluation_braided' [inst : ExactPairing X Y] :
   /- Whitney trick transcribed: https://mathoverflow.net/a/162729/493261 -/
   calc
     _ = 𝟙 X ⊗≫ X ◁ η_ X Y ⊗≫ (X ◁ (β_ Y X).inv ⊗≫ (β_ X Y).hom ▷ X) ⊗≫ ε_ X Y ▷ X ⊗≫ 𝟙 X := by
-      coherence
+      monoidal
     _ = 𝟙 X ⊗≫ X ◁ η_ X Y ⊗≫ (𝟙 (X ⊗ X ⊗ Y) ⊗≫ (β_ X X).hom ▷ Y ⊗≫ X ◁ (β_ X Y).hom
           ⊗≫ (β_ Y X).inv ▷ X ⊗≫ Y ◁ (β_ X X).inv ⊗≫ 𝟙 ((Y ⊗ X) ⊗ X)) ⊗≫ ε_ X Y ▷ X ⊗≫ 𝟙 X := by
       congr 3
-      simp only [monoidalComp, MonoidalCoherence.assoc'_hom, MonoidalCoherence.whiskerRight_hom,
-        MonoidalCoherence.refl_hom, whiskerRight_tensor, id_whiskerRight, id_comp, Iso.inv_hom_id,
-        MonoidalCoherence.assoc_hom, comp_id]
+      simp only [monoidalComp, MonoidalCoherence.assoc'_iso, MonoidalCoherence.whiskerRight_iso,
+        MonoidalCoherence.refl_iso, whiskerRightIso_refl, Iso.refl_trans, Iso.symm_hom,
+        MonoidalCoherence.assoc_iso, Iso.trans_refl, comp_id, id_comp]
       rw [← IsIso.eq_inv_comp]
       repeat rw [← assoc]
       iterate 5 rw [← IsIso.comp_inv_eq]
@@ -49,7 +49,7 @@ private theorem evaluation_coevaluation_braided' [inst : ExactPairing X Y] :
   rw [Iso.eq_comp_inv, ← Iso.inv_comp_eq_id]
   calc
     _ = 𝟙 Y ⊗≫ η_ X Y ▷ Y ⊗≫ ((β_ Y X).inv ▷ Y ⊗≫ Y ◁ (β_ X Y).hom) ≫ Y ◁ ε_ X Y ⊗≫ 𝟙 Y := by
-      coherence
+      monoidal
     _ = 𝟙 Y ⊗≫ η_ X Y ▷ Y ⊗≫ (𝟙 ((X ⊗ Y) ⊗ Y) ⊗≫ X ◁ (β_ Y Y).hom ⊗≫ (β_ X Y).hom ▷ Y
         ⊗≫ Y ◁ (β_ Y X).inv ⊗≫ (β_ Y Y).inv ▷ X ⊗≫ 𝟙 (Y ⊗ Y ⊗ X)) ⊗≫ Y ◁ ε_ X Y ⊗≫ 𝟙 Y := by
       congr 3
diff --git a/Mathlib/CategoryTheory/Monoidal/Rigid/FunctorCategory.lean b/Mathlib/CategoryTheory/Monoidal/Rigid/FunctorCategory.lean
index d458fd401af64..f520a501aff97 100644
--- a/Mathlib/CategoryTheory/Monoidal/Rigid/FunctorCategory.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Rigid/FunctorCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Rigid.Basic
 import Mathlib.CategoryTheory.Monoidal.FunctorCategory
diff --git a/Mathlib/CategoryTheory/Monoidal/Rigid/OfEquivalence.lean b/Mathlib/CategoryTheory/Monoidal/Rigid/OfEquivalence.lean
index 70e84ca5c9f1d..7eb07eb856049 100644
--- a/Mathlib/CategoryTheory/Monoidal/Rigid/OfEquivalence.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Rigid/OfEquivalence.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Rigid.Basic
 
diff --git a/Mathlib/CategoryTheory/Monoidal/Subcategory.lean b/Mathlib/CategoryTheory/Monoidal/Subcategory.lean
index bf51655e04ec4..5e77ff34ecfb7 100644
--- a/Mathlib/CategoryTheory/Monoidal/Subcategory.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Subcategory.lean
@@ -198,15 +198,16 @@ instance fullMonoidalClosedSubcategory : MonoidalClosed (FullSubcategory P) wher
     { rightAdj := FullSubcategory.lift P (fullSubcategoryInclusion P ⋙ ihom X.1)
         fun Y => prop_ihom X.2 Y.2
       adj :=
-        Adjunction.mkOfUnitCounit
         { unit :=
           { app := fun Y => (ihom.coev X.1).app Y.1
             naturality := fun Y Z f => ihom.coev_naturality X.1 f }
           counit :=
           { app := fun Y => (ihom.ev X.1).app Y.1
             naturality := fun Y Z f => ihom.ev_naturality X.1 f }
-          left_triangle := by ext Y; simp [FullSubcategory.comp_def, FullSubcategory.id_def]
-          right_triangle := by ext Y; simp [FullSubcategory.comp_def, FullSubcategory.id_def] } }
+          left_triangle_components := fun X ↦
+            by simp [FullSubcategory.comp_def, FullSubcategory.id_def]
+          right_triangle_components := fun Y ↦
+            by simp [FullSubcategory.comp_def, FullSubcategory.id_def] } }
 
 @[simp]
 theorem fullMonoidalClosedSubcategory_ihom_obj (X Y : FullSubcategory P) :
diff --git a/Mathlib/CategoryTheory/Monoidal/Tor.lean b/Mathlib/CategoryTheory/Monoidal/Tor.lean
index 4f9a2de2e1361..51db95353b832 100644
--- a/Mathlib/CategoryTheory/Monoidal/Tor.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Tor.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Abelian.LeftDerived
 import Mathlib.CategoryTheory.Monoidal.Preadditive
diff --git a/Mathlib/CategoryTheory/Monoidal/Transport.lean b/Mathlib/CategoryTheory/Monoidal/Transport.lean
index dd5bd2ff741f2..19d755f6f034b 100644
--- a/Mathlib/CategoryTheory/Monoidal/Transport.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Transport.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
 
@@ -87,7 +87,7 @@ The functor `F` must preserve all the data parts of the monoidal structure betwe
 categories.
 
 -/
-abbrev induced [MonoidalCategoryStruct D] (F : D ⥤ C) [F.Faithful]
+def induced [MonoidalCategoryStruct D] (F : D ⥤ C) [F.Faithful]
     (fData : InducingFunctorData F) :
     MonoidalCategory.{v₂} D where
   tensorHom_def {X₁ Y₁ X₂ Y₂} f g := F.map_injective <| by
@@ -135,7 +135,7 @@ def fromInduced [MonoidalCategoryStruct D] (F : D ⥤ C) [F.Faithful]
 
 /-- Transport a monoidal structure along an equivalence of (plain) categories.
 -/
-@[simps]
+@[simps (config := .lemmasOnly)]
 def transportStruct (e : C ≌ D) : MonoidalCategoryStruct.{v₂} D where
   tensorObj X Y := e.functor.obj (e.inverse.obj X ⊗ e.inverse.obj Y)
   whiskerLeft X _ _ f := e.functor.map (e.inverse.obj X ◁ e.inverse.map f)
@@ -144,16 +144,17 @@ def transportStruct (e : C ≌ D) : MonoidalCategoryStruct.{v₂} D where
   tensorUnit := e.functor.obj (𝟙_ C)
   associator X Y Z :=
     e.functor.mapIso
-      (((e.unitIso.app _).symm ⊗ Iso.refl _) ≪≫
+      (whiskerRightIso (e.unitIso.app _).symm _ ≪≫
         α_ (e.inverse.obj X) (e.inverse.obj Y) (e.inverse.obj Z) ≪≫
-        (Iso.refl _ ⊗ e.unitIso.app _))
+        whiskerLeftIso _ (e.unitIso.app _))
   leftUnitor X :=
-    e.functor.mapIso (((e.unitIso.app _).symm ⊗ Iso.refl _) ≪≫ λ_ (e.inverse.obj X)) ≪≫
+    e.functor.mapIso ((whiskerRightIso (e.unitIso.app _).symm _) ≪≫ λ_ (e.inverse.obj X)) ≪≫
       e.counitIso.app _
   rightUnitor X :=
-    e.functor.mapIso ((Iso.refl _ ⊗ (e.unitIso.app _).symm) ≪≫ ρ_ (e.inverse.obj X)) ≪≫
+    e.functor.mapIso ((whiskerLeftIso _ (e.unitIso.app _).symm) ≪≫ ρ_ (e.inverse.obj X)) ≪≫
       e.counitIso.app _
 
+attribute [local simp] transportStruct in
 /-- Transport a monoidal structure along an equivalence of (plain) categories.
 -/
 def transport (e : C ≌ D) : MonoidalCategory.{v₂} D :=
diff --git a/Mathlib/CategoryTheory/Monoidal/Types/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Types/Basic.lean
index 77a4726af04a5..9904eee364779 100644
--- a/Mathlib/CategoryTheory/Monoidal/Types/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Types/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison
+Authors: Michael Jendrusch, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Functor
 import Mathlib.CategoryTheory.ChosenFiniteProducts
diff --git a/Mathlib/CategoryTheory/Monoidal/Types/Coyoneda.lean b/Mathlib/CategoryTheory/Monoidal/Types/Coyoneda.lean
index 633d6872c7ad0..475acc453b566 100644
--- a/Mathlib/CategoryTheory/Monoidal/Types/Coyoneda.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Types/Coyoneda.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison
+Authors: Michael Jendrusch, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.Types.Basic
 import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
diff --git a/Mathlib/CategoryTheory/Monoidal/Types/Symmetric.lean b/Mathlib/CategoryTheory/Monoidal/Types/Symmetric.lean
index 69248ef1f0384..93ceaacca659f 100644
--- a/Mathlib/CategoryTheory/Monoidal/Types/Symmetric.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Types/Symmetric.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Michael Jendrusch. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Michael Jendrusch, Scott Morrison
+Authors: Michael Jendrusch, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Symmetric
 import Mathlib.CategoryTheory.Monoidal.Types.Basic
diff --git a/Mathlib/CategoryTheory/MorphismProperty/Basic.lean b/Mathlib/CategoryTheory/MorphismProperty/Basic.lean
index 2590d4b476d56..0a3a382da8016 100644
--- a/Mathlib/CategoryTheory/MorphismProperty/Basic.lean
+++ b/Mathlib/CategoryTheory/MorphismProperty/Basic.lean
@@ -13,8 +13,11 @@ import Mathlib.Order.CompleteBooleanAlgebra
 We provide the basic framework for talking about properties of morphisms.
 The following meta-property is defined
 
-* `RespectsIso`: `P` respects isomorphisms if `P f → P (e ≫ f)` and `P f → P (f ≫ e)`, where
-  `e` is an isomorphism.
+* `RespectsLeft P Q`: `P` respects the property `Q` on the left if `P f → P (i ≫ f)` where
+  `i` satisfies `Q`.
+* `RespectsRight P Q`: `P` respects the property `Q` on the right if `P f → P (f ≫ i)` where
+  `i` satisfies `Q`.
+* `Respects`: `P` respects `Q` if `P` respects `Q` both on the left and on the right.
 
 -/
 
@@ -83,7 +86,7 @@ lemma inverseImage_iff (P : MorphismProperty D) (F : C ⥤ D) {X Y : C} (f : X 
 
 /-- The image (up to isomorphisms) of a `MorphismProperty C` by a functor `C ⥤ D` -/
 def map (P : MorphismProperty C) (F : C ⥤ D) : MorphismProperty D := fun _ _ f =>
-  ∃ (X' Y' : C)  (f' : X' ⟶ Y') (_ : P f'), Nonempty (Arrow.mk (F.map f') ≅ Arrow.mk f)
+  ∃ (X' Y' : C) (f' : X' ⟶ Y') (_ : P f'), Nonempty (Arrow.mk (F.map f') ≅ Arrow.mk f)
 
 lemma map_mem_map (P : MorphismProperty C) (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) (hf : P f) :
     (P.map F) (F.map f) := ⟨X, Y, f, hf, ⟨Iso.refl _⟩⟩
@@ -93,22 +96,79 @@ lemma monotone_map (F : C ⥤ D) :
   intro P Q h X Y f ⟨X', Y', f', hf', ⟨e⟩⟩
   exact ⟨X', Y', f', h _ hf', ⟨e⟩⟩
 
-/-- A morphism property `RespectsIso` if it still holds when composed with an isomorphism -/
-class RespectsIso (P : MorphismProperty C) : Prop where
-  precomp {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) (hf : P f) : P (e.hom ≫ f)
-  postcomp {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) (hf : P f) : P (f ≫ e.hom)
+/-- A morphism property `P` satisfies `P.RespectsRight Q` if it is stable under post-composition
+with morphisms satisfying `Q`, i.e. whenever `P` holds for `f` and `Q` holds for `i` then `P`
+holds for `f ≫ i`. -/
+class RespectsRight (P Q : MorphismProperty C) : Prop where
+  postcomp {X Y Z : C} (i : Y ⟶ Z) (hi : Q i) (f : X ⟶ Y) (hf : P f) : P (f ≫ i)
 
-instance RespectsIso.op (P : MorphismProperty C) [h : RespectsIso P] : RespectsIso P.op :=
-  ⟨fun e f hf => h.2 e.unop f.unop hf, fun e f hf => h.1 e.unop f.unop hf⟩
+/-- A morphism property `P` satisfies `P.RespectsLeft Q` if it is stable under
+pre-composition with morphisms satisfying `Q`, i.e. whenever `P` holds for `f`
+and `Q` holds for `i` then `P` holds for `i ≫ f`. -/
+class RespectsLeft (P Q : MorphismProperty C) : Prop where
+  precomp {X Y Z : C} (i : X ⟶ Y) (hi : Q i) (f : Y ⟶ Z) (hf : P f) : P (i ≫ f)
 
-instance RespectsIso.unop (P : MorphismProperty Cᵒᵖ) [h : RespectsIso P] : RespectsIso P.unop :=
-  ⟨fun e f hf => h.2 e.op f.op hf, fun e f hf => h.1 e.op f.op hf⟩
+/-- A morphism property `P` satisfies `P.Respects Q` if it is stable under composition on the
+left and right by morphisms satisfying `Q`. -/
+class Respects (P Q : MorphismProperty C) extends P.RespectsLeft Q, P.RespectsRight Q : Prop where
 
-/-- The intersection of two isomorphism respecting morphism properties respects isomorphisms. -/
-instance RespectsIso.inf (P Q : MorphismProperty C) [RespectsIso P] [RespectsIso Q] :
-    RespectsIso (P ⊓ Q) where
-  precomp e f hf := ⟨RespectsIso.precomp e f hf.left, RespectsIso.precomp e f hf.right⟩
-  postcomp e f hf := ⟨RespectsIso.postcomp e f hf.left, RespectsIso.postcomp e f hf.right⟩
+instance (P Q : MorphismProperty C) [P.RespectsLeft Q] [P.RespectsRight Q] : P.Respects Q where
+
+instance (P Q : MorphismProperty C) [P.RespectsLeft Q] : P.op.RespectsRight Q.op where
+  postcomp i hi f hf := RespectsLeft.precomp (Q := Q) i.unop hi f.unop hf
+
+instance (P Q : MorphismProperty C) [P.RespectsRight Q] : P.op.RespectsLeft Q.op where
+  precomp i hi f hf := RespectsRight.postcomp (Q := Q) i.unop hi f.unop hf
+
+instance RespectsLeft.inf (P₁ P₂ Q : MorphismProperty C) [P₁.RespectsLeft Q]
+    [P₂.RespectsLeft Q] : (P₁ ⊓ P₂).RespectsLeft Q where
+  precomp i hi f hf := ⟨precomp i hi f hf.left, precomp i hi f hf.right⟩
+
+instance RespectsRight.inf (P₁ P₂ Q : MorphismProperty C) [P₁.RespectsRight Q]
+    [P₂.RespectsRight Q] : (P₁ ⊓ P₂).RespectsRight Q where
+  postcomp i hi f hf := ⟨postcomp i hi f hf.left, postcomp i hi f hf.right⟩
+
+variable (C)
+
+/-- The `MorphismProperty C` satisfied by isomorphisms in `C`. -/
+def isomorphisms : MorphismProperty C := fun _ _ f => IsIso f
+
+/-- The `MorphismProperty C` satisfied by monomorphisms in `C`. -/
+def monomorphisms : MorphismProperty C := fun _ _ f => Mono f
+
+/-- The `MorphismProperty C` satisfied by epimorphisms in `C`. -/
+def epimorphisms : MorphismProperty C := fun _ _ f => Epi f
+
+section
+
+variable {C}
+
+/-- `P` respects isomorphisms, if it respects the morphism property `isomorphisms C`, i.e.
+it is stable under pre- and postcomposition with isomorphisms. -/
+abbrev RespectsIso (P : MorphismProperty C) : Prop := P.Respects (isomorphisms C)
+
+lemma RespectsIso.mk (P : MorphismProperty C)
+    (hprecomp : ∀ {X Y Z : C} (e : X ≅ Y) (f : Y ⟶ Z) (_ : P f), P (e.hom ≫ f))
+    (hpostcomp : ∀ {X Y Z : C} (e : Y ≅ Z) (f : X ⟶ Y) (_ : P f), P (f ≫ e.hom)) :
+    P.RespectsIso where
+  precomp e (_ : IsIso e) f hf := hprecomp (asIso e) f hf
+  postcomp e (_ : IsIso e) f hf := hpostcomp (asIso e) f hf
+
+lemma RespectsIso.precomp (P : MorphismProperty C) [P.RespectsIso] {X Y Z : C} (e : X ⟶ Y)
+    [IsIso e] (f : Y ⟶ Z) (hf : P f) : P (e ≫ f) :=
+  RespectsLeft.precomp (Q := isomorphisms C) e ‹IsIso e› f hf
+
+lemma RespectsIso.postcomp (P : MorphismProperty C) [P.RespectsIso] {X Y Z : C} (e : Y ⟶ Z)
+    [IsIso e] (f : X ⟶ Y) (hf : P f) : P (f ≫ e) :=
+  RespectsRight.postcomp (Q := isomorphisms C) e ‹IsIso e› f hf
+
+instance RespectsIso.op (P : MorphismProperty C) [RespectsIso P] : RespectsIso P.op where
+  precomp e (_ : IsIso e) f hf := postcomp P e.unop f.unop hf
+  postcomp e (_ : IsIso e) f hf := precomp P e.unop f.unop hf
+
+instance RespectsIso.unop (P : MorphismProperty Cᵒᵖ) [RespectsIso P] : RespectsIso P.unop where
+  precomp e (_ : IsIso e) f hf := postcomp P e.op f.op hf
+  postcomp e (_ : IsIso e) f hf := precomp P e.op f.op hf
 
 /-- The closure by isomorphisms of a `MorphismProperty` -/
 def isoClosure (P : MorphismProperty C) : MorphismProperty C :=
@@ -119,10 +179,10 @@ lemma le_isoClosure (P : MorphismProperty C) : P ≤ P.isoClosure :=
 
 instance isoClosure_respectsIso (P : MorphismProperty C) :
     RespectsIso P.isoClosure where
-  precomp := fun e f ⟨_, _, f', hf', ⟨iso⟩⟩ => ⟨_, _, f', hf',
-      ⟨Arrow.isoMk (asIso iso.hom.left ≪≫ e.symm) (asIso iso.hom.right) (by simp)⟩⟩
-  postcomp := fun e f ⟨_, _, f', hf', ⟨iso⟩⟩ => ⟨_, _, f', hf',
-      ⟨Arrow.isoMk (asIso iso.hom.left) (asIso iso.hom.right ≪≫ e) (by simp)⟩⟩
+  precomp := fun e (he : IsIso e) f ⟨_, _, f', hf', ⟨iso⟩⟩ => ⟨_, _, f', hf',
+      ⟨Arrow.isoMk (asIso iso.hom.left ≪≫ asIso (inv e)) (asIso iso.hom.right) (by simp)⟩⟩
+  postcomp := fun e (he : IsIso e) f ⟨_, _, f', hf', ⟨iso⟩⟩ => ⟨_, _, f', hf',
+      ⟨Arrow.isoMk (asIso iso.hom.left) (asIso iso.hom.right ≪≫ asIso e) (by simp)⟩⟩
 
 lemma monotone_isoClosure : Monotone (isoClosure (C := C)) := by
   intro P Q h X Y f ⟨X', Y', f', hf', ⟨e⟩⟩
@@ -130,11 +190,11 @@ lemma monotone_isoClosure : Monotone (isoClosure (C := C)) := by
 
 theorem cancel_left_of_respectsIso (P : MorphismProperty C) [hP : RespectsIso P] {X Y Z : C}
     (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] : P (f ≫ g) ↔ P g :=
-  ⟨fun h => by simpa using hP.1 (asIso f).symm (f ≫ g) h, hP.1 (asIso f) g⟩
+  ⟨fun h => by simpa using RespectsIso.precomp P (inv f) (f ≫ g) h, RespectsIso.precomp P f g⟩
 
 theorem cancel_right_of_respectsIso (P : MorphismProperty C) [hP : RespectsIso P] {X Y Z : C}
     (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] : P (f ≫ g) ↔ P f :=
-  ⟨fun h => by simpa using hP.2 (asIso g).symm (f ≫ g) h, hP.2 (asIso g) f⟩
+  ⟨fun h => by simpa using RespectsIso.postcomp P (inv g) (f ≫ g) h, RespectsIso.postcomp P g f⟩
 
 theorem arrow_iso_iff (P : MorphismProperty C) [RespectsIso P] {f g : Arrow C}
     (e : f ≅ g) : P f.hom ↔ P g.hom := by
@@ -146,16 +206,13 @@ theorem arrow_mk_iso_iff (P : MorphismProperty C) [RespectsIso P] {W X Y Z : C}
   P.arrow_iso_iff e
 
 theorem RespectsIso.of_respects_arrow_iso (P : MorphismProperty C)
-    (hP : ∀ (f g : Arrow C) (_ : f ≅ g) (_ : P f.hom), P g.hom) : RespectsIso P := by
-  constructor
-  · intro X Y Z e f hf
-    refine hP (Arrow.mk f) (Arrow.mk (e.hom ≫ f)) (Arrow.isoMk e.symm (Iso.refl _) ?_) hf
-    dsimp
-    simp only [Iso.inv_hom_id_assoc, Category.comp_id]
-  · intro X Y Z e f hf
-    refine hP (Arrow.mk f) (Arrow.mk (f ≫ e.hom)) (Arrow.isoMk (Iso.refl _) e ?_) hf
-    dsimp
-    simp only [Category.id_comp]
+    (hP : ∀ (f g : Arrow C) (_ : f ≅ g) (_ : P f.hom), P g.hom) : RespectsIso P where
+  precomp {X Y Z} e (he : IsIso e) f hf := by
+    refine hP (Arrow.mk f) (Arrow.mk (e ≫ f)) (Arrow.isoMk (asIso (inv e)) (Iso.refl _) ?_) hf
+    simp
+  postcomp {X Y Z} e (he : IsIso e) f hf := by
+    refine hP (Arrow.mk f) (Arrow.mk (f ≫ e)) (Arrow.isoMk (Iso.refl _) (asIso e) ?_) hf
+    simp
 
 lemma isoClosure_eq_iff (P : MorphismProperty C) :
     P.isoClosure = P ↔ P.RespectsIso := by
@@ -227,12 +284,11 @@ lemma map_map (P : MorphismProperty C) (F : C ⥤ D) {E : Type*} [Category E] (G
     exact map_mem_map _ _ _ (map_mem_map _ _ _ hf)
 
 instance RespectsIso.inverseImage (P : MorphismProperty D) [RespectsIso P] (F : C ⥤ D) :
-    RespectsIso (P.inverseImage F) := by
-  constructor
-  all_goals
-    intro X Y Z e f hf
-    simpa [MorphismProperty.inverseImage, cancel_left_of_respectsIso,
-      cancel_right_of_respectsIso] using hf
+    RespectsIso (P.inverseImage F) where
+  precomp {X Y Z} e (he : IsIso e) f hf := by
+    simpa [MorphismProperty.inverseImage, cancel_left_of_respectsIso] using hf
+  postcomp {X Y Z} e (he : IsIso e) f hf := by
+    simpa [MorphismProperty.inverseImage, cancel_right_of_respectsIso] using hf
 
 lemma map_eq_of_iso (P : MorphismProperty C) {F G : C ⥤ D} (e : F ≅ G) :
     P.map F = P.map G := by
@@ -275,17 +331,7 @@ lemma inverseImage_map_eq_of_isEquivalence
   erw [((P.map F).inverseImage_equivalence_inverse_eq_map_functor (F.asEquivalence)), map_map,
     P.map_eq_of_iso F.asEquivalence.unitIso.symm, map_id]
 
-
-variable (C)
-
-/-- The `MorphismProperty C` satisfied by isomorphisms in `C`. -/
-def isomorphisms : MorphismProperty C := fun _ _ f => IsIso f
-
-/-- The `MorphismProperty C` satisfied by monomorphisms in `C`. -/
-def monomorphisms : MorphismProperty C := fun _ _ f => Mono f
-
-/-- The `MorphismProperty C` satisfied by epimorphisms in `C`. -/
-def epimorphisms : MorphismProperty C := fun _ _ f => Epi f
+end
 
 section
 
@@ -313,25 +359,25 @@ theorem epimorphisms.infer_property [hf : Epi f] : (epimorphisms C) f :=
 end
 
 instance RespectsIso.monomorphisms : RespectsIso (monomorphisms C) := by
-  constructor <;>
+  apply RespectsIso.mk <;>
     · intro X Y Z e f
       simp only [monomorphisms.iff]
       intro
       apply mono_comp
 
 instance RespectsIso.epimorphisms : RespectsIso (epimorphisms C) := by
-  constructor <;>
+  apply RespectsIso.mk <;>
     · intro X Y Z e f
       simp only [epimorphisms.iff]
       intro
       apply epi_comp
 
 instance RespectsIso.isomorphisms : RespectsIso (isomorphisms C) := by
-  constructor <;>
+  apply RespectsIso.mk <;>
     · intro X Y Z e f
       simp only [isomorphisms.iff]
       intro
-      infer_instance
+      exact IsIso.comp_isIso
 
 @[deprecated (since := "2024-07-02")] alias RespectsIso.cancel_left_isIso :=
   cancel_left_of_respectsIso
diff --git a/Mathlib/CategoryTheory/MorphismProperty/Composition.lean b/Mathlib/CategoryTheory/MorphismProperty/Composition.lean
index 056a5add8631c..aff2f87e7acc0 100644
--- a/Mathlib/CategoryTheory/MorphismProperty/Composition.lean
+++ b/Mathlib/CategoryTheory/MorphismProperty/Composition.lean
@@ -24,7 +24,7 @@ namespace MorphismProperty
 variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
 
 /-- Typeclass expressing that a morphism property contain identities. -/
-class ContainsIdentities (W : MorphismProperty C) : Prop :=
+class ContainsIdentities (W : MorphismProperty C) : Prop where
   /-- for all `X : C`, the identity of `X` satisfies the morphism property -/
   id_mem : ∀ (X : C), W (𝟙 X)
 
@@ -63,7 +63,7 @@ instance Pi.containsIdentities {J : Type w} {C : J → Type u}
 
 /-- A morphism property satisfies `IsStableUnderComposition` if the composition of
 two such morphisms still falls in the class. -/
-class IsStableUnderComposition (P : MorphismProperty C) : Prop :=
+class IsStableUnderComposition (P : MorphismProperty C) : Prop where
   comp_mem {X Y Z} (f : X ⟶ Y) (g : Y ⟶ Z) : P f → P g → P (f ≫ g)
 
 lemma comp_mem (W : MorphismProperty C) [W.IsStableUnderComposition]
@@ -91,9 +91,9 @@ theorem StableUnderInverse.unop {P : MorphismProperty Cᵒᵖ} (h : StableUnderI
 
 theorem respectsIso_of_isStableUnderComposition {P : MorphismProperty C}
     [P.IsStableUnderComposition] (hP : isomorphisms C ≤ P) :
-    RespectsIso P :=
-  ⟨fun _ _ hf => P.comp_mem _ _ (hP _ (isomorphisms.infer_property _)) hf,
-    fun _ _ hf => P.comp_mem _ _ hf (hP _ (isomorphisms.infer_property _))⟩
+    RespectsIso P := RespectsIso.mk _
+  (fun _ _ hf => P.comp_mem _ _ (hP _ (isomorphisms.infer_property _)) hf)
+    (fun _ _ hf => P.comp_mem _ _ hf (hP _ (isomorphisms.infer_property _)))
 
 instance IsStableUnderComposition.inverseImage {P : MorphismProperty D} [P.IsStableUnderComposition]
     (F : C ⥤ D) : (P.inverseImage F).IsStableUnderComposition where
@@ -129,7 +129,7 @@ end naturalityProperty
 /-- A morphism property is multiplicative if it contains identities and is stable by
 composition. -/
 class IsMultiplicative (W : MorphismProperty C)
-    extends W.ContainsIdentities, W.IsStableUnderComposition : Prop :=
+    extends W.ContainsIdentities, W.IsStableUnderComposition : Prop
 
 namespace IsMultiplicative
 
diff --git a/Mathlib/CategoryTheory/MorphismProperty/Limits.lean b/Mathlib/CategoryTheory/MorphismProperty/Limits.lean
index e41f911a77318..7352e3fdd289a 100644
--- a/Mathlib/CategoryTheory/MorphismProperty/Limits.lean
+++ b/Mathlib/CategoryTheory/MorphismProperty/Limits.lean
@@ -190,7 +190,7 @@ abbrev IsStableUnderProductsOfShape (J : Type*) := W.IsStableUnderLimitsOfShape
 lemma IsStableUnderProductsOfShape.mk (J : Type*)
     [W.RespectsIso] [HasProductsOfShape J C]
     (hW : ∀ (X₁ X₂ : J → C) (f : ∀ j, X₁ j ⟶ X₂ j) (_ : ∀ (j : J), W (f j)),
-      W (Pi.map f)) : W.IsStableUnderProductsOfShape J := by
+      W (Limits.Pi.map f)) : W.IsStableUnderProductsOfShape J := by
   intro X₁ X₂ c₁ c₂ hc₁ hc₂ f hf
   let φ := fun j => f.app (Discrete.mk j)
   have hf' := hW _ _ φ (fun j => hf (Discrete.mk j))
@@ -203,7 +203,7 @@ lemma IsStableUnderProductsOfShape.mk (J : Type*)
   simp
 
 /-- The condition that a property of morphisms is stable by finite products. -/
-class IsStableUnderFiniteProducts : Prop :=
+class IsStableUnderFiniteProducts : Prop where
   isStableUnderProductsOfShape (J : Type) [Finite J] : W.IsStableUnderProductsOfShape J
 
 lemma isStableUnderProductsOfShape_of_isStableUnderFiniteProducts
@@ -225,7 +225,7 @@ theorem diagonal_iff {X Y : C} {f : X ⟶ Y} : P.diagonal f ↔ P (pullback.diag
   Iff.rfl
 
 instance RespectsIso.diagonal [P.RespectsIso] : P.diagonal.RespectsIso := by
-  constructor
+  apply RespectsIso.mk
   · introv H
     rwa [diagonal_iff, pullback.diagonal_comp, P.cancel_left_of_respectsIso,
       P.cancel_left_of_respectsIso, ← P.cancel_right_of_respectsIso _
@@ -260,7 +260,7 @@ def universally (P : MorphismProperty C) : MorphismProperty C := fun X Y f =>
   ∀ ⦃X' Y' : C⦄ (i₁ : X' ⟶ X) (i₂ : Y' ⟶ Y) (f' : X' ⟶ Y') (_ : IsPullback f' i₁ i₂ f), P f'
 
 instance universally_respectsIso (P : MorphismProperty C) : P.universally.RespectsIso := by
-  constructor
+  apply RespectsIso.mk
   · intro X Y Z e f hf X' Z' i₁ i₂ f' H
     have : IsPullback (𝟙 _) (i₁ ≫ e.hom) i₁ e.inv :=
       IsPullback.of_horiz_isIso
diff --git a/Mathlib/CategoryTheory/MorphismProperty/Representable.lean b/Mathlib/CategoryTheory/MorphismProperty/Representable.lean
index 7c561da14382d..91bf09c44907e 100644
--- a/Mathlib/CategoryTheory/MorphismProperty/Representable.lean
+++ b/Mathlib/CategoryTheory/MorphismProperty/Representable.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Calle Sönne, Joël Riou, Ravi Vakil
 -/
 
-import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
+import Mathlib.CategoryTheory.MorphismProperty.Limits
 
 /-!
 
@@ -12,7 +12,7 @@ import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
 
 In this file we define and develop basic results about relatively representable morphisms.
 
-Clasically, a morphism `f : X ⟶ Y` of presheaves is said to be representable if for any morphism
+Classically, a morphism `f : X ⟶ Y` of presheaves is said to be representable if for any morphism
 `g : yoneda.obj X ⟶ G`, there exists a pullback square of the following form
 ```
   yoneda.obj Y --yoneda.map snd--> yoneda.obj X
@@ -32,9 +32,9 @@ representable in the classical case will then be given by `yoneda.relativelyRepr
 ## Main definitions
 Throughout this file, `F : C ⥤ D` is a functor between categories `C` and `D`.
 
-* We define `relativelyRepresentable` as a `MorphsimProperty`. A morphism `f : X ⟶ Y` in `D` is
-  said to be relatively representable if for any `g : F.obj a ⟶ Y`, there exists a pullback square
-  of the following form
+* We define `relativelyRepresentable` as a `MorphismProperty`. A morphism `f : X ⟶ Y` in `D` is
+  said to be relatively representable with respect to `F`, if for any `g : F.obj a ⟶ Y`, there
+  exists a pullback square of the following form
 ```
   F.obj b --F.map snd--> F.obj a
       |                     |
@@ -59,11 +59,19 @@ which is the preimage under `F` of `hf.fst g`.
 * `symmetry` and `symmetryIso` are variants of the fact that pullbacks are symmetric for
   representable morphisms, formulated internally to `C`. We assume that `F` is fully faithful here.
 
+## Main results
+
+* `relativelyRepresentable.isMultiplicative`: The class of relatively representable morphisms is
+  multiplicative.
+* `relativelyRepresentable.stableUnderBaseChange`: Being relatively representable is stable under
+  base change.
+* `relativelyRepresentable.of_isIso`: Isomorphisms are relatively representable.
+
 -/
 
 namespace CategoryTheory
 
-open Category Limits
+open Category Limits MorphismProperty
 
 universe v₁ v₂ u₁ u₂
 
@@ -249,6 +257,35 @@ instance [Full F] [Faithful F] : IsIso (hf'.symmetry hg) :=
 
 end
 
+/-- When `C` has pullbacks, then `F.map f` is representable with respect to `F` for any
+`f : a ⟶ b` in `C`. -/
+lemma map [Full F] [PreservesLimitsOfShape WalkingCospan F] [HasPullbacks C] {a b : C} (f : a ⟶ b) :
+    F.relativelyRepresentable (F.map f) := fun c g ↦ by
+  obtain ⟨g, rfl⟩ := F.map_surjective g
+  refine ⟨Limits.pullback f g, Limits.pullback.snd f g, F.map (Limits.pullback.fst f g), ?_⟩
+  apply F.map_isPullback <| IsPullback.of_hasPullback f g
+
+lemma of_isIso {X Y : D} (f : X ⟶ Y) [IsIso f] : F.relativelyRepresentable f :=
+  fun a g ↦ ⟨a, 𝟙 a, g ≫ CategoryTheory.inv f, IsPullback.of_vert_isIso ⟨by simp⟩⟩
+
+lemma isomorphisms_le : MorphismProperty.isomorphisms D ≤ F.relativelyRepresentable :=
+  fun _ _ f hf ↦ letI : IsIso f := hf; of_isIso F f
+
+instance isMultiplicative : IsMultiplicative F.relativelyRepresentable where
+  id_mem _ := of_isIso F _
+  comp_mem {F G H} f g hf hg := fun X h ↦
+    ⟨hf.pullback (hg.fst h), hf.snd (hg.fst h) ≫ hg.snd h, hf.fst (hg.fst h),
+      by simpa using IsPullback.paste_vert (hf.isPullback (hg.fst h)) (hg.isPullback h)⟩
+
+lemma stableUnderBaseChange : StableUnderBaseChange F.relativelyRepresentable := by
+  intro X Y Y' X' f g f' g' P₁ hg a h
+  refine ⟨hg.pullback (h ≫ f), hg.snd (h ≫ f), ?_, ?_⟩
+  · apply P₁.lift (hg.fst (h ≫ f)) (F.map (hg.snd (h ≫ f)) ≫ h) (by simpa using hg.w (h ≫ f))
+  · apply IsPullback.of_right' (hg.isPullback (h ≫ f)) P₁
+
+instance respectsIso : RespectsIso F.relativelyRepresentable :=
+  (stableUnderBaseChange F).respectsIso
+
 end Functor.relativelyRepresentable
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/NatIso.lean b/Mathlib/CategoryTheory/NatIso.lean
index f90d1af88c7d0..2899c82c95db6 100644
--- a/Mathlib/CategoryTheory/NatIso.lean
+++ b/Mathlib/CategoryTheory/NatIso.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.CategoryTheory.Functor.Category
 import Mathlib.CategoryTheory.Iso
diff --git a/Mathlib/CategoryTheory/NatTrans.lean b/Mathlib/CategoryTheory/NatTrans.lean
index 52013eb207f12..0c12f372eaeac 100644
--- a/Mathlib/CategoryTheory/NatTrans.lean
+++ b/Mathlib/CategoryTheory/NatTrans.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
+Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
 -/
 import Mathlib.Tactic.CategoryTheory.Reassoc
 
diff --git a/Mathlib/CategoryTheory/Noetherian.lean b/Mathlib/CategoryTheory/Noetherian.lean
index 1065af138e416..6b7a78e563ca6 100644
--- a/Mathlib/CategoryTheory/Noetherian.lean
+++ b/Mathlib/CategoryTheory/Noetherian.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.Lattice
 import Mathlib.CategoryTheory.EssentiallySmall
diff --git a/Mathlib/CategoryTheory/Opposites.lean b/Mathlib/CategoryTheory/Opposites.lean
index 8f3c692708837..6d4f00e441da3 100644
--- a/Mathlib/CategoryTheory/Opposites.lean
+++ b/Mathlib/CategoryTheory/Opposites.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison
+Authors: Stephen Morgan, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Equivalence
 
@@ -102,7 +102,7 @@ def unopUnop : Cᵒᵖᵒᵖ ⥤ C where
   obj X := unop (unop X)
   map f := f.unop.unop
 
-/-- The functor from a category to its double-opposite.  -/
+/-- The functor from a category to its double-opposite. -/
 @[simps]
 def opOp : C ⥤ Cᵒᵖᵒᵖ where
   obj X := op (op X)
diff --git a/Mathlib/CategoryTheory/PEmpty.lean b/Mathlib/CategoryTheory/PEmpty.lean
index 3f12752eb7f34..ee9546dc04437 100644
--- a/Mathlib/CategoryTheory/PEmpty.lean
+++ b/Mathlib/CategoryTheory/PEmpty.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.DiscreteCategory
 
diff --git a/Mathlib/CategoryTheory/PUnit.lean b/Mathlib/CategoryTheory/PUnit.lean
index 2f3d7472d33df..d3c89945af0f3 100644
--- a/Mathlib/CategoryTheory/PUnit.lean
+++ b/Mathlib/CategoryTheory/PUnit.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Functor.Const
 import Mathlib.CategoryTheory.DiscreteCategory
diff --git a/Mathlib/CategoryTheory/PathCategory.lean b/Mathlib/CategoryTheory/PathCategory.lean
index e27f6b40b20c9..4f0b3ec849790 100644
--- a/Mathlib/CategoryTheory/PathCategory.lean
+++ b/Mathlib/CategoryTheory/PathCategory.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.Quotient
@@ -59,10 +59,12 @@ def lift {C} [Category C] (φ : V ⥤q C) : Paths V ⥤ C where
       (fun _ f ihp => ihp ≫ φ.map f) Y f
   map_id X := rfl
   map_comp f g := by
-    induction' g with _ _ g' p ih _ _ _
-    · rw [Category.comp_id]
+    induction g with
+    | nil =>
+      rw [Category.comp_id]
       rfl
-    · have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
+    | cons g' p ih =>
+      have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
       rw [this]
       simp only at ih ⊢
       rw [ih, Category.assoc]
@@ -98,10 +100,12 @@ theorem lift_unique {C} [Category C] (φ : V ⥤q C) (Φ : Paths V ⥤ C)
     rfl
   · rintro X Y f
     dsimp [lift]
-    induction' f with _ _ p f' ih
-    · simp only [Category.comp_id]
+    induction f with
+    | nil =>
+      simp only [Category.comp_id]
       apply Functor.map_id
-    · simp only [Category.comp_id, Category.id_comp] at ih ⊢
+    | cons p f' ih =>
+      simp only [Category.comp_id, Category.id_comp] at ih ⊢
       -- Porting note: Had to do substitute `p.cons f'` and `f'.toPath` by their fully qualified
       -- versions in this `have` clause (elsewhere too).
       have : Φ.map (Quiver.Path.cons p f') = Φ.map p ≫ Φ.map (Quiver.Hom.toPath f') := by
diff --git a/Mathlib/CategoryTheory/Pi/Basic.lean b/Mathlib/CategoryTheory/Pi/Basic.lean
index caa1341b3caf6..166b83c21ef2a 100644
--- a/Mathlib/CategoryTheory/Pi/Basic.lean
+++ b/Mathlib/CategoryTheory/Pi/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Simon Hudon, Scott Morrison
+Authors: Simon Hudon, Kim Morrison
 -/
 import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.NatIso
diff --git a/Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean b/Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
index 1a22be0cc03ff..0dd92b9817e6a 100644
--- a/Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
+++ b/Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Adam Topaz, Scott Morrison
+Authors: Adam Topaz, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.ExactFunctor
 import Mathlib.CategoryTheory.Limits.Preserves.Finite
@@ -69,6 +69,8 @@ instance : Additive (𝟭 C) where
 instance {E : Type*} [Category E] [Preadditive E] (G : D ⥤ E) [Functor.Additive G] :
     Additive (F ⋙ G) where
 
+instance {J : Type*} [Category J] (j : J) : ((evaluation J C).obj j).Additive where
+
 @[simp]
 theorem map_neg {X Y : C} {f : X ⟶ Y} : F.map (-f) = -F.map f :=
   (F.mapAddHom : (X ⟶ Y) →+ (F.obj X ⟶ F.obj Y)).map_neg _
diff --git a/Mathlib/CategoryTheory/Preadditive/Biproducts.lean b/Mathlib/CategoryTheory/Preadditive/Biproducts.lean
index 095fb3ea4d5cf..5b06f3702ea3b 100644
--- a/Mathlib/CategoryTheory/Preadditive/Biproducts.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Biproducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Ext
 import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
diff --git a/Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean b/Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean
index 4ff63ae23affb..4dfeacbfed5e2 100644
--- a/Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean
+++ b/Mathlib/CategoryTheory/Preadditive/HomOrthogonal.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Linear.Basic
 import Mathlib.CategoryTheory.Preadditive.Biproducts
diff --git a/Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean b/Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean
index 4c932bdd5de89..9f756b5cdb943 100644
--- a/Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean
+++ b/Mathlib/CategoryTheory/Preadditive/InjectiveResolution.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jujian Zhang, Scott Morrison, Joël Riou
+Authors: Jujian Zhang, Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Homology.QuasiIso
 import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
diff --git a/Mathlib/CategoryTheory/Preadditive/Mat.lean b/Mathlib/CategoryTheory/Preadditive/Mat.lean
index 8d93ea7d3ea5b..d3d9783dc37c7 100644
--- a/Mathlib/CategoryTheory/Preadditive/Mat.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Mat.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Algebra.BigOperators.Pi
@@ -184,11 +184,11 @@ instance hasFiniteBiproducts : HasFiniteBiproducts (Mat_ C) where
               ext x y
               dsimp
               simp_rw [dite_comp, comp_dite]
-              simp only [ite_self, dite_eq_ite, dif_ctx_congr, Limits.comp_zero, Limits.zero_comp,
+              simp only [ite_self, dite_eq_ite, Limits.comp_zero, Limits.zero_comp,
                 eqToHom_trans, Finset.sum_congr]
               erw [Finset.sum_sigma]
               dsimp
-              simp only [if_congr, if_true, dif_ctx_congr, Finset.sum_dite_irrel, Finset.mem_univ,
+              simp only [if_true, Finset.sum_dite_irrel, Finset.mem_univ,
                 Finset.sum_const_zero, Finset.sum_congr, Finset.sum_dite_eq']
               split_ifs with h h'
               · substs h h'
diff --git a/Mathlib/CategoryTheory/Preadditive/Opposite.lean b/Mathlib/CategoryTheory/Preadditive/Opposite.lean
index ccc1021d809aa..115a2764163aa 100644
--- a/Mathlib/CategoryTheory/Preadditive/Opposite.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Opposite.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Adam Topaz, Johan Commelin, Joël Riou
+Authors: Kim Morrison, Adam Topaz, Johan Commelin, Joël Riou
 -/
 import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
 import Mathlib.Logic.Equiv.TransferInstance
diff --git a/Mathlib/CategoryTheory/Preadditive/Projective.lean b/Mathlib/CategoryTheory/Preadditive/Projective.lean
index 8059a72c9ca40..e0900aad10375 100644
--- a/Mathlib/CategoryTheory/Preadditive/Projective.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Projective.lean
@@ -1,14 +1,13 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
 import Mathlib.CategoryTheory.Adjunction.Limits
 import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
-import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
 import Mathlib.CategoryTheory.Limits.Preserves.Finite
-import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
+import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
 
 /-!
 # Projective objects and categories with enough projectives
diff --git a/Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean b/Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean
index 8d947ea5ca55e..1447ce251a876 100644
--- a/Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean
+++ b/Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Joël Riou
+Authors: Kim Morrison, Joël Riou
 -/
 import Mathlib.Algebra.Homology.QuasiIso
 import Mathlib.Algebra.Homology.SingleHomology
diff --git a/Mathlib/CategoryTheory/Preadditive/Schur.lean b/Mathlib/CategoryTheory/Preadditive/Schur.lean
index 5e012592ab269..2fd3b980f6f59 100644
--- a/Mathlib/CategoryTheory/Preadditive/Schur.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Schur.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.Algebra.Group.Ext
 import Mathlib.CategoryTheory.Simple
@@ -70,7 +70,7 @@ noncomputable instance [HasKernels C] {X : C} [Simple X] : DivisionRing (End X)
   qsmul := _
   qsmul_def := fun q a => rfl
 
-open FiniteDimensional
+open Module
 
 section
 
diff --git a/Mathlib/CategoryTheory/Preadditive/SingleObj.lean b/Mathlib/CategoryTheory/Preadditive/SingleObj.lean
index da6a0ed13304f..a2b5d1a38ddaf 100644
--- a/Mathlib/CategoryTheory/Preadditive/SingleObj.lean
+++ b/Mathlib/CategoryTheory/Preadditive/SingleObj.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.Basic
 import Mathlib.CategoryTheory.SingleObj
diff --git a/Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean b/Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean
index 96b1f0270625c..3e19f976a5b18 100644
--- a/Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Yoneda/Basic.lean
@@ -39,7 +39,7 @@ object `X` to the `End Y`-module of morphisms `X ⟶ Y`.
 @[simps]
 def preadditiveYonedaObj (Y : C) : Cᵒᵖ ⥤ ModuleCat.{v} (End Y) where
   obj X := ModuleCat.of _ (X.unop ⟶ Y)
-  map f := ModuleCat.ofHom
+  map f := ModuleCat.asHom
     { toFun := fun g => f.unop ≫ g
       map_add' := fun g g' => comp_add _ _ _ _ _ _
       map_smul' := fun r g => Eq.symm <| Category.assoc _ _ _ }
@@ -66,7 +66,7 @@ object `Y` to the `End X`-module of morphisms `X ⟶ Y`.
 @[simps]
 def preadditiveCoyonedaObj (X : Cᵒᵖ) : C ⥤ ModuleCat.{v} (End X) where
   obj Y := ModuleCat.of _ (unop X ⟶ Y)
-  map f := ModuleCat.ofHom
+  map f := ModuleCat.asHom
     { toFun := fun g => g ≫ f
       map_add' := fun g g' => add_comp _ _ _ _ _ _
       map_smul' := fun r g => Category.assoc _ _ _ }
diff --git a/Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean b/Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean
index d50174be5c41c..778a344fe5bbd 100644
--- a/Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Yoneda/Injective.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
 import Mathlib.CategoryTheory.Preadditive.Injective
diff --git a/Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean b/Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean
index 2434e2bdb0a20..3db3048dcfe74 100644
--- a/Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean
+++ b/Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
 import Mathlib.CategoryTheory.Preadditive.Projective
diff --git a/Mathlib/CategoryTheory/Products/Associator.lean b/Mathlib/CategoryTheory/Products/Associator.lean
index 232fe7f3c5ea2..041ecf6548890 100644
--- a/Mathlib/CategoryTheory/Products/Associator.lean
+++ b/Mathlib/CategoryTheory/Products/Associator.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison
+Authors: Stephen Morgan, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Products.Basic
 
diff --git a/Mathlib/CategoryTheory/Products/Basic.lean b/Mathlib/CategoryTheory/Products/Basic.lean
index 449595eab85d3..58958ff88a2de 100644
--- a/Mathlib/CategoryTheory/Products/Basic.lean
+++ b/Mathlib/CategoryTheory/Products/Basic.lean
@@ -1,9 +1,8 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison
+Authors: Stephen Morgan, Kim Morrison
 -/
-import Mathlib.CategoryTheory.EqToHom
 import Mathlib.CategoryTheory.Functor.Const
 import Mathlib.CategoryTheory.Opposites
 import Mathlib.Data.Prod.Basic
@@ -296,7 +295,18 @@ end Equivalence
 /-- `F.flip` composed with evaluation is the same as evaluating `F`. -/
 @[simps!]
 def flipCompEvaluation (F : A ⥤ B ⥤ C) (a) : F.flip ⋙ (evaluation _ _).obj a ≅ F.obj a :=
-  NatIso.ofComponents fun b => eqToIso rfl
+  NatIso.ofComponents fun b => Iso.refl _
+
+theorem flip_comp_evaluation (F : A ⥤ B ⥤ C) (a) : F.flip ⋙ (evaluation _ _).obj a = F.obj a :=
+  rfl
+
+/-- `F` composed with evaluation is the same as evaluating `F.flip`. -/
+@[simps!]
+def compEvaluation (F : A ⥤ B ⥤ C) (b) : F ⋙ (evaluation _ _).obj b ≅ F.flip.obj b :=
+  NatIso.ofComponents fun a => Iso.refl _
+
+theorem comp_evaluation (F : A ⥤ B ⥤ C) (b) : F ⋙ (evaluation _ _).obj b = F.flip.obj b :=
+  rfl
 
 variable (A B C)
 
diff --git a/Mathlib/CategoryTheory/Products/Bifunctor.lean b/Mathlib/CategoryTheory/Products/Bifunctor.lean
index 70e02805ce5ac..a0d335eb00794 100644
--- a/Mathlib/CategoryTheory/Products/Bifunctor.lean
+++ b/Mathlib/CategoryTheory/Products/Bifunctor.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison
+Authors: Stephen Morgan, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Products.Basic
 
diff --git a/Mathlib/CategoryTheory/Products/Unitor.lean b/Mathlib/CategoryTheory/Products/Unitor.lean
index f436f1074f993..7cb26c8bc22cf 100644
--- a/Mathlib/CategoryTheory/Products/Unitor.lean
+++ b/Mathlib/CategoryTheory/Products/Unitor.lean
@@ -42,7 +42,7 @@ def rightInverseUnitor : C ⥤ C × Discrete (PUnit : Type w) where
   obj X := ⟨X, ⟨PUnit.unit⟩⟩
   map f := ⟨f, 𝟙 _⟩
 
-/-- The equivalence of categories expressing left unity of products of categories.  -/
+/-- The equivalence of categories expressing left unity of products of categories. -/
 @[simps]
 def leftUnitorEquivalence : Discrete (PUnit : Type w) × C ≌ C where
   functor := leftUnitor C
@@ -50,7 +50,7 @@ def leftUnitorEquivalence : Discrete (PUnit : Type w) × C ≌ C where
   unitIso := Iso.refl _
   counitIso := Iso.refl _
 
-/-- The equivalence of categories expressing right unity of products of categories.  -/
+/-- The equivalence of categories expressing right unity of products of categories. -/
 @[simps]
 def rightUnitorEquivalence : C × Discrete (PUnit : Type w) ≌ C where
   functor := rightUnitor C
diff --git a/Mathlib/CategoryTheory/Shift/Basic.lean b/Mathlib/CategoryTheory/Shift/Basic.lean
index d830b0b79c97d..c8b2f4ddfaec9 100644
--- a/Mathlib/CategoryTheory/Shift/Basic.lean
+++ b/Mathlib/CategoryTheory/Shift/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin, Andrew Yang
+Authors: Kim Morrison, Johan Commelin, Andrew Yang
 -/
 import Mathlib.Algebra.Group.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
diff --git a/Mathlib/CategoryTheory/Shift/CommShift.lean b/Mathlib/CategoryTheory/Shift/CommShift.lean
index 5f75457aa59db..e75c095dca271 100644
--- a/Mathlib/CategoryTheory/Shift/CommShift.lean
+++ b/Mathlib/CategoryTheory/Shift/CommShift.lean
@@ -246,7 +246,7 @@ variable {C D E J : Type*} [Category C] [Category D] [Category E] [Category J]
 /-- If `τ : F₁ ⟶ F₂` is a natural transformation between two functors
 which commute with a shift by an additive monoid `A`, this typeclass
 asserts a compatibility of `τ` with these shifts. -/
-class CommShift : Prop :=
+class CommShift : Prop where
   comm' (a : A) : (F₁.commShiftIso a).hom ≫ whiskerRight τ _ =
     whiskerLeft _ τ ≫ (F₂.commShiftIso a).hom
 
diff --git a/Mathlib/CategoryTheory/Shift/Localization.lean b/Mathlib/CategoryTheory/Shift/Localization.lean
index f95e143399495..1f74884924d44 100644
--- a/Mathlib/CategoryTheory/Shift/Localization.lean
+++ b/Mathlib/CategoryTheory/Shift/Localization.lean
@@ -32,7 +32,7 @@ namespace MorphismProperty
 /-- A morphism property `W` on a category `C` is compatible with the shift by a
 monoid `A` when for all `a : A`, a morphism `f` belongs to `W`
 if and only if `f⟦a⟧'` does. -/
-class IsCompatibleWithShift : Prop :=
+class IsCompatibleWithShift : Prop where
   /-- the condition that for all `a : A`, the morphism property `W` is not changed when
   we take its inverse image by the shift functor by `a` -/
   condition : ∀ (a : A), W.inverseImage (shiftFunctor C a) = W
@@ -90,7 +90,7 @@ noncomputable def Functor.CommShift.localized :
 
 attribute [irreducible] HasShift.localized Functor.CommShift.localized
 
-/-- The localized category `W.Localization` is endowed with the induced shift.  -/
+/-- The localized category `W.Localization` is endowed with the induced shift. -/
 noncomputable instance HasShift.localization :
     HasShift W.Localization A :=
   HasShift.localized W.Q W A
@@ -104,7 +104,7 @@ attribute [irreducible] HasShift.localization MorphismProperty.commShift_Q
 
 variable [W.HasLocalization]
 
-/-- The localized category `W.Localization'` is endowed with the induced shift.  -/
+/-- The localized category `W.Localization'` is endowed with the induced shift. -/
 noncomputable instance HasShift.localization' :
     HasShift W.Localization' A :=
   HasShift.localized W.Q' W A
@@ -165,7 +165,7 @@ end commShiftOfLocalization
 
 /-- In the context of localization of categories, if a functor
 is induced by a functor which commutes with the shift, then
-this functor commutes with the shift.  -/
+this functor commutes with the shift. -/
 noncomputable def commShiftOfLocalization : F'.CommShift A where
   iso := commShiftOfLocalization.iso L W F F'
   zero := by
diff --git a/Mathlib/CategoryTheory/Shift/Quotient.lean b/Mathlib/CategoryTheory/Shift/Quotient.lean
index d029cd486c714..f789f82c2883c 100644
--- a/Mathlib/CategoryTheory/Shift/Quotient.lean
+++ b/Mathlib/CategoryTheory/Shift/Quotient.lean
@@ -32,7 +32,7 @@ namespace HomRel
 
 /-- A relation on morphisms is compatible with the shift by a monoid `A` when the
 relation if preserved by the shift. -/
-class IsCompatibleWithShift : Prop :=
+class IsCompatibleWithShift : Prop where
   /-- the condition that the relation is preserved by the shift -/
   condition : ∀ (a : A) ⦃X Y : C⦄ (f g : X ⟶ Y), r f g → r (f⟦a⟧') (g⟦a⟧')
 
diff --git a/Mathlib/CategoryTheory/Shift/ShiftedHom.lean b/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
index 9df10e3410095..09bea544345c6 100644
--- a/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
+++ b/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
@@ -28,7 +28,7 @@ variable {C : Type*} [Category C] {D : Type*} [Category D] {E : Type*} [Category
   {M : Type*} [AddMonoid M] [HasShift C M] [HasShift D M] [HasShift E M]
 
 /-- In a category `C` equipped with a shift by an additive monoid,
-this is the type of morphisms `X ⟶ (Y⟦n⟧)` for `m : M`.  -/
+this is the type of morphisms `X ⟶ (Y⟦n⟧)` for `m : M`. -/
 def ShiftedHom (X Y : C) (m : M) : Type _ := X ⟶ (Y⟦m⟧)
 
 instance [Preadditive C] (X Y : C) (n : M) : AddCommGroup (ShiftedHom X Y n) := by
diff --git a/Mathlib/CategoryTheory/Simple.lean b/Mathlib/CategoryTheory/Simple.lean
index 04c45673e542c..619a1bdfe7847 100644
--- a/Mathlib/CategoryTheory/Simple.lean
+++ b/Mathlib/CategoryTheory/Simple.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
 import Mathlib.CategoryTheory.Limits.Shapes.Kernels
@@ -56,7 +56,6 @@ theorem isIso_of_mono_of_nonzero {X Y : C} [Simple Y] {f : X ⟶ Y} [Mono f] (w
 
 theorem Simple.of_iso {X Y : C} [Simple Y] (i : X ≅ Y) : Simple X :=
   { mono_isIso_iff_nonzero := fun f m => by
-      haveI : Mono (f ≫ i.hom) := mono_comp _ _
       constructor
       · intro h w
         have j : IsIso (f ≫ i.hom) := by infer_instance
diff --git a/Mathlib/CategoryTheory/Sites/Adjunction.lean b/Mathlib/CategoryTheory/Sites/Adjunction.lean
index a2aa3e4aa6b95..334681a5529d0 100644
--- a/Mathlib/CategoryTheory/Sites/Adjunction.lean
+++ b/Mathlib/CategoryTheory/Sites/Adjunction.lean
@@ -3,6 +3,7 @@ Copyright (c) 2021 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Adam Topaz, Joël Riou
 -/
+import Mathlib.CategoryTheory.Adjunction.Restrict
 import Mathlib.CategoryTheory.Adjunction.Whiskering
 import Mathlib.CategoryTheory.Sites.PreservesSheafification
 
@@ -35,53 +36,36 @@ namespace Sheaf
 
 noncomputable section
 
-/-- An auxiliary definition to be used in defining `CategoryTheory.Sheaf.adjunction` below. -/
-@[simps]
-def composeEquiv [HasWeakSheafify J D] [HasSheafCompose J F] (adj : G ⊣ F)
-    (X : Sheaf J E) (Y : Sheaf J D) :
-    ((composeAndSheafify J G).obj X ⟶ Y) ≃ (X ⟶ (sheafCompose J F).obj Y) :=
-  let A := adj.whiskerRight Cᵒᵖ
-  { toFun := fun η => ⟨A.homEquiv _ _ (toSheafify J _ ≫ η.val)⟩
-    invFun := fun γ => ⟨sheafifyLift J ((A.homEquiv _ _).symm ((sheafToPresheaf _ _).map γ)) Y.2⟩
-    left_inv := by
-      intro η
-      ext1
-      dsimp
-      symm
-      apply sheafifyLift_unique
-      rw [Equiv.symm_apply_apply]
-    right_inv := by
-      intro γ
-      ext1
-      dsimp
-      -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-      erw [toSheafify_sheafifyLift, Equiv.apply_symm_apply] }
-
--- These lemmas have always been bad (#7657), but leanprover/lean4#2644 made `simp` start noticing
-attribute [nolint simpNF] CategoryTheory.Sheaf.composeEquiv_apply_val
-  CategoryTheory.Sheaf.composeEquiv_symm_apply_val
-
 /-- An adjunction `adj : G ⊣ F` with `F : D ⥤ E` and `G : E ⥤ D` induces an adjunction
 between `Sheaf J D` and `Sheaf J E`, in contexts where one can sheafify `D`-valued presheaves,
-and `F` preserves the correct limits. -/
-@[simps! unit_app_val counit_app_val]
+and postcomposing with `F` preserves the property of being a sheaf. -/
 def adjunction [HasWeakSheafify J D] [HasSheafCompose J F] (adj : G ⊣ F) :
     composeAndSheafify J G ⊣ sheafCompose J F :=
-  Adjunction.mkOfHomEquiv
-    { homEquiv := composeEquiv J adj
-      homEquiv_naturality_left_symm := fun f g => by
-        ext1
-        dsimp [composeEquiv]
-        rw [sheafifyMap_sheafifyLift]
-        erw [Adjunction.homEquiv_naturality_left_symm]
-        rw [whiskeringRight_obj_map]
-        rfl
-      homEquiv_naturality_right := fun f g => by
-        ext
-        dsimp [composeEquiv]
-        erw [Adjunction.homEquiv_unit, Adjunction.homEquiv_unit]
-        dsimp
-        simp }
+  Adjunction.restrictFullyFaithful ((adj.whiskerRight Cᵒᵖ).comp (sheafificationAdjunction J D))
+    (fullyFaithfulSheafToPresheaf J E) (Functor.FullyFaithful.id _) (Iso.refl _) (Iso.refl _)
+
+@[simp]
+lemma adjunction_unit_app_val [HasWeakSheafify J D] [HasSheafCompose J F] (adj : G ⊣ F)
+    (X : Sheaf J E) : ((adjunction J adj).unit.app X).val =
+      (adj.whiskerRight Cᵒᵖ).unit.app _ ≫ whiskerRight (toSheafify J (X.val ⋙ G)) F  := by
+  change (sheafToPresheaf _ _).map ((adjunction J adj).unit.app X) = _
+  simp only [Functor.id_obj, Functor.comp_obj, whiskeringRight_obj_obj, adjunction,
+    Adjunction.map_restrictFullyFaithful_unit_app, Adjunction.comp_unit_app,
+    sheafificationAdjunction_unit_app, whiskeringRight_obj_map, Iso.refl_hom, NatTrans.id_app,
+    Functor.comp_map, Functor.map_id, whiskerRight_id', Category.comp_id]
+  rfl
+
+@[simp]
+lemma adjunction_counit_app_val [HasWeakSheafify J D] [HasSheafCompose J F] (adj : G ⊣ F)
+    (Y : Sheaf J D) : ((adjunction J adj).counit.app Y).val =
+      sheafifyLift J (((adj.whiskerRight Cᵒᵖ).counit.app Y.val)) Y.cond := by
+  change ((𝟭 (Sheaf _ _)).map ((adjunction J adj).counit.app Y)).val = _
+  simp only [Functor.comp_obj, sheafToPresheaf_obj, sheafCompose_obj_val, whiskeringRight_obj_obj,
+    adjunction, Adjunction.map_restrictFullyFaithful_counit_app, Iso.refl_inv, NatTrans.id_app,
+    Functor.comp_map, whiskeringRight_obj_map, Adjunction.comp_counit_app,
+    instCategorySheaf_comp_val, instCategorySheaf_id_val, sheafificationAdjunction_counit_app_val,
+    sheafifyMap_sheafifyLift, Functor.id_obj, whiskerRight_id', Category.comp_id, Category.id_comp]
+
 
 instance [HasWeakSheafify J D] [F.IsRightAdjoint] : (sheafCompose J F).IsRightAdjoint :=
   (adjunction J (Adjunction.ofIsRightAdjoint F)).isRightAdjoint
@@ -100,8 +84,12 @@ lemma preservesSheafification_of_adjunction (adj : G ⊣ F) :
     convert (((adj.whiskerRight Cᵒᵖ).homEquiv Q R).trans
       (hf.homEquiv (R ⋙ F) ((sheafCompose J F).obj ⟨R, hR⟩).cond)).bijective
     ext g X
-    dsimp [Adjunction.whiskerRight, Adjunction.mkOfUnitCounit]
-    simp
+    -- The rest of this proof was
+    -- `dsimp [Adjunction.whiskerRight, Adjunction.mkOfUnitCounit]; simp` before #16317.
+    dsimp
+    rw [← NatTrans.comp_app]
+    congr
+    exact Adjunction.homEquiv_naturality_left _ _ _
 
 instance [G.IsLeftAdjoint] : J.PreservesSheafification G :=
   preservesSheafification_of_adjunction J (Adjunction.ofIsLeftAdjoint G)
@@ -126,8 +114,7 @@ theorem adjunctionToTypes_unit_app_val {G : Type max v₁ u₁ ⥤ D} (adj : G 
     ((adjunctionToTypes J adj).unit.app Y).val =
       (adj.whiskerRight _).unit.app ((sheafOfTypesToPresheaf J).obj Y) ≫
         whiskerRight (toSheafify J _) (forget D) := by
-  dsimp [adjunctionToTypes, Adjunction.comp]
-  simp
+  simp [adjunctionToTypes]
   rfl
 
 @[simp]
@@ -136,14 +123,8 @@ theorem adjunctionToTypes_counit_app_val {G : Type max v₁ u₁ ⥤ D} (adj : G
     ((adjunctionToTypes J adj).counit.app X).val =
       sheafifyLift J ((Functor.associator _ _ _).hom ≫ (adj.whiskerRight _).counit.app _) X.2 := by
   apply sheafifyLift_unique
-  dsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app,
-    instCategorySheaf_comp_val, instCategorySheaf_id_val]
-  rw [adjunction_counit_app_val]
-  erw [Category.id_comp, sheafifyMap_sheafifyLift, toSheafify_sheafifyLift]
   ext
-  dsimp [sheafEquivSheafOfTypes, Equivalence.symm, Equivalence.toAdjunction,
-    NatIso.ofComponents, Adjunction.whiskerRight, Adjunction.mkOfUnitCounit]
-  simp
+  simp [adjunctionToTypes, sheafEquivSheafOfTypes, Equivalence.symm]
 
 instance [(forget D).IsRightAdjoint] :
     (sheafForget.{_, _, _, _, max u₁ v₁} (D := D) J).IsRightAdjoint :=
diff --git a/Mathlib/CategoryTheory/Sites/Canonical.lean b/Mathlib/CategoryTheory/Sites/Canonical.lean
index 9c6a4291a3191..ec5fdd447f5e9 100644
--- a/Mathlib/CategoryTheory/Sites/Canonical.lean
+++ b/Mathlib/CategoryTheory/Sites/Canonical.lean
@@ -203,7 +203,7 @@ theorem isSheaf_yoneda_obj (X : C) : Presieve.IsSheaf (canonicalTopology C) (yon
   fun _ _ hS => sheaf_for_finestTopology _ (Set.mem_range_self _) _ hS
 
 /-- A representable functor is a sheaf for the canonical topology. -/
-theorem isSheaf_of_representable (P : Cᵒᵖ ⥤ Type v) [P.Representable] :
+theorem isSheaf_of_isRepresentable (P : Cᵒᵖ ⥤ Type v) [P.IsRepresentable] :
     Presieve.IsSheaf (canonicalTopology C) P :=
   Presieve.isSheaf_iso (canonicalTopology C) P.reprW (isSheaf_yoneda_obj _)
 
@@ -224,9 +224,9 @@ theorem of_yoneda_isSheaf (J : GrothendieckTopology C)
       apply h)
 
 /-- If `J` is subcanonical, then any representable is a `J`-sheaf. -/
-theorem isSheaf_of_representable {J : GrothendieckTopology C} (hJ : Subcanonical J)
-    (P : Cᵒᵖ ⥤ Type v) [P.Representable] : Presieve.IsSheaf J P :=
-  Presieve.isSheaf_of_le _ hJ (Sheaf.isSheaf_of_representable P)
+theorem isSheaf_of_isRepresentable {J : GrothendieckTopology C} (hJ : Subcanonical J)
+    (P : Cᵒᵖ ⥤ Type v) [P.IsRepresentable] : Presieve.IsSheaf J P :=
+  Presieve.isSheaf_of_le _ hJ (Sheaf.isSheaf_of_isRepresentable P)
 
 variable {J}
 
@@ -238,7 +238,7 @@ into the sheaf category.
 def yoneda (hJ : Subcanonical J) : C ⥤ Sheaf J (Type v) where
   obj X := ⟨CategoryTheory.yoneda.obj X, by
     rw [isSheaf_iff_isSheaf_of_type]
-    apply hJ.isSheaf_of_representable⟩
+    apply hJ.isSheaf_of_isRepresentable⟩
   map f := ⟨CategoryTheory.yoneda.map f⟩
 
 variable (hJ : Subcanonical J)
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean b/Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean
index 331d8853b1c58..e3d81579f0701 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/CoherentTopology.lean
@@ -85,13 +85,16 @@ theorem coherentTopology.mem_sieves_iff_hasEffectiveEpiFamily (S : Sieve X) :
         EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) )  := by
   constructor
   · intro h
-    induction' h with Y T hS Y Y R S _ _ a b
-    · obtain ⟨a, h, Y', π, h', _⟩ := hS
+    induction h with
+    | of Y T hS =>
+      obtain ⟨a, h, Y', π, h', _⟩ := hS
       refine ⟨a, h, Y', π, inferInstance, fun a' ↦ ?_⟩
       obtain ⟨rfl, _⟩ := h'
       exact ⟨Y' a', 𝟙 Y' a', π a', Presieve.ofArrows.mk a', by simp⟩
-    · exact ⟨Unit, inferInstance, fun _ => Y, fun _ => (𝟙 Y), inferInstance, by simp⟩
-    · obtain ⟨α, w, Y₁, π, ⟨h₁,h₂⟩⟩ := a
+    | top Y =>
+      exact ⟨Unit, inferInstance, fun _ => Y, fun _ => (𝟙 Y), inferInstance, by simp⟩
+    | transitive Y R S _ _ a b =>
+      obtain ⟨α, w, Y₁, π, ⟨h₁,h₂⟩⟩ := a
       choose β _ Y_n π_n H using fun a => b (h₂ a)
       exact ⟨(Σ a, β a), inferInstance, fun ⟨a,b⟩ => Y_n a b, fun ⟨a, b⟩ => (π_n a b) ≫ (π a),
         EffectiveEpiFamily.transitive_of_finite _ h₁ _ (fun a => (H a).1),
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean b/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean
index 7c93c78272de3..f0f439fb8e122 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean
@@ -78,9 +78,9 @@ theorem extensive_regular_generate_coherent [Preregular C] [FinitaryPreExtensive
           Set.mem_setOf_eq]
         exact Or.inr ⟨_, Sigma.desc f, ⟨rfl, inferInstance⟩⟩
       · rintro R g ⟨W, ψ, σ, ⟨⟩, rfl⟩
-        change _ ∈ sieves ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck _
+        change _ ∈ ((extensiveCoverage C) ⊔ (regularCoverage C)).toGrothendieck _ R
         rw [Sieve.pullback_comp]
-        apply pullback_stable'
+        apply pullback_stable
         have : generate (Presieve.ofArrows X fun (i : I) ↦ Sigma.ι X i) ≤
             (generate (Presieve.ofArrows X f)).pullback (Sigma.desc f) := by
           rintro Q q ⟨E, e, r, ⟨hq, rfl⟩⟩
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean b/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean
index e66a91cfd4c63..edc55d2e3f01e 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean
@@ -9,7 +9,7 @@ import Mathlib.CategoryTheory.Sites.Equivalence
 
 # Coherence and equivalence of categories
 
-This file proves that the coherent and regular topologies transfer nicely along equivalences of
+This file proves that the coherent and regular topologies transfer nicely along equivalences of
 categories.
 -/
 
@@ -27,7 +27,7 @@ section Coherent
 
 variable [Precoherent C]
 
-/-- `Precoherent` is preserved by equivalence of categories. -/
+/-- `Precoherent` is preserved by equivalence of categories. -/
 theorem precoherent (e : C ≌ D) : Precoherent D := e.inverse.reflects_precoherent
 
 instance [EssentiallySmall C] :
@@ -75,7 +75,7 @@ section Regular
 
 variable [Preregular C]
 
-/-- `Preregular` is preserved by equivalence of categories. -/
+/-- `Preregular` is preserved by equivalence of categories. -/
 theorem preregular (e : C ≌ D) : Preregular D := e.inverse.reflects_preregular
 
 instance [EssentiallySmall C] :
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
index 4030384a506a2..ae209ddc24c1e 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean
@@ -62,7 +62,7 @@ instance {α : Type} [Finite α] (Z : α → C) : (ofArrows Z (fun i ↦ Sigma.
 /-- Every Yoneda-presheaf is a sheaf for the extensive topology. -/
 theorem extensiveTopology.isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (extensiveTopology C)
     (yoneda.obj W) := by
-  erw [isSheaf_coverage]
+  rw [extensiveTopology, isSheaf_coverage]
   intro X R ⟨Y, α, Z, π, hR, hi⟩
   have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
   have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
@@ -75,19 +75,19 @@ theorem extensiveTopology.subcanonical : Sheaf.Subcanonical (extensiveTopology C
 variable [FinitaryExtensive C]
 
 /--
-A presheaf of sets on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite
+A presheaf of sets on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite
 products.
 -/
 theorem Presieve.isSheaf_iff_preservesFiniteProducts (F : Cᵒᵖ ⥤ Type w) :
     Presieve.IsSheaf (extensiveTopology C) F ↔
     Nonempty (PreservesFiniteProducts F) := by
   refine ⟨fun hF ↦ ⟨⟨fun α _ ↦ ⟨fun {K} ↦ ?_⟩⟩⟩, fun hF ↦ ?_⟩
-  · erw [Presieve.isSheaf_coverage] at hF
+  · rw [extensiveTopology, isSheaf_coverage] at hF
     let Z : α → C := fun i ↦ unop (K.obj ⟨i⟩)
-    have : (Presieve.ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks :=
-      (inferInstance : (Presieve.ofArrows Z (Sigma.ι Z)).hasPullbacks)
+    have : (ofArrows Z (Cofan.mk (∐ Z) (Sigma.ι Z)).inj).hasPullbacks :=
+      inferInstanceAs (ofArrows Z (Sigma.ι Z)).hasPullbacks
     have : ∀ (i : α), Mono (Cofan.inj (Cofan.mk (∐ Z) (Sigma.ι Z)) i) :=
-      (inferInstance : ∀ (i : α), Mono (Sigma.ι Z i))
+      inferInstanceAs <| ∀ (i : α), Mono (Sigma.ι Z i)
     let i : K ≅ Discrete.functor (fun i ↦ op (Z i)) := Discrete.natIsoFunctor
     let _ : PreservesLimit (Discrete.functor (fun i ↦ op (Z i))) F :=
         Presieve.preservesProductOfIsSheafFor F ?_ initialIsInitial _ (coproductIsCoproduct Z)
@@ -104,14 +104,14 @@ theorem Presieve.isSheaf_iff_preservesFiniteProducts (F : Cᵒᵖ ⥤ Type w) :
       ext
       simp
   · let _ := hF.some
-    erw [Presieve.isSheaf_coverage]
+    rw [extensiveTopology, Presieve.isSheaf_coverage]
     intro X R ⟨Y, α, Z, π, hR, hi⟩
     have : IsIso (Sigma.desc (Cofan.inj (Cofan.mk X π))) := hi
     have : R.Extensive := ⟨Y, α, Z, π, hR, ⟨Cofan.isColimitOfIsIsoSigmaDesc (Cofan.mk X π)⟩⟩
     exact isSheafFor_extensive_of_preservesFiniteProducts R F
 
 /--
-A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
+A presheaf on a category which is `FinitaryExtensive` is a sheaf iff it preserves finite products.
 -/
 theorem Presheaf.isSheaf_iff_preservesFiniteProducts (F : Cᵒᵖ ⥤ D) :
     IsSheaf (extensiveTopology C) F ↔ Nonempty (PreservesFiniteProducts F) := by
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean
index 0cf8285a9167f..1f7872d648175 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveTopology.lean
@@ -53,7 +53,8 @@ lemma extensiveTopology.mem_sieves_iff_contains_colimit_cofan {X : C} (S : Sieve
     apply (extensiveCoverage C).mem_toGrothendieck_sieves_of_superset (R := Presieve.ofArrows Y π)
     · exact fun _ _ hh ↦ by cases hh; exact h' _
     · refine ⟨α, inferInstance, Y, π, rfl, ?_⟩
-      erw [Limits.Cofan.isColimit_iff_isIso_sigmaDesc (c := Cofan.mk X π)]
+      rw [show IsIso (Sigma.desc π) ↔ _ from
+        Limits.Cofan.isColimit_iff_isIso_sigmaDesc (c := Cofan.mk X π)]
       exact h
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
index ef1204a732163..3a79842c88fc9 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean
@@ -16,14 +16,14 @@ and extensive topologies.
 ## Main results
 
 * `regularTopology.isLocallySurjective_iff` A morphism of presheaves `f : F ⟶ G` is locally
-  surjective for the regular topology iff for every object `X` of `C`, and every `y : G(X)`, there
-  is an effective epimorphism `φ : X' ⟶ X` and an `x : F(X)` such that `f_{X'}(x) = G(φ)(y)`.
+  surjective for the regular topology iff for every object `X` of `C`, and every `y : G(X)`, there
+  is an effective epimorphism `φ : X' ⟶ X` and an `x : F(X)` such that `f_{X'}(x) = G(φ)(y)`.
 
 * `coherentTopology.isLocallySurjective_iff` a morphism of sheaves for the coherent topology on a
   preregular finitary extensive category is locally surjective if and only if it is
   locally surjective for the regular topology.
 
-* `extensiveTopology.isLocallySurjective_iff` a morphism of sheaves for the extensive topology on a
+* `extensiveTopology.isLocallySurjective_iff` a morphism of sheaves for the extensive topology on a
   finitary extensive category is locally surjective iff it is objectwise surjective.
 -/
 
@@ -78,9 +78,9 @@ lemma extensiveTopology.surjective_of_isLocallySurjective_sheafOfTypes [Finitary
     Cofan.mk_ι_app]
   have : f.app ⟨Y a⟩ (y a) = G.map (π a).op x := (h' a).choose_spec
   change _ = G.map (π a).op x
-  erw [← this, ← NatTrans.naturality_apply (φ := f)]
-  apply congrArg
-  change (i.hom ≫ F.map (π a).op) y = _
+  rw [← this]
+  erw [← NatTrans.naturality_apply (φ := f)]
+  change f.app _ ((i.hom ≫ F.map (π a).op) y) = _
   erw [IsLimit.map_π]
   rfl
 
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean
index a42e9b7bca302..54bc130f05abb 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean
@@ -11,8 +11,8 @@ import Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology
 # Reflecting the property of being precoherent
 
 We prove that given a fully faithful functor `F : C ⥤ D` which preserves and reflects finite
-effective epimorphic families, such that for every object `X` of `D` there exists an object `W` of
-`C` with an effective epi `π : F.obj W ⟶ X`, the category `C` is `Precoherent` whenever `D` is.
+effective epimorphic families, such that for every object `X` of `D` there exists an object `W` of
+`C` with an effective epi `π : F.obj W ⟶ X`, the category `C` is `Precoherent` whenever `D` is.
 -/
 
 namespace CategoryTheory
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean
index 9b3784170505c..58eef6dd8e4c3 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean
@@ -11,8 +11,8 @@ import Mathlib.CategoryTheory.Sites.Coherent.RegularTopology
 # Reflecting the property of being preregular
 
 We prove that given a fully faithful functor `F : C ⥤ D`, with `Preregular D`, such that for every
-object `X` of `D` there exists an object `W` of `C` with an effective epi `π : F.obj W ⟶ X`, the
-category `C` is `Preregular`.
+object `X` of `D` there exists an object `W` of `C` with an effective epi `π : F.obj W ⟶ X`, the
+category `C` is `Preregular`.
 -/
 
 namespace CategoryTheory
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean
index 8292e7e2fadb1..e310d8b283782 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean
@@ -17,7 +17,7 @@ This file characterises sheaves for the regular topology.
 
 ## Main results
 
-* `equalizerCondition_iff_isSheaf`: In a preregular category with pullbacks, the sheaves for the
+* `equalizerCondition_iff_isSheaf`: In a preregular category with pullbacks, the sheaves for the
   regular topology are precisely the presheaves satisfying an equaliser condition with respect to
   effective epimorphisms.
 
@@ -46,7 +46,7 @@ lemma equalizerCondition_w (P : Cᵒᵖ ⥤ D) {X B : C} {π : X ⟶ B} (c : Pul
   simp only [← Functor.map_comp, ← op_comp, c.condition]
 
 /--
-A contravariant functor on `C` satisfies `SingleEqualizerCondition` with respect to a morphism `π`
+A contravariant functor on `C` satisfies `SingleEqualizerCondition` with respect to a morphism `π`
 if it takes its kernel pair to an equalizer diagram.
 -/
 def SingleEqualizerCondition (P : Cᵒᵖ ⥤ D) ⦃X B : C⦄ (π : X ⟶ B) : Prop :=
@@ -54,7 +54,7 @@ def SingleEqualizerCondition (P : Cᵒᵖ ⥤ D) ⦃X B : C⦄ (π : X ⟶ B) :
     Nonempty (IsLimit (Fork.ofι (P.map π.op) (equalizerCondition_w P c)))
 
 /--
-A contravariant functor on `C` satisfies `EqualizerCondition` if it takes kernel pairs of effective
+A contravariant functor on `C` satisfies `EqualizerCondition` if it takes kernel pairs of effective
 epimorphisms to equalizer diagrams.
 -/
 def EqualizerCondition (P : Cᵒᵖ ⥤ D) : Prop :=
@@ -148,7 +148,7 @@ theorem equalizerCondition_iff_isIso_lift (P : Cᵒᵖ ⥤ Type*) : EqualizerCon
     rw [mapToEqualizer_eq_comp, ← isIso_iff_bijective]
     infer_instance
 
-/-- `P` satisfies the equalizer condition iff its precomposition by an equivalence does. -/
+/-- `P` satisfies the equalizer condition iff its precomposition by an equivalence does. -/
 theorem equalizerCondition_iff_of_equivalence (P : Cᵒᵖ ⥤ D)
     (e : C ≌ E) : EqualizerCondition P ↔ EqualizerCondition (e.op.inverse ⋙ P) :=
   ⟨fun h ↦ equalizerCondition_precomp_of_preservesPullback P e.inverse h, fun h ↦
@@ -177,7 +177,7 @@ theorem parallelPair_pullback_initial {X B : C} (π : X ⟶ B)
     all_goals exact Comma.hom_ext _ _ (by erw [Over.comp_left]; simp [ij]) rfl
 
 /--
-Given a limiting pullback cone, the fork in `SingleEqualizerCondition` is limiting iff the diagram
+Given a limiting pullback cone, the fork in `SingleEqualizerCondition` is limiting iff the diagram
 in `Presheaf.isSheaf_iff_isLimit_coverage` is limiting.
 -/
 noncomputable def isLimit_forkOfι_equiv (P : Cᵒᵖ ⥤ D) {X B : C} (π : X ⟶ B)
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean b/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean
index 33f532e2e9241..7d62dc4654e53 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean
@@ -65,13 +65,15 @@ theorem mem_sieves_iff_hasEffectiveEpi (S : Sieve X) :
     ∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ (S.arrows π) := by
   constructor
   · intro h
-    induction' h with Y T hS Y Y R S _ _ a b
-    · rcases hS with ⟨Y', π, h'⟩
+    induction h with
+    | of Y T hS =>
+      rcases hS with ⟨Y', π, h'⟩
       refine ⟨Y', π, h'.2, ?_⟩
       rcases h' with ⟨rfl, _⟩
       exact ⟨Y', 𝟙 Y', π, Presieve.ofArrows.mk (), (by simp)⟩
-    · exact ⟨Y, (𝟙 Y), inferInstance, by simp only [Sieve.top_apply, forall_const]⟩
-    · rcases a with ⟨Y₁, π, ⟨h₁,h₂⟩⟩
+    | top Y => exact ⟨Y, (𝟙 Y), inferInstance, by simp only [Sieve.top_apply, forall_const]⟩
+    | transitive Y R S _ _ a b =>
+      rcases a with ⟨Y₁, π, ⟨h₁,h₂⟩⟩
       choose Y' π' _ H using b h₂
       exact ⟨Y', π' ≫ π, inferInstance, (by simpa using H)⟩
   · exact regularTopology.mem_sieves_of_hasEffectiveEpi S
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
index 8807986a7a2ee..a20deb1ceff5d 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
@@ -14,9 +14,9 @@ import Mathlib.CategoryTheory.Sites.Whiskering
 # Categories of coherent sheaves
 
 Given a fully faithful functor `F : C ⥤ D` into a precoherent category, which preserves and reflects
-finite effective epi families, and satisfies the property `F.EffectivelyEnough` (meaning that to
+finite effective epi families, and satisfies the property `F.EffectivelyEnough` (meaning that to
 every object in `C` there is an effective epi from an object in the image of `F`), the categories
-of coherent sheaves on `C` and `D` are equivalent (see
+of coherent sheaves on `C` and `D` are equivalent (see
 `CategoryTheory.coherentTopology.equivalence`).
 
 The main application of this equivalence is the characterisation of condensed sets as coherent
@@ -85,7 +85,15 @@ lemma eq_induced : haveI := F.reflects_precoherent
 
 instance : haveI := F.reflects_precoherent;
     F.IsDenseSubsite (coherentTopology C) (coherentTopology D) where
-  functorPushforward_mem_iff := by simp_rw [eq_induced F]; rfl
+  functorPushforward_mem_iff := by
+    rw [eq_induced F]
+    #adaptation_note
+    /--
+    This proof used to be `rfl`,
+    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
+    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
+    -/
+    exact Iff.rfl
 
 lemma coverPreserving : haveI := F.reflects_precoherent
     CoverPreserving (coherentTopology _) (coherentTopology _) F :=
@@ -101,7 +109,7 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
 
 /--
 The equivalence from coherent sheaves on `C` to coherent sheaves on `D`, given a fully faithful
-functor `F : C ⥤ D` to a precoherent category, which preserves and reflects effective epimorphic
+functor `F : C ⥤ D` to a precoherent category, which preserves and reflects effective epimorphic
 families, and satisfies `F.EffectivelyEnough`.
 -/
 noncomputable
@@ -124,7 +132,7 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
 
 /--
 The equivalence from coherent sheaves on `C` to coherent sheaves on `D`, given a fully faithful
-functor `F : C ⥤ D` to an extensive preregular category, which preserves and reflects effective
+functor `F : C ⥤ D` to an extensive preregular category, which preserves and reflects effective
 epimorphisms and satisfies `F.EffectivelyEnough`.
 -/
 noncomputable
@@ -181,7 +189,15 @@ lemma eq_induced : haveI := F.reflects_preregular
 
 instance : haveI := F.reflects_preregular;
     F.IsDenseSubsite (regularTopology C) (regularTopology D) where
-  functorPushforward_mem_iff := by simp_rw [eq_induced F]; rfl
+  functorPushforward_mem_iff := by
+    rw [eq_induced F]
+    #adaptation_note
+    /--
+    This proof used to be `rfl`,
+    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
+    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
+    -/
+    exact Iff.rfl
 
 lemma coverPreserving : haveI := F.reflects_preregular
     CoverPreserving (regularTopology _) (regularTopology _) F :=
@@ -197,7 +213,7 @@ variable {C : Type u₁} {D : Type u₂} [Category.{v₁} C] [Category.{v₂} D]
 
 /--
 The equivalence from regular sheaves on `C` to regular sheaves on `D`, given a fully faithful
-functor `F : C ⥤ D` to a preregular category, which preserves and reflects effective
+functor `F : C ⥤ D` to a preregular category, which preserves and reflects effective
 epimorphisms and satisfies `F.EffectivelyEnough`.
 -/
 noncomputable
@@ -246,7 +262,7 @@ theorem isSheaf_iff_extensiveSheaf_of_projective [Preregular C] [FinitaryExtensi
   rw [isSheaf_iff_preservesFiniteProducts_of_projective, isSheaf_iff_preservesFiniteProducts]
 
 /--
-The categories of coherent sheaves and extensive sheaves on `C` are equivalent if `C` is
+The categories of coherent sheaves and extensive sheaves on `C` are equivalent if `C` is
 preregular, finitary extensive, and every object is projective.
 -/
 @[simps]
diff --git a/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean b/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
index df5b4c90f4608..602fb9363514a 100644
--- a/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
+++ b/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean
@@ -96,7 +96,7 @@ def plusCompIso : J.plusObj P ⋙ F ≅ J.plusObj (P ⋙ F) :=
       ext
       dsimp
       simp only [Category.assoc]
-      erw [Multiequalizer.lift_ι, diagramCompIso_hom_ι, diagramCompIso_hom_ι, ← F.map_comp,
+      rw [Multiequalizer.lift_ι, diagramCompIso_hom_ι, diagramCompIso_hom_ι, ← F.map_comp,
         Multiequalizer.lift_ι])
 
 @[reassoc (attr := simp)]
diff --git a/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean b/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
index 3bff65deb65b1..8d46116dc8326 100644
--- a/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
+++ b/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean
@@ -153,7 +153,8 @@ theorem res_mk_eq_mk_pullback {Y X : C} {P : Cᵒᵖ ⥤ D} {S : J.Cover X} (x :
   ext i
   simp only [Functor.op_obj, unop_op, pullback_obj, diagram_obj, Functor.comp_obj,
     diagramPullback_app, Meq.equiv_apply, Meq.pullback_apply]
-  erw [← comp_apply, Multiequalizer.lift_ι, Meq.equiv_symm_eq_apply]
+  rw [← comp_apply, Multiequalizer.lift_ι]
+  erw [Meq.equiv_symm_eq_apply]
   cases i; rfl
 
 theorem toPlus_mk {X : C} {P : Cᵒᵖ ⥤ D} (S : J.Cover X) (x : P.obj (op X)) :
@@ -162,7 +163,8 @@ theorem toPlus_mk {X : C} {P : Cᵒᵖ ⥤ D} (S : J.Cover X) (x : P.obj (op X))
   let e : S ⟶ ⊤ := homOfLE (OrderTop.le_top _)
   rw [← colimit.w _ e.op]
   delta Cover.toMultiequalizer
-  erw [comp_apply, comp_apply]
+  rw [comp_apply]
+  erw [comp_apply]
   apply congr_arg
   dsimp [diagram]
   apply Concrete.multiequalizer_ext
@@ -347,7 +349,7 @@ theorem exists_of_sep (P : Cᵒᵖ ⥤ D)
   use mk w
   ext I
   dsimp [Meq.mk]
-  erw [ht, res_mk_eq_mk_pullback]
+  rw [ht, res_mk_eq_mk_pullback]
   -- Use the separatedness of `P⁺` to prove that this is indeed a gluing of our
   -- original local sections.
   apply sep P (T I)
diff --git a/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean b/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean
index 1fa5ab155f92b..46db78d658c1f 100644
--- a/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean
+++ b/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Dagur Asgeirsson
 -/
 import Mathlib.CategoryTheory.Sites.Sheafification
-
+import Mathlib.CategoryTheory.Sites.DenseSubsite
 /-!
 
 # The constant sheaf
@@ -12,55 +12,224 @@ import Mathlib.CategoryTheory.Sites.Sheafification
 We define the constant sheaf functor (the sheafification of the constant presheaf)
 `constantSheaf : D ⥤ Sheaf J D` and prove that it is left adjoint to evaluation at a terminal
 object (see `constantSheafAdj`).
+
+We also define a predicate on sheaves, `Sheaf.IsConstant`, saying that a sheaf is in the
+essential image of the constant sheaf functor.
+
+## Main results
+
+* `Sheaf.isConstant_iff_isIso_counit_app`: Provided that the constant sheaf functor is fully
+faithful, a sheaf is constant if and only if the counit of the constant sheaf adjunction applied to
+it is an isomorphism.
+
+* `Sheaf.isConstant_iff_of_equivalence` : The property of a sheaf of being constant is invariant
+under equivalence of sheaf categories.
+
+* `Sheaf.isConstant_iff_forget` : Given a "forgetful" functor `U : D ⥤ B` a sheaf `F : Sheaf J D` is
+constant if and only if the sheaf given by postcomposition with `U` is constant.
+
+## Future work
+
+* (Dagur) Use `Sheaf.isConstant_iff_forget` to prove that a condensed module is discrete if and
+only if its underlying condensed set is discrete.
 -/
 
 namespace CategoryTheory
 
-open Limits Opposite Category Functor Sheaf
+open Limits Opposite Category Functor Sheaf Adjunction
 
 variable {C : Type*} [Category C] (J : GrothendieckTopology C)
 variable (D : Type*) [Category D]
 
 /-- The constant presheaf functor is left adjoint to evaluation at a terminal object. -/
+@[simps! unit_app counit_app_app]
 noncomputable def constantPresheafAdj {T : C} (hT : IsTerminal T) :
-    Functor.const Cᵒᵖ ⊣ (evaluation Cᵒᵖ D).obj (op T) :=
-  Adjunction.mkOfUnitCounit {
-    unit := (Functor.constCompEvaluationObj D (op T)).hom
-    counit := {
-      app := fun F => {
-        app := fun ⟨X⟩ => F.map (IsTerminal.from hT X).op
-        naturality := fun _ _ _ => by
-          simp only [Functor.comp_obj, Functor.const_obj_obj, Functor.id_obj, Functor.const_obj_map,
-            Category.id_comp, ← Functor.map_comp]
-          congr
-          simp }
-      naturality := by intros; ext; simp /- Note: `aesop` works but is kind of slow -/ } }
+    Functor.const Cᵒᵖ ⊣ (evaluation Cᵒᵖ D).obj (op T) where
+  unit := (Functor.constCompEvaluationObj D (op T)).hom
+  counit := {
+    app := fun F => {
+      app := fun ⟨X⟩ => F.map (IsTerminal.from hT X).op
+      naturality := fun _ _ _ => by
+        simp only [Functor.comp_obj, Functor.const_obj_obj, Functor.id_obj, Functor.const_obj_map,
+          Category.id_comp, ← Functor.map_comp]
+        congr
+        simp }
+    naturality := by intros; ext; simp /- Note: `aesop` works but is kind of slow -/ }
 
 variable [HasWeakSheafify J D]
 
 /--
-The functor which maps an object of `D` to the constant sheaf at that object, i.e. the
+The functor which maps an object of `D` to the constant sheaf at that object, i.e. the
 sheafification of the constant presheaf.
 -/
 noncomputable def constantSheaf : D ⥤ Sheaf J D := Functor.const Cᵒᵖ ⋙ (presheafToSheaf J D)
 
 /-- The constant sheaf functor is left adjoint to evaluation at a terminal object. -/
+@[simps! counit_app]
 noncomputable def constantSheafAdj {T : C} (hT : IsTerminal T) :
     constantSheaf J D ⊣ (sheafSections J D).obj (op T) :=
   (constantPresheafAdj D hT).comp (sheafificationAdjunction J D)
 
-lemma constantSheafAdj_counit_app {T : C} (hT : IsTerminal T) (F : Sheaf J D) :
-    (constantSheafAdj J D hT).counit.app F =
-      (presheafToSheaf J D).map ((constantPresheafAdj D hT).counit.app F.val) ≫
-        (sheafificationAdjunction J D).counit.app F := by
+variable {D}
+
+namespace Sheaf
+
+/--
+A sheaf is constant if it is in the essential image of the constant sheaf functor.
+-/
+class IsConstant (F : Sheaf J D) : Prop where
+  mem_essImage : F ∈ (constantSheaf J D).essImage
+
+lemma mem_essImage_of_isConstant (F : Sheaf J D) [IsConstant J F] :
+    F ∈ (constantSheaf J D).essImage :=
+  IsConstant.mem_essImage
+
+lemma isConstant_congr {F G : Sheaf J D} (i : F ≅ G) [IsConstant J F] : IsConstant J G where
+  mem_essImage := essImage.ofIso i F.mem_essImage_of_isConstant
+
+lemma isConstant_of_iso {F : Sheaf J D} {X : D} (i : F ≅ (constantSheaf J D).obj X) :
+    IsConstant J F := ⟨_, ⟨i.symm⟩⟩
+
+lemma isConstant_iff_mem_essImage {L : D ⥤ Sheaf J D} {T : C} (hT : IsTerminal T)
+    (adj : L ⊣ (sheafSections J D).obj ⟨T⟩)
+    (F : Sheaf J D) : IsConstant J F ↔ F ∈ L.essImage := by
+  rw [essImage_eq_of_natIso (adj.leftAdjointUniq (constantSheafAdj J D hT))]
+  exact ⟨fun ⟨h⟩ ↦ h, fun h ↦ ⟨h⟩⟩
+
+lemma isConstant_of_isIso_counit_app (F : Sheaf J D) [HasTerminal C]
+    [IsIso <| (constantSheafAdj J D terminalIsTerminal).counit.app F] : IsConstant J F where
+  mem_essImage := ⟨_, ⟨asIso <| (constantSheafAdj J D terminalIsTerminal).counit.app F⟩⟩
+
+instance [(constantSheaf J D).Faithful] [(constantSheaf J D).Full] (F : Sheaf J D)
+    [IsConstant J F] {T : C} (hT : IsTerminal T) :
+    IsIso ((constantSheafAdj J D hT).counit.app F) := by
+  rw [isIso_counit_app_iff_mem_essImage]
+  exact F.mem_essImage_of_isConstant
+
+/--
+If the constant sheaf functor is fully faithful, then a sheaf is constant if and only if the
+counit of the constant sheaf adjunction applied to it is an isomorphism.
+-/
+lemma isConstant_iff_isIso_counit_app [(constantSheaf J D).Faithful] [(constantSheaf J D).Full]
+    (F : Sheaf J D) {T : C} (hT : IsTerminal T) :
+      IsConstant J F ↔ (IsIso <| (constantSheafAdj J D hT).counit.app F) :=
+  ⟨fun _ ↦ inferInstance, fun _ ↦ ⟨_, ⟨asIso <| (constantSheafAdj J D hT).counit.app F⟩⟩⟩
+
+/--
+A variant of `isConstant_iff_isIso_counit_app` for a general left adjoint to evaluation at a
+terminal object.
+-/
+lemma isConstant_iff_isIso_counit_app'  {L : D ⥤ Sheaf J D} {T : C} (hT : IsTerminal T)
+    (adj : L ⊣ (sheafSections J D).obj ⟨T⟩)
+    [L.Faithful] [L.Full] (F : Sheaf J D) : IsConstant J F ↔ IsIso (adj.counit.app F) :=
+  (isConstant_iff_mem_essImage J hT adj F).trans (isIso_counit_app_iff_mem_essImage adj).symm
+
+end Sheaf
+
+section Equivalence
+variable {C' : Type*} [Category C'] (K : GrothendieckTopology C') [HasWeakSheafify K D]
+variable (G : C ⥤ C') [∀ (X : (C')ᵒᵖ), HasLimitsOfShape (StructuredArrow X G.op) D]
+  [G.IsDenseSubsite J K] {T : C} (hT : IsTerminal T) (hT' : IsTerminal (G.obj T))
+
+open IsDenseSubsite
+
+variable (D) in
+/--
+The constant sheaf functor commutes up to isomorphism the equivalence of sheaf categories induced
+by a dense subsite.
+-/
+noncomputable def equivCommuteConstant :
+    constantSheaf J D ⋙ (sheafEquiv G J K D).functor ≅ constantSheaf K D :=
+  ((constantSheafAdj J D hT).comp (sheafEquiv G J K D).toAdjunction).leftAdjointUniq
+    (constantSheafAdj K D hT')
+
+variable (D) in
+/--
+The constant sheaf functor commutes up to isomorphism the inverse equivalence of sheaf categories
+induced by a dense subsite.
+-/
+noncomputable def equivCommuteConstant' :
+    constantSheaf J D ≅ constantSheaf K D ⋙ (sheafEquiv G J K D).inverse :=
+  isoWhiskerLeft (constantSheaf J D) (sheafEquiv G J K D).unitIso ≪≫
+    isoWhiskerRight (equivCommuteConstant J D K G hT hT') (sheafEquiv G J K D).inverse
+
+/- TODO: find suitable assumptions for proving generalizations of `equivCommuteConstant` and
+`equivCommuteConstant'` above, to commute `constantSheaf` with pullback/pushforward of sheaves. -/
+
+include hT hT' in
+/--
+The property of a sheaf of being constant is invariant under equivalence of sheaf
+categories.
+-/
+lemma Sheaf.isConstant_iff_of_equivalence (F : Sheaf K D) :
+    ((sheafEquiv G J K D).inverse.obj F).IsConstant J ↔ IsConstant K F := by
+  constructor
+  · exact fun ⟨Y, ⟨i⟩⟩ ↦ ⟨_, ⟨(equivCommuteConstant J D K G hT hT').symm.app _ ≪≫
+      (sheafEquiv G J K D).functor.mapIso i ≪≫ (sheafEquiv G J K D).counitIso.app _⟩⟩
+  · exact fun ⟨Y, ⟨i⟩⟩ ↦ ⟨_, ⟨(equivCommuteConstant' J D K G hT hT').app _ ≪≫
+      (sheafEquiv G J K D).inverse.mapIso i⟩⟩
+
+end Equivalence
+
+section Forget
+
+variable {B : Type*} [Category B] (U : D ⥤ B) [HasWeakSheafify J B]
+  [J.PreservesSheafification U] [J.HasSheafCompose U] (F : Sheaf J D)
+
+/--
+The constant sheaf functor commutes with `sheafCompose J U` up to isomorphism, provided that `U`
+preserves sheafification.
+-/
+noncomputable def constantCommuteCompose :
+    constantSheaf J D ⋙ sheafCompose J U ≅ U ⋙ constantSheaf J B :=
+  (isoWhiskerLeft (const Cᵒᵖ)
+    (sheafComposeNatIso J U (sheafificationAdjunction J D) (sheafificationAdjunction J B)).symm) ≪≫
+      isoWhiskerRight (compConstIso _ _).symm _
+
+lemma constantCommuteCompose_hom_app_val (X : D) : ((constantCommuteCompose J U).hom.app X).val =
+    (sheafifyComposeIso J U ((const Cᵒᵖ).obj X)).inv ≫ sheafifyMap J (constComp Cᵒᵖ X U).hom := rfl
+
+/-- The counit of `constantSheafAdj` factors through the isomorphism `constantCommuteCompose`. -/
+lemma constantSheafAdj_counit_w {T : C} (hT : IsTerminal T) :
+    ((constantCommuteCompose J U).hom.app (F.val.obj ⟨T⟩)) ≫
+      ((constantSheafAdj J B hT).counit.app ((sheafCompose J U).obj F)) =
+        ((sheafCompose J U).map ((constantSheafAdj J D hT).counit.app F)) := by
   apply Sheaf.hom_ext
-  apply sheafify_hom_ext _ _ _ F.cond
-  simp only [flip_obj_obj, sheafToPresheaf_obj, comp_obj, id_obj, constantSheafAdj, Adjunction.comp,
-    evaluation_obj_obj, constantPresheafAdj, Opposite.op_unop, Adjunction.mkOfUnitCounit_unit,
-    Adjunction.mkOfUnitCounit_counit, NatTrans.comp_app, associator_hom_app, whiskerLeft_app,
-    whiskerRight_app, instCategorySheaf_comp_val, instCategorySheaf_id_val,
-    sheafificationAdjunction_counit_app_val, sheafifyMap_sheafifyLift, comp_id,
-    toSheafify_sheafifyLift]
-  erw [id_comp, toSheafify_sheafifyLift]
+  rw [instCategorySheaf_comp_val, constantCommuteCompose_hom_app_val, assoc, Iso.inv_comp_eq]
+  apply sheafify_hom_ext _ _ _ ((sheafCompose J U).obj F).cond
+  ext
+  simp? says simp only [comp_obj, const_obj_obj, sheafCompose_obj_val, id_obj,
+      constantSheafAdj_counit_app, instCategorySheaf_comp_val,
+      sheafificationAdjunction_counit_app_val, sheafifyMap_sheafifyLift, comp_id,
+      toSheafify_sheafifyLift, NatTrans.comp_app, constComp_hom_app,
+      constantPresheafAdj_counit_app_app, Functor.comp_map, id_comp, flip_obj_obj,
+      sheafToPresheaf_obj, map_comp, sheafCompose_map_val, sheafComposeIso_hom_fac_assoc,
+      whiskerRight_app]
+  simp [← map_comp, ← NatTrans.comp_app]
+
+lemma Sheaf.isConstant_of_forget [constantSheaf J D |>.Faithful] [constantSheaf J D |>.Full]
+    [constantSheaf J B |>.Faithful] [constantSheaf J B |>.Full]
+    [(sheafCompose J U).ReflectsIsomorphisms] [((sheafCompose J U).obj F).IsConstant J]
+    {T : C} (hT : IsTerminal T) : F.IsConstant J := by
+  have : IsIso ((sheafCompose J U).map ((constantSheafAdj J D hT).counit.app F)) := by
+    rw [← constantSheafAdj_counit_w]
+    infer_instance
+  rw [F.isConstant_iff_isIso_counit_app (hT := hT)]
+  exact isIso_of_reflects_iso _ (sheafCompose J U)
+
+instance [h : F.IsConstant J] : ((sheafCompose J U).obj F).IsConstant J := by
+  obtain ⟨Y, ⟨i⟩⟩ := h
+  exact ⟨U.obj Y, ⟨(fullyFaithfulSheafToPresheaf _ _).preimageIso
+    (((sheafifyComposeIso J U ((const Cᵒᵖ).obj Y)).symm ≪≫
+      (presheafToSheaf J B ⋙ sheafToPresheaf J B).mapIso (constComp Cᵒᵖ Y U)).symm ≪≫
+        (sheafToPresheaf _ _).mapIso ((sheafCompose J U).mapIso i))⟩⟩
+
+lemma Sheaf.isConstant_iff_forget [constantSheaf J D |>.Faithful] [constantSheaf J D |>.Full]
+    [constantSheaf J B |>.Faithful] [constantSheaf J B |>.Full]
+      [(sheafCompose J U).ReflectsIsomorphisms] {T : C} (hT : IsTerminal T) :
+        F.IsConstant J ↔ ((sheafCompose J U).obj F).IsConstant J :=
+  ⟨fun _ ↦ inferInstance, fun _ ↦ Sheaf.isConstant_of_forget _ U F hT⟩
+
+end Forget
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Sites/Discrete.lean b/Mathlib/CategoryTheory/Sites/Discrete.lean
deleted file mode 100644
index 6998b9657e496..0000000000000
--- a/Mathlib/CategoryTheory/Sites/Discrete.lean
+++ /dev/null
@@ -1,283 +0,0 @@
-/-
-Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Dagur Asgeirsson
--/
-import Mathlib.CategoryTheory.Adjunction.FullyFaithful
-import Mathlib.CategoryTheory.Sites.ConstantSheaf
-import Mathlib.CategoryTheory.Sites.DenseSubsite
-import Mathlib.CategoryTheory.Sites.PreservesSheafification
-/-!
-
-# Discrete objects in sheaf categories.
-
-This file defines the notion of a discrete object in a sheaf category. A discrete sheaf in this
-context is a sheaf `F` such that the counit `(F(*))^cst ⟶ F` is an isomorphism. Here `*` denotes
-a particular chosen terminal object of the defining site, and `cst` denotes the constant sheaf.
-
-It is convenient to take an arbitrary terminal object; one might want to use this construction to
-talk about discrete sheaves on a site which has a particularly convenient terminal object, such as
-the one element space in `CompHaus`.
-
-## Main results
-
-* `isDiscrete_iff_mem_essImage` : A sheaf is discrete if and only if it is in the essential image
-of the constant sheaf functor.
-* `isDiscrete_iff_of_equivalence` : The property of a sheaf of being discrete is invariant under
-equivalence of sheaf categories.
-* `isDiscrete_iff_forget` : Given a "forgetful" functor `U : A ⥤ B` a sheaf `F : Sheaf J A` is
-discrete if and only if the sheaf given by postcomposition with `U` is discrete.
-
-## Future work
-
-* Use `isDiscrete_iff_forget` to prove that a condensed module is discrete if and only if its
-underlying condensed set is discrete.
--/
-
-open CategoryTheory Limits Functor Adjunction Opposite Category Functor
-
-namespace CategoryTheory.Sheaf
-
-variable {C : Type*} [Category C] (J : GrothendieckTopology C) {A : Type*} [Category A]
-  [HasWeakSheafify J A] {t : C} (ht : IsTerminal t)
-
-section
-variable [(constantSheaf J A).Faithful] [(constantSheaf J A).Full]
-
-/--
-A sheaf is discrete if it is a discrete object of the "underlying object" functor from the sheaf
-category to the target category.
--/
-abbrev IsDiscrete (F : Sheaf J A) : Prop := IsIso ((constantSheafAdj J A ht).counit.app F)
-
-lemma isDiscrete_of_iso {F : Sheaf J A} {X : A}
-    (i : F ≅ (constantSheaf J A).obj X) : IsDiscrete J ht F :=
-  isIso_counit_app_of_iso _ i
-
-lemma isDiscrete_iff_mem_essImage (F : Sheaf J A) :
-    F.IsDiscrete J ht ↔ F ∈ (constantSheaf J A).essImage :=
-  (constantSheafAdj J A ht).isIso_counit_app_iff_mem_essImage
-
-lemma isDiscrete_iff_mem_essImage' {L : A ⥤ Sheaf J A} (adj : L ⊣ (sheafSections J A).obj ⟨t⟩)
-    (F : Sheaf J A) :
-    IsDiscrete J ht F ↔ F ∈ L.essImage := by
-  let e : L ≅ constantSheaf J A := adj.leftAdjointUniq (constantSheafAdj _ _ ht)
-  refine ⟨fun h ↦ ⟨F.val.obj ⟨t⟩, ⟨?_⟩⟩, fun ⟨Y, ⟨i⟩⟩ ↦ ?_⟩
-  · exact e.app _ ≪≫ asIso ((constantSheafAdj _ _ ht).counit.app _)
-  · rw [isDiscrete_iff_mem_essImage]
-    exact ⟨Y, ⟨e.symm.app _ ≪≫ i⟩⟩
-
-lemma isDiscrete_iff_isIso_counit_app {L : A ⥤ Sheaf J A} (adj : L ⊣ (sheafSections J A).obj ⟨t⟩)
-    (F : Sheaf J A) :
-    IsDiscrete J ht F ↔ IsIso (adj.counit.app F) := by
-  have : L.Faithful := Functor.Faithful.of_iso (adj.leftAdjointUniq (constantSheafAdj _ _ ht)).symm
-  have : L.Full := Functor.Full.of_iso (adj.leftAdjointUniq (constantSheafAdj _ _ ht)).symm
-  rw [isIso_counit_app_iff_mem_essImage]
-  exact isDiscrete_iff_mem_essImage' _ _ adj _
-
-section Equivalence
-
-variable {D : Type*} [Category D] (K : GrothendieckTopology D) [HasWeakSheafify K A]
-variable (G : C ⥤ D)
-  [∀ (X : Dᵒᵖ), HasLimitsOfShape (StructuredArrow X G.op) A]
-  [G.IsDenseSubsite J K] (ht' : IsTerminal (G.obj t))
-
-open Functor.IsDenseSubsite
-
-noncomputable example :
-    let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-    e.inverse ⋙ (sheafSections J A).obj (op t) ≅ (sheafSections K A).obj (op (G.obj t)) :=
-  Iso.refl _
-
-variable (A) in
-/--
-The constant sheaf functor commutes up to isomorphism with any equivalence of sheaf categories.
-
-This is an auxiliary definition used to prove `Sheaf.isDiscrete_iff_of_equivalence` below, which
-says that the property of a sheaf of being a discrete object is invariant under equivalence of
-sheaf categories.
--/
-noncomputable def equivCommuteConstant :
-    let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-    constantSheaf J A ⋙ e.functor ≅ constantSheaf K A :=
-  let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-  (Adjunction.leftAdjointUniq ((constantSheafAdj J A ht).comp e.toAdjunction)
-    (constantSheafAdj K A ht'))
-
-variable (A) in
-/--
-The constant sheaf functor commutes up to isomorphism with any equivalence of sheaf categories.
-
-This is an auxiliary definition used to prove `Sheaf.isDiscrete_iff_of_equivalence` below, which
-says that the property of a sheaf of being a discrete object is invariant under equivalence of
-sheaf categories.
--/
-noncomputable def equivCommuteConstant' :
-    let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-    constantSheaf J A ≅ constantSheaf K A ⋙ e.inverse :=
-  let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-  isoWhiskerLeft (constantSheaf J A) e.unitIso ≪≫
-    isoWhiskerRight (equivCommuteConstant J A ht K G ht') e.inverse
-
-/--
-The property of a sheaf of being a discrete object is invariant under equivalence of sheaf
-categories.
--/
-lemma isDiscrete_iff_of_equivalence (F : Sheaf K A) :
-    let e : Sheaf J A ≌ Sheaf K A :=
-      sheafEquiv G J K A
-    haveI : (constantSheaf K A).Faithful :=
-      Functor.Faithful.of_iso (equivCommuteConstant J A ht K G ht')
-    haveI : (constantSheaf K A).Full :=
-      Functor.Full.of_iso (equivCommuteConstant J A ht K G ht')
-    (e.inverse.obj F).IsDiscrete J ht ↔ IsDiscrete K ht' F := by
-  intro e
-  have : (constantSheaf K A).Faithful :=
-      Functor.Faithful.of_iso (equivCommuteConstant J A ht K G ht')
-  have : (constantSheaf K A).Full :=
-    Functor.Full.of_iso (equivCommuteConstant J A ht K G ht')
-  simp only [isDiscrete_iff_mem_essImage]
-  constructor
-  · intro ⟨Y, ⟨i⟩⟩
-    let j : (constantSheaf K A).obj Y ≅ F :=
-      (equivCommuteConstant J A ht K G ht').symm.app _ ≪≫ e.functor.mapIso i ≪≫ e.counitIso.app _
-    exact ⟨_, ⟨j⟩⟩
-  · intro ⟨Y, ⟨i⟩⟩
-    let j : (constantSheaf J A).obj Y ≅ e.inverse.obj F :=
-      (equivCommuteConstant' J A ht K G ht').app _ ≪≫ e.inverse.mapIso i
-    exact ⟨_, ⟨j⟩⟩
-
-end Equivalence
-
-end
-
-section Forget
-
-variable {B : Type*} [Category B] (U : A ⥤ B) [HasWeakSheafify J B]
-  [J.PreservesSheafification U] [J.HasSheafCompose U] (F : Sheaf J A)
-
-open Limits
-
-/-- The constant sheaf functor commutes with `sheafCompose` up to isomorphism. -/
-@[simps!]
-noncomputable def constantCommuteCompose :
-    constantSheaf J A ⋙ sheafCompose J U ≅ U ⋙ constantSheaf J B :=
-  (isoWhiskerLeft (const Cᵒᵖ)
-    (sheafComposeNatIso J U (sheafificationAdjunction J A) (sheafificationAdjunction J B)).symm) ≪≫
-      isoWhiskerRight (compConstIso _ _).symm _
-
-lemma sheafComposeNatIso_app_counit (P : Sheaf J A) :
-    (sheafComposeNatIso J U (sheafificationAdjunction J A)
-      (sheafificationAdjunction J B)).hom.app _ ≫ (sheafCompose J U).map
-        ((sheafificationAdjunction J A).counit.app P) =
-          (sheafificationAdjunction J B).counit.app ((sheafCompose J U).obj P) := by
-  simp only [sheafToPresheaf_obj, Functor.comp_obj, whiskeringRight_obj_obj, Functor.id_obj,
-    sheafComposeNatIso, sheafComposeNatTrans, sheafCompose_obj_val,
-    sheafificationAdjunction_unit_app, asIso_hom]
-  erw [Adjunction.homEquiv_counit]
-  apply Sheaf.hom_ext
-  apply sheafify_hom_ext _ _ _ ((sheafCompose J U).obj P).cond
-  simp [← whiskerRight_comp]
-
-lemma constantCommuteComposeApp_comp_counit (F : Sheaf J A) :
-    ((constantCommuteCompose J U).app _).hom ≫
-      (constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F) =
-        (sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F) := by
-  simp only [← Iso.eq_inv_comp, constantSheafAdj_counit_app,
-    constantCommuteCompose, flip_obj_obj, sheafToPresheaf_obj, id_obj, NatIso.trans_app, comp_obj,
-    whiskeringRight_obj_obj, Iso.trans_inv, Iso.app_inv, isoWhiskerRight_inv, Iso.symm_inv,
-    whiskerRight_app, isoWhiskerLeft_inv, whiskerLeft_app, evaluation_obj_obj, Functor.map_comp,
-    assoc, sheafCompose_obj_val, ← sheafComposeNatIso_app_counit]
-  simp only [← assoc]
-  congr 1
-  have : (compConstIso Cᵒᵖ U).hom.app (F.val.obj ⟨t⟩) ≫
-      { app := fun Y ↦ (F.val ⋙ U).map (ht.from _).op
-        naturality := by intros; simp; rw [← Functor.map_comp, ← Functor.map_comp]; congr; simp } =
-      ((constantPresheafAdj B ht).counit.app (F.val ⋙ U)) := by ext; simp [constantPresheafAdj]
-  simp only [← this, assoc, Functor.map_comp]
-  congr 1
-  apply Sheaf.hom_ext
-  apply sheafify_hom_ext _ _ _ ((sheafCompose J U).obj ((presheafToSheaf J A).obj F.val)).cond
-  simp only [sheafCompose_obj_val, instCategorySheaf_comp_val, sheafCompose_map_val, comp_obj,
-    whiskeringRight_obj_obj, Functor.comp_map]
-  erw [← toSheafify_naturality_assoc, sheafComposeIso_hom_fac, sheafComposeIso_hom_fac_assoc]
-  ext
-  simp only [comp_obj, const_obj_obj, NatTrans.comp_app, whiskerRight_app, ← Functor.map_comp]
-  congr 1
-  simp only [constantPresheafAdj, comp_obj, evaluation_obj_obj, id_obj, Opposite.op_unop,
-    Adjunction.mkOfUnitCounit_counit, NatTrans.naturality]
-  erw [← NatTrans.comp_app, ← toSheafify_naturality]
-  simp only [NatTrans.comp_app, const_obj_obj, NatTrans.naturality]
-
-/-- Auxiliary lemma for `sheafCompose_reflects_discrete`. -/
-private lemma sheafifyComposeIso_comp_sheafCompose_map_constantSheafAdj_counit :
-  (sheafifyComposeIso J U ((const Cᵒᵖ).obj (F.val.obj { unop := t }))).hom ≫
-    ((sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F)).val =
-      ((presheafToSheaf J B ⋙ sheafToPresheaf J B).mapIso (constComp Cᵒᵖ _ U)).hom ≫
-        ((constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F)).val := by
-  apply sheafify_hom_ext _ _ _ ((sheafCompose J U).obj F).cond
-  simp only [sheafCompose_obj_val, id_obj, comp_obj, flip_obj_obj, sheafToPresheaf_obj,
-    sheafComposeIso_hom_fac_assoc, mapIso_hom, Functor.comp_map, sheafToPresheaf_map]
-  erw [Adjunction.unit_naturality_assoc]
-  simp only [const_obj_obj, const_obj_map, id_obj, constComp, comp_obj, sheafToPresheaf_obj,
-    sheafificationAdjunction_unit_app]
-  ext
-  simp only [comp_obj, const_obj_obj, NatTrans.comp_app, whiskerRight_app, Category.id_comp,
-    comp_obj, flip_obj_obj, sheafToPresheaf_obj, id_obj, constantSheafAdj,
-    Adjunction.comp, evaluation_obj_obj, NatTrans.comp_app, associator_hom_app, whiskerLeft_app,
-    whiskerRight_app, map_comp, instCategorySheaf_comp_val, sheafCompose_obj_val,
-    sheafCompose_map_val, instCategorySheaf_id_val, sheafificationAdjunction_counit_app_val,
-    NatTrans.id_app, sheafifyMap_sheafifyLift, Category.comp_id, Category.id_comp]
-  erw [Functor.map_id, Category.id_comp, ← NatTrans.comp_app]
-  simp only [toSheafify_sheafifyLift, ← Functor.map_comp, ← NatTrans.comp_app,
-    sheafifyMap_sheafifyLift, Category.comp_id,
-    constantPresheafAdj, comp_obj, evaluation_obj_obj, id_obj, op_unop,
-    mkOfUnitCounit_counit, Functor.comp_map]
-
-/-- Auxiliary lemma for `sheafCompose_reflects_discrete`. -/
-private lemma constantSheafAdj_counit_w :
-    ((sheafifyComposeIso J U ((const Cᵒᵖ).obj (F.val.obj ⟨t⟩))).symm ≪≫
-      (presheafToSheaf J B ⋙ sheafToPresheaf J B).mapIso (constComp Cᵒᵖ (F.val.obj ⟨t⟩) U)).hom ≫
-        ((constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F)).val =
-          ((sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F)).val := by
-  rw [← Iso.eq_inv_comp]
-  simp only [comp_obj, flip_obj_obj, sheafToPresheaf_obj, sheafCompose_obj_val, id_obj,
-    Iso.trans_inv, mapIso_inv, Functor.comp_map, sheafToPresheaf_map,
-    Iso.symm_inv, Category.assoc, sheafifyComposeIso_comp_sheafCompose_map_constantSheafAdj_counit,
-    mapIso_hom, ← instCategorySheaf_comp_val, Iso.map_inv_hom_id_assoc]
-
-lemma sheafCompose_reflects_discrete [(sheafCompose J U).ReflectsIsomorphisms]
-    [((sheafCompose J U).obj F).IsDiscrete J ht] :
-    F.IsDiscrete J ht := by
-  let f := (sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F)
-  have : IsIso ((sheafToPresheaf J B).map f) := by
-    simp only [comp_obj, flip_obj_obj, sheafToPresheaf_obj, sheafCompose_obj_val, id_obj,
-      sheafToPresheaf_map, f, ← constantSheafAdj_counit_w]
-    exact inferInstanceAs (IsIso (_ ≫ ((sheafToPresheaf J B).map
-      ((constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F)))))
-  have := isIso_of_reflects_iso f (sheafToPresheaf J B)
-  exact isIso_of_reflects_iso _ (sheafCompose J U)
-
-variable [(constantSheaf J A).Full] [(constantSheaf J A).Faithful]
-  [(constantSheaf J B).Full] [(constantSheaf J B).Faithful]
-
-instance [h : F.IsDiscrete J ht] : ((sheafCompose J U).obj F).IsDiscrete J ht := by
-  rw [isDiscrete_iff_mem_essImage] at h ⊢
-  obtain ⟨Y, ⟨i⟩⟩ := h
-  exact ⟨U.obj Y, ⟨(fullyFaithfulSheafToPresheaf _ _).preimageIso
-    (((sheafifyComposeIso J U ((const Cᵒᵖ).obj Y)).symm ≪≫
-      (presheafToSheaf J B ⋙ sheafToPresheaf J B).mapIso (constComp Cᵒᵖ Y U)).symm ≪≫
-        (sheafToPresheaf _ _).mapIso ((sheafCompose J U).mapIso i))⟩⟩
-
-lemma isDiscrete_iff_forget [(sheafCompose J U).ReflectsIsomorphisms] : F.IsDiscrete J ht ↔
-    ((sheafCompose J U).obj F).IsDiscrete J ht :=
-  ⟨fun _ ↦ inferInstance, fun _ ↦ sheafCompose_reflects_discrete _ _ U F⟩
-
-end Forget
-
-end CategoryTheory.Sheaf
diff --git a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
index 1466ccb25979a..44e39a7bba0c9 100644
--- a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
+++ b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
@@ -10,7 +10,7 @@ import Mathlib.Tactic.ApplyFun
 /-!
 # The equalizer diagram sheaf condition for a presieve
 
-In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a
+In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a
 sheaf *for* a particular presieve. In this file we provide equivalent conditions in terms of
 equalizer diagrams.
 
@@ -228,8 +228,8 @@ theorem compatible_iff (x : FirstObj P R) :
 See <https://stacks.math.columbia.edu/tag/00VM>.
 -/
 theorem sheaf_condition : R.IsSheafFor P ↔ Nonempty (IsLimit (Fork.ofι _ (w P R))) := by
-  rw [Types.type_equalizer_iff_unique]
-  erw [← Equiv.forall_congr_right (firstObjEqFamily P R).toEquiv.symm]
+  rw [Types.type_equalizer_iff_unique,
+    ← Equiv.forall_congr_right (firstObjEqFamily P R).toEquiv.symm]
   simp_rw [← compatible_iff, ← Iso.toEquiv_fun, Equiv.apply_symm_apply]
   apply forall₂_congr
   intro x _
@@ -252,7 +252,7 @@ open Presieve
 
 variable {B : C} {I : Type} (X : I → C) (π : (i : I) → X i ⟶ B)
     [(Presieve.ofArrows X π).hasPullbacks]
--- TODO: allow `I : Type w` 
+-- TODO: allow `I : Type w`
 
 /--
 The middle object of the fork diagram of <https://stacks.math.columbia.edu/tag/00VM>.
diff --git a/Mathlib/CategoryTheory/Sites/Equivalence.lean b/Mathlib/CategoryTheory/Sites/Equivalence.lean
index faf9debefee9f..2408d76270ca3 100644
--- a/Mathlib/CategoryTheory/Sites/Equivalence.lean
+++ b/Mathlib/CategoryTheory/Sites/Equivalence.lean
@@ -63,7 +63,8 @@ instance (priority := 900) [G.IsEquivalence] : IsCoverDense G J where
 instance : e.functor.IsDenseSubsite J (e.inverse.inducedTopology J) := by
   have : J = e.functor.inducedTopology (e.inverse.inducedTopology J) := by
     ext X S
-    erw [← GrothendieckTopology.pullback_mem_iff_of_isIso (i := e.unit.app X)]
+    rw [show S ∈ (e.functor.inducedTopology (e.inverse.inducedTopology J)) X ↔ _
+      from (GrothendieckTopology.pullback_mem_iff_of_isIso (i := e.unit.app X)).symm]
     congr!; ext Y f; simp
   nth_rw 1 [this]
   infer_instance
diff --git a/Mathlib/CategoryTheory/Sites/Grothendieck.lean b/Mathlib/CategoryTheory/Sites/Grothendieck.lean
index e42c500a941ed..0d3552925cc54 100644
--- a/Mathlib/CategoryTheory/Sites/Grothendieck.lean
+++ b/Mathlib/CategoryTheory/Sites/Grothendieck.lean
@@ -263,7 +263,13 @@ instance : InfSet (GrothendieckTopology C) where
 /-- See <https://stacks.math.columbia.edu/tag/00Z7> -/
 theorem isGLB_sInf (s : Set (GrothendieckTopology C)) : IsGLB s (sInf s) := by
   refine @IsGLB.of_image _ _ _ _ sieves ?_ _ _ ?_
-  · rfl
+  · #adaptation_note
+    /--
+    This proof used to be `rfl`,
+    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
+    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
+    -/
+    exact Iff.rfl
   · exact _root_.isGLB_sInf _
 
 /-- Construct a complete lattice from the `Inf`, but make the trivial and discrete topologies
@@ -487,7 +493,7 @@ def pullbackComp {X Y Z : C} (S : J.Cover X) (f : Z ⟶ Y) (g : Y ⟶ X) :
 /-- Combine a family of covers over a cover. -/
 def bind {X : C} (S : J.Cover X) (T : ∀ I : S.Arrow, J.Cover I.Y) : J.Cover X :=
   ⟨Sieve.bind S fun Y f hf => T ⟨Y, f, hf⟩,
-    J.bind_covering S.condition fun _ _ _ => (T _).condition⟩
+    J.bind_covering S.condition fun _ _ _ => (T { Y := _, f := _, hf := _ }).condition⟩
 
 /-- The canonical morphism from `S.bind T` to `T`. -/
 def bindToBase {X : C} (S : J.Cover X) (T : ∀ I : S.Arrow, J.Cover I.Y) : S.bind T ⟶ S :=
diff --git a/Mathlib/CategoryTheory/Sites/IsSheafFor.lean b/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
index ccd585a8efb58..136e553138ef3 100644
--- a/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
+++ b/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
@@ -123,7 +123,7 @@ In special cases, this condition can be simplified, see `pullbackCompatible_iff`
 This is referred to as a "compatible family" in Definition C2.1.2 of [Elephant], and on nlab:
 https://ncatlab.org/nlab/show/sheaf#GeneralDefinitionInComponents
 
-For a more explicit version in the case where `R` is of the form `Presieve.ofArrows`, see
+For a more explicit version in the case where `R` is of the form `Presieve.ofArrows`, see
 `CategoryTheory.Presieve.Arrows.Compatible`.
 -/
 def FamilyOfElements.Compatible (x : FamilyOfElements P R) : Prop :=
@@ -142,7 +142,7 @@ Equation (5). Viewing the type `FamilyOfElements` as the middle object of the fo
 https://stacks.math.columbia.edu/tag/00VM, this condition expresses that `pr₀* (x) = pr₁* (x)`,
 using the notation defined there.
 
-For a more explicit version in the case where `R` is of the form `Presieve.ofArrows`, see
+For a more explicit version in the case where `R` is of the form `Presieve.ofArrows`, see
 `CategoryTheory.Presieve.Arrows.PullbackCompatible`.
 -/
 def FamilyOfElements.PullbackCompatible (x : FamilyOfElements P R) [R.hasPullbacks] : Prop :=
@@ -663,7 +663,7 @@ section Arrows
 variable {B : C} {I : Type*} {X : I → C} (π : (i : I) → X i ⟶ B) (P)
 
 /--
-A more explicit version of `FamilyOfElements.Compatible` for a `Presieve.ofArrows`.
+A more explicit version of `FamilyOfElements.Compatible` for a `Presieve.ofArrows`.
 -/
 def Arrows.Compatible (x : (i : I) → P.obj (op (X i))) : Prop :=
   ∀ i j Z (gi : Z ⟶ X i) (gj : Z ⟶ X j), gi ≫ π i = gj ≫ π j →
@@ -720,7 +720,7 @@ theorem isSheafFor_arrows_iff : (ofArrows X π).IsSheafFor P ↔
 variable [(ofArrows X π).hasPullbacks]
 
 /--
-A more explicit version of `FamilyOfElements.PullbackCompatible` for a `Presieve.ofArrows`.
+A more explicit version of `FamilyOfElements.PullbackCompatible` for a `Presieve.ofArrows`.
 -/
 def Arrows.PullbackCompatible (x : (i : I) → P.obj (op (X i))) : Prop :=
   ∀ i j, P.map (pullback.fst (π i) (π j)).op (x i) =
diff --git a/Mathlib/CategoryTheory/Sites/LeftExact.lean b/Mathlib/CategoryTheory/Sites/LeftExact.lean
index c975c71f51e0c..d12346e4674e9 100644
--- a/Mathlib/CategoryTheory/Sites/LeftExact.lean
+++ b/Mathlib/CategoryTheory/Sites/LeftExact.lean
@@ -255,15 +255,15 @@ instance preservesfiniteLimits_presheafToSheaf [HasFiniteLimits D] :
 
 variable (J D)
 
-/-- `plusPlusSheaf` is isomorphic to an arbitrary choice of left adjoint. -/
+/-- `plusPlusSheaf` is isomorphic to an arbitrary choice of left adjoint. -/
 def plusPlusSheafIsoPresheafToSheaf : plusPlusSheaf J D ≅ presheafToSheaf J D :=
   (plusPlusAdjunction J D).leftAdjointUniq (sheafificationAdjunction J D)
 
-/-- `plusPlusFunctor` is isomorphic to `sheafification`. -/
+/-- `plusPlusFunctor` is isomorphic to `sheafification`. -/
 def plusPlusFunctorIsoSheafification : J.sheafification D ≅ sheafification J D :=
   isoWhiskerRight (plusPlusSheafIsoPresheafToSheaf J D) (sheafToPresheaf J D)
 
-/-- `plusPlus` is isomorphic to `sheafify`. -/
+/-- `plusPlus` is isomorphic to `sheafify`. -/
 def plusPlusIsoSheafify (P : Cᵒᵖ ⥤ D) : J.sheafify P ≅ sheafify J P :=
   (sheafToPresheaf J D).mapIso ((plusPlusSheafIsoPresheafToSheaf J D).app P)
 
diff --git a/Mathlib/CategoryTheory/Sites/Limits.lean b/Mathlib/CategoryTheory/Sites/Limits.lean
index d85824a7edd78..1c38d0e033185 100644
--- a/Mathlib/CategoryTheory/Sites/Limits.lean
+++ b/Mathlib/CategoryTheory/Sites/Limits.lean
@@ -107,7 +107,7 @@ def isLimitMultiforkOfIsLimit (F : K ⥤ Sheaf J D) (E : Cone (F ⋙ sheafToPres
       erw [(isLimitOfPreserves ((evaluation Cᵒᵖ D).obj (op X)) hE).fac
         (multiforkEvaluationCone F E X W S)]
       dsimp [multiforkEvaluationCone, Presheaf.isLimitOfIsSheaf]
-      erw [Presheaf.IsSheaf.amalgamate_map]
+      rw [Presheaf.IsSheaf.amalgamate_map]
       rfl)
     (by
       intro S m hm
diff --git a/Mathlib/CategoryTheory/Sites/LocallyInjective.lean b/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
index a537edefc713a..782d1b84f0591 100644
--- a/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
+++ b/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
@@ -115,7 +115,7 @@ lemma isLocallyInjective_iff_equalizerSieve_mem_imp :
     · intro Y f hf
       refine J.superset_covering (Sieve.le_pullback_bind S.1 T _ hf)
         (equalizerSieve_mem J φ _ _ ?_)
-      erw [NatTrans.naturality_apply, NatTrans.naturality_apply]
+      rw [NatTrans.naturality_apply, NatTrans.naturality_apply]
       exact hf
   · intro hφ
     exact ⟨fun {X} x y h => hφ x y (by simp [h])⟩
@@ -179,7 +179,7 @@ open GrothendieckTopology.Plus
 instance isLocallyInjective_toPlus (P : Cᵒᵖ ⥤ Type max u v) :
     IsLocallyInjective J (J.toPlus P) where
   equalizerSieve_mem {X} x y h := by
-    erw [toPlus_eq_mk, toPlus_eq_mk, eq_mk_iff_exists] at h
+    rw [toPlus_eq_mk, toPlus_eq_mk, eq_mk_iff_exists] at h
     obtain ⟨W, h₁, h₂, eq⟩ := h
     exact J.superset_covering (fun Y f hf => congr_fun (congr_arg Subtype.val eq) ⟨Y, f, hf⟩) W.2
 
diff --git a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
index 5d9ab1ddb8a9c..06a7da7304b54 100644
--- a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
+++ b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
@@ -60,7 +60,7 @@ theorem imageSieve_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s
 theorem imageSieve_app {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : F.obj (op U)) :
     imageSieve f (f.app _ s) = ⊤ := by
   ext V i
-  simp only [Sieve.top_apply, iff_true_iff, imageSieve_apply]
+  simp only [Sieve.top_apply, iff_true, imageSieve_apply]
   have := elementwise_of% (f.naturality i.op)
   exact ⟨F.map i.op s, this s⟩
 
@@ -94,7 +94,7 @@ instance {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsLocallySurjective J f] :
 theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
     IsLocallySurjective J f ↔ (imagePresheaf (whiskerRight f (forget A))).sheafify J = ⊤ := by
   simp only [Subpresheaf.ext_iff, Function.funext_iff, Set.ext_iff, top_subpresheaf_obj,
-    Set.top_eq_univ, Set.mem_univ, iff_true_iff]
+    Set.top_eq_univ, Set.mem_univ, iff_true]
   exact ⟨fun H _ => H.imageSieve_mem, fun H => ⟨H _⟩⟩
 
 theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) :
diff --git a/Mathlib/CategoryTheory/Sites/Plus.lean b/Mathlib/CategoryTheory/Sites/Plus.lean
index f8755d25759dd..9fe18afba43ba 100644
--- a/Mathlib/CategoryTheory/Sites/Plus.lean
+++ b/Mathlib/CategoryTheory/Sites/Plus.lean
@@ -67,24 +67,20 @@ theorem diagramNatTrans_id (X : C) (P : Cᵒᵖ ⥤ D) :
     J.diagramNatTrans (𝟙 P) X = 𝟙 (J.diagram P X) := by
   ext : 2
   refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
-  dsimp
-  simp only [limit.lift_π, Multifork.ofι_pt, Multifork.ofι_π_app, Category.id_comp]
-  erw [Category.comp_id]
+  simp
 
 @[simp]
 theorem diagramNatTrans_zero [Preadditive D] (X : C) (P Q : Cᵒᵖ ⥤ D) :
     J.diagramNatTrans (0 : P ⟶ Q) X = 0 := by
   ext : 2
   refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
-  dsimp
-  rw [zero_comp, Multiequalizer.lift_ι, comp_zero]
+  simp
 
 @[simp]
 theorem diagramNatTrans_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (X : C) :
     J.diagramNatTrans (η ≫ γ) X = J.diagramNatTrans η X ≫ J.diagramNatTrans γ X := by
   ext : 2
   refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
-  dsimp
   simp
 
 variable (D)
@@ -110,7 +106,7 @@ def plusObj : Cᵒᵖ ⥤ D where
     simp only [diagramPullback_app, colimit.ι_pre, ι_colimMap_assoc, Category.comp_id]
     let e := S.unop.pullbackId
     dsimp only [Functor.op, pullback_obj]
-    erw [← colimit.w _ e.inv.op, ← Category.assoc]
+    rw [← colimit.w _ e.inv.op, ← Category.assoc]
     convert Category.id_comp (colimit.ι (diagram J P (unop X)) S)
     refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)
     dsimp
@@ -127,7 +123,7 @@ def plusObj : Cᵒᵖ ⥤ D where
       Category.assoc]
     let e := S.unop.pullbackComp g.unop f.unop
     dsimp only [Functor.op, pullback_obj]
-    erw [← colimit.w _ e.inv.op, ← Category.assoc, ← Category.assoc]
+    rw [← colimit.w _ e.inv.op, ← Category.assoc, ← Category.assoc]
     congr 1
     refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)
     dsimp
diff --git a/Mathlib/CategoryTheory/Sites/Preserves.lean b/Mathlib/CategoryTheory/Sites/Preserves.lean
index c3df3fb2d4af6..233115606bae5 100644
--- a/Mathlib/CategoryTheory/Sites/Preserves.lean
+++ b/Mathlib/CategoryTheory/Sites/Preserves.lean
@@ -18,14 +18,14 @@ preserve "the corresponding products".
 More precisely, given a presheaf `F : Cᵒᵖ ⥤ Type*`, we have:
 
 * If `F` satisfies the sheaf condition with respect to the empty sieve on the initial object of `C`,
-  then `F` preserves terminal objects.
+  then `F` preserves terminal objects.
 See `preservesTerminalOfIsSheafForEmpty`.
 
-* If `F` furthermore satisfies the sheaf condition with respect to the presieve consisting of the
-  inclusion arrows in a coproduct in `C`, then `F` preserves the corresponding product.
+* If `F` furthermore satisfies the sheaf condition with respect to the presieve consisting of the
+  inclusion arrows in a coproduct in `C`, then `F` preserves the corresponding product.
 See `preservesProductOfIsSheafFor`.
 
-* If `F` preserves a product, then it satisfies the sheaf condition with respect to the
+* If `F` preserves a product, then it satisfies the sheaf condition with respect to the
   corresponding presieve of arrows.
 See `isSheafFor_of_preservesProduct`.
 -/
@@ -44,8 +44,8 @@ section Terminal
 
 variable (I) in
 /--
-If `F` is a presheaf which satisfies the sheaf condition with respect to the empty presieve on any
-object, then `F` takes that object to the terminal object.
+If `F` is a presheaf which satisfies the sheaf condition with respect to the empty presieve on any
+object, then `F` takes that object to the terminal object.
 -/
 noncomputable
 def isTerminal_of_isSheafFor_empty_presieve : IsTerminal (F.obj (op I)) := by
@@ -54,8 +54,8 @@ def isTerminal_of_isSheafFor_empty_presieve : IsTerminal (F.obj (op I)) := by
   exact ⟨⟨fun _ ↦ t⟩, fun a ↦ by ext; exact h.2 _ (by tauto)⟩
 
 /--
-If `F` is a presheaf which satisfies the sheaf condition with respect to the empty presieve on the
-initial object, then `F` preserves terminal objects.
+If `F` is a presheaf which satisfies the sheaf condition with respect to the empty presieve on the
+initial object, then `F` preserves terminal objects.
 -/
 noncomputable
 def preservesTerminalOfIsSheafForEmpty (hI : IsInitial I) : PreservesLimit (Functor.empty Cᵒᵖ) F :=
diff --git a/Mathlib/CategoryTheory/Sites/PreservesSheafification.lean b/Mathlib/CategoryTheory/Sites/PreservesSheafification.lean
index 88823548b8066..ee99ef8e05d4a 100644
--- a/Mathlib/CategoryTheory/Sites/PreservesSheafification.lean
+++ b/Mathlib/CategoryTheory/Sites/PreservesSheafification.lean
@@ -170,9 +170,7 @@ lemma sheafComposeNatTrans_fac (P : Cᵒᵖ ⥤ A) :
     adj₂.unit.app (P ⋙ F) ≫
       (sheafToPresheaf J B).map ((sheafComposeNatTrans J F adj₁ adj₂).app P) =
         whiskerRight (adj₁.unit.app P) F  := by
-  dsimp only [sheafComposeNatTrans]
-  erw [Adjunction.homEquiv_counit, Adjunction.unit_naturality_assoc,
-    adj₂.right_triangle_components, comp_id]
+  simp [sheafComposeNatTrans, -sheafToPresheaf_obj, -sheafToPresheaf_map]
 
 lemma sheafComposeNatTrans_app_uniq (P : Cᵒᵖ ⥤ A)
     (α : G₂.obj (P ⋙ F) ⟶ (sheafCompose J F).obj (G₁.obj P))
diff --git a/Mathlib/CategoryTheory/Sites/Sheaf.lean b/Mathlib/CategoryTheory/Sites/Sheaf.lean
index 14f206f8513f1..80f0edcd7ebfb 100644
--- a/Mathlib/CategoryTheory/Sites/Sheaf.lean
+++ b/Mathlib/CategoryTheory/Sites/Sheaf.lean
@@ -148,7 +148,7 @@ theorem isLimit_iff_isSheafFor :
   rw [Classical.nonempty_pi]; constructor
   · intro hu E x hx
     specialize hu hx.cone
-    erw [(homEquivAmalgamation hx).uniqueCongr.nonempty_congr] at hu
+    rw [(homEquivAmalgamation hx).uniqueCongr.nonempty_congr] at hu
     exact (unique_subtype_iff_exists_unique _).1 hu
   · rintro h ⟨E, π⟩
     let eqv := conesEquivSieveCompatibleFamily P S (op E)
@@ -575,7 +575,6 @@ theorem isSheaf_iff_multiequalizer [∀ (X : C) (S : J.Cover X), HasMultiequaliz
     · intro a
       symm
       erw [IsIso.inv_comp_eq]
-      dsimp
       simp
 
 end MultiequalizerConditions
diff --git a/Mathlib/CategoryTheory/Sites/SheafHom.lean b/Mathlib/CategoryTheory/Sites/SheafHom.lean
index aeec15f07d717..196eeab7477ac 100644
--- a/Mathlib/CategoryTheory/Sites/SheafHom.lean
+++ b/Mathlib/CategoryTheory/Sites/SheafHom.lean
@@ -82,7 +82,7 @@ def presheafHomSectionsEquiv : (presheafHom F G).sections ≃ (F ⟶ G) where
         dsimp
         refine Eq.trans ?_ ((s.1 ⟨X₁⟩).naturality
           (Over.homMk f : Over.mk f ⟶ Over.mk (𝟙 X₁)).op)
-        erw [← s.2 f.op, presheafHom_map_app_op_mk_id]
+        rw [← s.2 f.op, presheafHom_map_app_op_mk_id]
         rfl }
   invFun f := ⟨fun X => whiskerLeft _ f, fun _ => rfl⟩
   left_inv s := by
@@ -174,7 +174,7 @@ lemma presheafHom_isSheafFor  :
           rintro ⟨Z : Over Y₂.left, hZ⟩
           dsimp
           rw [assoc, assoc, app_cond hG x hx Y₂.hom Z.hom hZ, ← G.map_comp, ← op_comp]
-          erw [app_cond hG x hx Y₁.hom (Z.hom ≫ φ.left) (by simpa using hZ),
+          rw [app_cond hG x hx Y₁.hom (Z.hom ≫ φ.left) (by simpa using hZ),
             ← F.map_comp_assoc, op_comp]
           congr 3
           simp }, ?_⟩
diff --git a/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean b/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
index db8d37ff2bf28..c2d7b20d409ae 100644
--- a/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
+++ b/Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
@@ -13,7 +13,7 @@ Defines the notion of a sheaf of types (usually called a sheaf of sets by mathem
 on a category equipped with a Grothendieck topology, as well as a range of equivalent
 conditions useful in different situations.
 
-In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a
+In `Mathlib/CategoryTheory/Sites/IsSheafFor.lean` it is defined what it means for a presheaf to be a
 sheaf *for* a particular sieve. Given a Grothendieck topology `J`, `P` is a sheaf if it is a sheaf
 for every sieve in the topology. See `IsSheaf`.
 
diff --git a/Mathlib/CategoryTheory/Sites/Sheafification.lean b/Mathlib/CategoryTheory/Sites/Sheafification.lean
index f85396c4e3a8d..409605999eeb7 100644
--- a/Mathlib/CategoryTheory/Sites/Sheafification.lean
+++ b/Mathlib/CategoryTheory/Sites/Sheafification.lean
@@ -14,7 +14,7 @@ import Mathlib.CategoryTheory.Limits.Preserves.Finite
 Given a site `(C, J)` we define a typeclass `HasSheafify J A` saying that the inclusion functor from
 `A`-valued sheaves on `C` to presheaves admits a left exact left adjoint (sheafification).
 
-Note: to access the `HasSheafify` instance for suitable concrete categories, import the file
+Note: to access the `HasSheafify` instance for suitable concrete categories, import the file
 `Mathlib.CategoryTheory.Sites.LeftExact`.
 -/
 
@@ -37,7 +37,7 @@ abbrev HasWeakSheafify : Prop := (sheafToPresheaf J A).IsRightAdjoint
 left adjiont (sheafification).
 
 Given a finite limit preserving functor `F : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A` and an adjunction
-`adj : F ⊣ sheafToPresheaf J A`, use `HasSheafify.mk'` to construct a `HasSheafify` instance.
+`adj : F ⊣ sheafToPresheaf J A`, use `HasSheafify.mk'` to construct a `HasSheafify` instance.
 -/
 class HasSheafify : Prop where
   isRightAdjoint : HasWeakSheafify J A
@@ -141,7 +141,8 @@ theorem isIso_toSheafify {P : Cᵒᵖ ⥤ D} (hP : Presheaf.IsSheaf J P) : IsIso
     rfl
   · change (sheafToPresheaf _ _).map _ ≫ _ = _
     change _ ≫ (sheafificationAdjunction J D).unit.app ((sheafToPresheaf J D).obj ⟨P, hP⟩) = _
-    erw [← (sheafificationAdjunction J D).inv_counit_map  (X := ⟨P, hP⟩), comp_inv_eq_id]
+    rw [← (sheafificationAdjunction J D).inv_counit_map (X := ⟨P, hP⟩)]
+    simp
 
 /-- If `P` is a sheaf, then `P` is isomorphic to `sheafify J P`. -/
 noncomputable def isoSheafify {P : Cᵒᵖ ⥤ D} (hP : Presheaf.IsSheaf J P) : P ≅ sheafify J P :=
diff --git a/Mathlib/CategoryTheory/Sites/Sieves.lean b/Mathlib/CategoryTheory/Sites/Sieves.lean
index 149378759550f..172573aad9266 100644
--- a/Mathlib/CategoryTheory/Sites/Sieves.lean
+++ b/Mathlib/CategoryTheory/Sites/Sieves.lean
@@ -3,11 +3,7 @@ Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bhavik Mehta, Edward Ayers
 -/
-import Mathlib.CategoryTheory.Comma.Over
 import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
-import Mathlib.CategoryTheory.Yoneda
-import Mathlib.Data.Set.Lattice
-import Mathlib.Order.CompleteLattice
 
 /-!
 # Theory of sieves
@@ -82,11 +78,11 @@ theorem bind_comp {S : Presieve X} {R : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S f 
 -- Porting note: it seems the definition of `Presieve` must be unfolded in order to define
 --   this inductive type, it was thus renamed `singleton'`
 -- Note we can't make this into `HasSingleton` because of the out-param.
-/-- The singleton presieve.  -/
+/-- The singleton presieve. -/
 inductive singleton' : ⦃Y : C⦄ → (Y ⟶ X) → Prop
   | mk : singleton' f
 
-/-- The singleton presieve.  -/
+/-- The singleton presieve. -/
 def singleton : Presieve X := singleton' f
 
 lemma singleton.mk {f : Y ⟶ X} : singleton f f := singleton'.mk
diff --git a/Mathlib/CategoryTheory/Sites/Subsheaf.lean b/Mathlib/CategoryTheory/Sites/Subsheaf.lean
index ac8268e9caf70..198726f465d9b 100644
--- a/Mathlib/CategoryTheory/Sites/Subsheaf.lean
+++ b/Mathlib/CategoryTheory/Sites/Subsheaf.lean
@@ -118,7 +118,7 @@ theorem Subpresheaf.eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by
     infer_instance
   · intro H
     ext U x
-    apply iff_true_iff.mpr
+    apply (iff_of_eq (iff_true _)).mpr
     rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
     exact ((inv (G.ι.app U)) x).2
 
@@ -252,7 +252,7 @@ theorem Subpresheaf.sheafify_sheafify (h : Presieve.IsSheaf J F) :
     (G.sheafify J).sheafify J = G.sheafify J :=
   ((Subpresheaf.eq_sheafify_iff _ h).mpr <| G.sheafify_isSheaf h).symm
 
-/-- The lift of a presheaf morphism onto the sheafification subpresheaf.  -/
+/-- The lift of a presheaf morphism onto the sheafification subpresheaf. -/
 noncomputable def Subpresheaf.sheafifyLift (f : G.toPresheaf ⟶ F') (h : Presieve.IsSheaf J F') :
     (G.sheafify J).toPresheaf ⟶ F' where
   app U s := (h (G.sieveOfSection s.1) s.prop).amalgamate
@@ -292,7 +292,7 @@ theorem Subpresheaf.to_sheafify_lift_unique (h : Presieve.IsSheaf J F')
   apply (h _ hs).isSeparatedFor.ext
   rintro V i hi
   dsimp at hi
-  erw [← FunctorToTypes.naturality, ← FunctorToTypes.naturality]
+  rw [← FunctorToTypes.naturality, ← FunctorToTypes.naturality]
   exact (congr_fun (congr_app e <| op V) ⟨_, hi⟩ : _)
 
 theorem Subpresheaf.sheafify_le (h : G ≤ G') (hF : Presieve.IsSheaf J F)
@@ -305,7 +305,7 @@ theorem Subpresheaf.sheafify_le (h : G ≤ G') (hF : Presieve.IsSheaf J F)
     congr_arg (fun f : G.toPresheaf ⟶ G'.toPresheaf => (NatTrans.app f (op V) ⟨_, hi⟩).1)
       (G.to_sheafifyLift (Subpresheaf.homOfLe h) hG')
   convert this.symm
-  erw [← Subpresheaf.nat_trans_naturality]
+  rw [← Subpresheaf.nat_trans_naturality]
   rfl
 
 section Image
diff --git a/Mathlib/CategoryTheory/Sites/Whiskering.lean b/Mathlib/CategoryTheory/Sites/Whiskering.lean
index f80010696a0b9..17685ddf1e50a 100644
--- a/Mathlib/CategoryTheory/Sites/Whiskering.lean
+++ b/Mathlib/CategoryTheory/Sites/Whiskering.lean
@@ -129,10 +129,10 @@ instance hasSheafCompose_of_preservesMulticospan (F : A ⥤ B)
     exact ⟨Limits.IsLimit.postcomposeHomEquiv (S.multicospanComp F P) _ h⟩
 
 /--
-Composing a sheaf with a functor preserving limits of the same size as the hom sets in `C` yields a
+Composing a sheaf with a functor preserving limits of the same size as the hom sets in `C` yields a
 functor between sheaf categories.
 
-Note: the size of the limit that `F` is required to preserve in
+Note: the size of the limit that `F` is required to preserve in
 `hasSheafCompose_of_preservesMulticospan` is in general larger than this.
 -/
 instance hasSheafCompose_of_preservesLimitsOfSize [PreservesLimitsOfSize.{v₁, max u₁ v₁} F] :
diff --git a/Mathlib/CategoryTheory/Skeletal.lean b/Mathlib/CategoryTheory/Skeletal.lean
index 3260b8eb2fa9b..2ff548def5bb3 100644
--- a/Mathlib/CategoryTheory/Skeletal.lean
+++ b/Mathlib/CategoryTheory/Skeletal.lean
@@ -294,19 +294,17 @@ end
 variable {C}
 
 /-- An adjunction between thin categories gives an adjunction between their thin skeletons. -/
-def lowerAdjunction (R : D ⥤ C) (L : C ⥤ D) (h : L ⊣ R) : ThinSkeleton.map L ⊣ ThinSkeleton.map R :=
-  Adjunction.mkOfUnitCounit
-    { unit :=
-        {
-          app := fun X => by
-            letI := isIsomorphicSetoid C
-            exact Quotient.recOnSubsingleton X fun x => homOfLE ⟨h.unit.app x⟩ }
+def lowerAdjunction (R : D ⥤ C) (L : C ⥤ D) (h : L ⊣ R) :
+    ThinSkeleton.map L ⊣ ThinSkeleton.map R where
+  unit :=
+    { app := fun X => by
+        letI := isIsomorphicSetoid C
+        exact Quotient.recOnSubsingleton X fun x => homOfLE ⟨h.unit.app x⟩ }
       -- TODO: make quotient.rec_on_subsingleton' so the letI isn't needed
-      counit :=
-        {
-          app := fun X => by
-            letI := isIsomorphicSetoid D
-            exact Quotient.recOnSubsingleton X fun x => homOfLE ⟨h.counit.app x⟩ } }
+  counit :=
+    { app := fun X => by
+        letI := isIsomorphicSetoid D
+        exact Quotient.recOnSubsingleton X fun x => homOfLE ⟨h.counit.app x⟩ }
 
 end ThinSkeleton
 
diff --git a/Mathlib/CategoryTheory/SmallObject/Construction.lean b/Mathlib/CategoryTheory/SmallObject/Construction.lean
index 808c18354da20..fe1e1fa4599e9 100644
--- a/Mathlib/CategoryTheory/SmallObject/Construction.lean
+++ b/Mathlib/CategoryTheory/SmallObject/Construction.lean
@@ -143,7 +143,7 @@ lemma ιFunctorObj_πFunctorObj : ιFunctorObj f πX ≫ πFunctorObj f πX = π
 
 /-- The canonical morphism `∐ (functorObjSrcFamily f πX) ⟶ ∐ (functorObjSrcFamily f πY)`
 induced by a morphism in `φ : X ⟶ Y` such that `φ ≫ πX = πY`. -/
-noncomputable def functorMapSrc  (hφ : φ ≫ πY = πX) :
+noncomputable def functorMapSrc (hφ : φ ≫ πY = πX) :
     ∐ (functorObjSrcFamily f πX) ⟶ ∐ functorObjSrcFamily f πY :=
   Sigma.map' (fun x => FunctorObjIndex.mk x.i (x.t ≫ φ) x.b (by simp [hφ])) (fun _ => 𝟙 _)
 
@@ -168,7 +168,7 @@ lemma functorMapSrc_functorObjTop :
 
 /-- The canonical morphism `∐ functorObjTgtFamily f πX ⟶ ∐ functorObjTgtFamily f πY`
 induced by a morphism in `φ : X ⟶ Y` such that `φ ≫ πX = πY`. -/
-noncomputable def functorMapTgt  (hφ : φ ≫ πY = πX) :
+noncomputable def functorMapTgt (hφ : φ ≫ πY = πX) :
     ∐ functorObjTgtFamily f πX ⟶ ∐ functorObjTgtFamily f πY :=
   Sigma.map' (fun x => FunctorObjIndex.mk x.i (x.t ≫ φ) x.b (by simp [hφ])) (fun _ => 𝟙 _)
 
diff --git a/Mathlib/CategoryTheory/SmallObject/Iteration.lean b/Mathlib/CategoryTheory/SmallObject/Iteration.lean
index 4e9947245dd1b..a9470418b418d 100644
--- a/Mathlib/CategoryTheory/SmallObject/Iteration.lean
+++ b/Mathlib/CategoryTheory/SmallObject/Iteration.lean
@@ -123,7 +123,7 @@ lemma mapSucc_eq (i : J) (hi : i < j) :
 
 end
 
-variable (iter₁ iter₂ iter₃ : Φ.Iteration ε j)
+variable (iter₁ iter₂ : Φ.Iteration ε j)
 
 /-- A morphism between two objects `iter₁` and `iter₂` in the
 category `Φ.Iteration ε j` of `j`th iterations of a functor `Φ`
@@ -148,7 +148,7 @@ attribute [simp, reassoc] natTrans_app_zero
 def id : Hom iter₁ iter₁ where
   natTrans := 𝟙 _
 
-variable {iter₁ iter₂ iter₃}
+variable {iter₁ iter₂}
 
 -- Note: this is not made a global ext lemma because it is shown below
 -- that the type of morphisms is a subsingleton.
diff --git a/Mathlib/CategoryTheory/Subobject/Basic.lean b/Mathlib/CategoryTheory/Subobject/Basic.lean
index 076606a7a3d8f..34c23433816f0 100644
--- a/Mathlib/CategoryTheory/Subobject/Basic.lean
+++ b/Mathlib/CategoryTheory/Subobject/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.MonoOver
 import Mathlib.CategoryTheory.Skeletal
@@ -45,7 +45,7 @@ See also
 ## Notes
 
 This development originally appeared in Bhavik Mehta's "Topos theory for Lean" repository,
-and was ported to mathlib by Scott Morrison.
+and was ported to mathlib by Kim Morrison.
 
 ### Implementation note
 
diff --git a/Mathlib/CategoryTheory/Subobject/FactorThru.lean b/Mathlib/CategoryTheory/Subobject/FactorThru.lean
index 5ad249474de0f..d4f891f44b528 100644
--- a/Mathlib/CategoryTheory/Subobject/FactorThru.lean
+++ b/Mathlib/CategoryTheory/Subobject/FactorThru.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.Basic
 import Mathlib.CategoryTheory.Preadditive.Basic
diff --git a/Mathlib/CategoryTheory/Subobject/Lattice.lean b/Mathlib/CategoryTheory/Subobject/Lattice.lean
index ac8612b0e93bd..9fb683c1d2eaf 100644
--- a/Mathlib/CategoryTheory/Subobject/Lattice.lean
+++ b/Mathlib/CategoryTheory/Subobject/Lattice.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Functor.Currying
 import Mathlib.CategoryTheory.Subobject.FactorThru
@@ -379,18 +379,19 @@ theorem inf_arrow_factors_right {B : C} (X Y : Subobject B) : Y.Factors (X ⊓ Y
 theorem finset_inf_factors {I : Type*} {A B : C} {s : Finset I} {P : I → Subobject B} (f : A ⟶ B) :
     (s.inf P).Factors f ↔ ∀ i ∈ s, (P i).Factors f := by
   classical
-  induction' s using Finset.induction_on with _ _ _ ih
-  · simp [top_factors]
-  · simp [ih]
+  induction s using Finset.induction_on with
+  | empty => simp [top_factors]
+  | insert _ ih => simp [ih]
 
 -- `i` is explicit here because often we'd like to defer a proof of `m`
 theorem finset_inf_arrow_factors {I : Type*} {B : C} (s : Finset I) (P : I → Subobject B) (i : I)
     (m : i ∈ s) : (P i).Factors (s.inf P).arrow := by
   classical
   revert i m
-  induction' s using Finset.induction_on with _ _ _ ih
-  · rintro _ ⟨⟩
-  · intro _ m
+  induction s using Finset.induction_on with
+  | empty => rintro _ ⟨⟩
+  | insert _ ih =>
+    intro _ m
     rw [Finset.inf_insert]
     simp only [Finset.mem_insert] at m
     rcases m with (rfl | m)
@@ -472,9 +473,10 @@ theorem finset_sup_factors {I : Type*} {A B : C} {s : Finset I} {P : I → Subob
     (h : ∃ i ∈ s, (P i).Factors f) : (s.sup P).Factors f := by
   classical
   revert h
-  induction' s using Finset.induction_on with _ _ _ ih
-  · rintro ⟨_, ⟨⟨⟩, _⟩⟩
-  · rintro ⟨j, ⟨m, h⟩⟩
+  induction s using Finset.induction_on with
+  | empty => rintro ⟨_, ⟨⟨⟩, _⟩⟩
+  | insert _ ih =>
+    rintro ⟨j, ⟨m, h⟩⟩
     simp only [Finset.sup_insert]
     simp only [Finset.mem_insert] at m
     rcases m with (rfl | m)
diff --git a/Mathlib/CategoryTheory/Subobject/Limits.lean b/Mathlib/CategoryTheory/Subobject/Limits.lean
index 3fbcf2452701b..26f73bd07b7d2 100644
--- a/Mathlib/CategoryTheory/Subobject/Limits.lean
+++ b/Mathlib/CategoryTheory/Subobject/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.Lattice
 
@@ -333,8 +333,6 @@ section
 
 variable [HasEqualizers C]
 
-attribute [local instance] epi_comp
-
 /-- The morphism `imageSubobject (h ≫ f) ⟶ imageSubobject f`
 is an epimorphism when `h` is an epimorphism.
 In general this does not imply that `imageSubobject (h ≫ f) = imageSubobject f`,
diff --git a/Mathlib/CategoryTheory/Subobject/MonoOver.lean b/Mathlib/CategoryTheory/Subobject/MonoOver.lean
index 4236e96df2512..e2cfc4d68b330 100644
--- a/Mathlib/CategoryTheory/Subobject/MonoOver.lean
+++ b/Mathlib/CategoryTheory/Subobject/MonoOver.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Bhavik Mehta, Scott Morrison
+Authors: Bhavik Mehta, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.Over
 import Mathlib.CategoryTheory.Adjunction.Reflective
@@ -28,7 +28,7 @@ and prove their basic properties and relationships.
 ## Notes
 
 This development originally appeared in Bhavik Mehta's "Topos theory for Lean" repository,
-and was ported to mathlib by Scott Morrison.
+and was ported to mathlib by Kim Morrison.
 
 -/
 
@@ -219,13 +219,11 @@ end Pullback
 
 section Map
 
-attribute [instance] mono_comp
-
 /-- We can map monomorphisms over `X` to monomorphisms over `Y`
 by post-composition with a monomorphism `f : X ⟶ Y`.
 -/
 def map (f : X ⟶ Y) [Mono f] : MonoOver X ⥤ MonoOver Y :=
-  lift (Over.map f) fun g => by apply mono_comp g.arrow f
+  lift (Over.map f) fun g => mono_comp g.arrow f
 
 /-- `MonoOver.map` commutes with composition (up to a natural isomorphism). -/
 def mapComp (f : X ⟶ Y) (g : Y ⟶ Z) [Mono f] [Mono g] : map (f ≫ g) ≅ map f ⋙ map g :=
diff --git a/Mathlib/CategoryTheory/Subobject/Types.lean b/Mathlib/CategoryTheory/Subobject/Types.lean
index 54ebdb2816d00..cb2e06f94b8b1 100644
--- a/Mathlib/CategoryTheory/Subobject/Types.lean
+++ b/Mathlib/CategoryTheory/Subobject/Types.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.WellPowered
 import Mathlib.CategoryTheory.Types
diff --git a/Mathlib/CategoryTheory/Subobject/WellPowered.lean b/Mathlib/CategoryTheory/Subobject/WellPowered.lean
index 0a78979b8a9d0..399d029fc1211 100644
--- a/Mathlib/CategoryTheory/Subobject/WellPowered.lean
+++ b/Mathlib/CategoryTheory/Subobject/WellPowered.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Subobject.Basic
 import Mathlib.CategoryTheory.EssentiallySmall
diff --git a/Mathlib/CategoryTheory/Sums/Associator.lean b/Mathlib/CategoryTheory/Sums/Associator.lean
index 214fda16c81c1..52868ef851d64 100644
--- a/Mathlib/CategoryTheory/Sums/Associator.lean
+++ b/Mathlib/CategoryTheory/Sums/Associator.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Sums.Basic
 
diff --git a/Mathlib/CategoryTheory/Sums/Basic.lean b/Mathlib/CategoryTheory/Sums/Basic.lean
index b998f1052121e..51c5a75bafa70 100644
--- a/Mathlib/CategoryTheory/Sums/Basic.lean
+++ b/Mathlib/CategoryTheory/Sums/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Equivalence
 
@@ -31,7 +31,7 @@ section
 
 variable (C : Type u₁) [Category.{v₁} C] (D : Type u₁) [Category.{v₁} D]
 
-/- Porting note: `aesop_cat` not firing on `assoc` where autotac in Lean 3 did-/
+/- Porting note: `aesop_cat` not firing on `assoc` where autotac in Lean 3 did -/
 
 /-- `sum C D` gives the direct sum of two categories.
 -/
diff --git a/Mathlib/CategoryTheory/Thin.lean b/Mathlib/CategoryTheory/Thin.lean
index e44b1c223c030..a2c9cb1099b9c 100644
--- a/Mathlib/CategoryTheory/Thin.lean
+++ b/Mathlib/CategoryTheory/Thin.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison, Bhavik Mehta. All rights reserved.
+Copyright (c) 2019 Kim Morrison, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Bhavik Mehta
+Authors: Kim Morrison, Bhavik Mehta
 -/
 import Mathlib.CategoryTheory.Functor.Category
 import Mathlib.CategoryTheory.Iso
diff --git a/Mathlib/CategoryTheory/Triangulated/Basic.lean b/Mathlib/CategoryTheory/Triangulated/Basic.lean
index 308f5456a6a29..e76c664d5f1d3 100644
--- a/Mathlib/CategoryTheory/Triangulated/Basic.lean
+++ b/Mathlib/CategoryTheory/Triangulated/Basic.lean
@@ -260,9 +260,9 @@ variable {J : Type*} (T : J → Triangle C)
 /-- The product of a family of triangles. -/
 @[simps!]
 def productTriangle : Triangle C :=
-  Triangle.mk (Pi.map (fun j => (T j).mor₁))
-    (Pi.map (fun j => (T j).mor₂))
-    (Pi.map (fun j => (T j).mor₃) ≫ inv (piComparison _ _))
+  Triangle.mk (Limits.Pi.map (fun j => (T j).mor₁))
+    (Limits.Pi.map (fun j => (T j).mor₂))
+    (Limits.Pi.map (fun j => (T j).mor₃) ≫ inv (piComparison _ _))
 
 /-- A projection from the product of a family of triangles. -/
 @[simps]
diff --git a/Mathlib/CategoryTheory/Triangulated/Functor.lean b/Mathlib/CategoryTheory/Triangulated/Functor.lean
index 230768841d930..9507a7937e954 100644
--- a/Mathlib/CategoryTheory/Triangulated/Functor.lean
+++ b/Mathlib/CategoryTheory/Triangulated/Functor.lean
@@ -71,8 +71,12 @@ section Additive
 
 variable [Preadditive C] [Preadditive D] [F.Additive]
 
+#adaptation_note
+/--
+At nightly-2024-08-08 we needed to increase the maxHeartbeats here.
+-/
+set_option maxHeartbeats 400000 in
 /-- The functor `F.mapTriangle` commutes with the shift. -/
-@[simps!]
 noncomputable def mapTriangleCommShiftIso (n : ℤ) :
     Triangle.shiftFunctor C n ⋙ F.mapTriangle ≅ F.mapTriangle ⋙ Triangle.shiftFunctor D n :=
   NatIso.ofComponents (fun T => Triangle.isoMk _ _
@@ -85,6 +89,8 @@ noncomputable def mapTriangleCommShiftIso (n : ℤ) :
       simp only [comp_obj, assoc, Iso.inv_hom_id_app_assoc,
         ← Functor.map_comp, Iso.inv_hom_id_app, map_id, comp_id])) (by aesop_cat)
 
+attribute [simps!] mapTriangleCommShiftIso
+
 attribute [local simp] map_zsmul comp_zsmul zsmul_comp
   commShiftIso_zero commShiftIso_add commShiftIso_comp_hom_app
   shiftFunctorAdd'_eq_shiftFunctorAdd
@@ -304,6 +310,6 @@ lemma isTriangulated_of_essSurj_mapComposableArrows_two
   exact ⟨Octahedron.ofIso (e₁ := (e.app 0).symm) (e₂ := (e.app 1).symm) (e₃ := (e.app 2).symm)
     (comm₁₂ := ComposableArrows.naturality' e.inv 0 1)
     (comm₂₃ := ComposableArrows.naturality' e.inv 1 2)
-    (H := (someOctahedron rfl h₁₂' h₂₃' h₁₃').map F) _ _ _ _ _⟩
+    (H := (someOctahedron rfl h₁₂' h₂₃' h₁₃').map F) ..⟩
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean b/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
index 091de90c8c3c1..c3b954b379632 100644
--- a/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
+++ b/Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
@@ -563,7 +563,7 @@ lemma productTriangle_distinguished {J : Type*} (T : J → Triangle C)
     `φ'.hom₁` and `φ'.hom₂` are identities. Then, it suffices to show that
     `φ'.hom₃` is an isomorphism, which is achieved by using Yoneda's lemma
     and diagram chases. -/
-  let f₁ := Pi.map (fun j => (T j).mor₁)
+  let f₁ := Limits.Pi.map (fun j => (T j).mor₁)
   obtain ⟨Z, f₂, f₃, hT'⟩ := distinguished_cocone_triangle f₁
   let T' := Triangle.mk f₁ f₂ f₃
   change T' ∈ distTriang C at hT'
diff --git a/Mathlib/CategoryTheory/Triangulated/Subcategory.lean b/Mathlib/CategoryTheory/Triangulated/Subcategory.lean
index f4c688ba57647..22baf9e9ef9b8 100644
--- a/Mathlib/CategoryTheory/Triangulated/Subcategory.lean
+++ b/Mathlib/CategoryTheory/Triangulated/Subcategory.lean
@@ -152,16 +152,16 @@ lemma isoClosure_W : S.isoClosure.W = S.W := by
     exact ⟨Z, g, h, mem, le_isoClosure _ _ hZ⟩
 
 instance respectsIso_W : S.W.RespectsIso where
-  precomp := by
-    rintro X' X Y e f ⟨Z, g, h, mem, mem'⟩
-    refine ⟨Z, g, h ≫ e.inv⟦(1 : ℤ)⟧', isomorphic_distinguished _ mem _ ?_, mem'⟩
-    refine Triangle.isoMk _ _ e (Iso.refl _) (Iso.refl _) (by aesop_cat) (by aesop_cat) ?_
+  precomp {X' X Y} e (he : IsIso e) := by
+    rintro f ⟨Z, g, h, mem, mem'⟩
+    refine ⟨Z, g, h ≫ inv e⟦(1 : ℤ)⟧', isomorphic_distinguished _ mem _ ?_, mem'⟩
+    refine Triangle.isoMk _ _ (asIso e) (Iso.refl _) (Iso.refl _) (by aesop_cat) (by aesop_cat) ?_
     dsimp
-    simp only [assoc, ← Functor.map_comp, e.inv_hom_id, Functor.map_id, comp_id, id_comp]
-  postcomp := by
-    rintro X Y Y' e f ⟨Z, g, h, mem, mem'⟩
-    refine ⟨Z, e.inv ≫ g, h, isomorphic_distinguished _ mem _ ?_, mem'⟩
-    exact Triangle.isoMk _ _ (Iso.refl _) e.symm (Iso.refl _)
+    simp only [Functor.map_inv, assoc, IsIso.inv_hom_id, comp_id, id_comp]
+  postcomp {X Y Y'} e (he : IsIso e) := by
+    rintro f ⟨Z, g, h, mem, mem'⟩
+    refine ⟨Z, inv e ≫ g, h, isomorphic_distinguished _ mem _ ?_, mem'⟩
+    exact Triangle.isoMk _ _ (Iso.refl _) (asIso e).symm (Iso.refl _)
 
 instance : S.W.ContainsIdentities := by
   rw [← isoClosure_W]
diff --git a/Mathlib/CategoryTheory/Types.lean b/Mathlib/CategoryTheory/Types.lean
index b7cab7670d304..b2b29e06f39d9 100644
--- a/Mathlib/CategoryTheory/Types.lean
+++ b/Mathlib/CategoryTheory/Types.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl
+Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl
 -/
 import Mathlib.CategoryTheory.EpiMono
 import Mathlib.CategoryTheory.Functor.FullyFaithful
diff --git a/Mathlib/CategoryTheory/UnivLE.lean b/Mathlib/CategoryTheory/UnivLE.lean
index e6b8c95764011..0210136dcf7ea 100644
--- a/Mathlib/CategoryTheory/UnivLE.lean
+++ b/Mathlib/CategoryTheory/UnivLE.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.UnivLE
 import Mathlib.CategoryTheory.EssentialImage
diff --git a/Mathlib/CategoryTheory/Whiskering.lean b/Mathlib/CategoryTheory/Whiskering.lean
index a674d50bcf341..50cc4f617eec0 100644
--- a/Mathlib/CategoryTheory/Whiskering.lean
+++ b/Mathlib/CategoryTheory/Whiskering.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Iso
 import Mathlib.CategoryTheory.Functor.Category
@@ -113,6 +113,46 @@ def Functor.FullyFaithful.whiskeringRight {F : D ⥤ E} (hF : F.FullyFaithful)
         simp only [map_comp, map_preimage]
         apply f.naturality }
 
+theorem whiskeringLeft_obj_id : (whiskeringLeft C C E).obj (𝟭 _) = 𝟭 _ :=
+  rfl
+
+/-- The isomorphism between left-whiskering on the identity functor and the identity of the functor
+between the resulting functor categories. -/
+def whiskeringLeftObjIdIso : (whiskeringLeft C C E).obj (𝟭 _) ≅ 𝟭 _ :=
+  Iso.refl _
+
+theorem whiskeringLeft_obj_comp {D' : Type u₄} [Category.{v₄} D'] (F : C ⥤ D) (G : D ⥤ D') :
+    (whiskeringLeft C D' E).obj (F ⋙ G) =
+    (whiskeringLeft D D' E).obj G ⋙ (whiskeringLeft C D E).obj F :=
+  rfl
+
+/-- The isomorphism between left-whiskering on the composition of functors and the composition
+of two left-whiskering applications. -/
+def whiskeringLeftObjCompIso {D' : Type u₄} [Category.{v₄} D'] (F : C ⥤ D) (G : D ⥤ D') :
+    (whiskeringLeft C D' E).obj (F ⋙ G) ≅
+    (whiskeringLeft D D' E).obj G ⋙ (whiskeringLeft C D E).obj F :=
+  Iso.refl _
+
+theorem whiskeringRight_obj_id : (whiskeringRight E C C).obj (𝟭 _) = 𝟭 _ :=
+  rfl
+
+/-- The isomorphism between right-whiskering on the identity functor and the identity of the functor
+between the resulting functor categories. -/
+def wiskeringRightObjIdIso : (whiskeringRight E C C).obj (𝟭 _) ≅ 𝟭 _ :=
+  Iso.refl _
+
+theorem whiskeringRight_obj_comp {D' : Type u₄} [Category.{v₄} D'] (F : C ⥤ D) (G : D ⥤ D') :
+    (whiskeringRight E C D).obj F ⋙ (whiskeringRight E D D').obj G =
+    (whiskeringRight E C D').obj (F ⋙ G) :=
+  rfl
+
+/-- The isomorphism between right-whiskering on the composition of functors and the composition
+of two right-whiskering applications. -/
+def whiskeringRightObjCompIso {D' : Type u₄} [Category.{v₄} D'] (F : C ⥤ D) (G : D ⥤ D') :
+    (whiskeringRight E C D).obj F ⋙ (whiskeringRight E D D').obj G ≅
+    (whiskeringRight E C D').obj (F ⋙ G) :=
+  Iso.refl _
+
 instance full_whiskeringRight_obj {F : D ⥤ E} [F.Faithful] [F.Full] :
     ((whiskeringRight C D E).obj F).Full :=
   ((Functor.FullyFaithful.ofFullyFaithful F).whiskeringRight C).full
diff --git a/Mathlib/CategoryTheory/Widesubcategory.lean b/Mathlib/CategoryTheory/Widesubcategory.lean
index 9f60d4fe54f36..e225caffbde77 100644
--- a/Mathlib/CategoryTheory/Widesubcategory.lean
+++ b/Mathlib/CategoryTheory/Widesubcategory.lean
@@ -83,7 +83,7 @@ Structure for wide subcategories. Objects ignore the morphism property.
 -/
 @[ext, nolint unusedArguments]
 structure WideSubcategory (_P : MorphismProperty C) [IsMultiplicative _P] where
-  /-- The category of which this is a wide subcategory-/
+  /-- The category of which this is a wide subcategory -/
   obj : C
 
 instance WideSubcategory.category : Category.{v₁} (WideSubcategory P) :=
diff --git a/Mathlib/CategoryTheory/WithTerminal.lean b/Mathlib/CategoryTheory/WithTerminal.lean
index b7c8c24062abb..3c35d0efb590e 100644
--- a/Mathlib/CategoryTheory/WithTerminal.lean
+++ b/Mathlib/CategoryTheory/WithTerminal.lean
@@ -70,6 +70,7 @@ def Hom : WithTerminal C → WithTerminal C → Type v
   | of X, of Y => X ⟶ Y
   | star, of _ => PEmpty
   | _, star => PUnit
+attribute [nolint simpNF] Hom.eq_3
 
 /-- Identity morphisms for `WithTerminal C`. -/
 @[simp]
@@ -85,6 +86,8 @@ def comp : ∀ {X Y Z : WithTerminal C}, Hom X Y → Hom Y Z → Hom X Z
   | star, of _X, _ => fun f _g => PEmpty.elim f
   | _, star, of _Y => fun _f g => PEmpty.elim g
   | star, star, star => fun _ _ => PUnit.unit
+attribute [nolint simpNF] comp.eq_3
+attribute [nolint simpNF] comp.eq_4
 
 instance : Category.{v} (WithTerminal C) where
   Hom X Y := Hom X Y
@@ -371,6 +374,7 @@ def Hom : WithInitial C → WithInitial C → Type v
   | of X, of Y => X ⟶ Y
   | of _, _ => PEmpty
   | star, _ => PUnit
+attribute [nolint simpNF] Hom.eq_2
 
 /-- Identity morphisms for `WithInitial C`. -/
 @[simp]
@@ -386,6 +390,8 @@ def comp : ∀ {X Y Z : WithInitial C}, Hom X Y → Hom Y Z → Hom X Z
   | _, of _X, star => fun _f g => PEmpty.elim g
   | of _Y, star, _ => fun f _g => PEmpty.elim f
   | star, star, star => fun _ _ => PUnit.unit
+attribute [nolint simpNF] comp.eq_3
+attribute [nolint simpNF] comp.eq_4
 
 instance : Category.{v} (WithInitial C) where
   Hom X Y := Hom X Y
diff --git a/Mathlib/CategoryTheory/Yoneda.lean b/Mathlib/CategoryTheory/Yoneda.lean
index f3cbf94ce4ae3..f06afc7170c1c 100644
--- a/Mathlib/CategoryTheory/Yoneda.lean
+++ b/Mathlib/CategoryTheory/Yoneda.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Functor.Hom
 import Mathlib.CategoryTheory.Products.Basic
@@ -24,7 +24,7 @@ namespace CategoryTheory
 
 open Opposite
 
-universe v₁ u₁ u₂
+universe v v₁ u₁ u₂
 
 -- morphism levels before object levels. See note [CategoryTheory universes].
 variable {C : Type u₁} [Category.{v₁} C]
@@ -153,97 +153,228 @@ end Coyoneda
 
 namespace Functor
 
-/-- A functor `F : Cᵒᵖ ⥤ Type v₁` is representable if there is object `X` so `F ≅ yoneda.obj X`.
+/-- The data which expresses that a functor `F : Cᵒᵖ ⥤ Type v` is representable by `Y : C`. -/
+structure RepresentableBy (F : Cᵒᵖ ⥤ Type v) (Y : C) where
+  /-- the natural bijection `(X ⟶ Y) ≃ F.obj (op X)`. -/
+  homEquiv {X : C} : (X ⟶ Y) ≃ F.obj (op X)
+  homEquiv_comp {X X' : C} (f : X ⟶ X') (g : X' ⟶ Y) :
+    homEquiv (f ≫ g) = F.map f.op (homEquiv g)
+
+/-- If `F ≅ F'`, and `F` is representable, then `F'` is representable. -/
+def RepresentableBy.ofIso {F F' : Cᵒᵖ ⥤ Type v} {Y : C} (e : F.RepresentableBy Y) (e' : F ≅ F') :
+    F'.RepresentableBy Y where
+  homEquiv {X} := e.homEquiv.trans (e'.app _).toEquiv
+  homEquiv_comp {X X'} f g := by
+    dsimp
+    rw [e.homEquiv_comp]
+    apply congr_fun (e'.hom.naturality f.op)
+
+/-- The data which expresses that a functor `F : C ⥤ Type v` is corepresentable by `X : C`. -/
+structure CorepresentableBy (F : C ⥤ Type v) (X : C) where
+  /-- the natural bijection `(X ⟶ Y) ≃ F.obj Y`. -/
+  homEquiv {Y : C} : (X ⟶ Y) ≃ F.obj Y
+  homEquiv_comp {Y Y' : C} (g : Y ⟶ Y') (f : X ⟶ Y) :
+    homEquiv (f ≫ g) = F.map g (homEquiv f)
+
+/-- If `F ≅ F'`, and `F` is corepresentable, then `F'` is corepresentable. -/
+def CorepresentableBy.ofIso {F F' : C ⥤ Type v} {X : C} (e : F.CorepresentableBy X)
+    (e' : F ≅ F') :
+    F'.CorepresentableBy X where
+  homEquiv {X} := e.homEquiv.trans (e'.app _).toEquiv
+  homEquiv_comp {Y Y'} g f := by
+    dsimp
+    rw [e.homEquiv_comp]
+    apply congr_fun (e'.hom.naturality g)
+
+lemma RepresentableBy.homEquiv_eq {F : Cᵒᵖ ⥤ Type v} {Y : C} (e : F.RepresentableBy Y)
+    {X : C} (f : X ⟶ Y) :
+    e.homEquiv f = F.map f.op (e.homEquiv (𝟙 Y)) := by
+  conv_lhs => rw [← Category.comp_id f, e.homEquiv_comp]
+
+lemma CorepresentableBy.homEquiv_eq {F : C ⥤ Type v} {X : C} (e : F.CorepresentableBy X)
+    {Y : C} (f : X ⟶ Y) :
+    e.homEquiv f = F.map f (e.homEquiv (𝟙 X)) := by
+  conv_lhs => rw [← Category.id_comp f, e.homEquiv_comp]
+
+@[ext]
+lemma RepresentableBy.ext {F : Cᵒᵖ ⥤ Type v} {Y : C} {e e' : F.RepresentableBy Y}
+    (h : e.homEquiv (𝟙 Y) = e'.homEquiv (𝟙 Y)) : e = e' := by
+  have : ∀ {X : C} (f : X ⟶ Y), e.homEquiv f = e'.homEquiv f := fun {X} f ↦ by
+    rw [e.homEquiv_eq, e'.homEquiv_eq, h]
+  obtain ⟨e, he⟩ := e
+  obtain ⟨e', he'⟩ := e'
+  obtain rfl : @e = @e' := by ext; apply this
+  rfl
+
+@[ext]
+lemma CorepresentableBy.ext {F : C ⥤ Type v} {X : C} {e e' : F.CorepresentableBy X}
+    (h : e.homEquiv (𝟙 X) = e'.homEquiv (𝟙 X)) : e = e' := by
+  have : ∀ {Y : C} (f : X ⟶ Y), e.homEquiv f = e'.homEquiv f := fun {X} f ↦ by
+    rw [e.homEquiv_eq, e'.homEquiv_eq, h]
+  obtain ⟨e, he⟩ := e
+  obtain ⟨e', he'⟩ := e'
+  obtain rfl : @e = @e' := by ext; apply this
+  rfl
+
+/-- The obvious bijection `F.RepresentableBy Y ≃ (yoneda.obj Y ≅ F)`
+when `F : Cᵒᵖ ⥤ Type v₁` and `[Category.{v₁} C]`. -/
+def representableByEquiv {F : Cᵒᵖ ⥤ Type v₁} {Y : C} :
+    F.RepresentableBy Y ≃ (yoneda.obj Y ≅ F) where
+  toFun r := NatIso.ofComponents (fun _ ↦ r.homEquiv.toIso) (fun {X X'} f ↦ by
+    ext g
+    simp [r.homEquiv_comp])
+  invFun e :=
+    { homEquiv := (e.app _).toEquiv
+      homEquiv_comp := fun {X X'} f g ↦ congr_fun (e.hom.naturality f.op) g }
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+/-- The isomorphism `yoneda.obj Y ≅ F` induced by `e : F.RepresentableBy Y`. -/
+def RepresentableBy.toIso {F : Cᵒᵖ ⥤ Type v₁} {Y : C} (e : F.RepresentableBy Y) :
+    yoneda.obj Y ≅ F :=
+  representableByEquiv e
+
+/-- The obvious bijection `F.CorepresentableBy X ≃ (yoneda.obj Y ≅ F)`
+when `F : C ⥤ Type v₁` and `[Category.{v₁} C]`. -/
+def corepresentableByEquiv {F : C ⥤ Type v₁} {X : C} :
+    F.CorepresentableBy X ≃ (coyoneda.obj (op X) ≅ F) where
+  toFun r := NatIso.ofComponents (fun _ ↦ r.homEquiv.toIso) (fun {X X'} f ↦ by
+    ext g
+    simp [r.homEquiv_comp])
+  invFun e :=
+    { homEquiv := (e.app _).toEquiv
+      homEquiv_comp := fun {X X'} f g ↦ congr_fun (e.hom.naturality f) g }
+  left_inv _ := rfl
+  right_inv _ := rfl
+
+/-- The isomorphism `coyoneda.obj (op X) ≅ F` induced by `e : F.CorepresentableBy X`. -/
+def CorepresentableBy.toIso {F : C ⥤ Type v₁} {X : C} (e : F.CorepresentableBy X) :
+    coyoneda.obj (op X) ≅ F :=
+  corepresentableByEquiv e
+
+/-- A functor `F : Cᵒᵖ ⥤ Type v` is representable if there is oan bject `Y` with a structure
+`F.RepresentableBy Y`, i.e. there is a natural bijection `(X ⟶ Y) ≃ F.obj (op X)`,
+which may also be rephrased as a natural isomorphism `yoneda.obj X ≅ F` when `Category.{v} C`.
 
 See <https://stacks.math.columbia.edu/tag/001Q>.
 -/
-class Representable (F : Cᵒᵖ ⥤ Type v₁) : Prop where
-  /-- `Hom(-,X) ≅ F` via `f` -/
-  has_representation : ∃ (X : _), Nonempty (yoneda.obj X ≅ F)
+class IsRepresentable (F : Cᵒᵖ ⥤ Type v) : Prop where
+  has_representation : ∃ (Y : C), Nonempty (F.RepresentableBy Y)
+
+@[deprecated (since := "2024-10-03")] alias Representable := IsRepresentable
+
+lemma RepresentableBy.isRepresentable {F : Cᵒᵖ ⥤ Type v} {Y : C} (e : F.RepresentableBy Y) :
+    F.IsRepresentable where
+  has_representation := ⟨Y, ⟨e⟩⟩
 
-instance {X : C} : Representable (yoneda.obj X) where has_representation := ⟨X, ⟨Iso.refl _⟩⟩
+/-- Alternative constructure for `F.IsRepresentable`, which takes as an input an
+isomorphism `yoneda.obj X ≅ F`. -/
+lemma IsRepresentable.mk' {F : Cᵒᵖ ⥤ Type v₁} {X : C} (e : yoneda.obj X ≅ F) :
+    F.IsRepresentable :=
+  (representableByEquiv.symm e).isRepresentable
+
+instance {X : C} : IsRepresentable (yoneda.obj X) :=
+  IsRepresentable.mk' (Iso.refl _)
 
 /-- A functor `F : C ⥤ Type v₁` is corepresentable if there is object `X` so `F ≅ coyoneda.obj X`.
 
 See <https://stacks.math.columbia.edu/tag/001Q>.
 -/
-class Corepresentable (F : C ⥤ Type v₁) : Prop where
-  /-- `Hom(X,-) ≅ F` via `f` -/
-  has_corepresentation : ∃ (X : _), Nonempty (coyoneda.obj X ≅ F)
+class IsCorepresentable (F : C ⥤ Type v) : Prop where
+  has_corepresentation : ∃ (X : C), Nonempty (F.CorepresentableBy X)
+
+@[deprecated (since := "2024-10-03")] alias Corepresentable := IsCorepresentable
+
+lemma CorepresentableBy.isCorepresentable {F : C ⥤ Type v} {X : C} (e : F.CorepresentableBy X) :
+    F.IsCorepresentable where
+  has_corepresentation := ⟨X, ⟨e⟩⟩
+
+/-- Alternative constructure for `F.IsCorepresentable`, which takes as an input an
+isomorphism `coyoneda.obj (op X) ≅ F`. -/
+lemma IsCorepresentable.mk' {F : C ⥤ Type v₁} {X : C} (e : coyoneda.obj (op X) ≅ F) :
+    F.IsCorepresentable :=
+  (corepresentableByEquiv.symm e).isCorepresentable
 
-instance {X : Cᵒᵖ} : Corepresentable (coyoneda.obj X) where
-  has_corepresentation := ⟨X, ⟨Iso.refl _⟩⟩
+instance {X : Cᵒᵖ} : IsCorepresentable (coyoneda.obj X) :=
+  IsCorepresentable.mk' (Iso.refl _)
 
 -- instance : corepresentable (𝟭 (Type v₁)) :=
 -- corepresentable_of_nat_iso (op punit) coyoneda.punit_iso
 section Representable
 
-variable (F : Cᵒᵖ ⥤ Type v₁)
-variable [hF : F.Representable]
+variable (F : Cᵒᵖ ⥤ Type v) [hF : F.IsRepresentable]
 
 /-- The representing object for the representable functor `F`. -/
-noncomputable def reprX : C := hF.has_representation.choose
+noncomputable def reprX : C :=
+  hF.has_representation.choose
 
-/-- An isomorphism between a representable `F` and a functor of the
-form `C(-, F.reprX)`.  Note the components `F.reprW.app X`
-definitionally have type `(X.unop ⟶ F.repr_X) ≅ F.obj X`.
--/
-noncomputable def reprW : yoneda.obj F.reprX ≅ F :=
-  Representable.has_representation.choose_spec.some
+/-- A chosen term in `F.RepresentableBy (reprX F)` when `F.IsRepresentable` holds. -/
+noncomputable def representableBy : F.RepresentableBy F.reprX :=
+  hF.has_representation.choose_spec.some
 
 /-- The representing element for the representable functor `F`, sometimes called the universal
 element of the functor.
 -/
 noncomputable def reprx : F.obj (op F.reprX) :=
-  F.reprW.hom.app (op F.reprX) (𝟙 F.reprX)
+  F.representableBy.homEquiv (𝟙 _)
 
-theorem reprW_app_hom (X : Cᵒᵖ) (f : unop X ⟶ F.reprX) :
-    (F.reprW.app X).hom f = F.map f.op F.reprx := by
-  simp only [yoneda_obj_obj, Iso.app_hom, op_unop, reprx, ← FunctorToTypes.naturality,
-    yoneda_obj_map, unop_op, Quiver.Hom.unop_op, Category.comp_id]
+/-- An isomorphism between a representable `F` and a functor of the
+form `C(-, F.reprX)`.  Note the components `F.reprW.app X`
+definitionally have type `(X.unop ⟶ F.reprX) ≅ F.obj X`.
+-/
+noncomputable def reprW (F : Cᵒᵖ ⥤ Type v₁) [F.IsRepresentable] :
+    yoneda.obj F.reprX ≅ F := F.representableBy.toIso
+
+theorem reprW_hom_app (F : Cᵒᵖ ⥤ Type v₁) [F.IsRepresentable]
+    (X : Cᵒᵖ) (f : unop X ⟶ F.reprX) :
+    F.reprW.hom.app X f = F.map f.op F.reprx := by
+  apply RepresentableBy.homEquiv_eq
 
 end Representable
 
 section Corepresentable
 
-variable (F : C ⥤ Type v₁)
-variable [hF : F.Corepresentable]
+variable (F : C ⥤ Type v) [hF : F.IsCorepresentable]
 
 /-- The representing object for the corepresentable functor `F`. -/
 noncomputable def coreprX : C :=
-  hF.has_corepresentation.choose.unop
+  hF.has_corepresentation.choose
 
-/-- An isomorphism between a corepresnetable `F` and a functor of the form
-`C(F.corepr X, -)`. Note the components `F.coreprW.app X`
-definitionally have type `F.corepr_X ⟶ X ≅ F.obj X`.
--/
-noncomputable def coreprW : coyoneda.obj (op F.coreprX) ≅ F :=
+/-- A chosen term in `F.CorepresentableBy (coreprX F)` when `F.IsCorepresentable` holds. -/
+noncomputable def corepresentableBy : F.CorepresentableBy F.coreprX :=
   hF.has_corepresentation.choose_spec.some
 
 /-- The representing element for the corepresentable functor `F`, sometimes called the universal
 element of the functor.
 -/
 noncomputable def coreprx : F.obj F.coreprX :=
-  F.coreprW.hom.app F.coreprX (𝟙 F.coreprX)
+  F.corepresentableBy.homEquiv (𝟙 _)
+
+/-- An isomorphism between a corepresentable `F` and a functor of the form
+`C(F.corepr X, -)`. Note the components `F.coreprW.app X`
+definitionally have type `F.corepr_X ⟶ X ≅ F.obj X`.
+-/
+noncomputable def coreprW (F : C ⥤ Type v₁) [F.IsCorepresentable] :
+    coyoneda.obj (op F.coreprX) ≅ F :=
+  F.corepresentableBy.toIso
 
-theorem coreprW_app_hom (X : C) (f : F.coreprX ⟶ X) :
-    (F.coreprW.app X).hom f = F.map f F.coreprx := by
-  simp only [coyoneda_obj_obj, unop_op, Iso.app_hom, coreprx, ← FunctorToTypes.naturality,
-    coyoneda_obj_map, Category.id_comp]
+theorem coreprW_hom_app (F : C ⥤ Type v₁) [F.IsCorepresentable] (X : C) (f : F.coreprX ⟶ X) :
+    F.coreprW.hom.app X f = F.map f F.coreprx := by
+  apply CorepresentableBy.homEquiv_eq
 
 end Corepresentable
 
 end Functor
 
-theorem representable_of_natIso (F : Cᵒᵖ ⥤ Type v₁) {G} (i : F ≅ G) [F.Representable] :
-    G.Representable :=
-  { has_representation := ⟨F.reprX, ⟨F.reprW ≪≫ i⟩⟩ }
+theorem isRepresentable_of_natIso (F : Cᵒᵖ ⥤ Type v₁) {G} (i : F ≅ G) [F.IsRepresentable] :
+    G.IsRepresentable :=
+  (F.representableBy.ofIso i).isRepresentable
 
-theorem corepresentable_of_natIso (F : C ⥤ Type v₁) {G} (i : F ≅ G) [F.Corepresentable] :
-    G.Corepresentable :=
-  { has_corepresentation := ⟨op F.coreprX, ⟨F.coreprW ≪≫ i⟩⟩ }
+theorem corepresentable_of_natIso (F : C ⥤ Type v₁) {G} (i : F ≅ G) [F.IsCorepresentable] :
+    G.IsCorepresentable :=
+  (F.corepresentableBy.ofIso i).isCorepresentable
 
-instance : Functor.Corepresentable (𝟭 (Type v₁)) :=
+instance : Functor.IsCorepresentable (𝟭 (Type v₁)) :=
   corepresentable_of_natIso (coyoneda.obj (op PUnit)) Coyoneda.punitIso
 
 open Opposite
@@ -286,6 +417,7 @@ theorem yonedaEquiv_symm_app_apply {X : C} {F : Cᵒᵖ ⥤ Type v₁} (x : F.ob
     (f : Y.unop ⟶ X) : (yonedaEquiv.symm x).app Y f = F.map f.op x :=
   rfl
 
+/-- See also `yonedaEquiv_naturality'` for a more general version. -/
 lemma yonedaEquiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F)
     (g : Y ⟶ X) : F.map g.op (yonedaEquiv f) = yonedaEquiv (yoneda.map g ≫ f) := by
   change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g)
@@ -293,6 +425,9 @@ lemma yonedaEquiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.o
   dsimp
   simp
 
+/-- Variant of `yonedaEquiv_naturality` with general `g`. This is technically strictly more general
+    than `yonedaEquiv_naturality`, but `yonedaEquiv_naturality` is sometimes preferable because it
+    can avoid the "motive is not type correct" error. -/
 lemma yonedaEquiv_naturality' {X Y : Cᵒᵖ} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj (unop X) ⟶ F)
     (g : X ⟶ Y) : F.map g (yonedaEquiv f) = yonedaEquiv (yoneda.map g.unop ≫ f) :=
   yonedaEquiv_naturality _ _
@@ -305,6 +440,18 @@ lemma yonedaEquiv_yoneda_map {X Y : C} (f : X ⟶ Y) : yonedaEquiv (yoneda.map f
   rw [yonedaEquiv_apply]
   simp
 
+/-- See also `map_yonedaEquiv'` for a more general version. -/
+lemma map_yonedaEquiv {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F)
+    (g : Y ⟶ X) : F.map g.op (yonedaEquiv f) = f.app (op Y) g := by
+  rw [yonedaEquiv_naturality, yonedaEquiv_comp, yonedaEquiv_yoneda_map]
+
+/-- Variant of `map_yonedaEquiv` with general `g`. This is technically strictly more general
+    than `map_yonedaEquiv`, but `map_yonedaEquiv` is sometimes preferable because it
+    can avoid the "motive is not type correct" error. -/
+lemma map_yonedaEquiv' {X Y : Cᵒᵖ} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj (unop X) ⟶ F)
+    (g : X ⟶ Y) : F.map g (yonedaEquiv f) = f.app Y g.unop := by
+  rw [yonedaEquiv_naturality', yonedaEquiv_comp, yonedaEquiv_yoneda_map]
+
 lemma yonedaEquiv_symm_map {X Y : Cᵒᵖ} (f : X ⟶ Y) {F : Cᵒᵖ ⥤ Type v₁} (t : F.obj X) :
     yonedaEquiv.symm (F.map f t) = yoneda.map f.unop ≫ yonedaEquiv.symm t := by
   obtain ⟨u, rfl⟩ := yonedaEquiv.surjective t
@@ -481,6 +628,10 @@ lemma coyonedaEquiv_coyoneda_map {X Y : C} (f : X ⟶ Y) :
   rw [coyonedaEquiv_apply]
   simp
 
+lemma map_coyonedaEquiv {X Y : C} {F : C ⥤ Type v₁} (f : coyoneda.obj (op X) ⟶ F)
+    (g : X ⟶ Y) : F.map g (coyonedaEquiv f) = f.app Y g := by
+  rw [coyonedaEquiv_naturality, coyonedaEquiv_comp, coyonedaEquiv_coyoneda_map]
+
 lemma coyonedaEquiv_symm_map {X Y : C} (f : X ⟶ Y) {F : C ⥤ Type v₁} (t : F.obj X) :
     coyonedaEquiv.symm (F.map f t) = coyoneda.map f.op ≫ coyonedaEquiv.symm t := by
   obtain ⟨u, rfl⟩ := coyonedaEquiv.surjective t
diff --git a/Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean b/Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
index 9b660a299371d..076f7c450bcde 100644
--- a/Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
+++ b/Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean
@@ -356,7 +356,7 @@ theorem three_le_nValue (hN : 64 ≤ N) : 3 ≤ nValue N := by
     rw [rpow_natCast]
     exact (cast_le.2 hN).trans' (by norm_num1)
   apply lt_of_lt_of_le _ (log_le_log (rpow_pos_of_pos zero_lt_two _) this)
-  rw [log_rpow zero_lt_two, ← div_lt_iff']
+  rw [log_rpow zero_lt_two, ← div_lt_iff₀']
   · exact log_two_gt_d9.trans_le' (by norm_num1)
   · norm_num1
 
@@ -459,7 +459,7 @@ theorem roth_lower_bound_explicit (hN : 4096 ≤ N) :
 
 theorem exp_four_lt : exp 4 < 64 := by
   rw [show (64 : ℝ) = 2 ^ ((6 : ℕ) : ℝ) by rw [rpow_natCast]; norm_num1,
-    ← lt_log_iff_exp_lt (rpow_pos_of_pos zero_lt_two _), log_rpow zero_lt_two, ← div_lt_iff']
+    ← lt_log_iff_exp_lt (rpow_pos_of_pos zero_lt_two _), log_rpow zero_lt_two, ← div_lt_iff₀']
   · exact log_two_gt_d9.trans_le' (by norm_num1)
   · norm_num
 
diff --git a/Mathlib/Combinatorics/Additive/AP/Three/Defs.lean b/Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
index 30c00a8a00135..213220d6ea296 100644
--- a/Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
+++ b/Mathlib/Combinatorics/Additive/AP/Three/Defs.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 -/
 import Mathlib.Algebra.Order.Interval.Finset
+import Mathlib.Algebra.SMulWithZero
 import Mathlib.Combinatorics.Additive.FreimanHom
-import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.Order.Interval.Finset.Fin
 
 /-!
@@ -43,7 +43,7 @@ the size of the biggest 3AP-free subset of `{0, ..., n - 1}`.
 3AP-free, Salem-Spencer, Roth, arithmetic progression, average, three-free
 -/
 
-open Finset Function Nat
+open Finset Function
 open scoped Pointwise
 
 variable {F α β 𝕜 E : Type*}
@@ -103,18 +103,18 @@ end Monoid
 section CommMonoid
 variable [CommMonoid α] [CommMonoid β] {s A : Set α} {t B : Set β} {f : α → β} {a : α}
 
-/-- Arithmetic progressions of length three are preserved under `2`-Freiman homomorphisms. -/
+/-- Geometric progressions of length three are reflected under `2`-Freiman homomorphisms. -/
 @[to_additive
-"Arithmetic progressions of length three are preserved under `2`-Freiman homomorphisms."]
+"Arithmetic progressions of length three are reflected under `2`-Freiman homomorphisms."]
 lemma ThreeGPFree.of_image (hf : IsMulFreimanHom 2 s t f) (hf' : s.InjOn f) (hAs : A ⊆ s)
     (hA : ThreeGPFree (f '' A)) : ThreeGPFree A :=
   fun _ ha _ hb _ hc habc ↦ hf' (hAs ha) (hAs hb) <| hA (mem_image_of_mem _ ha)
     (mem_image_of_mem _ hb) (mem_image_of_mem _ hc) <|
     hf.mul_eq_mul (hAs ha) (hAs hc) (hAs hb) (hAs hb) habc
 
-/-- Arithmetic progressions of length three are preserved under `2`-Freiman isomorphisms. -/
+/-- Geometric progressions of length three are unchanged under `2`-Freiman isomorphisms. -/
 @[to_additive
-"Arithmetic progressions of length three are preserved under `2`-Freiman isomorphisms."]
+"Arithmetic progressions of length three are unchanged under `2`-Freiman isomorphisms."]
 lemma threeGPFree_image (hf : IsMulFreimanIso 2 s t f) (hAs : A ⊆ s) :
     ThreeGPFree (f '' A) ↔ ThreeGPFree A := by
   rw [ThreeGPFree, ThreeGPFree]
@@ -125,25 +125,23 @@ lemma threeGPFree_image (hf : IsMulFreimanIso 2 s t f) (hAs : A ⊆ s) :
 
 @[to_additive] alias ⟨_, ThreeGPFree.image⟩ := threeGPFree_image
 
-/-- Arithmetic progressions of length three are preserved under `2`-Freiman homomorphisms. -/
-@[to_additive]
+/-- Geometric progressions of length three are reflected under `2`-Freiman homomorphisms. -/
+@[to_additive
+"Arithmetic progressions of length three are reflected under `2`-Freiman homomorphisms."]
 lemma IsMulFreimanHom.threeGPFree (hf : IsMulFreimanHom 2 s t f) (hf' : s.InjOn f)
     (ht : ThreeGPFree t) : ThreeGPFree s :=
-  fun _ ha _ hb _ hc habc ↦ hf' ha hb <| ht (hf.mapsTo ha) (hf.mapsTo hb) (hf.mapsTo hc) <|
-    hf.mul_eq_mul ha hc hb hb habc
+  (ht.mono hf.mapsTo.image_subset).of_image hf hf' subset_rfl
 
-/-- Arithmetic progressions of length three are preserved under `2`-Freiman isomorphisms. -/
-@[to_additive]
+/-- Geometric progressions of length three are unchanged under `2`-Freiman isomorphisms. -/
+@[to_additive
+"Arithmetic progressions of length three are unchanged under `2`-Freiman isomorphisms."]
 lemma IsMulFreimanIso.threeGPFree_congr (hf : IsMulFreimanIso 2 s t f) :
-    ThreeGPFree s ↔ ThreeGPFree t where
-  mpr := hf.isMulFreimanHom.threeGPFree hf.bijOn.injOn
-  mp hs a hfa b hfb c hfc habc := by
-    obtain ⟨a, ha, rfl⟩ := hf.bijOn.surjOn hfa
-    obtain ⟨b, hb, rfl⟩ := hf.bijOn.surjOn hfb
-    obtain ⟨c, hc, rfl⟩ := hf.bijOn.surjOn hfc
-    exact congr_arg f <| hs ha hb hc <| (hf.mul_eq_mul ha hc hb hb).1 habc
+    ThreeGPFree s ↔ ThreeGPFree t := by
+  rw [← threeGPFree_image hf subset_rfl, hf.bijOn.image_eq]
 
-@[to_additive]
+/-- Geometric progressions of length three are preserved under semigroup homomorphisms. -/
+@[to_additive
+"Arithmetic progressions of length three are preserved under semigroup homomorphisms."]
 theorem ThreeGPFree.image' [FunLike F α β] [MulHomClass F α β] (f : F) (hf : (s * s).InjOn f)
     (h : ThreeGPFree s) : ThreeGPFree (f '' s) := by
   rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ habc
@@ -155,7 +153,7 @@ section CancelCommMonoid
 
 variable [CancelCommMonoid α] {s : Set α} {a : α}
 
-lemma ThreeGPFree.eq_right (hs : ThreeGPFree s) :
+@[to_additive] lemma ThreeGPFree.eq_right (hs : ThreeGPFree s) :
     ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ∀ ⦃c⦄, c ∈ s → a * c = b * b → b = c := by
   rintro a ha b hb c hc habc
   obtain rfl := hs ha hb hc habc
@@ -275,7 +273,7 @@ variable {s t} {n : ℕ}
 @[to_additive]
 theorem ThreeGPFree.le_mulRothNumber (hs : ThreeGPFree (s : Set α)) (h : s ⊆ t) :
     s.card ≤ mulRothNumber t :=
-  le_findGreatest (card_le_card h) ⟨s, h, rfl, hs⟩
+  Nat.le_findGreatest (card_le_card h) ⟨s, h, rfl, hs⟩
 
 @[to_additive]
 theorem ThreeGPFree.mulRothNumber_eq (hs : ThreeGPFree (s : Set α)) :
diff --git a/Mathlib/Combinatorics/Additive/Corner/Roth.lean b/Mathlib/Combinatorics/Additive/Corner/Roth.lean
index 69eec794bef92..73840313536fa 100644
--- a/Mathlib/Combinatorics/Additive/Corner/Roth.lean
+++ b/Mathlib/Combinatorics/Additive/Corner/Roth.lean
@@ -88,7 +88,7 @@ theorem corners_theorem (ε : ℝ) (hε : 0 < ε) (hG : cornersTheoremBound ε 
     rwa [mul_le_iff_le_one_left] at this
     positivity
   have := noAccidental hA
-  rw [Nat.floor_lt' (by positivity), inv_pos_lt_iff_one_lt_mul'] at hG
+  rw [Nat.floor_lt' (by positivity), inv_lt_iff_one_lt_mul₀'] at hG
   swap
   · have : ε / 9 ≤ 1 := by linarith
     positivity
diff --git a/Mathlib/Combinatorics/Additive/Dissociation.lean b/Mathlib/Combinatorics/Additive/Dissociation.lean
index bfb19d5419f00..f22b05d75f55d 100644
--- a/Mathlib/Combinatorics/Additive/Dissociation.lean
+++ b/Mathlib/Combinatorics/Additive/Dissociation.lean
@@ -4,9 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Group.Units.Equiv
 import Mathlib.Data.Fintype.Card
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Dissociation and span
diff --git a/Mathlib/Combinatorics/Additive/ETransform.lean b/Mathlib/Combinatorics/Additive/ETransform.lean
index c146fce4176ae..d7a7e6ffdcbc2 100644
--- a/Mathlib/Combinatorics/Additive/ETransform.lean
+++ b/Mathlib/Combinatorics/Additive/ETransform.lean
@@ -3,7 +3,7 @@ Copyright (c) 2023 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 
 /-!
 # e-transforms
diff --git a/Mathlib/Combinatorics/Additive/Energy.lean b/Mathlib/Combinatorics/Additive/Energy.lean
index 341b4509c4465..2cd35ab13760c 100644
--- a/Mathlib/Combinatorics/Additive/Energy.lean
+++ b/Mathlib/Combinatorics/Additive/Energy.lean
@@ -3,10 +3,10 @@ Copyright (c) 2022 Yaël Dillies, Ella Yu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Ella Yu
 -/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Algebra.Order.BigOperators.Ring.Finset
 import Mathlib.Data.Finset.Prod
 import Mathlib.Data.Fintype.Prod
-import Mathlib.Data.Finset.Pointwise.Basic
 
 /-!
 # Additive energy
@@ -164,7 +164,7 @@ variable [CommMonoid α]
 
 @[to_additive] lemma mulEnergy_comm (s t : Finset α) : Eₘ[s, t] = Eₘ[t, s] := by
   rw [mulEnergy, ← Finset.card_map (Equiv.prodComm _ _).toEmbedding, map_filter]
-  simp [-Finset.card_map, eq_comm, mulEnergy, mul_comm, map_eq_image, Function.comp]
+  simp [-Finset.card_map, eq_comm, mulEnergy, mul_comm, map_eq_image, Function.comp_def]
 
 end CommMonoid
 
@@ -183,7 +183,7 @@ lemma mulEnergy_univ_left : Eₘ[univ, t] = Fintype.card α * t.card ^ 2 := by
     rw [mul_right_cancel h.1]
   rw [← card_image_of_injOn this]
   congr with a
-  simp only [mem_filter, mem_product, mem_univ, true_and_iff, mem_image, exists_prop,
+  simp only [mem_filter, mem_product, mem_univ, true_and, mem_image, exists_prop,
     Prod.exists]
   refine ⟨fun h => ⟨a.1.1 * a.2.2⁻¹, _, _, h.1, by simp [f, mul_right_comm, h.2]⟩, ?_⟩
   rintro ⟨b, c, d, hcd, rfl⟩
diff --git a/Mathlib/Combinatorics/Additive/FreimanHom.lean b/Mathlib/Combinatorics/Additive/FreimanHom.lean
index ceb7bc28e54f4..01cf0ef4b01c3 100644
--- a/Mathlib/Combinatorics/Additive/FreimanHom.lean
+++ b/Mathlib/Combinatorics/Additive/FreimanHom.lean
@@ -1,13 +1,15 @@
 /-
 Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yaël Dillies
+Authors: Yaël Dillies, Bhavik Mehta
 -/
 import Mathlib.Algebra.BigOperators.Ring
 import Mathlib.Algebra.CharP.Defs
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
+import Mathlib.Algebra.Group.Submonoid.Operations
 import Mathlib.Algebra.Order.BigOperators.Group.Multiset
+import Mathlib.Algebra.Order.Ring.Nat
 import Mathlib.Data.ZMod.Defs
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Freiman homomorphisms
@@ -18,19 +20,29 @@ An `n`-Freiman homomorphism from `A` to `B` is a function `f : α → β` such t
 `f x₁ * ... * f xₙ = f y₁ * ... * f yₙ` for all `x₁, ..., xₙ, y₁, ..., yₙ ∈ A` such that
 `x₁ * ... * xₙ = y₁ * ... * yₙ`. In particular, any `MulHom` is a Freiman homomorphism.
 
+Note a `0`- or `1`-Freiman homomorphism is simply a map, thus a `2`-Freiman homomorphism is the
+first interesting case (and the most common). As `n` increases further, the property of being
+an `n`-Freiman homomorphism between abelian groups becomes increasingly stronger.
+
 An `n`-Freiman isomorphism from `A` to `B` is a function `f : α → β` bijective between `A` and `B`
 such that `f x₁ * ... * f xₙ = f y₁ * ... * f yₙ ↔ x₁ * ... * xₙ = y₁ * ... * yₙ` for all
 `x₁, ..., xₙ, y₁, ..., yₙ ∈ A`. In particular, any `MulEquiv` is a Freiman isomorphism.
 
 They are of interest in additive combinatorics.
 
-## Main declaration
+## Main declarations
 
 * `IsMulFreimanHom`: Predicate for a function to be a multiplicative Freiman homomorphism.
 * `IsAddFreimanHom`: Predicate for a function to be an additive Freiman homomorphism.
 * `IsMulFreimanIso`: Predicate for a function to be a multiplicative Freiman isomorphism.
 * `IsAddFreimanIso`: Predicate for a function to be an additive Freiman isomorphism.
 
+## Main results
+
+* `isMulFreimanHom_two`: Characterisation of `2`-Freiman homomorphisms.
+* `IsMulFreimanHom.mono`: If `m ≤ n` and `f` is an `n`-Freiman homomorphism, then it is also an
+  `m`-Freiman homomorphism.
+
 ## Implementation notes
 
 In the context of combinatorics, we are interested in Freiman homomorphisms over sets which are not
@@ -102,6 +114,20 @@ lemma IsMulFreimanIso.isMulFreimanHom (hf : IsMulFreimanIso n A B f) : IsMulFrei
   mapsTo := hf.bijOn.mapsTo
   map_prod_eq_map_prod _s _t hsA htA hs ht := (hf.map_prod_eq_map_prod hsA htA hs ht).2
 
+lemma IsMulFreimanHom.congr (hf₁ : IsMulFreimanHom n A B f₁) (h : EqOn f₁ f₂ A) :
+    IsMulFreimanHom n A B f₂ where
+  mapsTo := hf₁.mapsTo.congr h
+  map_prod_eq_map_prod s t hsA htA hs ht h' := by
+    rw [map_congr rfl fun x hx => (h (hsA hx)).symm, map_congr rfl fun x hx => (h (htA hx)).symm,
+      hf₁.map_prod_eq_map_prod hsA htA hs ht h']
+
+lemma IsMulFreimanIso.congr (hf₁ : IsMulFreimanIso n A B f₁) (h : EqOn f₁ f₂ A) :
+    IsMulFreimanIso n A B f₂ where
+  bijOn := hf₁.bijOn.congr h
+  map_prod_eq_map_prod s t hsA htA hs ht := by
+    rw [map_congr rfl fun x hx => h.symm (hsA hx), map_congr rfl fun x hx => h.symm (htA hx),
+      hf₁.map_prod_eq_map_prod hsA htA hs ht]
+
 @[to_additive]
 lemma IsMulFreimanHom.mul_eq_mul (hf : IsMulFreimanHom 2 A B f) {a b c d : α}
     (ha : a ∈ A) (hb : b ∈ A) (hc : c ∈ A) (hd : d ∈ A) (h : a * b = c * d) :
@@ -116,13 +142,21 @@ lemma IsMulFreimanIso.mul_eq_mul (hf : IsMulFreimanIso 2 A B f) {a b c d : α}
   simp_rw [← prod_pair]
   refine hf.map_prod_eq_map_prod ?_ ?_ (card_pair _ _) (card_pair _ _) <;> simp [ha, hb, hc, hd]
 
-/-- Characterisation of `2`-Freiman homs. -/
-@[to_additive "Characterisation of `2`-Freiman homs."]
+/-- Characterisation of `2`-Freiman homomorphisms. -/
+@[to_additive "Characterisation of `2`-Freiman homomorphisms."]
 lemma isMulFreimanHom_two :
     IsMulFreimanHom 2 A B f ↔ MapsTo f A B ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
       a * b = c * d → f a * f b = f c * f d where
   mp hf := ⟨hf.mapsTo, fun a ha b hb c hc d hd ↦ hf.mul_eq_mul ha hb hc hd⟩
-  mpr hf := ⟨hf.1, by aesop (add simp [Multiset.card_eq_two])⟩
+  mpr hf := ⟨hf.1, by aesop (add simp card_eq_two)⟩
+
+/-- Characterisation of `2`-Freiman homs. -/
+@[to_additive "Characterisation of `2`-Freiman isomorphisms."]
+lemma isMulFreimanIso_two :
+    IsMulFreimanIso 2 A B f ↔ BijOn f A B ∧ ∀ a ∈ A, ∀ b ∈ A, ∀ c ∈ A, ∀ d ∈ A,
+      f a * f b = f c * f d ↔ a * b = c * d where
+  mp hf := ⟨hf.bijOn, fun a ha b hb c hc d hd => hf.mul_eq_mul ha hb hc hd⟩
+  mpr hf := ⟨hf.1, by aesop (add simp card_eq_two)⟩
 
 @[to_additive] lemma isMulFreimanHom_id (hA : A₁ ⊆ A₂) : IsMulFreimanHom n A₁ A₂ id where
   mapsTo := hA
@@ -171,6 +205,27 @@ lemma isMulFreimanHom_const {b : β} (hb : b ∈ B) : IsMulFreimanHom n A B fun
   mapsTo _ _ := hb
   map_prod_eq_map_prod s t _ _ hs ht _ := by simp only [map_const', hs, prod_replicate, ht]
 
+@[to_additive (attr := simp)]
+lemma isMulFreimanHom_zero_iff : IsMulFreimanHom 0 A B f ↔ MapsTo f A B :=
+  ⟨fun h => h.mapsTo, fun h => ⟨h, by aesop⟩⟩
+
+@[to_additive (attr := simp)]
+lemma isMulFreimanIso_zero_iff : IsMulFreimanIso 0 A B f ↔ BijOn f A B :=
+  ⟨fun h => h.bijOn, fun h => ⟨h, by aesop⟩⟩
+
+@[to_additive (attr := simp) isAddFreimanHom_one_iff]
+lemma isMulFreimanHom_one_iff : IsMulFreimanHom 1 A B f ↔ MapsTo f A B :=
+  ⟨fun h => h.mapsTo, fun h => ⟨h, by aesop (add simp card_eq_one)⟩⟩
+
+@[to_additive (attr := simp) isAddFreimanIso_one_iff]
+lemma isMulFreimanIso_one_iff : IsMulFreimanIso 1 A B f ↔ BijOn f A B :=
+  ⟨fun h => h.bijOn, fun h => ⟨h, by aesop (add simp [card_eq_one, BijOn])⟩⟩
+
+@[to_additive (attr := simp)]
+lemma isMulFreimanHom_empty : IsMulFreimanHom n (∅ : Set α) B f where
+  mapsTo := mapsTo_empty f B
+  map_prod_eq_map_prod s t := by aesop (add simp eq_zero_of_forall_not_mem)
+
 @[to_additive (attr := simp)]
 lemma isMulFreimanIso_empty : IsMulFreimanIso n (∅ : Set α) (∅ : Set β) f where
   bijOn := bijOn_empty _
@@ -196,6 +251,11 @@ lemma isMulFreimanIso_empty : IsMulFreimanIso n (∅ : Set α) (∅ : Set β) f
   map_prod_eq_map_prod s t _ _ _ _ := by
     rw [← map_multiset_prod, ← map_multiset_prod, EquivLike.apply_eq_iff_eq]
 
+@[to_additive]
+lemma IsMulFreimanHom.subtypeVal {S : Type*} [SetLike S α] [SubmonoidClass S α] {s : S} :
+    IsMulFreimanHom n (univ : Set s) univ Subtype.val :=
+  MonoidHomClass.isMulFreimanHom (SubmonoidClass.subtype s) (mapsTo_univ ..)
+
 end CommMonoid
 
 section CancelCommMonoid
diff --git a/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean b/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
index 125742993fc6a..ae04f34d67a52 100644
--- a/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
+++ b/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
@@ -3,12 +3,12 @@ Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, George Shakan
 -/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
+import Mathlib.Algebra.Order.Field.Basic
+import Mathlib.Algebra.Order.Field.Rat
 import Mathlib.Algebra.Order.Ring.Basic
 import Mathlib.Combinatorics.Enumerative.DoubleCounting
-import Mathlib.Data.Finset.Pointwise.Basic
 import Mathlib.Tactic.GCongr
-import Mathlib.Algebra.Order.Field.Basic
-import Mathlib.Algebra.Order.Field.Rat
 
 /-!
 # The Plünnecke-Ruzsa inequality
diff --git a/Mathlib/Combinatorics/Additive/RuzsaCovering.lean b/Mathlib/Combinatorics/Additive/RuzsaCovering.lean
index 522ba740a3844..6e7183a4afdf7 100644
--- a/Mathlib/Combinatorics/Additive/RuzsaCovering.lean
+++ b/Mathlib/Combinatorics/Additive/RuzsaCovering.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.SetTheory.Cardinal.Finite
 
 /-!
diff --git a/Mathlib/Combinatorics/Colex.lean b/Mathlib/Combinatorics/Colex.lean
index 1f8a439741e9c..54e26b8a2cc4c 100644
--- a/Mathlib/Combinatorics/Colex.lean
+++ b/Mathlib/Combinatorics/Colex.lean
@@ -65,7 +65,7 @@ namespace Finset
 /-- Type synonym of `Finset α` equipped with the colexicographic order rather than the inclusion
 order. -/
 @[ext]
-structure Colex (α) :=
+structure Colex (α) where
   /-- `toColex` is the "identity" function between `Finset α` and `Finset.Colex α`. -/
   toColex ::
   /-- `ofColex` is the "identity" function between `Finset.Colex α` and `Finset α`. -/
@@ -197,7 +197,7 @@ lemma toColex_lt_singleton : toColex s < toColex {a} ↔ ∀ b ∈ s, b < a := b
   · refine h.2 <| eq_singleton_iff_unique_mem.2 ⟨hb, fun c hc ↦ (h.1 _ hc).2 hb⟩
   · simp at h
 
-/-- `{a} ≤ s` in colex iff `s` contains an element greated than or equal to `a`. -/
+/-- `{a} ≤ s` in colex iff `s` contains an element greater than or equal to `a`. -/
 lemma singleton_le_toColex : (toColex {a} : Colex α) ≤ toColex s ↔ ∃ x ∈ s, a ≤ x := by
   simp [toColex_le_toColex]; by_cases a ∈ s <;> aesop
 
diff --git a/Mathlib/Combinatorics/Configuration.lean b/Mathlib/Combinatorics/Configuration.lean
index d680b794a0d30..a703f011ed4da 100644
--- a/Mathlib/Combinatorics/Configuration.lean
+++ b/Mathlib/Combinatorics/Configuration.lean
@@ -3,10 +3,9 @@ Copyright (c) 2021 Thomas Browning. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Thomas Browning
 -/
-import Mathlib.Algebra.Order.BigOperators.Group.Finset
 import Mathlib.Combinatorics.Hall.Basic
-import Mathlib.Data.Fintype.BigOperators
-import Mathlib.SetTheory.Cardinal.Finite
+import Mathlib.Data.Matrix.Rank
+import Mathlib.LinearAlgebra.Projectivization.Constructions
 
 /-!
 # Configurations of Points and lines
@@ -57,7 +56,7 @@ instance [h : Fintype P] : Fintype (Dual P) :=
 -- Porting note (#11215): TODO: figure out if this is needed.
 set_option synthInstance.checkSynthOrder false in
 instance : Membership (Dual L) (Dual P) :=
-  ⟨Function.swap (Membership.mem : P → L → Prop)⟩
+  ⟨Function.swap (Membership.mem : L → P → Prop)⟩
 
 /-- A configuration is nondegenerate if:
   1) there does not exist a line that passes through all of the points,
@@ -129,7 +128,7 @@ theorem Nondegenerate.exists_injective_of_card_le [Nondegenerate P L] [Fintype P
     by_cases hs₁ : s.card = 1
     -- If `s = {l}`, then pick a point `p ∉ l`
     · obtain ⟨l, rfl⟩ := Finset.card_eq_one.mp hs₁
-      obtain ⟨p, hl⟩ := exists_point l
+      obtain ⟨p, hl⟩ := exists_point (P := P) l
       rw [Finset.card_singleton, Finset.singleton_biUnion, Nat.one_le_iff_ne_zero]
       exact Finset.card_ne_zero_of_mem (Set.mem_toFinset.mpr hl)
     suffices (s.biUnion t)ᶜ.card ≤ sᶜ.card by
@@ -196,9 +195,8 @@ theorem HasLines.pointCount_le_lineCount [HasLines P L] {p : P} {l : L} (h : p 
   exact
     Fintype.card_le_of_injective (fun p' => ⟨mkLine (this p'), (mkLine_ax (this p')).1⟩)
       fun p₁ p₂ hp =>
-      Subtype.ext
-        ((eq_or_eq p₁.2 p₂.2 (mkLine_ax (this p₁)).2
-              ((congr_arg _ (Subtype.ext_iff.mp hp)).mpr (mkLine_ax (this p₂)).2)).resolve_right
+      Subtype.ext ((eq_or_eq p₁.2 p₂.2 (mkLine_ax (this p₁)).2
+            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hp)).mpr (mkLine_ax (this p₂)).2)).resolve_right
           fun h' => (congr_arg (¬p ∈ ·) h').mp h (mkLine_ax (this p₁)).1)
 
 theorem HasPoints.lineCount_le_pointCount [HasPoints P L] {p : P} {l : L} (h : p ∉ l)
@@ -222,7 +220,7 @@ theorem HasLines.card_le [HasLines P L] [Fintype P] [Fintype L] :
       _ < ∑ p, lineCount L p := by
         obtain ⟨p, hp⟩ := not_forall.mp (mt (Fintype.card_le_of_surjective f) hc₂)
         refine sum_lt_sum_of_subset (subset_univ _) (mem_univ p) ?_ ?_ fun p _ _ ↦ zero_le _
-        · simpa only [Finset.mem_map, exists_prop, Finset.mem_univ, true_and_iff]
+        · simpa only [Finset.mem_map, exists_prop, Finset.mem_univ, true_and]
         · rw [lineCount, Nat.card_eq_fintype_card, Fintype.card_pos_iff]
           obtain ⟨l, _⟩ := @exists_line P L _ _ p
           exact
@@ -263,7 +261,7 @@ theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
           simp_rw [hf2, sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
           change pointCount P l • _ = lineCount L (f l) • _
           rw [hf2]
-      all_goals simp_rw [s, Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
+      all_goals simp_rw [s, Finset.mem_univ, true_and, Set.mem_toFinset]; exact fun p => Iff.rfl
     have step3 : ∑ i ∈ sᶜ, lineCount L i.1 = ∑ i ∈ sᶜ, pointCount P i.2 := by
       rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1
     rw [← Set.toFinset_compl] at step3
@@ -301,13 +299,12 @@ noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
       let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
         ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
       have hf : Function.Injective f := fun q₁ q₂ hq =>
-        Subtype.ext
-          ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
-                ((congr_arg _ (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
+        Subtype.ext ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
+            ((congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
             fun h => (congr_arg (¬p ∈ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
       have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
       obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
-      exact ⟨q, (congr_arg _ (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
+      exact ⟨q, (congr_arg (_ ∈ ·) (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
   { ‹HasLines P L› with
     mkPoint := fun {l₁ l₂} hl => Classical.choose (this l₁ l₂ hl)
     mkPoint_ax := fun {l₁ l₂} hl => Classical.choose_spec (this l₁ l₂ hl) }
@@ -368,7 +365,7 @@ theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p =
     or_not.elim (fun h₂ => ?_) fun h₂ => (HasLines.lineCount_eq_pointCount h h₂).trans hl₂
   refine or_not.elim (fun h₃ => ?_) fun h₃ => (HasLines.lineCount_eq_pointCount h h₃).trans hl₃
   rw [(eq_or_eq h₂ h₂₂ h₃ h₂₃).resolve_right fun h =>
-      h₃₃ ((congr_arg (Membership.mem p₃) h).mp h₃₂)]
+      h₃₃ ((congr_arg (p₃ ∈ ·) h).mp h₃₂)]
 
 variable (P) {L}
 
@@ -414,7 +411,7 @@ theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
   rw [← add_lt_add_iff_right 1, ← pointCount_eq _ l₂, pointCount, Nat.card_eq_fintype_card,
     Fintype.two_lt_card_iff]
   simp_rw [Ne, Subtype.ext_iff]
-  have h := mkPoint_ax fun h => h₂₁ ((congr_arg _ h).mpr h₂₂)
+  have h := mkPoint_ax (P := P) (L := L) fun h => h₂₁ ((congr_arg (p₂ ∈ ·) h).mpr h₂₂)
   exact
     ⟨⟨mkPoint _, h.2⟩, ⟨p₂, h₂₂⟩, ⟨p₃, h₃₂⟩, ne_of_mem_of_not_mem h.1 h₂₁,
       ne_of_mem_of_not_mem h.1 h₃₁, ne_of_mem_of_not_mem h₂₃ h₃₃⟩
@@ -464,4 +461,63 @@ theorem card_lines [Finite P] [Fintype L] : Fintype.card L = order P L ^ 2 + ord
 
 end ProjectivePlane
 
+namespace ofField
+
+variable {K : Type*} [Field K]
+
+open scoped LinearAlgebra.Projectivization
+
+open Matrix Projectivization
+
+instance : Membership (ℙ K (Fin 3 → K)) (ℙ K (Fin 3 → K)) :=
+  ⟨Function.swap orthogonal⟩
+
+lemma mem_iff (v w : ℙ K (Fin 3 → K)) : v ∈ w ↔ orthogonal v w :=
+  Iff.rfl
+
+-- This lemma can't be moved to the crossProduct file due to heavy imports
+lemma crossProduct_eq_zero_of_dotProduct_eq_zero {a b c d : Fin 3 → K} (hac : dotProduct a c = 0)
+    (hbc : dotProduct b c = 0) (had : dotProduct a d = 0) (hbd : dotProduct b d = 0) :
+    crossProduct a b = 0 ∨ crossProduct c d = 0 := by
+  by_contra h
+  simp_rw [not_or, ← ne_eq, crossProduct_ne_zero_iff_linearIndependent] at h
+  let A : Matrix (Fin 2) (Fin 3) K := ![a, b]
+  let B : Matrix (Fin 2) (Fin 3) K := ![c, d]
+  have hAB : A * B.transpose = 0 := by
+    ext i j
+    fin_cases i <;> fin_cases j <;> assumption
+  replace hAB := rank_add_rank_le_card_of_mul_eq_zero hAB
+  rw [rank_transpose, h.1.rank_matrix, h.2.rank_matrix, Fintype.card_fin, Fintype.card_fin] at hAB
+  contradiction
+
+lemma eq_or_eq_of_orthogonal {a b c d : ℙ K (Fin 3 → K)} (hac : a.orthogonal c)
+    (hbc : b.orthogonal c) (had : a.orthogonal d) (hbd : b.orthogonal d) :
+    a = b ∨ c = d := by
+  induction' a with a ha
+  induction' b with b hb
+  induction' c with c hc
+  induction' d with d hd
+  rw [mk_eq_mk_iff_crossProduct_eq_zero, mk_eq_mk_iff_crossProduct_eq_zero]
+  exact crossProduct_eq_zero_of_dotProduct_eq_zero hac hbc had hbd
+
+instance : Nondegenerate (ℙ K (Fin 3 → K)) (ℙ K (Fin 3 → K)) :=
+  { exists_point := exists_not_orthogonal_self
+    exists_line := exists_not_self_orthogonal
+    eq_or_eq := eq_or_eq_of_orthogonal }
+
+noncomputable instance [DecidableEq K] : ProjectivePlane (ℙ K (Fin 3 → K)) (ℙ K (Fin 3 → K)) :=
+  { mkPoint := by
+      intro v w _
+      exact cross v w
+    mkPoint_ax := fun h ↦ ⟨cross_orthogonal_left h, cross_orthogonal_right h⟩
+    mkLine := by
+      intro v w _
+      exact cross v w
+    mkLine_ax := fun h ↦ ⟨orthogonal_cross_left h, orthogonal_cross_right h⟩
+    exists_config := by
+      refine ⟨mk K ![0, 1, 1] ?_, mk K ![1, 0, 0] ?_, mk K ![1, 0, 1] ?_, mk K ![1, 0, 0] ?_,
+        mk K ![0, 1, 0] ?_, mk K ![0, 0, 1] ?_, ?_⟩ <;> simp [mem_iff, orthogonal_mk] }
+
+end ofField
+
 end Configuration
diff --git a/Mathlib/Combinatorics/Derangements/Finite.lean b/Mathlib/Combinatorics/Derangements/Finite.lean
index 31ea036e767de..210773f9903ec 100644
--- a/Mathlib/Combinatorics/Derangements/Finite.lean
+++ b/Mathlib/Combinatorics/Derangements/Finite.lean
@@ -42,12 +42,14 @@ theorem card_derangements_invariant {α β : Type*} [Fintype α] [DecidableEq α
 theorem card_derangements_fin_add_two (n : ℕ) :
     card (derangements (Fin (n + 2))) =
       (n + 1) * card (derangements (Fin n)) + (n + 1) * card (derangements (Fin (n + 1))) := by
-  -- get some basic results about the size of fin (n+1) plus or minus an element
+  -- get some basic results about the size of Fin (n+1) plus or minus an element
   have h1 : ∀ a : Fin (n + 1), card ({a}ᶜ : Set (Fin (n + 1))) = card (Fin n) := by
     intro a
-    simp only [Fintype.card_fin, Finset.card_fin, Fintype.card_ofFinset, Finset.filter_ne' _ a,
-      Set.mem_compl_singleton_iff, Finset.card_erase_of_mem (Finset.mem_univ a),
-      add_tsub_cancel_right]
+    simp only
+      [card_ofFinset (s := Finset.filter (fun x => x ∈ ({a}ᶜ : Set (Fin (n + 1)))) Finset.univ),
+      Set.mem_compl_singleton_iff, Finset.filter_ne' _ a,
+      Finset.card_erase_of_mem (Finset.mem_univ a), Finset.card_fin, add_tsub_cancel_right,
+      card_fin]
   have h2 : card (Fin (n + 2)) = card (Option (Fin (n + 1))) := by simp only [card_fin, card_option]
   -- rewrite the LHS and substitute in our fintype-level equivalence
   simp only [card_derangements_invariant h2,
diff --git a/Mathlib/Combinatorics/Enumerative/Composition.lean b/Mathlib/Combinatorics/Enumerative/Composition.lean
index cbe6dc1551920..917e4a7e41dbc 100644
--- a/Mathlib/Combinatorics/Enumerative/Composition.lean
+++ b/Mathlib/Combinatorics/Enumerative/Composition.lean
@@ -54,7 +54,7 @@ of `n`.
   blocks of `c`.
 * `join_splitWrtComposition` states that splitting a list and then joining it gives back the
   original list.
-* `joinSplitWrtComposition_join` states that joining a list of lists, and then splitting it back
+* `splitWrtComposition_join` states that joining a list of lists, and then splitting it back
   according to the right composition, gives back the original list of lists.
 
 We turn to the second viewpoint on compositions, that we realize as a finset of `Fin (n+1)`.
@@ -112,7 +112,7 @@ structure CompositionAsSet (n : ℕ) where
   boundaries : Finset (Fin n.succ)
   /-- Proof that `0` is a member of `boundaries`-/
   zero_mem : (0 : Fin n.succ) ∈ boundaries
-  /-- Last element of the composition-/
+  /-- Last element of the composition -/
   getLast_mem : Fin.last n ∈ boundaries
 
 instance {n : ℕ} : Inhabited (CompositionAsSet n) :=
@@ -168,6 +168,12 @@ theorem blocks_pos' (i : ℕ) (h : i < c.length) : 0 < c.blocks[i] :=
 theorem one_le_blocksFun (i : Fin c.length) : 1 ≤ c.blocksFun i :=
   c.one_le_blocks (c.blocksFun_mem_blocks i)
 
+theorem blocksFun_le {n} (c : Composition n) (i : Fin c.length) :
+    c.blocksFun i ≤ n := by
+  have := c.blocks_sum
+  have := List.le_sum_of_mem (c.blocksFun_mem_blocks i)
+  simp_all
+
 theorem length_le : c.length ≤ n := by
   conv_rhs => rw [← c.blocks_sum]
   exact length_le_sum_of_one_le _ fun i hi => c.one_le_blocks hi
@@ -759,7 +765,7 @@ def compositionAsSetEquiv (n : ℕ) : CompositionAsSet n ≃ Finset (Fin (n - 1)
       exact (zero_le i.val).trans_lt (i.2.trans_le (Nat.sub_le n 1))
     simp only [add_comm, Fin.ext_iff, Fin.val_zero, Fin.val_last, exists_prop, Set.toFinset_setOf,
       Finset.mem_univ, forall_true_left, Finset.mem_filter, add_eq_zero, and_false,
-      add_left_inj, false_or, true_and]
+      add_left_inj, false_or, true_and, reduceCtorEq]
     erw [Set.mem_setOf_eq]
     simp only [Finset.mem_val]
     constructor
diff --git a/Mathlib/Combinatorics/Enumerative/DoubleCounting.lean b/Mathlib/Combinatorics/Enumerative/DoubleCounting.lean
index 542deff1911e1..0264391a00385 100644
--- a/Mathlib/Combinatorics/Enumerative/DoubleCounting.lean
+++ b/Mathlib/Combinatorics/Enumerative/DoubleCounting.lean
@@ -39,7 +39,7 @@ namespace Finset
 
 section Bipartite
 
-variable (r : α → β → Prop) (s : Finset α) (t : Finset β) (a a' : α) (b b' : β)
+variable (r : α → β → Prop) (s : Finset α) (t : Finset β) (a : α) (b : β)
   [DecidablePred (r a)] [∀ a, Decidable (r a b)] {m n : ℕ}
 
 /-- Elements of `s` which are "below" `b` according to relation `r`. -/
@@ -58,7 +58,7 @@ theorem coe_bipartiteBelow : s.bipartiteBelow r b = ({a ∈ s | r a b} : Set α)
 @[simp, norm_cast]
 theorem coe_bipartiteAbove : t.bipartiteAbove r a = ({b ∈ t | r a b} : Set β) := coe_filter _ _
 
-variable {s t a a' b b'}
+variable {s t a b}
 
 @[simp]
 theorem mem_bipartiteBelow {a : α} : a ∈ s.bipartiteBelow r b ↔ a ∈ s ∧ r a b := mem_filter
@@ -72,7 +72,7 @@ theorem sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow [∀ a b, Decidable (
   exact sum_comm
 
 section OrderedSemiring
-variable [OrderedSemiring R] [DecidablePred (r a)] [∀ a, Decidable (r a b)] {m n : R}
+variable [OrderedSemiring R] {m n : R}
 
 /-- **Double counting** argument.
 
@@ -100,7 +100,7 @@ end OrderedSemiring
 
 section StrictOrderedSemiring
 variable [StrictOrderedSemiring R] (r : α → β → Prop) {s : Finset α} {t : Finset β}
-  (a a' : α) (b b' : β) [DecidablePred (r a)] [∀ a, Decidable (r a b)] {m n : R}
+  (a b) {m n : R}
 
 /-- **Double counting** argument.
 
diff --git a/Mathlib/Combinatorics/Enumerative/DyckWord.lean b/Mathlib/Combinatorics/Enumerative/DyckWord.lean
index 14697ac6ca6e0..220e3d4c4f141 100644
--- a/Mathlib/Combinatorics/Enumerative/DyckWord.lean
+++ b/Mathlib/Combinatorics/Enumerative/DyckWord.lean
@@ -3,8 +3,9 @@ Copyright (c) 2024 Jeremy Tan. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Tan
 -/
+import Batteries.Data.List.Count
 import Mathlib.Combinatorics.Enumerative.Catalan
-import Mathlib.Data.List.Indexes
+import Mathlib.Tactic.Positivity
 
 /-!
 # Dyck words
@@ -116,7 +117,7 @@ lemma head_eq_U (p : DyckWord) (h) : p.toList.head h = U := by
 lemma getLast_eq_D (p : DyckWord) (h) : p.toList.getLast h = D := by
   by_contra f; have s := p.count_U_eq_count_D
   rw [← dropLast_append_getLast h, (dichotomy _).resolve_right f] at s
-  simp_rw [dropLast_eq_take, count_append, count_singleton', ite_true, ite_false] at s
+  simp_rw [dropLast_eq_take, count_append, count_singleton', ite_true, reduceCtorEq, ite_false] at s
   have := p.count_D_le_count_U (p.toList.length - 1); omega
 
 include h in
@@ -165,7 +166,7 @@ def nest : DyckWord where
     apply add_le_add _ (p.count_D_le_count_U _)
     rcases i.eq_zero_or_pos with hi | hi; · simp [hi]
     rw [take_of_length_le (show [U].length ≤ i by rwa [length_singleton]), count_singleton']
-    simp only [ite_true, ite_false]
+    simp only [reduceCtorEq, ite_true, ite_false]
     rw [add_comm]
     exact add_le_add (zero_le _) ((count_le_length _ _).trans (by simp))
 
@@ -258,11 +259,18 @@ include h in
 lemma firstReturn_pos : 0 < p.firstReturn := by
   by_contra! f
   rw [Nat.le_zero, firstReturn, findIdx_eq] at f
-  · simp only [get_eq_getElem, getElem_range] at f
-    rw [← p.cons_tail_dropLast_concat h] at f
-    simp at f
+  #adaptation_note
+  /--
+  If we don't swap, then the second goal is dropped after completing the first goal.
+  What's going on?
+  -/
+  swap
   · rw [length_range, length_pos]
     exact toList_ne_nil.mpr h
+  · rw [getElem_range] at f
+    simp at f
+    rw [← p.cons_tail_dropLast_concat h] at f
+    simp at f
 
 include h in
 lemma firstReturn_lt_length : p.firstReturn < p.toList.length := by
@@ -277,13 +285,13 @@ lemma firstReturn_lt_length : p.firstReturn < p.toList.length := by
 include h in
 lemma count_take_firstReturn_add_one :
     (p.toList.take (p.firstReturn + 1)).count U = (p.toList.take (p.firstReturn + 1)).count D := by
-  have := findIdx_get (w := (length_range p.toList.length).symm ▸ firstReturn_lt_length h)
+  have := findIdx_getElem (w := (length_range p.toList.length).symm ▸ firstReturn_lt_length h)
   simpa using this
 
 lemma count_D_lt_count_U_of_lt_firstReturn {i : ℕ} (hi : i < p.firstReturn) :
     (p.toList.take (i + 1)).count D < (p.toList.take (i + 1)).count U := by
   have ne := not_of_lt_findIdx hi
-  rw [decide_eq_true_eq, ← ne_eq, get_eq_getElem, getElem_range] at ne
+  rw [decide_eq_false_iff_not, ← ne_eq, getElem_range] at ne
   exact lt_of_le_of_ne (p.count_D_le_count_U (i + 1)) ne.symm
 
 @[simp]
@@ -291,7 +299,7 @@ lemma firstReturn_add : (p + q).firstReturn = if p = 0 then q.firstReturn else p
   split_ifs with h; · simp [h]
   have u : (p + q).toList = p.toList ++ q.toList := rfl
   rw [firstReturn, findIdx_eq]
-  · simp_rw [get_eq_getElem, getElem_range, u, decide_eq_true_eq]
+  · simp_rw [u, decide_eq_true_eq, getElem_range]
     have v := firstReturn_lt_length h
     constructor
     · rw [take_append_eq_append_take, show p.firstReturn + 1 - p.toList.length = 0 by omega,
@@ -299,7 +307,7 @@ lemma firstReturn_add : (p + q).firstReturn = if p = 0 then q.firstReturn else p
     · intro j hj
       rw [take_append_eq_append_take, show j + 1 - p.toList.length = 0 by omega,
         take_zero, append_nil]
-      exact (count_D_lt_count_U_of_lt_firstReturn hj).ne'
+      simpa using (count_D_lt_count_U_of_lt_firstReturn hj).ne'
   · rw [length_range, u, length_append]
     exact Nat.lt_add_right _ (firstReturn_lt_length h)
 
@@ -307,14 +315,15 @@ lemma firstReturn_add : (p + q).firstReturn = if p = 0 then q.firstReturn else p
 lemma firstReturn_nest : p.nest.firstReturn = p.toList.length + 1 := by
   have u : p.nest.toList = U :: p.toList ++ [D] := rfl
   rw [firstReturn, findIdx_eq]
-  · simp_rw [get_eq_getElem, getElem_range, u, decide_eq_true_eq]
+  · simp_rw [u, decide_eq_true_eq, getElem_range]
     constructor
     · rw [take_of_length_le (by simp), ← u, p.nest.count_U_eq_count_D]
     · intro j hj
-      simp_rw [cons_append, take_cons, count_cons, beq_self_eq_true, ite_true,
-        beq_iff_eq, ite_false, take_append_eq_append_take,
+      simp_rw [cons_append, take_succ_cons, count_cons, beq_self_eq_true, ite_true,
+        beq_iff_eq, reduceCtorEq, ite_false, take_append_eq_append_take,
         show j - p.toList.length = 0 by omega, take_zero, append_nil]
       have := p.count_D_le_count_U j
+      simp only [add_zero, decide_eq_false_iff_not, ne_eq]
       omega
   · simp_rw [length_range, u, length_append, length_cons]
     exact Nat.lt_add_one _
diff --git a/Mathlib/Combinatorics/Enumerative/Partition.lean b/Mathlib/Combinatorics/Enumerative/Partition.lean
index d88c1fba8060d..94bcb8771017a 100644
--- a/Mathlib/Combinatorics/Enumerative/Partition.lean
+++ b/Mathlib/Combinatorics/Enumerative/Partition.lean
@@ -50,7 +50,7 @@ namespace Nat
 structure Partition (n : ℕ) where
   /-- positive integers summing to `n`-/
   parts : Multiset ℕ
-  /-- proof that the `parts` are positive-/
+  /-- proof that the `parts` are positive -/
   parts_pos : ∀ {i}, i ∈ parts → 0 < i
   /-- proof that the `parts` sum to `n`-/
   parts_sum : parts.sum = n
diff --git a/Mathlib/Combinatorics/HalesJewett.lean b/Mathlib/Combinatorics/HalesJewett.lean
index babe5c7b82742..f02b849012141 100644
--- a/Mathlib/Combinatorics/HalesJewett.lean
+++ b/Mathlib/Combinatorics/HalesJewett.lean
@@ -4,12 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Wärn
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Data.Countable.Small
 import Mathlib.Data.Fintype.Option
-import Mathlib.Data.Fintype.Pi
-import Mathlib.Data.Fintype.Prod
 import Mathlib.Data.Fintype.Shrink
-import Mathlib.Data.Fintype.Sum
 
 /-!
 # The Hales-Jewett theorem
@@ -154,7 +150,7 @@ def reindex (l : Subspace η α ι) (eη : η ≃ η') (eα : α ≃ α') (eι :
 
 protected lemma IsMono.reindex {eη : η ≃ η'} {eα : α ≃ α'} {eι : ι ≃ ι'} {C : (ι → α) → κ}
     (hl : l.IsMono C) : (l.reindex eη eα eι).IsMono fun x ↦ C <| eα.symm ∘ x ∘ eι := by
-  simp [reindex_isMono, Function.comp.assoc]; simpa [← Function.comp.assoc]
+  simp [reindex_isMono, Function.comp_assoc]; simpa [← Function.comp_assoc]
 
 end Subspace
 
diff --git a/Mathlib/Combinatorics/Hall/Basic.lean b/Mathlib/Combinatorics/Hall/Basic.lean
index 180890300f1c4..52a8937ef3082 100644
--- a/Mathlib/Combinatorics/Hall/Basic.lean
+++ b/Mathlib/Combinatorics/Hall/Basic.lean
@@ -57,7 +57,7 @@ universe u v
 
 /-- The set of matchings for `t` when restricted to a `Finset` of `ι`. -/
 def hallMatchingsOn {ι : Type u} {α : Type v} (t : ι → Finset α) (ι' : Finset ι) :=
-  { f : ι' → α | Function.Injective f ∧ ∀ x, f x ∈ t x }
+  { f : ι' → α | Function.Injective f ∧ ∀ (x : {x // x ∈ ι'}), f x ∈ t x }
 
 /-- Given a matching on a finset, construct the restriction of that matching to a subset. -/
 def hallMatchingsOn.restrict {ι : Type u} {α : Type v} (t : ι → Finset α) {ι' ι'' : Finset ι}
diff --git a/Mathlib/Combinatorics/Hindman.lean b/Mathlib/Combinatorics/Hindman.lean
index 0a41f5051974a..18555bba2e080 100644
--- a/Mathlib/Combinatorics/Hindman.lean
+++ b/Mathlib/Combinatorics/Hindman.lean
@@ -3,9 +3,10 @@ Copyright (c) 2021 David Wärn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Wärn
 -/
-import Mathlib.Topology.StoneCech
-import Mathlib.Topology.Algebra.Semigroup
+import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Data.Stream.Init
+import Mathlib.Topology.Algebra.Semigroup
+import Mathlib.Topology.StoneCech
 
 /-!
 # Hindman's theorem on finite sums
@@ -223,7 +224,7 @@ theorem FP.mul_two {M} [Semigroup M] (a : Stream' M) (i j : ℕ) (ij : i < j) :
   refine FP_drop_subset_FP _ i ?_
   rw [← Stream'.head_drop]
   apply FP.cons
-  rcases le_iff_exists_add.mp (Nat.succ_le_of_lt ij) with ⟨d, hd⟩
+  rcases Nat.exists_eq_add_of_le (Nat.succ_le_of_lt ij) with ⟨d, hd⟩
   -- Porting note: need to fix breakage of Set notation
   change _ ∈ FP _
   have := FP.singleton (a.drop i).tail d
@@ -245,7 +246,7 @@ theorem FP.finset_prod {M} [CommMonoid M] (a : Stream' M) (s : Finset ℕ) (hs :
     refine Set.mem_of_subset_of_mem ?_ (ih _ (Finset.erase_ssubset <| s.min'_mem hs) h)
     have : s.min' hs + 1 ≤ (s.erase (s.min' hs)).min' h :=
       Nat.succ_le_of_lt (Finset.min'_lt_of_mem_erase_min' _ _ <| Finset.min'_mem _ _)
-    cases' le_iff_exists_add.mp this with d hd
+    cases' Nat.exists_eq_add_of_le this with d hd
     rw [hd, add_comm, ← Stream'.drop_drop]
     apply FP_drop_subset_FP
 
diff --git a/Mathlib/Combinatorics/Optimization/ValuedCSP.lean b/Mathlib/Combinatorics/Optimization/ValuedCSP.lean
index eee23c4281a09..5063e0e98009e 100644
--- a/Mathlib/Combinatorics/Optimization/ValuedCSP.lean
+++ b/Mathlib/Combinatorics/Optimization/ValuedCSP.lean
@@ -134,7 +134,7 @@ lemma Function.HasMaxCutPropertyAt.rows_lt_aux
     {r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :
     f ![a, b] < f r := by
   rw [FractionalOperation.tt, Multiset.mem_map] at rin
-  rw [show r = ![r 0, r 1] from List.ofFn_inj.mp rfl]
+  rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]
   apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left
   intro equ
   have asymm : r 0 ≠ r 1 := by
@@ -146,7 +146,7 @@ lemma Function.HasMaxCutPropertyAt.rows_lt_aux
   apply asymm
   obtain ⟨o, in_omega, rfl⟩ := rin
   show o (fun j => ![![a, b], ![b, a]] j 0) = o (fun j => ![![a, b], ![b, a]] j 1)
-  convert symmega ![a, b] ![b, a] (List.Perm.swap b a []) o in_omega using 2 <;>
+  convert symmega ![a, b] ![b, a] (by simp [List.Perm.swap]) o in_omega using 2 <;>
     simp [Matrix.const_fin1_eq]
 
 lemma Function.HasMaxCutProperty.forbids_commutativeFractionalPolymorphism
diff --git a/Mathlib/Combinatorics/Quiver/Basic.lean b/Mathlib/Combinatorics/Quiver/Basic.lean
index 0000d08b93bfe..f0f77f70aa4b7 100644
--- a/Mathlib/Combinatorics/Quiver/Basic.lean
+++ b/Mathlib/Combinatorics/Quiver/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 David Wärn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: David Wärn, Scott Morrison
+Authors: David Wärn, Kim Morrison
 -/
 import Mathlib.Data.Opposite
 
diff --git a/Mathlib/Combinatorics/Quiver/Path.lean b/Mathlib/Combinatorics/Quiver/Path.lean
index 5466770a184d0..d87402e7fb551 100644
--- a/Mathlib/Combinatorics/Quiver/Path.lean
+++ b/Mathlib/Combinatorics/Quiver/Path.lean
@@ -1,10 +1,11 @@
 /-
 Copyright (c) 2021 David Wärn,. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: David Wärn, Scott Morrison
+Authors: David Wärn, Kim Morrison
 -/
 import Mathlib.Combinatorics.Quiver.Basic
 import Mathlib.Logic.Lemmas
+import Batteries.Data.List.Basic
 
 /-!
 # Paths in quivers
diff --git a/Mathlib/Combinatorics/Quiver/ReflQuiver.lean b/Mathlib/Combinatorics/Quiver/ReflQuiver.lean
new file mode 100644
index 0000000000000..b0f9c85ac7250
--- /dev/null
+++ b/Mathlib/Combinatorics/Quiver/ReflQuiver.lean
@@ -0,0 +1,131 @@
+/-
+Copyright (c) 2024 Mario Carneiro and Emily Riehl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro, Emily Riehl
+-/
+import Mathlib.Data.Set.Function
+import Mathlib.CategoryTheory.Category.Cat
+
+/-!
+# Reflexive Quivers
+
+This module defines reflexive quivers. A reflexive quiver, or "refl quiver" for short, extends
+a quiver with a specified endoarrow on each term in its type of objects.
+
+We also introduce morphisms between reflexive quivers, called reflexive prefunctors or "refl
+prefunctors" for short.
+
+Note: Currently Category does not extend ReflQuiver, although it could. (TODO: do this)
+-/
+namespace CategoryTheory
+universe v v₁ v₂ u u₁ u₂
+
+/-- A reflexive quiver extends a quiver with a specified arrow `id X : X ⟶ X` for each `X` in its
+type of objects. We denote these arrows by `id` since categories can be understood as an extension
+of refl quivers.
+-/
+class ReflQuiver (obj : Type u) extends Quiver.{v} obj : Type max u v where
+  /-- The identity morphism on an object. -/
+  id : ∀ X : obj, Hom X X
+
+/-- Notation for the identity morphism in a category. -/
+scoped notation "𝟙rq" => ReflQuiver.id  -- type as \b1
+
+instance catToReflQuiver {C : Type u} [inst : Category.{v} C] : ReflQuiver.{v+1, u} C :=
+  { inst with }
+
+@[simp] theorem ReflQuiver.id_eq_id {C : Type*} [Category C] (X : C) : 𝟙rq X = 𝟙 X := rfl
+
+/-- A morphism of reflexive quivers called a `ReflPrefunctor`. -/
+structure ReflPrefunctor (V : Type u₁) [ReflQuiver.{v₁} V] (W : Type u₂) [ReflQuiver.{v₂} W]
+    extends Prefunctor V W where
+  /-- A functor preserves identity morphisms. -/
+  map_id : ∀ X : V, map (𝟙rq X) = 𝟙rq (obj X) := by aesop_cat
+
+namespace ReflPrefunctor
+
+-- These lemmas can not be `@[simp]` because after `whnfR` they have a variable on the LHS.
+-- Nevertheless they are sometimes useful when building functors.
+lemma mk_obj {V W : Type*} [ReflQuiver V] [ReflQuiver W] {obj : V → W} {map} {X : V} :
+    (Prefunctor.mk obj map).obj X = obj X := rfl
+
+lemma mk_map {V W : Type*} [ReflQuiver V] [ReflQuiver W] {obj : V → W} {map} {X Y : V} {f : X ⟶ Y} :
+    (Prefunctor.mk obj map).map f = map f := rfl
+
+/-- Proving equality between reflexive prefunctors. This isn't an extensionality lemma,
+  because usually you don't really want to do this. -/
+theorem ext {V : Type u} [ReflQuiver.{v₁} V] {W : Type u₂} [ReflQuiver.{v₂} W]
+    {F G : ReflPrefunctor V W}
+    (h_obj : ∀ X, F.obj X = G.obj X)
+    (h_map : ∀ (X Y : V) (f : X ⟶ Y),
+      F.map f = Eq.recOn (h_obj Y).symm (Eq.recOn (h_obj X).symm (G.map f))) : F = G := by
+  obtain ⟨⟨F_obj⟩⟩ := F
+  obtain ⟨⟨G_obj⟩⟩ := G
+  obtain rfl : F_obj = G_obj := (Set.eqOn_univ F_obj G_obj).mp fun _ _ ↦ h_obj _
+  congr
+  funext X Y f
+  simpa using h_map X Y f
+
+/-- The identity morphism between reflexive quivers. -/
+@[simps!]
+def id (V : Type*) [ReflQuiver V] : ReflPrefunctor V V where
+  __ := Prefunctor.id _
+  map_id _ := rfl
+
+instance (V : Type*) [ReflQuiver V] : Inhabited (ReflPrefunctor V V) :=
+  ⟨id V⟩
+
+/-- Composition of morphisms between reflexive quivers. -/
+@[simps!]
+def comp {U : Type*} [ReflQuiver U] {V : Type*} [ReflQuiver V] {W : Type*} [ReflQuiver W]
+    (F : ReflPrefunctor U V) (G : ReflPrefunctor V W) : ReflPrefunctor U W where
+  __ := F.toPrefunctor.comp G.toPrefunctor
+  map_id _ := by simp [F.map_id, G.map_id]
+
+@[simp]
+theorem comp_id {U V : Type*} [ReflQuiver U] [ReflQuiver V] (F : ReflPrefunctor U V) :
+    F.comp (id _) = F := rfl
+
+@[simp]
+theorem id_comp {U V : Type*} [ReflQuiver U] [ReflQuiver V] (F : ReflPrefunctor U V) :
+    (id _).comp F = F := rfl
+
+@[simp]
+theorem comp_assoc {U V W Z : Type*} [ReflQuiver U] [ReflQuiver V] [ReflQuiver W] [ReflQuiver Z]
+    (F : ReflPrefunctor U V) (G : ReflPrefunctor V W) (H : ReflPrefunctor W Z) :
+    (F.comp G).comp H = F.comp (G.comp H) := rfl
+
+/-- Notation for a prefunctor between reflexive quivers. -/
+infixl:50 " ⥤rq " => ReflPrefunctor
+
+/-- Notation for composition of reflexive prefunctors. -/
+infixl:60 " ⋙rq " => ReflPrefunctor.comp
+
+/-- Notation for the identity prefunctor on a reflexive quiver. -/
+notation "𝟭rq" => id
+
+theorem congr_map {U V : Type*} [Quiver U] [Quiver V] (F : U ⥤q V) {X Y : U} {f g : X ⟶ Y}
+    (h : f = g) : F.map f = F.map g := congrArg F.map h
+
+end ReflPrefunctor
+
+/-- A functor has an underlying refl prefunctor.-/
+def Functor.toReflPrefunctor {C D} [Category C] [Category D] (F : C ⥤ D) : C ⥤rq D := { F with }
+
+@[simp]
+theorem Functor.toReflPrefunctor_toPrefunctor {C D : Cat} (F : C ⥤ D) :
+    (Functor.toReflPrefunctor F).toPrefunctor = F.toPrefunctor := rfl
+
+namespace ReflQuiver
+open Opposite
+
+/-- `Vᵒᵖ` reverses the direction of all arrows of `V`. -/
+instance opposite {V} [ReflQuiver V] : ReflQuiver Vᵒᵖ where
+   id X := op (𝟙rq X.unop)
+
+instance discreteReflQuiver (V : Type u) : ReflQuiver.{u+1} (Discrete V) :=
+  { discreteCategory V with }
+
+end ReflQuiver
+
+end CategoryTheory
diff --git a/Mathlib/Combinatorics/Quiver/Symmetric.lean b/Mathlib/Combinatorics/Quiver/Symmetric.lean
index f4fc13e6cfaa8..cf87c843a83f7 100644
--- a/Mathlib/Combinatorics/Quiver/Symmetric.lean
+++ b/Mathlib/Combinatorics/Quiver/Symmetric.lean
@@ -45,7 +45,7 @@ class HasReverse where
 def reverse {V} [Quiver.{v + 1} V] [HasReverse V] {a b : V} : (a ⟶ b) → (b ⟶ a) :=
   HasReverse.reverse'
 
-/-- A quiver `HasInvolutiveReverse` if reversing twice is the identity. -/
+/-- A quiver `HasInvolutiveReverse` if reversing twice is the identity. -/
 class HasInvolutiveReverse extends HasReverse V where
   /-- `reverse` is involutive -/
   inv' : ∀ {a b : V} (f : a ⟶ b), reverse (reverse f) = f
diff --git a/Mathlib/Combinatorics/Schnirelmann.lean b/Mathlib/Combinatorics/Schnirelmann.lean
index 71afde26c06ca..266093dff23ab 100644
--- a/Mathlib/Combinatorics/Schnirelmann.lean
+++ b/Mathlib/Combinatorics/Schnirelmann.lean
@@ -196,7 +196,7 @@ lemma schnirelmannDensity_finset (A : Finset ℕ) : schnirelmannDensity A = 0 :=
   let n : ℕ := ⌊A.card / ε⌋₊ + 1
   have hn : 0 < n := Nat.succ_pos _
   use n, hn
-  rw [div_lt_iff (Nat.cast_pos.2 hn), ← div_lt_iff' hε, Nat.cast_add_one]
+  rw [div_lt_iff₀ (Nat.cast_pos.2 hn), ← div_lt_iff₀' hε, Nat.cast_add_one]
   exact (Nat.lt_floor_add_one _).trans_le' <| by gcongr; simp [subset_iff]
 
 /-- The Schnirelmann density of any finite set is `0`. -/
@@ -259,7 +259,7 @@ lemma schnirelmannDensity_setOf_modeq_one {m : ℕ} :
   rw [← schnirelmannDensity_setOf_mod_eq_one hm]
   apply schnirelmannDensity_congr
   ext n
-  simp only [Set.mem_setOf_eq, Nat.ModEq, Nat.one_mod_of_ne_one hm]
+  simp only [Set.mem_setOf_eq, Nat.ModEq, Nat.one_mod_eq_one.mpr hm]
 
 lemma schnirelmannDensity_setOf_Odd : schnirelmannDensity (setOf Odd) = 2⁻¹ := by
   have h : setOf Odd = {n | n % 2 = 1} := Set.ext fun _ => Nat.odd_iff
diff --git a/Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean b/Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
index e27ea7862f429..ea08e4e2baeef 100644
--- a/Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
+++ b/Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
@@ -141,7 +141,7 @@ lemma map_truncatedSup [@DecidableRel β (· ≤ ·)] (e : α ≃o β) (s : Fins
   have : e a ∈ lowerClosure (s.map e.toEquiv.toEmbedding : Set β) ↔ a ∈ lowerClosure s := by simp
   simp_rw [truncatedSup, apply_dite e, map_finset_sup', map_top, this]
   congr with h
-  simp only [filter_map, Function.comp, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
+  simp only [filter_map, Function.comp_def, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
     OrderIso.le_iff_le, id]
   rw [sup'_map]
   -- TODO: Why can't `simp` use `Finset.sup'_map`?
@@ -216,7 +216,7 @@ lemma map_truncatedInf (e : α ≃o β) (s : Finset α) (a : α) :
   have : e a ∈ upperClosure (s.map e.toEquiv.toEmbedding) ↔ a ∈ upperClosure s := by simp
   simp_rw [truncatedInf, apply_dite e, map_finset_inf', map_bot, this]
   congr with h
-  simp only [filter_map, Function.comp, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
+  simp only [filter_map, Function.comp_def, Equiv.coe_toEmbedding, RelIso.coe_fn_toEquiv,
     OrderIso.le_iff_le, id, inf'_map]
 
 lemma truncatedInf_of_isAntichain (hs : IsAntichain (· ≤ ·) (s : Set α)) (ha : a ∈ s) :
diff --git a/Mathlib/Combinatorics/SetFamily/Compression/UV.lean b/Mathlib/Combinatorics/SetFamily/Compression/UV.lean
index 5ebbd630b7265..8bfa4e268eb8a 100644
--- a/Mathlib/Combinatorics/SetFamily/Compression/UV.lean
+++ b/Mathlib/Combinatorics/SetFamily/Compression/UV.lean
@@ -327,7 +327,7 @@ theorem shadow_compression_subset_compression_shadow (u v : Finset α)
         exact ⟨Or.inl hat, hav⟩
       · rw [← erase_sdiff_comm, sup_eq_union, erase_union_distrib, erase_eq_of_not_mem hau]
   intro s hs𝒜' hs𝒜
-  -- This is gonna be useful a couple of times so let's name it.
+  -- This is going to be useful a couple of times so let's name it.
   have m : ∀ y, y ∉ s → insert y s ∉ 𝒜 := fun y h a => hs𝒜 (mem_shadow_iff_insert_mem.2 ⟨y, h, a⟩)
   obtain ⟨x, _, _⟩ := mem_shadow_iff_insert_mem.1 hs𝒜'
   have hus : u ⊆ insert x s := le_of_mem_compression_of_not_mem ‹_ ∈ 𝒜'› (m _ ‹x ∉ s›)
diff --git a/Mathlib/Combinatorics/SetFamily/FourFunctions.lean b/Mathlib/Combinatorics/SetFamily/FourFunctions.lean
index 2c7b78ec4d1b5..d3bd37943cd19 100644
--- a/Mathlib/Combinatorics/SetFamily/FourFunctions.lean
+++ b/Mathlib/Combinatorics/SetFamily/FourFunctions.lean
@@ -219,7 +219,7 @@ lemma sum_collapse (h𝒜 : 𝒜 ⊆ (insert a u).powerset) (hu : a ∉ u) :
     _ = ∑ s ∈ u.powerset ∩ 𝒜, f s + ∑ s ∈ u.powerset.image (insert a) ∩ 𝒜, f s := ?_
     _ = ∑ s ∈ u.powerset ∩ 𝒜, f s + ∑ s ∈ ((insert a u).powerset \ u.powerset) ∩ 𝒜, f s := ?_
     _ = ∑ s ∈ 𝒜, f s := ?_
-  · rw [← sum_ite_mem, ← sum_ite_mem, sum_image, ← sum_add_distrib]
+  · rw [← Finset.sum_ite_mem, ← Finset.sum_ite_mem, sum_image, ← sum_add_distrib]
     · exact sum_congr rfl fun s hs ↦ collapse_eq (not_mem_mono (mem_powerset.1 hs) hu) _ _
     · exact (insert_erase_invOn.2.injOn).mono fun s hs ↦ not_mem_mono (mem_powerset.1 hs) hu
   · congr with s
diff --git a/Mathlib/Combinatorics/SetFamily/Kleitman.lean b/Mathlib/Combinatorics/SetFamily/Kleitman.lean
index 7fe6c323c2871..e534f30f51b45 100644
--- a/Mathlib/Combinatorics/SetFamily/Kleitman.lean
+++ b/Mathlib/Combinatorics/SetFamily/Kleitman.lean
@@ -79,5 +79,5 @@ theorem Finset.card_biUnion_le_of_intersecting (s : Finset ι) (f : ι → Finse
     (ih _ (fun i hi ↦ (hf₁ _ <| subset_cons _ hi).2.2)
     ((card_le_card <| subset_cons _).trans hs)) _).trans ?_
   rw [mul_tsub, two_mul, ← pow_succ',
-    ← add_tsub_assoc_of_le (pow_le_pow_right' (one_le_two : (1 : ℕ) ≤ 2) tsub_le_self),
+    ← add_tsub_assoc_of_le (pow_right_mono₀ (one_le_two : (1 : ℕ) ≤ 2) tsub_le_self),
     tsub_add_eq_add_tsub hs, card_cons, add_tsub_add_eq_tsub_right]
diff --git a/Mathlib/Combinatorics/SetFamily/KruskalKatona.lean b/Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
index c038ae31aa350..8b5ce0135fec7 100644
--- a/Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
+++ b/Mathlib/Combinatorics/SetFamily/KruskalKatona.lean
@@ -86,7 +86,7 @@ lemma shadow_initSeg [Fintype α] (hs : s.Nonempty) :
   -- if j < k, k is our colex witness for t ∪ {j} < s
   · refine Or.inr ⟨k, mem_of_mem_erase ‹_›, fun hk ↦ hkt <| mem_of_mem_insert_of_ne hk hjk.ne',
       fun x hx ↦ ?_⟩
-    simpa only [mem_insert, z hx, (hjk.trans hx).ne', mem_erase, Ne, false_or_iff,
+    simpa only [mem_insert, z hx, (hjk.trans hx).ne', mem_erase, Ne, false_or,
       and_iff_right_iff_imp] using fun _ ↦ ((min'_le _ _ <| mem_of_mem_erase hks).trans_lt hx).ne'
   -- if j = k, all of range k is in t so by sizes t ∪ {j} = s
   refine Or.inl (eq_of_subset_of_card_le (fun a ha ↦ ?_) hcard.ge).symm
@@ -142,7 +142,7 @@ private def UsefulCompression (U V : Finset α) : Prop :=
   Disjoint U V ∧ U.card = V.card ∧ ∃ (HU : U.Nonempty) (HV : V.Nonempty), max' U HU < max' V HV
 
 private instance UsefulCompression.instDecidableRel : @DecidableRel (Finset α) UsefulCompression :=
-  fun _U _V ↦ And.decidable
+  fun _ _ ↦ inferInstanceAs (Decidable (_ ∧ _))
 
 /-- Applying a good compression will decrease measure, keep cardinality, keep sizes and decrease
 shadow. In particular, 'good' means it's useful, and every smaller compression won't make a
diff --git a/Mathlib/Combinatorics/SetFamily/Shadow.lean b/Mathlib/Combinatorics/SetFamily/Shadow.lean
index bc4f6f690c062..a3b293b819c4c 100644
--- a/Mathlib/Combinatorics/SetFamily/Shadow.lean
+++ b/Mathlib/Combinatorics/SetFamily/Shadow.lean
@@ -75,7 +75,10 @@ theorem shadow_empty : ∂ (∅ : Finset (Finset α)) = ∅ :=
 theorem shadow_singleton_empty : ∂ ({∅} : Finset (Finset α)) = ∅ :=
   rfl
 
---TODO: Prove `∂ {{a}} = {∅}` quickly using `covers` and `GradeOrder`
+@[simp]
+theorem shadow_singleton (a : α) : ∂ {{a}} = {∅} := by
+  simp [shadow]
+
 /-- The shadow is monotone. -/
 @[mono]
 theorem shadow_monotone : Monotone (shadow : Finset (Finset α) → Finset (Finset α)) := fun _ _ =>
diff --git a/Mathlib/Combinatorics/SetFamily/Shatter.lean b/Mathlib/Combinatorics/SetFamily/Shatter.lean
index 6efd8f20ec42e..74f21f70fbc02 100644
--- a/Mathlib/Combinatorics/SetFamily/Shatter.lean
+++ b/Mathlib/Combinatorics/SetFamily/Shatter.lean
@@ -79,7 +79,7 @@ def shatterer (𝒜 : Finset (Finset α)) : Finset (Finset α) := (𝒜.biUnion
   simp_rw [mem_biUnion, mem_powerset]
   exact h.exists_superset
 
-lemma shatterer_mono (h : 𝒜 ⊆ ℬ) : 𝒜.shatterer ⊆ ℬ.shatterer :=
+@[gcongr] lemma shatterer_mono (h : 𝒜 ⊆ ℬ) : 𝒜.shatterer ⊆ ℬ.shatterer :=
   fun _ ↦ by simpa using Shatters.mono_left h
 
 lemma subset_shatterer (h : IsLowerSet (𝒜 : Set (Finset α))) : 𝒜 ⊆ 𝒜.shatterer :=
@@ -181,6 +181,8 @@ lemma shatterer_compress_subset_shatterer (a : α) (𝒜 : Finset (Finset α)) :
 /-- The Vapnik-Chervonenkis dimension of a set family is the maximal size of a set it shatters. -/
 def vcDim (𝒜 : Finset (Finset α)) : ℕ := 𝒜.shatterer.sup card
 
+@[gcongr] lemma vcDim_mono (h𝒜ℬ : 𝒜 ⊆ ℬ) : 𝒜.vcDim ≤ ℬ.vcDim := by unfold vcDim; gcongr
+
 lemma Shatters.card_le_vcDim (hs : 𝒜.Shatters s) : s.card ≤ 𝒜.vcDim := le_sup <| mem_shatterer.2 hs
 
 /-- Down-compressing decreases the VC-dimension. -/
diff --git a/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean b/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
index 6b8117d6925da..9782814a525cc 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
@@ -14,8 +14,8 @@ This module introduces *acyclic graphs* (a.k.a. *forests*) and *trees*.
 
 ## Main definitions
 
-* `SimpleGraph.IsAcyclic` is a predicate for a graph having no cyclic walks
-* `SimpleGraph.IsTree` is a predicate for a graph being a tree (a connected acyclic graph)
+* `SimpleGraph.IsAcyclic` is a predicate for a graph having no cyclic walks.
+* `SimpleGraph.IsTree` is a predicate for a graph being a tree (a connected acyclic graph).
 
 ## Main statements
 
@@ -114,7 +114,7 @@ theorem isAcyclic_of_path_unique (h : ∀ (v w : V) (p q : G.Path v w), p = q) :
   cases c with
   | nil => cases hc.2.1 rfl
   | cons ha c' =>
-    simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons, true_and_iff] at hc
+    simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons] at hc
     specialize h _ _ ⟨c', by simp only [Walk.isPath_def, hc.2]⟩ (Path.singleton ha.symm)
     rw [Path.singleton, Subtype.mk.injEq] at h
     simp [h] at hc
@@ -132,7 +132,7 @@ theorem isTree_iff_existsUnique_path :
     intro v w
     let q := (hc v w).some.toPath
     use q
-    simp only [true_and_iff, Path.isPath]
+    simp only [true_and, Path.isPath]
     intro p hp
     specialize hu ⟨p, hp⟩ q
     exact Subtype.ext_iff.mp hu
@@ -170,11 +170,12 @@ lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
     · exact (congrArg (·.fst) h)
     · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
         congrArg (·.snd) h
-      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
-      · rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
+      have h3 := congrArg length (hf' _ ((f _).tail.copy h1 rfl) ?_)
+      · rw [length_copy, ← add_left_inj 1,
+          length_tail_add_one (not_nil_of_ne (by simpa using ha))] at h3
         omega
       · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
-        exact (hf _).tail _
+        exact (hf _).tail (not_nil_of_ne (by simpa using ha))
   case surj =>
     simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
     intros x y h
@@ -188,7 +189,7 @@ lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
           length_cons, length_nil] at h'
       simp [Nat.le_zero, Nat.one_ne_zero] at h'
     rw [← hf' _ (.cons h.symm (f x)) ((cons_isPath_iff _ _).2 ⟨hf _, fun hy => ?contra⟩)]
-    · rfl
+    · simp only [firstDart_toProd, getVert_cons_succ, getVert_zero, Prod.swap_prod_mk]
     case contra =>
       suffices (f x).takeUntil y hy = .cons h .nil by
         rw [← take_spec _ hy] at h'
diff --git a/Mathlib/Combinatorics/SimpleGraph/Basic.lean b/Mathlib/Combinatorics/SimpleGraph/Basic.lean
index 6aaf4e25f4463..fd971bdfc268e 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Basic.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Basic.lean
@@ -15,13 +15,13 @@ This module defines simple graphs on a vertex type `V` as an irreflexive symmetr
 
 ## Main definitions
 
-* `SimpleGraph` is a structure for symmetric, irreflexive relations
+* `SimpleGraph` is a structure for symmetric, irreflexive relations.
 
-* `SimpleGraph.neighborSet` is the `Set` of vertices adjacent to a given vertex
+* `SimpleGraph.neighborSet` is the `Set` of vertices adjacent to a given vertex.
 
-* `SimpleGraph.commonNeighbors` is the intersection of the neighbor sets of two given vertices
+* `SimpleGraph.commonNeighbors` is the intersection of the neighbor sets of two given vertices.
 
-* `SimpleGraph.incidenceSet` is the `Set` of edges containing a given vertex
+* `SimpleGraph.incidenceSet` is the `Set` of edges containing a given vertex.
 
 * `CompleteAtomicBooleanAlgebra` instance: Under the subgraph relation, `SimpleGraph` forms a
   `CompleteAtomicBooleanAlgebra`. In other words, this is the complete lattice of spanning subgraphs
@@ -29,10 +29,10 @@ This module defines simple graphs on a vertex type `V` as an irreflexive symmetr
 
 ## TODO
 
-* This is the simplest notion of an unoriented graph.  This should
-  eventually fit into a more complete combinatorics hierarchy which
-  includes multigraphs and directed graphs.  We begin with simple graphs
-  in order to start learning what the combinatorics hierarchy should
+* This is the simplest notion of an unoriented graph.
+  This should eventually fit into a more complete combinatorics hierarchy which includes
+  multigraphs and directed graphs.
+  We begin with simple graphs in order to start learning what the combinatorics hierarchy should
   look like.
 -/
 
@@ -68,9 +68,8 @@ macro (name := aesop_graph?) "aesop_graph?" c:Aesop.tactic_clause* : tactic =>
       (rule_sets := [$(Lean.mkIdent `SimpleGraph):ident]))
 
 /--
-A variant of `aesop_graph` which does not fail if it is unable to solve the
-goal. Use this only for exploration! Nonterminal Aesop is even worse than
-nonterminal `simp`.
+A variant of `aesop_graph` which does not fail if it is unable to solve the goal.
+Use this only for exploration! Nonterminal Aesop is even worse than nonterminal `simp`.
 -/
 macro (name := aesop_graph_nonterminal) "aesop_graph_nonterminal" c:Aesop.tactic_clause* : tactic =>
   `(tactic|
@@ -123,6 +122,10 @@ instance {V : Type u} [Fintype V] [DecidableEq V] : Fintype (SimpleGraph V) wher
     · ext
       simp
 
+/-- There are finitely many simple graphs on a given finite type. -/
+instance SimpleGraph.instFinite {V : Type u} [Finite V] : Finite (SimpleGraph V) :=
+  .of_injective SimpleGraph.Adj fun _ _ ↦ SimpleGraph.ext
+
 /-- Construct the simple graph induced by the given relation. It
 symmetrizes the relation and makes it irreflexive. -/
 def SimpleGraph.fromRel {V : Type u} (r : V → V → Prop) : SimpleGraph V where
@@ -573,12 +576,12 @@ theorem fromEdgeSet_edgeSet : fromEdgeSet G.edgeSet = G := by
 @[simp]
 theorem fromEdgeSet_empty : fromEdgeSet (∅ : Set (Sym2 V)) = ⊥ := by
   ext v w
-  simp only [fromEdgeSet_adj, Set.mem_empty_iff_false, false_and_iff, bot_adj]
+  simp only [fromEdgeSet_adj, Set.mem_empty_iff_false, false_and, bot_adj]
 
 @[simp]
 theorem fromEdgeSet_univ : fromEdgeSet (Set.univ : Set (Sym2 V)) = ⊤ := by
   ext v w
-  simp only [fromEdgeSet_adj, Set.mem_univ, true_and_iff, top_adj]
+  simp only [fromEdgeSet_adj, Set.mem_univ, true_and, top_adj]
 
 @[simp]
 theorem fromEdgeSet_inter (s t : Set (Sym2 V)) :
@@ -603,7 +606,7 @@ theorem fromEdgeSet_sdiff (s t : Set (Sym2 V)) :
 theorem fromEdgeSet_mono {s t : Set (Sym2 V)} (h : s ⊆ t) : fromEdgeSet s ≤ fromEdgeSet t := by
   rintro v w
   simp (config := { contextual := true }) only [fromEdgeSet_adj, Ne, not_false_iff,
-    and_true_iff, and_imp]
+    and_true, and_imp]
   exact fun vws _ => h vws
 
 @[simp] lemma disjoint_fromEdgeSet : Disjoint G (fromEdgeSet s) ↔ Disjoint G.edgeSet s := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/Circulant.lean b/Mathlib/Combinatorics/SimpleGraph/Circulant.lean
index 1b44d179c5ede..121f720ec7a12 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Circulant.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Circulant.lean
@@ -3,8 +3,8 @@ Copyright (c) 2024 Iván Renison, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Iván Renison, Bhavik Mehta
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Combinatorics.SimpleGraph.Hasse
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Definition of circulant graphs
diff --git a/Mathlib/Combinatorics/SimpleGraph/Clique.lean b/Mathlib/Combinatorics/SimpleGraph/Clique.lean
index 8de6f50777720..4023561674eb6 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Clique.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Clique.lean
@@ -10,8 +10,8 @@ import Mathlib.Data.Finset.Pairwise
 /-!
 # Graph cliques
 
-This file defines cliques in simple graphs. A clique is a set of vertices that are pairwise
-adjacent.
+This file defines cliques in simple graphs.
+A clique is a set of vertices that are pairwise adjacent.
 
 ## Main declarations
 
@@ -270,9 +270,9 @@ theorem not_cliqueFree_of_top_embedding {n : ℕ} (f : (⊤ : SimpleGraph (Fin n
     ¬G.CliqueFree n := by
   simp only [CliqueFree, isNClique_iff, isClique_iff_induce_eq, not_forall, Classical.not_not]
   use Finset.univ.map f.toEmbedding
-  simp only [card_map, Finset.card_fin, eq_self_iff_true, and_true_iff]
+  simp only [card_map, Finset.card_fin, eq_self_iff_true, and_true]
   ext ⟨v, hv⟩ ⟨w, hw⟩
-  simp only [coe_map, Set.mem_image, coe_univ, Set.mem_univ, true_and_iff] at hv hw
+  simp only [coe_map, Set.mem_image, coe_univ, Set.mem_univ, true_and] at hv hw
   obtain ⟨v', rfl⟩ := hv
   obtain ⟨w', rfl⟩ := hw
   simp only [coe_sort_coe, RelEmbedding.coe_toEmbedding, comap_adj, Function.Embedding.coe_subtype,
diff --git a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
index 20c6684e3f809..9d98315f91068 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
@@ -13,11 +13,11 @@ import Mathlib.Data.Nat.Cast.Order.Ring
 /-!
 # Graph Coloring
 
-This module defines colorings of simple graphs (also known as proper
-colorings in the literature). A graph coloring is the attribution of
-"colors" to all of its vertices such that adjacent vertices have
-different colors. A coloring can be represented as a homomorphism into
-a complete graph, whose vertices represent the colors.
+This module defines colorings of simple graphs (also known as proper colorings in the literature).
+A graph coloring is the attribution of "colors" to all of its vertices such that adjacent vertices
+have different colors.
+A coloring can be represented as a homomorphism into a complete graph, whose vertices represent
+the colors.
 
 ## Main definitions
 
@@ -29,14 +29,12 @@ a complete graph, whose vertices represent the colors.
 * `G.Colorable n` is the proposition that `G` is `n`-colorable, which
   is whether there exists a coloring with at most *n* colors.
 
-* `G.chromaticNumber` is the minimal `n` such that `G` is
-  `n`-colorable, or `⊤` if it cannot be colored with finitely many
-  colors.
+* `G.chromaticNumber` is the minimal `n` such that `G` is `n`-colorable,
+  or `⊤` if it cannot be colored with finitely many colors.
   (Cardinal-valued chromatic numbers are more niche, so we stick to `ℕ∞`.)
   We write `G.chromaticNumber ≠ ⊤` to mean a graph is colorable with finitely many colors.
 
-* `C.colorClass c` is the set of vertices colored by `c : α` in the
-  coloring `C : G.Coloring α`.
+* `C.colorClass c` is the set of vertices colored by `c : α` in the coloring `C : G.Coloring α`.
 
 * `C.colorClasses` is the set containing all color classes.
 
@@ -371,7 +369,7 @@ lemma chromaticNumber_eq_iff_forall_surjective (hG : G.Colorable n) :
 
 theorem chromaticNumber_bot [Nonempty V] : (⊥ : SimpleGraph V).chromaticNumber = 1 := by
   have : (⊥ : SimpleGraph V).Colorable 1 := ⟨.mk 0 <| by simp⟩
-  exact this.chromaticNumber_le.antisymm <| ENat.one_le_iff_pos.2 <| chromaticNumber_pos this
+  exact this.chromaticNumber_le.antisymm <| Order.one_le_iff_pos.2 <| chromaticNumber_pos this
 
 @[simp]
 theorem chromaticNumber_top [Fintype V] : (⊤ : SimpleGraph V).chromaticNumber = Fintype.card V := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean b/Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
index 90abd52d115c4..33ece69cf7414 100644
--- a/Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/ConcreteColorings.lean
@@ -68,7 +68,7 @@ theorem Coloring.odd_length_iff_not_congr {α} {G : SimpleGraph α}
 theorem Walk.three_le_chromaticNumber_of_odd_loop {α} {G : SimpleGraph α} {u : α} (p : G.Walk u u)
     (hOdd : Odd p.length) : 3 ≤ G.chromaticNumber := Classical.by_contradiction <| by
   intro h
-  have h' : G.chromaticNumber ≤ 2 := ENat.le_of_lt_add_one <| not_le.mp h
+  have h' : G.chromaticNumber ≤ 2 := Order.le_of_lt_add_one <| not_le.mp h
   let c : G.Coloring (Fin 2) := (chromaticNumber_le_iff_colorable.mp h').some
   let c' : G.Coloring Bool := recolorOfEquiv G finTwoEquiv c
   have : ¬c' u ↔ c' u := (c'.odd_length_iff_not_congr p).mp hOdd
diff --git a/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean b/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
index e73b22eb31d5a..b19fdc540c1bb 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean
@@ -189,9 +189,9 @@ theorem toSubgraph_adj_getVert {u v} (w : G.Walk u v) {i : ℕ} (hi : i < w.leng
   | nil => cases hi
   | cons hxy i' ih =>
     cases i
-    · simp only [Walk.toSubgraph, Walk.getVert_zero, zero_add, cons_getVert_succ, Subgraph.sup_adj,
+    · simp only [Walk.toSubgraph, Walk.getVert_zero, zero_add, getVert_cons_succ, Subgraph.sup_adj,
       subgraphOfAdj_adj, true_or]
-    · simp only [Walk.toSubgraph, cons_getVert_succ, Subgraph.sup_adj, subgraphOfAdj_adj, Sym2.eq,
+    · simp only [Walk.toSubgraph, getVert_cons_succ, Subgraph.sup_adj, subgraphOfAdj_adj, Sym2.eq,
       Sym2.rel_iff', Prod.mk.injEq, Prod.swap_prod_mk]
       right
       exact ih (Nat.succ_lt_succ_iff.mp hi)
@@ -211,7 +211,7 @@ theorem toSubgraph_adj_iff {u v u' v'} (w : G.Walk u v) :
       cases hadj with
       | inl hl =>
         use 0
-        simp only [Walk.getVert_zero, zero_add, cons_getVert_succ]
+        simp only [Walk.getVert_zero, zero_add, getVert_cons_succ]
         refine ⟨?_, by simp only [length_cons, Nat.zero_lt_succ]⟩
         simp only [Sym2.eq, Sym2.rel_iff', Prod.mk.injEq, Prod.swap_prod_mk]
         cases hl with
@@ -220,7 +220,7 @@ theorem toSubgraph_adj_iff {u v u' v'} (w : G.Walk u v) :
       | inr hr =>
         obtain ⟨i, hi⟩ := (toSubgraph_adj_iff _).mp hr
         use i + 1
-        simp only [cons_getVert_succ]
+        simp only [getVert_cons_succ]
         constructor
         · exact hi.1
         · simp only [Walk.length_cons, add_lt_add_iff_right, Nat.add_lt_add_right hi.2 1]
diff --git a/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.lean b/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.lean
index 8a6d763769ed6..51818e2375813 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Connectivity/WalkCounting.lean
@@ -41,7 +41,7 @@ theorem set_walk_self_length_zero_eq (u : V) : {p : G.Walk u u | p.length = 0} =
 theorem set_walk_length_zero_eq_of_ne {u v : V} (h : u ≠ v) :
     {p : G.Walk u v | p.length = 0} = ∅ := by
   ext p
-  simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
+  simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]
   exact fun h' => absurd (Walk.eq_of_length_eq_zero h') h
 
 theorem set_walk_length_succ_eq (u v : V) (n : ℕ) :
@@ -107,12 +107,36 @@ theorem coe_finsetWalkLength_eq (n : ℕ) (u v : V) :
 
 variable {G}
 
-theorem Walk.mem_finsetWalkLength_iff_length_eq {n : ℕ} {u v : V} (p : G.Walk u v) :
+theorem mem_finsetWalkLength_iff {n : ℕ} {u v : V} {p : G.Walk u v} :
     p ∈ G.finsetWalkLength n u v ↔ p.length = n :=
   Set.ext_iff.mp (G.coe_finsetWalkLength_eq n u v) p
 
 variable (G)
 
+/-- The `Finset` of walks from `u` to `v` with length less than `n`. See `finsetWalkLength` for
+context. In particular, we use this definition for `SimpleGraph.Path.instFintype`. --/
+def finsetWalkLengthLT (n : ℕ) (u v : V) : Finset (G.Walk u v) :=
+  (Finset.range n).disjiUnion
+    (fun l ↦ G.finsetWalkLength l u v)
+    (fun l _ l' _ hne _ hsl hsl' p hp ↦
+      have hl : p.length = l := mem_finsetWalkLength_iff.mp (hsl hp)
+      have hl' : p.length = l' := mem_finsetWalkLength_iff.mp (hsl' hp)
+      False.elim <| hne <| hl.symm.trans hl')
+
+open Finset in
+theorem coe_finsetWalkLengthLT_eq (n : ℕ) (u v : V) :
+    (G.finsetWalkLengthLT n u v : Set (G.Walk u v)) = {p : G.Walk u v | p.length < n} := by
+  ext p
+  simp [finsetWalkLengthLT, mem_coe, mem_disjiUnion, mem_finsetWalkLength_iff]
+
+variable {G}
+
+theorem mem_finsetWalkLengthLT_iff {n : ℕ} {u v : V} {p : G.Walk u v} :
+    p ∈ G.finsetWalkLengthLT n u v ↔ p.length < n :=
+  Set.ext_iff.mp (G.coe_finsetWalkLengthLT_eq n u v) p
+
+variable (G)
+
 instance fintypeSetWalkLength (u v : V) (n : ℕ) : Fintype {p : G.Walk u v | p.length = n} :=
   Fintype.ofFinset (G.finsetWalkLength n u v) fun p => by
     rw [← Finset.mem_coe, coe_finsetWalkLength_eq]
@@ -132,14 +156,34 @@ theorem card_set_walk_length_eq (u v : V) (n : ℕ) :
   Fintype.card_ofFinset (G.finsetWalkLength n u v) fun p => by
     rw [← Finset.mem_coe, coe_finsetWalkLength_eq]
 
+instance fintypeSetWalkLengthLT (u v : V) (n : ℕ) : Fintype {p : G.Walk u v | p.length < n} :=
+  Fintype.ofFinset (G.finsetWalkLengthLT n u v) fun p ↦ by
+    rw [← Finset.mem_coe, coe_finsetWalkLengthLT_eq]
+
+instance fintypeSubtypeWalkLengthLT (u v : V) (n : ℕ) : Fintype {p : G.Walk u v // p.length < n} :=
+  fintypeSetWalkLengthLT G u v n
+
 instance fintypeSetPathLength (u v : V) (n : ℕ) :
     Fintype {p : G.Walk u v | p.IsPath ∧ p.length = n} :=
   Fintype.ofFinset ((G.finsetWalkLength n u v).filter Walk.IsPath) <| by
-    simp [Walk.mem_finsetWalkLength_iff_length_eq, and_comm]
+    simp [mem_finsetWalkLength_iff, and_comm]
+
+instance fintypeSubtypePathLength (u v : V) (n : ℕ) :
+    Fintype {p : G.Walk u v // p.IsPath ∧ p.length = n} :=
+  fintypeSetPathLength G u v n
+
+instance fintypeSetPathLengthLT (u v : V) (n : ℕ) :
+    Fintype {p : G.Walk u v | p.IsPath ∧ p.length < n} :=
+  Fintype.ofFinset ((G.finsetWalkLengthLT n u v).filter Walk.IsPath) <| by
+    simp [mem_finsetWalkLengthLT_iff, and_comm]
+
+instance fintypeSubtypePathLengthLT (u v : V) (n : ℕ) :
+    Fintype {p : G.Walk u v // p.IsPath ∧ p.length < n} :=
+  fintypeSetPathLengthLT G u v n
 
 end LocallyFinite
 
-section Finite
+section Fintype
 
 variable [DecidableEq V] [Fintype V] [DecidableRel G.Adj]
 
@@ -149,7 +193,7 @@ theorem reachable_iff_exists_finsetWalkLength_nonempty (u v : V) :
   · intro r
     refine r.elim_path fun p => ?_
     refine ⟨⟨_, p.isPath.length_lt⟩, p, ?_⟩
-    simp [Walk.mem_finsetWalkLength_iff_length_eq]
+    simp [mem_finsetWalkLength_iff]
   · rintro ⟨_, p, _⟩
     exact ⟨p⟩
 
@@ -166,6 +210,15 @@ instance : Decidable G.Connected := by
   rw [connected_iff, ← Finset.univ_nonempty_iff]
   infer_instance
 
+open Finset in
+instance Path.instFintype {u v : V} : Fintype (G.Path u v) where
+  elems := (univ (α := { p : G.Walk u v | p.IsPath ∧ p.length < Fintype.card V })).map
+    ⟨fun p ↦ { val := p.val, property := p.prop.left },
+     fun _ _ h ↦ SetCoe.ext <| Subtype.mk.injEq .. ▸ h⟩
+  complete p := mem_map.mpr ⟨
+    ⟨p.val, ⟨p.prop, p.prop.length_lt⟩⟩,
+    ⟨mem_univ _, rfl⟩⟩
+
 instance instDecidableMemSupp (c : G.ConnectedComponent) (v : V) : Decidable (v ∈ c.supp) :=
   c.recOn (fun w ↦ decidable_of_iff (G.Reachable v w) <| by simp)
     (fun _ _ _ _ ↦ Subsingleton.elim _ _)
@@ -179,18 +232,19 @@ lemma disjiUnion_supp_toFinset_eq_supp_toFinset {G' : SimpleGraph V} (h : G ≤
       c'.supp.toFinset :=
   Finset.coe_injective <| by simpa using ConnectedComponent.biUnion_supp_eq_supp h _
 
-lemma ConnectedComponent.odd_card_supp_iff_odd_subcomponents {G'}
+end Fintype
+
+lemma ConnectedComponent.odd_card_supp_iff_odd_subcomponents [Finite V] {G'}
     (h : G ≤ G') (c' : ConnectedComponent G') :
     Odd (Nat.card c'.supp) ↔ Odd (Nat.card
     ({c : ConnectedComponent G | c.supp ⊆ c'.supp ∧ Odd (Nat.card c.supp) })) := by
   classical
-  -- have := Fintype.ofFinite:
+  cases nonempty_fintype V
   rw [Nat.card_eq_card_toFinset, ← disjiUnion_supp_toFinset_eq_supp_toFinset h]
   simp only [Finset.card_disjiUnion, Set.toFinset_card]
   rw [Finset.odd_sum_iff_odd_card_odd, Nat.card_eq_fintype_card, Fintype.card_ofFinset]
-  simp only [Set.mem_setOf_eq, Nat.card_eq_fintype_card, Finset.filter_filter]
-
-end Finite
+  simp only [Nat.card_eq_fintype_card, Finset.filter_filter]
+  rfl
 
 lemma odd_card_iff_odd_components [Finite V] : Odd (Nat.card V) ↔
     Odd (Nat.card ({(c : ConnectedComponent G) | Odd (Nat.card c.supp)})) := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/Dart.lean b/Mathlib/Combinatorics/SimpleGraph/Dart.lean
index 34379dcf7bf71..8919e4cea77bf 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Dart.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Dart.lean
@@ -39,6 +39,14 @@ theorem Dart.ext (d₁ d₂ : G.Dart) (h : d₁.toProd = d₂.toProd) : d₁ = d
 -- Porting note: deleted `Dart.fst` and `Dart.snd` since they are now invalid declaration names,
 -- even though there is not actually a `SimpleGraph.Dart.fst` or `SimpleGraph.Dart.snd`.
 
+@[simp]
+theorem Dart.fst_ne_snd (d : G.Dart) : d.fst ≠ d.snd :=
+  fun h ↦ G.irrefl (h ▸ d.adj)
+
+@[simp]
+theorem Dart.snd_ne_fst (d : G.Dart) : d.snd ≠ d.fst :=
+  fun h ↦ G.irrefl (h ▸ d.adj)
+
 theorem Dart.toProd_injective : Function.Injective (Dart.toProd : G.Dart → V × V) :=
   Dart.ext
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean b/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean
index 7a78571cb9edc..4b09938bdf173 100644
--- a/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean
@@ -6,7 +6,7 @@ Authors: Kyle Miller
 import Mathlib.Algebra.BigOperators.Ring
 import Mathlib.Combinatorics.SimpleGraph.Dart
 import Mathlib.Combinatorics.SimpleGraph.Finite
-import Mathlib.Data.ZMod.Parity
+import Mathlib.Data.ZMod.Basic
 
 /-!
 # Degree-sum formula and handshaking lemma
@@ -51,7 +51,7 @@ variable [Fintype V] [DecidableRel G.Adj]
 theorem dart_fst_fiber [DecidableEq V] (v : V) :
     (univ.filter fun d : G.Dart => d.fst = v) = univ.image (G.dartOfNeighborSet v) := by
   ext d
-  simp only [mem_image, true_and_iff, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
+  simp only [mem_image, true_and, mem_filter, SetCoe.exists, mem_univ, exists_prop_of_true]
   constructor
   · rintro rfl
     exact ⟨_, d.adj, by ext <;> rfl⟩
@@ -121,7 +121,7 @@ theorem even_card_odd_degree_vertices [Fintype V] [DecidableRel G.Adj] :
       convert h
       exact ZMod.ne_zero_iff_odd.symm
     · intro v
-      simp only [true_and_iff, mem_filter, mem_univ, Ne]
+      simp only [true_and, mem_filter, mem_univ, Ne]
       rw [ZMod.eq_zero_iff_even, ZMod.eq_one_iff_odd, ← Nat.not_even_iff_odd, imp_self]
       trivial
 
@@ -131,7 +131,7 @@ theorem odd_card_odd_degree_vertices_ne [Fintype V] [DecidableEq V] [DecidableRe
   have hk : 0 < k := by
     have hh : (filter (fun v : V => Odd (G.degree v)) univ).Nonempty := by
       use v
-      simp only [true_and_iff, mem_filter, mem_univ]
+      simp only [true_and, mem_filter, mem_univ]
       exact h
     rwa [← card_pos, hg, ← two_mul, mul_pos_iff_of_pos_left] at hh
     exact zero_lt_two
@@ -144,7 +144,7 @@ theorem odd_card_odd_degree_vertices_ne [Fintype V] [DecidableEq V] [DecidableRe
     rw [add_assoc, one_add_one_eq_two, ← Nat.mul_succ, ← two_mul]
     congr
     omega
-  · simpa only [true_and_iff, mem_filter, mem_univ]
+  · simpa only [true_and, mem_filter, mem_univ]
 
 theorem exists_ne_odd_degree_of_exists_odd_degree [Fintype V] [DecidableRel G.Adj] (v : V)
     (h : Odd (G.degree v)) : ∃ w : V, w ≠ v ∧ Odd (G.degree w) := by
@@ -154,7 +154,7 @@ theorem exists_ne_odd_degree_of_exists_odd_degree [Fintype V] [DecidableRel G.Ad
     rw [hg]
     apply Nat.succ_pos
   rcases card_pos.mp hg' with ⟨w, hw⟩
-  simp only [true_and_iff, mem_filter, mem_univ, Ne] at hw
+  simp only [true_and, mem_filter, mem_univ, Ne] at hw
   exact ⟨w, hw⟩
 
 end SimpleGraph
diff --git a/Mathlib/Combinatorics/SimpleGraph/Density.lean b/Mathlib/Combinatorics/SimpleGraph/Density.lean
index 6f388c7ffd277..f71bb81713dbb 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Density.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Density.lean
@@ -122,7 +122,7 @@ theorem edgeDensity_nonneg (s : Finset α) (t : Finset β) : 0 ≤ edgeDensity r
   apply div_nonneg <;> exact mod_cast Nat.zero_le _
 
 theorem edgeDensity_le_one (s : Finset α) (t : Finset β) : edgeDensity r s t ≤ 1 := by
-  apply div_le_one_of_le
+  apply div_le_one_of_le₀
   · exact mod_cast card_interedges_le_mul r s t
   · exact mod_cast Nat.zero_le _
 
@@ -175,10 +175,10 @@ theorem edgeDensity_sub_edgeDensity_le_one_sub_mul (hs : s₂ ⊆ s₁) (ht : t
   refine (sub_le_sub_left (mul_edgeDensity_le_edgeDensity r hs ht hs₂ ht₂) _).trans ?_
   refine le_trans ?_ (mul_le_of_le_one_right ?_ (edgeDensity_le_one r s₂ t₂))
   · rw [sub_mul, one_mul]
-  refine sub_nonneg_of_le (mul_le_one ?_ ?_ ?_)
-  · exact div_le_one_of_le ((@Nat.cast_le ℚ).2 (card_le_card hs)) (Nat.cast_nonneg _)
+  refine sub_nonneg_of_le (mul_le_one₀ ?_ ?_ ?_)
+  · exact div_le_one_of_le₀ ((@Nat.cast_le ℚ).2 (card_le_card hs)) (Nat.cast_nonneg _)
   · apply div_nonneg <;> exact mod_cast Nat.zero_le _
-  · exact div_le_one_of_le ((@Nat.cast_le ℚ).2 (card_le_card ht)) (Nat.cast_nonneg _)
+  · exact div_le_one_of_le₀ ((@Nat.cast_le ℚ).2 (card_le_card ht)) (Nat.cast_nonneg _)
 
 theorem abs_edgeDensity_sub_edgeDensity_le_one_sub_mul (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁)
     (hs₂ : s₂.Nonempty) (ht₂ : t₂.Nonempty) :
diff --git a/Mathlib/Combinatorics/SimpleGraph/Diam.lean b/Mathlib/Combinatorics/SimpleGraph/Diam.lean
index 08b5cadd73ca5..7e6b6d1920a5b 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Diam.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Diam.lean
@@ -114,7 +114,7 @@ lemma ediam_bot [Nontrivial α] : (⊥ : SimpleGraph α).ediam = ⊤ :=
 
 @[simp]
 lemma ediam_top [Nontrivial α] : (⊤ : SimpleGraph α).ediam = 1 := by
-  apply le_antisymm ?_ <| ENat.one_le_iff_pos.mpr <| pos_iff_ne_zero.mpr ediam_ne_zero
+  apply le_antisymm ?_ <| Order.one_le_iff_pos.mpr <| pos_iff_ne_zero.mpr ediam_ne_zero
   apply ediam_def ▸ iSup_le_iff.mpr
   intro p
   by_cases h : (⊤ : SimpleGraph α).Adj p.1 p.2
@@ -129,7 +129,7 @@ lemma ediam_eq_one [Nontrivial α] : G.ediam = 1 ↔ G = ⊤ := by
   apply G.edist_pos_of_ne at h₂
   apply le_of_eq at h₁
   rw [ediam_def, iSup_le_iff] at h₁
-  exact edist_eq_one_iff_adj.mp <| le_antisymm (h₁ (u, v)) <| ENat.one_le_iff_pos.mpr h₂
+  exact edist_eq_one_iff_adj.mp <| le_antisymm (h₁ (u, v)) <| Order.one_le_iff_pos.mpr h₂
 
 end ediam
 
@@ -185,8 +185,7 @@ lemma diam_bot : (⊥ : SimpleGraph α).diam = 0 := by
 
 @[simp]
 lemma diam_top [Nontrivial α] : (⊤ : SimpleGraph α).diam = 1 := by
-  rw [diam, ediam_top]
-  rfl
+  rw [diam, ediam_top, ENat.toNat_one]
 
 @[simp]
 lemma diam_eq_zero : G.diam = 0 ↔ G.ediam = ⊤ ∨ Subsingleton α := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean b/Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
index 0947929af9f98..e5a841272ed81 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
@@ -255,7 +255,9 @@ def componentComplFunctor : (Finset V)ᵒᵖ ⥤ Type u where
   obj K := G.ComponentCompl K.unop
   map f := ComponentCompl.hom (le_of_op_hom f)
   map_id _ := funext fun C => C.hom_refl
-  map_comp h h' := funext fun C => C.hom_trans (le_of_op_hom h) (le_of_op_hom h')
+  map_comp {_ Y Z} h h' := funext fun C => by
+    convert C.hom_trans (le_of_op_hom h) (le_of_op_hom _)
+    exact h'
 
 /-- The end of a graph, defined as the sections of the functor `component_compl_functor` . -/
 protected def «end» :=
diff --git a/Mathlib/Combinatorics/SimpleGraph/Finite.lean b/Mathlib/Combinatorics/SimpleGraph/Finite.lean
index ed7de03fd9d74..bab7f40b821df 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Finite.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Finite.lean
@@ -92,6 +92,15 @@ theorem edgeFinset_inf [DecidableEq V] : (G₁ ⊓ G₂).edgeFinset = G₁.edgeF
 theorem edgeFinset_sdiff [DecidableEq V] :
     (G₁ \ G₂).edgeFinset = G₁.edgeFinset \ G₂.edgeFinset := by simp [edgeFinset]
 
+lemma disjoint_edgeFinset : Disjoint G₁.edgeFinset G₂.edgeFinset ↔ Disjoint G₁ G₂ := by
+  simp_rw [← Finset.disjoint_coe, coe_edgeFinset, disjoint_edgeSet]
+
+lemma edgeFinset_eq_empty : G.edgeFinset = ∅ ↔ G = ⊥ := by
+  rw [← edgeFinset_bot, edgeFinset_inj]
+
+lemma edgeFinset_nonempty : G.edgeFinset.Nonempty ↔ G ≠ ⊥ := by
+  rw [Finset.nonempty_iff_ne_empty, edgeFinset_eq_empty.ne]
+
 theorem edgeFinset_card : G.edgeFinset.card = Fintype.card G.edgeSet :=
   Set.toFinset_card _
 
@@ -242,7 +251,7 @@ theorem mem_incidenceFinset [DecidableEq V] (e : Sym2 V) :
   Set.mem_toFinset
 
 theorem incidenceFinset_eq_filter [DecidableEq V] [Fintype G.edgeSet] :
-    G.incidenceFinset v = G.edgeFinset.filter (Membership.mem v) := by
+    G.incidenceFinset v = G.edgeFinset.filter (v ∈ ·) := by
   ext e
   induction e
   simp [mk'_mem_incidenceSet_iff]
@@ -278,7 +287,7 @@ section Finite
 variable [Fintype V]
 
 instance neighborSetFintype [DecidableRel G.Adj] (v : V) : Fintype (G.neighborSet v) :=
-  @Subtype.fintype _ _
+  @Subtype.fintype _ (· ∈ G.neighborSet v)
     (by
       simp_rw [mem_neighborSet]
       infer_instance)
diff --git a/Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean b/Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean
index 769c8115eecfa..c5f35dfab6814 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean
@@ -36,7 +36,7 @@ lemma IsHamiltonian.map {H : SimpleGraph β} (f : G →g H) (hf : Bijective f) (
 
 /-- A hamiltonian path visits every vertex. -/
 @[simp] lemma IsHamiltonian.mem_support (hp : p.IsHamiltonian) (c : α) : c ∈ p.support := by
-  simp only [← List.count_pos_iff_mem, hp _, Nat.zero_lt_one]
+  simp only [← List.count_pos_iff, hp _, Nat.zero_lt_one]
 
 /-- Hamiltonian paths are paths. -/
 lemma IsHamiltonian.isPath (hp : p.IsHamiltonian) : p.IsPath :=
@@ -45,7 +45,7 @@ lemma IsHamiltonian.isPath (hp : p.IsHamiltonian) : p.IsPath :=
 /-- A path whose support contains every vertex is hamiltonian. -/
 lemma IsPath.isHamiltonian_of_mem (hp : p.IsPath) (hp' : ∀ w, w ∈ p.support) :
     p.IsHamiltonian := fun _ ↦
-  le_antisymm (List.nodup_iff_count_le_one.1 hp.support_nodup _) (List.count_pos_iff_mem.2 (hp' _))
+  le_antisymm (List.nodup_iff_count_le_one.1 hp.support_nodup _) (List.count_pos_iff.2 (hp' _))
 
 lemma IsPath.isHamiltonian_iff (hp : p.IsPath) : p.IsHamiltonian ↔ ∀ w, w ∈ p.support :=
   ⟨(·.mem_support), hp.isHamiltonian_of_mem⟩
@@ -66,8 +66,8 @@ lemma IsHamiltonian.length_eq (hp : p.IsHamiltonian) : p.length = Fintype.card 
 end
 
 /-- A hamiltonian cycle is a cycle that visits every vertex once. -/
-structure IsHamiltonianCycle (p : G.Walk a a) extends p.IsCycle : Prop :=
-  isHamiltonian_tail : (p.tail toIsCycle.not_nil).IsHamiltonian
+structure IsHamiltonianCycle (p : G.Walk a a) extends p.IsCycle : Prop where
+  isHamiltonian_tail : p.tail.IsHamiltonian
 
 variable {p : G.Walk a a}
 
@@ -78,28 +78,29 @@ lemma IsHamiltonianCycle.map {H : SimpleGraph β} (f : G →g H) (hf : Bijective
     (hp : p.IsHamiltonianCycle) : (p.map f).IsHamiltonianCycle where
   toIsCycle := hp.isCycle.map hf.injective
   isHamiltonian_tail := by
-    simp only [IsHamiltonian, support_tail, support_map, ne_eq, List.map_eq_nil, support_ne_nil,
+    simp only [IsHamiltonian, support_tail, support_map, ne_eq, List.map_eq_nil_iff, support_ne_nil,
       not_false_eq_true, List.count_tail, List.head_map, beq_iff_eq, hf.surjective.forall,
       hf.injective, List.count_map_of_injective]
     intro x
     rcases p with (_ | ⟨y, p⟩)
     · cases hp.ne_nil rfl
-    simp only [support_cons, List.count_cons, beq_iff_eq, List.head_cons, hf.injective.eq_iff,
-      add_tsub_cancel_right]
+    simp only [map_cons, getVert_cons_succ, tail_cons_eq, support_copy,support_map]
+    rw [List.count_map_of_injective _ _ hf.injective, ← support_copy, ← tail_cons_eq]
     exact hp.isHamiltonian_tail _
 
 lemma isHamiltonianCycle_isCycle_and_isHamiltonian_tail  :
-    p.IsHamiltonianCycle ↔ ∃ h : p.IsCycle, (p.tail h.not_nil).IsHamiltonian :=
+    p.IsHamiltonianCycle ↔ p.IsCycle ∧ p.tail.IsHamiltonian :=
   ⟨fun ⟨h, h'⟩ ↦ ⟨h, h'⟩, fun ⟨h, h'⟩ ↦ ⟨h, h'⟩⟩
 
 lemma isHamiltonianCycle_iff_isCycle_and_support_count_tail_eq_one :
     p.IsHamiltonianCycle ↔ p.IsCycle ∧ ∀ a, (support p).tail.count a = 1 := by
-  simp only [isHamiltonianCycle_isCycle_and_isHamiltonian_tail, IsHamiltonian, support_tail,
-    exists_prop]
+  simp (config := { contextual := true }) [isHamiltonianCycle_isCycle_and_isHamiltonian_tail,
+    IsHamiltonian, support_tail, IsCycle.not_nil, exists_prop]
 
 /-- A hamiltonian cycle visits every vertex. -/
 lemma IsHamiltonianCycle.mem_support (hp : p.IsHamiltonianCycle) (b : α) :
-    b ∈ p.support := List.mem_of_mem_tail <| support_tail p _ ▸ hp.isHamiltonian_tail.mem_support _
+    b ∈ p.support :=
+  List.mem_of_mem_tail <| support_tail p hp.1.not_nil ▸ hp.isHamiltonian_tail.mem_support _
 
 /-- The length of a hamiltonian cycle is the number of vertices. -/
 lemma IsHamiltonianCycle.length_eq [Fintype α] (hp : p.IsHamiltonianCycle) :
@@ -110,11 +111,11 @@ lemma IsHamiltonianCycle.length_eq [Fintype α] (hp : p.IsHamiltonianCycle) :
 
 lemma IsHamiltonianCycle.count_support_self (hp : p.IsHamiltonianCycle) :
     p.support.count a = 2 := by
-  rw [support_eq_cons, List.count_cons_self, ← support_tail, hp.isHamiltonian_tail]
+  rw [support_eq_cons, List.count_cons_self, ← support_tail _ hp.1.not_nil, hp.isHamiltonian_tail]
 
 lemma IsHamiltonianCycle.support_count_of_ne (hp : p.IsHamiltonianCycle) (h : a ≠ b) :
     p.support.count b = 1 := by
-  rw [← cons_support_tail p, List.count_cons_of_ne h.symm, hp.isHamiltonian_tail]
+  rw [← cons_support_tail p hp.1.not_nil, List.count_cons_of_ne h.symm, hp.isHamiltonian_tail]
 
 end Walk
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/Hasse.lean b/Mathlib/Combinatorics/SimpleGraph/Hasse.lean
index dd320fc0cd417..5e6d344ea80ad 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Hasse.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Hasse.lean
@@ -97,7 +97,7 @@ def pathGraph (n : ℕ) : SimpleGraph (Fin n) :=
 theorem pathGraph_adj {n : ℕ} {u v : Fin n} :
     (pathGraph n).Adj u v ↔ u.val + 1 = v.val ∨ v.val + 1 = u.val := by
   simp only [pathGraph, hasse]
-  simp_rw [← Fin.coe_covBy_iff, Nat.covBy_iff_succ_eq]
+  simp_rw [← Fin.coe_covBy_iff, covBy_iff_add_one_eq]
 
 theorem pathGraph_preconnected (n : ℕ) : (pathGraph n).Preconnected :=
   hasse_preconnected_of_succ _
@@ -107,6 +107,6 @@ theorem pathGraph_connected (n : ℕ) : (pathGraph (n + 1)).Connected :=
 
 theorem pathGraph_two_eq_top : pathGraph 2 = ⊤ := by
   ext u v
-  fin_cases u <;> fin_cases v <;> simp [pathGraph, ← Fin.coe_covBy_iff, Nat.covBy_iff_succ_eq]
+  fin_cases u <;> fin_cases v <;> simp [pathGraph, ← Fin.coe_covBy_iff, covBy_iff_add_one_eq]
 
 end SimpleGraph
diff --git a/Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean b/Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
index ec09c234563e3..8a8ce3e22216a 100644
--- a/Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
@@ -125,10 +125,10 @@ theorem sum_incMatrix_apply_of_mem_edgeSet [Fintype α] :
     intro a b h
     rw [mem_edgeSet] at h
     rw [← Nat.cast_two, ← card_pair h.ne]
-    simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]
+    simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h]
     congr 2
     ext e
-    simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]
+    simp only [mem_filter, mem_univ, true_and, mem_insert, mem_singleton]
 
 theorem sum_incMatrix_apply_of_not_mem_edgeSet [Fintype α] (h : e ∉ G.edgeSet) :
     ∑ a, G.incMatrix R a e = 0 :=
diff --git a/Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean b/Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean
index 570e89aa491e2..e609a7dee3f0e 100644
--- a/Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean
@@ -113,9 +113,9 @@ theorem lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable (x : V →
   rw [lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj]
   refine ⟨?_, fun h i j hA ↦ h i j hA.reachable⟩
   intro h i j ⟨w⟩
-  induction' w with w i j _ hA _ h'
-  · rfl
-  · exact (h i j hA).trans h'
+  induction w with
+  | nil => rfl
+  | cons hA _ h' => exact (h _ _ hA).trans h'
 
 theorem lapMatrix_toLin'_apply_eq_zero_iff_forall_reachable (x : V → ℝ) :
     Matrix.toLin' (G.lapMatrix ℝ) x = 0 ↔ ∀ i j : V, G.Reachable i j → x i = x j := by
@@ -156,9 +156,10 @@ lemma linearIndependent_lapMatrix_ker_basis_aux :
   rw [Subtype.ext_iff] at h0
   have h : ∑ c, g c • lapMatrix_ker_basis_aux G c = fun i ↦ g (connectedComponentMk G i) := by
     simp only [lapMatrix_ker_basis_aux, SetLike.mk_smul_mk, AddSubmonoid.coe_finset_sum]
-    conv_lhs => enter [2, c, j]; rw [Pi.smul_apply, smul_eq_mul, mul_ite, mul_one, mul_zero]
+    repeat rw [AddSubmonoid.coe_finset_sum]
     ext i
-    simp only [Finset.sum_apply, sum_ite_eq, mem_univ, ite_true]
+    simp only [Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_ite, mul_one, mul_zero, sum_ite_eq,
+      mem_univ, ↓reduceIte]
   rw [h] at h0
   intro c
   obtain ⟨i, h'⟩ : ∃ i : V, G.connectedComponentMk i = c := Quot.exists_rep c
@@ -171,9 +172,10 @@ lemma top_le_span_range_lapMatrix_ker_basis_aux :
   use Quot.lift x.val (by rw [← lapMatrix_toLin'_apply_eq_zero_iff_forall_reachable G x,
     LinearMap.map_coe_ker])
   ext j
-  simp only [lapMatrix_ker_basis_aux, AddSubmonoid.coe_finset_sum, Submodule.coe_toAddSubmonoid,
-    SetLike.val_smul, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_ite, mul_one, mul_zero,
-    sum_ite_eq, mem_univ, ite_true]
+  simp only [lapMatrix_ker_basis_aux]
+  rw [AddSubmonoid.coe_finset_sum]
+  simp only [SetLike.mk_smul_mk, Finset.sum_apply, Pi.smul_apply, smul_eq_mul, mul_ite, mul_one,
+    mul_zero, sum_ite_eq, mem_univ, ↓reduceIte]
   rfl
 
 /-- `lapMatrix_ker_basis G` is a basis of the nullspace indexed by its connected components,
@@ -187,8 +189,8 @@ end
 
 /-- The number of connected components in `G` is the dimension of the nullspace its Laplacian. -/
 theorem card_ConnectedComponent_eq_rank_ker_lapMatrix : Fintype.card G.ConnectedComponent =
-    FiniteDimensional.finrank ℝ (LinearMap.ker (Matrix.toLin' (G.lapMatrix ℝ))) := by
+    Module.finrank ℝ (LinearMap.ker (Matrix.toLin' (G.lapMatrix ℝ))) := by
   classical
-  rw [FiniteDimensional.finrank_eq_card_basis (lapMatrix_ker_basis G)]
+  rw [Module.finrank_eq_card_basis (lapMatrix_ker_basis G)]
 
 end SimpleGraph
diff --git a/Mathlib/Combinatorics/SimpleGraph/LineGraph.lean b/Mathlib/Combinatorics/SimpleGraph/LineGraph.lean
new file mode 100644
index 0000000000000..a1c37b83a30f6
--- /dev/null
+++ b/Mathlib/Combinatorics/SimpleGraph/LineGraph.lean
@@ -0,0 +1,39 @@
+/-
+Copyright (c) 2024 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+-/
+import Mathlib.Combinatorics.SimpleGraph.Basic
+
+/-!
+# LineGraph
+
+## Main definitions
+
+* `SimpleGraph.lineGraph` is the line graph of a simple graph `G`, with vertices as the edges of `G`
+  and two vertices of the line graph adjacent if the corresponding edges share a vertex in `G`.
+
+## Tags
+
+line graph
+-/
+
+namespace SimpleGraph
+
+variable {V : Type*} {G : SimpleGraph V}
+
+/--
+The line graph of a simple graph `G` has its vertex set as the edges of `G`, and two vertices of
+the line graph are adjacent if the corresponding edges share a vertex in `G`.
+-/
+def lineGraph {V : Type*} (G : SimpleGraph V) : SimpleGraph G.edgeSet where
+  Adj e₁ e₂ := e₁ ≠ e₂ ∧ (e₁ ∩ e₂ : Set V).Nonempty
+  symm e₁ e₂ := by intro h; rwa [ne_comm, Set.inter_comm]
+
+lemma lineGraph_adj_iff_exists {e₁ e₂ : G.edgeSet} :
+    (G.lineGraph).Adj e₁ e₂ ↔ e₁ ≠ e₂ ∧ ∃ v, v ∈ (e₁ : Sym2 V) ∧ v ∈ (e₂ : Sym2 V) := by
+  simp [Set.Nonempty, lineGraph]
+
+@[simp] lemma lineGraph_bot : (⊥ : SimpleGraph V).lineGraph = ⊥ := by aesop (add simp lineGraph)
+
+end SimpleGraph
diff --git a/Mathlib/Combinatorics/SimpleGraph/Matching.lean b/Mathlib/Combinatorics/SimpleGraph/Matching.lean
index 0969dbda00744..862f0f20756a5 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Matching.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Matching.lean
@@ -193,14 +193,21 @@ namespace ConnectedComponent
 
 section Finite
 
-variable [Fintype V]
-
-lemma even_card_of_isPerfectMatching [DecidableEq V] [DecidableRel G.Adj]
+lemma even_card_of_isPerfectMatching [Fintype V] [DecidableEq V] [DecidableRel G.Adj]
     (c : ConnectedComponent G) (hM : M.IsPerfectMatching) :
     Even (Fintype.card c.supp) := by
-  classical simpa using (hM.induce_connectedComponent_isMatching c).even_card
-
-lemma odd_matches_node_outside {u : Set V} {c : ConnectedComponent (Subgraph.deleteVerts ⊤ u).coe}
+  #adaptation_note
+  /--
+  After lean4#5020, some instances that use the chain of coercions
+  `[SetLike X], X → Set α → Sort _` are
+  blocked by the discrimination tree. This can be fixed by redeclaring the instance for `X`
+  using the double coercion but the proper fix seems to avoid the double coercion.
+  -/
+  letI : DecidablePred fun x ↦ x ∈ (M.induce c.supp).verts := fun a ↦ G.instDecidableMemSupp c a
+  simpa using (hM.induce_connectedComponent_isMatching c).even_card
+
+lemma odd_matches_node_outside [Finite V] {u : Set V}
+    {c : ConnectedComponent (Subgraph.deleteVerts ⊤ u).coe}
     (hM : M.IsPerfectMatching) (codd : Odd (Nat.card c.supp)) :
     ∃ᵉ (w ∈ u) (v : ((⊤ : G.Subgraph).deleteVerts u).verts), M.Adj v w ∧ v ∈ c.supp := by
   by_contra! h
@@ -216,7 +223,6 @@ lemma odd_matches_node_outside {u : Set V} {c : ConnectedComponent (Subgraph.del
       Subgraph.induce_adj, hwnu, not_false_eq_true, and_self, Subgraph.top_adj, M.adj_sub hw.1,
       and_true] at hv' ⊢
     trivial
-
   apply Nat.not_even_iff_odd.2 codd
   haveI : Fintype ↑(Subgraph.induce M (Subtype.val '' supp c)).verts := Fintype.ofFinite _
   classical
@@ -246,9 +252,9 @@ lemma IsMatchingFree.mono {G G' : SimpleGraph V} (h : G ≤ G') (hmf : G'.IsMatc
   simp only [Subgraph.map_verts, Hom.coe_ofLE, id_eq, Set.image_id']
   exact hc.2 v
 
-lemma exists_maximal_isMatchingFree [Fintype V] [DecidableEq V]
-    (h : G.IsMatchingFree) : ∃ Gmax : SimpleGraph V,
-    G ≤ Gmax ∧ Gmax.IsMatchingFree ∧ ∀ G', G' > Gmax → ∃ M : Subgraph G', M.IsPerfectMatching := by
+lemma exists_maximal_isMatchingFree [Finite V] (h : G.IsMatchingFree) :
+    ∃ Gmax : SimpleGraph V, G ≤ Gmax ∧ Gmax.IsMatchingFree ∧
+      ∀ G', G' > Gmax → ∃ M : Subgraph G', M.IsPerfectMatching := by
   simp_rw [← @not_forall_not _ Subgraph.IsPerfectMatching]
   obtain ⟨Gmax, hGmax⟩ := Finite.exists_le_maximal h
   exact ⟨Gmax, ⟨hGmax.1, ⟨hGmax.2.prop, fun _ h' ↦ hGmax.2.not_prop_of_gt h'⟩⟩⟩
diff --git a/Mathlib/Combinatorics/SimpleGraph/Metric.lean b/Mathlib/Combinatorics/SimpleGraph/Metric.lean
index 17d5d69884b4e..2f95492e595b1 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Metric.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Metric.lean
@@ -56,7 +56,7 @@ protected theorem Reachable.exists_walk_length_eq_edist (hr : G.Reachable u v) :
     ∃ p : G.Walk u v, p.length = G.edist u v :=
   csInf_mem <| Set.range_nonempty_iff_nonempty.mpr hr
 
-protected theorem Connected.exists_walk_length_eq_edist  (hconn : G.Connected) (u v : V) :
+protected theorem Connected.exists_walk_length_eq_edist (hconn : G.Connected) (u v : V) :
     ∃ p : G.Walk u v, p.length = G.edist u v :=
   (hconn u v).exists_walk_length_eq_edist
 
@@ -127,7 +127,7 @@ theorem edist_eq_one_iff_adj : G.edist u v = 1 ↔ G.Adj u v := by
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · obtain ⟨w, hw⟩ := exists_walk_of_edist_ne_top <| by rw [h]; simp
     exact w.adj_of_length_eq_one <| Nat.cast_eq_one.mp <| h ▸ hw
-  · exact le_antisymm (edist_le h.toWalk) (ENat.one_le_iff_pos.mpr <| edist_pos_of_ne h.ne)
+  · exact le_antisymm (edist_le h.toWalk) (Order.one_le_iff_pos.mpr <| edist_pos_of_ne h.ne)
 
 lemma edist_bot_of_ne (h : u ≠ v) : (⊥ : SimpleGraph V).edist u v = ⊤ := by
   rwa [ne_eq, ← reachable_bot.not, ← edist_ne_top_iff_reachable.not, not_not] at h
diff --git a/Mathlib/Combinatorics/SimpleGraph/Path.lean b/Mathlib/Combinatorics/SimpleGraph/Path.lean
index 4f6d0c3d95760..f40c73152000a 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Path.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Path.lean
@@ -178,6 +178,19 @@ theorem IsTrail.count_edges_eq_one [DecidableEq V] {u v : V} {p : G.Walk u v} (h
     {e : Sym2 V} (he : e ∈ p.edges) : p.edges.count e = 1 :=
   List.count_eq_one_of_mem h.edges_nodup he
 
+theorem IsTrail.length_le_card_edgeFinset [Fintype G.edgeSet] {u v : V}
+    {w : G.Walk u v} (h : w.IsTrail) : w.length ≤ G.edgeFinset.card := by
+  classical
+  let edges := w.edges.toFinset
+  have : edges.card = w.length := length_edges _ ▸ List.toFinset_card_of_nodup h.edges_nodup
+  rw [← this]
+  have : edges ⊆ G.edgeFinset := by
+    intro e h
+    refine mem_edgeFinset.mpr ?_
+    apply w.edges_subset_edgeSet
+    simpa [edges] using h
+  exact Finset.card_le_card this
+
 theorem IsPath.nil {u : V} : (nil : G.Walk u u).IsPath := by constructor <;> simp
 
 theorem IsPath.of_cons {u v w : V} {h : G.Adj u v} {p : G.Walk v w} :
@@ -236,7 +249,7 @@ theorem cons_isCycle_iff {u v : V} (p : G.Walk v u) (h : G.Adj u v) :
   have : p.support.Nodup → p.edges.Nodup := edges_nodup_of_support_nodup
   tauto
 
-lemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : ¬ p.Nil) : (p.tail hp').IsPath := by
+lemma IsPath.tail {p : G.Walk u v} (hp : p.IsPath) (hp' : ¬ p.Nil) : p.tail.IsPath := by
   rw [Walk.isPath_def] at hp ⊢
   rw [← cons_support_tail _ hp', List.nodup_cons] at hp
   exact hp.2
@@ -572,12 +585,12 @@ theorem mapEmbedding_injective (f : G ↪g G') (u v : V) :
 
 end Path
 
-/-! ### Transferring between graphs -/
+/-! ### Transferring between graphs -/
 
 namespace Walk
 
 variable {G} {p} {u v : V} {H : SimpleGraph V}
-variable (p : G.Walk u v)
+variable {p : G.Walk u v}
 
 protected theorem IsPath.transfer (hp) (pp : p.IsPath) :
     (p.transfer H hp).IsPath := by
@@ -1111,7 +1124,7 @@ theorem adj_and_reachable_delete_edges_iff_exists_cycle {v w : V} :
       rw [Sym2.eq_swap]
       intro h
       cases hp (Walk.edges_toPath_subset p h)
-    · simp only [Sym2.eq_swap, Walk.edges_cons, List.mem_cons, eq_self_iff_true, true_or_iff]
+    · simp only [Sym2.eq_swap, Walk.edges_cons, List.mem_cons, eq_self_iff_true, true_or]
   · rintro ⟨u, c, hc, he⟩
     refine ⟨c.adj_of_mem_edges he, ?_⟩
     by_contra! hb
@@ -1130,7 +1143,7 @@ theorem isBridge_iff_adj_and_forall_cycle_not_mem {v w : V} : G.IsBridge s(v, w)
   rw [← not_iff_not]
   push_neg
   rw [← adj_and_reachable_delete_edges_iff_exists_cycle]
-  simp only [h, true_and_iff]
+  simp only [h, true_and]
 
 theorem isBridge_iff_mem_and_forall_cycle_not_mem {e : Sym2 V} :
     G.IsBridge e ↔ e ∈ G.edgeSet ∧ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → e ∉ p.edges :=
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
index bd6d68eeeb2dd..5ee0f683c85af 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Bound.lean
@@ -117,7 +117,7 @@ theorem eps_pos (hPε : 100 ≤ (4 : ℝ) ^ P.parts.card * ε ^ 5) : 0 < ε :=
 
 theorem hundred_div_ε_pow_five_le_m [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α)
     (hPε : 100 ≤ (4 : ℝ) ^ P.parts.card * ε ^ 5) : 100 / ε ^ 5 ≤ m :=
-  (div_le_of_nonneg_of_le_mul (eps_pow_five_pos hPε).le (by positivity) hPε).trans <| by
+  (div_le_of_le_mul₀ (eps_pow_five_pos hPε).le (by positivity) hPε).trans <| by
     norm_cast
     rwa [Nat.le_div_iff_mul_le' (stepBound_pos (P.parts_nonempty <|
       univ_nonempty.ne_empty).card_pos), stepBound, mul_left_comm, ← mul_pow]
@@ -126,10 +126,10 @@ theorem hundred_le_m [Nonempty α] (hPα : P.parts.card * 16 ^ P.parts.card ≤
     (hPε : 100 ≤ (4 : ℝ) ^ P.parts.card * ε ^ 5) (hε : ε ≤ 1) : 100 ≤ m :=
   mod_cast
     (hundred_div_ε_pow_five_le_m hPα hPε).trans'
-      (le_div_self (by norm_num) (by sz_positivity) <| pow_le_one _ (by sz_positivity) hε)
+      (le_div_self (by norm_num) (by sz_positivity) <| pow_le_one₀ (by sz_positivity) hε)
 
 theorem a_add_one_le_four_pow_parts_card : a + 1 ≤ 4 ^ P.parts.card := by
-  have h : 1 ≤ 4 ^ P.parts.card := one_le_pow_of_one_le (by norm_num) _
+  have h : 1 ≤ 4 ^ P.parts.card := one_le_pow₀ (by norm_num)
   rw [stepBound, ← Nat.div_div_eq_div_mul]
   conv_rhs => rw [← Nat.sub_add_cancel h]
   rw [add_le_add_iff_right, tsub_le_iff_left, ← Nat.add_sub_assoc h]
@@ -175,8 +175,8 @@ theorem initialBound_pos : 0 < initialBound ε l :=
 
 theorem hundred_lt_pow_initialBound_mul {ε : ℝ} (hε : 0 < ε) (l : ℕ) :
     100 < ↑4 ^ initialBound ε l * ε ^ 5 := by
-  rw [← rpow_natCast 4, ← div_lt_iff (pow_pos hε 5), lt_rpow_iff_log_lt _ zero_lt_four, ←
-    div_lt_iff, initialBound, Nat.cast_max, Nat.cast_max]
+  rw [← rpow_natCast 4, ← div_lt_iff₀ (pow_pos hε 5), lt_rpow_iff_log_lt _ zero_lt_four, ←
+    div_lt_iff₀, initialBound, Nat.cast_max, Nat.cast_max]
   · push_cast
     exact lt_max_of_lt_right (lt_max_of_lt_right <| Nat.lt_floor_add_one _)
   · exact log_pos (by norm_num)
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
index 4ffe166e5b4b8..14c2df9eb7249 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean
@@ -114,7 +114,7 @@ private theorem card_nonuniformWitness_sdiff_biUnion_star (hV : V ∈ P.parts) (
   rw [sum_const]
   refine mul_le_mul_right' ?_ _
   have t := card_filter_atomise_le_two_pow (s := U) hX
-  refine t.trans (pow_le_pow_right (by norm_num) <| tsub_le_tsub_right ?_ _)
+  refine t.trans (pow_right_mono₀ (by norm_num) <| tsub_le_tsub_right ?_ _)
   exact card_image_le.trans (card_le_card <| filter_subset _ _)
 
 private theorem one_sub_eps_mul_card_nonuniformWitness_le_card_star (hV : V ∈ P.parts)
@@ -132,7 +132,7 @@ private theorem one_sub_eps_mul_card_nonuniformWitness_le_card_star (hV : V ∈
           ((2 : ℝ) * 2) ^ P.parts.card * m / U.card := by
         rw [mul_pow, ← mul_div_assoc, mul_assoc]
       _ = ↑4 ^ P.parts.card * m / U.card := by norm_num
-      _ ≤ 1 := div_le_one_of_le (pow_mul_m_le_card_part hP hU) (cast_nonneg _)
+      _ ≤ 1 := div_le_one_of_le₀ (pow_mul_m_le_card_part hP hU) (cast_nonneg _)
       _ ≤ ↑2 ^ P.parts.card * ε ^ 2 / 10 := by
         refine (one_le_sq_iff <| by positivity).1 ?_
         rw [div_pow, mul_pow, pow_right_comm, ← pow_mul ε,
@@ -240,7 +240,7 @@ private theorem m_add_one_div_m_le_one_add [Nonempty α]
     div_eq_mul_one_div _ (49 : ℝ), mul_div_left_comm (2 : ℝ), ← mul_sub_left_distrib, div_pow,
     div_le_iff₀ (show (0 : ℝ) < ↑100 ^ 2 by norm_num), mul_assoc, sq]
   refine mul_le_mul_of_nonneg_left ?_ (by sz_positivity)
-  exact (pow_le_one 5 (by sz_positivity) hε₁).trans (by norm_num)
+  exact (pow_le_one₀ (by sz_positivity) hε₁).trans (by norm_num)
 
 private theorem density_sub_eps_le_sum_density_div_card [Nonempty α]
     (hPα : P.parts.card * 16 ^ P.parts.card ≤ card α) (hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5)
@@ -269,7 +269,7 @@ private theorem density_sub_eps_le_sum_density_div_card [Nonempty α]
   rw [mul_mul_mul_comm, mul_comm (x.card : ℝ), mul_comm (y.card : ℝ), le_div_iff₀, mul_assoc]
   · refine mul_le_of_le_one_right (cast_nonneg _) ?_
     rw [div_mul_eq_mul_div, ← mul_assoc, mul_assoc]
-    refine div_le_one_of_le ?_ (by positivity)
+    refine div_le_one_of_le₀ ?_ (by positivity)
     refine (mul_le_mul_of_nonneg_right (one_sub_le_m_div_m_add_one_sq hPα hPε) ?_).trans ?_
     · exact mod_cast _root_.zero_le _
     rw [sq, mul_mul_mul_comm, mul_comm ((m : ℝ) / _), mul_comm ((m : ℝ) / _)]
@@ -379,9 +379,9 @@ private theorem eps_le_card_star_div [Nonempty α] (hPα : P.parts.card * 16 ^ P
     (hPε : ↑100 ≤ ↑4 ^ P.parts.card * ε ^ 5) (hε₁ : ε ≤ 1) (hU : U ∈ P.parts) (hV : V ∈ P.parts)
     (hUV : U ≠ V) (hunif : ¬G.IsUniform ε U V) :
     ↑4 / ↑5 * ε ≤ (star hP G ε hU V).card / ↑4 ^ P.parts.card := by
-  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one <| one_le_m_coe hPα)
+  have hm : (0 : ℝ) ≤ 1 - (↑m)⁻¹ := sub_nonneg_of_le (inv_le_one_of_one_le₀ <| one_le_m_coe hPα)
   have hε : 0 ≤ 1 - ε / 10 :=
-    sub_nonneg_of_le (div_le_one_of_le (hε₁.trans <| by norm_num) <| by norm_num)
+    sub_nonneg_of_le (div_le_one_of_le₀ (hε₁.trans <| by norm_num) <| by norm_num)
   have hε₀ : 0 < ε := by sz_positivity
   calc
     4 / 5 * ε = (1 - 1 / 10) * (1 - 9⁻¹) * ε := by norm_num
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean
index c2dc97dd66768..2691d38a6f4b2 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean
@@ -32,7 +32,7 @@ variable {α : Type*} [DecidableEq α] {s : Finset α} (P : Finpartition s) (G :
 namespace Finpartition
 
 /-- The energy of a partition, also known as index. Auxiliary quantity for Szemerédi's regularity
-lemma.  -/
+lemma. -/
 def energy : ℚ :=
   ((∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2) : ℚ) / (P.parts.card : ℚ) ^ 2
 
@@ -40,7 +40,7 @@ theorem energy_nonneg : 0 ≤ P.energy G := by
   exact div_nonneg (Finset.sum_nonneg fun _ _ => sq_nonneg _) <| sq_nonneg _
 
 theorem energy_le_one : P.energy G ≤ 1 :=
-  div_le_of_nonneg_of_le_mul (sq_nonneg _) zero_le_one <|
+  div_le_of_le_mul₀ (sq_nonneg _) zero_le_one <|
     calc
       ∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2 ≤ P.parts.offDiag.card • (1 : ℚ) :=
         sum_le_card_nsmul _ _ 1 fun uv _ =>
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
index d0c4c37b647fa..96d140587f9cc 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
@@ -45,8 +45,8 @@ theorem equitabilise_aux (hs : a * m + b * (m + 1) = s.card) :
   -- Get rid of the easy case `m = 0`
   obtain rfl | m_pos := m.eq_zero_or_pos
   · refine ⟨⊥, by simp, ?_, by simpa [Finset.filter_true_of_mem] using hs.symm⟩
-    simp only [le_zero_iff, card_eq_zero, mem_biUnion, exists_prop, mem_filter, id, and_assoc,
-      sdiff_eq_empty_iff_subset, subset_iff]
+    simp only [le_zero_iff, card_eq_zero, mem_biUnion, exists_prop, mem_filter, id,
+      and_assoc, sdiff_eq_empty_iff_subset, subset_iff]
     exact fun x hx a ha =>
       ⟨{a}, mem_map_of_mem _ (P.le hx ha), singleton_subset_iff.2 ha, mem_singleton_self _⟩
   -- Prove the case `m > 0` by strong induction on `s`
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean
index 9e7da73f754a5..4fe6e82a487fa 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean
@@ -93,7 +93,7 @@ private theorem distinctPairs_increment :
     P.parts.offDiag.attach.biUnion (distinctPairs hP G ε) ⊆ (increment hP G ε).parts.offDiag := by
   rintro ⟨Ui, Vj⟩
   simp only [distinctPairs, increment, mem_offDiag, bind_parts, mem_biUnion, Prod.exists,
-    exists_and_left, exists_prop, mem_product, mem_attach, true_and_iff, Subtype.exists, and_imp,
+    exists_and_left, exists_prop, mem_product, mem_attach, true_and, Subtype.exists, and_imp,
     mem_offDiag, forall_exists_index, exists₂_imp, Ne]
   refine fun U V hUV hUi hVj => ⟨⟨_, hUV.1, hUi⟩, ⟨_, hUV.2.1, hVj⟩, ?_⟩
   rintro rfl
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean
index 5e120205aeeb8..bee7ea6b54e48 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Lemma.lean
@@ -128,7 +128,7 @@ theorem szemeredi_regularity (hε : 0 < ε) (hl : l ≤ card α) :
   -- We gather a few numerical facts.
   have hεl' : 100 ≤ 4 ^ P.parts.card * ε ^ 5 :=
     (hundred_lt_pow_initialBound_mul hε l).le.trans
-      (mul_le_mul_of_nonneg_right (pow_le_pow_right (by norm_num) hP₂) <| by positivity)
+      (mul_le_mul_of_nonneg_right (pow_right_mono₀ (by norm_num) hP₂) <| by positivity)
   have hi : (i : ℝ) ≤ 4 / ε ^ 5 := by
     have hi : ε ^ 5 / 4 * ↑i ≤ 1 := hP₄.trans (mod_cast P.energy_le_one G)
     rw [div_mul_eq_mul_div, div_le_iff₀ (show (0 : ℝ) < 4 by norm_num)] at hi
diff --git a/Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean b/Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
index 005b689effd65..572c8e65f679c 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
@@ -7,7 +7,7 @@ import Mathlib.Algebra.BigOperators.Ring
 import Mathlib.Combinatorics.SimpleGraph.Density
 import Mathlib.Data.Nat.Cast.Order.Field
 import Mathlib.Order.Partition.Equipartition
-import Mathlib.SetTheory.Ordinal.Basic
+import Mathlib.SetTheory.Cardinal.Basic
 
 /-!
 # Graph uniformity and uniform partitions
@@ -276,7 +276,7 @@ lemma IsEquipartition.card_interedges_sparsePairs_le' (hP : P.IsEquipartition)
   · gcongr with UV hUV
     obtain ⟨U, V⟩ := UV
     simp [mk_mem_sparsePairs, ← card_interedges_div_card] at hUV
-    refine ((div_lt_iff ?_).1 hUV.2.2.2).le
+    refine ((div_lt_iff₀ ?_).1 hUV.2.2.2).le
     exact mul_pos (Nat.cast_pos.2 (P.nonempty_of_mem_parts hUV.1).card_pos)
       (Nat.cast_pos.2 (P.nonempty_of_mem_parts hUV.2.1).card_pos)
   norm_cast
diff --git a/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean b/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
index 68777353d2638..c0cc9e7d6ca21 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
@@ -133,6 +133,9 @@ theorem coe_adj_sub (G' : Subgraph G) (u v : G'.verts) (h : G'.coe.Adj u v) : G.
 protected theorem Adj.coe {H : G.Subgraph} {u v : V} (h : H.Adj u v) :
     H.coe.Adj ⟨u, H.edge_vert h⟩ ⟨v, H.edge_vert h.symm⟩ := h
 
+instance (G : SimpleGraph V) (H : Subgraph G) [DecidableRel H.Adj] : DecidableRel H.coe.Adj :=
+  fun a b ↦ ‹DecidableRel H.Adj› _ _
+
 /-- A subgraph is called a *spanning subgraph* if it contains all the vertices of `G`. -/
 def IsSpanning (G' : Subgraph G) : Prop :=
   ∀ v : V, v ∈ G'.verts
@@ -156,6 +159,8 @@ theorem Adj.of_spanningCoe {G' : Subgraph G} {u v : G'.verts} (h : G'.spanningCo
     G.Adj u v :=
   G'.adj_sub h
 
+lemma spanningCoe_le (G' : G.Subgraph) : G'.spanningCoe ≤ G := fun _ _ ↦ G'.3
+
 theorem spanningCoe_inj : G₁.spanningCoe = G₂.spanningCoe ↔ G₁.Adj = G₂.Adj := by
   simp [Subgraph.spanningCoe]
 
@@ -209,15 +214,27 @@ theorem edgeSet_subset (G' : Subgraph G) : G'.edgeSet ⊆ G.edgeSet :=
   Sym2.ind (fun _ _ ↦ G'.adj_sub)
 
 @[simp]
-theorem mem_edgeSet {G' : Subgraph G} {v w : V} : s(v, w) ∈ G'.edgeSet ↔ G'.Adj v w := Iff.rfl
+protected lemma mem_edgeSet {G' : Subgraph G} {v w : V} : s(v, w) ∈ G'.edgeSet ↔ G'.Adj v w := .rfl
+
+@[simp] lemma edgeSet_coe {G' : G.Subgraph} : G'.coe.edgeSet = Sym2.map (↑) ⁻¹' G'.edgeSet := by
+  ext e; induction' e using Sym2.ind with a b; simp
 
-theorem mem_verts_if_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet)
+lemma image_coe_edgeSet_coe (G' : G.Subgraph) : Sym2.map (↑) '' G'.coe.edgeSet = G'.edgeSet := by
+  rw [edgeSet_coe, Set.image_preimage_eq_iff]
+  rintro e he
+  induction' e using Sym2.ind with a b
+  rw [Subgraph.mem_edgeSet] at he
+  exact ⟨s(⟨a, edge_vert _ he⟩, ⟨b, edge_vert _ he.symm⟩), Sym2.map_pair_eq ..⟩
+
+theorem mem_verts_of_mem_edge {G' : Subgraph G} {e : Sym2 V} {v : V} (he : e ∈ G'.edgeSet)
     (hv : v ∈ e) : v ∈ G'.verts := by
   induction e
   rcases Sym2.mem_iff.mp hv with (rfl | rfl)
   · exact G'.edge_vert he
   · exact G'.edge_vert (G'.symm he)
 
+@[deprecated (since := "2024-10-01")] alias mem_verts_if_mem_edge := mem_verts_of_mem_edge
+
 /-- The `incidenceSet` is the set of edges incident to a given vertex. -/
 def incidenceSet (G' : Subgraph G) (v : V) : Set (Sym2 V) := {e ∈ G'.edgeSet | v ∈ e}
 
@@ -377,6 +394,18 @@ theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).v
 @[simp]
 theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts := by simp [iInf]
 
+@[simp] lemma coe_bot : (⊥ : G.Subgraph).coe = ⊥ := rfl
+
+@[simp] lemma IsInduced.top : (⊤ : G.Subgraph).IsInduced := fun _ _ ↦ id
+
+/-- The graph isomorphism between the top element of `G.subgraph` and `G`. -/
+def topIso : (⊤ : G.Subgraph).coe ≃g G where
+  toFun := (↑)
+  invFun a := ⟨a, Set.mem_univ _⟩
+  left_inv _ := Subtype.eta ..
+  right_inv _ := rfl
+  map_rel_iff' := .rfl
+
 theorem verts_spanningCoe_injective :
     (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by
   intro G₁ G₂ h
@@ -551,9 +580,12 @@ theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂
     Disjoint H₁.edgeSet H₂.edgeSet :=
   disjoint_iff_inf_le.mpr <| by simpa using edgeSet_mono h.le_bot
 
+section map
+variable {G' : SimpleGraph W} {f : G →g G'}
+
 /-- Graph homomorphisms induce a covariant function on subgraphs. -/
 @[simps]
-protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph where
+protected def map (f : G →g G') (H : G.Subgraph) : G'.Subgraph where
   verts := f '' H.verts
   Adj := Relation.Map H.Adj f f
   adj_sub := by
@@ -566,29 +598,26 @@ protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Sub
     rintro _ _ ⟨u, v, h, rfl, rfl⟩
     exact ⟨v, u, H.symm h, rfl, rfl⟩
 
-theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) := by
-  intro H H' h
+@[simp] lemma map_id (H : G.Subgraph) : H.map Hom.id = H := by ext <;> simp
+
+lemma map_comp {U : Type*} {G'' : SimpleGraph U} (H : G.Subgraph) (f : G →g G') (g : G' →g G'') :
+    H.map (g.comp f) = (H.map f).map g := by ext <;> simp [Subgraph.map]
+
+@[gcongr] lemma map_mono {H₁ H₂ : G.Subgraph} (hH : H₁ ≤ H₂) : H₁.map f ≤ H₂.map f := by
   constructor
   · intro
     simp only [map_verts, Set.mem_image, forall_exists_index, and_imp]
     rintro v hv rfl
-    exact ⟨_, h.1 hv, rfl⟩
+    exact ⟨_, hH.1 hv, rfl⟩
   · rintro _ _ ⟨u, v, ha, rfl, rfl⟩
-    exact ⟨_, _, h.2 ha, rfl, rfl⟩
-
-theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} :
-    (H ⊔ H').map f = H.map f ⊔ H'.map f := by
-  ext1
-  · simp only [Set.image_union, map_verts, verts_sup]
-  · ext
-    simp only [Relation.Map, map_adj, sup_adj]
-    constructor
-    · rintro ⟨a, b, h | h, rfl, rfl⟩
-      · exact Or.inl ⟨_, _, h, rfl, rfl⟩
-      · exact Or.inr ⟨_, _, h, rfl, rfl⟩
-    · rintro (⟨a, b, h, rfl, rfl⟩ | ⟨a, b, h, rfl, rfl⟩)
-      · exact ⟨_, _, Or.inl h, rfl, rfl⟩
-      · exact ⟨_, _, Or.inr h, rfl, rfl⟩
+    exact ⟨_, _, hH.2 ha, rfl, rfl⟩
+
+lemma map_monotone : Monotone (Subgraph.map f) := fun _ _ ↦ map_mono
+
+theorem map_sup (f : G →g G') (H₁ H₂ : G.Subgraph) : (H₁ ⊔ H₂).map f = H₁.map f ⊔ H₂.map f := by
+  ext <;> simp [Set.image_union, map_adj, sup_adj, Relation.Map, or_and_right, exists_or]
+
+end map
 
 /-- Graph homomorphisms induce a contravariant function on subgraphs. -/
 @[simps]
@@ -606,7 +635,7 @@ theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph
     simp only [comap_verts, Set.mem_preimage]
     apply h.1
   · intro v w
-    simp (config := { contextual := true }) only [comap_adj, and_imp, true_and_iff]
+    simp (config := { contextual := true }) only [comap_adj, and_imp, true_and]
     intro
     apply h.2
 
@@ -615,7 +644,7 @@ theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph
   refine ⟨fun h ↦ ⟨fun v hv ↦ ?_, fun v w hvw ↦ ?_⟩, fun h ↦ ⟨fun v ↦ ?_, fun v w ↦ ?_⟩⟩
   · simp only [comap_verts, Set.mem_preimage]
     exact h.1 ⟨v, hv, rfl⟩
-  · simp only [H.adj_sub hvw, comap_adj, true_and_iff]
+  · simp only [H.adj_sub hvw, comap_adj, true_and]
     exact h.2 ⟨v, w, hvw, rfl, rfl⟩
   · simp only [map_verts, Set.mem_image, forall_exists_index, and_imp]
     rintro w hw rfl
@@ -767,7 +796,7 @@ theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
   refine ⟨fun h ↦ by rw [h, singletonSubgraph_verts], fun h ↦ ?_⟩
   ext
   · rw [h, singletonSubgraph_verts]
-  · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false_iff]
+  · simp only [Prop.bot_eq_false, singletonSubgraph_adj, Pi.bot_apply, iff_false]
     intro ha
     have ha1 := ha.fst_mem
     have ha2 := ha.snd_mem
@@ -784,7 +813,7 @@ theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
     (G.subgraphOfAdj hvw).edgeSet = {s(v, w)} := by
   ext e
   refine e.ind ?_
-  simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj, iff_self_iff,
+  simp only [eq_comm, Set.mem_singleton_iff, Subgraph.mem_edgeSet, subgraphOfAdj_adj,
     forall₂_true_iff]
 
 lemma subgraphOfAdj_le_of_adj {v w : V} (H : G.Subgraph) (h : H.Adj v w) :
@@ -992,7 +1021,7 @@ theorem deleteEdges_le : G'.deleteEdges s ≤ G' := by
 theorem deleteEdges_le_of_le {s s' : Set (Sym2 V)} (h : s ⊆ s') :
     G'.deleteEdges s' ≤ G'.deleteEdges s := by
   constructor <;> simp (config := { contextual := true }) only [deleteEdges_verts, deleteEdges_adj,
-    true_and_iff, and_imp, subset_rfl]
+    true_and, and_imp, subset_rfl]
   exact fun _ _ _ hs' hs ↦ hs' (h hs)
 
 @[simp]
@@ -1044,7 +1073,7 @@ variable {G' G'' : G.Subgraph} {s s' : Set V}
 theorem induce_mono (hg : G' ≤ G'') (hs : s ⊆ s') : G'.induce s ≤ G''.induce s' := by
   constructor
   · simp [hs]
-  · simp (config := { contextual := true }) only [induce_adj, true_and_iff, and_imp]
+  · simp (config := { contextual := true }) only [induce_adj, and_imp]
     intro v w hv hw ha
     exact ⟨hs hv, hs hw, hg.2 ha⟩
 
@@ -1065,7 +1094,7 @@ theorem induce_self_verts : G'.induce G'.verts = G' := by
   ext
   · simp
   · constructor <;>
-      simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true_iff]
+      simp (config := { contextual := true }) only [induce_adj, imp_true_iff, and_true]
     exact fun ha ↦ ⟨G'.edge_vert ha, G'.edge_vert ha.symm⟩
 
 lemma le_induce_top_verts : G' ≤ (⊤ : G.Subgraph).induce G'.verts :=
diff --git a/Mathlib/Combinatorics/SimpleGraph/Trails.lean b/Mathlib/Combinatorics/SimpleGraph/Trails.lean
index 226fcd2ad45c6..174a67d7b2b8d 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Trails.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Trails.lean
@@ -58,18 +58,18 @@ theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail
     · rw [decide_eq_true_eq] at h
       obtain (rfl | rfl) := h
       · rw [Nat.even_add_one, ih]
-        simp only [huv.ne, imp_false, Ne, not_false_iff, true_and_iff, not_forall,
-          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
+        simp only [huv.ne, imp_false, Ne, not_false_iff, true_and, not_forall,
+          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and,
           and_iff_right_iff_imp]
         rintro rfl rfl
         exact G.loopless _ huv
       · rw [Nat.even_add_one, ih, ← not_iff_not]
-        simp only [huv.ne.symm, Ne, eq_self_iff_true, not_true, false_and_iff, not_forall,
-          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
+        simp only [huv.ne.symm, Ne, eq_self_iff_true, not_true, false_and, not_forall,
+          not_false_iff, exists_prop, and_true, Classical.not_not, true_and, iff_and_self]
         rintro rfl
         exact huv.ne
     · rw [decide_eq_true_eq, not_or] at h
-      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne] at ih ⊢
+      simp only [h.1, h.2, not_false_iff, true_and, add_zero, Ne] at ih ⊢
       rw [ih]
       constructor <;>
         · rintro h' h'' rfl
@@ -126,7 +126,7 @@ theorem IsEulerian.even_degree_iff {x u v : V} {p : G.Walk u v} (ht : p.IsEuleri
   rw [← Multiset.coe_countP, Multiset.countP_eq_card_filter, ← card_incidenceFinset_eq_degree]
   change Multiset.card _ = _
   congr 1
-  convert_to _ = (ht.isTrail.edgesFinset.filter (Membership.mem x)).val
+  convert_to _ = (ht.isTrail.edgesFinset.filter (x ∈ ·)).val
   have : Fintype G.edgeSet := fintypeEdgeSet ht
   rw [ht.edgesFinset_eq, G.incidenceFinset_eq_filter x]
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean b/Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
index 62b63093df2d8..96cd101472ef9 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
@@ -136,7 +136,8 @@ instance EdgeDisjointTriangles.instDecidable : Decidable G.EdgeDisjointTriangles
   decidable_of_iff ((G.cliqueFinset 3 : Set (Finset α)).Pairwise fun x y ↦ ((x ∩ y).card ≤ 1)) <| by
     simp only [coe_cliqueFinset, EdgeDisjointTriangles, Finset.card_le_one, ← coe_inter]; rfl
 
-instance LocallyLinear.instDecidable : Decidable G.LocallyLinear := And.decidable
+instance LocallyLinear.instDecidable : Decidable G.LocallyLinear :=
+  inferInstanceAs (Decidable (_ ∧ _))
 
 lemma EdgeDisjointTriangles.card_edgeFinset_le (hG : G.EdgeDisjointTriangles) :
     3 * (G.cliqueFinset 3).card ≤ G.edgeFinset.card := by
@@ -255,7 +256,7 @@ lemma FarFromTriangleFree.lt_half (hG : G.FarFromTriangleFree ε) : ε < 2⁻¹
   by_contra! hε
   refine lt_irrefl (ε * card α ^ 2) ?_
   have hε₀ : 0 < ε := hε.trans_lt' (by norm_num)
-  rw [inv_pos_le_iff_one_le_mul (zero_lt_two' 𝕜)] at hε
+  rw [inv_le_iff_one_le_mul₀ (zero_lt_two' 𝕜)] at hε
   calc
     _ ≤ (G.edgeFinset.card : 𝕜) := by
       simpa using hG.le_card_sub_card bot_le (cliqueFree_bot (le_succ _))
@@ -275,7 +276,7 @@ lemma FarFromTriangleFree.lt_half (hG : G.FarFromTriangleFree ε) : ε < 2⁻¹
   apply tsub_lt_self <;> positivity
 
 lemma FarFromTriangleFree.lt_one (hG : G.FarFromTriangleFree ε) : ε < 1 :=
-  hG.lt_half.trans <| inv_lt_one one_lt_two
+  hG.lt_half.trans two_inv_lt_one
 
 theorem FarFromTriangleFree.nonpos (h₀ : G.FarFromTriangleFree ε) (h₁ : G.CliqueFree 3) :
     ε ≤ 0 := by
diff --git a/Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean b/Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean
index 41df6e05b9aba..f1687fb68c6d4 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Triangle/Counting.lean
@@ -44,7 +44,7 @@ private lemma edgeDensity_badVertices_le (hε : 0 ≤ ε) (dst : 2 * ε ≤ G.ed
     G.edgeDensity (badVertices G ε s t) t ≤ G.edgeDensity s t - ε := by
   rw [edgeDensity_def]
   push_cast
-  refine div_le_of_nonneg_of_le_mul (by positivity) (sub_nonneg_of_le <| by linarith) ?_
+  refine div_le_of_le_mul₀ (by positivity) (sub_nonneg_of_le <| by linarith) ?_
   rw [mul_comm]
   exact G.card_interedges_badVertices_le
 
@@ -89,7 +89,7 @@ private lemma good_vertices_triangle_card [DecidableEq α] (dst : 2 * ε ≤ G.e
   rw [edgeDensity_def] at this
   push_cast at this
   have hε := utu.pos.le
-  refine le_trans ?_ (mul_le_of_nonneg_of_le_div (Nat.cast_nonneg _) (by positivity) this)
+  refine le_trans ?_ (mul_le_of_le_div₀ (Nat.cast_nonneg _) (by positivity) this)
   refine Eq.trans_le ?_
     (mul_le_mul_of_nonneg_left (mul_le_mul hY hZ (by positivity) (by positivity)) hε)
   ring
diff --git a/Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean b/Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean
index ceb4e3902e9d9..6bcfe21447840 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean
@@ -155,7 +155,7 @@ instance graph.instDecidableRelAdj : DecidableRel (graph t).Adj
   toFun x := {in₀ x.1, in₁ x.2.1, in₂ x.2.2}
   inj' := fun ⟨a, b, c⟩ ⟨a', b', c'⟩ ↦ by simpa only [Finset.Subset.antisymm_iff, Finset.subset_iff,
     mem_insert, mem_singleton, forall_eq_or_imp, forall_eq, Prod.mk.inj_iff, or_false, false_or,
-    in₀, in₁, in₂, Sum.inl.inj_iff, Sum.inr.inj_iff] using And.left
+    in₀, in₁, in₂, Sum.inl.inj_iff, Sum.inr.inj_iff, reduceCtorEq] using And.left
 
 lemma toTriangle_is3Clique (hx : x ∈ t) : (graph t).IsNClique 3 (toTriangle x) := by
   simp only [toTriangle_apply, is3Clique_triple_iff, in₀₁_iff, in₀₂_iff, in₁₂_iff]
diff --git a/Mathlib/Combinatorics/SimpleGraph/Walk.lean b/Mathlib/Combinatorics/SimpleGraph/Walk.lean
index 52c4b7eeca310..e2c574948f9c0 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Walk.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Walk.lean
@@ -177,16 +177,19 @@ theorem adj_getVert_succ {u v} (w : G.Walk u v) {i : ℕ} (hi : i < w.length) :
     · simp [getVert, hxy]
     · exact ih (Nat.succ_lt_succ_iff.1 hi)
 
+lemma getVert_cons_one {u v w} (q : G.Walk v w) (hadj : G.Adj u v) :
+    (q.cons hadj).getVert 1 = v := by
+  have : (q.cons hadj).getVert 1 = q.getVert 0 := rfl
+  simpa [getVert_zero] using this
+
 @[simp]
-lemma cons_getVert_succ {u v w n} (p : G.Walk v w) (h : G.Adj u v) :
+lemma getVert_cons_succ {u v w n} (p : G.Walk v w) (h : G.Adj u v) :
     (p.cons h).getVert (n + 1) = p.getVert n := rfl
 
-lemma cons_getVert {u v w n} (p : G.Walk v w) (h : G.Adj u v) (hn : n ≠ 0) :
+lemma getVert_cons {u v w n} (p : G.Walk v w) (h : G.Adj u v) (hn : n ≠ 0) :
     (p.cons h).getVert n = p.getVert (n - 1) := by
-  obtain ⟨i, hi⟩ : ∃ (i : ℕ), i.succ = n := by
-    use n - 1; exact Nat.succ_pred_eq_of_ne_zero hn
-  rw [← hi]
-  simp only [Nat.succ_eq_add_one, cons_getVert_succ, Nat.add_sub_cancel]
+  obtain ⟨n, rfl⟩ := Nat.exists_eq_add_one_of_ne_zero hn
+  rw [getVert_cons_succ, Nat.add_sub_cancel]
 
 @[simp]
 theorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w x) :
@@ -196,23 +199,21 @@ theorem cons_append {u v w x : V} (h : G.Adj u v) (p : G.Walk v w) (q : G.Walk w
 theorem cons_nil_append {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :
     (cons h nil).append p = cons h p := rfl
 
+@[simp]
+theorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p :=
+  rfl
+
 @[simp]
 theorem append_nil {u v : V} (p : G.Walk u v) : p.append nil = p := by
   induction p with
-  | nil => rfl
+  | nil => rw [nil_append]
   | cons _ _ ih => rw [cons_append, ih]
 
-@[simp]
-theorem nil_append {u v : V} (p : G.Walk u v) : nil.append p = p :=
-  rfl
-
 theorem append_assoc {u v w x : V} (p : G.Walk u v) (q : G.Walk v w) (r : G.Walk w x) :
     p.append (q.append r) = (p.append q).append r := by
   induction p with
-  | nil => rfl
-  | cons h p' ih =>
-    dsimp only [append]
-    rw [ih]
+  | nil => rw [nil_append, nil_append]
+  | cons h p' ih => rw [cons_append, cons_append, cons_append, ih]
 
 @[simp]
 theorem append_copy_copy {u v w u' v' w'} (p : G.Walk u v) (q : G.Walk v w)
@@ -494,9 +495,20 @@ theorem support_reverse {u v : V} (p : G.Walk u v) : p.reverse.support = p.suppo
 @[simp]
 theorem support_ne_nil {u v : V} (p : G.Walk u v) : p.support ≠ [] := by cases p <;> simp
 
+@[simp]
+theorem head_support {G : SimpleGraph V} {a b : V} (p : G.Walk a b) :
+    p.support.head (by simp) = a := by cases p <;> simp
+
+@[simp]
+theorem getLast_support {G : SimpleGraph V} {a b : V} (p : G.Walk a b) :
+    p.support.getLast (by simp) = b := by
+  induction p
+  · simp
+  · simpa
+
 theorem tail_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :
     (p.append p').support.tail = p.support.tail ++ p'.support.tail := by
-  rw [support_append, List.tail_append_of_ne_nil _ _ (support_ne_nil _)]
+  rw [support_append, List.tail_append_of_ne_nil (support_ne_nil _)]
 
 theorem support_eq_cons {u v : V} (p : G.Walk u v) : p.support = u :: p.support.tail := by
   cases p <;> simp
@@ -543,7 +555,7 @@ theorem subset_support_append_left {V : Type u} {G : SimpleGraph V} {u v w : V}
 theorem subset_support_append_right {V : Type u} {G : SimpleGraph V} {u v w : V}
     (p : G.Walk u v) (q : G.Walk v w) : q.support ⊆ (p.append q).support := by
   intro h
-  simp (config := { contextual := true }) only [mem_support_append_iff, or_true_iff, imp_true_iff]
+  simp (config := { contextual := true }) only [mem_support_append_iff, or_true, imp_true_iff]
 
 theorem coe_support {u v : V} (p : G.Walk u v) :
     (p.support : Multiset V) = {u} + p.support.tail := by cases p <;> rfl
@@ -636,6 +648,20 @@ theorem map_fst_darts_append {u v : V} (p : G.Walk u v) :
 theorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (·.fst) = p.support.dropLast := by
   simpa! using congr_arg List.dropLast (map_fst_darts_append p)
 
+@[simp]
+theorem head_darts_fst {G : SimpleGraph V} {a b : V} (p : G.Walk a b) (hp : p.darts ≠ []) :
+    (p.darts.head hp).fst = a := by
+  cases p
+  · contradiction
+  · simp
+
+@[simp]
+theorem getLast_darts_snd {G : SimpleGraph V} {a b : V} (p : G.Walk a b) (hp : p.darts ≠ []) :
+    (p.darts.getLast hp).snd = b := by
+  rw [← List.getLast_map (f := fun x : G.Dart ↦ x.snd)]
+  · simp_rw [p.map_snd_darts, List.getLast_tail, p.getLast_support]
+  · simpa
+
 @[simp]
 theorem edges_nil {u : V} : (nil : G.Walk u u).edges = [] := rfl
 
@@ -713,6 +739,19 @@ theorem edges_nodup_of_support_nodup {u v : V} {p : G.Walk u v} (h : p.support.N
     simp only [edges_cons, support_cons, List.nodup_cons] at h ⊢
     exact ⟨fun h' => h.1 (fst_mem_support_of_mem_edges p' h'), ih h.2⟩
 
+theorem edges_injective {u v : V} : Function.Injective (Walk.edges : G.Walk u v → List (Sym2 V))
+  | .nil, .nil, _ => rfl
+  | .nil, .cons _ _, h => by simp at h
+  | .cons _ _, .nil, h => by simp at h
+  | .cons' u v c h₁ w₁, .cons' _ v' _ h₂ w₂, h => by
+    have h₃ : u ≠ v' := by rintro rfl; exact G.loopless _ h₂
+    obtain ⟨rfl, h₃⟩ : v = v' ∧ w₁.edges = w₂.edges := by simpa [h₁, h₃] using h
+    obtain rfl := Walk.edges_injective h₃
+    rfl
+
+theorem darts_injective {u v : V} : Function.Injective (Walk.darts : G.Walk u v → List G.Dart) :=
+  edges_injective.of_comp
+
 /-- Predicate for the empty walk.
 
 Solves the dependent type problem where `p = G.Walk.nil` typechecks
@@ -741,6 +780,9 @@ lemma nil_iff_support_eq {p : G.Walk v w} : p.Nil ↔ p.support = [v] := by
 lemma nil_iff_length_eq {p : G.Walk v w} : p.Nil ↔ p.length = 0 := by
   cases p <;> simp
 
+lemma not_nil_iff_lt_length {p : G.Walk v w} : ¬ p.Nil ↔ 0 < p.length := by
+  cases p <;> simp
+
 lemma not_nil_iff {p : G.Walk v w} :
     ¬ p.Nil ↔ ∃ (u : V) (h : G.Adj v u) (q : G.Walk u w), p = cons h q := by
   cases p <;> simp [*]
@@ -765,61 +807,77 @@ lemma notNilRec_cons {motive : {u w : V} → (p : G.Walk u w) → ¬ p.Nil → S
     motive (q.cons h) Walk.not_nil_cons) (h' : G.Adj u v) (q' : G.Walk v w) :
     @Walk.notNilRec _ _ _ _ _ cons _ _ = cons h' q' := by rfl
 
-/-- The second vertex along a non-nil walk. -/
-def sndOfNotNil (p : G.Walk v w) (hp : ¬ p.Nil) : V :=
-  p.notNilRec (@fun _ u _ _ _ => u) hp
+@[simp] lemma adj_getVert_one {p : G.Walk v w} (hp : ¬ p.Nil) :
+    G.Adj v (p.getVert 1) := by
+  simpa using adj_getVert_succ p (by simpa [not_nil_iff_lt_length] using hp : 0 < p.length)
 
-@[simp] lemma adj_sndOfNotNil {p : G.Walk v w} (hp : ¬ p.Nil) :
-    G.Adj v (p.sndOfNotNil hp) :=
-  p.notNilRec (fun h _ => h) hp
+/-- The walk obtained by removing the first `n` darts of a walk. -/
+def drop {u v : V} (p : G.Walk u v) (n : ℕ) : G.Walk (p.getVert n) v :=
+  match p, n with
+  | .nil, _ => .nil
+  | p, 0 => p.copy (getVert_zero p).symm rfl
+  | .cons h q, (n + 1) => (q.drop n).copy (getVert_cons_succ _ h).symm rfl
 
 /-- The walk obtained by removing the first dart of a non-nil walk. -/
-def tail (p : G.Walk u v) (hp : ¬ p.Nil) : G.Walk (p.sndOfNotNil hp) v :=
-  p.notNilRec (fun _ q => q) hp
+def tail (p : G.Walk u v) : G.Walk (p.getVert 1) v := p.drop 1
+
+@[simp]
+lemma tail_cons_nil (h : G.Adj u v) : (Walk.cons h .nil).tail = .nil := by rfl
+
+lemma tail_cons_eq (h : G.Adj u v) (p : G.Walk v w) :
+    (p.cons h).tail = p.copy (getVert_zero p).symm rfl := by
+  match p with
+  | .nil => rfl
+  | .cons h q => rfl
 
 /-- The first dart of a walk. -/
 @[simps]
 def firstDart (p : G.Walk v w) (hp : ¬ p.Nil) : G.Dart where
   fst := v
-  snd := p.sndOfNotNil hp
-  adj := p.adj_sndOfNotNil hp
+  snd := p.getVert 1
+  adj := p.adj_getVert_one hp
 
 lemma edge_firstDart (p : G.Walk v w) (hp : ¬ p.Nil) :
-    (p.firstDart hp).edge = s(v, p.sndOfNotNil hp) := rfl
+    (p.firstDart hp).edge = s(v, p.getVert 1) := rfl
 
 variable {x y : V} -- TODO: rename to u, v, w instead?
 
-@[simp] lemma cons_tail_eq (p : G.Walk x y) (hp : ¬ p.Nil) :
-    cons (p.adj_sndOfNotNil hp) (p.tail hp) = p :=
-  p.notNilRec (fun _ _ => rfl) hp
+lemma cons_tail_eq (p : G.Walk x y) (hp : ¬ p.Nil) :
+    cons (p.adj_getVert_one hp) p.tail = p := by
+  cases p with
+  | nil => simp only [nil_nil, not_true_eq_false] at hp
+  | cons h q =>
+    simp only [getVert_cons_succ, tail_cons_eq, cons_copy, copy_rfl_rfl]
 
 @[simp] lemma cons_support_tail (p : G.Walk x y) (hp : ¬p.Nil) :
-    x :: (p.tail hp).support = p.support := by
-  rw [← support_cons, cons_tail_eq]
+    x :: p.tail.support = p.support := by
+  rw [← support_cons, cons_tail_eq _ hp]
 
 @[simp] lemma length_tail_add_one {p : G.Walk x y} (hp : ¬ p.Nil) :
-    (p.tail hp).length + 1 = p.length := by
-  rw [← length_cons, cons_tail_eq]
+    p.tail.length + 1 = p.length := by
+  rw [← length_cons, cons_tail_eq _ hp]
 
 @[simp] lemma nil_copy {x' y' : V} {p : G.Walk x y} (hx : x = x') (hy : y = y') :
     (p.copy hx hy).Nil = p.Nil := by
   subst_vars; rfl
 
-@[simp] lemma support_tail (p : G.Walk v v) (hp) :
-    (p.tail hp).support = p.support.tail := by
+@[simp] lemma support_tail (p : G.Walk v v) (hp : ¬ p.Nil) :
+    p.tail.support = p.support.tail := by
   rw [← cons_support_tail p hp, List.tail_cons]
 
 @[simp]
 lemma tail_cons {t u v} (p : G.Walk u v) (h : G.Adj t u) :
-    (p.cons h).tail not_nil_cons = p := by
-  unfold Walk.tail; simp only [notNilRec_cons]
+    (p.cons h).tail = p.copy (getVert_zero p).symm rfl := by
+  match p with
+  | .nil => rfl
+  | .cons h q => rfl
 
-lemma tail_support_eq_support_tail (p : G.Walk u v) (hnp : ¬p.Nil) :
-    (p.tail hnp).support = p.support.tail :=
-  p.notNilRec (by
-    intro u v w huv q
-    unfold Walk.tail
-    simp only [notNilRec_cons, Walk.support_cons, List.tail_cons]) hnp
+lemma support_tail_of_not_nil (p : G.Walk u v) (hnp : ¬p.Nil) :
+    p.tail.support = p.support.tail := by
+  match p with
+  | .nil => simp only [nil_nil, not_true_eq_false] at hnp
+  | .cons h q =>
+    simp only [tail_cons, getVert_cons_succ, support_copy, support_cons, List.tail_cons]
 
 /-! ### Walk decompositions -/
 
@@ -995,17 +1053,23 @@ theorem exists_boundary_dart {u v : V} (p : G.Walk u v) (S : Set V) (uS : u ∈
       exact ⟨d, List.Mem.tail _ hd, hcd⟩
     · exact ⟨⟨(x, y), a⟩, List.Mem.head _, uS, h⟩
 
-lemma getVert_tail {u v n} (p : G.Walk u v) (hnp: ¬ p.Nil) :
-    (p.tail hnp).getVert n = p.getVert (n + 1) :=
-  p.notNilRec (fun _ _ ↦ by simp only [tail_cons, cons_getVert_succ]) hnp
-
-@[simp]
-lemma cons_sndOfNotNil (q : G.Walk v w) (hadj : G.Adj u v) :
-    (q.cons hadj).sndOfNotNil not_nil_cons = v := by
-  unfold sndOfNotNil; simp only [notNilRec_cons]
-
-lemma getVert_one (p : G.Walk u v) (hnp : ¬ p.Nil) : p.getVert 1 = p.sndOfNotNil hnp :=
-  p.notNilRec (fun _ _ ↦ by simp only [cons_getVert_succ, getVert_zero, cons_sndOfNotNil]) hnp
+@[simp] lemma getVert_copy  {u v w x : V} (p : G.Walk u v) (i : ℕ) (h : u = w) (h' : v = x) :
+    (p.copy h h').getVert i = p.getVert i := by
+  subst_vars
+  match p, i with
+  | .nil, _ =>
+    rw [getVert_of_length_le _ (by simp only [length_nil, Nat.zero_le] : nil.length ≤ _)]
+    rw [getVert_of_length_le _ (by simp only [length_copy, length_nil, Nat.zero_le])]
+  | .cons hadj q, 0 => simp only [copy_rfl_rfl, getVert_zero]
+  | .cons hadj q, (n + 1) => simp only [copy_cons, getVert_cons_succ]; rfl
+
+@[simp] lemma getVert_tail {u v n} (p : G.Walk u v) (hnp: ¬ p.Nil) :
+    p.tail.getVert n = p.getVert (n + 1) := by
+  match p with
+  | .nil => rfl
+  | .cons h q =>
+    simp only [getVert_cons_succ, tail_cons_eq, getVert_cons]
+    exact getVert_copy q n (getVert_zero q).symm rfl
 
 /-- Given a walk `w` and a node in the support, there exists a natural `n`, such that given node
 is the `n`-th node (zero-indexed) in the walk. In addition, `n` is at most the length of the path.
@@ -1032,13 +1096,18 @@ theorem mem_support_iff_exists_getVert {u v w : V} {p : G.Walk v w} :
         rw [@nil_iff_length_eq]
         have : 1 ≤ p.length := by omega
         exact Nat.not_eq_zero_of_lt this
-      rw [← tail_support_eq_support_tail _ hnp]
+      rw [← support_tail_of_not_nil _ hnp]
       rw [mem_support_iff_exists_getVert]
       use n - 1
-      simp only [Nat.sub_le_iff_le_add, length_tail_add_one, getVert_tail]
-      have : n - 1 + 1 = n := by omega
+      simp only [Nat.sub_le_iff_le_add]
+      rw [getVert_tail _ hnp, length_tail_add_one hnp]
+      have : (n - 1 + 1) = n:= by omega
       rwa [this]
 termination_by p.length
+decreasing_by
+· simp_wf
+  rw [@Nat.lt_iff_add_one_le]
+  rw [length_tail_add_one hnp]
 
 end Walk
 
@@ -1126,7 +1195,7 @@ theorem map_injective_of_injective {f : G →g G'} (hinj : Function.Injective f)
     | cons _ _ =>
       simp only [map_cons, cons.injEq] at h
       cases hinj h.1
-      simp only [cons.injEq, heq_iff_eq, true_and_iff]
+      simp only [cons.injEq, heq_iff_eq, true_and]
       apply ih
       simpa using h.2
 
@@ -1134,7 +1203,7 @@ theorem map_injective_of_injective {f : G →g G'} (hinj : Function.Injective f)
 abbrev mapLe {G G' : SimpleGraph V} (h : G ≤ G') {u v : V} (p : G.Walk u v) : G'.Walk u v :=
   p.map (Hom.mapSpanningSubgraphs h)
 
-/-! ### Transferring between graphs -/
+/-! ### Transferring between graphs -/
 
 /-- The walk `p` transferred to lie in `H`, given that `H` contains its edges. -/
 @[simp]
diff --git a/Mathlib/Combinatorics/Young/YoungDiagram.lean b/Mathlib/Combinatorics/Young/YoungDiagram.lean
index 79b3e2892c891..fad6a3949ea28 100644
--- a/Mathlib/Combinatorics/Young/YoungDiagram.lean
+++ b/Mathlib/Combinatorics/Young/YoungDiagram.lean
@@ -67,7 +67,7 @@ namespace YoungDiagram
 
 instance : SetLike YoungDiagram (ℕ × ℕ) where
   -- Porting note (#11215): TODO: figure out how to do this correctly
-  coe := fun y => y.cells
+  coe y := y.cells
   coe_injective' μ ν h := by rwa [YoungDiagram.ext_iff, ← Finset.coe_inj]
 
 @[simp]
@@ -436,7 +436,7 @@ theorem rowLens_length_ofRowLens {w : List ℕ} {hw : w.Sorted (· ≥ ·)} (hpo
     (ofRowLens w hw).rowLens.length = w.length := by
   simp only [length_rowLens, colLen, Nat.find_eq_iff, mem_cells, mem_ofRowLens,
     lt_self_iff_false, IsEmpty.exists_iff, Classical.not_not]
-  exact ⟨not_false, fun n hn => ⟨hn, hpos _ (List.getElem_mem _ _ hn)⟩⟩
+  exact ⟨not_false, fun n hn => ⟨hn, hpos _ (List.getElem_mem hn)⟩⟩
 
 /-- The length of the `i`th row in `ofRowLens w hw` is the `i`th entry of `w` -/
 theorem rowLen_ofRowLens {w : List ℕ} {hw : w.Sorted (· ≥ ·)} (i : Fin w.length) :
diff --git a/Mathlib/Computability/Ackermann.lean b/Mathlib/Computability/Ackermann.lean
index 1b60a716c05f3..211e594b14670 100644
--- a/Mathlib/Computability/Ackermann.lean
+++ b/Mathlib/Computability/Ackermann.lean
@@ -3,7 +3,6 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Mathlib.Algebra.Order.Ring.Basic
 import Mathlib.Computability.Primrec
 import Mathlib.Tactic.Ring
 import Mathlib.Tactic.Linarith
@@ -75,26 +74,26 @@ theorem ack_succ_succ (m n : ℕ) : ack (m + 1) (n + 1) = ack m (ack (m + 1) n)
 @[simp]
 theorem ack_one (n : ℕ) : ack 1 n = n + 2 := by
   induction' n with n IH
-  · rfl
+  · simp
   · simp [IH]
 
 @[simp]
 theorem ack_two (n : ℕ) : ack 2 n = 2 * n + 3 := by
   induction' n with n IH
-  · rfl
+  · simp
   · simpa [mul_succ]
 
 -- Porting note: re-written to get rid of ack_three_aux
 @[simp]
 theorem ack_three (n : ℕ) : ack 3 n = 2 ^ (n + 3) - 3 := by
   induction' n with n IH
-  · rfl
+  · simp
   · rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,
         Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]
     have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_num
     apply H.trans
     rw [_root_.mul_le_mul_left two_pos]
-    exact pow_le_pow_right one_le_two (Nat.le_add_left 3 n)
+    exact pow_right_mono₀ one_le_two (Nat.le_add_left 3 n)
 
 theorem ack_pos : ∀ m n, 0 < ack m n
   | 0, n => by simp
diff --git a/Mathlib/Computability/AkraBazzi/AkraBazzi.lean b/Mathlib/Computability/AkraBazzi/AkraBazzi.lean
index 42babddc640ec..5f7d1c28e9623 100644
--- a/Mathlib/Computability/AkraBazzi/AkraBazzi.lean
+++ b/Mathlib/Computability/AkraBazzi/AkraBazzi.lean
@@ -266,7 +266,7 @@ lemma eventually_log_b_mul_pos : ∀ᶠ (n : ℕ) in atTop, ∀ i, 0 < log (b i
   exact h.eventually_gt_atTop 0
 
 @[aesop safe apply] lemma T_pos (n : ℕ) : 0 < T n := by
-  induction n using Nat.strongInductionOn with
+  induction n using Nat.strongRecOn with
   | ind n h_ind =>
     cases lt_or_le n R.n₀ with
     | inl hn => exact R.T_gt_zero' n hn -- n < R.n₀
@@ -508,7 +508,7 @@ lemma isTheta_smoothingFn_sub_self (i : α) :
 Every Akra-Bazzi recurrence has an associated exponent, denoted by `p : ℝ`, such that
 `∑ a_i b_i^p = 1`.  This section shows the existence and uniqueness of this exponent `p` for any
 `R : AkraBazziRecurrence`, and defines `R.asympBound` to be the asymptotic bound satisfied by `R`,
-namely `n^p (1 + ∑_{u < n} g(u) / u^(p+1))`.  -/
+namely `n^p (1 + ∑_{u < n} g(u) / u^(p+1))`. -/
 
 @[continuity]
 lemma continuous_sumCoeffsExp : Continuous (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) := by
@@ -1231,7 +1231,7 @@ lemma T_isBigO_smoothingFn_mul_asympBound :
   have h_one_sub_smoothingFn_pos' : 0 < 1 - ε n := h_smoothing_pos n hn
   rw [Real.norm_of_nonneg (R.T_nonneg n), Real.norm_of_nonneg (by positivity)]
   -- We now prove all other cases by induction
-  induction n using Nat.strongInductionOn with
+  induction n using Nat.strongRecOn with
   | ind n h_ind =>
     have b_mul_n₀_le_ri i : ⌊b' * ↑n₀⌋₊ ≤ r i n := by
       exact_mod_cast calc ⌊b' * (n₀ : ℝ)⌋₊ ≤ b' * n₀ := Nat.floor_le <| by positivity
@@ -1380,7 +1380,7 @@ lemma smoothingFn_mul_asympBound_isBigO_T :
   have h_one_sub_smoothingFn_pos' : 0 < 1 + ε n := h_smoothing_pos n hn
   rw [Real.norm_of_nonneg (R.T_nonneg n), Real.norm_of_nonneg (by positivity)]
   -- We now prove all other cases by induction
-  induction n using Nat.strongInductionOn with
+  induction n using Nat.strongRecOn with
   | ind n h_ind =>
     have b_mul_n₀_le_ri i : ⌊b' * ↑n₀⌋₊ ≤ r i n := by
       exact_mod_cast calc ⌊b' * ↑n₀⌋₊ ≤ b' * n₀ := Nat.floor_le <| by positivity
diff --git a/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean b/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
index 65daa0a486dab..aace8634bc1a8 100644
--- a/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
+++ b/Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean
@@ -201,7 +201,7 @@ lemma eventually_atTop_nonneg_or_nonpos (hf : GrowsPolynomially f) :
         have le_2n : max n₀ 2 ≤ (2 : ℝ)^n * max n₀ 2 := by
           nth_rewrite 1 [← one_mul (max n₀ 2)]
           gcongr
-          exact one_le_pow_of_one_le (by norm_num : (1 : ℝ) ≤ 2) _
+          exact one_le_pow₀ (by norm_num : (1 : ℝ) ≤ 2)
         have n₀_le_z : n₀ ≤ z := by
           calc n₀ ≤ max n₀ 2 := by simp
                 _ ≤ (2 : ℝ)^n * max n₀ 2 := le_2n
diff --git a/Mathlib/Computability/ContextFreeGrammar.lean b/Mathlib/Computability/ContextFreeGrammar.lean
index 4ee8b65304fa8..74ef24c357923 100644
--- a/Mathlib/Computability/ContextFreeGrammar.lean
+++ b/Mathlib/Computability/ContextFreeGrammar.lean
@@ -53,10 +53,10 @@ inductive Rewrites (r : ContextFreeRule T N) : List (Symbol T N) → List (Symbo
       r.Rewrites (x :: s₁) (x :: s₂)
 
 lemma Rewrites.exists_parts {r : ContextFreeRule T N} {u v : List (Symbol T N)}
-    (hyp : r.Rewrites u v) :
+    (hr : r.Rewrites u v) :
     ∃ p q : List (Symbol T N),
       u = p ++ [Symbol.nonterminal r.input] ++ q ∧ v = p ++ r.output ++ q := by
-  induction hyp with
+  induction hr with
   | head s =>
     use [], s
     simp
@@ -181,7 +181,7 @@ lemma Derives.append_left {v w : List (Symbol T g.NT)}
   | refl => rfl
   | tail _ last ih => exact ih.trans_produces <| last.append_left p
 
-/-- Add extra prefix to context-free deriving. -/
+/-- Add extra postfix to context-free deriving. -/
 lemma Derives.append_right {v w : List (Symbol T g.NT)}
     (hvw : g.Derives v w) (p : List (Symbol T g.NT)) :
     g.Derives (v ++ p) (w ++ p) := by
@@ -193,7 +193,10 @@ end ContextFreeGrammar
 
 /-- Context-free languages are defined by context-free grammars. -/
 def Language.IsContextFree (L : Language T) : Prop :=
-  ∃ g : ContextFreeGrammar.{uT} T, g.language = L
+  ∃ g : ContextFreeGrammar.{0} T, g.language = L
+
+proof_wanted Language.isContextFree_iff {L : Language T} :
+    L.IsContextFree ↔ ∃ g : ContextFreeGrammar.{uN} T, g.language = L
 
 section closure_reversal
 
diff --git a/Mathlib/Computability/DFA.lean b/Mathlib/Computability/DFA.lean
index 312e709b9fe77..48d4c11638987 100644
--- a/Mathlib/Computability/DFA.lean
+++ b/Mathlib/Computability/DFA.lean
@@ -28,11 +28,9 @@ Currently, there are two disjoint sets of simp lemmas: one for `DFA.eval`, and a
 - Should `mem_accepts` and `mem_acceptsFrom` be marked `@[simp]`?
 -/
 
-
-open Computability
-
 universe u v
 
+open Computability
 
 /-- A DFA is a set of states (`σ`), a transition function from state to state labelled by the
   alphabet (`step`), a starting state (`start`) and a set of acceptance states (`accept`). -/
@@ -247,3 +245,11 @@ theorem comap_reindex (f : α' → α) (g : σ ≃ σ') :
 end Maps
 
 end DFA
+
+/-- A regular language is a language that is defined by a DFA with finite states. -/
+def Language.IsRegular {T : Type u} (L : Language T) : Prop :=
+  ∃ σ : Type, ∃ _ : Fintype σ, ∃ M : DFA T σ, M.accepts = L
+
+proof_wanted Language.isRegular_iff {T : Type u} {L : Language T} :
+    L.IsRegular ↔ ∃ σ : Type v, ∃ _ : Fintype σ, ∃ M : DFA T σ, M.accepts = L
+-- probably needs `import Mathlib.Data.Countable.Small`
diff --git a/Mathlib/Computability/Halting.lean b/Mathlib/Computability/Halting.lean
index 407a9b60bf859..a2de5176d359b 100644
--- a/Mathlib/Computability/Halting.lean
+++ b/Mathlib/Computability/Halting.lean
@@ -173,6 +173,15 @@ protected theorem not {p : α → Prop} (hp : ComputablePred p) : ComputablePred
         simp only [Bool.not_eq_true]
         cases f n <;> rfl⟩
 
+/-- The computable functions are closed under if-then-else definitions
+with computable predicates. -/
+theorem ite {f₁ f₂ : ℕ → ℕ} (hf₁ : Computable f₁) (hf₂ : Computable f₂)
+    {c : ℕ → Prop} [DecidablePred c] (hc : ComputablePred c) :
+    Computable fun k ↦ if c k then f₁ k else f₂ k := by
+  simp_rw [← Bool.cond_decide]
+  obtain ⟨inst, hc⟩ := hc
+  convert hc.cond hf₁ hf₂
+
 theorem to_re {p : α → Prop} (hp : ComputablePred p) : RePred p := by
   obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp
   unfold RePred
@@ -214,7 +223,7 @@ theorem rice₂ (C : Set Code) (H : ∀ cf cg, eval cf = eval cg → (cf ∈ C 
                 (Partrec.nat_iff.1 <| eval_part.comp (const cg) Computable.id) ((hC _).1 fC),
         fun h => by {
           obtain rfl | rfl := h <;> simpa [ComputablePred, Set.mem_empty_iff_false] using
-            ⟨by infer_instance, Computable.const _⟩ }⟩
+            Computable.const _}⟩
 
 /-- The Halting problem is recursively enumerable -/
 theorem halting_problem_re (n) : RePred fun c => (eval c n).Dom :=
@@ -272,8 +281,6 @@ namespace Nat.Partrec'
 
 open Mathlib.Vector Partrec Computable
 
-open Nat (Partrec')
-
 open Nat.Partrec'
 
 theorem to_part {n f} (pf : @Partrec' n f) : _root_.Partrec f := by
diff --git a/Mathlib/Computability/Language.lean b/Mathlib/Computability/Language.lean
index c3cfcd963d09b..a8779cd1c9283 100644
--- a/Mathlib/Computability/Language.lean
+++ b/Mathlib/Computability/Language.lean
@@ -159,10 +159,10 @@ lemma mem_kstar_iff_exists_nonempty {x : List α} :
     x ∈ l∗ ↔ ∃ S : List (List α), x = S.join ∧ ∀ y ∈ S, y ∈ l ∧ y ≠ [] := by
   constructor
   · rintro ⟨S, rfl, h⟩
-    refine ⟨S.filter fun l ↦ !List.isEmpty l, by simp, fun y hy ↦ ?_⟩
+    refine ⟨S.filter fun l ↦ !List.isEmpty l, by simp [List.join_filter_not_isEmpty], fun y hy ↦ ?_⟩
     -- Porting note: The previous code was:
     -- rw [mem_filter, empty_iff_eq_nil] at hy
-    rw [mem_filter, Bool.not_eq_true', ← Bool.bool_iff_false, isEmpty_iff_eq_nil] at hy
+    rw [mem_filter, Bool.not_eq_true', ← Bool.bool_iff_false, List.isEmpty_iff] at hy
     exact ⟨h y hy.1, hy.2⟩
   · rintro ⟨S, hx, h⟩
     exact ⟨S, hx, fun y hy ↦ (h y hy).1⟩
diff --git a/Mathlib/Computability/PartrecCode.lean b/Mathlib/Computability/PartrecCode.lean
index 0bb7463dbe554..74b64e82859ab 100644
--- a/Mathlib/Computability/PartrecCode.lean
+++ b/Mathlib/Computability/PartrecCode.lean
@@ -170,7 +170,7 @@ private theorem encode_ofNatCode : ∀ n, encodeCode (ofNatCode n) = n
 instance instDenumerable : Denumerable Code :=
   mk'
     ⟨encodeCode, ofNatCode, fun c => by
-        induction c <;> try {rfl} <;> simp [encodeCode, ofNatCode, Nat.div2_val, *],
+        induction c <;> simp [encodeCode, ofNatCode, Nat.div2_val, *],
       encode_ofNatCode⟩
 
 theorem encodeCode_eq : encode = encodeCode :=
@@ -488,7 +488,7 @@ theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
   simp
 
 instance : Membership (ℕ →. ℕ) Code :=
-  ⟨fun f c => eval c = f⟩
+  ⟨fun c f => eval c = f⟩
 
 @[simp]
 theorem eval_const : ∀ n m, eval (Code.const n) m = Part.some n
@@ -900,7 +900,7 @@ private theorem hG : Primrec G := by
       Primrec.fst
 
 private theorem evaln_map (k c n) :
-    ((((List.range k)[n]?).map (evaln k c)).bind fun b => b) = evaln k c n := by
+    ((List.range k)[n]?.bind fun a ↦ evaln k c a) = evaln k c n := by
   by_cases kn : n < k
   · simp [List.getElem?_range kn]
   · rw [List.getElem?_len_le]
@@ -937,7 +937,7 @@ theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a
               (List.range n.unpair.1).map (evaln n.unpair.1 (ofNat Code n.unpair.2))) (k', c') n =
             evaln k' c' n := by
         intro k₁ c₁ n₁ hl
-        simp [lup, List.getElem?_range hl, evaln_map, Bind.bind]
+        simp [lup, List.getElem?_range hl, evaln_map, Bind.bind, Option.bind_map]
       cases' c with cf cg cf cg cf cg cf <;>
         simp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]
       · cases' encode_lt_pair cf cg with lf lg
@@ -969,7 +969,7 @@ theorem evaln_prim : Primrec fun a : (ℕ × Code) × ℕ => evaln a.1.1 a.1.2 a
   (Primrec.option_bind
     (Primrec.list_get?.comp (this.comp (_root_.Primrec.const ())
       (Primrec.encode_iff.2 Primrec.fst)) Primrec.snd) Primrec.snd.to₂).of_eq
-    fun ⟨⟨k, c⟩, n⟩ => by simp [evaln_map]
+    fun ⟨⟨k, c⟩, n⟩ => by simp [evaln_map, Option.bind_map]
 
 end
 
@@ -1015,4 +1015,15 @@ theorem fixed_point₂ {f : Code → ℕ →. ℕ} (hf : Partrec₂ f) : ∃ c :
 
 end
 
+/-- There are only countably many partial recursive partial functions `ℕ →. ℕ`. -/
+instance : Countable {f : ℕ →. ℕ // _root_.Partrec f} := by
+  apply Function.Surjective.countable (f := fun c => ⟨eval c, eval_part.comp (.const c) .id⟩)
+  intro ⟨f, hf⟩; simpa using exists_code.1 hf
+
+/-- There are only countably many computable functions `ℕ → ℕ`. -/
+instance : Countable {f : ℕ → ℕ // Computable f} :=
+  @Function.Injective.countable {f : ℕ → ℕ // Computable f} {f : ℕ →. ℕ // _root_.Partrec f} _
+    (fun f => ⟨f.val, f.2⟩)
+    (fun _ _ h => Subtype.val_inj.1 (PFun.lift_injective (by simpa using h)))
+
 end Nat.Partrec.Code
diff --git a/Mathlib/Computability/Primrec.lean b/Mathlib/Computability/Primrec.lean
index 38070a6702b20..350ee120e6376 100644
--- a/Mathlib/Computability/Primrec.lean
+++ b/Mathlib/Computability/Primrec.lean
@@ -275,7 +275,7 @@ end Primcodable
 
 namespace Primrec
 
-variable {α : Type*} {σ : Type*} [Primcodable α] [Primcodable σ]
+variable {α : Type*} [Primcodable α]
 
 open Nat.Primrec
 
@@ -457,8 +457,8 @@ end Primrec₂
 
 namespace Primrec
 
-variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} {σ : Type*}
-variable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable δ] [Primcodable σ]
+variable {α : Type*} {β : Type*} {σ : Type*}
+variable [Primcodable α] [Primcodable β] [Primcodable σ]
 
 theorem to₂ {f : α × β → σ} (hf : Primrec f) : Primrec₂ fun a b => f (a, b) :=
   hf.of_eq fun _ => rfl
@@ -651,7 +651,7 @@ theorem dom_fintype [Finite α] (f : α → σ) : Primrec f :=
   option_some_iff.1 <| by
     haveI := decidableEqOfEncodable α
     refine ((list_get?₁ (l.map f)).comp (list_indexOf₁ l)).of_eq fun a => ?_
-    rw [List.get?_eq_getElem?, List.getElem?_map, List.getElem?_indexOf  (m a), Option.map_some']
+    rw [List.get?_eq_getElem?, List.getElem?_map, List.getElem?_indexOf (m a), Option.map_some']
 
 -- Porting note: These are new lemmas
 -- I added it because it actually simplified the proofs
@@ -1088,8 +1088,7 @@ end Primrec
 
 namespace Primcodable
 
-variable {α : Type*} {β : Type*}
-variable [Primcodable α] [Primcodable β]
+variable {α : Type*} [Primcodable α]
 
 open Primrec
 
@@ -1139,8 +1138,8 @@ end Primcodable
 
 namespace Primrec
 
-variable {α : Type*} {β : Type*} {γ : Type*} {σ : Type*}
-variable [Primcodable α] [Primcodable β] [Primcodable γ] [Primcodable σ]
+variable {α : Type*} {β : Type*} {σ : Type*}
+variable [Primcodable α] [Primcodable β] [Primcodable σ]
 
 theorem subtype_val {p : α → Prop} [DecidablePred p] {hp : PrimrecPred p} :
     haveI := Primcodable.subtype hp
@@ -1215,7 +1214,7 @@ theorem vector_get {n} : Primrec₂ (@Vector.get α n) :=
 
 theorem list_ofFn :
     ∀ {n} {f : Fin n → α → σ}, (∀ i, Primrec (f i)) → Primrec fun a => List.ofFn fun i => f i a
-  | 0, _, _ => const []
+  | 0, _, _ => by simp only [List.ofFn_zero]; exact const []
   | n + 1, f, hf => by
     simpa [List.ofFn_succ] using list_cons.comp (hf 0) (list_ofFn fun i => hf i.succ)
 
diff --git a/Mathlib/Computability/Reduce.lean b/Mathlib/Computability/Reduce.lean
index 2d37bcabfb668..2b3eb7f41f3c9 100644
--- a/Mathlib/Computability/Reduce.lean
+++ b/Mathlib/Computability/Reduce.lean
@@ -369,7 +369,7 @@ private theorem le_antisymm {d₁ d₂ : ManyOneDegree} : d₁ ≤ d₂ → d₂
   induction d₁ using ManyOneDegree.ind_on
   induction d₂ using ManyOneDegree.ind_on
   intro hp hq
-  simp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and_iff]
+  simp_all only [ManyOneEquiv, of_le_of, of_eq_of, true_and]
 
 private theorem le_trans {d₁ d₂ d₃ : ManyOneDegree} : d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃ := by
   induction d₁ using ManyOneDegree.ind_on
diff --git a/Mathlib/Computability/RegularExpressions.lean b/Mathlib/Computability/RegularExpressions.lean
index 95abb4a8e3941..cbcdb86be9877 100644
--- a/Mathlib/Computability/RegularExpressions.lean
+++ b/Mathlib/Computability/RegularExpressions.lean
@@ -215,14 +215,15 @@ theorem char_rmatch_iff (a : α) (x : List α) : rmatch (char a) x ↔ x = [a] :
     · simp [List.singleton_inj]; tauto
   · rw [rmatch, rmatch, deriv]
     split_ifs with h
-    · simp only [deriv_one, zero_rmatch, cons.injEq, and_false]
-    · simp only [deriv_zero, zero_rmatch, cons.injEq, and_false]
+    · simp only [deriv_one, zero_rmatch, cons.injEq, and_false, reduceCtorEq]
+    · simp only [deriv_zero, zero_rmatch, cons.injEq, and_false, reduceCtorEq]
 
 theorem add_rmatch_iff (P Q : RegularExpression α) (x : List α) :
     (P + Q).rmatch x ↔ P.rmatch x ∨ Q.rmatch x := by
-  induction' x with _ _ ih generalizing P Q
-  · simp only [rmatch, matchEpsilon, Bool.or_eq_true_iff]
-  · repeat rw [rmatch]
+  induction x generalizing P Q with
+  | nil => simp only [rmatch, matchEpsilon, Bool.or_eq_true_iff]
+  | cons _ _ ih =>
+    repeat rw [rmatch]
     rw [deriv_add]
     exact ih _ _
 
@@ -294,7 +295,7 @@ theorem star_rmatch_iff (P : RegularExpression α) :
         · intro t' ht'
           cases ht' with
           | head ht' =>
-            simp only [ne_eq, not_false_iff, true_and, rmatch]
+            simp only [ne_eq, not_false_iff, true_and, rmatch, reduceCtorEq]
             exact ht
           | tail _ ht' => exact helem t' ht'
     · rintro ⟨S, hsum, helem⟩
@@ -305,7 +306,7 @@ theorem star_rmatch_iff (P : RegularExpression α) :
         · exact ⟨[], [], by tauto⟩
         · cases' t' with b t
           · simp only [forall_eq_or_imp, List.mem_cons] at helem
-            simp only [eq_self_iff_true, not_true, Ne, false_and_iff] at helem
+            simp only [eq_self_iff_true, not_true, Ne, false_and] at helem
           simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum
           refine ⟨t, U.join, hsum.2, ?_, ?_⟩
           · specialize helem (b :: t) (by simp)
diff --git a/Mathlib/Computability/TMToPartrec.lean b/Mathlib/Computability/TMToPartrec.lean
index b4e1c66e921db..79a322e54c754 100644
--- a/Mathlib/Computability/TMToPartrec.lean
+++ b/Mathlib/Computability/TMToPartrec.lean
@@ -248,7 +248,7 @@ theorem exists_code.comp {m n} {f : Vector ℕ n →. ℕ} {g : Fin n → Vector
   · obtain ⟨cf, hf⟩ := hf
     exact
       ⟨cf.comp cg, fun v => by
-        simp [hg, hf, map_bind, seq_bind_eq, Function.comp]
+        simp [hg, hf, map_bind, seq_bind_eq, Function.comp_def]
         rfl⟩
   clear hf f; induction' n with n IH
   · exact ⟨nil, fun v => by simp [Vector.mOfFn, Bind.bind]; rfl⟩
@@ -340,8 +340,8 @@ theorem exists_code {n} {f : Vector ℕ n →. ℕ} (hf : Nat.Partrec' f) :
       have := PFun.mem_fix_iff.1 h2
       simp only [hf, Part.bind_some] at this
       split_ifs at this with h
-      · simp only [List.headI_nil, List.headI_cons, exists_false, or_false_iff, Part.mem_some_iff,
-          List.tail_cons, false_and_iff, Sum.inl.injEq] at this
+      · simp only [List.headI_nil, List.headI_cons, exists_false, or_false, Part.mem_some_iff,
+          List.tail_cons, false_and, Sum.inl.injEq, reduceCtorEq] at this
         subst this
         exact ⟨_, ⟨h, @(hm)⟩, rfl⟩
       · refine IH (n.succ::v.val) (by simp_all) _ rfl fun m h' => ?_
@@ -511,10 +511,14 @@ def Cont.then : Cont → Cont → Cont
   | Cont.fix f k => fun k' => Cont.fix f (k.then k')
 
 theorem Cont.then_eval {k k' : Cont} {v} : (k.then k').eval v = k.eval v >>= k'.eval := by
-  induction' k with _ _ _ _ _ _ _ _ _ k_ih _ _ k_ih generalizing v <;>
-    simp only [Cont.eval, Cont.then, bind_assoc, pure_bind, *]
-  · simp only [← k_ih]
-  · split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]]
+  induction k generalizing v with
+  | halt => simp only [Cont.eval, Cont.then, pure_bind]
+  | cons₁ => simp only [Cont.eval, bind_assoc, *]
+  | cons₂ => simp only [Cont.eval, *]
+  | comp _ _ k_ih => simp only [Cont.eval, bind_assoc, ← k_ih]
+  | fix _ _ k_ih =>
+    simp only [Cont.eval, *]
+    split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]]
 
 /-- The `then k` function is a "configuration homomorphism". Its operation on states is to append
 `k` to the continuation of a `Cfg.ret` state, and to run `k` on `v` if we are in the `Cfg.halt v`
@@ -898,6 +902,7 @@ def natEnd : Γ' → Bool
   | Γ'.consₗ => true
   | Γ'.cons => true
   | _ => false
+attribute [nolint simpNF] natEnd.eq_3
 
 /-- Pop a value from the stack and place the result in local store. -/
 @[simp]
@@ -1233,7 +1238,7 @@ theorem move_ok {p k₁ k₂ q s L₁ o L₂} {S : K' → List Γ'} (h₁ : k₁
       rfl
     simp only [splitAtPred, Option.elim, List.head?, List.tail_cons, Option.iget_some] at e ⊢
     revert e; cases p a <;> intro e <;>
-      simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e ⊢
+      simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and, reduceCtorEq] at e ⊢
     simp only [e]
     rfl
   · refine TransGen.head rfl ?_
@@ -1288,7 +1293,7 @@ theorem clear_ok {p k q s L₁ o L₂} {S : K' → List Γ'} (e : splitAtPred p
       rfl
     simp only [splitAtPred, Option.elim, List.head?, List.tail_cons] at e ⊢
     revert e; cases p a <;> intro e <;>
-      simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and] at e ⊢
+      simp only [cond_false, cond_true, Prod.mk.injEq, true_and, false_and, reduceCtorEq] at e ⊢
     rcases e with ⟨e₁, e₂⟩
     rw [e₁, e₂]
   · refine TransGen.head rfl ?_
@@ -1405,7 +1410,7 @@ theorem succ_ok {q s n} {c d : List Γ'} :
         Reaches₁ (TM2.step tr) ⟨some q.succ, s, K'.elim (trPosNum a ++ [Γ'.cons]) l₁ c d⟩
           ⟨some (unrev q), s', K'.elim (l₂' ++ [Γ'.cons]) l₁' c d⟩ by
     obtain ⟨l₁', l₂', s', e, h⟩ := this []
-    simp? [List.reverseAux] at e says simp only [List.reverseAux] at e
+    simp? [List.reverseAux] at e says simp only [List.reverseAux, List.reverseAux_eq] at e
     refine h.trans ?_
     convert unrev_ok using 2
     simp [e, List.reverseAux_eq]
@@ -1597,10 +1602,13 @@ def trStmts₁ : Λ' → Finset Λ'
   | Q@(Λ'.ret _) => {Q}
 
 theorem trStmts₁_trans {q q'} : q' ∈ trStmts₁ q → trStmts₁ q' ⊆ trStmts₁ q := by
-  induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q₁ q₂ q₁_ih q₂_ih _ <;>
+  induction q with
+  | move _ _ _ q q_ih => _ | clear _ _ q q_ih => _ | copy q q_ih => _ | push _ _ q q_ih => _
+  | read q q_ih => _ | succ q q_ih => _ | pred q₁ q₂ q₁_ih q₂_ih => _ | ret => _ <;>
+  all_goals
     simp (config := { contextual := true }) only [trStmts₁, Finset.mem_insert, Finset.mem_union,
-      or_imp, Finset.mem_singleton, Finset.Subset.refl, imp_true_iff, true_and_iff]
-  repeat exact fun h => Finset.Subset.trans (q_ih h) (Finset.subset_insert _ _)
+      or_imp, Finset.mem_singleton, Finset.Subset.refl, imp_true_iff, true_and]
+    repeat exact fun h => Finset.Subset.trans (q_ih h) (Finset.subset_insert _ _)
   · simp
     intro s h x h'
     simp only [Finset.mem_biUnion, Finset.mem_univ, true_and, Finset.mem_insert]
@@ -1791,7 +1799,9 @@ theorem ret_supports {S k} (H₁ : contSupp k ⊆ S) : TM2.SupportsStmt S (tr (
 theorem trStmts₁_supports {S q} (H₁ : (q : Λ').Supports S) (HS₁ : trStmts₁ q ⊆ S) :
     Supports (trStmts₁ q) S := by
   have W := fun {q} => trStmts₁_self q
-  induction' q with _ _ _ q q_ih _ _ q q_ih q q_ih _ _ q q_ih q q_ih q q_ih q₁ q₂ q₁_ih q₂_ih _ <;>
+  induction q with
+  | move _ _ _ q q_ih => _ | clear _ _ q q_ih => _ | copy q q_ih => _ | push _ _ q q_ih => _
+  | read q q_ih => _ | succ q q_ih => _ | pred q₁ q₂ q₁_ih q₂_ih => _ | ret => _ <;>
     simp [trStmts₁, -Finset.singleton_subset_iff] at HS₁ ⊢
   any_goals
     cases' Finset.insert_subset_iff.1 HS₁ with h₁ h₂
@@ -1906,3 +1916,5 @@ end
 end PartrecToTM2
 
 end Turing
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/Computability/TuringMachine.lean b/Mathlib/Computability/TuringMachine.lean
index 99a30824f06d0..73b365b5200ee 100644
--- a/Mathlib/Computability/TuringMachine.lean
+++ b/Mathlib/Computability/TuringMachine.lean
@@ -263,7 +263,7 @@ def ListBlank.nth {Γ} [Inhabited Γ] (l : ListBlank Γ) (n : ℕ) : Γ := by
   rw [List.getI_eq_default _ h]
   rcases le_or_lt _ n with h₂ | h₂
   · rw [List.getI_eq_default _ h₂]
-  rw [List.getI_eq_get _ h₂, List.get_eq_getElem, List.getElem_append_right' h,
+  rw [List.getI_eq_get _ h₂, List.get_eq_getElem, List.getElem_append_right h,
     List.getElem_replicate]
 
 @[simp]
@@ -310,7 +310,7 @@ theorem ListBlank.nth_modifyNth {Γ} [Inhabited Γ] (f : Γ → Γ) (n i) (L : L
     (L.modifyNth f n).nth i = if i = n then f (L.nth i) else L.nth i := by
   induction' n with n IH generalizing i L
   · cases i <;> simp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth,
-      ListBlank.nth_succ, if_false, ListBlank.tail_cons]
+      ListBlank.nth_succ, if_false, ListBlank.tail_cons, reduceCtorEq]
   · cases i
     · rw [if_neg (Nat.succ_ne_zero _).symm]
       simp only [ListBlank.nth_zero, ListBlank.head_cons, ListBlank.modifyNth]
@@ -1383,7 +1383,7 @@ theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
     cases' q' with q' v'
     simp only [trStmts, Finset.mem_coe] at h₂ ⊢
     rw [Finset.mem_product] at h₂ ⊢
-    simp only [Finset.mem_univ, and_true_iff] at h₂ ⊢
+    simp only [Finset.mem_univ, and_true] at h₂ ⊢
     cases q'; · exact Multiset.mem_cons_self _ _
     simp only [tr, Option.mem_def] at h₁
     have := TM1.stmts_supportsStmt ss h₂
@@ -1614,7 +1614,8 @@ theorem stepAux_write (q : Stmt'₁) (v : σ) (a b : Γ) (L R : ListBlank Γ) :
   induction' l₂ with a l₂ IH generalizing l₁ l₂'
   · cases List.length_eq_zero.1 e
     rfl
-  cases' l₂' with b l₂' <;> simp only [List.length_nil, List.length_cons, Nat.succ_inj'] at e
+  cases' l₂' with b l₂' <;>
+    simp only [List.length_nil, List.length_cons, Nat.succ_inj', reduceCtorEq] at e
   rw [List.reverseAux, ← IH (a :: l₁) l₂' e]
   simp only [stepAux, ListBlank.append, Tape.write_mk', Tape.move_right_mk', ListBlank.head_cons,
     ListBlank.tail_cons]
@@ -1733,7 +1734,7 @@ theorem tr_supports [Inhabited Λ] {S : Finset Λ} (ss : Supports M S) :
       cases d <;> simp only [trNormal, iterate, supportsStmt_move, IH]
     | write f q IH =>
       unfold writes at hw ⊢
-      simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and_iff]
+      simp only [Finset.mem_image, Finset.mem_union, Finset.mem_univ, exists_prop, true_and]
         at hw ⊢
       replace IH := IH hs fun q hq ↦ hw q (Or.inr hq)
       refine ⟨supportsStmt_read _ fun a _ s ↦ hw _ (Or.inl ⟨_, rfl⟩), fun q' hq ↦ ?_⟩
@@ -2353,7 +2354,7 @@ theorem tr_respects_aux₂ [DecidableEq K] {k : K} {q : Stmt₂₁} {v : σ} {S
     · refine
         ⟨_, fun k' ↦ ?_, by
           erw [List.length_cons, Tape.move_right_n_head, Tape.mk'_nth_nat, addBottom_nth_succ_fst,
-            cond, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head,
+            cond_false, iterate_succ', Function.comp, Tape.move_right_left, Tape.move_right_n_head,
             Tape.mk'_nth_nat, Tape.write_move_right_n fun a : Γ' ↦ (a.1, update a.2 k none),
             addBottom_modifyNth fun a ↦ update a k none, addBottom_nth_snd,
             stk_nth_val _ (hL k), e,
@@ -2487,7 +2488,6 @@ theorem trCfg_init (k) (L : List (Γ k)) : TrCfg (TM2.init k L) (TM1.init (trIni
       rw [ListBlank.nth_mk, List.getI_eq_iget_get?, List.map, List.reverse_nil]
       cases L.reverse.get? i <;> rfl
   · rw [trInit, TM1.init]
-    dsimp only
     congr <;> cases L.reverse <;> try rfl
     simp only [List.map_map, List.tail_cons, List.map]
     rfl
@@ -2569,3 +2569,5 @@ end
 end TM2to1
 
 end Turing
+
+set_option linter.style.longFile 2700
diff --git a/Mathlib/Condensed/Basic.lean b/Mathlib/Condensed/Basic.lean
index fb904e7ecb227..8d384f0d7eb88 100644
--- a/Mathlib/Condensed/Basic.lean
+++ b/Mathlib/Condensed/Basic.lean
@@ -76,7 +76,7 @@ namespace CondensedSet
 -- Note: `simp` can prove this when stated for `Condensed C` for a concrete category `C`.
 -- However, it doesn't seem to see through the abbreviation `CondensedSet`
 @[simp]
-lemma hom_naturality_apply {X Y : CondensedSet.{u}} (f : X ⟶ Y)  {S T : CompHausᵒᵖ} (g : S ⟶ T)
+lemma hom_naturality_apply {X Y : CondensedSet.{u}} (f : X ⟶ Y) {S T : CompHausᵒᵖ} (g : S ⟶ T)
     (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) :=
   NatTrans.naturality_apply f.val g x
 
diff --git a/Mathlib/Condensed/Discrete/Basic.lean b/Mathlib/Condensed/Discrete/Basic.lean
index 5f2ed84dfda6a..d7964352317fa 100644
--- a/Mathlib/Condensed/Discrete/Basic.lean
+++ b/Mathlib/Condensed/Discrete/Basic.lean
@@ -12,13 +12,13 @@ import Mathlib.Condensed.Light.Basic
 # Discrete-underlying adjunction
 
 Given a category `C` with sheafification with respect to the coherent topology on compact Hausdorff
-spaces, we define a functor `C ⥤ Condensed C` which associates to an object of `C` the
+spaces, we define a functor `C ⥤ Condensed C` which associates to an object of `C` the
 corresponding "discrete" condensed object (see `Condensed.discrete`).
 
-In `Condensed.discreteUnderlyingAdj` we prove that this functor is left adjoint to the forgetful
+In `Condensed.discreteUnderlyingAdj` we prove that this functor is left adjoint to the forgetful
 functor from `Condensed C` to `C`.
 
-We also give the variant `LightCondensed.discreteUnderlyingAdj` for light condensed objects.
+We also give the variant `LightCondensed.discreteUnderlyingAdj` for light condensed objects.
 -/
 
 universe u v w
@@ -30,13 +30,13 @@ namespace Condensed
 variable (C : Type w) [Category.{u+1} C] [HasWeakSheafify (coherentTopology CompHaus.{u}) C]
 
 /--
-The discrete condensed object associated to an object of `C` is the constant sheaf at that object.
+The discrete condensed object associated to an object of `C` is the constant sheaf at that object.
 -/
 @[simps!]
 noncomputable def discrete : C ⥤ Condensed.{u} C := constantSheaf _ C
 
 /--
-The underlying object of a condensed object in `C` is the condensed object evaluated at a point.
+The underlying object of a condensed object in `C` is the condensed object evaluated at a point.
 This can be viewed as a sort of forgetful functor from `Condensed C` to `C`
 -/
 @[simps!]
@@ -45,7 +45,7 @@ noncomputable def underlying : Condensed.{u} C ⥤ C :=
 
 /--
 Discreteness is left adjoint to the forgetful functor. When `C` is `Type*`, this is analogous to
-`TopCat.adj₁ : TopCat.discrete ⊣ forget TopCat`.  
+`TopCat.adj₁ : TopCat.discrete ⊣ forget TopCat`.
 -/
 noncomputable def discreteUnderlyingAdj : discrete C ⊣ underlying C :=
   constantSheafAdj _ _ CompHaus.isTerminalPUnit
@@ -57,14 +57,14 @@ namespace LightCondensed
 variable (C : Type w) [Category.{u} C] [HasSheafify (coherentTopology LightProfinite.{u}) C]
 
 /--
-The discrete light condensed object associated to an object of `C` is the constant sheaf at that
+The discrete light condensed object associated to an object of `C` is the constant sheaf at that
 object.
 -/
 @[simps!]
 noncomputable def discrete : C ⥤ LightCondensed.{u} C := constantSheaf _ C
 
 /--
-The underlying object of a condensed object in `C` is the light condensed object evaluated at a
+The underlying object of a condensed object in `C` is the light condensed object evaluated at a
 point. This can be viewed as a sort of forgetful functor from `LightCondensed C` to `C`
 -/
 @[simps!]
@@ -73,7 +73,7 @@ noncomputable def underlying : LightCondensed.{u} C ⥤ C :=
 
 /--
 Discreteness is left adjoint to the forgetful functor. When `C` is `Type*`, this is analogous to
-`TopCat.adj₁ : TopCat.discrete ⊣ forget TopCat`.  
+`TopCat.adj₁ : TopCat.discrete ⊣ forget TopCat`.
 -/
 noncomputable def discreteUnderlyingAdj : discrete C ⊣ underlying C :=
   constantSheafAdj _ _ CompHausLike.isTerminalPUnit
diff --git a/Mathlib/Condensed/Discrete/LocallyConstant.lean b/Mathlib/Condensed/Discrete/LocallyConstant.lean
new file mode 100644
index 0000000000000..ca028c4a70293
--- /dev/null
+++ b/Mathlib/Condensed/Discrete/LocallyConstant.lean
@@ -0,0 +1,427 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Condensed.Discrete.Basic
+import Mathlib.Condensed.TopComparison
+import Mathlib.Topology.Category.CompHausLike.SigmaComparison
+import Mathlib.Topology.FiberPartition
+/-!
+
+# The sheaf of locally constant maps on `CompHausLike P`
+
+This file proves that under suitable conditions, the functor from the category of sets to the
+category of sheaves for the coherent topology on `CompHausLike P`, given by mapping a set to the
+sheaf of locally constant maps to it, is left adjoint to the "underlying set" functor (evaluation
+at the point).
+
+We apply this to prove that the constant sheaf functor into (light) condensed sets is isomorphic to
+the functor of sheaves of locally constant maps described above.
+
+## Proof sketch
+
+The hard part of this adjunction is to define the counit. Its components are defined as follows:
+
+Let `S : CompHausLike P` and let `Y` be a finite-product preserving presheaf on `CompHausLike P`
+(e.g. a sheaf for the coherent topology). We need to define a map `LocallyConstant S Y(*) ⟶ Y(S)`.
+Given a locally constant map `f : S → Y(*)`, let `S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding
+decomposition of `S` into the fibers. Let `yᵢ ∈ Y(*)` denote the value of `f` on `Sᵢ` and denote
+by `gᵢ` the canonical map `Y(*) → Y(Sᵢ)`. Our map then takes `f` to the image of
+`(g₁(y₁), ⋯, gₙ(yₙ))` under the isomorphism `Y(S₁) × ⋯ × Y(Sₙ) ≅ Y(S₁ ⊔ ⋯ ⊔ Sₙ) = Y(S)`.
+
+Now we need to prove that the counit is natural in `S : CompHausLike P` and
+`Y : Sheaf  (coherentTopology (CompHausLike P)) (Type _)`. There are two key lemmas in all
+naturality proofs in this file (both lemmas are in the `CompHausLike.LocallyConstant` namespace):
+
+* `presheaf_ext`: given `S`, `Y` and `f : LocallyConstant S Y(*)` like above, another presheaf
+  `X`, and two elements `x y : X(S)`, to prove that `x = y` it suffices to prove that for every
+  inclusion map `ιᵢ : Sᵢ ⟶ S`,  `X(ιᵢ)(x) = X(ιᵢ)(y)`.
+  Here it is important that we set everything up in such a way that the `Sᵢ` are literally subtypes
+  of `S`.
+
+* `incl_of_counitAppApp`: given  `S`, `Y` and `f : LocallyConstant S Y(*)` like above, we have
+  `Y(ιᵢ)(ε_{S, Y}(f)) = gᵢ(yᵢ)` where `ε` denotes the counit and the other notation is like above.
+
+## Main definitions
+
+* `CompHausLike.LocallyConstant.functor`: the functor from the category of sets to the category of
+  sheaves for the coherent topology on `CompHausLike P`, which takes a set `X` to
+  `LocallyConstant - X`
+  - `CondensedSet.LocallyConstant.functor` is the above functor in the case of condensed sets.
+  - `LightCondSet.LocallyConstant.functor` is the above functor in the case of light condensed sets.
+
+* `CompHausLike.LocallyConstant.adjunction`: the functor described above is left adjoint to the
+  "underlying set" functor `(sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u+1}⟩`, which takes
+  a sheaf `X` to the set `X(*)`.
+
+* `CondensedSet.LocallyConstant.iso`: the functor `CondensedSet.LocallyConstant.functor` is
+  isomorphic to the functor `Condensed.discrete (Type _)` (the constant sheaf functor from sets to
+  condensed sets).
+
+* `LightCondSet.LocallyConstant.iso`: the functor `LightCondSet.LocallyConstant.functor` is
+  isomorphic to the functor `LightCondensed.discrete (Type _)` (the constant sheaf functor from sets
+  to light condensed sets).
+
+-/
+
+universe u w
+
+open CategoryTheory Limits LocallyConstant TopologicalSpace.Fiber Opposite Function Fiber
+
+attribute [local instance] ConcreteCategory.instFunLike
+
+variable {P : TopCat.{u} → Prop}
+
+namespace CompHausLike.LocallyConstant
+
+/--
+The functor from the category of sets to presheaves on `CompHausLike P` given by locally constant
+maps.
+-/
+@[simps]
+def functorToPresheaves : Type (max u w) ⥤ ((CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w) where
+  obj X := {
+    obj := fun ⟨S⟩ ↦ LocallyConstant S X
+    map := fun f g ↦ g.comap f.unop }
+  map f := { app := fun S t ↦ t.map f }
+
+/--
+Locally constant maps are the same as continuous maps when the target is equipped with the discrete
+topology
+-/
+@[simps]
+def locallyConstantIsoContinuousMap (Y X : Type*) [TopologicalSpace Y] :
+    LocallyConstant Y X ≅ C(Y, TopCat.discrete.obj X) :=
+  letI : TopologicalSpace X := ⊥
+  haveI : DiscreteTopology X := ⟨rfl⟩
+  { hom := fun f ↦ (f : C(Y, X))
+    inv := fun f ↦ ⟨f, (IsLocallyConstant.iff_continuous f).mpr f.2⟩ }
+
+section Adjunction
+
+variable [∀ (S : CompHausLike.{u} P) (p : S → Prop), HasProp P (Subtype p)]
+
+section
+
+variable {Q : CompHausLike.{u} P} {Z : Type max u w} (r : LocallyConstant Q Z) (a : Fiber r)
+
+/-- A fiber of a locally constant map as a `CompHausLike P`. -/
+def fiber : CompHausLike.{u} P := CompHausLike.of P a.val
+
+instance : HasProp P (fiber r a) := inferInstanceAs (HasProp P (Subtype _))
+
+/-- The inclusion map from a component of the coproduct induced by `f` into `S`. -/
+def sigmaIncl : fiber r a ⟶ Q := TopologicalSpace.Fiber.sigmaIncl _ a
+
+/-- The canonical map from the coproduct induced by `f` to `S` as an isomorphism in
+`CompHausLike P`. -/
+noncomputable def sigmaIso [HasExplicitFiniteCoproducts.{u} P] : (finiteCoproduct (fiber r)) ≅ Q :=
+  isoOfBijective (sigmaIsoHom r) ⟨sigmaIsoHom_inj r, sigmaIsoHom_surj r⟩
+
+lemma sigmaComparison_comp_sigmaIso [HasExplicitFiniteCoproducts.{u} P]
+    (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w) :
+    (X.mapIso (sigmaIso r).op).hom ≫ sigmaComparison X (fun a ↦ (fiber r a).1) ≫
+      (fun g ↦ g a) = X.map (sigmaIncl r a).op := by
+  ext
+  simp only [Functor.mapIso_hom, Iso.op_hom, types_comp_apply, sigmaComparison, coe_of,
+    ← FunctorToTypes.map_comp_apply]
+  rfl
+
+end
+
+variable {S : CompHausLike.{u} P} {Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w}
+  [HasProp P PUnit.{u+1}] (f : LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u+1}))))
+
+/-- The projection of the counit. -/
+noncomputable def counitAppAppImage : (a : Fiber f) → Y.obj ⟨fiber f a⟩ :=
+  fun a ↦ Y.map (CompHausLike.isTerminalPUnit.from _).op a.image
+
+/--
+The counit is defined as follows: given a locally constant map `f : S → Y(*)`, let
+`S = S₁ ⊔ ⋯ ⊔ Sₙ` be the corresponding decomposition of `S` into the fibers. We need to provide an
+element of `Y(S)`. It suffices to provide an element of `Y(Sᵢ)` for all `i`. Let `yᵢ ∈ Y(*)` denote
+the value of `f` on `Sᵢ`. Our desired element is the image of `yᵢ` under the canonical map
+`Y(*) → Y(Sᵢ)`.
+-/
+noncomputable def counitAppApp (S : CompHausLike.{u} P) (Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w)
+    [PreservesFiniteProducts Y] [HasExplicitFiniteCoproducts.{u} P] :
+    LocallyConstant S (Y.obj (op (CompHausLike.of P PUnit.{u+1}))) ⟶ Y.obj ⟨S⟩ :=
+  fun r ↦ ((inv (sigmaComparison Y (fun a ↦ (fiber r a).1))) ≫
+    (Y.mapIso (sigmaIso r).op).inv) (counitAppAppImage r)
+
+-- This is the key lemma to prove naturality of the counit:
+/--
+To check equality of two elements of `X(S)`, it suffices to check equality after composing with
+each `X(S) → X(Sᵢ)`.
+-/
+lemma presheaf_ext (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w)
+    [PreservesFiniteProducts X] (x y : X.obj ⟨S⟩)
+    [HasExplicitFiniteCoproducts.{u} P]
+    (h : ∀ (a : Fiber f), X.map (sigmaIncl f a).op x = X.map (sigmaIncl f a).op y) : x = y := by
+  apply injective_of_mono (X.mapIso (sigmaIso f).op).hom
+  apply injective_of_mono (sigmaComparison X (fun a ↦ (fiber f a).1))
+  ext a
+  specialize h a
+  rw [← sigmaComparison_comp_sigmaIso] at h
+  exact h
+
+lemma incl_of_counitAppApp [PreservesFiniteProducts Y] [HasExplicitFiniteCoproducts.{u} P]
+    (a : Fiber f) : Y.map (sigmaIncl f a).op (counitAppApp S Y f) = counitAppAppImage f a := by
+  rw [← sigmaComparison_comp_sigmaIso, Functor.mapIso_hom, Iso.op_hom, types_comp_apply]
+  simp only [counitAppApp, Functor.mapIso_inv, ← Iso.op_hom, types_comp_apply,
+    ← FunctorToTypes.map_comp_apply, Iso.inv_hom_id, FunctorToTypes.map_id_apply]
+  exact congrFun (inv_hom_id_apply (asIso (sigmaComparison Y (fun a ↦ (fiber f a).1)))
+    (counitAppAppImage f)) _
+
+variable {T : CompHausLike.{u} P} (g : T ⟶ S)
+
+/--
+This is an auxiliary definition, the details do not matter. What's important is that this map exists
+so that the lemma `incl_comap` works.
+-/
+def componentHom (a : Fiber (f.comap g)) :
+    fiber _ a ⟶ fiber _ (Fiber.mk f (g a.preimage)) where
+  toFun x := ⟨g x.val, by
+    simp only [Fiber.mk, Set.mem_preimage, Set.mem_singleton_iff]
+    convert map_eq_image _ _ x
+    exact map_preimage_eq_image_map _ _ a⟩
+  continuous_toFun := by exact Continuous.subtype_mk (g.continuous.comp continuous_subtype_val) _
+    -- term mode gives "unknown free variable" error.
+
+lemma incl_comap {S T : (CompHausLike P)ᵒᵖ}
+    (f : LocallyConstant S.unop (Y.obj (op (CompHausLike.of P PUnit.{u+1}))))
+      (g : S ⟶ T) (a : Fiber (f.comap g.unop)) :
+        g ≫ (sigmaIncl (f.comap g.unop) a).op =
+          (sigmaIncl f _).op ≫ (componentHom f g.unop a).op :=
+  rfl
+
+/-- The counit is natural in `S : CompHausLike P` -/
+@[simps!]
+noncomputable def counitApp [HasExplicitFiniteCoproducts.{u} P]
+    (Y : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w) [PreservesFiniteProducts Y] :
+    (functorToPresheaves.obj (Y.obj (op (CompHausLike.of P PUnit.{u+1})))) ⟶ Y where
+  app := fun ⟨S⟩ ↦ counitAppApp S Y
+  naturality := by
+    intro S T g
+    ext f
+    apply presheaf_ext (f.comap g.unop)
+    intro a
+    simp only [op_unop, functorToPresheaves_obj_obj, types_comp_apply, functorToPresheaves_obj_map,
+      incl_of_counitAppApp, ← FunctorToTypes.map_comp_apply, incl_comap]
+    simp only [FunctorToTypes.map_comp_apply, incl_of_counitAppApp]
+    simp only [counitAppAppImage, ← FunctorToTypes.map_comp_apply, ← op_comp,
+      terminal.comp_from]
+    apply congrArg
+    exact image_eq_image_mk (g := g.unop) (a := a)
+
+variable (P) (X : TopCat.{max u w})
+    [HasExplicitFiniteCoproducts.{0} P] [HasExplicitPullbacks P]
+    (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f)
+
+/-- `locallyConstantIsoContinuousMap` is a natural isomorphism. -/
+noncomputable def functorToPresheavesIso (X : Type (max u w)) :
+    functorToPresheaves.{u, w}.obj X ≅ ((TopCat.discrete.obj X).toSheafCompHausLike P hs).val :=
+  NatIso.ofComponents (fun S ↦ locallyConstantIsoContinuousMap _ _)
+
+/-- `CompHausLike.LocallyConstant.functorToPresheaves` lands in sheaves. -/
+@[simps]
+def functor :
+    haveI := CompHausLike.preregular hs
+    Type (max u w) ⥤ Sheaf (coherentTopology (CompHausLike.{u} P)) (Type (max u w)) where
+  obj X := {
+    val := functorToPresheaves.{u, w}.obj X
+    cond := by
+      rw [Presheaf.isSheaf_of_iso_iff (functorToPresheavesIso P hs X)]
+      exact ((TopCat.discrete.obj X).toSheafCompHausLike P hs).cond }
+  map f := ⟨functorToPresheaves.{u, w}.map f⟩
+
+/--
+`CompHausLike.LocallyConstant.functor` is naturally isomorphic to the restriction of
+`topCatToSheafCompHausLike` to discrete topological spaces.
+-/
+noncomputable def functorIso :
+    functor.{u, w} P hs ≅ TopCat.discrete.{max w u} ⋙ topCatToSheafCompHausLike P hs :=
+  NatIso.ofComponents (fun X ↦ (fullyFaithfulSheafToPresheaf _ _).preimageIso
+    (functorToPresheavesIso P hs X))
+
+/-- The counit is natural in both `S : CompHausLike P` and
+`Y : Sheaf (coherentTopology (CompHausLike P)) (Type (max u w))` -/
+@[simps]
+noncomputable def counit [HasExplicitFiniteCoproducts.{u} P] : haveI := CompHausLike.preregular hs
+    (sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u+1}⟩ ⋙ functor.{u, w} P hs ⟶
+        𝟭 (Sheaf (coherentTopology (CompHausLike.{u} P)) (Type (max u w))) where
+  app X := haveI := CompHausLike.preregular hs
+    ⟨counitApp X.val⟩
+  naturality X Y g := by
+    have := CompHausLike.preregular hs
+    apply Sheaf.hom_ext
+    simp only [functor, id_eq, eq_mpr_eq_cast, Functor.comp_obj, Functor.flip_obj_obj,
+      sheafToPresheaf_obj, Functor.id_obj, Functor.comp_map, Functor.flip_obj_map,
+      sheafToPresheaf_map, Sheaf.instCategorySheaf_comp_val, Functor.id_map]
+    ext S (f : LocallyConstant _ _)
+    simp only [FunctorToTypes.comp, counitApp_app]
+    apply presheaf_ext (f.map (g.val.app (op (CompHausLike.of P PUnit.{u+1}))))
+    intro a
+    simp only [op_unop, functorToPresheaves_map_app, incl_of_counitAppApp]
+    apply presheaf_ext (f.comap (sigmaIncl _ _))
+    intro b
+    simp only [counitAppAppImage, ← FunctorToTypes.map_comp_apply, ← op_comp, CompHausLike.coe_of,
+      map_apply, IsTerminal.comp_from, ← map_preimage_eq_image_map]
+    change (_ ≫ Y.val.map _) _ = (_ ≫ Y.val.map _) _
+    simp only [← g.val.naturality,
+      show sigmaIncl (f.comap (sigmaIncl (f.map _) a)) b ≫ sigmaIncl (f.map _) a =
+        (sigmaInclIncl f _ a b) ≫ sigmaIncl f (Fiber.mk f _) from rfl]
+    simp only [op_comp, Functor.map_comp, types_comp_apply, incl_of_counitAppApp]
+    simp only [counitAppAppImage, ← FunctorToTypes.map_comp_apply, ← op_comp, terminal.comp_from]
+    rw [mk_image]
+    change (X.val.map _ ≫ _) _ = (X.val.map _ ≫ _) _
+    simp only [g.val.naturality]
+    simp only [types_comp_apply]
+    have := map_preimage_eq_image (f := g.val.app _ ∘ f) (a := a)
+    simp only [Function.comp_apply] at this
+    rw [this]
+    apply congrArg
+    symm
+    convert (b.preimage).prop
+    exact (mem_iff_eq_image (g.val.app _ ∘ f) _ _).symm
+
+/--
+The unit of the adjunciton is given by mapping each element to the corresponding constant map.
+-/
+@[simps]
+def unit : 𝟭 _ ⟶ functor P hs ⋙ (sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u+1}⟩ where
+  app X x := LocallyConstant.const _ x
+
+/-- The unit of the adjunction is an iso. -/
+noncomputable def unitIso : 𝟭 (Type max u w) ≅ functor.{u, w} P hs ⋙
+    (sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u+1}⟩ where
+  hom := unit P hs
+  inv := { app := fun X f ↦ f.toFun PUnit.unit }
+
+lemma adjunction_left_triangle [HasExplicitFiniteCoproducts.{u} P]
+    (X : Type max u w) : functorToPresheaves.{u, w}.map ((unit P hs).app X) ≫
+      ((counit P hs).app ((functor P hs).obj X)).val = 𝟙 (functorToPresheaves.obj X) := by
+  ext ⟨S⟩ (f : LocallyConstant _ X)
+  simp only [Functor.id_obj, Functor.comp_obj, FunctorToTypes.comp, NatTrans.id_app,
+    functorToPresheaves_obj_obj, types_id_apply]
+  simp only [counit, counitApp_app]
+  have := CompHausLike.preregular hs
+  apply presheaf_ext
+    (X := ((functor P hs).obj X).val) (Y := ((functor.{u, w} P hs).obj X).val)
+      (f.map ((unit P hs).app X))
+  intro a
+  erw [incl_of_counitAppApp]
+  simp only [functor_obj_val, functorToPresheaves_obj_obj, coe_of, Functor.id_obj,
+    counitAppAppImage, LocallyConstant.map_apply, functorToPresheaves_obj_map, Quiver.Hom.unop_op]
+  ext x
+  erw [← map_eq_image _ a x]
+  rfl
+
+/--
+`CompHausLike.LocallyConstant.functor` is left adjoint to the forgetful functor.
+-/
+@[simps]
+noncomputable def adjunction [HasExplicitFiniteCoproducts.{u} P] :
+    functor.{u, w} P hs ⊣ (sheafSections _ _).obj ⟨CompHausLike.of P PUnit.{u+1}⟩ where
+  unit := unit P hs
+  counit := counit P hs
+  left_triangle_components := by
+    intro X
+    simp only [Functor.comp_obj, Functor.id_obj, NatTrans.comp_app, Functor.flip_obj_obj,
+      sheafToPresheaf_obj, functor_obj_val, functorToPresheaves_obj_obj, coe_of, whiskerRight_app,
+      Functor.associator_hom_app, whiskerLeft_app, Category.id_comp, NatTrans.id_app']
+    apply Sheaf.hom_ext
+    rw [Sheaf.instCategorySheaf_comp_val, Sheaf.instCategorySheaf_id_val]
+    exact adjunction_left_triangle P hs X
+  right_triangle_components := by
+    intro X
+    ext (x : X.val.obj _)
+    simp only [Functor.comp_obj, Functor.id_obj, Functor.flip_obj_obj, sheafToPresheaf_obj,
+      FunctorToTypes.comp, whiskerLeft_app, unit_app, coe_of, Functor.associator_inv_app,
+      functor_obj_val, functorToPresheaves_obj_obj, types_id_apply, whiskerRight_app,
+      Functor.flip_obj_map, sheafToPresheaf_map, counit_app_val, counitApp_app, NatTrans.id_app']
+    have := CompHausLike.preregular hs
+    letI : PreservesFiniteProducts ((sheafToPresheaf (coherentTopology _) _).obj X) :=
+      inferInstanceAs (PreservesFiniteProducts (Sheaf.val _))
+    apply presheaf_ext ((unit P hs).app _ x)
+    intro a
+    erw [incl_of_counitAppApp]
+    simp only [sheafToPresheaf_obj, unit_app, coe_of, counitAppAppImage, coe_const]
+    erw [← map_eq_image _ a ⟨PUnit.unit, by simp [mem_iff_eq_image, ← map_preimage_eq_image]⟩]
+    rfl
+
+instance [HasExplicitFiniteCoproducts.{u} P] : IsIso (adjunction P hs).unit :=
+  inferInstanceAs (IsIso (unitIso P hs).hom)
+
+end Adjunction
+
+end CompHausLike.LocallyConstant
+
+section Condensed
+
+open Condensed CompHausLike
+
+namespace CondensedSet.LocallyConstant
+
+/-- The functor from sets to condensed sets given by locally constant maps into the set. -/
+abbrev functor : Type (u+1) ⥤ CondensedSet.{u} :=
+  CompHausLike.LocallyConstant.functor.{u, u+1} (P := fun _ ↦ True)
+    (hs := fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)
+
+/--
+`CondensedSet.LocallyConstant.functor` is isomorphic to `Condensed.discrete`
+(by uniqueness of adjoints).
+-/
+noncomputable def iso : functor ≅ discrete (Type (u+1)) :=
+  (LocallyConstant.adjunction _ _).leftAdjointUniq (discreteUnderlyingAdj _)
+
+/-- `CondensedSet.LocallyConstant.functor` is fully faithful. -/
+noncomputable def functorFullyFaithful : functor.FullyFaithful :=
+  (LocallyConstant.adjunction.{u, u+1} _ _).fullyFaithfulLOfIsIsoUnit
+
+noncomputable instance : functor.Faithful := functorFullyFaithful.faithful
+
+noncomputable instance : functor.Full := functorFullyFaithful.full
+
+instance : (discrete (Type _)).Faithful := Functor.Faithful.of_iso iso
+
+noncomputable instance : (discrete (Type _)).Full := Functor.Full.of_iso iso
+
+end CondensedSet.LocallyConstant
+
+namespace LightCondSet.LocallyConstant
+
+/-- The functor from sets to light condensed sets given by locally constant maps into the set. -/
+abbrev functor : Type u ⥤ LightCondSet.{u} :=
+  CompHausLike.LocallyConstant.functor.{u, u}
+    (P := fun X ↦ TotallyDisconnectedSpace X ∧ SecondCountableTopology X)
+    (hs := fun _ _ _ ↦ (LightProfinite.effectiveEpi_iff_surjective _).mp)
+
+instance (S : LightProfinite.{u}) (p : S → Prop) :
+    HasProp (fun X ↦ TotallyDisconnectedSpace X ∧ SecondCountableTopology X) (Subtype p) :=
+  ⟨⟨(inferInstance : TotallyDisconnectedSpace (Subtype p)),
+    (inferInstance : SecondCountableTopology {s | p s})⟩⟩
+
+/--
+`LightCondSet.LocallyConstant.functor` is isomorphic to `LightCondensed.discrete`
+(by uniqueness of adjoints).
+-/
+noncomputable def iso : functor ≅ LightCondensed.discrete (Type u) :=
+  (LocallyConstant.adjunction _ _).leftAdjointUniq (LightCondensed.discreteUnderlyingAdj _)
+
+/-- `LightCondSet.LocallyConstant.functor` is fully faithful. -/
+noncomputable def functorFullyFaithful : functor.{u}.FullyFaithful :=
+  (LocallyConstant.adjunction _ _).fullyFaithfulLOfIsIsoUnit
+
+instance : functor.{u}.Faithful := functorFullyFaithful.faithful
+
+instance : LightCondSet.LocallyConstant.functor.Full := functorFullyFaithful.full
+
+instance : (LightCondensed.discrete (Type u)).Faithful := Functor.Faithful.of_iso iso.{u}
+
+instance : (LightCondensed.discrete (Type u)).Full := Functor.Full.of_iso iso.{u}
+
+end LightCondSet.LocallyConstant
+
+end Condensed
diff --git a/Mathlib/Condensed/Discrete/Module.lean b/Mathlib/Condensed/Discrete/Module.lean
new file mode 100644
index 0000000000000..3bcf129729c30
--- /dev/null
+++ b/Mathlib/Condensed/Discrete/Module.lean
@@ -0,0 +1,281 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.CategoryTheory.Sites.ConstantSheaf
+import Mathlib.Condensed.Discrete.LocallyConstant
+import Mathlib.Condensed.Light.Module
+import Mathlib.Condensed.Module
+import Mathlib.Topology.LocallyConstant.Algebra
+/-!
+
+# Discrete condensed `R`-modules
+
+This file provides the necessary API to prove that a condensed `R`-module is discrete if and only
+if the underlying condensed set is (both for light condensed and condensed).
+
+That is, it defines the functor `CondensedMod.LocallyConstant.functor` which takes an `R`-module to
+the condensed `R`-modules given by locally constant maps to it, and proves that this functor is
+naturally isomorphic to the constant sheaf functor (and the analogues for light condensed modules).
+-/
+
+universe w u
+
+open CategoryTheory LocallyConstant CompHausLike Functor Category Functor Opposite
+
+attribute [local instance] ConcreteCategory.instFunLike
+
+variable {P : TopCat.{u} → Prop}
+
+namespace CompHausLike.LocallyConstantModule
+
+variable (R : Type (max u w)) [Ring R]
+
+/--
+The functor from the category of `R`-modules to presheaves on `CompHausLike P` given by locally
+constant maps.
+-/
+@[simps]
+def functorToPresheaves : ModuleCat.{max u w} R ⥤ ((CompHausLike.{u} P)ᵒᵖ ⥤ ModuleCat R) where
+  obj X := {
+    obj := fun ⟨S⟩ ↦ ModuleCat.of R (LocallyConstant S X)
+    map := fun f ↦ comapₗ R f.unop }
+  map f := { app := fun S ↦ mapₗ R f }
+
+variable [HasExplicitFiniteCoproducts.{0} P] [HasExplicitPullbacks.{u} P]
+  (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f)
+
+/-- `CompHausLike.LocallyConstantModule.functorToPresheaves` lands in sheaves. -/
+@[simps]
+def functor : haveI := CompHausLike.preregular hs
+    ModuleCat R ⥤ Sheaf (coherentTopology (CompHausLike.{u} P)) (ModuleCat R) where
+  obj X := {
+    val := (functorToPresheaves.{w, u} R).obj X
+    cond := by
+      have := CompHausLike.preregular hs
+      apply Presheaf.isSheaf_coherent_of_hasPullbacks_of_comp
+        (s := CategoryTheory.forget (ModuleCat R))
+      exact ((CompHausLike.LocallyConstant.functor P hs).obj _).cond }
+  map f := ⟨(functorToPresheaves.{w, u} R).map f⟩
+
+end CompHausLike.LocallyConstantModule
+
+namespace CondensedMod.LocallyConstant
+
+open Condensed
+
+variable (R : Type (u+1)) [Ring R]
+
+/-- `functorToPresheaves` in the case of `CompHaus`. -/
+abbrev functorToPresheaves : ModuleCat.{u+1} R ⥤ (CompHaus.{u}ᵒᵖ ⥤ ModuleCat R) :=
+  CompHausLike.LocallyConstantModule.functorToPresheaves.{u+1, u} R
+
+/-- `functorToPresheaves` as a functor to condensed modules. -/
+abbrev functor : ModuleCat R ⥤ CondensedMod.{u} R :=
+  CompHausLike.LocallyConstantModule.functor.{u+1, u} R
+    (fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteAux₁ (M : ModuleCat.{u+1} R) :
+    M ≅ (ModuleCat.of R (LocallyConstant (CompHaus.of PUnit.{u+1}) M)) where
+  hom := constₗ R
+  inv := evalₗ R PUnit.unit
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteAux₂ (M : ModuleCat R) :
+    (discrete _).obj M ≅ (discrete _).obj
+      (ModuleCat.of R (LocallyConstant (CompHaus.of PUnit.{u+1}) M)) :=
+  (discrete _).mapIso (functorIsoDiscreteAux₁ R M)
+
+instance (M : ModuleCat R) : IsIso ((forget R).map
+    ((discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj M))) := by
+  dsimp [Condensed.forget, discreteUnderlyingAdj]
+  rw [← constantSheafAdj_counit_w]
+  refine @IsIso.comp_isIso _ _ _ _ _ _ _ inferInstance ?_
+  have : (constantSheaf (coherentTopology CompHaus) (Type (u + 1))).Faithful :=
+    inferInstanceAs (discrete _).Faithful
+  have : (constantSheaf (coherentTopology CompHaus) (Type (u + 1))).Full :=
+    inferInstanceAs (discrete _).Full
+  rw [← Sheaf.isConstant_iff_isIso_counit_app]
+  constructor
+  change _ ∈ (discrete _).essImage
+  rw [essImage_eq_of_natIso CondensedSet.LocallyConstant.iso.symm]
+  exact obj_mem_essImage CondensedSet.LocallyConstant.functor M
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteComponents (M : ModuleCat R) :
+    (discrete _).obj M ≅ (functor R).obj M :=
+  have : (Condensed.forget R).ReflectsIsomorphisms :=
+    inferInstanceAs (sheafCompose _ _).ReflectsIsomorphisms
+  have : IsIso ((discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj M)) :=
+    isIso_of_reflects_iso _ (Condensed.forget R)
+  functorIsoDiscreteAux₂ R M ≪≫ asIso ((discreteUnderlyingAdj _).counit.app ((functor R).obj M))
+
+/--
+`CondensedMod.LocallyConstant.functor` is naturally isomorphic to the constant sheaf functor from
+`R`-modules to condensed `R`-modules.
+ -/
+noncomputable def functorIsoDiscrete : functor R ≅ discrete _ :=
+  NatIso.ofComponents (fun M ↦ (functorIsoDiscreteComponents R M).symm) fun f ↦ by
+    dsimp
+    rw [Iso.eq_inv_comp, ← Category.assoc, Iso.comp_inv_eq]
+    dsimp [functorIsoDiscreteComponents]
+    rw [assoc, ← Iso.eq_inv_comp,
+      ← (discreteUnderlyingAdj (ModuleCat R)).counit_naturality]
+    simp only [← assoc]
+    congr 1
+    rw [← Iso.comp_inv_eq]
+    apply Sheaf.hom_ext
+    simp [functorIsoDiscreteAux₂, ← Functor.map_comp]
+    rfl
+
+/--
+`CondensedMod.LocallyConstant.functor` is left adjoint to the forgetful functor from condensed
+`R`-modules to `R`-modules.
+-/
+noncomputable def adjunction : functor R ⊣ underlying (ModuleCat R) :=
+  Adjunction.ofNatIsoLeft (discreteUnderlyingAdj _) (functorIsoDiscrete R).symm
+
+/--
+`CondensedMod.LocallyConstant.functor` is fully faithful.
+-/
+noncomputable def fullyFaithfulFunctor : (functor R).FullyFaithful :=
+  (adjunction R).fullyFaithfulLOfCompIsoId
+    (NatIso.ofComponents fun M ↦ (functorIsoDiscreteAux₁ R _).symm)
+
+instance : (functor R).Faithful := (fullyFaithfulFunctor R).faithful
+
+instance : (functor R).Full := (fullyFaithfulFunctor R).full
+
+instance : (discrete (ModuleCat R)).Faithful :=
+  Functor.Faithful.of_iso (functorIsoDiscrete R)
+
+instance : (constantSheaf (coherentTopology CompHaus) (ModuleCat R)).Faithful :=
+  inferInstanceAs (discrete (ModuleCat R)).Faithful
+
+instance : (discrete (ModuleCat R)).Full :=
+  Functor.Full.of_iso (functorIsoDiscrete R)
+
+instance : (constantSheaf (coherentTopology CompHaus) (ModuleCat R)).Full :=
+  inferInstanceAs (discrete (ModuleCat R)).Full
+
+instance : (constantSheaf (coherentTopology CompHaus) (Type (u + 1))).Faithful :=
+  inferInstanceAs (discrete (Type (u + 1))).Faithful
+
+instance : (constantSheaf (coherentTopology CompHaus) (Type (u + 1))).Full :=
+  inferInstanceAs (discrete (Type (u + 1))).Full
+
+end CondensedMod.LocallyConstant
+
+namespace LightCondMod.LocallyConstant
+
+open LightCondensed
+
+variable (R : Type u) [Ring R]
+
+/-- `functorToPresheaves` in the case of `LightProfinite`. -/
+abbrev functorToPresheaves : ModuleCat.{u} R ⥤ (LightProfinite.{u}ᵒᵖ ⥤ ModuleCat R) :=
+  CompHausLike.LocallyConstantModule.functorToPresheaves.{u, u} R
+
+/-- `functorToPresheaves` as a functor to light condensed modules. -/
+abbrev functor : ModuleCat R ⥤ LightCondMod.{u} R :=
+  CompHausLike.LocallyConstantModule.functor.{u, u} R
+    (fun _ _ _ ↦ (LightProfinite.effectiveEpi_iff_surjective _).mp)
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteAux₁ (M : ModuleCat.{u} R) :
+    M ≅ (ModuleCat.of R (LocallyConstant (LightProfinite.of PUnit.{u+1}) M)) where
+  hom := constₗ R
+  inv := evalₗ R PUnit.unit
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteAux₂ (M : ModuleCat.{u} R) :
+    (discrete _).obj M ≅ (discrete _).obj
+      (ModuleCat.of R (LocallyConstant (LightProfinite.of PUnit.{u+1}) M)) :=
+  (discrete _).mapIso (functorIsoDiscreteAux₁ R M)
+
+-- Not stating this explicitly causes timeouts below.
+instance : HasSheafify (coherentTopology LightProfinite.{u}) (ModuleCat.{u} R) :=
+  inferInstance
+
+instance (M : ModuleCat R) :
+    IsIso ((LightCondensed.forget R).map
+    ((discreteUnderlyingAdj (ModuleCat R)).counit.app
+      ((functor R).obj M))) := by
+  dsimp [LightCondensed.forget, discreteUnderlyingAdj]
+  rw [← constantSheafAdj_counit_w]
+  refine @IsIso.comp_isIso _ _ _ _ _ _ _ inferInstance ?_
+  have : (constantSheaf (coherentTopology LightProfinite) (Type u)).Faithful :=
+    inferInstanceAs (discrete _).Faithful
+  have : (constantSheaf (coherentTopology LightProfinite) (Type u)).Full :=
+    inferInstanceAs (discrete _).Full
+  rw [← Sheaf.isConstant_iff_isIso_counit_app]
+  constructor
+  change _ ∈ (discrete _).essImage
+  rw [essImage_eq_of_natIso LightCondSet.LocallyConstant.iso.symm]
+  exact obj_mem_essImage LightCondSet.LocallyConstant.functor M
+
+/-- Auxiliary definition for `functorIsoDiscrete`. -/
+noncomputable def functorIsoDiscreteComponents (M : ModuleCat R) :
+    (discrete _).obj M ≅ (functor R).obj M :=
+  have : (LightCondensed.forget R).ReflectsIsomorphisms :=
+    inferInstanceAs (sheafCompose _ _).ReflectsIsomorphisms
+  have : IsIso ((discreteUnderlyingAdj (ModuleCat R)).counit.app ((functor R).obj M)) :=
+    isIso_of_reflects_iso _ (LightCondensed.forget R)
+  functorIsoDiscreteAux₂ R M ≪≫ asIso ((discreteUnderlyingAdj _).counit.app ((functor R).obj M))
+
+/--
+`LightCondMod.LocallyConstant.functor` is naturally isomorphic to the constant sheaf functor from
+`R`-modules to light condensed `R`-modules.
+ -/
+noncomputable def functorIsoDiscrete : functor R ≅ discrete _ :=
+  NatIso.ofComponents (fun M ↦ (functorIsoDiscreteComponents R M).symm) fun f ↦ by
+    dsimp
+    rw [Iso.eq_inv_comp, ← Category.assoc, Iso.comp_inv_eq]
+    dsimp [functorIsoDiscreteComponents]
+    rw [Category.assoc, ← Iso.eq_inv_comp,
+      ← (discreteUnderlyingAdj (ModuleCat R)).counit_naturality]
+    simp only [← assoc]
+    congr 1
+    rw [← Iso.comp_inv_eq]
+    apply Sheaf.hom_ext
+    simp [functorIsoDiscreteAux₂, ← Functor.map_comp]
+    rfl
+
+/--
+`LightCondMod.LocallyConstant.functor` is left adjoint to the forgetful functor from light condensed
+`R`-modules to `R`-modules.
+ -/
+noncomputable def adjunction : functor R ⊣ underlying (ModuleCat R) :=
+  Adjunction.ofNatIsoLeft (discreteUnderlyingAdj _) (functorIsoDiscrete R).symm
+
+/--
+`LightCondMod.LocallyConstant.functor` is fully faithful.
+-/
+noncomputable def fullyFaithfulFunctor : (functor R).FullyFaithful :=
+  (adjunction R).fullyFaithfulLOfCompIsoId
+    (NatIso.ofComponents fun M ↦ (functorIsoDiscreteAux₁ R _).symm)
+
+instance : (functor R).Faithful := (fullyFaithfulFunctor R).faithful
+
+instance : (functor R).Full := (fullyFaithfulFunctor R).full
+
+instance : (discrete.{u} (ModuleCat R)).Faithful := Functor.Faithful.of_iso (functorIsoDiscrete R)
+
+instance : (constantSheaf (coherentTopology LightProfinite.{u}) (ModuleCat.{u} R)).Faithful :=
+  inferInstanceAs (discrete.{u} (ModuleCat R)).Faithful
+
+instance : (discrete (ModuleCat.{u} R)).Full :=
+  Functor.Full.of_iso (functorIsoDiscrete R)
+
+instance : (constantSheaf (coherentTopology LightProfinite.{u}) (ModuleCat.{u} R)).Full :=
+  inferInstanceAs (discrete.{u} (ModuleCat.{u} R)).Full
+
+instance : (constantSheaf (coherentTopology LightProfinite) (Type u)).Faithful :=
+  inferInstanceAs (discrete (Type u)).Faithful
+
+instance : (constantSheaf (coherentTopology LightProfinite) (Type u)).Full :=
+  inferInstanceAs (discrete (Type u)).Full
+
+end LightCondMod.LocallyConstant
diff --git a/Mathlib/Condensed/Equivalence.lean b/Mathlib/Condensed/Equivalence.lean
index 579875bfae18a..abc50a1886387 100644
--- a/Mathlib/Condensed/Equivalence.lean
+++ b/Mathlib/Condensed/Equivalence.lean
@@ -20,7 +20,7 @@ Since Stonean spaces are the projective objects in `CompHaus`, which has enough
 and the notions of effective epimorphism, epimorphism and surjective continuous map are equivalent
 in `CompHaus` and `Stonean`, we can use the general setup in
 `Mathlib.CategoryTheory.Sites.Coherent.SheafComparison` to deduce the equivalence of categories.
-We give the corresponding statements for `Profinite` as well.
+We give the corresponding statements for `Profinite` as well.
 
 ## Main results
 
@@ -61,7 +61,7 @@ instance : Stonean.toProfinite.ReflectsEffectiveEpis where
     ((Stonean.effectiveEpi_tfae f).out 0 2).mpr (((Profinite.effectiveEpi_tfae _).out 0 2).mp h)
 
 /--
-An effective presentation of an `X : Profinite` with respect to the inclusion functor from `Stonean`
+An effective presentation of an `X : Profinite` with respect to the inclusion functor from `Stonean`
 -/
 noncomputable def stoneanToProfiniteEffectivePresentation (X : Profinite) :
     Stonean.toProfinite.EffectivePresentation X where
diff --git a/Mathlib/Condensed/Explicit.lean b/Mathlib/Condensed/Explicit.lean
index 15e67c5ac1557..541c0747e15fc 100644
--- a/Mathlib/Condensed/Explicit.lean
+++ b/Mathlib/Condensed/Explicit.lean
@@ -19,9 +19,9 @@ We give the following three explicit descriptions of condensed objects:
 * `Condensed.ofSheafStonean`: A finite-product-preserving presheaf on `CompHaus`, satisfying
   `EqualizerCondition`.
 
-The property `EqualizerCondition` is defined in `Mathlib/CategoryTheory/Sites/RegularSheaves.lean`
+The property `EqualizerCondition` is defined in `Mathlib/CategoryTheory/Sites/RegularSheaves.lean`
 and it says that for any effective epi `X ⟶ B` (in this case that is equivalent to being a
-continuous surjection), the presheaf `F` exhibits `F(B)` as the equalizer of the two maps
+continuous surjection), the presheaf `F` exhibits `F(B)` as the equalizer of the two maps
 `F(X) ⇉ F(X ×_B X)`
 
 We also give variants for condensed objects in concrete categories whose forgetful functor
@@ -150,17 +150,17 @@ end Condensed
 
 namespace CondensedSet
 
-/-- A `CondensedSet` version of `Condensed.ofSheafStonean`. -/
+/-- A `CondensedSet` version of `Condensed.ofSheafStonean`. -/
 noncomputable abbrev ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ Type (u+1)) [PreservesFiniteProducts F] :
     CondensedSet :=
   Condensed.ofSheafStonean F
 
-/-- A `CondensedSet` version of `Condensed.ofSheafProfinite`. -/
+/-- A `CondensedSet` version of `Condensed.ofSheafProfinite`. -/
 noncomputable abbrev ofSheafProfinite (F : Profinite.{u}ᵒᵖ ⥤ Type (u+1))
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : CondensedSet :=
   Condensed.ofSheafProfinite F hF
 
-/-- A `CondensedSet` version of `Condensed.ofSheafCompHaus`. -/
+/-- A `CondensedSet` version of `Condensed.ofSheafCompHaus`. -/
 noncomputable abbrev ofSheafCompHaus (F : CompHaus.{u}ᵒᵖ ⥤ Type (u+1))
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : CondensedSet :=
   Condensed.ofSheafCompHaus F hF
@@ -171,19 +171,19 @@ namespace CondensedMod
 
 variable (R : Type (u+1)) [Ring R]
 
-/-- A `CondensedMod` version of `Condensed.ofSheafStonean`. -/
+/-- A `CondensedMod` version of `Condensed.ofSheafStonean`. -/
 noncomputable abbrev ofSheafStonean (F : Stonean.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] : CondensedMod R :=
   haveI : HasLimitsOfSize.{u, u+1} (ModuleCat R) := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
   Condensed.ofSheafStonean F
 
-/-- A `CondensedMod` version of `Condensed.ofSheafProfinite`. -/
+/-- A `CondensedMod` version of `Condensed.ofSheafProfinite`. -/
 noncomputable abbrev ofSheafProfinite (F : Profinite.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : CondensedMod R :=
   haveI : HasLimitsOfSize.{u, u+1} (ModuleCat R) := hasLimitsOfSizeShrink.{u, u+1, u+1, u+1} _
   Condensed.ofSheafProfinite F hF
 
-/-- A `CondensedMod` version of `Condensed.ofSheafCompHaus`. -/
+/-- A `CondensedMod` version of `Condensed.ofSheafCompHaus`. -/
 noncomputable abbrev ofSheafCompHaus (F : CompHaus.{u}ᵒᵖ ⥤ ModuleCat.{u+1} R)
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : CondensedMod R :=
   Condensed.ofSheafCompHaus F hF
diff --git a/Mathlib/Condensed/Functors.lean b/Mathlib/Condensed/Functors.lean
index 73869ef6f22b8..c83e320059c0d 100644
--- a/Mathlib/Condensed/Functors.lean
+++ b/Mathlib/Condensed/Functors.lean
@@ -17,16 +17,16 @@ sets.
 
 ## Main definitions
 
-* `compHausToCondensed : CompHaus.{u} ⥤ CondensedSet.{u}` is essentially the yoneda presheaf
-  functor. We also define `profiniteToCondensed` and `stoneanToCondensed`.
+* `compHausToCondensed : CompHaus.{u} ⥤ CondensedSet.{u}` is essentially the yoneda presheaf
+  functor. We also define `profiniteToCondensed` and `stoneanToCondensed`.
 
 TODO (Dagur):
 
-* Define the analogues of `compHausToCondensed` for sheaves on `Profinite` and `Stonean` and provide
-  the relevant isomorphisms with `profiniteToCondensed` and `stoneanToCondensed`.
+* Define the analogues of `compHausToCondensed` for sheaves on `Profinite` and `Stonean` and provide
+  the relevant isomorphisms with `profiniteToCondensed` and `stoneanToCondensed`.
 
 * Define the functor `Type (u+1) ⥤ CondensedSet.{u}` which takes a set `X` to the presheaf given by
-  mapping a compact Hausdorff space `S` to `LocallyConstant S X`, along with the isomorphism with
+  mapping a compact Hausdorff space `S` to `LocallyConstant S X`, along with the isomorphism with
   the functor that goes through `TopCat.{u+1}`.
 
 -/
@@ -49,7 +49,7 @@ end Universes
 
 section Topology
 
-/-- The functor from `CompHaus` to `Condensed.{u} (Type u)` given by the Yoneda sheaf. -/
+/-- The functor from `CompHaus` to `Condensed.{u} (Type u)` given by the Yoneda sheaf. -/
 def compHausToCondensed' : CompHaus.{u} ⥤ Condensed.{u} (Type u) :=
   (coherentTopology.subcanonical CompHaus).yoneda
 
diff --git a/Mathlib/Condensed/Light/Basic.lean b/Mathlib/Condensed/Light/Basic.lean
index e4336e0a3ea2d..f5156d707ed0f 100644
--- a/Mathlib/Condensed/Light/Basic.lean
+++ b/Mathlib/Condensed/Light/Basic.lean
@@ -60,7 +60,7 @@ namespace LightCondSet
 -- Note: `simp` can prove this when stated for `LightCondensed C` for a concrete category `C`.
 -- However, it doesn't seem to see through the abbreviation `LightCondSet`
 @[simp]
-lemma hom_naturality_apply {X Y : LightCondSet.{u}} (f : X ⟶ Y)  {S T : LightProfiniteᵒᵖ}
+lemma hom_naturality_apply {X Y : LightCondSet.{u}} (f : X ⟶ Y) {S T : LightProfiniteᵒᵖ}
     (g : S ⟶ T) (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) :=
   NatTrans.naturality_apply f.val g x
 
diff --git a/Mathlib/Condensed/Light/Explicit.lean b/Mathlib/Condensed/Light/Explicit.lean
index 8967577b1c7d4..837d7f33f7fab 100644
--- a/Mathlib/Condensed/Light/Explicit.lean
+++ b/Mathlib/Condensed/Light/Explicit.lean
@@ -14,9 +14,9 @@ We give explicit description of light condensed sets:
 * `LightCondensed.ofSheafLightProfinite`: A finite-product-preserving presheaf on `LightProfinite`,
   satisfying `EqualizerCondition`.
 
-The property `EqualizerCondition` is defined in `Mathlib/CategoryTheory/Sites/RegularExtensive.lean`
+The property `EqualizerCondition` is defined in `Mathlib/CategoryTheory/Sites/RegularExtensive.lean`
 and it says that for any effective epi `X ⟶ B` (in this case that is equivalent to being a
-continuous surjection), the presheaf `F` exhibits `F(B)` as the equalizer of the two maps
+continuous surjection), the presheaf `F` exhibits `F(B)` as the equalizer of the two maps
 `F(X) ⇉ F(X ×_B X)`
 
 We also give variants for light condensed objects in concrete categories whose forgetful functor
@@ -71,7 +71,7 @@ end LightCondensed
 
 namespace LightCondSet
 
-/-- A `LightCondSet` version of `LightCondensed.ofSheafLightProfinite`. -/
+/-- A `LightCondSet` version of `LightCondensed.ofSheafLightProfinite`. -/
 noncomputable abbrev ofSheafLightProfinite (F : LightProfinite.{u}ᵒᵖ ⥤ Type u)
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : LightCondSet :=
   LightCondensed.ofSheafLightProfinite F hF
@@ -82,7 +82,7 @@ namespace LightCondMod
 
 variable (R : Type u) [Ring R]
 
-/-- A `LightCondAb` version of `LightCondensed.ofSheafLightProfinite`. -/
+/-- A `LightCondAb` version of `LightCondensed.ofSheafLightProfinite`. -/
 noncomputable abbrev ofSheafLightProfinite (F : LightProfinite.{u}ᵒᵖ ⥤ ModuleCat.{u} R)
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : LightCondMod.{u} R :=
   LightCondensed.ofSheafLightProfinite F hF
@@ -91,7 +91,7 @@ end LightCondMod
 
 namespace LightCondAb
 
-/-- A `LightCondAb` version of `LightCondensed.ofSheafLightProfinite`. -/
+/-- A `LightCondAb` version of `LightCondensed.ofSheafLightProfinite`. -/
 noncomputable abbrev ofSheafLightProfinite (F : LightProfiniteᵒᵖ ⥤ ModuleCat ℤ)
     [PreservesFiniteProducts F] (hF : EqualizerCondition F) : LightCondAb :=
   LightCondMod.ofSheafLightProfinite ℤ F hF
diff --git a/Mathlib/Condensed/Light/Functors.lean b/Mathlib/Condensed/Light/Functors.lean
index 191499d9bb0d8..7d58dbe10bb93 100644
--- a/Mathlib/Condensed/Light/Functors.lean
+++ b/Mathlib/Condensed/Light/Functors.lean
@@ -14,13 +14,13 @@ sets.
 
 ## Main definitions
 
-* `lightProfiniteToLightCondSet : LightProfinite.{u} ⥤ LightCondSet.{u}` 
+* `lightProfiniteToLightCondSet : LightProfinite.{u} ⥤ LightCondSet.{u}`
   is the yoneda presheaf functor.
 
 TODO (Dagur):
 
 * Define the functor `Type u ⥤ LightCondSet.{u}` which takes a set `X` to the presheaf given by
-  mapping a light profinite space `S` to `LocallyConstant S X`, along with the isomorphism with
+  mapping a light profinite space `S` to `LocallyConstant S X`, along with the isomorphism with
   the functor that goes through `TopCat.{u+1}`.
 
 -/
@@ -29,7 +29,7 @@ universe u v
 
 open CategoryTheory Limits
 
-/-- The functor from `LightProfinite.{u}` to `LightCondSet.{u}` given by the Yoneda sheaf. -/
+/-- The functor from `LightProfinite.{u}` to `LightCondSet.{u}` given by the Yoneda sheaf. -/
 def lightProfiniteToLightCondSet : LightProfinite.{u} ⥤ LightCondSet.{u} :=
   (coherentTopology.subcanonical LightProfinite).yoneda
 
diff --git a/Mathlib/Condensed/Light/Module.lean b/Mathlib/Condensed/Light/Module.lean
index d75eee03e1156..e4ba98cbb8812 100644
--- a/Mathlib/Condensed/Light/Module.lean
+++ b/Mathlib/Condensed/Light/Module.lean
@@ -68,7 +68,7 @@ namespace LightCondMod
 -- Note: `simp` can prove this when stated for `LightCondensed C` for a concrete category `C`.
 -- However, it doesn't seem to see through the abbreviation `LightCondMod`
 @[simp]
-lemma hom_naturality_apply {X Y : LightCondMod.{u} R} (f : X ⟶ Y)  {S T : LightProfiniteᵒᵖ}
+lemma hom_naturality_apply {X Y : LightCondMod.{u} R} (f : X ⟶ Y) {S T : LightProfiniteᵒᵖ}
     (g : S ⟶ T) (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) :=
   NatTrans.naturality_apply f.val g x
 
diff --git a/Mathlib/Condensed/Light/TopCatAdjunction.lean b/Mathlib/Condensed/Light/TopCatAdjunction.lean
index 2b67bf1e860bb..5b6ffa2fd3fa4 100644
--- a/Mathlib/Condensed/Light/TopCatAdjunction.lean
+++ b/Mathlib/Condensed/Light/TopCatAdjunction.lean
@@ -39,14 +39,14 @@ private def coinducingCoprod :
       X.val.obj ⟨LightProfinite.of PUnit⟩ :=
   fun ⟨⟨S, i⟩, s⟩ ↦ X.val.map (S.const s).op i
 
-/-- Let `X` be a light condensed set. We define a topology on `X(*)` as the quotient topology of
+/-- Let `X` be a light condensed set. We define a topology on `X(*)` as the quotient topology of
 all the maps from light profinite sets `S` to `X(*)`, corresponding to elements of `X(S)`.
 In other words, the topology coinduced by the map `LightCondSet.coinducingCoprod` above. -/
 local instance underlyingTopologicalSpace :
     TopologicalSpace (X.val.obj ⟨LightProfinite.of PUnit⟩) :=
   TopologicalSpace.coinduced (coinducingCoprod X) inferInstance
 
-/-- The object part of the functor `LightCondSet ⥤ TopCat`  -/
+/-- The object part of the functor `LightCondSet ⥤ TopCat` -/
 def toTopCat : TopCat.{u} := TopCat.of (X.val.obj ⟨LightProfinite.of PUnit⟩)
 
 lemma continuous_coinducingCoprod {S : LightProfinite.{u}} (x : X.val.obj ⟨S⟩) :
@@ -58,7 +58,7 @@ lemma continuous_coinducingCoprod {S : LightProfinite.{u}} (x : X.val.obj ⟨S
 
 variable {X} {Y : LightCondSet} (f : X ⟶ Y)
 
-/-- The map part of the functor `LightCondSet ⥤ TopCat`  -/
+/-- The map part of the functor `LightCondSet ⥤ TopCat` -/
 @[simps]
 def toTopCatMap : X.toTopCat ⟶ Y.toTopCat where
   toFun := f.val.app ⟨LightProfinite.of PUnit⟩
@@ -73,7 +73,7 @@ def toTopCatMap : X.toTopCat ⟶ Y.toTopCat where
     rw [this]
     exact continuous_coinducingCoprod _ _
 
-/-- The functor `LightCondSet ⥤ TopCat`  -/
+/-- The functor `LightCondSet ⥤ TopCat` -/
 @[simps]
 def _root_.lightCondSetToTopCat : LightCondSet.{u} ⥤ TopCat.{u} where
   obj X := X.toTopCat
@@ -87,7 +87,7 @@ def topCatAdjunctionCounit (X : TopCat.{u}) : X.toLightCondSet.toTopCat ⟶ X wh
     continuity
 
 /-- The counit of the adjunction `lightCondSetToTopCat ⊣ topCatToLightCondSet` is always bijective,
-but not an isomorphism in general (the inverse isn't continuous unless `X` is sequential).
+but not an isomorphism in general (the inverse isn't continuous unless `X` is sequential).
 -/
 def topCatAdjunctionCounitEquiv (X : TopCat.{u}) : X.toLightCondSet.toTopCat ≃ X where
   toFun := topCatAdjunctionCounit X
@@ -118,14 +118,13 @@ def topCatAdjunctionUnit (X : LightCondSet.{u}) : X ⟶ X.toTopCat.toLightCondSe
       rfl }
 
 /-- The adjunction `lightCondSetToTopCat ⊣ topCatToLightCondSet` -/
-noncomputable def topCatAdjunction : lightCondSetToTopCat.{u} ⊣ topCatToLightCondSet :=
-  Adjunction.mkOfUnitCounit {
-    unit := { app := topCatAdjunctionUnit }
-    counit := { app := topCatAdjunctionCounit }
-    left_triangle := by
-      ext Y
-      change Y.val.map (𝟙 _) _ = _
-      simp }
+noncomputable def topCatAdjunction : lightCondSetToTopCat.{u} ⊣ topCatToLightCondSet where
+  unit := { app := topCatAdjunctionUnit }
+  counit := { app := topCatAdjunctionCounit }
+  left_triangle_components Y := by
+    ext
+    change Y.val.map (𝟙 _) _ = _
+    simp
 
 instance (X : TopCat) : Epi (topCatAdjunction.counit.app X) := by
   rw [TopCat.epi_iff_surjective]
diff --git a/Mathlib/Condensed/Light/TopComparison.lean b/Mathlib/Condensed/Light/TopComparison.lean
index ee4dbcaabc7ae..bebf7bfe42841 100644
--- a/Mathlib/Condensed/Light/TopComparison.lean
+++ b/Mathlib/Condensed/Light/TopComparison.lean
@@ -26,7 +26,7 @@ noncomputable abbrev TopCat.toLightCondSet (X : TopCat.{u}) : LightCondSet.{u} :
   toSheafCompHausLike.{u} _ X (fun _ _ _ ↦ (LightProfinite.effectiveEpi_iff_surjective _).mp)
 
 /--
-`TopCat.toLightCondSet` yields a functor from `TopCat.{u}` to `LightCondSet.{u}`.
+`TopCat.toLightCondSet` yields a functor from `TopCat.{u}` to `LightCondSet.{u}`.
 -/
 noncomputable abbrev topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u} :=
   topCatToSheafCompHausLike.{u} _ (fun _ _ _ ↦ (LightProfinite.effectiveEpi_iff_surjective _).mp)
diff --git a/Mathlib/Condensed/Module.lean b/Mathlib/Condensed/Module.lean
index c9e76f9b22338..24a5079139cfa 100644
--- a/Mathlib/Condensed/Module.lean
+++ b/Mathlib/Condensed/Module.lean
@@ -74,7 +74,7 @@ namespace CondensedMod
 -- Note: `simp` can prove this when stated for `Condensed C` for a concrete category `C`.
 -- However, it doesn't seem to see through the abbreviation `CondensedMod`
 @[simp]
-lemma hom_naturality_apply {X Y : CondensedMod.{u} R} (f : X ⟶ Y)  {S T : CompHausᵒᵖ} (g : S ⟶ T)
+lemma hom_naturality_apply {X Y : CondensedMod.{u} R} (f : X ⟶ Y) {S T : CompHausᵒᵖ} (g : S ⟶ T)
     (x : X.val.obj S) : f.val.app T (X.val.map g x) = Y.val.map g (f.val.app S x) :=
   NatTrans.naturality_apply f.val g x
 
diff --git a/Mathlib/Condensed/TopCatAdjunction.lean b/Mathlib/Condensed/TopCatAdjunction.lean
index 13fedf0cb5e3d..732a1448149d9 100644
--- a/Mathlib/Condensed/TopCatAdjunction.lean
+++ b/Mathlib/Condensed/TopCatAdjunction.lean
@@ -34,13 +34,13 @@ private def CondensedSet.coinducingCoprod :
     (Σ (i : (S : CompHaus.{u}) × X.val.obj ⟨S⟩), i.fst) → X.val.obj ⟨CompHaus.of PUnit⟩ :=
   fun ⟨⟨S, i⟩, s⟩ ↦ X.val.map (S.const s).op i
 
-/-- Let `X` be a condensed set. We define a topology on `X(*)` as the quotient topology of
+/-- Let `X` be a condensed set. We define a topology on `X(*)` as the quotient topology of
 all the maps from compact Hausdorff `S` spaces to `X(*)`, corresponding to elements of `X(S)`.
 In other words, the topology coinduced by the map `CondensedSet.coinducingCoprod` above. -/
 local instance : TopologicalSpace (X.val.obj ⟨CompHaus.of PUnit⟩) :=
   TopologicalSpace.coinduced (coinducingCoprod X) inferInstance
 
-/-- The object part of the functor `CondensedSet ⥤ TopCat`  -/
+/-- The object part of the functor `CondensedSet ⥤ TopCat`  -/
 def CondensedSet.toTopCat : TopCat.{u+1} := TopCat.of (X.val.obj ⟨CompHaus.of PUnit⟩)
 
 namespace CondensedSet
@@ -54,7 +54,7 @@ lemma continuous_coinducingCoprod {S : CompHaus.{u}} (x : X.val.obj ⟨S⟩) :
 
 variable {X} {Y : CondensedSet} (f : X ⟶ Y)
 
-/-- The map part of the functor `CondensedSet ⥤ TopCat`  -/
+/-- The map part of the functor `CondensedSet ⥤ TopCat`  -/
 @[simps]
 def toTopCatMap : X.toTopCat ⟶ Y.toTopCat where
   toFun := f.val.app ⟨CompHaus.of PUnit⟩
@@ -71,7 +71,7 @@ def toTopCatMap : X.toTopCat ⟶ Y.toTopCat where
 
 end CondensedSet
 
-/-- The functor `CondensedSet ⥤ TopCat`  -/
+/-- The functor `CondensedSet ⥤ TopCat`  -/
 @[simps]
 def condensedSetToTopCat : CondensedSet.{u} ⥤ TopCat.{u+1} where
   obj X := X.toTopCat
@@ -88,7 +88,7 @@ def topCatAdjunctionCounit (X : TopCat.{u+1}) : X.toCondensedSet.toTopCat ⟶ X
     continuity
 
 /-- The counit of the adjunction `condensedSetToTopCat ⊣ topCatToCondensedSet` is always bijective,
-but not an isomorphism in general (the inverse isn't continuous unless `X` is compactly generated).
+but not an isomorphism in general (the inverse isn't continuous unless `X` is compactly generated).
 -/
 def topCatAdjunctionCounitEquiv (X : TopCat.{u+1}) : X.toCondensedSet.toTopCat ≃ X where
   toFun := topCatAdjunctionCounit X
@@ -119,14 +119,13 @@ def topCatAdjunctionUnit (X : CondensedSet.{u}) : X ⟶ X.toTopCat.toCondensedSe
       rfl }
 
 /-- The adjunction `condensedSetToTopCat ⊣ topCatToCondensedSet` -/
-noncomputable def topCatAdjunction : condensedSetToTopCat.{u} ⊣ topCatToCondensedSet :=
-  Adjunction.mkOfUnitCounit {
-    unit := { app := topCatAdjunctionUnit }
-    counit := { app := topCatAdjunctionCounit }
-    left_triangle := by
-      ext Y
-      change Y.val.map (𝟙 _) _ = _
-      simp }
+noncomputable def topCatAdjunction : condensedSetToTopCat.{u} ⊣ topCatToCondensedSet where
+  unit := { app := topCatAdjunctionUnit }
+  counit := { app := topCatAdjunctionCounit }
+  left_triangle_components Y := by
+    ext
+    change Y.val.map (𝟙 _) _ = _
+    simp
 
 instance (X : TopCat) : Epi (topCatAdjunction.counit.app X) := by
   rw [TopCat.epi_iff_surjective]
diff --git a/Mathlib/Condensed/TopComparison.lean b/Mathlib/Condensed/TopComparison.lean
index 290e5f0d3433f..c8679d14fddbc 100644
--- a/Mathlib/Condensed/TopComparison.lean
+++ b/Mathlib/Condensed/TopComparison.lean
@@ -12,7 +12,7 @@ import Mathlib.Topology.Category.TopCat.Yoneda
 
 # The functor from topological spaces to condensed sets
 
-This file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has
+This file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has
 nice properties, like preserving pullbacks and finite coproducts, then this Yoneda presheaf
 satisfies the sheaf condition for the regular and extensive topologies respectively.
 
@@ -31,7 +31,7 @@ variable {C : Type u} [Category.{v} C] (G : C ⥤ TopCat.{w})
   (X : Type w') [TopologicalSpace X]
 
 /--
-An auxiliary lemma to that allows us to use `QuotientMap.lift` in the proof of
+An auxiliary lemma to that allows us to use `QuotientMap.lift` in the proof of
 `equalizerCondition_yonedaPresheaf`.
 -/
 theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullback π π]
@@ -56,7 +56,7 @@ theorem factorsThrough_of_pullbackCondition {Z B : C} {π : Z ⟶ B} [HasPullbac
 
 /--
 If `G` preserves the relevant pullbacks and every effective epi in `C` is a quotient map (which is
-the case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer
+the case when `C` is `CompHaus` or `Profinite`), then `yonedaPresheaf` satisfies the equalizer
 condition which is required to be a sheaf for the regular topology.
 -/
 theorem equalizerCondition_yonedaPresheaf
@@ -83,7 +83,7 @@ theorem equalizerCondition_yonedaPresheaf
     exact DFunLike.ext'_iff.mp ((hq Z B π).lift_comp a (factorsThrough_of_pullbackCondition G X ha))
 
 /--
-If `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or
+If `G` preserves finite coproducts (which is the case when `C` is `CompHaus`, `Profinite` or
 `Stonean`), then `yonedaPresheaf` preserves finite products, which is required to be a sheaf for
 the extensive topology.
 -/
@@ -99,7 +99,7 @@ variable (P : TopCat.{u} → Prop) (X : TopCat.{max u w})
     (hs : ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y), EffectiveEpi f → Function.Surjective f)
 
 /--
-The sheaf on `CompHausLike P` of continuous maps to a topological space.
+The sheaf on `CompHausLike P` of continuous maps to a topological space.
 -/
 @[simps! val_obj val_map]
 def TopCat.toSheafCompHausLike :
@@ -136,7 +136,7 @@ noncomputable abbrev TopCat.toCondensedSet (X : TopCat.{u+1}) : CondensedSet.{u}
   toSheafCompHausLike.{u+1} _ X (fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)
 
 /--
-`TopCat.toCondensedSet` yields a functor from `TopCat.{u+1}` to `CondensedSet.{u}`.
+`TopCat.toCondensedSet` yields a functor from `TopCat.{u+1}` to `CondensedSet.{u}`.
 -/
 noncomputable abbrev topCatToCondensedSet : TopCat.{u+1} ⥤ CondensedSet.{u} :=
   topCatToSheafCompHausLike.{u+1} _ (fun _ _ _ ↦ ((CompHaus.effectiveEpi_tfae _).out 0 2).mp)
diff --git a/Mathlib/Control/Applicative.lean b/Mathlib/Control/Applicative.lean
index 50670dad3673d..3b6bb33ac8951 100644
--- a/Mathlib/Control/Applicative.lean
+++ b/Mathlib/Control/Applicative.lean
@@ -5,6 +5,7 @@ Authors: Simon Hudon
 -/
 import Mathlib.Algebra.Group.Defs
 import Mathlib.Control.Functor
+import Mathlib.Control.Basic
 
 /-!
 # `applicative` instances
@@ -28,7 +29,7 @@ variable {α β γ σ : Type u}
 
 theorem Applicative.map_seq_map (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :
     f <$> x <*> g <$> y = ((· ∘ g) ∘ f) <$> x <*> y := by
-  simp [flip, functor_norm]
+  simp [flip, functor_norm, Function.comp_def]
 
 theorem Applicative.pure_seq_eq_map' (f : α → β) : ((pure f : F (α → β)) <*> ·) = (f <$> ·) := by
   ext; simp [functor_norm]
@@ -82,11 +83,11 @@ theorem map_pure (f : α → β) (x : α) : (f <$> pure x : Comp F G β) = pure
   Comp.ext <| by simp
 
 theorem seq_pure (f : Comp F G (α → β)) (x : α) : f <*> pure x = (fun g : α → β => g x) <$> f :=
-  Comp.ext <| by simp [(· ∘ ·), functor_norm]
+  Comp.ext <| by simp [comp_def, functor_norm]
 
 theorem seq_assoc (x : Comp F G α) (f : Comp F G (α → β)) (g : Comp F G (β → γ)) :
     g <*> (f <*> x) = @Function.comp α β γ <$> g <*> f <*> x :=
-  Comp.ext <| by simp [(· ∘ ·), functor_norm]
+  Comp.ext <| by simp [comp_def, functor_norm]
 
 theorem pure_seq_eq_map (f : α → β) (x : Comp F G α) : pure f <*> x = f <$> x :=
   Comp.ext <| by simp [Applicative.pure_seq_eq_map', functor_norm]
@@ -119,7 +120,7 @@ instance {f : Type u → Type w} {g : Type v → Type u} [Applicative f] [Applic
   commutative_prod _ _ := by
     simp! [map, Seq.seq]
     rw [commutative_map]
-    simp only [mk, flip, seq_map_assoc, Function.comp, map_map]
+    simp only [mk, flip, seq_map_assoc, Function.comp_def, map_map]
     congr
     funext x y
     rw [commutative_map]
diff --git a/Mathlib/Control/Basic.lean b/Mathlib/Control/Basic.lean
index 8796014b34cc6..e6e6aa94abeee 100644
--- a/Mathlib/Control/Basic.lean
+++ b/Mathlib/Control/Basic.lean
@@ -18,12 +18,7 @@ variable {α β γ : Type u}
 
 section Functor
 
-variable {f : Type u → Type v} [Functor f] [LawfulFunctor f]
-@[functor_norm]
-theorem Functor.map_map (m : α → β) (g : β → γ) (x : f α) : g <$> m <$> x = (g ∘ m) <$> x :=
-  (comp_map _ _ _).symm
--- order of implicits
--- order of implicits
+attribute [functor_norm] Functor.map_map
 
 end Functor
 
@@ -54,6 +49,7 @@ theorem seq_map_assoc (x : F (α → β)) (f : γ → α) (y : F γ) :
   simp only [← pure_seq]
   simp only [seq_assoc, Function.comp, seq_pure, ← comp_map]
   simp [pure_seq]
+  rfl
 
 @[functor_norm]
 theorem map_seq (f : β → γ) (x : F (α → β)) (y : F α) :
@@ -66,10 +62,6 @@ section Monad
 
 variable {m : Type u → Type v} [Monad m] [LawfulMonad m]
 
-theorem map_bind (x : m α) {g : α → m β} {f : β → γ} :
-    f <$> (x >>= g) = x >>= fun a => f <$> g a := by
-  rw [← bind_pure_comp, bind_assoc]; simp [bind_pure_comp]
-
 theorem seq_bind_eq (x : m α) {g : β → m γ} {f : α → β} :
     f <$> x >>= g = x >>= g ∘ f :=
   show bind (f <$> x) g = bind x (g ∘ f) by
@@ -213,8 +205,6 @@ class CommApplicative (m : Type u → Type v) [Applicative m] extends LawfulAppl
 
 open Functor
 
-variable {m}
-
 theorem CommApplicative.commutative_map {m : Type u → Type v} [h : Applicative m]
     [CommApplicative m] {α β γ} (a : m α) (b : m β) {f : α → β → γ} :
   f <$> a <*> b = flip f <$> b <*> a :=
@@ -224,3 +214,4 @@ theorem CommApplicative.commutative_map {m : Type u → Type v} [h : Applicative
     _ = (fun b a => f a b) <$> b <*> a := by
       rw [@CommApplicative.commutative_prod m h]
       simp [seq_map_assoc, map_seq, seq_assoc, seq_pure, map_map, (· ∘ ·)]
+      rfl
diff --git a/Mathlib/Control/Bitraversable/Instances.lean b/Mathlib/Control/Bitraversable/Instances.lean
index ad3106d4f11c4..0627c13d3bbac 100644
--- a/Mathlib/Control/Bitraversable/Instances.lean
+++ b/Mathlib/Control/Bitraversable/Instances.lean
@@ -93,7 +93,7 @@ instance (priority := 10) Bitraversable.isLawfulTraversable [LawfulBitraversable
   constructor <;> intros <;>
     simp [traverse, comp_tsnd, functor_norm]
   · simp [tsnd_eq_snd_id, (· <$> ·), id.mk]
-  · simp [tsnd, binaturality, Function.comp, functor_norm]
+  · simp [tsnd, binaturality, Function.comp_def, functor_norm]
 
 end
 
diff --git a/Mathlib/Control/Bitraversable/Lemmas.lean b/Mathlib/Control/Bitraversable/Lemmas.lean
index 085f0781b4305..a13f3aa61a6b2 100644
--- a/Mathlib/Control/Bitraversable/Lemmas.lean
+++ b/Mathlib/Control/Bitraversable/Lemmas.lean
@@ -66,27 +66,27 @@ theorem id_tsnd : ∀ {α β} (x : t α β), tsnd (F := Id) pure x = pure x :=
 theorem comp_tfst {α₀ α₁ α₂ β} (f : α₀ → F α₁) (f' : α₁ → G α₂) (x : t α₀ β) :
     Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk ∘ map f' ∘ f) x := by
   rw [← comp_bitraverse]
-  simp only [Function.comp, tfst, map_pure, Pure.pure]
+  simp only [Function.comp_def, tfst, map_pure, Pure.pure]
 
 @[higher_order tfst_comp_tsnd]
 theorem tfst_tsnd {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
     Comp.mk (tfst f <$> tsnd f' x)
       = bitraverse (Comp.mk ∘ pure ∘ f) (Comp.mk ∘ map pure ∘ f') x := by
   rw [← comp_bitraverse]
-  simp only [Function.comp, map_pure]
+  simp only [Function.comp_def, map_pure]
 
 @[higher_order tsnd_comp_tfst]
 theorem tsnd_tfst {α₀ α₁ β₀ β₁} (f : α₀ → F α₁) (f' : β₀ → G β₁) (x : t α₀ β₀) :
     Comp.mk (tsnd f' <$> tfst f x)
       = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x := by
   rw [← comp_bitraverse]
-  simp only [Function.comp, map_pure]
+  simp only [Function.comp_def, map_pure]
 
 @[higher_order tsnd_comp_tsnd]
 theorem comp_tsnd {α β₀ β₁ β₂} (g : β₀ → F β₁) (g' : β₁ → G β₂) (x : t α β₀) :
     Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk ∘ map g' ∘ g) x := by
   rw [← comp_bitraverse]
-  simp only [Function.comp, map_pure]
+  simp only [Function.comp_def, map_pure]
   rfl
 
 open Bifunctor
diff --git a/Mathlib/Control/EquivFunctor.lean b/Mathlib/Control/EquivFunctor.lean
index 05e6050b3e91e..ecbf36b36a8a5 100644
--- a/Mathlib/Control/EquivFunctor.lean
+++ b/Mathlib/Control/EquivFunctor.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Equiv.Defs
 import Mathlib.Tactic.Convert
diff --git a/Mathlib/Control/EquivFunctor/Instances.lean b/Mathlib/Control/EquivFunctor/Instances.lean
index 87b89a8ea37f9..13987af2679b9 100644
--- a/Mathlib/Control/EquivFunctor/Instances.lean
+++ b/Mathlib/Control/EquivFunctor/Instances.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Fintype.Basic
 import Mathlib.Control.EquivFunctor
diff --git a/Mathlib/Control/Fold.lean b/Mathlib/Control/Fold.lean
index 5d51cf1544185..71a4d437faaa3 100644
--- a/Mathlib/Control/Fold.lean
+++ b/Mathlib/Control/Fold.lean
@@ -315,7 +315,8 @@ theorem toList_spec (xs : t α) : toList xs = FreeMonoid.toList (foldMap FreeMon
               Function.comp_apply]
 
 theorem foldMap_map [Monoid γ] (f : α → β) (g : β → γ) (xs : t α) :
-    foldMap g (f <$> xs) = foldMap (g ∘ f) xs := by simp only [foldMap, traverse_map, Function.comp]
+    foldMap g (f <$> xs) = foldMap (g ∘ f) xs := by
+  simp only [foldMap, traverse_map, Function.comp_def]
 
 theorem foldl_toList (f : α → β → α) (xs : t β) (x : α) :
     foldl f x xs = List.foldl f x (toList xs) := by
@@ -331,18 +332,19 @@ theorem foldr_toList (f : α → β → β) (xs : t α) (x : β) :
 
 theorem toList_map (f : α → β) (xs : t α) : toList (f <$> xs) = f <$> toList xs := by
   simp only [toList_spec, Free.map_eq_map, foldMap_hom, foldMap_map, FreeMonoid.ofList_toList,
-    FreeMonoid.map_of, (· ∘ ·)]
+    FreeMonoid.map_of, Function.comp_def]
 
 @[simp]
 theorem foldl_map (g : β → γ) (f : α → γ → α) (a : α) (l : t β) :
     foldl f a (g <$> l) = foldl (fun x y => f x (g y)) a l := by
   #adaptation_note /-- nightly-2024-03-16: simp was
   simp only [foldl, foldMap_map, (· ∘ ·), flip] -/
-  simp only [foldl, foldMap_map, (· ∘ ·), Function.flip_def]
+  simp only [foldl, foldMap_map, Function.comp_def, Function.flip_def]
 
 @[simp]
 theorem foldr_map (g : β → γ) (f : γ → α → α) (a : α) (l : t β) :
-    foldr f a (g <$> l) = foldr (f ∘ g) a l := by simp only [foldr, foldMap_map, (· ∘ ·), flip]
+    foldr f a (g <$> l) = foldr (f ∘ g) a l := by
+  simp only [foldr, foldMap_map, Function.comp_def, flip]
 
 @[simp]
 theorem toList_eq_self {xs : List α} : toList xs = xs := by
@@ -357,9 +359,9 @@ theorem length_toList {xs : t α} : length xs = List.length (toList xs) := by
   generalize toList xs = ys
   rw [← Nat.add_zero ys.length]
   generalize 0 = n
-  induction' ys with _ _ ih generalizing n
-  · simp
-  · simp_arith [ih]
+  induction ys generalizing n with
+  | nil => simp
+  | cons _ _ ih => simp_arith [ih]
 
 variable {m : Type u → Type u} [Monad m] [LawfulMonad m]
 
@@ -382,11 +384,12 @@ theorem foldlm_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β)
     foldlm f a (g <$> l) = foldlm (fun x y => f x (g y)) a l := by
   #adaptation_note /-- nightly-2024-03-16: simp was
   simp only [foldlm, foldMap_map, (· ∘ ·), flip] -/
-  simp only [foldlm, foldMap_map, (· ∘ ·), Function.flip_def]
+  simp only [foldlm, foldMap_map, Function.comp_def, Function.flip_def]
 
 @[simp]
 theorem foldrm_map (g : β → γ) (f : γ → α → m α) (a : α) (l : t β) :
-    foldrm f a (g <$> l) = foldrm (f ∘ g) a l := by simp only [foldrm, foldMap_map, (· ∘ ·), flip]
+    foldrm f a (g <$> l) = foldrm (f ∘ g) a l := by
+  simp only [foldrm, foldMap_map, Function.comp_def, flip]
 
 end Equalities
 
diff --git a/Mathlib/Control/Functor.lean b/Mathlib/Control/Functor.lean
index 59ebe2553d456..305a88d2cc83f 100644
--- a/Mathlib/Control/Functor.lean
+++ b/Mathlib/Control/Functor.lean
@@ -3,7 +3,7 @@ Copyright (c) 2017 Simon Hudon. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Simon Hudon
 -/
-import Mathlib.Control.Basic
+import Mathlib.Tactic.Attr.Register
 import Mathlib.Data.Set.Defs
 import Mathlib.Tactic.TypeStar
 import Batteries.Tactic.Lint
@@ -181,7 +181,7 @@ protected theorem id_map : ∀ x : Comp F G α, Comp.map id x = x
 
 protected theorem comp_map (g' : α → β) (h : β → γ) :
     ∀ x : Comp F G α, Comp.map (h ∘ g') x = Comp.map h (Comp.map g' x)
-  | Comp.mk x => by simp [Comp.map, Comp.mk, Functor.map_comp_map, functor_norm]
+  | Comp.mk x => by simp [Comp.map, Comp.mk, Functor.map_comp_map, functor_norm, Function.comp_def]
   -- Porting note: `Comp.mk` wasn't needed in mathlib3
 
 instance lawfulFunctor : LawfulFunctor (Comp F G) where
diff --git a/Mathlib/Control/Functor/Multivariate.lean b/Mathlib/Control/Functor/Multivariate.lean
index b2d82ecaeea4d..993d066487aa9 100644
--- a/Mathlib/Control/Functor/Multivariate.lean
+++ b/Mathlib/Control/Functor/Multivariate.lean
@@ -213,7 +213,7 @@ theorem LiftR_RelLast_iff (x y : F (α ::: β)) :
     -- Porting note: proof was
     -- rw [MvFunctor.map_map, MvFunctor.map_map, (· ⊚ ·), (· ⊚ ·)]
     -- congr <;> ext i ⟨x, _⟩ <;> cases i <;> rfl
-    suffices  (fun i t => t.val.fst) = ((fun i x => (MvFunctor.f' rr n α i x).val.fst))
+    suffices (fun i t => t.val.fst) = ((fun i x => (MvFunctor.f' rr n α i x).val.fst))
             ∧ (fun i t => t.val.snd) = ((fun i x => (MvFunctor.f' rr n α i x).val.snd)) by
       rw [this.1, this.2]
     constructor <;> ext i ⟨x, _⟩ <;> cases i <;> rfl
diff --git a/Mathlib/Control/Lawful.lean b/Mathlib/Control/Lawful.lean
index 769ec2655f4ec..f2cd84ab5da94 100644
--- a/Mathlib/Control/Lawful.lean
+++ b/Mathlib/Control/Lawful.lean
@@ -48,7 +48,7 @@ end StateT
 
 namespace ExceptT
 
-variable {α β ε : Type u} {m : Type u → Type v} (x : ExceptT ε m α)
+variable {α ε : Type u} {m : Type u → Type v} (x : ExceptT ε m α)
 
 -- Porting note: This is proven by proj reduction in Lean 3.
 @[simp]
@@ -73,7 +73,6 @@ namespace ReaderT
 
 section
 
-variable {ρ : Type u}
 variable {m : Type u → Type v}
 variable {α σ : Type u}
 
diff --git a/Mathlib/Control/LawfulFix.lean b/Mathlib/Control/LawfulFix.lean
index 82375eb1c0037..f22207cffe5c1 100644
--- a/Mathlib/Control/LawfulFix.lean
+++ b/Mathlib/Control/LawfulFix.lean
@@ -35,11 +35,12 @@ halting problem. Instead, this requirement is limited to only functions that are
 sense of `ω`-complete partial orders, which excludes the example because it is not monotone
 (making the input argument less defined can make `f` more defined). -/
 class LawfulFix (α : Type*) [OmegaCompletePartialOrder α] extends Fix α where
-  fix_eq : ∀ {f : α →o α}, Continuous f → Fix.fix f = f (Fix.fix f)
+  fix_eq : ∀ {f : α → α}, ωScottContinuous f → Fix.fix f = f (Fix.fix f)
 
+@[deprecated LawfulFix.fix_eq (since := "2024-08-26")]
 theorem LawfulFix.fix_eq' {α} [OmegaCompletePartialOrder α] [LawfulFix α] {f : α → α}
-    (hf : Continuous' f) : Fix.fix f = f (Fix.fix f) :=
-  LawfulFix.fix_eq (hf.to_bundled _)
+    (hf : ωScottContinuous f) : Fix.fix f = f (Fix.fix f) :=
+  LawfulFix.fix_eq hf
 
 namespace Part
 
@@ -79,7 +80,7 @@ theorem mem_iff (a : α) (b : β a) : b ∈ Part.fix f a ↔ ∃ i, b ∈ approx
     · rcases le_total i j with H | H <;> [skip; symm] <;> apply_assumption <;> assumption
     replace hh := approx_mono f case _ _ hh
     apply Part.mem_unique h₁ hh
-  · simp only [fix_def' (⇑f) h₀, not_exists, false_iff_iff, not_mem_none]
+  · simp only [fix_def' (⇑f) h₀, not_exists, false_iff, not_mem_none]
     simp only [dom_iff_mem, not_exists] at h₀
     intro; apply h₀
 
@@ -107,7 +108,7 @@ def approxChain : Chain ((a : _) → Part <| β a) :=
   ⟨approx f, approx_mono f⟩
 
 theorem le_f_of_mem_approx {x} : x ∈ approxChain f → x ≤ f x := by
-  simp only [(· ∈ ·), forall_exists_index]
+  simp only [Membership.mem, forall_exists_index]
   rintro i rfl
   apply approx_mono'
 
@@ -157,8 +158,30 @@ theorem fix_le {X : (a : _) → Part <| β a} (hX : f X ≤ X) : Part.fix f ≤
     · apply f.monotone i_ih
     · apply hX
 
+variable {g : ((a : _) → Part <| β a) → (a : _) → Part <| β a}
+
+theorem fix_eq_ωSup_of_ωScottContinuous (hc : ωScottContinuous g) : Part.fix g =
+    ωSup (approxChain (⟨g,hc.monotone⟩ : ((a : _) → Part <| β a) →o (a : _) → Part <| β a)) := by
+  rw [← fix_eq_ωSup]
+  rfl
+
+theorem fix_eq_of_ωScottContinuous (hc : ωScottContinuous g) :
+    Part.fix g = g (Part.fix g) := by
+  rw [fix_eq_ωSup_of_ωScottContinuous hc, hc.map_ωSup]
+  apply le_antisymm
+  · apply ωSup_le_ωSup_of_le _
+    intro i
+    exists i
+    intro x
+    apply le_f_of_mem_approx _ ⟨i, rfl⟩
+  · apply ωSup_le_ωSup_of_le _
+    intro i
+    exists i.succ
+
 variable {f}
 
+set_option linter.deprecated false in
+@[deprecated fix_eq_of_ωScottContinuous (since := "2024-08-26")]
 theorem fix_eq (hc : Continuous f) : Part.fix f = f (Part.fix f) := by
   rw [fix_eq_ωSup f, hc]
   apply le_antisymm
@@ -182,6 +205,14 @@ def toUnitMono (f : Part α →o Part α) : (Unit → Part α) →o Unit → Par
   toFun x u := f (x u)
   monotone' x y (h : x ≤ y) u := f.monotone <| h u
 
+theorem ωScottContinuous_toUnitMono (f : Part α → Part α) (hc : ωScottContinuous f) :
+    ωScottContinuous (toUnitMono ⟨f,hc.monotone⟩) := .of_map_ωSup_of_orderHom fun _ => by
+  ext ⟨⟩ : 1
+  dsimp [OmegaCompletePartialOrder.ωSup]
+  erw [hc.map_ωSup, Chain.map_comp]; rfl
+
+set_option linter.deprecated false in
+@[deprecated ωScottContinuous_toUnitMono (since := "2024-08-26")]
 theorem to_unit_cont (f : Part α →o Part α) (hc : Continuous f) : Continuous (toUnitMono f)
   | _ => by
     ext ⟨⟩ : 1
@@ -189,8 +220,8 @@ theorem to_unit_cont (f : Part α →o Part α) (hc : Continuous f) : Continuous
     erw [hc, Chain.map_comp]; rfl
 
 instance lawfulFix : LawfulFix (Part α) :=
-  ⟨fun {f : Part α →o Part α} hc ↦ show Part.fix (toUnitMono f) () = _ by
-    rw [Part.fix_eq (to_unit_cont f hc)]; rfl⟩
+  ⟨fun {f : Part α → Part α} hc ↦ show Part.fix (toUnitMono ⟨f,hc.monotone⟩) () = _ by
+    rw [Part.fix_eq_of_ωScottContinuous (ωScottContinuous_toUnitMono f hc)]; rfl⟩
 
 end Part
 
@@ -199,7 +230,7 @@ open Sigma
 namespace Pi
 
 instance lawfulFix {β} : LawfulFix (α → Part β) :=
-  ⟨fun {_f} ↦ Part.fix_eq⟩
+  ⟨fun {_f} ↦ Part.fix_eq_of_ωScottContinuous⟩
 
 variable {γ : ∀ a : α, β a → Type*}
 
@@ -225,12 +256,32 @@ variable [(x y : _) → OmegaCompletePartialOrder <| γ x y]
 
 open OmegaCompletePartialOrder.Chain
 
+theorem ωScottContinuous_curry :
+    ωScottContinuous (monotoneCurry α β γ) :=
+  ωScottContinuous.of_map_ωSup_of_orderHom fun c ↦ by
+    ext x y
+    dsimp [curry, ωSup]
+    rw [map_comp, map_comp]
+    rfl
+
+set_option linter.deprecated false in
+@[deprecated ωScottContinuous_curry (since := "2024-08-26")]
 theorem continuous_curry : Continuous <| monotoneCurry α β γ := fun c ↦ by
   ext x y
   dsimp [curry, ωSup]
   rw [map_comp, map_comp]
   rfl
 
+theorem ωScottContinuous_uncurry :
+    ωScottContinuous (monotoneUncurry α β γ) :=
+    .of_map_ωSup_of_orderHom fun c ↦ by
+  ext ⟨x, y⟩
+  dsimp [uncurry, ωSup]
+  rw [map_comp, map_comp]
+  rfl
+
+set_option linter.deprecated false in
+@[deprecated ωScottContinuous_uncurry  (since := "2024-08-26")]
 theorem continuous_uncurry : Continuous <| monotoneUncurry α β γ := fun c ↦ by
   ext ⟨x, y⟩
   dsimp [uncurry, ωSup]
@@ -248,9 +299,18 @@ variable [∀ x y, OmegaCompletePartialOrder <| γ x y]
 
 section Curry
 
-variable {f : ((x : _) → (y : β x) → γ x y) →o (x : _) → (y : β x) → γ x y}
+variable {f : (∀ a b, γ a b) → ∀ a b, γ a b}
+
+theorem uncurry_curry_ωScottContinuous (hc : ωScottContinuous f) :
+    ωScottContinuous <| (monotoneUncurry α β γ).comp <|
+      (⟨f,hc.monotone⟩ : ((x : _) → (y : β x) → γ x y) →o (x : _) → (y : β x) → γ x y).comp <|
+      monotoneCurry α β γ :=
+  (ωScottContinuous_uncurry _ _ _).comp (hc.comp (ωScottContinuous_curry _ _ _))
 
-theorem uncurry_curry_continuous (hc : Continuous f) :
+set_option linter.deprecated false in
+@[deprecated uncurry_curry_ωScottContinuous  (since := "2024-08-26")]
+theorem uncurry_curry_continuous {f : ((x : _) → (y : β x) → γ x y) →o (x : _) → (y : β x) → γ x y}
+    (hc : Continuous f) :
     Continuous <| (monotoneUncurry α β γ).comp <| f.comp <| monotoneCurry α β γ :=
   continuous_comp _ _ (continuous_comp _ _ (continuous_curry _ _ _) hc) (continuous_uncurry _ _ _)
 
@@ -260,7 +320,7 @@ instance lawfulFix' [LawfulFix <| (x : Sigma β) → γ x.1 x.2] :
     LawfulFix ((x y : _) → γ x y) where
   fix_eq {_f} hc := by
     dsimp [fix]
-    conv_lhs => erw [LawfulFix.fix_eq (uncurry_curry_continuous hc)]
+    conv_lhs => erw [LawfulFix.fix_eq (uncurry_curry_ωScottContinuous hc)]
     rfl
 
 end Pi
diff --git a/Mathlib/Control/Monad/Cont.lean b/Mathlib/Control/Monad/Cont.lean
index 4a719782d7e74..29ed930cc7cb5 100644
--- a/Mathlib/Control/Monad/Cont.lean
+++ b/Mathlib/Control/Monad/Cont.lean
@@ -49,7 +49,7 @@ namespace ContT
 
 export MonadCont (Label goto)
 
-variable {r : Type u} {m : Type u → Type v} {α β γ ω : Type w}
+variable {r : Type u} {m : Type u → Type v} {α β : Type w}
 
 def run : ContT r m α → (α → m r) → m r :=
   id
diff --git a/Mathlib/Control/Monad/Writer.lean b/Mathlib/Control/Monad/Writer.lean
index fcbf590189c9f..a9b71af0c73a6 100644
--- a/Mathlib/Control/Monad/Writer.lean
+++ b/Mathlib/Control/Monad/Writer.lean
@@ -57,7 +57,7 @@ protected def runThe (ω : Type u) (cmd : WriterT ω M α) : M (α × ω) := cmd
 @[ext]
 protected theorem ext {ω : Type u} (x x' : WriterT ω M α) (h : x.run = x'.run) : x = x' := h
 
-variable {ω : Type u} {α β : Type u} [Monad M]
+variable [Monad M]
 
 /-- Creates an instance of `Monad`, with explicitly given `empty` and `append` operations.
 
diff --git a/Mathlib/Control/Random.lean b/Mathlib/Control/Random.lean
index 05198d61db646..c83ecbaba193b 100644
--- a/Mathlib/Control/Random.lean
+++ b/Mathlib/Control/Random.lean
@@ -28,18 +28,18 @@ defining objects that can be created randomly.
 
 -/
 
-set_option autoImplicit true
+set_option autoImplicit true -- Note: this file uses `autoImplicit` pervasively
 
 /-- A monad transformer to generate random objects using the generic generator type `g` -/
 abbrev RandGT (g : Type) := StateT (ULift g)
-/-- A monad to generate random objects using the generator type `g`.  -/
+/-- A monad to generate random objects using the generator type `g`. -/
 abbrev RandG (g : Type) := RandGT g Id
 
 /-- A monad transformer to generate random objects using the generator type `StdGen`.
 `RandT m α` should be thought of a random value in `m α`. -/
 abbrev RandT := RandGT StdGen
 
-/-- A monad to generate random objects using the generator type `StdGen`.  -/
+/-- A monad to generate random objects using the generator type `StdGen`. -/
 abbrev Rand := RandG StdGen
 
 instance [MonadLift m n] : MonadLiftT (RandGT g m) (RandGT g n) where
diff --git a/Mathlib/Control/Traversable/Basic.lean b/Mathlib/Control/Traversable/Basic.lean
index 6c338a28da53e..eff2d5afd7056 100644
--- a/Mathlib/Control/Traversable/Basic.lean
+++ b/Mathlib/Control/Traversable/Basic.lean
@@ -6,6 +6,7 @@ Authors: Simon Hudon
 import Mathlib.Data.Option.Defs
 import Mathlib.Control.Functor
 import Batteries.Data.List.Basic
+import Mathlib.Control.Basic
 
 /-!
 # Traversable type class
@@ -60,8 +61,8 @@ universe u v w
 
 section ApplicativeTransformation
 
-variable (F : Type u → Type v) [Applicative F] [LawfulApplicative F]
-variable (G : Type u → Type w) [Applicative G] [LawfulApplicative G]
+variable (F : Type u → Type v) [Applicative F]
+variable (G : Type u → Type w) [Applicative G]
 
 /-- A transformation between applicative functors.  It is a natural
 transformation such that `app` preserves the `Pure.pure` and
@@ -203,8 +204,7 @@ export Traversable (traverse)
 section Functions
 
 variable {t : Type u → Type u}
-variable {m : Type u → Type v} [Applicative m]
-variable {α β : Type u}
+variable {α : Type u}
 variable {f : Type u → Type u} [Applicative f]
 
 /-- A traversable functor commutes with all applicative functors. -/
@@ -221,7 +221,7 @@ satisfy a naturality condition with respect to applicative
 transformations. -/
 class LawfulTraversable (t : Type u → Type u) [Traversable t] extends LawfulFunctor t :
     Prop where
-  /-- `traverse` plays well with `pure` of the identity monad-/
+  /-- `traverse` plays well with `pure` of the identity monad -/
   id_traverse : ∀ {α} (x : t α), traverse (pure : α → Id α) x = x
   /-- `traverse` plays well with composition of applicative functors. -/
   comp_traverse :
@@ -249,8 +249,6 @@ instance : LawfulTraversable Id where
 
 section
 
-variable {F : Type u → Type v} [Applicative F]
-
 instance : Traversable Option :=
   ⟨Option.traverse⟩
 
diff --git a/Mathlib/Control/Traversable/Equiv.lean b/Mathlib/Control/Traversable/Equiv.lean
index 6b7276bafe9a7..db32b26b14bb2 100644
--- a/Mathlib/Control/Traversable/Equiv.lean
+++ b/Mathlib/Control/Traversable/Equiv.lean
@@ -43,7 +43,7 @@ protected def map {α β : Type u} (f : α → β) (x : t' α) : t' β :=
   eqv β <| map f ((eqv α).symm x)
 
 /-- The function `Equiv.map` transfers the functoriality of `t` to
-`t'` using the equivalences `eqv`.  -/
+`t'` using the equivalences `eqv`. -/
 protected def functor : Functor t' where map := Equiv.map eqv
 
 variable [LawfulFunctor t]
@@ -53,7 +53,7 @@ protected theorem id_map {α : Type u} (x : t' α) : Equiv.map eqv id x = x := b
 
 protected theorem comp_map {α β γ : Type u} (g : α → β) (h : β → γ) (x : t' α) :
     Equiv.map eqv (h ∘ g) x = Equiv.map eqv h (Equiv.map eqv g x) := by
-  simpa [Equiv.map] using comp_map ..
+  simp [Equiv.map, Function.comp_def]
 
 protected theorem lawfulFunctor : @LawfulFunctor _ (Equiv.functor eqv) :=
   -- Porting note: why is `_inst` required here?
diff --git a/Mathlib/Control/Traversable/Instances.lean b/Mathlib/Control/Traversable/Instances.lean
index 9e4afad602947..a91d6a6202f43 100644
--- a/Mathlib/Control/Traversable/Instances.lean
+++ b/Mathlib/Control/Traversable/Instances.lean
@@ -32,7 +32,7 @@ theorem Option.id_traverse {α} (x : Option α) : Option.traverse (pure : α →
 theorem Option.comp_traverse {α β γ} (f : β → F γ) (g : α → G β) (x : Option α) :
     Option.traverse (Comp.mk ∘ (f <$> ·) ∘ g) x =
       Comp.mk (Option.traverse f <$> Option.traverse g x) := by
-  cases x <;> simp! [functor_norm] <;> rfl
+  cases x <;> (simp! [functor_norm] <;> rfl)
 
 theorem Option.traverse_eq_map_id {α β} (f : α → β) (x : Option α) :
     Option.traverse ((pure : _ → Id _) ∘ f) x = (pure : _ → Id _) (f <$> x) := by cases x <;> rfl
@@ -148,7 +148,7 @@ variable [LawfulApplicative G]
 protected theorem comp_traverse {α β γ : Type u} (f : β → F γ) (g : α → G β) (x : σ ⊕ α) :
     Sum.traverse (Comp.mk ∘ (f <$> ·) ∘ g) x =
     Comp.mk.{u} (Sum.traverse f <$> Sum.traverse g x) := by
-  cases x <;> simp! [Sum.traverse, map_id, functor_norm] <;> rfl
+  cases x <;> (simp! [Sum.traverse, map_id, functor_norm] <;> rfl)
 
 protected theorem traverse_eq_map_id {α β} (f : α → β) (x : σ ⊕ α) :
     Sum.traverse ((pure : _ → Id _) ∘ f) x = (pure : _ → Id _) (f <$> x) := by
diff --git a/Mathlib/Control/Traversable/Lemmas.lean b/Mathlib/Control/Traversable/Lemmas.lean
index 25ada54df0f22..7b98b4bf635c3 100644
--- a/Mathlib/Control/Traversable/Lemmas.lean
+++ b/Mathlib/Control/Traversable/Lemmas.lean
@@ -39,7 +39,6 @@ variable [Applicative F] [LawfulApplicative F]
 variable [Applicative G] [LawfulApplicative G]
 variable {α β γ : Type u}
 variable (g : α → F β)
-variable (h : β → G γ)
 variable (f : β → γ)
 
 /-- The natural applicative transformation from the identity functor
@@ -56,7 +55,7 @@ def PureTransformation :
 theorem pureTransformation_apply {α} (x : id α) : PureTransformation F x = pure x :=
   rfl
 
-variable {F G} (x : t β)
+variable {F G}
 
 -- Porting note: need to specify `m/F/G := Id` because `id` no longer has a `Monad` instance
 theorem map_eq_traverse_id : map (f := t) f = traverse (m := Id) (pure ∘ f) :=
diff --git a/Mathlib/Control/ULift.lean b/Mathlib/Control/ULift.lean
index 785d0d187c5d7..2482e0095fe06 100644
--- a/Mathlib/Control/ULift.lean
+++ b/Mathlib/Control/ULift.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Jannis Limperg
+Authors: Kim Morrison, Jannis Limperg
 -/
 
 import Mathlib.Init
@@ -78,7 +78,7 @@ end PLift
 
 namespace ULift
 
-variable {α : Type u} {β : Type v} {f : α → β}
+variable {α : Type u} {β : Type v}
 
 /-- Functorial action. -/
 protected def map (f : α → β) (a : ULift.{u'} α) : ULift.{v'} β := ULift.up.{v'} (f a.down)
diff --git a/Mathlib/Data/Analysis/Topology.lean b/Mathlib/Data/Analysis/Topology.lean
index f8db5f20dd999..6da9c694593d9 100644
--- a/Mathlib/Data/Analysis/Topology.lean
+++ b/Mathlib/Data/Analysis/Topology.lean
@@ -208,7 +208,7 @@ end Ctop.Realizer
 
 /-- A `LocallyFinite.Realizer F f` is a realization that `f` is locally finite, namely it is a
 choice of open sets from the basis of `F` such that they intersect only finitely many of the values
-of `f`.  -/
+of `f`. -/
 structure LocallyFinite.Realizer [TopologicalSpace α] (F : Ctop.Realizer α) (f : β → Set α) where
   bas : ∀ a, { s // a ∈ F.F s }
   sets : ∀ x : α, Fintype { i | (f i ∩ F.F (bas x)).Nonempty }
@@ -237,7 +237,7 @@ instance [TopologicalSpace α] [Finite β] (F : Ctop.Realizer α) (f : β → Se
   (locallyFinite_iff_exists_realizer _).1 <| locallyFinite_of_finite _
 
 /-- A `Compact.Realizer s` is a realization that `s` is compact, namely it is a
-choice of finite open covers for each set family covering `s`.  -/
+choice of finite open covers for each set family covering `s`. -/
 def Compact.Realizer [TopologicalSpace α] (s : Set α) :=
   ∀ {f : Filter α} (F : f.Realizer) (x : F.σ), f ≠ ⊥ → F.F x ⊆ s → { a // a ∈ s ∧ 𝓝 a ⊓ f ≠ ⊥ }
 
diff --git a/Mathlib/Data/Array/ExtractLemmas.lean b/Mathlib/Data/Array/ExtractLemmas.lean
index bc66fc0660fbc..b27b5245af41d 100644
--- a/Mathlib/Data/Array/ExtractLemmas.lean
+++ b/Mathlib/Data/Array/ExtractLemmas.lean
@@ -27,7 +27,7 @@ theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
   · simp only [size_extract, size_append]
     omega
   · intro h1 h2 h3
-    rw [get_extract, get_append_left, get_extract]
+    rw [getElem_extract, getElem_append_left, getElem_extract]
 
 theorem extract_append_right {a b : Array α} {i j : Nat} (h : a.size ≤ i) :
     (a ++ b).extract i j = b.extract (i - a.size) (j - a.size) := by
@@ -35,8 +35,8 @@ theorem extract_append_right {a b : Array α} {i j : Nat} (h : a.size ≤ i) :
   · rw [size_extract, size_extract, size_append]
     omega
   · intro k hi h2
-    rw [get_extract, get_extract,
-      get_append_right (show size a ≤ i + k by omega)]
+    rw [getElem_extract, getElem_extract,
+      getElem_append_right (show size a ≤ i + k by omega)]
     congr
     omega
 
@@ -50,6 +50,6 @@ theorem extract_extract {s1 e2 e1 s2 : Nat} {a : Array α} (h : s1 + e2 ≤ e1)
   · simp only [size_extract]
     omega
   · intro i h1 h2
-    simp only [get_extract, Nat.add_assoc]
+    simp only [getElem_extract, Nat.add_assoc]
 
 end Array
diff --git a/Mathlib/Data/Bool/Basic.lean b/Mathlib/Data/Bool/Basic.lean
index da700043cf44d..047c5c7807e38 100644
--- a/Mathlib/Data/Bool/Basic.lean
+++ b/Mathlib/Data/Bool/Basic.lean
@@ -3,9 +3,8 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Jeremy Avigad
 -/
-import Batteries.Tactic.Init
+import Mathlib.Logic.Basic
 import Mathlib.Logic.Function.Defs
-import Mathlib.Order.Defs
 
 /-!
 # Booleans
@@ -120,21 +119,6 @@ alias of_decide_iff := decide_eq_true_iff
 
 theorem dichotomy (b : Bool) : b = false ∨ b = true := by cases b <;> simp
 
-theorem forall_bool' {p : Bool → Prop} (b : Bool) : (∀ x, p x) ↔ p b ∧ p !b :=
-  ⟨fun h ↦ ⟨h _, h _⟩, fun ⟨h₁, h₂⟩ x ↦ by cases b <;> cases x <;> assumption⟩
-
-@[simp]
-theorem forall_bool {p : Bool → Prop} : (∀ b, p b) ↔ p false ∧ p true :=
-  forall_bool' false
-
-theorem exists_bool' {p : Bool → Prop} (b : Bool) : (∃ x, p x) ↔ p b ∨ p !b :=
-  ⟨fun ⟨x, hx⟩ ↦ by cases x <;> cases b <;> first | exact .inl ‹_› | exact .inr ‹_›,
-    fun h ↦ by cases h <;> exact ⟨_, ‹_›⟩⟩
-
-@[simp]
-theorem exists_bool {p : Bool → Prop} : (∃ b, p b) ↔ p false ∨ p true :=
-  exists_bool' false
-
 theorem not_ne_id : not ≠ id := fun h ↦ false_ne_true <| congrFun h true
 
 @[deprecated (since := "2024-06-07")] alias eq_true_of_ne_false := eq_true_of_ne_false
@@ -181,7 +165,7 @@ attribute [simp] xor_assoc
 
 theorem xor_iff_ne : ∀ {x y : Bool}, xor x y = true ↔ x ≠ y := by decide
 
-/-! ### De Morgan's laws for booleans-/
+/-! ### De Morgan's laws for booleans -/
 
 instance linearOrder : LinearOrder Bool where
   le_refl := by decide
@@ -221,8 +205,8 @@ def ofNat (n : Nat) : Bool :=
 
 @[simp] lemma toNat_beq_zero (b : Bool) : (b.toNat == 0) = !b := by cases b <;> rfl
 @[simp] lemma toNat_bne_zero (b : Bool) : (b.toNat != 0) =  b := by simp [bne]
-@[simp] lemma toNat_beq_one  (b : Bool) : (b.toNat == 1) =  b := by cases b <;> rfl
-@[simp] lemma toNat_bne_one  (b : Bool) : (b.toNat != 1) = !b := by simp [bne]
+@[simp] lemma toNat_beq_one (b : Bool) : (b.toNat == 1) =  b := by cases b <;> rfl
+@[simp] lemma toNat_bne_one (b : Bool) : (b.toNat != 1) = !b := by simp [bne]
 
 theorem ofNat_le_ofNat {n m : Nat} (h : n ≤ m) : ofNat n ≤ ofNat m := by
   simp only [ofNat, ne_eq, _root_.decide_not]
@@ -243,7 +227,10 @@ theorem ofNat_toNat (b : Bool) : ofNat (toNat b) = b := by
 theorem injective_iff {α : Sort*} {f : Bool → α} : Function.Injective f ↔ f false ≠ f true :=
   ⟨fun Hinj Heq ↦ false_ne_true (Hinj Heq), fun H x y hxy ↦ by
     cases x <;> cases y
-    exacts [rfl, (H hxy).elim, (H hxy.symm).elim, rfl]⟩
+    · rfl
+    · exact (H hxy).elim
+    · exact (H hxy.symm).elim
+    · rfl⟩
 
 /-- **Kaminski's Equation** -/
 theorem apply_apply_apply (f : Bool → Bool) (x : Bool) : f (f (f x)) = f x := by
diff --git a/Mathlib/Data/ByteArray.lean b/Mathlib/Data/ByteArray.lean
index f7c249fb03484..c199be41b3f7c 100644
--- a/Mathlib/Data/ByteArray.lean
+++ b/Mathlib/Data/ByteArray.lean
@@ -88,20 +88,6 @@ def ByteSliceT.toSlice : ByteSliceT → ByteSlice
 /-- Convert a byte array into a byte slice. -/
 def ByteArray.toSlice (arr : ByteArray) : ByteSlice := ⟨arr, 0, arr.size⟩
 
-/-- Convert a string of assumed-ASCII characters into a byte array.
-(If any characters are non-ASCII they will be reduced modulo 256.) -/
-@[deprecated (since := "2024-08-19")]
-def String.toAsciiByteArray (s : String) : ByteArray :=
-  let rec loop (p : Pos) (out : ByteArray) : ByteArray :=
-    if h : s.atEnd p then out else
-    let c := s.get p
-    have : utf8ByteSize s - (next s p).byteIdx < utf8ByteSize s - p.byteIdx :=
-      Nat.sub_lt_sub_left (Nat.lt_of_not_le <| mt decide_eq_true h)
-        (Nat.lt_add_of_pos_right (Char.utf8Size_pos _))
-    loop (s.next p) (out.push c.toUInt8)
-    termination_by utf8ByteSize s - p.byteIdx
-  loop 0 ByteArray.empty
-
 /-- Convert a byte slice into a string. This does not handle non-ASCII characters correctly:
 every byte will become a unicode character with codepoint < 256. -/
 def ByteSlice.toString (bs : ByteSlice) : String := Id.run do
diff --git a/Mathlib/Data/Complex/Basic.lean b/Mathlib/Data/Complex/Basic.lean
index 6483469eea3bc..bc2017ba047a1 100644
--- a/Mathlib/Data/Complex/Basic.lean
+++ b/Mathlib/Data/Complex/Basic.lean
@@ -442,6 +442,15 @@ lemma re_ofNat (n : ℕ) [n.AtLeastTwo] : (no_index (OfNat.ofNat n) : ℂ).re =
 @[simp, norm_cast] lemma ratCast_re (q : ℚ) : (q : ℂ).re = q := rfl
 @[simp, norm_cast] lemma ratCast_im (q : ℚ) : (q : ℂ).im = 0 := rfl
 
+lemma re_nsmul (n : ℕ) (z : ℂ) : (n • z).re = n • z.re := smul_re ..
+lemma im_nsmul (n : ℕ) (z : ℂ) : (n • z).im = n • z.im := smul_im ..
+lemma re_zsmul (n : ℤ) (z : ℂ) : (n • z).re = n • z.re := smul_re ..
+lemma im_zsmul (n : ℤ) (z : ℂ) : (n • z).im = n • z.im := smul_im ..
+@[simp] lemma re_nnqsmul (q : ℚ≥0) (z : ℂ) : (q • z).re = q • z.re := smul_re ..
+@[simp] lemma im_nnqsmul (q : ℚ≥0) (z : ℂ) : (q • z).im = q • z.im := smul_im ..
+@[simp] lemma re_qsmul (q : ℚ) (z : ℂ) : (q • z).re = q • z.re := smul_re ..
+@[simp] lemma im_qsmul (q : ℚ) (z : ℂ) : (q • z).im = q • z.im := smul_im ..
+
 @[deprecated (since := "2024-04-17")]
 alias rat_cast_im := ratCast_im
 
@@ -628,6 +637,28 @@ def ofReal : ℝ →+* ℂ where
 theorem ofReal_eq_coe (r : ℝ) : ofReal r = r :=
   rfl
 
+variable {α : Type*}
+
+@[simp] lemma ofReal_comp_add (f g : α → ℝ) : ofReal' ∘ (f + g) = ofReal' ∘ f + ofReal' ∘ g :=
+  map_comp_add ofReal ..
+
+@[simp] lemma ofReal_comp_sub (f g : α → ℝ) : ofReal' ∘ (f - g) = ofReal' ∘ f - ofReal' ∘ g :=
+  map_comp_sub ofReal ..
+
+@[simp] lemma ofReal_comp_neg (f : α → ℝ) : ofReal' ∘ (-f) = -(ofReal' ∘ f) := map_comp_neg ofReal _
+
+lemma ofReal_comp_nsmul (n : ℕ) (f : α → ℝ) : ofReal' ∘ (n • f) = n • (ofReal' ∘ f) :=
+  map_comp_nsmul ofReal ..
+
+lemma ofReal_comp_zsmul (n : ℤ) (f : α → ℝ) : ofReal' ∘ (n • f) = n • (ofReal' ∘ f) :=
+  map_comp_zsmul ofReal ..
+
+@[simp] lemma ofReal_comp_mul (f g : α → ℝ) : ofReal' ∘ (f * g) = ofReal' ∘ f * ofReal' ∘ g :=
+  map_comp_mul ofReal ..
+
+@[simp] lemma ofReal_comp_pow (f : α → ℝ) (n : ℕ) : ofReal' ∘ (f ^ n) = (ofReal' ∘ f) ^ n :=
+  map_comp_pow ofReal ..
+
 @[simp]
 theorem I_sq : I ^ 2 = -1 := by rw [sq, I_mul_I]
 
diff --git a/Mathlib/Data/Complex/BigOperators.lean b/Mathlib/Data/Complex/BigOperators.lean
index 43c85d7750bfa..df65eec1fe310 100644
--- a/Mathlib/Data/Complex/BigOperators.lean
+++ b/Mathlib/Data/Complex/BigOperators.lean
@@ -3,14 +3,16 @@ Copyright (c) 2017 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard, Mario Carneiro
 -/
-import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Algebra.BigOperators.Balance
 import Mathlib.Data.Complex.Basic
 
 /-!
 # Finite sums and products of complex numbers
-
 -/
 
+open Fintype
+open scoped BigOperators
+
 namespace Complex
 
 variable {α : Type*} (s : Finset α)
@@ -23,12 +25,45 @@ theorem ofReal_prod (f : α → ℝ) : ((∏ i ∈ s, f i : ℝ) : ℂ) = ∏ i
 theorem ofReal_sum (f : α → ℝ) : ((∑ i ∈ s, f i : ℝ) : ℂ) = ∑ i ∈ s, (f i : ℂ) :=
   map_sum ofReal _ _
 
+@[simp, norm_cast]
+lemma ofReal_expect (f : α → ℝ) : (𝔼 i ∈ s, f i : ℝ) = 𝔼 i ∈ s, (f i : ℂ) :=
+  map_expect ofReal ..
+
+@[simp, norm_cast]
+lemma ofReal_balance [Fintype α] (f : α → ℝ) (a : α) :
+    ((balance f a : ℝ) : ℂ) = balance ((↑) ∘ f) a := by simp [balance]
+
+@[simp] lemma ofReal_comp_balance {ι : Type*} [Fintype ι] (f : ι → ℝ) :
+    ofReal ∘ balance f = balance (ofReal ∘ f : ι → ℂ) := funext <| ofReal_balance _
+
 @[simp]
 theorem re_sum (f : α → ℂ) : (∑ i ∈ s, f i).re = ∑ i ∈ s, (f i).re :=
   map_sum reAddGroupHom f s
 
+@[simp]
+lemma re_expect (f : α → ℂ) : (𝔼 i ∈ s, f i).re = 𝔼 i ∈ s, (f i).re :=
+  map_expect (LinearMap.mk reAddGroupHom.toAddHom (by simp)) f s
+
+@[simp]
+lemma re_balance [Fintype α] (f : α → ℂ) (a : α) : re (balance f a) = balance (re ∘ f) a := by
+  simp [balance]
+
+@[simp] lemma re_comp_balance {ι : Type*} [Fintype ι] (f : ι → ℂ) :
+    re ∘ balance f = balance (re ∘ f) := funext <| re_balance _
+
 @[simp]
 theorem im_sum (f : α → ℂ) : (∑ i ∈ s, f i).im = ∑ i ∈ s, (f i).im :=
   map_sum imAddGroupHom f s
 
+@[simp]
+lemma im_expect (f : α → ℂ) : (𝔼 i ∈ s, f i).im = 𝔼 i ∈ s, (f i).im :=
+  map_expect (LinearMap.mk imAddGroupHom.toAddHom (by simp)) f s
+
+@[simp]
+lemma im_balance [Fintype α] (f : α → ℂ) (a : α) : im (balance f a) = balance (im ∘ f) a := by
+  simp [balance]
+
+@[simp] lemma im_comp_balance {ι : Type*} [Fintype ι] (f : ι → ℂ) :
+    im ∘ balance f = balance (im ∘ f) := funext <| im_balance _
+
 end Complex
diff --git a/Mathlib/Data/Complex/Exponential.lean b/Mathlib/Data/Complex/Exponential.lean
index 0f6cf22a89718..990585b33e180 100644
--- a/Mathlib/Data/Complex/Exponential.lean
+++ b/Mathlib/Data/Complex/Exponential.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
 -/
 import Mathlib.Algebra.Order.CauSeq.BigOperators
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Data.Complex.Abs
 import Mathlib.Data.Complex.BigOperators
 import Mathlib.Data.Nat.Choose.Sum
@@ -28,7 +28,7 @@ theorem isCauSeq_abs_exp (z : ℂ) :
   let ⟨n, hn⟩ := exists_nat_gt (abs z)
   have hn0 : (0 : ℝ) < n := lt_of_le_of_lt (abs.nonneg _) hn
   IsCauSeq.series_ratio_test n (abs z / n) (div_nonneg (abs.nonneg _) (le_of_lt hn0))
-    (by rwa [div_lt_iff hn0, one_mul]) fun m hm => by
+    (by rwa [div_lt_iff₀ hn0, one_mul]) fun m hm => by
       rw [abs_abs, abs_abs, Nat.factorial_succ, pow_succ', mul_comm m.succ, Nat.cast_mul, ← div_div,
         mul_div_assoc, mul_div_right_comm, map_mul, map_div₀, abs_natCast]
       gcongr
@@ -186,6 +186,7 @@ theorem exp_add : exp (x + y) = exp x * exp y := by
 
 -- Porting note (#11445): new definition
 /-- the exponential function as a monoid hom from `Multiplicative ℂ` to `ℂ` -/
+@[simps]
 noncomputable def expMonoidHom : MonoidHom (Multiplicative ℂ) ℂ :=
   { toFun := fun z => exp (Multiplicative.toAdd z),
     map_one' := by simp,
@@ -689,6 +690,7 @@ nonrec theorem exp_add : exp (x + y) = exp x * exp y := by simp [exp_add, exp]
 
 -- Porting note (#11445): new definition
 /-- the exponential function as a monoid hom from `Multiplicative ℝ` to `ℝ` -/
+@[simps]
 noncomputable def expMonoidHom : MonoidHom (Multiplicative ℝ) ℝ :=
   { toFun := fun x => exp (Multiplicative.toAdd x),
     map_one' := by simp,
@@ -1108,7 +1110,7 @@ theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
       simp_rw [map_mul, map_pow, map_div₀, abs_natCast]
       gcongr
       rw [abv_pow abs]
-      exact pow_le_one _ (abs.nonneg _) hx
+      exact pow_le_one₀ (abs.nonneg _) hx
     _ = abs x ^ n * ∑ m ∈ (range j).filter fun k => n ≤ k, (1 / m.factorial : ℝ) := by
       simp [abs_mul, abv_pow abs, abs_div, ← mul_sum]
     _ ≤ abs x ^ n * (n.succ * (n.factorial * n : ℝ)⁻¹) := by
@@ -1332,9 +1334,9 @@ theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
           (calc
             |x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
                   gcongr
-                  · exact pow_le_one _ (abs_nonneg _) hx
+                  · exact pow_le_one₀ (abs_nonneg _) hx
                   · rw [sq, ← abs_mul_self, abs_mul]
-                    exact mul_le_one hx (abs_nonneg _) hx
+                    exact mul_le_one₀ hx (abs_nonneg _) hx
             _ < 1 := by norm_num)
     _ ≤ cos x := sub_le_comm.1 (abs_sub_le_iff.1 (cos_bound hx)).2
 
@@ -1396,7 +1398,7 @@ theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
       repeat erw [Finset.sum_range_succ]
       norm_num [Nat.factorial]
       nlinarith
-    _ < 1 / (1 - x) := by rw [lt_div_iff] <;> nlinarith
+    _ < 1 / (1 - x) := by rw [lt_div_iff₀] <;> nlinarith
 
 theorem exp_bound_div_one_sub_of_interval {x : ℝ} (h1 : 0 ≤ x) (h2 : x < 1) :
     Real.exp x ≤ 1 / (1 - x) := by
@@ -1410,7 +1412,7 @@ theorem add_one_lt_exp {x : ℝ} (hx : x ≠ 0) : x + 1 < Real.exp x := by
   obtain h' | h' := le_or_lt 1 (-x)
   · linarith [x.exp_pos]
   have hx' : 0 < x + 1 := by linarith
-  simpa [add_comm, exp_neg, inv_lt_inv (exp_pos _) hx']
+  simpa [add_comm, exp_neg, inv_lt_inv₀ (exp_pos _) hx']
     using exp_bound_div_one_sub_of_interval' (neg_pos.2 hx) h'
 
 theorem add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x := by
diff --git a/Mathlib/Data/Complex/ExponentialBounds.lean b/Mathlib/Data/Complex/ExponentialBounds.lean
index cb4022b3c3fe8..9531022b50196 100644
--- a/Mathlib/Data/Complex/ExponentialBounds.lean
+++ b/Mathlib/Data/Complex/ExponentialBounds.lean
@@ -36,16 +36,14 @@ theorem exp_one_lt_d9 : exp 1 < 2.7182818286 :=
   lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) (by norm_num)
 
 theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by
-  rw [exp_neg, lt_inv _ (exp_pos _)]
+  rw [exp_neg, lt_inv_comm₀ _ (exp_pos _)]
   · refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
     norm_num
   · norm_num
 
 theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by
-  rw [exp_neg, inv_lt (exp_pos _)]
-  · refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
-    norm_num
-  · norm_num
+  rw [exp_neg, inv_lt_comm₀ (exp_pos _) (by norm_num)]
+  exact lt_of_lt_of_le (by norm_num) (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
 
 theorem log_two_near_10 : |log 2 - 287209 / 414355| ≤ 1 / 10 ^ 10 := by
   suffices |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34) by
diff --git a/Mathlib/Data/Complex/FiniteDimensional.lean b/Mathlib/Data/Complex/FiniteDimensional.lean
index 5fbde35674c53..f989ee2a1847c 100644
--- a/Mathlib/Data/Complex/FiniteDimensional.lean
+++ b/Mathlib/Data/Complex/FiniteDimensional.lean
@@ -15,12 +15,11 @@ This file contains the `FiniteDimensional ℝ ℂ` instance, as well as some res
 (`finrank` and `Module.rank`).
 -/
 
-open FiniteDimensional
+open Module
 
 namespace Complex
 
-instance : FiniteDimensional ℝ ℂ :=
-  of_fintype_basis basisOneI
+instance : FiniteDimensional ℝ ℂ := .of_fintype_basis basisOneI
 
 @[simp]
 theorem finrank_real_complex : finrank ℝ ℂ = 2 := by
@@ -50,8 +49,8 @@ theorem rank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :
     simp only [Cardinal.lift_id']
 
 theorem finrank_real_of_complex (E : Type*) [AddCommGroup E] [Module ℂ E] :
-    FiniteDimensional.finrank ℝ E = 2 * FiniteDimensional.finrank ℂ E := by
-  rw [← FiniteDimensional.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]
+    Module.finrank ℝ E = 2 * Module.finrank ℂ E := by
+  rw [← Module.finrank_mul_finrank ℝ ℂ E, Complex.finrank_real_complex]
 
 section Rational
 
diff --git a/Mathlib/Data/Complex/Module.lean b/Mathlib/Data/Complex/Module.lean
index f13e8b88cc1a3..e8fc5c5c392dc 100644
--- a/Mathlib/Data/Complex/Module.lean
+++ b/Mathlib/Data/Complex/Module.lean
@@ -291,7 +291,7 @@ def liftAux (I' : A) (hf : I' * I' = -1) : ℂ →ₐ[ℝ] A :=
       congr 1
       -- equate "real" and "imaginary" parts
       · let inst : SMulCommClass ℝ A A := by infer_instance  -- Porting note: added
-        rw [smul_mul_smul, hf, smul_neg, ← Algebra.algebraMap_eq_smul_one, ← sub_eq_add_neg, ←
+        rw [smul_mul_smul_comm, hf, smul_neg, ← Algebra.algebraMap_eq_smul_one, ← sub_eq_add_neg, ←
           RingHom.map_mul, ← RingHom.map_sub]
       · rw [Algebra.smul_def, Algebra.smul_def, Algebra.smul_def, ← Algebra.right_comm _ x₂, ←
           mul_assoc, ← add_mul, ← RingHom.map_mul, ← RingHom.map_mul, ← RingHom.map_add]
diff --git a/Mathlib/Data/Complex/Order.lean b/Mathlib/Data/Complex/Order.lean
index f43e1f5666884..a9bc019f2b0bf 100644
--- a/Mathlib/Data/Complex/Order.lean
+++ b/Mathlib/Data/Complex/Order.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Complex.Abs
 
diff --git a/Mathlib/Data/Countable/Basic.lean b/Mathlib/Data/Countable/Basic.lean
index e312c883233a8..89694dd120b07 100644
--- a/Mathlib/Data/Countable/Basic.lean
+++ b/Mathlib/Data/Countable/Basic.lean
@@ -65,12 +65,19 @@ instance Sum.uncountable_inl [Uncountable α] : Uncountable (α ⊕ β) :=
 instance Sum.uncountable_inr [Uncountable β] : Uncountable (α ⊕ β) :=
   inr_injective.uncountable
 
-instance [Countable α] : Countable (Option α) :=
-  Countable.of_equiv _ (Equiv.optionEquivSumPUnit.{_, 0} α).symm
+instance Option.instCountable [Countable α] : Countable (Option α) :=
+  Countable.of_equiv _ (Equiv.optionEquivSumPUnit.{0, _} α).symm
+
+instance WithTop.instCountable [Countable α] : Countable (WithTop α) := Option.instCountable
+instance WithBot.instCountable [Countable α] : Countable (WithBot α) := Option.instCountable
+instance ENat.instCountable : Countable ℕ∞ := Option.instCountable
 
 instance Option.instUncountable [Uncountable α] : Uncountable (Option α) :=
   Injective.uncountable fun _ _ ↦ Option.some_inj.1
 
+instance WithTop.instUncountable [Uncountable α] : Uncountable (WithTop α) := Option.instUncountable
+instance WithBot.instUncountable [Uncountable α] : Uncountable (WithBot α) := Option.instUncountable
+
 instance [Countable α] [Countable β] : Countable (α × β) := by
   rcases exists_injective_nat α with ⟨f, hf⟩
   rcases exists_injective_nat β with ⟨g, hg⟩
diff --git a/Mathlib/Data/Countable/Small.lean b/Mathlib/Data/Countable/Small.lean
index c41b36c6d5183..72387f0c4a096 100644
--- a/Mathlib/Data/Countable/Small.lean
+++ b/Mathlib/Data/Countable/Small.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Basic
 import Mathlib.Data.Countable.Defs
diff --git a/Mathlib/Data/DFinsupp/Basic.lean b/Mathlib/Data/DFinsupp/Basic.lean
index 2a240cc29359d..11111a2e35ca6 100644
--- a/Mathlib/Data/DFinsupp/Basic.lean
+++ b/Mathlib/Data/DFinsupp/Basic.lean
@@ -3,6 +3,7 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Kenny Lau
 -/
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
 import Mathlib.Algebra.Group.Action.Prod
 import Mathlib.Algebra.Group.Submonoid.Membership
@@ -11,7 +12,6 @@ import Mathlib.Algebra.Module.LinearMap.Defs
 import Mathlib.Data.Finset.Preimage
 import Mathlib.Data.Fintype.Quotient
 import Mathlib.Data.Set.Finite
-import Mathlib.GroupTheory.GroupAction.BigOperators
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 
 /-!
@@ -1300,7 +1300,7 @@ instance distribMulAction₂ [Monoid γ] [∀ i j, AddMonoid (δ i j)]
 
 variable [DecidableEq ι]
 
-/-- The natural map between `Π₀ (i : Σ i, α i), δ i.1 i.2` and `Π₀ i (j : α i), δ i j`.  -/
+/-- The natural map between `Π₀ (i : Σ i, α i), δ i.1 i.2` and `Π₀ i (j : α i), δ i j`. -/
 def sigmaCurry [∀ i j, Zero (δ i j)] (f : Π₀ (i : Σ _, _), δ i.1 i.2) :
     Π₀ (i) (j), δ i j where
   toFun := fun i ↦
@@ -2093,3 +2093,5 @@ instance DFinsupp.infinite_of_right {ι : Sort _} {π : ι → Sort _} [∀ i, I
   DFinsupp.infinite_of_exists_right (Classical.arbitrary ι)
 
 end FiniteInfinite
+
+set_option linter.style.longFile 2200
diff --git a/Mathlib/Data/DFinsupp/Interval.lean b/Mathlib/Data/DFinsupp/Interval.lean
index 460f8e07408e8..8588c90c87525 100644
--- a/Mathlib/Data/DFinsupp/Interval.lean
+++ b/Mathlib/Data/DFinsupp/Interval.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Data.Fintype.BigOperators
 import Mathlib.Data.DFinsupp.Order
 import Mathlib.Order.Interval.Finset.Basic
diff --git a/Mathlib/Data/DFinsupp/Order.lean b/Mathlib/Data/DFinsupp/Order.lean
index db1c8bc2bdd5f..ebb4970f7a3ce 100644
--- a/Mathlib/Data/DFinsupp/Order.lean
+++ b/Mathlib/Data/DFinsupp/Order.lean
@@ -44,7 +44,14 @@ lemma le_def : f ≤ g ↔ ∀ i, f i ≤ g i := Iff.rfl
 def orderEmbeddingToFun : (Π₀ i, α i) ↪o ∀ i, α i where
   toFun := DFunLike.coe
   inj' := DFunLike.coe_injective
-  map_rel_iff' := by rfl
+  map_rel_iff' :=
+    #adaptation_note
+    /--
+    This proof used to be `rfl`,
+    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
+    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
+    -/
+    Iff.rfl
 
 @[simp, norm_cast]
 lemma coe_orderEmbeddingToFun : ⇑(orderEmbeddingToFun (α := α)) = DFunLike.coe := rfl
diff --git a/Mathlib/Data/DFinsupp/WellFounded.lean b/Mathlib/Data/DFinsupp/WellFounded.lean
index e10cffb7ca8e8..4262697dac419 100644
--- a/Mathlib/Data/DFinsupp/WellFounded.lean
+++ b/Mathlib/Data/DFinsupp/WellFounded.lean
@@ -6,8 +6,8 @@ Authors: Junyan Xu
 import Mathlib.Data.DFinsupp.Lex
 import Mathlib.Order.GameAdd
 import Mathlib.Order.Antisymmetrization
-import Mathlib.SetTheory.Ordinal.Basic
 import Mathlib.Tactic.AdaptationNote
+import Mathlib.SetTheory.Cardinal.Basic
 
 /-!
 # Well-foundedness of the lexicographic and product orders on `DFinsupp` and `Pi`
diff --git a/Mathlib/Data/ENNReal/Basic.lean b/Mathlib/Data/ENNReal/Basic.lean
index d1633df0c8af5..11fe99146aa06 100644
--- a/Mathlib/Data/ENNReal/Basic.lean
+++ b/Mathlib/Data/ENNReal/Basic.lean
@@ -214,6 +214,8 @@ theorem ofReal_eq_coe_nnreal {x : ℝ} (h : 0 ≤ x) :
     ENNReal.ofReal x = ofNNReal ⟨x, h⟩ :=
   (coe_nnreal_eq ⟨x, h⟩).symm
 
+theorem ofNNReal_toNNReal (x : ℝ) : (Real.toNNReal x : ℝ≥0∞) = ENNReal.ofReal x := rfl
+
 @[simp] theorem ofReal_coe_nnreal : ENNReal.ofReal p = p := (coe_nnreal_eq p).symm
 
 @[simp, norm_cast] theorem coe_zero : ↑(0 : ℝ≥0) = (0 : ℝ≥0∞) := rfl
@@ -709,8 +711,3 @@ def evalENNRealOfNNReal : PositivityExt where eval {u α} _zα _pα e := do
   | _, _, _ => throwError "not ENNReal.ofNNReal"
 
 end Mathlib.Meta.Positivity
-
-@[deprecated (since := "2023-12-23")] protected alias
-ENNReal.le_inv_smul_iff_of_pos := le_inv_smul_iff_of_pos
-@[deprecated (since := "2023-12-23")] protected alias
-ENNReal.inv_smul_le_iff_of_pos := inv_smul_le_iff_of_pos
diff --git a/Mathlib/Data/ENNReal/Inv.lean b/Mathlib/Data/ENNReal/Inv.lean
index d0e3217ac181c..1fc1f2eb0733b 100644
--- a/Mathlib/Data/ENNReal/Inv.lean
+++ b/Mathlib/Data/ENNReal/Inv.lean
@@ -438,9 +438,7 @@ protected theorem half_lt_self (hz : a ≠ 0) (ht : a ≠ ∞) : a / 2 < a := by
 protected theorem half_le_self : a / 2 ≤ a :=
   le_add_self.trans_eq <| ENNReal.add_halves _
 
-theorem sub_half (h : a ≠ ∞) : a - a / 2 = a / 2 := by
-  lift a to ℝ≥0 using h
-  exact sub_eq_of_add_eq (mul_ne_top coe_ne_top <| by simp) (ENNReal.add_halves a)
+theorem sub_half (h : a ≠ ∞) : a - a / 2 = a / 2 := ENNReal.sub_eq_of_eq_add' h a.add_halves.symm
 
 @[simp]
 theorem one_sub_inv_two : (1 : ℝ≥0∞) - 2⁻¹ = 2⁻¹ := by
@@ -576,13 +574,13 @@ theorem Ioo_zero_top_eq_iUnion_Ico_zpow {y : ℝ≥0∞} (hy : 1 < y) (h'y : y 
 theorem zpow_le_of_le {x : ℝ≥0∞} (hx : 1 ≤ x) {a b : ℤ} (h : a ≤ b) : x ^ a ≤ x ^ b := by
   induction' a with a a <;> induction' b with b b
   · simp only [Int.ofNat_eq_coe, zpow_natCast]
-    exact pow_le_pow_right hx (Int.le_of_ofNat_le_ofNat h)
+    exact pow_right_mono₀ hx (Int.le_of_ofNat_le_ofNat h)
   · apply absurd h (not_le_of_gt _)
     exact lt_of_lt_of_le (Int.negSucc_lt_zero _) (Int.ofNat_nonneg _)
   · simp only [zpow_negSucc, Int.ofNat_eq_coe, zpow_natCast]
     refine (ENNReal.inv_le_one.2 ?_).trans ?_ <;> exact one_le_pow_of_one_le' hx _
   · simp only [zpow_negSucc, ENNReal.inv_le_inv]
-    apply pow_le_pow_right hx
+    apply pow_right_mono₀ hx
     simpa only [← Int.ofNat_le, neg_le_neg_iff, Int.ofNat_add, Int.ofNat_one, Int.negSucc_eq] using
       h
 
@@ -603,5 +601,233 @@ protected theorem zpow_sub {x : ℝ≥0∞} (x_ne_zero : x ≠ 0) (x_ne_top : x
     x ^ (m - n) = (x ^ m) * (x ^ n)⁻¹ := by
   rw [sub_eq_add_neg, ENNReal.zpow_add x_ne_zero x_ne_top, ENNReal.zpow_neg x_ne_zero x_ne_top n]
 
+variable {ι κ : Sort*} {f g : ι → ℝ≥0∞} {s : Set ℝ≥0∞} {a : ℝ≥0∞}
+
+@[simp] lemma iSup_eq_zero : ⨆ i, f i = 0 ↔ ∀ i, f i = 0 := iSup_eq_bot
+
+@[simp] lemma iSup_zero_eq_zero : ⨆ _ : ι, (0 : ℝ≥0∞) = 0 := by simp
+
+lemma iSup_natCast : ⨆ n : ℕ, (n : ℝ≥0∞) = ∞ :=
+  (iSup_eq_top _).2 fun _b hb => ENNReal.exists_nat_gt (lt_top_iff_ne_top.1 hb)
+
+@[simp] lemma iSup_lt_eq_self (a : ℝ≥0∞) : ⨆ b, ⨆ _ : b < a, b = a := by
+  refine le_antisymm (iSup₂_le fun b hb ↦ hb.le) ?_
+  refine le_of_forall_lt fun c hca ↦ ?_
+  obtain ⟨d, hcd, hdb⟩ := exists_between hca
+  exact hcd.trans_le <| le_iSup₂_of_le d hdb le_rfl
+
+lemma isUnit_iff : IsUnit a ↔ a ≠ 0 ∧ a ≠ ∞ := by
+  refine ⟨fun ha ↦ ⟨ha.ne_zero, ?_⟩,
+    fun ha ↦ ⟨⟨a, a⁻¹, ENNReal.mul_inv_cancel ha.1 ha.2, ENNReal.inv_mul_cancel ha.1 ha.2⟩, rfl⟩⟩
+  obtain ⟨u, rfl⟩ := ha
+  rintro hu
+  have := congr($hu * u⁻¹)
+  norm_cast at this
+  simp [mul_inv_cancel] at this
+
+/-- Left multiplication by a nonzero finite `a` as an order isomorphism. -/
+@[simps! toEquiv apply symm_apply]
+def mulLeftOrderIso (a  : ℝ≥0∞) (ha : IsUnit a) : ℝ≥0∞ ≃o ℝ≥0∞ where
+  toEquiv := ha.unit.mulLeft
+  map_rel_iff' := by simp [ENNReal.mul_le_mul_left, ha.ne_zero, (isUnit_iff.1 ha).2]
+
+/-- Right multiplication by a nonzero finite `a` as an order isomorphism. -/
+@[simps! toEquiv apply symm_apply]
+def mulRightOrderIso (a  : ℝ≥0∞) (ha : IsUnit a) : ℝ≥0∞ ≃o ℝ≥0∞ where
+  toEquiv := ha.unit.mulRight
+  map_rel_iff' := by simp [ENNReal.mul_le_mul_right, ha.ne_zero, (isUnit_iff.1 ha).2]
+
+lemma mul_iSup (a : ℝ≥0∞) (f : ι → ℝ≥0∞) : a * ⨆ i, f i = ⨆ i, a * f i := by
+  by_cases hf : ∀ i, f i = 0
+  · simp [hf]
+  obtain rfl | ha₀ := eq_or_ne a 0
+  · simp
+  obtain rfl | ha := eq_or_ne a ∞
+  · obtain ⟨i, hi⟩ := not_forall.1 hf
+    simpa [iSup_eq_zero.not.2 hf, eq_comm (a := ⊤)] using le_iSup_of_le i (top_mul hi).ge
+  · exact (mulLeftOrderIso _ <| isUnit_iff.2 ⟨ha₀, ha⟩).map_iSup _
+
+lemma iSup_mul (f : ι → ℝ≥0∞) (a : ℝ≥0∞) : (⨆ i, f i) * a = ⨆ i, f i * a := by
+  simp [mul_comm, mul_iSup]
+
+lemma mul_sSup {a : ℝ≥0∞} : a * sSup s = ⨆ b ∈ s, a * b := by
+  simp only [sSup_eq_iSup, mul_iSup]
+
+lemma sSup_mul {a : ℝ≥0∞} : sSup s * a = ⨆ b ∈ s, b * a := by
+  simp only [sSup_eq_iSup, iSup_mul]
+
+lemma iSup_div (f : ι → ℝ≥0∞) (a : ℝ≥0∞) : iSup f / a = ⨆ i, f i / a := iSup_mul ..
+lemma sSup_div (s : Set ℝ≥0∞) (a : ℝ≥0∞) : sSup s / a = ⨆ b ∈ s, b / a := sSup_mul ..
+
+/-- Very general version for distributivity of multiplication over an infimum.
+
+See `ENNReal.mul_iInf_of_ne` for the special case assuming `a ≠ 0` and `a ≠ ∞`, and
+`ENNReal.mul_iInf` for the special case assuming `Nonempty ι`. -/
+lemma mul_iInf' (hinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) (h₀ : a = 0 → Nonempty ι) :
+    a * ⨅ i, f i = ⨅ i, a * f i := by
+  obtain rfl | ha₀ := eq_or_ne a 0
+  · simp [h₀ rfl]
+  obtain rfl | ha := eq_or_ne a ∞
+  · obtain ⟨i, hi⟩ | hf := em (∃ i, f i = 0)
+    · rw [(iInf_eq_bot _).2, (iInf_eq_bot _).2, bot_eq_zero, mul_zero] <;>
+        exact fun _ _↦ ⟨i, by simpa [hi]⟩
+    · rw [top_mul (mt (hinfty rfl) hf), eq_comm, iInf_eq_top]
+      exact fun i ↦ top_mul fun hi ↦ hf ⟨i, hi⟩
+  · exact (mulLeftOrderIso _ <| isUnit_iff.2 ⟨ha₀, ha⟩).map_iInf _
+
+/-- Very general version for distributivity of multiplication over an infimum.
+
+See `ENNReal.iInf_mul_of_ne` for the special case assuming `a ≠ 0` and `a ≠ ∞`, and
+`ENNReal.iInf_mul` for the special case assuming `Nonempty ι`. -/
+lemma iInf_mul' (hinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) (h₀ : a = 0 → Nonempty ι) :
+    (⨅ i, f i) * a = ⨅ i, f i * a := by simpa only [mul_comm a] using mul_iInf' hinfty h₀
+
+/-- If `a ≠ 0` and `a ≠ ∞`, then right multiplication by `a` maps infimum to infimum.
+
+See `ENNReal.mul_iInf'` for the general case, and `ENNReal.iInf_mul` for another special case that
+assumes `Nonempty ι` but does not require `a ≠ 0`, and `ENNReal`. -/
+lemma mul_iInf_of_ne (ha₀ : a ≠ 0) (ha : a ≠ ∞) : a * ⨅ i, f i = ⨅ i, a * f i :=
+  mul_iInf' (by simp [ha]) (by simp [ha₀])
+
+/-- If `a ≠ 0` and `a ≠ ∞`, then right multiplication by `a` maps infimum to infimum.
+
+See `ENNReal.iInf_mul'` for the general case, and `ENNReal.iInf_mul` for another special case that
+assumes `Nonempty ι` but does not require `a ≠ 0`. -/
+lemma iInf_mul_of_ne (ha₀ : a ≠ 0) (ha : a ≠ ∞) : (⨅ i, f i) * a = ⨅ i, f i * a :=
+  iInf_mul' (by simp [ha]) (by simp [ha₀])
+
+/-- See `ENNReal.mul_iInf'` for the general case, and `ENNReal.mul_iInf_of_ne` for another special
+case that assumes `a ≠ 0` but does not require `Nonempty ι`. -/
+lemma mul_iInf [Nonempty ι] (hinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) :
+    a * ⨅ i, f i = ⨅ i, a * f i := mul_iInf' hinfty fun _ ↦ ‹Nonempty ι›
+
+/-- See `ENNReal.iInf_mul'` for the general case, and `ENNReal.iInf_mul_of_ne` for another special
+case that assumes `a ≠ 0` but does not require `Nonempty ι`. -/
+lemma iInf_mul [Nonempty ι] (hinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) :
+    (⨅ i, f i) * a = ⨅ i, f i * a := iInf_mul' hinfty fun _ ↦ ‹Nonempty ι›
+
+/-- Very general version for distributivity of division over an infimum.
+
+See `ENNReal.iInf_div_of_ne` for the special case assuming `a ≠ 0` and `a ≠ ∞`, and
+`ENNReal.iInf_div` for the special case assuming `Nonempty ι`. -/
+lemma iInf_div' (hinfty : a = 0 → ⨅ i, f i = 0 → ∃ i, f i = 0) (h₀ : a = ∞ → Nonempty ι) :
+    (⨅ i, f i) / a = ⨅ i, f i / a := iInf_mul' (by simpa) (by simpa)
+
+/-- If `a ≠ 0` and `a ≠ ∞`, then division by `a` maps infimum to infimum.
+
+See `ENNReal.iInf_div'` for the general case, and `ENNReal.iInf_div` for another special case that
+assumes `Nonempty ι` but does not require `a ≠ ∞`. -/
+lemma iInf_div_of_ne (ha₀ : a ≠ 0) (ha : a ≠ ∞) : (⨅ i, f i) / a = ⨅ i, f i / a :=
+  iInf_div' (by simp [ha₀]) (by simp [ha])
+
+/-- See `ENNReal.iInf_div'` for the general case, and `ENNReal.iInf_div_of_ne` for another special
+case that assumes `a ≠ ∞` but does not require `Nonempty ι`. -/
+lemma iInf_div [Nonempty ι] (hinfty : a = 0 → ⨅ i, f i = 0 → ∃ i, f i = 0) :
+    (⨅ i, f i) / a = ⨅ i, f i / a := iInf_div' hinfty fun _ ↦ ‹Nonempty ι›
+
+lemma inv_iInf (f : ι → ℝ≥0∞) : (⨅ i, f i)⁻¹ = ⨆ i, (f i)⁻¹ := OrderIso.invENNReal.map_iInf _
+lemma inv_iSup (f : ι → ℝ≥0∞) : (⨆ i, f i)⁻¹ = ⨅ i, (f i)⁻¹ := OrderIso.invENNReal.map_iSup _
+
+lemma inv_sInf (s : Set ℝ≥0∞) : (sInf s)⁻¹ = ⨆ a ∈ s, a⁻¹ := by simp [sInf_eq_iInf, inv_iInf]
+lemma inv_sSup (s : Set ℝ≥0∞) : (sSup s)⁻¹ = ⨅ a ∈ s, a⁻¹ := by simp [sSup_eq_iSup, inv_iSup]
+
+lemma add_iSup [Nonempty ι] (f : ι → ℝ≥0∞) : a + ⨆ i, f i = ⨆ i, a + f i := by
+  obtain rfl | ha := eq_or_ne a ∞
+  · simp
+  refine le_antisymm ?_ <| iSup_le fun i ↦ add_le_add_left (le_iSup ..) _
+  refine add_le_of_le_tsub_left_of_le (le_iSup_of_le (Classical.arbitrary _) le_self_add) ?_
+  exact iSup_le fun i ↦ ENNReal.le_sub_of_add_le_left ha <| le_iSup (a + f ·) i
+
+lemma iSup_add [Nonempty ι] (f : ι → ℝ≥0∞) : (⨆ i, f i) + a = ⨆ i, f i + a := by
+  simp [add_comm, add_iSup]
+
+lemma add_biSup' {p : ι → Prop} (h : ∃ i, p i) (f : ι → ℝ≥0∞) :
+    a + ⨆ i, ⨆ _ : p i, f i = ⨆ i, ⨆ _ : p i, a + f i := by
+  haveI : Nonempty {i // p i} := nonempty_subtype.2 h
+  simp only [iSup_subtype', add_iSup]
+
+lemma biSup_add' {p : ι → Prop} (h : ∃ i, p i) (f : ι → ℝ≥0∞) :
+    (⨆ i, ⨆ _ : p i, f i) + a = ⨆ i, ⨆ _ : p i, f i + a := by simp only [add_comm, add_biSup' h]
+
+lemma add_biSup {ι : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → ℝ≥0∞) :
+    a + ⨆ i ∈ s, f i = ⨆ i ∈ s, a + f i := add_biSup' hs _
+
+lemma biSup_add {ι : Type*} {s : Set ι} (hs : s.Nonempty) (f : ι → ℝ≥0∞) :
+    (⨆ i ∈ s, f i) + a = ⨆ i ∈ s, f i + a := biSup_add' hs _
+
+lemma add_sSup (hs : s.Nonempty) : a + sSup s = ⨆ b ∈ s, a + b := by
+  rw [sSup_eq_iSup, add_biSup hs]
+
+lemma sSup_add (hs : s.Nonempty) : sSup s + a = ⨆ b ∈ s, b + a := by
+  rw [sSup_eq_iSup, biSup_add hs]
+
+lemma iSup_add_iSup_le [Nonempty ι] [Nonempty κ] {g : κ → ℝ≥0∞} (h : ∀ i j, f i + g j ≤ a) :
+    iSup f + iSup g ≤ a := by simp_rw [iSup_add, add_iSup]; exact iSup₂_le h
+
+lemma biSup_add_biSup_le' {p : ι → Prop} {q : κ → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j)
+    {g : κ → ℝ≥0∞} (h : ∀ i, p i → ∀ j, q j → f i + g j ≤ a) :
+    (⨆ i, ⨆ _ : p i, f i) + ⨆ j, ⨆ _ : q j, g j ≤ a := by
+  simp_rw [biSup_add' hp, add_biSup' hq]
+  exact iSup₂_le fun i hi => iSup₂_le (h i hi)
+
+lemma biSup_add_biSup_le {ι κ : Type*} {s : Set ι} {t : Set κ} (hs : s.Nonempty) (ht : t.Nonempty)
+    {f : ι → ℝ≥0∞} {g : κ → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i ∈ s, ∀ j ∈ t, f i + g j ≤ a) :
+    (⨆ i ∈ s, f i) + ⨆ j ∈ t, g j ≤ a := biSup_add_biSup_le' hs ht h
+
+lemma iSup_add_iSup (h : ∀ i j, ∃ k, f i + g j ≤ f k + g k) : iSup f + iSup g = ⨆ i, f i + g i := by
+  cases isEmpty_or_nonempty ι
+  · simp only [iSup_of_empty, bot_eq_zero, zero_add]
+  · refine le_antisymm ?_ (iSup_le fun a => add_le_add (le_iSup _ _) (le_iSup _ _))
+    refine iSup_add_iSup_le fun i j => ?_
+    rcases h i j with ⟨k, hk⟩
+    exact le_iSup_of_le k hk
+
+lemma iSup_add_iSup_of_monotone {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)] {f g : ι → ℝ≥0∞}
+    (hf : Monotone f) (hg : Monotone g) : iSup f + iSup g = ⨆ a, f a + g a :=
+  iSup_add_iSup fun i j ↦ (exists_ge_ge i j).imp fun _k ⟨hi, hj⟩ ↦ by gcongr <;> apply_rules
+
+lemma finsetSum_iSup {α ι : Type*} {s : Finset α} {f : α → ι → ℝ≥0∞}
+    (hf : ∀ i j, ∃ k, ∀ a, f a i ≤ f a k ∧ f a j ≤ f a k) :
+    ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i := by
+  induction' s using Finset.cons_induction with a s ha ihs
+  · simp
+  simp_rw [Finset.sum_cons, ihs]
+  refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_
+  gcongr
+  exacts [(hk a).1, (hk _).2]
+
+lemma finsetSum_iSup_of_monotone {α ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)] {s : Finset α}
+    {f : α → ι → ℝ≥0∞} (hf : ∀ a, Monotone (f a)) : (∑ a ∈ s, iSup (f a)) = ⨆ n, ∑ a ∈ s, f a n :=
+  finsetSum_iSup fun i j ↦ (exists_ge_ge i j).imp fun _k ⟨hi, hj⟩ a ↦ ⟨hf a hi, hf a hj⟩
+
+@[deprecated (since := "2024-07-14")]
+alias finset_sum_iSup_nat := finsetSum_iSup_of_monotone
+
+lemma smul_iSup {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (f : ι → ℝ≥0∞) (c : R) :
+    c • ⨆ i, f i = ⨆ i, c • f i := by
+  simp only [← smul_one_mul c (f _), ← smul_one_mul c (iSup _), ENNReal.mul_iSup]
+
+lemma smul_sSup {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (s : Set ℝ≥0∞) (c : R) :
+    c • sSup s = ⨆ a ∈ s, c • a := by
+  simp_rw [← smul_one_mul c (sSup s), ENNReal.mul_sSup, smul_one_mul]
+
+lemma sub_iSup [Nonempty ι] (ha : a ≠ ∞) : a - ⨆ i, f i = ⨅ i, a - f i := by
+  obtain ⟨i, hi⟩ | h := em (∃ i, a < f i)
+  · rw [tsub_eq_zero_iff_le.2 <| le_iSup_of_le _ hi.le, (iInf_eq_bot _).2, bot_eq_zero]
+    exact fun x hx ↦ ⟨i, by simpa [hi.le]⟩
+  simp_rw [not_exists, not_lt] at h
+  refine le_antisymm (le_iInf fun i ↦ tsub_le_tsub_left (le_iSup ..) _) <|
+    ENNReal.le_sub_of_add_le_left (ne_top_of_le_ne_top ha <| iSup_le h) <|
+    add_le_of_le_tsub_right_of_le (iInf_le_of_le (Classical.arbitrary _) tsub_le_self) <|
+    iSup_le fun i ↦ ?_
+  rw [← sub_sub_cancel ha (h _)]
+  exact tsub_le_tsub_left (iInf_le (a - f ·) i) _
+
+-- TODO: Prove the two one-side versions
+lemma exists_lt_add_of_lt_add {x y z : ℝ≥0∞} (h : x < y + z) (hy : y ≠ 0) (hz : z ≠ 0) :
+    ∃ y' < y, ∃ z' < z, x < y' + z' := by
+  contrapose! h;
+  simpa using biSup_add_biSup_le' (by exact ⟨0, hy.bot_lt⟩) (by exact ⟨0, hz.bot_lt⟩) h
+
 end Inv
 end ENNReal
diff --git a/Mathlib/Data/ENNReal/Operations.lean b/Mathlib/Data/ENNReal/Operations.lean
index 75da1428393a1..473c3d6c65d78 100644
--- a/Mathlib/Data/ENNReal/Operations.lean
+++ b/Mathlib/Data/ENNReal/Operations.lean
@@ -5,6 +5,7 @@ Authors: Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.BigOperators.WithTop
 import Mathlib.Algebra.GroupWithZero.Divisibility
+import Mathlib.Algebra.Module.Basic
 import Mathlib.Data.ENNReal.Basic
 
 /-!
@@ -255,6 +256,7 @@ section Cancel
 -- Porting note (#11215): TODO: generalize to `WithTop`
 /-- An element `a` is `AddLECancellable` if `a + b ≤ a + c` implies `b ≤ c` for all `b` and `c`.
   This is true in `ℝ≥0∞` for all elements except `∞`. -/
+@[simp]
 theorem addLECancellable_iff_ne {a : ℝ≥0∞} : AddLECancellable a ↔ a ≠ ∞ := by
   constructor
   · rintro h rfl
@@ -293,11 +295,13 @@ theorem sub_eq_sInf {a b : ℝ≥0∞} : a - b = sInf { d | a ≤ d + b } :=
   le_antisymm (le_sInf fun _ h => tsub_le_iff_right.mpr h) <| sInf_le <| mem_setOf.2 le_tsub_add
 
 /-- This is a special case of `WithTop.coe_sub` in the `ENNReal` namespace -/
-@[simp] theorem coe_sub : (↑(r - p) : ℝ≥0∞) = ↑r - ↑p := WithTop.coe_sub
+@[simp, norm_cast] theorem coe_sub : (↑(r - p) : ℝ≥0∞) = ↑r - ↑p := WithTop.coe_sub
 
 /-- This is a special case of `WithTop.top_sub_coe` in the `ENNReal` namespace -/
 @[simp] theorem top_sub_coe : ∞ - ↑r = ∞ := WithTop.top_sub_coe
 
+@[simp] lemma top_sub (ha : a ≠ ∞) : ∞ - a = ∞ := by lift a to ℝ≥0 using ha; exact top_sub_coe
+
 /-- This is a special case of `WithTop.sub_top` in the `ENNReal` namespace -/
 theorem sub_top : a - ∞ = 0 := WithTop.sub_top
 
@@ -313,15 +317,37 @@ theorem natCast_sub (m n : ℕ) : ↑(m - n) = (m - n : ℝ≥0∞) := by
 @[deprecated (since := "2024-04-17")]
 alias nat_cast_sub := natCast_sub
 
+/-- See `ENNReal.sub_eq_of_eq_add'` for a version assuming that `a = c + b` itself is finite rather
+than `b`. -/
 protected theorem sub_eq_of_eq_add (hb : b ≠ ∞) : a = c + b → a - b = c :=
   (cancel_of_ne hb).tsub_eq_of_eq_add
 
+/-- Weaker version of `ENNReal.sub_eq_of_eq_add` assuming that `a = c + b` itself is finite rather
+han `b`. -/
+protected lemma sub_eq_of_eq_add' (ha : a ≠ ∞) : a = c + b → a - b = c :=
+  (cancel_of_ne ha).tsub_eq_of_eq_add'
+
+/-- See `ENNReal.eq_sub_of_add_eq'` for a version assuming that `b = a + c` itself is finite rather
+than `c`. -/
 protected theorem eq_sub_of_add_eq (hc : c ≠ ∞) : a + c = b → a = b - c :=
   (cancel_of_ne hc).eq_tsub_of_add_eq
 
+/-- Weaker version of `ENNReal.eq_sub_of_add_eq` assuming that `b = a + c` itself is finite rather
+than `c`. -/
+protected lemma eq_sub_of_add_eq' (hb : b ≠ ∞) : a + c = b → a = b - c :=
+  (cancel_of_ne hb).eq_tsub_of_add_eq'
+
+/-- See `ENNReal.sub_eq_of_eq_add_rev'` for a version assuming that `a = b + c` itself is finite
+rather than `b`. -/
 protected theorem sub_eq_of_eq_add_rev (hb : b ≠ ∞) : a = b + c → a - b = c :=
   (cancel_of_ne hb).tsub_eq_of_eq_add_rev
 
+/-- Weaker version of `ENNReal.sub_eq_of_eq_add_rev` assuming that `a = b + c` itself is finite
+rather than `b`. -/
+protected lemma sub_eq_of_eq_add_rev' (ha : a ≠ ∞) : a = b + c → a - b = c :=
+  (cancel_of_ne ha).tsub_eq_of_eq_add_rev'
+
+@[deprecated ENNReal.sub_eq_of_eq_add (since := "2024-09-30")]
 theorem sub_eq_of_add_eq (hb : b ≠ ∞) (hc : a + b = c) : c - b = a :=
   ENNReal.sub_eq_of_eq_add hb hc.symm
 
@@ -337,7 +363,7 @@ protected theorem sub_add_eq_add_sub (hab : b ≤ a) (b_ne_top : b ≠ ∞) :
     a - b + c = a + c - b := by
   by_cases c_top : c = ∞
   · simpa [c_top] using ENNReal.eq_sub_of_add_eq b_ne_top rfl
-  refine (sub_eq_of_add_eq b_ne_top ?_).symm
+  refine ENNReal.eq_sub_of_add_eq b_ne_top ?_
   simp only [add_assoc, add_comm c b]
   simpa only [← add_assoc] using (add_left_inj c_top).mpr <| tsub_add_cancel_of_le hab
 
@@ -523,6 +549,15 @@ theorem smul_top {R} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ
   -- Porting note: need the primed version of `one_ne_zero` now
   simp_rw [smul_eq_zero, or_iff_left (one_ne_zero' ℝ≥0∞)]
 
+lemma nnreal_smul_lt_top {x : ℝ≥0} {y : ℝ≥0∞} (hy : y < ⊤) : x • y < ⊤ := mul_lt_top (by simp) hy
+lemma nnreal_smul_ne_top {x : ℝ≥0} {y : ℝ≥0∞} (hy : y ≠ ⊤) : x • y ≠ ⊤ := mul_ne_top (by simp) hy
+
+lemma nnreal_smul_ne_top_iff {x : ℝ≥0} {y : ℝ≥0∞} (hx : x ≠ 0) : x • y ≠ ⊤ ↔ y ≠ ⊤ :=
+  ⟨by rintro h rfl; simp [smul_top, hx] at h, nnreal_smul_ne_top⟩
+
+lemma nnreal_smul_lt_top_iff {x : ℝ≥0} {y : ℝ≥0∞} (hx : x ≠ 0) : x • y < ⊤ ↔ y < ⊤ := by
+  rw [lt_top_iff_ne_top, lt_top_iff_ne_top, nnreal_smul_ne_top_iff hx]
+
 end Actions
 
 end ENNReal
diff --git a/Mathlib/Data/ENNReal/Real.lean b/Mathlib/Data/ENNReal/Real.lean
index 4b6df926ef130..e9cc940e48612 100644
--- a/Mathlib/Data/ENNReal/Real.lean
+++ b/Mathlib/Data/ENNReal/Real.lean
@@ -235,7 +235,7 @@ lemma ofNat_le_ofReal {n : ℕ} [n.AtLeastTwo] {p : ℝ} :
     no_index (OfNat.ofNat n) ≤ ENNReal.ofReal p ↔ OfNat.ofNat n ≤ p :=
   natCast_le_ofReal (NeZero.ne n)
 
-@[simp]
+@[simp, norm_cast]
 lemma ofReal_le_natCast {r : ℝ} {n : ℕ} : ENNReal.ofReal r ≤ n ↔ r ≤ n :=
   coe_le_coe.trans Real.toNNReal_le_natCast
 
@@ -508,6 +508,19 @@ theorem toReal_sSup (s : Set ℝ≥0∞) (hf : ∀ r ∈ s, r ≠ ∞) :
     (sSup s).toReal = sSup (ENNReal.toReal '' s) := by
   simp only [ENNReal.toReal, toNNReal_sSup s hf, NNReal.coe_sSup, Set.image_image]
 
+@[simp] lemma ofReal_iInf [Nonempty ι] (f : ι → ℝ) :
+    ENNReal.ofReal (⨅ i, f i) = ⨅ i, ENNReal.ofReal (f i) := by
+  obtain ⟨i, hi⟩ | h := em (∃ i, f i ≤ 0)
+  · rw [(iInf_eq_bot _).2 fun _ _ ↦ ⟨i, by simpa [ofReal_of_nonpos hi]⟩]
+    simp [Real.iInf_nonpos' ⟨i, hi⟩]
+  replace h i : 0 ≤ f i := le_of_not_le fun hi ↦ h ⟨i, hi⟩
+  refine eq_of_forall_le_iff fun a ↦ ?_
+  obtain rfl | ha := eq_or_ne a ∞
+  · simp
+  rw [le_iInf_iff, le_ofReal_iff_toReal_le ha, le_ciInf_iff ⟨0, by simpa [mem_lowerBounds]⟩]
+  · exact forall_congr' fun i ↦ (le_ofReal_iff_toReal_le ha (h _)).symm
+  · exact Real.iInf_nonneg h
+
 theorem iInf_add : iInf f + a = ⨅ i, f i + a :=
   le_antisymm (le_iInf fun _ => add_le_add (iInf_le _ _) <| le_rfl)
     (tsub_le_iff_right.1 <| le_iInf fun _ => tsub_le_iff_right.2 <| iInf_le _ _)
@@ -545,51 +558,12 @@ theorem iInf_sum {α : Type*} {f : ι → α → ℝ≥0∞} {s : Finset α} [No
     rw [Finset.forall_mem_cons] at hk
     exact add_le_add hk.1.1 (Finset.sum_le_sum fun a ha => (hk.2 a ha).2)
 
-/-- If `x ≠ 0` and `x ≠ ∞`, then right multiplication by `x` maps infimum to infimum.
-See also `ENNReal.iInf_mul` that assumes `[Nonempty ι]` but does not require `x ≠ 0`. -/
-theorem iInf_mul_of_ne {ι} {f : ι → ℝ≥0∞} {x : ℝ≥0∞} (h0 : x ≠ 0) (h : x ≠ ∞) :
-    iInf f * x = ⨅ i, f i * x :=
-  le_antisymm mul_right_mono.map_iInf_le
-    ((ENNReal.div_le_iff_le_mul (Or.inl h0) <| Or.inl h).mp <|
-      le_iInf fun _ => (ENNReal.div_le_iff_le_mul (Or.inl h0) <| Or.inl h).mpr <| iInf_le _ _)
-
-/-- If `x ≠ ∞`, then right multiplication by `x` maps infimum over a nonempty type to infimum. See
-also `ENNReal.iInf_mul_of_ne` that assumes `x ≠ 0` but does not require `[Nonempty ι]`. -/
-theorem iInf_mul {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {x : ℝ≥0∞} (h : x ≠ ∞) :
-    iInf f * x = ⨅ i, f i * x := by
-  by_cases h0 : x = 0
-  · simp only [h0, mul_zero, iInf_const]
-  · exact iInf_mul_of_ne h0 h
-
-/-- If `x ≠ ∞`, then left multiplication by `x` maps infimum over a nonempty type to infimum. See
-also `ENNReal.mul_iInf_of_ne` that assumes `x ≠ 0` but does not require `[Nonempty ι]`. -/
-theorem mul_iInf {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {x : ℝ≥0∞} (h : x ≠ ∞) :
-    x * iInf f = ⨅ i, x * f i := by simpa only [mul_comm] using iInf_mul h
-
-/-- If `x ≠ 0` and `x ≠ ∞`, then left multiplication by `x` maps infimum to infimum.
-See also `ENNReal.mul_iInf` that assumes `[Nonempty ι]` but does not require `x ≠ 0`. -/
-theorem mul_iInf_of_ne {ι} {f : ι → ℝ≥0∞} {x : ℝ≥0∞} (h0 : x ≠ 0) (h : x ≠ ∞) :
-    x * iInf f = ⨅ i, x * f i := by simpa only [mul_comm] using iInf_mul_of_ne h0 h
-
-/-! `supr_mul`, `mul_supr` and variants are in `Topology.Instances.ENNReal`. -/
-
 end iInf
 
 section iSup
-
-@[simp]
-theorem iSup_eq_zero {ι : Sort*} {f : ι → ℝ≥0∞} : ⨆ i, f i = 0 ↔ ∀ i, f i = 0 :=
-  iSup_eq_bot
-
-@[simp]
-theorem iSup_zero_eq_zero {ι : Sort*} : ⨆ _ : ι, (0 : ℝ≥0∞) = 0 := by simp
-
 theorem sup_eq_zero {a b : ℝ≥0∞} : a ⊔ b = 0 ↔ a = 0 ∧ b = 0 :=
   sup_eq_bot_iff
 
-theorem iSup_natCast : ⨆ n : ℕ, (n : ℝ≥0∞) = ∞ :=
-  (iSup_eq_top _).2 fun _b hb => ENNReal.exists_nat_gt (lt_top_iff_ne_top.1 hb)
-
 @[deprecated (since := "2024-04-05")] alias iSup_coe_nat := iSup_natCast
 
 end iSup
diff --git a/Mathlib/Data/ENat/Basic.lean b/Mathlib/Data/ENat/Basic.lean
index 5854a9a681ac5..3486a11a1cef2 100644
--- a/Mathlib/Data/ENat/Basic.lean
+++ b/Mathlib/Data/ENat/Basic.lean
@@ -51,6 +51,9 @@ variable {m n : ℕ∞}
 `ℕ → ℕ∞` is `Nat.cast`. -/
 @[simp] theorem some_eq_coe : (WithTop.some : ℕ → ℕ∞) = Nat.cast := rfl
 
+instance : SuccAddOrder ℕ∞ where
+  succ_eq_add_one x := by cases x <;> simp [SuccOrder.succ]
+
 -- Porting note: `simp` and `norm_cast` can prove it
 --@[simp, norm_cast]
 theorem coe_zero : ((0 : ℕ) : ℕ∞) = 0 :=
@@ -101,6 +104,14 @@ lemma toNatHom_apply (n : ℕ) : toNatHom n = toNat n := rfl
 theorem toNat_coe (n : ℕ) : toNat n = n :=
   rfl
 
+@[simp]
+theorem toNat_zero : toNat 0 = 0 :=
+  rfl
+
+@[simp]
+theorem toNat_one : toNat 1 = 1 :=
+  rfl
+
 -- See note [no_index around OfNat.ofNat]
 @[simp]
 theorem toNat_ofNat (n : ℕ) [n.AtLeastTwo] : toNat (no_index (OfNat.ofNat n)) = n :=
@@ -202,32 +213,32 @@ lemma toNat_le_toNat {m n : ℕ∞} (h : m ≤ n) (hn : n ≠ ⊤) : toNat m ≤
   toNat_le_of_le_coe <| h.trans_eq (coe_toNat hn).symm
 
 @[simp]
-theorem succ_def (m : ℕ∞) : Order.succ m = m + 1 := by
-  cases m
-  · rfl
-  · change ite .. = _
-    simp
+theorem succ_def (m : ℕ∞) : Order.succ m = m + 1 :=
+  Order.succ_eq_add_one m
 
+@[deprecated Order.add_one_le_of_lt (since := "2024-09-04")]
 theorem add_one_le_of_lt (h : m < n) : m + 1 ≤ n :=
-  m.succ_def ▸ Order.succ_le_of_lt h
+  Order.add_one_le_of_lt h
 
 theorem add_one_le_iff (hm : m ≠ ⊤) : m + 1 ≤ n ↔ m < n :=
-  m.succ_def ▸ (Order.succ_le_iff_of_not_isMax <| by rwa [isMax_iff_eq_top])
+  Order.add_one_le_iff_of_not_isMax (not_isMax_iff_ne_top.mpr hm)
 
+@[deprecated Order.one_le_iff_pos (since := "2024-09-04")]
 theorem one_le_iff_pos : 1 ≤ n ↔ 0 < n :=
-  add_one_le_iff WithTop.zero_ne_top
+  Order.one_le_iff_pos
 
 theorem one_le_iff_ne_zero : 1 ≤ n ↔ n ≠ 0 :=
-  one_le_iff_pos.trans pos_iff_ne_zero
+  Order.one_le_iff_pos.trans pos_iff_ne_zero
 
 lemma lt_one_iff_eq_zero : n < 1 ↔ n = 0 :=
   not_le.symm.trans one_le_iff_ne_zero.not_left
 
+@[deprecated Order.le_of_lt_add_one (since := "2024-09-04")]
 theorem le_of_lt_add_one (h : m < n + 1) : m ≤ n :=
-  Order.le_of_lt_succ <| n.succ_def.symm ▸ h
+  Order.le_of_lt_add_one h
 
 theorem lt_add_one_iff (hm : n ≠ ⊤) : m < n + 1 ↔ m ≤ n :=
-  n.succ_def ▸ Order.lt_succ_iff_of_not_isMax (not_isMax_iff_ne_top.mpr hm)
+  Order.lt_add_one_iff_of_not_isMax (not_isMax_iff_ne_top.mpr hm)
 
 theorem le_coe_iff {n : ℕ∞} {k : ℕ} : n ≤ ↑k ↔ ∃ (n₀ : ℕ), n = n₀ ∧ n₀ ≤ k :=
   WithTop.le_coe_iff
@@ -248,4 +259,26 @@ theorem nat_induction {P : ℕ∞ → Prop} (a : ℕ∞) (h0 : P 0) (hsuc : ∀
   · exact htop A
   · exact A _
 
+lemma add_one_nat_le_withTop_of_lt {m : ℕ} {n : WithTop ℕ∞} (h : m < n) : (m + 1 : ℕ) ≤ n := by
+  match n with
+  | ⊤ => exact le_top
+  | (⊤ : ℕ∞) => exact WithTop.coe_le_coe.2 (OrderTop.le_top _)
+  | (n : ℕ) => simpa only [Nat.cast_le, ge_iff_le, Nat.cast_lt] using h
+
+@[simp] lemma coe_top_add_one : ((⊤ : ℕ∞) : WithTop ℕ∞) + 1 = (⊤ : ℕ∞) := rfl
+
+@[simp] lemma add_one_eq_coe_top_iff (n : WithTop ℕ∞) :
+    n + 1 = (⊤ : ℕ∞) ↔ n = (⊤ : ℕ∞) := by
+  match n with
+  | ⊤ => exact Iff.rfl
+  | (⊤ : ℕ∞) => exact Iff.rfl
+  | (n : ℕ) => norm_cast; simp only [coe_ne_top, iff_false, ne_eq]
+
+@[simp] lemma nat_ne_coe_top (n : ℕ) : (n : WithTop ℕ∞) ≠ (⊤ : ℕ∞) := ne_of_beq_false rfl
+
+lemma one_le_iff_ne_zero_withTop {n : WithTop ℕ∞} :
+    1 ≤ n ↔ n ≠ 0 :=
+  ⟨fun h ↦ (zero_lt_one.trans_le h).ne',
+    fun h ↦ add_one_nat_le_withTop_of_lt (pos_iff_ne_zero.mpr h)⟩
+
 end ENat
diff --git a/Mathlib/Data/Fin/Basic.lean b/Mathlib/Data/Fin/Basic.lean
index 12906e1a342c1..e0de8283c150d 100644
--- a/Mathlib/Data/Fin/Basic.lean
+++ b/Mathlib/Data/Fin/Basic.lean
@@ -5,9 +5,11 @@ Authors: Robert Y. Lewis, Keeley Hoek
 -/
 import Mathlib.Algebra.NeZero
 import Mathlib.Data.Nat.Defs
+import Mathlib.Data.Int.DivMod
 import Mathlib.Logic.Embedding.Basic
 import Mathlib.Logic.Equiv.Set
 import Mathlib.Tactic.Common
+import Mathlib.Tactic.Attr.Register
 
 /-!
 # The finite type with `n` elements
@@ -181,6 +183,8 @@ protected theorem heq_fun₂_iff {α : Sort*} {k l k' l' : ℕ} (h : k = l) (h'
   subst h'
   simp [Function.funext_iff]
 
+/-- Two elements of `Fin k` and `Fin l` are heq iff their values in `ℕ` coincide. This requires
+`k = l`. For the left implication without this assumption, see `val_eq_val_of_heq`. -/
 protected theorem heq_ext_iff {k l : ℕ} (h : k = l) {i : Fin k} {j : Fin l} :
     HEq i j ↔ (i : ℕ) = (j : ℕ) := by
   subst h
@@ -188,6 +192,7 @@ protected theorem heq_ext_iff {k l : ℕ} (h : k = l) {i : Fin k} {j : Fin l} :
 
 end coe
 
+
 section Order
 
 /-!
@@ -270,6 +275,12 @@ theorem pos_iff_ne_zero' [NeZero n] (a : Fin n) : 0 < a ↔ a ≠ 0 := by
 
 @[simp] lemma cast_eq_self (a : Fin n) : cast rfl a = a := rfl
 
+@[simp] theorem cast_eq_zero {k l : ℕ} [NeZero k] [NeZero l]
+    (h : k = l) (x : Fin k) : Fin.cast h x = 0 ↔ x = 0 := by simp [← val_eq_val]
+
+lemma cast_injective {k l : ℕ} (h : k = l) : Injective (Fin.cast h) :=
+  fun a b hab ↦ by simpa [← val_eq_val] using hab
+
 theorem rev_involutive : Involutive (rev : Fin n → Fin n) := rev_rev
 
 /-- `Fin.rev` as an `Equiv.Perm`, the antitone involution `Fin n → Fin n` given by
@@ -324,6 +335,58 @@ theorem one_lt_last [NeZero n] : 1 < last (n + 1) := by
 
 end Order
 
+/-! ### Coercions to `ℤ` and the `fin_omega` tactic. -/
+
+open Int
+
+theorem coe_int_sub_eq_ite {n : Nat} (u v : Fin n) :
+    ((u - v : Fin n) : Int) = if v ≤ u then (u - v : Int) else (u - v : Int) + n := by
+  rw [Fin.sub_def]
+  split
+  · rw [ofNat_emod, Int.emod_eq_sub_self_emod, Int.emod_eq_of_lt] <;> omega
+  · rw [ofNat_emod, Int.emod_eq_of_lt] <;> omega
+
+theorem coe_int_sub_eq_mod {n : Nat} (u v : Fin n) :
+    ((u - v : Fin n) : Int) = ((u : Int) - (v : Int)) % n := by
+  rw [coe_int_sub_eq_ite]
+  split
+  · rw [Int.emod_eq_of_lt] <;> omega
+  · rw [Int.emod_eq_add_self_emod, Int.emod_eq_of_lt] <;> omega
+
+theorem coe_int_add_eq_ite {n : Nat} (u v : Fin n) :
+    ((u + v : Fin n) : Int) = if (u + v : ℕ) < n then (u + v : Int) else (u + v : Int) - n := by
+  rw [Fin.add_def]
+  split
+  · rw [ofNat_emod, Int.emod_eq_of_lt] <;> omega
+  · rw [ofNat_emod, Int.emod_eq_sub_self_emod, Int.emod_eq_of_lt] <;> omega
+
+theorem coe_int_add_eq_mod {n : Nat} (u v : Fin n) :
+    ((u + v : Fin n) : Int) = ((u : Int) + (v : Int)) % n := by
+  rw [coe_int_add_eq_ite]
+  split
+  · rw [Int.emod_eq_of_lt] <;> omega
+  · rw [Int.emod_eq_sub_self_emod, Int.emod_eq_of_lt] <;> omega
+
+-- Write `a + b` as `if (a + b : ℕ) < n then (a + b : ℤ) else (a + b : ℤ) - n` and
+-- similarly `a - b` as `if (b : ℕ) ≤ a then (a - b : ℤ) else (a - b : ℤ) + n`.
+attribute [fin_omega] coe_int_sub_eq_ite coe_int_add_eq_ite
+
+-- Rewrite inequalities in `Fin` to inequalities in `ℕ`
+attribute [fin_omega] Fin.lt_iff_val_lt_val Fin.le_iff_val_le_val
+
+-- Rewrite `1 : Fin (n + 2)` to `1 : ℤ`
+attribute [fin_omega] val_one
+
+/--
+Preprocessor for `omega` to handle inequalities in `Fin`.
+Note that this involves a lot of case splitting, so may be slow.
+-/
+-- Further adjustment to the simp set can probably make this more powerful.
+-- Please experiment and PR updates!
+macro "fin_omega" : tactic => `(tactic|
+  { try simp only [fin_omega, ← Int.ofNat_lt, ← Int.ofNat_le] at *
+    omega })
+
 section Add
 
 /-!
@@ -351,16 +414,6 @@ section Monoid
 protected theorem add_zero [NeZero n] (k : Fin n) : k + 0 = k := by
   simp only [add_def, val_zero', Nat.add_zero, mod_eq_of_lt (is_lt k)]
 
--- Porting note (#10618): removing `simp`, `simp` can prove it with AddCommMonoid instance
-protected theorem zero_add [NeZero n] (k : Fin n) : 0 + k = k := by
-  simp [Fin.ext_iff, add_def, mod_eq_of_lt (is_lt k)]
-
-instance {a : ℕ} [NeZero n] : OfNat (Fin n) a where
-  ofNat := Fin.ofNat' a n.pos_of_neZero
-
-instance inhabited (n : ℕ) [NeZero n] : Inhabited (Fin n) :=
-  ⟨0⟩
-
 instance inhabitedFinOneAdd (n : ℕ) : Inhabited (Fin (1 + n)) :=
   haveI : NeZero (1 + n) := by rw [Nat.add_comm]; infer_instance
   inferInstance
@@ -371,8 +424,8 @@ theorem default_eq_zero (n : ℕ) [NeZero n] : (default : Fin n) = 0 :=
 
 section from_ad_hoc
 
-@[simp] lemma ofNat'_zero {h : 0 < n} [NeZero n] : (Fin.ofNat' 0 h : Fin n) = 0 := rfl
-@[simp] lemma ofNat'_one {h : 0 < n} [NeZero n] : (Fin.ofNat' 1 h : Fin n) = 1 := rfl
+@[simp] lemma ofNat'_zero [NeZero n] : (Fin.ofNat' n 0) = 0 := rfl
+@[simp] lemma ofNat'_one [NeZero n] : (Fin.ofNat' n 1) = 1 := rfl
 
 end from_ad_hoc
 
@@ -402,12 +455,12 @@ alias val_nat_cast := val_natCast
 
 -- Porting note: is this the right name for things involving `Nat.cast`?
 /-- Converting an in-range number to `Fin (n + 1)` produces a result
-whose value is the original number.  -/
+whose value is the original number. -/
 theorem val_cast_of_lt {n : ℕ} [NeZero n] {a : ℕ} (h : a < n) : (a : Fin n).val = a :=
   Nat.mod_eq_of_lt h
 
 /-- If `n` is non-zero, converting the value of a `Fin n` to `Fin n` results
-in the same value.  -/
+in the same value. -/
 @[simp] theorem cast_val_eq_self {n : ℕ} [NeZero n] (a : Fin n) : (a.val : Fin n) = a :=
   Fin.ext <| val_cast_of_lt a.isLt
 
@@ -415,7 +468,8 @@ in the same value.  -/
 
 -- Porting note: this is syntactically the same as `cast_val_of_lt`
 
-@[simp] lemma natCast_self (n : ℕ) [NeZero n] : (n : Fin n) = 0 := by ext; simp
+-- This is a special case of `CharP.cast_eq_zero` that doesn't require typeclass search
+@[simp high] lemma natCast_self (n : ℕ) [NeZero n] : (n : Fin n) = 0 := by ext; simp
 
 @[deprecated (since := "2024-04-17")]
 alias nat_cast_self := natCast_self
@@ -452,13 +506,6 @@ lemma natCast_strictMono (hbn : b ≤ n) (hab : a < b) : (a : Fin (n + 1)) < b :
 
 end OfNatCoe
 
-@[simp]
-theorem one_eq_zero_iff [NeZero n] : (1 : Fin n) = 0 ↔ n = 1 := by
-  obtain _ | _ | n := n <;> simp [Fin.ext_iff]
-
-@[simp]
-theorem zero_eq_one_iff [NeZero n] : (0 : Fin n) = 1 ↔ n = 1 := by rw [eq_comm, one_eq_zero_iff]
-
 end Add
 
 section Succ
@@ -514,10 +561,6 @@ This one instead uses a `NeZero n` typeclass hypothesis.
 theorem le_zero_iff' {n : ℕ} [NeZero n] {k : Fin n} : k ≤ 0 ↔ k = 0 :=
   ⟨fun h => Fin.ext <| by rw [Nat.eq_zero_of_le_zero h]; rfl, by rintro rfl; exact Nat.le_refl _⟩
 
--- Move to Batteries?
-@[simp] theorem cast_refl {n : Nat} (h : n = n) :
-    Fin.cast h = id := rfl
-
 -- TODO: Move to Batteries
 @[simp] lemma castLE_inj {hmn : m ≤ n} {a b : Fin m} : castLE hmn a = castLE hmn b ↔ a = b := by
   simp [Fin.ext_iff]
@@ -533,7 +576,7 @@ lemma castAdd_injective (m n : ℕ) : Injective (@Fin.castAdd m n) := castLE_inj
 
 lemma castSucc_injective (n : ℕ) : Injective (@Fin.castSucc n) := castAdd_injective _ _
 
-/-- `Fin.castLE` as an `Embedding`, `castLEEmb h i` embeds `i` into a larger `Fin` type.  -/
+/-- `Fin.castLE` as an `Embedding`, `castLEEmb h i` embeds `i` into a larger `Fin` type. -/
 @[simps! apply]
 def castLEEmb (h : n ≤ m) : Fin n ↪ Fin m where
   toFun := castLE h
@@ -711,7 +754,7 @@ theorem castSucc_ne_zero_of_lt {p i : Fin n} (h : p < i) : castSucc i ≠ 0 := b
     exact ((zero_le _).trans_lt h).ne'
 
 theorem succ_ne_last_iff (a : Fin (n + 1)) : succ a ≠ last (n + 1) ↔ a ≠ last n :=
-  not_iff_not.mpr <| succ_eq_last_succ a
+  not_iff_not.mpr <| succ_eq_last_succ
 
 theorem succ_ne_last_of_lt {p i : Fin n} (h : i < p) : succ i ≠ last n := by
   cases n
@@ -774,7 +817,7 @@ theorem le_pred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ 0) : j ≤ pred i
   rw [← succ_le_succ_iff, succ_pred]
 
 theorem castSucc_pred_eq_pred_castSucc {a : Fin (n + 1)} (ha : a ≠ 0)
-    (ha' := a.castSucc_ne_zero_iff.mpr ha) :
+    (ha' := castSucc_ne_zero_iff.mpr ha) :
     (a.pred ha).castSucc = (castSucc a).pred ha' := rfl
 
 theorem castSucc_pred_add_one_eq {a : Fin (n + 1)} (ha : a ≠ 0) :
@@ -1427,11 +1470,11 @@ theorem eq_zero (n : Fin 1) : n = 0 := Subsingleton.elim _ _
 instance uniqueFinOne : Unique (Fin 1) where
   uniq _ := Subsingleton.elim _ _
 
-@[simp]
+@[deprecated val_eq_zero (since := "2024-09-18")]
 theorem coe_fin_one (a : Fin 1) : (a : ℕ) = 0 := by simp [Subsingleton.elim a 0]
 
-lemma eq_one_of_neq_zero (i : Fin 2) (hi : i ≠ 0) : i = 1 :=
-  fin_two_eq_of_eq_zero_iff (by simpa only [one_eq_zero_iff, succ.injEq, iff_false] using hi)
+lemma eq_one_of_neq_zero (i : Fin 2) (hi : i ≠ 0) : i = 1 := by
+  fin_omega
 
 @[simp]
 theorem coe_neg_one : ↑(-1 : Fin (n + 1)) = n := by
@@ -1444,15 +1487,7 @@ theorem last_sub (i : Fin (n + 1)) : last n - i = Fin.rev i :=
   Fin.ext <| by rw [coe_sub_iff_le.2 i.le_last, val_last, val_rev, Nat.succ_sub_succ_eq_sub]
 
 theorem add_one_le_of_lt {n : ℕ} {a b : Fin (n + 1)} (h : a < b) : a + 1 ≤ b := by
-  cases' a with a ha
-  cases' b with b hb
-  cases n
-  · simp only [Nat.zero_add, Nat.lt_one_iff] at ha hb
-    simp [ha, hb]
-  simp only [le_iff_val_le_val, val_add, lt_iff_val_lt_val, val_mk, val_one] at h ⊢
-  rwa [Nat.mod_eq_of_lt, Nat.succ_le_iff]
-  rw [Nat.succ_lt_succ_iff]
-  exact h.trans_le (Nat.le_of_lt_succ hb)
+  cases n <;> fin_omega
 
 theorem exists_eq_add_of_le {n : ℕ} {a b : Fin n} (h : a ≤ b) : ∃ k ≤ b, b = a + k := by
   obtain ⟨k, hk⟩ : ∃ k : ℕ, (b : ℕ) = a + k := Nat.exists_eq_add_of_le h
@@ -1463,28 +1498,32 @@ theorem exists_eq_add_of_le {n : ℕ} {a b : Fin n} (h : a ≤ b) : ∃ k ≤ b,
 theorem exists_eq_add_of_lt {n : ℕ} {a b : Fin (n + 1)} (h : a < b) :
     ∃ k < b, k + 1 ≤ b ∧ b = a + k + 1 := by
   cases n
-  · cases' a with a ha
-    cases' b with b hb
-    simp only [Nat.zero_add, Nat.lt_one_iff] at ha hb
-    simp [ha, hb] at h
+  · omega
   obtain ⟨k, hk⟩ : ∃ k : ℕ, (b : ℕ) = a + k + 1 := Nat.exists_eq_add_of_lt h
   have hkb : k < b := by omega
-  refine ⟨⟨k, hkb.trans b.is_lt⟩, hkb, ?_, ?_⟩
-  · rw [Fin.le_iff_val_le_val, Fin.val_add_one]
-    split_ifs <;> simp [Nat.succ_le_iff, hkb]
+  refine ⟨⟨k, hkb.trans b.is_lt⟩, hkb, by fin_omega, ?_⟩
   simp [Fin.ext_iff, Fin.val_add, ← hk, Nat.mod_eq_of_lt b.is_lt]
 
 lemma pos_of_ne_zero {n : ℕ} {a : Fin (n + 1)} (h : a ≠ 0) :
     0 < a :=
   Nat.pos_of_ne_zero (val_ne_of_ne h)
 
+lemma sub_succ_le_sub_of_le {n : ℕ} {u v : Fin (n + 2)} (h : u < v) : v - (u + 1) < v - u := by
+  fin_omega
+
 end AddGroup
 
 @[simp]
-theorem coe_ofNat_eq_mod (m n : ℕ) [NeZero m] :
+theorem coe_natCast_eq_mod (m n : ℕ) [NeZero m] :
     ((n : Fin m) : ℕ) = n % m :=
   rfl
 
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem coe_ofNat_eq_mod (m n : ℕ) [NeZero m] :
+    ((no_index OfNat.ofNat n : Fin m) : ℕ) = OfNat.ofNat n % m :=
+  rfl
+
 section Mul
 
 /-!
@@ -1508,14 +1547,6 @@ protected theorem zero_mul' [NeZero n] (k : Fin n) : (0 : Fin n) * k = 0 := by
 
 end Mul
 
-open Qq in
-instance toExpr (n : ℕ) : Lean.ToExpr (Fin n) where
-  toTypeExpr := q(Fin $n)
-  toExpr := match n with
-    | 0 => finZeroElim
-    | k + 1 => fun i => show Q(Fin $n) from
-      have i : Q(Nat) := Lean.mkRawNatLit i -- raw literal to avoid ofNat-double-wrapping
-      have : Q(NeZero $n) := haveI : $n =Q $k + 1 := ⟨⟩; by exact q(NeZero.succ)
-      q(OfNat.ofNat $i)
-
 end Fin
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/Fin/Tuple/Basic.lean b/Mathlib/Data/Fin/Tuple/Basic.lean
index d0ef565be0f00..e4cb2370272a7 100644
--- a/Mathlib/Data/Fin/Tuple/Basic.lean
+++ b/Mathlib/Data/Fin/Tuple/Basic.lean
@@ -5,7 +5,6 @@ Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
 -/
 import Mathlib.Data.Fin.Basic
 import Mathlib.Data.Nat.Find
-import Batteries.Data.Fin.Lemmas
 
 /-!
 # Operation on tuples
@@ -94,13 +93,13 @@ example (α : Fin 0 → Sort u) : Unique (∀ i : Fin 0, α i) := by infer_insta
 theorem tuple0_le {α : Fin 0 → Type*} [∀ i, Preorder (α i)] (f g : ∀ i, α i) : f ≤ g :=
   finZeroElim
 
-variable {α : Fin (n + 1) → Type u} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ) (i : Fin n)
+variable {α : Fin (n + 1) → Sort u} (x : α 0) (q : ∀ i, α i) (p : ∀ i : Fin n, α i.succ) (i : Fin n)
   (y : α i.succ) (z : α 0)
 
 /-- The tail of an `n+1` tuple, i.e., its last `n` entries. -/
 def tail (q : ∀ i, α i) : ∀ i : Fin n, α i.succ := fun i ↦ q i.succ
 
-theorem tail_def {n : ℕ} {α : Fin (n + 1) → Type*} {q : ∀ i, α i} :
+theorem tail_def {n : ℕ} {α : Fin (n + 1) → Sort*} {q : ∀ i, α i} :
     (tail fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q k.succ :=
   rfl
 
@@ -118,7 +117,7 @@ theorem cons_succ : cons x p i.succ = p i := by simp [cons]
 theorem cons_zero : cons x p 0 = x := by simp [cons]
 
 @[simp]
-theorem cons_one {α : Fin (n + 2) → Type*} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) :
+theorem cons_one {α : Fin (n + 2) → Sort*} (x : α 0) (p : ∀ i : Fin n.succ, α i.succ) :
     cons x p 1 = p 0 := by
   rw [← cons_succ x p]; rfl
 
@@ -193,7 +192,7 @@ theorem consCases_cons {P : (∀ i : Fin n.succ, α i) → Sort v} (h : ∀ x₀
 
 /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.cons`. -/
 @[elab_as_elim]
-def consInduction {α : Type*} {P : ∀ {n : ℕ}, (Fin n → α) → Sort v} (h0 : P Fin.elim0)
+def consInduction {α : Sort*} {P : ∀ {n : ℕ}, (Fin n → α) → Sort v} (h0 : P Fin.elim0)
     (h : ∀ {n} (x₀) (x : Fin n → α), P x → P (Fin.cons x₀ x)) : ∀ {n : ℕ} (x : Fin n → α), P x
   | 0, x => by convert h0
   | n + 1, x => consCases (fun x₀ x ↦ h _ _ <| consInduction h0 h _) x
@@ -255,7 +254,7 @@ theorem tail_update_succ : tail (update q i.succ y) = update (tail q) i y := by
     simp [tail]
   · simp [tail, (Fin.succ_injective n).ne h, h]
 
-theorem comp_cons {α : Type*} {β : Type*} (g : α → β) (y : α) (q : Fin n → α) :
+theorem comp_cons {α : Sort*} {β : Sort*} (g : α → β) (y : α) (q : Fin n → α) :
     g ∘ cons y q = cons (g y) (g ∘ q) := by
   ext j
   by_cases h : j = 0
@@ -265,11 +264,15 @@ theorem comp_cons {α : Type*} {β : Type*} (g : α → β) (y : α) (q : Fin n
     have : j'.succ = j := succ_pred j h
     rw [← this, cons_succ, comp_apply, comp_apply, cons_succ]
 
-theorem comp_tail {α : Type*} {β : Type*} (g : α → β) (q : Fin n.succ → α) :
+theorem comp_tail {α : Sort*} {β : Sort*} (g : α → β) (q : Fin n.succ → α) :
     g ∘ tail q = tail (g ∘ q) := by
   ext j
   simp [tail]
 
+section Preorder
+
+variable {α : Fin (n + 1) → Type*}
+
 theorem le_cons [∀ i, Preorder (α i)] {x : α 0} {q : ∀ i, α i} {p : ∀ i : Fin n, α i.succ} :
     q ≤ cons x p ↔ q 0 ≤ x ∧ tail q ≤ p :=
   forall_fin_succ.trans <| and_congr Iff.rfl <| forall_congr' fun j ↦ by simp [tail]
@@ -282,33 +285,37 @@ theorem cons_le_cons [∀ i, Preorder (α i)] {x₀ y₀ : α 0} {x y : ∀ i :
     cons x₀ x ≤ cons y₀ y ↔ x₀ ≤ y₀ ∧ x ≤ y :=
   forall_fin_succ.trans <| and_congr_right' <| by simp only [cons_succ, Pi.le_def]
 
+end Preorder
+
 theorem range_fin_succ {α} (f : Fin (n + 1) → α) :
     Set.range f = insert (f 0) (Set.range (Fin.tail f)) :=
   Set.ext fun _ ↦ exists_fin_succ.trans <| eq_comm.or Iff.rfl
 
 @[simp]
-theorem range_cons {α : Type*} {n : ℕ} (x : α) (b : Fin n → α) :
+theorem range_cons {α} {n : ℕ} (x : α) (b : Fin n → α) :
     Set.range (Fin.cons x b : Fin n.succ → α) = insert x (Set.range b) := by
   rw [range_fin_succ, cons_zero, tail_cons]
 
 section Append
 
+variable {α : Sort*}
+
 /-- Append a tuple of length `m` to a tuple of length `n` to get a tuple of length `m + n`.
 This is a non-dependent version of `Fin.add_cases`. -/
-def append {α : Type*} (a : Fin m → α) (b : Fin n → α) : Fin (m + n) → α :=
+def append (a : Fin m → α) (b : Fin n → α) : Fin (m + n) → α :=
   @Fin.addCases _ _ (fun _ => α) a b
 
 @[simp]
-theorem append_left {α : Type*} (u : Fin m → α) (v : Fin n → α) (i : Fin m) :
+theorem append_left (u : Fin m → α) (v : Fin n → α) (i : Fin m) :
     append u v (Fin.castAdd n i) = u i :=
   addCases_left _
 
 @[simp]
-theorem append_right {α : Type*} (u : Fin m → α) (v : Fin n → α) (i : Fin n) :
+theorem append_right (u : Fin m → α) (v : Fin n → α) (i : Fin n) :
     append u v (natAdd m i) = v i :=
   addCases_right _
 
-theorem append_right_nil {α : Type*} (u : Fin m → α) (v : Fin n → α) (hv : n = 0) :
+theorem append_right_nil (u : Fin m → α) (v : Fin n → α) (hv : n = 0) :
     append u v = u ∘ Fin.cast (by rw [hv, Nat.add_zero]) := by
   refine funext (Fin.addCases (fun l => ?_) fun r => ?_)
   · rw [append_left, Function.comp_apply]
@@ -317,11 +324,11 @@ theorem append_right_nil {α : Type*} (u : Fin m → α) (v : Fin n → α) (hv
   · exact (Fin.cast hv r).elim0
 
 @[simp]
-theorem append_elim0 {α : Type*} (u : Fin m → α) :
+theorem append_elim0 (u : Fin m → α) :
     append u Fin.elim0 = u ∘ Fin.cast (Nat.add_zero _) :=
   append_right_nil _ _ rfl
 
-theorem append_left_nil {α : Type*} (u : Fin m → α) (v : Fin n → α) (hu : m = 0) :
+theorem append_left_nil (u : Fin m → α) (v : Fin n → α) (hu : m = 0) :
     append u v = v ∘ Fin.cast (by rw [hu, Nat.zero_add]) := by
   refine funext (Fin.addCases (fun l => ?_) fun r => ?_)
   · exact (Fin.cast hu l).elim0
@@ -330,11 +337,11 @@ theorem append_left_nil {α : Type*} (u : Fin m → α) (v : Fin n → α) (hu :
     simp [hu]
 
 @[simp]
-theorem elim0_append {α : Type*} (v : Fin n → α) :
+theorem elim0_append (v : Fin n → α) :
     append Fin.elim0 v = v ∘ Fin.cast (Nat.zero_add _) :=
   append_left_nil _ _ rfl
 
-theorem append_assoc {p : ℕ} {α : Type*} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) :
+theorem append_assoc {p : ℕ} (a : Fin m → α) (b : Fin n → α) (c : Fin p → α) :
     append (append a b) c = append a (append b c) ∘ Fin.cast (Nat.add_assoc ..) := by
   ext i
   rw [Function.comp_apply]
@@ -349,7 +356,7 @@ theorem append_assoc {p : ℕ} {α : Type*} (a : Fin m → α) (b : Fin n → α
     simp [← natAdd_natAdd]
 
 /-- Appending a one-tuple to the left is the same as `Fin.cons`. -/
-theorem append_left_eq_cons {α : Type*} {n : ℕ} (x₀ : Fin 1 → α) (x : Fin n → α) :
+theorem append_left_eq_cons {n : ℕ} (x₀ : Fin 1 → α) (x : Fin n → α) :
     Fin.append x₀ x = Fin.cons (x₀ 0) x ∘ Fin.cast (Nat.add_comm ..) := by
   ext i
   refine Fin.addCases ?_ ?_ i <;> clear i
@@ -361,21 +368,21 @@ theorem append_left_eq_cons {α : Type*} {n : ℕ} (x₀ : Fin 1 → α) (x : Fi
     exact Fin.cons_succ _ _ _
 
 /-- `Fin.cons` is the same as appending a one-tuple to the left. -/
-theorem cons_eq_append {α : Type*} (x : α) (xs : Fin n → α) :
+theorem cons_eq_append (x : α) (xs : Fin n → α) :
     cons x xs = append (cons x Fin.elim0) xs ∘ Fin.cast (Nat.add_comm ..) := by
   funext i; simp [append_left_eq_cons]
 
-@[simp] lemma append_cast_left {n m} {α : Type*} (xs : Fin n → α) (ys : Fin m → α) (n' : ℕ)
+@[simp] lemma append_cast_left {n m} (xs : Fin n → α) (ys : Fin m → α) (n' : ℕ)
     (h : n' = n) :
     Fin.append (xs ∘ Fin.cast h) ys = Fin.append xs ys ∘ (Fin.cast <| by rw [h]) := by
   subst h; simp
 
-@[simp] lemma append_cast_right {n m} {α : Type*} (xs : Fin n → α) (ys : Fin m → α) (m' : ℕ)
+@[simp] lemma append_cast_right {n m} (xs : Fin n → α) (ys : Fin m → α) (m' : ℕ)
     (h : m' = m) :
     Fin.append xs (ys ∘ Fin.cast h) = Fin.append xs ys ∘ (Fin.cast <| by rw [h]) := by
   subst h; simp
 
-lemma append_rev {m n} {α : Type*} (xs : Fin m → α) (ys : Fin n → α) (i : Fin (m + n)) :
+lemma append_rev {m n} (xs : Fin m → α) (ys : Fin n → α) (i : Fin (m + n)) :
     append xs ys (rev i) = append (ys ∘ rev) (xs ∘ rev) (cast (Nat.add_comm ..) i) := by
   rcases rev_surjective i with ⟨i, rfl⟩
   rw [rev_rev]
@@ -383,7 +390,7 @@ lemma append_rev {m n} {α : Type*} (xs : Fin m → α) (ys : Fin n → α) (i :
   · simp [rev_castAdd]
   · simp [cast_rev, rev_addNat]
 
-lemma append_comp_rev {m n} {α : Type*} (xs : Fin m → α) (ys : Fin n → α) :
+lemma append_comp_rev {m n} (xs : Fin m → α) (ys : Fin n → α) :
     append xs ys ∘ rev = append (ys ∘ rev) (xs ∘ rev) ∘ cast (Nat.add_comm ..) :=
   funext <| append_rev xs ys
 
@@ -391,31 +398,33 @@ end Append
 
 section Repeat
 
+variable {α : Sort*}
+
 /-- Repeat `a` `m` times. For example `Fin.repeat 2 ![0, 3, 7] = ![0, 3, 7, 0, 3, 7]`. -/
 -- Porting note: removed @[simp]
-def «repeat» {α : Type*} (m : ℕ) (a : Fin n → α) : Fin (m * n) → α
+def «repeat» (m : ℕ) (a : Fin n → α) : Fin (m * n) → α
   | i => a i.modNat
 
 -- Porting note: added (leanprover/lean4#2042)
 @[simp]
-theorem repeat_apply {α : Type*} (a : Fin n → α) (i : Fin (m * n)) :
+theorem repeat_apply (a : Fin n → α) (i : Fin (m * n)) :
     Fin.repeat m a i = a i.modNat :=
   rfl
 
 @[simp]
-theorem repeat_zero {α : Type*} (a : Fin n → α) :
+theorem repeat_zero (a : Fin n → α) :
     Fin.repeat 0 a = Fin.elim0 ∘ cast (Nat.zero_mul _) :=
   funext fun x => (cast (Nat.zero_mul _) x).elim0
 
 @[simp]
-theorem repeat_one {α : Type*} (a : Fin n → α) : Fin.repeat 1 a = a ∘ cast (Nat.one_mul _) := by
+theorem repeat_one (a : Fin n → α) : Fin.repeat 1 a = a ∘ cast (Nat.one_mul _) := by
   generalize_proofs h
   apply funext
   rw [(Fin.rightInverse_cast h.symm).surjective.forall]
   intro i
   simp [modNat, Nat.mod_eq_of_lt i.is_lt]
 
-theorem repeat_succ {α : Type*} (a : Fin n → α) (m : ℕ) :
+theorem repeat_succ (a : Fin n → α) (m : ℕ) :
     Fin.repeat m.succ a =
       append a (Fin.repeat m a) ∘ cast ((Nat.succ_mul _ _).trans (Nat.add_comm ..)) := by
   generalize_proofs h
@@ -426,7 +435,7 @@ theorem repeat_succ {α : Type*} (a : Fin n → α) (m : ℕ) :
   · simp [modNat]
 
 @[simp]
-theorem repeat_add {α : Type*} (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repeat (m₁ + m₂) a =
+theorem repeat_add (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repeat (m₁ + m₂) a =
     append (Fin.repeat m₁ a) (Fin.repeat m₂ a) ∘ cast (Nat.add_mul ..) := by
   generalize_proofs h
   apply funext
@@ -435,11 +444,11 @@ theorem repeat_add {α : Type*} (a : Fin n → α) (m₁ m₂ : ℕ) : Fin.repea
   · simp [modNat, Nat.mod_eq_of_lt l.is_lt]
   · simp [modNat, Nat.add_mod]
 
-theorem repeat_rev {α : Type*} (a : Fin n → α) (k : Fin (m * n)) :
+theorem repeat_rev (a : Fin n → α) (k : Fin (m * n)) :
     Fin.repeat m a k.rev = Fin.repeat m (a ∘ Fin.rev) k :=
   congr_arg a k.modNat_rev
 
-theorem repeat_comp_rev {α} (a : Fin n → α) :
+theorem repeat_comp_rev (a : Fin n → α) :
     Fin.repeat m a ∘ Fin.rev = Fin.repeat m (a ∘ Fin.rev) :=
   funext <| repeat_rev a
 
@@ -457,14 +466,14 @@ several places. -/
 
 -- Porting note: `i.castSucc` does not work like it did in Lean 3;
 -- `(castSucc i)` must be used.
-variable {α : Fin (n + 1) → Type u} (x : α (last n)) (q : ∀ i, α i)
+variable {α : Fin (n + 1) → Sort*} (x : α (last n)) (q : ∀ i, α i)
   (p : ∀ i : Fin n, α (castSucc i)) (i : Fin n) (y : α (castSucc i)) (z : α (last n))
 
 /-- The beginning of an `n+1` tuple, i.e., its first `n` entries -/
 def init (q : ∀ i, α i) (i : Fin n) : α (castSucc i) :=
   q (castSucc i)
 
-theorem init_def {n : ℕ} {α : Fin (n + 1) → Type*} {q : ∀ i, α i} :
+theorem init_def {q : ∀ i, α i} :
     (init fun k : Fin (n + 1) ↦ q k) = fun k : Fin n ↦ q (castSucc k) :=
   rfl
 
@@ -486,21 +495,21 @@ theorem snoc_castSucc : snoc p x (castSucc i) = p i := by
   convert cast_eq rfl (p i)
 
 @[simp]
-theorem snoc_comp_castSucc {n : ℕ} {α : Sort _} {a : α} {f : Fin n → α} :
+theorem snoc_comp_castSucc {α : Sort*} {a : α} {f : Fin n → α} :
     (snoc f a : Fin (n + 1) → α) ∘ castSucc = f :=
   funext fun i ↦ by rw [Function.comp_apply, snoc_castSucc]
 
 @[simp]
 theorem snoc_last : snoc p x (last n) = x := by simp [snoc]
 
-lemma snoc_zero {α : Type*} (p : Fin 0 → α) (x : α) :
+lemma snoc_zero {α : Sort*} (p : Fin 0 → α) (x : α) :
     Fin.snoc p x = fun _ ↦ x := by
   ext y
   have : Subsingleton (Fin (0 + 1)) := Fin.subsingleton_one
   simp only [Subsingleton.elim y (Fin.last 0), snoc_last]
 
 @[simp]
-theorem snoc_comp_nat_add {n m : ℕ} {α : Sort _} (f : Fin (m + n) → α) (a : α) :
+theorem snoc_comp_nat_add {n m : ℕ} {α : Sort*} (f : Fin (m + n) → α) (a : α) :
     (snoc f a : Fin _ → α) ∘ (natAdd m : Fin (n + 1) → Fin (m + n + 1)) =
       snoc (f ∘ natAdd m) a := by
   ext i
@@ -511,13 +520,13 @@ theorem snoc_comp_nat_add {n m : ℕ} {α : Sort _} (f : Fin (m + n) → α) (a
     rw [natAdd_castSucc, snoc_castSucc]
 
 @[simp]
-theorem snoc_cast_add {α : Fin (n + m + 1) → Type*} (f : ∀ i : Fin (n + m), α (castSucc i))
+theorem snoc_cast_add {α : Fin (n + m + 1) → Sort*} (f : ∀ i : Fin (n + m), α (castSucc i))
     (a : α (last (n + m))) (i : Fin n) : (snoc f a) (castAdd (m + 1) i) = f (castAdd m i) :=
   dif_pos _
 
 -- Porting note: Had to `unfold comp`
 @[simp]
-theorem snoc_comp_cast_add {n m : ℕ} {α : Sort _} (f : Fin (n + m) → α) (a : α) :
+theorem snoc_comp_cast_add {n m : ℕ} {α : Sort*} (f : Fin (n + m) → α) (a : α) :
     (snoc f a : Fin _ → α) ∘ castAdd (m + 1) = f ∘ castAdd m :=
   funext (by unfold comp; exact snoc_cast_add _ _)
 
@@ -543,7 +552,7 @@ theorem snoc_update : snoc (update p i y) x = update (snoc p x) (castSucc i) y :
         · simp [h, h']
         · exact heq_of_cast_eq C2 rfl
       rw [E1, E2]
-      exact eq_rec_compose (Eq.trans C2.symm C1) C2 y
+      rfl
     · have : ¬castLT j h = i := by
         intro E
         apply h'
@@ -588,14 +597,14 @@ theorem init_update_castSucc : init (update q (castSucc i) y) = update (init q)
 
 /-- `tail` and `init` commute. We state this lemma in a non-dependent setting, as otherwise it
 would involve a cast to convince Lean that the two types are equal, making it harder to use. -/
-theorem tail_init_eq_init_tail {β : Type*} (q : Fin (n + 2) → β) :
+theorem tail_init_eq_init_tail {β : Sort*} (q : Fin (n + 2) → β) :
     tail (init q) = init (tail q) := by
   ext i
   simp [tail, init, castSucc_fin_succ]
 
 /-- `cons` and `snoc` commute. We state this lemma in a non-dependent setting, as otherwise it
 would involve a cast to convince Lean that the two types are equal, making it harder to use. -/
-theorem cons_snoc_eq_snoc_cons {β : Type*} (a : β) (q : Fin n → β) (b : β) :
+theorem cons_snoc_eq_snoc_cons {β : Sort*} (a : β) (q : Fin n → β) (b : β) :
     @cons n.succ (fun _ ↦ β) a (snoc q b) = snoc (cons a q) b := by
   ext i
   by_cases h : i = 0
@@ -613,7 +622,7 @@ theorem cons_snoc_eq_snoc_cons {β : Type*} (a : β) (q : Fin n → β) (b : β)
   rw [eq_last_of_not_lt h', succ_last]
   simp
 
-theorem comp_snoc {α : Type*} {β : Type*} (g : α → β) (q : Fin n → α) (y : α) :
+theorem comp_snoc {α : Sort*} {β : Sort*} (g : α → β) (q : Fin n → α) (y : α) :
     g ∘ snoc q y = snoc (g ∘ q) (g y) := by
   ext j
   by_cases h : j.val < n
@@ -622,7 +631,7 @@ theorem comp_snoc {α : Type*} {β : Type*} (g : α → β) (q : Fin n → α) (
     simp
 
 /-- Appending a one-tuple to the right is the same as `Fin.snoc`. -/
-theorem append_right_eq_snoc {α : Type*} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) :
+theorem append_right_eq_snoc {α : Sort*} {n : ℕ} (x : Fin n → α) (x₀ : Fin 1 → α) :
     Fin.append x x₀ = Fin.snoc x (x₀ 0) := by
   ext i
   refine Fin.addCases ?_ ?_ i <;> clear i
@@ -634,21 +643,21 @@ theorem append_right_eq_snoc {α : Type*} {n : ℕ} (x : Fin n → α) (x₀ : F
     exact (@snoc_last _ (fun _ => α) _ _).symm
 
 /-- `Fin.snoc` is the same as appending a one-tuple -/
-theorem snoc_eq_append {α : Type*} (xs : Fin n → α) (x : α) :
+theorem snoc_eq_append {α : Sort*} (xs : Fin n → α) (x : α) :
     snoc xs x = append xs (cons x Fin.elim0) :=
   (append_right_eq_snoc xs (cons x Fin.elim0)).symm
 
-theorem append_left_snoc {n m} {α : Type*} (xs : Fin n → α) (x : α) (ys : Fin m → α) :
+theorem append_left_snoc {n m} {α : Sort*} (xs : Fin n → α) (x : α) (ys : Fin m → α) :
     Fin.append (Fin.snoc xs x) ys =
       Fin.append xs (Fin.cons x ys) ∘ Fin.cast (Nat.succ_add_eq_add_succ ..) := by
   rw [snoc_eq_append, append_assoc, append_left_eq_cons, append_cast_right]; rfl
 
-theorem append_right_cons {n m} {α : Type*} (xs : Fin n → α) (y : α) (ys : Fin m → α) :
+theorem append_right_cons {n m} {α : Sort*} (xs : Fin n → α) (y : α) (ys : Fin m → α) :
     Fin.append xs (Fin.cons y ys) =
       Fin.append (Fin.snoc xs y) ys ∘ Fin.cast (Nat.succ_add_eq_add_succ ..).symm := by
   rw [append_left_snoc]; rfl
 
-theorem append_cons {α} (a : α) (as : Fin n → α) (bs : Fin m → α) :
+theorem append_cons {α : Sort*} (a : α) (as : Fin n → α) (bs : Fin m → α) :
     Fin.append (cons a as) bs
     = cons a (Fin.append as bs) ∘ (Fin.cast <| Nat.add_right_comm n 1 m) := by
   funext i
@@ -662,7 +671,7 @@ theorem append_cons {α} (a : α) (as : Fin n → α) (bs : Fin m → α) :
     · have : ¬i < n := Nat.not_le.mpr <| Nat.lt_succ.mp <| Nat.not_le.mp h
       simp [addCases, this]
 
-theorem append_snoc {α} (as : Fin n → α) (bs : Fin m → α) (b : α) :
+theorem append_snoc {α : Sort*} (as : Fin n → α) (bs : Fin m → α) (b : α) :
     Fin.append as (snoc bs b) = snoc (Fin.append as bs) b := by
   funext i
   rcases i with ⟨i, isLt⟩
@@ -676,7 +685,7 @@ theorem append_snoc {α} (as : Fin n → α) (bs : Fin m → α) (b : α) :
   · have := Nat.sub_lt_left_of_lt_add (Nat.not_lt.mp lt_n) lt_add
     contradiction
 
-theorem comp_init {α : Type*} {β : Type*} (g : α → β) (q : Fin n.succ → α) :
+theorem comp_init {α : Sort*} {β : Sort*} (g : α → β) (q : Fin n.succ → α) :
     g ∘ init q = init (g ∘ q) := by
   ext j
   simp [init]
@@ -696,7 +705,7 @@ def snocCases {P : (∀ i : Fin n.succ, α i) → Sort*}
 
 /-- Recurse on a tuple by splitting into `Fin.elim0` and `Fin.snoc`. -/
 @[elab_as_elim]
-def snocInduction {α : Type*}
+def snocInduction {α : Sort*}
     {P : ∀ {n : ℕ}, (Fin n → α) → Sort*}
     (h0 : P Fin.elim0)
     (h : ∀ {n} (x : Fin n → α) (x₀), P x → P (Fin.snoc x x₀)) : ∀ {n : ℕ} (x : Fin n → α), P x
@@ -707,7 +716,7 @@ end TupleRight
 
 section InsertNth
 
-variable {α : Fin (n + 1) → Type u} {β : Type v}
+variable {α : Fin (n + 1) → Sort*} {β : Sort*}
 
 /- Porting note: Lean told me `(fun x x_1 ↦ α x)` was an invalid motive, but disabling
 automatic insertion and specifying that motive seems to work. -/
@@ -779,13 +788,13 @@ theorem insertNth_apply_succAbove (i : Fin (n + 1)) (x : α i) (p : ∀ j, α (i
     generalize hk : castPred ((succAbove i) j) H₁ = k
     rw [castPred_succAbove _ _ hlt] at hk; cases hk
     intro; rfl
-  · generalize_proofs H₁ H₂; revert H₂
+  · generalize_proofs H₀ H₁ H₂; revert H₂
     generalize hk : pred (succAbove i j) H₁ = k
     erw [pred_succAbove _ _ (Fin.not_lt.1 hlt)] at hk; cases hk
     intro; rfl
 
 @[simp]
-theorem succAbove_cases_eq_insertNth : @succAboveCases.{u + 1} = @insertNth.{u} :=
+theorem succAbove_cases_eq_insertNth : @succAboveCases = @insertNth :=
   rfl
 
 @[simp] lemma removeNth_insertNth (p : Fin (n + 1)) (a : α p) (f : ∀ i, α (succAbove p i)) :
@@ -855,7 +864,7 @@ theorem insertNth_last (x : α (last n)) (p : ∀ j : Fin n, α ((last n).succAb
 theorem insertNth_last' (x : β) (p : Fin n → β) :
     @insertNth _ (fun _ ↦ β) (last n) x p = snoc p x := by simp [insertNth_last]
 
-lemma insertNth_rev {α : Type*} (i : Fin (n + 1)) (a : α) (f : Fin n → α) (j : Fin (n + 1)) :
+lemma insertNth_rev {α : Sort*} (i : Fin (n + 1)) (a : α) (f : Fin n → α) (j : Fin (n + 1)) :
     insertNth (α := fun _ ↦ α) i a f (rev j) = insertNth (α := fun _ ↦ α) i.rev a (f ∘ rev) j := by
   induction j using Fin.succAboveCases
   · exact rev i
@@ -889,8 +898,9 @@ theorem insertNth_binop (op : ∀ j, α j → α j → α j) (i : Fin (n + 1)) (
       op j (i.insertNth x p j) (i.insertNth y q j) :=
   insertNth_eq_iff.2 <| by unfold removeNth; simp
 
-section
-variable [∀ i, Preorder (α i)]
+section Preorder
+
+variable {α : Fin (n + 1) → Type*} [∀ i, Preorder (α i)]
 
 theorem insertNth_le_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)} {q : ∀ j, α j} :
     i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j ↦ q (i.succAbove j) := by
@@ -900,7 +910,7 @@ theorem le_insertNth_iff {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbov
     q ≤ i.insertNth x p ↔ q i ≤ x ∧ (fun j ↦ q (i.succAbove j)) ≤ p := by
   simp [Pi.le_def, forall_iff_succAbove i]
 
-end
+end Preorder
 
 open Set
 
@@ -916,7 +926,7 @@ lemma insertNth_self_removeNth (p : Fin (n + 1)) (f : ∀ j, α j) :
 /-- Separates an `n+1`-tuple, returning a selected index and then the rest of the tuple.
 Functional form of `Equiv.piFinSuccAbove`. -/
 @[deprecated removeNth (since := "2024-06-19")]
-def extractNth (i : Fin (n + 1)) (f : (∀ j, α j)) :
+def extractNth {α : Fin (n + 1) → Type*} (i : Fin (n + 1)) (f : (∀ j, α j)) :
     α i × ∀ j, α (i.succAbove j) :=
   (f i, removeNth i f)
 
@@ -1030,7 +1040,7 @@ end Find
 
 section ContractNth
 
-variable {α : Type*}
+variable {α : Sort*}
 
 /-- Sends `(g₀, ..., gₙ)` to `(g₀, ..., op gⱼ gⱼ₊₁, ..., gₙ)`. -/
 def contractNth (j : Fin (n + 1)) (op : α → α → α) (g : Fin (n + 1) → α) (k : Fin n) : α :=
diff --git a/Mathlib/Data/Fin/Tuple/Finset.lean b/Mathlib/Data/Fin/Tuple/Finset.lean
index 5d845ab373afb..d376279817e25 100644
--- a/Mathlib/Data/Fin/Tuple/Finset.lean
+++ b/Mathlib/Data/Fin/Tuple/Finset.lean
@@ -3,34 +3,87 @@ Copyright (c) 2023 Bolton Bailey. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bolton Bailey
 -/
-import Mathlib.Data.Fin.Tuple.Basic
+import Mathlib.Data.Finset.Prod
 import Mathlib.Data.Fintype.Pi
+import Mathlib.Logic.Equiv.Fin
 
 /-!
 # Fin-indexed tuples of finsets
 -/
 
-open Fintype
+open Fin Fintype
 
 namespace Fin
-variable {n : ℕ} {α : Fin (n + 1) → Type*}
+variable {n : ℕ} {α : Fin (n + 1) → Type*} {f : ∀ i, α i} {s : ∀ i, Finset (α i)} {p : Fin (n + 1)}
+
+open Fintype
+
+lemma mem_piFinset_iff_zero_tail :
+    f ∈ Fintype.piFinset s ↔ f 0 ∈ s 0 ∧ tail f ∈ piFinset (tail s) := by
+  simp only [Fintype.mem_piFinset, forall_fin_succ, tail]
 
-lemma mem_piFinset_succ {x : ∀ i, α i} {s : ∀ i, Finset (α i)} :
-    x ∈ piFinset s ↔ x 0 ∈ s 0 ∧ tail x ∈ piFinset (tail s) := by
-  simp only [mem_piFinset, forall_iff_succ, tail]
+lemma mem_piFinset_iff_last_init :
+    f ∈ piFinset s ↔ f (last n) ∈ s (last n) ∧ init f ∈ piFinset (init s) := by
+  simp only [Fintype.mem_piFinset, forall_fin_succ', init, and_comm]
 
-lemma mem_piFinset_succ' {x : ∀ i, α i} {s : ∀ i, Finset (α i)} :
-    x ∈ piFinset s ↔ x (last n) ∈ s (last n) ∧ init x ∈ piFinset (init s) := by
-  simp only [mem_piFinset, forall_iff_castSucc, init]
+lemma mem_piFinset_iff_pivot_removeNth (p : Fin (n + 1)) :
+    f ∈ piFinset s ↔ f p ∈ s p ∧ removeNth p f ∈ piFinset (removeNth p s) := by
+  simp only [Fintype.mem_piFinset, forall_iff_succAbove p, removeNth]
 
-lemma cons_mem_piFinset_cons {x₀ : α 0} {x : ∀ i : Fin n, α i.succ}
-    {s₀ : Finset (α 0)} {s : ∀ i : Fin n, Finset (α i.succ)} :
-    cons x₀ x ∈ piFinset (cons s₀ s) ↔ x₀ ∈ s₀ ∧ x ∈ piFinset s := by
-  simp_rw [mem_piFinset_succ, cons_zero, tail_cons]
+@[deprecated (since := "2024-09-20")] alias mem_piFinset_succ := mem_piFinset_iff_zero_tail
+@[deprecated (since := "2024-09-20")] alias mem_piFinset_succ' := mem_piFinset_iff_last_init
 
-lemma snoc_mem_piFinset_snoc {x : ∀ i : Fin n, α i.castSucc} {xₙ : α (.last n)}
-    {s : ∀ i : Fin n, Finset (α i.castSucc)} {sₙ : Finset (α <| .last n)} :
-    snoc x xₙ ∈ piFinset (snoc s sₙ) ↔ xₙ ∈ sₙ ∧ x ∈ piFinset s := by
-  simp_rw [mem_piFinset_succ', init_snoc, snoc_last]
+lemma cons_mem_piFinset_cons {x_zero : α 0} {x_tail : (i : Fin n) → α i.succ}
+    {s_zero : Finset (α 0)} {s_tail : (i : Fin n) → Finset (α i.succ)} :
+    cons x_zero x_tail ∈ piFinset (cons s_zero s_tail) ↔
+      x_zero ∈ s_zero ∧ x_tail ∈ piFinset s_tail := by
+  simp_rw [mem_piFinset_iff_zero_tail, cons_zero, tail_cons]
+
+lemma snoc_mem_piFinset_snoc {x_last : α (last n)} {x_init : (i : Fin n) → α i.castSucc}
+    {s_last : Finset (α (last n))} {s_init : (i : Fin n) → Finset (α i.castSucc)} :
+    snoc x_init x_last ∈ piFinset (snoc s_init s_last) ↔
+      x_last ∈ s_last ∧ x_init ∈ piFinset s_init := by
+  simp_rw [mem_piFinset_iff_last_init, init_snoc, snoc_last]
+
+lemma insertNth_mem_piFinset_insertNth {x_pivot : α p} {x_remove : ∀ i, α (succAbove p i)}
+    {s_pivot : Finset (α p)} {s_remove : ∀ i, Finset (α (succAbove p i))} :
+    insertNth p x_pivot x_remove ∈ piFinset (insertNth p s_pivot s_remove) ↔
+      x_pivot ∈ s_pivot ∧ x_remove ∈ piFinset s_remove := by
+  simp [mem_piFinset_iff_pivot_removeNth p]
 
 end Fin
+
+namespace Finset
+variable {n : ℕ} {α : Fin (n + 1) → Type*} {p : Fin (n + 1)} (S : ∀ i, Finset (α i))
+
+lemma map_consEquiv_filter_piFinset (P : (∀ i, α (succ i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| tail r).map (consEquiv α).symm.toEmbedding =
+      S 0 ×ˢ (piFinset fun x ↦ S <| succ x).filter P := by
+  unfold tail; ext; simp [Fin.forall_iff_succ, and_assoc]
+
+lemma map_snocEquiv_filter_piFinset (P : (∀ i, α (castSucc i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| init r).map (snocEquiv α).symm.toEmbedding =
+      S (last _) ×ˢ (piFinset <| init S).filter P := by
+  unfold init; ext; simp [Fin.forall_iff_castSucc, and_assoc]
+
+lemma map_insertNthEquiv_filter_piFinset (P : (∀ i, α (p.succAbove i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| p.removeNth r).map (p.insertNthEquiv α).symm.toEmbedding =
+      S p ×ˢ (piFinset <| p.removeNth S).filter P := by
+  unfold removeNth; ext; simp [Fin.forall_iff_succAbove p, and_assoc]
+
+lemma card_consEquiv_filter_piFinset (P : (∀ i, α (succ i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| tail r).card =
+      (S 0).card * ((piFinset fun x ↦ S <| succ x).filter P).card := by
+  rw [← card_product, ← map_consEquiv_filter_piFinset, card_map]
+
+lemma card_snocEquiv_filter_piFinset (P : (∀ i, α (castSucc i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| init r).card =
+      (S (last _)).card * ((piFinset <| init S).filter P).card := by
+  rw [← card_product, ← map_snocEquiv_filter_piFinset, card_map]
+
+lemma card_insertNthEquiv_filter_piFinset (P : (∀ i, α (p.succAbove i)) → Prop) [DecidablePred P] :
+    ((piFinset S).filter fun r ↦ P <| p.removeNth r).card =
+      (S p).card * ((piFinset <| p.removeNth S).filter P).card := by
+  rw [← card_product, ← map_insertNthEquiv_filter_piFinset, card_map]
+
+end Finset
diff --git a/Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean b/Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
index 4cef85d6ad0d3..7f311baa1744c 100644
--- a/Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
+++ b/Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean
@@ -123,7 +123,7 @@ theorem antidiagonalTuple_one (n : ℕ) : antidiagonalTuple 1 n = [![n]] := by
     Nat.sub_self, List.bind_append, List.bind_singleton, List.bind_map]
   conv_rhs => rw [← List.nil_append [![n]]]
   congr 1
-  simp_rw [List.bind_eq_nil, List.mem_range, List.map_eq_nil]
+  simp_rw [List.bind_eq_nil_iff, List.mem_range, List.map_eq_nil_iff]
   intro x hx
   obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_lt hx
   rw [add_assoc, add_tsub_cancel_left, antidiagonalTuple_zero_succ]
@@ -143,8 +143,8 @@ theorem antidiagonalTuple_pairwise_pi_lex :
     simp_rw [antidiagonalTuple, List.pairwise_bind, List.pairwise_map, List.mem_map,
       forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
     simp only [mem_antidiagonal, Prod.forall, and_imp, forall_apply_eq_imp_iff₂]
-    simp only [Fin.pi_lex_lt_cons_cons, eq_self_iff_true, true_and_iff, lt_self_iff_false,
-      false_or_iff]
+    simp only [Fin.pi_lex_lt_cons_cons, eq_self_iff_true, true_and, lt_self_iff_false,
+      false_or]
     refine ⟨fun _ _ _ => antidiagonalTuple_pairwise_pi_lex k _, ?_⟩
     induction' n with n n_ih
     · rw [antidiagonal_zero]
diff --git a/Mathlib/Data/Fin/Tuple/Sort.lean b/Mathlib/Data/Fin/Tuple/Sort.lean
index d53751d03f1f5..6dfe8a11de7e8 100644
--- a/Mathlib/Data/Fin/Tuple/Sort.lean
+++ b/Mathlib/Data/Fin/Tuple/Sort.lean
@@ -178,7 +178,7 @@ theorem comp_sort_eq_comp_iff_monotone : f ∘ σ = f ∘ sort f ↔ Monotone (f
 
 /-- The sorted versions of a tuple `f` and of any permutation of `f` agree. -/
 theorem comp_perm_comp_sort_eq_comp_sort : (f ∘ σ) ∘ sort (f ∘ σ) = f ∘ sort f := by
-  rw [Function.comp.assoc, ← Equiv.Perm.coe_mul]
+  rw [Function.comp_assoc, ← Equiv.Perm.coe_mul]
   exact unique_monotone (monotone_sort (f ∘ σ)) (monotone_sort f)
 
 /-- If a permutation `f ∘ σ` of the tuple `f` is not the same as `f ∘ sort f`, then `f ∘ σ`
diff --git a/Mathlib/Data/FinEnum.lean b/Mathlib/Data/FinEnum.lean
index 82688911d453d..c2d94fcfa23ad 100644
--- a/Mathlib/Data/FinEnum.lean
+++ b/Mathlib/Data/FinEnum.lean
@@ -76,7 +76,7 @@ noncomputable def ofInjective {α β} (f : α → β) [DecidableEq α] [FinEnum
   ofList ((toList β).filterMap (partialInv f))
     (by
       intro x
-      simp only [mem_toList, true_and_iff, List.mem_filterMap]
+      simp only [mem_toList, true_and, List.mem_filterMap]
       use f x
       simp only [h, Function.partialInv_left])
 
@@ -107,41 +107,19 @@ def Finset.enum [DecidableEq α] : List α → List (Finset α)
   | [] => [∅]
   | x :: xs => do
     let r ← Finset.enum xs
-    [r, {x} ∪ r]
+    [r, insert x r]
 
 @[simp]
 theorem Finset.mem_enum [DecidableEq α] (s : Finset α) (xs : List α) :
     s ∈ Finset.enum xs ↔ ∀ x ∈ s, x ∈ xs := by
-  induction' xs with xs_hd generalizing s <;> simp [*, Finset.enum]
-  · simp [Finset.eq_empty_iff_forall_not_mem]
-  · constructor
-    · rintro ⟨a, h, h'⟩ x hx
-      cases' h' with _ h' a b
-      · right
-        apply h
-        subst a
-        exact hx
-      · simp only [h', mem_union, mem_singleton] at hx ⊢
-        cases' hx with hx hx'
-        · exact Or.inl hx
-        · exact Or.inr (h _ hx')
-    · intro h
-      exists s \ ({xs_hd} : Finset α)
-      simp only [and_imp, mem_sdiff, mem_singleton]
-      simp only [or_iff_not_imp_left] at h
-      exists h
-      by_cases h : xs_hd ∈ s
-      · have : {xs_hd} ⊆ s := by
-          simp only [HasSubset.Subset, *, forall_eq, mem_singleton]
-        simp only [union_sdiff_of_subset this, or_true_iff, Finset.union_sdiff_of_subset,
-          eq_self_iff_true]
-      · left
-        symm
-        simp only [sdiff_eq_self]
-        intro a
-        simp only [and_imp, mem_inter, mem_singleton]
-        rintro h₀ rfl
-        exact (h h₀).elim
+  induction xs generalizing s with
+  | nil => simp [enum, eq_empty_iff_forall_not_mem]
+  | cons x xs ih =>
+      simp only [enum, List.bind_eq_bind, List.mem_bind, List.mem_cons, List.mem_singleton,
+        List.not_mem_nil, or_false, ih]
+      refine ⟨by aesop, fun hs => ⟨s.erase x, ?_⟩⟩
+      simp only [or_iff_not_imp_left] at hs
+      simp (config := { contextual := true }) [eq_comm (a := s), or_iff_not_imp_left, hs]
 
 instance Finset.finEnum [FinEnum α] : FinEnum (Finset α) :=
   ofList (Finset.enum (toList α)) (by intro; simp)
@@ -172,6 +150,8 @@ instance PSigma.finEnumPropProp {α : Prop} {β : α → Prop} [Decidable α] [
   if h : ∃ a, β a then ofList [⟨h.fst, h.snd⟩] (by rintro ⟨⟩; simp)
   else ofList [] fun a => (h ⟨a.fst, a.snd⟩).elim
 
+instance [DecidableEq α] (xs : List α) : FinEnum { x : α // x ∈ xs } := ofList xs.attach (by simp)
+
 instance (priority := 100) [FinEnum α] : Fintype α where
   elems := univ.map (equiv).symm.toEmbedding
   complete := by intros; simp
diff --git a/Mathlib/Data/Finmap.lean b/Mathlib/Data/Finmap.lean
index fb4a761948449..770bb27ffee01 100644
--- a/Mathlib/Data/Finmap.lean
+++ b/Mathlib/Data/Finmap.lean
@@ -17,7 +17,7 @@ open List
 
 variable {α : Type u} {β : α → Type v}
 
-/-! ### Multisets of sigma types-/
+/-! ### Multisets of sigma types -/
 
 namespace Multiset
 
@@ -151,7 +151,7 @@ theorem ext_iff' {s t : Finmap β} : s.entries = t.entries ↔ s = t :=
 
 /-- The predicate `a ∈ s` means that `s` has a value associated to the key `a`. -/
 instance : Membership α (Finmap β) :=
-  ⟨fun a s => a ∈ s.entries.keys⟩
+  ⟨fun s a => a ∈ s.entries.keys⟩
 
 theorem mem_def {a : α} {s : Finmap β} : a ∈ s ↔ a ∈ s.entries.keys :=
   Iff.rfl
@@ -213,7 +213,7 @@ theorem keys_singleton (a : α) (b : β a) : (singleton a b).keys = {a} :=
 
 @[simp]
 theorem mem_singleton (x y : α) (b : β y) : x ∈ singleton y b ↔ x = y := by
-  simp only [singleton]; erw [mem_cons, mem_nil_iff, or_false_iff]
+  simp only [singleton]; erw [mem_cons, mem_nil_iff, or_false]
 
 section
 
@@ -341,7 +341,8 @@ end
 /-- Fold a commutative function over the key-value pairs in the map -/
 def foldl {δ : Type w} (f : δ → ∀ a, β a → δ)
     (H : ∀ d a₁ b₁ a₂ b₂, f (f d a₁ b₁) a₂ b₂ = f (f d a₂ b₂) a₁ b₁) (d : δ) (m : Finmap β) : δ :=
-  m.entries.foldl (fun d s => f d s.1 s.2) (fun _ _ _ => H _ _ _ _ _) d
+  letI : RightCommutative fun d (s : Sigma β) ↦ f d s.1 s.2 := ⟨fun _ _ _ ↦ H _ _ _ _ _⟩
+  m.entries.foldl (fun d s => f d s.1 s.2) d
 
 /-- `any f s` returns `true` iff there exists a value `v` in `s` such that `f v = true`. -/
 def any (f : ∀ x, β x → Bool) (s : Finmap β) : Bool :=
diff --git a/Mathlib/Data/Finset/Basic.lean b/Mathlib/Data/Finset/Basic.lean
index 4f229d81672f8..e86ab424aa5fb 100644
--- a/Mathlib/Data/Finset/Basic.lean
+++ b/Mathlib/Data/Finset/Basic.lean
@@ -125,7 +125,7 @@ assert_not_exists CompleteLattice
 
 assert_not_exists OrderedCommMonoid
 
-open Multiset Subtype Nat Function
+open Multiset Subtype Function
 
 universe u
 
@@ -164,7 +164,7 @@ instance decidableEq [DecidableEq α] : DecidableEq (Finset α)
 
 
 instance : Membership α (Finset α) :=
-  ⟨fun a s => a ∈ s.1⟩
+  ⟨fun s a => a ∈ s.1⟩
 
 theorem mem_def {a : α} {s : Finset α} : a ∈ s ↔ a ∈ s.1 :=
   Iff.rfl
@@ -773,9 +773,13 @@ theorem mk_cons {s : Multiset α} (h : (a ::ₘ s).Nodup) :
 theorem cons_empty (a : α) : cons a ∅ (not_mem_empty _) = {a} := rfl
 
 @[simp, aesop safe apply (rule_sets := [finsetNonempty])]
-theorem nonempty_cons (h : a ∉ s) : (cons a s h).Nonempty :=
+theorem cons_nonempty (h : a ∉ s) : (cons a s h).Nonempty :=
   ⟨a, mem_cons.2 <| Or.inl rfl⟩
 
+@[deprecated (since := "2024-09-19")] alias nonempty_cons := cons_nonempty
+
+@[simp] theorem cons_ne_empty (h : a ∉ s) : cons a s h ≠ ∅ := (cons_nonempty _).ne_empty
+
 @[simp]
 theorem nonempty_mk {m : Multiset α} {hm} : (⟨m, hm⟩ : Finset α).Nonempty ↔ m ≠ 0 := by
   induction m using Multiset.induction_on <;> simp
@@ -803,6 +807,11 @@ theorem ssubset_iff_exists_cons_subset : s ⊂ t ↔ ∃ (a : _) (h : a ∉ s),
   obtain ⟨a, hs, ht⟩ := not_subset.1 h.2
   exact ⟨a, ht, cons_subset.2 ⟨hs, h.subset⟩⟩
 
+theorem cons_swap (hb : b ∉ s) (ha : a ∉ s.cons b hb) :
+    (s.cons b hb).cons a ha = (s.cons a fun h ↦ ha (mem_cons.mpr (.inr h))).cons b fun h ↦
+      ha (mem_cons.mpr (.inl ((mem_cons.mp h).elim symm (fun h ↦ False.elim (hb h))))) :=
+  eq_of_veq <| Multiset.cons_swap a b s.val
+
 end Cons
 
 /-! ### disjoint -/
@@ -1038,7 +1047,7 @@ theorem insert_subset_insert (a : α) {s t : Finset α} (h : s ⊆ t) : insert a
   simp_rw [← coe_subset]; simp [-coe_subset, ha]
 
 theorem insert_inj (ha : a ∉ s) : insert a s = insert b s ↔ a = b :=
-  ⟨fun h => eq_of_not_mem_of_mem_insert (h.subst <| mem_insert_self _ _) ha, congr_arg (insert · s)⟩
+  ⟨fun h => eq_of_not_mem_of_mem_insert (h ▸ mem_insert_self _ _) ha, congr_arg (insert · s)⟩
 
 theorem insert_inj_on (s : Finset α) : Set.InjOn (fun a => insert a s) sᶜ := fun _ h _ _ =>
   (insert_inj h).1
@@ -1098,7 +1107,7 @@ obtained by inserting an element in `t`. -/
 @[elab_as_elim]
 theorem Nonempty.cons_induction {α : Type*} {p : ∀ s : Finset α, s.Nonempty → Prop}
     (singleton : ∀ a, p {a} (singleton_nonempty _))
-    (cons : ∀ a s (h : a ∉ s) (hs), p s hs → p (Finset.cons a s h) (nonempty_cons h))
+    (cons : ∀ a s (h : a ∉ s) (hs), p s hs → p (Finset.cons a s h) (cons_nonempty h))
     {s : Finset α} (hs : s.Nonempty) : p s hs := by
   induction s using Finset.cons_induction with
   | empty => exact (not_nonempty_empty hs).elim
@@ -1427,7 +1436,7 @@ theorem insert_inter_of_not_mem {s₁ s₂ : Finset α} {a : α} (h : a ∉ s₂
     insert a s₁ ∩ s₂ = s₁ ∩ s₂ :=
   ext fun x => by
     have : ¬(x = a ∧ x ∈ s₂) := by rintro ⟨rfl, H⟩; exact h H
-    simp only [mem_inter, mem_insert, or_and_right, this, false_or_iff]
+    simp only [mem_inter, mem_insert, or_and_right, this, false_or]
 
 @[simp]
 theorem inter_insert_of_not_mem {s₁ s₂ : Finset α} {a : α} (h : a ∉ s₁) :
@@ -1479,7 +1488,7 @@ instance : DistribLattice (Finset α) :=
   { le_sup_inf := fun a b c => by
       simp (config := { contextual := true }) only
         [sup_eq_union, inf_eq_inter, le_eq_subset, subset_iff, mem_inter, mem_union, and_imp,
-        or_imp, true_or_iff, imp_true_iff, true_and_iff, or_true_iff] }
+        or_imp, true_or, imp_true_iff, true_and, or_true] }
 
 @[simp]
 theorem union_left_idem (s t : Finset α) : s ∪ (s ∪ t) = s ∪ t := sup_left_idem _ _
@@ -1637,7 +1646,7 @@ theorem erase_eq_self : s.erase a = s ↔ a ∉ s :=
 theorem erase_insert_eq_erase (s : Finset α) (a : α) : (insert a s).erase a = s.erase a :=
   ext fun x => by
     simp (config := { contextual := true }) only [mem_erase, mem_insert, and_congr_right_iff,
-      false_or_iff, iff_self_iff, imp_true_iff]
+      false_or, iff_self, imp_true_iff]
 
 theorem erase_insert {a : α} {s : Finset α} (h : a ∉ s) : erase (insert a s) a = s := by
   rw [erase_insert_eq_erase, erase_eq_of_not_mem h]
@@ -1654,7 +1663,7 @@ theorem erase_cons_of_ne {a b : α} {s : Finset α} (ha : a ∉ s) (hb : a ≠ b
 
 @[simp] theorem insert_erase (h : a ∈ s) : insert a (erase s a) = s :=
   ext fun x => by
-    simp only [mem_insert, mem_erase, or_and_left, dec_em, true_and_iff]
+    simp only [mem_insert, mem_erase, or_and_left, dec_em, true_and]
     apply or_iff_right_of_imp
     rintro rfl
     exact h
@@ -1772,11 +1781,11 @@ instance : GeneralizedBooleanAlgebra (Finset α) :=
       simp only [Finset.ext_iff, mem_union, mem_sdiff, inf_eq_inter, sup_eq_union, mem_inter,
         ← and_or_left, em, and_true, implies_true]
     inf_inf_sdiff := fun x y => by
-      simp only [Finset.ext_iff, inter_sdiff_self, inter_empty, inter_assoc, false_iff_iff,
+      simp only [Finset.ext_iff, inter_sdiff_self, inter_empty, inter_assoc, false_iff,
         inf_eq_inter, not_mem_empty, bot_eq_empty, not_false_iff, implies_true] }
 
 theorem not_mem_sdiff_of_mem_right (h : a ∈ t) : a ∉ s \ t := by
-  simp only [mem_sdiff, h, not_true, not_false_iff, and_false_iff]
+  simp only [mem_sdiff, h, not_true, not_false_iff, and_false]
 
 theorem not_mem_sdiff_of_not_mem_left (h : a ∉ s) : a ∉ s \ t := by simp [h]
 
@@ -1912,7 +1921,7 @@ theorem union_sdiff_self (s t : Finset α) : (s ∪ t) \ t = s \ t :=
 
 -- TODO: Do we want to delete this lemma and `Finset.disjUnion_singleton`,
 -- or instead add `Finset.union_singleton`/`Finset.singleton_union`?
-theorem sdiff_singleton_eq_erase (a : α) (s : Finset α) : s \ singleton a = erase s a := by
+theorem sdiff_singleton_eq_erase (a : α) (s : Finset α) : s \ {a} = erase s a := by
   ext
   rw [mem_erase, mem_sdiff, mem_singleton, and_comm]
 
@@ -1968,9 +1977,9 @@ theorem erase_union_of_mem (ha : a ∈ t) (s : Finset α) : s.erase a ∪ t = s
 theorem union_erase_of_mem (ha : a ∈ s) (t : Finset α) : s ∪ t.erase a = s ∪ t := by
   rw [← insert_erase (mem_union_left t ha), erase_union_distrib, ← insert_union, insert_erase ha]
 
-@[simp]
-theorem sdiff_singleton_eq_self (ha : a ∉ s) : s \ {a} = s :=
-  sdiff_eq_self_iff_disjoint.2 <| by simp [ha]
+@[simp, deprecated erase_eq_of_not_mem (since := "2024-10-01")]
+theorem sdiff_singleton_eq_self (ha : a ∉ s) : s \ {a} = s := by
+  rw [← erase_eq, erase_eq_of_not_mem ha]
 
 theorem Nontrivial.sdiff_singleton_nonempty {c : α} {s : Finset α} (hS : s.Nontrivial) :
     (s \ {c}).Nonempty := by
@@ -2034,6 +2043,8 @@ theorem disjoint_sdiff_inter (s t : Finset α) : Disjoint (s \ t) (s ∩ t) :=
 theorem sdiff_eq_self_iff_disjoint : s \ t = s ↔ Disjoint s t :=
   sdiff_eq_self_iff_disjoint'
 
+@[deprecated (since := "2024-10-01")] alias sdiff_eq_self := sdiff_eq_self_iff_disjoint
+
 theorem sdiff_eq_self_of_disjoint (h : Disjoint s t) : s \ t = s :=
   sdiff_eq_self_iff_disjoint.2 h
 
@@ -2089,10 +2100,11 @@ theorem mem_attach (s : Finset α) : ∀ x, x ∈ s.attach :=
 theorem attach_empty : attach (∅ : Finset α) = ∅ :=
   rfl
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem attach_nonempty_iff {s : Finset α} : s.attach.Nonempty ↔ s.Nonempty := by
   simp [Finset.Nonempty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected alias ⟨_, Nonempty.attach⟩ := attach_nonempty_iff
 
 @[simp]
@@ -2310,7 +2322,7 @@ theorem filter_singleton (a : α) : filter p {a} = if p a then {a} else ∅ := b
     split_ifs with h <;> by_cases h' : x = a <;> simp [h, h']
 
 theorem filter_cons_of_pos (a : α) (s : Finset α) (ha : a ∉ s) (hp : p a) :
-    filter p (cons a s ha) = cons a (filter p s) (mem_filter.not.mpr <| mt And.left ha) :=
+    filter p (cons a s ha) = cons a (filter p s) ((mem_of_mem_filter _).mt ha) :=
   eq_of_veq <| Multiset.filter_cons_of_pos s.val hp
 
 theorem filter_cons_of_neg (a : α) (s : Finset α) (ha : a ∉ s) (hp : ¬p a) :
@@ -2339,6 +2351,8 @@ theorem disjoint_filter_filter_neg (s t : Finset α) (p : α → Prop)
     Disjoint (s.filter p) (t.filter fun a => ¬p a) :=
   disjoint_filter_filter' s t disjoint_compl_right
 
+@[deprecated (since := "2024-10-01")] alias filter_inter_filter_neg_eq := disjoint_filter_filter_neg
+
 theorem filter_disj_union (s : Finset α) (t : Finset α) (h : Disjoint s t) :
     filter p (disjUnion s t h) = (filter p s).disjUnion (filter p t) (disjoint_filter_filter h) :=
   eq_of_veq <| Multiset.filter_add _ _ _
@@ -2352,15 +2366,10 @@ lemma _root_.Set.pairwiseDisjoint_filter [DecidableEq β] (f : α → β) (s : S
 
 theorem filter_cons {a : α} (s : Finset α) (ha : a ∉ s) :
     filter p (cons a s ha) =
-      (if p a then {a} else ∅ : Finset α).disjUnion (filter p s)
-        (by
-          split_ifs
-          · rw [disjoint_singleton_left]
-            exact mem_filter.not.mpr <| mt And.left ha
-          · exact disjoint_empty_left _) := by
+      if p a then cons a (filter p s) ((mem_of_mem_filter _).mt ha) else filter p s := by
   split_ifs with h
-  · rw [filter_cons_of_pos _ _ _ ha h, singleton_disjUnion]
-  · rw [filter_cons_of_neg _ _ _ ha h, empty_disjUnion]
+  · rw [filter_cons_of_pos _ _ _ ha h]
+  · rw [filter_cons_of_neg _ _ _ ha h]
 
 section
 variable [DecidableEq α]
@@ -2393,7 +2402,7 @@ theorem filter_insert (a : α) (s : Finset α) :
 
 theorem filter_erase (a : α) (s : Finset α) : filter p (erase s a) = erase (filter p s) a := by
   ext x
-  simp only [and_assoc, mem_filter, iff_self_iff, mem_erase]
+  simp only [and_assoc, mem_filter, iff_self, mem_erase]
 
 theorem filter_or (s : Finset α) : (s.filter fun a => p a ∨ q a) = s.filter p ∪ s.filter q :=
   ext fun _ => by simp [mem_filter, mem_union, and_or_left]
@@ -2413,9 +2422,6 @@ lemma filter_and_not (s : Finset α) (p q : α → Prop) [DecidablePred p] [Deci
 theorem sdiff_eq_filter (s₁ s₂ : Finset α) : s₁ \ s₂ = filter (· ∉ s₂) s₁ :=
   ext fun _ => by simp [mem_sdiff, mem_filter]
 
-theorem sdiff_eq_self (s₁ s₂ : Finset α) : s₁ \ s₂ = s₁ ↔ s₁ ∩ s₂ ⊆ ∅ := by
-  simp [Subset.antisymm_iff, disjoint_iff_inter_eq_empty]
-
 theorem subset_union_elim {s : Finset α} {t₁ t₂ : Set α} (h : ↑s ⊆ t₁ ∪ t₂) :
     ∃ s₁ s₂ : Finset α, s₁ ∪ s₂ = s ∧ ↑s₁ ⊆ t₁ ∧ ↑s₂ ⊆ t₂ \ t₁ := by
   classical
@@ -2463,7 +2469,7 @@ theorem filter_eq [DecidableEq β] (s : Finset β) (b : β) :
     rintro rfl
     exact ⟨h, rfl⟩
   · ext
-    simp only [mem_filter, not_and, iff_false_iff, not_mem_empty, decide_eq_true_eq]
+    simp only [mem_filter, not_and, iff_false, not_mem_empty, decide_eq_true_eq]
     rintro m rfl
     exact h m
 
@@ -2484,10 +2490,6 @@ theorem filter_ne [DecidableEq β] (s : Finset β) (b : β) :
 theorem filter_ne' [DecidableEq β] (s : Finset β) (b : β) : (s.filter fun a => a ≠ b) = s.erase b :=
   _root_.trans (filter_congr fun _ _ => by simp_rw [@ne_comm _ b]) (filter_ne s b)
 
-theorem filter_inter_filter_neg_eq (s t : Finset α) :
-    (s.filter p ∩ t.filter fun a => ¬p a) = ∅ := by
-  simpa using (disjoint_filter_filter_neg s t p).eq_bot
-
 theorem filter_union_filter_of_codisjoint (s : Finset α) (h : Codisjoint p q) :
     s.filter p ∪ s.filter q = s :=
   (filter_or _ _ _).symm.trans <| filter_true_of_mem fun x _ => h.top_le x trivial
@@ -2498,6 +2500,8 @@ theorem filter_union_filter_neg_eq [∀ x, Decidable (¬p x)] (s : Finset α) :
 
 end
 
+variable {p q}
+
 lemma filter_inj : s.filter p = t.filter p ↔ ∀ ⦃a⦄, p a → (a ∈ s ↔ a ∈ t) := by
   simp [Finset.ext_iff]
 
@@ -2511,6 +2515,8 @@ end Filter
 
 section Range
 
+open Nat
+
 variable {n m l : ℕ}
 
 /-- `range n` is the set of natural numbers less than `n`. -/
@@ -2568,15 +2574,19 @@ theorem mem_range_le {n x : ℕ} (hx : x ∈ range n) : x ≤ n :=
 theorem mem_range_sub_ne_zero {n x : ℕ} (hx : x ∈ range n) : n - x ≠ 0 :=
   _root_.ne_of_gt <| Nat.sub_pos_of_lt <| mem_range.1 hx
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem nonempty_range_iff : (range n).Nonempty ↔ n ≠ 0 :=
   ⟨fun ⟨k, hk⟩ => (k.zero_le.trans_lt <| mem_range.1 hk).ne',
    fun h => ⟨0, mem_range.2 <| Nat.pos_iff_ne_zero.2 h⟩⟩
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+protected alias ⟨_, Aesop.range_nonempty⟩ := nonempty_range_iff
+
 @[simp]
 theorem range_eq_empty_iff : range n = ∅ ↔ n = 0 := by
   rw [← not_nonempty_iff_eq_empty, nonempty_range_iff, not_not]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 theorem nonempty_range_succ : (range <| n + 1).Nonempty :=
   nonempty_range_iff.2 n.succ_ne_zero
 
@@ -2591,11 +2601,15 @@ lemma range_nontrivial {n : ℕ} (hn : 1 < n) : (Finset.range n).Nontrivial := b
   rw [Finset.Nontrivial, Finset.coe_range]
   exact ⟨0, Nat.zero_lt_one.trans hn, 1, hn, Nat.zero_ne_one⟩
 
+theorem exists_nat_subset_range (s : Finset ℕ) : ∃ n : ℕ, s ⊆ range n :=
+  s.induction_on (by simp)
+    fun a s _ ⟨n, hn⟩ => ⟨max (a + 1) n, insert_subset (by simp) (hn.trans (by simp))⟩
+
 end Range
 
 -- useful rules for calculations with quantifiers
 theorem exists_mem_empty_iff (p : α → Prop) : (∃ x, x ∈ (∅ : Finset α) ∧ p x) ↔ False := by
-  simp only [not_mem_empty, false_and_iff, exists_false]
+  simp only [not_mem_empty, false_and, exists_false]
 
 theorem exists_mem_insert [DecidableEq α] (a : α) (s : Finset α) (p : α → Prop) :
     (∃ x, x ∈ insert a s ∧ p x) ↔ p a ∨ ∃ x, x ∈ s ∧ p x := by
@@ -2712,10 +2726,13 @@ theorem toFinset_union (s t : Multiset α) : (s ∪ t).toFinset = s.toFinset ∪
 theorem toFinset_eq_empty {m : Multiset α} : m.toFinset = ∅ ↔ m = 0 :=
   Finset.val_inj.symm.trans Multiset.dedup_eq_zero
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem toFinset_nonempty : s.toFinset.Nonempty ↔ s ≠ 0 := by
   simp only [toFinset_eq_empty, Ne, Finset.nonempty_iff_ne_empty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+protected alias ⟨_, Aesop.toFinset_nonempty_of_ne⟩ := toFinset_nonempty
+
 @[simp]
 theorem toFinset_subset : s.toFinset ⊆ t.toFinset ↔ s ⊆ t := by
   simp only [Finset.subset_iff, Multiset.subset_iff, Multiset.mem_toFinset]
@@ -2845,10 +2862,13 @@ theorem toFinset_inter (l l' : List α) : (l ∩ l').toFinset = l.toFinset ∩ l
 theorem toFinset_eq_empty_iff (l : List α) : l.toFinset = ∅ ↔ l = nil := by
   cases l <;> simp
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem toFinset_nonempty_iff (l : List α) : l.toFinset.Nonempty ↔ l ≠ [] := by
   simp [Finset.nonempty_iff_ne_empty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.toFinset_nonempty_of_ne⟩ := toFinset_nonempty_iff
+
 @[simp]
 theorem toFinset_filter (s : List α) (p : α → Bool) :
     (s.filter p).toFinset = s.toFinset.filter (p ·) := by
@@ -2897,6 +2917,11 @@ theorem toList_toFinset [DecidableEq α] (s : Finset α) : s.toList.toFinset = s
   ext
   simp
 
+theorem _root_.List.toFinset_toList [DecidableEq α] {s : List α} (hs : s.Nodup) :
+    s.toFinset.toList.Perm s := by
+  apply List.perm_of_nodup_nodup_toFinset_eq (nodup_toList _) hs
+  rw [toList_toFinset]
+
 @[simp]
 theorem toList_eq_singleton_iff {a : α} {s : Finset α} : s.toList = [a] ↔ s = {a} := by
   rw [toList, Multiset.toList_eq_singleton_iff, val_eq_singleton_iff]
@@ -3046,6 +3071,8 @@ You can add lemmas to the rule-set by tagging them with either:
 * `aesop safe apply (rule_sets := [finsetNonempty])` if they are always a good idea to follow or
 * `aesop unsafe apply (rule_sets := [finsetNonempty])` if they risk directing the search to a blind
   alley.
+
+TODO: should some of the lemmas be `aesop safe simp` instead?
 -/
 def proveFinsetNonempty {u : Level} {α : Q(Type u)} (s : Q(Finset $α)) :
     MetaM (Option Q(Finset.Nonempty $s)) := do
@@ -3070,3 +3097,5 @@ def proveFinsetNonempty {u : Level} {α : Q(Type u)} (s : Q(Finset $α)) :
   Lean.getExprMVarAssignment? mvar
 
 end Mathlib.Meta
+
+set_option linter.style.longFile 3200
diff --git a/Mathlib/Data/Finset/Card.lean b/Mathlib/Data/Finset/Card.lean
index ee90c07098ad8..6cd9aef810c64 100644
--- a/Mathlib/Data/Finset/Card.lean
+++ b/Mathlib/Data/Finset/Card.lean
@@ -296,7 +296,7 @@ theorem card_eq_of_bijective (f : ∀ i, i < n → α) (hf : ∀ a ∈ s, ∃ i,
   have : s = (range n).attach.image fun i => f i.1 (mem_range.1 i.2) := by
     ext a
     suffices _ : a ∈ s ↔ ∃ (i : _) (hi : i ∈ range n), f i (mem_range.1 hi) = a by
-      simpa only [mem_image, mem_attach, true_and_iff, Subtype.exists]
+      simpa only [mem_image, mem_attach, true_and, Subtype.exists]
     constructor
     · intro ha; obtain ⟨i, hi, rfl⟩ := hf a ha; use i, mem_range.2 hi
     · rintro ⟨i, hi, rfl⟩; apply hf'
@@ -683,6 +683,11 @@ lemma exists_of_one_lt_card_pi {ι : Type*} {α : ι → Type*} [∀ i, Decidabl
   obtain rfl | hne := eq_or_ne (a2 i) ai
   exacts [⟨a1, h1, hne⟩, ⟨a2, h2, hne⟩]
 
+theorem card_eq_succ_iff_cons :
+    s.card = n + 1 ↔ ∃ a t, ∃ (h : a ∉ t), cons a t h = s ∧ t.card = n :=
+  ⟨cons_induction_on s (by simp) fun a s _ _ _ => ⟨a, s, by simp_all⟩,
+   fun ⟨a, t, _, hs, _⟩ => by simpa [← hs]⟩
+
 section DecidableEq
 variable [DecidableEq α]
 
@@ -826,6 +831,4 @@ theorem lt_wf {α} : WellFounded (@LT.lt (Finset α) _) :=
     card_lt_card hxy
   Subrelation.wf H <| InvImage.wf _ <| (Nat.lt_wfRel).2
 
-@[deprecated (since := "2023-12-27")] alias card_le_of_subset := card_le_card
-
 end Finset
diff --git a/Mathlib/Data/Finset/Density.lean b/Mathlib/Data/Finset/Density.lean
index 43125f7fcc4f3..002513c40e808 100644
--- a/Mathlib/Data/Finset/Density.lean
+++ b/Mathlib/Data/Finset/Density.lean
@@ -110,23 +110,23 @@ lemma dens_image [Fintype β] [DecidableEq β] {f : α → β} (hf : Bijective f
     (s.image f).dens = s.dens := by
   simpa [map_eq_image, -dens_map_equiv] using dens_map_equiv (.ofBijective f hf)
 
-lemma card_mul_dens (s : Finset α) : Fintype.card α * s.dens = s.card := by
+@[simp] lemma card_mul_dens (s : Finset α) : Fintype.card α * s.dens = s.card := by
   cases isEmpty_or_nonempty α
   · simp [Subsingleton.elim s ∅]
   rw [dens, mul_div_cancel₀]
   exact mod_cast Fintype.card_ne_zero
 
-lemma dens_mul_card (s : Finset α) : s.dens * Fintype.card α = s.card := by
+@[simp] lemma dens_mul_card (s : Finset α) : s.dens * Fintype.card α = s.card := by
   rw [mul_comm, card_mul_dens]
 
 section Semifield
 variable [Semifield 𝕜] [CharZero 𝕜]
 
-lemma natCast_card_mul_nnratCast_dens (s : Finset α) : (Fintype.card α * s.dens : 𝕜) = s.card :=
-  mod_cast s.card_mul_dens
+@[simp] lemma natCast_card_mul_nnratCast_dens (s : Finset α) :
+    (Fintype.card α * s.dens : 𝕜) = s.card := mod_cast s.card_mul_dens
 
-lemma nnratCast_dens_mul_natCast_card (s : Finset α) : s.dens * Fintype.card α = s.card :=
-  mod_cast s.dens_mul_card
+@[simp] lemma nnratCast_dens_mul_natCast_card (s : Finset α) :
+    (s.dens * Fintype.card α : 𝕜) = s.card := mod_cast s.dens_mul_card
 
 @[norm_cast] lemma nnratCast_dens (s : Finset α) : (s.dens : 𝕜) = s.card / Fintype.card α := by
   simp [dens]
diff --git a/Mathlib/Data/Finset/Fin.lean b/Mathlib/Data/Finset/Fin.lean
index fbe5e4dd8b999..03b2036cefe0c 100644
--- a/Mathlib/Data/Finset/Fin.lean
+++ b/Mathlib/Data/Finset/Fin.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Scott Morrison, Johan Commelin
+Authors: Chris Hughes, Kim Morrison, Johan Commelin
 -/
 import Mathlib.Data.Finset.Card
 
diff --git a/Mathlib/Data/Finset/Finsupp.lean b/Mathlib/Data/Finset/Finsupp.lean
index 08de380f2d4b9..3d7111a8e4605 100644
--- a/Mathlib/Data/Finset/Finsupp.lean
+++ b/Mathlib/Data/Finset/Finsupp.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
 import Mathlib.Algebra.BigOperators.Finsupp
-import Mathlib.Data.Finset.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.Data.Finsupp.Indicator
 import Mathlib.Data.Fintype.BigOperators
 
diff --git a/Mathlib/Data/Finset/Functor.lean b/Mathlib/Data/Finset/Functor.lean
index 2ae78966a4cd7..1359446c44d7e 100644
--- a/Mathlib/Data/Finset/Functor.lean
+++ b/Mathlib/Data/Finset/Functor.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yaël Dillies, Scott Morrison
+Authors: Yaël Dillies, Kim Morrison
 -/
 import Mathlib.Data.Finset.Lattice
 import Mathlib.Data.Finset.NAry
@@ -198,11 +198,16 @@ theorem map_comp_coe (h : α → β) :
     Functor.map h ∘ Multiset.toFinset = Multiset.toFinset ∘ Functor.map h :=
   funext fun _ => image_toFinset
 
+@[simp]
+theorem map_comp_coe_apply (h : α → β) (s : Multiset α) :
+    s.toFinset.image h = (h <$> s).toFinset :=
+  congrFun (map_comp_coe h) s
+
 theorem map_traverse (g : α → G β) (h : β → γ) (s : Finset α) :
     Functor.map h <$> traverse g s = traverse (Functor.map h ∘ g) s := by
   unfold traverse
-  simp only [map_comp_coe, functor_norm]
-  rw [LawfulFunctor.comp_map, Multiset.map_traverse]
+  simp only [Functor.map_map, fmap_def, map_comp_coe_apply, Multiset.fmap_def, ←
+    Multiset.map_traverse]
 
 end Traversable
 
diff --git a/Mathlib/Data/Finset/Image.lean b/Mathlib/Data/Finset/Image.lean
index 0594188ab0c3b..88939058da20c 100644
--- a/Mathlib/Data/Finset/Image.lean
+++ b/Mathlib/Data/Finset/Image.lean
@@ -129,7 +129,7 @@ theorem map_map (f : α ↪ β) (g : β ↪ γ) (s : Finset α) : (s.map f).map
 
 theorem map_comm {β'} {f : β ↪ γ} {g : α ↪ β} {f' : α ↪ β'} {g' : β' ↪ γ}
     (h_comm : ∀ a, f (g a) = g' (f' a)) : (s.map g).map f = (s.map f').map g' := by
-  simp_rw [map_map, Embedding.trans, Function.comp, h_comm]
+  simp_rw [map_map, Embedding.trans, Function.comp_def, h_comm]
 
 theorem _root_.Function.Semiconj.finset_map {f : α ↪ β} {ga : α ↪ α} {gb : β ↪ β}
     (h : Function.Semiconj f ga gb) : Function.Semiconj (map f) (map ga) (map gb) := fun _ =>
@@ -186,7 +186,7 @@ theorem filter_map {p : β → Prop} [DecidablePred p] :
 lemma map_filter' (p : α → Prop) [DecidablePred p] (f : α ↪ β) (s : Finset α)
     [DecidablePred (∃ a, p a ∧ f a = ·)] :
     (s.filter p).map f = (s.map f).filter fun b => ∃ a, p a ∧ f a = b := by
-  simp [(· ∘ ·), filter_map, f.injective.eq_iff]
+  simp [Function.comp_def, filter_map, f.injective.eq_iff]
 
 lemma filter_attach' [DecidableEq α] (s : Finset α) (p : s → Prop) [DecidablePred p] :
     s.attach.filter p =
@@ -201,7 +201,7 @@ lemma filter_attach (p : α → Prop) [DecidablePred p] (s : Finset α) :
 
 theorem map_filter {f : α ≃ β} {p : α → Prop} [DecidablePred p] :
     (s.filter p).map f.toEmbedding = (s.map f.toEmbedding).filter (p ∘ f.symm) := by
-  simp only [filter_map, Function.comp, Equiv.toEmbedding_apply, Equiv.symm_apply_apply]
+  simp only [filter_map, Function.comp_def, Equiv.toEmbedding_apply, Equiv.symm_apply_apply]
 
 @[simp]
 theorem disjoint_map {s t : Finset α} (f : α ↪ β) :
@@ -242,10 +242,11 @@ theorem map_cons (f : α ↪ β) (a : α) (s : Finset α) (ha : a ∉ s) :
 @[simp]
 theorem map_eq_empty : s.map f = ∅ ↔ s = ∅ := (map_injective f).eq_iff' (map_empty f)
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem map_nonempty : (s.map f).Nonempty ↔ s.Nonempty :=
   mod_cast Set.image_nonempty (f := f) (s := s)
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected alias ⟨_, Nonempty.map⟩ := map_nonempty
 
 @[simp]
@@ -344,25 +345,16 @@ theorem _root_.Function.Injective.mem_finset_image (hf : Injective f) :
   obtain ⟨y, hy, heq⟩ := mem_image.1 h
   exact hf heq ▸ hy
 
-theorem filter_mem_image_eq_image (f : α → β) (s : Finset α) (t : Finset β) (h : ∀ x ∈ s, f x ∈ t) :
-    (t.filter fun y => y ∈ s.image f) = s.image f := by
-  ext
-  simp only [mem_filter, mem_image, decide_eq_true_eq, and_iff_right_iff_imp, forall_exists_index,
-    and_imp]
-  rintro x xel rfl
-  exact h _ xel
-
-theorem fiber_nonempty_iff_mem_image (f : α → β) (s : Finset α) (y : β) :
-    (s.filter fun x => f x = y).Nonempty ↔ y ∈ s.image f := by simp [Finset.Nonempty]
 
 @[simp, norm_cast]
 theorem coe_image : ↑(s.image f) = f '' ↑s :=
   Set.ext <| by simp only [mem_coe, mem_image, Set.mem_image, implies_true]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma image_nonempty : (s.image f).Nonempty ↔ s.Nonempty :=
   mod_cast Set.image_nonempty (f := f) (s := (s : Set α))
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected theorem Nonempty.image (h : s.Nonempty) (f : α → β) : (s.image f).Nonempty :=
   image_nonempty.2 h
 
@@ -392,7 +384,7 @@ theorem image_image [DecidableEq γ] {g : β → γ} : (s.image f).image g = s.i
 
 theorem image_comm {β'} [DecidableEq β'] [DecidableEq γ] {f : β → γ} {g : α → β} {f' : α → β'}
     {g' : β' → γ} (h_comm : ∀ a, f (g a) = g' (f' a)) :
-    (s.image g).image f = (s.image f').image g' := by simp_rw [image_image, comp, h_comm]
+    (s.image g).image f = (s.image f').image g' := by simp_rw [image_image, comp_def, h_comm]
 
 theorem _root_.Function.Semiconj.finset_image [DecidableEq α] {f : α → β} {ga : α → α} {gb : β → β}
     (h : Function.Semiconj f ga gb) : Function.Semiconj (image f) (image ga) (image gb) := fun _ =>
@@ -439,6 +431,14 @@ theorem filter_image {p : β → Prop} [DecidablePred p] :
       ⟨by rintro ⟨⟨x, h1, rfl⟩, h2⟩; exact ⟨x, ⟨h1, h2⟩, rfl⟩,
        by rintro ⟨x, ⟨h1, h2⟩, rfl⟩; exact ⟨⟨x, h1, rfl⟩, h2⟩⟩
 
+@[deprecated filter_mem_eq_inter (since := "2024-09-15")]
+theorem filter_mem_image_eq_image (f : α → β) (s : Finset α) (t : Finset β) (h : ∀ x ∈ s, f x ∈ t) :
+    (t.filter fun y => y ∈ s.image f) = s.image f := by
+  rwa [filter_mem_eq_inter, inter_eq_right, image_subset_iff]
+
+theorem fiber_nonempty_iff_mem_image {y : β} : (s.filter (f · = y)).Nonempty ↔ y ∈ s.image f := by
+  simp [Finset.Nonempty]
+
 theorem image_union [DecidableEq α] {f : α → β} (s₁ s₂ : Finset α) :
     (s₁ ∪ s₂).image f = s₁.image f ∪ s₂.image f :=
   mod_cast Set.image_union f s₁ s₂
@@ -697,9 +697,23 @@ theorem fin_mono {n} : Monotone (Finset.fin n) := fun s t h x => by simpa using
 theorem fin_map {n} {s : Finset ℕ} : (s.fin n).map Fin.valEmbedding = s.filter (· < n) := by
   simp [Finset.fin, Finset.map_map]
 
+/--
+If a finset `t` is a subset of the image of another finset `s` under `f`, then it is equal to the
+image of a subset of `s`.
+
+For the version where `s` is a set, see `subset_set_image_iff`.
+-/
+theorem subset_image_iff [DecidableEq β] {s : Finset α} {t : Finset β} {f : α → β} :
+    t ⊆ s.image f ↔ ∃ s' : Finset α, s' ⊆ s ∧ s'.image f = t := by
+  refine ⟨fun ht => ?_, fun ⟨s', hs', h⟩ => h ▸ image_subset_image hs'⟩
+  refine ⟨s.filter (f · ∈ t), filter_subset _ _, le_antisymm (by simp [image_subset_iff]) ?_⟩
+  intro x hx
+  specialize ht hx
+  aesop
+
 /-- If a `Finset` is a subset of the image of a `Set` under `f`,
 then it is equal to the `Finset.image` of a `Finset` subset of that `Set`. -/
-theorem subset_image_iff [DecidableEq β] {s : Set α} {t : Finset β} {f : α → β} :
+theorem subset_set_image_iff [DecidableEq β] {s : Set α} {t : Finset β} {f : α → β} :
     ↑t ⊆ f '' s ↔ ∃ s' : Finset α, ↑s' ⊆ s ∧ s'.image f = t := by
   constructor; swap
   · rintro ⟨t, ht, rfl⟩
@@ -757,9 +771,3 @@ theorem finsetCongr_toEmbedding (e : α ≃ β) :
   rfl
 
 end Equiv
-
-namespace Finset
-
-@[deprecated (since := "2023-12-27")] alias image_filter := filter_image
-
-end Finset
diff --git a/Mathlib/Data/Finset/Lattice.lean b/Mathlib/Data/Finset/Lattice.lean
index c36b539aba616..b41a111701bda 100644
--- a/Mathlib/Data/Finset/Lattice.lean
+++ b/Mathlib/Data/Finset/Lattice.lean
@@ -5,17 +5,22 @@ Authors: Mario Carneiro
 -/
 import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
 import Mathlib.Data.Finset.Fold
-import Mathlib.Data.Finset.Option
 import Mathlib.Data.Finset.Pi
 import Mathlib.Data.Finset.Prod
 import Mathlib.Data.Multiset.Lattice
 import Mathlib.Data.Set.Lattice
 import Mathlib.Order.Hom.Lattice
-import Mathlib.Order.Minimal
 import Mathlib.Order.Nat
 
 /-!
 # Lattice operations on finsets
+
+This file is concerned with folding binary lattice operations over finsets.
+
+For the special case of maximum and minimum of a finset, see Max.lean.
+
+See also SetLattice.lean, which is instead concerned with how big lattice or set operations behave
+when indexed by a finset.
 -/
 
 assert_not_exists OrderedCommMonoid
@@ -204,9 +209,6 @@ theorem _root_.List.foldr_sup_eq_sup_toFinset [DecidableEq α] (l : List α) :
 theorem subset_range_sup_succ (s : Finset ℕ) : s ⊆ range (s.sup id).succ := fun _ hn =>
   mem_range.2 <| Nat.lt_succ_of_le <| @le_sup _ _ _ _ _ id _ hn
 
-theorem exists_nat_subset_range (s : Finset ℕ) : ∃ n : ℕ, s ⊆ range n :=
-  ⟨_, s.subset_range_sup_succ⟩
-
 theorem sup_induction {p : α → Prop} (hb : p ⊥) (hp : ∀ a₁, p a₁ → ∀ a₂, p a₂ → p (a₁ ⊔ a₂))
     (hs : ∀ b ∈ s, p (f b)) : p (s.sup f) := by
   induction s using Finset.cons_induction with
@@ -220,7 +222,7 @@ theorem sup_le_of_le_directed {α : Type*} [SemilatticeSup α] [OrderBot α] (s
     (∀ x ∈ t, ∃ y ∈ s, x ≤ y) → ∃ x ∈ s, t.sup id ≤ x := by
   classical
     induction' t using Finset.induction_on with a r _ ih h
-    · simpa only [forall_prop_of_true, and_true_iff, forall_prop_of_false, bot_le, not_false_iff,
+    · simpa only [forall_prop_of_true, and_true, forall_prop_of_false, bot_le, not_false_iff,
         sup_empty, forall_true_iff, not_mem_empty]
     · intro h
       have incs : (r : Set α) ⊆ ↑(insert a r) := by
@@ -520,7 +522,7 @@ theorem inf_sup {κ : ι → Type*} (s : Finset ι) (t : ∀ i, Finset (κ i)) (
   -- Porting note: `simpa` doesn't support placeholders in proof terms
   have := h (fun j hj => if hji : j = i then cast (congr_arg κ hji.symm) a
       else g _ <| mem_of_mem_insert_of_ne hj hji) (fun j hj => ?_)
-  · simpa only [cast_eq, dif_pos, Function.comp, Subtype.coe_mk, dif_neg, aux] using this
+  · simpa only [cast_eq, dif_pos, Function.comp_def, Subtype.coe_mk, dif_neg, aux] using this
   rw [mem_insert] at hj
   obtain (rfl | hj) := hj
   · simpa
@@ -704,7 +706,7 @@ theorem coe_sup' : ((s.sup' H f : α) : WithBot α) = s.sup ((↑) ∘ f) := by
 
 @[simp]
 theorem sup'_cons {b : β} {hb : b ∉ s} :
-    (cons b s hb).sup' (nonempty_cons hb) f = f b ⊔ s.sup' H f := by
+    (cons b s hb).sup' (cons_nonempty hb) f = f b ⊔ s.sup' H f := by
   rw [← WithBot.coe_eq_coe]
   simp [WithBot.coe_sup]
 
@@ -873,7 +875,7 @@ theorem coe_inf' : ((s.inf' H f : α) : WithTop α) = s.inf ((↑) ∘ f) :=
 
 @[simp]
 theorem inf'_cons {b : β} {hb : b ∉ s} :
-    (cons b s hb).inf' (nonempty_cons hb) f = f b ⊓ s.inf' H f :=
+    (cons b s hb).inf' (cons_nonempty hb) f = f b ⊓ s.inf' H f :=
   @sup'_cons αᵒᵈ _ _ _ H f _ _
 
 @[simp]
@@ -1155,529 +1157,6 @@ theorem exists_mem_eq_inf [OrderTop α] (s : Finset ι) (h : s.Nonempty) (f : ι
 
 end LinearOrder
 
-/-! ### max and min of finite sets -/
-
-
-section MaxMin
-
-variable [LinearOrder α]
-
-/-- Let `s` be a finset in a linear order. Then `s.max` is the maximum of `s` if `s` is not empty,
-and `⊥` otherwise. It belongs to `WithBot α`. If you want to get an element of `α`, see
-`s.max'`. -/
-protected def max (s : Finset α) : WithBot α :=
-  sup s (↑)
-
-theorem max_eq_sup_coe {s : Finset α} : s.max = s.sup (↑) :=
-  rfl
-
-theorem max_eq_sup_withBot (s : Finset α) : s.max = sup s (↑) :=
-  rfl
-
-@[simp]
-theorem max_empty : (∅ : Finset α).max = ⊥ :=
-  rfl
-
-@[simp]
-theorem max_insert {a : α} {s : Finset α} : (insert a s).max = max ↑a s.max :=
-  fold_insert_idem
-
-@[simp]
-theorem max_singleton {a : α} : Finset.max {a} = (a : WithBot α) := by
-  rw [← insert_emptyc_eq]
-  exact max_insert
-
-theorem max_of_mem {s : Finset α} {a : α} (h : a ∈ s) : ∃ b : α, s.max = b := by
-  obtain ⟨b, h, _⟩ := le_sup (α := WithBot α) h _ rfl
-  exact ⟨b, h⟩
-
-theorem max_of_nonempty {s : Finset α} (h : s.Nonempty) : ∃ a : α, s.max = a :=
-  let ⟨_, h⟩ := h
-  max_of_mem h
-
-theorem max_eq_bot {s : Finset α} : s.max = ⊥ ↔ s = ∅ :=
-  ⟨fun h ↦ s.eq_empty_or_nonempty.elim id fun H ↦ by
-      obtain ⟨a, ha⟩ := max_of_nonempty H
-      rw [h] at ha; cases ha; , -- the `;` is needed since the `cases` syntax allows `cases a, b`
-    fun h ↦ h.symm ▸ max_empty⟩
-
-theorem mem_of_max {s : Finset α} : ∀ {a : α}, s.max = a → a ∈ s := by
-  induction' s using Finset.induction_on with b s _ ih
-  · intro _ H; cases H
-  · intro a h
-    by_cases p : b = a
-    · induction p
-      exact mem_insert_self b s
-    · cases' max_choice (↑b) s.max with q q <;> rw [max_insert, q] at h
-      · cases h
-        cases p rfl
-      · exact mem_insert_of_mem (ih h)
-
-theorem le_max {a : α} {s : Finset α} (as : a ∈ s) : ↑a ≤ s.max :=
-  le_sup as
-
-theorem not_mem_of_max_lt_coe {a : α} {s : Finset α} (h : s.max < a) : a ∉ s :=
-  mt le_max h.not_le
-
-theorem le_max_of_eq {s : Finset α} {a b : α} (h₁ : a ∈ s) (h₂ : s.max = b) : a ≤ b :=
-  WithBot.coe_le_coe.mp <| (le_max h₁).trans h₂.le
-
-theorem not_mem_of_max_lt {s : Finset α} {a b : α} (h₁ : b < a) (h₂ : s.max = ↑b) : a ∉ s :=
-  Finset.not_mem_of_max_lt_coe <| h₂.trans_lt <| WithBot.coe_lt_coe.mpr h₁
-
-@[gcongr]
-theorem max_mono {s t : Finset α} (st : s ⊆ t) : s.max ≤ t.max :=
-  sup_mono st
-
-protected theorem max_le {M : WithBot α} {s : Finset α} (st : ∀ a ∈ s, (a : WithBot α) ≤ M) :
-    s.max ≤ M :=
-  Finset.sup_le st
-
-@[simp]
-protected lemma max_le_iff {m : WithBot α} {s : Finset α} : s.max ≤ m ↔ ∀ a ∈ s, a ≤ m :=
-  Finset.sup_le_iff
-
-@[simp]
-protected lemma max_eq_top [OrderTop α] {s : Finset α} : s.max = ⊤ ↔ ⊤ ∈ s :=
-  Finset.sup_eq_top_iff.trans <| by simp
-
-/-- Let `s` be a finset in a linear order. Then `s.min` is the minimum of `s` if `s` is not empty,
-and `⊤` otherwise. It belongs to `WithTop α`. If you want to get an element of `α`, see
-`s.min'`. -/
-protected def min (s : Finset α) : WithTop α :=
-  inf s (↑)
-
-theorem min_eq_inf_withTop (s : Finset α) : s.min = inf s (↑) :=
-  rfl
-
-@[simp]
-theorem min_empty : (∅ : Finset α).min = ⊤ :=
-  rfl
-
-@[simp]
-theorem min_insert {a : α} {s : Finset α} : (insert a s).min = min (↑a) s.min :=
-  fold_insert_idem
-
-@[simp]
-theorem min_singleton {a : α} : Finset.min {a} = (a : WithTop α) := by
-  rw [← insert_emptyc_eq]
-  exact min_insert
-
-theorem min_of_mem {s : Finset α} {a : α} (h : a ∈ s) : ∃ b : α, s.min = b := by
-  obtain ⟨b, h, _⟩ := inf_le (α := WithTop α) h _ rfl
-  exact ⟨b, h⟩
-
-theorem min_of_nonempty {s : Finset α} (h : s.Nonempty) : ∃ a : α, s.min = a :=
-  let ⟨_, h⟩ := h
-  min_of_mem h
-
-@[simp]
-theorem min_eq_top {s : Finset α} : s.min = ⊤ ↔ s = ∅ := by
-  simp [Finset.min, eq_empty_iff_forall_not_mem]
-
-theorem mem_of_min {s : Finset α} : ∀ {a : α}, s.min = a → a ∈ s :=
-  @mem_of_max αᵒᵈ _ s
-
-theorem min_le {a : α} {s : Finset α} (as : a ∈ s) : s.min ≤ a :=
-  inf_le as
-
-theorem not_mem_of_coe_lt_min {a : α} {s : Finset α} (h : ↑a < s.min) : a ∉ s :=
-  mt min_le h.not_le
-
-theorem min_le_of_eq {s : Finset α} {a b : α} (h₁ : b ∈ s) (h₂ : s.min = a) : a ≤ b :=
-  WithTop.coe_le_coe.mp <| h₂.ge.trans (min_le h₁)
-
-theorem not_mem_of_lt_min {s : Finset α} {a b : α} (h₁ : a < b) (h₂ : s.min = ↑b) : a ∉ s :=
-  Finset.not_mem_of_coe_lt_min <| (WithTop.coe_lt_coe.mpr h₁).trans_eq h₂.symm
-
-@[gcongr]
-theorem min_mono {s t : Finset α} (st : s ⊆ t) : t.min ≤ s.min :=
-  inf_mono st
-
-protected theorem le_min {m : WithTop α} {s : Finset α} (st : ∀ a : α, a ∈ s → m ≤ a) : m ≤ s.min :=
-  Finset.le_inf st
-
-@[simp]
-protected theorem le_min_iff {m : WithTop α} {s : Finset α} : m ≤ s.min ↔ ∀ a ∈ s, m ≤ a :=
-  Finset.le_inf_iff
-
-@[simp]
-protected theorem min_eq_bot [OrderBot α] {s : Finset α} : s.min = ⊥ ↔ ⊥ ∈ s :=
-  Finset.max_eq_top (α := αᵒᵈ)
-
-/-- Given a nonempty finset `s` in a linear order `α`, then `s.min' H` is its minimum, as an
-element of `α`, where `H` is a proof of nonemptiness. Without this assumption, use instead `s.min`,
-taking values in `WithTop α`. -/
-def min' (s : Finset α) (H : s.Nonempty) : α :=
-  inf' s H id
-
-/-- Given a nonempty finset `s` in a linear order `α`, then `s.max' H` is its maximum, as an
-element of `α`, where `H` is a proof of nonemptiness. Without this assumption, use instead `s.max`,
-taking values in `WithBot α`. -/
-def max' (s : Finset α) (H : s.Nonempty) : α :=
-  sup' s H id
-
-variable (s : Finset α) (H : s.Nonempty) {x : α}
-
-theorem min'_mem : s.min' H ∈ s :=
-  mem_of_min <| by simp only [Finset.min, min', id_eq, coe_inf']; rfl
-
-theorem min'_le (x) (H2 : x ∈ s) : s.min' ⟨x, H2⟩ ≤ x :=
-  min_le_of_eq H2 (WithTop.coe_untop _ _).symm
-
-theorem le_min' (x) (H2 : ∀ y ∈ s, x ≤ y) : x ≤ s.min' H :=
-  H2 _ <| min'_mem _ _
-
-theorem isLeast_min' : IsLeast (↑s) (s.min' H) :=
-  ⟨min'_mem _ _, min'_le _⟩
-
-@[simp]
-theorem le_min'_iff {x} : x ≤ s.min' H ↔ ∀ y ∈ s, x ≤ y :=
-  le_isGLB_iff (isLeast_min' s H).isGLB
-
-/-- `{a}.min' _` is `a`. -/
-@[simp]
-theorem min'_singleton (a : α) : ({a} : Finset α).min' (singleton_nonempty _) = a := by simp [min']
-
-theorem max'_mem : s.max' H ∈ s :=
-  mem_of_max <| by simp only [max', Finset.max, id_eq, coe_sup']; rfl
-
-theorem le_max' (x) (H2 : x ∈ s) : x ≤ s.max' ⟨x, H2⟩ :=
-  le_max_of_eq H2 (WithBot.coe_unbot _ _).symm
-
-theorem max'_le (x) (H2 : ∀ y ∈ s, y ≤ x) : s.max' H ≤ x :=
-  H2 _ <| max'_mem _ _
-
-theorem isGreatest_max' : IsGreatest (↑s) (s.max' H) :=
-  ⟨max'_mem _ _, le_max' _⟩
-
-@[simp]
-theorem max'_le_iff {x} : s.max' H ≤ x ↔ ∀ y ∈ s, y ≤ x :=
-  isLUB_le_iff (isGreatest_max' s H).isLUB
-
-@[simp]
-theorem max'_lt_iff {x} : s.max' H < x ↔ ∀ y ∈ s, y < x :=
-  ⟨fun Hlt y hy => (s.le_max' y hy).trans_lt Hlt, fun H => H _ <| s.max'_mem _⟩
-
-@[simp]
-theorem lt_min'_iff : x < s.min' H ↔ ∀ y ∈ s, x < y :=
-  @max'_lt_iff αᵒᵈ _ _ H _
-
-theorem max'_eq_sup' : s.max' H = s.sup' H id := rfl
-
-theorem min'_eq_inf' : s.min' H = s.inf' H id := rfl
-
-/-- `{a}.max' _` is `a`. -/
-@[simp]
-theorem max'_singleton (a : α) : ({a} : Finset α).max' (singleton_nonempty _) = a := by simp [max']
-
-theorem min'_lt_max' {i j} (H1 : i ∈ s) (H2 : j ∈ s) (H3 : i ≠ j) :
-    s.min' ⟨i, H1⟩ < s.max' ⟨i, H1⟩ :=
-  isGLB_lt_isLUB_of_ne (s.isLeast_min' _).isGLB (s.isGreatest_max' _).isLUB H1 H2 H3
-
-/-- If there's more than 1 element, the min' is less than the max'. An alternate version of
-`min'_lt_max'` which is sometimes more convenient.
--/
-theorem min'_lt_max'_of_card (h₂ : 1 < card s) :
-    s.min' (Finset.card_pos.1 <| by omega) < s.max' (Finset.card_pos.1 <| by omega) := by
-  rcases one_lt_card.1 h₂ with ⟨a, ha, b, hb, hab⟩
-  exact s.min'_lt_max' ha hb hab
-
-theorem map_ofDual_min (s : Finset αᵒᵈ) : s.min.map ofDual = (s.image ofDual).max := by
-  rw [max_eq_sup_withBot, sup_image]
-  exact congr_fun Option.map_id _
-
-theorem map_ofDual_max (s : Finset αᵒᵈ) : s.max.map ofDual = (s.image ofDual).min := by
-  rw [min_eq_inf_withTop, inf_image]
-  exact congr_fun Option.map_id _
-
-theorem map_toDual_min (s : Finset α) : s.min.map toDual = (s.image toDual).max := by
-  rw [max_eq_sup_withBot, sup_image]
-  exact congr_fun Option.map_id _
-
-theorem map_toDual_max (s : Finset α) : s.max.map toDual = (s.image toDual).min := by
-  rw [min_eq_inf_withTop, inf_image]
-  exact congr_fun Option.map_id _
-
--- Porting note: new proofs without `convert` for the next four theorems.
-
-theorem ofDual_min' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
-    ofDual (min' s hs) = max' (s.image ofDual) (hs.image _) := by
-  rw [← WithBot.coe_eq_coe]
-  simp only [min'_eq_inf', id_eq, ofDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',
-    sup_image]
-  rfl
-
-theorem ofDual_max' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
-    ofDual (max' s hs) = min' (s.image ofDual) (hs.image _) := by
-  rw [← WithTop.coe_eq_coe]
-  simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',
-    inf_image]
-  rfl
-
-theorem toDual_min' {s : Finset α} (hs : s.Nonempty) :
-    toDual (min' s hs) = max' (s.image toDual) (hs.image _) := by
-  rw [← WithBot.coe_eq_coe]
-  simp only [min'_eq_inf', id_eq, toDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',
-    sup_image]
-  rfl
-
-theorem toDual_max' {s : Finset α} (hs : s.Nonempty) :
-    toDual (max' s hs) = min' (s.image toDual) (hs.image _) := by
-  rw [← WithTop.coe_eq_coe]
-  simp only [max'_eq_sup', id_eq, toDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',
-    inf_image]
-  rfl
-
-theorem max'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
-    s.max' H ≤ t.max' (H.mono hst) :=
-  le_max' _ _ (hst (s.max'_mem H))
-
-theorem min'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
-    t.min' (H.mono hst) ≤ s.min' H :=
-  min'_le _ _ (hst (s.min'_mem H))
-
-theorem max'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
-    (insert a s).max' (s.insert_nonempty a) = max (s.max' H) a :=
-  (isGreatest_max' _ _).unique <| by
-    rw [coe_insert, max_comm]
-    exact (isGreatest_max' _ _).insert _
-
-theorem min'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
-    (insert a s).min' (s.insert_nonempty a) = min (s.min' H) a :=
-  (isLeast_min' _ _).unique <| by
-    rw [coe_insert, min_comm]
-    exact (isLeast_min' _ _).insert _
-
-theorem lt_max'_of_mem_erase_max' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.max' H)) :
-    a < s.max' H :=
-  lt_of_le_of_ne (le_max' _ _ (mem_of_mem_erase ha)) <| ne_of_mem_of_not_mem ha <| not_mem_erase _ _
-
-theorem min'_lt_of_mem_erase_min' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.min' H)) :
-    s.min' H < a :=
-  @lt_max'_of_mem_erase_max' αᵒᵈ _ s H _ a ha
-
-/-- To rewrite from right to left, use `Monotone.map_finset_max'`. -/
-@[simp]
-theorem max'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
-    (h : (s.image f).Nonempty) : (s.image f).max' h = f (s.max' h.of_image) := by
-  simp only [max', sup'_image]
-  exact .symm <| comp_sup'_eq_sup'_comp _ _ fun _ _ ↦ hf.map_max
-
-/-- A version of `Finset.max'_image` with LHS and RHS reversed.
-Also, this version assumes that `s` is nonempty, not its image. -/
-lemma _root_.Monotone.map_finset_max' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
-    (h : s.Nonempty) : f (s.max' h) = (s.image f).max' (h.image f) :=
-  .symm <| max'_image hf ..
-
-/-- To rewrite from right to left, use `Monotone.map_finset_min'`. -/
-@[simp]
-theorem min'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
-    (h : (s.image f).Nonempty) : (s.image f).min' h = f (s.min' h.of_image) := by
-  simp only [min', inf'_image]
-  exact .symm <| comp_inf'_eq_inf'_comp _ _ fun _ _ ↦ hf.map_min
-
-/-- A version of `Finset.min'_image` with LHS and RHS reversed.
-Also, this version assumes that `s` is nonempty, not its image. -/
-lemma _root_.Monotone.map_finset_min' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
-    (h : s.Nonempty) : f (s.min' h) = (s.image f).min' (h.image f) :=
-  .symm <| min'_image hf ..
-
-theorem coe_max' {s : Finset α} (hs : s.Nonempty) : ↑(s.max' hs) = s.max :=
-  coe_sup' hs id
-
-theorem coe_min' {s : Finset α} (hs : s.Nonempty) : ↑(s.min' hs) = s.min :=
-  coe_inf' hs id
-
-theorem max_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
-    s.max ∈ (s.image (↑) : Finset (WithBot α)) :=
-  mem_image.2 ⟨max' s hs, max'_mem _ _, coe_max' hs⟩
-
-theorem min_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
-    s.min ∈ (s.image (↑) : Finset (WithTop α)) :=
-  mem_image.2 ⟨min' s hs, min'_mem _ _, coe_min' hs⟩
-
-theorem max_mem_insert_bot_image_coe (s : Finset α) :
-    s.max ∈ (insert ⊥ (s.image (↑)) : Finset (WithBot α)) :=
-  mem_insert.2 <| s.eq_empty_or_nonempty.imp max_eq_bot.2 max_mem_image_coe
-
-theorem min_mem_insert_top_image_coe (s : Finset α) :
-    s.min ∈ (insert ⊤ (s.image (↑)) : Finset (WithTop α)) :=
-  mem_insert.2 <| s.eq_empty_or_nonempty.imp min_eq_top.2 min_mem_image_coe
-
-theorem max'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).max' s0 ≠ x :=
-  ne_of_mem_erase (max'_mem _ s0)
-
-theorem min'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).min' s0 ≠ x :=
-  ne_of_mem_erase (min'_mem _ s0)
-
-theorem max_erase_ne_self {s : Finset α} : (s.erase x).max ≠ x := by
-  by_cases s0 : (s.erase x).Nonempty
-  · refine ne_of_eq_of_ne (coe_max' s0).symm ?_
-    exact WithBot.coe_eq_coe.not.mpr (max'_erase_ne_self _)
-  · rw [not_nonempty_iff_eq_empty.mp s0, max_empty]
-    exact WithBot.bot_ne_coe
-
-theorem min_erase_ne_self {s : Finset α} : (s.erase x).min ≠ x := by
-  -- Porting note: old proof `convert @max_erase_ne_self αᵒᵈ _ _ _`
-  convert @max_erase_ne_self αᵒᵈ _ (toDual x) (s.map toDual.toEmbedding) using 1
-  apply congr_arg -- Porting note: forces unfolding to see `Finset.min` is `Finset.max`
-  congr!
-  ext; simp only [mem_map_equiv]; exact Iff.rfl
-
-theorem exists_next_right {x : α} {s : Finset α} (h : ∃ y ∈ s, x < y) :
-    ∃ y ∈ s, x < y ∧ ∀ z ∈ s, x < z → y ≤ z :=
-  have Hne : (s.filter (x < ·)).Nonempty := h.imp fun y hy => mem_filter.2 (by simpa)
-  have aux := mem_filter.1 (min'_mem _ Hne)
-  ⟨min' _ Hne, aux.1, by simp, fun z hzs hz => min'_le _ _ <| mem_filter.2 ⟨hzs, by simpa⟩⟩
-
-theorem exists_next_left {x : α} {s : Finset α} (h : ∃ y ∈ s, y < x) :
-    ∃ y ∈ s, y < x ∧ ∀ z ∈ s, z < x → z ≤ y :=
-  @exists_next_right αᵒᵈ _ x s h
-
-/-- If finsets `s` and `t` are interleaved, then `Finset.card s ≤ Finset.card t + 1`. -/
-theorem card_le_of_interleaved {s t : Finset α}
-    (h : ∀ᵉ (x ∈ s) (y ∈ s),
-        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
-    s.card ≤ t.card + 1 := by
-  replace h : ∀ᵉ (x ∈ s) (y ∈ s), x < y → ∃ z ∈ t, x < z ∧ z < y := by
-    intro x hx y hy hxy
-    rcases exists_next_right ⟨y, hy, hxy⟩ with ⟨a, has, hxa, ha⟩
-    rcases h x hx a has hxa fun z hzs hz => hz.2.not_le <| ha _ hzs hz.1 with ⟨b, hbt, hxb, hba⟩
-    exact ⟨b, hbt, hxb, hba.trans_le <| ha _ hy hxy⟩
-  set f : α → WithTop α := fun x => (t.filter fun y => x < y).min
-  have f_mono : StrictMonoOn f s := by
-    intro x hx y hy hxy
-    rcases h x hx y hy hxy with ⟨a, hat, hxa, hay⟩
-    calc
-      f x ≤ a := min_le (mem_filter.2 ⟨hat, by simpa⟩)
-      _ < f y :=
-        (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>
-          WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2)
-
-  calc
-    s.card = (s.image f).card := (card_image_of_injOn f_mono.injOn).symm
-    _ ≤ (insert ⊤ (t.image (↑)) : Finset (WithTop α)).card :=
-      card_mono <| image_subset_iff.2 fun x _ =>
-          insert_subset_insert _ (image_subset_image <| filter_subset _ _)
-            (min_mem_insert_top_image_coe _)
-    _ ≤ t.card + 1 := (card_insert_le _ _).trans (Nat.add_le_add_right card_image_le _)
-
-/-- If finsets `s` and `t` are interleaved, then `Finset.card s ≤ Finset.card (t \ s) + 1`. -/
-theorem card_le_diff_of_interleaved {s t : Finset α}
-    (h :
-      ∀ᵉ (x ∈ s) (y ∈ s),
-        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
-    s.card ≤ (t \ s).card + 1 :=
-  card_le_of_interleaved fun x hx y hy hxy hs =>
-    let ⟨z, hzt, hxz, hzy⟩ := h x hx y hy hxy hs
-    ⟨z, mem_sdiff.2 ⟨hzt, fun hzs => hs z hzs ⟨hxz, hzy⟩⟩, hxz, hzy⟩
-
-/-- Induction principle for `Finset`s in a linearly ordered type: a predicate is true on all
-`s : Finset α` provided that:
-
-* it is true on the empty `Finset`,
-* for every `s : Finset α` and an element `a` strictly greater than all elements of `s`, `p s`
-  implies `p (insert a s)`. -/
-@[elab_as_elim]
-theorem induction_on_max [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
-    (step : ∀ a s, (∀ x ∈ s, x < a) → p s → p (insert a s)) : p s := by
-  induction' s using Finset.strongInductionOn with s ihs
-  rcases s.eq_empty_or_nonempty with (rfl | hne)
-  · exact h0
-  · have H : s.max' hne ∈ s := max'_mem s hne
-    rw [← insert_erase H]
-    exact step _ _ (fun x => s.lt_max'_of_mem_erase_max' hne) (ihs _ <| erase_ssubset H)
-
-/-- Induction principle for `Finset`s in a linearly ordered type: a predicate is true on all
-`s : Finset α` provided that:
-
-* it is true on the empty `Finset`,
-* for every `s : Finset α` and an element `a` strictly less than all elements of `s`, `p s`
-  implies `p (insert a s)`. -/
-@[elab_as_elim]
-theorem induction_on_min [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
-    (step : ∀ a s, (∀ x ∈ s, a < x) → p s → p (insert a s)) : p s :=
-  @induction_on_max αᵒᵈ _ _ _ s h0 step
-
-end MaxMin
-
-section MaxMinInductionValue
-
-variable [LinearOrder α] [LinearOrder β]
-
-/-- Induction principle for `Finset`s in any type from which a given function `f` maps to a linearly
-ordered type : a predicate is true on all `s : Finset α` provided that:
-
-* it is true on the empty `Finset`,
-* for every `s : Finset α` and an element `a` such that for elements of `s` denoted by `x` we have
-  `f x ≤ f a`, `p s` implies `p (insert a s)`. -/
-@[elab_as_elim]
-theorem induction_on_max_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
-    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f x ≤ f a) → p s → p (insert a s)) : p s := by
-  induction' s using Finset.strongInductionOn with s ihs
-  rcases (s.image f).eq_empty_or_nonempty with (hne | hne)
-  · simp only [image_eq_empty] at hne
-    simp only [hne, h0]
-  · have H : (s.image f).max' hne ∈ s.image f := max'_mem (s.image f) hne
-    simp only [mem_image, exists_prop] at H
-    rcases H with ⟨a, has, hfa⟩
-    rw [← insert_erase has]
-    refine step _ _ (not_mem_erase a s) (fun x hx => ?_) (ihs _ <| erase_ssubset has)
-    rw [hfa]
-    exact le_max' _ _ (mem_image_of_mem _ <| mem_of_mem_erase hx)
-
-/-- Induction principle for `Finset`s in any type from which a given function `f` maps to a linearly
-ordered type : a predicate is true on all `s : Finset α` provided that:
-
-* it is true on the empty `Finset`,
-* for every `s : Finset α` and an element `a` such that for elements of `s` denoted by `x` we have
-  `f a ≤ f x`, `p s` implies `p (insert a s)`. -/
-@[elab_as_elim]
-theorem induction_on_min_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
-    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f a ≤ f x) → p s → p (insert a s)) : p s :=
-  @induction_on_max_value αᵒᵈ ι _ _ _ _ s h0 step
-
-end MaxMinInductionValue
-
-section ExistsMaxMin
-
-variable [LinearOrder α]
-
-theorem exists_max_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
-    ∃ x ∈ s, ∀ x' ∈ s, f x' ≤ f x := by
-  cases' max_of_nonempty (h.image f) with y hy
-  rcases mem_image.mp (mem_of_max hy) with ⟨x, hx, rfl⟩
-  exact ⟨x, hx, fun x' hx' => le_max_of_eq (mem_image_of_mem f hx') hy⟩
-
-theorem exists_min_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
-    ∃ x ∈ s, ∀ x' ∈ s, f x ≤ f x' :=
-  @exists_max_image αᵒᵈ β _ s f h
-
-end ExistsMaxMin
-
-theorem isGLB_iff_isLeast [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
-    IsGLB (s : Set α) i ↔ IsLeast (↑s) i := by
-  refine ⟨fun his => ?_, IsLeast.isGLB⟩
-  suffices i = min' s hs by
-    rw [this]
-    exact isLeast_min' s hs
-  rw [IsGLB, IsGreatest, mem_lowerBounds, mem_upperBounds] at his
-  exact le_antisymm (his.1 (Finset.min' s hs) (Finset.min'_mem s hs)) (his.2 _ (Finset.min'_le s))
-
-theorem isLUB_iff_isGreatest [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
-    IsLUB (s : Set α) i ↔ IsGreatest (↑s) i :=
-  @isGLB_iff_isLeast αᵒᵈ _ i s hs
-
-theorem isGLB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsGLB (s : Set α) i)
-    (hs : s.Nonempty) : i ∈ s := by
-  rw [← mem_coe]
-  exact ((isGLB_iff_isLeast i s hs).mp his).1
-
-theorem isLUB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsLUB (s : Set α) i)
-    (hs : s.Nonempty) : i ∈ s :=
-  @isGLB_mem αᵒᵈ _ i s his hs
-
 end Finset
 
 namespace Multiset
@@ -1726,228 +1205,3 @@ theorem sup_singleton' [DecidableEq α] (s : Finset α) : s.sup singleton = s :=
   (s.sup_singleton'' _).trans image_id
 
 end Finset
-
-section Lattice
-
-variable {ι' : Sort*} [CompleteLattice α]
-
-/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : Finset ι` of suprema
-`⨆ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `iSup_eq_iSup_finset'` for a version
-that works for `ι : Sort*`. -/
-theorem iSup_eq_iSup_finset (s : ι → α) : ⨆ i, s i = ⨆ t : Finset ι, ⨆ i ∈ t, s i := by
-  classical
-  refine le_antisymm ?_ ?_
-  · exact iSup_le fun b => le_iSup_of_le {b} <| le_iSup_of_le b <| le_iSup_of_le (by simp) <| le_rfl
-  · exact iSup_le fun t => iSup_le fun b => iSup_le fun _ => le_iSup _ _
-
-/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : Finset ι` of suprema
-`⨆ i ∈ t, s i`. This version works for `ι : Sort*`. See `iSup_eq_iSup_finset` for a version
-that assumes `ι : Type*` but has no `PLift`s. -/
-theorem iSup_eq_iSup_finset' (s : ι' → α) :
-    ⨆ i, s i = ⨆ t : Finset (PLift ι'), ⨆ i ∈ t, s (PLift.down i) := by
-  rw [← iSup_eq_iSup_finset, ← Equiv.plift.surjective.iSup_comp]; rfl
-
-/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : Finset ι` of infima
-`⨅ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `iInf_eq_iInf_finset'` for a version
-that works for `ι : Sort*`. -/
-theorem iInf_eq_iInf_finset (s : ι → α) : ⨅ i, s i = ⨅ (t : Finset ι) (i ∈ t), s i :=
-  @iSup_eq_iSup_finset αᵒᵈ _ _ _
-
-/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : Finset ι` of infima
-`⨅ i ∈ t, s i`. This version works for `ι : Sort*`. See `iInf_eq_iInf_finset` for a version
-that assumes `ι : Type*` but has no `PLift`s. -/
-theorem iInf_eq_iInf_finset' (s : ι' → α) :
-    ⨅ i, s i = ⨅ t : Finset (PLift ι'), ⨅ i ∈ t, s (PLift.down i) :=
-  @iSup_eq_iSup_finset' αᵒᵈ _ _ _
-
-end Lattice
-
-namespace Set
-
-variable {ι' : Sort*}
-
-/-- Union of an indexed family of sets `s : ι → Set α` is equal to the union of the unions
-of finite subfamilies. This version assumes `ι : Type*`. See also `iUnion_eq_iUnion_finset'` for
-a version that works for `ι : Sort*`. -/
-theorem iUnion_eq_iUnion_finset (s : ι → Set α) : ⋃ i, s i = ⋃ t : Finset ι, ⋃ i ∈ t, s i :=
-  iSup_eq_iSup_finset s
-
-/-- Union of an indexed family of sets `s : ι → Set α` is equal to the union of the unions
-of finite subfamilies. This version works for `ι : Sort*`. See also `iUnion_eq_iUnion_finset` for
-a version that assumes `ι : Type*` but avoids `PLift`s in the right hand side. -/
-theorem iUnion_eq_iUnion_finset' (s : ι' → Set α) :
-    ⋃ i, s i = ⋃ t : Finset (PLift ι'), ⋃ i ∈ t, s (PLift.down i) :=
-  iSup_eq_iSup_finset' s
-
-/-- Intersection of an indexed family of sets `s : ι → Set α` is equal to the intersection of the
-intersections of finite subfamilies. This version assumes `ι : Type*`. See also
-`iInter_eq_iInter_finset'` for a version that works for `ι : Sort*`. -/
-theorem iInter_eq_iInter_finset (s : ι → Set α) : ⋂ i, s i = ⋂ t : Finset ι, ⋂ i ∈ t, s i :=
-  iInf_eq_iInf_finset s
-
-/-- Intersection of an indexed family of sets `s : ι → Set α` is equal to the intersection of the
-intersections of finite subfamilies. This version works for `ι : Sort*`. See also
-`iInter_eq_iInter_finset` for a version that assumes `ι : Type*` but avoids `PLift`s in the right
-hand side. -/
-theorem iInter_eq_iInter_finset' (s : ι' → Set α) :
-    ⋂ i, s i = ⋂ t : Finset (PLift ι'), ⋂ i ∈ t, s (PLift.down i) :=
-  iInf_eq_iInf_finset' s
-
-end Set
-
-namespace Finset
-
-section minimal
-
-variable [DecidableEq α] {P : Finset α → Prop} {s : Finset α}
-
-theorem maximal_iff_forall_insert (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
-    Maximal P s ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s) := by
-  simp only [Maximal, and_congr_right_iff]
-  exact fun _ ↦ ⟨fun h x hxs hx ↦ hxs <| h hx (subset_insert _ _) (mem_insert_self x s),
-    fun h t ht hst x hxt ↦ by_contra fun hxs ↦ h x hxs (hP ht (insert_subset hxt hst))⟩
-
-theorem minimal_iff_forall_diff_singleton (hP : ∀ ⦃s t⦄, P t → t ⊆ s → P s) :
-    Minimal P s ↔ P s ∧ ∀ x ∈ s, ¬ P (s.erase x) where
-  mp h := ⟨h.prop, fun x hxs hx ↦ by simpa using h.le_of_le hx (erase_subset _ _) hxs⟩
-  mpr h := ⟨h.1, fun t ht hts x hxs ↦ by_contra fun hxt ↦
-    h.2 x hxs <| hP ht (subset_erase.2 ⟨hts, hxt⟩)⟩
-
-end minimal
-
-/-! ### Interaction with big lattice/set operations -/
-
-section Lattice
-
-theorem iSup_coe [SupSet β] (f : α → β) (s : Finset α) : ⨆ x ∈ (↑s : Set α), f x = ⨆ x ∈ s, f x :=
-  rfl
-
-theorem iInf_coe [InfSet β] (f : α → β) (s : Finset α) : ⨅ x ∈ (↑s : Set α), f x = ⨅ x ∈ s, f x :=
-  rfl
-
-variable [CompleteLattice β]
-
-theorem iSup_singleton (a : α) (s : α → β) : ⨆ x ∈ ({a} : Finset α), s x = s a := by simp
-
-theorem iInf_singleton (a : α) (s : α → β) : ⨅ x ∈ ({a} : Finset α), s x = s a := by simp
-
-theorem iSup_option_toFinset (o : Option α) (f : α → β) : ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x := by
-  simp
-
-theorem iInf_option_toFinset (o : Option α) (f : α → β) : ⨅ x ∈ o.toFinset, f x = ⨅ x ∈ o, f x :=
-  @iSup_option_toFinset _ βᵒᵈ _ _ _
-
-variable [DecidableEq α]
-
-theorem iSup_union {f : α → β} {s t : Finset α} :
-    ⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x := by simp [iSup_or, iSup_sup_eq]
-
-theorem iInf_union {f : α → β} {s t : Finset α} :
-    ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x :=
-  @iSup_union α βᵒᵈ _ _ _ _ _
-
-theorem iSup_insert (a : α) (s : Finset α) (t : α → β) :
-    ⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x := by
-  rw [insert_eq]
-  simp only [iSup_union, Finset.iSup_singleton]
-
-theorem iInf_insert (a : α) (s : Finset α) (t : α → β) :
-    ⨅ x ∈ insert a s, t x = t a ⊓ ⨅ x ∈ s, t x :=
-  @iSup_insert α βᵒᵈ _ _ _ _ _
-
-theorem iSup_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
-    ⨆ x ∈ s.image f, g x = ⨆ y ∈ s, g (f y) := by rw [← iSup_coe, coe_image, iSup_image, iSup_coe]
-
-theorem iInf_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
-    ⨅ x ∈ s.image f, g x = ⨅ y ∈ s, g (f y) := by rw [← iInf_coe, coe_image, iInf_image, iInf_coe]
-
-theorem iSup_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
-    ⨆ i ∈ insert x t, Function.update f x s i = s ⊔ ⨆ i ∈ t, f i := by
-  simp only [Finset.iSup_insert, update_same]
-  rcongr (i hi); apply update_noteq; rintro rfl; exact hx hi
-
-theorem iInf_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
-    ⨅ i ∈ insert x t, update f x s i = s ⊓ ⨅ i ∈ t, f i :=
-  @iSup_insert_update α βᵒᵈ _ _ _ _ f _ hx
-
-theorem iSup_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
-    ⨆ y ∈ s.biUnion t, f y = ⨆ (x ∈ s) (y ∈ t x), f y := by simp [@iSup_comm _ α, iSup_and]
-
-theorem iInf_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
-    ⨅ y ∈ s.biUnion t, f y = ⨅ (x ∈ s) (y ∈ t x), f y :=
-  @iSup_biUnion _ βᵒᵈ _ _ _ _ _ _
-
-end Lattice
-
-theorem set_biUnion_coe (s : Finset α) (t : α → Set β) : ⋃ x ∈ (↑s : Set α), t x = ⋃ x ∈ s, t x :=
-  rfl
-
-theorem set_biInter_coe (s : Finset α) (t : α → Set β) : ⋂ x ∈ (↑s : Set α), t x = ⋂ x ∈ s, t x :=
-  rfl
-
-theorem set_biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Finset α), s x = s a :=
-  iSup_singleton a s
-
-theorem set_biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Finset α), s x = s a :=
-  iInf_singleton a s
-
-@[simp]
-theorem set_biUnion_preimage_singleton (f : α → β) (s : Finset β) :
-    ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s :=
-  Set.biUnion_preimage_singleton f s
-
-theorem set_biUnion_option_toFinset (o : Option α) (f : α → Set β) :
-    ⋃ x ∈ o.toFinset, f x = ⋃ x ∈ o, f x :=
-  iSup_option_toFinset o f
-
-theorem set_biInter_option_toFinset (o : Option α) (f : α → Set β) :
-    ⋂ x ∈ o.toFinset, f x = ⋂ x ∈ o, f x :=
-  iInf_option_toFinset o f
-
-theorem subset_set_biUnion_of_mem {s : Finset α} {f : α → Set β} {x : α} (h : x ∈ s) :
-    f x ⊆ ⋃ y ∈ s, f y :=
-  show f x ≤ ⨆ y ∈ s, f y from le_iSup_of_le x <| by simp only [h, iSup_pos, le_refl]
-
-variable [DecidableEq α]
-
-theorem set_biUnion_union (s t : Finset α) (u : α → Set β) :
-    ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
-  iSup_union
-
-theorem set_biInter_inter (s t : Finset α) (u : α → Set β) :
-    ⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
-  iInf_union
-
-theorem set_biUnion_insert (a : α) (s : Finset α) (t : α → Set β) :
-    ⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x :=
-  iSup_insert a s t
-
-theorem set_biInter_insert (a : α) (s : Finset α) (t : α → Set β) :
-    ⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x :=
-  iInf_insert a s t
-
-theorem set_biUnion_finset_image {f : γ → α} {g : α → Set β} {s : Finset γ} :
-    ⋃ x ∈ s.image f, g x = ⋃ y ∈ s, g (f y) :=
-  iSup_finset_image
-
-theorem set_biInter_finset_image {f : γ → α} {g : α → Set β} {s : Finset γ} :
-    ⋂ x ∈ s.image f, g x = ⋂ y ∈ s, g (f y) :=
-  iInf_finset_image
-
-theorem set_biUnion_insert_update {x : α} {t : Finset α} (f : α → Set β) {s : Set β} (hx : x ∉ t) :
-    ⋃ i ∈ insert x t, @update _ _ _ f x s i = s ∪ ⋃ i ∈ t, f i :=
-  iSup_insert_update f hx
-
-theorem set_biInter_insert_update {x : α} {t : Finset α} (f : α → Set β) {s : Set β} (hx : x ∉ t) :
-    ⋂ i ∈ insert x t, @update _ _ _ f x s i = s ∩ ⋂ i ∈ t, f i :=
-  iInf_insert_update f hx
-
-theorem set_biUnion_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → Set β) :
-    ⋃ y ∈ s.biUnion t, f y = ⋃ (x ∈ s) (y ∈ t x), f y :=
-  iSup_biUnion s t f
-
-theorem set_biInter_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → Set β) :
-    ⋂ y ∈ s.biUnion t, f y = ⋂ (x ∈ s) (y ∈ t x), f y :=
-  iInf_biUnion s t f
-
-end Finset
diff --git a/Mathlib/Data/Finset/Max.lean b/Mathlib/Data/Finset/Max.lean
new file mode 100644
index 0000000000000..a972df5dc3ddc
--- /dev/null
+++ b/Mathlib/Data/Finset/Max.lean
@@ -0,0 +1,543 @@
+/-
+Copyright (c) 2018 Mario Carneiro. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro
+-/
+import Mathlib.Data.Finset.Lattice
+
+/-!
+# Maximum and minimum of finite sets
+-/
+
+assert_not_exists OrderedCommMonoid
+assert_not_exists MonoidWithZero
+
+open Function Multiset OrderDual
+
+variable {F α β γ ι κ : Type*}
+
+namespace Finset
+
+/-! ### max and min of finite sets -/
+
+section MaxMin
+
+variable [LinearOrder α]
+
+/-- Let `s` be a finset in a linear order. Then `s.max` is the maximum of `s` if `s` is not empty,
+and `⊥` otherwise. It belongs to `WithBot α`. If you want to get an element of `α`, see
+`s.max'`. -/
+protected def max (s : Finset α) : WithBot α :=
+  sup s (↑)
+
+theorem max_eq_sup_coe {s : Finset α} : s.max = s.sup (↑) :=
+  rfl
+
+theorem max_eq_sup_withBot (s : Finset α) : s.max = sup s (↑) :=
+  rfl
+
+@[simp]
+theorem max_empty : (∅ : Finset α).max = ⊥ :=
+  rfl
+
+@[simp]
+theorem max_insert {a : α} {s : Finset α} : (insert a s).max = max ↑a s.max :=
+  fold_insert_idem
+
+@[simp]
+theorem max_singleton {a : α} : Finset.max {a} = (a : WithBot α) := by
+  rw [← insert_emptyc_eq]
+  exact max_insert
+
+theorem max_of_mem {s : Finset α} {a : α} (h : a ∈ s) : ∃ b : α, s.max = b := by
+  obtain ⟨b, h, _⟩ := le_sup (α := WithBot α) h _ rfl
+  exact ⟨b, h⟩
+
+theorem max_of_nonempty {s : Finset α} (h : s.Nonempty) : ∃ a : α, s.max = a :=
+  let ⟨_, h⟩ := h
+  max_of_mem h
+
+theorem max_eq_bot {s : Finset α} : s.max = ⊥ ↔ s = ∅ :=
+  ⟨fun h ↦ s.eq_empty_or_nonempty.elim id fun H ↦ by
+      obtain ⟨a, ha⟩ := max_of_nonempty H
+      rw [h] at ha; cases ha; , -- the `;` is needed since the `cases` syntax allows `cases a, b`
+    fun h ↦ h.symm ▸ max_empty⟩
+
+theorem mem_of_max {s : Finset α} : ∀ {a : α}, s.max = a → a ∈ s := by
+  induction' s using Finset.induction_on with b s _ ih
+  · intro _ H; cases H
+  · intro a h
+    by_cases p : b = a
+    · induction p
+      exact mem_insert_self b s
+    · cases' max_choice (↑b) s.max with q q <;> rw [max_insert, q] at h
+      · cases h
+        cases p rfl
+      · exact mem_insert_of_mem (ih h)
+
+theorem le_max {a : α} {s : Finset α} (as : a ∈ s) : ↑a ≤ s.max :=
+  le_sup as
+
+theorem not_mem_of_max_lt_coe {a : α} {s : Finset α} (h : s.max < a) : a ∉ s :=
+  mt le_max h.not_le
+
+theorem le_max_of_eq {s : Finset α} {a b : α} (h₁ : a ∈ s) (h₂ : s.max = b) : a ≤ b :=
+  WithBot.coe_le_coe.mp <| (le_max h₁).trans h₂.le
+
+theorem not_mem_of_max_lt {s : Finset α} {a b : α} (h₁ : b < a) (h₂ : s.max = ↑b) : a ∉ s :=
+  Finset.not_mem_of_max_lt_coe <| h₂.trans_lt <| WithBot.coe_lt_coe.mpr h₁
+
+@[gcongr]
+theorem max_mono {s t : Finset α} (st : s ⊆ t) : s.max ≤ t.max :=
+  sup_mono st
+
+protected theorem max_le {M : WithBot α} {s : Finset α} (st : ∀ a ∈ s, (a : WithBot α) ≤ M) :
+    s.max ≤ M :=
+  Finset.sup_le st
+
+@[simp]
+protected lemma max_le_iff {m : WithBot α} {s : Finset α} : s.max ≤ m ↔ ∀ a ∈ s, a ≤ m :=
+  Finset.sup_le_iff
+
+@[simp]
+protected lemma max_eq_top [OrderTop α] {s : Finset α} : s.max = ⊤ ↔ ⊤ ∈ s :=
+  Finset.sup_eq_top_iff.trans <| by simp
+
+/-- Let `s` be a finset in a linear order. Then `s.min` is the minimum of `s` if `s` is not empty,
+and `⊤` otherwise. It belongs to `WithTop α`. If you want to get an element of `α`, see
+`s.min'`. -/
+protected def min (s : Finset α) : WithTop α :=
+  inf s (↑)
+
+theorem min_eq_inf_withTop (s : Finset α) : s.min = inf s (↑) :=
+  rfl
+
+@[simp]
+theorem min_empty : (∅ : Finset α).min = ⊤ :=
+  rfl
+
+@[simp]
+theorem min_insert {a : α} {s : Finset α} : (insert a s).min = min (↑a) s.min :=
+  fold_insert_idem
+
+@[simp]
+theorem min_singleton {a : α} : Finset.min {a} = (a : WithTop α) := by
+  rw [← insert_emptyc_eq]
+  exact min_insert
+
+theorem min_of_mem {s : Finset α} {a : α} (h : a ∈ s) : ∃ b : α, s.min = b := by
+  obtain ⟨b, h, _⟩ := inf_le (α := WithTop α) h _ rfl
+  exact ⟨b, h⟩
+
+theorem min_of_nonempty {s : Finset α} (h : s.Nonempty) : ∃ a : α, s.min = a :=
+  let ⟨_, h⟩ := h
+  min_of_mem h
+
+@[simp]
+theorem min_eq_top {s : Finset α} : s.min = ⊤ ↔ s = ∅ := by
+  simp [Finset.min, eq_empty_iff_forall_not_mem]
+
+theorem mem_of_min {s : Finset α} : ∀ {a : α}, s.min = a → a ∈ s :=
+  @mem_of_max αᵒᵈ _ s
+
+theorem min_le {a : α} {s : Finset α} (as : a ∈ s) : s.min ≤ a :=
+  inf_le as
+
+theorem not_mem_of_coe_lt_min {a : α} {s : Finset α} (h : ↑a < s.min) : a ∉ s :=
+  mt min_le h.not_le
+
+theorem min_le_of_eq {s : Finset α} {a b : α} (h₁ : b ∈ s) (h₂ : s.min = a) : a ≤ b :=
+  WithTop.coe_le_coe.mp <| h₂.ge.trans (min_le h₁)
+
+theorem not_mem_of_lt_min {s : Finset α} {a b : α} (h₁ : a < b) (h₂ : s.min = ↑b) : a ∉ s :=
+  Finset.not_mem_of_coe_lt_min <| (WithTop.coe_lt_coe.mpr h₁).trans_eq h₂.symm
+
+@[gcongr]
+theorem min_mono {s t : Finset α} (st : s ⊆ t) : t.min ≤ s.min :=
+  inf_mono st
+
+protected theorem le_min {m : WithTop α} {s : Finset α} (st : ∀ a : α, a ∈ s → m ≤ a) : m ≤ s.min :=
+  Finset.le_inf st
+
+@[simp]
+protected theorem le_min_iff {m : WithTop α} {s : Finset α} : m ≤ s.min ↔ ∀ a ∈ s, m ≤ a :=
+  Finset.le_inf_iff
+
+@[simp]
+protected theorem min_eq_bot [OrderBot α] {s : Finset α} : s.min = ⊥ ↔ ⊥ ∈ s :=
+  Finset.max_eq_top (α := αᵒᵈ)
+
+/-- Given a nonempty finset `s` in a linear order `α`, then `s.min' H` is its minimum, as an
+element of `α`, where `H` is a proof of nonemptiness. Without this assumption, use instead `s.min`,
+taking values in `WithTop α`. -/
+def min' (s : Finset α) (H : s.Nonempty) : α :=
+  inf' s H id
+
+/-- Given a nonempty finset `s` in a linear order `α`, then `s.max' H` is its maximum, as an
+element of `α`, where `H` is a proof of nonemptiness. Without this assumption, use instead `s.max`,
+taking values in `WithBot α`. -/
+def max' (s : Finset α) (H : s.Nonempty) : α :=
+  sup' s H id
+
+variable (s : Finset α) (H : s.Nonempty) {x : α}
+
+theorem min'_mem : s.min' H ∈ s :=
+  mem_of_min <| by simp only [Finset.min, min', id_eq, coe_inf']; rfl
+
+theorem min'_le (x) (H2 : x ∈ s) : s.min' ⟨x, H2⟩ ≤ x :=
+  min_le_of_eq H2 (WithTop.coe_untop _ _).symm
+
+theorem le_min' (x) (H2 : ∀ y ∈ s, x ≤ y) : x ≤ s.min' H :=
+  H2 _ <| min'_mem _ _
+
+theorem isLeast_min' : IsLeast (↑s) (s.min' H) :=
+  ⟨min'_mem _ _, min'_le _⟩
+
+@[simp]
+theorem le_min'_iff {x} : x ≤ s.min' H ↔ ∀ y ∈ s, x ≤ y :=
+  le_isGLB_iff (isLeast_min' s H).isGLB
+
+/-- `{a}.min' _` is `a`. -/
+@[simp]
+theorem min'_singleton (a : α) : ({a} : Finset α).min' (singleton_nonempty _) = a := by simp [min']
+
+theorem max'_mem : s.max' H ∈ s :=
+  mem_of_max <| by simp only [max', Finset.max, id_eq, coe_sup']; rfl
+
+theorem le_max' (x) (H2 : x ∈ s) : x ≤ s.max' ⟨x, H2⟩ :=
+  le_max_of_eq H2 (WithBot.coe_unbot _ _).symm
+
+theorem max'_le (x) (H2 : ∀ y ∈ s, y ≤ x) : s.max' H ≤ x :=
+  H2 _ <| max'_mem _ _
+
+theorem isGreatest_max' : IsGreatest (↑s) (s.max' H) :=
+  ⟨max'_mem _ _, le_max' _⟩
+
+@[simp]
+theorem max'_le_iff {x} : s.max' H ≤ x ↔ ∀ y ∈ s, y ≤ x :=
+  isLUB_le_iff (isGreatest_max' s H).isLUB
+
+@[simp]
+theorem max'_lt_iff {x} : s.max' H < x ↔ ∀ y ∈ s, y < x :=
+  ⟨fun Hlt y hy => (s.le_max' y hy).trans_lt Hlt, fun H => H _ <| s.max'_mem _⟩
+
+@[simp]
+theorem lt_min'_iff : x < s.min' H ↔ ∀ y ∈ s, x < y :=
+  @max'_lt_iff αᵒᵈ _ _ H _
+
+theorem max'_eq_sup' : s.max' H = s.sup' H id := rfl
+
+theorem min'_eq_inf' : s.min' H = s.inf' H id := rfl
+
+/-- `{a}.max' _` is `a`. -/
+@[simp]
+theorem max'_singleton (a : α) : ({a} : Finset α).max' (singleton_nonempty _) = a := by simp [max']
+
+theorem min'_lt_max' {i j} (H1 : i ∈ s) (H2 : j ∈ s) (H3 : i ≠ j) :
+    s.min' ⟨i, H1⟩ < s.max' ⟨i, H1⟩ :=
+  isGLB_lt_isLUB_of_ne (s.isLeast_min' _).isGLB (s.isGreatest_max' _).isLUB H1 H2 H3
+
+/-- If there's more than 1 element, the min' is less than the max'. An alternate version of
+`min'_lt_max'` which is sometimes more convenient.
+-/
+theorem min'_lt_max'_of_card (h₂ : 1 < card s) :
+    s.min' (Finset.card_pos.1 <| by omega) < s.max' (Finset.card_pos.1 <| by omega) := by
+  rcases one_lt_card.1 h₂ with ⟨a, ha, b, hb, hab⟩
+  exact s.min'_lt_max' ha hb hab
+
+theorem map_ofDual_min (s : Finset αᵒᵈ) : s.min.map ofDual = (s.image ofDual).max := by
+  rw [max_eq_sup_withBot, sup_image]
+  exact congr_fun Option.map_id _
+
+theorem map_ofDual_max (s : Finset αᵒᵈ) : s.max.map ofDual = (s.image ofDual).min := by
+  rw [min_eq_inf_withTop, inf_image]
+  exact congr_fun Option.map_id _
+
+theorem map_toDual_min (s : Finset α) : s.min.map toDual = (s.image toDual).max := by
+  rw [max_eq_sup_withBot, sup_image]
+  exact congr_fun Option.map_id _
+
+theorem map_toDual_max (s : Finset α) : s.max.map toDual = (s.image toDual).min := by
+  rw [min_eq_inf_withTop, inf_image]
+  exact congr_fun Option.map_id _
+
+-- Porting note: new proofs without `convert` for the next four theorems.
+
+theorem ofDual_min' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
+    ofDual (min' s hs) = max' (s.image ofDual) (hs.image _) := by
+  rw [← WithBot.coe_eq_coe]
+  simp only [min'_eq_inf', id_eq, ofDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',
+    sup_image]
+  rfl
+
+theorem ofDual_max' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
+    ofDual (max' s hs) = min' (s.image ofDual) (hs.image _) := by
+  rw [← WithTop.coe_eq_coe]
+  simp only [max'_eq_sup', id_eq, ofDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',
+    inf_image]
+  rfl
+
+theorem toDual_min' {s : Finset α} (hs : s.Nonempty) :
+    toDual (min' s hs) = max' (s.image toDual) (hs.image _) := by
+  rw [← WithBot.coe_eq_coe]
+  simp only [min'_eq_inf', id_eq, toDual_inf', Function.comp_apply, coe_sup', max'_eq_sup',
+    sup_image]
+  rfl
+
+theorem toDual_max' {s : Finset α} (hs : s.Nonempty) :
+    toDual (max' s hs) = min' (s.image toDual) (hs.image _) := by
+  rw [← WithTop.coe_eq_coe]
+  simp only [max'_eq_sup', id_eq, toDual_sup', Function.comp_apply, coe_inf', min'_eq_inf',
+    inf_image]
+  rfl
+
+theorem max'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
+    s.max' H ≤ t.max' (H.mono hst) :=
+  le_max' _ _ (hst (s.max'_mem H))
+
+theorem min'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
+    t.min' (H.mono hst) ≤ s.min' H :=
+  min'_le _ _ (hst (s.min'_mem H))
+
+theorem max'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
+    (insert a s).max' (s.insert_nonempty a) = max (s.max' H) a :=
+  (isGreatest_max' _ _).unique <| by
+    rw [coe_insert, max_comm]
+    exact (isGreatest_max' _ _).insert _
+
+theorem min'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
+    (insert a s).min' (s.insert_nonempty a) = min (s.min' H) a :=
+  (isLeast_min' _ _).unique <| by
+    rw [coe_insert, min_comm]
+    exact (isLeast_min' _ _).insert _
+
+theorem lt_max'_of_mem_erase_max' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.max' H)) :
+    a < s.max' H :=
+  lt_of_le_of_ne (le_max' _ _ (mem_of_mem_erase ha)) <| ne_of_mem_of_not_mem ha <| not_mem_erase _ _
+
+theorem min'_lt_of_mem_erase_min' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.min' H)) :
+    s.min' H < a :=
+  @lt_max'_of_mem_erase_max' αᵒᵈ _ s H _ a ha
+
+/-- To rewrite from right to left, use `Monotone.map_finset_max'`. -/
+@[simp]
+theorem max'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
+    (h : (s.image f).Nonempty) : (s.image f).max' h = f (s.max' h.of_image) := by
+  simp only [max', sup'_image]
+  exact .symm <| comp_sup'_eq_sup'_comp _ _ fun _ _ ↦ hf.map_max
+
+/-- A version of `Finset.max'_image` with LHS and RHS reversed.
+Also, this version assumes that `s` is nonempty, not its image. -/
+lemma _root_.Monotone.map_finset_max' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
+    (h : s.Nonempty) : f (s.max' h) = (s.image f).max' (h.image f) :=
+  .symm <| max'_image hf ..
+
+/-- To rewrite from right to left, use `Monotone.map_finset_min'`. -/
+@[simp]
+theorem min'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
+    (h : (s.image f).Nonempty) : (s.image f).min' h = f (s.min' h.of_image) := by
+  simp only [min', inf'_image]
+  exact .symm <| comp_inf'_eq_inf'_comp _ _ fun _ _ ↦ hf.map_min
+
+/-- A version of `Finset.min'_image` with LHS and RHS reversed.
+Also, this version assumes that `s` is nonempty, not its image. -/
+lemma _root_.Monotone.map_finset_min' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
+    (h : s.Nonempty) : f (s.min' h) = (s.image f).min' (h.image f) :=
+  .symm <| min'_image hf ..
+
+theorem coe_max' {s : Finset α} (hs : s.Nonempty) : ↑(s.max' hs) = s.max :=
+  coe_sup' hs id
+
+theorem coe_min' {s : Finset α} (hs : s.Nonempty) : ↑(s.min' hs) = s.min :=
+  coe_inf' hs id
+
+theorem max_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
+    s.max ∈ (s.image (↑) : Finset (WithBot α)) :=
+  mem_image.2 ⟨max' s hs, max'_mem _ _, coe_max' hs⟩
+
+theorem min_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
+    s.min ∈ (s.image (↑) : Finset (WithTop α)) :=
+  mem_image.2 ⟨min' s hs, min'_mem _ _, coe_min' hs⟩
+
+theorem max_mem_insert_bot_image_coe (s : Finset α) :
+    s.max ∈ (insert ⊥ (s.image (↑)) : Finset (WithBot α)) :=
+  mem_insert.2 <| s.eq_empty_or_nonempty.imp max_eq_bot.2 max_mem_image_coe
+
+theorem min_mem_insert_top_image_coe (s : Finset α) :
+    s.min ∈ (insert ⊤ (s.image (↑)) : Finset (WithTop α)) :=
+  mem_insert.2 <| s.eq_empty_or_nonempty.imp min_eq_top.2 min_mem_image_coe
+
+theorem max'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).max' s0 ≠ x :=
+  ne_of_mem_erase (max'_mem _ s0)
+
+theorem min'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).min' s0 ≠ x :=
+  ne_of_mem_erase (min'_mem _ s0)
+
+theorem max_erase_ne_self {s : Finset α} : (s.erase x).max ≠ x := by
+  by_cases s0 : (s.erase x).Nonempty
+  · refine ne_of_eq_of_ne (coe_max' s0).symm ?_
+    exact WithBot.coe_eq_coe.not.mpr (max'_erase_ne_self _)
+  · rw [not_nonempty_iff_eq_empty.mp s0, max_empty]
+    exact WithBot.bot_ne_coe
+
+theorem min_erase_ne_self {s : Finset α} : (s.erase x).min ≠ x := by
+  -- Porting note: old proof `convert @max_erase_ne_self αᵒᵈ _ _ _`
+  convert @max_erase_ne_self αᵒᵈ _ (toDual x) (s.map toDual.toEmbedding) using 1
+  apply congr_arg -- Porting note: forces unfolding to see `Finset.min` is `Finset.max`
+  congr!
+  ext; simp only [mem_map_equiv]; exact Iff.rfl
+
+theorem exists_next_right {x : α} {s : Finset α} (h : ∃ y ∈ s, x < y) :
+    ∃ y ∈ s, x < y ∧ ∀ z ∈ s, x < z → y ≤ z :=
+  have Hne : (s.filter (x < ·)).Nonempty := h.imp fun y hy => mem_filter.2 (by simpa)
+  have aux := mem_filter.1 (min'_mem _ Hne)
+  ⟨min' _ Hne, aux.1, by simp, fun z hzs hz => min'_le _ _ <| mem_filter.2 ⟨hzs, by simpa⟩⟩
+
+theorem exists_next_left {x : α} {s : Finset α} (h : ∃ y ∈ s, y < x) :
+    ∃ y ∈ s, y < x ∧ ∀ z ∈ s, z < x → z ≤ y :=
+  @exists_next_right αᵒᵈ _ x s h
+
+/-- If finsets `s` and `t` are interleaved, then `Finset.card s ≤ Finset.card t + 1`. -/
+theorem card_le_of_interleaved {s t : Finset α}
+    (h : ∀ᵉ (x ∈ s) (y ∈ s),
+        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
+    s.card ≤ t.card + 1 := by
+  replace h : ∀ᵉ (x ∈ s) (y ∈ s), x < y → ∃ z ∈ t, x < z ∧ z < y := by
+    intro x hx y hy hxy
+    rcases exists_next_right ⟨y, hy, hxy⟩ with ⟨a, has, hxa, ha⟩
+    rcases h x hx a has hxa fun z hzs hz => hz.2.not_le <| ha _ hzs hz.1 with ⟨b, hbt, hxb, hba⟩
+    exact ⟨b, hbt, hxb, hba.trans_le <| ha _ hy hxy⟩
+  set f : α → WithTop α := fun x => (t.filter fun y => x < y).min
+  have f_mono : StrictMonoOn f s := by
+    intro x hx y hy hxy
+    rcases h x hx y hy hxy with ⟨a, hat, hxa, hay⟩
+    calc
+      f x ≤ a := min_le (mem_filter.2 ⟨hat, by simpa⟩)
+      _ < f y :=
+        (Finset.lt_inf_iff <| WithTop.coe_lt_top a).2 fun b hb =>
+          WithTop.coe_lt_coe.2 <| hay.trans (by simpa using (mem_filter.1 hb).2)
+
+  calc
+    s.card = (s.image f).card := (card_image_of_injOn f_mono.injOn).symm
+    _ ≤ (insert ⊤ (t.image (↑)) : Finset (WithTop α)).card :=
+      card_mono <| image_subset_iff.2 fun x _ =>
+          insert_subset_insert _ (image_subset_image <| filter_subset _ _)
+            (min_mem_insert_top_image_coe _)
+    _ ≤ t.card + 1 := (card_insert_le _ _).trans (Nat.add_le_add_right card_image_le _)
+
+/-- If finsets `s` and `t` are interleaved, then `Finset.card s ≤ Finset.card (t \ s) + 1`. -/
+theorem card_le_diff_of_interleaved {s t : Finset α}
+    (h :
+      ∀ᵉ (x ∈ s) (y ∈ s),
+        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
+    s.card ≤ (t \ s).card + 1 :=
+  card_le_of_interleaved fun x hx y hy hxy hs =>
+    let ⟨z, hzt, hxz, hzy⟩ := h x hx y hy hxy hs
+    ⟨z, mem_sdiff.2 ⟨hzt, fun hzs => hs z hzs ⟨hxz, hzy⟩⟩, hxz, hzy⟩
+
+/-- Induction principle for `Finset`s in a linearly ordered type: a predicate is true on all
+`s : Finset α` provided that:
+
+* it is true on the empty `Finset`,
+* for every `s : Finset α` and an element `a` strictly greater than all elements of `s`, `p s`
+  implies `p (insert a s)`. -/
+@[elab_as_elim]
+theorem induction_on_max [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
+    (step : ∀ a s, (∀ x ∈ s, x < a) → p s → p (insert a s)) : p s := by
+  induction' s using Finset.strongInductionOn with s ihs
+  rcases s.eq_empty_or_nonempty with (rfl | hne)
+  · exact h0
+  · have H : s.max' hne ∈ s := max'_mem s hne
+    rw [← insert_erase H]
+    exact step _ _ (fun x => s.lt_max'_of_mem_erase_max' hne) (ihs _ <| erase_ssubset H)
+
+/-- Induction principle for `Finset`s in a linearly ordered type: a predicate is true on all
+`s : Finset α` provided that:
+
+* it is true on the empty `Finset`,
+* for every `s : Finset α` and an element `a` strictly less than all elements of `s`, `p s`
+  implies `p (insert a s)`. -/
+@[elab_as_elim]
+theorem induction_on_min [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
+    (step : ∀ a s, (∀ x ∈ s, a < x) → p s → p (insert a s)) : p s :=
+  @induction_on_max αᵒᵈ _ _ _ s h0 step
+
+end MaxMin
+
+section MaxMinInductionValue
+
+variable [LinearOrder α] [LinearOrder β]
+
+/-- Induction principle for `Finset`s in any type from which a given function `f` maps to a linearly
+ordered type : a predicate is true on all `s : Finset α` provided that:
+
+* it is true on the empty `Finset`,
+* for every `s : Finset α` and an element `a` such that for elements of `s` denoted by `x` we have
+  `f x ≤ f a`, `p s` implies `p (insert a s)`. -/
+@[elab_as_elim]
+theorem induction_on_max_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
+    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f x ≤ f a) → p s → p (insert a s)) : p s := by
+  induction' s using Finset.strongInductionOn with s ihs
+  rcases (s.image f).eq_empty_or_nonempty with (hne | hne)
+  · simp only [image_eq_empty] at hne
+    simp only [hne, h0]
+  · have H : (s.image f).max' hne ∈ s.image f := max'_mem (s.image f) hne
+    simp only [mem_image, exists_prop] at H
+    rcases H with ⟨a, has, hfa⟩
+    rw [← insert_erase has]
+    refine step _ _ (not_mem_erase a s) (fun x hx => ?_) (ihs _ <| erase_ssubset has)
+    rw [hfa]
+    exact le_max' _ _ (mem_image_of_mem _ <| mem_of_mem_erase hx)
+
+/-- Induction principle for `Finset`s in any type from which a given function `f` maps to a linearly
+ordered type : a predicate is true on all `s : Finset α` provided that:
+
+* it is true on the empty `Finset`,
+* for every `s : Finset α` and an element `a` such that for elements of `s` denoted by `x` we have
+  `f a ≤ f x`, `p s` implies `p (insert a s)`. -/
+@[elab_as_elim]
+theorem induction_on_min_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
+    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f a ≤ f x) → p s → p (insert a s)) : p s :=
+  @induction_on_max_value αᵒᵈ ι _ _ _ _ s h0 step
+
+end MaxMinInductionValue
+
+section ExistsMaxMin
+
+variable [LinearOrder α]
+
+theorem exists_max_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
+    ∃ x ∈ s, ∀ x' ∈ s, f x' ≤ f x := by
+  cases' max_of_nonempty (h.image f) with y hy
+  rcases mem_image.mp (mem_of_max hy) with ⟨x, hx, rfl⟩
+  exact ⟨x, hx, fun x' hx' => le_max_of_eq (mem_image_of_mem f hx') hy⟩
+
+theorem exists_min_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
+    ∃ x ∈ s, ∀ x' ∈ s, f x ≤ f x' :=
+  @exists_max_image αᵒᵈ β _ s f h
+
+end ExistsMaxMin
+
+theorem isGLB_iff_isLeast [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
+    IsGLB (s : Set α) i ↔ IsLeast (↑s) i := by
+  refine ⟨fun his => ?_, IsLeast.isGLB⟩
+  suffices i = min' s hs by
+    rw [this]
+    exact isLeast_min' s hs
+  rw [IsGLB, IsGreatest, mem_lowerBounds, mem_upperBounds] at his
+  exact le_antisymm (his.1 (Finset.min' s hs) (Finset.min'_mem s hs)) (his.2 _ (Finset.min'_le s))
+
+theorem isLUB_iff_isGreatest [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
+    IsLUB (s : Set α) i ↔ IsGreatest (↑s) i :=
+  @isGLB_iff_isLeast αᵒᵈ _ i s hs
+
+theorem isGLB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsGLB (s : Set α) i)
+    (hs : s.Nonempty) : i ∈ s := by
+  rw [← mem_coe]
+  exact ((isGLB_iff_isLeast i s hs).mp his).1
+
+theorem isLUB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsLUB (s : Set α) i)
+    (hs : s.Nonempty) : i ∈ s :=
+  @isGLB_mem αᵒᵈ _ i s his hs
+
+end Finset
diff --git a/Mathlib/Data/Finset/MulAntidiagonal.lean b/Mathlib/Data/Finset/MulAntidiagonal.lean
index 683d02323b1ff..534b98c618c6b 100644
--- a/Mathlib/Data/Finset/MulAntidiagonal.lean
+++ b/Mathlib/Data/Finset/MulAntidiagonal.lean
@@ -3,8 +3,8 @@ Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Yaël Dillies
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.Monoid.Defs
-import Mathlib.Data.Set.Pointwise.Basic
 import Mathlib.Data.Set.MulAntidiagonal
 
 /-! # Multiplication antidiagonal as a `Finset`.
diff --git a/Mathlib/Data/Finset/NAry.lean b/Mathlib/Data/Finset/NAry.lean
index 74ad2d6a47ad7..4db70840b348d 100644
--- a/Mathlib/Data/Finset/NAry.lean
+++ b/Mathlib/Data/Finset/NAry.lean
@@ -94,11 +94,12 @@ theorem image₂_subset_iff_left : image₂ f s t ⊆ u ↔ ∀ a ∈ s, (t.imag
 theorem image₂_subset_iff_right : image₂ f s t ⊆ u ↔ ∀ b ∈ t, (s.image fun a => f a b) ⊆ u := by
   simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_swap α]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem image₂_nonempty_iff : (image₂ f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := by
   rw [← coe_nonempty, coe_image₂]
   exact image2_nonempty_iff
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 theorem Nonempty.image₂ (hs : s.Nonempty) (ht : t.Nonempty) : (image₂ f s t).Nonempty :=
   image₂_nonempty_iff.2 ⟨hs, ht⟩
 
@@ -453,9 +454,9 @@ theorem card_dvd_card_image₂_left (hf : ∀ b ∈ t, Injective fun a => f a b)
 
 /-- If a `Finset` is a subset of the image of two `Set`s under a binary operation,
 then it is a subset of the `Finset.image₂` of two `Finset` subsets of these `Set`s. -/
-theorem subset_image₂ {s : Set α} {t : Set β} (hu : ↑u ⊆ image2 f s t) :
+theorem subset_set_image₂ {s : Set α} {t : Set β} (hu : ↑u ⊆ image2 f s t) :
     ∃ (s' : Finset α) (t' : Finset β), ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ image₂ f s' t' := by
-  rw [← Set.image_prod, subset_image_iff] at hu
+  rw [← Set.image_prod, subset_set_image_iff] at hu
   rcases hu with ⟨u, hu, rfl⟩
   classical
   use u.image Prod.fst, u.image Prod.snd
@@ -463,6 +464,8 @@ theorem subset_image₂ {s : Set α} {t : Set β} (hu : ↑u ⊆ image2 f s t) :
     image_subset_iff]
   exact ⟨fun _ h ↦ (hu h).1, fun _ h ↦ (hu h).2, fun x hx ↦ mem_image₂_of_mem hx hx⟩
 
+@[deprecated (since := "2024-09-22")] alias subset_image₂ := subset_set_image₂
+
 end
 section UnionInter
 
diff --git a/Mathlib/Data/Finset/NatDivisors.lean b/Mathlib/Data/Finset/NatDivisors.lean
index 07fa594f4b6a3..cbc7aa896c2f5 100644
--- a/Mathlib/Data/Finset/NatDivisors.lean
+++ b/Mathlib/Data/Finset/NatDivisors.lean
@@ -3,8 +3,8 @@ Copyright (c) 2023 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Yury Kudryashov
 -/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
 import Mathlib.NumberTheory.Divisors
-import Mathlib.Data.Finset.Pointwise.Basic
 
 /-!
 #  `Nat.divisors` as a multiplicative homomorpism
diff --git a/Mathlib/Data/Finset/NoncommProd.lean b/Mathlib/Data/Finset/NoncommProd.lean
index 16af175533226..6074576260a47 100644
--- a/Mathlib/Data/Finset/NoncommProd.lean
+++ b/Mathlib/Data/Finset/NoncommProd.lean
@@ -39,16 +39,16 @@ namespace Multiset
 on all elements `x ∈ s`. -/
 def noncommFoldr (s : Multiset α)
     (comm : { x | x ∈ s }.Pairwise fun x y => ∀ b, f x (f y b) = f y (f x b)) (b : β) : β :=
-  s.attach.foldr (f ∘ Subtype.val)
-    (fun ⟨_, hx⟩ ⟨_, hy⟩ =>
+  letI : LeftCommutative (α := { x // x ∈ s }) (f ∘ Subtype.val) :=
+    ⟨fun ⟨_, hx⟩ ⟨_, hy⟩ =>
       haveI : IsRefl α fun x y => ∀ b, f x (f y b) = f y (f x b) := ⟨fun _ _ => rfl⟩
-      comm.of_refl hx hy)
-    b
+      comm.of_refl hx hy⟩
+  s.attach.foldr (f ∘ Subtype.val) b
 
 @[simp]
 theorem noncommFoldr_coe (l : List α) (comm) (b : β) :
     noncommFoldr f (l : Multiset α) comm b = l.foldr f b := by
-  simp only [noncommFoldr, coe_foldr, coe_attach, List.attach, List.attachWith, Function.comp]
+  simp only [noncommFoldr, coe_foldr, coe_attach, List.attach, List.attachWith, Function.comp_def]
   rw [← List.foldr_map]
   simp [List.map_pmap]
 
@@ -61,8 +61,8 @@ theorem noncommFoldr_cons (s : Multiset α) (a : α) (h h') (b : β) :
   induction s using Quotient.inductionOn
   simp
 
-theorem noncommFoldr_eq_foldr (s : Multiset α) (h : LeftCommutative f) (b : β) :
-    noncommFoldr f s (fun x _ y _ _ => h x y) b = foldr f h b s := by
+theorem noncommFoldr_eq_foldr (s : Multiset α) [h : LeftCommutative f] (b : β) :
+    noncommFoldr f s (fun x _ y _ _ => h.left_comm x y) b = foldr f b s := by
   induction s using Quotient.inductionOn
   simp
 
diff --git a/Mathlib/Data/Finset/Pi.lean b/Mathlib/Data/Finset/Pi.lean
index 4415777515d4a..9751986c4519c 100644
--- a/Mathlib/Data/Finset/Pi.lean
+++ b/Mathlib/Data/Finset/Pi.lean
@@ -85,10 +85,17 @@ theorem Pi.cons_injective {a : α} {b : δ a} {s : Finset α} (hs : a ∉ s) :
 theorem pi_empty {t : ∀ a : α, Finset (β a)} : pi (∅ : Finset α) t = singleton (Pi.empty β) :=
   rfl
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma pi_nonempty : (s.pi t).Nonempty ↔ ∀ a ∈ s, (t a).Nonempty := by
   simp [Finset.Nonempty, Classical.skolem]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, pi_nonempty_of_forall_nonempty⟩ := pi_nonempty
+
+@[simp]
+lemma pi_eq_empty : s.pi t = ∅ ↔ ∃ a ∈ s, t a = ∅ := by
+  simp [← not_nonempty_iff_eq_empty]
+
 @[simp]
 theorem pi_insert [∀ a, DecidableEq (β a)] {s : Finset α} {t : ∀ a : α, Finset (β a)} {a : α}
     (ha : a ∉ s) : pi (insert a s) t = (t a).biUnion fun b => (pi s t).image (Pi.cons s a b) := by
@@ -146,5 +153,30 @@ def piDiag (s : Finset α) (ι : Type*) [DecidableEq (ι → α)] : Finset (ι 
 @[simp] lemma card_piDiag (s : Finset α) (ι : Type*) [DecidableEq (ι → α)] [Nonempty ι] :
     (s.piDiag ι).card = s.card := by rw [piDiag, card_image_of_injective _ const_injective]
 
+/-! ### Restriction -/
+
+variable {π : ι → Type*}
+
+/-- Restrict domain of a function `f` to a finite set `s`. -/
+@[simp]
+def restrict (s : Finset ι) (f : (i : ι) → π i) : (i : s) → π i := fun x ↦ f x
+
+theorem restrict_def (s : Finset ι) : s.restrict (π := π) = fun f x ↦ f x := rfl
+
+/-- If a function `f` is restricted to a finite set `t`, and `s ⊆ t`,
+this is the restriction to `s`. -/
+@[simp]
+def restrict₂ {s t : Finset ι} (hst : s ⊆ t) (f : (i : t) → π i) : (i : s) → π i :=
+  fun x ↦ f ⟨x.1, hst x.2⟩
+
+theorem restrict₂_def {s t : Finset ι} (hst : s ⊆ t) :
+    restrict₂ (π := π) hst = fun f x ↦ f ⟨x.1, hst x.2⟩ := rfl
+
+theorem restrict₂_comp_restrict {s t : Finset ι} (hst : s ⊆ t) :
+    (restrict₂ (π := π) hst) ∘ t.restrict = s.restrict := rfl
+
+theorem restrict₂_comp_restrict₂ {s t u : Finset ι} (hst : s ⊆ t) (htu : t ⊆ u) :
+    (restrict₂ (π := π) hst) ∘ (restrict₂ htu) = restrict₂ (hst.trans htu) := rfl
+
 end Pi
 end Finset
diff --git a/Mathlib/Data/Finset/PiInduction.lean b/Mathlib/Data/Finset/PiInduction.lean
index d040789ab2d0d..14801945c227c 100644
--- a/Mathlib/Data/Finset/PiInduction.lean
+++ b/Mathlib/Data/Finset/PiInduction.lean
@@ -3,6 +3,7 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
+import Mathlib.Data.Finset.Max
 import Mathlib.Data.Finset.Sigma
 import Mathlib.Data.Fintype.Card
 
@@ -65,7 +66,7 @@ maps provided that it is true on `fun _ ↦ ∅` and for any function `g : ∀ i
 `i : ι`, and `x ∉ g i`, `p g` implies `p (update g i (insert x (g i)))`.
 
 See also `Finset.induction_on_pi_max` and `Finset.induction_on_pi_min` for specialized versions
-that require `∀ i, LinearOrder (α i)`.  -/
+that require `∀ i, LinearOrder (α i)`. -/
 theorem induction_on_pi {p : (∀ i, Finset (α i)) → Prop} (f : ∀ i, Finset (α i)) (h0 : p fun _ ↦ ∅)
     (step : ∀ (g : ∀ i, Finset (α i)) (i : ι), ∀ x ∉ g i, p g → p (update g i (insert x (g i)))) :
     p f :=
diff --git a/Mathlib/Data/Finset/Powerset.lean b/Mathlib/Data/Finset/Powerset.lean
index ca7a34c9e0cbb..6e8275f105d31 100644
--- a/Mathlib/Data/Finset/Powerset.lean
+++ b/Mathlib/Data/Finset/Powerset.lean
@@ -234,10 +234,13 @@ theorem powersetCard_succ_insert [DecidableEq α] {x : α} {s : Finset α} (h :
   have : x ∉ t := fun H => h (ht H)
   simp [card_insert_of_not_mem this, Nat.succ_inj']
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma powersetCard_nonempty : (powersetCard n s).Nonempty ↔ n ≤ s.card := by
   aesop (add simp [Finset.Nonempty, exists_subset_card_eq, card_le_card])
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, powersetCard_nonempty_of_le⟩ := powersetCard_nonempty
+
 @[simp]
 theorem powersetCard_self (s : Finset α) : powersetCard s.card s = {s} := by
   ext
diff --git a/Mathlib/Data/Finset/Preimage.lean b/Mathlib/Data/Finset/Preimage.lean
index d5c3ba82dd978..657acee065919 100644
--- a/Mathlib/Data/Finset/Preimage.lean
+++ b/Mathlib/Data/Finset/Preimage.lean
@@ -21,7 +21,7 @@ namespace Finset
 
 section Preimage
 
-/-- Preimage of `s : Finset β` under a map `f` injective on `f ⁻¹' s` as a `Finset`.  -/
+/-- Preimage of `s : Finset β` under a map `f` injective on `f ⁻¹' s` as a `Finset`. -/
 noncomputable def preimage (s : Finset β) (f : α → β) (hf : Set.InjOn f (f ⁻¹' ↑s)) : Finset α :=
   (s.finite_toSet.preimage hf).toFinset
 
@@ -80,6 +80,10 @@ theorem map_subset_iff_subset_preimage {f : α ↪ β} {s : Finset α} {t : Fins
     s.map f ⊆ t ↔ s ⊆ t.preimage f f.injective.injOn := by
   classical rw [map_eq_image, image_subset_iff_subset_preimage]
 
+lemma card_preimage (s : Finset β) (f : α → β) (hf) [DecidablePred (· ∈ Set.range f)] :
+    (s.preimage f hf).card = {x ∈ s | x ∈ Set.range f}.card :=
+  card_nbij f (by simp) (by simpa) (fun b hb ↦ by aesop)
+
 theorem image_preimage [DecidableEq β] (f : α → β) (s : Finset β) [∀ x, Decidable (x ∈ Set.range f)]
     (hf : Set.InjOn f (f ⁻¹' ↑s)) : image f (preimage s f hf) = s.filter fun x => x ∈ Set.range f :=
   Finset.coe_inj.1 <| by
@@ -96,7 +100,7 @@ theorem preimage_subset {f : α ↪ β} {s : Finset β} {t : Finset α} (hs : s
 theorem subset_map_iff {f : α ↪ β} {s : Finset β} {t : Finset α} :
     s ⊆ t.map f ↔ ∃ u ⊆ t, s = u.map f := by
   classical
-  simp_rw [← coe_subset, coe_map, subset_image_iff, map_eq_image, eq_comm]
+  simp_rw [map_eq_image, subset_image_iff, eq_comm]
 
 theorem sigma_preimage_mk {β : α → Type*} [DecidableEq α] (s : Finset (Σa, β a)) (t : Finset α) :
     (t.sigma fun a => s.preimage (Sigma.mk a) sigma_mk_injective.injOn) =
@@ -114,5 +118,13 @@ theorem sigma_image_fst_preimage_mk {β : α → Type*} [DecidableEq α] (s : Fi
       s :=
   s.sigma_preimage_mk_of_subset (Subset.refl _)
 
+@[simp] lemma preimage_inl (s : Finset (α ⊕ β)) :
+    s.preimage Sum.inl Sum.inl_injective.injOn = s.toLeft := by
+  ext x; simp
+
+@[simp] lemma preimage_inr (s : Finset (α ⊕ β)) :
+    s.preimage Sum.inr Sum.inr_injective.injOn = s.toRight := by
+  ext x; simp
+
 end Preimage
 end Finset
diff --git a/Mathlib/Data/Finset/Prod.lean b/Mathlib/Data/Finset/Prod.lean
index b0304921a947b..5af6d692ec6e5 100644
--- a/Mathlib/Data/Finset/Prod.lean
+++ b/Mathlib/Data/Finset/Prod.lean
@@ -168,6 +168,7 @@ theorem empty_product (t : Finset β) : (∅ : Finset α) ×ˢ t = ∅ :=
 theorem product_empty (s : Finset α) : s ×ˢ (∅ : Finset β) = ∅ :=
   eq_empty_of_forall_not_mem fun _ h => not_mem_empty _ (Finset.mem_product.1 h).2
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 theorem Nonempty.product (hs : s.Nonempty) (ht : t.Nonempty) : (s ×ˢ t).Nonempty :=
   let ⟨x, hx⟩ := hs
   let ⟨y, hy⟩ := ht
@@ -181,7 +182,7 @@ theorem Nonempty.snd (h : (s ×ˢ t).Nonempty) : t.Nonempty :=
   let ⟨xy, hxy⟩ := h
   ⟨xy.2, (mem_product.1 hxy).2⟩
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem nonempty_product : (s ×ˢ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   ⟨fun h => ⟨h.fst, h.snd⟩, fun h => h.1.product h.2⟩
 
diff --git a/Mathlib/Data/Finset/SMulAntidiagonal.lean b/Mathlib/Data/Finset/SMulAntidiagonal.lean
index 86b0c882b6255..53d0c959253ec 100644
--- a/Mathlib/Data/Finset/SMulAntidiagonal.lean
+++ b/Mathlib/Data/Finset/SMulAntidiagonal.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Scott Carnahan. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Carnahan
 -/
-import Mathlib.Data.Set.Pointwise.SMul
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Data.Set.SMulAntidiagonal
 
 /-!
diff --git a/Mathlib/Data/Finset/Sigma.lean b/Mathlib/Data/Finset/Sigma.lean
index 5be327d9e74de..9be90160317cf 100644
--- a/Mathlib/Data/Finset/Sigma.lean
+++ b/Mathlib/Data/Finset/Sigma.lean
@@ -5,6 +5,7 @@ Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
 -/
 import Mathlib.Data.Finset.Lattice
 import Mathlib.Data.Set.Sigma
+import Mathlib.Order.CompleteLattice.Finset
 
 /-!
 # Finite sets in a sigma type
@@ -51,9 +52,12 @@ theorem coe_sigma (s : Finset ι) (t : ∀ i, Finset (α i)) :
     (s.sigma t : Set (Σ i, α i)) = (s : Set ι).sigma fun i ↦ (t i : Set (α i)) :=
   Set.ext fun _ => mem_sigma
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem sigma_nonempty : (s.sigma t).Nonempty ↔ ∃ i ∈ s, (t i).Nonempty := by simp [Finset.Nonempty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.sigma_nonempty_of_exists_nonempty⟩ := sigma_nonempty
+
 @[simp]
 theorem sigma_eq_empty : s.sigma t = ∅ ↔ ∀ i ∈ s, t i = ∅ := by
   simp only [← not_nonempty_iff_eq_empty, sigma_nonempty, not_exists, not_and]
diff --git a/Mathlib/Data/Finset/Sort.lean b/Mathlib/Data/Finset/Sort.lean
index a8cf2ef2b5057..361e4f3721a7f 100644
--- a/Mathlib/Data/Finset/Sort.lean
+++ b/Mathlib/Data/Finset/Sort.lean
@@ -6,7 +6,7 @@ Authors: Mario Carneiro
 import Mathlib.Order.RelIso.Set
 import Mathlib.Data.Multiset.Sort
 import Mathlib.Data.List.NodupEquivFin
-import Mathlib.Data.Finset.Lattice
+import Mathlib.Data.Finset.Max
 import Mathlib.Data.Fintype.Card
 
 /-!
@@ -32,6 +32,10 @@ variable (r : α → α → Prop) [DecidableRel r] [IsTrans α r] [IsAntisymm α
 def sort (s : Finset α) : List α :=
   Multiset.sort r s.1
 
+@[simp]
+theorem sort_val (s : Finset α) : Multiset.sort r s.val = sort r s :=
+  rfl
+
 @[simp]
 theorem sort_sorted (s : Finset α) : List.Sorted r (sort r s) :=
   Multiset.sort_sorted _ _
@@ -64,11 +68,27 @@ theorem sort_empty : sort r ∅ = [] :=
 theorem sort_singleton (a : α) : sort r {a} = [a] :=
   Multiset.sort_singleton r a
 
+theorem sort_cons {a : α} {s : Finset α} (h₁ : ∀ b ∈ s, r a b) (h₂ : a ∉ s) :
+    sort r (cons a s h₂) = a :: sort r s := by
+  rw [sort, cons_val, Multiset.sort_cons r a _ h₁, sort_val]
+
+theorem sort_insert [DecidableEq α] {a : α} {s : Finset α} (h₁ : ∀ b ∈ s, r a b) (h₂ : a ∉ s) :
+    sort r (insert a s) = a :: sort r s := by
+  rw [← cons_eq_insert _ _ h₂, sort_cons r h₁]
+
 open scoped List in
 theorem sort_perm_toList (s : Finset α) : sort r s ~ s.toList := by
   rw [← Multiset.coe_eq_coe]
   simp only [coe_toList, sort_eq]
 
+theorem _root_.List.toFinset_sort [DecidableEq α] {l : List α} (hl : l.Nodup) :
+    sort r l.toFinset = l ↔ l.Sorted r := by
+  refine ⟨?_, List.eq_of_perm_of_sorted ((sort_perm_toList r _).trans (List.toFinset_toList hl))
+    (sort_sorted r _)⟩
+  intro h
+  rw [← h]
+  exact sort_sorted r _
+
 end sort
 
 section SortLinearOrder
@@ -188,8 +208,8 @@ theorem orderEmbOfFin_singleton (a : α) (i : Fin 1) :
 the increasing bijection `orderEmbOfFin s h`. -/
 theorem orderEmbOfFin_unique {s : Finset α} {k : ℕ} (h : s.card = k) {f : Fin k → α}
     (hfs : ∀ x, f x ∈ s) (hmono : StrictMono f) : f = s.orderEmbOfFin h := by
-  apply Fin.strictMono_unique hmono (s.orderEmbOfFin h).strictMono
-  rw [range_orderEmbOfFin, ← Set.image_univ, ← coe_univ, ← coe_image, coe_inj]
+  rw [← hmono.range_inj (s.orderEmbOfFin h).strictMono, range_orderEmbOfFin, ← Set.image_univ,
+    ← coe_univ, ← coe_image, coe_inj]
   refine eq_of_subset_of_card_le (fun x hx => ?_) ?_
   · rcases mem_image.1 hx with ⟨x, _, rfl⟩
     exact hfs x
diff --git a/Mathlib/Data/Finset/Sum.lean b/Mathlib/Data/Finset/Sum.lean
index f9863829696b5..678aaccf6ad61 100644
--- a/Mathlib/Data/Finset/Sum.lean
+++ b/Mathlib/Data/Finset/Sum.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yaël Dillies
+Authors: Yaël Dillies, Bhavik Mehta
 -/
 import Mathlib.Data.Multiset.Sum
 import Mathlib.Data.Finset.Card
@@ -15,6 +15,8 @@ the `Finset.sum` operation which computes the additive sum.
 ## Main declarations
 
 * `Finset.disjSum`: `s.disjSum t` is the disjoint sum of `s` and `t`.
+* `Finset.toLeft`: Given a finset of elements `α ⊕ β`, extracts all the elements of the form `α`.
+* `Finset.toRight`: Given a finset of elements `α ⊕ β`, extracts all the elements of the form `β`.
 -/
 
 
@@ -94,4 +96,109 @@ theorem disj_sum_strictMono_right (s : Finset α) :
     StrictMono (s.disjSum : Finset β → Finset (α ⊕ β)) := fun _ _ =>
   disjSum_ssubset_disjSum_of_subset_of_ssubset Subset.rfl
 
+@[simp] lemma disjSum_inj {α β : Type*} {s₁ s₂ : Finset α} {t₁ t₂ : Finset β} :
+    s₁.disjSum t₁ = s₂.disjSum t₂ ↔ s₁ = s₂ ∧ t₁ = t₂ := by
+  simp [Finset.ext_iff]
+
+lemma Injective2_disjSum {α β : Type*} : Function.Injective2 (@disjSum α β) :=
+  fun _ _ _ _ => by simp [Finset.ext_iff]
+
+/--
+Given a finset of elements `α ⊕ β`, extract all the elements of the form `α`. This
+forms a quasi-inverse to `disjSum`, in that it recovers its left input.
+
+See also `List.partitionMap`.
+-/
+def toLeft (s : Finset (α ⊕ β)) : Finset α :=
+  s.disjiUnion (Sum.elim singleton (fun _ => ∅)) <| by
+    simp [Set.PairwiseDisjoint, Set.Pairwise, Function.onFun, eq_comm]
+
+/--
+Given a finset of elements `α ⊕ β`, extract all the elements of the form `β`. This
+forms a quasi-inverse to `disjSum`, in that it recovers its right input.
+
+See also `List.partitionMap`.
+-/
+def toRight (s : Finset (α ⊕ β)) : Finset β :=
+  s.disjiUnion (Sum.elim (fun _ => ∅) singleton) <| by
+    simp [Set.PairwiseDisjoint, Set.Pairwise, Function.onFun, eq_comm]
+
+variable {u v : Finset (α ⊕ β)}
+
+@[simp] lemma mem_toLeft {x : α} : x ∈ u.toLeft ↔ inl x ∈ u := by
+  simp [toLeft]
+
+@[simp] lemma mem_toRight {x : β} : x ∈ u.toRight ↔ inr x ∈ u := by
+  simp [toRight]
+
+@[gcongr]
+lemma toLeft_subset_toLeft : u ⊆ v → u.toLeft ⊆ v.toLeft :=
+  fun h _ => by simpa only [mem_toLeft] using @h _
+
+@[gcongr]
+lemma toRight_subset_toRight : u ⊆ v → u.toRight ⊆ v.toRight :=
+  fun h _ => by simpa only [mem_toRight] using @h _
+
+lemma toLeft_monotone : Monotone (@toLeft α β) := fun _ _ => toLeft_subset_toLeft
+lemma toRight_monotone : Monotone (@toRight α β) := fun _ _ => toRight_subset_toRight
+
+lemma toLeft_disjSum_toRight : u.toLeft.disjSum u.toRight = u := by
+  ext (x | x) <;> simp
+
+lemma card_toLeft_add_card_toRight : u.toLeft.card + u.toRight.card = u.card := by
+  rw [← card_disjSum, toLeft_disjSum_toRight]
+
+lemma card_toLeft_le : u.toLeft.card ≤ u.card :=
+  (Nat.le_add_right _ _).trans_eq card_toLeft_add_card_toRight
+
+lemma card_toRight_le : u.toRight.card ≤ u.card :=
+  (Nat.le_add_left _ _).trans_eq card_toLeft_add_card_toRight
+
+@[simp] lemma toLeft_disjSum : (s.disjSum t).toLeft = s := by ext x; simp
+
+@[simp] lemma toRight_disjSum : (s.disjSum t).toRight = t := by ext x; simp
+
+lemma disjSum_eq_iff : s.disjSum t = u ↔ s = u.toLeft ∧ t = u.toRight :=
+  ⟨fun h => by simp [← h], fun h => by simp [h, toLeft_disjSum_toRight]⟩
+
+lemma eq_disjSum_iff : u = s.disjSum t ↔ u.toLeft = s ∧ u.toRight = t :=
+  ⟨fun h => by simp [h], fun h => by simp [← h, toLeft_disjSum_toRight]⟩
+
+@[simp] lemma toLeft_map_sumComm : (u.map (Equiv.sumComm _ _).toEmbedding).toLeft = u.toRight := by
+  ext x; simp
+
+@[simp] lemma toRight_map_sumComm : (u.map (Equiv.sumComm _ _).toEmbedding).toRight = u.toLeft := by
+  ext x; simp
+
+@[simp] lemma toLeft_cons_inl (ha) :
+    (cons (inl a) u ha).toLeft = cons a u.toLeft (by simpa) := by ext y; simp
+@[simp] lemma toLeft_cons_inr (hb) :
+    (cons (inr b) u hb).toLeft = u.toLeft := by ext y; simp
+@[simp] lemma toRight_cons_inl (ha) :
+    (cons (inl a) u ha).toRight = u.toRight := by ext y; simp
+@[simp] lemma toRight_cons_inr (hb) :
+    (cons (inr b) u hb).toRight = cons b u.toRight (by simpa) := by ext y; simp
+
+variable [DecidableEq α] [DecidableEq β]
+
+lemma toLeft_image_swap : (u.image Sum.swap).toLeft = u.toRight := by
+  ext x; simp
+
+lemma toRight_image_swap : (u.image Sum.swap).toRight = u.toLeft := by
+  ext x; simp
+
+@[simp] lemma toLeft_insert_inl : (insert (inl a) u).toLeft = insert a u.toLeft := by ext y; simp
+@[simp] lemma toLeft_insert_inr : (insert (inr b) u).toLeft = u.toLeft := by ext y; simp
+@[simp] lemma toRight_insert_inl : (insert (inl a) u).toRight = u.toRight := by ext y; simp
+@[simp] lemma toRight_insert_inr : (insert (inr b) u).toRight = insert b u.toRight := by ext y; simp
+
+lemma toLeft_inter : (u ∩ v).toLeft = u.toLeft ∩ v.toLeft := by ext x; simp
+lemma toRight_inter : (u ∩ v).toRight = u.toRight ∩ v.toRight := by ext x; simp
+
+lemma toLeft_union : (u ∪ v).toLeft = u.toLeft ∪ v.toLeft := by ext x; simp
+lemma toRight_union : (u ∪ v).toRight = u.toRight ∪ v.toRight := by ext x; simp
+
+lemma toLeft_sdiff : (u \ v).toLeft = u.toLeft \ v.toLeft := by ext x; simp
+lemma toRight_sdiff : (u \ v).toRight = u.toRight \ v.toRight := by ext x; simp
+
 end Finset
diff --git a/Mathlib/Data/Finset/Sups.lean b/Mathlib/Data/Finset/Sups.lean
index 31bed739a4441..5188385616bb7 100644
--- a/Mathlib/Data/Finset/Sups.lean
+++ b/Mathlib/Data/Finset/Sups.lean
@@ -99,10 +99,11 @@ theorem forall_sups_iff {p : α → Prop} : (∀ c ∈ s ⊻ t, p c) ↔ ∀ a 
 theorem sups_subset_iff : s ⊻ t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a ⊔ b ∈ u :=
   image₂_subset_iff
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem sups_nonempty : (s ⊻ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected theorem Nonempty.sups : s.Nonempty → t.Nonempty → (s ⊻ t).Nonempty :=
   Nonempty.image₂
 
@@ -145,7 +146,7 @@ theorem sups_inter_subset_right : s ⊻ (t₁ ∩ t₂) ⊆ s ⊻ t₁ ∩ s ⊻
 
 theorem subset_sups {s t : Set α} :
     ↑u ⊆ s ⊻ t → ∃ s' t' : Finset α, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' ⊻ t' :=
-  subset_image₂
+  subset_set_image₂
 
 lemma image_sups (f : F) (s t : Finset α) : image f (s ⊻ t) = image f s ⊻ image f t :=
   image_image₂_distrib <| map_sup f
@@ -245,10 +246,11 @@ theorem forall_infs_iff {p : α → Prop} : (∀ c ∈ s ⊼ t, p c) ↔ ∀ a 
 theorem infs_subset_iff : s ⊼ t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a ⊓ b ∈ u :=
   image₂_subset_iff
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem infs_nonempty : (s ⊼ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
   image₂_nonempty_iff
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected theorem Nonempty.infs : s.Nonempty → t.Nonempty → (s ⊼ t).Nonempty :=
   Nonempty.image₂
 
@@ -291,7 +293,7 @@ theorem infs_inter_subset_right : s ⊼ (t₁ ∩ t₂) ⊆ s ⊼ t₁ ∩ s ⊼
 
 theorem subset_infs {s t : Set α} :
     ↑u ⊆ s ⊼ t → ∃ s' t' : Finset α, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' ⊼ t' :=
-  subset_image₂
+  subset_set_image₂
 
 lemma image_infs (f : F) (s t : Finset α) : image f (s ⊼ t) = image f s ⊼ image f t :=
   image_image₂_distrib <| map_inf f
@@ -480,6 +482,8 @@ theorem disjSups_comm : s ○ t = t ○ s := by
     rw [sup_comm] at hs
     exact ⟨b, hb, a, ha, hd, hs⟩
 
+instance : @Std.Commutative (Finset α) (· ○ ·) := ⟨disjSups_comm⟩
+
 end DisjSups
 
 open FinsetFamily
@@ -490,12 +494,14 @@ variable [DecidableEq α]
 variable [DistribLattice α] [OrderBot α] [@DecidableRel α Disjoint] (s t u v : Finset α)
 
 theorem disjSups_assoc : ∀ s t u : Finset α, s ○ t ○ u = s ○ (t ○ u) := by
-  refine associative_of_commutative_of_le disjSups_comm ?_
+  refine (associative_of_commutative_of_le inferInstance ?_).assoc
   simp only [le_eq_subset, disjSups_subset_iff, mem_disjSups]
   rintro s t u _ ⟨a, ha, b, hb, hab, rfl⟩ c hc habc
   rw [disjoint_sup_left] at habc
   exact ⟨a, ha, _, ⟨b, hb, c, hc, habc.2, rfl⟩, hab.sup_right habc.1, (sup_assoc ..).symm⟩
 
+instance : @Std.Associative (Finset α) (· ○ ·) := ⟨disjSups_assoc⟩
+
 theorem disjSups_left_comm : s ○ (t ○ u) = t ○ (s ○ u) := by
   simp_rw [← disjSups_assoc, disjSups_comm s]
 
@@ -551,9 +557,10 @@ lemma forall_mem_diffs {p : α → Prop} : (∀ c ∈ s \\ t, p c) ↔ ∀ a ∈
 
 @[simp] lemma diffs_subset_iff : s \\ t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a \ b ∈ u := image₂_subset_iff
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma diffs_nonempty : (s \\ t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := image₂_nonempty_iff
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected lemma Nonempty.diffs : s.Nonempty → t.Nonempty → (s \\ t).Nonempty := Nonempty.image₂
 
 lemma Nonempty.of_diffs_left : (s \\ t).Nonempty → s.Nonempty := Nonempty.of_image₂_left
@@ -575,7 +582,7 @@ lemma diffs_inter_subset_right : s \\ (t₁ ∩ t₂) ⊆ s \\ t₁ ∩ s \\ t
 
 lemma subset_diffs {s t : Set α} :
     ↑u ⊆ Set.image2 (· \ ·) s t → ∃ s' t' : Finset α, ↑s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' \\ t' :=
-  subset_image₂
+  subset_set_image₂
 
 variable (s t u)
 
@@ -625,10 +632,11 @@ lemma exists_compls_iff {p : α → Prop} : (∃ a ∈ sᶜˢ, p a) ↔ ∃ a 
 
 lemma compls_subset_iff : sᶜˢ ⊆ t ↔ s ⊆ tᶜˢ := by rw [← compls_subset_compls, compls_compls]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma compls_nonempty : sᶜˢ.Nonempty ↔ s.Nonempty := map_nonempty
 
 protected alias ⟨Nonempty.of_compls, Nonempty.compls⟩ := compls_nonempty
+attribute [aesop safe apply (rule_sets := [finsetNonempty])] Nonempty.compls
 
 @[simp] lemma compls_empty : (∅ : Finset α)ᶜˢ = ∅ := map_empty _
 @[simp] lemma compls_eq_empty : sᶜˢ = ∅ ↔ s = ∅ := map_eq_empty
diff --git a/Mathlib/Data/Finset/Sym.lean b/Mathlib/Data/Finset/Sym.lean
index 951449e44c9db..c591056cea7ff 100644
--- a/Mathlib/Data/Finset/Sym.lean
+++ b/Mathlib/Data/Finset/Sym.lean
@@ -109,11 +109,12 @@ theorem sym2_empty : (∅ : Finset α).sym2 = ∅ := rfl
 theorem sym2_eq_empty : s.sym2 = ∅ ↔ s = ∅ := by
   rw [← val_eq_zero, sym2_val, Multiset.sym2_eq_zero_iff, val_eq_zero]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem sym2_nonempty : s.sym2.Nonempty ↔ s.Nonempty := by
   rw [← not_iff_not]
   simp_rw [not_nonempty_iff_eq_empty, sym2_eq_empty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
 protected alias ⟨_, Nonempty.sym2⟩ := sym2_nonempty
 
 @[simp]
diff --git a/Mathlib/Data/Finset/Union.lean b/Mathlib/Data/Finset/Union.lean
index b54499d3e58d9..3df81fb5d2be1 100644
--- a/Mathlib/Data/Finset/Union.lean
+++ b/Mathlib/Data/Finset/Union.lean
@@ -52,7 +52,7 @@ lemma disjiUnion_val (s : Finset α) (t : α → Finset β) (h) :
 
 @[simp, norm_cast]
 lemma coe_disjiUnion {h} : (s.disjiUnion t h : Set β) = ⋃ x ∈ (s : Set α), t x := by
-  simp [Set.ext_iff, mem_disjiUnion, Set.mem_iUnion, iff_self_iff, mem_coe, imp_true_iff]
+  simp [Set.ext_iff, mem_disjiUnion, Set.mem_iUnion, mem_coe, imp_true_iff]
 
 @[simp] lemma disjiUnion_cons (a : α) (s : Finset α) (ha : a ∉ s) (f : α → Finset β) (H) :
     disjiUnion (cons a s ha) f H =
@@ -119,7 +119,7 @@ protected def biUnion (s : Finset α) (t : α → Finset β) : Finset β :=
 
 @[simp, norm_cast]
 lemma coe_biUnion : (s.biUnion t : Set β) = ⋃ x ∈ (s : Set α), t x := by
-  simp [Set.ext_iff, mem_biUnion, Set.mem_iUnion, iff_self_iff, mem_coe, imp_true_iff]
+  simp [Set.ext_iff, mem_biUnion, Set.mem_iUnion, mem_coe, imp_true_iff]
 
 @[simp]
 lemma biUnion_insert [DecidableEq α] {a : α} : (insert a s).biUnion t = t a ∪ s.biUnion t :=
diff --git a/Mathlib/Data/Finset/Update.lean b/Mathlib/Data/Finset/Update.lean
index a0a7aa87054bf..d4dfb055c18ea 100644
--- a/Mathlib/Data/Finset/Update.lean
+++ b/Mathlib/Data/Finset/Update.lean
@@ -56,8 +56,7 @@ theorem updateFinset_updateFinset {s t : Finset ι} (hst : Disjoint s t)
   set e := Equiv.Finset.union s t hst
   congr with i
   by_cases his : i ∈ s <;> by_cases hit : i ∈ t <;>
-    simp only [updateFinset, his, hit, dif_pos, dif_neg, Finset.mem_union, true_or_iff,
-      false_or_iff, not_false_iff]
+    simp only [updateFinset, his, hit, dif_pos, dif_neg, Finset.mem_union, false_or, not_false_iff]
   · exfalso; exact Finset.disjoint_left.mp hst his hit
   · exact piCongrLeft_sum_inl (fun b : ↥(s ∪ t) => π b) e y z ⟨i, his⟩ |>.symm
   · exact piCongrLeft_sum_inr (fun b : ↥(s ∪ t) => π b) e y z ⟨i, hit⟩ |>.symm
diff --git a/Mathlib/Data/Finsupp/Basic.lean b/Mathlib/Data/Finsupp/Basic.lean
index 3b6f3678beba3..a9e39269baa78 100644
--- a/Mathlib/Data/Finsupp/Basic.lean
+++ b/Mathlib/Data/Finsupp/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Scott Morrison
+Authors: Johannes Hölzl, Kim Morrison
 -/
 import Mathlib.Algebra.BigOperators.Finsupp
 import Mathlib.Algebra.Group.Action.Basic
@@ -85,7 +85,8 @@ theorem not_mem_graph_snd_zero (a : α) (f : α →₀ M) : (a, (0 : M)) ∉ f.g
 
 @[simp]
 theorem image_fst_graph [DecidableEq α] (f : α →₀ M) : f.graph.image Prod.fst = f.support := by
-  classical simp only [graph, map_eq_image, image_image, Embedding.coeFn_mk, (· ∘ ·), image_id']
+  classical
+  simp only [graph, map_eq_image, image_image, Embedding.coeFn_mk, Function.comp_def, image_id']
 
 theorem graph_injective (α M) [Zero M] : Injective (@graph α M _) := by
   intro f g h
@@ -621,7 +622,7 @@ theorem sum_comapDomain [Zero M] [AddCommMonoid N] (f : α → β) (l : β →
 theorem eq_zero_of_comapDomain_eq_zero [AddCommMonoid M] (f : α → β) (l : β →₀ M)
     (hf : Set.BijOn f (f ⁻¹' ↑l.support) ↑l.support) : comapDomain f l hf.injOn = 0 → l = 0 := by
   rw [← support_eq_empty, ← support_eq_empty, comapDomain]
-  simp only [Finset.ext_iff, Finset.not_mem_empty, iff_false_iff, mem_preimage]
+  simp only [Finset.ext_iff, Finset.not_mem_empty, iff_false, mem_preimage]
   intro h a ha
   cases' hf.2.2 ha with b hb
   exact h b (hb.2.symm ▸ ha)
@@ -1702,3 +1703,5 @@ theorem sigmaFinsuppAddEquivPiFinsupp_apply {α : Type*} {ιs : η → Type*} [A
 end Sigma
 
 end Finsupp
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/Data/Finsupp/Defs.lean b/Mathlib/Data/Finsupp/Defs.lean
index 1f5d59d93695e..11717314f657f 100644
--- a/Mathlib/Data/Finsupp/Defs.lean
+++ b/Mathlib/Data/Finsupp/Defs.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Scott Morrison
+Authors: Johannes Hölzl, Kim Morrison
 -/
 import Mathlib.Algebra.Group.Indicator
 import Mathlib.Algebra.Group.Submonoid.Basic
@@ -23,7 +23,7 @@ Functions with finite support are used (at least) in the following parts of the
 
 * the linear combination of a family of vectors `v i` with coefficients `f i` (as used, e.g., to
   define linearly independent family `LinearIndependent`) is defined as a map
-  `Finsupp.total : (ι → M) → (ι →₀ R) →ₗ[R] M`.
+  `Finsupp.linearCombination : (ι → M) → (ι →₀ R) →ₗ[R] M`.
 
 Some other constructions are naturally equivalent to `α →₀ M` with some `α` and `M` but are defined
 in a different way in the library:
@@ -756,7 +756,7 @@ def embDomain (f : α ↪ β) (v : α →₀ M) : β →₀ M where
   mem_support_toFun a₂ := by
     dsimp
     split_ifs with h
-    · simp only [h, true_iff_iff, Ne]
+    · simp only [h, true_iff, Ne]
       rw [← not_mem_support_iff, not_not]
       classical apply Finset.choose_mem
     · simp only [h, Ne, ne_self_iff_false, not_true_eq_false]
@@ -910,6 +910,20 @@ theorem support_add_eq [DecidableEq α] {g₁ g₂ : α →₀ M} (h : Disjoint
 theorem single_add (a : α) (b₁ b₂ : M) : single a (b₁ + b₂) = single a b₁ + single a b₂ :=
   (zipWith_single_single _ _ _ _ _).symm
 
+theorem support_single_add {a : α} {b : M} {f : α →₀ M} (ha : a ∉ f.support) (hb : b ≠ 0) :
+    support (single a b + f) = cons a f.support ha := by
+  classical
+  have H := support_single_ne_zero a hb
+  rw [support_add_eq, H, cons_eq_insert, insert_eq]
+  rwa [H, disjoint_singleton_left]
+
+theorem support_add_single {a : α} {b : M} {f : α →₀ M} (ha : a ∉ f.support) (hb : b ≠ 0) :
+    support (f + single a b) = cons a f.support ha := by
+  classical
+  have H := support_single_ne_zero a hb
+  rw [support_add_eq, H, union_comm, cons_eq_insert, insert_eq]
+  rwa [H, disjoint_singleton_right]
+
 instance instAddZeroClass : AddZeroClass (α →₀ M) :=
   DFunLike.coe_injective.addZeroClass _ coe_zero coe_add
 
@@ -1034,6 +1048,60 @@ theorem induction_linear {p : (α →₀ M) → Prop} (f : α →₀ M) (h0 : p
     (hadd : ∀ f g : α →₀ M, p f → p g → p (f + g)) (hsingle : ∀ a b, p (single a b)) : p f :=
   induction₂ f h0 fun _a _b _f _ _ w => hadd _ _ w (hsingle _ _)
 
+section LinearOrder
+
+variable [LinearOrder α] {p : (α →₀ M) → Prop}
+
+/-- A finitely supported function can be built by adding up `single a b` for increasing `a`.
+
+The theorem `induction_on_max₂` swaps the argument order in the sum. -/
+theorem induction_on_max (f : α →₀ M) (h0 : p 0)
+    (ha : ∀ (a b) (f : α →₀ M), (∀ c ∈ f.support, c < a) → b ≠ 0 → p f → p (single a b + f)) :
+    p f := by
+  suffices ∀ (s) (f : α →₀ M), f.support = s → p f from this _ _ rfl
+  refine fun s => s.induction_on_max (fun f h => ?_) (fun a s hm hf f hs => ?_)
+  · rwa [support_eq_empty.1 h]
+  · have hs' : (erase a f).support = s := by
+      rw [support_erase, hs, erase_insert (fun ha => (hm a ha).false)]
+    rw [← single_add_erase a f]
+    refine ha _ _ _ (fun c hc => hm _ <| hs'.symm ▸ hc) ?_ (hf _ hs')
+    rw [← mem_support_iff, hs]
+    exact mem_insert_self a s
+
+/-- A finitely supported function can be built by adding up `single a b` for decreasing `a`.
+
+The theorem `induction_on_min₂` swaps the argument order in the sum. -/
+theorem induction_on_min (f : α →₀ M) (h0 : p 0)
+    (ha : ∀ (a b) (f : α →₀ M), (∀ c ∈ f.support, a < c) → b ≠ 0 → p f → p (single a b + f)) :
+    p f :=
+  induction_on_max (α := αᵒᵈ) f h0 ha
+
+/-- A finitely supported function can be built by adding up `single a b` for increasing `a`.
+
+The theorem `induction_on_max` swaps the argument order in the sum. -/
+theorem induction_on_max₂ (f : α →₀ M) (h0 : p 0)
+    (ha : ∀ (a b) (f : α →₀ M), (∀ c ∈ f.support, c < a) → b ≠ 0 → p f → p (f + single a b)) :
+    p f := by
+  suffices ∀ (s) (f : α →₀ M), f.support = s → p f from this _ _ rfl
+  refine fun s => s.induction_on_max (fun f h => ?_) (fun a s hm hf f hs => ?_)
+  · rwa [support_eq_empty.1 h]
+  · have hs' : (erase a f).support = s := by
+      rw [support_erase, hs, erase_insert (fun ha => (hm a ha).false)]
+    rw [← erase_add_single a f]
+    refine ha _ _ _ (fun c hc => hm _ <| hs'.symm ▸ hc) ?_ (hf _ hs')
+    rw [← mem_support_iff, hs]
+    exact mem_insert_self a s
+
+/-- A finitely supported function can be built by adding up `single a b` for decreasing `a`.
+
+The theorem `induction_on_min` swaps the argument order in the sum. -/
+theorem induction_on_min₂ (f : α →₀ M) (h0 : p 0)
+    (ha : ∀ (a b) (f : α →₀ M), (∀ c ∈ f.support, a < c) → b ≠ 0 → p f → p (f + single a b)) :
+    p f :=
+  induction_on_max₂ (α := αᵒᵈ) f h0 ha
+
+end LinearOrder
+
 @[simp]
 theorem add_closure_setOf_eq_single :
     AddSubmonoid.closure { f : α →₀ M | ∃ a b, f = single a b } = ⊤ :=
diff --git a/Mathlib/Data/Finsupp/Order.lean b/Mathlib/Data/Finsupp/Order.lean
index c77303cc05d77..2d4dbe742d79c 100644
--- a/Mathlib/Data/Finsupp/Order.lean
+++ b/Mathlib/Data/Finsupp/Order.lean
@@ -3,7 +3,9 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Aaron Anderson
 -/
+import Mathlib.Algebra.Order.BigOperators.Group.Finset
 import Mathlib.Algebra.Order.Module.Defs
+import Mathlib.Algebra.Order.Pi
 import Mathlib.Data.Finsupp.Basic
 
 /-!
@@ -21,7 +23,7 @@ noncomputable section
 
 open Finset
 
-variable {ι α β : Type*}
+variable {ι κ α β : Type*}
 
 namespace Finsupp
 
@@ -32,6 +34,15 @@ section Zero
 
 variable [Zero α]
 
+section OrderedAddCommMonoid
+variable [OrderedAddCommMonoid β] {f : ι →₀ α} {h₁ h₂ : ι → α → β}
+
+@[gcongr]
+lemma sum_le_sum (h : ∀ i ∈ f.support, h₁ i (f i) ≤ h₂ i (f i)) : f.sum h₁ ≤ f.sum h₂ :=
+  Finset.sum_le_sum h
+
+end OrderedAddCommMonoid
+
 section LE
 variable [LE α] {f g : ι →₀ α}
 
@@ -58,7 +69,7 @@ theorem orderEmbeddingToFun_apply {f : ι →₀ α} {i : ι} : orderEmbeddingTo
 end LE
 
 section Preorder
-variable [Preorder α] {f g : ι →₀ α}
+variable [Preorder α] {f g : ι →₀ α} {i : ι} {a b : α}
 
 instance preorder : Preorder (ι →₀ α) :=
   { Finsupp.instLEFinsupp with
@@ -72,6 +83,26 @@ lemma coe_mono : Monotone (Finsupp.toFun : (ι →₀ α) → ι → α) := fun
 
 lemma coe_strictMono : Monotone (Finsupp.toFun : (ι →₀ α) → ι → α) := fun _ _ ↦ id
 
+@[simp] lemma single_le_single : single i a ≤ single i b ↔ a ≤ b := by
+  classical exact Pi.single_le_single
+
+lemma single_mono : Monotone (single i : α → ι →₀ α) := fun _ _ ↦ single_le_single.2
+
+@[gcongr] protected alias ⟨_, GCongr.single_mono⟩ := single_le_single
+
+@[simp] lemma single_nonneg : 0 ≤ single i a ↔ 0 ≤ a := by classical exact Pi.single_nonneg
+@[simp] lemma single_nonpos : single i a ≤ 0 ↔ a ≤ 0 := by classical exact Pi.single_nonpos
+
+variable [OrderedAddCommMonoid β]
+
+lemma sum_le_sum_index [DecidableEq ι] {f₁ f₂ : ι →₀ α} {h : ι → α → β} (hf : f₁ ≤ f₂)
+    (hh : ∀ i ∈ f₁.support ∪ f₂.support, Monotone (h i))
+    (hh₀ : ∀ i ∈ f₁.support ∪ f₂.support, h i 0 = 0) : f₁.sum h ≤ f₂.sum h := by
+  classical
+  rw [sum_of_support_subset _ Finset.subset_union_left _ hh₀,
+    sum_of_support_subset _ Finset.subset_union_right _ hh₀]
+  exact Finset.sum_le_sum fun i hi ↦ hh _ hi <| hf _
+
 end Preorder
 
 instance partialorder [PartialOrder α] : PartialOrder (ι →₀ α) :=
@@ -117,11 +148,24 @@ end Zero
 
 /-! ### Algebraic order structures -/
 
+section OrderedAddCommMonoid
+variable [OrderedAddCommMonoid α] {i : ι} {f : ι → κ} {g g₁ g₂ : ι →₀ α}
 
-instance orderedAddCommMonoid [OrderedAddCommMonoid α] : OrderedAddCommMonoid (ι →₀ α) :=
+instance orderedAddCommMonoid : OrderedAddCommMonoid (ι →₀ α) :=
   { Finsupp.instAddCommMonoid, Finsupp.partialorder with
     add_le_add_left := fun _a _b h c s => add_le_add_left (h s) (c s) }
 
+lemma mapDomain_mono : Monotone (mapDomain f : (ι →₀ α) → (κ →₀ α)) := by
+  classical exact fun g₁ g₂ h ↦ sum_le_sum_index h (fun _ _ ↦ single_mono) (by simp)
+
+@[gcongr] protected lemma GCongr.mapDomain_mono (hg : g₁ ≤ g₂) : g₁.mapDomain f ≤ g₂.mapDomain f :=
+  mapDomain_mono hg
+
+lemma mapDomain_nonneg (hg : 0 ≤ g) : 0 ≤ g.mapDomain f := by simpa using mapDomain_mono hg
+lemma mapDomain_nonpos (hg : g ≤ 0) : g.mapDomain f ≤ 0 := by simpa using mapDomain_mono hg
+
+end OrderedAddCommMonoid
+
 instance orderedCancelAddCommMonoid [OrderedCancelAddCommMonoid α] :
     OrderedCancelAddCommMonoid (ι →₀ α) :=
   { Finsupp.orderedAddCommMonoid with
diff --git a/Mathlib/Data/Finsupp/Pointwise.lean b/Mathlib/Data/Finsupp/Pointwise.lean
index 609f2a8f9b146..059bf76ad5eed 100644
--- a/Mathlib/Data/Finsupp/Pointwise.lean
+++ b/Mathlib/Data/Finsupp/Pointwise.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Module.Defs
 import Mathlib.Algebra.Ring.Pi
diff --git a/Mathlib/Data/Finsupp/Weight.lean b/Mathlib/Data/Finsupp/Weight.lean
index 34223547e511d..ed767ee615ac8 100644
--- a/Mathlib/Data/Finsupp/Weight.lean
+++ b/Mathlib/Data/Finsupp/Weight.lean
@@ -24,7 +24,7 @@ We fix a type `σ` and an `AddCommMonoid M`, as well as a function `w : σ → M
 
 - `Finsupp.weight` of a finitely supported function `f : σ →₀ ℕ`
 with respect to `w`: it is the sum `∑ (f i) • (w i)`.
-It is an `AddMonoidHom` map defined using `Finsupp.total`.
+It is an `AddMonoidHom` map defined using `Finsupp.linearCombination`.
 
 - `Finsupp.le_weight`says that `f s ≤ f.weight w` when `M = ℕ``
 
@@ -72,7 +72,7 @@ variable [AddCommMonoid M]
 /-- The `weight` of the finitely supported function `f : σ →₀ ℕ`
 with respect to `w : σ → M` is the sum `∑(f i)•(w i)`. -/
 noncomputable def weight : (σ →₀ ℕ) →+ M :=
-  (Finsupp.total ℕ w).toAddMonoidHom
+  (Finsupp.linearCombination ℕ w).toAddMonoidHom
 
 @[deprecated weight (since := "2024-07-20")]
 alias _root_.MvPolynomial.weightedDegree := weight
@@ -145,7 +145,7 @@ variable {M : Type*} [CanonicallyOrderedAddCommMonoid M] (w : σ → M)
 
 theorem le_weight_of_ne_zero' {s : σ} {f : σ →₀ ℕ} (hs : f s ≠ 0) :
     w s ≤ weight w f :=
-  le_weight_of_ne_zero w (fun _ ↦ zero_le _) hs
+  le_weight_of_ne_zero (fun _ ↦ zero_le _) hs
 
 /-- If `M` is a `CanonicallyOrderedAddCommMonoid`, then `weight f` is zero iff `f=0. -/
 theorem weight_eq_zero_iff_eq_zero
@@ -157,7 +157,7 @@ theorem weight_eq_zero_iff_eq_zero
     ext s
     simp only [Finsupp.coe_zero, Pi.zero_apply]
     by_contra hs
-    apply NonTorsionWeight.ne_zero w _
+    apply NonTorsionWeight.ne_zero w s
     rw [← nonpos_iff_eq_zero, ← h]
     exact le_weight_of_ne_zero' w hs
   · intro h
diff --git a/Mathlib/Data/Fintype/Basic.lean b/Mathlib/Data/Fintype/Basic.lean
index 93e4e1e02e765..0ff3b973f0141 100644
--- a/Mathlib/Data/Fintype/Basic.lean
+++ b/Mathlib/Data/Fintype/Basic.lean
@@ -5,7 +5,6 @@ Authors: Mario Carneiro
 -/
 import Mathlib.Data.Finset.Image
 import Mathlib.Data.List.FinRange
-import Mathlib.Init.Data.Nat.Lemmas
 
 /-!
 # Finite types
@@ -94,7 +93,7 @@ theorem Nonempty.eq_univ [Subsingleton α] : s.Nonempty → s = univ := by
 theorem univ_nonempty_iff : (univ : Finset α).Nonempty ↔ Nonempty α := by
   rw [← coe_nonempty, coe_univ, Set.nonempty_iff_univ_nonempty]
 
-@[aesop unsafe apply (rule_sets := [finsetNonempty])]
+@[simp, aesop unsafe apply (rule_sets := [finsetNonempty])]
 theorem univ_nonempty [Nonempty α] : (univ : Finset α).Nonempty :=
   univ_nonempty_iff.2 ‹_›
 
@@ -271,7 +270,7 @@ theorem compl_erase : (s.erase a)ᶜ = insert a sᶜ := by
 @[simp]
 theorem compl_insert : (insert a s)ᶜ = sᶜ.erase a := by
   ext
-  simp only [not_or, mem_insert, iff_self_iff, mem_compl, mem_erase]
+  simp only [not_or, mem_insert, mem_compl, mem_erase]
 
 theorem insert_compl_insert (ha : a ∉ s) : insert a (insert a s)ᶜ = sᶜ := by
   simp_rw [compl_insert, insert_erase (mem_compl.2 ha)]
@@ -493,7 +492,7 @@ This function computes by checking all terms `a : α` to find the `f a = b`, so
 -/
 def invOfMemRange : Set.range f → α := fun b =>
   Finset.choose (fun a => f a = b) Finset.univ
-    ((existsUnique_congr (by simp)).mp (hf.exists_unique_of_mem_range b.property))
+    ((existsUnique_congr (by simp)).mp (hf.existsUnique_of_mem_range b.property))
 
 theorem left_inv_of_invOfMemRange (b : Set.range f) : f (hf.invOfMemRange b) = b :=
   (Finset.choose_spec (fun a => f a = b) _ _).right
@@ -593,7 +592,7 @@ namespace Set
 
 variable {s t : Set α}
 
-/-- Construct a finset enumerating a set `s`, given a `Fintype` instance.  -/
+/-- Construct a finset enumerating a set `s`, given a `Fintype` instance. -/
 def toFinset (s : Set α) [Fintype s] : Finset α :=
   (@Finset.univ s _).map <| Function.Embedding.subtype _
 
@@ -622,10 +621,13 @@ def decidableMemOfFintype [DecidableEq α] (s : Set α) [Fintype s] (a) : Decida
 theorem coe_toFinset (s : Set α) [Fintype s] : (↑s.toFinset : Set α) = s :=
   Set.ext fun _ => mem_toFinset
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem toFinset_nonempty {s : Set α} [Fintype s] : s.toFinset.Nonempty ↔ s.Nonempty := by
   rw [← Finset.coe_nonempty, coe_toFinset]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.toFinset_nonempty_of_nonempty⟩ := toFinset_nonempty
+
 @[simp]
 theorem toFinset_inj {s t : Set α} [Fintype s] [Fintype t] : s.toFinset = t.toFinset ↔ s = t :=
   ⟨fun h => by rw [← s.coe_toFinset, h, t.coe_toFinset], fun h => by simp [h]⟩
diff --git a/Mathlib/Data/Fintype/BigOperators.lean b/Mathlib/Data/Fintype/BigOperators.lean
index c1ae6185decf8..b772a62fd0956 100644
--- a/Mathlib/Data/Fintype/BigOperators.lean
+++ b/Mathlib/Data/Fintype/BigOperators.lean
@@ -224,26 +224,31 @@ theorem Fintype.prod_sum_type (f : α₁ ⊕ α₂ → M) :
     ∏ x, f x = (∏ a₁, f (Sum.inl a₁)) * ∏ a₂, f (Sum.inr a₂) :=
   prod_disj_sum _ _ _
 
-@[to_additive (attr := simp) Fintype.sum_prod_type]
-theorem Fintype.prod_prod_type [CommMonoid γ] {f : α₁ × α₂ → γ} :
+/-- The product over a product type equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Fintype.prod_prod_type'`. -/
+@[to_additive Fintype.sum_prod_type "The sum over a product type equals the sum of fiberwise sums.
+For rewriting in the reverse direction, use `Fintype.sum_prod_type'`."]
+theorem Fintype.prod_prod_type [CommMonoid γ] (f : α₁ × α₂ → γ) :
     ∏ x, f x = ∏ x, ∏ y, f (x, y) :=
-  Finset.prod_product
+  Finset.prod_product ..
 
-/-- An uncurried version of `Finset.prod_prod_type`. -/
-@[to_additive Fintype.sum_prod_type' "An uncurried version of `Finset.sum_prod_type`"]
-theorem Fintype.prod_prod_type' [CommMonoid γ] {f : α₁ → α₂ → γ} :
+/-- The product over a product type equals the product of the fiberwise products. For rewriting
+in the reverse direction, use `Fintype.prod_prod_type`. -/
+@[to_additive Fintype.sum_prod_type' "The sum over a product type equals the sum of fiberwise sums.
+For rewriting in the reverse direction, use `Fintype.sum_prod_type`."]
+theorem Fintype.prod_prod_type' [CommMonoid γ] (f : α₁ → α₂ → γ) :
     ∏ x : α₁ × α₂, f x.1 x.2 = ∏ x, ∏ y, f x y :=
-  Finset.prod_product'
+  Finset.prod_product' ..
 
 @[to_additive Fintype.sum_prod_type_right]
-theorem Fintype.prod_prod_type_right [CommMonoid γ] {f : α₁ × α₂ → γ} :
+theorem Fintype.prod_prod_type_right [CommMonoid γ] (f : α₁ × α₂ → γ) :
     ∏ x, f x = ∏ y, ∏ x, f (x, y) :=
-  Finset.prod_product_right
+  Finset.prod_product_right ..
 
 /-- An uncurried version of `Finset.prod_prod_type_right`. -/
 @[to_additive Fintype.sum_prod_type_right' "An uncurried version of `Finset.sum_prod_type_right`"]
-theorem Fintype.prod_prod_type_right' [CommMonoid γ] {f : α₁ → α₂ → γ} :
+theorem Fintype.prod_prod_type_right' [CommMonoid γ] (f : α₁ → α₂ → γ) :
     ∏ x : α₁ × α₂, f x.1 x.2 = ∏ y, ∏ x, f x y :=
-  Finset.prod_product_right'
+  Finset.prod_product_right' ..
 
 end
diff --git a/Mathlib/Data/Fintype/Card.lean b/Mathlib/Data/Fintype/Card.lean
index c2ac99212e5f5..45c0a90562ef5 100644
--- a/Mathlib/Data/Fintype/Card.lean
+++ b/Mathlib/Data/Fintype/Card.lean
@@ -48,8 +48,6 @@ assert_not_exists MulAction
 
 open Function
 
-open Nat
-
 universe u v
 
 variable {α β γ : Type*}
@@ -295,6 +293,10 @@ equality of types, using it should be avoided if possible. -/
 theorem fin_injective : Function.Injective Fin := fun m n h =>
   (Fintype.card_fin m).symm.trans <| (Fintype.card_congr <| Equiv.cast h).trans (Fintype.card_fin n)
 
+theorem Fin.val_eq_val_of_heq {k l : ℕ} {i : Fin k} {j : Fin l} (h : HEq i j) :
+    (i : ℕ) = (j : ℕ) :=
+  (Fin.heq_ext_iff (fin_injective (type_eq_of_heq h))).1 h
+
 /-- A reversed version of `Fin.cast_eq_cast` that is easier to rewrite with. -/
 theorem Fin.cast_eq_cast' {n m : ℕ} (h : Fin n = Fin m) :
     _root_.cast h = Fin.cast (fin_injective h) := by
@@ -542,7 +544,7 @@ theorem one_lt_card_iff : 1 < card α ↔ ∃ a b : α, a ≠ b :=
   one_lt_card_iff_nontrivial.trans nontrivial_iff
 
 nonrec theorem two_lt_card_iff : 2 < card α ↔ ∃ a b c : α, a ≠ b ∧ a ≠ c ∧ b ≠ c := by
-  simp_rw [← Finset.card_univ, two_lt_card_iff, mem_univ, true_and_iff]
+  simp_rw [← Finset.card_univ, two_lt_card_iff, mem_univ, true_and]
 
 theorem card_of_bijective {f : α → β} (hf : Bijective f) : card α = card β :=
   card_congr (Equiv.ofBijective f hf)
@@ -815,7 +817,7 @@ theorem wellFounded_of_trans_of_irrefl (r : α → α → Prop) [IsTrans α r] [
     fun x y hxy =>
     Finset.card_lt_card <| by
       simp only [Finset.lt_iff_ssubset.symm, lt_iff_le_not_le, Finset.le_iff_subset,
-          Finset.subset_iff, mem_filter, true_and_iff, mem_univ, hxy]
+          Finset.subset_iff, mem_filter, true_and, mem_univ, hxy]
       exact
         ⟨fun z hzx => _root_.trans hzx hxy,
           not_forall_of_exists_not ⟨x, Classical.not_imp.2 ⟨hxy, irrefl x⟩⟩⟩
diff --git a/Mathlib/Data/Fintype/Fin.lean b/Mathlib/Data/Fintype/Fin.lean
index bdd198e51a477..9c66b04ae7c46 100644
--- a/Mathlib/Data/Fintype/Fin.lean
+++ b/Mathlib/Data/Fintype/Fin.lean
@@ -37,8 +37,7 @@ theorem Iio_last_eq_map : Iio (Fin.last n) = Finset.univ.map Fin.castSuccEmb :=
 @[simp]
 theorem Ioi_succ (i : Fin n) : Ioi i.succ = (Ioi i).map (Fin.succEmb _) := by
   ext i
-  simp only [mem_filter, mem_Ioi, mem_map, mem_univ, true_and_iff, Function.Embedding.coeFn_mk,
-    exists_true_left]
+  simp only [mem_filter, mem_Ioi, mem_map, mem_univ, Function.Embedding.coeFn_mk, exists_true_left]
   constructor
   · refine cases ?_ ?_ i
     · rintro ⟨⟨⟩⟩
@@ -53,21 +52,20 @@ theorem Iio_castSucc (i : Fin n) : Iio (castSucc i) = (Iio i).map Fin.castSuccEm
   rw [Finset.map_map, Fin.map_valEmbedding_Iio]
   exact (Fin.map_valEmbedding_Iio i).symm
 
-theorem card_filter_univ_succ' (p : Fin (n + 1) → Prop) [DecidablePred p] :
-    (univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p ∘ Fin.succ)).card := by
-  rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]
-  split_ifs <;> simp
-
 theorem card_filter_univ_succ (p : Fin (n + 1) → Prop) [DecidablePred p] :
     (univ.filter p).card =
-    if p 0 then (univ.filter (p ∘ Fin.succ)).card + 1 else (univ.filter (p ∘ Fin.succ)).card :=
-  (card_filter_univ_succ' p).trans (by split_ifs <;> simp [add_comm 1])
+    if p 0 then (univ.filter (p ∘ Fin.succ)).card + 1 else (univ.filter (p ∘ Fin.succ)).card := by
+  rw [Fin.univ_succ, filter_cons, apply_ite Finset.card, card_cons, filter_map, card_map]; rfl
+
+theorem card_filter_univ_succ' (p : Fin (n + 1) → Prop) [DecidablePred p] :
+    (univ.filter p).card = ite (p 0) 1 0 + (univ.filter (p ∘ Fin.succ)).card := by
+  rw [card_filter_univ_succ]; split_ifs <;> simp [add_comm]
 
 theorem card_filter_univ_eq_vector_get_eq_count [DecidableEq α] (a : α) (v : Vector α n) :
     (univ.filter fun i => v.get i = a).card = v.toList.count a := by
   induction' v with n x xs hxs
   · simp
-  · simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp,
+  · simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp_def,
       Vector.get_cons_succ, hxs, List.count_cons, add_comm (ite (x = a) 1 0), beq_iff_eq]
 
 end Fin
diff --git a/Mathlib/Data/Fintype/Lattice.lean b/Mathlib/Data/Fintype/Lattice.lean
index 27835fd7979ce..64865d35bb0c7 100644
--- a/Mathlib/Data/Fintype/Lattice.lean
+++ b/Mathlib/Data/Fintype/Lattice.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 import Mathlib.Data.Fintype.Card
-import Mathlib.Data.Finset.Lattice
+import Mathlib.Data.Finset.Max
 
 /-!
 # Lemmas relating fintypes and order/lattice structure.
diff --git a/Mathlib/Data/Fintype/Order.lean b/Mathlib/Data/Fintype/Order.lean
index 439e738f0439c..6fa3b6c7e32fd 100644
--- a/Mathlib/Data/Fintype/Order.lean
+++ b/Mathlib/Data/Fintype/Order.lean
@@ -156,15 +156,37 @@ end Fintype
 
 /-! ### Properties for PartialOrders -/
 
-lemma Finite.exists_ge_minimal {α} [Finite α] [PartialOrder α] {a : α} {p : α → Prop} (h : p a) :
-    ∃ b, b ≤ a ∧ Minimal p b := by
+section PartialOrder
+
+variable {α : Type*} [PartialOrder α] {a : α} {p : α → Prop}
+
+lemma Finite.exists_minimal_le [Finite α] (h : p a) : ∃ b, b ≤ a ∧ Minimal p b := by
   obtain ⟨b, ⟨hba, hb⟩, hbmin⟩ :=
     Set.Finite.exists_minimal_wrt id {x | x ≤ a ∧ p x} (Set.toFinite _) ⟨a, rfl.le, h⟩
   exact ⟨b, hba, hb, fun x hx hxb ↦ (hbmin x ⟨hxb.trans hba, hx⟩ hxb).le⟩
 
-lemma Finite.exists_le_maximal {α} [Finite α] [PartialOrder α] {a : α} {p : α → Prop} (h : p a) :
-    ∃ b, a ≤ b ∧ Maximal p b :=
-  Finite.exists_ge_minimal (α := αᵒᵈ) h
+@[deprecated (since := "2024-09-23")] alias Finite.exists_ge_minimal := Finite.exists_minimal_le
+
+lemma Finite.exists_le_maximal [Finite α] (h : p a) : ∃ b, a ≤ b ∧ Maximal p b :=
+  Finite.exists_minimal_le (α := αᵒᵈ) h
+
+lemma Finset.exists_minimal_le (s : Finset α) (h : a ∈ s) : ∃ b, b ≤ a ∧ Minimal (· ∈ s) b := by
+  obtain ⟨⟨b, _⟩, lb, minb⟩ := @Finite.exists_minimal_le s _ ⟨a, h⟩ (·.1 ∈ s) _ h
+  use b, lb; rwa [minimal_subtype, inf_idem] at minb
+
+lemma Finset.exists_le_maximal (s : Finset α) (h : a ∈ s) : ∃ b, a ≤ b ∧ Maximal (· ∈ s) b :=
+  s.exists_minimal_le (α := αᵒᵈ) h
+
+lemma Set.Finite.exists_minimal_le {s : Set α} (hs : s.Finite) (h : a ∈ s) :
+    ∃ b, b ≤ a ∧ Minimal (· ∈ s) b := by
+  obtain ⟨b, lb, minb⟩ := hs.toFinset.exists_minimal_le (hs.mem_toFinset.mpr h)
+  use b, lb; simpa using minb
+
+lemma Set.Finite.exists_le_maximal {s : Set α} (hs : s.Finite) (h : a ∈ s) :
+    ∃ b, a ≤ b ∧ Maximal (· ∈ s) b :=
+  hs.exists_minimal_le (α := αᵒᵈ) h
+
+end PartialOrder
 
 /-! ### Concrete instances -/
 
@@ -188,7 +210,7 @@ variable {α : Type*} {r : α → α → Prop} [IsTrans α r] {β γ : Type*} [N
 
 theorem Directed.finite_set_le (D : Directed r f) {s : Set γ} (hs : s.Finite) :
     ∃ z, ∀ i ∈ s, r (f i) (f z) := by
-  convert D.finset_le hs.toFinset; rw [Set.Finite.mem_toFinset]
+  convert D.finset_le hs.toFinset using 3; rw [Set.Finite.mem_toFinset]
 
 theorem Directed.finite_le (D : Directed r f) (g : β → γ) : ∃ z, ∀ i, r (f (g i)) (f z) := by
   classical
diff --git a/Mathlib/Data/Fintype/Perm.lean b/Mathlib/Data/Fintype/Perm.lean
index 4e2bc19dc273d..47dbd625b6d70 100644
--- a/Mathlib/Data/Fintype/Perm.lean
+++ b/Mathlib/Data/Fintype/Perm.lean
@@ -36,7 +36,7 @@ theorem length_permsOfList : ∀ l : List α, length (permsOfList l) = l.length
   | [] => rfl
   | a :: l => by
     rw [length_cons, Nat.factorial_succ]
-    simp only [permsOfList, length_append, length_permsOfList, length_bind, comp,
+    simp only [permsOfList, length_append, length_permsOfList, length_bind, comp_def,
      length_map, map_const', sum_replicate, smul_eq_mul, succ_mul]
     ring
 
@@ -46,7 +46,7 @@ theorem mem_permsOfList_of_mem {l : List α} {f : Perm α} (h : ∀ x, f x ≠ x
   | nil =>
     -- Porting note: applied `not_mem_nil` because it is no longer true definitionally.
     simp only [not_mem_nil] at h
-    exact List.mem_singleton.2 (Equiv.ext fun x => Decidable.by_contradiction <| h x)
+    exact List.mem_singleton.2 (Equiv.ext fun x => Decidable.byContradiction <| h x)
   | cons a l IH =>
   by_cases hfa : f a = a
   · refine mem_append_left _ (IH fun x hx => mem_of_ne_of_mem ?_ (h x hx))
diff --git a/Mathlib/Data/Fintype/Pi.lean b/Mathlib/Data/Fintype/Pi.lean
index a216886258a7e..99a88bda7004a 100644
--- a/Mathlib/Data/Fintype/Pi.lean
+++ b/Mathlib/Data/Fintype/Pi.lean
@@ -49,12 +49,16 @@ theorem piFinset_subset (t₁ t₂ : ∀ a, Finset (δ a)) (h : ∀ a, t₁ a 
     piFinset t₁ ⊆ piFinset t₂ := fun _ hg => mem_piFinset.2 fun a => h a <| mem_piFinset.1 hg a
 
 @[simp]
-theorem piFinset_empty [Nonempty α] : piFinset (fun _ => ∅ : ∀ i, Finset (δ i)) = ∅ :=
-  eq_empty_of_forall_not_mem fun _ => by simp
+theorem piFinset_eq_empty : piFinset s = ∅ ↔ ∃ i, s i = ∅ := by simp [piFinset]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
-lemma piFinset_nonempty : (piFinset s).Nonempty ↔ ∀ a, (s a).Nonempty := by
-  simp [Finset.Nonempty, Classical.skolem]
+@[simp]
+theorem piFinset_empty [Nonempty α] : piFinset (fun _ => ∅ : ∀ i, Finset (δ i)) = ∅ := by simp
+
+@[simp]
+lemma piFinset_nonempty : (piFinset s).Nonempty ↔ ∀ a, (s a).Nonempty := by simp [piFinset]
+
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.piFinset_nonempty_of_forall_nonempty⟩ := piFinset_nonempty
 
 lemma _root_.Finset.Nonempty.piFinset_const {ι : Type*} [Fintype ι] [DecidableEq ι] {s : Finset β}
     (hs : s.Nonempty) : (piFinset fun _ : ι ↦ s).Nonempty := piFinset_nonempty.2 fun _ ↦ hs
diff --git a/Mathlib/Data/Fintype/Prod.lean b/Mathlib/Data/Fintype/Prod.lean
index 0d91984495e7f..2a5dfcdbbbf59 100644
--- a/Mathlib/Data/Fintype/Prod.lean
+++ b/Mathlib/Data/Fintype/Prod.lean
@@ -14,8 +14,6 @@ import Mathlib.Data.Finset.Prod
 
 open Function
 
-open Nat
-
 universe u v
 
 variable {α β γ : Type*}
diff --git a/Mathlib/Data/Fintype/Quotient.lean b/Mathlib/Data/Fintype/Quotient.lean
index 83851ba7b8a46..1675f05ea0a49 100644
--- a/Mathlib/Data/Fintype/Quotient.lean
+++ b/Mathlib/Data/Fintype/Quotient.lean
@@ -63,7 +63,7 @@ def Quotient.finChoice {ι : Type*} [DecidableEq ι] [Fintype ι] {α : ι → T
         (⟦fun (i : ι) (_ : i ∈ a) => Quotient.out (f i)⟧ : Quotient (by infer_instance))
       apply eq_of_heq
       trans (g a)
-      · exact eq_rec_heq (φ := fun l : Multiset ι => @Quotient (∀ i ∈ l, α i) (by infer_instance))
+      · exact eqRec_heq (φ := fun l : Multiset ι => @Quotient (∀ i ∈ l, α i) (by infer_instance))
           (Quotient.sound h) (g a)
       · change HEq (g a) (g b); congr 1; exact Quotient.sound h))
     (fun f => ⟦fun i => f i (Finset.mem_univ _)⟧) (fun a b h => Quotient.sound fun i => by apply h)
diff --git a/Mathlib/Data/FunLike/Basic.lean b/Mathlib/Data/FunLike/Basic.lean
index f23481b549544..3a04435a85ca6 100644
--- a/Mathlib/Data/FunLike/Basic.lean
+++ b/Mathlib/Data/FunLike/Basic.lean
@@ -17,7 +17,7 @@ There is the "D"ependent version `DFunLike` and the non-dependent version `FunLi
 
 A typical type of morphisms should be declared as:
 ```
-structure MyHom (A B : Type*) [MyClass A] [MyClass B] :=
+structure MyHom (A B : Type*) [MyClass A] [MyClass B] where
   (toFun : A → B)
   (map_op' : ∀ (x y : A), toFun (MyClass.op x y) = MyClass.op (toFun x) (toFun y))
 
@@ -79,7 +79,7 @@ The second step is to add instances of your new `MyHomClass` for all types exten
 Typically, you can just declare a new class analogous to `MyHomClass`:
 
 ```
-structure CoolerHom (A B : Type*) [CoolClass A] [CoolClass B] extends MyHom A B :=
+structure CoolerHom (A B : Type*) [CoolClass A] [CoolClass B] extends MyHom A B where
   (map_cool' : toFun CoolClass.cool = CoolClass.cool)
 
 class CoolerHomClass (F : Type*) (A B : outParam Type*) [CoolClass A] [CoolClass B]
diff --git a/Mathlib/Data/FunLike/Embedding.lean b/Mathlib/Data/FunLike/Embedding.lean
index b65a26da5cc05..a8551af72e079 100644
--- a/Mathlib/Data/FunLike/Embedding.lean
+++ b/Mathlib/Data/FunLike/Embedding.lean
@@ -14,7 +14,7 @@ This typeclass is primarily for use by embeddings such as `RelEmbedding`.
 
 A typical type of embeddings should be declared as:
 ```
-structure MyEmbedding (A B : Type*) [MyClass A] [MyClass B] :=
+structure MyEmbedding (A B : Type*) [MyClass A] [MyClass B] where
   (toFun : A → B)
   (injective' : Function.Injective toFun)
   (map_op' : ∀ (x y : A), toFun (MyClass.op x y) = MyClass.op (toFun x) (toFun y))
@@ -58,8 +58,8 @@ Continuing the example above:
 You should extend this class when you extend `MyEmbedding`. -/
 class MyEmbeddingClass (F : Type*) (A B : outParam Type*) [MyClass A] [MyClass B]
     [FunLike F A B]
-    extends EmbeddingLike F A B :=
-  (map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))
+    extends EmbeddingLike F A B where
+  map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y)
 
 @[simp]
 lemma map_op {F A B : Type*} [MyClass A] [MyClass B] [FunLike F A B] [MyEmbeddingClass F A B]
@@ -84,12 +84,12 @@ The second step is to add instances of your new `MyEmbeddingClass` for all types
 Typically, you can just declare a new class analogous to `MyEmbeddingClass`:
 
 ```
-structure CoolerEmbedding (A B : Type*) [CoolClass A] [CoolClass B] extends MyEmbedding A B :=
+structure CoolerEmbedding (A B : Type*) [CoolClass A] [CoolClass B] extends MyEmbedding A B where
   (map_cool' : toFun CoolClass.cool = CoolClass.cool)
 
 class CoolerEmbeddingClass (F : Type*) (A B : outParam Type*) [CoolClass A] [CoolClass B]
     [FunLike F A B]
-    extends MyEmbeddingClass F A B :=
+    extends MyEmbeddingClass F A B where
   (map_cool : ∀ (f : F), f CoolClass.cool = CoolClass.cool)
 
 @[simp]
diff --git a/Mathlib/Data/FunLike/Equiv.lean b/Mathlib/Data/FunLike/Equiv.lean
index d2a640eddc593..92881dc5ec3a9 100644
--- a/Mathlib/Data/FunLike/Equiv.lean
+++ b/Mathlib/Data/FunLike/Equiv.lean
@@ -14,7 +14,7 @@ This typeclass is primarily for use by isomorphisms like `MonoidEquiv` and `Line
 
 A typical type of isomorphisms should be declared as:
 ```
-structure MyIso (A B : Type*) [MyClass A] [MyClass B] extends Equiv A B :=
+structure MyIso (A B : Type*) [MyClass A] [MyClass B] extends Equiv A B where
   (map_op' : ∀ (x y : A), toFun (MyClass.op x y) = MyClass.op (toFun x) (toFun y))
 
 namespace MyIso
@@ -77,12 +77,12 @@ The second step is to add instances of your new `MyIsoClass` for all types exten
 Typically, you can just declare a new class analogous to `MyIsoClass`:
 
 ```
-structure CoolerIso (A B : Type*) [CoolClass A] [CoolClass B] extends MyIso A B :=
+structure CoolerIso (A B : Type*) [CoolClass A] [CoolClass B] extends MyIso A B where
   (map_cool' : toFun CoolClass.cool = CoolClass.cool)
 
 class CoolerIsoClass (F : Type*) (A B : outParam Type*) [CoolClass A] [CoolClass B]
     [EquivLike F A B]
-    extends MyIsoClass F A B :=
+    extends MyIsoClass F A B where
   (map_cool : ∀ (f : F), f CoolClass.cool = CoolClass.cool)
 
 @[simp] lemma map_cool {F A B : Type*} [CoolClass A] [CoolClass B]
diff --git a/Mathlib/Data/Int/Bitwise.lean b/Mathlib/Data/Int/Bitwise.lean
index 646ee6284ae3e..664c817a6e578 100644
--- a/Mathlib/Data/Int/Bitwise.lean
+++ b/Mathlib/Data/Int/Bitwise.lean
@@ -24,7 +24,7 @@ def div2 : ℤ → ℤ
   | (n : ℕ) => n.div2
   | -[n +1] => negSucc n.div2
 
-/-- `bodd n` returns `true` if `n` is odd-/
+/-- `bodd n` returns `true` if `n` is odd -/
 def bodd : ℤ → Bool
   | (n : ℕ) => n.bodd
   | -[n +1] => not (n.bodd)
@@ -81,7 +81,7 @@ def ldiff : ℤ → ℤ → ℤ
   | -[m +1], -[n +1] => Nat.ldiff n m
 
 -- Porting note: I don't know why `Nat.xor'` got the prime, but I'm matching this change here
-/-- `xor` computes the bitwise `xor` of two natural numbers-/
+/-- `xor` computes the bitwise `xor` of two natural numbers -/
 protected def xor : ℤ → ℤ → ℤ
   | (m : ℕ), (n : ℕ) => (m ^^^ n)
   | (m : ℕ), -[n +1] => -[(m ^^^ n) +1]
diff --git a/Mathlib/Data/Int/CardIntervalMod.lean b/Mathlib/Data/Int/CardIntervalMod.lean
index 045f7ee4c7aa8..b5725e3f43eb6 100644
--- a/Mathlib/Data/Int/CardIntervalMod.lean
+++ b/Mathlib/Data/Int/CardIntervalMod.lean
@@ -43,14 +43,14 @@ include hr
 lemma Ico_filter_dvd_eq : (Ico a b).filter (r ∣ ·) =
     (Ico ⌈a / (r : ℚ)⌉ ⌈b / (r : ℚ)⌉).map ⟨(· * r), mul_left_injective₀ hr.ne'⟩ := by
   ext x
-  simp only [mem_map, mem_filter, mem_Ico, ceil_le, lt_ceil, div_le_iff₀, lt_div_iff,
+  simp only [mem_map, mem_filter, mem_Ico, ceil_le, lt_ceil, div_le_iff₀, lt_div_iff₀,
     dvd_iff_exists_eq_mul_left, cast_pos.2 hr, ← cast_mul, cast_lt, cast_le]
   aesop
 
 lemma Ioc_filter_dvd_eq : (Ioc a b).filter (r ∣ ·) =
     (Ioc ⌊a / (r : ℚ)⌋ ⌊b / (r : ℚ)⌋).map ⟨(· * r), mul_left_injective₀ hr.ne'⟩ := by
   ext x
-  simp only [mem_map, mem_filter, mem_Ioc, floor_lt, le_floor, div_lt_iff, le_div_iff₀,
+  simp only [mem_map, mem_filter, mem_Ioc, floor_lt, le_floor, div_lt_iff₀, le_div_iff₀,
     dvd_iff_exists_eq_mul_left, cast_pos.2 hr, ← cast_mul, cast_lt, cast_le]
   aesop
 
@@ -125,7 +125,7 @@ theorem count_modEq_card_eq_ceil (v : ℕ) :
     rw [← div_add_mod v r, cast_add, cast_mul, add_comm]
     tactic => simp_rw [← sub_sub, sub_div (_ - _), mul_div_cancel_left₀ _ hr'.ne', ceil_sub_nat]
     rw [sub_sub_sub_cancel_right, cast_zero, zero_sub]
-  rw [sub_eq_self, ceil_eq_zero_iff, Set.mem_Ioc, div_le_iff₀ hr', lt_div_iff hr', neg_one_mul,
+  rw [sub_eq_self, ceil_eq_zero_iff, Set.mem_Ioc, div_le_iff₀ hr', lt_div_iff₀ hr', neg_one_mul,
     zero_mul, neg_lt_neg_iff, cast_lt]
   exact ⟨mod_lt _ hr, by simp⟩
 
@@ -139,10 +139,10 @@ theorem count_modEq_card (v : ℕ) :
     mul_div_cancel_left₀ _ hr'.ne', add_comm, Int.ceil_add_nat, add_comm]
   rw [add_right_inj]
   split_ifs with h
-  · rw [← cast_sub h.le, Int.ceil_eq_iff, div_le_iff₀ hr', lt_div_iff hr', cast_one, Int.cast_one,
+  · rw [← cast_sub h.le, Int.ceil_eq_iff, div_le_iff₀ hr', lt_div_iff₀ hr', cast_one, Int.cast_one,
       sub_self, zero_mul, cast_pos, tsub_pos_iff_lt, one_mul, cast_le, tsub_le_iff_right]
     exact ⟨h, ((mod_lt _ hr).trans_le (by simp)).le⟩
-  · rw [cast_zero, ceil_eq_zero_iff, Set.mem_Ioc, div_le_iff₀ hr', lt_div_iff hr', zero_mul,
+  · rw [cast_zero, ceil_eq_zero_iff, Set.mem_Ioc, div_le_iff₀ hr', lt_div_iff₀ hr', zero_mul,
       tsub_nonpos, ← neg_eq_neg_one_mul, neg_lt_sub_iff_lt_add, ← cast_add, cast_lt, cast_le]
     exact ⟨(mod_lt _ hr).trans_le (by simp), not_lt.mp h⟩
 
diff --git a/Mathlib/Data/Int/Defs.lean b/Mathlib/Data/Int/Defs.lean
index 31f5d7d622d04..6bf06c9ff023a 100644
--- a/Mathlib/Data/Int/Defs.lean
+++ b/Mathlib/Data/Int/Defs.lean
@@ -30,7 +30,6 @@ namespace Int
 variable {a b c d m n : ℤ}
 
 section Order
-variable {a b c : ℤ}
 
 protected lemma le_rfl : a ≤ a := a.le_refl
 protected lemma lt_or_lt_of_ne : a ≠ b → a < b ∨ b < a := Int.lt_or_gt_of_ne
@@ -43,6 +42,7 @@ protected lemma le_antisymm_iff : a = b ↔ a ≤ b ∧ b ≤ a :=
   ⟨fun h ↦ ⟨Int.le_of_eq h, Int.ge_of_eq h⟩, fun h ↦ Int.le_antisymm h.1 h.2⟩
 protected lemma le_iff_eq_or_lt : a ≤ b ↔ a = b ∨ a < b := by
   rw [Int.le_antisymm_iff, Int.lt_iff_le_not_le, ← and_or_left]; simp [em]
+
 protected lemma le_iff_lt_or_eq : a ≤ b ↔ a < b ∨ a = b := by rw [Int.le_iff_eq_or_lt, or_comm]
 
 end Order
@@ -234,7 +234,7 @@ where
     conv => rhs; exact b.add_zero.symm
     rw [Int.add_lt_add_iff_left]; apply negSucc_lt_zero
 
-variable (b) {z b b H0 Hs Hp}
+variable {z b H0 Hs Hp}
 
 lemma inductionOn'_self : b.inductionOn' b H0 Hs Hp = H0 :=
   cast_eq_iff_heq.mpr <| .symm <| by rw [b.sub_self, ← cast_eq_iff_heq]; rfl
@@ -562,13 +562,12 @@ lemma lt_of_toNat_lt {a b : ℤ} (h : toNat a < toNat b) : a < b :=
   (toNat_lt_toNat <| lt_toNat.1 <| Nat.lt_of_le_of_lt (Nat.zero_le _) h).1 h
 
 @[simp] lemma toNat_pred_coe_of_pos {i : ℤ} (h : 0 < i) : ((i.toNat - 1 : ℕ) : ℤ) = i - 1 := by
-  simp [h, Int.le_of_lt h, push_cast]
+  simp only [lt_toNat, Nat.cast_ofNat_Int, h, natCast_pred_of_pos, Int.le_of_lt h, toNat_of_nonneg]
 
 @[simp] lemma toNat_eq_zero : ∀ {n : ℤ}, n.toNat = 0 ↔ n ≤ 0
   | (n : ℕ) => by simp
   | -[n+1] => by simpa [toNat] using Int.le_of_lt (negSucc_lt_zero n)
 
-@[simp]
 theorem toNat_sub_of_le {a b : ℤ} (h : b ≤ a) : (toNat (a - b) : ℤ) = a - b :=
   Int.toNat_of_nonneg (Int.sub_nonneg_of_le h)
 
diff --git a/Mathlib/Data/Int/DivMod.lean b/Mathlib/Data/Int/DivMod.lean
new file mode 100644
index 0000000000000..e2d8db47293a8
--- /dev/null
+++ b/Mathlib/Data/Int/DivMod.lean
@@ -0,0 +1,20 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+
+/-!
+# Basic lemmas about division and modulo for integers
+
+-/
+
+namespace Int
+
+/-! ### `emod` -/
+
+theorem emod_eq_sub_self_emod {a b : Int} : a % b = (a - b) % b :=
+  (emod_sub_cancel a b).symm
+
+theorem emod_eq_add_self_emod {a b : Int} : a % b = (a + b) % b :=
+  add_emod_self.symm
diff --git a/Mathlib/Data/Int/GCD.lean b/Mathlib/Data/Int/GCD.lean
index 038b354fda39a..6c4e2b4ad9ed9 100644
--- a/Mathlib/Data/Int/GCD.lean
+++ b/Mathlib/Data/Int/GCD.lean
@@ -322,7 +322,7 @@ theorem gcd_least_linear {a b : ℤ} (ha : a ≠ 0) :
     IsLeast { n : ℕ | 0 < n ∧ ∃ x y : ℤ, ↑n = a * x + b * y } (a.gcd b) := by
   simp_rw [← gcd_dvd_iff]
   constructor
-  · simpa [and_true_iff, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha
+  · simpa [and_true, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha
   · simp only [lowerBounds, and_imp, Set.mem_setOf_eq]
     exact fun n hn_pos hn => Nat.le_of_dvd hn_pos hn
 
diff --git a/Mathlib/Data/Int/Lemmas.lean b/Mathlib/Data/Int/Lemmas.lean
index 9300c6b89865c..61ccb22c02adb 100644
--- a/Mathlib/Data/Int/Lemmas.lean
+++ b/Mathlib/Data/Int/Lemmas.lean
@@ -6,6 +6,7 @@ Authors: Jeremy Avigad
 import Mathlib.Data.Int.Bitwise
 import Mathlib.Data.Int.Order.Lemmas
 import Mathlib.Data.Set.Function
+import Mathlib.Data.Set.Monotone
 import Mathlib.Order.Interval.Set.Basic
 
 /-!
diff --git a/Mathlib/Data/Int/Log.lean b/Mathlib/Data/Int/Log.lean
index adc07b2f77f74..ac78731f26da4 100644
--- a/Mathlib/Data/Int/Log.lean
+++ b/Mathlib/Data/Int/Log.lean
@@ -94,7 +94,7 @@ theorem zpow_log_le_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) : (b : R) ^
     rw [zpow_natCast, ← Nat.cast_pow, ← Nat.le_floor_iff hr.le]
     exact Nat.pow_log_le_self b (Nat.floor_pos.mpr hr1).ne'
   · rw [log_of_right_le_one _ hr1, zpow_neg, zpow_natCast, ← Nat.cast_pow]
-    exact inv_le_of_inv_le hr (Nat.ceil_le.1 <| Nat.le_pow_clog hb _)
+    exact inv_le_of_inv_le₀ hr (Nat.ceil_le.1 <| Nat.le_pow_clog hb _)
 
 theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1) := by
   rcases le_or_lt r 0 with hr | hr
@@ -106,11 +106,11 @@ theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (lo
     apply Nat.lt_of_floor_lt
     exact Nat.lt_pow_succ_log_self hb _
   · rw [log_of_right_le_one _ hr1.le]
-    have hcri : 1 < r⁻¹ := one_lt_inv hr hr1
+    have hcri : 1 < r⁻¹ := (one_lt_inv₀ hr).2 hr1
     have : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊ :=
       Nat.succ_le_of_lt (Nat.clog_pos hb <| Nat.one_lt_cast.1 <| hcri.trans_le (Nat.le_ceil _))
     rw [neg_add_eq_sub, ← neg_sub, ← Int.ofNat_one, ← Int.ofNat_sub this, zpow_neg, zpow_natCast,
-      lt_inv hr (pow_pos (Nat.cast_pos.mpr <| zero_lt_one.trans hb) _), ← Nat.cast_pow]
+      lt_inv_comm₀ hr (pow_pos (Nat.cast_pos.mpr <| zero_lt_one.trans hb) _), ← Nat.cast_pow]
     refine Nat.lt_ceil.1 ?_
     exact Nat.pow_pred_clog_lt_self hb <| Nat.one_lt_cast.1 <| hcri.trans_le <| Nat.le_ceil _
 
@@ -122,6 +122,16 @@ theorem log_zero_right (b : ℕ) : log b (0 : R) = 0 :=
 theorem log_one_right (b : ℕ) : log b (1 : R) = 0 := by
   rw [log_of_one_le_right _ le_rfl, Nat.floor_one, Nat.log_one_right, Int.ofNat_zero]
 
+@[simp]
+theorem log_zero_left (r : R) : log 0 r = 0 := by
+  simp only [log, Nat.log_zero_left, Nat.cast_zero, Nat.clog_zero_left, neg_zero, ite_self]
+
+@[simp]
+theorem log_one_left (r : R) : log 1 r = 0 := by
+  by_cases hr : 1 ≤ r
+  · simp_all only [log, ↓reduceIte, Nat.log_one_left, Nat.cast_zero]
+  · simp only [log, Nat.log_one_left, Nat.cast_zero, Nat.clog_one_left, neg_zero, ite_self]
+
 -- Porting note: needed to replace b ^ z with (b : R) ^ z in the below
 theorem log_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : log b ((b : R) ^ z : R) = z := by
   obtain ⟨n, rfl | rfl⟩ := Int.eq_nat_or_neg z
@@ -136,7 +146,7 @@ theorem log_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : log b ((b : R) ^ z : R) = z
 theorem log_mono_right {b : ℕ} {r₁ r₂ : R} (h₀ : 0 < r₁) (h : r₁ ≤ r₂) : log b r₁ ≤ log b r₂ := by
   rcases le_total r₁ 1 with h₁ | h₁ <;> rcases le_total r₂ 1 with h₂ | h₂
   · rw [log_of_right_le_one _ h₁, log_of_right_le_one _ h₂, neg_le_neg_iff, Int.ofNat_le]
-    exact Nat.clog_mono_right _ (Nat.ceil_mono <| inv_le_inv_of_le h₀ h)
+    exact Nat.clog_mono_right _ (Nat.ceil_mono <| inv_anti₀ h₀ h)
   · rw [log_of_right_le_one _ h₁, log_of_one_le_right _ h₂]
     exact (neg_nonpos.mpr (Int.natCast_nonneg _)).trans (Int.natCast_nonneg _)
   · obtain rfl := le_antisymm h (h₂.trans h₁)
@@ -193,8 +203,8 @@ theorem clog_of_right_le_zero (b : ℕ) {r : R} (hr : r ≤ 0) : clog b r = 0 :=
 theorem clog_inv (b : ℕ) (r : R) : clog b r⁻¹ = -log b r := by
   cases' lt_or_le 0 r with hrp hrp
   · obtain hr | hr := le_total 1 r
-    · rw [clog_of_right_le_one _ (inv_le_one hr), log_of_one_le_right _ hr, inv_inv]
-    · rw [clog_of_one_le_right _ (one_le_inv hrp hr), log_of_right_le_one _ hr, neg_neg]
+    · rw [clog_of_right_le_one _ (inv_le_one_of_one_le₀ hr), log_of_one_le_right _ hr, inv_inv]
+    · rw [clog_of_one_le_right _ ((one_le_inv₀ hrp).2 hr), log_of_right_le_one _ hr, neg_neg]
   · rw [clog_of_right_le_zero _ (inv_nonpos.mpr hrp), log_of_right_le_zero _ hrp, neg_zero]
 
 @[simp]
@@ -225,13 +235,13 @@ theorem self_le_zpow_clog {b : ℕ} (hb : 1 < b) (r : R) : r ≤ (b : R) ^ clog
   rcases le_or_lt r 0 with hr | hr
   · rw [clog_of_right_le_zero _ hr, zpow_zero]
     exact hr.trans zero_le_one
-  rw [← neg_log_inv_eq_clog, zpow_neg, le_inv hr (zpow_pos_of_pos _ _)]
+  rw [← neg_log_inv_eq_clog, zpow_neg, le_inv_comm₀ hr (zpow_pos_of_pos _ _)]
   · exact zpow_log_le_self hb (inv_pos.mpr hr)
   · exact Nat.cast_pos.mpr (zero_le_one.trans_lt hb)
 
 theorem zpow_pred_clog_lt_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) :
     (b : R) ^ (clog b r - 1) < r := by
-  rw [← neg_log_inv_eq_clog, ← neg_add', zpow_neg, inv_lt _ hr]
+  rw [← neg_log_inv_eq_clog, ← neg_add', zpow_neg, inv_lt_comm₀ _ hr]
   · exact lt_zpow_succ_log_self hb _
   · exact zpow_pos_of_pos (Nat.cast_pos.mpr <| zero_le_one.trans_lt hb) _
 
@@ -243,6 +253,16 @@ theorem clog_zero_right (b : ℕ) : clog b (0 : R) = 0 :=
 theorem clog_one_right (b : ℕ) : clog b (1 : R) = 0 := by
   rw [clog_of_one_le_right _ le_rfl, Nat.ceil_one, Nat.clog_one_right, Int.ofNat_zero]
 
+@[simp]
+theorem clog_zero_left (r : R) : clog 0 r = 0 := by
+  by_cases hr : 1 ≤ r
+  · simp only [clog, Nat.clog_zero_left, Nat.cast_zero, Nat.log_zero_left, neg_zero, ite_self]
+  · simp only [clog, hr, ite_cond_eq_false, Nat.log_zero_left, Nat.cast_zero, neg_zero]
+
+@[simp]
+theorem clog_one_left (r : R) : clog 1 r = 0 := by
+  simp only [clog, Nat.log_one_left, Nat.cast_zero, Nat.clog_one_left, neg_zero, ite_self]
+
 -- Porting note: needed to replace b ^ z with (b : R) ^ z in the below
 theorem clog_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : clog b ((b : R) ^ z : R) = z := by
   rw [← neg_log_inv_eq_clog, ← zpow_neg, log_zpow hb, neg_neg]
@@ -251,7 +271,7 @@ theorem clog_zpow {b : ℕ} (hb : 1 < b) (z : ℤ) : clog b ((b : R) ^ z : R) =
 theorem clog_mono_right {b : ℕ} {r₁ r₂ : R} (h₀ : 0 < r₁) (h : r₁ ≤ r₂) :
     clog b r₁ ≤ clog b r₂ := by
   rw [← neg_log_inv_eq_clog, ← neg_log_inv_eq_clog, neg_le_neg_iff]
-  exact log_mono_right (inv_pos.mpr <| h₀.trans_le h) (inv_le_inv_of_le h₀ h)
+  exact log_mono_right (inv_pos.mpr <| h₀.trans_le h) (inv_anti₀ h₀ h)
 
 variable (R)
 
diff --git a/Mathlib/Data/Int/ModEq.lean b/Mathlib/Data/Int/ModEq.lean
index bdd7e1559edbd..d171639d57b4d 100644
--- a/Mathlib/Data/Int/ModEq.lean
+++ b/Mathlib/Data/Int/ModEq.lean
@@ -4,8 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
 import Mathlib.Data.Nat.ModEq
-import Mathlib.Tactic.Abel
-import Mathlib.Tactic.GCongr.Core
 
 /-!
 
@@ -92,8 +90,7 @@ theorem mod_modEq (a n) : a % n ≡ a [ZMOD n] :=
 
 @[simp]
 theorem neg_modEq_neg : -a ≡ -b [ZMOD n] ↔ a ≡ b [ZMOD n] := by
--- Porting note: Restore old proof once #3309 is through
-  simp [-sub_neg_eq_add, neg_sub_neg, modEq_iff_dvd, dvd_sub_comm]
+  simp only [modEq_iff_dvd, (by omega : -b - -a = -(b - a)), Int.dvd_neg]
 
 @[simp]
 theorem modEq_neg : a ≡ b [ZMOD -n] ↔ a ≡ b [ZMOD n] := by simp [modEq_iff_dvd]
@@ -105,9 +102,9 @@ protected theorem of_dvd (d : m ∣ n) (h : a ≡ b [ZMOD n]) : a ≡ b [ZMOD m]
 
 protected theorem mul_left' (h : a ≡ b [ZMOD n]) : c * a ≡ c * b [ZMOD c * n] := by
   obtain hc | rfl | hc := lt_trichotomy c 0
-  · rw [← neg_modEq_neg, ← modEq_neg, ← neg_mul, ← neg_mul, ← neg_mul]
+  · rw [← neg_modEq_neg, ← modEq_neg, ← Int.neg_mul, ← Int.neg_mul, ← Int.neg_mul]
     simp only [ModEq, mul_emod_mul_of_pos _ _ (neg_pos.2 hc), h.eq]
-  · simp only [zero_mul, ModEq.rfl]
+  · simp only [Int.zero_mul, ModEq.rfl]
   · simp only [ModEq, mul_emod_mul_of_pos _ _ hc, h.eq]
 
 protected theorem mul_right' (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD n * c] := by
@@ -115,7 +112,7 @@ protected theorem mul_right' (h : a ≡ b [ZMOD n]) : a * c ≡ b * c [ZMOD n *
 
 @[gcongr]
 protected theorem add (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a + c ≡ b + d [ZMOD n] :=
-  modEq_iff_dvd.2 <| by convert dvd_add h₁.dvd h₂.dvd using 1; abel
+  modEq_iff_dvd.2 <| by convert Int.dvd_add h₁.dvd h₂.dvd using 1; omega
 
 @[gcongr] protected theorem add_left (c : ℤ) (h : a ≡ b [ZMOD n]) : c + a ≡ c + b [ZMOD n] :=
   ModEq.rfl.add h
@@ -125,10 +122,10 @@ protected theorem add (h₁ : a ≡ b [ZMOD n]) (h₂ : c ≡ d [ZMOD n]) : a +
 
 protected theorem add_left_cancel (h₁ : a ≡ b [ZMOD n]) (h₂ : a + c ≡ b + d [ZMOD n]) :
     c ≡ d [ZMOD n] :=
-  have : d - c = b + d - (a + c) - (b - a) := by abel
+  have : d - c = b + d - (a + c) - (b - a) := by omega
   modEq_iff_dvd.2 <| by
     rw [this]
-    exact dvd_sub h₂.dvd h₁.dvd
+    exact Int.dvd_sub h₂.dvd h₁.dvd
 
 protected theorem add_left_cancel' (c : ℤ) (h : c + a ≡ c + b [ZMOD n]) : a ≡ b [ZMOD n] :=
   ModEq.rfl.add_left_cancel h
@@ -183,7 +180,7 @@ theorem cancel_right_div_gcd (hm : 0 < m) (h : a * c ≡ b * c [ZMOD m]) :
   rw [modEq_iff_dvd] at h ⊢
   -- Porting note: removed `show` due to leanprover-community/mathlib4#3305
   refine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d ∣ c / d * (b - a)) ?_
-  · rw [mul_comm, ← Int.mul_ediv_assoc (b - a) gcd_dvd_right, sub_mul]
+  · rw [mul_comm, ← Int.mul_ediv_assoc (b - a) gcd_dvd_right, Int.sub_mul]
     exact Int.ediv_dvd_ediv gcd_dvd_left h
   · rw [gcd_div gcd_dvd_left gcd_dvd_right, natAbs_ofNat,
       Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')]
@@ -233,7 +230,7 @@ theorem modEq_add_fac {a b n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a + n *
     _ ≡ b [ZMOD n] := by rw [add_zero]
 
 theorem modEq_sub_fac {a b n : ℤ} (c : ℤ) (ha : a ≡ b [ZMOD n]) : a - n * c ≡ b [ZMOD n] := by
-  convert Int.modEq_add_fac (-c) ha using 1; rw [mul_neg, sub_eq_add_neg]
+  convert Int.modEq_add_fac (-c) ha using 1; rw [Int.mul_neg, sub_eq_add_neg]
 
 theorem modEq_add_fac_self {a t n : ℤ} : a + n * t ≡ a [ZMOD n] :=
   modEq_add_fac _ ModEq.rfl
diff --git a/Mathlib/Data/Int/Order/Lemmas.lean b/Mathlib/Data/Int/Order/Lemmas.lean
index e7bc9e148fc42..d664a5b284aca 100644
--- a/Mathlib/Data/Int/Order/Lemmas.lean
+++ b/Mathlib/Data/Int/Order/Lemmas.lean
@@ -34,15 +34,4 @@ theorem natAbs_le_iff_mul_self_le {a b : ℤ} : a.natAbs ≤ b.natAbs ↔ a * a
   rw [← abs_le_iff_mul_self_le, abs_eq_natAbs, abs_eq_natAbs]
   exact Int.ofNat_le.symm
 
-/-! ### units -/
-
-
-theorem eq_zero_of_abs_lt_dvd {m x : ℤ} (h1 : m ∣ x) (h2 : |x| < m) : x = 0 := by
-  obtain rfl | hm := eq_or_ne m 0
-  · exact Int.zero_dvd.1 h1
-  rcases h1 with ⟨d, rfl⟩
-  apply mul_eq_zero_of_right
-  rw [← abs_lt_one_iff, ← mul_lt_iff_lt_one_right (abs_pos.mpr hm), ← abs_mul]
-  exact lt_of_lt_of_le h2 (le_abs_self m)
-
 end Int
diff --git a/Mathlib/Data/Int/Star.lean b/Mathlib/Data/Int/Star.lean
index 5f30de5022d52..b4e7290794d36 100644
--- a/Mathlib/Data/Int/Star.lean
+++ b/Mathlib/Data/Int/Star.lean
@@ -3,11 +3,11 @@ Copyright (c) 2024 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Algebra.Star.Order
 import Mathlib.Algebra.Order.Group.Abs
 import Mathlib.Algebra.Order.Monoid.Submonoid
 import Mathlib.Algebra.Order.Ring.Basic
 import Mathlib.Algebra.Order.Ring.Int
+import Mathlib.Algebra.Order.Star.Basic
 
 /-!
 # Star ordered ring structure on `ℤ`
diff --git a/Mathlib/Data/Int/SuccPred.lean b/Mathlib/Data/Int/SuccPred.lean
index d16d87c88e2dd..51f82b5b2556a 100644
--- a/Mathlib/Data/Int/SuccPred.lean
+++ b/Mathlib/Data/Int/SuccPred.lean
@@ -21,6 +21,8 @@ namespace Int
 @[instance] abbrev instSuccOrder : SuccOrder ℤ :=
   { SuccOrder.ofSuccLeIff succ fun {_ _} => Iff.rfl with succ := succ }
 
+instance instSuccAddOrder : SuccAddOrder ℤ := ⟨fun _ => rfl⟩
+
 -- so that Lean reads `Int.pred` through `PredOrder.pred`
 @[instance] abbrev instPredOrder : PredOrder ℤ where
   pred := pred
@@ -28,6 +30,8 @@ namespace Int
   min_of_le_pred ha := ((sub_one_lt_of_le le_rfl).not_le ha).elim
   le_pred_of_lt {_ _} := le_sub_one_of_lt
 
+instance instPredSubOrder : PredSubOrder ℤ := ⟨fun _ => rfl⟩
+
 @[simp]
 theorem succ_eq_succ : Order.succ = succ :=
   rfl
@@ -36,46 +40,44 @@ theorem succ_eq_succ : Order.succ = succ :=
 theorem pred_eq_pred : Order.pred = pred :=
   rfl
 
+@[deprecated Order.one_le_iff_pos (since := "2024-09-04")]
 theorem pos_iff_one_le {a : ℤ} : 0 < a ↔ 1 ≤ a :=
   Order.succ_le_iff.symm
 
-theorem succ_iterate (a : ℤ) : ∀ n, succ^[n] a = a + n
-  | 0 => (add_zero a).symm
-  | n + 1 => by
-    rw [Function.iterate_succ', Int.ofNat_succ, ← add_assoc]
-    exact congr_arg _ (succ_iterate a n)
+@[deprecated Order.succ_iterate (since := "2024-09-04")]
+protected theorem succ_iterate (a : ℤ) : ∀ n, succ^[n] a = a + n :=
+  Order.succ_iterate a
 
-theorem pred_iterate (a : ℤ) : ∀ n, pred^[n] a = a - n
-  | 0 => (sub_zero a).symm
-  | n + 1 => by
-    rw [Function.iterate_succ', Int.ofNat_succ, ← sub_sub]
-    exact congr_arg _ (pred_iterate a n)
+@[deprecated Order.pred_iterate (since := "2024-09-04")]
+protected theorem pred_iterate (a : ℤ) : ∀ n, pred^[n] a = a - n :=
+  Order.pred_iterate a
 
 instance : IsSuccArchimedean ℤ :=
   ⟨fun {a b} h =>
-    ⟨(b - a).toNat, by
-      rw [succ_eq_succ, succ_iterate, toNat_sub_of_le h, ← add_sub_assoc, add_sub_cancel_left]⟩⟩
+    ⟨(b - a).toNat, by rw [succ_iterate, toNat_sub_of_le h, ← add_sub_assoc, add_sub_cancel_left]⟩⟩
 
 instance : IsPredArchimedean ℤ :=
   ⟨fun {a b} h =>
-    ⟨(b - a).toNat, by rw [pred_eq_pred, pred_iterate, toNat_sub_of_le h, sub_sub_cancel]⟩⟩
+    ⟨(b - a).toNat, by rw [pred_iterate, toNat_sub_of_le h, sub_sub_cancel]⟩⟩
 
 /-! ### Covering relation -/
 
 
+@[deprecated Order.covBy_iff_add_one_eq (since := "2024-09-04")]
 protected theorem covBy_iff_succ_eq {m n : ℤ} : m ⋖ n ↔ m + 1 = n :=
   succ_eq_iff_covBy.symm
 
-@[simp]
-theorem sub_one_covBy (z : ℤ) : z - 1 ⋖ z := by rw [Int.covBy_iff_succ_eq, sub_add_cancel]
+@[deprecated Order.sub_one_covBy (since := "2024-09-04")]
+theorem sub_one_covBy (z : ℤ) : z - 1 ⋖ z :=
+  Order.sub_one_covBy z
 
-@[simp]
+@[deprecated Order.covBy_add_one (since := "2024-09-04")]
 theorem covBy_add_one (z : ℤ) : z ⋖ z + 1 :=
-  Int.covBy_iff_succ_eq.mpr rfl
+  Order.covBy_add_one z
 
 @[simp, norm_cast]
 theorem natCast_covBy {a b : ℕ} : (a : ℤ) ⋖ b ↔ a ⋖ b := by
-  rw [Nat.covBy_iff_succ_eq, Int.covBy_iff_succ_eq]
+  rw [Order.covBy_iff_add_one_eq, Order.covBy_iff_add_one_eq]
   exact Int.natCast_inj
 
 end Int
diff --git a/Mathlib/Data/Int/WithZero.lean b/Mathlib/Data/Int/WithZero.lean
new file mode 100644
index 0000000000000..3cf41e5f88e38
--- /dev/null
+++ b/Mathlib/Data/Int/WithZero.lean
@@ -0,0 +1,91 @@
+/-
+Copyright (c) 2024 María Inés de Frutos-Fernández, Filippo A. E. Nuccio. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: María Inés de Frutos-Fernández, Filippo A. E. Nuccio
+-/
+import Mathlib.Data.NNReal.Basic
+
+/-!
+# WithZero
+
+In this file we provide some basic API lemmas for the `WithZero` construction and we define
+the morphism `WithZeroMultInt.toNNReal`.
+
+## Main Definitions
+
+* `WithZeroMultInt.toNNReal` : The `MonoidWithZeroHom` from `ℤₘ₀ → ℝ≥0` sending `0 ↦ 0` and
+  `x ↦ e^(Multiplicative.toAdd (WithZero.unzero hx)` when `x ≠ 0`, for a nonzero `e : ℝ≥0`.
+
+## Main Results
+
+* `WithZeroMultInt.toNNReal_strictMono` : The map `withZeroMultIntToNNReal` is strictly
+   monotone whenever `1 < e`.
+
+## Tags
+
+WithZero, multiplicative, nnreal
+-/
+
+noncomputable section
+
+open scoped NNReal
+
+open Multiplicative WithZero
+
+namespace WithZeroMulInt
+
+/-- Given a nonzero `e : ℝ≥0`, this is the map `ℤₘ₀ → ℝ≥0` sending `0 ↦ 0` and
+  `x ↦ e^(Multiplicative.toAdd (WithZero.unzero hx)` when `x ≠ 0` as a `MonoidWithZeroHom`. -/
+def toNNReal {e : NNReal} (he : e ≠ 0) : ℤₘ₀ →*₀ ℝ≥0 where
+  toFun := fun x ↦ if hx : x = 0 then 0 else e ^ Multiplicative.toAdd (WithZero.unzero hx)
+  map_zero' := rfl
+  map_one' := by
+    simp only [dif_neg one_ne_zero]
+    erw [toAdd_one, zpow_zero]
+  map_mul' x y := by
+    simp only
+    by_cases hxy : x * y = 0
+    · cases' zero_eq_mul.mp (Eq.symm hxy) with hx hy
+      --either x = 0 or y = 0
+      · rw [dif_pos hxy, dif_pos hx, MulZeroClass.zero_mul]
+      · rw [dif_pos hxy, dif_pos hy, MulZeroClass.mul_zero]
+    · cases' mul_ne_zero_iff.mp hxy with hx hy
+      --  x Equiv≠ 0 and y ≠ 0
+      rw [dif_neg hxy, dif_neg hx, dif_neg hy, ← zpow_add' (Or.inl he), ← toAdd_mul]
+      congr
+      rw [← WithZero.coe_inj, WithZero.coe_mul, coe_unzero hx, coe_unzero hy, coe_unzero hxy]
+
+theorem toNNReal_pos_apply {e : NNReal} (he : e ≠ 0) {x : ℤₘ₀} (hx : x = 0) :
+    toNNReal he x = 0 := by
+  simp only [toNNReal, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
+  split_ifs; rfl
+
+theorem toNNReal_neg_apply {e : NNReal} (he : e ≠ 0) {x : ℤₘ₀} (hx : x ≠ 0) :
+    toNNReal he x = e ^ Multiplicative.toAdd (WithZero.unzero hx) := by
+  simp only [toNNReal, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
+  split_ifs
+  · tauto
+  · rfl
+
+/-- `toNNReal` sends nonzero elements to nonzero elements. -/
+theorem toNNReal_ne_zero {e : NNReal} {m : ℤₘ₀} (he : e ≠ 0) (hm : m ≠ 0) : toNNReal he m ≠ 0 := by
+  simp only [ne_eq, map_eq_zero, hm, not_false_eq_true]
+
+/-- `toNNReal` sends nonzero elements to positive elements. -/
+theorem toNNReal_pos {e : NNReal} {m : ℤₘ₀} (he : e ≠ 0) (hm : m ≠ 0) : 0 < toNNReal he m :=
+  lt_of_le_of_ne zero_le' (toNNReal_ne_zero he hm).symm
+
+/-- The map `toNNReal` is strictly monotone whenever `1 < e`. -/
+theorem toNNReal_strictMono {e : NNReal} (he : 1 < e) :
+    StrictMono (toNNReal (ne_zero_of_lt he)) := by
+  intro x y hxy
+  simp only [toNNReal, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
+  split_ifs with hx hy hy
+  · simp only [hy, not_lt_zero'] at hxy
+  · exact NNReal.zpow_pos (ne_zero_of_lt he) _
+  · simp only [hy, not_lt_zero'] at hxy
+  · rw [zpow_lt_iff_lt he, Multiplicative.toAdd_lt, ← WithZero.coe_lt_coe, WithZero.coe_unzero hx,
+      WithZero.coe_unzero hy]
+    exact hxy
+
+end WithZeroMulInt
diff --git a/Mathlib/Data/LazyList/Basic.lean b/Mathlib/Data/LazyList/Basic.lean
index ad67f3d3f60aa..7a9b820336505 100644
--- a/Mathlib/Data/LazyList/Basic.lean
+++ b/Mathlib/Data/LazyList/Basic.lean
@@ -47,25 +47,29 @@ instance : Traversable LazyList where
 @[deprecated (since := "2024-07-22")]
 instance : LawfulTraversable LazyList := by
   apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
-  · induction' xs using LazyList.rec with _ _ _ _ ih
-    · simp only [Functor.map, LazyList.traverse, pure, Equiv.map, listEquivLazyList,
+  · induction xs using LazyList.rec with
+    | nil =>
+      simp only [Functor.map, LazyList.traverse, pure, Equiv.map, listEquivLazyList,
         Equiv.coe_fn_symm_mk, toList, Equiv.coe_fn_mk, ofList]
-    · simpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk,
+    | cons =>
+      simpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk,
         LazyList.traverse, Seq.seq, toList, ofList, cons.injEq, true_and]
-    · ext; apply ih
+    | mk _ ih => ext; apply ih
   · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
       Functor.mapConst]
-    induction' xs using LazyList.rec with _ _ _ _ ih
-    · simp only [LazyList.traverse, pure, Functor.map, toList, ofList]
-    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
-    · congr; apply ih
+    induction xs using LazyList.rec with
+    | nil => simp only [LazyList.traverse, pure, Functor.map, toList, ofList]
+    | cons =>
+      simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
+    | mk _ ih => congr; apply ih
   · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
-    induction' xs using LazyList.rec with _ tl ih _ ih
-    · simp only [LazyList.traverse, toList, List.traverse, map_pure, ofList]
-    · replace ih : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih
+    induction xs using LazyList.rec with
+    | nil => simp only [LazyList.traverse, toList, List.traverse, map_pure, ofList]
+    | cons _ tl ih =>
+      replace ih : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih
       simp [traverse.eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq,
-        Function.comp, Thunk.pure, ofList]
-    · apply ih
+        Function.comp_def, Thunk.pure, ofList]
+    | mk _ ih => apply ih
 
 @[deprecated (since := "2024-07-22"), simp]
 theorem bind_singleton {α} (x : LazyList α) : x.bind singleton = x := by
@@ -91,10 +95,10 @@ instance : LawfulMonad LazyList := LawfulMonad.mk'
     apply append_nil)
   (bind_assoc := by
     intro _ _ _ xs _ _
-    induction' xs using LazyList.rec with _ _ _ _ ih
-    · simp only [bind, LazyList.bind]
-    · simp only [bind, LazyList.bind, append_bind]; congr
-    · congr; funext; apply ih)
+    induction xs using LazyList.rec with
+    | nil => simp only [bind, LazyList.bind]
+    | cons => simp only [bind, LazyList.bind, append_bind]; congr
+    | mk _ ih => congr; funext; apply ih)
   (bind_pure_comp := by
     intro _ _ f xs
     simp only [bind, Functor.map, pure, singleton]
diff --git a/Mathlib/Data/List/AList.lean b/Mathlib/Data/List/AList.lean
index 1498c2afd3a09..cb4442a55a060 100644
--- a/Mathlib/Data/List/AList.lean
+++ b/Mathlib/Data/List/AList.lean
@@ -78,7 +78,7 @@ theorem keys_nodup (s : AList β) : s.keys.Nodup :=
 
 /-- The predicate `a ∈ s` means that `s` has a value associated to the key `a`. -/
 instance : Membership α (AList β) :=
-  ⟨fun a s => a ∈ s.keys⟩
+  ⟨fun s a => a ∈ s.keys⟩
 
 theorem mem_keys {a : α} {s : AList β} : a ∈ s ↔ a ∈ s.keys :=
   Iff.rfl
diff --git a/Mathlib/Data/List/Basic.lean b/Mathlib/Data/List/Basic.lean
index 5fdbd0133fd78..d3d29ac80d33d 100644
--- a/Mathlib/Data/List/Basic.lean
+++ b/Mathlib/Data/List/Basic.lean
@@ -3,15 +3,15 @@ Copyright (c) 2014 Parikshit Khanna. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
 -/
+import Mathlib.Control.Basic
 import Mathlib.Data.Nat.Defs
 import Mathlib.Data.Option.Basic
 import Mathlib.Data.List.Defs
 import Mathlib.Data.List.Monad
-import Mathlib.Init.Data.List.Lemmas
+import Mathlib.Logic.OpClass
 import Mathlib.Logic.Unique
 import Mathlib.Order.Basic
 import Mathlib.Tactic.Common
-import Batteries.Data.List.Perm
 
 /-!
 # Basic properties of lists
@@ -36,15 +36,9 @@ variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : Lis
 @[deprecated (since := "2024-07-27")]
 theorem le_eq_not_gt [LT α] : ∀ l₁ l₂ : List α, (l₁ ≤ l₂) = ¬l₂ < l₁ := fun _ _ => rfl
 
-@[deprecated (since := "2024-06-07")] alias toArray_data := Array.data_toArray
-
 -- Porting note: Delete this attribute
 -- attribute [inline] List.head!
 
-theorem getElem?_eq (l : List α) (i : Nat) :
-    l[i]? = if h : i < l.length then some l[i] else none := by
-  split <;> simp_all
-
 /-- There is only one list of an empty type -/
 instance uniqueOfIsEmpty [IsEmpty α] : Unique (List α) :=
   { instInhabitedList with
@@ -64,9 +58,6 @@ instance : Std.Associative (α := List α) Append.append where
 
 theorem singleton_injective : Injective fun a : α => [a] := fun _ _ h => (cons_eq_cons.1 h).1
 
-theorem singleton_inj {a b : α} : [a] = [b] ↔ a = b :=
-  singleton_injective.eq_iff
-
 theorem set_of_mem_cons (l : List α) (a : α) : { x | x ∈ a :: l } = insert a { x | x ∈ l } :=
   Set.ext fun _ => mem_cons
 
@@ -193,8 +184,6 @@ theorem append_subset_of_subset_of_subset {l₁ l₂ l : List α} (l₁subl : l
     l₁ ++ l₂ ⊆ l :=
   fun _ h ↦ (mem_append.1 h).elim (@l₁subl _) (@l₂subl _)
 
-alias ⟨eq_nil_of_subset_nil, _⟩ := subset_nil
-
 theorem map_subset_iff {l₁ l₂ : List α} (f : α → β) (h : Injective f) :
     map f l₁ ⊆ map f l₂ ↔ l₁ ⊆ l₂ := by
   refine ⟨?_, map_subset f⟩; intro h2 x hx
@@ -206,26 +195,10 @@ theorem map_subset_iff {l₁ l₂ : List α} (f : α → β) (h : Injective f) :
 theorem append_eq_has_append {L₁ L₂ : List α} : List.append L₁ L₂ = L₁ ++ L₂ :=
   rfl
 
-@[deprecated (since := "2024-03-24")] alias append_eq_cons_iff := append_eq_cons
-
-@[deprecated (since := "2024-03-24")] alias cons_eq_append_iff := cons_eq_append
-
 @[deprecated (since := "2024-01-18")] alias append_left_cancel := append_cancel_left
 
 @[deprecated (since := "2024-01-18")] alias append_right_cancel := append_cancel_right
 
-@[simp] theorem append_left_eq_self {x y : List α} : x ++ y = y ↔ x = [] := by
-  rw [← append_left_inj (s₁ := x), nil_append]
-
-@[simp] theorem self_eq_append_left {x y : List α} : y = x ++ y ↔ x = [] := by
-  rw [eq_comm, append_left_eq_self]
-
-@[simp] theorem append_right_eq_self {x y : List α} : x ++ y = x ↔ y = [] := by
-  rw [← append_right_inj (t₁ := y), append_nil]
-
-@[simp] theorem self_eq_append_right {x y : List α} : x = x ++ y ↔ y = [] := by
-  rw [eq_comm, append_right_eq_self]
-
 theorem append_right_injective (s : List α) : Injective fun t ↦ s ++ t :=
   fun _ _ ↦ append_cancel_left
 
@@ -248,13 +221,10 @@ theorem replicate_subset_singleton (n) (a : α) : replicate n a ⊆ [a] := fun _
   mem_singleton.2 (eq_of_mem_replicate h)
 
 theorem subset_singleton_iff {a : α} {L : List α} : L ⊆ [a] ↔ ∃ n, L = replicate n a := by
-  simp only [eq_replicate, subset_def, mem_singleton, exists_eq_left']
-
-@[simp] theorem tail_replicate (a : α) (n) :
-    tail (replicate n a) = replicate (n - 1) a := by cases n <;> rfl
+  simp only [eq_replicate_iff, subset_def, mem_singleton, exists_eq_left']
 
 theorem replicate_right_injective {n : ℕ} (hn : n ≠ 0) : Injective (@replicate α n) :=
-  fun _ _ h => (eq_replicate.1 h).2 _ <| mem_replicate.2 ⟨hn, rfl⟩
+  fun _ _ h => (eq_replicate_iff.1 h).2 _ <| mem_replicate.2 ⟨hn, rfl⟩
 
 theorem replicate_right_inj {a b : α} {n : ℕ} (hn : n ≠ 0) :
     replicate n a = replicate n b ↔ a = b :=
@@ -310,10 +280,6 @@ theorem reverse_surjective : Surjective (@reverse α) :=
 theorem reverse_bijective : Bijective (@reverse α) :=
   reverse_involutive.bijective
 
-@[simp]
-theorem reverse_inj {l₁ l₂ : List α} : reverse l₁ = reverse l₂ ↔ l₁ = l₂ :=
-  reverse_injective.eq_iff
-
 theorem concat_eq_reverse_cons (a : α) (l : List α) : concat l a = reverse (a :: reverse l) := by
   simp only [concat_eq_append, reverse_cons, reverse_reverse]
 
@@ -323,9 +289,7 @@ theorem map_reverseAux (f : α → β) (l₁ l₂ : List α) :
 
 /-! ### empty -/
 
-theorem isEmpty_iff_eq_nil {l : List α} : l.isEmpty ↔ l = [] := by cases l <;> simp [isEmpty]
-
-/-! ### dropLast -/
+@[deprecated (since := "2024-08-15")] alias isEmpty_iff_eq_nil := isEmpty_iff
 
 /-! ### getLast -/
 
@@ -338,11 +302,9 @@ theorem getLast_append_singleton {a : α} (l : List α) :
 -- Porting note: name should be fixed upstream
 theorem getLast_append' (l₁ l₂ : List α) (h : l₂ ≠ []) :
     getLast (l₁ ++ l₂) (append_ne_nil_of_right_ne_nil l₁ h) = getLast l₂ h := by
-  induction' l₁ with _ _ ih
-  · simp
-  · simp only [cons_append]
-    rw [List.getLast_cons]
-    exact ih
+  induction l₁ with
+  | nil => simp
+  | cons _ _ ih => simp only [cons_append]; rw [List.getLast_cons]; exact ih
 
 theorem getLast_concat' {a : α} (l : List α) : getLast (concat l a) (concat_ne_nil a l) = a := by
   simp
@@ -390,24 +352,10 @@ lemma getLast_filter {p : α → Bool} :
 
 /-! ### getLast? -/
 
-@[simp]
-theorem getLast?_cons_cons (a b : α) (l : List α) :
-    getLast? (a :: b :: l) = getLast? (b :: l) := rfl
-
-@[simp]
-theorem getLast?_eq_none : ∀ {l : List α}, getLast? l = none ↔ l = []
-  | [] => by simp
-  | [a] => by simp
-  | a :: b :: l => by simp [@getLast?_eq_none (b :: l)]
+@[deprecated (since := "2024-09-06")] alias getLast?_eq_none := getLast?_eq_none_iff
 
 @[deprecated (since := "2024-06-20")] alias getLast?_isNone := getLast?_eq_none
 
-@[simp]
-theorem getLast?_isSome : ∀ {l : List α}, l.getLast?.isSome ↔ l ≠ []
-  | [] => by simp
-  | [a] => by simp
-  | a :: b :: l => by simp [@getLast?_isSome (b :: l)]
-
 theorem mem_getLast?_eq_getLast : ∀ {l : List α} {x : α}, x ∈ l.getLast? → ∃ h, x = getLast l h
   | [], x, hx => False.elim <| by simp at hx
   | [a], x, hx =>
@@ -428,10 +376,6 @@ theorem mem_getLast?_cons {x y : α} : ∀ {l : List α}, x ∈ l.getLast? → x
   | [], _ => by contradiction
   | _ :: _, h => h
 
-theorem mem_of_mem_getLast? {l : List α} {a : α} (ha : a ∈ l.getLast?) : a ∈ l :=
-  let ⟨_, h₂⟩ := mem_getLast?_eq_getLast ha
-  h₂.symm ▸ getLast_mem _
-
 theorem dropLast_append_getLast? : ∀ {l : List α}, ∀ a ∈ l.getLast?, dropLast l ++ [a] = l
   | [], a, ha => (Option.not_mem_none a ha).elim
   | [a], _, rfl => rfl
@@ -474,6 +418,10 @@ theorem head!_nil [Inhabited α] : ([] : List α).head! = default := rfl
 @[simp] theorem head_cons_tail (x : List α) (h : x ≠ []) : x.head h :: x.tail = x := by
   cases x <;> simp at h ⊢
 
+theorem head_eq_getElem_zero {l : List α} (hl : l ≠ []) :
+    l.head hl = l[0]'(length_pos.2 hl) :=
+  (getElem_zero _).symm
+
 theorem head!_eq_head? [Inhabited α] (l : List α) : head! l = (head? l).iget := by cases l <;> rfl
 
 theorem surjective_head! [Inhabited α] : Surjective (@head! α _) := fun x => ⟨[x], rfl⟩
@@ -491,9 +439,6 @@ theorem eq_cons_of_mem_head? {x : α} : ∀ {l : List α}, x ∈ l.head? → l =
     simp only [head?, Option.mem_def, Option.some_inj] at h
     exact h ▸ rfl
 
-theorem mem_of_mem_head? {x : α} {l : List α} (h : x ∈ l.head?) : x ∈ l :=
-  (eq_cons_of_mem_head? h).symm ▸ mem_cons_self _ _
-
 @[simp] theorem head!_cons [Inhabited α] (a : α) (l : List α) : head! (a :: l) = a := rfl
 
 @[simp]
@@ -536,33 +481,34 @@ theorem head!_mem_self [Inhabited α] {l : List α} (h : l ≠ nil) : l.head! 
   have h' := mem_cons_self l.head! l.tail
   rwa [cons_head!_tail h] at h'
 
-theorem tail_append_of_ne_nil (l l' : List α) (h : l ≠ []) : (l ++ l').tail = l.tail ++ l' := by
-  cases l
-  · contradiction
-  · simp
-
 theorem get_eq_get? (l : List α) (i : Fin l.length) :
     l.get i = (l.get? i).get (by simp [getElem?_eq_getElem]) := by
-  simp [getElem_eq_iff]
+  simp
+
+theorem exists_mem_iff_getElem {l : List α} {p : α → Prop} :
+    (∃ x ∈ l, p x) ↔ ∃ (i : ℕ) (_ : i < l.length), p l[i] := by
+  simp only [mem_iff_getElem]
+  exact ⟨fun ⟨_x, ⟨i, hi, hix⟩, hxp⟩ ↦ ⟨i, hi, hix ▸ hxp⟩, fun ⟨i, hi, hp⟩ ↦ ⟨_, ⟨i, hi, rfl⟩, hp⟩⟩
+
+theorem forall_mem_iff_getElem {l : List α} {p : α → Prop} :
+    (∀ x ∈ l, p x) ↔ ∀ (i : ℕ) (_ : i < l.length), p l[i] := by
+  simp [mem_iff_getElem, @forall_swap α]
+
+theorem getElem_cons {l : List α} {a : α} {n : ℕ} (h : n < (a :: l).length) :
+    (a :: l)[n] = if hn : n = 0 then a else l[n - 1]'(by rw [length_cons] at h; omega) := by
+  cases n <;> simp
 
 theorem get_tail (l : List α) (i) (h : i < l.tail.length)
     (h' : i + 1 < l.length := (by simp only [length_tail] at h; omega)) :
     l.tail.get ⟨i, h⟩ = l.get ⟨i + 1, h'⟩ := by
   cases l <;> [cases h; rfl]
 
+@[deprecated (since := "2024-08-22")]
 theorem get_cons {l : List α} {a : α} {n} (hl) :
     (a :: l).get ⟨n, hl⟩ = if hn : n = 0 then a else
-      l.get ⟨n - 1, by contrapose! hl; rw [length_cons]; omega⟩ := by
-  split_ifs with h
-  · simp [h]
-  cases l
-  · rw [length_singleton, Nat.lt_succ_iff] at hl
-    omega
-  cases n
-  · contradiction
-  rfl
+      l.get ⟨n - 1, by contrapose! hl; rw [length_cons]; omega⟩ :=
+  getElem_cons hl
 
-@[simp 1100]
 theorem modifyHead_modifyHead (l : List α) (f g : α → α) :
     (l.modifyHead f).modifyHead g = l.modifyHead (g ∘ f) := by cases l <;> simp
 
@@ -691,26 +637,11 @@ lemma cons_sublist_cons' {a b : α} : a :: l₁ <+ b :: l₂ ↔ a :: l₁ <+ l
 
 theorem sublist_cons_of_sublist (a : α) (h : l₁ <+ l₂) : l₁ <+ a :: l₂ := h.cons _
 
-theorem tail_sublist : ∀ l : List α, tail l <+ l
-  | [] => .slnil
-  | a::l => sublist_cons_self a l
-
-@[gcongr] protected theorem Sublist.tail : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → tail l₁ <+ tail l₂
-  | _, _, slnil => .slnil
-  | _, _, Sublist.cons _ h => (tail_sublist _).trans h
-  | _, _, Sublist.cons₂ _ h => h
-
-theorem Sublist.of_cons_cons {l₁ l₂ : List α} {a b : α} (h : a :: l₁ <+ b :: l₂) : l₁ <+ l₂ :=
-  h.tail
-
 @[deprecated (since := "2024-04-07")]
 theorem sublist_of_cons_sublist_cons {a} (h : a :: l₁ <+ a :: l₂) : l₁ <+ l₂ := h.of_cons_cons
 
 @[deprecated (since := "2024-04-07")] alias cons_sublist_cons_iff := cons_sublist_cons
 
-theorem eq_nil_of_sublist_nil {l : List α} (s : l <+ []) : l = [] :=
-  eq_nil_of_subset_nil <| s.subset
-
 -- Porting note: this lemma seems to have been renamed on the occasion of its move to Batteries
 alias sublist_nil_iff_eq_nil := sublist_nil
 
@@ -771,7 +702,7 @@ theorem indexOf_eq_length {a : α} {l : List α} : indexOf a l = length l ↔ a
   rw [cond_eq_if]
   split_ifs with h <;> simp at h
   · exact iff_of_false (by rintro ⟨⟩) fun H => H <| Or.inl h.symm
-  · simp only [Ne.symm h, false_or_iff]
+  · simp only [Ne.symm h, false_or]
     rw [← ih]
     exact succ_inj'
 
@@ -787,7 +718,7 @@ theorem indexOf_le_length {a : α} {l : List α} : indexOf a l ≤ length l := b
   · rw [if_neg h]; exact succ_le_succ ih
 
 theorem indexOf_lt_length {a} {l : List α} : indexOf a l < length l ↔ a ∈ l :=
-  ⟨fun h => Decidable.by_contradiction fun al => Nat.ne_of_lt h <| indexOf_eq_length.2 al,
+  ⟨fun h => Decidable.byContradiction fun al => Nat.ne_of_lt h <| indexOf_eq_length.2 al,
    fun al => (lt_of_le_of_ne indexOf_le_length) fun h => indexOf_eq_length.1 h al⟩
 
 theorem indexOf_append_of_mem {a : α} (h : a ∈ l₁) : indexOf a (l₁ ++ l₂) = indexOf a l₁ := by
@@ -882,14 +813,6 @@ theorem getElem?_indexOf [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) :
 theorem indexOf_get? [DecidableEq α] {a : α} {l : List α} (h : a ∈ l) :
     get? l (indexOf a l) = some a := by simp [h]
 
-@[deprecated (since := "2023-01-05")]
-theorem get_reverse_aux₁ :
-    ∀ (l r : List α) (i h1 h2), get (reverseAux l r) ⟨i + length l, h1⟩ = get r ⟨i, h2⟩
-  | [], r, i => fun h1 _ => rfl
-  | a :: l, r, i => by
-    rw [show i + length (a :: l) = i + 1 + length l from Nat.add_right_comm i (length l) 1]
-    exact fun h1 h2 => get_reverse_aux₁ l (a :: r) (i + 1) h1 (succ_lt_succ h2)
-
 theorem indexOf_inj [DecidableEq α] {l : List α} {x y : α} (hx : x ∈ l) (hy : y ∈ l) :
     indexOf x l = indexOf y l ↔ x = y :=
   ⟨fun h => by
@@ -899,33 +822,13 @@ theorem indexOf_inj [DecidableEq α] {l : List α} {x y : α} (hx : x ∈ l) (hy
       simp only [h]
     simp only [indexOf_get] at x_eq_y; exact x_eq_y, fun h => by subst h; rfl⟩
 
-theorem getElem_reverse_aux₂ :
-    ∀ (l r : List α) (i : Nat) (h1) (h2),
-      (reverseAux l r)[length l - 1 - i]'h1 = l[i]'h2
-  | [], r, i, h1, h2 => absurd h2 (Nat.not_lt_zero _)
-  | a :: l, r, 0, h1, _ => by
-    have aux := get_reverse_aux₁ l (a :: r) 0
-    rw [Nat.zero_add] at aux
-    exact aux _ (zero_lt_succ _)
-  | a :: l, r, i + 1, h1, h2 => by
-    have aux := getElem_reverse_aux₂ l (a :: r) i
-    have heq : length (a :: l) - 1 - (i + 1) = length l - 1 - i := by rw [length]; omega
-    rw [← heq] at aux
-    apply aux
-
-@[simp] theorem getElem_reverse (l : List α) (i : Nat) (h1 h2) :
-    (reverse l)[length l - 1 - i]'h1 = l[i]'h2 :=
-  getElem_reverse_aux₂ _ _ _ _ _
-
-@[deprecated getElem_reverse_aux₂ (since := "2024-06-12")]
-theorem get_reverse_aux₂ (l r : List α) (i : Nat) (h1) (h2) :
-    get (reverseAux l r) ⟨length l - 1 - i, h1⟩ = get l ⟨i, h2⟩ := by
-  simp [getElem_reverse_aux₂, h1, h2]
-
 @[deprecated getElem_reverse (since := "2024-06-12")]
 theorem get_reverse (l : List α) (i : Nat) (h1 h2) :
-    get (reverse l) ⟨length l - 1 - i, h1⟩ = get l ⟨i, h2⟩ :=
-  get_reverse_aux₂ _ _ _ _ _
+    get (reverse l) ⟨length l - 1 - i, h1⟩ = get l ⟨i, h2⟩ := by
+  rw [get_eq_getElem, get_eq_getElem, getElem_reverse]
+  congr
+  dsimp
+  omega
 
 theorem get_reverse' (l : List α) (n) (hn') :
     l.reverse.get n = l.get ⟨l.length - 1 - n, hn'⟩ := by
@@ -969,18 +872,6 @@ theorem modifyNth_eq_set (f : α → α) :
   | n + 1, b :: l =>
     (congr_arg (cons b) (modifyNth_eq_set f n l)).trans <| by cases h : l[n]? <;> simp [h]
 
-theorem length_modifyNthTail (f : List α → List α) (H : ∀ l, length (f l) = length l) :
-    ∀ n l, length (modifyNthTail f n l) = length l
-  | 0, _ => H _
-  | _ + 1, [] => rfl
-  | _ + 1, _ :: _ => @congr_arg _ _ _ _ (· + 1) (length_modifyNthTail _ H _ _)
-
--- Porting note: Duplicate of `modify_get?_length`
--- (but with a substantially better name?)
--- @[simp]
-theorem length_modifyNth (f : α → α) : ∀ n l, length (modifyNth f n l) = length l :=
-  modify_get?_length f
-
 @[simp]
 theorem getElem_set_of_ne {l : List α} {i j : ℕ} (h : i ≠ j) (a : α)
     (hj : j < (l.set i a).length) :
@@ -1018,9 +909,6 @@ theorem infix_bind_of_mem {a : α} {as : List α} (h : a ∈ as) (f : α → Lis
 theorem map_eq_map {α β} (f : α → β) (l : List α) : f <$> l = map f l :=
   rfl
 
-@[simp]
-theorem map_tail (f : α → β) (l) : map f (tail l) = tail (map f l) := by cases l <;> rfl
-
 /-- A single `List.map` of a composition of functions is equal to
 composing a `List.map` with another `List.map`, fully applied.
 This is the reverse direction of `List.map_map`.
@@ -1116,13 +1004,6 @@ theorem zipWith_flip (f : α → β → γ) : ∀ as bs, zipWith (flip f) bs as
 
 /-! ### take, drop -/
 
-theorem take_cons (n) (a : α) (l : List α) : take (succ n) (a :: l) = a :: take n l :=
-  rfl
-
-@[simp]
-theorem drop_tail (l : List α) (n : ℕ) : l.tail.drop n = l.drop (n + 1) := by
-  rw [← drop_drop, drop_one]
-
 theorem cons_getElem_drop_succ {l : List α} {n : Nat} {h : n < l.length} :
     l[n] :: l.drop (n + 1) = l.drop n :=
   (drop_eq_getElem_cons h).symm
@@ -1227,11 +1108,10 @@ theorem foldr_fixed {b : β} : ∀ l : List α, foldr (fun _ b => b) b l = b :=
 -- Porting note (#10618): simp can prove this
 -- @[simp]
 theorem foldr_eta : ∀ l : List α, foldr cons [] l = l := by
-  simp only [foldr_self_append, append_nil, forall_const]
+  simp only [foldr_cons_eq_append, append_nil, forall_const]
 
-@[simp]
 theorem reverse_foldl {l : List α} : reverse (foldl (fun t h => h :: t) [] l) = l := by
-  rw [← foldr_reverse]; simp only [foldr_self_append, append_nil, reverse_reverse]
+  rw [← foldr_reverse]; simp only [foldr_cons_eq_append, append_nil, reverse_reverse]
 
 theorem foldl_hom₂ (l : List ι) (f : α → β → γ) (op₁ : α → ι → α) (op₂ : β → ι → β)
     (op₃ : γ → ι → γ) (a : α) (b : β) (h : ∀ a b i, f (op₁ a i) (op₂ b i) = op₃ (f a b) i) :
@@ -1255,52 +1135,6 @@ theorem injective_foldl_comp {l : List (α → α)} {f : α → α}
     apply Function.Injective.comp hf
     apply hl _ (List.mem_cons_self _ _)
 
-/-- Induction principle for values produced by a `foldr`: if a property holds
-for the seed element `b : β` and for all incremental `op : α → β → β`
-performed on the elements `(a : α) ∈ l`. The principle is given for
-a `Sort`-valued predicate, i.e., it can also be used to construct data. -/
-def foldrRecOn {C : β → Sort*} (l : List α) (op : α → β → β) (b : β) (hb : C b)
-    (hl : ∀ b, C b → ∀ a ∈ l, C (op a b)) : C (foldr op b l) := by
-  induction l with
-  | nil => exact hb
-  | cons hd tl IH =>
-    refine hl _ ?_ hd (mem_cons_self hd tl)
-    refine IH ?_
-    intro y hy x hx
-    exact hl y hy x (mem_cons_of_mem hd hx)
-
-/-- Induction principle for values produced by a `foldl`: if a property holds
-for the seed element `b : β` and for all incremental `op : β → α → β`
-performed on the elements `(a : α) ∈ l`. The principle is given for
-a `Sort`-valued predicate, i.e., it can also be used to construct data. -/
-def foldlRecOn {C : β → Sort*} (l : List α) (op : β → α → β) (b : β) (hb : C b)
-    (hl : ∀ b, C b → ∀ a ∈ l, C (op b a)) : C (foldl op b l) := by
-  induction l generalizing b with
-  | nil => exact hb
-  | cons hd tl IH =>
-    refine IH _ ?_ ?_
-    · exact hl b hb hd (mem_cons_self hd tl)
-    · intro y hy x hx
-      exact hl y hy x (mem_cons_of_mem hd hx)
-
-@[simp]
-theorem foldrRecOn_nil {C : β → Sort*} (op : α → β → β) (b) (hb : C b) (hl) :
-    foldrRecOn [] op b hb hl = hb :=
-  rfl
-
-@[simp]
-theorem foldrRecOn_cons {C : β → Sort*} (x : α) (l : List α) (op : α → β → β) (b) (hb : C b)
-    (hl : ∀ b, C b → ∀ a ∈ x :: l, C (op a b)) :
-    foldrRecOn (x :: l) op b hb hl =
-      hl _ (foldrRecOn l op b hb fun b hb a ha => hl b hb a (mem_cons_of_mem _ ha)) x
-        (mem_cons_self _ _) :=
-  rfl
-
-@[simp]
-theorem foldlRecOn_nil {C : β → Sort*} (op : β → α → β) (b) (hb : C b) (hl) :
-    foldlRecOn [] op b hb hl = hb :=
-  rfl
-
 /-- Consider two lists `l₁` and `l₂` with designated elements `a₁` and `a₂` somewhere in them:
 `l₁ = x₁ ++ [a₁] ++ z₁` and `l₂ = x₂ ++ [a₂] ++ z₂`.
 Assume the designated element `a₂` is present in neither `x₁` nor `z₁`.
@@ -1310,7 +1144,7 @@ lemma append_cons_inj_of_not_mem {x₁ x₂ z₁ z₂ : List α} {a₁ a₂ : α
     (notin_x : a₂ ∉ x₁) (notin_z : a₂ ∉ z₁) :
     x₁ ++ a₁ :: z₁ = x₂ ++ a₂ :: z₂ ↔ x₁ = x₂ ∧ a₁ = a₂ ∧ z₁ = z₂ := by
   constructor
-  · simp only [append_eq_append_iff, cons_eq_append, cons_eq_cons]
+  · simp only [append_eq_append_iff, cons_eq_append_iff, cons_eq_cons]
     rintro (⟨c, rfl, ⟨rfl, rfl, rfl⟩ | ⟨d, rfl, rfl⟩⟩ |
       ⟨c, rfl, ⟨rfl, rfl, rfl⟩ | ⟨d, rfl, rfl⟩⟩) <;> simp_all
   · rintro ⟨rfl, rfl, rfl⟩
@@ -1365,24 +1199,30 @@ theorem get?_succ_scanl {i : ℕ} : (scanl f b l).get? (i + 1) =
     · simp
     · simp only [hl, get?]
 
-theorem get_succ_scanl {i : ℕ} {h : i + 1 < (scanl f b l).length} :
-    (scanl f b l).get ⟨i + 1, h⟩ =
-      f ((scanl f b l).get ⟨i, Nat.lt_of_succ_lt h⟩)
-        (l.get ⟨i, Nat.lt_of_succ_lt_succ (lt_of_lt_of_le h (le_of_eq (length_scanl b l)))⟩) := by
+theorem getElem_succ_scanl {i : ℕ} (h : i + 1 < (scanl f b l).length) :
+    (scanl f b l)[i + 1] =
+      f ((scanl f b l)[i]'(Nat.lt_of_succ_lt h))
+        (l[i]'(Nat.lt_of_succ_lt_succ (h.trans_eq (length_scanl b l)))) := by
   induction i generalizing b l with
   | zero =>
     cases l
     · simp only [length, zero_eq, lt_self_iff_false] at h
     · simp
   | succ i hi =>
-    cases l with
-    | nil =>
-      simp only [length] at h
+    cases l
+    · simp only [length] at h
       exact absurd h (by omega)
-    | cons head tail =>
-      simp_rw [get_of_eq scanl_cons, get_eq_getElem]; rw [getElem_append_right']
-      · simp_rw [length_singleton, Nat.add_one_sub_one]; exact hi
-      · rw [length_singleton]; omega
+    · simp_rw [scanl_cons]
+      rw [getElem_append_right]
+      · simp only [length, Nat.zero_add 1, succ_add_sub_one, hi]; rfl
+      · simp only [length_singleton]; omega
+
+@[deprecated getElem_succ_scanl (since := "2024-08-22")]
+theorem get_succ_scanl {i : ℕ} {h : i + 1 < (scanl f b l).length} :
+    (scanl f b l).get ⟨i + 1, h⟩ =
+      f ((scanl f b l).get ⟨i, Nat.lt_of_succ_lt h⟩)
+        (l.get ⟨i, Nat.lt_of_succ_lt_succ (lt_of_lt_of_le h (le_of_eq (length_scanl b l)))⟩) :=
+  getElem_succ_scanl h
 
 end Scanl
 
@@ -1404,25 +1244,27 @@ section FoldlEqFoldr
 -- foldl and foldr coincide when f is commutative and associative
 variable {f : α → α → α}
 
-theorem foldl1_eq_foldr1 (hassoc : Associative f) :
+theorem foldl1_eq_foldr1 [hassoc : Std.Associative f] :
     ∀ a b l, foldl f a (l ++ [b]) = foldr f b (a :: l)
   | a, b, nil => rfl
   | a, b, c :: l => by
-    simp only [cons_append, foldl_cons, foldr_cons, foldl1_eq_foldr1 hassoc _ _ l]; rw [hassoc]
+    simp only [cons_append, foldl_cons, foldr_cons, foldl1_eq_foldr1 _ _ l]
+    rw [hassoc.assoc]
 
-theorem foldl_eq_of_comm_of_assoc (hcomm : Commutative f) (hassoc : Associative f) :
+theorem foldl_eq_of_comm_of_assoc [hcomm : Std.Commutative f] [hassoc : Std.Associative f] :
     ∀ a b l, foldl f a (b :: l) = f b (foldl f a l)
-  | a, b, nil => hcomm a b
+  | a, b, nil => hcomm.comm a b
   | a, b, c :: l => by
     simp only [foldl_cons]
-    rw [← foldl_eq_of_comm_of_assoc hcomm hassoc .., right_comm _ hcomm hassoc]; rfl
+    have : RightCommutative f := inferInstance
+    rw [← foldl_eq_of_comm_of_assoc .., this.right_comm]; rfl
 
-theorem foldl_eq_foldr (hcomm : Commutative f) (hassoc : Associative f) :
+theorem foldl_eq_foldr [Std.Commutative f] [Std.Associative f] :
     ∀ a l, foldl f a l = foldr f a l
   | a, nil => rfl
   | a, b :: l => by
-    simp only [foldr_cons, foldl_eq_of_comm_of_assoc hcomm hassoc]
-    rw [foldl_eq_foldr hcomm hassoc a l]
+    simp only [foldr_cons, foldl_eq_of_comm_of_assoc]
+    rw [foldl_eq_foldr a l]
 
 end FoldlEqFoldr
 
@@ -1464,13 +1306,6 @@ local notation a " ⋆ " b => op a b
 /-- Notation for `foldl op a l`. -/
 local notation l " <*> " a => foldl op a l
 
-theorem foldl_assoc : ∀ {l : List α} {a₁ a₂}, (l <*> a₁ ⋆ a₂) = a₁ ⋆ l <*> a₂
-  | [], a₁, a₂ => rfl
-  | a :: l, a₁, a₂ =>
-    calc
-      ((a :: l) <*> a₁ ⋆ a₂) = l <*> a₁ ⋆ a₂ ⋆ a := by simp only [foldl_cons, ha.assoc]
-      _ = a₁ ⋆ (a :: l) <*> a₂ := by rw [foldl_assoc, foldl_cons]
-
 theorem foldl_op_eq_op_foldr_assoc :
     ∀ {l : List α} {a₁ a₂}, ((l <*> a₁) ⋆ a₂) = a₁ ⋆ l.foldr (· ⋆ ·) a₂
   | [], a₁, a₂ => rfl
@@ -1523,7 +1358,7 @@ theorem intersperse_cons_cons (a b c : α) (tl : List α) :
 
 section SplitAtOn
 
-variable (p : α → Bool) (xs ys : List α) (ls : List (List α)) (f : List α → List α)
+variable (p : α → Bool) (xs : List α) (ls : List (List α))
 
 attribute [simp] splitAt_eq
 
@@ -1653,18 +1488,19 @@ theorem modifyLast.go_append_one (f : α → α) (a : α) (tl : List α) (r : Ar
     rw [modifyLast.go, modifyLast.go]
     case x_3 | x_3 => exact append_ne_nil_of_right_ne_nil tl (cons_ne_nil a [])
     rw [modifyLast.go_append_one _ _ tl _, modifyLast.go_append_one _ _ tl (Array.push #[] hd)]
-    simp only [Array.toListAppend_eq, Array.push_data, Array.data_toArray, nil_append, append_assoc]
+    simp only [Array.toListAppend_eq, Array.push_toList, Array.toList_toArray, nil_append,
+      append_assoc]
 
 theorem modifyLast_append_one (f : α → α) (a : α) (l : List α) :
     modifyLast f (l ++ [a]) = l ++ [f a] := by
   cases l with
   | nil =>
-    simp only [nil_append, modifyLast, modifyLast.go, Array.toListAppend_eq, Array.data_toArray]
+    simp only [nil_append, modifyLast, modifyLast.go, Array.toListAppend_eq, Array.toList_toArray]
   | cons _ tl =>
     simp only [cons_append, modifyLast]
     rw [modifyLast.go]
     case x_3 => exact append_ne_nil_of_right_ne_nil tl (cons_ne_nil a [])
-    rw [modifyLast.go_append_one, Array.toListAppend_eq, Array.push_data, Array.data_toArray,
+    rw [modifyLast.go_append_one, Array.toListAppend_eq, Array.push_toList, Array.toList_toArray,
       nil_append, cons_append, nil_append, cons_inj_right]
     exact modifyLast_append_one _ _ tl
 
@@ -1699,8 +1535,6 @@ theorem sizeOf_lt_sizeOf_of_mem [SizeOf α] {x : α} {l : List α} (hx : x ∈ l
 
 section find?
 
-variable {p : α → Bool} {l : List α} {a : α}
-
 @[deprecated (since := "2024-05-05")] alias find?_mem := mem_of_find?_eq_some
 
 end find?
@@ -1715,11 +1549,13 @@ variable (f : α → Option α)
 theorem lookmap.go_append (l : List α) (acc : Array α) :
     lookmap.go f l acc = acc.toListAppend (lookmap f l) := by
   cases l with
-  | nil => rfl
+  | nil => simp [go, lookmap]
   | cons hd tl =>
     rw [lookmap, go, go]
     cases f hd with
-    | none => simp only [go_append tl _, Array.toListAppend_eq, append_assoc, Array.push_data]; rfl
+    | none =>
+      simp only [go_append tl _, Array.toListAppend_eq, append_assoc, Array.push_toList]
+      rfl
     | some a => rfl
 
 @[simp]
@@ -1729,13 +1565,13 @@ theorem lookmap_nil : [].lookmap f = [] :=
 @[simp]
 theorem lookmap_cons_none {a : α} (l : List α) (h : f a = none) :
     (a :: l).lookmap f = a :: l.lookmap f := by
-  simp only [lookmap, lookmap.go, Array.toListAppend_eq, Array.data_toArray, nil_append]
+  simp only [lookmap, lookmap.go, Array.toListAppend_eq, Array.toList_toArray, nil_append]
   rw [lookmap.go_append, h]; rfl
 
 @[simp]
 theorem lookmap_cons_some {a b : α} (l : List α) (h : f a = some b) :
     (a :: l).lookmap f = b :: l := by
-  simp only [lookmap, lookmap.go, Array.toListAppend_eq, Array.data_toArray, nil_append]
+  simp only [lookmap, lookmap.go, Array.toListAppend_eq, Array.toList_toArray, nil_append]
   rw [h]
 
 theorem lookmap_some : ∀ l : List α, l.lookmap some = l
@@ -1878,25 +1714,25 @@ theorem monotone_filter_right (l : List α) ⦃p q : α → Bool⦄
 lemma map_filter' {f : α → β} (hf : Injective f) (l : List α)
     [DecidablePred fun b => ∃ a, p a ∧ f a = b] :
     (l.filter p).map f = (l.map f).filter fun b => ∃ a, p a ∧ f a = b := by
-  simp [(· ∘ ·), filter_map, hf.eq_iff]
+  simp [comp_def, filter_map, hf.eq_iff]
 
 lemma filter_attach' (l : List α) (p : {a // a ∈ l} → Bool) [DecidableEq α] :
     l.attach.filter p =
       (l.filter fun x => ∃ h, p ⟨x, h⟩).attach.map (Subtype.map id fun x => mem_of_mem_filter) := by
   classical
   refine map_injective_iff.2 Subtype.coe_injective ?_
-  simp [(· ∘ ·), map_filter' _ Subtype.coe_injective]
+  simp [comp_def, map_filter' _ Subtype.coe_injective]
 
 -- Porting note: `Lean.Internal.coeM` forces us to type-ascript `{x // x ∈ l}`
 lemma filter_attach (l : List α) (p : α → Bool) :
     (l.attach.filter fun x => p x : List {x // x ∈ l}) =
       (l.filter p).attach.map (Subtype.map id fun x => mem_of_mem_filter) :=
   map_injective_iff.2 Subtype.coe_injective <| by
-    simp_rw [map_map, (· ∘ ·), Subtype.map, id, ← Function.comp_apply (g := Subtype.val),
+    simp_rw [map_map, comp_def, Subtype.map, id, ← Function.comp_apply (g := Subtype.val),
       ← filter_map, attach_map_subtype_val]
 
 lemma filter_comm (q) (l : List α) : filter p (filter q l) = filter q (filter p l) := by
-  simp [and_comm]
+  simp [Bool.and_comm]
 
 @[simp]
 theorem filter_true (l : List α) :
@@ -1928,8 +1764,7 @@ theorem dropWhile_get_zero_not (l : List α) (hl : 0 < (l.dropWhile p).length) :
       simp_all only [dropWhile_cons_of_pos]
     · simp [hp]
 
-@[deprecated (since := "2024-08-19")] alias nthLe_tail := get_tail
-@[deprecated (since := "2024-08-19")] alias nthLe_cons := get_cons
+@[deprecated (since := "2024-08-19")] alias nthLe_cons := getElem_cons
 @[deprecated (since := "2024-08-19")] alias dropWhile_nthLe_zero_not := dropWhile_get_zero_not
 
 variable {p} {l : List α}
@@ -2019,7 +1854,7 @@ theorem erase_getElem [DecidableEq ι] {l : List ι} {i : ℕ} (hi : i < l.lengt
     | succ i =>
       have hi' : i < l.length := by simpa using hi
       if ha : a = l[i] then
-        simpa [ha] using .trans (perm_cons_erase (l.getElem_mem i _)) (.cons _ (IH hi'))
+        simpa [ha] using .trans (perm_cons_erase (getElem_mem _)) (.cons _ (IH hi'))
       else
         simpa [ha] using IH hi'
 
@@ -2314,7 +2149,7 @@ variable {p q : α → Prop} {l : List α}
 
 @[simp]
 theorem forall_cons (p : α → Prop) (x : α) : ∀ l : List α, Forall p (x :: l) ↔ p x ∧ Forall p l
-  | [] => (and_true_iff _).symm
+  | [] => (and_iff_left_of_imp fun _ ↦ trivial).symm
   | _ :: _ => Iff.rfl
 
 theorem forall_iff_forall_mem : ∀ {l : List α}, Forall p l ↔ ∀ x ∈ l, p x
@@ -2339,16 +2174,8 @@ end Forall
 
 /-! ### Miscellaneous lemmas -/
 
-@[simp]
-theorem getElem_attach (L : List α) (i : Nat) (h : i < L.attach.length) :
-    L.attach[i].1 = L[i]'(length_attach L ▸ h) :=
-  calc
-    L.attach[i].1 = (L.attach.map Subtype.val)[i]'(by simpa using h) := by
-      rw [getElem_map]
-    _ = L[i]'_ := by congr 2; simp
-
 theorem get_attach (L : List α) (i) :
-    (L.attach.get i).1 = L.get ⟨i, length_attach L ▸ i.2⟩ := by simp
+    (L.attach.get i).1 = L.get ⟨i, length_attach (L := L) ▸ i.2⟩ := by simp
 
 @[simp 1100]
 theorem mem_map_swap (x : α) (y : β) (xs : List (α × β)) :
@@ -2415,6 +2242,15 @@ theorem disjoint_map {f : α → β} {s t : List α} (hf : Function.Injective f)
   rw [← pmap_eq_map _ _ _ (fun _ _ ↦ trivial), ← pmap_eq_map _ _ _ (fun _ _ ↦ trivial)]
   exact disjoint_pmap _ _ (fun _ _ _ _ h' ↦ hf h') h
 
+alias Disjoint.map := disjoint_map
+
+theorem Disjoint.of_map {f : α → β} {s t : List α} (h : Disjoint (s.map f) (t.map f)) :
+    Disjoint s t := fun _a has hat ↦
+  h (mem_map_of_mem f has) (mem_map_of_mem f hat)
+
+theorem Disjoint.map_iff {f : α → β} {s t : List α} (hf : Function.Injective f) :
+    Disjoint (s.map f) (t.map f) ↔ Disjoint s t :=
+  ⟨fun h ↦ h.of_map, fun h ↦ h.map hf⟩
 
 theorem Perm.disjoint_left {l₁ l₂ l : List α} (p : List.Perm l₁ l₂) :
     Disjoint l₁ l ↔ Disjoint l₂ l := by
@@ -2435,8 +2271,7 @@ theorem disjoint_reverse_right {l₁ l₂ : List α} : Disjoint l₁ l₂.revers
 end Disjoint
 
 section lookup
-
-variable {α β : Type*} [BEq α] [LawfulBEq α]
+variable [BEq α] [LawfulBEq α]
 
 lemma lookup_graph (f : α → β) {a : α} {as : List α} (h : a ∈ as) :
     lookup a (as.map fun x => (x, f x)) = some (f a) := by
@@ -2449,3 +2284,5 @@ lemma lookup_graph (f : α → β) {a : α} {as : List α} (h : a ∈ as) :
 end lookup
 
 end List
+
+set_option linter.style.longFile 2700
diff --git a/Mathlib/Data/List/Chain.lean b/Mathlib/Data/List/Chain.lean
index a6c2410414b7d..965aee596b838 100644
--- a/Mathlib/Data/List/Chain.lean
+++ b/Mathlib/Data/List/Chain.lean
@@ -6,7 +6,6 @@ Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
 import Mathlib.Logic.Relation
 import Mathlib.Data.List.Forall2
 import Mathlib.Data.List.Lex
-import Mathlib.Data.List.Infix
 
 /-!
 # Relation chain
@@ -37,18 +36,21 @@ theorem Chain.iff {S : α → α → Prop} (H : ∀ a b, R a b ↔ S a b) {a : 
 theorem Chain.iff_mem {a : α} {l : List α} :
     Chain R a l ↔ Chain (fun x y => x ∈ a :: l ∧ y ∈ l ∧ R x y) a l :=
   ⟨fun p => by
-    induction' p with _ a b l r _ IH <;> constructor <;>
-      [exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩;
-      exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩],
+    induction p with
+    | nil => exact nil
+    | @cons _ _ _ r _ IH =>
+      constructor
+      · exact ⟨mem_cons_self _ _, mem_cons_self _ _, r⟩
+      · exact IH.imp fun a b ⟨am, bm, h⟩ => ⟨mem_cons_of_mem _ am, mem_cons_of_mem _ bm, h⟩,
     Chain.imp fun a b h => h.2.2⟩
 
 theorem chain_singleton {a b : α} : Chain R a [b] ↔ R a b := by
-  simp only [chain_cons, Chain.nil, and_true_iff]
+  simp only [chain_cons, Chain.nil, and_true]
 
 theorem chain_split {a b : α} {l₁ l₂ : List α} :
     Chain R a (l₁ ++ b :: l₂) ↔ Chain R a (l₁ ++ [b]) ∧ Chain R b l₂ := by
   induction' l₁ with x l₁ IH generalizing a <;>
-    simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true_iff, and_assoc]
+    simp only [*, nil_append, cons_append, Chain.nil, chain_cons, and_true, and_assoc]
 
 @[simp]
 theorem chain_append_cons_cons {a b c : α} {l₁ l₂ : List α} :
@@ -97,7 +99,7 @@ protected theorem Chain.pairwise [IsTrans α R] :
   | a, _, @Chain.cons _ _ _ b l h hb =>
     hb.pairwise.cons
       (by
-        simp only [mem_cons, forall_eq_or_imp, h, true_and_iff]
+        simp only [mem_cons, forall_eq_or_imp, h, true_and]
         exact fun c hc => _root_.trans h (rel_of_pairwise_cons hb.pairwise hc))
 
 theorem chain_iff_pairwise [IsTrans α R] {a : α} {l : List α} : Chain R a l ↔ Pairwise R (a :: l) :=
@@ -136,6 +138,19 @@ theorem chain_iff_get {R} : ∀ {a : α} {l : List α}, Chain R a l ↔
     intro i w
     exact h (i+1) (by simp only [length_cons]; omega)
 
+theorem chain_replicate_of_rel (n : ℕ) {a : α} (h : r a a) : Chain r a (replicate n a) :=
+  match n with
+  | 0 => Chain.nil
+  | n + 1 => Chain.cons h (chain_replicate_of_rel n h)
+
+theorem chain_eq_iff_eq_replicate {a : α} {l : List α} :
+    Chain (· = ·) a l ↔ l = replicate l.length a :=
+  match l with
+  | [] => by simp
+  | b :: l => by
+    rw [chain_cons]
+    simp (config := {contextual := true}) [eq_comm, replicate_succ, chain_eq_iff_eq_replicate]
+
 theorem Chain'.imp {S : α → α → Prop} (H : ∀ a b, R a b → S a b) {l : List α} (p : Chain' R l) :
     Chain' S l := by cases l <;> [trivial; exact Chain.imp H p]
 
@@ -230,8 +245,7 @@ theorem chain'_append :
   | [], l => by simp
   | [a], l => by simp [chain'_cons', and_comm]
   | a :: b :: l₁, l₂ => by
-    rw [cons_append, cons_append, chain'_cons, chain'_cons, ← cons_append, chain'_append,
-      and_assoc]
+    rw [cons_append, cons_append, chain'_cons, chain'_cons, ← cons_append, chain'_append, and_assoc]
     simp
 
 theorem Chain'.append (h₁ : Chain' R l₁) (h₂ : Chain' R l₂)
@@ -264,7 +278,7 @@ theorem Chain'.take (h : Chain' R l) (n : ℕ) : Chain' R (take n l) :=
   h.prefix (take_prefix _ _)
 
 theorem chain'_pair {x y} : Chain' R [x, y] ↔ R x y := by
-  simp only [chain'_singleton, chain'_cons, and_true_iff]
+  simp only [chain'_singleton, chain'_cons, and_true]
 
 theorem Chain'.imp_head {x y} (h : ∀ {z}, R x z → R y z) {l} (hl : Chain' R (x :: l)) :
     Chain' R (y :: l) :=
@@ -318,31 +332,53 @@ theorem exists_chain_of_relationReflTransGen (h : Relation.ReflTransGen r a b) :
     refine ⟨d :: l, Chain.cons e hl₁, ?_⟩
     rwa [getLast_cons_cons]
 
+/-- Given a chain from `a` to `b`, and a predicate true at `a`, if `r x y → p x → p y` then
+the predicate is true everywhere in the chain.
+That is, we can propagate the predicate down the chain.
+-/
+theorem Chain.induction (p : α → Prop) (l : List α) (h : Chain r a l)
+    (carries : ∀ ⦃x y : α⦄, r x y → p x → p y) (initial : p a) : ∀ i ∈ l, p i := by
+  induction h with
+  | nil => simp
+  | @cons a b t hab _ h_ind =>
+    simp only [mem_cons, forall_eq_or_imp]
+    exact ⟨carries hab initial, h_ind (carries hab initial)⟩
+
+/-- A version of `List.Chain.induction` for `List.Chain'`
+-/
+theorem Chain'.induction (p : α → Prop) (l : List α) (h : Chain' r l)
+    (carries : ∀ ⦃x y : α⦄, r x y → p x → p y) (initial : (lne : l ≠ []) → p (l.head lne)) :
+    ∀ i ∈ l, p i := by
+  unfold Chain' at h
+  split at h
+  · simp
+  · simp_all only [ne_eq, not_false_eq_true, head_cons, true_implies, mem_cons, forall_eq_or_imp,
+      true_and, reduceCtorEq]
+    exact h.induction p _ carries initial
+
 /-- Given a chain from `a` to `b`, and a predicate true at `b`, if `r x y → p y → p x` then
 the predicate is true everywhere in the chain and at `a`.
 That is, we can propagate the predicate up the chain.
 -/
-theorem Chain.induction (p : α → Prop) (l : List α) (h : Chain r a l)
+theorem Chain.backwards_induction (p : α → Prop) (l : List α) (h : Chain r a l)
     (hb : getLast (a :: l) (cons_ne_nil _ _) = b) (carries : ∀ ⦃x y : α⦄, r x y → p y → p x)
     (final : p b) : ∀ i ∈ a :: l, p i := by
-  induction' l with _ _ l_ih generalizing a
-  · cases hb
-    simpa using final
-  · rw [chain_cons] at h
-    simp only [mem_cons]
-    rintro _ (rfl | H)
-    · apply carries h.1 (l_ih h.2 hb _ (mem_cons.2 (Or.inl rfl)))
-    · apply l_ih h.2 hb _ (mem_cons.2 H)
+  have : Chain' (flip (flip r)) (a :: l) := by simpa [Chain']
+  replace this := chain'_reverse.mpr this
+  simp_rw (config := {singlePass := true}) [← List.mem_reverse]
+  apply this.induction _ _ (fun _ _ h ↦ carries h)
+  simpa only [ne_eq, reverse_eq_nil_iff, not_false_eq_true, head_reverse, forall_true_left, hb,
+    reduceCtorEq]
 
 /-- Given a chain from `a` to `b`, and a predicate true at `b`, if `r x y → p y → p x` then
 the predicate is true at `a`.
 That is, we can propagate the predicate all the way up the chain.
 -/
 @[elab_as_elim]
-theorem Chain.induction_head (p : α → Prop) (l : List α) (h : Chain r a l)
+theorem Chain.backwards_induction_head (p : α → Prop) (l : List α) (h : Chain r a l)
     (hb : getLast (a :: l) (cons_ne_nil _ _) = b) (carries : ∀ ⦃x y : α⦄, r x y → p y → p x)
     (final : p b) : p a :=
-  (Chain.induction p l h hb carries final) _ (mem_cons_self _ _)
+  (Chain.backwards_induction p l h hb carries final) _ (mem_cons_self _ _)
 
 /--
 If there is an `r`-chain starting from `a` and ending at `b`, then `a` and `b` are related by the
@@ -350,7 +386,7 @@ reflexive transitive closure of `r`. The converse of `exists_chain_of_relationRe
 -/
 theorem relationReflTransGen_of_exists_chain (l : List α) (hl₁ : Chain r a l)
     (hl₂ : getLast (a :: l) (cons_ne_nil _ _) = b) : Relation.ReflTransGen r a b :=
-  Chain.induction_head _ l hl₁ hl₂ (fun _ _ => Relation.ReflTransGen.head)
+  Chain.backwards_induction_head _ l hl₁ hl₂ (fun _ _ => Relation.ReflTransGen.head)
     Relation.ReflTransGen.refl
 
 theorem Chain'.cons_of_le [LinearOrder α] {a : α} {as m : List α}
@@ -362,7 +398,7 @@ theorem Chain'.cons_of_le [LinearOrder α] {a : α} {as m : List α}
     apply hm.cons
     cases as with
     | nil =>
-      simp only [le_iff_lt_or_eq, or_false] at hmas
+      simp only [le_iff_lt_or_eq, reduceCtorEq, or_false] at hmas
       exact (List.Lex.not_nil_right (·<·) _ hmas).elim
     | cons a' as =>
       rw [List.chain'_cons] at ha
@@ -376,6 +412,36 @@ theorem Chain'.cons_of_le [LinearOrder α] {a : α} {as m : List α}
         exact (List.lt_iff_lex_lt _ _).mp (List.lt.head _ _ hh)
       · simp_all only [List.cons.injEq, le_refl]
 
+lemma Chain'.chain {α : Type*} {R : α → α → Prop} {l : List α} {v : α}
+    (hl : l.Chain' R) (hv : (lne : l ≠ []) → R v (l.head lne)) : l.Chain R v := by
+  rw [List.chain_iff_get]
+  constructor
+  · intro h
+    rw [List.get_mk_zero]
+    apply hv
+  · exact List.chain'_iff_get.mp hl
+
+lemma Chain'.iterate_eq_of_apply_eq {α : Type*} {f : α → α} {l : List α}
+    (hl : l.Chain' (fun x y ↦ f x = y)) (i : ℕ) (hi : i < l.length) :
+    f^[i] l[0] = l[i] := by
+  induction' i with i h
+  · rfl
+  · rw [Function.iterate_succ', Function.comp_apply, h (by omega)]
+    rw [List.chain'_iff_get] at hl
+    apply hl
+    omega
+
+theorem chain'_replicate_of_rel (n : ℕ) {a : α} (h : r a a) : Chain' r (replicate n a) :=
+  match n with
+  | 0 => chain'_nil
+  | n + 1 => chain_replicate_of_rel n h
+
+theorem chain'_eq_iff_eq_replicate {l : List α} :
+    Chain' (· = ·) l ↔ ∀ a ∈ l.head?, l = replicate l.length a :=
+  match l with
+  | [] => by simp
+  | a :: l => by simp [Chain', chain_eq_iff_eq_replicate, replicate_succ]
+
 end List
 
 
diff --git a/Mathlib/Data/List/Count.lean b/Mathlib/Data/List/Count.lean
index 1e128c3d42e2b..89475cec9fe1e 100644
--- a/Mathlib/Data/List/Count.lean
+++ b/Mathlib/Data/List/Count.lean
@@ -3,7 +3,8 @@ Copyright (c) 2014 Parikshit Khanna. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
 -/
-import Mathlib.Data.Nat.Defs
+import Mathlib.Logic.Function.Basic
+import Mathlib.Tactic.Common
 
 /-!
 # Counting in lists
@@ -19,7 +20,7 @@ assert_not_exists Ring
 
 open Nat
 
-variable {α : Type*} {l : List α}
+variable {α : Type*}
 
 namespace List
 
@@ -30,15 +31,9 @@ section Count
 @[simp]
 theorem count_map_of_injective {β} [DecidableEq α] [DecidableEq β] (l : List α) (f : α → β)
     (hf : Function.Injective f) (x : α) : count (f x) (map f l) = count x l := by
-  simp only [count, countP_map, (· ∘ ·), hf.beq_eq]
-
-variable [DecidableEq α]
-
-@[deprecated (since := "2023-08-23")]
-theorem count_cons' (a b : α) (l : List α) :
-    count a (b :: l) = count a l + if a = b then 1 else 0 := by
-  simp only [count, beq_iff_eq, countP_cons, Nat.add_right_inj]
-  simp only [eq_comm]
+  simp only [count, countP_map]
+  unfold Function.comp
+  simp only [hf.beq_eq]
 
 end Count
 
diff --git a/Mathlib/Data/List/Cycle.lean b/Mathlib/Data/List/Cycle.lean
index d24bebc4562cc..9c8cbaa143209 100644
--- a/Mathlib/Data/List/Cycle.lean
+++ b/Mathlib/Data/List/Cycle.lean
@@ -165,7 +165,7 @@ theorem next_getLast_cons (h : x ∈ l) (y : α) (h : x ∈ y :: l) (hy : x ≠
   intro H
   obtain ⟨⟨_ | k, hk⟩, hk'⟩ := get_of_mem H
   · rw [← Option.some_inj] at hk'
-    rw [← get?_eq_get, dropLast_eq_take, get?_eq_getElem?, getElem?_take, getElem?_cons_zero,
+    rw [← get?_eq_get, dropLast_eq_take, get?_eq_getElem?, getElem?_take_of_lt, getElem?_cons_zero,
       Option.some_inj] at hk'
     · exact hy (Eq.symm hk')
     rw [length_cons]
@@ -178,7 +178,7 @@ theorem next_getLast_cons (h : x ∈ l) (y : α) (h : x ∈ y :: l) (hy : x ≠
     refine Fin.val_eq_of_eq <| @hl ⟨k, Nat.lt_of_succ_lt <| by simpa using hk⟩
       ⟨tl.length, by simp⟩ ?_
     rw [← Option.some_inj] at hk'
-    rw [← get?_eq_get, dropLast_eq_take, get?_eq_getElem?, getElem?_take, getElem?_cons_succ,
+    rw [← get?_eq_get, dropLast_eq_take, get?_eq_getElem?, getElem?_take_of_lt, getElem?_cons_succ,
       getElem?_eq_getElem, Option.some_inj] at hk'
     · rw [get_eq_getElem, hk']
       simp only [getLast_eq_getElem, length_cons, Nat.succ_eq_add_one, Nat.succ_sub_succ_eq_sub,
@@ -334,7 +334,7 @@ theorem prev_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
   obtain ⟨⟨n, hn⟩, rfl⟩ := get_of_mem hx
   simp only [next_get, prev_get, h, Nat.mod_add_mod]
   cases' l with hd tl
-  · simp at hx
+  · simp at hn
   · have : (n + 1 + length tl) % (length tl + 1) = n := by
       rw [length_cons] at hn
       rw [add_assoc, add_comm 1, Nat.add_mod_right, Nat.mod_eq_of_lt hn]
@@ -345,7 +345,7 @@ theorem next_prev (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
   obtain ⟨⟨n, hn⟩, rfl⟩ := get_of_mem hx
   simp only [next_get, prev_get, h, Nat.mod_add_mod]
   cases' l with hd tl
-  · simp at hx
+  · simp at hn
   · have : (n + length tl + 1) % (length tl + 1) = n := by
       rw [length_cons] at hn
       rw [add_assoc, Nat.add_mod_right, Nat.mod_eq_of_lt hn]
@@ -357,14 +357,13 @@ theorem prev_reverse_eq_next (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l)
   have lpos : 0 < l.length := k.zero_le.trans_lt hk
   have key : l.length - 1 - k < l.length := by omega
   rw [← getElem_pmap l.next (fun _ h => h) (by simpa using hk)]
-  simp_rw [← getElem_reverse l k (key.trans_le (by simp)), pmap_next_eq_rotate_one _ h]
+  simp_rw [getElem_eq_getElem_reverse (l := l), pmap_next_eq_rotate_one _ h]
   rw [← getElem_pmap l.reverse.prev fun _ h => h]
   · simp_rw [pmap_prev_eq_rotate_length_sub_one _ (nodup_reverse.mpr h), rotate_reverse,
       length_reverse, Nat.mod_eq_of_lt (Nat.sub_lt lpos Nat.succ_pos'),
       Nat.sub_sub_self (Nat.succ_le_of_lt lpos)]
-    rw [← getElem_reverse]
+    rw [getElem_eq_getElem_reverse]
     · simp [Nat.sub_sub_self (Nat.le_sub_one_of_lt hk)]
-    · simpa using (Nat.sub_le _ _).trans_lt (Nat.sub_lt lpos Nat.succ_pos')
   · simpa
 
 theorem next_reverse_eq_prev (l : List α) (h : Nodup l) (x : α) (hx : x ∈ l) :
@@ -456,7 +455,7 @@ theorem induction_on {C : Cycle α → Prop} (s : Cycle α) (H0 : C nil)
     assumption'
 
 /-- For `x : α`, `s : Cycle α`, `x ∈ s` indicates that `x` occurs at least once in `s`. -/
-def Mem (a : α) (s : Cycle α) : Prop :=
+def Mem (s : Cycle α) (a : α) : Prop :=
   Quot.liftOn s (fun l => a ∈ l) fun _ _ e => propext <| e.mem_iff
 
 instance : Membership α (Cycle α) :=
@@ -529,7 +528,7 @@ theorem Subsingleton.congr {s : Cycle α} (h : Subsingleton s) :
     ∀ ⦃x⦄ (_hx : x ∈ s) ⦃y⦄ (_hy : y ∈ s), x = y := by
   induction' s using Quot.inductionOn with l
   simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,
-    length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h
+    length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or] at h
   rcases h with (rfl | ⟨z, rfl⟩) <;> simp
 
 /-- A `s : Cycle α` that is made up of at least two unique elements. -/
@@ -544,7 +543,7 @@ theorem nontrivial_coe_nodup_iff {l : List α} (hl : l.Nodup) :
   · simp
   · simp
   · simp only [mem_cons, exists_prop, mem_coe_iff, List.length, Ne, Nat.succ_le_succ_iff,
-      Nat.zero_le, iff_true_iff]
+      Nat.zero_le, iff_true]
     refine ⟨hd, hd', ?_, by simp⟩
     simp only [not_or, mem_cons, nodup_cons] at hl
     exact hl.left.left
@@ -881,7 +880,7 @@ theorem chain_iff_pairwise [IsTrans α r] : Chain r s ↔ ∀ a ∈ s, ∀ b ∈
     intro hs b hb c hc
     rw [Cycle.chain_coe_cons, List.chain_iff_pairwise] at hs
     simp only [pairwise_append, pairwise_cons, mem_append, mem_singleton, List.not_mem_nil,
-      IsEmpty.forall_iff, imp_true_iff, Pairwise.nil, forall_eq, true_and_iff] at hs
+      IsEmpty.forall_iff, imp_true_iff, Pairwise.nil, forall_eq, true_and] at hs
     simp only [mem_coe_iff, mem_cons] at hb hc
     rcases hb with (rfl | hb) <;> rcases hc with (rfl | hc)
     · exact hs.1 c (Or.inr rfl)
diff --git a/Mathlib/Data/List/Dedup.lean b/Mathlib/Data/List/Dedup.lean
index d75adc40ad06c..55576850f682a 100644
--- a/Mathlib/Data/List/Dedup.lean
+++ b/Mathlib/Data/List/Dedup.lean
@@ -5,6 +5,7 @@ Authors: Mario Carneiro
 -/
 import Mathlib.Data.List.Nodup
 import Mathlib.Data.List.Lattice
+import Batteries.Data.List.Pairwise
 
 /-!
 # Erasure of duplicates in a list
@@ -81,7 +82,7 @@ theorem dedup_eq_cons (l : List α) (a : α) (l' : List α) :
     l.dedup = a :: l' ↔ a ∈ l ∧ a ∉ l' ∧ l.dedup.tail = l' := by
   refine ⟨fun h => ?_, fun h => ?_⟩
   · refine ⟨mem_dedup.1 (h.symm ▸ mem_cons_self _ _), fun ha => ?_, by rw [h, tail_cons]⟩
-    have := count_pos_iff_mem.2 ha
+    have := count_pos_iff.2 ha
     have : count a l.dedup ≤ 1 := nodup_iff_count_le_one.1 (nodup_dedup l) a
     rw [h, count_cons_self] at this
     omega
@@ -95,7 +96,7 @@ theorem dedup_eq_nil (l : List α) : l.dedup = [] ↔ l = [] := by
   induction' l with a l hl
   · exact Iff.rfl
   · by_cases h : a ∈ l
-    · simp only [List.dedup_cons_of_mem h, hl, List.ne_nil_of_mem h]
+    · simp only [List.dedup_cons_of_mem h, hl, List.ne_nil_of_mem h, reduceCtorEq]
     · simp only [List.dedup_cons_of_not_mem h, List.cons_ne_nil]
 
 protected theorem Nodup.dedup {l : List α} (h : l.Nodup) : l.dedup = l :=
diff --git a/Mathlib/Data/List/Defs.lean b/Mathlib/Data/List/Defs.lean
index fc6eaa2b034dc..f0aad1fcd0f00 100644
--- a/Mathlib/Data/List/Defs.lean
+++ b/Mathlib/Data/List/Defs.lean
@@ -10,6 +10,7 @@ import Mathlib.Util.CompileInductive
 import Batteries.Tactic.Lint.Basic
 import Batteries.Data.List.Lemmas
 import Batteries.Data.RBMap.Basic
+import Batteries.Logic
 
 /-!
 ## Definitions on lists
@@ -446,4 +447,64 @@ theorem iterate_eq_iterateTR : @iterate = @iterateTR := by
   funext α f a n
   exact Eq.symm <| iterateTR_loop_eq f a n []
 
+section MapAccumr
+
+/-- Runs a function over a list returning the intermediate results and a final result. -/
+def mapAccumr (f : α → γ → γ × β) : List α → γ → γ × List β
+  | [], c => (c, [])
+  | y :: yr, c =>
+    let r := mapAccumr f yr c
+    let z := f y r.1
+    (z.1, z.2 :: r.2)
+
+/-- Length of the list obtained by `mapAccumr`. -/
+@[simp]
+theorem length_mapAccumr :
+    ∀ (f : α → γ → γ × β) (x : List α) (s : γ), length (mapAccumr f x s).2 = length x
+  | f, _ :: x, s => congr_arg succ (length_mapAccumr f x s)
+  | _, [], _ => rfl
+
+/-- Runs a function over two lists returning the intermediate results and a final result. -/
+def mapAccumr₂ (f : α → β → γ → γ × δ) : List α → List β → γ → γ × List δ
+  | [], _, c => (c, [])
+  | _, [], c => (c, [])
+  | x :: xr, y :: yr, c =>
+    let r := mapAccumr₂ f xr yr c
+    let q := f x y r.1
+    (q.1, q.2 :: r.2)
+
+/-- Length of a list obtained using `mapAccumr₂`. -/
+@[simp]
+theorem length_mapAccumr₂ :
+    ∀ (f : α → β → γ → γ × δ) (x y c), length (mapAccumr₂ f x y c).2 = min (length x) (length y)
+  | f, _ :: x, _ :: y, c =>
+    calc
+      succ (length (mapAccumr₂ f x y c).2) = succ (min (length x) (length y)) :=
+        congr_arg succ (length_mapAccumr₂ f x y c)
+      _ = min (succ (length x)) (succ (length y)) := Eq.symm (succ_min_succ (length x) (length y))
+  | _, _ :: _, [], _ => rfl
+  | _, [], _ :: _, _ => rfl
+  | _, [], [], _ => rfl
+
+end MapAccumr
+
+section Deprecated
+
+@[deprecated List.mem_cons (since := "2024-08-10")]
+theorem mem_cons_eq (a y : α) (l : List α) : (a ∈ y :: l) = (a = y ∨ a ∈ l) :=
+  propext List.mem_cons
+
+alias ⟨eq_or_mem_of_mem_cons, _⟩ := mem_cons
+
+@[deprecated List.not_mem_nil (since := "2024-08-10")]
+theorem not_exists_mem_nil (p : α → Prop) : ¬∃ x ∈ @nil α, p x :=
+  fun ⟨_, hx, _⟩ => List.not_mem_nil _ hx
+
+@[deprecated (since := "2024-03-23")] alias not_bex_nil := not_exists_mem_nil
+@[deprecated (since := "2024-03-23")] alias bex_cons := exists_mem_cons
+
+@[deprecated (since := "2024-08-10")] alias length_le_of_sublist := Sublist.length_le
+
+end Deprecated
+
 end List
diff --git a/Mathlib/Data/List/DropRight.lean b/Mathlib/Data/List/DropRight.lean
index 1642572f92b56..10638f80de52e 100644
--- a/Mathlib/Data/List/DropRight.lean
+++ b/Mathlib/Data/List/DropRight.lean
@@ -3,8 +3,7 @@ Copyright (c) 2022 Yakov Pechersky. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yakov Pechersky
 -/
-import Mathlib.Data.List.Infix
-
+import Mathlib.Data.List.Basic
 /-!
 
 # Dropping or taking from lists on the right
@@ -199,7 +198,7 @@ theorem rtakeWhile_eq_nil_iff : rtakeWhile p l = [] ↔ ∀ hl : l ≠ [], ¬p (
   · simp only [rtakeWhile, takeWhile, reverse_nil, true_iff]
     intro f; contradiction
   · simp only [rtakeWhile, reverse_append, takeWhile, ne_eq, not_false_eq_true,
-      getLast_append_of_ne_nil, getLast_singleton]
+      getLast_append_of_ne_nil, getLast_singleton, reduceCtorEq]
     refine ⟨fun h => ?_ , fun h => ?_⟩
     · split at h <;> simp_all
     · simp [h]
diff --git a/Mathlib/Data/List/Duplicate.lean b/Mathlib/Data/List/Duplicate.lean
index 0a020d73eafca..9023118d20cb8 100644
--- a/Mathlib/Data/List/Duplicate.lean
+++ b/Mathlib/Data/List/Duplicate.lean
@@ -39,9 +39,9 @@ theorem Duplicate.duplicate_cons (h : x ∈+ l) (y : α) : x ∈+ y :: l :=
   Duplicate.cons_duplicate h
 
 theorem Duplicate.mem (h : x ∈+ l) : x ∈ l := by
-  induction' h with l' _ y l' _ hm
-  · exact mem_cons_self _ _
-  · exact mem_cons_of_mem _ hm
+  induction h with
+  | cons_mem => exact mem_cons_self _ _
+  | cons_duplicate _ hm => exact mem_cons_of_mem _ hm
 
 theorem Duplicate.mem_cons_self (h : x ∈+ x :: l) : x ∈ l := by
   cases' h with _ h _ _ h
diff --git a/Mathlib/Data/List/EditDistance/Bounds.lean b/Mathlib/Data/List/EditDistance/Bounds.lean
index 5ad1aafb05254..7a20e9e681c9a 100644
--- a/Mathlib/Data/List/EditDistance/Bounds.lean
+++ b/Mathlib/Data/List/EditDistance/Bounds.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Order.Monoid.Canonical.Defs
 import Mathlib.Data.List.Infix
diff --git a/Mathlib/Data/List/EditDistance/Defs.lean b/Mathlib/Data/List/EditDistance/Defs.lean
index ed13538ca6e25..a52cf793066d8 100644
--- a/Mathlib/Data/List/EditDistance/Defs.lean
+++ b/Mathlib/Data/List/EditDistance/Defs.lean
@@ -1,9 +1,10 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Defs
+import Batteries.Data.List.Basic
 
 /-!
 # Levenshtein distances
diff --git a/Mathlib/Data/List/EditDistance/Estimator.lean b/Mathlib/Data/List/EditDistance/Estimator.lean
index 056d119f6af71..e000b2bc0d02b 100644
--- a/Mathlib/Data/List/EditDistance/Estimator.lean
+++ b/Mathlib/Data/List/EditDistance/Estimator.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger
+Authors: Kim Morrison
 -/
 import Mathlib.Data.List.EditDistance.Bounds
 import Mathlib.Order.Estimator
diff --git a/Mathlib/Data/List/Enum.lean b/Mathlib/Data/List/Enum.lean
index 244f19ab4893c..d4b73fc6bec78 100644
--- a/Mathlib/Data/List/Enum.lean
+++ b/Mathlib/Data/List/Enum.lean
@@ -3,33 +3,43 @@ Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
 -/
-import Batteries.Tactic.Alias
-import Mathlib.Tactic.TypeStar
-import Mathlib.Data.Nat.Notation
+import Mathlib.Data.List.Basic
 
 /-!
 # Properties of `List.enum`
+
+## Deprecation note
+
+Many lemmas in this file have been replaced by theorems in Lean4,
+in terms of `xs[i]?` and `xs[i]` rather than `get` and `get?`.
+
+The deprecated results here are unused in Mathlib.
+Any downstream users who can not easily adapt may remove the deprecations as needed.
 -/
 
 namespace List
 
 variable {α β : Type*}
 
+@[deprecated getElem?_enumFrom (since := "2024-08-15")]
 theorem get?_enumFrom (n) (l : List α) (m) :
     get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a) := by
   simp
 
 @[deprecated (since := "2024-04-06")] alias enumFrom_get? := get?_enumFrom
 
+@[deprecated getElem?_enum (since := "2024-08-15")]
 theorem get?_enum (l : List α) (n) : get? (enum l) n = (get? l n).map fun a => (n, a) := by
   simp
 
 @[deprecated (since := "2024-04-06")] alias enum_get? := get?_enum
 
+@[deprecated getElem_enumFrom (since := "2024-08-15")]
 theorem get_enumFrom (l : List α) (n) (i : Fin (l.enumFrom n).length) :
     (l.enumFrom n).get i = (n + i, l.get (i.cast enumFrom_length)) := by
   simp
 
+@[deprecated getElem_enum (since := "2024-08-15")]
 theorem get_enum (l : List α) (i : Fin l.enum.length) :
     l.enum.get i = (i.1, l.get (i.cast enum_length)) := by
   simp
@@ -44,12 +54,29 @@ theorem mk_mem_enumFrom_iff_le_and_get?_sub {n i : ℕ} {x : α} {l : List α} :
     (i, x) ∈ enumFrom n l ↔ n ≤ i ∧ l.get? (i - n) = x := by
   simp [mk_mem_enumFrom_iff_le_and_getElem?_sub]
 
-@[deprecated mem_enum_iff_getElem? (since := "2024-08-12")]
+@[deprecated mk_mem_enum_iff_getElem? (since := "2024-08-15")]
 theorem mk_mem_enum_iff_get? {i : ℕ} {x : α} {l : List α} : (i, x) ∈ enum l ↔ l.get? i = x := by
   simp [enum, mk_mem_enumFrom_iff_le_and_getElem?_sub]
 
-@[deprecated mk_mem_enum_iff_getElem? (since := "2024-08-12")]
-theorem mem_enum_iff_get? {x : ℕ × α} {l : List α} : x ∈ enum l ↔ l.get? x.1 = x.2 := by
-  simp [mem_enum_iff_getElem?]
+set_option linter.deprecated false in
+@[deprecated mem_enum_iff_getElem? (since := "2024-08-15")]
+theorem mem_enum_iff_get? {x : ℕ × α} {l : List α} : x ∈ enum l ↔ l.get? x.1 = x.2 :=
+  mk_mem_enum_iff_get?
+
+theorem forall_mem_enumFrom {l : List α} {n : ℕ} {p : ℕ × α → Prop} :
+    (∀ x ∈ l.enumFrom n, p x) ↔ ∀ (i : ℕ) (_ : i < length l), p (n + i, l[i]) := by
+  simp only [forall_mem_iff_getElem, getElem_enumFrom, enumFrom_length]
+
+theorem forall_mem_enum {l : List α} {p : ℕ × α → Prop} :
+    (∀ x ∈ l.enum, p x) ↔ ∀ (i : ℕ) (_ : i < length l), p (i, l[i]) :=
+  forall_mem_enumFrom.trans <| by simp
+
+theorem exists_mem_enumFrom {l : List α} {n : ℕ} {p : ℕ × α → Prop} :
+    (∃ x ∈ l.enumFrom n, p x) ↔ ∃ (i : ℕ) (_ : i < length l), p (n + i, l[i]) := by
+  simp only [exists_mem_iff_getElem, getElem_enumFrom, enumFrom_length]
+
+theorem exists_mem_enum {l : List α} {p : ℕ × α → Prop} :
+    (∃ x ∈ l.enum, p x) ↔ ∃ (i : ℕ) (_ : i < length l), p (i, l[i]) :=
+  exists_mem_enumFrom.trans <| by simp
 
 end List
diff --git a/Mathlib/Data/List/FinRange.lean b/Mathlib/Data/List/FinRange.lean
index 3ca008028c7c7..62d8122999597 100644
--- a/Mathlib/Data/List/FinRange.lean
+++ b/Mathlib/Data/List/FinRange.lean
@@ -1,10 +1,11 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Kenny Lau, Scott Morrison, Alex Keizer
+Authors: Mario Carneiro, Kenny Lau, Kim Morrison, Alex Keizer
 -/
 import Mathlib.Data.List.OfFn
 import Mathlib.Data.List.Range
+import Batteries.Data.List.Perm
 
 /-!
 # Lists of elements of `Fin n`
@@ -29,7 +30,7 @@ theorem finRange_succ_eq_map (n : ℕ) : finRange n.succ = 0 :: (finRange n).map
   apply map_injective_iff.mpr Fin.val_injective
   rw [map_cons, map_coe_finRange, range_succ_eq_map, Fin.val_zero, ← map_coe_finRange, map_map,
     map_map]
-  simp only [Function.comp, Fin.val_succ]
+  simp only [Function.comp_def, Fin.val_succ]
 
 theorem finRange_succ (n : ℕ) :
     finRange n.succ = (finRange n |>.map Fin.castSucc |>.concat (.last _)) := by
@@ -41,7 +42,7 @@ theorem finRange_succ (n : ℕ) :
 theorem ofFn_eq_pmap {n} {f : Fin n → α} :
     ofFn f = pmap (fun i hi => f ⟨i, hi⟩) (range n) fun _ => mem_range.1 := by
   rw [pmap_eq_map_attach]
-  exact ext_getElem (by simp) fun i hi1 hi2 => by simp [getElem_ofFn f i hi1]
+  exact ext_getElem (by simp) fun i hi1 hi2 => by simp [List.getElem_ofFn f i hi1]
 
 theorem ofFn_id (n) : ofFn id = finRange n :=
   ofFn_eq_pmap
@@ -71,7 +72,7 @@ open List
 theorem Equiv.Perm.map_finRange_perm {n : ℕ} (σ : Equiv.Perm (Fin n)) :
     map σ (finRange n) ~ finRange n := by
   rw [perm_ext_iff_of_nodup ((nodup_finRange n).map σ.injective) <| nodup_finRange n]
-  simpa [mem_map, mem_finRange, true_and_iff, iff_true_iff] using σ.surjective
+  simpa [mem_map, mem_finRange] using σ.surjective
 
 /-- The list obtained from a permutation of a tuple `f` is permutation equivalent to
 the list obtained from `f`. -/
diff --git a/Mathlib/Data/List/Forall2.lean b/Mathlib/Data/List/Forall2.lean
index df7cdc09f1e48..20808976c60ab 100644
--- a/Mathlib/Data/List/Forall2.lean
+++ b/Mathlib/Data/List/Forall2.lean
@@ -84,15 +84,18 @@ theorem forall₂_cons_right_iff {b l u} :
     match u, h with
     | _, ⟨_, _, h₁, h₂, rfl⟩ => Forall₂.cons h₁ h₂
 
+#adaptation_note
+/--
+After nightly-2024-09-06 we can remove the `_root_` prefixes below.
+-/
 theorem forall₂_and_left {p : α → Prop} :
     ∀ l u, Forall₂ (fun a b => p a ∧ R a b) l u ↔ (∀ a ∈ l, p a) ∧ Forall₂ R l u
   | [], u => by
-    simp only [forall₂_nil_left_iff, forall_prop_of_false (not_mem_nil _), imp_true_iff,
-      true_and_iff]
+    simp only [forall₂_nil_left_iff, forall_prop_of_false (not_mem_nil _), imp_true_iff, true_and]
   | a :: l, u => by
-    simp only [forall₂_and_left l, forall₂_cons_left_iff, forall_mem_cons, and_assoc,
+    simp only [forall₂_and_left l, forall₂_cons_left_iff, forall_mem_cons, _root_.and_assoc,
       @and_comm _ (p a), @and_left_comm _ (p a), exists_and_left]
-    simp only [and_comm, and_assoc, and_left_comm, ← exists_and_right]
+    simp only [_root_.and_comm, _root_.and_assoc, and_left_comm, ← exists_and_right]
 
 @[simp]
 theorem forall₂_map_left_iff {f : γ → α} :
@@ -244,7 +247,7 @@ theorem rel_filter {p : α → Bool} {q : β → Bool}
     dsimp [LiftFun] at hpq
     by_cases h : p a
     · have : q b := by rwa [← hpq h₁]
-      simp only [filter_cons_of_pos h, filter_cons_of_pos this, forall₂_cons, h₁, true_and_iff,
+      simp only [filter_cons_of_pos h, filter_cons_of_pos this, forall₂_cons, h₁, true_and,
         rel_filter hpq h₂]
     · have : ¬q b := by rwa [← hpq h₁]
       simp only [filter_cons_of_neg h, filter_cons_of_neg this, rel_filter hpq h₂]
@@ -268,19 +271,25 @@ inductive SublistForall₂ (R : α → β → Prop) : List α → List β → Pr
 theorem sublistForall₂_iff {l₁ : List α} {l₂ : List β} :
     SublistForall₂ R l₁ l₂ ↔ ∃ l, Forall₂ R l₁ l ∧ l <+ l₂ := by
   constructor <;> intro h
-  · induction' h with _ a b l1 l2 rab _ ih b l1 l2 _ ih
-    · exact ⟨nil, Forall₂.nil, nil_sublist _⟩
-    · obtain ⟨l, hl1, hl2⟩ := ih
+  · induction h with
+    | nil => exact ⟨nil, Forall₂.nil, nil_sublist _⟩
+    | @cons a b l1 l2 rab _ ih =>
+      obtain ⟨l, hl1, hl2⟩ := ih
       exact ⟨b :: l, Forall₂.cons rab hl1, hl2.cons_cons b⟩
-    · obtain ⟨l, hl1, hl2⟩ := ih
+    | cons_right _ ih =>
+      obtain ⟨l, hl1, hl2⟩ := ih
       exact ⟨l, hl1, hl2.trans (Sublist.cons _ (Sublist.refl _))⟩
   · obtain ⟨l, hl1, hl2⟩ := h
     revert l₁
-    induction' hl2 with _ _ _ _ ih _ _ _ _ ih <;> intro l₁ hl1
-    · rw [forall₂_nil_right_iff.1 hl1]
+    induction hl2 with
+    | slnil =>
+      intro l₁ hl1
+      rw [forall₂_nil_right_iff.1 hl1]
       exact SublistForall₂.nil
-    · exact SublistForall₂.cons_right (ih hl1)
-    · cases' hl1 with _ _ _ _ hr hl _
+    | cons _ _ ih => intro l₁ hl1; exact SublistForall₂.cons_right (ih hl1)
+    | cons₂ _ _ ih =>
+      intro l₁ hl1
+      cases' hl1 with _ _ _ _ hr hl _
       exact SublistForall₂.cons hr (ih hl)
 
 instance SublistForall₂.is_refl [IsRefl α Rₐ] : IsRefl (List α) (SublistForall₂ Rₐ) :=
@@ -289,11 +298,13 @@ instance SublistForall₂.is_refl [IsRefl α Rₐ] : IsRefl (List α) (SublistFo
 instance SublistForall₂.is_trans [IsTrans α Rₐ] : IsTrans (List α) (SublistForall₂ Rₐ) :=
   ⟨fun a b c => by
     revert a b
-    induction' c with _ _ ih
-    · rintro _ _ h1 h2
+    induction c with
+    | nil =>
+      rintro _ _ h1 h2
       cases h2
       exact h1
-    · rintro a b h1 h2
+    | cons _ _ ih =>
+      rintro a b h1 h2
       cases' h2 with _ _ _ _ _ hbc tbc _ _ y1 btc
       · cases h1
         exact SublistForall₂.nil
@@ -310,4 +321,22 @@ theorem Sublist.sublistForall₂ {l₁ l₂ : List α} (h : l₁ <+ l₂) [IsRef
 theorem tail_sublistForall₂_self [IsRefl α Rₐ] (l : List α) : SublistForall₂ Rₐ l.tail l :=
   l.tail_sublist.sublistForall₂
 
+@[simp]
+theorem sublistForall₂_map_left_iff {f : γ → α} {l₁ : List γ} {l₂ : List β} :
+    SublistForall₂ R (map f l₁) l₂ ↔ SublistForall₂ (fun c b => R (f c) b) l₁ l₂ := by
+  simp [sublistForall₂_iff]
+
+@[simp]
+theorem sublistForall₂_map_right_iff {f : γ → β} {l₁ : List α} {l₂ : List γ} :
+    SublistForall₂ R l₁ (map f l₂) ↔ SublistForall₂ (fun a c => R a (f c)) l₁ l₂ := by
+  simp only [sublistForall₂_iff]
+  constructor
+  · rintro ⟨l1, h1, h2⟩
+    obtain ⟨l', hl1, rfl⟩ := sublist_map_iff.mp h2
+    use l'
+    simpa [hl1] using h1
+  · rintro ⟨l1, h1, h2⟩
+    use l1.map f
+    simp [h1, h2.map]
+
 end List
diff --git a/Mathlib/Data/List/GetD.lean b/Mathlib/Data/List/GetD.lean
index bc144f95b4785..e16ce04be7e33 100644
--- a/Mathlib/Data/List/GetD.lean
+++ b/Mathlib/Data/List/GetD.lean
@@ -72,16 +72,13 @@ alias getD_replicate_default_eq := getElem?_getD_replicate_default_eq
 theorem getD_append (l l' : List α) (d : α) (n : ℕ) (h : n < l.length) :
     (l ++ l').getD n d = l.getD n d := by
   rw [getD_eq_getElem _ _ (Nat.lt_of_lt_of_le h (length_append _ _ ▸ Nat.le_add_right _ _)),
-    getElem_append _ h, getD_eq_getElem]
+    getElem_append_left h, getD_eq_getElem]
 
 theorem getD_append_right (l l' : List α) (d : α) (n : ℕ) (h : l.length ≤ n) :
     (l ++ l').getD n d = l'.getD (n - l.length) d := by
   cases Nat.lt_or_ge n (l ++ l').length with
   | inl h' =>
-    rw [getD_eq_getElem (l ++ l') d h', getElem_append_right, getD_eq_getElem]
-    · rw [length_append] at h'
-      exact Nat.sub_lt_left_of_lt_add h h'
-    · exact Nat.not_lt_of_le h
+    rw [getD_eq_getElem (l ++ l') d h', getElem_append_right h, getD_eq_getElem]
   | inr h' =>
     rw [getD_eq_default _ _ h', getD_eq_default]
     rwa [Nat.le_sub_iff_add_le' h, ← length_append]
diff --git a/Mathlib/Data/List/Indexes.lean b/Mathlib/Data/List/Indexes.lean
index a6c5eba666fed..7e53018e89d4b 100644
--- a/Mathlib/Data/List/Indexes.lean
+++ b/Mathlib/Data/List/Indexes.lean
@@ -55,9 +55,9 @@ theorem mapIdxGo_append : ∀ (f : ℕ → α → β) (l₁ l₂ : List α) (arr
       cases l₂
       · rfl
       · rw [List.length_append] at h; contradiction
-    rw [l₁_nil, l₂_nil]; simp only [mapIdx.go, Array.toList_eq, Array.toArray_data]
+    rw [l₁_nil, l₂_nil]; simp only [mapIdx.go, List.toArray_toList]
   · cases' l₁ with head tail <;> simp only [mapIdx.go]
-    · simp only [nil_append, Array.toList_eq, Array.toArray_data]
+    · simp only [nil_append, List.toArray_toList]
     · simp only [List.append_eq]
       rw [ih]
       · simp only [cons_append, length_cons, length_append, Nat.succ.injEq] at h
@@ -67,7 +67,7 @@ theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array
     length (mapIdx.go f l arr) = length l + arr.size := by
   intro f l
   induction' l with head tail ih
-  · intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
+  · intro; simp only [mapIdx.go, length_nil, Nat.zero_add]
   · intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons]
     simp only [Nat.add_succ, Fin.add_zero, Nat.add_comm]
 
@@ -77,7 +77,7 @@ theorem mapIdx_append_one : ∀ (f : ℕ → α → β) (l : List α) (e : α),
   unfold mapIdx
   rw [mapIdxGo_append f l [e]]
   simp only [mapIdx.go, Array.size_toArray, mapIdxGo_length, length_nil, Nat.add_zero,
-    Array.toList_eq, Array.push_data, Array.data_toArray]
+    Array.push_toList, Array.toList_toArray]
 
 @[local simp]
 theorem map_enumFrom_eq_zipWith : ∀ (l : List α) (n : ℕ) (f : ℕ → α → β),
@@ -119,15 +119,16 @@ theorem getElem?_mapIdx_go (f : ℕ → α → β) : ∀ (l : List α) (arr : Ar
     (mapIdx.go f l arr)[i]? =
       if h : i < arr.size then some arr[i] else Option.map (f i) l[i - arr.size]?
   | [], arr, i => by
-    simp [mapIdx.go, getElem?_eq, Array.getElem_eq_data_getElem]
+    simp only [mapIdx.go, Array.toListImpl_eq, getElem?_eq, Array.length_toList,
+      Array.getElem_eq_getElem_toList, length_nil, Nat.not_lt_zero, ↓reduceDIte, Option.map_none']
   | a :: l, arr, i => by
     rw [mapIdx.go, getElem?_mapIdx_go]
     simp only [Array.size_push]
     split <;> split
     · simp only [Option.some.injEq]
-      rw [Array.getElem_eq_data_getElem]
-      simp only [Array.push_data]
-      rw [getElem_append_left, Array.getElem_eq_data_getElem]
+      rw [Array.getElem_eq_getElem_toList]
+      simp only [Array.push_toList]
+      rw [getElem_append_left, Array.getElem_eq_getElem_toList]
     · have : i = arr.size := by omega
       simp_all
     · omega
@@ -158,7 +159,7 @@ theorem mapIdx_append (K L : List α) (f : ℕ → α → β) :
 
 @[simp]
 theorem mapIdx_eq_nil {f : ℕ → α → β} {l : List α} : List.mapIdx f l = [] ↔ l = [] := by
-  rw [List.mapIdx_eq_enum_map, List.map_eq_nil, List.enum_eq_nil]
+  rw [List.mapIdx_eq_enum_map, List.map_eq_nil_iff, List.enum_eq_nil]
 
 theorem get_mapIdx (l : List α) (f : ℕ → α → β) (i : ℕ) (h : i < l.length)
     (h' : i < (l.mapIdx f).length := h.trans_le (l.length_mapIdx f).ge) :
@@ -282,73 +283,6 @@ theorem findIdxs_eq_map_indexesValues (p : α → Prop) [DecidablePred p] (as :
     map_filter_eq_foldr, findIdxs, uncurry, foldrIdx_eq_foldr_enum, decide_eq_true_eq, comp_apply,
     Bool.cond_decide]
 
-section FindIdx -- TODO: upstream to Batteries
-
-theorem findIdx_eq_length {p : α → Bool} {xs : List α} :
-    xs.findIdx p = xs.length ↔ ∀ x ∈ xs, ¬p x := by
-  induction xs with
-  | nil => simp_all
-  | cons x xs ih =>
-    rw [findIdx_cons, length_cons]
-    constructor <;> intro h
-    · have : ¬p x := by contrapose h; simp_all
-      simp_all
-    · simp_rw [h x (mem_cons_self x xs), cond_false, Nat.succ.injEq, ih]
-      exact fun y hy ↦ h y <| mem_cons.mpr (Or.inr hy)
-
-theorem findIdx_le_length (p : α → Bool) {xs : List α} : xs.findIdx p ≤ xs.length := by
-  by_cases e : ∃ x ∈ xs, p x
-  · exact (findIdx_lt_length_of_exists e).le
-  · push_neg at e; exact (findIdx_eq_length.mpr e).le
-
-theorem findIdx_lt_length {p : α → Bool} {xs : List α} :
-    xs.findIdx p < xs.length ↔ ∃ x ∈ xs, p x := by
-  rw [← not_iff_not, not_lt]
-  have := @le_antisymm_iff _ _ (xs.findIdx p) xs.length
-  simp only [findIdx_le_length, true_and] at this
-  rw [← this, findIdx_eq_length, not_exists]
-  simp only [Bool.not_eq_true, not_and]
-
-/-- `p` does not hold for elements with indices less than `xs.findIdx p`. -/
-theorem not_of_lt_findIdx {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.findIdx p) :
-    ¬p (xs.get ⟨i, h.trans_le (findIdx_le_length p)⟩) := by
-  revert i
-  induction xs with
-  | nil => intro i h; rw [findIdx_nil] at h; omega
-  | cons x xs ih =>
-    intro i h
-    have ho := h
-    rw [findIdx_cons] at h
-    have npx : ¬p x := by by_contra y; rw [y, cond_true] at h; omega
-    simp_rw [npx, cond_false] at h
-    cases' i.eq_zero_or_pos with e e
-    · simpa only [e, Fin.zero_eta, get_cons_zero]
-    · have ipm := Nat.succ_pred_eq_of_pos e
-      have ilt := ho.trans_le (findIdx_le_length p)
-      rw [(Fin.mk_eq_mk (h' := ipm ▸ ilt)).mpr ipm.symm, get_cons_succ]
-      rw [← ipm, Nat.succ_lt_succ_iff] at h
-      exact ih h
-
-theorem le_findIdx_of_not {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length)
-    (h2 : ∀ j (hji : j < i), ¬p (xs.get ⟨j, hji.trans h⟩)) : i ≤ xs.findIdx p := by
-  by_contra! f
-  exact absurd (@findIdx_get _ p xs (f.trans h)) (h2 (xs.findIdx p) f)
-
-theorem lt_findIdx_of_not {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length)
-    (h2 : ∀ j (hji : j ≤ i), ¬p (xs.get ⟨j, hji.trans_lt h⟩)) : i < xs.findIdx p := by
-  by_contra! f
-  exact absurd (@findIdx_get _ p xs (f.trans_lt h)) (h2 (xs.findIdx p) f)
-
-theorem findIdx_eq {p : α → Bool} {xs : List α} {i : ℕ} (h : i < xs.length) :
-    xs.findIdx p = i ↔ p (xs.get ⟨i, h⟩) ∧ ∀ j (hji : j < i), ¬p (xs.get ⟨j, hji.trans h⟩) := by
-  refine ⟨fun f ↦ ⟨f ▸ (@findIdx_get _ p xs (f ▸ h)), fun _ hji ↦ not_of_lt_findIdx (f ▸ hji)⟩,
-    fun ⟨h1, h2⟩ ↦ ?_⟩
-  apply Nat.le_antisymm _ (le_findIdx_of_not h h2)
-  contrapose! h1
-  exact not_of_lt_findIdx h1
-
-end FindIdx
-
 section FoldlIdx
 
 -- Porting note: Changed argument order of `foldlIdxSpec` to align better with `foldlIdx`.
@@ -423,12 +357,12 @@ theorem mapIdxMGo_eq_mapIdxMAuxSpec
       congr
       conv => { lhs; intro x; rw [ih _ _ h]; }
       funext x
-      simp only [Array.toList_eq, Array.push_data, append_assoc, singleton_append, Array.size_push,
+      simp only [Array.push_toList, append_assoc, singleton_append, Array.size_push,
         map_eq_pure_bind]
 
 theorem mapIdxM_eq_mmap_enum [LawfulMonad m] {β} (f : ℕ → α → m β) (as : List α) :
     as.mapIdxM f = List.traverse (uncurry f) (enum as) := by
-  simp only [mapIdxM, mapIdxMGo_eq_mapIdxMAuxSpec, Array.toList_eq, Array.data_toArray,
+  simp only [mapIdxM, mapIdxMGo_eq_mapIdxMAuxSpec, Array.toList_toArray,
     nil_append, mapIdxMAuxSpec, Array.size_toArray, length_nil, id_map', enum]
 
 end MapIdxM
diff --git a/Mathlib/Data/List/Infix.lean b/Mathlib/Data/List/Infix.lean
index 10b247e344ea9..a399857f66da2 100644
--- a/Mathlib/Data/List/Infix.lean
+++ b/Mathlib/Data/List/Infix.lean
@@ -25,8 +25,6 @@ All those (except `insert`) are defined in `Mathlib.Data.List.Defs`.
 * `l₁ <:+: l₂`: `l₁` is an infix of `l₂`.
 -/
 
-open Nat
-
 variable {α β : Type*}
 
 namespace List
@@ -35,41 +33,12 @@ variable {l l₁ l₂ l₃ : List α} {a b : α} {m n : ℕ}
 
 /-! ### prefix, suffix, infix -/
 
-
 section Fix
 
-theorem prefix_rfl : l <+: l :=
-  prefix_refl _
-
-theorem suffix_rfl : l <:+ l :=
-  suffix_refl _
-
-theorem infix_rfl : l <:+: l :=
-  infix_refl _
-
-theorem prefix_concat (a : α) (l) : l <+: concat l a := by simp
-
-theorem prefix_concat_iff {l₁ l₂ : List α} {a : α} :
-    l₁ <+: l₂ ++ [a] ↔ l₁ = l₂ ++ [a] ∨ l₁ <+: l₂ := by
-  simpa only [← reverse_concat', reverse_inj, reverse_suffix] using
-    suffix_cons_iff (l₁ := l₁.reverse) (l₂ := l₂.reverse)
-
-protected alias ⟨_, IsSuffix.reverse⟩ := reverse_prefix
-
-protected alias ⟨_, IsPrefix.reverse⟩ := reverse_suffix
-
-protected alias ⟨_, IsInfix.reverse⟩ := reverse_infix
-
 @[deprecated IsSuffix.reverse (since := "2024-08-12")] alias isSuffix.reverse := IsSuffix.reverse
 @[deprecated IsPrefix.reverse (since := "2024-08-12")] alias isPrefix.reverse := IsPrefix.reverse
 @[deprecated IsInfix.reverse (since := "2024-08-12")] alias isInfix.reverse := IsInfix.reverse
 
-alias ⟨eq_nil_of_infix_nil, _⟩ := infix_nil
-
-alias ⟨eq_nil_of_prefix_nil, _⟩ := prefix_nil
-
-alias ⟨eq_nil_of_suffix_nil, _⟩ := suffix_nil
-
 @[deprecated IsInfix.eq_of_length (since := "2024-08-12")]
 theorem eq_of_infix_of_length_eq (h : l₁ <:+: l₂) : l₁.length = l₂.length → l₁ = l₂ :=
   h.eq_of_length
@@ -94,67 +63,13 @@ lemma dropSlice_subset (n m : ℕ) (l : List α) : l.dropSlice n m ⊆ l :=
 lemma mem_of_mem_dropSlice {n m : ℕ} {l : List α} {a : α} (h : a ∈ l.dropSlice n m) : a ∈ l :=
   dropSlice_subset n m l h
 
-theorem takeWhile_prefix (p : α → Bool) : l.takeWhile p <+: l :=
-  ⟨l.dropWhile p, takeWhile_append_dropWhile p l⟩
-
-theorem dropWhile_suffix (p : α → Bool) : l.dropWhile p <:+ l :=
-  ⟨l.takeWhile p, takeWhile_append_dropWhile p l⟩
-
-theorem dropLast_prefix : ∀ l : List α, l.dropLast <+: l
-  | [] => ⟨nil, by rw [dropLast, List.append_nil]⟩
-  | a :: l => ⟨_, dropLast_append_getLast (cons_ne_nil a l)⟩
-
-theorem tail_suffix (l : List α) : tail l <:+ l := by rw [← drop_one]; apply drop_suffix
-
-theorem dropLast_sublist (l : List α) : l.dropLast <+ l :=
-  (dropLast_prefix l).sublist
-
-@[gcongr]
-theorem drop_sublist_drop_left (l : List α) {m n : ℕ} (h : m ≤ n) : drop n l <+ drop m l := by
-  rw [← Nat.sub_add_cancel h, ← drop_drop]
-  apply drop_sublist
-
-theorem dropLast_subset (l : List α) : l.dropLast ⊆ l :=
-  (dropLast_sublist l).subset
-
 theorem tail_subset (l : List α) : tail l ⊆ l :=
   (tail_sublist l).subset
 
 theorem mem_of_mem_dropLast (h : a ∈ l.dropLast) : a ∈ l :=
   dropLast_subset l h
 
-theorem mem_of_mem_tail (h : a ∈ l.tail) : a ∈ l :=
-  tail_subset l h
-
-@[gcongr]
-protected theorem Sublist.drop : ∀ {l₁ l₂ : List α}, l₁ <+ l₂ → ∀ n, l₁.drop n <+ l₂.drop n
-  | _, _, h, 0 => h
-  | _, _, h, n + 1 => by rw [← drop_tail, ← drop_tail]; exact h.tail.drop n
-
-theorem prefix_iff_eq_append : l₁ <+: l₂ ↔ l₁ ++ drop (length l₁) l₂ = l₂ :=
-  ⟨by rintro ⟨r, rfl⟩; rw [drop_left], fun e => ⟨_, e⟩⟩
-
-theorem suffix_iff_eq_append : l₁ <:+ l₂ ↔ take (length l₂ - length l₁) l₂ ++ l₁ = l₂ :=
-  ⟨by rintro ⟨r, rfl⟩; simp only [length_append, Nat.add_sub_cancel_right, take_left], fun e =>
-    ⟨_, e⟩⟩
-
-theorem prefix_iff_eq_take : l₁ <+: l₂ ↔ l₁ = take (length l₁) l₂ :=
-  ⟨fun h => append_cancel_right <| (prefix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
-    fun e => e.symm ▸ take_prefix _ _⟩
-
-theorem prefix_take_iff {x y : List α} {n : ℕ} : x <+: y.take n ↔ x <+: y ∧ x.length ≤ n := by
-  constructor
-  · intro h
-    constructor
-    · exact List.IsPrefix.trans h <| List.take_prefix n y
-    · replace h := h.length_le
-      rw [length_take, Nat.le_min] at h
-      exact h.left
-  · intro ⟨hp, hl⟩
-    have hl' := hp.length_le
-    rw [List.prefix_iff_eq_take] at *
-    rw [hp, List.take_take]
-    simp [min_eq_left, hl, hl']
+attribute [gcongr] Sublist.drop
 
 theorem concat_get_prefix {x y : List α} (h : x <+: y) (hl : x.length < y.length) :
     x ++ [y.get ⟨x.length, hl⟩] <+: y := by
@@ -163,75 +78,18 @@ theorem concat_get_prefix {x y : List α} (h : x <+: y) (hl : x.length < y.lengt
   convert List.take_append_drop (x.length + 1) y using 2
   rw [← List.take_concat_get, List.concat_eq_append]; rfl
 
-theorem suffix_iff_eq_drop : l₁ <:+ l₂ ↔ l₁ = drop (length l₂ - length l₁) l₂ :=
-  ⟨fun h => append_cancel_left <| (suffix_iff_eq_append.1 h).trans (take_append_drop _ _).symm,
-    fun e => e.symm ▸ drop_suffix _ _⟩
-
-instance decidablePrefix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <+: l₂)
-  | [], l₂ => isTrue ⟨l₂, rfl⟩
-  | a :: l₁, [] => isFalse fun ⟨t, te⟩ => List.noConfusion te
-  | a :: l₁, b :: l₂ =>
-    if h : a = b then
-      @decidable_of_decidable_of_iff _ _ (decidablePrefix l₁ l₂) (by rw [← h, prefix_cons_inj])
-    else
-      isFalse fun ⟨t, te⟩ => h <| by injection te
-
--- Alternatively, use mem_tails
-instance decidableSuffix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <:+ l₂)
-  | [], l₂ => isTrue ⟨l₂, append_nil _⟩
-  | a :: l₁, [] => isFalse <| mt (Sublist.length_le ∘ IsSuffix.sublist) (by simp)
-  | l₁, b :: l₂ =>
-    @decidable_of_decidable_of_iff _ _
-      (@instDecidableOr _ _ _ (l₁.decidableSuffix l₂))
-      suffix_cons_iff.symm
-
 instance decidableInfix [DecidableEq α] : ∀ l₁ l₂ : List α, Decidable (l₁ <:+: l₂)
   | [], l₂ => isTrue ⟨[], l₂, rfl⟩
   | a :: l₁, [] => isFalse fun ⟨s, t, te⟩ => by simp at te
   | l₁, b :: l₂ =>
-    @decidable_of_decidable_of_iff _ _
-      (@instDecidableOr _ _ (l₁.decidablePrefix (b :: l₂)) (l₁.decidableInfix l₂))
+    letI := l₁.decidableInfix l₂
+    @decidable_of_decidable_of_iff (l₁ <+: b :: l₂ ∨ l₁ <:+: l₂) _ _
       infix_cons_iff.symm
 
-theorem prefix_take_le_iff {L : List α} (hm : m < L.length) :
-    L.take m <+: L.take n ↔ m ≤ n := by
-  simp only [prefix_iff_eq_take, length_take]
-  induction m generalizing L n with
-  | zero => simp [min_eq_left, eq_self_iff_true, Nat.zero_le, take]
-  | succ m IH =>
-    cases L with
-    | nil => simp_all
-    | cons l ls =>
-      cases n with
-      | zero =>
-        simp
-      | succ n =>
-        simp only [length_cons, succ_eq_add_one, Nat.add_lt_add_iff_right] at hm
-        simp [← @IH n ls hm, Nat.min_eq_left, Nat.le_of_lt hm]
-
 @[deprecated cons_prefix_cons (since := "2024-08-14")]
 theorem cons_prefix_iff : a :: l₁ <+: b :: l₂ ↔ a = b ∧ l₁ <+: l₂ := by
   simp
 
-protected theorem IsPrefix.map (h : l₁ <+: l₂) (f : α → β) : l₁.map f <+: l₂.map f := by
-  induction' l₁ with hd tl hl generalizing l₂
-  · simp only [nil_prefix, map_nil]
-  · cases' l₂ with hd₂ tl₂
-    · simpa only using eq_nil_of_prefix_nil h
-    · rw [cons_prefix_cons] at h
-      simp only [List.map_cons, h, prefix_cons_inj, hl, map]
-
-protected theorem IsPrefix.filterMap (h : l₁ <+: l₂) (f : α → Option β) :
-    l₁.filterMap f <+: l₂.filterMap f := by
-  induction' l₁ with hd₁ tl₁ hl generalizing l₂
-  · simp only [nil_prefix, filterMap_nil]
-  · cases' l₂ with hd₂ tl₂
-    · simpa only using eq_nil_of_prefix_nil h
-    · rw [cons_prefix_cons] at h
-      rw [← @singleton_append _ hd₁ _, ← @singleton_append _ hd₂ _, filterMap_append,
-        filterMap_append, h.left, prefix_append_right_inj]
-      exact hl h.right
-
 @[deprecated (since := "2024-03-26")] alias IsPrefix.filter_map := IsPrefix.filterMap
 
 protected theorem IsPrefix.reduceOption {l₁ l₂ : List (Option α)} (h : l₁ <+: l₂) :
@@ -239,17 +97,17 @@ protected theorem IsPrefix.reduceOption {l₁ l₂ : List (Option α)} (h : l₁
   h.filterMap id
 
 instance : IsPartialOrder (List α) (· <+: ·) where
-  refl := prefix_refl
+  refl _ := prefix_rfl
   trans _ _ _ := IsPrefix.trans
   antisymm _ _ h₁ h₂ := h₁.eq_of_length <| h₁.length_le.antisymm h₂.length_le
 
 instance : IsPartialOrder (List α) (· <:+ ·) where
-  refl := suffix_refl
+  refl _ := suffix_rfl
   trans _ _ _ := IsSuffix.trans
   antisymm _ _ h₁ h₂ := h₁.eq_of_length <| h₁.length_le.antisymm h₂.length_le
 
 instance : IsPartialOrder (List α) (· <:+: ·) where
-  refl := infix_refl
+  refl _ := infix_rfl
   trans _ _ _ := IsInfix.trans
   antisymm _ _ h₁ h₂ := h₁.eq_of_length <| h₁.length_le.antisymm h₂.length_le
 
@@ -261,7 +119,7 @@ section InitsTails
 theorem mem_inits : ∀ s t : List α, s ∈ inits t ↔ s <+: t
   | s, [] =>
     suffices s = nil ↔ s <+: nil by simpa only [inits, mem_singleton]
-    ⟨fun h => h.symm ▸ prefix_refl [], eq_nil_of_prefix_nil⟩
+    ⟨fun h => h.symm ▸ prefix_rfl, eq_nil_of_prefix_nil⟩
   | s, a :: t =>
     suffices (s = nil ∨ ∃ l ∈ inits t, a :: l = s) ↔ s <+: a :: t by simpa
     ⟨fun o =>
@@ -299,11 +157,21 @@ theorem inits_cons (a : α) (l : List α) : inits (a :: l) = [] :: l.inits.map f
 
 theorem tails_cons (a : α) (l : List α) : tails (a :: l) = (a :: l) :: l.tails := by simp
 
-@[simp]
+#adaptation_note
+/--
+This can be removed after nightly-2024-09-07.
+-/
+attribute [-simp] map_tail
+
+#adaptation_note
+/--
+`nolint simpNF` should be removed after nightly-2024-09-07.
+-/
+@[simp, nolint simpNF]
 theorem inits_append : ∀ s t : List α, inits (s ++ t) = s.inits ++ t.inits.tail.map fun l => s ++ l
   | [], [] => by simp
-  | [], a :: t => by simp [· ∘ ·]
-  | a :: s, t => by simp [inits_append s t, · ∘ ·]
+  | [], a :: t => by simp
+  | a :: s, t => by simp [inits_append s t, Function.comp_def]
 
 @[simp]
 theorem tails_append :
@@ -402,24 +270,13 @@ theorem subset_insert (a : α) (l : List α) : l ⊆ l.insert a :=
 
 end Insert
 
-theorem mem_of_mem_suffix (hx : a ∈ l₁) (hl : l₁ <:+ l₂) : a ∈ l₂ :=
-  hl.subset hx
-
-theorem IsPrefix.ne_nil {x y : List α} (h : x <+: y) (hx : x ≠ []) : y ≠ [] := by
-  rintro rfl; exact hx <| List.prefix_nil.mp h
-
-theorem IsPrefix.getElem {x y : List α} (h : x <+: y) {n} (hn : n < x.length) :
-    x[n] = y[n]'(hn.trans_le h.length_le) := by
-  obtain ⟨_, rfl⟩ := h
-  exact (List.getElem_append n hn).symm
+@[deprecated (since := "2024-08-15")] alias mem_of_mem_suffix := IsSuffix.mem
 
+@[deprecated IsPrefix.getElem (since := "2024-08-15")]
 theorem IsPrefix.get_eq {x y : List α} (h : x <+: y) {n} (hn : n < x.length) :
     x.get ⟨n, hn⟩ = y.get ⟨n, hn.trans_le h.length_le⟩ := by
   simp only [get_eq_getElem, IsPrefix.getElem h hn]
 
-theorem IsPrefix.head_eq {x y : List α} (h : x <+: y) (hx : x ≠ []) :
-    x.head hx = y.head (h.ne_nil hx) := by
-  cases x <;> cases y <;> simp only [head_cons, ne_eq, not_true_eq_false] at hx ⊢
-  all_goals (obtain ⟨_, h⟩ := h; injection h)
+@[deprecated (since := "2024-08-15")] alias IsPrefix.head_eq := IsPrefix.head
 
 end List
diff --git a/Mathlib/Data/List/InsertNth.lean b/Mathlib/Data/List/InsertNth.lean
index 99646ea33a4c5..b6b147595d8d8 100644
--- a/Mathlib/Data/List/InsertNth.lean
+++ b/Mathlib/Data/List/InsertNth.lean
@@ -83,13 +83,17 @@ theorem insertNth_comm (a b : α) :
     simp only [insertNth_succ_cons, cons.injEq, true_and]
     exact insertNth_comm a b i j l (Nat.le_of_succ_le_succ h₀) (Nat.le_of_succ_le_succ h₁)
 
+#adaptation_note
+/--
+After nightly-2024-09-06 we can remove the `_root_` prefixes below.
+-/
 theorem mem_insertNth {a b : α} :
     ∀ {n : ℕ} {l : List α} (_ : n ≤ l.length), a ∈ l.insertNth n b ↔ a = b ∨ a ∈ l
   | 0, as, _ => by simp
   | n + 1, [], h => (Nat.not_succ_le_zero _ h).elim
   | n + 1, a' :: as, h => by
     rw [List.insertNth_succ_cons, mem_cons, mem_insertNth (Nat.le_of_succ_le_succ h),
-      ← or_assoc, @or_comm (a = a'), or_assoc, mem_cons]
+      ← _root_.or_assoc, @or_comm (a = a'), _root_.or_assoc, mem_cons]
 
 theorem insertNth_of_length_lt (l : List α) (x : α) (n : ℕ) (h : l.length < n) :
     insertNth n x l = l := by
diff --git a/Mathlib/Data/List/Intervals.lean b/Mathlib/Data/List/Intervals.lean
index fdfa018a9e884..870f434107df2 100644
--- a/Mathlib/Data/List/Intervals.lean
+++ b/Mathlib/Data/List/Intervals.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.List.Lattice
 import Mathlib.Data.Bool.Basic
@@ -135,7 +135,7 @@ theorem filter_lt_of_top_le {n m l : ℕ} (hml : m ≤ l) :
     simp only [(lt_of_lt_of_le (mem.1 hk).2 hml), decide_True]
 
 theorem filter_lt_of_le_bot {n m l : ℕ} (hln : l ≤ n) : ((Ico n m).filter fun x => x < l) = [] :=
-  filter_eq_nil.2 fun k hk => by
+  filter_eq_nil_iff.2 fun k hk => by
      simp only [decide_eq_true_eq, not_lt]
      apply le_trans hln
      exact (mem.1 hk).1
@@ -161,7 +161,7 @@ theorem filter_le_of_le_bot {n m l : ℕ} (hln : l ≤ n) :
     exact le_trans hln (mem.1 hk).1
 
 theorem filter_le_of_top_le {n m l : ℕ} (hml : m ≤ l) : ((Ico n m).filter fun x => l ≤ x) = [] :=
-  filter_eq_nil.2 fun k hk => by
+  filter_eq_nil_iff.2 fun k hk => by
     rw [decide_eq_true_eq]
     exact not_le_of_gt (lt_of_lt_of_le (mem.1 hk).2 hml)
 
diff --git a/Mathlib/Data/List/Join.lean b/Mathlib/Data/List/Join.lean
index 4f91751d1ece8..69f2b6f90ab34 100644
--- a/Mathlib/Data/List/Join.lean
+++ b/Mathlib/Data/List/Join.lean
@@ -19,31 +19,8 @@ variable {α β : Type*}
 
 namespace List
 
--- Porting note (#10618): simp can prove this
--- @[simp]
-theorem join_singleton (l : List α) : [l].join = l := by rw [join, join, append_nil]
-
-@[deprecated join_eq_nil_iff (since := "2024-07-10")]
-theorem join_eq_nil : ∀ {L : List (List α)}, join L = [] ↔ ∀ l ∈ L, l = [] := join_eq_nil_iff
-
-
-@[simp]
-theorem join_filter_not_isEmpty  :
-    ∀ {L : List (List α)}, join (L.filter fun l => !l.isEmpty) = L.join
-  | [] => rfl
-  | [] :: L => by
-      simp [join_filter_not_isEmpty (L := L), isEmpty_iff_eq_nil]
-  | (a :: l) :: L => by
-      simp [join_filter_not_isEmpty (L := L)]
-
 @[deprecated (since := "2024-02-25")] alias join_filter_isEmpty_eq_false := join_filter_not_isEmpty
 
-@[simp]
-theorem join_filter_ne_nil [DecidablePred fun l : List α => l ≠ []] {L : List (List α)} :
-    join (L.filter fun l => l ≠ []) = L.join := by
-  simp only [ne_eq, ← isEmpty_iff_eq_nil, Bool.not_eq_true, Bool.decide_eq_false,
-    join_filter_not_isEmpty]
-
 /-- See `List.length_join` for the corresponding statement using `List.sum`. -/
 lemma length_join' (L : List (List α)) : length (join L) = Nat.sum (map length L) := by
   induction L <;> [rfl; simp only [*, join, map, Nat.sum_cons, length_append]]
@@ -70,11 +47,6 @@ lemma countP_bind' (p : β → Bool) (l : List α) (f : α → List β) :
 lemma count_bind' [BEq β] (l : List α) (f : α → List β) (x : β) :
     count x (l.bind f) = Nat.sum (map (count x ∘ f) l) := countP_bind' _ _ _
 
-@[simp]
-theorem bind_eq_nil {l : List α} {f : α → List β} : List.bind l f = [] ↔ ∀ x ∈ l, f x = [] :=
-  join_eq_nil_iff.trans <| by
-    simp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
-
 /-- In a join, taking the first elements up to an index which is the sum of the lengths of the
 first `i` sublists, is the same as taking the join of the first `i` sublists.
 
@@ -129,18 +101,6 @@ theorem drop_take_succ_join_eq_get' (L : List (List α)) (i : Fin L.length) :
       get L i := by
    simp [drop_take_succ_join_eq_getElem']
 
-/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the
-sublists. -/
-theorem eq_iff_join_eq (L L' : List (List α)) :
-    L = L' ↔ L.join = L'.join ∧ map length L = map length L' := by
-  refine ⟨fun H => by simp [H], ?_⟩
-  rintro ⟨join_eq, length_eq⟩
-  apply ext_getElem
-  · have : length (map length L) = length (map length L') := by rw [length_eq]
-    simpa using this
-  · intro n h₁ h₂
-    rw [← drop_take_succ_join_eq_getElem', ← drop_take_succ_join_eq_getElem', join_eq, length_eq]
-
 theorem join_drop_length_sub_one {L : List (List α)} (h : L ≠ []) :
     (L.drop (L.length - 1)).join = L.getLast h := by
   induction L using List.reverseRecOn
@@ -151,25 +111,10 @@ theorem join_drop_length_sub_one {L : List (List α)} (h : L ≠ []) :
 `(x ++ l₁) ++ (x ++ l₂) ++ ... ++ (x ++ lₙ) ++ x` where `L = [l₁, l₂, ..., lₙ]`. -/
 theorem append_join_map_append (L : List (List α)) (x : List α) :
     x ++ (L.map (· ++ x)).join = (L.map (x ++ ·)).join ++ x := by
-  induction' L with _ _ ih
-  · rw [map_nil, join, append_nil, map_nil, join, nil_append]
-  · rw [map_cons, join, map_cons, join, append_assoc, ih, append_assoc, append_assoc]
-
-
-/-- Any member of `L : List (List α))` is a sublist of `L.join` -/
-lemma sublist_join (L : List (List α)) {s : List α} (hs : s ∈ L) :
-    s.Sublist L.join := by
   induction L with
-  | nil =>
-    exfalso
-    exact not_mem_nil s hs
-  | cons t m ht =>
-    cases mem_cons.mp hs with
-    | inl h =>
-      rw [h]
-      simp only [join_cons, sublist_append_left]
-    | inr h =>
-      simp only [join_cons]
-      exact sublist_append_of_sublist_right (ht h)
+  | nil => rw [map_nil, join, append_nil, map_nil, join, nil_append]
+  | cons _ _ ih => rw [map_cons, join, map_cons, join, append_assoc, ih, append_assoc, append_assoc]
+
+@[deprecated (since := "2024-08-15")] alias sublist_join := sublist_join_of_mem
 
 end List
diff --git a/Mathlib/Data/List/Lattice.lean b/Mathlib/Data/List/Lattice.lean
index 181f14ce99536..5a0ae92ccc6c0 100644
--- a/Mathlib/Data/List/Lattice.lean
+++ b/Mathlib/Data/List/Lattice.lean
@@ -2,7 +2,7 @@
 Copyright (c) 2014 Parikshit Khanna. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro,
-Scott Morrison
+Kim Morrison
 -/
 import Mathlib.Data.List.Basic
 
@@ -29,7 +29,7 @@ open Nat
 
 namespace List
 
-variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
+variable {α : Type*} {l₁ l₂ : List α} {p : α → Prop} {a : α}
 
 /-! ### `Disjoint` -/
 
@@ -178,13 +178,13 @@ theorem cons_bagInter_of_neg (l₁ : List α) (h : a ∉ l₂) :
 
 @[simp]
 theorem mem_bagInter {a : α} : ∀ {l₁ l₂ : List α}, a ∈ l₁.bagInter l₂ ↔ a ∈ l₁ ∧ a ∈ l₂
-  | [], l₂ => by simp only [nil_bagInter, not_mem_nil, false_and_iff]
+  | [], l₂ => by simp only [nil_bagInter, not_mem_nil, false_and]
   | b :: l₁, l₂ => by
     by_cases h : b ∈ l₂
     · rw [cons_bagInter_of_pos _ h, mem_cons, mem_cons, mem_bagInter]
       by_cases ba : a = b
-      · simp only [ba, h, eq_self_iff_true, true_or_iff, true_and_iff]
-      · simp only [mem_erase_of_ne ba, ba, false_or_iff]
+      · simp only [ba, h, eq_self_iff_true, true_or, true_and]
+      · simp only [mem_erase_of_ne ba, ba, false_or]
     · rw [cons_bagInter_of_neg _ h, mem_bagInter, mem_cons, or_and_right]
       symm
       apply or_iff_right_of_imp
@@ -203,7 +203,7 @@ theorem count_bagInter {a : α} :
       by_cases ba : b = a
       · simp only [beq_iff_eq]
         rw [if_pos ba, Nat.sub_add_cancel]
-        rwa [succ_le_iff, count_pos_iff_mem, ← ba]
+        rwa [succ_le_iff, count_pos_iff, ← ba]
       · simp only [beq_iff_eq]
         rw [if_neg ba, Nat.sub_zero, Nat.add_zero, Nat.add_zero]
     · rw [cons_bagInter_of_neg _ hb, count_bagInter]
diff --git a/Mathlib/Data/List/Lemmas.lean b/Mathlib/Data/List/Lemmas.lean
index 02849f7d446ba..b055f1b187222 100644
--- a/Mathlib/Data/List/Lemmas.lean
+++ b/Mathlib/Data/List/Lemmas.lean
@@ -5,7 +5,6 @@ Authors: Yakov Pechersky, Yury Kudryashov
 -/
 import Mathlib.Data.Set.Image
 import Mathlib.Data.List.InsertNth
-import Mathlib.Init.Data.List.Lemmas
 
 /-! # Some lemmas about lists involving sets
 
@@ -18,36 +17,20 @@ variable {α β γ : Type*}
 
 namespace List
 
--- TODO: Replace `List.getElem_reverse`
-theorem getElem_reverse' (l : List α) (i : Nat) (h1 h2) :
-    (reverse l)[i]'h1 = l[length l - 1 - i]'h2 := by
-  rw [← getElem_reverse l _ (by omega) (by omega)]
-  congr
-  simp at h1
-  omega
+@[deprecated (since := "2024-08-20")] alias getElem_reverse' := getElem_reverse
 
 theorem tail_reverse_eq_reverse_dropLast (l : List α) :
     l.reverse.tail = l.dropLast.reverse := by
   ext i v; by_cases hi : i < l.length - 1
   · simp only [← drop_one]
     rw [getElem?_eq_getElem (by simpa), getElem?_eq_getElem (by simpa),
-      ← getElem_drop _, getElem_reverse', getElem_reverse', getElem_dropLast]
-    simp [show l.length - 1 - (1 + i) = l.length - 1 - 1 - i by omega]
+      ← getElem_drop' _, getElem_reverse, getElem_reverse, getElem_dropLast]
+    · simp [show l.length - 1 - (1 + i) = l.length - 1 - 1 - i by omega]
     all_goals ((try simp); omega)
   · rw [getElem?_eq_none, getElem?_eq_none]
     all_goals (simp; omega)
 
-theorem getLast_tail (l : List α) (hl : l.tail ≠ []) :
-    l.tail.getLast hl = l.getLast (by intro h; rw [h] at hl; simp at hl) := by
-  simp only [← drop_one, ne_eq, drop_eq_nil_iff_le,
-    not_le, getLast_eq_getElem, length_drop] at hl |-
-  rw [← getElem_drop]
-  simp [show 1 + (l.length - 1 - 1) = l.length - 1 by omega]
-  omega
-
-lemma getElem_tail {i} (L : List α) (hi : i < L.tail.length) :
-    L.tail[i] = L[i + 1]'(by simp at *; omega) := by
-  induction L <;> simp at hi |-
+@[deprecated (since := "2024-08-19")] alias nthLe_tail := getElem_tail
 
 theorem injOn_insertNth_index_of_not_mem (l : List α) (x : α) (hx : x ∉ l) :
     Set.InjOn (fun k => insertNth k x l) { n | n ≤ l.length } := by
@@ -61,7 +44,7 @@ theorem injOn_insertNth_index_of_not_mem (l : List α) (x : α) (hx : x ∉ l) :
     · rfl
     · simp [hx.left] at h
     · simp [Ne.symm hx.left] at h
-    · simp only [true_and_iff, eq_self_iff_true, insertNth_succ_cons] at h
+    · simp only [true_and, eq_self_iff_true, insertNth_succ_cons] at h
       rw [Nat.succ_inj']
       refine IH hx.right ?_ ?_ (by injection h)
       · simpa [Nat.succ_le_succ_iff] using hn
diff --git a/Mathlib/Data/List/MinMax.lean b/Mathlib/Data/List/MinMax.lean
index c20935506c371..52a8f59fb7167 100644
--- a/Mathlib/Data/List/MinMax.lean
+++ b/Mathlib/Data/List/MinMax.lean
@@ -433,25 +433,31 @@ theorem minimum_of_length_pos_le_getElem {i : ℕ} (w : i < l.length) (h := (Nat
     l.minimum_of_length_pos h ≤ l[i] :=
   getElem_le_maximum_of_length_pos (α := αᵒᵈ) w
 
-lemma getD_maximum?_eq_unbot'_maximum (l : List α) (d : α) :
-    l.maximum?.getD d = l.maximum.unbot' d := by
+lemma getD_max?_eq_unbot'_maximum (l : List α) (d : α) :
+    l.max?.getD d = l.maximum.unbot' d := by
   cases hy : l.maximum with
   | bot => simp [List.maximum_eq_bot.mp hy]
   | coe y =>
     rw [List.maximum_eq_coe_iff] at hy
     simp only [WithBot.unbot'_coe]
-    cases hz : l.maximum? with
-    | none => simp [List.maximum?_eq_none_iff.mp hz] at hy
+    cases hz : l.max? with
+    | none => simp [List.max?_eq_none_iff.mp hz] at hy
     | some z =>
       have : Antisymm (α := α) (· ≤ ·) := ⟨_root_.le_antisymm⟩
-      rw [List.maximum?_eq_some_iff] at hz
+      rw [List.max?_eq_some_iff] at hz
       · rw [Option.getD_some]
         exact _root_.le_antisymm (hy.right _ hz.left) (hz.right _ hy.left)
       all_goals simp [le_total]
 
-lemma getD_minimum?_eq_untop'_minimum (l : List α) (d : α) :
-    l.minimum?.getD d = l.minimum.untop' d :=
-  getD_maximum?_eq_unbot'_maximum (α := αᵒᵈ) _ _
+@[deprecated (since := "2024-09-29")]
+alias getD_maximum?_eq_unbot'_maximum := getD_max?_eq_unbot'_maximum
+
+lemma getD_min?_eq_untop'_minimum (l : List α) (d : α) :
+    l.min?.getD d = l.minimum.untop' d :=
+  getD_max?_eq_unbot'_maximum (α := αᵒᵈ) _ _
+
+@[deprecated (since := "2024-09-29")]
+alias getD_minimum?_eq_untop'_minimum := getD_min?_eq_untop'_minimum
 
 end LinearOrder
 
diff --git a/Mathlib/Data/List/Monad.lean b/Mathlib/Data/List/Monad.lean
index a89b5f05d5272..04d1477cbd14d 100644
--- a/Mathlib/Data/List/Monad.lean
+++ b/Mathlib/Data/List/Monad.lean
@@ -9,11 +9,11 @@ import Mathlib.Init
 # Monad instances for `List`
 -/
 
-universe u v w
+universe u
 
 namespace List
 
-variable {α : Type u} {β : Type v} {γ : Type w}
+variable {α : Type u}
 
 instance instMonad : Monad List.{u} where
   pure := @List.pure
diff --git a/Mathlib/Data/List/NatAntidiagonal.lean b/Mathlib/Data/List/NatAntidiagonal.lean
index fbbf06a9c47bb..3d11150f01d9e 100644
--- a/Mathlib/Data/List/NatAntidiagonal.lean
+++ b/Mathlib/Data/List/NatAntidiagonal.lean
@@ -59,7 +59,7 @@ theorem nodup_antidiagonal (n : ℕ) : Nodup (antidiagonal n) :=
 @[simp]
 theorem antidiagonal_succ {n : ℕ} :
     antidiagonal (n + 1) = (0, n + 1) :: (antidiagonal n).map (Prod.map Nat.succ id) := by
-  simp only [antidiagonal, range_succ_eq_map, map_cons, true_and_iff, Nat.add_succ_sub_one,
+  simp only [antidiagonal, range_succ_eq_map, map_cons, Nat.add_succ_sub_one,
     Nat.add_zero, id, eq_self_iff_true, Nat.sub_zero, map_map, Prod.map_mk]
   apply congr rfl (congr rfl _)
   ext; simp
diff --git a/Mathlib/Data/List/Nodup.lean b/Mathlib/Data/List/Nodup.lean
index 8a52d78c4594f..f1a4e94793afc 100644
--- a/Mathlib/Data/List/Nodup.lean
+++ b/Mathlib/Data/List/Nodup.lean
@@ -16,9 +16,9 @@ predicate.
 
 universe u v
 
-open Nat Function
+open Function
 
-variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α → Prop} {a b : α}
+variable {α : Type u} {β : Type v} {l l₁ l₂ : List α} {r : α → α → Prop} {a : α}
 
 namespace List
 
@@ -140,14 +140,14 @@ theorem nodup_iff_count_le_one [DecidableEq α] {l : List α} : Nodup l ↔ ∀
 
 theorem nodup_iff_count_eq_one [DecidableEq α] : Nodup l ↔ ∀ a ∈ l, count a l = 1 :=
   nodup_iff_count_le_one.trans <| forall_congr' fun _ =>
-    ⟨fun H h => H.antisymm (count_pos_iff_mem.mpr h),
+    ⟨fun H h => H.antisymm (count_pos_iff.mpr h),
      fun H => if h : _ then (H h).le else (count_eq_zero.mpr h).trans_le (Nat.zero_le 1)⟩
 
 
 @[simp]
 theorem count_eq_one_of_mem [DecidableEq α] {a : α} {l : List α} (d : Nodup l) (h : a ∈ l) :
     count a l = 1 :=
-  _root_.le_antisymm (nodup_iff_count_le_one.1 d a) (Nat.succ_le_of_lt (count_pos_iff_mem.2 h))
+  _root_.le_antisymm (nodup_iff_count_le_one.1 d a) (Nat.succ_le_of_lt (count_pos_iff.2 h))
 
 theorem count_eq_of_nodup [DecidableEq α] {a : α} {l : List α} (d : Nodup l) :
     count a l = if a ∈ l then 1 else 0 := by
@@ -244,8 +244,8 @@ theorem Nodup.erase_getElem [DecidableEq α] {l : List α} (hl : l.Nodup)
       · simp [IH hl.2]
       · rw [beq_iff_eq]
         simp only [getElem_cons_succ]
-        simp only [length_cons, succ_eq_add_one, Nat.add_lt_add_iff_right] at h
-        exact mt (· ▸ l.getElem_mem i h) hl.1
+        simp only [length_cons, Nat.succ_eq_add_one, Nat.add_lt_add_iff_right] at h
+        exact mt (· ▸ getElem_mem h) hl.1
 
 theorem Nodup.erase_get [DecidableEq α] {l : List α} (hl : l.Nodup) (i : Fin l.length) :
     l.erase (l.get i) = l.eraseIdx ↑i := by
@@ -262,8 +262,8 @@ theorem nodup_join {L : List (List α)} :
 theorem nodup_bind {l₁ : List α} {f : α → List β} :
     Nodup (l₁.bind f) ↔
       (∀ x ∈ l₁, Nodup (f x)) ∧ Pairwise (fun a b : α => Disjoint (f a) (f b)) l₁ := by
-  simp only [List.bind, nodup_join, pairwise_map, and_comm, and_left_comm, mem_map, exists_imp,
-      and_imp]
+  simp only [List.bind, nodup_join, pairwise_map, and_comm, and_left_comm, mem_map,
+    exists_imp, and_imp]
   rw [show (∀ (l : List β) (x : α), f x = l → x ∈ l₁ → Nodup l) ↔ ∀ x : α, x ∈ l₁ → Nodup (f x)
       from forall_swap.trans <| forall_congr' fun _ => forall_eq']
 
@@ -304,13 +304,12 @@ theorem Nodup.union [DecidableEq α] (l₁ : List α) (h : Nodup l₂) : (l₁ 
 theorem Nodup.inter [DecidableEq α] (l₂ : List α) : Nodup l₁ → Nodup (l₁ ∩ l₂) :=
   Nodup.filter _
 
-theorem Nodup.diff_eq_filter [DecidableEq α] :
+theorem Nodup.diff_eq_filter [BEq α] [LawfulBEq α] :
     ∀ {l₁ l₂ : List α} (_ : l₁.Nodup), l₁.diff l₂ = l₁.filter (· ∉ l₂)
   | l₁, [], _ => by simp
   | l₁, a :: l₂, hl₁ => by
     rw [diff_cons, (hl₁.erase _).diff_eq_filter, hl₁.erase_eq_filter, filter_filter]
-    simp only [decide_not, Bool.not_eq_true', decide_eq_false_iff_not, bne_iff_ne, ne_eq, and_comm,
-      Bool.decide_and, mem_cons, not_or]
+    simp only [decide_not, bne, Bool.and_comm, mem_cons, not_or, decide_mem_cons, Bool.not_or]
 
 theorem Nodup.mem_diff_iff [DecidableEq α] (hl₁ : l₁.Nodup) : a ∈ l₁.diff l₂ ↔ a ∈ l₁ ∧ a ∉ l₂ := by
   rw [hl₁.diff_eq_filter, mem_filter, decide_eq_true_iff]
@@ -366,7 +365,7 @@ theorem Nodup.take_eq_filter_mem [DecidableEq α] :
   | [], n, _ => by simp
   | b::l, 0, _ => by simp
   | b::l, n+1, hl => by
-    rw [take_cons, Nodup.take_eq_filter_mem (Nodup.of_cons hl), List.filter_cons_of_pos (by simp)]
+    rw [take_succ_cons, Nodup.take_eq_filter_mem (Nodup.of_cons hl), filter_cons_of_pos (by simp)]
     congr 1
     refine List.filter_congr ?_
     intro x hx
diff --git a/Mathlib/Data/List/NodupEquivFin.lean b/Mathlib/Data/List/NodupEquivFin.lean
index 20600bbc8e828..0dcf11280caae 100644
--- a/Mathlib/Data/List/NodupEquivFin.lean
+++ b/Mathlib/Data/List/NodupEquivFin.lean
@@ -126,7 +126,7 @@ theorem sublist_of_orderEmbedding_get?_eq {l l' : List α} (f : ℕ ↪o ℕ)
     exact ix.succ_pos
   rw [← List.take_append_drop (f 0 + 1) l', ← List.singleton_append]
   apply List.Sublist.append _ (IH _ this)
-  rw [List.singleton_sublist, ← h, l'.getElem_take _ (Nat.lt_succ_self _)]
+  rw [List.singleton_sublist, ← h, l'.getElem_take' _ (Nat.lt_succ_self _)]
   apply List.get_mem
 
 /-- A `l : List α` is `Sublist l l'` for `l' : List α` iff
@@ -137,12 +137,14 @@ theorem sublist_iff_exists_orderEmbedding_get?_eq {l l' : List α} :
     l <+ l' ↔ ∃ f : ℕ ↪o ℕ, ∀ ix : ℕ, l.get? ix = l'.get? (f ix) := by
   constructor
   · intro H
-    induction' H with xs ys y _H IH xs ys x _H IH
-    · simp
-    · obtain ⟨f, hf⟩ := IH
+    induction H with
+    | slnil => simp
+    | cons _ _ IH =>
+      obtain ⟨f, hf⟩ := IH
       refine ⟨f.trans (OrderEmbedding.ofStrictMono (· + 1) fun _ => by simp), ?_⟩
       simpa using hf
-    · obtain ⟨f, hf⟩ := IH
+    | cons₂ _ _ IH =>
+      obtain ⟨f, hf⟩ := IH
       refine
         ⟨OrderEmbedding.ofMapLEIff (fun ix : ℕ => if ix = 0 then 0 else (f ix.pred).succ) ?_, ?_⟩
       · rintro ⟨_ | a⟩ ⟨_ | b⟩ <;> simp [Nat.succ_le_succ_iff]
diff --git a/Mathlib/Data/List/OfFn.lean b/Mathlib/Data/List/OfFn.lean
index 164cd8f7a1fc5..9aa74790b948b 100644
--- a/Mathlib/Data/List/OfFn.lean
+++ b/Mathlib/Data/List/OfFn.lean
@@ -3,6 +3,7 @@ Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
+import Batteries.Data.List.OfFn
 import Batteries.Data.List.Pairwise
 import Mathlib.Data.Fin.Tuple.Basic
 
@@ -16,7 +17,6 @@ of length `n`.
 
 The main statements pertain to lists generated using `List.ofFn`
 
-- `List.length_ofFn`, which tells us the length of such a list
 - `List.get?_ofFn`, which tells us the nth element of such a list
 - `List.equivSigmaTuple`, which is an `Equiv` between lists and the functions that generate them
   via `List.ofFn`.
@@ -32,44 +32,9 @@ open Nat
 
 namespace List
 
-@[simp]
-theorem length_ofFn_go {n} (f : Fin n → α) (i j h) : length (ofFn.go f i j h) = i := by
-  induction i generalizing j <;> simp_all [ofFn.go]
-
-/-- The length of a list converted from a function is the size of the domain. -/
-@[simp]
-theorem length_ofFn {n} (f : Fin n → α) : length (ofFn f) = n := by
-  simp [ofFn, length_ofFn_go]
-
-theorem getElem_ofFn_go {n} (f : Fin n → α) (i j h) (k) (hk : k < (ofFn.go f i j h).length) :
-    (ofFn.go f i j h)[k] = f ⟨j + k, by simp at hk; omega⟩ := by
-  let i+1 := i
-  cases k <;> simp [ofFn.go, getElem_ofFn_go (i := i)]
-  congr 2; omega
-
-theorem get_ofFn_go {n} (f : Fin n → α) (i j h) (k) (hk) :
-    get (ofFn.go f i j h) ⟨k, hk⟩ = f ⟨j + k, by simp at hk; omega⟩ := by
-  simp [getElem_ofFn_go]
-
-@[simp]
-theorem getElem_ofFn {n} (f : Fin n → α) (i : Nat) (h : i < (ofFn f).length) :
-    (ofFn f)[i] = f ⟨i, by simp_all⟩ := by
-  simp [ofFn, getElem_ofFn_go]
-
 theorem get_ofFn {n} (f : Fin n → α) (i) : get (ofFn f) i = f (Fin.cast (by simp) i) := by
   simp; congr
 
-/-- The `n`th element of a list -/
-@[simp]
-theorem getElem?_ofFn {n} (f : Fin n → α) (i) : (ofFn f)[i]? = ofFnNthVal f i :=
-  if h : i < (ofFn f).length
-  then by
-    rw [getElem?_eq_getElem h, getElem_ofFn]
-    · simp only [length_ofFn] at h; simp [ofFnNthVal, h]
-  else by
-    rw [ofFnNthVal, dif_neg] <;>
-    simpa using h
-
 /-- The `n`th element of a list -/
 theorem get?_ofFn {n} (f : Fin n → α) (i) : get? (ofFn f) i = ofFnNthVal f i := by
   simp
@@ -117,7 +82,7 @@ theorem ofFn_succ' {n} (f : Fin (succ n) → α) :
 
 @[simp]
 theorem ofFn_eq_nil_iff {n : ℕ} {f : Fin n → α} : ofFn f = [] ↔ n = 0 := by
-  cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero]
+  cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero, reduceCtorEq]
 
 theorem last_ofFn {n : ℕ} (f : Fin n → α) (h : ofFn f ≠ [])
     (hn : n - 1 < n := Nat.pred_lt <| ofFn_eq_nil_iff.not.mp h) :
diff --git a/Mathlib/Data/List/Pairwise.lean b/Mathlib/Data/List/Pairwise.lean
index 81b73d9be2e19..1a562e854c8c7 100644
--- a/Mathlib/Data/List/Pairwise.lean
+++ b/Mathlib/Data/List/Pairwise.lean
@@ -57,30 +57,6 @@ theorem Pairwise.set_pairwise (hl : Pairwise R l) (hr : Symmetric R) : { x | x 
     simp only [map, pairwise_cons, mem_map, forall_exists_index, and_imp,
       forall_apply_eq_imp_iff₂, pairwise_map]
 
-theorem pairwise_pmap {p : β → Prop} {f : ∀ b, p b → α} {l : List β} (h : ∀ x ∈ l, p x) :
-    Pairwise R (l.pmap f h) ↔
-      Pairwise (fun b₁ b₂ => ∀ (h₁ : p b₁) (h₂ : p b₂), R (f b₁ h₁) (f b₂ h₂)) l := by
-  induction' l with a l ihl
-  · simp
-  obtain ⟨_, hl⟩ : p a ∧ ∀ b, b ∈ l → p b := by simpa using h
-  simp only [ihl hl, pairwise_cons, exists₂_imp, pmap, and_congr_left_iff, mem_pmap]
-  refine fun _ => ⟨fun H b hb _ hpb => H _ _ hb rfl, ?_⟩
-  rintro H _ b hb rfl
-  exact H b hb _ _
-
-theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
-    (h : ∀ x ∈ l, p x) {S : β → β → Prop}
-    (hS : ∀ ⦃x⦄ (hx : p x) ⦃y⦄ (hy : p y), R x y → S (f x hx) (f y hy)) :
-    Pairwise S (l.pmap f h) := by
-  refine (pairwise_pmap h).2 (Pairwise.imp_of_mem ?_ hl)
-  intros; apply hS; assumption
-
-theorem pairwise_of_forall_mem_list {l : List α} {r : α → α → Prop} (h : ∀ a ∈ l, ∀ b ∈ l, r a b) :
-    l.Pairwise r := by
-  rw [pairwise_iff_forall_sublist]
-  intro a b hab
-  apply h <;> (apply hab.subset; simp)
-
 theorem pairwise_of_reflexive_of_forall_ne {l : List α} {r : α → α → Prop} (hr : Reflexive r)
     (h : ∀ a ∈ l, ∀ b ∈ l, a ≠ b → r a b) : l.Pairwise r := by
   rw [pairwise_iff_forall_sublist]
diff --git a/Mathlib/Data/List/Perm.lean b/Mathlib/Data/List/Perm.lean
index 0d30675ef3e1e..14611136f55f6 100644
--- a/Mathlib/Data/List/Perm.lean
+++ b/Mathlib/Data/List/Perm.lean
@@ -10,6 +10,7 @@ import Mathlib.Data.List.InsertNth
 import Mathlib.Data.List.Lattice
 import Mathlib.Data.List.Permutation
 import Mathlib.Data.Nat.Factorial.Basic
+import Batteries.Data.List.Perm
 
 /-!
 # List Permutations
@@ -144,17 +145,17 @@ lemma count_eq_count_filter_add [DecidableEq α] (P : α → Prop) [DecidablePre
   convert countP_eq_countP_filter_add l _ P
   simp only [decide_not]
 
-theorem Perm.foldl_eq {f : β → α → β} {l₁ l₂ : List α} (rcomm : RightCommutative f) (p : l₁ ~ l₂) :
+theorem Perm.foldl_eq {f : β → α → β} {l₁ l₂ : List α} [rcomm : RightCommutative f] (p : l₁ ~ l₂) :
     ∀ b, foldl f b l₁ = foldl f b l₂ :=
-  p.foldl_eq' fun x _hx y _hy z => rcomm z x y
+  p.foldl_eq' fun x _hx y _hy z => rcomm.right_comm z x y
 
-theorem Perm.foldr_eq {f : α → β → β} {l₁ l₂ : List α} (lcomm : LeftCommutative f) (p : l₁ ~ l₂) :
+theorem Perm.foldr_eq {f : α → β → β} {l₁ l₂ : List α} [lcomm : LeftCommutative f] (p : l₁ ~ l₂) :
     ∀ b, foldr f b l₁ = foldr f b l₂ := by
   intro b
   induction p using Perm.recOnSwap' generalizing b with
   | nil => rfl
   | cons _ _ r  => simp [r b]
-  | swap' _ _ _ r => simp only [foldr_cons]; rw [lcomm, r b]
+  | swap' _ _ _ r => simp only [foldr_cons]; rw [lcomm.left_comm, r b]
   | trans _ _ r₁ r₂ => exact Eq.trans (r₁ b) (r₂ b)
 
 section
@@ -165,8 +166,13 @@ local notation a " * " b => op a b
 
 local notation l " <*> " a => foldl op a l
 
-theorem Perm.fold_op_eq {l₁ l₂ : List α} {a : α} (h : l₁ ~ l₂) : (l₁ <*> a) = l₂ <*> a :=
-  h.foldl_eq (right_comm _ IC.comm IA.assoc) _
+theorem Perm.foldl_op_eq {l₁ l₂ : List α} {a : α} (h : l₁ ~ l₂) : (l₁ <*> a) = l₂ <*> a :=
+  h.foldl_eq _
+
+theorem Perm.foldr_op_eq {l₁ l₂ : List α} {a : α} (h : l₁ ~ l₂) : l₁.foldr op a = l₂.foldr op a :=
+  h.foldr_eq _
+
+@[deprecated (since := "2024-09-28")] alias Perm.fold_op_eq := Perm.foldl_op_eq
 
 end
 
@@ -216,16 +222,18 @@ variable [DecidableEq α]
 
 theorem Perm.bagInter_right {l₁ l₂ : List α} (t : List α) (h : l₁ ~ l₂) :
     l₁.bagInter t ~ l₂.bagInter t := by
-  induction' h with x _ _ _ _ x y _ _ _ _ _ _ ih_1 ih_2 generalizing t; · simp
-  · by_cases x ∈ t <;> simp [*, Perm.cons]
-  · by_cases h : x = y
+  induction h generalizing t with
+  | nil => simp
+  | cons x => by_cases x ∈ t <;> simp [*, Perm.cons]
+  | swap x y =>
+    by_cases h : x = y
     · simp [h]
     by_cases xt : x ∈ t <;> by_cases yt : y ∈ t
     · simp [xt, yt, mem_erase_of_ne h, mem_erase_of_ne (Ne.symm h), erase_comm, swap]
     · simp [xt, yt, mt mem_of_mem_erase, Perm.cons]
     · simp [xt, yt, mt mem_of_mem_erase, Perm.cons]
     · simp [xt, yt]
-  · exact (ih_1 _).trans (ih_2 _)
+  | trans _ _ ih_1 ih_2 => exact (ih_1 _).trans (ih_2 _)
 
 theorem Perm.bagInter_left (l : List α) {t₁ t₂ : List α} (p : t₁ ~ t₂) :
     l.bagInter t₁ = l.bagInter t₂ := by
@@ -263,7 +271,7 @@ theorem Perm.inter_append {l t₁ t₂ : List α} (h : Disjoint t₁ t₂) :
     · have h₂ : x ∉ t₂ := h h₁
       simp [*]
     by_cases h₂ : x ∈ t₂
-    · simp only [*, inter_cons_of_not_mem, false_or_iff, mem_append, inter_cons_of_mem,
+    · simp only [*, inter_cons_of_not_mem, false_or, mem_append, inter_cons_of_mem,
         not_false_iff]
       refine Perm.trans (Perm.cons _ l_ih) ?_
       change [x] ++ xs ∩ t₁ ++ xs ∩ t₂ ~ xs ∩ t₁ ++ ([x] ++ xs ∩ t₂)
@@ -335,10 +343,9 @@ theorem Perm.drop_inter [DecidableEq α] {xs ys : List α} (n : ℕ) (h : xs ~ y
   by_cases h'' : n ≤ xs.length
   · let n' := xs.length - n
     have h₀ : n = xs.length - n' := by rwa [Nat.sub_sub_self]
-    have h₁ : n' ≤ xs.length := Nat.sub_le ..
-    have h₂ : xs.drop n = (xs.reverse.take n').reverse := by
-      rw [take_reverse h₁, h₀, reverse_reverse]
-    rw [h₂]
+    have h₁ : xs.drop n = (xs.reverse.take n').reverse := by
+      rw [take_reverse, h₀, reverse_reverse]
+    rw [h₁]
     apply (reverse_perm _).trans
     rw [inter_reverse]
     apply Perm.take_inter _ _ h'
@@ -453,14 +460,16 @@ theorem perm_permutations'Aux_comm (a b : α) (l : List α) :
   exact perm_append_comm.append (ih.map _)
 
 theorem Perm.permutations' {s t : List α} (p : s ~ t) : permutations' s ~ permutations' t := by
-  induction' p with a s t _ IH a b l s t u _ _ IH₁ IH₂; · simp
-  · exact IH.bind_right _
-  · dsimp
+  induction p with
+  | nil => simp
+  | cons _ _ IH => exact IH.bind_right _
+  | swap =>
+    dsimp
     rw [bind_assoc, bind_assoc]
     apply Perm.bind_left
     intro l' _
     apply perm_permutations'Aux_comm
-  · exact IH₁.trans IH₂
+  | trans _ _ IH₁ IH₂ => exact IH₁.trans IH₂
 
 theorem permutations_perm_permutations' (ts : List α) : ts.permutations ~ ts.permutations' := by
   obtain ⟨n, h⟩ : ∃ n, length ts < n := ⟨_, Nat.lt_succ_self _⟩
diff --git a/Mathlib/Data/List/Permutation.lean b/Mathlib/Data/List/Permutation.lean
index 873cf8aacfaee..dfa4674956933 100644
--- a/Mathlib/Data/List/Permutation.lean
+++ b/Mathlib/Data/List/Permutation.lean
@@ -182,7 +182,7 @@ theorem mem_foldr_permutationsAux2 {t : α} {ts : List α} {r L : List (List α)
 theorem length_foldr_permutationsAux2 (t : α) (ts : List α) (r L : List (List α)) :
     length (foldr (fun y r => (permutationsAux2 t ts r y id).2) r L) =
       Nat.sum (map length L) + length r := by
-  simp [foldr_permutationsAux2, (· ∘ ·), length_permutationsAux2, length_bind']
+  simp [foldr_permutationsAux2, Function.comp_def, length_permutationsAux2, length_bind']
 
 theorem length_foldr_permutationsAux2' (t : α) (ts : List α) (r L : List (List α)) (n)
     (H : ∀ l ∈ L, length l = n) :
diff --git a/Mathlib/Data/List/Pi.lean b/Mathlib/Data/List/Pi.lean
index 25244772bc6f4..629d8a1e4f855 100644
--- a/Mathlib/Data/List/Pi.lean
+++ b/Mathlib/Data/List/Pi.lean
@@ -24,12 +24,12 @@ def nil (α : ι → Sort*) : (∀ i ∈ ([] : List ι), α i) :=
 
 variable {i : ι} {l : List ι}
 
-/-- Given `f` a function whose domain is `i :: l`, get its value at `i`.  -/
+/-- Given `f` a function whose domain is `i :: l`, get its value at `i`. -/
 def head (f : ∀ j ∈ i :: l, α j) : α i :=
   f i (mem_cons_self _ _)
 
 /-- Given `f` a function whose domain is `i :: l`, produce a function whose domain
-is restricted to `l`.  -/
+is restricted to `l`. -/
 def tail (f : ∀ j ∈ i :: l, α j) : ∀ j ∈ l, α j :=
   fun j hj ↦ f j (mem_cons_of_mem _ hj)
 
diff --git a/Mathlib/Data/List/Prime.lean b/Mathlib/Data/List/Prime.lean
index e5812cb86b262..45c8f957e4d93 100644
--- a/Mathlib/Data/List/Prime.lean
+++ b/Mathlib/Data/List/Prime.lean
@@ -43,7 +43,7 @@ end CommMonoidWithZero
 
 section CancelCommMonoidWithZero
 
-variable {M : Type*} [CancelCommMonoidWithZero M] [Unique (Units M)]
+variable {M : Type*} [CancelCommMonoidWithZero M] [Subsingleton Mˣ]
 
 theorem mem_list_primes_of_dvd_prod {p : M} (hp : Prime p) {L : List M} (hL : ∀ q ∈ L, Prime q)
     (hpL : p ∣ L.prod) : p ∈ L := by
@@ -54,10 +54,10 @@ theorem perm_of_prod_eq_prod :
     ∀ {l₁ l₂ : List M}, l₁.prod = l₂.prod → (∀ p ∈ l₁, Prime p) → (∀ p ∈ l₂, Prime p) → Perm l₁ l₂
   | [], [], _, _, _ => Perm.nil
   | [], a :: l, h₁, _, h₃ =>
-    have ha : a ∣ 1 := @prod_nil M _ ▸ h₁.symm ▸ (@prod_cons _ _ l a).symm ▸ dvd_mul_right _ _
+    have ha : a ∣ 1 := prod_nil (M := M) ▸ h₁.symm ▸ (prod_cons (l := l)).symm ▸ dvd_mul_right _ _
     absurd ha (Prime.not_dvd_one (h₃ a (mem_cons_self _ _)))
   | a :: l, [], h₁, h₂, _ =>
-    have ha : a ∣ 1 := @prod_nil M _ ▸ h₁ ▸ (@prod_cons _ _ l a).symm ▸ dvd_mul_right _ _
+    have ha : a ∣ 1 := prod_nil (M := M) ▸ h₁ ▸ (prod_cons (l := l)).symm ▸ dvd_mul_right _ _
     absurd ha (Prime.not_dvd_one (h₂ a (mem_cons_self _ _)))
   | a :: l₁, b :: l₂, h, hl₁, hl₂ => by
     classical
diff --git a/Mathlib/Data/List/Range.lean b/Mathlib/Data/List/Range.lean
index 8ec0dcdc8cf4a..63ca906ece471 100644
--- a/Mathlib/Data/List/Range.lean
+++ b/Mathlib/Data/List/Range.lean
@@ -1,11 +1,10 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Kenny Lau, Scott Morrison
+Authors: Mario Carneiro, Kenny Lau, Kim Morrison
 -/
 import Mathlib.Data.List.Chain
 import Mathlib.Data.List.Nodup
-import Mathlib.Data.List.Pairwise
 
 /-!
 # Ranges of naturals as lists
@@ -35,8 +34,7 @@ theorem chain'_range_succ (r : ℕ → ℕ → Prop) (n : ℕ) :
   induction' n with n hn
   · simp
   · rw [range_succ]
-    simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton,
-      and_true_iff]
+    simp only [append_assoc, singleton_append, chain'_append_cons_cons, chain'_singleton, and_true]
     rw [hn, forall_lt_succ]
 
 theorem chain_range_succ (r : ℕ → ℕ → Prop) (n a : ℕ) :
diff --git a/Mathlib/Data/List/ReduceOption.lean b/Mathlib/Data/List/ReduceOption.lean
index f86d28a812369..8994967357edb 100644
--- a/Mathlib/Data/List/ReduceOption.lean
+++ b/Mathlib/Data/List/ReduceOption.lean
@@ -34,8 +34,7 @@ theorem reduceOption_map {l : List (Option α)} {f : α → β} :
   induction' l with hd tl hl
   · simp only [reduceOption_nil, map_nil]
   · cases hd <;>
-      simpa [true_and_iff, Option.map_some', map, eq_self_iff_true,
-        reduceOption_cons_of_some] using hl
+      simpa [Option.map_some', map, eq_self_iff_true, reduceOption_cons_of_some] using hl
 
 theorem reduceOption_append (l l' : List (Option α)) :
     (l ++ l').reduceOption = l.reduceOption ++ l'.reduceOption :=
diff --git a/Mathlib/Data/List/Rotate.lean b/Mathlib/Data/List/Rotate.lean
index ccec73856febc..8b6479898785b 100644
--- a/Mathlib/Data/List/Rotate.lean
+++ b/Mathlib/Data/List/Rotate.lean
@@ -116,7 +116,7 @@ theorem length_rotate (l : List α) (n : ℕ) : (l.rotate n).length = l.length :
 
 @[simp]
 theorem rotate_replicate (a : α) (n : ℕ) (k : ℕ) : (replicate n a).rotate k = replicate n a :=
-  eq_replicate.2 ⟨by rw [length_rotate, length_replicate], fun b hb =>
+  eq_replicate_iff.2 ⟨by rw [length_rotate, length_replicate], fun b hb =>
     eq_of_mem_replicate <| mem_rotate.1 hb⟩
 
 theorem rotate_eq_drop_append_take {l : List α} {n : ℕ} :
@@ -168,7 +168,6 @@ theorem rotate_eq_nil_iff {l : List α} {n : ℕ} : l.rotate n = [] ↔ l = [] :
     · simp
     · simp [rotate_cons_succ, hn]
 
-@[simp]
 theorem nil_eq_rotate_iff {l : List α} {n : ℕ} : [] = l.rotate n ↔ [] = l := by
   rw [eq_comm, rotate_eq_nil_iff, eq_comm]
 
@@ -195,11 +194,11 @@ theorem getElem?_rotate {l : List α} {n m : ℕ} (hml : m < l.length) :
     (l.rotate n)[m]? = l[(m + n) % l.length]? := by
   rw [rotate_eq_drop_append_take_mod]
   rcases lt_or_le m (l.drop (n % l.length)).length with hm | hm
-  · rw [getElem?_append hm, getElem?_drop, ← add_mod_mod]
+  · rw [getElem?_append_left hm, getElem?_drop, ← add_mod_mod]
     rw [length_drop, Nat.lt_sub_iff_add_lt] at hm
     rw [mod_eq_of_lt hm, Nat.add_comm]
   · have hlt : n % length l < length l := mod_lt _ (m.zero_le.trans_lt hml)
-    rw [getElem?_append_right hm, getElem?_take, length_drop]
+    rw [getElem?_append_right hm, getElem?_take_of_lt, length_drop]
     · congr 1
       rw [length_drop] at hm
       have hm' := Nat.sub_le_iff_le_add'.1 hm
@@ -474,7 +473,7 @@ theorem IsRotated.dropLast_tail {α}
   | [] => by simp
   | [_] => by simp
   | a :: b :: L => by
-    simp at hL' |-
+    simp only [head_cons, ne_eq, reduceCtorEq, not_false_eq_true, getLast_cons] at hL'
     simp [hL', IsRotated.cons_getLast_dropLast]
 
 /-- List of all cyclic permutations of `l`.
diff --git a/Mathlib/Data/List/Sections.lean b/Mathlib/Data/List/Sections.lean
index 5fc1305b9cafa..c604e4d75e47c 100644
--- a/Mathlib/Data/List/Sections.lean
+++ b/Mathlib/Data/List/Sections.lean
@@ -25,7 +25,7 @@ theorem mem_sections {L : List (List α)} {f} : f ∈ sections L ↔ Forall₂ (
       exact Forall₂.nil
     simp only [sections, bind_eq_bind, mem_bind, mem_map] at h
     rcases h with ⟨_, _, _, _, rfl⟩
-    simp only [*, forall₂_cons, true_and_iff]
+    simp only [*, forall₂_cons, true_and]
   · induction' h with a l f L al fL fs
     · simp only [sections, mem_singleton]
     simp only [sections, bind_eq_bind, mem_bind, mem_map]
diff --git a/Mathlib/Data/List/Sigma.lean b/Mathlib/Data/List/Sigma.lean
index 400351de48dff..ba2c572fe9480 100644
--- a/Mathlib/Data/List/Sigma.lean
+++ b/Mathlib/Data/List/Sigma.lean
@@ -217,7 +217,7 @@ theorem dlookup_map (l : List (Sigma β))
 theorem dlookup_map₁ {β : Type v} (l : List (Σ _ : α, β))
     {f : α → α'} (hf : Function.Injective f) (a : α) :
     (l.map fun x => ⟨f x.1, x.2⟩ : List (Σ _ : α', β)).dlookup (f a) = l.dlookup a := by
-  rw [dlookup_map (β' := fun _ => β) l hf (fun _ x => x) a, Option.map_id'']
+  rw [dlookup_map (β' := fun _ => β) l hf (fun _ x => x) a, Option.map_id']
 
 theorem dlookup_map₂ {γ δ : α → Type*} {l : List (Σ a, γ a)} {f : ∀ a, γ a → δ a} (a : α) :
     (l.map fun x => ⟨x.1, f _ x.2⟩ : List (Σ a, δ a)).dlookup a = (l.dlookup a).map (f a) :=
@@ -250,7 +250,7 @@ theorem lookupAll_eq_nil {a : α} :
     by_cases h : a = a'
     · subst a'
       simp only [lookupAll_cons_eq, mem_cons, Sigma.mk.inj_iff, heq_eq_eq, true_and, not_or,
-        false_iff, not_forall, not_and, not_not]
+        false_iff, not_forall, not_and, not_not, reduceCtorEq]
       use b
       simp
     · simp [h, lookupAll_eq_nil]
@@ -379,7 +379,9 @@ theorem kerase_cons_ne {a} {s : Sigma β} {l : List (Sigma β)} (h : a ≠ s.1)
 
 @[simp]
 theorem kerase_of_not_mem_keys {a} {l : List (Sigma β)} (h : a ∉ l.keys) : kerase a l = l := by
-  induction' l with _ _ ih <;> [rfl; (simp [not_or] at h; simp [h.1, ih h.2])]
+  induction l with
+  | nil => rfl
+  | cons _ _ ih => simp [not_or] at h; simp [h.1, ih h.2]
 
 theorem kerase_sublist (a : α) (l : List (Sigma β)) : kerase a l <+ l :=
   eraseP_sublist _
@@ -417,7 +419,7 @@ theorem mem_keys_kerase_of_ne {a₁ a₂} {l : List (Sigma β)} (h : a₁ ≠ a
     else by simp [q, p]
 
 theorem keys_kerase {a} {l : List (Sigma β)} : (kerase a l).keys = l.keys.erase a := by
-  rw [keys, kerase, erase_eq_eraseP, eraseP_map, Function.comp]
+  rw [keys, kerase, erase_eq_eraseP, eraseP_map, Function.comp_def]
   simp only [beq_eq_decide]
   congr
   funext
diff --git a/Mathlib/Data/List/Sort.lean b/Mathlib/Data/List/Sort.lean
index 8e144c9536908..be93169d29b88 100644
--- a/Mathlib/Data/List/Sort.lean
+++ b/Mathlib/Data/List/Sort.lean
@@ -5,8 +5,8 @@ Authors: Jeremy Avigad
 -/
 import Mathlib.Data.List.OfFn
 import Mathlib.Data.List.Nodup
-import Mathlib.Data.List.Infix
 import Mathlib.Order.Fin.Basic
+import Batteries.Data.List.Perm
 
 /-!
 # Sorting algorithms on lists
@@ -14,7 +14,25 @@ import Mathlib.Order.Fin.Basic
 In this file we define `List.Sorted r l` to be an alias for `List.Pairwise r l`.
 This alias is preferred in the case that `r` is a `<` or `≤`-like relation.
 Then we define two sorting algorithms:
-`List.insertionSort` and `List.mergeSort`, and prove their correctness.
+`List.insertionSort` and `List.mergeSort'`, and prove their correctness.
+-/
+
+#adaptation_note
+/--
+`List.mergeSort` has now been implemented in Lean4.
+It improves on the one here by being a "stable" sort
+(in the sense that a sorted sublist of the original list remains a sublist of the result),
+and is also marginally faster.
+
+However we haven't yet replaced `List.mergeSort'` here.
+The obstacle is that `mergeSort'` is written using `r : α → α → Prop` with `[DecidableRel r]`,
+while `mergeSort` uses `r : α → α → Bool`. This is hardly insurmountable,
+but it's a bit of work that hasn't been done yet.
+
+`List.mergeSort'` is only used in Mathlib to sort multisets for printing, so this is not critical.
+
+A pull request cleaning up here, and ideally deprecating or deleting `List.mergeSort'`,
+would be welcome.
 -/
 
 
@@ -105,8 +123,8 @@ theorem eq_of_perm_of_sorted [IsAntisymm α r] {l₁ l₂ : List α} (hp : l₁
     congr
     have : ∀ x ∈ u₂, x = a := fun x m =>
       antisymm ((pairwise_append.1 hs₂).2.2 _ m a (mem_cons_self _ _)) (h₁ _ (by simp [m]))
-    rw [(@eq_replicate _ a (length u₂ + 1) (a :: u₂)).2,
-        (@eq_replicate _ a (length u₂ + 1) (u₂ ++ [a])).2] <;>
+    rw [(@eq_replicate_iff _ a (length u₂ + 1) (a :: u₂)).2,
+        (@eq_replicate_iff _ a (length u₂ + 1) (u₂ ++ [a])).2] <;>
         constructor <;>
       simp [iff_true_intro this, or_comm]
 
@@ -131,7 +149,7 @@ theorem Sorted.rel_of_mem_take_of_mem_drop {l : List α} (h : List.Sorted r l) {
     (hx : x ∈ List.take k l) (hy : y ∈ List.drop k l) : r x y := by
   obtain ⟨iy, hiy, rfl⟩ := getElem_of_mem hy
   obtain ⟨ix, hix, rfl⟩ := getElem_of_mem hx
-  rw [getElem_take', getElem_drop']
+  rw [getElem_take, getElem_drop]
   rw [length_take] at hix
   exact h.rel_get_of_lt (Nat.lt_add_right _ (Nat.lt_min.mp hix).left)
 
@@ -180,6 +198,10 @@ def orderedInsert (a : α) : List α → List α
   | [] => [a]
   | b :: l => if a ≼ b then a :: b :: l else b :: orderedInsert a l
 
+theorem orderedInsert_of_le {a b : α} (l : List α) (h : a ≼ b) :
+    orderedInsert r a (b :: l) = a :: b :: l :=
+  dif_pos h
+
 /-- `insertionSort l` returns `l` sorted using the insertion sort algorithm. -/
 @[simp]
 def insertionSort : List α → List α
@@ -263,6 +285,17 @@ theorem mem_insertionSort {l : List α} {x : α} : x ∈ l.insertionSort r ↔ x
 theorem length_insertionSort (l : List α) : (insertionSort r l).length = l.length :=
   (perm_insertionSort r _).length_eq
 
+theorem insertionSort_cons {a : α} {l : List α} (h : ∀ b ∈ l, r a b) :
+    insertionSort r (a :: l) = a :: insertionSort r l := by
+  rw [insertionSort]
+  cases hi : insertionSort r l with
+  | nil => rfl
+  | cons b m =>
+    rw [orderedInsert_of_le]
+    apply h b <| (mem_insertionSort r).1 _
+    rw [hi]
+    exact mem_cons_self b m
+
 theorem map_insertionSort (f : α → β) (l : List α) (hl : ∀ a ∈ l, ∀ b ∈ l, a ≼ b ↔ f a ≼ f b) :
     (l.insertionSort r).map f = (l.map f).insertionSort s := by
   induction l with
@@ -327,6 +360,40 @@ theorem sublist_orderedInsert (x : α) (xs : List α) : xs <+ xs.orderedInsert r
   refine Sublist.trans ?_ (.append_left (.cons _ (.refl _)) _)
   rw [takeWhile_append_dropWhile]
 
+theorem cons_sublist_orderedInsert {l c : List α} {a : α} (hl : c <+ l) (ha : ∀ a' ∈ c, a ≼ a') :
+    a :: c <+ orderedInsert r a l := by
+  induction l with
+  | nil         => simp_all only [sublist_nil, orderedInsert, Sublist.refl]
+  | cons _ _ ih =>
+    unfold orderedInsert
+    split_ifs with hr
+    · exact .cons₂ _ hl
+    · cases hl with
+      | cons _ h => exact .cons _ <| ih h
+      | cons₂    => exact absurd (ha _ <| mem_cons_self ..) hr
+
+theorem Sublist.orderedInsert_sublist [IsTrans α r] {as bs} (x) (hs : as <+ bs) (hb : bs.Sorted r) :
+    orderedInsert r x as <+ orderedInsert r x bs := by
+  cases as with
+  | nil => simp
+  | cons a as =>
+    cases bs with
+    | nil => contradiction
+    | cons b bs =>
+      unfold orderedInsert
+      cases hs <;> split_ifs with hr
+      · exact .cons₂ _ <| .cons _ ‹a :: as <+ bs›
+      · have ih := orderedInsert_sublist x ‹a :: as <+ bs›  hb.of_cons
+        simp only [hr, orderedInsert, ite_true] at ih
+        exact .trans ih <| .cons _ (.refl _)
+      · have hba := pairwise_cons.mp hb |>.left _ (mem_of_cons_sublist ‹a :: as <+ bs›)
+        exact absurd (trans_of _ ‹r x b› hba) hr
+      · have ih := orderedInsert_sublist x ‹a :: as <+ bs› hb.of_cons
+        rw [orderedInsert, if_neg hr] at ih
+        exact .cons _ ih
+      · simp_all only [sorted_cons, cons_sublist_cons]
+      · exact .cons₂ _ <| orderedInsert_sublist x ‹as <+ bs› hb.of_cons
+
 section TotalAndTransitive
 
 variable [IsTotal α r] [IsTrans α r]
@@ -356,6 +423,59 @@ theorem sorted_insertionSort : ∀ l, Sorted r (insertionSort r l)
 
 end TotalAndTransitive
 
+/--
+If `c` is a sorted sublist of `l`, then `c` is still a sublist of `insertionSort r l`.
+-/
+theorem sublist_insertionSort {l c : List α} (hr : c.Pairwise r) (hc : c <+ l) :
+    c <+ insertionSort r l := by
+  induction l generalizing c with
+  | nil         => simp_all only [sublist_nil, insertionSort, Sublist.refl]
+  | cons _ _ ih =>
+    cases hc with
+    | cons  _ h => exact ih hr h |>.trans (sublist_orderedInsert ..)
+    | cons₂ _ h =>
+      obtain ⟨hr, hp⟩ := pairwise_cons.mp hr
+      exact cons_sublist_orderedInsert (ih hp h) hr
+
+/--
+Another statement of stability of insertion sort.
+If a pair `[a, b]` is a sublist of `l` and `r a b`,
+then `[a, b]` is still a sublist of `insertionSort r l`.
+-/
+theorem pair_sublist_insertionSort {a b : α} {l : List α} (hab : r a b) (h : [a, b] <+ l) :
+    [a, b] <+ insertionSort r l :=
+  sublist_insertionSort (pairwise_pair.mpr hab) h
+
+variable [IsAntisymm α r] [IsTotal α r] [IsTrans α r]
+
+/--
+A version of `insertionSort_stable` which only assumes `c <+~ l` (instead of `c <+ l`), but
+additionally requires `IsAntisymm α r`, `IsTotal α r` and `IsTrans α r`.
+-/
+theorem sublist_insertionSort' {l c : List α} (hs : c.Sorted r) (hc : c <+~ l) :
+    c <+ insertionSort r l := by
+  classical
+  obtain ⟨d, hc, hd⟩ := hc
+  induction l generalizing c d with
+  | nil         => simp_all only [sublist_nil, insertionSort, nil_perm]
+  | cons a _ ih =>
+    cases hd with
+    | cons  _ h => exact ih hs _ hc h |>.trans (sublist_orderedInsert ..)
+    | cons₂ _ h =>
+      specialize ih (hs.erase _) _ (erase_cons_head a ‹List _› ▸ hc.erase a) h
+      have hm := hc.mem_iff.mp <| mem_cons_self ..
+      have he := orderedInsert_erase _ _ hm hs
+      exact he ▸ Sublist.orderedInsert_sublist _ ih (sorted_insertionSort ..)
+
+/--
+Another statement of stability of insertion sort.
+If a pair `[a, b]` is a sublist of a permutation of `l` and `a ≼ b`,
+then `[a, b]` is still a sublist of `insertionSort r l`.
+-/
+theorem pair_sublist_insertionSort' {a b : α} {l : List α} (hab : a ≼ b) (h : [a, b] <+~ l) :
+    [a, b] <+ insertionSort r l :=
+  sublist_insertionSort' (pairwise_pair.mpr hab) h
+
 end Correctness
 
 end InsertionSort
@@ -421,7 +541,7 @@ theorem perm_split : ∀ {l l₁ l₂ : List α}, split l = (l₁, l₂) → l ~
     exact ((perm_split e).trans perm_append_comm).cons a
 
 /-- Implementation of a merge sort algorithm to sort a list. -/
-def mergeSort : List α → List α
+def mergeSort' : List α → List α
   | [] => []
   | [a] => [a]
   | a :: b :: l => by
@@ -429,47 +549,47 @@ def mergeSort : List α → List α
     let ls := (split (a :: b :: l))
     have := length_split_fst_le l
     have := length_split_snd_le l
-    exact merge (r · ·) (mergeSort ls.1) (mergeSort ls.2)
+    exact merge (mergeSort' ls.1) (mergeSort' ls.2) (r · ·)
   termination_by l => length l
 
 @[nolint unusedHavesSuffices] -- Porting note: false positive
-theorem mergeSort_cons_cons {a b} {l l₁ l₂ : List α} (h : split (a :: b :: l) = (l₁, l₂)) :
-    mergeSort r (a :: b :: l) = merge (r · ·) (mergeSort r l₁) (mergeSort r l₂) := by
-  simp only [mergeSort, h]
+theorem mergeSort'_cons_cons {a b} {l l₁ l₂ : List α} (h : split (a :: b :: l) = (l₁, l₂)) :
+    mergeSort' r (a :: b :: l) = merge (mergeSort' r l₁) (mergeSort' r l₂) (r · ·) := by
+  simp only [mergeSort', h]
 
 section Correctness
 
-theorem perm_mergeSort : ∀ l : List α, mergeSort r l ~ l
-  | [] => by simp [mergeSort]
-  | [a] => by simp [mergeSort]
+theorem perm_mergeSort' : ∀ l : List α, mergeSort' r l ~ l
+  | [] => by simp [mergeSort']
+  | [a] => by simp [mergeSort']
   | a :: b :: l => by
     cases' e : split (a :: b :: l) with l₁ l₂
     cases' length_split_lt e with h₁ h₂
-    rw [mergeSort_cons_cons r e]
+    rw [mergeSort'_cons_cons r e]
     apply (perm_merge (r · ·) _ _).trans
     exact
-      ((perm_mergeSort l₁).append (perm_mergeSort l₂)).trans (perm_split e).symm
+      ((perm_mergeSort' l₁).append (perm_mergeSort' l₂)).trans (perm_split e).symm
   termination_by l => length l
 
 @[simp]
-theorem mem_mergeSort {l : List α} {x : α} : x ∈ l.mergeSort r ↔ x ∈ l :=
-  (perm_mergeSort r l).mem_iff
+theorem mem_mergeSort' {l : List α} {x : α} : x ∈ l.mergeSort' r ↔ x ∈ l :=
+  (perm_mergeSort' r l).mem_iff
 
 @[simp]
-theorem length_mergeSort (l : List α) : (mergeSort r l).length = l.length :=
-  (perm_mergeSort r _).length_eq
+theorem length_mergeSort' (l : List α) : (mergeSort' r l).length = l.length :=
+  (perm_mergeSort' r _).length_eq
 
 section TotalAndTransitive
 
 variable {r} [IsTotal α r] [IsTrans α r]
 
-theorem Sorted.merge : ∀ {l l' : List α}, Sorted r l → Sorted r l' → Sorted r (merge (r · ·) l l')
+theorem Sorted.merge : ∀ {l l' : List α}, Sorted r l → Sorted r l' → Sorted r (merge l l' (r · ·) )
   | [], [], _, _ => by simp
   | [], b :: l', _, h₂ => by simpa using h₂
   | a :: l, [], h₁, _ => by simpa using h₁
   | a :: l, b :: l', h₁, h₂ => by
     by_cases h : a ≼ b
-    · suffices ∀ b' ∈ List.merge (r · ·) l (b :: l'), r a b' by
+    · suffices ∀ b' ∈ List.merge l (b :: l') (r · ·) , r a b' by
         simpa [h, h₁.of_cons.merge h₂]
       intro b' bm
       rcases show b' = b ∨ b' ∈ l ∨ b' ∈ l' by
@@ -479,7 +599,7 @@ theorem Sorted.merge : ∀ {l l' : List α}, Sorted r l → Sorted r l' → Sort
         assumption
       · exact rel_of_sorted_cons h₁ _ bl
       · exact _root_.trans h (rel_of_sorted_cons h₂ _ bl')
-    · suffices ∀ b' ∈ List.merge (r · ·) (a :: l) l', r b b' by
+    · suffices ∀ b' ∈ List.merge (a :: l) l' (r · ·) , r b b' by
         simpa [h, h₁.merge h₂.of_cons]
       intro b' bm
       have ba : b ≼ a := (total_of r _ _).resolve_left h
@@ -492,54 +612,36 @@ theorem Sorted.merge : ∀ {l l' : List α}, Sorted r l → Sorted r l' → Sort
 
 variable (r)
 
-theorem sorted_mergeSort : ∀ l : List α, Sorted r (mergeSort r l)
-  | [] => by simp [mergeSort]
-  | [a] => by simp [mergeSort]
+theorem sorted_mergeSort' : ∀ l : List α, Sorted r (mergeSort' r l)
+  | [] => by simp [mergeSort']
+  | [a] => by simp [mergeSort']
   | a :: b :: l => by
     cases' e : split (a :: b :: l) with l₁ l₂
     cases' length_split_lt e with h₁ h₂
-    rw [mergeSort_cons_cons r e]
-    exact (sorted_mergeSort l₁).merge (sorted_mergeSort l₂)
+    rw [mergeSort'_cons_cons r e]
+    exact (sorted_mergeSort' l₁).merge (sorted_mergeSort' l₂)
   termination_by l => length l
 
-theorem mergeSort_eq_self [IsAntisymm α r] {l : List α} : Sorted r l → mergeSort r l = l :=
-  eq_of_perm_of_sorted (perm_mergeSort _ _) (sorted_mergeSort _ _)
+theorem mergeSort'_eq_self [IsAntisymm α r] {l : List α} : Sorted r l → mergeSort' r l = l :=
+  eq_of_perm_of_sorted (perm_mergeSort' _ _) (sorted_mergeSort' _ _)
 
-theorem mergeSort_eq_insertionSort [IsAntisymm α r] (l : List α) :
-    mergeSort r l = insertionSort r l :=
-  eq_of_perm_of_sorted ((perm_mergeSort r l).trans (perm_insertionSort r l).symm)
-    (sorted_mergeSort r l) (sorted_insertionSort r l)
+theorem mergeSort'_eq_insertionSort [IsAntisymm α r] (l : List α) :
+    mergeSort' r l = insertionSort r l :=
+  eq_of_perm_of_sorted ((perm_mergeSort' r l).trans (perm_insertionSort r l).symm)
+    (sorted_mergeSort' r l) (sorted_insertionSort r l)
 
 end TotalAndTransitive
 
 end Correctness
 
 @[simp]
-theorem mergeSort_nil : [].mergeSort r = [] := by rw [List.mergeSort]
+theorem mergeSort'_nil : [].mergeSort' r = [] := by rw [List.mergeSort']
 
 @[simp]
-theorem mergeSort_singleton (a : α) : [a].mergeSort r = [a] := by rw [List.mergeSort]
-
-theorem map_merge (f : α → β) (r : α → α → Bool) (s : β → β → Bool) (l l' : List α)
-    (hl : ∀ a ∈ l, ∀ b ∈ l', r a b = s (f a) (f b)) :
-    (l.merge r l').map f = (l.map f).merge s (l'.map f) := by
-  match l, l' with
-  | [], x' => simp
-  | x, [] => simp
-  | x :: xs, x' :: xs' =>
-    simp_rw [List.forall_mem_cons, forall_and] at hl
-    simp_rw [List.map, List.cons_merge_cons]
-    rw [← hl.1.1]
-    split
-    · rw [List.map, map_merge _ r s, List.map]
-      simp_rw [List.forall_mem_cons, forall_and]
-      exact ⟨hl.2.1, hl.2.2⟩
-    · rw [List.map, map_merge _ r s, List.map]
-      simp_rw [List.forall_mem_cons]
-      exact ⟨hl.1.2, hl.2.2⟩
-
-theorem map_mergeSort (f : α → β) (l : List α) (hl : ∀ a ∈ l, ∀ b ∈ l, a ≼ b ↔ f a ≼ f b) :
-    (l.mergeSort r).map f = (l.map f).mergeSort s :=
+theorem mergeSort'_singleton (a : α) : [a].mergeSort' r = [a] := by rw [List.mergeSort']
+
+theorem map_mergeSort' (f : α → β) (l : List α) (hl : ∀ a ∈ l, ∀ b ∈ l, a ≼ b ↔ f a ≼ f b) :
+    (l.mergeSort' r).map f = (l.map f).mergeSort' s :=
   match l with
   | [] => by simp
   | [x] => by simp
@@ -553,9 +655,9 @@ theorem map_mergeSort (f : α → β) (l : List α) (hl : ∀ a ∈ l, ∀ b ∈
     have := length_split_fst_le l
     have := length_split_snd_le l
     simp_rw [List.map]
-    rw [List.mergeSort_cons_cons _ e, List.mergeSort_cons_cons _ fe,
-      map_merge _ (r · ·) (s · ·), map_mergeSort _ l₁ hl.1.1, map_mergeSort _ l₂ hl.2.2]
-    simp_rw [mem_mergeSort, decide_eq_decide]
+    rw [List.mergeSort'_cons_cons _ e, List.mergeSort'_cons_cons _ fe,
+      map_merge, map_mergeSort' _ l₁ hl.1.1, map_mergeSort' _ l₂ hl.2.2]
+    simp_rw [mem_mergeSort', decide_eq_decide]
     exact hl.1.2
   termination_by length l
 
@@ -565,5 +667,5 @@ end sort
 
 -- try them out!
 --#eval insertionSort (fun m n : ℕ => m ≤ n) [5, 27, 221, 95, 17, 43, 7, 2, 98, 567, 23, 12]
---#eval mergeSort     (fun m n : ℕ => m ≤ n) [5, 27, 221, 95, 17, 43, 7, 2, 98, 567, 23, 12]
+--#eval mergeSort'    (fun m n : ℕ => m ≤ n) [5, 27, 221, 95, 17, 43, 7, 2, 98, 567, 23, 12]
 end List
diff --git a/Mathlib/Data/List/Sublists.lean b/Mathlib/Data/List/Sublists.lean
index 7df5ef9099e10..3d8300bd2d98a 100644
--- a/Mathlib/Data/List/Sublists.lean
+++ b/Mathlib/Data/List/Sublists.lean
@@ -44,7 +44,7 @@ theorem sublists'Aux_eq_array_foldl (a : α) : ∀ (r₁ r₂ : List (List α)),
     sublists'Aux a r₁ r₂ = ((r₁.toArray).foldl (init := r₂.toArray)
       (fun r l => r.push (a :: l))).toList := by
   intro r₁ r₂
-  rw [sublists'Aux, Array.foldl_eq_foldl_data]
+  rw [sublists'Aux, Array.foldl_eq_foldl_toList]
   have := List.foldl_hom Array.toList (fun r l => r.push (a :: l))
     (fun r l => r ++ [a :: l]) r₁ r₂.toArray (by simp)
   simpa using this
@@ -53,8 +53,7 @@ theorem sublists'_eq_sublists'Aux (l : List α) :
     sublists' l = l.foldr (fun a r => sublists'Aux a r r) [[]] := by
   simp only [sublists', sublists'Aux_eq_array_foldl]
   rw [← List.foldr_hom Array.toList]
-  · rfl
-  · intros _ _; congr <;> simp
+  · intros _ _; congr
 
 theorem sublists'Aux_eq_map (a : α) (r₁ : List (List α)) : ∀ (r₂ : List (List α)),
     sublists'Aux a r₁ r₂ = r₂ ++ map (cons a) r₁ :=
@@ -107,7 +106,7 @@ theorem sublistsAux_eq_array_foldl :
       (r.toArray.foldl (init := #[])
         fun r l => (r.push l).push (a :: l)).toList := by
   funext a r
-  simp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]
+  simp only [sublistsAux, Array.foldl_eq_foldl_toList, Array.mkEmpty]
   have := foldl_hom Array.toList (fun r l => (r.push l).push (a :: l))
     (fun (r : List (List α)) l => r ++ [l, a :: l]) r #[]
     (by simp)
@@ -126,10 +125,9 @@ theorem sublistsAux_eq_bind :
   ext α l : 2
   trans l.foldr sublistsAux [[]]
   · rw [sublistsAux_eq_bind, sublists]
-  · simp only [sublistsFast, sublistsAux_eq_array_foldl, Array.foldr_eq_foldr_data]
+  · simp only [sublistsFast, sublistsAux_eq_array_foldl, Array.foldr_eq_foldr_toList]
     rw [← foldr_hom Array.toList]
-    · rfl
-    · intros _ _; congr <;> simp
+    · intros _ _; congr
 
 theorem sublists_append (l₁ l₂ : List α) :
     sublists (l₁ ++ l₂) = (sublists l₂) >>= (fun x => (sublists l₁).map (· ++ x)) := by
@@ -138,7 +136,7 @@ theorem sublists_append (l₁ l₂ : List α) :
   | nil => simp
   | cons a l₁ ih =>
     rw [foldr_cons, ih]
-    simp [List.bind, join_join, Function.comp]
+    simp [List.bind, join_join, Function.comp_def]
 
 theorem sublists_cons (a : α) (l : List α) :
     sublists (a :: l) = sublists l >>= (fun x => [x, a :: x]) :=
@@ -155,13 +153,13 @@ theorem sublists_concat (l : List α) (a : α) :
 theorem sublists_reverse (l : List α) : sublists (reverse l) = map reverse (sublists' l) := by
   induction' l with hd tl ih <;> [rfl;
     simp only [reverse_cons, sublists_append, sublists'_cons, map_append, ih, sublists_singleton,
-      map_eq_map, bind_eq_bind, map_map, bind_cons, append_nil, bind_nil, (· ∘ ·)]]
+      map_eq_map, bind_eq_bind, map_map, bind_cons, append_nil, bind_nil, Function.comp_def]]
 
 theorem sublists_eq_sublists' (l : List α) : sublists l = map reverse (sublists' (reverse l)) := by
   rw [← sublists_reverse, reverse_reverse]
 
 theorem sublists'_reverse (l : List α) : sublists' (reverse l) = map reverse (sublists l) := by
-  simp only [sublists_eq_sublists', map_map, map_id'' reverse_reverse, Function.comp]
+  simp only [sublists_eq_sublists', map_map, map_id'' reverse_reverse, Function.comp_def]
 
 theorem sublists'_eq_sublists (l : List α) : sublists' l = map reverse (sublists (reverse l)) := by
   rw [← sublists'_reverse, reverse_reverse]
diff --git a/Mathlib/Data/List/Sym.lean b/Mathlib/Data/List/Sym.lean
index 80954c1ad9dc7..1d6228b9258e3 100644
--- a/Mathlib/Data/List/Sym.lean
+++ b/Mathlib/Data/List/Sym.lean
@@ -237,7 +237,7 @@ theorem sym_one_eq : xs.sym 1 = xs.map (· ::ₛ .nil) := by
 
 theorem sym2_eq_sym_two : xs.sym2.map (Sym2.equivSym α) = xs.sym 2 := by
   induction xs with
-  | nil => simp only [List.sym, map_eq_nil, sym2_eq_nil_iff]
+  | nil => simp only [List.sym, map_eq_nil_iff, sym2_eq_nil_iff]
   | cons x xs ih =>
     rw [List.sym, ← ih, sym_one_eq, map_map, List.sym2, map_append, map_map]
     rfl
diff --git a/Mathlib/Data/List/Zip.lean b/Mathlib/Data/List/Zip.lean
index b53a34837076a..cb8b7daa47501 100644
--- a/Mathlib/Data/List/Zip.lean
+++ b/Mathlib/Data/List/Zip.lean
@@ -92,7 +92,7 @@ theorem zipWith3_same_right (f : α → β → β → γ) :
   | _ :: _, [] => rfl
   | _ :: as, _ :: bs => congr_arg (cons _) <| zipWith3_same_right f as bs
 
-instance (f : α → α → β) [IsSymmOp α β f] : IsSymmOp (List α) (List β) (zipWith f) :=
+instance (f : α → α → β) [IsSymmOp f] : IsSymmOp (zipWith f) :=
   ⟨zipWith_comm_of_comm f IsSymmOp.symm_op⟩
 
 @[simp]
diff --git a/Mathlib/Data/MLList/BestFirst.lean b/Mathlib/Data/MLList/BestFirst.lean
index 04d8b676c5644..1eceab8e6500c 100644
--- a/Mathlib/Data/MLList/BestFirst.lean
+++ b/Mathlib/Data/MLList/BestFirst.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Batteries.Data.MLList.Basic
 import Mathlib.Data.Prod.Lex
diff --git a/Mathlib/Data/MLList/Dedup.lean b/Mathlib/Data/MLList/Dedup.lean
index 880fe43d4416e..f608048a51890 100644
--- a/Mathlib/Data/MLList/Dedup.lean
+++ b/Mathlib/Data/MLList/Dedup.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Batteries.Data.MLList.Basic
@@ -9,8 +9,14 @@ import Batteries.Data.HashMap.Basic
 
 /-!
 # Lazy deduplication of lazy lists
+
+## Deprecation
+
+This material has been moved out of Mathlib to https://github.com/semorrison/lean-monadic-list.
 -/
 
+set_option linter.deprecated false
+
 open Batteries
 
 namespace MLList
@@ -19,6 +25,7 @@ variable {α β : Type} {m : Type → Type} [Monad m] [BEq β] [Hashable β]
 
 /-- Lazily deduplicate a lazy list, using a stored `HashMap`. -/
 -- We choose `HashMap` here instead of `RBSet` as the initial application is `Expr`.
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def dedupBy (L : MLList m α) (f : α → m β) : MLList m α :=
   ((L.liftM : MLList (StateT (HashMap β Unit) m) α) >>= fun a => do
       let b ← f a
@@ -28,6 +35,7 @@ def dedupBy (L : MLList m α) (f : α → m β) : MLList m α :=
   |>.runState' ∅
 
 /-- Lazily deduplicate a lazy list, using a stored `HashMap`. -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def dedup (L : MLList m β) : MLList m β :=
   L.dedupBy (fun b => pure b)
 
diff --git a/Mathlib/Data/MLList/DepthFirst.lean b/Mathlib/Data/MLList/DepthFirst.lean
index 0e0b0e96a5e2b..cc407b0ef5c2d 100644
--- a/Mathlib/Data/MLList/DepthFirst.lean
+++ b/Mathlib/Data/MLList/DepthFirst.lean
@@ -1,11 +1,11 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
-import Lean.Data.HashSet
 import Batteries.Data.MLList.Basic
 import Mathlib.Control.Combinators
+import Std.Data.HashSet.Basic
 
 /-!
 # Depth first search
@@ -17,8 +17,14 @@ or a lazy list `α → MLList MetaM α`.
 This is useful in meta code for searching for solutions in the presence of alternatives.
 It can be nice to represent the choices via a lazy list,
 so the later choices don't need to be evaluated while we do depth first search on earlier choices.
+
+## Deprecation
+
+This material has been moved out of Mathlib to https://github.com/semorrison/lean-monadic-list.
 -/
 
+set_option linter.deprecated false
+
 universe u
 variable {α : Type u} {m : Type u → Type u}
 
@@ -28,6 +34,7 @@ variable [Monad m] [Alternative m]
 
 /-- A generalisation of `depthFirst`, which allows the generation function to know the current
 depth, and to count the depth starting from a specified value. -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 partial def depthFirst' (f : Nat → α → m α) (n : Nat) (a : α) : m α :=
 pure a <|> joinM ((f n a) <&> (depthFirst' f (n+1)))
 
@@ -43,6 +50,7 @@ The option `maxDepth` limits the search depth.
 Note that if the graph is not a tree then elements will be visited multiple times.
 (See `depthFirstRemovingDuplicates`)
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def depthFirst (f : α → m α) (a : α) (maxDepth : Option Nat := none) : m α :=
   match maxDepth with
   | some d => depthFirst' (fun n a => if n ≤ d then f a else failure) 0 a
@@ -66,15 +74,16 @@ avoiding duplication up to equality or isomorphism,
 use Brendan McKay's method of "generation by canonical construction path".
 -/
 -- TODO can you make this work in `List` and `MLList m` simultaneously, by being tricky with monads?
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def depthFirstRemovingDuplicates {α : Type u} [BEq α] [Hashable α]
     (f : α → MLList m α) (a : α) (maxDepth : Option Nat := none) : MLList m α :=
-let f' : α → MLList (StateT.{u} (HashSet α) m) α := fun a =>
+let f' : α → MLList (StateT.{u} (Std.HashSet α) m) α := fun a =>
   (f a).liftM >>= fun b => do
     let s ← get
     if s.contains b then failure
     set <| s.insert b
     pure b
-(depthFirst f' a maxDepth).runState' (HashSet.empty.insert a)
+(depthFirst f' a maxDepth).runState' (Std.HashSet.empty.insert a)
 
 /--
 Variant of `depthFirst`,
@@ -82,6 +91,7 @@ using an internal `HashSet` to record and avoid already visited nodes.
 
 This version describes the graph using `α → List α`, and returns the list of nodes visited in order.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def depthFirstRemovingDuplicates' [BEq α] [Hashable α]
     (f : α → List α) (a : α) (maxDepth : Option Nat := none) : List α :=
 depthFirstRemovingDuplicates
diff --git a/Mathlib/Data/MLList/IO.lean b/Mathlib/Data/MLList/IO.lean
index 8f0a5da58d9d2..caec83d6d5838 100644
--- a/Mathlib/Data/MLList/IO.lean
+++ b/Mathlib/Data/MLList/IO.lean
@@ -1,20 +1,26 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Batteries.Data.MLList.Basic
 
 /-!
 # Reading from handles, files, and processes as lazy lists.
+
+## Deprecation
+
+This material has been moved out of Mathlib to https://github.com/semorrison/lean-monadic-list.
 -/
 
 open System IO.FS
+set_option linter.deprecated false
 
 namespace MLList
 
 /-- Read lines of text from a handle, as a lazy list in `IO`. -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def linesFromHandle (h : Handle) : MLList IO String :=
   MLList.iterate (do
     let line ← h.getLine
@@ -31,6 +37,7 @@ def linesFromHandle (h : Handle) : MLList IO String :=
   |>.takeWhile (·.isSome) |>.map (fun o => o.getD "")
 
 /-- Read lines of text from a file, as a lazy list in `IO`. -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def lines (f : FilePath) : MLList IO String := .squash fun _ => do
   return linesFromHandle (← Handle.mk f Mode.read)
 
@@ -39,6 +46,7 @@ open IO.Process in
 Run a command with given input on `stdio`,
 returning `stdout` as a lazy list in `IO`.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def runCmd (cmd : String) (args : Array String) (input : String := "") : MLList IO String := do
   let child ← spawn
     { cmd := cmd, args := args, stdin := .piped, stdout := .piped, stderr := .piped }
diff --git a/Mathlib/Data/MLList/Split.lean b/Mathlib/Data/MLList/Split.lean
index 1aaf2fbbd9927..2705ad27826e6 100644
--- a/Mathlib/Data/MLList/Split.lean
+++ b/Mathlib/Data/MLList/Split.lean
@@ -1,15 +1,21 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Batteries.Data.MLList.Basic
 import Mathlib.Data.ULift
 
 /-!
 # Functions for splitting monadic lazy lists.
+
+## Deprecation
+
+This material has been moved out of Mathlib to https://github.com/semorrison/lean-monadic-list.
 -/
 
+set_option linter.deprecated false
+
 namespace MLList
 
 universe u
@@ -20,6 +26,7 @@ Extract the prefix of a lazy list consisting of elements up to and including
 the first element satisfying a monadic predicate.
 Return (in the monad) the prefix as a `List`, along with the remaining elements as a `MLList`.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 partial def getUpToFirstM (L : MLList m α) (p : α → m (ULift Bool)) : m (List α × MLList m α) := do
   match ← L.uncons with
   | none => return ([], nil)
@@ -34,6 +41,7 @@ Extract the prefix of a lazy list consisting of elements up to and including
 the first element satisfying a predicate.
 Return (in the monad) the prefix as a `List`, along with the remaining elements as a `MLList`.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def getUpToFirst (L : MLList m α) (p : α → Bool) : m (List α × MLList m α) :=
   L.getUpToFirstM fun a => pure (.up (p a))
 
@@ -45,6 +53,7 @@ Return (in the monad) the prefix as a `List`, along with the remaining elements
 (Note that the first element *not* satisfying the predicate will be generated,
 and pushed back on to the remaining lazy list.)
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 partial def splitWhileM (L : MLList m α) (p : α → m (ULift Bool)) :
     m (List α × MLList m α) := do
   match ← L.uncons with
@@ -63,6 +72,7 @@ Return (in the monad) the prefix as a `List`, along with the remaining elements
 (Note that the first element *not* satisfying the predicate will be generated,
 and pushed back on to the remaining lazy list.)
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def splitWhile (L : MLList m α) (p : α → Bool) : m (List α × MLList m α) :=
   L.splitWhileM fun a => pure (.up (p a))
 
@@ -72,6 +82,7 @@ a monadic function.
 Return a lazy lists of pairs, consisting of a value under that function,
 and a maximal list of elements having that value.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 partial def groupByM [DecidableEq β] (L : MLList m α) (f : α → m β) : MLList m (β × List α) :=
   L.cases (fun _ => nil) fun a t => squash fun _ => do
     let b ← f a
@@ -83,6 +94,7 @@ Splits a lazy list into contiguous sublists of elements with the same value unde
 Return a lazy lists of pairs, consisting of a value under that function,
 and a maximal list of elements having that value.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def groupBy [DecidableEq β] (L : MLList m α) (f : α → β) : MLList m (β × List α) :=
   L.groupByM fun a => pure (f a)
 
@@ -93,6 +105,7 @@ def groupBy [DecidableEq β] (L : MLList m α) (f : α → β) : MLList m (β ×
 Split a lazy list into contiguous sublists,
 starting a new sublist each time a monadic predicate changes from `false` to `true`.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 partial def splitAtBecomesTrueM (L : MLList m α) (p : α → m (ULift Bool)) : MLList m (List α) :=
   aux (L.groupByM p)
 where aux (M : MLList m (ULift.{u} Bool × List α)) : MLList m (List α) :=
@@ -106,6 +119,7 @@ where aux (M : MLList m (ULift.{u} Bool × List α)) : MLList m (List α) :=
 Split a lazy list into contiguous sublists,
 starting a new sublist each time a predicate changes from `false` to `true`.
 -/
+@[deprecated "See deprecation note in module documentation." (since := "2024-08-22")]
 def splitAtBecomesTrue (L : MLList m α) (p : α → Bool) : MLList m (List α) :=
   L.splitAtBecomesTrueM fun a => pure (.up (p a))
 
diff --git a/Mathlib/Data/Matrix/Basic.lean b/Mathlib/Data/Matrix/Basic.lean
index 3d645f33d8402..63bc543ba910b 100644
--- a/Mathlib/Data/Matrix/Basic.lean
+++ b/Mathlib/Data/Matrix/Basic.lean
@@ -5,6 +5,7 @@ Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu
 -/
 import Mathlib.Algebra.Algebra.Opposite
 import Mathlib.Algebra.Algebra.Pi
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.Algebra.BigOperators.Pi
 import Mathlib.Algebra.BigOperators.Ring
 import Mathlib.Algebra.BigOperators.RingEquiv
@@ -13,7 +14,6 @@ import Mathlib.Algebra.Star.BigOperators
 import Mathlib.Algebra.Star.Module
 import Mathlib.Algebra.Star.Pi
 import Mathlib.Data.Fintype.BigOperators
-import Mathlib.GroupTheory.GroupAction.BigOperators
 
 /-!
 # Matrices
@@ -492,6 +492,11 @@ theorem diagonal_conjTranspose [AddMonoid α] [StarAddMonoid α] (v : n → α)
   rw [conjTranspose, diagonal_transpose, diagonal_map (star_zero _)]
   rfl
 
+theorem diagonal_unique [Unique m] [DecidableEq m] [Zero α] (d : m → α) :
+    diagonal d = of fun _ _ => d default := by
+  ext i j
+  rw [Subsingleton.elim i default, Subsingleton.elim j default, diagonal_apply_eq _ _, of_apply]
+
 section One
 
 variable [Zero α] [One α]
@@ -733,7 +738,7 @@ theorem dotProduct_comp_equiv_symm (e : n ≃ m) : u ⬝ᵥ x ∘ e.symm = u ∘
 @[simp]
 theorem comp_equiv_dotProduct_comp_equiv (e : m ≃ n) : x ∘ e ⬝ᵥ y ∘ e = x ⬝ᵥ y := by
   -- Porting note: was `simp only` with all three lemmas
-  rw [← dotProduct_comp_equiv_symm]; simp only [Function.comp, Equiv.apply_symm_apply]
+  rw [← dotProduct_comp_equiv_symm]; simp only [Function.comp_def, Equiv.apply_symm_apply]
 
 end NonUnitalNonAssocSemiring
 
@@ -2093,7 +2098,6 @@ variants which this lemma would not apply to:
 * `Matrix.conjTranspose_intCast_smul`
 * `Matrix.conjTranspose_inv_natCast_smul`
 * `Matrix.conjTranspose_inv_intCast_smul`
-* `Matrix.conjTranspose_rat_smul`
 * `Matrix.conjTranspose_ratCast_smul`
 -/
 @[simp]
@@ -2161,7 +2165,6 @@ theorem conjTranspose_ratCast_smul [DivisionRing R] [AddCommGroup α] [StarAddMo
     (c : ℚ) (M : Matrix m n α) : ((c : R) • M)ᴴ = (c : R) • Mᴴ :=
   Matrix.ext <| by simp
 
-@[simp]
 theorem conjTranspose_rat_smul [AddCommGroup α] [StarAddMonoid α] [Module ℚ α] (c : ℚ)
     (M : Matrix m n α) : (c • M)ᴴ = c • Mᴴ :=
   Matrix.ext <| by simp
@@ -2526,10 +2529,12 @@ theorem map_dotProduct [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S)
 
 theorem map_vecMul [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (M : Matrix n m R)
     (v : n → R) (i : m) : f ((v ᵥ* M) i) =  ((f ∘ v) ᵥ* M.map f) i := by
-  simp only [Matrix.vecMul, Matrix.map_apply, RingHom.map_dotProduct, Function.comp]
+  simp only [Matrix.vecMul, Matrix.map_apply, RingHom.map_dotProduct, Function.comp_def]
 
 theorem map_mulVec [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (M : Matrix m n R)
     (v : n → R) (i : m) : f ((M *ᵥ v) i) = (M.map f *ᵥ (f ∘ v)) i := by
-  simp only [Matrix.mulVec, Matrix.map_apply, RingHom.map_dotProduct, Function.comp]
+  simp only [Matrix.mulVec, Matrix.map_apply, RingHom.map_dotProduct, Function.comp_def]
 
 end RingHom
+
+set_option linter.style.longFile 2700
diff --git a/Mathlib/Data/Matrix/Basis.lean b/Mathlib/Data/Matrix/Basis.lean
index 1204f7bb9f157..01c7a8cf445b6 100644
--- a/Mathlib/Data/Matrix/Basis.lean
+++ b/Mathlib/Data/Matrix/Basis.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Jalex Stark. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
+Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
 -/
 import Mathlib.Data.Matrix.Basic
 import Mathlib.LinearAlgebra.Matrix.Trace
diff --git a/Mathlib/Data/Matrix/ColumnRowPartitioned.lean b/Mathlib/Data/Matrix/ColumnRowPartitioned.lean
index 8d0ba9f8ba139..1fef44043ba64 100644
--- a/Mathlib/Data/Matrix/ColumnRowPartitioned.lean
+++ b/Mathlib/Data/Matrix/ColumnRowPartitioned.lean
@@ -74,11 +74,11 @@ lemma toRows₂_apply (A : Matrix (m₁ ⊕ m₂) n R) (i : m₂) (j : n) :
     (toRows₂ A) i j = A (Sum.inr i) j := rfl
 
 @[simp]
-lemma toRows₁_fromRows  (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :
+lemma toRows₁_fromRows (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :
     toRows₁ (fromRows A₁ A₂) = A₁ := rfl
 
 @[simp]
-lemma toRows₂_fromRows  (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :
+lemma toRows₂_fromRows (A₁ : Matrix m₁ n R) (A₂ : Matrix m₂ n R) :
     toRows₂ (fromRows A₁ A₂) = A₂ := rfl
 
 @[simp]
@@ -258,17 +258,8 @@ lemma fromColumns_mul_fromRows_eq_one_comm
     [Fintype n₁] [Fintype n₂] [Fintype n] [DecidableEq n] [DecidableEq n₁] [DecidableEq n₂]
     (e : n ≃ n₁ ⊕ n₂)
     (A₁ : Matrix n n₁ R) (A₂ : Matrix n n₂ R) (B₁ : Matrix n₁ n R) (B₂ : Matrix n₂ n R) :
-    fromColumns A₁ A₂ * fromRows B₁ B₂ = 1 ↔ fromRows B₁ B₂ * fromColumns A₁ A₂ = 1 := by
-  calc fromColumns A₁ A₂ * fromRows B₁ B₂ = 1
-  _ ↔ submatrix (fromColumns A₁ A₂) id e * submatrix (fromRows B₁ B₂) e id = 1 := by
-    simp
-  _ ↔ submatrix (fromRows B₁ B₂) e id * submatrix (fromColumns A₁ A₂) id e = 1 :=
-    mul_eq_one_comm
-  _ ↔ reindex e.symm e.symm (fromRows B₁ B₂ * fromColumns A₁ A₂) = reindex e.symm e.symm 1 := by
-    simp only [reindex_apply, Equiv.symm_symm, submatrix_one_equiv,
-        submatrix_mul (he₂ := Function.bijective_id)]
-  _ ↔ fromRows B₁ B₂ * fromColumns A₁ A₂ = 1 :=
-    (reindex _ _).injective.eq_iff
+    fromColumns A₁ A₂ * fromRows B₁ B₂ = 1 ↔ fromRows B₁ B₂ * fromColumns A₁ A₂ = 1 :=
+  mul_eq_one_comm_of_equiv e
 
 /-- The lemma `fromColumns_mul_fromRows_eq_one_comm` specialized to the case where the index sets n₁
 and n₂, are the result of subtyping by a predicate and its complement. -/
diff --git a/Mathlib/Data/Matrix/Composition.lean b/Mathlib/Data/Matrix/Composition.lean
index 31de61ff426dd..1970544baa99b 100644
--- a/Mathlib/Data/Matrix/Composition.lean
+++ b/Mathlib/Data/Matrix/Composition.lean
@@ -24,7 +24,7 @@ Semiring, and Algebra over a CommSemiring K.
 
 namespace Matrix
 
-variable  (I J K L R : Type*)
+variable (I J K L R : Type*)
 
 /-- I by J matrix where each entry is a K by L matrix is equivalent to
     I × K by J × L matrix -/
@@ -55,9 +55,7 @@ variable [Semiring R] [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J]
 @[simps!]
 def compRingEquiv : Matrix I I (Matrix J J R) ≃+* Matrix (I × J) (I × J) R where
   __ := Matrix.compAddEquiv I I J J R
-  map_mul' _ _ := by
-    ext _ _
-    exact (Matrix.sum_apply _ _ _ _).trans <| Eq.symm Fintype.sum_prod_type
+  map_mul' _ _ := by ext; exact (Matrix.sum_apply ..).trans <| .symm <| Fintype.sum_prod_type ..
 
 end Semiring
 
diff --git a/Mathlib/Data/Matrix/DMatrix.lean b/Mathlib/Data/Matrix/DMatrix.lean
index 64fcb5711e21b..2b482b6e0ee61 100644
--- a/Mathlib/Data/Matrix/DMatrix.lean
+++ b/Mathlib/Data/Matrix/DMatrix.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Hom.Defs
 
@@ -20,7 +20,7 @@ In most applications `m` and `n` are finite types. -/
 def DMatrix (m : Type u) (n : Type u') (α : m → n → Type v) : Type max u u' v :=
   ∀ i j, α i j
 
-variable {l m n o : Type*}
+variable {m n : Type*}
 variable {α : m → n → Type v}
 
 namespace DMatrix
diff --git a/Mathlib/Data/Matrix/DoublyStochastic.lean b/Mathlib/Data/Matrix/DoublyStochastic.lean
new file mode 100644
index 0000000000000..8bc91f3996116
--- /dev/null
+++ b/Mathlib/Data/Matrix/DoublyStochastic.lean
@@ -0,0 +1,139 @@
+/-
+Copyright (c) 2024 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+-/
+
+import Mathlib.Analysis.Convex.Basic
+import Mathlib.LinearAlgebra.Matrix.Permutation
+
+/-!
+# Doubly stochastic matrices
+
+## Main definitions
+
+* `doublyStochastic`: a square matrix is doubly stochastic if all entries are nonnegative, and left
+  or right multiplication by the vector of all 1s gives the vector of all 1s. Equivalently, all
+  row and column sums are equal to 1.
+
+## Main statements
+
+* `convex_doublyStochastic`: The set of doubly stochastic matrices is convex.
+* `permMatrix_mem_doublyStochastic`: Any permutation matrix is doubly stochastic.
+
+## TODO
+
+Define the submonoids of row-stochastic and column-stochastic matrices.
+Show that the submonoid of doubly stochastic matrices is the meet of them, or redefine it as such.
+
+## Tags
+
+Doubly stochastic, Birkhoff's theorem, Birkhoff-von Neumann theorem
+-/
+
+open Finset Function Matrix
+
+variable {R n : Type*} [Fintype n] [DecidableEq n]
+
+section OrderedSemiring
+variable [OrderedSemiring R] {M : Matrix n n R}
+
+/--
+A square matrix is doubly stochastic iff all entries are nonnegative, and left or right
+multiplication by the vector of all 1s gives the vector of all 1s.
+-/
+def doublyStochastic (R n : Type*) [Fintype n] [DecidableEq n] [OrderedSemiring R] :
+    Submonoid (Matrix n n R) where
+  carrier := {M | (∀ i j, 0 ≤ M i j) ∧ M *ᵥ 1 = 1 ∧ 1 ᵥ* M = 1 }
+  mul_mem' {M N} hM hN := by
+    refine ⟨fun i j => sum_nonneg fun i _ => mul_nonneg (hM.1 _ _) (hN.1 _ _), ?_, ?_⟩
+    next => rw [← mulVec_mulVec, hN.2.1, hM.2.1]
+    next => rw [← vecMul_vecMul, hM.2.2, hN.2.2]
+  one_mem' := by simp [zero_le_one_elem]
+
+lemma mem_doublyStochastic :
+    M ∈ doublyStochastic R n ↔ (∀ i j, 0 ≤ M i j) ∧ M *ᵥ 1 = 1 ∧ 1 ᵥ* M = 1 :=
+  Iff.rfl
+
+lemma mem_doublyStochastic_iff_sum :
+    M ∈ doublyStochastic R n ↔
+      (∀ i j, 0 ≤ M i j) ∧ (∀ i, ∑ j, M i j = 1) ∧ ∀ j, ∑ i, M i j = 1 := by
+  simp [funext_iff, doublyStochastic, mulVec, vecMul, dotProduct]
+
+/-- Every entry of a doubly stochastic matrix is nonnegative. -/
+lemma nonneg_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) {i j : n} : 0 ≤ M i j :=
+  hM.1 _ _
+
+/-- Each row sum of a doubly stochastic matrix is 1. -/
+lemma sum_row_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) (i : n) : ∑ j, M i j = 1 :=
+  (mem_doublyStochastic_iff_sum.1 hM).2.1 _
+
+/-- Each column sum of a doubly stochastic matrix is 1. -/
+lemma sum_col_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) (j : n) : ∑ i, M i j = 1 :=
+  (mem_doublyStochastic_iff_sum.1 hM).2.2 _
+
+/-- A doubly stochastic matrix multiplied with the all-ones column vector is 1. -/
+lemma mulVec_one_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) : M *ᵥ 1 = 1 :=
+  (mem_doublyStochastic.1 hM).2.1
+
+/-- The all-ones row vector multiplied with a doubly stochastic matrix is 1. -/
+lemma one_vecMul_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) : 1 ᵥ* M = 1 :=
+  (mem_doublyStochastic.1 hM).2.2
+
+/-- Every entry of a doubly stochastic matrix is less than or equal to 1. -/
+lemma le_one_of_mem_doublyStochastic (hM : M ∈ doublyStochastic R n) {i j : n} :
+    M i j ≤ 1 := by
+  rw [← sum_row_of_mem_doublyStochastic hM i]
+  exact single_le_sum (fun k _ => hM.1 _ k) (mem_univ j)
+
+/-- The set of doubly stochastic matrices is convex. -/
+lemma convex_doublyStochastic : Convex R (doublyStochastic R n : Set (Matrix n n R)) := by
+  intro x hx y hy a b ha hb h
+  simp only [SetLike.mem_coe, mem_doublyStochastic_iff_sum] at hx hy ⊢
+  simp [add_nonneg, ha, hb, mul_nonneg, hx, hy, sum_add_distrib, ← mul_sum, h]
+
+/-- Any permutation matrix is doubly stochastic. -/
+lemma permMatrix_mem_doublyStochastic {σ : Equiv.Perm n} :
+    σ.permMatrix R ∈ doublyStochastic R n := by
+  rw [mem_doublyStochastic_iff_sum]
+  refine ⟨fun i j => ?g1, ?g2, ?g3⟩
+  case g1 => aesop
+  case g2 => simp [Equiv.toPEquiv_apply]
+  case g3 => simp [Equiv.toPEquiv_apply, ← Equiv.eq_symm_apply]
+
+end OrderedSemiring
+
+section LinearOrderedSemifield
+
+variable [LinearOrderedSemifield R] {M : Matrix n n R}
+
+/--
+A matrix is `s` times a doubly stochastic matrix iff all entries are nonnegative, and all row and
+column sums are equal to `s`.
+
+This lemma is useful for the proof of Birkhoff's theorem - in particular because it allows scaling
+by nonnegative factors rather than positive ones only.
+-/
+lemma exists_mem_doublyStochastic_eq_smul_iff {M : Matrix n n R} {s : R} (hs : 0 ≤ s) :
+    (∃ M' ∈ doublyStochastic R n, M = s • M') ↔
+      (∀ i j, 0 ≤ M i j) ∧ (∀ i, ∑ j, M i j = s) ∧ (∀ j, ∑ i, M i j = s) := by
+  classical
+  constructor
+  case mp =>
+    rintro ⟨M', hM', rfl⟩
+    rw [mem_doublyStochastic_iff_sum] at hM'
+    simp only [smul_apply, smul_eq_mul, ← mul_sum]
+    exact ⟨fun i j => mul_nonneg hs (hM'.1 _ _), by simp [hM']⟩
+  rcases eq_or_lt_of_le hs with rfl | hs
+  case inl =>
+    simp only [zero_smul, exists_and_right, and_imp]
+    intro h₁ h₂ _
+    refine ⟨⟨1, Submonoid.one_mem _⟩, ?_⟩
+    ext i j
+    specialize h₂ i
+    rw [sum_eq_zero_iff_of_nonneg (by simp [h₁ i])] at h₂
+    exact h₂ _ (by simp)
+  rintro ⟨hM₁, hM₂, hM₃⟩
+  exact ⟨s⁻¹ • M, by simp [mem_doublyStochastic_iff_sum, ← mul_sum, hs.ne', inv_mul_cancel₀, *]⟩
+
+end LinearOrderedSemifield
diff --git a/Mathlib/Data/Matrix/Invertible.lean b/Mathlib/Data/Matrix/Invertible.lean
index c0923041fcdb8..d5581bd17d0de 100644
--- a/Mathlib/Data/Matrix/Invertible.lean
+++ b/Mathlib/Data/Matrix/Invertible.lean
@@ -1,9 +1,10 @@
 /-
 Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Eric Wieser
+Authors: Eric Wieser, Ahmad Alkhalawi
 -/
 import Mathlib.Data.Matrix.Basic
+import Mathlib.Tactic.Abel
 
 /-! # Extra lemmas about invertible matrices
 
@@ -31,22 +32,31 @@ namespace Matrix
 section Semiring
 variable [Semiring α]
 
-/-- A copy of `invOf_mul_self_assoc` for rectangular matrices. -/
-protected theorem invOf_mul_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
+/-- A copy of `invOf_mul_cancel_left` for rectangular matrices. -/
+protected theorem invOf_mul_cancel_left (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
     ⅟ A * (A * B) = B := by rw [← Matrix.mul_assoc, invOf_mul_self, Matrix.one_mul]
 
-/-- A copy of `mul_invOf_self_assoc` for rectangular matrices. -/
-protected theorem mul_invOf_self_assoc (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
+/-- A copy of `mul_invOf_cancel_left` for rectangular matrices. -/
+protected theorem mul_invOf_cancel_left (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
     A * (⅟ A * B) = B := by rw [← Matrix.mul_assoc, mul_invOf_self, Matrix.one_mul]
 
-/-- A copy of `mul_invOf_mul_self_cancel` for rectangular matrices. -/
-protected theorem mul_invOf_mul_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
+/-- A copy of `invOf_mul_cancel_right` for rectangular matrices. -/
+protected theorem invOf_mul_cancel_right (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
     A * ⅟ B * B = A := by rw [Matrix.mul_assoc, invOf_mul_self, Matrix.mul_one]
 
-/-- A copy of `mul_mul_invOf_self_cancel` for rectangular matrices. -/
-protected theorem mul_mul_invOf_self_cancel (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
+/-- A copy of `mul_invOf_cancel_right` for rectangular matrices. -/
+protected theorem mul_invOf_cancel_right (A : Matrix m n α) (B : Matrix n n α) [Invertible B] :
     A * B * ⅟ B = A := by rw [Matrix.mul_assoc, mul_invOf_self, Matrix.mul_one]
 
+@[deprecated (since := "2024-09-07")]
+protected alias invOf_mul_self_assoc := Matrix.invOf_mul_cancel_left
+@[deprecated (since := "2024-09-07")]
+protected alias mul_invOf_self_assoc := Matrix.mul_invOf_cancel_left
+@[deprecated (since := "2024-09-07")]
+protected alias mul_invOf_mul_self_cancel := Matrix.invOf_mul_cancel_right
+@[deprecated (since := "2024-09-07")]
+protected alias mul_mul_invOf_self_cancel := Matrix.mul_invOf_cancel_right
+
 section ConjTranspose
 variable [StarRing α] (A : Matrix n n α)
 
@@ -101,4 +111,68 @@ def transposeInvertibleEquivInvertible : Invertible Aᵀ ≃ Invertible A where
 
 end CommSemiring
 
+section Ring
+
+section Woodbury
+
+variable [Fintype m] [DecidableEq m] [Ring α]
+    (A : Matrix n n α) (U : Matrix n m α) (C : Matrix m m α) (V : Matrix m n α)
+    [Invertible A] [Invertible C] [Invertible (⅟C + V * ⅟A * U)]
+
+-- No spaces around multiplication signs for better clarity
+lemma add_mul_mul_invOf_mul_eq_one :
+    (A + U*C*V)*(⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A) = 1 := by
+  calc
+    (A + U*C*V)*(⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A)
+    _ = A*⅟A - A*⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A + U*C*V*⅟A - U*C*V*⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A := by
+      simp_rw [add_sub_assoc, add_mul, mul_sub, Matrix.mul_assoc]
+    _ = (1 + U*C*V*⅟A) - (U*⅟(⅟C + V*⅟A*U)*V*⅟A + U*C*V*⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A) := by
+      rw [mul_invOf_self, Matrix.one_mul]
+      abel
+    _ = 1 + U*C*V*⅟A - (U + U*C*V*⅟A*U)*⅟(⅟C + V*⅟A*U)*V*⅟A := by
+      rw [sub_right_inj, Matrix.add_mul, Matrix.add_mul, Matrix.add_mul]
+    _ = 1 + U*C*V*⅟A - U*C*(⅟C + V*⅟A*U)*⅟(⅟C + V*⅟A*U)*V*⅟A := by
+      congr
+      simp only [Matrix.mul_add, Matrix.mul_invOf_cancel_right, ← Matrix.mul_assoc]
+    _ = 1 := by
+      rw [Matrix.mul_invOf_cancel_right]
+      abel
+
+/-- Like `add_mul_mul_invOf_mul_eq_one`, but with multiplication reversed. -/
+lemma add_mul_mul_invOf_mul_eq_one' :
+    (⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A)*(A + U*C*V) = 1 := by
+  calc
+    (⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A)*(A + U*C*V)
+    _ = ⅟A*A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*A + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*U*C*V := by
+      simp_rw [add_sub_assoc, _root_.mul_add, _root_.sub_mul, Matrix.mul_assoc]
+    _ = (1 + ⅟A*U*C*V) - (⅟A*U*⅟(⅟C + V*⅟A*U)*V + ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A*U*C*V) := by
+      rw [invOf_mul_self, Matrix.invOf_mul_cancel_right]
+      abel
+    _ = 1 + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*(V + V*⅟A*U*C*V) := by
+      rw [sub_right_inj, Matrix.mul_add]
+      simp_rw [Matrix.mul_assoc]
+    _ = 1 + ⅟A*U*C*V - ⅟A*U*⅟(⅟C + V*⅟A*U)*(⅟C + V*⅟A*U)*C*V := by
+      congr 1
+      simp only [Matrix.mul_add, Matrix.add_mul, ← Matrix.mul_assoc,
+        Matrix.invOf_mul_cancel_right]
+    _ = 1 := by
+      rw [Matrix.invOf_mul_cancel_right]
+      abel
+
+/-- If matrices `A`, `C`, and `C⁻¹ + V * A⁻¹ * U` are invertible, then so is `A + U * C * V`-/
+def invertibleAddMulMul : Invertible (A + U*C*V) where
+  invOf := ⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A
+  invOf_mul_self := add_mul_mul_invOf_mul_eq_one' _ _ _ _
+  mul_invOf_self := add_mul_mul_invOf_mul_eq_one _ _ _ _
+
+/-- The **Woodbury Identity** (`⅟` version). -/
+theorem invOf_add_mul_mul [Invertible (A + U*C*V)] :
+    ⅟(A + U*C*V) = ⅟A - ⅟A*U*⅟(⅟C + V*⅟A*U)*V*⅟A := by
+  letI := invertibleAddMulMul A U C V
+  convert (rfl : ⅟(A + U*C*V) = _)
+
+end Woodbury
+
+end Ring
+
 end Matrix
diff --git a/Mathlib/Data/Matrix/Rank.lean b/Mathlib/Data/Matrix/Rank.lean
index 8ae0a21d0de4a..13db0d03e2bc9 100644
--- a/Mathlib/Data/Matrix/Rank.lean
+++ b/Mathlib/Data/Matrix/Rank.lean
@@ -26,7 +26,7 @@ open Matrix
 
 namespace Matrix
 
-open FiniteDimensional
+open Module
 
 variable {l m n o R : Type*} [Fintype n] [Fintype o]
 
@@ -168,7 +168,7 @@ variable [Field R]
 /-- The rank of a diagnonal matrix is the count of non-zero elements on its main diagonal -/
 theorem rank_diagonal [Fintype m] [DecidableEq m] [DecidableEq R] (w : m → R) :
     (diagonal w).rank = Fintype.card {i // (w i) ≠ 0} := by
-  rw [Matrix.rank, ← Matrix.toLin'_apply', FiniteDimensional.finrank, ← LinearMap.rank,
+  rw [Matrix.rank, ← Matrix.toLin'_apply', Module.finrank, ← LinearMap.rank,
     LinearMap.rank_diagonal, Cardinal.toNat_natCast]
 
 end Field
@@ -265,4 +265,23 @@ theorem rank_eq_finrank_span_row [Field R] [Finite m] (A : Matrix m n R) :
   cases nonempty_fintype m
   rw [← rank_transpose, rank_eq_finrank_span_cols, transpose_transpose]
 
+theorem _root_.LinearIndependent.rank_matrix [Field R] [Fintype m]
+    {M : Matrix m n R} (h : LinearIndependent R M) : M.rank = Fintype.card m := by
+  rw [M.rank_eq_finrank_span_row, linearIndependent_iff_card_eq_finrank_span.mp h, Set.finrank]
+
+lemma rank_add_rank_le_card_of_mul_eq_zero [Field R] [Finite l] [Fintype m]
+    {A : Matrix l m R} {B : Matrix m n R} (hAB : A * B = 0) :
+    A.rank + B.rank ≤ Fintype.card m := by
+  classical
+  let el : Basis l R (l → R) := Pi.basisFun R l
+  let em : Basis m R (m → R) := Pi.basisFun R m
+  let en : Basis n R (n → R) := Pi.basisFun R n
+  rw [Matrix.rank_eq_finrank_range_toLin A el em,
+      Matrix.rank_eq_finrank_range_toLin B em en,
+      ← Module.finrank_fintype_fun_eq_card R,
+      ← LinearMap.finrank_range_add_finrank_ker (Matrix.toLin em el A),
+      add_le_add_iff_left]
+  apply Submodule.finrank_mono
+  rw [LinearMap.range_le_ker_iff, ← Matrix.toLin_mul, hAB, map_zero]
+
 end Matrix
diff --git a/Mathlib/Data/Matrix/RowCol.lean b/Mathlib/Data/Matrix/RowCol.lean
index 2529d88fd19ee..6ce7feb0bcaa5 100644
--- a/Mathlib/Data/Matrix/RowCol.lean
+++ b/Mathlib/Data/Matrix/RowCol.lean
@@ -136,6 +136,16 @@ theorem row_mulVec [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α
   ext
   rfl
 
+theorem row_mulVec_eq_const [Fintype m] [NonUnitalNonAssocSemiring α]  (v w : m → α) :
+    Matrix.row ι v *ᵥ w = Function.const _ (v ⬝ᵥ w) := rfl
+
+theorem mulVec_col_eq_const [Fintype m] [NonUnitalNonAssocSemiring α] (v w : m → α) :
+    v ᵥ* Matrix.col ι w = Function.const _ (v ⬝ᵥ w) := rfl
+
+theorem row_mul_col [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) :
+    row ι v * col ι w = of fun _ _ => v ⬝ᵥ w :=
+  rfl
+
 @[simp]
 theorem row_mul_col_apply [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) (i j) :
     (row ι v * col ι w) i j = v ⬝ᵥ w :=
diff --git a/Mathlib/Data/Matroid/Basic.lean b/Mathlib/Data/Matroid/Basic.lean
index d2f447b1e4ab5..5f098b5a7e053 100644
--- a/Mathlib/Data/Matroid/Basic.lean
+++ b/Mathlib/Data/Matroid/Basic.lean
@@ -169,7 +169,7 @@ def Matroid.ExchangeProperty {α : Type _} (P : Set α → Prop) : Prop :=
   ∀ X Y, P X → P Y → ∀ a ∈ X \ Y, ∃ b ∈ Y \ X, P (insert b (X \ {a}))
 
 /-- A set `X` has the maximal subset property for a predicate `P` if every subset of `X` satisfying
-  `P` is contained in a maximal subset of `X` satisfying `P`.  -/
+  `P` is contained in a maximal subset of `X` satisfying `P`. -/
 def Matroid.ExistsMaximalSubsetProperty {α : Type _} (P : Set α → Prop) (X : Set α) : Prop :=
   ∀ I, P I → I ⊆ X → ∃ J, I ⊆ J ∧ Maximal (fun K ↦ P K ∧ K ⊆ X) J
 
@@ -713,7 +713,7 @@ def Basis (M : Matroid α) (I X : Set α) : Prop :=
   Maximal (fun A ↦ M.Indep A ∧ A ⊆ X) I ∧ X ⊆ M.E
 
 /-- A `Basis'` is a basis without the requirement that `X ⊆ M.E`. This is convenient for some
-  API building, especially when working with rank and closure.  -/
+  API building, especially when working with rank and closure. -/
 def Basis' (M : Matroid α) (I X : Set α) : Prop :=
   Maximal (fun A ↦ M.Indep A ∧ A ⊆ X) I
 
diff --git a/Mathlib/Data/Matroid/Map.lean b/Mathlib/Data/Matroid/Map.lean
index b231376efdada..bf22fe80c5b92 100644
--- a/Mathlib/Data/Matroid/Map.lean
+++ b/Mathlib/Data/Matroid/Map.lean
@@ -501,7 +501,7 @@ end map
 section mapSetEquiv
 
 /-- Map `M : Matroid α` to a `Matroid β` with ground set `E` using an equivalence `M.E ≃ E`.
-Defined using `Matroid.ofExistsMatroid` for better defeq.  -/
+Defined using `Matroid.ofExistsMatroid` for better defeq. -/
 def mapSetEquiv (M : Matroid α) {E : Set β} (e : M.E ≃ E) : Matroid β :=
   Matroid.ofExistsMatroid E (fun I ↦ (M.Indep ↑(e.symm '' (E ↓∩ I)) ∧ I ⊆ E))
   ⟨M.mapSetEmbedding (e.toEmbedding.trans <| Function.Embedding.subtype _), by
diff --git a/Mathlib/Data/Matroid/Restrict.lean b/Mathlib/Data/Matroid/Restrict.lean
index 46d88f70b7cf7..013d3c6c68740 100644
--- a/Mathlib/Data/Matroid/Restrict.lean
+++ b/Mathlib/Data/Matroid/Restrict.lean
@@ -36,7 +36,7 @@ It also proves some `Basis` analogues of `Base` lemmas that, while they could be
 ## Implementation Notes
 
 Since `R` and `M.E` are both terms in `Set α`, to define the restriction `M ↾ R`,
-we need to either insist that `R ⊆ M.E`, or to say what happens when `R` contains the junk
+we need to either insist that `R ⊆ M.E`, or to say what happens when `R` contains the junk
 outside `M.E`.
 
 It turns out that `R ⊆ M.E` is just an unnecessary hypothesis; if we say the restriction
@@ -115,7 +115,7 @@ section restrict
 
 /-- Change the ground set of a matroid to some `R : Set α`. The independent sets of the restriction
   are the independent subsets of the new ground set. Most commonly used when `R ⊆ M.E`,
-  but it is convenient not to require this. The elements of `R \ M.E` become 'loops'.  -/
+  but it is convenient not to require this. The elements of `R \ M.E` become 'loops'. -/
 def restrict (M : Matroid α) (R : Set α) : Matroid α := (M.restrictIndepMatroid R).matroid
 
 /-- `M ↾ R` means `M.restrict R`. -/
diff --git a/Mathlib/Data/Multiset/Basic.lean b/Mathlib/Data/Multiset/Basic.lean
index 84d90d90e8178..0c4fb024e449e 100644
--- a/Mathlib/Data/Multiset/Basic.lean
+++ b/Mathlib/Data/Multiset/Basic.lean
@@ -26,7 +26,7 @@ open List Subtype Nat Function
 variable {α : Type*} {β : Type v} {γ : Type*}
 
 /-- `Multiset α` is the quotient of `List α` by list permutation. The result
-  is a type of finite sets with duplicates allowed.  -/
+  is a type of finite sets with duplicates allowed. -/
 def Multiset.{u} (α : Type u) : Type u :=
   Quotient (List.isSetoid α)
 
@@ -112,7 +112,7 @@ theorem coe_eq_zero (l : List α) : (l : Multiset α) = 0 ↔ l = [] :=
   Iff.trans coe_eq_coe perm_nil
 
 theorem coe_eq_zero_iff_isEmpty (l : List α) : (l : Multiset α) = 0 ↔ l.isEmpty :=
-  Iff.trans (coe_eq_zero l) isEmpty_iff_eq_nil.symm
+  Iff.trans (coe_eq_zero l) isEmpty_iff.symm
 
 /-! ### `Multiset.cons` -/
 
@@ -148,7 +148,7 @@ theorem cons_inj_right (a : α) : ∀ {s t : Multiset α}, a ::ₘ s = a ::ₘ t
 @[elab_as_elim]
 protected theorem induction {p : Multiset α → Prop} (empty : p 0)
     (cons : ∀ (a : α) (s : Multiset α), p s → p (a ::ₘ s)) : ∀ s, p s := by
-  rintro ⟨l⟩; induction' l with _ _ ih <;> [exact empty; exact cons _ _ ih]
+  rintro ⟨l⟩; induction l with | nil => exact empty | cons _ _ ih => exact cons _ _ ih
 
 @[elab_as_elim]
 protected theorem induction_on {p : Multiset α → Prop} (s : Multiset α) (empty : p 0)
@@ -203,7 +203,7 @@ end Rec
 section Mem
 
 /-- `a ∈ s` means that `a` has nonzero multiplicity in `s`. -/
-def Mem (a : α) (s : Multiset α) : Prop :=
+def Mem (s : Multiset α) (a : α) : Prop :=
   Quot.liftOn s (fun l => a ∈ l) fun l₁ l₂ (e : l₁ ~ l₂) => propext <| e.mem_iff
 
 instance : Membership α (Multiset α) :=
@@ -306,7 +306,7 @@ theorem coe_singleton (a : α) : ([a] : Multiset α) = {a} :=
 
 @[simp]
 theorem mem_singleton {a b : α} : b ∈ ({a} : Multiset α) ↔ b = a := by
-  simp only [← cons_zero, mem_cons, iff_self_iff, or_false_iff, not_mem_zero]
+  simp only [← cons_zero, mem_cons, iff_self, or_false, not_mem_zero]
 
 theorem mem_singleton_self (a : α) : a ∈ ({a} : Multiset α) := by
   rw [← cons_zero]
@@ -528,6 +528,11 @@ theorem le_cons_of_not_mem (m : a ∉ s) : s ≤ a ::ₘ t ↔ s ≤ t := by
     perm_middle.subperm_left.2
       ((subperm_cons _).2 <| ((sublist_or_mem_of_sublist s).resolve_right m₁).subperm)
 
+theorem cons_le_of_not_mem (hs : a ∉ s) : a ::ₘ s ≤ t ↔ a ∈ t ∧ s ≤ t := by
+  apply Iff.intro (fun h ↦ ⟨subset_of_le h (mem_cons_self a s), le_trans (le_cons_self s a) h⟩)
+  rintro ⟨h₁, h₂⟩; rcases exists_cons_of_mem h₁ with ⟨_, rfl⟩
+  exact cons_le_cons _ ((le_cons_of_not_mem hs).mp h₂)
+
 @[simp]
 theorem singleton_ne_zero (a : α) : ({a} : Multiset α) ≠ 0 :=
   ne_of_gt (lt_cons_self _ _)
@@ -601,6 +606,10 @@ theorem le_add_right (s t : Multiset α) : s ≤ s + t := by simpa using add_le_
 
 theorem le_add_left (s t : Multiset α) : s ≤ t + s := by simpa using add_le_add_right (zero_le t) s
 
+lemma subset_add_left {s t : Multiset α} : s ⊆ s + t := subset_of_le <| le_add_right s t
+
+lemma subset_add_right {s t : Multiset α} : s ⊆ t + s := subset_of_le <| le_add_left s t
+
 theorem le_iff_exists_add {s t : Multiset α} : s ≤ t ↔ ∃ u, t = s + u :=
   ⟨fun h =>
     leInductionOn h fun s =>
@@ -961,6 +970,8 @@ theorem mem_of_mem_erase {a b : α} {s : Multiset α} : a ∈ s.erase b → a 
 theorem erase_comm (s : Multiset α) (a b : α) : (s.erase a).erase b = (s.erase b).erase a :=
   Quot.inductionOn s fun l => congr_arg _ <| l.erase_comm a b
 
+instance : RightCommutative erase (α := α) := ⟨erase_comm⟩
+
 theorem erase_le_erase {s t : Multiset α} (a : α) (h : s ≤ t) : s.erase a ≤ t.erase a :=
   leInductionOn h fun h => (h.erase _).subperm
 
@@ -1189,92 +1200,104 @@ theorem map_surjective_of_surjective {f : α → β} (hf : Function.Surjective f
 /-! ### `Multiset.fold` -/
 
 
+section foldl
+
 /-- `foldl f H b s` is the lift of the list operation `foldl f b l`,
   which folds `f` over the multiset. It is well defined when `f` is right-commutative,
   that is, `f (f b a₁) a₂ = f (f b a₂) a₁`. -/
-def foldl (f : β → α → β) (H : RightCommutative f) (b : β) (s : Multiset α) : β :=
-  Quot.liftOn s (fun l => List.foldl f b l) fun _l₁ _l₂ p => p.foldl_eq H b
+def foldl (f : β → α → β) [RightCommutative f] (b : β) (s : Multiset α) : β :=
+  Quot.liftOn s (fun l => List.foldl f b l) fun _l₁ _l₂ p => p.foldl_eq b
+
+variable (f : β → α → β) [RightCommutative f]
 
 @[simp]
-theorem foldl_zero (f : β → α → β) (H b) : foldl f H b 0 = b :=
+theorem foldl_zero (b) : foldl f b 0 = b :=
   rfl
 
 @[simp]
-theorem foldl_cons (f : β → α → β) (H b a s) : foldl f H b (a ::ₘ s) = foldl f H (f b a) s :=
+theorem foldl_cons (b a s) : foldl f b (a ::ₘ s) = foldl f (f b a) s :=
   Quot.inductionOn s fun _l => rfl
 
 @[simp]
-theorem foldl_add (f : β → α → β) (H b s t) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=
+theorem foldl_add (b s t) : foldl f b (s + t) = foldl f (foldl f b s) t :=
   Quotient.inductionOn₂ s t fun _l₁ _l₂ => foldl_append _ _ _ _
 
+end foldl
+
+section foldr
+
 /-- `foldr f H b s` is the lift of the list operation `foldr f b l`,
   which folds `f` over the multiset. It is well defined when `f` is left-commutative,
   that is, `f a₁ (f a₂ b) = f a₂ (f a₁ b)`. -/
-def foldr (f : α → β → β) (H : LeftCommutative f) (b : β) (s : Multiset α) : β :=
-  Quot.liftOn s (fun l => List.foldr f b l) fun _l₁ _l₂ p => p.foldr_eq H b
+def foldr (f : α → β → β) [LeftCommutative f] (b : β) (s : Multiset α) : β :=
+  Quot.liftOn s (fun l => List.foldr f b l) fun _l₁ _l₂ p => p.foldr_eq b
+
+variable (f : α → β → β) [LeftCommutative f]
 
 @[simp]
-theorem foldr_zero (f : α → β → β) (H b) : foldr f H b 0 = b :=
+theorem foldr_zero (b) : foldr f b 0 = b :=
   rfl
 
 @[simp]
-theorem foldr_cons (f : α → β → β) (H b a s) : foldr f H b (a ::ₘ s) = f a (foldr f H b s) :=
+theorem foldr_cons (b a s) : foldr f b (a ::ₘ s) = f a (foldr f b s) :=
   Quot.inductionOn s fun _l => rfl
 
 @[simp]
-theorem foldr_singleton (f : α → β → β) (H b a) : foldr f H b ({a} : Multiset α) = f a b :=
+theorem foldr_singleton (b a) : foldr f b ({a} : Multiset α) = f a b :=
   rfl
 
 @[simp]
-theorem foldr_add (f : α → β → β) (H b s t) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=
+theorem foldr_add (b s t) : foldr f b (s + t) = foldr f (foldr f b t) s :=
   Quotient.inductionOn₂ s t fun _l₁ _l₂ => foldr_append _ _ _ _
 
+end foldr
+
 @[simp]
-theorem coe_foldr (f : α → β → β) (H : LeftCommutative f) (b : β) (l : List α) :
-    foldr f H b l = l.foldr f b :=
+theorem coe_foldr (f : α → β → β) [LeftCommutative f] (b : β) (l : List α) :
+    foldr f b l = l.foldr f b :=
   rfl
 
 @[simp]
-theorem coe_foldl (f : β → α → β) (H : RightCommutative f) (b : β) (l : List α) :
-    foldl f H b l = l.foldl f b :=
+theorem coe_foldl (f : β → α → β) [RightCommutative f] (b : β) (l : List α) :
+    foldl f b l = l.foldl f b :=
   rfl
 
-theorem coe_foldr_swap (f : α → β → β) (H : LeftCommutative f) (b : β) (l : List α) :
-    foldr f H b l = l.foldl (fun x y => f y x) b :=
-  (congr_arg (foldr f H b) (coe_reverse l)).symm.trans <| foldr_reverse _ _ _
+theorem coe_foldr_swap (f : α → β → β) [LeftCommutative f] (b : β) (l : List α) :
+    foldr f b l = l.foldl (fun x y => f y x) b :=
+  (congr_arg (foldr f b) (coe_reverse l)).symm.trans <| foldr_reverse _ _ _
 
-theorem foldr_swap (f : α → β → β) (H : LeftCommutative f) (b : β) (s : Multiset α) :
-    foldr f H b s = foldl (fun x y => f y x) (fun _x _y _z => (H _ _ _).symm) b s :=
-  Quot.inductionOn s fun _l => coe_foldr_swap _ _ _ _
+theorem foldr_swap (f : α → β → β) [LeftCommutative f] (b : β) (s : Multiset α) :
+    foldr f b s = foldl (fun x y => f y x) b s :=
+  Quot.inductionOn s fun _l => coe_foldr_swap _ _ _
 
-theorem foldl_swap (f : β → α → β) (H : RightCommutative f) (b : β) (s : Multiset α) :
-    foldl f H b s = foldr (fun x y => f y x) (fun _x _y _z => (H _ _ _).symm) b s :=
-  (foldr_swap _ _ _ _).symm
+theorem foldl_swap (f : β → α → β) [RightCommutative f] (b : β) (s : Multiset α) :
+    foldl f b s = foldr (fun x y => f y x) b s :=
+  (foldr_swap _ _ _).symm
 
-theorem foldr_induction' (f : α → β → β) (H : LeftCommutative f) (x : β) (q : α → Prop)
+theorem foldr_induction' (f : α → β → β) [LeftCommutative f] (x : β) (q : α → Prop)
     (p : β → Prop) (s : Multiset α) (hpqf : ∀ a b, q a → p b → p (f a b)) (px : p x)
-    (q_s : ∀ a ∈ s, q a) : p (foldr f H x s) := by
+    (q_s : ∀ a ∈ s, q a) : p (foldr f x s) := by
   induction s using Multiset.induction with
   | empty => simpa
   | cons a s ihs =>
     simp only [forall_mem_cons, foldr_cons] at q_s ⊢
     exact hpqf _ _ q_s.1 (ihs q_s.2)
 
-theorem foldr_induction (f : α → α → α) (H : LeftCommutative f) (x : α) (p : α → Prop)
+theorem foldr_induction (f : α → α → α) [LeftCommutative f] (x : α) (p : α → Prop)
     (s : Multiset α) (p_f : ∀ a b, p a → p b → p (f a b)) (px : p x) (p_s : ∀ a ∈ s, p a) :
-    p (foldr f H x s) :=
-  foldr_induction' f H x p p s p_f px p_s
+    p (foldr f x s) :=
+  foldr_induction' f x p p s p_f px p_s
 
-theorem foldl_induction' (f : β → α → β) (H : RightCommutative f) (x : β) (q : α → Prop)
+theorem foldl_induction' (f : β → α → β) [RightCommutative f] (x : β) (q : α → Prop)
     (p : β → Prop) (s : Multiset α) (hpqf : ∀ a b, q a → p b → p (f b a)) (px : p x)
-    (q_s : ∀ a ∈ s, q a) : p (foldl f H x s) := by
+    (q_s : ∀ a ∈ s, q a) : p (foldl f x s) := by
   rw [foldl_swap]
-  exact foldr_induction' (fun x y => f y x) (fun x y z => (H _ _ _).symm) x q p s hpqf px q_s
+  exact foldr_induction' (fun x y => f y x) x q p s hpqf px q_s
 
-theorem foldl_induction (f : α → α → α) (H : RightCommutative f) (x : α) (p : α → Prop)
+theorem foldl_induction (f : α → α → α) [RightCommutative f] (x : α) (p : α → Prop)
     (s : Multiset α) (p_f : ∀ a b, p a → p b → p (f b a)) (px : p x) (p_s : ∀ a ∈ s, p a) :
-    p (foldl f H x s) :=
-  foldl_induction' f H x p p s p_f px p_s
+    p (foldl f x s) :=
+  foldl_induction' f x p p s p_f px p_s
 
 /-! ### Map for partial functions -/
 
@@ -1327,8 +1350,7 @@ theorem pmap_eq_map (p : α → Prop) (f : α → β) (s : Multiset α) :
 
 theorem pmap_congr {p q : α → Prop} {f : ∀ a, p a → β} {g : ∀ a, q a → β} (s : Multiset α) :
     ∀ {H₁ H₂}, (∀ a ∈ s, ∀ (h₁ h₂), f a h₁ = g a h₂) → pmap f s H₁ = pmap g s H₂ :=
-  @(Quot.inductionOn s (fun l _H₁ _H₂ h => congr_arg _ <| List.pmap_congr l h))
-
+  @(Quot.inductionOn s (fun l _H₁ _H₂ h => congr_arg _ <| List.pmap_congr_left l h))
 
 theorem map_pmap {p : α → Prop} (g : β → γ) (f : ∀ a, p a → β) (s) :
     ∀ H, map g (pmap f s H) = pmap (fun a h => g (f a h)) s H :=
@@ -1373,7 +1395,7 @@ theorem attach_cons (a : α) (m : Multiset α) :
   Quotient.inductionOn m fun l =>
     congr_arg _ <|
       congr_arg (List.cons _) <| by
-        rw [List.map_pmap]; exact List.pmap_congr _ fun _ _ _ _ => Subtype.eq rfl
+        rw [List.map_pmap]; exact List.pmap_congr_left _ fun _ _ _ _ => Subtype.eq rfl
 
 section DecidablePiExists
 
@@ -1464,9 +1486,9 @@ instance : ExistsAddOfLE (Multiset α) where
 theorem cons_sub_of_le (a : α) {s t : Multiset α} (h : t ≤ s) : a ::ₘ s - t = a ::ₘ (s - t) := by
   rw [← singleton_add, ← singleton_add, add_tsub_assoc_of_le h]
 
-theorem sub_eq_fold_erase (s t : Multiset α) : s - t = foldl erase erase_comm s t :=
+theorem sub_eq_fold_erase (s t : Multiset α) : s - t = foldl erase s t :=
   Quotient.inductionOn₂ s t fun l₁ l₂ => by
-    show ofList (l₁.diff l₂) = foldl erase erase_comm l₁ l₂
+    show ofList (l₁.diff l₂) = foldl erase l₁ l₂
     rw [diff_eq_foldl l₁ l₂]
     symm
     exact foldl_hom _ _ _ _ _ fun x y => rfl
@@ -1739,13 +1761,12 @@ theorem mem_filter_of_mem {a : α} {l} (m : a ∈ l) (h : p a) : a ∈ filter p
 
 theorem filter_eq_self {s} : filter p s = s ↔ ∀ a ∈ s, p a :=
   Quot.inductionOn s fun _l =>
-    Iff.trans ⟨fun h => (filter_sublist _).eq_of_length (@congr_arg _ _ _ _ card h),
+    Iff.trans ⟨fun h => (filter_sublist _).eq_of_length (congr_arg card h),
       congr_arg ofList⟩ <| by simp
 
 theorem filter_eq_nil {s} : filter p s = 0 ↔ ∀ a ∈ s, ¬p a :=
   Quot.inductionOn s fun _l =>
-    Iff.trans ⟨fun h => eq_nil_of_length_eq_zero (@congr_arg _ _ _ _ card h), congr_arg ofList⟩ <|
-      by simpa using List.filter_eq_nil (p := (p ·))
+    Iff.trans ⟨fun h => eq_nil_of_length_eq_zero (congr_arg card h), congr_arg ofList⟩ (by simp)
 
 theorem le_filter {s t} : s ≤ filter p t ↔ s ≤ t ∧ ∀ a ∈ s, p a :=
   ⟨fun h => ⟨le_trans h (filter_le _ _), fun _a m => of_mem_filter (mem_of_le h m)⟩, fun ⟨h, al⟩ =>
@@ -1834,7 +1855,7 @@ theorem filter_map (f : β → α) (s : Multiset β) : filter p (map f s) = map
 lemma map_filter' {f : α → β} (hf : Injective f) (s : Multiset α)
     [DecidablePred fun b => ∃ a, p a ∧ f a = b] :
     (s.filter p).map f = (s.map f).filter fun b => ∃ a, p a ∧ f a = b := by
-  simp [(· ∘ ·), filter_map, hf.eq_iff]
+  simp [comp_def, filter_map, hf.eq_iff]
 
 lemma card_filter_le_iff (s : Multiset α) (P : α → Prop) [DecidablePred P] (n : ℕ) :
     card (s.filter P) ≤ n ↔ ∀ s' ≤ s, n < card s' → ∃ a ∈ s', ¬ P a := by
@@ -2000,11 +2021,11 @@ theorem countP_eq_countP_filter_add (s) (p q : α → Prop) [DecidablePred p] [D
 
 @[simp]
 theorem countP_True {s : Multiset α} : countP (fun _ => True) s = card s :=
-  Quot.inductionOn s fun _l => List.countP_true
+  Quot.inductionOn s fun _l => congrFun List.countP_true _
 
 @[simp]
 theorem countP_False {s : Multiset α} : countP (fun _ => False) s = 0 :=
-  Quot.inductionOn s fun _l => List.countP_false
+  Quot.inductionOn s fun _l => congrFun List.countP_false _
 
 theorem countP_map (f : α → β) (s : Multiset α) (p : β → Prop) [DecidablePred p] :
     countP p (map f s) = card (s.filter fun a => p (f a)) := by
@@ -2033,7 +2054,7 @@ lemma filter_attach (s : Multiset α) (p : α → Prop) [DecidablePred p] :
 variable {p}
 
 theorem countP_pos {s} : 0 < countP p s ↔ ∃ a ∈ s, p a :=
-  Quot.inductionOn s fun _l => by simpa using List.countP_pos (p ·)
+  Quot.inductionOn s fun _l => by simp
 
 theorem countP_eq_zero {s} : countP p s = 0 ↔ ∀ a ∈ s, ¬p a :=
   Quot.inductionOn s fun _l => by simp [List.countP_eq_zero]
@@ -2146,8 +2167,8 @@ theorem count_replicate_self (a : α) (n : ℕ) : count a (replicate n a) = n :=
 
 theorem count_replicate (a b : α) (n : ℕ) : count a (replicate n b) = if b = a then n else 0 := by
   convert List.count_replicate a b n
-  rw [← coe_count, coe_replicate]
-  simp
+  · rw [← coe_count, coe_replicate]
+  · simp
 
 @[simp]
 theorem count_erase_self (a : α) (s : Multiset α) : count a (erase s a) = count a s - 1 :=
@@ -2333,6 +2354,24 @@ theorem map_count_True_eq_filter_card (s : Multiset α) (p : α → Prop) [Decid
   simp only [count_eq_card_filter_eq, filter_map, card_map, Function.id_comp,
     eq_true_eq_id, Function.comp_apply]
 
+@[simp] theorem sub_singleton [DecidableEq α] (a : α) (s : Multiset α) : s - {a} = s.erase a := by
+  ext
+  simp only [count_sub, count_singleton]
+  split <;> simp_all
+
+theorem mem_sub [DecidableEq α] {a : α} {s t : Multiset α} :
+    a ∈ s - t ↔ t.count a < s.count a := by
+  rw [← count_pos, count_sub, Nat.sub_pos_iff_lt]
+
+theorem inter_add_sub_of_add_eq_add [DecidableEq α] {M N P Q : Multiset α} (h : M + N = P + Q) :
+    (N ∩ Q) + (P - M) = N := by
+  ext x
+  rw [Multiset.count_add, Multiset.count_inter, Multiset.count_sub]
+  have h0 : M.count x + N.count x = P.count x + Q.count x := by
+    rw [Multiset.ext] at h
+    simp_all only [Multiset.mem_add, Multiset.count_add]
+  omega
+
 /-! ### Lift a relation to `Multiset`s -/
 
 
@@ -2733,7 +2772,6 @@ theorem coe_subsingletonEquiv [Subsingleton α] :
     (subsingletonEquiv α : List α → Multiset α) = ofList :=
   rfl
 
-@[deprecated (since := "2023-12-27")] alias card_le_of_le := card_le_card
-@[deprecated (since := "2023-12-27")] alias card_lt_of_lt := card_lt_card
-
 end Multiset
+
+set_option linter.style.longFile 2900
diff --git a/Mathlib/Data/Multiset/Fintype.lean b/Mathlib/Data/Multiset/Fintype.lean
index eef7f8002adbf..ee4962cb23b82 100644
--- a/Mathlib/Data/Multiset/Fintype.lean
+++ b/Mathlib/Data/Multiset/Fintype.lean
@@ -188,8 +188,7 @@ theorem Multiset.map_univ_coeEmbedding (m : Multiset α) :
   ext ⟨x, i⟩
   simp only [Fin.exists_iff, Finset.mem_map, Finset.mem_univ, Multiset.coeEmbedding_apply,
     Prod.mk.inj_iff, exists_true_left, Multiset.exists_coe, Multiset.coe_mk, Fin.val_mk,
-    exists_prop, exists_eq_right_right, exists_eq_right, Multiset.mem_toEnumFinset, iff_self_iff,
-    true_and_iff]
+    exists_prop, exists_eq_right_right, exists_eq_right, Multiset.mem_toEnumFinset, true_and]
 
 @[simp]
 theorem Multiset.map_univ_coe (m : Multiset α) :
diff --git a/Mathlib/Data/Multiset/Fold.lean b/Mathlib/Data/Multiset/Fold.lean
index 6cced7eebd0e1..5ec3691c966d1 100644
--- a/Mathlib/Data/Multiset/Fold.lean
+++ b/Mathlib/Data/Multiset/Fold.lean
@@ -26,10 +26,10 @@ local notation a " * " b => op a b
 /-- `fold op b s` folds a commutative associative operation `op` over
   the multiset `s`. -/
 def fold : α → Multiset α → α :=
-  foldr op (left_comm _ hc.comm ha.assoc)
+  foldr op
 
 theorem fold_eq_foldr (b : α) (s : Multiset α) :
-    fold op b s = foldr op (left_comm _ hc.comm ha.assoc) b s :=
+    fold op b s = foldr op b s :=
   rfl
 
 @[simp]
@@ -37,10 +37,10 @@ theorem coe_fold_r (b : α) (l : List α) : fold op b l = l.foldr op b :=
   rfl
 
 theorem coe_fold_l (b : α) (l : List α) : fold op b l = l.foldl op b :=
-  (coe_foldr_swap op _ b l).trans <| by simp [hc.comm]
+  (coe_foldr_swap op b l).trans <| by simp [hc.comm]
 
 theorem fold_eq_foldl (b : α) (s : Multiset α) :
-    fold op b s = foldl op (right_comm _ hc.comm ha.assoc) b s :=
+    fold op b s = foldl op b s :=
   Quot.inductionOn s fun _ => coe_fold_l _ _ _
 
 @[simp]
@@ -49,7 +49,7 @@ theorem fold_zero (b : α) : (0 : Multiset α).fold op b = b :=
 
 @[simp]
 theorem fold_cons_left : ∀ (b a : α) (s : Multiset α), (a ::ₘ s).fold op b = a * s.fold op b :=
-  foldr_cons _ _
+  foldr_cons _
 
 theorem fold_cons_right (b a : α) (s : Multiset α) : (a ::ₘ s).fold op b = s.fold op b * a := by
   simp [hc.comm]
@@ -74,7 +74,7 @@ theorem fold_bind {ι : Type*} (s : Multiset ι) (t : ι → Multiset α) (b : 
   · rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih]
 
 theorem fold_singleton (b a : α) : ({a} : Multiset α).fold op b = a * b :=
-  foldr_singleton _ _ _ _
+  foldr_singleton _ _ _
 
 theorem fold_distrib {f g : β → α} (u₁ u₂ : α) (s : Multiset β) :
     (s.map fun x => f x * g x).fold op (u₁ * u₂) = (s.map f).fold op u₁ * (s.map g).fold op u₂ :=
diff --git a/Mathlib/Data/Multiset/Functor.lean b/Mathlib/Data/Multiset/Functor.lean
index 05b29adb37aa7..81575f5f445c4 100644
--- a/Mathlib/Data/Multiset/Functor.lean
+++ b/Mathlib/Data/Multiset/Functor.lean
@@ -55,7 +55,7 @@ def traverse : Multiset α' → F (Multiset β') := by
       congr
       funext a b l
       simpa [flip] using Perm.swap a b l
-    simp [(· ∘ ·), this, functor_norm, Coe.coe]
+    simp [Function.comp_def, this, functor_norm, Coe.coe]
   | trans => simp [*]
 
 instance : Monad Multiset :=
@@ -99,7 +99,6 @@ theorem comp_traverse {G H : Type _ → Type _} [Applicative G] [Applicative H]
   intro
   simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,
     functor_norm]
-  simp only [Function.comp, lift_coe]
 
 theorem map_traverse {G : Type* → Type _} [Applicative G] [CommApplicative G] {α β γ : Type _}
     (g : α → G β) (h : β → γ) (x : Multiset α) :
@@ -107,7 +106,8 @@ theorem map_traverse {G : Type* → Type _} [Applicative G] [CommApplicative G]
   refine Quotient.inductionOn x ?_
   intro
   simp only [traverse, quot_mk_to_coe, lift_coe, Function.comp_apply, Functor.map_map, map_comp_coe]
-  rw [LawfulFunctor.comp_map, Traversable.map_traverse']
+  rw [Traversable.map_traverse']
+  simp only [fmap_def, Function.comp_apply, Functor.map_map, List.map_eq_map]
   rfl
 
 theorem traverse_map {G : Type* → Type _} [Applicative G] [CommApplicative G] {α β γ : Type _}
diff --git a/Mathlib/Data/Multiset/Lattice.lean b/Mathlib/Data/Multiset/Lattice.lean
index ebc72e3ce307b..194cae4b5a776 100644
--- a/Mathlib/Data/Multiset/Lattice.lean
+++ b/Mathlib/Data/Multiset/Lattice.lean
@@ -79,9 +79,9 @@ theorem nodup_sup_iff {α : Type*} [DecidableEq α] {m : Multiset (Multiset α)}
     m.sup.Nodup ↔ ∀ a : Multiset α, a ∈ m → a.Nodup := by
   -- Porting note: this was originally `apply m.induction_on`, which failed due to
   -- `failed to elaborate eliminator, expected type is not available`
-  induction' m using Multiset.induction_on with _ _ h
-  · simp
-  · simp [h]
+  induction m using Multiset.induction_on with
+  | empty => simp
+  | cons _ _ h => simp [h]
 
 end Sup
 
diff --git a/Mathlib/Data/Multiset/Powerset.lean b/Mathlib/Data/Multiset/Powerset.lean
index e344eddc1b02f..924f768a2ffbb 100644
--- a/Mathlib/Data/Multiset/Powerset.lean
+++ b/Mathlib/Data/Multiset/Powerset.lean
@@ -50,16 +50,18 @@ theorem powersetAux'_cons (a : α) (l : List α) :
   simp [powersetAux']
 
 theorem powerset_aux'_perm {l₁ l₂ : List α} (p : l₁ ~ l₂) : powersetAux' l₁ ~ powersetAux' l₂ := by
-  induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ _ _ IH₁ IH₂
-  · simp
-  · simp only [powersetAux'_cons]
+  induction p with
+  | nil => simp
+  | cons _ _ IH =>
+    simp only [powersetAux'_cons]
     exact IH.append (IH.map _)
-  · simp only [powersetAux'_cons, map_append, List.map_map, append_assoc]
+  | swap a b =>
+    simp only [powersetAux'_cons, map_append, List.map_map, append_assoc]
     apply Perm.append_left
     rw [← append_assoc, ← append_assoc,
       (by funext s; simp [cons_swap] : cons b ∘ cons a = cons a ∘ cons b)]
     exact perm_append_comm.append_right _
-  · exact IH₁.trans IH₂
+  | trans _ _ IH₁ IH₂ => exact IH₁.trans IH₂
 
 theorem powersetAux_perm {l₁ l₂ : List α} (p : l₁ ~ l₂) : powersetAux l₁ ~ powersetAux l₂ :=
   powersetAux_perm_powersetAux'.trans <|
@@ -179,11 +181,13 @@ theorem powersetCardAux_perm {n} {l₁ l₂ : List α} (p : l₁ ~ l₂) :
     powersetCardAux n l₁ ~ powersetCardAux n l₂ := by
   induction' n with n IHn generalizing l₁ l₂
   · simp
-  induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ _ _ IH₁ IH₂
-  · rfl
-  · simp only [powersetCardAux_cons]
+  induction p with
+  | nil => rfl
+  | cons _ p IH =>
+    simp only [powersetCardAux_cons]
     exact IH.append ((IHn p).map _)
-  · simp only [powersetCardAux_cons, append_assoc]
+  | swap a b =>
+    simp only [powersetCardAux_cons, append_assoc]
     apply Perm.append_left
     cases n
     · simp [Perm.swap]
@@ -191,7 +195,7 @@ theorem powersetCardAux_perm {n} {l₁ l₂ : List α} (p : l₁ ~ l₂) :
     rw [← append_assoc, ← append_assoc,
       (by funext s; simp [cons_swap] : cons b ∘ cons a = cons a ∘ cons b)]
     exact perm_append_comm.append_right _
-  · exact IH₁.trans IH₂
+  | trans _ _ IH₁ IH₂ => exact IH₁.trans IH₂
 
 /-- `powersetCard n s` is the multiset of all submultisets of `s` of length `n`. -/
 def powersetCard (n : ℕ) (s : Multiset α) : Multiset (Multiset α) :=
@@ -219,7 +223,7 @@ theorem powersetCard_cons (n : ℕ) (a : α) (s) :
 
 theorem powersetCard_one (s : Multiset α) : powersetCard 1 s = s.map singleton :=
   Quotient.inductionOn s fun l ↦ by
-    simp [powersetCard_coe, sublistsLen_one, map_reverse, Function.comp]
+    simp [powersetCard_coe, sublistsLen_one, map_reverse, Function.comp_def]
 
 @[simp]
 theorem mem_powersetCard {n : ℕ} {s t : Multiset α} : s ∈ powersetCard n t ↔ s ≤ t ∧ card s = n :=
diff --git a/Mathlib/Data/Multiset/Sections.lean b/Mathlib/Data/Multiset/Sections.lean
index 8a09ff3a33675..4cc916de7c550 100644
--- a/Mathlib/Data/Multiset/Sections.lean
+++ b/Mathlib/Data/Multiset/Sections.lean
@@ -42,7 +42,7 @@ theorem coe_sections :
   | a :: l => by
     simp only [List.map_cons, List.sections]
     rw [← cons_coe, sections_cons, bind_map_comm, coe_sections l]
-    simp [List.sections, (· ∘ ·), List.bind]
+    simp [List.sections, Function.comp_def, List.bind]
 
 @[simp]
 theorem sections_add (s t : Multiset (Multiset α)) :
diff --git a/Mathlib/Data/Multiset/Sort.lean b/Mathlib/Data/Multiset/Sort.lean
index b40162019f9c4..997af93b548c0 100644
--- a/Mathlib/Data/Multiset/Sort.lean
+++ b/Mathlib/Data/Multiset/Sort.lean
@@ -25,42 +25,47 @@ variable (r' : β → β → Prop) [DecidableRel r'] [IsTrans β r'] [IsAntisymm
 /-- `sort s` constructs a sorted list from the multiset `s`.
   (Uses merge sort algorithm.) -/
 def sort (s : Multiset α) : List α :=
-  Quot.liftOn s (mergeSort r) fun _ _ h =>
-    eq_of_perm_of_sorted ((perm_mergeSort _ _).trans <| h.trans (perm_mergeSort _ _).symm)
-      (sorted_mergeSort r _) (sorted_mergeSort r _)
+  Quot.liftOn s (mergeSort' r) fun _ _ h =>
+    eq_of_perm_of_sorted ((perm_mergeSort' _ _).trans <| h.trans (perm_mergeSort' _ _).symm)
+      (sorted_mergeSort' r _) (sorted_mergeSort' r _)
 
 @[simp]
-theorem coe_sort (l : List α) : sort r l = mergeSort r l :=
+theorem coe_sort (l : List α) : sort r l = mergeSort' r l :=
   rfl
 
 @[simp]
 theorem sort_sorted (s : Multiset α) : Sorted r (sort r s) :=
-  Quot.inductionOn s fun _l => sorted_mergeSort r _
+  Quot.inductionOn s fun _l => sorted_mergeSort' r _
 
 @[simp]
 theorem sort_eq (s : Multiset α) : ↑(sort r s) = s :=
-  Quot.inductionOn s fun _ => Quot.sound <| perm_mergeSort _ _
+  Quot.inductionOn s fun _ => Quot.sound <| perm_mergeSort' _ _
 
 @[simp]
 theorem mem_sort {s : Multiset α} {a : α} : a ∈ sort r s ↔ a ∈ s := by rw [← mem_coe, sort_eq]
 
 @[simp]
 theorem length_sort {s : Multiset α} : (sort r s).length = card s :=
-  Quot.inductionOn s <| length_mergeSort _
+  Quot.inductionOn s <| length_mergeSort' _
 
 @[simp]
 theorem sort_zero : sort r 0 = [] :=
-  List.mergeSort_nil r
+  List.mergeSort'_nil r
 
 @[simp]
 theorem sort_singleton (a : α) : sort r {a} = [a] :=
-  List.mergeSort_singleton r a
+  List.mergeSort'_singleton r a
 
 theorem map_sort (f : α → β) (s : Multiset α)
     (hs : ∀ a ∈ s, ∀ b ∈ s, r a b ↔ r' (f a) (f b)) :
     (s.sort r).map f = (s.map f).sort r' := by
   revert s
-  exact Quot.ind fun _ => List.map_mergeSort _ _ _ _
+  exact Quot.ind fun _ => List.map_mergeSort' _ _ _ _
+
+theorem sort_cons (a : α) (s : Multiset α) :
+    (∀ b ∈ s, r a b) → sort r (a ::ₘ s) = a :: sort r s := by
+  refine Quot.inductionOn s fun l => ?_
+  simpa [mergeSort'_eq_insertionSort] using insertionSort_cons r
 
 end sort
 
diff --git a/Mathlib/Data/NNReal/Basic.lean b/Mathlib/Data/NNReal/Basic.lean
index 8148b3562ef7b..0b975180adee6 100644
--- a/Mathlib/Data/NNReal/Basic.lean
+++ b/Mathlib/Data/NNReal/Basic.lean
@@ -3,16 +3,13 @@ Copyright (c) 2018 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Algebra.Algebra.Defs
+import Mathlib.Algebra.BigOperators.Expect
 import Mathlib.Algebra.Order.BigOperators.Ring.Finset
 import Mathlib.Algebra.Order.Field.Canonical.Basic
-import Mathlib.Algebra.Order.Nonneg.Field
 import Mathlib.Algebra.Order.Nonneg.Floor
+import Mathlib.Algebra.Ring.Regular
 import Mathlib.Data.Real.Pointwise
 import Mathlib.Order.ConditionallyCompleteLattice.Group
-import Mathlib.Tactic.Bound.Attribute
-import Mathlib.Tactic.GCongr.Core
-import Mathlib.Algebra.Ring.Regular
 
 /-!
 # Nonnegative real numbers
@@ -55,6 +52,7 @@ This file defines `ℝ≥0` as a localized notation for `NNReal`.
 assert_not_exists Star
 
 open Function
+open scoped BigOperators
 
 -- to ensure these instances are computable
 /-- Nonnegative real numbers. -/
@@ -155,6 +153,9 @@ protected theorem coe_injective : Injective ((↑) : ℝ≥0 → ℝ) := Subtype
 
 @[simp, norm_cast] lemma coe_one : ((1 : ℝ≥0) : ℝ) = 1 := rfl
 
+@[simp] lemma mk_zero : (⟨0, le_rfl⟩ : ℝ≥0) = 0 := rfl
+@[simp] lemma mk_one : (⟨1, zero_le_one⟩ : ℝ≥0) = 1 := rfl
+
 @[simp, norm_cast]
 protected theorem coe_add (r₁ r₂ : ℝ≥0) : ((r₁ + r₂ : ℝ≥0) : ℝ) = r₁ + r₂ :=
   rfl
@@ -275,27 +276,32 @@ theorem coe_multiset_sum (s : Multiset ℝ≥0) : ((s.sum : ℝ≥0) : ℝ) = (s
 theorem coe_multiset_prod (s : Multiset ℝ≥0) : ((s.prod : ℝ≥0) : ℝ) = (s.map (↑)).prod :=
   map_multiset_prod toRealHom s
 
+variable {ι : Type*} {s : Finset ι} {f : ι → ℝ}
+
 @[simp, norm_cast]
-theorem coe_sum {α} {s : Finset α} {f : α → ℝ≥0} : ↑(∑ a ∈ s, f a) = ∑ a ∈ s, (f a : ℝ) :=
+theorem coe_sum (s : Finset ι) (f : ι → ℝ≥0) : ∑ i ∈ s, f i = ∑ i ∈ s, (f i : ℝ) :=
   map_sum toRealHom _ _
 
-theorem _root_.Real.toNNReal_sum_of_nonneg {α} {s : Finset α} {f : α → ℝ}
-    (hf : ∀ a, a ∈ s → 0 ≤ f a) :
+@[simp, norm_cast]
+lemma coe_expect (s : Finset ι) (f : ι → ℝ≥0) : 𝔼 i ∈ s, f i = 𝔼 i ∈ s, (f i : ℝ) :=
+  map_expect toRealHom ..
+
+theorem _root_.Real.toNNReal_sum_of_nonneg (hf : ∀ i ∈ s, 0 ≤ f i) :
     Real.toNNReal (∑ a ∈ s, f a) = ∑ a ∈ s, Real.toNNReal (f a) := by
   rw [← coe_inj, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)]
   exact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]
 
 @[simp, norm_cast]
-theorem coe_prod {α} {s : Finset α} {f : α → ℝ≥0} : ↑(∏ a ∈ s, f a) = ∏ a ∈ s, (f a : ℝ) :=
+theorem coe_prod (s : Finset ι) (f : ι → ℝ≥0) : ↑(∏ a ∈ s, f a) = ∏ a ∈ s, (f a : ℝ) :=
   map_prod toRealHom _ _
 
-theorem _root_.Real.toNNReal_prod_of_nonneg {α} {s : Finset α} {f : α → ℝ}
-    (hf : ∀ a, a ∈ s → 0 ≤ f a) :
+theorem _root_.Real.toNNReal_prod_of_nonneg (hf : ∀ a, a ∈ s → 0 ≤ f a) :
     Real.toNNReal (∏ a ∈ s, f a) = ∏ a ∈ s, Real.toNNReal (f a) := by
   rw [← coe_inj, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]
   exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]
 
 @[simp, norm_cast] lemma coe_nsmul (r : ℝ≥0) (n : ℕ) : ↑(n • r) = n • (r : ℝ) := rfl
+@[simp, norm_cast] lemma coe_nnqsmul (q : ℚ≥0) (x : ℝ≥0) : ↑(q • x) = (q • x : ℝ) := rfl
 
 @[simp, norm_cast]
 protected theorem coe_natCast (n : ℕ) : (↑(↑n : ℝ≥0) : ℝ) = n :=
@@ -428,7 +434,7 @@ theorem orderIsoIccZeroCoe_symm_apply_coe (a : ℝ≥0) (b : Set.Iic a) :
 
 -- note we need the `@` to make the `Membership.mem` have a sensible type
 theorem coe_image {s : Set ℝ≥0} :
-    (↑) '' s = { x : ℝ | ∃ h : 0 ≤ x, @Membership.mem ℝ≥0 _ _ ⟨x, h⟩ s } :=
+    (↑) '' s = { x : ℝ | ∃ h : 0 ≤ x, @Membership.mem ℝ≥0 _ _ s ⟨x, h⟩ } :=
   Subtype.coe_image
 
 theorem bddAbove_coe {s : Set ℝ≥0} : BddAbove (((↑) : ℝ≥0 → ℝ) '' s) ↔ BddAbove s :=
@@ -835,25 +841,25 @@ theorem div_le_of_le_mul' {a b c : ℝ≥0} (h : a ≤ b * c) : a / b ≤ c :=
 
 @[deprecated le_div_iff₀ (since := "2024-08-21")]
 protected lemma le_div_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ a * r ≤ b :=
-  le_div_iff₀ <| pos_iff_ne_zero.2 hr
+  le_div_iff₀ hr.bot_lt
 
-nonrec theorem le_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ r * a ≤ b :=
-  le_div_iff₀' <| pos_iff_ne_zero.2 hr
+@[deprecated le_div_iff₀' (since := "2024-10-02")]
+theorem le_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a ≤ b / r ↔ r * a ≤ b := le_div_iff₀' hr.bot_lt
 
-theorem div_lt_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < b * r :=
-  lt_iff_lt_of_le_iff_le (le_div_iff₀ (pos_iff_ne_zero.2 hr))
+@[deprecated div_lt_iff₀ (since := "2024-10-02")]
+theorem div_lt_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < b * r := div_lt_iff₀ hr.bot_lt
 
-theorem div_lt_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < r * b :=
-  lt_iff_lt_of_le_iff_le (le_div_iff₀' (pos_iff_ne_zero.2 hr))
+@[deprecated div_lt_iff₀' (since := "2024-10-02")]
+theorem div_lt_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a / r < b ↔ a < r * b := div_lt_iff₀' hr.bot_lt
 
-theorem lt_div_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ a * r < b :=
-  lt_iff_lt_of_le_iff_le (div_le_iff₀ (pos_iff_ne_zero.2 hr))
+@[deprecated lt_div_iff₀ (since := "2024-10-02")]
+theorem lt_div_iff {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ a * r < b := lt_div_iff₀ hr.bot_lt
 
-theorem lt_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ r * a < b :=
-  lt_iff_lt_of_le_iff_le (div_le_iff₀' (pos_iff_ne_zero.2 hr))
+@[deprecated lt_div_iff₀' (since := "2024-10-02")]
+theorem lt_div_iff' {a b r : ℝ≥0} (hr : r ≠ 0) : a < b / r ↔ r * a < b := lt_div_iff₀' hr.bot_lt
 
 theorem mul_lt_of_lt_div {a b r : ℝ≥0} (h : a < b / r) : a * r < b :=
-  (lt_div_iff fun hr => False.elim <| by simp [hr] at h).1 h
+  (lt_div_iff₀ <| pos_iff_ne_zero.2 fun hr => False.elim <| by simp [hr] at h).1 h
 
 theorem div_le_div_left_of_le {a b c : ℝ≥0} (c0 : c ≠ 0) (cb : c ≤ b) :
     a / b ≤ a / c :=
@@ -878,8 +884,7 @@ nonrec theorem half_lt_self {a : ℝ≥0} (h : a ≠ 0) : a / 2 < a :=
   half_lt_self h.bot_lt
 
 theorem div_lt_one_of_lt {a b : ℝ≥0} (h : a < b) : a / b < 1 := by
-  rwa [div_lt_iff, one_mul]
-  exact ne_of_gt (lt_of_le_of_lt (zero_le _) h)
+  rwa [div_lt_iff₀ h.bot_lt, one_mul]
 
 theorem _root_.Real.toNNReal_inv {x : ℝ} : Real.toNNReal x⁻¹ = (Real.toNNReal x)⁻¹ := by
   rcases le_total 0 x with hx | hx
@@ -896,13 +901,13 @@ theorem _root_.Real.toNNReal_div' {x y : ℝ} (hy : 0 ≤ y) :
   rw [div_eq_inv_mul, div_eq_inv_mul, Real.toNNReal_mul (inv_nonneg.2 hy), Real.toNNReal_inv]
 
 theorem inv_lt_one_iff {x : ℝ≥0} (hx : x ≠ 0) : x⁻¹ < 1 ↔ 1 < x := by
-  rw [← one_div, div_lt_iff hx, one_mul]
+  rw [← one_div, div_lt_iff₀ hx.bot_lt, one_mul]
 
 theorem zpow_pos {x : ℝ≥0} (hx : x ≠ 0) (n : ℤ) : 0 < x ^ n :=
   zpow_pos_of_pos hx.bot_lt _
 
 theorem inv_lt_inv {x y : ℝ≥0} (hx : x ≠ 0) (h : x < y) : y⁻¹ < x⁻¹ :=
-  inv_lt_inv_of_lt hx.bot_lt h
+  inv_strictAnti₀ hx.bot_lt h
 
 end Inv
 
@@ -940,7 +945,7 @@ theorem iInf_empty [IsEmpty ι] (f : ι → ℝ≥0) : ⨅ i, f i = 0 := by
 @[simp]
 theorem iInf_const_zero {α : Sort*} : ⨅ _ : α, (0 : ℝ≥0) = 0 := by
   rw [← coe_inj, coe_iInf]
-  exact Real.ciInf_const_zero
+  exact Real.iInf_const_zero
 
 theorem iInf_mul (f : ι → ℝ≥0) (a : ℝ≥0) : iInf f * a = ⨅ i, f i * a := by
   rw [← coe_inj, NNReal.coe_mul, coe_iInf, coe_iInf]
diff --git a/Mathlib/Data/Nat/BitIndices.lean b/Mathlib/Data/Nat/BitIndices.lean
index 2b9034e73c7b8..79bd2eb583ef4 100644
--- a/Mathlib/Data/Nat/BitIndices.lean
+++ b/Mathlib/Data/Nat/BitIndices.lean
@@ -7,8 +7,8 @@ import Mathlib.Data.List.Sort
 import Mathlib.Data.Nat.Bitwise
 import Mathlib.Algebra.BigOperators.Ring.List
 import Mathlib.Algebra.Order.BigOperators.Group.List
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Algebra.Order.Sub.Defs
-import Mathlib.Algebra.Star.Order
 
 /-!
 # Bit Indices
@@ -35,9 +35,9 @@ elements of `s` in increasing order. -/
 def bitIndices (n : ℕ) : List ℕ :=
   @binaryRec (fun _ ↦ List ℕ) [] (fun b _ s ↦ b.casesOn (s.map (· + 1)) (0 :: s.map (· + 1))) n
 
-@[simp] theorem bitIndices_zero : bitIndices 0 = [] := by rfl
+@[simp] theorem bitIndices_zero : bitIndices 0 = [] := by simp [bitIndices]
 
-@[simp] theorem bitIndices_one : bitIndices 1 = [0] := by rfl
+@[simp] theorem bitIndices_one : bitIndices 1 = [0] := by simp [bitIndices]
 
 theorem bitIndices_bit_true (n : ℕ) :
     bitIndices (bit true n) = 0 :: ((bitIndices n).map (· + 1)) :=
diff --git a/Mathlib/Data/Nat/Bits.lean b/Mathlib/Data/Nat/Bits.lean
index 5995d94fe4549..753381e7318d1 100644
--- a/Mathlib/Data/Nat/Bits.lean
+++ b/Mathlib/Data/Nat/Bits.lean
@@ -25,7 +25,7 @@ and `Nat.digits`.
 
 -- Once we're in the `Nat` namespace, `xor` will inconveniently resolve to `Nat.xor`.
 /-- `bxor` denotes the `xor` function i.e. the exclusive-or function on type `Bool`. -/
-local notation "bxor" => _root_.xor
+local notation "bxor" => xor
 
 namespace Nat
 universe u
@@ -43,12 +43,12 @@ def boddDiv2 : ℕ → Bool × ℕ
 /-- `div2 n = ⌊n/2⌋` the greatest integer smaller than `n/2`-/
 def div2 (n : ℕ) : ℕ := (boddDiv2 n).2
 
-/-- `bodd n` returns `true` if `n` is odd-/
+/-- `bodd n` returns `true` if `n` is odd -/
 def bodd (n : ℕ) : Bool := (boddDiv2 n).1
 
 @[simp] lemma bodd_zero : bodd 0 = false := rfl
 
-lemma bodd_one : bodd 1 = true := rfl
+@[simp] lemma bodd_one : bodd 1 = true := rfl
 
 lemma bodd_two : bodd 2 = false := rfl
 
@@ -88,12 +88,12 @@ lemma mod_two_of_bodd (n : ℕ) : n % 2 = cond (bodd n) 1 0 := by
 
 @[simp] lemma div2_zero : div2 0 = 0 := rfl
 
-lemma div2_one : div2 1 = 0 := rfl
+@[simp] lemma div2_one : div2 1 = 0 := rfl
 
 lemma div2_two : div2 2 = 1 := rfl
 
 @[simp]
-lemma div2_succ (n : ℕ) : div2 (succ n) = cond (bodd n) (succ (div2 n)) (div2 n) := by
+lemma div2_succ (n : ℕ) : div2 (n + 1) = cond (bodd n) (succ (div2 n)) (div2 n) := by
   simp only [bodd, boddDiv2, div2]
   rcases boddDiv2 n with ⟨_|_, _⟩ <;> simp
 
@@ -132,7 +132,7 @@ lemma bit_zero : bit false 0 = 0 :=
 
 /-- `shiftLeft' b m n` performs a left shift of `m` `n` times
  and adds the bit `b` as the least significant bit each time.
- Returns the corresponding natural number-/
+ Returns the corresponding natural number -/
 def shiftLeft' (b : Bool) (m : ℕ) : ℕ → ℕ
   | 0 => m
   | n + 1 => bit b (shiftLeft' b m n)
@@ -195,6 +195,12 @@ lemma binaryRec_zero {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit
   rw [binaryRec]
   rfl
 
+@[simp]
+lemma binaryRec_one {C : Nat → Sort u} (z : C 0) (f : ∀ b n, C n → C (bit b n)) :
+    binaryRec z f 1 = f true 0 z := by
+  rw [binaryRec]
+  simp
+
 /-! bitwise ops -/
 
 lemma bodd_bit (b n) : bodd (bit b n) = b := by
@@ -291,7 +297,7 @@ theorem bit_ne_zero (b) {n} (h : n ≠ 0) : bit b n ≠ 0 := by
 @[simp]
 theorem bitCasesOn_bit {C : ℕ → Sort u} (H : ∀ b n, C (bit b n)) (b : Bool) (n : ℕ) :
     bitCasesOn (bit b n) H = H b n :=
-  eq_of_heq <| (eq_rec_heq _ _).trans <| by rw [bodd_bit, div2_bit]
+  eq_of_heq <| (eqRec_heq _ _).trans <| by rw [bodd_bit, div2_bit]
 
 @[simp]
 theorem bitCasesOn_bit0 {C : ℕ → Sort u} (H : ∀ b n, C (bit b n)) (n : ℕ) :
@@ -340,7 +346,7 @@ theorem binaryRec_eq' {C : ℕ → Sort*} {z : C 0} {f : ∀ b n, C n → C (bit
   split_ifs with h'
   · rcases bit_eq_zero_iff.mp h' with ⟨rfl, rfl⟩
     rw [binaryRec_zero]
-    simp only [imp_false, or_false_iff, eq_self_iff_true, not_true] at h
+    simp only [imp_false, or_false, eq_self_iff_true, not_true, reduceCtorEq] at h
     exact h.symm
   · dsimp only []
     generalize_proofs e
@@ -391,6 +397,7 @@ theorem bit1_bits (n : ℕ) : (2 * n + 1).bits = true :: n.bits :=
 @[simp]
 theorem one_bits : Nat.bits 1 = [true] := by
   convert bit1_bits 0
+  simp
 
 -- TODO Find somewhere this can live.
 -- example : bits 3423 = [true, true, true, true, true, false, true, false, true, false, true, true]
diff --git a/Mathlib/Data/Nat/Bitwise.lean b/Mathlib/Data/Nat/Bitwise.lean
index 0d5dc3611afd5..98bc466c54c8d 100644
--- a/Mathlib/Data/Nat/Bitwise.lean
+++ b/Mathlib/Data/Nat/Bitwise.lean
@@ -9,6 +9,7 @@ import Mathlib.Algebra.Ring.Nat
 import Mathlib.Order.Basic
 import Mathlib.Tactic.AdaptationNote
 import Mathlib.Tactic.Common
+import Mathlib.Algebra.NeZero
 
 /-!
 # Bitwise operations on natural numbers
@@ -246,34 +247,11 @@ theorem lt_of_testBit {n m : ℕ} (i : ℕ) (hn : testBit n i = false) (hm : tes
         _ ≤ 2 * m := mul_le_mul_left 2 this
     · exact Nat.succ_lt_succ this'
 
-@[simp]
-theorem testBit_two_pow_self (n : ℕ) : testBit (2 ^ n) n = true := by
-  rw [testBit, shiftRight_eq_div_pow, Nat.div_self (Nat.pow_pos Nat.zero_lt_two)]
-  simp
-
-theorem testBit_two_pow_of_ne {n m : ℕ} (hm : n ≠ m) : testBit (2 ^ n) m = false := by
-  rw [testBit, shiftRight_eq_div_pow]
-  cases' hm.lt_or_lt with hm hm
-  · rw [Nat.div_eq_of_lt]
-    · simp
-    · exact Nat.pow_lt_pow_right Nat.one_lt_two hm
-  · rw [Nat.pow_div hm.le Nat.two_pos, ← Nat.sub_add_cancel (succ_le_of_lt <| Nat.sub_pos_of_lt hm)]
-    -- Porting note: XXX why does this make it work?
-    rw [(rfl : succ 0 = 1)]
-    simp [pow_succ, and_one_is_mod, mul_mod_left]
-
-theorem testBit_two_pow (n m : ℕ) : testBit (2 ^ n) m = (n = m) := by
-  by_cases h : n = m
-  · cases h
-    simp
-  · rw [testBit_two_pow_of_ne h]
-    simp [h]
-
 theorem bitwise_swap {f : Bool → Bool → Bool} :
     bitwise (Function.swap f) = Function.swap (bitwise f) := by
   funext m n
   simp only [Function.swap]
-  induction' m using Nat.strongInductionOn with m ih generalizing n
+  induction' m using Nat.strongRecOn with m ih generalizing n
   cases' m with m
   <;> cases' n with n
   <;> try rw [bitwise_zero_left, bitwise_zero_right]
@@ -295,9 +273,6 @@ theorem lor_comm (n m : ℕ) : n ||| m = m ||| n :=
 theorem land_comm (n m : ℕ) : n &&& m = m &&& n :=
   bitwise_comm Bool.and_comm n m
 
-protected lemma xor_comm (n m : ℕ) : n ^^^ m = m ^^^ n :=
-  bitwise_comm (Bool.bne_eq_xor ▸ Bool.xor_comm) n m
-
 lemma and_two_pow (n i : ℕ) : n &&& 2 ^ i = (n.testBit i).toNat * 2 ^ i := by
   refine eq_of_testBit_eq fun j => ?_
   obtain rfl | hij := Decidable.eq_or_ne i j <;> cases' h : n.testBit i
@@ -309,13 +284,6 @@ lemma and_two_pow (n i : ℕ) : n &&& 2 ^ i = (n.testBit i).toNat * 2 ^ i := by
 lemma two_pow_and (n i : ℕ) : 2 ^ i &&& n = 2 ^ i * (n.testBit i).toNat := by
   rw [mul_comm, land_comm, and_two_pow]
 
-@[simp]
-theorem zero_xor (n : ℕ) : 0 ^^^ n = n := by simp [HXor.hXor, Xor.xor, xor]
-
-@[simp]
-theorem xor_zero (n : ℕ) : n ^^^ 0 = n := by simp [HXor.hXor, Xor.xor, xor]
-
-
 /-- Proving associativity of bitwise operations in general essentially boils down to a huge case
     distinction, so it is shorter to use this tactic instead of proving it in the general case. -/
 macro "bitwise_assoc_tac" : tactic => set_option hygiene false in `(tactic| (
@@ -328,22 +296,16 @@ macro "bitwise_assoc_tac" : tactic => set_option hygiene false in `(tactic| (
   -- This is necessary because these are simp lemmas in mathlib
   <;> simp [hn, Bool.or_assoc, Bool.and_assoc, Bool.bne_eq_xor]))
 
-protected lemma xor_assoc (n m k : ℕ) : (n ^^^ m) ^^^ k = n ^^^ (m ^^^ k) := by bitwise_assoc_tac
-
 theorem land_assoc (n m k : ℕ) : (n &&& m) &&& k = n &&& (m &&& k) := by bitwise_assoc_tac
 
 theorem lor_assoc (n m k : ℕ) : (n ||| m) ||| k = n ||| (m ||| k) := by bitwise_assoc_tac
 
-@[simp]
-theorem xor_self (n : ℕ) : n ^^^ n = 0 :=
-  zero_of_testBit_eq_false fun i => by simp
-
 -- These lemmas match `mul_inv_cancel_right` and `mul_inv_cancel_left`.
 theorem xor_cancel_right (n m : ℕ) : (m ^^^ n) ^^^ n = m := by
-  rw [Nat.xor_assoc, xor_self, xor_zero]
+  rw [Nat.xor_assoc, Nat.xor_self, xor_zero]
 
 theorem xor_cancel_left (n m : ℕ) : n ^^^ (n ^^^ m) = m := by
-  rw [← Nat.xor_assoc, xor_self, zero_xor]
+  rw [← Nat.xor_assoc, Nat.xor_self, zero_xor]
 
 theorem xor_right_injective {n : ℕ} : Function.Injective (HXor.hXor n : ℕ → ℕ) := fun m m' h => by
   rw [← xor_cancel_left n m, ← xor_cancel_left n m', h]
@@ -362,45 +324,44 @@ theorem xor_left_inj {n m m' : ℕ} : m ^^^ n = m' ^^^ n ↔ m = m' :=
 
 @[simp]
 theorem xor_eq_zero {n m : ℕ} : n ^^^ m = 0 ↔ n = m := by
-  rw [← xor_self n, xor_right_inj, eq_comm]
+  rw [← Nat.xor_self n, xor_right_inj, eq_comm]
 
 theorem xor_ne_zero {n m : ℕ} : n ^^^ m ≠ 0 ↔ n ≠ m :=
   xor_eq_zero.not
 
-theorem xor_trichotomy {a b c : ℕ} (h : a ≠ b ^^^ c) :
-    b ^^^ c < a ∨ a ^^^ c < b ∨ a ^^^ b < c := by
-  set v := a ^^^ (b ^^^ c) with hv
+theorem xor_trichotomy {a b c : ℕ} (h : a ^^^ b ^^^ c ≠ 0) :
+    b ^^^ c < a ∨ c ^^^ a < b ∨ a ^^^ b < c := by
+  set v := a ^^^ b ^^^ c with hv
   -- The xor of any two of `a`, `b`, `c` is the xor of `v` and the third.
   have hab : a ^^^ b = c ^^^ v := by
-    rw [hv]
-    conv_rhs =>
-      rw [Nat.xor_comm]
-      simp [Nat.xor_assoc]
-  have hac : a ^^^ c = b ^^^ v := by
-    rw [hv]
-    conv_rhs =>
-      right
-      rw [← Nat.xor_comm]
-    rw [← Nat.xor_assoc, ← Nat.xor_assoc, xor_self, zero_xor, Nat.xor_comm]
-  have hbc : b ^^^ c = a ^^^ v := by simp [hv, ← Nat.xor_assoc]
+    rw [Nat.xor_comm c, xor_cancel_right]
+  have hbc : b ^^^ c = a ^^^ v := by
+    rw [← Nat.xor_assoc, xor_cancel_left]
+  have hca : c ^^^ a = b ^^^ v := by
+    rw [hv, Nat.xor_assoc, Nat.xor_comm a, ← Nat.xor_assoc, xor_cancel_left]
   -- If `i` is the position of the most significant bit of `v`, then at least one of `a`, `b`, `c`
   -- has a one bit at position `i`.
-  obtain ⟨i, ⟨hi, hi'⟩⟩ := exists_most_significant_bit (xor_ne_zero.2 h)
-  have : testBit a i = true ∨ testBit b i = true ∨ testBit c i = true := by
+  obtain ⟨i, ⟨hi, hi'⟩⟩ := exists_most_significant_bit h
+  have : testBit a i ∨ testBit b i ∨ testBit c i := by
     contrapose! hi
-    simp only [Bool.eq_false_eq_not_eq_true, Ne, testBit_xor, Bool.bne_eq_xor] at hi ⊢
-    rw [hi.1, hi.2.1, hi.2.2, Bool.xor_false, Bool.xor_false]
+    simp_rw [Bool.eq_false_eq_not_eq_true] at hi ⊢
+    rw [testBit_xor, testBit_xor, hi.1, hi.2.1, hi.2.2]
+    rfl
   -- If, say, `a` has a one bit at position `i`, then `a xor v` has a zero bit at position `i`, but
   -- the same bits as `a` in positions greater than `j`, so `a xor v < a`.
-  rcases this with (h | h | h)
+  obtain h | h | h := this
   on_goal 1 => left; rw [hbc]
-  on_goal 2 => right; left; rw [hac]
+  on_goal 2 => right; left; rw [hca]
   on_goal 3 => right; right; rw [hab]
   all_goals
-    exact lt_of_testBit i (by simp [h, hi]) h fun j hj => by simp [hi' _ hj]
-
-theorem lt_xor_cases {a b c : ℕ} (h : a < b ^^^ c) : a ^^^ c < b ∨ a ^^^ b < c :=
-  (or_iff_right fun h' => (h.asymm h').elim).1 <| xor_trichotomy h.ne
+    refine lt_of_testBit i ?_ h fun j hj => ?_
+    · rw [testBit_xor, h, hi]
+      rfl
+    · simp only [testBit_xor, hi' _ hj, Bool.bne_false]
+
+theorem lt_xor_cases {a b c : ℕ} (h : a < b ^^^ c) : a ^^^ c < b ∨ a ^^^ b < c := by
+  obtain ha | hb | hc := xor_trichotomy <| Nat.xor_assoc _ _ _ ▸ xor_ne_zero.2 h.ne
+  exacts [(h.asymm ha).elim, Or.inl <| Nat.xor_comm _ _ ▸ hb, Or.inr hc]
 
 @[simp] theorem bit_lt_two_pow_succ_iff {b x n} : bit b x < 2 ^ (n + 1) ↔ x < 2 ^ n := by
   cases b <;> simp <;> omega
diff --git a/Mathlib/Data/Nat/Cast/Commute.lean b/Mathlib/Data/Nat/Cast/Commute.lean
index 4d8306384d75f..9d1e9bc89094b 100644
--- a/Mathlib/Data/Nat/Cast/Commute.lean
+++ b/Mathlib/Data/Nat/Cast/Commute.lean
@@ -11,7 +11,7 @@ import Mathlib.Algebra.Ring.Commute
 
 -/
 
-variable {α β : Type*}
+variable {α : Type*}
 
 namespace Nat
 
diff --git a/Mathlib/Data/Nat/Cast/Defs.lean b/Mathlib/Data/Nat/Cast/Defs.lean
index 0a184083269a5..937ea2f0f088c 100644
--- a/Mathlib/Data/Nat/Cast/Defs.lean
+++ b/Mathlib/Data/Nat/Cast/Defs.lean
@@ -83,7 +83,7 @@ class AddMonoidWithOne (R : Type*) extends NatCast R, AddMonoid R, One R where
   /-- The canonical map `ℕ → R` is a homomorphism. -/
   natCast_succ : ∀ n, natCast (n + 1) = natCast n + 1 := by intros; rfl
 
-/-- An `AddCommMonoidWithOne` is an `AddMonoidWithOne` satisfying `a + b = b + a`.  -/
+/-- An `AddCommMonoidWithOne` is an `AddMonoidWithOne` satisfying `a + b = b + a`. -/
 class AddCommMonoidWithOne (R : Type*) extends AddMonoidWithOne R, AddCommMonoid R
 
 library_note "coercion into rings"
@@ -149,8 +149,7 @@ protected def binCast [Zero R] [One R] [Add R] : ℕ → R
 @[simp]
 theorem binCast_eq [AddMonoidWithOne R] (n : ℕ) :
     (Nat.binCast n : R) = ((n : ℕ) : R) := by
-  apply Nat.strongInductionOn n
-  intros k hk
+  induction n using Nat.strongRecOn with | ind k hk => ?_
   cases k with
   | zero => rw [Nat.binCast, Nat.cast_zero]
   | succ k =>
diff --git a/Mathlib/Data/Nat/Cast/NeZero.lean b/Mathlib/Data/Nat/Cast/NeZero.lean
index d49f78b3b3095..c3f00eb8b8820 100644
--- a/Mathlib/Data/Nat/Cast/NeZero.lean
+++ b/Mathlib/Data/Nat/Cast/NeZero.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Gabriel Ebner
 -/
 import Mathlib.Data.Nat.Cast.Defs
-import Mathlib.Algebra.NeZero
 
 /-!
 # Lemmas about nonzero elements of an `AddMonoidWithOne`
diff --git a/Mathlib/Data/Nat/Cast/Order/Field.lean b/Mathlib/Data/Nat/Cast/Order/Field.lean
index 8b16df559d51a..bf5cfd6d2e238 100644
--- a/Mathlib/Data/Nat/Cast/Order/Field.lean
+++ b/Mathlib/Data/Nat/Cast/Order/Field.lean
@@ -22,7 +22,7 @@ variable {α : Type*} [LinearOrderedSemifield α]
 
 lemma cast_inv_le_one : ∀ n : ℕ, (n⁻¹ : α) ≤ 1
   | 0 => by simp
-  | n + 1 => inv_le_one <| by simp [Nat.cast_nonneg]
+  | n + 1 => inv_le_one_of_one_le₀ <| by simp [Nat.cast_nonneg]
 
 /-- Natural division is always less than division in the field. -/
 theorem cast_div_le {m n : ℕ} : ((m / n : ℕ) : α) ≤ m / n := by
diff --git a/Mathlib/Data/Nat/Cast/Synonym.lean b/Mathlib/Data/Nat/Cast/Synonym.lean
index fad1703cc9e63..075d62d0d87cf 100644
--- a/Mathlib/Data/Nat/Cast/Synonym.lean
+++ b/Mathlib/Data/Nat/Cast/Synonym.lean
@@ -22,7 +22,7 @@ the natural numbers into an additive monoid with a one (`Nat.cast`).
 -- where `simp [map_zero]` should suffice. (Similarly for `map_one`.)
 -- See https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/simp.20regression.20with.20MonoidHomClass
 
-variable {α β : Type*}
+variable {α : Type*}
 
 /-! ### Order dual -/
 
diff --git a/Mathlib/Data/Nat/Choose/Basic.lean b/Mathlib/Data/Nat/Choose/Basic.lean
index af9347cfba776..e3c7af108183b 100644
--- a/Mathlib/Data/Nat/Choose/Basic.lean
+++ b/Mathlib/Data/Nat/Choose/Basic.lean
@@ -60,6 +60,21 @@ theorem choose_succ_succ (n k : ℕ) : choose (succ n) (succ k) = choose n k + c
 theorem choose_succ_succ' (n k : ℕ) : choose (n + 1) (k + 1) = choose n k + choose n (k + 1) :=
   rfl
 
+theorem choose_succ_left (n k : ℕ) (hk : 0 < k) :
+    choose (n + 1) k = choose n (k - 1) + choose n k := by
+  obtain ⟨l, rfl⟩ : ∃ l, k = l + 1 := Nat.exists_eq_add_of_le' hk
+  rfl
+
+theorem choose_succ_right (n k : ℕ) (hn : 0 < n) :
+    choose n (k + 1) = choose (n - 1) k + choose (n - 1) (k + 1) := by
+  obtain ⟨l, rfl⟩ : ∃ l, n = l + 1 := Nat.exists_eq_add_of_le' hn
+  rfl
+
+theorem choose_eq_choose_pred_add {n k : ℕ} (hn : 0 < n) (hk : 0 < k) :
+    choose n k = choose (n - 1) (k - 1) + choose (n - 1) k := by
+  obtain ⟨l, rfl⟩ : ∃ l, k = l + 1 := Nat.exists_eq_add_of_le' hk
+  rw [choose_succ_right _ _ hn, Nat.add_one_sub_one]
+
 theorem choose_eq_zero_of_lt : ∀ {n k}, n < k → choose n k = 0
   | _, 0, hk => absurd hk (Nat.not_lt_zero _)
   | 0, k + 1, _ => choose_zero_succ _
diff --git a/Mathlib/Data/Nat/Choose/Bounds.lean b/Mathlib/Data/Nat/Choose/Bounds.lean
index 22dc4a8bb5371..d4b21ddf606d9 100644
--- a/Mathlib/Data/Nat/Choose/Bounds.lean
+++ b/Mathlib/Data/Nat/Choose/Bounds.lean
@@ -15,7 +15,7 @@ bounds `n^r/r^r ≤ n.choose r ≤ e^r n^r/r^r` in the future.
 
 ## Main declarations
 
-* `Nat.choose_le_pow`: `n.choose r ≤ n^r / r!`
+* `Nat.choose_le_pow_div`: `n.choose r ≤ n^r / r!`
 * `Nat.pow_le_choose`: `(n + 1 - r)^r / r! ≤ n.choose r`. Beware of the fishy ℕ-subtraction.
 -/
 
@@ -26,13 +26,22 @@ variable {α : Type*} [LinearOrderedSemifield α]
 
 namespace Nat
 
-theorem choose_le_pow (r n : ℕ) : (n.choose r : α) ≤ (n ^ r : α) / r ! := by
+theorem choose_le_pow_div (r n : ℕ) : (n.choose r : α) ≤ (n ^ r : α) / r ! := by
   rw [le_div_iff₀']
   · norm_cast
     rw [← Nat.descFactorial_eq_factorial_mul_choose]
     exact n.descFactorial_le_pow r
   exact mod_cast r.factorial_pos
 
+lemma choose_le_descFactorial (n k : ℕ) : n.choose k ≤ n.descFactorial k := by
+  rw [choose_eq_descFactorial_div_factorial]
+  exact Nat.div_le_self _ _
+
+/-- This lemma was changed on 2024/08/29, the old statement is available
+in `Nat.choose_le_pow_div`. -/
+lemma choose_le_pow (n k : ℕ) : n.choose k ≤ n ^ k :=
+  (choose_le_descFactorial n k).trans (descFactorial_le_pow n k)
+
 -- horrific casting is due to ℕ-subtraction
 theorem pow_le_choose (r n : ℕ) : ((n + 1 - r : ℕ) ^ r : α) / r ! ≤ n.choose r := by
   rw [div_le_iff₀']
diff --git a/Mathlib/Data/Nat/Choose/Factorization.lean b/Mathlib/Data/Nat/Choose/Factorization.lean
index 71966a049eb3a..258c4eb897817 100644
--- a/Mathlib/Data/Nat/Choose/Factorization.lean
+++ b/Mathlib/Data/Nat/Choose/Factorization.lean
@@ -78,7 +78,7 @@ theorem factorization_choose_of_lt_three_mul (hp' : p ≠ 2) (hk : p ≤ k) (hk'
         n < 3 * p := hn
         _ ≤ p * p := mul_le_mul_right' this p
         _ = p ^ 2 := (sq p).symm
-        _ ≤ p ^ i := pow_le_pow_right hp.one_lt.le hi
+        _ ≤ p ^ i := pow_right_mono₀ hp.one_lt.le hi
     rwa [mod_eq_of_lt (lt_of_le_of_lt hkn hn), mod_eq_of_lt (lt_of_le_of_lt tsub_le_self hn),
       add_tsub_cancel_of_le hkn]
 
diff --git a/Mathlib/Data/Nat/Choose/Sum.lean b/Mathlib/Data/Nat/Choose/Sum.lean
index 0ab264a555cb6..f339da88c6daf 100644
--- a/Mathlib/Data/Nat/Choose/Sum.lean
+++ b/Mathlib/Data/Nat/Choose/Sum.lean
@@ -74,6 +74,17 @@ theorem add_pow [CommSemiring R] (x y : R) (n : ℕ) :
     (x + y) ^ n = ∑ m ∈ range (n + 1), x ^ m * y ^ (n - m) * n.choose m :=
   (Commute.all x y).add_pow n
 
+/-- A special case of the **binomial theorem** -/
+theorem sub_pow [CommRing R] (x y : R) (n : ℕ) :
+    (x - y) ^ n = ∑ m ∈ range (n + 1), (-1) ^ (m + n) * x ^ m * y ^ (n - m) * n.choose m := by
+  rw [sub_eq_add_neg, add_pow]
+  congr! 1 with m hm
+  have : (-1 : R) ^ (n - m) = (-1) ^ (n + m) := by
+    rw [mem_range] at hm
+    simp [show n + m = n - m + 2 * m by omega, pow_add]
+  rw [neg_pow, this]
+  ring
+
 namespace Nat
 
 /-- The sum of entries in a row of Pascal's triangle -/
@@ -116,7 +127,7 @@ theorem four_pow_le_two_mul_add_one_mul_central_binom (n : ℕ) :
     _ ≤ ∑ m ∈ range (2 * n + 1), (2 * n).choose (2 * n / 2) := by gcongr; apply choose_le_middle
     _ = (2 * n + 1) * choose (2 * n) n := by simp
 
-/-- **Zhu Shijie's identity** aka hockey-stick identity. -/
+/-- **Zhu Shijie's identity** aka hockey-stick identity, version with `Icc`. -/
 theorem sum_Icc_choose (n k : ℕ) : ∑ m ∈ Icc k n, m.choose k = (n + 1).choose (k + 1) := by
   rcases lt_or_le n k with h | h
   · rw [choose_eq_zero_of_lt (by omega), Icc_eq_empty_of_lt h, sum_empty]
@@ -126,6 +137,20 @@ theorem sum_Icc_choose (n k : ℕ) : ∑ m ∈ Icc k n, m.choose k = (n + 1).cho
       rw [← Ico_insert_right (by omega), sum_insert (by simp),
         show Ico k (n + 1) = Icc k n by rfl, ih, choose_succ_succ' (n + 1)]
 
+/-- **Zhu Shijie's identity** aka hockey-stick identity, version with `range`.
+Summing `(i + k).choose k` for `i ∈ [0, n]` gives `(n + k + 1).choose (k + 1)`.
+
+Combinatorial interpretation: `(i + k).choose k` is the number of decompositions of `[0, i)` in
+`k + 1` (possibly empty) intervals (this follows from a stars and bars description). In particular,
+`(n + k + 1).choose (k + 1)` corresponds to decomposing `[0, n)` into `k + 2` intervals.
+By putting away the last interval (of some length `n - i`),
+we have to decompose the remaining interval `[0, i)` into `k + 1` intervals, hence the sum. -/
+lemma sum_range_add_choose (n k : ℕ) :
+    ∑ i ∈ Finset.range (n + 1), (i + k).choose k = (n + k + 1).choose (k + 1) := by
+  rw [← sum_Icc_choose, range_eq_Ico]
+  convert (sum_map _ (addRightEmbedding k) (·.choose k)).symm using 2
+  rw [map_add_right_Ico, zero_add, add_right_comm, Nat.Ico_succ_right]
+
 end Nat
 
 theorem Int.alternating_sum_range_choose {n : ℕ} :
diff --git a/Mathlib/Data/Nat/Count.lean b/Mathlib/Data/Nat/Count.lean
index c3e60e57f4397..cb2e2160ea3aa 100644
--- a/Mathlib/Data/Nat/Count.lean
+++ b/Mathlib/Data/Nat/Count.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
+Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
 -/
 import Mathlib.SetTheory.Cardinal.Basic
 
diff --git a/Mathlib/Data/Nat/Defs.lean b/Mathlib/Data/Nat/Defs.lean
index 23dc7e10ab9c7..3bff75138a170 100644
--- a/Mathlib/Data/Nat/Defs.lean
+++ b/Mathlib/Data/Nat/Defs.lean
@@ -5,7 +5,7 @@ Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro
 -/
 import Mathlib.Logic.Function.Basic
 import Mathlib.Logic.Nontrivial.Defs
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 import Mathlib.Tactic.PushNeg
 import Mathlib.Util.AssertExists
 
@@ -57,7 +57,7 @@ assert_not_exists Monoid
 open Function
 
 namespace Nat
-variable {a b c d m n k : ℕ} {p q : ℕ → Prop}
+variable {a b c d m n k : ℕ} {p : ℕ → Prop}
 
 -- TODO: Move the `LinearOrder ℕ` instance to `Order.Nat` (#13092).
 instance instLinearOrder : LinearOrder ℕ where
@@ -137,10 +137,20 @@ lemma one_lt_iff_ne_zero_and_ne_one : ∀ {n : ℕ}, 1 < n ↔ n ≠ 0 ∧ n ≠
 
 lemma le_one_iff_eq_zero_or_eq_one : ∀ {n : ℕ}, n ≤ 1 ↔ n = 0 ∨ n = 1 := by simp [le_succ_iff]
 
-@[simp] lemma lt_one_iff : n < 1 ↔ n = 0 := Nat.lt_succ_iff.trans <| by rw [le_zero_eq]
-
 lemma one_le_of_lt (h : a < b) : 1 ≤ b := Nat.lt_of_le_of_lt (Nat.zero_le _) h
 
+protected lemma min_left_comm (a b c : ℕ) : min a (min b c) = min b (min a c) := by
+  rw [← Nat.min_assoc, ← Nat.min_assoc, b.min_comm]
+
+protected lemma max_left_comm (a b c : ℕ) : max a (max b c) = max b (max a c) := by
+  rw [← Nat.max_assoc, ← Nat.max_assoc, b.max_comm]
+
+protected lemma min_right_comm (a b c : ℕ) : min (min a b) c = min (min a c) b := by
+  rw [Nat.min_assoc, Nat.min_assoc, b.min_comm]
+
+protected lemma max_right_comm (a b c : ℕ) : max (max a b) c = max (max a c) b := by
+  rw [Nat.max_assoc, Nat.max_assoc, b.max_comm]
+
 @[simp] lemma min_eq_zero_iff : min m n = 0 ↔ m = 0 ∨ n = 0 := by omega
 @[simp] lemma max_eq_zero_iff : max m n = 0 ↔ m = 0 ∧ n = 0 := by omega
 
@@ -155,8 +165,13 @@ lemma pred_eq_of_eq_succ (H : m = n.succ) : m.pred = n := by simp [H]
 @[simp] lemma pred_eq_succ_iff : n - 1 = m + 1 ↔ n = m + 2 := by
   cases n <;> constructor <;> rintro ⟨⟩ <;> rfl
 
+#adaptation_note
+/--
+After nightly-2024-09-06 we can remove both the `_root_` prefixes below.
+-/
 lemma forall_lt_succ : (∀ m < n + 1, p m) ↔ (∀ m < n, p m) ∧ p n := by
-  simp only [Nat.lt_succ_iff, Nat.le_iff_lt_or_eq, or_comm, forall_eq_or_imp, and_comm]
+  simp only [Nat.lt_succ_iff, Nat.le_iff_lt_or_eq, _root_.or_comm, forall_eq_or_imp,
+    _root_.and_comm]
 
 lemma exists_lt_succ : (∃ m < n + 1, p m) ↔ (∃ m < n, p m) ∨ p n := by
   rw [← not_iff_not]
@@ -208,8 +223,7 @@ attribute [simp] le_add_left le_add_right Nat.lt_add_left_iff_pos Nat.lt_add_rig
 
 -- Sometimes a bare `Nat.add` or similar appears as a consequence of unfolding during pattern
 -- matching. These lemmas package them back up as typeclass mediated operations.
--- TODO: This is a duplicate of `Nat.add_eq`
-@[simp] lemma add_def : Nat.add m n = m + n := rfl
+@[deprecated (since := "2024-04-05")] alias add_def := add_eq
 
 -- We want to use these two lemmas earlier than the lemmas simp can prove them with
 @[simp, nolint simpNF] protected lemma add_eq_left : a + b = a ↔ b = 0 := by omega
@@ -298,11 +312,11 @@ lemma two_mul_ne_two_mul_add_one : 2 * n ≠ 2 * m + 1 :=
 
 -- TODO: Replace `Nat.mul_right_cancel_iff` with `Nat.mul_left_inj`
 protected lemma mul_left_inj (ha : a ≠ 0) : b * a = c * a ↔ b = c :=
-  Nat.mul_right_cancel_iff (Nat.pos_iff_ne_zero.2 ha) _ _
+  Nat.mul_right_cancel_iff (Nat.pos_iff_ne_zero.2 ha)
 
 -- TODO: Replace `Nat.mul_left_cancel_iff` with `Nat.mul_right_inj`
 protected lemma mul_right_inj (ha : a ≠ 0) : a * b = a * c ↔ b = c :=
-  Nat.mul_left_cancel_iff (Nat.pos_iff_ne_zero.2 ha) _ _
+  Nat.mul_left_cancel_iff (Nat.pos_iff_ne_zero.2 ha)
 
 protected lemma mul_ne_mul_left (ha : a ≠ 0) : b * a ≠ c * a ↔ b ≠ c :=
   not_congr (Nat.mul_left_inj ha)
@@ -586,9 +600,6 @@ protected lemma pow_le_pow_iff_left {n : ℕ} (hn : n ≠ 0) : a ^ n ≤ b ^ n 
 protected lemma pow_lt_pow_iff_left (hn : n ≠ 0) : a ^ n < b ^ n ↔ a < b := by
   simp only [← Nat.not_le, Nat.pow_le_pow_iff_left hn]
 
-@[deprecated (since := "2023-12-23")] alias pow_lt_pow_of_lt_left := Nat.pow_lt_pow_left
-@[deprecated (since := "2023-12-23")] alias pow_le_iff_le_left := Nat.pow_le_pow_iff_left
-
 lemma pow_left_injective (hn : n ≠ 0) : Injective (fun a : ℕ ↦ a ^ n) := by
   simp [Injective, le_antisymm_iff, Nat.pow_le_pow_iff_left hn]
 
@@ -822,7 +833,23 @@ This is an alias of `Nat.leRec`, specialized to `Prop`. -/
 @[elab_as_elim]
 lemma le_induction {m : ℕ} {P : ∀ n, m ≤ n → Prop} (base : P m m.le_refl)
     (succ : ∀ n hmn, P n hmn → P (n + 1) (le_succ_of_le hmn)) : ∀ n hmn, P n hmn :=
-  @Nat.leRec (motive := P) base succ
+  @Nat.leRec (motive := P) _ base succ
+
+/-- Induction principle deriving the next case from the two previous ones. -/
+def twoStepInduction {P : ℕ → Sort*} (zero : P 0) (one : P 1)
+    (more : ∀ n, P n → P (n + 1) → P (n + 2)) : ∀ a, P a
+  | 0 => zero
+  | 1 => one
+  | _ + 2 => more _ (twoStepInduction zero one more _) (twoStepInduction zero one more _)
+
+@[elab_as_elim]
+protected theorem strong_induction_on {p : ℕ → Prop} (n : ℕ)
+    (h : ∀ n, (∀ m, m < n → p m) → p n) : p n :=
+  Nat.strongRecOn n h
+
+protected theorem case_strong_induction_on {p : ℕ → Prop} (a : ℕ) (hz : p 0)
+    (hi : ∀ n, (∀ m, m ≤ n → p m) → p (n + 1)) : p a :=
+  Nat.caseStrongRecOn a hz hi
 
 /-- Decreasing induction: if `P (k+1)` implies `P k` for all `k < n`, then `P n` implies `P m` for
 all `m ≤ n`.
@@ -990,9 +1017,6 @@ lemma div_ne_zero_iff_of_dvd (hba : b ∣ a) : a / b ≠ 0 ↔ a ≠ 0 ∧ b ≠
 @[simp] lemma mul_mod_mod (a b c : ℕ) : (a * (b % c)) % c = a * b % c := by
   rw [mul_mod, mod_mod, ← mul_mod]
 
-@[simp] lemma mod_mul_mod (a b c : ℕ) : (a % c * b) % c = a * b % c := by
-  rw [mul_mod, mod_mod, ← mul_mod]
-
 lemma pow_mod (a b n : ℕ) : a ^ b % n = (a % n) ^ b % n := by
   induction b with
   | zero => rfl
@@ -1062,8 +1086,11 @@ lemma sub_mod_eq_zero_of_mod_eq (h : m % k = n % k) : (m - n) % k = 0 := by
 @[simp] lemma one_mod (n : ℕ) : 1 % (n + 2) = 1 :=
   Nat.mod_eq_of_lt (Nat.add_lt_add_right n.succ_pos 1)
 
-lemma one_mod_of_ne_one : ∀ {n : ℕ}, n ≠ 1 → 1 % n = 1
-  | 0, _ | (n + 2), _ => by simp
+lemma one_mod_eq_one : ∀ {n : ℕ}, 1 % n = 1 ↔ n ≠ 1
+  | 0 | 1 | n + 2 => by simp
+
+@[deprecated (since := "2024-08-28")]
+lemma one_mod_of_ne_one  : ∀ {n : ℕ}, n ≠ 1 → 1 % n = 1 := one_mod_eq_one.mpr
 
 lemma dvd_sub_mod (k : ℕ) : n ∣ k - k % n :=
   ⟨k / n, Nat.sub_eq_of_eq_add (Nat.div_add_mod k n).symm⟩
diff --git a/Mathlib/Data/Nat/Digits.lean b/Mathlib/Data/Nat/Digits.lean
index 3465ed5369aa7..23cd40d657cba 100644
--- a/Mathlib/Data/Nat/Digits.lean
+++ b/Mathlib/Data/Nat/Digits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
+Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
 -/
 import Mathlib.Algebra.BigOperators.Intervals
 import Mathlib.Algebra.BigOperators.Ring.List
@@ -246,9 +246,7 @@ theorem ofDigits_digits (b n : ℕ) : ofDigits b (digits b n) = n := by
       · rfl
       · rw [Nat.zero_add] at ih ⊢
         simp only [ih, add_comm 1, ofDigits_one_cons, Nat.cast_id, digits_one_succ]
-    · apply Nat.strongInductionOn n _
-      clear n
-      intro n h
+    · induction n using Nat.strongRecOn with | ind n h => ?_
       cases n
       · rw [digits_zero]
         rfl
@@ -258,9 +256,9 @@ theorem ofDigits_digits (b n : ℕ) : ofDigits b (digits b n) = n := by
         rw [Nat.mod_add_div]
 
 theorem ofDigits_one (L : List ℕ) : ofDigits 1 L = L.sum := by
-  induction' L with _ _ ih
-  · rfl
-  · simp [ofDigits, List.sum_cons, ih]
+  induction L with
+  | nil => rfl
+  | cons _ _ ih => simp [ofDigits, List.sum_cons, ih]
 
 /-!
 ### Properties
@@ -328,8 +326,8 @@ theorem getLast_digit_ne_zero (b : ℕ) {m : ℕ} (hm : m ≠ 0) :
     simp only [zero_add, digits_one, List.getLast_replicate_succ m 1]
     exact Nat.one_ne_zero
   revert hm
-  apply Nat.strongInductionOn m
-  intro n IH hn
+  induction m using Nat.strongRecOn with | ind n IH => ?_
+  intro hn
   by_cases hnb : n < b + 2
   · simpa only [digits_of_lt (b + 2) n hn hnb]
   · rw [digits_getLast n (le_add_left 2 b)]
@@ -345,7 +343,7 @@ theorem mul_ofDigits (n : ℕ) {b : ℕ} {l : List ℕ} :
     rw [List.map_cons, ofDigits_cons, ofDigits_cons, ← ih]
     ring
 
-/-- The addition of ofDigits of two lists is equal to ofDigits of digit-wise addition of them-/
+/-- The addition of ofDigits of two lists is equal to ofDigits of digit-wise addition of them -/
 theorem ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq {b : ℕ} {l1 l2 : List ℕ}
     (h : l1.length = l2.length) :
     ofDigits b l1 + ofDigits b l2 = ofDigits b (l1.zipWith (· + ·) l2) := by
@@ -362,8 +360,8 @@ theorem ofDigits_add_ofDigits_eq_ofDigits_zipWith_of_length_eq {b : ℕ} {l1 l2
 
 /-- The digits in the base b+2 expansion of n are all less than b+2 -/
 theorem digits_lt_base' {b m : ℕ} : ∀ {d}, d ∈ digits (b + 2) m → d < b + 2 := by
-  apply Nat.strongInductionOn m
-  intro n IH d hd
+  induction m using Nat.strongRecOn with | ind n IH => ?_
+  intro d hd
   cases' n with n
   · rw [digits_zero] at hd
     cases hd
@@ -440,9 +438,10 @@ theorem le_digits_len_le (b n m : ℕ) (h : n ≤ m) : (digits b n).length ≤ (
 
 @[mono]
 theorem ofDigits_monotone {p q : ℕ} (L : List ℕ) (h : p ≤ q) : ofDigits p L ≤ ofDigits q L := by
-  induction' L with _ _ hi
-  · rfl
-  · simp only [ofDigits, cast_id, add_le_add_iff_left]
+  induction L with
+  | nil => rfl
+  | cons _ _ hi =>
+    simp only [ofDigits, cast_id, add_le_add_iff_left]
     exact Nat.mul_le_mul h hi
 
 theorem sum_le_ofDigits {p : ℕ} (L : List ℕ) (h : 1 ≤ p) : L.sum ≤ ofDigits p L :=
@@ -571,7 +570,7 @@ theorem sub_one_mul_sum_log_div_pow_eq_sub_sum_digits {p : ℕ} (n : ℕ) :
 theorem digits_two_eq_bits (n : ℕ) : digits 2 n = n.bits.map fun b => cond b 1 0 := by
   induction' n using Nat.binaryRecFromOne with b n h ih
   · simp
-  · rfl
+  · simp
   rw [bits_append_bit _ _ fun hn => absurd hn h]
   cases b
   · rw [digits_def' one_lt_two]
diff --git a/Mathlib/Data/Nat/Dist.lean b/Mathlib/Data/Nat/Dist.lean
index 8151b3d7c07f2..b207d03af6224 100644
--- a/Mathlib/Data/Nat/Dist.lean
+++ b/Mathlib/Data/Nat/Dist.lean
@@ -89,7 +89,7 @@ theorem dist_succ_succ {i j : Nat} : dist (succ i) (succ j) = dist i j := by
   simp [dist, succ_sub_succ]
 
 theorem dist_pos_of_ne {i j : Nat} : i ≠ j → 0 < dist i j := fun hne =>
-  Nat.ltByCases
+  ltByCases i j
     (fun h : i < j => by rw [dist_eq_sub_of_le (le_of_lt h)]; apply tsub_pos_of_lt h)
     (fun h : i = j => by contradiction) fun h : i > j => by
     rw [dist_eq_sub_of_le_right (le_of_lt h)]; apply tsub_pos_of_lt h
diff --git a/Mathlib/Data/Nat/Factorial/Basic.lean b/Mathlib/Data/Nat/Factorial/Basic.lean
index cacc9997b8c7e..79539c43a7625 100644
--- a/Mathlib/Data/Nat/Factorial/Basic.lean
+++ b/Mathlib/Data/Nat/Factorial/Basic.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
 -/
 import Mathlib.Data.Nat.Defs
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 import Mathlib.Tactic.Common
 import Mathlib.Tactic.Monotonicity.Attr
 
diff --git a/Mathlib/Data/Nat/Factorization/Basic.lean b/Mathlib/Data/Nat/Factorization/Basic.lean
index 62885ed5181d6..27ddcf0d3a424 100644
--- a/Mathlib/Data/Nat/Factorization/Basic.lean
+++ b/Mathlib/Data/Nat/Factorization/Basic.lean
@@ -13,7 +13,7 @@ import Mathlib.Tactic.IntervalCases
 # Basic lemmas on prime factorizations
 -/
 
-open Nat Finset List Finsupp
+open Finset List Finsupp
 
 namespace Nat
 variable {a b m n p : ℕ}
@@ -309,7 +309,7 @@ theorem dvd_iff_prime_pow_dvd_dvd (n d : ℕ) :
   rcases eq_or_ne n 0 with (rfl | hn)
   · simp
   rcases eq_or_ne d 0 with (rfl | hd)
-  · simp only [zero_dvd_iff, hn, false_iff_iff, not_forall]
+  · simp only [zero_dvd_iff, hn, false_iff, not_forall]
     exact ⟨2, n, prime_two, dvd_zero _, mt (le_of_dvd hn.bot_lt) (lt_two_pow n).not_le⟩
   refine ⟨fun h p k _ hpkd => dvd_trans hpkd h, ?_⟩
   rw [← factorization_prime_le_iff_dvd hd hn]
@@ -388,7 +388,7 @@ lemma factorizationLCMRight_pos :
   rw [factorizationLCMRight, Finsupp.prod_ne_zero_iff]
   intro p _ H
   by_cases h : b.factorization p ≤ a.factorization p
-  · simp only [h, reduceIte, pow_eq_zero_iff', ne_eq] at H
+  · simp only [h, reduceIte, pow_eq_zero_iff', ne_eq, reduceCtorEq] at H
   · simp only [h, ↓reduceIte, pow_eq_zero_iff', ne_eq] at H
     simpa [H.1] using H.2
 
@@ -530,7 +530,7 @@ theorem prod_pow_prime_padicValNat (n : Nat) (hn : n ≠ 0) (m : Nat) (pr : n <
 
 -- TODO: Port lemmas from `Data/Nat/Multiplicity` to here, re-written in terms of `factorization`
 /-- Exactly `n / p` naturals in `[1, n]` are multiples of `p`.
-See `Nat.card_multiples'` for an alternative spelling of the statement.  -/
+See `Nat.card_multiples'` for an alternative spelling of the statement. -/
 theorem card_multiples (n p : ℕ) : card ((Finset.range n).filter fun e => p ∣ e + 1) = n / p := by
   induction' n with n hn
   · simp
diff --git a/Mathlib/Data/Nat/Factorization/Defs.lean b/Mathlib/Data/Nat/Factorization/Defs.lean
index fbabc1edc00fd..fb64f3dd9d2d6 100644
--- a/Mathlib/Data/Nat/Factorization/Defs.lean
+++ b/Mathlib/Data/Nat/Factorization/Defs.lean
@@ -86,7 +86,7 @@ alias factorization_eq_factors_multiset := factorization_eq_primeFactorsList_mul
 
 theorem Prime.factorization_pos_of_dvd {n p : ℕ} (hp : p.Prime) (hn : n ≠ 0) (h : p ∣ n) :
     0 < n.factorization p := by
-    rwa [← primeFactorsList_count_eq, count_pos_iff_mem, mem_primeFactorsList_iff_dvd hn hp]
+    rwa [← primeFactorsList_count_eq, count_pos_iff, mem_primeFactorsList_iff_dvd hn hp]
 
 theorem multiplicity_eq_factorization {n p : ℕ} (pp : p.Prime) (hn : n ≠ 0) :
     multiplicity p n = n.factorization p := by
@@ -172,7 +172,7 @@ theorem factorization_prod {α : Type*} {S : Finset α} {g : α → ℕ} (hS : 
     · simp
     · intro x T hxS hTS hxT IH
       have hT : T.prod g ≠ 0 := prod_ne_zero_iff.mpr fun x hx => hS x (hTS hx)
-      simp [prod_insert hxT, sum_insert hxT, ← IH, factorization_mul (hS x hxS) hT]
+      simp [prod_insert hxT, sum_insert hxT, IH, factorization_mul (hS x hxS) hT]
 
 /-- For any `p`, the power of `p` in `n^k` is `k` times the power in `n` -/
 @[simp]
diff --git a/Mathlib/Data/Nat/Factorization/Induction.lean b/Mathlib/Data/Nat/Factorization/Induction.lean
index 1add41840d66f..df145576e8514 100644
--- a/Mathlib/Data/Nat/Factorization/Induction.lean
+++ b/Mathlib/Data/Nat/Factorization/Induction.lean
@@ -22,7 +22,7 @@ we can define `P` for all natural numbers. -/
 @[elab_as_elim]
 def recOnPrimePow {P : ℕ → Sort*} (h0 : P 0) (h1 : P 1)
     (h : ∀ a p n : ℕ, p.Prime → ¬p ∣ a → 0 < n → P a → P (p ^ n * a)) : ∀ a : ℕ, P a := fun a =>
-  Nat.strongRecOn a fun n =>
+  Nat.strongRecOn' a fun n =>
     match n with
     | 0 => fun _ => h0
     | 1 => fun _ => h1
diff --git a/Mathlib/Data/Nat/Factorization/PrimePow.lean b/Mathlib/Data/Nat/Factorization/PrimePow.lean
index 5f35804b888b8..37333d9e42ecf 100644
--- a/Mathlib/Data/Nat/Factorization/PrimePow.lean
+++ b/Mathlib/Data/Nat/Factorization/PrimePow.lean
@@ -137,6 +137,6 @@ theorem Nat.mul_divisors_filter_prime_pow {a b : ℕ} (hab : a.Coprime b) :
   · simp only [Nat.coprime_zero_right] at hab
     simp [hab, Finset.filter_singleton, not_isPrimePow_one]
   ext n
-  simp only [ha, hb, Finset.mem_union, Finset.mem_filter, Nat.mul_eq_zero, and_true_iff, Ne,
+  simp only [ha, hb, Finset.mem_union, Finset.mem_filter, Nat.mul_eq_zero, and_true, Ne,
     and_congr_left_iff, not_false_iff, Nat.mem_divisors, or_self_iff]
   apply hab.isPrimePow_dvd_mul
diff --git a/Mathlib/Data/Nat/Fib/Basic.lean b/Mathlib/Data/Nat/Fib/Basic.lean
index fb9601268e67e..4d5698a009a29 100644
--- a/Mathlib/Data/Nat/Fib/Basic.lean
+++ b/Mathlib/Data/Nat/Fib/Basic.lean
@@ -5,10 +5,8 @@ Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
 import Mathlib.Data.Finset.NatAntidiagonal
-import Mathlib.Data.Nat.GCD.Basic
 import Mathlib.Data.Nat.Bits
-import Mathlib.Init.Data.Nat.Lemmas
-import Mathlib.Logic.Function.Iterate
+import Mathlib.Data.Nat.GCD.Basic
 import Mathlib.Tactic.Ring
 import Mathlib.Tactic.Zify
 
diff --git a/Mathlib/Data/Nat/Find.lean b/Mathlib/Data/Nat/Find.lean
index e37f555306458..84ee3406a3e5a 100644
--- a/Mathlib/Data/Nat/Find.lean
+++ b/Mathlib/Data/Nat/Find.lean
@@ -11,7 +11,7 @@ import Batteries.WF
 # `Nat.find` and `Nat.findGreatest`
 -/
 
-variable {a b c d m n k : ℕ} {p q : ℕ → Prop}
+variable {m n k : ℕ} {p q : ℕ → Prop}
 
 namespace Nat
 
diff --git a/Mathlib/Data/Nat/GCD/Basic.lean b/Mathlib/Data/Nat/GCD/Basic.lean
index 5b4fc3d84d354..8d4d667ffa251 100644
--- a/Mathlib/Data/Nat/GCD/Basic.lean
+++ b/Mathlib/Data/Nat/GCD/Basic.lean
@@ -5,7 +5,6 @@ Authors: Jeremy Avigad, Leonardo de Moura
 -/
 import Mathlib.Algebra.GroupWithZero.Divisibility
 import Mathlib.Algebra.Ring.Nat
-import Mathlib.Init.Data.Nat.Lemmas
 
 /-!
 # Properties of `Nat.gcd`, `Nat.lcm`, and `Nat.Coprime`
@@ -235,7 +234,7 @@ theorem gcd_mul_of_coprime_of_dvd {a b c : ℕ} (hac : Coprime a c) (b_dvd_c : b
 
 theorem Coprime.eq_of_mul_eq_zero {m n : ℕ} (h : m.Coprime n) (hmn : m * n = 0) :
     m = 0 ∧ n = 1 ∨ m = 1 ∧ n = 0 :=
-  (Nat.eq_zero_of_mul_eq_zero hmn).imp (fun hm => ⟨hm, n.coprime_zero_left.mp <| hm ▸ h⟩) fun hn =>
+  (Nat.mul_eq_zero.mp hmn).imp (fun hm => ⟨hm, n.coprime_zero_left.mp <| hm ▸ h⟩) fun hn =>
     let eq := hn ▸ h.symm
     ⟨m.coprime_zero_left.mp <| eq, hn⟩
 
diff --git a/Mathlib/Data/Nat/Lattice.lean b/Mathlib/Data/Nat/Lattice.lean
index dadaa5b902c92..46be67ef9f99d 100644
--- a/Mathlib/Data/Nat/Lattice.lean
+++ b/Mathlib/Data/Nat/Lattice.lean
@@ -45,9 +45,9 @@ theorem _root_.Set.Infinite.Nat.sSup_eq_zero {s : Set ℕ} (h : s.Infinite) : sS
 theorem sInf_eq_zero {s : Set ℕ} : sInf s = 0 ↔ 0 ∈ s ∨ s = ∅ := by
   cases eq_empty_or_nonempty s with
   | inl h => subst h
-             simp only [or_true_iff, eq_self_iff_true, iff_true_iff, iInf, InfSet.sInf,
+             simp only [or_true, eq_self_iff_true, iInf, InfSet.sInf,
                         mem_empty_iff_false, exists_false, dif_neg, not_false_iff]
-  | inr h => simp only [h.ne_empty, or_false_iff, Nat.sInf_def, h, Nat.find_eq_zero]
+  | inr h => simp only [h.ne_empty, or_false, Nat.sInf_def, h, Nat.find_eq_zero]
 
 @[simp]
 theorem sInf_empty : sInf ∅ = 0 := by
diff --git a/Mathlib/Data/Nat/Log.lean b/Mathlib/Data/Nat/Log.lean
index 461879596638b..f2cf3df6eca07 100644
--- a/Mathlib/Data/Nat/Log.lean
+++ b/Mathlib/Data/Nat/Log.lean
@@ -3,8 +3,6 @@ Copyright (c) 2020 Simon Hudon. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Simon Hudon, Yaël Dillies
 -/
-import Mathlib.Data.Nat.Defs
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Order.Interval.Set.Basic
 import Mathlib.Tactic.Bound.Attribute
 import Mathlib.Tactic.Monotonicity.Attr
@@ -111,14 +109,23 @@ theorem log_lt_of_lt_pow {b x y : ℕ} (hy : y ≠ 0) : y < b ^ x → log b y <
 theorem lt_pow_of_log_lt {b x y : ℕ} (hb : 1 < b) : log b y < x → y < b ^ x :=
   lt_imp_lt_of_le_imp_le (le_log_of_pow_le hb)
 
+lemma log_lt_self (b : ℕ) {x : ℕ} (hx : x ≠ 0) : log b x < x :=
+  match le_or_lt b 1 with
+  | .inl h => log_of_left_le_one h x ▸ Nat.pos_iff_ne_zero.2 hx
+  | .inr h => log_lt_of_lt_pow hx <| lt_pow_self h _
+
+lemma log_le_self (b x : ℕ) : log b x ≤ x :=
+  if hx : x = 0 then by simp [hx]
+  else (log_lt_self b hx).le
+
 theorem lt_pow_succ_log_self {b : ℕ} (hb : 1 < b) (x : ℕ) : x < b ^ (log b x).succ :=
   lt_pow_of_log_lt hb (lt_succ_self _)
 
 theorem log_eq_iff {b m n : ℕ} (h : m ≠ 0 ∨ 1 < b ∧ n ≠ 0) :
     log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1) := by
   rcases em (1 < b ∧ n ≠ 0) with (⟨hb, hn⟩ | hbn)
-  · rw [le_antisymm_iff, ← Nat.lt_succ_iff, ← pow_le_iff_le_log, ← lt_pow_iff_log_lt, and_comm] <;>
-      assumption
+  · rw [le_antisymm_iff, ← Nat.lt_succ_iff, ← pow_le_iff_le_log, ← lt_pow_iff_log_lt,
+      and_comm] <;> assumption
   have hm : m ≠ 0 := h.resolve_right hbn
   rw [not_and_or, not_lt, Ne, not_not] at hbn
   rcases hbn with (hb | rfl)
@@ -199,6 +206,13 @@ theorem add_pred_div_lt {b n : ℕ} (hb : 1 < b) (hn : 2 ≤ n) : (n + b - 1) /
     succ_pred_eq_of_pos (by omega)]
   exact Nat.add_le_mul hn hb
 
+lemma log2_eq_log_two {n : ℕ} : Nat.log2 n = Nat.log 2 n := by
+  rcases eq_or_ne n 0 with rfl | hn
+  · rw [log2_zero, log_zero_right]
+  apply eq_of_forall_le_iff
+  intro m
+  rw [Nat.le_log2 hn, ← Nat.pow_le_iff_le_log Nat.one_lt_two hn]
+
 /-! ### Ceil logarithm -/
 
 
diff --git a/Mathlib/Data/Nat/ModEq.lean b/Mathlib/Data/Nat/ModEq.lean
index 5877782fc1102..29ff2fed4da9d 100644
--- a/Mathlib/Data/Nat/ModEq.lean
+++ b/Mathlib/Data/Nat/ModEq.lean
@@ -3,10 +3,8 @@ Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
-import Mathlib.Algebra.Ring.Regular
+import Mathlib.Algebra.Order.Group.Unbundled.Int
 import Mathlib.Data.Int.GCD
-import Mathlib.Data.Int.Order.Lemmas
-import Mathlib.Tactic.NormNum.Basic
 
 /-!
 # Congruences modulo a natural number
@@ -24,6 +22,7 @@ and proves basic properties about it such as the Chinese Remainder Theorem
 ModEq, congruence, mod, MOD, modulo
 -/
 
+assert_not_exists OrderedAddCommMonoid
 assert_not_exists Function.support
 
 namespace Nat
@@ -37,8 +36,8 @@ notation:50 a " ≡ " b " [MOD " n "]" => ModEq n a b
 
 variable {m n a b c d : ℕ}
 
--- Porting note: This instance should be derivable automatically
-instance : Decidable (ModEq n a b) := decEq (a % n) (b % n)
+-- Since `ModEq` is semi-reducible, we need to provide the decidable instance manually
+instance : Decidable (ModEq n a b) := inferInstanceAs <| Decidable (a % n = b % n)
 
 namespace ModEq
 
@@ -91,7 +90,7 @@ theorem mod_modEq (a n) : a % n ≡ a [MOD n] :=
 namespace ModEq
 
 lemma of_dvd (d : m ∣ n) (h : a ≡ b [MOD n]) : a ≡ b [MOD m] :=
-  modEq_of_dvd <| d.natCast.trans h.dvd
+  modEq_of_dvd <| Int.ofNat_dvd.mpr d |>.trans h.dvd
 
 protected theorem mul_left' (c : ℕ) (h : a ≡ b [MOD n]) : c * a ≡ c * b [MOD c * n] := by
   unfold ModEq at *; rw [mul_mod_mul_left, mul_mod_mul_left, h]
@@ -122,7 +121,7 @@ protected theorem pow (m : ℕ) (h : a ≡ b [MOD n]) : a ^ m ≡ b ^ m [MOD n]
 @[gcongr]
 protected theorem add (h₁ : a ≡ b [MOD n]) (h₂ : c ≡ d [MOD n]) : a + c ≡ b + d [MOD n] := by
   rw [modEq_iff_dvd, Int.ofNat_add, Int.ofNat_add, add_sub_add_comm]
-  exact dvd_add h₁.dvd h₂.dvd
+  exact Int.dvd_add h₁.dvd h₂.dvd
 
 @[gcongr]
 protected theorem add_left (c : ℕ) (h : a ≡ b [MOD n]) : c + a ≡ c + b [MOD n] :=
@@ -136,7 +135,7 @@ protected theorem add_left_cancel (h₁ : a ≡ b [MOD n]) (h₂ : a + c ≡ b +
     c ≡ d [MOD n] := by
   simp only [modEq_iff_dvd, Int.ofNat_add] at *
   rw [add_sub_add_comm] at h₂
-  convert _root_.dvd_sub h₂ h₁ using 1
+  convert Int.dvd_sub h₂ h₁ using 1
   rw [add_sub_cancel_left]
 
 protected theorem add_left_cancel' (c : ℕ) (h : c + a ≡ c + b [MOD n]) : a ≡ b [MOD n] :=
@@ -155,7 +154,8 @@ protected theorem add_right_cancel' (c : ℕ) (h : a + c ≡ b + c [MOD n]) : a
 For cancelling left multiplication in the modulus, see `Nat.ModEq.of_mul_left`. -/
 protected theorem mul_left_cancel' {a b c m : ℕ} (hc : c ≠ 0) :
     c * a ≡ c * b [MOD c * m] → a ≡ b [MOD m] := by
-  simp [modEq_iff_dvd, ← mul_sub, mul_dvd_mul_iff_left (by simp [hc] : (c : ℤ) ≠ 0)]
+  simp only [modEq_iff_dvd, Int.natCast_mul, ← Int.mul_sub]
+  exact fun h => (Int.dvd_of_mul_dvd_mul_left (Int.ofNat_ne_zero.mpr hc) h)
 
 protected theorem mul_left_cancel_iff' {a b c m : ℕ} (hc : c ≠ 0) :
     c * a ≡ c * b [MOD c * m] ↔ a ≡ b [MOD m] :=
@@ -166,7 +166,8 @@ protected theorem mul_left_cancel_iff' {a b c m : ℕ} (hc : c ≠ 0) :
 For cancelling right multiplication in the modulus, see `Nat.ModEq.of_mul_right`. -/
 protected theorem mul_right_cancel' {a b c m : ℕ} (hc : c ≠ 0) :
     a * c ≡ b * c [MOD m * c] → a ≡ b [MOD m] := by
-  simp [modEq_iff_dvd, ← sub_mul, mul_dvd_mul_iff_right (by simp [hc] : (c : ℤ) ≠ 0)]
+  simp only [modEq_iff_dvd, Int.natCast_mul, ← Int.sub_mul]
+  exact fun h => (Int.dvd_of_mul_dvd_mul_right (Int.ofNat_ne_zero.mpr hc) h)
 
 protected theorem mul_right_cancel_iff' {a b c m : ℕ} (hc : c ≠ 0) :
     a * c ≡ b * c [MOD m * c] ↔ a ≡ b [MOD m] :=
@@ -204,10 +205,10 @@ namespace ModEq
 theorem le_of_lt_add (h1 : a ≡ b [MOD m]) (h2 : a < b + m) : a ≤ b :=
   (le_total a b).elim id fun h3 =>
     Nat.le_of_sub_eq_zero
-      (eq_zero_of_dvd_of_lt ((modEq_iff_dvd' h3).mp h1.symm) ((tsub_lt_iff_left h3).mpr h2))
+      (eq_zero_of_dvd_of_lt ((modEq_iff_dvd' h3).mp h1.symm) (by omega))
 
 theorem add_le_of_lt (h1 : a ≡ b [MOD m]) (h2 : a < b) : a + m ≤ b :=
-  le_of_lt_add (add_modEq_right.trans h1) (add_lt_add_right h2 m)
+  le_of_lt_add (add_modEq_right.trans h1) (by omega)
 
 theorem dvd_iff (h : a ≡ b [MOD m]) (hdm : d ∣ m) : d ∣ a ↔ d ∣ b := by
   simp only [← modEq_zero_iff_dvd]
@@ -227,9 +228,7 @@ lemma eq_of_abs_lt (h : a ≡ b [MOD m]) (h2 : |(b : ℤ) - a| < m) : a = b := b
   exact Int.eq_zero_of_abs_lt_dvd h.dvd h2
 
 lemma eq_of_lt_of_lt (h : a ≡ b [MOD m]) (ha : a < m) (hb : b < m) : a = b :=
-  h.eq_of_abs_lt <| abs_sub_lt_iff.2
-    ⟨(sub_le_self _ <| Int.natCast_nonneg _).trans_lt <| Int.ofNat_lt.2 hb,
-    (sub_le_self _ <| Int.natCast_nonneg _).trans_lt <| Int.ofNat_lt.2 ha⟩
+  h.eq_of_abs_lt <| Int.abs_sub_lt_of_lt_lt ha hb
 
 /-- To cancel a common factor `c` from a `ModEq` we must divide the modulus `m` by `gcd m c` -/
 lemma cancel_left_div_gcd (hm : 0 < m) (h : c * a ≡ c * b [MOD m]) :  a ≡ b [MOD m / gcd m c] := by
@@ -241,7 +240,7 @@ lemma cancel_left_div_gcd (hm : 0 < m) (h : c * a ≡ c * b [MOD m]) :  a ≡ b
   · show (m / d : ℤ) ∣ c / d * (b - a)
     rw [mul_comm, ← Int.mul_ediv_assoc (b - a) (Int.natCast_dvd_natCast.mpr hcd), mul_comm]
     apply Int.ediv_dvd_ediv (Int.natCast_dvd_natCast.mpr hmd)
-    rw [mul_sub]
+    rw [Int.mul_sub]
     exact modEq_iff_dvd.mp h
   · show Int.gcd (m / d) (c / d) = 1
     simp only [← Int.natCast_div, Int.gcd_natCast_natCast (m / d) (c / d), gcd_div hmd hcd,
@@ -299,18 +298,18 @@ def chineseRemainder' (h : a ≡ b [MOD gcd n m]) : { k // k ≡ a [MOD n] ∧ k
         have hcoedvd : ∀ t, (gcd n m : ℤ) ∣ t * (b - a) := fun t => h.dvd.mul_left _
         have := gcd_eq_gcd_ab n m
         constructor <;> rw [Int.emod_def, ← sub_add] <;>
-            refine dvd_add ?_ (dvd_mul_of_dvd_left ?_ _) <;>
+            refine Int.dvd_add ?_ (dvd_mul_of_dvd_left ?_ _) <;>
           try norm_cast
         · rw [← sub_eq_iff_eq_add'] at this
-          rw [← this, sub_mul, ← add_sub_assoc, add_comm, add_sub_assoc, ← mul_sub,
+          rw [← this, Int.sub_mul, ← add_sub_assoc, add_comm, add_sub_assoc, ← Int.mul_sub,
             Int.add_ediv_of_dvd_left, Int.mul_ediv_cancel_left _ hnonzero,
-            Int.mul_ediv_assoc _ h.dvd, ← sub_sub, sub_self, zero_sub, dvd_neg, mul_assoc]
+            Int.mul_ediv_assoc _ h.dvd, ← sub_sub, sub_self, zero_sub, Int.dvd_neg, mul_assoc]
           · exact dvd_mul_right _ _
           norm_cast
           exact dvd_mul_right _ _
         · exact dvd_lcm_left n m
         · rw [← sub_eq_iff_eq_add] at this
-          rw [← this, sub_mul, sub_add, ← mul_sub, Int.sub_ediv_of_dvd,
+          rw [← this, Int.sub_mul, sub_add, ← Int.mul_sub, Int.sub_ediv_of_dvd,
             Int.mul_ediv_cancel_left _ hnonzero, Int.mul_ediv_assoc _ h.dvd, ← sub_add, sub_self,
             zero_add, mul_assoc]
           · exact dvd_mul_right _ _
@@ -407,7 +406,7 @@ protected theorem add_div_of_dvd_right {a b c : ℕ} (hca : c ∣ a) : (a + b) /
     add_div_eq_of_add_mod_lt
       (by
         rw [Nat.mod_eq_zero_of_dvd hca, zero_add]
-        exact Nat.mod_lt _ (pos_iff_ne_zero.mpr h))
+        exact Nat.mod_lt _ (zero_lt_of_ne_zero h))
 
 protected theorem add_div_of_dvd_left {a b c : ℕ} (hca : c ∣ b) : (a + b) / c = a / c + b / c := by
   rwa [add_comm, Nat.add_div_of_dvd_right, add_comm]
@@ -430,27 +429,24 @@ theorem odd_mul_odd {n m : ℕ} : n % 2 = 1 → m % 2 = 1 → n * m % 2 = 1 := b
 
 theorem odd_mul_odd_div_two {m n : ℕ} (hm1 : m % 2 = 1) (hn1 : n % 2 = 1) :
     m * n / 2 = m * (n / 2) + m / 2 :=
-  have hm0 : 0 < m := Nat.pos_of_ne_zero fun h => by simp_all
   have hn0 : 0 < n := Nat.pos_of_ne_zero fun h => by simp_all
   mul_right_injective₀ two_ne_zero <| by
     dsimp
     rw [mul_add, two_mul_odd_div_two hm1, mul_left_comm, two_mul_odd_div_two hn1,
-      two_mul_odd_div_two (Nat.odd_mul_odd hm1 hn1), mul_tsub, mul_one, ←
-      add_tsub_assoc_of_le (succ_le_of_lt hm0),
-      tsub_add_cancel_of_le (le_mul_of_one_le_right (Nat.zero_le _) hn0)]
+      two_mul_odd_div_two (Nat.odd_mul_odd hm1 hn1), Nat.mul_sub, mul_one, ←
+      Nat.add_sub_assoc (by omega), Nat.sub_add_cancel (Nat.le_mul_of_pos_right m hn0)]
 
 theorem odd_of_mod_four_eq_one {n : ℕ} : n % 4 = 1 → n % 2 = 1 := by
-  simpa [ModEq, show 2 * 2 = 4 by norm_num] using @ModEq.of_mul_left 2 n 1 2
+  simpa [ModEq] using @ModEq.of_mul_left 2 n 1 2
 
 theorem odd_of_mod_four_eq_three {n : ℕ} : n % 4 = 3 → n % 2 = 1 := by
-  simpa [ModEq, show 2 * 2 = 4 by norm_num, show 3 % 4 = 3 by norm_num] using
-    @ModEq.of_mul_left 2 n 3 2
+  simpa [ModEq] using @ModEq.of_mul_left 2 n 3 2
 
 /-- A natural number is odd iff it has residue `1` or `3` mod `4`-/
 theorem odd_mod_four_iff {n : ℕ} : n % 2 = 1 ↔ n % 4 = 1 ∨ n % 4 = 3 :=
   have help : ∀ m : ℕ, m < 4 → m % 2 = 1 → m = 1 ∨ m = 3 := by decide
   ⟨fun hn =>
-    help (n % 4) (mod_lt n (by norm_num)) <| (mod_mod_of_dvd n (by decide : 2 ∣ 4)).trans hn,
+    help (n % 4) (mod_lt n (by omega)) <| (mod_mod_of_dvd n (by decide : 2 ∣ 4)).trans hn,
     fun h => Or.elim h odd_of_mod_four_eq_one odd_of_mod_four_eq_three⟩
 
 lemma mod_eq_of_modEq {a b n} (h : a ≡ b [MOD n]) (hb : b < n) : a % n = b :=
diff --git a/Mathlib/Data/Nat/Notation.lean b/Mathlib/Data/Nat/Notation.lean
index 582c017ea9161..729e9b67b0f52 100644
--- a/Mathlib/Data/Nat/Notation.lean
+++ b/Mathlib/Data/Nat/Notation.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Leonardo de Moura
 -/
 
-import Mathlib.Init
 /-!
 # Notation `ℕ` for the natural numbers.
 -/
diff --git a/Mathlib/Data/Nat/Nth.lean b/Mathlib/Data/Nat/Nth.lean
index e31e9bcb4390d..13b3543cb9adc 100644
--- a/Mathlib/Data/Nat/Nth.lean
+++ b/Mathlib/Data/Nat/Nth.lean
@@ -1,13 +1,14 @@
 /-
 Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
+Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
 -/
 import Mathlib.Data.List.GetD
 import Mathlib.Data.Nat.Count
 import Mathlib.Data.Nat.SuccPred
 import Mathlib.Order.Interval.Set.Monotone
 import Mathlib.Order.OrderIsoNat
+import Mathlib.Order.WellFounded
 
 /-!
 # The `n`th Number Satisfying a Predicate
@@ -237,7 +238,7 @@ theorem nth_eq_zero {n} :
     exacts [nth_zero_of_zero h₀, nth_of_card_le hf hle]
 
 theorem nth_eq_zero_mono (h₀ : ¬p 0) {a b : ℕ} (hab : a ≤ b) (ha : nth p a = 0) : nth p b = 0 := by
-  simp only [nth_eq_zero, h₀, false_and_iff, false_or_iff] at ha ⊢
+  simp only [nth_eq_zero, h₀, false_and, false_or] at ha ⊢
   exact ha.imp fun hf hle => hle.trans hab
 
 theorem le_nth_of_lt_nth_succ {k a : ℕ} (h : a < nth p (k + 1)) (ha : p a) : a ≤ nth p k := by
@@ -302,6 +303,10 @@ theorem count_nth_of_lt_card_finite {n : ℕ} (hp : (setOf p).Finite) (hlt : n <
 theorem count_nth_of_infinite (hp : (setOf p).Infinite) (n : ℕ) : count p (nth p n) = n :=
   count_nth fun hf => absurd hf hp
 
+theorem surjective_count_of_infinite_setOf (h : {n | p n}.Infinite) :
+    Function.Surjective (Nat.count p) :=
+  fun n => ⟨nth p n, count_nth_of_infinite h n⟩
+
 theorem count_nth_succ {n : ℕ} (hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card) :
     count p (nth p n + 1) = n + 1 := by rw [count_succ, count_nth hn, if_pos (nth_mem _ hn)]
 
diff --git a/Mathlib/Data/Nat/PartENat.lean b/Mathlib/Data/Nat/PartENat.lean
index 1111f4009e1b7..cc13db102cac5 100644
--- a/Mathlib/Data/Nat/PartENat.lean
+++ b/Mathlib/Data/Nat/PartENat.lean
@@ -88,8 +88,8 @@ instance addCommMonoid : AddCommMonoid PartENat where
   add := (· + ·)
   zero := 0
   add_comm x y := Part.ext' and_comm fun _ _ => add_comm _ _
-  zero_add x := Part.ext' (true_and_iff _) fun _ _ => zero_add _
-  add_zero x := Part.ext' (and_true_iff _) fun _ _ => add_zero _
+  zero_add x := Part.ext' (iff_of_eq (true_and _)) fun _ _ => zero_add _
+  add_zero x := Part.ext' (iff_of_eq (and_true _)) fun _ _ => add_zero _
   add_assoc x y z := Part.ext' and_assoc fun _ _ => add_assoc _ _ _
   nsmul := nsmulRec
 
@@ -98,7 +98,7 @@ instance : AddCommMonoidWithOne PartENat :=
     one := 1
     natCast := some
     natCast_zero := rfl
-    natCast_succ := fun _ => Part.ext' (true_and_iff _).symm fun _ _ => rfl }
+    natCast_succ := fun _ => Part.ext' (iff_of_eq (true_and _)).symm fun _ _ => rfl }
 
 theorem some_eq_natCast (n : ℕ) : some n = n :=
   rfl
@@ -157,7 +157,7 @@ protected theorem casesOn {P : PartENat → Prop} : ∀ a : PartENat, P ⊤ →
 
 -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later
 theorem top_add (x : PartENat) : ⊤ + x = ⊤ :=
-  Part.ext' (false_and_iff _) fun h => h.left.elim
+  Part.ext' (iff_of_eq (false_and _)) fun h => h.left.elim
 
 -- not a simp lemma as we will provide a `LinearOrderedAddCommMonoidWithTop` instance later
 theorem add_top (x : PartENat) : x + ⊤ = ⊤ := by rw [add_comm, top_add]
@@ -379,7 +379,7 @@ theorem eq_top_iff_forall_le (x : PartENat) : x = ⊤ ↔ ∀ n : ℕ, (n : Part
 
 theorem pos_iff_one_le {x : PartENat} : 0 < x ↔ 1 ≤ x :=
   PartENat.casesOn x
-    (by simp only [iff_true_iff, le_top, natCast_lt_top, ← @Nat.cast_zero PartENat])
+    (by simp only [le_top, natCast_lt_top, ← @Nat.cast_zero PartENat])
     fun n => by
       rw [← Nat.cast_zero, ← Nat.cast_one, PartENat.coe_lt_coe, PartENat.coe_le_coe]
       rfl
diff --git a/Mathlib/Data/Nat/Periodic.lean b/Mathlib/Data/Nat/Periodic.lean
index 5836ce4f952c9..1e2e335bec976 100644
--- a/Mathlib/Data/Nat/Periodic.lean
+++ b/Mathlib/Data/Nat/Periodic.lean
@@ -24,7 +24,7 @@ theorem periodic_gcd (a : ℕ) : Periodic (gcd a) a := by
   simp only [forall_const, gcd_add_self_right, eq_self_iff_true, Periodic]
 
 theorem periodic_coprime (a : ℕ) : Periodic (Coprime a) a := by
-  simp only [coprime_add_self_right, forall_const, iff_self_iff, eq_iff_iff, Periodic]
+  simp only [coprime_add_self_right, forall_const, eq_iff_iff, Periodic]
 
 theorem periodic_mod (a : ℕ) : Periodic (fun n => n % a) a := by
   simp only [forall_const, eq_self_iff_true, add_mod_right, Periodic]
diff --git a/Mathlib/Data/Nat/Prime/Basic.lean b/Mathlib/Data/Nat/Prime/Basic.lean
index e75aca751b05e..06a9ca4221721 100644
--- a/Mathlib/Data/Nat/Prime/Basic.lean
+++ b/Mathlib/Data/Nat/Prime/Basic.lean
@@ -156,7 +156,7 @@ theorem Prime.eq_one_of_pow {x n : ℕ} (h : (x ^ n).Prime) : n = 1 :=
 theorem Prime.pow_eq_iff {p a k : ℕ} (hp : p.Prime) : a ^ k = p ↔ a = p ∧ k = 1 := by
   refine ⟨fun h => ?_, fun h => by rw [h.1, h.2, pow_one]⟩
   rw [← h] at hp
-  rw [← h, hp.eq_one_of_pow, eq_self_iff_true, and_true_iff, pow_one]
+  rw [← h, hp.eq_one_of_pow, eq_self_iff_true, _root_.and_true, pow_one]
 
 theorem pow_minFac {n k : ℕ} (hk : k ≠ 0) : (n ^ k).minFac = n.minFac := by
   rcases eq_or_ne n 1 with (rfl | hn)
@@ -247,7 +247,7 @@ theorem ne_one_iff_exists_prime_dvd : ∀ {n}, n ≠ 1 ↔ ∃ p : ℕ, p.Prime
   | n + 2 => by
     let a := n + 2
     let ha : a ≠ 1 := Nat.succ_succ_ne_one n
-    simp only [true_iff_iff, Ne, not_false_iff, ha]
+    simp only [true_iff, Ne, not_false_iff, ha]
     exact ⟨a.minFac, Nat.minFac_prime ha, a.minFac_dvd⟩
 
 theorem eq_one_iff_not_exists_prime_dvd {n : ℕ} : n = 1 ↔ ∀ p : ℕ, p.Prime → ¬p ∣ n := by
@@ -287,37 +287,6 @@ lemma Prime.pow_inj {p q m n : ℕ} (hp : p.Prime) (hq : q.Prime)
     (Prime.dvd_of_dvd_pow hq <| h.symm ▸ dvd_pow_self q (succ_ne_zero n))
   exact ⟨H, succ_inj'.mp <| Nat.pow_right_injective hq.two_le (H ▸ h)⟩
 
-theorem exists_pow_lt_factorial (c : ℕ) : ∃ n0 > 1, ∀ n ≥ n0, c ^ n < (n - 1)! := by
-  refine ⟨2 * (c ^ 2 + 1), ?_, ?_⟩
-  · omega
-  intro n hn
-  obtain ⟨d, rfl⟩ := Nat.exists_eq_add_of_le hn
-  obtain (rfl | c0) := c.eq_zero_or_pos
-  · simp [Nat.factorial_pos]
-  refine (Nat.le_mul_of_pos_right _ (Nat.pow_pos (n := d) c0)).trans_lt ?_
-  convert_to (c ^ 2) ^ (c ^ 2 + d + 1) < (c ^ 2 + (c ^ 2 + d + 1))!
-  · rw [← pow_mul, ← pow_add]
-    congr 1
-    omega
-  · congr
-    omega
-  refine lt_of_lt_of_le ?_ Nat.factorial_mul_pow_le_factorial
-  rw [← one_mul (_ ^ _ : ℕ)]
-  exact Nat.mul_lt_mul_of_le_of_lt (Nat.one_le_of_lt (Nat.factorial_pos _))
-    (Nat.pow_lt_pow_left (Nat.lt_succ_self _) (Nat.succ_ne_zero _)) (Nat.factorial_pos _)
-
-theorem exists_mul_pow_lt_factorial (a : ℕ) (c : ℕ) : ∃ n0, ∀ n ≥ n0, a * c ^ n < (n - 1)! := by
-  obtain ⟨n0, hn, h⟩ := Nat.exists_pow_lt_factorial (a * c)
-  refine ⟨n0, fun n hn => lt_of_le_of_lt ?_ (h n hn)⟩
-  rw [mul_pow]
-  refine Nat.mul_le_mul_right _ (Nat.le_self_pow ?_ _)
-  omega
-
-theorem exists_prime_mul_pow_lt_factorial (n a c : ℕ) : ∃ p > n, p.Prime ∧ a * c ^ p < (p - 1)! :=
-  have ⟨n0, h⟩ := Nat.exists_mul_pow_lt_factorial a c
-  have ⟨p, hp, prime_p⟩ := (max (n + 1) n0).exists_infinite_primes
-  ⟨p, (le_max_left _ _).trans hp, prime_p, h _ <| le_of_max_le_right hp⟩
-
 end Nat
 
 namespace Int
diff --git a/Mathlib/Data/Nat/Prime/Defs.lean b/Mathlib/Data/Nat/Prime/Defs.lean
index 8cff72a3c2aa9..a7b24cb865b8f 100644
--- a/Mathlib/Data/Nat/Prime/Defs.lean
+++ b/Mathlib/Data/Nat/Prime/Defs.lean
@@ -161,7 +161,7 @@ theorem Prime.not_dvd_one {p : ℕ} (pp : Prime p) : ¬p ∣ 1 :=
   Irreducible.not_dvd_one pp
 
 theorem prime_mul_iff {a b : ℕ} : Nat.Prime (a * b) ↔ a.Prime ∧ b = 1 ∨ b.Prime ∧ a = 1 := by
-  simp only [iff_self_iff, irreducible_mul_iff, ← irreducible_iff_nat_prime, Nat.isUnit_iff]
+  simp only [irreducible_mul_iff, ← irreducible_iff_nat_prime, Nat.isUnit_iff]
 
 theorem not_prime_mul {a b : ℕ} (a1 : a ≠ 1) (b1 : b ≠ 1) : ¬Prime (a * b) := by
   simp [prime_mul_iff, _root_.not_or, *]
@@ -337,7 +337,7 @@ theorem minFac_le_div {n : ℕ} (pos : 0 < n) (np : ¬Prime n) : minFac n ≤ n
   | ⟨0, h0⟩ => absurd pos <| by rw [h0, mul_zero]; decide
   | ⟨1, h1⟩ => by
     rw [mul_one] at h1
-    rw [prime_def_minFac, not_and_or, ← h1, eq_self_iff_true, _root_.not_true, or_false_iff,
+    rw [prime_def_minFac, not_and_or, ← h1, eq_self_iff_true, _root_.not_true, _root_.or_false,
       not_le] at np
     rw [le_antisymm (le_of_lt_succ np) (succ_le_of_lt pos), minFac_one, Nat.div_one]
   | ⟨x + 2, hx⟩ => by
diff --git a/Mathlib/Data/Nat/Squarefree.lean b/Mathlib/Data/Nat/Squarefree.lean
index 4f5bc89c79f72..d11d003343934 100644
--- a/Mathlib/Data/Nat/Squarefree.lean
+++ b/Mathlib/Data/Nat/Squarefree.lean
@@ -52,7 +52,7 @@ theorem _root_.Squarefree.natFactorization_le_one {n : ℕ} (p : ℕ) (hn : Squa
   rw [multiplicity.squarefree_iff_multiplicity_le_one] at hn
   by_cases hp : p.Prime
   · have := hn p
-    simp only [multiplicity_eq_factorization hp hn', Nat.isUnit_iff, hp.ne_one, or_false_iff]
+    simp only [multiplicity_eq_factorization hp hn', Nat.isUnit_iff, hp.ne_one, or_false]
       at this
     exact mod_cast this
   · rw [factorization_eq_zero_of_non_prime _ hp]
@@ -87,7 +87,7 @@ theorem Squarefree.ext_iff {n m : ℕ} (hn : Squarefree n) (hm : Squarefree m) :
     · rwa [h₂, eq_comm, ← h₁]
     · rw [h₂, h₃.resolve_left]
       rw [← h₁, h₂]
-      simp only [Nat.one_ne_zero, not_false_iff]
+      simp only [Nat.one_ne_zero, not_false_iff, reduceCtorEq]
   rw [factorization_eq_zero_of_non_prime _ hp, factorization_eq_zero_of_non_prime _ hp]
 
 theorem squarefree_pow_iff {n k : ℕ} (hn : n ≠ 1) (hk : k ≠ 0) :
diff --git a/Mathlib/Data/Nat/SuccPred.lean b/Mathlib/Data/Nat/SuccPred.lean
index ba369b1c0fb26..8f04e28fb1012 100644
--- a/Mathlib/Data/Nat/SuccPred.lean
+++ b/Mathlib/Data/Nat/SuccPred.lean
@@ -4,8 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
 import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Algebra.Order.SuccPred
 import Mathlib.Data.Fin.Basic
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 
 /-!
 # Successors and predecessors of naturals
@@ -23,6 +24,8 @@ variable {m n : ℕ}
 @[instance] abbrev instSuccOrder : SuccOrder ℕ :=
   SuccOrder.ofSuccLeIff succ Nat.succ_le
 
+instance instSuccAddOrder : SuccAddOrder ℕ := ⟨fun _ => rfl⟩
+
 -- so that Lean reads `Nat.pred` through `pred_order.pred`
 @[instance] abbrev instPredOrder : PredOrder ℕ where
   pred := pred
@@ -36,6 +39,8 @@ variable {m n : ℕ}
     · exact (a.not_lt_zero h).elim
     · exact le_of_succ_le_succ h
 
+instance instPredSubOrder : PredSubOrder ℕ := ⟨fun _ => rfl⟩
+
 @[simp]
 theorem succ_eq_succ : Order.succ = succ :=
   rfl
@@ -44,25 +49,22 @@ theorem succ_eq_succ : Order.succ = succ :=
 theorem pred_eq_pred : Order.pred = pred :=
   rfl
 
-theorem succ_iterate (a : ℕ) : ∀ n, succ^[n] a = a + n
-  | 0 => rfl
-  | n + 1 => by
-    rw [Function.iterate_succ', add_succ]
-    exact congr_arg _ (succ_iterate a n)
+protected theorem succ_iterate (a : ℕ) : ∀ n, succ^[n] a = a + n :=
+  Order.succ_iterate a
 
-theorem pred_iterate (a : ℕ) : ∀ n, pred^[n] a = a - n
+protected theorem pred_iterate (a : ℕ) : ∀ n, pred^[n] a = a - n
   | 0 => rfl
   | n + 1 => by
     rw [Function.iterate_succ', sub_succ]
-    exact congr_arg _ (pred_iterate a n)
+    exact congr_arg _ (Nat.pred_iterate a n)
 
 lemma le_succ_iff_eq_or_le : m ≤ n.succ ↔ m = n.succ ∨ m ≤ n := Order.le_succ_iff_eq_or_le
 
 instance : IsSuccArchimedean ℕ :=
-  ⟨fun {a} {b} h => ⟨b - a, by rw [succ_eq_succ, succ_iterate, add_tsub_cancel_of_le h]⟩⟩
+  ⟨fun {a} {b} h => ⟨b - a, by rw [succ_eq_succ, Nat.succ_iterate, add_tsub_cancel_of_le h]⟩⟩
 
 instance : IsPredArchimedean ℕ :=
-  ⟨fun {a} {b} h => ⟨b - a, by rw [pred_eq_pred, pred_iterate, tsub_tsub_cancel_of_le h]⟩⟩
+  ⟨fun {a} {b} h => ⟨b - a, by rw [pred_eq_pred, Nat.pred_iterate, tsub_tsub_cancel_of_le h]⟩⟩
 
 lemma forall_ne_zero_iff (P : ℕ → Prop) :
     (∀ i, i ≠ 0 → P i) ↔ (∀ i, P (i + 1)) :=
@@ -71,8 +73,9 @@ lemma forall_ne_zero_iff (P : ℕ → Prop) :
 /-! ### Covering relation -/
 
 
+@[deprecated Order.covBy_iff_add_one_eq (since := "2024-09-04")]
 protected theorem covBy_iff_succ_eq {m n : ℕ} : m ⋖ n ↔ m + 1 = n :=
-  succ_eq_iff_covBy.symm
+  covBy_iff_add_one_eq
 
 end Nat
 
diff --git a/Mathlib/Data/Nat/Totient.lean b/Mathlib/Data/Nat/Totient.lean
index 5602a10425923..ec08e85c219d9 100644
--- a/Mathlib/Data/Nat/Totient.lean
+++ b/Mathlib/Data/Nat/Totient.lean
@@ -229,7 +229,7 @@ theorem prime_iff_card_units (p : ℕ) [Fintype (ZMod p)ˣ] :
     p.Prime ↔ Fintype.card (ZMod p)ˣ = p - 1 := by
   cases' eq_zero_or_neZero p with hp hp
   · subst hp
-    simp only [ZMod, not_prime_zero, false_iff_iff, zero_tsub]
+    simp only [ZMod, not_prime_zero, false_iff, zero_tsub]
     -- the subst created a non-defeq but subsingleton instance diamond; resolve it
     suffices Fintype.card ℤˣ ≠ 0 by convert this
     simp
@@ -245,7 +245,7 @@ theorem totient_eq_one_iff : ∀ {n : ℕ}, n.totient = 1 ↔ n = 1 ∨ n = 2
   | 2 => by simp
   | n + 3 => by
     have : 3 ≤ n + 3 := le_add_self
-    simp only [succ_succ_ne_one, false_or_iff]
+    simp only [succ_succ_ne_one, false_or]
     exact ⟨fun h => not_even_one.elim <| h ▸ totient_even this, by rintro ⟨⟩⟩
 
 theorem dvd_two_of_totient_le_one {a : ℕ} (han : 0 < a) (ha : a.totient ≤ 1) : a ∣ 2 := by
diff --git a/Mathlib/Data/Nat/Upto.lean b/Mathlib/Data/Nat/Upto.lean
index 47b6802fa1c14..f62ea9e9062d1 100644
--- a/Mathlib/Data/Nat/Upto.lean
+++ b/Mathlib/Data/Nat/Upto.lean
@@ -28,7 +28,7 @@ no `j` less than `i` satisfies `p`. This is an initial segment of the
 natural numbers, up to and including the first value satisfying `p`.
 
 We will be particularly interested in the case where there exists a value
-satisfying `p`, because in this case the `>` relation is well-founded.  -/
+satisfying `p`, because in this case the `>` relation is well-founded. -/
 abbrev Upto (p : ℕ → Prop) : Type :=
   { i : ℕ // ∀ j < i, ¬p j }
 
diff --git a/Mathlib/Data/Num/Basic.lean b/Mathlib/Data/Num/Basic.lean
index 1353fdd100589..5e35ca5550ad8 100644
--- a/Mathlib/Data/Num/Basic.lean
+++ b/Mathlib/Data/Num/Basic.lean
@@ -169,33 +169,27 @@ section
 
 variable {α : Type*} [One α] [Add α]
 
-section deprecated
-set_option linter.deprecated false
-
 /-- `castPosNum` casts a `PosNum` into any type which has `1` and `+`. -/
-@[deprecated (since := "2022-11-18"), coe]
+@[coe]
 def castPosNum : PosNum → α
   | 1 => 1
   | PosNum.bit0 a => castPosNum a + castPosNum a
   | PosNum.bit1 a => castPosNum a + castPosNum a + 1
 
 /-- `castNum` casts a `Num` into any type which has `0`, `1` and `+`. -/
-@[deprecated (since := "2022-11-18"), coe]
+@[coe]
 def castNum [Zero α] : Num → α
   | 0 => 0
   | Num.pos p => castPosNum p
 
 -- see Note [coercion into rings]
-@[deprecated (since := "2023-03-31")] instance (priority := 900) posNumCoe : CoeHTCT PosNum α :=
+instance (priority := 900) posNumCoe : CoeHTCT PosNum α :=
   ⟨castPosNum⟩
 
 -- see Note [coercion into rings]
-@[deprecated (since := "2023-03-31")]
 instance (priority := 900) numNatCoe [Zero α] : CoeHTCT Num α :=
   ⟨castNum⟩
 
-end deprecated
-
 instance : Repr PosNum :=
   ⟨fun n _ => repr (n : ℕ)⟩
 
@@ -593,19 +587,17 @@ def gcd (a b : ZNum) : Num :=
 end ZNum
 
 section
-
-set_option linter.deprecated false
 variable {α : Type*} [Zero α] [One α] [Add α] [Neg α]
 
 /-- `castZNum` casts a `ZNum` into any type which has `0`, `1`, `+` and `neg` -/
-@[deprecated (since := "2022-11-18"), coe]
+@[coe]
 def castZNum : ZNum → α
   | 0 => 0
   | ZNum.pos p => p
   | ZNum.neg p => -p
 
 -- see Note [coercion into rings]
-@[deprecated (since := "2023-03-31")] instance (priority := 900) znumCoe : CoeHTCT ZNum α :=
+instance (priority := 900) znumCoe : CoeHTCT ZNum α :=
   ⟨castZNum⟩
 
 instance : Repr ZNum :=
diff --git a/Mathlib/Data/Num/Lemmas.lean b/Mathlib/Data/Num/Lemmas.lean
index c3669ef62772a..66c8cae9f974a 100644
--- a/Mathlib/Data/Num/Lemmas.lean
+++ b/Mathlib/Data/Num/Lemmas.lean
@@ -867,7 +867,7 @@ theorem castNum_testBit (m n) : testBit m n = Nat.testBit m n := by
     · rfl
     · rw [PosNum.cast_bit1, ← two_mul, ← congr_fun Nat.bit_true, Nat.testBit_bit_zero]
     · rw [PosNum.cast_bit0, ← two_mul, ← congr_fun Nat.bit_false, Nat.testBit_bit_zero]
-    · simp
+    · simp [Nat.testBit_add_one]
     · rw [PosNum.cast_bit1, ← two_mul, ← congr_fun Nat.bit_true, Nat.testBit_bit_succ, IH]
     · rw [PosNum.cast_bit0, ← two_mul, ← congr_fun Nat.bit_false, Nat.testBit_bit_succ, IH]
 
@@ -1576,3 +1576,5 @@ instance SNum.lt : LT SNum :=
 
 instance SNum.le : LE SNum :=
   ⟨fun a b => (a : ℤ) ≤ b⟩
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/Opposite.lean b/Mathlib/Data/Opposite.lean
index 9fedb11939f4b..c9f9e120603ca 100644
--- a/Mathlib/Data/Opposite.lean
+++ b/Mathlib/Data/Opposite.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Reid Barton, Simon Hudon, Kenny Lau
+Authors: Kim Morrison, Reid Barton, Simon Hudon, Kenny Lau
 -/
 import Mathlib.Logic.Equiv.Defs
 
@@ -30,7 +30,7 @@ variable (α : Sort u)
   both `unop (op X) = X` and `op (unop X) = X` are definitional equalities.
 
 -/
-structure Opposite :=
+structure Opposite where
   /-- The canonical map `α → αᵒᵖ`. -/
   op ::
   /-- The canonical map `αᵒᵖ → α`. -/
diff --git a/Mathlib/Data/Option/Basic.lean b/Mathlib/Data/Option/Basic.lean
index 533abe5270de7..b0bab527487d9 100644
--- a/Mathlib/Data/Option/Basic.lean
+++ b/Mathlib/Data/Option/Basic.lean
@@ -137,29 +137,12 @@ variable {p : α → Prop} (f : ∀ a : α, p a → β) (x : Option α)
 theorem pbind_eq_bind (f : α → Option β) (x : Option α) : (x.pbind fun a _ ↦ f a) = x.bind f := by
   cases x <;> simp only [pbind, none_bind', some_bind']
 
-theorem map_bind {α β γ} (f : β → γ) (x : Option α) (g : α → Option β) :
-    Option.map f (x >>= g) = x >>= fun a ↦ Option.map f (g a) := by
-  simp only [← map_eq_map, ← bind_pure_comp, LawfulMonad.bind_assoc]
-
 theorem map_bind' (f : β → γ) (x : Option α) (g : α → Option β) :
     Option.map f (x.bind g) = x.bind fun a ↦ Option.map f (g a) := by cases x <;> simp
 
-theorem map_pbind (f : β → γ) (x : Option α) (g : ∀ a, a ∈ x → Option β) :
-    Option.map f (x.pbind g) = x.pbind fun a H ↦ Option.map f (g a H) := by
-  cases x <;> simp only [pbind, map_none']
-
 theorem pbind_map (f : α → β) (x : Option α) (g : ∀ b : β, b ∈ x.map f → Option γ) :
     pbind (Option.map f x) g = x.pbind fun a h ↦ g (f a) (mem_map_of_mem _ h) := by cases x <;> rfl
 
-@[simp]
-theorem pmap_none (f : ∀ a : α, p a → β) {H} : pmap f (@none α) H = none :=
-  rfl
-
-@[simp]
-theorem pmap_some (f : ∀ a : α, p a → β) {x : α} (h : p x) :
-    pmap f (some x) = fun _ ↦ some (f x h) :=
-  rfl
-
 theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by
   rw [mem_def] at ha ⊢
   subst ha
@@ -194,40 +177,20 @@ theorem pbind_eq_none {f : ∀ a : α, a ∈ x → Option β}
     (h' : ∀ a (H : a ∈ x), f a H = none → x = none) : x.pbind f = none ↔ x = none := by
   cases x
   · simp
-  · simp only [pbind, iff_false]
+  · simp only [pbind, iff_false, reduceCtorEq]
     intro h
     cases h' _ rfl h
 
 theorem pbind_eq_some {f : ∀ a : α, a ∈ x → Option β} {y : β} :
     x.pbind f = some y ↔ ∃ (z : α) (H : z ∈ x), f z H = some y := by
   rcases x with (_|x)
-  · simp only [pbind, false_iff, not_exists]
-    intro z h
-    simp at h
+  · simp
   · simp only [pbind]
     refine ⟨fun h ↦ ⟨x, rfl, h⟩, ?_⟩
     rintro ⟨z, H, hz⟩
     simp only [mem_def, Option.some_inj] at H
     simpa [H] using hz
 
--- Porting note: Can't simp tag this anymore because `pmap` simplifies
--- @[simp]
-theorem pmap_eq_none_iff {h} : pmap f x h = none ↔ x = none := by cases x <;> simp
-
--- Porting note: Can't simp tag this anymore because `pmap` simplifies
--- @[simp]
-theorem pmap_eq_some_iff {hf} {y : β} :
-    pmap f x hf = some y ↔ ∃ (a : α) (H : x = some a), f a (hf a H) = y := by
-  rcases x with (_|x)
-  · simp only [not_mem_none, exists_false, pmap, not_false_iff, exists_prop_of_false]
-  · constructor
-    · intro h
-      simp only [pmap, Option.some_inj] at h
-      exact ⟨x, rfl, h⟩
-    · rintro ⟨a, H, rfl⟩
-      simp only [mem_def, Option.some_inj] at H
-      simp only [H, pmap]
-
 -- Porting note: Can't simp tag this anymore because `join` and `pmap` simplify
 -- @[simp]
 theorem join_pmap_eq_pmap_join {f : ∀ a, p a → β} {x : Option (Option α)} (H) :
@@ -253,11 +216,6 @@ theorem orElse_none' (x : Option α) : x.orElse (fun _ ↦ none) = x := by cases
 theorem exists_ne_none {p : Option α → Prop} : (∃ x ≠ none, p x) ↔ (∃ x : α, p x) := by
   simp only [← exists_prop, bex_ne_none]
 
-@[simp]
-theorem get_map (f : α → β) {o : Option α} (h : isSome (o.map f)) :
-    (o.map f).get h = f (o.get (by rwa [← isSome_map'])) := by
-  cases o <;> [simp at h; rfl]
-
 theorem iget_mem [Inhabited α] : ∀ {o : Option α}, isSome o → o.iget ∈ o
   | some _, _ => rfl
 
@@ -311,8 +269,8 @@ compile_inductive% Option
 theorem orElse_eq_some (o o' : Option α) (x : α) :
     (o <|> o') = some x ↔ o = some x ∨ o = none ∧ o' = some x := by
   cases o
-  · simp only [true_and, false_or, eq_self_iff_true, none_orElse]
-  · simp only [some_orElse, or_false, false_and]
+  · simp only [true_and, false_or, eq_self_iff_true, none_orElse, reduceCtorEq]
+  · simp only [some_orElse, or_false, false_and, reduceCtorEq]
 
 
 theorem orElse_eq_some' (o o' : Option α) (x : α) :
@@ -323,7 +281,7 @@ theorem orElse_eq_some' (o o' : Option α) (x : α) :
 theorem orElse_eq_none (o o' : Option α) : (o <|> o') = none ↔ o = none ∧ o' = none := by
   cases o
   · simp only [true_and, none_orElse, eq_self_iff_true]
-  · simp only [some_orElse, false_and]
+  · simp only [some_orElse, reduceCtorEq, false_and]
 
 @[simp]
 theorem orElse_eq_none' (o o' : Option α) : o.orElse (fun _ ↦ o') = none ↔ o = none ∧ o' = none :=
diff --git a/Mathlib/Data/Option/Defs.lean b/Mathlib/Data/Option/Defs.lean
index e140c43e996c9..c15bedf853135 100644
--- a/Mathlib/Data/Option/Defs.lean
+++ b/Mathlib/Data/Option/Defs.lean
@@ -75,10 +75,6 @@ abbrev iget [Inhabited α] : Option α → α
 theorem iget_some [Inhabited α] {a : α} : (some a).iget = a :=
   rfl
 
-@[simp]
-theorem mem_toList {a : α} {o : Option α} : a ∈ toList o ↔ a ∈ o := by
-  cases o <;> simp [toList, eq_comm]
-
 instance liftOrGet_isCommutative (f : α → α → α) [Std.Commutative f] :
     Std.Commutative (liftOrGet f) :=
   ⟨fun a b ↦ by cases a <;> cases b <;> simp [liftOrGet, Std.Commutative.comm]⟩
diff --git a/Mathlib/Data/Ordering/Basic.lean b/Mathlib/Data/Ordering/Basic.lean
index 47a83a56ad26f..45c2f5778c1d4 100644
--- a/Mathlib/Data/Ordering/Basic.lean
+++ b/Mathlib/Data/Ordering/Basic.lean
@@ -3,9 +3,10 @@ Copyright (c) 2016 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
+import Batteries.Tactic.Alias
+import Mathlib.Tactic.Lemma
+import Mathlib.Tactic.TypeStar
 
-
-import Mathlib.Init
 /-!
 # Helper definitions and instances for `Ordering`
 -/
@@ -16,18 +17,26 @@ deriving instance Repr for Ordering
 
 namespace Ordering
 
-/-- Combine two `Ordering`s lexicographically. -/
-@[inline]
-def orElse : Ordering → Ordering → Ordering
-  | lt, _ => lt
-  | eq, o => o
-  | gt, _ => gt
-
-/-- The relation corresponding to each `Ordering` constructor (e.g. `.lt.toProp a b` is `a < b`). -/
-def toRel {α : Type u} [LT α] : Ordering → α → α → Prop
-  | .lt => (· < ·)
-  | .eq => Eq
-  | .gt => (· > ·)
+variable {α : Type*}
+
+@[deprecated (since := "2024-09-13")] alias orElse := «then»
+
+/-- `Compares o a b` means that `a` and `b` have the ordering relation `o` between them, assuming
+that the relation `a < b` is defined. -/
+-- Porting note: we have removed `@[simp]` here in favour of separate simp lemmas,
+-- otherwise this definition will unfold to a match.
+def Compares [LT α] : Ordering → α → α → Prop
+  | lt, a, b => a < b
+  | eq, a, b => a = b
+  | gt, a, b => a > b
+
+@[deprecated (since := "2024-09-13")] alias toRel := Compares
+
+@[simp] lemma compares_lt [LT α] (a b : α) : Compares lt a b = (a < b) := rfl
+
+@[simp] lemma compares_eq [LT α] (a b : α) : Compares eq a b = (a = b) := rfl
+
+@[simp] lemma compares_gt [LT α] (a b : α) : Compares gt a b = (a > b) := rfl
 
 end Ordering
 
diff --git a/Mathlib/Data/Ordering/Lemmas.lean b/Mathlib/Data/Ordering/Lemmas.lean
index b2a248ef9a866..7b9714aa85ed5 100644
--- a/Mathlib/Data/Ordering/Lemmas.lean
+++ b/Mathlib/Data/Ordering/Lemmas.lean
@@ -39,12 +39,12 @@ attribute [local simp] cmpUsing
 
 @[simp]
 theorem cmpUsing_eq_lt (a b : α) : (cmpUsing lt a b = Ordering.lt) = lt a b := by
-  simp only [cmpUsing, Ordering.ite_eq_lt_distrib, ite_self, if_false_right, and_true]
+  simp only [cmpUsing, Ordering.ite_eq_lt_distrib, ite_self, if_false_right, and_true, reduceCtorEq]
 
 @[simp]
 theorem cmpUsing_eq_gt [IsStrictOrder α lt] (a b : α) : cmpUsing lt a b = Ordering.gt ↔ lt b a := by
   simp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right, and_true, if_false_left,
-    and_iff_right_iff_imp]
+    and_iff_right_iff_imp, reduceCtorEq]
   exact fun hba hab ↦ (irrefl a) (_root_.trans hab hba)
 
 @[simp]
diff --git a/Mathlib/Data/Ordmap/Ordnode.lean b/Mathlib/Data/Ordmap/Ordnode.lean
index 360bb33039b14..31245ed62ad9a 100644
--- a/Mathlib/Data/Ordmap/Ordnode.lean
+++ b/Mathlib/Data/Ordmap/Ordnode.lean
@@ -313,11 +313,11 @@ def All (P : α → Prop) : Ordnode α → Prop
   | node _ l x r => All P l ∧ P x ∧ All P r
 
 instance All.decidable {P : α → Prop} : (t : Ordnode α) → [DecidablePred P] → Decidable (All P t)
-  | nil => decidableTrue
-  | node _ l _ r =>
+  | nil => isTrue trivial
+  | node _ l m r =>
     have : Decidable (All P l) := All.decidable l
     have : Decidable (All P r) := All.decidable r
-    And.decidable
+    inferInstanceAs <| Decidable (All P l ∧ P m ∧ All P r)
 
 /-- O(n). Does any element of the map satisfy property `P`?
 
@@ -328,11 +328,11 @@ def Any (P : α → Prop) : Ordnode α → Prop
   | node _ l x r => Any P l ∨ P x ∨ Any P r
 
 instance Any.decidable {P : α → Prop} : (t : Ordnode α ) → [DecidablePred P] → Decidable (Any P t)
-  | nil => decidableFalse
-  | node _ l _ r =>
+  | nil => isFalse id
+  | node _ l m r =>
     have : Decidable (Any P l) := Any.decidable l
     have : Decidable (Any P r) := Any.decidable r
-    Or.decidable
+    inferInstanceAs <| Decidable (Any P l ∨ P m ∨ Any P r)
 
 /-- O(n). Exact membership in the set. This is useful primarily for stating
 correctness properties; use `∈` for a version that actually uses the BST property
@@ -604,7 +604,8 @@ instance [Std.ToFormat α] : Std.ToFormat (Ordnode α) where
 def Equiv (t₁ t₂ : Ordnode α) : Prop :=
   t₁.size = t₂.size ∧ t₁.toList = t₂.toList
 
-instance [DecidableEq α] : DecidableRel (@Equiv α) := fun _ _ => And.decidable
+instance [DecidableEq α] : DecidableRel (@Equiv α) := fun x y =>
+  inferInstanceAs (Decidable (x.size = y.size ∧ x.toList = y.toList))
 
 /-- O(2^n). Constructs the powerset of a given set, that is, the set of all subsets.
 
@@ -845,7 +846,7 @@ def find (x : α) : Ordnode α → Option α
     | Ordering.gt => find x r
 
 instance : Membership α (Ordnode α) :=
-  ⟨fun x t => t.mem x⟩
+  ⟨fun t x => t.mem x⟩
 
 instance mem.decidable (x : α) (t : Ordnode α) : Decidable (x ∈ t) :=
   Bool.decEq _ _
diff --git a/Mathlib/Data/Ordmap/Ordset.lean b/Mathlib/Data/Ordmap/Ordset.lean
index fc25de56b3862..06b25c58fc953 100644
--- a/Mathlib/Data/Ordmap/Ordset.lean
+++ b/Mathlib/Data/Ordmap/Ordset.lean
@@ -151,7 +151,7 @@ and nothing on the other. -/
 def BalancedSz (l r : ℕ) : Prop :=
   l + r ≤ 1 ∨ l ≤ delta * r ∧ r ≤ delta * l
 
-instance BalancedSz.dec : DecidableRel BalancedSz := fun _ _ => Or.decidable
+instance BalancedSz.dec : DecidableRel BalancedSz := fun _ _ => inferInstanceAs (Decidable (_ ∨ _))
 
 /-- The `Balanced t` asserts that the tree `t` satisfies the balance invariants
 (at every level). -/
@@ -378,7 +378,7 @@ theorem Sized.rotateR_size {l x r} (hl : Sized l) :
   rw [← size_dual, dual_rotateR, hl.dual.rotateL_size, size_dual, size_dual, add_comm (size l)]
 
 theorem Sized.balance' {l x r} (hl : @Sized α l) (hr : Sized r) : Sized (balance' l x r) := by
-  unfold balance'; split_ifs
+  unfold Ordnode.balance'; split_ifs
   · exact hl.node' hr
   · exact hl.rotateL hr
   · exact hl.rotateR hr
@@ -1258,7 +1258,7 @@ theorem Valid'.glue_aux {l r o₁ o₂} (hl : Valid' o₁ l o₂) (hr : Valid' o
       suffices H : _ by
         refine ⟨Valid'.balanceL (hl.of_lt ?_ ?_) v H, ?_⟩
         · refine @findMin'_all (P := fun a : α => Bounded nil o₁ (a : WithBot α))
-            rl rx (sep.2.1.1.imp ?_) hr.1.1.to_nil
+            _ rl rx (sep.2.1.1.imp ?_) hr.1.1.to_nil
           exact fun y h => hl.1.1.to_nil.mono_right (le_of_lt h)
         · exact
             @findMin'_all _ (fun a => All (· < a) (.node ls ll lx lr)) rl rx
@@ -1556,7 +1556,7 @@ def find (x : α) (s : Ordset α) : Option α :=
   Ordnode.find x s.val
 
 instance instMembership : Membership α (Ordset α) :=
-  ⟨fun x s => mem x s⟩
+  ⟨fun s x => mem x s⟩
 
 instance mem.decidable (x : α) (s : Ordset α) : Decidable (x ∈ s) :=
   instDecidableEqBool _ _
@@ -1578,3 +1578,5 @@ def map {β} [Preorder β] (f : α → β) (f_strict_mono : StrictMono f) (s : O
   ⟨Ordnode.map f s.val, Ordnode.map.valid f_strict_mono s.property⟩
 
 end Ordset
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/PEquiv.lean b/Mathlib/Data/PEquiv.lean
index 046339273cdd6..920084d283b41 100644
--- a/Mathlib/Data/PEquiv.lean
+++ b/Mathlib/Data/PEquiv.lean
@@ -201,7 +201,7 @@ theorem mem_ofSet_iff {s : Set α} [DecidablePred (· ∈ s)] {a b : α} :
   · simp only [mem_def, eq_comm, some.injEq, iff_self_and]
     rintro rfl
     exact h
-  · simp only [mem_def, false_iff, not_and]
+  · simp only [mem_def, false_iff, not_and, reduceCtorEq]
     rintro rfl
     exact h
 
@@ -294,9 +294,9 @@ def single (a : α) (b : β) :
     dsimp only
     split_ifs with h1 h2
     · simp [*]
-    · simp only [mem_def, some.injEq, iff_false] at *
+    · simp only [mem_def, some.injEq, iff_false, reduceCtorEq] at *
       exact Ne.symm h2
-    · simp only [mem_def, some.injEq, false_iff] at *
+    · simp only [mem_def, some.injEq, false_iff, reduceCtorEq] at *
       exact Ne.symm h1
     · simp
 
@@ -343,7 +343,7 @@ theorem trans_single_of_eq_none {b : β} (c : γ) {f : δ ≃. β} (h : f.symm b
   ext
   simp only [eq_none_iff_forall_not_mem, Option.mem_def, f.eq_some_iff] at h
   dsimp [PEquiv.trans, single]
-  simp only [mem_def, bind_eq_some, iff_false, not_exists, not_and]
+  simp only [mem_def, bind_eq_some, iff_false, not_exists, not_and, reduceCtorEq]
   intros
   split_ifs <;> simp_all
 
@@ -390,7 +390,7 @@ instance [DecidableEq α] [DecidableEq β] : SemilatticeInf (α ≃. β) :=
           · contrapose! h2
             rw [h2]
             rw [← h1, hf, h2] at hg
-            simp only [mem_def, true_iff_iff, eq_self_iff_true] at hg
+            simp only [mem_def, true_iff, eq_self_iff_true] at hg
             rw [hg]
           · contrapose! h1
             rw [h1] at hf h2
diff --git a/Mathlib/Data/PFun.lean b/Mathlib/Data/PFun.lean
index 3e342591b8463..376018c747f3e 100644
--- a/Mathlib/Data/PFun.lean
+++ b/Mathlib/Data/PFun.lean
@@ -58,7 +58,7 @@ open Function
 def PFun (α β : Type*) :=
   α → Part β
 
-/-- `α →. β` is notation for the type `PFun α β` of partial functions from `α` to `β`.  -/
+/-- `α →. β` is notation for the type `PFun α β` of partial functions from `α` to `β`. -/
 infixr:25 " →. " => PFun
 
 namespace PFun
@@ -286,7 +286,7 @@ def fixInduction {C : α → Sort*} {f : α →. β ⊕ α} {b : β} {a : α} (h
   have h₂ := (Part.mem_assert_iff.1 h).snd
   generalize_proofs at h₂
   clear h
-  induction' ‹Acc _ _› with a ha IH
+  induction ‹Acc _ _› with | intro a ha IH => _
   have h : b ∈ f.fix a := Part.mem_assert_iff.2 ⟨⟨a, ha⟩, h₂⟩
   exact H a h fun a' fa' => IH a' fa' (Part.mem_assert_iff.1 (fix_fwd h fa')).snd
 
@@ -545,7 +545,7 @@ theorem mem_prodLift {f : α →. β} {g : α →. γ} {x : α} {y : β × γ} :
   trans ∃ hp hq, (f x).get hp = y.1 ∧ (g x).get hq = y.2
   · simp only [prodLift, Part.mem_mk_iff, And.exists, Prod.ext_iff]
   -- Porting note: was just `[exists_and_left, exists_and_right]`
-  · simp only [exists_and_left, exists_and_right, (· ∈ ·), Part.Mem]
+  · simp only [exists_and_left, exists_and_right, Membership.mem, Part.Mem]
 
 /-- Product of partial functions. -/
 def prodMap (f : α →. γ) (g : β →. δ) : α × β →. γ × δ := fun x =>
@@ -569,7 +569,7 @@ theorem mem_prodMap {f : α →. γ} {g : β →. δ} {x : α × β} {y : γ ×
     y ∈ f.prodMap g x ↔ y.1 ∈ f x.1 ∧ y.2 ∈ g x.2 := by
   trans ∃ hp hq, (f x.1).get hp = y.1 ∧ (g x.2).get hq = y.2
   · simp only [prodMap, Part.mem_mk_iff, And.exists, Prod.ext_iff]
-  · simp only [exists_and_left, exists_and_right, (· ∈ ·), Part.Mem]
+  · simp only [exists_and_left, exists_and_right, Membership.mem, Part.Mem]
 
 @[simp]
 theorem prodLift_fst_comp_snd_comp (f : α →. γ) (g : β →. δ) :
diff --git a/Mathlib/Data/PFunctor/Multivariate/M.lean b/Mathlib/Data/PFunctor/Multivariate/M.lean
index 8a40846b7c6fc..cb6ed15fd11ac 100644
--- a/Mathlib/Data/PFunctor/Multivariate/M.lean
+++ b/Mathlib/Data/PFunctor/Multivariate/M.lean
@@ -58,7 +58,7 @@ variable {n : ℕ} (P : MvPFunctor.{u} (n + 1))
 
 /-- A path from the root of a tree to one of its node -/
 inductive M.Path : P.last.M → Fin2 n → Type u
-  | root  (x : P.last.M)
+  | root (x : P.last.M)
           (a : P.A)
           (f : P.last.B a → P.last.M)
           (h : PFunctor.M.dest x = ⟨a, f⟩)
@@ -268,21 +268,21 @@ theorem M.bisim₀ {α : TypeVec n} (R : P.M α → P.M α → Prop) (h₀ : Equ
   intro i
   replace h₁ := congr_fun (congr_fun h₁ Fin2.fz) i
   simp only [TypeVec.comp, appendFun, splitFun] at h₁
-  replace h₁ := Quot.exact _ h₁
+  replace h₁ := Quot.eqvGen_exact h₁
   rw [h₀.eqvGen_iff] at h₁
   exact h₁
 
 theorem M.bisim' {α : TypeVec n} (R : P.M α → P.M α → Prop)
     (h : ∀ x y, R x y → (id ::: Quot.mk R) <$$> M.dest _ x = (id ::: Quot.mk R) <$$> M.dest _ y)
     (x y) (r : R x y) : x = y := by
-  have := M.bisim₀ P (EqvGen R) ?_ ?_
-  · solve_by_elim [EqvGen.rel]
-  · apply EqvGen.is_equivalence
+  have := M.bisim₀ P (Relation.EqvGen R) ?_ ?_
+  · solve_by_elim [Relation.EqvGen.rel]
+  · apply Relation.EqvGen.is_equivalence
   · clear r x y
     introv Hr
-    have : ∀ x y, R x y → EqvGen R x y := @EqvGen.rel _ R
+    have : ∀ x y, R x y → Relation.EqvGen R x y := @Relation.EqvGen.rel _ R
     induction Hr
-    · rw [← Quot.factor_mk_eq R (EqvGen R) this]
+    · rw [← Quot.factor_mk_eq R (Relation.EqvGen R) this]
       rwa [appendFun_comp_id, ← MvFunctor.map_map, ← MvFunctor.map_map, h]
     all_goals aesop
 
diff --git a/Mathlib/Data/PFunctor/Multivariate/W.lean b/Mathlib/Data/PFunctor/Multivariate/W.lean
index 449da825496bd..369a12fd42178 100644
--- a/Mathlib/Data/PFunctor/Multivariate/W.lean
+++ b/Mathlib/Data/PFunctor/Multivariate/W.lean
@@ -169,7 +169,7 @@ theorem wRec_eq {α : TypeVec n} {C : Type*}
     (g : ∀ a : P.A, P.drop.B a ⟹ α → (P.last.B a → P.W α) → (P.last.B a → C) → C) (a : P.A)
     (f' : P.drop.B a ⟹ α) (f : P.last.B a → P.W α) :
     P.wRec g (P.wMk a f' f) = g a f' f fun i => P.wRec g (f i) := by
-  rw [wMk, wRec]; dsimp; rw [wpRec_eq]
+  rw [wMk, wRec]; rw [wpRec_eq]
   dsimp only [wPathDestLeft_wPathCasesOn, wPathDestRight_wPathCasesOn]
   congr
 
diff --git a/Mathlib/Data/PFunctor/Univariate/M.lean b/Mathlib/Data/PFunctor/Univariate/M.lean
index f0339ae9be2d7..73fe251a4dda5 100644
--- a/Mathlib/Data/PFunctor/Univariate/M.lean
+++ b/Mathlib/Data/PFunctor/Univariate/M.lean
@@ -91,7 +91,7 @@ theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n))
   · rfl
   · -- cases' h with _ _ _ _ _ h₀ h₁
     cases h
-    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
+    simp only [truncate, Function.comp_def, eq_self_iff_true, heq_iff_eq]
     -- Porting note: used to be `ext y`
     rename_i n_ih a f y h₁
     suffices (fun x => truncate (y x)) = f
@@ -281,9 +281,7 @@ theorem mk_dest (x : M F) : M.mk (dest x) = x := by
   cases' h : x.approx (succ n) with _ hd ch
   have h' : hd = head' (x.approx 1) := by
     rw [← head_succ' n, h, head']
-    · split
-      injections
-    · apply x.consistent
+    apply x.consistent
   revert ch
   rw [h']
   intros ch h
@@ -481,8 +479,8 @@ theorem ext_aux [Inhabited (M F)] [DecidableEq F.A] {n : ℕ} (x y z : M F) (hx
     induction y using PFunctor.M.casesOn'
     simp only [iselect_nil] at hrec
     subst hrec
-    simp only [approx_mk, true_and_iff, eq_self_iff_true, heq_iff_eq, zero_eq, CofixA.intro.injEq,
-                heq_eq_eq, eq_iff_true_of_subsingleton, and_self]
+    simp only [approx_mk, eq_self_iff_true, heq_iff_eq, zero_eq, CofixA.intro.injEq,
+      heq_eq_eq, eq_iff_true_of_subsingleton, and_self]
   · cases hx
     cases hy
     induction x using PFunctor.M.casesOn'
@@ -490,7 +488,7 @@ theorem ext_aux [Inhabited (M F)] [DecidableEq F.A] {n : ℕ} (x y z : M F) (hx
     subst z
     iterate 3 (have := mk_inj ‹_›; cases this)
     rename_i n_ih a f₃ f₂ hAgree₂ _ _ h₂ _ _ f₁ h₁ hAgree₁ clr
-    simp only [approx_mk, true_and_iff, eq_self_iff_true, heq_iff_eq]
+    simp only [approx_mk, eq_self_iff_true, heq_iff_eq]
 
     have := mk_inj h₁
     cases this; clear h₁
diff --git a/Mathlib/Data/PNat/Defs.lean b/Mathlib/Data/PNat/Defs.lean
index 3004d33fc9df7..6eb644184ac56 100644
--- a/Mathlib/Data/PNat/Defs.lean
+++ b/Mathlib/Data/PNat/Defs.lean
@@ -3,7 +3,6 @@ Copyright (c) 2017 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Neil Strickland
 -/
-import Mathlib.Algebra.NeZero
 import Mathlib.Data.Nat.Defs
 import Mathlib.Order.Basic
 import Mathlib.Order.TypeTags
diff --git a/Mathlib/Data/PNat/Interval.lean b/Mathlib/Data/PNat/Interval.lean
index 4a310ffe979ce..f10b5ea85500c 100644
--- a/Mathlib/Data/PNat/Interval.lean
+++ b/Mathlib/Data/PNat/Interval.lean
@@ -53,35 +53,19 @@ theorem map_subtype_embedding_uIcc : (uIcc a b).map (Embedding.subtype _) = uIcc
 
 @[simp]
 theorem card_Icc : (Icc a b).card = b + 1 - a := by
-  rw [← Nat.card_Icc]
-  -- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-  -- https://github.com/leanprover-community/mathlib4/issues/5164
-  erw [← Finset.map_subtype_embedding_Icc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
-  rw [card_map]
+  rw [← Nat.card_Icc, ← map_subtype_embedding_Icc, card_map]
 
 @[simp]
 theorem card_Ico : (Ico a b).card = b - a := by
-  rw [← Nat.card_Ico]
-  -- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-  -- https://github.com/leanprover-community/mathlib4/issues/5164
-  erw [← Finset.map_subtype_embedding_Ico _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
-  rw [card_map]
+  rw [← Nat.card_Ico, ← map_subtype_embedding_Ico, card_map]
 
 @[simp]
 theorem card_Ioc : (Ioc a b).card = b - a := by
-  rw [← Nat.card_Ioc]
-  -- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-  -- https://github.com/leanprover-community/mathlib4/issues/5164
-  erw [← Finset.map_subtype_embedding_Ioc _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
-  rw [card_map]
+  rw [← Nat.card_Ioc, ← map_subtype_embedding_Ioc, card_map]
 
 @[simp]
 theorem card_Ioo : (Ioo a b).card = b - a - 1 := by
-  rw [← Nat.card_Ioo]
-  -- Porting note: I had to change this to `erw` *and* provide the proof, yuck.
-  -- https://github.com/leanprover-community/mathlib4/issues/5164
-  erw [← Finset.map_subtype_embedding_Ioo _ a b (fun c x _ hx _ hc _ => hc.trans_le hx)]
-  rw [card_map]
+  rw [← Nat.card_Ioo, ← map_subtype_embedding_Ioo, card_map]
 
 @[simp]
 theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 := by
diff --git a/Mathlib/Data/PNat/Xgcd.lean b/Mathlib/Data/PNat/Xgcd.lean
index 03c2d2cb0227d..f5c3a61a1fe5f 100644
--- a/Mathlib/Data/PNat/Xgcd.lean
+++ b/Mathlib/Data/PNat/Xgcd.lean
@@ -202,7 +202,7 @@ theorem flip_v : (flip u).v = u.v.swap := by
   · simp only
     ring
 
-/-- Properties of division with remainder for a / b.  -/
+/-- Properties of division with remainder for a / b. -/
 theorem rq_eq : u.r + (u.bp + 1) * u.q = u.ap + 1 :=
   Nat.mod_add_div (u.ap + 1) (u.bp + 1)
 
diff --git a/Mathlib/Data/PSigma/Order.lean b/Mathlib/Data/PSigma/Order.lean
index 097dd92225da5..fe0d6b8051e5b 100644
--- a/Mathlib/Data/PSigma/Order.lean
+++ b/Mathlib/Data/PSigma/Order.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Minchao Wu
+Authors: Kim Morrison, Minchao Wu
 -/
 import Mathlib.Data.Sigma.Lex
 import Mathlib.Order.BoundedOrder
diff --git a/Mathlib/Data/Part.lean b/Mathlib/Data/Part.lean
index cdbd018a1d088..c4903f9bf5b4a 100644
--- a/Mathlib/Data/Part.lean
+++ b/Mathlib/Data/Part.lean
@@ -81,7 +81,7 @@ theorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o
   | ⟨_, _⟩ => rfl
 
 /-- `a ∈ o` means that `o` is defined and equal to `a` -/
-protected def Mem (a : α) (o : Part α) : Prop :=
+protected def Mem (o : Part α) (a : α) : Prop :=
   ∃ h, o.get h = a
 
 instance : Membership α (Part α) :=
diff --git a/Mathlib/Data/Prod/Basic.lean b/Mathlib/Data/Prod/Basic.lean
index d070ee6df7f46..125f987461805 100644
--- a/Mathlib/Data/Prod/Basic.lean
+++ b/Mathlib/Data/Prod/Basic.lean
@@ -28,14 +28,6 @@ def mk.injArrow {x₁ : α} {y₁ : β} {x₂ : α} {y₂ : β} :
 theorem mk.eta : ∀ {p : α × β}, (p.1, p.2) = p
   | (_, _) => rfl
 
-@[simp]
-theorem «forall» {p : α × β → Prop} : (∀ x, p x) ↔ ∀ a b, p (a, b) :=
-  ⟨fun h a b ↦ h (a, b), fun h ⟨a, b⟩ ↦ h a b⟩
-
-@[simp]
-theorem «exists» {p : α × β → Prop} : (∃ x, p x) ↔ ∃ a b, p (a, b) :=
-  ⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩
-
 theorem forall' {p : α → β → Prop} : (∀ x : α × β, p x.1 x.2) ↔ ∀ a b, p a b :=
   Prod.forall
 
@@ -194,12 +186,12 @@ theorem snd_eq_iff : ∀ {p : α × β} {x : β}, p.2 = x ↔ p = (p.1, x)
 
 variable {r : α → α → Prop} {s : β → β → Prop} {x y : α × β}
 
-lemma lex_iff : Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2 := lex_def _ _
+lemma lex_iff : Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2 := lex_def
 
 instance Lex.decidable [DecidableEq α]
     (r : α → α → Prop) (s : β → β → Prop) [DecidableRel r] [DecidableRel s] :
     DecidableRel (Prod.Lex r s) :=
-  fun _ _ ↦ decidable_of_decidable_of_iff (lex_def r s).symm
+  fun _ _ ↦ decidable_of_decidable_of_iff lex_def.symm
 
 @[refl]
 theorem Lex.refl_left (r : α → α → Prop) (s : β → β → Prop) [IsRefl α r] : ∀ x, Prod.Lex r s x x
diff --git a/Mathlib/Data/Prod/Lex.lean b/Mathlib/Data/Prod/Lex.lean
index 6aa1e4d947f7b..b0271fecee021 100644
--- a/Mathlib/Data/Prod/Lex.lean
+++ b/Mathlib/Data/Prod/Lex.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Minchao Wu
+Authors: Kim Morrison, Minchao Wu
 -/
 import Mathlib.Order.BoundedOrder
 
@@ -29,30 +29,26 @@ Related files are:
 -/
 
 
-variable {α β γ : Type*}
+variable {α β : Type*}
 
 namespace Prod.Lex
 
-@[inherit_doc] notation:35 α " ×ₗ " β:34 => Lex (Prod α β)
-
-instance decidableEq (α β : Type*) [DecidableEq α] [DecidableEq β] : DecidableEq (α ×ₗ β) :=
-  instDecidableEqProd
+open Batteries
 
-instance inhabited (α β : Type*) [Inhabited α] [Inhabited β] : Inhabited (α ×ₗ β) :=
-  instInhabitedProd
+@[inherit_doc] notation:35 α " ×ₗ " β:34 => Lex (Prod α β)
 
-/-- Dictionary / lexicographic ordering on pairs.  -/
+/-- Dictionary / lexicographic ordering on pairs. -/
 instance instLE (α β : Type*) [LT α] [LE β] : LE (α ×ₗ β) where le := Prod.Lex (· < ·) (· ≤ ·)
 
 instance instLT (α β : Type*) [LT α] [LT β] : LT (α ×ₗ β) where lt := Prod.Lex (· < ·) (· < ·)
 
 theorem le_iff [LT α] [LE β] (a b : α × β) :
     toLex a ≤ toLex b ↔ a.1 < b.1 ∨ a.1 = b.1 ∧ a.2 ≤ b.2 :=
-  Prod.lex_def (· < ·) (· ≤ ·)
+  Prod.lex_def
 
 theorem lt_iff [LT α] [LT β] (a b : α × β) :
     toLex a < toLex b ↔ a.1 < b.1 ∨ a.1 = b.1 ∧ a.2 < b.2 :=
-  Prod.lex_def (· < ·) (· < ·)
+  Prod.lex_def
 
 example (x : α) (y : β) : toLex (x, y) = toLex (x, y) := rfl
 
@@ -124,16 +120,36 @@ instance partialOrder (α β : Type*) [PartialOrder α] [PartialOrder β] : Part
     haveI : IsAntisymm β (· ≤ ·) := ⟨fun _ _ => le_antisymm⟩
     exact antisymm (r := Prod.Lex _ _)
 
+instance instOrdLexProd [Ord α] [Ord β] : Ord (α ×ₗ β) := lexOrd
+
+theorem compare_def [Ord α] [Ord β] : @compare (α ×ₗ β) _ =
+    compareLex (compareOn fun x => (ofLex x).1) (compareOn fun x => (ofLex x).2) := rfl
+
+theorem _root_.lexOrd_eq [Ord α] [Ord β] : @lexOrd α β _ _ = instOrdLexProd := rfl
+
+theorem _root_.Ord.lex_eq [oα : Ord α] [oβ : Ord β] : Ord.lex oα oβ = instOrdLexProd := rfl
+
+instance [Ord α] [Ord β] [OrientedOrd α] [OrientedOrd β] : OrientedOrd (α ×ₗ β) :=
+  inferInstanceAs (OrientedCmp (compareLex _ _))
+
+instance [Ord α] [Ord β] [TransOrd α] [TransOrd β] : TransOrd (α ×ₗ β) :=
+  inferInstanceAs (TransCmp (compareLex _ _))
+
 /-- Dictionary / lexicographic linear order for pairs. -/
 instance linearOrder (α β : Type*) [LinearOrder α] [LinearOrder β] : LinearOrder (α ×ₗ β) :=
   { Prod.Lex.partialOrder α β with
-    le_total := total_of (Prod.Lex _ _),
-    decidableLE := Prod.Lex.decidable _ _,
-    decidableLT := Prod.Lex.decidable _ _,
-    decidableEq := Lex.decidableEq _ _, }
-
-instance [Ord α] [Ord β] : Ord (α ×ₗ β) where
-  compare := compareLex (compareOn (·.1)) (compareOn (·.2))
+    le_total := total_of (Prod.Lex _ _)
+    decidableLE := Prod.Lex.decidable _ _
+    decidableLT := Prod.Lex.decidable _ _
+    decidableEq := instDecidableEqLex _
+    compare_eq_compareOfLessAndEq := fun a b => by
+      have : DecidableRel (· < · : α ×ₗ β → α ×ₗ β → Prop) := Prod.Lex.decidable _ _
+      have : BEqOrd (α ×ₗ β) := ⟨by
+        simp [compare_def, compareLex, compareOn, Ordering.then_eq_eq, compare_eq_iff_eq]⟩
+      have : LTOrd (α ×ₗ β) := ⟨by
+        simp [compare_def, compareLex, compareOn, Ordering.then_eq_lt, lt_iff,
+          compare_lt_iff_lt, compare_eq_iff_eq]⟩
+      convert LTCmp.eq_compareOfLessAndEq (cmp := compare) a b }
 
 instance orderBot [PartialOrder α] [Preorder β] [OrderBot α] [OrderBot β] : OrderBot (α ×ₗ β) where
   bot := toLex ⊥
diff --git a/Mathlib/Data/QPF/Multivariate/Basic.lean b/Mathlib/Data/QPF/Multivariate/Basic.lean
index a0d0fd66952f8..9c13a82349a20 100644
--- a/Mathlib/Data/QPF/Multivariate/Basic.lean
+++ b/Mathlib/Data/QPF/Multivariate/Basic.lean
@@ -69,7 +69,7 @@ matched because they preserve the properties of QPF. The latter example,
 each proves that some operations on functors preserves the QPF structure
 -/
 
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 /-!
 ## Reference
 
diff --git a/Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean b/Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
index 7a6f0204617b0..284cbecc9b925 100644
--- a/Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
+++ b/Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
@@ -315,7 +315,7 @@ def Fix.drec {β : Fix F α → Type u}
       rhs
       rw [← ih]
     rw [MvFunctor.map_map, ← appendFun_comp, id_comp]
-    simp only [Function.comp]
+    simp only [Function.comp_def]
   cast (by rw [this]) y.2
 
 end MvQPF
diff --git a/Mathlib/Data/QPF/Univariate/Basic.lean b/Mathlib/Data/QPF/Univariate/Basic.lean
index 3f3dbf6911b6b..b767d995bd81b 100644
--- a/Mathlib/Data/QPF/Univariate/Basic.lean
+++ b/Mathlib/Data/QPF/Univariate/Basic.lean
@@ -176,7 +176,7 @@ theorem recF_eq_of_Wequiv {α : Type u} (u : F α → α) (x y : q.P.W) :
     Wequiv x y → recF u x = recF u y := by
   intro h
   induction h with
-  | ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp, ih]
+  | ind a f f' _ ih => simp only [recF_eq', PFunctor.map_eq, Function.comp_def, ih]
   | abs a f a' f' h => simp only [recF_eq', abs_map, h]
   | trans x y z _ _ ih₁ ih₂ => exact Eq.trans ih₁ ih₂
 
diff --git a/Mathlib/Data/Quot.lean b/Mathlib/Data/Quot.lean
index 45370bbed8764..6bee31e7d530c 100644
--- a/Mathlib/Data/Quot.lean
+++ b/Mathlib/Data/Quot.lean
@@ -3,8 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Mathlib.Init.Data.Quot
-import Mathlib.Logic.Relator
+import Mathlib.Logic.Relation
 import Mathlib.Logic.Unique
 import Mathlib.Util.Notation3
 
@@ -275,8 +274,8 @@ instance (q₁ : Quotient sa) (q₂ : Quotient sb) (f : α → β → Prop)
 end Quotient
 
 theorem Quot.eq {α : Type*} {r : α → α → Prop} {x y : α} :
-    Quot.mk r x = Quot.mk r y ↔ EqvGen r x y :=
-  ⟨Quot.exact r, Quot.EqvGen_sound⟩
+    Quot.mk r x = Quot.mk r y ↔ Relation.EqvGen r x y :=
+  ⟨Quot.eqvGen_exact, Quot.eqvGen_sound⟩
 
 @[simp]
 theorem Quotient.eq [r : Setoid α] {x y : α} : Quotient.mk r x = ⟦y⟧ ↔ x ≈ y :=
diff --git a/Mathlib/Data/Rat/Cast/CharZero.lean b/Mathlib/Data/Rat/Cast/CharZero.lean
index e24e6bf537738..f9e4f204e7d9b 100644
--- a/Mathlib/Data/Rat/Cast/CharZero.lean
+++ b/Mathlib/Data/Rat/Cast/CharZero.lean
@@ -73,8 +73,8 @@ variable [DivisionSemiring α] [CharZero α] {p q : ℚ≥0}
 lemma cast_injective : Injective ((↑) : ℚ≥0 → α) := by
   rintro p q hpq
   rw [NNRat.cast_def, NNRat.cast_def, Commute.div_eq_div_iff] at hpq
-  rw [← p.num_div_den, ← q.num_div_den, div_eq_div_iff]
-  norm_cast at hpq ⊢
+  on_goal 1 => rw [← p.num_div_den, ← q.num_div_den, div_eq_div_iff]
+  · norm_cast at hpq ⊢
   any_goals norm_cast
   any_goals apply den_ne_zero
   exact Nat.cast_commute ..
diff --git a/Mathlib/Data/Rat/Cast/Defs.lean b/Mathlib/Data/Rat/Cast/Defs.lean
index 643f4426c88c1..739dbc17cb0ce 100644
--- a/Mathlib/Data/Rat/Cast/Defs.lean
+++ b/Mathlib/Data/Rat/Cast/Defs.lean
@@ -19,10 +19,6 @@ import Mathlib.Data.Rat.Lemmas
 We define the canonical injection from ℚ into an arbitrary division ring and prove various
 casting lemmas showing the well-behavedness of this injection.
 
-## Notations
-
-- `/.` is infix notation for `Rat.divInt`.
-
 ## Tags
 
 rat, rationals, field, ℚ, numerator, denominator, num, denom, cast, coercion, casting
diff --git a/Mathlib/Data/Rat/Lemmas.lean b/Mathlib/Data/Rat/Lemmas.lean
index f684d39ce96de..0aa562b90bce1 100644
--- a/Mathlib/Data/Rat/Lemmas.lean
+++ b/Mathlib/Data/Rat/Lemmas.lean
@@ -53,7 +53,7 @@ theorem num_mk (n d : ℤ) : (n /. d).num = d.sign * n / n.gcd d := by
   have (m : ℕ) : Int.natAbs (m + 1) = m + 1 := by
     rw [← Nat.cast_one, ← Nat.cast_add, Int.natAbs_cast]
   rcases d with ((_ | _) | _) <;>
-  rw [← Int.div_eq_ediv_of_dvd] <;>
+  rw [← Int.tdiv_eq_ediv_of_dvd] <;>
   simp [divInt, mkRat, Rat.normalize, Nat.succPNat, Int.sign, Int.gcd,
     Int.zero_ediv, Int.ofNat_dvd_left, Nat.gcd_dvd_left, this]
 
@@ -195,7 +195,7 @@ theorem div_int_inj {a b c d : ℤ} (hb0 : 0 < b) (hd0 : 0 < d) (h1 : Nat.Coprim
 theorem intCast_div_self (n : ℤ) : ((n / n : ℤ) : ℚ) = n / n := by
   by_cases hn : n = 0
   · subst hn
-    simp only [Int.cast_zero, Int.zero_div, zero_div, Int.ediv_zero]
+    simp only [Int.cast_zero, Int.zero_tdiv, zero_div, Int.ediv_zero]
   · have : (n : ℚ) ≠ 0 := by rwa [← coe_int_inj] at hn
     simp only [Int.ediv_self hn, Int.cast_one, Ne, not_false_iff, div_self this]
 
@@ -247,9 +247,9 @@ theorem inv_intCast_num (a : ℤ) : (a : ℚ)⁻¹.num = Int.sign a := by
   rcases lt_trichotomy a 0 with lt | rfl | gt
   · obtain ⟨a, rfl⟩ : ∃ b, -b = a := ⟨-a, a.neg_neg⟩
     simp at lt
-    simp [Rat.inv_neg, inv_intCast_num_of_pos lt, (Int.sign_eq_one_iff_pos _).mpr lt]
-  · rfl
-  · simp [inv_intCast_num_of_pos gt, (Int.sign_eq_one_iff_pos _).mpr gt]
+    simp [Rat.inv_neg, inv_intCast_num_of_pos lt, Int.sign_eq_one_iff_pos.mpr lt]
+  · simp
+  · simp [inv_intCast_num_of_pos gt, Int.sign_eq_one_iff_pos.mpr gt]
 
 @[simp]
 theorem inv_natCast_num (a : ℕ) : (a : ℚ)⁻¹.num = Int.sign a :=
@@ -268,7 +268,7 @@ theorem inv_intCast_den (a : ℤ) : (a : ℚ)⁻¹.den = if a = 0 then 1 else a.
     rw [if_neg (by omega)]
     simp only [Int.cast_neg, Rat.inv_neg, neg_den, inv_intCast_den_of_pos lt, Int.natAbs_neg]
     exact Int.eq_natAbs_of_zero_le (by omega)
-  · rfl
+  · simp
   · rw [if_neg (by omega)]
     simp only [inv_intCast_den_of_pos gt]
     exact Int.eq_natAbs_of_zero_le (by omega)
diff --git a/Mathlib/Data/Rat/Star.lean b/Mathlib/Data/Rat/Star.lean
index 8a89379fa0770..8f4b38afcab0e 100644
--- a/Mathlib/Data/Rat/Star.lean
+++ b/Mathlib/Data/Rat/Star.lean
@@ -5,7 +5,7 @@ Authors: Jireh Loreaux, Yaël Dillies
 -/
 import Mathlib.Algebra.GroupWithZero.Commute
 import Mathlib.Algebra.Order.Ring.Abs
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Data.NNRat.Lemmas
 import Mathlib.Algebra.Order.Monoid.Submonoid
 import Mathlib.Tactic.FieldSimp
diff --git a/Mathlib/Data/Real/Archimedean.lean b/Mathlib/Data/Real/Archimedean.lean
index 320d601909bab..5c7543769551c 100644
--- a/Mathlib/Data/Real/Archimedean.lean
+++ b/Mathlib/Data/Real/Archimedean.lean
@@ -3,8 +3,8 @@ Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Floris van Doorn
 -/
-import Mathlib.Algebra.Bounds
 import Mathlib.Algebra.Order.Archimedean.Basic
+import Mathlib.Algebra.Order.Group.Pointwise.Bounds
 import Mathlib.Data.Real.Basic
 import Mathlib.Order.Interval.Set.Disjoint
 
@@ -17,6 +17,7 @@ open scoped Classical
 open Pointwise CauSeq
 
 namespace Real
+variable {ι : Sort*} {f : ι → ℝ} {s : Set ℝ} {a : ℝ}
 
 instance instArchimedean : Archimedean ℝ :=
   archimedean_iff_rat_le.2 fun x =>
@@ -49,9 +50,9 @@ theorem exists_floor (x : ℝ) : ∃ ub : ℤ, (ub : ℝ) ≤ x ∧ ∀ z : ℤ,
     (let ⟨n, hn⟩ := exists_int_lt x
     ⟨n, le_of_lt hn⟩)
 
-theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by
+theorem exists_isLUB (hne : s.Nonempty) (hbdd : BddAbove s) : ∃ x, IsLUB s x := by
   rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
-  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
+  have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ s, (m : ℝ) ≤ y * d } := by
     cases' exists_int_gt U with k hk
     refine fun d => ⟨k * d, fun z h => ?_⟩
     rcases h with ⟨y, yS, hy⟩
@@ -60,14 +61,14 @@ theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃
     exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg
   choose f hf using fun d : ℕ =>
     Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩
-  have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
+  have hf₁ : ∀ n > 0, ∃ y ∈ s, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>
     let ⟨y, yS, hy⟩ := (hf n).1
     ⟨y, yS, by simpa using (div_le_iff₀ (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩
-  have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
+  have hf₂ : ∀ n > 0, ∀ y ∈ s, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by
     intro n n0 y yS
     have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)
     simp only [Rat.cast_div, Rat.cast_intCast, Rat.cast_natCast, gt_iff_lt]
-    rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, inv_mul_cancel₀]
+    rwa [lt_div_iff₀ (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, inv_mul_cancel₀]
     exact ne_of_gt (Nat.cast_pos.2 n0)
   have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by
     intro ε ε0
@@ -81,7 +82,7 @@ theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃
     have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)
     have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)
     rcases hf₁ _ j0 with ⟨y, yS, hy⟩
-    refine lt_of_lt_of_le ((Rat.cast_lt (K := ℝ)).1 ?_) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)
+    refine lt_of_lt_of_le ((Rat.cast_lt (K := ℝ)).1 ?_) ((inv_le_comm₀ ε0 (Nat.cast_pos.2 k0)).1 ik)
     simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
   let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
   refine ⟨mk g, ⟨fun x xS => ?_, fun y h => ?_⟩⟩
@@ -92,7 +93,7 @@ theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃
     replace hK := hK.le.trans (Nat.cast_le.2 nK)
     have n0 : 0 < n := Nat.cast_pos.1 ((inv_pos.2 xz).trans_le hK)
     refine le_trans ?_ (hf₂ _ n0 _ xS).le
-    rwa [le_sub_comm, inv_le (Nat.cast_pos.2 n0 : (_ : ℝ) < _) xz]
+    rwa [le_sub_comm, inv_le_comm₀ (Nat.cast_pos.2 n0 : (_ : ℝ) < _) xz]
   · exact
       mk_le_of_forall_le
         ⟨1, fun n n1 =>
@@ -100,56 +101,52 @@ theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃
           le_trans hx (h xS)⟩
 
 /-- A nonempty, bounded below set of real numbers has a greatest lower bound. -/
-theorem exists_isGLB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddBelow S) : ∃ x, IsGLB S x := by
-  have hne' : (-S).Nonempty := Set.nonempty_neg.mpr hne
-  have hbdd' : BddAbove (-S) := bddAbove_neg.mpr hbdd
+theorem exists_isGLB (hne : s.Nonempty) (hbdd : BddBelow s) : ∃ x, IsGLB s x := by
+  have hne' : (-s).Nonempty := Set.nonempty_neg.mpr hne
+  have hbdd' : BddAbove (-s) := bddAbove_neg.mpr hbdd
   use -Classical.choose (Real.exists_isLUB hne' hbdd')
   rw [← isLUB_neg]
   exact Classical.choose_spec (Real.exists_isLUB hne' hbdd')
 
 noncomputable instance : SupSet ℝ :=
-  ⟨fun S => if h : S.Nonempty ∧ BddAbove S then Classical.choose (exists_isLUB h.1 h.2) else 0⟩
+  ⟨fun s => if h : s.Nonempty ∧ BddAbove s then Classical.choose (exists_isLUB h.1 h.2) else 0⟩
 
-theorem sSup_def (S : Set ℝ) :
-    sSup S = if h : S.Nonempty ∧ BddAbove S then Classical.choose (exists_isLUB h.1 h.2) else 0 :=
+theorem sSup_def (s : Set ℝ) :
+    sSup s = if h : s.Nonempty ∧ BddAbove s then Classical.choose (exists_isLUB h.1 h.2) else 0 :=
   rfl
 
-protected theorem isLUB_sSup (S : Set ℝ) (h₁ : S.Nonempty) (h₂ : BddAbove S) :
-    IsLUB S (sSup S) := by
+protected theorem isLUB_sSup (h₁ : s.Nonempty) (h₂ : BddAbove s) : IsLUB s (sSup s) := by
   simp only [sSup_def, dif_pos (And.intro h₁ h₂)]
   apply Classical.choose_spec
 
 noncomputable instance : InfSet ℝ :=
-  ⟨fun S => -sSup (-S)⟩
+  ⟨fun s => -sSup (-s)⟩
 
-theorem sInf_def (S : Set ℝ) : sInf S = -sSup (-S) :=
-  rfl
+theorem sInf_def (s : Set ℝ) : sInf s = -sSup (-s) := rfl
 
-protected theorem is_glb_sInf (S : Set ℝ) (h₁ : S.Nonempty) (h₂ : BddBelow S) :
-    IsGLB S (sInf S) := by
+protected theorem isGLB_sInf (h₁ : s.Nonempty) (h₂ : BddBelow s) : IsGLB s (sInf s) := by
   rw [sInf_def, ← isLUB_neg', neg_neg]
-  exact Real.isLUB_sSup _ h₁.neg h₂.neg
-
-noncomputable instance : ConditionallyCompleteLinearOrder ℝ :=
-  { Real.linearOrder, Real.lattice with
-    sSup := SupSet.sSup
-    sInf := InfSet.sInf
-    le_csSup := fun s a hs ha => (Real.isLUB_sSup s ⟨a, ha⟩ hs).1 ha
-    csSup_le := fun s a hs ha => (Real.isLUB_sSup s hs ⟨a, ha⟩).2 ha
-    csInf_le := fun s a hs ha => (Real.is_glb_sInf s ⟨a, ha⟩ hs).1 ha
-    le_csInf := fun s a hs ha => (Real.is_glb_sInf s hs ⟨a, ha⟩).2 ha
-    csSup_of_not_bddAbove := fun s hs ↦ by simp [hs, sSup_def]
-    csInf_of_not_bddBelow := fun s hs ↦ by simp [hs, sInf_def, sSup_def] }
-
-theorem lt_sInf_add_pos {s : Set ℝ} (h : s.Nonempty) {ε : ℝ} (hε : 0 < ε) :
-    ∃ a ∈ s, a < sInf s + ε :=
+  exact Real.isLUB_sSup h₁.neg h₂.neg
+
+@[deprecated (since := "2024-10-02")] alias is_glb_sInf := isGLB_sInf
+
+noncomputable instance : ConditionallyCompleteLinearOrder ℝ where
+  __ := Real.linearOrder
+  __ := Real.lattice
+  le_csSup s a hs ha := (Real.isLUB_sSup ⟨a, ha⟩ hs).1 ha
+  csSup_le s a hs ha := (Real.isLUB_sSup hs ⟨a, ha⟩).2 ha
+  csInf_le s a hs ha := (Real.isGLB_sInf ⟨a, ha⟩ hs).1 ha
+  le_csInf s a hs ha := (Real.isGLB_sInf hs ⟨a, ha⟩).2 ha
+  csSup_of_not_bddAbove s hs := by simp [hs, sSup_def]
+  csInf_of_not_bddBelow s hs := by simp [hs, sInf_def, sSup_def]
+
+theorem lt_sInf_add_pos (h : s.Nonempty) {ε : ℝ} (hε : 0 < ε) : ∃ a ∈ s, a < sInf s + ε :=
   exists_lt_of_csInf_lt h <| lt_add_of_pos_right _ hε
 
-theorem add_neg_lt_sSup {s : Set ℝ} (h : s.Nonempty) {ε : ℝ} (hε : ε < 0) :
-    ∃ a ∈ s, sSup s + ε < a :=
+theorem add_neg_lt_sSup (h : s.Nonempty) {ε : ℝ} (hε : ε < 0) : ∃ a ∈ s, sSup s + ε < a :=
   exists_lt_of_lt_csSup h <| add_lt_iff_neg_left.2 hε
 
-theorem sInf_le_iff {s : Set ℝ} (h : BddBelow s) (h' : s.Nonempty) {a : ℝ} :
+theorem sInf_le_iff (h : BddBelow s) (h' : s.Nonempty) :
     sInf s ≤ a ↔ ∀ ε, 0 < ε → ∃ x ∈ s, x < a + ε := by
   rw [le_iff_forall_pos_lt_add]
   constructor <;> intro H ε ε_pos
@@ -157,7 +154,7 @@ theorem sInf_le_iff {s : Set ℝ} (h : BddBelow s) (h' : s.Nonempty) {a : ℝ} :
   · rcases H ε ε_pos with ⟨x, x_in, hx⟩
     exact csInf_lt_of_lt h x_in hx
 
-theorem le_sSup_iff {s : Set ℝ} (h : BddAbove s) (h' : s.Nonempty) {a : ℝ} :
+theorem le_sSup_iff (h : BddAbove s) (h' : s.Nonempty) :
     a ≤ sSup s ↔ ∀ ε, ε < 0 → ∃ x ∈ s, a + ε < x := by
   rw [le_iff_forall_pos_lt_add]
   refine ⟨fun H ε ε_neg => ?_, fun H ε ε_pos => ?_⟩
@@ -169,102 +166,128 @@ theorem le_sSup_iff {s : Set ℝ} (h : BddAbove s) (h' : s.Nonempty) {a : ℝ} :
 theorem sSup_empty : sSup (∅ : Set ℝ) = 0 :=
   dif_neg <| by simp
 
-@[simp] lemma iSup_of_isEmpty {α : Sort*} [IsEmpty α] (f : α → ℝ) : ⨆ i, f i = 0 := by
+@[simp] lemma iSup_of_isEmpty [IsEmpty ι] (f : ι → ℝ) : ⨆ i, f i = 0 := by
   dsimp [iSup]
   convert Real.sSup_empty
   rw [Set.range_eq_empty_iff]
   infer_instance
 
 @[simp]
-theorem ciSup_const_zero {α : Sort*} : ⨆ _ : α, (0 : ℝ) = 0 := by
-  cases isEmpty_or_nonempty α
+theorem iSup_const_zero : ⨆ _ : ι, (0 : ℝ) = 0 := by
+  cases isEmpty_or_nonempty ι
   · exact Real.iSup_of_isEmpty _
   · exact ciSup_const
 
-theorem sSup_of_not_bddAbove {s : Set ℝ} (hs : ¬BddAbove s) : sSup s = 0 :=
-  dif_neg fun h => hs h.2
-
-theorem iSup_of_not_bddAbove {α : Sort*} {f : α → ℝ} (hf : ¬BddAbove (Set.range f)) :
-    ⨆ i, f i = 0 :=
-  sSup_of_not_bddAbove hf
+lemma sSup_of_not_bddAbove (hs : ¬BddAbove s) : sSup s = 0 := dif_neg fun h => hs h.2
+lemma iSup_of_not_bddAbove (hf : ¬BddAbove (Set.range f)) : ⨆ i, f i = 0 := sSup_of_not_bddAbove hf
 
 theorem sSup_univ : sSup (@Set.univ ℝ) = 0 := Real.sSup_of_not_bddAbove not_bddAbove_univ
 
 @[simp]
 theorem sInf_empty : sInf (∅ : Set ℝ) = 0 := by simp [sInf_def, sSup_empty]
 
-@[simp] nonrec lemma iInf_of_isEmpty {α : Sort*} [IsEmpty α] (f : α → ℝ) : ⨅ i, f i = 0 := by
+@[simp] nonrec lemma iInf_of_isEmpty [IsEmpty ι] (f : ι → ℝ) : ⨅ i, f i = 0 := by
   rw [iInf_of_isEmpty, sInf_empty]
 
 @[simp]
-theorem ciInf_const_zero {α : Sort*} : ⨅ _ : α, (0 : ℝ) = 0 := by
-  cases isEmpty_or_nonempty α
+theorem iInf_const_zero : ⨅ _ : ι, (0 : ℝ) = 0 := by
+  cases isEmpty_or_nonempty ι
   · exact Real.iInf_of_isEmpty _
   · exact ciInf_const
 
-theorem sInf_of_not_bddBelow {s : Set ℝ} (hs : ¬BddBelow s) : sInf s = 0 :=
+theorem sInf_of_not_bddBelow (hs : ¬BddBelow s) : sInf s = 0 :=
   neg_eq_zero.2 <| sSup_of_not_bddAbove <| mt bddAbove_neg.1 hs
 
-theorem iInf_of_not_bddBelow {α : Sort*} {f : α → ℝ} (hf : ¬BddBelow (Set.range f)) :
-    ⨅ i, f i = 0 :=
+theorem iInf_of_not_bddBelow (hf : ¬BddBelow (Set.range f)) : ⨅ i, f i = 0 :=
   sInf_of_not_bddBelow hf
 
-/--
-As `0` is the default value for `Real.sSup` of the empty set or sets which are not bounded above, it
-suffices to show that `S` is bounded below by `0` to show that `0 ≤ sSup S`.
--/
-theorem sSup_nonneg (S : Set ℝ) (hS : ∀ x ∈ S, (0 : ℝ) ≤ x) : 0 ≤ sSup S := by
-  rcases S.eq_empty_or_nonempty with (rfl | ⟨y, hy⟩)
-  · exact sSup_empty.ge
-  · apply dite _ (fun h => le_csSup_of_le h hy <| hS y hy) fun h => (sSup_of_not_bddAbove h).ge
+/-- As `sSup s = 0` when `s` is an empty set of reals, it suffices to show that all elements of `s`
+are at most some nonnegative number `a` to show that `sSup s ≤ a`.
 
-/--
-As `0` is the default value for `Real.sSup` of the empty set or sets which are not bounded above, it
-suffices to show that `f i` is nonnegative to show that `0 ≤ ⨆ i, f i`.
--/
-protected theorem iSup_nonneg {ι : Sort*} {f : ι → ℝ} (hf : ∀ i, 0 ≤ f i) : 0 ≤ ⨆ i, f i :=
-  sSup_nonneg _ <| Set.forall_mem_range.2 hf
+See also `csSup_le`. -/
+protected lemma sSup_le (hs : ∀ x ∈ s, x ≤ a) (ha : 0 ≤ a) : sSup s ≤ a := by
+  obtain rfl | hs' := s.eq_empty_or_nonempty
+  exacts [sSup_empty.trans_le ha, csSup_le hs' hs]
 
-/--
-As `0` is the default value for `Real.sSup` of the empty set or sets which are not bounded above, it
-suffices to show that all elements of `S` are bounded by a nonnegative number to show that `sSup S`
-is bounded by this number.
--/
-protected theorem sSup_le {S : Set ℝ} {a : ℝ} (hS : ∀ x ∈ S, x ≤ a) (ha : 0 ≤ a) : sSup S ≤ a := by
-  rcases S.eq_empty_or_nonempty with (rfl | hS₂)
-  exacts [sSup_empty.trans_le ha, csSup_le hS₂ hS]
+/-- As `⨆ i, f i = 0` when the domain of the real-valued function `f` is empty, it suffices to show
+that all values of `f` are at most some nonnegative number `a` to show that `⨆ i, f i ≤ a`.
 
-protected theorem iSup_le {ι : Sort*} {f : ι → ℝ} {a : ℝ} (hS : ∀ i, f i ≤ a) (ha : 0 ≤ a) :
-    ⨆ i, f i ≤ a :=
-  Real.sSup_le (Set.forall_mem_range.2 hS) ha
+See also `ciSup_le`. -/
+protected lemma iSup_le (hf : ∀ i, f i ≤ a) (ha : 0 ≤ a) : ⨆ i, f i ≤ a :=
+  Real.sSup_le (Set.forall_mem_range.2 hf) ha
 
-/-- As `0` is the default value for `Real.sSup` of the empty set, it suffices to show that `S` is
-bounded above by `0` to show that `sSup S ≤ 0`.
--/
-theorem sSup_nonpos (S : Set ℝ) (hS : ∀ x ∈ S, x ≤ (0 : ℝ)) : sSup S ≤ 0 :=
-  Real.sSup_le hS le_rfl
+/-- As `sInf s = 0` when `s` is an empty set of reals, it suffices to show that all elements of `s`
+are at least some nonpositive number `a` to show that `a ≤ sInf s`.
 
-/-- As `0` is the default value for `Real.sInf` of the empty set, it suffices to show that `S` is
-bounded below by `0` to show that `0 ≤ sInf S`.
--/
-theorem sInf_nonneg (S : Set ℝ) (hS : ∀ x ∈ S, (0 : ℝ) ≤ x) : 0 ≤ sInf S := by
-  rcases S.eq_empty_or_nonempty with (rfl | hS₂)
-  exacts [sInf_empty.ge, le_csInf hS₂ hS]
+See also `le_csInf`. -/
+protected lemma le_sInf (hs : ∀ x ∈ s, a ≤ x) (ha : a ≤ 0) : a ≤ sInf s := by
+  obtain rfl | hs' := s.eq_empty_or_nonempty
+  exacts [ha.trans_eq sInf_empty.symm, le_csInf hs' hs]
 
-/-- As `0` is the default value for `Real.sInf` of the empty set, it suffices to show that `f i` is
-bounded below by `0` to show that `0 ≤ iInf f`.
--/
-theorem iInf_nonneg {ι} {f : ι → ℝ} (hf : ∀ i, 0 ≤ f i) : 0 ≤ iInf f :=
-  sInf_nonneg _ <| Set.forall_mem_range.2 hf
+/-- As `⨅ i, f i = 0` when the domain of the real-valued function `f` is empty, it suffices to show
+that all values of `f` are at least some nonpositive number `a` to show that `a ≤ ⨅ i, f i`.
 
-/--
-As `0` is the default value for `Real.sInf` of the empty set or sets which are not bounded below, it
-suffices to show that `S` is bounded above by `0` to show that `sInf S ≤ 0`.
--/
-theorem sInf_nonpos (S : Set ℝ) (hS : ∀ x ∈ S, x ≤ (0 : ℝ)) : sInf S ≤ 0 := by
-  rcases S.eq_empty_or_nonempty with (rfl | ⟨y, hy⟩)
+See also `le_ciInf`. -/
+protected lemma le_iInf (hf : ∀ i, a ≤ f i) (ha : a ≤ 0) : a ≤ ⨅ i, f i :=
+  Real.le_sInf (Set.forall_mem_range.2 hf) ha
+
+/-- As `sSup s = 0` when `s` is an empty set of reals, it suffices to show that all elements of `s`
+are nonpositive to show that `sSup s ≤ 0`. -/
+lemma sSup_nonpos (hs : ∀ x ∈ s, x ≤ 0) : sSup s ≤ 0 := Real.sSup_le hs le_rfl
+
+/-- As `⨆ i, f i = 0` when the domain of the real-valued function `f` is empty,
+it suffices to show that all values of `f` are nonpositive to show that `⨆ i, f i ≤ 0`. -/
+lemma iSup_nonpos (hf : ∀ i, f i ≤ 0) : ⨆ i, f i ≤ 0 := Real.iSup_le hf le_rfl
+
+/-- As `sInf s = 0` when `s` is an empty set of reals, it suffices to show that all elements of `s`
+are nonnegative to show that `0 ≤ sInf s`. -/
+lemma sInf_nonneg (hs : ∀ x ∈ s, 0 ≤ x) : 0 ≤ sInf s := Real.le_sInf hs le_rfl
+
+/-- As `⨅ i, f i = 0` when the domain of the real-valued function `f` is empty,
+it suffices to show that all values of `f` are nonnegative to show that `0 ≤ ⨅ i, f i`. -/
+lemma iInf_nonneg (hf : ∀ i, 0 ≤ f i) : 0 ≤ iInf f := Real.le_iInf hf le_rfl
+
+/-- As `sSup s = 0` when `s` is a set of reals that's unbounded above, it suffices to show that `s`
+contains a nonnegative element to show that `0 ≤ sSup s`. -/
+lemma sSup_nonneg' (hs : ∃ x ∈ s, 0 ≤ x) : 0 ≤ sSup s := by
+  obtain ⟨x, hxs, hx⟩ := hs
+  exact dite _ (fun h ↦ le_csSup_of_le h hxs hx) fun h ↦ (sSup_of_not_bddAbove h).ge
+
+/-- As `⨆ i, f i = 0` when the real-valued function `f` is unbounded above,
+it suffices to show that `f` takes a nonnegative value to show that `0 ≤ ⨆ i, f i`. -/
+lemma iSup_nonneg' (hf : ∃ i, 0 ≤ f i) : 0 ≤ ⨆ i, f i := sSup_nonneg' <| Set.exists_range_iff.2 hf
+
+/-- As `sInf s = 0` when `s` is a set of reals that's unbounded below, it suffices to show that `s`
+contains a nonpositive element to show that `sInf s ≤ 0`. -/
+lemma sInf_nonpos' (hs : ∃ x ∈ s, x ≤ 0) : sInf s ≤ 0 := by
+  obtain ⟨x, hxs, hx⟩ := hs
+  exact dite _ (fun h ↦ csInf_le_of_le h hxs hx) fun h ↦ (sInf_of_not_bddBelow h).le
+
+/-- As `⨅ i, f i = 0` when the real-valued function `f` is unbounded below,
+it suffices to show that `f` takes a nonpositive value to show that `0 ≤ ⨅ i, f i`. -/
+lemma iInf_nonpos' (hf : ∃ i, f i ≤ 0) : ⨅ i, f i ≤ 0 := sInf_nonpos' <| Set.exists_range_iff.2 hf
+
+/-- As `sSup s = 0` when `s` is a set of reals that's either empty or unbounded above,
+it suffices to show that all elements of `s` are nonnegative to show that `0 ≤ sSup s`. -/
+lemma sSup_nonneg (hs : ∀ x ∈ s, 0 ≤ x) : 0 ≤ sSup s := by
+  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
+  · exact sSup_empty.ge
+  · exact sSup_nonneg' ⟨x, hx, hs _ hx⟩
+
+/-- As `⨆ i, f i = 0` when the domain of the real-valued function `f` is empty or unbounded above,
+it suffices to show that all values of `f` are nonnegative to show that `0 ≤ ⨆ i, f i`. -/
+lemma iSup_nonneg (hf : ∀ i, 0 ≤ f i) : 0 ≤ ⨆ i, f i := sSup_nonneg <| Set.forall_mem_range.2 hf
+
+/-- As `sInf s = 0` when `s` is a set of reals that's either empty or unbounded below,
+it suffices to show that all elements of `s` are nonpositive to show that `sInf s ≤ 0`. -/
+lemma sInf_nonpos (hs : ∀ x ∈ s, x ≤ 0) : sInf s ≤ 0 := by
+  obtain rfl | ⟨x, hx⟩ := s.eq_empty_or_nonempty
   · exact sInf_empty.le
-  · apply dite _ (fun h => csInf_le_of_le h hy <| hS y hy) fun h => (sInf_of_not_bddBelow h).le
+  · exact sInf_nonpos' ⟨x, hx, hs _ hx⟩
+
+/-- As `⨅ i, f i = 0` when the domain of the real-valued function `f` is empty or unbounded below,
+it suffices to show that all values of `f` are nonpositive to show that `0 ≤ ⨅ i, f i`. -/
+lemma iInf_nonpos (hf : ∀ i, f i ≤ 0) : ⨅ i, f i ≤ 0 := sInf_nonpos <| Set.forall_mem_range.2 hf
 
 theorem sInf_le_sSup (s : Set ℝ) (h₁ : BddBelow s) (h₂ : BddAbove s) : sInf s ≤ sSup s := by
   rcases s.eq_empty_or_nonempty with (rfl | hne)
@@ -272,12 +295,12 @@ theorem sInf_le_sSup (s : Set ℝ) (h₁ : BddBelow s) (h₂ : BddAbove s) : sIn
   · exact csInf_le_csSup h₁ h₂ hne
 
 theorem cauSeq_converges (f : CauSeq ℝ abs) : ∃ x, f ≈ const abs x := by
-  let S := { x : ℝ | const abs x < f }
-  have lb : ∃ x, x ∈ S := exists_lt f
-  have ub' : ∀ x, f < const abs x → ∀ y ∈ S, y ≤ x := fun x h y yS =>
+  let s := {x : ℝ | const abs x < f}
+  have lb : ∃ x, x ∈ s := exists_lt f
+  have ub' : ∀ x, f < const abs x → ∀ y ∈ s, y ≤ x := fun x h y yS =>
     le_of_lt <| const_lt.1 <| CauSeq.lt_trans yS h
-  have ub : ∃ x, ∀ y ∈ S, y ≤ x := (exists_gt f).imp ub'
-  refine ⟨sSup S, ((lt_total _ _).resolve_left fun h => ?_).resolve_right fun h => ?_⟩
+  have ub : ∃ x, ∀ y ∈ s, y ≤ x := (exists_gt f).imp ub'
+  refine ⟨sSup s, ((lt_total _ _).resolve_left fun h => ?_).resolve_right fun h => ?_⟩
   · rcases h with ⟨ε, ε0, i, ih⟩
     refine (csSup_le lb (ub' _ ?_)).not_lt (sub_lt_self _ (half_pos ε0))
     refine ⟨_, half_pos ε0, i, fun j ij => ?_⟩
@@ -336,8 +359,7 @@ theorem iInter_Iic_rat : ⋂ r : ℚ, Iic (r : ℝ) = ∅ := by
   exact iInter_Iic_eq_empty_iff.mpr not_bddBelow_coe
 
 /-- Exponentiation is eventually larger than linear growth. -/
-lemma exists_natCast_add_one_lt_pow_of_one_lt {a : ℝ} (ha : 1 < a) :
-    ∃ m : ℕ, (m + 1 : ℝ) < a ^ m := by
+lemma exists_natCast_add_one_lt_pow_of_one_lt (ha : 1 < a) : ∃ m : ℕ, (m + 1 : ℝ) < a ^ m := by
   obtain ⟨k, posk, hk⟩ : ∃ k : ℕ, 0 < k ∧ 1 / k + 1 < a := by
     contrapose! ha
     refine le_of_forall_lt_rat_imp_le ?_
diff --git a/Mathlib/Data/Real/Basic.lean b/Mathlib/Data/Real/Basic.lean
index 8de982a5ba82f..7cec019cb7dbe 100644
--- a/Mathlib/Data/Real/Basic.lean
+++ b/Mathlib/Data/Real/Basic.lean
@@ -600,3 +600,10 @@ def IsNonarchimedean {A : Type*} [Add A] (f : A → ℝ) : Prop :=
 `f (r ^ n) = (f r) ^ n`. -/
 def IsPowMul {R : Type*} [Pow R ℕ] (f : R → ℝ) :=
   ∀ (a : R) {n : ℕ}, 1 ≤ n → f (a ^ n) = f a ^ n
+
+/-- A ring homomorphism `f : α →+* β` is bounded with respect to the functions `nα : α → ℝ` and
+  `nβ : β → ℝ` if there exists a positive constant `C` such that for all `x` in `α`,
+  `nβ (f x) ≤ C * nα x`. -/
+def RingHom.IsBoundedWrt {α : Type*} [Ring α] {β : Type*} [Ring β] (nα : α → ℝ) (nβ : β → ℝ)
+    (f : α →+* β) : Prop :=
+  ∃ C : ℝ, 0 < C ∧ ∀ x : α, nβ (f x) ≤ C * nα x
diff --git a/Mathlib/Data/Real/ConjExponents.lean b/Mathlib/Data/Real/ConjExponents.lean
index 5323170031fd4..04bfc20da0ec1 100644
--- a/Mathlib/Data/Real/ConjExponents.lean
+++ b/Mathlib/Data/Real/ConjExponents.lean
@@ -18,6 +18,8 @@ analysis, especially when dealing with `L^p` spaces.
 * `Real.conjExponent`: Conjugate exponent of a real number.
 * `NNReal.IsConjExponent`: Predicate for two nonnegative real numbers to be conjugate.
 * `NNReal.conjExponent`: Conjugate exponent of a nonnegative real number.
+* `ENNReal.IsConjExponent`: Predicate for two extended nonnegative real numbers to be conjugate.
+* `ENNReal.conjExponent`: Conjugate exponent of an extended nonnegative real number.
 
 ## TODO
 
@@ -27,7 +29,7 @@ analysis, especially when dealing with `L^p` spaces.
 
 noncomputable section
 
-open scoped ENNReal
+open scoped ENNReal NNReal
 
 namespace Real
 
@@ -106,7 +108,7 @@ theorem inv_add_inv_conj_ennreal : (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)
 end
 
 protected lemma inv_inv (ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a⁻¹.IsConjExponent b⁻¹ :=
-  ⟨one_lt_inv ha <| by linarith, by simpa only [inv_inv]⟩
+  ⟨(one_lt_inv₀ ha).2 <| by linarith, by simpa only [inv_inv]⟩
 
 lemma inv_one_sub_inv (ha₀ : 0 < a) (ha₁ : a < 1) : a⁻¹.IsConjExponent (1 - a)⁻¹ :=
   .inv_inv ha₀ (sub_pos_of_lt ha₁) <| add_tsub_cancel_of_le ha₁.le
@@ -116,6 +118,8 @@ lemma one_sub_inv_inv (ha₀ : 0 < a) (ha₁ : a < 1) : (1 - a)⁻¹.IsConjExpon
 
 end IsConjExponent
 
+lemma isConjExponent_comm : p.IsConjExponent q ↔ q.IsConjExponent p := ⟨.symm, .symm⟩
+
 lemma isConjExponent_iff_eq_conjExponent (hp : 1 < p) : p.IsConjExponent q ↔ q = p / (p - 1) :=
   ⟨IsConjExponent.conj_eq, fun h ↦ ⟨hp, by field_simp [h]⟩⟩
 
@@ -195,7 +199,7 @@ end
 
 protected lemma inv_inv (ha : a ≠ 0) (hb : b ≠ 0) (hab : a + b = 1) :
     a⁻¹.IsConjExponent b⁻¹ :=
-  ⟨one_lt_inv ha.bot_lt <| by rw [← hab]; exact lt_add_of_pos_right _ hb.bot_lt, by
+  ⟨(one_lt_inv₀ ha.bot_lt).2 <| by rw [← hab]; exact lt_add_of_pos_right _ hb.bot_lt, by
     simpa only [inv_inv] using hab⟩
 
 lemma inv_one_sub_inv (ha₀ : a ≠ 0) (ha₁ : a < 1) : a⁻¹.IsConjExponent (1 - a)⁻¹ :=
@@ -206,6 +210,8 @@ lemma one_sub_inv_inv (ha₀ : a ≠ 0) (ha₁ : a < 1) : (1 - a)⁻¹.IsConjExp
 
 end IsConjExponent
 
+lemma isConjExponent_comm : p.IsConjExponent q ↔ q.IsConjExponent p := ⟨.symm, .symm⟩
+
 lemma isConjExponent_iff_eq_conjExponent (h : 1 < p) : p.IsConjExponent q ↔ q = p / (p - 1) := by
   rw [← isConjExponent_coe, Real.isConjExponent_iff_eq_conjExponent (mod_cast h), ← coe_inj,
     NNReal.coe_div, NNReal.coe_sub h.le, coe_one]
@@ -220,3 +226,120 @@ protected lemma Real.IsConjExponent.toNNReal {p q : ℝ} (hpq : p.IsConjExponent
   one_lt := by simpa using hpq.one_lt
   inv_add_inv_conj := by rw [← toNNReal_inv, ← toNNReal_inv, ← toNNReal_add hpq.inv_nonneg
     hpq.symm.inv_nonneg, hpq.inv_add_inv_conj, toNNReal_one]
+
+namespace ENNReal
+
+/-- Two extended nonnegative real exponents `p, q` are conjugate and satisfy the equality
+`1/p + 1/q = 1`. This condition shows up in many theorems in analysis, notably related to `L^p`
+norms. Note that we permit one of the exponents to be `∞` and the other `1`. -/
+@[mk_iff]
+structure IsConjExponent (p q : ℝ≥0∞) : Prop where
+  inv_add_inv_conj : p⁻¹ + q⁻¹ = 1
+
+/-- The conjugate exponent of `p` is `q = 1 + (p - 1)⁻¹`, so that `1/p + 1/q = 1`. -/
+noncomputable def conjExponent (p : ℝ≥0∞) : ℝ≥0∞ := 1 + (p - 1)⁻¹
+
+lemma coe_conjExponent {p : ℝ≥0} (hp : 1 < p) : p.conjExponent = conjExponent p := by
+  rw [NNReal.conjExponent, conjExponent]
+  norm_cast
+  rw [← coe_inv (tsub_pos_of_lt hp).ne']
+  norm_cast
+  field_simp [(tsub_pos_of_lt hp).ne']
+  rw [tsub_add_cancel_of_le hp.le]
+
+variable {a b p q : ℝ≥0∞} (h : p.IsConjExponent q)
+
+@[simp, norm_cast] lemma isConjExponent_coe {p q : ℝ≥0} :
+    IsConjExponent p q ↔ p.IsConjExponent q := by
+  simp only [isConjExponent_iff, NNReal.isConjExponent_iff]
+  refine ⟨fun h ↦ ⟨?_, ?_⟩, ?_⟩
+  · simpa using (ENNReal.lt_add_right (fun hp ↦ by simp [hp] at h) <| by simp).trans_eq h
+  · rw [← coe_inv, ← coe_inv] at h
+    · norm_cast at h
+    all_goals rintro rfl; simp at h
+  · rintro ⟨hp, h⟩
+    rw [← coe_inv (zero_lt_one.trans hp).ne', ← coe_inv, ← coe_add, h, coe_one]
+    rintro rfl
+    simp [hp.ne'] at h
+
+alias ⟨_, _root_.NNReal.IsConjExponent.coe_ennreal⟩ := isConjExponent_coe
+
+namespace IsConjExponent
+
+protected lemma conjExponent (hp : 1 ≤ p) : p.IsConjExponent (conjExponent p) := by
+  have : p ≠ 0 := (zero_lt_one.trans_le hp).ne'
+  rw [isConjExponent_iff, conjExponent, add_comm]
+  refine (AddLECancellable.eq_tsub_iff_add_eq_of_le (α := ℝ≥0∞) (by simpa) (by simpa)).1 ?_
+  rw [inv_eq_iff_eq_inv]
+  obtain rfl | hp₁ := hp.eq_or_lt
+  · simp
+  obtain rfl | hp := eq_or_ne p ∞
+  · simp
+  calc
+    1 + (p - 1)⁻¹ = (p - 1 + 1) / (p - 1) := by
+      rw [ENNReal.add_div, ENNReal.div_self ((tsub_pos_of_lt hp₁).ne') (sub_ne_top hp), one_div]
+    _ = (1 - p⁻¹)⁻¹ := by
+      rw [tsub_add_cancel_of_le, ← inv_eq_iff_eq_inv, div_eq_mul_inv, ENNReal.mul_inv, inv_inv,
+        ENNReal.mul_sub, ENNReal.inv_mul_cancel, mul_one] <;> simp [*]
+
+section
+include h
+
+@[symm]
+protected lemma symm : q.IsConjExponent p where
+  inv_add_inv_conj := by simpa [add_comm] using h.inv_add_inv_conj
+
+lemma one_le : 1 ≤ p := ENNReal.inv_le_one.1 <| by
+  rw [← add_zero p⁻¹, ← h.inv_add_inv_conj]; gcongr; positivity
+
+lemma pos : 0 < p := zero_lt_one.trans_le h.one_le
+lemma ne_zero : p ≠ 0 := h.pos.ne'
+
+lemma one_sub_inv : 1 - p⁻¹ = q⁻¹ :=
+  ENNReal.sub_eq_of_eq_add_rev' one_ne_top h.inv_add_inv_conj.symm
+
+lemma conjExponent_eq : conjExponent p = q := by
+  have hp : 1 ≤ p := h.one_le
+  have : p⁻¹ ≠ ∞ := by simpa using h.ne_zero
+  simpa [ENNReal.add_right_inj, *] using
+    (IsConjExponent.conjExponent hp).inv_add_inv_conj.trans h.inv_add_inv_conj.symm
+
+lemma conj_eq : q = 1 + (p - 1)⁻¹ := h.conjExponent_eq.symm
+
+lemma mul_eq_add : p * q = p + q := by
+  obtain rfl | hp := eq_or_ne p ∞
+  · simp [h.symm.ne_zero]
+  obtain rfl | hq := eq_or_ne q ∞
+  · simp [h.ne_zero]
+  rw [← mul_one (_ * _), ← h.inv_add_inv_conj, mul_add, mul_right_comm,
+    ENNReal.mul_inv_cancel h.ne_zero hp, one_mul, mul_assoc,
+    ENNReal.mul_inv_cancel h.symm.ne_zero hq, mul_one, add_comm]
+
+lemma div_conj_eq_sub_one : p / q = p - 1 := by
+  obtain rfl | hq := eq_or_ne q ∞
+  · simp [h.symm.conj_eq]
+  refine ENNReal.eq_sub_of_add_eq one_ne_top ?_
+  rw [← ENNReal.div_self h.symm.ne_zero hq, ← ENNReal.add_div, ← h.mul_eq_add, mul_div_assoc,
+    ENNReal.div_self h.symm.ne_zero hq, mul_one]
+
+end
+
+protected lemma inv_inv (hab : a + b = 1) : a⁻¹.IsConjExponent b⁻¹ where
+  inv_add_inv_conj := by simpa only [inv_inv] using hab
+
+lemma inv_one_sub_inv (ha : a ≤ 1) : a⁻¹.IsConjExponent (1 - a)⁻¹ :=
+  .inv_inv <| add_tsub_cancel_of_le ha
+
+lemma one_sub_inv_inv (ha : a ≤ 1) : (1 - a)⁻¹.IsConjExponent a⁻¹ := (inv_one_sub_inv ha).symm
+
+lemma top_one : IsConjExponent ∞ 1 := ⟨by simp⟩
+lemma one_top : IsConjExponent 1 ∞ := ⟨by simp⟩
+
+end IsConjExponent
+
+lemma isConjExponent_comm : p.IsConjExponent q ↔ q.IsConjExponent p := ⟨.symm, .symm⟩
+
+lemma isConjExponent_iff_eq_conjExponent (hp : 1 ≤ p) : p.IsConjExponent q ↔ q = 1 + (p - 1)⁻¹ :=
+  ⟨fun h ↦ h.conj_eq, by rintro rfl; exact .conjExponent hp⟩
+
+end ENNReal
diff --git a/Mathlib/Data/Real/EReal.lean b/Mathlib/Data/Real/EReal.lean
index 3de259cbaf4d5..4ff49b8b9459b 100644
--- a/Mathlib/Data/Real/EReal.lean
+++ b/Mathlib/Data/Real/EReal.lean
@@ -929,6 +929,64 @@ lemma neg_sub {x y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊥) (h2 : x ≠ ⊤ ∨ y
     - (x - y) = - x + y := by
   rw [sub_eq_add_neg, neg_add _ _, sub_eq_add_neg, neg_neg] <;> simp_all
 
+/-! ### Addition and order -/
+
+lemma le_of_forall_lt_iff_le {x y : EReal} : (∀ z : ℝ, x < z → y ≤ z) ↔ y ≤ x := by
+  refine ⟨fun h ↦ WithBot.le_of_forall_lt_iff_le.1 ?_, fun h _ x_z ↦ h.trans x_z.le⟩
+  rw [WithTop.forall]
+  aesop
+
+lemma ge_of_forall_gt_iff_ge {x y : EReal} : (∀ z : ℝ, z < y → z ≤ x) ↔ y ≤ x := by
+  refine ⟨fun h ↦ WithBot.ge_of_forall_gt_iff_ge.1 ?_, fun h _ x_z ↦ x_z.le.trans h⟩
+  rw [WithTop.forall]
+  aesop
+
+/-- This lemma is superseded by `add_le_of_forall_add_le`. -/
+private lemma top_add_le_of_forall_add_le {a b : EReal} (h : ∀ c < ⊤, ∀ d < a, c + d ≤ b) :
+    ⊤ + a ≤ b := by
+  induction a with
+  | h_bot => exact add_bot ⊤ ▸ bot_le
+  | h_real a =>
+    refine top_add_coe a ▸ le_of_forall_lt_iff_le.1 fun c b_c ↦ ?_
+    specialize h (c - a + 1) (coe_lt_top (c - a + 1)) (a - 1)
+    rw [← coe_one, ← coe_sub, ← coe_sub, ← coe_add, ← coe_add, add_add_sub_cancel, sub_add_cancel,
+      EReal.coe_lt_coe_iff] at h
+    exact (not_le_of_lt b_c (h (sub_one_lt a))).rec
+  | h_top =>
+    refine top_add_top ▸ le_of_forall_lt_iff_le.1 fun c b_c ↦ ?_
+    specialize h c (coe_lt_top c) 0 zero_lt_top
+    rw [add_zero] at h
+    exact (not_le_of_lt b_c h).rec
+
+lemma add_le_of_forall_add_le {a b c : EReal} (h : ∀ d < a, ∀ e < b, d + e ≤ c) : a + b ≤ c := by
+  induction a with
+  | h_bot => exact bot_add b ▸ bot_le
+  | h_real a => induction b with
+    | h_bot => exact add_bot (a : EReal) ▸ bot_le
+    | h_real b =>
+      refine (@ge_of_forall_gt_iff_ge c (a+b)).1 fun d d_ab ↦ ?_
+      rw [← coe_add, EReal.coe_lt_coe_iff] at d_ab
+      rcases exists_between d_ab with ⟨e, e_d, e_ab⟩
+      have key₁ : (a + d - e : ℝ) < (a : EReal) := by apply EReal.coe_lt_coe_iff.2; linarith
+      have key₂ : (e - a : ℝ) < (b : EReal) := by apply EReal.coe_lt_coe_iff.2; linarith
+      apply le_of_eq_of_le _ (h (a + d - e) key₁ (e - a) key₂)
+      rw [← coe_add, ← coe_sub,  ← coe_sub, ← coe_add, sub_add_sub_cancel, add_sub_cancel_left]
+    | h_top =>
+      rw [add_comm (a : EReal) ⊤]
+      exact top_add_le_of_forall_add_le fun d d_top e e_a ↦ (add_comm d e ▸ h e e_a d d_top)
+  | h_top => exact top_add_le_of_forall_add_le h
+
+lemma le_add_of_forall_le_add {a b c : EReal} (h₁ : a ≠ ⊥ ∨ b ≠ ⊤) (h₂ : a ≠ ⊤ ∨ b ≠ ⊥)
+    (h : ∀ d > a, ∀ e > b, c ≤ d + e) :
+    c ≤ a + b := by
+  rw [← neg_le_neg_iff, neg_add h₁ h₂]
+  refine add_le_of_forall_add_le fun d d_a e e_b ↦ ?_
+  have h₃ : d ≠ ⊥ ∨ e ≠ ⊤ := Or.inr (ne_top_of_lt e_b)
+  have h₄ : d ≠ ⊤ ∨ e ≠ ⊥ := Or.inl (ne_top_of_lt d_a)
+  rw [← neg_neg d, neg_lt_iff_neg_lt, neg_neg a] at d_a
+  rw [← neg_neg e, neg_lt_iff_neg_lt, neg_neg b] at e_b
+  exact le_neg_of_le_neg <| neg_add h₃ h₄ ▸ h (- d) d_a (- e) e_b
+
 /-!
 ### Subtraction
 
@@ -1169,52 +1227,6 @@ lemma left_distrib_of_nonneg {a b c : EReal} (ha : 0 ≤ a) (hb : 0 ≤ b) :
   nth_rewrite 1 [EReal.mul_comm]; nth_rewrite 2 [EReal.mul_comm]; nth_rewrite 3 [EReal.mul_comm]
   exact right_distrib_of_nonneg ha hb
 
-lemma le_iff_le_forall_real_gt (x y : EReal) : (∀ z : ℝ, x < z → y ≤ z) ↔ y ≤ x := by
-  symm
-  refine ⟨fun h z x_lt_z ↦ le_trans h (le_of_lt x_lt_z), ?_⟩
-  intro h
-  induction x
-  · apply le_of_eq ((eq_bot_iff_forall_lt y).2 _)
-    intro z
-    specialize h (z-1) (bot_lt_coe (z-1))
-    apply lt_of_le_of_lt h
-    rw [EReal.coe_lt_coe_iff]
-    exact sub_one_lt z
-  · induction y
-    · exact bot_le
-    · norm_cast
-      norm_cast at h
-      by_contra x_lt_y
-      rcases exists_between (lt_of_not_le x_lt_y) with ⟨z, x_lt_z, z_lt_y⟩
-      specialize h z x_lt_z
-      exact not_le_of_lt z_lt_y h
-    · exfalso
-      specialize h (_+ 1) (EReal.coe_lt_coe_iff.2 (lt_add_one _))
-      exact not_le_of_lt (coe_lt_top (_ + 1)) h
-  · exact le_top
-
-lemma ge_iff_le_forall_real_lt (x y : EReal) : (∀ z : ℝ, z < y → z ≤ x) ↔ y ≤ x := by
-  refine ⟨fun h ↦ ?_, fun h z z_lt_y ↦ le_trans (le_of_lt z_lt_y) h⟩
-  induction x with
-  | h_bot =>
-    refine ((eq_bot_iff_forall_lt y).2 fun z ↦ ?_).le
-    refine lt_of_not_le fun z_le_y ↦ (not_le_of_lt (bot_lt_coe (z - 1)) (h (z - 1)
-      (lt_of_lt_of_le ?_ z_le_y)))
-    exact_mod_cast sub_one_lt z
-  | h_real x =>
-    induction y with
-    | h_bot => exact bot_le
-    | h_real y =>
-      norm_cast at h ⊢
-      by_contra! x_lt_y
-      rcases exists_between x_lt_y with ⟨z, x_lt_z, z_lt_y⟩
-      exact not_le_of_lt x_lt_z (h z z_lt_y)
-    | h_top =>
-      exfalso
-      norm_cast at h
-      exact not_le_of_lt (lt_add_one x) <| h (x + 1) (coe_lt_top (x + 1))
-  | h_top => exact le_top
-
 /-! ### Absolute value -/
 
 -- Porting note (#11215): TODO: use `Real.nnabs` for the case `(x : ℝ)`
@@ -1548,7 +1560,7 @@ lemma div_right_distrib_of_nonneg {a b c : EReal} (h : 0 ≤ a) (h' : 0 ≤ b) :
     (a + b) / c = (a / c) + (b / c) :=
   EReal.right_distrib_of_nonneg h h'
 
-/-! #### Division and Order s-/
+/-! #### Division and Order s -/
 
 lemma monotone_div_right_of_nonneg {b : EReal} (h : 0 ≤ b) : Monotone fun a ↦ a / b :=
   fun _ _ h' ↦ mul_le_mul_of_nonneg_right h' (inv_nonneg_of_nonneg h)
@@ -1668,3 +1680,5 @@ unsafe def positivity_coe_ennreal_ereal : expr → tactic strictness
 
 end Tactic
 -/
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Data/Real/GoldenRatio.lean b/Mathlib/Data/Real/GoldenRatio.lean
index 63905e911911d..42e9c2559ca88 100644
--- a/Mathlib/Data/Real/GoldenRatio.lean
+++ b/Mathlib/Data/Real/GoldenRatio.lean
@@ -95,7 +95,7 @@ theorem gold_ne_zero : φ ≠ 0 :=
 
 theorem one_lt_gold : 1 < φ := by
   refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
-  simp [← sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
+  simp [← sq, gold_pos, zero_lt_one]
 
 theorem gold_lt_two : φ < 2 := by calc
   (1 + sqrt 5) / 2 < (1 + 3) / 2 := by gcongr; rw [sqrt_lt'] <;> norm_num
@@ -109,7 +109,7 @@ theorem goldConj_ne_zero : ψ ≠ 0 :=
 
 theorem neg_one_lt_goldConj : -1 < ψ := by
   rw [neg_lt, ← inv_gold]
-  exact inv_lt_one one_lt_gold
+  exact inv_lt_one_of_one_lt₀ one_lt_gold
 
 /-!
 ## Irrationality
@@ -120,8 +120,7 @@ theorem neg_one_lt_goldConj : -1 < ψ := by
 theorem gold_irrational : Irrational φ := by
   have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
   have := this.rat_add 1
-  have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
-  convert this
+  convert this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
   norm_num
   field_simp
 
@@ -129,8 +128,7 @@ theorem gold_irrational : Irrational φ := by
 theorem goldConj_irrational : Irrational ψ := by
   have := Nat.Prime.irrational_sqrt (show Nat.Prime 5 by norm_num)
   have := this.rat_sub 1
-  have := this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
-  convert this
+  convert this.rat_mul (show (0.5 : ℚ) ≠ 0 by norm_num)
   norm_num
   field_simp
 
@@ -170,12 +168,12 @@ theorem fib_isSol_fibRec : fibRec.IsSolution (fun x => x.fib : ℕ → α) := by
 /-- The geometric sequence `fun n ↦ φ^n` is a solution of `fibRec`. -/
 theorem geom_gold_isSol_fibRec : fibRec.IsSolution (φ ^ ·) := by
   rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
-  simp [sub_eq_zero, - div_pow] -- Porting note: Added `- div_pow`
+  simp [sub_eq_zero]
 
 /-- The geometric sequence `fun n ↦ ψ^n` is a solution of `fibRec`. -/
 theorem geom_goldConj_isSol_fibRec : fibRec.IsSolution (ψ ^ ·) := by
   rw [fibRec.geom_sol_iff_root_charPoly, fibRec_charPoly_eq]
-  simp [sub_eq_zero, - div_pow] -- Porting note: Added `- div_pow`
+  simp [sub_eq_zero]
 
 end Fibrec
 
diff --git a/Mathlib/Data/Real/Hyperreal.lean b/Mathlib/Data/Real/Hyperreal.lean
index 3ec6b4835c02e..6ffec46ad6c83 100644
--- a/Mathlib/Data/Real/Hyperreal.lean
+++ b/Mathlib/Data/Real/Hyperreal.lean
@@ -127,11 +127,11 @@ theorem ofSeq_surjective : Function.Surjective ofSeq := Quot.exists_rep
 theorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n :=
   Germ.coe_lt
 
-/-- A sample infinitesimal hyperreal-/
+/-- A sample infinitesimal hyperreal -/
 noncomputable def epsilon : ℝ* :=
   ofSeq fun n => n⁻¹
 
-/-- A sample infinite hyperreal-/
+/-- A sample infinite hyperreal -/
 noncomputable def omega : ℝ* := ofSeq Nat.cast
 
 @[inherit_doc] scoped notation "ε" => Hyperreal.epsilon
@@ -599,12 +599,12 @@ theorem infinitePos_iff_infinitesimal_inv_pos {x : ℝ*} :
   ⟨fun hip =>
     ⟨infinitesimal_def.mpr fun r hr =>
         ⟨lt_trans (coe_lt_coe.2 (neg_neg_of_pos hr)) (inv_pos.2 (hip 0)),
-          (inv_lt (coe_lt_coe.2 hr) (hip 0)).mp (by convert hip r⁻¹)⟩,
+          inv_lt_of_inv_lt₀ (coe_lt_coe.2 hr) (by convert hip r⁻¹)⟩,
       inv_pos.2 <| hip 0⟩,
     fun ⟨hi, hp⟩ r =>
     @_root_.by_cases (r = 0) (↑r < x) (fun h => Eq.substr h (inv_pos.mp hp)) fun h =>
       lt_of_le_of_lt (coe_le_coe.2 (le_abs_self r))
-        ((inv_lt_inv (inv_pos.mp hp) (coe_lt_coe.2 (abs_pos.2 h))).mp
+        ((inv_lt_inv₀ (inv_pos.mp hp) (coe_lt_coe.2 (abs_pos.2 h))).mp
           ((infinitesimal_def.mp hi) |r|⁻¹ (inv_pos.2 (abs_pos.2 h))).2)⟩
 
 theorem infiniteNeg_iff_infinitesimal_inv_neg {x : ℝ*} :
diff --git a/Mathlib/Data/Real/Pi/Bounds.lean b/Mathlib/Data/Real/Pi/Bounds.lean
index 20ca11d6273d2..131af1b4136f2 100644
--- a/Mathlib/Data/Real/Pi/Bounds.lean
+++ b/Mathlib/Data/Real/Pi/Bounds.lean
@@ -11,41 +11,36 @@ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
 This file contains lemmas which establish bounds on `Real.pi`.
 Notably, these include `pi_gt_sqrtTwoAddSeries` and `pi_lt_sqrtTwoAddSeries`,
 which bound `π` using series;
-numerical bounds on `π` such as `pi_gt_314`and `pi_lt_315` (more precise versions are given, too).
+numerical bounds on `π` such as `pi_gt_d2` and `pi_lt_d2` (more precise versions are given, too).
 
 See also `Mathlib/Data/Real/Pi/Leibniz.lean` and `Mathlib/Data/Real/Pi/Wallis.lean` for infinite
 formulas for `π`.
 -/
 
--- Porting note: needed to add a lot of type ascriptions for lean to interpret numbers as reals.
-
 open scoped Real
 
 namespace Real
 
-theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
-    (2 : ℝ) ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) < π := by
-  have : √(2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
-    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]
+theorem pi_gt_sqrtTwoAddSeries (n : ℕ) : 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) < π := by
+  have : √(2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π := by
+    rw [← lt_div_iff₀, ← sin_pi_over_two_pow_succ]
     focus
       apply sin_lt
       apply div_pos pi_pos
     all_goals apply pow_pos; norm_num
-  apply lt_of_le_of_lt (le_of_eq _) this
+  refine lt_of_le_of_lt (le_of_eq ?_) this
   rw [pow_succ' _ (n + 1), ← mul_assoc, div_mul_cancel₀, mul_comm]; norm_num
 
 theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
-    π < (2 : ℝ) ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / (4 : ℝ) ^ n := by
-  have : π <
-      (√(2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) + (1 : ℝ) / ((2 : ℝ) ^ n) ^ 3 / 4) *
-      (2 : ℝ) ^ (n + 2) := by
-    rw [← div_lt_iff (by norm_num), ← sin_pi_over_two_pow_succ]
+    π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n := by
+  have : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * (2 : ℝ) ^ (n + 2) := by
+    rw [← div_lt_iff₀ (by norm_num), ← sin_pi_over_two_pow_succ]
     refine lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube ?_ ?_)) ?_
     · apply div_pos pi_pos; apply pow_pos; norm_num
     · rw [div_le_iff₀']
       · refine le_trans pi_le_four ?_
         simp only [show (4 : ℝ) = (2 : ℝ) ^ 2 by norm_num, mul_one]
-        apply pow_le_pow_right (by norm_num)
+        apply pow_right_mono₀ (by norm_num)
         apply le_add_of_nonneg_left; apply Nat.zero_le
       · apply pow_pos; norm_num
     apply add_le_add_left; rw [div_le_div_right (by norm_num)]
@@ -57,10 +52,8 @@ theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
     · rw [← le_div_iff₀ (by norm_num)]
       refine le_trans ((div_le_div_right ?_).mpr pi_le_four) ?_
       · apply pow_pos; norm_num
-      · simp only [pow_succ', ← div_div, one_div]
-      -- Porting note: removed `convert le_rfl`
-        norm_num
-  apply lt_of_lt_of_le this (le_of_eq _); rw [add_mul]; congr 1
+      · ring_nf; rfl
+  refine lt_of_lt_of_le this (le_of_eq ?_); rw [add_mul]; congr 1
   · ring
   simp only [show (4 : ℝ) = 2 ^ 2 by norm_num, ← pow_mul, div_div, ← pow_add]
   rw [one_div, one_div, inv_mul_eq_iff_eq_mul₀, eq_comm, mul_inv_eq_iff_eq_mul₀, ← pow_add]
@@ -74,7 +67,7 @@ theorem pi_lower_bound_start (n : ℕ) {a}
     (h : sqrtTwoAddSeries ((0 : ℕ) / (1 : ℕ)) n ≤ (2 : ℝ) - (a / (2 : ℝ) ^ (n + 1)) ^ 2) :
     a < π := by
   refine lt_of_le_of_lt ?_ (pi_gt_sqrtTwoAddSeries n); rw [mul_comm]
-  refine (div_le_iff₀ (pow_pos (by norm_num) _ : (0 : ℝ) < _)).mp (le_sqrt_of_sq_le ?_)
+  refine (div_le_iff₀ (pow_pos (by norm_num) _)).mp (le_sqrt_of_sq_le ?_)
   rwa [le_sub_comm, show (0 : ℝ) = (0 : ℕ) / (1 : ℕ) by rw [Nat.cast_zero, zero_div]]
 
 theorem sqrtTwoAddSeries_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ} (hz : sqrtTwoAddSeries (c / d) n ≤ z)
@@ -87,35 +80,6 @@ theorem sqrtTwoAddSeries_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ} (hz : sqrtT
     add_div_eq_mul_add_div _ _ (ne_of_gt hb'), div_le_div_iff hb' (pow_pos hd' _)]
   exact mod_cast h
 
-section Tactic
-
-open Lean Elab Tactic
-
-/-- `numDen stx` takes a syntax expression `stx` and
-* if it is of the form `a / b`, then it returns `some (a, b)`;
-* otherwise it returns `none`.
--/
-private def numDen : Syntax → Option (Syntax.Term × Syntax.Term)
-  | `($a / $b) => some (a, b)
-  | _          => none
-
-/-- Create a proof of `a < π` for a fixed rational number `a`, given a witness, which is a
-sequence of rational numbers `√2 < r 1 < r 2 < ... < r n < 2` satisfying the property that
-`√(2 + r i) ≤ r(i+1)`, where `r 0 = 0` and `√(2 - r n) ≥ a/2^(n+1)`. -/
-elab "pi_lower_bound " "[" l:term,* "]" : tactic => do
-  let rat_sep := l.elemsAndSeps
-  let sep := rat_sep.getD 1 .missing
-  let ratStx := rat_sep.filter (· != sep)
-  let n := ← (toExpr ratStx.size).toSyntax
-  let els := (ratStx.map numDen).reduceOption
-  evalTactic (← `(tactic| apply pi_lower_bound_start $n))
-  let _ := ← els.mapM fun (x, y) => do
-      evalTactic (← `(tactic| apply sqrtTwoAddSeries_step_up $x $y))
-  evalTactic (← `(tactic| simp [sqrtTwoAddSeries]))
-  allGoals (evalTactic (← `(tactic| norm_num1)))
-
-end Tactic
-
 /-- From a lower bound on `sqrtTwoAddSeries 0 n = 2 cos (π / 2 ^ (n+1))` of the form
 `2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrtTwoAddSeries 0 n`, one can deduce the upper bound
 `π < a` thanks to basic trigonometric formulas as expressed in `pi_lt_sqrtTwoAddSeries`. -/
@@ -141,52 +105,133 @@ theorem sqrtTwoAddSeries_step_down (a b : ℕ) {c d n : ℕ} {z : ℝ}
 
 section Tactic
 
-open Lean Elab Tactic
+open Lean Elab Tactic Qq
+
+/-- Create a proof of `a < π` for a fixed rational number `a`, given a witness, which is a
+sequence of rational numbers `√2 < r 1 < r 2 < ... < r n < 2` satisfying the property that
+`√(2 + r i) ≤ r(i+1)`, where `r 0 = 0` and `√(2 - r n) ≥ a/2^(n+1)`. -/
+elab "pi_lower_bound " "[" l:term,* "]" : tactic => do
+  have els := l.getElems
+  let n := quote els.size
+  evalTactic (← `(tactic| apply pi_lower_bound_start $n))
+  for l in els do
+    let {num, den, ..} ← unsafe Meta.evalExpr ℚ q(ℚ) (← Term.elabTermAndSynthesize l (some q(ℚ)))
+    evalTactic (← `(tactic| apply sqrtTwoAddSeries_step_up $(quote num.toNat) $(quote den)))
+  evalTactic (← `(tactic| simp [sqrtTwoAddSeries]))
+  allGoals <| evalTactic (← `(tactic| norm_num1))
 
 /-- Create a proof of `π < a` for a fixed rational number `a`, given a witness, which is a
 sequence of rational numbers `√2 < r 1 < r 2 < ... < r n < 2` satisfying the property that
-`√(2 + r i) ≥ r(i+1)`, where `r 0 = 0` and `√(2 - r n) ≥ (a - 1/4^n) / 2^(n+1)`. -/
+`√(2 + r i) ≥ r(i+1)`, where `r 0 = 0` and `√(2 - r n) ≤ (a - 1/4^n) / 2^(n+1)`. -/
 elab "pi_upper_bound " "[" l:term,* "]" : tactic => do
-  let rat_sep := l.elemsAndSeps
-  let sep := rat_sep.getD 1 .missing
-  let ratStx := rat_sep.filter (· != sep)
-  let n := ← (toExpr ratStx.size).toSyntax
-  let els := (ratStx.map numDen).reduceOption
+  have els := l.getElems
+  let n := quote els.size
   evalTactic (← `(tactic| apply pi_upper_bound_start $n))
-  let _ := ← els.mapM fun (x, y) => do
-      evalTactic (← `(tactic| apply sqrtTwoAddSeries_step_down $x $y))
+  for l in els do
+    let {num, den, ..} ← unsafe Meta.evalExpr ℚ q(ℚ) (← Term.elabTermAndSynthesize l (some q(ℚ)))
+    evalTactic (← `(tactic| apply sqrtTwoAddSeries_step_down $(quote num.toNat) $(quote den)))
   evalTactic (← `(tactic| simp [sqrtTwoAddSeries]))
-  allGoals (evalTactic (← `(tactic| norm_num1)))
+  allGoals <| evalTactic (← `(tactic| norm_num1))
 
 end Tactic
 
-theorem pi_gt_three : 3 < π := by
-  pi_lower_bound [23/16]
-
-theorem pi_gt_314 : 3.14 < π := by
-  pi_lower_bound [99 / 70, 874 / 473, 1940 / 989, 1447 / 727]
-
-theorem pi_lt_315 : π < 3.15 := by
-  pi_upper_bound [140 / 99, 279 / 151, 51 / 26, 412 / 207]
-
-theorem pi_gt_31415 : 3.1415 < π := by
-  pi_lower_bound
-        [11482 / 8119, 5401 / 2923, 2348 / 1197, 11367 / 5711, 25705 / 12868, 23235 / 11621]
-
-theorem pi_lt_31416 : π < 3.1416 := by
-    pi_upper_bound
-        [4756 / 3363, 101211 / 54775, 505534 / 257719, 83289 / 41846, 411278 / 205887,
-          438142 / 219137, 451504 / 225769, 265603 / 132804, 849938 / 424971]
-
-theorem pi_gt_3141592 : 3.141592 < π := by
-    pi_lower_bound
-        [11482 / 8119, 7792 / 4217, 54055 / 27557, 949247 / 476920, 3310126 / 1657059,
-          2635492 / 1318143, 1580265 / 790192, 1221775 / 610899, 3612247 / 1806132, 849943 / 424972]
+/-!
+The below witnesses were generated using the following Mathematica script:
+```mathematica
+bound[a_, Iters -> n_, Rounding -> extra_, Precision -> prec_] := Module[{r0, r, r2, diff, sign},
+  On[Assert];
+  sign = If[a >= \[Pi], Print["upper"]; 1, Print["lower"]; -1];
+  r0 = 2 - ((a - (sign + 1)/2/4^n)/2^(n + 1))^2;
+  r = Log[2 - NestList[#^2 - 2 &, N[r0, prec], n - 1]];
+  diff = (r[[-1]] - Log[2 - Sqrt[2]])/(Length[r] + 1);
+  If[sign diff <= 0, Return["insufficient iterations"]];
+  r2 = Log[Rationalize[Exp[#], extra (Exp[#] - Exp[# - sign diff])] &
+    /@ (r - diff Range[1, Length[r]])];
+  Assert[sign (2 - Exp@r2[[1]] - r0) >= 0];
+  Assert[And @@ Table[
+    sign (Sqrt@(4 - Exp@r2[[i + 1]]) - (2 - Exp@r2[[i]])) >= 0, {i, 1, Length[r2] - 1}]];
+  Assert[sign (Exp@r2[[-1]] - (2 - Sqrt[2])) >= 0];
+  With[{s1 = ToString@InputForm[2 - #], s2 = ToString@InputForm[#]},
+    If[StringLength[s1] <= StringLength[s2] + 2, s1, "2-" <> s2]] & /@ Exp@Reverse@r2
+];
+```
+-/
 
-theorem pi_lt_3141593 : π < 3.141593 := by
-    pi_upper_bound
-        [27720 / 19601, 56935 / 30813, 49359 / 25163, 258754 / 130003, 113599 / 56868,
-          1101994 / 551163, 8671537 / 4336095, 3877807 / 1938940, 52483813 / 26242030,
-          56946167 / 28473117, 23798415 / 11899211]
+theorem pi_gt_three : 3 < π := by
+  -- bound[3, Iters -> 1, Rounding -> 2, Precision -> 3]
+  pi_lower_bound [23 / 16]
+
+theorem pi_lt_four : π < 4 := by
+  -- bound[4, Iters -> 1, Rounding -> 1, Precision -> 1]
+  pi_upper_bound [4 / 3]
+
+theorem pi_gt_d2 : 3.14 < π := by
+  -- bound[314*^-2, Iters -> 4, Rounding -> 1.5, Precision -> 8]
+  pi_lower_bound [338 / 239, 704 / 381, 1940 / 989, 1447 / 727]
+@[deprecated (since := "2024-09-19")] alias pi_gt_314 := pi_gt_d2
+
+theorem pi_lt_d2 : π < 3.15 := by
+  -- bound[315*^-2, Iters -> 4, Rounding -> 1.4, Precision -> 7]
+  pi_upper_bound [41 / 29, 109 / 59, 865 / 441, 412 / 207]
+@[deprecated (since := "2024-09-19")] alias pi_lt_315 := pi_lt_d2
+
+theorem pi_gt_d4 : 3.1415 < π := by
+  -- bound[31415*^-4, Iters -> 6, Rounding -> 1.1, Precision -> 10]
+  pi_lower_bound [
+    1970 / 1393, 3010 / 1629, 11689 / 5959, 10127 / 5088, 33997 / 17019, 23235 / 11621]
+@[deprecated (since := "2024-09-19")] alias pi_gt_31415 := pi_gt_d4
+
+theorem pi_lt_d4 : π < 3.1416 := by
+  -- bound[31416*^-4, Iters -> 9, Rounding -> .9, Precision -> 16]
+  pi_upper_bound [
+    4756/3363, 14965/8099, 21183/10799, 49188/24713, 2-53/22000, 2-71/117869, 2-47/312092,
+    2-17/451533, 2-4/424971]
+@[deprecated (since := "2024-09-19")] alias pi_lt_31416 := pi_lt_d4
+
+theorem pi_gt_d6 : 3.141592 < π := by
+  -- bound[3141592*^-6, Iters -> 10, Rounding -> .8, Precision -> 16]
+  pi_lower_bound [
+    11482/8119, 7792/4217, 54055/27557, 2-623/64690, 2-337/139887, 2-208/345307, 2-167/1108925,
+    2-64/1699893, 2-31/3293535, 2-48/20398657]
+@[deprecated (since := "2024-09-19")] alias pi_gt_3141592 := pi_gt_d6
+
+theorem pi_lt_d6 : π < 3.141593 := by
+  -- bound[3141593*^-6, Iters -> 11, Rounding -> .5, Precision -> 17]
+  pi_upper_bound [
+    35839/25342, 49143/26596, 145729/74292, 294095/147759, 2-137/56868, 2-471/781921, 2-153/1015961,
+    2-157/4170049, 2-28/2974805, 2-9/3824747, 2-7/11899211]
+@[deprecated (since := "2024-09-19")] alias pi_lt_3141593 := pi_lt_d6
+
+theorem pi_gt_d20 : 3.14159265358979323846 < π := by
+  -- bound[314159265358979323846*^-20, Iters -> 34, Rounding -> .6, Precision -> 46]
+  pi_lower_bound [
+    671574048197/474874563549, 58134718954/31462283181, 3090459598621/1575502640777,
+    2-7143849599/741790664068, 8431536490061/4220852446654, 2-2725579171/4524814682468,
+    2-2494895647/16566776788806, 2-608997841/16175484287402, 2-942567063/100141194694075,
+    2-341084060/144951150987041, 2-213717653/363295959742218, 2-71906926/488934711121807,
+    2-29337101/797916288104986, 2-45326311/4931175952730065, 2-7506877/3266776448781479,
+    2-5854787/10191338039232571, 2-4538642/31601378399861717, 2-276149/7691013341581098,
+    2-350197/39013283396653714, 2-442757/197299283738495963, 2-632505/1127415566199968707,
+    2-1157/8249230030392285, 2-205461/5859619883403334178, 2-33721/3846807755987625852,
+    2-11654/5317837263222296743, 2-8162/14897610345776687857, 2-731/5337002285107943372,
+    2-1320/38549072592845336201, 2-707/82588467645883795866, 2-53/24764858756615791675,
+    2-237/442963888703240952920, 2-128/956951523274512100791, 2-32/956951523274512100783,
+    2-27/3229711391051478340136]
+
+theorem pi_lt_d20 : π < 3.14159265358979323847 := by
+  -- bound[314159265358979323847*^-20, Iters -> 34, Rounding -> .5, Precision -> 46]
+  pi_upper_bound [
+    215157040700/152139002499, 936715022285/506946517009, 1760670193473/897581880893,
+    2-6049918861/628200981455, 2-8543385003/3546315642356, 2-2687504973/4461606579043,
+    2-1443277808/9583752057175, 2-546886849/14525765179168, 2-650597193/69121426717657,
+    2-199969519/84981432264454, 2-226282901/384655467333100, 2-60729699/412934601558121,
+    2-25101251/682708800188252, 2-7156464/778571703825145, 2-7524725/3274543383827551,
+    2-4663362/8117442793616861, 2-1913009/13319781840326041, 2-115805/3225279830894912,
+    2-708749/78957345705688293, 2-131255/58489233342660393, 2-101921/181670219085488669,
+    2-44784/319302953916238627, 2-82141/2342610212364552264, 2-4609/525783249231842696,
+    2-4567/2083967975041722089, 2-2273/4148770928197796067, 2-563/4110440884426500846,
+    2-784/22895812812720260289, 2-1717/200571992854289218531, 2-368/171952226838388893139,
+    2-149/278487845640434185590, 2-207/1547570041545500037992, 2-20/598094702046570062987,
+    2-7/837332582865198088180]
 
 end Real
diff --git a/Mathlib/Data/Real/Pi/Irrational.lean b/Mathlib/Data/Real/Pi/Irrational.lean
new file mode 100644
index 0000000000000..e3884b774698d
--- /dev/null
+++ b/Mathlib/Data/Real/Pi/Irrational.lean
@@ -0,0 +1,306 @@
+/-
+Copyright (c) 2022 Bhavik Mehta. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Bhavik Mehta
+-/
+import Mathlib.Analysis.SpecialFunctions.Integrals
+import Mathlib.Data.Real.Irrational
+import Mathlib.Topology.Algebra.Order.Floor
+
+/-!
+# `Real.pi` is irrational
+
+The main result of this file is `irrational_pi`.
+
+The proof is adapted from https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#Cartwright's_proof.
+
+The proof idea is as follows.
+* Define a sequence of integrals `I n θ = ∫ x in (-1)..1, (1 - x ^ 2) ^ n * cos (x * θ)`.
+* Give a recursion formula for `I (n + 2) θ * θ ^ 2` in terms of `I n θ` and `I (n + 1) θ`.
+  Note we do not find it helpful to define `J` as in the above proof, and instead work directly
+  with `I`.
+* Define polynomials with integer coefficients `sinPoly n` and `cosPoly n` such that
+  `I n θ * θ ^ (2 * n + 1) = n ! * (sinPoly n θ * sin θ + cosPoly n θ * cos θ)`.
+  Note that in the informal proof, these polynomials are not defined explicitly, but we find it
+  useful to define them by recursion.
+* Show that both these polynomials have degree bounded by `n`.
+* Show that `0 < I n (π / 2) ≤ 2` for all `n`.
+* Now we can finish: if `π / 2` is rational, write it as `a / b` with `a, b > 0`. Then
+  `b ^ (2 * n + 1) * sinPoly n (a / b)` is a positive integer by the degree bound. But it is equal
+  to `a ^ (2 * n + 1) / n ! * I n (π / 2) ≤ 2 * a * (2 * n + 1) / n !`, which converges to 0 as
+  `n → ∞`.
+
+-/
+
+noncomputable section
+
+open intervalIntegral MeasureTheory.MeasureSpace Set Polynomial Real
+open scoped Nat
+
+/-- The sequence of integrals used for Cartwright's proof of irrationality of `π`. -/
+private def I (n : ℕ) (θ : ℝ) : ℝ := ∫ x in (-1)..1, (1 - x ^ 2) ^ n * cos (x * θ)
+
+variable {n : ℕ} {θ : ℝ}
+
+private lemma I_zero : I 0 θ * θ = 2 * sin θ := by
+  rw [mul_comm, I]
+  simp [mul_integral_comp_mul_right, two_mul]
+
+/--
+Auxiliary for the proof that `π` is irrational.
+While it is most natural to give the recursive formula for `I (n + 2) θ`, as well as give the second
+base case of `I 1 θ`, it is in fact more convenient to give the recursive formula for `I (n + 1) θ`
+in terms of `I n θ` and `I (n - 1) θ` (note the natural subtraction!).
+Despite the usually inconvenient subtraction, this in fact allows deducing both of the above facts
+with significantly fewer analysis computations.
+In addition, note the `0 ^ n` on the right hand side - this is intentional, and again allows
+combining the proof of the "usual" recursion formula and the base case `I 1 θ`.
+-/
+private lemma recursion' (n : ℕ) :
+    I (n + 1) θ * θ ^ 2 = - (2 * 2 * ((n + 1) * (0 ^ n * cos θ))) +
+      2 * (n + 1) * (2 * n + 1) * I n θ - 4 * (n + 1) * n * I (n - 1) θ := by
+  rw [I]
+  let f (x : ℝ) : ℝ := 1 - x ^ 2
+  let u₁ (x : ℝ) : ℝ := f x ^ (n + 1)
+  let u₁' (x : ℝ) : ℝ := - (2 * (n + 1) * x * f x ^ n)
+  let v₁ (x : ℝ) : ℝ := sin (x * θ)
+  let v₁' (x : ℝ) : ℝ := cos (x * θ) * θ
+  let u₂ (x : ℝ) : ℝ := x * (f x) ^ n
+  let u₂' (x : ℝ) : ℝ := (f x) ^ n - 2 * n * x ^ 2 * (f x) ^ (n - 1)
+  let v₂ (x : ℝ) : ℝ := cos (x * θ)
+  let v₂' (x : ℝ) : ℝ := -sin (x * θ) * θ
+  have hfd : Continuous f := by fun_prop
+  have hu₁d : Continuous u₁' := by fun_prop
+  have hv₁d : Continuous v₁' := by fun_prop
+  have hu₂d : Continuous u₂' := by fun_prop
+  have hv₂d : Continuous v₂' := by fun_prop
+  have hu₁_eval_one : u₁ 1 = 0 := by simp only [u₁, f]; simp
+  have hu₁_eval_neg_one : u₁ (-1) = 0 := by simp only [u₁, f]; simp
+  have t : u₂ 1 * v₂ 1 - u₂ (-1) * v₂ (-1) = 2 * (0 ^ n * cos θ) := by simp [u₂, v₂, f, ← two_mul]
+  have hf (x) : HasDerivAt f (- 2 * x) x := by
+    convert (hasDerivAt_pow 2 x).const_sub 1 using 1
+    simp
+  have hu₁ (x) : HasDerivAt u₁ (u₁' x) x := by
+    convert (hf x).pow _ using 1
+    simp only [Nat.add_succ_sub_one, u₁', Nat.cast_add_one]
+    ring
+  have hv₁ (x) : HasDerivAt v₁ (v₁' x) x := (hasDerivAt_mul_const θ).sin
+  have hu₂ (x) : HasDerivAt u₂ (u₂' x) x := by
+    convert (hasDerivAt_id' x).mul ((hf x).pow _) using 1
+    simp only [u₂']
+    ring
+  have hv₂ (x) : HasDerivAt v₂ (v₂' x) x := (hasDerivAt_mul_const θ).cos
+  convert_to (∫ (x : ℝ) in (-1)..1, u₁ x * v₁' x) * θ = _ using 1
+  · simp_rw [u₁, v₁', ← intervalIntegral.integral_mul_const, sq θ, mul_assoc]
+  rw [integral_mul_deriv_eq_deriv_mul (fun x _ => hu₁ x) (fun x _ => hv₁ x)
+    (hu₁d.intervalIntegrable _ _) (hv₁d.intervalIntegrable _ _), hu₁_eval_one, hu₁_eval_neg_one,
+    zero_mul, zero_mul, sub_zero, zero_sub, ← integral_neg, ← integral_mul_const]
+  convert_to ((-2 : ℝ) * (n + 1)) * ∫ (x : ℝ) in (-1)..1, (u₂ x * v₂' x) = _ using 1
+  · rw [← integral_const_mul]
+    congr 1 with x
+    dsimp [u₁', v₁, u₂, v₂']
+    ring
+  rw [integral_mul_deriv_eq_deriv_mul (fun x _ => hu₂ x) (fun x _ => hv₂ x)
+    (hu₂d.intervalIntegrable _ _) (hv₂d.intervalIntegrable _ _),
+    mul_sub, t, neg_mul, neg_mul, neg_mul, sub_neg_eq_add]
+  have (x) : u₂' x = (2 * n + 1) * f x ^ n - 2 * n * f x ^ (n - 1) := by
+    cases n with
+    | zero => simp [u₂']
+    | succ n => ring!
+  simp_rw [this, sub_mul, mul_assoc _ _ (v₂ _)]
+  have : Continuous v₂ := by fun_prop
+  rw [mul_mul_mul_comm, integral_sub, mul_sub, add_sub_assoc]
+  · congr 1
+    simp_rw [integral_const_mul]
+    ring!
+  all_goals exact Continuous.intervalIntegrable (by fun_prop) _ _
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The recursive formula for `I (n + 2) θ * θ ^ 2` in terms of `I n θ` and `I (n + 1) θ`.
+-/
+private lemma recursion (n : ℕ) :
+    I (n + 2) θ * θ ^ 2 =
+      2 * (n + 2) * (2 * n + 3) * I (n + 1) θ - 4 * (n + 2) * (n + 1) * I n θ := by
+  rw [recursion' (n + 1)]
+  simp
+  ring!
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The second base case for the induction on `n`, giving an explicit formula for `I 1 θ`.
+-/
+private lemma I_one : I 1 θ * θ ^ 3 = 4 * sin θ - 4 * θ * cos θ := by
+  rw [_root_.pow_succ, ← mul_assoc, recursion' 0, sub_mul, add_mul, mul_assoc _ (I 0 θ), I_zero]
+  ring
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The first of the two integer-coefficient polynomials that describe the behaviour of the
+sequence of integrals `I`.
+While not given in the informal proof, these are easy to deduce from the recursion formulae.
+-/
+private def sinPoly : ℕ → ℤ[X]
+  | 0 => C 2
+  | 1 => C 4
+  | (n+2) => ((2 : ℤ) * (2 * n + 3)) • sinPoly (n + 1) + monomial 2 (-4) * sinPoly n
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The second of the two integer-coefficient polynomials that describe the behaviour of the
+sequence of integrals `I`.
+While not given in the informal proof, these are easy to deduce from the recursion formulae.
+-/
+private def cosPoly : ℕ → ℤ[X]
+  | 0 => 0
+  | 1 => monomial 1 (-4)
+  | (n+2) => ((2 : ℤ) * (2 * n + 3)) • cosPoly (n + 1) + monomial 2 (-4) * cosPoly n
+
+/--
+Auxiliary for the proof that `π` is irrational.
+Prove a degree bound for `sinPoly n` by induction. Note this is where we find the value in an
+explicit description of `sinPoly`.
+-/
+private lemma sinPoly_natDegree_le : ∀ n : ℕ, (sinPoly n).natDegree ≤ n
+  | 0 => by simp [sinPoly]
+  | 1 => by simp only [natDegree_C, mul_one, zero_le', sinPoly]
+  | n + 2 => by
+      rw [sinPoly]
+      refine natDegree_add_le_of_degree_le ((natDegree_smul_le _ _).trans ?_) ?_
+      · exact (sinPoly_natDegree_le (n + 1)).trans (by simp)
+      refine natDegree_mul_le.trans ?_
+      simpa [add_comm 2] using sinPoly_natDegree_le n
+
+/--
+Auxiliary for the proof that `π` is irrational.
+Prove a degree bound for `cosPoly n` by induction. Note this is where we find the value in an
+explicit description of `cosPoly`.
+-/
+private lemma cosPoly_natDegree_le : ∀ n : ℕ, (cosPoly n).natDegree ≤ n
+  | 0 => by simp [cosPoly]
+  | 1 => (natDegree_monomial_le _).trans (by simp)
+  | n + 2 => by
+      rw [cosPoly]
+      refine natDegree_add_le_of_degree_le ((natDegree_smul_le _ _).trans ?_) ?_
+      · exact (cosPoly_natDegree_le (n + 1)).trans (by simp)
+      exact natDegree_mul_le.trans (by simp [add_comm 2, cosPoly_natDegree_le n])
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The key lemma: the sequence of integrals `I` can be written as a linear combination of `sin` and
+`cos`, with coefficients given by the polynomials `sinPoly` and `cosPoly`.
+-/
+private lemma sinPoly_add_cosPoly_eval (θ : ℝ) :
+    ∀ n : ℕ,
+      I n θ * θ ^ (2 * n + 1) = n ! * ((sinPoly n).eval₂ (Int.castRingHom _) θ * sin θ +
+        (cosPoly n).eval₂ (Int.castRingHom _) θ * cos θ)
+  | 0 => by simp [sinPoly, cosPoly, I_zero]
+  | 1 => by simp [I_one, sinPoly, cosPoly, sub_eq_add_neg]
+  | n + 2 => by
+      calc I (n + 2) θ * θ ^ (2 * (n + 2) + 1) = I (n + 2) θ * θ ^ 2 * θ ^ (2 * n + 3) := by ring
+        _ = 2 * (n + 2) * (2 * n + 3) * (I (n + 1) θ * θ ^ (2 * (n + 1) + 1)) -
+            4 * (n + 2) * (n + 1) * θ ^ 2 * (I n θ * θ ^ (2 * n + 1)) := by rw [recursion]; ring
+        _ = _ := by simp [sinPoly_add_cosPoly_eval, sinPoly, cosPoly, Nat.factorial_succ]; ring
+
+/--
+Auxiliary for the proof that `π` is irrational.
+For a polynomial `p` with natural degree `≤ k` and integer coefficients, evaluating `p` at a
+rational `a / b` gives a rational of the form `z / b ^ k`.
+TODO: should this be moved elsewhere? It uses none of the pi-specific definitions.
+-/
+private lemma is_integer {p : ℤ[X]} (a b : ℤ) {k : ℕ} (hp : p.natDegree ≤ k) :
+    ∃ z : ℤ, p.eval₂ (Int.castRingHom ℝ) (a / b) * b ^ k = z := by
+  rcases eq_or_ne b 0 with rfl | hb
+  · rcases k.eq_zero_or_pos with rfl | hk
+    · exact ⟨p.coeff 0, by simp⟩
+    exact ⟨0, by simp [hk.ne']⟩
+  refine ⟨∑ i in p.support, p.coeff i * a ^ i * b ^ (k - i), ?_⟩
+  conv => lhs; rw [← sum_monomial_eq p]
+  rw [eval₂_sum, sum, Finset.sum_mul, Int.cast_sum]
+  simp only [eval₂_monomial, eq_intCast, div_pow, Int.cast_mul, Int.cast_pow]
+  refine Finset.sum_congr rfl (fun i hi => ?_)
+  have ik := (le_natDegree_of_mem_supp i hi).trans hp
+  rw [mul_assoc, div_mul_comm, ← Int.cast_pow, ← Int.cast_pow, ← Int.cast_pow,
+    ← pow_sub_mul_pow b ik, ← Int.cast_div_charZero, Int.mul_ediv_cancel _ (pow_ne_zero _ hb),
+    ← mul_assoc, mul_right_comm, ← Int.cast_pow]
+  exact dvd_mul_left _ _
+
+open Filter
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The integrand in the definition of `I` is nonnegative and takes a positive value at least one point,
+so the integral is positive.
+-/
+private lemma I_pos : 0 < I n (π / 2) := by
+  refine integral_pos (by norm_num) (Continuous.continuousOn (by continuity)) ?_ ⟨0, by simp⟩
+  refine fun x hx => mul_nonneg (pow_nonneg ?_ _) ?_
+  · rw [sub_nonneg, sq_le_one_iff_abs_le_one, abs_le]
+    exact ⟨hx.1.le, hx.2⟩
+  refine cos_nonneg_of_neg_pi_div_two_le_of_le ?_ ?_ <;>
+  nlinarith [hx.1, hx.2, pi_pos]
+
+/--
+Auxiliary for the proof that `π` is irrational.
+The integrand in the definition of `I` is bounded by 1 and the interval has length 2, so the
+integral is bounded above by `2`.
+-/
+private lemma I_le (n : ℕ) : I n (π / 2) ≤ 2 := by
+  rw [← norm_of_nonneg I_pos.le]
+  refine (norm_integral_le_of_norm_le_const ?_).trans (show (1 : ℝ) * _ ≤ _ by norm_num)
+  intros x hx
+  simp only [uIoc_of_le, neg_le_self_iff, zero_le_one, mem_Ioc] at hx
+  rw [norm_eq_abs, abs_mul, abs_pow]
+  refine mul_le_one₀ (pow_le_one₀ (abs_nonneg _) ?_) (abs_nonneg _) (abs_cos_le_one _)
+  rw [abs_le]
+  constructor <;> nlinarith
+
+/--
+Auxiliary for the proof that `π` is irrational.
+For any real `a`, we have that `a ^ (2n+1) / n!` tends to `0` as `n → ∞`.  This is just a
+reformulation of tendsto_pow_div_factorial_atTop, which asserts the same for `a ^ n / n!`
+-/
+private lemma tendsto_pow_div_factorial_at_top_aux (a : ℝ) :
+    Tendsto (fun n => (a : ℝ) ^ (2 * n + 1) / n !) atTop (nhds 0) := by
+  rw [← mul_zero a]
+  refine ((FloorSemiring.tendsto_pow_div_factorial_atTop (a ^ 2)).const_mul a).congr (fun x => ?_)
+  rw [← pow_mul, mul_div_assoc', _root_.pow_succ']
+
+/-- If `x` is rational, it can be written as `a / b` with `a : ℤ` and `b : ℕ` satisfying `b > 0`. -/
+private lemma not_irrational_exists_rep {x : ℝ} :
+    ¬Irrational x → ∃ (a : ℤ) (b : ℕ), 0 < b ∧ x = a / b := by
+  rw [Irrational, not_not, mem_range]
+  rintro ⟨q, rfl⟩
+  exact ⟨q.num, q.den, q.pos, by exact_mod_cast (Rat.num_div_den _).symm⟩
+
+@[simp] theorem irrational_pi : Irrational π := by
+  apply Irrational.of_div_nat 2
+  rw [Nat.cast_two]
+  by_contra h'
+  obtain ⟨a, b, hb, h⟩ := not_irrational_exists_rep h'
+  have ha : (0 : ℝ) < a := by
+    have : 0 < (a : ℝ) / b := h ▸ pi_div_two_pos
+    rwa [lt_div_iff₀ (by positivity), zero_mul] at this
+  have k (n : ℕ) : 0 < (a : ℝ) ^ (2 * n + 1) / n ! := by positivity
+  have j : ∀ᶠ n : ℕ in atTop, (a : ℝ) ^ (2 * n + 1) / n ! * I n (π / 2) < 1 := by
+    have := eventually_lt_of_tendsto_lt (show (0 : ℝ) < 1 / 2 by norm_num)
+              (tendsto_pow_div_factorial_at_top_aux a)
+    filter_upwards [this] with n hn
+    rw [lt_div_iff₀ (zero_lt_two : (0 : ℝ) < 2)] at hn
+    exact hn.trans_le' (mul_le_mul_of_nonneg_left (I_le _) (by positivity))
+  obtain ⟨n, hn⟩ := j.exists
+  have hn' : 0 < a ^ (2 * n + 1) / n ! * I n (π / 2) := mul_pos (k _) I_pos
+  obtain ⟨z, hz⟩ : ∃ z : ℤ, (sinPoly n).eval₂ (Int.castRingHom ℝ) (a / b) * b ^ (2 * n + 1) = z :=
+    is_integer a b ((sinPoly_natDegree_le _).trans (by linarith))
+  have e := sinPoly_add_cosPoly_eval (π / 2) n
+  rw [cos_pi_div_two, sin_pi_div_two, mul_zero, mul_one, add_zero] at e
+  have : a ^ (2 * n + 1) / n ! * I n (π / 2) =
+      eval₂ (Int.castRingHom ℝ) (π / 2) (sinPoly n) * b ^ (2 * n + 1) := by
+    nth_rw 2 [h] at e
+    field_simp at e ⊢
+    linear_combination e
+  have : (0 : ℝ) < z ∧ (z : ℝ) < 1 := by simp [← hz, ← h, ← this, hn', hn]
+  norm_cast at this
+  omega
+
+end
diff --git a/Mathlib/Data/Real/Pi/Wallis.lean b/Mathlib/Data/Real/Pi/Wallis.lean
index 3b407345328a5..23aca82d62f0e 100644
--- a/Mathlib/Data/Real/Pi/Wallis.lean
+++ b/Mathlib/Data/Real/Pi/Wallis.lean
@@ -84,9 +84,7 @@ theorem le_W (k : ℕ) : ((2 : ℝ) * k + 1) / (2 * k + 2) * (π / 2) ≤ W k :=
   rw [W_eq_integral_sin_pow_div_integral_sin_pow, le_div_iff₀ (integral_sin_pow_pos _)]
   convert integral_sin_pow_succ_le (2 * k + 1)
   rw [integral_sin_pow (2 * k)]
-  simp only [sin_zero, ne_eq, add_eq_zero, and_false, not_false_eq_true, zero_pow, cos_zero,
-    mul_one, sin_pi, cos_pi, mul_neg, neg_zero, sub_self, zero_div, zero_add]
-  norm_cast
+  simp
 
 theorem tendsto_W_nhds_pi_div_two : Tendsto W atTop (𝓝 <| π / 2) := by
   refine tendsto_of_tendsto_of_tendsto_of_le_of_le ?_ tendsto_const_nhds le_W W_le
diff --git a/Mathlib/Data/Real/Pointwise.lean b/Mathlib/Data/Real/Pointwise.lean
index bb83de2135f9c..8672dab314d36 100644
--- a/Mathlib/Data/Real/Pointwise.lean
+++ b/Mathlib/Data/Real/Pointwise.lean
@@ -6,6 +6,7 @@ Authors: Yaël Dillies, Eric Wieser
 import Mathlib.Algebra.Order.Module.OrderedSMul
 import Mathlib.Algebra.Order.Module.Pointwise
 import Mathlib.Data.Real.Archimedean
+import Mathlib.Data.Set.Pointwise.SMul
 
 /-!
 # Pointwise operations on sets of reals
diff --git a/Mathlib/Data/Real/Sqrt.lean b/Mathlib/Data/Real/Sqrt.lean
index 6780b6ca40871..48f8aad052901 100644
--- a/Mathlib/Data/Real/Sqrt.lean
+++ b/Mathlib/Data/Real/Sqrt.lean
@@ -127,8 +127,7 @@ theorem continuous_sqrt : Continuous (√· : ℝ → ℝ) :=
 
 theorem sqrt_eq_zero_of_nonpos (h : x ≤ 0) : sqrt x = 0 := by simp [sqrt, Real.toNNReal_eq_zero.2 h]
 
-theorem sqrt_nonneg (x : ℝ) : 0 ≤ √x :=
-  NNReal.coe_nonneg _
+@[simp] theorem sqrt_nonneg (x : ℝ) : 0 ≤ √x := NNReal.coe_nonneg _
 
 @[simp]
 theorem mul_self_sqrt (h : 0 ≤ x) : √x * √x = x := by
diff --git a/Mathlib/Data/Real/StarOrdered.lean b/Mathlib/Data/Real/StarOrdered.lean
index 3bb4a4ddd017d..ef1ad46ac2800 100644
--- a/Mathlib/Data/Real/StarOrdered.lean
+++ b/Mathlib/Data/Real/StarOrdered.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kim Morrison
 -/
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Data.Real.Sqrt
 
 /-! # `ℝ` and `ℝ≥0` are *-ordered rings. -/
diff --git a/Mathlib/Data/Rel.lean b/Mathlib/Data/Rel.lean
index a0fbadb670218..27adb8a69b26d 100644
--- a/Mathlib/Data/Rel.lean
+++ b/Mathlib/Data/Rel.lean
@@ -333,7 +333,7 @@ theorem graph_injective : Injective (graph : (α → β) → Rel α β) := by
 
 @[simp] lemma graph_inj {f g : α → β} : f.graph = g.graph ↔ f = g := graph_injective.eq_iff
 
-theorem graph_id : graph id = @Eq α := by simp  (config := { unfoldPartialApp := true }) [graph]
+theorem graph_id : graph id = @Eq α := by simp (config := { unfoldPartialApp := true }) [graph]
 
 theorem graph_comp {f : β → γ} {g : α → β} : graph (f ∘ g) = Rel.comp (graph g) (graph f) := by
   ext x y
diff --git a/Mathlib/Data/Semiquot.lean b/Mathlib/Data/Semiquot.lean
index 5557146ba9262..fb1f34b2cb8c3 100644
--- a/Mathlib/Data/Semiquot.lean
+++ b/Mathlib/Data/Semiquot.lean
@@ -34,7 +34,7 @@ namespace Semiquot
 variable {α : Type*} {β : Type*}
 
 instance : Membership α (Semiquot α) :=
-  ⟨fun a q => a ∈ q.s⟩
+  ⟨fun q a => a ∈ q.s⟩
 
 /-- Construct a `Semiquot α` from `h : a ∈ s` where `s : Set α`. -/
 def mk {a : α} {s : Set α} (h : a ∈ s) : Semiquot α :=
diff --git a/Mathlib/Data/Seq/Computation.lean b/Mathlib/Data/Seq/Computation.lean
index 9b4a265a6475b..9ff96e3dd47ef 100644
--- a/Mathlib/Data/Seq/Computation.lean
+++ b/Mathlib/Data/Seq/Computation.lean
@@ -2,8 +2,6 @@
 Copyright (c) 2017 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
-
-Coinductive formalization of unbounded computations.
 -/
 import Mathlib.Data.Nat.Find
 import Mathlib.Data.Stream.Init
@@ -236,18 +234,19 @@ section Bisim
 
 variable (R : Computation α → Computation α → Prop)
 
-/-- bisimilarity relation-/
+/-- bisimilarity relation -/
 local infixl:50 " ~ " => R
 
-/-- Bisimilarity over a sum of `Computation`s-/
+/-- Bisimilarity over a sum of `Computation`s -/
 def BisimO : α ⊕ (Computation α) → α ⊕ (Computation α) → Prop
   | Sum.inl a, Sum.inl a' => a = a'
   | Sum.inr s, Sum.inr s' => R s s'
   | _, _ => False
 
 attribute [simp] BisimO
+attribute [nolint simpNF] BisimO.eq_3
 
-/-- Attribute expressing bisimilarity over two `Computation`s-/
+/-- Attribute expressing bisimilarity over two `Computation`s -/
 def IsBisimulation :=
   ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → BisimO R (destruct s₁) (destruct s₂)
 
@@ -280,7 +279,7 @@ end Bisim
 -- It's more of a stretch to use ∈ for this relation, but it
 -- asserts that the computation limits to the given value.
 /-- Assertion that a `Computation` limits to a given value-/
-protected def Mem (a : α) (s : Computation α) :=
+protected def Mem (s : Computation α) (a : α) :=
   some a ∈ s.1
 
 instance : Membership α (Computation α) :=
@@ -504,7 +503,8 @@ theorem length_thinkN (s : Computation α) [_h : Terminates s] (n) :
 
 theorem eq_thinkN {s : Computation α} {a n} (h : Results s a n) : s = thinkN (pure a) n := by
   revert s
-  induction' n with n IH <;> intro s <;> apply recOn s (fun a' => _) fun s => _ <;> intro a h
+  induction n with | zero => _ | succ n IH => _ <;>
+  (intro s; apply recOn s (fun a' => _) fun s => _) <;> intro a h
   · rw [← eq_of_pure_mem h.mem]
     rfl
   · cases' of_results_think h with n h
@@ -519,7 +519,7 @@ theorem eq_thinkN' (s : Computation α) [_h : Terminates s] :
     s = thinkN (pure (get s)) (length s) :=
   eq_thinkN (results_of_terminates _)
 
-/-- Recursor based on membership-/
+/-- Recursor based on membership -/
 def memRecOn {C : Computation α → Sort v} {a s} (M : a ∈ s) (h1 : C (pure a))
     (h2 : ∀ s, C s → C (think s)) : C s := by
   haveI T := terminates_of_mem M
@@ -694,7 +694,8 @@ theorem length_bind (s : Computation α) (f : α → Computation β) [_T1 : Term
 
 theorem of_results_bind {s : Computation α} {f : α → Computation β} {b k} :
     Results (bind s f) b k → ∃ a m n, Results s a m ∧ Results (f a) b n ∧ k = n + m := by
-  induction' k with n IH generalizing s <;> apply recOn s (fun a => _) fun s' => _ <;> intro e h
+  induction k generalizing s with | zero => _ | succ n IH => _
+    <;> apply recOn s (fun a => _) fun s' => _ <;> intro e h
   · simp only [ret_bind] at h
     exact ⟨e, _, _, results_pure _, h, rfl⟩
   · have := congr_arg head (eq_thinkN h)
@@ -753,7 +754,7 @@ theorem exists_of_mem_map {f : α → β} {b : β} {s : Computation α} (h : b 
   exact ⟨a, as, mem_unique (ret_mem _) fb⟩
 
 instance terminates_map (f : α → β) (s : Computation α) [Terminates s] : Terminates (map f s) := by
-  rw [← bind_pure]; exact terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))
+  rw [← bind_pure]; exact terminates_of_mem (mem_bind (get_mem s) (get_mem (α := β) (f (get s))))
 
 theorem terminates_map_iff (f : α → β) (s : Computation α) : Terminates (map f s) ↔ Terminates s :=
   ⟨fun ⟨⟨_, h⟩⟩ =>
@@ -820,7 +821,7 @@ theorem orElse_empty (c : Computation α) : (c <|> empty α) = c := by
 def Equiv (c₁ c₂ : Computation α) : Prop :=
   ∀ a, a ∈ c₁ ↔ a ∈ c₂
 
-/-- equivalence relation for computations-/
+/-- equivalence relation for computations -/
 scoped infixl:50 " ~ " => Equiv
 
 @[refl]
@@ -1041,7 +1042,7 @@ theorem map_congr {s1 s2 : Computation α} {f : α → β}
   rw [← lift_eq_iff_equiv]
   exact liftRel_map Eq _ ((lift_eq_iff_equiv _ _).2 h1) fun {a} b => congr_arg _
 
-/-- Alternate definition of `LiftRel` over relations between `Computation`s-/
+/-- Alternate definition of `LiftRel` over relations between `Computation`s -/
 def LiftRelAux (R : α → β → Prop) (C : Computation α → Computation β → Prop) :
     α ⊕ (Computation α) → β ⊕ (Computation β) → Prop
   | Sum.inl a, Sum.inl b => R a b
diff --git a/Mathlib/Data/Seq/Parallel.lean b/Mathlib/Data/Seq/Parallel.lean
index 95a1d67f43238..ee4bdac0f33db 100644
--- a/Mathlib/Data/Seq/Parallel.lean
+++ b/Mathlib/Data/Seq/Parallel.lean
@@ -128,7 +128,6 @@ theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : T
     · have C : corec parallel.aux1 (l, S) = pure a := by
         apply destruct_eq_pure
         rw [corec_eq, parallel.aux1]
-        dsimp only []
         rw [h]
         simp only [rmap]
       rw [C]
@@ -146,7 +145,6 @@ theorem terminates_parallel {S : WSeq (Computation α)} {c} (h : c ∈ S) [T : T
     · have C : corec parallel.aux1 (l, S) = pure a := by
         apply destruct_eq_pure
         rw [corec_eq, parallel.aux1]
-        dsimp only []
         rw [h]
         simp only [rmap]
       rw [C]
diff --git a/Mathlib/Data/Seq/Seq.lean b/Mathlib/Data/Seq/Seq.lean
index 974c780ddf181..e1c08ea1f9bdc 100644
--- a/Mathlib/Data/Seq/Seq.lean
+++ b/Mathlib/Data/Seq/Seq.lean
@@ -126,7 +126,7 @@ def tail (s : Seq α) : Seq α :=
     exact al n'⟩
 
 /-- member definition for `Seq`-/
-protected def Mem (a : α) (s : Seq α) :=
+protected def Mem (s : Seq α) (a : α) :=
   some a ∈ s.1
 
 instance : Membership α (Seq α) :=
@@ -258,7 +258,7 @@ theorem mem_rec_on {C : Seq α → Prop} {a s} (M : a ∈ s)
     rw [h_eq] at e
     apply h1 _ _ (Or.inr (IH e))
 
-/-- Corecursor over pairs of `Option` values-/
+/-- Corecursor over pairs of `Option` values -/
 def Corec.f (f : β → Option (α × β)) : Option β → Option α × Option β
   | none => (none, none)
   | some b =>
@@ -315,8 +315,9 @@ def BisimO : Option (Seq1 α) → Option (Seq1 α) → Prop
   | _, _ => False
 
 attribute [simp] BisimO
+attribute [nolint simpNF] BisimO.eq_3
 
-/-- a relation is bisimilar if it meets the `BisimO` test-/
+/-- a relation is bisimilar if it meets the `BisimO` test -/
 def IsBisimulation :=
   ∀ ⦃s₁ s₂⦄, s₁ ~ s₂ → BisimO R (destruct s₁) (destruct s₂)
 
diff --git a/Mathlib/Data/Seq/WSeq.lean b/Mathlib/Data/Seq/WSeq.lean
index 9d86f9ce4e599..4817d9024ec51 100644
--- a/Mathlib/Data/Seq/WSeq.lean
+++ b/Mathlib/Data/Seq/WSeq.lean
@@ -106,8 +106,8 @@ def recOn {C : WSeq α → Sort v} (s : WSeq α) (h1 : C nil) (h2 : ∀ x s, C (
   Seq.recOn s h1 fun o => Option.recOn o h3 h2
 
 /-- membership for weak sequences-/
-protected def Mem (a : α) (s : WSeq α) :=
-  Seq.Mem (some a) s
+protected def Mem (s : WSeq α) (a : α) :=
+  Seq.Mem s (some a)
 
 instance membership : Membership α (WSeq α) :=
   ⟨WSeq.Mem⟩
@@ -394,6 +394,7 @@ def LiftRelO (R : α → β → Prop) (C : WSeq α → WSeq β → Prop) :
   | none, none => True
   | some (a, s), some (b, t) => R a b ∧ C s t
   | _, _ => False
+attribute [nolint simpNF] LiftRelO.eq_3
 
 theorem LiftRelO.imp {R S : α → β → Prop} {C D : WSeq α → WSeq β → Prop} (H1 : ∀ a b, R a b → S a b)
     (H2 : ∀ s t, C s t → D s t) : ∀ {o p}, LiftRelO R C o p → LiftRelO S D o p
@@ -406,7 +407,7 @@ theorem LiftRelO.imp_right (R : α → β → Prop) {C D : WSeq α → WSeq β 
     (H : ∀ s t, C s t → D s t) {o p} : LiftRelO R C o p → LiftRelO R D o p :=
   LiftRelO.imp (fun _ _ => id) H
 
-/-- Definition of bisimilarity for weak sequences-/
+/-- Definition of bisimilarity for weak sequences -/
 @[simp]
 def BisimO (R : WSeq α → WSeq α → Prop) : Option (α × WSeq α) → Option (α × WSeq α) → Prop :=
   LiftRelO (· = ·) R
@@ -684,7 +685,7 @@ theorem append_nil (s : WSeq α) : append s nil = s :=
 theorem append_assoc (s t u : WSeq α) : append (append s t) u = append s (append t u) :=
   Seq.append_assoc _ _ _
 
-/-- auxiliary definition of tail over weak sequences-/
+/-- auxiliary definition of tail over weak sequences -/
 @[simp]
 def tail.aux : Option (α × WSeq α) → Computation (Option (α × WSeq α))
   | none => Computation.pure none
@@ -694,7 +695,7 @@ theorem destruct_tail (s : WSeq α) : destruct (tail s) = destruct s >>= tail.au
   simp only [tail, destruct_flatten, tail.aux]; rw [← bind_pure_comp, LawfulMonad.bind_assoc]
   apply congr_arg; ext1 (_ | ⟨a, s⟩) <;> apply (@pure_bind Computation _ _ _ _ _ _).trans _ <;> simp
 
-/-- auxiliary definition of drop over weak sequences-/
+/-- auxiliary definition of drop over weak sequences -/
 @[simp]
 def drop.aux : ℕ → Option (α × WSeq α) → Computation (Option (α × WSeq α))
   | 0 => Computation.pure
@@ -716,12 +717,9 @@ theorem head_terminates_of_head_tail_terminates (s : WSeq α) [T : Terminates (h
     Terminates (head s) :=
   (head_terminates_iff _).2 <| by
     rcases (head_terminates_iff _).1 T with ⟨⟨a, h⟩⟩
-    simp? [tail] at h says simp only [tail, destruct_flatten] at h
+    simp? [tail] at h says simp only [tail, destruct_flatten, bind_map_left] at h
     rcases exists_of_mem_bind h with ⟨s', h1, _⟩
-    unfold Functor.map at h1
-    exact
-      let ⟨t, h3, _⟩ := Computation.exists_of_mem_map h1
-      Computation.terminates_of_mem h3
+    exact terminates_of_mem h1
 
 theorem destruct_some_of_destruct_tail_some {s : WSeq α} {a} (h : some a ∈ destruct (tail s)) :
     ∃ a', some a' ∈ destruct s := by
@@ -816,7 +814,7 @@ theorem eq_or_mem_iff_mem {s : WSeq α} {a a' s'} :
   · cases' this with i1 i2
     rw [i1, i2]
     cases' s' with f al
-    dsimp only [cons, (· ∈ ·), WSeq.Mem, Seq.Mem, Seq.cons]
+    dsimp only [cons, Membership.mem, WSeq.Mem, Seq.Mem, Seq.cons]
     have h_a_eq_a' : a = a' ↔ some (some a) = some (some a') := by simp
     rw [h_a_eq_a']
     refine ⟨Stream'.eq_or_mem_of_mem_cons, fun o => ?_⟩
@@ -1330,7 +1328,7 @@ theorem liftRel_map {δ} (R : α → β → Prop) (S : γ → δ → Prop) {s1 :
 theorem map_congr (f : α → β) {s t : WSeq α} (h : s ~ʷ t) : map f s ~ʷ map f t :=
   liftRel_map _ _ h fun {_ _} => congr_arg _
 
-/-- auxiliary definition of `destruct_append` over weak sequences-/
+/-- auxiliary definition of `destruct_append` over weak sequences -/
 @[simp]
 def destruct_append.aux (t : WSeq α) : Option (α × WSeq α) → Computation (Option (α × WSeq α))
   | none => destruct t
@@ -1349,7 +1347,7 @@ theorem destruct_append (s t : WSeq α) :
     · refine ⟨nil, t, ?_, ?_⟩ <;> simp
   · exact ⟨s, t, rfl, rfl⟩
 
-/-- auxiliary definition of `destruct_join` over weak sequences-/
+/-- auxiliary definition of `destruct_join` over weak sequences -/
 @[simp]
 def destruct_join.aux : Option (WSeq α × WSeq (WSeq α)) → Computation (Option (α × WSeq α))
   | none => Computation.pure none
@@ -1407,7 +1405,7 @@ theorem liftRel_join.lem (R : α → β → Prop) {S T} {U : WSeq α → WSeq β
           U s1 s2)
     {a} (ma : a ∈ destruct (join S)) : ∃ b, b ∈ destruct (join T) ∧ LiftRelO R U a b := by
   cases' exists_results_of_mem ma with n h; clear ma; revert S T ST a
-  induction' n using Nat.strongInductionOn with n IH
+  induction' n using Nat.strongRecOn with n IH
   intro S T ST a ra; simp only [destruct_join] at ra
   exact
     let ⟨o, m, k, rs1, rs2, en⟩ := of_results_bind ra
@@ -1622,3 +1620,5 @@ instance lawfulMonad : LawfulMonad WSeq :=
 end WSeq
 
 end Stream'
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Data/Set/Basic.lean b/Mathlib/Data/Set/Basic.lean
index 040bc43d60d9f..691836860b538 100644
--- a/Mathlib/Data/Set/Basic.lean
+++ b/Mathlib/Data/Set/Basic.lean
@@ -435,6 +435,11 @@ theorem Nonempty.to_type : s.Nonempty → Nonempty α := fun ⟨x, _⟩ => ⟨x
 instance univ.nonempty [Nonempty α] : Nonempty (↥(Set.univ : Set α)) :=
   Set.univ_nonempty.to_subtype
 
+-- Redeclare for refined keys
+-- `Nonempty (@Subtype _ (@Membership.mem _ (Set _) _ (@Top.top (Set _) _)))`
+instance instNonemptyTop [Nonempty α] : Nonempty (⊤ : Set α) :=
+  inferInstanceAs (Nonempty (univ : Set α))
+
 theorem nonempty_of_nonempty_subtype [Nonempty (↥s)] : s.Nonempty :=
   nonempty_subtype.mp ‹_›
 
@@ -617,11 +622,11 @@ theorem union_self (a : Set α) : a ∪ a = a :=
 
 @[simp]
 theorem union_empty (a : Set α) : a ∪ ∅ = a :=
-  ext fun _ => or_false_iff _
+  ext fun _ => iff_of_eq (or_false _)
 
 @[simp]
 theorem empty_union (a : Set α) : ∅ ∪ a = a :=
-  ext fun _ => false_or_iff _
+  ext fun _ => iff_of_eq (false_or _)
 
 theorem union_comm (a b : Set α) : a ∪ b = b ∪ a :=
   ext fun _ => or_comm
@@ -735,11 +740,11 @@ theorem inter_self (a : Set α) : a ∩ a = a :=
 
 @[simp]
 theorem inter_empty (a : Set α) : a ∩ ∅ = ∅ :=
-  ext fun _ => and_false_iff _
+  ext fun _ => iff_of_eq (and_false _)
 
 @[simp]
 theorem empty_inter (a : Set α) : ∅ ∩ a = ∅ :=
-  ext fun _ => false_and_iff _
+  ext fun _ => iff_of_eq (false_and _)
 
 theorem inter_comm (a b : Set α) : a ∩ b = b ∩ a :=
   ext fun _ => and_comm
@@ -962,7 +967,7 @@ theorem insert_union_distrib (a : α) (s t : Set α) : insert a (s ∪ t) = inse
   ext fun _ => or_or_distrib_left
 
 theorem insert_inj (ha : a ∉ s) : insert a s = insert b s ↔ a = b :=
-  ⟨fun h => eq_of_not_mem_of_mem_insert (h.subst <| mem_insert a s) ha,
+  ⟨fun h => eq_of_not_mem_of_mem_insert (h ▸ mem_insert a s) ha,
     congr_arg (fun x => insert x s)⟩
 
 -- useful in proofs by induction
@@ -1129,7 +1134,7 @@ theorem sep_eq_self_iff_mem_true : { x ∈ s | p x } = s ↔ ∀ x ∈ s, p x :=
 
 @[simp]
 theorem sep_eq_empty_iff_mem_false : { x ∈ s | p x } = ∅ ↔ ∀ x ∈ s, ¬p x := by
-  simp_rw [Set.ext_iff, mem_sep_iff, mem_empty_iff_false, iff_false_iff, not_and]
+  simp_rw [Set.ext_iff, mem_sep_iff, mem_empty_iff_false, iff_false, not_and]
 
 --Porting note (#10618): removed `simp` attribute because `simp` can prove it
 theorem sep_true : { x ∈ s | True } = s :=
@@ -1182,7 +1187,7 @@ theorem Nonempty.subset_singleton_iff (h : s.Nonempty) : s ⊆ {a} ↔ s = {a} :
   subset_singleton_iff_eq.trans <| or_iff_right h.ne_empty
 
 theorem ssubset_singleton_iff {s : Set α} {x : α} : s ⊂ {x} ↔ s = ∅ := by
-  rw [ssubset_iff_subset_ne, subset_singleton_iff_eq, or_and_right, and_not_self_iff, or_false_iff,
+  rw [ssubset_iff_subset_ne, subset_singleton_iff_eq, or_and_right, and_not_self_iff, or_false,
     and_iff_left_iff_imp]
   exact fun h => h ▸ (singleton_ne_empty _).symm
 
@@ -1576,7 +1581,7 @@ theorem insert_diff_self_of_not_mem {a : α} {s : Set α} (h : a ∉ s) : insert
 theorem insert_diff_eq_singleton {a : α} {s : Set α} (h : a ∉ s) : insert a s \ s = {a} := by
   ext
   rw [Set.mem_diff, Set.mem_insert_iff, Set.mem_singleton_iff, or_and_right, and_not_self_iff,
-    or_false_iff, and_iff_left_iff_imp]
+    or_false, and_iff_left_iff_imp]
   rintro rfl
   exact h
 
@@ -1953,7 +1958,7 @@ open Set
 
 namespace Function
 
-variable {ι : Sort*} {α : Type*} {β : Type*} {f : α → β}
+variable {α : Type*} {β : Type*}
 
 theorem Injective.nonempty_apply_iff {f : Set α → Set β} (hf : Injective f) (h2 : f ∅ = ∅)
     {s : Set α} : (f s).Nonempty ↔ s.Nonempty := by
@@ -1973,9 +1978,8 @@ section Inclusion
 variable {α : Type*} {s t u : Set α}
 
 /-- `inclusion` is the "identity" function between two subsets `s` and `t`, where `s ⊆ t` -/
-def inclusion (h : s ⊆ t) : s → t := fun x : s => (⟨x, h x.2⟩ : t)
+abbrev inclusion (h : s ⊆ t) : s → t := fun x : s => (⟨x, h x.2⟩ : t)
 
-@[simp]
 theorem inclusion_self (x : s) : inclusion Subset.rfl x = x := by
   cases x
   rfl
@@ -2012,7 +2016,6 @@ theorem val_comp_inclusion (h : s ⊆ t) : Subtype.val ∘ inclusion h = Subtype
 theorem inclusion_injective (h : s ⊆ t) : Injective (inclusion h)
   | ⟨_, _⟩, ⟨_, _⟩ => Subtype.ext_iff_val.2 ∘ Subtype.ext_iff_val.1
 
-@[simp]
 theorem inclusion_inj (h : s ⊆ t) {x y : s} : inclusion h x = inclusion h y ↔ x = y :=
   (inclusion_injective h).eq_iff
 
@@ -2022,11 +2025,9 @@ theorem eq_of_inclusion_surjective {s t : Set α} {h : s ⊆ t}
   obtain ⟨y, hy⟩ := h_surj ⟨x, hx⟩
   exact mem_of_eq_of_mem (congr_arg Subtype.val hy).symm y.prop
 
-@[simp]
 theorem inclusion_le_inclusion [Preorder α] {s t : Set α} (h : s ⊆ t) {x y : s} :
     inclusion h x ≤ inclusion h y ↔ x ≤ y := Iff.rfl
 
-@[simp]
 theorem inclusion_lt_inclusion [Preorder α] {s t : Set α} (h : s ⊆ t) {x y : s} :
     inclusion h x < inclusion h y ↔ x < y := Iff.rfl
 
@@ -2140,7 +2141,7 @@ end Monotone
 
 /-! ### Disjoint sets -/
 
-variable {α β : Type*} {s t u : Set α} {f : α → β}
+variable {α : Type*} {s t u : Set α}
 
 namespace Disjoint
 
@@ -2172,3 +2173,5 @@ end Disjoint
 
 @[simp] theorem Prop.compl_singleton (p : Prop) : ({p}ᶜ : Set Prop) = {¬p} :=
   ext fun q ↦ by simpa [@Iff.comm q] using not_iff
+
+set_option linter.style.longFile 2300
diff --git a/Mathlib/Data/Set/Card.lean b/Mathlib/Data/Set/Card.lean
index d19e63e1fa630..57aba3430a761 100644
--- a/Mathlib/Data/Set/Card.lean
+++ b/Mathlib/Data/Set/Card.lean
@@ -544,7 +544,7 @@ theorem nonempty_of_ncard_ne_zero (hs : s.ncard ≠ 0) : s.Nonempty := by
   rw [nonempty_iff_ne_empty]; rintro rfl; simp at hs
 
 @[simp] theorem ncard_singleton (a : α) : ({a} : Set α).ncard = 1 := by
-  simpa [ncard, encard_singleton] using ENat.toNat_coe 1
+  simp [ncard]
 
 theorem ncard_singleton_inter (a : α) (s : Set α) : ({a} ∩ s).ncard ≤ 1 := by
   rw [← Nat.cast_le (α := ℕ∞), (toFinite _).cast_ncard_eq, Nat.cast_one]
@@ -616,6 +616,15 @@ theorem ncard_exchange' {a b : α} (ha : a ∉ s) (hb : b ∈ s) :
   rw [← ncard_exchange ha hb, ← singleton_union, ← singleton_union, union_diff_distrib,
     @diff_singleton_eq_self _ b {a} fun h ↦ ha (by rwa [← mem_singleton_iff.mp h])]
 
+lemma odd_card_insert_iff {a : α} (hs : s.Finite := by toFinite_tac) (ha : a ∉ s) :
+    Odd (insert a s).ncard ↔ Even s.ncard := by
+  rw [ncard_insert_of_not_mem ha hs, Nat.odd_add]
+  simp only [Nat.odd_add, ← Nat.not_even_iff_odd, Nat.not_even_one, iff_false, Decidable.not_not]
+
+lemma even_card_insert_iff {a : α} (hs : s.Finite := by toFinite_tac) (ha : a ∉ s) :
+    Even (insert a s).ncard ↔ Odd s.ncard := by
+  rw [ncard_insert_of_not_mem ha hs, Nat.even_add_one, Nat.not_even_iff_odd]
+
 end InsertErase
 
 variable {f : α → β}
@@ -753,7 +762,7 @@ theorem surj_on_of_inj_on_of_ncard_le {t : Set β} (f : ∀ a ∈ s, β) (hf : 
   have hft := ht.fintype
   have hft' := Fintype.ofInjective f' finj
   set f'' : ∀ a, a ∈ s.toFinset → β := fun a h ↦ f a (by simpa using h)
-  convert @Finset.surj_on_of_inj_on_of_card_le _ _ _ t.toFinset f'' _ _ _ _ (by simpa)
+  convert @Finset.surj_on_of_inj_on_of_card_le _ _ _ t.toFinset f'' _ _ _ _ (by simpa) using 1
   · simp
   · simp [hf]
   · intros a₁ a₂ ha₁ ha₂ h
@@ -947,7 +956,8 @@ theorem exists_eq_insert_iff_ncard (hs : s.Finite := by toFinite_tac) :
     convert Iff.rfl using 2; simp only [Finite.mem_toFinset]
     ext x
     simp [Finset.ext_iff, Set.ext_iff]
-  simp only [ht.ncard, exists_prop, add_eq_zero, and_false, iff_false, not_exists, not_and]
+  simp only [ht.ncard, exists_prop, add_eq_zero, and_false, iff_false, not_exists, not_and,
+    reduceCtorEq]
   rintro x - rfl
   exact ht (hs.insert x)
 
@@ -991,6 +1001,12 @@ theorem one_lt_ncard_iff (hs : s.Finite := by toFinite_tac) :
   rw [one_lt_ncard hs]
   simp only [exists_prop, exists_and_left]
 
+lemma one_lt_ncard_of_nonempty_of_even (hs : Set.Finite s) (hn : Set.Nonempty s := by toFinite_tac)
+    (he : Even (s.ncard)) : 1 < s.ncard := by
+  rw [← Set.ncard_pos hs] at hn
+  have : s.ncard ≠ 1 := fun h ↦ by simp [h] at he
+  omega
+
 theorem two_lt_ncard_iff (hs : s.Finite := by toFinite_tac) :
     2 < s.ncard ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c := by
   simp_rw [ncard_eq_toFinset_card _ hs, Finset.two_lt_card_iff, Finite.mem_toFinset]
@@ -1034,7 +1050,4 @@ theorem ncard_eq_three : s.ncard = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y 
   simp [h]
 
 end ncard
-
-@[deprecated (since := "2023-12-27")] alias ncard_le_of_subset := ncard_le_ncard
-
 end Set
diff --git a/Mathlib/Data/Set/Constructions.lean b/Mathlib/Data/Set/Constructions.lean
index 9011537864b94..6690c61075cd1 100644
--- a/Mathlib/Data/Set/Constructions.lean
+++ b/Mathlib/Data/Set/Constructions.lean
@@ -59,12 +59,14 @@ theorem finiteInter_mem (cond : FiniteInter S) (F : Finset (Set α)) :
 
 theorem finiteInterClosure_insert {A : Set α} (cond : FiniteInter S) (P)
     (H : P ∈ finiteInterClosure (insert A S)) : P ∈ S ∨ ∃ Q ∈ S, P = A ∩ Q := by
-  induction' H with S h T1 T2 _ _ h1 h2
-  · cases h
+  induction H with
+  | basic h =>
+    cases h
     · exact Or.inr ⟨Set.univ, cond.univ_mem, by simpa⟩
     · exact Or.inl (by assumption)
-  · exact Or.inl cond.univ_mem
-  · rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)
+  | univ => exact Or.inl cond.univ_mem
+  | @inter T1 T2 _ _ h1 h2 =>
+    rcases h1 with (h | ⟨Q, hQ, rfl⟩) <;> rcases h2 with (i | ⟨R, hR, rfl⟩)
     · exact Or.inl (cond.inter_mem h i)
     · exact
         Or.inr ⟨T1 ∩ R, cond.inter_mem h hR, by simp only [← Set.inter_assoc, Set.inter_comm _ A]⟩
diff --git a/Mathlib/Data/Set/Countable.lean b/Mathlib/Data/Set/Countable.lean
index bd1f3720faffe..4add89c35bb25 100644
--- a/Mathlib/Data/Set/Countable.lean
+++ b/Mathlib/Data/Set/Countable.lean
@@ -101,8 +101,8 @@ lemma range_enumerateCountable_of_mem {s : Set α} (h : s.Countable) {default :
 lemma enumerateCountable_mem {s : Set α} (h : s.Countable) {default : α} (h_mem : default ∈ s)
     (n : ℕ) :
     enumerateCountable h default n ∈ s := by
-  conv_rhs => rw [← range_enumerateCountable_of_mem h h_mem]
-  exact mem_range_self n
+  convert mem_range_self n
+  exact (range_enumerateCountable_of_mem h h_mem).symm
 
 end Enumerate
 
@@ -226,7 +226,7 @@ theorem Countable.of_diff {s t : Set α} (h : (s \ t).Countable) (ht : t.Countab
 
 @[simp]
 theorem countable_insert {s : Set α} {a : α} : (insert a s).Countable ↔ s.Countable := by
-  simp only [insert_eq, countable_union, countable_singleton, true_and_iff]
+  simp only [insert_eq, countable_union, countable_singleton, true_and]
 
 protected theorem Countable.insert {s : Set α} (a : α) (h : s.Countable) : (insert a s).Countable :=
   countable_insert.2 h
@@ -257,6 +257,10 @@ theorem countable_setOf_finite_subset {s : Set α} (hs : s.Countable) :
   lift t to Finset s using ht.of_finite_image Subtype.val_injective.injOn
   exact mem_range_self _
 
+/-- The set of finite sets in a countable type is countable. -/
+theorem Countable.setOf_finite [Countable α] : {s : Set α | s.Finite}.Countable := by
+  simpa using countable_setOf_finite_subset countable_univ
+
 theorem countable_univ_pi {π : α → Type*} [Finite α] {s : ∀ a, Set (π a)}
     (hs : ∀ a, (s a).Countable) : (pi univ s).Countable :=
   have := fun a ↦ (hs a).to_subtype; .of_equiv _ (Equiv.Set.univPi s).symm
@@ -287,9 +291,9 @@ theorem countable_setOf_nonempty_of_disjoint {f : β → Set α}
   have A : Injective F := by
     rintro ⟨t, ht⟩ ⟨t', ht'⟩ htt'
     have A : (f t ∩ f t').Nonempty := by
-      refine ⟨F ⟨t, ht⟩, hF _, ?_⟩
+      refine ⟨F ⟨t, ht⟩, hF ⟨t, _⟩, ?_⟩
       rw [htt']
-      exact hF _
+      exact hF ⟨t', _⟩
     simp only [Subtype.mk.injEq]
     by_contra H
     exact not_disjoint_iff_nonempty_inter.2 A (hf H)
diff --git a/Mathlib/Data/Set/Defs.lean b/Mathlib/Data/Set/Defs.lean
index 86d21566248d4..4ffdc4697e45f 100644
--- a/Mathlib/Data/Set/Defs.lean
+++ b/Mathlib/Data/Set/Defs.lean
@@ -50,7 +50,7 @@ def setOf {α : Type u} (p : α → Prop) : Set α :=
 namespace Set
 
 /-- Membership in a set -/
-protected def Mem (a : α) (s : Set α) : Prop :=
+protected def Mem (s : Set α) (a : α) : Prop :=
   s a
 
 instance : Membership α (Set α) :=
diff --git a/Mathlib/Data/Set/Enumerate.lean b/Mathlib/Data/Set/Enumerate.lean
index c6313777fa313..fd4f8dbb791b1 100644
--- a/Mathlib/Data/Set/Enumerate.lean
+++ b/Mathlib/Data/Set/Enumerate.lean
@@ -88,7 +88,8 @@ theorem enumerate_inj {n₁ n₂ : ℕ} {a : α} {s : Set α} (h_sel : ∀ s a,
       cases h : sel s with
       /- Porting note: The original covered both goals with just `simp_all <;> tauto` -/
       | none =>
-        simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h]
+        simp_all only [add_comm, self_eq_add_left, Nat.add_succ, enumerate_eq_none_of_sel _ h,
+          reduceCtorEq]
       | some =>
         simp_all only [add_comm, self_eq_add_left, enumerate, Option.some.injEq,
                        Nat.add_succ, Nat.succ.injEq]
diff --git a/Mathlib/Data/Set/Finite.lean b/Mathlib/Data/Set/Finite.lean
index 933b71188dd78..cf7a2bfbb02d2 100644
--- a/Mathlib/Data/Set/Finite.lean
+++ b/Mathlib/Data/Set/Finite.lean
@@ -3,8 +3,8 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Kyle Miller
 -/
-import Mathlib.Data.Finset.Basic
 import Mathlib.Data.Finite.Basic
+import Mathlib.Data.Finset.Max
 import Mathlib.Data.Set.Functor
 import Mathlib.Data.Set.Lattice
 
@@ -280,6 +280,9 @@ section FintypeInstances
 instance fintypeUniv [Fintype α] : Fintype (@univ α) :=
   Fintype.ofEquiv α (Equiv.Set.univ α).symm
 
+-- Redeclared with appropriate keys
+instance fintypeTop [Fintype α] : Fintype (⊤ : Set α) := inferInstanceAs (Fintype (univ : Set α))
+
 /-- If `(Set.univ : Set α)` is finite then `α` is a finite type. -/
 noncomputable def fintypeOfFiniteUniv (H : (univ (α := α)).Finite) : Fintype α :=
   @Fintype.ofEquiv _ (univ : Set α) H.fintype (Equiv.Set.univ _)
@@ -894,7 +897,7 @@ theorem exists_subset_image_finite_and {f : α → β} {s : Set α} {p : Set β
     (∃ t ⊆ f '' s, t.Finite ∧ p t) ↔ ∃ t ⊆ s, t.Finite ∧ p (f '' t) := by
   classical
   simp_rw [@and_comm (_ ⊆ _), and_assoc, exists_finite_iff_finset, @and_comm (p _),
-    Finset.subset_image_iff]
+    Finset.subset_set_image_iff]
   aesop
 
 section Pi
@@ -1348,9 +1351,10 @@ theorem exists_upper_bound_image [Nonempty α] [LinearOrder β] (s : Set α) (f
 theorem Finite.iSup_biInf_of_monotone {ι ι' α : Type*} [Preorder ι'] [Nonempty ι']
     [IsDirected ι' (· ≤ ·)] [Order.Frame α] {s : Set ι} (hs : s.Finite) {f : ι → ι' → α}
     (hf : ∀ i ∈ s, Monotone (f i)) : ⨆ j, ⨅ i ∈ s, f i j = ⨅ i ∈ s, ⨆ j, f i j := by
-  induction' s, hs using Set.Finite.dinduction_on with a s _ _ ihs hf
-  · simp [iSup_const]
-  · rw [forall_mem_insert] at hf
+  induction s, hs using Set.Finite.dinduction_on with
+  | H0 => simp [iSup_const]
+  | H1 _ _ ihs =>
+    rw [forall_mem_insert] at hf
     simp only [iInf_insert, ← ihs hf.2]
     exact iSup_inf_of_monotone hf.1 fun j₁ j₂ hj => iInf₂_mono fun i hi => hf.2 i hi hj
 
@@ -1524,6 +1528,7 @@ protected theorem bddBelow [SemilatticeInf α] [Nonempty α] (s : Finset α) : B
 
 end Finset
 
+section LinearOrder
 variable [LinearOrder α] {s : Set α}
 
 /-- If a linear order does not contain any triple of elements `x < y < z`, then this type
@@ -1563,3 +1568,20 @@ theorem DirectedOn.exists_mem_subset_of_finset_subset_biUnion {α ι : Type*} {f
   rw [Set.biUnion_eq_iUnion] at hs
   haveI := hn.coe_sort
   simpa using (directed_comp.2 hc.directed_val).exists_mem_subset_of_finset_subset_biUnion hs
+
+end LinearOrder
+
+namespace List
+variable (α) [Finite α] (n : ℕ)
+
+lemma finite_length_eq : {l : List α | l.length = n}.Finite := Vector.finite
+
+lemma finite_length_lt : {l : List α | l.length < n}.Finite := by
+  convert (Finset.range n).finite_toSet.biUnion fun i _ ↦ finite_length_eq α i; ext; simp
+
+lemma finite_length_le : {l : List α | l.length ≤ n}.Finite := by
+  simpa [Nat.lt_succ_iff] using finite_length_lt α (n + 1)
+
+end List
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/Set/Function.lean b/Mathlib/Data/Set/Function.lean
index 41e0b2257e3c0..3810aaead4d4d 100644
--- a/Mathlib/Data/Set/Function.lean
+++ b/Mathlib/Data/Set/Function.lean
@@ -44,11 +44,13 @@ section restrict
 takes an argument `↥s` instead of `Subtype s`. -/
 def restrict (s : Set α) (f : ∀ a : α, π a) : ∀ a : s, π a := fun x => f x
 
+theorem restrict_def (s : Set α) : s.restrict (π := π) = fun f x ↦ f x := rfl
+
 theorem restrict_eq (f : α → β) (s : Set α) : s.restrict f = f ∘ Subtype.val :=
   rfl
 
 @[simp]
-theorem restrict_apply (f : α → β) (s : Set α) (x : s) : s.restrict f x = f x :=
+theorem restrict_apply (f : (a : α) → π a) (s : Set α) (x : s) : s.restrict f x = f x :=
   rfl
 
 theorem restrict_eq_iff {f : ∀ a, π a} {s : Set α} {g : ∀ a : s, π a} :
@@ -110,6 +112,20 @@ theorem restrict_extend_compl_range (f : α → β) (g : α → γ) (g' : β →
   classical
   exact restrict_dite_compl _ _
 
+/-- If a function `f` is restricted to a set `t`, and `s ⊆ t`, this is the restriction to `s`. -/
+@[simp]
+def restrict₂ {s t : Set α} (hst : s ⊆ t) (f : ∀ a : t, π a) : ∀ a : s, π a :=
+  fun x => f ⟨x.1, hst x.2⟩
+
+theorem restrict₂_def {s t : Set α} (hst : s ⊆ t) :
+    restrict₂ (π := π) hst = fun f x ↦ f ⟨x.1, hst x.2⟩ := rfl
+
+theorem restrict₂_comp_restrict {s t : Set α} (hst : s ⊆ t) :
+    (restrict₂ (π := π) hst) ∘ t.restrict = s.restrict := rfl
+
+theorem restrict₂_comp_restrict₂ {s t u : Set α} (hst : s ⊆ t) (htu : t ⊆ u) :
+    (restrict₂ (π := π) hst) ∘ (restrict₂ htu) = restrict₂ (hst.trans htu) := rfl
+
 theorem range_extend_subset (f : α → β) (g : α → γ) (g' : β → γ) :
     range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ := by
   classical
@@ -150,8 +166,7 @@ end restrict
 /-! ### Equality on a set -/
 section equality
 
-variable {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {p : Set γ} {f f₁ f₂ f₃ : α → β} {g g₁ g₂ : β → γ}
-  {f' f₁' f₂' : β → α} {g' : γ → β} {a : α} {b : β}
+variable {s s₁ s₂ : Set α} {f₁ f₂ f₃ : α → β} {g : β → γ} {a : α}
 
 @[simp]
 theorem eqOn_empty (f₁ f₂ : α → β) : EqOn f₁ f₂ ∅ := fun _ => False.elim
@@ -217,78 +232,7 @@ alias ⟨EqOn.comp_eq, _⟩ := eqOn_range
 
 end equality
 
-/-! ### Congruence lemmas for monotonicity and antitonicity -/
-section Order
-
-variable {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β]
-
-theorem _root_.MonotoneOn.congr (h₁ : MonotoneOn f₁ s) (h : s.EqOn f₁ f₂) : MonotoneOn f₂ s := by
-  intro a ha b hb hab
-  rw [← h ha, ← h hb]
-  exact h₁ ha hb hab
-
-theorem _root_.AntitoneOn.congr (h₁ : AntitoneOn f₁ s) (h : s.EqOn f₁ f₂) : AntitoneOn f₂ s :=
-  h₁.dual_right.congr h
-
-theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
-    StrictMonoOn f₂ s := by
-  intro a ha b hb hab
-  rw [← h ha, ← h hb]
-  exact h₁ ha hb hab
-
-theorem _root_.StrictAntiOn.congr (h₁ : StrictAntiOn f₁ s) (h : s.EqOn f₁ f₂) : StrictAntiOn f₂ s :=
-  h₁.dual_right.congr h
-
-theorem EqOn.congr_monotoneOn (h : s.EqOn f₁ f₂) : MonotoneOn f₁ s ↔ MonotoneOn f₂ s :=
-  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
-
-theorem EqOn.congr_antitoneOn (h : s.EqOn f₁ f₂) : AntitoneOn f₁ s ↔ AntitoneOn f₂ s :=
-  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
-
-theorem EqOn.congr_strictMonoOn (h : s.EqOn f₁ f₂) : StrictMonoOn f₁ s ↔ StrictMonoOn f₂ s :=
-  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
-
-theorem EqOn.congr_strictAntiOn (h : s.EqOn f₁ f₂) : StrictAntiOn f₁ s ↔ StrictAntiOn f₂ s :=
-  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
-
-end Order
-
-/-! ### Monotonicity lemmas-/
-section Mono
-
-variable {s s₁ s₂ : Set α} {f f₁ f₂ : α → β} [Preorder α] [Preorder β]
-
-theorem _root_.MonotoneOn.mono (h : MonotoneOn f s) (h' : s₂ ⊆ s) : MonotoneOn f s₂ :=
-  fun _ hx _ hy => h (h' hx) (h' hy)
-
-theorem _root_.AntitoneOn.mono (h : AntitoneOn f s) (h' : s₂ ⊆ s) : AntitoneOn f s₂ :=
-  fun _ hx _ hy => h (h' hx) (h' hy)
-
-theorem _root_.StrictMonoOn.mono (h : StrictMonoOn f s) (h' : s₂ ⊆ s) : StrictMonoOn f s₂ :=
-  fun _ hx _ hy => h (h' hx) (h' hy)
-
-theorem _root_.StrictAntiOn.mono (h : StrictAntiOn f s) (h' : s₂ ⊆ s) : StrictAntiOn f s₂ :=
-  fun _ hx _ hy => h (h' hx) (h' hy)
-
-protected theorem _root_.MonotoneOn.monotone (h : MonotoneOn f s) :
-    Monotone (f ∘ Subtype.val : s → β) :=
-  fun x y hle => h x.coe_prop y.coe_prop hle
-
-protected theorem _root_.AntitoneOn.monotone (h : AntitoneOn f s) :
-    Antitone (f ∘ Subtype.val : s → β) :=
-  fun x y hle => h x.coe_prop y.coe_prop hle
-
-protected theorem _root_.StrictMonoOn.strictMono (h : StrictMonoOn f s) :
-    StrictMono (f ∘ Subtype.val : s → β) :=
-  fun x y hlt => h x.coe_prop y.coe_prop hlt
-
-protected theorem _root_.StrictAntiOn.strictAnti (h : StrictAntiOn f s) :
-    StrictAnti (f ∘ Subtype.val : s → β) :=
-  fun x y hlt => h x.coe_prop y.coe_prop hlt
-
-end Mono
-
-variable {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {p : Set γ} {f f₁ f₂ f₃ : α → β} {g g₁ g₂ : β → γ}
+variable {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {p : Set γ} {f f₁ f₂ : α → β} {g g₁ g₂ : β → γ}
   {f' f₁' f₂' : β → α} {g' : γ → β} {a : α} {b : β}
 
 section MapsTo
@@ -417,6 +361,9 @@ theorem mapsTo_union : MapsTo f (s₁ ∪ s₂) t ↔ MapsTo f s₁ t ∧ MapsTo
 theorem MapsTo.inter (h₁ : MapsTo f s t₁) (h₂ : MapsTo f s t₂) : MapsTo f s (t₁ ∩ t₂) := fun _ hx =>
   ⟨h₁ hx, h₂ hx⟩
 
+lemma MapsTo.insert (h : MapsTo f s t) (x : α) : MapsTo f (insert x s) (insert (f x) t) := by
+  simpa [← singleton_union] using h.mono_right subset_union_right
+
 theorem MapsTo.inter_inter (h₁ : MapsTo f s₁ t₁) (h₂ : MapsTo f s₂ t₂) :
     MapsTo f (s₁ ∩ s₂) (t₁ ∩ t₂) := fun _ hx => ⟨h₁ hx.1, h₂ hx.2⟩
 
@@ -437,11 +384,6 @@ theorem mapsTo_image_iff {f : α → β} {g : γ → α} {s : Set γ} {t : Set 
     MapsTo f (g '' s) t ↔ MapsTo (f ∘ g) s t :=
   ⟨fun h c hc => h ⟨c, hc, rfl⟩, fun h _ ⟨_, hc⟩ => hc.2 ▸ h hc.1⟩
 
-@[deprecated (since := "2023-12-25")]
-lemma maps_image_to (f : α → β) (g : γ → α) (s : Set γ) (t : Set β) :
-    MapsTo f (g '' s) t ↔ MapsTo (f ∘ g) s t :=
-  mapsTo_image_iff
-
 lemma MapsTo.comp_left (g : β → γ) (hf : MapsTo f s t) : MapsTo (g ∘ f) s (g '' t) :=
   fun x hx ↦ ⟨f x, hf hx, rfl⟩
 
@@ -452,10 +394,6 @@ lemma MapsTo.comp_right {s : Set β} {t : Set γ} (hg : MapsTo g s t) (f : α 
 lemma mapsTo_univ_iff : MapsTo f univ t ↔ ∀ x, f x ∈ t :=
   ⟨fun h _ => h (mem_univ _), fun h x _ => h x⟩
 
-@[deprecated (since := "2023-12-25")]
-theorem maps_univ_to (f : α → β) (s : Set β) : MapsTo f univ s ↔ ∀ a, f a ∈ s :=
-  mapsTo_univ_iff
-
 @[simp]
 lemma mapsTo_range_iff {g : ι → α} : MapsTo f (range g) t ↔ ∀ i, f (g i) ∈ t :=
   forall_mem_range
@@ -502,8 +440,6 @@ theorem preimage_restrictPreimage {u : Set t} :
   rw [← preimage_preimage (g := f) (f := Subtype.val), ← image_val_preimage_restrictPreimage,
     preimage_image_eq _ Subtype.val_injective]
 
-variable {U : ι → Set β}
-
 lemma restrictPreimage_injective (hf : Injective f) : Injective (t.restrictPreimage f) :=
   fun _ _ e => Subtype.coe_injective <| hf <| Subtype.mk.inj e
 
@@ -689,8 +625,15 @@ theorem InjOn.imageFactorization_injective (h : InjOn f s) :
 end injOn
 
 section graphOn
+variable {x : α × β}
+
+@[simp] lemma mem_graphOn : x ∈ s.graphOn f ↔ x.1 ∈ s ∧ f x.1 = x.2 := by aesop (add simp graphOn)
 
 @[simp] lemma graphOn_empty (f : α → β) : graphOn f ∅ = ∅ := image_empty _
+@[simp] lemma graphOn_eq_empty : graphOn f s = ∅ ↔ s = ∅ := image_eq_empty
+@[simp] lemma graphOn_nonempty : (s.graphOn f).Nonempty ↔ s.Nonempty := image_nonempty
+
+protected alias ⟨_, Nonempty.graphOn⟩ := graphOn_nonempty
 
 @[simp]
 lemma graphOn_union (f : α → β) (s t : Set α) : graphOn f (s ∪ t) = graphOn f s ∪ graphOn f t :=
@@ -709,6 +652,24 @@ lemma graphOn_insert (f : α → β) (x : α) (s : Set α) :
 lemma image_fst_graphOn (f : α → β) (s : Set α) : Prod.fst '' graphOn f s = s := by
   simp [graphOn, image_image]
 
+@[simp] lemma image_snd_graphOn (f : α → β) : Prod.snd '' s.graphOn f = f '' s := by ext x; simp
+
+lemma fst_injOn_graph : (s.graphOn f).InjOn Prod.fst := by aesop (add simp InjOn)
+
+lemma graphOn_comp (s : Set α) (f : α → β) (g : β → γ) :
+    s.graphOn (g ∘ f) = (fun x ↦ (x.1, g x.2)) '' s.graphOn f := by
+  simpa using image_comp (fun x ↦ (x.1, g x.2)) (fun x ↦ (x, f x)) _
+
+lemma graphOn_univ_eq_range : univ.graphOn f = range fun x ↦ (x, f x) := image_univ
+
+@[simp] lemma graphOn_inj {g : α → β} : s.graphOn f = s.graphOn g ↔ s.EqOn f g := by
+  simp [Set.ext_iff, funext_iff, forall_swap, EqOn]
+
+lemma graphOn_univ_inj {g : α → β} : univ.graphOn f = univ.graphOn g ↔ f = g := by simp
+
+lemma graphOn_univ_injective : Injective (univ.graphOn : (α → β) → Set (α × β)) :=
+  fun _f _g ↦ graphOn_univ_inj.1
+
 lemma exists_eq_graphOn_image_fst [Nonempty β] {s : Set (α × β)} :
     (∃ f : α → β, s = graphOn f (Prod.fst '' s)) ↔ InjOn Prod.fst s := by
   refine ⟨?_, fun h ↦ ?_⟩
@@ -1312,23 +1273,6 @@ lemma bijOn_comm {g : β → α} (h : InvOn f g t s) : BijOn f s t ↔ BijOn g t
 
 end Set
 
-/-! ### Monotone -/
-namespace Monotone
-
-variable [Preorder α] [Preorder β] {f : α → β}
-
-protected theorem restrict (h : Monotone f) (s : Set α) : Monotone (s.restrict f) := fun _ _ hxy =>
-  h hxy
-
-protected theorem codRestrict (h : Monotone f) {s : Set β} (hs : ∀ x, f x ∈ s) :
-    Monotone (s.codRestrict f hs) :=
-  h
-
-protected theorem rangeFactorization (h : Monotone f) : Monotone (Set.rangeFactorization f) :=
-  h
-
-end Monotone
-
 /-! ### Piecewise defined function -/
 namespace Set
 
@@ -1351,10 +1295,6 @@ theorem piecewise_insert_self {j : α} [∀ i, Decidable (i ∈ insert j s)] :
 
 variable [∀ j, Decidable (j ∈ s)]
 
--- TODO: move!
-instance Compl.decidableMem (j : α) : Decidable (j ∈ sᶜ) :=
-  instDecidableNot
-
 theorem piecewise_insert [DecidableEq α] (j : α) [∀ i, Decidable (i ∈ insert j s)] :
     (insert j s).piecewise f g = Function.update (s.piecewise f g) j (f j) := by
   simp (config := { unfoldPartialApp := true }) only [piecewise, mem_insert_iff]
@@ -1395,11 +1335,14 @@ theorem le_piecewise {δ : α → Type*} [∀ i, Preorder (δ i)] {s : Set α} [
     g ≤ s.piecewise f₁ f₂ :=
   @piecewise_le α (fun i => (δ i)ᵒᵈ) _ s _ _ _ _ h₁ h₂
 
-theorem piecewise_le_piecewise {δ : α → Type*} [∀ i, Preorder (δ i)] {s : Set α}
+@[gcongr]
+theorem piecewise_mono {δ : α → Type*} [∀ i, Preorder (δ i)] {s : Set α}
     [∀ j, Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : ∀ i, δ i} (h₁ : ∀ i ∈ s, f₁ i ≤ g₁ i)
     (h₂ : ∀ i ∉ s, f₂ i ≤ g₂ i) : s.piecewise f₁ f₂ ≤ s.piecewise g₁ g₂ := by
   apply piecewise_le <;> intros <;> simp [*]
 
+@[deprecated (since := "2024-10-06")] alias piecewise_le_piecewise := piecewise_mono
+
 @[simp]
 theorem piecewise_insert_of_ne {i j : α} (h : i ≠ j) [∀ i, Decidable (i ∈ insert j s)] :
     (insert j s).piecewise f g i = s.piecewise f g i := by simp [piecewise, h]
@@ -1494,46 +1437,6 @@ theorem univ_pi_piecewise_univ {ι : Type*} {α : ι → Type*} (s : Set ι) (t
 
 end Set
 
-section strictMono
-
-theorem StrictMonoOn.injOn [LinearOrder α] [Preorder β] {f : α → β} {s : Set α}
-    (H : StrictMonoOn f s) : s.InjOn f := fun x hx y hy hxy =>
-  show Ordering.eq.Compares x y from (H.compares hx hy).1 hxy
-
-theorem StrictAntiOn.injOn [LinearOrder α] [Preorder β] {f : α → β} {s : Set α}
-    (H : StrictAntiOn f s) : s.InjOn f :=
-  @StrictMonoOn.injOn α βᵒᵈ _ _ f s H
-
-theorem StrictMonoOn.comp [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α}
-    {t : Set β} (hg : StrictMonoOn g t) (hf : StrictMonoOn f s) (hs : Set.MapsTo f s t) :
-    StrictMonoOn (g ∘ f) s := fun _x hx _y hy hxy => hg (hs hx) (hs hy) <| hf hx hy hxy
-
-theorem StrictMonoOn.comp_strictAntiOn [Preorder α] [Preorder β] [Preorder γ] {g : β → γ}
-    {f : α → β} {s : Set α} {t : Set β} (hg : StrictMonoOn g t) (hf : StrictAntiOn f s)
-    (hs : Set.MapsTo f s t) : StrictAntiOn (g ∘ f) s := fun _x hx _y hy hxy =>
-  hg (hs hy) (hs hx) <| hf hx hy hxy
-
-theorem StrictAntiOn.comp [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α}
-    {t : Set β} (hg : StrictAntiOn g t) (hf : StrictAntiOn f s) (hs : Set.MapsTo f s t) :
-    StrictMonoOn (g ∘ f) s := fun _x hx _y hy hxy => hg (hs hy) (hs hx) <| hf hx hy hxy
-
-theorem StrictAntiOn.comp_strictMonoOn [Preorder α] [Preorder β] [Preorder γ] {g : β → γ}
-    {f : α → β} {s : Set α} {t : Set β} (hg : StrictAntiOn g t) (hf : StrictMonoOn f s)
-    (hs : Set.MapsTo f s t) : StrictAntiOn (g ∘ f) s := fun _x hx _y hy hxy =>
-  hg (hs hx) (hs hy) <| hf hx hy hxy
-
-@[simp]
-theorem strictMono_restrict [Preorder α] [Preorder β] {f : α → β} {s : Set α} :
-    StrictMono (s.restrict f) ↔ StrictMonoOn f s := by simp [Set.restrict, StrictMono, StrictMonoOn]
-
-alias ⟨_root_.StrictMono.of_restrict, _root_.StrictMonoOn.restrict⟩ := strictMono_restrict
-
-theorem StrictMono.codRestrict [Preorder α] [Preorder β] {f : α → β} (hf : StrictMono f)
-    {s : Set β} (hs : ∀ x, f x ∈ s) : StrictMono (Set.codRestrict f s hs) :=
-  hf
-
-end strictMono
-
 namespace Function
 
 open Set
@@ -1621,21 +1524,9 @@ theorem update_comp_eq_of_not_mem_range {α : Sort*} {β : Type*} {γ : Sort*} [
 theorem insert_injOn (s : Set α) : sᶜ.InjOn fun a => insert a s := fun _a ha _ _ =>
   (insert_inj ha).1
 
-theorem monotoneOn_of_rightInvOn_of_mapsTo {α β : Type*} [PartialOrder α] [LinearOrder β]
-    {φ : β → α} {ψ : α → β} {t : Set β} {s : Set α} (hφ : MonotoneOn φ t)
-    (φψs : Set.RightInvOn ψ φ s) (ψts : Set.MapsTo ψ s t) : MonotoneOn ψ s := by
-  rintro x xs y ys l
-  rcases le_total (ψ x) (ψ y) with (ψxy|ψyx)
-  · exact ψxy
-  · have := hφ (ψts ys) (ψts xs) ψyx
-    rw [φψs.eq ys, φψs.eq xs] at this
-    induction le_antisymm l this
-    exact le_refl _
-
-theorem antitoneOn_of_rightInvOn_of_mapsTo [PartialOrder α] [LinearOrder β]
-    {φ : β → α} {ψ : α → β} {t : Set β} {s : Set α} (hφ : AntitoneOn φ t)
-    (φψs : Set.RightInvOn ψ φ s) (ψts : Set.MapsTo ψ s t) : AntitoneOn ψ s :=
-  (monotoneOn_of_rightInvOn_of_mapsTo hφ.dual_left φψs ψts).dual_right
+lemma apply_eq_of_range_eq_singleton {f : α → β} {b : β} (h : range f = {b}) (a : α) :
+    f a = b := by
+  simpa only [h, mem_singleton_iff] using mem_range_self (f := f) a
 
 end Function
 
@@ -1741,3 +1632,5 @@ lemma bijOn_swap (ha : a ∈ s) (hb : b ∈ s) : BijOn (swap a b) s s :=
     simp [*, swap_apply_of_ne_of_ne]
 
 end Equiv
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Data/Set/Image.lean b/Mathlib/Data/Set/Image.lean
index 09b79da91cf44..9b390e67bad19 100644
--- a/Mathlib/Data/Set/Image.lean
+++ b/Mathlib/Data/Set/Image.lean
@@ -987,24 +987,24 @@ end Range
 
 section Subsingleton
 
-variable {s : Set α}
+variable {s : Set α} {f : α → β}
 
 /-- The image of a subsingleton is a subsingleton. -/
 theorem Subsingleton.image (hs : s.Subsingleton) (f : α → β) : (f '' s).Subsingleton :=
   fun _ ⟨_, hx, Hx⟩ _ ⟨_, hy, Hy⟩ => Hx ▸ Hy ▸ congr_arg f (hs hx hy)
 
 /-- The preimage of a subsingleton under an injective map is a subsingleton. -/
-theorem Subsingleton.preimage {s : Set β} (hs : s.Subsingleton) {f : α → β}
+theorem Subsingleton.preimage {s : Set β} (hs : s.Subsingleton)
     (hf : Function.Injective f) : (f ⁻¹' s).Subsingleton := fun _ ha _ hb => hf <| hs ha hb
 
 /-- If the image of a set under an injective map is a subsingleton, the set is a subsingleton. -/
-theorem subsingleton_of_image {f : α → β} (hf : Function.Injective f) (s : Set α)
+theorem subsingleton_of_image (hf : Function.Injective f) (s : Set α)
     (hs : (f '' s).Subsingleton) : s.Subsingleton :=
   (hs.preimage hf).anti <| subset_preimage_image _ _
 
 /-- If the preimage of a set under a surjective map is a subsingleton,
 the set is a subsingleton. -/
-theorem subsingleton_of_preimage {f : α → β} (hf : Function.Surjective f) (s : Set β)
+theorem subsingleton_of_preimage (hf : Function.Surjective f) (s : Set β)
     (hs : (f ⁻¹' s).Subsingleton) : s.Subsingleton := fun fx hx fy hy => by
   rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
   exact congr_arg f (hs hx hy)
@@ -1013,29 +1013,39 @@ theorem subsingleton_range {α : Sort*} [Subsingleton α] (f : α → β) : (ran
   forall_mem_range.2 fun x => forall_mem_range.2 fun y => congr_arg f (Subsingleton.elim x y)
 
 /-- The preimage of a nontrivial set under a surjective map is nontrivial. -/
-theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial) {f : α → β}
+theorem Nontrivial.preimage {s : Set β} (hs : s.Nontrivial)
     (hf : Function.Surjective f) : (f ⁻¹' s).Nontrivial := by
   rcases hs with ⟨fx, hx, fy, hy, hxy⟩
   rcases hf fx, hf fy with ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩
   exact ⟨x, hx, y, hy, mt (congr_arg f) hxy⟩
 
 /-- The image of a nontrivial set under an injective map is nontrivial. -/
-theorem Nontrivial.image (hs : s.Nontrivial) {f : α → β} (hf : Function.Injective f) :
+theorem Nontrivial.image (hs : s.Nontrivial) (hf : Function.Injective f) :
     (f '' s).Nontrivial :=
   let ⟨x, hx, y, hy, hxy⟩ := hs
   ⟨f x, mem_image_of_mem f hx, f y, mem_image_of_mem f hy, hf.ne hxy⟩
 
+theorem Nontrivial.image_of_injOn (hs : s.Nontrivial) (hf : s.InjOn f) :
+    (f '' s).Nontrivial := by
+  obtain ⟨x, hx, y, hy, hxy⟩ := hs
+  exact ⟨f x, mem_image_of_mem _ hx, f y, mem_image_of_mem _ hy, (hxy <| hf hx hy ·)⟩
+
 /-- If the image of a set is nontrivial, the set is nontrivial. -/
 theorem nontrivial_of_image (f : α → β) (s : Set α) (hs : (f '' s).Nontrivial) : s.Nontrivial :=
   let ⟨_, ⟨x, hx, rfl⟩, _, ⟨y, hy, rfl⟩, hxy⟩ := hs
   ⟨x, hx, y, hy, mt (congr_arg f) hxy⟩
 
 @[simp]
-theorem image_nontrivial {f : α → β} (hf : f.Injective) : (f '' s).Nontrivial ↔ s.Nontrivial :=
+theorem image_nontrivial (hf : f.Injective) : (f '' s).Nontrivial ↔ s.Nontrivial :=
   ⟨nontrivial_of_image f s, fun h ↦ h.image hf⟩
 
+@[simp]
+theorem InjOn.image_nontrivial_iff (hf : s.InjOn f) :
+    (f '' s).Nontrivial ↔ s.Nontrivial :=
+  ⟨nontrivial_of_image f s, fun h ↦ h.image_of_injOn hf⟩
+
 /-- If the preimage of a set under an injective map is nontrivial, the set is nontrivial. -/
-theorem nontrivial_of_preimage {f : α → β} (hf : Function.Injective f) (s : Set β)
+theorem nontrivial_of_preimage (hf : Function.Injective f) (s : Set β)
     (hs : (f ⁻¹' s).Nontrivial) : s.Nontrivial :=
   (hs.image hf).mono <| image_preimage_subset _ _
 
@@ -1080,6 +1090,9 @@ theorem Injective.image_injective (hf : Injective f) : Injective (image f) := by
   intro s t h
   rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, h]
 
+lemma Injective.image_strictMono (inj : Function.Injective f) : StrictMono (image f) :=
+  monotone_image.strictMono_of_injective inj.image_injective
+
 theorem Surjective.preimage_subset_preimage_iff {s t : Set β} (hf : Surjective f) :
     f ⁻¹' s ⊆ f ⁻¹' t ↔ s ⊆ t := by
   apply Set.preimage_subset_preimage_iff
@@ -1090,13 +1103,17 @@ theorem Surjective.range_comp {ι' : Sort*} {f : ι → ι'} (hf : Surjective f)
     range (g ∘ f) = range g :=
   ext fun y => (@Surjective.exists _ _ _ hf fun x => g x = y).symm
 
-theorem Injective.mem_range_iff_exists_unique (hf : Injective f) {b : β} :
+theorem Injective.mem_range_iff_existsUnique (hf : Injective f) {b : β} :
     b ∈ range f ↔ ∃! a, f a = b :=
   ⟨fun ⟨a, h⟩ => ⟨a, h, fun _ ha => hf (ha.trans h.symm)⟩, ExistsUnique.exists⟩
 
-theorem Injective.exists_unique_of_mem_range (hf : Injective f) {b : β} (hb : b ∈ range f) :
-    ∃! a, f a = b :=
-  hf.mem_range_iff_exists_unique.mp hb
+alias ⟨Injective.existsUnique_of_mem_range, _⟩ := Injective.mem_range_iff_existsUnique
+
+@[deprecated (since := "2024-09-25")]
+alias Injective.mem_range_iff_exists_unique := Injective.mem_range_iff_existsUnique
+
+@[deprecated (since := "2024-09-25")]
+alias Injective.exists_unique_of_mem_range := Injective.existsUnique_of_mem_range
 
 theorem Injective.compl_image_eq (hf : Injective f) (s : Set α) :
     (f '' s)ᶜ = f '' sᶜ ∪ (range f)ᶜ := by
diff --git a/Mathlib/Data/Set/Lattice.lean b/Mathlib/Data/Set/Lattice.lean
index 44297c6dee25b..b6f29fa406fdf 100644
--- a/Mathlib/Data/Set/Lattice.lean
+++ b/Mathlib/Data/Set/Lattice.lean
@@ -1243,6 +1243,14 @@ theorem image_sInter_subset (S : Set (Set α)) (f : α → β) : f '' ⋂₀ S 
   rw [sInter_eq_biInter]
   apply image_iInter₂_subset
 
+theorem image2_sInter_right_subset (t : Set α) (S : Set (Set β)) (f : α → β → γ) :
+    image2 f t (⋂₀ S) ⊆ ⋂ s ∈ S, image2 f t s := by
+  aesop
+
+theorem image2_sInter_left_subset (S : Set (Set α)) (t : Set β)  (f : α → β → γ) :
+    image2 f (⋂₀ S) t ⊆ ⋂ s ∈ S, image2 f s t := by
+  aesop
+
 /-! ### `restrictPreimage` -/
 
 
@@ -1575,6 +1583,14 @@ theorem image2_iUnion_right (s : Set α) (t : ι → Set β) :
     image2 f s (⋃ i, t i) = ⋃ i, image2 f s (t i) := by
   simp only [← image_prod, prod_iUnion, image_iUnion]
 
+theorem image2_sUnion_left (S : Set (Set α)) (t : Set β) :
+    image2 f (⋃₀ S) t = ⋃ s ∈ S, image2 f s t := by
+  aesop
+
+theorem image2_sUnion_right (s : Set α) (T : Set (Set β)) :
+    image2 f s (⋃₀ T) = ⋃ t ∈ T, image2 f s t := by
+  aesop
+
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:107:6: warning: expanding binder group (i j) -/
 theorem image2_iUnion₂_left (s : ∀ i, κ i → Set α) (t : Set β) :
@@ -1610,6 +1626,16 @@ theorem image2_iInter₂_subset_right (s : Set α) (t : ∀ i, κ i → Set β)
   simp_rw [image2_subset_iff, mem_iInter]
   exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)
 
+theorem image2_sInter_subset_left (S : Set (Set α)) (t : Set β) :
+    image2 f (⋂₀ S) t ⊆ ⋂ s ∈ S, image2 f s t := by
+  rw [sInter_eq_biInter]
+  exact image2_iInter₂_subset_left ..
+
+theorem image2_sInter_subset_right (s : Set α) (T : Set (Set β)) :
+    image2 f s (⋂₀ T) ⊆ ⋂ t ∈ T, image2 f s t := by
+  rw [sInter_eq_biInter]
+  exact image2_iInter₂_subset_right ..
+
 theorem prod_eq_biUnion_left : s ×ˢ t = ⋃ a ∈ s, (fun b => (a, b)) '' t := by
   rw [iUnion_image_left, image2_mk_eq_prod]
 
@@ -1935,3 +1961,5 @@ lemma forall_sUnion {S : Set (Set α)} {p : α → Prop} :
 lemma exists_sUnion {S : Set (Set α)} {p : α → Prop} :
     (∃ x ∈ ⋃₀ S, p x) ↔ ∃ s ∈ S, ∃ x ∈ s, p x := by
   simp_rw [← exists_prop, ← iSup_Prop_eq, iSup_sUnion]
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/Data/Set/List.lean b/Mathlib/Data/Set/List.lean
index 6ce4a3c2a793f..60839e4f416e6 100644
--- a/Mathlib/Data/Set/List.lean
+++ b/Mathlib/Data/Set/List.lean
@@ -48,7 +48,7 @@ theorem range_list_get? : range l.get? = insert none (some '' { x | x ∈ l }) :
 theorem range_list_getD (d : α) : (range fun n : Nat => l[n]?.getD d) = insert d { x | x ∈ l } :=
   calc
     (range fun n => l[n]?.getD d) = (fun o : Option α => o.getD d) '' range l.get? := by
-      simp [← range_comp, (· ∘ ·)]
+      simp [← range_comp, Function.comp_def]
     _ = insert d { x | x ∈ l } := by
       simp only [range_list_get?, image_insert_eq, Option.getD, image_image, image_id']
 
diff --git a/Mathlib/Data/Set/MemPartition.lean b/Mathlib/Data/Set/MemPartition.lean
index 25b578031b0f6..5a13d62ad7baf 100644
--- a/Mathlib/Data/Set/MemPartition.lean
+++ b/Mathlib/Data/Set/MemPartition.lean
@@ -118,7 +118,6 @@ lemma memPartitionSet_succ (f : ℕ → Set α) (n : ℕ) (a : α) [Decidable (a
     memPartitionSet f (n + 1) a
       = if a ∈ f n then memPartitionSet f n a ∩ f n else memPartitionSet f n a \ f n := by
   simp [memPartitionSet]
-  congr
 
 lemma memPartitionSet_mem (f : ℕ → Set α) (n : ℕ) (a : α) :
     memPartitionSet f n a ∈ memPartition f n := by
diff --git a/Mathlib/Data/Set/Monotone.lean b/Mathlib/Data/Set/Monotone.lean
new file mode 100644
index 0000000000000..5bd60bb9fb1c5
--- /dev/null
+++ b/Mathlib/Data/Set/Monotone.lean
@@ -0,0 +1,193 @@
+/-
+Copyright (c) 2014 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
+-/
+import Mathlib.Data.Set.Function
+
+/-!
+# Monotone functions over sets
+-/
+
+variable {α β γ : Type*}
+
+open Equiv Equiv.Perm Function
+
+namespace Set
+
+
+/-! ### Congruence lemmas for monotonicity and antitonicity -/
+section Order
+
+variable {s : Set α} {f₁ f₂ : α → β} [Preorder α] [Preorder β]
+
+theorem _root_.MonotoneOn.congr (h₁ : MonotoneOn f₁ s) (h : s.EqOn f₁ f₂) : MonotoneOn f₂ s := by
+  intro a ha b hb hab
+  rw [← h ha, ← h hb]
+  exact h₁ ha hb hab
+
+theorem _root_.AntitoneOn.congr (h₁ : AntitoneOn f₁ s) (h : s.EqOn f₁ f₂) : AntitoneOn f₂ s :=
+  h₁.dual_right.congr h
+
+theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
+    StrictMonoOn f₂ s := by
+  intro a ha b hb hab
+  rw [← h ha, ← h hb]
+  exact h₁ ha hb hab
+
+theorem _root_.StrictAntiOn.congr (h₁ : StrictAntiOn f₁ s) (h : s.EqOn f₁ f₂) : StrictAntiOn f₂ s :=
+  h₁.dual_right.congr h
+
+theorem EqOn.congr_monotoneOn (h : s.EqOn f₁ f₂) : MonotoneOn f₁ s ↔ MonotoneOn f₂ s :=
+  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
+
+theorem EqOn.congr_antitoneOn (h : s.EqOn f₁ f₂) : AntitoneOn f₁ s ↔ AntitoneOn f₂ s :=
+  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
+
+theorem EqOn.congr_strictMonoOn (h : s.EqOn f₁ f₂) : StrictMonoOn f₁ s ↔ StrictMonoOn f₂ s :=
+  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
+
+theorem EqOn.congr_strictAntiOn (h : s.EqOn f₁ f₂) : StrictAntiOn f₁ s ↔ StrictAntiOn f₂ s :=
+  ⟨fun h₁ => h₁.congr h, fun h₂ => h₂.congr h.symm⟩
+
+end Order
+
+/-! ### Monotonicity lemmas -/
+section Mono
+
+variable {s s₂ : Set α} {f : α → β} [Preorder α] [Preorder β]
+
+theorem _root_.MonotoneOn.mono (h : MonotoneOn f s) (h' : s₂ ⊆ s) : MonotoneOn f s₂ :=
+  fun _ hx _ hy => h (h' hx) (h' hy)
+
+theorem _root_.AntitoneOn.mono (h : AntitoneOn f s) (h' : s₂ ⊆ s) : AntitoneOn f s₂ :=
+  fun _ hx _ hy => h (h' hx) (h' hy)
+
+theorem _root_.StrictMonoOn.mono (h : StrictMonoOn f s) (h' : s₂ ⊆ s) : StrictMonoOn f s₂ :=
+  fun _ hx _ hy => h (h' hx) (h' hy)
+
+theorem _root_.StrictAntiOn.mono (h : StrictAntiOn f s) (h' : s₂ ⊆ s) : StrictAntiOn f s₂ :=
+  fun _ hx _ hy => h (h' hx) (h' hy)
+
+protected theorem _root_.MonotoneOn.monotone (h : MonotoneOn f s) :
+    Monotone (f ∘ Subtype.val : s → β) :=
+  fun x y hle => h x.coe_prop y.coe_prop hle
+
+protected theorem _root_.AntitoneOn.monotone (h : AntitoneOn f s) :
+    Antitone (f ∘ Subtype.val : s → β) :=
+  fun x y hle => h x.coe_prop y.coe_prop hle
+
+protected theorem _root_.StrictMonoOn.strictMono (h : StrictMonoOn f s) :
+    StrictMono (f ∘ Subtype.val : s → β) :=
+  fun x y hlt => h x.coe_prop y.coe_prop hlt
+
+protected theorem _root_.StrictAntiOn.strictAnti (h : StrictAntiOn f s) :
+    StrictAnti (f ∘ Subtype.val : s → β) :=
+  fun x y hlt => h x.coe_prop y.coe_prop hlt
+
+lemma MonotoneOn_insert_iff {a : α} :
+    MonotoneOn f (insert a s) ↔
+       (∀ b ∈ s, b ≤ a → f b ≤ f a) ∧ (∀ b ∈ s, a ≤ b → f a ≤ f b) ∧ MonotoneOn f s := by
+  simp [MonotoneOn, forall_and]
+
+lemma AntitoneOn_insert_iff {a : α} :
+    AntitoneOn f (insert a s) ↔
+       (∀ b ∈ s, b ≤ a → f a ≤ f b) ∧ (∀ b ∈ s, a ≤ b → f b ≤ f a) ∧ AntitoneOn f s :=
+  @MonotoneOn_insert_iff α βᵒᵈ _ _ _ _ _
+
+end Mono
+
+end Set
+
+
+
+open Function
+
+/-! ### Monotone -/
+namespace Monotone
+
+variable [Preorder α] [Preorder β] {f : α → β}
+
+protected theorem restrict (h : Monotone f) (s : Set α) : Monotone (s.restrict f) := fun _ _ hxy =>
+  h hxy
+
+protected theorem codRestrict (h : Monotone f) {s : Set β} (hs : ∀ x, f x ∈ s) :
+    Monotone (s.codRestrict f hs) :=
+  h
+
+protected theorem rangeFactorization (h : Monotone f) : Monotone (Set.rangeFactorization f) :=
+  h
+
+end Monotone
+
+section strictMono
+
+theorem StrictMonoOn.injOn [LinearOrder α] [Preorder β] {f : α → β} {s : Set α}
+    (H : StrictMonoOn f s) : s.InjOn f := fun x hx y hy hxy =>
+  show Ordering.eq.Compares x y from (H.compares hx hy).1 hxy
+
+theorem StrictAntiOn.injOn [LinearOrder α] [Preorder β] {f : α → β} {s : Set α}
+    (H : StrictAntiOn f s) : s.InjOn f :=
+  @StrictMonoOn.injOn α βᵒᵈ _ _ f s H
+
+theorem StrictMonoOn.comp [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α}
+    {t : Set β} (hg : StrictMonoOn g t) (hf : StrictMonoOn f s) (hs : Set.MapsTo f s t) :
+    StrictMonoOn (g ∘ f) s := fun _x hx _y hy hxy => hg (hs hx) (hs hy) <| hf hx hy hxy
+
+theorem StrictMonoOn.comp_strictAntiOn [Preorder α] [Preorder β] [Preorder γ] {g : β → γ}
+    {f : α → β} {s : Set α} {t : Set β} (hg : StrictMonoOn g t) (hf : StrictAntiOn f s)
+    (hs : Set.MapsTo f s t) : StrictAntiOn (g ∘ f) s := fun _x hx _y hy hxy =>
+  hg (hs hy) (hs hx) <| hf hx hy hxy
+
+theorem StrictAntiOn.comp [Preorder α] [Preorder β] [Preorder γ] {g : β → γ} {f : α → β} {s : Set α}
+    {t : Set β} (hg : StrictAntiOn g t) (hf : StrictAntiOn f s) (hs : Set.MapsTo f s t) :
+    StrictMonoOn (g ∘ f) s := fun _x hx _y hy hxy => hg (hs hy) (hs hx) <| hf hx hy hxy
+
+theorem StrictAntiOn.comp_strictMonoOn [Preorder α] [Preorder β] [Preorder γ] {g : β → γ}
+    {f : α → β} {s : Set α} {t : Set β} (hg : StrictAntiOn g t) (hf : StrictMonoOn f s)
+    (hs : Set.MapsTo f s t) : StrictAntiOn (g ∘ f) s := fun _x hx _y hy hxy =>
+  hg (hs hx) (hs hy) <| hf hx hy hxy
+
+@[simp]
+theorem strictMono_restrict [Preorder α] [Preorder β] {f : α → β} {s : Set α} :
+    StrictMono (s.restrict f) ↔ StrictMonoOn f s := by simp [Set.restrict, StrictMono, StrictMonoOn]
+
+alias ⟨_root_.StrictMono.of_restrict, _root_.StrictMonoOn.restrict⟩ := strictMono_restrict
+
+theorem StrictMono.codRestrict [Preorder α] [Preorder β] {f : α → β} (hf : StrictMono f)
+    {s : Set β} (hs : ∀ x, f x ∈ s) : StrictMono (Set.codRestrict f s hs) :=
+  hf
+
+lemma strictMonoOn_insert_iff [Preorder α] [Preorder β] {f : α → β} {s : Set α} {a : α} :
+    StrictMonoOn f (insert a s) ↔
+       (∀ b ∈ s, b < a → f b < f a) ∧ (∀ b ∈ s, a < b → f a < f b) ∧ StrictMonoOn f s := by
+  simp [StrictMonoOn, forall_and]
+
+lemma strictAntiOn_insert_iff [Preorder α] [Preorder β] {f : α → β} {s : Set α} {a : α} :
+    StrictAntiOn f (insert a s) ↔
+       (∀ b ∈ s, b < a → f a < f b) ∧ (∀ b ∈ s, a < b → f b < f a) ∧ StrictAntiOn f s :=
+  @strictMonoOn_insert_iff α βᵒᵈ _ _ _ _ _
+
+end strictMono
+
+namespace Function
+
+open Set
+
+theorem monotoneOn_of_rightInvOn_of_mapsTo {α β : Type*} [PartialOrder α] [LinearOrder β]
+    {φ : β → α} {ψ : α → β} {t : Set β} {s : Set α} (hφ : MonotoneOn φ t)
+    (φψs : Set.RightInvOn ψ φ s) (ψts : Set.MapsTo ψ s t) : MonotoneOn ψ s := by
+  rintro x xs y ys l
+  rcases le_total (ψ x) (ψ y) with (ψxy|ψyx)
+  · exact ψxy
+  · have := hφ (ψts ys) (ψts xs) ψyx
+    rw [φψs.eq ys, φψs.eq xs] at this
+    induction le_antisymm l this
+    exact le_refl _
+
+theorem antitoneOn_of_rightInvOn_of_mapsTo [PartialOrder α] [LinearOrder β]
+    {φ : β → α} {ψ : α → β} {t : Set β} {s : Set α} (hφ : AntitoneOn φ t)
+    (φψs : Set.RightInvOn ψ φ s) (ψts : Set.MapsTo ψ s t) : AntitoneOn ψ s :=
+  (monotoneOn_of_rightInvOn_of_mapsTo hφ.dual_left φψs ψts).dual_right
+
+end Function
diff --git a/Mathlib/Data/Set/Operations.lean b/Mathlib/Data/Set/Operations.lean
index 620c7afe1b44c..bed6dc9156d8b 100644
--- a/Mathlib/Data/Set/Operations.lean
+++ b/Mathlib/Data/Set/Operations.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2016 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad, Johannes Hölzl, Reid Barton, Scott Morrison, Patrick Massot, Kyle Miller,
+Authors: Jeremy Avigad, Johannes Hölzl, Reid Barton, Kim Morrison, Patrick Massot, Kyle Miller,
 Minchao Wu, Yury Kudryashov, Floris van Doorn
 -/
 import Mathlib.Data.SProd
diff --git a/Mathlib/Data/Set/Pairwise/Basic.lean b/Mathlib/Data/Set/Pairwise/Basic.lean
index 64e482ceb8858..a40768a7e768d 100644
--- a/Mathlib/Data/Set/Pairwise/Basic.lean
+++ b/Mathlib/Data/Set/Pairwise/Basic.lean
@@ -39,7 +39,7 @@ variable {f g : ι → α} {s t u : Set α} {a b : α}
 theorem pairwise_on_bool (hr : Symmetric r) {a b : α} :
     Pairwise (r on fun c => cond c a b) ↔ r a b := by simpa [Pairwise, Function.onFun] using @hr a b
 
-theorem pairwise_disjoint_on_bool [SemilatticeInf α] [OrderBot α] {a b : α} :
+theorem pairwise_disjoint_on_bool [PartialOrder α] [OrderBot α] {a b : α} :
     Pairwise (Disjoint on fun c => cond c a b) ↔ Disjoint a b :=
   pairwise_on_bool Disjoint.symm
 
@@ -47,14 +47,24 @@ theorem Symmetric.pairwise_on [LinearOrder ι] (hr : Symmetric r) (f : ι → α
     Pairwise (r on f) ↔ ∀ ⦃m n⦄, m < n → r (f m) (f n) :=
   ⟨fun h _m _n hmn => h hmn.ne, fun h _m _n hmn => hmn.lt_or_lt.elim (@h _ _) fun h' => hr (h h')⟩
 
-theorem pairwise_disjoint_on [SemilatticeInf α] [OrderBot α] [LinearOrder ι] (f : ι → α) :
+theorem pairwise_disjoint_on [PartialOrder α] [OrderBot α] [LinearOrder ι] (f : ι → α) :
     Pairwise (Disjoint on f) ↔ ∀ ⦃m n⦄, m < n → Disjoint (f m) (f n) :=
   Symmetric.pairwise_on Disjoint.symm f
 
-theorem pairwise_disjoint_mono [SemilatticeInf α] [OrderBot α] (hs : Pairwise (Disjoint on f))
+theorem pairwise_disjoint_mono [PartialOrder α] [OrderBot α] (hs : Pairwise (Disjoint on f))
     (h : g ≤ f) : Pairwise (Disjoint on g) :=
   hs.mono fun i j hij => Disjoint.mono (h i) (h j) hij
 
+theorem Pairwise.disjoint_extend_bot [PartialOrder γ] [OrderBot γ]
+    {e : α → β} {f : α → γ} (hf : Pairwise (Disjoint on f)) (he : FactorsThrough f e) :
+    Pairwise (Disjoint on extend e f ⊥) := by
+  intro b₁ b₂ hne
+  rcases em (∃ a₁, e a₁ = b₁) with ⟨a₁, rfl⟩ | hb₁
+  · rcases em (∃ a₂, e a₂ = b₂) with ⟨a₂, rfl⟩ | hb₂
+    · simpa only [onFun, he.extend_apply] using hf (ne_of_apply_ne e hne)
+    · simpa only [onFun, extend_apply' _ _ _ hb₂] using disjoint_bot_right
+  · simpa only [onFun, extend_apply' _ _ _ hb₁] using disjoint_bot_left
+
 namespace Set
 
 theorem Pairwise.mono (h : t ⊆ s) (hs : s.Pairwise r) : t.Pairwise r :=
@@ -129,8 +139,7 @@ theorem pairwise_union_of_symmetric (hr : Symmetric r) :
 
 theorem pairwise_insert :
     (insert a s).Pairwise r ↔ s.Pairwise r ∧ ∀ b ∈ s, a ≠ b → r a b ∧ r b a := by
-  simp only [insert_eq, pairwise_union, pairwise_singleton, true_and_iff, mem_singleton_iff,
-    forall_eq]
+  simp only [insert_eq, pairwise_union, pairwise_singleton, true_and, mem_singleton_iff, forall_eq]
 
 theorem pairwise_insert_of_not_mem (ha : a ∉ s) :
     (insert a s).Pairwise r ↔ s.Pairwise r ∧ ∀ b ∈ s, r a b ∧ r b a :=
diff --git a/Mathlib/Data/Set/Pairwise/Lattice.lean b/Mathlib/Data/Set/Pairwise/Lattice.lean
index 9830f4e15681f..6e82a9acdc83d 100644
--- a/Mathlib/Data/Set/Pairwise/Lattice.lean
+++ b/Mathlib/Data/Set/Pairwise/Lattice.lean
@@ -71,7 +71,7 @@ theorem PairwiseDisjoint.biUnion {s : Set ι'} {g : ι' → Set ι} {f : ι →
   obtain ⟨c, hc, ha⟩ := ha
   obtain ⟨d, hd, hb⟩ := hb
   obtain hcd | hcd := eq_or_ne (g c) (g d)
-  · exact hg d hd (hcd.subst ha) hb hab
+  · exact hg d hd (hcd ▸ ha) hb hab
   -- Porting note: the elaborator couldn't figure out `f` here.
   · exact (hs hc hd <| ne_of_apply_ne _ hcd).mono
       (le_iSup₂ (f := fun i (_ : i ∈ g c) => f i) a ha)
diff --git a/Mathlib/Data/Set/Pointwise/BigOperators.lean b/Mathlib/Data/Set/Pointwise/BigOperators.lean
index 8ad41832d91c2..7772e8e35548f 100644
--- a/Mathlib/Data/Set/Pointwise/BigOperators.lean
+++ b/Mathlib/Data/Set/Pointwise/BigOperators.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
 import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Data.Set.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 
 /-!
 # Results about pointwise operations on sets and big operators.
diff --git a/Mathlib/Data/Set/Pointwise/BoundedMul.lean b/Mathlib/Data/Set/Pointwise/BoundedMul.lean
index b98cbbde6ca66..2d630b685665e 100644
--- a/Mathlib/Data/Set/Pointwise/BoundedMul.lean
+++ b/Mathlib/Data/Set/Pointwise/BoundedMul.lean
@@ -3,8 +3,8 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury KudryashovJ
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.Monoid.Defs
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 
diff --git a/Mathlib/Data/Set/Pointwise/Finite.lean b/Mathlib/Data/Set/Pointwise/Finite.lean
index 02e69997156ee..370ad1550f6b6 100644
--- a/Mathlib/Data/Set/Pointwise/Finite.lean
+++ b/Mathlib/Data/Set/Pointwise/Finite.lean
@@ -3,8 +3,9 @@ Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Floris van Doorn
 -/
+import Mathlib.Algebra.Group.Action.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Data.Set.Finite
-import Mathlib.Data.Set.Pointwise.SMul
 
 /-! # Finiteness lemmas for pointwise operations on sets -/
 
@@ -25,16 +26,6 @@ theorem finite_one : (1 : Set α).Finite :=
 
 end One
 
-section InvolutiveInv
-
-variable [InvolutiveInv α] {s : Set α}
-
-@[to_additive]
-theorem Finite.inv (hs : s.Finite) : s⁻¹.Finite :=
-  hs.preimage inv_injective.injOn
-
-end InvolutiveInv
-
 section Mul
 
 variable [Mul α] {s t : Set α}
@@ -108,20 +99,52 @@ section Cancel
 variable [Mul α] [IsLeftCancelMul α] [IsRightCancelMul α] {s t : Set α}
 
 @[to_additive]
-theorem infinite_mul : (s * t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty :=
-  infinite_image2 (fun _ _ => (mul_left_injective _).injOn) fun _ _ =>
-    (mul_right_injective _).injOn
+lemma finite_mul : (s * t).Finite ↔ s.Finite ∧ t.Finite ∨ s = ∅ ∨ t = ∅ :=
+  finite_image2 (fun _ _ ↦ (mul_left_injective _).injOn) fun _ _ ↦ (mul_right_injective _).injOn
 
 @[to_additive]
-lemma finite_mul : (s * t).Finite ↔ s.Finite ∧ t.Finite ∨ s = ∅ ∨ t = ∅ :=
-  finite_image2  (fun _ _ ↦ (mul_left_injective _).injOn)
-    fun _ _ ↦ (mul_right_injective _).injOn
+lemma infinite_mul : (s * t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty :=
+  infinite_image2 (fun _ _ => (mul_left_injective _).injOn) fun _ _ => (mul_right_injective _).injOn
 
 end Cancel
 
+section InvolutiveInv
+variable [InvolutiveInv α] {s : Set α}
+
+@[to_additive (attr := simp)] lemma finite_inv : s⁻¹.Finite ↔ s.Finite := by
+  rw [← image_inv, finite_image_iff inv_injective.injOn]
+
+@[to_additive (attr := simp)] lemma infinite_inv : s⁻¹.Infinite ↔ s.Infinite := finite_inv.not
+
+@[to_additive] alias ⟨Finite.of_inv, Finite.inv⟩ := finite_inv
+
+end InvolutiveInv
+
+section Div
+variable [Div α] {s t : Set α}
+
+@[to_additive] lemma Finite.div : s.Finite → t.Finite → (s / t).Finite := .image2 _
+
+/-- Division preserves finiteness. -/
+@[to_additive "Subtraction preserves finiteness."]
+def fintypeDiv [DecidableEq α] (s t : Set α) [Fintype s] [Fintype t] : Fintype (s / t) :=
+  Set.fintypeImage2 _ _ _
+
+end Div
+
 section Group
 
-variable [Group α] [MulAction α β] {a : α} {s : Set β}
+variable [Group α] {s t : Set α}
+
+@[to_additive]
+lemma finite_div : (s / t).Finite ↔ s.Finite ∧ t.Finite ∨ s = ∅ ∨ t = ∅ :=
+  finite_image2 (fun _ _ ↦ div_left_injective.injOn) fun _ _ ↦ div_right_injective.injOn
+
+@[to_additive]
+lemma infinite_div : (s / t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty :=
+  infinite_image2 (fun _ _ ↦ div_left_injective.injOn) fun _ _ ↦ div_right_injective.injOn
+
+variable [MulAction α β] {a : α} {s : Set β}
 
 @[to_additive (attr := simp)]
 theorem finite_smul_set : (a • s).Finite ↔ s.Finite :=
@@ -131,11 +154,8 @@ theorem finite_smul_set : (a • s).Finite ↔ s.Finite :=
 theorem infinite_smul_set : (a • s).Infinite ↔ s.Infinite :=
   infinite_image_iff (MulAction.injective _).injOn
 
-alias ⟨Finite.of_smul_set, _⟩ := finite_smul_set
-
-alias ⟨_, Infinite.smul_set⟩ := infinite_smul_set
-
-attribute [to_additive] Finite.of_smul_set Infinite.smul_set
+@[to_additive] alias ⟨Finite.of_smul_set, _⟩ := finite_smul_set
+@[to_additive] alias ⟨_, Infinite.smul_set⟩ := infinite_smul_set
 
 end Group
 
diff --git a/Mathlib/Data/Set/Pointwise/Interval.lean b/Mathlib/Data/Set/Pointwise/Interval.lean
index 3200f9fcb68fd..96c008f454acd 100644
--- a/Mathlib/Data/Set/Pointwise/Interval.lean
+++ b/Mathlib/Data/Set/Pointwise/Interval.lean
@@ -3,11 +3,11 @@ Copyright (c) 2020 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Patrick Massot
 -/
-import Mathlib.Order.Interval.Set.UnorderedInterval
-import Mathlib.Algebra.Order.Interval.Set.Monoid
-import Mathlib.Data.Set.Pointwise.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.Field.Basic
 import Mathlib.Algebra.Order.Group.MinMax
+import Mathlib.Algebra.Order.Interval.Set.Monoid
+import Mathlib.Order.Interval.Set.UnorderedInterval
 
 /-!
 # (Pre)images of intervals
@@ -509,12 +509,12 @@ variable [LinearOrderedField α] {a : α}
 @[simp]
 theorem preimage_mul_const_Iio (a : α) {c : α} (h : 0 < c) :
     (fun x => x * c) ⁻¹' Iio a = Iio (a / c) :=
-  ext fun _x => (lt_div_iff h).symm
+  ext fun _x => (lt_div_iff₀ h).symm
 
 @[simp]
 theorem preimage_mul_const_Ioi (a : α) {c : α} (h : 0 < c) :
     (fun x => x * c) ⁻¹' Ioi a = Ioi (a / c) :=
-  ext fun _x => (div_lt_iff h).symm
+  ext fun _x => (div_lt_iff₀ h).symm
 
 @[simp]
 theorem preimage_mul_const_Iic (a : α) {c : α} (h : 0 < c) :
@@ -582,11 +582,11 @@ theorem preimage_mul_const_Icc_of_neg (a b : α) {c : α} (h : c < 0) :
 
 @[simp]
 theorem preimage_const_mul_Iio (a : α) {c : α} (h : 0 < c) : (c * ·) ⁻¹' Iio a = Iio (a / c) :=
-  ext fun _x => (lt_div_iff' h).symm
+  ext fun _x => (lt_div_iff₀' h).symm
 
 @[simp]
 theorem preimage_const_mul_Ioi (a : α) {c : α} (h : 0 < c) : (c * ·) ⁻¹' Ioi a = Ioi (a / c) :=
-  ext fun _x => (div_lt_iff' h).symm
+  ext fun _x => (div_lt_iff₀' h).symm
 
 @[simp]
 theorem preimage_const_mul_Iic (a : α) {c : α} (h : 0 < c) : (c * ·) ⁻¹' Iic a = Iic (a / c) :=
@@ -718,9 +718,9 @@ theorem image_mul_left_Ioo {a : α} (h : 0 < a) (b c : α) :
 theorem inv_Ioo_0_left {a : α} (ha : 0 < a) : (Ioo 0 a)⁻¹ = Ioi a⁻¹ := by
   ext x
   exact
-    ⟨fun h => inv_inv x ▸ (inv_lt_inv ha h.1).2 h.2, fun h =>
+    ⟨fun h => inv_inv x ▸ (inv_lt_inv₀ ha h.1).2 h.2, fun h =>
       ⟨inv_pos.2 <| (inv_pos.2 ha).trans h,
-        inv_inv a ▸ (inv_lt_inv ((inv_pos.2 ha).trans h)
+        inv_inv a ▸ (inv_lt_inv₀ ((inv_pos.2 ha).trans h)
           (inv_pos.2 ha)).2 h⟩⟩
 
 theorem inv_Ioi {a : α} (ha : 0 < a) : (Ioi a)⁻¹ = Ioo 0 a⁻¹ := by
diff --git a/Mathlib/Data/Set/Pointwise/ListOfFn.lean b/Mathlib/Data/Set/Pointwise/ListOfFn.lean
index c87a75ed34d9d..21e7712a88181 100644
--- a/Mathlib/Data/Set/Pointwise/ListOfFn.lean
+++ b/Mathlib/Data/Set/Pointwise/ListOfFn.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
 import Mathlib.Algebra.BigOperators.Group.List
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Data.List.OfFn
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Pointwise operations with lists of sets
diff --git a/Mathlib/Data/Set/Pointwise/SMul.lean b/Mathlib/Data/Set/Pointwise/SMul.lean
index f5b21fa76aef9..e3ed79c944b46 100644
--- a/Mathlib/Data/Set/Pointwise/SMul.lean
+++ b/Mathlib/Data/Set/Pointwise/SMul.lean
@@ -4,28 +4,16 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Floris van Doorn
 -/
 import Mathlib.Algebra.Group.Pi.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.GroupWithZero.Action.Basic
 import Mathlib.Algebra.Module.Defs
 import Mathlib.Data.Set.Pairwise.Basic
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
-# Pointwise operations of sets
+# Pointwise action on sets
 
-This file defines pointwise algebraic operations on sets.
-
-## Main declarations
-
-For sets `s` and `t` and scalar `a`:
-* `s • t`: Scalar multiplication, set of all `x • y` where `x ∈ s` and `y ∈ t`.
-* `s +ᵥ t`: Scalar addition, set of all `x +ᵥ y` where `x ∈ s` and `y ∈ t`.
-* `s -ᵥ t`: Scalar subtraction, set of all `x -ᵥ y` where `x ∈ s` and `y ∈ t`.
-* `a • s`: Scaling, set of all `a • x` where `x ∈ s`.
-* `a +ᵥ s`: Translation, set of all `a +ᵥ x` where `x ∈ s`.
-
-For `α` a semigroup/monoid, `Set α` is a semigroup/monoid.
-
-Appropriate definitions and results are also transported to the additive theory via `to_additive`.
+This file proves that several kinds of actions of a type `α` on another type `β` transfer to actions
+of `α`/`Set α` on `Set β`.
 
 ## Implementation notes
 
@@ -49,255 +37,6 @@ open Pointwise
 
 section SMul
 
-/-- The dilation of set `x • s` is defined as `{x • y | y ∈ s}` in locale `Pointwise`. -/
-@[to_additive
-      "The translation of set `x +ᵥ s` is defined as `{x +ᵥ y | y ∈ s}` in
-      locale `Pointwise`."]
-protected def smulSet [SMul α β] : SMul α (Set β) :=
-  ⟨fun a ↦ image (a • ·)⟩
-
-/-- The pointwise scalar multiplication of sets `s • t` is defined as `{x • y | x ∈ s, y ∈ t}` in
-locale `Pointwise`. -/
-@[to_additive
-      "The pointwise scalar addition of sets `s +ᵥ t` is defined as
-      `{x +ᵥ y | x ∈ s, y ∈ t}` in locale `Pointwise`."]
-protected def smul [SMul α β] : SMul (Set α) (Set β) :=
-  ⟨image2 (· • ·)⟩
-
-scoped[Pointwise] attribute [instance] Set.smulSet Set.smul
-
-scoped[Pointwise] attribute [instance] Set.vaddSet Set.vadd
-
-section SMul
-
-variable {ι : Sort*} {κ : ι → Sort*} [SMul α β] {s s₁ s₂ : Set α} {t t₁ t₂ u : Set β} {a : α}
-  {b : β}
-
-@[to_additive (attr := simp)]
-theorem image2_smul : image2 SMul.smul s t = s • t :=
-  rfl
-
-@[to_additive vadd_image_prod]
-theorem image_smul_prod : (fun x : α × β ↦ x.fst • x.snd) '' s ×ˢ t = s • t :=
-  image_prod _
-
-@[to_additive]
-theorem mem_smul : b ∈ s • t ↔ ∃ x ∈ s, ∃ y ∈ t, x • y = b :=
-  Iff.rfl
-
-@[to_additive]
-theorem smul_mem_smul : a ∈ s → b ∈ t → a • b ∈ s • t :=
-  mem_image2_of_mem
-
-@[to_additive (attr := simp)]
-theorem empty_smul : (∅ : Set α) • t = ∅ :=
-  image2_empty_left
-
-@[to_additive (attr := simp)]
-theorem smul_empty : s • (∅ : Set β) = ∅ :=
-  image2_empty_right
-
-@[to_additive (attr := simp)]
-theorem smul_eq_empty : s • t = ∅ ↔ s = ∅ ∨ t = ∅ :=
-  image2_eq_empty_iff
-
-@[to_additive (attr := simp)]
-theorem smul_nonempty : (s • t).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
-  image2_nonempty_iff
-
-@[to_additive]
-theorem Nonempty.smul : s.Nonempty → t.Nonempty → (s • t).Nonempty :=
-  Nonempty.image2
-
-@[to_additive]
-theorem Nonempty.of_smul_left : (s • t).Nonempty → s.Nonempty :=
-  Nonempty.of_image2_left
-
-@[to_additive]
-theorem Nonempty.of_smul_right : (s • t).Nonempty → t.Nonempty :=
-  Nonempty.of_image2_right
-
-@[to_additive (attr := simp low+1)]
-theorem smul_singleton : s • ({b} : Set β) = (· • b) '' s :=
-  image2_singleton_right
-
-@[to_additive (attr := simp low+1)]
-theorem singleton_smul : ({a} : Set α) • t = a • t :=
-  image2_singleton_left
-
-@[to_additive (attr := simp high)]
-theorem singleton_smul_singleton : ({a} : Set α) • ({b} : Set β) = {a • b} :=
-  image2_singleton
-
-@[to_additive (attr := mono)]
-theorem smul_subset_smul : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ • t₁ ⊆ s₂ • t₂ :=
-  image2_subset
-
-@[to_additive]
-theorem smul_subset_smul_left : t₁ ⊆ t₂ → s • t₁ ⊆ s • t₂ :=
-  image2_subset_left
-
-@[to_additive]
-theorem smul_subset_smul_right : s₁ ⊆ s₂ → s₁ • t ⊆ s₂ • t :=
-  image2_subset_right
-
-@[to_additive]
-theorem smul_subset_iff : s • t ⊆ u ↔ ∀ a ∈ s, ∀ b ∈ t, a • b ∈ u :=
-  image2_subset_iff
-
-
-@[to_additive]
-theorem union_smul : (s₁ ∪ s₂) • t = s₁ • t ∪ s₂ • t :=
-  image2_union_left
-
-@[to_additive]
-theorem smul_union : s • (t₁ ∪ t₂) = s • t₁ ∪ s • t₂ :=
-  image2_union_right
-
-@[to_additive]
-theorem inter_smul_subset : (s₁ ∩ s₂) • t ⊆ s₁ • t ∩ s₂ • t :=
-  image2_inter_subset_left
-
-@[to_additive]
-theorem smul_inter_subset : s • (t₁ ∩ t₂) ⊆ s • t₁ ∩ s • t₂ :=
-  image2_inter_subset_right
-
-@[to_additive]
-theorem inter_smul_union_subset_union : (s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ :=
-  image2_inter_union_subset_union
-
-@[to_additive]
-theorem union_smul_inter_subset_union : (s₁ ∪ s₂) • (t₁ ∩ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂ :=
-  image2_union_inter_subset_union
-
-@[to_additive]
-theorem iUnion_smul_left_image : ⋃ a ∈ s, a • t = s • t :=
-  iUnion_image_left _
-
-@[to_additive]
-theorem iUnion_smul_right_image : ⋃ a ∈ t, (· • a) '' s = s • t :=
-  iUnion_image_right _
-
-@[to_additive]
-theorem iUnion_smul (s : ι → Set α) (t : Set β) : (⋃ i, s i) • t = ⋃ i, s i • t :=
-  image2_iUnion_left _ _ _
-
-@[to_additive]
-theorem smul_iUnion (s : Set α) (t : ι → Set β) : (s • ⋃ i, t i) = ⋃ i, s • t i :=
-  image2_iUnion_right _ _ _
-
-@[to_additive]
-theorem iUnion₂_smul (s : ∀ i, κ i → Set α) (t : Set β) :
-    (⋃ (i) (j), s i j) • t = ⋃ (i) (j), s i j • t :=
-  image2_iUnion₂_left _ _ _
-
-@[to_additive]
-theorem smul_iUnion₂ (s : Set α) (t : ∀ i, κ i → Set β) :
-    (s • ⋃ (i) (j), t i j) = ⋃ (i) (j), s • t i j :=
-  image2_iUnion₂_right _ _ _
-
-@[to_additive]
-theorem iInter_smul_subset (s : ι → Set α) (t : Set β) : (⋂ i, s i) • t ⊆ ⋂ i, s i • t :=
-  image2_iInter_subset_left _ _ _
-
-@[to_additive]
-theorem smul_iInter_subset (s : Set α) (t : ι → Set β) : (s • ⋂ i, t i) ⊆ ⋂ i, s • t i :=
-  image2_iInter_subset_right _ _ _
-
-@[to_additive]
-theorem iInter₂_smul_subset (s : ∀ i, κ i → Set α) (t : Set β) :
-    (⋂ (i) (j), s i j) • t ⊆ ⋂ (i) (j), s i j • t :=
-  image2_iInter₂_subset_left _ _ _
-
-@[to_additive]
-theorem smul_iInter₂_subset (s : Set α) (t : ∀ i, κ i → Set β) :
-    (s • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s • t i j :=
-  image2_iInter₂_subset_right _ _ _
-
-@[to_additive]
-theorem smul_set_subset_smul {s : Set α} : a ∈ s → a • t ⊆ s • t :=
-  image_subset_image2_right
-
-@[to_additive (attr := simp)]
-theorem iUnion_smul_set (s : Set α) (t : Set β) : ⋃ a ∈ s, a • t = s • t :=
-  iUnion_image_left _
-
-end SMul
-
-section SMulSet
-
-variable {ι : Sort*} {κ : ι → Sort*} [SMul α β] {s t t₁ t₂ : Set β} {a : α} {b : β} {x y : β}
-
-@[to_additive]
-theorem image_smul : (fun x ↦ a • x) '' t = a • t :=
-  rfl
-
-scoped[Pointwise] attribute [simp] Set.image_smul Set.image_vadd
-
-@[to_additive]
-theorem mem_smul_set : x ∈ a • t ↔ ∃ y, y ∈ t ∧ a • y = x :=
-  Iff.rfl
-
-@[to_additive]
-theorem smul_mem_smul_set : b ∈ s → a • b ∈ a • s :=
-  mem_image_of_mem _
-
-@[to_additive (attr := simp)]
-theorem smul_set_empty : a • (∅ : Set β) = ∅ :=
-  image_empty _
-
-@[to_additive (attr := simp)]
-theorem smul_set_eq_empty : a • s = ∅ ↔ s = ∅ :=
-  image_eq_empty
-
-@[to_additive (attr := simp)]
-theorem smul_set_nonempty : (a • s).Nonempty ↔ s.Nonempty :=
-  image_nonempty
-
-@[to_additive (attr := simp)]
-theorem smul_set_singleton : a • ({b} : Set β) = {a • b} :=
-  image_singleton
-
-@[to_additive]
-theorem smul_set_mono : s ⊆ t → a • s ⊆ a • t :=
-  image_subset _
-
-@[to_additive]
-theorem smul_set_subset_iff : a • s ⊆ t ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ t :=
-  image_subset_iff
-
-@[to_additive]
-theorem smul_set_union : a • (t₁ ∪ t₂) = a • t₁ ∪ a • t₂ :=
-  image_union _ _ _
-
-@[to_additive]
-theorem smul_set_inter_subset : a • (t₁ ∩ t₂) ⊆ a • t₁ ∩ a • t₂ :=
-  image_inter_subset _ _ _
-
-@[to_additive]
-theorem smul_set_iUnion (a : α) (s : ι → Set β) : (a • ⋃ i, s i) = ⋃ i, a • s i :=
-  image_iUnion
-
-@[to_additive]
-theorem smul_set_iUnion₂ (a : α) (s : ∀ i, κ i → Set β) :
-    (a • ⋃ (i) (j), s i j) = ⋃ (i) (j), a • s i j :=
-  image_iUnion₂ _ _
-
-@[to_additive]
-theorem smul_set_iInter_subset (a : α) (t : ι → Set β) : (a • ⋂ i, t i) ⊆ ⋂ i, a • t i :=
-  image_iInter_subset _ _
-
-@[to_additive]
-theorem smul_set_iInter₂_subset (a : α) (t : ∀ i, κ i → Set β) :
-    (a • ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), a • t i j :=
-  image_iInter₂_subset _ _
-
-@[to_additive]
-theorem Nonempty.smul_set : s.Nonempty → (a • s).Nonempty :=
-  Nonempty.image _
-
-end SMulSet
-
 section Mul
 
 variable [Mul α] {s t u : Set α} {a : α}
@@ -327,19 +66,6 @@ end Mul
 
 variable {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
 
-@[to_additive]
-theorem range_smul_range {ι κ : Type*} [SMul α β] (b : ι → α) (c : κ → β) :
-    range b • range c = range fun p : ι × κ ↦ b p.1 • c p.2 :=
-  image2_range ..
-
-@[to_additive]
-theorem smul_set_range [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
-    a • range f = range fun i ↦ a • f i :=
-  (range_comp _ _).symm
-
-@[to_additive] lemma range_smul [SMul α β] {ι : Sort*} (a : α) (f : ι → β) :
-    range (fun i ↦ a • f i) = a • range f := (smul_set_range ..).symm
-
 @[to_additive] lemma range_mul [Mul α] {ι : Sort*} (a : α) (f : ι → α) :
     range (fun i ↦ a * f i) = a • range f := range_smul a f
 
@@ -458,141 +184,8 @@ instance [Zero α] [Mul α] [NoZeroDivisors α] : NoZeroDivisors (Set α) :=
 
 end SMul
 
-section VSub
-
-variable {ι : Sort*} {κ : ι → Sort*} [VSub α β] {s s₁ s₂ t t₁ t₂ : Set β} {u : Set α} {a : α}
-  {b c : β}
-
-instance vsub : VSub (Set α) (Set β) :=
-  ⟨image2 (· -ᵥ ·)⟩
-
-@[simp]
-theorem image2_vsub : (image2 VSub.vsub s t : Set α) = s -ᵥ t :=
-  rfl
-
-theorem image_vsub_prod : (fun x : β × β ↦ x.fst -ᵥ x.snd) '' s ×ˢ t = s -ᵥ t :=
-  image_prod _
-
-theorem mem_vsub : a ∈ s -ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x -ᵥ y = a :=
-  Iff.rfl
-
-theorem vsub_mem_vsub (hb : b ∈ s) (hc : c ∈ t) : b -ᵥ c ∈ s -ᵥ t :=
-  mem_image2_of_mem hb hc
-
-@[simp]
-theorem empty_vsub (t : Set β) : ∅ -ᵥ t = ∅ :=
-  image2_empty_left
-
-@[simp]
-theorem vsub_empty (s : Set β) : s -ᵥ ∅ = ∅ :=
-  image2_empty_right
-
-@[simp]
-theorem vsub_eq_empty : s -ᵥ t = ∅ ↔ s = ∅ ∨ t = ∅ :=
-  image2_eq_empty_iff
-
-@[simp]
-theorem vsub_nonempty : (s -ᵥ t : Set α).Nonempty ↔ s.Nonempty ∧ t.Nonempty :=
-  image2_nonempty_iff
-
-theorem Nonempty.vsub : s.Nonempty → t.Nonempty → (s -ᵥ t : Set α).Nonempty :=
-  Nonempty.image2
-
-theorem Nonempty.of_vsub_left : (s -ᵥ t : Set α).Nonempty → s.Nonempty :=
-  Nonempty.of_image2_left
-
-theorem Nonempty.of_vsub_right : (s -ᵥ t : Set α).Nonempty → t.Nonempty :=
-  Nonempty.of_image2_right
-
-@[simp low+1]
-theorem vsub_singleton (s : Set β) (b : β) : s -ᵥ {b} = (· -ᵥ b) '' s :=
-  image2_singleton_right
-
-@[simp low+1]
-theorem singleton_vsub (t : Set β) (b : β) : {b} -ᵥ t = (b -ᵥ ·) '' t :=
-  image2_singleton_left
-
-@[simp high]
-theorem singleton_vsub_singleton : ({b} : Set β) -ᵥ {c} = {b -ᵥ c} :=
-  image2_singleton
-
-@[mono]
-theorem vsub_subset_vsub : s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ -ᵥ t₁ ⊆ s₂ -ᵥ t₂ :=
-  image2_subset
-
-theorem vsub_subset_vsub_left : t₁ ⊆ t₂ → s -ᵥ t₁ ⊆ s -ᵥ t₂ :=
-  image2_subset_left
-
-theorem vsub_subset_vsub_right : s₁ ⊆ s₂ → s₁ -ᵥ t ⊆ s₂ -ᵥ t :=
-  image2_subset_right
-
-theorem vsub_subset_iff : s -ᵥ t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x -ᵥ y ∈ u :=
-  image2_subset_iff
-
-theorem vsub_self_mono (h : s ⊆ t) : s -ᵥ s ⊆ t -ᵥ t :=
-  vsub_subset_vsub h h
-
-theorem union_vsub : s₁ ∪ s₂ -ᵥ t = s₁ -ᵥ t ∪ (s₂ -ᵥ t) :=
-  image2_union_left
-
-theorem vsub_union : s -ᵥ (t₁ ∪ t₂) = s -ᵥ t₁ ∪ (s -ᵥ t₂) :=
-  image2_union_right
-
-theorem inter_vsub_subset : s₁ ∩ s₂ -ᵥ t ⊆ (s₁ -ᵥ t) ∩ (s₂ -ᵥ t) :=
-  image2_inter_subset_left
-
-theorem vsub_inter_subset : s -ᵥ t₁ ∩ t₂ ⊆ (s -ᵥ t₁) ∩ (s -ᵥ t₂) :=
-  image2_inter_subset_right
-
-theorem inter_vsub_union_subset_union : s₁ ∩ s₂ -ᵥ (t₁ ∪ t₂) ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) :=
-  image2_inter_union_subset_union
-
-theorem union_vsub_inter_subset_union : s₁ ∪ s₂ -ᵥ t₁ ∩ t₂ ⊆ s₁ -ᵥ t₁ ∪ (s₂ -ᵥ t₂) :=
-  image2_union_inter_subset_union
-
-theorem iUnion_vsub_left_image : ⋃ a ∈ s, (a -ᵥ ·) '' t = s -ᵥ t :=
-  iUnion_image_left _
-
-theorem iUnion_vsub_right_image : ⋃ a ∈ t, (· -ᵥ a) '' s = s -ᵥ t :=
-  iUnion_image_right _
-
-theorem iUnion_vsub (s : ι → Set β) (t : Set β) : (⋃ i, s i) -ᵥ t = ⋃ i, s i -ᵥ t :=
-  image2_iUnion_left _ _ _
-
-theorem vsub_iUnion (s : Set β) (t : ι → Set β) : (s -ᵥ ⋃ i, t i) = ⋃ i, s -ᵥ t i :=
-  image2_iUnion_right _ _ _
-
-theorem iUnion₂_vsub (s : ∀ i, κ i → Set β) (t : Set β) :
-    (⋃ (i) (j), s i j) -ᵥ t = ⋃ (i) (j), s i j -ᵥ t :=
-  image2_iUnion₂_left _ _ _
-
-theorem vsub_iUnion₂ (s : Set β) (t : ∀ i, κ i → Set β) :
-    (s -ᵥ ⋃ (i) (j), t i j) = ⋃ (i) (j), s -ᵥ t i j :=
-  image2_iUnion₂_right _ _ _
-
-theorem iInter_vsub_subset (s : ι → Set β) (t : Set β) : (⋂ i, s i) -ᵥ t ⊆ ⋂ i, s i -ᵥ t :=
-  image2_iInter_subset_left _ _ _
-
-theorem vsub_iInter_subset (s : Set β) (t : ι → Set β) : (s -ᵥ ⋂ i, t i) ⊆ ⋂ i, s -ᵥ t i :=
-  image2_iInter_subset_right _ _ _
-
-theorem iInter₂_vsub_subset (s : ∀ i, κ i → Set β) (t : Set β) :
-    (⋂ (i) (j), s i j) -ᵥ t ⊆ ⋂ (i) (j), s i j -ᵥ t :=
-  image2_iInter₂_subset_left _ _ _
-
-theorem vsub_iInter₂_subset (s : Set β) (t : ∀ i, κ i → Set β) :
-    (s -ᵥ ⋂ (i) (j), t i j) ⊆ ⋂ (i) (j), s -ᵥ t i j :=
-  image2_iInter₂_subset_right _ _ _
-
-end VSub
-
 open Pointwise
 
-@[to_additive]
-theorem image_smul_comm [SMul α β] [SMul α γ] (f : β → γ) (a : α) (s : Set β) :
-    (∀ b, f (a • b) = a • f b) → f '' (a • s) = a • f '' s :=
-  image_comm
-
 @[to_additive]
 theorem image_smul_distrib [MulOneClass α] [MulOneClass β] [FunLike F α β] [MonoidHomClass F α β]
     (f : F) (a : α) (s : Set α) :
diff --git a/Mathlib/Data/Set/Prod.lean b/Mathlib/Data/Set/Prod.lean
index 0db6873c91241..d147798baa0d9 100644
--- a/Mathlib/Data/Set/Prod.lean
+++ b/Mathlib/Data/Set/Prod.lean
@@ -9,13 +9,15 @@ import Mathlib.Data.SProd
 /-!
 # Sets in product and pi types
 
-This file defines the product of sets in `α × β` and in `Π i, α i` along with the diagonal of a
-type.
+This file proves basic properties of product of sets in `α × β` and in `Π i, α i`, and of the
+diagonal of a type.
 
 ## Main declarations
 
+This file contains basic results on the following notions, which are defined in `Set.Operations`.
+
 * `Set.prod`: Binary product of sets. For `s : Set α`, `t : Set β`, we have
-  `s.prod t : Set (α × β)`.
+  `s.prod t : Set (α × β)`. Denoted by `s ×ˢ t`.
 * `Set.diagonal`: Diagonal of a type. `Set.diagonal α = {(x, x) | x : α}`.
 * `Set.offDiag`: Off-diagonal. `s ×ˢ s` without the diagonal.
 * `Set.pi`: Arbitrary product of sets.
@@ -38,7 +40,7 @@ theorem Subsingleton.prod (hs : s.Subsingleton) (ht : t.Subsingleton) :
   Prod.ext (hs hx.1 hy.1) (ht hx.2 hy.2)
 
 noncomputable instance decidableMemProd [DecidablePred (· ∈ s)] [DecidablePred (· ∈ t)] :
-    DecidablePred (· ∈ s ×ˢ t) := fun _ => And.decidable
+    DecidablePred (· ∈ s ×ˢ t) := fun x => inferInstanceAs (Decidable (x.1 ∈ s ∧ x.2 ∈ t))
 
 @[gcongr]
 theorem prod_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : s₁ ×ˢ t₁ ⊆ s₂ ×ˢ t₂ :=
@@ -72,17 +74,17 @@ theorem exists_prod_set {p : α × β → Prop} : (∃ x ∈ s ×ˢ t, p x) ↔
 @[simp]
 theorem prod_empty : s ×ˢ (∅ : Set β) = ∅ := by
   ext
-  exact and_false_iff _
+  exact iff_of_eq (and_false _)
 
 @[simp]
 theorem empty_prod : (∅ : Set α) ×ˢ t = ∅ := by
   ext
-  exact false_and_iff _
+  exact iff_of_eq (false_and _)
 
 @[simp, mfld_simps]
 theorem univ_prod_univ : @univ α ×ˢ @univ β = univ := by
   ext
-  exact true_and_iff _
+  exact iff_of_eq (true_and _)
 
 theorem univ_prod {t : Set β} : (univ : Set α) ×ˢ t = Prod.snd ⁻¹' t := by simp [prod_eq]
 
@@ -299,6 +301,9 @@ theorem fst_image_prod (s : Set β) {t : Set α} (ht : t.Nonempty) : Prod.fst ''
     let ⟨x, hx⟩ := ht
     ⟨(y, x), ⟨hy, hx⟩, rfl⟩
 
+lemma mapsTo_fst_prod {s : Set α} {t : Set β} : MapsTo Prod.fst (s ×ˢ t) s :=
+  fun _ hx ↦ (mem_prod.1 hx).1
+
 theorem prod_subset_preimage_snd (s : Set α) (t : Set β) : s ×ˢ t ⊆ Prod.snd ⁻¹' t :=
   inter_subset_right
 
@@ -310,6 +315,9 @@ theorem snd_image_prod {s : Set α} (hs : s.Nonempty) (t : Set β) : Prod.snd ''
     let ⟨x, x_in⟩ := hs
     ⟨(x, y), ⟨x_in, y_in⟩, rfl⟩
 
+lemma mapsTo_snd_prod {s : Set α} {t : Set β} : MapsTo Prod.snd (s ×ˢ t) t :=
+  fun _ hx ↦ (mem_prod.1 hx).2
+
 theorem prod_diff_prod : s ×ˢ t \ s₁ ×ˢ t₁ = s ×ˢ (t \ t₁) ∪ (s \ s₁) ×ˢ t := by
   ext x
   by_cases h₁ : x.1 ∈ s₁ <;> by_cases h₂ : x.2 ∈ t₁ <;> simp [*]
@@ -326,7 +334,7 @@ theorem prod_subset_prod_iff : s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t
   · have := image_subset (Prod.snd : α × β → β) H
     rwa [snd_image_prod st.1, snd_image_prod (h.mono H).fst] at this
   · intro H
-    simp only [st.1.ne_empty, st.2.ne_empty, or_false_iff] at H
+    simp only [st.1.ne_empty, st.2.ne_empty, or_false] at H
     exact prod_mono H.1 H.2
 
 theorem prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t).Nonempty) :
@@ -335,7 +343,7 @@ theorem prod_eq_prod_iff_of_nonempty (h : (s ×ˢ t).Nonempty) :
   · intro heq
     have h₁ : (s₁ ×ˢ t₁ : Set _).Nonempty := by rwa [← heq]
     rw [prod_nonempty_iff] at h h₁
-    rw [← fst_image_prod s h.2, ← fst_image_prod s₁ h₁.2, heq, eq_self_iff_true, true_and_iff, ←
+    rw [← fst_image_prod s h.2, ← fst_image_prod s₁ h₁.2, heq, eq_self_iff_true, true_and, ←
       snd_image_prod h.1 t, ← snd_image_prod h₁.1 t₁, heq]
   · rintro ⟨rfl, rfl⟩
     rfl
@@ -344,18 +352,17 @@ theorem prod_eq_prod_iff :
     s ×ˢ t = s₁ ×ˢ t₁ ↔ s = s₁ ∧ t = t₁ ∨ (s = ∅ ∨ t = ∅) ∧ (s₁ = ∅ ∨ t₁ = ∅) := by
   symm
   rcases eq_empty_or_nonempty (s ×ˢ t) with h | h
-  · simp_rw [h, @eq_comm _ ∅, prod_eq_empty_iff, prod_eq_empty_iff.mp h, true_and_iff,
+  · simp_rw [h, @eq_comm _ ∅, prod_eq_empty_iff, prod_eq_empty_iff.mp h, true_and,
       or_iff_right_iff_imp]
     rintro ⟨rfl, rfl⟩
     exact prod_eq_empty_iff.mp h
   rw [prod_eq_prod_iff_of_nonempty h]
   rw [nonempty_iff_ne_empty, Ne, prod_eq_empty_iff] at h
-  simp_rw [h, false_and_iff, or_false_iff]
+  simp_rw [h, false_and, or_false]
 
 @[simp]
 theorem prod_eq_iff_eq (ht : t.Nonempty) : s ×ˢ t = s₁ ×ˢ t ↔ s = s₁ := by
-  simp_rw [prod_eq_prod_iff, ht.ne_empty, and_true_iff, or_iff_left_iff_imp,
-    or_false_iff]
+  simp_rw [prod_eq_prod_iff, ht.ne_empty, and_true, or_iff_left_iff_imp, or_false]
   rintro ⟨rfl, rfl⟩
   rfl
 
@@ -631,7 +638,7 @@ theorem pi_congr (h : s₁ = s₂) (h' : ∀ i ∈ s₁, t₁ i = t₂ i) : s₁
 
 theorem pi_eq_empty (hs : i ∈ s) (ht : t i = ∅) : s.pi t = ∅ := by
   ext f
-  simp only [mem_empty_iff_false, not_forall, iff_false_iff, mem_pi, Classical.not_imp]
+  simp only [mem_empty_iff_false, not_forall, iff_false, mem_pi, Classical.not_imp]
   exact ⟨i, hs, by simp [ht]⟩
 
 theorem univ_pi_eq_empty (ht : t i = ∅) : pi univ t = ∅ :=
@@ -679,15 +686,6 @@ theorem disjoint_pi : Disjoint (s.pi t₁) (s.pi t₂) ↔ ∃ i ∈ s, Disjoint
 
 end Nonempty
 
--- Porting note: Removing `simp` - LHS does not simplify
-theorem range_dcomp (f : ∀ i, α i → β i) :
-    (range fun g : ∀ i, α i => fun i => f i (g i)) = pi univ fun i => range (f i) := by
-  refine Subset.antisymm ?_ fun x hx => ?_
-  · rintro _ ⟨x, rfl⟩ i -
-    exact ⟨x i, rfl⟩
-  · choose y hy using hx
-    exact ⟨fun i => y i trivial, funext fun i => hy i trivial⟩
-
 @[simp]
 theorem insert_pi (i : ι) (s : Set ι) (t : ∀ i, Set (α i)) :
     pi (insert i s) t = eval i ⁻¹' t i ∩ pi s t := by
@@ -802,8 +800,8 @@ theorem eval_image_univ_pi (ht : (pi univ t).Nonempty) :
     (fun f : ∀ i, α i => f i) '' pi univ t = t i :=
   eval_image_pi (mem_univ i) ht
 
-theorem dcomp_image_pi {f : ∀ i, α i → β i} (hf : ∀ i ∉ s, Surjective (f i)) (t : ∀ i, Set (α i)) :
-    (f _ ∘' ·) '' s.pi t = s.pi fun i ↦ f i '' t i := by
+theorem piMap_image_pi {f : ∀ i, α i → β i} (hf : ∀ i ∉ s, Surjective (f i)) (t : ∀ i, Set (α i)) :
+    Pi.map f '' s.pi t = s.pi fun i ↦ f i '' t i := by
   refine Subset.antisymm (image_subset_iff.2 fun a ha i hi ↦ mem_image_of_mem _ (ha _ hi)) ?_
   intro b hb
   have : ∀ i, ∃ a, f i a = b i ∧ (i ∈ s → a ∈ t i) := by
@@ -815,9 +813,19 @@ theorem dcomp_image_pi {f : ∀ i, α i → β i} (hf : ∀ i ∉ s, Surjective
   choose a hab hat using this
   exact ⟨a, hat, funext hab⟩
 
-theorem dcomp_image_univ_pi (f : ∀ i, α i → β i) (t : ∀ i, Set (α i)) :
-    (f _ ∘' ·) '' univ.pi t = univ.pi fun i ↦ f i '' t i :=
-  dcomp_image_pi (by simp) t
+@[deprecated (since := "2024-10-06")] alias dcomp_image_pi := piMap_image_pi
+
+theorem piMap_image_univ_pi (f : ∀ i, α i → β i) (t : ∀ i, Set (α i)) :
+    Pi.map f '' univ.pi t = univ.pi fun i ↦ f i '' t i :=
+  piMap_image_pi (by simp) t
+
+@[deprecated (since := "2024-10-06")] alias dcomp_image_univ_pi := piMap_image_univ_pi
+
+@[simp]
+theorem range_piMap (f : ∀ i, α i → β i) : range (Pi.map f) = pi univ fun i ↦ range (f i) := by
+  simp only [← image_univ, ← piMap_image_univ_pi, pi_univ]
+
+@[deprecated (since := "2024-10-06")] alias range_dcomp := range_piMap
 
 theorem pi_subset_pi_iff : pi s t₁ ⊆ pi s t₂ ↔ (∀ i ∈ s, t₁ i ⊆ t₂ i) ∨ pi s t₁ = ∅ := by
   refine
@@ -854,7 +862,10 @@ theorem update_preimage_pi [DecidableEq ι] {f : ∀ i, α i} (hi : i ∈ s)
 theorem update_image [DecidableEq ι] (x : (i : ι) → β i) (i : ι) (s : Set (β i)) :
     update x i '' s = Set.univ.pi (update (fun j ↦ {x j}) i s) := by
   ext y
-  simp [update_eq_iff, and_left_comm (a := _ ∈ s), forall_update_iff, eq_comm (a := y _)]
+  simp only [mem_image, update_eq_iff, ne_eq, and_left_comm (a := _ ∈ s), exists_eq_left, mem_pi,
+    mem_univ, true_implies]
+  rw [forall_update_iff (p := fun x s => y x ∈ s)]
+  simp [eq_comm]
 
 theorem update_preimage_univ_pi [DecidableEq ι] {f : ∀ i, α i} (hf : ∀ j ≠ i, f j ∈ t j) :
     update f i ⁻¹' pi univ t = t i :=
diff --git a/Mathlib/Data/Set/Semiring.lean b/Mathlib/Data/Set/Semiring.lean
index 9acda7f2291a3..e6ff5cf305201 100644
--- a/Mathlib/Data/Set/Semiring.lean
+++ b/Mathlib/Data/Set/Semiring.lean
@@ -3,9 +3,9 @@ Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Order.Kleene
 import Mathlib.Algebra.Order.Ring.Canonical
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Sets as a semiring under union
diff --git a/Mathlib/Data/Set/Sigma.lean b/Mathlib/Data/Set/Sigma.lean
index b10a51f01db79..7201429b16acc 100644
--- a/Mathlib/Data/Set/Sigma.lean
+++ b/Mathlib/Data/Set/Sigma.lean
@@ -68,15 +68,17 @@ theorem exists_sigma_iff {p : (Σi, α i) → Prop} :
     (∃ x ∈ s.sigma t, p x) ↔ ∃ i ∈ s, ∃ a ∈ t i, p ⟨i, a⟩ :=
   ⟨fun ⟨⟨i, a⟩, ha, h⟩ ↦ ⟨i, ha.1, a, ha.2, h⟩, fun ⟨i, hi, a, ha, h⟩ ↦ ⟨⟨i, a⟩, ⟨hi, ha⟩, h⟩⟩
 
-@[simp] theorem sigma_empty : s.sigma (fun i ↦ (∅ : Set (α i))) = ∅ := ext fun _ ↦ and_false_iff _
+@[simp] theorem sigma_empty : s.sigma (fun i ↦ (∅ : Set (α i))) = ∅ :=
+  ext fun _ ↦ iff_of_eq (and_false _)
 
-@[simp] theorem empty_sigma : (∅ : Set ι).sigma t = ∅ := ext fun _ ↦ false_and_iff _
+@[simp] theorem empty_sigma : (∅ : Set ι).sigma t = ∅ := ext fun _ ↦ iff_of_eq (false_and _)
 
-theorem univ_sigma_univ : (@univ ι).sigma (fun _ ↦ @univ (α i)) = univ := ext fun _ ↦ true_and_iff _
+theorem univ_sigma_univ : (@univ ι).sigma (fun _ ↦ @univ (α i)) = univ :=
+  ext fun _ ↦ iff_of_eq (true_and _)
 
 @[simp]
 theorem sigma_univ : s.sigma (fun _ ↦ univ : ∀ i, Set (α i)) = Sigma.fst ⁻¹' s :=
-  ext fun _ ↦ and_true_iff _
+  ext fun _ ↦ iff_of_eq (and_true _)
 
 @[simp] theorem univ_sigma_preimage_mk (s : Set (Σ i, α i)) :
     (univ : Set ι).sigma (fun i ↦ Sigma.mk i ⁻¹' s) = s :=
diff --git a/Mathlib/Data/Set/Subsingleton.lean b/Mathlib/Data/Set/Subsingleton.lean
index b31273026245f..fa736da6540c2 100644
--- a/Mathlib/Data/Set/Subsingleton.lean
+++ b/Mathlib/Data/Set/Subsingleton.lean
@@ -215,7 +215,7 @@ theorem Nontrivial.ne_singleton {x} (hs : s.Nontrivial) : s ≠ {x} := fun H =>
   exact not_nontrivial_singleton hs
 
 theorem Nontrivial.not_subset_singleton {x} (hs : s.Nontrivial) : ¬s ⊆ {x} :=
-  (not_congr subset_singleton_iff_eq).2 (not_or_of_not hs.ne_empty hs.ne_singleton)
+  (not_congr subset_singleton_iff_eq).2 (not_or_intro hs.ne_empty hs.ne_singleton)
 
 theorem nontrivial_univ [Nontrivial α] : (univ : Set α).Nontrivial :=
   let ⟨x, y, hxy⟩ := exists_pair_ne α
diff --git a/Mathlib/Data/Set/UnionLift.lean b/Mathlib/Data/Set/UnionLift.lean
index 9b8184a95693c..cd2d6ffd31433 100644
--- a/Mathlib/Data/Set/UnionLift.lean
+++ b/Mathlib/Data/Set/UnionLift.lean
@@ -62,7 +62,6 @@ variable {S : ι → Set α} {f : ∀ i, S i → β}
 theorem iUnionLift_mk {i : ι} (x : S i) (hx : (x : α) ∈ T) :
     iUnionLift S f hf T hT ⟨x, hx⟩ = f i x := hf _ i x _ _
 
-@[simp]
 theorem iUnionLift_inclusion {i : ι} (x : S i) (h : S i ⊆ T) :
     iUnionLift S f hf T hT (Set.inclusion h x) = f i x :=
   iUnionLift_mk x _
@@ -139,7 +138,6 @@ theorem iUnionLift_binary (dir : Directed (· ≤ ·) S) (op : T → T → T) (o
   have hxy : (Set.inclusion (Set.subset_iUnion S k) (opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩) : α) ∈ S k :=
     (opi k ⟨x, hik hi⟩ ⟨y, hjk hj⟩).prop
   conv_lhs => rw [hx, hy, ← hopi, iUnionLift_of_mem _ hxy]
-  rfl
 
 end UnionLift
 
diff --git a/Mathlib/Data/SetLike/Basic.lean b/Mathlib/Data/SetLike/Basic.lean
index bd1f5ca66f08e..2a07db5d06280 100644
--- a/Mathlib/Data/SetLike/Basic.lean
+++ b/Mathlib/Data/SetLike/Basic.lean
@@ -28,7 +28,7 @@ boilerplate for every `SetLike`: a `coe_sort`, a `coe` to set, a
 
 A typical subobject should be declared as:
 ```
-structure MySubobject (X : Type*) [ObjectTypeclass X] :=
+structure MySubobject (X : Type*) [ObjectTypeclass X] where
   (carrier : Set X)
   (op_mem' : ∀ {x : X}, x ∈ carrier → sorry ∈ carrier)
 
@@ -60,7 +60,7 @@ end MySubobject
 
 An alternative to `SetLike` could have been an extensional `Membership` typeclass:
 ```
-class ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β :=
+class ExtMembership (α : out_param <| Type u) (β : Type v) extends Membership α β where
   (ext_iff : ∀ {s t : β}, s = t ↔ ∀ (x : α), x ∈ s ↔ x ∈ t)
 ```
 While this is equivalent, `SetLike` conveniently uses a carrier set projection directly.
@@ -106,7 +106,7 @@ variable {A : Type*} {B : Type*} [i : SetLike A B]
 instance : CoeTC A (Set B) where coe := SetLike.coe
 
 instance (priority := 100) instMembership : Membership B A :=
-  ⟨fun x p => x ∈ (p : Set B)⟩
+  ⟨fun p x => x ∈ (p : Set B)⟩
 
 instance (priority := 100) : CoeSort A (Type _) :=
   ⟨fun p => { x : B // x ∈ p }⟩
@@ -121,9 +121,9 @@ uses the `SetLike.instMembership` instance. -/
 def delabSubtypeSetLike : Delab := whenPPOption getPPNotation do
   let #[_, .lam n _ body _] := (← getExpr).getAppArgs | failure
   guard <| body.isAppOf ``Membership.mem
-  let #[_, _, inst, .bvar 0, _] := body.getAppArgs | failure
+  let #[_, _, inst, _, .bvar 0] := body.getAppArgs | failure
   guard <| inst.isAppOfArity ``instMembership 3
-  let S ← withAppArg <| withBindingBody n <| withNaryArg 4 delab
+  let S ← withAppArg <| withBindingBody n <| withNaryArg 3 delab
   `(↥$S)
 
 end Delab
@@ -191,17 +191,15 @@ instance (priority := 100) instPartialOrder : PartialOrder A :=
 theorem le_def {S T : A} : S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T :=
   Iff.rfl
 
-@[simp, norm_cast]
-theorem coe_subset_coe {S T : A} : (S : Set B) ⊆ T ↔ S ≤ T :=
-  Iff.rfl
+@[simp, norm_cast] lemma coe_subset_coe {S T : A} : (S : Set B) ⊆ T ↔ S ≤ T := .rfl
+@[simp, norm_cast] lemma coe_ssubset_coe {S T : A} : (S : Set B) ⊂ T ↔ S < T := .rfl
+
+@[gcongr] protected alias ⟨_, GCongr.coe_subset_coe⟩ := coe_subset_coe
+@[gcongr] protected alias ⟨_, GCongr.coe_ssubset_coe⟩ := coe_ssubset_coe
 
 @[mono]
 theorem coe_mono : Monotone (SetLike.coe : A → Set B) := fun _ _ => coe_subset_coe.mpr
 
-@[simp, norm_cast]
-theorem coe_ssubset_coe {S T : A} : (S : Set B) ⊂ T ↔ S < T :=
-  Iff.rfl
-
 @[mono]
 theorem coe_strictMono : StrictMono (SetLike.coe : A → Set B) := fun _ _ => coe_ssubset_coe.mpr
 
diff --git a/Mathlib/Data/Setoid/Basic.lean b/Mathlib/Data/Setoid/Basic.lean
index 294f426f6d013..ee631cca59242 100644
--- a/Mathlib/Data/Setoid/Basic.lean
+++ b/Mathlib/Data/Setoid/Basic.lean
@@ -111,6 +111,49 @@ protected def prod (r : Setoid α) (s : Setoid β) :
     ⟨fun x => ⟨r.refl' x.1, s.refl' x.2⟩, fun h => ⟨r.symm' h.1, s.symm' h.2⟩,
       fun h₁ h₂ => ⟨r.trans' h₁.1 h₂.1, s.trans' h₁.2 h₂.2⟩⟩
 
+lemma prod_apply {r : Setoid α} {s : Setoid β} {x₁ x₂ : α} {y₁ y₂ : β} :
+    @Setoid.r _ (r.prod s) (x₁, y₁) (x₂, y₂) ↔ (@Setoid.r _ r x₁ x₂ ∧ @Setoid.r _ s y₁ y₂) :=
+  Iff.rfl
+
+lemma piSetoid_apply {ι : Sort*} {α : ι → Sort*} {r : ∀ i, Setoid (α i)} {x y : ∀ i, α i} :
+    @Setoid.r _ (@piSetoid _ _ r) x y ↔ ∀ i, @Setoid.r _ (r i) (x i) (y i) :=
+  Iff.rfl
+
+/-- A bijection between the product of two quotients and the quotient by the product of the
+equivalence relations. -/
+@[simps]
+def prodQuotientEquiv (r : Setoid α) (s : Setoid β) :
+    Quotient r × Quotient s ≃ Quotient (r.prod s) where
+  toFun := fun (x, y) ↦ Quotient.map₂' Prod.mk (fun _ _ hx _ _ hy ↦ ⟨hx, hy⟩) x y
+  invFun := fun q ↦ Quotient.liftOn' q (fun xy ↦ (Quotient.mk'' xy.1, Quotient.mk'' xy.2))
+    fun x y hxy ↦ Prod.ext (by simpa using hxy.1) (by simpa using hxy.2)
+  left_inv := fun q ↦ by
+    rcases q with ⟨qa, qb⟩
+    exact Quotient.inductionOn₂' qa qb fun _ _ ↦ rfl
+  right_inv := fun q ↦ by
+    simp only
+    refine Quotient.inductionOn' q fun _ ↦ rfl
+
+/-- A bijection between an indexed product of quotients and the quotient by the product of the
+equivalence relations. -/
+@[simps]
+noncomputable def piQuotientEquiv {ι : Sort*} {α : ι → Sort*} (r : ∀ i, Setoid (α i)) :
+    (∀ i, Quotient (r i)) ≃ Quotient (@piSetoid _ _ r) where
+  toFun := fun x ↦ Quotient.mk'' fun i ↦ (x i).out'
+  invFun := fun q ↦ Quotient.liftOn' q (fun x i ↦ Quotient.mk'' (x i)) fun x y hxy ↦ by
+    ext i
+    simpa using hxy i
+  left_inv := fun q ↦ by
+    ext i
+    simp
+  right_inv := fun q ↦ by
+    refine Quotient.inductionOn' q fun _ ↦ ?_
+    simp only [Quotient.liftOn'_mk'', Quotient.eq'']
+    intro i
+    change Setoid.r _ _
+    rw [← Quotient.eq'']
+    simp
+
 /-- The infimum of two equivalence relations. -/
 instance : Inf (Setoid α) :=
   ⟨fun r s =>
@@ -192,10 +235,14 @@ def map_sInf {S : Set (Setoid α)} {s : Setoid α} (h : s ∈ S) :
     Quotient (sInf S) → Quotient s :=
   Setoid.map_of_le fun _ _ a ↦ a s h
 
+section EqvGen
+
+open Relation
+
 /-- The inductively defined equivalence closure of a binary relation r is the infimum
     of the set of all equivalence relations containing r. -/
 theorem eqvGen_eq (r : α → α → Prop) :
-    EqvGen.Setoid r = sInf { s : Setoid α | ∀ ⦃x y⦄, r x y → s.Rel x y } :=
+    EqvGen.setoid r = sInf { s : Setoid α | ∀ ⦃x y⦄, r x y → s.Rel x y } :=
   le_antisymm
     (fun _ _ H =>
       EqvGen.rec (fun _ _ h _ hs => hs h) (refl' _) (fun _ _ _ => symm' _)
@@ -205,20 +252,20 @@ theorem eqvGen_eq (r : α → α → Prop) :
 /-- The supremum of two equivalence relations r and s is the equivalence closure of the binary
     relation `x is related to y by r or s`. -/
 theorem sup_eq_eqvGen (r s : Setoid α) :
-    r ⊔ s = EqvGen.Setoid fun x y => r.Rel x y ∨ s.Rel x y := by
+    r ⊔ s = EqvGen.setoid fun x y => r.Rel x y ∨ s.Rel x y := by
   rw [eqvGen_eq]
   apply congr_arg sInf
   simp only [le_def, or_imp, ← forall_and]
 
 /-- The supremum of 2 equivalence relations r and s is the equivalence closure of the
     supremum of the underlying binary operations. -/
-theorem sup_def {r s : Setoid α} : r ⊔ s = EqvGen.Setoid (r.Rel ⊔ s.Rel) := by
+theorem sup_def {r s : Setoid α} : r ⊔ s = EqvGen.setoid (r.Rel ⊔ s.Rel) := by
   rw [sup_eq_eqvGen]; rfl
 
 /-- The supremum of a set S of equivalence relations is the equivalence closure of the binary
     relation `there exists r ∈ S relating x and y`. -/
 theorem sSup_eq_eqvGen (S : Set (Setoid α)) :
-    sSup S = EqvGen.Setoid fun x y => ∃ r : Setoid α, r ∈ S ∧ r.Rel x y := by
+    sSup S = EqvGen.setoid fun x y => ∃ r : Setoid α, r ∈ S ∧ r.Rel x y := by
   rw [eqvGen_eq]
   apply congr_arg sInf
   simp only [upperBounds, le_def, and_imp, exists_imp]
@@ -227,39 +274,41 @@ theorem sSup_eq_eqvGen (S : Set (Setoid α)) :
 
 /-- The supremum of a set of equivalence relations is the equivalence closure of the
     supremum of the set's image under the map to the underlying binary operation. -/
-theorem sSup_def {s : Set (Setoid α)} : sSup s = EqvGen.Setoid (sSup (Rel '' s)) := by
+theorem sSup_def {s : Set (Setoid α)} : sSup s = EqvGen.setoid (sSup (Rel '' s)) := by
   rw [sSup_eq_eqvGen, sSup_image]
   congr with (x y)
   simp only [iSup_apply, iSup_Prop_eq, exists_prop]
 
 /-- The equivalence closure of an equivalence relation r is r. -/
 @[simp]
-theorem eqvGen_of_setoid (r : Setoid α) : EqvGen.Setoid r.r = r :=
+theorem eqvGen_of_setoid (r : Setoid α) : EqvGen.setoid r.r = r :=
   le_antisymm (by rw [eqvGen_eq]; exact sInf_le fun _ _ => id) EqvGen.rel
 
 /-- Equivalence closure is idempotent. -/
 @[simp]
-theorem eqvGen_idem (r : α → α → Prop) : EqvGen.Setoid (EqvGen.Setoid r).Rel = EqvGen.Setoid r :=
+theorem eqvGen_idem (r : α → α → Prop) : EqvGen.setoid (EqvGen.setoid r).Rel = EqvGen.setoid r :=
   eqvGen_of_setoid _
 
 /-- The equivalence closure of a binary relation r is contained in any equivalence
     relation containing r. -/
 theorem eqvGen_le {r : α → α → Prop} {s : Setoid α} (h : ∀ x y, r x y → s.Rel x y) :
-    EqvGen.Setoid r ≤ s := by rw [eqvGen_eq]; exact sInf_le h
+    EqvGen.setoid r ≤ s := by rw [eqvGen_eq]; exact sInf_le h
 
 /-- Equivalence closure of binary relations is monotone. -/
 theorem eqvGen_mono {r s : α → α → Prop} (h : ∀ x y, r x y → s x y) :
-    EqvGen.Setoid r ≤ EqvGen.Setoid s :=
+    EqvGen.setoid r ≤ EqvGen.setoid s :=
   eqvGen_le fun _ _ hr => EqvGen.rel _ _ <| h _ _ hr
 
 /-- There is a Galois insertion of equivalence relations on α into binary relations
     on α, with equivalence closure the lower adjoint. -/
-def gi : @GaloisInsertion (α → α → Prop) (Setoid α) _ _ EqvGen.Setoid Rel where
-  choice r _ := EqvGen.Setoid r
+def gi : @GaloisInsertion (α → α → Prop) (Setoid α) _ _ EqvGen.setoid Rel where
+  choice r _ := EqvGen.setoid r
   gc _ s := ⟨fun H _ _ h => H <| EqvGen.rel _ _ h, fun H => eqvGen_of_setoid s ▸ eqvGen_mono H⟩
   le_l_u x := (eqvGen_of_setoid x).symm ▸ le_refl x
   choice_eq _ _ := rfl
 
+end EqvGen
+
 open Function
 
 /-- A function from α to β is injective iff its kernel is the bottom element of the complete lattice
@@ -335,7 +384,7 @@ variable {r f}
     closure of the relation on `f`'s image defined by '`x ≈ y` iff the elements of `f⁻¹(x)` are
     related to the elements of `f⁻¹(y)` by `r`.' -/
 def map (r : Setoid α) (f : α → β) : Setoid β :=
-  EqvGen.Setoid fun x y => ∃ a b, f a = x ∧ f b = y ∧ r.Rel a b
+  Relation.EqvGen.setoid fun x y => ∃ a b, f a = x ∧ f b = y ∧ r.Rel a b
 
 /-- Given a surjective function f whose kernel is contained in an equivalence relation r, the
     equivalence relation on f's codomain defined by x ≈ y ↔ the elements of f⁻¹(x) are related to
@@ -414,7 +463,7 @@ def sigmaQuotientEquivOfLe {r s : Setoid α} (hle : r ≤ s) :
     (Σ q : Quotient s, Quotient (r.comap (Subtype.val : Quotient.mk s ⁻¹' {q} → α))) ≃
       Quotient r :=
   .trans (.symm <| .sigmaCongrRight fun _ ↦ .subtypeQuotientEquivQuotientSubtype
-      (s₁ := r) (s₂ := r.comap Subtype.val) _ (fun _ ↦ Iff.rfl) fun _ _ ↦ Iff.rfl)
+      (s₁ := r) (s₂ := r.comap Subtype.val) _ _ (fun _ ↦ Iff.rfl) fun _ _ ↦ Iff.rfl)
     (.sigmaFiberEquiv fun a ↦ a.lift (Quotient.mk s) fun _ _ h ↦ Quotient.sound <| hle h)
 
 end Setoid
@@ -428,7 +477,8 @@ theorem Quotient.subsingleton_iff {s : Setoid α} : Subsingleton (Quotient s) 
   simp_rw [Prop.top_eq_true, true_implies, Quotient.eq']
   rfl
 
-theorem Quot.subsingleton_iff (r : α → α → Prop) : Subsingleton (Quot r) ↔ EqvGen r = ⊤ := by
+theorem Quot.subsingleton_iff (r : α → α → Prop) :
+    Subsingleton (Quot r) ↔ Relation.EqvGen r = ⊤ := by
   simp only [_root_.subsingleton_iff, _root_.eq_top_iff, Pi.le_def, Pi.top_apply, forall_const]
   refine (surjective_quot_mk _).forall.trans (forall_congr' fun a => ?_)
   refine (surjective_quot_mk _).forall.trans (forall_congr' fun b => ?_)
diff --git a/Mathlib/Data/Setoid/Partition.lean b/Mathlib/Data/Setoid/Partition.lean
index c002dca9c38fe..7813b2de0bebe 100644
--- a/Mathlib/Data/Setoid/Partition.lean
+++ b/Mathlib/Data/Setoid/Partition.lean
@@ -301,7 +301,7 @@ This type is primarily useful for definitional control of `s` - if this is not n
 structure IndexedPartition {ι α : Type*} (s : ι → Set α) where
   /-- two indexes are equal if they are equal in membership  -/
   eq_of_mem : ∀ {x i j}, x ∈ s i → x ∈ s j → i = j
-  /-- sends an index to an element of the corresponding set-/
+  /-- sends an index to an element of the corresponding set -/
   some : ι → α
   /-- membership invariance for `some`-/
   some_mem : ∀ i, some i ∈ s i
diff --git a/Mathlib/Data/Sigma/Basic.lean b/Mathlib/Data/Sigma/Basic.lean
index b7c1a5a690c6d..552e884b81259 100644
--- a/Mathlib/Data/Sigma/Basic.lean
+++ b/Mathlib/Data/Sigma/Basic.lean
@@ -243,16 +243,11 @@ protected theorem eq {α : Sort*} {β : α → Sort*} : ∀ {p₁ p₂ : Σ' a,
     (Eq.recOn h₁ p₁.2 : β p₂.1) = p₂.2 → p₁ = p₂
   | ⟨_, _⟩, _, rfl, rfl => rfl
 
-@[simp]
+-- This should not be a simp lemma, since its discrimination tree key would just be `→`.
 theorem «forall» {p : (Σ'a, β a) → Prop} : (∀ x, p x) ↔ ∀ a b, p ⟨a, b⟩ :=
   ⟨fun h a b ↦ h ⟨a, b⟩, fun h ⟨a, b⟩ ↦ h a b⟩
 
-#adaptation_note
-/--
-This should be renamed back to `exists` after `nightly-2024-07-31`.
--/
-@[simp]
-theorem exists' {p : (Σ'a, β a) → Prop} : (∃ x, p x) ↔ ∃ a b, p ⟨a, b⟩ :=
+@[simp] lemma «exists» {p : (Σ' a, β a) → Prop} : (∃ x, p x) ↔ ∃ a b, p ⟨a, b⟩ :=
   ⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩
 
 /-- A specialized ext lemma for equality of `PSigma` types over an indexed subtype. -/
diff --git a/Mathlib/Data/Sigma/Lex.lean b/Mathlib/Data/Sigma/Lex.lean
index 6b5e6be4f0f6d..db66010da8c5d 100644
--- a/Mathlib/Data/Sigma/Lex.lean
+++ b/Mathlib/Data/Sigma/Lex.lean
@@ -137,7 +137,7 @@ end Sigma
 
 namespace PSigma
 
-variable {ι : Sort*} {α : ι → Sort*} {r r₁ r₂ : ι → ι → Prop} {s s₁ s₂ : ∀ i, α i → α i → Prop}
+variable {ι : Sort*} {α : ι → Sort*} {r : ι → ι → Prop} {s : ∀ i, α i → α i → Prop}
 
 theorem lex_iff {a b : Σ' i, α i} :
     Lex r s a b ↔ r a.1 b.1 ∨ ∃ h : a.1 = b.1, s b.1 (h.rec a.2) b.2 := by
diff --git a/Mathlib/Data/Stream/Defs.lean b/Mathlib/Data/Stream/Defs.lean
index 2a596ea48dac9..ce2dc694fd4fc 100644
--- a/Mathlib/Data/Stream/Defs.lean
+++ b/Mathlib/Data/Stream/Defs.lean
@@ -48,7 +48,7 @@ def Any (p : α → Prop) (s : Stream' α) := ∃ n, p (get s n)
 
 /-- `a ∈ s` means that `a = Stream'.get n s` for some `n`. -/
 instance : Membership α (Stream' α) :=
-  ⟨fun a s => Any (fun b => a = b) s⟩
+  ⟨fun s a => Any (fun b => a = b) s⟩
 
 /-- Apply a function `f` to all elements of a stream `s`. -/
 def map (f : α → β) (s : Stream' α) : Stream' β := fun n => f (get s n)
diff --git a/Mathlib/Data/Stream/Init.lean b/Mathlib/Data/Stream/Init.lean
index 7f714ab54b4df..c90ad62ac925d 100644
--- a/Mathlib/Data/Stream/Init.lean
+++ b/Mathlib/Data/Stream/Init.lean
@@ -12,7 +12,7 @@ import Mathlib.Data.List.Basic
 
 Porting note:
 This file used to be in the core library. It was moved to `mathlib` and renamed to `init` to avoid
-name clashes.  -/
+name clashes. -/
 
 open Nat Function Option
 
@@ -493,7 +493,7 @@ theorem length_take (n : ℕ) (s : Stream' α) : (take n s).length = n := by
 theorem take_take {s : Stream' α} : ∀ {m n}, (s.take n).take m = s.take (min n m)
   | 0, n => by rw [Nat.min_zero, List.take_zero, take_zero]
   | m, 0 => by rw [Nat.zero_min, take_zero, List.take_nil]
-  | m+1, n+1 => by rw [take_succ, List.take_cons, Nat.succ_min_succ, take_succ, take_take]
+  | m+1, n+1 => by rw [take_succ, List.take_succ_cons, Nat.succ_min_succ, take_succ, take_take]
 
 @[simp] theorem concat_take_get {n : ℕ} {s : Stream' α} : s.take n ++ [s.get n] = s.take (n+1) :=
   (take_succ' n).symm
diff --git a/Mathlib/Data/String/Basic.lean b/Mathlib/Data/String/Basic.lean
index 1208ee3912785..d30479b15e611 100644
--- a/Mathlib/Data/String/Basic.lean
+++ b/Mathlib/Data/String/Basic.lean
@@ -3,6 +3,7 @@ Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
+import Batteries.Data.String.Lemmas
 import Mathlib.Data.List.Lex
 import Mathlib.Data.Char
 import Mathlib.Tactic.AdaptationNote
@@ -60,7 +61,7 @@ theorem ltb_cons_addChar (c : Char) (cs₁ cs₂ : List Char) (i₁ i₂ : Pos)
   intro ⟨cs₁⟩ ⟨cs₂⟩ i₁ i₂ <;>
   intros <;>
   (conv => lhs; unfold ltb) <;> (conv => rhs; unfold ltb) <;>
-  simp only [Iterator.hasNext_cons_addChar, ite_false, ite_true, *]
+  simp only [Iterator.hasNext_cons_addChar, ite_false, ite_true, *, reduceCtorEq]
   · rename_i h₂ h₁ heq ih
     simp only [Iterator.next, next, heq, Iterator.curr, get_cons_addChar, ite_true] at ih ⊢
     repeat rw [Pos.addChar_right_comm _ c]
diff --git a/Mathlib/Data/Subtype.lean b/Mathlib/Data/Subtype.lean
index 48aaa0d08f165..0583ba40e98cc 100644
--- a/Mathlib/Data/Subtype.lean
+++ b/Mathlib/Data/Subtype.lean
@@ -207,7 +207,7 @@ end Subtype
 namespace Subtype
 
 /-! Some facts about sets, which require that `α` is a type. -/
-variable {α β γ : Type*} {p : α → Prop}
+variable {α : Type*}
 
 @[simp]
 theorem coe_prop {S : Set α} (a : { a // a ∈ S }) : ↑a ∈ S :=
diff --git a/Mathlib/Data/Sum/Basic.lean b/Mathlib/Data/Sum/Basic.lean
index 9265b7b297c2d..d65e9ce7874fe 100644
--- a/Mathlib/Data/Sum/Basic.lean
+++ b/Mathlib/Data/Sum/Basic.lean
@@ -35,7 +35,7 @@ theorem sum_rec_congr (P : α ⊕ β → Sort*) (f : ∀ i, P (inl i)) (g : ∀
 
 section get
 
-variable {x y : α ⊕ β}
+variable {x : α ⊕ β}
 
 theorem eq_left_iff_getLeft_eq {a : α} : x = inl a ↔ ∃ h, x.getLeft h = a := by
   cases x <;> simp
@@ -133,7 +133,7 @@ theorem isRight_right (h : LiftRel r s (inr b) y) : y.isRight := by cases h; rfl
 theorem exists_of_isLeft_left (h₁ : LiftRel r s x y) (h₂ : x.isLeft) :
     ∃ a c, r a c ∧ x = inl a ∧ y = inl c := by
   rcases isLeft_iff.mp h₂ with ⟨_, rfl⟩
-  simp only [liftRel_iff, false_and, and_false, exists_false, or_false] at h₁
+  simp only [liftRel_iff, false_and, and_false, exists_false, or_false, reduceCtorEq] at h₁
   exact h₁
 
 theorem exists_of_isLeft_right (h₁ : LiftRel r s x y) (h₂ : y.isLeft) :
@@ -142,7 +142,7 @@ theorem exists_of_isLeft_right (h₁ : LiftRel r s x y) (h₂ : y.isLeft) :
 theorem exists_of_isRight_left (h₁ : LiftRel r s x y) (h₂ : x.isRight) :
     ∃ b d, s b d ∧ x = inr b ∧ y = inr d := by
   rcases isRight_iff.mp h₂ with ⟨_, rfl⟩
-  simp only [liftRel_iff, false_and, and_false, exists_false, false_or] at h₁
+  simp only [liftRel_iff, false_and, and_false, exists_false, false_or, reduceCtorEq] at h₁
   exact h₁
 
 theorem exists_of_isRight_right (h₁ : LiftRel r s x y) (h₂ : y.isRight) :
diff --git a/Mathlib/Data/Sum/Interval.lean b/Mathlib/Data/Sum/Interval.lean
index fa3850967d3d3..75478d551bcc8 100644
--- a/Mathlib/Data/Sum/Interval.lean
+++ b/Mathlib/Data/Sum/Interval.lean
@@ -300,7 +300,7 @@ local elab "simp_lex" : tactic => do
         map_empty, not_mem_empty, true_and, inl_mem_disjSum, inr_mem_disjSum, and_true, ofLex_toLex,
         mem_map, Embedding.coeFn_mk, exists_prop, exists_eq_right, Embedding.inl_apply,
         -- Porting note: added
-        inl.injEq, inr.injEq]
+        inl.injEq, inr.injEq, reduceCtorEq]
   )
 
 instance locallyFiniteOrder : LocallyFiniteOrder (α ⊕ₗ β) where
diff --git a/Mathlib/Data/Sym/Basic.lean b/Mathlib/Data/Sym/Basic.lean
index aa85d9183ee69..4378b439653ed 100644
--- a/Mathlib/Data/Sym/Basic.lean
+++ b/Mathlib/Data/Sym/Basic.lean
@@ -142,7 +142,7 @@ theorem card_coe : Multiset.card (s : Multiset α) = n := s.prop
 /-- `α ∈ s` means that `a` appears as one of the factors in `s`.
 -/
 instance : Membership α (Sym α n) :=
-  ⟨fun a s => a ∈ s.1⟩
+  ⟨fun s a => a ∈ s.1⟩
 
 instance decidableMem [DecidableEq α] (a : α) (s : Sym α n) : Decidable (a ∈ s) :=
   s.1.decidableMem _
diff --git a/Mathlib/Data/Sym/Sym2.lean b/Mathlib/Data/Sym/Sym2.lean
index 4e94151a85335..945179cd192f3 100644
--- a/Mathlib/Data/Sym/Sym2.lean
+++ b/Mathlib/Data/Sym/Sym2.lean
@@ -299,8 +299,7 @@ instance : SetLike (Sym2 α) α where
     induction' z with x y
     induction' z' with x' y'
     have hx := h x; have hy := h y; have hx' := h x'; have hy' := h y'
-    simp only [mem_iff', eq_self_iff_true, or_true_iff, iff_true_iff,
-      true_or_iff, true_iff_iff] at hx hy hx' hy'
+    simp only [mem_iff', eq_self_iff_true] at hx hy hx' hy'
     aesop
 
 @[simp]
@@ -318,7 +317,7 @@ theorem mem_mk_left (x y : α) : x ∈ s(x, y) :=
   ⟨y, rfl⟩
 
 theorem mem_mk_right (x y : α) : y ∈ s(x, y) :=
-  eq_swap.subst <| mem_mk_left y x
+  eq_swap ▸ mem_mk_left y x
 
 @[simp, aesop norm (rule_sets := [Sym2])]
 theorem mem_iff {a b c : α} : a ∈ s(b, c) ↔ a = b ∨ a = c :=
diff --git a/Mathlib/Data/TypeVec.lean b/Mathlib/Data/TypeVec.lean
index 104d68020ce33..ed974cab4a625 100644
--- a/Mathlib/Data/TypeVec.lean
+++ b/Mathlib/Data/TypeVec.lean
@@ -448,16 +448,16 @@ end
 @[simp]
 theorem prod_fst_mk {α β : TypeVec n} (i : Fin2 n) (a : α i) (b : β i) :
     TypeVec.prod.fst i (prod.mk i a b) = a := by
-  induction' i with _ _ _ i_ih
-  · simp_all only [prod.fst, prod.mk]
-  apply i_ih
+  induction i with
+  | fz => simp_all only [prod.fst, prod.mk]
+  | fs _ i_ih => apply i_ih
 
 @[simp]
 theorem prod_snd_mk {α β : TypeVec n} (i : Fin2 n) (a : α i) (b : β i) :
     TypeVec.prod.snd i (prod.mk i a b) = b := by
-  induction' i with _ _ _ i_ih
-  · simp_all [prod.snd, prod.mk]
-  apply i_ih
+  induction i with
+  | fz => simp_all [prod.snd, prod.mk]
+  | fs _ i_ih => apply i_ih
 
 /-- `prod` is functorial -/
 protected def prod.map : ∀ {n} {α α' β β' : TypeVec.{u} n}, α ⟹ β → α' ⟹ β' → α ⊗ α' ⟹ β ⊗ β'
@@ -493,9 +493,9 @@ theorem snd_diag {α : TypeVec n} : TypeVec.prod.snd ⊚ (prod.diag : α ⟹ _)
 
 theorem repeatEq_iff_eq {α : TypeVec n} {i x y} :
     ofRepeat (repeatEq α i (prod.mk _ x y)) ↔ x = y := by
-  induction' i with _ _ _ i_ih
-  · rfl
-  erw [repeatEq, i_ih]
+  induction i with
+  | fz => rfl
+  | fs _ i_ih => erw [repeatEq, i_ih]
 
 /-- given a predicate vector `p` over vector `α`, `Subtype_ p` is the type of vectors
 that contain an `α` that satisfies `p` -/
@@ -547,17 +547,16 @@ theorem subtypeVal_nil {α : TypeVec.{u} 0} (ps : α ⟹ «repeat» 0 Prop) :
 
 theorem diag_sub_val {n} {α : TypeVec.{u} n} : subtypeVal (repeatEq α) ⊚ diagSub = prod.diag := by
   ext i x
-  induction' i with _ _ _ i_ih
-  · simp only [comp, subtypeVal, repeatEq.eq_2, diagSub, prod.diag]
-  apply @i_ih (drop α)
+  induction i with
+  | fz => simp only [comp, subtypeVal, repeatEq.eq_2, diagSub, prod.diag]
+  | fs _ i_ih => apply @i_ih (drop α)
 
 theorem prod_id : ∀ {n} {α β : TypeVec.{u} n}, (id ⊗' id) = (id : α ⊗ β ⟹ _) := by
   intros
   ext i a
-  induction' i with _ _ _ i_ih
-  · cases a
-    rfl
-  · apply i_ih
+  induction i with
+  | fz => cases a; rfl
+  | fs _ i_ih => apply i_ih
 
 theorem append_prod_appendFun {n} {α α' β β' : TypeVec.{u} n} {φ φ' ψ ψ' : Type u}
     {f₀ : α ⟹ α'} {g₀ : β ⟹ β'} {f₁ : φ → φ'} {g₁ : ψ → ψ'} :
@@ -655,9 +654,10 @@ theorem prod_map_id {α β : TypeVec n} : (@TypeVec.id _ α ⊗' @TypeVec.id _ 
 @[simp]
 theorem subtypeVal_diagSub {α : TypeVec n} : subtypeVal (repeatEq α) ⊚ diagSub = prod.diag := by
   ext i x
-  induction' i with _ _ _ i_ih
-  · simp [comp, diagSub, subtypeVal, prod.diag]
-  · simp only [comp, subtypeVal, diagSub, prod.diag] at *
+  induction i with
+  | fz => simp [comp, diagSub, subtypeVal, prod.diag]
+  | fs _ i_ih =>
+    simp only [comp, subtypeVal, diagSub, prod.diag] at *
     apply i_ih
 
 @[simp]
diff --git a/Mathlib/Data/Vector/Basic.lean b/Mathlib/Data/Vector/Basic.lean
index 909cbc5b72f6e..5db96436f76b0 100644
--- a/Mathlib/Data/Vector/Basic.lean
+++ b/Mathlib/Data/Vector/Basic.lean
@@ -636,7 +636,7 @@ protected theorem comp_traverse (f : β → F γ) (g : α → G β) (x : Vector
   · simp! [cast, *, functor_norm]
     rfl
   · rw [Vector.traverse_def, ih]
-    simp [functor_norm, (· ∘ ·)]
+    simp [functor_norm, Function.comp_def]
 
 protected theorem traverse_eq_map_id {α β} (f : α → β) :
     ∀ x : Vector α n, x.traverse ((pure : _ → Id _) ∘ f) = (pure : _ → Id _) (map f x) := by
diff --git a/Mathlib/Data/Vector/Defs.lean b/Mathlib/Data/Vector/Defs.lean
index 46d770bdb44cf..8c08c0254b502 100644
--- a/Mathlib/Data/Vector/Defs.lean
+++ b/Mathlib/Data/Vector/Defs.lean
@@ -3,7 +3,7 @@ Copyright (c) 2016 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
-import Mathlib.Init.Data.List.Lemmas
+import Mathlib.Data.List.Defs
 import Mathlib.Tactic.Common
 
 /-!
@@ -52,7 +52,7 @@ def head : Vector α (Nat.succ n) → α
 theorem head_cons (a : α) : ∀ v : Vector α n, head (cons a v) = a
   | ⟨_, _⟩ => rfl
 
-/-- The tail of a vector, with an empty vector having empty tail.  -/
+/-- The tail of a vector, with an empty vector having empty tail. -/
 def tail : Vector α n → Vector α (n - 1)
   | ⟨[], h⟩ => ⟨[], congrArg pred h⟩
   | ⟨_ :: v, h⟩ => ⟨v, congrArg pred h⟩
@@ -119,7 +119,7 @@ def take (i : ℕ) : Vector α n → Vector α (min i n)
 
 /-- Remove the element at position `i` from a vector of length `n`. -/
 def eraseIdx (i : Fin n) : Vector α n → Vector α (n - 1)
-  | ⟨l, p⟩ => ⟨List.eraseIdx l i.1, by rw [l.length_eraseIdx] <;> rw [p]; exact i.2⟩
+  | ⟨l, p⟩ => ⟨List.eraseIdx l i.1, by rw [l.length_eraseIdx_of_lt] <;> rw [p]; exact i.2⟩
 
 @[deprecated (since := "2024-05-04")] alias removeNth := eraseIdx
 
@@ -154,7 +154,7 @@ def mapAccumr₂ (f : α → β → σ → σ × φ) : Vector α n → Vector β
 
 end Accum
 
-/-! ### Shift Primitives-/
+/-! ### Shift Primitives -/
 section Shift
 
 /-- `shiftLeftFill v i` is the vector obtained by left-shifting `v` `i` times and padding with the
@@ -181,7 +181,7 @@ protected theorem eq_nil (v : Vector α 0) : v = nil :=
   v.eq nil (List.eq_nil_of_length_eq_zero v.2)
 
 /-- Vector of length from a list `v`
-with witness that `v` has length `n` maps to `v` under `toList`.  -/
+with witness that `v` has length `n` maps to `v` under `toList`. -/
 @[simp]
 theorem toList_mk (v : List α) (P : List.length v = n) : toList (Subtype.mk v P) = v :=
   rfl
diff --git a/Mathlib/Data/Vector/Zip.lean b/Mathlib/Data/Vector/Zip.lean
index eee02c2029439..f8c69ad4814b4 100644
--- a/Mathlib/Data/Vector/Zip.lean
+++ b/Mathlib/Data/Vector/Zip.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Vector.Basic
 
diff --git a/Mathlib/Data/Vector3.lean b/Mathlib/Data/Vector3.lean
index 4e0562d15422e..9ec3dea934529 100644
--- a/Mathlib/Data/Vector3.lean
+++ b/Mathlib/Data/Vector3.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 import Mathlib.Data.Fin.Fin2
-import Mathlib.Init.Logic
 import Mathlib.Util.Notation3
 import Mathlib.Tactic.TypeStar
 
@@ -236,7 +235,7 @@ theorem vectorAllP_singleton (p : α → Prop) (x : α) : VectorAllP p (cons x [
 @[simp]
 theorem vectorAllP_cons (p : α → Prop) (x : α) (v : Vector3 α n) :
     VectorAllP p (x :: v) ↔ p x ∧ VectorAllP p v :=
-  Vector3.recOn v (and_true_iff _).symm fun _ _ _ => Iff.rfl
+  Vector3.recOn v (iff_of_eq (and_true _)).symm fun _ _ _ => Iff.rfl
 
 theorem vectorAllP_iff_forall (p : α → Prop) (v : Vector3 α n) :
     VectorAllP p v ↔ ∀ i, p (v i) := by
diff --git a/Mathlib/Data/ZMod/Basic.lean b/Mathlib/Data/ZMod/Basic.lean
index db8d87e23d3e4..8c9d56fcedfec 100644
--- a/Mathlib/Data/ZMod/Basic.lean
+++ b/Mathlib/Data/ZMod/Basic.lean
@@ -435,7 +435,7 @@ theorem castHom_bijective [Fintype R] (h : Fintype.card R = n) :
       intro hn
       rw [hn] at h
       exact (Fintype.card_eq_zero_iff.mp h).elim' 0⟩
-  rw [Fintype.bijective_iff_injective_and_card, ZMod.card, h, eq_self_iff_true, and_true_iff]
+  rw [Fintype.bijective_iff_injective_and_card, ZMod.card, h, eq_self_iff_true, and_true]
   apply ZMod.castHom_injective
 
 /-- The unique ring isomorphism between `ZMod n` and a ring `R`
@@ -446,7 +446,7 @@ noncomputable def ringEquiv [Fintype R] (h : Fintype.card R = n) : ZMod n ≃+*
 /-- The unique ring isomorphism between `ZMod p` and a ring `R` of cardinality a prime `p`.
 
 If you need any property of this isomorphism, first of all use `ringEquivOfPrime_eq_ringEquiv`
-below (after `have : CharP R p := ...`) and deduce it by the results about `ZMod.ringEquiv`.  -/
+below (after `have : CharP R p := ...`) and deduce it by the results about `ZMod.ringEquiv`. -/
 noncomputable def ringEquivOfPrime [Fintype R] {p : ℕ} (hp : p.Prime) (hR : Fintype.card R = p) :
     ZMod p ≃+* R :=
   have : Nontrivial R := Fintype.one_lt_card_iff_nontrivial.1 (hR ▸ hp.one_lt)
@@ -752,6 +752,12 @@ theorem val_mul_of_lt {n : ℕ} {a b : ZMod n} (h : a.val * b.val < n) :
   rw [val_mul]
   apply Nat.mod_eq_of_lt h
 
+theorem val_mul_iff_lt {n : ℕ} [NeZero n] (a b : ZMod n) :
+    (a * b).val = a.val * b.val ↔ a.val * b.val < n := by
+  constructor <;> intro h
+  · rw [← h]; apply ZMod.val_lt
+  · apply ZMod.val_mul_of_lt h
+
 instance nontrivial (n : ℕ) [Fact (1 < n)] : Nontrivial (ZMod n) :=
   ⟨⟨0, 1, fun h =>
       zero_ne_one <|
@@ -820,6 +826,17 @@ theorem eq_iff_modEq_nat (n : ℕ) {a b : ℕ} : (a : ZMod n) = b ↔ a ≡ b [M
   · rw [Fin.ext_iff, Nat.ModEq, ← val_natCast, ← val_natCast]
     exact Iff.rfl
 
+theorem eq_zero_iff_even {n : ℕ} : (n : ZMod 2) = 0 ↔ Even n :=
+  (CharP.cast_eq_zero_iff (ZMod 2) 2 n).trans even_iff_two_dvd.symm
+
+theorem eq_one_iff_odd {n : ℕ} : (n : ZMod 2) = 1 ↔ Odd n := by
+  rw [← @Nat.cast_one (ZMod 2), ZMod.eq_iff_modEq_nat, Nat.odd_iff, Nat.ModEq]
+
+theorem ne_zero_iff_odd {n : ℕ} : (n : ZMod 2) ≠ 0 ↔ Odd n := by
+  constructor <;>
+    · contrapose
+      simp [eq_zero_iff_even]
+
 theorem coe_mul_inv_eq_one {n : ℕ} (x : ℕ) (h : Nat.Coprime x n) :
     ((x : ZMod n) * (x : ZMod n)⁻¹) = 1 := by
   rw [Nat.Coprime, Nat.gcd_comm, Nat.gcd_rec] at h
@@ -988,6 +1005,14 @@ theorem natAbs_mod_two (a : ℤ) : (a.natAbs : ZMod 2) = a := by
 theorem val_ne_zero {n : ℕ} (a : ZMod n) : a.val ≠ 0 ↔ a ≠ 0 :=
   (val_eq_zero a).not
 
+theorem val_pos {n : ℕ} {a : ZMod n} : 0 < a.val ↔ a ≠ 0 := by
+  simp [pos_iff_ne_zero]
+
+theorem val_eq_one : ∀ {n : ℕ} (_ : 1 < n) (a : ZMod n), a.val = 1 ↔ a = 1
+  | 0, hn, _
+  | 1, hn, _ => by simp at hn
+  | n + 2, _, _ => by simp only [val, ZMod, Fin.ext_iff, Fin.val_one]
+
 theorem neg_eq_self_iff {n : ℕ} (a : ZMod n) : -a = a ↔ a = 0 ∨ 2 * a.val = n := by
   rw [neg_eq_iff_add_eq_zero, ← two_mul]
   cases n
@@ -1018,6 +1043,17 @@ theorem neg_eq_self_iff {n : ℕ} (a : ZMod n) : -a = a ↔ a = 0 ∨ 2 * a.val
 theorem val_cast_of_lt {n : ℕ} {a : ℕ} (h : a < n) : (a : ZMod n).val = a := by
   rw [val_natCast, Nat.mod_eq_of_lt h]
 
+theorem val_cast_zmod_lt {m : ℕ} [NeZero m] (n : ℕ) [NeZero n] (a : ZMod m) :
+    (a.cast : ZMod n).val < m := by
+  rcases m with (⟨⟩|⟨m⟩); · cases NeZero.ne 0 rfl
+  by_cases h : m < n
+  · rcases n with (⟨⟩|⟨n⟩); · simp at h
+    rw [← natCast_val, val_cast_of_lt]
+    · apply a.val_lt
+    apply lt_of_le_of_lt (Nat.le_of_lt_succ (ZMod.val_lt a)) h
+  · rw [not_lt] at h
+    apply lt_of_lt_of_le (ZMod.val_lt _) (le_trans h (Nat.le_succ m))
+
 theorem neg_val' {n : ℕ} [NeZero n] (a : ZMod n) : (-a).val = (n - a.val) % n :=
   calc
     (-a).val = val (-a) % n := by rw [Nat.mod_eq_of_lt (-a).val_lt]
@@ -1063,6 +1099,28 @@ theorem cast_cast_zmod_of_le {m n : ℕ} [hm : NeZero m] (h : m ≤ n) (a : ZMod
   have : NeZero n := ⟨((Nat.zero_lt_of_ne_zero hm.out).trans_le h).ne'⟩
   rw [cast_eq_val, val_cast_eq_val_of_lt (a.val_lt.trans_le h), natCast_zmod_val]
 
+theorem val_pow {m n : ℕ} {a : ZMod n} [ilt : Fact (1 < n)] (h : a.val ^ m < n) :
+    (a ^ m).val = a.val ^ m := by
+  induction m with
+  | zero => simp [ZMod.val_one]
+  | succ m ih =>
+    have : a.val ^ m < n := by
+      obtain rfl | ha := eq_or_ne a 0
+      · by_cases hm : m = 0
+        · cases hm; simp [ilt.out]
+        · simp only [val_zero, ne_eq, hm, not_false_eq_true, zero_pow, Nat.zero_lt_of_lt h]
+      · exact lt_of_le_of_lt
+         (Nat.pow_le_pow_of_le_right (by rwa [gt_iff_lt, ZMod.val_pos]) (Nat.le_succ m)) h
+    rw [pow_succ, ZMod.val_mul, ih this, ← pow_succ, Nat.mod_eq_of_lt h]
+
+theorem val_pow_le {m n : ℕ} [Fact (1 < n)] {a : ZMod n} : (a ^ m).val ≤ a.val ^ m := by
+  induction m with
+  | zero => simp [ZMod.val_one]
+  | succ m ih =>
+    rw [pow_succ, pow_succ]
+    apply le_trans (ZMod.val_mul_le _ _)
+    apply Nat.mul_le_mul_right _ ih
+
 /-- `valMinAbs x` returns the integer in the same equivalence class as `x` that is closest to `0`,
   The result will be in the interval `(-n/2, n/2]`. -/
 def valMinAbs : ∀ {n : ℕ}, ZMod n → ℤ
@@ -1429,6 +1487,8 @@ lemma pow_pow_zmod_val_inv (hn : (Nat.card α).Coprime n) (a : α) :
 
 end Group
 
+open ZMod
+
 /-- The range of `(m * · + k)` on natural numbers is the set of elements `≥ k` in the
 residue class of `k` mod `m`. -/
 lemma Nat.range_mul_add (m k : ℕ) :
@@ -1442,3 +1502,17 @@ lemma Nat.range_mul_add (m k : ℕ) :
     simp only [ha, Nat.cast_add, add_right_eq_self, ZMod.natCast_zmod_eq_zero_iff_dvd] at H₁
     obtain ⟨b, rfl⟩ := H₁
     exact ⟨b, ha⟩
+
+/-- Equivalence between `ℕ` and `ZMod N × ℕ`, sending `n` to `(n mod N, n / N)`. -/
+def Nat.residueClassesEquiv (N : ℕ) [NeZero N] : ℕ ≃ ZMod N × ℕ where
+  toFun n := (↑n, n / N)
+  invFun p := p.1.val + N * p.2
+  left_inv n := by simpa only [val_natCast] using mod_add_div n N
+  right_inv p := by
+    ext1
+    · simp only [add_comm p.1.val, cast_add, cast_mul, natCast_self, zero_mul, natCast_val,
+        cast_id', id_eq, zero_add]
+    · simp only [add_comm p.1.val, mul_add_div (NeZero.pos _),
+        (Nat.div_eq_zero_iff <| (NeZero.pos _)).2 p.1.val_lt, add_zero]
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/ZMod/Defs.lean b/Mathlib/Data/ZMod/Defs.lean
index 324dec3252219..76e27012548a5 100644
--- a/Mathlib/Data/ZMod/Defs.lean
+++ b/Mathlib/Data/ZMod/Defs.lean
@@ -5,6 +5,7 @@ Authors: Eric Rodriguez
 -/
 import Mathlib.Algebra.Group.Fin.Basic
 import Mathlib.Algebra.NeZero
+import Mathlib.Algebra.Ring.Int
 import Mathlib.Data.Nat.ModEq
 import Mathlib.Data.Fintype.Card
 
@@ -103,7 +104,7 @@ namespace ZMod
 instance instUnique : Unique (ZMod 1) := Fin.uniqueFinOne
 
 instance fintype : ∀ (n : ℕ) [NeZero n], Fintype (ZMod n)
-  | 0, h => (h.ne rfl).elim
+  | 0, h => (h.ne _ rfl).elim
   | n + 1, _ => Fin.fintype (n + 1)
 
 instance infinite : Infinite (ZMod 0) :=
diff --git a/Mathlib/Data/ZMod/Parity.lean b/Mathlib/Data/ZMod/Parity.lean
deleted file mode 100644
index b6686f1e23ec5..0000000000000
--- a/Mathlib/Data/ZMod/Parity.lean
+++ /dev/null
@@ -1,33 +0,0 @@
-/-
-Copyright (c) 2020 Kyle Miller. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kyle Miller
--/
-import Mathlib.Algebra.Order.Ring.Abs
-import Mathlib.Data.ZMod.Basic
-
-/-!
-# Relating parity to natural numbers mod 2
-
-This module provides lemmas relating `ZMod 2` to `Even` and `Odd`.
-
-## Tags
-
-parity, zmod, even, odd
--/
-
-
-namespace ZMod
-
-theorem eq_zero_iff_even {n : ℕ} : (n : ZMod 2) = 0 ↔ Even n :=
-  (CharP.cast_eq_zero_iff (ZMod 2) 2 n).trans even_iff_two_dvd.symm
-
-theorem eq_one_iff_odd {n : ℕ} : (n : ZMod 2) = 1 ↔ Odd n := by
-  rw [← @Nat.cast_one (ZMod 2), ZMod.eq_iff_modEq_nat, Nat.odd_iff, Nat.ModEq]
-
-theorem ne_zero_iff_odd {n : ℕ} : (n : ZMod 2) ≠ 0 ↔ Odd n := by
-  constructor <;>
-    · contrapose
-      simp [eq_zero_iff_even]
-
-end ZMod
diff --git a/Mathlib/Data/ZMod/Units.lean b/Mathlib/Data/ZMod/Units.lean
index 319a7d55013c4..3d4def6281661 100644
--- a/Mathlib/Data/ZMod/Units.lean
+++ b/Mathlib/Data/ZMod/Units.lean
@@ -42,7 +42,7 @@ theorem unitsMap_surjective [hm : NeZero m] (h : n ∣ m) :
     have ⟨k, hk⟩ := this x.val.val (val_coe_unit_coprime x)
     refine ⟨unitOfCoprime _ hk, Units.ext ?_⟩
     have : NeZero n := ⟨fun hn ↦ hm.out (eq_zero_of_zero_dvd (hn ▸ h))⟩
-    simp [unitsMap_def]
+    simp [unitsMap_def, - castHom_apply]
   intro x hx
   let ps := m.primeFactors.filter (fun p ↦ ¬p ∣ x)
   use ps.prod id
diff --git a/Mathlib/Deprecated/Group.lean b/Mathlib/Deprecated/Group.lean
index d2d59cec9f5b0..539782f49e725 100644
--- a/Mathlib/Deprecated/Group.lean
+++ b/Mathlib/Deprecated/Group.lean
@@ -359,7 +359,7 @@ end Units
 
 namespace IsUnit
 
-variable {M : Type*} {N : Type*} [Monoid M] [Monoid N] {x : M}
+variable {M : Type*} {N : Type*} [Monoid M] [Monoid N]
 
 theorem map' {f : M → N} (hf : IsMonoidHom f) {x : M} (h : IsUnit x) : IsUnit (f x) :=
   h.map (MonoidHom.of hf)
diff --git a/Mathlib/Deprecated/HashMap.lean b/Mathlib/Deprecated/HashMap.lean
index a70ec4665c767..b5c815bd6e5a0 100644
--- a/Mathlib/Deprecated/HashMap.lean
+++ b/Mathlib/Deprecated/HashMap.lean
@@ -1,12 +1,9 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
-
-As `HashMap` has been completely reimplemented in `Batteries`,
-nothing from the mathlib3 file `data.hash_map` is reflected here.
-The porting header is just here to mark that no further work on `data.hash_map` is desired.
+Authors: Kim Morrison
 -/
+
 import Mathlib.Init
 import Mathlib.Tactic.TypeStar
 import Batteries.Data.HashMap.Basic
@@ -15,7 +12,9 @@ import Batteries.Data.RBMap.Basic
 /-!
 # Additional API for `HashMap` and `RBSet`.
 
-These should be replaced by proper implementations in Batteries.
+As `HashMap` has been completely reimplemented in `Batteries`,
+nothing from the mathlib3 file `data.hash_map` is reflected here.
+The porting header is just here to mark that no further work on `data.hash_map` is desired.
 -/
 
 variable {α β : Type*}
diff --git a/Mathlib/Deprecated/Subgroup.lean b/Mathlib/Deprecated/Subgroup.lean
index 17691db27d6d4..f30ed0f0e3ca3 100644
--- a/Mathlib/Deprecated/Subgroup.lean
+++ b/Mathlib/Deprecated/Subgroup.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,
+Authors: Johannes Hölzl, Mitchell Rowett, Kim Morrison, Johan Commelin, Mario Carneiro,
   Michael Howes
 -/
 import Mathlib.Algebra.Group.Subgroup.Basic
@@ -453,7 +453,8 @@ theorem closure_subgroup {s : Set G} (hs : IsSubgroup s) : closure s = s :=
 @[to_additive]
 theorem exists_list_of_mem_closure {s : Set G} {a : G} (h : a ∈ closure s) :
     ∃ l : List G, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = a :=
-  InClosure.recOn h (fun {x} hxs => ⟨[x], List.forall_mem_singleton.2 <| Or.inl hxs, one_mul _⟩)
+  InClosure.recOn h
+    (fun {x} hxs => ⟨[x], List.forall_mem_singleton.2 <| Or.inl hxs, List.prod_singleton⟩)
     ⟨[], List.forall_mem_nil _, rfl⟩
     (fun {x} _ ⟨L, HL1, HL2⟩ =>
       ⟨L.reverse.map Inv.inv, fun x hx =>
@@ -589,11 +590,11 @@ theorem normalClosure.is_normal : IsNormalSubgroup (normalClosure s) :=
 /-- The normal closure of s is the smallest normal subgroup containing s. -/
 theorem normalClosure_subset {s t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) :
     normalClosure s ⊆ t := fun a w => by
-  induction' w with x hx x _ ihx x y _ _ ihx ihy
-  · exact conjugatesOfSet_subset' ht h <| hx
-  · exact ht.toIsSubgroup.toIsSubmonoid.one_mem
-  · exact ht.toIsSubgroup.inv_mem ihx
-  · exact ht.toIsSubgroup.toIsSubmonoid.mul_mem ihx ihy
+  induction w with
+  | basic hx => exact conjugatesOfSet_subset' ht h <| hx
+  | one => exact ht.toIsSubgroup.toIsSubmonoid.one_mem
+  | inv _ ihx => exact ht.toIsSubgroup.inv_mem ihx
+  | mul _ _ ihx ihy => exact ht.toIsSubgroup.toIsSubmonoid.mul_mem ihx ihy
 
 theorem normalClosure_subset_iff {s t : Set G} (ht : IsNormalSubgroup t) :
     s ⊆ t ↔ normalClosure s ⊆ t :=
diff --git a/Mathlib/Deprecated/Subring.lean b/Mathlib/Deprecated/Subring.lean
index 6a652a4c798a3..20fc691ea4b93 100644
--- a/Mathlib/Deprecated/Subring.lean
+++ b/Mathlib/Deprecated/Subring.lean
@@ -97,14 +97,15 @@ theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
     ∃ L : List (List R), (∀ l ∈ L, ∀ x ∈ l, x ∈ s ∨ x = (-1 : R)) ∧ (L.map List.prod).sum = a :=
   AddGroup.InClosure.recOn h
     fun {x} hx ↦ match x, Monoid.exists_list_of_mem_closure hx with
-    | _, ⟨L, h1, rfl⟩ => ⟨[L], List.forall_mem_singleton.2 fun r hr ↦ Or.inl (h1 r hr), zero_add _⟩
+    | _, ⟨L, h1, rfl⟩ =>
+      ⟨[L], List.forall_mem_singleton.2 fun r hr ↦ Or.inl (h1 r hr), List.sum_singleton⟩
     ⟨[], List.forall_mem_nil _, rfl⟩
     fun {b} _ ih ↦ match b, ih with
     | _, ⟨L1, h1, rfl⟩ =>
       ⟨L1.map (List.cons (-1)),
         fun L2 h2 ↦ match L2, List.mem_map.1 h2 with
         | _, ⟨L3, h3, rfl⟩ => List.forall_mem_cons.2 ⟨Or.inr rfl, h1 L3 h3⟩, by
-        simp only [List.map_map, (· ∘ ·), List.prod_cons, neg_one_mul]
+        simp only [List.map_map, Function.comp_def, List.prod_cons, neg_one_mul]
         refine List.recOn L1 neg_zero.symm fun hd tl ih ↦ ?_
         rw [List.map_cons, List.sum_cons, ih, List.map_cons, List.sum_cons, neg_add]⟩
     fun {r1 r2} _ _ ih1 ih2 ↦ match r1, r2, ih1, ih2 with
diff --git a/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean b/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
index dd5ee918f3497..bf93db2355a6d 100644
--- a/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
+++ b/Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
@@ -565,7 +565,7 @@ theorem translationNumber_eq_of_tendsto_aux {τ' : ℝ} (h : Tendsto f.transnumA
 theorem translationNumber_eq_of_tendsto₀ {τ' : ℝ}
     (h : Tendsto (fun n : ℕ => f^[n] 0 / n) atTop (𝓝 τ')) : τ f = τ' :=
   f.translationNumber_eq_of_tendsto_aux <| by
-    simpa [(· ∘ ·), transnumAuxSeq_def, coe_pow] using
+    simpa [Function.comp_def, transnumAuxSeq_def, coe_pow] using
       h.comp (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two)
 
 theorem translationNumber_eq_of_tendsto₀' {τ' : ℝ}
@@ -852,8 +852,8 @@ theorem semiconj_of_group_action_of_forall_translationNumber_eq {G : Type*} [Gro
   have hF₁ : ∀ g, ⇑(F₁ g) = f₁ g := fun _ => rfl
   have hF₂ : ∀ g, ⇑(F₂ g) = f₂ g := fun _ => rfl
   -- Now we apply `csSup_div_semiconj` and go back to `f₁` and `f₂`.
-  refine ⟨⟨⟨_, fun x y hxy => ?_⟩, fun x => ?_⟩, csSup_div_semiconj F₂ F₁ fun x => ?_⟩ <;>
-    simp only [hF₁, hF₂, ← map_inv, coe_mk]
+  refine ⟨⟨⟨fun x ↦ ⨆ g', (F₂ g')⁻¹ (F₁ g' x), fun x y hxy => ?_⟩, fun x => ?_⟩,
+    csSup_div_semiconj F₂ F₁ fun x => ?_⟩ <;> simp only [hF₁, hF₂, ← map_inv, coe_mk]
   · exact ciSup_mono (this y) fun g => mono _ (mono _ hxy)
   · simp only [map_add_one]
     exact (Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const)
diff --git a/Mathlib/Dynamics/Ergodic/Action/Basic.lean b/Mathlib/Dynamics/Ergodic/Action/Basic.lean
index 06740aa5a6bad..8dfe2a8de714c 100644
--- a/Mathlib/Dynamics/Ergodic/Action/Basic.lean
+++ b/Mathlib/Dynamics/Ergodic/Action/Basic.lean
@@ -85,12 +85,10 @@ theorem ergodicSMul_iterateMulAct {f : α → α} (hf : Measurable f) :
   simp only [ergodicSMul_iff, smulInvariantMeasure_iterateMulAct, hf]
   refine ⟨fun ⟨h₁, h₂⟩ ↦ ⟨h₁, ⟨?_⟩⟩, fun h ↦ ⟨h.1, ?_⟩⟩
   · intro s hm hs
-    rw [← eventuallyConst_set']
     refine h₂ hm fun n ↦ ?_
     nth_rewrite 2 [← Function.IsFixedPt.preimage_iterate hs n.val]
     rfl
   · intro s hm hs
-    rw [eventuallyConst_set']
-    exact h.quasiErgodic.ae_empty_or_univ' hm <| hs (.mk 1)
+    exact h.quasiErgodic.aeconst_set₀ hm.nullMeasurableSet <| hs (.mk 1)
 
 end MeasureTheory
diff --git a/Mathlib/Dynamics/Ergodic/Action/OfMinimal.lean b/Mathlib/Dynamics/Ergodic/Action/OfMinimal.lean
new file mode 100644
index 0000000000000..266a45e4f70f7
--- /dev/null
+++ b/Mathlib/Dynamics/Ergodic/Action/OfMinimal.lean
@@ -0,0 +1,241 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Dynamics.Ergodic.Action.Regular
+import Mathlib.MeasureTheory.Measure.ContinuousPreimage
+import Mathlib.MeasureTheory.Measure.Haar.Unique
+
+/-!
+# Ergodicity from minimality
+
+In this file we prove that the left shift `(a * ·)` on a compact topological group `G`
+is ergodic with respect to the Haar measure if and only if its minimal,
+i.e., the powers `a ^ n` are dense in `G`.
+
+The proof of the more difficult "if minimal, then ergodic" implication
+is based on the ergodicity of the left action of a group on itself
+and the following fact that we prove in `ergodic_smul_of_denseRange_pow` below:
+
+If a monoid `M` continuously acts on an R₁ topological space `X`,
+`g` is an element of `M such that its natural powers are dense in `M`,
+and `μ` is a finite inner regular measure on `X` which is ergodic with respect to the action of `M`,
+then the scalar multiplication by `g` is an ergodic map.
+
+We also prove that a continuous monoid homomorphism `f : G →* G` is ergodic,
+if it is surjective and the preimages of `1` under iterations of `f` are dense in the group.
+This theorem applies, e.g., to the map `z ↦ n • z` on the additive circle or a torus.
+-/
+
+open MeasureTheory Filter Set Function
+open scoped Pointwise Topology
+
+section SMul
+
+variable {M : Type*} [TopologicalSpace M]
+  {X : Type*} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X]
+  [SMul M X] [ContinuousSMul M X]
+  {μ : Measure X} [IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul M X μ] {s : Set X}
+
+/-- Let `M` act continuously on an R₁ topological space `X`.
+Let `μ` be a finite inner regular measure on `X` which is ergodic with respect to this action.
+If a null measurable set `s` is a.e. equal
+to its preimages under the action of a dense set of elements of `M`,
+then it is either null or conull. -/
+@[to_additive "Let `M` act continuously on an R₁ topological space `X`.
+Let `μ` be a finite inner regular measure on `X` which is ergodic with respect to this action.
+If a null measurable set `s` is a.e. equal
+to its preimages under the action of a dense set of elements of `M`,
+then it is either null or conull."]
+theorem aeconst_of_dense_setOf_preimage_smul_ae (hsm : NullMeasurableSet s μ)
+    (hd : Dense {g : M | (g • ·) ⁻¹' s =ᵐ[μ] s}) : EventuallyConst s (ae μ) := by
+  borelize M
+  refine aeconst_of_forall_preimage_smul_ae_eq M hsm ?_
+  rwa [dense_iff_closure_eq, IsClosed.closure_eq, eq_univ_iff_forall] at hd
+  let f : C(M × X, X) := ⟨(· • ·).uncurry, continuous_smul⟩
+  exact isClosed_setOf_preimage_ae_eq f.curry.continuous (measurePreserving_smul · μ) _ hsm
+    (measure_ne_top _ _)
+
+@[to_additive]
+theorem aeconst_of_dense_setOf_preimage_smul_eq (hsm : NullMeasurableSet s μ)
+    (hd : Dense {g : M | (g • ·) ⁻¹' s = s}) : EventuallyConst s (ae μ) :=
+  aeconst_of_dense_setOf_preimage_smul_ae hsm <| hd.mono fun _ h ↦ mem_setOf.2 <| .of_eq h
+
+/-- If a monoid `M` continuously acts on an R₁ topological space `X`,
+`g` is an element of `M such that its natural powers are dense in `M`,
+and `μ` is a finite inner regular measure on `X` which is ergodic with respect to the action of `M`,
+then the scalar multiplication by `g` is an ergodic map. -/
+@[to_additive "If an additive monoid `M` continuously acts on an R₁ topological space `X`,
+`g` is an element of `M such that its natural multiples are dense in `M`,
+and `μ` is a finite inner regular measure on `X` which is ergodic with respect to the action of `M`,
+then the vector addition of `g` is an ergodic map."]
+theorem ergodic_smul_of_denseRange_pow {M : Type*} [Monoid M] [TopologicalSpace M]
+    [MulAction M X] [ContinuousSMul M X] {g : M} (hg : DenseRange (g ^ · : ℕ → M))
+    (μ : Measure X) [IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul M X μ] :
+    Ergodic (g • ·) μ := by
+  borelize M
+  refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
+  refine aeconst_of_dense_setOf_preimage_smul_eq hsm.nullMeasurableSet (hg.mono ?_)
+  refine range_subset_iff.2 fun n ↦ ?_
+  rw [mem_setOf, ← smul_iterate, preimage_iterate_eq, iterate_fixed hs]
+
+end SMul
+
+section IsScalarTower
+
+variable {M X : Type*} [Monoid M] [SMul M X]
+  [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X]
+  (μ : Measure X) [IsFiniteMeasure μ] [μ.InnerRegular]
+
+/-- If `N` acts continuously and ergodically on `X` and `M` acts minimally on `N`,
+then the corresponding action of `M` on `X` is ergodic. -/
+@[to_additive
+  "If `N` acts additively continuously and ergodically on `X` and `M` acts minimally on `N`,
+then the corresponding action of `M` on `X` is ergodic."]
+theorem ErgodicSMul.trans_isMinimal (N : Type*) [MulAction M N]
+    [Monoid N] [TopologicalSpace N] [MulAction.IsMinimal M N]
+    [MulAction N X] [IsScalarTower M N X] [ContinuousSMul N X] [ErgodicSMul N X μ] :
+    ErgodicSMul M X μ where
+  measure_preimage_smul c s hsm := by
+    simpa only [smul_one_smul] using SMulInvariantMeasure.measure_preimage_smul (c • 1 : N) hsm
+  aeconst_of_forall_preimage_smul_ae_eq {s} hsm hs := by
+    refine aeconst_of_dense_setOf_preimage_smul_ae (M := N) hsm.nullMeasurableSet ?_
+    refine (MulAction.dense_orbit M 1).mono ?_
+    rintro _ ⟨g, rfl⟩
+    simpa using hs g
+
+end IsScalarTower
+
+section MulActionGroup
+
+variable {G : Type*} [Group G] [TopologicalSpace G] [ContinuousInv G]
+  {X : Type*} [TopologicalSpace X] [R1Space X] [MeasurableSpace X] [BorelSpace X]
+  [MulAction G X] [ContinuousSMul G X]
+  {μ : Measure X} [IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul G X μ] {s : Set X}
+
+@[to_additive]
+theorem aeconst_of_dense_aestabilizer_smul (hsm : NullMeasurableSet s μ)
+    (hd : Dense (MulAction.aestabilizer G μ s : Set G)) : EventuallyConst s (ae μ) :=
+  aeconst_of_dense_setOf_preimage_smul_ae hsm <| (hd.preimage (isOpenMap_inv _)).mono fun g hg ↦ by
+    simpa only [preimage_smul] using hg
+
+/-- If a monoid `M` continuously acts on an R₁ topological space `X`,
+`g` is an element of `M such that its integer powers are dense in `M`,
+and `μ` is a finite inner regular measure on `X` which is ergodic with respect to the action of `M`,
+then the scalar multiplication by `g` is an ergodic map. -/
+@[to_additive "If an additive monoid `M` continuously acts on an R₁ topological space `X`,
+`g` is an element of `M such that its integer multiples are dense in `M`,
+and `μ` is a finite inner regular measure on `X` which is ergodic with respect to the action of `M`,
+then the vector addition of `g` is an ergodic map."]
+theorem ergodic_smul_of_denseRange_zpow {g : G} (hg : DenseRange (g ^ · : ℤ → G))
+    (μ : Measure X) [IsFiniteMeasure μ] [μ.InnerRegular] [ErgodicSMul G X μ] :
+    Ergodic (g • ·) μ := by
+  borelize G
+  refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
+  refine aeconst_of_dense_aestabilizer_smul hsm.nullMeasurableSet (hg.mono ?_)
+  rw [← Subgroup.coe_zpowers, SetLike.coe_subset_coe, ← Subgroup.zpowers_inv, Subgroup.zpowers_le,
+    MulAction.mem_aestabilizer, ← preimage_smul, hs]
+
+ end MulActionGroup
+
+section TopologicalGroup
+
+variable {G : Type*} [Group G] [TopologicalSpace G] [TopologicalGroup G] [MeasurableSpace G]
+
+/-- If the left multiplication by `g` is ergodic
+with respect to a measure which is positive on nonempty open sets,
+then the integer powers of `g` are dense in `G`. -/
+@[to_additive "If the left addition of `g` is ergodic
+with respect to a measure which is positive on nonempty open sets,
+then the integer multiples of `g` are dense in `G`."]
+theorem DenseRange.zpow_of_ergodic_mul_left [OpensMeasurableSpace G]
+    {μ : Measure G} [μ.IsOpenPosMeasure] {g : G} (hg : Ergodic (g * ·) μ) :
+    DenseRange (g ^ · : ℤ → G) := by
+  intro a
+  by_contra h
+  obtain ⟨V, hV₁, hVo, hV⟩ :
+      ∃ V : Set G, 1 ∈ V ∧ IsOpen V ∧ ∀ x ∈ V, ∀ y ∈ V, ∀ m : ℤ, g ^ m ≠ a * x / y := by
+    rw [← mem_compl_iff, ← interior_compl, mem_interior_iff_mem_nhds] at h
+    have : Tendsto (fun (x, y) ↦ a * x / y) (𝓝 1) (𝓝 a) :=
+      Continuous.tendsto' (by fun_prop) _ _ (by simp)
+    rw [nhds_prod_eq] at this
+    simpa [(nhds_basis_opens (1 : G)).prod_self.mem_iff, prod_subset_iff, and_assoc] using this h
+  set s := ⋃ m : ℤ, g ^ m • V
+  have hso : IsOpen s := isOpen_iUnion fun m ↦ hVo.smul _
+  have hsne : s.Nonempty := ⟨1, mem_iUnion.2 ⟨0, by simpa⟩⟩
+  have hd : Disjoint s (a • V) := by
+    simp_rw [s, disjoint_iUnion_left, disjoint_left]
+    rintro m _ ⟨x, hx, rfl⟩ ⟨y, hy, hxy⟩
+    apply hV y hy x hx m
+    simp_all
+  have hgs : (g * ·) ⁻¹' s = s := by
+    simp only [s, preimage_iUnion, ← smul_eq_mul, preimage_smul]
+    refine iUnion_congr_of_surjective _ (add_left_surjective (-1)) fun m ↦ ?_
+    simp [zpow_add, mul_smul]
+  cases hg.measure_self_or_compl_eq_zero hso.measurableSet hgs with
+  | inl h => exact hso.measure_ne_zero _ hsne h
+  | inr h =>
+    refine (hVo.smul a).measure_ne_zero μ (.image _ ⟨1, hV₁⟩) (measure_mono_null ?_ h)
+    rwa [disjoint_right] at hd
+
+variable [SecondCountableTopology G] [BorelSpace G] {g : G}
+
+@[to_additive]
+theorem ergodic_mul_left_of_denseRange_pow (hg : DenseRange (g ^ · : ℕ → G))
+    (μ : Measure G) [IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant] :
+    Ergodic (g * ·) μ :=
+  ergodic_smul_of_denseRange_pow hg μ
+
+@[to_additive]
+theorem ergodic_mul_left_of_denseRange_zpow (hg : DenseRange (g ^ · : ℤ → G))
+    (μ : Measure G) [IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant] :
+    Ergodic (g * ·) μ :=
+  ergodic_smul_of_denseRange_zpow hg μ
+
+@[to_additive]
+theorem ergodic_mul_left_iff_denseRange_zpow (μ : Measure G) [IsFiniteMeasure μ]
+    [μ.InnerRegular] [μ.IsMulLeftInvariant] [NeZero μ] :
+    Ergodic (g * ·) μ ↔ DenseRange (g ^ · : ℤ → G) :=
+  ⟨.zpow_of_ergodic_mul_left, (ergodic_mul_left_of_denseRange_zpow · μ)⟩
+
+end TopologicalGroup
+
+namespace MonoidHom
+
+variable {G : Type*} [Group G] [TopologicalSpace G]
+  [TopologicalGroup G] [SecondCountableTopology G] [MeasurableSpace G] [BorelSpace G]
+
+/-- Let `f : G →* G` be a group endomorphism of a topological group with second countable topology.
+If the preimages of `1` under the iterations of `f` are dense,
+then it is preergodic with respect to any finite inner regular left invariant measure. -/
+@[to_additive "Let `f : G →+ G` be an additive group endomorphism
+of a topological additive group with second countable topology.
+If the preimages of `0` under the iterations of `f` are dense,
+then it is preergodic with respect to any finite inner regular left invariant measure."]
+theorem preErgodic_of_dense_iUnion_preimage_one
+    {μ : Measure G} [IsFiniteMeasure μ] [μ.InnerRegular] [μ.IsMulLeftInvariant]
+    (f : G →* G) (hf : Dense (⋃ n, f^[n] ⁻¹' 1)) : PreErgodic f μ := by
+  refine ⟨fun s hsm hs ↦ aeconst_of_dense_setOf_preimage_smul_eq (M := G) hsm.nullMeasurableSet ?_⟩
+  refine hf.mono <| iUnion_subset fun n x hx ↦ ?_
+  have hsn : f^[n] ⁻¹' s = s := by
+    rw [preimage_iterate_eq, iterate_fixed hs]
+  rw [mem_preimage, Set.mem_one] at hx
+  rw [mem_setOf, ← hsn]
+  ext y
+  simp [hx]
+
+/-- Let `f : G →* G` be a continuous surjective group endomorphism
+of a compact topological group with second countable topology.
+If the preimages of `1` under the iterations of `f` are dense,
+then `f` is ergodic with respect to any finite inner regular left invariant measure. -/
+@[to_additive "Let `f : G →+ G` be a continuous surjective additive group endomorphism
+of a compact topological additive group with second countable topology.
+If the preimages of `0` under the iterations of `f` are dense,
+then `f` is ergodic with respect to any finite inner regular left invariant measure."]
+theorem ergodic_of_dense_iUnion_preimage_one [CompactSpace G] {μ : Measure G} [μ.IsHaarMeasure]
+    (f : G →* G) (hf : Dense (⋃ n, f^[n] ⁻¹' 1)) (hcont : Continuous f) (hsurj : Surjective f) :
+    Ergodic f μ :=
+  ⟨f.measurePreserving hcont hsurj rfl, f.preErgodic_of_dense_iUnion_preimage_one hf⟩
+
+end MonoidHom
diff --git a/Mathlib/Dynamics/Ergodic/AddCircle.lean b/Mathlib/Dynamics/Ergodic/AddCircle.lean
index 5bd20dcdead3d..d19dedb1a3963 100644
--- a/Mathlib/Dynamics/Ergodic/AddCircle.lean
+++ b/Mathlib/Dynamics/Ergodic/AddCircle.lean
@@ -100,7 +100,7 @@ theorem ae_empty_or_univ_of_forall_vadd_ae_eq_self {s : Set <| AddCircle T}
 
 theorem ergodic_zsmul {n : ℤ} (hn : 1 < |n|) : Ergodic fun y : AddCircle T => n • y :=
   { measurePreserving_zsmul volume (abs_pos.mp <| lt_trans zero_lt_one hn) with
-    ae_empty_or_univ := fun s hs hs' => by
+    aeconst_set := fun s hs hs' => by
       let u : ℕ → AddCircle T := fun j => ↑((↑1 : ℝ) / ↑(n.natAbs ^ j) * T)
       replace hn : 1 < n.natAbs := by rwa [Int.abs_eq_natAbs, Nat.one_lt_cast] at hn
       have hu₀ : ∀ j, addOrderOf (u j) = n.natAbs ^ j := fun j => by
@@ -114,6 +114,7 @@ theorem ergodic_zsmul {n : ℤ} (hn : 1 < |n|) : Ergodic fun y : AddCircle T =>
         rw [vadd_eq_self_of_preimage_zsmul_eq_self hs' (hnu j)]
       have hu₂ : Tendsto (fun j => addOrderOf <| u j) atTop atTop := by
         simp_rw [hu₀]; exact Nat.tendsto_pow_atTop_atTop_of_one_lt hn
+      rw [eventuallyConst_set']
       exact ae_empty_or_univ_of_forall_vadd_ae_eq_self hs.nullMeasurableSet hu₁ hu₂ }
 
 theorem ergodic_nsmul {n : ℕ} (hn : 1 < n) : Ergodic fun y : AddCircle T => n • y :=
diff --git a/Mathlib/Dynamics/Ergodic/Conservative.lean b/Mathlib/Dynamics/Ergodic/Conservative.lean
index 4853d8c9c1bfc..f89c15dd99f1b 100644
--- a/Mathlib/Dynamics/Ergodic/Conservative.lean
+++ b/Mathlib/Dynamics/Ergodic/Conservative.lean
@@ -67,6 +67,17 @@ protected theorem id (μ : Measure α) : Conservative id μ :=
     exists_mem_iterate_mem' := fun _ _ h0 => by
       simpa [exists_ne] using nonempty_of_measure_ne_zero h0 }
 
+theorem of_absolutelyContinuous {ν : Measure α} (h : Conservative f μ) (hν : ν ≪ μ)
+    (h' : QuasiMeasurePreserving f ν ν) : Conservative f ν :=
+  ⟨h', fun _ hsm h0 ↦ h.exists_mem_iterate_mem' hsm (mt (@hν _) h0)⟩
+
+/-- Restriction of a conservative system to an invariant set is a conservative system,
+formulated in terms of the restriction of the measure. -/
+theorem measureRestrict (h : Conservative f μ) (hs : MapsTo f s s) :
+    Conservative f (μ.restrict s) :=
+  .of_absolutelyContinuous h (absolutelyContinuous_of_le restrict_le_self) <|
+    h.toQuasiMeasurePreserving.restrict hs
+
 /-- If `f` is a conservative self-map and `s` is a null measurable set of nonzero measure,
 then there exists a point `x ∈ s` that returns to `s` under a non-zero iteration of `f`. -/
 theorem exists_mem_iterate_mem (hf : Conservative f μ)
@@ -147,7 +158,7 @@ theorem measure_inter_frequently_image_mem_eq (hf : Conservative f μ) (hs : Nul
 
 /-- Poincaré recurrence theorem: if `f` is a conservative dynamical system and `s` is a measurable
 set, then for `μ`-a.e. `x`, if the orbit of `x` visits `s` at least once, then it visits `s`
-infinitely many times.  -/
+infinitely many times. -/
 theorem ae_forall_image_mem_imp_frequently_image_mem (hf : Conservative f μ)
     (hs : NullMeasurableSet s μ) : ∀ᵐ x ∂μ, ∀ k, f^[k] x ∈ s → ∃ᶠ n in atTop, f^[n] x ∈ s := by
   refine ae_all_iff.2 fun k => ?_
diff --git a/Mathlib/Dynamics/Ergodic/Ergodic.lean b/Mathlib/Dynamics/Ergodic/Ergodic.lean
index 4316c1c557a29..eb2fbac4d06d4 100644
--- a/Mathlib/Dynamics/Ergodic/Ergodic.lean
+++ b/Mathlib/Dynamics/Ergodic/Ergodic.lean
@@ -33,36 +33,40 @@ open Set Function Filter MeasureTheory MeasureTheory.Measure
 
 open ENNReal
 
-variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
+variable {α : Type*} {m : MeasurableSpace α} {s : Set α}
 
 /-- A map `f : α → α` is said to be pre-ergodic with respect to a measure `μ` if any measurable
 strictly invariant set is either almost empty or full. -/
-structure PreErgodic (μ : Measure α := by volume_tac) : Prop where
-  ae_empty_or_univ : ∀ ⦃s⦄, MeasurableSet s → f ⁻¹' s = s → s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ
+structure PreErgodic (f : α → α) (μ : Measure α := by volume_tac) : Prop where
+  aeconst_set ⦃s⦄ : MeasurableSet s → f ⁻¹' s = s → EventuallyConst s (ae μ)
 
 /-- A map `f : α → α` is said to be ergodic with respect to a measure `μ` if it is measure
 preserving and pre-ergodic. -/
 -- porting note (#5171): removed @[nolint has_nonempty_instance]
-structure Ergodic (μ : Measure α := by volume_tac) extends
+structure Ergodic (f : α → α) (μ : Measure α := by volume_tac) extends
   MeasurePreserving f μ μ, PreErgodic f μ : Prop
 
 /-- A map `f : α → α` is said to be quasi ergodic with respect to a measure `μ` if it is quasi
 measure preserving and pre-ergodic. -/
 -- porting note (#5171): removed @[nolint has_nonempty_instance]
-structure QuasiErgodic (μ : Measure α := by volume_tac) extends
+structure QuasiErgodic (f : α → α) (μ : Measure α := by volume_tac) extends
   QuasiMeasurePreserving f μ μ, PreErgodic f μ : Prop
 
-variable {f} {μ : Measure α}
+variable {f : α → α} {μ : Measure α}
 
 namespace PreErgodic
 
+theorem ae_empty_or_univ (hf : PreErgodic f μ) (hs : MeasurableSet s) (hfs : f ⁻¹' s = s) :
+    s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by
+  simpa only [eventuallyConst_set'] using hf.aeconst_set hs hfs
+
 theorem measure_self_or_compl_eq_zero (hf : PreErgodic f μ) (hs : MeasurableSet s)
     (hs' : f ⁻¹' s = s) : μ s = 0 ∨ μ sᶜ = 0 := by
   simpa using hf.ae_empty_or_univ hs hs'
 
 theorem ae_mem_or_ae_nmem (hf : PreErgodic f μ) (hsm : MeasurableSet s) (hs : f ⁻¹' s = s) :
     (∀ᵐ x ∂μ, x ∈ s) ∨ ∀ᵐ x ∂μ, x ∉ s :=
-  (hf.ae_empty_or_univ hsm hs).symm.imp eventuallyEq_univ.1 eventuallyEq_empty.1
+  eventuallyConst_set.1 <| hf.aeconst_set hsm hs
 
 /-- On a probability space, the (pre)ergodicity condition is a zero one law. -/
 theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s)
@@ -70,7 +74,7 @@ theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs
   simpa [hs] using hf.measure_self_or_compl_eq_zero hs hs'
 
 theorem of_iterate (n : ℕ) (hf : PreErgodic f^[n] μ) : PreErgodic f μ :=
-  ⟨fun _ hs hs' => hf.ae_empty_or_univ hs <| IsFixedPt.preimage_iterate hs' n⟩
+  ⟨fun _ hs hs' => hf.aeconst_set hs <| IsFixedPt.preimage_iterate hs' n⟩
 
 end PreErgodic
 
@@ -79,23 +83,18 @@ namespace MeasureTheory.MeasurePreserving
 variable {β : Type*} {m' : MeasurableSpace β} {μ' : Measure β} {s' : Set β} {g : α → β}
 
 theorem preErgodic_of_preErgodic_conjugate (hg : MeasurePreserving g μ μ') (hf : PreErgodic f μ)
-    {f' : β → β} (h_comm : g ∘ f = f' ∘ g) : PreErgodic f' μ' :=
-  ⟨by
-    intro s hs₀ hs₁
-    replace hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s := by rw [← preimage_comp, h_comm, preimage_comp, hs₁]
-    cases' hf.ae_empty_or_univ (hg.measurable hs₀) hs₁ with hs₂ hs₂ <;> [left; right]
-    · simpa only [ae_eq_empty, hg.measure_preimage hs₀.nullMeasurableSet] using hs₂
-    · simpa only [ae_eq_univ, ← preimage_compl,
-        hg.measure_preimage hs₀.compl.nullMeasurableSet] using hs₂⟩
+    {f' : β → β} (h_comm : Semiconj g f f') : PreErgodic f' μ' where
+  aeconst_set s hs₀ hs₁ := by
+    rw [← hg.aeconst_preimage hs₀.nullMeasurableSet]
+    apply hf.aeconst_set (hg.measurable hs₀)
+    rw [← preimage_comp, h_comm.comp_eq, preimage_comp, hs₁]
 
 theorem preErgodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
     PreErgodic (e ∘ f ∘ e.symm) μ' ↔ PreErgodic f μ := by
   refine ⟨fun hf => preErgodic_of_preErgodic_conjugate (h.symm e) hf ?_,
       fun hf => preErgodic_of_preErgodic_conjugate h hf ?_⟩
-  · change (e.symm ∘ e) ∘ f ∘ e.symm = f ∘ e.symm
-    rw [MeasurableEquiv.symm_comp_self, id_comp]
-  · change e ∘ f = e ∘ f ∘ e.symm ∘ e
-    rw [MeasurableEquiv.symm_comp_self, comp_id]
+  · simp [Semiconj]
+  · simp [Semiconj]
 
 theorem ergodic_conjugate_iff {e : α ≃ᵐ β} (h : MeasurePreserving e μ μ') :
     Ergodic (e ∘ f ∘ e.symm) μ' ↔ Ergodic f μ := by
@@ -109,25 +108,26 @@ end MeasureTheory.MeasurePreserving
 
 namespace QuasiErgodic
 
-/-- For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are
-still either almost empty or full. -/
-theorem ae_empty_or_univ₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ)
-    (hs : f ⁻¹' s =ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
+theorem aeconst_set₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ) (hs : f ⁻¹' s =ᵐ[μ] s) :
+    EventuallyConst s (ae μ) :=
   let ⟨_t, h₀, h₁, h₂⟩ := hf.toQuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae hsm hs
-  (hf.ae_empty_or_univ h₀ h₂).imp h₁.symm.trans h₁.symm.trans
+  (hf.aeconst_set h₀ h₂).congr h₁
 
 /-- For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are
 still either almost empty or full. -/
-theorem ae_empty_or_univ' (hf : QuasiErgodic f μ) (hs : MeasurableSet s) (hs' : f ⁻¹' s =ᵐ[μ] s) :
+theorem ae_empty_or_univ₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ)
+    (hs : f ⁻¹' s =ᵐ[μ] s) :
     s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
-  ae_empty_or_univ₀ hf hs.nullMeasurableSet hs'
+  eventuallyConst_set'.mp <| hf.aeconst_set₀ hsm hs
+
+@[deprecated (since := "2024-07-21")] alias ae_empty_or_univ' := ae_empty_or_univ₀
 
 /-- For a quasi ergodic map, sets that are almost invariant (rather than strictly invariant) are
 still either almost empty or full. -/
 theorem ae_mem_or_ae_nmem₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ)
     (hs : f ⁻¹' s =ᵐ[μ] s) :
     (∀ᵐ x ∂μ, x ∈ s) ∨ ∀ᵐ x ∂μ, x ∉ s :=
-  (hf.ae_empty_or_univ₀ hsm hs).symm.imp (by simp [mem_ae_iff]) (by simp [ae_iff])
+  eventuallyConst_set.mp <| hf.aeconst_set₀ hsm hs
 
 end QuasiErgodic
 
@@ -138,23 +138,21 @@ theorem quasiErgodic (hf : Ergodic f μ) : QuasiErgodic f μ :=
   { hf.toPreErgodic, hf.toMeasurePreserving.quasiMeasurePreserving with }
 
 /-- See also `Ergodic.ae_empty_or_univ_of_preimage_ae_le`. -/
-theorem ae_empty_or_univ_of_preimage_ae_le' (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_preimage_ae_le' (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : f ⁻¹' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by
-  refine hf.quasiErgodic.ae_empty_or_univ' hs ?_
-  refine ae_eq_of_ae_subset_of_measure_ge hs'
-    (hf.measure_preimage hs.nullMeasurableSet).symm.le ?_ h_fin
-  exact measurableSet_preimage hf.measurable hs
+  refine hf.quasiErgodic.ae_empty_or_univ₀ hs ?_
+  refine ae_eq_of_ae_subset_of_measure_ge hs' (hf.measure_preimage hs).ge ?_ h_fin
+  exact hs.preimage hf.quasiMeasurePreserving
 
 /-- See also `Ergodic.ae_empty_or_univ_of_ae_le_preimage`. -/
-theorem ae_empty_or_univ_of_ae_le_preimage' (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_ae_le_preimage' (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : s ≤ᵐ[μ] f ⁻¹' s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by
-  replace h_fin : μ (f ⁻¹' s) ≠ ∞ := by rwa [hf.measure_preimage hs.nullMeasurableSet]
-  refine hf.quasiErgodic.ae_empty_or_univ' hs ?_
-  exact (ae_eq_of_ae_subset_of_measure_ge hs'
-    (hf.measure_preimage hs.nullMeasurableSet).le hs h_fin).symm
+  replace h_fin : μ (f ⁻¹' s) ≠ ∞ := by rwa [hf.measure_preimage hs]
+  refine hf.quasiErgodic.ae_empty_or_univ₀ hs ?_
+  exact (ae_eq_of_ae_subset_of_measure_ge hs' (hf.measure_preimage hs).le hs h_fin).symm
 
 /-- See also `Ergodic.ae_empty_or_univ_of_image_ae_le`. -/
-theorem ae_empty_or_univ_of_image_ae_le' (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_image_ae_le' (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : f '' s ≤ᵐ[μ] s) (h_fin : μ s ≠ ∞) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ := by
   replace hs' : s ≤ᵐ[μ] f ⁻¹' s :=
     (HasSubset.Subset.eventuallyLE (subset_preimage_image f s)).trans
@@ -165,15 +163,15 @@ section IsFiniteMeasure
 
 variable [IsFiniteMeasure μ]
 
-theorem ae_empty_or_univ_of_preimage_ae_le (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_preimage_ae_le (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : f ⁻¹' s ≤ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
   ae_empty_or_univ_of_preimage_ae_le' hf hs hs' <| measure_ne_top μ s
 
-theorem ae_empty_or_univ_of_ae_le_preimage (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_ae_le_preimage (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : s ≤ᵐ[μ] f ⁻¹' s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
   ae_empty_or_univ_of_ae_le_preimage' hf hs hs' <| measure_ne_top μ s
 
-theorem ae_empty_or_univ_of_image_ae_le (hf : Ergodic f μ) (hs : MeasurableSet s)
+theorem ae_empty_or_univ_of_image_ae_le (hf : Ergodic f μ) (hs : NullMeasurableSet s μ)
     (hs' : f '' s ≤ᵐ[μ] s) : s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
   ae_empty_or_univ_of_image_ae_le' hf hs hs' <| measure_ne_top μ s
 
diff --git a/Mathlib/Dynamics/Ergodic/MeasurePreserving.lean b/Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
index 3a4d7509a3c58..aeb910c97b9ae 100644
--- a/Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
+++ b/Mathlib/Dynamics/Ergodic/MeasurePreserving.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.MeasureTheory.Measure.AEMeasurable
+import Mathlib.Order.Filter.EventuallyConst
 
 /-!
 # Measure preserving maps
@@ -101,13 +102,13 @@ protected theorem comp_left_iff {g : α → β} {e : β ≃ᵐ γ} (h : MeasureP
     MeasurePreserving (e ∘ g) μa μc ↔ MeasurePreserving g μa μb := by
   refine ⟨fun hg => ?_, fun hg => h.comp hg⟩
   convert (MeasurePreserving.symm e h).comp hg
-  simp [← Function.comp.assoc e.symm e g]
+  simp [← Function.comp_assoc e.symm e g]
 
 protected theorem comp_right_iff {g : α → β} {e : γ ≃ᵐ α} (h : MeasurePreserving e μc μa) :
     MeasurePreserving (g ∘ e) μc μb ↔ MeasurePreserving g μa μb := by
   refine ⟨fun hg => ?_, fun hg => hg.comp h⟩
   convert hg.comp (MeasurePreserving.symm e h)
-  simp [Function.comp.assoc g e e.symm]
+  simp [Function.comp_assoc g e e.symm]
 
 protected theorem sigmaFinite {f : α → β} (hf : MeasurePreserving f μa μb) [SigmaFinite μb] :
     SigmaFinite μa :=
@@ -126,13 +127,25 @@ theorem measure_preimage_equiv {f : α ≃ᵐ β} (hf : MeasurePreserving f μa
     μa (f ⁻¹' s) = μb s :=
   measure_preimage_emb hf f.measurableEmbedding s
 
-protected theorem iterate {f : α → α} (hf : MeasurePreserving f μa μa) :
-    ∀ n, MeasurePreserving f^[n] μa μa
-  | 0 => MeasurePreserving.id μa
-  | n + 1 => (MeasurePreserving.iterate hf n).comp hf
+theorem aeconst_comp [MeasurableSingletonClass γ] {f : α → β} (hf : MeasurePreserving f μa μb)
+    {g : β → γ} (hg : NullMeasurable g μb) :
+    Filter.EventuallyConst (g ∘ f) (ae μa) ↔ Filter.EventuallyConst g (ae μb) :=
+  exists_congr fun s ↦ and_congr_left fun hs ↦ by
+    simp only [Filter.mem_map, mem_ae_iff, ← hf.measure_preimage (hg hs.measurableSet).compl,
+      preimage_comp, preimage_compl]
+
+theorem aeconst_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β}
+    (hs : NullMeasurableSet s μb) :
+    Filter.EventuallyConst (f ⁻¹' s) (ae μa) ↔ Filter.EventuallyConst s (ae μb) :=
+  aeconst_comp hf hs.mem
 
 variable {μ : Measure α} {f : α → α} {s : Set α}
 
+protected theorem iterate (hf : MeasurePreserving f μ μ) :
+    ∀ n, MeasurePreserving f^[n] μ μ
+  | 0 => .id μ
+  | n + 1 => (MeasurePreserving.iterate hf n).comp hf
+
 open scoped symmDiff in
 lemma measure_symmDiff_preimage_iterate_le
     (hf : MeasurePreserving f μ μ) (hs : NullMeasurableSet s μ) (n : ℕ) :
diff --git a/Mathlib/Dynamics/OmegaLimit.lean b/Mathlib/Dynamics/OmegaLimit.lean
index 61f82cc579837..7de977bddf6ba 100644
--- a/Mathlib/Dynamics/OmegaLimit.lean
+++ b/Mathlib/Dynamics/OmegaLimit.lean
@@ -86,8 +86,8 @@ theorem mapsTo_omegaLimit' {α' β' : Type*} [TopologicalSpace β'] {f : Filter
   intro y hy u hu
   refine map_mem_closure hgc (hy _ (inter_mem hu hg)) (forall_image2_iff.2 fun t ht x hx ↦ ?_)
   calc
-    gb (ϕ t x) = ϕ' t (ga x) := ht.2 hx
-    _ ∈ image2 ϕ' u s' := mem_image2_of_mem ht.1 (hs hx)
+    ϕ' t (ga x) ∈ image2 ϕ' u s' := mem_image2_of_mem ht.1 (hs hx)
+    _ = gb (ϕ t x) := ht.2 hx |>.symm
 
 theorem mapsTo_omegaLimit {α' β' : Type*} [TopologicalSpace β'] {f : Filter τ} {ϕ : τ → α → β}
     {ϕ' : τ → α' → β'} {ga : α → α'} {s' : Set α'} (hs : MapsTo ga s s') {gb : β → β'}
@@ -192,6 +192,9 @@ theorem omegaLimit_subset_closure_fw_image {u : Set τ} (hu : u ∈ f) :
   rw [mem_iInter] at hx
   exact hx ⟨u, hu⟩
 
+-- An instance with better keys
+instance : Inhabited f.sets := Filter.inhabitedMem
+
 /-!
 ### ω-limits and compactness
 -/
diff --git a/Mathlib/Dynamics/PeriodicPts.lean b/Mathlib/Dynamics/PeriodicPts.lean
index 3f47192c4cc47..160a6606f5e0f 100644
--- a/Mathlib/Dynamics/PeriodicPts.lean
+++ b/Mathlib/Dynamics/PeriodicPts.lean
@@ -338,7 +338,7 @@ theorem not_isPeriodicPt_of_pos_of_lt_minimalPeriod :
 theorem IsPeriodicPt.minimalPeriod_dvd (hx : IsPeriodicPt f n x) : minimalPeriod f x ∣ n :=
   (eq_or_lt_of_le <| n.zero_le).elim (fun hn0 => hn0 ▸ dvd_zero _) fun hn0 =>
     -- Porting note: `Nat.dvd_iff_mod_eq_zero` gained explicit arguments
-    (Nat.dvd_iff_mod_eq_zero _ _).2 <|
+    Nat.dvd_iff_mod_eq_zero.2 <|
       (hx.mod <| isPeriodicPt_minimalPeriod f x).eq_zero_of_lt_minimalPeriod <|
         Nat.mod_lt _ <| hx.minimalPeriod_pos hn0
 
@@ -433,7 +433,7 @@ theorem periodicOrbit_length : (periodicOrbit f x).length = minimalPeriod f x :=
 @[simp]
 theorem periodicOrbit_eq_nil_iff_not_periodic_pt :
     periodicOrbit f x = Cycle.nil ↔ x ∉ periodicPts f := by
-  simp only [periodicOrbit.eq_1, Cycle.coe_eq_nil, List.map_eq_nil, List.range_eq_nil]
+  simp only [periodicOrbit.eq_1, Cycle.coe_eq_nil, List.map_eq_nil_iff, List.range_eq_nil]
   exact minimalPeriod_eq_zero_iff_nmem_periodicPts
 
 theorem periodicOrbit_eq_nil_of_not_periodic_pt (h : x ∉ periodicPts f) :
diff --git a/Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean b/Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean
index 0333f199aa230..2b8f40fded2bf 100644
--- a/Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean
+++ b/Mathlib/Dynamics/TopologicalEntropy/CoverEntropy.lean
@@ -19,7 +19,7 @@ A notable choice is that we define the topological entropy of a subset `F` of th
 Usually, one defines the entropy of an invariant subset `F` as the entropy of the restriction of the
 transformation to `F`. We avoid the latter definition as it would involve frequent manipulation of
 subtypes. Our version directly gives a meaning to the topological entropy of a subsystem, and a
-single theorem (`subset_restriction_entropy` in `TopologicalEntropy.Morphism`) will give the
+single theorem (`subset_restriction_entropy` in `TopologicalEntropy.Semiconj`) will give the
 equivalence between both versions.
 
 Another choice is to give a meaning to the entropy of `∅` (it must be `-∞` to stay coherent) and to
@@ -28,11 +28,11 @@ reals `[-∞, +∞]`. The consequence is that we use `ℕ∞`, `ℝ≥0∞` and
 
 ## Main definitions
 - `IsDynCoverOf`: property that dynamical balls centered on a subset `s` cover a subset `F`.
-- `coverMincard`: minimal cardinal of a dynamical cover. Takes values in `ℕ∞`.
-- `coverEntropyInfEnt`/`coverEntropyEnt`: exponential growth of `coverMincard`. The former is
-defined with a `liminf`, the later with a `limsup`. Take values in `EReal`.
-- `coverEntropyInf`/`coverEntropy`: supremum of `coverEntropyInfEnt`/`coverEntropyEnt` over
-all entourages (or limit as the entourages go to the diagonal). These are Bowen-Dinaburg's
+- `coverMincard`: minimal cardinality of a dynamical cover. Takes values in `ℕ∞`.
+- `coverEntropyInfEntourage`/`coverEntropyEntourage`: exponential growth of `coverMincard`.
+The former is defined with a `liminf`, the later with a `limsup`. Take values in `EReal`.
+- `coverEntropyInf`/`coverEntropy`: supremum of `coverEntropyInfEntourage`/`coverEntropyEntourage`
+over all entourages (or limit as the entourages go to the diagonal). These are Bowen-Dinaburg's
 versions of the topological entropy with covers. Take values in `EReal`.
 
 ## Implementation notes
@@ -46,8 +46,8 @@ using only `coverEntropy`.
 ## Main results
 - `IsDynCoverOf.iterate_le_pow`: given a dynamical cover at time `n`, creates dynamical covers
 at all iterates `n * m` with controlled cardinality.
-- `IsDynCoverOf.coverEntropyEnt_le_log_card_div`: upper bound on `coverEntropyEnt` given any
-dynamical cover.
+- `IsDynCoverOf.coverEntropyEntourage_le_log_card_div`: upper bound on `coverEntropyEntourage`
+given any dynamical cover.
 - `coverEntropyInf_eq_coverEntropy`: equality between the notions of topological entropy defined
 with a `liminf` and a `limsup`.
 
@@ -56,11 +56,11 @@ cover, entropy
 
 ## TODO
 The most painful part of many manipulations involving topological entropy is going from
-`coverMincard` to `coverEntropyInfEnt`/`coverEntropyEnt`. It involves a logarithm, a division, a
-`liminf`/`limsup`, and multiple coercions. The best thing to do would be to write a file on
-"exponential growth" to make a clean pathway from estimates on `coverMincard` to estimates on
-`coverEntropyInf`/`coverEntropy`. It would also be useful in other similar contexts, including the
-definition of entropy using nets.
+`coverMincard` to `coverEntropyInfEntourage`/`coverEntropyEntourage`. It involves a logarithm,
+a division, a `liminf`/`limsup`, and multiple coercions. The best thing to do would be to write
+a file on "exponential growth" to make a clean pathway from estimates on `coverMincard`
+to estimates on `coverEntropyInf`/`coverEntropy`. It would also be useful
+in other similar contexts, including the definition of entropy using nets.
 
 Get versions of the topological entropy on (pseudo-e)metric spaces.
 -/
@@ -133,8 +133,8 @@ lemma IsDynCoverOf.nonempty_inter {T : X → X} {F : Set X} {U : Set (X × X)} {
 /-- From a dynamical cover `s` with entourage `U` and time `m`, we construct covers with entourage
 `U ○ U` and any multiple `m * n` of `m` with controlled cardinality. This lemma is the first step
 in a submultiplicative-like property of `coverMincard`, with consequences such as explicit bounds
-for the topological entropy (`coverEntropyInfEnt_le_card_div`) and an equality between two notions
-of topological entropy (`coverEntropyInf_eq_coverEntropySup_of_inv`).-/
+for the topological entropy (`coverEntropyInfEntourage_le_card_div`) and an equality between
+two notions of topological entropy (`coverEntropyInf_eq_coverEntropySup_of_inv`).-/
 lemma IsDynCoverOf.iterate_le_pow {T : X → X} {F : Set X} (F_inv : MapsTo T F F) {U : Set (X × X)}
     (U_symm : SymmetricRel U) {m : ℕ} (n : ℕ) {s : Finset X} (h : IsDynCoverOf T F U m s) :
     ∃ t : Finset X, IsDynCoverOf T F (U ○ U) (m * n) t ∧ t.card ≤ s.card ^ n := by
@@ -234,8 +234,8 @@ lemma exists_isDynCoverOf_of_isCompact_invariant [UniformSpace X] {T : X → X}
 noncomputable def coverMincard (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) : ℕ∞ :=
   ⨅ (s : Finset X) (_ : IsDynCoverOf T F U n s), (s.card : ℕ∞)
 
-lemma coverMincard_le_card {T : X → X} {F : Set X} {U : Set (X × X)} {n : ℕ} {s : Finset X}
-    (h : IsDynCoverOf T F U n s) :
+lemma IsDynCoverOf.coverMincard_le_card {T : X → X} {F : Set X} {U : Set (X × X)} {n : ℕ}
+    {s : Finset X} (h : IsDynCoverOf T F U n s) :
     coverMincard T F U n ≤ s.card := iInf₂_le s h
 
 lemma coverMincard_monotone_time (T : X → X) (F : Set X) (U : Set (X × X)) :
@@ -290,7 +290,7 @@ lemma coverMincard_zero (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X 
   rcases h with ⟨x, _⟩
   have := isDynCoverOf_zero T F U (singleton_nonempty x)
   rw [← Finset.coe_singleton] at this
-  apply le_of_le_of_eq (coverMincard_le_card this)
+  apply this.coverMincard_le_card.trans_eq
   rw [Finset.card_singleton, Nat.cast_one]
 
 lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
@@ -299,7 +299,7 @@ lemma coverMincard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
   rcases h with ⟨x, _⟩
   have := isDynCoverOf_univ T F n (singleton_nonempty x)
   rw [← Finset.coe_singleton] at this
-  apply le_of_le_of_eq (coverMincard_le_card this)
+  apply this.coverMincard_le_card.trans_eq
   rw [Finset.card_singleton, Nat.cast_one]
 
 lemma coverMincard_mul_le_pow {T : X → X} {F : Set X} (F_inv : MapsTo T F F) {U : Set (X × X)}
@@ -310,16 +310,16 @@ lemma coverMincard_mul_le_pow {T : X → X} {F : Set X} (F_inv : MapsTo T F F) {
   rcases n.eq_zero_or_pos with rfl | n_pos
   · rw [mul_zero, coverMincard_zero T F_nonempty (U ○ U), pow_zero]
   rcases eq_top_or_lt_top (coverMincard T F U m) with h | h
-  · exact h ▸ le_of_le_of_eq (le_top (α := ℕ∞)) (ENat.top_pow n_pos).symm
+  · exact h ▸ (le_top (α := ℕ∞)).trans_eq (ENat.top_pow n_pos).symm
   · rcases (coverMincard_finite_iff T F U m).1 h with ⟨s, s_cover, s_coverMincard⟩
     rcases s_cover.iterate_le_pow F_inv U_symm n with ⟨t, t_cover, t_le_sn⟩
     rw [← s_coverMincard]
-    exact (coverMincard_le_card t_cover).trans (WithTop.coe_le_coe.2 t_le_sn)
+    exact t_cover.coverMincard_le_card.trans (WithTop.coe_le_coe.2 t_le_sn)
 
 lemma coverMincard_le_pow {T : X → X} {F : Set X} (F_inv : MapsTo T F F) {U : Set (X × X)}
     (U_symm : SymmetricRel U) {m : ℕ} (m_pos : 0 < m) (n : ℕ) :
     coverMincard T F (U ○ U) n ≤ coverMincard T F U m ^ (n / m + 1) :=
-  (coverMincard_monotone_time T F (U ○ U) (le_of_lt (Nat.lt_mul_div_succ n m_pos))).trans
+  (coverMincard_monotone_time T F (U ○ U) (Nat.lt_mul_div_succ n m_pos).le).trans
     (coverMincard_mul_le_pow F_inv U_symm m (n / m + 1))
 
 lemma coverMincard_finite_of_isCompact_uniformContinuous [UniformSpace X] {T : X → X}
@@ -327,13 +327,13 @@ lemma coverMincard_finite_of_isCompact_uniformContinuous [UniformSpace X] {T : X
     (n : ℕ) :
     coverMincard T F U n < ⊤ := by
   rcases exists_isDynCoverOf_of_isCompact_uniformContinuous F_comp h U_uni n with ⟨s, s_cover⟩
-  exact (coverMincard_le_card s_cover).trans_lt (WithTop.coe_lt_top s.card)
+  exact s_cover.coverMincard_le_card.trans_lt (WithTop.coe_lt_top s.card)
 
 lemma coverMincard_finite_of_isCompact_invariant [UniformSpace X] {T : X → X} {F : Set X}
     (F_comp : IsCompact F) (F_inv : MapsTo T F F) {U : Set (X × X)} (U_uni : U ∈ 𝓤 X) (n : ℕ) :
     coverMincard T F U n < ⊤ := by
   rcases exists_isDynCoverOf_of_isCompact_invariant F_comp F_inv U_uni n with ⟨s, s_cover⟩
-  exact (coverMincard_le_card s_cover).trans_lt (WithTop.coe_lt_top s.card)
+  exact s_cover.coverMincard_le_card.trans_lt (WithTop.coe_lt_top s.card)
 
 /-- All dynamical balls of a minimal dynamical cover of `F` intersect `F`. This lemma is the key
   to relate Bowen-Dinaburg's definition of topological entropy with covers and their definition
@@ -356,7 +356,7 @@ lemma nonempty_inter_of_coverMincard {T : X → X} {F : Set X} {U : Set (X × X)
     rw [← ball_empt]
     rw [z_x] at hz
     exact mem_inter y_F hz
-  apply not_lt_of_le (coverMincard_le_card smaller_cover)
+  apply smaller_cover.coverMincard_le_card.not_lt
   rw [← h']
   exact_mod_cast Finset.card_erase_lt_of_mem x_s
 
@@ -391,83 +391,82 @@ lemma log_coverMincard_le_add {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
   have h_nm : (0 : EReal) ≤ (n / m : ℕ) := Nat.cast_nonneg' (n / m)
   have h_log := log_coverMincard_nonneg T F_nemp U m
   have n_div_n := EReal.div_self (natCast_ne_bot n) (natCast_ne_top n)
-    (ne_of_gt (Nat.cast_pos'.2 n_pos))
+    (Nat.cast_pos'.2 n_pos).ne.symm
   apply le_trans <| div_le_div_right_of_nonneg (Nat.cast_pos'.2 n_pos).le
     (log_monotone (ENat.toENNReal_le.2 (coverMincard_le_pow F_inv U_symm m_pos n)))
   rw [ENat.toENNReal_pow, log_pow, Nat.cast_add, Nat.cast_one, right_distrib_of_nonneg h_nm
     zero_le_one, one_mul, div_right_distrib_of_nonneg (Left.mul_nonneg h_nm h_log) h_log, mul_comm,
     ← EReal.mul_div, div_eq_mul_inv _ (m : EReal)]
   apply add_le_add_right (mul_le_mul_of_nonneg_left _ h_log)
-  apply le_of_le_of_eq <| div_le_div_right_of_nonneg (Nat.cast_pos'.2 n_pos).le (natCast_div_le n m)
+  apply (div_le_div_right_of_nonneg (Nat.cast_pos'.2 n_pos).le (natCast_div_le n m)).trans_eq
   rw [EReal.div_div, mul_comm, ← EReal.div_div, n_div_n, one_div (m : EReal)]
 
 /-! ### Cover entropy of entourages -/
 
 open Filter
 
-/-- The entropy of an entourage `U` (`Ent` stands for "entourage"), defined as the exponential rate
-  of growth of the size of the smallest `(U, n)`-refined cover of `F`. Takes values in the space of
-  extended real numbers `[-∞, +∞]`. This first version uses a `limsup`, and is chosen as the
-  default definition.-/
-noncomputable def coverEntropyEnt (T : X → X) (F : Set X) (U : Set (X × X)) :=
+/-- The entropy of an entourage `U`, defined as the exponential rate of growth of the size
+  of the smallest `(U, n)`-refined cover of `F`. Takes values in the space of extended real numbers
+  `[-∞, +∞]`. This first version uses a `limsup`, and is chosen as the default definition.-/
+noncomputable def coverEntropyEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :=
   atTop.limsup fun n : ℕ ↦ log (coverMincard T F U n) / n
 
-/-- The entropy of an entourage `U` (`Ent` stands for "entourage"), defined as the exponential rate
-  of growth of the size of the smallest `(U, n)`-refined cover of `F`. Takes values in the space of
-  extended real numbers `[-∞, +∞]`. This second version uses a `liminf`, and is chosen as an
-  alternative definition.-/
-noncomputable def coverEntropyInfEnt (T : X → X) (F : Set X) (U : Set (X × X)) :=
+/-- The entropy of an entourage `U`, defined as the exponential rate of growth of the size
+  of the smallest `(U, n)`-refined cover of `F`. Takes values in the space of extended real numbers
+  `[-∞, +∞]`. This second version uses a `liminf`, and is chosen as an alternative definition.-/
+noncomputable def coverEntropyInfEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :=
   atTop.liminf fun n : ℕ ↦ log (coverMincard T F U n) / n
 
-lemma coverEntropyInfEnt_antitone (T : X → X) (F : Set X) :
-    Antitone (fun U : Set (X × X) ↦ coverEntropyInfEnt T F U) :=
+lemma coverEntropyInfEntourage_antitone (T : X → X) (F : Set X) :
+    Antitone (fun U : Set (X × X) ↦ coverEntropyInfEntourage T F U) :=
   fun _ _ U_V ↦ (liminf_le_liminf) <| Eventually.of_forall
     fun n ↦ monotone_div_right_of_nonneg (Nat.cast_nonneg' n)
     <| log_monotone (ENat.toENNReal_mono (coverMincard_antitone T F n U_V))
 
-lemma coverEntropyEnt_antitone (T : X → X) (F : Set X) :
-    Antitone (fun U : Set (X × X) ↦ coverEntropyEnt T F U) :=
+lemma coverEntropyEntourage_antitone (T : X → X) (F : Set X) :
+    Antitone (fun U : Set (X × X) ↦ coverEntropyEntourage T F U) :=
   fun _ _ U_V ↦ (limsup_le_limsup) <| Eventually.of_forall
     fun n ↦ monotone_div_right_of_nonneg (Nat.cast_nonneg' n)
     <| log_monotone (ENat.toENNReal_mono (coverMincard_antitone T F n U_V))
 
-lemma coverEntropyInfEnt_le_coverEntropyEnt (T : X → X) (F : Set X) (U : Set (X × X)) :
-    coverEntropyInfEnt T F U ≤ coverEntropyEnt T F U := liminf_le_limsup
+lemma coverEntropyInfEntourage_le_coverEntropyEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :
+    coverEntropyInfEntourage T F U ≤ coverEntropyEntourage T F U := liminf_le_limsup
 
 @[simp]
-lemma coverEntropyEnt_empty {T : X → X} {U : Set (X × X)} :
-    coverEntropyEnt T ∅ U = ⊥ := by
+lemma coverEntropyEntourage_empty {T : X → X} {U : Set (X × X)} :
+    coverEntropyEntourage T ∅ U = ⊥ := by
   suffices h : ∀ᶠ n : ℕ in atTop, log (coverMincard T ∅ U n) / n = ⊥ by
-    rw [coverEntropyEnt]
+    rw [coverEntropyEntourage]
     exact limsup_congr h ▸ limsup_const ⊥
   · simp only [coverMincard_empty, ENat.toENNReal_zero, log_zero, eventually_atTop]
     exact ⟨1, fun n n_pos ↦ bot_div_of_pos_ne_top (Nat.cast_pos'.2 n_pos) (natCast_ne_top n)⟩
 
 @[simp]
-lemma coverEntropyInfEnt_empty {T : X → X} {U : Set (X × X)} :
-    coverEntropyInfEnt T ∅ U = ⊥ :=
-  eq_bot_mono (coverEntropyInfEnt_le_coverEntropyEnt T ∅ U) coverEntropyEnt_empty
+lemma coverEntropyInfEntourage_empty {T : X → X} {U : Set (X × X)} :
+    coverEntropyInfEntourage T ∅ U = ⊥ :=
+  eq_bot_mono (coverEntropyInfEntourage_le_coverEntropyEntourage T ∅ U) coverEntropyEntourage_empty
 
-lemma coverEntropyInfEnt_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
-    0 ≤ coverEntropyInfEnt T F U :=
+lemma coverEntropyInfEntourage_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
+    0 ≤ coverEntropyInfEntourage T F U :=
   (le_iInf fun n ↦ div_nonneg (log_coverMincard_nonneg T h U n) (Nat.cast_nonneg' n)).trans
     iInf_le_liminf
 
-lemma coverEntropyEnt_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
-    0 ≤ coverEntropyEnt T F U :=
-  (coverEntropyInfEnt_nonneg T h U).trans (coverEntropyInfEnt_le_coverEntropyEnt T F U)
+lemma coverEntropyEntourage_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
+    0 ≤ coverEntropyEntourage T F U :=
+  (coverEntropyInfEntourage_nonneg T h U).trans
+    (coverEntropyInfEntourage_le_coverEntropyEntourage T F U)
 
-lemma coverEntropyEnt_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
-    coverEntropyEnt T F univ = 0 := by
-  simp [coverEntropyEnt, coverMincard_univ T h]
+lemma coverEntropyEntourage_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
+    coverEntropyEntourage T F univ = 0 := by
+  simp [coverEntropyEntourage, coverMincard_univ T h]
 
-lemma coverEntropyInfEnt_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
-    coverEntropyInfEnt T F univ = 0 := by
-  simp [coverEntropyInfEnt, coverMincard_univ T h]
+lemma coverEntropyInfEntourage_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
+    coverEntropyInfEntourage T F univ = 0 := by
+  simp [coverEntropyInfEntourage, coverMincard_univ T h]
 
-lemma coverEntropyEnt_le_log_coverMincard_div {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
+lemma coverEntropyEntourage_le_log_coverMincard_div {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
     {U : Set (X × X)} (U_symm : SymmetricRel U) {n : ℕ} (n_pos : 0 < n) :
-    coverEntropyEnt T F (U ○ U) ≤ log (coverMincard T F U n) / n := by
+    coverEntropyEntourage T F (U ○ U) ≤ log (coverMincard T F U n) / n := by
   -- Deal with the edge cases: `F = ∅` or `F` has no finite cover.
   rcases eq_or_ne (log (coverMincard T F U n)) ⊥ with logm_bot | logm_nneg
   · rw [log_eq_bot_iff, ← ENat.toENNReal_zero, ENat.toENNReal_coe_eq_iff,
@@ -487,46 +486,48 @@ lemma coverEntropyEnt_le_log_coverMincard_div {T : X → X} {F : Set X} (F_inv :
     have := @limsup_add_le_add_limsup ℕ atTop u v
     rw [h, add_zero] at this
     specialize this (Or.inr EReal.zero_ne_top) (Or.inr EReal.zero_ne_bot)
-    exact le_of_le_of_eq this (limsup_const (log (coverMincard T F U n) / n))
+    exact this.trans_eq (limsup_const (log (coverMincard T F U n) / n))
   exact Tendsto.limsup_eq (EReal.tendsto_const_div_atTop_nhds_zero_nat logm_nneg logm_fin)
 
-lemma IsDynCoverOf.coverEntropyEnt_le_log_card_div {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
-    {U : Set (X × X)} (U_symm : SymmetricRel U) {n : ℕ} (n_pos : 0 < n) {s : Finset X}
-    (h : IsDynCoverOf T F U n s) :
-    coverEntropyEnt T F (U ○ U) ≤ log s.card / n := by
-  apply (coverEntropyEnt_le_log_coverMincard_div F_inv U_symm n_pos).trans
+lemma IsDynCoverOf.coverEntropyEntourage_le_log_card_div {T : X → X} {F : Set X}
+    (F_inv : MapsTo T F F) {U : Set (X × X)} (U_symm : SymmetricRel U) {n : ℕ} (n_pos : 0 < n)
+    {s : Finset X} (h : IsDynCoverOf T F U n s) :
+    coverEntropyEntourage T F (U ○ U) ≤ log s.card / n := by
+  apply (coverEntropyEntourage_le_log_coverMincard_div F_inv U_symm n_pos).trans
   apply monotone_div_right_of_nonneg (Nat.cast_nonneg' n) (log_monotone _)
   exact_mod_cast coverMincard_le_card h
 
-lemma coverEntropyEnt_le_coverEntropyInfEnt {T : X → X} {F : Set X} (F_inv : MapsTo T F F)
-    {U : Set (X × X)} (U_symm : SymmetricRel U) :
-    coverEntropyEnt T F (U ○ U) ≤ coverEntropyInfEnt T F U :=
+lemma coverEntropyEntourage_le_coverEntropyInfEntourage {T : X → X} {F : Set X}
+    (F_inv : MapsTo T F F) {U : Set (X × X)} (U_symm : SymmetricRel U) :
+    coverEntropyEntourage T F (U ○ U) ≤ coverEntropyInfEntourage T F U :=
   (le_liminf_of_le) (eventually_atTop.2
-    ⟨1, fun m m_pos ↦ coverEntropyEnt_le_log_coverMincard_div F_inv U_symm m_pos⟩)
+    ⟨1, fun m m_pos ↦ coverEntropyEntourage_le_log_coverMincard_div F_inv U_symm m_pos⟩)
 
-lemma coverEntropyEnt_finite_of_isCompact_invariant [UniformSpace X] {T : X → X} {F : Set X}
+lemma coverEntropyEntourage_finite_of_isCompact_invariant [UniformSpace X] {T : X → X} {F : Set X}
     (F_comp : IsCompact F) (F_inv : MapsTo T F F) {U : Set (X × X)} (U_uni : U ∈ 𝓤 X) :
-    coverEntropyEnt T F U < ⊤ := by
+    coverEntropyEntourage T F U < ⊤ := by
   rcases comp_symm_mem_uniformity_sets U_uni with ⟨V, V_uni, V_symm, V_U⟩
   rcases exists_isDynCoverOf_of_isCompact_invariant F_comp F_inv V_uni 1 with ⟨s, s_cover⟩
-  apply (coverEntropyEnt_antitone T F V_U).trans_lt
-  apply (s_cover.coverEntropyEnt_le_log_card_div F_inv V_symm zero_lt_one).trans_lt
+  apply (coverEntropyEntourage_antitone T F V_U).trans_lt
+  apply (s_cover.coverEntropyEntourage_le_log_card_div F_inv V_symm zero_lt_one).trans_lt
   rw [Nat.cast_one, div_one, log_lt_top_iff, ← ENat.toENNReal_top]
   exact_mod_cast Ne.lt_top (ENat.coe_ne_top (Finset.card s))
 
 /-! ### Cover entropy -/
 
-/-- The entropy of `T` restricted to `F`, obtained by taking the supremum over entourages.
-  Note that this supremum is approached by taking small entourages. This first version uses a
-  `limsup`, and is chosen as the default definition for topological entropy.-/
+/-- The entropy of `T` restricted to `F`, obtained by taking the supremum
+  of `coverEntropyEntourage` over entourages. Note that this supremum is approached by taking small
+  entourages. This first version uses a `limsup`, and is chosen as the default definition
+  for topological entropy.-/
 noncomputable def coverEntropy [UniformSpace X] (T : X → X) (F : Set X) :=
-  ⨆ U ∈ 𝓤 X, coverEntropyEnt T F U
+  ⨆ U ∈ 𝓤 X, coverEntropyEntourage T F U
 
-/-- The entropy of `T` restricted to `F`, obtained by taking the supremum over entourages.
-  Note that this supremum is approached by taking small entourages. This second version uses a
-  `liminf`, and is chosen as an alternative definition for topological entropy.-/
+/-- The entropy of `T` restricted to `F`, obtained by taking the supremum
+  of `coverEntropyInfEntourage` over entourages. Note that this supremum is approached by taking
+  small entourages. This second version uses a `liminf`, and is chosen as an alternative
+  definition for topological entropy.-/
 noncomputable def coverEntropyInf [UniformSpace X] (T : X → X) (F : Set X) :=
-  ⨆ U ∈ 𝓤 X, coverEntropyInfEnt T F U
+  ⨆ U ∈ 𝓤 X, coverEntropyInfEntourage T F U
 
 lemma coverEntropyInf_antitone (T : X → X) (F : Set X) :
     Antitone fun (u : UniformSpace X) ↦ @coverEntropyInf X u T F :=
@@ -538,49 +539,51 @@ lemma coverEntropy_antitone (T : X → X) (F : Set X) :
 
 variable [UniformSpace X]
 
-lemma coverEntropyEnt_le_coverEntropy (T : X → X) (F : Set X) {U : Set (X × X)} (h : U ∈ 𝓤 X) :
-    coverEntropyEnt T F U ≤ coverEntropy T F :=
-  le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ coverEntropyEnt T F U) U h
+lemma coverEntropyEntourage_le_coverEntropy (T : X → X) (F : Set X) {U : Set (X × X)}
+    (h : U ∈ 𝓤 X) :
+    coverEntropyEntourage T F U ≤ coverEntropy T F :=
+  le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ coverEntropyEntourage T F U) U h
 
-lemma coverEntropyInfEnt_le_coverEntropyInf (T : X → X) (F : Set X) {U : Set (X × X)}
+lemma coverEntropyInfEntourage_le_coverEntropyInf (T : X → X) (F : Set X) {U : Set (X × X)}
     (h : U ∈ 𝓤 X) :
-    coverEntropyInfEnt T F U ≤ coverEntropyInf T F :=
-  le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ coverEntropyInfEnt T F U) U h
+    coverEntropyInfEntourage T F U ≤ coverEntropyInf T F :=
+  le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ coverEntropyInfEntourage T F U) U h
 
 lemma coverEntropy_eq_iSup_basis {ι : Sort*} {p : ι → Prop} {s : ι → Set (X × X)}
     (h : (𝓤 X).HasBasis p s) (T : X → X) (F : Set X) :
-    coverEntropy T F = ⨆ (i : ι) (_ : p i), coverEntropyEnt T F (s i) := by
+    coverEntropy T F = ⨆ (i : ι) (_ : p i), coverEntropyEntourage T F (s i) := by
   refine (iSup₂_le fun U U_uni ↦ ?_).antisymm
     (iSup₂_mono' fun i h_i ↦ ⟨s i, HasBasis.mem_of_mem h h_i, le_refl _⟩)
   rcases (HasBasis.mem_iff h).1 U_uni with ⟨i, h_i, si_U⟩
-  exact (coverEntropyEnt_antitone T F si_U).trans
-    (le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ coverEntropyEnt T F (s i)) i h_i)
+  exact (coverEntropyEntourage_antitone T F si_U).trans
+    (le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ coverEntropyEntourage T F (s i)) i h_i)
 
 lemma coverEntropyInf_eq_iSup_basis {ι : Sort*} {p : ι → Prop} {s : ι → Set (X × X)}
     (h : (𝓤 X).HasBasis p s) (T : X → X) (F : Set X) :
-    coverEntropyInf T F = ⨆ (i : ι) (_ : p i), coverEntropyInfEnt T F (s i) := by
+    coverEntropyInf T F = ⨆ (i : ι) (_ : p i), coverEntropyInfEntourage T F (s i) := by
   refine (iSup₂_le fun U U_uni ↦ ?_).antisymm
     (iSup₂_mono' fun i h_i ↦ ⟨s i, HasBasis.mem_of_mem h h_i, le_refl _⟩)
   rcases (HasBasis.mem_iff h).1 U_uni with ⟨i, h_i, si_U⟩
-  exact (coverEntropyInfEnt_antitone T F si_U).trans
-    (le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ coverEntropyInfEnt T F (s i)) i h_i)
+  exact (coverEntropyInfEntourage_antitone T F si_U).trans
+    (le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ coverEntropyInfEntourage T F (s i)) i h_i)
 
 lemma coverEntropyInf_le_coverEntropy (T : X → X) (F : Set X) :
     coverEntropyInf T F ≤ coverEntropy T F :=
-  iSup₂_mono fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ coverEntropyInfEnt_le_coverEntropyEnt T F U
+  iSup₂_mono fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦
+    coverEntropyInfEntourage_le_coverEntropyEntourage T F U
 
 @[simp]
 lemma coverEntropy_empty {T : X → X} : coverEntropy T ∅ = ⊥ := by
-  simp only [coverEntropy, coverEntropyEnt_empty, iSup_bot]
+  simp only [coverEntropy, coverEntropyEntourage_empty, iSup_bot]
 
 @[simp]
 lemma coverEntropyInf_empty {T : X → X} : coverEntropyInf T ∅ = ⊥ := by
-  simp only [coverEntropyInf, coverEntropyInfEnt_empty, iSup_bot]
+  simp only [coverEntropyInf, coverEntropyInfEntourage_empty, iSup_bot]
 
 lemma coverEntropyInf_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) :
     0 ≤ coverEntropyInf T F :=
-  le_of_eq_of_le (coverEntropyInfEnt_univ T h).symm
-    (coverEntropyInfEnt_le_coverEntropyInf T F univ_mem)
+  (coverEntropyInfEntourage_le_coverEntropyInf T F univ_mem).trans_eq'
+    (coverEntropyInfEntourage_univ T h).symm
 
 lemma coverEntropy_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) :
     0 ≤ coverEntropy T F :=
@@ -590,7 +593,7 @@ lemma coverEntropyInf_eq_coverEntropy (T : X → X) {F : Set X} (h : MapsTo T F
     coverEntropyInf T F = coverEntropy T F := by
   refine le_antisymm (coverEntropyInf_le_coverEntropy T F) (iSup₂_le fun U U_uni ↦ ?_)
   rcases comp_symm_mem_uniformity_sets U_uni with ⟨V, V_uni, V_symm, V_U⟩
-  exact (coverEntropyEnt_antitone T F V_U).trans
-    (le_iSup₂_of_le V V_uni (coverEntropyEnt_le_coverEntropyInfEnt h V_symm))
+  exact (coverEntropyEntourage_antitone T F V_U).trans
+    (le_iSup₂_of_le V V_uni (coverEntropyEntourage_le_coverEntropyInfEntourage h V_symm))
 
 end Dynamics
diff --git a/Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.lean b/Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.lean
index ad6c52b6be339..08b6c75d0d4c0 100644
--- a/Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.lean
+++ b/Mathlib/Dynamics/TopologicalEntropy/DynamicalEntourage.lean
@@ -88,7 +88,7 @@ lemma _root_.isOpen.dynEntourage [TopologicalSpace X] {T : X → X} (T_cont : Co
     IsOpen (dynEntourage T U n) := by
   rw [dynEntourage_eq_inter_Ico T U n]
   refine isOpen_iInter_of_finite fun k ↦ ?_
-  exact continuous_def.1 ((T_cont.prod_map T_cont).iterate k) U U_open
+  exact U_open.preimage ((T_cont.prodMap T_cont).iterate k)
 
 lemma dynEntourage_monotone (T : X → X) (n : ℕ) :
     Monotone (fun U : Set (X × X) ↦ dynEntourage T U n) :=
diff --git a/Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean b/Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean
new file mode 100644
index 0000000000000..7720c694f1ce9
--- /dev/null
+++ b/Mathlib/Dynamics/TopologicalEntropy/NetEntropy.lean
@@ -0,0 +1,408 @@
+/-
+Copyright (c) 2024 Damien Thomine. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Damien Thomine, Pietro Monticone
+-/
+import Mathlib.Dynamics.TopologicalEntropy.CoverEntropy
+
+/-!
+# Topological entropy via nets
+We implement Bowen-Dinaburg's definitions of the topological entropy, via nets.
+
+The major design decisions are the same as in `Mathlib.Dynamics.TopologicalEntropy.CoverEntropy`,
+and are explained in detail there: use of uniform spaces, definition of the topological entropy of
+a subset, and values taken in `EReal`.
+
+Given a map `T : X → X` and a subset `F ⊆ X`, the topological entropy is loosely defined using
+nets as the exponential growth (in `n`) of the number of distinguishable orbits of length `n`
+starting from `F`. More precisely, given an entourage `U`, two orbits of length `n` can be
+distinguished if there exists some index `k < n` such that `T^[k] x` and `T^[k] y` are far enough
+(i.e. `(T^[k] x, T^[k] y)` is not in `U`). The maximal number of distinguishable orbits of
+length `n` is `netMaxcard T F U n`, and its exponential growth `netEntropyEntourage T F U`. This
+quantity increases when `U` decreases, and a definition of the topological entropy is
+`⨆ U ∈ 𝓤 X, netEntropyInfEntourage T F U`.
+
+The definition of topological entropy using nets coincides with the definition using covers.
+Instead of defining a new notion of topological entropy, we prove that
+`coverEntropy` coincides with `⨆ U ∈ 𝓤 X, netEntropyEntourage T F U`.
+
+## Main definitions
+- `IsDynNetIn`: property that dynamical balls centered on a subset `s` of `F` are disjoint.
+- `netMaxcard`: maximal cardinality of a dynamical net. Takes values in `ℕ∞`.
+- `netEntropyInfEntourage`/`netEntropyEntourage`: exponential growth of `netMaxcard`. The former is
+defined with a `liminf`, the latter with a `limsup`. Take values in `EReal`.
+
+## Implementation notes
+As when using covers, there are two competing definitions `netEntropyInfEntourage` and
+`netEntropyEntourage` in this file: one uses a `liminf`, the other a `limsup`. When using covers,
+we chose the `limsup` definition as the default.
+
+## Main results
+- `coverEntropy_eq_iSup_netEntropyEntourage`: equality between the notions of topological entropy
+defined with covers and with nets. Has a variant for `coverEntropyInf`.
+
+## Tags
+net, entropy
+
+## TODO
+Get versions of the topological entropy on (pseudo-e)metric spaces.
+-/
+
+namespace Dynamics
+
+open Set Uniformity UniformSpace
+
+variable {X : Type*}
+
+/-! ### Dynamical nets -/
+
+/-- Given a subset `F`, an entourage `U` and an integer `n`, a subset `s` of `F` is a
+`(U, n)`-dynamical net of `F` if no two orbits of length `n` of points in `s` shadow each other.-/
+def IsDynNetIn (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) (s : Set X) : Prop :=
+  s ⊆ F ∧ s.PairwiseDisjoint (fun x : X ↦ ball x (dynEntourage T U n))
+
+lemma IsDynNetIn.of_le {T : X → X} {F : Set X} {U : Set (X × X)} {m n : ℕ} (m_n : m ≤ n) {s : Set X}
+    (h : IsDynNetIn T F U m s) :
+    IsDynNetIn T F U n s :=
+  ⟨h.1, PairwiseDisjoint.mono h.2 (fun x ↦ ball_mono (dynEntourage_antitone T U m_n) x)⟩
+
+lemma IsDynNetIn.of_entourage_subset {T : X → X} {F : Set X} {U V : Set (X × X)} (U_V : U ⊆ V)
+    {n : ℕ} {s : Set X} (h : IsDynNetIn T F V n s) :
+    IsDynNetIn T F U n s :=
+  ⟨h.1, PairwiseDisjoint.mono h.2 (fun x ↦ ball_mono (dynEntourage_monotone T n U_V) x)⟩
+
+lemma isDynNetIn_empty {T : X → X} {F : Set X} {U : Set (X × X)} {n : ℕ} :
+    IsDynNetIn T F U n ∅ :=
+  ⟨empty_subset F, pairwise_empty _⟩
+
+lemma isDynNetIn_singleton (T : X → X) {F : Set X} (U : Set (X × X)) (n : ℕ) {x : X} (h : x ∈ F) :
+    IsDynNetIn T F U n {x} :=
+  ⟨singleton_subset_iff.2 h, pairwise_singleton x _⟩
+
+/-- Given an entourage `U` and a time `n`, a dynamical net has a smaller cardinality than
+  a dynamical cover. This lemma is the first of two key results to compare two versions of
+  topological entropy: with cover and with nets, the second being `coverMincard_le_netMaxcard`.-/
+lemma IsDynNetIn.card_le_card_of_isDynCoverOf {T : X → X} {F : Set X} {U : Set (X × X)}
+    (U_symm : SymmetricRel U) {n : ℕ} {s t : Finset X} (hs : IsDynNetIn T F U n s)
+    (ht : IsDynCoverOf T F U n t) :
+    s.card ≤ t.card := by
+  have (x : X) (x_s : x ∈ s) : ∃ z ∈ t, x ∈ ball z (dynEntourage T U n) := by
+    specialize ht (hs.1 x_s)
+    simp only [Finset.coe_sort_coe, mem_iUnion, Subtype.exists, exists_prop] at ht
+    exact ht
+  choose! F s_t using this
+  simp only [mem_ball_symmetry (U_symm.dynEntourage T n)] at s_t
+  apply Finset.card_le_card_of_injOn F (fun x x_s ↦ (s_t x x_s).1)
+  exact fun x x_s y y_s Fx_Fy ↦
+    PairwiseDisjoint.elim_set hs.2 x_s y_s (F x) (s_t x x_s).2 (Fx_Fy ▸ (s_t y y_s).2)
+
+/-! ### Maximal cardinality of dynamical nets -/
+
+/-- The largest cardinality of a `(U, n)`-dynamical net of `F`. Takes values in `ℕ∞`, and is
+infinite if and only if `F` admits nets of arbitrarily large size.-/
+noncomputable def netMaxcard (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) : ℕ∞ :=
+  ⨆ (s : Finset X) (_ : IsDynNetIn T F U n s), (s.card : ℕ∞)
+
+lemma IsDynNetIn.card_le_netMaxcard {T : X → X} {F : Set X} {U : Set (X × X)} {n : ℕ} {s : Finset X}
+    (h : IsDynNetIn T F U n s) :
+    s.card ≤ netMaxcard T F U n :=
+  le_iSup₂ (α := ℕ∞) s h
+
+lemma netMaxcard_monotone_time (T : X → X) (F : Set X) (U : Set (X × X)) :
+    Monotone (fun n : ℕ ↦ netMaxcard T F U n) :=
+  fun _ _ m_n ↦ biSup_mono (fun _ h ↦ h.of_le m_n)
+
+lemma netMaxcard_antitone (T : X → X) (F : Set X) (n : ℕ) :
+    Antitone (fun U : Set (X × X) ↦ netMaxcard T F U n) :=
+  fun _ _ U_V ↦ biSup_mono (fun _ h ↦ h.of_entourage_subset U_V)
+
+lemma netMaxcard_finite_iff (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) :
+    netMaxcard T F U n < ⊤ ↔
+    ∃ s : Finset X, IsDynNetIn T F U n s ∧ (s.card : ℕ∞) = netMaxcard T F U n := by
+  apply Iff.intro <;> intro h
+  · rcases WithTop.ne_top_iff_exists.1 h.ne with ⟨k, k_max⟩
+    rw [← k_max]
+    simp only [ENat.some_eq_coe, Nat.cast_inj]
+    -- The criterion we want to use is `Nat.sSup_mem`. We rewrite `netMaxcard` with an `sSup`,
+    -- then check its `BddAbove` and `Nonempty` hypotheses.
+    have : netMaxcard T F U n
+      = sSup (WithTop.some '' (Finset.card '' {s : Finset X | IsDynNetIn T F U n s})) := by
+      rw [netMaxcard, ← image_comp, sSup_image]
+      simp only [mem_setOf_eq, ENat.some_eq_coe, Function.comp_apply]
+    rw [this] at k_max
+    have h_bdda : BddAbove (Finset.card '' {s : Finset X | IsDynNetIn T F U n s}) := by
+      refine ⟨k, mem_upperBounds.2 ?_⟩
+      simp only [mem_image, mem_setOf_eq, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+      intro s h
+      rw [← WithTop.coe_le_coe, k_max]
+      apply le_sSup
+      simp only [ENat.some_eq_coe, mem_image, mem_setOf_eq, Nat.cast_inj, exists_eq_right]
+      exact Filter.frequently_principal.mp fun a ↦ a h rfl
+    have h_nemp : (Finset.card '' {s : Finset X | IsDynNetIn T F U n s}).Nonempty := by
+      refine ⟨0, ?_⟩
+      simp only [mem_image, mem_setOf_eq, Finset.card_eq_zero, exists_eq_right, Finset.coe_empty]
+      exact isDynNetIn_empty
+    rw [← WithTop.coe_sSup' h_bdda, ENat.some_eq_coe, Nat.cast_inj] at k_max
+    have key := Nat.sSup_mem h_nemp h_bdda
+    rw [← k_max, mem_image] at key
+    simp only [mem_setOf_eq] at key
+    exact key
+  · rcases h with ⟨s, _, s_netMaxcard⟩
+    rw [← s_netMaxcard]
+    exact WithTop.coe_lt_top s.card
+
+@[simp]
+lemma netMaxcard_empty {T : X → X} {U : Set (X × X)} {n : ℕ} : netMaxcard T ∅ U n = 0 := by
+  rw [netMaxcard, ← bot_eq_zero, iSup₂_eq_bot]
+  intro s s_net
+  replace s_net := subset_empty_iff.1 s_net.1
+  norm_cast at s_net
+  rw [s_net, Finset.card_empty, CharP.cast_eq_zero, bot_eq_zero']
+
+lemma netMaxcard_eq_zero_iff (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) :
+    netMaxcard T F U n = 0 ↔ F = ∅ := by
+  refine Iff.intro (fun h ↦ ?_) (fun h ↦ by rw [h, netMaxcard_empty])
+  rw [eq_empty_iff_forall_not_mem]
+  intro x x_F
+  have key := isDynNetIn_singleton T U n x_F
+  rw [← Finset.coe_singleton] at key
+  replace key := key.card_le_netMaxcard
+  rw [Finset.card_singleton, Nat.cast_one, h] at key
+  exact key.not_lt zero_lt_one
+
+lemma one_le_netMaxcard_iff (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) :
+    1 ≤ netMaxcard T F U n ↔ F.Nonempty := by
+  rw [ENat.one_le_iff_ne_zero, nonempty_iff_ne_empty]
+  exact not_iff_not.2 (netMaxcard_eq_zero_iff T F U n)
+
+lemma netMaxcard_zero (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
+    netMaxcard T F U 0 = 1 := by
+  apply (iSup₂_le _).antisymm ((one_le_netMaxcard_iff T F U 0).2 h)
+  intro s ⟨_, s_net⟩
+  simp only [ball, dynEntourage_zero, preimage_univ] at s_net
+  norm_cast
+  refine Finset.card_le_one.2 (fun x x_s y y_s ↦ ?_)
+  exact PairwiseDisjoint.elim_set s_net x_s y_s x (mem_univ x) (mem_univ x)
+
+lemma netMaxcard_univ (T : X → X) {F : Set X} (h : F.Nonempty) (n : ℕ) :
+    netMaxcard T F univ n = 1 := by
+  apply (iSup₂_le _).antisymm ((one_le_netMaxcard_iff T F univ n).2 h)
+  intro s ⟨_, s_net⟩
+  simp only [ball, dynEntourage_univ, preimage_univ] at s_net
+  norm_cast
+  refine Finset.card_le_one.2 (fun x x_s y y_s ↦ ?_)
+  exact PairwiseDisjoint.elim_set s_net x_s y_s x (mem_univ x) (mem_univ x)
+
+lemma netMaxcard_infinite_iff (T : X → X) (F : Set X) (U : Set (X × X)) (n : ℕ) :
+    netMaxcard T F U n = ⊤ ↔ ∀ k : ℕ, ∃ s : Finset X, IsDynNetIn T F U n s ∧ k ≤ s.card := by
+  apply Iff.intro <;> intro h
+  · intro k
+    rw [netMaxcard, iSup_subtype', iSup_eq_top] at h
+    specialize h k (ENat.coe_lt_top k)
+    simp only [Nat.cast_lt, Subtype.exists, exists_prop] at h
+    rcases h with ⟨s, s_net, s_k⟩
+    exact ⟨s, ⟨s_net, s_k.le⟩⟩
+  · refine WithTop.forall_gt_iff_eq_top.1 fun k ↦ ?_
+    specialize h (k + 1)
+    rcases h with ⟨s, s_net, s_card⟩
+    apply s_net.card_le_netMaxcard.trans_lt'
+    rw [ENat.some_eq_coe, Nat.cast_lt]
+    exact (lt_add_one k).trans_le s_card
+
+lemma netMaxcard_le_coverMincard (T : X → X) (F : Set X) {U : Set (X × X)} (U_symm : SymmetricRel U)
+    (n : ℕ) :
+    netMaxcard T F U n ≤ coverMincard T F U n := by
+  rcases eq_top_or_lt_top (coverMincard T F U n) with h | h
+  · exact h ▸ le_top
+  · rcases ((coverMincard_finite_iff T F U n).1 h) with ⟨t, t_cover, t_mincard⟩
+    rw [← t_mincard]
+    exact iSup₂_le (fun s s_net ↦ Nat.cast_le.2 (s_net.card_le_card_of_isDynCoverOf U_symm t_cover))
+
+/-- Given an entourage `U` and a time `n`, a minimal dynamical cover by `U ○ U` has a smaller
+  cardinality than a maximal dynamical net by `U`. This lemma is the second of two key results to
+  compare two versions topological entropy: with cover and with nets.-/
+lemma coverMincard_le_netMaxcard (T : X → X) (F : Set X) {U : Set (X × X)} (U_rfl : idRel ⊆ U)
+    (U_symm : SymmetricRel U) (n : ℕ) :
+    coverMincard T F (U ○ U) n ≤ netMaxcard T F U n := by
+  classical
+  -- WLOG, there exists a maximal dynamical net `s`.
+  rcases (eq_top_or_lt_top (netMaxcard T F U n)) with h | h
+  · exact h ▸ le_top
+  rcases ((netMaxcard_finite_iff T F U n).1 h) with ⟨s, s_net, s_netMaxcard⟩
+  rw [← s_netMaxcard]
+  apply IsDynCoverOf.coverMincard_le_card
+  --  We have to check that `s` is a cover for `dynEntourage T F (U ○ U) n`.
+  -- If `s` is not a cover, then we can add to `s` a point `x` which is not covered
+  -- and get a new net. This contradicts the maximality of `s`.
+  by_contra h
+  rcases not_subset.1 h with ⟨x, x_F, x_uncov⟩
+  simp only [Finset.mem_coe, mem_iUnion, exists_prop, not_exists, not_and] at x_uncov
+  have larger_net : IsDynNetIn T F U n (insert x s) :=
+    And.intro (insert_subset x_F s_net.1) (pairwiseDisjoint_insert.2 (And.intro s_net.2
+      (fun y y_s _ ↦ (disjoint_left.2 (fun z z_x z_y ↦ x_uncov y y_s
+        (mem_ball_dynEntourage_comp T n U_symm x y (nonempty_of_mem ⟨z_x, z_y⟩)))))))
+  rw [← Finset.coe_insert x s] at larger_net
+  apply larger_net.card_le_netMaxcard.not_lt
+  rw [← s_netMaxcard, Nat.cast_lt]
+  refine (lt_add_one s.card).trans_eq (Finset.card_insert_of_not_mem fun x_s ↦ ?_).symm
+  apply x_uncov x x_s (ball_mono (dynEntourage_monotone T n (subset_comp_self U_rfl)) x
+    (ball_mono (idRel_subset_dynEntourage T U_rfl n) x _))
+  simp only [ball, mem_preimage, mem_idRel]
+
+open ENNReal EReal
+
+lemma log_netMaxcard_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) (n : ℕ) :
+    0 ≤ log (netMaxcard T F U n) := by
+  apply zero_le_log_iff.2
+  rw [← ENat.toENNReal_one, ENat.toENNReal_le]
+  exact (one_le_netMaxcard_iff T F U n).2 h
+
+/-! ### Net entropy of entourages -/
+
+open Filter
+
+/-- The entropy of an entourage `U`, defined as the exponential rate of growth of the size of the
+largest `(U, n)`-dynamical net of `F`. Takes values in the space of extended real numbers
+`[-∞,+∞]`. This version uses a `limsup`, and is chosen as the default definition.-/
+noncomputable def netEntropyEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :=
+  atTop.limsup fun n : ℕ ↦ log (netMaxcard T F U n) / n
+
+/-- The entropy of an entourage `U`, defined as the exponential rate of growth of the size of the
+largest `(U, n)`-dynamical net of `F`. Takes values in the space of extended real numbers
+`[-∞,+∞]`. This version uses a `liminf`, and is an alternative definition.-/
+noncomputable def netEntropyInfEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :=
+  atTop.liminf fun n : ℕ ↦ log (netMaxcard T F U n) / n
+
+lemma netEntropyInfEntourage_antitone (T : X → X) (F : Set X) :
+    Antitone (fun U : Set (X × X) ↦ netEntropyInfEntourage T F U) :=
+  fun _ _ U_V ↦ (liminf_le_liminf) (Eventually.of_forall
+    fun n ↦ monotone_div_right_of_nonneg (Nat.cast_nonneg' n)
+      (log_monotone (ENat.toENNReal_mono (netMaxcard_antitone T F n U_V))))
+
+lemma netEntropyEntourage_antitone (T : X → X) (F : Set X) :
+    Antitone (fun U : Set (X × X) ↦ netEntropyEntourage T F U) :=
+  fun _ _ U_V ↦ (limsup_le_limsup) (Eventually.of_forall
+    fun n ↦ (monotone_div_right_of_nonneg (Nat.cast_nonneg' n)
+      (log_monotone (ENat.toENNReal_mono (netMaxcard_antitone T F n U_V)))))
+
+lemma netEntropyInfEntourage_le_netEntropyEntourage (T : X → X) (F : Set X) (U : Set (X × X)) :
+    netEntropyInfEntourage T F U ≤ netEntropyEntourage T F U := liminf_le_limsup
+
+@[simp]
+lemma netEntropyEntourage_empty {T : X → X} {U : Set (X × X)} : netEntropyEntourage T ∅ U = ⊥ := by
+  suffices h : ∀ᶠ n : ℕ in atTop, log (netMaxcard T ∅ U n) / n = ⊥ by
+    rw [netEntropyEntourage, limsup_congr h]
+    exact limsup_const ⊥
+  simp only [netMaxcard_empty, ENat.toENNReal_zero, log_zero, eventually_atTop]
+  exact ⟨1, fun n n_pos ↦ bot_div_of_pos_ne_top (Nat.cast_pos'.2 n_pos) (natCast_ne_top n)⟩
+
+@[simp]
+lemma netEntropyInfEntourage_empty {T : X → X} {U : Set (X × X)} :
+    netEntropyInfEntourage T ∅ U = ⊥ :=
+  eq_bot_mono (netEntropyInfEntourage_le_netEntropyEntourage T ∅ U) netEntropyEntourage_empty
+
+lemma netEntropyInfEntourage_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
+    0 ≤ netEntropyInfEntourage T F U :=
+  (le_iInf fun n ↦ div_nonneg (log_netMaxcard_nonneg T h U n) (Nat.cast_nonneg' n)).trans
+    iInf_le_liminf
+
+lemma netEntropyEntourage_nonneg (T : X → X) {F : Set X} (h : F.Nonempty) (U : Set (X × X)) :
+    0 ≤ netEntropyEntourage T F U :=
+  (netEntropyInfEntourage_nonneg T h U).trans (netEntropyInfEntourage_le_netEntropyEntourage T F U)
+
+lemma netEntropyInfEntourage_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
+    netEntropyInfEntourage T F univ = 0 := by simp [netEntropyInfEntourage, netMaxcard_univ T h]
+
+lemma netEntropyEntourage_univ (T : X → X) {F : Set X} (h : F.Nonempty) :
+    netEntropyEntourage T F univ = 0 := by simp [netEntropyEntourage, netMaxcard_univ T h]
+
+lemma netEntropyInfEntourage_le_coverEntropyInfEntourage (T : X → X) (F : Set X) {U : Set (X × X)}
+    (U_symm : SymmetricRel U) :
+    netEntropyInfEntourage T F U ≤ coverEntropyInfEntourage T F U :=
+  (liminf_le_liminf) (Eventually.of_forall fun n ↦ (div_le_div_right_of_nonneg (Nat.cast_nonneg' n)
+    (log_monotone (ENat.toENNReal_le.2 (netMaxcard_le_coverMincard T F U_symm n)))))
+
+lemma coverEntropyInfEntourage_le_netEntropyInfEntourage (T : X → X) (F : Set X) {U : Set (X × X)}
+    (U_rfl : idRel ⊆ U) (U_symm : SymmetricRel U) :
+    coverEntropyInfEntourage T F (U ○ U) ≤ netEntropyInfEntourage T F U := by
+  refine (liminf_le_liminf) (Eventually.of_forall fun n ↦ ?_)
+  apply div_le_div_right_of_nonneg (Nat.cast_nonneg' n) (log_monotone _)
+  exact ENat.toENNReal_le.2 (coverMincard_le_netMaxcard T F U_rfl U_symm n)
+
+lemma netEntropyEntourage_le_coverEntropyEntourage (T : X → X) (F : Set X) {U : Set (X × X)}
+    (U_symm : SymmetricRel U) :
+    netEntropyEntourage T F U ≤ coverEntropyEntourage T F U := by
+  refine (limsup_le_limsup) (Eventually.of_forall fun n ↦ ?_)
+  apply div_le_div_right_of_nonneg (Nat.cast_nonneg' n) (log_monotone _)
+  exact ENat.toENNReal_le.2 (netMaxcard_le_coverMincard T F U_symm n)
+
+lemma coverEntropyEntourage_le_netEntropyEntourage (T : X → X) (F : Set X) {U : Set (X × X)}
+    (U_rfl : idRel ⊆ U) (U_symm : SymmetricRel U) :
+    coverEntropyEntourage T F (U ○ U) ≤ netEntropyEntourage T F U := by
+  refine (limsup_le_limsup) (Eventually.of_forall fun n ↦ ?_)
+  apply div_le_div_right_of_nonneg (Nat.cast_nonneg' n) (log_monotone _)
+  exact ENat.toENNReal_le.2 (coverMincard_le_netMaxcard T F U_rfl U_symm n)
+
+/-! ### Relationship with entropy via covers -/
+
+variable [UniformSpace X] (T : X → X) (F : Set X)
+
+/-- Bowen-Dinaburg's definition of topological entropy using nets is
+  `⨆ U ∈ 𝓤 X, netEntropyEntourage T F U`. This quantity is the same as the topological entropy using
+  covers, so there is no need to define a new notion of topological entropy. This version of the
+  theorem relates the `liminf` versions of topological entropy.-/
+theorem coverEntropyInf_eq_iSup_netEntropyInfEntourage :
+    coverEntropyInf T F = ⨆ U ∈ 𝓤 X, netEntropyInfEntourage T F U := by
+  apply le_antisymm <;> refine iSup₂_le fun U U_uni ↦ ?_
+  · rcases (comp_symm_mem_uniformity_sets U_uni) with ⟨V, V_uni, V_symm, V_comp_U⟩
+    apply (coverEntropyInfEntourage_antitone T F V_comp_U).trans (le_iSup₂_of_le V V_uni _)
+    exact coverEntropyInfEntourage_le_netEntropyInfEntourage T F (refl_le_uniformity V_uni) V_symm
+  · apply (netEntropyInfEntourage_antitone T F (symmetrizeRel_subset_self U)).trans
+    apply (le_iSup₂ (symmetrizeRel U) (symmetrize_mem_uniformity U_uni)).trans'
+    exact netEntropyInfEntourage_le_coverEntropyInfEntourage T F (symmetric_symmetrizeRel U)
+
+/-- Bowen-Dinaburg's definition of topological entropy using nets is
+  `⨆ U ∈ 𝓤 X, netEntropyEntourage T F U`. This quantity is the same as the topological entropy using
+  covers, so there is no need to define a new notion of topological entropy. This version of the
+  theorem relates the `limsup` versions of topological entropy.-/
+theorem coverEntropy_eq_iSup_netEntropyEntourage :
+    coverEntropy T F = ⨆ U ∈ 𝓤 X, netEntropyEntourage T F U := by
+  apply le_antisymm <;> refine iSup₂_le fun U U_uni ↦ ?_
+  · rcases (comp_symm_mem_uniformity_sets U_uni) with ⟨V, V_uni, V_symm, V_comp_U⟩
+    apply (coverEntropyEntourage_antitone T F V_comp_U).trans (le_iSup₂_of_le V V_uni _)
+    exact coverEntropyEntourage_le_netEntropyEntourage T F (refl_le_uniformity V_uni) V_symm
+  · apply (netEntropyEntourage_antitone T F (symmetrizeRel_subset_self U)).trans
+    apply (le_iSup₂ (symmetrizeRel U) (symmetrize_mem_uniformity U_uni)).trans'
+    exact netEntropyEntourage_le_coverEntropyEntourage T F (symmetric_symmetrizeRel U)
+
+lemma coverEntropyInf_eq_iSup_basis_netEntropyInfEntourage {ι : Sort*} {p : ι → Prop}
+    {s : ι → Set (X × X)} (h : (𝓤 X).HasBasis p s) (T : X → X) (F : Set X) :
+    coverEntropyInf T F = ⨆ (i : ι) (_ : p i), netEntropyInfEntourage T F (s i) := by
+  rw [coverEntropyInf_eq_iSup_netEntropyInfEntourage T F]
+  apply (iSup₂_mono' fun i h_i ↦ ⟨s i, HasBasis.mem_of_mem h h_i, le_refl _⟩).antisymm'
+  refine iSup₂_le fun U U_uni ↦ ?_
+  rcases (HasBasis.mem_iff h).1 U_uni with ⟨i, h_i, si_U⟩
+  apply (netEntropyInfEntourage_antitone T F si_U).trans
+  exact le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ netEntropyInfEntourage T F (s i)) i h_i
+
+lemma coverEntropy_eq_iSup_basis_netEntropyEntourage {ι : Sort*} {p : ι → Prop}
+    {s : ι → Set (X × X)} (h : (𝓤 X).HasBasis p s) (T : X → X) (F : Set X) :
+    coverEntropy T F = ⨆ (i : ι) (_ : p i), netEntropyEntourage T F (s i) := by
+  rw [coverEntropy_eq_iSup_netEntropyEntourage T F]
+  apply (iSup₂_mono' fun i h_i ↦ ⟨s i, HasBasis.mem_of_mem h h_i, le_refl _⟩).antisymm'
+  refine iSup₂_le fun U U_uni ↦ ?_
+  rcases (HasBasis.mem_iff h).1 U_uni with ⟨i, h_i, si_U⟩
+  apply (netEntropyEntourage_antitone T F si_U).trans _
+  exact le_iSup₂ (f := fun (i : ι) (_ : p i) ↦ netEntropyEntourage T F (s i)) i h_i
+
+lemma netEntropyInfEntourage_le_coverEntropyInf {U : Set (X × X)} (h : U ∈ 𝓤 X) :
+    netEntropyInfEntourage T F U ≤ coverEntropyInf T F :=
+  coverEntropyInf_eq_iSup_netEntropyInfEntourage T F ▸
+    le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ netEntropyInfEntourage T F U) U h
+
+lemma netEntropyEntourage_le_coverEntropy {U : Set (X × X)} (h : U ∈ 𝓤 X) :
+    netEntropyEntourage T F U ≤ coverEntropy T F :=
+  coverEntropy_eq_iSup_netEntropyEntourage T F ▸
+    le_iSup₂ (f := fun (U : Set (X × X)) (_ : U ∈ 𝓤 X) ↦ netEntropyEntourage T F U) U h
+
+end Dynamics
diff --git a/Mathlib/FieldTheory/Adjoin.lean b/Mathlib/FieldTheory/Adjoin.lean
index ba316a0f73367..c83073349810a 100644
--- a/Mathlib/FieldTheory/Adjoin.lean
+++ b/Mathlib/FieldTheory/Adjoin.lean
@@ -8,6 +8,7 @@ import Mathlib.FieldTheory.IntermediateField.Algebraic
 import Mathlib.FieldTheory.Separable
 import Mathlib.FieldTheory.SplittingField.IsSplittingField
 import Mathlib.RingTheory.TensorProduct.Basic
+import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
 
 /-!
 # Adjoining Elements to Fields
@@ -27,7 +28,7 @@ For example, `Algebra.adjoin K {x}` might not include `x⁻¹`.
  - `F⟮α⟯`: adjoin a single element `α` to `F` (in scope `IntermediateField`).
 -/
 
-open FiniteDimensional Polynomial
+open Module Polynomial
 
 namespace IntermediateField
 
@@ -251,6 +252,14 @@ theorem map_iSup {ι : Sort*} (f : E →ₐ[F] K) (s : ι → IntermediateField
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+theorem map_inf (s t : IntermediateField F E) (f : E →ₐ[F] K) :
+    (s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter f.injective)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : E →ₐ[F] K) (s : ι → IntermediateField F E) :
+    (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective f.injective).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem _root_.AlgHom.fieldRange_eq_map (f : E →ₐ[F] K) :
     f.fieldRange = IntermediateField.map f ⊤ :=
   SetLike.ext' Set.image_univ.symm
@@ -356,12 +365,12 @@ theorem adjoin_adjoin_left (T : Set E) :
   apply Set.eq_of_subset_of_subset <;> rw [adjoin_subset_adjoin_iff] <;> constructor
   · rintro _ ⟨⟨x, hx⟩, rfl⟩; exact adjoin.mono _ _ _ Set.subset_union_left hx
   · exact subset_adjoin_of_subset_right _ _ Set.subset_union_right
--- Porting note: orginal proof times out
+-- Porting note: original proof times out
   · rintro x ⟨f, rfl⟩
     refine Subfield.subset_closure ?_
     left
     exact ⟨f, rfl⟩
--- Porting note: orginal proof times out
+-- Porting note: original proof times out
   · refine Set.union_subset (fun x hx => Subfield.subset_closure ?_)
       (fun x hx => Subfield.subset_closure ?_)
     · left
@@ -863,7 +872,7 @@ theorem adjoin_natCast (n : ℕ) : F⟮(n : E)⟯ = ⊥ :=
 
 section AdjoinRank
 
-open FiniteDimensional Module
+open Module Module
 
 variable {K L : IntermediateField F E}
 
@@ -997,6 +1006,11 @@ theorem adjoinRootEquivAdjoin_apply_root (h : IsIntegral F α) :
     adjoinRootEquivAdjoin F h (AdjoinRoot.root (minpoly F α)) = AdjoinSimple.gen F α :=
   AdjoinRoot.lift_root (aeval_gen_minpoly F α)
 
+@[simp]
+theorem adjoinRootEquivAdjoin_symm_apply_gen (h : IsIntegral F α) :
+    (adjoinRootEquivAdjoin F h).symm (AdjoinSimple.gen F α) = AdjoinRoot.root (minpoly F α) := by
+  rw [AlgEquiv.symm_apply_eq, adjoinRootEquivAdjoin_apply_root]
+
 theorem adjoin_root_eq_top (p : K[X]) [Fact (Irreducible p)] : K⟮AdjoinRoot.root p⟯ = ⊤ :=
   (eq_adjoin_of_eq_algebra_adjoin K _ ⊤ (AdjoinRoot.adjoinRoot_eq_top (f := p)).symm).symm
 
@@ -1029,7 +1043,7 @@ theorem isAlgebraic_adjoin_simple {x : L} (hx : IsIntegral K x) : Algebra.IsAlge
   have := adjoin.finiteDimensional hx; Algebra.IsAlgebraic.of_finite K K⟮x⟯
 
 theorem adjoin.finrank {x : L} (hx : IsIntegral K x) :
-    FiniteDimensional.finrank K K⟮x⟯ = (minpoly K x).natDegree := by
+    Module.finrank K K⟮x⟯ = (minpoly K x).natDegree := by
   rw [PowerBasis.finrank (adjoin.powerBasis hx : _)]
   rfl
 
@@ -1069,6 +1083,8 @@ theorem adjoin_minpoly_coeff_of_exists_primitive_element
     ((g.monic_toSubring _ _).mpr <| (minpoly.monic <| .of_finite K α).map _).ne_zero using 1
   rw [natDegree_toSubring, natDegree_map]
 
+instance : Module.Finite F (⊥ : IntermediateField F E) := Subalgebra.finite_bot
+
 variable {F} in
 /-- If `E / F` is an infinite algebraic extension, then there exists an intermediate field
 `L / F` with arbitrarily large finite extension degree. -/
@@ -1098,6 +1114,14 @@ theorem _root_.minpoly.degree_le (x : L) [FiniteDimensional K L] :
     (minpoly K x).degree ≤ finrank K L :=
   degree_le_of_natDegree_le (minpoly.natDegree_le x)
 
+/-- If `x : L` is an integral element in a field extension `L` over `K`, then the degree of the
+  minimal polynomial of `x` over `K` divides `[L : K]`.-/
+theorem _root_.minpoly.degree_dvd {x : L} (hx : IsIntegral K x) :
+    (minpoly K x).natDegree ∣ finrank K L := by
+  rw [dvd_iff_exists_eq_mul_left, ← IntermediateField.adjoin.finrank hx]
+  use finrank K⟮x⟯ L
+  rw [mul_comm, finrank_mul_finrank]
+
 -- TODO: generalize to `Sort`
 /-- A compositum of algebraic extensions is algebraic -/
 theorem isAlgebraic_iSup {ι : Type*} {t : ι → IntermediateField K L}
@@ -1153,7 +1177,7 @@ theorem card_algHom_adjoin_integral (h : IsIntegral F α) (h_sep : IsSeparable F
   exact h_sep
 
 -- Apparently `K⟮root f⟯ →+* K⟮root f⟯` is expensive to unify during instance synthesis.
-open FiniteDimensional AdjoinRoot in
+open Module AdjoinRoot in
 /-- Let `f, g` be monic polynomials over `K`. If `f` is irreducible, and `g(x) - α` is irreducible
 in `K⟮α⟯` with `α` a root of `f`, then `f(g(x))` is irreducible. -/
 theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g.Monic)
@@ -1201,7 +1225,7 @@ theorem _root_.Polynomial.irreducible_comp {f g : K[X]} (hfm : f.Monic) (hgm : g
       rw [← finrank_top', ← this, adjoin.finrank]
       exact IsIntegral.of_finite _ _
     · simp [← key₂]
-  have := FiniteDimensional.finrank_mul_finrank K K⟮aeval (root p) g⟯ Kx
+  have := Module.finrank_mul_finrank K K⟮aeval (root p) g⟯ Kx
   rwa [key₁', key₂', (AdjoinRoot.powerBasis hp₁.ne_zero).finrank, powerBasis_dim, eq_comm] at this
 
 end AdjoinIntegralElement
diff --git a/Mathlib/FieldTheory/Cardinality.lean b/Mathlib/FieldTheory/Cardinality.lean
index 498e3b8d326c7..045db9322dcbf 100644
--- a/Mathlib/FieldTheory/Cardinality.lean
+++ b/Mathlib/FieldTheory/Cardinality.lean
@@ -43,8 +43,8 @@ theorem Fintype.isPrimePow_card_of_field {α} [Fintype α] [Field α] : IsPrimeP
   let b := IsNoetherian.finsetBasis (ZMod p) α
   rw [Module.card_fintype b, ZMod.card, isPrimePow_pow_iff]
   · exact hp.1.isPrimePow
-  rw [← FiniteDimensional.finrank_eq_card_basis b]
-  exact FiniteDimensional.finrank_pos.ne'
+  rw [← Module.finrank_eq_card_basis b]
+  exact Module.finrank_pos.ne'
 
 /-- A `Fintype` can be given a field structure iff its cardinality is a prime power. -/
 theorem Fintype.nonempty_field_iff {α} [Fintype α] : Nonempty (Field α) ↔ IsPrimePow ‖α‖ := by
@@ -75,5 +75,5 @@ theorem Field.nonempty_iff {α : Type u} : Nonempty (Field α) ↔ IsPrimePow #
   rw [Cardinal.isPrimePow_iff]
   cases' fintypeOrInfinite α with h h
   · simpa only [Cardinal.mk_fintype, Nat.cast_inj, exists_eq_left',
-      (Cardinal.nat_lt_aleph0 _).not_le, false_or_iff] using Fintype.nonempty_field_iff
-  · simpa only [← Cardinal.infinite_iff, h, true_or_iff, iff_true_iff] using Infinite.nonempty_field
+      (Cardinal.nat_lt_aleph0 _).not_le, false_or] using Fintype.nonempty_field_iff
+  · simpa only [← Cardinal.infinite_iff, h, true_or, iff_true] using Infinite.nonempty_field
diff --git a/Mathlib/FieldTheory/Differential/Basic.lean b/Mathlib/FieldTheory/Differential/Basic.lean
new file mode 100644
index 0000000000000..05c4742401057
--- /dev/null
+++ b/Mathlib/FieldTheory/Differential/Basic.lean
@@ -0,0 +1,92 @@
+/-
+Copyright (c) 2024 Daniel Weber. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Daniel Weber
+-/
+import Mathlib.Algebra.EuclideanDomain.Field
+import Mathlib.RingTheory.Derivation.DifferentialRing
+import Mathlib.RingTheory.LocalRing.Basic
+import Mathlib.Tactic.FieldSimp
+
+/-!
+# Differential Fields
+
+This file defines the logarithmic derivative `Differential.logDeriv` and proves properties of it.
+This is defined algebraically, compared to `logDeriv` which is analytical.
+-/
+
+namespace Differential
+
+open algebraMap
+
+variable {R : Type*} [Field R] [Differential R] (a b : R)
+
+/--
+The logarithmic derivative of a is a′ / a.
+-/
+def logDeriv : R := a′ / a
+
+@[simp]
+lemma logDeriv_zero : logDeriv (0 : R) = 0 := by
+  simp [logDeriv]
+
+@[simp]
+lemma logDeriv_one : logDeriv (1 : R) = 0 := by
+  simp [logDeriv]
+
+lemma logDeriv_mul (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a * b) = logDeriv a + logDeriv b := by
+  unfold logDeriv
+  field_simp
+  ring
+
+lemma logDeriv_div (ha : a ≠ 0) (hb : b ≠ 0) : logDeriv (a / b) = logDeriv a - logDeriv b := by
+  unfold logDeriv
+  field_simp [Derivation.leibniz_div, smul_sub]
+  ring
+
+@[simp]
+lemma logDeriv_pow (n : ℕ) (a : R) : logDeriv (a ^ n) = n * logDeriv a := by
+  induction n with
+  | zero => simp
+  | succ n h2 =>
+    obtain rfl | hb := eq_or_ne a 0
+    · simp
+    · rw [Nat.cast_add, Nat.cast_one, add_mul, one_mul, ← h2, pow_succ, logDeriv_mul] <;>
+      simp [hb]
+
+lemma logDeriv_eq_zero : logDeriv a = 0 ↔ a′ = 0 :=
+  ⟨fun h ↦ by simp only [logDeriv, div_eq_zero_iff] at h; rcases h with h|h <;> simp [h],
+  fun h ↦ by unfold logDeriv at *; simp [h]⟩
+
+lemma logDeriv_multisetProd {ι : Type*} (s : Multiset ι) {f : ι → R} (h : ∀ x ∈ s, f x ≠ 0) :
+    logDeriv (s.map f).prod = (s.map fun x ↦ logDeriv (f x)).sum := by
+  induction s using Multiset.induction_on
+  · simp
+  · rename_i h₂
+    simp only [Function.comp_apply, Multiset.map_cons, Multiset.sum_cons, Multiset.prod_cons]
+    rw [← h₂]
+    · apply logDeriv_mul
+      · simp [h]
+      · simp_all
+    · simp_all
+
+lemma logDeriv_prod (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x ≠ 0) :
+    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := logDeriv_multisetProd _ h
+
+lemma logDeriv_prod_of_eq_zero (ι : Type*) (s : Finset ι) (f : ι → R) (h : ∀ x ∈ s, f x = 0) :
+    logDeriv (∏ x ∈ s, f x) = ∑ x ∈ s, logDeriv (f x) := by
+  unfold logDeriv
+  simp_all
+
+lemma logDeriv_algebraMap {F K : Type*} [Field F] [Field K] [Differential F] [Differential K]
+    [Algebra F K] [DifferentialAlgebra F K]
+    (a : F) : logDeriv (algebraMap F K a) = algebraMap F K (logDeriv a) := by
+  unfold logDeriv
+  simp [deriv_algebraMap]
+
+@[norm_cast]
+lemma _root_.algebraMap.coe_logDeriv {F K : Type*} [Field F] [Field K] [Differential F]
+    [Differential K] [Algebra F K] [DifferentialAlgebra F K]
+    (a : F) : logDeriv a = logDeriv (a : K) := (logDeriv_algebraMap a).symm
+
+end Differential
diff --git a/Mathlib/FieldTheory/Extension.lean b/Mathlib/FieldTheory/Extension.lean
index 4fca6eb496228..abfd3dbf451d7 100644
--- a/Mathlib/FieldTheory/Extension.lean
+++ b/Mathlib/FieldTheory/Extension.lean
@@ -144,7 +144,7 @@ end
 
 variable (hK : ∀ s ∈ S, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
   (hK' : ∀ s : E, IsIntegral F s ∧ (minpoly F s).Splits (algebraMap F K))
-  {L : IntermediateField F E}  (f : L →ₐ[F] K) (hL : L ≤ adjoin F S) {x : E} {y : K}
+  {L : IntermediateField F E} (f : L →ₐ[F] K) (hL : L ≤ adjoin F S) {x : E} {y : K}
 
 section
 include hK
diff --git a/Mathlib/FieldTheory/Finite/Basic.lean b/Mathlib/FieldTheory/Finite/Basic.lean
index dad50e120b86c..715387ba0669b 100644
--- a/Mathlib/FieldTheory/Finite/Basic.lean
+++ b/Mathlib/FieldTheory/Finite/Basic.lean
@@ -145,9 +145,8 @@ theorem sum_subgroup_units_eq_zero [Ring K] [NoZeroDivisors K]
   have h_sum_map := Finset.univ.sum_map a_mul_emb fun x => ((x : Kˣ) : K)
   -- ... and the former is the sum of x over G.
   -- By algebraic manipulation, we have Σ G, x = ∑ G, a x = a ∑ G, x
-  simp only [a_mul_emb, h_unchanged, Function.Embedding.coeFn_mk, Function.Embedding.toFun_eq_coe,
-    mulLeftEmbedding_apply, Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul,
-    ← Finset.mul_sum] at h_sum_map
+  simp only [h_unchanged, mulLeftEmbedding_apply, Subgroup.coe_mul, Units.val_mul, ← mul_sum,
+    a_mul_emb] at h_sum_map
   -- thus one of (a - 1) or ∑ G, x is zero
   have hzero : (((a : Kˣ) : K) - 1) = 0 ∨ ∑ x : ↥G, ((x : Kˣ) : K) = 0 := by
     rw [← mul_eq_zero, sub_mul, ← h_sum_map, one_mul, sub_self]
@@ -163,8 +162,9 @@ theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
     ∑ x : G, (x.val : K) = if G = ⊥ then 1 else 0 := by
   by_cases G_bot : G = ⊥
   · subst G_bot
-    simp only [ite_true, Subgroup.mem_bot, Fintype.card_ofSubsingleton, Nat.cast_ite, Nat.cast_one,
-      Nat.cast_zero, univ_unique, Set.default_coe_singleton, sum_singleton, Units.val_one]
+    simp only [univ_unique, sum_singleton, ↓reduceIte, Units.val_eq_one, OneMemClass.coe_eq_one]
+    rw [Set.default_coe_singleton]
+    rfl
   · simp only [G_bot, ite_false]
     exact sum_subgroup_units_eq_zero G_bot
 
@@ -184,7 +184,7 @@ theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
       (fun x : ↥G => (((x : Kˣ) : K) * ((a : Kˣ) : K)) ^ k)
         = (fun x : ↥G => ((x : Kˣ) : K) ^ k) ∘ fun x : ↥G => x * a := by
       funext x
-      simp only [Function.comp_apply, Submonoid.coe_mul, Subgroup.coe_toSubmonoid, Units.val_mul]
+      simp only [Function.comp_apply, Subgroup.coe_mul, Units.val_mul]
     rw [as_comp, ← Multiset.map_map]
     congr
     rw [eq_comm]
@@ -293,7 +293,7 @@ theorem sum_pow_lt_card_sub_one (i : ℕ) (h : i < q - 1) : ∑ x : K, x ^ i = 0
     let φ : Kˣ ↪ K := ⟨fun x ↦ x, Units.ext⟩
     have : univ.map φ = univ \ {0} := by
       ext x
-      simpa only [mem_map, mem_univ, Function.Embedding.coeFn_mk, true_and_iff, mem_sdiff,
+      simpa only [mem_map, mem_univ, Function.Embedding.coeFn_mk, true_and, mem_sdiff,
         mem_singleton, φ] using isUnit_iff_ne_zero
     calc
       ∑ x : K, x ^ i = ∑ x ∈ univ \ {(0 : K)}, x ^ i := by
@@ -455,9 +455,9 @@ variable {V : Type*} [Fintype K] [DivisionRing K] [AddCommGroup V] [Module K V]
 -- should this go in a namespace?
 -- finite_dimensional would be natural,
 -- but we don't assume it...
-theorem card_eq_pow_finrank [Fintype V] : Fintype.card V = q ^ FiniteDimensional.finrank K V := by
+theorem card_eq_pow_finrank [Fintype V] : Fintype.card V = q ^ Module.finrank K V := by
   let b := IsNoetherian.finsetBasis K V
-  rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b]
+  rw [Module.card_fintype b, ← Module.finrank_eq_card_basis b]
 
 end
 
diff --git a/Mathlib/FieldTheory/Finite/GaloisField.lean b/Mathlib/FieldTheory/Finite/GaloisField.lean
index c2262e4cf2be6..89b627d0328c8 100644
--- a/Mathlib/FieldTheory/Finite/GaloisField.lean
+++ b/Mathlib/FieldTheory/Finite/GaloisField.lean
@@ -3,10 +3,10 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson, Alex J. Best, Johan Commelin, Eric Rodriguez, Ruben Van de Velde
 -/
+import Mathlib.Algebra.Algebra.ZMod
 import Mathlib.Algebra.CharP.Algebra
-import Mathlib.Data.ZMod.Algebra
 import Mathlib.FieldTheory.Finite.Basic
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.FieldTheory.SplittingField.IsSplittingField
 
 /-!
@@ -88,11 +88,11 @@ instance : Fintype (GaloisField p n) := by
   dsimp only [GaloisField]
   exact FiniteDimensional.fintypeOfFintype (ZMod p) (GaloisField p n)
 
-theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n := by
+theorem finrank {n} (h : n ≠ 0) : Module.finrank (ZMod p) (GaloisField p n) = n := by
   set g_poly := (X ^ p ^ n - X : (ZMod p)[X])
   have hp : 1 < p := h_prime.out.one_lt
   have aux : g_poly ≠ 0 := FiniteField.X_pow_card_pow_sub_X_ne_zero _ h hp
-  -- Porting note: in the statment of `key`, replaced `g_poly` by its value otherwise the
+  -- Porting note: in the statement of `key`, replaced `g_poly` by its value otherwise the
   -- proof fails
   have key : Fintype.card (g_poly.rootSet (GaloisField p n)) = g_poly.natDegree :=
     card_rootSet_eq_natDegree (galois_poly_separable p _ (dvd_pow (dvd_refl p) h))
@@ -139,7 +139,7 @@ theorem finrank {n} (h : n ≠ 0) : FiniteDimensional.finrank (ZMod p) (GaloisFi
 
 theorem card (h : n ≠ 0) : Fintype.card (GaloisField p n) = p ^ n := by
   let b := IsNoetherian.finsetBasis (ZMod p) (GaloisField p n)
-  rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b, ZMod.card, finrank p h]
+  rw [Module.card_fintype b, ← Module.finrank_eq_card_basis b, ZMod.card, finrank p h]
 
 theorem splits_zmod_X_pow_sub_X : Splits (RingHom.id (ZMod p)) (X ^ p - X) := by
   have hp : 1 < p := h_prime.out.one_lt
@@ -189,7 +189,7 @@ namespace FiniteField
 variable {K : Type*} [Field K] [Fintype K] {K' : Type*} [Field K'] [Fintype K']
 
 /-- Uniqueness of finite fields:
-  Any two finite fields of the same cardinality are (possibly non canonically) isomorphic-/
+  Any two finite fields of the same cardinality are (possibly non canonically) isomorphic -/
 def algEquivOfCardEq (p : ℕ) [h_prime : Fact p.Prime] [Algebra (ZMod p) K] [Algebra (ZMod p) K']
     (hKK' : Fintype.card K = Fintype.card K') : K ≃ₐ[ZMod p] K' := by
   have : CharP K p := by rw [← Algebra.charP_iff (ZMod p) K p]; exact ZMod.charP p
@@ -203,7 +203,7 @@ def algEquivOfCardEq (p : ℕ) [h_prime : Fact p.Prime] [Algebra (ZMod p) K] [Al
   exact AlgEquiv.trans hGalK hK'Gal
 
 /-- Uniqueness of finite fields:
-  Any two finite fields of the same cardinality are (possibly non canonically) isomorphic-/
+  Any two finite fields of the same cardinality are (possibly non canonically) isomorphic -/
 def ringEquivOfCardEq (hKK' : Fintype.card K = Fintype.card K') : K ≃+* K' := by
   choose p _char_p_K using CharP.exists K
   choose p' _char_p'_K' using CharP.exists K'
diff --git a/Mathlib/FieldTheory/Finite/Polynomial.lean b/Mathlib/FieldTheory/Finite/Polynomial.lean
index 8207de7d06c14..00e92545eba5b 100644
--- a/Mathlib/FieldTheory/Finite/Polynomial.lean
+++ b/Mathlib/FieldTheory/Finite/Polynomial.lean
@@ -215,8 +215,8 @@ instance [Finite σ] : FiniteDimensional K (R σ K) := by
         simpa only [rank_R] using Cardinal.nat_lt_aleph0 (Fintype.card (σ → K)))
 
 open Classical in
-theorem finrank_R [Fintype σ] : FiniteDimensional.finrank K (R σ K) = Fintype.card (σ → K) :=
-  FiniteDimensional.finrank_eq_of_rank_eq (rank_R σ K)
+theorem finrank_R [Fintype σ] : Module.finrank K (R σ K) = Fintype.card (σ → K) :=
+  Module.finrank_eq_of_rank_eq (rank_R σ K)
 
 -- Porting note: was `(evalᵢ σ K).range`.
 theorem range_evalᵢ [Finite σ] : range (evalᵢ σ K) = ⊤ := by
@@ -228,7 +228,7 @@ theorem ker_evalₗ [Finite σ] : ker (evalᵢ σ K) = ⊥ := by
   cases nonempty_fintype σ
   refine (ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank ?_).mpr (range_evalᵢ σ K)
   classical
-  rw [FiniteDimensional.finrank_fintype_fun_eq_card, finrank_R]
+  rw [Module.finrank_fintype_fun_eq_card, finrank_R]
 
 theorem eq_zero_of_eval_eq_zero [Finite σ] (p : MvPolynomial σ K) (h : ∀ v : σ → K, eval v p = 0)
     (hp : p ∈ restrictDegree σ K (Fintype.card K - 1)) : p = 0 :=
diff --git a/Mathlib/FieldTheory/Finiteness.lean b/Mathlib/FieldTheory/Finiteness.lean
index 6c2e5621a9272..d6cc69e04c7ae 100644
--- a/Mathlib/FieldTheory/Finiteness.lean
+++ b/Mathlib/FieldTheory/Finiteness.lean
@@ -72,7 +72,7 @@ theorem coeSort_finsetBasisIndex [IsNoetherian K V] :
 /-- In a noetherian module over a division ring, there exists a finite basis.
 This is indexed by the `Finset` `IsNoetherian.finsetBasisIndex`.
 This is in contrast to the result `finite_basis_index (Basis.ofVectorSpace K V)`,
-which provides a set and a `Set.finite`.
+which provides a set and a `Set.Finite`.
 -/
 noncomputable def finsetBasis [IsNoetherian K V] : Basis (finsetBasisIndex K V) K V :=
   (Basis.ofVectorSpace K V).reindex (by rw [coeSort_finsetBasisIndex])
diff --git a/Mathlib/FieldTheory/Fixed.lean b/Mathlib/FieldTheory/Fixed.lean
index 11e8c66077f0c..965f23a35eefe 100644
--- a/Mathlib/FieldTheory/Fixed.lean
+++ b/Mathlib/FieldTheory/Fixed.lean
@@ -8,7 +8,7 @@ import Mathlib.Algebra.Ring.Action.Field
 import Mathlib.Algebra.Ring.Action.Invariant
 import Mathlib.FieldTheory.Normal
 import Mathlib.FieldTheory.Separable
-import Mathlib.FieldTheory.Tower
+import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
 
 /-!
 # Fixed field under a group action.
@@ -30,7 +30,7 @@ element of `G`, where `G` is a group that acts on `F`.
 
 noncomputable section
 
-open MulAction Finset FiniteDimensional
+open MulAction Finset Module
 
 universe u v w
 
@@ -118,8 +118,8 @@ theorem linearIndependent_smul_of_linearIndependent {s : Finset F} :
   rw [coe_insert] at hs ⊢
   rw [linearIndependent_insert (mt mem_coe.1 has)] at hs
   rw [linearIndependent_insert' (mt mem_coe.1 has)]; refine ⟨ih hs.1, fun ha => ?_⟩
-  rw [Finsupp.mem_span_image_iff_total] at ha; rcases ha with ⟨l, hl, hla⟩
-  rw [Finsupp.total_apply_of_mem_supported F hl] at hla
+  rw [Finsupp.mem_span_image_iff_linearCombination] at ha; rcases ha with ⟨l, hl, hla⟩
+  rw [Finsupp.linearCombination_apply_of_mem_supported F hl] at hla
   suffices ∀ i ∈ s, l i ∈ FixedPoints.subfield G F by
     replace hla := (sum_apply _ _ fun i => l i • toFun G F i).symm.trans (congr_fun hla 1)
     simp_rw [Pi.smul_apply, toFun_apply, one_smul] at hla
@@ -166,7 +166,8 @@ def minpoly : Polynomial (FixedPoints.subfield G F) :=
 namespace minpoly
 
 theorem monic : (minpoly G F x).Monic := by
-  simp only [minpoly, Polynomial.monic_toSubring]
+  simp only [minpoly]
+  rw [Polynomial.monic_toSubring]
   exact prodXSubSMul.monic G F x
 
 theorem eval₂ :
diff --git a/Mathlib/FieldTheory/Galois.lean b/Mathlib/FieldTheory/Galois/Basic.lean
similarity index 96%
rename from Mathlib/FieldTheory/Galois.lean
rename to Mathlib/FieldTheory/Galois/Basic.lean
index 97c0878478581..3f571ef51fa06 100644
--- a/Mathlib/FieldTheory/Galois.lean
+++ b/Mathlib/FieldTheory/Galois/Basic.lean
@@ -24,7 +24,7 @@ In this file we define Galois extensions as extensions which are both separable
 
 - `IntermediateField.fixingSubgroup_fixedField` : If `E/F` is finite dimensional (but not
   necessarily Galois) then `fixingSubgroup (fixedField H) = H`
-- `IntermediateField.fixedField_fixingSubgroup`: If `E/F` is finite dimensional and Galois
+- `IsGalois.fixedField_fixingSubgroup`: If `E/F` is finite dimensional and Galois
   then `fixedField (fixingSubgroup K) = K`
 
 Together, these two results prove the Galois correspondence.
@@ -35,7 +35,7 @@ Together, these two results prove the Galois correspondence.
 
 open scoped Polynomial IntermediateField
 
-open FiniteDimensional AlgEquiv
+open Module AlgEquiv
 
 section
 
@@ -380,32 +380,23 @@ theorem of_separable_splitting_field [sp : p.IsSplittingField F E] (hp : p.Separ
     rw [IntermediateField.adjoin_zero]
   intro K x hx hK
   simp only [P] at *
-  -- Porting note: need to specify two implicit arguments of `finrank_mul_finrank`
-  letI := K⟮x⟯.module
-  letI := K⟮x⟯.isScalarTower (R := F)
   rw [of_separable_splitting_field_aux hp K (Multiset.mem_toFinset.mp hx), hK, finrank_mul_finrank]
   symm
   refine LinearEquiv.finrank_eq ?_
   rfl
 
-/-- Equivalent characterizations of a Galois extension of finite degree-/
+/-- Equivalent characterizations of a Galois extension of finite degree -/
 theorem tfae [FiniteDimensional F E] : List.TFAE [
     IsGalois F E,
     IntermediateField.fixedField (⊤ : Subgroup (E ≃ₐ[F] E)) = ⊥,
     Fintype.card (E ≃ₐ[F] E) = finrank F E,
     ∃ p : F[X], p.Separable ∧ p.IsSplittingField F E] := by
-  tfae_have 1 → 2
-  · exact fun h => OrderIso.map_bot (@intermediateFieldEquivSubgroup F _ E _ _ _ h).symm
-  tfae_have 1 → 3
-  · intro; exact card_aut_eq_finrank F E
-  tfae_have 1 → 4
-  · intro; exact is_separable_splitting_field F E
-  tfae_have 2 → 1
-  · exact of_fixedField_eq_bot F E
-  tfae_have 3 → 1
-  · exact of_card_aut_eq_finrank F E
-  tfae_have 4 → 1
-  · rintro ⟨h, hp1, _⟩; exact of_separable_splitting_field hp1
+  tfae_have 1 → 2 := fun h ↦ OrderIso.map_bot (@intermediateFieldEquivSubgroup F _ E _ _ _ h).symm
+  tfae_have 1 → 3 := fun _ ↦ card_aut_eq_finrank F E
+  tfae_have 1 → 4 := fun _ ↦ is_separable_splitting_field F E
+  tfae_have 2 → 1 := of_fixedField_eq_bot F E
+  tfae_have 3 → 1 := of_card_aut_eq_finrank F E
+  tfae_have 4 → 1 := fun ⟨h, hp1, _⟩ ↦ of_separable_splitting_field hp1
   tfae_finish
 
 end IsGalois
diff --git a/Mathlib/FieldTheory/IntermediateField/Algebraic.lean b/Mathlib/FieldTheory/IntermediateField/Algebraic.lean
index c546c92eb2c36..55ecc96148306 100644
--- a/Mathlib/FieldTheory/IntermediateField/Algebraic.lean
+++ b/Mathlib/FieldTheory/IntermediateField/Algebraic.lean
@@ -7,12 +7,13 @@ import Mathlib.FieldTheory.IntermediateField.Basic
 import Mathlib.RingTheory.Algebraic
 import Mathlib.FieldTheory.Tower
 import Mathlib.FieldTheory.Minpoly.Basic
+import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
 
 /-!
 # Results on finite dimensionality and algebraicity of intermediate fields.
 -/
 
-open FiniteDimensional
+open Module
 
 variable {K : Type*} {L : Type*} [Field K] [Field L] [Algebra K L]
   {S : IntermediateField K L}
@@ -23,11 +24,8 @@ section FiniteDimensional
 
 variable (F E : IntermediateField K L)
 
-instance finiteDimensional_left [FiniteDimensional K L] : FiniteDimensional K F :=
-  left K F L
-
-instance finiteDimensional_right [FiniteDimensional K L] : FiniteDimensional F L :=
-  right K F L
+instance finiteDimensional_left [FiniteDimensional K L] : FiniteDimensional K F := .left K F L
+instance finiteDimensional_right [FiniteDimensional K L] : FiniteDimensional F L := .right K F L
 
 @[simp]
 theorem rank_eq_rank_subalgebra : Module.rank K F.toSubalgebra = Module.rank K F :=
@@ -96,7 +94,7 @@ theorem minpoly_eq (x : S) : minpoly K x = minpoly K (x : L) :=
 
 end IntermediateField
 
-/-- If `L/K` is algebraic, the `K`-subalgebras of `L` are all fields.  -/
+/-- If `L/K` is algebraic, the `K`-subalgebras of `L` are all fields. -/
 def subalgebraEquivIntermediateField [Algebra.IsAlgebraic K L] :
     Subalgebra K L ≃o IntermediateField K L where
   toFun S := S.toIntermediateField fun x hx => S.inv_mem_of_algebraic
diff --git a/Mathlib/FieldTheory/IntermediateField/Basic.lean b/Mathlib/FieldTheory/IntermediateField/Basic.lean
index 580f2d03ea7f6..27a9fb8cd57cd 100644
--- a/Mathlib/FieldTheory/IntermediateField/Basic.lean
+++ b/Mathlib/FieldTheory/IntermediateField/Basic.lean
@@ -307,29 +307,31 @@ theorem coe_prod {ι : Type*} [Fintype ι] (f : ι → S) : (↑(∏ i, f i) : L
 
 /-! `IntermediateField`s inherit structure from their `Subalgebra` coercions. -/
 
+instance toAlgebra : Algebra S L :=
+  inferInstanceAs (Algebra S.toSubalgebra L)
 
 instance module' {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] : Module R S :=
-  S.toSubalgebra.module'
+  inferInstanceAs (Module R S.toSubalgebra)
 
-instance module : Module K S :=
-  inferInstanceAs (Module K S.toSubsemiring)
+instance algebra' {R' K L : Type*} [Field K] [Field L] [Algebra K L] (S : IntermediateField K L)
+    [CommSemiring R'] [SMul R' K] [Algebra R' L] [IsScalarTower R' K L] : Algebra R' S :=
+  inferInstanceAs (Algebra R' S.toSubalgebra)
 
 instance isScalarTower {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] :
     IsScalarTower R K S :=
-  inferInstanceAs (IsScalarTower R K S.toSubsemiring)
+  inferInstanceAs (IsScalarTower R K S.toSubalgebra)
 
 @[simp]
 theorem coe_smul {R} [Semiring R] [SMul R K] [Module R L] [IsScalarTower R K L] (r : R) (x : S) :
     ↑(r • x : S) = (r • (x : L)) :=
   rfl
 
-instance algebra : Algebra K S :=
-  inferInstanceAs (Algebra K S.toSubsemiring)
-
 @[simp] lemma algebraMap_apply (x : S) : algebraMap S L x = x := rfl
 
 @[simp] lemma coe_algebraMap_apply (x : K) : ↑(algebraMap K S x) = algebraMap K L x := rfl
 
+instance {R : Type*} [Semiring R] [Algebra L R] : SMul S R := S.instSMulSubtypeMem
+
 instance isScalarTower_bot {R : Type*} [Semiring R] [Algebra L R] : IsScalarTower S L R :=
   IsScalarTower.subalgebra _ _ _ S.toSubalgebra
 
@@ -341,6 +343,8 @@ instance isScalarTower_mid {R : Type*} [Semiring R] [Algebra L R] [Algebra K R]
 instance isScalarTower_mid' : IsScalarTower K S L :=
   S.isScalarTower_mid
 
+instance {E} [Semiring E] [Algebra L E] : Algebra S E := inferInstanceAs (Algebra S.toSubalgebra E)
+
 section shortcut_instances
 variable {E} [Field E] [Algebra L E] (T : IntermediateField S E) {S}
 instance : Algebra S T := T.algebra
diff --git a/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean b/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean
index bef0b0e2de8f4..787429c5e3827 100644
--- a/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean
+++ b/Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean
@@ -73,13 +73,13 @@ theorem toSplittingField_evalXSelf {s : Finset (MonicIrreducible k)} {f} (hf : f
 
 theorem spanEval_ne_top : spanEval k ≠ ⊤ := by
   rw [Ideal.ne_top_iff_one, spanEval, Ideal.span, ← Set.image_univ,
-    Finsupp.mem_span_image_iff_total]
+    Finsupp.mem_span_image_iff_linearCombination]
   rintro ⟨v, _, hv⟩
   replace hv := congr_arg (toSplittingField k v.support) hv
-  rw [map_one, Finsupp.total_apply, Finsupp.sum, map_sum, Finset.sum_eq_zero] at hv
+  rw [map_one, Finsupp.linearCombination_apply, Finsupp.sum, map_sum, Finset.sum_eq_zero] at hv
   · exact zero_ne_one hv
   intro j hj
-  rw [smul_eq_mul, map_mul, toSplittingField_evalXSelf (s := v.support) hj,
+  rw [smul_eq_mul, map_mul, toSplittingField_evalXSelf _ (s := v.support) hj,
     mul_zero]
 
 /-- A random maximal ideal that contains `spanEval k` -/
@@ -250,8 +250,7 @@ theorem Step.isIntegral (n) : ∀ z : Step k n, IsIntegral k z := by
     apply @RingHom.IsIntegral.trans (Step k 0) (Step k a) (Step k (a + 1)) _ _ _
         (toStepOfLE k 0 a (a.zero_le : 0 ≤ a)) (toStepSucc k a) _
     · intro z
-      have := AdjoinMonic.isIntegral (Step k a) (z : Step k (a + 1))
-      convert this
+      convert AdjoinMonic.isIntegral (Step k a) (z : Step k (a + 1))
     · convert h -- Porting note: This times out at 500000
 
 instance toStepOfLE.directedSystem : DirectedSystem (Step k) fun i j h => toStepOfLE k i j h :=
diff --git a/Mathlib/FieldTheory/IsAlgClosed/Classification.lean b/Mathlib/FieldTheory/IsAlgClosed/Classification.lean
index 1a9f0ce2972f9..2f13ff3833e58 100644
--- a/Mathlib/FieldTheory/IsAlgClosed/Classification.lean
+++ b/Mathlib/FieldTheory/IsAlgClosed/Classification.lean
@@ -3,9 +3,9 @@ Copyright (c) 2022 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
-import Mathlib.Algebra.Polynomial.Cardinal
+import Mathlib.Algebra.Algebra.ZMod
 import Mathlib.Algebra.MvPolynomial.Cardinal
-import Mathlib.Data.ZMod.Algebra
+import Mathlib.Algebra.Polynomial.Cardinal
 import Mathlib.FieldTheory.IsAlgClosed.Basic
 import Mathlib.RingTheory.AlgebraicIndependent
 
@@ -54,7 +54,7 @@ theorem cardinal_mk_le_sigma_polynomial :
       intro h
       simp? at h says simp only [Set.coe_setOf, ne_eq, Set.mem_setOf_eq, Sigma.mk.inj_iff] at h
       refine (Subtype.heq_iff_coe_eq ?_).1 h.2
-      simp only [h.1, iff_self_iff, forall_true_iff]
+      simp only [h.1, forall_true_iff]
 
 /-- The cardinality of an algebraic extension is at most the maximum of the cardinality
 of the base ring or `ℵ₀` -/
diff --git a/Mathlib/FieldTheory/IsPerfectClosure.lean b/Mathlib/FieldTheory/IsPerfectClosure.lean
index 70d3708472eb5..743199124c340 100644
--- a/Mathlib/FieldTheory/IsPerfectClosure.lean
+++ b/Mathlib/FieldTheory/IsPerfectClosure.lean
@@ -60,7 +60,7 @@ perfect ring, perfect closure, purely inseparable
 
 -/
 
-open FiniteDimensional Polynomial IntermediateField Field
+open Module Polynomial IntermediateField Field
 
 noncomputable section
 
diff --git a/Mathlib/FieldTheory/IsSepClosed.lean b/Mathlib/FieldTheory/IsSepClosed.lean
index a986c4144c374..1a11e8d38dff7 100644
--- a/Mathlib/FieldTheory/IsSepClosed.lean
+++ b/Mathlib/FieldTheory/IsSepClosed.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jz Pan
 -/
 import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 
 /-!
 # Separably Closed Field
diff --git a/Mathlib/FieldTheory/KrullTopology.lean b/Mathlib/FieldTheory/KrullTopology.lean
index 18d9b02736673..7f7f53c3109a7 100644
--- a/Mathlib/FieldTheory/KrullTopology.lean
+++ b/Mathlib/FieldTheory/KrullTopology.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Sebastian Monnet. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sebastian Monnet
 -/
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.Topology.Algebra.FilterBasis
 import Mathlib.Topology.Algebra.OpenSubgroup
 import Mathlib.Tactic.ByContra
diff --git a/Mathlib/FieldTheory/KummerExtension.lean b/Mathlib/FieldTheory/KummerExtension.lean
index 65eb3d88e45c4..671a4b1bb374a 100644
--- a/Mathlib/FieldTheory/KummerExtension.lean
+++ b/Mathlib/FieldTheory/KummerExtension.lean
@@ -5,7 +5,7 @@ Authors: Andrew Yang
 -/
 import Mathlib.RingTheory.RootsOfUnity.Basic
 import Mathlib.RingTheory.AdjoinRoot
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.LinearAlgebra.Eigenspace.Minpoly
 import Mathlib.RingTheory.Norm.Basic
 /-!
@@ -272,6 +272,11 @@ theorem Polynomial.separable_X_pow_sub_C_of_irreducible : (X ^ n - C a).Separabl
     AdjoinRoot.algebraMap_eq,
     X_pow_sub_C_eq_prod (hζ.map_of_injective (algebraMap K _).injective) hn
     (root_X_pow_sub_C_pow n a), separable_prod_X_sub_C_iff']
+  #adaptation_note
+  /--
+  After https://github.com/leanprover/lean4/pull/5376 we need to provide this helper instance.
+  -/
+  have : MonoidHomClass (K →+* K[n√a]) K K[n√a] := inferInstance
   exact (hζ.map_of_injective (algebraMap K K[n√a]).injective).injOn_pow_mul
     (root_X_pow_sub_C_ne_zero (lt_of_le_of_ne (show 1 ≤ n from hn) (Ne.symm hn')) _)
 
@@ -530,7 +535,7 @@ lemma isGalois_of_isSplittingField_X_pow_sub_C : IsGalois K L :=
   IsGalois.of_separable_splitting_field (separable_X_pow_sub_C_of_irreducible hζ a H)
 
 include hζ H in
-lemma finrank_of_isSplittingField_X_pow_sub_C : FiniteDimensional.finrank K L = n := by
+lemma finrank_of_isSplittingField_X_pow_sub_C : Module.finrank K L = n := by
   have := Polynomial.IsSplittingField.finiteDimensional L (X ^ n - C a)
   have := isGalois_of_isSplittingField_X_pow_sub_C hζ H L
   have hn := Nat.pos_iff_ne_zero.mpr (ne_zero_of_irreducible_X_pow_sub_C H)
@@ -545,9 +550,9 @@ end IsSplittingField
 section IsCyclic
 
 variable {L} [Field L] [Algebra K L] [FiniteDimensional K L]
-variable (hK : (primitiveRoots (FiniteDimensional.finrank K L) K).Nonempty)
+variable (hK : (primitiveRoots (Module.finrank K L) K).Nonempty)
 
-open FiniteDimensional
+open Module
 variable (K L)
 
 include hK in
@@ -623,7 +628,7 @@ lemma isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top
 
 end IsCyclic
 
-open FiniteDimensional in
+open Module in
 /--
 Suppose `L/K` is a finite extension of dimension `n`, and `K` contains all `n`-th roots of unity.
 Then `L/K` is cyclic iff
@@ -631,21 +636,18 @@ Then `L/K` is cyclic iff
 `L = K[α]` for some `αⁿ ∈ K`.
 -/
 lemma isCyclic_tfae (K L) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
-    (hK : (primitiveRoots (FiniteDimensional.finrank K L) K).Nonempty) :
+    (hK : (primitiveRoots (Module.finrank K L) K).Nonempty) :
     List.TFAE [
       IsGalois K L ∧ IsCyclic (L ≃ₐ[K] L),
       ∃ a : K, Irreducible (X ^ (finrank K L) - C a) ∧
         IsSplittingField K L (X ^ (finrank K L) - C a),
       ∃ (α : L), α ^ (finrank K L) ∈ Set.range (algebraMap K L) ∧ K⟮α⟯ = ⊤] := by
   tfae_have 1 → 3
-  · intro ⟨inst₁, inst₂⟩
-    exact exists_root_adjoin_eq_top_of_isCyclic K L hK
+  | ⟨inst₁, inst₂⟩ => exists_root_adjoin_eq_top_of_isCyclic K L hK
   tfae_have 3 → 2
-  · intro ⟨α, ⟨a, ha⟩, hα⟩
-    exact ⟨a, irreducible_X_pow_sub_C_of_root_adjoin_eq_top ha.symm hα,
+  | ⟨α, ⟨a, ha⟩, hα⟩ => ⟨a, irreducible_X_pow_sub_C_of_root_adjoin_eq_top ha.symm hα,
       isSplittingField_X_pow_sub_C_of_root_adjoin_eq_top hK ha.symm hα⟩
   tfae_have 2 → 1
-  · intro ⟨a, H, inst⟩
-    exact ⟨isGalois_of_isSplittingField_X_pow_sub_C hK H L,
+  | ⟨a, H, inst⟩ => ⟨isGalois_of_isSplittingField_X_pow_sub_C hK H L,
       isCyclic_of_isSplittingField_X_pow_sub_C hK H L⟩
   tfae_finish
diff --git a/Mathlib/FieldTheory/Minpoly/Basic.lean b/Mathlib/FieldTheory/Minpoly/Basic.lean
index bff61c9cc012d..ee4583757d618 100644
--- a/Mathlib/FieldTheory/Minpoly/Basic.lean
+++ b/Mathlib/FieldTheory/Minpoly/Basic.lean
@@ -60,8 +60,8 @@ theorem eq_zero (hx : ¬IsIntegral A x) : minpoly A x = 0 :=
 
 theorem algHom_eq (f : B →ₐ[A] B') (hf : Function.Injective f) (x : B) :
     minpoly A (f x) = minpoly A x := by
-  refine dif_ctx_congr (isIntegral_algHom_iff _ hf) (fun _ => ?_) fun _ => rfl
-  simp_rw [← Polynomial.aeval_def, aeval_algHom, AlgHom.comp_apply, _root_.map_eq_zero_iff f hf]
+  simp_rw [minpoly, isIntegral_algHom_iff _ hf, ← Polynomial.aeval_def, aeval_algHom,
+    AlgHom.comp_apply, _root_.map_eq_zero_iff f hf]
 
 theorem algebraMap_eq {B} [CommRing B] [Algebra A B] [Algebra B B'] [IsScalarTower A B B']
     (h : Function.Injective (algebraMap B B')) (x : B) :
diff --git a/Mathlib/FieldTheory/Minpoly/Field.lean b/Mathlib/FieldTheory/Minpoly/Field.lean
index 32d3a549998d0..b129b8a0ecbc5 100644
--- a/Mathlib/FieldTheory/Minpoly/Field.lean
+++ b/Mathlib/FieldTheory/Minpoly/Field.lean
@@ -127,22 +127,47 @@ theorem eq_of_irreducible [Nontrivial B] {p : A[X]} (hp1 : Irreducible p)
   · rw [aeval_mul, hp2, zero_mul]
   · rwa [Polynomial.Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel₀]
 
-theorem add_algebraMap {B : Type*} [CommRing B] [Algebra A B] {x : B} (hx : IsIntegral A x)
+theorem add_algebraMap {B : Type*} [CommRing B] [Algebra A B] (x : B)
     (a : A) : minpoly A (x + algebraMap A B a) = (minpoly A x).comp (X - C a) := by
-  refine (minpoly.unique _ _ ((minpoly.monic hx).comp_X_sub_C _) ?_ fun q qmo hq => ?_).symm
-  · simp [aeval_comp]
-  · have : (Polynomial.aeval x) (q.comp (X + C a)) = 0 := by simpa [aeval_comp] using hq
-    have H := minpoly.min A x (qmo.comp_X_add_C _) this
-    rw [degree_eq_natDegree qmo.ne_zero,
-      degree_eq_natDegree ((minpoly.monic hx).comp_X_sub_C _).ne_zero, natDegree_comp,
-      natDegree_X_sub_C, mul_one]
-    rwa [degree_eq_natDegree (minpoly.ne_zero hx),
-      degree_eq_natDegree (qmo.comp_X_add_C _).ne_zero, natDegree_comp,
-      natDegree_X_add_C, mul_one] at H
-
-theorem sub_algebraMap {B : Type*} [CommRing B] [Algebra A B] {x : B} (hx : IsIntegral A x)
+  by_cases hx : IsIntegral A x
+  · refine (minpoly.unique _ _ ((minpoly.monic hx).comp_X_sub_C _) ?_ fun q qmo hq => ?_).symm
+    · simp [aeval_comp]
+    · have : (Polynomial.aeval x) (q.comp (X + C a)) = 0 := by simpa [aeval_comp] using hq
+      have H := minpoly.min A x (qmo.comp_X_add_C _) this
+      rw [degree_eq_natDegree qmo.ne_zero,
+        degree_eq_natDegree ((minpoly.monic hx).comp_X_sub_C _).ne_zero, natDegree_comp,
+        natDegree_X_sub_C, mul_one]
+      rwa [degree_eq_natDegree (minpoly.ne_zero hx),
+        degree_eq_natDegree (qmo.comp_X_add_C _).ne_zero, natDegree_comp,
+        natDegree_X_add_C, mul_one] at H
+  · rw [minpoly.eq_zero hx, minpoly.eq_zero, zero_comp]
+    refine fun h ↦ hx ?_
+    simpa only [add_sub_cancel_right] using IsIntegral.sub h (isIntegral_algebraMap (x := a))
+
+theorem sub_algebraMap {B : Type*} [CommRing B] [Algebra A B] {x : B}
     (a : A) : minpoly A (x - algebraMap A B a) = (minpoly A x).comp (X + C a) := by
-  simpa [sub_eq_add_neg] using add_algebraMap hx (-a)
+  simpa [sub_eq_add_neg] using add_algebraMap x (-a)
+
+theorem neg {B : Type*} [CommRing B] [Algebra A B] (x : B) :
+    minpoly A (- x) = (-1) ^ (natDegree (minpoly A x)) * (minpoly A x).comp (- X) := by
+  by_cases hx : IsIntegral A x
+  · refine (minpoly.unique _ _ ((minpoly.monic hx).neg_one_pow_natDegree_mul_comp_neg_X)
+        ?_ fun q qmo hq => ?_).symm
+    · simp [aeval_comp]
+    · have : (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (- X)) = 0 := by
+        simpa [aeval_comp] using hq
+      have H := minpoly.min A x qmo.neg_one_pow_natDegree_mul_comp_neg_X this
+      have n1 := ((minpoly.monic hx).neg_one_pow_natDegree_mul_comp_neg_X).ne_zero
+      have n2 := qmo.neg_one_pow_natDegree_mul_comp_neg_X.ne_zero
+      rw [degree_eq_natDegree qmo.ne_zero,
+        degree_eq_natDegree n1, natDegree_mul (by simp) (right_ne_zero_of_mul n1), natDegree_comp]
+      rw [degree_eq_natDegree (minpoly.ne_zero hx),
+        degree_eq_natDegree qmo.neg_one_pow_natDegree_mul_comp_neg_X.ne_zero,
+        natDegree_mul (by simp) (right_ne_zero_of_mul n2), natDegree_comp] at H
+      simpa using H
+  · rw [minpoly.eq_zero hx, minpoly.eq_zero, zero_comp]
+    · simp only [natDegree_zero, pow_zero, mul_zero]
+    · exact IsIntegral.neg_iff.not.mpr hx
 
 section AlgHomFintype
 
@@ -158,7 +183,7 @@ variable (F E K : Type*) [Field F] [Ring E] [CommRing K] [IsDomain K] [Algebra F
 -- though it isn't very computable in practice (since neither `finrank` nor `finBasis` are).
 /-- Function from Hom_K(E,L) to pi type Π (x : basis), roots of min poly of x -/
 def rootsOfMinPolyPiType (φ : E →ₐ[F] K)
-    (x : range (FiniteDimensional.finBasis F E : _ → E)) :
+    (x : range (Module.finBasis F E : _ → E)) :
     { l : K // l ∈ (minpoly F x.1).aroots K } :=
   ⟨φ x, by
     rw [mem_roots_map (minpoly.ne_zero_of_finite F x.val),
@@ -169,14 +194,14 @@ theorem aux_inj_roots_of_min_poly : Injective (rootsOfMinPolyPiType F E K) := by
   -- needs explicit coercion on the RHS
   suffices (f : E →ₗ[F] K) = (g : E →ₗ[F] K) by rwa [DFunLike.ext'_iff] at this ⊢
   rw [funext_iff] at h
-  exact LinearMap.ext_on (FiniteDimensional.finBasis F E).span_eq fun e he =>
+  exact LinearMap.ext_on (Module.finBasis F E).span_eq fun e he =>
     Subtype.ext_iff.mp (h ⟨e, he⟩)
 
 /-- Given field extensions `E/F` and `K/F`, with `E/F` finite, there are finitely many `F`-algebra
   homomorphisms `E →ₐ[K] K`. -/
 noncomputable instance AlgHom.fintype : Fintype (E →ₐ[F] K) :=
   @Fintype.ofInjective _ _
-    (Fintype.subtypeProd (finite_range (FiniteDimensional.finBasis F E)) fun e =>
+    (Fintype.subtypeProd (finite_range (Module.finBasis F E)) fun e =>
       (minpoly F e).aroots K)
     _ (aux_inj_roots_of_min_poly F E K)
 
diff --git a/Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean b/Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean
index 5abd2d46ef12c..3db3147d29b05 100644
--- a/Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean
+++ b/Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean
@@ -186,4 +186,24 @@ theorem _root_.PowerBasis.ofGenMemAdjoin'_gen (B : PowerBasis R S) (hint : IsInt
 
 end AdjoinRoot
 
+section Subring
+
+variable {K L : Type*} [Field K] [Field L] [Algebra K L]
+
+variable (A : Subring K) [IsIntegrallyClosed A] [IsFractionRing A K]
+
+-- Implementation note: `inferInstance` does not work for these.
+instance : Algebra A (integralClosure A L) := Subalgebra.algebra (integralClosure A L)
+instance : SMul A (integralClosure A L) := Algebra.toSMul
+instance : IsScalarTower A ((integralClosure A L)) L :=
+  IsScalarTower.subalgebra' A L L (integralClosure A L)
+
+/-- The minimal polynomial of `x : L` over `K` agrees with its minimal polynomial over the
+integrally closed subring `A`. -/
+theorem ofSubring (x : integralClosure A L) :
+    Polynomial.map (algebraMap A K) (minpoly A x) = minpoly K (x : L) :=
+  eq_comm.mpr (isIntegrallyClosed_eq_field_fractions K L (IsIntegralClosure.isIntegral A L x))
+
+end Subring
+
 end minpoly
diff --git a/Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean b/Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
index 317ae7f335968..fae1bec1703c0 100644
--- a/Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
+++ b/Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean
@@ -20,7 +20,7 @@ See `traceForm_dualBasis_powerBasis_eq`.
 - `span_coeff_minpolyDiv`: The coefficients of `minpolyDiv` spans `R<x>`.
 -/
 
-open Polynomial FiniteDimensional
+open Polynomial Module
 
 variable (R K) {L S} [CommRing R] [Field K] [Field L] [CommRing S] [Algebra R S] [Algebra K L]
 variable (x : S)
@@ -164,8 +164,8 @@ lemma span_coeff_minpolyDiv :
       Submodule.span_le]
     simp only [Finset.coe_image, Finset.coe_range, Set.image_subset_iff]
     intro i
-    apply Nat.strongInductionOn i
-    intro i hi hi'
+    induction i using Nat.strongRecOn with | ind i hi => ?_
+    intro hi'
     have : coeff (minpolyDiv R x) (natDegree (minpolyDiv R x) - i) ∈
         Submodule.span R (Set.range (coeff (minpolyDiv R x))) :=
       Submodule.subset_span (Set.mem_range_self _)
diff --git a/Mathlib/FieldTheory/NormalClosure.lean b/Mathlib/FieldTheory/NormalClosure.lean
index 128ea47d5bfd0..680dc94bf8709 100644
--- a/Mathlib/FieldTheory/NormalClosure.lean
+++ b/Mathlib/FieldTheory/NormalClosure.lean
@@ -6,6 +6,7 @@ Authors: Thomas Browning
 
 import Mathlib.FieldTheory.Normal
 import Mathlib.Order.Closure
+import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
 /-!
 # Normal closures
 
@@ -173,7 +174,7 @@ variable [Algebra K L] [IsScalarTower F K L]
 
 noncomputable instance algebra :
     Algebra K (normalClosure F K L) :=
-  IntermediateField.algebra
+  IntermediateField.algebra'
     { ⨆ f : K →ₐ[F] L, f.fieldRange with
       algebraMap_mem' := fun r ↦ (toAlgHom F K L).fieldRange_le_normalClosure ⟨r, rfl⟩ }
 
diff --git a/Mathlib/FieldTheory/Perfect.lean b/Mathlib/FieldTheory/Perfect.lean
index f0725115ada01..6e4aecef5c66e 100644
--- a/Mathlib/FieldTheory/Perfect.lean
+++ b/Mathlib/FieldTheory/Perfect.lean
@@ -376,7 +376,7 @@ variable [PerfectRing R p]
 a bijection from the set of roots of `Polynomial.expand R p f` to the set of roots of `f`.
 It's given by `x ↦ x ^ p`, see `rootsExpandEquivRoots_apply`. -/
 noncomputable def rootsExpandEquivRoots : (expand R p f).roots.toFinset ≃ f.roots.toFinset :=
-  ((frobeniusEquiv R p).image _).trans <| .Set.ofEq <| show _ '' (setOf _) = setOf _ by
+  ((frobeniusEquiv R p).image _).trans <| .Set.ofEq <| show _ '' setOf (· ∈ _) = setOf (· ∈ _) by
     classical simp_rw [← roots_expand_image_frobenius (p := p) (f := f), Finset.mem_val,
       Finset.setOf_mem, Finset.coe_image, RingEquiv.toEquiv_eq_coe, EquivLike.coe_coe,
       frobeniusEquiv_apply]
@@ -389,7 +389,8 @@ a bijection from the set of roots of `Polynomial.expand R (p ^ n) f` to the set
 It's given by `x ↦ x ^ (p ^ n)`, see `rootsExpandPowEquivRoots_apply`. -/
 noncomputable def rootsExpandPowEquivRoots (n : ℕ) :
     (expand R (p ^ n) f).roots.toFinset ≃ f.roots.toFinset :=
-  ((iterateFrobeniusEquiv R p n).image _).trans <| .Set.ofEq <| show _ '' (setOf _) = setOf _ by
+  ((iterateFrobeniusEquiv R p n).image _).trans <|
+    .Set.ofEq <| show _ '' (setOf (· ∈ _)) = setOf (· ∈ _) by
     classical simp_rw [← roots_expand_image_iterateFrobenius (p := p) (f := f) (n := n),
       Finset.mem_val, Finset.setOf_mem, Finset.coe_image, RingEquiv.toEquiv_eq_coe,
       EquivLike.coe_coe, iterateFrobeniusEquiv_apply]
diff --git a/Mathlib/FieldTheory/PerfectClosure.lean b/Mathlib/FieldTheory/PerfectClosure.lean
index 06db163e5553c..df62d0fc00813 100644
--- a/Mathlib/FieldTheory/PerfectClosure.lean
+++ b/Mathlib/FieldTheory/PerfectClosure.lean
@@ -317,7 +317,7 @@ theorem mk_eq_iff (x y : ℕ × K) :
     mk K p x = mk K p y ↔ ∃ z, (frobenius K p)^[y.1 + z] x.2 = (frobenius K p)^[x.1 + z] y.2 := by
   constructor
   · intro H
-    replace H := Quot.exact _ H
+    replace H := Quot.eqvGen_exact H
     induction H with
     | rel x y H => cases' H with n x; exact ⟨0, rfl⟩
     | refl H => exact ⟨0, rfl⟩
diff --git a/Mathlib/FieldTheory/PolynomialGaloisGroup.lean b/Mathlib/FieldTheory/PolynomialGaloisGroup.lean
index 3c936d1dc75f4..ef2d383b8d52b 100644
--- a/Mathlib/FieldTheory/PolynomialGaloisGroup.lean
+++ b/Mathlib/FieldTheory/PolynomialGaloisGroup.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Thomas Browning, Patrick Lutz
 -/
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 
 /-!
 # Galois Groups of Polynomials
@@ -41,7 +41,7 @@ noncomputable section
 
 open scoped Polynomial
 
-open FiniteDimensional
+open Module
 
 namespace Polynomial
 
@@ -253,7 +253,7 @@ theorem restrictDvd_surjective (hpq : p ∣ q) (hq : q ≠ 0) :
 
 variable (p q)
 
-/-- The Galois group of a product maps into the product of the Galois groups.  -/
+/-- The Galois group of a product maps into the product of the Galois groups. -/
 def restrictProd : (p * q).Gal →* p.Gal × q.Gal :=
   MonoidHom.prod (restrictDvd (dvd_mul_right p q)) (restrictDvd (dvd_mul_left q p))
 
@@ -384,10 +384,10 @@ theorem prime_degree_dvd_card [CharZero F] (p_irr : Irreducible p) (p_deg : p.na
   let α : p.SplittingField :=
     rootOfSplits (algebraMap F p.SplittingField) (SplittingField.splits p) hp
   have hα : IsIntegral F α := .of_finite F α
-  use FiniteDimensional.finrank F⟮α⟯ p.SplittingField
+  use Module.finrank F⟮α⟯ p.SplittingField
   suffices (minpoly F α).natDegree = p.natDegree by
     letI _ : AddCommGroup F⟮α⟯ := Ring.toAddCommGroup
-    rw [← FiniteDimensional.finrank_mul_finrank F F⟮α⟯ p.SplittingField,
+    rw [← Module.finrank_mul_finrank F F⟮α⟯ p.SplittingField,
       IntermediateField.adjoin.finrank hα, this]
   suffices minpoly F α ∣ p by
     have key := (minpoly.irreducible hα).dvd_symm p_irr this
diff --git a/Mathlib/FieldTheory/PrimitiveElement.lean b/Mathlib/FieldTheory/PrimitiveElement.lean
index 65b6d1707f19d..da215b5d8dbf4 100644
--- a/Mathlib/FieldTheory/PrimitiveElement.lean
+++ b/Mathlib/FieldTheory/PrimitiveElement.lean
@@ -36,7 +36,7 @@ exists_adjoin_simple_eq_top
 
 noncomputable section
 
-open FiniteDimensional Polynomial IntermediateField
+open Module Polynomial IntermediateField
 
 namespace Field
 
@@ -63,7 +63,7 @@ theorem exists_primitive_element_of_finite_top [Finite E] : ∃ α : E, F⟮α
 /-- Primitive element theorem for finite dimensional extension of a finite field. -/
 theorem exists_primitive_element_of_finite_bot [Finite F] [FiniteDimensional F E] :
     ∃ α : E, F⟮α⟯ = ⊤ :=
-  haveI : Finite E := finite_of_finite F E
+  haveI : Finite E := FiniteDimensional.finite_of_finite F E
   exists_primitive_element_of_finite_top F E
 
 end PrimitiveElementFinite
@@ -367,7 +367,7 @@ section iff
 
 namespace Field
 
-open FiniteDimensional IntermediateField Polynomial Algebra Set
+open Module IntermediateField Polynomial Algebra Set
 
 variable (F : Type*) {E : Type*} [Field F] [Field E] [Algebra F E] [FiniteDimensional F E]
 
diff --git a/Mathlib/FieldTheory/PurelyInseparable.lean b/Mathlib/FieldTheory/PurelyInseparable.lean
index a2d8f90ef45d9..93c3226de53d7 100644
--- a/Mathlib/FieldTheory/PurelyInseparable.lean
+++ b/Mathlib/FieldTheory/PurelyInseparable.lean
@@ -127,19 +127,22 @@ separable degree, degree, separable closure, purely inseparable
 
 -/
 
-open FiniteDimensional Polynomial IntermediateField Field
+open Module Polynomial IntermediateField Field Finsupp
 
 noncomputable section
 
 universe u v w
 
-variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
-variable (K : Type w) [Field K] [Algebra F K]
-
 section IsPurelyInseparable
 
+variable (F : Type u) (E : Type v) [CommRing F] [Ring E] [Algebra F E]
+variable (K : Type w) [Ring K] [Algebra F K]
+
 /-- Typeclass for purely inseparable field extensions: an algebraic extension `E / F` is purely
-inseparable if and only if the minimal polynomial of every element of `E ∖ F` is not separable. -/
+inseparable if and only if the minimal polynomial of every element of `E ∖ F` is not separable.
+
+We define this for general (commutative) rings and only assume `F` and `E` are fields
+if this is needed for a proof. -/
 class IsPurelyInseparable : Prop where
   isIntegral : Algebra.IsIntegral F E
   inseparable' (x : E) : IsSeparable F x → x ∈ (algebraMap F E).range
@@ -150,7 +153,7 @@ variable {E} in
 theorem IsPurelyInseparable.isIntegral' [IsPurelyInseparable F E] (x : E) : IsIntegral F x :=
   Algebra.IsIntegral.isIntegral _
 
-theorem IsPurelyInseparable.isAlgebraic [IsPurelyInseparable F E] :
+theorem IsPurelyInseparable.isAlgebraic [Nontrivial F] [IsPurelyInseparable F E] :
     Algebra.IsAlgebraic F E := inferInstance
 
 variable {E}
@@ -163,7 +166,7 @@ variable {F K}
 
 theorem isPurelyInseparable_iff : IsPurelyInseparable F E ↔ ∀ x : E,
     IsIntegral F x ∧ (IsSeparable F x → x ∈ (algebraMap F E).range) :=
-  ⟨fun h x ↦ ⟨h.isIntegral' x, h.inseparable' x⟩, fun h ↦ ⟨⟨fun x ↦ (h x).1⟩, fun x ↦ (h x).2⟩⟩
+  ⟨fun h x ↦ ⟨h.isIntegral' _ x, h.inseparable' x⟩, fun h ↦ ⟨⟨fun x ↦ (h x).1⟩, fun x ↦ (h x).2⟩⟩
 
 /-- Transfer `IsPurelyInseparable` across an `AlgEquiv`. -/
 theorem AlgEquiv.isPurelyInseparable (e : K ≃ₐ[F] E) [IsPurelyInseparable F K] :
@@ -179,7 +182,9 @@ theorem AlgEquiv.isPurelyInseparable_iff (e : K ≃ₐ[F] E) :
 
 /-- If `E / F` is an algebraic extension, `F` is separably closed,
 then `E / F` is purely inseparable. -/
-theorem Algebra.IsAlgebraic.isPurelyInseparable_of_isSepClosed [Algebra.IsAlgebraic F E]
+theorem Algebra.IsAlgebraic.isPurelyInseparable_of_isSepClosed
+    {F : Type u} {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E]
+    [Algebra.IsAlgebraic F E]
     [IsSepClosed F] : IsPurelyInseparable F E :=
   ⟨inferInstance, fun x h ↦ minpoly.mem_range_of_degree_eq_one F x <|
     IsSepClosed.degree_eq_one_of_irreducible F (minpoly.irreducible
@@ -194,27 +199,37 @@ theorem IsPurelyInseparable.surjective_algebraMap_of_isSeparable
 
 /-- If `E / F` is both purely inseparable and separable, then `algebraMap F E` is bijective. -/
 theorem IsPurelyInseparable.bijective_algebraMap_of_isSeparable
+    [Nontrivial E] [NoZeroSMulDivisors F E]
     [IsPurelyInseparable F E] [Algebra.IsSeparable F E] : Function.Bijective (algebraMap F E) :=
-  ⟨(algebraMap F E).injective, surjective_algebraMap_of_isSeparable F E⟩
+  ⟨NoZeroSMulDivisors.algebraMap_injective F E, surjective_algebraMap_of_isSeparable F E⟩
 
 variable {F E} in
+/-- If a subalgebra of `E / F` is both purely inseparable and separable, then it is equal
+to `F`. -/
+theorem Subalgebra.eq_bot_of_isPurelyInseparable_of_isSeparable (L : Subalgebra F E)
+    [IsPurelyInseparable F L] [Algebra.IsSeparable F L] : L = ⊥ := bot_unique fun x hx ↦ by
+  obtain ⟨y, hy⟩ := IsPurelyInseparable.surjective_algebraMap_of_isSeparable F L ⟨x, hx⟩
+  exact ⟨y, congr_arg (Subalgebra.val _) hy⟩
+
 /-- If an intermediate field of `E / F` is both purely inseparable and separable, then it is equal
 to `F`. -/
-theorem IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable (L : IntermediateField F E)
+theorem IntermediateField.eq_bot_of_isPurelyInseparable_of_isSeparable
+    {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] (L : IntermediateField F E)
     [IsPurelyInseparable F L] [Algebra.IsSeparable F L] : L = ⊥ := bot_unique fun x hx ↦ by
   obtain ⟨y, hy⟩ := IsPurelyInseparable.surjective_algebraMap_of_isSeparable F L ⟨x, hx⟩
   exact ⟨y, congr_arg (algebraMap L E) hy⟩
 
 /-- If `E / F` is purely inseparable, then the separable closure of `F` in `E` is
 equal to `F`. -/
-theorem separableClosure.eq_bot_of_isPurelyInseparable [IsPurelyInseparable F E] :
+theorem separableClosure.eq_bot_of_isPurelyInseparable
+    (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E] [IsPurelyInseparable F E] :
     separableClosure F E = ⊥ :=
   bot_unique fun x h ↦ IsPurelyInseparable.inseparable F x (mem_separableClosure_iff.1 h)
 
-variable {F E} in
 /-- If `E / F` is an algebraic extension, then the separable closure of `F` in `E` is
 equal to `F` if and only if `E / F` is purely inseparable. -/
-theorem separableClosure.eq_bot_iff [Algebra.IsAlgebraic F E] :
+theorem separableClosure.eq_bot_iff
+    {F : Type u} {E : Type v} [Field F] [Field E] [Algebra F E] [Algebra.IsAlgebraic F E] :
     separableClosure F E = ⊥ ↔ IsPurelyInseparable F E :=
   ⟨fun h ↦ isPurelyInseparable_iff.2 fun x ↦ ⟨Algebra.IsIntegral.isIntegral x, fun hs ↦ by
     simpa only [h] using mem_separableClosure_iff.2 hs⟩, fun _ ↦ eq_bot_of_isPurelyInseparable F E⟩
@@ -222,12 +237,15 @@ theorem separableClosure.eq_bot_iff [Algebra.IsAlgebraic F E] :
 instance isPurelyInseparable_self : IsPurelyInseparable F F :=
   ⟨inferInstance, fun x _ ↦ ⟨x, rfl⟩⟩
 
-variable {E}
+section
+
+variable (F : Type u) {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E]
+variable (q : ℕ) [ExpChar F q] (x : E)
 
 /-- A field extension `E / F` of exponential characteristic `q` is purely inseparable
 if and only if for every element `x` of `E`, there exists a natural number `n` such that
 `x ^ (q ^ n)` is contained in `F`. -/
-theorem isPurelyInseparable_iff_pow_mem (q : ℕ) [ExpChar F q] :
+theorem isPurelyInseparable_iff_pow_mem :
     IsPurelyInseparable F E ↔ ∀ x : E, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by
   rw [isPurelyInseparable_iff]
   refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
@@ -238,16 +256,19 @@ theorem isPurelyInseparable_iff_pow_mem (q : ℕ) [ExpChar F q] :
   have halg : IsIntegral F x := by_contra fun h' ↦ by
     simp only [minpoly.eq_zero h', natSepDegree_zero, zero_ne_one] at hdeg
   refine ⟨halg, fun hsep ↦ ?_⟩
-  rw [hsep.natSepDegree_eq_natDegree, ← adjoin.finrank halg,
-    IntermediateField.finrank_eq_one_iff] at hdeg
-  simpa only [hdeg] using mem_adjoin_simple_self F x
+  rwa [hsep.natSepDegree_eq_natDegree, minpoly.natDegree_eq_one_iff] at hdeg
 
-theorem IsPurelyInseparable.pow_mem (q : ℕ) [ExpChar F q] [IsPurelyInseparable F E] (x : E) :
+theorem IsPurelyInseparable.pow_mem [IsPurelyInseparable F E] :
     ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range :=
   (isPurelyInseparable_iff_pow_mem F q).1 ‹_› x
 
+end
+
 end IsPurelyInseparable
 
+variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
+variable (K : Type w) [Field K] [Algebra F K]
+
 section perfectClosure
 
 /-- The relative perfect closure of `F` in `E`, consists of the elements `x` of `E` such that there
@@ -282,7 +303,7 @@ theorem mem_perfectClosure_iff_pow_mem (q : ℕ) [ExpChar F q] {x : E} :
     x ∈ perfectClosure F E ↔ ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by
   rw [mem_perfectClosure_iff, ringExpChar.eq F q]
 
-/-- An element is contained in the relative perfect closure if and only if its mininal polynomial
+/-- An element is contained in the relative perfect closure if and only if its minimal polynomial
 has separable degree one. -/
 theorem mem_perfectClosure_iff_natSepDegree_eq_one {x : E} :
     x ∈ perfectClosure F E ↔ (minpoly F x).natSepDegree = 1 := by
@@ -583,7 +604,7 @@ if `E` is purely inseparable over it. -/
 theorem separableClosure_le (L : IntermediateField F E)
     [h : IsPurelyInseparable L E] : separableClosure F E ≤ L := fun x hx ↦ by
   obtain ⟨y, rfl⟩ := h.inseparable' _ <|
-    IsSeparable.of_isScalarTower L (mem_separableClosure_iff.1 hx)
+    IsSeparable.tower_top L (mem_separableClosure_iff.1 hx)
   exact y.2
 
 /-- If `E / F` is algebraic, then an intermediate field of `E / F` contains the
@@ -629,7 +650,7 @@ namespace IntermediateField
 instance isPurelyInseparable_bot : IsPurelyInseparable F (⊥ : IntermediateField F E) :=
   (botEquiv F E).symm.isPurelyInseparable
 
-/-- `F⟮x⟯ / F` is a purely inseparable extension if and only if the mininal polynomial of `x`
+/-- `F⟮x⟯ / F` is a purely inseparable extension if and only if the minimal polynomial of `x`
 has separable degree one. -/
 theorem isPurelyInseparable_adjoin_simple_iff_natSepDegree_eq_one {x : E} :
     IsPurelyInseparable F F⟮x⟯ ↔ (minpoly F x).natSepDegree = 1 := by
@@ -735,13 +756,14 @@ private theorem LinearIndependent.map_pow_expChar_pow_of_fd_isSeparable
   have h' := h.coe_range
   let ι' := h'.extend (Set.range v).subset_univ
   let b : Basis ι' F E := Basis.extend h'
-  letI : Fintype ι' := fintypeBasisIndex b
+  letI : Fintype ι' := FiniteDimensional.fintypeBasisIndex b
   have H := linearIndependent_of_top_le_span_of_card_eq_finrank
     (span_map_pow_expChar_pow_eq_top_of_isSeparable q n b.span_eq).ge
     (finrank_eq_card_basis b).symm
   let f (i : ι) : ι' := ⟨v i, h'.subset_extend _ ⟨i, rfl⟩⟩
   convert H.comp f fun _ _ heq ↦ h.injective (by simpa only [f, Subtype.mk.injEq] using heq)
-  simp_rw [Function.comp_apply, b, Basis.extend_apply_self]
+  simp_rw [Function.comp_apply, b]
+  rw [Basis.extend_apply_self]
 
 /-- If `E / F` is a separable extension of exponential characteristic `q`, if `{ u_i }` is a
 family of elements of `E` which is `F`-linearly independent, then `{ u_i ^ (q ^ n) }` is also
@@ -920,8 +942,8 @@ theorem LinearIndependent.map_of_isPurelyInseparable_of_isSeparable [IsPurelyIns
     rw [hlF, Finsupp.not_mem_support_iff.1 hs, zero_pow this]
   replace h := linearIndependent_iff.1 (h.map_pow_expChar_pow_of_isIntegral' q n hsep) lF₀ <| by
     replace hl := congr($hl ^ q ^ n)
-    rw [Finsupp.total_apply, Finsupp.sum, sum_pow_char_pow, zero_pow this] at hl
-    rw [← hl, Finsupp.total_apply, Finsupp.onFinset_sum _ (fun _ ↦ by exact zero_smul _ _)]
+    rw [linearCombination_apply, Finsupp.sum, sum_pow_char_pow, zero_pow this] at hl
+    rw [← hl, linearCombination_apply, onFinset_sum _ (fun _ ↦ by exact zero_smul _ _)]
     refine Finset.sum_congr rfl fun i _ ↦ ?_
     simp_rw [Algebra.smul_def, mul_pow, IsScalarTower.algebraMap_apply F E K, hlF, map_pow]
   refine pow_eq_zero ((hlF _).symm.trans ?_)
@@ -1049,7 +1071,7 @@ theorem minpoly.map_eq_of_isSeparable_of_isPurelyInseparable (x : K)
   have hi' : IsIntegral E x := IsIntegral.tower_top hi
   refine eq_of_monic_of_dvd_of_natDegree_le (monic hi') ((monic hi).map (algebraMap F E))
     (dvd_map_of_isScalarTower F E x) (le_of_eq ?_)
-  have hsep' := IsSeparable.of_isScalarTower E hsep
+  have hsep' := IsSeparable.tower_top E hsep
   haveI := (isSeparable_adjoin_simple_iff_isSeparable _ _).2 hsep
   haveI := (isSeparable_adjoin_simple_iff_isSeparable _ _).2 hsep'
   have := Algebra.IsSeparable.isAlgebraic F F⟮x⟯
diff --git a/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean b/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
index 12d1f729cadc0..14a02db0ca588 100644
--- a/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
+++ b/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
@@ -213,7 +213,7 @@ end Polynomial
 
 namespace RatFunc
 
-open scoped DiscreteValuation
+open scoped Multiplicative
 
 open Polynomial
 
diff --git a/Mathlib/FieldTheory/RatFunc/Basic.lean b/Mathlib/FieldTheory/RatFunc/Basic.lean
index abc89f3acb54f..477f652402ff1 100644
--- a/Mathlib/FieldTheory/RatFunc/Basic.lean
+++ b/Mathlib/FieldTheory/RatFunc/Basic.lean
@@ -638,16 +638,8 @@ theorem algebraMap_injective : Function.Injective (algebraMap K[X] (RatFunc K))
   rw [← ofFractionRing_comp_algebraMap]
   exact ofFractionRing_injective.comp (IsFractionRing.injective _ _)
 
-@[simp]
-theorem algebraMap_eq_zero_iff {x : K[X]} : algebraMap K[X] (RatFunc K) x = 0 ↔ x = 0 :=
-  ⟨(injective_iff_map_eq_zero _).mp (algebraMap_injective K) _, fun hx => by
-    rw [hx, RingHom.map_zero]⟩
-
 variable {K}
 
-theorem algebraMap_ne_zero {x : K[X]} (hx : x ≠ 0) : algebraMap K[X] (RatFunc K) x ≠ 0 :=
-  mt (algebraMap_eq_zero_iff K).mp hx
-
 section LiftAlgHom
 
 variable {L R S : Type*} [Field L] [CommRing R] [IsDomain R] [CommSemiring S] [Algebra S K[X]]
@@ -712,8 +704,15 @@ instance : IsFractionRing K[X] (RatFunc K) where
     simp only [← ofFractionRing_algebraMap, Function.comp_apply, ← ofFractionRing_mul]
     rw [ofFractionRing.injEq]  -- Porting note: added
 
+@[deprecated "Use NoZeroSMulDivisors.algebraMap_eq_zero_iff instead." (since := "2024-09-08")]
+theorem algebraMap_eq_zero_iff {x : K[X]} : algebraMap K[X] (RatFunc K) x = 0 ↔ x = 0 := by
+  simp
+
 variable {K}
 
+theorem algebraMap_ne_zero {x : K[X]} (hx : x ≠ 0) : algebraMap K[X] (RatFunc K) x ≠ 0 := by
+  simpa
+
 @[simp]
 theorem liftOn_div {P : Sort v} (p q : K[X]) (f : K[X] → K[X] → P) (f0 : ∀ p, f p 0 = f 0 1)
     (H' : ∀ {p q p' q'} (_hq : q ≠ 0) (_hq' : q' ≠ 0), q' * p = q * p' → f p q = f p' q')
diff --git a/Mathlib/FieldTheory/RatFunc/Defs.lean b/Mathlib/FieldTheory/RatFunc/Defs.lean
index b1ffa6109ae6d..b45d1838b3eb7 100644
--- a/Mathlib/FieldTheory/RatFunc/Defs.lean
+++ b/Mathlib/FieldTheory/RatFunc/Defs.lean
@@ -63,7 +63,7 @@ the maps between `RatFunc K` and another field of fractions of `K[X]`,
 especially `FractionRing K[X]`, are given by `IsLocalization.algEquiv`.
 -/
 structure RatFunc [CommRing K] : Type u where ofFractionRing ::
-/-- the coercion to the fraction ring of the polynomial ring-/
+/-- the coercion to the fraction ring of the polynomial ring -/
   toFractionRing : FractionRing K[X]
 
 namespace RatFunc
diff --git a/Mathlib/FieldTheory/Separable.lean b/Mathlib/FieldTheory/Separable.lean
index fddc5fbc1dcee..11f14b5edd03f 100644
--- a/Mathlib/FieldTheory/Separable.lean
+++ b/Mathlib/FieldTheory/Separable.lean
@@ -22,7 +22,7 @@ properties about separable polynomials here.
 * `IsSeparable K x`: an element `x` is separable over `K` iff the minimal polynomial of `x`
 over `K` is separable.
 * `Algebra.IsSeparable K L`: `L` is separable over `K` iff every element in `L` is separable
-over `K`
+over `K`.
 
 -/
 
@@ -337,8 +337,7 @@ theorem separable_or {f : F[X]} (hf : Irreducible f) :
     exact
       Or.inr
         ⟨by rw [separable_iff_derivative_ne_zero hf, Classical.not_not, H], contract p f,
-          of_irreducible_map (expand F p : F[X] →+* F[X])
-            (by rwa [← expand_contract p H hp.ne'] at hf),
+          Irreducible.of_map (by rwa [← expand_contract p H hp.ne'] at hf),
           expand_contract p H hp.ne'⟩
   else Or.inl <| (separable_iff_derivative_ne_zero hf).2 H
 
@@ -503,7 +502,7 @@ open Polynomial
 
 section CommRing
 
-variable (F K : Type*) [CommRing F] [Ring K] [Algebra F K]
+variable (F L K : Type*) [CommRing F] [Ring K] [Algebra F K]
 
 -- TODO: refactor to allow transcendental extensions?
 -- See: https://en.wikipedia.org/wiki/Separable_extension#Separability_of_transcendental_extensions
@@ -514,7 +513,7 @@ variable (F K : Type*) [CommRing F] [Ring K] [Algebra F K]
 variable {K} in
 /--
 An element `x` of an algebra `K` over a commutative ring `F` is said to be *separable*, if its
-minimal polynamial over `K` is separable. Note that the minimal polynomial of any element not
+minimal polynomial over `K` is separable. Note that the minimal polynomial of any element not
 integral over `F` is defined to be `0`, which is not a separable polynomial.
 -/
 def IsSeparable (x : K) : Prop := Polynomial.Separable (minpoly F x)
@@ -547,6 +546,11 @@ theorem IsSeparable.isIntegral {x : K} (h : IsSeparable F x) : IsIntegral F x :=
 theorem Algebra.IsSeparable.isIntegral [Algebra.IsSeparable F K] : ∀ x : K, IsIntegral F x :=
   fun x ↦ _root_.IsSeparable.isIntegral (Algebra.IsSeparable.isSeparable F x)
 
+variable (K) in
+instance Algebra.IsSeparable.isAlgebraic [Nontrivial F] [Algebra.IsSeparable F K] :
+    Algebra.IsAlgebraic F K :=
+  ⟨fun x ↦ (Algebra.IsSeparable.isIntegral F x).isAlgebraic⟩
+
 variable {F}
 
 theorem Algebra.isSeparable_iff :
@@ -554,72 +558,132 @@ theorem Algebra.isSeparable_iff :
   ⟨fun _ x => ⟨Algebra.IsSeparable.isIntegral F x, Algebra.IsSeparable.isSeparable F x⟩,
     fun h => ⟨fun x => (h x).2⟩⟩
 
-variable {E : Type*} [Ring E] [Algebra F E] (e : K ≃ₐ[F] E)
+variable {E : Type*}
+
+section AlgEquiv
+
+variable [Ring E] [Algebra F E] (e : K ≃ₐ[F] E)
 include e
 
+/-- Transfer `IsSeparable` across an `AlgEquiv`. -/
+theorem AlgEquiv.isSeparable_iff {x : K} : IsSeparable F (e x) ↔ IsSeparable F x := by
+  simp only [IsSeparable, minpoly.algEquiv_eq e x]
+
 /-- Transfer `Algebra.IsSeparable` across an `AlgEquiv`. -/
-theorem AlgEquiv.isSeparable [Algebra.IsSeparable F K] : Algebra.IsSeparable F E :=
-  ⟨fun _ ↦
-    by rw [IsSeparable, ← minpoly.algEquiv_eq e.symm]; exact Algebra.IsSeparable.isSeparable F _⟩
+theorem AlgEquiv.Algebra.isSeparable [Algebra.IsSeparable F K] : Algebra.IsSeparable F E :=
+  ⟨fun _ ↦ e.symm.isSeparable_iff.mp (Algebra.IsSeparable.isSeparable _ _)⟩
 
-theorem AlgEquiv.isSeparable_iff : Algebra.IsSeparable F K ↔ Algebra.IsSeparable F E :=
-  ⟨fun _ ↦ e.isSeparable, fun _ ↦ e.symm.isSeparable⟩
+@[deprecated (since := "2024-08-06")]
+alias AlgEquiv.isSeparable := AlgEquiv.Algebra.isSeparable
 
-variable (F K)
+theorem AlgEquiv.Algebra.isSeparable_iff : Algebra.IsSeparable F K ↔ Algebra.IsSeparable F E :=
+  ⟨fun _ ↦ AlgEquiv.Algebra.isSeparable e, fun _ ↦ AlgEquiv.Algebra.isSeparable e.symm⟩
 
-instance Algebra.IsSeparable.isAlgebraic [Nontrivial F] [Algebra.IsSeparable F K] :
-    Algebra.IsAlgebraic F K :=
-  ⟨fun x ↦ (Algebra.IsSeparable.isIntegral F x).isAlgebraic⟩
+end AlgEquiv
+
+section IsScalarTower
+
+variable [Field L] [CommRing E] [Algebra F L]
+    [Algebra F E] [Algebra L E] [IsScalarTower F L E]
+
+/-- If `E / L / F` is a scalar tower and `x : E` is separable over `F`, then it's also separable
+over `L`. -/
+theorem IsSeparable.tower_top
+    {x : E} (h : IsSeparable F x) : IsSeparable L x :=
+  h.map.of_dvd (minpoly.dvd_map_of_isScalarTower _ _ _)
+
+variable (F E) in
+theorem Algebra.isSeparable_tower_top_of_isSeparable [Algebra.IsSeparable F E] :
+    Algebra.IsSeparable L E :=
+  ⟨fun x ↦ IsSeparable.tower_top _ (Algebra.IsSeparable.isSeparable F x)⟩
+
+@[deprecated (since := "2024-08-06")]
+alias IsSeparable.of_isScalarTower := Algebra.isSeparable_tower_top_of_isSeparable
+
+end IsScalarTower
 
 end CommRing
 
-instance Algebra.isSeparable_self (F : Type*) [Field F] : Algebra.IsSeparable F F :=
-  ⟨fun x => by
-    rw [IsSeparable, minpoly.eq_X_sub_C']
-    exact separable_X_sub_C⟩
+section Field
+
+variable (F : Type*) [Field F] {K E E' : Type*}
+
+section IsIntegral
+
+variable [Ring K] [Algebra F K]
+
+variable {F} in
+theorem isSeparable_algebraMap (x : F) : IsSeparable F (algebraMap F K x) :=
+  Polynomial.Separable.of_dvd (Polynomial.separable_X_sub_C (x := x))
+    (minpoly.dvd F (algebraMap F K x) (by simp only [map_sub, aeval_X, aeval_C, sub_self]))
+
+instance Algebra.isSeparable_self : Algebra.IsSeparable F F :=
+  ⟨isSeparable_algebraMap⟩
+
+variable [IsDomain K] [Algebra.IsIntegral F K] [CharZero F]
+
+theorem IsSeparable.of_integral (x : K) : IsSeparable F x :=
+  (minpoly.irreducible <| Algebra.IsIntegral.isIntegral x).separable
 
 -- See note [lower instance priority]
-/-- A finite field extension in characteristic 0 is separable. -/
-instance (priority := 100) Algebra.IsSeparable.of_finite (F K : Type*) [Field F] [Field K]
-    [Algebra F K] [FiniteDimensional F K] [CharZero F] : Algebra.IsSeparable F K :=
-  ⟨fun x => (minpoly.irreducible <| .of_finite F x).separable⟩
-
-section IsSeparableTower
-
-/-- If `R / K / A` is an extension tower, `x : R` is separable over `A`, then it's also separable
-over `K`. -/
-theorem IsSeparable.of_isScalarTower {A : Type*} [CommRing A]
-    (K : Type*) [Field K] [Algebra A K] {R : Type*} [CommRing R] [Algebra A R] [Algebra K R]
-    [IsScalarTower A K R] {x : R} (h : IsSeparable A x) : IsSeparable K x :=
-  h.map.of_dvd (minpoly.dvd_map_of_isScalarTower _ _ _)
+variable (K) in
+/-- A integral field extension in characteristic 0 is separable. -/
+protected instance (priority := 100) Algebra.IsSeparable.of_integral : Algebra.IsSeparable F K :=
+  ⟨_root_.IsSeparable.of_integral _⟩
 
-variable (F K E : Type*) [Field F] [Field K] [Field E] [Algebra F K] [Algebra F E] [Algebra K E]
-  [IsScalarTower F K E]
+end IsIntegral
 
-theorem Algebra.isSeparable_tower_top_of_isSeparable [Algebra.IsSeparable F E] :
-    Algebra.IsSeparable K E :=
-  ⟨fun x ↦ IsSeparable.of_isScalarTower _ (Algebra.IsSeparable.isSeparable F x)⟩
+section IsScalarTower
 
-theorem Algebra.isSeparable_tower_bot_of_isSeparable [h : Algebra.IsSeparable F E] :
-    Algebra.IsSeparable F K :=
-  ⟨fun x ↦
+variable [Field K] [Ring E] [Algebra F K] [Algebra F E] [Algebra K E]
+  [Nontrivial E] [IsScalarTower F K E]
+
+variable {F} in
+/-- If `E / K / F` is a scalar tower and `algebraMap K E x` is separable over `F`, then `x` is
+``
+also separable over `F`. -/
+theorem IsSeparable.tower_bot {x : K} (h : IsSeparable F (algebraMap K E x)) : IsSeparable F x :=
     have ⟨_q, hq⟩ :=
       minpoly.dvd F x
         ((aeval_algebraMap_eq_zero_iff _ _ _).mp (minpoly.aeval F ((algebraMap K E) x)))
-    (Eq.mp (congrArg Separable hq) (h.isSeparable _)).of_mul_left⟩
+    (Eq.mp (congrArg Separable hq) h).of_mul_left
+
+variable (K E) in
+theorem Algebra.isSeparable_tower_bot_of_isSeparable [h : Algebra.IsSeparable F E] :
+    Algebra.IsSeparable F K :=
+  ⟨fun _ ↦ IsSeparable.tower_bot (h.isSeparable _ _)⟩
 
-variable {E}
+end IsScalarTower
 
-theorem Algebra.IsSeparable.of_algHom (E' : Type*) [Field E'] [Algebra F E'] (f : E →ₐ[F] E')
-    [Algebra.IsSeparable F E'] : Algebra.IsSeparable F E := by
-  letI : Algebra E E' := RingHom.toAlgebra f.toRingHom
+section
+
+variable [Field E] [Field E'] [Algebra F E] [Algebra F E']
+    (f : E →ₐ[F] E')
+include f
+
+variable {F} in
+theorem IsSeparable.of_algHom {x : E} (h : IsSeparable F (f x)) : IsSeparable F x := by
+  let _ : Algebra E E' := RingHom.toAlgebra f.toRingHom
   haveI : IsScalarTower F E E' := IsScalarTower.of_algebraMap_eq fun x => (f.commutes x).symm
-  exact Algebra.isSeparable_tower_bot_of_isSeparable F E E'
+  exact h.tower_bot
+
+
+variable (E') in
+theorem Algebra.IsSeparable.of_algHom [Algebra.IsSeparable F E'] : Algebra.IsSeparable F E :=
+  ⟨fun x => (Algebra.IsSeparable.isSeparable F (f x)).of_algHom⟩
+
+end
 
-lemma Algebra.IsSeparable.of_equiv_equiv {A₁ B₁ A₂ B₂ : Type*} [Field A₁] [Field B₁]
+end Field
+
+section AlgEquiv
+
+variable {A₁ B₁ A₂ B₂ : Type*} [Field A₁] [Field B₁]
     [Field A₂] [Field B₂] [Algebra A₁ B₁] [Algebra A₂ B₂] (e₁ : A₁ ≃+* A₂) (e₂ : B₁ ≃+* B₂)
     (he : RingHom.comp (algebraMap A₂ B₂) ↑e₁ = RingHom.comp ↑e₂ (algebraMap A₁ B₁))
-    [Algebra.IsSeparable A₁ B₁] : Algebra.IsSeparable A₂ B₂ := by
+include e₁ e₂ he
+
+lemma IsSeparable.of_equiv_equiv {x : B₁} (h : IsSeparable A₁ x) : IsSeparable A₂ (e₂ x) :=
   letI := e₁.toRingHom.toAlgebra
   letI := ((algebraMap A₁ B₁).comp e₁.symm.toRingHom).toAlgebra
   haveI : IsScalarTower A₁ A₂ B₁ := IsScalarTower.of_algebraMap_eq
@@ -628,10 +692,15 @@ lemma Algebra.IsSeparable.of_equiv_equiv {A₁ B₁ A₂ B₂ : Type*} [Field A
     { e₂ with
       commutes' := fun r ↦ by
         simpa [RingHom.algebraMap_toAlgebra] using DFunLike.congr_fun he.symm (e₁.symm r) }
-  haveI := Algebra.isSeparable_tower_top_of_isSeparable A₁ A₂ B₁
-  exact Algebra.IsSeparable.of_algHom _ _ e.symm.toAlgHom
+  have := IsSeparable.tower_top A₂ h
+  IsSeparable.of_algHom e.symm ((e₂.symm_apply_apply x).symm ▸ this)
+
+lemma Algebra.IsSeparable.of_equiv_equiv
+    [Algebra.IsSeparable A₁ B₁] : Algebra.IsSeparable A₂ B₂ :=
+  ⟨fun x ↦ (e₂.apply_symm_apply x) ▸ _root_.IsSeparable.of_equiv_equiv e₁ e₂ he
+    (Algebra.IsSeparable.isSeparable _ _)⟩
 
-end IsSeparableTower
+end AlgEquiv
 
 section CardAlgHom
 
diff --git a/Mathlib/FieldTheory/SeparableClosure.lean b/Mathlib/FieldTheory/SeparableClosure.lean
index 35439789c6f2a..68110800b109d 100644
--- a/Mathlib/FieldTheory/SeparableClosure.lean
+++ b/Mathlib/FieldTheory/SeparableClosure.lean
@@ -59,7 +59,7 @@ separable degree, degree, separable closure
 
 -/
 
-open FiniteDimensional Polynomial IntermediateField Field
+open Module Polynomial IntermediateField Field
 
 noncomputable section
 
@@ -115,7 +115,7 @@ theorem separableClosure.map_eq_of_separableClosure_eq_bot [Algebra E K] [IsScal
     (separableClosure F E).map (IsScalarTower.toAlgHom F E K) = separableClosure F K := by
   refine le_antisymm (map_le_of_algHom _) (fun x hx ↦ ?_)
   obtain ⟨y, rfl⟩ := mem_bot.1 <| h ▸ mem_separableClosure_iff.2
-    (IsSeparable.of_isScalarTower E <| mem_separableClosure_iff.1 hx)
+    (IsSeparable.tower_top E <| mem_separableClosure_iff.1 hx)
   exact ⟨y, (map_mem_separableClosure_iff <| IsScalarTower.toAlgHom F E K).mp hx, rfl⟩
 
 /-- If `i` is an `F`-algebra isomorphism of `E` and `K`, then the image of `separableClosure F E`
@@ -170,7 +170,8 @@ theorem separableClosure.separableClosure_eq_bot :
 theorem separableClosure.normalClosure_eq_self :
     normalClosure F (separableClosure F E) E = separableClosure F E :=
   le_antisymm (normalClosure_le_iff.2 fun i ↦
-    haveI : Algebra.IsSeparable F i.fieldRange := (AlgEquiv.ofInjectiveField i).isSeparable
+    have : Algebra.IsSeparable F i.fieldRange :=
+      (AlgEquiv.Algebra.isSeparable (AlgEquiv.ofInjectiveField i))
     le_separableClosure F E _) (le_normalClosure _)
 
 /-- If `E` is normal over `F`, then the separable closure of `F` in `E` is Galois (i.e.
@@ -219,7 +220,7 @@ theorem separableClosure.eq_top_iff : separableClosure F E = ⊤ ↔ Algebra.IsS
 `separableClosure E K`. -/
 theorem separableClosure.le_restrictScalars [Algebra E K] [IsScalarTower F E K] :
     separableClosure F K ≤ (separableClosure E K).restrictScalars F :=
-  fun _ h ↦ IsSeparable.of_isScalarTower E h
+  fun _ ↦ IsSeparable.tower_top E
 
 /-- If `K / E / F` is a field extension tower, such that `E / F` is separable, then
 `separableClosure F K` is equal to `separableClosure E K`. -/
diff --git a/Mathlib/FieldTheory/SeparableDegree.lean b/Mathlib/FieldTheory/SeparableDegree.lean
index e9e5e31f32d7d..fb1ec8ab2843c 100644
--- a/Mathlib/FieldTheory/SeparableDegree.lean
+++ b/Mathlib/FieldTheory/SeparableDegree.lean
@@ -67,7 +67,7 @@ This file contains basics about the separable degree of a field extension.
   if `K / E / F` is a field extension tower, such that `K / E` is algebraic,
   then there is a non-canonical bijection `Field.Emb F E × Field.Emb E K ≃ Field.Emb F K`.
   In particular, the separable degrees satisfy the tower law: $[E:F]_s [K:E]_s = [K:F]_s$
-  (see also `FiniteDimensional.finrank_mul_finrank`).
+  (see also `Module.finrank_mul_finrank`).
 
 - `Polynomial.natSepDegree_le_natDegree`: the separable degree of a polynomial is smaller than
   its degree.
@@ -118,7 +118,7 @@ separable degree, degree, polynomial
 
 -/
 
-open FiniteDimensional Polynomial IntermediateField Field
+open Module Polynomial IntermediateField Field
 
 noncomputable section
 
@@ -208,7 +208,7 @@ def embEquivOfAdjoinSplits {S : Set E} (hS : adjoin F S = ⊤)
     (hS ▸ isAlgebraic_adjoin (S := S) fun x hx ↦ (hK x hx).1)
   have halg := (topEquiv (F := F) (E := E)).isAlgebraic
   Classical.choice <| Function.Embedding.antisymm
-    (halg.algHomEmbeddingOfSplits (fun _ ↦ splits_of_mem_adjoin F (S := S) hK (hS ▸ mem_top)) _)
+    (halg.algHomEmbeddingOfSplits (fun _ ↦ splits_of_mem_adjoin F E (S := S) hK (hS ▸ mem_top)) _)
     (halg.algHomEmbeddingOfSplits (fun _ ↦ IsAlgClosed.splits_codomain _) _)
 
 /-- The `Field.finSepDegree F E` is equal to the cardinality of `E →ₐ[F] K`
@@ -246,7 +246,7 @@ def embProdEmbOfIsAlgebraic [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgeb
 
 /-- If `K / E / F` is a field extension tower, such that `K / E` is algebraic, then their
 separable degrees satisfy the tower law
-$[E:F]_s [K:E]_s = [K:F]_s$. See also `FiniteDimensional.finrank_mul_finrank`. -/
+$[E:F]_s [K:E]_s = [K:F]_s$. See also `Module.finrank_mul_finrank`. -/
 theorem finSepDegree_mul_finSepDegree_of_isAlgebraic
     [Algebra E K] [IsScalarTower F E K] [Algebra.IsAlgebraic E K] :
     finSepDegree F E * finSepDegree E K = finSepDegree F K := by
@@ -589,7 +589,7 @@ end Polynomial
 
 namespace minpoly
 
-variable {F E}
+variable {F : Type u} {E : Type v} [Field F] [Ring E] [IsDomain E] [Algebra F E]
 variable (q : ℕ) [hF : ExpChar F q] {x : E}
 
 /-- The minimal polynomial of an element of `E / F` of exponential characteristic `q` has
@@ -632,17 +632,18 @@ separable degree one if and only if the minimal polynomial is of the form
 theorem natSepDegree_eq_one_iff_eq_X_sub_C_pow : (minpoly F x).natSepDegree = 1 ↔
     ∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n := by
   haveI := expChar_of_injective_algebraMap (algebraMap F E).injective q
-  haveI := expChar_of_injective_algebraMap (NoZeroSMulDivisors.algebraMap_injective E E[X]) q
+  haveI := expChar_of_injective_ringHom (C_injective (R := E)) q
   refine ⟨fun h ↦ ?_, fun ⟨n, h⟩ ↦ (natSepDegree_eq_one_iff_pow_mem q).2 ?_⟩
   · obtain ⟨n, y, h⟩ := (natSepDegree_eq_one_iff_eq_X_pow_sub_C q).1 h
     have hx := congr_arg (Polynomial.aeval x) h.symm
     rw [minpoly.aeval, map_sub, map_pow, aeval_X, aeval_C, sub_eq_zero, eq_comm] at hx
     use n
-    rw [h, Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, hx, map_pow, ← sub_pow_expChar_pow]
+    rw [h, Polynomial.map_sub, Polynomial.map_pow, map_X, map_C, hx, map_pow,
+      ← sub_pow_expChar_pow_of_commute E[X] X (C x) (commute_X _)]
   apply_fun constantCoeff at h
   simp_rw [map_pow, map_sub, constantCoeff_apply, coeff_map, coeff_X_zero, coeff_C_zero] at h
   rw [zero_sub, neg_pow, ExpChar.neg_one_pow_expChar_pow] at h
-  exact ⟨n, -(minpoly F x).coeff 0, by rw [map_neg, h]; ring1⟩
+  exact ⟨n, -(minpoly F x).coeff 0, by rw [map_neg, h, neg_mul, one_mul, neg_neg]⟩
 
 end minpoly
 
@@ -700,7 +701,7 @@ theorem finSepDegree_dvd_finrank : finSepDegree F E ∣ finrank F E := by
     set M := L⟮x⟯
     have := Algebra.IsAlgebraic.of_finite L M
     rwa [finSepDegree_mul_finSepDegree_of_isAlgebraic F L M,
-      FiniteDimensional.finrank_mul_finrank F L M] at hdvd
+      Module.finrank_mul_finrank F L M] at hdvd
   rw [finrank_of_infinite_dimensional hfd]
   exact dvd_zero _
 
@@ -730,11 +731,11 @@ theorem finSepDegree_eq_finrank_of_isSeparable [Algebra.IsSeparable F E] :
   simp only at h ⊢
   have heq : _ * _ = _ * _ := congr_arg₂ (· * ·) h <|
     (finSepDegree_adjoin_simple_eq_finrank_iff L E x (IsAlgebraic.of_finite L x)).2 <|
-      IsSeparable.of_isScalarTower L (Algebra.IsSeparable.isSeparable F x)
+      IsSeparable.tower_top L (Algebra.IsSeparable.isSeparable F x)
   set M := L⟮x⟯
   have := Algebra.IsAlgebraic.of_finite L M
   rwa [finSepDegree_mul_finSepDegree_of_isAlgebraic F L M,
-    FiniteDimensional.finrank_mul_finrank F L M] at heq
+    Module.finrank_mul_finrank F L M] at heq
 
 alias Algebra.IsSeparable.finSepDegree_eq := finSepDegree_eq_finrank_of_isSeparable
 
@@ -748,7 +749,7 @@ theorem finSepDegree_eq_finrank_iff [FiniteDimensional F E] :
       (finSepDegree_adjoin_simple_le_finrank F E x halg) <| le_of_not_lt fun h ↦ ?_
     have := Nat.mul_lt_mul_of_lt_of_le' h (finSepDegree_le_finrank F⟮x⟯ E) Fin.size_pos'
     rw [finSepDegree_mul_finSepDegree_of_isAlgebraic F F⟮x⟯ E,
-      FiniteDimensional.finrank_mul_finrank F F⟮x⟯ E] at this
+      Module.finrank_mul_finrank F F⟮x⟯ E] at this
     linarith only [heq, this]⟩, fun _ ↦ finSepDegree_eq_finrank_of_isSeparable F E⟩
 
 end Field
@@ -795,7 +796,7 @@ theorem IsSeparable.of_algebra_isSeparable_of_isSeparable [Algebra E K] [IsScala
   have := finSepDegree_mul_finSepDegree_of_isAlgebraic F E' E'⟮x⟯
   rw [finSepDegree_eq_finrank_of_isSeparable F E',
     finSepDegree_eq_finrank_of_isSeparable E' E'⟮x⟯,
-    FiniteDimensional.finrank_mul_finrank F E' E'⟮x⟯,
+    Module.finrank_mul_finrank F E' E'⟮x⟯,
     eq_comm, finSepDegree_eq_finrank_iff F E'⟮x⟯] at this
   change Algebra.IsSeparable F (restrictScalars F E'⟮x⟯) at this
   exact isSeparable_of_mem_isSeparable F K hx
@@ -812,7 +813,7 @@ theorem IntermediateField.isSeparable_adjoin_pair_of_isSeparable {x y : E}
     (hx : IsSeparable F x) (hy : IsSeparable F y) :
     Algebra.IsSeparable F F⟮x, y⟯ := by
   rw [← adjoin_simple_adjoin_simple]
-  replace hy := IsSeparable.of_isScalarTower F⟮x⟯ hy
+  replace hy := IsSeparable.tower_top F⟮x⟯ hy
   rw [← isSeparable_adjoin_simple_iff_isSeparable] at hx hy
   exact Algebra.IsSeparable.trans F F⟮x⟯ F⟮x⟯⟮y⟯
 
@@ -841,6 +842,19 @@ theorem isSeparable_add {x y : E} (hx : IsSeparable F x) (hy : IsSeparable F y)
   isSeparable_of_mem_isSeparable F E <| F⟮x, y⟯.add_mem (subset_adjoin F _ (.inl rfl))
     (subset_adjoin F _ (.inr rfl))
 
+/-- If `x` is a separable elements, then `-x` is also a separable element. -/
+theorem isSeparable_neg {x : E} (hx : IsSeparable F x) :
+    IsSeparable F (-x) :=
+  haveI := (isSeparable_adjoin_simple_iff_isSeparable F E).2 hx
+  isSeparable_of_mem_isSeparable F E <| F⟮x⟯.neg_mem <| mem_adjoin_simple_self F x
+
+/-- If `x` and `y` are both separable elements, then `x - y` is also a separable element. -/
+theorem isSeparable_sub {x y : E} (hx : IsSeparable F x) (hy : IsSeparable F y) :
+    IsSeparable F (x - y) :=
+  haveI := isSeparable_adjoin_pair_of_isSeparable F E hx hy
+  isSeparable_of_mem_isSeparable F E <| F⟮x, y⟯.sub_mem (subset_adjoin F _ (.inl rfl))
+    (subset_adjoin F _ (.inr rfl))
+
 /-- If `x` is a separable element, then `x⁻¹` is also a separable element. -/
 theorem isSeparable_inv {x : E} (hx : IsSeparable F x) : IsSeparable F x⁻¹ :=
   haveI := (isSeparable_adjoin_simple_iff_isSeparable F E).2 hx
diff --git a/Mathlib/FieldTheory/Tower.lean b/Mathlib/FieldTheory/Tower.lean
index 160ecd9442513..297650824b8be 100644
--- a/Mathlib/FieldTheory/Tower.lean
+++ b/Mathlib/FieldTheory/Tower.lean
@@ -3,27 +3,19 @@ Copyright (c) 2020 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
-import Mathlib.Data.Nat.Prime.Defs
 import Mathlib.RingTheory.AlgebraTower
-import Mathlib.LinearAlgebra.FiniteDimensional.Defs
-import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
-import Mathlib.RingTheory.LocalRing.Basic
+import Mathlib.RingTheory.Noetherian
 
 /-!
-# Tower of field extensions
+# Finiteness of `IsScalarTower`
 
-In this file we prove the tower law for arbitrary extensions and finite extensions.
-Suppose `L` is a field extension of `K` and `K` is a field extension of `F`.
-Then `[L:F] = [L:K] [K:F]` where `[E₁:E₂]` means the `E₂`-dimension of `E₁`.
+We prove that given `IsScalarTower F K A`, if `A` is finite as a module over `F` then
+`A` is finite over `K`, and
+(as long as `A` is Noetherian over `F` and we have `NoZeroSMulDivisors K A`) `K` is finite over `F`.
 
-In fact we generalize it to rings and modules, where `L` is not necessarily a field,
-but just a free module over `K`.
+In particular these conditions hold when `A`, `F`, and `K` are fields.
 
-## Implementation notes
-
-We prove two versions, since there are two notions of dimensions: `Module.rank` which gives
-the dimension of an arbitrary vector space as a cardinal, and `FiniteDimensional.finrank` which
-gives the dimension of a finite-dimensional vector space as a natural number.
+The formulas for the dimensions are given elsewhere by `Module.finrank_mul_finrank`.
 
 ## Tags
 
@@ -38,49 +30,31 @@ open Cardinal Submodule
 
 variable (F : Type u) (K : Type v) (A : Type w)
 
-section Field
-
-variable [DivisionRing F] [DivisionRing K] [AddCommGroup A]
-variable [Module F K] [Module K A] [Module F A] [IsScalarTower F K A]
-
-namespace FiniteDimensional
-
-open IsNoetherian
+namespace Module.Finite
 
+variable [Ring F] [Ring K] [Module F K]
+  [AddCommGroup A] [Module K A] [NoZeroSMulDivisors K A]
+  [Module F A] [IsNoetherian F A] [IsScalarTower F K A] in
 /-- In a tower of field extensions `A / K / F`, if `A / F` is finite, so is `K / F`.
 
 (In fact, it suffices that `A` is a nontrivial ring.)
 
 Note this cannot be an instance as Lean cannot infer `A`.
 -/
-theorem left [Nontrivial A] [FiniteDimensional F A] : FiniteDimensional F K :=
+theorem left [Nontrivial A] : Module.Finite F K :=
   let ⟨x, hx⟩ := exists_ne (0 : A)
-  FiniteDimensional.of_injective
+  Module.Finite.of_injective
     (LinearMap.ringLmapEquivSelf K ℕ A |>.symm x |>.restrictScalars F) (smul_left_injective K hx)
 
-theorem right [hf : FiniteDimensional F A] : FiniteDimensional K A :=
+variable [Semiring F] [Semiring K] [Module F K]
+  [AddCommMonoid A] [Module K A] [Module F A] [IsScalarTower F K A] in
+theorem right [hf : Module.Finite F A] : Module.Finite K A :=
   let ⟨⟨b, hb⟩⟩ := hf
   ⟨⟨b, Submodule.restrictScalars_injective F _ _ <| by
     rw [Submodule.restrictScalars_top, eq_top_iff, ← hb, Submodule.span_le]
     exact Submodule.subset_span⟩⟩
 
-theorem Subalgebra.isSimpleOrder_of_finrank_prime (F A) [Field F] [Ring A] [IsDomain A]
-    [Algebra F A] (hp : (finrank F A).Prime) : IsSimpleOrder (Subalgebra F A) :=
-  { toNontrivial :=
-      ⟨⟨⊥, ⊤, fun he =>
-          Nat.not_prime_one ((Subalgebra.bot_eq_top_iff_finrank_eq_one.1 he).subst hp)⟩⟩
-    eq_bot_or_eq_top := fun K => by
-      haveI : FiniteDimensional _ _ := .of_finrank_pos hp.pos
-      letI := divisionRingOfFiniteDimensional F K
-      refine (hp.eq_one_or_self_of_dvd _ ⟨_, (finrank_mul_finrank F K A).symm⟩).imp ?_ fun h => ?_
-      · exact fun h' => Subalgebra.eq_bot_of_finrank_one h'
-      · exact
-          Algebra.toSubmodule_eq_top.1 (eq_top_of_finrank_eq <| K.finrank_toSubmodule.trans h) }
--- TODO: `IntermediateField` version
-
-@[deprecated (since := "2024-01-12")]
-alias finrank_linear_map' := FiniteDimensional.finrank_linearMap_self
-
-end FiniteDimensional
-
-end Field
+end Module.Finite
+
+alias FiniteDimensional.left := Module.Finite.left
+alias FiniteDimensional.right := Module.Finite.right
diff --git a/Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean b/Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
index 1a4c7dfdf186b..6e6e991b9b0b4 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean
@@ -22,7 +22,7 @@ This file defines oriented angles in Euclidean affine spaces.
 
 noncomputable section
 
-open FiniteDimensional Complex
+open Module Complex
 
 open scoped Affine EuclideanGeometry Real RealInnerProductSpace ComplexConjugate
 
@@ -622,7 +622,7 @@ theorem _root_.Collinear.oangle_sign_of_sameRay_vsub {p₁ p₂ p₃ p₄ : P} (
     have hf : ContinuousOn (fun p : P × P × P => ∡ p.1 p.2.1 p.2.2) s := by
       refine ContinuousAt.continuousOn fun p hp => continuousAt_oangle ?_ ?_
       all_goals
-        simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp
+        simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_univ, true_and, Prod.ext_iff] at hp
         obtain ⟨q₁, q₅, q₂⟩ := p
         dsimp only at hp ⊢
         obtain ⟨⟨⟨q, hq⟩, v⟩, hv, rfl, rfl, rfl⟩ := hp
@@ -638,7 +638,7 @@ theorem _root_.Collinear.oangle_sign_of_sameRay_vsub {p₁ p₂ p₃ p₄ : P} (
         exact smul_vsub_rev_mem_vectorSpan_pair _ _ _
     have hsp : ∀ p : P × P × P, p ∈ s → ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ π := by
       intro p hp
-      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,
+      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and,
         Prod.ext_iff] at hp
       obtain ⟨q₁, q₅, q₂⟩ := p
       dsimp only at hp ⊢
@@ -656,13 +656,13 @@ theorem _root_.Collinear.oangle_sign_of_sameRay_vsub {p₁ p₂ p₃ p₄ : P} (
         rw [direction_affineSpan]
         exact smul_vsub_rev_mem_vectorSpan_pair _ _ _
     have hp₁p₂s : (p₁, p₅, p₂) ∈ s := by
-      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,
+      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and,
         Prod.ext_iff]
       refine ⟨⟨⟨p₁, left_mem_affineSpan_pair ℝ _ _⟩, p₂ -ᵥ p₁⟩,
         ⟨SameRay.rfl, vsub_ne_zero.2 hp₁p₂.symm⟩, ?_⟩
       simp
     have hp₃p₄s : (p₃, p₅, p₄) ∈ s := by
-      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff,
+      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and,
         Prod.ext_iff]
       refine ⟨⟨⟨p₃, hc.mem_affineSpan_of_mem_of_ne (Set.mem_insert _ _)
         (Set.mem_insert_of_mem _ (Set.mem_insert _ _))
diff --git a/Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean b/Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
index fb641332e085a..3f2938cdce408 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
@@ -31,7 +31,7 @@ modulo `2 * π` as equalities of `(2 : ℤ) • θ`.
 
 noncomputable section
 
-open FiniteDimensional Complex
+open Module Complex
 
 open scoped Real RealInnerProductSpace ComplexConjugate
 
diff --git a/Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean b/Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
index d634919578060..8ca1f21c9b812 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
@@ -25,7 +25,7 @@ open scoped RealInnerProductSpace
 
 namespace Orientation
 
-open FiniteDimensional
+open Module
 
 variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
 variable [hd2 : Fact (finrank ℝ V = 2)] (o : Orientation ℝ V (Fin 2))
@@ -519,7 +519,7 @@ end Orientation
 
 namespace EuclideanGeometry
 
-open FiniteDimensional
+open Module
 
 variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
   [NormedAddTorsor V P] [hd2 : Fact (finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)]
diff --git a/Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean b/Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
index b933762e6b4a6..734899c867f50 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
@@ -20,7 +20,7 @@ This file defines rotations by oriented angles in real inner product spaces.
 
 noncomputable section
 
-open FiniteDimensional Complex
+open Module Complex
 
 open scoped Real RealInnerProductSpace ComplexConjugate
 
@@ -65,9 +65,8 @@ def rotation (θ : Real.Angle) : V ≃ₗᵢ[ℝ] V :=
       · simp only [o.rightAngleRotation_rightAngleRotation, o.rotationAux_apply,
           Function.comp_apply, id, LinearEquiv.coe_coe, LinearIsometry.coe_toLinearMap,
           LinearIsometryEquiv.coe_toLinearEquiv, map_smul, map_sub, LinearMap.coe_comp,
-          LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply, ← mul_smul, add_smul,
-          smul_add, smul_neg, smul_sub, mul_comm, sq]
-        abel
+          LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply]
+        module
       · simp)
     (by
       ext x
@@ -75,10 +74,8 @@ def rotation (θ : Real.Angle) : V ≃ₗᵢ[ℝ] V :=
       · simp only [o.rightAngleRotation_rightAngleRotation, o.rotationAux_apply,
           Function.comp_apply, id, LinearEquiv.coe_coe, LinearIsometry.coe_toLinearMap,
           LinearIsometryEquiv.coe_toLinearEquiv, map_add, map_smul, LinearMap.coe_comp,
-          LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply,
-          add_smul, smul_neg, smul_sub, smul_smul]
-        ring_nf
-        abel
+          LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply]
+        module
       · simp)
 
 theorem rotation_apply (θ : Real.Angle) (x : V) :
@@ -104,8 +101,7 @@ theorem rotation_eq_matrix_toLin (θ : Real.Angle) {x : V} (hx : x ≠ 0) :
 /-- The determinant of `rotation` (as a linear map) is equal to `1`. -/
 @[simp]
 theorem det_rotation (θ : Real.Angle) : LinearMap.det (o.rotation θ).toLinearMap = 1 := by
-  haveI : Nontrivial V :=
-    FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
+  haveI : Nontrivial V := nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
   obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
   rw [o.rotation_eq_matrix_toLin θ hx]
   simpa [sq] using θ.cos_sq_add_sin_sq
@@ -146,11 +142,9 @@ theorem rotation_pi_div_two : o.rotation (π / 2 : ℝ) = J := by
 @[simp]
 theorem rotation_rotation (θ₁ θ₂ : Real.Angle) (x : V) :
     o.rotation θ₁ (o.rotation θ₂ x) = o.rotation (θ₁ + θ₂) x := by
-  simp only [o.rotation_apply, ← mul_smul, Real.Angle.cos_add, Real.Angle.sin_add, add_smul,
-    sub_smul, LinearIsometryEquiv.trans_apply, smul_add, LinearIsometryEquiv.map_add,
-    LinearIsometryEquiv.map_smul, rightAngleRotation_rightAngleRotation, smul_neg]
-  ring_nf
-  abel
+  simp only [o.rotation_apply, Real.Angle.cos_add, Real.Angle.sin_add, LinearIsometryEquiv.map_add,
+    LinearIsometryEquiv.trans_apply, map_smul, rightAngleRotation_rightAngleRotation]
+  module
 
 /-- Rotating twice is equivalent to rotating by the sum of the angles. -/
 @[simp]
@@ -338,8 +332,7 @@ theorem oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero {x y : V} (θ : Real.Angle
 theorem exists_linearIsometryEquiv_eq_of_det_pos {f : V ≃ₗᵢ[ℝ] V}
     (hd : 0 < LinearMap.det (f.toLinearEquiv : V →ₗ[ℝ] V)) :
     ∃ θ : Real.Angle, f = o.rotation θ := by
-  haveI : Nontrivial V :=
-    FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
+  haveI : Nontrivial V := nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
   obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
   use o.oangle x (f x)
   apply LinearIsometryEquiv.toLinearEquiv_injective
diff --git a/Mathlib/Geometry/Euclidean/Angle/Sphere.lean b/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
index dd2a20922dbdb..17315a1b24430 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Sphere.lean
@@ -16,7 +16,7 @@ This file proves results about angles in circles and spheres.
 
 noncomputable section
 
-open FiniteDimensional Complex
+open Module Complex
 
 open scoped EuclideanGeometry Real RealInnerProductSpace ComplexConjugate
 
diff --git a/Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean b/Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
index 9dcdde2ba5730..b58876fe73b9d 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
@@ -350,7 +350,7 @@ theorem angle_eq_zero_iff_ne_and_wbtw {p₁ p₂ p₃ : P} :
   · rw [angle, angle_eq_zero_iff]
     rintro ⟨hp₁p₂, r, hr0, hp₃p₂⟩
     rcases le_or_lt 1 r with (hr1 | hr1)
-    · refine Or.inl ⟨vsub_ne_zero.1 hp₁p₂, r⁻¹, ⟨(inv_pos.2 hr0).le, inv_le_one hr1⟩, ?_⟩
+    · refine Or.inl ⟨vsub_ne_zero.1 hp₁p₂, r⁻¹, ⟨(inv_pos.2 hr0).le, inv_le_one_of_one_le₀ hr1⟩, ?_⟩
       rw [AffineMap.lineMap_apply, hp₃p₂, smul_smul, inv_mul_cancel₀ hr0.ne.symm, one_smul,
         vsub_vadd]
     · refine Or.inr ⟨?_, r, ⟨hr0.le, hr1.le⟩, ?_⟩
diff --git a/Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean b/Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
index a85b10c3cdb58..03de613c6af8b 100644
--- a/Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
+++ b/Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
@@ -121,7 +121,7 @@ theorem cos_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
     Real.cos (angle x (x + y)) = ‖x‖ / ‖x + y‖ := by
   rw [angle_add_eq_arccos_of_inner_eq_zero h,
     Real.cos_arccos (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
-      (div_le_one_of_le _ (norm_nonneg _))]
+      (div_le_one_of_le₀ _ (norm_nonneg _))]
   rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
     norm_add_sq_eq_norm_sq_add_norm_sq_real h]
   exact le_add_of_nonneg_right (mul_self_nonneg _)
@@ -131,7 +131,7 @@ theorem sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x 
     Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖ := by
   rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,
     Real.sin_arcsin (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))
-      (div_le_one_of_le _ (norm_nonneg _))]
+      (div_le_one_of_le₀ _ (norm_nonneg _))]
   rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),
     norm_add_sq_eq_norm_sq_add_norm_sq_real h]
   exact le_add_of_nonneg_left (mul_self_nonneg _)
diff --git a/Mathlib/Geometry/Euclidean/Basic.lean b/Mathlib/Geometry/Euclidean/Basic.lean
index 3431ffc86bbca..b9807b923e2cb 100644
--- a/Mathlib/Geometry/Euclidean/Basic.lean
+++ b/Mathlib/Geometry/Euclidean/Basic.lean
@@ -113,8 +113,9 @@ another point. -/
 theorem dist_smul_vadd_eq_dist {v : V} (p₁ p₂ : P) (hv : v ≠ 0) (r : ℝ) :
     dist (r • v +ᵥ p₁) p₂ = dist p₁ p₂ ↔ r = 0 ∨ r = -2 * ⟪v, p₁ -ᵥ p₂⟫ / ⟪v, v⟫ := by
   conv_lhs =>
-    rw [← mul_self_inj_of_nonneg dist_nonneg dist_nonneg, dist_smul_vadd_sq, ← sub_eq_zero,
-      add_sub_assoc, dist_eq_norm_vsub V p₁ p₂, ← real_inner_self_eq_norm_mul_norm, sub_self]
+    rw [← mul_self_inj_of_nonneg dist_nonneg dist_nonneg, dist_smul_vadd_sq, mul_assoc,
+      ← sub_eq_zero, add_sub_assoc, dist_eq_norm_vsub V p₁ p₂, ← real_inner_self_eq_norm_mul_norm,
+      sub_self]
   have hvi : ⟪v, v⟫ ≠ 0 := by simpa using hv
   have hd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 = 2 * ⟪v, p₁ -ᵥ p₂⟫ * (2 * ⟪v, p₁ -ᵥ p₂⟫) := by
     rw [discrim]
@@ -124,7 +125,7 @@ theorem dist_smul_vadd_eq_dist {v : V} (p₁ p₂ : P) (hv : v ≠ 0) (r : ℝ)
     mul_div_assoc]
   norm_num
 
-open AffineSubspace FiniteDimensional
+open AffineSubspace Module
 
 /-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
 most two points `p₁` `p₂` in a two-dimensional subspace containing those points
@@ -150,7 +151,7 @@ theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace 
       · rw [real_inner_comm]
         exact ho
   have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
-    refine eq_of_le_of_finrank_eq ?_ ?_
+    refine Submodule.eq_of_le_of_finrank_eq ?_ ?_
     · rw [Submodule.span_le, Set.range_subset_iff]
       intro i
       fin_cases i
@@ -171,13 +172,13 @@ theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace 
     exact ⟨t₁, t₂, hv⟩
   rcases hv (p -ᵥ p₁) (vsub_mem_direction hps hp₁s) with ⟨t₁, t₂, hpt⟩
   simp only [hpt, inner_add_right, inner_smul_right, ho, mul_zero, add_zero,
-    mul_eq_zero, inner_self_eq_zero, vsub_eq_zero_iff_eq, hc.symm, or_false_iff] at hop
+    mul_eq_zero, inner_self_eq_zero, vsub_eq_zero_iff_eq, hc.symm, or_false] at hop
   rw [hop, zero_smul, zero_add, ← eq_vadd_iff_vsub_eq] at hpt
   subst hpt
   have hp' : (p₂ -ᵥ p₁ : V) ≠ 0 := by simp [hp.symm]
   have hp₂ : dist ((1 : ℝ) • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁ := by simp [hp₂c₁]
   rw [← hp₁c₁, dist_smul_vadd_eq_dist _ _ hp'] at hpc₁ hp₂
-  simp only [one_ne_zero, false_or_iff] at hp₂
+  simp only [one_ne_zero, false_or] at hp₂
   rw [hp₂.symm] at hpc₁
   cases' hpc₁ with hpc₁ hpc₁ <;> simp [hpc₁]
 
diff --git a/Mathlib/Geometry/Euclidean/Circumcenter.lean b/Mathlib/Geometry/Euclidean/Circumcenter.lean
index 6ab81a7fe36a9..ab715d6d83899 100644
--- a/Mathlib/Geometry/Euclidean/Circumcenter.lean
+++ b/Mathlib/Geometry/Euclidean/Circumcenter.lean
@@ -316,11 +316,11 @@ theorem eq_circumradius_of_dist_eq {n : ℕ} (s : Simplex ℝ P n) {p : P}
     r = s.circumradius := by
   have h := s.circumsphere_unique_dist_eq.2 ⟨p, r⟩
   simp only [hp, hr, forall_const, eq_self_iff_true, subset_sphere, Sphere.ext_iff,
-    Set.forall_mem_range, mem_sphere, true_and_iff] at h
+    Set.forall_mem_range, mem_sphere] at h
   -- Porting note: added the next three lines (`simp` less powerful)
   rw [subset_sphere (s := ⟨p, r⟩)] at h
   simp only [hp, hr, forall_const, eq_self_iff_true, subset_sphere, Sphere.ext_iff,
-    Set.forall_mem_range, mem_sphere, true_and_iff] at h
+    Set.forall_mem_range, mem_sphere, true_and] at h
   exact h.2
 
 /-- The circumradius is non-negative. -/
@@ -427,7 +427,7 @@ theorem dist_circumcenter_sq_eq_sq_sub_circumradius {n : ℕ} {r : ℝ} (s : Sim
 
 /-- If there exists a distance that a point has from all vertices of a
 simplex, the orthogonal projection of that point onto the subspace
-spanned by that simplex is its circumcenter.  -/
+spanned by that simplex is its circumcenter. -/
 theorem orthogonalProjection_eq_circumcenter_of_exists_dist_eq {n : ℕ} (s : Simplex ℝ P n) {p : P}
     (hr : ∃ r, ∀ i, dist (s.points i) p = r) :
     ↑(s.orthogonalProjectionSpan p) = s.circumcenter := by
@@ -445,7 +445,7 @@ theorem orthogonalProjection_eq_circumcenter_of_exists_dist_eq {n : ℕ} (s : Si
 
 /-- If a point has the same distance from all vertices of a simplex,
 the orthogonal projection of that point onto the subspace spanned by
-that simplex is its circumcenter.  -/
+that simplex is its circumcenter. -/
 theorem orthogonalProjection_eq_circumcenter_of_dist_eq {n : ℕ} (s : Simplex ℝ P n) {p : P} {r : ℝ}
     (hr : ∀ i, dist (s.points i) p = r) : ↑(s.orthogonalProjectionSpan p) = s.circumcenter :=
   s.orthogonalProjection_eq_circumcenter_of_exists_dist_eq ⟨r, hr⟩
@@ -667,7 +667,7 @@ end Affine
 
 namespace EuclideanGeometry
 
-open Affine AffineSubspace FiniteDimensional
+open Affine AffineSubspace Module
 
 variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
   [NormedAddTorsor V P]
@@ -866,7 +866,7 @@ theorem eq_or_eq_reflection_of_dist_eq {n : ℕ} {s : Simplex ℝ P n} {p p₁ p
   by_cases hp : p = s.orthogonalProjectionSpan p
   · rw [Simplex.orthogonalProjectionSpan] at hp
     rw [hp₁, hp₂, ← hp]
-    simp only [true_or_iff, eq_self_iff_true, smul_zero, vsub_self]
+    simp only [true_or, eq_self_iff_true, smul_zero, vsub_self]
   · have hz : ⟪p -ᵥ orthogonalProjection span_s p, p -ᵥ orthogonalProjection span_s p⟫ ≠ 0 := by
       simpa only [Ne, vsub_eq_zero_iff_eq, inner_self_eq_zero] using hp
     rw [mul_left_inj' hz, mul_self_eq_mul_self_iff] at hd₁
diff --git a/Mathlib/Geometry/Euclidean/MongePoint.lean b/Mathlib/Geometry/Euclidean/MongePoint.lean
index 6343ce224c3da..fbf488bef44b3 100644
--- a/Mathlib/Geometry/Euclidean/MongePoint.lean
+++ b/Mathlib/Geometry/Euclidean/MongePoint.lean
@@ -344,7 +344,7 @@ theorem vectorSpan_isOrtho_altitude_direction {n : ℕ} (s : Simplex ℝ P (n +
   rw [direction_altitude]
   exact (Submodule.isOrtho_orthogonal_right _).mono_right inf_le_left
 
-open FiniteDimensional
+open Module
 
 /-- An altitude is finite-dimensional. -/
 instance finiteDimensional_direction_altitude {n : ℕ} (s : Simplex ℝ P (n + 1)) (i : Fin (n + 2)) :
@@ -392,7 +392,7 @@ theorem affineSpan_pair_eq_altitude_iff {n : ℕ} (s : Simplex ℝ P (n + 1)) (i
     rw [vectorSpan_eq_span_vsub_set_left_ne ℝ (Set.mem_insert _ _),
       Set.insert_diff_of_mem _ (Set.mem_singleton _),
       Set.diff_singleton_eq_self fun h => hne (Set.mem_singleton_iff.1 h), Set.image_singleton]
-    refine eq_of_le_of_finrank_eq ?_ ?_
+    refine Submodule.eq_of_le_of_finrank_eq ?_ ?_
     · rw [Submodule.span_le]
       simpa using h
     · rw [finrank_direction_altitude, finrank_span_set_eq_card]
@@ -404,7 +404,7 @@ end Simplex
 
 namespace Triangle
 
-open EuclideanGeometry Finset Simplex AffineSubspace FiniteDimensional
+open EuclideanGeometry Finset Simplex AffineSubspace Module
 
 variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
   [NormedAddTorsor V P]
@@ -548,7 +548,8 @@ theorem altitude_replace_orthocenter_eq_affineSpan {t₁ t₂ : Triangle ℝ P}
   have he : affineSpan ℝ (Set.range t₂.points) = affineSpan ℝ (Set.range t₁.points) := by
     refine ext_of_direction_eq ?_
       ⟨t₁.points i₃, mem_affineSpan ℝ ⟨j₃, h₃⟩, mem_affineSpan ℝ (Set.mem_range_self _)⟩
-    refine eq_of_le_of_finrank_eq (direction_le (spanPoints_subset_coe_of_subset_coe ?_)) ?_
+    refine Submodule.eq_of_le_of_finrank_eq (direction_le (spanPoints_subset_coe_of_subset_coe ?_))
+      ?_
     · have hu : (Finset.univ : Finset (Fin 3)) = {j₁, j₂, j₃} := by
         clear h₁ h₂ h₃
         -- Porting note (#11043): was `decide!`
@@ -602,7 +603,7 @@ end Affine
 
 namespace EuclideanGeometry
 
-open Affine AffineSubspace FiniteDimensional
+open Affine AffineSubspace Module
 
 variable {V : Type*} {P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
   [NormedAddTorsor V P]
@@ -706,7 +707,7 @@ theorem affineSpan_of_orthocentricSystem {s : Set P} (ho : OrthocentricSystem s)
     ⟨p 0, mem_affineSpan ℝ (Set.mem_range_self _), mem_affineSpan ℝ (hps (Set.mem_range_self _))⟩
   have hfd : FiniteDimensional ℝ (affineSpan ℝ s).direction := by rw [hs]; infer_instance
   haveI := hfd
-  refine eq_of_le_of_finrank_eq (direction_le (affineSpan_mono ℝ hps)) ?_
+  refine Submodule.eq_of_le_of_finrank_eq (direction_le (affineSpan_mono ℝ hps)) ?_
   rw [hs, direction_affineSpan, direction_affineSpan, ha.finrank_vectorSpan (Fintype.card_fin _),
     t.independent.finrank_vectorSpan (Fintype.card_fin _)]
 
diff --git a/Mathlib/Geometry/Euclidean/Sphere/Basic.lean b/Mathlib/Geometry/Euclidean/Sphere/Basic.lean
index 0b37ec458a9f3..6c5067aad4c23 100644
--- a/Mathlib/Geometry/Euclidean/Sphere/Basic.lean
+++ b/Mathlib/Geometry/Euclidean/Sphere/Basic.lean
@@ -33,7 +33,7 @@ namespace EuclideanGeometry
 
 variable {V : Type*} (P : Type*)
 
-open FiniteDimensional
+open Module
 
 /-- A `Sphere P` bundles a `center` and `radius`. This definition does not require the radius to
 be positive; that should be given as a hypothesis to lemmas that require it. -/
@@ -57,7 +57,7 @@ instance : Coe (Sphere P) (Set P) :=
   ⟨fun s => Metric.sphere s.center s.radius⟩
 
 instance : Membership P (Sphere P) :=
-  ⟨fun p s => p ∈ (s : Set P)⟩
+  ⟨fun s p => p ∈ (s : Set P)⟩
 
 theorem Sphere.mk_center (c : P) (r : ℝ) : (⟨c, r⟩ : Sphere P).center = c :=
   rfl
diff --git a/Mathlib/Geometry/Euclidean/Triangle.lean b/Mathlib/Geometry/Euclidean/Triangle.lean
index 5a28ab76286ee..9e2ab6a89582d 100644
--- a/Mathlib/Geometry/Euclidean/Triangle.lean
+++ b/Mathlib/Geometry/Euclidean/Triangle.lean
@@ -295,7 +295,7 @@ theorem angle_add_angle_add_angle_eq_pi {p1 p2 p3 : P} (h2 : p2 ≠ p1) (h3 : p3
 /-- The **sum of the angles of a triangle** (possibly degenerate, where the triangle is a line),
 oriented angles at point. -/
 theorem oangle_add_oangle_add_oangle_eq_pi [Module.Oriented ℝ V (Fin 2)]
-    [Fact (FiniteDimensional.finrank ℝ V = 2)] {p1 p2 p3 : P} (h21 : p2 ≠ p1) (h32 : p3 ≠ p2)
+    [Fact (Module.finrank ℝ V = 2)] {p1 p2 p3 : P} (h21 : p2 ≠ p1) (h32 : p3 ≠ p2)
     (h13 : p1 ≠ p3) : ∡ p1 p2 p3 + ∡ p2 p3 p1 + ∡ p3 p1 p2 = π := by
   simpa only [neg_vsub_eq_vsub_rev] using
     positiveOrientation.oangle_add_cyc3_neg_left (vsub_ne_zero.mpr h21) (vsub_ne_zero.mpr h32)
diff --git a/Mathlib/Geometry/Manifold/AnalyticManifold.lean b/Mathlib/Geometry/Manifold/AnalyticManifold.lean
index 0da289761a98b..8ef606a4540a1 100644
--- a/Mathlib/Geometry/Manifold/AnalyticManifold.lean
+++ b/Mathlib/Geometry/Manifold/AnalyticManifold.lean
@@ -3,10 +3,7 @@ Copyright (c) 2023 Michael Lee. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Lee, Geoffrey Irving
 -/
-import Mathlib.Analysis.Analytic.Composition
 import Mathlib.Analysis.Analytic.Constructions
-import Mathlib.Analysis.Analytic.Linear
-import Mathlib.Analysis.Analytic.Within
 import Mathlib.Analysis.Calculus.FDeriv.Analytic
 import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
 
@@ -20,7 +17,7 @@ interior and smooth everywhere (including at the boundary).  The definition mirr
 analytic manifolds are smooth manifolds.
 
 Completeness is required throughout, but this is nonessential: it is due to many of the lemmas about
-AnalyticWithinOn` requiring completeness for ease of proof.
+AnalyticOn` requiring completeness for ease of proof.
 -/
 
 noncomputable section
@@ -31,7 +28,7 @@ open scoped Manifold Filter Topology
 
 variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
 
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [CompleteSpace E] {H : Type*}
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*}
   [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H) {M : Type*} [TopologicalSpace M]
 
 /-!
@@ -45,10 +42,10 @@ analytic on the interior, and map the interior to itself.  This allows us to def
 section analyticGroupoid
 
 /-- Given a model with corners `(E, H)`, we define the pregroupoid of analytic transformations of
-`H` as the maps that are `AnalyticWithinOn` when read in `E` through `I`.  Using `AnalyticWithinOn`
-rather than `AnalyticOn` gives us meaningful definitions at boundary points. -/
+`H` as the maps that are `AnalyticOn` when read in `E` through `I`.  Using `AnalyticOn`
+rather than `AnalyticOnNhd` gives us meaningful definitions at boundary points. -/
 def analyticPregroupoid : Pregroupoid H where
-  property f s := AnalyticWithinOn 𝕜 (I ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I)
+  property f s := AnalyticOn 𝕜 (I ∘ f ∘ I.symm) (I.symm ⁻¹' s ∩ range I)
   comp {f g u v} hf hg _ _ _ := by
     have : I ∘ (g ∘ f) ∘ I.symm = (I ∘ g ∘ I.symm) ∘ I ∘ f ∘ I.symm := by ext x; simp
     simp only [this]
@@ -57,12 +54,12 @@ def analyticPregroupoid : Pregroupoid H where
     · rintro x ⟨hx1, _⟩
       simpa only [mfld_simps] using hx1.2
   id_mem := by
-    apply (analyticOn_id 𝕜).analyticWithinOn.congr
+    apply analyticOn_id.congr
     rintro x ⟨_, hx2⟩
     obtain ⟨y, hy⟩ := mem_range.1 hx2
     simp only [mfld_simps, ← hy]
   locality {f u} _ H := by
-    apply analyticWithinOn_of_locally_analyticWithinOn
+    apply analyticOn_of_locally_analyticOn
     rintro y ⟨hy1, hy2⟩
     obtain ⟨x, hx⟩ := mem_range.1 hy2
     simp only [mfld_simps, ← hx] at hy1 ⊢
@@ -78,8 +75,8 @@ def analyticPregroupoid : Pregroupoid H where
     rw [fg _ hy1]
 
 /-- Given a model with corners `(E, H)`, we define the groupoid of analytic transformations of
-`H` as the maps that are `AnalyticWithinOn` when read in `E` through `I`.  Using `AnalyticWithinOn`
-rather than `AnalyticOn` gives us meaningful definitions at boundary points. -/
+`H` as the maps that are `AnalyticOn` when read in `E` through `I`.  Using `AnalyticOn`
+rather than `AnalyticOnNhd` gives us meaningful definitions at boundary points. -/
 def analyticGroupoid : StructureGroupoid H :=
   (analyticPregroupoid I).groupoid
 
@@ -87,10 +84,10 @@ def analyticGroupoid : StructureGroupoid H :=
 theorem ofSet_mem_analyticGroupoid {s : Set H} (hs : IsOpen s) :
     PartialHomeomorph.ofSet s hs ∈ analyticGroupoid I := by
   rw [analyticGroupoid, mem_groupoid_of_pregroupoid]
-  suffices h : AnalyticWithinOn 𝕜 (I ∘ I.symm) (I.symm ⁻¹' s ∩ range I) by
+  suffices h : AnalyticOn 𝕜 (I ∘ I.symm) (I.symm ⁻¹' s ∩ range I) by
     simp [h, analyticPregroupoid]
-  have hi : AnalyticWithinOn 𝕜 id (univ : Set E) := (analyticOn_id _).analyticWithinOn
-  exact (hi.mono (subset_univ _)).congr (fun x hx ↦ (I.right_inv hx.2).symm)
+  have hi : AnalyticOn 𝕜 id (univ : Set E) := analyticOn_id
+  exact (hi.mono (subset_univ _)).congr (fun x hx ↦ I.right_inv hx.2)
 
 /-- The composition of a partial homeomorphism from `H` to `M` and its inverse belongs to
 the analytic groupoid. -/
@@ -111,35 +108,35 @@ instance : ClosedUnderRestriction (analyticGroupoid I) :=
 /-- `f ∈ analyticGroupoid` iff it and its inverse are analytic within `range I`. -/
 lemma mem_analyticGroupoid {I : ModelWithCorners 𝕜 E H} {f : PartialHomeomorph H H} :
     f ∈ analyticGroupoid I ↔
-      AnalyticWithinOn 𝕜 (I ∘ f ∘ I.symm) (I.symm ⁻¹' f.source ∩ range I) ∧
-      AnalyticWithinOn 𝕜 (I ∘ f.symm ∘ I.symm) (I.symm ⁻¹' f.target ∩ range I) := by
+      AnalyticOn 𝕜 (I ∘ f ∘ I.symm) (I.symm ⁻¹' f.source ∩ range I) ∧
+      AnalyticOn 𝕜 (I ∘ f.symm ∘ I.symm) (I.symm ⁻¹' f.target ∩ range I) := by
   rfl
 
 /-- The analytic groupoid on a boundaryless charted space modeled on a complete vector space
 consists of the partial homeomorphisms which are analytic and have analytic inverse. -/
 theorem mem_analyticGroupoid_of_boundaryless [I.Boundaryless] (e : PartialHomeomorph H H) :
-    e ∈ analyticGroupoid I ↔ AnalyticOn 𝕜 (I ∘ e ∘ I.symm) (I '' e.source) ∧
-      AnalyticOn 𝕜 (I ∘ e.symm ∘ I.symm) (I '' e.target) := by
+    e ∈ analyticGroupoid I ↔ AnalyticOnNhd 𝕜 (I ∘ e ∘ I.symm) (I '' e.source) ∧
+      AnalyticOnNhd 𝕜 (I ∘ e.symm ∘ I.symm) (I '' e.target) := by
   simp only [mem_analyticGroupoid, I.range_eq_univ, inter_univ, I.image_eq]
-  rw [IsOpen.analyticWithinOn_iff_analyticOn, IsOpen.analyticWithinOn_iff_analyticOn]
+  rw [IsOpen.analyticOn_iff_analyticOnNhd, IsOpen.analyticOn_iff_analyticOnNhd]
   · exact I.continuous_symm.isOpen_preimage _ e.open_target
   · exact I.continuous_symm.isOpen_preimage _ e.open_source
 
 /-- `analyticGroupoid` is closed under products -/
 theorem analyticGroupoid_prod {E A : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-    [CompleteSpace E] [TopologicalSpace A] {F B : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-    [CompleteSpace F] [TopologicalSpace B] {I : ModelWithCorners 𝕜 E A} {J : ModelWithCorners 𝕜 F B}
+    [TopologicalSpace A] {F B : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+    [TopologicalSpace B] {I : ModelWithCorners 𝕜 E A} {J : ModelWithCorners 𝕜 F B}
     {f : PartialHomeomorph A A} {g : PartialHomeomorph B B}
     (fa : f ∈ analyticGroupoid I) (ga : g ∈ analyticGroupoid J) :
     f.prod g ∈ analyticGroupoid (I.prod J) := by
   have pe : range (I.prod J) = (range I).prod (range J) := I.range_prod
   simp only [mem_analyticGroupoid, Function.comp, image_subset_iff] at fa ga ⊢
-  exact ⟨AnalyticWithinOn.prod
-      (fa.1.comp (analyticOn_fst _).analyticWithinOn fun _ m ↦ ⟨m.1.1, (pe.subst m.2).1⟩)
-      (ga.1.comp (analyticOn_snd _).analyticWithinOn fun _ m ↦ ⟨m.1.2, (pe.subst m.2).2⟩),
-    AnalyticWithinOn.prod
-      (fa.2.comp (analyticOn_fst _).analyticWithinOn fun _ m ↦ ⟨m.1.1, (pe.subst m.2).1⟩)
-      (ga.2.comp (analyticOn_snd _).analyticWithinOn fun _ m ↦ ⟨m.1.2, (pe.subst m.2).2⟩)⟩
+  exact ⟨AnalyticOn.prod
+      (fa.1.comp analyticOn_fst fun _ m ↦ ⟨m.1.1, (pe ▸ m.2).1⟩)
+      (ga.1.comp analyticOn_snd fun _ m ↦ ⟨m.1.2, (pe ▸ m.2).2⟩),
+    AnalyticOn.prod
+      (fa.2.comp analyticOn_fst fun _ m ↦ ⟨m.1.1, (pe ▸ m.2).1⟩)
+      (ga.2.comp analyticOn_snd fun _ m ↦ ⟨m.1.2, (pe ▸ m.2).2⟩)⟩
 
 end analyticGroupoid
 
@@ -155,8 +152,8 @@ instance AnalyticManifold.self : AnalyticManifold 𝓘(𝕜, E) E where
 
 /-- `M × N` is an analytic manifold if `M` and `N` are -/
 instance AnalyticManifold.prod {E A : Type} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-    [CompleteSpace E] [TopologicalSpace A] {F B : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-    [CompleteSpace F] [TopologicalSpace B] {I : ModelWithCorners 𝕜 E A} {J : ModelWithCorners 𝕜 F B}
+    [TopologicalSpace A] {F B : Type} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+    [TopologicalSpace B] {I : ModelWithCorners 𝕜 E A} {J : ModelWithCorners 𝕜 F B}
     {M : Type} [TopologicalSpace M] [ChartedSpace A M] [m : AnalyticManifold I M]
     {N : Type} [TopologicalSpace N] [ChartedSpace B N] [n : AnalyticManifold J N] :
     AnalyticManifold (I.prod J) (M × N) where
@@ -167,7 +164,8 @@ instance AnalyticManifold.prod {E A : Type} [NormedAddCommGroup E] [NormedSpace
       (n.toHasGroupoid.compatible hf2 hg2)
 
 /-- Analytic manifolds are smooth manifolds. -/
-instance AnalyticManifold.smoothManifoldWithCorners [ChartedSpace H M] [cm : AnalyticManifold I M] :
+instance AnalyticManifold.smoothManifoldWithCorners [ChartedSpace H M]
+    [cm : AnalyticManifold I M] [CompleteSpace E] :
     SmoothManifoldWithCorners I M where
   compatible := by
     intro f g hf hg
diff --git a/Mathlib/Geometry/Manifold/BumpFunction.lean b/Mathlib/Geometry/Manifold/BumpFunction.lean
index d32ef29050ef2..16f7d52a36180 100644
--- a/Mathlib/Geometry/Manifold/BumpFunction.lean
+++ b/Mathlib/Geometry/Manifold/BumpFunction.lean
@@ -34,7 +34,7 @@ variable {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E]
   {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M]
   [ChartedSpace H M]
 
-open Function Filter FiniteDimensional Set Metric
+open Function Filter Module Set Metric
 
 open scoped Topology Manifold
 
diff --git a/Mathlib/Geometry/Manifold/ChartedSpace.lean b/Mathlib/Geometry/Manifold/ChartedSpace.lean
index c1ce4ba8838db..d62d89afae12c 100644
--- a/Mathlib/Geometry/Manifold/ChartedSpace.lean
+++ b/Mathlib/Geometry/Manifold/ChartedSpace.lean
@@ -176,7 +176,7 @@ structure StructureGroupoid (H : Type u) [TopologicalSpace H] where
 variable [TopologicalSpace H]
 
 instance : Membership (PartialHomeomorph H H) (StructureGroupoid H) :=
-  ⟨fun (e : PartialHomeomorph H H) (G : StructureGroupoid H) ↦ e ∈ G.members⟩
+  ⟨fun (G : StructureGroupoid H) (e : PartialHomeomorph H H) ↦ e ∈ G.members⟩
 
 instance (H : Type u) [TopologicalSpace H] :
     SetLike (StructureGroupoid H) (PartialHomeomorph H H) where
@@ -1161,15 +1161,15 @@ theorem chartAt_subtype_val_symm_eventuallyEq (U : Opens M) {x : U} :
   exact Filter.eventuallyEq_of_mem heUx_nhds (e.subtypeRestr_symm_eqOn ⟨x⟩)
 
 theorem chartAt_inclusion_symm_eventuallyEq {U V : Opens M} (hUV : U ≤ V) {x : U} :
-    (chartAt H (Set.inclusion hUV x)).symm
-    =ᶠ[𝓝 (chartAt H (Set.inclusion hUV x) (Set.inclusion hUV x))]
-    Set.inclusion hUV ∘ (chartAt H x).symm := by
+    (chartAt H (Opens.inclusion hUV x)).symm
+    =ᶠ[𝓝 (chartAt H (Opens.inclusion hUV x) (Set.inclusion hUV x))]
+    Opens.inclusion hUV ∘ (chartAt H x).symm := by
   set e := chartAt H (x : M)
   have heUx_nhds : (e.subtypeRestr ⟨x⟩).target ∈ 𝓝 (e x) := by
     apply (e.subtypeRestr ⟨x⟩).open_target.mem_nhds
     exact e.map_subtype_source ⟨x⟩ (mem_chart_source _ _)
   exact Filter.eventuallyEq_of_mem heUx_nhds <| e.subtypeRestr_symm_eqOn_of_le ⟨x⟩
-    ⟨Set.inclusion hUV x⟩ hUV
+    ⟨Opens.inclusion hUV x⟩ hUV
 end TopologicalSpace.Opens
 
 /-- Restricting a chart of `M` to an open subset `s` yields a chart in the maximal atlas of `s`.
@@ -1289,7 +1289,13 @@ theorem StructureGroupoid.restriction_mem_maximalAtlas_subtype
   rw [PartialHomeomorph.subtypeRestr_def, PartialHomeomorph.trans_refl]
   let goal := e.toHomeomorphSourceTarget.toPartialHomeomorph ≫ₕ (t.partialHomeomorphSubtypeCoe this)
   have : goal ≈ e.subtypeRestr (s := s) hs :=
-    (goal.eqOnSource_iff (e.subtypeRestr (s := s) hs)).mpr ⟨by simp [s, goal], by intro _ _; rfl⟩
+    (goal.eqOnSource_iff (e.subtypeRestr (s := s) hs)).mpr
+      ⟨by
+        simp only [trans_toPartialEquiv, PartialEquiv.trans_source,
+          Homeomorph.toPartialHomeomorph_source, toFun_eq_coe, Homeomorph.toPartialHomeomorph_apply,
+          Opens.partialHomeomorphSubtypeCoe_source, preimage_univ, inter_self, subtypeRestr_source,
+          goal, s]
+        exact Subtype.coe_preimage_self _ |>.symm, by intro _ _; rfl⟩
   exact G.mem_maximalAtlas_of_eqOnSource (M := s) this (G.restriction_in_maximalAtlas he hs)
 
 /-- Each chart of a charted space is a structomorphism between its source and target. -/
diff --git a/Mathlib/Geometry/Manifold/Complex.lean b/Mathlib/Geometry/Manifold/Complex.lean
index a0110600277bb..a9ea94cdde69c 100644
--- a/Mathlib/Geometry/Manifold/Complex.lean
+++ b/Mathlib/Geometry/Manifold/Complex.lean
@@ -90,7 +90,7 @@ theorem norm_eqOn_of_isPreconnected_of_isMaxOn {f : M → F} {U : Set M} {c : M}
     replace hm : IsLocalMax (‖f ·‖) x :=
       mem_of_superset (ho.mem_nhds hx.1) fun z hz ↦ (hm hz).out.trans_eq hx.2.symm
     replace hd : ∀ᶠ y in 𝓝 x, MDifferentiableAt I 𝓘(ℂ, F) f y :=
-      (eventually_mem_nhds.2 (ho.mem_nhds hx.1)).mono fun z ↦ hd.mdifferentiableAt
+      (eventually_mem_nhds_iff.2 (ho.mem_nhds hx.1)).mono fun z ↦ hd.mdifferentiableAt
     exact (Complex.norm_eventually_eq_of_mdifferentiableAt_of_isLocalMax hd hm).mono fun _ ↦
       (Eq.trans · hx.2)
   have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, rfl⟩
diff --git a/Mathlib/Geometry/Manifold/ContMDiff/Basic.lean b/Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
index a8f5c4d77931c..c8a972fc9462a 100644
--- a/Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiff/Basic.lean
@@ -64,13 +64,13 @@ theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' → M''} (x : M)
     filter_upwards [hf.1.tendsto (extChartAt_source_mem_nhds I' (f x)),
       inter_mem_nhdsWithin s (extChartAt_source_mem_nhds I x)]
     rintro x' (hfx' : f x' ∈ e'.source) ⟨hx's, hx'⟩
-    simp only [e.map_source hx', true_and_iff, e.left_inv hx', st hx's, *]
+    simp only [e.map_source hx', true_and, e.left_inv hx', st hx's, *]
   refine ((hg.2.comp _ (hf.2.mono inter_subset_right) inter_subset_left).mono_of_mem
     (inter_mem ?_ self_mem_nhdsWithin)).congr_of_eventuallyEq ?_ ?_
   · filter_upwards [A]
     rintro x' ⟨ht, hfx'⟩
     simp only [*, mem_preimage, writtenInExtChartAt, (· ∘ ·), mem_inter_iff, e'.left_inv,
-      true_and_iff]
+      true_and]
     exact mem_range_self _
   · filter_upwards [A]
     rintro x' ⟨-, hfx'⟩
@@ -217,7 +217,7 @@ variable {c : M'}
 theorem contMDiff_const : ContMDiff I I' n fun _ : M => c := by
   intro x
   refine ⟨continuousWithinAt_const, ?_⟩
-  simp only [ContDiffWithinAtProp, (· ∘ ·)]
+  simp only [ContDiffWithinAtProp, Function.comp_def]
   exact contDiffWithinAt_const
 
 @[to_additive]
@@ -322,12 +322,13 @@ theorem ContMDiff.extend_one [T2Space M] [One M'] {n : ℕ∞} {U : Opens M} {f
   refine contMDiff_of_mulTSupport (fun x h ↦ ?_) _
   lift x to U using Subtype.coe_image_subset _ _
     (supp.mulTSupport_extend_one_subset continuous_subtype_val h)
-  rw [← contMdiffAt_subtype_iff, ← comp_def]
+  rw [← contMdiffAt_subtype_iff]
+  simp_rw [← comp_def]
   erw [extend_comp Subtype.val_injective]
   exact diff.contMDiffAt
 
 theorem contMDiff_inclusion {n : ℕ∞} {U V : Opens M} (h : U ≤ V) :
-    ContMDiff I I n (Set.inclusion h : U → V) := by
+    ContMDiff I I n (Opens.inclusion h : U → V) := by
   rintro ⟨x, hx : x ∈ U⟩
   apply (contDiffWithinAt_localInvariantProp I I n).liftProp_inclusion
   intro y
@@ -345,7 +346,8 @@ theorem Smooth.extend_one [T2Space M] [One M'] {U : Opens M} {f : U → M'}
     (supp : HasCompactMulSupport f) (diff : Smooth I I' f) : Smooth I I' (Subtype.val.extend f 1) :=
   ContMDiff.extend_one supp diff
 
-theorem smooth_inclusion {U V : Opens M} (h : U ≤ V) : Smooth I I (Set.inclusion h : U → V) :=
+theorem smooth_inclusion {U V : Opens M} (h : U ≤ V) :
+    Smooth I I (Opens.inclusion h : U → V) :=
   contMDiff_inclusion h
 
 end Inclusion
diff --git a/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean b/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
index 3028f6318c3cd..cde9ee6782c47 100644
--- a/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
@@ -769,7 +769,7 @@ theorem contMDiffAt_iff_contMDiffAt_nhds
   refine ⟨?_, fun h => h.self_of_nhds⟩
   rw [contMDiffAt_iff_contMDiffOn_nhds]
   rintro ⟨u, hu, h⟩
-  refine (eventually_mem_nhds.mpr hu).mono fun x' hx' => ?_
+  refine (eventually_mem_nhds_iff.mpr hu).mono fun x' hx' => ?_
   exact (h x' <| mem_of_mem_nhds hx').contMDiffAt hx'
 
 /-! ### Congruence lemmas -/
diff --git a/Mathlib/Geometry/Manifold/ContMDiff/Product.lean b/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
index 6cd5c2919d09b..5208fce1bdcc3 100644
--- a/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiff/Product.lean
@@ -337,7 +337,7 @@ theorem contMDiffWithinAt_pi_space :
     ContMDiffWithinAt I 𝓘(𝕜, ∀ i, Fi i) n φ s x ↔
       ∀ i, ContMDiffWithinAt I 𝓘(𝕜, Fi i) n (fun x => φ x i) s x := by
   simp only [contMDiffWithinAt_iff, continuousWithinAt_pi, contDiffWithinAt_pi, forall_and,
-    writtenInExtChartAt, extChartAt_model_space_eq_id, (· ∘ ·), PartialEquiv.refl_coe, id]
+    writtenInExtChartAt, extChartAt_model_space_eq_id, Function.comp_def, PartialEquiv.refl_coe, id]
 
 theorem contMDiffOn_pi_space :
     ContMDiffOn I 𝓘(𝕜, ∀ i, Fi i) n φ s ↔ ∀ i, ContMDiffOn I 𝓘(𝕜, Fi i) n (fun x => φ x i) s :=
diff --git a/Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean b/Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
index 2daa9b0a934c0..0b4140f98c392 100644
--- a/Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean
@@ -94,9 +94,9 @@ protected theorem ContMDiffAt.mfderiv {x₀ : N} (f : N → M → M') (g : N →
           (range I) (extChartAt I (g x₀) (g ((extChartAt J x₀).symm x))))
       (range J) (extChartAt J x₀ x₀) := by
     rw [contMDiffAt_iff] at hf hg
-    simp_rw [Function.comp, uncurry, extChartAt_prod, PartialEquiv.prod_coe_symm,
+    simp_rw [Function.comp_def, uncurry, extChartAt_prod, PartialEquiv.prod_coe_symm,
       ModelWithCorners.range_prod] at hf ⊢
-    refine ContDiffWithinAt.fderivWithin ?_ hg.2 I.unique_diff hmn (mem_range_self _) ?_
+    refine ContDiffWithinAt.fderivWithin ?_ hg.2 I.uniqueDiffOn hmn (mem_range_self _) ?_
     · simp_rw [extChartAt_to_inv]; exact hf.2
     · rw [← image_subset_iff]
       rintro _ ⟨x, -, rfl⟩
@@ -108,7 +108,7 @@ protected theorem ContMDiffAt.mfderiv {x₀ : N} (f : N → M → M') (g : N →
           (extChartAt I (g x₀) (g x)))
       x₀ := by
     simp_rw [contMDiffAt_iff_source_of_mem_source (mem_chart_source G x₀),
-      contMDiffWithinAt_iff_contDiffWithinAt, Function.comp]
+      contMDiffWithinAt_iff_contDiffWithinAt, Function.comp_def]
     exact this
   have :
     ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m
@@ -165,7 +165,7 @@ protected theorem ContMDiffAt.mfderiv {x₀ : N} (f : N → M → M') (g : N →
             PartialEquiv.mem_symm_trans_source _ (mem_extChartAt_source I' (f x₂ (g x₂)))
               h3x₂).differentiableWithinAt le_top
     have h3f := (h2x₂.mdifferentiableAt le_rfl).differentiableWithinAt_writtenInExtChartAt
-    refine fderivWithin.comp₃ _ hI' h3f hI ?_ ?_ ?_ ?_ (I.unique_diff _ <| mem_range_self _)
+    refine fderivWithin.comp₃ _ hI' h3f hI ?_ ?_ ?_ ?_ (I.uniqueDiffOn _ <| mem_range_self _)
     · exact fun x _ => mem_range_self _
     · exact fun x _ => mem_range_self _
     · simp_rw [writtenInExtChartAt, Function.comp_apply,
@@ -577,7 +577,7 @@ theorem tangentMap_tangentBundle_pure [Is : SmoothManifoldWithCorners I M] (p :
     · simp
     · exact differentiableAt_id'
     · exact differentiableAt_const _
-    · exact ModelWithCorners.unique_diff_at_image I
+    · exact ModelWithCorners.uniqueDiffWithinAt_image I
     · exact differentiableAt_id'.prod (differentiableAt_const _)
   simp (config := { unfoldPartialApp := true }) only [Bundle.zeroSection, tangentMap, mfderiv, A,
     if_pos, chartAt, FiberBundle.chartedSpace_chartAt, TangentBundle.trivializationAt_apply,
diff --git a/Mathlib/Geometry/Manifold/Diffeomorph.lean b/Mathlib/Geometry/Manifold/Diffeomorph.lean
index 45a273cc452b0..07f8ab47bb49f 100644
--- a/Mathlib/Geometry/Manifold/Diffeomorph.lean
+++ b/Mathlib/Geometry/Manifold/Diffeomorph.lean
@@ -282,7 +282,7 @@ theorem contMDiffWithinAt_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N)
   constructor
   · intro Hfh
     rw [← h.symm_apply_apply x] at Hfh
-    simpa only [(· ∘ ·), h.apply_symm_apply] using
+    simpa only [Function.comp_def, h.apply_symm_apply] using
       Hfh.comp (h x) (h.symm.contMDiffWithinAt.of_le hm) (mapsTo_preimage _ _)
   · rw [← h.image_eq_preimage]
     exact fun hf => hf.comp x (h.contMDiffWithinAt.of_le hm) (mapsTo_image _ _)
@@ -308,7 +308,7 @@ theorem contMDiff_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N 
 theorem contMDiffWithinAt_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n)
     {s x} : ContMDiffWithinAt I' J m (h ∘ f) s x ↔ ContMDiffWithinAt I' I m f s x :=
   ⟨fun Hhf => by
-    simpa only [(· ∘ ·), h.symm_apply_apply] using
+    simpa only [Function.comp_def, h.symm_apply_apply] using
       (h.symm.contMDiffAt.of_le hm).comp_contMDiffWithinAt _ Hhf,
     fun Hf => (h.contMDiffAt.of_le hm).comp_contMDiffWithinAt _ Hf⟩
 
@@ -447,7 +447,7 @@ variable (I) (e : E ≃ₘ[𝕜] E')
 def transDiffeomorph (I : ModelWithCorners 𝕜 E H) (e : E ≃ₘ[𝕜] E') : ModelWithCorners 𝕜 E' H where
   toPartialEquiv := I.toPartialEquiv.trans e.toEquiv.toPartialEquiv
   source_eq := by simp
-  unique_diff' := by simp [range_comp e, I.unique_diff]
+  uniqueDiffOn' := by simp [range_comp e, I.uniqueDiffOn]
   continuous_toFun := e.continuous.comp I.continuous
   continuous_invFun := I.continuous_symm.comp e.symm.continuous
 
diff --git a/Mathlib/Geometry/Manifold/Instances/Real.lean b/Mathlib/Geometry/Manifold/Instances/Real.lean
index 32997ffec1950..43cbf79926a58 100644
--- a/Mathlib/Geometry/Manifold/Instances/Real.lean
+++ b/Mathlib/Geometry/Manifold/Instances/Real.lean
@@ -136,7 +136,7 @@ def modelWithCornersEuclideanHalfSpace (n : ℕ) [NeZero n] :
     exact ⟨max_eq_left xprop, fun i _ => rfl⟩
   right_inv' x hx := update_eq_iff.2 ⟨max_eq_left hx, fun i _ => rfl⟩
   source_eq := rfl
-  unique_diff' := by
+  uniqueDiffOn' := by
     have : UniqueDiffOn ℝ _ :=
       UniqueDiffOn.pi (Fin n) (fun _ => ℝ) _ _ fun i (_ : i ∈ ({0} : Set (Fin n))) =>
         uniqueDiffOn_Ici 0
@@ -151,7 +151,7 @@ model for manifolds with corners -/
 def modelWithCornersEuclideanQuadrant (n : ℕ) :
     ModelWithCorners ℝ (EuclideanSpace ℝ (Fin n)) (EuclideanQuadrant n) where
   toFun := Subtype.val
-  invFun x := ⟨fun i => max (x i) 0, fun i => by simp only [le_refl, or_true_iff, le_max_iff]⟩
+  invFun x := ⟨fun i => max (x i) 0, fun i => by simp only [le_refl, or_true, le_max_iff]⟩
   source := univ
   target := { x | ∀ i, 0 ≤ x i }
   map_source' x _ := x.property
@@ -159,7 +159,7 @@ def modelWithCornersEuclideanQuadrant (n : ℕ) :
   left_inv' x _ := by ext i; simp only [Subtype.coe_mk, x.2 i, max_eq_left]
   right_inv' x hx := by ext1 i; simp only [hx i, max_eq_left]
   source_eq := rfl
-  unique_diff' := by
+  uniqueDiffOn' := by
     have this : UniqueDiffOn ℝ _ :=
       UniqueDiffOn.univ_pi (Fin n) (fun _ => ℝ) _ fun _ => uniqueDiffOn_Ici 0
     simpa only [pi_univ_Ici] using this
diff --git a/Mathlib/Geometry/Manifold/Instances/Sphere.lean b/Mathlib/Geometry/Manifold/Instances/Sphere.lean
index 41a32a5f566cb..f5d59a6f08222 100644
--- a/Mathlib/Geometry/Manifold/Instances/Sphere.lean
+++ b/Mathlib/Geometry/Manifold/Instances/Sphere.lean
@@ -11,6 +11,7 @@ import Mathlib.Analysis.InnerProductSpace.PiL2
 import Mathlib.Geometry.Manifold.Algebra.LieGroup
 import Mathlib.Geometry.Manifold.Instances.Real
 import Mathlib.Geometry.Manifold.MFDeriv.Basic
+import Mathlib.Tactic.Module
 
 /-!
 # Manifold structure on the sphere
@@ -66,7 +67,7 @@ variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
 noncomputable section
 
-open Metric FiniteDimensional Function
+open Metric Module Function
 
 open scoped Manifold
 
@@ -161,7 +162,7 @@ theorem contDiff_stereoInvFunAux : ContDiff ℝ ⊤ (stereoInvFunAux v) := by
   have h₁ : ContDiff ℝ ⊤ fun w : E => (‖w‖ ^ 2 + 4)⁻¹ := by
     refine (h₀.add contDiff_const).inv ?_
     intro x
-    nlinarith
+    positivity
   have h₂ : ContDiff ℝ ⊤ fun w => (4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v := by
     refine (contDiff_const.smul contDiff_id).add ?_
     exact (h₀.sub contDiff_const).smul contDiff_const
@@ -177,15 +178,16 @@ theorem stereoInvFun_apply (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) :
     (stereoInvFun hv w : E) = (‖w‖ ^ 2 + 4)⁻¹ • ((4 : ℝ) • w + (‖w‖ ^ 2 - 4) • v) :=
   rfl
 
+open scoped InnerProductSpace in
 theorem stereoInvFun_ne_north_pole (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) :
     stereoInvFun hv w ≠ (⟨v, by simp [hv]⟩ : sphere (0 : E) 1) := by
   refine Subtype.coe_ne_coe.1 ?_
   rw [← inner_lt_one_iff_real_of_norm_one _ hv]
   · have hw : ⟪v, w⟫_ℝ = 0 := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2
     have hw' : (‖(w : E)‖ ^ 2 + 4)⁻¹ * (‖(w : E)‖ ^ 2 - 4) < 1 := by
-      refine (inv_mul_lt_iff' ?_).mpr ?_
-      · nlinarith
-      linarith
+      rw [inv_mul_lt_iff₀']
+      · linarith
+      positivity
     simpa [real_inner_comm, inner_add_right, inner_smul_right, real_inner_self_eq_norm_mul_norm, hw,
       hv] using hw'
   · simpa using stereoInvFunAux_mem hv w.2
@@ -193,6 +195,8 @@ theorem stereoInvFun_ne_north_pole (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) :
 theorem continuous_stereoInvFun (hv : ‖v‖ = 1) : Continuous (stereoInvFun hv) :=
   continuous_induced_rng.2 (contDiff_stereoInvFunAux.continuous.comp continuous_subtype_val)
 
+open scoped InnerProductSpace in
+attribute [-simp] AddSubgroupClass.coe_norm Submodule.coe_norm in
 theorem stereo_left_inv (hv : ‖v‖ = 1) {x : sphere (0 : E) 1} (hx : (x : E) ≠ v) :
     stereoInvFun hv (stereoToFun v x) = x := by
   ext
@@ -210,57 +214,35 @@ theorem stereo_left_inv (hv : ‖v‖ = 1) {x : sphere (0 : E) 1} (hx : (x : E)
     · simp [← split]
     · simp [norm_smul, hv, ← sq, sq_abs]
     · exact sq _
-  -- two facts which will be helpful for clearing denominators in the main calculation
-  have ha : 1 - a ≠ 0 := by
+  -- a fact which will be helpful for clearing denominators in the main calculation
+  have ha : 0 < 1 - a := by
     have : a < 1 := (inner_lt_one_iff_real_of_norm_one hv (by simp)).mpr hx.symm
     linarith
-  -- the core of the problem is these two algebraic identities:
-  have h₁ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * 4 * (2 / (1 - a)) = 1 := by
-    -- TODO(#15486): used to be `field_simp`, but was really slow
-    -- replaced by `simp only ...` to speed up. Reinstate `field_simp` once it is faster.
-    simp (disch := field_simp_discharge) only [AddSubgroupClass.coe_norm, div_mul_eq_mul_div,
-      div_add', inv_div, mul_div_assoc', div_div, div_eq_iff, one_mul]
-    simp only [Submodule.coe_norm] at *; nlinarith only [pythag]
-  have h₂ : (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 + 4)⁻¹ * (2 ^ 2 / (1 - a) ^ 2 * ‖y‖ ^ 2 - 4) = a := by
-    -- TODO(#15486): used to be `field_simp`, but was really slow
-    -- replaced by `simp only ...` to speed up. Reinstate `field_simp` once it is faster.
-    simp (disch := field_simp_discharge) only [AddSubgroupClass.coe_norm, div_mul_eq_mul_div,
-      div_add', inv_div, div_sub', mul_div_assoc', div_div, div_eq_iff]
-    transitivity (1 - a) ^ 2 * (a * (2 ^ 2 * ‖y‖ ^ 2 + 4 * (1 - a) ^ 2))
-    · congr
-      simp only [Submodule.coe_norm] at *
-      nlinarith only [pythag]
-    ring!
-  convert
-    congr_arg₂ Add.add (congr_arg (fun t => t • (y : E)) h₁) (congr_arg (fun t => t • v) h₂) using 1
-  · simp only [innerSL_apply, norm_smul, norm_div, RCLike.norm_ofNat, Real.norm_eq_abs,
-      AddSubgroupClass.coe_norm, mul_pow, div_pow, sq_abs, SetLike.val_smul, mul_smul, a]
-    -- Porting note: used to be simp only [split, add_comm] but get maxRec errors
-    rw [split, add_comm]
-    ac_rfl
-  -- Porting note: this branch did not exit in ml3
-  · rw [split, add_comm]
-    congr
-    dsimp
-    rw [one_smul]
+  rw [split, Submodule.coe_smul_of_tower]
+  simp only [norm_smul, norm_div, RCLike.norm_ofNat, Real.norm_eq_abs, abs_of_nonneg ha.le]
+  match_scalars
+  · field_simp
+    linear_combination 4 * (1 - a) * pythag
+  · field_simp
+    linear_combination 4 * (a - 1) ^ 3 * pythag
 
 theorem stereo_right_inv (hv : ‖v‖ = 1) (w : (ℝ ∙ v)ᗮ) : stereoToFun v (stereoInvFun hv w) = w := by
-  have : 2 / (1 - (‖(w : E)‖ ^ 2 + 4)⁻¹ * (‖(w : E)‖ ^ 2 - 4)) * (‖(w : E)‖ ^ 2 + 4)⁻¹ * 4 = 1 := by
-    -- TODO(#15486): used to be `field_simp`, but was really slow
-    -- replaced by `simp only ...` to speed up. Reinstate `field_simp` once it is faster.
-    simp (disch := field_simp_discharge) only [inv_eq_one_div, div_mul_eq_mul_div, one_mul,
-      sub_div', add_sub_sub_cancel, div_div_eq_mul_div, mul_div_assoc', mul_one, div_div,
-      div_eq_iff]
-    ring
-  convert congr_arg (· • w) this
-  · have h₁ : orthogonalProjection (ℝ ∙ v)ᗮ v = 0 :=
-      orthogonalProjection_orthogonalComplement_singleton_eq_zero v
-    -- Porting note: was innerSL _ and now just inner
-    have h₃ : inner v w = (0 : ℝ) := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2
-    -- Porting note: was innerSL _ and now just inner
-    have h₄ : inner v v = (1 : ℝ) := by simp [real_inner_self_eq_norm_mul_norm, hv]
-    simp [h₁, h₃, h₄, ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, mul_smul]
-  · simp
+  simp only [stereoToFun, stereoInvFun, stereoInvFunAux, smul_add, map_add, map_smul, innerSL_apply,
+    orthogonalProjection_mem_subspace_eq_self]
+  have h₁ : orthogonalProjection (ℝ ∙ v)ᗮ v = 0 :=
+    orthogonalProjection_orthogonalComplement_singleton_eq_zero v
+  -- Porting note: was innerSL _ and now just inner
+  have h₂ : inner v w = (0 : ℝ) := Submodule.mem_orthogonal_singleton_iff_inner_right.mp w.2
+  -- Porting note: was innerSL _ and now just inner
+  have h₃ : inner v v = (1 : ℝ) := by simp [real_inner_self_eq_norm_mul_norm, hv]
+  rw [h₁, h₂, h₃]
+  match_scalars
+  -- TODO(#15486): used to be `field_simp`, but was really slow
+  -- replaced by `simp only ...` to speed up. Reinstate `field_simp` once it is faster.
+  simp (disch := field_simp_discharge) only [add_div', add_sub_sub_cancel, div_div,
+    div_div_eq_mul_div, div_eq_iff, div_mul_eq_mul_div, inv_eq_one_div,
+    mul_div_assoc', mul_one, mul_zero, one_mul, smul_eq_mul, sub_div', zero_add, zero_div, zero_mul]
+  ring
 
 /-- Stereographic projection from the unit sphere in `E`, centred at a unit vector `v` in `E`;
 this is the version as a partial homeomorphism. -/
@@ -372,6 +354,8 @@ end ChartedSpace
 
 section SmoothManifold
 
+open scoped InnerProductSpace
+
 theorem sphere_ext_iff (u v : sphere (0 : E) 1) : u = v ↔ ⟪(u : E), v⟫_ℝ = 1 := by
   simp [Subtype.ext_iff, inner_eq_one_iff_of_norm_one]
 
@@ -421,7 +405,7 @@ instance (n : ℕ) :
   haveI := Fact.mk (@finrank_euclideanSpace_fin ℝ _ (n + 1))
   EuclideanSpace.instSmoothManifoldWithCornersSphere
 
-/-- The inclusion map (i.e., `coe`) from the sphere in `E` to `E` is smooth.  -/
+/-- The inclusion map (i.e., `coe`) from the sphere in `E` to `E` is smooth. -/
 theorem contMDiff_coe_sphere {n : ℕ} [Fact (finrank ℝ E = n + 1)] :
     ContMDiff (𝓡 n) 𝓘(ℝ, E) ∞ ((↑) : sphere (0 : E) 1 → E) := by
   -- Porting note: trouble with filling these implicit variables in the instance
diff --git a/Mathlib/Geometry/Manifold/IntegralCurve.lean b/Mathlib/Geometry/Manifold/IntegralCurve.lean
index beb8c885abb24..76b954315da82 100644
--- a/Mathlib/Geometry/Manifold/IntegralCurve.lean
+++ b/Mathlib/Geometry/Manifold/IntegralCurve.lean
@@ -171,7 +171,7 @@ lemma IsIntegralCurveOn.hasDerivAt (hγ : IsIntegralCurveOn γ v s) {t : ℝ} (h
 lemma IsIntegralCurveAt.eventually_hasDerivAt (hγ : IsIntegralCurveAt γ v t₀) :
     ∀ᶠ t in 𝓝 t₀, HasDerivAt ((extChartAt I (γ t₀)) ∘ γ)
       (tangentCoordChange I (γ t) (γ t₀) (γ t) (v (γ t))) t := by
-  apply eventually_mem_nhds.mpr
+  apply eventually_mem_nhds_iff.mpr
     (hγ.continuousAt.preimage_mem_nhds (extChartAt_source_mem_nhds I _)) |>.and hγ |>.mono
   rintro t ⟨ht1, ht2⟩
   have hsrc := mem_of_mem_nhds ht1
@@ -266,7 +266,7 @@ lemma IsIntegralCurveAt.comp_mul_ne_zero (hγ : IsIntegralCurveAt γ v t₀) {a
   convert h.comp_mul a
   ext t
   rw [mem_setOf_eq, Metric.mem_ball, Metric.mem_ball, Real.dist_eq, Real.dist_eq,
-    lt_div_iff (abs_pos.mpr ha), ← abs_mul, sub_mul, div_mul_cancel₀ _ ha]
+    lt_div_iff₀ (abs_pos.mpr ha), ← abs_mul, sub_mul, div_mul_cancel₀ _ ha]
 
 lemma isIntegralCurveAt_comp_mul_ne_zero {a : ℝ} (ha : a ≠ 0) :
     IsIntegralCurveAt γ v t₀ ↔ IsIntegralCurveAt (γ ∘ (· * a)) (a • v) (t₀ / a) := by
@@ -325,7 +325,7 @@ theorem exists_isIntegralCurveAt_of_contMDiffAt [CompleteSpace E]
   rw [continuousAt_def, hf1] at hcont
   have hnhds : f ⁻¹' (interior (extChartAt I x₀).target) ∈ 𝓝 t₀ :=
     hcont _ (isOpen_interior.mem_nhds ((I.isInteriorPoint_iff).mp hx))
-  rw [← eventually_mem_nhds] at hnhds
+  rw [← eventually_mem_nhds_iff] at hnhds
   -- obtain a neighbourhood `s` so that the above conditions both hold in `s`
   obtain ⟨s, hs, haux⟩ := (hf2.and hnhds).exists_mem
   -- prove that `γ := (extChartAt I x₀).symm ∘ f` is a desired integral curve
@@ -391,7 +391,7 @@ theorem isIntegralCurveAt_eventuallyEq_of_contMDiffAt (hγt₀ : I.IsInteriorPoi
   have hlip (t : ℝ) : LipschitzOnWith K ((fun _ ↦ v') t) ((fun _ ↦ s) t) := hlip
   -- internal lemmas to reduce code duplication
   have hsrc {g} (hg : IsIntegralCurveAt g v t₀) :
-    ∀ᶠ t in 𝓝 t₀, g ⁻¹' (extChartAt I (g t₀)).source ∈ 𝓝 t := eventually_mem_nhds.mpr <|
+    ∀ᶠ t in 𝓝 t₀, g ⁻¹' (extChartAt I (g t₀)).source ∈ 𝓝 t := eventually_mem_nhds_iff.mpr <|
       continuousAt_def.mp hg.continuousAt _ <| extChartAt_source_mem_nhds I (g t₀)
   have hmem {g : ℝ → M} {t} (ht : g ⁻¹' (extChartAt I (g t₀)).source ∈ 𝓝 t) :
     g t ∈ (extChartAt I (g t₀)).source := mem_preimage.mp <| mem_of_mem_nhds ht
@@ -492,7 +492,7 @@ theorem isIntegralCurve_eq_of_contMDiff (hγt : ∀ t, I.IsInteriorPoint (γ t))
     obtain ⟨hS₂, hS₃⟩ := abs_lt.mp hS₂
     exact ⟨T + S, by constructor <;> constructor <;> linarith⟩
   exact isIntegralCurveOn_Ioo_eqOn_of_contMDiff ht (fun t _ ↦ hγt t) hv
-    ((hγ.isIntegralCurveOn _).mono  (subset_univ _))
+    ((hγ.isIntegralCurveOn _).mono (subset_univ _))
     ((hγ'.isIntegralCurveOn _).mono (subset_univ _)) h ht₀
 
 theorem isIntegralCurve_Ioo_eq_of_contMDiff_boundaryless [BoundarylessManifold I M]
diff --git a/Mathlib/Geometry/Manifold/InteriorBoundary.lean b/Mathlib/Geometry/Manifold/InteriorBoundary.lean
index 1bfcd8ec40979..ae3c3a754f493 100644
--- a/Mathlib/Geometry/Manifold/InteriorBoundary.lean
+++ b/Mathlib/Geometry/Manifold/InteriorBoundary.lean
@@ -21,10 +21,12 @@ Define the interior and boundary of a manifold.
 - `ModelWithCorners.univ_eq_interior_union_boundary`: `M` is the union of its interior and boundary
 - `ModelWithCorners.interior_boundary_disjoint`: interior and boundary of `M` are disjoint
 - `BoundarylessManifold.isInteriorPoint`: if `M` is boundaryless, every point is an interior point
-
+- `ModelWithCorners.Boundaryless.boundary_eq_empty` and `of_boundary_eq_empty`:
+`M` is boundaryless if and only if its boundary is empty
 - `ModelWithCorners.interior_prod`: the interior of `M × N` is the product of the interiors
 of `M` and `N`.
 - `ModelWithCorners.boundary_prod`: the boundary of `M × N` is `∂M × N ∪ (M × ∂N)`.
+- `ModelWithCorners.BoundarylessManifold.prod`: if `M` and `N` are boundaryless, so is `M × N`
 
 ## Tags
 manifold, interior, boundary
@@ -96,10 +98,14 @@ lemma disjoint_interior_boundary : Disjoint (I.interior M) (I.boundary M) := by
   exact ⟨h1, h2⟩
 
 /-- The boundary is the complement of the interior. -/
-lemma boundary_eq_complement_interior : I.boundary M = (I.interior M)ᶜ := by
-  apply (compl_unique ?_ I.interior_union_boundary_eq_univ).symm
+lemma compl_interior : (I.interior M)ᶜ = I.boundary M:= by
+  apply compl_unique ?_ I.interior_union_boundary_eq_univ
   exact disjoint_iff_inter_eq_empty.mp (I.disjoint_interior_boundary)
 
+/-- The interior is the complement of the boundary. -/
+lemma compl_boundary : (I.boundary M)ᶜ = I.interior M:= by
+  rw [← compl_interior, compl_compl]
+
 variable {I} in
 lemma _root_.range_mem_nhds_isInteriorPoint {x : M} (h : I.IsInteriorPoint x) :
     range I ∈ 𝓝 (extChartAt I x x) := by
@@ -123,36 +129,47 @@ instance : BoundarylessManifold I M where
     let r := ((chartAt H x).isOpen_extend_target I).interior_eq
     have : extChartAt I x = (chartAt H x).extend I := rfl
     rw [← this] at r
-    rw [ModelWithCorners.isInteriorPoint_iff, r]
+    rw [isInteriorPoint_iff, r]
     exact PartialEquiv.map_source _ (mem_extChartAt_source _ _)
 
 end Boundaryless
 
 section BoundarylessManifold
-variable [BoundarylessManifold I M]
 
 /-- The empty manifold is boundaryless. -/
 instance BoundarylessManifold.of_empty [IsEmpty M] : BoundarylessManifold I M where
   isInteriorPoint' x := (IsEmpty.false x).elim
 
-lemma _root_.BoundarylessManifold.isInteriorPoint {x : M} :
+lemma _root_.BoundarylessManifold.isInteriorPoint {x : M} [BoundarylessManifold I M] :
     IsInteriorPoint I x := BoundarylessManifold.isInteriorPoint' x
 
 /-- If `I` is boundaryless, `M` has full interior. -/
-lemma interior_eq_univ : I.interior M = univ :=
+lemma interior_eq_univ [BoundarylessManifold I M] : I.interior M = univ :=
   eq_univ_of_forall fun _ => BoundarylessManifold.isInteriorPoint I
 
 /-- Boundaryless manifolds have empty boundary. -/
-lemma Boundaryless.boundary_eq_empty : I.boundary M = ∅ := by
-  rw [I.boundary_eq_complement_interior, I.interior_eq_univ, compl_empty_iff]
+lemma Boundaryless.boundary_eq_empty [BoundarylessManifold I M] : I.boundary M = ∅ := by
+  rw [← I.compl_interior, I.interior_eq_univ, compl_empty_iff]
 
 instance [BoundarylessManifold I M] : IsEmpty (I.boundary M) :=
   isEmpty_coe_sort.mpr (Boundaryless.boundary_eq_empty I)
 
+/-- `M` is boundaryless iff its boundary is empty. -/
+lemma Boundaryless.iff_boundary_eq_empty : I.boundary M = ∅ ↔ BoundarylessManifold I M := by
+  refine ⟨fun h ↦ { isInteriorPoint' := ?_ }, fun a ↦ boundary_eq_empty I⟩
+  intro x
+  show x ∈ I.interior M
+  rw [← compl_interior, compl_empty_iff] at h
+  rw [h]
+  trivial
+
+/-- Manifolds with empty boundary are boundaryless. -/
+lemma Boundaryless.of_boundary_eq_empty (h : I.boundary M = ∅) : BoundarylessManifold I M :=
+  (Boundaryless.iff_boundary_eq_empty (I := I)).mp h
+
 end BoundarylessManifold
-end ModelWithCorners
 
--- Interior and boundary of the product of two manifolds.
+/-! Interior and boundary of the product of two manifolds. -/
 section prod
 
 variable {I}
@@ -162,43 +179,54 @@ variable {I}
   (J : ModelWithCorners 𝕜 E' H') {x : M} {y : N}
 
 /-- The interior of `M × N` is the product of the interiors of `M` and `N`. -/
-lemma ModelWithCorners.interior_prod :
+lemma interior_prod :
     (I.prod J).interior (M × N) = (I.interior M) ×ˢ (J.interior N) := by
   ext p
   have aux : (interior (range ↑I)) ×ˢ (interior (range J)) = interior (range (I.prod J)) := by
     rw [← interior_prod_eq, ← Set.range_prod_map, modelWithCorners_prod_coe]
   constructor <;> intro hp
   · replace hp : (I.prod J).IsInteriorPoint p := hp
-    rw [ModelWithCorners.IsInteriorPoint, ← aux] at hp
+    rw [IsInteriorPoint, ← aux] at hp
     exact hp
-  · obtain ⟨h₁, h₂⟩ := Set.mem_prod.mp hp
-    rw [ModelWithCorners.interior] at h₁ h₂
-    show (I.prod J).IsInteriorPoint p
-    rw [ModelWithCorners.IsInteriorPoint, ← aux, mem_prod]
-    exact ⟨h₁, h₂⟩
+  · show (I.prod J).IsInteriorPoint p
+    rw [IsInteriorPoint, ← aux, mem_prod]
+    obtain h := Set.mem_prod.mp hp
+    rw [ModelWithCorners.interior] at h
+    exact h
 
 /-- The boundary of `M × N` is `∂M × N ∪ (M × ∂N)`. -/
-lemma ModelWithCorners.boundary_prod :
+lemma boundary_prod :
     (I.prod J).boundary (M × N) = Set.prod univ (J.boundary N) ∪ Set.prod (I.boundary M) univ := by
   let h := calc (I.prod J).boundary (M × N)
-    _ = ((I.prod J).interior (M × N))ᶜ := (I.prod J).boundary_eq_complement_interior
-    _ = ((I.interior M) ×ˢ (J.interior N))ᶜ := by rw [ModelWithCorners.interior_prod]
+    _ = ((I.prod J).interior (M × N))ᶜ := (I.prod J).compl_interior.symm
+    _ = ((I.interior M) ×ˢ (J.interior N))ᶜ := by rw [interior_prod]
     _ = (I.interior M)ᶜ ×ˢ univ ∪ univ ×ˢ (J.interior N)ᶜ := by rw [compl_prod_eq_union]
-  rw [h, I.boundary_eq_complement_interior, J.boundary_eq_complement_interior, union_comm]
+  rw [h, I.compl_interior, J.compl_interior, union_comm]
   rfl
 
 /-- If `M` is boundaryless, `∂(M×N) = M × ∂N`. -/
-lemma boundary_of_boundaryless_left [I.Boundaryless] :
+lemma boundary_of_boundaryless_left [BoundarylessManifold I M] :
     (I.prod J).boundary (M × N) = Set.prod (univ : Set M) (J.boundary N) := by
-  rw [ModelWithCorners.boundary_prod, ModelWithCorners.Boundaryless.boundary_eq_empty I]
+  rw [boundary_prod, Boundaryless.boundary_eq_empty I]
   have : Set.prod (∅ : Set M) (univ : Set N) = ∅ := Set.empty_prod
   rw [this, union_empty]
 
 /-- If `N` is boundaryless, `∂(M×N) = ∂M × N`. -/
-lemma boundary_of_boundaryless_right [J.Boundaryless] :
+lemma boundary_of_boundaryless_right [BoundarylessManifold J N] :
     (I.prod J).boundary (M × N) = Set.prod (I.boundary M) (univ : Set N) := by
-  rw [ModelWithCorners.boundary_prod, ModelWithCorners.Boundaryless.boundary_eq_empty J]
+  rw [boundary_prod, Boundaryless.boundary_eq_empty J]
   have : Set.prod (univ : Set M) (∅ : Set N) = ∅ := Set.prod_empty
   rw [this, empty_union]
 
+/-- The product of two boundaryless manifolds is boundaryless. -/
+instance BoundarylessManifold.prod [BoundarylessManifold I M] [BoundarylessManifold J N] :
+    BoundarylessManifold (I.prod J) (M × N) := by
+  apply Boundaryless.of_boundary_eq_empty
+  simp only [boundary_prod, Boundaryless.boundary_eq_empty, union_empty_iff]
+  -- These are simp lemmas, but `simp` does not apply them on its own:
+  -- presumably because of the distinction between `Prod` and `ModelProd`
+  exact ⟨Set.prod_empty, Set.empty_prod⟩
+
 end prod
+
+end ModelWithCorners
diff --git a/Mathlib/Geometry/Manifold/LocalInvariantProperties.lean b/Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
index c464a37a39429..5ea4b4a42e6b8 100644
--- a/Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
+++ b/Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
@@ -68,7 +68,7 @@ structure LocalInvariantProp (P : (H → H') → Set H → H → Prop) : Prop wh
   left_invariance' : ∀ {s x f} {e' : PartialHomeomorph H' H'},
     e' ∈ G' → s ⊆ f ⁻¹' e'.source → f x ∈ e'.source → P f s x → P (e' ∘ f) s x
 
-variable {G G'} {P : (H → H') → Set H → H → Prop} {s t u : Set H} {x : H}
+variable {G G'} {P : (H → H') → Set H → H → Prop}
 variable (hG : G.LocalInvariantProp G' P)
 include hG
 
@@ -272,8 +272,8 @@ theorem liftPropWithinAt_indep_chart_aux (he : e ∈ G.maximalAtlas M) (xe : x 
     (xf : g x ∈ f.source) (hf' : f' ∈ G'.maximalAtlas M') (xf' : g x ∈ f'.source)
     (hgs : ContinuousWithinAt g s x) :
     P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := by
-  rw [← Function.comp.assoc, hG.liftPropWithinAt_indep_chart_source_aux (f ∘ g) he xe he' xe',
-    Function.comp.assoc, hG.liftPropWithinAt_indep_chart_target_aux xe' hf xf hf' xf' hgs]
+  rw [← Function.comp_assoc, hG.liftPropWithinAt_indep_chart_source_aux (f ∘ g) he xe he' xe',
+    Function.comp_assoc, hG.liftPropWithinAt_indep_chart_target_aux xe' hf xf hf' xf' hgs]
 
 theorem liftPropWithinAt_indep_chart [HasGroupoid M G] [HasGroupoid M' G']
     (he : e ∈ G.maximalAtlas M) (xe : x ∈ e.source) (hf : f ∈ G'.maximalAtlas M')
@@ -292,9 +292,9 @@ theorem liftPropWithinAt_indep_chart_source [HasGroupoid M G] (he : e ∈ G.maxi
   rw [liftPropWithinAt_self_source, liftPropWithinAt_iff',
     e.symm.continuousWithinAt_iff_continuousWithinAt_comp_right xe, e.symm_symm]
   refine and_congr Iff.rfl ?_
-  rw [Function.comp_apply, e.left_inv xe, ← Function.comp.assoc,
+  rw [Function.comp_apply, e.left_inv xe, ← Function.comp_assoc,
     hG.liftPropWithinAt_indep_chart_source_aux (chartAt _ (g x) ∘ g) (chart_mem_maximalAtlas G x)
-      (mem_chart_source _ x) he xe, Function.comp.assoc]
+      (mem_chart_source _ x) he xe, Function.comp_assoc]
 
 /-- A version of `liftPropWithinAt_indep_chart`, only for the target. -/
 theorem liftPropWithinAt_indep_chart_target [HasGroupoid M' G'] (hf : f ∈ G'.maximalAtlas M')
@@ -302,7 +302,7 @@ theorem liftPropWithinAt_indep_chart_target [HasGroupoid M' G'] (hf : f ∈ G'.m
     LiftPropWithinAt P g s x ↔ ContinuousWithinAt g s x ∧ LiftPropWithinAt P (f ∘ g) s x := by
   rw [liftPropWithinAt_self_target, liftPropWithinAt_iff', and_congr_right_iff]
   intro hg
-  simp_rw [(f.continuousAt xf).comp_continuousWithinAt hg, true_and_iff]
+  simp_rw [(f.continuousAt xf).comp_continuousWithinAt hg, true_and]
   exact hG.liftPropWithinAt_indep_chart_target_aux (mem_chart_source _ _)
     (chart_mem_maximalAtlas _ _) (mem_chart_source _ _) hf xf hg
 
@@ -430,7 +430,7 @@ theorem liftPropOn_of_liftProp (mono : ∀ ⦃s x t⦄ ⦃f : H → H'⦄, t ⊆
 theorem liftPropAt_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
     (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G) (hx : x ∈ e.source) : LiftPropAt Q e x := by
   simp_rw [LiftPropAt, hG.liftPropWithinAt_indep_chart he hx G.id_mem_maximalAtlas (mem_univ _),
-    (e.continuousAt hx).continuousWithinAt, true_and_iff]
+    (e.continuousAt hx).continuousWithinAt, true_and]
   exact hG.congr' (e.eventually_right_inverse' hx) (hQ _)
 
 theorem liftPropOn_of_mem_maximalAtlas [HasGroupoid M G] (hG : G.LocalInvariantProp G Q)
@@ -482,7 +482,7 @@ theorem liftPropOn_of_mem_groupoid (hG : G.LocalInvariantProp G Q) (hQ : ∀ y,
 
 theorem liftProp_id (hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) :
     LiftProp Q (id : M → M) := by
-  simp_rw [liftProp_iff, continuous_id, true_and_iff]
+  simp_rw [liftProp_iff, continuous_id, true_and]
   exact fun x ↦ hG.congr' ((chartAt H x).eventually_right_inverse <| mem_chart_target H x) (hQ _)
 
 theorem liftPropAt_iff_comp_subtype_val (hG : LocalInvariantProp G G' P) {U : Opens M}
@@ -515,9 +515,9 @@ theorem liftProp_subtype_val {Q : (H → H) → Set H → H → Prop} (hG : Loca
 
 theorem liftProp_inclusion {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q)
     (hQ : ∀ y, Q id univ y) {U V : Opens M} (hUV : U ≤ V) :
-    LiftProp Q (Set.inclusion hUV : U → V) := by
+    LiftProp Q (Opens.inclusion hUV : U → V) := by
   intro x
-  show LiftPropAt Q (id ∘ inclusion hUV) x
+  show LiftPropAt Q (id ∘ Opens.inclusion hUV) x
   rw [← hG.liftPropAt_iff_comp_inclusion hUV]
   apply hG.liftProp_id hQ
 
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean b/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
index 5161781aac7d7..908659feaf54a 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/Atlas.lean
@@ -156,7 +156,7 @@ theorem tangentMap_chart_symm {p : TangentBundle I M} {q : TangentBundle I H}
 lemma mfderiv_chartAt_eq_tangentCoordChange {x y : M} (hsrc : x ∈ (chartAt H y).source) :
     mfderiv I I (chartAt H y) x = tangentCoordChange I x y x := by
   have := mdifferentiableAt_atlas I (ChartedSpace.chart_mem_atlas _) hsrc
-  simp [mfderiv, if_pos this, Function.comp.assoc]
+  simp [mfderiv, if_pos this, Function.comp_assoc]
 
 end Charts
 
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/Basic.lean b/Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
index f9b96b110f3fc..a5e16af8e340f 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/Basic.lean
@@ -44,7 +44,7 @@ variable
 theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x := by
   unfold UniqueMDiffWithinAt
   simp only [preimage_univ, univ_inter]
-  exact I.unique_diff _ (mem_range_self _)
+  exact I.uniqueDiffOn _ (mem_range_self _)
 
 variable {I}
 
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean b/Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
index efe86094c46e5..5e0909b901206 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean
@@ -158,7 +158,7 @@ variable {c : M'}
 theorem hasMFDerivAt_const (c : M') (x : M) :
     HasMFDerivAt I I' (fun _ : M => c) x (0 : TangentSpace I x →L[𝕜] TangentSpace I' c) := by
   refine ⟨continuous_const.continuousAt, ?_⟩
-  simp only [writtenInExtChartAt, (· ∘ ·), hasFDerivWithinAt_const]
+  simp only [writtenInExtChartAt, Function.comp_def, hasFDerivWithinAt_const]
 
 theorem hasMFDerivWithinAt_const (c : M') (s : Set M) (x : M) :
     HasMFDerivWithinAt I I' (fun _ : M => c) s x (0 : TangentSpace I x →L[𝕜] TangentSpace I' c) :=
@@ -324,7 +324,7 @@ theorem MDifferentiableAt.mfderiv_prod {f : M → M'} {g : M → M''} {x : M}
   classical
   simp_rw [mfderiv, if_pos (hf.prod_mk hg), if_pos hf, if_pos hg]
   exact hf.differentiableWithinAt_writtenInExtChartAt.fderivWithin_prod
-    hg.differentiableWithinAt_writtenInExtChartAt (I.unique_diff _ (mem_range_self _))
+    hg.differentiableWithinAt_writtenInExtChartAt (I.uniqueDiffOn _ (mem_range_self _))
 
 variable (I I' I'')
 
diff --git a/Mathlib/Geometry/Manifold/PartitionOfUnity.lean b/Mathlib/Geometry/Manifold/PartitionOfUnity.lean
index 23d818e090332..d0c29cb8d57a6 100644
--- a/Mathlib/Geometry/Manifold/PartitionOfUnity.lean
+++ b/Mathlib/Geometry/Manifold/PartitionOfUnity.lean
@@ -57,7 +57,7 @@ smooth bump function, partition of unity
 
 universe uι uE uH uM uF
 
-open Function Filter FiniteDimensional Set
+open Function Filter Module Set
 open scoped Topology Manifold
 
 noncomputable section
@@ -752,7 +752,7 @@ theorem exists_msmooth_support_eq_eq_one_iff
   · exact f_diff.div₀ (f_diff.add g_diff) (fun x ↦ ne_of_gt (A x))
   -- show that the range is included in `[0, 1]`
   · refine range_subset_iff.2 (fun x ↦ ⟨div_nonneg (f_pos x) (A x).le, ?_⟩)
-    apply div_le_one_of_le _ (A x).le
+    apply div_le_one_of_le₀ _ (A x).le
     simpa only [le_add_iff_nonneg_right] using g_pos x
   -- show that the support is `s`
   · have B : support (fun x ↦ f x + g x) = univ := eq_univ_of_forall (fun x ↦ (A x).ne')
diff --git a/Mathlib/Geometry/Manifold/Sheaf/Basic.lean b/Mathlib/Geometry/Manifold/Sheaf/Basic.lean
index 1eacc748c2ec9..2368052ad3acf 100644
--- a/Mathlib/Geometry/Manifold/Sheaf/Basic.lean
+++ b/Mathlib/Geometry/Manifold/Sheaf/Basic.lean
@@ -55,7 +55,7 @@ def StructureGroupoid.LocalInvariantProp.localPredicate (hG : LocalInvariantProp
   res := by
     intro U V i f h x
     have hUV : U ≤ V := CategoryTheory.leOfHom i
-    show ChartedSpace.LiftPropAt P (f ∘ Set.inclusion hUV) x
+    show ChartedSpace.LiftPropAt P (f ∘ Opens.inclusion hUV) x
     rw [← hG.liftPropAt_iff_comp_inclusion hUV]
     apply h
   locality := by
@@ -63,7 +63,7 @@ def StructureGroupoid.LocalInvariantProp.localPredicate (hG : LocalInvariantProp
     obtain ⟨U, hxU, i, hU : ChartedSpace.LiftProp P (f ∘ i)⟩ := h x
     let x' : U := ⟨x, hxU⟩
     have hUV : U ≤ V := CategoryTheory.leOfHom i
-    have : ChartedSpace.LiftPropAt P f (inclusion hUV x') := by
+    have : ChartedSpace.LiftPropAt P f (Opens.inclusion hUV x') := by
       rw [hG.liftPropAt_iff_comp_inclusion hUV]
       exact hU x'
     convert this
diff --git a/Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean b/Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
index f051aa094ef52..1b1e09a291612 100644
--- a/Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
+++ b/Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
@@ -138,7 +138,7 @@ structure ModelWithCorners (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Ty
     [NormedAddCommGroup E] [NormedSpace 𝕜 E] (H : Type*) [TopologicalSpace H] extends
     PartialEquiv H E where
   source_eq : source = univ
-  unique_diff' : UniqueDiffOn 𝕜 toPartialEquiv.target
+  uniqueDiffOn' : UniqueDiffOn 𝕜 toPartialEquiv.target
   continuous_toFun : Continuous toFun := by continuity
   continuous_invFun : Continuous invFun := by continuity
 
@@ -149,7 +149,7 @@ def modelWithCornersSelf (𝕜 : Type*) [NontriviallyNormedField 𝕜] (E : Type
     [NormedAddCommGroup E] [NormedSpace 𝕜 E] : ModelWithCorners 𝕜 E E where
   toPartialEquiv := PartialEquiv.refl E
   source_eq := rfl
-  unique_diff' := uniqueDiffOn_univ
+  uniqueDiffOn' := uniqueDiffOn_univ
   continuous_toFun := continuous_id
   continuous_invFun := continuous_id
 
@@ -236,8 +236,11 @@ theorem target_eq : I.target = range (I : H → E) := by
   rw [← image_univ, ← I.source_eq]
   exact I.image_source_eq_target.symm
 
-protected theorem unique_diff : UniqueDiffOn 𝕜 (range I) :=
-  I.target_eq ▸ I.unique_diff'
+protected theorem uniqueDiffOn : UniqueDiffOn 𝕜 (range I) :=
+  I.target_eq ▸ I.uniqueDiffOn'
+
+@[deprecated (since := "2024-09-30")]
+protected alias unique_diff := ModelWithCorners.uniqueDiffOn
 
 @[simp, mfld_simps]
 protected theorem left_inv (x : H) : I.symm (I x) = x := by refine I.left_inv' ?_; simp
@@ -290,17 +293,26 @@ theorem symm_map_nhdsWithin_image {x : H} {s : Set H} : map I.symm (𝓝[I '' s]
 theorem symm_map_nhdsWithin_range (x : H) : map I.symm (𝓝[range I] I x) = 𝓝 x := by
   rw [← I.map_nhds_eq, map_map, I.symm_comp_self, map_id]
 
-theorem unique_diff_preimage {s : Set H} (hs : IsOpen s) :
+theorem uniqueDiffOn_preimage {s : Set H} (hs : IsOpen s) :
     UniqueDiffOn 𝕜 (I.symm ⁻¹' s ∩ range I) := by
   rw [inter_comm]
-  exact I.unique_diff.inter (hs.preimage I.continuous_invFun)
+  exact I.uniqueDiffOn.inter (hs.preimage I.continuous_invFun)
+
+@[deprecated (since := "2024-09-30")]
+alias unique_diff_preimage := uniqueDiffOn_preimage
 
-theorem unique_diff_preimage_source {β : Type*} [TopologicalSpace β] {e : PartialHomeomorph H β} :
+theorem uniqueDiffOn_preimage_source {β : Type*} [TopologicalSpace β] {e : PartialHomeomorph H β} :
     UniqueDiffOn 𝕜 (I.symm ⁻¹' e.source ∩ range I) :=
-  I.unique_diff_preimage e.open_source
+  I.uniqueDiffOn_preimage e.open_source
+
+@[deprecated (since := "2024-09-30")]
+alias unique_diff_preimage_source := uniqueDiffOn_preimage_source
+
+theorem uniqueDiffWithinAt_image {x : H} : UniqueDiffWithinAt 𝕜 (range I) (I x) :=
+  I.uniqueDiffOn _ (mem_range_self _)
 
-theorem unique_diff_at_image {x : H} : UniqueDiffWithinAt 𝕜 (range I) (I x) :=
-  I.unique_diff _ (mem_range_self _)
+@[deprecated (since := "2024-09-30")]
+alias unique_diff_at_image := uniqueDiffWithinAt_image
 
 theorem symm_continuousWithinAt_comp_right_iff {X} [TopologicalSpace X] {f : H → X} {s : Set H}
     {x : H} :
@@ -309,7 +321,7 @@ theorem symm_continuousWithinAt_comp_right_iff {X} [TopologicalSpace X] {f : H 
   · have := h.comp I.continuousWithinAt (mapsTo_preimage _ _)
     simp_rw [preimage_inter, preimage_preimage, I.left_inv, preimage_id', preimage_range,
       inter_univ] at this
-    rwa [Function.comp.assoc, I.symm_comp_self] at this
+    rwa [Function.comp_assoc, I.symm_comp_self] at this
   · rw [← I.left_inv x] at h; exact h.comp I.continuousWithinAt_symm inter_subset_left
 
 protected theorem locallyCompactSpace [LocallyCompactSpace E] (I : ModelWithCorners 𝕜 E H) :
@@ -369,9 +381,9 @@ def ModelWithCorners.prod {𝕜 : Type u} [NontriviallyNormedField 𝕜] {E : Ty
     invFun := fun x => (I.symm x.1, I'.symm x.2)
     source := { x | x.1 ∈ I.source ∧ x.2 ∈ I'.source }
     source_eq := by simp only [setOf_true, mfld_simps]
-    unique_diff' := I.unique_diff'.prod I'.unique_diff'
-    continuous_toFun := I.continuous_toFun.prod_map I'.continuous_toFun
-    continuous_invFun := I.continuous_invFun.prod_map I'.continuous_invFun }
+    uniqueDiffOn' := I.uniqueDiffOn'.prod I'.uniqueDiffOn'
+    continuous_toFun := I.continuous_toFun.prodMap I'.continuous_toFun
+    continuous_invFun := I.continuous_invFun.prodMap I'.continuous_invFun }
 
 /-- Given a finite family of `ModelWithCorners` `I i` on `(E i, H i)`, we define the model with
 corners `pi I` on `(Π i, E i, ModelPi H)`. See note [Manifold type tags] for explanation about
@@ -382,7 +394,7 @@ def ModelWithCorners.pi {𝕜 : Type u} [NontriviallyNormedField 𝕜] {ι : Typ
     ModelWithCorners 𝕜 (∀ i, E i) (ModelPi H) where
   toPartialEquiv := PartialEquiv.pi fun i => (I i).toPartialEquiv
   source_eq := by simp only [pi_univ, mfld_simps]
-  unique_diff' := UniqueDiffOn.pi ι E _ _ fun i _ => (I i).unique_diff'
+  uniqueDiffOn' := UniqueDiffOn.pi ι E _ _ fun i _ => (I i).uniqueDiffOn'
   continuous_toFun := continuous_pi fun i => (I i).continuous.comp (continuous_apply i)
   continuous_invFun := continuous_pi fun i => (I i).continuous_symm.comp (continuous_apply i)
 
@@ -427,7 +439,7 @@ section Boundaryless
 
 /-- Property ensuring that the model with corners `I` defines manifolds without boundary. This
   differs from the more general `BoundarylessManifold`, which requires every point on the manifold
-  to be an interior point.  -/
+  to be an interior point. -/
 class ModelWithCorners.Boundaryless {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*}
     [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
     (I : ModelWithCorners 𝕜 E H) : Prop where
@@ -1057,7 +1069,7 @@ theorem extChartAt_target (x : M) :
 
 theorem uniqueDiffOn_extChartAt_target (x : M) : UniqueDiffOn 𝕜 (extChartAt I x).target := by
   rw [extChartAt_target]
-  exact I.unique_diff_preimage (chartAt H x).open_target
+  exact I.uniqueDiffOn_preimage (chartAt H x).open_target
 
 theorem uniqueDiffWithinAt_extChartAt_target (x : M) :
     UniqueDiffWithinAt 𝕜 (extChartAt I x).target (extChartAt I x x) :=
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean b/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
index b4db2a8aa2149..02abb186cc8f9 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/Basic.lean
@@ -175,7 +175,7 @@ theorem contMDiffWithinAt_totalSpace (f : M → TotalSpace F E) {s : Set M} {x
   simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
   rw [and_and_and_comm, ← FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
   intro hf
-  simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
+  simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp_def,
     PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEquiv.refl_coe,
     extChartAt_self_apply, modelWithCornersSelf_coe, Function.id_def, ← chartedSpaceSelf_prod]
   refine (contMDiffWithinAt_prod_iff _).trans (and_congr ?_ Iff.rfl)
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean b/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
index 4b107cd53d598..2a0e077c860aa 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean
@@ -122,7 +122,7 @@ theorem SmoothFiberwiseLinear.locality_aux₁ (e : PartialHomeomorph (B × F) (B
   have hu' : ∀ p : e.source, (p : B × F).fst ∈ u p := by
     intro p
     have : (p : B × F) ∈ e.source ∩ s p := ⟨p.prop, hsp p⟩
-    simpa only [hesu, mem_prod, mem_univ, and_true_iff] using this
+    simpa only [hesu, mem_prod, mem_univ, and_true] using this
   have heu : ∀ p : e.source, ∀ q : B × F, q.fst ∈ u p → q ∈ e.source := by
     intro p q hq
     have : q ∈ u p ×ˢ (univ : Set F) := ⟨hq, trivial⟩
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean b/Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean
index ff5840676376f..b8beaedc2623e 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth, Floris van Doorn
 -/
 import Mathlib.Geometry.Manifold.MFDeriv.Basic
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Geometry.Manifold.Algebra.LieGroup
 
 /-!
diff --git a/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean b/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
index fed71bc4299df..df7663e67c7ac 100644
--- a/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
+++ b/Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean
@@ -55,7 +55,7 @@ theorem contDiffOn_fderiv_coord_change (i j : atlas H M) :
   have h : ((i.1.extend I).symm ≫ j.1.extend I).source ⊆ range I := by
     rw [i.1.extend_coord_change_source]; apply image_subset_range
   intro x hx
-  refine (ContDiffWithinAt.fderivWithin_right ?_ I.unique_diff le_top <| h hx).mono h
+  refine (ContDiffWithinAt.fderivWithin_right ?_ I.uniqueDiffOn le_top <| h hx).mono h
   refine (PartialHomeomorph.contDiffOn_extend_coord_change I (subset_maximalAtlas I j.2)
     (subset_maximalAtlas I i.2) x hx).mono_of_mem ?_
   exact i.1.extend_coord_change_source_mem_nhdsWithin j.1 I hx
@@ -84,7 +84,7 @@ def tangentBundleCore : VectorBundleCore 𝕜 M E (atlas H M) where
   coordChange_self i x hx v := by
     simp only
     rw [Filter.EventuallyEq.fderivWithin_eq, fderivWithin_id', ContinuousLinearMap.id_apply]
-    · exact I.unique_diff_at_image
+    · exact I.uniqueDiffWithinAt_image
     · filter_upwards [i.1.extend_target_mem_nhdsWithin I hx] with y hy
       exact (i.1.extend I).right_inv hy
     · simp_rw [Function.comp_apply, i.1.extend_left_inv I hx]
@@ -105,7 +105,7 @@ def tangentBundleCore : VectorBundleCore 𝕜 M E (atlas H M) where
     · exact (contDiffWithinAt_extend_coord_change' I (subset_maximalAtlas I j.2)
         (subset_maximalAtlas I i.2) hxj hxi).differentiableWithinAt le_top
     · intro x _; exact mem_range_self _
-    · exact I.unique_diff_at_image
+    · exact I.uniqueDiffWithinAt_image
     · rw [Function.comp_apply, i.1.extend_left_inv I hxi]
 
 -- Porting note: moved to a separate `simp high` lemma b/c `simp` can simplify the LHS
diff --git a/Mathlib/Geometry/Manifold/WhitneyEmbedding.lean b/Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
index e99bb7ea3797f..3e3f2e4bb92fd 100644
--- a/Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
+++ b/Mathlib/Geometry/Manifold/WhitneyEmbedding.lean
@@ -31,7 +31,7 @@ variable {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E
   [FiniteDimensional ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
   {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [SmoothManifoldWithCorners I M]
 
-open Function Filter FiniteDimensional Set
+open Function Filter Module Set
 open scoped Manifold
 
 noncomputable section
diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
index 75e798757aa2f..80753af5d8fda 100644
--- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
+++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean
@@ -89,10 +89,11 @@ noncomputable def Hom.stalkMap {X Y : LocallyRingedSpace.{u}} (f : Hom X Y) (x :
     Y.presheaf.stalk (f.1.1 x) ⟶ X.presheaf.stalk x :=
   f.val.stalkMap x
 
-instance {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X) : IsLocalRingHom (f.stalkMap x) :=
+instance isLocalRingHomStalkMap {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X) :
+    IsLocalRingHom (f.stalkMap x) :=
   f.2 x
 
-instance {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X) :
+instance isLocalRingHomValStalkMap {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X) :
     IsLocalRingHom (f.val.stalkMap x) :=
   f.2 x
 
@@ -108,7 +109,7 @@ instance (X : LocallyRingedSpace.{u}) : Inhabited (Hom X X) :=
 def comp {X Y Z : LocallyRingedSpace.{u}} (f : Hom X Y) (g : Hom Y Z) : Hom X Z :=
   ⟨f.val ≫ g.val, fun x => by
     erw [PresheafedSpace.stalkMap.comp]
-    exact @isLocalRingHom_comp _ _ _ _ _ _ _ _ (f.2 _) (g.2 _)⟩
+    exact @RingHom.isLocalRingHom_comp _ _ _ _ _ _ _ _ (f.2 _) (g.2 _)⟩
 
 /-- The category of locally ringed spaces. -/
 instance : Category LocallyRingedSpace.{u} where
@@ -271,7 +272,7 @@ theorem preimage_basicOpen {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) {U : Ope
   · rintro ⟨y, hy : IsUnit _, rfl⟩
     erw [RingedSpace.mem_basicOpen _ _ ⟨f.1.base y.1, y.2⟩]
     erw [← PresheafedSpace.stalkMap_germ_apply] at hy
-    exact (isUnit_map_iff (f.val.stalkMap _) _).mp hy
+    exact (isUnit_map_iff (f.stalkMap _) _).mp hy
 
 -- This actually holds for all ringed spaces with nontrivial stalks.
 theorem basicOpen_zero (X : LocallyRingedSpace.{u}) (U : Opens X.carrier) :
diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
index b94c92ea5a9a2..48bad4266f46f 100644
--- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
+++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
@@ -247,6 +247,8 @@ noncomputable def coequalizer : LocallyRingedSpace where
   localRing x := by
     obtain ⟨y, rfl⟩ :=
       (TopCat.epi_iff_surjective (coequalizer.π f.val g.val).base).mp inferInstance x
+    -- TODO: this instance was found automatically before #6045
+    have _ : IsLocalRingHom ((coequalizer.π f.val g.val).stalkMap y) := inferInstance
     exact ((coequalizer.π f.val g.val : _).stalkMap y).domain_localRing
 
 /-- The explicit coequalizer cofork of locally ringed spaces. -/
@@ -275,9 +277,12 @@ noncomputable def coequalizerCoforkIsColimit : IsColimit (coequalizerCofork f g)
     -- but this is no longer possible
     set h := _
     change IsLocalRingHom h
-    suffices IsLocalRingHom (((coequalizerCofork f g).π.val.stalkMap _).comp h) from
-      isLocalRingHom_of_comp _ ((coequalizerCofork f g).π.val.stalkMap _)
-    change IsLocalRingHom (_ ≫ (coequalizerCofork f g).π.val.stalkMap y)
+    suffices _ : IsLocalRingHom (((coequalizerCofork f g).π.1.stalkMap _).comp h) by
+      apply isLocalRingHom_of_comp _ ((coequalizerCofork f g).π.1.stalkMap _)
+    -- note to reviewers: this `change` is now more brittle because it now has to fully resolve
+    -- the type to be able to search for `MonoidHomClass`, even though of course all homs in
+    -- `CommRingCat` are clearly such
+    change IsLocalRingHom (h ≫ (coequalizerCofork f g).π.val.stalkMap y)
     erw [← PresheafedSpace.stalkMap.comp]
     apply isLocalRingHom_stalkMap_congr _ _ (coequalizer.π_desc s.π.1 e).symm y
     infer_instance
diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
index e4395a78b6708..0baaf07ca911e 100644
--- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
+++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
@@ -77,7 +77,15 @@ lemma Γevaluation_ne_zero_iff_mem_basicOpen (x : X) (f : X.presheaf.obj (op ⊤
     X.Γevaluation x f ≠ 0 ↔ x ∈ X.toRingedSpace.basicOpen f :=
   evaluation_ne_zero_iff_mem_basicOpen X ⟨x, show x ∈ ⊤ by trivial⟩ f
 
-variable {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y)
+variable {X Y : LocallyRingedSpace.{u}} (f : X ⟶ Y) (x : X)
+
+-- TODO: This instance is found before #6045.
+-- We need this strange instance for `residueFieldMap`, the type of `F` must be fixed
+-- like this. The instance `IsLocalRingHom (f.stalkMap x)` already exists, but does not work for
+-- `residueFieldMap`.
+instance : IsLocalRingHom (F := Y.presheaf.stalk (f.val.base x) →+* X.presheaf.stalk x)
+    (f.stalkMap x) :=
+  f.2 x
 
 /-- If `X ⟶ Y` is a morphism of locally ringed spaces and `x` a point of `X`, we obtain
 a morphism of residue fields in the other direction. -/
@@ -101,6 +109,8 @@ lemma residueFieldMap_comp {Z : LocallyRingedSpace.{u}} (g : Y ⟶ Z) (x : X) :
   simp only [comp_val, SheafedSpace.comp_base, Function.comp_apply, residueFieldMap]
   simp_rw [stalkMap_comp]
   haveI : IsLocalRingHom (g.stalkMap (f.val.base x)) := inferInstance
+  -- TODO: This instance is found before #6045.
+  haveI : IsLocalRingHom (f.stalkMap x) := inferInstance
   apply LocalRing.ResidueField.map_comp
 
 @[reassoc]
diff --git a/Mathlib/Geometry/RingedSpace/OpenImmersion.lean b/Mathlib/Geometry/RingedSpace/OpenImmersion.lean
index 06448493da24d..011fbc8d6b3f1 100644
--- a/Mathlib/Geometry/RingedSpace/OpenImmersion.lean
+++ b/Mathlib/Geometry/RingedSpace/OpenImmersion.lean
@@ -69,7 +69,7 @@ class PresheafedSpace.IsOpenImmersion {X Y : PresheafedSpace C} (f : X ⟶ Y) :
   /-- the underlying continuous map of underlying spaces from the source to an open subset of the
     target. -/
   base_open : OpenEmbedding f.base
-  /-- the underlying sheaf morphism is an isomorphism on each open subset-/
+  /-- the underlying sheaf morphism is an isomorphism on each open subset -/
   c_iso : ∀ U : Opens X, IsIso (f.c.app (op (base_open.isOpenMap.functor.obj U)))
 
 /-- A morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism
@@ -167,7 +167,7 @@ noncomputable def invApp (U : Opens X) :
 
 @[simp, reassoc]
 theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
-    X.presheaf.map i ≫ H.invApp (unop V) =
+    X.presheaf.map i ≫ H.invApp _ (unop V) =
       invApp f (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) := by
   simp only [invApp, ← Category.assoc]
   rw [IsIso.comp_inv_eq]
@@ -179,11 +179,11 @@ theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
 instance (U : Opens X) : IsIso (invApp f U) := by delta invApp; infer_instance
 
 theorem inv_invApp (U : Opens X) :
-    inv (H.invApp U) =
+    inv (H.invApp _ U) =
       f.c.app (op (opensFunctor f |>.obj U)) ≫
         X.presheaf.map
           (eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.inj])) := by
-  rw [← cancel_epi (H.invApp U), IsIso.hom_inv_id]
+  rw [← cancel_epi (H.invApp _ U), IsIso.hom_inv_id]
   delta invApp
   simp [← Functor.map_comp]
 
@@ -195,7 +195,7 @@ theorem invApp_app (U : Opens X) :
 
 @[simp, reassoc]
 theorem app_invApp (U : Opens Y) :
-    f.c.app (op U) ≫ H.invApp ((Opens.map f.base).obj U) =
+    f.c.app (op U) ≫ H.invApp _ ((Opens.map f.base).obj U) =
       Y.presheaf.map
         ((homOfLE (Set.image_preimage_subset f.base U.1)).op :
           op U ⟶ op (opensFunctor f |>.obj ((Opens.map f.base).obj U))) := by
@@ -244,7 +244,7 @@ instance ofRestrict {X : TopCat} (Y : PresheafedSpace C) {f : X ⟶ Y.carrier}
 @[elementwise, simp]
 theorem ofRestrict_invApp {C : Type*} [Category C] (X : PresheafedSpace C) {Y : TopCat}
     {f : Y ⟶ TopCat.of X.carrier} (h : OpenEmbedding f) (U : Opens (X.restrict h).carrier) :
-    (PresheafedSpace.IsOpenImmersion.ofRestrict X h).invApp U = 𝟙 _ := by
+    (PresheafedSpace.IsOpenImmersion.ofRestrict X h).invApp _ U = 𝟙 _ := by
   delta invApp
   rw [IsIso.comp_inv_eq, Category.id_comp]
   change X.presheaf.map _ = X.presheaf.map _
@@ -290,7 +290,7 @@ def pullbackConeOfLeftFst :
   base := pullback.fst _ _
   c :=
     { app := fun U =>
-        hf.invApp (unop U) ≫
+        hf.invApp _ (unop U) ≫
           g.c.app (op (hf.base_open.isOpenMap.functor.obj (unop U))) ≫
             Y.presheaf.map
               (eqToHom
@@ -772,21 +772,21 @@ noncomputable def invApp (U : Opens X) :
 
 @[reassoc (attr := simp)]
 theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
-    X.presheaf.map i ≫ H.invApp (unop V) =
-      H.invApp (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) :=
+    X.presheaf.map i ≫ H.invApp _ (unop V) =
+      H.invApp _ (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) :=
   PresheafedSpace.IsOpenImmersion.inv_naturality f i
 
-instance (U : Opens X) : IsIso (H.invApp U) := by delta invApp; infer_instance
+instance (U : Opens X) : IsIso (H.invApp _ U) := by delta invApp; infer_instance
 
 theorem inv_invApp (U : Opens X) :
-    inv (H.invApp U) =
+    inv (H.invApp _ U) =
       f.c.app (op (opensFunctor f |>.obj U)) ≫
         X.presheaf.map (eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=
   PresheafedSpace.IsOpenImmersion.inv_invApp f U
 
 @[reassoc (attr := simp)]
 theorem invApp_app (U : Opens X) :
-    H.invApp U ≫ f.c.app (op (opensFunctor f |>.obj U)) =
+    H.invApp _ U ≫ f.c.app (op (opensFunctor f |>.obj U)) =
       X.presheaf.map (eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=
   PresheafedSpace.IsOpenImmersion.invApp_app f U
 
@@ -794,7 +794,7 @@ attribute [elementwise] invApp_app
 
 @[reassoc (attr := simp)]
 theorem app_invApp (U : Opens Y) :
-    f.c.app (op U) ≫ H.invApp ((Opens.map f.base).obj U) =
+    f.c.app (op U) ≫ H.invApp _ ((Opens.map f.base).obj U) =
       Y.presheaf.map
         ((homOfLE (Set.image_preimage_subset f.base U.1)).op :
           op U ⟶ op (opensFunctor f |>.obj ((Opens.map f.base).obj U))) :=
@@ -818,7 +818,7 @@ instance ofRestrict {X : TopCat} (Y : SheafedSpace C) {f : X ⟶ Y.carrier}
 @[elementwise, simp]
 theorem ofRestrict_invApp {C : Type*} [Category C] (X : SheafedSpace C) {Y : TopCat}
     {f : Y ⟶ TopCat.of X.carrier} (h : OpenEmbedding f) (U : Opens (X.restrict h).carrier) :
-    (SheafedSpace.IsOpenImmersion.ofRestrict X h).invApp U = 𝟙 _ :=
+    (SheafedSpace.IsOpenImmersion.ofRestrict X h).invApp _ U = 𝟙 _ :=
   PresheafedSpace.IsOpenImmersion.ofRestrict_invApp _ h U
 
 /-- An open immersion is an iso if the underlying continuous map is epi. -/
@@ -946,6 +946,8 @@ instance mono : Mono f :=
 instance : SheafedSpace.IsOpenImmersion (LocallyRingedSpace.forgetToSheafedSpace.map f) :=
   H
 
+-- note to reviewers: is there a `count_heartbeats` for this?
+set_option synthInstance.maxHeartbeats 30000 in
 /-- An explicit pullback cone over `cospan f g` if `f` is an open immersion. -/
 def pullbackConeOfLeft : PullbackCone f g := by
   refine PullbackCone.mk ?_
@@ -964,6 +966,7 @@ def pullbackConeOfLeft : PullbackCone f g := by
 instance : LocallyRingedSpace.IsOpenImmersion (pullbackConeOfLeft f g).snd :=
   show PresheafedSpace.IsOpenImmersion (Y.toPresheafedSpace.ofRestrict _) by infer_instance
 
+set_option synthInstance.maxHeartbeats 80000 in
 /-- The constructed `pullbackConeOfLeft` is indeed limiting. -/
 def pullbackConeOfLeftIsLimit : IsLimit (pullbackConeOfLeft f g) :=
   PullbackCone.isLimitAux' _ fun s => by
@@ -982,7 +985,8 @@ def pullbackConeOfLeftIsLimit : IsLimit (pullbackConeOfLeft f g) :=
       change _ = _ ≫ s.snd.1.stalkMap x at this
       rw [PresheafedSpace.stalkMap.comp, ← IsIso.eq_inv_comp] at this
       rw [this]
-      infer_instance
+      -- TODO: This instance is found by `infer_instance` before #6045.
+      apply CommRingCat.isLocalRingHom_comp
     · intro m _ h₂
       rw [← cancel_mono (pullbackConeOfLeft f g).snd]
       exact h₂.trans <| LocallyRingedSpace.Hom.ext
@@ -1154,7 +1158,7 @@ is an open immersion iff every stalk map is an iso.
 theorem of_stalk_iso {X Y : LocallyRingedSpace} (f : X ⟶ Y) (hf : OpenEmbedding f.1.base)
     [stalk_iso : ∀ x : X.1, IsIso (f.stalkMap x)] :
     LocallyRingedSpace.IsOpenImmersion f :=
-  SheafedSpace.IsOpenImmersion.of_stalk_iso hf (H := stalk_iso)
+  SheafedSpace.IsOpenImmersion.of_stalk_iso _ hf (H := stalk_iso)
 
 end OfStalkIso
 
@@ -1180,21 +1184,21 @@ noncomputable def invApp (U : Opens X) :
 
 @[reassoc (attr := simp)]
 theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
-    X.presheaf.map i ≫ H.invApp (unop V) =
-      H.invApp (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) :=
+    X.presheaf.map i ≫ H.invApp _ (unop V) =
+      H.invApp _ (unop U) ≫ Y.presheaf.map (opensFunctor f |>.op.map i) :=
   PresheafedSpace.IsOpenImmersion.inv_naturality f.1 i
 
-instance (U : Opens X) : IsIso (H.invApp U) := by delta invApp; infer_instance
+instance (U : Opens X) : IsIso (H.invApp _ U) := by delta invApp; infer_instance
 
 theorem inv_invApp (U : Opens X) :
-    inv (H.invApp U) =
+    inv (H.invApp _ U) =
       f.1.c.app (op (opensFunctor f |>.obj U)) ≫
         X.presheaf.map (eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=
   PresheafedSpace.IsOpenImmersion.inv_invApp f.1 U
 
 @[reassoc (attr := simp)]
 theorem invApp_app (U : Opens X) :
-    H.invApp U ≫ f.1.c.app (op (opensFunctor f |>.obj U)) =
+    H.invApp _ U ≫ f.1.c.app (op (opensFunctor f |>.obj U)) =
       X.presheaf.map (eqToHom (by simp [Opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=
   PresheafedSpace.IsOpenImmersion.invApp_app f.1 U
 
@@ -1202,7 +1206,7 @@ attribute [elementwise] invApp_app
 
 @[reassoc (attr := simp)]
 theorem app_invApp (U : Opens Y) :
-    f.1.c.app (op U) ≫ H.invApp ((Opens.map f.1.base).obj U) =
+    f.1.c.app (op U) ≫ H.invApp _ ((Opens.map f.1.base).obj U) =
       Y.presheaf.map
         ((homOfLE (Set.image_preimage_subset f.1.base U.1)).op :
           op U ⟶ op (opensFunctor f |>.obj ((Opens.map f.1.base).obj U))) :=
@@ -1211,7 +1215,7 @@ theorem app_invApp (U : Opens Y) :
 /-- A variant of `app_inv_app` that gives an `eqToHom` instead of `homOfLe`. -/
 @[reassoc]
 theorem app_inv_app' (U : Opens Y) (hU : (U : Set Y) ⊆ Set.range f.1.base) :
-    f.1.c.app (op U) ≫ H.invApp ((Opens.map f.1.base).obj U) =
+    f.1.c.app (op U) ≫ H.invApp _ ((Opens.map f.1.base).obj U) =
       Y.presheaf.map
         (eqToHom <|
             le_antisymm (Set.image_preimage_subset f.1.base U.1) <|
@@ -1226,7 +1230,7 @@ instance ofRestrict {X : TopCat} (Y : LocallyRingedSpace) {f : X ⟶ Y.carrier}
 @[elementwise, simp]
 theorem ofRestrict_invApp (X : LocallyRingedSpace) {Y : TopCat}
     {f : Y ⟶ TopCat.of X.carrier} (h : OpenEmbedding f) (U : Opens (X.restrict h).carrier) :
-    (LocallyRingedSpace.IsOpenImmersion.ofRestrict X h).invApp U = 𝟙 _ :=
+    (LocallyRingedSpace.IsOpenImmersion.ofRestrict X h).invApp _ U = 𝟙 _ :=
   PresheafedSpace.IsOpenImmersion.ofRestrict_invApp _ h U
 
 instance stalk_iso (x : X) : IsIso (f.stalkMap x) :=
diff --git a/Mathlib/Geometry/RingedSpace/PresheafedSpace.lean b/Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
index b94f7d8b249c7..43d40b52b040c 100644
--- a/Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
+++ b/Mathlib/Geometry/RingedSpace/PresheafedSpace.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.Presheaf
 import Mathlib.CategoryTheory.Adjunction.FullyFaithful
@@ -58,7 +58,7 @@ attribute [coe] PresheafedSpace.carrier
 
 -- Porting note: we add this instance, as Lean does not reliably use the `CoeOut` instance above
 -- in downstream files.
-instance : CoeSort (PresheafedSpace C) Type* where coe := fun X => X.carrier
+instance : CoeSort (PresheafedSpace C) Type* where coe X := X.carrier
 
 -- Porting note: the following lemma is removed because it is a syntactic tauto
 /-@[simp]
@@ -340,7 +340,7 @@ theorem restrict_top_presheaf (X : PresheafedSpace C) :
     (X.restrict (Opens.openEmbedding ⊤)).presheaf =
       (Opens.inclusionTopIso X.carrier).inv _* X.presheaf := by
   dsimp
-  rw [Opens.inclusion_top_functor X.carrier]
+  rw [Opens.inclusion'_top_functor X.carrier]
   rfl
 
 theorem ofRestrict_top_c (X : PresheafedSpace C) :
diff --git a/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean b/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
index b5ac976467db8..2bd1fd993c02a 100644
--- a/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
+++ b/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
@@ -149,7 +149,7 @@ theorem pullback_base (i j k : D.J) (S : Set (D.V (i, j)).carrier) :
 /-- The red and the blue arrows in ![this diagram](https://i.imgur.com/0GiBUh6.png) commute. -/
 @[simp, reassoc]
 theorem f_invApp_f_app (i j k : D.J) (U : Opens (D.V (i, j)).carrier) :
-    (D.f_open i j).invApp U ≫ (D.f i k).c.app _ =
+    (D.f_open i j).invApp _ U ≫ (D.f i k).c.app _ =
       (π₁ i, j, k).c.app (op U) ≫
         (π₂⁻¹ i, j, k) (unop _) ≫
           (D.V _).presheaf.map
@@ -161,7 +161,7 @@ theorem f_invApp_f_app (i j k : D.J) (U : Opens (D.V (i, j)).carrier) :
                 apply pullback_base)) := by
   have := PresheafedSpace.congr_app (@pullback.condition _ _ _ _ _ (D.f i j) (D.f i k) _)
   dsimp only [comp_c_app] at this
-  rw [← cancel_epi (inv ((D.f_open i j).invApp U)), IsIso.inv_hom_id_assoc,
+  rw [← cancel_epi (inv ((D.f_open i j).invApp _ U)), IsIso.inv_hom_id_assoc,
     IsOpenImmersion.inv_invApp]
   simp_rw [Category.assoc]
   erw [(π₁ i, j, k).c.naturality_assoc, reassoc_of% this, ← Functor.map_comp_assoc,
@@ -271,7 +271,7 @@ def opensImagePreimageMap (i j : D.J) (U : Opens (D.U i).carrier) :
       (Opens.map (𝖣.ι j).base).obj ((D.ι_openEmbedding i).isOpenMap.functor.obj U)) :=
   (D.f i j).c.app (op U) ≫
     (D.t j i).c.app _ ≫
-      (D.f_open j i).invApp (unop _) ≫
+      (D.f_open j i).invApp _ (unop _) ≫
         (𝖣.U j).presheaf.map (eqToHom (D.ι_image_preimage_eq i j U)).op
 
 theorem opensImagePreimageMap_app' (i j k : D.J) (U : Opens (D.U i).carrier) :
@@ -334,9 +334,9 @@ def ιInvAppπApp {i : D.J} (U : Opens (D.U i).carrier) (j) :
     rw [Set.preimage_preimage]
     change (D.f j k ≫ 𝖣.ι j).base ⁻¹' _ = _
     -- Porting note: used to be `congr 3`
-    refine congr_arg (· ⁻¹' _) ?_
-    convert congr_arg (ContinuousMap.toFun (α := D.V ⟨j, k⟩) (β := D.glued) ·) ?_
-    refine congr_arg (PresheafedSpace.Hom.base (C := C) ·) ?_
+    suffices D.f j k ≫ D.ι j = colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k)) by
+      rw [this]
+      rfl
     exact colimit.w 𝖣.diagram.multispan (WalkingMultispan.Hom.fst (j, k))
   · exact D.opensImagePreimageMap i j U
 
@@ -436,11 +436,27 @@ abbrev ιInvAppπEqMap {i : D.J} (U : Opens (D.U i).carrier) :=
 theorem π_ιInvApp_π (i j : D.J) (U : Opens (D.U i).carrier) :
     D.diagramOverOpenπ U i ≫ D.ιInvAppπEqMap U ≫ D.ιInvApp U ≫ D.diagramOverOpenπ U j =
       D.diagramOverOpenπ U j := by
-  -- Porting note: originally, the proof of monotonicity was left a blank and proved in the end
-  -- but Lean 4 doesn't like this any more, so the proof is restructured
-  rw [← @cancel_mono (f := (componentwiseDiagram 𝖣.diagram.multispan _).map
-    (Quiver.Hom.op (WalkingMultispan.Hom.snd (i, j))) ≫ 𝟙 _) _ _ (by
-    rw [Category.comp_id]
+  rw [← @cancel_mono
+          (f := (componentwiseDiagram 𝖣.diagram.multispan _).map
+            (Quiver.Hom.op (WalkingMultispan.Hom.snd (i, j))) ≫ 𝟙 _) ..]
+  · simp_rw [Category.assoc]
+    rw [limit.w_assoc]
+    erw [limit.lift_π_assoc]
+    rw [Category.comp_id, Category.comp_id]
+    change _ ≫ _ ≫ (_ ≫ _) ≫ _ = _
+    rw [congr_app (D.t_id _), id_c_app]
+    simp_rw [Category.assoc]
+    rw [← Functor.map_comp_assoc]
+    -- Porting note (#11224): change `rw` to `erw`
+    erw [IsOpenImmersion.inv_naturality_assoc]
+    erw [IsOpenImmersion.app_invApp_assoc]
+    iterate 3 rw [← Functor.map_comp_assoc]
+    rw [NatTrans.naturality_assoc]
+    erw [← (D.V (i, j)).presheaf.map_comp]
+    convert
+      limit.w (componentwiseDiagram 𝖣.diagram.multispan _)
+        (Quiver.Hom.op (WalkingMultispan.Hom.fst (i, j)))
+  · rw [Category.comp_id]
     apply (config := { allowSynthFailures := true }) mono_comp
     change Mono ((_ ≫ D.f j i).c.app _)
     rw [comp_c_app]
@@ -448,24 +464,7 @@ theorem π_ιInvApp_π (i j : D.J) (U : Opens (D.U i).carrier) :
     · erw [D.ι_image_preimage_eq i j U]
       infer_instance
     · have : IsIso (D.t i j).c := by apply c_isIso_of_iso
-      infer_instance)]
-  simp_rw [Category.assoc]
-  rw [limit.w_assoc]
-  erw [limit.lift_π_assoc]
-  rw [Category.comp_id, Category.comp_id]
-  change _ ≫ _ ≫ (_ ≫ _) ≫ _ = _
-  rw [congr_app (D.t_id _), id_c_app]
-  simp_rw [Category.assoc]
-  rw [← Functor.map_comp_assoc]
-  -- Porting note (#11224): change `rw` to `erw`
-  erw [IsOpenImmersion.inv_naturality_assoc]
-  erw [IsOpenImmersion.app_invApp_assoc]
-  iterate 3 rw [← Functor.map_comp_assoc]
-  rw [NatTrans.naturality_assoc]
-  erw [← (D.V (i, j)).presheaf.map_comp]
-  convert
-    limit.w (componentwiseDiagram 𝖣.diagram.multispan _)
-      (Quiver.Hom.op (WalkingMultispan.Hom.fst (i, j)))
+      infer_instance
 
 /-- `ιInvApp` is the inverse of `D.ι i` on `U`. -/
 theorem π_ιInvApp_eq_id (i : D.J) (U : Opens (D.U i).carrier) :
diff --git a/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean b/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
index 1a3dd8fdd7585..3ce2bceb89768 100644
--- a/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
+++ b/Mathlib/Geometry/RingedSpace/PresheafedSpace/HasColimits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Geometry.RingedSpace.PresheafedSpace
 import Mathlib.Topology.Category.TopCat.Limits.Basic
@@ -97,7 +97,7 @@ def pushforwardDiagramToColimit (F : J ⥤ PresheafedSpace.{_, _, v} C) :
   obj j := op (colimit.ι (F ⋙ PresheafedSpace.forget C) j _* (F.obj j).presheaf)
   map {j j'} f :=
     ((pushforward C (colimit.ι (F ⋙ PresheafedSpace.forget C) j')).map (F.map f).c ≫
-      (Pushforward.comp  ((F ⋙ PresheafedSpace.forget C).map f)
+      (Pushforward.comp ((F ⋙ PresheafedSpace.forget C).map f)
         (colimit.ι (F ⋙ PresheafedSpace.forget C) j') (F.obj j).presheaf).inv ≫
       (pushforwardEq (colimit.w (F ⋙ PresheafedSpace.forget C) f) (F.obj j).presheaf).hom).op
   map_id j := by
diff --git a/Mathlib/Geometry/RingedSpace/SheafedSpace.lean b/Mathlib/Geometry/RingedSpace/SheafedSpace.lean
index 5ec030cd1628c..11a937b20ac78 100644
--- a/Mathlib/Geometry/RingedSpace/SheafedSpace.lean
+++ b/Mathlib/Geometry/RingedSpace/SheafedSpace.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
 import Mathlib.Geometry.RingedSpace.Stalks
@@ -47,7 +47,7 @@ namespace SheafedSpace
 instance coeCarrier : CoeOut (SheafedSpace C) TopCat where coe X := X.carrier
 
 instance coeSort : CoeSort (SheafedSpace C) Type* where
-  coe := fun X => X.1
+  coe X := X.1
 
 /-- Extract the `sheaf C (X : Top)` from a `SheafedSpace C`. -/
 def sheaf (X : SheafedSpace C) : Sheaf C (X : TopCat) :=
diff --git a/Mathlib/Geometry/RingedSpace/Stalks.lean b/Mathlib/Geometry/RingedSpace/Stalks.lean
index 2bff8fe9dea93..858441ca519fa 100644
--- a/Mathlib/Geometry/RingedSpace/Stalks.lean
+++ b/Mathlib/Geometry/RingedSpace/Stalks.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Geometry.RingedSpace.PresheafedSpace
 import Mathlib.CategoryTheory.Limits.Final
diff --git a/Mathlib/GroupTheory/Abelianization.lean b/Mathlib/GroupTheory/Abelianization.lean
index 51b0cfca00034..5ee8a835a60c4 100644
--- a/Mathlib/GroupTheory/Abelianization.lean
+++ b/Mathlib/GroupTheory/Abelianization.lean
@@ -125,7 +125,7 @@ theorem commutator_subset_ker : commutator G ≤ f.ker := by
 /-- If `f : G → A` is a group homomorphism to an abelian group, then `lift f` is the unique map
   from the abelianization of a `G` to `A` that factors through `f`. -/
 def lift : (G →* A) ≃ (Abelianization G →* A) where
-  toFun f := QuotientGroup.lift _ f fun _ h => f.mem_ker.2 <| commutator_subset_ker _ h
+  toFun f := QuotientGroup.lift _ f fun _ h => MonoidHom.mem_ker.2 <| commutator_subset_ker _ h
   invFun F := F.comp of
   left_inv _ := MonoidHom.ext fun _ => rfl
   right_inv _ := MonoidHom.ext fun x => QuotientGroup.induction_on x fun _ => rfl
diff --git a/Mathlib/GroupTheory/Archimedean.lean b/Mathlib/GroupTheory/Archimedean.lean
index 77957e91b3685..4105f66700e0c 100644
--- a/Mathlib/GroupTheory/Archimedean.lean
+++ b/Mathlib/GroupTheory/Archimedean.lean
@@ -25,70 +25,81 @@ is cyclic.  (There are several other methods one could use to prove this fact, i
 algebraic methods, but none seem to exist in mathlib as of writing.  The closest is
 `Subgroup.is_cyclic`, but that has not been transferred to `AddSubgroup`.)
 
+The file also supports multiplicative groups via `MulArchimedean`.
+
 The result is also used in `Topology.Instances.Real` as an ingredient in the classification of
 subgroups of `ℝ`.
 -/
 
 open Set
-variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
+variable {G : Type*} [LinearOrderedCommGroup G] [MulArchimedean G]
 
-/-- Given a subgroup `H` of a decidable linearly ordered archimedean abelian group `G`, if there
-exists a minimal element `a` of `H ∩ G_{>0}` then `H` is generated by `a`. -/
-theorem AddSubgroup.cyclic_of_min {H : AddSubgroup G} {a : G}
-    (ha : IsLeast { g : G | g ∈ H ∧ 0 < g } a) : H = AddSubgroup.closure {a} := by
+/-- Given a subgroup `H` of a decidable linearly ordered mul-archimedean abelian group `G`, if there
+exists a minimal element `a` of `H ∩ G_{>1}` then `H` is generated by `a`. -/
+@[to_additive AddSubgroup.cyclic_of_min "Given a subgroup `H` of a decidable linearly ordered
+archimedean abelian group `G`, if there exists a minimal element `a` of `H ∩ G_{>0}` then `H` is
+generated by `a`. "]
+theorem Subgroup.cyclic_of_min {H : Subgroup G} {a : G}
+    (ha : IsLeast { g : G | g ∈ H ∧ 1 < g } a) : H = closure {a} := by
   obtain ⟨⟨a_in, a_pos⟩, a_min⟩ := ha
   refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
   intro g g_in
-  obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zsmul_near_of_pos' a_pos g
-  have h_zero : g - k • a = 0 := by
+  obtain ⟨k, ⟨nonneg, lt⟩, _⟩ := existsUnique_zpow_near_of_one_lt a_pos g
+  have h_zero : g / (a ^ k) = 1 := by
     by_contra h
-    have h : a ≤ g - k • a := by
+    have h : a ≤ g / (a ^ k) := by
       refine a_min ⟨?_, ?_⟩
-      · exact AddSubgroup.sub_mem H g_in (AddSubgroup.zsmul_mem H a_in k)
-      · exact lt_of_le_of_ne nonneg (Ne.symm h)
-    have h' : ¬a ≤ g - k • a := not_le.mpr lt
+      · exact Subgroup.div_mem H g_in (Subgroup.zpow_mem H a_in k)
+      · exact lt_of_le_of_ne (by simpa using nonneg) (Ne.symm h)
+    have h' : ¬a ≤ g / (a ^ k) := not_le.mpr (by simpa [zpow_add_one, div_lt_iff_lt_mul'] using lt)
     contradiction
-  simp [sub_eq_zero.mp h_zero, AddSubgroup.mem_closure_singleton]
+  simp [div_eq_one.mp h_zero, mem_closure_singleton]
 
-/-- If a nontrivial additive subgroup of a linear ordered additive commutative group is disjoint
-with the interval `Set.Ioo 0 a` for some positive `a`, then the set of positive elements of this
+/-- If a nontrivial subgroup of a linear ordered commutative group is disjoint
+with the interval `Set.Ioo 1 a` for some `1 < a`, then the set of elements greater than 1 of this
 group admits the least element. -/
-theorem AddSubgroup.exists_isLeast_pos {H : AddSubgroup G} (hbot : H ≠ ⊥) {a : G} (h₀ : 0 < a)
-    (hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, IsLeast { g : G | g ∈ H ∧ 0 < g } b := by
+@[to_additive "If a nontrivial additive subgroup of a linear ordered additive commutative group is
+disjoint with the interval `Set.Ioo 0 a` for some positive `a`, then the set of positive elements of
+this group admits the least element."]
+theorem Subgroup.exists_isLeast_one_lt {H : Subgroup G} (hbot : H ≠ ⊥) {a : G} (h₀ : 1 < a)
+    (hd : Disjoint (H : Set G) (Ioo 1 a)) : ∃ b, IsLeast { g : G | g ∈ H ∧ 1 < g } b := by
   -- todo: move to a lemma?
-  have hex : ∀ g > 0, ∃ n : ℕ, g ∈ Ioc (n • a) ((n + 1) • a) := fun g hg => by
-    rcases existsUnique_add_zsmul_mem_Ico h₀ 0 (g - a) with ⟨m, ⟨hm, hm'⟩, -⟩
-    simp only [zero_add, sub_le_iff_le_add, sub_add_cancel, ← add_one_zsmul] at hm hm'
+  have hex : ∀ g > 1, ∃ n : ℕ, g ∈ Ioc (a ^ n) (a ^ (n + 1)) := fun g hg => by
+    rcases existsUnique_mul_zpow_mem_Ico h₀ 1 (g / a) with ⟨m, ⟨hm, hm'⟩, -⟩
+    simp only [one_mul, div_le_iff_le_mul, div_mul_cancel, ← zpow_add_one] at hm hm'
     lift m to ℕ
-    · rw [← Int.lt_add_one_iff, ← zsmul_lt_zsmul_iff h₀, zero_zsmul]
+    · rw [← Int.lt_add_one_iff, ← zpow_lt_zpow_iff h₀, zpow_zero]
       exact hg.trans_le hm
-    · simp only [← Nat.cast_succ, natCast_zsmul] at hm hm'
+    · simp only [← Nat.cast_succ, zpow_natCast] at hm hm'
       exact ⟨m, hm', hm⟩
-  have : ∃ n : ℕ, Set.Nonempty (H ∩ Ioc (n • a) ((n + 1) • a)) := by
-    rcases (bot_or_exists_ne_zero H).resolve_left hbot with ⟨g, hgH, hg₀⟩
-    rcases hex |g| (abs_pos.2 hg₀) with ⟨n, hn⟩
-    exact ⟨n, _, (@abs_mem_iff (AddSubgroup G) G _ _).2 hgH, hn⟩
+  have : ∃ n : ℕ, Set.Nonempty (H ∩ Ioc (a ^ n) (a ^ (n + 1))) := by
+    rcases (bot_or_exists_ne_one H).resolve_left hbot with ⟨g, hgH, hg₀⟩
+    rcases hex |g|ₘ (one_lt_mabs.2 hg₀) with ⟨n, hn⟩
+    exact ⟨n, _, (@mabs_mem_iff (Subgroup G) G _ _).2 hgH, hn⟩
   classical rcases Nat.findX this with ⟨n, ⟨x, hxH, hnx, hxn⟩, hmin⟩
   by_contra hxmin
   simp only [IsLeast, not_and, mem_setOf_eq, mem_lowerBounds, not_exists, not_forall,
     not_le] at hxmin
-  rcases hxmin x ⟨hxH, (nsmul_nonneg h₀.le _).trans_lt hnx⟩ with ⟨y, ⟨hyH, hy₀⟩, hxy⟩
+  rcases hxmin x ⟨hxH, (one_le_pow_of_one_le'  h₀.le _).trans_lt hnx⟩ with ⟨y, ⟨hyH, hy₀⟩, hxy⟩
   rcases hex y hy₀ with ⟨m, hm⟩
   cases' lt_or_le m n with hmn hnm
   · exact hmin m hmn ⟨y, hyH, hm⟩
-  · refine disjoint_left.1 hd (sub_mem hxH hyH) ⟨sub_pos.2 hxy, sub_lt_iff_lt_add'.2 ?_⟩
-    calc x ≤ (n + 1) • a := hxn
-    _ ≤ (m + 1) • a := nsmul_le_nsmul_left h₀.le (add_le_add_right hnm _)
-    _ = m • a + a := succ_nsmul _ _
-    _ < y + a := add_lt_add_right hm.1 _
+  · refine disjoint_left.1 hd (div_mem hxH hyH) ⟨one_lt_div'.2 hxy, div_lt_iff_lt_mul'.2 ?_⟩
+    calc x ≤ a^ (n + 1) := hxn
+    _ ≤ a ^ (m + 1) := pow_le_pow_right' h₀.le (add_le_add_right hnm _)
+    _ = a ^ m * a := pow_succ _ _
+    _ < y * a := mul_lt_mul_right' hm.1 _
 
-/-- If an additive subgroup of a linear ordered additive commutative group is disjoint with the
-interval `Set.Ioo 0 a` for some positive `a`, then this is a cyclic subgroup. -/
-theorem AddSubgroup.cyclic_of_isolated_zero {H : AddSubgroup G} {a : G} (h₀ : 0 < a)
-    (hd : Disjoint (H : Set G) (Ioo 0 a)) : ∃ b, H = closure {b} := by
+/-- If a subgroup of a linear ordered commutative group is disjoint with the
+interval `Set.Ioo 1 a` for some `1 < a`, then this is a cyclic subgroup. -/
+@[to_additive AddSubgroup.cyclic_of_isolated_zero "If an additive subgroup of a linear ordered
+additive commutative group is disjoint with the interval `Set.Ioo 0 a` for some positive `a`, then
+this is a cyclic subgroup."]
+theorem Subgroup.cyclic_of_isolated_one {H : Subgroup G} {a : G} (h₀ : 1 < a)
+    (hd : Disjoint (H : Set G) (Ioo 1 a)) : ∃ b, H = closure {b} := by
   rcases eq_or_ne H ⊥ with rfl | hbot
-  · exact ⟨0, closure_singleton_zero.symm⟩
-  · exact (exists_isLeast_pos hbot h₀ hd).imp fun _ => cyclic_of_min
+  · exact ⟨1, closure_singleton_one.symm⟩
+  · exact (exists_isLeast_one_lt hbot h₀ hd).imp fun _ => cyclic_of_min
 
 /-- Every subgroup of `ℤ` is cyclic. -/
 theorem Int.subgroup_cyclic (H : AddSubgroup ℤ) : ∃ a, H = AddSubgroup.closure {a} :=
diff --git a/Mathlib/GroupTheory/ArchimedeanDensely.lean b/Mathlib/GroupTheory/ArchimedeanDensely.lean
new file mode 100644
index 0000000000000..8b00a52cd33de
--- /dev/null
+++ b/Mathlib/GroupTheory/ArchimedeanDensely.lean
@@ -0,0 +1,235 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+import Mathlib.GroupTheory.Archimedean
+import Mathlib.Algebra.Group.Equiv.TypeTags
+import Mathlib.Algebra.Group.Subgroup.Pointwise
+import Mathlib.Algebra.Order.Group.TypeTags
+import Mathlib.Algebra.Order.Hom.Monoid
+
+/-!
+# Archimedean groups are either discrete or densely ordered
+
+This file proves a few additional facts about linearly ordered additive groups which satisfy the
+  `Archimedean` property --
+  they are either order-isomorphic and additvely isomorphic to the integers,
+  or they are densely ordered.
+
+They are placed here in a separate file (rather than incorporated as a continuation of
+`GroupTheory.Archimedean`) because they rely on some imports from pointwise lemmas.
+-/
+
+open Multiplicative Set
+
+-- no earlier file imports the necessary requirements for the next two
+
+/-- The subgroup generated by an element of a group equals the set of
+integer powers of the element, such that each power is a unique element.
+This is the stronger version of `Subgroup.mem_closure_singleton`. -/
+@[to_additive "The additive subgroup generated by an element of an additive group equals the set of
+integer multiples of the element, such that each multiple is a unique element.
+This is the stronger version of `AddSubgroup.mem_closure_singleton`."]
+lemma Subgroup.mem_closure_singleton_iff_existsUnique_zpow {G : Type*}
+    [LinearOrderedCommGroup G] {a b : G} (ha : a ≠ 1) :
+    b ∈ closure {a} ↔ ∃! k : ℤ, a ^ k = b := by
+  rw [mem_closure_singleton]
+  constructor
+  · suffices Function.Injective (a ^ · : ℤ → G) by
+      rintro ⟨m, rfl⟩
+      exact ⟨m, rfl, fun k hk ↦ this hk⟩
+    rcases ha.lt_or_lt with ha | ha
+    · exact (zpow_right_strictAnti ha).injective
+    · exact (zpow_right_strictMono ha).injective
+  · exact fun h ↦ h.exists
+
+open Subgroup in
+/-- In two linearly ordered groups, the closure of an element of one group
+is isomorphic (and order-isomorphic) to the closure of an element in the other group. -/
+@[to_additive "In two linearly ordered additive groups, the closure of an element of one group
+is isomorphic (and order-isomorphic) to the closure of an element in the other group."]
+noncomputable def LinearOrderedCommGroup.closure_equiv_closure {G G' : Type*}
+    [LinearOrderedCommGroup G] [LinearOrderedCommGroup G'] (x : G) (y : G') (hxy : x = 1 ↔ y = 1) :
+    closure ({x} : Set G) ≃*o closure ({y} : Set G') :=
+  if hx : x = 1 then by
+    refine ⟨⟨⟨fun _ ↦ ⟨1, by simp [hxy.mp hx]⟩, fun _ ↦ ⟨1, by simp [hx]⟩, ?_, ?_⟩, ?_⟩, ?_⟩
+    · intro ⟨a, ha⟩
+      simpa [hx, closure_singleton_one, eq_comm] using ha
+    · intro ⟨a, ha⟩
+      simpa [hxy.mp hx, closure_singleton_one, eq_comm] using ha
+    · intros
+      simp
+    · intro ⟨a, ha⟩ ⟨b, hb⟩
+      simp only [hx, closure_singleton_one, mem_bot] at ha hb
+      simp [ha, hb]
+  else by
+    set x' := max x x⁻¹ with hx'
+    have xpos : 1 < x' := by
+      simp [hx', eq_comm, hx]
+    set y' := max y y⁻¹ with hy'
+    have ypos : 1 < y' := by
+      simp [hy', eq_comm, ← hxy, hx]
+    have hxc : closure {x} = closure {x'} := by
+      rcases max_cases x x⁻¹ with H|H <;>
+      simp [hx', H.left]
+    have hyc : closure {y} = closure {y'} := by
+      rcases max_cases y y⁻¹ with H|H <;>
+      simp [hy', H.left]
+    refine ⟨⟨⟨
+      fun a ↦ ⟨y' ^ ((mem_closure_singleton).mp
+        (by simpa [hxc] using a.prop)).choose, ?_⟩,
+      fun a ↦ ⟨x' ^ ((mem_closure_singleton).mp
+        (by simpa [hyc] using a.prop)).choose, ?_⟩,
+        ?_, ?_⟩, ?_⟩, ?_⟩
+    · rw [hyc, mem_closure_singleton]
+      exact ⟨_, rfl⟩
+    · rw [hxc, mem_closure_singleton]
+      exact ⟨_, rfl⟩
+    · intro a
+      generalize_proofs A B C D
+      rw [Subtype.ext_iff, ← (C a).choose_spec, zpow_right_inj xpos,
+          ← zpow_right_inj ypos, (A ⟨_, D a⟩).choose_spec]
+    · intro a
+      generalize_proofs A B C D
+      rw [Subtype.ext_iff, ← (C a).choose_spec, zpow_right_inj ypos,
+          ← zpow_right_inj xpos, (A ⟨_, D a⟩).choose_spec]
+    · intro a b
+      generalize_proofs A B C D E F
+      simp only [Submonoid.coe_mul, coe_toSubmonoid, Submonoid.mk_mul_mk, Subtype.mk.injEq,
+                 coe_mul, MulMemClass.mk_mul_mk, Subtype.ext_iff]
+      rw [← zpow_add, zpow_right_inj ypos, ← zpow_right_inj xpos, zpow_add,
+          (A a).choose_spec, (A b).choose_spec, (A (a * b)).choose_spec]
+      simp
+    · intro a b
+      simp only [MulEquiv.coe_mk, Equiv.coe_fn_mk, Subtype.mk_le_mk]
+      generalize_proofs A B C D
+      simp [zpow_le_zpow_iff ypos, ← zpow_le_zpow_iff xpos, A.choose_spec, B.choose_spec]
+
+variable {G : Type*} [LinearOrderedCommGroup G] [MulArchimedean G]
+
+@[to_additive]
+lemma Subgroup.isLeast_of_closure_iff_eq_mabs {a b : G} :
+    IsLeast {y : G | y ∈ closure ({a} : Set G) ∧ 1 < y} b ↔ b = |a|ₘ ∧ 1 < b := by
+  constructor <;> intro h
+  · have := Subgroup.cyclic_of_min h
+    have ha : a ∈ closure ({b} : Set G) := by
+      simp [← this]
+    rw [mem_closure_singleton] at ha
+    obtain ⟨n, rfl⟩ := ha
+    have := h.left
+    simp only [mem_closure_singleton, mem_setOf_eq, ← mul_zsmul] at this
+    obtain ⟨m, hm⟩ := this.left
+    have key : m * n = 1 := by
+      rw [← zpow_right_inj this.right, zpow_mul', hm, zpow_one]
+    rw [Int.mul_eq_one_iff_eq_one_or_neg_one] at key
+    rw [eq_comm]
+    rcases key with ⟨rfl, rfl⟩|⟨rfl, rfl⟩ <;>
+    simp [this.right.le, this.right, mabs]
+  · wlog ha : 1 ≤ a generalizing a
+    · convert @this (a⁻¹) ?_ (by simpa using le_of_not_le ha) using 4
+      · simp
+      · rwa [mabs_inv]
+    rw [mabs, sup_eq_left.mpr ((inv_le_one'.mpr ha).trans ha)] at h
+    rcases h with ⟨rfl, h⟩
+    refine ⟨?_, ?_⟩
+    · simp [h]
+    · intro x
+      simp only [mem_closure_singleton, mem_setOf_eq, and_imp, forall_exists_index]
+      rintro k rfl hk
+      rw [← zpow_one b, ← zpow_mul, one_mul, zpow_le_zpow_iff h, ← zero_add 1,
+          ← Int.lt_iff_add_one_le]
+      contrapose! hk
+      rw [← Left.one_le_inv_iff, ← zpow_neg]
+      exact one_le_zpow ha (by simp [hk])
+
+/-- If an element of a linearly ordered archimedean additive group is the least positive element,
+then the whole group is isomorphic (and order-isomorphic) to the integers. -/
+noncomputable def LinearOrderedAddCommGroup.int_orderAddMonoidIso_of_isLeast_pos {G : Type*}
+    [LinearOrderedAddCommGroup G] [Archimedean G] {x : G}
+    (h : IsLeast {y : G | 0 < y} x) : G ≃+o ℤ := by
+  have : IsLeast {y : G | y ∈ (⊤ : AddSubgroup G) ∧ 0 < y} x := by simpa using h
+  replace this := AddSubgroup.cyclic_of_min this
+  let e : G ≃+o (⊤ : AddSubgroup G) := ⟨AddSubsemigroup.topEquiv.symm,
+    (AddEquiv.strictMono_symm AddSubsemigroup.strictMono_topEquiv).le_iff_le⟩
+  let e' : (⊤ : AddSubgroup G) ≃+o AddSubgroup.closure {x} :=
+    ⟨AddEquiv.subsemigroupCongr (by simp [this]),
+     (AddEquiv.strictMono_subsemigroupCongr _).le_iff_le⟩
+  let g : (⊤ : AddSubgroup ℤ) ≃+o ℤ := ⟨AddSubsemigroup.topEquiv,
+    (AddSubsemigroup.strictMono_topEquiv).le_iff_le⟩
+  let g' : AddSubgroup.closure ({1} : Set ℤ) ≃+o (⊤ : AddSubgroup ℤ) :=
+    ⟨(.subsemigroupCongr (by simp [AddSubgroup.closure_singleton_int_one_eq_top])),
+     (AddEquiv.strictMono_subsemigroupCongr _).le_iff_le⟩
+  let f := closure_equiv_closure x (1 : ℤ) (by simp [h.left.ne'])
+  exact ((((e.trans e').trans f).trans g').trans g : G ≃+o ℤ)
+
+/-- If an element of a linearly ordered mul-archimedean group is the least element greater than 1,
+then the whole group is isomorphic (and order-isomorphic) to the multiplicative integers. -/
+@[to_additive existing LinearOrderedAddCommGroup.int_orderAddMonoidIso_of_isLeast_pos]
+noncomputable def LinearOrderedCommGroup.multiplicative_int_orderMonoidIso_of_isLeast_one_lt
+    {x : G} (h : IsLeast {y : G | 1 < y} x) : G ≃*o Multiplicative ℤ := by
+  have : IsLeast {y : Additive G | 0 < y} (.ofMul x) := h
+  let f' := LinearOrderedAddCommGroup.int_orderAddMonoidIso_of_isLeast_pos (G := Additive G) this
+  exact ⟨AddEquiv.toMultiplicative' f', by simp⟩
+
+/-- Any linearly ordered archimedean additive group is either isomorphic (and order-isomorphic)
+to the integers, or is densely ordered. -/
+lemma LinearOrderedAddCommGroup.discrete_or_denselyOrdered (G : Type*)
+    [LinearOrderedAddCommGroup G] [Archimedean G] :
+    Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G := by
+  by_cases H : ∃ x, IsLeast {y : G | 0 < y} x
+  · obtain ⟨x, hx⟩ := H
+    exact Or.inl ⟨(int_orderAddMonoidIso_of_isLeast_pos hx)⟩
+  · push_neg at H
+    refine Or.inr ⟨?_⟩
+    intro x y hxy
+    specialize H (y - x)
+    obtain ⟨z, hz⟩ : ∃ z : G, 0 < z ∧ z < y - x := by
+      contrapose! H
+      refine ⟨by simp [hxy], fun _ ↦ H _⟩
+    refine ⟨x + z, ?_, ?_⟩
+    · simp [hz.left]
+    · simpa [lt_sub_iff_add_lt'] using hz.right
+
+variable (G) in
+/-- Any linearly ordered mul-archimedean group is either isomorphic (and order-isomorphic)
+to the multiplicative integers, or is densely ordered. -/
+@[to_additive existing]
+lemma LinearOrderedCommGroup.discrete_or_denselyOrdered :
+    Nonempty (G ≃*o Multiplicative ℤ) ∨ DenselyOrdered G := by
+  refine (LinearOrderedAddCommGroup.discrete_or_denselyOrdered (Additive G)).imp ?_ id
+  rintro ⟨f, hf⟩
+  exact ⟨AddEquiv.toMultiplicative' f, hf⟩
+
+/-- Any nontrivial (has other than 0 and 1) linearly ordered mul-archimedean group with zero is
+either isomorphic (and order-isomorphic) to `ℤₘ₀`, or is densely ordered. -/
+lemma LinearOrderedCommGroupWithZero.discrete_or_denselyOrdered (G : Type*)
+    [LinearOrderedCommGroupWithZero G] [Nontrivial Gˣ] [MulArchimedean G] :
+    Nonempty (G ≃*o ℤₘ₀) ∨ DenselyOrdered G := by
+  classical
+  refine (LinearOrderedCommGroup.discrete_or_denselyOrdered Gˣ).imp ?_ ?_
+  · intro ⟨f⟩
+    refine ⟨OrderMonoidIso.trans
+      ⟨WithZero.withZeroUnitsEquiv.symm, ?_⟩ ⟨f.withZero, ?_⟩⟩
+    · intro
+      simp only [WithZero.withZeroUnitsEquiv, MulEquiv.symm_mk,
+        MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, MulEquiv.coe_mk,
+        Equiv.coe_fn_symm_mk ]
+      split_ifs <;>
+      simp_all [← Units.val_le_val]
+    · intro a b
+      induction a <;> induction b <;>
+      simp [MulEquiv.withZero]
+  · intro H
+    refine ⟨fun x y h ↦ ?_⟩
+    rcases (zero_le' (a := x)).eq_or_lt with rfl|hx
+    · lift y to Gˣ using h.ne'.isUnit
+      obtain ⟨z, hz⟩ := exists_ne (1 : Gˣ)
+      refine ⟨(y * |z|ₘ⁻¹ : Gˣ), ?_, ?_⟩
+      · simp [zero_lt_iff]
+      · rw [Units.val_lt_val]
+        simp [hz]
+    · obtain ⟨z, hz, hz'⟩ := H.dense (Units.mk0 x hx.ne') (Units.mk0 y (hx.trans h).ne')
+        (by simp [← Units.val_lt_val, h])
+      refine ⟨z, ?_, ?_⟩ <;>
+      simpa [← Units.val_lt_val]
diff --git a/Mathlib/GroupTheory/CommutingProbability.lean b/Mathlib/GroupTheory/CommutingProbability.lean
index 50ea33f499459..dc3739fd84c36 100644
--- a/Mathlib/GroupTheory/CommutingProbability.lean
+++ b/Mathlib/GroupTheory/CommutingProbability.lean
@@ -68,19 +68,19 @@ theorem commProb_pos [h : Nonempty M] : 0 < commProb M :=
       (pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2)
 
 theorem commProb_le_one : commProb M ≤ 1 := by
-  refine div_le_one_of_le ?_ (sq_nonneg (Nat.card M : ℚ))
+  refine div_le_one_of_le₀ ?_ (sq_nonneg (Nat.card M : ℚ))
   rw [← Nat.cast_pow, Nat.cast_le, sq, ← Nat.card_prod]
   apply Finite.card_subtype_le
 
 variable {M}
 
 theorem commProb_eq_one_iff [h : Nonempty M] :
-    commProb M = 1 ↔ Commutative ((· * ·) : M → M → M) := by
+    commProb M = 1 ↔ Std.Commutative ((· * ·) : M → M → M) := by
   haveI := Fintype.ofFinite M
   rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
   rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
     set_fintype_card_eq_univ_iff, Set.eq_univ_iff_forall]
-  · exact ⟨fun h x y ↦ h (x, y), fun h x ↦ h x.1 x.2⟩
+  · exact ⟨fun h ↦ ⟨fun x y ↦ h (x, y)⟩, fun h x ↦ h.comm x.1 x.2⟩
   · exact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)
 
 variable (G : Type*) [Group G]
@@ -118,8 +118,8 @@ theorem Subgroup.commProb_quotient_le [H.Normal] : commProb (G ⧸ H) ≤ commPr
 variable (G)
 
 theorem inv_card_commutator_le_commProb : (↑(Nat.card (commutator G)))⁻¹ ≤ commProb G :=
-  (inv_pos_le_iff_one_le_mul (Nat.cast_pos.mpr Finite.card_pos)).mpr
-    (le_trans (ge_of_eq (commProb_eq_one_iff.mpr (Abelianization.commGroup G).mul_comm))
+  (inv_le_iff_one_le_mul₀ (Nat.cast_pos.mpr Finite.card_pos)).mpr
+    (le_trans (ge_of_eq (commProb_eq_one_iff.mpr ⟨(Abelianization.commGroup G).mul_comm⟩))
       (commutator G).commProb_quotient_le)
 
 -- Construction of group with commuting probability 1/n
@@ -131,6 +131,7 @@ lemma commProb_odd {n : ℕ} (hn : Odd n) :
   qify [show 2 ∣ n + 3 by rw [Nat.dvd_iff_mod_eq_zero, Nat.add_mod, Nat.odd_iff.mp hn]]
   rw [div_div, ← mul_assoc]
   congr
+  norm_num
 
 private lemma div_two_lt {n : ℕ} (h0 : n ≠ 0) : n / 2 < n :=
   Nat.div_lt_self (Nat.pos_of_ne_zero h0) (lt_add_one 1)
diff --git a/Mathlib/GroupTheory/Congruence/Basic.lean b/Mathlib/GroupTheory/Congruence/Basic.lean
index f9137be6f7410..fe6cfe5261d0a 100644
--- a/Mathlib/GroupTheory/Congruence/Basic.lean
+++ b/Mathlib/GroupTheory/Congruence/Basic.lean
@@ -110,12 +110,11 @@ variable [Mul M] [Mul N] [Mul P] (c : Con M)
 instance : Inhabited (Con M) :=
   ⟨conGen EmptyRelation⟩
 
--- Porting note: upgraded to FunLike
 /-- A coercion from a congruence relation to its underlying binary relation. -/
 @[to_additive "A coercion from an additive congruence relation to its underlying binary relation."]
 instance : FunLike (Con M) M (M → Prop) where
   coe c := c.r
-  coe_injective' := fun x y h => by
+  coe_injective' x y h := by
     rcases x with ⟨⟨x, _⟩, _⟩
     rcases y with ⟨⟨y, _⟩, _⟩
     have : x = y := h
@@ -151,7 +150,7 @@ theorem rel_mk {s : Setoid M} {h a b} : Con.mk s h a b ↔ r a b :=
 @[to_additive "Given a type `M` with an addition, `x, y ∈ M`, and an additive congruence relation
 `c` on `M`, `(x, y) ∈ M × M` iff `x` is related to `y` by `c`."]
 instance : Membership (M × M) (Con M) :=
-  ⟨fun x c => c x.1 x.2⟩
+  ⟨fun c x => c x.1 x.2⟩
 
 variable {c}
 
@@ -377,7 +376,7 @@ theorem coe_sInf (S : Set (Con M)) :
 
 @[to_additive (attr := simp, norm_cast)]
 theorem coe_iInf {ι : Sort*} (f : ι → Con M) : ⇑(iInf f) = ⨅ i, ⇑(f i) := by
-  rw [iInf, coe_sInf, ← Set.range_comp, sInf_range, Function.comp]
+  rw [iInf, coe_sInf, ← Set.range_comp, sInf_range, Function.comp_def]
 
 @[to_additive]
 instance : PartialOrder (Con M) where
diff --git a/Mathlib/GroupTheory/CoprodI.lean b/Mathlib/GroupTheory/CoprodI.lean
index 71732e3b60cb4..67d5fdc6738bd 100644
--- a/Mathlib/GroupTheory/CoprodI.lean
+++ b/Mathlib/GroupTheory/CoprodI.lean
@@ -364,7 +364,7 @@ theorem rcons_inj {i} : Function.Injective (rcons : Pair M i → Word M) := by
     rw [← he] at h'
     exact h' rfl
   · have : m = m' ∧ w.toList = w'.toList := by
-      simpa [cons, rcons, dif_neg hm, dif_neg hm', true_and_iff, eq_self_iff_true, Subtype.mk_eq_mk,
+      simpa [cons, rcons, dif_neg hm, dif_neg hm', eq_self_iff_true, Subtype.mk_eq_mk,
         heq_iff_eq, ← Subtype.ext_iff_val] using he
     rcases this with ⟨rfl, h⟩
     congr
@@ -772,9 +772,10 @@ theorem mulHead_head {i j : ι} (w : NeWord M i j) (x : M i) (hnotone : x * w.he
 theorem mulHead_prod {i j : ι} (w : NeWord M i j) (x : M i) (hnotone : x * w.head ≠ 1) :
     (mulHead w x hnotone).prod = of x * w.prod := by
   unfold mulHead
-  induction' w with _ _ _ _ _ _ _ _ _ _ w_ih_w₁ w_ih_w₂
-  · simp [mulHead, replaceHead]
-  · specialize w_ih_w₁ _ hnotone
+  induction w with
+  | singleton => simp [mulHead, replaceHead]
+  | append _ _ _ w_ih_w₁ w_ih_w₂ =>
+    specialize w_ih_w₁ _ hnotone
     clear w_ih_w₂
     simp? [replaceHead, ← mul_assoc] at * says
       simp only [replaceHead, head, append_prod, ← mul_assoc] at *
diff --git a/Mathlib/GroupTheory/Coset/Basic.lean b/Mathlib/GroupTheory/Coset/Basic.lean
index a827eae726011..44bb4b0f21528 100644
--- a/Mathlib/GroupTheory/Coset/Basic.lean
+++ b/Mathlib/GroupTheory/Coset/Basic.lean
@@ -1,10 +1,9 @@
 /-
 Copyright (c) 2018 Mitchell Rowett. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mitchell Rowett, Scott Morrison
+Authors: Mitchell Rowett, Kim Morrison
 -/
 import Mathlib.Algebra.Quotient
-import Mathlib.Algebra.Group.Subgroup.Actions
 import Mathlib.Algebra.Group.Subgroup.MulOpposite
 import Mathlib.GroupTheory.GroupAction.Basic
 
@@ -403,6 +402,21 @@ theorem out_eq' (a : α ⧸ s) : mk a.out' = a :=
 
 variable (s)
 
+/-- Given a subgroup `s`, the function that sends a subgroup `t` to the pair consisting of
+its intersection with `s` and its image in the quotient `α ⧸ s` is strictly monotone, even though
+it is not injective in general. -/
+@[to_additive QuotientAddGroup.strictMono_comap_prod_image "Given an additive subgroup `s`,
+the function that sends an additive subgroup `t` to the pair consisting of
+its intersection with `s` and its image in the quotient `α ⧸ s`
+is strictly monotone, even though it is not injective in general."]
+theorem strictMono_comap_prod_image :
+    StrictMono fun t : Subgroup α ↦ (t.comap s.subtype, mk (s := s) '' t) := by
+  refine fun t₁ t₂ h ↦ ⟨⟨Subgroup.comap_mono h.1, Set.image_mono h.1⟩,
+    mt (fun ⟨le1, le2⟩ a ha ↦ ?_) h.2⟩
+  obtain ⟨a', h', eq⟩ := le2 ⟨_, ha, rfl⟩
+  convert ← t₁.mul_mem h' (@le1 ⟨_, QuotientGroup.eq.1 eq⟩ <| t₂.mul_mem (t₂.inv_mem <| h.1 h') ha)
+  apply mul_inv_cancel_left
+
 /- It can be useful to write `obtain ⟨h, H⟩ := mk_out'_eq_mul ...`, and then `rw [H]` or
   `simp_rw [H]` or `simp only [H]`. In order for `simp_rw` and `simp only` to work, this lemma is
   stated in terms of an arbitrary `h : s`, rather than the specific `h = g⁻¹ * (mk g).out'`. -/
@@ -438,6 +452,12 @@ theorem preimage_image_mk_eq_iUnion_image (N : Subgroup α) (s : Set α) :
   rw [preimage_image_mk, iUnion_congr_of_surjective (·⁻¹) inv_surjective]
   exact fun x ↦ image_mul_right'
 
+@[to_additive]
+theorem preimage_image_mk_eq_mul (N : Subgroup α) (s : Set α) :
+    mk ⁻¹' ((mk : α → α ⧸ N) '' s) = s * N := by
+  rw [preimage_image_mk_eq_iUnion_image, iUnion_subtype, ← image2_mul, ← iUnion_image_right]
+  simp only [SetLike.mem_coe]
+
 end QuotientGroup
 
 namespace Subgroup
diff --git a/Mathlib/GroupTheory/Coset/Card.lean b/Mathlib/GroupTheory/Coset/Card.lean
index 79fd105a82f88..67a686a06e9b1 100644
--- a/Mathlib/GroupTheory/Coset/Card.lean
+++ b/Mathlib/GroupTheory/Coset/Card.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Mitchell Rowett. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mitchell Rowett, Scott Morrison
+Authors: Mitchell Rowett, Kim Morrison
 -/
 import Mathlib.GroupTheory.Coset.Basic
 import Mathlib.SetTheory.Cardinal.Finite
diff --git a/Mathlib/GroupTheory/CosetCover.lean b/Mathlib/GroupTheory/CosetCover.lean
index c9cefdcb06558..03164c5973c5e 100644
--- a/Mathlib/GroupTheory/CosetCover.lean
+++ b/Mathlib/GroupTheory/CosetCover.lean
@@ -343,7 +343,7 @@ theorem exists_index_le_card_of_leftCoset_cover :
     | inl hindex =>
       rwa [hindex, Nat.cast_zero, inv_zero, inv_pos, Nat.cast_pos]
     | inr hindex =>
-      exact inv_lt_inv_of_lt (by exact_mod_cast hs') (by exact_mod_cast h i hi ⟨hindex⟩)
+      exact inv_strictAnti₀ (by exact_mod_cast hs') (by exact_mod_cast h i hi ⟨hindex⟩)
   apply (Finset.sum_lt_sum_of_nonempty hs hlt).trans_eq
   rw [Finset.sum_const, nsmul_eq_mul, mul_inv_cancel₀ (Nat.cast_ne_zero.mpr hs'.ne')]
 
diff --git a/Mathlib/GroupTheory/Coxeter/Inversion.lean b/Mathlib/GroupTheory/Coxeter/Inversion.lean
index 4901cbdf6ff8b..18946d5bd27dc 100644
--- a/Mathlib/GroupTheory/Coxeter/Inversion.lean
+++ b/Mathlib/GroupTheory/Coxeter/Inversion.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mitchell Lee
 -/
 import Mathlib.GroupTheory.Coxeter.Length
-import Mathlib.Data.ZMod.Parity
 import Mathlib.Data.List.GetD
 
 /-!
@@ -300,13 +299,11 @@ theorem rightInvSeq_drop (ω : List B) (j : ℕ) :
 
 theorem leftInvSeq_take (ω : List B) (j : ℕ) :
     lis (ω.take j) = (lis ω).take j := by
-  obtain le | ge := Nat.le_or_ge j ω.length
-  · simp only [leftInvSeq_eq_reverse_rightInvSeq_reverse]
-    rw [List.take_reverse (by simpa)]
-    nth_rw 1 [← List.reverse_reverse ω]
-    rw [List.take_reverse (by simpa)]
-    simp [rightInvSeq_drop]
-  · rw [take_of_length_le ge, take_of_length_le (by simpa)]
+  simp only [leftInvSeq_eq_reverse_rightInvSeq_reverse]
+  rw [List.take_reverse]
+  nth_rw 1 [← List.reverse_reverse ω]
+  rw [List.take_reverse]
+  simp [rightInvSeq_drop]
 
 theorem isReflection_of_mem_rightInvSeq (ω : List B) {t : W} (ht : t ∈ ris ω) :
     cs.IsReflection t := by
diff --git a/Mathlib/GroupTheory/DoubleCoset.lean b/Mathlib/GroupTheory/DoubleCoset.lean
index 1d710162e3e55..771be71a00bb9 100644
--- a/Mathlib/GroupTheory/DoubleCoset.lean
+++ b/Mathlib/GroupTheory/DoubleCoset.lean
@@ -74,8 +74,8 @@ def Quotient (H K : Set G) : Type _ :=
 theorem rel_iff {H K : Subgroup G} {x y : G} :
     (setoid ↑H ↑K).Rel x y ↔ ∃ a ∈ H, ∃ b ∈ K, y = a * x * b :=
   Iff.trans
-    ⟨fun hxy => (congr_arg _ hxy).mpr (mem_doset_self H K y), fun hxy => (doset_eq_of_mem hxy).symm⟩
-    mem_doset
+    ⟨fun (hxy : doset x H K = doset y H K) => hxy ▸ mem_doset_self H K y,
+      fun hxy => (doset_eq_of_mem hxy).symm⟩ mem_doset
 
 theorem bot_rel_eq_leftRel (H : Subgroup G) :
     (setoid ↑(⊥ : Subgroup G) ↑H).Rel = (QuotientGroup.leftRel H).Rel := by
@@ -131,7 +131,7 @@ theorem mk_eq_of_doset_eq {H K : Subgroup G} {a b : G} (h : doset a H K = doset
   rw [eq]
   exact mem_doset.mp (h.symm ▸ mem_doset_self H K b)
 
-theorem disjoint_out' {H K : Subgroup G} {a b : Quotient H.1 K} :
+theorem disjoint_out' {H K : Subgroup G} {a b : Quotient H K} :
     a ≠ b → Disjoint (doset a.out' H K) (doset b.out' (H : Set G) K) := by
   contrapose!
   intro h
@@ -140,7 +140,7 @@ theorem disjoint_out' {H K : Subgroup G} {a b : Quotient H.1 K} :
 theorem union_quotToDoset (H K : Subgroup G) : ⋃ q, quotToDoset H K q = Set.univ := by
   ext x
   simp only [Set.mem_iUnion, quotToDoset, mem_doset, SetLike.mem_coe, exists_prop, Set.mem_univ,
-    iff_true_iff]
+    iff_true]
   use mk H K x
   obtain ⟨h, k, h3, h4, h5⟩ := mk_out'_eq_mul H K x
   refine ⟨h⁻¹, H.inv_mem h3, k⁻¹, K.inv_mem h4, ?_⟩
@@ -172,19 +172,17 @@ theorem doset_union_leftCoset (H K : Subgroup G) (a : G) :
     simp only [hxy, ← mul_assoc, hy, one_mul, inv_mul_cancel, Subgroup.coe_mk, inv_mul_cancel_right]
 
 theorem left_bot_eq_left_quot (H : Subgroup G) :
-    Quotient (⊥ : Subgroup G).1 (H : Set G) = (G ⧸ H) := by
+    Quotient (⊥ : Subgroup G) (H : Set G) = (G ⧸ H) := by
   unfold Quotient
   congr
   ext
   simp_rw [← bot_rel_eq_leftRel H]
-  rfl
 
 theorem right_bot_eq_right_quot (H : Subgroup G) :
-    Quotient (H.1 : Set G) (⊥ : Subgroup G) = _root_.Quotient (QuotientGroup.rightRel H) := by
+    Quotient (H : Set G) (⊥ : Subgroup G) = _root_.Quotient (QuotientGroup.rightRel H) := by
   unfold Quotient
   congr
   ext
   simp_rw [← rel_bot_eq_right_group_rel H]
-  rfl
 
 end Doset
diff --git a/Mathlib/GroupTheory/Exponent.lean b/Mathlib/GroupTheory/Exponent.lean
index 94ce55d4cf069..f408bce7cb363 100644
--- a/Mathlib/GroupTheory/Exponent.lean
+++ b/Mathlib/GroupTheory/Exponent.lean
@@ -452,7 +452,7 @@ theorem exists_orderOf_eq_exponent (hG : ExponentExists G) : ∃ g : G, orderOf
   rw [(Commute.all _ g).orderOf_mul_eq_mul_orderOf_of_coprime hcoprime, hpk',
     hg, ha, hk, pow_add, pow_add, pow_one, ← mul_assoc, ← mul_assoc,
     Nat.div_mul_cancel, mul_assoc, lt_mul_iff_one_lt_right <| hG.orderOf_pos t, ← pow_succ]
-  · exact one_lt_pow hp.one_lt a.succ_ne_zero
+  · exact one_lt_pow₀ hp.one_lt a.succ_ne_zero
   · exact hpk
 
 @[to_additive]
diff --git a/Mathlib/GroupTheory/FiniteAbelian.lean b/Mathlib/GroupTheory/FiniteAbelian.lean
index 0e74393883b9e..8ac8ceb36aec7 100644
--- a/Mathlib/GroupTheory/FiniteAbelian.lean
+++ b/Mathlib/GroupTheory/FiniteAbelian.lean
@@ -148,7 +148,7 @@ lemma equiv_directSum_zmod_of_finite' (G : Type*) [AddCommGroup G] [Finite G] :
   refine ⟨{i : ι // n i ≠ 0}, inferInstance, fun i ↦ p i ^ n i, ?_,
     ⟨e.trans (directSumNeZeroMulEquiv ι _ _).symm⟩⟩
   rintro ⟨i, hi⟩
-  exact one_lt_pow (hp _).one_lt hi
+  exact one_lt_pow₀ (hp _).one_lt hi
 
 theorem finite_of_fg_torsion [hG' : AddGroup.FG G] (hG : AddMonoid.IsTorsion G) : Finite G :=
   @Module.finite_of_fg_torsion _ _ _ (Module.Finite.iff_addGroup_fg.mpr hG') <|
diff --git a/Mathlib/GroupTheory/FixedPointFree.lean b/Mathlib/GroupTheory/FixedPointFree.lean
index 2f9fb7c2c40f9..a91326561faf5 100644
--- a/Mathlib/GroupTheory/FixedPointFree.lean
+++ b/Mathlib/GroupTheory/FixedPointFree.lean
@@ -54,8 +54,8 @@ theorem prod_pow_eq_one (hφ : FixedPointFree φ) {n : ℕ} (hn : φ^[n] = _root
 
 theorem coe_eq_inv_of_sq_eq_one (hφ : FixedPointFree φ) (h2 : φ^[2] = _root_.id) : ⇑φ = (·⁻¹) := by
   ext g
-  have key : 1 * g * φ g = 1 := hφ.prod_pow_eq_one h2 g
-  rwa [one_mul, ← inv_eq_iff_mul_eq_one, eq_comm] at key
+  have key : g * φ g = 1 := by simpa [List.range_succ] using hφ.prod_pow_eq_one h2 g
+  rwa [← inv_eq_iff_mul_eq_one, eq_comm] at key
 
 section Involutive
 
diff --git a/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean b/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
index 8cc24301aaf7c..477433e03185a 100644
--- a/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
+++ b/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
@@ -153,7 +153,7 @@ theorem support_zsmul (k : ℤ) (h : k ≠ 0) (a : FreeAbelianGroup X) :
     support (k • a) = support a := by
   ext x
   simp only [mem_support_iff, AddMonoidHom.map_zsmul]
-  simp only [h, zsmul_int_int, false_or_iff, Ne, mul_eq_zero]
+  simp only [h, zsmul_int_int, false_or, Ne, mul_eq_zero]
 
 @[simp]
 theorem support_nsmul (k : ℕ) (h : k ≠ 0) (a : FreeAbelianGroup X) :
diff --git a/Mathlib/GroupTheory/FreeGroup/Basic.lean b/Mathlib/GroupTheory/FreeGroup/Basic.lean
index 74da573c0472e..4dfea2fbf13f6 100644
--- a/Mathlib/GroupTheory/FreeGroup/Basic.lean
+++ b/Mathlib/GroupTheory/FreeGroup/Basic.lean
@@ -128,7 +128,7 @@ theorem not_step_nil : ¬Step [] L := by
   generalize h' : [] = L'
   intro h
   cases' h with L₁ L₂
-  simp [List.nil_eq_append] at h'
+  simp [List.nil_eq_append_iff] at h'
 
 @[to_additive]
 theorem Step.cons_left_iff {a : α} {b : Bool} :
@@ -287,7 +287,8 @@ theorem red_iff_irreducible {x1 b1 x2 b2} (h : (x1, b1) ≠ (x2, b2)) :
   generalize eq : [(x1, not b1), (x2, b2)] = L'
   intro L h'
   cases h'
-  simp [List.cons_eq_append, List.nil_eq_append] at eq
+  simp only [List.cons_eq_append_iff, List.cons.injEq, Prod.mk.injEq, and_false,
+    List.nil_eq_append_iff, exists_const, or_self, or_false, List.cons_ne_nil] at eq
   rcases eq with ⟨rfl, ⟨rfl, rfl⟩, ⟨rfl, rfl⟩, rfl⟩
   simp at h
 
@@ -312,8 +313,8 @@ theorem inv_of_red_of_ne {x1 b1 x2 b2} (H1 : (x1, b1) ≠ (x2, b2))
 open List -- for <+ notation
 
 @[to_additive]
-theorem Step.sublist (H : Red.Step L₁ L₂) : Sublist L₂ L₁ := by
-  cases H; simp; constructor; constructor; rfl
+theorem Step.sublist (H : Red.Step L₁ L₂) : L₂ <+ L₁ := by
+  cases H; simp
 
 /-- If `w₁ w₂` are words such that `w₁` reduces to `w₂`, then `w₂` is a sublist of `w₁`. -/
 @[to_additive "If `w₁ w₂` are words such that `w₁` reduces to `w₂`, then `w₂` is a sublist of
@@ -383,8 +384,8 @@ theorem eqvGen_step_iff_join_red : EqvGen Red.Step L₁ L₂ ↔ Join Red L₁ L
     (fun h =>
       have : EqvGen (Join Red) L₁ L₂ := h.mono fun _ _ => join_red_of_step
       equivalence_join_red.eqvGen_iff.1 this)
-    (join_of_equivalence (EqvGen.is_equivalence _) fun _ _ =>
-      reflTransGen_of_equivalence (EqvGen.is_equivalence _) EqvGen.rel)
+    (join_of_equivalence (Relation.EqvGen.is_equivalence _) fun _ _ =>
+      reflTransGen_of_equivalence (Relation.EqvGen.is_equivalence _) EqvGen.rel)
 
 end FreeGroup
 
@@ -462,7 +463,7 @@ theorem invRev_length : (invRev L₁).length = L₁.length := by simp [invRev]
 
 @[to_additive (attr := simp)]
 theorem invRev_invRev : invRev (invRev L₁) = L₁ := by
-  simp [invRev, List.map_reverse, (· ∘ ·)]
+  simp [invRev, List.map_reverse, Function.comp_def]
 
 @[to_additive (attr := simp)]
 theorem invRev_empty : invRev ([] : List (α × Bool)) = [] :=
@@ -538,7 +539,7 @@ def of (x : α) : FreeGroup α :=
 @[to_additive]
 theorem Red.exact : mk L₁ = mk L₂ ↔ Join Red L₁ L₂ :=
   calc
-    mk L₁ = mk L₂ ↔ EqvGen Red.Step L₁ L₂ := Iff.intro (Quot.exact _) Quot.EqvGen_sound
+    mk L₁ = mk L₂ ↔ EqvGen Red.Step L₁ L₂ := Iff.intro Quot.eqvGen_exact Quot.eqvGen_sound
     _ ↔ Join Red L₁ L₂ := eqvGen_step_iff_join_red
 
 /-- The canonical map from the type to the free group is an injection. -/
@@ -571,7 +572,7 @@ def lift : (α → β) ≃ (FreeGroup α →* β) where
     MonoidHom.mk' (Quot.lift (Lift.aux f) fun L₁ L₂ => Red.Step.lift) <| by
       rintro ⟨L₁⟩ ⟨L₂⟩; simp [Lift.aux]
   invFun g := g ∘ of
-  left_inv f := one_mul _
+  left_inv f := List.prod_singleton
   right_inv g :=
     MonoidHom.ext <| by
       rintro ⟨L⟩
@@ -592,7 +593,7 @@ theorem lift.mk : lift f (mk L) = List.prod (L.map fun x => cond x.2 (f x.1) (f
 
 @[to_additive (attr := simp)]
 theorem lift.of {x} : lift f (of x) = f x :=
-  one_mul _
+  List.prod_singleton
 
 @[to_additive]
 theorem lift.unique (g : FreeGroup α →* β) (hg : ∀ x, g (FreeGroup.of x) = f x) {x} :
@@ -669,7 +670,7 @@ theorem map.id' (x : FreeGroup α) : map (fun z => z) x = x :=
 @[to_additive]
 theorem map.comp {γ : Type w} (f : α → β) (g : β → γ) (x) :
     map g (map f x) = map (g ∘ f) x := by
-  rcases x with ⟨L⟩; simp [(· ∘ ·)]
+  rcases x with ⟨L⟩; simp [Function.comp_def]
 
 @[to_additive (attr := simp)]
 theorem map.of {x} : map f (of x) = of (f x) :=
diff --git a/Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean b/Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean
index ff82a3c54c8d9..5999f4a20b721 100644
--- a/Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean
+++ b/Mathlib/GroupTheory/FreeGroup/IsFreeGroup.lean
@@ -149,7 +149,7 @@ def ofLift {G : Type u} [Group G] (X : Type u) (of : X → G)
 
 /-- If a group satisfies the universal property of a free group with respect to a given type, then
 it admits a free group basis based on this type. Here
-the universal property is expressed as in `IsFreeGroup.unique_lift`.  -/
+the universal property is expressed as in `IsFreeGroup.unique_lift`. -/
 def ofUniqueLift {G : Type u} [Group G] (X : Type u) (of : X → G)
     (h : ∀ {H : Type u} [Group H] (f : X → H), ∃! F : G →* H, ∀ a, F (of a) = f a) :
     FreeGroupBasis X G :=
@@ -212,7 +212,7 @@ theorem ext_hom ⦃f g : G →* H⦄ (h : ∀ a : Generators G, f (of a) = g (of
 group extends in a unique way to a homomorphism from `G`.
 
 Note that since `IsFreeGroup.lift` is expressed as a bijection, it already
-expresses the universal property.  -/
+expresses the universal property. -/
 theorem unique_lift (f : Generators G → H) : ∃! F : G →* H, ∀ a, F (of a) = f a := by
   simpa only [Function.funext_iff] using lift.symm.bijective.existsUnique f
 
@@ -226,7 +226,7 @@ lemma ofLift {G : Type u} [Group G] (X : Type u) (of : X → G)
   (FreeGroupBasis.ofLift X of lift lift_of).isFreeGroup
 
 /-- If a group satisfies the universal property of a free group with respect to a given type, then
-it is free. Here the universal property is expressed as in `IsFreeGroup.unique_lift`.  -/
+it is free. Here the universal property is expressed as in `IsFreeGroup.unique_lift`. -/
 lemma ofUniqueLift {G : Type u} [Group G] (X : Type u) (of : X → G)
     (h : ∀ {H : Type u} [Group H] (f : X → H), ∃! F : G →* H, ∀ a, F (of a) = f a) :
     IsFreeGroup G :=
diff --git a/Mathlib/GroupTheory/GroupAction/Basic.lean b/Mathlib/GroupTheory/GroupAction/Basic.lean
index d7f65f24f03ac..cc847556ec7f8 100644
--- a/Mathlib/GroupTheory/GroupAction/Basic.lean
+++ b/Mathlib/GroupTheory/GroupAction/Basic.lean
@@ -327,6 +327,10 @@ theorem mem_orbit_smul (g : G) (a : α) : a ∈ orbit G (g • a) := by
 theorem smul_mem_orbit_smul (g h : G) (a : α) : g • a ∈ orbit G (h • a) := by
   simp only [orbit_smul, mem_orbit]
 
+@[to_additive]
+instance instMulAction (H : Subgroup G) : MulAction H α :=
+  inferInstanceAs (MulAction H.toSubmonoid α)
+
 @[to_additive]
 lemma orbit_subgroup_subset (H : Subgroup G) (a : α) : orbit H a ⊆ orbit G a :=
   orbit_submonoid_subset H.toSubmonoid a
@@ -345,11 +349,10 @@ lemma mem_subgroup_orbit_iff {H : Subgroup G} {x : α} {a b : orbit G x} :
     a ∈ MulAction.orbit H b ↔ (a : α) ∈ MulAction.orbit H (b : α) := by
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · rcases h with ⟨g, rfl⟩
-    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, orbit.coe_smul, ← Submonoid.smul_def]
     exact MulAction.mem_orbit _ g
   · rcases h with ⟨g, h⟩
-    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, ← orbit.coe_smul,
-             ← Submonoid.smul_def, ← Subtype.ext_iff] at h
+    dsimp at h
+    erw [← orbit.coe_smul, ← Subtype.ext_iff] at h
     subst h
     exact MulAction.mem_orbit _ g
 
@@ -563,12 +566,11 @@ lemma orbitRel.Quotient.mem_subgroup_orbit_iff {H : Subgroup G} {x : orbitRel.Qu
     {a b : x.orbit} : (a : α) ∈ MulAction.orbit H (b : α) ↔ a ∈ MulAction.orbit H b := by
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · rcases h with ⟨g, h⟩
-    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, ← orbit.coe_smul,
-      ← Submonoid.smul_def, ← Subtype.ext_iff] at h
+    dsimp at h
+    erw [← orbit.coe_smul, ← Subtype.ext_iff] at h
     subst h
     exact MulAction.mem_orbit _ g
   · rcases h with ⟨g, rfl⟩
-    simp_rw [Submonoid.smul_def, Subgroup.coe_toSubmonoid, orbit.coe_smul, ← Submonoid.smul_def]
     exact MulAction.mem_orbit _ g
 
 @[to_additive]
@@ -770,6 +772,7 @@ theorem le_stabilizer_iff_smul_le (s : Set α) (H : Subgroup G) :
 theorem mem_stabilizer_of_finite_iff_smul_le (s : Set α) (hs : s.Finite) (g : G) :
     g ∈ stabilizer G s ↔ g • s ⊆ s := by
   haveI : Fintype s := Set.Finite.fintype hs
+  haveI : Finite (g • s : Set α) := Finite.Set.finite_image ..
   haveI : Fintype (g • s : Set α) := Fintype.ofFinite _
   rw [mem_stabilizer_iff]
   constructor
diff --git a/Mathlib/GroupTheory/GroupAction/BigOperators.lean b/Mathlib/GroupTheory/GroupAction/BigOperators.lean
deleted file mode 100644
index 7ad7cb7f697d9..0000000000000
--- a/Mathlib/GroupTheory/GroupAction/BigOperators.lean
+++ /dev/null
@@ -1,69 +0,0 @@
-/-
-Copyright (c) 2020 Yury Kudryashov. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov
--/
-import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Algebra.GroupWithZero.Action.Defs
-import Mathlib.Data.Finset.Basic
-import Mathlib.Data.Multiset.Basic
-import Mathlib.Algebra.BigOperators.Finprod
-
-/-!
-# Lemmas about group actions on big operators
-
-Note that analogous lemmas for `Module`s like `Finset.sum_smul` appear in other files.
--/
-
-
-variable {α β γ : Type*}
-
-section
-
-variable [AddMonoid β] [DistribSMul α β]
-
-theorem List.smul_sum {r : α} {l : List β} : r • l.sum = (l.map (r • ·)).sum :=
-  map_list_sum (DistribSMul.toAddMonoidHom β r) l
-
-end
-
-section
-
-variable [Monoid α] [Monoid β] [MulDistribMulAction α β]
-
-theorem List.smul_prod {r : α} {l : List β} : r • l.prod = (l.map (r • ·)).prod :=
-  map_list_prod (MulDistribMulAction.toMonoidHom β r) l
-
-end
-
-section
-
-variable [AddCommMonoid β] [DistribSMul α β]
-
-theorem Multiset.smul_sum {r : α} {s : Multiset β} : r • s.sum = (s.map (r • ·)).sum :=
-  (DistribSMul.toAddMonoidHom β r).map_multiset_sum s
-
-theorem Finset.smul_sum {r : α} {f : γ → β} {s : Finset γ} :
-    (r • ∑ x ∈ s, f x) = ∑ x ∈ s, r • f x :=
-  map_sum (DistribSMul.toAddMonoidHom β r) f s
-
-end
-
-section
-
-variable [Monoid α] [CommMonoid β] [MulDistribMulAction α β]
-
-theorem Multiset.smul_prod {r : α} {s : Multiset β} : r • s.prod = (s.map (r • ·)).prod :=
-  (MulDistribMulAction.toMonoidHom β r).map_multiset_prod s
-
-theorem Finset.smul_prod {r : α} {f : γ → β} {s : Finset γ} :
-    (r • ∏ x ∈ s, f x) = ∏ x ∈ s, r • f x :=
-  map_prod (MulDistribMulAction.toMonoidHom β r) f s
-
-theorem smul_finprod {ι : Sort*} [Finite ι] {f : ι → β} (r : α) :
-    r • ∏ᶠ x : ι, f x = ∏ᶠ x : ι, r • (f x) := by
-  cases nonempty_fintype (PLift ι)
-  simp only [finprod_eq_prod_plift_of_mulSupport_subset (s := Finset.univ) (by simp),
-    finprod_eq_prod_of_fintype, Finset.smul_prod]
-
-end
diff --git a/Mathlib/GroupTheory/GroupAction/Blocks.lean b/Mathlib/GroupTheory/GroupAction/Blocks.lean
index c82066c332d33..9fbae888310e9 100644
--- a/Mathlib/GroupTheory/GroupAction/Blocks.lean
+++ b/Mathlib/GroupTheory/GroupAction/Blocks.lean
@@ -8,6 +8,7 @@ import Mathlib.Data.Setoid.Partition
 import Mathlib.GroupTheory.GroupAction.Basic
 import Mathlib.GroupTheory.GroupAction.Pointwise
 import Mathlib.GroupTheory.GroupAction.SubMulAction
+import Mathlib.Algebra.Group.Subgroup.Actions
 
 /-! # Blocks
 
@@ -135,7 +136,8 @@ theorem IsBlock.def_one {B : Set X} :
     rw [Set.disjoint_iff] at h ⊢
     rintro x hx
     suffices g'⁻¹ • x ∈ (g'⁻¹ * g) • B ∩ B by apply h this
-    simp only [Set.mem_inter_iff, ← Set.mem_smul_set_iff_inv_smul_mem, ← smul_smul, smul_inv_smul]
+    rw [Set.mem_inter_iff]
+    simp only [← smul_smul, ← Set.mem_smul_set_iff_inv_smul_mem, smul_inv_smul]
     exact hx
 
 theorem IsBlock.mk_notempty_one {B : Set X} :
@@ -356,11 +358,11 @@ lemma smul_orbit_eq_orbit_smul (N : Subgroup G) [nN : N.Normal] (a : X) (g : G)
   constructor
   · rintro ⟨⟨k, hk⟩, rfl⟩
     use ⟨g * k * g⁻¹, nN.conj_mem k hk g⟩
-    simp only [Submonoid.mk_smul]
+    simp only [Subgroup.mk_smul]
     rw [smul_smul, inv_mul_cancel_right, ← smul_smul]
   · rintro ⟨⟨k, hk⟩, rfl⟩
     use ⟨g⁻¹ * k * g, nN.conj_mem' k hk g⟩
-    simp only [Submonoid.mk_smul]
+    simp only [Subgroup.mk_smul]
     simp only [← mul_assoc, ← smul_smul, smul_inv_smul, inv_inv]
 
 /-- An orbit of a normal subgroup is a block -/
@@ -396,7 +398,7 @@ theorem IsBlock.of_orbit {H : Subgroup G} {a : X} (hH : stabilizer G a ≤ H) :
   simp_rw [IsBlock.def_one, or_iff_not_imp_right, Set.not_disjoint_iff]
   rintro g ⟨-, ⟨-, ⟨h₁, rfl⟩, h⟩, ⟨h₂, rfl⟩⟩
   suffices g ∈ H by
-    rw [← Subgroup.coe_mk H g this, ← H.smul_def, smul_orbit (⟨g, this⟩ : H) a]
+    rw [← Subgroup.coe_mk H g this, ← H.toSubmonoid.smul_def, smul_orbit (⟨g, this⟩ : H) a]
   rw [← mul_mem_cancel_left h₂⁻¹.2, ← mul_mem_cancel_right h₁.2]
   apply hH
   simp only [mem_stabilizer_iff, InvMemClass.coe_inv, mul_smul, inv_smul_eq_iff]
@@ -419,7 +421,7 @@ theorem IsBlock.orbit_stabilizer_eq
   ext x
   constructor
   · rintro ⟨⟨k, k_mem⟩, rfl⟩
-    simp only [Submonoid.mk_smul]
+    simp only [Subgroup.mk_smul]
     rw [← k_mem, Set.smul_mem_smul_set_iff]
     exact ha
   · intro hx
@@ -434,11 +436,11 @@ theorem stabilizer_orbit_eq {a : X} {H : Subgroup G} (hH : stabilizer G a ≤ H)
   constructor
   · intro hg
     obtain ⟨-, ⟨b, rfl⟩, h⟩ := hg.symm ▸ mem_orbit_self a
-    simp_rw [H.smul_def, ← mul_smul, ← mem_stabilizer_iff] at h
+    simp_rw [H.toSubmonoid.smul_def, ← mul_smul, ← mem_stabilizer_iff] at h
     exact (mul_mem_cancel_right b.2).mp (hH h)
   · intro hg
-    rw [mem_stabilizer_iff, ← Subgroup.coe_mk H g hg, ← Submonoid.smul_def]
-    apply smul_orbit
+    rw [mem_stabilizer_iff, ← Subgroup.coe_mk H g hg, ← Submonoid.smul_def (S := H.toSubmonoid)]
+    apply smul_orbit (G := H)
 
 variable (G)
 
diff --git a/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean b/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
index 4fd82a9683bf7..afcac247ac820 100644
--- a/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
+++ b/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
@@ -3,7 +3,6 @@ Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir
 -/
-import Mathlib.Algebra.Group.Subgroup.Actions
 import Mathlib.GroupTheory.GroupAction.Basic
 import Mathlib.GroupTheory.GroupAction.FixedPoints
 
diff --git a/Mathlib/GroupTheory/GroupAction/Hom.lean b/Mathlib/GroupTheory/GroupAction/Hom.lean
index d843581eba0b4..2a9db62dd963a 100644
--- a/Mathlib/GroupTheory/GroupAction/Hom.lean
+++ b/Mathlib/GroupTheory/GroupAction/Hom.lean
@@ -59,7 +59,7 @@ variable (Z : Type*) [SMul P Z]
 
 /-- Equivariant functions :
 When `φ : M → N` is a function, and types `X` and `Y` are endowed with actions of `M` and `N`,
-a function `f : X → Y` is `φ`-equivariant if `f (m • x) = (φ m) • (f x)`.  -/
+a function `f : X → Y` is `φ`-equivariant if `f (m • x) = (φ m) • (f x)`. -/
 -- Porting note(#5171): this linter isn't ported yet.
 -- @[nolint has_nonempty_instance]
 structure MulActionHom where
diff --git a/Mathlib/GroupTheory/GroupAction/Pointwise.lean b/Mathlib/GroupTheory/GroupAction/Pointwise.lean
index 091782116de91..4a0bf768bba53 100644
--- a/Mathlib/GroupTheory/GroupAction/Pointwise.lean
+++ b/Mathlib/GroupTheory/GroupAction/Pointwise.lean
@@ -4,8 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
   Frédéric Dupuis, Heather Macbeth, Antoine Chambert-Loir
 -/
-
-import Mathlib.Data.Set.Pointwise.SMul
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.GroupTheory.GroupAction.Hom
 
 /-!
diff --git a/Mathlib/GroupTheory/GroupAction/Quotient.lean b/Mathlib/GroupTheory/GroupAction/Quotient.lean
index 8a484cbcfe58d..3cd5bab40dafd 100644
--- a/Mathlib/GroupTheory/GroupAction/Quotient.lean
+++ b/Mathlib/GroupTheory/GroupAction/Quotient.lean
@@ -9,6 +9,7 @@ import Mathlib.GroupTheory.GroupAction.ConjAct
 import Mathlib.GroupTheory.GroupAction.Hom
 import Mathlib.GroupTheory.Coset.Basic
 import Mathlib.GroupTheory.Commutator.Basic
+import Mathlib.Algebra.Group.Subgroup.Actions
 
 /-!
 # Properties of group actions involving quotient groups
@@ -362,6 +363,48 @@ instance finite_quotient_of_finite_quotient_of_finite_quotient {H : Subgroup α}
   rw [(equivSubgroupOrbits β H).finite_iff]
   infer_instance
 
+/-- Given a group acting freely and transitively, an equivalence between the orbits under the
+action of a subgroup and the quotient group. -/
+@[to_additive "Given an additive group acting freely and transitively, an equivalence between the
+orbits under the action of an additive subgroup and the quotient group."]
+noncomputable def equivSubgroupOrbitsQuotientGroup [IsPretransitive α β]
+    (free : ∀ y : β, MulAction.stabilizer α y = ⊥) (H : Subgroup α) :
+    orbitRel.Quotient H β ≃ α ⧸ H where
+  toFun := fun q ↦ q.liftOn' (fun y ↦ (exists_smul_eq α y x).choose) (by
+    intro y₁ y₂ h
+    rw [orbitRel_r_apply] at h
+    rw [Quotient.eq'', leftRel_eq]
+    dsimp only
+    rcases h with ⟨g, rfl⟩
+    dsimp only
+    suffices (exists_smul_eq α (g • y₂) x).choose = (exists_smul_eq α y₂ x).choose * g⁻¹ by
+      simp [this]
+    rw [← inv_mul_eq_one, ← Subgroup.mem_bot, ← free ((g : α) • y₂)]
+    simp only [mem_stabilizer_iff, smul_smul, mul_assoc, InvMemClass.coe_inv, inv_mul_cancel,
+               mul_one]
+    rw [← smul_smul, (exists_smul_eq α y₂ x).choose_spec, inv_smul_eq_iff,
+        (exists_smul_eq α ((g : α) • y₂) x).choose_spec])
+  invFun := fun q ↦ q.liftOn' (fun g ↦ ⟦g⁻¹ • x⟧) (by
+    intro g₁ g₂ h
+    rw [leftRel_eq] at h
+    simp only
+    rw [← @Quotient.mk''_eq_mk, Quotient.eq'', orbitRel_r_apply]
+    exact ⟨⟨_, h⟩, by simp [mul_smul]⟩)
+  left_inv := fun y ↦ by
+    induction' y using Quotient.inductionOn' with y
+    simp only [Quotient.liftOn'_mk'']
+    rw [← @Quotient.mk''_eq_mk, Quotient.eq'', orbitRel_r_apply]
+    convert mem_orbit_self _
+    rw [inv_smul_eq_iff, (exists_smul_eq α y x).choose_spec]
+  right_inv := fun g ↦ by
+    induction' g using Quotient.inductionOn' with g
+    simp only [Quotient.liftOn'_mk'', Quotient.liftOn'_mk, QuotientGroup.mk]
+    rw [Quotient.eq'', leftRel_eq]
+    simp only
+    convert one_mem H
+    · rw [inv_mul_eq_one, eq_comm, ← inv_mul_eq_one, ← Subgroup.mem_bot, ← free (g⁻¹ • x),
+        mem_stabilizer_iff, mul_smul, (exists_smul_eq α (g⁻¹ • x) x).choose_spec]
+
 end MulAction
 
 theorem ConjClasses.card_carrier {G : Type*} [Group G] [Fintype G] (g : G)
diff --git a/Mathlib/GroupTheory/GroupAction/Ring.lean b/Mathlib/GroupTheory/GroupAction/Ring.lean
index 1d669771f20c3..8b637c59b69f7 100644
--- a/Mathlib/GroupTheory/GroupAction/Ring.lean
+++ b/Mathlib/GroupTheory/GroupAction/Ring.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Algebra.Group.Action.Defs
 import Mathlib.Algebra.Ring.Defs
+import Mathlib.Algebra.SMulWithZero
 
 /-!
 # Commutativity and associativity of action of integers on rings
@@ -21,6 +21,9 @@ open scoped Int
 
 variable {α : Type*}
 
+instance NonUnitalNonAssocSemiring.toDistribSMul [NonUnitalNonAssocSemiring α] :
+    DistribSMul α α where smul_add := mul_add
+
 /-- Note that `AddMonoid.nat_smulCommClass` requires stronger assumptions on `α`. -/
 instance NonUnitalNonAssocSemiring.nat_smulCommClass [NonUnitalNonAssocSemiring α] :
     SMulCommClass ℕ α α where
diff --git a/Mathlib/GroupTheory/GroupAction/SubMulAction.lean b/Mathlib/GroupTheory/GroupAction/SubMulAction.lean
index 0d6267151258e..6f4c858afeb02 100644
--- a/Mathlib/GroupTheory/GroupAction/SubMulAction.lean
+++ b/Mathlib/GroupTheory/GroupAction/SubMulAction.lean
@@ -153,7 +153,7 @@ end OfTower
 
 end SetLike
 
-/-- A SubMulAction is a set which is closed under scalar multiplication.  -/
+/-- A SubMulAction is a set which is closed under scalar multiplication. -/
 structure SubMulAction (R : Type u) (M : Type v) [SMul R M] : Type v where
   /-- The underlying set of a `SubMulAction`. -/
   carrier : Set M
@@ -243,7 +243,7 @@ variable [Monoid R] [MulAction R M] {A : Type*} [SetLike A M]
 variable [hA : SMulMemClass A R M] (S' : A)
 
 -- Prefer subclasses of `MulAction` over `SMulMemClass`.
-/-- A `SubMulAction` of a `MulAction` is a `MulAction`.  -/
+/-- A `SubMulAction` of a `MulAction` is a `MulAction`. -/
 instance (priority := 75) toMulAction : MulAction R S' :=
   Subtype.coe_injective.mulAction Subtype.val (SetLike.val_smul S')
 
diff --git a/Mathlib/GroupTheory/GroupAction/Support.lean b/Mathlib/GroupTheory/GroupAction/Support.lean
index d0f3000e1c542..392881eb03ad9 100644
--- a/Mathlib/GroupTheory/GroupAction/Support.lean
+++ b/Mathlib/GroupTheory/GroupAction/Support.lean
@@ -3,7 +3,8 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Data.Set.Pointwise.SMul
+import Mathlib.Algebra.Group.Action.Basic
+import Mathlib.Algebra.Group.Pointwise.Set.Basic
 
 /-!
 # Support of an element under an action action
diff --git a/Mathlib/GroupTheory/HNNExtension.lean b/Mathlib/GroupTheory/HNNExtension.lean
index 477f83de90efd..2ba02b333a426 100644
--- a/Mathlib/GroupTheory/HNNExtension.lean
+++ b/Mathlib/GroupTheory/HNNExtension.lean
@@ -42,7 +42,7 @@ def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
 /-- The HNN Extension of a group `G`, `HNNExtension G A B φ`. Given a group `G`, subgroups `A` and
 `B` and an isomorphism `φ` of `A` and `B`, we adjoin a letter `t` to `G`, such that for
 any `a ∈ A`, the conjugate of `of a` by `t` is `of (φ a)`, where `of` is the canonical
-map from `G` into the `HNNExtension`.  -/
+map from `G` into the `HNNExtension`. -/
 def HNNExtension (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) : Type _ :=
   (HNNExtension.con G A B φ).Quotient
 
@@ -174,7 +174,7 @@ namespace NormalWord
 variable (G A B)
 /-- To put word in the HNN Extension into a normal form, we must choose an element of each right
 coset of both `A` and `B`, such that the chosen element of the subgroup itself is `1`. -/
-structure TransversalPair : Type _ :=
+structure TransversalPair : Type _ where
   /-- The transversal of each subgroup -/
   set : ℤˣ → Set G
   /-- We have exactly one element of each coset of the subgroup -/
@@ -187,7 +187,7 @@ instance TransversalPair.nonempty : Nonempty (TransversalPair G A B) := by
 /-- A reduced word is a `head`, which is an element of `G`, followed by the product list of pairs.
 There should also be no sequences of the form `t^u * g * t^-u`, where `g` is in
 `toSubgroup A B u` This is a less strict condition than required for `NormalWord`. -/
-structure ReducedWord : Type _ :=
+structure ReducedWord : Type _ where
   /-- Every `ReducedWord` is the product of an element of the group and a word made up
   of letters each of which is in the transversal. `head` is that element of the base group. -/
   head : G
@@ -215,7 +215,7 @@ The normal form is a `head`, which is an element of `G`, followed by the product
 `toSubgroup A B u`. There should also be no sequences of the form `t^u * g * t^-u`
 where `g ∈ toSubgroup A B u` -/
 structure _root_.HNNExtension.NormalWord (d : TransversalPair G A B)
-    extends ReducedWord G A B : Type _ :=
+    extends ReducedWord G A B : Type _ where
   /-- Every element `g : G` in the list is the chosen element of its coset -/
   mem_set : ∀ (u : ℤˣ) (g : G), (u, g) ∈ toList → g ∈ d.set u
 
@@ -281,7 +281,7 @@ def cons (g : G) (u : ℤˣ) (w : NormalWord d) (h1 : w.head ∈ d.set u)
       rintro ⟨u', g'⟩ hu' hw1
       exact h2 _ (by simp_all) hw1 }
 
-/-- A recursor to induct on a `NormalWord`, by proving the propert is preserved under `cons` -/
+/-- A recursor to induct on a `NormalWord`, by proving the property is preserved under `cons` -/
 @[elab_as_elim]
 def consRecOn {motive : NormalWord d → Sort*} (w : NormalWord d)
     (ofGroup : ∀g, motive (ofGroup g))
@@ -347,7 +347,7 @@ theorem unitsSMulGroup_snd (u : ℤˣ) (g : G) :
 
 variable {d}
 
-/-- `Cancels u w` is a predicate expressing whether `t^u` cancels with some occurence
+/-- `Cancels u w` is a predicate expressing whether `t^u` cancels with some occurrence
 of `t^-u` when we multiply `t^u` by `w`. -/
 def Cancels (u : ℤˣ) (w : NormalWord d) : Prop :=
   (w.head ∈ (toSubgroup A B u : Subgroup G)) ∧ w.toList.head?.map Prod.fst = some (-u)
@@ -473,9 +473,8 @@ theorem unitsSMul_one_group_smul (g : A) (w : NormalWord d) :
     dsimp
     congr 1
     · conv_lhs => erw [IsComplement.equiv_mul_left]
-      simp? says
-        simp only [toSubgroup_one, SetLike.coe_sort_coe, map_mul, Submonoid.coe_mul,
-          coe_toSubmonoid]
+      simp_rw [toSubgroup_one]
+      simp only [SetLike.coe_sort_coe, map_mul, Subgroup.coe_mul]
     conv_lhs => erw [IsComplement.equiv_mul_left]
     rfl
 
@@ -653,9 +652,9 @@ theorem exists_normalWord_prod_eq
       simp [mul_assoc, unitsSMulGroup, (d.compl _).coe_equiv_snd_eq_one_iff_mem]
 
 /-- Two reduced words representing the same element of the `HNNExtension G A B φ` have the same
-length corresponding list, with the same pattern of occurences of `t^1` and `t^(-1)`,
+length corresponding list, with the same pattern of occurrences of `t^1` and `t^(-1)`,
 and also the `head` is in the same left coset of `toSubgroup A B (-u)`, where `u : ℤˣ`
-is the exponent of the first occurence of `t` in the word. -/
+is the exponent of the first occurrence of `t` in the word. -/
 theorem map_fst_eq_and_of_prod_eq {w₁ w₂ : ReducedWord G A B}
     (hprod : w₁.prod φ = w₂.prod φ) :
     w₁.toList.map Prod.fst = w₂.toList.map Prod.fst ∧
@@ -674,7 +673,7 @@ theorem map_fst_eq_and_of_prod_eq {w₁ w₂ : ReducedWord G A B}
   rwa [← List.head?_map, ← hw₂'2, hw₁'2, List.head?_map]
 
 /-- **Britton's Lemma**. Any reduced word whose product is an element of `G`, has no
-occurences of `t`.  -/
+occurrences of `t`. -/
 theorem toList_eq_nil_of_mem_of_range (w : ReducedWord G A B)
     (hw : w.prod φ ∈ (of.range : Subgroup (HNNExtension G A B φ))) :
     w.toList = [] := by
diff --git a/Mathlib/GroupTheory/Index.lean b/Mathlib/GroupTheory/Index.lean
index ba3d6289d737f..99284c35f6952 100644
--- a/Mathlib/GroupTheory/Index.lean
+++ b/Mathlib/GroupTheory/Index.lean
@@ -157,9 +157,9 @@ theorem index_eq_two_iff : H.index = 2 ↔ ∃ a, ∀ b, Xor' (b * a ∈ H) (b 
 
 @[to_additive]
 theorem mul_mem_iff_of_index_two (h : H.index = 2) {a b : G} : a * b ∈ H ↔ (a ∈ H ↔ b ∈ H) := by
-  by_cases ha : a ∈ H; · simp only [ha, true_iff_iff, mul_mem_cancel_left ha]
-  by_cases hb : b ∈ H; · simp only [hb, iff_true_iff, mul_mem_cancel_right hb]
-  simp only [ha, hb, iff_self_iff, iff_true_iff]
+  by_cases ha : a ∈ H; · simp only [ha, true_iff, mul_mem_cancel_left ha]
+  by_cases hb : b ∈ H; · simp only [hb, iff_true, mul_mem_cancel_right hb]
+  simp only [ha, hb, iff_true]
   rcases index_eq_two_iff.1 h with ⟨c, hc⟩
   refine (hc _).or.resolve_left ?_
   rwa [mul_assoc, mul_mem_cancel_right ((hc _).or.resolve_right hb)]
diff --git a/Mathlib/GroupTheory/MonoidLocalization/Basic.lean b/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
index 988928ee6d456..b5db348da57a7 100644
--- a/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
+++ b/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
@@ -548,7 +548,7 @@ theorem mk'_spec' (x) (y : S) : f.toMap y * f.mk' x y = f.toMap x := by rw [mul_
 
 @[to_additive]
 theorem eq_mk'_iff_mul_eq {x} {y : S} {z} : z = f.mk' x y ↔ z * f.toMap y = f.toMap x :=
-  ⟨fun H ↦ by rw [H, mk'_spec], fun H ↦ by erw [mul_inv_right, H]⟩
+  ⟨fun H ↦ by rw [H, mk'_spec], fun H ↦ by rw [mk', mul_inv_right, H]⟩
 
 @[to_additive]
 theorem mk'_eq_iff_eq_mul {x} {y : S} {z} : f.mk' x y = z ↔ f.toMap x = z * f.toMap y := by
@@ -773,9 +773,7 @@ theorem lift_comp : (f.lift hg).comp f.toMap = g := by ext; exact f.lift_eq hg _
 @[to_additive (attr := simp)]
 theorem lift_of_comp (j : N →* P) : f.lift (f.isUnit_comp j) = j := by
   ext
-  rw [lift_spec]
-  show j _ = j _ * _
-  erw [← j.map_mul, sec_spec']
+  simp_rw [lift_spec, MonoidHom.comp_apply, ← j.map_mul, sec_spec']
 
 @[to_additive]
 theorem epic_of_localizationMap {j k : N →* P} (h : ∀ a, j.comp f.toMap a = k.comp f.toMap a) :
@@ -834,8 +832,8 @@ theorem lift_surjective_iff :
     obtain ⟨z, hz⟩ := H v
     obtain ⟨x, hx⟩ := f.surj z
     use x
-    rw [← hz, f.eq_mk'_iff_mul_eq.2 hx, lift_mk', mul_assoc, mul_comm _ (g ↑x.2)]
-    erw [IsUnit.mul_liftRight_inv (g.restrict S) hg, mul_one]
+    rw [← hz, f.eq_mk'_iff_mul_eq.2 hx, lift_mk', mul_assoc, mul_comm _ (g ↑x.2),
+      ← MonoidHom.restrict_apply, IsUnit.mul_liftRight_inv (g.restrict S) hg, mul_one]
   · intro H v
     obtain ⟨x, hx⟩ := H v
     use f.mk' x.1 x.2
@@ -1131,9 +1129,9 @@ of `AddCommMonoid`s, `k ∘ f` is a Localization map for `M` at `S`."]
 def ofMulEquivOfLocalizations (k : N ≃* P) : LocalizationMap S P :=
   (k.toMonoidHom.comp f.toMap).toLocalizationMap (fun y ↦ isUnit_comp f k.toMonoidHom y)
     (fun v ↦
-      let ⟨z, hz⟩ := k.toEquiv.surjective v
+      let ⟨z, hz⟩ := k.surjective v
       let ⟨x, hx⟩ := f.surj z
-      ⟨x, show v * k _ = k _ by rw [← hx, map_mul, ← hz]; rfl⟩)
+      ⟨x, show v * k _ = k _ by rw [← hx, map_mul, ← hz]⟩)
     fun x y ↦ (k.apply_eq_iff_eq.trans f.eq_iff_exists).1
 
 @[to_additive (attr := simp)]
@@ -1203,18 +1201,17 @@ def ofMulEquivOfDom {k : P ≃* M} (H : T.map k.toMonoidHom = S) : LocalizationM
       ⟨z, hz⟩)
     (fun z ↦
       let ⟨x, hx⟩ := f.surj z
-      let ⟨v, hv⟩ := k.toEquiv.surjective x.1
-      let ⟨w, hw⟩ := k.toEquiv.surjective x.2
-      ⟨(v, ⟨w, H' ▸ show k w ∈ S from hw.symm ▸ x.2.2⟩),
-        show z * f.toMap (k.toEquiv w) = f.toMap (k.toEquiv v) by erw [hv, hw, hx]⟩)
-    fun x y ↦
-    show f.toMap _ = f.toMap _ → _ by
-      erw [f.eq_iff_exists]
-      exact
-        fun ⟨c, hc⟩ ↦
-          let ⟨d, hd⟩ := k.toEquiv.surjective c
-          ⟨⟨d, H' ▸ show k d ∈ S from hd.symm ▸ c.2⟩, by
-            erw [← hd, ← map_mul k, ← map_mul k] at hc; exact k.toEquiv.injective hc⟩
+      let ⟨v, hv⟩ := k.surjective x.1
+      let ⟨w, hw⟩ := k.surjective x.2
+      ⟨(v, ⟨w, H' ▸ show k w ∈ S from hw.symm ▸ x.2.2⟩), by
+        simp_rw [MonoidHom.comp_apply, MulEquiv.toMonoidHom_eq_coe, MonoidHom.coe_coe, hv, hw, hx]⟩)
+    fun x y ↦ by
+      rw [MonoidHom.comp_apply, MonoidHom.comp_apply, MulEquiv.toMonoidHom_eq_coe,
+        MonoidHom.coe_coe, f.eq_iff_exists]
+      rintro ⟨c, hc⟩
+      let ⟨d, hd⟩ := k.surjective c
+      refine ⟨⟨d, H' ▸ show k d ∈ S from hd.symm ▸ c.2⟩, ?_⟩
+      rw [← hd, ← map_mul k, ← map_mul k] at hc; exact k.injective hc
 
 @[to_additive (attr := simp)]
 theorem ofMulEquivOfDom_apply {k : P ≃* M} (H : T.map k.toMonoidHom = S) (x) :
diff --git a/Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean b/Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
index b0c5d7e2cac4a..3c6c5f38a066e 100644
--- a/Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
+++ b/Mathlib/GroupTheory/MonoidLocalization/MonoidWithZero.lean
@@ -42,7 +42,7 @@ namespace Submonoid
 
 variable {S N} in
 /-- If `S` contains `0` then the localization at `S` is trivial. -/
-theorem LocalizationMap.subsingleton  (f : Submonoid.LocalizationMap S N) (h : 0 ∈ S) :
+theorem LocalizationMap.subsingleton (f : Submonoid.LocalizationMap S N) (h : 0 ∈ S) :
     Subsingleton N := by
   refine ⟨fun a b ↦ ?_⟩
   rw [← LocalizationMap.mk'_sec f a, ← LocalizationMap.mk'_sec f b, LocalizationMap.eq]
@@ -152,7 +152,7 @@ theorem leftCancelMulZero_of_le_isLeftRegular
 
 /-- Given a Localization map `f : M →*₀ N` for a Submonoid `S ⊆ M`,
 if `M` is a cancellative monoid with zero, and all elements of `S` are
-regular, then N is a cancellative monoid with zero.  -/
+regular, then N is a cancellative monoid with zero. -/
 theorem isLeftRegular_of_le_isCancelMulZero (f : LocalizationWithZeroMap S N)
     [IsCancelMulZero M] (h : ∀ ⦃x⦄, x ∈ S → IsRegular x) : IsCancelMulZero N := by
   have : IsLeftCancelMulZero N :=
diff --git a/Mathlib/GroupTheory/Nilpotent.lean b/Mathlib/GroupTheory/Nilpotent.lean
index a4b435fb7cba6..90196dde62882 100644
--- a/Mathlib/GroupTheory/Nilpotent.lean
+++ b/Mathlib/GroupTheory/Nilpotent.lean
@@ -285,7 +285,7 @@ instance lowerCentralSeries_normal (n : ℕ) : Normal (lowerCentralSeries G n) :
 theorem lowerCentralSeries_antitone : Antitone (lowerCentralSeries G) := by
   refine antitone_nat_of_succ_le fun n x hx => ?_
   simp only [mem_lowerCentralSeries_succ_iff, exists_prop, mem_top, exists_true_left,
-    true_and_iff] at hx
+    true_and] at hx
   refine
     closure_induction hx ?_ (Subgroup.one_mem _) (@Subgroup.mul_mem _ _ _) (@Subgroup.inv_mem _ _ _)
   rintro y ⟨z, hz, a, ha⟩
@@ -811,16 +811,15 @@ theorem isNilpotent_of_finite_tfae :
         ∀ (p : ℕ) (_hp : Fact p.Prime) (P : Sylow p G), (↑P : Subgroup G).Normal,
         Nonempty
           ((∀ p : (Nat.card G).primeFactors, ∀ P : Sylow p G, (↑P : Subgroup G)) ≃* G)] := by
-  tfae_have 1 → 2
-  · exact @normalizerCondition_of_isNilpotent _ _
+  tfae_have 1 → 2 := @normalizerCondition_of_isNilpotent _ _
   tfae_have 2 → 3
-  · exact fun h H => NormalizerCondition.normal_of_coatom H h
+  | h, H => NormalizerCondition.normal_of_coatom H h
   tfae_have 3 → 4
-  · intro h p _ P; exact Sylow.normal_of_all_max_subgroups_normal h _
+  | h, p, _, P => Sylow.normal_of_all_max_subgroups_normal h _
   tfae_have 4 → 5
-  · exact fun h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
+  | h => Nonempty.intro (Sylow.directProductOfNormal fun {p hp hP} => h p hp hP)
   tfae_have 5 → 1
-  · rintro ⟨e⟩; exact isNilpotent_of_product_of_sylow_group e
+  | ⟨e⟩ => isNilpotent_of_product_of_sylow_group e
   tfae_finish
 
 @[deprecated (since := "2024-06-05")] alias isNilpotent_of_finite_tFAE := isNilpotent_of_finite_tfae
diff --git a/Mathlib/GroupTheory/OrderOfElement.lean b/Mathlib/GroupTheory/OrderOfElement.lean
index fa52929c1c355..9e33d48b21996 100644
--- a/Mathlib/GroupTheory/OrderOfElement.lean
+++ b/Mathlib/GroupTheory/OrderOfElement.lean
@@ -717,7 +717,7 @@ automatic in the case of a finite cancellative monoid. -/
 `addOrderOf_nsmul` but with one assumption less which is automatic in the case of a
 finite cancellative additive monoid."]
 theorem orderOf_pow (x : G) : orderOf (x ^ n) = orderOf x / gcd (orderOf x) n :=
-  (isOfFinOrder_of_finite _).orderOf_pow _
+  (isOfFinOrder_of_finite _).orderOf_pow ..
 
 @[to_additive]
 theorem mem_powers_iff_mem_range_orderOf [DecidableEq G] :
@@ -745,7 +745,7 @@ end Finite
 variable [Fintype G] {x : G}
 
 @[to_additive]
-lemma orderOf_eq_card_powers : orderOf x = Fintype.card (powers x : Set G) :=
+lemma orderOf_eq_card_powers : orderOf x = Fintype.card (powers x : Submonoid G) :=
   (Fintype.card_fin (orderOf x)).symm.trans <|
     Fintype.card_eq.2 ⟨finEquivPowers x <| isOfFinOrder_of_finite _⟩
 
@@ -821,6 +821,7 @@ variable [Fintype G] {x : G} {n : ℕ}
 /-- See also `Nat.card_addSubgroupZPowers`. -/
 @[to_additive "See also `Nat.card_subgroup`."]
 theorem Fintype.card_zpowers : Fintype.card (zpowers x) = orderOf x :=
+  letI : Fintype (zpowers x) := (Subgroup.zpowers x).instFintypeSubtypeMemOfDecidablePred
   (Fintype.card_eq.2 ⟨finEquivZPowers x <| isOfFinOrder_of_finite _⟩).symm.trans <|
     Fintype.card_fin (orderOf x)
 
@@ -948,7 +949,9 @@ theorem inf_eq_bot_of_coprime {G : Type*} [Group G] {H K : Subgroup G}
 /- TODO: Generalise to `Submonoid.powers`. -/
 @[to_additive]
 theorem image_range_orderOf [DecidableEq G] :
+    letI : Fintype (zpowers x) := (Subgroup.zpowers x).instFintypeSubtypeMemOfDecidablePred
     Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = (zpowers x : Set G).toFinset := by
+  letI : Fintype (zpowers x) := (Subgroup.zpowers x).instFintypeSubtypeMemOfDecidablePred
   ext x
   rw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]
 
@@ -1028,8 +1031,8 @@ theorem orderOf_abs_ne_one (h : |x| ≠ 1) : orderOf x = 0 := by
   intro n hn hx
   replace hx : |x| ^ n = 1 := by simpa only [abs_one, abs_pow] using congr_arg abs hx
   cases' h.lt_or_lt with h h
-  · exact ((pow_lt_one (abs_nonneg x) h hn.ne').ne hx).elim
-  · exact ((one_lt_pow h hn.ne').ne' hx).elim
+  · exact ((pow_lt_one₀ (abs_nonneg x) h hn.ne').ne hx).elim
+  · exact ((one_lt_pow₀ h hn.ne').ne' hx).elim
 
 theorem LinearOrderedRing.orderOf_le_two : orderOf x ≤ 2 := by
   cases' ne_or_eq |x| 1 with h h
diff --git a/Mathlib/GroupTheory/PGroup.lean b/Mathlib/GroupTheory/PGroup.lean
index ea820704dd164..52fcaf2fb7a60 100644
--- a/Mathlib/GroupTheory/PGroup.lean
+++ b/Mathlib/GroupTheory/PGroup.lean
@@ -136,7 +136,7 @@ theorem nontrivial_iff_card [Finite G] : Nontrivial G ↔ ∃ n > 0, Nat.card G
       hk⟩,
     fun ⟨k, hk0, hk⟩ =>
     Finite.one_lt_card_iff_nontrivial.1 <|
-      hk.symm ▸ one_lt_pow (Fact.out (p := p.Prime)).one_lt (ne_of_gt hk0)⟩
+      hk.symm ▸ one_lt_pow₀ (Fact.out (p := p.Prime)).one_lt (ne_of_gt hk0)⟩
 
 variable {α : Type*} [MulAction G α]
 
@@ -337,7 +337,7 @@ theorem cyclic_center_quotient_of_card_eq_prime_sq (hG : Nat.card G = p ^ 2) :
 /-- A group of order `p ^ 2` is commutative. See also `IsPGroup.commutative_of_card_eq_prime_sq`
 for just the proof that `∀ a b, a * b = b * a` -/
 def commGroupOfCardEqPrimeSq (hG : Nat.card G = p ^ 2) : CommGroup G :=
-  @commGroupOfCycleCenterQuotient _ _ _ _ (cyclic_center_quotient_of_card_eq_prime_sq hG) _
+  @commGroupOfCyclicCenterQuotient _ _ _ _ (cyclic_center_quotient_of_card_eq_prime_sq hG) _
     (QuotientGroup.ker_mk' (center G)).le
 
 /-- A group of order `p ^ 2` is commutative. See also `IsPGroup.commGroupOfCardEqPrimeSq`
diff --git a/Mathlib/GroupTheory/Perm/Basic.lean b/Mathlib/GroupTheory/Perm/Basic.lean
index 4ff40a3a4a271..5dc3ff87f4586 100644
--- a/Mathlib/GroupTheory/Perm/Basic.lean
+++ b/Mathlib/GroupTheory/Perm/Basic.lean
@@ -396,7 +396,7 @@ theorem ofSubtype_apply_of_not_mem (f : Perm (Subtype p)) (ha : ¬p a) : ofSubty
 theorem mem_iff_ofSubtype_apply_mem (f : Perm (Subtype p)) (x : α) :
     p x ↔ p ((ofSubtype f : α → α) x) :=
   if h : p x then by
-    simpa only [h, true_iff_iff, MonoidHom.coe_mk, ofSubtype_apply_of_mem f h] using (f ⟨x, h⟩).2
+    simpa only [h, true_iff, MonoidHom.coe_mk, ofSubtype_apply_of_mem f h] using (f ⟨x, h⟩).2
   else by simp [h, ofSubtype_apply_of_not_mem f h]
 
 @[simp]
@@ -404,6 +404,16 @@ theorem subtypePerm_ofSubtype (f : Perm (Subtype p)) :
     subtypePerm (ofSubtype f) (mem_iff_ofSubtype_apply_mem f) = f :=
   Equiv.ext fun x => Subtype.coe_injective (ofSubtype_apply_coe f x)
 
+theorem ofSubtype_subtypePerm_of_mem {p : α → Prop} [DecidablePred p]
+    {g : Perm α} (hg : ∀ (x : α), p x ↔ p (g x))
+    {a : α} (ha : p a) : (ofSubtype (g.subtypePerm hg)) a = g a :=
+  ofSubtype_apply_of_mem (g.subtypePerm hg) ha
+
+theorem ofSubtype_subtypePerm_of_not_mem {p : α → Prop} [DecidablePred p]
+    {g : Perm α} (hg : ∀ (x : α), p x ↔ p (g x))
+    {a : α} (ha : ¬ p a) : (ofSubtype (g.subtypePerm hg)) a = a :=
+  ofSubtype_apply_of_not_mem (g.subtypePerm hg) ha
+
 /-- Permutations on a subtype are equivalent to permutations on the original type that fix pointwise
 the rest. -/
 @[simps]
diff --git a/Mathlib/GroupTheory/Perm/Closure.lean b/Mathlib/GroupTheory/Perm/Closure.lean
index 6fa1105deb178..9564bafaf6fcc 100644
--- a/Mathlib/GroupTheory/Perm/Closure.lean
+++ b/Mathlib/GroupTheory/Perm/Closure.lean
@@ -40,7 +40,7 @@ theorem closure_isCycle : closure { σ : Perm β | IsCycle σ } = ⊤ := by
 
 variable [DecidableEq α] [Fintype α]
 
-theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.support = ⊤) (x : α) :
+theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.support = univ) (x : α) :
     closure ({σ, swap x (σ x)} : Set (Perm α)) = ⊤ := by
   let H := closure ({σ, swap x (σ x)} : Set (Perm α))
   have h3 : σ ∈ H := subset_closure (Set.mem_insert σ _)
@@ -51,8 +51,7 @@ theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.su
     | zero => exact subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))
     | succ n ih =>
       convert H.mul_mem (H.mul_mem h3 ih) (H.inv_mem h3)
-      simp_rw [mul_swap_eq_swap_mul, mul_inv_cancel_right, pow_succ']
-      rfl
+      simp_rw [mul_swap_eq_swap_mul, mul_inv_cancel_right, pow_succ', coe_mul, comp_apply]
   have step2 : ∀ n : ℕ, swap x ((σ ^ n) x) ∈ H := by
     intro n
     induction n with
@@ -70,9 +69,9 @@ theorem closure_cycle_adjacent_swap {σ : Perm α} (h1 : IsCycle σ) (h2 : σ.su
       exact H.mul_mem (H.mul_mem (step1 n) ih) (step1 n)
   have step3 : ∀ y : α, swap x y ∈ H := by
     intro y
-    have hx : x ∈ (⊤ : Finset α) := Finset.mem_univ x
+    have hx : x ∈ univ := Finset.mem_univ x
     rw [← h2, mem_support] at hx
-    have hy : y ∈ (⊤ : Finset α) := Finset.mem_univ y
+    have hy : y ∈ univ := Finset.mem_univ y
     rw [← h2, mem_support] at hy
     cases' IsCycle.exists_pow_eq h1 hx hy with n hn
     rw [← hn]
@@ -97,7 +96,7 @@ theorem closure_cycle_coprime_swap {n : ℕ} {σ : Perm α} (h0 : Nat.Coprime n
     closure ({σ, swap x ((σ ^ n) x)} : Set (Perm α)) = ⊤ := by
   rw [← Finset.card_univ, ← h2, ← h1.orderOf] at h0
   cases' exists_pow_eq_self_of_coprime h0 with m hm
-  have h2' : (σ ^ n).support = ⊤ := Eq.trans (support_pow_coprime h0) h2
+  have h2' : (σ ^ n).support = univ := Eq.trans (support_pow_coprime h0) h2
   have h1' : IsCycle ((σ ^ n) ^ (m : ℤ)) := by rwa [← hm] at h1
   replace h1' : IsCycle (σ ^ n) :=
     h1'.of_pow (le_trans (support_pow_le σ n) (ge_of_eq (congr_arg support hm)))
diff --git a/Mathlib/GroupTheory/Perm/ConjAct.lean b/Mathlib/GroupTheory/Perm/ConjAct.lean
new file mode 100644
index 0000000000000..33786abcf6552
--- /dev/null
+++ b/Mathlib/GroupTheory/Perm/ConjAct.lean
@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Antoine Chambert-Loir
+-/
+import Mathlib.Algebra.Group.Pointwise.Finset.Basic
+import Mathlib.GroupTheory.GroupAction.ConjAct
+import Mathlib.GroupTheory.Perm.Cycle.Basic
+import Mathlib.GroupTheory.Perm.Cycle.Factors
+import Mathlib.GroupTheory.Perm.Support
+/-! # Some lemmas pertaining to the action of `ConjAct (Perm α)` on `Perm α`
+
+We prove some lemmas related to the action of `ConjAct (Perm α)` on `Perm α`:
+
+Let `α` be a decidable fintype.
+
+* `conj_support_eq` relates the support of `k • g` with that of `g`
+
+* `cycleFactorsFinset_conj_eq`, `mem_cycleFactorsFinset_conj'`
+  and `cycleFactorsFinset_conj` relate the set of cycles of `g`,  `g.cycleFactorsFinset`,
+  with that for `k • g`
+
+-/
+
+namespace Equiv.Perm
+
+open scoped Pointwise
+
+variable {α : Type*} [DecidableEq α] [Fintype α]
+
+/-- `a : α` belongs to the support of `k • g` iff
+  `k⁻¹ * a` belongs to the support of `g` -/
+theorem mem_conj_support (k : ConjAct (Perm α)) (g : Perm α) (a : α) :
+    a ∈ (k • g).support ↔ ConjAct.ofConjAct k⁻¹ a ∈ g.support := by
+  simp only [mem_support, ConjAct.smul_def, not_iff_not, coe_mul,
+    Function.comp_apply, ConjAct.ofConjAct_inv]
+  apply Equiv.apply_eq_iff_eq_symm_apply
+
+theorem cycleFactorsFinset_conj (g k : Perm α) :
+    (ConjAct.toConjAct k • g).cycleFactorsFinset =
+      Finset.map (MulAut.conj k).toEquiv.toEmbedding g.cycleFactorsFinset := by
+  ext c
+  rw [ConjAct.smul_def, ConjAct.ofConjAct_toConjAct, Finset.mem_map_equiv,
+    ← mem_cycleFactorsFinset_conj g k]
+  simp only [MulEquiv.toEquiv_eq_coe, MulEquiv.coe_toEquiv_symm, MulAut.conj_symm_apply]
+  group
+
+/-- A permutation `c` is a cycle of `g` iff `k • c` is a cycle of `k • g` -/
+@[simp]
+theorem mem_cycleFactorsFinset_conj'
+    (k : ConjAct (Perm α)) (g c : Perm α) :
+    k • c ∈ (k • g).cycleFactorsFinset ↔ c ∈ g.cycleFactorsFinset := by
+  simp only [ConjAct.smul_def]
+  apply mem_cycleFactorsFinset_conj g k
+
+theorem cycleFactorsFinset_conj_eq
+    (k : ConjAct (Perm α)) (g : Perm α) :
+    cycleFactorsFinset (k • g) = k • cycleFactorsFinset g := by
+  ext c
+  rw [← mem_cycleFactorsFinset_conj' k⁻¹ (k • g) c]
+  simp only [inv_smul_smul]
+  exact Finset.inv_smul_mem_iff
+
+end Equiv.Perm
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Basic.lean b/Mathlib/GroupTheory/Perm/Cycle/Basic.lean
index e57ccd1bc59bb..a13b28ec141ca 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Basic.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Basic.lean
@@ -359,9 +359,10 @@ theorem isCycle_swap_mul_aux₁ {α : Type*} [DecidableEq α] :
     ∀ (n : ℕ) {b x : α} {f : Perm α} (_ : (swap x (f x) * f) b ≠ b) (_ : (f ^ n) (f x) = b),
       ∃ i : ℤ, ((swap x (f x) * f) ^ i) (f x) = b := by
   intro n
-  induction' n with n hn
-  · exact fun _ h => ⟨0, h⟩
-  · intro b x f hb h
+  induction n with
+  | zero => exact fun _ h => ⟨0, h⟩
+  | succ n hn =>
+    intro b x f hb h
     exact if hfbx : f x = b then ⟨0, hfbx⟩
       else
         have : f b ≠ b ∧ b ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hb
@@ -379,9 +380,10 @@ theorem isCycle_swap_mul_aux₂ {α : Type*} [DecidableEq α] :
     ∀ (n : ℤ) {b x : α} {f : Perm α} (_ : (swap x (f x) * f) b ≠ b) (_ : (f ^ n) (f x) = b),
       ∃ i : ℤ, ((swap x (f x) * f) ^ i) (f x) = b := by
   intro n
-  induction' n with n n
-  · exact isCycle_swap_mul_aux₁ n
-  · intro b x f hb h
+  induction n with
+  | ofNat n => exact isCycle_swap_mul_aux₁ n
+  | negSucc n =>
+    intro b x f hb h
     exact if hfbx' : f x = b then ⟨0, hfbx'⟩
       else
         have : f b ≠ b ∧ b ≠ x := ne_and_ne_of_swap_mul_apply_ne_self hb
@@ -985,3 +987,140 @@ theorem sum_mul_sum_eq_sum_perm (hσ : σ.IsCycleOn s) (f g : ι → α) :
   sum_smul_sum_eq_sum_perm hσ f g
 
 end Finset
+
+namespace Equiv.Perm
+
+theorem subtypePerm_apply_pow_of_mem {g : Perm α} {s : Finset α}
+    (hs : ∀ x : α, x ∈ s ↔ g x ∈ s) {n : ℕ} {x : α} (hx : x ∈ s) :
+    ((g.subtypePerm hs ^ n) (⟨x, hx⟩ : s) : α) = (g ^ n) x := by
+  simp only [subtypePerm_pow, subtypePerm_apply]
+
+theorem subtypePerm_apply_zpow_of_mem {g : Perm α} {s : Finset α}
+    (hs : ∀ x : α, x ∈ s ↔ g x ∈ s) {i : ℤ} {x : α} (hx : x ∈ s) :
+    ((g.subtypePerm hs ^ i) (⟨x, hx⟩ : s) : α) = (g ^ i) x := by
+  simp only [subtypePerm_zpow, subtypePerm_apply]
+
+variable [Fintype α] [DecidableEq α]
+
+/-- Restrict a permutation to its support -/
+def subtypePermOfSupport (c : Perm α) : Perm c.support :=
+  subtypePerm c fun _ : α => apply_mem_support.symm
+
+/-- Restrict a permutation to a Finset containing its support -/
+def subtypePerm_of_support_le (c : Perm α) {s : Finset α}
+    (hcs : c.support ⊆ s) : Equiv.Perm s :=
+  subtypePerm c (isInvariant_of_support_le hcs)
+
+/-- Support of a cycle is nonempty -/
+theorem IsCycle.nonempty_support {g : Perm α} (hg : g.IsCycle) :
+    g.support.Nonempty := by
+  rw [Finset.nonempty_iff_ne_empty, ne_eq, support_eq_empty_iff]
+  exact IsCycle.ne_one hg
+
+/-- Centralizer of a cycle is a power of that cycle on the cycle -/
+theorem IsCycle.commute_iff' {g c : Perm α} (hc : c.IsCycle) :
+    Commute g c ↔
+      ∃ hc' : ∀ x : α, x ∈ c.support ↔ g x ∈ c.support,
+        subtypePerm g hc' ∈ Subgroup.zpowers c.subtypePermOfSupport := by
+  constructor
+  · intro hgc
+    have hgc' := mem_support_iff_of_commute hgc
+    use hgc'
+    obtain ⟨a, ha⟩ := IsCycle.nonempty_support hc
+    obtain ⟨i, hi⟩ := hc.sameCycle (mem_support.mp ha) (mem_support.mp ((hgc' a).mp ha))
+    use i
+    ext ⟨x, hx⟩
+    simp only [subtypePermOfSupport, Subtype.coe_mk, subtypePerm_apply]
+    rw [subtypePerm_apply_zpow_of_mem]
+    obtain ⟨j, rfl⟩ := hc.sameCycle (mem_support.mp ha) (mem_support.mp hx)
+    simp only [← mul_apply, Commute.eq (Commute.zpow_right hgc j)]
+    rw [← zpow_add, add_comm i j, zpow_add]
+    simp only [mul_apply, EmbeddingLike.apply_eq_iff_eq]
+    exact hi
+  · rintro ⟨hc', ⟨i, hi⟩⟩
+    ext x
+    simp only [coe_mul, Function.comp_apply]
+    by_cases hx : x ∈ c.support
+    · suffices hi' : ∀ x ∈ c.support, g x = (c ^ i) x by
+        rw [hi' x hx, hi' (c x) (apply_mem_support.mpr hx)]
+        simp only [← mul_apply, ← zpow_add_one, ← zpow_one_add, add_comm]
+      intro x hx
+      have hix := Perm.congr_fun hi ⟨x, hx⟩
+      simp only [← Subtype.coe_inj, subtypePermOfSupport, Subtype.coe_mk, subtypePerm_apply,
+        subtypePerm_apply_zpow_of_mem] at hix
+      exact hix.symm
+    · rw [not_mem_support.mp hx, eq_comm, ← not_mem_support]
+      contrapose! hx
+      exact (hc' x).mpr hx
+
+/-- A permutation `g` commutes with a cycle `c` if and only if
+  `c.support` is invariant under `g`, and `g` acts on it as a power of `c`. -/
+theorem IsCycle.commute_iff {g c : Perm α} (hc : c.IsCycle) :
+    Commute g c ↔
+      ∃ hc' : ∀ x : α, x ∈ c.support ↔ g x ∈ c.support,
+        ofSubtype (subtypePerm g hc') ∈ Subgroup.zpowers c := by
+  simp_rw [hc.commute_iff', Subgroup.mem_zpowers_iff]
+  refine exists_congr fun hc' => exists_congr fun k => ?_
+  rw [subtypePermOfSupport, subtypePerm_zpow c k]
+  simp only [Perm.ext_iff, subtypePerm_apply, Subtype.mk.injEq, Subtype.forall]
+  apply forall_congr'
+  intro a
+  by_cases ha : a ∈ c.support
+  · rw [imp_iff_right ha, ofSubtype_subtypePerm_of_mem hc' ha]
+  · rw [iff_true_left (fun b ↦ (ha b).elim), ofSubtype_apply_of_not_mem, ← not_mem_support]
+    · exact Finset.not_mem_mono (support_zpow_le c k) ha
+    · exact ha
+
+theorem zpow_eq_ofSubtype_subtypePerm_iff
+    {g c : Equiv.Perm α} {s : Finset α}
+    (hg : ∀ x, x ∈ s ↔ g x ∈ s) (hc : c.support ⊆ s) (n : ℤ) :
+    c ^ n = ofSubtype (g.subtypePerm hg) ↔
+      c.subtypePerm (isInvariant_of_support_le hc) ^ n = g.subtypePerm hg := by
+  constructor
+  · intro h
+    ext ⟨x, hx⟩
+    simp only [Perm.congr_fun h x, subtypePerm_apply_zpow_of_mem, Subtype.coe_mk, subtypePerm_apply]
+    rw [ofSubtype_apply_of_mem]
+    · simp only [Subtype.coe_mk, subtypePerm_apply]
+    · exact hx
+  · intro h; ext x
+    rw [← h]
+    by_cases hx : x ∈ s
+    · rw [ofSubtype_apply_of_mem (subtypePerm c _ ^ n) hx,
+        subtypePerm_zpow, subtypePerm_apply]
+    · rw [ofSubtype_apply_of_not_mem (subtypePerm c _ ^ n) hx,
+        ← not_mem_support]
+      exact fun hx' ↦ hx (hc (support_zpow_le _ _ hx'))
+
+theorem cycle_zpow_mem_support_iff {g : Perm α}
+    (hg : g.IsCycle) {n : ℤ} {x : α} (hx : g x ≠ x) :
+    (g ^ n) x = x ↔ n % g.support.card = 0 := by
+  set q := n / g.support.card
+  set r := n % g.support.card
+  have div_euc : r + g.support.card * q = n ∧ 0 ≤ r ∧ r < g.support.card := by
+    rw [← Int.ediv_emod_unique _]
+    · exact ⟨rfl, rfl⟩
+    simp only [Int.natCast_pos]
+    apply lt_of_lt_of_le _ (IsCycle.two_le_card_support hg); norm_num
+  simp only [← hg.orderOf] at div_euc
+  obtain ⟨m, hm⟩ := Int.eq_ofNat_of_zero_le div_euc.2.1
+  simp only [hm, Nat.cast_nonneg, Nat.cast_lt, true_and] at div_euc
+  rw [← div_euc.1, zpow_add g]
+  simp only [hm, Nat.cast_eq_zero, zpow_natCast, coe_mul, comp_apply,zpow_mul,
+    pow_orderOf_eq_one, one_zpow, coe_one, id_eq]
+  have : (g ^ m) x = x ↔ g ^ m = 1 := by
+    constructor
+    · intro hgm
+      simp only [IsCycle.pow_eq_one_iff hg]
+      use x
+    · intro hgm
+      simp only [hgm, coe_one, id_eq]
+  rw [this]
+  by_cases hm0 : m = 0
+  · simp only [hm0, pow_zero, Nat.cast_zero]
+  · simp only [Nat.cast_eq_zero, hm0, iff_false]
+    exact pow_ne_one_of_lt_orderOf hm0 div_euc.2
+
+end Perm
+
+end Equiv
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
index 57afc2d245faf..c16d73ee548bd 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
@@ -316,7 +316,7 @@ theorem toList_formPerm_nontrivial (l : List α) (hl : 2 ≤ l.length) (hn : Nod
   · refine ext_getElem (by simp) fun k hk hk' => ?_
     simp only [get_eq_getElem, formPerm_pow_apply_getElem _ hn, zero_add, getElem_map,
       getElem_range, Nat.mod_eq_of_lt hk']
-  · simpa [hs] using get_mem _ _ _
+  · simp [hs]
 
 theorem toList_formPerm_isRotated_self (l : List α) (hl : 2 ≤ l.length) (hn : Nodup l) (x : α)
     (hx : x ∈ l) : toList (formPerm l) x ~r l := by
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Factors.lean b/Mathlib/GroupTheory/Perm/Cycle/Factors.lean
index 87dff651c4a05..e2fc3224e97be 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Factors.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Factors.lean
@@ -225,7 +225,7 @@ theorem mem_support_cycleOf_iff [DecidableEq α] [Fintype α] :
     simp [hx]
   · rw [mem_support, cycleOf_apply]
     split_ifs with hy
-    · simp only [hx, hy, iff_true_iff, Ne, not_false_iff, and_self_iff, mem_support]
+    · simp only [hx, hy, Ne, not_false_iff, and_self_iff, mem_support]
       rcases hy with ⟨k, rfl⟩
       rw [← not_mem_support]
       simpa using hx
@@ -291,6 +291,19 @@ theorem SameCycle.exists_pow_eq [DecidableEq α] [Fintype α] (f : Perm α) (h :
     rw [not_mem_support] at hx
     rw [pow_apply_eq_self_of_apply_eq_self hx, zpow_apply_eq_self_of_apply_eq_self hx]
 
+theorem zpow_eq_zpow_on_iff [DecidableEq α] [Fintype α]
+    (g : Perm α) {m n : ℤ} {x : α} (hx : g x ≠ x) :
+    (g ^ m) x = (g ^ n) x ↔
+      m % (g.cycleOf x).support.card = n % (g.cycleOf x).support.card := by
+  rw [Int.emod_eq_emod_iff_emod_sub_eq_zero]
+  conv_lhs => rw [← Int.sub_add_cancel m n, Int.add_comm, zpow_add]
+  simp only [coe_mul, Function.comp_apply, EmbeddingLike.apply_eq_iff_eq]
+  rw [← Int.dvd_iff_emod_eq_zero]
+  rw [← cycleOf_zpow_apply_self g x, cycle_zpow_mem_support_iff]
+  · rw [← Int.dvd_iff_emod_eq_zero]
+  · exact isCycle_cycleOf g hx
+  · simp only [mem_support, cycleOf_apply_self]; exact hx
+
 end CycleOf
 
 
@@ -303,14 +316,12 @@ section cycleFactors
 open scoped List in
 /-- Given a list `l : List α` and a permutation `f : Perm α` whose nonfixed points are all in `l`,
   recursively factors `f` into cycles. -/
-def cycleFactorsAux [DecidableEq α] [Fintype α] :
-    ∀ (l : List α) (f : Perm α),
-      (∀ {x}, f x ≠ x → x ∈ l) →
-        { l : List (Perm α) // l.prod = f ∧ (∀ g ∈ l, IsCycle g) ∧ l.Pairwise Disjoint } := by
-  intro l f h
-  exact match l with
+def cycleFactorsAux [DecidableEq α] [Fintype α] (l : List α) (f : Perm α)
+    (h : ∀ {x}, f x ≠ x → x ∈ l) :
+    { l : List (Perm α) // l.prod = f ∧ (∀ g ∈ l, IsCycle g) ∧ l.Pairwise Disjoint } :=
+  match l with
   | [] => ⟨[], by
-      { simp only [imp_false, List.Pairwise.nil, List.not_mem_nil, forall_const, and_true_iff,
+      { simp only [imp_false, List.Pairwise.nil, List.not_mem_nil, forall_const, and_true,
           forall_prop_of_false, Classical.not_not, not_false_iff, List.prod_nil] at *
         ext
         simp [*]}⟩
@@ -318,7 +329,7 @@ def cycleFactorsAux [DecidableEq α] [Fintype α] :
     if hx : f x = x then cycleFactorsAux l f (by
         intro y hy; exact List.mem_of_ne_of_mem (fun h => hy (by rwa [h])) (h hy))
     else
-      let ⟨m, hm₁, hm₂, hm₃⟩ :=
+      let ⟨m, hm⟩ :=
         cycleFactorsAux l ((cycleOf f x)⁻¹ * f) (by
         intro y hy
         exact List.mem_of_ne_of_mem
@@ -328,10 +339,8 @@ def cycleFactorsAux [DecidableEq α] [Fintype α] :
             (h fun h : f y = y => by
               rw [mul_apply, h, Ne, inv_eq_iff_eq, cycleOf_apply] at hy
               split_ifs at hy <;> tauto))
-      ⟨cycleOf f x::m, by
-        rw [List.prod_cons, hm₁]
-        simp,
-        fun g hg ↦ ((List.mem_cons).1 hg).elim (fun hg => hg.symm ▸ isCycle_cycleOf _ hx) (hm₂ g),
+      ⟨cycleOf f x :: m, by simp [List.prod_cons, hm.1],
+        fun g hg ↦ ((List.mem_cons).1 hg).elim (fun hg => hg ▸ isCycle_cycleOf _ hx) (hm.2.1 g),
         List.pairwise_cons.2
           ⟨fun g hg y =>
             or_iff_not_imp_left.2 fun hfy =>
@@ -340,16 +349,17 @@ def cycleFactorsAux [DecidableEq α] [Fintype α] :
               have hgm : (g::m.erase g) ~ m :=
                 List.cons_perm_iff_perm_erase.2 ⟨hg, List.Perm.refl _⟩
               have : ∀ h ∈ m.erase g, Disjoint g h :=
-                (List.pairwise_cons.1 ((hgm.pairwise_iff Disjoint.symm).2 hm₃)).1
+                (List.pairwise_cons.1 ((hgm.pairwise_iff Disjoint.symm).2 hm.2.2)).1
               by_cases id fun hgy : g y ≠ y =>
                 (disjoint_prod_right _ this y).resolve_right <| by
                   have hsc : SameCycle f⁻¹ x (f y) := by
                     rwa [sameCycle_inv, sameCycle_apply_right]
-                  rw [disjoint_prod_perm hm₃ hgm.symm, List.prod_cons,
+                  have hm₁ := hm.1
+                  rw [disjoint_prod_perm hm.2.2 hgm.symm, List.prod_cons,
                       ← eq_inv_mul_iff_mul_eq] at hm₁
                   rwa [hm₁, mul_apply, mul_apply, cycleOf_inv, hsc.cycleOf_apply, inv_apply_self,
                     inv_eq_iff_eq, eq_comm],
-            hm₃⟩⟩
+            hm.2.2⟩⟩
 
 theorem mem_list_cycles_iff {α : Type*} [Finite α] {l : List (Perm α)}
     (h1 : ∀ σ : Perm α, σ ∈ l → σ.IsCycle) (h2 : l.Pairwise Disjoint) {σ : Perm α} :
@@ -492,6 +502,11 @@ theorem cycleOf_mem_cycleFactorsFinset_iff {f : Perm α} {x : α} :
     · rw [cycleOf_apply_of_not_sameCycle H] at hy
       contradiction
 
+lemma cycleOf_ne_one_iff_mem_cycleFactorsFinset {g : Equiv.Perm α} {x : α} :
+    g.cycleOf x ≠ 1 ↔ g.cycleOf x ∈ g.cycleFactorsFinset := by
+  rw [Equiv.Perm.cycleOf_mem_cycleFactorsFinset_iff, Equiv.Perm.mem_support,
+        ne_eq, Equiv.Perm.cycleOf_eq_one_iff]
+
 theorem mem_cycleFactorsFinset_support_le {p f : Perm α} (h : p ∈ cycleFactorsFinset f) :
     p.support ≤ f.support := by
   rw [mem_cycleFactorsFinset_iff] at h
@@ -578,6 +593,73 @@ theorem cycle_is_cycleOf {f c : Equiv.Perm α} {a : α} (ha : a ∈ c.support)
     Equiv.Perm.not_mem_support.mp
       (Finset.disjoint_left.mp (Equiv.Perm.Disjoint.disjoint_support hfc) ha)
 
+
+theorem eq_cycleOf_of_mem_cycleFactorsFinset_iff
+    (g c : Perm α) (hc : c ∈ g.cycleFactorsFinset) (x : α) :
+    c = g.cycleOf x ↔ x ∈ c.support := by
+  refine ⟨?_, (cycle_is_cycleOf · hc)⟩
+  rintro rfl
+  rw [mem_support, cycleOf_apply_self, ne_eq, ← cycleOf_eq_one_iff]
+  exact (mem_cycleFactorsFinset_iff.mp hc).left.ne_one
+
+/-- A permutation `c` is a cycle of `g` iff `k * c * k⁻¹` is a cycle of `k * g * k⁻¹` -/
+theorem mem_cycleFactorsFinset_conj (g k c : Perm α) :
+    k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset ↔ c ∈ g.cycleFactorsFinset := by
+  suffices imp_lemma : ∀ {g k c : Perm α},
+      c ∈ g.cycleFactorsFinset → k * c * k⁻¹ ∈ (k * g * k⁻¹).cycleFactorsFinset by
+    refine ⟨fun h ↦ ?_, imp_lemma⟩
+    have aux : ∀ h : Perm α, h = k⁻¹ * (k * h * k⁻¹) * k := fun _ ↦ by group
+    rw [aux g, aux c]
+    exact imp_lemma h
+  intro g k c
+  simp only [mem_cycleFactorsFinset_iff]
+  apply And.imp IsCycle.conj
+  intro hc a ha
+  simp only [coe_mul, Function.comp_apply, EmbeddingLike.apply_eq_iff_eq]
+  apply hc
+  rw [mem_support] at ha ⊢
+  contrapose! ha
+  simp only [mul_smul, ← Perm.smul_def] at ha ⊢
+  rw [ha]
+  simp only [Perm.smul_def, apply_inv_self]
+
+/-- If a permutation commutes with every cycle of `g`, then it commutes with `g`
+
+NB. The converse is false. Commuting with every cycle of `g` means that we belong
+to the kernel of the action of `Equiv.Perm α` on `g.cycleFactorsFinset` -/
+theorem commute_of_mem_cycleFactorsFinset_commute (k g : Perm α)
+    (hk : ∀ c ∈ g.cycleFactorsFinset, Commute k c) :
+    Commute k g := by
+  rw [← cycleFactorsFinset_noncommProd g (cycleFactorsFinset_mem_commute g)]
+  apply Finset.noncommProd_commute
+  simpa only [id_eq] using hk
+
+/-- The cycles of a permutation commute with it -/
+theorem self_mem_cycle_factors_commute {g c : Perm α}
+    (hc : c ∈ g.cycleFactorsFinset) : Commute c g := by
+  apply commute_of_mem_cycleFactorsFinset_commute
+  intro c' hc'
+  by_cases hcc' : c = c'
+  · rw [hcc']
+  · apply g.cycleFactorsFinset_mem_commute hc hc'; exact hcc'
+
+/-- If `c` and `d` are cycles of `g`, then `d` stabilizes the support of `c` -/
+theorem mem_support_cycle_of_cycle {g d c : Perm α}
+    (hc : c ∈ g.cycleFactorsFinset) (hd : d ∈ g.cycleFactorsFinset) :
+    ∀ x : α, x ∈ c.support ↔ d x ∈ c.support := by
+  intro x
+  simp only [mem_support, not_iff_not]
+  by_cases h : c = d
+  · rw [← h, EmbeddingLike.apply_eq_iff_eq]
+  · rw [← Perm.mul_apply,
+      Commute.eq (cycleFactorsFinset_mem_commute g hc hd h),
+      mul_apply, EmbeddingLike.apply_eq_iff_eq]
+
+/-- If a permutation is a cycle of `g`, then its support is invariant under `g`-/
+theorem mem_cycleFactorsFinset_support {g c : Perm α} (hc : c ∈ g.cycleFactorsFinset) (a : α) :
+    a ∈ c.support ↔ g a ∈ c.support :=
+  mem_support_iff_of_commute (self_mem_cycle_factors_commute hc).symm a
+
 end CycleFactorsFinset
 
 @[elab_as_elim]
@@ -647,6 +729,39 @@ theorem cycleFactorsFinset_mul_inv_mem_eq_sdiff [DecidableEq α] [Fintype α] {f
       · exact fun H =>
         not_mem_empty _ (hd.disjoint_cycleFactorsFinset.le_bot (mem_inter_of_mem H hf))
 
+theorem IsCycle.forall_commute_iff [DecidableEq α] [Fintype α] (g z : Perm α) :
+    (∀ c ∈ g.cycleFactorsFinset, Commute z c) ↔
+      ∀ c ∈ g.cycleFactorsFinset,
+      ∃ (hc : ∀ x : α, x ∈ c.support ↔ z x ∈ c.support),
+        ofSubtype (subtypePerm z hc) ∈ Subgroup.zpowers c := by
+  apply forall_congr'
+  intro c
+  apply imp_congr_right
+  intro hc
+  exact IsCycle.commute_iff (mem_cycleFactorsFinset_iff.mp hc).1
+
+/-- A permutation restricted to the support of a cycle factor is that cycle factor -/
+theorem subtypePerm_on_cycleFactorsFinset [DecidableEq α] [Fintype α]
+    {g c : Perm α} (hc : c ∈ g.cycleFactorsFinset) :
+    g.subtypePerm (mem_cycleFactorsFinset_support hc) = c.subtypePermOfSupport := by
+  ext ⟨x, hx⟩
+  simp only [subtypePerm_apply, Subtype.coe_mk, subtypePermOfSupport]
+  exact ((mem_cycleFactorsFinset_iff.mp hc).2 x hx).symm
+
+theorem commute_iff_of_mem_cycleFactorsFinset [DecidableEq α] [Fintype α]{g k c : Equiv.Perm α}
+    (hc : c ∈ g.cycleFactorsFinset) :
+    Commute k c ↔
+      ∃ hc' : ∀ x : α, x ∈ c.support ↔ k x ∈ c.support,
+        k.subtypePerm hc' ∈ Subgroup.zpowers
+          (g.subtypePerm (mem_cycleFactorsFinset_support hc)) := by
+  rw [IsCycle.commute_iff' (mem_cycleFactorsFinset_iff.mp hc).1]
+  apply exists_congr
+  intro hc'
+  simp only [Subgroup.mem_zpowers_iff]
+  apply exists_congr
+  intro n
+  rw [Equiv.Perm.subtypePerm_on_cycleFactorsFinset hc]
+
 end cycleFactors
 
 end Perm
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Type.lean b/Mathlib/GroupTheory/Perm/Cycle/Type.lean
index de73b2a5937e7..5a823f43577e4 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Type.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Type.lean
@@ -64,7 +64,7 @@ theorem cycleType_eq {σ : Perm α} (l : List (Perm α)) (h0 : l.prod = σ)
     σ.cycleType = l.map (Finset.card ∘ support) := by
   have hl : l.Nodup := nodup_of_pairwise_disjoint_cycles h1 h2
   rw [cycleType_eq' l.toFinset]
-  · simp [List.dedup_eq_self.mpr hl, (· ∘ ·)]
+  · simp [List.dedup_eq_self.mpr hl, Function.comp_def]
   · simpa using h1
   · simpa [hl] using h2
   · simp [hl, h0]
@@ -134,6 +134,11 @@ theorem sum_cycleType (σ : Perm α) : σ.cycleType.sum = σ.support.card := by
   | base_cycles σ hσ => rw [hσ.cycleType, sum_coe, List.sum_singleton]
   | induction_disjoint σ τ hd _ hσ hτ => rw [hd.cycleType, sum_add, hσ, hτ, hd.card_support_mul]
 
+theorem card_fixedPoints (σ : Equiv.Perm α) :
+    Fintype.card (Function.fixedPoints σ) = Fintype.card α - σ.cycleType.sum := by
+  rw [Equiv.Perm.sum_cycleType, ← Finset.card_compl, Fintype.card_ofFinset]
+  congr; aesop
+
 theorem sign_of_cycleType' (σ : Perm α) :
     sign σ = (σ.cycleType.map fun n => -(-1 : ℤˣ) ^ n).prod := by
   induction σ using cycle_induction_on with
diff --git a/Mathlib/GroupTheory/Perm/DomMulAct.lean b/Mathlib/GroupTheory/Perm/DomMulAct.lean
index 716f2084d8364..4bfd4c758ab0d 100644
--- a/Mathlib/GroupTheory/Perm/DomMulAct.lean
+++ b/Mathlib/GroupTheory/Perm/DomMulAct.lean
@@ -6,9 +6,9 @@ Authors: Junyan Xu, Antoine Chambert-Loir
 import Mathlib.Algebra.Group.Subgroup.Basic
 import Mathlib.GroupTheory.GroupAction.DomAct.Basic
 import Mathlib.GroupTheory.GroupAction.Basic
-
 import Mathlib.Data.Fintype.Basic
 import Mathlib.Data.Fintype.Perm
+import Mathlib.Data.Set.Card
 import Mathlib.SetTheory.Cardinal.Finite
 
 /-!  Subgroup of `Equiv.Perm α` preserving a function
@@ -29,6 +29,8 @@ Let `α` and `ι` by types and let `f : α → ι`
   the cardinality of the type of permutations preserving `p` :
   `Fintype.card {g : Perm α // f ∘ g = f} = ∏ i, (Fintype.card {a // f a = i})!`.
 
+* Without `Fintype ι`, `DomMulAct.stabilizer_card' p` gives an equivalent
+  formula, where the product is restricted to `Finset.univ.image f`.
 -/
 
 variable {α ι : Type*} {f : α → ι}
@@ -86,22 +88,62 @@ lemma stabilizerMulEquiv_apply (g : (stabilizer (Perm α)ᵈᵐᵃ f)ᵐᵒᵖ)
 
 section Fintype
 
-variable [Fintype α] [Fintype ι] [DecidableEq α] [DecidableEq ι]
+variable [Fintype α]
 
 open Nat
 
 variable (f)
 
 /-- The cardinality of the type of permutations preserving a function -/
-theorem stabilizer_card :
+theorem stabilizer_card [DecidableEq α] [DecidableEq ι] [Fintype ι] :
     Fintype.card {g : Perm α // f ∘ g = f} = ∏ i, (Fintype.card {a // f a = i})! := by
   -- rewriting via Nat.card because Fintype instance is not found
-  rw [← Nat.card_eq_fintype_card, Nat.card_congr (subtypeEquiv mk fun _ ↦ ?_),
+  rw [← Nat.card_eq_fintype_card,
+    Nat.card_congr (subtypeEquiv mk fun _ ↦ ?_),
     Nat.card_congr MulOpposite.opEquiv,
     Nat.card_congr (DomMulAct.stabilizerMulEquiv f).toEquiv, Nat.card_pi]
   · exact Finset.prod_congr rfl fun i _ ↦ by rw [Nat.card_eq_fintype_card, Fintype.card_perm]
   · rfl
 
+/-- The cardinality of the set of permutations preserving a function -/
+theorem stabilizer_ncard [Fintype ι] :
+    Set.ncard {g : Perm α | f ∘ g = f} = ∏ i, (Set.ncard {a | f a = i})! := by
+  classical
+  simp only [← Set.Nat.card_coe_set_eq, Set.coe_setOf, card_eq_fintype_card]
+  exact stabilizer_card f
+
+variable [DecidableEq α] [DecidableEq ι]
+
+/-- The cardinality of the type of permutations preserving a function
+  (without the finiteness assumption on target)-/
+theorem stabilizer_card':
+    Fintype.card {g : Perm α // f ∘ g = f} =
+      ∏ i in Finset.univ.image f, (Fintype.card ({a // f a = i}))! := by
+  set φ : α → Finset.univ.image f :=
+    Set.codRestrict f (Finset.univ.image f) (fun a => by simp)
+  suffices ∀ g : Perm α, f ∘ g = f ↔ φ ∘ g = φ by
+    simp only [this, stabilizer_card]
+    apply Finset.prod_bij (fun g _ => g.val)
+    · exact fun g _ => Finset.coe_mem g
+    · exact fun g _ g' _ =>  SetCoe.ext
+    · exact fun g hg => by
+        rw [Finset.mem_image] at hg
+        obtain ⟨a, _, rfl⟩ := hg
+        use ⟨f a, by simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]⟩
+        simp only [Finset.univ_eq_attach, Finset.mem_attach, exists_const]
+    · intro i _
+      apply congr_arg
+      apply Fintype.card_congr
+      apply Equiv.subtypeEquiv (Equiv.refl α)
+      intro a
+      rw [refl_apply, ← Subtype.coe_inj]
+      simp only [φ, Set.val_codRestrict_apply]
+  · intro g
+    simp only [Function.funext_iff]
+    apply forall_congr'
+    intro a
+    simp only [Function.comp_apply, φ, ← Subtype.coe_inj, Set.val_codRestrict_apply]
+
 end Fintype
 
 end DomMulAct
diff --git a/Mathlib/GroupTheory/Perm/Sign.lean b/Mathlib/GroupTheory/Perm/Sign.lean
index 1f4c096e2337d..269920ef0aab3 100644
--- a/Mathlib/GroupTheory/Perm/Sign.lean
+++ b/Mathlib/GroupTheory/Perm/Sign.lean
@@ -47,7 +47,7 @@ def modSwap (i j : α) : Setoid (Perm α) :=
 
 noncomputable instance {α : Type*} [Fintype α] [DecidableEq α] (i j : α) :
     DecidableRel (modSwap i j).r :=
-  fun _ _ => Or.decidable
+  fun _ _ => inferInstanceAs (Decidable (_ ∨ _))
 
 /-- Given a list `l : List α` and a permutation `f : Perm α` such that the nonfixed points of `f`
   are in `l`, recursively factors `f` as a product of transpositions. -/
@@ -159,7 +159,7 @@ def finPairsLT (n : ℕ) : Finset (Σ_ : Fin n, Fin n) :=
   (univ : Finset (Fin n)).sigma fun a => (range a).attachFin fun _ hm => (mem_range.1 hm).trans a.2
 
 theorem mem_finPairsLT {n : ℕ} {a : Σ_ : Fin n, Fin n} : a ∈ finPairsLT n ↔ a.2 < a.1 := by
-  simp only [finPairsLT, Fin.lt_iff_val_lt_val, true_and_iff, mem_attachFin, mem_range, mem_univ,
+  simp only [finPairsLT, Fin.lt_iff_val_lt_val, true_and, mem_attachFin, mem_range, mem_univ,
     mem_sigma]
 
 /-- `signAux σ` is the sign of a permutation on `Fin n`, defined as the parity of the number of
@@ -255,7 +255,7 @@ private theorem signAux_swap_zero_one' (n : ℕ) : signAux (swap (0 : Fin (n + 2
     rcases a₁.zero_le.eq_or_lt with (rfl | H)
     · exact absurd a₂.zero_le ha₁.not_le
     rcases a₂.zero_le.eq_or_lt with (rfl | H')
-    · simp only [and_true_iff, eq_self_iff_true, heq_iff_eq, mem_singleton, Sigma.mk.inj_iff] at ha₂
+    · simp only [and_true, eq_self_iff_true, heq_iff_eq, mem_singleton, Sigma.mk.inj_iff] at ha₂
       have : 1 < a₁ := lt_of_le_of_ne (Nat.succ_le_of_lt ha₁)
         (Ne.symm (by intro h; apply ha₂; simp [h]))
       have h01 : Equiv.swap (0 : Fin (n + 2)) 1 0 = 1 := by simp
@@ -417,7 +417,7 @@ theorem sign_trans_trans_symm [DecidableEq β] [Fintype β] (f : Perm β) (e : 
 theorem sign_prod_list_swap {l : List (Perm α)} (hl : ∀ g ∈ l, IsSwap g) :
     sign l.prod = (-1) ^ l.length := by
   have h₁ : l.map sign = List.replicate l.length (-1) :=
-    List.eq_replicate.2
+    List.eq_replicate_iff.2
       ⟨by simp, fun u hu =>
         let ⟨g, hg⟩ := List.mem_map.1 hu
         hg.2 ▸ (hl _ hg.1).sign_eq⟩
@@ -526,7 +526,7 @@ theorem prod_prodExtendRight {α : Type*} [DecidableEq α] (σ : α → Perm β)
   · rw [← ha'] at *
     refine Or.inl ⟨l.mem_cons_self a, ?_⟩
     rw [prodExtendRight_apply_eq]
-  · refine Or.inr ⟨fun h => not_or_of_not ha' not_mem_l ((List.mem_cons).mp h), ?_⟩
+  · refine Or.inr ⟨fun h => not_or_intro ha' not_mem_l ((List.mem_cons).mp h), ?_⟩
     rw [prodExtendRight_apply_ne _ ha']
 
 section congr
@@ -549,7 +549,7 @@ theorem sign_prodCongrRight (σ : α → Perm β) : sign (prodCongrRight σ) = 
     exact List.mem_toFinset.mpr (mem_l b)
   rw [← prod_prodExtendRight σ hl mem_l, map_list_prod sign, List.map_map, ← l_to_finset,
     List.prod_toFinset _ hl]
-  simp_rw [← fun a => sign_prodExtendRight a (σ a), Function.comp]
+  simp_rw [← fun a => sign_prodExtendRight a (σ a), Function.comp_def]
 
 theorem sign_prodCongrLeft (σ : α → Perm β) : sign (prodCongrLeft σ) = ∏ k, sign (σ k) := by
   refine (sign_eq_sign_of_equiv _ _ (prodComm β α) ?_).trans (sign_prodCongrRight σ)
diff --git a/Mathlib/GroupTheory/Perm/Support.lean b/Mathlib/GroupTheory/Perm/Support.lean
index 63593ca5a6715..f95ba42476e2a 100644
--- a/Mathlib/GroupTheory/Perm/Support.lean
+++ b/Mathlib/GroupTheory/Perm/Support.lean
@@ -279,7 +279,7 @@ theorem coe_support_eq_set_support (f : Perm α) : (f.support : Set α) = { x |
 
 @[simp]
 theorem support_eq_empty_iff {σ : Perm α} : σ.support = ∅ ↔ σ = 1 := by
-  simp_rw [Finset.ext_iff, mem_support, Finset.not_mem_empty, iff_false_iff, not_not,
+  simp_rw [Finset.ext_iff, mem_support, Finset.not_mem_empty, iff_false, not_not,
     Equiv.Perm.ext_iff, one_apply]
 
 @[simp]
@@ -296,6 +296,12 @@ theorem support_congr (h : f.support ⊆ g.support) (h' : ∀ x ∈ g.support, f
   · rw [not_mem_support.mp hx, ← not_mem_support]
     exact fun H => hx (h H)
 
+/-- If g and c commute, then g stabilizes the support of c -/
+theorem mem_support_iff_of_commute {g c : Perm α} (hgc : Commute g c) (x : α) :
+    x ∈ c.support ↔ g x ∈ c.support := by
+  simp only [mem_support, not_iff_not, ← mul_apply]
+  rw [← hgc, mul_apply, Equiv.apply_eq_iff_eq]
+
 theorem support_mul_le (f g : Perm α) : (f * g).support ≤ f.support ⊔ g.support := fun x => by
   simp only [sup_eq_union]
   rw [mem_union, mem_support, mem_support, mem_support, mul_apply, ← not_and_or, not_imp_not]
@@ -326,6 +332,37 @@ theorem support_inv (σ : Perm α) : support σ⁻¹ = σ.support := by
 theorem apply_mem_support {x : α} : f x ∈ f.support ↔ x ∈ f.support := by
   rw [mem_support, mem_support, Ne, Ne, apply_eq_iff_eq]
 
+/-- The support of a permutation is invariant -/
+theorem isInvariant_of_support_le {c : Perm α} {s : Finset α} (hcs : c.support ≤ s) (x : α) :
+    x ∈ s ↔ c x ∈ s := by
+  by_cases hx' : x ∈ c.support
+  · simp only [hcs hx', true_iff, hcs (apply_mem_support.mpr hx')]
+  · rw [not_mem_support.mp hx']
+
+/-- A permutation c is the extension of a restriction of g to s
+  iff its support is contained in s and its restriction is that of g -/
+lemma ofSubtype_eq_iff {g c : Equiv.Perm α} {s : Finset α}
+    (hg : ∀ x, x ∈ s ↔ g x ∈ s) :
+    ofSubtype (g.subtypePerm hg) = c ↔
+      c.support ≤ s ∧
+      ∀ (hc' : ∀ x, x ∈ s ↔ c x ∈ s), c.subtypePerm hc' = g.subtypePerm hg := by
+  simp only [Equiv.ext_iff, subtypePerm_apply, Subtype.mk.injEq, Subtype.forall]
+  constructor
+  · intro h
+    constructor
+    · intro a ha
+      by_contra ha'
+      rw [mem_support, ← h a, ofSubtype_apply_of_not_mem (p := (· ∈ s)) _ ha'] at ha
+      exact ha rfl
+    · intro _ a ha
+      rw [← h a, ofSubtype_apply_of_mem (p := (· ∈ s)) _ ha, subtypePerm_apply]
+  · rintro ⟨hc, h⟩ a
+    specialize h (isInvariant_of_support_le hc)
+    by_cases ha : a ∈ s
+    · rw [h a ha, ofSubtype_apply_of_mem (p := (· ∈ s)) _ ha, subtypePerm_apply]
+    · rw [ofSubtype_apply_of_not_mem (p := (· ∈ s)) _ ha, eq_comm, ← not_mem_support]
+      exact Finset.not_mem_mono hc ha
+
 -- @[simp] -- Porting note (#10618): simp can prove this
 theorem pow_apply_mem_support {n : ℕ} {x : α} : (f ^ n) x ∈ f.support ↔ x ∈ f.support := by
   simp only [mem_support, ne_eq, apply_pow_apply_eq_iff]
@@ -390,7 +427,7 @@ theorem support_swap_iff (x y : α) : support (swap x y) = {x, y} ↔ x ≠ y :=
 
 theorem support_swap_mul_swap {x y z : α} (h : List.Nodup [x, y, z]) :
     support (swap x y * swap y z) = {x, y, z} := by
-  simp only [List.not_mem_nil, and_true_iff, List.mem_cons, not_false_iff, List.nodup_cons,
+  simp only [List.not_mem_nil, and_true, List.mem_cons, not_false_iff, List.nodup_cons,
     List.mem_singleton, and_self_iff, List.nodup_nil] at h
   push_neg at h
   apply le_antisymm
@@ -487,7 +524,7 @@ theorem support_extend_domain (f : α ≃ Subtype p) {g : Perm α} :
       rw [eq_symm_apply]
       exact Subtype.coe_injective ha
   · rw [extendDomain_apply_not_subtype _ _ pb]
-    simp only [not_exists, false_iff_iff, not_and, eq_self_iff_true, not_true]
+    simp only [not_exists, false_iff, not_and, eq_self_iff_true, not_true]
     rintro a _ rfl
     exact pb (Subtype.prop _)
 
diff --git a/Mathlib/GroupTheory/PresentedGroup.lean b/Mathlib/GroupTheory/PresentedGroup.lean
index 492c52f336a3f..35916a0da95d5 100644
--- a/Mathlib/GroupTheory/PresentedGroup.lean
+++ b/Mathlib/GroupTheory/PresentedGroup.lean
@@ -66,11 +66,11 @@ local notation "F" => FreeGroup.lift f
 
 theorem closure_rels_subset_ker (h : ∀ r ∈ rels, FreeGroup.lift f r = 1) :
     Subgroup.normalClosure rels ≤ MonoidHom.ker F :=
-  Subgroup.normalClosure_le_normal fun x w ↦ (MonoidHom.mem_ker _).2 (h x w)
+  Subgroup.normalClosure_le_normal fun x w ↦ MonoidHom.mem_ker.2 (h x w)
 
 theorem to_group_eq_one_of_mem_closure (h : ∀ r ∈ rels, FreeGroup.lift f r = 1) :
     ∀ x ∈ Subgroup.normalClosure rels, F x = 1 :=
-  fun _ w ↦ (MonoidHom.mem_ker _).1 <| closure_rels_subset_ker h w
+  fun _ w ↦ MonoidHom.mem_ker.1 <| closure_rels_subset_ker h w
 
 /-- The extension of a map `f : α → G` that satisfies the given relations to a group homomorphism
 from `PresentedGroup rels → G`. -/
diff --git a/Mathlib/GroupTheory/PushoutI.lean b/Mathlib/GroupTheory/PushoutI.lean
index 6f3baeb1be190..ab561721ed4f6 100644
--- a/Mathlib/GroupTheory/PushoutI.lean
+++ b/Mathlib/GroupTheory/PushoutI.lean
@@ -301,7 +301,7 @@ theorem prod_empty : (empty : NormalWord d).prod = 1 := by
   simp [prod, empty]
 
 /-- A constructor that multiplies a `NormalWord` by an element, with condition to make
-sure the underlying list does get longer.  -/
+sure the underlying list does get longer. -/
 @[simps!]
 noncomputable def cons {i} (g : G i) (w : NormalWord d) (hmw : w.fstIdx ≠ some i)
     (hgr : g ∉ (φ i).range) : NormalWord d :=
@@ -328,7 +328,7 @@ variable [DecidableEq ι] [∀ i, DecidableEq (G i)]
 
 /-- Given a word in `CoprodI`, if every letter is in the transversal and when
 we multiply by an element of the base group it still has this property,
-then the element of the base group we multiplied by was one.  -/
+then the element of the base group we multiplied by was one. -/
 theorem eq_one_of_smul_normalized (w : CoprodI.Word G) {i : ι} (h : H)
     (hw : ∀ i g, ⟨i, g⟩ ∈ w.toList → g ∈ d.set i)
     (hφw : ∀ j g, ⟨j, g⟩ ∈ (CoprodI.of (φ i h) • w).toList → g ∈ d.set j) :
@@ -355,8 +355,9 @@ theorem eq_one_of_smul_normalized (w : CoprodI.Word G) {i : ι} (h : H)
         equiv_mul_left_of_mem (d.compl i) ⟨_, rfl⟩ , hhead, Subtype.ext_iff,
         Prod.ext_iff, Subgroup.coe_mul] at h
       rcases h with ⟨h₁, h₂⟩
-      rw [h₂, equiv_one (d.compl i) (one_mem _) (d.one_mem _), mul_one,
-        ((injective_iff_map_eq_one' _).1 (d.injective i))] at h₁
+      rw [h₂, equiv_one (d.compl i) (one_mem _) (d.one_mem _)] at h₁
+      erw [mul_one] at h₁
+      simp only [((injective_iff_map_eq_one' _).1 (d.injective i))] at h₁
       contradiction
     · rw [Word.equivPair_head]
       dsimp
diff --git a/Mathlib/GroupTheory/QuotientGroup/Basic.lean b/Mathlib/GroupTheory/QuotientGroup/Basic.lean
index 7b310fdd932b2..2172f44a0ecee 100644
--- a/Mathlib/GroupTheory/QuotientGroup/Basic.lean
+++ b/Mathlib/GroupTheory/QuotientGroup/Basic.lean
@@ -2,9 +2,9 @@
 Copyright (c) 2018 Kevin Buzzard, Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard, Patrick Massot
-
-This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.
 -/
+-- This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.
+
 import Mathlib.Algebra.Group.Subgroup.Pointwise
 import Mathlib.GroupTheory.Congruence.Basic
 import Mathlib.GroupTheory.Coset.Basic
@@ -51,13 +51,13 @@ variable {G : Type u} [Group G] (N : Subgroup G) [nN : N.Normal] {H : Type v} [G
 @[to_additive "The additive congruence relation generated by a normal additive subgroup."]
 protected def con : Con G where
   toSetoid := leftRel N
-  mul' := @fun a b c d hab hcd => by
+  mul' := fun {a b c d} hab hcd => by
     rw [leftRel_eq] at hab hcd ⊢
     dsimp only
     calc
-      (a * c)⁻¹ * (b * d) = c⁻¹ * (a⁻¹ * b) * c⁻¹⁻¹ * (c⁻¹ * d) := by
+      c⁻¹ * (a⁻¹ * b) * c⁻¹⁻¹ * (c⁻¹ * d) ∈ N := N.mul_mem (nN.conj_mem _ hab _) hcd
+      _ = (a * c)⁻¹ * (b * d) := by
         simp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left]
-      _ ∈ N := N.mul_mem (nN.conj_mem _ hab _) hcd
 
 @[to_additive]
 instance Quotient.group : Group (G ⧸ N) :=
@@ -167,6 +167,11 @@ theorem mk_prod {G ι : Type*} [CommGroup G] (N : Subgroup G) (s : Finset ι) {f
 
 @[to_additive (attr := simp)] lemma map_mk'_self : N.map (mk' N) = ⊥ := by aesop
 
+@[to_additive QuotientAddGroup.strictMono_comap_prod_map]
+theorem strictMono_comap_prod_map :
+    StrictMono fun H : Subgroup G ↦ (H.comap N.subtype, H.map (mk' N)) :=
+  strictMono_comap_prod_image N
+
 /-- A group homomorphism `φ : G →* M` with `N ⊆ ker(φ)` descends (i.e. `lift`s) to a
 group homomorphism `G/N →* M`. -/
 @[to_additive "An `AddGroup` homomorphism `φ : G →+ M` with `N ⊆ ker(φ)` descends (i.e. `lift`s)
@@ -320,7 +325,7 @@ open MonoidHom
 /-- The induced map from the quotient by the kernel to the codomain. -/
 @[to_additive "The induced map from the quotient by the kernel to the codomain."]
 def kerLift : G ⧸ ker φ →* H :=
-  lift _ φ fun _g => φ.mem_ker.mp
+  lift _ φ fun _g => mem_ker.mp
 
 @[to_additive (attr := simp)]
 theorem kerLift_mk (g : G) : (kerLift φ) g = φ g :=
@@ -340,7 +345,7 @@ theorem kerLift_injective : Injective (kerLift φ) := fun a b =>
 /-- The induced map from the quotient by the kernel to the range. -/
 @[to_additive "The induced map from the quotient by the kernel to the range."]
 def rangeKerLift : G ⧸ ker φ →* φ.range :=
-  lift _ φ.rangeRestrict fun g hg => (mem_ker _).mp <| by rwa [ker_rangeRestrict]
+  lift _ φ.rangeRestrict fun g hg => mem_ker.mp <| by rwa [ker_rangeRestrict]
 
 @[to_additive]
 theorem rangeKerLift_injective : Injective (rangeKerLift φ) := fun a b =>
diff --git a/Mathlib/GroupTheory/QuotientGroup/Finite.lean b/Mathlib/GroupTheory/QuotientGroup/Finite.lean
index 83c6e3b290b14..76877f51f87b7 100644
--- a/Mathlib/GroupTheory/QuotientGroup/Finite.lean
+++ b/Mathlib/GroupTheory/QuotientGroup/Finite.lean
@@ -2,9 +2,9 @@
 Copyright (c) 2018 Kevin Buzzard, Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard, Patrick Massot
-
-This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.
 -/
+-- This file is to a certain extent based on `quotient_module.lean` by Johannes Hölzl.
+
 import Mathlib.Algebra.Group.Subgroup.Finite
 import Mathlib.GroupTheory.QuotientGroup.Basic
 
diff --git a/Mathlib/GroupTheory/SchurZassenhaus.lean b/Mathlib/GroupTheory/SchurZassenhaus.lean
index 1cb524d82f333..a143963c7cc26 100644
--- a/Mathlib/GroupTheory/SchurZassenhaus.lean
+++ b/Mathlib/GroupTheory/SchurZassenhaus.lean
@@ -262,8 +262,8 @@ private theorem exists_right_complement'_of_coprime_aux' [Finite G] (hG : Nat.ca
     {N : Subgroup G} [N.Normal] (hN : Nat.Coprime (Nat.card N) N.index) :
     ∃ H : Subgroup G, IsComplement' N H := by
   revert G
-  apply Nat.strongInductionOn n
-  rintro n ih G _ _ rfl N _ hN
+  induction n using Nat.strongRecOn with | ind n ih => ?_
+  rintro G _ _ rfl N _ hN
   refine not_forall_not.mp fun h3 => ?_
   haveI := SchurZassenhausInduction.step7 hN (fun G' _ _ hG' => by apply ih _ hG'; rfl) h3
   exact not_exists_of_forall_not h3 (exists_right_complement'_of_coprime_aux hN)
diff --git a/Mathlib/GroupTheory/SemidirectProduct.lean b/Mathlib/GroupTheory/SemidirectProduct.lean
index 904d910b72cfa..6996df7ab8182 100644
--- a/Mathlib/GroupTheory/SemidirectProduct.lean
+++ b/Mathlib/GroupTheory/SemidirectProduct.lean
@@ -222,7 +222,7 @@ section Map
 variable {N₁ : Type*} {G₁ : Type*} [Group N₁] [Group G₁] {φ₁ : G₁ →* MulAut N₁}
 
 /-- Define a map from `N ⋊[φ] G` to `N₁ ⋊[φ₁] G₁` given maps `N →* N₁` and `G →* G₁` that
-  satisfy a commutativity condition `∀ n g, f₁ (φ g n) = φ₁ (f₂ g) (f₁ n)`.  -/
+  satisfy a commutativity condition `∀ n g, f₁ (φ g n) = φ₁ (f₂ g) (f₁ n)`. -/
 def map (f₁ : N →* N₁) (f₂ : G →* G₁)
     (h : ∀ g : G, f₁.comp (φ g).toMonoidHom = (φ₁ (f₂ g)).toMonoidHom.comp f₁) :
     N ⋊[φ] G →* N₁ ⋊[φ₁] G₁ where
diff --git a/Mathlib/GroupTheory/Solvable.lean b/Mathlib/GroupTheory/Solvable.lean
index 769f40a7f6ef5..f3783500b961a 100644
--- a/Mathlib/GroupTheory/Solvable.lean
+++ b/Mathlib/GroupTheory/Solvable.lean
@@ -180,7 +180,7 @@ theorem not_solvable_of_mem_derivedSeries {g : G} (h1 : g ≠ 1)
     (h2 : ∀ n : ℕ, g ∈ derivedSeries G n) : ¬IsSolvable G :=
   mt (isSolvable_def _).mp
     (not_exists_of_forall_not fun n h =>
-      h1 (Subgroup.mem_bot.mp ((congr_arg (Membership.mem g) h).mp (h2 n))))
+      h1 (Subgroup.mem_bot.mp ((congr_arg (g ∈ ·) h).mp (h2 n))))
 
 theorem Equiv.Perm.fin_5_not_solvable : ¬IsSolvable (Equiv.Perm (Fin 5)) := by
   let x : Equiv.Perm (Fin 5) := ⟨![1, 2, 0, 3, 4], ![2, 0, 1, 3, 4], by decide, by decide⟩
diff --git a/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean b/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
index 965253cc75dc7..38b5ab4ee0b07 100644
--- a/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
+++ b/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
@@ -109,7 +109,7 @@ theorem isCyclic_of_orderOf_eq_card [Fintype α] (x : α) (hx : orderOf x = Fint
   use x
   rw [← Set.range_iff_surjective, ← coe_zpowers]
   rw [← Fintype.card_congr (Equiv.Set.univ α), ← Fintype.card_zpowers] at hx
-  exact Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
+  convert Set.eq_of_subset_of_card_le (Set.subset_univ _) (ge_of_eq hx)
 @[deprecated (since := "2024-02-21")]
 alias isAddCyclic_of_orderOf_eq_card := isAddCyclic_of_addOrderOf_eq_card
 
@@ -428,7 +428,7 @@ private theorem card_orderOf_eq_totient_aux₁ :
     refine Finset.sum_congr rfl fun m hm => ?_
     simp only [mem_filter, mem_range, mem_properDivisors] at hm
     refine IH m hm.2 (hm.1.trans hd) (Finset.card_pos.2 ⟨a ^ (d / m), ?_⟩)
-    simp only [mem_filter, mem_univ, orderOf_pow a, ha, true_and_iff,
+    simp only [mem_filter, mem_univ, orderOf_pow a, ha, true_and,
       Nat.gcd_eq_right (div_dvd_of_dvd hm.1), Nat.div_div_self hm.1 hd0]
   have h2 :
     (∑ m ∈ d.divisors, (univ.filter fun a : α => orderOf a = m).card) =
@@ -512,10 +512,10 @@ open Subgroup
 variable {G : Type*} {H : Type*} [Group G] [Group H]
 
 /-- A group is commutative if the quotient by the center is cyclic.
-  Also see `commGroup_of_cycle_center_quotient` for the `CommGroup` instance. -/
+  Also see `commGroupOfCyclicCenterQuotient` for the `CommGroup` instance. -/
 @[to_additive
       "A group is commutative if the quotient by the center is cyclic.
-      Also see `addCommGroup_of_cycle_center_quotient` for the `AddCommGroup` instance."]
+      Also see `addCommGroupOfCyclicCenterQuotient` for the `AddCommGroup` instance."]
 theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G)
     (a b : G) : a * b = b * a :=
   let ⟨⟨x, y, (hxy : f y = x)⟩, (hx : ∀ a : f.range, a ∈ zpowers _)⟩ :=
@@ -539,9 +539,9 @@ theorem commutative_of_cyclic_center_quotient [IsCyclic H] (f : G →* H) (hf :
 alias commutative_of_add_cyclic_center_quotient := commutative_of_addCyclic_center_quotient
 
 /-- A group is commutative if the quotient by the center is cyclic. -/
-@[to_additive commutativeOfAddCycleCenterQuotient
+@[to_additive
       "A group is commutative if the quotient by the center is cyclic."]
-def commGroupOfCycleCenterQuotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G) :
+def commGroupOfCyclicCenterQuotient [IsCyclic H] (f : G →* H) (hf : f.ker ≤ center G) :
     CommGroup G :=
   { show Group G by infer_instance with mul_comm := commutative_of_cyclic_center_quotient f hf }
 
@@ -656,7 +656,7 @@ lemma not_isCyclic_iff_exponent_eq_prime [Group α] {p : ℕ} (hp : p.Prime)
   let _inst : Fintype α := @Fintype.ofFinite α <| Nat.finite_of_card_ne_zero <| by aesop
   have hα' : Fintype.card α = p ^ 2 := by simpa using hα
   have := (Fintype.one_lt_card_iff_nontrivial (α := α)).mp <|
-    hα' ▸ one_lt_pow hp.one_lt two_ne_zero
+    hα' ▸ one_lt_pow₀ hp.one_lt two_ne_zero
   /- in the forward direction, we apply `exponent_eq_prime_iff`, and the reverse direction follows
   immediately because if `α` has exponent `p`, it has no element of order `p ^ 2`. -/
   refine ⟨fun h_cyc ↦ (Monoid.exponent_eq_prime_iff hp).mpr fun g hg ↦ ?_, fun h_exp h_cyc ↦ by
diff --git a/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean b/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
index 8cbe12005aff0..53949cd3a109b 100644
--- a/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
+++ b/Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
@@ -111,7 +111,7 @@ instance : Infinite (DihedralGroup 0) :=
   DihedralGroup.fintypeHelper.infinite_iff.mp inferInstance
 
 instance : Nontrivial (DihedralGroup n) :=
-  ⟨⟨r 0, sr 0, by simp_rw [ne_eq, not_false_eq_true]⟩⟩
+  ⟨⟨r 0, sr 0, by simp_rw [ne_eq, reduceCtorEq, not_false_eq_true]⟩⟩
 
 /-- If `0 < n`, then `DihedralGroup n` has `2n` elements.
 -/
@@ -150,7 +150,7 @@ theorem orderOf_sr (i : ZMod n) : orderOf (sr i) = 2 := by
   · rw [sq, sr_mul_self]
   · -- Porting note: Previous proof was `decide`
     revert n
-    simp_rw [one_def, ne_eq, forall_const, not_false_eq_true]
+    simp_rw [one_def, ne_eq, reduceCtorEq, forall_const, not_false_eq_true]
 
 /-- If `0 < n`, then `r 1` has order `n`.
 -/
diff --git a/Mathlib/GroupTheory/SpecificGroups/Quaternion.lean b/Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
index be7d8cba6dabf..0db9e8be9fe21 100644
--- a/Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
+++ b/Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
@@ -204,7 +204,7 @@ theorem orderOf_xa [NeZero n] (i : ZMod (2 * n)) : orderOf (xa i) = 4 := by
     apply_fun ZMod.val at h'
     apply_fun (· / n) at h'
     simp only [ZMod.val_natCast, ZMod.val_zero, Nat.zero_div, Nat.mod_mul_left_div_self,
-      Nat.div_self (NeZero.pos n)] at h'
+      Nat.div_self (NeZero.pos n), reduceCtorEq] at h'
   · norm_num
 
 /-- In the special case `n = 1`, `Quaternion 1` is a cyclic group (of order `4`). -/
diff --git a/Mathlib/GroupTheory/Sylow.lean b/Mathlib/GroupTheory/Sylow.lean
index 45a76037e7714..a534e244575fe 100644
--- a/Mathlib/GroupTheory/Sylow.lean
+++ b/Mathlib/GroupTheory/Sylow.lean
@@ -332,7 +332,8 @@ theorem Sylow.conj_eq_normalizer_conj_of_mem_centralizer [Fact p.Prime] [Finite
 theorem Sylow.conj_eq_normalizer_conj_of_mem [Fact p.Prime] [Finite (Sylow p G)] (P : Sylow p G)
     [_hP : (P : Subgroup G).IsCommutative] (x g : G) (hx : x ∈ P) (hy : g⁻¹ * x * g ∈ P) :
     ∃ n ∈ (P : Subgroup G).normalizer, g⁻¹ * x * g = n⁻¹ * x * n :=
-  P.conj_eq_normalizer_conj_of_mem_centralizer x g (le_centralizer P hx) (le_centralizer P hy)
+  P.conj_eq_normalizer_conj_of_mem_centralizer x g
+    (le_centralizer (P : Subgroup G) hx) (le_centralizer (P : Subgroup G) hy)
 
 /-- Sylow `p`-subgroups are in bijection with cosets of the normalizer of a Sylow `p`-subgroup -/
 noncomputable def Sylow.equivQuotientNormalizer [Fact p.Prime] [Finite (Sylow p G)]
@@ -458,7 +459,7 @@ def fixedPointsMulLeftCosetsEquivQuotient (H : Subgroup G) [Finite (H : Set G)]
       rfl)
 
 /-- If `H` is a `p`-subgroup of `G`, then the index of `H` inside its normalizer is congruent
-  mod `p` to the index of `H`.  -/
+  mod `p` to the index of `H`. -/
 theorem card_quotient_normalizer_modEq_card_quotient [Finite G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime]
     {H : Subgroup G} (hH : Nat.card H = p ^ n) :
     Nat.card (normalizer H ⧸ Subgroup.comap ((normalizer H).subtype : normalizer H →* G) H) ≡
@@ -467,7 +468,7 @@ theorem card_quotient_normalizer_modEq_card_quotient [Finite G] {p : ℕ} {n : 
   exact ((IsPGroup.of_card hH).card_modEq_card_fixedPoints _).symm
 
 /-- If `H` is a subgroup of `G` of cardinality `p ^ n`, then the cardinality of the
-  normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`.  -/
+  normalizer of `H` is congruent mod `p ^ (n + 1)` to the cardinality of `G`. -/
 theorem card_normalizer_modEq_card [Finite G] {p : ℕ} {n : ℕ} [hp : Fact p.Prime] {H : Subgroup G}
     (hH : Nat.card H = p ^ n) : Nat.card (normalizer H) ≡ Nat.card G [MOD p ^ (n + 1)] := by
   have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
diff --git a/Mathlib/GroupTheory/Torsion.lean b/Mathlib/GroupTheory/Torsion.lean
index ec5b8b580bb9e..08d223f5db3ba 100644
--- a/Mathlib/GroupTheory/Torsion.lean
+++ b/Mathlib/GroupTheory/Torsion.lean
@@ -91,7 +91,7 @@ theorem IsTorsion.of_surjective {f : G →* H} (hf : Function.Surjective f) (tG
 theorem IsTorsion.extension_closed {f : G →* H} (hN : N = f.ker) (tH : IsTorsion H)
     (tN : IsTorsion N) : IsTorsion G := fun g => by
   obtain ⟨ngn, ngnpos, hngn⟩ := (tH <| f g).exists_pow_eq_one
-  have hmem := f.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
+  have hmem := MonoidHom.mem_ker.mpr ((f.map_pow g ngn).trans hngn)
   lift g ^ ngn to N using hN.symm ▸ hmem with gn h
   obtain ⟨nn, nnpos, hnn⟩ := (tN gn).exists_pow_eq_one
   exact isOfFinOrder_iff_pow_eq_one.mpr <| ⟨ngn * nn, mul_pos ngnpos nnpos, by
@@ -201,7 +201,10 @@ variable {G} {p}
 @[to_additive primaryComponent.exists_orderOf_eq_prime_nsmul
   "Elements of the `p`-primary component have additive order `p^n` for some `n`"]
 theorem primaryComponent.exists_orderOf_eq_prime_pow (g : CommMonoid.primaryComponent G p) :
-    ∃ n : ℕ, orderOf g = p ^ n := by simpa [primaryComponent] using g.property
+    ∃ n : ℕ, orderOf g = p ^ n := by
+      obtain ⟨_, hn⟩ := g.property
+      rw [orderOf_submonoid g] at hn
+      exact ⟨_, hn⟩
 
 /-- The `p`- and `q`-primary components are disjoint for `p ≠ q`. -/
 @[to_additive "The `p`- and `q`-primary components are disjoint for `p ≠ q`."]
@@ -294,7 +297,7 @@ end CommGroup
 end CommGroup
 
 namespace Monoid
-
+section Monoid
 variable (G) [Monoid G]
 
 /-- A predicate on a monoid saying that only 1 is of finite order. -/
@@ -323,22 +326,17 @@ lemma isTorsionFree_iff_torsion_eq_bot {G} [CommGroup G] :
 end Monoid
 
 section Group
-
-open Monoid
-
 variable [Group G]
 
 /-- A nontrivial torsion group is not torsion-free. -/
-@[to_additive AddMonoid.IsTorsion.not_torsion_free
-      "A nontrivial additive torsion group is not torsion-free."]
+@[to_additive "A nontrivial additive torsion group is not torsion-free."]
 theorem IsTorsion.not_torsion_free [hN : Nontrivial G] : IsTorsion G → ¬IsTorsionFree G := fun tG =>
   not_isTorsionFree_iff.mpr <| by
     obtain ⟨x, hx⟩ := (nontrivial_iff_exists_ne (1 : G)).mp hN
     exact ⟨x, hx, tG x⟩
 
 /-- A nontrivial torsion-free group is not torsion. -/
-@[to_additive AddMonoid.IsTorsionFree.not_torsion
-      "A nontrivial torsion-free additive group is not torsion."]
+@[to_additive "A nontrivial torsion-free additive group is not torsion."]
 theorem IsTorsionFree.not_torsion [hN : Nontrivial G] : IsTorsionFree G → ¬IsTorsion G := fun tfG =>
   (not_isTorsion_iff _).mpr <| by
     obtain ⟨x, hx⟩ := (nontrivial_iff_exists_ne (1 : G)).mp hN
@@ -368,8 +366,8 @@ open CommGroup (torsion)
 variable (G) [CommGroup G]
 
 /-- Quotienting a group by its torsion subgroup yields a torsion free group. -/
-@[to_additive AddIsTorsionFree.quotient_torsion
-      "Quotienting a group by its additive torsion subgroup yields an additive torsion free group."]
+@[to_additive
+"Quotienting a group by its additive torsion subgroup yields an additive torsion free group."]
 theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun g hne hfin =>
   hne <| by
     induction' g using QuotientGroup.induction_on with g
@@ -380,21 +378,26 @@ theorem IsTorsionFree.quotient_torsion : IsTorsionFree <| G ⧸ torsion G := fun
         (isOfFinOrder_iff_pow_eq_one.mpr ⟨m * n, mul_pos mpos npos, (pow_mul g m n).symm ▸ hn⟩)
 
 end CommGroup
+end Monoid
+
+namespace AddMonoid
 
 lemma isTorsionFree_iff_noZeroSMulDivisors_nat {M : Type*} [AddMonoid M] :
-    AddMonoid.IsTorsionFree M ↔ NoZeroSMulDivisors ℕ M := by
+    IsTorsionFree M ↔ NoZeroSMulDivisors ℕ M := by
   simp_rw [AddMonoid.IsTorsionFree, isOfFinAddOrder_iff_nsmul_eq_zero, not_exists, not_and,
     pos_iff_ne_zero, noZeroSMulDivisors_iff, forall_swap (β := ℕ)]
   exact forall₂_congr fun _ _ ↦ by tauto
 
 lemma isTorsionFree_iff_noZeroSMulDivisors_int [AddGroup G] :
-    AddMonoid.IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G := by
+    IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G := by
   simp_rw [AddMonoid.IsTorsionFree, isOfFinAddOrder_iff_zsmul_eq_zero, not_exists, not_and,
     noZeroSMulDivisors_iff, forall_swap (β := ℤ)]
   exact forall₂_congr fun _ _ ↦ by tauto
 
-@[deprecated (since := "2024-02-29")]
-alias AddMonoid.IsTorsionFree_iff_noZeroSMulDivisors := isTorsionFree_iff_noZeroSMulDivisors_int
-
 lemma IsTorsionFree.of_noZeroSMulDivisors {M : Type*} [AddMonoid M] [NoZeroSMulDivisors ℕ M] :
-    AddMonoid.IsTorsionFree M := isTorsionFree_iff_noZeroSMulDivisors_nat.2 ‹_›
+    IsTorsionFree M := isTorsionFree_iff_noZeroSMulDivisors_nat.2 ‹_›
+
+alias ⟨IsTorsionFree.noZeroSMulDivisors_nat, _⟩ := isTorsionFree_iff_noZeroSMulDivisors_nat
+alias ⟨IsTorsionFree.noZeroSMulDivisors_int, _⟩ := isTorsionFree_iff_noZeroSMulDivisors_int
+
+end AddMonoid
diff --git a/Mathlib/InformationTheory/Hamming.lean b/Mathlib/InformationTheory/Hamming.lean
index 5f6eb9c5030e9..63b352144a998 100644
--- a/Mathlib/InformationTheory/Hamming.lean
+++ b/Mathlib/InformationTheory/Hamming.lean
@@ -262,12 +262,12 @@ instance (α) [Semiring α] (β : ι → Type*) [∀ i, AddCommMonoid (β i)] [
 /-! API to/from the type synonym. -/
 
 
-/-- `Hamming.toHamming` is the identity function to the `Hamming` of a type.  -/
+/-- `Hamming.toHamming` is the identity function to the `Hamming` of a type. -/
 @[match_pattern]
 def toHamming : (∀ i, β i) ≃ Hamming β :=
   Equiv.refl _
 
-/-- `Hamming.ofHamming` is the identity function from the `Hamming` of a type.  -/
+/-- `Hamming.ofHamming` is the identity function from the `Hamming` of a type. -/
 @[match_pattern]
 def ofHamming : Hamming β ≃ ∀ i, β i :=
   Equiv.refl _
diff --git a/Mathlib/Init.lean b/Mathlib/Init.lean
index c06095f8af584..dff297db373ea 100644
--- a/Mathlib/Init.lean
+++ b/Mathlib/Init.lean
@@ -1,6 +1,15 @@
+import Mathlib.Tactic.Linter.DocPrime
+import Mathlib.Tactic.Linter.HashCommandLinter
+import Mathlib.Tactic.Linter.GlobalAttributeIn
+-- This file imports Batteries.Tactic.Lint, where the `env_linter` attribute is defined.
+import Mathlib.Tactic.Linter.Lint
+import Mathlib.Tactic.Linter.OldObtain
+import Mathlib.Tactic.Linter.RefineLinter
+import Mathlib.Tactic.Linter.UnusedTactic
+import Mathlib.Tactic.Linter.Style
+
 /-!
 This is the root file in Mathlib: it is imported by virtually *all* Mathlib files.
-
 For this reason, the imports of this files are carefully curated.
 Any modification involving a change in the imports of this file should be discussed beforehand.
 
@@ -9,4 +18,17 @@ Here are some general guidelines:
 * every import needs to have a comment explaining why the import is there;
 * strong preference for avoiding files that themselves have imports beyond `Lean`, and
   any exception to this rule should by accompanied by a comment explaining the transitive imports.
+
+## Linters
+
+All syntax linters defined in Mathlib which are active by default are imported here.
+Syntax linters need to be imported to take effect, hence we would like them to be imported
+as early as possible.
+
+All linters imported here have no bulk imports;
+**Not** imported in this file are
+- the text-based linters in `Linters/TextBased.lean`, as they can be imported later
+- the `minImports` linter, as that is still disabled by default.
+- the `haveLet` linter, as it is currently disabled by default
+
 -/
diff --git a/Mathlib/Init/Algebra/Classes.lean b/Mathlib/Init/Algebra/Classes.lean
index e5149ae4efa16..bb4475917a4fd 100644
--- a/Mathlib/Init/Algebra/Classes.lean
+++ b/Mathlib/Init/Algebra/Classes.lean
@@ -3,7 +3,6 @@ Copyright (c) 2017 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
-import Mathlib.Init.Logic
 import Mathlib.Order.Defs
 
 /-!
@@ -14,143 +13,28 @@ The files in `Mathlib/Init` are leftovers from the port from Mathlib3.
 We intend to move all the content of these files out into the main `Mathlib` directory structure.
 Contributions assisting with this are appreciated.
 
-(Yaël): The only things of interest in this file now are the unbundled algebra classes
-
 # Unbundled algebra classes
 
-These classes are part of an incomplete refactor described
+These classes were part of an incomplete refactor described
 [here on the github Wiki](https://github.com/leanprover/lean/wiki/Refactoring-structures#encoding-the-algebraic-hierarchy-1).
 However a subset of them are widely used in mathlib3,
 and it has been tricky to clean this up as this file was in core Lean 3.
-
-By themselves, these classes are not good replacements for the `Monoid` / `Group` etc structures
-provided by mathlib, as they are not discoverable by `simp` unlike the current lemmas due to there
-being little to index on. The Wiki page linked above describes an algebraic normalizer, but it was
-never implemented in Lean 3.
-
-## Porting note:
-This file is ancient, and it would be good to replace it with a clean version
-that provides what mathlib4 actually needs.
-
-I've omitted all the `@[algebra]` attributes, as they are not used elsewhere.
-
-The section `StrictWeakOrder` has been omitted, but I've left the mathport output in place.
-Please delete if cleaning up.
-
-I've commented out some classes which we think are completely unused in mathlib.
-
-I've added many of the declarations to `nolints.json`.
-If you clean up this file, please add documentation to classes that we are keeping.
-
-Mario made the following analysis of uses in mathlib3:
-* `is_symm_op`: unused except for some instances
-* `is_commutative`: used a fair amount via some theorems about folds
-  (also assuming `is_associative`)
-* `is_associative`: ditto, also used in `noncomm_fold`
-* `is_left_id`, `is_right_id`: unused except in the mathlib class `is_unital` and in `mono`
-  (which looks like it could use `is_unital`)
-* `is_left_null`, `is_right_null`: unused
-* `is_left_cancel`, `is_right_cancel`: unused except for instances
-* `is_idempotent`: this one is actually used to prove things not directly about `is_idempotent`
-* `is_left_distrib`, `is_right_distrib`, `is_left_inv`, `is_right_inv`, `is_cond_left_inv`,
-  `is_cond_right_inv`: unused
-* `is_distinct`: unused (although we reinvented this one as `nontrivial`)
-* `is_irrefl`, `is_refl`, `is_symm`, `is_trans`: significant usage
-* `is_asymm`, `is_antisymm`, `is_total`, `is_strict_order`: a lot of uses but all in order theory
-  and it's unclear how much could not be transferred to another typeclass
-* `is_preorder`: unused except for instances
-  (except `antisymmetrization`, maybe it could be transferred)
-* `is_total_preorder`, `is_partial_order`: unused except for instances
-* `is_linear_order`: unused except for instances
-* `is_equiv`: unused except for instances (most uses can use `equivalence` instead)
-* `is_per`: unused
-* `is_incomp_trans`: unused
-* `is_strict_weak_order`: significant usage (most of it on `rbmap`, could be transferred)
-* `is_trichotomous`: some usage
-* `is_strict_total_order`: looks like the only usage is in `rbmap` again
 -/
 
 set_option linter.deprecated false
 
 universe u v
 
--- Porting note: removed `outParam`
-class IsSymmOp (α : Sort u) (β : Sort v) (op : α → α → β) : Prop where
-  symm_op : ∀ a b, op a b = op b a
-
-/-- A commutative binary operation. -/
-@[deprecated Std.Commutative (since := "2024-02-02")]
-abbrev IsCommutative (α : Sort u) (op : α → α → α) := Std.Commutative op
-
-instance (priority := 100) isSymmOp_of_isCommutative (α : Sort u) (op : α → α → α)
-    [Std.Commutative op] : IsSymmOp α α op where symm_op := Std.Commutative.comm
-
-/-- An associative binary operation. -/
-@[deprecated Std.Associative (since := "2024-02-02")]
-abbrev IsAssociative (α : Sort u) (op : α → α → α) := Std.Associative op
-
-/-- A binary operation with a left identity. -/
-@[deprecated Std.LawfulLeftIdentity (since := "2024-02-02")]
-abbrev IsLeftId (α : Sort u) (op : α → α → α) (o : outParam α) := Std.LawfulLeftIdentity op o
-
-/-- A binary operation with a right identity. -/
-@[deprecated Std.LawfulRightIdentity (since := "2024-02-02")]
-abbrev IsRightId (α : Sort u) (op : α → α → α) (o : outParam α) := Std.LawfulRightIdentity op o
-
--- class IsLeftNull (α : Sort u) (op : α → α → α) (o : outParam α) : Prop where
---   left_null : ∀ a, op o a = o
-
--- class IsRightNull (α : Sort u) (op : α → α → α) (o : outParam α) : Prop where
---   right_null : ∀ a, op a o = o
+variable {α : Sort u}
 
+@[deprecated (since := "2024-09-11")]
 class IsLeftCancel (α : Sort u) (op : α → α → α) : Prop where
   left_cancel : ∀ a b c, op a b = op a c → b = c
 
+@[deprecated (since := "2024-09-11")]
 class IsRightCancel (α : Sort u) (op : α → α → α) : Prop where
   right_cancel : ∀ a b c, op a b = op c b → a = c
 
-@[deprecated Std.IdempotentOp (since := "2024-02-02")]
-abbrev IsIdempotent (α : Sort u) (op : α → α → α) := Std.IdempotentOp op
-
--- class IsLeftDistrib (α : Sort u) (op₁ : α → α → α) (op₂ : outParam <| α → α → α) : Prop where
---   left_distrib : ∀ a b c, op₁ a (op₂ b c) = op₂ (op₁ a b) (op₁ a c)
-
--- class IsRightDistrib (α : Sort u) (op₁ : α → α → α) (op₂ : outParam <| α → α → α) : Prop where
---   right_distrib : ∀ a b c, op₁ (op₂ a b) c = op₂ (op₁ a c) (op₁ b c)
-
--- class IsLeftInv (α : Sort u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α) :
---     Prop where
---   left_inv : ∀ a, op (inv a) a = o
-
--- class IsRightInv (α : Sort u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α) :
---     Prop where
---   right_inv : ∀ a, op a (inv a) = o
-
--- class IsCondLeftInv (α : Sort u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α)
---     (p : outParam <| α → Prop) : Prop where
---   left_inv : ∀ a, p a → op (inv a) a = o
-
--- class IsCondRightInv (α : Sort u) (op : α → α → α) (inv : outParam <| α → α) (o : outParam α)
---     (p : outParam <| α → Prop) : Prop where
---   right_inv : ∀ a, p a → op a (inv a) = o
-
--- class IsDistinct (α : Sort u) (a : α) (b : α) : Prop where
---   distinct : a ≠ b
-
-/-
--- The following type class doesn't seem very useful, a regular simp lemma should work for this.
-class is_inv (α : Sort u) (β : Sort v) (f : α → β) (g : out β → α) : Prop :=
-(inv : ∀ a, g (f a) = a)
-
--- The following one can also be handled using a regular simp lemma
-class is_idempotent (α : Sort u) (f : α → α) : Prop :=
-(idempotent : ∀ a, f (f a) = f a)
--/
-
-/-- The opposite of a symmetric relation is symmetric. -/
-instance (priority := 100) isSymmOp_of_isSymm (α : Sort u) (r : α → α → Prop) [IsSymm α r] :
-    IsSymmOp α Prop r where symm_op a b := propext <| Iff.intro (IsSymm.symm a b) (IsSymm.symm b a)
-
 /-- `IsTotalPreorder X r` means that the binary relation `r` on `X` is total and a preorder. -/
 @[deprecated (since := "2024-07-30")]
 class IsTotalPreorder (α : Sort u) (r : α → α → Prop) extends IsTrans α r, IsTotal α r : Prop
@@ -161,10 +45,6 @@ instance (priority := 100) isTotalPreorder_isPreorder (α : Sort u) (r : α →
   trans := s.trans
   refl a := Or.elim (@IsTotal.total _ r _ a a) id id
 
--- /-- `IsPer X r` means that the binary relation `r` on `X` is a partial equivalence relation, that
--- is, `IsSymm X r` and `IsTrans X r`. -/
--- class IsPer (α : Sort u) (r : α → α → Prop) extends IsSymm α r, IsTrans α r : Prop
-
 /-- `IsIncompTrans X lt` means that for `lt` a binary relation on `X`, the incomparable relation
 `fun a b => ¬ lt a b ∧ ¬ lt b a` is transitive. -/
 @[deprecated (since := "2024-07-30")]
@@ -177,7 +57,7 @@ instance (priority := 100) (α : Sort u) (lt : α → α → Prop) [IsStrictWeak
 
 section
 
-variable {α : Sort u} {r : α → α → Prop}
+variable {r : α → α → Prop}
 
 local infixl:50 " ≺ " => r
 
@@ -203,7 +83,7 @@ namespace StrictWeakOrder
 
 section
 
-variable {α : Sort u} {r : α → α → Prop}
+variable {r : α → α → Prop}
 
 local infixl:50 " ≺ " => r
 
@@ -241,7 +121,7 @@ instance isEquiv : IsEquiv α (@Equiv _ r) where
 end
 
 /-- The equivalence relation induced by `lt` -/
-notation:50 a " ≈[" lt "]" b:50 => @Equiv _ lt a b--Equiv (r := lt) a b
+notation:50 a " ≈[" lt "]" b:50 => @Equiv _ lt a b
 
 end StrictWeakOrder
 
@@ -276,7 +156,6 @@ instance : IsTotalPreorder α (· ≤ ·) where
   trans := @le_trans _ _
   total := le_total
 
--- TODO(Leo): decide whether we should keep this instance or not
 set_option linter.deprecated false in
 @[deprecated (since := "2024-07-30")]
 instance isStrictWeakOrder_of_linearOrder : IsStrictWeakOrder α (· < ·) :=
@@ -290,7 +169,7 @@ theorem lt_of_lt_of_incomp {α : Sort u} {lt : α → α → Prop} [IsStrictWeak
     [DecidableRel lt] : ∀ {a b c}, lt a b → ¬lt b c ∧ ¬lt c b → lt a c :=
   @fun a b c hab ⟨nbc, ncb⟩ =>
   have nca : ¬lt c a := fun hca => absurd (trans_of lt hca hab) ncb
-  Decidable.by_contradiction fun nac : ¬lt a c =>
+  Decidable.byContradiction fun nac : ¬lt a c =>
     have : ¬lt a b ∧ ¬lt b a := incomp_trans_of lt ⟨nac, nca⟩ ⟨ncb, nbc⟩
     absurd hab this.1
 
@@ -299,7 +178,7 @@ theorem lt_of_incomp_of_lt {α : Sort u} {lt : α → α → Prop} [IsStrictWeak
     [DecidableRel lt] : ∀ {a b c}, ¬lt a b ∧ ¬lt b a → lt b c → lt a c :=
   @fun a b c ⟨nab, nba⟩ hbc =>
   have nca : ¬lt c a := fun hca => absurd (trans_of lt hbc hca) nba
-  Decidable.by_contradiction fun nac : ¬lt a c =>
+  Decidable.byContradiction fun nac : ¬lt a c =>
     have : ¬lt b c ∧ ¬lt c b := incomp_trans_of lt ⟨nba, nab⟩ ⟨nac, nca⟩
     absurd hbc this.1
 
diff --git a/Mathlib/Init/Data/List/Lemmas.lean b/Mathlib/Init/Data/List/Lemmas.lean
deleted file mode 100644
index cc9edf558aca9..0000000000000
--- a/Mathlib/Init/Data/List/Lemmas.lean
+++ /dev/null
@@ -1,97 +0,0 @@
-/-
-Copyright (c) 2014 Parikshit Khanna. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn
--/
-import Batteries.Logic
-
-/-!
-# Note about `Mathlib/Init/`
-The files in `Mathlib/Init` are leftovers from the port from Mathlib3.
-(They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)
-
-We intend to move all the content of these files out into the main `Mathlib` directory structure.
-Contributions assisting with this are appreciated.
-
-# Lemmas for `List` not (yet) in `Batteries`
--/
-
-universe u v w w₁ w₂
-
-variable {α : Type u} {β : Type v} {γ : Type w}
-
-namespace List
-
-open Nat
-
-/-! mem -/
-
-@[deprecated List.mem_cons (since := "2024-08-10")]
-theorem mem_cons_eq (a y : α) (l : List α) : (a ∈ y :: l) = (a = y ∨ a ∈ l) :=
-  propext List.mem_cons
-
-alias ⟨eq_or_mem_of_mem_cons, _⟩ := mem_cons
-
-@[deprecated List.not_mem_nil (since := "2024-08-10")]
-theorem not_exists_mem_nil (p : α → Prop) : ¬∃ x ∈ @nil α, p x :=
-  fun ⟨_, hx, _⟩ => List.not_mem_nil _ hx
-
-@[deprecated (since := "2024-03-23")] alias not_bex_nil := not_exists_mem_nil
-@[deprecated (since := "2024-03-23")] alias bex_cons := exists_mem_cons
-
-/-! sublists -/
-
-@[deprecated (since := "2024-08-10")] alias length_le_of_sublist := Sublist.length_le
-
-/-! map_accumr -/
-
-section MapAccumr
-
-variable {φ : Type w₁} {σ : Type w₂}
-
-/-- Runs a function over a list returning the intermediate results and a final result. -/
-def mapAccumr (f : α → σ → σ × β) : List α → σ → σ × List β
-  | [], c => (c, [])
-  | y :: yr, c =>
-    let r := mapAccumr f yr c
-    let z := f y r.1
-    (z.1, z.2 :: r.2)
-
-/-- Length of the list obtained by `mapAccumr`. -/
-@[simp]
-theorem length_mapAccumr :
-    ∀ (f : α → σ → σ × β) (x : List α) (s : σ), length (mapAccumr f x s).2 = length x
-  | f, _ :: x, s => congr_arg succ (length_mapAccumr f x s)
-  | _, [], _ => rfl
-
-end MapAccumr
-
-section MapAccumr₂
-
-variable {φ : Type w₁} {σ : Type w₂}
-
-/-- Runs a function over two lists returning the intermediate results and a final result. -/
-def mapAccumr₂ (f : α → β → σ → σ × φ) : List α → List β → σ → σ × List φ
-  | [], _, c => (c, [])
-  | _, [], c => (c, [])
-  | x :: xr, y :: yr, c =>
-    let r := mapAccumr₂ f xr yr c
-    let q := f x y r.1
-    (q.1, q.2 :: r.2)
-
-/-- Length of a list obtained using `mapAccumr₂`. -/
-@[simp]
-theorem length_mapAccumr₂ :
-    ∀ (f : α → β → σ → σ × φ) (x y c), length (mapAccumr₂ f x y c).2 = min (length x) (length y)
-  | f, _ :: x, _ :: y, c =>
-    calc
-      succ (length (mapAccumr₂ f x y c).2) = succ (min (length x) (length y)) :=
-        congr_arg succ (length_mapAccumr₂ f x y c)
-      _ = min (succ (length x)) (succ (length y)) := Eq.symm (succ_min_succ (length x) (length y))
-  | _, _ :: _, [], _ => rfl
-  | _, [], _ :: _, _ => rfl
-  | _, [], [], _ => rfl
-
-end MapAccumr₂
-
-end List
diff --git a/Mathlib/Init/Data/Nat/Lemmas.lean b/Mathlib/Init/Data/Nat/Lemmas.lean
index 0117ea849a1e2..fe8bd94cacac2 100644
--- a/Mathlib/Init/Data/Nat/Lemmas.lean
+++ b/Mathlib/Init/Data/Nat/Lemmas.lean
@@ -24,22 +24,13 @@ namespace Nat
 
 /-! multiplication -/
 
-theorem eq_zero_of_mul_eq_zero {n m : ℕ} (h : n * m = 0) : n = 0 ∨ m = 0 :=
-  mul_eq_zero.mp h
-
-/-! properties of inequality -/
-
-protected def ltGeByCases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C :=
-  Decidable.byCases h₁ fun h => h₂ (Or.elim (Nat.lt_or_ge a b) (fun a => absurd a h) fun a => a)
-
-protected def ltByCases {a b : ℕ} {C : Sort u} (h₁ : a < b → C) (h₂ : a = b → C) (h₃ : b < a → C) :
-    C :=
-  Nat.ltGeByCases h₁ fun h₁ => Nat.ltGeByCases h₃ fun h => h₂ (Nat.le_antisymm h h₁)
+@[deprecated (since := "2024-08-23")] alias ⟨eq_zero_of_mul_eq_zero, _⟩ := mul_eq_zero
 
 -- TODO: there are four variations, depending on which variables we assume to be nonneg
 
 /-! successor and predecessor -/
 
+@[deprecated (since := "2024-08-23")]
 def discriminate {B : Sort u} {n : ℕ} (H1 : n = 0 → B) (H2 : ∀ m, n = succ m → B) : B := by
   induction n with
   | zero => exact H1 rfl
@@ -53,12 +44,6 @@ theorem one_eq_succ_zero : 1 = succ 0 :=
 
 /-! induction principles -/
 
-def twoStepInduction {P : ℕ → Sort u} (H1 : P 0) (H2 : P 1)
-    (H3 : ∀ (n : ℕ) (_IH1 : P n) (_IH2 : P (succ n)), P (succ (succ n))) : ∀ a : ℕ, P a
-  | 0 => H1
-  | 1 => H2
-  | succ (succ _n) => H3 _ (twoStepInduction H1 H2 H3 _) (twoStepInduction H1 H2 H3 _)
-
 -- Unused in Mathlib;
 -- if downstream projects find this essential please copy it or remove the deprecation.
 @[deprecated (since := "2024-07-27")]
@@ -68,19 +53,6 @@ def subInduction {P : ℕ → ℕ → Sort u} (H1 : ∀ m, P 0 m) (H2 : ∀ n, P
   | succ _n, 0 => H2 _
   | succ n, succ m => H3 _ _ (subInduction H1 H2 H3 n m)
 
--- Porting note: added `elab_as_elim`
-@[elab_as_elim]
-protected theorem strong_induction_on {p : Nat → Prop} (n : Nat)
-    (h : ∀ n, (∀ m, m < n → p m) → p n) : p n :=
-  Nat.strongRecOn n h
-
-protected theorem case_strong_induction_on {p : Nat → Prop} (a : Nat) (hz : p 0)
-    (hi : ∀ n, (∀ m, m ≤ n → p m) → p (succ n)) : p a :=
-  Nat.strong_induction_on a fun n =>
-    match n with
-    | 0 => fun _ => hz
-    | n + 1 => fun h₁ => hi n fun _m h₂ => h₁ _ (lt_succ_of_le h₂)
-
 /-! mod -/
 
 -- Unused in Mathlib;
diff --git a/Mathlib/Init/Data/Quot.lean b/Mathlib/Init/Data/Quot.lean
deleted file mode 100644
index 891d304ebce19..0000000000000
--- a/Mathlib/Init/Data/Quot.lean
+++ /dev/null
@@ -1,69 +0,0 @@
-/-
-Copyright (c) 2015 Microsoft Corporation. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Leonardo de Moura
--/
-
-import Mathlib.Init
-/-!
-# Note about `Mathlib/Init/`
-The files in `Mathlib/Init` are leftovers from the port from Mathlib3.
-(They contain content moved from lean3 itself that Mathlib needed but was not moved to lean4.)
-
-We intend to move all the content of these files out into the main `Mathlib` directory structure.
-Contributions assisting with this are appreciated.
-
-# Quotient types
-
-These are ported from the Lean 3 standard library file `init/data/quot.lean`.
--/
-
-universe u
-
-section
-
-variable {α : Type u}
-variable (r : α → α → Prop)
-
-/-- `EqvGen r` is the equivalence relation generated by `r`. -/
-inductive EqvGen : α → α → Prop
-  | rel : ∀ x y, r x y → EqvGen x y
-  | refl : ∀ x, EqvGen x x
-  | symm : ∀ x y, EqvGen x y → EqvGen y x
-  | trans : ∀ x y z, EqvGen x y → EqvGen y z → EqvGen x z
-
-theorem EqvGen.is_equivalence : Equivalence (@EqvGen α r) :=
-  Equivalence.mk EqvGen.refl (EqvGen.symm _ _) (EqvGen.trans _ _ _)
-
-/-- `EqvGen.Setoid r` is the setoid generated by a relation `r`.
-
-The motivation for this definition is that `Quot r` behaves like `Quotient (EqvGen.Setoid r)`,
-see for example `Quot.exact` and `Quot.EqvGen_sound`.
--/
-def EqvGen.Setoid : Setoid α :=
-  Setoid.mk _ (EqvGen.is_equivalence r)
-
-theorem Quot.exact {a b : α} (H : Quot.mk r a = Quot.mk r b) : EqvGen r a b :=
-  @Quotient.exact _ (EqvGen.Setoid r) a b (congrArg
-    (Quot.lift (Quotient.mk (EqvGen.Setoid r)) (fun x y h ↦ Quot.sound (EqvGen.rel x y h))) H)
-
-theorem Quot.EqvGen_sound {r : α → α → Prop} {a b : α} (H : EqvGen r a b) :
-    Quot.mk r a = Quot.mk r b :=
-  EqvGen.rec
-    (fun _ _ h ↦ Quot.sound h)
-    (fun _ ↦ rfl)
-    (fun _ _ _ IH ↦ Eq.symm IH)
-    (fun _ _ _ _ _ IH₁ IH₂ ↦ Eq.trans IH₁ IH₂)
-    H
-
-end
-
-open Decidable
-instance Quotient.decidableEq {α : Sort u} {s : Setoid α} [d : ∀ a b : α, Decidable (a ≈ b)] :
-    DecidableEq (Quotient s) :=
-  fun q₁ q₂ : Quotient s ↦
-    Quotient.recOnSubsingleton₂ q₁ q₂
-      (fun a₁ a₂ ↦
-        match (d a₁ a₂) with
-        | (isTrue h₁)  => isTrue (Quotient.sound h₁)
-        | (isFalse h₂) => isFalse (fun h ↦ absurd (Quotient.exact h) h₂))
diff --git a/Mathlib/Init/Logic.lean b/Mathlib/Init/Logic.lean
index f6d926751096d..5fae3e2475809 100644
--- a/Mathlib/Init/Logic.lean
+++ b/Mathlib/Init/Logic.lean
@@ -3,11 +3,7 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
 -/
-import Mathlib.Tactic.Lemma
-import Mathlib.Tactic.Relation.Trans
-import Mathlib.Tactic.ProjectionNotation
 import Batteries.Tactic.Alias
-import Batteries.Tactic.Lint.Misc
 
 /-!
 # Note about `Mathlib/Init/`
@@ -18,222 +14,157 @@ We intend to move all the content of these files out into the main `Mathlib` dir
 Contributions assisting with this are appreciated.
 -/
 
+set_option linter.deprecated false
+
 universe u v
 variable {α : Sort u}
 
-/- Eq -/
+section Binary
 
-theorem not_of_eq_false {p : Prop} (h : p = False) : ¬p := fun hp ↦ h ▸ hp
+variable {α : Type u} {β : Type v} (f : α → α → α) (inv : α → α) (one : α)
 
-theorem cast_proof_irrel {β : Sort u} (h₁ h₂ : α = β) (a : α) : cast h₁ a = cast h₂ a := rfl
+/-- Local notation for `f`, high priority to avoid ambiguity with `HMul.hMul`. -/
+local infix:70 (priority := high) " * " => f
 
-attribute [symm] Eq.symm
+/-- Local notation for `inv`, high priority to avoid ambiguity with `Inv.inv`. -/
+local postfix:100 (priority := high) "⁻¹" => inv
 
-/- Ne -/
+variable (g : α → α → α)
 
-attribute [symm] Ne.symm
+/-- Local notation for `g`, high priority to avoid ambiguity with `HAdd.hAdd`. -/
+local infix:65 (priority := high) " + " => g
 
-/- HEq -/
+@[deprecated Std.Commutative (since := "2024-09-13")]
+def Commutative       := ∀ a b, a * b = b * a
+@[deprecated Std.Associative (since := "2024-09-13")]
+def Associative       := ∀ a b c, (a * b) * c = a * (b * c)
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def LeftIdentity      := ∀ a, one * a = a
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def RightIdentity     := ∀ a, a * one = a
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def RightInverse      := ∀ a, a * a⁻¹ = one
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def LeftCancelative   := ∀ a b c, a * b = a * c → b = c
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def RightCancelative  := ∀ a b c, a * b = c * b → a = c
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def LeftDistributive  := ∀ a b c, a * (b + c) = a * b + a * c
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+def RightDistributive := ∀ a b c, (a + b) * c = a * c + b * c
 
-alias eq_rec_heq := eqRec_heq
+end Binary
 
--- FIXME This is still rejected after #857
--- attribute [refl] HEq.refl
-attribute [symm] HEq.symm
-attribute [trans] HEq.trans
-attribute [trans] heq_of_eq_of_heq
+@[deprecated (since := "2024-09-03")] alias not_of_eq_false := of_eq_false
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
+theorem cast_proof_irrel {β : Sort u} (h₁ h₂ : α = β) (a : α) : cast h₁ a = cast h₂ a := rfl
+@[deprecated (since := "2024-09-03")] alias eq_rec_heq := eqRec_heq
+@[deprecated (since := "2024-09-03")] alias heq_prop := proof_irrel_heq
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem heq_of_eq_rec_left {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} :
     (e : a = a') → (h₂ : Eq.rec (motive := fun a _ ↦ φ a) p₁ e = p₂) → HEq p₁ p₂
   | rfl, rfl => HEq.rfl
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem heq_of_eq_rec_right {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} :
     (e : a' = a) → (h₂ : p₁ = Eq.rec (motive := fun a _ ↦ φ a) p₂ e) → HEq p₁ p₂
   | rfl, rfl => HEq.rfl
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem of_heq_true {a : Prop} (h : HEq a True) : a := of_eq_true (eq_of_heq h)
 
+@[deprecated (since := "2024-09-03")]
 theorem eq_rec_compose {α β φ : Sort u} :
     ∀ (p₁ : β = φ) (p₂ : α = β) (a : α),
       (Eq.recOn p₁ (Eq.recOn p₂ a : β) : φ) = Eq.recOn (Eq.trans p₂ p₁) a
   | rfl, rfl, _ => rfl
 
-theorem heq_prop {P Q : Prop} (p : P) (q : Q) : HEq p q :=
-  Subsingleton.helim (propext <| iff_of_true p q) _ _
-
-/- and -/
-
-variable {a b c d : Prop}
+-- unused in Mathlib
+@[deprecated (since := "2024-09-11")] alias ⟨not_of_not_not_not, _⟩ := not_not_not
 
-/- xor -/
+variable {a : Prop} (p : Prop)
 
-def Xor' (a b : Prop) := (a ∧ ¬ b) ∨ (b ∧ ¬ a)
-
-/- iff -/
-
-attribute [refl] Iff.refl
-attribute [trans] Iff.trans
-attribute [symm] Iff.symm
-
-alias ⟨not_of_not_not_not, _⟩ := not_not_not
-
-variable (p)
-
--- FIXME: remove _iff and add _eq for the lean 4 core versions
+@[deprecated and_true (since := "2024-09-12")]
 theorem and_true_iff : p ∧ True ↔ p := iff_of_eq (and_true _)
+@[deprecated true_and (since := "2024-09-12")]
 theorem true_and_iff : True ∧ p ↔ p := iff_of_eq (true_and _)
+@[deprecated and_false (since := "2024-09-12")]
 theorem and_false_iff : p ∧ False ↔ False := iff_of_eq (and_false _)
+@[deprecated false_and (since := "2024-09-12")]
 theorem false_and_iff : False ∧ p ↔ False := iff_of_eq (false_and _)
-
-theorem true_or_iff : True ∨ p ↔ True := iff_of_eq (true_or _)
+@[deprecated or_true (since := "2024-09-12")]
 theorem or_true_iff : p ∨ True ↔ True := iff_of_eq (or_true _)
-theorem false_or_iff : False ∨ p ↔ p := iff_of_eq (false_or _)
+@[deprecated true_or (since := "2024-09-12")]
+theorem true_or_iff : True ∨ p ↔ True := iff_of_eq (true_or _)
+@[deprecated or_false (since := "2024-09-12")]
 theorem or_false_iff : p ∨ False ↔ p := iff_of_eq (or_false _)
-
-theorem not_or_of_not : ¬a → ¬b → ¬(a ∨ b) := fun h1 h2 ↦ not_or.2 ⟨h1, h2⟩
-
+@[deprecated false_or (since := "2024-09-12")]
+theorem false_or_iff : False ∨ p ↔ p := iff_of_eq (false_or _)
+@[deprecated iff_true (since := "2024-09-12")]
 theorem iff_true_iff : (a ↔ True) ↔ a := iff_of_eq (iff_true _)
+@[deprecated true_iff (since := "2024-09-12")]
 theorem true_iff_iff : (True ↔ a) ↔ a := iff_of_eq (true_iff _)
-
+@[deprecated iff_false (since := "2024-09-12")]
 theorem iff_false_iff : (a ↔ False) ↔ ¬a := iff_of_eq (iff_false _)
-
+@[deprecated false_iff (since := "2024-09-12")]
 theorem false_iff_iff : (False ↔ a) ↔ ¬a := iff_of_eq (false_iff _)
-
+@[deprecated iff_self (since := "2024-09-12")]
 theorem iff_self_iff (a : Prop) : (a ↔ a) ↔ True := iff_of_eq (iff_self _)
-
-/- exists unique -/
-
-def ExistsUnique (p : α → Prop) := ∃ x, p x ∧ ∀ y, p y → y = x
-
-namespace Mathlib.Notation
-open Lean
-
-/--
-Checks to see that `xs` has only one binder.
--/
-def isExplicitBinderSingular (xs : TSyntax ``explicitBinders) : Bool :=
-  match xs with
-  | `(explicitBinders| $_:binderIdent $[: $_]?) => true
-  | `(explicitBinders| ($_:binderIdent : $_)) => true
-  | _ => false
-
-open TSyntax.Compat in
-/--
-`∃! x : α, p x` means that there exists a unique `x` in `α` such that `p x`.
-This is notation for `ExistsUnique (fun (x : α) ↦ p x)`.
-
-This notation does not allow multiple binders like `∃! (x : α) (y : β), p x y`
-as a shorthand for `∃! (x : α), ∃! (y : β), p x y` since it is liable to be misunderstood.
-Often, the intended meaning is instead `∃! q : α × β, p q.1 q.2`.
--/
-macro "∃!" xs:explicitBinders ", " b:term : term => do
-  if !isExplicitBinderSingular xs then
-    Macro.throwErrorAt xs "\
-      The `ExistsUnique` notation should not be used with more than one binder.\n\
-      \n\
-      The reason for this is that `∃! (x : α), ∃! (y : β), p x y` has a completely different \
-      meaning from `∃! q : α × β, p q.1 q.2`. \
-      To prevent confusion, this notation requires that you be explicit \
-      and use one with the correct interpretation."
-  expandExplicitBinders ``ExistsUnique xs b
-
-/--
-Pretty-printing for `ExistsUnique`, following the same pattern as pretty printing for `Exists`.
-However, it does *not* merge binders.
--/
-@[app_unexpander ExistsUnique] def unexpandExistsUnique : Lean.PrettyPrinter.Unexpander
-  | `($(_) fun $x:ident ↦ $b)                      => `(∃! $x:ident, $b)
-  | `($(_) fun ($x:ident : $t) ↦ $b)               => `(∃! $x:ident : $t, $b)
-  | _                                               => throw ()
-
-/--
-`∃! x ∈ s, p x` means `∃! x, x ∈ s ∧ p x`, which is to say that there exists a unique `x ∈ s`
-such that `p x`.
-Similarly, notations such as `∃! x ≤ n, p n` are supported,
-using any relation defined using the `binder_predicate` command.
--/
-syntax "∃! " binderIdent binderPred ", " term : term
-
-macro_rules
-  | `(∃! $x:ident $p:binderPred, $b) => `(∃! $x:ident, satisfies_binder_pred% $x $p ∧ $b)
-  | `(∃! _ $p:binderPred, $b) => `(∃! x, satisfies_binder_pred% x $p ∧ $b)
-
-end Mathlib.Notation
-
--- @[intro] -- TODO
-theorem ExistsUnique.intro {p : α → Prop} (w : α)
-    (h₁ : p w) (h₂ : ∀ y, p y → y = w) : ∃! x, p x := ⟨w, h₁, h₂⟩
-
-theorem ExistsUnique.elim {α : Sort u} {p : α → Prop} {b : Prop}
-    (h₂ : ∃! x, p x) (h₁ : ∀ x, p x → (∀ y, p y → y = x) → b) : b :=
-  Exists.elim h₂ (fun w hw ↦ h₁ w (And.left hw) (And.right hw))
-
-theorem exists_unique_of_exists_of_unique {α : Sort u} {p : α → Prop}
-    (hex : ∃ x, p x) (hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) : ∃! x, p x :=
-  Exists.elim hex (fun x px ↦ ExistsUnique.intro x px (fun y (h : p y) ↦ hunique y x h px))
-
-theorem ExistsUnique.exists {p : α → Prop} : (∃! x, p x) → ∃ x, p x | ⟨x, h, _⟩ => ⟨x, h⟩
-
-theorem ExistsUnique.unique {α : Sort u} {p : α → Prop}
-    (h : ∃! x, p x) {y₁ y₂ : α} (py₁ : p y₁) (py₂ : p y₂) : y₁ = y₂ :=
-  let ⟨_, _, hy⟩ := h; (hy _ py₁).trans (hy _ py₂).symm
-
-/- exists, forall, exists unique congruences -/
-
--- TODO
--- attribute [congr] forall_congr'
--- attribute [congr] exists_congr'
-
--- @[congr]
-theorem existsUnique_congr {p q : α → Prop} (h : ∀ a, p a ↔ q a) : (∃! a, p a) ↔ ∃! a, q a :=
-  exists_congr fun _ ↦ and_congr (h _) <| forall_congr' fun _ ↦ imp_congr_left (h _)
+@[deprecated (since := "2024-09-12")] alias not_or_of_not := not_or_intro
 
 /- decidable -/
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem decide_True' (h : Decidable True) : decide True = true := by simp
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem decide_False' (h : Decidable False) : decide False = false := by simp
 
 namespace Decidable
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def recOn_true [h : Decidable p] {h₁ : p → Sort u} {h₂ : ¬p → Sort u}
     (h₃ : p) (h₄ : h₁ h₃) : Decidable.recOn h h₂ h₁ :=
   cast (by match h with | .isTrue _ => rfl) h₄
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def recOn_false [h : Decidable p] {h₁ : p → Sort u} {h₂ : ¬p → Sort u} (h₃ : ¬p) (h₄ : h₂ h₃) :
     Decidable.recOn h h₂ h₁ :=
   cast (by match h with | .isFalse _ => rfl) h₄
 
-alias by_cases := byCases
-alias by_contradiction := byContradiction
+@[deprecated (since := "2024-09-03")] alias by_cases := byCases
+@[deprecated (since := "2024-09-03")] alias by_contradiction := byContradiction
 @[deprecated (since := "2024-07-27")] alias not_not_iff := not_not
 
 end Decidable
 
-alias Or.decidable := instDecidableOr
-alias And.decidable := instDecidableAnd
-alias Not.decidable := instDecidableNot
-alias Iff.decidable := instDecidableIff
-alias decidableTrue := instDecidableTrue
-alias decidableFalse := instDecidableFalse
-
-instance {q : Prop} [Decidable p] [Decidable q] : Decidable (Xor' p q) :=
-    inferInstanceAs (Decidable (Or ..))
+@[deprecated (since := "2024-09-03")] alias Or.decidable := instDecidableOr
+@[deprecated (since := "2024-09-03")] alias And.decidable := instDecidableAnd
+@[deprecated (since := "2024-09-03")] alias Not.decidable := instDecidableNot
+@[deprecated (since := "2024-09-03")] alias Iff.decidable := instDecidableIff
+@[deprecated (since := "2024-09-03")] alias decidableTrue := instDecidableTrue
+@[deprecated (since := "2024-09-03")] alias decidableFalse := instDecidableFalse
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def IsDecEq {α : Sort u} (p : α → α → Bool) : Prop := ∀ ⦃x y : α⦄, p x y = true → x = y
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def IsDecRefl {α : Sort u} (p : α → α → Bool) : Prop := ∀ x, p x x = true
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def decidableEq_of_bool_pred {α : Sort u} {p : α → α → Bool} (h₁ : IsDecEq p)
     (h₂ : IsDecRefl p) : DecidableEq α
   | x, y =>
     if hp : p x y = true then isTrue (h₁ hp)
     else isFalse (fun hxy : x = y ↦ absurd (h₂ y) (by rwa [hxy] at hp))
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem decidableEq_inl_refl {α : Sort u} [h : DecidableEq α] (a : α) :
     h a a = isTrue (Eq.refl a) :=
   match h a a with
   | isTrue _ => rfl
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem decidableEq_inr_neg {α : Sort u} [h : DecidableEq α] {a b : α}
     (n : a ≠ b) : h a b = isFalse n :=
   match h a b with
@@ -241,6 +172,7 @@ theorem decidableEq_inr_neg {α : Sort u} [h : DecidableEq α] {a b : α}
 
 /- subsingleton -/
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem rec_subsingleton {p : Prop} [h : Decidable p] {h₁ : p → Sort u} {h₂ : ¬p → Sort u}
     [h₃ : ∀ h : p, Subsingleton (h₁ h)] [h₄ : ∀ h : ¬p, Subsingleton (h₂ h)] :
     Subsingleton (Decidable.recOn h h₂ h₁) :=
@@ -248,24 +180,27 @@ theorem rec_subsingleton {p : Prop} [h : Decidable p] {h₁ : p → Sort u} {h
   | isTrue h => h₃ h
   | isFalse h => h₄ h
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem imp_of_if_pos {c t e : Prop} [Decidable c] (h : ite c t e) (hc : c) : t :=
   (if_pos hc ▸ h :)
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem imp_of_if_neg {c t e : Prop} [Decidable c] (h : ite c t e) (hnc : ¬c) : e :=
   (if_neg hnc ▸ h :)
 
-theorem if_ctx_congr {α : Sort u} {b c : Prop} [dec_b : Decidable b] [dec_c : Decidable c]
-    {x y u v : α} (h_c : b ↔ c) (h_t : c → x = u) (h_e : ¬c → y = v) : ite b x y = ite c u v :=
+@[deprecated (since := "2024-09-11")]
+theorem dif_ctx_congr {α : Sort u} {b c : Prop} [dec_b : Decidable b] [dec_c : Decidable c]
+    {x : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
+    (h_c : b ↔ c) (h_t : ∀ h : c, x (Iff.mpr h_c h) = u h)
+    (h_e : ∀ h : ¬c, y (Iff.mpr (not_congr h_c) h) = v h) :
+    @dite α b dec_b x y = @dite α c dec_c u v :=
   match dec_b, dec_c with
-  | isFalse _,  isFalse h₂ => h_e h₂
-  | isTrue _,   isTrue h₂  => h_t h₂
-  | isFalse h₁, isTrue h₂  => absurd h₂ (Iff.mp (not_congr h_c) h₁)
-  | isTrue h₁,  isFalse h₂ => absurd h₁ (Iff.mpr (not_congr h_c) h₂)
-
-theorem if_congr {α : Sort u} {b c : Prop} [Decidable b] [Decidable c]
-    {x y u v : α} (h_c : b ↔ c) (h_t : x = u) (h_e : y = v) : ite b x y = ite c u v :=
-  if_ctx_congr h_c (fun _ ↦ h_t) (fun _ ↦ h_e)
+  | isFalse _, isFalse h₂ => h_e h₂
+  | isTrue _, isTrue h₂ => h_t h₂
+  | isFalse h₁, isTrue h₂ => absurd h₂ (Iff.mp (not_congr h_c) h₁)
+  | isTrue h₁, isFalse h₂ => absurd h₁ (Iff.mpr (not_congr h_c) h₂)
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem if_ctx_congr_prop {b c x y u v : Prop} [dec_b : Decidable b] [dec_c : Decidable c]
     (h_c : b ↔ c) (h_t : c → (x ↔ u)) (h_e : ¬c → (y ↔ v)) : ite b x y ↔ ite c u v :=
   match dec_b, dec_c with
@@ -275,10 +210,12 @@ theorem if_ctx_congr_prop {b c x y u v : Prop} [dec_b : Decidable b] [dec_c : De
   | isTrue h₁,  isFalse h₂ => absurd h₁ (Iff.mpr (not_congr h_c) h₂)
 
 -- @[congr]
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem if_congr_prop {b c x y u v : Prop} [Decidable b] [Decidable c] (h_c : b ↔ c) (h_t : x ↔ u)
     (h_e : y ↔ v) : ite b x y ↔ ite c u v :=
   if_ctx_congr_prop h_c (fun _ ↦ h_t) (fun _ ↦ h_e)
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem if_ctx_simp_congr_prop {b c x y u v : Prop} [Decidable b] (h_c : b ↔ c) (h_t : c → (x ↔ u))
     -- FIXME: after https://github.com/leanprover/lean4/issues/1867 is fixed,
     -- this should be changed back to:
@@ -286,6 +223,7 @@ theorem if_ctx_simp_congr_prop {b c x y u v : Prop} [Decidable b] (h_c : b ↔ c
     (h_e : ¬c → (y ↔ v)) : ite b x y ↔ @ite _ c (decidable_of_decidable_of_iff h_c) u v :=
   if_ctx_congr_prop (dec_c := decidable_of_decidable_of_iff h_c) h_c h_t h_e
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem if_simp_congr_prop {b c x y u v : Prop} [Decidable b] (h_c : b ↔ c) (h_t : x ↔ u)
     -- FIXME: after https://github.com/leanprover/lean4/issues/1867 is fixed,
     -- this should be changed back to:
@@ -293,18 +231,7 @@ theorem if_simp_congr_prop {b c x y u v : Prop} [Decidable b] (h_c : b ↔ c) (h
     (h_e : y ↔ v) : ite b x y ↔ (@ite _ c (decidable_of_decidable_of_iff h_c) u v) :=
   if_ctx_simp_congr_prop h_c (fun _ ↦ h_t) (fun _ ↦ h_e)
 
--- @[congr]
-theorem dif_ctx_congr {α : Sort u} {b c : Prop} [dec_b : Decidable b] [dec_c : Decidable c]
-    {x : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
-    (h_c : b ↔ c) (h_t : ∀ h : c, x (Iff.mpr h_c h) = u h)
-    (h_e : ∀ h : ¬c, y (Iff.mpr (not_congr h_c) h) = v h) :
-    @dite α b dec_b x y = @dite α c dec_c u v :=
-  match dec_b, dec_c with
-  | isFalse _, isFalse h₂ => h_e h₂
-  | isTrue _, isTrue h₂ => h_t h₂
-  | isFalse h₁, isTrue h₂ => absurd h₂ (Iff.mp (not_congr h_c) h₁)
-  | isTrue h₁, isFalse h₂ => absurd h₁ (Iff.mpr (not_congr h_c) h₂)
-
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem dif_ctx_simp_congr {α : Sort u} {b c : Prop} [Decidable b]
     {x : b → α} {u : c → α} {y : ¬b → α} {v : ¬c → α}
     (h_c : b ↔ c) (h_t : ∀ h : c, x (Iff.mpr h_c h) = u h)
@@ -315,111 +242,31 @@ theorem dif_ctx_simp_congr {α : Sort u} {b c : Prop} [Decidable b]
     dite b x y = @dite _ c (decidable_of_decidable_of_iff h_c) u v :=
   dif_ctx_congr (dec_c := decidable_of_decidable_of_iff h_c) h_c h_t h_e
 
+@[deprecated (since := "2024-09-03")]
 def AsTrue (c : Prop) [Decidable c] : Prop := if c then True else False
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 def AsFalse (c : Prop) [Decidable c] : Prop := if c then False else True
 
+@[deprecated (since := "2024-09-03")]
 theorem AsTrue.get {c : Prop} [h₁ : Decidable c] (_ : AsTrue c) : c :=
   match h₁ with
   | isTrue h_c => h_c
 
 /- Equalities for rewriting let-expressions -/
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem let_value_eq {α : Sort u} {β : Sort v} {a₁ a₂ : α} (b : α → β)
     (h : a₁ = a₂) : (let x : α := a₁; b x) = (let x : α := a₂; b x) := congrArg b h
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem let_value_heq {α : Sort v} {β : α → Sort u} {a₁ a₂ : α} (b : ∀ x : α, β x)
     (h : a₁ = a₂) : HEq (let x : α := a₁; b x) (let x : α := a₂; b x) := by cases h; rfl
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem let_body_eq {α : Sort v} {β : α → Sort u} (a : α) {b₁ b₂ : ∀ x : α, β x}
     (h : ∀ x, b₁ x = b₂ x) : (let x : α := a; b₁ x) = (let x : α := a; b₂ x) := by exact h _ ▸ rfl
 
+@[deprecated (since := "2024-09-03")] -- unused in Mathlib
 theorem let_eq {α : Sort v} {β : Sort u} {a₁ a₂ : α} {b₁ b₂ : α → β}
     (h₁ : a₁ = a₂) (h₂ : ∀ x, b₁ x = b₂ x) :
     (let x : α := a₁; b₁ x) = (let x : α := a₂; b₂ x) := by simp [h₁, h₂]
-
-section Relation
-
-variable {α : Sort u} {β : Sort v} (r : β → β → Prop)
-
-/-- Local notation for an arbitrary binary relation `r`. -/
-local infix:50 " ≺ " => r
-
-/-- A reflexive relation relates every element to itself. -/
-def Reflexive := ∀ x, x ≺ x
-
-/-- A relation is symmetric if `x ≺ y` implies `y ≺ x`. -/
-def Symmetric := ∀ ⦃x y⦄, x ≺ y → y ≺ x
-
-/-- A relation is transitive if `x ≺ y` and `y ≺ z` together imply `x ≺ z`. -/
-def Transitive := ∀ ⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z
-
-lemma Equivalence.reflexive {r : β → β → Prop} (h : Equivalence r) : Reflexive r := h.refl
-
-lemma Equivalence.symmetric {r : β → β → Prop} (h : Equivalence r) : Symmetric r := fun _ _ ↦ h.symm
-
-lemma Equivalence.transitive {r : β → β → Prop} (h : Equivalence r) : Transitive r :=
-  fun _ _ _ ↦ h.trans
-
-/-- A relation is total if for all `x` and `y`, either `x ≺ y` or `y ≺ x`. -/
-def Total := ∀ x y, x ≺ y ∨ y ≺ x
-
-/-- Irreflexive means "not reflexive". -/
-def Irreflexive := ∀ x, ¬ x ≺ x
-
-/-- A relation is antisymmetric if `x ≺ y` and `y ≺ x` together imply that `x = y`. -/
-def AntiSymmetric := ∀ ⦃x y⦄, x ≺ y → y ≺ x → x = y
-
-/-- An empty relation does not relate any elements. -/
-@[nolint unusedArguments]
-def EmptyRelation := fun _ _ : α ↦ False
-
-theorem InvImage.trans (f : α → β) (h : Transitive r) : Transitive (InvImage r f) :=
-  fun (a₁ a₂ a₃ : α) (h₁ : InvImage r f a₁ a₂) (h₂ : InvImage r f a₂ a₃) ↦ h h₁ h₂
-
-theorem InvImage.irreflexive (f : α → β) (h : Irreflexive r) : Irreflexive (InvImage r f) :=
-  fun (a : α) (h₁ : InvImage r f a a) ↦ h (f a) h₁
-
-end Relation
-
-section Binary
-
-variable {α : Type u} {β : Type v} (f : α → α → α) (inv : α → α) (one : α)
-
-/-- Local notation for `f`, high priority to avoid ambiguity with `HMul.hMul`. -/
-local infix:70 (priority := high) " * " => f
-
-/-- Local notation for `inv`, high priority to avoid ambiguity with `Inv.inv`. -/
-local postfix:100 (priority := high) "⁻¹" => inv
-
-variable (g : α → α → α)
-
-/-- Local notation for `g`, high priority to avoid ambiguity with `HAdd.hAdd`. -/
-local infix:65 (priority := high) " + " => g
-
-def Commutative       := ∀ a b, a * b = b * a
-def Associative       := ∀ a b c, (a * b) * c = a * (b * c)
-def LeftIdentity      := ∀ a, one * a = a
-def RightIdentity     := ∀ a, a * one = a
-def RightInverse      := ∀ a, a * a⁻¹ = one
-def LeftCancelative   := ∀ a b c, a * b = a * c → b = c
-def RightCancelative  := ∀ a b c, a * b = c * b → a = c
-def LeftDistributive  := ∀ a b c, a * (b + c) = a * b + a * c
-def RightDistributive := ∀ a b c, (a + b) * c = a * c + b * c
-def RightCommutative (h : β → α → β) := ∀ b a₁ a₂, h (h b a₁) a₂ = h (h b a₂) a₁
-def LeftCommutative  (h : α → β → β) := ∀ a₁ a₂ b, h a₁ (h a₂ b) = h a₂ (h a₁ b)
-
-theorem left_comm : Commutative f → Associative f → LeftCommutative f :=
-  fun hcomm hassoc a b c ↦
-    calc  a*(b*c)
-      _ = (a*b)*c := Eq.symm (hassoc a b c)
-      _ = (b*a)*c := hcomm a b ▸ rfl
-      _ = b*(a*c) := hassoc b a c
-
-theorem right_comm : Commutative f → Associative f → RightCommutative f :=
-  fun hcomm hassoc a b c ↦
-    calc  (a*b)*c
-      _ = a*(b*c) := hassoc a b c
-      _ = a*(c*b) := hcomm b c ▸ rfl
-      _ = (a*c)*b := Eq.symm (hassoc a c b)
-
-end Binary
diff --git a/Mathlib/Lean/CoreM.lean b/Mathlib/Lean/CoreM.lean
index 9a1cceb3fcd0a..475e0a4f41f08 100644
--- a/Mathlib/Lean/CoreM.lean
+++ b/Mathlib/Lean/CoreM.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Tactic.ToExpr
 
diff --git a/Mathlib/Lean/Expr.lean b/Mathlib/Lean/Expr.lean
index f95a20189c59f..fa0ec85aea5d0 100644
--- a/Mathlib/Lean/Expr.lean
+++ b/Mathlib/Lean/Expr.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Robert Y. Lewis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Simon Hudon, Scott Morrison, Keeley Hoek, Robert Y. Lewis, Floris van Doorn
+Authors: Mario Carneiro, Simon Hudon, Kim Morrison, Keeley Hoek, Robert Y. Lewis, Floris van Doorn
 -/
 import Mathlib.Lean.Expr.Basic
 import Mathlib.Lean.Expr.ReplaceRec
diff --git a/Mathlib/Lean/Expr/Basic.lean b/Mathlib/Lean/Expr/Basic.lean
index 0a3a8fa1db16f..5991b42748293 100644
--- a/Mathlib/Lean/Expr/Basic.lean
+++ b/Mathlib/Lean/Expr/Basic.lean
@@ -1,14 +1,15 @@
 /-
 Copyright (c) 2019 Robert Y. Lewis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Simon Hudon, Scott Morrison, Keeley Hoek, Robert Y. Lewis,
+Authors: Mario Carneiro, Simon Hudon, Kim Morrison, Keeley Hoek, Robert Y. Lewis,
 Floris van Doorn, Edward Ayers, Arthur Paulino
 -/
 import Mathlib.Init
 import Lean.Meta.Tactic.Rewrite
 import Batteries.Lean.Expr
 import Batteries.Data.Rat.Basic
-import Batteries.Data.List.Basic
+import Batteries.Tactic.Alias
+import Lean.Elab.Binders
 
 /-!
 # Additional operations on Expr and related types
@@ -323,6 +324,13 @@ otherwise, it returns `none`. -/
   let (type, _, lhs, rhs) ← p.app4? ``LE.le
   return (type, lhs, rhs)
 
+/-- `Lean.Expr.lt? e` takes `e : Expr` as input.
+If `e` represents `a < b`, then it returns `some (t, a, b)`, where `t` is the Type of `a`,
+otherwise, it returns `none`. -/
+@[inline] def lt? (p : Expr) : Option (Expr × Expr × Expr) := do
+  let (type, _, lhs, rhs) ← p.app4? ``LT.lt
+  return (type, lhs, rhs)
+
 /-- Given a proposition `ty` that is an `Eq`, `Iff`, or `HEq`, returns `(tyLhs, lhs, tyRhs, rhs)`,
 where `lhs : tyLhs` and `rhs : tyRhs`,
 and where `lhs` is related to `rhs` by the respective relation.
diff --git a/Mathlib/Lean/Expr/ReplaceRec.lean b/Mathlib/Lean/Expr/ReplaceRec.lean
index 80f1f08324276..185dbe0ba8785 100644
--- a/Mathlib/Lean/Expr/ReplaceRec.lean
+++ b/Mathlib/Lean/Expr/ReplaceRec.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Robert Y. Lewis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Simon Hudon, Scott Morrison, Keeley Hoek, Robert Y. Lewis,
+Authors: Mario Carneiro, Simon Hudon, Kim Morrison, Keeley Hoek, Robert Y. Lewis,
 Floris van Doorn, Edward Ayers
 -/
 import Lean.Expr
diff --git a/Mathlib/Lean/LocalContext.lean b/Mathlib/Lean/LocalContext.lean
index 5bc891c5cdf0b..3c410ec6b63e6 100644
--- a/Mathlib/Lean/LocalContext.lean
+++ b/Mathlib/Lean/LocalContext.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.LocalContext
diff --git a/Mathlib/Lean/Meta/Basic.lean b/Mathlib/Lean/Meta/Basic.lean
index a8066da2f4022..71d36a1cdcc99 100644
--- a/Mathlib/Lean/Meta/Basic.lean
+++ b/Mathlib/Lean/Meta/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Meta.AppBuilder
diff --git a/Mathlib/Lean/Meta/CongrTheorems.lean b/Mathlib/Lean/Meta/CongrTheorems.lean
index fdc29cb221ec7..a8dafe249c20f 100644
--- a/Mathlib/Lean/Meta/CongrTheorems.lean
+++ b/Mathlib/Lean/Meta/CongrTheorems.lean
@@ -284,8 +284,8 @@ where
     let rec loop (i : Nat)
         (ftyx ftyy : Expr) (xs ys : Array Expr) (fixed' : Array Bool) : MetaM α := do
       if i < numVars then
-        let ftyx ← whnf ftyx
-        let ftyy ← whnf ftyy
+        let ftyx ← whnfD ftyx
+        let ftyy ← whnfD ftyy
         unless ftyx.isForall do
           throwError "doubleTelescope: function doesn't have enough parameters"
         withLocalDeclD ftyx.bindingName! ftyx.bindingDomain! fun fvarx => do
diff --git a/Mathlib/Lean/Meta/DiscrTree.lean b/Mathlib/Lean/Meta/DiscrTree.lean
index c20618862b96c..4698847cefc5a 100644
--- a/Mathlib/Lean/Meta/DiscrTree.lean
+++ b/Mathlib/Lean/Meta/DiscrTree.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Meta.DiscrTree
diff --git a/Mathlib/Lean/Meta/Simp.lean b/Mathlib/Lean/Meta/Simp.lean
index 19d5c973616f2..5f38f103a0dcc 100644
--- a/Mathlib/Lean/Meta/Simp.lean
+++ b/Mathlib/Lean/Meta/Simp.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Gabriel Ebner, Floris van Doorn
+Authors: Kim Morrison, Gabriel Ebner, Floris van Doorn
 -/
 import Mathlib.Init
 import Lean.Elab.Tactic.Simp
diff --git a/Mathlib/Lean/Name.lean b/Mathlib/Lean/Name.lean
index 268d785610641..4f3b4e06b54e3 100644
--- a/Mathlib/Lean/Name.lean
+++ b/Mathlib/Lean/Name.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Meta.Match.MatcherInfo
diff --git a/Mathlib/Lean/PrettyPrinter/Delaborator.lean b/Mathlib/Lean/PrettyPrinter/Delaborator.lean
index bcc547afa2239..557bdf947198a 100644
--- a/Mathlib/Lean/PrettyPrinter/Delaborator.lean
+++ b/Mathlib/Lean/PrettyPrinter/Delaborator.lean
@@ -14,13 +14,6 @@ namespace Lean.PrettyPrinter.Delaborator
 
 open Lean.Meta Lean.SubExpr SubExpr
 
-namespace SubExpr
-
-variable {α : Type} [Inhabited α]
-variable {m : Type → Type} [Monad m]
-
-end SubExpr
-
 /-- Assuming the current expression in a lambda or pi,
 descend into the body using an unused name generated from the binder's name.
 Provides `d` with both `Syntax` for the bound name as an identifier
diff --git a/Mathlib/Lean/Thunk.lean b/Mathlib/Lean/Thunk.lean
index e9d504dff0852..9ec29601a07ca 100644
--- a/Mathlib/Lean/Thunk.lean
+++ b/Mathlib/Lean/Thunk.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Simon Hudon
 -/
 import Mathlib.Init
-import Batteries.Data.Thunk
 
 /-!
 # Basic facts about `Thunk`.
diff --git a/Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean b/Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
index b66ea39b78a5a..eb8a41bbe9312 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
@@ -565,6 +565,13 @@ theorem lineMap_vsub_lineMap (p₁ p₂ p₃ p₄ : P1) (c : k) :
     lineMap p₁ p₂ c -ᵥ lineMap p₃ p₄ c = lineMap (p₁ -ᵥ p₃) (p₂ -ᵥ p₄) c :=
   ((fst : P1 × P1 →ᵃ[k] P1) -ᵥ (snd : P1 × P1 →ᵃ[k] P1)).apply_lineMap (_, _) (_, _) c
 
+@[simp] lemma lineMap_lineMap_right (p₀ p₁ : P1) (c d : k) :
+    lineMap p₀ (lineMap p₀ p₁ c) d = lineMap p₀ p₁ (d * c) := by simp [lineMap_apply, mul_smul]
+
+@[simp] lemma lineMap_lineMap_left (p₀ p₁ : P1) (c d : k) :
+    lineMap (lineMap p₀ p₁ c) p₁ d = lineMap p₀ p₁ (1 - (1 - d) * (1 - c)) := by
+  simp_rw [lineMap_apply_one_sub, ← lineMap_apply_one_sub p₁, lineMap_lineMap_right]
+
 /-- Decomposition of an affine map in the special case when the point space and vector space
 are the same. -/
 theorem decomp (f : V1 →ᵃ[k] V2) : (f : V1 → V2) = ⇑f.linear + fun _ => f 0 := by
diff --git a/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean b/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
index 3dff1b6db9f72..ab4b6063b2c23 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
@@ -230,7 +230,6 @@ theorem vadd_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s
   rw [hv]
   convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp
   rw [one_smul]
-  exact s.mem_coe k P _
 
 /-- Subtracting two points in the subspace produces a vector in the direction. -/
 theorem vsub_mem_direction {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
@@ -732,8 +731,10 @@ theorem card_pos_of_affineSpan_eq_top {ι : Type*} [Fintype ι] {p : ι → P}
 
 attribute [local instance] toAddTorsor
 
-/-- The top affine subspace is linearly equivalent to the affine space.
+-- An instance with better keys for the context
+instance : Nonempty (⊤ : AffineSubspace k P) := inferInstanceAs (Nonempty (⊤ : Set P))
 
+/-- The top affine subspace is linearly equivalent to the affine space.
 This is the affine version of `Submodule.topEquiv`. -/
 @[simps! linear apply symm_apply_coe]
 def topEquiv : (⊤ : AffineSubspace k P) ≃ᵃ[k] P where
@@ -1691,3 +1692,5 @@ theorem affineSpan_pair_parallel_iff_vectorSpan_eq {p₁ p₂ p₃ p₄ : P} :
     not_nonempty_iff_eq_empty]
 
 end AffineSubspace
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Basis.lean b/Mathlib/LinearAlgebra/AffineSpace/Basis.lean
index 878ac8fe1f82c..ee7f13ec2dba6 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Basis.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Basis.lean
@@ -311,7 +311,7 @@ instance [SMul G G'] [IsScalarTower G G' V] : IsScalarTower G G' (AffineBasis ι
 
 @[simp] lemma coord_smul (a : G) (b : AffineBasis ι k V) (i : ι) :
     (a • b).coord i = (b.coord i).comp (DistribMulAction.toLinearEquiv _ _ a).symm.toAffineMap := by
-  ext v; simp [coord]
+  ext v; simp [map_sub, coord]
 
 /-- TODO: generalize to include `SMul (P ≃ᵃ[k] P) (AffineBasis ι k P)`, which acts on `P` with a
 `VAdd` version of a `DistribMulAction`. -/
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Combination.lean b/Mathlib/LinearAlgebra/AffineSpace/Combination.lean
index ae32ab4bf5651..74aad257f2367 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Combination.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Combination.lean
@@ -903,13 +903,13 @@ theorem weightedVSub_mem_vectorSpan {s : Finset ι} {w : ι → k} (h : ∑ i 
     rcases isEmpty_or_nonempty ι with (hι | ⟨⟨i0⟩⟩)
     · simp [Finset.eq_empty_of_isEmpty s]
     · rw [vectorSpan_range_eq_span_range_vsub_right k p i0, ← Set.image_univ,
-        Finsupp.mem_span_image_iff_total,
+        Finsupp.mem_span_image_iff_linearCombination,
         Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero s w p h (p i0),
         Finset.weightedVSubOfPoint_apply]
       let w' := Set.indicator (↑s) w
       have hwx : ∀ i, w' i ≠ 0 → i ∈ s := fun i => Set.mem_of_indicator_ne_zero
       use Finsupp.onFinset s w' hwx, Set.subset_univ _
-      rw [Finsupp.total_apply, Finsupp.onFinset_sum hwx]
+      rw [Finsupp.linearCombination_apply, Finsupp.onFinset_sum hwx]
       · apply Finset.sum_congr rfl
         intro i hi
         simp [w', Set.indicator_apply, if_pos hi]
@@ -954,7 +954,7 @@ theorem mem_vectorSpan_iff_eq_weightedVSub {v : V} {p : ι → P} :
     · rcases isEmpty_or_nonempty ι with (hι | ⟨⟨i0⟩⟩)
       swap
       · rw [vectorSpan_range_eq_span_range_vsub_right k p i0, ← Set.image_univ,
-          Finsupp.mem_span_image_iff_total]
+          Finsupp.mem_span_image_iff_linearCombination]
         rintro ⟨l, _, hv⟩
         use insert i0 l.support
         set w :=
@@ -972,7 +972,7 @@ theorem mem_vectorSpan_iff_eq_weightedVSub {v : V} {p : ι → P} :
         have hz : w i0 • (p i0 -ᵥ p i0 : V) = 0 := (vsub_self (p i0)).symm ▸ smul_zero _
         change (fun i => w i • (p i -ᵥ p i0 : V)) i0 = 0 at hz
         rw [Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero _ w p hw (p i0),
-          Finset.weightedVSubOfPoint_apply, ← hv, Finsupp.total_apply,
+          Finset.weightedVSubOfPoint_apply, ← hv, Finsupp.linearCombination_apply,
           @Finset.sum_insert_zero _ _ l.support i0 _ _ _ hz]
         change (∑ i ∈ l.support, l i • _) = _
         congr with i
diff --git a/Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean b/Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean
index 546f23a3b7e1e..12ffbd6ace838 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean
@@ -179,7 +179,7 @@ theorem symm_symm (e : P₁ ≃ᵃL[k] P₂) : e.symm.symm = e := rfl
 theorem symm_symm_apply (e : P₁ ≃ᵃL[k] P₂) (x : P₁) : e.symm.symm x = e x :=
   rfl
 
-theorem symm_apply_eq (e : P₁ ≃ᵃL[k] P₂)  {x y} : e.symm x = y ↔ x = e y :=
+theorem symm_apply_eq (e : P₁ ≃ᵃL[k] P₂) {x y} : e.symm x = y ↔ x = e y :=
   e.toAffineEquiv.symm_apply_eq
 
 theorem eq_symm_apply (e : P₁ ≃ᵃL[k] P₂) {x y} : y = e.symm x ↔ e y = x :=
diff --git a/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean b/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
index e59a26e0c1ce5..935dc82abfb3b 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
@@ -29,14 +29,14 @@ section AffineSpace'
 variable (k : Type*) {V : Type*} {P : Type*}
 variable {ι : Type*}
 
-open AffineSubspace FiniteDimensional Module
+open AffineSubspace Module
 
 variable [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
 
 /-- The `vectorSpan` of a finite set is finite-dimensional. -/
 theorem finiteDimensional_vectorSpan_of_finite {s : Set P} (h : Set.Finite s) :
     FiniteDimensional k (vectorSpan k s) :=
-  span_of_finite k <| h.vsub h
+  .span_of_finite k <| h.vsub h
 
 /-- The `vectorSpan` of a family indexed by a `Fintype` is
 finite-dimensional. -/
@@ -202,7 +202,7 @@ theorem finrank_vectorSpan_le_iff_not_affineIndependent [Fintype ι] (p : ι →
 variable {k}
 
 lemma AffineIndependent.card_le_finrank_succ [Fintype ι] {p : ι → P} (hp : AffineIndependent k p) :
-    Fintype.card ι ≤ FiniteDimensional.finrank k (vectorSpan k (Set.range p)) + 1 := by
+    Fintype.card ι ≤ Module.finrank k (vectorSpan k (Set.range p)) + 1 := by
   cases isEmpty_or_nonempty ι
   · simp [Fintype.card_eq_zero]
   rw [← tsub_le_iff_right]
@@ -224,7 +224,7 @@ lemma AffineIndependent.card_le_card_of_subset_affineSpan {s t : Finset V}
   have direction_le := AffineSubspace.direction_le (affineSpan_mono k hst)
   rw [AffineSubspace.affineSpan_coe, direction_affineSpan, direction_affineSpan,
     ← @Subtype.range_coe _ (s : Set V), ← @Subtype.range_coe _ (t : Set V)] at direction_le
-  have finrank_le := add_le_add_right (Submodule.finrank_le_finrank_of_le direction_le) 1
+  have finrank_le := add_le_add_right (Submodule.finrank_mono direction_le) 1
   -- We use `erw` to elide the difference between `↥s` and `↥(s : Set V)}`
   erw [hs.finrank_vectorSpan_add_one] at finrank_le
   simpa using finrank_le.trans <| finrank_vectorSpan_range_add_one_le _ _
@@ -257,7 +257,7 @@ theorem AffineIndependent.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_ad
     (hi : AffineIndependent k p) {s : Finset ι} {sm : Submodule k V} [FiniteDimensional k sm]
     (hle : vectorSpan k (s.image p : Set P) ≤ sm) (hc : Finset.card s = finrank k sm + 1) :
     vectorSpan k (s.image p : Set P) = sm :=
-  eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan_image_finset hc
+  Submodule.eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan_image_finset hc
 
 /-- If the `vectorSpan` of a finite affinely independent
 family lies in a submodule with dimension one less than its
@@ -266,7 +266,7 @@ theorem AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one [Fintyp
     (hi : AffineIndependent k p) {sm : Submodule k V} [FiniteDimensional k sm]
     (hle : vectorSpan k (Set.range p) ≤ sm) (hc : Fintype.card ι = finrank k sm + 1) :
     vectorSpan k (Set.range p) = sm :=
-  eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan hc
+  Submodule.eq_of_le_of_finrank_eq hle <| hi.finrank_vectorSpan hc
 
 /-- If the `affineSpan` of a finite subset of an affinely independent
 family lies in an affine subspace whose direction has dimension one
@@ -371,7 +371,7 @@ alias ⟨Collinear.finrank_le_one, _⟩ := collinear_iff_finrank_le_one
 
 /-- A subset of a collinear set is collinear. -/
 theorem Collinear.subset {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Collinear k s₂) : Collinear k s₁ :=
-  (rank_le_of_submodule (vectorSpan k s₁) (vectorSpan k s₂) (vectorSpan_mono k hs)).trans h
+  (Submodule.rank_mono (vectorSpan_mono k hs)).trans h
 
 /-- The `vectorSpan` of collinear points is finite-dimensional. -/
 theorem Collinear.finiteDimensional_vectorSpan {s : Set P} (h : Collinear k s) :
@@ -634,7 +634,7 @@ alias ⟨Coplanar.finrank_le_two, _⟩ := coplanar_iff_finrank_le_two
 
 /-- A subset of a coplanar set is coplanar. -/
 theorem Coplanar.subset {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Coplanar k s₂) : Coplanar k s₁ :=
-  (rank_le_of_submodule (vectorSpan k s₁) (vectorSpan k s₂) (vectorSpan_mono k hs)).trans h
+  (Submodule.rank_mono (vectorSpan_mono k hs)).trans h
 
 /-- Collinear points are coplanar. -/
 theorem Collinear.coplanar {s : Set P} (h : Collinear k s) : Coplanar k s :=
@@ -669,7 +669,7 @@ section DivisionRing
 
 variable {k : Type*} {V : Type*} {P : Type*}
 
-open AffineSubspace FiniteDimensional Module
+open AffineSubspace Module Module
 
 variable [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
 
@@ -764,12 +764,12 @@ protected theorem finite_set [FiniteDimensional k V] {s : Set ι} (b : AffineBas
   finite_set_of_fin_dim_affineIndependent k b.ind
 
 theorem card_eq_finrank_add_one [Fintype ι] (b : AffineBasis ι k P) :
-    Fintype.card ι = FiniteDimensional.finrank k V + 1 :=
+    Fintype.card ι = Module.finrank k V + 1 :=
   have : FiniteDimensional k V := b.finiteDimensional
   b.ind.affineSpan_eq_top_iff_card_eq_finrank_add_one.mp b.tot
 
 theorem exists_affineBasis_of_finiteDimensional [Fintype ι] [FiniteDimensional k V]
-    (h : Fintype.card ι = FiniteDimensional.finrank k V + 1) : Nonempty (AffineBasis ι k P) := by
+    (h : Fintype.card ι = Module.finrank k V + 1) : Nonempty (AffineBasis ι k P) := by
   obtain ⟨s, b, hb⟩ := AffineBasis.exists_affineBasis k V P
   lift s to Finset P using b.finite_set
   refine ⟨b.reindex <| Fintype.equivOfCardEq ?_⟩
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Matrix.lean b/Mathlib/LinearAlgebra/AffineSpace/Matrix.lean
index 5ee4cb759178f..3ed1a0ba08f47 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Matrix.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Matrix.lean
@@ -61,7 +61,7 @@ theorem affineIndependent_of_toMatrix_right_inv [Fintype ι] [Finite ι'] [Decid
     ext j
     change (∑ i, w₁ i • b.coord j (p i)) = ∑ i, w₂ i • b.coord j (p i)
     -- Porting note: Added `u` because `∘` was causing trouble
-    have u : (fun i => b.coord j (p i)) = b.coord j ∘ p := by simp only [(· ∘ ·)]
+    have u : (fun i => b.coord j (p i)) = b.coord j ∘ p := by simp only [Function.comp_def]
     rw [← Finset.univ.affineCombination_eq_linear_combination _ _ hw₁,
       ← Finset.univ.affineCombination_eq_linear_combination _ _ hw₂, u,
       ← Finset.univ.map_affineCombination p w₁ hw₁, ← Finset.univ.map_affineCombination p w₂ hw₂,
diff --git a/Mathlib/LinearAlgebra/AffineSpace/Ordered.lean b/Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
index bc28ee54454cb..9f3c4c81c015f 100644
--- a/Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
+++ b/Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
@@ -134,15 +134,13 @@ theorem lineMap_le_right_iff_le (h : r < 1) : lineMap a b r ≤ b ↔ a ≤ b :=
   Iff.trans (by rw [lineMap_apply_one]) (lineMap_le_lineMap_iff_of_lt h)
 
 @[simp]
-theorem midpoint_le_right : midpoint k a b ≤ b ↔ a ≤ b :=
-  lineMap_le_right_iff_le <| inv_lt_one one_lt_two
+theorem midpoint_le_right : midpoint k a b ≤ b ↔ a ≤ b := lineMap_le_right_iff_le two_inv_lt_one
 
 theorem right_le_lineMap_iff_le (h : r < 1) : b ≤ lineMap a b r ↔ b ≤ a :=
   lineMap_le_right_iff_le (E := Eᵒᵈ) h
 
 @[simp]
-theorem right_le_midpoint : b ≤ midpoint k a b ↔ b ≤ a :=
-  right_le_lineMap_iff_le <| inv_lt_one one_lt_two
+theorem right_le_midpoint : b ≤ midpoint k a b ↔ b ≤ a := right_le_lineMap_iff_le two_inv_lt_one
 
 end
 
diff --git a/Mathlib/LinearAlgebra/Alternating/Basic.lean b/Mathlib/LinearAlgebra/Alternating/Basic.lean
index 90b2803f45d8e..007a27d357368 100644
--- a/Mathlib/LinearAlgebra/Alternating/Basic.lean
+++ b/Mathlib/LinearAlgebra/Alternating/Basic.lean
@@ -88,7 +88,7 @@ section Coercions
 
 instance instFunLike : FunLike (M [⋀^ι]→ₗ[R] N) (ι → M) N where
   coe f := f.toFun
-  coe_injective' := fun f g h ↦ by
+  coe_injective' f g h := by
     rcases f with ⟨⟨_, _, _⟩, _⟩
     rcases g with ⟨⟨_, _, _⟩, _⟩
     congr
@@ -603,8 +603,8 @@ theorem map_perm [DecidableEq ι] [Fintype ι] (v : ι → M) (σ : Equiv.Perm 
   -- Porting note: `apply` → `induction'`
   induction' σ using Equiv.Perm.swap_induction_on' with s x y hxy hI
   · simp
-  · -- Porting note: `← Function.comp.assoc` & `-Equiv.Perm.sign_swap'` are required.
-    simpa [← Function.comp.assoc, g.map_swap (v ∘ s) hxy,
+  · -- Porting note: `← Function.comp_assoc` & `-Equiv.Perm.sign_swap'` are required.
+    simpa [← Function.comp_assoc, g.map_swap (v ∘ s) hxy,
       Equiv.Perm.sign_swap hxy, -Equiv.Perm.sign_swap'] using hI
 
 theorem map_congr_perm [DecidableEq ι] [Fintype ι] (σ : Equiv.Perm ι) :
@@ -656,10 +656,10 @@ def domDomCongrEquiv (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃+ M [⋀^ι']
   invFun := domDomCongr σ.symm
   left_inv f := by
     ext
-    simp [Function.comp]
+    simp [Function.comp_def]
   right_inv m := by
     ext
-    simp [Function.comp]
+    simp [Function.comp_def]
   map_add' := domDomCongr_add σ
 
 section DomDomLcongr
@@ -671,8 +671,8 @@ variable (S : Type*) [Semiring S] [Module S N] [SMulCommClass R S N]
 def domDomCongrₗ (σ : ι ≃ ι') : M [⋀^ι]→ₗ[R] N ≃ₗ[S] M [⋀^ι']→ₗ[R] N where
   toFun := domDomCongr σ
   invFun := domDomCongr σ.symm
-  left_inv f := by ext; simp [Function.comp]
-  right_inv m := by ext; simp [Function.comp]
+  left_inv f := by ext; simp [Function.comp_def]
+  right_inv m := by ext; simp [Function.comp_def]
   map_add' := domDomCongr_add σ
   map_smul' := domDomCongr_smul σ
 
diff --git a/Mathlib/LinearAlgebra/Alternating/DomCoprod.lean b/Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
index ce50529aa2ab6..9f90f39e9ee1f 100644
--- a/Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
+++ b/Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
@@ -226,15 +226,15 @@ theorem MultilinearMap.domCoprod_alternization [DecidableEq ιa] [DecidableEq ι
   -- unfold the quotient mess left by `Finset.sum_partition`
   -- Porting note: Was `conv in .. => ..`.
   erw
-    [@Finset.filter_congr _ _ (fun a => @Quotient.decidableEq _ _
+    [@Finset.filter_congr _ _ _ (fun a => @Quotient.decidableEq _ _
       (QuotientGroup.leftRelDecidable (MonoidHom.range (Perm.sumCongrHom ιa ιb)))
       (Quotient.mk (QuotientGroup.leftRel (MonoidHom.range (Perm.sumCongrHom ιa ιb))) a)
       (Quotient.mk'' σ)) _ (s := Finset.univ)
     fun x _ => QuotientGroup.eq (s := MonoidHom.range (Perm.sumCongrHom ιa ιb)) (a := x) (b := σ)]
   -- eliminate a multiplication
   rw [← Finset.map_univ_equiv (Equiv.mulLeft σ), Finset.filter_map, Finset.sum_map]
-  simp_rw [Equiv.coe_toEmbedding, Equiv.coe_mulLeft, (· ∘ ·), mul_inv_rev, inv_mul_cancel_right,
-    Subgroup.inv_mem_iff, MonoidHom.mem_range, Finset.univ_filter_exists,
+  simp_rw [Equiv.coe_toEmbedding, Equiv.coe_mulLeft, Function.comp_def, mul_inv_rev,
+    inv_mul_cancel_right, Subgroup.inv_mem_iff, MonoidHom.mem_range, Finset.univ_filter_exists,
     Finset.sum_image Perm.sumCongrHom_injective.injOn]
   -- now we're ready to clean up the RHS, pulling out the summation
   rw [domCoprod.summand_mk'', MultilinearMap.domCoprod_alternization_coe, ← Finset.sum_product',
diff --git a/Mathlib/LinearAlgebra/Basis/Basic.lean b/Mathlib/LinearAlgebra/Basis/Basic.lean
index 2574523287e2d..46ed8282bcd9a 100644
--- a/Mathlib/LinearAlgebra/Basis/Basic.lean
+++ b/Mathlib/LinearAlgebra/Basis/Basic.lean
@@ -22,7 +22,7 @@ noncomputable section
 
 universe u
 
-open Function Set Submodule
+open Function Set Submodule Finsupp
 
 variable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {K : Type*}
 variable {M : Type*} {M' M'' : Type*} {V : Type u} {V' : Type*}
@@ -51,7 +51,7 @@ end Coord
 protected theorem linearIndependent : LinearIndependent R b :=
   linearIndependent_iff.mpr fun l hl =>
     calc
-      l = b.repr (Finsupp.total _ b l) := (b.repr_total l).symm
+      l = b.repr (Finsupp.linearCombination _ b l) := (b.repr_linearCombination l).symm
       _ = 0 := by rw [hl, LinearEquiv.map_zero]
 
 protected theorem ne_zero [Nontrivial R] (i) : b i ≠ 0 :=
@@ -125,7 +125,7 @@ protected theorem noZeroSMulDivisors [NoZeroDivisors R] (b : Basis ι R M) :
     NoZeroSMulDivisors R M :=
   ⟨fun {c x} hcx => by
     exact or_iff_not_imp_right.mpr fun hx => by
-      rw [← b.total_repr x, ← LinearMap.map_smul] at hcx
+      rw [← b.linearCombination_repr x, ← LinearMap.map_smul] at hcx
       have := linearIndependent_iff.mp b.linearIndependent (c • b.repr x) hcx
       rw [smul_eq_zero] at this
       exact this.resolve_right fun hr => hx (b.repr.map_eq_zero_iff.mp hr)⟩
@@ -203,17 +203,18 @@ theorem maximal [Nontrivial R] (b : Basis ι R M) : b.linearIndependent.Maximal
   intro x p
   by_contra q
   -- and write it in terms of the basis.
-  have e := b.total_repr x
+  have e := b.linearCombination_repr x
   -- This then expresses `x` as a linear combination
   -- of elements of `w` which are in the range of `b`,
   let u : ι ↪ w :=
     ⟨fun i => ⟨b i, h ⟨i, rfl⟩⟩, fun i i' r =>
       b.injective (by simpa only [Subtype.mk_eq_mk] using r)⟩
-  simp_rw [Finsupp.total_apply] at e
+  simp_rw [Finsupp.linearCombination_apply] at e
   change ((b.repr x).sum fun (i : ι) (a : R) ↦ a • (u i : M)) = ((⟨x, p⟩ : w) : M) at e
-  rw [← Finsupp.sum_embDomain (f := u) (g := fun x r ↦ r • (x : M)), ← Finsupp.total_apply] at e
+  rw [← Finsupp.sum_embDomain (f := u) (g := fun x r ↦ r • (x : M)),
+      ← Finsupp.linearCombination_apply] at e
   -- Now we can contradict the linear independence of `hi`
-  refine hi.total_ne_of_not_mem_support _ ?_ e
+  refine hi.linearCombination_ne_of_not_mem_support _ ?_ e
   simp only [Finset.mem_map, Finsupp.support_embDomain]
   rintro ⟨j, -, W⟩
   simp only [u, Embedding.coeFn_mk, Subtype.mk_eq_mk] at W
@@ -227,8 +228,8 @@ variable (hli : LinearIndependent R v) (hsp : ⊤ ≤ span R (range v))
 protected noncomputable def mk : Basis ι R M :=
   .ofRepr
     { hli.repr.comp (LinearMap.id.codRestrict _ fun _ => hsp Submodule.mem_top) with
-      invFun := Finsupp.total _ v
-      left_inv := fun x => hli.total_repr ⟨x, _⟩
+      invFun := Finsupp.linearCombination _ v
+      left_inv := fun x => hli.linearCombination_repr ⟨x, _⟩
       right_inv := fun _ => hli.repr_eq rfl }
 
 @[simp]
@@ -236,7 +237,7 @@ theorem mk_repr : (Basis.mk hli hsp).repr x = hli.repr ⟨x, hsp Submodule.mem_t
   rfl
 
 theorem mk_apply (i : ι) : Basis.mk hli hsp i = v i :=
-  show Finsupp.total _ v _ = v i by simp
+  show Finsupp.linearCombination _ v _ = v i by simp
 
 @[simp]
 theorem coe_mk : ⇑(Basis.mk hli hsp) = v :=
@@ -279,7 +280,7 @@ protected noncomputable def span : Basis ι R (span R (range v)) :=
       rfl
     have h₂ : map (Submodule.subtype (span R (range v))) (span R (range fun i => ⟨v i, this i⟩)) =
         span R (range v) := by
-      rw [← span_image, Submodule.coeSubtype]
+      rw [← span_image, Submodule.coe_subtype]
       -- Porting note: why doesn't `rw [h₁]` work here?
       exact congr_arg _ h₁
     have h₃ : (x : M) ∈ map (Submodule.subtype (span R (range v)))
@@ -492,28 +493,31 @@ lemma Basis.mem_center_iff {A}
     constructor
     case comm =>
       intro y
-      rw [← b.total_repr y, Finsupp.total_apply, Finsupp.sum, Finset.sum_mul, Finset.mul_sum]
+      rw [← b.linearCombination_repr y, linearCombination_apply, sum, Finset.sum_mul,
+          Finset.mul_sum]
       simp_rw [mul_smul_comm, smul_mul_assoc, (h.1 _).eq]
     case left_assoc =>
       intro c d
-      rw [← b.total_repr c, ← b.total_repr d, Finsupp.total_apply, Finsupp.total_apply, Finsupp.sum,
-        Finsupp.sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
+      rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
+          linearCombination_apply, sum, sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum,
+          Finset.mul_sum]
       simp_rw [smul_mul_assoc, Finset.mul_sum, Finset.sum_mul, mul_smul_comm, Finset.mul_sum,
         Finset.smul_sum, smul_mul_assoc, mul_smul_comm, (h.2 _ _).1,
         (@SMulCommClass.smul_comm R R A)]
       rw [Finset.sum_comm]
     case mid_assoc =>
       intro c d
-      rw [← b.total_repr c, ← b.total_repr d, Finsupp.total_apply, Finsupp.total_apply, Finsupp.sum,
-        Finsupp.sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum, Finset.mul_sum]
+      rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
+          linearCombination_apply, sum, sum, Finset.sum_mul, Finset.mul_sum, Finset.mul_sum,
+          Finset.mul_sum]
       simp_rw [smul_mul_assoc, Finset.sum_mul, mul_smul_comm, smul_mul_assoc, (h.2 _ _).2.1]
     case right_assoc =>
       intro c d
-      rw [← b.total_repr c, ← b.total_repr d, Finsupp.total_apply, Finsupp.total_apply, Finsupp.sum,
-        Finsupp.sum, Finset.sum_mul]
+      rw [← b.linearCombination_repr c, ← b.linearCombination_repr d, linearCombination_apply,
+          linearCombination_apply, sum, Finsupp.sum, Finset.sum_mul]
       simp_rw [smul_mul_assoc, Finset.mul_sum, Finset.sum_mul, mul_smul_comm, Finset.mul_sum,
-        Finset.smul_sum, smul_mul_assoc, mul_smul_comm, Finset.sum_mul, smul_mul_assoc,
-        (h.2 _ _).2.2]
+               Finset.smul_sum, smul_mul_assoc, mul_smul_comm, Finset.sum_mul, smul_mul_assoc,
+               (h.2 _ _).2.2]
 
 section RestrictScalars
 
@@ -553,7 +557,7 @@ theorem Basis.mem_span_iff_repr_mem (m : M) :
   refine
     ⟨fun hm i => ⟨(b.restrictScalars R).repr ⟨m, hm⟩ i, b.restrictScalars_repr_apply R ⟨m, hm⟩ i⟩,
       fun h => ?_⟩
-  rw [← b.total_repr m, Finsupp.total_apply S _]
+  rw [← b.linearCombination_repr m, Finsupp.linearCombination_apply S _]
   refine sum_mem fun i _ => ?_
   obtain ⟨_, h⟩ := h i
   simp_rw [← h, algebraMap_smul]
diff --git a/Mathlib/LinearAlgebra/Basis/Bilinear.lean b/Mathlib/LinearAlgebra/Basis/Bilinear.lean
index dd756b57e5821..ec749b417a941 100644
--- a/Mathlib/LinearAlgebra/Basis/Bilinear.lean
+++ b/Mathlib/LinearAlgebra/Basis/Bilinear.lean
@@ -41,8 +41,8 @@ Version for semi-bilinear maps, see `sum_repr_mul_repr_mul` for the bilinear ver
 theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) :
     ((b₁.repr x).sum fun i xi => (b₂.repr y).sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) =
       B x y := by
-  conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y]
-  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
+  conv_rhs => rw [← b₁.linearCombination_repr x, ← b₂.linearCombination_repr y]
+  simp_rw [Finsupp.linearCombination_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
     LinearMap.map_smulₛₗ]
 
 /-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis.
@@ -51,8 +51,8 @@ Version for bilinear maps, see `sum_repr_mul_repr_mulₛₗ` for the semi-biline
 theorem sum_repr_mul_repr_mul {B : Mₗ →ₗ[Rₗ] Nₗ →ₗ[Rₗ] Pₗ} (x y) :
     ((b₁'.repr x).sum fun i xi => (b₂'.repr y).sum fun j yj => xi • yj • B (b₁' i) (b₂' j)) =
       B x y := by
-  conv_rhs => rw [← b₁'.total_repr x, ← b₂'.total_repr y]
-  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smul₂,
+  conv_rhs => rw [← b₁'.linearCombination_repr x, ← b₂'.linearCombination_repr y]
+  simp_rw [Finsupp.linearCombination_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smul₂,
     LinearMap.map_smul]
 
 end AddCommMonoid
diff --git a/Mathlib/LinearAlgebra/Basis/Cardinality.lean b/Mathlib/LinearAlgebra/Basis/Cardinality.lean
index 496f90d3f98e8..284ad5ed30aa4 100644
--- a/Mathlib/LinearAlgebra/Basis/Cardinality.lean
+++ b/Mathlib/LinearAlgebra/Basis/Cardinality.lean
@@ -3,7 +3,8 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Kim Morrison
 -/
-import Mathlib.LinearAlgebra.Basis.Basic
+import Mathlib.LinearAlgebra.Basis.Defs
+import Mathlib.LinearAlgebra.LinearIndependent
 import Mathlib.SetTheory.Cardinal.Cofinality
 
 /-!
@@ -12,13 +13,15 @@ import Mathlib.SetTheory.Cardinal.Cofinality
 
 section Finite
 
-open Basis Cardinal Set Submodule
+open Basis Cardinal Set Submodule Finsupp
 
-universe u v v' v'' u₁' w w'
+universe u v w w'
 
-variable {R : Type u} {M M₁ : Type v} {M' : Type v'} {ι : Type w}
-variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁] [Nontrivial R]
-variable [Module R M] [Module R M'] [Module R M₁]
+variable {R : Type u} {M : Type v}
+
+section Semiring
+
+variable [Semiring R] [AddCommGroup M] [Nontrivial R] [Module R M]
 
 -- One might hope that a finite spanning set implies that any linearly independent set is finite.
 -- While this is true over a division ring
@@ -49,9 +52,9 @@ lemma basis_finite_of_finite_spans (w : Set M) (hw : w.Finite) (s : span R w = 
   let bS : Set M := b '' S
   have h : ∀ x ∈ w, x ∈ span R bS := by
     intro x m
-    rw [← b.total_repr x, Finsupp.span_image_eq_map_total, Submodule.mem_map]
+    rw [← b.linearCombination_repr x, span_image_eq_map_linearCombination, Submodule.mem_map]
     use b.repr x
-    simp only [and_true_iff, eq_self_iff_true, Finsupp.mem_supported]
+    simp only [and_true, eq_self_iff_true, Finsupp.mem_supported]
     rw [Finset.coe_subset, ← Finset.le_iff_subset]
     exact Finset.le_sup (f := fun x : w ↦ (b.repr ↑x).support) (Finset.mem_univ (⟨x, m⟩ : w))
   -- Thus this finite subset of the basis elements spans the entire module.
@@ -63,7 +66,14 @@ lemma basis_finite_of_finite_spans (w : Set M) (hw : w.Finite) (s : span R w = 
     rw [k]
     exact mem_top
   -- giving the desire contradiction.
-  exact b.linearIndependent.not_mem_span_image nm k'
+  simp only [self_mem_span_image, Finset.mem_coe, bS] at k'
+  exact nm k'
+
+end Semiring
+
+section Ring
+
+variable [Ring R] [AddCommGroup M] [Nontrivial R] [Module R M]
 
 -- From [Les familles libres maximales d'un module ont-elles le meme cardinal?][lazarus1973]
 /-- Over any ring `R`, if `b` is a basis for a module `M`,
@@ -101,9 +111,9 @@ theorem union_support_maximal_linearIndependent_eq_range_basis {ι : Type w} (b
     apply LinearIndependent.to_subtype_range
     rw [linearIndependent_iff]
     intro l z
-    rw [Finsupp.total_option] at z
+    rw [Finsupp.linearCombination_option] at z
     simp only [v', Option.elim'] at z
-    change _ + Finsupp.total R v l.some = 0 at z
+    change _ + Finsupp.linearCombination R v l.some = 0 at z
     -- We have some linear combination of `b b'` and the `v i`, which we want to show is trivial.
     -- We'll first show the coefficient of `b b'` is zero,
     -- by expressing the `v i` in the basis `b`, and using that the `v i` have no `b b'` term.
@@ -113,8 +123,9 @@ theorem union_support_maximal_linearIndependent_eq_range_basis {ι : Type w} (b
       apply_fun fun x => b.repr x b' at z
       simp only [repr_self, map_smul, mul_one, Finsupp.single_eq_same, Pi.neg_apply,
         Finsupp.smul_single', map_neg, Finsupp.coe_neg] at z
-      erw [DFunLike.congr_fun (Finsupp.apply_total R (b.repr : M →ₗ[R] ι →₀ R) v l.some) b'] at z
-      simpa [Finsupp.total_apply, w] using z
+      erw [DFunLike.congr_fun (apply_linearCombination R (b.repr : M →ₗ[R] ι →₀ R) v l.some) b']
+        at z
+      simpa [Finsupp.linearCombination_apply, w] using z
     -- Then all the other coefficients are zero, because `v` is linear independent.
     have l₁ : l.some = 0 := by
       rw [l₀, zero_smul, zero_add] at z
@@ -152,4 +163,6 @@ theorem infinite_basis_le_maximal_linearIndependent {ι : Type w} (b : Basis ι
     {κ : Type w} (v : κ → M) (i : LinearIndependent R v) (m : i.Maximal) : #ι ≤ #κ :=
   Cardinal.lift_le.mp (infinite_basis_le_maximal_linearIndependent' b v i m)
 
+end Ring
+
 end Finite
diff --git a/Mathlib/LinearAlgebra/Basis/Defs.lean b/Mathlib/LinearAlgebra/Basis/Defs.lean
index f75109e649df3..c731874f77c56 100644
--- a/Mathlib/LinearAlgebra/Basis/Defs.lean
+++ b/Mathlib/LinearAlgebra/Basis/Defs.lean
@@ -59,7 +59,7 @@ noncomputable section
 
 universe u
 
-open Function Set Submodule
+open Function Set Submodule Finsupp
 
 variable {ι : Type*} {ι' : Type*} {R : Type*} {R₂ : Type*} {K : Type*}
 variable {M : Type*} {M' M'' : Type*} {V : Type u} {V' : Type*}
@@ -134,36 +134,42 @@ theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then
   rw [repr_self, Finsupp.single_apply]
 
 @[simp]
-theorem repr_symm_apply (v) : b.repr.symm v = Finsupp.total R b v :=
+theorem repr_symm_apply (v) : b.repr.symm v = Finsupp.linearCombination R b v :=
   calc
     b.repr.symm v = b.repr.symm (v.sum Finsupp.single) := by simp
     _ = v.sum fun i vi => b.repr.symm (Finsupp.single i vi) := map_finsupp_sum ..
-    _ = Finsupp.total R b v := by simp only [repr_symm_single, Finsupp.total_apply]
+    _ = Finsupp.linearCombination R b v := by simp only [repr_symm_single,
+                                                         Finsupp.linearCombination_apply]
 
 @[simp]
-theorem coe_repr_symm : ↑b.repr.symm = Finsupp.total R b :=
+theorem coe_repr_symm : ↑b.repr.symm = Finsupp.linearCombination R b :=
   LinearMap.ext fun v => b.repr_symm_apply v
 
 @[simp]
-theorem repr_total (v) : b.repr (Finsupp.total _ b v) = v := by
+theorem repr_linearCombination (v) : b.repr (Finsupp.linearCombination _ b v) = v := by
   rw [← b.coe_repr_symm]
   exact b.repr.apply_symm_apply v
 
+@[deprecated (since := "2024-08-29")] alias repr_total := repr_linearCombination
+
 @[simp]
-theorem total_repr : Finsupp.total _ b (b.repr x) = x := by
+theorem linearCombination_repr : Finsupp.linearCombination _ b (b.repr x) = x := by
   rw [← b.coe_repr_symm]
   exact b.repr.symm_apply_apply x
 
+@[deprecated (since := "2024-08-29")] alias total_repr := linearCombination_repr
+
 theorem repr_range : LinearMap.range (b.repr : M →ₗ[R] ι →₀ R) = Finsupp.supported R R univ := by
   rw [LinearEquiv.range, Finsupp.supported_univ]
 
 theorem mem_span_repr_support (m : M) : m ∈ span R (b '' (b.repr m).support) :=
-  (Finsupp.mem_span_image_iff_total _).2 ⟨b.repr m, by simp [Finsupp.mem_supported_support]⟩
+  (Finsupp.mem_span_image_iff_linearCombination _).2
+    ⟨b.repr m, by simp [Finsupp.mem_supported_support]⟩
 
 theorem repr_support_subset_of_mem_span (s : Set ι) {m : M}
     (hm : m ∈ span R (b '' s)) : ↑(b.repr m).support ⊆ s := by
-  rcases (Finsupp.mem_span_image_iff_total _).1 hm with ⟨l, hl, rfl⟩
-  rwa [repr_total, ← Finsupp.mem_supported R l]
+  rcases (Finsupp.mem_span_image_iff_linearCombination _).1 hm with ⟨l, hl, rfl⟩
+  rwa [repr_linearCombination, ← Finsupp.mem_supported R l]
 
 theorem mem_span_image {m : M} {s : Set ι} : m ∈ span R (b '' s) ↔ ↑(b.repr m).support ⊆ s :=
   ⟨repr_support_subset_of_mem_span _ _, fun h ↦
@@ -219,7 +225,7 @@ theorem dvd_coord_smul (i : ι) (m : M) (r : R) : r ∣ b.coord i (r • m) :=
 
 theorem coord_repr_symm (b : Basis ι R M) (i : ι) (f : ι →₀ R) :
     b.coord i (b.repr.symm f) = f i := by
-  simp only [repr_symm_apply, coord_apply, repr_total]
+  simp only [repr_symm_apply, coord_apply, repr_linearCombination]
 
 end Coord
 
@@ -232,13 +238,13 @@ variable {M₁ : Type*} [AddCommMonoid M₁] [Module R₁ M₁]
 /-- Two linear maps are equal if they are equal on basis vectors. -/
 theorem ext {f₁ f₂ : M →ₛₗ[σ] M₁} (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ := by
   ext x
-  rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]
+  rw [← b.linearCombination_repr x, Finsupp.linearCombination_apply, Finsupp.sum]
   simp only [map_sum, LinearMap.map_smulₛₗ, h]
 
 /-- Two linear equivs are equal if they are equal on basis vectors. -/
 theorem ext' {f₁ f₂ : M ≃ₛₗ[σ] M₁} (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ := by
   ext x
-  rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]
+  rw [← b.linearCombination_repr x, Finsupp.linearCombination_apply, Finsupp.sum]
   simp only [map_sum, LinearEquiv.map_smulₛₗ, h]
 
 /-- Two elements are equal iff their coordinates are equal. -/
@@ -532,8 +538,8 @@ theorem mem_submodule_iff {P : Submodule R M} (b : Basis ι R P) {x : M} :
     x ∈ P ↔ ∃ c : ι →₀ R, x = Finsupp.sum c fun i x => x • (b i : M) := by
   conv_lhs =>
     rw [← P.range_subtype, ← Submodule.map_top, ← b.span_eq, Submodule.map_span, ← Set.range_comp,
-        ← Finsupp.range_total]
-  simp [@eq_comm _ x, Function.comp, Finsupp.total_apply]
+        ← Finsupp.range_linearCombination]
+  simp [@eq_comm _ x, Function.comp, Finsupp.linearCombination_apply]
 
 section Constr
 
@@ -551,7 +557,7 @@ you can recover an `AddEquiv` by setting `S := ℕ`.
 See library note [bundled maps over different rings].
 -/
 def constr : (ι → M') ≃ₗ[S] M →ₗ[R] M' where
-  toFun f := (Finsupp.total R id).comp <| Finsupp.lmapDomain R R f ∘ₗ ↑b.repr
+  toFun f := (Finsupp.linearCombination R id).comp <| Finsupp.lmapDomain R R f ∘ₗ ↑b.repr
   invFun f i := f (b i)
   left_inv f := by
     ext
@@ -567,12 +573,12 @@ def constr : (ι → M') ≃ₗ[S] M →ₗ[R] M' where
     simp
 
 theorem constr_def (f : ι → M') :
-    constr (M' := M') b S f = Finsupp.total R id ∘ₗ Finsupp.lmapDomain R R f ∘ₗ ↑b.repr :=
+    constr (M' := M') b S f = linearCombination R id ∘ₗ Finsupp.lmapDomain R R f ∘ₗ ↑b.repr :=
   rfl
 
 theorem constr_apply (f : ι → M') (x : M) :
     constr (M' := M') b S f x = (b.repr x).sum fun b a => a • f b := by
-  simp only [constr_def, LinearMap.comp_apply, Finsupp.lmapDomain_apply, Finsupp.total_apply]
+  simp only [constr_def, LinearMap.comp_apply, lmapDomain_apply, linearCombination_apply]
   rw [Finsupp.sum_mapDomain_index] <;> simp [add_smul]
 
 @[simp]
@@ -590,7 +596,7 @@ theorem constr_range {f : ι → M'} :
     LinearMap.range (constr (M' := M') b S f) = span R (range f) := by
   rw [b.constr_def S f, LinearMap.range_comp, LinearMap.range_comp, LinearEquiv.range, ←
     Finsupp.supported_univ, Finsupp.lmapDomain_supported, ← Set.image_univ, ←
-    Finsupp.span_image_eq_map_total, Set.image_id]
+    Finsupp.span_image_eq_map_linearCombination, Set.image_id]
 
 @[simp]
 theorem constr_comp (f : M' →ₗ[R] M') (v : ι → M') :
@@ -704,7 +710,7 @@ a function `x : ι → R` to the linear combination `∑_i x i • v i`. -/
 @[simp]
 theorem Basis.equivFun_symm_apply [Fintype ι] (b : Basis ι R M) (x : ι → R) :
     b.equivFun.symm x = ∑ i, x i • b i := by
-  simp [Basis.equivFun, Finsupp.total_apply, Finsupp.sum_fintype, Finsupp.equivFunOnFinite]
+  simp [Basis.equivFun, Finsupp.linearCombination_apply, sum_fintype, equivFunOnFinite]
 
 @[simp]
 theorem Basis.equivFun_apply [Finite ι] (b : Basis ι R M) (u : M) : b.equivFun u = b.repr u :=
diff --git a/Mathlib/LinearAlgebra/Basis/VectorSpace.lean b/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
index 5ffcc1cc36b43..e7a219e569dab 100644
--- a/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
+++ b/Mathlib/LinearAlgebra/Basis/VectorSpace.lean
@@ -86,6 +86,61 @@ theorem subset_extend {s : Set V} (hs : LinearIndependent K ((↑) : s → V)) :
     s ⊆ hs.extend (Set.subset_univ _) :=
   hs.subset_extend _
 
+/-- If `s` is a family of linearly independent vectors contained in a set `t` spanning `V`,
+then one can get a basis of `V` containing `s` and contained in `t`. -/
+noncomputable def extendLe (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) :
+    Basis (hs.extend hst) K V :=
+  Basis.mk
+    (@LinearIndependent.restrict_of_comp_subtype _ _ _ id _ _ _ _ (hs.linearIndependent_extend _))
+    (le_trans ht <| Submodule.span_le.2 <| by simpa using hs.subset_span_extend hst)
+
+theorem extendLe_apply_self (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) (x : hs.extend hst) :
+    Basis.extendLe hs hst ht x = x :=
+  Basis.mk_apply _ _ _
+
+@[simp]
+theorem coe_extendLe (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) : ⇑(Basis.extendLe hs hst ht) = ((↑) : _ → _) :=
+  funext (extendLe_apply_self hs hst ht)
+
+theorem range_extendLe (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) :
+    range (Basis.extendLe hs hst ht) = hs.extend hst := by
+  rw [coe_extendLe, Subtype.range_coe_subtype, setOf_mem_eq]
+
+theorem subset_extendLe (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) :
+    s ⊆ range (Basis.extendLe hs hst ht) :=
+  (range_extendLe hs hst ht).symm ▸ hs.subset_extend hst
+
+theorem extendLe_subset (hs : LinearIndependent K ((↑) : s → V))
+    (hst : s ⊆ t) (ht : ⊤ ≤ span K t) :
+    range (Basis.extendLe hs hst ht) ⊆ t :=
+  (range_extendLe hs hst ht).symm ▸ hs.extend_subset hst
+
+/-- If a set `s` spans the space, this is a basis contained in `s`. -/
+noncomputable def ofSpan (hs : ⊤ ≤ span K s) :
+    Basis ((linearIndependent_empty K V).extend (empty_subset s)) K V :=
+  extendLe (linearIndependent_empty K V) (empty_subset s) hs
+
+theorem ofSpan_apply_self (hs : ⊤ ≤ span K s)
+    (x : (linearIndependent_empty K V).extend (empty_subset s)) :
+    Basis.ofSpan hs x = x :=
+  extendLe_apply_self (linearIndependent_empty K V) (empty_subset s) hs x
+
+@[simp]
+theorem coe_ofSpan (hs : ⊤ ≤ span K s) : ⇑(ofSpan hs) = ((↑) : _ → _) :=
+  funext (ofSpan_apply_self hs)
+
+theorem range_ofSpan (hs : ⊤ ≤ span K s) :
+    range (ofSpan hs) = (linearIndependent_empty K V).extend (empty_subset s) := by
+  rw [coe_ofSpan, Subtype.range_coe_subtype, setOf_mem_eq]
+
+theorem ofSpan_subset (hs : ⊤ ≤ span K s) : range (ofSpan hs) ⊆ s :=
+  extendLe_subset (linearIndependent_empty K V) (empty_subset s) hs
+
 section
 
 variable (K V)
diff --git a/Mathlib/LinearAlgebra/BilinearForm/Hom.lean b/Mathlib/LinearAlgebra/BilinearForm/Hom.lean
index 9e99d5b63ae90..f32f28e8aea31 100644
--- a/Mathlib/LinearAlgebra/BilinearForm/Hom.lean
+++ b/Mathlib/LinearAlgebra/BilinearForm/Hom.lean
@@ -347,8 +347,8 @@ theorem ext_basis (h : ∀ i j, B (b i) (b j) = F₂ (b i) (b j)) : B = F₂ :=
 /-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis. -/
 theorem sum_repr_mul_repr_mul (x y : M) :
     ((b.repr x).sum fun i xi => (b.repr y).sum fun j yj => xi • yj • B (b i) (b j)) = B x y := by
-  conv_rhs => rw [← b.total_repr x, ← b.total_repr y]
-  simp_rw [Finsupp.total_apply, Finsupp.sum, sum_left, sum_right, smul_left, smul_right,
+  conv_rhs => rw [← b.linearCombination_repr x, ← b.linearCombination_repr y]
+  simp_rw [Finsupp.linearCombination_apply, Finsupp.sum, sum_left, sum_right, smul_left, smul_right,
     smul_eq_mul]
 
 end Basis
diff --git a/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean b/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
index fb7036771789e..bff969f2c534c 100644
--- a/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
+++ b/Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean
@@ -217,7 +217,7 @@ theorem iIsOrtho.not_isOrtho_basis_self_of_nondegenerate {n : Type w} [Nontrivia
   intro ho
   refine v.ne_zero i (hB (v i) fun m => ?_)
   obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m
-  rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, sum_right]
+  rw [Basis.repr_symm_apply, Finsupp.linearCombination_apply, Finsupp.sum, sum_right]
   apply Finset.sum_eq_zero
   rintro j -
   rw [smul_right]
@@ -237,7 +237,8 @@ theorem iIsOrtho.nondegenerate_iff_not_isOrtho_basis_self {n : Type w} [Nontrivi
   ext i
   rw [Finsupp.zero_apply]
   specialize hB (v i)
-  simp_rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, sum_left, smul_left] at hB
+  simp_rw [Basis.repr_symm_apply, Finsupp.linearCombination_apply, Finsupp.sum, sum_left,
+           smul_left] at hB
   rw [Finset.sum_eq_single i] at hB
   · exact eq_zero_of_ne_zero_of_mul_right_eq_zero (ho i) hB
   · intro j _ hij
@@ -282,7 +283,7 @@ lemma ker_restrict_eq_of_codisjoint {p q : Submodule R M} (hpq : Codisjoint p q)
     {B : LinearMap.BilinForm R M} (hB : ∀ x ∈ p, ∀ y ∈ q, B x y = 0) :
     LinearMap.ker (B.restrict p) = (LinearMap.ker B).comap p.subtype := by
   ext ⟨z, hz⟩
-  simp only [LinearMap.mem_ker, Submodule.mem_comap, Submodule.coeSubtype]
+  simp only [LinearMap.mem_ker, Submodule.mem_comap, Submodule.coe_subtype]
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · ext w
     obtain ⟨x, hx, y, hy, rfl⟩ := Submodule.exists_add_eq_of_codisjoint hpq w
@@ -293,7 +294,7 @@ lemma ker_restrict_eq_of_codisjoint {p q : Submodule R M} (hpq : Codisjoint p q)
 lemma inf_orthogonal_self_le_ker_restrict {W : Submodule R M} (b₁ : B.IsRefl) :
     W ⊓ B.orthogonal W ≤ (LinearMap.ker <| B.restrict W).map W.subtype := by
   rintro v ⟨hv : v ∈ W, hv' : v ∈ B.orthogonal W⟩
-  simp only [Submodule.mem_map, mem_ker, restrict_apply, Submodule.coeSubtype, Subtype.exists,
+  simp only [Submodule.mem_map, mem_ker, restrict_apply, Submodule.coe_subtype, Subtype.exists,
     exists_and_left, exists_prop, exists_eq_right_right]
   refine ⟨?_, hv⟩
   ext ⟨w, hw⟩
@@ -301,7 +302,7 @@ lemma inf_orthogonal_self_le_ker_restrict {W : Submodule R M} (b₁ : B.IsRefl)
 
 variable [FiniteDimensional K V]
 
-open FiniteDimensional Submodule
+open Module Submodule
 
 variable {B : BilinForm K V}
 
diff --git a/Mathlib/LinearAlgebra/BilinearMap.lean b/Mathlib/LinearAlgebra/BilinearMap.lean
index b9e49faf0456c..10c008423626e 100644
--- a/Mathlib/LinearAlgebra/BilinearMap.lean
+++ b/Mathlib/LinearAlgebra/BilinearMap.lean
@@ -37,16 +37,15 @@ variable {R : Type*} [Semiring R] {S : Type*} [Semiring S]
 variable {R₂ : Type*} [Semiring R₂] {S₂ : Type*} [Semiring S₂]
 variable {M : Type*} {N : Type*} {P : Type*}
 variable {M₂ : Type*} {N₂ : Type*} {P₂ : Type*}
-variable {Nₗ : Type*} {Pₗ : Type*}
-variable {M' : Type*} {N' : Type*} {P' : Type*}
+variable {Pₗ : Type*}
+variable {M' : Type*} {P' : Type*}
 variable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]
-variable [AddCommMonoid M₂] [AddCommMonoid N₂] [AddCommMonoid P₂]
-variable [AddCommMonoid Nₗ] [AddCommMonoid Pₗ]
-variable [AddCommGroup M'] [AddCommGroup N'] [AddCommGroup P']
+variable [AddCommMonoid M₂] [AddCommMonoid N₂] [AddCommMonoid P₂] [AddCommMonoid Pₗ]
+variable [AddCommGroup M'] [AddCommGroup P']
 variable [Module R M] [Module S N] [Module R₂ P] [Module S₂ P]
 variable [Module R M₂] [Module S N₂] [Module R P₂] [Module S₂ P₂]
 variable [Module R Pₗ] [Module S Pₗ]
-variable [Module R M'] [Module S N'] [Module R₂ P'] [Module S₂ P']
+variable [Module R M'] [Module R₂ P'] [Module S₂ P']
 variable [SMulCommClass S₂ R₂ P] [SMulCommClass S R Pₗ] [SMulCommClass S₂ R₂ P']
 variable [SMulCommClass S₂ R P₂]
 variable {ρ₁₂ : R →+* R₂} {σ₁₂ : S →+* S₂}
@@ -376,14 +375,11 @@ end CommSemiring
 
 section CommRing
 
-variable {R R₂ S S₂ M N P : Type*}
-variable {Mₗ Nₗ Pₗ : Type*}
-variable [CommRing R] [CommRing S] [CommRing R₂] [CommRing S₂]
+variable {R M : Type*} [CommRing R]
 
 section AddCommGroup
 
-variable [AddCommGroup M] [AddCommGroup N] [AddCommGroup P]
-variable [Module R M] [Module S N] [Module R₂ P] [Module S₂ P]
+variable [AddCommGroup M] [Module R M]
 
 theorem lsmul_injective [NoZeroSMulDivisors R M] {x : R} (hx : x ≠ 0) :
     Function.Injective (lsmul R M x) :=
diff --git a/Mathlib/LinearAlgebra/Charpoly/Basic.lean b/Mathlib/LinearAlgebra/Charpoly/Basic.lean
index eff887d7c175d..a54a424f7f68e 100644
--- a/Mathlib/LinearAlgebra/Charpoly/Basic.lean
+++ b/Mathlib/LinearAlgebra/Charpoly/Basic.lean
@@ -51,7 +51,7 @@ section Coeff
 theorem charpoly_monic : f.charpoly.Monic :=
   Matrix.charpoly_monic _
 
-open FiniteDimensional in
+open Module in
 lemma charpoly_natDegree [Nontrivial R] [StrongRankCondition R] :
     natDegree (charpoly f) = finrank R M := by
   rw [charpoly, Matrix.charpoly_natDegree_eq_dim, finrank_eq_card_chooseBasisIndex]
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
index 64eab0f1101ed..30687d5b35594 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean
@@ -358,7 +358,7 @@ variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
 
 theorem ι_mul_ι (r₁ r₂) : ι (0 : QuadraticForm R R) r₁ * ι (0 : QuadraticForm R R) r₂ = 0 := by
   rw [← mul_one r₁, ← mul_one r₂, ← smul_eq_mul R, ← smul_eq_mul R, LinearMap.map_smul,
-    LinearMap.map_smul, smul_mul_smul, ι_sq_scalar, QuadraticMap.zero_apply, RingHom.map_zero,
+    LinearMap.map_smul, smul_mul_smul_comm, ι_sq_scalar, QuadraticMap.zero_apply, RingHom.map_zero,
     smul_zero]
 
 /-- The clifford algebra over a 1-dimensional vector space with 0 quadratic form is isomorphic to
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
index ab5a28a5dcedc..76bff36c9a4d1 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/Even.lean
@@ -125,7 +125,7 @@ variable {Q}
 namespace even.lift
 
 /-- An auxiliary submodule used to store the half-applied values of `f`.
-This is the span of elements `f'` such that `∃ x m₂, ∀ m₁, f' m₁ = f m₁ m₂ * x`.  -/
+This is the span of elements `f'` such that `∃ x m₂, ∀ m₁, f' m₁ = f m₁ m₂ * x`. -/
 private def S : Submodule R (M →ₗ[R] A) :=
   Submodule.span R
     {f' | ∃ x m₂, f' = LinearMap.lcomp R _ (f.bilin.flip m₂) (LinearMap.mulRight R x)}
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean
index ff16afc3d7f70..d699e29ff8b2d 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean
@@ -187,7 +187,7 @@ theorem toEven_comp_ofEven : (toEven Q).comp (ofEven Q) = AlgHom.id R _ :=
                 have h2 : -(r₂ • e0 Q * v Q m₁) = v Q m₁ * r₂ • e0 Q := by
                   rw [mul_smul_comm, smul_mul_assoc, ← smul_neg, neg_e0_mul_v]
                 have h3 : -algebraMap R _ (r₁ * r₂) = r₁ • e0 Q * r₂ • e0 Q := by
-                  rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul, e0_mul_e0, smul_neg]
+                  rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul_comm, e0_mul_e0, smul_neg]
                 rw [sub_eq_add_neg, sub_eq_add_neg, h1, h2, h3]
               _ = ι (Q' Q) (m₁, r₁) * ι (Q' Q) (m₂, r₂) := by
                 rw [ι_eq_v_add_smul_e0, ι_eq_v_add_smul_e0, mul_add, add_mul, add_mul, add_assoc]
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
index 4359ba5b328d0..e5f6c2be85383 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
@@ -154,7 +154,7 @@ theorem evenOdd_induction (n : ZMod 2) {motive : ∀ x, x ∈ evenOdd Q n → Pr
           motive (ι Q m₁ * ι Q m₂ * x)
             (zero_add n ▸ SetLike.mul_mem_graded (ι_mul_ι_mem_evenOdd_zero Q m₁ m₂) hx))
     (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q n) : motive x hx := by
-  apply Submodule.iSup_induction' (C := motive) _ (range_ι_pow 0 (Submodule.zero_mem _)) add
+  apply Submodule.iSup_induction' (C := motive) _ _ (range_ι_pow 0 (Submodule.zero_mem _)) add
   refine Subtype.rec ?_
   simp_rw [ZMod.natCast_eq_iff, add_comm n.val]
   rintro n' ⟨k, rfl⟩ xv
@@ -197,7 +197,7 @@ theorem even_induction {motive : ∀ x, x ∈ evenOdd Q 0 → Prop}
           motive (ι Q m₁ * ι Q m₂ * x)
             (zero_add (0 : ZMod 2) ▸ SetLike.mul_mem_graded (ι_mul_ι_mem_evenOdd_zero Q m₁ m₂) hx))
     (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 0) : motive x hx := by
-  refine evenOdd_induction (motive := motive) (fun rx => ?_) add ι_mul_ι_mul x hx
+  refine evenOdd_induction _ _ (motive := motive) (fun rx => ?_) add ι_mul_ι_mul x hx
   rintro ⟨r, rfl⟩
   exact algebraMap r
 
@@ -213,7 +213,7 @@ theorem odd_induction {P : ∀ x, x ∈ evenOdd Q 1 → Prop}
           P (CliffordAlgebra.ι Q m₁ * CliffordAlgebra.ι Q m₂ * x)
             (zero_add (1 : ZMod 2) ▸ SetLike.mul_mem_graded (ι_mul_ι_mem_evenOdd_zero Q m₁ m₂) hx))
     (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 1) : P x hx := by
-  refine evenOdd_induction (motive := P) (fun ιv => ?_) add ι_mul_ι_mul x hx
+  refine evenOdd_induction _ _ (motive := P) (fun ιv => ?_) add ι_mul_ι_mul x hx
   -- Porting note: was `simp_rw [ZMod.val_one, pow_one]`, lean4#1926
   intro h; rw [ZMod.val_one, pow_one] at h; revert h
   rintro ⟨v, rfl⟩
diff --git a/Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean b/Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean
index 1e9ee43c0a118..e06d940aa2396 100644
--- a/Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean
+++ b/Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean
@@ -116,7 +116,7 @@ def ofProd : CliffordAlgebra (Q₁.prod Q₂) →ₐ[R] (evenOdd Q₁ ᵍ⊗[R]
           ∘ₗ (evenOdd Q₂ 1).subtype ∘ₗ (ι Q₂).codRestrict _ (ι_mem_evenOdd_one Q₂)),
     fun m => by
       simp_rw [LinearMap.coprod_apply, LinearMap.coe_comp, Function.comp_apply,
-        AlgHom.toLinearMap_apply, QuadraticMap.prod_apply, Submodule.coeSubtype,
+        AlgHom.toLinearMap_apply, QuadraticMap.prod_apply, Submodule.coe_subtype,
         GradedTensorProduct.includeLeft_apply, GradedTensorProduct.includeRight_apply, map_add,
         add_mul, mul_add, GradedTensorProduct.algebraMap_def,
         GradedTensorProduct.tmul_one_mul_one_tmul, GradedTensorProduct.tmul_one_mul_coe_tmul,
diff --git a/Mathlib/LinearAlgebra/Coevaluation.lean b/Mathlib/LinearAlgebra/Coevaluation.lean
index bd0df67099a43..31b64ef5ce5f8 100644
--- a/Mathlib/LinearAlgebra/Coevaluation.lean
+++ b/Mathlib/LinearAlgebra/Coevaluation.lean
@@ -25,7 +25,7 @@ noncomputable section
 
 section coevaluation
 
-open TensorProduct FiniteDimensional
+open TensorProduct Module
 
 open TensorProduct
 
diff --git a/Mathlib/LinearAlgebra/Contraction.lean b/Mathlib/LinearAlgebra/Contraction.lean
index 8ba1276a30496..0e7e5c811a68a 100644
--- a/Mathlib/LinearAlgebra/Contraction.lean
+++ b/Mathlib/LinearAlgebra/Contraction.lean
@@ -201,7 +201,7 @@ section CommRing
 variable [CommRing R]
 variable [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [AddCommGroup Q]
 variable [Module R M] [Module R N] [Module R P] [Module R Q]
-variable [Free R M] [Finite R M] [Free R N] [Finite R N]
+variable [Free R M] [Module.Finite R M] [Free R N] [Module.Finite R N]
 
 /-- When `M` is a finite free module, the map `lTensorHomToHomLTensor` is an equivalence. Note
 that `lTensorHomEquivHomLTensor` is not defined directly in terms of
diff --git a/Mathlib/LinearAlgebra/CrossProduct.lean b/Mathlib/LinearAlgebra/CrossProduct.lean
index f6ba8092be836..8fb6567b4ec21 100644
--- a/Mathlib/LinearAlgebra/CrossProduct.lean
+++ b/Mathlib/LinearAlgebra/CrossProduct.lean
@@ -141,3 +141,27 @@ theorem jacobi_cross (u v w : Fin 3 → R) : u ×₃ (v ×₃ w) + v ×₃ (w ×
   lie_jacobi u v w
 
 end LeibnizProperties
+
+-- this can also be proved via `Matrix.dotProduct_eq_zero_iff` and `triple_product_eq_det`, but
+-- that would require much heavier imports.
+lemma crossProduct_ne_zero_iff_linearIndependent {F : Type*} [Field F] {v w : Fin 3 → F} :
+    crossProduct v w ≠ 0 ↔ LinearIndependent F ![v, w] := by
+  rw [not_iff_comm]
+  by_cases hv : v = 0
+  · rw [hv, map_zero, LinearMap.zero_apply, eq_self, iff_true]
+    exact fun h ↦ h.ne_zero 0 rfl
+  constructor
+  · rw [LinearIndependent.pair_iff' hv, not_forall_not]
+    rintro ⟨a, rfl⟩
+    rw [LinearMap.map_smul, cross_self, smul_zero]
+  have hv' : v = ![v 0, v 1, v 2] := by simp [← List.ofFn_inj]
+  have hw' : w = ![w 0, w 1, w 2] := by simp [← List.ofFn_inj]
+  intro h1 h2
+  simp_rw [cross_apply, cons_eq_zero_iff, zero_empty, and_true, sub_eq_zero] at h1
+  have h20 := LinearIndependent.pair_iff.mp h2 (- w 0) (v 0)
+  have h21 := LinearIndependent.pair_iff.mp h2 (- w 1) (v 1)
+  have h22 := LinearIndependent.pair_iff.mp h2 (- w 2) (v 2)
+  rw [neg_smul, neg_add_eq_zero, hv', hw', smul_vec3, smul_vec3, ← hv', ← hw'] at h20 h21 h22
+  simp only [smul_eq_mul, mul_comm (w 0), mul_comm (w 1), mul_comm (w 2), h1] at h20 h21 h22
+  rw [hv', cons_eq_zero_iff, cons_eq_zero_iff, cons_eq_zero_iff, zero_empty] at hv
+  exact hv ⟨(h20 trivial).2, (h21 trivial).2, (h22 trivial).2, rfl⟩
diff --git a/Mathlib/LinearAlgebra/DFinsupp.lean b/Mathlib/LinearAlgebra/DFinsupp.lean
index 1f4cf5ef07267..7f631f041c8c3 100644
--- a/Mathlib/LinearAlgebra/DFinsupp.lean
+++ b/Mathlib/LinearAlgebra/DFinsupp.lean
@@ -299,7 +299,7 @@ theorem biSup_eq_range_dfinsupp_lsum (p : ι → Prop) [DecidablePred p] (S : ι
   apply le_antisymm
   · refine iSup₂_le fun i hi y hy => ⟨DFinsupp.single i ⟨y, hy⟩, ?_⟩
     rw [LinearMap.comp_apply, filterLinearMap_apply, filter_single_pos _ _ hi]
-    simp only [lsum_apply_apply, sumAddHom_single, LinearMap.toAddMonoidHom_coe, coeSubtype]
+    simp only [lsum_apply_apply, sumAddHom_single, LinearMap.toAddMonoidHom_coe, coe_subtype]
   · rintro x ⟨v, rfl⟩
     refine dfinsupp_sumAddHom_mem _ _ _ fun i _ => ?_
     refine mem_iSup_of_mem i ?_
@@ -322,7 +322,7 @@ theorem mem_iSup_iff_exists_dfinsupp' (p : ι → Submodule R N) [∀ (i) (x : p
     (x : N) : x ∈ iSup p ↔ ∃ f : Π₀ i, p i, (f.sum fun i xi => ↑xi) = x := by
   rw [mem_iSup_iff_exists_dfinsupp]
   simp_rw [DFinsupp.lsum_apply_apply, DFinsupp.sumAddHom_apply,
-    LinearMap.toAddMonoidHom_coe, coeSubtype]
+    LinearMap.toAddMonoidHom_coe, coe_subtype]
 
 theorem mem_biSup_iff_exists_dfinsupp (p : ι → Prop) [DecidablePred p] (S : ι → Submodule R N)
     (x : N) :
@@ -340,7 +340,7 @@ lemma mem_iSup_iff_exists_finsupp (p : ι → Submodule R N) (x : N) :
   refine ⟨fun ⟨f, hf⟩ ↦ ⟨⟨f.support, fun i ↦ (f i : N), by simp⟩, by simp, hf⟩, ?_⟩
   rintro ⟨f, hf, rfl⟩
   refine ⟨DFinsupp.mk f.support fun i ↦ ⟨f i, hf i⟩, Finset.sum_congr ?_ fun i hi ↦ ?_⟩
-  · ext; simp
+  · ext; simp [mk_eq_zero]
   · simp [Finsupp.mem_support_iff.mp hi]
 
 theorem mem_iSup_finset_iff_exists_sum {s : Finset ι} (p : ι → Submodule R N) (a : N) :
@@ -422,14 +422,15 @@ theorem independent_of_dfinsupp_sumAddHom_injective (p : ι → AddSubmonoid N)
   rw [← independent_map_orderIso_iff (AddSubmonoid.toNatSubmodule : AddSubmonoid N ≃o _)]
   exact independent_of_dfinsupp_lsum_injective _ h
 
-/-- Combining `DFinsupp.lsum` with `LinearMap.toSpanSingleton` is the same as `Finsupp.total` -/
+/-- Combining `DFinsupp.lsum` with `LinearMap.toSpanSingleton` is the same as
+`Finsupp.linearCombination` -/
 theorem lsum_comp_mapRange_toSpanSingleton [∀ m : R, Decidable (m ≠ 0)] (p : ι → Submodule R N)
     {v : ι → N} (hv : ∀ i : ι, v i ∈ p i) :
     (lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype : _ →ₗ[R] _).comp
         ((mapRange.linearMap fun i => LinearMap.toSpanSingleton R (↥(p i)) ⟨v i, hv i⟩ :
               _ →ₗ[R] _).comp
           (finsuppLequivDFinsupp R : (ι →₀ R) ≃ₗ[R] _).toLinearMap) =
-      Finsupp.total R v := by
+      Finsupp.linearCombination R v := by
   ext
   simp
 
diff --git a/Mathlib/LinearAlgebra/Determinant.lean b/Mathlib/LinearAlgebra/Determinant.lean
index 984b62e9b92a4..8f8edaca16e32 100644
--- a/Mathlib/LinearAlgebra/Determinant.lean
+++ b/Mathlib/LinearAlgebra/Determinant.lean
@@ -231,15 +231,15 @@ theorem det_id : LinearMap.det (LinearMap.id : M →ₗ[A] M) = 1 :=
 @[simp]
 theorem det_smul {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M] (c : 𝕜)
     (f : M →ₗ[𝕜] M) :
-    LinearMap.det (c • f) = c ^ FiniteDimensional.finrank 𝕜 M * LinearMap.det f := by
+    LinearMap.det (c • f) = c ^ Module.finrank 𝕜 M * LinearMap.det f := by
   by_cases H : ∃ s : Finset M, Nonempty (Basis s 𝕜 M)
   · have : FiniteDimensional 𝕜 M := by
       rcases H with ⟨s, ⟨hs⟩⟩
       exact FiniteDimensional.of_fintype_basis hs
-    simp only [← det_toMatrix (FiniteDimensional.finBasis 𝕜 M), LinearEquiv.map_smul,
+    simp only [← det_toMatrix (Module.finBasis 𝕜 M), LinearEquiv.map_smul,
       Fintype.card_fin, Matrix.det_smul]
   · classical
-      have : FiniteDimensional.finrank 𝕜 M = 0 := finrank_eq_zero_of_not_exists_basis H
+      have : Module.finrank 𝕜 M = 0 := finrank_eq_zero_of_not_exists_basis H
       simp [coe_det, H, this]
 
 theorem det_zero' {ι : Type*} [Finite ι] [Nonempty ι] (b : Basis ι A M) :
@@ -253,7 +253,7 @@ and `0` otherwise. We give a formula that also works in infinite dimension, wher
 the determinant to be `1`. -/
 @[simp]
 theorem det_zero {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M] :
-    LinearMap.det (0 : M →ₗ[𝕜] M) = (0 : 𝕜) ^ FiniteDimensional.finrank 𝕜 M := by
+    LinearMap.det (0 : M →ₗ[𝕜] M) = (0 : 𝕜) ^ Module.finrank 𝕜 M := by
   simp only [← zero_smul 𝕜 (1 : M →ₗ[𝕜] M), det_smul, mul_one, MonoidHom.map_one]
 
 theorem det_eq_one_of_subsingleton [Subsingleton M] (f : M →ₗ[R] M) :
@@ -263,14 +263,14 @@ theorem det_eq_one_of_subsingleton [Subsingleton M] (f : M →ₗ[R] M) :
   exact Matrix.det_isEmpty
 
 theorem det_eq_one_of_finrank_eq_zero {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M]
-    [Module 𝕜 M] (h : FiniteDimensional.finrank 𝕜 M = 0) (f : M →ₗ[𝕜] M) :
+    [Module 𝕜 M] (h : Module.finrank 𝕜 M = 0) (f : M →ₗ[𝕜] M) :
     LinearMap.det (f : M →ₗ[𝕜] M) = 1 := by
   classical
     refine @LinearMap.det_cases M _ 𝕜 _ _ _ (fun t => t = 1) f ?_ rfl
     intro s b
     have : IsEmpty s := by
       rw [← Fintype.card_eq_zero_iff]
-      exact (FiniteDimensional.finrank_eq_card_basis b).symm.trans h
+      exact (Module.finrank_eq_card_basis b).symm.trans h
     exact Matrix.det_isEmpty
 
 /-- Conjugating a linear map by a linear equiv does not change its determinant. -/
@@ -423,8 +423,8 @@ theorem LinearEquiv.coe_ofIsUnitDet {f : M →ₗ[R] M'} {v : Basis ι R M} {v'
 determinant is nonzero. -/
 abbrev LinearMap.equivOfDetNeZero {𝕜 : Type*} [Field 𝕜] {M : Type*} [AddCommGroup M] [Module 𝕜 M]
     [FiniteDimensional 𝕜 M] (f : M →ₗ[𝕜] M) (hf : LinearMap.det f ≠ 0) : M ≃ₗ[𝕜] M :=
-  have : IsUnit (LinearMap.toMatrix (FiniteDimensional.finBasis 𝕜 M)
-      (FiniteDimensional.finBasis 𝕜 M) f).det := by
+  have : IsUnit (LinearMap.toMatrix (Module.finBasis 𝕜 M)
+      (Module.finBasis 𝕜 M) f).det := by
     rw [LinearMap.det_toMatrix]
     exact isUnit_iff_ne_zero.2 hf
   LinearEquiv.ofIsUnitDet this
@@ -558,7 +558,7 @@ theorem Basis.det_reindex' {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Ba
 
 theorem Basis.det_reindex_symm {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (b : Basis ι R M)
     (v : ι → M) (e : ι' ≃ ι) : (b.reindex e.symm).det (v ∘ e) = b.det v := by
-  rw [Basis.det_reindex, Function.comp.assoc, e.self_comp_symm, Function.comp_id]
+  rw [Basis.det_reindex, Function.comp_assoc, e.self_comp_symm, Function.comp_id]
 
 @[simp]
 theorem Basis.det_map (b : Basis ι R M) (f : M ≃ₗ[R] M') (v : ι → M') :
diff --git a/Mathlib/LinearAlgebra/Dimension/Basic.lean b/Mathlib/LinearAlgebra/Dimension/Basic.lean
index 673047ce643a6..6da291d1211aa 100644
--- a/Mathlib/LinearAlgebra/Dimension/Basic.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison
+Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
 -/
 import Mathlib.LinearAlgebra.LinearIndependent
 
@@ -93,15 +93,6 @@ theorem cardinal_le_rank' {s : Set M}
 
 end LinearIndependent
 
-@[deprecated (since := "2023-12-27")]
-alias cardinal_lift_le_rank_of_linearIndependent := LinearIndependent.cardinal_lift_le_rank
-@[deprecated (since := "2023-12-27")]
-alias cardinal_lift_le_rank_of_linearIndependent' := LinearIndependent.cardinal_lift_le_rank
-@[deprecated (since := "2023-12-27")]
-alias cardinal_le_rank_of_linearIndependent := LinearIndependent.cardinal_le_rank
-@[deprecated (since := "2023-12-27")]
-alias cardinal_le_rank_of_linearIndependent' := LinearIndependent.cardinal_le_rank'
-
 section SurjectiveInjective
 
 section Module
@@ -285,11 +276,12 @@ theorem lift_rank_map_le (f : M →ₗ[R] M') (p : Submodule R M) :
 theorem rank_map_le (f : M →ₗ[R] M₁) (p : Submodule R M) :
     Module.rank R (p.map f) ≤ Module.rank R p := by simpa using lift_rank_map_le f p
 
-theorem rank_le_of_submodule (s t : Submodule R M) (h : s ≤ t) :
-    Module.rank R s ≤ Module.rank R t :=
+lemma Submodule.rank_mono {s t : Submodule R M} (h : s ≤ t) : Module.rank R s ≤ Module.rank R t :=
   (Submodule.inclusion h).rank_le_of_injective fun ⟨x, _⟩ ⟨y, _⟩ eq =>
     Subtype.eq <| show x = y from Subtype.ext_iff_val.1 eq
 
+@[deprecated (since := "2024-09-30")] alias rank_le_of_submodule := Submodule.rank_mono
+
 /-- Two linearly equivalent vector spaces have the same dimension, a version with different
 universes. -/
 theorem LinearEquiv.lift_rank_eq (f : M ≃ₗ[R] M') :
@@ -331,9 +323,11 @@ theorem rank_range_of_surjective (f : M →ₗ[R] M') (h : Surjective f) :
     Module.rank R (LinearMap.range f) = Module.rank R M' := by
   rw [LinearMap.range_eq_top.2 h, rank_top]
 
-theorem rank_submodule_le (s : Submodule R M) : Module.rank R s ≤ Module.rank R M := by
+theorem Submodule.rank_le (s : Submodule R M) : Module.rank R s ≤ Module.rank R M := by
   rw [← rank_top R M]
-  exact rank_le_of_submodule _ _ le_top
+  exact rank_mono le_top
+
+@[deprecated (since := "2024-10-02")] alias rank_submodule_le := Submodule.rank_le
 
 theorem LinearMap.lift_rank_le_of_surjective (f : M →ₗ[R] M') (h : Surjective f) :
     lift.{v} (Module.rank R M') ≤ lift.{v'} (Module.rank R M) := by
diff --git a/Mathlib/LinearAlgebra/Dimension/Constructions.lean b/Mathlib/LinearAlgebra/Dimension/Constructions.lean
index 3724ba1ee3ccd..9a7b997644973 100644
--- a/Mathlib/LinearAlgebra/Dimension/Constructions.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Constructions.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
+Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
 -/
 import Mathlib.LinearAlgebra.Dimension.Free
 import Mathlib.Algebra.Module.Torsion
@@ -35,7 +35,7 @@ universe u v v' u₁' w w'
 variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}
 variable {ι : Type w} {ι' : Type w'} {η : Type u₁'} {φ : η → Type*}
 
-open Cardinal Basis Submodule Function Set FiniteDimensional DirectSum
+open Basis Cardinal DirectSum Function Module Set Submodule
 
 variable [Ring R] [CommRing S] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M₁]
 variable [Module R M]
@@ -70,7 +70,7 @@ theorem rank_quotient_add_rank_le [Nontrivial R] (M' : Submodule R M) :
   refine ciSup_le fun ⟨s, hs⟩ ↦ ciSup_le fun ⟨t, ht⟩ ↦ ?_
   choose f hf using Quotient.mk_surjective M'
   simpa [add_comm] using (LinearIndependent.sum_elim_of_quotient ht (fun (i : s) ↦ f i)
-    (by simpa [Function.comp, hf] using hs)).cardinal_le_rank
+    (by simpa [Function.comp_def, hf] using hs)).cardinal_le_rank
 
 theorem rank_quotient_le (p : Submodule R M) : Module.rank R (M ⧸ p) ≤ Module.rank R M :=
   (mkQ p).rank_le_of_surjective (surjective_quot_mk _)
@@ -142,7 +142,7 @@ theorem rank_prod' : Module.rank R (M × M₁) = Module.rank R M + Module.rank R
 
 /-- The finrank of `M × M'` is `(finrank R M) + (finrank R M')`. -/
 @[simp]
-theorem FiniteDimensional.finrank_prod [Module.Finite R M] [Module.Finite R M'] :
+theorem Module.finrank_prod [Module.Finite R M] [Module.Finite R M'] :
     finrank R (M × M') = finrank R M + finrank R M' := by
   simp [finrank, rank_lt_aleph0 R M, rank_lt_aleph0 R M']
 
@@ -209,7 +209,7 @@ theorem rank_matrix'' (m n : Type u) [Finite m] [Finite n] :
 
 open Fintype
 
-namespace FiniteDimensional
+namespace Module
 
 @[simp]
 theorem finrank_finsupp {ι : Type v} [Fintype ι] : finrank R (ι →₀ M) = card ι * finrank R M := by
@@ -234,7 +234,7 @@ theorem finrank_directSum {ι : Type v} [Fintype ι] (M : ι → Type w) [∀ i
 theorem finrank_matrix (m n : Type*) [Fintype m] [Fintype n] :
     finrank R (Matrix m n R) = card m * card n := by simp [finrank]
 
-end FiniteDimensional
+end Module
 
 end Finsupp
 
@@ -260,13 +260,13 @@ theorem rank_pi [Finite η] : Module.rank R (∀ i, φ i) =
 variable (R)
 
 /-- The finrank of `(ι → R)` is `Fintype.card ι`. -/
-theorem FiniteDimensional.finrank_pi {ι : Type v} [Fintype ι] :
+theorem Module.finrank_pi {ι : Type v} [Fintype ι] :
     finrank R (ι → R) = Fintype.card ι := by
   simp [finrank]
 
 --TODO: this should follow from `LinearEquiv.finrank_eq`, that is over a field.
 /-- The finrank of a finite product is the sum of the finranks. -/
-theorem FiniteDimensional.finrank_pi_fintype
+theorem Module.finrank_pi_fintype
     {ι : Type v} [Fintype ι] {M : ι → Type w} [∀ i : ι, AddCommGroup (M i)]
     [∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
     finrank R (∀ i, M i) = ∑ i, finrank R (M i) := by
@@ -294,12 +294,12 @@ variable (R)
 
 /-- The vector space of functions on a `Fintype ι` has finrank equal to the cardinality of `ι`. -/
 @[simp]
-theorem FiniteDimensional.finrank_fintype_fun_eq_card : finrank R (η → R) = Fintype.card η :=
+theorem Module.finrank_fintype_fun_eq_card : finrank R (η → R) = Fintype.card η :=
   finrank_eq_of_rank_eq rank_fun'
 
 /-- The vector space of functions on `Fin n` has finrank equal to `n`. -/
 -- @[simp] -- Porting note (#10618): simp already proves this
-theorem FiniteDimensional.finrank_fin_fun {n : ℕ} : finrank R (Fin n → R) = n := by simp
+theorem Module.finrank_fin_fun {n : ℕ} : finrank R (Fin n → R) = n := by simp
 
 variable {R}
 
@@ -343,7 +343,7 @@ theorem rank_tensorProduct' :
 
 /-- The `S`-finrank of `M ⊗[R] M'` is `(finrank S M) * (finrank R M')`. -/
 @[simp]
-theorem FiniteDimensional.finrank_tensorProduct :
+theorem Module.finrank_tensorProduct :
     finrank R (M ⊗[S] M') = finrank R M * finrank S M' := by simp [finrank]
 
 end TensorProduct
@@ -352,7 +352,7 @@ section SubmoduleRank
 
 section
 
-open FiniteDimensional
+open Module
 
 namespace Submodule
 
@@ -372,7 +372,7 @@ variable [StrongRankCondition R]
 /-- The dimension of a submodule is bounded by the dimension of the ambient space. -/
 theorem Submodule.finrank_le [Module.Finite R M] (s : Submodule R M) :
     finrank R s ≤ finrank R M :=
-  toNat_le_toNat (rank_submodule_le s) (rank_lt_aleph0 _ _)
+  toNat_le_toNat (Submodule.rank_le s) (rank_lt_aleph0 _ _)
 
 /-- The dimension of a quotient is bounded by the dimension of the ambient space. -/
 theorem Submodule.finrank_quotient_le [Module.Finite R M] (s : Submodule R M) :
@@ -386,12 +386,12 @@ theorem Submodule.finrank_map_le
     finrank R (p.map f) ≤ finrank R p :=
   finrank_le_finrank_of_rank_le_rank (lift_rank_map_le _ _) (rank_lt_aleph0 _ _)
 
-theorem Submodule.finrank_le_finrank_of_le {s t : Submodule R M} [Module.Finite R t] (hst : s ≤ t) :
+theorem Submodule.finrank_mono {s t : Submodule R M} [Module.Finite R t] (hst : s ≤ t) :
     finrank R s ≤ finrank R t :=
-  calc
-    finrank R s = finrank R (s.comap t.subtype) :=
-      (Submodule.comapSubtypeEquivOfLe hst).finrank_eq.symm
-    _ ≤ finrank R t := Submodule.finrank_le _
+  Cardinal.toNat_le_toNat (Submodule.rank_mono hst) (rank_lt_aleph0 R ↥t)
+
+@[deprecated (since := "2024-09-30")]
+alias Submodule.finrank_le_finrank_of_le := Submodule.finrank_mono
 
 end
 
@@ -402,7 +402,7 @@ section Span
 variable [StrongRankCondition R]
 
 theorem rank_span_le (s : Set M) : Module.rank R (span R s) ≤ #s := by
-  rw [Finsupp.span_eq_range_total, ← lift_strictMono.le_iff_le]
+  rw [Finsupp.span_eq_range_linearCombination, ← lift_strictMono.le_iff_le]
   refine (lift_rank_range_le _).trans ?_
   rw [rank_finsupp_self]
   simp only [lift_lift, le_refl]
@@ -413,7 +413,7 @@ theorem rank_span_finset_le (s : Finset M) : Module.rank R (span R (s : Set M))
 theorem rank_span_of_finset (s : Finset M) : Module.rank R (span R (s : Set M)) < ℵ₀ :=
   (rank_span_finset_le s).trans_lt (Cardinal.nat_lt_aleph0 _)
 
-open Submodule FiniteDimensional
+open Submodule Module
 
 variable (R)
 
diff --git a/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean b/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
index 6b0db8a0b52de..cf1afda61d172 100644
--- a/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
+++ b/Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
@@ -2,7 +2,7 @@
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen,
-Scott Morrison, Chris Hughes, Anne Baanen, Junyan Xu
+Kim Morrison, Chris Hughes, Anne Baanen, Junyan Xu
 -/
 import Mathlib.LinearAlgebra.Basis.VectorSpace
 import Mathlib.LinearAlgebra.Dimension.Finite
@@ -12,7 +12,7 @@ import Mathlib.LinearAlgebra.Dimension.RankNullity
 /-!
 # Dimension of vector spaces
 
-In this file we provide results about `Module.rank` and `FiniteDimensional.finrank` of vector spaces
+In this file we provide results about `Module.rank` and `Module.finrank` of vector spaces
 over division rings.
 
 ## Main statements
@@ -112,7 +112,7 @@ end Module
 
 section Basis
 
-open FiniteDimensional
+open Module
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
@@ -180,9 +180,7 @@ theorem linearIndependent_iff_card_eq_finrank_span {ι : Type*} [Fintype ι] {b
         simp [f, Set.mem_image, Set.mem_range]
       rw [hf] at h
       have hx : (x : V) ∈ span K (Set.range b) := x.property
-      conv at hx =>
-        arg 2
-        rw [h]
+      simp_rw [h] at hx
       simpa [f, mem_map] using hx
     have hi : LinearMap.ker f = ⊥ := ker_subtype _
     convert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi
@@ -249,8 +247,8 @@ theorem max_aleph0_card_le_rank_fun_nat : max ℵ₀ #K ≤ Module.rank K (ℕ 
     contrapose! card_K
     exact (power_lt_aleph0 card_K <| nat_lt_aleph0 _).le
   obtain ⟨e⟩ := lift_mk_le'.mp (card_ιL.trans_eq (lift_uzero #ιL).symm)
-  have rep_e := bK.total_repr (bL ∘ e)
-  rw [Finsupp.total_apply, Finsupp.sum] at rep_e
+  have rep_e := bK.linearCombination_repr (bL ∘ e)
+  rw [Finsupp.linearCombination_apply, Finsupp.sum] at rep_e
   set c := bK.repr (bL ∘ e)
   set s := c.support
   let f i (j : s) : L := ⟨bK j i, Subfield.subset_closure ⟨(j, i), rfl⟩⟩
diff --git a/Mathlib/LinearAlgebra/Dimension/Finite.lean b/Mathlib/LinearAlgebra/Dimension/Finite.lean
index c3175f4462e68..90c82db1cfcc3 100644
--- a/Mathlib/LinearAlgebra/Dimension/Finite.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Finite.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Scott Morrison
+Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
 -/
 import Mathlib.Algebra.Module.Torsion
 import Mathlib.SetTheory.Cardinal.Cofinality
@@ -26,7 +26,7 @@ variable [Module R M] [Module R M'] [Module R M₁]
 
 attribute [local instance] nontrivial_of_invariantBasisNumber
 
-open Cardinal Basis Submodule Function Set FiniteDimensional
+open Basis Cardinal Function Module Set Submodule
 
 theorem rank_le {n : ℕ}
     (H : ∀ s : Finset M, (LinearIndependent R fun i : s => (i : M)) → s.card ≤ n) :
@@ -48,7 +48,7 @@ lemma rank_eq_zero_iff :
     rw [← Cardinal.one_le_iff_ne_zero]
     have : LinearIndependent R (fun _ : Unit ↦ x) :=
       linearIndependent_iff.mpr (fun l hl ↦ Finsupp.unique_ext <| not_not.mp fun H ↦
-        hx _ H ((Finsupp.total_unique _ _ _).symm.trans hl))
+        hx _ H ((Finsupp.linearCombination_unique _ _ _).symm.trans hl))
     simpa using this.cardinal_lift_le_rank
   · intro h
     rw [← le_zero_iff, Module.rank_def]
@@ -191,15 +191,6 @@ theorem setFinite [Module.Finite R M] {b : Set M}
 
 end LinearIndependent
 
-@[deprecated (since := "2023-12-27")]
-alias cardinal_mk_le_finrank_of_linearIndependent := LinearIndependent.cardinal_mk_le_finrank
-@[deprecated (since := "2023-12-27")]
-alias fintype_card_le_finrank_of_linearIndependent := LinearIndependent.fintype_card_le_finrank
-@[deprecated (since := "2023-12-27")]
-alias finset_card_le_finrank_of_linearIndependent := LinearIndependent.finset_card_le_finrank
-@[deprecated (since := "2023-12-27")]
-alias Module.Finite.lt_aleph0_of_linearIndependent := LinearIndependent.lt_aleph0_of_finite
-
 lemma exists_set_linearIndependent_of_lt_rank {n : Cardinal} (hn : n < Module.rank R M) :
     ∃ s : Set M, #s = n ∧ LinearIndependent R ((↑) : s → M) := by
   obtain ⟨⟨s, hs⟩, hs'⟩ := exists_lt_of_lt_ciSup' (hn.trans_eq (Module.rank_def R M))
@@ -214,11 +205,11 @@ lemma exists_finset_linearIndependent_of_le_rank {n : ℕ} (hn : n ≤ Module.ra
       (Cardinal.bddAbove_range.{v, v} _) _ (h.trans (Module.rank_def R M)).symm
     have : Finite s := lt_aleph0_iff_finite.mp (hs' ▸ nat_lt_aleph0 n)
     cases nonempty_fintype s
-    exact ⟨s.toFinset, by simpa using hs', by convert hs <;> exact Set.mem_toFinset⟩
+    exact ⟨s.toFinset, by simpa using hs', by convert hs using 3 <;> exact Set.mem_toFinset⟩
   · obtain ⟨s, hs, hs'⟩ := exists_set_linearIndependent_of_lt_rank h
     have : Finite s := lt_aleph0_iff_finite.mp (hs ▸ nat_lt_aleph0 n)
     cases nonempty_fintype s
-    exact ⟨s.toFinset, by simpa using hs, by convert hs' <;> exact Set.mem_toFinset⟩
+    exact ⟨s.toFinset, by simpa using hs, by convert hs' using 3 <;> exact Set.mem_toFinset⟩
 
 lemma exists_linearIndependent_of_le_rank {n : ℕ} (hn : n ≤ Module.rank R M) :
     ∃ f : Fin n → M, LinearIndependent R f :=
@@ -363,7 +354,7 @@ variable [Nontrivial R]
 
 /-- A (finite dimensional) space that is a subsingleton has zero `finrank`. -/
 @[nontriviality]
-theorem FiniteDimensional.finrank_zero_of_subsingleton [Subsingleton M] :
+theorem Module.finrank_zero_of_subsingleton [Subsingleton M] :
     finrank R M = 0 := by
   rw [finrank, rank_subsingleton', _root_.map_zero]
 
@@ -374,12 +365,12 @@ section
 variable [NoZeroSMulDivisors R M]
 
 /-- A finite dimensional space is nontrivial if it has positive `finrank`. -/
-theorem FiniteDimensional.nontrivial_of_finrank_pos (h : 0 < finrank R M) : Nontrivial M :=
+theorem Module.nontrivial_of_finrank_pos (h : 0 < finrank R M) : Nontrivial M :=
   rank_pos_iff_nontrivial.mp (lt_rank_of_lt_finrank h)
 
 /-- A finite dimensional space is nontrivial if it has `finrank` equal to the successor of a
 natural number. -/
-theorem FiniteDimensional.nontrivial_of_finrank_eq_succ {n : ℕ}
+theorem Module.nontrivial_of_finrank_eq_succ {n : ℕ}
     (hn : finrank R M = n.succ) : Nontrivial M :=
   nontrivial_of_finrank_pos (R := R) (by rw [hn]; exact n.succ_pos)
 
@@ -398,31 +389,31 @@ section StrongRankCondition
 variable [StrongRankCondition R] [Module.Finite R M]
 
 /-- A finite rank torsion-free module has positive `finrank` iff it has a nonzero element. -/
-theorem FiniteDimensional.finrank_pos_iff_exists_ne_zero [NoZeroSMulDivisors R M] :
+theorem Module.finrank_pos_iff_exists_ne_zero [NoZeroSMulDivisors R M] :
     0 < finrank R M ↔ ∃ x : M, x ≠ 0 := by
   rw [← @rank_pos_iff_exists_ne_zero R M, ← finrank_eq_rank]
   norm_cast
 
 /-- An `R`-finite torsion-free module has positive `finrank` iff it is nontrivial. -/
-theorem FiniteDimensional.finrank_pos_iff [NoZeroSMulDivisors R M] :
+theorem Module.finrank_pos_iff [NoZeroSMulDivisors R M] :
     0 < finrank R M ↔ Nontrivial M := by
   rw [← rank_pos_iff_nontrivial (R := R), ← finrank_eq_rank]
   norm_cast
 
 /-- A nontrivial finite dimensional space has positive `finrank`. -/
-theorem FiniteDimensional.finrank_pos [NoZeroSMulDivisors R M] [h : Nontrivial M] :
+theorem Module.finrank_pos [NoZeroSMulDivisors R M] [h : Nontrivial M] :
     0 < finrank R M :=
   finrank_pos_iff.mpr h
 
-/-- See `FiniteDimensional.finrank_zero_iff`
+/-- See `Module.finrank_zero_iff`
   for the stronger version with `NoZeroSMulDivisors R M`. -/
-theorem FiniteDimensional.finrank_eq_zero_iff :
+theorem Module.finrank_eq_zero_iff :
     finrank R M = 0 ↔ ∀ x : M, ∃ a : R, a ≠ 0 ∧ a • x = 0 := by
   rw [← rank_eq_zero_iff (R := R), ← finrank_eq_rank]
   norm_cast
 
 /-- The `StrongRankCondition` is automatic. See `commRing_strongRankCondition`. -/
-theorem FiniteDimensional.finrank_eq_zero_iff_isTorsion {R} [CommRing R] [StrongRankCondition R]
+theorem Module.finrank_eq_zero_iff_isTorsion {R} [CommRing R] [StrongRankCondition R]
     [IsDomain R] [Module R M] [Module.Finite R M] :
     finrank R M = 0 ↔ Module.IsTorsion R M := by
   rw [← rank_eq_zero_iff_isTorsion (R := R), ← finrank_eq_rank]
@@ -430,14 +421,14 @@ theorem FiniteDimensional.finrank_eq_zero_iff_isTorsion {R} [CommRing R] [Strong
 
 /-- A finite dimensional space has zero `finrank` iff it is a subsingleton.
 This is the `finrank` version of `rank_zero_iff`. -/
-theorem FiniteDimensional.finrank_zero_iff [NoZeroSMulDivisors R M] :
+theorem Module.finrank_zero_iff [NoZeroSMulDivisors R M] :
     finrank R M = 0 ↔ Subsingleton M := by
   rw [← rank_zero_iff (R := R), ← finrank_eq_rank]
   norm_cast
 
 end StrongRankCondition
 
-theorem FiniteDimensional.finrank_eq_zero_of_rank_eq_zero (h : Module.rank R M = 0) :
+theorem Module.finrank_eq_zero_of_rank_eq_zero (h : Module.rank R M = 0) :
     finrank R M = 0 := by
   delta finrank
   rw [h, zero_toNat]
diff --git a/Mathlib/LinearAlgebra/Dimension/Finrank.lean b/Mathlib/LinearAlgebra/Dimension/Finrank.lean
index 9bf3954c81775..69668ef8b667a 100644
--- a/Mathlib/LinearAlgebra/Dimension/Finrank.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Finrank.lean
@@ -13,7 +13,7 @@ Definition of the rank of a module, or dimension of a vector space, as a natural
 
 ## Main definitions
 
-Defined is `FiniteDimensional.finrank`, the dimension of a finite dimensional space, returning a
+Defined is `Module.finrank`, the dimension of a finite dimensional space, returning a
 `Nat`, as opposed to `Module.rank`, which returns a `Cardinal`. When the space has infinite
 dimension, its `finrank` is by convention set to `0`.
 
@@ -38,7 +38,7 @@ open Cardinal Submodule Module Function
 variable {R : Type u} {M : Type v} {N : Type w}
 variable [Ring R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
 
-namespace FiniteDimensional
+namespace Module
 
 section Ring
 
@@ -52,6 +52,8 @@ of `M` over `R`.
 noncomputable def finrank (R M : Type*) [Semiring R] [AddCommGroup M] [Module R M] : ℕ :=
   Cardinal.toNat (Module.rank R M)
 
+@[deprecated (since := "2024-10-01")] protected alias _root_.FiniteDimensional.finrank := finrank
+
 theorem finrank_eq_of_rank_eq {n : ℕ} (h : Module.rank R M = ↑n) : finrank R M = n := by
   apply_fun toNat at h
   rw [toNat_natCast] at h
@@ -92,9 +94,9 @@ theorem finrank_le_finrank_of_rank_le_rank
 
 end Ring
 
-end FiniteDimensional
+end Module
 
-open FiniteDimensional
+open Module
 
 namespace LinearEquiv
 
diff --git a/Mathlib/LinearAlgebra/Dimension/Free.lean b/Mathlib/LinearAlgebra/Dimension/Free.lean
index 3b06f6f965e75..af3007b4da717 100644
--- a/Mathlib/LinearAlgebra/Dimension/Free.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Free.lean
@@ -14,7 +14,7 @@ import Mathlib.SetTheory.Cardinal.Finsupp
 ## Main result
 - `LinearEquiv.nonempty_equiv_iff_lift_rank_eq`:
   Two free modules are isomorphic iff they have the same dimension.
-- `FiniteDimensional.finBasis`:
+- `Module.finBasis`:
   An arbitrary basis of a finite free module indexed by `Fin n` given `finrank R M = n`.
 
 -/
@@ -24,7 +24,7 @@ noncomputable section
 
 universe u v v' w
 
-open Cardinal Basis Submodule Function Set DirectSum FiniteDimensional
+open Cardinal Basis Submodule Function Set DirectSum Module
 
 section Tower
 
@@ -57,7 +57,7 @@ theorem rank_mul_rank (A : Type v) [AddCommGroup A]
 
 /-- Tower law: if `A` is a `K`-module and `K` is an extension of `F` then
 $\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$. -/
-theorem FiniteDimensional.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by
+theorem Module.finrank_mul_finrank : finrank F K * finrank K A = finrank F A := by
   simp_rw [finrank]
   rw [← toNat_lift.{w} (Module.rank F K), ← toNat_lift.{v} (Module.rank K A), ← toNat_mul,
     lift_rank_mul_lift_rank, toNat_lift]
@@ -79,7 +79,7 @@ theorem rank_eq_card_chooseBasisIndex : Module.rank R M = #(ChooseBasisIndex R M
   (chooseBasis R M).mk_eq_rank''.symm
 
 /-- The finrank of a free module `M` over `R` is the cardinality of `ChooseBasisIndex R M`. -/
-theorem _root_.FiniteDimensional.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
+theorem _root_.Module.finrank_eq_card_chooseBasisIndex [Module.Finite R M] :
     finrank R M = Fintype.card (ChooseBasisIndex R M) := by
   simp [finrank, rank_eq_card_chooseBasisIndex]
 
@@ -161,35 +161,30 @@ noncomputable def LinearEquiv.ofFinrankEq [Module.Finite R M] [Module.Finite R M
 
 variable {M M'}
 
+namespace Module
+
 /-- See `rank_lt_aleph0` for the inverse direction without `Module.Free R M`. -/
-lemma Module.rank_lt_alpeh0_iff :
-    Module.rank R M < ℵ₀ ↔ Module.Finite R M := by
+lemma rank_lt_aleph0_iff : Module.rank R M < ℵ₀ ↔ Module.Finite R M := by
   rw [Free.rank_eq_card_chooseBasisIndex, mk_lt_aleph0_iff]
   exact ⟨fun h ↦ Finite.of_basis (Free.chooseBasis R M),
     fun I ↦ Finite.of_fintype (Free.ChooseBasisIndex R M)⟩
 
-theorem FiniteDimensional.finrank_of_not_finite
-    (h : ¬Module.Finite R M) :
-    finrank R M = 0 := by
-  rw [finrank, toNat_eq_zero, ← not_lt, Module.rank_lt_alpeh0_iff]
+theorem finrank_of_not_finite (h : ¬Module.Finite R M) : finrank R M = 0 := by
+  rw [finrank, toNat_eq_zero, ← not_lt, Module.rank_lt_aleph0_iff]
   exact .inr h
 
-theorem Module.finite_of_finrank_pos (h : 0 < finrank R M) :
-    Module.Finite R M := by
+theorem finite_of_finrank_pos (h : 0 < finrank R M) : Module.Finite R M := by
   contrapose h
   simp [finrank_of_not_finite h]
 
-theorem Module.finite_of_finrank_eq_succ {n : ℕ}
-    (hn : finrank R M = n.succ) : Module.Finite R M :=
-  Module.finite_of_finrank_pos <| by rw [hn]; exact n.succ_pos
+theorem finite_of_finrank_eq_succ {n : ℕ} (hn : finrank R M = n.succ) : Module.Finite R M :=
+  finite_of_finrank_pos <| by rw [hn]; exact n.succ_pos
 
-theorem Module.finite_iff_of_rank_eq_nsmul {W} [AddCommGroup W]
-    [Module R W] [Module.Free R W] {n : ℕ} (hn : n ≠ 0)
-    (hVW : Module.rank R M = n • Module.rank R W) :
+theorem finite_iff_of_rank_eq_nsmul {W} [AddCommGroup W] [Module R W] [Module.Free R W] {n : ℕ}
+    (hn : n ≠ 0) (hVW : Module.rank R M = n • Module.rank R W) :
     Module.Finite R M ↔ Module.Finite R W := by
-  simp only [← rank_lt_alpeh0_iff, hVW, nsmul_lt_aleph0_iff_of_ne_zero hn]
+  simp only [← rank_lt_aleph0_iff, hVW, nsmul_lt_aleph0_iff_of_ne_zero hn]
 
-namespace FiniteDimensional
 variable (R M)
 
 /-- A finite rank free module has a basis indexed by `Fin (finrank R M)`. -/
@@ -220,4 +215,4 @@ theorem basisUnique_repr_eq_zero_iff {ι : Type*} [Unique ι]
     (basisUnique ι h).repr.map_eq_zero_iff.mp (Finsupp.ext fun j => Subsingleton.elim i j ▸ hv),
     fun hv => by rw [hv, LinearEquiv.map_zero, Finsupp.zero_apply]⟩
 
-end FiniteDimensional
+end Module
diff --git a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
index 883252149ac34..17254b66084e3 100644
--- a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
+++ b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean
@@ -17,7 +17,7 @@ and `Mathlib/LinearAlgebra/FiniteDimensional.lean`.
 
 -/
 
-open Cardinal Submodule Set FiniteDimensional
+open Cardinal Module Module Set Submodule
 
 universe u v
 
@@ -27,7 +27,7 @@ variable {K : Type u} {V : Type v} [Ring K] [StrongRankCondition K] [AddCommGrou
 
 /-- The `ι` indexed basis on `V`, where `ι` is an empty type and `V` is zero-dimensional.
 
-See also `FiniteDimensional.finBasis`.
+See also `Module.finBasis`.
 -/
 noncomputable def Basis.ofRankEqZero [Module.Free K V] {ι : Type*} [IsEmpty ι]
     (hV : Module.rank K V = 0) : Basis ι K V :=
@@ -139,8 +139,10 @@ its span. -/
 theorem rank_submodule_eq_one_iff (s : Submodule K V) [Module.Free K s] :
     Module.rank K s = 1 ↔ ∃ v₀ ∈ s, v₀ ≠ 0 ∧ s ≤ K ∙ v₀ := by
   simp_rw [rank_eq_one_iff, le_span_singleton_iff]
-  refine ⟨fun ⟨⟨v₀, hv₀⟩, H, h⟩ ↦ ⟨v₀, hv₀, fun h' ↦ by simp [h'] at H, fun v hv ↦ ?_⟩,
-    fun ⟨v₀, hv₀, H, h⟩ ↦ ⟨⟨v₀, hv₀⟩, fun h' ↦ H (by simpa using h'), fun ⟨v, hv⟩ ↦ ?_⟩⟩
+  refine ⟨fun ⟨⟨v₀, hv₀⟩, H, h⟩ ↦ ⟨v₀, hv₀, fun h' ↦ by
+    simp only [h', ne_eq] at H; exact H rfl, fun v hv ↦ ?_⟩,
+    fun ⟨v₀, hv₀, H, h⟩ ↦ ⟨⟨v₀, hv₀⟩,
+      fun h' ↦ H (by rwa [AddSubmonoid.mk_eq_zero] at h'), fun ⟨v, hv⟩ ↦ ?_⟩⟩
   · obtain ⟨r, hr⟩ := h ⟨v, hv⟩
     exact ⟨r, by rwa [Subtype.ext_iff, coe_smul] at hr⟩
   · obtain ⟨r, hr⟩ := h v hv
@@ -184,7 +186,7 @@ theorem finrank_eq_one_iff [Module.Free K V] (ι : Type*) [Unique ι] :
     finrank K V = 1 ↔ Nonempty (Basis ι K V) := by
   constructor
   · intro h
-    exact ⟨basisUnique ι h⟩
+    exact ⟨Module.basisUnique ι h⟩
   · rintro ⟨b⟩
     simpa using finrank_eq_card_basis b
 
diff --git a/Mathlib/LinearAlgebra/Dimension/LinearMap.lean b/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
index 418403e43f094..ca2cb869a8ec5 100644
--- a/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
+++ b/Mathlib/LinearAlgebra/Dimension/LinearMap.lean
@@ -34,7 +34,7 @@ abbrev rank (f : V →ₗ[K] V') : Cardinal :=
   Module.rank K (LinearMap.range f)
 
 theorem rank_le_range (f : V →ₗ[K] V') : rank f ≤ Module.rank K V' :=
-  rank_submodule_le _
+  Submodule.rank_le _
 
 theorem rank_le_domain (f : V →ₗ[K] V₁) : rank f ≤ Module.rank K V :=
   rank_range_le _
@@ -46,7 +46,7 @@ theorem rank_zero [Nontrivial K] : rank (0 : V →ₗ[K] V') = 0 := by
 variable [AddCommGroup V''] [Module K V'']
 
 theorem rank_comp_le_left (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') : rank (f.comp g) ≤ rank f := by
-  refine rank_le_of_submodule _ _ ?_
+  refine Submodule.rank_mono ?_
   rw [LinearMap.range_comp]
   exact LinearMap.map_le_range
 
@@ -82,7 +82,7 @@ variable [AddCommGroup V'] [Module K V']
 theorem rank_add_le (f g : V →ₗ[K] V') : rank (f + g) ≤ rank f + rank g :=
   calc
     rank (f + g) ≤ Module.rank K (LinearMap.range f ⊔ LinearMap.range g : Submodule K V') := by
-      refine rank_le_of_submodule _ _ ?_
+      refine Submodule.rank_mono ?_
       exact LinearMap.range_le_iff_comap.2 <| eq_top_iff'.2 fun x =>
         show f x + g x ∈ (LinearMap.range f ⊔ LinearMap.range g : Submodule K V') from
         mem_sup.2 ⟨_, ⟨x, rfl⟩, _, ⟨x, rfl⟩, rfl⟩
diff --git a/Mathlib/LinearAlgebra/Dimension/Localization.lean b/Mathlib/LinearAlgebra/Dimension/Localization.lean
index 215eca3c7a657..b6445fadfe529 100644
--- a/Mathlib/LinearAlgebra/Dimension/Localization.lean
+++ b/Mathlib/LinearAlgebra/Dimension/Localization.lean
@@ -15,9 +15,9 @@ import Mathlib.RingTheory.OreLocalization.OreSet
 
 - `IsLocalizedModule.lift_rank_eq`: `rank_Rₚ Mₚ = rank R M`.
 - `rank_quotient_add_rank_of_isDomain`: The **rank-nullity theorem** for commutative domains.
-
 -/
-open Cardinal nonZeroDivisors
+
+open Cardinal Module nonZeroDivisors
 
 section CommRing
 
diff --git a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
index 39b863dc84a22..2e6cb6ba25096 100644
--- a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
+++ b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean
@@ -104,7 +104,7 @@ theorem exists_linearIndependent_of_lt_rank [StrongRankCondition R]
       ← rank_quotient_add_rank (Submodule.span R s), add_comm, rank_span_set hs]
     exact HasLeftInverse.injective ⟨Submodule.Quotient.mk, hsec⟩
   · apply LinearIndependent.union_of_quotient Submodule.subset_span hs
-    rwa [Function.comp, linearIndependent_image (hsec'.symm ▸ injective_id).injOn.image_of_comp,
+    rwa [Function.comp_def, linearIndependent_image (hsec'.symm ▸ injective_id).injOn.image_of_comp,
       ← image_comp, hsec', image_id]
 
 /-- Given a family of `n` linearly independent vectors in a space of dimension `> n`, one may extend
@@ -136,7 +136,7 @@ theorem exists_linearIndependent_pair_of_one_lt_rank [StrongRankCondition R]
     [NoZeroSMulDivisors R M] (h : 1 < Module.rank R M) {x : M} (hx : x ≠ 0) :
     ∃ y, LinearIndependent R ![x, y] := by
   obtain ⟨y, hy⟩ := exists_linearIndependent_snoc_of_lt_rank (linearIndependent_unique ![x] hx) h
-  have : Fin.snoc ![x] y = ![x, y] := Iff.mp List.ofFn_inj rfl
+  have : Fin.snoc ![x] y = ![x, y] := by simp [Fin.snoc, ← List.ofFn_inj]
   rw [this] at hy
   exact ⟨y, hy⟩
 
@@ -171,7 +171,7 @@ theorem Submodule.rank_add_le_rank_add_rank (s t : Submodule R M) :
 
 section Finrank
 
-open Submodule FiniteDimensional
+open Submodule Module
 
 variable [StrongRankCondition R]
 
diff --git a/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean b/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
index e76b36cfe1fb8..26dc00965d15b 100644
--- a/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
+++ b/Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.LinearAlgebra.Dimension.Finrank
 import Mathlib.LinearAlgebra.InvariantBasisNumber
@@ -106,10 +106,10 @@ theorem Basis.le_span'' {ι : Type*} [Fintype ι] (b : Basis ι R M) {w : Set M}
   -- We construct a surjective linear map `(w → R) →ₗ[R] (ι → R)`,
   -- by expressing a linear combination in `w` as a linear combination in `ι`.
   fapply card_le_of_surjective' R
-  · exact b.repr.toLinearMap.comp (Finsupp.total R (↑))
+  · exact b.repr.toLinearMap.comp (Finsupp.linearCombination R (↑))
   · apply Surjective.comp (g := b.repr.toLinearMap)
     · apply LinearEquiv.surjective
-    rw [← LinearMap.range_eq_top, Finsupp.range_total]
+    rw [← LinearMap.range_eq_top, Finsupp.range_linearCombination]
     simpa using s
 
 /--
@@ -162,7 +162,7 @@ section StrongRankCondition
 
 variable [StrongRankCondition R]
 
-open Submodule
+open Submodule Finsupp
 
 -- An auxiliary lemma for `linearIndependent_le_span'`,
 -- with the additional assumption that the linearly independent family is finite.
@@ -174,11 +174,12 @@ theorem linearIndependent_le_span_aux' {ι : Type*} [Fintype ι] (v : ι → M)
   -- and expressing that (using the axiom of choice) as a linear combination over `w`.
   -- We can do this linearly by constructing the map on a basis.
   fapply card_le_of_injective' R
-  · apply Finsupp.total
+  · apply Finsupp.linearCombination
     exact fun i => Span.repr R w ⟨v i, s (mem_range_self i)⟩
   · intro f g h
-    apply_fun Finsupp.total R ((↑) : w → M) at h
-    simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h
+    apply_fun linearCombination R ((↑) : w → M) at h
+    simp only [linearCombination_linearCombination, Submodule.coe_mk,
+               Span.finsupp_linearCombination_repr] at h
     rw [← sub_eq_zero, ← LinearMap.map_sub] at h
     exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h)
 
@@ -375,14 +376,12 @@ theorem Ideal.rank_eq {R S : Type*} [CommRing R] [StrongRankCondition R] [Ring S
       ((LinearMap.ker_eq_bot (f := (Submodule.subtype I : I →ₗ[R] S))).mpr Subtype.coe_injective)))
     (c.card_le_card_of_linearIndependent this)
 
-open FiniteDimensional
+namespace Module
 
 theorem finrank_eq_nat_card_basis (h : Basis ι R M) :
     finrank R M = Nat.card ι := by
   rw [Nat.card, ← toNat_lift.{v}, h.mk_eq_rank, toNat_lift, finrank]
 
-namespace FiniteDimensional
-
 /-- If a vector space (or module) has a finite basis, then its dimension (or rank) is equal to the
 cardinality of the basis. -/
 theorem finrank_eq_card_basis {ι : Type w} [Fintype ι] (h : Basis ι R M) :
@@ -391,8 +390,8 @@ theorem finrank_eq_card_basis {ι : Type w} [Fintype ι] (h : Basis ι R M) :
 
 /-- If a free module is of finite rank, then the cardinality of any basis is equal to its
 `finrank`. -/
-theorem _root_.Module.mk_finrank_eq_card_basis [Module.Finite R M]
-    {ι : Type w} (h : Basis ι R M) : (finrank R M : Cardinal.{w}) = #ι := by
+theorem mk_finrank_eq_card_basis [Module.Finite R M] {ι : Type w} (h : Basis ι R M) :
+    (finrank R M : Cardinal.{w}) = #ι := by
   cases @nonempty_fintype _ (Module.Finite.finite_basis h)
   rw [Cardinal.mk_fintype, finrank_eq_card_basis h]
 
@@ -401,10 +400,6 @@ cardinality of the basis. This lemma uses a `Finset` instead of indexed types. -
 theorem finrank_eq_card_finset_basis {ι : Type w} {b : Finset ι} (h : Basis b R M) :
     finrank R M = Finset.card b := by rw [finrank_eq_card_basis h, Fintype.card_coe]
 
-end FiniteDimensional
-
-open FiniteDimensional
-
 variable (R)
 
 @[simp]
@@ -414,15 +409,15 @@ theorem rank_self : Module.rank R R = 1 := by
 /-- A ring satisfying `StrongRankCondition` (such as a `DivisionRing`) is one-dimensional as a
 module over itself. -/
 @[simp]
-theorem FiniteDimensional.finrank_self : finrank R R = 1 :=
+theorem finrank_self : finrank R R = 1 :=
   finrank_eq_of_rank_eq (by simp)
 
 /-- Given a basis of a ring over itself indexed by a type `ι`, then `ι` is `Unique`. -/
-noncomputable def Basis.unique {ι : Type*} (b : Basis ι R R) : Unique ι := by
-  have A : Cardinal.mk ι = ↑(FiniteDimensional.finrank R R) :=
+noncomputable def _root_.Basis.unique {ι : Type*} (b : Basis ι R R) : Unique ι := by
+  have A : Cardinal.mk ι = ↑(Module.finrank R R) :=
     (Module.mk_finrank_eq_card_basis b).symm
   -- Porting note: replace `algebraMap.coe_one` with `Nat.cast_one`
-  simp only [Cardinal.eq_one_iff_unique, FiniteDimensional.finrank_self, Nat.cast_one] at A
+  simp only [Cardinal.eq_one_iff_unique, Module.finrank_self, Nat.cast_one] at A
   exact Nonempty.some ((unique_iff_subsingleton_and_nonempty _).2 A)
 
 variable (M)
@@ -435,19 +430,23 @@ theorem rank_lt_aleph0 [Module.Finite R M] : Module.rank R M < ℵ₀ := by
   refine (ciSup_le' fun i => ?_).trans_lt (nat_lt_aleph0 S.card)
   exact linearIndependent_le_span_finset _ i.prop S hS
 
-@[deprecated (since := "2024-01-01")]
-protected alias FiniteDimensional.rank_lt_aleph0 := rank_lt_aleph0
+noncomputable instance {R M : Type*} [DivisionRing R] [AddCommGroup M] [Module R M]
+    {s t : Set M} [Module.Finite R (span R t)]
+    (hs : LinearIndependent R ((↑) : s → M)) (hst : s ⊆ t) :
+    Fintype (hs.extend hst) := by
+  refine Classical.choice (Cardinal.lt_aleph0_iff_fintype.1 ?_)
+  rw [← rank_span_set (hs.linearIndependent_extend hst), hs.span_extend_eq_span]
+  exact Module.rank_lt_aleph0 ..
 
 /-- If `M` is finite, `finrank M = rank M`. -/
 @[simp]
-theorem finrank_eq_rank [Module.Finite R M] :
-    ↑(FiniteDimensional.finrank R M) = Module.rank R M := by
-  rw [FiniteDimensional.finrank, cast_toNat_of_lt_aleph0 (rank_lt_aleph0 R M)]
+theorem finrank_eq_rank [Module.Finite R M] : ↑(finrank R M) = Module.rank R M := by
+  rw [Module.finrank, cast_toNat_of_lt_aleph0 (rank_lt_aleph0 R M)]
+
+end Module
 
-@[deprecated (since := "2024-01-01")]
-protected alias FiniteDimensional.finrank_eq_rank := finrank_eq_rank
+open Module
 
-variable {R M}
 variable {M'} [AddCommGroup M'] [Module R M']
 
 theorem LinearMap.finrank_le_finrank_of_injective [Module.Finite R M'] {f : M →ₗ[R] M'}
diff --git a/Mathlib/LinearAlgebra/Dual.lean b/Mathlib/LinearAlgebra/Dual.lean
index 8e1d2e6a50520..5d556906e958a 100644
--- a/Mathlib/LinearAlgebra/Dual.lean
+++ b/Mathlib/LinearAlgebra/Dual.lean
@@ -91,6 +91,8 @@ The dual space of an $R$-module $M$ is the $R$-module of $R$-linear maps $M \to
     splitting of `V₁`.
 -/
 
+open Module Submodule
+
 noncomputable section
 
 namespace Module
@@ -181,7 +183,7 @@ def LinearMap.dualMap (f : M₁ →ₗ[R] M₂) : Dual R M₂ →ₗ[R] Dual R M
 -- Porting note: with reducible def need to specify some parameters to transpose explicitly
   Module.Dual.transpose (R := R) f
 
-lemma LinearMap.dualMap_eq_lcomp (f : M₁ →ₗ[R] M₂) : f.dualMap = f.lcomp R := rfl
+lemma LinearMap.dualMap_eq_lcomp (f : M₁ →ₗ[R] M₂) : f.dualMap = f.lcomp R R := rfl
 
 -- Porting note: with reducible def need to specify some parameters to transpose explicitly
 theorem LinearMap.dualMap_def (f : M₁ →ₗ[R] M₂) : f.dualMap = Module.Dual.transpose (R := R) f :=
@@ -280,29 +282,34 @@ theorem toDual_apply (i j : ι) : b.toDual (b i) (b j) = if i = j then 1 else 0
   simp only [eq_comm]
 
 @[simp]
-theorem toDual_total_left (f : ι →₀ R) (i : ι) :
-    b.toDual (Finsupp.total R b f) (b i) = f i := by
-  rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
+theorem toDual_linearCombination_left (f : ι →₀ R) (i : ι) :
+    b.toDual (Finsupp.linearCombination R b f) (b i) = f i := by
+  rw [Finsupp.linearCombination_apply, Finsupp.sum, _root_.map_sum, LinearMap.sum_apply]
   simp_rw [LinearMap.map_smul, LinearMap.smul_apply, toDual_apply, smul_eq_mul, mul_boole,
     Finset.sum_ite_eq']
   split_ifs with h
   · rfl
   · rw [Finsupp.not_mem_support_iff.mp h]
 
+@[deprecated (since := "2024-08-29")] alias toDual_total_left := toDual_linearCombination_left
+
 @[simp]
-theorem toDual_total_right (f : ι →₀ R) (i : ι) :
-    b.toDual (b i) (Finsupp.total R b f) = f i := by
-  rw [Finsupp.total_apply, Finsupp.sum, _root_.map_sum]
+theorem toDual_linearCombination_right (f : ι →₀ R) (i : ι) :
+    b.toDual (b i) (Finsupp.linearCombination R b f) = f i := by
+  rw [Finsupp.linearCombination_apply, Finsupp.sum, _root_.map_sum]
   simp_rw [LinearMap.map_smul, toDual_apply, smul_eq_mul, mul_boole, Finset.sum_ite_eq]
   split_ifs with h
   · rfl
   · rw [Finsupp.not_mem_support_iff.mp h]
 
+@[deprecated (since := "2024-08-29")] alias toDual_total_right :=
+  toDual_linearCombination_right
+
 theorem toDual_apply_left (m : M) (i : ι) : b.toDual m (b i) = b.repr m i := by
-  rw [← b.toDual_total_left, b.total_repr]
+  rw [← b.toDual_linearCombination_left, b.linearCombination_repr]
 
 theorem toDual_apply_right (i : ι) (m : M) : b.toDual (b i) m = b.repr m i := by
-  rw [← b.toDual_total_right, b.total_repr]
+  rw [← b.toDual_linearCombination_right, b.linearCombination_repr]
 
 theorem coe_toDual_self (i : ι) : b.toDual (b i) = b.coord i := by
   ext
@@ -335,8 +342,8 @@ theorem toDual_ker : LinearMap.ker b.toDual = ⊥ :=
 theorem toDual_range [Finite ι] : LinearMap.range b.toDual = ⊤ := by
   refine eq_top_iff'.2 fun f => ?_
   let lin_comb : ι →₀ R := Finsupp.equivFunOnFinite.symm fun i => f (b i)
-  refine ⟨Finsupp.total R b lin_comb, b.ext fun i => ?_⟩
-  rw [b.toDual_eq_repr _ i, repr_total b]
+  refine ⟨Finsupp.linearCombination R b lin_comb, b.ext fun i => ?_⟩
+  rw [b.toDual_eq_repr _ i, repr_linearCombination b]
   rfl
 
 end CommSemiring
@@ -379,7 +386,7 @@ theorem toDualEquiv_apply (m : M) : b.toDualEquiv m = b.toDual m :=
 theorem linearEquiv_dual_iff_finiteDimensional [Field K] [AddCommGroup V] [Module K V] :
     Nonempty (V ≃ₗ[K] Dual K V) ↔ FiniteDimensional K V := by
   refine ⟨fun ⟨e⟩ ↦ ?_, fun h ↦ ⟨(Module.Free.chooseBasis K V).toDualEquiv⟩⟩
-  rw [FiniteDimensional, ← Module.rank_lt_alpeh0_iff]
+  rw [FiniteDimensional, ← Module.rank_lt_aleph0_iff]
   by_contra!
   apply (lift_rank_lt_rank_dual this).ne
   have := e.lift_rank_eq
@@ -395,17 +402,19 @@ theorem dualBasis_apply_self (i j : ι) : b.dualBasis i (b j) =
   convert b.toDual_apply i j using 2
   rw [@eq_comm _ j i]
 
-theorem total_dualBasis (f : ι →₀ R) (i : ι) :
-    Finsupp.total R b.dualBasis f (b i) = f i := by
+theorem linearCombination_dualBasis (f : ι →₀ R) (i : ι) :
+    Finsupp.linearCombination R b.dualBasis f (b i) = f i := by
   cases nonempty_fintype ι
-  rw [Finsupp.total_apply, Finsupp.sum_fintype, LinearMap.sum_apply]
+  rw [Finsupp.linearCombination_apply, Finsupp.sum_fintype, LinearMap.sum_apply]
   · simp_rw [LinearMap.smul_apply, smul_eq_mul, dualBasis_apply_self, mul_boole,
     Finset.sum_ite_eq, if_pos (Finset.mem_univ i)]
   · intro
     rw [zero_smul]
 
+@[deprecated (since := "2024-08-29")] alias total_dualBasis := linearCombination_dualBasis
+
 theorem dualBasis_repr (l : Dual R M) (i : ι) : b.dualBasis.repr l i = l (b i) := by
-  rw [← total_dualBasis b, Basis.total_repr b.dualBasis l]
+  rw [← linearCombination_dualBasis b, Basis.linearCombination_repr b.dualBasis l]
 
 theorem dualBasis_apply (i : ι) (m : M) : b.dualBasis i m = b.repr m i :=
   b.toDual_apply_right i m
@@ -442,24 +451,26 @@ theorem eval_range {ι : Type*} [Finite ι] (b : Basis ι R M) :
 
 section
 
-variable [Finite R M] [Free R M]
+variable [Module.Finite R M] [Free R M]
 
 instance dual_free : Free R (Dual R M) :=
   Free.of_basis (Free.chooseBasis R M).dualBasis
 
-instance dual_finite : Finite R (Dual R M) :=
+instance dual_finite : Module.Finite R (Dual R M) :=
   Finite.of_basis (Free.chooseBasis R M).dualBasis
 
 end
 
 end CommRing
 
-/-- `simp` normal form version of `total_dualBasis` -/
+/-- `simp` normal form version of `linearCombination_dualBasis` -/
 @[simp]
-theorem total_coord [CommRing R] [AddCommGroup M] [Module R M] [Finite ι] (b : Basis ι R M)
-    (f : ι →₀ R) (i : ι) : Finsupp.total R b.coord f (b i) = f i := by
+theorem linearCombination_coord [CommRing R] [AddCommGroup M] [Module R M] [Finite ι]
+    (b : Basis ι R M) (f : ι →₀ R) (i : ι) : Finsupp.linearCombination R b.coord f (b i) = f i := by
   haveI := Classical.decEq ι
-  rw [← coe_dualBasis, total_dualBasis]
+  rw [← coe_dualBasis, linearCombination_dualBasis]
+
+@[deprecated (since := "2024-08-29")] alias total_coord := linearCombination_coord
 
 theorem dual_rank_eq [CommRing K] [AddCommGroup V] [Module K V] [Finite ι] (b : Basis ι K V) :
     Cardinal.lift.{uK,uV} (Module.rank K V) = Module.rank K (Dual K V) := by
@@ -473,7 +484,7 @@ universe uK uV
 variable {K : Type uK} {V : Type uV}
 variable [CommRing K] [AddCommGroup V] [Module K V] [Module.Free K V]
 
-open Module Module.Dual Submodule LinearMap Cardinal Basis FiniteDimensional
+open Module Module.Dual Submodule LinearMap Cardinal Basis Module
 
 section
 
@@ -531,7 +542,7 @@ theorem nontrivial_dual_iff :
 instance instNontrivialDual [Nontrivial V] : Nontrivial (Dual K V) :=
   (nontrivial_dual_iff K).mpr inferInstance
 
-theorem finite_dual_iff : Finite K (Dual K V) ↔ Finite K V := by
+theorem finite_dual_iff : Module.Finite K (Dual K V) ↔ Module.Finite K V := by
   constructor <;> intro h
   · obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := K) (M := V)
     nontriviality K
@@ -569,7 +580,8 @@ class IsReflexive : Prop where
 lemma bijective_dual_eval [IsReflexive R M] : Bijective (Dual.eval R M) :=
   IsReflexive.bijective_dual_eval'
 
-instance IsReflexive.of_finite_of_free [Finite R M] [Free R M] : IsReflexive R M where
+/-- See also `Module.instFiniteDimensionalOfIsReflexive` for the converse over a field. -/
+instance IsReflexive.of_finite_of_free [Module.Finite R M] [Free R M] : IsReflexive R M where
   bijective_dual_eval' := ⟨LinearMap.ker_eq_bot.mp (Free.chooseBasis R M).eval_ker,
                            LinearMap.range_eq_top.mp (Free.chooseBasis R M).eval_range⟩
 
@@ -639,6 +651,22 @@ instance _root_.MulOpposite.instModuleIsReflexive : IsReflexive R (MulOpposite M
 instance _root_.ULift.instModuleIsReflexive.{w} : IsReflexive R (ULift.{w} M) :=
   equiv ULift.moduleEquiv.symm
 
+instance instFiniteDimensionalOfIsReflexive (K V : Type*)
+    [Field K] [AddCommGroup V] [Module K V] [IsReflexive K V] :
+    FiniteDimensional K V := by
+  rw [FiniteDimensional, ← rank_lt_aleph0_iff]
+  by_contra! contra
+  suffices lift (Module.rank K V) < Module.rank K (Dual K (Dual K V)) by
+    have heq := lift_rank_eq_of_equiv_equiv (R := K) (R' := K) (M := V) (M' := Dual K (Dual K V))
+      (ZeroHom.id K) (evalEquiv K V) bijective_id (fun r v ↦ (evalEquiv K V).map_smul _ _)
+    rw [← lift_umax, heq, lift_id'] at this
+    exact lt_irrefl _ this
+  have h₁ : lift (Module.rank K V) < Module.rank K (Dual K V) := lift_rank_lt_rank_dual contra
+  have h₂ : Module.rank K (Dual K V) < Module.rank K (Dual K (Dual K V)) := by
+    convert lift_rank_lt_rank_dual <| le_trans (by simpa) h₁.le
+    rw [lift_id']
+  exact lt_trans h₁ h₂
+
 end IsReflexive
 
 end Module
@@ -662,6 +690,47 @@ theorem exists_dual_map_eq_bot_of_lt_top (hp : p < ⊤) (hp' : Free R (M ⧸ p))
   obtain ⟨f, hf, hf'⟩ := p.exists_dual_map_eq_bot_of_nmem hx hp'
   exact ⟨f, by aesop, hf'⟩
 
+variable {ι 𝕜 E : Type*} [Field 𝕜] [AddCommGroup E] [Module 𝕜 E]
+
+open LinearMap Set FiniteDimensional
+
+theorem _root_.FiniteDimensional.mem_span_of_iInf_ker_le_ker [FiniteDimensional 𝕜 E]
+    {L : ι → E →ₗ[𝕜] 𝕜} {K : E →ₗ[𝕜] 𝕜}
+    (h : ⨅ i, LinearMap.ker (L i) ≤ ker K) : K ∈ span 𝕜 (range L) := by
+  by_contra hK
+  rcases exists_dual_map_eq_bot_of_nmem hK inferInstance with ⟨φ, φne, hφ⟩
+  let φs := (Module.evalEquiv 𝕜 E).symm φ
+  have : K φs = 0 := by
+    refine h <| (Submodule.mem_iInf _).2 fun i ↦ (mem_bot 𝕜).1 ?_
+    rw [← hφ, Submodule.mem_map]
+    exact ⟨L i, Submodule.subset_span ⟨i, rfl⟩, (apply_evalEquiv_symm_apply 𝕜 E _ φ).symm⟩
+  simp only [apply_evalEquiv_symm_apply, φs, φne] at this
+
+/-- Given some linear forms $L_1, ..., L_n, K$ over a vector space $E$, if
+$\bigcap_{i=1}^n \mathrm{ker}(L_i) \subseteq \mathrm{ker}(K)$, then $K$ is in the space generated
+by $L_1, ..., L_n$. -/
+theorem _root_.mem_span_of_iInf_ker_le_ker [Finite ι] {L : ι → E →ₗ[𝕜] 𝕜} {K : E →ₗ[𝕜] 𝕜}
+    (h : ⨅ i, ker (L i) ≤ ker K) : K ∈ span 𝕜 (range L) := by
+  have _ := Fintype.ofFinite ι
+  let φ : E →ₗ[𝕜] ι → 𝕜 := LinearMap.pi L
+  let p := ⨅ i, ker (L i)
+  have p_eq : p = ker φ := (ker_pi L).symm
+  let ψ : (E ⧸ p) →ₗ[𝕜] ι → 𝕜 := p.liftQ φ p_eq.le
+  have _ : FiniteDimensional 𝕜 (E ⧸ p) := of_injective ψ (ker_eq_bot.1 (ker_liftQ_eq_bot' p φ p_eq))
+  let L' i : (E ⧸ p) →ₗ[𝕜] 𝕜 := p.liftQ (L i) (iInf_le _ i)
+  let K' : (E ⧸ p) →ₗ[𝕜] 𝕜 := p.liftQ K h
+  have : ⨅ i, ker (L' i) ≤ ker K' := by
+    simp_rw [← ker_pi, L', pi_liftQ_eq_liftQ_pi, ker_liftQ_eq_bot' p φ p_eq]
+    exact bot_le
+  obtain ⟨c, hK'⟩ :=
+    (mem_span_range_iff_exists_fun 𝕜).1 (FiniteDimensional.mem_span_of_iInf_ker_le_ker this)
+  refine (mem_span_range_iff_exists_fun 𝕜).2 ⟨c, ?_⟩
+  conv_lhs => enter [2]; intro i; rw [← p.liftQ_mkQ (L i) (iInf_le _ i)]
+  rw [← p.liftQ_mkQ K h]
+  ext x
+  convert LinearMap.congr_fun hK' (p.mkQ x)
+  simp only [coeFn_sum, Finset.sum_apply, smul_apply, coe_comp, Function.comp_apply, smul_eq_mul]
+
 end Submodule
 
 section DualBases
@@ -708,11 +777,11 @@ theorem coeffs_apply (h : DualBases e ε) (m : M) (i : ι) : h.coeffs m i = ε i
   rfl
 
 /-- linear combinations of elements of `e`.
-This is a convenient abbreviation for `Finsupp.total R e l` -/
+This is a convenient abbreviation for `Finsupp.linearCombination R e l` -/
 def lc {ι} (e : ι → M) (l : ι →₀ R) : M :=
   l.sum fun (i : ι) (a : R) => a • e i
 
-theorem lc_def (e : ι → M) (l : ι →₀ R) : lc e l = Finsupp.total R e l :=
+theorem lc_def (e : ι → M) (l : ι →₀ R) : lc e l = Finsupp.linearCombination R e l :=
   rfl
 
 open Module
@@ -767,7 +836,7 @@ theorem coe_basis : ⇑h.basis = e := by
 theorem mem_of_mem_span {H : Set ι} {x : M} (hmem : x ∈ Submodule.span R (e '' H)) :
     ∀ i : ι, ε i x ≠ 0 → i ∈ H := by
   intro i hi
-  rcases (Finsupp.mem_span_image_iff_total _).mp hmem with ⟨l, supp_l, rfl⟩
+  rcases (Finsupp.mem_span_image_iff_linearCombination _).mp hmem with ⟨l, supp_l, rfl⟩
   apply not_imp_comm.mp ((Finsupp.mem_supported' _ _).mp supp_l i)
   rwa [← lc_def, h.dual_lc] at hi
 
@@ -1048,7 +1117,7 @@ theorem dualEquivDual_apply (φ : Module.Dual K W) :
 
 section
 
-open FiniteDimensional
+open FiniteDimensional Module
 
 instance instModuleDualFiniteDimensional [FiniteDimensional K V] :
     FiniteDimensional K (Module.Dual K V) := by
@@ -1070,7 +1139,7 @@ theorem dualAnnihilator_dualAnnihilator_eq (W : Subspace K V) :
   rw [dualCoannihilator, ← Module.mapEvalEquiv_symm_apply] at this
   rwa [← OrderIso.symm_apply_eq]
 
-/-- The quotient by the dual is isomorphic to its dual annihilator.  -/
+/-- The quotient by the dual is isomorphic to its dual annihilator. -/
 -- Porting note (#11036): broken dot notation lean4#1910 LinearMap.range
 noncomputable def quotDualEquivAnnihilator (W : Subspace K V) :
     (Module.Dual K V ⧸ LinearMap.range W.dualLift) ≃ₗ[K] W.dualAnnihilator :=
@@ -1087,7 +1156,7 @@ noncomputable def quotEquivAnnihilator (W : Subspace K V) : (V ⧸ W) ≃ₗ[K]
   -- refine' LinearEquiv.quot_equiv_of_equiv _ (Basis.ofVectorSpace K V).toDualEquiv
   -- exact (Basis.ofVectorSpace K W).toDualEquiv.trans W.dual_equiv_dual
 
-open FiniteDimensional
+open Module
 
 @[simp]
 theorem finrank_dualCoannihilator_eq {Φ : Subspace K (Module.Dual K V)} :
@@ -1155,12 +1224,12 @@ def dualCopairing (W : Submodule R M) : W.dualAnnihilator →ₗ[R] M ⧸ W →
         exact (mem_dualAnnihilator φ).mp hφ w hw)
 
 -- Porting note: helper instance
-instance (W : Submodule R M) : FunLike (W.dualAnnihilator) M R :=
-  { coe := fun φ => φ.val,
-    coe_injective' := fun φ ψ h => by
-      ext
-      simp only [Function.funext_iff] at h
-      exact h _ }
+instance (W : Submodule R M) : FunLike (W.dualAnnihilator) M R where
+  coe φ := φ.val
+  coe_injective' φ ψ h := by
+    ext
+    simp only [Function.funext_iff] at h
+    exact h _
 
 @[simp]
 theorem dualCopairing_apply {W : Submodule R M} (φ : W.dualAnnihilator) (x : M) :
@@ -1222,7 +1291,7 @@ theorem dualQuotEquivDualAnnihilator_symm_apply_mk (W : Submodule R M) (φ : W.d
   rfl
 
 theorem finite_dualAnnihilator_iff {W : Submodule R M} [Free R (M ⧸ W)] :
-    Finite R W.dualAnnihilator ↔ Finite R (M ⧸ W) :=
+    Module.Finite R W.dualAnnihilator ↔ Module.Finite R (M ⧸ W) :=
   (Finite.equiv_iff W.dualQuotEquivDualAnnihilator.symm).trans (finite_dual_iff R)
 
 open LinearMap in
@@ -1315,7 +1384,6 @@ lemma range_eq_top_of_ne_zero :
   rw [eq_top_iff]
   exact fun x _ ↦ ⟨x • (f v)⁻¹ • v, by simp [inv_mul_cancel₀ hv]⟩
 
-open FiniteDimensional
 variable [FiniteDimensional K V₁]
 
 lemma finrank_ker_add_one_of_ne_zero :
@@ -1470,7 +1538,7 @@ end Subspace
 
 section FiniteDimensional
 
-open FiniteDimensional LinearMap
+open Module LinearMap
 
 namespace LinearMap
 
@@ -1601,7 +1669,7 @@ theorem dualAnnihilator_dualAnnihilator_eq_map (W : Subspace K V) [FiniteDimensi
   haveI := e1.finiteDimensional
   let e2 := (Free.chooseBasis K _).toDualEquiv ≪≫ₗ W.dualAnnihilator.dualQuotEquivDualAnnihilator
   haveI := LinearEquiv.finiteDimensional (V₂ := W.dualAnnihilator.dualAnnihilator) e2
-  rw [FiniteDimensional.eq_of_le_of_finrank_eq (map_le_dualAnnihilator_dualAnnihilator W)]
+  rw [eq_of_le_of_finrank_eq (map_le_dualAnnihilator_dualAnnihilator W)]
   rw [← (equivMapOfInjective _ (eval_apply_injective K (V := V)) W).finrank_eq, e1.finrank_eq]
   exact e2.finrank_eq
 
@@ -1738,3 +1806,5 @@ noncomputable def dualDistribEquiv : Dual R M ⊗[R] Dual R N ≃ₗ[R] Dual R (
   dualDistribEquivOfBasis (Module.Free.chooseBasis R M) (Module.Free.chooseBasis R N)
 
 end TensorProduct
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Basic.lean b/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
index b10d365990732..e2e7aa9fcb4c9 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Basic.lean
@@ -53,209 +53,530 @@ namespace Module
 
 namespace End
 
-open FiniteDimensional Set
+open Module Set
 
 variable {K R : Type v} {V M : Type w} [CommRing R] [AddCommGroup M] [Module R M] [Field K]
   [AddCommGroup V] [Module K V]
 
+/-- The submodule `unifEigenspace f μ k` for a linear map `f`, a scalar `μ`,
+and a number `k : ℕ∞` is the kernel of `(f - μ • id) ^ k` if `k` is a natural number,
+or the union of all these kernels if `k = ∞`. -/
+def unifEigenspace (f : End R M) (μ : R) : ℕ∞ →o Submodule R M where
+  toFun k := ⨆ l : ℕ, ⨆ _ : l ≤ k, LinearMap.ker ((f - μ • 1) ^ l)
+  monotone' _ _ hkl := biSup_mono fun _ hi ↦ hi.trans hkl
+
+lemma mem_unifEigenspace {f : End R M} {μ : R} {k : ℕ∞} {x : M} :
+    x ∈ f.unifEigenspace μ k ↔ ∃ l : ℕ, l ≤ k ∧ x ∈ LinearMap.ker ((f - μ • 1) ^ l) := by
+  have : Nonempty {l : ℕ // l ≤ k} := ⟨⟨0, zero_le _⟩⟩
+  have : Directed (ι := { i : ℕ // i ≤ k }) (· ≤ ·) fun i ↦ LinearMap.ker ((f - μ • 1) ^ (i : ℕ)) :=
+    Monotone.directed_le fun m n h ↦ by simpa using (f - μ • 1).iterateKer.monotone h
+  simp_rw [unifEigenspace, OrderHom.coe_mk, LinearMap.mem_ker, iSup_subtype',
+    Submodule.mem_iSup_of_directed _ this, LinearMap.mem_ker, Subtype.exists, exists_prop]
+
+lemma unifEigenspace_directed {f : End R M} {μ : R} {k : ℕ∞} :
+    Directed (· ≤ ·) (fun l : {l : ℕ // l ≤ k} ↦ f.unifEigenspace μ l) := by
+  have aux : Monotone ((↑) : {l : ℕ // l ≤ k} → ℕ∞) := fun x y h ↦ by simpa using h
+  exact ((unifEigenspace f μ).monotone.comp aux).directed_le
+
+lemma mem_unifEigenspace_nat {f : End R M} {μ : R} {k : ℕ} {x : M} :
+    x ∈ f.unifEigenspace μ k ↔ x ∈ LinearMap.ker ((f - μ • 1) ^ k) := by
+  rw [mem_unifEigenspace]
+  constructor
+  · rintro ⟨l, hl, hx⟩
+    simp only [Nat.cast_le] at hl
+    exact (f - μ • 1).iterateKer.monotone hl hx
+  · intro hx
+    exact ⟨k, le_rfl, hx⟩
+
+lemma mem_unifEigenspace_top {f : End R M} {μ : R} {x : M} :
+    x ∈ f.unifEigenspace μ ⊤ ↔ ∃ k : ℕ, x ∈ LinearMap.ker ((f - μ • 1) ^ k) := by
+  simp [mem_unifEigenspace]
+
+lemma unifEigenspace_nat {f : End R M} {μ : R} {k : ℕ} :
+    f.unifEigenspace μ k = LinearMap.ker ((f - μ • 1) ^ k) := by
+  ext; simp [mem_unifEigenspace_nat]
+
+lemma unifEigenspace_eq_iSup_unifEigenspace_nat (f : End R M) (μ : R) (k : ℕ∞) :
+    f.unifEigenspace μ k = ⨆ l : {l : ℕ // l ≤ k}, f.unifEigenspace μ l := by
+  simp_rw [unifEigenspace_nat, unifEigenspace, OrderHom.coe_mk, iSup_subtype]
+
+lemma unifEigenspace_top (f : End R M) (μ : R) :
+    f.unifEigenspace μ ⊤ = ⨆ k : ℕ, f.unifEigenspace μ k := by
+  rw [unifEigenspace_eq_iSup_unifEigenspace_nat, iSup_subtype]
+  simp only [le_top, iSup_pos, OrderHom.coe_mk]
+
+lemma unifEigenspace_one {f : End R M} {μ : R} :
+    f.unifEigenspace μ 1 = LinearMap.ker (f - μ • 1) := by
+  rw [← Nat.cast_one, unifEigenspace_nat, pow_one]
+
+@[simp]
+lemma mem_unifEigenspace_one {f : End R M} {μ : R} {x : M} :
+    x ∈ f.unifEigenspace μ 1 ↔ f x = μ • x := by
+  rw [unifEigenspace_one, LinearMap.mem_ker, LinearMap.sub_apply,
+    sub_eq_zero, LinearMap.smul_apply, LinearMap.one_apply]
+
+-- `simp` can prove this using `unifEigenspace_zero`
+lemma mem_unifEigenspace_zero {f : End R M} {μ : R} {x : M} :
+    x ∈ f.unifEigenspace μ 0 ↔ x = 0 := by
+  rw [← Nat.cast_zero, mem_unifEigenspace_nat, pow_zero, LinearMap.mem_ker, LinearMap.one_apply]
+
+@[simp]
+lemma unifEigenspace_zero {f : End R M} {μ : R} :
+    f.unifEigenspace μ 0 = ⊥ := by
+  ext; apply mem_unifEigenspace_zero
+
+@[simp]
+lemma unifEigenspace_zero_nat (f : End R M) (k : ℕ) :
+    f.unifEigenspace 0 k = LinearMap.ker (f ^ k) := by
+  ext; simp [mem_unifEigenspace_nat]
+
+/-- Let `M` be an `R`-module, and `f` an `R`-linear endomorphism of `M`,
+and let `μ : R` and `k : ℕ∞` be given.
+Then `x : M` satisfies `HasUnifEigenvector f μ k x` if
+`x ∈ f.unifEigenspace μ k` and `x ≠ 0`.
+
+For `k = 1`, this means that `x` is an eigenvector of `f` with eigenvalue `μ`. -/
+def HasUnifEigenvector (f : End R M) (μ : R) (k : ℕ∞) (x : M) : Prop :=
+  x ∈ f.unifEigenspace μ k ∧ x ≠ 0
+
+/-- Let `M` be an `R`-module, and `f` an `R`-linear endomorphism of `M`.
+Then `μ : R` and `k : ℕ∞` satisfy `HasUnifEigenvalue f μ k` if
+`f.unifEigenspace μ k ≠ ⊥`.
+
+For `k = 1`, this means that `μ` is an eigenvalue of `f`. -/
+def HasUnifEigenvalue (f : End R M) (μ : R) (k : ℕ∞) : Prop :=
+  f.unifEigenspace μ k ≠ ⊥
+
+/-- Let `M` be an `R`-module, and `f` an `R`-linear endomorphism of `M`.
+For `k : ℕ∞`, we define `UnifEigenvalues f k` to be the type of all
+`μ : R` that satisfy `f.HasUnifEigenvalue μ k`.
+
+For `k = 1` this is the type of all eigenvalues of `f`. -/
+def UnifEigenvalues (f : End R M) (k : ℕ∞) : Type _ :=
+  { μ : R // f.HasUnifEigenvalue μ k }
+
+/-- The underlying value of a bundled eigenvalue. -/
+@[coe]
+def UnifEigenvalues.val (f : Module.End R M) (k : ℕ∞) : UnifEigenvalues f k → R := Subtype.val
+
+instance UnifEigenvalues.instCoeOut {f : Module.End R M} (k : ℕ∞) :
+    CoeOut (UnifEigenvalues f k) R where
+  coe := UnifEigenvalues.val f k
+
+instance UnivEigenvalues.instDecidableEq [DecidableEq R] (f : Module.End R M) (k : ℕ∞) :
+    DecidableEq (UnifEigenvalues f k) :=
+  inferInstanceAs (DecidableEq (Subtype (fun x : R ↦ f.HasUnifEigenvalue x k)))
+
+lemma HasUnifEigenvector.hasUnifEigenvalue {f : End R M} {μ : R} {k : ℕ∞} {x : M}
+    (h : f.HasUnifEigenvector μ k x) : f.HasUnifEigenvalue μ k := by
+  rw [HasUnifEigenvalue, Submodule.ne_bot_iff]
+  use x; exact h
+
+lemma HasUnifEigenvector.apply_eq_smul {f : End R M} {μ : R} {x : M}
+    (hx : f.HasUnifEigenvector μ 1 x) : f x = μ • x :=
+  mem_unifEigenspace_one.mp hx.1
+
+lemma HasUnifEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasUnifEigenvector μ 1 v)
+    (n : ℕ) : (f ^ n) v = μ ^ n • v := by
+  induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
+
+theorem HasUnifEigenvalue.exists_hasUnifEigenvector
+    {f : End R M} {μ : R} {k : ℕ∞} (hμ : f.HasUnifEigenvalue μ k) :
+    ∃ v, f.HasUnifEigenvector μ k v :=
+  Submodule.exists_mem_ne_zero_of_ne_bot hμ
+
+lemma HasUnifEigenvalue.pow {f : End R M} {μ : R} (h : f.HasUnifEigenvalue μ 1) (n : ℕ) :
+    (f ^ n).HasUnifEigenvalue (μ ^ n) 1 := by
+  rw [HasUnifEigenvalue, Submodule.ne_bot_iff]
+  obtain ⟨m : M, hm⟩ := h.exists_hasUnifEigenvector
+  exact ⟨m, by simpa [mem_unifEigenspace_one] using hm.pow_apply n, hm.2⟩
+
+/-- A nilpotent endomorphism has nilpotent eigenvalues.
+
+See also `LinearMap.isNilpotent_trace_of_isNilpotent`. -/
+lemma HasUnifEigenvalue.isNilpotent_of_isNilpotent [NoZeroSMulDivisors R M] {f : End R M}
+    (hfn : IsNilpotent f) {μ : R} (hf : f.HasUnifEigenvalue μ 1) :
+    IsNilpotent μ := by
+  obtain ⟨m : M, hm⟩ := hf.exists_hasUnifEigenvector
+  obtain ⟨n : ℕ, hn : f ^ n = 0⟩ := hfn
+  exact ⟨n, by simpa [hn, hm.2, eq_comm (a := (0 : M))] using hm.pow_apply n⟩
+
+lemma HasUnifEigenvalue.mem_spectrum {f : End R M} {μ : R} (hμ : HasUnifEigenvalue f μ 1) :
+    μ ∈ spectrum R f := by
+  refine spectrum.mem_iff.mpr fun h_unit ↦ ?_
+  set f' := LinearMap.GeneralLinearGroup.toLinearEquiv h_unit.unit
+  rcases hμ.exists_hasUnifEigenvector with ⟨v, hv⟩
+  refine hv.2 ((LinearMap.ker_eq_bot'.mp f'.ker) v (?_ : μ • v - f v = 0))
+  rw [hv.apply_eq_smul, sub_self]
+
+lemma hasUnifEigenvalue_iff_mem_spectrum [FiniteDimensional K V] {f : End K V} {μ : K} :
+    f.HasUnifEigenvalue μ 1 ↔ μ ∈ spectrum K f := by
+  rw [spectrum.mem_iff, IsUnit.sub_iff, LinearMap.isUnit_iff_ker_eq_bot,
+    HasUnifEigenvalue, unifEigenspace_one, ne_eq, not_iff_not]
+  simp [Submodule.ext_iff, LinearMap.mem_ker]
+
+alias ⟨_, HasUnifEigenvalue.of_mem_spectrum⟩ := hasUnifEigenvalue_iff_mem_spectrum
+
+lemma unifEigenspace_div (f : End K V) (a b : K) (hb : b ≠ 0) :
+    unifEigenspace f (a / b) 1 = LinearMap.ker (b • f - a • 1) :=
+  calc
+    unifEigenspace f (a / b) 1 = unifEigenspace f (b⁻¹ * a) 1 := by rw [div_eq_mul_inv, mul_comm]
+    _ = LinearMap.ker (f - (b⁻¹ * a) • 1)     := by rw [unifEigenspace_one]
+    _ = LinearMap.ker (f - b⁻¹ • a • 1)       := by rw [smul_smul]
+    _ = LinearMap.ker (b • (f - b⁻¹ • a • 1)) := by rw [LinearMap.ker_smul _ b hb]
+    _ = LinearMap.ker (b • f - a • 1)         := by rw [smul_sub, smul_inv_smul₀ hb]
+
+/-- The generalized eigenrange for a linear map `f`, a scalar `μ`, and an exponent `k ∈ ℕ∞`
+is the range of `(f - μ • id) ^ k` if `k` is a natural number,
+or the infimum of these ranges if `k = ∞`. -/
+def unifEigenrange (f : End R M) (μ : R) (k : ℕ∞) : Submodule R M :=
+  ⨅ l : ℕ, ⨅ (_ : l ≤ k), LinearMap.range ((f - μ • 1) ^ l)
+
+lemma unifEigenrange_nat {f : End R M} {μ : R} {k : ℕ} :
+    f.unifEigenrange μ k = LinearMap.range ((f - μ • 1) ^ k) := by
+  ext x
+  simp only [unifEigenrange, Nat.cast_le, Submodule.mem_iInf, LinearMap.mem_range]
+  constructor
+  · intro h
+    exact h _ le_rfl
+  · rintro ⟨x, rfl⟩ i hi
+    have : k = i + (k - i) := by omega
+    rw [this, pow_add]
+    exact ⟨_, rfl⟩
+
+/-- The exponent of a generalized eigenvalue is never 0. -/
+lemma HasUnifEigenvalue.exp_ne_zero {f : End R M} {μ : R} {k : ℕ}
+    (h : f.HasUnifEigenvalue μ k) : k ≠ 0 := by
+  rintro rfl
+  simp [HasUnifEigenvalue, Nat.cast_zero, unifEigenspace_zero] at h
+
+/-- If there exists a natural number `k` such that the kernel of `(f - μ • id) ^ k` is the
+maximal generalized eigenspace, then this value is the least such `k`. If not, this value is not
+meaningful. -/
+noncomputable def maxUnifEigenspaceIndex (f : End R M) (μ : R) :=
+  monotonicSequenceLimitIndex <| (f.unifEigenspace μ).comp <| WithTop.coeOrderHom.toOrderHom
+
+/-- For an endomorphism of a Noetherian module, the maximal eigenspace is always of the form kernel
+`(f - μ • id) ^ k` for some `k`. -/
+lemma unifEigenspace_top_eq_maxUnifEigenspaceIndex [h : IsNoetherian R M] (f : End R M) (μ : R) :
+    unifEigenspace f μ ⊤ = f.unifEigenspace μ (maxUnifEigenspaceIndex f μ) := by
+  rw [isNoetherian_iff] at h
+  have := WellFounded.iSup_eq_monotonicSequenceLimit h <|
+    (f.unifEigenspace μ).comp <| WithTop.coeOrderHom.toOrderHom
+  convert this using 1
+  simp only [unifEigenspace, OrderHom.coe_mk, le_top, iSup_pos, OrderHom.comp_coe,
+    Function.comp_def]
+  rw [iSup_prod', iSup_subtype', ← sSup_range, ← sSup_range]
+  congr
+  aesop
+
+lemma unifEigenspace_le_unifEigenspace_maxUnifEigenspaceIndex [IsNoetherian R M] (f : End R M)
+    (μ : R) (k : ℕ∞) :
+    f.unifEigenspace μ k ≤ f.unifEigenspace μ (maxUnifEigenspaceIndex f μ) := by
+  rw [← unifEigenspace_top_eq_maxUnifEigenspaceIndex]
+  exact (f.unifEigenspace μ).monotone le_top
+
+/-- Generalized eigenspaces for exponents at least `finrank K V` are equal to each other. -/
+theorem unifEigenspace_eq_unifEigenspace_maxUnifEigenspaceIndex_of_le [IsNoetherian R M]
+    (f : End R M) (μ : R) {k : ℕ} (hk : maxUnifEigenspaceIndex f μ ≤ k) :
+    f.unifEigenspace μ k = f.unifEigenspace μ (maxUnifEigenspaceIndex f μ) :=
+  le_antisymm
+    (unifEigenspace_le_unifEigenspace_maxUnifEigenspaceIndex _ _ _)
+    ((f.unifEigenspace μ).monotone <| by simpa using hk)
+
+/-- A generalized eigenvalue for some exponent `k` is also
+    a generalized eigenvalue for exponents larger than `k`. -/
+lemma HasUnifEigenvalue.le {f : End R M} {μ : R} {k m : ℕ∞}
+    (hm : k ≤ m) (hk : f.HasUnifEigenvalue μ k) :
+    f.HasUnifEigenvalue μ m := by
+  unfold HasUnifEigenvalue at *
+  contrapose! hk
+  rw [← le_bot_iff, ← hk]
+  exact (f.unifEigenspace _).monotone hm
+
+/-- A generalized eigenvalue for some exponent `k` is also
+    a generalized eigenvalue for positive exponents. -/
+lemma HasUnifEigenvalue.lt {f : End R M} {μ : R} {k m : ℕ∞}
+    (hm : 0 < m) (hk : f.HasUnifEigenvalue μ k) :
+    f.HasUnifEigenvalue μ m := by
+  apply HasUnifEigenvalue.le (k := 1) (Order.one_le_iff_pos.mpr hm)
+  intro contra; apply hk
+  rw [unifEigenspace_one, LinearMap.ker_eq_bot] at contra
+  rw [eq_bot_iff]
+  intro x hx
+  rw [mem_unifEigenspace] at hx
+  rcases hx with ⟨l, -, hx⟩
+  rwa [LinearMap.ker_eq_bot.mpr] at hx
+  rw [LinearMap.coe_pow (f - μ • 1) l]
+  exact Function.Injective.iterate contra l
+
+/-- Generalized eigenvalues are actually just eigenvalues. -/
+@[simp]
+lemma hasUnifEigenvalue_iff_hasUnifEigenvalue_one {f : End R M} {μ : R} {k : ℕ∞} (hk : 0 < k) :
+    f.HasUnifEigenvalue μ k ↔ f.HasUnifEigenvalue μ 1 :=
+  ⟨HasUnifEigenvalue.lt zero_lt_one, HasUnifEigenvalue.lt hk⟩
+
+lemma maxUnifEigenspaceIndex_le_finrank [FiniteDimensional K V] (f : End K V) (μ : K) :
+    maxUnifEigenspaceIndex f μ ≤ finrank K V := by
+  apply Nat.sInf_le
+  intro n hn
+  apply le_antisymm
+  · exact (f.unifEigenspace μ).monotone <| WithTop.coeOrderHom.monotone hn
+  · show (f.unifEigenspace μ) n ≤ (f.unifEigenspace μ) (finrank K V)
+    rw [unifEigenspace_nat, unifEigenspace_nat]
+    apply ker_pow_le_ker_pow_finrank
+
+/-- Every generalized eigenvector is a generalized eigenvector for exponent `finrank K V`.
+    (Lemma 8.11 of [axler2015]) -/
+lemma unifEigenspace_le_unifEigenspace_finrank [FiniteDimensional K V] (f : End K V)
+    (μ : K) (k : ℕ∞) : f.unifEigenspace μ k ≤ f.unifEigenspace μ (finrank K V) := by
+  calc f.unifEigenspace μ k
+      ≤ f.unifEigenspace μ ⊤ := (f.unifEigenspace _).monotone le_top
+    _ ≤ f.unifEigenspace μ (finrank K V) := by
+      rw [unifEigenspace_top_eq_maxUnifEigenspaceIndex]
+      exact (f.unifEigenspace _).monotone <| by simpa using maxUnifEigenspaceIndex_le_finrank f μ
+
+/-- Generalized eigenspaces for exponents at least `finrank K V` are equal to each other. -/
+theorem unifEigenspace_eq_unifEigenspace_finrank_of_le [FiniteDimensional K V]
+    (f : End K V) (μ : K) {k : ℕ} (hk : finrank K V ≤ k) :
+    f.unifEigenspace μ k = f.unifEigenspace μ (finrank K V) :=
+  le_antisymm
+    (unifEigenspace_le_unifEigenspace_finrank _ _ _)
+    ((f.unifEigenspace μ).monotone <| by simpa using hk)
+
+lemma mapsTo_unifEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) (k : ℕ∞) :
+    MapsTo g (f.unifEigenspace μ k) (f.unifEigenspace μ k) := by
+  intro x hx
+  simp only [SetLike.mem_coe, mem_unifEigenspace, LinearMap.mem_ker] at hx ⊢
+  rcases hx with ⟨l, hl, hx⟩
+  replace h : Commute ((f - μ • (1 : End R M)) ^ l) g :=
+    (h.sub_left <| Algebra.commute_algebraMap_left μ g).pow_left l
+  use l, hl
+  rw [← LinearMap.comp_apply, ← LinearMap.mul_eq_comp, h.eq, LinearMap.mul_eq_comp,
+    LinearMap.comp_apply, hx, map_zero]
+
+/-- The restriction of `f - μ • 1` to the `k`-fold generalized `μ`-eigenspace is nilpotent. -/
+lemma isNilpotent_restrict_unifEigenspace_nat (f : End R M) (μ : R) (k : ℕ)
+    (h : MapsTo (f - μ • (1 : End R M))
+      (f.unifEigenspace μ k) (f.unifEigenspace μ k) :=
+      mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ k) :
+    IsNilpotent ((f - μ • 1).restrict h) := by
+  use k
+  ext ⟨x, hx⟩
+  rw [mem_unifEigenspace_nat] at hx
+  rw [LinearMap.zero_apply, ZeroMemClass.coe_zero, ZeroMemClass.coe_eq_zero,
+    LinearMap.pow_restrict, LinearMap.restrict_apply]
+  ext
+  simpa
+
+/-- The restriction of `f - μ • 1` to the generalized `μ`-eigenspace is nilpotent. -/
+lemma isNilpotent_restrict_unifEigenspace_top [IsNoetherian R M] (f : End R M) (μ : R)
+    (h : MapsTo (f - μ • (1 : End R M))
+      (f.unifEigenspace μ ⊤) (f.unifEigenspace μ ⊤) :=
+      mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ _) :
+    IsNilpotent ((f - μ • 1).restrict h) := by
+  apply isNilpotent_restrict_of_le
+  swap; apply isNilpotent_restrict_unifEigenspace_nat f μ (maxUnifEigenspaceIndex f μ)
+  rw [unifEigenspace_top_eq_maxUnifEigenspaceIndex]
+
 /-- The submodule `eigenspace f μ` for a linear map `f` and a scalar `μ` consists of all vectors `x`
     such that `f x = μ • x`. (Def 5.36 of [axler2015])-/
-def eigenspace (f : End R M) (μ : R) : Submodule R M :=
-  LinearMap.ker (f - algebraMap R (End R M) μ)
+abbrev eigenspace (f : End R M) (μ : R) : Submodule R M :=
+  f.unifEigenspace μ 1
+
+lemma eigenspace_def {f : End R M} {μ : R} :
+    f.eigenspace μ = LinearMap.ker (f - μ • 1) := by
+  rw [eigenspace, unifEigenspace_one]
 
 @[simp]
-theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by simp [eigenspace]
+theorem eigenspace_zero (f : End R M) : f.eigenspace 0 = LinearMap.ker f := by
+  simp only [eigenspace, ← Nat.cast_one (R := ℕ∞), unifEigenspace_zero_nat, pow_one]
 
 /-- A nonzero element of an eigenspace is an eigenvector. (Def 5.7 of [axler2015]) -/
-def HasEigenvector (f : End R M) (μ : R) (x : M) : Prop :=
-  x ∈ eigenspace f μ ∧ x ≠ 0
+abbrev HasEigenvector (f : End R M) (μ : R) (x : M) : Prop :=
+  HasUnifEigenvector f μ 1 x
+
+lemma hasEigenvector_iff {f : End R M} {μ : R} {x : M} :
+    f.HasEigenvector μ x ↔ x ∈ f.eigenspace μ ∧ x ≠ 0 := Iff.rfl
 
 /-- A scalar `μ` is an eigenvalue for a linear map `f` if there are nonzero vectors `x`
     such that `f x = μ • x`. (Def 5.5 of [axler2015]) -/
-def HasEigenvalue (f : End R M) (a : R) : Prop :=
-  eigenspace f a ≠ ⊥
+abbrev HasEigenvalue (f : End R M) (a : R) : Prop :=
+  HasUnifEigenvalue f a 1
+
+lemma hasEigenvalue_iff {f : End R M} {μ : R} :
+    f.HasEigenvalue μ ↔ f.eigenspace μ ≠ ⊥ := Iff.rfl
 
 /-- The eigenvalues of the endomorphism `f`, as a subtype of `R`. -/
-def Eigenvalues (f : End R M) : Type _ :=
-  { μ : R // f.HasEigenvalue μ }
+abbrev Eigenvalues (f : End R M) : Type _ :=
+  UnifEigenvalues f 1
 
 @[coe]
-def Eigenvalues.val (f : Module.End R M) : Eigenvalues f → R := Subtype.val
-
-instance Eigenvalues.instCoeOut {f : Module.End R M} : CoeOut (Eigenvalues f) R where
-  coe := Eigenvalues.val f
-
-instance Eigenvalues.instDecidableEq [DecidableEq R] (f : Module.End R M) :
-    DecidableEq (Eigenvalues f) :=
-  inferInstanceAs (DecidableEq (Subtype (fun x : R => HasEigenvalue f x)))
+abbrev Eigenvalues.val (f : Module.End R M) : Eigenvalues f → R := UnifEigenvalues.val f 1
 
 theorem hasEigenvalue_of_hasEigenvector {f : End R M} {μ : R} {x : M} (h : HasEigenvector f μ x) :
-    HasEigenvalue f μ := by
-  rw [HasEigenvalue, Submodule.ne_bot_iff]
-  use x; exact h
+    HasEigenvalue f μ :=
+  h.hasUnifEigenvalue
 
-theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x := by
-  rw [eigenspace, LinearMap.mem_ker, LinearMap.sub_apply, algebraMap_end_apply, sub_eq_zero]
+theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x :=
+  mem_unifEigenspace_one
 
+nonrec
 theorem HasEigenvector.apply_eq_smul {f : End R M} {μ : R} {x : M} (hx : f.HasEigenvector μ x) :
     f x = μ • x :=
-  mem_eigenspace_iff.mp hx.1
+  hx.apply_eq_smul
 
+nonrec
 theorem HasEigenvector.pow_apply {f : End R M} {μ : R} {v : M} (hv : f.HasEigenvector μ v) (n : ℕ) :
-    (f ^ n) v = μ ^ n • v := by
-  induction n <;> simp [*, pow_succ f, hv.apply_eq_smul, smul_smul, pow_succ' μ]
+    (f ^ n) v = μ ^ n • v :=
+  hv.pow_apply n
 
 theorem HasEigenvalue.exists_hasEigenvector {f : End R M} {μ : R} (hμ : f.HasEigenvalue μ) :
     ∃ v, f.HasEigenvector μ v :=
   Submodule.exists_mem_ne_zero_of_ne_bot hμ
 
+nonrec
 lemma HasEigenvalue.pow {f : End R M} {μ : R} (h : f.HasEigenvalue μ) (n : ℕ) :
-    (f ^ n).HasEigenvalue (μ ^ n) := by
-  rw [HasEigenvalue, Submodule.ne_bot_iff]
-  obtain ⟨m : M, hm⟩ := h.exists_hasEigenvector
-  exact ⟨m, by simpa [mem_eigenspace_iff] using hm.pow_apply n, hm.2⟩
+    (f ^ n).HasEigenvalue (μ ^ n) :=
+  h.pow n
 
 /-- A nilpotent endomorphism has nilpotent eigenvalues.
 
 See also `LinearMap.isNilpotent_trace_of_isNilpotent`. -/
+nonrec
 lemma HasEigenvalue.isNilpotent_of_isNilpotent [NoZeroSMulDivisors R M] {f : End R M}
     (hfn : IsNilpotent f) {μ : R} (hf : f.HasEigenvalue μ) :
-    IsNilpotent μ := by
-  obtain ⟨m : M, hm⟩ := hf.exists_hasEigenvector
-  obtain ⟨n : ℕ, hn : f ^ n = 0⟩ := hfn
-  exact ⟨n, by simpa [hn, hm.2, eq_comm (a := (0 : M))] using hm.pow_apply n⟩
+    IsNilpotent μ :=
+  hf.isNilpotent_of_isNilpotent hfn
 
+nonrec
 theorem HasEigenvalue.mem_spectrum {f : End R M} {μ : R} (hμ : HasEigenvalue f μ) :
-    μ ∈ spectrum R f := by
-  refine spectrum.mem_iff.mpr fun h_unit => ?_
-  set f' := LinearMap.GeneralLinearGroup.toLinearEquiv h_unit.unit
-  rcases hμ.exists_hasEigenvector with ⟨v, hv⟩
-  refine hv.2 ((LinearMap.ker_eq_bot'.mp f'.ker) v (?_ : μ • v - f v = 0))
-  rw [hv.apply_eq_smul, sub_self]
+    μ ∈ spectrum R f :=
+  hμ.mem_spectrum
 
 theorem hasEigenvalue_iff_mem_spectrum [FiniteDimensional K V] {f : End K V} {μ : K} :
-    f.HasEigenvalue μ ↔ μ ∈ spectrum K f := by
-  rw [spectrum.mem_iff, IsUnit.sub_iff, LinearMap.isUnit_iff_ker_eq_bot, HasEigenvalue, eigenspace]
+    f.HasEigenvalue μ ↔ μ ∈ spectrum K f :=
+  hasUnifEigenvalue_iff_mem_spectrum
 
 alias ⟨_, HasEigenvalue.of_mem_spectrum⟩ := hasEigenvalue_iff_mem_spectrum
 
 theorem eigenspace_div (f : End K V) (a b : K) (hb : b ≠ 0) :
     eigenspace f (a / b) = LinearMap.ker (b • f - algebraMap K (End K V) a) :=
-  calc
-    eigenspace f (a / b) = eigenspace f (b⁻¹ * a) := by rw [div_eq_mul_inv, mul_comm]
-    _ = LinearMap.ker (f - (b⁻¹ * a) • LinearMap.id) := by rw [eigenspace]; rfl
-    _ = LinearMap.ker (f - b⁻¹ • a • LinearMap.id) := by rw [smul_smul]
-    _ = LinearMap.ker (f - b⁻¹ • algebraMap K (End K V) a) := rfl
-    _ = LinearMap.ker (b • (f - b⁻¹ • algebraMap K (End K V) a)) := by
-        rw [LinearMap.ker_smul _ b hb]
-    _ = LinearMap.ker (b • f - algebraMap K (End K V) a) := by rw [smul_sub, smul_inv_smul₀ hb]
+  unifEigenspace_div f a b hb
 
 /-- The generalized eigenspace for a linear map `f`, a scalar `μ`, and an exponent `k ∈ ℕ` is the
 kernel of `(f - μ • id) ^ k`. (Def 8.10 of [axler2015]). Furthermore, a generalized eigenspace for
 some exponent `k` is contained in the generalized eigenspace for exponents larger than `k`. -/
 def genEigenspace (f : End R M) (μ : R) : ℕ →o Submodule R M where
-  toFun k := LinearMap.ker ((f - algebraMap R (End R M) μ) ^ k)
-  monotone' k m hm := by
-    simp only [← pow_sub_mul_pow _ hm]
-    exact
-      LinearMap.ker_le_ker_comp ((f - algebraMap R (End R M) μ) ^ k)
-        ((f - algebraMap R (End R M) μ) ^ (m - k))
+  toFun k := f.unifEigenspace μ k
+  monotone' k l hkl := (f.unifEigenspace μ).monotone <| by simpa
+
+lemma genEigenspace_def (f : End R M) (μ : R) (k : ℕ) :
+    f.genEigenspace μ k = LinearMap.ker ((f - μ • 1) ^ k) := by
+  rw [genEigenspace, OrderHom.coe_mk, unifEigenspace_nat]
 
 @[simp]
 theorem mem_genEigenspace (f : End R M) (μ : R) (k : ℕ) (m : M) :
-    m ∈ f.genEigenspace μ k ↔ ((f - μ • (1 : End R M)) ^ k) m = 0 := Iff.rfl
+    m ∈ f.genEigenspace μ k ↔ ((f - μ • (1 : End R M)) ^ k) m = 0 :=
+  mem_unifEigenspace_nat
 
 @[simp]
 theorem genEigenspace_zero (f : End R M) (k : ℕ) :
-    f.genEigenspace 0 k = LinearMap.ker (f ^ k) := by
-  simp [Module.End.genEigenspace]
+    f.genEigenspace 0 k = LinearMap.ker (f ^ k) :=
+  unifEigenspace_zero_nat _ _
 
 /-- A nonzero element of a generalized eigenspace is a generalized eigenvector.
     (Def 8.9 of [axler2015])-/
-def HasGenEigenvector (f : End R M) (μ : R) (k : ℕ) (x : M) : Prop :=
-  x ≠ 0 ∧ x ∈ genEigenspace f μ k
+abbrev HasGenEigenvector (f : End R M) (μ : R) (k : ℕ) (x : M) : Prop :=
+  HasUnifEigenvector f μ k x
+
+lemma hasGenEigenvector_iff {f : End R M} {μ : R} {k : ℕ} {x : M} :
+    f.HasGenEigenvector μ k x ↔ x ∈ f.genEigenspace μ k ∧ x ≠ 0 := Iff.rfl
 
 /-- A scalar `μ` is a generalized eigenvalue for a linear map `f` and an exponent `k ∈ ℕ` if there
     are generalized eigenvectors for `f`, `k`, and `μ`. -/
-def HasGenEigenvalue (f : End R M) (μ : R) (k : ℕ) : Prop :=
-  genEigenspace f μ k ≠ ⊥
+abbrev HasGenEigenvalue (f : End R M) (μ : R) (k : ℕ) : Prop :=
+  HasUnifEigenvalue f μ k
+
+lemma hasGenEigenvalue_iff {f : End R M} {μ : R} {k : ℕ} :
+    f.HasGenEigenvalue μ k ↔ f.genEigenspace μ k ≠ ⊥ := Iff.rfl
 
 /-- The generalized eigenrange for a linear map `f`, a scalar `μ`, and an exponent `k ∈ ℕ` is the
     range of `(f - μ • id) ^ k`. -/
-def genEigenrange (f : End R M) (μ : R) (k : ℕ) : Submodule R M :=
-  LinearMap.range ((f - algebraMap R (End R M) μ) ^ k)
+abbrev genEigenrange (f : End R M) (μ : R) (k : ℕ) : Submodule R M :=
+  unifEigenrange f μ k
+
+lemma genEigenrange_def {f : End R M} {μ : R} {k : ℕ} :
+    f.genEigenrange μ k = LinearMap.range ((f - μ • 1) ^ k) := by
+  rw [genEigenrange, unifEigenrange_nat]
 
 /-- The exponent of a generalized eigenvalue is never 0. -/
 theorem exp_ne_zero_of_hasGenEigenvalue {f : End R M} {μ : R} {k : ℕ}
-    (h : f.HasGenEigenvalue μ k) : k ≠ 0 := by
-  rintro rfl
-  exact h LinearMap.ker_id
+    (h : f.HasGenEigenvalue μ k) : k ≠ 0 :=
+  HasUnifEigenvalue.exp_ne_zero h
 
 /-- The union of the kernels of `(f - μ • id) ^ k` over all `k`. -/
-def maxGenEigenspace (f : End R M) (μ : R) : Submodule R M :=
-  ⨆ k, f.genEigenspace μ k
+abbrev maxGenEigenspace (f : End R M) (μ : R) : Submodule R M :=
+  unifEigenspace f μ ⊤
+
+lemma maxGenEigenspace_def (f : End R M) (μ : R) :
+    f.maxGenEigenspace μ = ⨆ k, f.genEigenspace μ k := by
+  simp_rw [maxGenEigenspace, unifEigenspace_top, genEigenspace, OrderHom.coe_mk]
 
 theorem genEigenspace_le_maximal (f : End R M) (μ : R) (k : ℕ) :
     f.genEigenspace μ k ≤ f.maxGenEigenspace μ :=
-  le_iSup _ _
+  (f.unifEigenspace μ).monotone le_top
 
 @[simp]
 theorem mem_maxGenEigenspace (f : End R M) (μ : R) (m : M) :
-    m ∈ f.maxGenEigenspace μ ↔ ∃ k : ℕ, ((f - μ • (1 : End R M)) ^ k) m = 0 := by
-  simp only [maxGenEigenspace, ← mem_genEigenspace, Submodule.mem_iSup_of_chain]
+    m ∈ f.maxGenEigenspace μ ↔ ∃ k : ℕ, ((f - μ • (1 : End R M)) ^ k) m = 0 :=
+  mem_unifEigenspace_top
 
 /-- If there exists a natural number `k` such that the kernel of `(f - μ • id) ^ k` is the
 maximal generalized eigenspace, then this value is the least such `k`. If not, this value is not
 meaningful. -/
-noncomputable def maxGenEigenspaceIndex (f : End R M) (μ : R) :=
-  monotonicSequenceLimitIndex (f.genEigenspace μ)
+noncomputable abbrev maxGenEigenspaceIndex (f : End R M) (μ : R) :=
+  maxUnifEigenspaceIndex f μ
 
 /-- For an endomorphism of a Noetherian module, the maximal eigenspace is always of the form kernel
 `(f - μ • id) ^ k` for some `k`. -/
-theorem maxGenEigenspace_eq [h : IsNoetherian R M] (f : End R M) (μ : R) :
+theorem maxGenEigenspace_eq [IsNoetherian R M] (f : End R M) (μ : R) :
     maxGenEigenspace f μ = f.genEigenspace μ (maxGenEigenspaceIndex f μ) :=
-  h.wf.iSup_eq_monotonicSequenceLimit (f.genEigenspace μ)
+  unifEigenspace_top_eq_maxUnifEigenspaceIndex _ _
 
 /-- A generalized eigenvalue for some exponent `k` is also
     a generalized eigenvalue for exponents larger than `k`. -/
 theorem hasGenEigenvalue_of_hasGenEigenvalue_of_le {f : End R M} {μ : R} {k : ℕ}
     {m : ℕ} (hm : k ≤ m) (hk : f.HasGenEigenvalue μ k) :
-    f.HasGenEigenvalue μ m := by
-  unfold HasGenEigenvalue at *
-  contrapose! hk
-  rw [← le_bot_iff, ← hk]
-  exact (f.genEigenspace μ).monotone hm
+    f.HasGenEigenvalue μ m :=
+  hk.le <| by simpa using hm
 
 /-- The eigenspace is a subspace of the generalized eigenspace. -/
 theorem eigenspace_le_genEigenspace {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k) :
     f.eigenspace μ ≤ f.genEigenspace μ k :=
-  (f.genEigenspace μ).monotone (Nat.succ_le_of_lt hk)
+  (f.unifEigenspace _).monotone <| by simpa using Nat.succ_le_of_lt hk
 
 /-- All eigenvalues are generalized eigenvalues. -/
 theorem hasGenEigenvalue_of_hasEigenvalue {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k)
-    (hμ : f.HasEigenvalue μ) : f.HasGenEigenvalue μ k := by
-  apply hasGenEigenvalue_of_hasGenEigenvalue_of_le hk
-  rw [HasGenEigenvalue, genEigenspace, OrderHom.coe_mk, pow_one]
-  exact hμ
+    (hμ : f.HasEigenvalue μ) : f.HasGenEigenvalue μ k :=
+  hμ.lt <| by simpa using hk
 
 /-- All generalized eigenvalues are eigenvalues. -/
 theorem hasEigenvalue_of_hasGenEigenvalue {f : End R M} {μ : R} {k : ℕ}
-    (hμ : f.HasGenEigenvalue μ k) : f.HasEigenvalue μ := by
-  intro contra; apply hμ
-  erw [LinearMap.ker_eq_bot] at contra ⊢; rw [LinearMap.coe_pow]
-  exact Function.Injective.iterate contra k
+    (hμ : f.HasGenEigenvalue μ k) : f.HasEigenvalue μ :=
+  hμ.lt zero_lt_one
 
 /-- Generalized eigenvalues are actually just eigenvalues. -/
 @[simp]
 theorem hasGenEigenvalue_iff_hasEigenvalue {f : End R M} {μ : R} {k : ℕ} (hk : 0 < k) :
     f.HasGenEigenvalue μ k ↔ f.HasEigenvalue μ :=
-  ⟨hasEigenvalue_of_hasGenEigenvalue, hasGenEigenvalue_of_hasEigenvalue hk⟩
+  hasUnifEigenvalue_iff_hasUnifEigenvalue_one <| by simpa using hk
 
 /-- Every generalized eigenvector is a generalized eigenvector for exponent `finrank K V`.
     (Lemma 8.11 of [axler2015]) -/
 theorem genEigenspace_le_genEigenspace_finrank [FiniteDimensional K V] (f : End K V)
     (μ : K) (k : ℕ) : f.genEigenspace μ k ≤ f.genEigenspace μ (finrank K V) :=
-  ker_pow_le_ker_pow_finrank _ _
+  unifEigenspace_le_unifEigenspace_finrank _ _ _
 
 @[simp] theorem iSup_genEigenspace_eq_genEigenspace_finrank
     [FiniteDimensional K V] (f : End K V) (μ : K) :
@@ -266,7 +587,7 @@ theorem genEigenspace_le_genEigenspace_finrank [FiniteDimensional K V] (f : End
 theorem genEigenspace_eq_genEigenspace_finrank_of_le [FiniteDimensional K V]
     (f : End K V) (μ : K) {k : ℕ} (hk : finrank K V ≤ k) :
     f.genEigenspace μ k = f.genEigenspace μ (finrank K V) :=
-  ker_pow_eq_ker_pow_finrank_of_le hk
+  unifEigenspace_eq_unifEigenspace_finrank_of_le f μ hk
 
 lemma mapsTo_genEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) (k : ℕ) :
     MapsTo g (f.genEigenspace μ k) (f.genEigenspace μ k) := by
@@ -277,21 +598,43 @@ lemma mapsTo_genEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) (k
   rw [← LinearMap.comp_apply, ← LinearMap.mul_eq_comp, h.eq, LinearMap.mul_eq_comp,
     LinearMap.comp_apply, hx, map_zero]
 
-lemma mapsTo_iSup_genEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) :
-    MapsTo g ↑(⨆ k, f.genEigenspace μ k) ↑(⨆ k, f.genEigenspace μ k) := by
+lemma iSup_genEigenspace_eq (f : End R M) (μ : R) :
+    ⨆ k, (f.genEigenspace μ) k = f.unifEigenspace μ ⊤ := by
+  rw [unifEigenspace_eq_iSup_unifEigenspace_nat]
+  ext
+  simp only [iSup_subtype, le_top, iSup_pos]
+  rfl
+
+lemma mapsTo_maxGenEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) :
+    MapsTo g ↑(f.maxGenEigenspace μ) ↑(f.maxGenEigenspace μ) := by
+  rw [maxGenEigenspace_def]
   simp only [MapsTo, Submodule.coe_iSup_of_chain, mem_iUnion, SetLike.mem_coe]
   rintro x ⟨k, hk⟩
   exact ⟨k, f.mapsTo_genEigenspace_of_comm h μ k hk⟩
 
+lemma mapsTo_iSup_genEigenspace_of_comm {f g : End R M} (h : Commute f g) (μ : R) :
+    MapsTo g ↑(⨆ k, f.genEigenspace μ k) ↑(⨆ k, f.genEigenspace μ k) := by
+  rw [← maxGenEigenspace_def]
+  apply mapsTo_maxGenEigenspace_of_comm h
+
 /-- The restriction of `f - μ • 1` to the `k`-fold generalized `μ`-eigenspace is nilpotent. -/
 lemma isNilpotent_restrict_sub_algebraMap (f : End R M) (μ : R) (k : ℕ)
     (h : MapsTo (f - algebraMap R (End R M) μ)
       (f.genEigenspace μ k) (f.genEigenspace μ k) :=
       mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ k) :
+    IsNilpotent ((f - algebraMap R (End R M) μ).restrict h) :=
+  isNilpotent_restrict_unifEigenspace_nat _ _ _
+
+/-- The restriction of `f - μ • 1` to the generalized `μ`-eigenspace is nilpotent. -/
+lemma isNilpotent_restrict_maxGenEigenspace_sub_algebraMap [IsNoetherian R M] (f : End R M) (μ : R)
+    (h : MapsTo (f - algebraMap R (End R M) μ)
+      ↑(f.maxGenEigenspace μ) ↑(f.maxGenEigenspace μ) :=
+      mapsTo_maxGenEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ) :
     IsNilpotent ((f - algebraMap R (End R M) μ).restrict h) := by
-  use k
-  ext
-  simp [LinearMap.restrict_apply, LinearMap.pow_restrict _]
+  apply isNilpotent_restrict_of_le (q := f.unifEigenspace μ (maxUnifEigenspaceIndex f μ))
+    _ (isNilpotent_restrict_unifEigenspace_nat f μ (maxUnifEigenspaceIndex f μ))
+  rw [maxGenEigenspace_eq]
+  exact le_rfl
 
 /-- The restriction of `f - μ • 1` to the generalized `μ`-eigenspace is nilpotent. -/
 lemma isNilpotent_restrict_iSup_sub_algebraMap [IsNoetherian R M] (f : End R M) (μ : R)
@@ -299,42 +642,53 @@ lemma isNilpotent_restrict_iSup_sub_algebraMap [IsNoetherian R M] (f : End R M)
       ↑(⨆ k, f.genEigenspace μ k) ↑(⨆ k, f.genEigenspace μ k) :=
       mapsTo_iSup_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ) μ) :
     IsNilpotent ((f - algebraMap R (End R M) μ).restrict h) := by
-  obtain ⟨l, hl⟩ : ∃ l, ⨆ k, f.genEigenspace μ k = f.genEigenspace μ l :=
-    ⟨_, maxGenEigenspace_eq f μ⟩
-  use l
-  ext ⟨x, hx⟩
-  simpa [hl, LinearMap.restrict_apply, LinearMap.pow_restrict _] using hx
-
-lemma disjoint_genEigenspace [NoZeroSMulDivisors R M]
-    (f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) (k l : ℕ) :
-    Disjoint (f.genEigenspace μ₁ k) (f.genEigenspace μ₂ l) := by
+  apply isNilpotent_restrict_of_le (q := f.unifEigenspace μ (maxUnifEigenspaceIndex f μ))
+    _ (isNilpotent_restrict_unifEigenspace_nat f μ (maxUnifEigenspaceIndex f μ))
+  apply iSup_le
+  intro k
+  apply unifEigenspace_le_unifEigenspace_maxUnifEigenspaceIndex
+
+lemma disjoint_unifEigenspace [NoZeroSMulDivisors R M]
+    (f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) (k l : ℕ∞) :
+    Disjoint (f.unifEigenspace μ₁ k) (f.unifEigenspace μ₂ l) := by
+  rw [unifEigenspace_eq_iSup_unifEigenspace_nat, unifEigenspace_eq_iSup_unifEigenspace_nat]
+  simp_rw [unifEigenspace_directed.disjoint_iSup_left, unifEigenspace_directed.disjoint_iSup_right]
+  rintro ⟨k, -⟩ ⟨l, -⟩
   nontriviality M
   have := NoZeroSMulDivisors.isReduced R M
   rw [disjoint_iff]
-  set p := f.genEigenspace μ₁ k ⊓ f.genEigenspace μ₂ l
+  set p := f.unifEigenspace μ₁ k ⊓ f.unifEigenspace μ₂ l
   by_contra hp
   replace hp : Nontrivial p := Submodule.nontrivial_iff_ne_bot.mpr hp
   let f₁ : End R p := (f - algebraMap R (End R M) μ₁).restrict <| MapsTo.inter_inter
-    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₁ k)
-    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₂ l)
+    (mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₁ k)
+    (mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₁) μ₂ l)
   let f₂ : End R p := (f - algebraMap R (End R M) μ₂).restrict <| MapsTo.inter_inter
-    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₂) μ₁ k)
-    (mapsTo_genEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₂) μ₂ l)
+    (mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₂) μ₁ k)
+    (mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f μ₂) μ₂ l)
   have : IsNilpotent (f₂ - f₁) := by
-    apply Commute.isNilpotent_sub (x := f₂) (y := f₁) _ ⟨l, ?_⟩ ⟨k, ?_⟩
+    apply Commute.isNilpotent_sub (x := f₂) (y := f₁) _
+      (isNilpotent_restrict_of_le inf_le_right _)
+      (isNilpotent_restrict_of_le inf_le_left _)
     · ext; simp [f₁, f₂, smul_sub, sub_sub, smul_comm μ₁, add_sub_left_comm]
-    all_goals ext ⟨x, _, _⟩; simpa [LinearMap.restrict_apply, LinearMap.pow_restrict _] using ‹_›
+    apply mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f _)
+    apply isNilpotent_restrict_unifEigenspace_nat
+    apply mapsTo_unifEigenspace_of_comm (Algebra.mul_sub_algebraMap_commutes f _)
+    apply isNilpotent_restrict_unifEigenspace_nat
   have hf₁₂ : f₂ - f₁ = algebraMap R (End R p) (μ₁ - μ₂) := by ext; simp [f₁, f₂, sub_smul]
   rw [hf₁₂, IsNilpotent.map_iff (NoZeroSMulDivisors.algebraMap_injective R (End R p)),
     isNilpotent_iff_eq_zero, sub_eq_zero] at this
   contradiction
 
+lemma disjoint_genEigenspace [NoZeroSMulDivisors R M]
+    (f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) (k l : ℕ) :
+    Disjoint (f.genEigenspace μ₁ k) (f.genEigenspace μ₂ l) :=
+  disjoint_unifEigenspace f hμ k l
+
 lemma disjoint_iSup_genEigenspace [NoZeroSMulDivisors R M]
     (f : End R M) {μ₁ μ₂ : R} (hμ : μ₁ ≠ μ₂) :
     Disjoint (⨆ k, f.genEigenspace μ₁ k) (⨆ k, f.genEigenspace μ₂ k) := by
-  simp_rw [(f.genEigenspace μ₁).mono.directed_le.disjoint_iSup_left,
-    (f.genEigenspace μ₂).mono.directed_le.disjoint_iSup_right]
-  exact disjoint_genEigenspace f hμ
+  simpa only [iSup_genEigenspace_eq] using disjoint_unifEigenspace f hμ ⊤ ⊤
 
 lemma injOn_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
     InjOn (⨆ k, f.genEigenspace · k) {μ | ⨆ k, f.genEigenspace μ k ≠ ⊥} := by
@@ -343,12 +697,14 @@ lemma injOn_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
   apply hμ₂
   simpa only [hμ₁₂, disjoint_self] using f.disjoint_iSup_genEigenspace contra
 
-theorem independent_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
-    CompleteLattice.Independent (fun μ ↦ ⨆ k, f.genEigenspace μ k) := by
+theorem independent_maxGenEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
+    CompleteLattice.Independent f.maxGenEigenspace := by
   classical
   suffices ∀ μ (s : Finset R), μ ∉ s → Disjoint (⨆ k, f.genEigenspace μ k)
       (s.sup fun μ ↦ ⨆ k, f.genEigenspace μ k) by
-    simp_rw [CompleteLattice.independent_iff_supIndep_of_injOn f.injOn_genEigenspace,
+    show CompleteLattice.Independent (f.maxGenEigenspace ·)
+    simp_rw [maxGenEigenspace_def,
+      CompleteLattice.independent_iff_supIndep_of_injOn f.injOn_genEigenspace,
       Finset.supIndep_iff_disjoint_erase]
     exact fun s μ _ ↦ this _ _ (s.not_mem_erase μ)
   intro μ₁ s
@@ -382,6 +738,11 @@ theorem independent_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
   simp_rw [Submodule.mem_iSup_of_chain, mem_genEigenspace]
   exact ⟨k, hyz⟩
 
+theorem independent_genEigenspace [NoZeroSMulDivisors R M] (f : End R M) :
+    CompleteLattice.Independent (fun μ ↦ ⨆ k, f.genEigenspace μ k) := by
+  simp_rw [← maxGenEigenspace_def]
+  apply independent_maxGenEigenspace
+
 /-- The eigenspaces of a linear operator form an independent family of subspaces of `M`.  That is,
 any eigenspace has trivial intersection with the span of all the other eigenspaces. -/
 theorem eigenspaces_independent [NoZeroSMulDivisors R M] (f : End R M) :
@@ -394,7 +755,7 @@ theorem eigenvectors_linearIndependent' {ι : Type*} [NoZeroSMulDivisors R M]
     (f : End R M) (μ : ι → R) (hμ : Function.Injective μ) (v : ι → M)
     (h_eigenvec : ∀ i, f.HasEigenvector (μ i) (v i)) : LinearIndependent R v :=
   f.eigenspaces_independent.comp hμ |>.linearIndependent _
-    (fun i => h_eigenvec i |>.left) (fun i => h_eigenvec i |>.right)
+    (fun i ↦ h_eigenvec i |>.left) (fun i ↦ h_eigenvec i |>.right)
 
 /-- Eigenvectors corresponding to distinct eigenvalues of a linear operator are linearly
     independent. (Lemma 5.10 of [axler2015])
@@ -405,7 +766,7 @@ theorem eigenvectors_linearIndependent' {ι : Type*} [NoZeroSMulDivisors R M]
 theorem eigenvectors_linearIndependent [NoZeroSMulDivisors R M]
     (f : End R M) (μs : Set R) (xs : μs → M)
     (h_eigenvec : ∀ μ : μs, f.HasEigenvector μ (xs μ)) : LinearIndependent R xs :=
-  f.eigenvectors_linearIndependent' (fun μ : μs => μ) Subtype.coe_injective _ h_eigenvec
+  f.eigenvectors_linearIndependent' (fun μ : μs ↦ μ) Subtype.coe_injective _ h_eigenvec
 
 /-- If `f` maps a subspace `p` into itself, then the generalized eigenspace of the restriction
     of `f` to `p` is the part of the generalized eigenspace of `f` that lies in `p`. -/
@@ -413,7 +774,7 @@ theorem genEigenspace_restrict (f : End R M) (p : Submodule R M) (k : ℕ) (μ :
     (hfp : ∀ x : M, x ∈ p → f x ∈ p) :
     genEigenspace (LinearMap.restrict f hfp) μ k =
       Submodule.comap p.subtype (f.genEigenspace μ k) := by
-  simp only [genEigenspace, OrderHom.coe_mk, ← LinearMap.ker_comp]
+  simp only [genEigenspace_def, OrderHom.coe_mk, ← LinearMap.ker_comp]
   induction' k with k ih
   · rw [pow_zero, pow_zero, LinearMap.one_eq_id]
     apply (Submodule.ker_subtype _).symm
@@ -426,6 +787,50 @@ lemma _root_.Submodule.inf_genEigenspace (f : End R M) (p : Submodule R M) {k :
       (genEigenspace (LinearMap.restrict f hfp) μ k).map p.subtype := by
   rw [f.genEigenspace_restrict _ _ _ hfp, Submodule.map_comap_eq, Submodule.range_subtype]
 
+/-- Given a family of endomorphisms `i ↦ f i`, a family of candidate eigenvalues `i ↦ μ i`, and a
+submodule `p` which is invariant wrt every `f i`, the intersection of `p` with the simultaneous
+maximal generalised eigenspace (taken over all `i`), is the same as the simultaneous maximal
+generalised eigenspace of the `f i` restricted to `p`. -/
+lemma _root_.Submodule.inf_iInf_maxGenEigenspace_of_forall_mapsTo {ι : Type*} {μ : ι → R}
+    (f : ι → End R M) (p : Submodule R M) (hfp : ∀ i, MapsTo (f i) p p) :
+    p ⊓ ⨅ i, (f i).maxGenEigenspace (μ i) =
+      (⨅ i, maxGenEigenspace ((f i).restrict (hfp i)) (μ i)).map p.subtype := by
+  cases isEmpty_or_nonempty ι
+  · simp [iInf_of_isEmpty]
+  · simp_rw [inf_iInf, maxGenEigenspace_def, ((f _).genEigenspace _).mono.directed_le.inf_iSup_eq,
+      p.inf_genEigenspace _ (hfp _), ← Submodule.map_iSup, Submodule.map_iInf _ p.injective_subtype]
+
+/-- Given a family of endomorphisms `i ↦ f i`, a family of candidate eigenvalues `i ↦ μ i`, and a
+distinguished index `i` whose maximal generalised `μ i`-eigenspace is invariant wrt every `f j`,
+taking simultaneous maximal generalised eigenspaces is unaffected by first restricting to the
+distinguished generalised `μ i`-eigenspace. -/
+lemma iInf_maxGenEigenspace_restrict_map_subtype_eq
+    {ι : Type*} {μ : ι → R} (i : ι) (f : ι → End R M)
+    (h : ∀ j, MapsTo (f j) ((f i).maxGenEigenspace (μ i)) ((f i).maxGenEigenspace (μ i))) :
+    letI p := (f i).maxGenEigenspace (μ i)
+    letI q (j : ι) := maxGenEigenspace ((f j).restrict (h j)) (μ j)
+    (⨅ j, q j).map p.subtype = ⨅ j, (f j).maxGenEigenspace (μ j) := by
+  have : Nonempty ι := ⟨i⟩
+  set p := (f i).maxGenEigenspace (μ i)
+  have : ⨅ j, (f j).maxGenEigenspace (μ j) = p ⊓ ⨅ j, (f j).maxGenEigenspace (μ j) := by
+    refine le_antisymm ?_ inf_le_right
+    simpa only [le_inf_iff, le_refl, and_true] using iInf_le _ _
+  rw [Submodule.map_iInf _ p.injective_subtype, this, Submodule.inf_iInf]
+  simp_rw [maxGenEigenspace_def, Submodule.map_iSup,
+    ((f _).genEigenspace _).mono.directed_le.inf_iSup_eq, p.inf_genEigenspace (f _) (h _)]
+
+lemma mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo
+    {p : Submodule R M} (f g : End R M) (hf : MapsTo f p p) (hg : MapsTo g p p) {μ₁ μ₂ : R}
+    (h : MapsTo f (g.maxGenEigenspace μ₁) (g.maxGenEigenspace μ₂)) :
+    MapsTo (f.restrict hf)
+      (maxGenEigenspace (g.restrict hg) μ₁)
+      (maxGenEigenspace (g.restrict hg) μ₂) := by
+  intro x hx
+  simp_rw [SetLike.mem_coe, mem_maxGenEigenspace, ← LinearMap.restrict_smul_one _,
+    LinearMap.restrict_sub _, LinearMap.pow_restrict _, LinearMap.restrict_apply,
+    Submodule.mk_eq_zero, ← mem_maxGenEigenspace] at hx ⊢
+  exact h hx
+
 /-- If `p` is an invariant submodule of an endomorphism `f`, then the `μ`-eigenspace of the
 restriction of `f` to `p` is a submodule of the `μ`-eigenspace of `f`. -/
 theorem eigenspace_restrict_le_eigenspace (f : End R M) {p : Submodule R M} (hfp : ∀ x ∈ p, f x ∈ p)
@@ -440,16 +845,18 @@ theorem generalized_eigenvec_disjoint_range_ker [FiniteDimensional K V] (f : End
       (f.genEigenspace μ (finrank K V)) := by
   have h :=
     calc
-      Submodule.comap ((f - algebraMap _ _ μ) ^ finrank K V)
+      Submodule.comap ((f - μ • 1) ^ finrank K V)
         (f.genEigenspace μ (finrank K V)) =
           LinearMap.ker ((f - algebraMap _ _ μ) ^ finrank K V *
             (f - algebraMap K (End K V) μ) ^ finrank K V) := by
-              rw [genEigenspace, OrderHom.coe_mk, ← LinearMap.ker_comp]; rfl
-      _ = f.genEigenspace μ (finrank K V + finrank K V) := by rw [← pow_add]; rfl
+              rw [genEigenspace, OrderHom.coe_mk, unifEigenspace_nat, ← LinearMap.ker_comp]; rfl
+      _ = f.genEigenspace μ (finrank K V + finrank K V) := by
+              rw [← pow_add, genEigenspace, OrderHom.coe_mk, unifEigenspace_nat]; rfl
       _ = f.genEigenspace μ (finrank K V) := by
-        rw [genEigenspace_eq_genEigenspace_finrank_of_le]; omega
-  rw [disjoint_iff_inf_le, genEigenrange, LinearMap.range_eq_map,
-    Submodule.map_inf_eq_map_inf_comap, top_inf_eq, h]
+              rw [genEigenspace_eq_genEigenspace_finrank_of_le]; omega
+  rw [disjoint_iff_inf_le, genEigenrange, unifEigenrange_nat, LinearMap.range_eq_map,
+    Submodule.map_inf_eq_map_inf_comap, top_inf_eq, h,
+    genEigenspace, OrderHom.coe_mk, unifEigenspace_nat]
   apply Submodule.map_comap_le
 
 /-- If an invariant subspace `p` of an endomorphism `f` is disjoint from the `μ`-eigenspace of `f`,
@@ -476,11 +883,13 @@ theorem map_genEigenrange_le {f : End K V} {μ : K} {n : ℕ} :
   calc
     Submodule.map f (f.genEigenrange μ n) =
       LinearMap.range (f * (f - algebraMap _ _ μ) ^ n) := by
-        rw [genEigenrange]; exact (LinearMap.range_comp _ _).symm
+        rw [genEigenrange, unifEigenrange_nat]; exact (LinearMap.range_comp _ _).symm
     _ = LinearMap.range ((f - algebraMap _ _ μ) ^ n * f) := by
         rw [Algebra.mul_sub_algebraMap_pow_commutes]
     _ = Submodule.map ((f - algebraMap _ _ μ) ^ n) (LinearMap.range f) := LinearMap.range_comp _ _
-    _ ≤ f.genEigenrange μ n := LinearMap.map_le_range
+    _ ≤ f.genEigenrange μ n := by
+      rw [genEigenrange, unifEigenrange_nat]
+      apply LinearMap.map_le_range
 
 lemma iSup_genEigenspace_le_smul (f : Module.End R M) (μ t : R) :
     (⨆ k, f.genEigenspace μ k) ≤ ⨆ k, (t • f).genEigenspace (t * μ) k := by
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean b/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean
index 76346aad28f59..7ae63cc2e199b 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean
@@ -36,7 +36,7 @@ lemma hasEigenvector_toLin_diagonal (d : n → R) (i : n) (b : Basis n R M) :
 /--  Standard basis vectors are eigenvectors of any associated diagonal linear operator. -/
 lemma hasEigenvector_toLin'_diagonal (d : n → R) (i : n) :
     HasEigenvector (toLin' (diagonal d)) (d i) (Pi.basisFun R n i)  :=
-  hasEigenvector_toLin_diagonal ..
+  hasEigenvector_toLin_diagonal _ _ (Pi.basisFun R n)
 
 /-- Eigenvalues of a diagonal linear operator are the diagonal entries. -/
 lemma hasEigenvalue_toLin_diagonal_iff (d : n → R) {μ : R} [NoZeroSMulDivisors R M]
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean b/Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
index 95510728e054f..70c17ee7b2097 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean
@@ -21,7 +21,7 @@ namespace Module
 
 namespace End
 
-open Polynomial FiniteDimensional
+open Polynomial Module
 
 open scoped Polynomial
 
@@ -68,7 +68,7 @@ variable {f} {μ : K}
 
 theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue μ := by
   cases' dvd_iff_isRoot.2 h with p hp
-  rw [HasEigenvalue, eigenspace]
+  rw [hasEigenvalue_iff, eigenspace_def]
   intro con
   cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu
   have p_ne_0 : p ≠ 0 := by
@@ -78,7 +78,7 @@ theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue
   have : (aeval f) p = 0 := by
     have h_aeval := minpoly.aeval K f
     revert h_aeval
-    simp [hp, ← hu]
+    simp [hp, ← hu, Algebra.algebraMap_eq_smul_one]
   have h_deg := minpoly.degree_le_of_ne_zero K f p_ne_0 this
   rw [hp, degree_mul, degree_X_sub_C, Polynomial.degree_eq_natDegree p_ne_0] at h_deg
   norm_cast at h_deg
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean b/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
index 03cddde5f1423..aefced273795c 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Semisimple.lean
@@ -55,7 +55,7 @@ lemma IsSemisimple.genEigenspace_eq_eigenspace
 lemma IsSemisimple.maxGenEigenspace_eq_eigenspace
     (hf : f.IsSemisimple) (μ : R) :
     f.maxGenEigenspace μ = f.eigenspace μ := by
-  simp_rw [maxGenEigenspace, ← (f.genEigenspace μ).monotone.iSup_nat_add 1,
+  simp_rw [maxGenEigenspace_def, ← (f.genEigenspace μ).monotone.iSup_nat_add 1,
     hf.genEigenspace_eq_eigenspace μ (Nat.zero_lt_succ _), ciSup_const]
 
 end Module.End
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
index a8726197a4438..03e4ac0132b66 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
@@ -5,6 +5,7 @@ Authors: Alexander Bentkamp
 -/
 import Mathlib.LinearAlgebra.Eigenspace.Basic
 import Mathlib.FieldTheory.IsAlgClosed.Spectrum
+import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
 
 /-!
 # Triangularizable linear endomorphisms
@@ -37,7 +38,7 @@ generalized eigenspaces span the whole space.
 eigenspace, eigenvector, eigenvalue, eigen
 -/
 
-open Set Function Module FiniteDimensional
+open Set Function Module Module
 
 variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
    {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
@@ -52,7 +53,7 @@ theorem exists_hasEigenvalue_of_iSup_genEigenspace_eq_top [Nontrivial M] {f : En
   intro μ
   replace contra : ∀ k, f.genEigenspace μ k = ⊥ := fun k ↦ by
     have hk : ¬ f.HasGenEigenvalue μ k := fun hk ↦ contra μ (f.hasEigenvalue_of_hasGenEigenvalue hk)
-    rwa [HasGenEigenvalue, not_not] at hk
+    rwa [hasGenEigenvalue_iff, not_not] at hk
   simp [contra]
 
 -- This is Lemma 5.21 of [axler2015], although we are no longer following that proof.
@@ -98,6 +99,8 @@ theorem iSup_genEigenspace_eq_top [IsAlgClosed K] [FiniteDimensional K V] (f : E
       apply pos_finrank_genEigenspace_of_hasEigenvalue hμ₀ (Nat.zero_lt_succ n)
     -- and the dimensions of `ES` and `ER` add up to `finrank K V`.
     have h_dim_add : finrank K ER + finrank K ES = finrank K V := by
+      dsimp only [ER, ES]
+      rw [Module.End.genEigenspace_def, Module.End.genEigenrange_def]
       apply LinearMap.finrank_range_add_finrank_ker
     -- Therefore the dimension `ER` mus be smaller than `finrank K V`.
     have h_dim_ER : finrank K ER < n.succ := by linarith
@@ -167,7 +170,8 @@ theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈
     · rw [hμμ']
     replace hm₂ : ((f - algebraMap K (End K V) μ') ^ finrank K V) (m μ') = 0 := by
       obtain ⟨k, hk⟩ := (mem_iSup_of_chain _ _).mp (hm₂ μ')
-      exact Module.End.genEigenspace_le_genEigenspace_finrank _ _ k hk
+      simpa only [End.mem_genEigenspace] using
+        Module.End.genEigenspace_le_genEigenspace_finrank _ _ k hk
     have : _ = g := (m.support.erase μ).noncommProd_erase_mul (Finset.mem_erase.mpr ⟨hμμ', hμ'⟩)
       (fun μ ↦ (f - algebraMap K (End K V) μ) ^ finrank K V) (fun μ₁ _ μ₂ _ _ ↦ h_comm μ₁ μ₂)
     rw [← this, LinearMap.mul_apply, hm₂, _root_.map_zero]
@@ -184,9 +188,10 @@ theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈
     apply LinearMap.injOn_of_disjoint_ker (subset_refl _)
     have this := f.independent_genEigenspace
     simp_rw [f.iSup_genEigenspace_eq_genEigenspace_finrank] at this ⊢
-    rw [LinearMap.ker_noncommProd_eq_of_supIndep_ker _ _ <| this.supIndep' (m.support.erase μ),
-      ← Finset.sup_eq_iSup]
-    exact Finset.supIndep_iff_disjoint_erase.mp (this.supIndep' m.support) μ hμ
+    rw [LinearMap.ker_noncommProd_eq_of_supIndep_ker, ← Finset.sup_eq_iSup]
+    · simpa only [End.genEigenspace_def] using
+        Finset.supIndep_iff_disjoint_erase.mp (this.supIndep' m.support) μ hμ
+    · simpa only [End.genEigenspace_def] using this.supIndep' (m.support.erase μ)
   have hg₄ : SurjOn g
       ↑(p ⊓ ⨆ k, f.genEigenspace μ k) ↑(p ⊓ ⨆ k, f.genEigenspace μ k) := by
     have : MapsTo g
@@ -217,3 +222,54 @@ theorem Module.End.iSup_genEigenspace_restrict_eq_top
   simp_rw [Submodule.inf_genEigenspace f p h, Submodule.comap_subtype_self,
     ← Submodule.map_iSup, Submodule.comap_map_eq_of_injective h_inj] at this
   exact this.symm
+
+/-- Given a family of endomorphisms `i ↦ f i` which are compatible in the sense that every maximal
+generalised eigenspace of `f i` is invariant wrt `f j`, if each `f i` is triangularizable, the
+family is simultaneously triangularizable. -/
+lemma Module.End.iSup_iInf_maxGenEigenspace_eq_top_of_forall_mapsTo
+    {ι : Type*} [FiniteDimensional K V]
+    (f : ι → End K V)
+    (h : ∀ i j φ, MapsTo (f i) ((f j).maxGenEigenspace φ) ((f j).maxGenEigenspace φ))
+    (h' : ∀ i, ⨆ μ, (f i).maxGenEigenspace μ = ⊤) :
+    ⨆ χ : ι → K, ⨅ i, (f i).maxGenEigenspace (χ i) = ⊤ := by
+  generalize h_dim : finrank K V = n
+  induction n using Nat.strongRecOn generalizing V with | ind n ih => ?_
+  obtain this | ⟨i : ι, hy : ¬ ∃ φ, (f i).maxGenEigenspace φ = ⊤⟩ :=
+    forall_or_exists_not (fun j : ι ↦ ∃ φ : K, (f j).maxGenEigenspace φ = ⊤)
+  · choose χ hχ using this
+    replace hχ : ⨅ i, (f i).maxGenEigenspace (χ i) = ⊤ := by simpa
+    simp_rw [eq_top_iff] at hχ ⊢
+    exact le_trans hχ <| le_iSup (fun χ : ι → K ↦ ⨅ i, (f i).maxGenEigenspace (χ i)) χ
+  · replace hy : ∀ φ, finrank K ((f i).maxGenEigenspace φ) < n := fun φ ↦ by
+      simp_rw [not_exists, ← lt_top_iff_ne_top] at hy; exact h_dim ▸ Submodule.finrank_lt (hy φ)
+    have hi (j : ι) (φ : K) :
+        MapsTo (f j) ((f i).maxGenEigenspace φ) ((f i).maxGenEigenspace φ) := by
+      exact h j i φ
+    replace ih (φ : K) :
+        ⨆ χ : ι → K, ⨅ j, maxGenEigenspace ((f j).restrict (hi j φ)) (χ j) = ⊤ := by
+      apply ih _ (hy φ)
+      · intro j k μ
+        exact mapsTo_restrict_maxGenEigenspace_restrict_of_mapsTo (f j) (f k) _ _ (h j k μ)
+      · simp_rw [maxGenEigenspace_def] at h' ⊢
+        exact fun j ↦ Module.End.iSup_genEigenspace_restrict_eq_top _ (h' j)
+      · rfl
+    replace ih (φ : K) :
+        ⨆ (χ : ι → K) (_ : χ i = φ), ⨅ j, maxGenEigenspace ((f j).restrict (hi j φ)) (χ j) = ⊤ := by
+      suffices ∀ χ : ι → K, χ i ≠ φ → ⨅ j, maxGenEigenspace ((f j).restrict (hi j φ)) (χ j) = ⊥ by
+        specialize ih φ; rw [iSup_split, biSup_congr this] at ih; simpa using ih
+      intro χ hχ
+      rw [eq_bot_iff, ← ((f i).maxGenEigenspace φ).ker_subtype, LinearMap.ker,
+        ← Submodule.map_le_iff_le_comap, ← Submodule.inf_iInf_maxGenEigenspace_of_forall_mapsTo,
+        ← disjoint_iff_inf_le]
+      simp_rw [maxGenEigenspace_def]
+      exact ((f i).disjoint_iSup_genEigenspace hχ.symm).mono_right (iInf_le _ i)
+    replace ih (φ : K) :
+        ⨆ (χ : ι → K) (_ : χ i = φ), ⨅ j, maxGenEigenspace (f j) (χ j) =
+        maxGenEigenspace (f i) φ := by
+      have (χ : ι → K) (hχ : χ i = φ) : ⨅ j, maxGenEigenspace (f j) (χ j) =
+          (⨅ j, maxGenEigenspace ((f j).restrict (hi j φ)) (χ j)).map
+            ((f i).maxGenEigenspace φ).subtype := by
+        rw [← hχ, iInf_maxGenEigenspace_restrict_map_subtype_eq]
+      simp_rw [biSup_congr this, ← Submodule.map_iSup, ih, Submodule.map_top,
+        Submodule.range_subtype]
+    simpa only [← ih, iSup_comm (ι := K), iSup_iSup_eq_right] using h' i
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Zero.lean b/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
index 4cc93bd2b2b32..30837353e1b0a 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Zero.lean
@@ -33,7 +33,7 @@ variable {R K M : Type*} [CommRing R] [IsDomain R] [Field K] [AddCommGroup M]
 variable [Module R M] [Module.Finite R M] [Module.Free R M]
 variable [Module K M] [Module.Finite K M]
 
-open FiniteDimensional Module.Free Polynomial
+open Module Module.Free Polynomial
 
 lemma IsNilpotent.charpoly_eq_X_pow_finrank (φ : Module.End R M) (h : IsNilpotent φ) :
     φ.charpoly = X ^ finrank R M := by
@@ -52,15 +52,14 @@ lemma charpoly_nilpotent_tfae [IsNoetherian R M] (φ : Module.End R M) :
       φ.charpoly = X ^ finrank R M,
       ∀ m : M, ∃ (n : ℕ), (φ ^ n) m = 0,
       natTrailingDegree φ.charpoly = finrank R M ] := by
-  tfae_have 1 → 2
-  · apply IsNilpotent.charpoly_eq_X_pow_finrank
+  tfae_have 1 → 2 := IsNilpotent.charpoly_eq_X_pow_finrank _
   tfae_have 2 → 3
-  · intro h m
+  | h, m => by
     use finrank R M
     suffices φ ^ finrank R M = 0 by simp only [this, LinearMap.zero_apply]
     simpa only [h, map_pow, aeval_X] using φ.aeval_self_charpoly
   tfae_have 3 → 1
-  · intro h
+  | h => by
     obtain ⟨n, hn⟩ := Filter.eventually_atTop.mp <| φ.eventually_iSup_ker_pow_eq
     use n
     ext x
@@ -68,8 +67,8 @@ lemma charpoly_nilpotent_tfae [IsNoetherian R M] (φ : Module.End R M) :
     obtain ⟨k, hk⟩ := h x
     rw [← mem_ker] at hk
     exact Submodule.mem_iSup_of_mem _ hk
-  tfae_have 2 ↔ 4
-  · rw [← φ.charpoly_natDegree, φ.charpoly_monic.eq_X_pow_iff_natTrailingDegree_eq_natDegree]
+  tfae_have 2 ↔ 4 := by
+    rw [← φ.charpoly_natDegree, φ.charpoly_monic.eq_X_pow_iff_natTrailingDegree_eq_natDegree]
   tfae_finish
 
 lemma charpoly_eq_X_pow_iff [IsNoetherian R M] (φ : Module.End R M) :
@@ -85,27 +84,25 @@ lemma hasEigenvalue_zero_tfae (φ : Module.End K M) :
       LinearMap.det φ = 0,
       ⊥ < ker φ,
       ∃ (m : M), m ≠ 0 ∧ φ m = 0 ] := by
-  tfae_have 1 ↔ 2
-  · exact Module.End.hasEigenvalue_iff_isRoot
-  tfae_have 2 → 3
-  · obtain ⟨F, hF⟩ := minpoly_dvd_charpoly φ
+  tfae_have 1 ↔ 2 := Module.End.hasEigenvalue_iff_isRoot
+  tfae_have 2 → 3 := by
+    obtain ⟨F, hF⟩ := minpoly_dvd_charpoly φ
     simp only [IsRoot.def, constantCoeff_apply, coeff_zero_eq_eval_zero, hF, eval_mul]
     intro h; rw [h, zero_mul]
-  tfae_have 3 → 4
-  · rw [← LinearMap.det_toMatrix (chooseBasis K M), Matrix.det_eq_sign_charpoly_coeff,
+  tfae_have 3 → 4 := by
+    rw [← LinearMap.det_toMatrix (chooseBasis K M), Matrix.det_eq_sign_charpoly_coeff,
       constantCoeff_apply, charpoly]
     intro h; rw [h, mul_zero]
-  tfae_have 4 → 5
-  · exact bot_lt_ker_of_det_eq_zero
-  tfae_have 5 → 6
-  · contrapose!
+  tfae_have 4 → 5 := bot_lt_ker_of_det_eq_zero
+  tfae_have 5 → 6 := by
+    contrapose!
     simp only [not_bot_lt_iff, eq_bot_iff]
     intro h x
     simp only [mem_ker, Submodule.mem_bot]
     contrapose!
     apply h
   tfae_have 6 → 1
-  · rintro ⟨x, h1, h2⟩
+  | ⟨x, h1, h2⟩ => by
     apply Module.End.hasEigenvalue_of_hasEigenvector ⟨_, h1⟩
     simpa only [Module.End.eigenspace_zero, mem_ker] using h2
   tfae_finish
@@ -135,7 +132,7 @@ lemma finrank_maxGenEigenspace (φ : Module.End K M) :
     finrank K (φ.maxGenEigenspace 0) = natTrailingDegree (φ.charpoly) := by
   set V := φ.maxGenEigenspace 0
   have hV : V = ⨆ (n : ℕ), ker (φ ^ n) := by
-    simp [V, Module.End.maxGenEigenspace, Module.End.genEigenspace]
+    simp [V, Module.End.maxGenEigenspace_def, Module.End.genEigenspace_def]
   let W := ⨅ (n : ℕ), LinearMap.range (φ ^ n)
   have hVW : IsCompl V W := by
     rw [hV]
@@ -163,7 +160,7 @@ lemma finrank_maxGenEigenspace (φ : Module.End K M) :
     apply b.ext
     simp only [Basis.prod_apply, coe_inl, coe_inr, prodMap_apply, LinearEquiv.conj_apply,
       LinearEquiv.symm_symm, Submodule.coe_prodEquivOfIsCompl, coe_comp, LinearEquiv.coe_coe,
-      Function.comp_apply, coprod_apply, Submodule.coeSubtype, map_add, Sum.forall, Sum.elim_inl,
+      Function.comp_apply, coprod_apply, Submodule.coe_subtype, map_add, Sum.forall, Sum.elim_inl,
       map_zero, ZeroMemClass.coe_zero, add_zero, LinearEquiv.eq_symm_apply, and_self,
       Submodule.coe_prodEquivOfIsCompl', restrict_coe_apply, implies_true, Sum.elim_inr, zero_add,
       e, V, W, ψ, F, G, b]
diff --git a/Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean b/Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
index 311feb0bf3305..cd48d0d7611d4 100644
--- a/Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
+++ b/Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean
@@ -164,6 +164,9 @@ theorem algebraMap_eq_zero_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x =
 theorem algebraMap_eq_one_iff (x : R) : algebraMap R (ExteriorAlgebra R M) x = 1 ↔ x = 1 :=
   map_eq_one_iff (algebraMap _ _) (algebraMap_leftInverse _).injective
 
+instance isLocalRingHom_algebraMap : IsLocalRingHom (algebraMap R (ExteriorAlgebra R M)) :=
+  isLocalRingHom_of_leftInverse _ (algebraMap_leftInverse M)
+
 theorem isUnit_algebraMap (r : R) : IsUnit (algebraMap R (ExteriorAlgebra R M) r) ↔ IsUnit r :=
   isUnit_map_of_leftInverse _ (algebraMap_leftInverse M)
 
@@ -299,8 +302,8 @@ theorem ιMulti_apply {n : ℕ} (v : Fin n → M) : ιMulti R n v = (List.ofFn f
   rfl
 
 @[simp]
-theorem ιMulti_zero_apply (v : Fin 0 → M) : ιMulti R 0 v = 1 :=
-  rfl
+theorem ιMulti_zero_apply (v : Fin 0 → M) : ιMulti R 0 v = 1 := by
+  simp [ιMulti]
 
 @[simp]
 theorem ιMulti_succ_apply {n : ℕ} (v : Fin n.succ → M) :
@@ -372,7 +375,7 @@ theorem map_apply_ι (f : M →ₗ[R] N) (m : M) : map f (ι R m) = ι R (f m) :
 theorem map_apply_ιMulti {n : ℕ} (f : M →ₗ[R] N) (m : Fin n → M) :
     map f (ιMulti R n m) = ιMulti R n (f ∘ m) := by
   rw [ιMulti_apply, ιMulti_apply, map_list_prod]
-  simp only [List.map_ofFn, Function.comp, map_apply_ι]
+  simp only [List.map_ofFn, Function.comp_def, map_apply_ι]
 
 @[simp]
 theorem map_comp_ιMulti {n : ℕ} (f : M →ₗ[R] N) :
diff --git a/Mathlib/LinearAlgebra/FiniteDimensional.lean b/Mathlib/LinearAlgebra/FiniteDimensional.lean
index 8aec07c6ff9dd..7b2f68f6cdcdd 100644
--- a/Mathlib/LinearAlgebra/FiniteDimensional.lean
+++ b/Mathlib/LinearAlgebra/FiniteDimensional.lean
@@ -27,7 +27,7 @@ variable {K : Type u} {V : Type v}
 
 namespace Submodule
 
-open IsNoetherian FiniteDimensional
+open IsNoetherian Module
 
 section DivisionRing
 
@@ -116,7 +116,7 @@ end FiniteDimensional
 
 namespace LinearMap
 
-open FiniteDimensional
+open Module
 
 section DivisionRing
 
@@ -142,7 +142,7 @@ end DivisionRing
 
 end LinearMap
 
-open FiniteDimensional
+open Module
 
 namespace LinearMap
 
diff --git a/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean b/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
index a172d496e574e..f7ab907b35f9c 100644
--- a/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
+++ b/Mathlib/LinearAlgebra/FiniteDimensional/Defs.lean
@@ -27,7 +27,7 @@ that all these points of view are equivalent, with the following lemmas
 
 - `fintypeBasisIndex` states that a finite-dimensional
   vector space has a finite basis
-- `FiniteDimensional.finBasis` and `FiniteDimensional.finBasisOfFinrankEq`
+- `Module.finBasis` and `Module.finBasisOfFinrankEq`
   are bases for finite dimensional vector spaces, where the index type
   is `Fin` (in `Mathlib.LinearAlgebra.Dimension.Free`)
 - `of_fintype_basis` states that the existence of a basis indexed by a
@@ -70,7 +70,7 @@ Plenty of the results hold for general fg modules or notherian modules, and they
 
 universe u v v' w
 
-open Cardinal Submodule Module Function
+open Cardinal Function IsNoetherian Module Submodule
 
 /-- `FiniteDimensional` vector spaces are defined to be finite modules.
 Use `FiniteDimensional.of_fintype_basis` to prove finite dimension from another definition. -/
@@ -80,11 +80,6 @@ abbrev FiniteDimensional (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module
 variable {K : Type u} {V : Type v}
 
 namespace FiniteDimensional
-
-open IsNoetherian
-
-section DivisionRing
-
 variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
   [Module K V₂]
 
@@ -145,10 +140,8 @@ theorem of_finite_basis {ι : Type w} {s : Set ι} (h : Basis s K V) (hs : Set.F
 instance finiteDimensional_submodule [FiniteDimensional K V] (S : Submodule K V) :
     FiniteDimensional K S := by
   letI : IsNoetherian K V := iff_fg.2 ?_
-  · exact
-      iff_fg.1
-        (IsNoetherian.iff_rank_lt_aleph0.2
-          (lt_of_le_of_lt (rank_submodule_le _) (_root_.rank_lt_aleph0 K V)))
+  · exact iff_fg.1 <| IsNoetherian.iff_rank_lt_aleph0.2 <|
+      (Submodule.rank_le _).trans_lt (rank_lt_aleph0 K V)
   · infer_instance
 
 /-- A quotient of a finite-dimensional space is also finite-dimensional. -/
@@ -156,18 +149,6 @@ instance finiteDimensional_quotient [FiniteDimensional K V] (S : Submodule K V)
     FiniteDimensional K (V ⧸ S) :=
   Module.Finite.quotient K S
 
-variable (K V)
-
-/-- In a finite-dimensional space, its dimension (seen as a cardinal) coincides with its
-`finrank`. This is a copy of `finrank_eq_rank _ _` which creates easier typeclass searches. -/
-theorem finrank_eq_rank' [FiniteDimensional K V] : (finrank K V : Cardinal.{v}) = Module.rank K V :=
-  finrank_eq_rank _ _
-
-variable {K V}
-
-theorem finrank_of_infinite_dimensional (h : ¬FiniteDimensional K V) : finrank K V = 0 :=
-  FiniteDimensional.finrank_of_not_finite h
-
 theorem of_finrank_pos (h : 0 < finrank K V) : FiniteDimensional K V :=
   Module.finite_of_finrank_pos h
 
@@ -181,6 +162,24 @@ theorem of_fact_finrank_eq_succ (n : ℕ) [hn : Fact (finrank K V = n + 1)] :
     FiniteDimensional K V :=
   of_finrank_eq_succ hn.out
 
+end FiniteDimensional
+
+namespace Module
+
+variable (K V)
+variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
+  [Module K V₂]
+
+/-- In a finite-dimensional space, its dimension (seen as a cardinal) coincides with its
+`finrank`. This is a copy of `finrank_eq_rank _ _` which creates easier typeclass searches. -/
+theorem finrank_eq_rank' [FiniteDimensional K V] : (finrank K V : Cardinal.{v}) = Module.rank K V :=
+  finrank_eq_rank _ _
+
+variable {K V}
+
+theorem finrank_of_infinite_dimensional (h : ¬FiniteDimensional K V) : finrank K V = 0 :=
+  Module.finrank_of_not_finite h
+
 theorem finiteDimensional_iff_of_rank_eq_nsmul {W} [AddCommGroup W] [Module K W] {n : ℕ}
     (hn : n ≠ 0) (hVW : Module.rank K V = n • Module.rank K W) :
     FiniteDimensional K V ↔ FiniteDimensional K W :=
@@ -192,11 +191,16 @@ theorem finrank_eq_card_basis' [FiniteDimensional K V] {ι : Type w} (h : Basis
     (finrank K V : Cardinal.{w}) = #ι :=
   Module.mk_finrank_eq_card_basis h
 
+end Module
+
+namespace FiniteDimensional
+section DivisionRing
+variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
+  [Module K V₂]
+
 theorem _root_.LinearIndependent.lt_aleph0_of_finiteDimensional {ι : Type w} [FiniteDimensional K V]
     {v : ι → V} (h : LinearIndependent K v) : #ι < ℵ₀ :=
   h.lt_aleph0_of_finite
-@[deprecated (since := "2023-12-27")]
-alias lt_aleph0_of_linearIndependent := LinearIndependent.lt_aleph0_of_finiteDimensional
 
 /-- If a submodule has maximal dimension in a finite dimensional space, then it is equal to the
 whole space. -/
@@ -221,7 +225,7 @@ theorem _root_.Submodule.eq_top_of_finrank_eq [FiniteDimensional K V] {S : Submo
       (by
         rw [Set.card_image_of_injective _ Subtype.coe_injective, ← finrank_eq_card_basis bS, ←
             finrank_eq_card_basis b, h])
-  rw [← b.span_eq, b_eq, Basis.coe_extend, Subtype.range_coe, ← this, ← Submodule.coeSubtype,
+  rw [← b.span_eq, b_eq, Basis.coe_extend, Subtype.range_coe, ← this, ← Submodule.coe_subtype,
     span_image]
   have := bS.span_eq
   rw [bS_eq, Basis.coe_ofVectorSpace, Subtype.range_coe] at this
@@ -254,8 +258,6 @@ section
 
 open Finset
 
-section
-
 variable {L : Type*} [LinearOrderedField L]
 variable {W : Type v} [AddCommGroup W] [Module L W]
 
@@ -271,16 +273,14 @@ theorem exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card [Finite
   exact ⟨f, sum, total, exists_pos_of_sum_zero_of_exists_nonzero f total nonzero⟩
 
 
-end
-
 end
 
 /-- In a vector space with dimension 1, each set {v} is a basis for `v ≠ 0`. -/
 @[simps repr_apply]
 noncomputable def basisSingleton (ι : Type*) [Unique ι] (h : finrank K V = 1) (v : V)
     (hv : v ≠ 0) : Basis ι K V :=
-  let b := FiniteDimensional.basisUnique ι h
-  let h : b.repr v default ≠ 0 := mt FiniteDimensional.basisUnique_repr_eq_zero_iff.mp hv
+  let b := Module.basisUnique ι h
+  let h : b.repr v default ≠ 0 := mt Module.basisUnique_repr_eq_zero_iff.mp hv
   Basis.ofRepr
     { toFun := fun w => Finsupp.single default (b.repr w default / b.repr v default)
       invFun := fun f => f default • v
@@ -326,8 +326,6 @@ section ZeroRank
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
-open FiniteDimensional
-
 theorem FiniteDimensional.of_rank_eq_nat {n : ℕ} (h : Module.rank K V = n) :
     FiniteDimensional K V :=
   Module.finite_of_rank_eq_nat h
@@ -350,7 +348,7 @@ alias finiteDimensional_of_rank_eq_one := FiniteDimensional.of_rank_eq_one
 variable (K V)
 
 instance finiteDimensional_bot : FiniteDimensional K (⊥ : Submodule K V) :=
-  of_rank_eq_zero <| by simp
+  .of_rank_eq_zero <| by simp
 
 variable {K V}
 
@@ -358,7 +356,7 @@ end ZeroRank
 
 namespace Submodule
 
-open IsNoetherian FiniteDimensional
+open IsNoetherian Module
 
 section DivisionRing
 
@@ -374,8 +372,7 @@ theorem finiteDimensional_of_le {S₁ S₂ : Submodule K V} [FiniteDimensional K
     FiniteDimensional K S₁ :=
   haveI : IsNoetherian K S₂ := iff_fg.2 inferInstance
   iff_fg.1
-    (IsNoetherian.iff_rank_lt_aleph0.2
-      (lt_of_le_of_lt (rank_le_of_submodule _ _ h) (rank_lt_aleph0 K S₂)))
+    (IsNoetherian.iff_rank_lt_aleph0.2 ((Submodule.rank_mono h).trans_lt (rank_lt_aleph0 K S₂)))
 
 /-- The inf of two submodules, the first finite-dimensional, is
 finite-dimensional. -/
@@ -424,7 +421,7 @@ end Submodule
 
 namespace LinearEquiv
 
-open FiniteDimensional
+open Module
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
   [Module K V₂]
@@ -449,10 +446,7 @@ instance finiteDimensional_finsupp {ι : Type*} [Finite ι] [FiniteDimensional K
 
 end
 
-namespace FiniteDimensional
-
-section DivisionRing
-
+namespace Submodule
 variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
   [Module K V₂]
 
@@ -470,33 +464,29 @@ theorem eq_of_le_of_finrank_eq {S₁ S₂ : Submodule K V} [FiniteDimensional K
     (hd : finrank K S₁ = finrank K S₂) : S₁ = S₂ :=
   eq_of_le_of_finrank_le hle hd.ge
 
-section Subalgebra
+end Submodule
+
+namespace Subalgebra
 
 variable {K L : Type*} [Field K] [Ring L] [Algebra K L] {F E : Subalgebra K L}
   [hfin : FiniteDimensional K E]
 
 /-- If a subalgebra is contained in a finite-dimensional
 subalgebra with the same or smaller dimension, they are equal. -/
-theorem _root_.Subalgebra.eq_of_le_of_finrank_le (h_le : F ≤ E)
-    (h_finrank : finrank K E ≤ finrank K F) : F = E :=
+theorem eq_of_le_of_finrank_le (h_le : F ≤ E) (h_finrank : finrank K E ≤ finrank K F) : F = E :=
   haveI : Module.Finite K (Subalgebra.toSubmodule E) := hfin
-  Subalgebra.toSubmodule_injective <| FiniteDimensional.eq_of_le_of_finrank_le h_le h_finrank
+  toSubmodule_injective <| Submodule.eq_of_le_of_finrank_le h_le h_finrank
 
 /-- If a subalgebra is contained in a finite-dimensional
 subalgebra with the same dimension, they are equal. -/
-theorem _root_.Subalgebra.eq_of_le_of_finrank_eq (h_le : F ≤ E)
-    (h_finrank : finrank K F = finrank K E) : F = E :=
-  Subalgebra.eq_of_le_of_finrank_le h_le h_finrank.ge
+theorem eq_of_le_of_finrank_eq (h_le : F ≤ E) (h_finrank : finrank K F = finrank K E) : F = E :=
+  eq_of_le_of_finrank_le h_le h_finrank.ge
 
 end Subalgebra
 
-end DivisionRing
-
-end FiniteDimensional
-
 namespace LinearMap
 
-open FiniteDimensional
+open Module
 
 section DivisionRing
 
@@ -600,7 +590,7 @@ end LinearMap
 
 namespace LinearEquiv
 
-open FiniteDimensional
+open Module
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 variable [FiniteDimensional K V]
@@ -647,14 +637,14 @@ theorem isUnit_iff_range_eq_top [FiniteDimensional K V] (f : V →ₗ[K] V) :
 
 end LinearMap
 
-open Module FiniteDimensional
+open FiniteDimensional Module
 
 section
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
 
 theorem finrank_zero_iff_forall_zero [FiniteDimensional K V] : finrank K V = 0 ↔ ∀ x : V, x = 0 :=
-  FiniteDimensional.finrank_zero_iff.trans (subsingleton_iff_forall_eq 0)
+  Module.finrank_zero_iff.trans (subsingleton_iff_forall_eq 0)
 
 /-- If `ι` is an empty type and `V` is zero-dimensional, there is a unique `ι`-indexed basis. -/
 noncomputable def basisOfFinrankZero [FiniteDimensional K V] {ι : Type*} [IsEmpty ι]
@@ -679,7 +669,7 @@ noncomputable def divisionRingOfFiniteDimensional (F K : Type*) [Field F] [Ring
   inv x :=
     letI := Classical.decEq K
     if H : x = 0 then 0 else Classical.choose <| FiniteDimensional.exists_mul_eq_one F H
-  mul_inv_cancel x hx := show x * dite _ (h := _) _ = _ by
+  mul_inv_cancel x hx := show x * dite _ (h := _) _ _ = _ by
     rw [dif_neg hx]
     exact (Classical.choose_spec (FiniteDimensional.exists_mul_eq_one F hx):)
   inv_zero := dif_pos rfl
@@ -694,21 +684,6 @@ noncomputable def fieldOfFiniteDimensional (F K : Type*) [Field F] [h : CommRing
   { divisionRingOfFiniteDimensional F K with
     toCommRing := h }
 end
-
-namespace Submodule
-
-section DivisionRing
-
-variable [DivisionRing K] [AddCommGroup V] [Module K V] {V₂ : Type v'} [AddCommGroup V₂]
-  [Module K V₂]
-
-theorem finrank_mono [FiniteDimensional K V] : Monotone fun s : Submodule K V => finrank K s :=
-  fun _ _ => finrank_le_finrank_of_le
-
-end DivisionRing
-
-end Submodule
-
 section DivisionRing
 
 variable [DivisionRing K] [AddCommGroup V] [Module K V]
@@ -722,6 +697,7 @@ theorem finrank_span_singleton {v : V} (hv : v ≠ 0) : finrank K (K ∙ v) = 1
   · exact finrank_span_le_card ({v} : Set V)
   · rw [Nat.succ_le_iff, finrank_pos_iff]
     use ⟨v, mem_span_singleton_self v⟩, 0
+    apply Subtype.coe_ne_coe.mp
     simp [hv]
 
 /-- In a one-dimensional space, any vector is a multiple of any nonzero vector -/
diff --git a/Mathlib/LinearAlgebra/Finsupp.lean b/Mathlib/LinearAlgebra/Finsupp.lean
index 563a9e33c8aad..810b8a01afebd 100644
--- a/Mathlib/LinearAlgebra/Finsupp.lean
+++ b/Mathlib/LinearAlgebra/Finsupp.lean
@@ -23,10 +23,10 @@ interpreted as a submodule of `α →₀ M`. We also define `LinearMap` versions
   linear map;
 * `Finsupp.restrictDom`: `Finsupp.filter` as a linear map to `Finsupp.supported s`;
 * `Finsupp.lsum`: `Finsupp.sum` or `Finsupp.liftAddHom` as a `LinearMap`;
-* `Finsupp.total R (v : ι → M)`: sends `l : ι → R` to the linear combination of `v i` with
-  coefficients `l i`;
-* `Finsupp.totalOn`: a restricted version of `Finsupp.total` with domain `Finsupp.supported R R s`
-  and codomain `Submodule.span R (v '' s)`;
+* `Finsupp.linearCombination R (v : ι → M)`: sends `l : ι → R` to the linear combination of
+  `v i` with coefficients `l i`;
+* `Finsupp.linearCombinationOn`: a restricted version of `Finsupp.linearCombination` with domain
+  `Finsupp.supported R R s` and codomain `Submodule.span R (v '' s)`;
 * `Finsupp.supportedEquivFinsupp`: a linear equivalence between the functions `α →₀ M` supported
   on `s` and the functions `s →₀ M`;
 * `Finsupp.lmapDomain`: a linear map version of `Finsupp.mapDomain`;
@@ -365,7 +365,7 @@ theorem supported_iUnion {δ : Type*} (s : δ → Set α) :
   · exact zero_mem _
   · refine fun x a l _ _ => add_mem ?_
     by_cases h : ∃ i, x ∈ s i
-    · simp only [mem_comap, coe_comp, coeSubtype, Function.comp_apply, restrictDom_apply,
+    · simp only [mem_comap, coe_comp, coe_subtype, Function.comp_apply, restrictDom_apply,
         mem_iUnion, h, filter_single_of_pos]
       cases' h with i hi
       exact le_iSup (fun i => supported M R (s i)) i (single_mem_supported R _ hi)
@@ -587,75 +587,103 @@ def lcomapDomain (f : α → β) (hf : Function.Injective f) : (β →₀ M) →
 
 end LComapDomain
 
-section Total
+section LinearCombination
 
 variable (R)
 variable {α' : Type*} {M' : Type*} [AddCommMonoid M'] [Module R M'] (v : α → M) {v' : α' → M'}
 
 /-- Interprets (l : α →₀ R) as a linear combination of the elements in the family (v : α → M) and
     evaluates this linear combination. -/
-protected def total : (α →₀ R) →ₗ[R] M :=
+def linearCombination : (α →₀ R) →ₗ[R] M :=
   Finsupp.lsum ℕ fun i => LinearMap.id.smulRight (v i)
 
+@[deprecated (since := "2024-08-29")] noncomputable alias total := linearCombination
+
 variable {v}
 
-theorem total_apply (l : α →₀ R) : Finsupp.total R v l = l.sum fun i a => a • v i :=
+theorem linearCombination_apply (l : α →₀ R) : linearCombination R v l = l.sum fun i a => a • v i :=
   rfl
 
-theorem total_apply_of_mem_supported {l : α →₀ R} {s : Finset α}
-    (hs : l ∈ supported R R (↑s : Set α)) : Finsupp.total R v l = s.sum fun i => l i • v i :=
+@[deprecated (since := "2024-08-29")] alias total_apply := linearCombination_apply
+
+theorem linearCombination_apply_of_mem_supported {l : α →₀ R} {s : Finset α}
+    (hs : l ∈ supported R R (↑s : Set α)) : linearCombination R v l = s.sum fun i => l i • v i :=
   Finset.sum_subset hs fun x _ hxg =>
     show l x • v x = 0 by rw [not_mem_support_iff.1 hxg, zero_smul]
 
+@[deprecated (since := "2024-08-29")] alias total_apply_of_mem_supported :=
+  linearCombination_apply_of_mem_supported
+
 @[simp]
-theorem total_single (c : R) (a : α) : Finsupp.total R v (single a c) = c • v a := by
-  simp [total_apply, sum_single_index]
+theorem linearCombination_single (c : R) (a : α) :
+    linearCombination R v (single a c) = c • v a := by
+  simp [linearCombination_apply, sum_single_index]
+
+@[deprecated (since := "2024-08-29")] alias total_single := linearCombination_single
+
+theorem linearCombination_zero_apply (x : α →₀ R) : (linearCombination R (0 : α → M)) x = 0 := by
+  simp [linearCombination_apply]
 
-theorem total_zero_apply (x : α →₀ R) : (Finsupp.total R (0 : α → M)) x = 0 := by
-  simp [Finsupp.total_apply]
+@[deprecated (since := "2024-08-29")] alias total_zero_apply := linearCombination_zero_apply
 
 variable (α M)
 
 @[simp]
-theorem total_zero : Finsupp.total R (0 : α → M) = 0 :=
-  LinearMap.ext (total_zero_apply R)
+theorem linearCombination_zero : linearCombination R (0 : α → M) = 0 :=
+  LinearMap.ext (linearCombination_zero_apply R)
+
+@[deprecated (since := "2024-08-29")] alias total_zero := linearCombination_zero
 
 variable {α M}
 
-theorem apply_total (f : M →ₗ[R] M') (v) (l : α →₀ R) :
-    f (Finsupp.total R v l) = Finsupp.total R (f ∘ v) l := by
+theorem apply_linearCombination (f : M →ₗ[R] M') (v) (l : α →₀ R) :
+    f (linearCombination R v l) = linearCombination R (f ∘ v) l := by
   apply Finsupp.induction_linear l <;> simp (config := { contextual := true })
 
-theorem apply_total_id (f : M →ₗ[R] M') (l : M →₀ R) :
-    f (Finsupp.total R _root_.id l) = Finsupp.total R f l :=
-  apply_total ..
+@[deprecated (since := "2024-08-29")] alias apply_total := apply_linearCombination
+
+theorem apply_linearCombination_id (f : M →ₗ[R] M') (l : M →₀ R) :
+    f (linearCombination R _root_.id l) = linearCombination R f l :=
+  apply_linearCombination ..
+
+@[deprecated (since := "2024-08-29")] alias apply_total_id := apply_linearCombination_id
 
-theorem total_unique [Unique α] (l : α →₀ R) (v : α → M) :
-    Finsupp.total R v l = l default • v default := by rw [← total_single, ← unique_single l]
+theorem linearCombination_unique [Unique α] (l : α →₀ R) (v : α → M) :
+    linearCombination R v l = l default • v default := by
+  rw [← linearCombination_single, ← unique_single l]
 
-theorem total_surjective (h : Function.Surjective v) :
-    Function.Surjective (Finsupp.total R v) := by
+@[deprecated (since := "2024-08-29")] alias total_unique := linearCombination_unique
+
+theorem linearCombination_surjective (h : Function.Surjective v) :
+    Function.Surjective (linearCombination R v) := by
   intro x
   obtain ⟨y, hy⟩ := h x
   exact ⟨Finsupp.single y 1, by simp [hy]⟩
 
-theorem total_range (h : Function.Surjective v) : LinearMap.range (Finsupp.total R v) = ⊤ :=
-  range_eq_top.2 <| total_surjective R h
+@[deprecated (since := "2024-08-29")] alias total_surjective := linearCombination_surjective
+
+theorem linearCombination_range (h : Function.Surjective v) :
+    LinearMap.range (linearCombination R v) = ⊤ :=
+  range_eq_top.2 <| linearCombination_surjective R h
+
+@[deprecated (since := "2024-08-29")] alias total_range := linearCombination_range
 
 /-- Any module is a quotient of a free module. This is stated as surjectivity of
-`Finsupp.total R id : (M →₀ R) →ₗ[R] M`. -/
-theorem total_id_surjective (M) [AddCommMonoid M] [Module R M] :
-    Function.Surjective (Finsupp.total R (id : M → M)) :=
-  total_surjective R Function.surjective_id
+`Finsupp.linearCombination R id : (M →₀ R) →ₗ[R] M`. -/
+theorem linearCombination_id_surjective (M) [AddCommMonoid M] [Module R M] :
+    Function.Surjective (linearCombination R (id : M → M)) :=
+  linearCombination_surjective R Function.surjective_id
+
+@[deprecated (since := "2024-08-29")] alias total_id_surjective := linearCombination_id_surjective
 
-theorem range_total : LinearMap.range (Finsupp.total R v) = span R (range v) := by
+theorem range_linearCombination : LinearMap.range (linearCombination R v) = span R (range v) := by
   ext x
   constructor
   · intro hx
     rw [LinearMap.mem_range] at hx
     rcases hx with ⟨l, hl⟩
     rw [← hl]
-    rw [Finsupp.total_apply]
+    rw [linearCombination_apply]
     exact sum_mem fun i _ => Submodule.smul_mem _ _ (subset_span (mem_range_self i))
   · apply span_le.2
     intro x hx
@@ -664,50 +692,68 @@ theorem range_total : LinearMap.range (Finsupp.total R v) = span R (range v) :=
     use Finsupp.single i 1
     simp [hi]
 
-theorem lmapDomain_total (f : α → α') (g : M →ₗ[R] M') (h : ∀ i, g (v i) = v' (f i)) :
-    (Finsupp.total R v').comp (lmapDomain R R f) = g.comp (Finsupp.total R v) := by
+@[deprecated (since := "2024-08-29")] alias range_total := range_linearCombination
+
+theorem lmapDomain_linearCombination (f : α → α') (g : M →ₗ[R] M') (h : ∀ i, g (v i) = v' (f i)) :
+    (linearCombination R v').comp (lmapDomain R R f) = g.comp (linearCombination R v) := by
   ext l
-  simp [total_apply, Finsupp.sum_mapDomain_index, add_smul, h]
+  simp [linearCombination_apply, Finsupp.sum_mapDomain_index, add_smul, h]
 
-theorem total_comp_lmapDomain (f : α → α') :
-    (Finsupp.total R v').comp (Finsupp.lmapDomain R R f) = Finsupp.total R (v' ∘ f) := by
+@[deprecated (since := "2024-08-29")] alias lmapDomain_total := lmapDomain_linearCombination
+
+theorem linearCombination_comp_lmapDomain (f : α → α') :
+    (linearCombination R v').comp (Finsupp.lmapDomain R R f) = linearCombination R (v' ∘ f) := by
   ext
   simp
 
+@[deprecated (since := "2024-08-29")] alias total_comp_lmapDomain :=
+  linearCombination_comp_lmapDomain
+
 @[simp]
-theorem total_embDomain (f : α ↪ α') (l : α →₀ R) :
-    (Finsupp.total R v') (embDomain f l) = (Finsupp.total R (v' ∘ f)) l := by
-  simp [total_apply, Finsupp.sum, support_embDomain, embDomain_apply]
+theorem linearCombination_embDomain (f : α ↪ α') (l : α →₀ R) :
+    (linearCombination R v') (embDomain f l) = (linearCombination R (v' ∘ f)) l := by
+  simp [linearCombination_apply, Finsupp.sum, support_embDomain, embDomain_apply]
+
+@[deprecated (since := "2024-08-29")] alias total_embDomain := linearCombination_embDomain
 
 @[simp]
-theorem total_mapDomain (f : α → α') (l : α →₀ R) :
-    (Finsupp.total R v') (mapDomain f l) = (Finsupp.total R (v' ∘ f)) l :=
-  LinearMap.congr_fun (total_comp_lmapDomain _ _) l
+theorem linearCombination_mapDomain (f : α → α') (l : α →₀ R) :
+    (linearCombination R v') (mapDomain f l) = (linearCombination R (v' ∘ f)) l :=
+  LinearMap.congr_fun (linearCombination_comp_lmapDomain _ _) l
+
+@[deprecated (since := "2024-08-29")] alias total_mapDomain := linearCombination_mapDomain
 
 @[simp]
-theorem total_equivMapDomain (f : α ≃ α') (l : α →₀ R) :
-    (Finsupp.total R v') (equivMapDomain f l) = (Finsupp.total R (v' ∘ f)) l := by
-  rw [equivMapDomain_eq_mapDomain, total_mapDomain]
+theorem linearCombination_equivMapDomain (f : α ≃ α') (l : α →₀ R) :
+    (linearCombination R v') (equivMapDomain f l) = (linearCombination R (v' ∘ f)) l := by
+  rw [equivMapDomain_eq_mapDomain, linearCombination_mapDomain]
+
+@[deprecated (since := "2024-08-29")] alias total_equivMapDomain := linearCombination_equivMapDomain
+
+/-- A version of `Finsupp.range_linearCombination` which is useful for going in the other
+direction -/
+theorem span_eq_range_linearCombination (s : Set M) :
+    span R s = LinearMap.range (linearCombination R ((↑) : s → M)) := by
+  rw [range_linearCombination, Subtype.range_coe_subtype, Set.setOf_mem_eq]
 
-/-- A version of `Finsupp.range_total` which is useful for going in the other direction -/
-theorem span_eq_range_total (s : Set M) :
-    span R s = LinearMap.range (Finsupp.total R ((↑) : s → M)) := by
-  rw [range_total, Subtype.range_coe_subtype, Set.setOf_mem_eq]
+@[deprecated (since := "2024-08-29")] alias span_eq_range_total := span_eq_range_linearCombination
 
-theorem mem_span_iff_total (s : Set M) (x : M) :
-    x ∈ span R s ↔ ∃ l : s →₀ R, Finsupp.total R (↑) l = x :=
-  (SetLike.ext_iff.1 <| span_eq_range_total _ _) x
+theorem mem_span_iff_linearCombination (s : Set M) (x : M) :
+    x ∈ span R s ↔ ∃ l : s →₀ R, linearCombination R (↑) l = x :=
+  (SetLike.ext_iff.1 <| span_eq_range_linearCombination _ _) x
+
+@[deprecated (since := "2024-08-29")] alias mem_span_iff_total := mem_span_iff_linearCombination
 
 variable {R}
 
 theorem mem_span_range_iff_exists_finsupp {v : α → M} {x : M} :
     x ∈ span R (range v) ↔ ∃ c : α →₀ R, (c.sum fun i a => a • v i) = x := by
-  simp only [← Finsupp.range_total, LinearMap.mem_range, Finsupp.total_apply]
+  simp only [← Finsupp.range_linearCombination, LinearMap.mem_range, linearCombination_apply]
 
 variable (R)
 
-theorem span_image_eq_map_total (s : Set α) :
-    span R (v '' s) = Submodule.map (Finsupp.total R v) (supported R R s) := by
+theorem span_image_eq_map_linearCombination (s : Set α) :
+    span R (v '' s) = Submodule.map (linearCombination R v) (supported R R s) := by
   apply span_eq_of_le
   · intro x hx
     rw [Set.mem_image] at hx
@@ -722,75 +768,99 @@ theorem span_image_eq_map_total (s : Set α) :
       · exact smul_mem _ _ (subset_span (Set.mem_image_of_mem _ h))
       · simp [(Finsupp.mem_supported' R _).1 hz _ h]
     -- Porting note: `rw` is required to infer metavariables in `sum_mem`.
-    rw [mem_comap, total_apply]
+    rw [mem_comap, linearCombination_apply]
     refine sum_mem ?_
     simp [this]
 
-theorem mem_span_image_iff_total {s : Set α} {x : M} :
-    x ∈ span R (v '' s) ↔ ∃ l ∈ supported R R s, Finsupp.total R v l = x := by
-  rw [span_image_eq_map_total]
+@[deprecated (since := "2024-08-29")] alias span_image_eq_map_total :=
+  span_image_eq_map_linearCombination
+
+theorem mem_span_image_iff_linearCombination {s : Set α} {x : M} :
+    x ∈ span R (v '' s) ↔ ∃ l ∈ supported R R s, linearCombination R v l = x := by
+  rw [span_image_eq_map_linearCombination]
   simp
 
-theorem total_option (v : Option α → M) (f : Option α →₀ R) :
-    Finsupp.total R v f =
-      f none • v none + Finsupp.total R (v ∘ Option.some) f.some := by
-  rw [total_apply, sum_option_index_smul, total_apply]; simp
+@[deprecated (since := "2024-08-29")] alias mem_span_image_iff_total :=
+  mem_span_image_iff_linearCombination
+
+theorem linearCombination_option (v : Option α → M) (f : Option α →₀ R) :
+    linearCombination R v f =
+      f none • v none + linearCombination R (v ∘ Option.some) f.some := by
+  rw [linearCombination_apply, sum_option_index_smul, linearCombination_apply]; simp
 
-theorem total_total {α β : Type*} (A : α → M) (B : β → α →₀ R) (f : β →₀ R) :
-    Finsupp.total R A (Finsupp.total R B f) =
-      Finsupp.total R (fun b => Finsupp.total R A (B b)) f := by
+@[deprecated (since := "2024-08-29")] alias total_option := linearCombination_option
+
+theorem linearCombination_linearCombination {α β : Type*} (A : α → M) (B : β → α →₀ R)
+    (f : β →₀ R) : linearCombination R A (linearCombination R B f) =
+      linearCombination R (fun b => linearCombination R A (B b)) f := by
   classical
-  simp only [total_apply]
+  simp only [linearCombination_apply]
   apply induction_linear f
   · simp only [sum_zero_index]
   · intro f₁ f₂ h₁ h₂
     simp [sum_add_index, h₁, h₂, add_smul]
   · simp [sum_single_index, sum_smul_index, smul_sum, mul_smul]
 
+@[deprecated (since := "2024-08-29")] alias total_total := linearCombination_linearCombination
+
 @[simp]
-theorem total_fin_zero (f : Fin 0 → M) : Finsupp.total R f = 0 := by
+theorem linearCombination_fin_zero (f : Fin 0 → M) : linearCombination R f = 0 := by
   ext i
   apply finZeroElim i
 
+@[deprecated (since := "2024-08-29")] alias total_fin_zero := linearCombination_fin_zero
+
 variable (α) (M) (v)
 
-/-- `Finsupp.totalOn M v s` interprets `p : α →₀ R` as a linear combination of a
+/-- `Finsupp.linearCombinationOn M v s` interprets `p : α →₀ R` as a linear combination of a
 subset of the vectors in `v`, mapping it to the span of those vectors.
 
 The subset is indicated by a set `s : Set α` of indices.
 -/
-protected def totalOn (s : Set α) : supported R R s →ₗ[R] span R (v '' s) :=
-  LinearMap.codRestrict _ ((Finsupp.total _ v).comp (Submodule.subtype (supported R R s)))
-    fun ⟨l, hl⟩ => (mem_span_image_iff_total _).2 ⟨l, hl, rfl⟩
+def linearCombinationOn (s : Set α) : supported R R s →ₗ[R] span R (v '' s) :=
+  LinearMap.codRestrict _ ((linearCombination _ v).comp (Submodule.subtype (supported R R s)))
+    fun ⟨l, hl⟩ => (mem_span_image_iff_linearCombination _).2 ⟨l, hl, rfl⟩
+
+@[deprecated (since := "2024-08-29")] noncomputable alias totalOn := linearCombinationOn
 
 variable {α} {M} {v}
 
-theorem totalOn_range (s : Set α) : LinearMap.range (Finsupp.totalOn α M R v s) = ⊤ := by
-  rw [Finsupp.totalOn, LinearMap.range_eq_map, LinearMap.map_codRestrict,
+theorem linearCombinationOn_range (s : Set α) :
+    LinearMap.range (linearCombinationOn α M R v s) = ⊤ := by
+  rw [linearCombinationOn, LinearMap.range_eq_map, LinearMap.map_codRestrict,
     ← LinearMap.range_le_iff_comap, range_subtype, Submodule.map_top, LinearMap.range_comp,
     range_subtype]
-  exact (span_image_eq_map_total _ _).le
+  exact (span_image_eq_map_linearCombination _ _).le
+
+@[deprecated (since := "2024-08-29")] alias totalOn_range := linearCombinationOn_range
 
-theorem total_comp (f : α' → α) :
-    Finsupp.total R (v ∘ f) = (Finsupp.total R v).comp (lmapDomain R R f) := by
+theorem linearCombination_comp (f : α' → α) :
+    linearCombination R (v ∘ f) = (linearCombination R v).comp (lmapDomain R R f) := by
   ext
-  simp [total_apply]
+  simp [linearCombination_apply]
 
-theorem total_comapDomain (f : α → α') (l : α' →₀ R) (hf : Set.InjOn f (f ⁻¹' ↑l.support)) :
-    Finsupp.total R v (Finsupp.comapDomain f l hf) =
+@[deprecated (since := "2024-08-29")] alias total_comp := linearCombination_comp
+
+theorem linearCombination_comapDomain (f : α → α') (l : α' →₀ R)
+    (hf : Set.InjOn f (f ⁻¹' ↑l.support)) : linearCombination R v (Finsupp.comapDomain f l hf) =
       (l.support.preimage f hf).sum fun i => l (f i) • v i := by
-  rw [Finsupp.total_apply]; rfl
+  rw [linearCombination_apply]; rfl
+
+@[deprecated (since := "2024-08-29")] alias total_comapDomain := linearCombination_comapDomain
 
-theorem total_onFinset {s : Finset α} {f : α → R} (g : α → M) (hf : ∀ a, f a ≠ 0 → a ∈ s) :
-    Finsupp.total R g (Finsupp.onFinset s f hf) = Finset.sum s fun x : α => f x • g x := by
+theorem linearCombination_onFinset {s : Finset α} {f : α → R} (g : α → M)
+    (hf : ∀ a, f a ≠ 0 → a ∈ s) :
+    linearCombination R g (Finsupp.onFinset s f hf) = Finset.sum s fun x : α => f x • g x := by
   classical
-  simp only [Finsupp.total_apply, Finsupp.sum, Finsupp.onFinset_apply, Finsupp.support_onFinset]
+  simp only [linearCombination_apply, Finsupp.sum, Finsupp.onFinset_apply, Finsupp.support_onFinset]
   rw [Finset.sum_filter_of_ne]
   intro x _ h
   contrapose! h
   simp [h]
 
-end Total
+@[deprecated (since := "2024-08-29")] alias total_onFinset := linearCombination_onFinset
+
+end LinearCombination
 
 /-- An equivalence of domains induces a linear equivalence of finitely supported functions.
 
@@ -1037,13 +1107,13 @@ variable {α M : Type*} (R : Type*) [Fintype α] [Semiring R] [AddCommMonoid M]
 variable (S : Type*) [Semiring S] [Module S M] [SMulCommClass R S M]
 variable (v : α → M)
 
-/-- `Fintype.total R S v f` is the linear combination of vectors in `v` with weights in `f`.
-This variant of `Finsupp.total` is defined on fintype indexed vectors.
+/-- `Fintype.linearCombination R S v f` is the linear combination of vectors in `v` with weights
+in `f`. This variant of `Finsupp.linearCombination` is defined on fintype indexed vectors.
 
 This map is linear in `v` if `R` is commutative, and always linear in `f`.
 See note [bundled maps over different rings] for why separate `R` and `S` semirings are used.
 -/
-protected def Fintype.total : (α → M) →ₗ[S] (α → R) →ₗ[R] M where
+protected def Fintype.linearCombination : (α → M) →ₗ[S] (α → R) →ₗ[R] M where
   toFun v :=
     { toFun := fun f => ∑ i, f i • v i
       map_add' := fun f g => by simp_rw [← Finset.sum_add_distrib, ← add_smul]; rfl
@@ -1051,39 +1121,55 @@ protected def Fintype.total : (α → M) →ₗ[S] (α → R) →ₗ[R] M where
   map_add' u v := by ext; simp [Finset.sum_add_distrib, Pi.add_apply, smul_add]
   map_smul' r v := by ext; simp [Finset.smul_sum, smul_comm]
 
+@[deprecated (since := "2024-08-29")] alias Fintype.total := Fintype.linearCombination
+
 variable {S}
 
-theorem Fintype.total_apply (f) : Fintype.total R S v f = ∑ i, f i • v i :=
+theorem Fintype.linearCombination_apply (f) : Fintype.linearCombination R S v f = ∑ i, f i • v i :=
   rfl
 
+@[deprecated (since := "2024-08-29")] alias Fintype.total_apply := Fintype.linearCombination_apply
+
 @[simp]
-theorem Fintype.total_apply_single [DecidableEq α] (i : α) (r : R) :
-    Fintype.total R S v (Pi.single i r) = r • v i := by
-  simp_rw [Fintype.total_apply, Pi.single_apply, ite_smul, zero_smul]
+theorem Fintype.linearCombination_apply_single [DecidableEq α] (i : α) (r : R) :
+    Fintype.linearCombination R S v (Pi.single i r) = r • v i := by
+  simp_rw [Fintype.linearCombination_apply, Pi.single_apply, ite_smul, zero_smul]
   rw [Finset.sum_ite_eq', if_pos (Finset.mem_univ _)]
 
+@[deprecated (since := "2024-08-29")] alias Fintype.total_apply_single :=
+  Fintype.linearCombination_apply_single
+
 variable (S)
 
-theorem Finsupp.total_eq_fintype_total_apply (x : α → R) : Finsupp.total R v
-    ((Finsupp.linearEquivFunOnFinite R R α).symm x) = Fintype.total R S v x := by
+theorem Finsupp.linearCombination_eq_fintype_linearCombination_apply (x : α → R) :
+    linearCombination R v ((Finsupp.linearEquivFunOnFinite R R α).symm x) =
+      Fintype.linearCombination R S v x := by
   apply Finset.sum_subset
   · exact Finset.subset_univ _
   · intro x _ hx
     rw [Finsupp.not_mem_support_iff.mp hx]
     exact zero_smul _ _
 
-theorem Finsupp.total_eq_fintype_total :
-    (Finsupp.total R v).comp (Finsupp.linearEquivFunOnFinite R R α).symm.toLinearMap =
-      Fintype.total R S v :=
-  LinearMap.ext <| Finsupp.total_eq_fintype_total_apply R S v
+@[deprecated (since := "2024-08-29")] alias Finsupp.total_eq_fintype_total_apply :=
+  Finsupp.linearCombination_eq_fintype_linearCombination_apply
+
+theorem Finsupp.linearCombination_eq_fintype_linearCombination :
+    (linearCombination R v).comp (Finsupp.linearEquivFunOnFinite R R α).symm.toLinearMap =
+      Fintype.linearCombination R S v :=
+  LinearMap.ext <| linearCombination_eq_fintype_linearCombination_apply R S v
+
+@[deprecated (since := "2024-08-29")] alias Finsupp.total_eq_fintype_total :=
+  Finsupp.linearCombination_eq_fintype_linearCombination
 
 variable {S}
 
 @[simp]
-theorem Fintype.range_total :
-    LinearMap.range (Fintype.total R S v) = Submodule.span R (Set.range v) := by
-  rw [← Finsupp.total_eq_fintype_total, LinearMap.range_comp, LinearEquiv.range,
-    Submodule.map_top, Finsupp.range_total]
+theorem Fintype.range_linearCombination :
+    LinearMap.range (Fintype.linearCombination R S v) = Submodule.span R (Set.range v) := by
+  rw [← Finsupp.linearCombination_eq_fintype_linearCombination, LinearMap.range_comp,
+      LinearEquiv.range, Submodule.map_top, Finsupp.range_linearCombination]
+
+@[deprecated (since := "2024-08-29")] alias Fintype.range_total := Fintype.range_linearCombination
 
 section SpanRange
 
@@ -1113,32 +1199,40 @@ end Fintype
 variable {R : Type*} {M : Type*} {N : Type*}
 variable [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
 
+open Finsupp
+
 section
 
 variable (R)
 
 /-- Pick some representation of `x : span R w` as a linear combination in `w`,
-using the axiom of choice.
+  ((Finsupp.mem_span_iff_linearCombination _ _ _).mp x.2).choose
 -/
 irreducible_def Span.repr (w : Set M) (x : span R w) : w →₀ R :=
-  ((Finsupp.mem_span_iff_total _ _ _).mp x.2).choose
+  ((Finsupp.mem_span_iff_linearCombination _ _ _).mp x.2).choose
 
 @[simp]
-theorem Span.finsupp_total_repr {w : Set M} (x : span R w) :
-    Finsupp.total R ((↑) : w → M) (Span.repr R w x) = x := by
+theorem Span.finsupp_linearCombination_repr {w : Set M} (x : span R w) :
+    Finsupp.linearCombination R ((↑) : w → M) (Span.repr R w x) = x := by
   rw [Span.repr_def]
-  exact ((Finsupp.mem_span_iff_total _ _ _).mp x.2).choose_spec
+  exact ((Finsupp.mem_span_iff_linearCombination _ _ _).mp x.2).choose_spec
 
+@[deprecated (since := "2024-08-29")] alias Span.finsupp_total_repr :=
+  Span.finsupp_linearCombination_repr
 end
 
 protected theorem Submodule.finsupp_sum_mem {ι β : Type*} [Zero β] (S : Submodule R M) (f : ι →₀ β)
     (g : ι → β → M) (h : ∀ c, f c ≠ 0 → g c (f c) ∈ S) : f.sum g ∈ S :=
   AddSubmonoidClass.finsupp_sum_mem S f g h
 
-theorem LinearMap.map_finsupp_total (f : M →ₗ[R] N) {ι : Type*} {g : ι → M} (l : ι →₀ R) :
-    f (Finsupp.total R g l) = Finsupp.total R (f ∘ g) l := by
+theorem LinearMap.map_finsupp_linearCombination (f : M →ₗ[R] N) {ι : Type*} {g : ι → M}
+    (l : ι →₀ R) : f (linearCombination R g l) = linearCombination R (f ∘ g) l := by
   -- Porting note: `(· ∘ ·)` is required.
-  simp only [Finsupp.total_apply, Finsupp.total_apply, Finsupp.sum, map_sum, map_smul, (· ∘ ·)]
+  simp only [linearCombination_apply, linearCombination_apply, Finsupp.sum, map_sum, map_smul,
+             (· ∘ ·)]
+
+@[deprecated (since := "2024-08-29")] alias LinearMap.map_finsupp_total :=
+  LinearMap.map_finsupp_linearCombination
 
 theorem Submodule.exists_finset_of_mem_iSup {ι : Sort _} (p : ι → Submodule R M) {m : M}
     (hm : m ∈ ⨆ i, p i) : ∃ s : Finset ι, m ∈ ⨆ i ∈ s, p i := by
@@ -1169,9 +1263,9 @@ theorem mem_span_finset {s : Finset M} {x : M} :
     x ∈ span R (↑s : Set M) ↔ ∃ f : M → R, ∑ i ∈ s, f i • i = x :=
   ⟨fun hx =>
     let ⟨v, hvs, hvx⟩ :=
-      (Finsupp.mem_span_image_iff_total _).1
+      (Finsupp.mem_span_image_iff_linearCombination _).1
         (show x ∈ span R (_root_.id '' (↑s : Set M)) by rwa [Set.image_id])
-    ⟨v, hvx ▸ (Finsupp.total_apply_of_mem_supported _ hvs).symm⟩,
+    ⟨v, hvx ▸ (linearCombination_apply_of_mem_supported _ hvs).symm⟩,
     fun ⟨f, hf⟩ => hf ▸ sum_mem fun i hi => smul_mem _ _ <| subset_span hi⟩
 
 /-- An element `m ∈ M` is contained in the `R`-submodule spanned by a set `s ⊆ M`, if and only if
@@ -1181,7 +1275,7 @@ theorem mem_span_set {m : M} {s : Set M} :
     m ∈ Submodule.span R s ↔
       ∃ c : M →₀ R, (c.support : Set M) ⊆ s ∧ (c.sum fun mi r => r • mi) = m := by
   conv_lhs => rw [← Set.image_id s]
-  exact Finsupp.mem_span_image_iff_total R (v := _root_.id (α := M))
+  exact Finsupp.mem_span_image_iff_linearCombination R (v := _root_.id (α := M))
 
 /-- An element `m ∈ M` is contained in the `R`-submodule spanned by a set `s ⊆ M`, if and only if
 `m` can be written as a finite `R`-linear combination of elements of `s`.
diff --git a/Mathlib/LinearAlgebra/FinsuppVectorSpace.lean b/Mathlib/LinearAlgebra/FinsuppVectorSpace.lean
index 6c96e6b6ad6ae..a25ac55d3ce5c 100644
--- a/Mathlib/LinearAlgebra/FinsuppVectorSpace.lean
+++ b/Mathlib/LinearAlgebra/FinsuppVectorSpace.lean
@@ -110,8 +110,7 @@ theorem coe_basis {φ : ι → Type*} (b : ∀ i, Basis (φ i) R M) :
           Finsupp.single_apply_left sigma_mk_injective]
       · have : Sigma.mk i x ≠ Sigma.mk j y := fun h' => h <| congrArg (fun s => s.fst) h'
         -- Porting note: previously `this` not needed
-        simp only [basis_repr, single_apply, h, this, false_and_iff, if_false, LinearEquiv.map_zero,
-        zero_apply]
+        simp only [basis_repr, single_apply, h, this, if_false, LinearEquiv.map_zero, zero_apply]
 
 /-- The basis on `ι →₀ M` with basis vectors `fun i ↦ single i 1`. -/
 @[simps]
diff --git a/Mathlib/LinearAlgebra/FreeAlgebra.lean b/Mathlib/LinearAlgebra/FreeAlgebra.lean
index 5b221a8d639dd..6853792914e06 100644
--- a/Mathlib/LinearAlgebra/FreeAlgebra.lean
+++ b/Mathlib/LinearAlgebra/FreeAlgebra.lean
@@ -5,6 +5,7 @@ Authors: Eric Wieser
 -/
 import Mathlib.LinearAlgebra.Basis.Cardinality
 import Mathlib.Algebra.FreeAlgebra
+import Mathlib.LinearAlgebra.FreeModule.Basic
 import Mathlib.LinearAlgebra.FinsuppVectorSpace
 import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
 import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
diff --git a/Mathlib/LinearAlgebra/FreeModule/Basic.lean b/Mathlib/LinearAlgebra/FreeModule/Basic.lean
index d93fc0610a6b4..ae0a984945266 100644
--- a/Mathlib/LinearAlgebra/FreeModule/Basic.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/Basic.lean
@@ -13,7 +13,8 @@ import Mathlib.LinearAlgebra.TensorProduct.Basis
 We introduce a class `Module.Free R M`, for `R` a `Semiring` and `M` an `R`-module and we provide
 several basic instances for this class.
 
-Use `Finsupp.total_id_surjective` to prove that any module is the quotient of a free module.
+Use `Finsupp.linearCombination_id_surjective` to prove that any module is the quotient of a free
+module.
 
 ## Main definition
 
diff --git a/Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean b/Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean
index b79eba46aca11..fd4961dd7d2fc 100644
--- a/Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/Finite/Basic.lean
@@ -20,16 +20,9 @@ We provide some instances for finite and free modules.
 
 universe u v w
 
-variable (R : Type u) (M : Type v) (N : Type w)
-
-namespace Module.Free
-
-section Ring
-
-variable [Ring R] [AddCommGroup M] [Module R M] [Module.Free R M]
-
 /-- If a free module is finite, then the arbitrary basis is finite. -/
-noncomputable instance ChooseBasisIndex.fintype [Module.Finite R M] :
+noncomputable instance Module.Free.ChooseBasisIndex.fintype (R : Type u) (M : Type v)
+    [Semiring R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] :
     Fintype (Module.Free.ChooseBasisIndex R M) := by
   refine @Fintype.ofFinite _ ?_
   cases subsingleton_or_nontrivial R
@@ -38,28 +31,17 @@ noncomputable instance ChooseBasisIndex.fintype [Module.Finite R M] :
     infer_instance
   · exact Module.Finite.finite_basis (chooseBasis _ _)
 
-end Ring
-
-section CommRing
-
-variable [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M]
-variable [AddCommGroup N] [Module R N] [Module.Free R N]
-variable {R}
-
 /-- A free module with a basis indexed by a `Fintype` is finite. -/
-theorem _root_.Module.Finite.of_basis {R M ι : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
+theorem Module.Finite.of_basis {R M ι : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
     [_root_.Finite ι] (b : Basis ι R M) : Module.Finite R M := by
   cases nonempty_fintype ι
   classical
     refine ⟨⟨Finset.univ.image b, ?_⟩⟩
     simp only [Set.image_univ, Finset.coe_univ, Finset.coe_image, Basis.span_eq]
 
-instance _root_.Module.Finite.matrix {ι₁ ι₂ : Type*} [_root_.Finite ι₁] [_root_.Finite ι₂] :
+instance Module.Finite.matrix {R : Type u} [Semiring R]
+    {ι₁ ι₂ : Type*} [_root_.Finite ι₁] [_root_.Finite ι₂] :
     Module.Finite R (Matrix ι₁ ι₂ R) := by
   cases nonempty_fintype ι₁
   cases nonempty_fintype ι₂
   exact Module.Finite.of_basis (Pi.basis fun _ => Pi.basisFun R _)
-
-end CommRing
-
-end Module.Free
diff --git a/Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean b/Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean
index 7be847207c3de..27e40eb067043 100644
--- a/Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean
@@ -26,7 +26,7 @@ variable (R : Type u) (S : Type u') (M : Type v) (N : Type w)
 
 open Module.Free (chooseBasis ChooseBasisIndex)
 
-open FiniteDimensional (finrank)
+open Module (finrank)
 
 section Ring
 
@@ -46,25 +46,28 @@ instance Module.Finite.linearMap [Module.Finite S N] : Module.Finite S (M →ₗ
 variable [StrongRankCondition R] [StrongRankCondition S] [Module.Free S N]
 
 open Cardinal
-theorem FiniteDimensional.rank_linearMap :
+theorem Module.rank_linearMap :
     Module.rank S (M →ₗ[R] N) = lift.{w} (Module.rank R M) * lift.{v} (Module.rank S N) := by
   rw [(linearMapEquivFun R S M N).rank_eq, rank_fun_eq_lift_mul,
     ← finrank_eq_card_chooseBasisIndex, ← finrank_eq_rank R, lift_natCast]
 
 /-- The finrank of `M →ₗ[R] N` as an `S`-module is `(finrank R M) * (finrank S N)`. -/
-theorem FiniteDimensional.finrank_linearMap :
+theorem Module.finrank_linearMap :
     finrank S (M →ₗ[R] N) = finrank R M * finrank S N := by
   simp_rw [finrank, rank_linearMap, toNat_mul, toNat_lift]
 
 variable [Module R S] [SMulCommClass R S S]
 
-theorem FiniteDimensional.rank_linearMap_self :
+theorem Module.rank_linearMap_self :
     Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) := by
   rw [rank_linearMap, rank_self, lift_one, mul_one]
 
-theorem FiniteDimensional.finrank_linearMap_self : finrank S (M →ₗ[R] S) = finrank R M := by
+theorem Module.finrank_linearMap_self : finrank S (M →ₗ[R] S) = finrank R M := by
   rw [finrank_linearMap, finrank_self, mul_one]
 
+@[deprecated (since := "2024-01-12")]
+alias Module.finrank_linear_map' := Module.finrank_linearMap_self
+
 end Ring
 
 section AlgHom
@@ -81,12 +84,12 @@ theorem cardinal_mk_algHom_le_rank : #(M →ₐ[K] L) ≤ lift.{v} (Module.rank
   convert (linearIndependent_algHom_toLinearMap K M L).cardinal_lift_le_rank
   · rw [lift_id]
   · have := Module.nontrivial K L
-    rw [lift_id, FiniteDimensional.rank_linearMap_self]
+    rw [lift_id, Module.rank_linearMap_self]
 
 theorem card_algHom_le_finrank : Nat.card (M →ₐ[K] L) ≤ finrank K M := by
   convert toNat_le_toNat (cardinal_mk_algHom_le_rank K M L) ?_
   · rw [toNat_lift, finrank]
-  · rw [lift_lt_aleph0]; have := Module.nontrivial K L; apply rank_lt_aleph0
+  · rw [lift_lt_aleph0]; have := Module.nontrivial K L; apply Module.rank_lt_aleph0
 
 end AlgHom
 
diff --git a/Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean b/Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean
index 44870709f726a..054d145f3347d 100644
--- a/Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/IdealQuotient.lean
@@ -114,9 +114,9 @@ noncomputable def quotientEquivDirectSum :
 theorem finrank_quotient_eq_sum {ι} [Fintype ι] (b : Basis ι R S) [Nontrivial F]
     [∀ i, Module.Free F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))]
     [∀ i, Module.Finite F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))] :
-    FiniteDimensional.finrank F (S ⧸ I) =
-      ∑ i, FiniteDimensional.finrank F (R ⧸ span ({I.smithCoeffs b hI i} : Set R)) := by
+    Module.finrank F (S ⧸ I) =
+      ∑ i, Module.finrank F (R ⧸ span ({I.smithCoeffs b hI i} : Set R)) := by
   -- slow, and dot notation doesn't work
-  rw [LinearEquiv.finrank_eq <| quotientEquivDirectSum F b hI, FiniteDimensional.finrank_directSum]
+  rw [LinearEquiv.finrank_eq <| quotientEquivDirectSum F b hI, Module.finrank_directSum]
 
 end Ideal
diff --git a/Mathlib/LinearAlgebra/FreeModule/Norm.lean b/Mathlib/LinearAlgebra/FreeModule/Norm.lean
index 39bb414d9ce5c..5332fcc328e26 100644
--- a/Mathlib/LinearAlgebra/FreeModule/Norm.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/Norm.lean
@@ -71,7 +71,7 @@ instance (b : Basis ι F[X] S) {I : Ideal S} (hI : I ≠ ⊥) (i : ι) :
 `F`-vector space is the degree of the norm of `f` relative to `F[X]`. -/
 theorem finrank_quotient_span_eq_natDegree_norm [Algebra F S] [IsScalarTower F F[X] S]
     (b : Basis ι F[X] S) {f : S} (hf : f ≠ 0) :
-    FiniteDimensional.finrank F (S ⧸ span ({f} : Set S)) = (Algebra.norm F[X] f).natDegree := by
+    Module.finrank F (S ⧸ span ({f} : Set S)) = (Algebra.norm F[X] f).natDegree := by
   haveI := Fintype.ofFinite ι
   have h := span_singleton_eq_bot.not.2 hf
   rw [natDegree_eq_of_degree_eq
diff --git a/Mathlib/LinearAlgebra/FreeModule/PID.lean b/Mathlib/LinearAlgebra/FreeModule/PID.lean
index 4d837a3b56384..3e5251f600539 100644
--- a/Mathlib/LinearAlgebra/FreeModule/PID.lean
+++ b/Mathlib/LinearAlgebra/FreeModule/PID.lean
@@ -234,7 +234,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
     rw [LinearMap.mem_ker] at hx'
     have hc' : (c • ⟨y', y'M⟩ + ⟨x, xM⟩ : M) = 0 := by exact @Subtype.coe_injective O (· ∈ M) _ _ hc
     simpa only [LinearMap.map_add, LinearMap.map_zero, LinearMap.map_smul, smul_eq_mul, add_zero,
-      mul_eq_zero, ϕy'_ne_zero, hx', or_false_iff] using congr_arg ϕ hc'
+      mul_eq_zero, ϕy'_ne_zero, hx', or_false] using congr_arg ϕ hc'
   -- And `a • y'` is orthogonal to `N'`.
   have ay'_ortho_N' : ∀ (c : R), ∀ z ∈ N', c • a • y' + z = 0 → c = 0 := by
     intro c z zN' hc
@@ -251,7 +251,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
       refine ⟨-b, Submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, ?_, ?_⟩⟩
       · refine LinearMap.mem_ker.mpr (show ϕ (⟨z, N_le_M zN⟩ - b • ⟨y, N_le_M yN⟩) = 0 from ?_)
         rw [LinearMap.map_sub, LinearMap.map_smul, hb, ϕy_eq, smul_eq_mul, mul_comm, sub_self]
-      · simp only [sub_eq_add_neg, neg_smul, coeSubtype]
+      · simp only [sub_eq_add_neg, neg_smul, coe_subtype]
   -- And extend a basis for `M'` with `y'`
   intro m' hn'm' bM'
   refine ⟨Nat.succ_le_succ hn'm', ?_, ?_⟩
@@ -270,7 +270,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
   · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm
   · rw [Fin.castLE_succ]
-    simp only [Fin.cons_succ, Function.comp_apply, coe_inclusion, map_coe, coeSubtype, h i]
+    simp only [Fin.cons_succ, Function.comp_apply, coe_inclusion, map_coe, coe_subtype, h i]
 
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
@@ -369,9 +369,8 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
         apply this
       intro i
       calc
-        (∏ j, a j) • s i = (∏ j ∈ {i}ᶜ, a j) • a i • s i := by
-          rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
-        _ ∈ N := N.smul_mem _ (ha' i)
+        (∏ j ∈ {i}ᶜ, a j) • a i • s i ∈ N := N.smul_mem _ (ha' i)
+        _ = (∏ j, a j) • s i := by rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
 
     -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
     obtain ⟨n, b : Basis (Fin n) R (LinearMap.range φ)⟩ := Submodule.basisOfPidOfLE this sI_basis
diff --git a/Mathlib/LinearAlgebra/InvariantBasisNumber.lean b/Mathlib/LinearAlgebra/InvariantBasisNumber.lean
index e339f797284f7..1a9f732fb0a95 100644
--- a/Mathlib/LinearAlgebra/InvariantBasisNumber.lean
+++ b/Mathlib/LinearAlgebra/InvariantBasisNumber.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel, Scott Morrison
+Authors: Markus Himmel, Kim Morrison
 -/
 import Mathlib.RingTheory.OrzechProperty
 import Mathlib.RingTheory.Ideal.Quotient
diff --git a/Mathlib/LinearAlgebra/Isomorphisms.lean b/Mathlib/LinearAlgebra/Isomorphisms.lean
index 7a9bb31cd43fa..54a39b6372052 100644
--- a/Mathlib/LinearAlgebra/Isomorphisms.lean
+++ b/Mathlib/LinearAlgebra/Isomorphisms.lean
@@ -83,7 +83,7 @@ theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
   rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
   rintro ⟨x, hx⟩; rcases mem_sup.1 hx with ⟨y, hy, z, hz, rfl⟩
   use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
-  simp only [mem_comap, map_sub, coeSubtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
+  simp only [mem_comap, map_sub, coe_subtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
 
 /--
 Second Isomorphism Law : the canonical map from `p/(p ∩ p')` to `(p+p')/p'` as a linear isomorphism.
@@ -122,7 +122,7 @@ theorem quotientInfEquivSupQuotient_symm_apply_eq_zero_iff {p p' : Submodule R M
     (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x) = 0 ↔ (x : M) ∈ p' :=
   (LinearEquiv.symm_apply_eq _).trans <| by
     -- porting note (#10745): was `simp`.
-    rw [_root_.map_zero, Quotient.mk_eq_zero, mem_comap, Submodule.coeSubtype]
+    rw [_root_.map_zero, Quotient.mk_eq_zero, mem_comap, Submodule.coe_subtype]
 
 theorem quotientInfEquivSupQuotient_symm_apply_right (p p' : Submodule R M) {x : ↥(p ⊔ p')}
     (hx : (x : M) ∈ p') : (quotientInfEquivSupQuotient p p').symm (Submodule.Quotient.mk x)
diff --git a/Mathlib/LinearAlgebra/Lagrange.lean b/Mathlib/LinearAlgebra/Lagrange.lean
index 982898e6d7590..ab7d725e81277 100644
--- a/Mathlib/LinearAlgebra/Lagrange.lean
+++ b/Mathlib/LinearAlgebra/Lagrange.lean
@@ -145,7 +145,7 @@ theorem basisDivisor_self : basisDivisor x x = 0 := by
   simp only [basisDivisor, sub_self, inv_zero, map_zero, zero_mul]
 
 theorem basisDivisor_inj (hxy : basisDivisor x y = 0) : x = y := by
-  simp_rw [basisDivisor, mul_eq_zero, X_sub_C_ne_zero, or_false_iff, C_eq_zero, inv_eq_zero,
+  simp_rw [basisDivisor, mul_eq_zero, X_sub_C_ne_zero, or_false, C_eq_zero, inv_eq_zero,
     sub_eq_zero] at hxy
   exact hxy
 
diff --git a/Mathlib/LinearAlgebra/LinearDisjoint.lean b/Mathlib/LinearAlgebra/LinearDisjoint.lean
index b611bef80c0f4..f290e4e777d86 100644
--- a/Mathlib/LinearAlgebra/LinearDisjoint.lean
+++ b/Mathlib/LinearAlgebra/LinearDisjoint.lean
@@ -208,15 +208,15 @@ theorem of_basis_right' {ι : Type*} (n : Basis ι R N)
 
 /-- If `{ m_i }` is an `R`-basis of `M`, if `{ n_i }` is an `R`-basis of `N`,
 such that the family `{ m_i * n_j }` in `S` is `R`-linearly independent
-(in this result it is stated as the relevant `Finsupp.total` is injective),
+(in this result it is stated as the relevant `Finsupp.linearCombination` is injective),
 then `M` and `N` are linearly disjoint. -/
 theorem of_basis_mul' {κ ι : Type*} (m : Basis κ R M) (n : Basis ι R N)
-    (H : Function.Injective (Finsupp.total R fun i : κ × ι ↦ (m i.1 * n i.2 : S))) :
+    (H : Function.Injective (Finsupp.linearCombination R fun i : κ × ι ↦ (m i.1 * n i.2 : S))) :
     M.LinearDisjoint N := by
   let i0 := (finsuppTensorFinsupp' R κ ι).symm
   let i1 := TensorProduct.congr m.repr n.repr
   let i := mulMap M N ∘ₗ (i0.trans i1.symm).toLinearMap
-  have : i = Finsupp.total R fun i : κ × ι ↦ (m i.1 * n i.2 : S) := by
+  have : i = Finsupp.linearCombination R fun i : κ × ι ↦ (m i.1 * n i.2 : S) := by
     ext x
     simp [i, i0, i1, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
   simp_rw [← this, i, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.injective_comp] at H
@@ -311,7 +311,7 @@ theorem linearIndependent_left_of_flat (H : M.LinearDisjoint N) [Module.Flat R N
     {ι : Type*} {m : ι → M} (hm : LinearIndependent R m) : LinearMap.ker (mulLeftMap N m) = ⊥ := by
   refine LinearMap.ker_eq_bot_of_injective ?_
   classical simp_rw [mulLeftMap_eq_mulMap_comp, LinearMap.coe_comp, LinearEquiv.coe_coe,
-    ← Function.comp.assoc, EquivLike.injective_comp]
+    ← Function.comp_assoc, EquivLike.injective_comp]
   rw [LinearIndependent, LinearMap.ker_eq_bot] at hm
   exact H.injective.comp (Module.Flat.rTensor_preserves_injective_linearMap (M := N) _ hm)
 
@@ -332,7 +332,7 @@ theorem linearIndependent_right_of_flat (H : M.LinearDisjoint N) [Module.Flat R
     {ι : Type*} {n : ι → N} (hn : LinearIndependent R n) : LinearMap.ker (mulRightMap M n) = ⊥ := by
   refine LinearMap.ker_eq_bot_of_injective ?_
   classical simp_rw [mulRightMap_eq_mulMap_comp, LinearMap.coe_comp, LinearEquiv.coe_coe,
-    ← Function.comp.assoc, EquivLike.injective_comp]
+    ← Function.comp_assoc, EquivLike.injective_comp]
   rw [LinearIndependent, LinearMap.ker_eq_bot] at hn
   exact H.injective.comp (Module.Flat.lTensor_preserves_injective_linearMap (M := M) _ hn)
 
@@ -354,13 +354,13 @@ theorem linearIndependent_mul_of_flat_left (H : M.LinearDisjoint N) [Module.Flat
     (hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 := by
   rw [LinearIndependent, LinearMap.ker_eq_bot] at hm hn ⊢
   let i0 := (finsuppTensorFinsupp' R κ ι).symm
-  let i1 := LinearMap.rTensor (ι →₀ R) (Finsupp.total R m)
-  let i2 := LinearMap.lTensor M (Finsupp.total R n)
+  let i1 := LinearMap.rTensor (ι →₀ R) (Finsupp.linearCombination R m)
+  let i2 := LinearMap.lTensor M (Finsupp.linearCombination R n)
   let i := mulMap M N ∘ₗ i2 ∘ₗ i1 ∘ₗ i0.toLinearMap
   have h1 : Function.Injective i1 := Module.Flat.rTensor_preserves_injective_linearMap _ hm
   have h2 : Function.Injective i2 := Module.Flat.lTensor_preserves_injective_linearMap _ hn
   have h : Function.Injective i := H.injective.comp h2 |>.comp h1 |>.comp i0.injective
-  have : i = Finsupp.total R fun i ↦ (m i.1).1 * (n i.2).1 := by
+  have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
     ext x
     simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
   rwa [this] at h
@@ -375,13 +375,13 @@ theorem linearIndependent_mul_of_flat_right (H : M.LinearDisjoint N) [Module.Fla
     (hn : LinearIndependent R n) : LinearIndependent R fun (i : κ × ι) ↦ (m i.1).1 * (n i.2).1 := by
   rw [LinearIndependent, LinearMap.ker_eq_bot] at hm hn ⊢
   let i0 := (finsuppTensorFinsupp' R κ ι).symm
-  let i1 := LinearMap.lTensor (κ →₀ R) (Finsupp.total R n)
-  let i2 := LinearMap.rTensor N (Finsupp.total R m)
+  let i1 := LinearMap.lTensor (κ →₀ R) (Finsupp.linearCombination R n)
+  let i2 := LinearMap.rTensor N (Finsupp.linearCombination R m)
   let i := mulMap M N ∘ₗ i2 ∘ₗ i1 ∘ₗ i0.toLinearMap
   have h1 : Function.Injective i1 := Module.Flat.lTensor_preserves_injective_linearMap _ hn
   have h2 : Function.Injective i2 := Module.Flat.rTensor_preserves_injective_linearMap _ hm
   have h : Function.Injective i := H.injective.comp h2 |>.comp h1 |>.comp i0.injective
-  have : i = Finsupp.total R fun i ↦ (m i.1).1 * (n i.2).1 := by
+  have : i = Finsupp.linearCombination R fun i ↦ (m i.1).1 * (n i.2).1 := by
     ext x
     simp [i, i0, i1, i2, finsuppTensorFinsupp'_symm_single_eq_single_one_tmul]
   rwa [this] at h
@@ -477,8 +477,8 @@ theorem not_linearIndependent_pair_of_commute_of_flat_left [Module.Flat R M]
   have hm : mulRightMap M n m = 0 := by simp [m, n, show _ * _ = _ * _ from hc]
   rw [← LinearMap.mem_ker, H.linearIndependent_right_of_flat hn, mem_bot] at hm
   simp only [Fin.isValue, sub_eq_zero, Finsupp.single_eq_single_iff, zero_ne_one, Subtype.mk.injEq,
-    SetLike.coe_eq_coe, false_and, AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero,
-    false_or, m] at hm
+    SetLike.coe_eq_coe, false_and, false_or, m] at hm
+  repeat rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hm
   exact h.ne_zero 0 hm.2
 
 /-- If `M` and `N` are linearly disjoint, if `N` is flat, then any two commutative
@@ -493,8 +493,8 @@ theorem not_linearIndependent_pair_of_commute_of_flat_right [Module.Flat R N]
   have hn : mulLeftMap N m n = 0 := by simp [m, n, show _ * _ = _ * _ from hc]
   rw [← LinearMap.mem_ker, H.linearIndependent_left_of_flat hm, mem_bot] at hn
   simp only [Fin.isValue, sub_eq_zero, Finsupp.single_eq_single_iff, zero_ne_one, Subtype.mk.injEq,
-    SetLike.coe_eq_coe, false_and, AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero,
-    false_or, n] at hn
+    SetLike.coe_eq_coe, false_and, false_or, n] at hn
+  repeat rw [AddSubmonoid.mk_eq_zero, ZeroMemClass.coe_eq_zero] at hn
   exact h.ne_zero 0 hn.2
 
 /-- If `M` and `N` are linearly disjoint, if one of `M` and `N` is flat, then any two commutative
@@ -512,7 +512,7 @@ if any two elements of `↥(M ⊓ N)` are commutative, then the rank of `↥(M 
 theorem rank_inf_le_one_of_commute_of_flat (hf : Module.Flat R M ∨ Module.Flat R N)
     (hc : ∀ (m n : ↥(M ⊓ N)), Commute m.1 n.1) : Module.rank R ↥(M ⊓ N) ≤ 1 := by
   nontriviality R
-  refine rank_le fun s h ↦ ?_
+  refine _root_.rank_le fun s h ↦ ?_
   by_contra hs
   rw [not_le, ← Fintype.card_coe, Fintype.one_lt_card_iff_nontrivial] at hs
   obtain ⟨a, b, hab⟩ := hs.exists_pair_ne
diff --git a/Mathlib/LinearAlgebra/LinearIndependent.lean b/Mathlib/LinearAlgebra/LinearIndependent.lean
index 06ba0e1bd59ec..9ce4ca2f473b7 100644
--- a/Mathlib/LinearAlgebra/LinearIndependent.lean
+++ b/Mathlib/LinearAlgebra/LinearIndependent.lean
@@ -11,7 +11,8 @@ import Mathlib.Tactic.FinCases
 import Mathlib.Tactic.LinearCombination
 import Mathlib.Lean.Expr.ExtraRecognizers
 import Mathlib.Data.Set.Subsingleton
-import Mathlib.Tactic.Abel
+import Mathlib.Tactic.Module
+import Mathlib.Tactic.NoncommRing
 
 /-!
 
@@ -21,11 +22,11 @@ This file defines linear independence in a module or vector space.
 
 It is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light.
 
-We define `LinearIndependent R v` as `ker (Finsupp.total R v) = ⊥`. Here `Finsupp.total` is the
-linear map sending a function `f : ι →₀ R` with finite support to the linear combination of vectors
-from `v` with these coefficients. Then we prove that several other statements are equivalent to this
-one, including injectivity of `Finsupp.total R v` and some versions with explicitly written
-linear combinations.
+We define `LinearIndependent R v` as `ker (Finsupp.linearCombination R v) = ⊥`. Here
+`Finsupp.linearCombination` is the linear map sending a function `f : ι →₀ R` with finite support to
+the linear combination of vectors from `v` with these coefficients. Then we prove that several other
+statements are equivalent to this one, including injectivity of `Finsupp.linearCombination R v` and
+some versions with explicitly written linear combinations.
 
 ## Main definitions
 All definitions are given for families of vectors, i.e. `v : ι → M` where `M` is the module or
@@ -97,7 +98,7 @@ variable (R) (v)
 
 /-- `LinearIndependent R v` states the family of vectors `v` is linearly independent over `R`. -/
 def LinearIndependent : Prop :=
-  LinearMap.ker (Finsupp.total R v) = ⊥
+  LinearMap.ker (Finsupp.linearCombination R v) = ⊥
 
 open Lean PrettyPrinter.Delaborator SubExpr in
 /-- Delaborator for `LinearIndependent` that suggests pretty printing with type hints
@@ -122,7 +123,7 @@ def delabLinearIndependent : Delab :=
 variable {R} {v}
 
 theorem linearIndependent_iff :
-    LinearIndependent R v ↔ ∀ l, Finsupp.total R v l = 0 → l = 0 := by
+    LinearIndependent R v ↔ ∀ l, Finsupp.linearCombination R v l = 0 → l = 0 := by
   simp [LinearIndependent, LinearMap.ker_eq_bot']
 
 theorem linearIndependent_iff' :
@@ -132,7 +133,7 @@ theorem linearIndependent_iff' :
     ⟨fun hf s g hg i his =>
       have h :=
         hf (∑ i ∈ s, Finsupp.single i (g i)) <| by
-          simpa only [map_sum, Finsupp.total_single] using hg
+          simpa only [map_sum, Finsupp.linearCombination_single] using hg
       calc
         g i = (Finsupp.lapply i : (ι →₀ R) →ₗ[R] R) (Finsupp.single i (g i)) := by
           { rw [Finsupp.lapply_apply, Finsupp.single_eq_same] }
@@ -220,7 +221,7 @@ lemma LinearIndependent.pair_iff {x y : M} :
 linearly independent family. -/
 theorem LinearIndependent.comp (h : LinearIndependent R v) (f : ι' → ι) (hf : Injective f) :
     LinearIndependent R (v ∘ f) := by
-  rw [linearIndependent_iff, Finsupp.total_comp]
+  rw [linearIndependent_iff, Finsupp.linearCombination_comp]
   intro l hl
   have h_map_domain : ∀ x, (Finsupp.mapDomain f l) (f x) = 0 := by
     rw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl]; simp
@@ -244,24 +245,24 @@ disjoint with the submodule spanned by the vectors of `v`, then `f ∘ v` is a l
 family of vectors. See also `LinearIndependent.map'` for a special case assuming `ker f = ⊥`. -/
 theorem LinearIndependent.map (hv : LinearIndependent R v) {f : M →ₗ[R] M'}
     (hf_inj : Disjoint (span R (range v)) (LinearMap.ker f)) : LinearIndependent R (f ∘ v) := by
-  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_total,
+  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_linearCombination,
     map_inf_eq_map_inf_comap, map_le_iff_le_comap, comap_bot, Finsupp.supported_univ, top_inf_eq]
       at hf_inj
   unfold LinearIndependent at hv ⊢
   rw [hv, le_bot_iff] at hf_inj
   haveI : Inhabited M := ⟨0⟩
-  rw [Finsupp.total_comp, Finsupp.lmapDomain_total _ _ f, LinearMap.ker_comp,
-    hf_inj]
+  rw [Finsupp.linearCombination_comp, Finsupp.lmapDomain_linearCombination _ _ f,
+      LinearMap.ker_comp, hf_inj]
   exact fun _ => rfl
 
-/-- If `v` is an injective family of vectors such that `f ∘ v` is linearly independent, then `v`
+/-- If `v` is an injective family of vectors such that `f ∘ v` is linearly independent, then `v`
     spans a submodule disjoint from the kernel of `f` -/
 theorem Submodule.range_ker_disjoint {f : M →ₗ[R] M'}
     (hv : LinearIndependent R (f ∘ v)) :
     Disjoint (span R (range v)) (LinearMap.ker f) := by
-  rw [LinearIndependent, Finsupp.total_comp, Finsupp.lmapDomain_total R _ f (fun _ ↦ rfl),
-    LinearMap.ker_comp] at hv
-  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_total,
+  rw [LinearIndependent, Finsupp.linearCombination_comp,
+      Finsupp.lmapDomain_linearCombination R _ f (fun _ ↦ rfl), LinearMap.ker_comp] at hv
+  rw [disjoint_iff_inf_le, ← Set.image_univ, Finsupp.span_image_eq_map_linearCombination,
     map_inf_eq_map_inf_comap, hv, inf_bot_eq, map_bot]
 
 /-- An injective linear map sends linearly independent families of vectors to linearly independent
@@ -406,13 +407,13 @@ section Subtype
 
 theorem linearIndependent_comp_subtype {s : Set ι} :
     LinearIndependent R (v ∘ (↑) : s → M) ↔
-      ∀ l ∈ Finsupp.supported R R s, (Finsupp.total R v) l = 0 → l = 0 := by
-  simp only [linearIndependent_iff, (· ∘ ·), Finsupp.mem_supported, Finsupp.total_apply,
+      ∀ l ∈ Finsupp.supported R R s, (Finsupp.linearCombination R v) l = 0 → l = 0 := by
+  simp only [linearIndependent_iff, (· ∘ ·), Finsupp.mem_supported, Finsupp.linearCombination_apply,
     Set.subset_def, Finset.mem_coe]
   constructor
   · intro h l hl₁ hl₂
-    have := h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl₁).trans hl₂)
-    exact (Finsupp.subtypeDomain_eq_zero_iff hl₁).1 this
+    exact (Finsupp.subtypeDomain_eq_zero_iff hl₁).1 <|
+      h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl₁).trans hl₂)
   · intro h l hl
     refine Finsupp.embDomain_eq_zero.1 (h (l.embDomain <| Function.Embedding.subtype s) ?_ ?_)
     · suffices ∀ i hi, ¬l ⟨i, hi⟩ = 0 → i ∈ s by simpa
@@ -423,10 +424,10 @@ theorem linearIndependent_comp_subtype {s : Set ι} :
 
 theorem linearDependent_comp_subtype' {s : Set ι} :
     ¬LinearIndependent R (v ∘ (↑) : s → M) ↔
-      ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ Finsupp.total R v f = 0 ∧ f ≠ 0 := by
+      ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ Finsupp.linearCombination R v f = 0 ∧ f ≠ 0 := by
   simp [linearIndependent_comp_subtype, and_left_comm]
 
-/-- A version of `linearDependent_comp_subtype'` with `Finsupp.total` unfolded. -/
+/-- A version of `linearDependent_comp_subtype'` with `Finsupp.linearCombination` unfolded. -/
 theorem linearDependent_comp_subtype {s : Set ι} :
     ¬LinearIndependent R (v ∘ (↑) : s → M) ↔
       ∃ f : ι →₀ R, f ∈ Finsupp.supported R R s ∧ ∑ i ∈ f.support, f i • v i = 0 ∧ f ≠ 0 :=
@@ -434,25 +435,28 @@ theorem linearDependent_comp_subtype {s : Set ι} :
 
 theorem linearIndependent_subtype {s : Set M} :
     LinearIndependent R (fun x => x : s → M) ↔
-      ∀ l ∈ Finsupp.supported R R s, (Finsupp.total R id) l = 0 → l = 0 := by
+      ∀ l ∈ Finsupp.supported R R s, (Finsupp.linearCombination R id) l = 0 → l = 0 := by
   apply linearIndependent_comp_subtype (v := id)
 
 theorem linearIndependent_comp_subtype_disjoint {s : Set ι} :
     LinearIndependent R (v ∘ (↑) : s → M) ↔
-      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.total R v) := by
+      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.linearCombination R v) := by
   rw [linearIndependent_comp_subtype, LinearMap.disjoint_ker]
 
 theorem linearIndependent_subtype_disjoint {s : Set M} :
     LinearIndependent R (fun x => x : s → M) ↔
-      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.total R id) := by
+      Disjoint (Finsupp.supported R R s) (LinearMap.ker <| Finsupp.linearCombination R id) := by
   apply linearIndependent_comp_subtype_disjoint (v := id)
 
-theorem linearIndependent_iff_totalOn {s : Set M} :
+theorem linearIndependent_iff_linearCombinationOn {s : Set M} :
     LinearIndependent R (fun x => x : s → M) ↔
-    (LinearMap.ker <| Finsupp.totalOn M M R id s) = ⊥ := by
-  rw [Finsupp.totalOn, LinearMap.ker, LinearMap.comap_codRestrict, Submodule.map_bot, comap_bot,
-    LinearMap.ker_comp, linearIndependent_subtype_disjoint, disjoint_iff_inf_le, ←
-    map_comap_subtype, map_le_iff_le_comap, comap_bot, ker_subtype, le_bot_iff]
+    (LinearMap.ker <| Finsupp.linearCombinationOn M M R id s) = ⊥ := by
+  rw [Finsupp.linearCombinationOn, LinearMap.ker, LinearMap.comap_codRestrict, Submodule.map_bot,
+      comap_bot, LinearMap.ker_comp, linearIndependent_subtype_disjoint, disjoint_iff_inf_le,
+      ← map_comap_subtype, map_le_iff_le_comap, comap_bot, ker_subtype, le_bot_iff]
+
+@[deprecated (since := "2024-08-29")] alias linearIndependent_iff_totalOn :=
+  linearIndependent_iff_linearCombinationOn
 
 theorem LinearIndependent.restrict_of_comp_subtype {s : Set ι}
     (hs : LinearIndependent R (v ∘ (↑) : s → M)) : LinearIndependent R (s.restrict v) :=
@@ -515,18 +519,26 @@ variable [Ring R] [AddCommGroup M] [AddCommGroup M'] [AddCommGroup M'']
 variable [Module R M] [Module R M'] [Module R M'']
 variable {a b : R} {x y : M}
 
-theorem linearIndependent_iff_injective_total :
-    LinearIndependent R v ↔ Function.Injective (Finsupp.total R v) :=
+theorem linearIndependent_iff_injective_linearCombination :
+    LinearIndependent R v ↔ Function.Injective (Finsupp.linearCombination R v) :=
   linearIndependent_iff.trans
-    (injective_iff_map_eq_zero (Finsupp.total R v).toAddMonoidHom).symm
+    (injective_iff_map_eq_zero (Finsupp.linearCombination R v).toAddMonoidHom).symm
+
+@[deprecated (since := "2024-08-29")] alias linearIndependent_iff_injective_total :=
+  linearIndependent_iff_injective_linearCombination
+
+alias ⟨LinearIndependent.injective_linearCombination, _⟩ :=
 
-alias ⟨LinearIndependent.injective_total, _⟩ := linearIndependent_iff_injective_total
+  linearIndependent_iff_injective_linearCombination
+
+@[deprecated (since := "2024-08-29")] alias LinearIndependent.injective_total :=
+  LinearIndependent.injective_linearCombination
 
 theorem LinearIndependent.injective [Nontrivial R] (hv : LinearIndependent R v) : Injective v := by
   intro i j hij
   let l : ι →₀ R := Finsupp.single i (1 : R) - Finsupp.single j 1
-  have h_total : Finsupp.total R v l = 0 := by
-    simp_rw [l, LinearMap.map_sub, Finsupp.total_apply]
+  have h_total : Finsupp.linearCombination R v l = 0 := by
+    simp_rw [l, LinearMap.map_sub, Finsupp.linearCombination_apply]
     simp [hij]
   have h_single_eq : Finsupp.single i (1 : R) = Finsupp.single j 1 := by
     rw [linearIndependent_iff] at hv
@@ -580,15 +592,13 @@ theorem LinearIndependent.units_smul {v : ι → M} (hv : LinearIndependent R v)
     exact (hgs i hi).symm ▸ zero_smul _ _
   · rw [← hsum, Finset.sum_congr rfl _]
     intros
-    erw [Pi.smul_apply, smul_assoc]
-    rfl
+    rw [Pi.smul_apply', smul_assoc, Units.smul_def]
 
 lemma LinearIndependent.eq_of_pair {x y : M} (h : LinearIndependent R ![x, y])
     {s t s' t' : R} (h' : s • x + t • y = s' • x + t' • y) : s = s' ∧ t = t' := by
   have : (s - s') • x + (t - t') • y = 0 := by
-    rw [← sub_eq_zero_of_eq h', ← sub_eq_zero]
-    simp only [sub_smul]
-    abel
+    rw [← sub_eq_zero_of_eq h']
+    match_scalars <;> noncomm_ring
   simpa [sub_eq_zero] using h.eq_zero_of_pair this
 
 lemma LinearIndependent.eq_zero_of_pair' {x y : M} (h : LinearIndependent R ![x, y])
@@ -606,8 +616,7 @@ lemma LinearIndependent.linear_combination_pair_of_det_ne_zero {R M : Type*} [Co
   apply LinearIndependent.pair_iff.2 (fun s t hst ↦ ?_)
   have H : (s * a + t * c) • x + (s * b + t * d) • y = 0 := by
     convert hst using 1
-    simp only [_root_.add_smul, smul_add, smul_smul]
-    abel
+    module
   have I1 : s * a + t * c = 0 := (h.eq_zero_of_pair H).1
   have I2 : s * b + t * d = 0 := (h.eq_zero_of_pair H).2
   have J1 : (a * d - b * c) * s = 0 := by linear_combination d * I1 - c * I2
@@ -657,8 +666,8 @@ theorem LinearIndependent.eq_of_smul_apply_eq_smul_apply {M : Type*} [AddCommGro
     {v : ι → M} (li : LinearIndependent R v) (c d : R) (i j : ι) (hc : c ≠ 0)
     (h : c • v i = d • v j) : i = j := by
   let l : ι →₀ R := Finsupp.single i c - Finsupp.single j d
-  have h_total : Finsupp.total R v l = 0 := by
-    simp_rw [l, LinearMap.map_sub, Finsupp.total_apply]
+  have h_total : Finsupp.linearCombination R v l = 0 := by
+    simp_rw [l, LinearMap.map_sub, Finsupp.linearCombination_apply]
     simp [h]
   have h_single_eq : Finsupp.single i c = Finsupp.single j d := by
     rw [linearIndependent_iff] at li
@@ -673,9 +682,9 @@ section Subtype
 
 theorem LinearIndependent.disjoint_span_image (hv : LinearIndependent R v) {s t : Set ι}
     (hs : Disjoint s t) : Disjoint (Submodule.span R <| v '' s) (Submodule.span R <| v '' t) := by
-  simp only [disjoint_def, Finsupp.mem_span_image_iff_total]
+  simp only [disjoint_def, Finsupp.mem_span_image_iff_linearCombination]
   rintro _ ⟨l₁, hl₁, rfl⟩ ⟨l₂, hl₂, H⟩
-  rw [hv.injective_total.eq_iff] at H; subst l₂
+  rw [hv.injective_linearCombination.eq_iff] at H; subst l₂
   have : l₁ = 0 := Submodule.disjoint_def.mp (Finsupp.disjoint_supported_supported hs) _ hl₁ hl₂
   simp [this]
 
@@ -689,16 +698,20 @@ theorem LinearIndependent.not_mem_span_image [Nontrivial R] (hv : LinearIndepend
   refine disjoint_def.1 (hv.disjoint_span_image ?_) (v x) h' w
   simpa using h
 
-theorem LinearIndependent.total_ne_of_not_mem_support [Nontrivial R] (hv : LinearIndependent R v)
-    {x : ι} (f : ι →₀ R) (h : x ∉ f.support) : Finsupp.total R v f ≠ v x := by
+theorem LinearIndependent.linearCombination_ne_of_not_mem_support [Nontrivial R]
+    (hv : LinearIndependent R v) {x : ι} (f : ι →₀ R) (h : x ∉ f.support) :
+    Finsupp.linearCombination R v f ≠ v x := by
   replace h : x ∉ (f.support : Set ι) := h
   have p := hv.not_mem_span_image h
   intro w
   rw [← w] at p
-  rw [Finsupp.span_image_eq_map_total] at p
+  rw [Finsupp.span_image_eq_map_linearCombination] at p
   simp only [not_exists, not_and, mem_map] at p -- Porting note: `mem_map` isn't currently triggered
   exact p f (f.mem_supported_support R) rfl
 
+@[deprecated (since := "2024-08-29")] alias LinearIndependent.total_ne_of_not_mem_support :=
+  LinearIndependent.linearCombination_ne_of_not_mem_support
+
 theorem linearIndependent_sum {v : ι ⊕ ι' → M} :
     LinearIndependent R v ↔
       LinearIndependent R (v ∘ Sum.inl) ∧
@@ -801,41 +814,54 @@ variable (hv : LinearIndependent R v)
 /-- Canonical isomorphism between linear combinations and the span of linearly independent vectors.
 -/
 @[simps (config := { rhsMd := default }) symm_apply]
-def LinearIndependent.totalEquiv (hv : LinearIndependent R v) :
+def LinearIndependent.linearCombinationEquiv (hv : LinearIndependent R v) :
     (ι →₀ R) ≃ₗ[R] span R (range v) := by
-  apply LinearEquiv.ofBijective (LinearMap.codRestrict (span R (range v)) (Finsupp.total R v) _)
+  apply LinearEquiv.ofBijective (LinearMap.codRestrict (span R (range v))
+                                 (Finsupp.linearCombination R v) _)
   constructor
   · rw [← LinearMap.ker_eq_bot, LinearMap.ker_codRestrict]
     · apply hv
     · intro l
-      rw [← Finsupp.range_total]
+      rw [← Finsupp.range_linearCombination]
       rw [LinearMap.mem_range]
       apply mem_range_self l
   · rw [← LinearMap.range_eq_top, LinearMap.range_eq_map, LinearMap.map_codRestrict, ←
       LinearMap.range_le_iff_comap, range_subtype, Submodule.map_top]
-    rw [Finsupp.range_total]
+    rw [Finsupp.range_linearCombination]
+
+@[deprecated (since := "2024-08-29")] noncomputable alias LinearIndependent.totalEquiv :=
+  LinearIndependent.linearCombinationEquiv
 
 -- Porting note: The original theorem generated by `simps` was
 --               different from the theorem on Lean 3, and not simp-normal form.
 @[simp]
-theorem LinearIndependent.totalEquiv_apply_coe (hv : LinearIndependent R v) (l : ι →₀ R) :
-    hv.totalEquiv l = Finsupp.total R v l := rfl
+theorem LinearIndependent.linearCombinationEquiv_apply_coe (hv : LinearIndependent R v)
+    (l : ι →₀ R) : hv.linearCombinationEquiv l = Finsupp.linearCombination R v l := rfl
 
+@[deprecated (since := "2024-08-29")] alias LinearIndependent.totalEquiv_apply_coe :=
+  LinearIndependent.linearCombinationEquiv_apply_coe
 /-- Linear combination representing a vector in the span of linearly independent vectors.
 
 Given a family of linearly independent vectors, we can represent any vector in their span as
 a linear combination of these vectors. These are provided by this linear map.
-It is simply one direction of `LinearIndependent.total_equiv`. -/
+It is simply one direction of `LinearIndependent.linearCombinationEquiv`. -/
 def LinearIndependent.repr (hv : LinearIndependent R v) : span R (range v) →ₗ[R] ι →₀ R :=
-  hv.totalEquiv.symm
+  hv.linearCombinationEquiv.symm
 
 @[simp]
-theorem LinearIndependent.total_repr (x) : Finsupp.total R v (hv.repr x) = x :=
-  Subtype.ext_iff.1 (LinearEquiv.apply_symm_apply hv.totalEquiv x)
+theorem LinearIndependent.linearCombination_repr (x) :
+    Finsupp.linearCombination R v (hv.repr x) = x :=
+  Subtype.ext_iff.1 (LinearEquiv.apply_symm_apply hv.linearCombinationEquiv x)
+
+@[deprecated (since := "2024-08-29")] alias LinearIndependent.total_repr :=
+  LinearIndependent.linearCombination_repr
 
-theorem LinearIndependent.total_comp_repr :
-    (Finsupp.total R v).comp hv.repr = Submodule.subtype _ :=
-  LinearMap.ext <| hv.total_repr
+theorem LinearIndependent.linearCombination_comp_repr :
+    (Finsupp.linearCombination R v).comp hv.repr = Submodule.subtype _ :=
+  LinearMap.ext <| hv.linearCombination_repr
+
+@[deprecated (since := "2024-08-29")] alias LinearIndependent.total_comp_repr :=
+  LinearIndependent.linearCombination_comp_repr
 
 theorem LinearIndependent.repr_ker : LinearMap.ker hv.repr = ⊥ := by
   rw [LinearIndependent.repr, LinearEquiv.ker]
@@ -844,22 +870,22 @@ theorem LinearIndependent.repr_range : LinearMap.range hv.repr = ⊤ := by
   rw [LinearIndependent.repr, LinearEquiv.range]
 
 theorem LinearIndependent.repr_eq {l : ι →₀ R} {x : span R (range v)}
-    (eq : Finsupp.total R v l = ↑x) : hv.repr x = l := by
+    (eq : Finsupp.linearCombination R v l = ↑x) : hv.repr x = l := by
   have :
-    ↑((LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l) =
-      Finsupp.total R v l :=
+    ↑((LinearIndependent.linearCombinationEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l) =
+      Finsupp.linearCombination R v l :=
     rfl
-  have : (LinearIndependent.totalEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l = x := by
+  have : (LinearIndependent.linearCombinationEquiv hv : (ι →₀ R) →ₗ[R] span R (range v)) l = x := by
     rw [eq] at this
     exact Subtype.ext_iff.2 this
-  rw [← LinearEquiv.symm_apply_apply hv.totalEquiv l]
+  rw [← LinearEquiv.symm_apply_apply hv.linearCombinationEquiv l]
   rw [← this]
   rfl
 
 theorem LinearIndependent.repr_eq_single (i) (x : span R (range v)) (hx : ↑x = v i) :
     hv.repr x = Finsupp.single i 1 := by
   apply hv.repr_eq
-  simp [Finsupp.total_single, hx]
+  simp [Finsupp.linearCombination_single, hx]
 
 theorem LinearIndependent.span_repr_eq [Nontrivial R] (x) :
     Span.repr R (Set.range v) x =
@@ -867,17 +893,18 @@ theorem LinearIndependent.span_repr_eq [Nontrivial R] (x) :
   have p :
     (Span.repr R (Set.range v) x).equivMapDomain (Equiv.ofInjective _ hv.injective).symm =
       hv.repr x := by
-    apply (LinearIndependent.totalEquiv hv).injective
+    apply (LinearIndependent.linearCombinationEquiv hv).injective
     ext
-    simp only [LinearIndependent.totalEquiv_apply_coe, Equiv.self_comp_ofInjective_symm,
-      LinearIndependent.total_repr, Finsupp.total_equivMapDomain, Span.finsupp_total_repr]
+    simp only [LinearIndependent.linearCombinationEquiv_apply_coe, Equiv.self_comp_ofInjective_symm,
+      LinearIndependent.linearCombination_repr, Finsupp.linearCombination_equivMapDomain,
+      Span.finsupp_linearCombination_repr]
   ext ⟨_, ⟨i, rfl⟩⟩
   simp [← p]
 
 theorem linearIndependent_iff_not_smul_mem_span :
     LinearIndependent R v ↔ ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \ {i})) → a = 0 :=
   ⟨fun hv i a ha => by
-    rw [Finsupp.span_image_eq_map_total, mem_map] at ha
+    rw [Finsupp.span_image_eq_map_linearCombination, mem_map] at ha
     rcases ha with ⟨l, hl, e⟩
     rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl
     by_contra hn
@@ -886,7 +913,7 @@ theorem linearIndependent_iff_not_smul_mem_span :
       ext i; simp only [Finsupp.zero_apply]
       by_contra hn
       refine hn (H i _ ?_)
-      refine (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, ?_, ?_⟩
+      refine (Finsupp.mem_span_image_iff_linearCombination R).2 ⟨Finsupp.single i (l i) - l, ?_, ?_⟩
       · rw [Finsupp.mem_supported']
         intro j hj
         have hij : j = i :=
@@ -979,13 +1006,14 @@ theorem surjective_of_linearIndependent_of_span [Nontrivial R] (hv : LinearIndep
   intro i
   let repr : (span R (range (v ∘ f)) : Type _) → ι' →₀ R := (hv.comp f f.injective).repr
   let l := (repr ⟨v i, hss (mem_range_self i)⟩).mapDomain f
-  have h_total_l : Finsupp.total R v l = v i := by
+  have h_total_l : Finsupp.linearCombination R v l = v i := by
     dsimp only [l]
-    rw [Finsupp.total_mapDomain]
-    rw [(hv.comp f f.injective).total_repr]
+    rw [Finsupp.linearCombination_mapDomain]
+    rw [(hv.comp f f.injective).linearCombination_repr]
     -- Porting note: `rfl` isn't necessary.
-  have h_total_eq : (Finsupp.total R v) l = (Finsupp.total R v) (Finsupp.single i 1) := by
-    rw [h_total_l, Finsupp.total_single, one_smul]
+  have h_total_eq : Finsupp.linearCombination R v l = Finsupp.linearCombination R v
+       (Finsupp.single i 1) := by
+    rw [h_total_l, Finsupp.linearCombination_single, one_smul]
   have l_eq : l = _ := LinearMap.ker_eq_bot.1 hv h_total_eq
   dsimp only [l] at l_eq
   rw [← Finsupp.embDomain_eq_mapDomain] at l_eq
@@ -1000,7 +1028,7 @@ theorem eq_of_linearIndependent_of_span_subtype [Nontrivial R] {s t : Set M}
     ⟨fun x => ⟨x.1, h x.2⟩, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)⟩
   have h_surj : Surjective f := by
     apply surjective_of_linearIndependent_of_span hs f _
-    convert hst <;> simp [f, comp]
+    convert hst <;> simp [f, comp_def]
   show s = t
   apply Subset.antisymm _ h
   intro x hx
@@ -1082,11 +1110,10 @@ theorem linearIndependent_monoidHom (G : Type*) [Monoid G] (L : Type*) [CommRing
                   rw [Finset.sum_insert has, Finset.sum_insert has]
                 _ =
                     (∑ i ∈ insert a s, g i * i (x * y)) -
-                      ∑ i ∈ insert a s, a x * (g i * i y) :=
-                  congr
-                    (congr_arg Sub.sub
-                      (Finset.sum_congr rfl fun i _ => by rw [i.map_mul, mul_assoc]))
-                    (Finset.sum_congr rfl fun _ _ => by rw [mul_assoc, mul_left_comm])
+                      ∑ i ∈ insert a s, a x * (g i * i y) := by
+                  congrm ∑ i ∈ insert a s, ?_ - ∑ i ∈ insert a s, ?_
+                  · rw [map_mul, mul_assoc]
+                  · rw [mul_assoc, mul_left_comm]
                 _ =
                     (∑ i ∈ insert a s, (g i • (i : G → L))) (x * y) -
                       a x * (∑ i ∈ insert a s, (g i • (i : G → L))) y := by
@@ -1164,7 +1191,7 @@ variable {v : ι → M} {s t : Set M} {x y z : M}
 
 theorem linearIndependent_unique_iff (v : ι → M) [Unique ι] :
     LinearIndependent R v ↔ v default ≠ 0 := by
-  simp only [linearIndependent_iff, Finsupp.total_unique, smul_eq_zero]
+  simp only [linearIndependent_iff, Finsupp.linearCombination_unique, smul_eq_zero]
   refine ⟨fun h hv => ?_, fun hv l hl => Finsupp.unique_ext <| hl.resolve_right hv⟩
   have := h (Finsupp.single default 1) (Or.inr hv)
   exact one_ne_zero (Finsupp.single_eq_zero.1 this)
@@ -1202,7 +1229,7 @@ theorem mem_span_insert_exchange :
   have a0 : a ≠ 0 := by
     rintro rfl
     simp_all
-  simp [a0, smul_add, smul_smul]
+  match_scalars <;> simp [a0]
 
 theorem linearIndependent_iff_not_mem_span :
     LinearIndependent K v ↔ ∀ i, v i ∉ span K (v '' (univ \ {i})) := by
@@ -1225,7 +1252,7 @@ theorem linearIndependent_option' :
     LinearIndependent K (fun o => Option.casesOn' o x v : Option ι → V) ↔
       LinearIndependent K v ∧ x ∉ Submodule.span K (range v) := by
   -- Porting note: Explicit universe level is required in `Equiv.optionEquivSumPUnit`.
-  rw [← linearIndependent_equiv (Equiv.optionEquivSumPUnit.{_, u'} ι).symm, linearIndependent_sum,
+  rw [← linearIndependent_equiv (Equiv.optionEquivSumPUnit.{u', _} ι).symm, linearIndependent_sum,
     @range_unique _ PUnit, @linearIndependent_unique_iff PUnit, disjoint_span_singleton]
   dsimp [(· ∘ ·)]
   refine ⟨fun h => ⟨h.1, fun hx => h.2.1 <| h.2.2 hx⟩, fun h => ⟨h.1, ?_, fun hx => (h.2 hx).elim⟩⟩
@@ -1250,7 +1277,7 @@ theorem linearIndependent_insert' {ι} {s : Set ι} {a : ι} {f : ι → V} (has
     linearIndependent_option]
   -- Porting note: `simp [(· ∘ ·), range_comp f]` → `simp [(· ∘ ·)]; erw [range_comp f ..]; simp`
   -- https://github.com/leanprover-community/mathlib4/issues/5164
-  simp only [(· ∘ ·)]
+  simp only [Function.comp_def]
   erw [range_comp f ((↑) : s → ι)]
   simp
 
@@ -1276,8 +1303,8 @@ theorem LinearIndependent.pair_iff' {x y : V} (hx : x ≠ 0) :
     by_cases ht : t = 0
     · exact ⟨by simpa [ht, hx] using hst, ht⟩
     apply_fun (t⁻¹ • ·) at hst
-    simp only [smul_add, smul_smul, inv_mul_cancel₀ ht, one_smul, smul_zero] at hst
-    cases H (-(t⁻¹ * s)) (by rwa [neg_smul, neg_eq_iff_eq_neg, eq_neg_iff_add_eq_zero])
+    simp only [smul_add, smul_smul, inv_mul_cancel₀ ht] at hst
+    cases H (-(t⁻¹ * s)) <| by linear_combination (norm := match_scalars <;> noncomm_ring) -hst
 
 theorem linearIndependent_fin_cons {n} {v : Fin n → V} :
     LinearIndependent K (Fin.cons x v : Fin (n + 1) → V) ↔
@@ -1374,6 +1401,10 @@ theorem LinearIndependent.subset_span_extend (hs : LinearIndependent K (fun x =>
   let ⟨_hbt, _hsb, htb, _hli⟩ := Classical.choose_spec (exists_linearIndependent_extension hs hst)
   htb
 
+theorem LinearIndependent.span_extend_eq_span (hs : LinearIndependent K (fun x => x : s → V))
+    (hst : s ⊆ t) : span K (hs.extend hst) = span K t :=
+  le_antisymm (span_mono (hs.extend_subset hst)) (span_le.2 (hs.subset_span_extend hst))
+
 theorem LinearIndependent.linearIndependent_extend (hs : LinearIndependent K (fun x => x : s → V))
     (hst : s ⊆ t) : LinearIndependent K ((↑) : hs.extend hst → V) :=
   let ⟨_hbt, _hsb, _htb, hli⟩ := Classical.choose_spec (exists_linearIndependent_extension hs hst)
diff --git a/Mathlib/LinearAlgebra/LinearPMap.lean b/Mathlib/LinearAlgebra/LinearPMap.lean
index a22821e0e63c5..bae8ae629f0f7 100644
--- a/Mathlib/LinearAlgebra/LinearPMap.lean
+++ b/Mathlib/LinearAlgebra/LinearPMap.lean
@@ -847,7 +847,7 @@ theorem le_graph_of_le {f g : E →ₗ.[R] F} (h : f ≤ g) : f.graph ≤ g.grap
   rw [mem_graph_iff] at hx ⊢
   cases' hx with y hx
   use ⟨y, h.1 y.2⟩
-  simp only [hx, Submodule.coe_mk, eq_self_iff_true, true_and_iff]
+  simp only [hx, Submodule.coe_mk, eq_self_iff_true, true_and]
   convert hx.2 using 1
   refine (h.2 ?_).symm
   simp only [hx.1, Submodule.coe_mk]
diff --git a/Mathlib/LinearAlgebra/Matrix/Adjugate.lean b/Mathlib/LinearAlgebra/Matrix/Adjugate.lean
index 86afb594ac8cb..84ce3839bb755 100644
--- a/Mathlib/LinearAlgebra/Matrix/Adjugate.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Adjugate.lean
@@ -158,7 +158,7 @@ theorem cramer_submatrix_equiv (A : Matrix m m α) (e : n ≃ m) (b : n → α)
     cramer (A.submatrix e e) b = cramer A (b ∘ e.symm) ∘ e := by
   ext i
   simp_rw [Function.comp_apply, cramer_apply, updateColumn_submatrix_equiv,
-    det_submatrix_equiv_self e, Function.comp]
+    det_submatrix_equiv_self e, Function.comp_def]
 
 theorem cramer_reindex (e : m ≃ n) (A : Matrix m m α) (b : n → α) :
     cramer (reindex e e A) b = cramer A (b ∘ e) ∘ e.symm :=
@@ -495,8 +495,8 @@ theorem adjugate_adjugate (A : Matrix n n α) (h : Fintype.card n ≠ 1) :
   let A' := mvPolynomialX n n ℤ
   suffices adjugate (adjugate A') = det A' ^ (Fintype.card n - 2) • A' by
     rw [← mvPolynomialX_mapMatrix_aeval ℤ A, ← AlgHom.map_adjugate, ← AlgHom.map_adjugate, this,
-      ← AlgHom.map_det, ← map_pow (MvPolynomial.aeval _), AlgHom.mapMatrix_apply,
-      AlgHom.mapMatrix_apply, Matrix.map_smul' _ _ _ (_root_.map_mul _)]
+      ← AlgHom.map_det, ← map_pow (MvPolynomial.aeval fun p : n × n ↦ A p.1 p.2),
+      AlgHom.mapMatrix_apply, AlgHom.mapMatrix_apply, Matrix.map_smul' _ _ _ (_root_.map_mul _)]
   have h_card' : Fintype.card n - 2 + 1 = Fintype.card n - 1 := by simp [h_card]
   have is_reg : IsSMulRegular (MvPolynomial (n × n) ℤ) (det A') := fun x y =>
     mul_left_cancel₀ (det_mvPolynomialX_ne_zero n ℤ)
diff --git a/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean b/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
index 1c8964c5c90ae..f6e6c15a38732 100644
--- a/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
+++ b/Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
@@ -3,10 +3,6 @@ Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen, Kexing Ying
 -/
-import Mathlib.LinearAlgebra.Matrix.Basis
-import Mathlib.LinearAlgebra.Matrix.Nondegenerate
-import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
-import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
 import Mathlib.LinearAlgebra.BilinearForm.Properties
 import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
 
diff --git a/Mathlib/LinearAlgebra/Matrix/Block.lean b/Mathlib/LinearAlgebra/Matrix/Block.lean
index c998bad25a879..8432d56b10b42 100644
--- a/Mathlib/LinearAlgebra/Matrix/Block.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Block.lean
@@ -62,7 +62,7 @@ theorem blockTriangular_reindex_iff {b : n → α} {e : m ≃ n} :
   · convert h.submatrix
     simp only [reindex_apply, submatrix_submatrix, submatrix_id_id, Equiv.symm_comp_self]
   · convert h.submatrix
-    simp only [comp.assoc b e e.symm, Equiv.self_comp_symm, comp_id]
+    simp only [comp_assoc b e e.symm, Equiv.self_comp_symm, comp_id]
 
 protected theorem BlockTriangular.transpose :
     M.BlockTriangular b → Mᵀ.BlockTriangular (toDual ∘ b) :=
diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean
index cffadee4baad5..83b4e6ed3bec2 100644
--- a/Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.LinearAlgebra.Matrix.Adjugate
 import Mathlib.RingTheory.PolynomialAlgebra
diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
index ae025df9cfe66..4a0d9f53997f4 100644
--- a/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean
@@ -150,7 +150,7 @@ theorem matPolyEquiv_symm_map_eval (M : (Matrix n n R)[X]) (r : R) :
     DFunLike.congr_fun this M
   ext : 1
   · ext M : 1
-    simp [Function.comp]
+    simp [Function.comp_def]
   · simp [smul_eq_diagonal_mul]
 
 theorem matPolyEquiv_eval_eq_map (M : Matrix n n R[X]) (r : R) :
@@ -204,7 +204,7 @@ lemma derivative_det_one_add_X_smul_aux {n} (M : Matrix (Fin n) (Fin n) R) :
         rw [det_eq_zero_of_column_eq_zero 0, eval_zero, mul_zero]
         intro j
         rw [submatrix_apply, Fin.succAbove_of_castSucc_lt, one_apply_ne]
-        · exact (bne_iff_ne (Fin.succ j) (Fin.castSucc 0)).mp rfl
+        · exact (bne_iff_ne (a := Fin.succ j) (b := Fin.castSucc 0)).mp rfl
         · rw [Fin.castSucc_zero]; exact lt_of_le_of_ne (Fin.zero_le _) hi.symm
     · exact fun H ↦ (H <| Finset.mem_univ _).elim
 
@@ -214,7 +214,7 @@ lemma derivative_det_one_add_X_smul (M : Matrix n n R) :
   let e := Matrix.reindexLinearEquiv R R (Fintype.equivFin n) (Fintype.equivFin n)
   rw [← Matrix.det_reindexLinearEquiv_self R[X] (Fintype.equivFin n)]
   convert derivative_det_one_add_X_smul_aux (e M)
-  · ext; simp [e]
+  · ext; simp [map_add, e]
   · delta trace
     rw [← (Fintype.equivFin n).symm.sum_comp]
     simp_rw [e, reindexLinearEquiv_apply, reindex_apply, diag_apply, submatrix_apply]
@@ -326,7 +326,8 @@ lemma reverse_charpoly (M : Matrix n n R) :
       ← mul_one (Fintype.card n : ℤ), ← T_pow, map_pow, invert_T, mul_comm]
   rw [← det_smul, smul_sub, scalar_apply, ← diagonal_smul, Pi.smul_def, smul_eq_mul, ht,
     diagonal_one, invert.map_det]
-  simp [t, map_smul', smul_eq_diagonal_mul]
+  simp [map_sub, _root_.map_one, _root_.map_mul, t, map_smul', smul_eq_diagonal_mul]
+
 
 @[simp] lemma eval_charpolyRev :
     eval 0 M.charpolyRev = 1 := by
diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean
index ccd5c26310966..4eb330442505f 100644
--- a/Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean
@@ -58,7 +58,7 @@ namespace Matrix
 theorem det_eq_prod_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
     A.det = (Matrix.charpoly A).roots.prod := by
   rw [det_eq_sign_charpoly_coeff, ← charpoly_natDegree_eq_dim A,
-    Polynomial.prod_roots_eq_coeff_zero_of_monic_of_split A.charpoly_monic hAps, ← mul_assoc,
+    Polynomial.prod_roots_eq_coeff_zero_of_monic_of_splits A.charpoly_monic hAps, ← mul_assoc,
     ← pow_two, pow_right_comm, neg_one_sq, one_pow, one_mul]
 
 theorem trace_eq_sum_roots_charpoly_of_splits (hAps : A.charpoly.Splits (RingHom.id R)) :
diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
index 45a4644bc6fc5..73a35fa98fa85 100644
--- a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
@@ -28,41 +28,41 @@ variable (b : ι → M)
 open Polynomial Matrix
 
 /-- The composition of a matrix (as an endomorphism of `ι → R`) with the projection
-`(ι → R) →ₗ[R] M`.  -/
+`(ι → R) →ₗ[R] M`. -/
 def PiToModule.fromMatrix [DecidableEq ι] : Matrix ι ι R →ₗ[R] (ι → R) →ₗ[R] M :=
-  (LinearMap.llcomp R _ _ _ (Fintype.total R R b)).comp algEquivMatrix'.symm.toLinearMap
+  (LinearMap.llcomp R _ _ _ (Fintype.linearCombination R R b)).comp algEquivMatrix'.symm.toLinearMap
 
 theorem PiToModule.fromMatrix_apply [DecidableEq ι] (A : Matrix ι ι R) (w : ι → R) :
-    PiToModule.fromMatrix R b A w = Fintype.total R R b (A *ᵥ w) :=
+    PiToModule.fromMatrix R b A w = Fintype.linearCombination R R b (A *ᵥ w) :=
   rfl
 
 theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
     PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by
-  rw [PiToModule.fromMatrix_apply, Fintype.total_apply, Matrix.mulVec_single]
+  rw [PiToModule.fromMatrix_apply, Fintype.linearCombination_apply, Matrix.mulVec_single]
   simp_rw [mul_one]
 
 /-- The endomorphisms of `M` acts on `(ι → R) →ₗ[R] M`, and takes the projection
 to a `(ι → R) →ₗ[R] M`. -/
 def PiToModule.fromEnd : Module.End R M →ₗ[R] (ι → R) →ₗ[R] M :=
-  LinearMap.lcomp _ _ (Fintype.total R R b)
+  LinearMap.lcomp _ _ (Fintype.linearCombination R R b)
 
 theorem PiToModule.fromEnd_apply (f : Module.End R M) (w : ι → R) :
-    PiToModule.fromEnd R b f w = f (Fintype.total R R b w) :=
+    PiToModule.fromEnd R b f w = f (Fintype.linearCombination R R b w) :=
   rfl
 
 theorem PiToModule.fromEnd_apply_single_one [DecidableEq ι] (f : Module.End R M) (i : ι) :
     PiToModule.fromEnd R b f (Pi.single i 1) = f (b i) := by
   rw [PiToModule.fromEnd_apply]
   congr
-  convert Fintype.total_apply_single (S := R) R b i (1 : R)
+  convert Fintype.linearCombination_apply_single (S := R) R b i (1 : R)
   rw [one_smul]
 
 theorem PiToModule.fromEnd_injective (hb : Submodule.span R (Set.range b) = ⊤) :
     Function.Injective (PiToModule.fromEnd R b) := by
   intro x y e
   ext m
-  obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.total R R b) := by
-    rw [(Fintype.range_total R b).trans hb]
+  obtain ⟨m, rfl⟩ : m ∈ LinearMap.range (Fintype.linearCombination R R b) := by
+    rw [(Fintype.range_linearCombination R b).trans hb]
     exact Submodule.mem_top
   exact (LinearMap.congr_fun e m : _)
 
@@ -71,18 +71,19 @@ section
 variable {R} [DecidableEq ι]
 
 /-- We say that a matrix represents an endomorphism of `M` if the matrix acting on `ι → R` is
-equal to `f` via the projection `(ι → R) →ₗ[R] M` given by a fixed (spanning) set.  -/
+equal to `f` via the projection `(ι → R) →ₗ[R] M` given by a fixed (spanning) set. -/
 def Matrix.Represents (A : Matrix ι ι R) (f : Module.End R M) : Prop :=
   PiToModule.fromMatrix R b A = PiToModule.fromEnd R b f
 
 variable {b}
 
 theorem Matrix.Represents.congr_fun {A : Matrix ι ι R} {f : Module.End R M} (h : A.Represents b f)
-    (x) : Fintype.total R R b (A *ᵥ x) = f (Fintype.total R R b x) :=
+    (x) : Fintype.linearCombination R R b (A *ᵥ x) = f (Fintype.linearCombination R R b x) :=
   LinearMap.congr_fun h x
 
 theorem Matrix.represents_iff {A : Matrix ι ι R} {f : Module.End R M} :
-    A.Represents b f ↔ ∀ x, Fintype.total R R b (A *ᵥ x) = f (Fintype.total R R b x) :=
+    A.Represents b f ↔
+      ∀ x, Fintype.linearCombination R R b (A *ᵥ x) = f (Fintype.linearCombination R R b x) :=
   ⟨fun e x => e.congr_fun x, fun H => LinearMap.ext fun x => H x⟩
 
 theorem Matrix.represents_iff' {A : Matrix ι ι R} {f : Module.End R M} :
diff --git a/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean b/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
index 85a32f9416529..c33327a8858ae 100644
--- a/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean
@@ -136,7 +136,7 @@ theorem det_mul (M N : Matrix n n R) : det (M * N) = det M * det N :=
           ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (p i) * N (p i) i :=
       (Eq.symm <|
         sum_subset (filter_subset _ _) fun f _ hbij =>
-          det_mul_aux <| by simpa only [true_and_iff, mem_filter, mem_univ] using hbij)
+          det_mul_aux <| by simpa only [true_and, mem_filter, mem_univ] using hbij)
     _ = ∑ τ : Perm n, ∑ σ : Perm n, ε σ * ∏ i, M (σ i) (τ i) * N (τ i) i :=
       sum_bij (fun p h ↦ Equiv.ofBijective p (mem_filter.1 h).2) (fun _ _ ↦ mem_univ _)
         (fun _ _ _ _ h ↦ by injection h)
@@ -183,14 +183,14 @@ theorem det_mul_left_comm (M N P : Matrix m m R) : det (M * (N * P)) = det (N *
 theorem det_mul_right_comm (M N P : Matrix m m R) : det (M * N * P) = det (M * P * N) := by
   rw [Matrix.mul_assoc, Matrix.mul_assoc, det_mul, det_mul_comm N P, ← det_mul]
 
--- TODO(mathlib4#6607): fix elaboration so that the ascription isn't needed
+-- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
 theorem det_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
-    det ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = det N := by
+    det (M.val * N * M⁻¹.val) = det N := by
   rw [det_mul_right_comm, Units.mul_inv, one_mul]
 
--- TODO(mathlib4#6607): fix elaboration so that the ascription isn't needed
+-- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
 theorem det_units_conj' (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
-    det ((↑M⁻¹ : Matrix _ _ _) * N * (↑M : Matrix _ _ _)) = det N :=
+    det (M⁻¹.val * N * ↑M.val) = det N :=
   det_units_conj M⁻¹ N
 
 /-- Transposing a matrix preserves the determinant. -/
@@ -747,7 +747,7 @@ theorem det_fin_one_of (a : R) : det !![a] = a :=
 theorem det_fin_two (A : Matrix (Fin 2) (Fin 2) R) : det A = A 0 0 * A 1 1 - A 0 1 * A 1 0 := by
   simp only [det_succ_row_zero, det_unique, Fin.default_eq_zero, submatrix_apply,
     Fin.succ_zero_eq_one, Fin.sum_univ_succ, Fin.val_zero, Fin.zero_succAbove, univ_unique,
-    Fin.val_succ, Fin.coe_fin_one, Fin.succ_succAbove_zero, sum_singleton]
+    Fin.val_succ, Fin.val_eq_zero, Fin.succ_succAbove_zero, sum_singleton]
   ring
 
 @[simp]
@@ -763,7 +763,7 @@ theorem det_fin_three (A : Matrix (Fin 3) (Fin 3) R) :
   simp only [det_succ_row_zero, ← Nat.not_even_iff_odd, submatrix_apply, Fin.succ_zero_eq_one,
     submatrix_submatrix, det_unique, Fin.default_eq_zero, comp_apply, Fin.succ_one_eq_two,
     Fin.sum_univ_succ, Fin.val_zero, Fin.zero_succAbove, univ_unique, Fin.val_succ,
-    Fin.coe_fin_one, Fin.succ_succAbove_zero, sum_singleton, Fin.succ_succAbove_one, even_add_self]
+    Fin.val_eq_zero, Fin.succ_succAbove_zero, sum_singleton, Fin.succ_succAbove_one, even_add_self]
   ring
 
 end Matrix
diff --git a/Mathlib/LinearAlgebra/Matrix/DotProduct.lean b/Mathlib/LinearAlgebra/Matrix/DotProduct.lean
index a549bac8773ce..d943a52a81685 100644
--- a/Mathlib/LinearAlgebra/Matrix/DotProduct.lean
+++ b/Mathlib/LinearAlgebra/Matrix/DotProduct.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
 -/
 import Mathlib.Algebra.Ring.Regular
-import Mathlib.Algebra.Star.Order
+import Mathlib.Algebra.Order.Star.Basic
 import Mathlib.Data.Matrix.Basic
 import Mathlib.LinearAlgebra.StdBasis
 import Mathlib.Algebra.Order.BigOperators.Group.Finset
diff --git a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean
index 0f2bd256cdc60..d7867798bc310 100644
--- a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean
+++ b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Card.lean
@@ -27,7 +27,7 @@ variable {K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V]
 variable [Fintype K] [Finite V]
 
 local notation "q" => Fintype.card K
-local notation "n" => FiniteDimensional.finrank K V
+local notation "n" => Module.finrank K V
 
 attribute [local instance] Fintype.ofFinite in
 open Fintype in
@@ -38,8 +38,8 @@ theorem card_linearIndependent {k : ℕ} (hk : k ≤ n) :
       ∏ i : Fin k, (q ^ n - q ^ i.val) := by
   rw [Nat.card_eq_fintype_card]
   induction k with
-  | zero => simp only [LinearIndependent, Finsupp.total_fin_zero, ker_zero, card_ofSubsingleton,
-      Finset.univ_eq_empty, Finset.prod_empty]
+  | zero => simp only [LinearIndependent, Finsupp.linearCombination_fin_zero, ker_zero,
+      card_ofSubsingleton, Finset.univ_eq_empty, Finset.prod_empty]
   | succ k ih =>
       have (s : { s : Fin k → V // LinearIndependent K s }) :
           card ((Submodule.span K (Set.range (s : Fin k → V)))ᶜ : Set (V)) =
@@ -86,8 +86,8 @@ theorem card_GL_field :
   rcases Nat.eq_zero_or_pos n with rfl | hn
   · simp [Nat.card_eq_fintype_card]
   · rw [Nat.card_congr (equiv_GL_linearindependent n hn), card_linearIndependent,
-    FiniteDimensional.finrank_fintype_fun_eq_card, Fintype.card_fin]
-    simp only [FiniteDimensional.finrank_fintype_fun_eq_card, Fintype.card_fin, le_refl]
+    Module.finrank_fintype_fun_eq_card, Fintype.card_fin]
+    simp only [Module.finrank_fintype_fun_eq_card, Fintype.card_fin, le_refl]
 
 end field
 
diff --git a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
index 301e6932dcf83..5ae49bf1b77c9 100644
--- a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
+++ b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
@@ -37,7 +37,7 @@ open LinearMap
 attribute [-instance] SpecialLinearGroup.instCoeFun
 
 /-- `GL n R` is the group of `n` by `n` `R`-matrices with unit determinant.
-Defined as a subtype of matrices-/
+Defined as a subtype of matrices -/
 abbrev GeneralLinearGroup (n : Type u) (R : Type v) [DecidableEq n] [Fintype n] [CommRing R] :
     Type _ :=
   (Matrix n n R)ˣ
@@ -69,7 +69,7 @@ def det : GL n R →* Rˣ where
   map_one' := Units.ext det_one
   map_mul' A B := Units.ext <| det_mul _ _
 
-/-- The `GL n R` and `Matrix.GeneralLinearGroup R n` groups are multiplicatively equivalent-/
+/-- The `GL n R` and `Matrix.GeneralLinearGroup R n` groups are multiplicatively equivalent -/
 def toLin : GL n R ≃* LinearMap.GeneralLinearGroup R (n → R) :=
   Units.mapEquiv toLinAlgEquiv'.toMulEquiv
 
@@ -137,6 +137,10 @@ def map (f : R →+* S) : GL n R →* GL n S := Units.map <| (RingHom.mapMatrix
 theorem map_id : map (RingHom.id R) = MonoidHom.id (GL n R) :=
   rfl
 
+@[simp]
+protected lemma map_apply (f : R →+* S) (i j : n) (g : GL n R) : map f g i j = f (g i j) := by
+  rfl
+
 @[simp]
 theorem map_comp (f : T →+* R) (g : R →+* S) :
     map (g.comp f) = (map g).comp (map (n := n) f) :=
@@ -147,6 +151,44 @@ theorem map_comp_apply (f : T →+* R) (g : R →+* S) (x : GL n T) :
     (map g).comp (map f) x = map g (map f x) :=
   rfl
 
+variable (f : R →+* S)
+
+@[simp]
+protected lemma map_one : map f (1 : GL n R) = 1 := by
+  ext
+  simp only [_root_.map_one, Units.val_one]
+
+protected lemma map_mul (g h : GL n R) : map f (g * h) = map f g * map f h := by
+  ext
+  simp only [_root_.map_mul, Units.val_mul]
+
+protected lemma map_inv (g : GL n R) : map f g⁻¹ = (map f g)⁻¹ := by
+  ext
+  simp only [_root_.map_inv, coe_units_inv]
+
+protected lemma map_det (g : GL n R) : Matrix.GeneralLinearGroup.det (map f g) =
+    Units.map f (Matrix.GeneralLinearGroup.det g) := by
+  ext
+  simp only [map, RingHom.mapMatrix_apply, Units.inv_eq_val_inv, Matrix.coe_units_inv,
+    Matrix.GeneralLinearGroup.val_det_apply, Units.coe_map, MonoidHom.coe_coe]
+  exact Eq.symm (RingHom.map_det f g.1)
+
+lemma map_mul_map_inv (g : GL n R) : map f g * map f g⁻¹ = 1 := by
+  simp only [map_inv, mul_inv_cancel]
+
+lemma map_inv_mul_map (g : GL n R) : map f g⁻¹ * map f g = 1 := by
+  simp only [map_inv, inv_mul_cancel]
+
+@[simp]
+lemma coe_map_mul_map_inv (g : GL n R) : g.val.map f * g.val⁻¹.map f = 1 := by
+  rw [← Matrix.map_mul]
+  simp only [isUnits_det_units, mul_nonsing_inv, map_zero, _root_.map_one, Matrix.map_one]
+
+@[simp]
+lemma coe_map_inv_mul_map (g : GL n R) : g.val⁻¹.map f * g.val.map f = 1 := by
+  rw [← Matrix.map_mul]
+  simp only [isUnits_det_units, nonsing_inv_mul, map_zero, _root_.map_one, Matrix.map_one]
+
 end GeneralLinearGroup
 
 namespace SpecialLinearGroup
diff --git a/Mathlib/LinearAlgebra/Matrix/Gershgorin.lean b/Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
index 39a98afe5cabe..06f6a65c35c43 100644
--- a/Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Gershgorin.lean
@@ -37,7 +37,7 @@ theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toL
       refine (h_i ▸ Finset.le_sup' (fun i => ‖v i‖) (Finset.mem_univ j)).trans ?_
       exact norm_le_zero_iff.mpr h_nz
     have h_le : ∀ j, ‖v j * (v i)⁻¹‖ ≤ 1 := fun j => by
-      rw [norm_mul, norm_inv, mul_inv_le_iff' (norm_pos_iff.mpr h_nz), one_mul]
+      rw [norm_mul, norm_inv, mul_inv_le_iff₀ (norm_pos_iff.mpr h_nz), one_mul]
       exact h_i ▸ Finset.le_sup' (fun i => ‖v i‖) (Finset.mem_univ j)
     simp_rw [mem_closedBall_iff_norm']
     refine ⟨i, ?_⟩
@@ -62,7 +62,7 @@ theorem det_ne_zero_of_sum_row_lt_diag (h : ∀ k, ∑ j ∈ Finset.univ.erase k
   suffices ∃ k, 0 ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) by
     exact this.imp (fun a h ↦ by rwa [mem_closedBall_iff_norm', sub_zero] at h)
   refine eigenvalue_mem_ball ?_
-  rw [Module.End.HasEigenvalue,  Module.End.eigenspace_zero, ne_comm]
+  rw [Module.End.hasEigenvalue_iff, Module.End.eigenspace_zero, ne_comm]
   exact ne_of_lt (LinearMap.bot_lt_ker_of_det_eq_zero (by rwa [LinearMap.det_toLin']))
 
 /-- If `A` is a column strictly dominant diagonal matrix, then it's determinant is nonzero. -/
diff --git a/Mathlib/LinearAlgebra/Matrix/HermitianFunctionalCalculus.lean b/Mathlib/LinearAlgebra/Matrix/HermitianFunctionalCalculus.lean
index aedad97e0580a..ff884d2e289ad 100644
--- a/Mathlib/LinearAlgebra/Matrix/HermitianFunctionalCalculus.lean
+++ b/Mathlib/LinearAlgebra/Matrix/HermitianFunctionalCalculus.lean
@@ -7,7 +7,7 @@ Authors: Jon Bannon, Jireh Loreaux
 import Mathlib.LinearAlgebra.Matrix.Spectrum
 import Mathlib.LinearAlgebra.Eigenspace.Matrix
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique
-import Mathlib.Topology.ContinuousFunction.Units
+import Mathlib.Topology.ContinuousMap.Units
 
 /-!
 # Continuous Functional Calculus for Hermitian Matrices
@@ -76,7 +76,7 @@ noncomputable def cfcAux : C(spectrum ℝ A, ℝ) →⋆ₐ[ℝ] (Matrix n n 
     congr! with i
     simp
   commutes' r := by
-    simp only [Function.comp, algebraMap_apply, smul_eq_mul, mul_one]
+    simp only [Function.comp_def, algebraMap_apply, smul_eq_mul, mul_one]
     rw [← mul_one (algebraMap _ _ _), ← unitary.coe_mul_star_self hA.eigenvectorUnitary,
       ← Algebra.left_comm, unitary.coe_star, mul_assoc]
     congr!
@@ -134,7 +134,7 @@ instance instContinuousFunctionalCalculus :
       simp only [isSelfAdjoint_iff, cfcAux_apply, mul_assoc, star_mul, star_star]
       rw [star_eq_conjTranspose, diagonal_conjTranspose]
       congr!
-      simp [Pi.star_def, Function.comp]
+      simp [Pi.star_def, Function.comp_def]
   predicate_zero := .zero _
 
 instance instUniqueContinuousFunctionalCalculus :
@@ -157,7 +157,8 @@ lemma cfc_eq (f : ℝ → ℝ) : cfc f A = hA.cfc f := by
   have := cfcHom_eq_of_continuous_of_map_id hA' hA.cfcAux hA.closedEmbedding_cfcAux.continuous
     hA.cfcAux_id
   rw [cfc_apply f A hA' (by rw [continuousOn_iff_continuous_restrict]; fun_prop), this]
-  simp only [cfcAux_apply, ContinuousMap.coe_mk, Function.comp, Set.restrict_apply, IsHermitian.cfc]
+  simp only [cfcAux_apply, ContinuousMap.coe_mk, Function.comp_def, Set.restrict_apply,
+    IsHermitian.cfc]
 
 end IsHermitian
 end Matrix
diff --git a/Mathlib/LinearAlgebra/Matrix/InvariantBasisNumber.lean b/Mathlib/LinearAlgebra/Matrix/InvariantBasisNumber.lean
index 5509163c9fd47..a424543278ed7 100644
--- a/Mathlib/LinearAlgebra/Matrix/InvariantBasisNumber.lean
+++ b/Mathlib/LinearAlgebra/Matrix/InvariantBasisNumber.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.LinearAlgebra.Matrix.ToLin
 import Mathlib.LinearAlgebra.InvariantBasisNumber
diff --git a/Mathlib/LinearAlgebra/Matrix/LDL.lean b/Mathlib/LinearAlgebra/Matrix/LDL.lean
index 46bbe801bd60b..277c2bddd5c89 100644
--- a/Mathlib/LinearAlgebra/Matrix/LDL.lean
+++ b/Mathlib/LinearAlgebra/Matrix/LDL.lean
@@ -107,7 +107,7 @@ noncomputable def LDL.lower :=
   (LDL.lowerInv hS)⁻¹
 
 /-- **LDL decomposition**: any positive definite matrix `S` can be
-decomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal matrix.  -/
+decomposed as `S = LDLᴴ` where `L` is a lower-triangular matrix and `D` is a diagonal matrix. -/
 theorem LDL.lower_conj_diag : LDL.lower hS * LDL.diag hS * (LDL.lower hS)ᴴ = S := by
   rw [LDL.lower, conjTranspose_nonsing_inv, Matrix.mul_assoc,
     Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),
diff --git a/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean b/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
index e91010d1bc343..41e80c3f6b369 100644
--- a/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
+++ b/Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
@@ -175,6 +175,20 @@ theorem det_ne_zero_of_right_inverse [Nontrivial α] (h : A * B = 1) : A.det ≠
 
 end Invertible
 
+
+section
+
+variable [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α]
+
+/-- A version of `mul_eq_one_comm` that works for square matrices with rectangular types. -/
+theorem mul_eq_one_comm_of_equiv {A : Matrix m n α} {B : Matrix n m α} (e : m ≃ n) :
+    A * B = 1 ↔ B * A = 1 := by
+  refine (reindex e e).injective.eq_iff.symm.trans ?_
+  rw [reindex_apply, reindex_apply, submatrix_one_equiv, ← submatrix_mul_equiv _ _ _ (.refl _),
+    mul_eq_one_comm, submatrix_mul_equiv, coe_refl, submatrix_id_id]
+
+end
+
 section Inv
 
 variable [Fintype n] [DecidableEq n] [CommRing α]
@@ -456,6 +470,10 @@ theorem nonsing_inv_nonsing_inv (h : IsUnit A.det) : A⁻¹⁻¹ = A :=
 theorem isUnit_nonsing_inv_det_iff {A : Matrix n n α} : IsUnit A⁻¹.det ↔ IsUnit A.det := by
   rw [Matrix.det_nonsing_inv, isUnit_ring_inverse]
 
+@[simp]
+theorem isUnit_nonsing_inv_iff {A : Matrix n n α} : IsUnit A⁻¹ ↔ IsUnit A := by
+  simp_rw [isUnit_iff_isUnit_det, isUnit_nonsing_inv_det_iff]
+
 -- `IsUnit.invertible` lifts the proposition `IsUnit A` to a constructive inverse of `A`.
 /-- A version of `Matrix.invertibleOfDetInvertible` with the inverse defeq to `A⁻¹` that is
 therefore noncomputable. -/
@@ -593,6 +611,42 @@ theorem inv_diagonal (v : n → α) : (diagonal v)⁻¹ = diagonal (Ring.inverse
 
 end Diagonal
 
+/-- The inverse of a 1×1 or 0×0 matrix is always diagonal.
+
+While we could write this as `of fun _ _ => Ring.inverse (A default default)` on the RHS, this is
+less useful because:
+
+* It wouldn't work for 0×0 matrices.
+* More things are true about diagonal matrices than constant matrices, and so more lemmas exist.
+
+`Matrix.diagonal_unique` can be used to reach this form, while `Ring.inverse_eq_inv` can be used
+to replace `Ring.inverse` with `⁻¹`.
+-/
+@[simp]
+theorem inv_subsingleton [Subsingleton m] [Fintype m] [DecidableEq m] (A : Matrix m m α) :
+    A⁻¹ = diagonal fun i => Ring.inverse (A i i) := by
+  rw [inv_def, adjugate_subsingleton, smul_one_eq_diagonal]
+  congr! with i
+  exact det_eq_elem_of_subsingleton _ _
+
+section Woodbury
+
+variable [Fintype m] [DecidableEq m]
+variable (A : Matrix n n α) (U : Matrix n m α) (C : Matrix m m α) (V : Matrix m n α)
+
+/-- The **Woodbury Identity** (`⁻¹` version). -/
+theorem add_mul_mul_inv_eq_sub (hA : IsUnit A) (hC : IsUnit C) (hAC : IsUnit (C⁻¹ + V * A⁻¹ * U)) :
+    (A + U * C * V)⁻¹ = A⁻¹ - A⁻¹ * U * (C⁻¹ + V * A⁻¹ * U)⁻¹ * V * A⁻¹ := by
+  obtain ⟨_⟩ := hA.nonempty_invertible
+  obtain ⟨_⟩ := hC.nonempty_invertible
+  obtain ⟨iAC⟩ := hAC.nonempty_invertible
+  simp only [← invOf_eq_nonsing_inv] at iAC
+  letI := invertibleAddMulMul A U C V
+  simp only [← invOf_eq_nonsing_inv]
+  apply invOf_add_mul_mul
+
+end Woodbury
+
 @[simp]
 theorem inv_inv_inv (A : Matrix n n α) : A⁻¹⁻¹⁻¹ = A⁻¹ := by
   by_cases h : IsUnit A.det
@@ -722,4 +776,21 @@ theorem det_conj' {M : Matrix m m α} (h : IsUnit M) (N : Matrix m m α) :
 
 end Det
 
+/-! ### More results about traces -/
+
+
+section trace
+
+variable [Fintype m] [DecidableEq m]
+
+/-- A variant of `Matrix.trace_units_conj`. -/
+theorem trace_conj {M : Matrix m m α} (h : IsUnit M) (N : Matrix m m α) :
+    trace (M * N * M⁻¹) = trace N := by rw [← h.unit_spec, ← coe_units_inv, trace_units_conj]
+
+/-- A variant of `Matrix.trace_units_conj'`. -/
+theorem trace_conj' {M : Matrix m m α} (h : IsUnit M) (N : Matrix m m α) :
+    trace (M⁻¹ * N * M) = trace N := by rw [← h.unit_spec, ← coe_units_inv, trace_units_conj']
+
+end trace
+
 end Matrix
diff --git a/Mathlib/LinearAlgebra/Matrix/Polynomial.lean b/Mathlib/LinearAlgebra/Matrix/Polynomial.lean
index 72e1f37352e62..64469ff133624 100644
--- a/Mathlib/LinearAlgebra/Matrix/Polynomial.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Polynomial.lean
@@ -38,7 +38,7 @@ theorem natDegree_det_X_add_C_le (A B : Matrix n n α) :
   rw [det_apply]
   refine (natDegree_sum_le _ _).trans ?_
   refine Multiset.max_le_of_forall_le _ _ ?_
-  simp only [forall_apply_eq_imp_iff, true_and_iff, Function.comp_apply, Multiset.map_map,
+  simp only [forall_apply_eq_imp_iff, true_and, Function.comp_apply, Multiset.map_map,
     Multiset.mem_map, exists_imp, Finset.mem_univ_val]
   intro g
   calc
diff --git a/Mathlib/LinearAlgebra/Matrix/PosDef.lean b/Mathlib/LinearAlgebra/Matrix/PosDef.lean
index a982d1b30fdeb..9343bc8070623 100644
--- a/Mathlib/LinearAlgebra/Matrix/PosDef.lean
+++ b/Mathlib/LinearAlgebra/Matrix/PosDef.lean
@@ -18,7 +18,7 @@ of quadratic forms. Most results require `𝕜 = ℝ` or `ℂ`.
 * `Matrix.PosSemidef` : a matrix `M : Matrix n n 𝕜` is positive semidefinite if it is hermitian
   and `xᴴMx` is nonnegative for all `x`.
 
-## Main results
+## Main results
 
 * `Matrix.posSemidef_iff_eq_transpose_mul_self` : a matrix `M : Matrix n n 𝕜` is positive
   semidefinite iff it has the form `Bᴴ * B` for some `B`.
@@ -450,6 +450,25 @@ theorem det_pos [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : 0 < det
   intro i _
   simpa using hM.eigenvalues_pos i
 
+theorem isUnit [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : IsUnit M :=
+  isUnit_iff_isUnit_det _ |>.2 <| hM.det_pos.ne'.isUnit
+
+protected theorem inv [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) : M⁻¹.PosDef := by
+  refine ⟨hM.isHermitian.inv, fun x hx => ?_⟩
+  have := hM.2 (M⁻¹ *ᵥ x) ((Matrix.mulVec_injective_iff_isUnit.mpr ?_ |>.ne_iff' ?_).2 hx)
+  · let _inst := hM.isUnit.invertible
+    rwa [star_mulVec, mulVec_mulVec, Matrix.mul_inv_of_invertible, one_mulVec,
+      ← star_pos_iff, ← star_mulVec, ← star_dotProduct] at this
+  · simpa using hM.isUnit
+  · simp
+
+@[simp]
+theorem _root_.Matrix.posDef_inv_iff [DecidableEq n] {M : Matrix n n 𝕜} :
+    M⁻¹.PosDef ↔ M.PosDef :=
+  ⟨fun h =>
+    letI := (Matrix.isUnit_nonsing_inv_iff.1 <| h.isUnit).invertible
+    Matrix.inv_inv_of_invertible M ▸ h.inv, (·.inv)⟩
+
 end PosDef
 
 end Matrix
diff --git a/Mathlib/LinearAlgebra/Matrix/SchurComplement.lean b/Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
index ea9e82931381e..3c18c28e836bd 100644
--- a/Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
+++ b/Mathlib/LinearAlgebra/Matrix/SchurComplement.lean
@@ -53,8 +53,8 @@ theorem fromBlocks_eq_of_invertible₁₁ (A : Matrix m m α) (B : Matrix m n α
       fromBlocks 1 0 (C * ⅟ A) 1 * fromBlocks A 0 0 (D - C * ⅟ A * B) *
         fromBlocks 1 (⅟ A * B) 0 1 := by
   simp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
-    Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_self_assoc,
-    Matrix.mul_invOf_mul_self_cancel, Matrix.mul_assoc, add_sub_cancel]
+    Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_cancel_left,
+    Matrix.invOf_mul_cancel_right, Matrix.mul_assoc, add_sub_cancel]
 
 /-- LDU decomposition of a block matrix with an invertible bottom-right corner, using the
 Schur complement. -/
@@ -78,7 +78,7 @@ def fromBlocksZero₂₁Invertible (A : Matrix m m α) (B : Matrix m n α) (D :
     [Invertible A] [Invertible D] : Invertible (fromBlocks A B 0 D) :=
   invertibleOfLeftInverse _ (fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 (⅟ D)) <| by
     simp_rw [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, zero_add, add_zero,
-      Matrix.neg_mul, invOf_mul_self, Matrix.mul_invOf_mul_self_cancel, add_neg_cancel,
+      Matrix.neg_mul, invOf_mul_self, Matrix.invOf_mul_cancel_right, add_neg_cancel,
       fromBlocks_one]
 
 /-- A lower-block-triangular matrix is invertible if its diagonal is. -/
@@ -88,7 +88,7 @@ def fromBlocksZero₁₂Invertible (A : Matrix m m α) (C : Matrix n m α) (D :
       (fromBlocks (⅟ A) 0 (-(⅟ D * C * ⅟ A))
         (⅟ D)) <| by -- a symmetry argument is more work than just copying the proof
     simp_rw [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, zero_add, add_zero,
-      Matrix.neg_mul, invOf_mul_self, Matrix.mul_invOf_mul_self_cancel, neg_add_cancel,
+      Matrix.neg_mul, invOf_mul_self, Matrix.invOf_mul_cancel_right, neg_add_cancel,
       fromBlocks_one]
 
 theorem invOf_fromBlocks_zero₂₁_eq (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α)
@@ -224,7 +224,7 @@ end Triangular
 
 section Block
 
-/-! #### General 2×2 block matrices-/
+/-! #### General 2×2 block matrices -/
 
 
 /-- A block matrix is invertible if the bottom right corner and the corresponding schur complement
diff --git a/Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean b/Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
index 44fdd5263d583..8e620f838f3f0 100644
--- a/Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
+++ b/Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
@@ -485,7 +485,7 @@ end
 
 end ToMatrix
 
-/-! ### Adjoint pairs-/
+/-! ### Adjoint pairs -/
 
 
 section MatrixAdjoints
@@ -559,8 +559,7 @@ theorem Matrix.isAdjointPair_equiv (P : Matrix n n R) (h : IsUnit P) :
   let v := Pᵀ.nonsingInvUnit (P.isUnit_det_transpose h')
   let x := A₁ᵀ * Pᵀ * J
   let y := J * P * A₂
-  -- TODO(mathlib4#6607): fix elaboration so `val` isn't needed
-  suffices x * u.val = v.val * y ↔ (v⁻¹).val * x = y * (u⁻¹).val by
+  suffices x * u = v * y ↔ v⁻¹ * x = y * u⁻¹ by
     dsimp only [Matrix.IsAdjointPair]
     simp only [Matrix.transpose_mul]
     simp only [← mul_assoc, P.transpose_nonsing_inv]
@@ -623,7 +622,7 @@ end MatrixAdjoints
 
 namespace LinearMap
 
-/-! ### Nondegenerate bilinear forms-/
+/-! ### Nondegenerate bilinear forms -/
 
 
 section Det
diff --git a/Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean b/Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
index bf7e0f5c040f9..abb3fa721cfad 100644
--- a/Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
+++ b/Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
@@ -285,7 +285,7 @@ def center_equiv_rootsOfUnity' (i : n) :
   map_mul' A B := by
     dsimp
     ext
-    simp only [Submonoid.coe_mul, coe_mul, rootsOfUnity.val_mkOfPowEq_coe, Units.val_mul]
+    simp only [rootsOfUnity.val_mkOfPowEq_coe, Subgroup.coe_mul, Units.val_mul]
     rw [← scalar_eq_coe_self_center A i, ← scalar_eq_coe_self_center B i]
     simp
 
diff --git a/Mathlib/LinearAlgebra/Matrix/Spectrum.lean b/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
index d69134ff4820b..e20e0a672cfc3 100644
--- a/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
@@ -17,7 +17,7 @@ the spectral theorem for linear maps (`LinearMap.IsSymmetric.eigenvectorBasis_ap
 
 ## Tags
 
-spectral theorem, diagonalization theorem-/
+spectral theorem, diagonalization theorem -/
 
 namespace Matrix
 
diff --git a/Mathlib/LinearAlgebra/Matrix/ToLin.lean b/Mathlib/LinearAlgebra/Matrix/ToLin.lean
index f7a4aba7c8c89..803a75b7be149 100644
--- a/Mathlib/LinearAlgebra/Matrix/ToLin.lean
+++ b/Mathlib/LinearAlgebra/Matrix/ToLin.lean
@@ -942,7 +942,7 @@ If `M₁` and `M₂` are modules with basis `b₁` and `b₂` respectively index
 by finite types `ι₁` and `ι₂`,
 then `Basis.linearMap b₁ b₂` is the basis of `M₁ →ₗ[R] M₂` indexed by `ι₂ × ι₁`
 where `(i, j)` indexes the linear map that sends `b j` to `b i`
-and sends all other basis vectors to `0`.  -/
+and sends all other basis vectors to `0`. -/
 @[simps! (config := .lemmasOnly) repr_apply repr_symm_apply]
 noncomputable
 def linearMap (b₁ : Basis ι₁ R M₁) (b₂ : Basis ι₂ R M₂) :
@@ -968,7 +968,7 @@ induced by a basis of the module.
 If `M` is a module with basis `b` indexed by a finite type `ι`,
 then `Basis.end b` is the basis of `Module.End R M` indexed by `ι × ι`
 where `(i, j)` indexes the linear map that sends `b j` to `b i`
-and sends all other basis vectors to `0`.  -/
+and sends all other basis vectors to `0`. -/
 @[simps! (config := .lemmasOnly) repr_apply repr_symm_apply]
 noncomputable
 abbrev _root_.Basis.end (b : Basis ι R M) : Basis (ι × ι) R (Module.End R M) :=
diff --git a/Mathlib/LinearAlgebra/Matrix/Trace.lean b/Mathlib/LinearAlgebra/Matrix/Trace.lean
index 1e018cf22f592..40da0be3f44c0 100644
--- a/Mathlib/LinearAlgebra/Matrix/Trace.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Trace.lean
@@ -109,7 +109,7 @@ theorem _root_.AddMonoidHom.map_trace [AddCommMonoid S] (f : R →+ S) (A : Matr
 
 lemma trace_blockDiagonal [DecidableEq p] (M : p → Matrix n n R) :
     trace (blockDiagonal M) = ∑ i, trace (M i) := by
-  simp [blockDiagonal, trace, Finset.sum_comm (γ := n)]
+  simp [blockDiagonal, trace, Finset.sum_comm (γ := n), Fintype.sum_prod_type]
 
 lemma trace_blockDiagonal' [DecidableEq p] {m : p → Type*} [∀ i, Fintype (m i)]
     (M : ∀ i, Matrix (m i) (m i) R) :
@@ -175,6 +175,23 @@ lemma trace_submatrix_succ {n : ℕ} [NonUnitalNonAssocSemiring R]
   rw [← (finSuccEquiv n).symm.sum_comp]
   simp
 
+section CommSemiring
+
+variable [DecidableEq m] [CommSemiring R]
+
+-- TODO(mathlib4#6607): fix elaboration so that the ascription isn't needed
+theorem trace_units_conj (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
+    trace ((M : Matrix _ _ _) * N * (↑M⁻¹ : Matrix _ _ _)) = trace N := by
+  rw [trace_mul_cycle, Units.inv_mul, one_mul]
+
+set_option linter.docPrime false in
+-- TODO(mathlib4#6607): fix elaboration so that the ascription isn't needed
+theorem trace_units_conj' (M : (Matrix m m R)ˣ) (N : Matrix m m R) :
+    trace ((↑M⁻¹ : Matrix _ _ _) * N * (↑M : Matrix _ _ _)) = trace N :=
+  trace_units_conj M⁻¹ N
+
+end CommSemiring
+
 section Fin
 
 variable [AddCommMonoid R]
diff --git a/Mathlib/LinearAlgebra/Matrix/Transvection.lean b/Mathlib/LinearAlgebra/Matrix/Transvection.lean
index 5197d7684d30d..9a13c74d20d49 100644
--- a/Mathlib/LinearAlgebra/Matrix/Transvection.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Transvection.lean
@@ -101,7 +101,7 @@ theorem updateRow_eq_transvection [Finite n] (c : R) :
         StdBasisMatrix.apply_of_ne]
   · simp only [updateRow_ne, transvection, ha, Ne.symm ha, StdBasisMatrix.apply_of_ne, add_zero,
       Algebra.id.smul_eq_mul, Ne, not_false_iff, DMatrix.add_apply, Pi.smul_apply,
-      mul_zero, false_and_iff, add_apply]
+      mul_zero, false_and, add_apply]
 
 variable [Fintype n]
 
@@ -424,7 +424,7 @@ theorem listTransvecCol_mul_last_col (hM : M (inr unit) (inr unit) ≠ 0) (i : F
       rcases le_or_lt (n + 1) i with (hi | hi)
       · simp only [hi, n.le_succ.trans hi, if_true]
       · rw [if_neg, if_neg]
-        · simpa only [hni.symm, not_le, or_false_iff] using Nat.lt_succ_iff_lt_or_eq.1 hi
+        · simpa only [hni.symm, not_le, or_false] using Nat.lt_succ_iff_lt_or_eq.1 hi
         · simpa only [not_le] using hi
   | self =>
     simp only [length_listTransvecCol, le_refl, List.drop_eq_nil_of_le, List.prod_nil,
@@ -447,7 +447,7 @@ theorem mul_listTransvecRow_last_col_take (i : Fin r ⊕ Unit) {k : ℕ} (hk : k
     simp only [List.take_succ, ← Matrix.mul_assoc, this, List.prod_append, Matrix.mul_one,
       List.prod_cons, List.prod_nil, Option.toList_some]
     rw [mul_transvection_apply_of_ne, IH hkr.le]
-    simp only [Ne, not_false_iff]
+    simp only [Ne, not_false_iff, reduceCtorEq]
 
 /-- Multiplying by all the matrices in `listTransvecRow M` does not change the last column. -/
 theorem mul_listTransvecRow_last_col (i : Fin r ⊕ Unit) :
@@ -500,7 +500,7 @@ theorem mul_listTransvecRow_last_row (hM : M (inr unit) (inr unit) ≠ 0) (i : F
       · simp [hi, n.le_succ.trans hi, if_true]
       · rw [if_neg, if_neg]
         · simpa only [not_le] using hi
-        · simpa only [hni.symm, not_le, or_false_iff] using Nat.lt_succ_iff_lt_or_eq.1 hi
+        · simpa only [hni.symm, not_le, or_false] using Nat.lt_succ_iff_lt_or_eq.1 hi
 
 /-- Multiplying by all the matrices either in `listTransvecCol M` and `listTransvecRow M` kills
 all the coefficients in the last row but the last one. -/
@@ -549,8 +549,8 @@ theorem exists_isTwoBlockDiagonal_of_ne_zero (hM : M (inr unit) (inr unit) ≠ 0
     List.ofFn fun i : Fin r =>
       ⟨inr unit, inl i, by simp, -M (inr unit) (inl i) / M (inr unit) (inr unit)⟩
   refine ⟨L, L', ?_⟩
-  have A : L.map toMatrix = listTransvecCol M := by simp [L, listTransvecCol, (· ∘ ·)]
-  have B : L'.map toMatrix = listTransvecRow M := by simp [L', listTransvecRow, (· ∘ ·)]
+  have A : L.map toMatrix = listTransvecCol M := by simp [L, listTransvecCol, Function.comp_def]
+  have B : L'.map toMatrix = listTransvecRow M := by simp [L', listTransvecRow, Function.comp_def]
   rw [A, B]
   exact isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow M hM
 
diff --git a/Mathlib/LinearAlgebra/Matrix/ZPow.lean b/Mathlib/LinearAlgebra/Matrix/ZPow.lean
index d28a1a1448f30..dd11fe3101db7 100644
--- a/Mathlib/LinearAlgebra/Matrix/ZPow.lean
+++ b/Mathlib/LinearAlgebra/Matrix/ZPow.lean
@@ -110,12 +110,14 @@ theorem _root_.IsUnit.det_zpow {A : M} (h : IsUnit A.det) (n : ℤ) : IsUnit (A
   · simpa using h.pow n.succ
 
 theorem isUnit_det_zpow_iff {A : M} {z : ℤ} : IsUnit (A ^ z).det ↔ IsUnit A.det ∨ z = 0 := by
-  induction' z using Int.induction_on with z _ z _
-  · simp
-  · rw [← Int.ofNat_succ, zpow_natCast, det_pow, isUnit_pow_succ_iff, ← Int.ofNat_zero,
+  induction z using Int.induction_on with
+  | hz => simp
+  | hp z =>
+    rw [← Int.ofNat_succ, zpow_natCast, det_pow, isUnit_pow_succ_iff, ← Int.ofNat_zero,
       Int.ofNat_inj]
     simp
-  · rw [← neg_add', ← Int.ofNat_succ, zpow_neg_natCast, isUnit_nonsing_inv_det_iff, det_pow,
+  | hn z =>
+    rw [← neg_add', ← Int.ofNat_succ, zpow_neg_natCast, isUnit_nonsing_inv_det_iff, det_pow,
       isUnit_pow_succ_iff, neg_eq_zero, ← Int.ofNat_zero, Int.ofNat_inj]
     simp
 
diff --git a/Mathlib/LinearAlgebra/Multilinear/Basic.lean b/Mathlib/LinearAlgebra/Multilinear/Basic.lean
index b9fed90e5e9ec..25022902c14a5 100644
--- a/Mathlib/LinearAlgebra/Multilinear/Basic.lean
+++ b/Mathlib/LinearAlgebra/Multilinear/Basic.lean
@@ -109,7 +109,7 @@ variable [Semiring R] [∀ i, AddCommMonoid (M i)] [∀ i, AddCommMonoid (M₁ i
 -- Porting note: Replaced CoeFun with FunLike instance
 instance : FunLike (MultilinearMap R M₁ M₂) (∀ i, M₁ i) M₂ where
   coe f := f.toFun
-  coe_injective' := fun f g h ↦ by cases f; cases g; cases h; rfl
+  coe_injective' f g h := by cases f; cases g; cases h; rfl
 
 initialize_simps_projections MultilinearMap (toFun → apply)
 
@@ -453,14 +453,9 @@ theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i,
   induction' n using Nat.strong_induction_on with n IH generalizing A
   -- If one of the sets is empty, then all the sums are zero
   by_cases Ai_empty : ∃ i, A i = ∅
-  · rcases Ai_empty with ⟨i, hi⟩
-    have : ∑ j ∈ A i, g i j = 0 := by rw [hi, Finset.sum_empty]
-    rw [f.map_coord_zero i this]
-    have : piFinset A = ∅ := by
-      refine Finset.eq_empty_of_forall_not_mem fun r hr => ?_
-      have : r i ∈ A i := mem_piFinset.mp hr i
-      simp [hi] at this
-    rw [this, Finset.sum_empty]
+  · obtain ⟨i, hi⟩ : ∃ i, ∑ j ∈ A i, g i j = 0 := Ai_empty.imp fun i hi ↦ by simp [hi]
+    have hpi : piFinset A = ∅ := by simpa
+    rw [f.map_coord_zero i hi, hpi, Finset.sum_empty]
   push_neg at Ai_empty
   -- Otherwise, if all sets are at most singletons, then they are exactly singletons and the result
   -- is again straightforward
@@ -522,7 +517,7 @@ theorem map_sum_finset_aux [DecidableEq ι] [Fintype ι] {n : ℕ} (h : (∑ i,
         simpa [C] using hj
       rw [this]
       simp only [B, mem_sdiff, eq_self_iff_true, not_true, not_false_iff, Finset.mem_singleton,
-        update_same, and_false_iff]
+        update_same, and_false]
     · simp [hi]
   have Beq :
     Function.update (fun i => ∑ j ∈ A i, g i j) i₀ (∑ j ∈ B i₀, g i₀ j) = fun i =>
@@ -1432,25 +1427,21 @@ theorem MultilinearMap.uncurry_curryLeft (f : MultilinearMap R M M₂) :
 
 variable (R M M₂)
 
-/-- The space of multilinear maps on `∀ (i : Fin (n+1)), M i` is canonically isomorphic to
+/-- The space of multilinear maps on `Π (i : Fin (n+1)), M i` is canonically isomorphic to
 the space of linear maps from `M 0` to the space of multilinear maps on
-`∀ (i : Fin n), M i.succ`, by separating the first variable. We register this isomorphism as a
+`Π (i : Fin n), M i.succ`, by separating the first variable. We register this isomorphism as a
 linear isomorphism in `multilinearCurryLeftEquiv R M M₂`.
 
-The direct and inverse maps are given by `f.uncurryLeft` and `f.curryLeft`. Use these
+The direct and inverse maps are given by `f.curryLeft` and `f.uncurryLeft`. Use these
 unless you need the full framework of linear equivs. -/
 def multilinearCurryLeftEquiv :
-    (M 0 →ₗ[R] MultilinearMap R (fun i : Fin n => M i.succ) M₂) ≃ₗ[R] MultilinearMap R M M₂ where
-  toFun := LinearMap.uncurryLeft
-  map_add' f₁ f₂ := by
-    ext m
-    rfl
-  map_smul' c f := by
-    ext m
-    rfl
-  invFun := MultilinearMap.curryLeft
-  left_inv := LinearMap.curry_uncurryLeft
-  right_inv := MultilinearMap.uncurry_curryLeft
+    MultilinearMap R M M₂ ≃ₗ[R] (M 0 →ₗ[R] MultilinearMap R (fun i : Fin n => M i.succ) M₂) where
+  toFun := MultilinearMap.curryLeft
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+  invFun := LinearMap.uncurryLeft
+  left_inv := MultilinearMap.uncurry_curryLeft
+  right_inv := LinearMap.curry_uncurryLeft
 
 variable {R M M₂}
 
@@ -1542,27 +1533,22 @@ theorem MultilinearMap.uncurry_curryRight (f : MultilinearMap R M M₂) :
 
 variable (R M M₂)
 
-/-- The space of multilinear maps on `∀ (i : Fin (n+1)), M i` is canonically isomorphic to
-the space of linear maps from the space of multilinear maps on `∀ (i : Fin n), M (castSucc i)` to
+/-- The space of multilinear maps on `Π (i : Fin (n+1)), M i` is canonically isomorphic to
+the space of linear maps from the space of multilinear maps on `Π (i : Fin n), M (castSucc i)` to
 the space of linear maps on `M (last n)`, by separating the last variable. We register this
 isomorphism as a linear isomorphism in `multilinearCurryRightEquiv R M M₂`.
 
-The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
+The direct and inverse maps are given by `f.curryRight` and `f.uncurryRight`. Use these
 unless you need the full framework of linear equivs. -/
 def multilinearCurryRightEquiv :
-    MultilinearMap R (fun i : Fin n => M (castSucc i)) (M (last n) →ₗ[R] M₂) ≃ₗ[R]
-      MultilinearMap R M M₂ where
-  toFun := MultilinearMap.uncurryRight
-  map_add' f₁ f₂ := by
-    ext m
-    rfl
-  map_smul' c f := by
-    ext m
-    rw [smul_apply]
-    rfl
-  invFun := MultilinearMap.curryRight
-  left_inv := MultilinearMap.curry_uncurryRight
-  right_inv := MultilinearMap.uncurry_curryRight
+    MultilinearMap R M M₂ ≃ₗ[R]
+      MultilinearMap R (fun i : Fin n => M (castSucc i)) (M (last n) →ₗ[R] M₂) where
+  toFun := MultilinearMap.curryRight
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+  invFun := MultilinearMap.uncurryRight
+  left_inv := MultilinearMap.uncurry_curryRight
+  right_inv := MultilinearMap.curry_uncurryRight
 
 namespace MultilinearMap
 
@@ -1771,3 +1757,5 @@ def range [Nonempty ι] (f : MultilinearMap R M₁ M₂) : SubMulAction R M₂ :
 end Submodule
 
 end MultilinearMap
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean b/Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean
index 918241ffc7e57..5127794e5fa7f 100644
--- a/Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean
+++ b/Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean
@@ -40,8 +40,8 @@ private theorem free_and_finite_fin (n : ℕ) (N : Fin n → Type*) [∀ i, AddC
         Module.Finite R (N 0 →ₗ[R] MultilinearMap R (fun i : Fin n => N i.succ) M₂) by
       cases this
       exact
-        ⟨Module.Free.of_equiv (multilinearCurryLeftEquiv R N M₂),
-          Module.Finite.equiv (multilinearCurryLeftEquiv R N M₂)⟩
+        ⟨Module.Free.of_equiv (multilinearCurryLeftEquiv R N M₂).symm,
+          Module.Finite.equiv (multilinearCurryLeftEquiv R N M₂).symm⟩
     cases ih fun i => N i.succ
     exact ⟨Module.Free.linearMap _ _ _ _, Module.Finite.linearMap _ _ _ _⟩
 
diff --git a/Mathlib/LinearAlgebra/Orientation.lean b/Mathlib/LinearAlgebra/Orientation.lean
index 68891b346f5df..549ff52629c7b 100644
--- a/Mathlib/LinearAlgebra/Orientation.lean
+++ b/Mathlib/LinearAlgebra/Orientation.lean
@@ -326,7 +326,7 @@ namespace Orientation
 
 variable [Fintype ι]
 
-open FiniteDimensional
+open FiniteDimensional Module
 
 /-- If the index type has cardinality equal to the finite dimension, any two orientations are
 equal or negations. -/
diff --git a/Mathlib/LinearAlgebra/PerfectPairing.lean b/Mathlib/LinearAlgebra/PerfectPairing.lean
index 74a47d2060113..dfec7b49779cd 100644
--- a/Mathlib/LinearAlgebra/PerfectPairing.lean
+++ b/Mathlib/LinearAlgebra/PerfectPairing.lean
@@ -34,7 +34,7 @@ open Function Module
 variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
 
 /-- A perfect pairing of two (left) modules over a commutative ring. -/
-structure PerfectPairing :=
+structure PerfectPairing where
   toLin : M →ₗ[R] N →ₗ[R] R
   bijectiveLeft : Bijective toLin
   bijectiveRight : Bijective toLin.flip
@@ -49,6 +49,10 @@ instance instFunLike : FunLike (PerfectPairing R M N) M (N →ₗ[R] R) where
   coe f := f.toLin
   coe_injective' x y h := by cases x; cases y; simpa using h
 
+@[simp]
+lemma toLin_apply (p : PerfectPairing R M N) {x : M} : p.toLin x = p x := by
+  rfl
+
 variable (p : PerfectPairing R M N)
 
 /-- Given a perfect pairing between `M` and `N`, we may interchange the roles of `M` and `N`. -/
@@ -57,7 +61,13 @@ protected def flip : PerfectPairing R N M where
   bijectiveLeft := p.bijectiveRight
   bijectiveRight := p.bijectiveLeft
 
-@[simp] lemma flip_flip : p.flip.flip = p := rfl
+@[simp]
+lemma flip_apply_apply {x : M} {y : N} : p.flip y x = p x y :=
+  rfl
+
+@[simp]
+lemma flip_flip : p.flip.flip = p :=
+  rfl
 
 /-- The linear equivalence from `M` to `Dual R N` induced by a perfect pairing. -/
 noncomputable def toDualLeft : M ≃ₗ[R] Dual R N :=
@@ -130,6 +140,11 @@ def IsReflexive.toPerfectPairingDual : PerfectPairing R (Dual R M) M where
   bijectiveLeft := bijective_id
   bijectiveRight := bijective_dual_eval R M
 
+@[simp]
+lemma IsReflexive.toPerfectPairingDual_apply {f : Dual R M} {x : M} :
+    IsReflexive.toPerfectPairingDual (R := R) f x = f x :=
+  rfl
+
 variable (e : N ≃ₗ[R] Dual R M)
 
 namespace LinearEquiv
diff --git a/Mathlib/LinearAlgebra/Pi.lean b/Mathlib/LinearAlgebra/Pi.lean
index d1342d034d19e..18abf808fda35 100644
--- a/Mathlib/LinearAlgebra/Pi.lean
+++ b/Mathlib/LinearAlgebra/Pi.lean
@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
 -/
 import Mathlib.Algebra.Group.Fin.Tuple
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
 import Mathlib.Algebra.BigOperators.Pi
 import Mathlib.Algebra.Module.Prod
 import Mathlib.Algebra.Module.Submodule.Ker
 import Mathlib.Algebra.Module.Submodule.Range
 import Mathlib.Algebra.Module.Equiv.Basic
-import Mathlib.GroupTheory.GroupAction.BigOperators
 import Mathlib.Logic.Equiv.Fin
 
 /-!
diff --git a/Mathlib/LinearAlgebra/Prod.lean b/Mathlib/LinearAlgebra/Prod.lean
index 8a58b466b4060..fb36914029e72 100644
--- a/Mathlib/LinearAlgebra/Prod.lean
+++ b/Mathlib/LinearAlgebra/Prod.lean
@@ -449,7 +449,7 @@ theorem ker_coprod_of_disjoint_range {M₂ : Type*} [AddCommGroup M₂] [Module
   rintro ⟨y, z⟩ h
   simp only [mem_ker, mem_prod, coprod_apply] at h ⊢
   have : f y ∈ (range f) ⊓ (range g) := by
-    simp only [true_and_iff, mem_range, mem_inf, exists_apply_eq_apply]
+    simp only [true_and, mem_range, mem_inf, exists_apply_eq_apply]
     use -z
     rwa [eq_comm, map_neg, ← sub_eq_zero, sub_neg_eq_add]
   rw [hd.eq_bot, mem_bot] at this
@@ -525,10 +525,9 @@ def fstEquiv : Submodule.fst R M M₂ ≃ₗ[R] M where
   -- Porting note: proofs were `tidy` or `simp`
   toFun x := x.1.1
   invFun m := ⟨⟨m, 0⟩, by simp only [fst, comap_bot, mem_ker, snd_apply]⟩
-  map_add' := by simp only [AddSubmonoid.coe_add, coe_toAddSubmonoid, Prod.fst_add, Subtype.forall,
-    implies_true, Prod.forall, forall_const]
+  map_add' := by simp only [coe_add, Prod.fst_add, implies_true]
   map_smul' := by simp only [SetLike.val_smul, Prod.smul_fst, RingHom.id_apply, Subtype.forall,
-    implies_true, Prod.forall, forall_const]
+    implies_true]
   left_inv := by
     rintro ⟨⟨x, y⟩, hy⟩
     simp only [fst, comap_bot, mem_ker, snd_apply] at hy
@@ -557,10 +556,9 @@ def sndEquiv : Submodule.snd R M M₂ ≃ₗ[R] M₂ where
   -- Porting note: proofs were `tidy` or `simp`
   toFun x := x.1.2
   invFun n := ⟨⟨0, n⟩, by simp only [snd, comap_bot, mem_ker, fst_apply]⟩
-  map_add' := by simp only [AddSubmonoid.coe_add, coe_toAddSubmonoid, Prod.snd_add, Subtype.forall,
-    implies_true, Prod.forall, forall_const]
+  map_add' := by simp only [coe_add, Prod.snd_add, implies_true]
   map_smul' := by simp only [SetLike.val_smul, Prod.smul_snd, RingHom.id_apply, Subtype.forall,
-    implies_true, Prod.forall, forall_const]
+    implies_true]
   left_inv := by
     rintro ⟨⟨x, y⟩, hx⟩
     simp only [snd, comap_bot, mem_ker, fst_apply] at hx
diff --git a/Mathlib/LinearAlgebra/Projection.lean b/Mathlib/LinearAlgebra/Projection.lean
index e730906ab556c..d7766cbbf641f 100644
--- a/Mathlib/LinearAlgebra/Projection.lean
+++ b/Mathlib/LinearAlgebra/Projection.lean
@@ -393,10 +393,10 @@ theorem eq_conj_prod_map' {f : E →ₗ[R] E} (h : IsProj p f) :
         prodMap id 0 ∘ₗ (p.prodEquivOfIsCompl (ker f) h.isCompl).symm.toLinearMap := by
   rw [← LinearMap.comp_assoc, LinearEquiv.eq_comp_toLinearMap_symm]
   ext x
-  · simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inl, coprod_apply, coeSubtype,
+  · simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inl, coprod_apply, coe_subtype,
       _root_.map_zero, add_zero, h.map_id x x.2, prodMap_apply, id_apply]
   · simp only [coe_prodEquivOfIsCompl, comp_apply, coe_inr, coprod_apply, _root_.map_zero,
-      coeSubtype, zero_add, map_coe_ker, prodMap_apply, zero_apply, add_zero]
+      coe_subtype, zero_add, map_coe_ker, prodMap_apply, zero_apply, add_zero]
 
 end IsProj
 
diff --git a/Mathlib/LinearAlgebra/Projectivization/Basic.lean b/Mathlib/LinearAlgebra/Projectivization/Basic.lean
index fd5d3d6aa6db7..94d4e7925bb43 100644
--- a/Mathlib/LinearAlgebra/Projectivization/Basic.lean
+++ b/Mathlib/LinearAlgebra/Projectivization/Basic.lean
@@ -78,7 +78,7 @@ theorem rep_nonzero (v : ℙ K V) : v.rep ≠ 0 :=
 @[simp]
 theorem mk_rep (v : ℙ K V) : mk K v.rep v.rep_nonzero = v := Quotient.out_eq' _
 
-open FiniteDimensional
+open Module
 
 /-- Consider an element of the projectivization as a submodule of `V`. -/
 protected def submodule (v : ℙ K V) : Submodule K V :=
diff --git a/Mathlib/LinearAlgebra/Projectivization/Constructions.lean b/Mathlib/LinearAlgebra/Projectivization/Constructions.lean
new file mode 100644
index 0000000000000..fd7470292b540
--- /dev/null
+++ b/Mathlib/LinearAlgebra/Projectivization/Constructions.lean
@@ -0,0 +1,135 @@
+/-
+Copyright (c) 2024 Thomas Browning. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Thomas Browning
+-/
+import Mathlib.LinearAlgebra.CrossProduct
+import Mathlib.LinearAlgebra.Matrix.DotProduct
+import Mathlib.LinearAlgebra.Projectivization.Basic
+
+/-!
+
+# Dot Product and Cross Product on Projective Spaces
+
+This file defines the dot product and cross product on projective spaces.
+
+## Definitions
+- `Projectivization.orthogonal v w` is defined as vanishing of the dot product.
+- `Projectivization.cross v w` for `v w : ℙ F (Fin 3 → F)` is defined as the cross product of
+  `v` and `w` provided that `v ≠ w`. If `v = w`, then the cross product would be zero, so we
+  instead define `cross v v = v`.
+
+-/
+
+variable {F : Type*} [Field F] {m : Type*} [Fintype m]
+
+namespace Projectivization
+
+open scoped LinearAlgebra.Projectivization
+
+section DotProduct
+
+/-- Orthogonality on the projective plane. -/
+def orthogonal : ℙ F (m → F) → ℙ F (m → F) → Prop :=
+  Quotient.lift₂ (fun v w ↦ Matrix.dotProduct v.1 w.1 = 0) (fun _ _ _ _ ⟨_, h1⟩ ⟨_, h2⟩ ↦ by
+    simp_rw [← h1, ← h2, Matrix.dotProduct_smul, Matrix.smul_dotProduct, smul_smul,
+      smul_eq_zero_iff_eq])
+
+lemma orthogonal_mk {v w : m → F} (hv : v ≠ 0) (hw : w ≠ 0) :
+    orthogonal (mk F v hv) (mk F w hw) ↔ Matrix.dotProduct v w = 0 :=
+  Iff.rfl
+
+lemma orthogonal_comm {v w : ℙ F (m → F)} : orthogonal v w ↔ orthogonal w v := by
+  induction' v with v hv
+  induction' w with w hw
+  rw [orthogonal_mk hv hw, orthogonal_mk hw hv, Matrix.dotProduct_comm]
+
+lemma exists_not_self_orthogonal (v : ℙ F (m → F)) : ∃ w, ¬ orthogonal v w := by
+  induction' v with v hv
+  rw [ne_eq, ← Matrix.dotProduct_eq_zero_iff, not_forall] at hv
+  obtain ⟨w, hw⟩ := hv
+  exact ⟨mk F w fun h ↦ hw (by rw [h, Matrix.dotProduct_zero]), hw⟩
+
+lemma exists_not_orthogonal_self (v : ℙ F (m → F)) : ∃ w, ¬ orthogonal w v := by
+  simp only [orthogonal_comm]
+  exact exists_not_self_orthogonal v
+
+end DotProduct
+
+section CrossProduct
+
+lemma mk_eq_mk_iff_crossProduct_eq_zero {v w : Fin 3 → F} (hv : v ≠ 0) (hw : w ≠ 0) :
+    mk F v hv = mk F w hw ↔ crossProduct v w = 0 := by
+  rw [← not_iff_not, mk_eq_mk_iff', not_exists, ← LinearIndependent.pair_iff' hw,
+    ← crossProduct_ne_zero_iff_linearIndependent, ← cross_anticomm, neg_ne_zero]
+
+variable [DecidableEq F]
+
+/-- Cross product on the projective plane. -/
+def cross : ℙ F (Fin 3 → F) → ℙ F (Fin 3 → F) → ℙ F (Fin 3 → F) :=
+  Quotient.map₂' (fun v w ↦ if h : crossProduct v.1 w.1 = 0 then v else ⟨crossProduct v.1 w.1, h⟩)
+    (fun _ _ ⟨a, ha⟩ _ _ ⟨b, hb⟩ ↦ by
+      simp_rw [← ha, ← hb, LinearMap.map_smul_of_tower, LinearMap.smul_apply, smul_smul,
+        mul_comm b a, smul_eq_zero_iff_eq]
+      split_ifs
+      · use a
+      · use a * b)
+
+lemma cross_mk {v w : Fin 3 → F} (hv : v ≠ 0) (hw : w ≠ 0) :
+    cross (mk F v hv) (mk F w hw) =
+      if h : crossProduct v w = 0 then mk F v hv else mk F (crossProduct v w) h := by
+  change Quotient.mk'' _ = _
+  split_ifs with h <;> simp only [h] <;> rfl
+
+lemma cross_mk_of_cross_eq_zero {v w : Fin 3 → F} (hv : v ≠ 0) (hw : w ≠ 0)
+    (h : crossProduct v w = 0) :
+    cross (mk F v hv) (mk F w hw) = mk F v hv := by
+  rw [cross_mk, dif_pos h]
+
+lemma cross_mk_of_cross_ne_zero {v w : Fin 3 → F} (hv : v ≠ 0) (hw : w ≠ 0)
+    (h : crossProduct v w ≠ 0) :
+    cross (mk F v hv) (mk F w hw) = mk F (crossProduct v w) h := by
+  rw [cross_mk, dif_neg h]
+
+lemma cross_self (v : ℙ F (Fin 3 → F)) : cross v v = v := by
+  induction' v with v hv
+  rw [cross_mk_of_cross_eq_zero]
+  rw [← mk_eq_mk_iff_crossProduct_eq_zero hv]
+
+lemma cross_mk_of_ne {v w : Fin 3 → F} (hv : v ≠ 0) (hw : w ≠ 0) (h : mk F v hv ≠ mk F w hw) :
+    cross (mk F v hv) (mk F w hw) = mk F (crossProduct v w)
+      (mt (mk_eq_mk_iff_crossProduct_eq_zero hv hw).mpr h) := by
+  rw [cross_mk_of_cross_ne_zero]
+
+lemma cross_comm (v w : ℙ F (Fin 3 → F)) : cross v w = cross w v := by
+  rcases eq_or_ne v w with rfl | h
+  · rfl
+  · induction' v with v hv
+    induction' w with w hw
+    rw [cross_mk_of_ne hv hw h, cross_mk_of_ne hw hv h.symm, mk_eq_mk_iff_crossProduct_eq_zero,
+      ← cross_anticomm v w, map_neg, _root_.cross_self, neg_zero]
+
+theorem cross_orthogonal_left {v w : ℙ F (Fin 3 → F)} (h : v ≠ w) :
+    (cross v w).orthogonal v := by
+  induction' v with v hv
+  induction' w with w hw
+  rw [cross_mk_of_ne hv hw h, orthogonal_mk, Matrix.dotProduct_comm, dot_self_cross]
+
+theorem cross_orthogonal_right {v w : ℙ F (Fin 3 → F)} (h : v ≠ w) :
+    (cross v w).orthogonal w := by
+  rw [cross_comm]
+  exact cross_orthogonal_left h.symm
+
+theorem orthogonal_cross_left {v w : ℙ F (Fin 3 → F)} (h : v ≠ w) :
+    v.orthogonal (cross v w) := by
+  rw [orthogonal_comm]
+  exact cross_orthogonal_left h
+
+lemma orthogonal_cross_right {v w : ℙ F (Fin 3 → F)} (h : v ≠ w) :
+    w.orthogonal (cross v w) := by
+  rw [orthogonal_comm]
+  exact cross_orthogonal_right h
+
+end CrossProduct
+
+end Projectivization
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean b/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
index 0600057bbd9fc..fe2fe2b66dfe2 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
@@ -145,7 +145,6 @@ section QuadraticForm
 
 variable (R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M]
 
-variable (R M) in
 /-- A quadratic form on a module. -/
 abbrev QuadraticForm : Type _ := QuadraticMap R M R
 
@@ -855,15 +854,15 @@ theorem associated_toQuadraticMap (B : BilinMap R M R) (x y : M) :
 
 theorem associated_left_inverse (h : B₁.IsSymm) : associatedHom S B₁.toQuadraticMap = B₁ :=
   LinearMap.ext₂ fun x y => by
-    rw [associated_toQuadraticMap, ← h.eq x y, RingHom.id_apply, ← two_mul, ← smul_mul_assoc,
-      smul_eq_mul, invOf_mul_self, one_mul]
+    rw [associated_toQuadraticMap, ← h.eq x y, RingHom.id_apply]
+    match_scalars
+    linear_combination invOf_mul_self' (2:R)
 
 -- Porting note: moved from below to golf the next theorem
 theorem associated_eq_self_apply (x : M) : associatedHom S Q x x = Q x := by
-  rw [associated_apply, map_add_self, ← three_add_one_eq_four, ← two_add_one_eq_three, add_smul,
-    add_smul, one_smul, add_sub_cancel_right, add_sub_cancel_right, two_smul, ← two_smul R,
-    ← smul_assoc]
-  simp only [smul_eq_mul, invOf_mul_self', one_smul]
+  rw [associated_apply, map_add_self]
+  match_scalars
+  linear_combination invOf_mul_self' (2:R)
 
 theorem toQuadraticMap_associated : (associatedHom S Q).toQuadraticMap = Q :=
   QuadraticMap.ext <| associated_eq_self_apply S Q
@@ -1181,7 +1180,7 @@ theorem exists_bilinForm_self_ne_zero [htwo : Invertible (2 : R)] {B : BilinForm
   obtain ⟨x, hx⟩ := QuadraticMap.exists_quadraticForm_ne_zero hB₁
   exact ⟨x, fun h => hx (Q.associated_eq_self_apply ℕ x ▸ h)⟩
 
-open FiniteDimensional
+open Module
 
 variable {V : Type u} {K : Type v} [Field K] [AddCommGroup V] [Module K V]
 variable [FiniteDimensional K V]
@@ -1195,7 +1194,7 @@ theorem exists_orthogonal_basis [hK : Invertible (2 : K)] {B : LinearMap.BilinFo
   haveI := finrank_pos_iff.1 (hd.symm ▸ Nat.succ_pos d : 0 < finrank K V)
   -- either the bilinear form is trivial or we can pick a non-null `x`
   obtain rfl | hB₁ := eq_or_ne B 0
-  · let b := FiniteDimensional.finBasis K V
+  · let b := Module.finBasis K V
     rw [hd] at b
     exact ⟨b, fun i j _ => rfl⟩
   obtain ⟨x, hx⟩ := exists_bilinForm_self_ne_zero hB₁ hB₂
@@ -1289,8 +1288,7 @@ theorem basisRepr_eq_of_iIsOrtho {R M} [CommRing R] [AddCommGroup M] [Module R M
       smul_eq_mul, smul_eq_mul]
     ring_nf
   · intro i _ hij
-    rw [LinearMap.map_smul, LinearMap.map_smul₂,
-      show associatedHom R Q (v i) (v j) = 0 from hv₂ hij, smul_eq_mul, smul_eq_mul,
-      mul_zero, mul_zero]
+    rw [LinearMap.map_smul, LinearMap.map_smul₂, hv₂ hij]
+    module
 
 end QuadraticMap
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Basis.lean b/Mathlib/LinearAlgebra/QuadraticForm/Basis.lean
index 0c19cad735241..c2c99925d7bb1 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Basis.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Basis.lean
@@ -35,8 +35,8 @@ theorem toBilin_apply (Q : QuadraticMap R M N) (bm : Basis ι R M) (i j : ι) :
 theorem toQuadraticMap_toBilin (Q : QuadraticMap R M N) (bm : Basis ι R M) :
     (Q.toBilin bm).toQuadraticMap = Q := by
   ext x
-  rw [← bm.total_repr x, LinearMap.BilinMap.toQuadraticMap_apply, Finsupp.total_apply,
-    Finsupp.sum]
+  rw [← bm.linearCombination_repr x, LinearMap.BilinMap.toQuadraticMap_apply,
+      Finsupp.linearCombination_apply, Finsupp.sum]
   simp_rw [LinearMap.map_sum₂, map_sum, LinearMap.map_smul₂, _root_.map_smul, toBilin_apply,
     smul_ite, smul_zero, ← Finset.sum_product', ← Finset.diag_union_offDiag,
     Finset.sum_union (Finset.disjoint_diag_offDiag _), Finset.sum_diag, if_true]
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Complex.lean b/Mathlib/LinearAlgebra/QuadraticForm/Complex.lean
index 3285feacba1f0..8f1a7a9b3e50d 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Complex.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Complex.lean
@@ -70,7 +70,7 @@ noncomputable def isometryEquivSumSquaresUnits (w : ι → Units ℂ) :
 the sum of squares, i.e. `weightedSumSquares` with weight `fun (i : ι) => 1`. -/
 theorem equivalent_sum_squares {M : Type*} [AddCommGroup M] [Module ℂ M] [FiniteDimensional ℂ M]
     (Q : QuadraticForm ℂ M) (hQ : (associated (R := ℂ) Q).SeparatingLeft) :
-    Equivalent Q (weightedSumSquares ℂ (1 : Fin (FiniteDimensional.finrank ℂ M) → ℂ)) :=
+    Equivalent Q (weightedSumSquares ℂ (1 : Fin (Module.finrank ℂ M) → ℂ)) :=
   let ⟨w, ⟨hw₁⟩⟩ := Q.equivalent_weightedSumSquares_units_of_nondegenerate' hQ
   ⟨hw₁.trans (isometryEquivSumSquaresUnits w)⟩
 
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean b/Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean
index ff5c68946f2d8..567af287573fc 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/IsometryEquiv.lean
@@ -141,7 +141,7 @@ variable [Field K] [Invertible (2 : K)] [AddCommGroup V] [Module K V]
 /-- Given an orthogonal basis, a quadratic form is isometrically equivalent with a weighted sum of
 squares. -/
 noncomputable def isometryEquivWeightedSumSquares (Q : QuadraticForm K V)
-    (v : Basis (Fin (FiniteDimensional.finrank K V)) K V)
+    (v : Basis (Fin (Module.finrank K V)) K V)
     (hv₁ : (associated (R := K) Q).IsOrthoᵢ v) :
     Q.IsometryEquiv (weightedSumSquares K fun i => Q (v i)) := by
   let iso := Q.isometryEquivBasisRepr v
@@ -154,13 +154,13 @@ variable [FiniteDimensional K V]
 open LinearMap.BilinForm
 
 theorem equivalent_weightedSumSquares (Q : QuadraticForm K V) :
-    ∃ w : Fin (FiniteDimensional.finrank K V) → K, Equivalent Q (weightedSumSquares K w) :=
+    ∃ w : Fin (Module.finrank K V) → K, Equivalent Q (weightedSumSquares K w) :=
   let ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm _ Q)
   ⟨_, ⟨Q.isometryEquivWeightedSumSquares v hv₁⟩⟩
 
 theorem equivalent_weightedSumSquares_units_of_nondegenerate' (Q : QuadraticForm K V)
     (hQ : (associated (R := K) Q).SeparatingLeft) :
-    ∃ w : Fin (FiniteDimensional.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
+    ∃ w : Fin (Module.finrank K V) → Kˣ, Equivalent Q (weightedSumSquares K w) := by
   obtain ⟨v, hv₁⟩ := exists_orthogonal_basis (associated_isSymm K Q)
   have hv₂ := hv₁.not_isOrtho_basis_self_of_separatingLeft hQ
   simp_rw [LinearMap.IsOrtho, associated_eq_self_apply] at hv₂
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean b/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean
index f316c983c328a..c0d2eb6ac8a7a 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat.lean
@@ -16,7 +16,7 @@ universe v u
 
 variable (R : Type u) [CommRing R]
 
-/-- The category of quadratic modules; modules with an associated quadratic form-/
+/-- The category of quadratic modules; modules with an associated quadratic form -/
 structure QuadraticModuleCat extends ModuleCat.{v} R where
   /-- The quadratic form associated with the module. -/
   form : QuadraticForm R carrier
@@ -43,7 +43,7 @@ def of {X : Type v} [AddCommGroup X] [Module R X] (Q : QuadraticForm R X) :
 /-- A type alias for `QuadraticForm.LinearIsometry` to avoid confusion between the categorical and
 algebraic spellings of composition. -/
 @[ext]
-structure Hom (V W : QuadraticModuleCat.{v} R) :=
+structure Hom (V W : QuadraticModuleCat.{v} R) where
   /-- The underlying isometry -/
   toIsometry : V.form →qᵢ W.form
 
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Monoidal.lean b/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Monoidal.lean
index a63bb0629a1d4..f7b52334b7167 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Monoidal.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/QuadraticModuleCat/Monoidal.lean
@@ -86,22 +86,22 @@ noncomputable instance instMonoidalCategory : MonoidalCategory (QuadraticModuleC
       εIso := Iso.refl _
       leftUnitor_eq := fun X => by
         simp only [forget₂_obj, forget₂_map, Iso.refl_symm, Iso.trans_assoc, Iso.trans_hom,
-          Iso.refl_hom, tensorIso_hom, MonoidalCategory.tensorHom_id]
+          Iso.refl_hom, MonoidalCategory.tensorIso_hom, MonoidalCategory.tensorHom_id]
         dsimp only [toModuleCat_tensor, ModuleCat.of_coe]
         erw [MonoidalCategory.id_whiskerRight]
         simp
         rfl
       rightUnitor_eq := fun X => by
         simp only [forget₂_obj, forget₂_map, Iso.refl_symm, Iso.trans_assoc, Iso.trans_hom,
-          Iso.refl_hom, tensorIso_hom, MonoidalCategory.id_tensorHom]
+          Iso.refl_hom, MonoidalCategory.tensorIso_hom, MonoidalCategory.id_tensorHom]
         dsimp only [toModuleCat_tensor, ModuleCat.of_coe]
         erw [MonoidalCategory.whiskerLeft_id]
         simp
         rfl
       associator_eq := fun X Y Z => by
         dsimp only [forget₂_obj, forget₂_map_associator_hom]
-        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom, tensorIso_hom,
-          Iso.refl_hom, MonoidalCategory.tensor_id]
+        simp only [eqToIso_refl, Iso.refl_trans, Iso.refl_symm, Iso.trans_hom,
+          MonoidalCategory.tensorIso_hom, Iso.refl_hom, MonoidalCategory.tensor_id]
         dsimp only [toModuleCat_tensor, ModuleCat.of_coe]
         rw [Category.id_comp, Category.id_comp, Category.comp_id, MonoidalCategory.tensor_id,
           Category.id_comp] }
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/Real.lean b/Mathlib/LinearAlgebra/QuadraticForm/Real.lean
index 7ec3a6f4947d8..b9fe9bdf90ae3 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/Real.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/Real.lean
@@ -53,7 +53,7 @@ noncomputable def isometryEquivSignWeightedSumSquares (w : ι → ℝ) :
 sum of squares with the weights being ±1, `SignType` version. -/
 theorem equivalent_sign_ne_zero_weighted_sum_squared {M : Type*} [AddCommGroup M] [Module ℝ M]
     [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) (hQ : (associated (R := ℝ) Q).SeparatingLeft) :
-    ∃ w : Fin (FiniteDimensional.finrank ℝ M) → SignType,
+    ∃ w : Fin (Module.finrank ℝ M) → SignType,
       (∀ i, w i ≠ 0) ∧ Equivalent Q (weightedSumSquares ℝ fun i ↦ (w i : ℝ)) :=
   let ⟨w, ⟨hw₁⟩⟩ := Q.equivalent_weightedSumSquares_units_of_nondegenerate' hQ
   ⟨sign ∘ ((↑) : ℝˣ → ℝ) ∘ w, fun i => sign_ne_zero.2 (w i).ne_zero,
@@ -63,7 +63,7 @@ theorem equivalent_sign_ne_zero_weighted_sum_squared {M : Type*} [AddCommGroup M
 sum of squares with the weights being ±1. -/
 theorem equivalent_one_neg_one_weighted_sum_squared {M : Type*} [AddCommGroup M] [Module ℝ M]
     [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) (hQ : (associated (R := ℝ) Q).SeparatingLeft) :
-    ∃ w : Fin (FiniteDimensional.finrank ℝ M) → ℝ,
+    ∃ w : Fin (Module.finrank ℝ M) → ℝ,
       (∀ i, w i = -1 ∨ w i = 1) ∧ Equivalent Q (weightedSumSquares ℝ w) :=
   let ⟨w, hw₀, hw⟩ := Q.equivalent_sign_ne_zero_weighted_sum_squared hQ
   ⟨(w ·), fun i ↦ by cases hi : w i <;> simp_all, hw⟩
@@ -72,7 +72,7 @@ theorem equivalent_one_neg_one_weighted_sum_squared {M : Type*} [AddCommGroup M]
 sum of squares with the weights being ±1 or 0, `SignType` version. -/
 theorem equivalent_signType_weighted_sum_squared {M : Type*} [AddCommGroup M] [Module ℝ M]
     [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) :
-    ∃ w : Fin (FiniteDimensional.finrank ℝ M) → SignType,
+    ∃ w : Fin (Module.finrank ℝ M) → SignType,
       Equivalent Q (weightedSumSquares ℝ fun i ↦ (w i : ℝ)) :=
   let ⟨w, ⟨hw₁⟩⟩ := Q.equivalent_weightedSumSquares
   ⟨sign ∘ w, ⟨hw₁.trans (isometryEquivSignWeightedSumSquares w)⟩⟩
@@ -81,7 +81,7 @@ theorem equivalent_signType_weighted_sum_squared {M : Type*} [AddCommGroup M] [M
 sum of squares with the weights being ±1 or 0. -/
 theorem equivalent_one_zero_neg_one_weighted_sum_squared {M : Type*} [AddCommGroup M] [Module ℝ M]
     [FiniteDimensional ℝ M] (Q : QuadraticForm ℝ M) :
-    ∃ w : Fin (FiniteDimensional.finrank ℝ M) → ℝ,
+    ∃ w : Fin (Module.finrank ℝ M) → ℝ,
       (∀ i, w i = -1 ∨ w i = 0 ∨ w i = 1) ∧ Equivalent Q (weightedSumSquares ℝ w) :=
   let ⟨w, hw⟩ := Q.equivalent_signType_weighted_sum_squared
   ⟨(w ·), fun i ↦ by cases h : w i <;> simp [h], hw⟩
diff --git a/Mathlib/LinearAlgebra/Quotient.lean b/Mathlib/LinearAlgebra/Quotient.lean
index 071665a1e5ff9..ece10805d5505 100644
--- a/Mathlib/LinearAlgebra/Quotient.lean
+++ b/Mathlib/LinearAlgebra/Quotient.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
 -/
 import Mathlib.LinearAlgebra.Span
+import Mathlib.LinearAlgebra.Pi
 import Mathlib.Algebra.Module.Equiv.Basic
 import Mathlib.GroupTheory.QuotientGroup.Basic
 import Mathlib.SetTheory.Cardinal.Finite
@@ -284,9 +285,14 @@ def mkQ : M →ₗ[R] M ⧸ p where
 theorem mkQ_apply (x : M) : p.mkQ x = (Quotient.mk x : M ⧸ p) :=
   rfl
 
-theorem mkQ_surjective (A : Submodule R M) : Function.Surjective A.mkQ := by
+theorem mkQ_surjective : Function.Surjective p.mkQ := by
   rintro ⟨x⟩; exact ⟨x, rfl⟩
 
+theorem strictMono_comap_prod_map :
+    StrictMono fun m : Submodule R M ↦ (m.comap p.subtype, m.map p.mkQ) :=
+  fun m₁ m₂ ↦ QuotientAddGroup.strictMono_comap_prod_map
+    p.toAddSubgroup (a := m₁.toAddSubgroup) (b := m₂.toAddSubgroup)
+
 end
 
 variable {R₂ M₂ : Type*} [Ring R₂] [AddCommGroup M₂] [Module R₂ M₂] {τ₁₂ : R →+* R₂}
@@ -312,6 +318,14 @@ theorem liftQ_apply (f : M →ₛₗ[τ₁₂] M₂) {h} (x : M) : p.liftQ f h (
 @[simp]
 theorem liftQ_mkQ (f : M →ₛₗ[τ₁₂] M₂) (h) : (p.liftQ f h).comp p.mkQ = f := by ext; rfl
 
+theorem pi_liftQ_eq_liftQ_pi {ι : Type*} {N : ι → Type*}
+    [∀ i, AddCommGroup (N i)] [∀ i, Module R (N i)]
+    (f : (i : ι) → M →ₗ[R] (N i)) {p : Submodule R M} (h : ∀ i, p ≤ ker (f i)) :
+    LinearMap.pi (fun i ↦ p.liftQ (f i) (h i)) =
+      p.liftQ (LinearMap.pi f) (LinearMap.ker_pi f ▸ le_iInf h) := by
+  ext x i
+  simp
+
 /-- Special case of `submodule.liftQ` when `p` is the span of `x`. In this case, the condition on
 `f` simply becomes vanishing at `x`. -/
 def liftQSpanSingleton (x : M) (f : M →ₛₗ[τ₁₂] M₂) (h : f x = 0) : (M ⧸ R ∙ x) →ₛₗ[τ₁₂] M₂ :=
diff --git a/Mathlib/LinearAlgebra/Ray.lean b/Mathlib/LinearAlgebra/Ray.lean
index 02d99609f467d..547a7c8540f4b 100644
--- a/Mathlib/LinearAlgebra/Ray.lean
+++ b/Mathlib/LinearAlgebra/Ray.lean
@@ -3,10 +3,10 @@ Copyright (c) 2021 Joseph Myers. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joseph Myers
 -/
-import Mathlib.Algebra.Group.Subgroup.Actions
 import Mathlib.Algebra.Order.Module.Algebra
 import Mathlib.LinearAlgebra.LinearIndependent
 import Mathlib.Algebra.Ring.Subring.Units
+import Mathlib.Tactic.Positivity
 
 /-!
 # Rays in modules
@@ -107,7 +107,7 @@ lemma sameRay_nonneg_smul_right (v : M) (h : 0 ≤ a) : SameRay R v (a • v) :=
   · rw [← algebraMap_smul R a v, h, zero_smul]
     exact zero_right _
   · refine Or.inr <| Or.inr ⟨algebraMap S R a, 1, h, by nontriviality R; exact zero_lt_one, ?_⟩
-    rw [algebraMap_smul, one_smul]
+    module
 
 /-- A nonnegative multiple of a vector is in the same ray as that vector. -/
 lemma sameRay_nonneg_smul_left (v : M) (ha : 0 ≤ a) : SameRay R (a • v) v :=
@@ -171,9 +171,8 @@ theorem add_left (hx : SameRay R x z) (hy : SameRay R y z) : SameRay R (x + y) z
   rcases hx.exists_pos hx₀ hz₀ with ⟨rx, rz₁, hrx, hrz₁, Hx⟩
   rcases hy.exists_pos hy₀ hz₀ with ⟨ry, rz₂, hry, hrz₂, Hy⟩
   refine Or.inr (Or.inr ⟨rx * ry, ry * rz₁ + rx * rz₂, mul_pos hrx hry, ?_, ?_⟩)
-  · apply_rules [add_pos, mul_pos]
-  · simp only [mul_smul, smul_add, add_smul, ← Hx, ← Hy]
-    rw [smul_comm]
+  · positivity
+  · convert congr(ry • $Hx + rx • $Hy) using 1 <;> module
 
 /-- If `y` and `z` are on the same ray as `x`, then so is `y + z`. -/
 theorem add_right (hy : SameRay R x y) (hz : SameRay R x z) : SameRay R x (y + z) :=
@@ -467,7 +466,7 @@ theorem sameRay_smul_right_iff {v : M} {r : R} : SameRay R v (r • v) ↔ 0 ≤
 is positive. -/
 theorem sameRay_smul_right_iff_of_ne {v : M} (hv : v ≠ 0) {r : R} (hr : r ≠ 0) :
     SameRay R v (r • v) ↔ 0 < r := by
-  simp only [sameRay_smul_right_iff, hv, or_false_iff, hr.symm.le_iff_lt]
+  simp only [sameRay_smul_right_iff, hv, or_false, hr.symm.le_iff_lt]
 
 @[simp]
 theorem sameRay_smul_left_iff {v : M} {r : R} : SameRay R (r • v) v ↔ 0 ≤ r ∨ v = 0 :=
@@ -485,7 +484,7 @@ theorem sameRay_neg_smul_right_iff {v : M} {r : R} : SameRay R (-v) (r • v) 
 
 theorem sameRay_neg_smul_right_iff_of_ne {v : M} {r : R} (hv : v ≠ 0) (hr : r ≠ 0) :
     SameRay R (-v) (r • v) ↔ r < 0 := by
-  simp only [sameRay_neg_smul_right_iff, hv, or_false_iff, hr.le_iff_lt]
+  simp only [sameRay_neg_smul_right_iff, hv, or_false, hr.le_iff_lt]
 
 @[simp]
 theorem sameRay_neg_smul_left_iff {v : M} {r : R} : SameRay R (r • v) (-v) ↔ r ≤ 0 ∨ v = 0 :=
@@ -532,11 +531,11 @@ theorem sameRay_or_sameRay_neg_iff_not_linearIndependent {x y : M} :
       rcases lt_trichotomy (m 1) 0 with (hm1 | hm1 | hm1)
     · refine
         Or.inr (Or.inr (Or.inr ⟨-m 0, -m 1, Left.neg_pos_iff.2 hm0, Left.neg_pos_iff.2 hm1, ?_⟩))
-      rw [neg_smul_neg, neg_smul, hm, neg_neg]
+      linear_combination (norm := module) -hm
     · exfalso
       simp [hm1, hx, hm0.ne] at hm
     · refine Or.inl (Or.inr (Or.inr ⟨-m 0, m 1, Left.neg_pos_iff.2 hm0, hm1, ?_⟩))
-      rw [neg_smul, hm, neg_neg]
+      linear_combination (norm := module) -hm
     · exfalso
       simp [hm0, hy, hm1.ne] at hm
     · rw [Fin.exists_fin_two] at hmne
diff --git a/Mathlib/LinearAlgebra/Reflection.lean b/Mathlib/LinearAlgebra/Reflection.lean
index e2f754a64ed62..d807bcfeec42e 100644
--- a/Mathlib/LinearAlgebra/Reflection.lean
+++ b/Mathlib/LinearAlgebra/Reflection.lean
@@ -70,7 +70,7 @@ lemma preReflection_apply_self (h : f x = 2) :
 
 lemma involutive_preReflection (h : f x = 2) :
     Involutive (preReflection x f) :=
-  fun y ↦ by simp [h, smul_sub, two_smul, preReflection_apply]
+  fun y ↦ by simp [map_sub, h, smul_sub, two_smul, preReflection_apply]
 
 lemma preReflection_preReflection (g : Dual R M) (h : f x = 2) :
     preReflection (preReflection x f y) (preReflection f (Dual.eval R M x) g) =
@@ -161,13 +161,19 @@ lemma Dual.eq_of_preReflection_mapsTo' [CharZero R] [NoZeroSMulDivisors R M]
   set Φ' : Set (span R Φ) := range (inclusion <| Submodule.subset_span (R := R) (s := Φ))
   rw [← finite_coe_iff] at hΦ₁
   have hΦ'₁ : Φ'.Finite := finite_range (inclusion Submodule.subset_span)
-  have hΦ'₂ : span R Φ' = ⊤ := by simp [Φ']
+  have hΦ'₂ : span R Φ' = ⊤ := by
+    simp only [Φ']
+    rw [range_inclusion]
+    simp
   let x' : span R Φ := ⟨x, hx'⟩
   have hx' : x' ≠ 0 := Subtype.coe_ne_coe.1 hx
   have this : ∀ {F : Dual R M}, MapsTo (preReflection x F) Φ Φ →
       MapsTo (preReflection x' ((span R Φ).subtype.dualMap F)) Φ' Φ' := by
     intro F hF ⟨y, hy⟩ hy'
-    simp only [Φ', range_inclusion, SetLike.coe_sort_coe, mem_setOf_eq] at hy' ⊢
+    simp only [Φ'] at hy' ⊢
+    rw [range_inclusion] at hy'
+    simp only [SetLike.coe_sort_coe, mem_setOf_eq] at hy' ⊢
+    rw [range_inclusion]
     exact hF hy'
   exact eq_of_preReflection_mapsTo hx' hΦ'₁ hΦ'₂ hf₁ (this hf₂) hg₁ (this hg₂)
 
diff --git a/Mathlib/LinearAlgebra/RootSystem/Basic.lean b/Mathlib/LinearAlgebra/RootSystem/Basic.lean
index 8ca70db9d0e55..4aba11c944d5e 100644
--- a/Mathlib/LinearAlgebra/RootSystem/Basic.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/Basic.lean
@@ -48,7 +48,7 @@ variable (p : PerfectPairing R M N) (root : ι ↪ M) (coroot : ι ↪ N) (i j :
 include h
 
 private theorem exist_eq_reflection_of_mapsTo  :
-    ∃ k, root k = (preReflection (root i) (p.toLin.flip (coroot i))) (root j) :=
+    ∃ k, root k = (preReflection (root i) (p.flip (coroot i))) (root j) :=
   h i (mem_range_self j)
 
 variable (hp : ∀ i, p.toLin (root i) (coroot i) = 2)
@@ -112,7 +112,7 @@ Formally, the point is that the hypothesis `hc` depends only on the range of the
 @[ext]
 protected lemma ext [CharZero R] [NoZeroSMulDivisors R M]
     {P₁ P₂ : RootPairing ι R M N}
-    (he : P₁.toLin = P₂.toLin)
+    (he : P₁.toPerfectPairing = P₂.toPerfectPairing)
     (hr : P₁.root = P₂.root)
     (hc : range P₁.coroot = range P₂.coroot) :
     P₁ = P₂ := by
@@ -120,8 +120,8 @@ protected lemma ext [CharZero R] [NoZeroSMulDivisors R M]
     ext i j
     refine P₁.root.injective ?_
     conv_rhs => rw [hr]
-    rw [root_reflection_perm, root_reflection_perm]
-    simp only [hr, he, hc', reflection_apply]
+    simp only [root_reflection_perm, reflection_apply, coroot']
+    simp only [hr, he, hc']
   suffices P₁.coroot = P₂.coroot by
     cases' P₁ with p₁; cases' P₂ with p₂; cases p₁; cases p₂; congr; exact hp this
   have := NoZeroSMulDivisors.int_of_charZero R M
@@ -139,34 +139,35 @@ private lemma coroot_eq_coreflection_of_root_eq' [CharZero R] [NoZeroSMulDivisor
     (p : PerfectPairing R M N)
     (root : ι ↪ M)
     (coroot : ι ↪ N)
-    (hp : ∀ i, p.toLin (root i) (coroot i) = 2)
-    (hr : ∀ i, MapsTo (preReflection (root i) (p.toLin.flip (coroot i))) (range root) (range root))
-    (hc : ∀ i, MapsTo (preReflection (coroot i) (p.toLin (root i))) (range coroot) (range coroot))
-    {i j k : ι} (hk : root k = preReflection (root i) (p.toLin.flip (coroot i)) (root j)) :
-    coroot k = preReflection (coroot i) (p.toLin (root i)) (coroot j) := by
+    (hp : ∀ i, p (root i) (coroot i) = 2)
+    (hr : ∀ i, MapsTo (preReflection (root i) (p.flip (coroot i))) (range root) (range root))
+    (hc : ∀ i, MapsTo (preReflection (coroot i) (p (root i))) (range coroot) (range coroot))
+    {i j k : ι} (hk : root k = preReflection (root i) (p.flip (coroot i)) (root j)) :
+    coroot k = preReflection (coroot i) (p (root i)) (coroot j) := by
   set α := root i
   set β := root j
   set α' := coroot i
   set β' := coroot j
-  set sα := preReflection α (p.toLin.flip α')
-  set sβ := preReflection β (p.toLin.flip β')
-  let sα' := preReflection α' (p.toLin α)
-  have hij : preReflection (sα β) (p.toLin.flip (sα' β')) = sα ∘ₗ sβ ∘ₗ sα := by
+  set sα := preReflection α (p.flip α')
+  set sβ := preReflection β (p.flip β')
+  let sα' := preReflection α' (p α)
+  have hij : preReflection (sα β) (p.flip (sα' β')) = sα ∘ₗ sβ ∘ₗ sα := by
     ext
-    have hpi : (p.toLin.flip (coroot i)) (root i) = 2 := by rw [LinearMap.flip_apply, hp i]
-    simp [α, β, α', β', sα, sβ, sα', ← preReflection_preReflection β (p.toLin.flip β') hpi,
+    have hpi : (p.flip (coroot i)) (root i) = 2 := by rw [PerfectPairing.flip_apply_apply, hp i]
+    simp [α, β, α', β', sα, sβ, sα', ← preReflection_preReflection β (p.flip β') hpi,
       preReflection_apply]
-  have hk₀ : root k ≠ 0 := fun h ↦ by simpa [h] using hp k
+  have hk₀ : root k ≠ 0 := fun h ↦ by simpa [h, ← PerfectPairing.toLin_apply] using hp k
   obtain ⟨l, hl⟩ := hc i (mem_range_self j)
   rw [← hl]
-  have hkl : (p.toLin.flip (coroot l)) (root k) = 2 := by
-    simp [hl, hk, preReflection_apply, mul_sub, mul_two, β', α, α', β, sα, hp i, hp j]
-    rw [mul_comm (p.toLin (root i) (coroot j))]
+  have hkl : (p.flip (coroot l)) (root k) = 2 := by
+    simp only [hl, preReflection_apply, hk, PerfectPairing.flip_apply_apply, map_sub, hp j,
+      map_smul, smul_eq_mul, hp i, mul_sub, sα, α, α', β, mul_two, mul_add]
+    rw [mul_comm (p (root i) (coroot j))]
     abel
-  suffices p.toLin.flip (coroot k) = p.toLin.flip (coroot l) from p.bijectiveRight.injective this
+  suffices p.flip (coroot k) = p.flip (coroot l) from p.bijectiveRight.injective this
   have _i : NoZeroSMulDivisors ℤ M := NoZeroSMulDivisors.int_of_charZero R M
   have := injOn_dualMap_subtype_span_range_range (finite_range root)
-    (c := p.toLin.flip ∘ coroot) hp hr
+    (c := p.flip ∘ coroot) hp hr
   apply this (mem_range_self k) (mem_range_self l)
   refine Dual.eq_of_preReflection_mapsTo' hk₀ (finite_range root)
     (Submodule.subset_span <| mem_range_self k) (hp k) (hr k) hkl ?_
@@ -179,9 +180,9 @@ def mk' [Finite ι] [CharZero R] [NoZeroSMulDivisors R M]
     (p : PerfectPairing R M N)
     (root : ι ↪ M)
     (coroot : ι ↪ N)
-    (hp : ∀ i, p.toLin (root i) (coroot i) = 2)
-    (hr : ∀ i, MapsTo (preReflection (root i) (p.toLin.flip (coroot i))) (range root) (range root))
-    (hc : ∀ i, MapsTo (preReflection (coroot i) (p.toLin (root i))) (range coroot) (range coroot)) :
+    (hp : ∀ i, p (root i) (coroot i) = 2)
+    (hr : ∀ i, MapsTo (preReflection (root i) (p.flip (coroot i))) (range root) (range root))
+    (hc : ∀ i, MapsTo (preReflection (coroot i) (p (root i))) (range coroot) (range coroot)) :
     RootPairing ι R M N where
   toPerfectPairing := p
   root := root
@@ -190,7 +191,7 @@ def mk' [Finite ι] [CharZero R] [NoZeroSMulDivisors R M]
   reflection_perm i := RootPairing.equiv_of_mapsTo p root coroot i hr hp
   reflection_perm_root i j := by
     rw [equiv_of_mapsTo_apply, (exist_eq_reflection_of_mapsTo p root coroot i j hr).choose_spec,
-      preReflection_apply, LinearMap.flip_apply]
+      preReflection_apply, PerfectPairing.flip_apply_apply]
   reflection_perm_coroot i j := by
     refine (coroot_eq_coreflection_of_root_eq' p root coroot hp hr hc ?_).symm
     rw [equiv_of_mapsTo_apply, (exist_eq_reflection_of_mapsTo p root coroot i j hr).choose_spec]
@@ -208,11 +209,11 @@ its roots. -/
 @[ext]
 protected lemma ext [CharZero R] [NoZeroSMulDivisors R M]
     {P₁ P₂ : RootSystem ι R M N}
-    (he : P₁.toLin = P₂.toLin)
+    (he : P₁.toPerfectPairing = P₂.toPerfectPairing)
     (hr : P₁.root = P₂.root) :
     P₁ = P₂ := by
-  suffices ∀ P₁ P₂ : RootSystem ι R M N, P₁.toLin = P₂.toLin → P₁.root = P₂.root →
-      range P₁.coroot ⊆ range P₂.coroot by
+  suffices ∀ P₁ P₂ : RootSystem ι R M N, P₁.toPerfectPairing = P₂.toPerfectPairing →
+      P₁.root = P₂.root → range P₁.coroot ⊆ range P₂.coroot by
     have h₁ := this P₁ P₂ he hr
     have h₂ := this P₂ P₁ he.symm hr.symm
     cases' P₁ with P₁
@@ -233,28 +234,28 @@ private lemma coroot_eq_coreflection_of_root_eq_of_span_eq_top [CharZero R] [NoZ
     (p : PerfectPairing R M N)
     (root : ι ↪ M)
     (coroot : ι ↪ N)
-    (hp : ∀ i, p.toLin (root i) (coroot i) = 2)
-    (hs : ∀ i, MapsTo (preReflection (root i) (p.toLin.flip (coroot i))) (range root) (range root))
+    (hp : ∀ i, p (root i) (coroot i) = 2)
+    (hs : ∀ i, MapsTo (preReflection (root i) (p.flip (coroot i))) (range root) (range root))
     (hsp : span R (range root) = ⊤)
-    {i j k : ι} (hk : root k = preReflection (root i) (p.toLin.flip (coroot i)) (root j)) :
-    coroot k = preReflection (coroot i) (p.toLin (root i)) (coroot j) := by
+    {i j k : ι} (hk : root k = preReflection (root i) (p.flip (coroot i)) (root j)) :
+    coroot k = preReflection (coroot i) (p (root i)) (coroot j) := by
   set α := root i
   set β := root j
   set α' := coroot i
   set β' := coroot j
-  set sα := preReflection α (p.toLin.flip α')
-  set sβ := preReflection β (p.toLin.flip β')
-  let sα' := preReflection α' (p.toLin α)
+  set sα := preReflection α (p.flip α')
+  set sβ := preReflection β (p.flip β')
+  let sα' := preReflection α' (p α)
   have hij : preReflection (sα β) (p.toLin.flip (sα' β')) = sα ∘ₗ sβ ∘ₗ sα := by
     ext
-    have hpi : (p.toLin.flip (coroot i)) (root i) = 2 := by rw [LinearMap.flip_apply, hp i]
-    simp [α, β, α', β', sα, sβ, sα', ← preReflection_preReflection β (p.toLin.flip β') hpi,
+    have hpi : (p.flip (coroot i)) (root i) = 2 := by rw [PerfectPairing.flip_apply_apply, hp i]
+    simp [α, β, α', β', sα, sβ, sα', ← preReflection_preReflection β (p.flip β') hpi,
       preReflection_apply] -- v4.7.0-rc1 issues
-  have hk₀ : root k ≠ 0 := fun h ↦ by simpa [h] using hp k
+  have hk₀ : root k ≠ 0 := fun h ↦ by simpa [h, ← PerfectPairing.toLin_apply] using hp k
   apply p.bijectiveRight.injective
   apply Dual.eq_of_preReflection_mapsTo hk₀ (finite_range root) hsp (hp k) (hs k)
-  · simp [α, β, α', β', sα, sβ, sα', hk, preReflection_apply, hp i, hp j, mul_two,
-      mul_comm (p.toLin α β')]
+  · simp [map_sub, α, β, α', β', sα, sβ, sα', hk, preReflection_apply, hp i, hp j, mul_two,
+      mul_comm (p α β')]
     ring -- v4.7.0-rc1 issues
   · rw [hk, hij]
     exact (hs i).comp <| (hs j).comp (hs i)
diff --git a/Mathlib/LinearAlgebra/RootSystem/Defs.lean b/Mathlib/LinearAlgebra/RootSystem/Defs.lean
index b375be2df788c..eac40333d52fc 100644
--- a/Mathlib/LinearAlgebra/RootSystem/Defs.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/Defs.lean
@@ -32,7 +32,6 @@ This file contains basic definitions for root systems and root data.
 ## TODO
 
  * Base change of root pairings (may need flatness; perhaps should go in a different file).
- * Isomorphism of root pairings.
  * Crystallographic root systems are isomorphic to base changes of root systems over `ℤ`: Take
    `M₀` and `N₀` to be the `ℤ`-span of roots and coroots.
 
@@ -51,7 +50,6 @@ between roots and coroots is (implicitly) included and the coroots are included
 Empirically this seems to be by far the most convenient design and by providing extensionality
 lemmas expressing the uniqueness we expect to get the best of both worlds.
 
-
 Furthermore, we require roots and coroots to be injections from a base indexing type `ι` rather than
 subsets of their codomains. This design was chosen to avoid the bijection between roots and coroots
 being a dependently-typed object. A third option would be to have the roots and coroots be subsets
@@ -88,7 +86,7 @@ evaluates to `2`, and the permutation attached to each element of `ι` is compat
 reflections on the corresponding roots and coroots.
 
 It exists to allow for a convenient unification of the theories of root systems and root data. -/
-structure RootPairing extends PerfectPairing R M N :=
+structure RootPairing extends PerfectPairing R M N where
   /-- A parametrized family of vectors, called roots. -/
   root : ι ↪ M
   /-- A parametrized family of dual vectors, called coroots. -/
@@ -97,9 +95,9 @@ structure RootPairing extends PerfectPairing R M N :=
   /-- A parametrized family of permutations, induced by reflection. -/
   reflection_perm : ι → (ι ≃ ι)
   reflection_perm_root : ∀ i j,
-    root j - toLin (root j) (coroot i) • root i = root (reflection_perm i j)
+    root j - toPerfectPairing (root j) (coroot i) • root i = root (reflection_perm i j)
   reflection_perm_coroot : ∀ i j,
-    coroot j - toLin (root i) (coroot j) • coroot i = coroot (reflection_perm i j)
+    coroot j - toPerfectPairing (root i) (coroot j) • coroot i = coroot (reflection_perm i j)
 
 /-- A root datum is a root pairing with coefficients in the integers and for which the root and
 coroot spaces are finitely-generated free Abelian groups.
@@ -112,7 +110,7 @@ abbrev RootDatum (X₁ X₂ : Type*) [AddCommGroup X₁] [AddCommGroup X₂] :=
 
 Note that this is slightly more general than the usual definition in the sense that `N` is not
 required to be the dual of `M`. -/
-structure RootSystem extends RootPairing ι R M N :=
+structure RootSystem extends RootPairing ι R M N where
   span_eq_top : span R (range root) = ⊤
 
 attribute [simp] RootSystem.span_eq_top
@@ -123,11 +121,15 @@ variable {ι R M N}
 variable (P : RootPairing ι R M N) (i j : ι)
 
 lemma ne_zero [CharZero R] : (P.root i : M) ≠ 0 :=
-  fun h ↦ by simpa [h] using P.root_coroot_two i
+  fun h ↦ by simpa [h, map_zero] using P.root_coroot_two i
 
 lemma ne_zero' [CharZero R] : (P.coroot i : N) ≠ 0 :=
   fun h ↦ by simpa [h] using P.root_coroot_two i
 
+@[simp]
+lemma toLin_toPerfectPairing (x : M) (y : N) : P.toLin x y = P.toPerfectPairing x y :=
+  rfl
+
 /-- If we interchange the roles of `M` and `N`, we still have a root pairing. -/
 protected def flip : RootPairing ι R N M :=
   { P.toPerfectPairing.flip with
@@ -142,11 +144,25 @@ protected def flip : RootPairing ι R N M :=
 lemma flip_flip : P.flip.flip = P :=
   rfl
 
+/-- Roots written as functionals on the coweight space. -/
+abbrev root' (i : ι) : Dual R N := P.toPerfectPairing (P.root i)
+
+/-- Coroots written as functionals on the weight space. -/
+abbrev coroot' (i : ι) : Dual R M := P.toPerfectPairing.flip (P.coroot i)
+
 /-- This is the pairing between roots and coroots. -/
-def pairing : R := P.toLin (P.root i) (P.coroot j)
+def pairing : R := P.root' i (P.coroot j)
 
 @[simp]
-lemma root_coroot_eq_pairing : P.toLin (P.root i) (P.coroot j) = P.pairing i j :=
+lemma root_coroot_eq_pairing : P.toPerfectPairing (P.root i) (P.coroot j) = P.pairing i j :=
+  rfl
+
+@[simp]
+lemma root'_coroot_eq_pairing : P.root' i (P.coroot j) = P.pairing i j :=
+  rfl
+
+@[simp]
+lemma root_coroot'_eq_pairing : P.coroot' i (P.root j) = P.pairing j i :=
   rfl
 
 lemma coroot_root_eq_pairing : P.toLin.flip (P.coroot i) (P.root j) = P.pairing j i := by
@@ -174,7 +190,7 @@ theorem mapsTo_reflection_root :
   exact P.root_reflection_perm i j ▸ mem_range_self (P.reflection_perm i j)
 
 lemma reflection_apply (x : M) :
-    P.reflection i x = x - (P.toLin x (P.coroot i)) • P.root i :=
+    P.reflection i x = x - (P.coroot' i x) • P.root i :=
   rfl
 
 lemma reflection_apply_root :
@@ -205,7 +221,7 @@ lemma reflection_sq :
 lemma reflection_perm_sq :
     P.reflection_perm i ^ 2 = 1 := by
   ext j
-  refine Embedding.injective P.root ?_
+  apply P.root.injective
   simp only [sq, Equiv.Perm.mul_apply, root_reflection_perm, reflection_same, Equiv.Perm.one_apply]
 
 @[simp]
@@ -213,6 +229,18 @@ lemma reflection_perm_inv :
     (P.reflection_perm i)⁻¹ = P.reflection_perm i :=
   (mul_eq_one_iff_eq_inv.mp <| P.reflection_perm_sq i).symm
 
+@[simp]
+lemma reflection_perm_self : P.reflection_perm i (P.reflection_perm i j) = j := by
+  apply P.root.injective
+  simp only [root_reflection_perm, reflection_same]
+
+lemma reflection_perm_involutive : Involutive (P.reflection_perm i) :=
+  involutive_iff_iter_2_eq_id.mpr (by ext; simp)
+
+@[simp]
+lemma reflection_perm_symm : (P.reflection_perm i).symm = P.reflection_perm i :=
+  Involutive.symm_eq_self_of_involutive (P.reflection_perm i) <| P.reflection_perm_involutive i
+
 lemma bijOn_reflection_root :
     BijOn (P.reflection i) (range P.root) (range P.root) :=
   Module.bijOn_reflection_of_mapsTo _ <| P.mapsTo_reflection_root i
@@ -237,7 +265,7 @@ theorem mapsTo_coreflection_coroot :
   exact P.coroot_reflection_perm i j ▸ mem_range_self (P.reflection_perm i j)
 
 lemma coreflection_apply (f : N) :
-    P.coreflection i f = f - (P.toLin (P.root i) f) • P.coroot i :=
+    P.coreflection i f = f - (P.root' i) f • P.coroot i :=
   rfl
 
 lemma coreflection_apply_coroot :
@@ -279,7 +307,7 @@ lemma coreflection_eq_flip_reflection :
 lemma reflection_dualMap_eq_coreflection :
     (P.reflection i).dualMap ∘ₗ P.toLin.flip = P.toLin.flip ∘ₗ P.coreflection i := by
   ext n m
-  simp [coreflection_apply, reflection_apply, mul_comm (P.toLin m (P.coroot i))]
+  simp [map_sub, coreflection_apply, reflection_apply, mul_comm (P.toPerfectPairing m (P.coroot i))]
 
 lemma coroot_eq_coreflection_of_root_eq
     {i j k : ι} (hk : P.root k = P.reflection i (P.root j)) :
@@ -287,10 +315,42 @@ lemma coroot_eq_coreflection_of_root_eq
   rw [← P.root_reflection_perm, EmbeddingLike.apply_eq_iff_eq] at hk
   rw [← P.coroot_reflection_perm, hk]
 
+lemma coroot'_reflection_perm {i j : ι} :
+    P.coroot' (P.reflection_perm i j) = P.coroot' j ∘ₗ P.reflection i := by
+  ext y
+  simp [coreflection_apply_coroot, reflection_apply, map_sub, mul_comm]
+
+lemma coroot'_reflection {i j : ι} (y : M) :
+    P.coroot' j (P.reflection i y) = P.coroot' (P.reflection_perm i j) y :=
+  (LinearMap.congr_fun P.coroot'_reflection_perm y).symm
+
+lemma pairing_reflection_perm (i j k : ι) :
+    P.pairing j (P.reflection_perm i k) = P.pairing (P.reflection_perm i j) k := by
+  simp only [pairing, root', coroot_reflection_perm, root_reflection_perm]
+  simp only [coreflection_apply_coroot, map_sub, map_smul, smul_eq_mul,
+    reflection_apply_root]
+  simp only [← toLin_toPerfectPairing, map_smul, LinearMap.smul_apply, map_sub, map_smul,
+    LinearMap.sub_apply, smul_eq_mul]
+  simp only [PerfectPairing.toLin_apply, root'_coroot_eq_pairing, sub_right_inj, mul_comm]
+
+@[simp]
+lemma pairing_reflection_perm_self_left (P : RootPairing ι R M N) (i j : ι) :
+    P.pairing (P.reflection_perm i i) j = - P.pairing i j := by
+  rw [pairing, root', ← reflection_perm_root, root'_coroot_eq_pairing, pairing_same, two_smul,
+    sub_add_cancel_left, ← toLin_toPerfectPairing, LinearMap.map_neg₂, toLin_toPerfectPairing,
+    root'_coroot_eq_pairing]
+
+@[simp]
+lemma pairing_reflection_perm_self_right (i j : ι) :
+    P.pairing i (P.reflection_perm j j) = - P.pairing i j := by
+  rw [pairing, ← reflection_perm_coroot, root_coroot_eq_pairing, pairing_same, two_smul,
+    sub_add_cancel_left, ← toLin_toPerfectPairing, map_neg, toLin_toPerfectPairing,
+    root_coroot_eq_pairing]
+
 /-- A root pairing is said to be crystallographic if the pairing between a root and coroot is
 always an integer. -/
 def IsCrystallographic : Prop :=
-  ∀ i, MapsTo (P.toLin (P.root i)) (range P.coroot) (zmultiples (1 : R))
+  ∀ i, MapsTo (P.root' i) (range P.coroot) (zmultiples (1 : R))
 
 lemma isCrystallographic_iff :
     P.IsCrystallographic ↔ ∀ i j, ∃ z : ℤ, z = P.pairing i j := by
@@ -417,7 +477,7 @@ lemma two_nsmul_reflection_eq_of_perm_eq (hij : P.reflection_perm i = P.reflecti
     2 • ⇑(P.reflection i) = 2 • P.reflection j := by
   ext x
   suffices 2 • P.toLin x (P.coroot i) • P.root i = 2 • P.toLin x (P.coroot j) • P.root j by
-    simpa [reflection_apply]
+    simpa [reflection_apply, smul_sub]
   calc 2 • P.toLin x (P.coroot i) • P.root i
       = P.toLin x (P.coroot i) • ((2 : R) • P.root i) := ?_
     _ = P.toLin x (P.coroot i) • (P.pairing i j • P.root j) := ?_
@@ -472,12 +532,12 @@ lemma IsOrthogonal.symm : IsOrthogonal P i j ↔ IsOrthogonal P j i := by
   simp only [IsOrthogonal, and_comm]
 
 lemma isOrthogonal_comm (h : IsOrthogonal P i j) : Commute (P.reflection i) (P.reflection j) := by
-  rw [Commute, SemiconjBy]
+  rw [commute_iff_eq]
   ext v
   replace h : P.pairing i j = 0 ∧ P.pairing j i = 0 := by simpa [IsOrthogonal] using h
   erw [LinearMap.mul_apply, LinearMap.mul_apply]
-  simp only [LinearEquiv.coe_coe, reflection_apply, map_sub, map_smul, root_coroot_eq_pairing,
-    zero_smul, sub_zero, h]
+  simp only [LinearEquiv.coe_coe, reflection_apply, PerfectPairing.flip_apply_apply, map_sub,
+    map_smul, root_coroot_eq_pairing, h, zero_smul, sub_zero]
   abel
 
 end RootPairing
diff --git a/Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean b/Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean
new file mode 100644
index 0000000000000..d495f1b08bbcc
--- /dev/null
+++ b/Mathlib/LinearAlgebra/RootSystem/Finite/CanonicalBilinear.lean
@@ -0,0 +1,137 @@
+/-
+Copyright (c) 2024 Scott Carnahan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Carnahan
+-/
+import Mathlib.LinearAlgebra.RootSystem.Defs
+import Mathlib.Algebra.Ring.SumsOfSquares
+
+/-!
+# The canonical bilinear form on a finite root pairing
+Given a finite root pairing, we define a canonical map from weight space to coweight space, and the
+corresponding bilinear form. This form is symmetric and Weyl-invariant, and if the base ring is
+linearly ordered, then the form is root-positive, positive-semidefinite on the weight space, and
+positive-definite on the span of roots.
+From these facts, it is easy to show that Coxeter weights in a finite root pairing are bounded
+above by 4. Thus, the pairings of roots and coroots in a crystallographic root pairing are
+restricted to a small finite set of possibilities.
+Another application is to the faithfulness of the Weyl group action on roots, and finiteness of the
+Weyl group.
+
+## Main definitions:
+ * `Polarization`: A distinguished linear map from the weight space to the coweight space.
+ * `RootForm` : The bilinear form on weight space corresponding to `Polarization`.
+
+## References:
+ * SGAIII Exp. XXI
+ * Bourbaki, Lie groups and Lie algebras
+
+## Main results:
+ * `polarization_self_sum_of_squares` : The inner product of any weight vector is a sum of squares.
+ * `rootForm_reflection_reflection_apply` : `RootForm` is invariant with respect
+   to reflections.
+ * `rootForm_self_smul_coroot`: The inner product of a root with itself times the
+   corresponding coroot is equal to two times Polarization applied to the root.
+
+## TODO (possibly in other files)
+ * Positivity and nondegeneracy
+ * Weyl-invariance
+ * Faithfulness of Weyl group action, and finiteness of Weyl group, for finite root systems.
+ * Relation to Coxeter weight.  In particular, positivity constraints for finite root pairings mean
+  we restrict to weights between 0 and 4.
+-/
+
+open Function
+open Module hiding reflection
+
+noncomputable section
+
+variable {ι R M N : Type*}
+
+namespace RootPairing
+
+section CommRing
+
+variable [Fintype ι] [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+(P : RootPairing ι R M N)
+
+/-- An invariant linear map from weight space to coweight space. -/
+def Polarization : M →ₗ[R] N :=
+  ∑ i, LinearMap.toSpanSingleton R N (P.coroot i) ∘ₗ P.coroot' i
+
+@[simp]
+lemma Polarization_apply (x : M) :
+    P.Polarization x = ∑ i, P.coroot' i x • P.coroot i := by
+  simp [Polarization]
+
+/-- An invariant linear map from coweight space to weight space. -/
+def CoPolarization : N →ₗ[R] M :=
+  ∑ i, LinearMap.toSpanSingleton R M (P.root i) ∘ₗ P.root' i
+
+@[simp]
+lemma CoPolarization_apply (x : N) :
+    P.CoPolarization x = ∑ i, P.root' i x • P.root i := by
+  simp [CoPolarization]
+
+lemma CoPolarization_eq : P.CoPolarization = P.flip.Polarization :=
+  rfl
+
+/-- An invariant inner product on the weight space. -/
+def RootForm : LinearMap.BilinForm R M :=
+  ∑ i, (P.coroot' i).smulRight (P.coroot' i)
+
+/-- An invariant inner product on the coweight space. -/
+def CorootForm : LinearMap.BilinForm R N :=
+  ∑ i, (P.root' i).smulRight (P.root' i)
+
+lemma rootForm_apply_apply (x y : M) : P.RootForm x y =
+    ∑ (i : ι), P.coroot' i x * P.coroot' i y := by
+  simp [RootForm]
+
+lemma rootForm_symmetric :
+    LinearMap.IsSymm P.RootForm := by
+  simp [LinearMap.IsSymm, mul_comm, rootForm_apply_apply]
+
+@[simp]
+lemma rootForm_reflection_reflection_apply (i : ι) (x y : M) :
+    P.RootForm (P.reflection i x) (P.reflection i y) = P.RootForm x y := by
+  simp only [rootForm_apply_apply, coroot'_reflection]
+  exact Fintype.sum_equiv (P.reflection_perm i)
+    (fun j ↦ (P.coroot' (P.reflection_perm i j) x) * (P.coroot' (P.reflection_perm i j) y))
+    (fun j ↦ P.coroot' j x * P.coroot' j y) (congrFun rfl)
+
+/-- This is SGA3 XXI Lemma 1.2.1 (10), key for proving nondegeneracy and positivity. -/
+lemma rootForm_self_smul_coroot (i : ι) :
+    (P.RootForm (P.root i) (P.root i)) • P.coroot i = 2 • P.Polarization (P.root i) := by
+  have hP : P.Polarization (P.root i) =
+      ∑ j : ι, P.pairing i (P.reflection_perm i j) • P.coroot (P.reflection_perm i j) := by
+    simp_rw [Polarization_apply, root_coroot'_eq_pairing]
+    exact (Fintype.sum_equiv (P.reflection_perm i)
+          (fun j ↦ P.pairing i (P.reflection_perm i j) • P.coroot (P.reflection_perm i j))
+          (fun j ↦ P.pairing i j • P.coroot j) (congrFun rfl)).symm
+  rw [two_nsmul]
+  nth_rw 2 [hP]
+  rw [Polarization_apply]
+  simp only [root_coroot'_eq_pairing, pairing_reflection_perm, pairing_reflection_perm_self_left,
+    ← reflection_perm_coroot, smul_sub, neg_smul, sub_neg_eq_add]
+  rw [Finset.sum_add_distrib, ← add_assoc, ← sub_eq_iff_eq_add]
+  simp only [rootForm_apply_apply, LinearMap.coe_comp, comp_apply, Polarization_apply,
+    root_coroot_eq_pairing, map_sum, LinearMapClass.map_smul, Finset.sum_neg_distrib, ← smul_assoc]
+  rw [Finset.sum_smul, add_neg_eq_zero.mpr rfl]
+  exact sub_eq_zero_of_eq rfl
+
+lemma corootForm_self_smul_root (i : ι) :
+    (P.CorootForm (P.coroot i) (P.coroot i)) • P.root i = 2 • P.CoPolarization (P.coroot i) :=
+  rootForm_self_smul_coroot (P.flip) i
+
+lemma rootForm_self_sum_of_squares (x : M) :
+    IsSumSq (P.RootForm x x) :=
+  P.rootForm_apply_apply x x ▸ isSumSq_sum_mul_self Finset.univ _
+
+lemma rootForm_root_self (j : ι) :
+    P.RootForm (P.root j) (P.root j) = ∑ (i : ι), (P.pairing j i) * (P.pairing j i) := by
+  simp [rootForm_apply_apply]
+
+end CommRing
+
+end RootPairing
diff --git a/Mathlib/LinearAlgebra/RootSystem/Hom.lean b/Mathlib/LinearAlgebra/RootSystem/Hom.lean
new file mode 100644
index 0000000000000..5d58b28dabbf4
--- /dev/null
+++ b/Mathlib/LinearAlgebra/RootSystem/Hom.lean
@@ -0,0 +1,125 @@
+/-
+Copyright (c) 2024 Scott Carnahan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Carnahan
+-/
+import Mathlib.LinearAlgebra.RootSystem.Defs
+
+/-!
+# Morphisms of root pairings
+This file defines morphisms of root pairings, following the definition of morphisms of root data
+given in SGA III Exp. 21 Section 6.
+
+## Main definitions:
+ * `Hom`: A morphism of root data is a linear map of weight spaces, its transverse on coweight
+   spaces, and a bijection on the set that indexes roots and coroots.
+ * `Hom.id`: The identity morphism.
+ * `Hom.comp`: The composite of two morphisms.
+
+## TODO
+
+ * Special types of morphisms: Isogenies, weight/coweight space embeddings
+ * Weyl group reimplementation?
+
+-/
+
+open Set Function
+
+noncomputable section
+
+variable {ι R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+
+namespace RootPairing
+
+/-- A morphism of root pairings is a pair of mutually transposed maps of weight and coweight spaces
+that preserves roots and coroots.  We make the map of indexing sets explicit. -/
+@[ext]
+structure Hom {ι₂ M₂ N₂ : Type*}
+    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]
+    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) where
+  /-- A linear map on weight space. -/
+  weightMap : M →ₗ[R] M₂
+  /-- A contravariant linear map on coweight space. -/
+  coweightMap : N₂ →ₗ[R] N
+  /-- A bijection on index sets. -/
+  indexEquiv : ι ≃ ι₂
+  weight_coweight_transpose : weightMap.dualMap ∘ₗ Q.toDualRight = P.toDualRight ∘ₗ coweightMap
+  root_weightMap : weightMap ∘ P.root = Q.root ∘ indexEquiv
+  coroot_coweightMap : coweightMap ∘ Q.coroot = P.coroot ∘ indexEquiv.symm
+
+/-- The identity morphism of a root pairing. -/
+@[simps!]
+def Hom.id (P : RootPairing ι R M N) : Hom P P where
+  weightMap := LinearMap.id
+  coweightMap := LinearMap.id
+  indexEquiv := Equiv.refl ι
+  weight_coweight_transpose := by simp
+  root_weightMap := by simp
+  coroot_coweightMap := by simp
+
+/-- Composition of morphisms -/
+@[simps!]
+def Hom.comp {ι₁ M₁ N₁ ι₂ M₂ N₂ : Type*} [AddCommGroup M₁] [Module R M₁] [AddCommGroup N₁]
+    [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]
+    {P : RootPairing ι R M N} {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂}
+    (g : Hom P₁ P₂) (f : Hom P P₁) : Hom P P₂ where
+  weightMap := g.weightMap ∘ₗ f.weightMap
+  coweightMap := f.coweightMap ∘ₗ g.coweightMap
+  indexEquiv := f.indexEquiv.trans g.indexEquiv
+  weight_coweight_transpose := by
+    ext φ x
+    rw [← LinearMap.dualMap_comp_dualMap, ← LinearMap.comp_assoc _ f.coweightMap,
+      ← f.weight_coweight_transpose, LinearMap.comp_assoc g.coweightMap,
+      ← g.weight_coweight_transpose, ← LinearMap.comp_assoc]
+  root_weightMap := by
+    ext i
+    simp only [LinearMap.coe_comp, Equiv.coe_trans]
+    rw [comp_assoc, f.root_weightMap, ← comp_assoc, g.root_weightMap, comp_assoc]
+  coroot_coweightMap := by
+    ext i
+    simp only [LinearMap.coe_comp, Equiv.symm_trans_apply]
+    rw [comp_assoc, g.coroot_coweightMap, ← comp_assoc, f.coroot_coweightMap, comp_assoc]
+    simp
+
+@[simp]
+lemma Hom.id_comp {ι₂ M₂ N₂ : Type*}
+    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]
+    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :
+    Hom.comp f (Hom.id P) = f := by
+  ext x <;> simp
+
+@[simp]
+lemma Hom.comp_id {ι₂ M₂ N₂ : Type*}
+    [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]
+    (P : RootPairing ι R M N) (Q : RootPairing ι₂ R M₂ N₂) (f : Hom P Q) :
+    Hom.comp (Hom.id Q) f = f := by
+  ext x <;> simp
+
+@[simp]
+lemma Hom.comp_assoc {ι₁ M₁ N₁ ι₂ M₂ N₂ ι₃ M₃ N₃ : Type*} [AddCommGroup M₁] [Module R M₁]
+    [AddCommGroup N₁] [Module R N₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup N₂] [Module R N₂]
+    [AddCommGroup M₃] [Module R M₃] [AddCommGroup N₃] [Module R N₃] {P : RootPairing ι R M N}
+    {P₁ : RootPairing ι₁ R M₁ N₁} {P₂ : RootPairing ι₂ R M₂ N₂} {P₃ : RootPairing ι₃ R M₃ N₃}
+    (h : Hom P₂ P₃) (g : Hom P₁ P₂) (f : Hom P P₁) :
+    Hom.comp (Hom.comp h g) f = Hom.comp h (Hom.comp g f) := by
+  ext <;> simp
+
+/-- The endomorphism of a root pairing given by a reflection. -/
+@[simps!]
+def Hom.reflection (P : RootPairing ι R M N) (i : ι) : Hom P P where
+  weightMap := P.reflection i
+  coweightMap := P.coreflection i
+  indexEquiv := P.reflection_perm i
+  weight_coweight_transpose := by
+    ext f x
+    simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, comp_apply,
+      PerfectPairing.toDualRight_apply, LinearMap.dualMap_apply, PerfectPairing.flip_apply_apply,
+      LinearEquiv.comp_coe, LinearEquiv.trans_apply]
+    rw [RootPairing.reflection_apply, RootPairing.coreflection_apply]
+    simp only [← PerfectPairing.toLin_apply, map_sub, map_smul, LinearMap.sub_apply,
+      toLin_toPerfectPairing, LinearMap.smul_apply, smul_eq_mul, sub_right_inj]
+    simp only [PerfectPairing.toLin_apply, PerfectPairing.flip_apply_apply, mul_comm]
+  root_weightMap := by ext; simp
+  coroot_coweightMap := by ext; simp
+
+end RootPairing
diff --git a/Mathlib/LinearAlgebra/RootSystem/OfBilinear.lean b/Mathlib/LinearAlgebra/RootSystem/OfBilinear.lean
new file mode 100644
index 0000000000000..b339fa3a131cc
--- /dev/null
+++ b/Mathlib/LinearAlgebra/RootSystem/OfBilinear.lean
@@ -0,0 +1,182 @@
+/-
+Copyright (c) 2024 Scott Carnahan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Carnahan
+-/
+import Mathlib.LinearAlgebra.RootSystem.Defs
+
+/-!
+# Root pairings made from bilinear forms
+A common construction of root systems is given by taking the set of all vectors in an integral
+lattice for which reflection yields an automorphism of the lattice.  In this file, we generalize
+this construction, replacing the ring of integers with an arbitrary commutative ring and the
+integral lattice with an arbitrary reflexive module equipped with a bilinear form.
+
+## Main definitions:
+ * `LinearMap.IsReflective`: Length is a regular value of `R`, and reflection is definable.
+ * `LinearMap.IsReflective.coroot`: The coroot corresponding to a reflective vector.
+ * `RootPairing.of_Bilinear`: The root pairing whose roots are reflective vectors.
+
+## TODO
+ * properties
+-/
+
+open Set Function Module
+
+noncomputable section
+
+variable {ι R M N : Type*} [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+
+namespace LinearMap
+
+/-- A vector `x` is reflective with respect to a bilinear form if multiplication by its norm is
+injective, and for any vector `y`, the norm of `x` divides twice the inner product of `x` and `y`.
+These conditions are what we need when describing reflection as a map taking `y` to
+`y - 2 • (B x y) / (B x x) • x`. -/
+structure IsReflective (B : M →ₗ[R] M →ₗ[R] R) (x : M) : Prop where
+  regular : IsRegular (B x x)
+  dvd_two_mul : ∀ y, B x x ∣ 2 * B x y
+
+variable (B : M →ₗ[R] M →ₗ[R] R) {x : M}
+
+namespace IsReflective
+
+lemma of_dvd_two [IsDomain R] [NeZero (2 : R)] (hx : B x x ∣ 2) :
+    IsReflective B x where
+  regular := isRegular_of_ne_zero <| fun contra ↦ by simp [contra, two_ne_zero (α := R)] at hx
+  dvd_two_mul y := hx.mul_right (B x y)
+
+variable (hx : IsReflective B x)
+
+/-- The coroot attached to a reflective vector. -/
+def coroot : M →ₗ[R] R where
+  toFun y := (hx.2 y).choose
+  map_add' a b := by
+    refine hx.1.1 ?_
+    simp only
+    rw [← (hx.2 (a + b)).choose_spec, mul_add, ← (hx.2 a).choose_spec, ← (hx.2 b).choose_spec,
+      map_add, mul_add]
+  map_smul' r a := by
+    refine hx.1.1 ?_
+    simp only [RingHom.id_apply]
+    rw [← (hx.2 (r • a)).choose_spec, smul_eq_mul, mul_left_comm, ← (hx.2 a).choose_spec, map_smul,
+      two_mul, smul_eq_mul, two_mul, mul_add]
+
+@[simp]
+lemma apply_self_mul_coroot_apply {y : M} : B x x * coroot B hx y = 2 * B x y :=
+  (hx.dvd_two_mul y).choose_spec.symm
+
+@[simp]
+lemma smul_coroot : B x x • coroot B hx = 2 • B x := by
+  ext y
+  simp [smul_apply, smul_eq_mul, nsmul_eq_mul, Nat.cast_ofNat, apply_self_mul_coroot_apply]
+
+@[simp]
+lemma coroot_apply_self : coroot B hx x = 2 :=
+  hx.regular.left <| by simp [mul_comm _ (B x x)]
+
+lemma isOrthogonal_reflection (hSB : LinearMap.IsSymm B) :
+    B.IsOrthogonal (Module.reflection (coroot_apply_self B hx)) := by
+  intro y z
+  simp only [LinearEquiv.coe_coe, reflection_apply, LinearMap.map_sub, map_smul, sub_apply,
+    smul_apply, smul_eq_mul]
+  refine hx.1.1 ?_
+  simp only [mul_sub, ← mul_assoc, apply_self_mul_coroot_apply]
+  rw [sub_eq_iff_eq_add, ← hSB x y, RingHom.id_apply, mul_assoc _ _ (B x x), mul_comm _ (B x x),
+    apply_self_mul_coroot_apply]
+  ring
+
+lemma reflective_reflection (hSB : LinearMap.IsSymm B) {y : M}
+    (hx : IsReflective B x) (hy : IsReflective B y) :
+    IsReflective B (Module.reflection (coroot_apply_self B hx) y) := by
+  constructor
+  · rw [← LinearEquiv.coe_coe, isOrthogonal_reflection B hx hSB]
+    exact hy.1
+  · intro z
+    have hz : Module.reflection (coroot_apply_self B hx)
+        (Module.reflection (coroot_apply_self B hx) z) = z := by
+      exact (LinearEquiv.eq_symm_apply (Module.reflection (coroot_apply_self B hx))).mp rfl
+    rw [← hz, ← LinearEquiv.coe_coe, isOrthogonal_reflection B hx hSB,
+      isOrthogonal_reflection B hx hSB]
+    exact hy.2 _
+
+end IsReflective
+
+end LinearMap
+
+namespace RootPairing
+
+open LinearMap IsReflective
+
+/-- The root pairing given by all reflective vectors for a bilinear form. -/
+def ofBilinear [IsReflexive R M] (B : M →ₗ[R] M →ₗ[R] R) (hNB : LinearMap.Nondegenerate B)
+    (hSB : LinearMap.IsSymm B) (h2 : IsRegular (2 : R)) :
+    RootPairing {x : M | IsReflective B x} R M (Dual R M) where
+  toPerfectPairing := (IsReflexive.toPerfectPairingDual (R := R) (M := M)).flip
+  root := Embedding.subtype fun x ↦ IsReflective B x
+  coroot :=
+    { toFun := fun x => IsReflective.coroot B x.2
+      inj' := by
+        intro x y hxy
+        simp only [mem_setOf_eq] at hxy -- x* = y*
+        have h1 : ∀ z, IsReflective.coroot B x.2 z = IsReflective.coroot B y.2 z :=
+          fun z => congrFun (congrArg DFunLike.coe hxy) z
+        have h2x : ∀ z, B x x * IsReflective.coroot B x.2 z =
+            B x x * IsReflective.coroot B y.2 z :=
+          fun z => congrArg (HMul.hMul ((B x) x)) (h1 z)
+        have h2y : ∀ z, B y y * IsReflective.coroot B x.2 z =
+            B y y * IsReflective.coroot B y.2 z :=
+          fun z => congrArg (HMul.hMul ((B y) y)) (h1 z)
+        simp_rw [apply_self_mul_coroot_apply B x.2] at h2x -- 2(x,z) = (x,x)y*(z)
+        simp_rw [apply_self_mul_coroot_apply B y.2] at h2y -- (y,y)x*(z) = 2(y,z)
+        have h2xy : B x x = B y y := by
+          refine h2.1 ?_
+          dsimp only
+          specialize h2x y
+          rw [coroot_apply_self] at h2x
+          specialize h2y x
+          rw [coroot_apply_self] at h2y
+          rw [mul_comm, ← h2x, ← hSB, RingHom.id_apply, ← h2y, mul_comm]
+        rw [Subtype.ext_iff_val, ← sub_eq_zero]
+        refine hNB.1 _ (fun z => ?_)
+        rw [map_sub, LinearMap.sub_apply, sub_eq_zero]
+        refine h2.1 ?_
+        dsimp only
+        rw [h2x z, ← h2y z, hxy, h2xy] }
+  root_coroot_two x := by
+    dsimp only [coe_setOf, Embedding.coe_subtype, PerfectPairing.toLin_apply, mem_setOf_eq, id_eq,
+      eq_mp_eq_cast, RingHom.id_apply, eq_mpr_eq_cast, cast_eq, LinearMap.sub_apply,
+      Embedding.coeFn_mk, PerfectPairing.flip_apply_apply]
+    exact coroot_apply_self B x.2
+  reflection_perm x :=
+    { toFun := fun y => ⟨(Module.reflection (coroot_apply_self B x.2) y),
+        reflective_reflection B hSB x.2 y.2⟩
+      invFun := fun y => ⟨(Module.reflection (coroot_apply_self B x.2) y),
+        reflective_reflection B hSB x.2 y.2⟩
+      left_inv := by
+        intro y
+        simp [involutive_reflection (coroot_apply_self B x.2) y]
+      right_inv := by
+        intro y
+        simp [involutive_reflection (coroot_apply_self B x.2) y] }
+  reflection_perm_root x y := by
+    simp [Module.reflection_apply]
+  reflection_perm_coroot x y := by
+    simp only [coe_setOf, mem_setOf_eq, Embedding.coeFn_mk, Embedding.coe_subtype,
+      PerfectPairing.flip_apply_apply, IsReflexive.toPerfectPairingDual_apply, Equiv.coe_fn_mk]
+    ext z
+    simp only [LinearMap.sub_apply, LinearMap.smul_apply, smul_eq_mul]
+    refine y.2.1.1 ?_
+    simp only [mem_setOf_eq, PerfectPairing.flip_apply_apply, mul_sub,
+      apply_self_mul_coroot_apply B y.2, ← mul_assoc]
+    rw [← isOrthogonal_reflection B x.2 hSB y y, apply_self_mul_coroot_apply, ← hSB z, ← hSB z,
+      RingHom.id_apply, RingHom.id_apply, LinearEquiv.coe_coe, Module.reflection_apply, map_sub,
+      mul_sub, sub_eq_sub_iff_comm, sub_left_inj]
+    refine x.2.1.1 ?_
+    simp only [mem_setOf_eq, map_smul, smul_eq_mul]
+    rw [← mul_assoc _ _ (B z x), ← mul_assoc _ _ (B z x), mul_left_comm,
+      apply_self_mul_coroot_apply B x.2, mul_left_comm (B x x), apply_self_mul_coroot_apply B x.2,
+      ← hSB x y, RingHom.id_apply, ← hSB x z, RingHom.id_apply]
+    ring
+
+end RootPairing
diff --git a/Mathlib/LinearAlgebra/RootSystem/RootPairingCat.lean b/Mathlib/LinearAlgebra/RootSystem/RootPairingCat.lean
new file mode 100644
index 0000000000000..662e12e5e940c
--- /dev/null
+++ b/Mathlib/LinearAlgebra/RootSystem/RootPairingCat.lean
@@ -0,0 +1,61 @@
+/-
+Copyright (c) 2024 Scott Carnahan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Carnahan
+-/
+import Mathlib.LinearAlgebra.RootSystem.Hom
+import Mathlib.CategoryTheory.Category.Basic
+
+/-!
+# The category of root pairings
+This file defines the category of root pairings, following the definition of category of root data
+given in SGA III Exp. 21 Section 6.
+
+## Main definitions:
+ * `RootPairingCat`: Objects are root pairings.
+
+## TODO
+
+ * Forgetful functors
+ * Functions passing between module maps and root pairing homs
+
+## Implementation details
+
+This is mostly copied from `ModuleCat`.
+
+-/
+
+open Set Function CategoryTheory
+
+noncomputable section
+
+universe v u
+
+variable {R : Type u} [CommRing R]
+
+/-- Objects in the category of root pairings. -/
+structure RootPairingCat (R : Type u) [CommRing R] where
+  /-- The weight space of a root pairing. -/
+  weight : Type v
+  [weightIsAddCommGroup : AddCommGroup weight]
+  [weightIsModule : Module R weight]
+  /-- The coweight space of a root pairing. -/
+  coweight : Type v
+  [coweightIsAddCommGroup : AddCommGroup coweight]
+  [coweightIsModule : Module R coweight]
+  /-- The set that indexes roots and coroots. -/
+  index : Type v
+  /-- The root pairing structure. -/
+  pairing : RootPairing index R weight coweight
+
+attribute [instance] RootPairingCat.weightIsAddCommGroup RootPairingCat.weightIsModule
+attribute [instance] RootPairingCat.coweightIsAddCommGroup RootPairingCat.coweightIsModule
+
+namespace RootPairingCat
+
+instance category : Category.{v, max (v+1) u} (RootPairingCat.{v} R) where
+  Hom P Q := RootPairing.Hom P.pairing Q.pairing
+  id P := RootPairing.Hom.id P.pairing
+  comp f g := RootPairing.Hom.comp g f
+
+end RootPairingCat
diff --git a/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean b/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
index ccf50330e9a81..e9d55f92f8209 100644
--- a/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
+++ b/Mathlib/LinearAlgebra/RootSystem/RootPositive.lean
@@ -55,7 +55,7 @@ lemma two_mul_apply_root_root :
     2 * B (P.root i) (P.root j) = P.pairing i j * B (P.root j) (P.root j) := by
   rw [two_mul, ← eq_sub_iff_add_eq]
   nth_rw 1 [← IsRootPositive.apply_reflection_eq (P := P) (B := B) j (P.root i) (P.root j)]
-  rw [reflection_apply, reflection_apply_self, root_coroot_eq_pairing, LinearMap.map_sub₂,
+  rw [reflection_apply, reflection_apply_self, root_coroot'_eq_pairing, LinearMap.map_sub₂,
     LinearMap.map_smul₂, smul_eq_mul, LinearMap.map_neg, LinearMap.map_neg, mul_neg, neg_sub_neg]
 
 @[simp]
diff --git a/Mathlib/LinearAlgebra/Semisimple.lean b/Mathlib/LinearAlgebra/Semisimple.lean
index ea02d0bd4374e..3aeb94be63ed6 100644
--- a/Mathlib/LinearAlgebra/Semisimple.lean
+++ b/Mathlib/LinearAlgebra/Semisimple.lean
@@ -80,7 +80,7 @@ lemma eq_zero_of_isNilpotent_isSemisimple (hn : IsNilpotent f) (hs : f.IsSemisim
   have ⟨n, h0⟩ := hn
   rw [← aeval_X (R := R) f]; rw [← aeval_X_pow (R := R) f] at h0
   rw [← RingHom.mem_ker, ← AEval.annihilator_eq_ker_aeval (M := M)] at h0 ⊢
-  exact hs.annihilator_isRadical ⟨n, h0⟩
+  exact hs.annihilator_isRadical _ _ ⟨n, h0⟩
 
 @[simp]
 lemma isSemisimple_sub_algebraMap_iff {μ : R} :
@@ -145,7 +145,7 @@ theorem IsSemisimple.minpoly_squarefree : Squarefree (minpoly K f) :=
 
 protected theorem IsSemisimple.aeval (p : K[X]) : (aeval f p).IsSemisimple :=
   let R := K[X] ⧸ Ideal.span {minpoly K f}
-  have : Finite K R :=
+  have : Module.Finite K R :=
     (AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f).finite
   have : IsReduced R := (Ideal.isRadical_iff_quotient_reduced _).mp <|
     span_minpoly_eq_annihilator K f ▸ hf.annihilator_isRadical
@@ -174,9 +174,9 @@ theorem IsSemisimple.of_mem_adjoin_pair {a : End K M} (ha : a ∈ Algebra.adjoin
     a.IsSemisimple := by
   let R := K[X] ⧸ Ideal.span {minpoly K f}
   let S := AdjoinRoot ((minpoly K g).map <| algebraMap K R)
-  have : Finite K R :=
+  have : Module.Finite K R :=
     (AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f).finite
-  have : Finite R S :=
+  have : Module.Finite R S :=
     (AdjoinRoot.powerBasis' <| (minpoly.monic <| Algebra.IsIntegral.isIntegral g).map _).finite
   #adaptation_note
   /--
@@ -187,7 +187,7 @@ theorem IsSemisimple.of_mem_adjoin_pair {a : End K M} (ha : a ∈ Algebra.adjoin
   -/
   set_option maxSynthPendingDepth 2 in
   have : IsScalarTower K R S := .of_algebraMap_eq fun _ ↦ rfl
-  have : Finite K S := .trans R S
+  have : Module.Finite K S := .trans R S
   have : IsArtinianRing R := .of_finite K R
   have : IsReduced R := (Ideal.isRadical_iff_quotient_reduced _).mp <|
     span_minpoly_eq_annihilator K f ▸ hf.annihilator_isRadical
diff --git a/Mathlib/LinearAlgebra/SesquilinearForm.lean b/Mathlib/LinearAlgebra/SesquilinearForm.lean
index f97b86492d2e9..99617d933657d 100644
--- a/Mathlib/LinearAlgebra/SesquilinearForm.lean
+++ b/Mathlib/LinearAlgebra/SesquilinearForm.lean
@@ -458,9 +458,40 @@ theorem IsAdjointPair.smul (c : R) (h : IsAdjointPair B B' f g) :
 
 end AddCommGroup
 
+section OrthogonalMap
+
+variable {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
+  (B : LinearMap.BilinForm R M) (f : Module.End R M)
+
+/-- A linear transformation `f` is orthogonal with respect to a bilinear form `B` if `B` is
+bi-invariant with respect to `f`. -/
+def IsOrthogonal : Prop :=
+  ∀ x y, B (f x) (f y) = B x y
+
+variable {B f}
+
+@[simp]
+lemma _root_.LinearEquiv.isAdjointPair_symm_iff {f : M ≃ₗ[R] M} :
+    LinearMap.IsAdjointPair B B f f.symm ↔ B.IsOrthogonal f :=
+  ⟨fun hf x y ↦ by simpa using hf x (f y), fun hf x y ↦ by simpa using hf x (f.symm y)⟩
+
+lemma isOrthogonal_of_forall_apply_same
+    (h : IsLeftRegular (2 : R)) (hB : B.IsSymm) (hf : ∀ x, B (f x) (f x) = B x x) :
+    B.IsOrthogonal f := by
+  intro x y
+  suffices 2 * B (f x) (f y) = 2 * B x y from h this
+  have := hf (x + y)
+  simp only [map_add, add_apply, hf x, hf y, show B y x = B x y from hB.eq y x] at this
+  rw [show B (f y) (f x) = B (f x) (f y) from hB.eq (f y) (f x)] at this
+  simp only [add_assoc, add_right_inj] at this
+  simp only [← add_assoc, add_left_inj] at this
+  simpa only [← two_mul] using this
+
+end OrthogonalMap
+
 end AdjointPair
 
-/-! ### Self-adjoint pairs-/
+/-! ### Self-adjoint pairs -/
 
 
 section SelfadjointPair
@@ -704,7 +735,7 @@ theorem IsOrthoᵢ.not_isOrtho_basis_self_of_separatingLeft [Nontrivial R]
   intro ho
   refine v.ne_zero i (hB (v i) fun m ↦ ?_)
   obtain ⟨vi, rfl⟩ := v.repr.symm.surjective m
-  rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, map_sum]
+  rw [Basis.repr_symm_apply, Finsupp.linearCombination_apply, Finsupp.sum, map_sum]
   apply Finset.sum_eq_zero
   rintro j -
   rw [map_smulₛₗ]
@@ -733,7 +764,8 @@ theorem IsOrthoᵢ.separatingLeft_of_not_isOrtho_basis_self [NoZeroSMulDivisors
   ext i
   rw [Finsupp.zero_apply]
   specialize hB (v i)
-  simp_rw [Basis.repr_symm_apply, Finsupp.total_apply, Finsupp.sum, map_sum₂, map_smulₛₗ₂] at hB
+  simp_rw [Basis.repr_symm_apply, Finsupp.linearCombination_apply, Finsupp.sum, map_sum₂,
+           map_smulₛₗ₂] at hB
   rw [Finset.sum_eq_single i] at hB
   · exact (smul_eq_zero.mp hB).elim _root_.id (h i).elim
   · intro j _hj hij
diff --git a/Mathlib/LinearAlgebra/Span.lean b/Mathlib/LinearAlgebra/Span.lean
index e6fec62b7d0d8..460f7003ad94b 100644
--- a/Mathlib/LinearAlgebra/Span.lean
+++ b/Mathlib/LinearAlgebra/Span.lean
@@ -74,7 +74,7 @@ theorem subset_span : s ⊆ span R s := fun _ h => mem_span.2 fun _ hp => hp h
 theorem span_le {p} : span R s ≤ p ↔ s ⊆ p :=
   ⟨Subset.trans subset_span, fun ss _ h => mem_span.1 h _ ss⟩
 
-theorem span_mono (h : s ⊆ t) : span R s ≤ span R t :=
+@[gcongr] theorem span_mono (h : s ⊆ t) : span R s ≤ span R t :=
   span_le.2 <| Subset.trans h subset_span
 
 theorem span_monotone : Monotone (span R : Set M → Submodule R M) := fun _ _ => span_mono
@@ -362,8 +362,9 @@ theorem coe_iSup_of_chain (a : ℕ →o Submodule R M) : (↑(⨆ k, a k) : Set
 /-- We can regard `coe_iSup_of_chain` as the statement that `(↑) : (Submodule R M) → Set M` is
 Scott continuous for the ω-complete partial order induced by the complete lattice structures. -/
 theorem coe_scott_continuous :
-    OmegaCompletePartialOrder.Continuous' ((↑) : Submodule R M → Set M) :=
-  ⟨SetLike.coe_mono, coe_iSup_of_chain⟩
+    OmegaCompletePartialOrder.ωScottContinuous ((↑) : Submodule R M → Set M) :=
+  OmegaCompletePartialOrder.ωScottContinuous.of_monotone_map_ωSup
+    ⟨SetLike.coe_mono, coe_iSup_of_chain⟩
 
 @[simp]
 theorem mem_iSup_of_chain (a : ℕ →o Submodule R M) (m : M) : (m ∈ ⨆ k, a k) ↔ ∃ k, m ∈ a k :=
diff --git a/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean b/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean
index 7ca2ffdc1829e..24e2067ed86e6 100644
--- a/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean
+++ b/Mathlib/LinearAlgebra/TensorAlgebra/Basis.lean
@@ -35,7 +35,7 @@ with its index. -/
 noncomputable def equivFreeAlgebra (b : Basis κ R M) :
     TensorAlgebra R M ≃ₐ[R] FreeAlgebra R κ :=
   AlgEquiv.ofAlgHom
-    (TensorAlgebra.lift _ (Finsupp.total _ (FreeAlgebra.ι _) ∘ₗ b.repr.toLinearMap))
+    (TensorAlgebra.lift _ (Finsupp.linearCombination _ (FreeAlgebra.ι _) ∘ₗ b.repr.toLinearMap))
     (FreeAlgebra.lift _ (ι R ∘ b))
     (by ext; simp)
     (hom_ext <| b.ext fun i => by simp)
diff --git a/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean b/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean
index bb0ee53dd453c..e37eea1ee4fdd 100644
--- a/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean
+++ b/Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean
@@ -32,8 +32,8 @@ theorem toTensorAlgebra_tprod {n} (x : Fin n → M) :
 
 @[simp]
 theorem toTensorAlgebra_gOne :
-    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 :=
-  TensorPower.toTensorAlgebra_tprod _
+    TensorPower.toTensorAlgebra (@GradedMonoid.GOne.one _ (fun n => ⨂[R]^n M) _ _) = 1 := by
+  simp [GradedMonoid.GOne.one, TensorPower.toTensorAlgebra_tprod]
 
 @[simp]
 theorem toTensorAlgebra_gMul {i j} (a : (⨂[R]^i) M) (b : (⨂[R]^j) M) :
@@ -128,18 +128,18 @@ theorem _root_.TensorPower.list_prod_gradedMonoid_mk_single (n : ℕ) (x : Fin n
     rfl
   · intro n x₀ x ih
     rw [List.finRange_succ_eq_map, List.map_cons, List.prod_cons, List.map_map]
-    simp_rw [Function.comp, Fin.cons_zero, Fin.cons_succ]
+    simp_rw [Function.comp_def, Fin.cons_zero, Fin.cons_succ]
     rw [ih, GradedMonoid.mk_mul_mk, TensorPower.tprod_mul_tprod]
     refine TensorPower.gradedMonoid_eq_of_cast (add_comm _ _) ?_
     dsimp only [GradedMonoid.mk]
     rw [TensorPower.cast_tprod]
-    simp_rw [Fin.append_left_eq_cons, Function.comp]
+    simp_rw [Fin.append_left_eq_cons, Function.comp_def]
     congr 1 with i
 
 theorem toDirectSum_tensorPower_tprod {n} (x : Fin n → M) :
     toDirectSum (tprod R M n x) = DirectSum.of _ n (PiTensorProduct.tprod R x) := by
   rw [tprod_apply, map_list_prod, List.map_ofFn]
-  simp_rw [Function.comp, toDirectSum_ι]
+  simp_rw [Function.comp_def, toDirectSum_ι]
   rw [DirectSum.list_prod_ofFn_of_eq_dProd]
   apply DirectSum.of_eq_of_gradedMonoid_eq
   rw [GradedMonoid.mk_list_dProd]
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Basic.lean b/Mathlib/LinearAlgebra/TensorProduct/Basic.lean
index 6bbe92a9b4b88..98e1a776a8717 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Basic.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Basic.lean
@@ -449,7 +449,7 @@ variable (R M N)
 
 /-- The simple (aka pure) elements span the tensor product. -/
 theorem span_tmul_eq_top : Submodule.span R { t : M ⊗[R] N | ∃ m n, m ⊗ₜ n = t } = ⊤ := by
-  ext t; simp only [Submodule.mem_top, iff_true_iff]
+  ext t; simp only [Submodule.mem_top, iff_true]
   refine t.induction_on ?_ ?_ ?_
   · exact Submodule.zero_mem _
   · intro m n
@@ -526,6 +526,13 @@ theorem ext' {g h : M ⊗[R] N →ₗ[R] P} (H : ∀ x y, g (x ⊗ₜ y) = h (x
     TensorProduct.induction_on z (by simp_rw [LinearMap.map_zero]) H fun x y ihx ihy => by
       rw [g.map_add, h.map_add, ihx, ihy]
 
+theorem ext₃ {g h : (M ⊗[R] N) ⊗[R] P →ₗ[R] Q}
+    (H : ∀ x y z, g (x ⊗ₜ y ⊗ₜ z) = h (x ⊗ₜ y ⊗ₜ z)) : g = h :=
+  ext' fun x => TensorProduct.induction_on x
+    (fun x => by simp only [zero_tmul, map_zero])
+    (fun x y => H x y)
+    (fun x y ihx ihy z => by rw [add_tmul, g.map_add, h.map_add, ihx, ihy])
+
 theorem lift.unique {g : M ⊗[R] N →ₗ[R] P} (H : ∀ x y, g (x ⊗ₜ y) = f x y) : g = lift f :=
   ext' fun m n => by rw [H, lift.tmul]
 
@@ -669,7 +676,7 @@ theorem comm_tmul (m : M) (n : N) : (TensorProduct.comm R M N) (m ⊗ₜ n) = n
 theorem comm_symm_tmul (m : M) (n : N) : (TensorProduct.comm R M N).symm (n ⊗ₜ m) = m ⊗ₜ n :=
   rfl
 
-lemma lift_comp_comm_eq  (f : M →ₗ[R] N →ₗ[R] P) :
+lemma lift_comp_comm_eq (f : M →ₗ[R] N →ₗ[R] P) :
     lift f ∘ₗ TensorProduct.comm R N M = lift f.flip :=
   ext rfl
 end
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean b/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
index 58c1cf353dbaa..18c2623b09806 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean
@@ -5,7 +5,6 @@ Authors: Eric Wieser
 -/
 import Mathlib.LinearAlgebra.TensorProduct.Graded.External
 import Mathlib.RingTheory.GradedAlgebra.Basic
-import Mathlib.GroupTheory.GroupAction.Ring
 
 /-!
 # Graded tensor products over graded algebras
@@ -221,7 +220,7 @@ theorem tmul_coe_mul_one_tmul {j₁ : ι} (a₁ : A) (b₁ : ℬ j₁) (b₂ : B
 theorem tmul_one_mul_one_tmul (a₁ : A) (b₂ : B) :
     (a₁ ᵍ⊗ₜ[R] (1 : B) * (1 : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) = (a₁ : A) ᵍ⊗ₜ (b₂ : B) := by
   convert tmul_coe_mul_zero_coe_tmul 𝒜 ℬ
-    a₁ (@GradedMonoid.GOne.one _ (ℬ ·) _ _) (@GradedMonoid.GOne.one _ (𝒜 ·) _ _) b₂
+    a₁ (GradedMonoid.GOne.one (A := (ℬ ·))) (GradedMonoid.GOne.one (A := (𝒜 ·))) b₂
   · rw [SetLike.coe_gOne, mul_one]
   · rw [SetLike.coe_gOne, one_mul]
 
diff --git a/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean b/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
index a0d3f02f948ef..14d7396e79842 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean
@@ -6,6 +6,7 @@ Authors: Antoine Chambert-Loir
 
 import Mathlib.Algebra.Exact
 import Mathlib.RingTheory.TensorProduct.Basic
+import Mathlib.RingTheory.Ideal.Quotient
 
 /-! # Right-exactness properties of tensor product
 
@@ -546,7 +547,7 @@ lemma Ideal.map_includeLeft_eq (I : Ideal A) :
           simp only [map_zero, smul_eq_mul, mul_zero]
         | tmul x y =>
           use (a • x) ⊗ₜ[R] (b * y)
-          simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype, smul_eq_mul, tmul_mul_tmul]
+          simp only [LinearMap.lTensor_tmul, Submodule.coe_subtype, smul_eq_mul, tmul_mul_tmul]
           with_unfolding_all rfl
         | add x y hx hy =>
           obtain ⟨x', hx'⟩ := hx
@@ -565,7 +566,7 @@ lemma Ideal.map_includeLeft_eq (I : Ideal A) :
         rw [map_zero]
         apply zero_mem
     | tmul a b =>
-        simp only [LinearMap.rTensor_tmul, Submodule.coeSubtype]
+        simp only [LinearMap.rTensor_tmul, Submodule.coe_subtype]
         suffices (a : A) ⊗ₜ[R] b = ((1 : A) ⊗ₜ[R] b) * ((a : A) ⊗ₜ[R] (1 : B)) by
           simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
             Submodule.mem_toAddSubmonoid, Submodule.restrictScalars_mem]
@@ -614,7 +615,7 @@ lemma Ideal.map_includeRight_eq (I : Ideal B) :
           simp only [map_zero, smul_eq_mul, mul_zero]
         | tmul x y =>
           use (a * x) ⊗ₜ[R] (b •y)
-          simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype, smul_eq_mul, tmul_mul_tmul]
+          simp only [LinearMap.lTensor_tmul, Submodule.coe_subtype, smul_eq_mul, tmul_mul_tmul]
           rfl
         | add x y hx hy =>
           obtain ⟨x', hx'⟩ := hx
@@ -633,7 +634,7 @@ lemma Ideal.map_includeRight_eq (I : Ideal B) :
         rw [map_zero]
         apply zero_mem
     | tmul a b =>
-        simp only [LinearMap.lTensor_tmul, Submodule.coeSubtype]
+        simp only [LinearMap.lTensor_tmul, Submodule.coe_subtype]
         suffices a ⊗ₜ[R] (b : B) = (a ⊗ₜ[R] (1 : B)) * ((1 : A) ⊗ₜ[R] (b : B)) by
           rw [this]
           simp only [AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup,
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean b/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean
index 5e2c363f1116d..83b52e58279fd 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean
@@ -3,9 +3,10 @@ Copyright (c) 2024 Jz Pan. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jz Pan
 -/
+import Mathlib.LinearAlgebra.Dimension.Constructions
+import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
 import Mathlib.LinearAlgebra.TensorProduct.Submodule
 import Mathlib.RingTheory.TensorProduct.Basic
-import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
 
 /-!
 
@@ -35,7 +36,7 @@ mainly used in the definition of linearly disjointness.
 
 open scoped TensorProduct
 
-open FiniteDimensional
+open Module
 
 noncomputable section
 
@@ -193,7 +194,7 @@ theorem rank_sup_le_of_free [Module.Free R A] [Module.Free R B] :
   exact rank_range_le (A.mulMap B).toLinearMap
 
 /-- If `A` and `B` are subalgebras of a commutative `R`-algebra `S`, both of them are
-free `R`-algebras, then the `FiniteDimensional.finrank` of `A ⊔ B` is less than or equal to
+free `R`-algebras, then the `Module.finrank` of `A ⊔ B` is less than or equal to
 the product of that of `A` and `B`. -/
 theorem finrank_sup_le_of_free [Module.Free R A] [Module.Free R B] :
     finrank R ↥(A ⊔ B) ≤ finrank R A * finrank R B := by
@@ -205,7 +206,7 @@ theorem finrank_sup_le_of_free [Module.Free R A] [Module.Free R B] :
   wlog hA : ¬ Module.Finite R A generalizing A B
   · have := this B A (fun h' ↦ h h'.symm) (not_and.1 h (of_not_not hA))
     rwa [sup_comm, mul_comm] at this
-  rw [← Module.rank_lt_alpeh0_iff, not_lt] at hA
+  rw [← Module.rank_lt_aleph0_iff, not_lt] at hA
   have := LinearMap.rank_le_of_injective _ <| Submodule.inclusion_injective <|
     show toSubmodule A ≤ toSubmodule (A ⊔ B) by simp
   rw [show finrank R A = 0 from Cardinal.toNat_apply_of_aleph0_le hA,
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean b/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean
index 9e69bcafac312..838f15025ec00 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean
@@ -234,13 +234,13 @@ theorem comm_trans_rTensorOne :
 
 variable {M} in
 theorem mulLeftMap_eq_mulMap_comp {ι : Type*} [DecidableEq ι] (m : ι → M) :
-    mulLeftMap N m = mulMap M N ∘ₗ LinearMap.rTensor N (Finsupp.total R m) ∘ₗ
+    mulLeftMap N m = mulMap M N ∘ₗ LinearMap.rTensor N (Finsupp.linearCombination R m) ∘ₗ
       (TensorProduct.finsuppScalarLeft R N ι).symm.toLinearMap := by
   ext; simp
 
 variable {N} in
 theorem mulRightMap_eq_mulMap_comp {ι : Type*} [DecidableEq ι] (n : ι → N) :
-    mulRightMap M n = mulMap M N ∘ₗ LinearMap.lTensor M (Finsupp.total R n) ∘ₗ
+    mulRightMap M n = mulMap M N ∘ₗ LinearMap.lTensor M (Finsupp.linearCombination R n) ∘ₗ
       (TensorProduct.finsuppScalarRight R M ι).symm.toLinearMap := by
   ext; simp
 
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Tower.lean b/Mathlib/LinearAlgebra/TensorProduct/Tower.lean
index da36cbf10939d..5ed4fbb0c5ba5 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Tower.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Tower.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin, Eric Wieser
+Authors: Kim Morrison, Johan Commelin, Eric Wieser
 -/
 import Mathlib.Algebra.Algebra.Tower
 import Mathlib.LinearAlgebra.TensorProduct.Basic
diff --git a/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean b/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
index e0b4960e75dd4..8a6dcf8f11f00 100644
--- a/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
+++ b/Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean
@@ -60,7 +60,7 @@ variable (R : Type u) [CommRing R]
 variable {M : Type u} [AddCommGroup M] [Module R M]
 variable {N : Type u} [AddCommGroup N] [Module R N]
 
-open DirectSum LinearMap Function Submodule
+open DirectSum LinearMap Function Submodule Finsupp
 
 namespace TensorProduct
 
@@ -102,7 +102,7 @@ vanishes, then it vanishes trivially. -/
 theorem vanishesTrivially_of_sum_tmul_eq_zero (hm : Submodule.span R (Set.range m) = ⊤)
     (hmn : ∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N)) : VanishesTrivially R m n := by
   -- Define a map $G \colon R^\iota \to M$ whose matrix entries are the $m_i$. It is surjective.
-  set G : (ι →₀ R) →ₗ[R] M := Finsupp.total R m with hG
+  set G : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R m with hG
   have G_basis_eq (i : ι) : G (Finsupp.single i 1) = m i := by simp [hG, toModule_lof]
   have G_surjective : Surjective G := by
     apply LinearMap.range_eq_top.mp
@@ -146,7 +146,7 @@ theorem vanishesTrivially_of_sum_tmul_eq_zero (hm : Submodule.span R (Set.range
     symm at hkn
     simp only [map_sum, finsuppScalarLeft_apply_tmul, zero_smul, Finsupp.single_zero,
       Finsupp.sum_single_index, one_smul, Finsupp.finset_sum_apply, Finsupp.single_apply,
-      Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, rTensor_tmul, coeSubtype, Finsupp.sum_apply,
+      Finset.sum_ite_eq', Finset.mem_univ, ↓reduceIte, rTensor_tmul, coe_subtype, Finsupp.sum_apply,
       Finsupp.sum_ite_eq', Finsupp.mem_support_iff, ne_eq, ite_not, en] at hkn
     simp only [Finset.univ_eq_attach, Finset.sum_attach ma (fun x ↦ (x.1 : ι →₀ R) i • x.2)]
     convert hkn using 2 with x _
@@ -154,7 +154,7 @@ theorem vanishesTrivially_of_sum_tmul_eq_zero (hm : Submodule.span R (Set.range
     · next h'x => rw [h'x, zero_smul]
     · rfl
   · rintro ⟨⟨⟨k, hk⟩, _⟩, _⟩
-    simpa only [hG, Finsupp.total_apply, zero_smul, implies_true, Finsupp.sum_fintype] using
+    simpa only [hG, linearCombination_apply, zero_smul, implies_true, Finsupp.sum_fintype] using
       mem_ker.mp hk
 
 /-- **Equational criterion for vanishing**
@@ -181,11 +181,11 @@ theorem vanishesTrivially_of_sum_tmul_eq_zero_of_rTensor_injective
   set m' : ι → span R (Set.range m) := Subtype.coind m mem_M' with m'_eq
   have hm' : span R (Set.range m') = ⊤ := by
     apply map_injective_of_injective (injective_subtype (span R (Set.range m)))
-    rw [Submodule.map_span, Submodule.map_top, range_subtype, coeSubtype, ← Set.range_comp]
+    rw [Submodule.map_span, Submodule.map_top, range_subtype, coe_subtype, ← Set.range_comp]
     rfl
   have hm'n : ∑ i, m' i ⊗ₜ n i = (0 : span R (Set.range m) ⊗[R] N) := by
     apply hm
-    simp only [m'_eq, map_sum, rTensor_tmul, coeSubtype, Subtype.coind_coe, _root_.map_zero, hmn]
+    simp only [m'_eq, map_sum, rTensor_tmul, coe_subtype, Subtype.coind_coe, _root_.map_zero, hmn]
   have : VanishesTrivially R m' n := vanishesTrivially_of_sum_tmul_eq_zero R hm' hm'n
   unfold VanishesTrivially at this ⊢
   convert this with κ _ a y j
@@ -218,7 +218,7 @@ theorem rTensor_injective_of_forall_vanishesTrivially
   obtain ⟨s, rfl⟩ := exists_finset x
   rw [← Finset.sum_attach]
   apply sum_tmul_eq_zero_of_vanishesTrivially
-  simp only [map_sum, rTensor_tmul, coeSubtype] at hx
+  simp only [map_sum, rTensor_tmul, coe_subtype] at hx
   have := hMN ((Finset.sum_attach s _).trans hx)
   unfold VanishesTrivially at this ⊢
   convert this with κ _ a y j
diff --git a/Mathlib/LinearAlgebra/Trace.lean b/Mathlib/LinearAlgebra/Trace.lean
index d8dfa8fb4f30b..81598681a3b50 100644
--- a/Mathlib/LinearAlgebra/Trace.lean
+++ b/Mathlib/LinearAlgebra/Trace.lean
@@ -25,7 +25,7 @@ universe u v w
 namespace LinearMap
 
 open scoped Matrix
-open FiniteDimensional TensorProduct
+open Module TensorProduct
 
 section
 
diff --git a/Mathlib/LinearAlgebra/Vandermonde.lean b/Mathlib/LinearAlgebra/Vandermonde.lean
index 91ce552c56948..b8de8431c2653 100644
--- a/Mathlib/LinearAlgebra/Vandermonde.lean
+++ b/Mathlib/LinearAlgebra/Vandermonde.lean
@@ -35,7 +35,7 @@ namespace Matrix
 
 /-- `vandermonde v` is the square matrix with `i`th row equal to `1, v i, v i ^ 2, v i ^ 3, ...`.
 -/
-def vandermonde {n : ℕ} (v : Fin n → R) : Matrix (Fin n) (Fin n) R := fun i j => v i ^ (j : ℕ)
+def vandermonde {n : ℕ} (v : Fin n → R) : Matrix (Fin n) (Fin n) R := .of fun i j => v i ^ (j : ℕ)
 
 @[simp]
 theorem vandermonde_apply {n : ℕ} (v : Fin n → R) (i j) : vandermonde v i j = v i ^ (j : ℕ) :=
@@ -52,7 +52,7 @@ theorem vandermonde_cons {n : ℕ} (v0 : R) (v : Fin n → R) :
   simp [pow_succ']
 
 theorem vandermonde_succ {n : ℕ} (v : Fin n.succ → R) :
-    vandermonde v =
+    vandermonde v = .of
       Fin.cons (fun (j : Fin n.succ) => v 0 ^ (j : ℕ)) fun i =>
         Fin.cons 1 fun j => v i.succ * vandermonde (Fin.tail v) i j := by
   conv_lhs => rw [← Fin.cons_self_tail v, vandermonde_cons]
@@ -102,11 +102,12 @@ theorem det_vandermonde {n : ℕ} (v : Fin n → R) :
             ∑ k ∈ Finset.range (j + 1 : ℕ), v i.succ ^ k * v 0 ^ (j - k : ℕ) :=
       (det_mul_column (fun i => v (Fin.succ i) - v 0) _)
     _ = (∏ i ∈ Finset.univ, (v (Fin.succ i) - v 0)) *
-    det fun i j : Fin n => v (Fin.succ i) ^ (j : ℕ) := congr_arg _ ?_
+    det (of fun i j : Fin n => v (Fin.succ i) ^ (j : ℕ)) := congr_arg _ ?_
     _ = ∏ i : Fin n.succ, ∏ j ∈ Ioi i, (v j - v i) := by
       simp_rw [Fin.prod_univ_succ, Fin.prod_Ioi_zero, Fin.prod_Ioi_succ]
-      have h := ih (v ∘ Fin.succ)
-      unfold Function.comp at h
+      have h : (of fun i j : Fin n ↦ v i.succ ^ (j : ℕ)).det =
+          ∏ x : Fin n, ∏ y ∈ Ioi x, (v y.succ - v x.succ) := by
+        simpa using ih (v ∘ Fin.succ)
       rw [h]
 
   · intro i j
@@ -187,6 +188,7 @@ theorem eval_matrixOfPolynomials_eq_vandermonde_mul_matrixOfPolynomials {n : ℕ
   congr
   ext k
   rw [mul_comm, Matrix.of_apply, RingHom.id_apply]
+  rfl
 
 theorem det_eval_matrixOfPolynomials_eq_det_vandermonde {n : ℕ} (v : Fin n → R) (p : Fin n → R[X])
     (h_deg : ∀ i, (p i).natDegree = i) (h_monic : ∀ i, Monic <| p i) :
diff --git a/Mathlib/Logic/Basic.lean b/Mathlib/Logic/Basic.lean
index 2be615051376d..538530f266a43 100644
--- a/Mathlib/Logic/Basic.lean
+++ b/Mathlib/Logic/Basic.lean
@@ -37,11 +37,10 @@ section Miscellany
 --   And.decidable Or.decidable Decidable.false Xor.decidable Iff.decidable Decidable.true
 --   Implies.decidable Not.decidable Ne.decidable Bool.decidableEq Decidable.toBool
 
+-- attribute [refl] HEq.refl -- FIXME This is still rejected after #857
+attribute [trans] Iff.trans HEq.trans heq_of_eq_of_heq
 attribute [simp] cast_heq
 
--- This can be removed once we move to Lean v4.11
-attribute [simp] insert_emptyc_eq
-
 /-- An identity function with its main argument implicit. This will be printed as `hidden` even
 if it is applied to a large term, so it can be used for elision,
 as done in the `elide` and `unelide` tactics. -/
@@ -171,7 +170,7 @@ theorem eq_or_ne {α : Sort*} (x y : α) : x = y ∨ x ≠ y := em <| x = y
 
 theorem ne_or_eq {α : Sort*} (x y : α) : x ≠ y ∨ x = y := em' <| x = y
 
-theorem by_contradiction {p : Prop} : (¬p → False) → p := Decidable.by_contradiction
+theorem by_contradiction {p : Prop} : (¬p → False) → p := Decidable.byContradiction
 
 theorem by_cases {p q : Prop} (hpq : p → q) (hnpq : ¬p → q) : q :=
 if hp : p then hpq hp else hnpq hp
@@ -237,6 +236,11 @@ lemma Iff.ne_right {α β : Sort*} {a b : α} {c d : β} : (a ≠ b ↔ c = d) 
 
 /-! ### Declarations about `Xor'` -/
 
+/-- `Xor' a b` is the exclusive-or of propositions. -/
+def Xor' (a b : Prop) := (a ∧ ¬b) ∨ (b ∧ ¬a)
+
+instance [Decidable a] [Decidable b] : Decidable (Xor' a b) := inferInstanceAs (Decidable (Or ..))
+
 @[simp] theorem xor_true : Xor' True = Not := by
   simp (config := { unfoldPartialApp := true }) [Xor']
 
@@ -425,14 +429,20 @@ theorem rec_heq_iff_heq {α β : Sort _} {a b : α} {C : α → Sort*} {x : C a}
 theorem heq_rec_iff_heq {α β : Sort _} {a b : α} {C : α → Sort*} {x : β} {y : C a} {e : a = b} :
     HEq x (e ▸ y) ↔ HEq x y := by subst e; rfl
 
+universe u
+variable {α β : Sort u} {e : β = α} {a : α} {b : β}
+
+lemma heq_of_eq_cast (e : β = α) : a = cast e b → HEq a b := by rintro rfl; simp
+
+lemma eq_cast_iff_heq : a = cast e b ↔ HEq a b := ⟨heq_of_eq_cast _, fun h ↦ by cases h; rfl⟩
+
 end Equality
 
 /-! ### Declarations about quantifiers -/
 section Quantifiers
 section Dependent
 
-variable {α : Sort*} {β : α → Sort*} {γ : ∀ a, β a → Sort*} {δ : ∀ a b, γ a b → Sort*}
-  {ε : ∀ a b c, δ a b c → Sort*}
+variable {α : Sort*} {β : α → Sort*} {γ : ∀ a, β a → Sort*}
 
 theorem pi_congr {β' : α → Sort _} (h : ∀ a, β a = β' a) : (∀ a, β a) = ∀ a, β' a :=
   (funext h : β = β') ▸ rfl
@@ -458,7 +468,7 @@ theorem Exists₃.imp {p q : ∀ a b, γ a b → Prop} (h : ∀ a b c, p a b c 
 
 end Dependent
 
-variable {α β : Sort*} {p q : α → Prop}
+variable {α β : Sort*} {p : α → Prop}
 
 theorem forall_swap {p : α → β → Prop} : (∀ x y, p x y) ↔ ∀ y x, p x y :=
   ⟨fun f x y ↦ f y x, fun f x y ↦ f y x⟩
@@ -508,15 +518,6 @@ theorem forall₂_true_iff {β : α → Sort*} : (∀ a, β a → True) ↔ True
 theorem forall₃_true_iff {β : α → Sort*} {γ : ∀ a, β a → Sort*} :
     (∀ (a) (b : β a), γ a b → True) ↔ True := by simp
 
-@[simp] theorem exists_unique_iff_exists [Subsingleton α] {p : α → Prop} :
-    (∃! x, p x) ↔ ∃ x, p x :=
-  ⟨fun h ↦ h.exists, Exists.imp fun x hx ↦ ⟨hx, fun y _ ↦ Subsingleton.elim y x⟩⟩
-
--- forall_forall_const is no longer needed
-
-theorem exists_unique_const {b : Prop} (α : Sort*) [i : Nonempty α] [Subsingleton α] :
-    (∃! _ : α, b) ↔ b := by simp
-
 theorem Decidable.and_forall_ne [DecidableEq α] (a : α) {p : α → Prop} :
     (p a ∧ ∀ b, b ≠ a → p b) ↔ ∀ b, p b := by
   simp only [← @forall_eq _ p a, ← forall_and, ← or_imp, Decidable.em, forall_const]
@@ -527,12 +528,6 @@ theorem and_forall_ne (a : α) : (p a ∧ ∀ b, b ≠ a → p b) ↔ ∀ b, p b
 theorem Ne.ne_or_ne {x y : α} (z : α) (h : x ≠ y) : x ≠ z ∨ y ≠ z :=
   not_and_or.1 <| mt (and_imp.2 (· ▸ ·)) h.symm
 
-@[simp] theorem exists_unique_eq {a' : α} : ∃! a, a = a' := by
-  simp only [eq_comm, ExistsUnique, and_self, forall_eq', exists_eq']
-
-@[simp] theorem exists_unique_eq' {a' : α} : ∃! a, a' = a := by
-  simp only [ExistsUnique, and_self, forall_eq', exists_eq']
-
 @[simp]
 theorem exists_apply_eq_apply' (f : α → β) (a' : α) : ∃ a, f a' = f a := ⟨a', rfl⟩
 
@@ -611,10 +606,6 @@ protected theorem Decidable.forall_or_right {q} {p : α → Prop} [Decidable q]
 theorem forall_or_right {q} {p : α → Prop} : (∀ x, p x ∨ q) ↔ (∀ x, p x) ∨ q :=
   Decidable.forall_or_right
 
-theorem exists_unique_prop {p q : Prop} : (∃! _ : p, q) ↔ p ∧ q := by simp
-
-@[simp] theorem exists_unique_false : ¬∃! _ : α, False := fun ⟨_, h, _⟩ ↦ h
-
 theorem Exists.fst {b : Prop} {p : b → Prop} : Exists p → b
   | ⟨h, _⟩ => h
 
@@ -631,9 +622,6 @@ theorem Prop.forall_iff {p : Prop → Prop} : (∀ h, p h) ↔ p False ∧ p Tru
 theorem exists_iff_of_forall {p : Prop} {q : p → Prop} (h : ∀ h, q h) : (∃ h, q h) ↔ p :=
   ⟨Exists.fst, fun H ↦ ⟨H, h H⟩⟩
 
-theorem exists_unique_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃! h' : p, q h') ↔ q h :=
-  @exists_unique_const (q h) p ⟨h⟩ _
-
 theorem exists_prop_of_false {p : Prop} {q : p → Prop} : ¬p → ¬∃ h' : p, q h' :=
   mt Exists.fst
 
@@ -672,29 +660,6 @@ lemma iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) := propext ⟨propext, Eq.to_
 @[simp] theorem forall_true_left (p : True → Prop) : (∀ x, p x) ↔ p True.intro :=
   forall_prop_of_true _
 
-theorem ExistsUnique.elim₂ {α : Sort*} {p : α → Sort*} [∀ x, Subsingleton (p x)]
-    {q : ∀ (x) (_ : p x), Prop} {b : Prop} (h₂ : ∃! x, ∃! h : p x, q x h)
-    (h₁ : ∀ (x) (h : p x), q x h → (∀ (y) (hy : p y), q y hy → y = x) → b) : b := by
-  simp only [exists_unique_iff_exists] at h₂
-  apply h₂.elim
-  exact fun x ⟨hxp, hxq⟩ H ↦ h₁ x hxp hxq fun y hyp hyq ↦ H y ⟨hyp, hyq⟩
-
-theorem ExistsUnique.intro₂ {α : Sort*} {p : α → Sort*} [∀ x, Subsingleton (p x)]
-    {q : ∀ (x : α) (_ : p x), Prop} (w : α) (hp : p w) (hq : q w hp)
-    (H : ∀ (y) (hy : p y), q y hy → y = w) : ∃! x, ∃! hx : p x, q x hx := by
-  simp only [exists_unique_iff_exists]
-  exact ExistsUnique.intro w ⟨hp, hq⟩ fun y ⟨hyp, hyq⟩ ↦ H y hyp hyq
-
-theorem ExistsUnique.exists₂ {α : Sort*} {p : α → Sort*} {q : ∀ (x : α) (_ : p x), Prop}
-    (h : ∃! x, ∃! hx : p x, q x hx) : ∃ (x : _) (hx : p x), q x hx :=
-  h.exists.imp fun _ hx ↦ hx.exists
-
-theorem ExistsUnique.unique₂ {α : Sort*} {p : α → Sort*} [∀ x, Subsingleton (p x)]
-    {q : ∀ (x : α) (_ : p x), Prop} (h : ∃! x, ∃! hx : p x, q x hx) {y₁ y₂ : α}
-    (hpy₁ : p y₁) (hqy₁ : q y₁ hpy₁) (hpy₂ : p y₂) (hqy₂ : q y₂ hpy₂) : y₁ = y₂ := by
-  simp only [exists_unique_iff_exists] at h
-  exact h.unique ⟨hpy₁, hqy₁⟩ ⟨hpy₂, hqy₂⟩
-
 end Quantifiers
 
 /-! ### Classical lemmas -/
@@ -705,7 +670,7 @@ namespace Classical
 /-- Any prop `p` is decidable classically. A shorthand for `Classical.propDecidable`. -/
 noncomputable def dec (p : Prop) : Decidable p := by infer_instance
 
-variable {α : Sort*} {p : α → Prop}
+variable {α : Sort*}
 
 /-- Any predicate `p` is decidable classically. -/
 noncomputable def decPred (p : α → Prop) : DecidablePred p := by infer_instance
@@ -786,7 +751,7 @@ noncomputable def Exists.classicalRecOn {α : Sort*} {p : α → Prop} (h : ∃
 /-! ### Declarations about bounded quantifiers -/
 section BoundedQuantifiers
 
-variable {α : Sort*} {r p q : α → Prop} {P Q : ∀ x, p x → Prop} {b : Prop}
+variable {α : Sort*} {r p q : α → Prop} {P Q : ∀ x, p x → Prop}
 
 theorem bex_def : (∃ (x : _) (_ : p x), q x) ↔ ∃ x, p x ∧ q x :=
   ⟨fun ⟨x, px, qx⟩ ↦ ⟨x, px, qx⟩, fun ⟨x, px, qx⟩ ↦ ⟨x, px, qx⟩⟩
@@ -828,8 +793,6 @@ theorem exists_mem_of_exists (H : ∀ x, p x) : (∃ x, q x) → ∃ (x : _) (_
 theorem exists_of_exists_mem : (∃ (x : _) (_ : p x), q x) → ∃ x, q x
   | ⟨x, _, hq⟩ => ⟨x, hq⟩
 
-theorem exists₂_imp : (∃ x h, P x h) → b ↔ ∀ x h, P x h → b := by simp
-
 @[deprecated (since := "2024-03-23")] alias bex_of_exists := exists_mem_of_exists
 @[deprecated (since := "2024-03-23")] alias exists_of_bex := exists_of_exists_mem
 @[deprecated (since := "2024-03-23")] alias bex_imp := exists₂_imp
@@ -963,6 +926,9 @@ variable [Decidable Q]
 theorem ite_and : ite (P ∧ Q) a b = ite P (ite Q a b) b := by
   by_cases hp : P <;> by_cases hq : Q <;> simp [hp, hq]
 
+theorem ite_or : ite (P ∨ Q) a b = ite P a (ite Q a b) := by
+  by_cases hp : P <;> by_cases hq : Q <;> simp [hp, hq]
+
 theorem dite_dite_comm {B : Q → α} {C : ¬P → ¬Q → α} (h : P → ¬Q) :
     (if p : P then A p else if q : Q then B q else C p q) =
      if q : Q then B q else if p : P then A p else C p q :=
@@ -994,13 +960,29 @@ theorem dite_prop_iff_and {Q : P → Prop} {R : ¬P → Prop} :
     dite P Q R ↔ (∀ h, Q h) ∧ (∀ h, R h) := by
   by_cases h : P <;> simp [h, forall_prop_of_false, forall_prop_of_true]
 
+section congr
+
+variable [Decidable Q] {x y u v : α}
+
+theorem if_ctx_congr (h_c : P ↔ Q) (h_t : Q → x = u) (h_e : ¬Q → y = v) : ite P x y = ite Q u v :=
+  match ‹Decidable P›, ‹Decidable Q› with
+  | isFalse _,  isFalse h₂ => by simp_all
+  | isTrue _,   isTrue h₂  => by simp_all
+  | isFalse h₁, isTrue h₂  => absurd h₂ (Iff.mp (not_congr h_c) h₁)
+  | isTrue h₁,  isFalse h₂ => absurd h₁ (Iff.mpr (not_congr h_c) h₂)
+
+theorem if_congr (h_c : P ↔ Q) (h_t : x = u) (h_e : y = v) : ite P x y = ite Q u v :=
+  if_ctx_congr h_c (fun _ ↦ h_t) (fun _ ↦ h_e)
+
+end congr
+
 end ite
 
 theorem not_beq_of_ne {α : Type*} [BEq α] [LawfulBEq α] {a b : α} (ne : a ≠ b) : ¬(a == b) :=
   fun h => ne (eq_of_beq h)
 
 theorem beq_eq_decide {α : Type*} [BEq α] [LawfulBEq α] {a b : α} : (a == b) = decide (a = b) := by
-  rw [← beq_iff_eq a b]
+  rw [← beq_iff_eq (a := a) (b := b)]
   cases a == b <;> simp
 
 @[simp] lemma beq_eq_beq {α β : Type*} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {a₁ a₂ : α}
diff --git a/Mathlib/Logic/Denumerable.lean b/Mathlib/Logic/Denumerable.lean
index 94274fd7d2531..cf0e64c97bd1d 100644
--- a/Mathlib/Logic/Denumerable.lean
+++ b/Mathlib/Logic/Denumerable.lean
@@ -294,7 +294,7 @@ private theorem right_inverse_aux : ∀ n, toFunAux (ofNat s n) = n
       simp only [Finset.ext_iff, mem_insert, mem_range, mem_filter]
       exact fun m =>
         ⟨fun h => by
-          simp only [h.2, and_true_iff]
+          simp only [h.2, and_true]
           exact Or.symm (lt_or_eq_of_le ((@lt_succ_iff_le _ _ _ ⟨m, h.2⟩ _).1 h.1)),
          fun h =>
           h.elim (fun h => h.symm ▸ ⟨lt_succ_self _, (ofNat s n).prop⟩) fun h =>
diff --git a/Mathlib/Logic/Embedding/Set.lean b/Mathlib/Logic/Embedding/Set.lean
index bff9f99b31ff2..0c0ce8045a169 100644
--- a/Mathlib/Logic/Embedding/Set.lean
+++ b/Mathlib/Logic/Embedding/Set.lean
@@ -90,7 +90,7 @@ variable {α : Type*}
 subtypes `{x // p x} ⊕ {x // q x}` such that `¬ p x` is sent to the right, when
 `Disjoint p q`.
 
-See also `Equiv.sumCompl`, for when `IsCompl p q`.  -/
+See also `Equiv.sumCompl`, for when `IsCompl p q`. -/
 @[simps apply]
 def subtypeOrEquiv (p q : α → Prop) [DecidablePred p] (h : Disjoint p q) :
     { x // p x ∨ q x } ≃ { x // p x } ⊕ { x // q x } where
diff --git a/Mathlib/Logic/Encodable/Basic.lean b/Mathlib/Logic/Encodable/Basic.lean
index d6cac938c8892..60412ec201292 100644
--- a/Mathlib/Logic/Encodable/Basic.lean
+++ b/Mathlib/Logic/Encodable/Basic.lean
@@ -47,7 +47,7 @@ class Encodable (α : Type*) where
   --explicitly in Lean4
   /-- Decoding from ℕ to Option α-/
   decode : ℕ → Option α
-  /-- Invariant relationship between encoding and decoding-/
+  /-- Invariant relationship between encoding and decoding -/
   encodek : ∀ a, decode (encode a) = some a
 
 attribute [simp] Encodable.encodek
diff --git a/Mathlib/Logic/Encodable/Lattice.lean b/Mathlib/Logic/Encodable/Lattice.lean
index d5457863486ba..61e01cc0053a4 100644
--- a/Mathlib/Logic/Encodable/Lattice.lean
+++ b/Mathlib/Logic/Encodable/Lattice.lean
@@ -39,7 +39,7 @@ theorem iUnion_decode₂_cases {f : β → Set α} {C : Set α → Prop} (H0 : C
     C (⋃ b ∈ decode₂ β n, f b) :=
   match decode₂ β n with
   | none => by
-    simp only [Option.mem_def, iUnion_of_empty, iUnion_empty]
+    simp only [Option.mem_def, iUnion_of_empty, iUnion_empty, reduceCtorEq]
     apply H0
   | some b => by
     convert H1 b
diff --git a/Mathlib/Logic/Equiv/Array.lean b/Mathlib/Logic/Equiv/Array.lean
index cd9c6fbef7d1f..eb98b0edd323b 100644
--- a/Mathlib/Logic/Equiv/Array.lean
+++ b/Mathlib/Logic/Equiv/Array.lean
@@ -42,7 +42,7 @@ namespace Equiv
 
 /-- The natural equivalence between arrays and lists. -/
 def arrayEquivList (α : Type*) : Array α ≃ List α :=
-  ⟨Array.data, Array.mk, fun _ => rfl, fun _ => rfl⟩
+  ⟨Array.toList, Array.mk, fun _ => rfl, fun _ => rfl⟩
 
 end Equiv
 
diff --git a/Mathlib/Logic/Equiv/Basic.lean b/Mathlib/Logic/Equiv/Basic.lean
index 7366d0704cfe7..198c48db494a9 100644
--- a/Mathlib/Logic/Equiv/Basic.lean
+++ b/Mathlib/Logic/Equiv/Basic.lean
@@ -9,6 +9,7 @@ import Mathlib.Data.Prod.Basic
 import Mathlib.Data.Sigma.Basic
 import Mathlib.Data.Subtype
 import Mathlib.Data.Sum.Basic
+import Mathlib.Init.Algebra.Classes
 import Mathlib.Logic.Equiv.Defs
 import Mathlib.Logic.Function.Conjugate
 import Mathlib.Tactic.Coe
@@ -51,7 +52,7 @@ universe u v w z
 open Function
 
 -- Unless required to be `Type*`, all variables in this file are `Sort*`
-variable {α α₁ α₂ β β₁ β₂ γ γ₁ γ₂ δ : Sort*}
+variable {α α₁ α₂ β β₁ β₂ γ δ : Sort*}
 
 namespace Equiv
 
@@ -367,6 +368,26 @@ theorem sumAssoc_symm_apply_inr_inl {α β γ} (b) :
 theorem sumAssoc_symm_apply_inr_inr {α β γ} (c) : (sumAssoc α β γ).symm (inr (inr c)) = inr c :=
   rfl
 
+/-- Four-way commutativity of `sum`. The name matches `add_add_add_comm`. -/
+@[simps apply]
+def sumSumSumComm (α β γ δ) : (α ⊕ β) ⊕ γ ⊕ δ ≃ (α ⊕ γ) ⊕ β ⊕ δ where
+  toFun :=
+    (sumAssoc (α ⊕ γ) β δ) ∘ (Sum.map (sumAssoc α γ β).symm (@id δ))
+      ∘ (Sum.map (Sum.map (@id α) (sumComm β γ)) (@id δ))
+      ∘ (Sum.map (sumAssoc α β γ) (@id δ))
+      ∘ (sumAssoc (α ⊕ β) γ δ).symm
+  invFun :=
+    (sumAssoc (α ⊕ β) γ δ) ∘ (Sum.map (sumAssoc α β γ).symm (@id δ))
+      ∘ (Sum.map (Sum.map (@id α) (sumComm β γ).symm) (@id δ))
+      ∘ (Sum.map (sumAssoc α γ β) (@id δ))
+      ∘ (sumAssoc (α ⊕ γ) β δ).symm
+  left_inv x := by rcases x with ((a | b) | (c | d)) <;> simp
+  right_inv x := by rcases x with ((a | c) | (b | d)) <;> simp
+
+@[simp]
+theorem sumSumSumComm_symm (α β γ δ) : (sumSumSumComm α β γ δ).symm = sumSumSumComm α γ β δ :=
+  rfl
+
 /-- Sum with `IsEmpty` is equivalent to the original type. -/
 @[simps symm_apply]
 def sumEmpty (α β) [IsEmpty β] : α ⊕ β ≃ α where
@@ -474,7 +495,7 @@ the sum of the two subtypes `{a // p a}` and its complement `{a // ¬ p a}`
 is naturally equivalent to `α`.
 
 See `subtypeOrEquiv` for sum types over subtypes `{x // p x}` and `{x // q x}`
-that are not necessarily `IsCompl p q`.  -/
+that are not necessarily `IsCompl p q`. -/
 def sumCompl {α : Type*} (p : α → Prop) [DecidablePred p] :
     { a // p a } ⊕ { a // ¬p a } ≃ α where
   toFun := Sum.elim Subtype.val Subtype.val
@@ -608,7 +629,7 @@ section
 /-- A family of equivalences `∀ a, β₁ a ≃ β₂ a` generates an equivalence between `∀ a, β₁ a` and
 `∀ a, β₂ a`. -/
 def piCongrRight {β₁ β₂ : α → Sort*} (F : ∀ a, β₁ a ≃ β₂ a) : (∀ a, β₁ a) ≃ (∀ a, β₂ a) :=
-  ⟨fun H a => F a (H a), fun H a => (F a).symm (H a), fun H => funext <| by simp,
+  ⟨Pi.map fun a ↦ F a, Pi.map fun a ↦ (F a).symm, fun H => funext <| by simp,
     fun H => funext <| by simp⟩
 
 /-- Given `φ : α → β → Sort*`, we have an equivalence between `∀ a b, φ a b` and `∀ b a, φ a b`.
@@ -1559,8 +1580,6 @@ namespace Equiv
 
 section
 
-variable (P : α → Sort w) (e : α ≃ β)
-
 /-- Transport dependent functions through an equivalence of the base space.
 -/
 @[simps apply, simps (config := .lemmasOnly) symm_apply]
@@ -1669,7 +1688,7 @@ theorem piCongr_symm_apply (f : ∀ b, Z b) :
 
 @[simp]
 theorem piCongr_apply_apply (f : ∀ a, W a) (a : α) : h₁.piCongr h₂ f (h₁ a) = h₂ a (f a) := by
-  simp only [piCongr, piCongrRight, trans_apply, coe_fn_mk, piCongrLeft_apply_apply]
+  simp only [piCongr, piCongrRight, trans_apply, coe_fn_mk, piCongrLeft_apply_apply, Pi.map_apply]
 
 end
 
@@ -1713,9 +1732,13 @@ instance [Std.Associative f] : Std.Associative (e.arrowCongr (e.arrowCongr e) f)
 instance [Std.IdempotentOp f] : Std.IdempotentOp (e.arrowCongr (e.arrowCongr e) f) :=
   (e.semiconj₂_conj f).isIdempotent_right e.surjective
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-09-11")]
 instance [IsLeftCancel α₁ f] : IsLeftCancel β₁ (e.arrowCongr (e.arrowCongr e) f) :=
   ⟨e.surjective.forall₃.2 fun x y z => by simpa using @IsLeftCancel.left_cancel _ f _ x y z⟩
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-09-11")]
 instance [IsRightCancel α₁ f] : IsRightCancel β₁ (e.arrowCongr (e.arrowCongr e) f) :=
   ⟨e.surjective.forall₃.2 fun x y z => by simpa using @IsRightCancel.right_cancel _ f _ x y z⟩
 
@@ -1812,3 +1835,5 @@ theorem piCongrLeft'_symm_update [DecidableEq α] [DecidableEq β] (P : α → S
   simp [(e.piCongrLeft' P).symm_apply_eq, piCongrLeft'_update]
 
 end Function
+
+set_option linter.style.longFile 2000
diff --git a/Mathlib/Logic/Equiv/Defs.lean b/Mathlib/Logic/Equiv/Defs.lean
index 5cbf17cceb9f4..79b92f29b0e1b 100644
--- a/Mathlib/Logic/Equiv/Defs.lean
+++ b/Mathlib/Logic/Equiv/Defs.lean
@@ -494,6 +494,23 @@ theorem eq_symm_comp {α β γ} (e : α ≃ β) (f : γ → α) (g : γ → β)
 theorem symm_comp_eq {α β γ} (e : α ≃ β) (f : γ → α) (g : γ → β) : e.symm ∘ g = f ↔ g = e ∘ f :=
   ((Equiv.refl γ).arrowCongr e).symm_apply_eq
 
+theorem trans_eq_refl_iff_eq_symm {f : α ≃ β} {g : β ≃ α} :
+    f.trans g = Equiv.refl α ↔ f = g.symm := by
+  rw [← Equiv.coe_inj, coe_trans, coe_refl, ← eq_symm_comp, comp_id, Equiv.coe_inj]
+
+theorem trans_eq_refl_iff_symm_eq {f : α ≃ β} {g : β ≃ α} :
+    f.trans g = Equiv.refl α ↔ f.symm = g := by
+  rw [trans_eq_refl_iff_eq_symm]
+  exact ⟨fun h ↦ h ▸ rfl, fun h ↦ h ▸ rfl⟩
+
+theorem eq_symm_iff_trans_eq_refl {f : α ≃ β} {g : β ≃ α} :
+    f = g.symm ↔ f.trans g = Equiv.refl α :=
+  trans_eq_refl_iff_eq_symm.symm
+
+theorem symm_eq_iff_trans_eq_refl {f : α ≃ β} {g : β ≃ α} :
+    f.symm = g ↔ f.trans g = Equiv.refl α :=
+  trans_eq_refl_iff_symm_eq.symm
+
 /-- `PUnit` sorts in any two universes are equivalent. -/
 def punitEquivPUnit : PUnit.{v} ≃ PUnit.{w} :=
   ⟨fun _ => .unit, fun _ => .unit, fun ⟨⟩ => rfl, fun ⟨⟩ => rfl⟩
@@ -611,7 +628,7 @@ theorem sigmaCongrRight_symm {α} {β₁ β₂ : α → Type*} (F : ∀ a, β₁
 theorem sigmaCongrRight_refl {α} {β : α → Type*} :
     (sigmaCongrRight fun a => Equiv.refl (β a)) = Equiv.refl (Σ a, β a) := rfl
 
-/-- A `PSigma` with `Prop` fibers is equivalent to the subtype.  -/
+/-- A `PSigma` with `Prop` fibers is equivalent to the subtype. -/
 def psigmaEquivSubtype {α : Type v} (P : α → Prop) : (Σ' i, P i) ≃ Subtype P where
   toFun x := ⟨x.1, x.2⟩
   invFun x := ⟨x.1, x.2⟩
diff --git a/Mathlib/Logic/Equiv/Embedding.lean b/Mathlib/Logic/Equiv/Embedding.lean
index 5d383980cd4ea..10d04e64a9425 100644
--- a/Mathlib/Logic/Equiv/Embedding.lean
+++ b/Mathlib/Logic/Equiv/Embedding.lean
@@ -26,7 +26,7 @@ def sumEmbeddingEquivProdEmbeddingDisjoint {α β γ : Type*} :
       rintro _ ⟨a, h⟩ ⟨b, rfl⟩
       simp only [trans_apply, inl_apply, inr_apply] at h
       have : Sum.inl a = Sum.inr b := f.injective h
-      simp only at this⟩
+      simp only [reduceCtorEq] at this⟩
   invFun := fun ⟨⟨f, g⟩, disj⟩ =>
     ⟨fun x =>
       match x with
diff --git a/Mathlib/Logic/Equiv/Fin.lean b/Mathlib/Logic/Equiv/Fin.lean
index 1d07d2d1b2b25..3453d19eac020 100644
--- a/Mathlib/Logic/Equiv/Fin.lean
+++ b/Mathlib/Logic/Equiv/Fin.lean
@@ -470,7 +470,7 @@ def Int.divModEquiv (n : ℕ) [NeZero n] : ℤ ≃ ℤ × Fin n where
   toFun a := (a / n, ↑(a.natMod n))
   invFun p := p.1 * n + ↑p.2
   left_inv a := by
-    simp_rw [Fin.coe_ofNat_eq_mod, natCast_mod, natMod,
+    simp_rw [Fin.coe_natCast_eq_mod, natCast_mod, natMod,
       toNat_of_nonneg (emod_nonneg _ <| natCast_eq_zero.not.2 (NeZero.ne n)), emod_emod,
       ediv_add_emod']
   right_inv := fun ⟨q, r, hrn⟩ => by
diff --git a/Mathlib/Logic/Equiv/Functor.lean b/Mathlib/Logic/Equiv/Functor.lean
index e0f6ec21b8efa..203a282c37bf9 100644
--- a/Mathlib/Logic/Equiv/Functor.lean
+++ b/Mathlib/Logic/Equiv/Functor.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Simon Hudon, Scott Morrison
+Authors: Johan Commelin, Simon Hudon, Kim Morrison
 -/
 import Mathlib.Control.Bifunctor
 import Mathlib.Logic.Equiv.Defs
diff --git a/Mathlib/Logic/Equiv/List.lean b/Mathlib/Logic/Equiv/List.lean
index 2aaafa70d6bc5..7a050c68661b5 100644
--- a/Mathlib/Logic/Equiv/List.lean
+++ b/Mathlib/Logic/Equiv/List.lean
@@ -289,7 +289,7 @@ instance multiset : Denumerable (Multiset α) :=
         raise_lower (List.sorted_cons.2 ⟨fun n _ => Nat.zero_le n, (s.map encode).sort_sorted _⟩)
       simp [-Multiset.map_coe, this],
      fun n => by
-      simp [-Multiset.map_coe, List.mergeSort_eq_self _ (raise_sorted _ _), lower_raise]⟩
+      simp [-Multiset.map_coe, List.mergeSort'_eq_self _ (raise_sorted _ _), lower_raise]⟩
 
 end Multiset
 
@@ -344,7 +344,7 @@ instance finset : Denumerable (Finset α) :=
           raise_lower' (fun n _ => Nat.zero_le n) (Finset.sort_sorted_lt _)],
       fun n => by
       simp [-Multiset.map_coe, Finset.map, raise'Finset, Finset.sort,
-        List.mergeSort_eq_self (· ≤ ·) ((raise'_sorted _ _).imp (@le_of_lt _ _)), lower_raise']⟩
+        List.mergeSort'_eq_self (· ≤ ·) ((raise'_sorted _ _).imp (@le_of_lt _ _)), lower_raise']⟩
 
 end Finset
 
diff --git a/Mathlib/Logic/Equiv/Option.lean b/Mathlib/Logic/Equiv/Option.lean
index 68a899c718422..9c0c3d02003ee 100644
--- a/Mathlib/Logic/Equiv/Option.lean
+++ b/Mathlib/Logic/Equiv/Option.lean
@@ -130,7 +130,7 @@ theorem some_removeNone_iff {x : α} : some (removeNone e x) = e none ↔ e.symm
   · rw [removeNone_some _ ⟨a, h⟩]
     have h1 := congr_arg e.symm h
     rw [symm_apply_apply] at h1
-    simp only [false_iff_iff, apply_eq_iff_eq]
+    simp only [apply_eq_iff_eq, reduceCtorEq]
     simp [h1, apply_eq_iff_eq]
 
 @[simp]
diff --git a/Mathlib/Logic/Equiv/PartialEquiv.lean b/Mathlib/Logic/Equiv/PartialEquiv.lean
index ccfde61a6ffb6..2ea2f2a455748 100644
--- a/Mathlib/Logic/Equiv/PartialEquiv.lean
+++ b/Mathlib/Logic/Equiv/PartialEquiv.lean
@@ -719,7 +719,7 @@ theorem EqOnSource.restr {e e' : PartialEquiv α β} (he : e ≈ e') (s : Set α
 theorem EqOnSource.source_inter_preimage_eq {e e' : PartialEquiv α β} (he : e ≈ e') (s : Set β) :
     e.source ∩ e ⁻¹' s = e'.source ∩ e' ⁻¹' s := by rw [he.eqOn.inter_preimage_eq, source_eq he]
 
-/-- Composition of a partial equivlance and its inverse is equivalent to
+/-- Composition of a partial equivalence and its inverse is equivalent to
 the restriction of the identity to the source. -/
 theorem self_trans_symm : e.trans e.symm ≈ ofSet e.source := by
   have A : (e.trans e.symm).source = e.source := by mfld_set_tac
@@ -843,8 +843,8 @@ variable {ι : Type*} {αi βi γi : ι → Type*}
 /-- The product of a family of partial equivalences, as a partial equivalence on the pi type. -/
 @[simps (config := mfld_cfg) apply source target]
 protected def pi (ei : ∀ i, PartialEquiv (αi i) (βi i)) : PartialEquiv (∀ i, αi i) (∀ i, βi i) where
-  toFun f i := ei i (f i)
-  invFun f i := (ei i).symm (f i)
+  toFun := Pi.map fun i ↦ ei i
+  invFun := Pi.map fun i ↦ (ei i).symm
   source := pi univ fun i => (ei i).source
   target := pi univ fun i => (ei i).target
   map_source' _ hf i hi := (ei i).map_source (hf i hi)
diff --git a/Mathlib/Logic/Equiv/Set.lean b/Mathlib/Logic/Equiv/Set.lean
index fb51576130691..660731bfc4b8a 100644
--- a/Mathlib/Logic/Equiv/Set.lean
+++ b/Mathlib/Logic/Equiv/Set.lean
@@ -239,6 +239,9 @@ TODO: this is the same as `Equiv.setCongr`! -/
 protected def ofEq {α : Type u} {s t : Set α} (h : s = t) : s ≃ t :=
   Equiv.setCongr h
 
+lemma Equiv.strictMono_setCongr {α : Type*} [PartialOrder α] {S T : Set α} (h : S = T) :
+    StrictMono (setCongr h) := fun _ _ ↦ id
+
 /-- If `a ∉ s`, then `insert a s` is equivalent to `s ⊕ PUnit`. -/
 protected def insert {α} {s : Set.{u} α} [DecidablePred (· ∈ s)] {a : α} (H : a ∉ s) :
     (insert a s : Set α) ≃ s ⊕ PUnit.{u + 1} :=
diff --git a/Mathlib/Logic/Equiv/TransferInstance.lean b/Mathlib/Logic/Equiv/TransferInstance.lean
index 4c39c97683377..f4afabe33c38e 100644
--- a/Mathlib/Logic/Equiv/TransferInstance.lean
+++ b/Mathlib/Logic/Equiv/TransferInstance.lean
@@ -677,11 +677,13 @@ namespace Finite
 
 attribute [-instance] Fin.instMul
 
-/-- Any finite group in universe `u` is equivalent to some finite group in universe `0`. -/
-lemma exists_type_zero_nonempty_mulEquiv (G : Type u) [Group G] [Finite G] :
-    ∃ (G' : Type) (_ : Group G') (_ : Fintype G'), Nonempty (G ≃* G') := by
+/-- Any finite group in universe `u` is equivalent to some finite group in universe `v`. -/
+lemma exists_type_univ_nonempty_mulEquiv (G : Type u) [Group G] [Finite G] :
+    ∃ (G' : Type v) (_ : Group G') (_ : Fintype G'), Nonempty (G ≃* G') := by
   obtain ⟨n, ⟨e⟩⟩ := Finite.exists_equiv_fin G
-  letI groupH : Group (Fin n) := Equiv.group e.symm
-  exact ⟨Fin n, inferInstance, inferInstance, ⟨MulEquiv.symm <| Equiv.mulEquiv e.symm⟩⟩
+  let f : Fin n ≃ ULift (Fin n) := Equiv.ulift.symm
+  let e : G ≃ ULift (Fin n) := e.trans f
+  letI groupH : Group (ULift (Fin n)) := e.symm.group
+  exact ⟨ULift (Fin n), groupH, inferInstance, ⟨MulEquiv.symm <| e.symm.mulEquiv⟩⟩
 
 end Finite
diff --git a/Mathlib/Logic/ExistsUnique.lean b/Mathlib/Logic/ExistsUnique.lean
new file mode 100644
index 0000000000000..c41ef54d3dac5
--- /dev/null
+++ b/Mathlib/Logic/ExistsUnique.lean
@@ -0,0 +1,140 @@
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
+-/
+import Mathlib.Tactic.TypeStar
+
+/-!
+# `ExistsUnique`
+
+This file defines the `ExistsUnique` predicate, notated as `∃!`, and proves some of its
+basic properties.
+-/
+
+variable {α : Sort*}
+
+/-- For `p : α → Prop`, `ExistsUnique p` means that there exists a unique `x : α` with `p x`. -/
+def ExistsUnique (p : α → Prop) := ∃ x, p x ∧ ∀ y, p y → y = x
+
+namespace Mathlib.Notation
+open Lean
+
+/-- Checks to see that `xs` has only one binder. -/
+def isExplicitBinderSingular (xs : TSyntax ``explicitBinders) : Bool :=
+  match xs with
+  | `(explicitBinders| $_:binderIdent $[: $_]?) => true
+  | `(explicitBinders| ($_:binderIdent : $_)) => true
+  | _ => false
+
+open TSyntax.Compat in
+/--
+`∃! x : α, p x` means that there exists a unique `x` in `α` such that `p x`.
+This is notation for `ExistsUnique (fun (x : α) ↦ p x)`.
+
+This notation does not allow multiple binders like `∃! (x : α) (y : β), p x y`
+as a shorthand for `∃! (x : α), ∃! (y : β), p x y` since it is liable to be misunderstood.
+Often, the intended meaning is instead `∃! q : α × β, p q.1 q.2`.
+-/
+macro "∃!" xs:explicitBinders ", " b:term : term => do
+  if !isExplicitBinderSingular xs then
+    Macro.throwErrorAt xs "\
+      The `ExistsUnique` notation should not be used with more than one binder.\n\
+      \n\
+      The reason for this is that `∃! (x : α), ∃! (y : β), p x y` has a completely different \
+      meaning from `∃! q : α × β, p q.1 q.2`. \
+      To prevent confusion, this notation requires that you be explicit \
+      and use one with the correct interpretation."
+  expandExplicitBinders ``ExistsUnique xs b
+
+/--
+Pretty-printing for `ExistsUnique`, following the same pattern as pretty printing for `Exists`.
+However, it does *not* merge binders.
+-/
+@[app_unexpander ExistsUnique] def unexpandExistsUnique : Lean.PrettyPrinter.Unexpander
+  | `($(_) fun $x:ident ↦ $b)                      => `(∃! $x:ident, $b)
+  | `($(_) fun ($x:ident : $t) ↦ $b)               => `(∃! $x:ident : $t, $b)
+  | _                                               => throw ()
+
+/--
+`∃! x ∈ s, p x` means `∃! x, x ∈ s ∧ p x`, which is to say that there exists a unique `x ∈ s`
+such that `p x`.
+Similarly, notations such as `∃! x ≤ n, p n` are supported,
+using any relation defined using the `binder_predicate` command.
+-/
+syntax "∃! " binderIdent binderPred ", " term : term
+
+macro_rules
+  | `(∃! $x:ident $p:binderPred, $b) => `(∃! $x:ident, satisfies_binder_pred% $x $p ∧ $b)
+  | `(∃! _ $p:binderPred, $b) => `(∃! x, satisfies_binder_pred% x $p ∧ $b)
+
+end Mathlib.Notation
+
+-- @[intro] -- TODO
+theorem ExistsUnique.intro {p : α → Prop} (w : α)
+    (h₁ : p w) (h₂ : ∀ y, p y → y = w) : ∃! x, p x := ⟨w, h₁, h₂⟩
+
+theorem ExistsUnique.elim {p : α → Prop} {b : Prop}
+    (h₂ : ∃! x, p x) (h₁ : ∀ x, p x → (∀ y, p y → y = x) → b) : b :=
+  Exists.elim h₂ (fun w hw ↦ h₁ w (And.left hw) (And.right hw))
+
+theorem exists_unique_of_exists_of_unique {p : α → Prop}
+    (hex : ∃ x, p x) (hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) : ∃! x, p x :=
+  Exists.elim hex (fun x px ↦ ExistsUnique.intro x px (fun y (h : p y) ↦ hunique y x h px))
+
+theorem ExistsUnique.exists {p : α → Prop} : (∃! x, p x) → ∃ x, p x | ⟨x, h, _⟩ => ⟨x, h⟩
+
+theorem ExistsUnique.unique {p : α → Prop}
+    (h : ∃! x, p x) {y₁ y₂ : α} (py₁ : p y₁) (py₂ : p y₂) : y₁ = y₂ :=
+  let ⟨_, _, hy⟩ := h; (hy _ py₁).trans (hy _ py₂).symm
+
+-- TODO
+-- attribute [congr] forall_congr'
+-- attribute [congr] exists_congr'
+
+-- @[congr]
+theorem existsUnique_congr {p q : α → Prop} (h : ∀ a, p a ↔ q a) : (∃! a, p a) ↔ ∃! a, q a :=
+  exists_congr fun _ ↦ and_congr (h _) <| forall_congr' fun _ ↦ imp_congr_left (h _)
+
+@[simp] theorem exists_unique_iff_exists [Subsingleton α] {p : α → Prop} :
+    (∃! x, p x) ↔ ∃ x, p x :=
+  ⟨fun h ↦ h.exists, Exists.imp fun x hx ↦ ⟨hx, fun y _ ↦ Subsingleton.elim y x⟩⟩
+
+theorem exists_unique_const {b : Prop} (α : Sort*) [i : Nonempty α] [Subsingleton α] :
+    (∃! _ : α, b) ↔ b := by simp
+
+@[simp] theorem exists_unique_eq {a' : α} : ∃! a, a = a' := by
+  simp only [eq_comm, ExistsUnique, and_self, forall_eq', exists_eq']
+
+@[simp] theorem exists_unique_eq' {a' : α} : ∃! a, a' = a := by
+  simp only [ExistsUnique, and_self, forall_eq', exists_eq']
+
+theorem exists_unique_prop {p q : Prop} : (∃! _ : p, q) ↔ p ∧ q := by simp
+
+@[simp] theorem exists_unique_false : ¬∃! _ : α, False := fun ⟨_, h, _⟩ ↦ h
+
+theorem exists_unique_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃! h' : p, q h') ↔ q h :=
+  @exists_unique_const (q h) p ⟨h⟩ _
+
+theorem ExistsUnique.elim₂ {p : α → Sort*} [∀ x, Subsingleton (p x)]
+    {q : ∀ (x) (_ : p x), Prop} {b : Prop} (h₂ : ∃! x, ∃! h : p x, q x h)
+    (h₁ : ∀ (x) (h : p x), q x h → (∀ (y) (hy : p y), q y hy → y = x) → b) : b := by
+  simp only [exists_unique_iff_exists] at h₂
+  apply h₂.elim
+  exact fun x ⟨hxp, hxq⟩ H ↦ h₁ x hxp hxq fun y hyp hyq ↦ H y ⟨hyp, hyq⟩
+
+theorem ExistsUnique.intro₂ {p : α → Sort*} [∀ x, Subsingleton (p x)]
+    {q : ∀ (x : α) (_ : p x), Prop} (w : α) (hp : p w) (hq : q w hp)
+    (H : ∀ (y) (hy : p y), q y hy → y = w) : ∃! x, ∃! hx : p x, q x hx := by
+  simp only [exists_unique_iff_exists]
+  exact ExistsUnique.intro w ⟨hp, hq⟩ fun y ⟨hyp, hyq⟩ ↦ H y hyp hyq
+
+theorem ExistsUnique.exists₂ {p : α → Sort*} {q : ∀ (x : α) (_ : p x), Prop}
+    (h : ∃! x, ∃! hx : p x, q x hx) : ∃ (x : _) (hx : p x), q x hx :=
+  h.exists.imp fun _ hx ↦ hx.exists
+
+theorem ExistsUnique.unique₂ {p : α → Sort*} [∀ x, Subsingleton (p x)]
+    {q : ∀ (x : α) (_ : p x), Prop} (h : ∃! x, ∃! hx : p x, q x hx) {y₁ y₂ : α}
+    (hpy₁ : p y₁) (hqy₁ : q y₁ hpy₁) (hpy₂ : p y₂) (hqy₂ : q y₂ hpy₂) : y₁ = y₂ := by
+  simp only [exists_unique_iff_exists] at h
+  exact h.unique ⟨hpy₁, hqy₁⟩ ⟨hpy₂, hqy₂⟩
diff --git a/Mathlib/Logic/Function/Basic.lean b/Mathlib/Logic/Function/Basic.lean
index 35aaecce63413..1af46bf00e7fa 100644
--- a/Mathlib/Logic/Function/Basic.lean
+++ b/Mathlib/Logic/Function/Basic.lean
@@ -3,9 +3,9 @@ Copyright (c) 2016 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathlib.Init.Algebra.Classes
 import Mathlib.Data.Set.Defs
 import Mathlib.Logic.Basic
+import Mathlib.Logic.ExistsUnique
 import Mathlib.Logic.Nonempty
 import Batteries.Tactic.Init
 
@@ -118,10 +118,15 @@ theorem Injective.of_comp_iff' (f : α → β) {g : γ → α} (hg : Bijective g
     Injective (f ∘ g) ↔ Injective f :=
   ⟨fun I ↦ I.of_comp_right hg.2, fun h ↦ h.comp hg.injective⟩
 
+theorem Injective.piMap {ι : Sort*} {α β : ι → Sort*} {f : ∀ i, α i → β i}
+    (hf : ∀ i, Injective (f i)) : Injective (Pi.map f) := fun _ _ h ↦
+  funext fun i ↦ hf i <| congrFun h _
+
+@[deprecated (since := "2024-10-06")] alias injective_pi_map := Injective.piMap
+
 /-- Composition by an injective function on the left is itself injective. -/
-theorem Injective.comp_left {g : β → γ} (hg : Function.Injective g) :
-    Function.Injective (g ∘ · : (α → β) → α → γ) :=
-  fun _ _ hgf ↦ funext fun i ↦ hg <| (congr_fun hgf i : _)
+theorem Injective.comp_left {g : β → γ} (hg : Injective g) : Injective (g ∘ · : (α → β) → α → γ) :=
+  .piMap fun _ ↦ hg
 
 theorem injective_of_subsingleton [Subsingleton α] (f : α → β) : Injective f :=
   fun _ _ _ ↦ Subsingleton.elim _ _
@@ -329,7 +334,7 @@ theorem LeftInverse.eq_rightInverse {f : α → β} {g₁ g₂ : β → α} (h
     (h₂ : RightInverse g₂ f) : g₁ = g₂ :=
   calc
     g₁ = g₁ ∘ f ∘ g₂ := by rw [h₂.comp_eq_id, comp_id]
-     _ = g₂ := by rw [← comp.assoc, h₁.comp_eq_id, id_comp]
+     _ = g₂ := by rw [← comp_assoc, h₁.comp_eq_id, id_comp]
 
 attribute [local instance] Classical.propDecidable
 
@@ -359,7 +364,7 @@ end
 
 section InvFun
 
-variable {α β : Sort*} [Nonempty α] {f : α → β} {a : α} {b : β}
+variable {α β : Sort*} [Nonempty α] {f : α → β} {b : β}
 
 attribute [local instance] Classical.propDecidable
 
@@ -442,10 +447,22 @@ theorem surjective_to_subsingleton [na : Nonempty α] [Subsingleton β] (f : α
     Surjective f :=
   fun _ ↦ let ⟨a⟩ := na; ⟨a, Subsingleton.elim _ _⟩
 
+theorem Surjective.piMap {ι : Sort*} {α β : ι → Sort*} {f : ∀ i, α i → β i}
+    (hf : ∀ i, Surjective (f i)) : Surjective (Pi.map f) := fun g ↦
+  ⟨fun i ↦ surjInv (hf i) (g i), funext fun _ ↦ rightInverse_surjInv _ _⟩
+
+@[deprecated (since := "2024-10-06")] alias surjective_pi_map := Surjective.piMap
+
 /-- Composition by a surjective function on the left is itself surjective. -/
 theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
-    Surjective (g ∘ · : (α → β) → α → γ) := fun f ↦
-  ⟨surjInv hg ∘ f, funext fun _ ↦ rightInverse_surjInv _ _⟩
+    Surjective (g ∘ · : (α → β) → α → γ) :=
+  .piMap fun _ ↦ hg
+
+theorem Bijective.piMap {ι : Sort*} {α β : ι → Sort*} {f : ∀ i, α i → β i}
+    (hf : ∀ i, Bijective (f i)) : Bijective (Pi.map f) :=
+  ⟨.piMap fun i ↦ (hf i).1, .piMap fun i ↦ (hf i).2⟩
+
+@[deprecated (since := "2024-10-06")] alias bijective_pi_map := Bijective.piMap
 
 /-- Composition by a bijective function on the left is itself bijective. -/
 theorem Bijective.comp_left {g : β → γ} (hg : Bijective g) :
@@ -457,7 +474,7 @@ end SurjInv
 section Update
 
 variable {α : Sort u} {β : α → Sort v} {α' : Sort w} [DecidableEq α]
-  {f g : (a : α) → β a} {a : α} {b : β a}
+  {f : (a : α) → β a} {a : α} {b : β a}
 
 
 /-- Replacing the value of a function at a given point by a given value. -/
@@ -653,6 +670,10 @@ lemma FactorsThrough.extend_comp {g : α → γ} (e' : β → γ) (hf : FactorsT
     extend f g e' ∘ f = g :=
   funext fun a => hf.extend_apply e' a
 
+@[simp]
+lemma extend_const (f : α → β) (c : γ) : extend f (fun _ ↦ c) (fun _ ↦ c) = fun _ ↦ c :=
+  funext fun _ ↦ ite_id _
+
 @[simp]
 theorem extend_comp (hf : Injective f) (g : α → γ) (e' : β → γ) : extend f g e' ∘ f = g :=
   funext fun a ↦ hf.extend_apply g e' a
@@ -668,7 +689,7 @@ theorem Injective.surjective_comp_right [Nonempty γ] (hf : Injective f) :
 theorem Bijective.comp_right (hf : Bijective f) : Bijective fun g : β → γ ↦ g ∘ f :=
   ⟨hf.surjective.injective_comp_right, fun g ↦
     ⟨g ∘ surjInv hf.surjective,
-     by simp only [comp.assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp_id]⟩⟩
+     by simp only [comp_assoc g _ f, (leftInverse_surjInv hf).comp_eq_id, comp_id]⟩⟩
 
 end Extend
 
@@ -932,7 +953,7 @@ theorem Function.LeftInverse.eq_rec_eq {γ : β → Sort v} {f : α → β} {g :
     (h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
     -- TODO: mathlib3 uses `(congr_arg f (h a)).rec (C (g (f a)))` for LHS
     @Eq.rec β (f (g (f a))) (fun x _ ↦ γ x) (C (g (f a))) (f a) (congr_arg f (h a)) = C a :=
-  eq_of_heq <| (eq_rec_heq _ _).trans <| by rw [h]
+  eq_of_heq <| (eqRec_heq _ _).trans <| by rw [h]
 
 theorem Function.LeftInverse.eq_rec_on_eq {γ : β → Sort v} {f : α → β} {g : β → α}
     (h : Function.LeftInverse g f) (C : ∀ a : α, γ (f a)) (a : α) :
@@ -950,12 +971,9 @@ if for each pair of distinct points there is a function taking different values
 def Set.SeparatesPoints {α β : Type*} (A : Set (α → β)) : Prop :=
   ∀ ⦃x y : α⦄, x ≠ y → ∃ f ∈ A, (f x : β) ≠ f y
 
-theorem IsSymmOp.flip_eq (op) [IsSymmOp α β op] : flip op = op :=
-  funext fun a ↦ funext fun b ↦ (IsSymmOp.symm_op a b).symm
-
 theorem InvImage.equivalence {α : Sort u} {β : Sort v} (r : β → β → Prop) (f : α → β)
     (h : Equivalence r) : Equivalence (InvImage r f) :=
-  ⟨fun _ ↦ h.1 _, fun w ↦ h.symm w, fun h₁ h₂ ↦ InvImage.trans r f (fun _ _ _ ↦ h.trans) h₁ h₂⟩
+  ⟨fun _ ↦ h.1 _, h.symm, h.trans⟩
 
 instance {α β : Type*} {r : α → β → Prop} {x : α × β} [Decidable (r x.1 x.2)] :
   Decidable (uncurry r x) :=
diff --git a/Mathlib/Logic/Function/CompTypeclasses.lean b/Mathlib/Logic/Function/CompTypeclasses.lean
index 7fcd8ca32286c..69d8ee1e0294f 100644
--- a/Mathlib/Logic/Function/CompTypeclasses.lean
+++ b/Mathlib/Logic/Function/CompTypeclasses.lean
@@ -24,7 +24,7 @@ TODO :
 section CompTriple
 
 /-- Class of composing triples -/
-class CompTriple  {M N P : Type*} (φ : M → N) (ψ : N → P) (χ : outParam (M → P)) : Prop where
+class CompTriple {M N P : Type*} (φ : M → N) (ψ : N → P) (χ : outParam (M → P)) : Prop where
   /-- The maps form a commuting triangle -/
   comp_eq : ψ.comp φ = χ
 
diff --git a/Mathlib/Logic/Function/Conjugate.lean b/Mathlib/Logic/Function/Conjugate.lean
index 76dc5be2b2999..69098eb142cbf 100644
--- a/Mathlib/Logic/Function/Conjugate.lean
+++ b/Mathlib/Logic/Function/Conjugate.lean
@@ -31,7 +31,7 @@ def Semiconj (f : α → β) (ga : α → α) (gb : β → β) : Prop :=
 
 namespace Semiconj
 
-variable {f fab : α → β} {fbc : β → γ} {ga ga' : α → α} {gb gb' : β → β} {gc gc' : γ → γ}
+variable {f fab : α → β} {fbc : β → γ} {ga ga' : α → α} {gb gb' : β → β} {gc : γ → γ}
 
 /-- Definition of `Function.Semiconj` in terms of functional equality. -/
 lemma _root_.Function.semiconj_iff_comp_eq : Semiconj f ga gb ↔ f ∘ ga = gb ∘ f := funext_iff.symm
diff --git a/Mathlib/Logic/Function/Defs.lean b/Mathlib/Logic/Function/Defs.lean
index 57c703c66bfff..ab1505baf76fa 100644
--- a/Mathlib/Logic/Function/Defs.lean
+++ b/Mathlib/Logic/Function/Defs.lean
@@ -88,12 +88,9 @@ theorem comp_id (f : α → β) : f ∘ id = f := rfl
 @[deprecated (since := "2024-01-14")] alias right_id := comp_id
 @[deprecated (since := "2024-01-14")] alias comp.right_id := comp_id
 
-theorem comp.assoc (f : φ → δ) (g : β → φ) (h : α → β) : (f ∘ g) ∘ h = f ∘ g ∘ h :=
+theorem comp_assoc (f : φ → δ) (g : β → φ) (h : α → β) : (f ∘ g) ∘ h = f ∘ g ∘ h :=
   rfl
-
-@[simp] theorem const_comp {γ : Sort*} (f : α → β) (c : γ) : const β c ∘ f = const α c := rfl
-
-@[simp] theorem comp_const (f : β → φ) (b : β) : f ∘ const α b = const α (f b) := rfl
+@[deprecated (since := "2024-09-24")] alias comp.assoc := comp_assoc
 
 @[deprecated (since := "2024-01-14")] alias comp_const_right := comp_const
 
@@ -206,3 +203,16 @@ protected theorem RightInverse.id {g : β → α} {f : α → β} (h : RightInve
 def IsFixedPt (f : α → α) (x : α) := f x = x
 
 end Function
+
+namespace Pi
+
+variable {ι : Sort*} {α β : ι → Sort*}
+
+/-- Sends a dependent function `a : ∀ i, α i` to a dependent function `Pi.map f a : ∀ i, β i`
+by applying `f i` to `i`-th component. -/
+protected def map (f : ∀ i, α i → β i) : (∀ i, α i) → (∀ i, β i) := fun a i ↦ f i (a i)
+
+@[simp]
+lemma map_apply (f : ∀ i, α i → β i) (a : ∀ i, α i) (i : ι) : Pi.map f a i = f i (a i) := rfl
+
+end Pi
diff --git a/Mathlib/Logic/Function/FiberPartition.lean b/Mathlib/Logic/Function/FiberPartition.lean
new file mode 100644
index 0000000000000..e1463b5d0e46b
--- /dev/null
+++ b/Mathlib/Logic/Function/FiberPartition.lean
@@ -0,0 +1,71 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Data.Set.Basic
+/-!
+
+This file defines the type `f.Fiber` of fibers of a function `f : Y → Z`, and provides some API
+to work with and construct terms of this type.
+
+Note: this API is designed to be useful when defining the counit of the adjunction between
+the functor which takes a set to the condensed set corresponding to locally constant maps to that
+set, and the forgetful functor from the category of condensed sets to the category of sets
+(see PR #14027).
+-/
+
+variable {X Y Z : Type*}
+
+namespace Function
+
+/-- The indexing set of the partition. -/
+def Fiber (f : Y → Z) : Type _ := Set.range (fun (x : Set.range f) ↦ f ⁻¹' {x.val})
+
+namespace Fiber
+
+/--
+Any `a : Fiber f` is of the form `f ⁻¹' {x}` for some `x` in the image of `f`. We define `a.image`
+as an arbitrary such `x`.
+-/
+noncomputable def image (f : Y → Z) (a : Fiber f) : Z := a.2.choose.1
+
+lemma eq_fiber_image  (f : Y → Z) (a : Fiber f) : a.1 = f ⁻¹' {a.image} := a.2.choose_spec.symm
+
+/--
+Given `y : Y`, `Fiber.mk f y` is the fiber of `f` that `y` belongs to, as an element of `Fiber f`.
+-/
+def mk (f : Y → Z) (y : Y) : Fiber f := ⟨f ⁻¹' {f y}, by simp⟩
+
+/-- `y : Y` as a term of the type `Fiber.mk f y` -/
+def mkSelf (f : Y → Z) (y : Y) : (mk f y).val := ⟨y, rfl⟩
+
+lemma map_eq_image (f : Y → Z) (a : Fiber f) (x : a.1) : f x = a.image := by
+  have := a.2.choose_spec
+  rw [← Set.mem_singleton_iff, ← Set.mem_preimage]
+  convert x.prop
+
+lemma mk_image (f : Y → Z) (y : Y) : (Fiber.mk f y).image = f y :=
+  (map_eq_image (x := mkSelf f y)).symm
+
+lemma mem_iff_eq_image (f : Y → Z) (y : Y) (a : Fiber f) : y ∈ a.val ↔ f y = a.image :=
+  ⟨fun h ↦ a.map_eq_image _ ⟨y, h⟩, fun h ↦ by rw [a.eq_fiber_image]; exact h⟩
+
+/-- An arbitrary element of `a : Fiber f`. -/
+noncomputable def preimage (f : Y → Z) (a : Fiber f) : Y := a.2.choose.2.choose
+
+lemma map_preimage_eq_image (f : Y → Z) (a : Fiber f) : f a.preimage = a.image :=
+  a.2.choose.2.choose_spec
+
+lemma fiber_nonempty (f : Y → Z) (a : Fiber f) : Set.Nonempty a.val := by
+  refine ⟨preimage f a, ?_⟩
+  rw [mem_iff_eq_image, ← map_preimage_eq_image]
+
+lemma map_preimage_eq_image_map {W : Type*} (f : Y → Z) (g : Z → W) (a : Fiber (g ∘ f)) :
+    g (f a.preimage) = a.image := by rw [← map_preimage_eq_image, comp_apply]
+
+lemma image_eq_image_mk (f : Y → Z) (g : X → Y) (a : Fiber (f ∘ g)) :
+    a.image = (Fiber.mk f (g (a.preimage _))).image := by
+  rw [← map_preimage_eq_image_map _ _ a, mk_image]
+
+end Function.Fiber
diff --git a/Mathlib/Logic/Godel/GodelBetaFunction.lean b/Mathlib/Logic/Godel/GodelBetaFunction.lean
index 4fe1e162e2c5a..52ef094f4d92b 100644
--- a/Mathlib/Logic/Godel/GodelBetaFunction.lean
+++ b/Mathlib/Logic/Godel/GodelBetaFunction.lean
@@ -76,8 +76,7 @@ private lemma pairwise_coprime_coprimes (a : Fin m → ℕ) : Pairwise (Coprime
   have hja : j < supOfSeq a := lt_of_lt_of_le j.prop (le_step (le_max_left _ _))
   exact coprime_mul_succ
     (Nat.succ_le_succ <| le_of_lt ltij)
-    (Nat.dvd_factorial
-      (by simp [Nat.succ_sub_succ, ltij])
+    (Nat.dvd_factorial (by omega)
       (by simpa only [Nat.succ_sub_succ] using le_of_lt (lt_of_le_of_lt (sub_le j i) hja)))
 
 /-- Gödel's Beta Function. This is similar to `(Encodable.decodeList).get i`, but it is easier to
diff --git a/Mathlib/Logic/IsEmpty.lean b/Mathlib/Logic/IsEmpty.lean
index a7c0a41156632..b46a933890f1c 100644
--- a/Mathlib/Logic/IsEmpty.lean
+++ b/Mathlib/Logic/IsEmpty.lean
@@ -138,7 +138,7 @@ theorem not_isEmpty_iff : ¬IsEmpty α ↔ Nonempty α :=
 
 @[simp]
 theorem isEmpty_Prop {p : Prop} : IsEmpty p ↔ ¬p := by
-  simp only [← not_nonempty_iff, nonempty_Prop]
+  simp only [← not_nonempty_iff, nonempty_prop]
 
 @[simp]
 theorem isEmpty_pi {π : α → Sort*} : IsEmpty (∀ a, π a) ↔ ∃ a, IsEmpty (π a) := by
diff --git a/Mathlib/Logic/Lemmas.lean b/Mathlib/Logic/Lemmas.lean
index 1056a7ffc156f..3e92a9dc21060 100644
--- a/Mathlib/Logic/Lemmas.lean
+++ b/Mathlib/Logic/Lemmas.lean
@@ -24,7 +24,7 @@ theorem iff_right_comm {a b c : Prop} : ((a ↔ b) ↔ c) ↔ ((a ↔ c) ↔ b)
 
 protected alias ⟨HEq.eq, Eq.heq⟩ := heq_iff_eq
 
-variable {α : Sort*} {p q r : Prop} [Decidable p] [Decidable q] {a b c : α}
+variable {α : Sort*} {p q : Prop} [Decidable p] [Decidable q] {a b c : α}
 
 theorem dite_dite_distrib_left {a : p → α} {b : ¬p → q → α} {c : ¬p → ¬q → α} :
     (dite p a fun hp ↦ dite q (b hp) (c hp)) =
diff --git a/Mathlib/Logic/Nonempty.lean b/Mathlib/Logic/Nonempty.lean
index 67d8bd928a527..1c8198b9475d3 100644
--- a/Mathlib/Logic/Nonempty.lean
+++ b/Mathlib/Logic/Nonempty.lean
@@ -32,9 +32,7 @@ theorem Nonempty.exists {α} {p : Nonempty α → Prop} : (∃ h : Nonempty α,
 theorem exists_true_iff_nonempty {α : Sort*} : (∃ _ : α, True) ↔ Nonempty α :=
   Iff.intro (fun ⟨a, _⟩ ↦ ⟨a⟩) fun ⟨a⟩ ↦ ⟨a, trivial⟩
 
-@[simp]
-theorem nonempty_Prop {p : Prop} : Nonempty p ↔ p :=
-  Iff.intro (fun ⟨h⟩ ↦ h) fun h ↦ ⟨h⟩
+@[deprecated (since := "2024-08-30")] alias nonempty_Prop := nonempty_prop
 
 theorem Nonempty.imp {α} {p : Prop} : (Nonempty α → p) ↔ (α → p) :=
   Nonempty.forall
@@ -101,13 +99,6 @@ theorem Nonempty.elim_to_inhabited {α : Sort*} [h : Nonempty α] {p : Prop} (f
     p :=
   h.elim <| f ∘ Inhabited.mk
 
-protected instance Prod.instNonempty {α β} [h : Nonempty α] [h2 : Nonempty β] : Nonempty (α × β) :=
-  h.elim fun g ↦ h2.elim fun g2 ↦ ⟨⟨g, g2⟩⟩
-
-protected instance Pi.instNonempty {ι : Sort*} {α : ι → Sort*} [∀ i, Nonempty (α i)] :
-    Nonempty (∀ i, α i) :=
-  ⟨fun _ ↦ Classical.arbitrary _⟩
-
 theorem Classical.nonempty_pi {ι} {α : ι → Sort*} : Nonempty (∀ i, α i) ↔ ∀ i, Nonempty (α i) :=
   ⟨fun ⟨f⟩ a ↦ ⟨f a⟩, @Pi.instNonempty _ _⟩
 
diff --git a/Mathlib/Logic/Nontrivial/Defs.lean b/Mathlib/Logic/Nontrivial/Defs.lean
index 51b62d4be17b3..e0ac42a17d027 100644
--- a/Mathlib/Logic/Nontrivial/Defs.lean
+++ b/Mathlib/Logic/Nontrivial/Defs.lean
@@ -3,7 +3,6 @@ Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Init.Logic
 import Mathlib.Logic.Function.Defs
 import Mathlib.Tactic.TypeStar
 
diff --git a/Mathlib/Logic/OpClass.lean b/Mathlib/Logic/OpClass.lean
new file mode 100644
index 0000000000000..2f65fe0b3d85e
--- /dev/null
+++ b/Mathlib/Logic/OpClass.lean
@@ -0,0 +1,60 @@
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+
+/-!
+# Typeclasses for commuting heterogeneous operations
+
+The three classes in this file are for two-argument functions where one input is of type `α`,
+the output is of type `β` and the other input is of type `α` or `β`.
+They express the property that permuting arguments of type `α` does not change the result.
+
+## Main definitions
+
+* `IsSymmOp`: for `op : α → α → β`, `op a b = op b a`.
+* `LeftCommutative`: for `op : α → β → β`, `op a₁ (op a₂ b) = op a₂ (op a₁ b)`.
+* `RightCommutative`: for `op : β → α → β`, `op (op b a₁) a₂ = op (op b a₂) a₁`.
+-/
+
+universe u v
+
+variable {α : Sort u} {β : Sort v}
+
+/-- `IsSymmOp op` where `op : α → α → β` says that `op` is a symmetric operation,
+i.e. `op a b = op b a`.
+It is the natural generalisation of `Std.Commutative` (`β = α`) and `IsSymm` (`β = Prop`). -/
+class IsSymmOp (op : α → α → β) : Prop where
+  /-- A symmetric operation satisfies `op a b = op b a`. -/
+  symm_op : ∀ a b, op a b = op b a
+
+/-- `LeftCommutative op` where `op : α → β → β` says that `op` is a left-commutative operation,
+i.e. `op a₁ (op a₂ b) = op a₂ (op a₁ b)`. -/
+class LeftCommutative (op : α → β → β) : Prop where
+  /-- A left-commutative operation satisfies `op a₁ (op a₂ b) = op a₂ (op a₁ b)`. -/
+  left_comm : (a₁ a₂ : α) → (b : β) → op a₁ (op a₂ b) = op a₂ (op a₁ b)
+
+/-- `RightCommutative op` where `op : β → α → β` says that `op` is a right-commutative operation,
+i.e. `op (op b a₁) a₂ = op (op b a₂) a₁`. -/
+class RightCommutative (op : β → α → β) : Prop where
+  /-- A right-commutative operation satisfies `op (op b a₁) a₂ = op (op b a₂) a₁`. -/
+  right_comm : (b : β) → (a₁ a₂ : α) → op (op b a₁) a₂ = op (op b a₂) a₁
+
+instance (priority := 100) isSymmOp_of_isCommutative (α : Sort u) (op : α → α → α)
+    [Std.Commutative op] : IsSymmOp op where symm_op := Std.Commutative.comm
+
+theorem IsSymmOp.flip_eq (op : α → α → β) [IsSymmOp op] : flip op = op :=
+  funext fun a ↦ funext fun b ↦ (IsSymmOp.symm_op a b).symm
+
+instance {f : α → β → β} [h : LeftCommutative f] : RightCommutative (fun x y ↦ f y x) :=
+  ⟨fun _ _ _ ↦ (h.left_comm _ _ _).symm⟩
+
+instance {f : β → α → β} [h : RightCommutative f] : LeftCommutative (fun x y ↦ f y x) :=
+  ⟨fun _ _ _ ↦ (h.right_comm _ _ _).symm⟩
+
+instance {f : α → α → α} [hc : Std.Commutative f] [ha : Std.Associative f] : LeftCommutative f :=
+  ⟨fun a b c ↦ by rw [← ha.assoc, hc.comm a, ha.assoc]⟩
+
+instance {f : α → α → α} [hc : Std.Commutative f] [ha : Std.Associative f] : RightCommutative f :=
+  ⟨fun a b c ↦ by rw [ha.assoc, hc.comm b, ha.assoc]⟩
diff --git a/Mathlib/Logic/Pairwise.lean b/Mathlib/Logic/Pairwise.lean
index 26dd58b7b3c9e..13ae772203bbe 100644
--- a/Mathlib/Logic/Pairwise.lean
+++ b/Mathlib/Logic/Pairwise.lean
@@ -21,11 +21,11 @@ This file defines pairwise relations.
 
 open Set Function
 
-variable {α β γ ι ι' : Type*} {r p q : α → α → Prop}
+variable {α β ι : Type*} {r p : α → α → Prop}
 
 section Pairwise
 
-variable {f g : ι → α} {s t u : Set α} {a b : α}
+variable {f : ι → α} {s : Set α} {a b : α}
 
 /-- A relation `r` holds pairwise if `r i j` for all `i ≠ j`. -/
 def Pairwise (r : α → α → Prop) :=
diff --git a/Mathlib/Logic/Relation.lean b/Mathlib/Logic/Relation.lean
index db02735b7e8ed..2a90c042f467f 100644
--- a/Mathlib/Logic/Relation.lean
+++ b/Mathlib/Logic/Relation.lean
@@ -5,7 +5,6 @@ Authors: Johannes Hölzl
 -/
 import Mathlib.Logic.Function.Basic
 import Mathlib.Logic.Relator
-import Mathlib.Init.Data.Quot
 import Mathlib.Tactic.Use
 import Mathlib.Tactic.MkIffOfInductiveProp
 import Mathlib.Tactic.SimpRw
@@ -13,8 +12,8 @@ import Mathlib.Tactic.SimpRw
 /-!
 # Relation closures
 
-This file defines the reflexive, transitive, and reflexive transitive closures of relations.
-It also proves some basic results on definitions such as `EqvGen`.
+This file defines the reflexive, transitive, reflexive transitive and equivalence closures
+of relations and proves some basic results on them.
 
 Note that this is about unbundled relations, that is terms of types of the form `α → β → Prop`. For
 the bundled version, see `Rel`.
@@ -31,6 +30,8 @@ the bundled version, see `Rel`.
   the reflexive closure of the transitive closure, or the transitive closure of the reflexive
   closure. In terms of rewriting systems, this means that `a` can be rewritten to `b` in a number of
   rewrites.
+* `Relation.EqvGen`: Equivalence closure. `EqvGen r` relates everything `ReflTransGen r` relates,
+  plus for all related pairs it relates them in the opposite order.
 * `Relation.Comp`:  Relation composition. We provide notation `∘r`. For `r : α → β → Prop` and
   `s : β → γ → Prop`, `r ∘r s`relates `a : α` and `c : γ` iff there exists `b : β` that's related to
   both.
@@ -99,7 +100,7 @@ theorem Transitive.comap (h : Transitive r) (f : α → β) : Transitive (r on f
   fun _ _ _ hab hbc ↦ h hab hbc
 
 theorem Equivalence.comap (h : Equivalence r) (f : α → β) : Equivalence (r on f) :=
-  ⟨h.reflexive.comap f, @(h.symmetric.comap f), @(h.transitive.comap f)⟩
+  ⟨fun a ↦ h.refl (f a), h.symm, h.trans⟩
 
 end Comap
 
@@ -212,7 +213,7 @@ instance [Decidable (∃ a b, r a b ∧ f a = c ∧ g b = d)] : Decidable (Relat
 
 end Map
 
-variable {r : α → α → Prop} {a b c d : α}
+variable {r : α → α → Prop} {a b c : α}
 
 /-- `ReflTransGen r`: reflexive transitive closure of `r` -/
 @[mk_iff ReflTransGen.cases_tail_iff]
@@ -228,9 +229,16 @@ inductive ReflGen (r : α → α → Prop) (a : α) : α → Prop
   | refl : ReflGen r a a
   | single {b} : r a b → ReflGen r a b
 
-attribute [mk_iff] TransGen
-
+variable (r) in
+/-- `EqvGen r`: equivalence closure of `r`. -/
+@[mk_iff]
+inductive EqvGen : α → α → Prop
+  | rel x y : r x y → EqvGen x y
+  | refl x : EqvGen x x
+  | symm x y : EqvGen x y → EqvGen y x
+  | trans x y z : EqvGen x y → EqvGen y z → EqvGen x z
 
+attribute [mk_iff] TransGen
 attribute [refl] ReflGen.refl
 
 namespace ReflGen
@@ -335,9 +343,7 @@ theorem trans_left (hab : TransGen r a b) (hbc : ReflTransGen r b c) : TransGen
 instance : Trans (TransGen r) (ReflTransGen r) (TransGen r) :=
   ⟨trans_left⟩
 
-@[trans]
-theorem trans (hab : TransGen r a b) (hbc : TransGen r b c) : TransGen r a c :=
-  trans_left hab hbc.to_reflTransGen
+attribute [trans] trans
 
 instance : Trans (TransGen r) (TransGen r) (TransGen r) :=
   ⟨trans⟩
@@ -560,6 +566,34 @@ lemma reflTransGen_eq_reflGen (hr : Transitive r) :
 
 end ReflTransGen
 
+namespace EqvGen
+
+variable (r)
+
+theorem is_equivalence : Equivalence (@EqvGen α r) :=
+  Equivalence.mk EqvGen.refl (EqvGen.symm _ _) (EqvGen.trans _ _ _)
+
+/-- `EqvGen.setoid r` is the setoid generated by a relation `r`.
+
+The motivation for this definition is that `Quot r` behaves like `Quotient (EqvGen.setoid r)`,
+see for example `Quot.eqvGen_exact` and `Quot.eqvGen_sound`. -/
+def setoid : Setoid α :=
+  Setoid.mk _ (EqvGen.is_equivalence r)
+
+theorem mono {r p : α → α → Prop} (hrp : ∀ a b, r a b → p a b) (h : EqvGen r a b) :
+    EqvGen p a b := by
+  induction h with
+  | rel a b h => exact EqvGen.rel _ _ (hrp _ _ h)
+  | refl => exact EqvGen.refl _
+  | symm a b _ ih => exact EqvGen.symm _ _ ih
+  | trans a b c _ _ hab hbc => exact EqvGen.trans _ _ _ hab hbc
+
+@[deprecated (since := "2024-09-01")] alias _root_.EqvGen.is_equivalence := is_equivalence
+@[deprecated (since := "2024-09-01")] alias _root_.EqvGen.Setoid := setoid
+@[deprecated (since := "2024-09-01")] alias _root_.EqvGen.mono := mono
+
+end EqvGen
+
 /-- The join of a relation on a single type is a new relation for which
 pairs of terms are related if there is a third term they are both
 related to.  For example, if `r` is a relation representing rewrites
@@ -639,8 +673,31 @@ end Relation
 
 section EqvGen
 
+open Relation
+
 variable {r : α → α → Prop} {a b : α}
 
+theorem Quot.eqvGen_exact (H : Quot.mk r a = Quot.mk r b) : EqvGen r a b :=
+  @Quotient.exact _ (EqvGen.setoid r) a b (congrArg
+    (Quot.lift (Quotient.mk (EqvGen.setoid r)) (fun x y h ↦ Quot.sound (EqvGen.rel x y h))) H)
+
+theorem Quot.eqvGen_sound (H : EqvGen r a b) : Quot.mk r a = Quot.mk r b :=
+  EqvGen.rec
+    (fun _ _ h ↦ Quot.sound h)
+    (fun _ ↦ rfl)
+    (fun _ _ _ IH ↦ Eq.symm IH)
+    (fun _ _ _ _ _ IH₁ IH₂ ↦ Eq.trans IH₁ IH₂)
+    H
+
+instance Quotient.decidableEq {α : Sort*} {s : Setoid α} [d : ∀ a b : α, Decidable (a ≈ b)] :
+    DecidableEq (Quotient s) :=
+  fun q₁ q₂ : Quotient s ↦
+    Quotient.recOnSubsingleton₂ q₁ q₂
+      (fun a₁ a₂ ↦
+        match (d a₁ a₂) with
+        | (isTrue h₁)  => isTrue (Quotient.sound h₁)
+        | (isFalse h₂) => isFalse (fun h ↦ absurd (Quotient.exact h) h₂))
+
 theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
   Iff.intro
     (by
@@ -655,12 +712,7 @@ theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
 theorem Equivalence.eqvGen_eq (h : Equivalence r) : EqvGen r = r :=
   funext fun _ ↦ funext fun _ ↦ propext <| h.eqvGen_iff
 
-theorem EqvGen.mono {r p : α → α → Prop} (hrp : ∀ a b, r a b → p a b) (h : EqvGen r a b) :
-    EqvGen p a b := by
-  induction h with
-  | rel a b h => exact EqvGen.rel _ _ (hrp _ _ h)
-  | refl => exact EqvGen.refl _
-  | symm a b _ ih => exact EqvGen.symm _ _ ih
-  | trans a b c _ _ hab hbc => exact EqvGen.trans _ _ _ hab hbc
+@[deprecated (since := "2024-08-29")] alias Quot.exact := Quot.eqvGen_exact
+@[deprecated (since := "2024-08-29")] alias Quot.EqvGen_sound := Quot.eqvGen_sound
 
 end EqvGen
diff --git a/Mathlib/Logic/Relator.lean b/Mathlib/Logic/Relator.lean
index 7e0f0e054c1ac..17ab1660850b2 100644
--- a/Mathlib/Logic/Relator.lean
+++ b/Mathlib/Logic/Relator.lean
@@ -96,8 +96,8 @@ lemma rel_not : (Iff ⇒ Iff) Not Not :=
 lemma bi_total_eq {α : Type u₁} : Relator.BiTotal (@Eq α) :=
   { left := fun a => ⟨a, rfl⟩, right := fun a => ⟨a, rfl⟩ }
 
-variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
-variable {r : α → β → Prop} {p : β → γ → Prop} {q : γ → δ → Prop}
+variable {α : Type*} {β : Type*} {γ : Type*}
+variable {r : α → β → Prop}
 
 lemma LeftUnique.flip (h : LeftUnique r) : RightUnique (flip r) :=
   fun _ _ _ h₁ h₂ => h h₁ h₂
@@ -116,7 +116,7 @@ lemma rel_eq {r : α → β → Prop} (hr : BiUnique r) : (r ⇒ r ⇒ (·↔·)
 
 open Function
 
-variable {α : Type*} {r₁₁ : α → α → Prop} {r₁₂ : α → β → Prop} {r₂₁ : β → α → Prop}
+variable {r₁₁ : α → α → Prop} {r₁₂ : α → β → Prop} {r₂₁ : β → α → Prop}
   {r₂₃ : β → γ → Prop} {r₁₃ : α → γ → Prop}
 
 namespace LeftTotal
diff --git a/Mathlib/Logic/Small/Basic.lean b/Mathlib/Logic/Small/Basic.lean
index 902d79ca3fdf2..7024044988faa 100644
--- a/Mathlib/Logic/Small/Basic.lean
+++ b/Mathlib/Logic/Small/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Defs
 import Mathlib.Logic.Equiv.Set
@@ -16,6 +16,10 @@ universe u w v v'
 
 section
 
+
+-- TODO(timotree3): lower the priority on this instance?
+-- This instance applies to every synthesis problem of the form `Small ↥s` for some set `s`,
+-- but we have lots of instances of `Small` for specific set constructions.
 instance small_subtype (α : Type v) [Small.{w} α] (P : α → Prop) : Small.{w} { x // P x } :=
   small_map (equivShrink α).subtypeEquivOfSubtype'
 
diff --git a/Mathlib/Logic/Small/Defs.lean b/Mathlib/Logic/Small/Defs.lean
index c93c2ba60d7d1..51d7142ca561b 100644
--- a/Mathlib/Logic/Small/Defs.lean
+++ b/Mathlib/Logic/Small/Defs.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Equiv.Defs
 import Mathlib.Tactic.MkIffOfInductiveProp
diff --git a/Mathlib/Logic/Small/Group.lean b/Mathlib/Logic/Small/Group.lean
index 3ec5b0486b284..4242a307e776b 100644
--- a/Mathlib/Logic/Small/Group.lean
+++ b/Mathlib/Logic/Small/Group.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Defs
 import Mathlib.Logic.Equiv.TransferInstance
diff --git a/Mathlib/Logic/Small/List.lean b/Mathlib/Logic/Small/List.lean
index ef015f7089a1f..13b6fd0b0469a 100644
--- a/Mathlib/Logic/Small/List.lean
+++ b/Mathlib/Logic/Small/List.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Basic
 import Mathlib.Data.Vector.Basic
diff --git a/Mathlib/Logic/Small/Module.lean b/Mathlib/Logic/Small/Module.lean
index 4280027faf576..8dd1444204daa 100644
--- a/Mathlib/Logic/Small/Module.lean
+++ b/Mathlib/Logic/Small/Module.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Group
 import Mathlib.Logic.Small.Ring
diff --git a/Mathlib/Logic/Small/Ring.lean b/Mathlib/Logic/Small/Ring.lean
index 2c55cfd01e1c5..d64f33557a2c1 100644
--- a/Mathlib/Logic/Small/Ring.lean
+++ b/Mathlib/Logic/Small/Ring.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Defs
 import Mathlib.Logic.Equiv.TransferInstance
diff --git a/Mathlib/Logic/Small/Set.lean b/Mathlib/Logic/Small/Set.lean
index deae0a2f8ca8c..069f25c85eaa0 100644
--- a/Mathlib/Logic/Small/Set.lean
+++ b/Mathlib/Logic/Small/Set.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Markus Himmel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Markus Himmel
+Authors: Markus Himmel, Timothy Carlin-Burns
 -/
 import Mathlib.Data.Set.Lattice
 import Mathlib.Logic.Small.Basic
@@ -10,29 +10,111 @@ import Mathlib.Logic.Small.Basic
 # Results about `Small` on coerced sets
 -/
 
-universe u v w
+universe u u1 u2 u3 u4
 
-theorem small_subset {α : Type v} {s t : Set α} (hts : t ⊆ s) [Small.{u} s] : Small.{u} t :=
-  let f : t → s := fun x => ⟨x, hts x.prop⟩
-  @small_of_injective _ _ _ f fun _ _ hxy => Subtype.ext (Subtype.mk.inj hxy)
+variable {α : Type u1} {β : Type u2} {γ : Type u3} {ι : Type u4}
 
-instance small_range {α : Type v} {β : Type w} (f : α → β) [Small.{u} α] :
+theorem small_subset {s t : Set α} (hts : t ⊆ s) [Small.{u} s] : Small.{u} t :=
+  small_of_injective (Set.inclusion_injective hts)
+
+instance small_powerset (s : Set α) [Small.{u} s] : Small.{u} (𝒫 s) :=
+  small_map (Equiv.Set.powerset s)
+
+instance small_setProd (s : Set α) (t : Set β) [Small.{u} s] [Small.{u} t] :
+    Small.{u} (s ×ˢ t : Set (α × β)) :=
+  small_of_injective (Equiv.Set.prod s t).injective
+
+instance small_setPi {β : α → Type u2} (s : (a : α) → Set (β a))
+    [Small.{u} α] [∀ a, Small.{u} (s a)] : Small.{u} (Set.pi Set.univ s) :=
+  small_of_injective (Equiv.Set.univPi s).injective
+
+instance small_range (f : α → β) [Small.{u} α] :
     Small.{u} (Set.range f) :=
   small_of_surjective Set.surjective_onto_range
 
-instance small_image {α : Type v} {β : Type w} (f : α → β) (S : Set α) [Small.{u} S] :
-    Small.{u} (f '' S) :=
+instance small_image (f : α → β) (s : Set α) [Small.{u} s] :
+    Small.{u} (f '' s) :=
   small_of_surjective Set.surjective_onto_image
 
-instance small_union {α : Type v} (s t : Set α) [Small.{u} s] [Small.{u} t] :
+instance small_image2 (f : α → β → γ) (s : Set α) (t : Set β) [Small.{u} s] [Small.{u} t] :
+    Small.{u} (Set.image2 f s t) := by
+  rw [← Set.image_uncurry_prod]
+  infer_instance
+
+theorem small_univ_iff : Small.{u} (@Set.univ α) ↔ Small.{u} α :=
+  small_congr <| Equiv.Set.univ α
+
+instance small_univ [h : Small.{u} α] : Small.{u} (@Set.univ α) :=
+  small_univ_iff.2 h
+
+instance small_union (s t : Set α) [Small.{u} s] [Small.{u} t] :
     Small.{u} (s ∪ t : Set α) := by
   rw [← Subtype.range_val (s := s), ← Subtype.range_val (s := t), ← Set.Sum.elim_range]
   infer_instance
 
-instance small_iUnion {α : Type v} {ι : Type w} [Small.{u} ι] (s : ι → Set α)
+instance small_iUnion [Small.{u} ι] (s : ι → Set α)
     [∀ i, Small.{u} (s i)] : Small.{u} (⋃ i, s i) :=
   small_of_surjective <| Set.sigmaToiUnion_surjective _
 
-instance small_sUnion {α : Type v} (s : Set (Set α)) [Small.{u} s] [∀ t : s, Small.{u} t] :
+instance small_sUnion (s : Set (Set α)) [Small.{u} s] [∀ t : s, Small.{u} t] :
     Small.{u} (⋃₀ s) :=
   Set.sUnion_eq_iUnion ▸ small_iUnion _
+
+instance small_biUnion (s : Set ι) [Small.{u} s]
+    (f : (i : ι) → i ∈ s → Set α) [∀ i hi, Small.{u} (f i hi)] : Small.{u} (⋃ i, ⋃ hi, f i hi) :=
+  Set.biUnion_eq_iUnion s f ▸ small_iUnion _
+
+instance small_insert (x : α) (s : Set α) [Small.{u} s] :
+    Small.{u} (insert x s : Set α) :=
+  Set.insert_eq x s ▸ small_union.{u} {x} s
+
+instance small_diff (s t : Set α) [Small.{u} s] : Small.{u} (s \ t : Set α) :=
+  small_subset (Set.diff_subset)
+
+instance small_sep (s : Set α) (P : α → Prop) [Small.{u} s] :
+    Small.{u} { x | x ∈ s ∧ P x} :=
+  small_subset (Set.sep_subset s P)
+
+instance small_inter_of_left (s t : Set α) [Small.{u} s] :
+    Small.{u} (s ∩ t : Set α) :=
+  small_subset Set.inter_subset_left
+
+instance small_inter_of_right (s t : Set α) [Small.{u} t] :
+    Small.{u} (s ∩ t : Set α) :=
+  small_subset Set.inter_subset_right
+
+theorem small_iInter (s : ι → Set α) (i : ι)
+    [Small.{u} (s i)] : Small.{u} (⋂ i, s i) :=
+  small_subset (Set.iInter_subset s i)
+
+instance small_iInter' [Nonempty ι] (s : ι → Set α)
+    [∀ i, Small.{u} (s i)] : Small.{u} (⋂ i, s i) :=
+  let ⟨i⟩ : Nonempty ι := inferInstance
+  small_iInter s i
+
+theorem small_sInter {s : Set (Set α)} {t : Set α} (ht : t ∈ s)
+    [Small.{u} t] : Small.{u} (⋂₀ s) :=
+  Set.sInter_eq_iInter ▸ small_iInter _ ⟨t, ht⟩
+
+instance small_sInter' {s : Set (Set α)} [Nonempty s]
+    [∀ t : s, Small.{u} t] : Small.{u} (⋂₀ s) :=
+  let ⟨t⟩ : Nonempty s := inferInstance
+  small_sInter t.prop
+
+theorem small_biInter {s : Set ι} {i : ι} (hi : i ∈ s)
+    (f : (i : ι) → i ∈ s → Set α) [Small.{u} (f i hi)] : Small.{u} (⋂ i, ⋂ hi, f i hi) :=
+  Set.biInter_eq_iInter s f ▸ small_iInter _ ⟨i, hi⟩
+
+instance small_biInter' (s : Set ι) [Nonempty s]
+    (f : (i : ι) → i ∈ s → Set α) [∀ i hi, Small.{u} (f i hi)] : Small.{u} (⋂ i, ⋂ hi, f i hi) :=
+  let ⟨t⟩ : Nonempty s := inferInstance
+  small_biInter t.prop f
+
+theorem small_empty : Small.{u} (∅ : Set α) :=
+  inferInstance
+
+theorem small_single (x : α) : Small.{u} ({x} : Set α) :=
+  inferInstance
+
+theorem small_pair (x y : α) : Small.{u} ({x, y} : Set α) :=
+  inferInstance
diff --git a/Mathlib/Logic/Unique.lean b/Mathlib/Logic/Unique.lean
index 8588f1676e164..678d68e2ff016 100644
--- a/Mathlib/Logic/Unique.lean
+++ b/Mathlib/Logic/Unique.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
 import Mathlib.Logic.IsEmpty
-import Mathlib.Init.Logic
 import Mathlib.Tactic.Inhabit
 
 /-!
@@ -248,7 +247,6 @@ instance {α} [IsEmpty α] : Unique (Option α) :=
 end Option
 
 section Subtype
-variable {α : Sort u}
 
 instance Unique.subtypeEq (y : α) : Unique { x // x = y } where
   default := ⟨y, rfl⟩
diff --git a/Mathlib/Logic/UnivLE.lean b/Mathlib/Logic/UnivLE.lean
index aad869948aea4..d9f43bcd66b60 100644
--- a/Mathlib/Logic/UnivLE.lean
+++ b/Mathlib/Logic/UnivLE.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Logic.Small.Defs
 
@@ -14,7 +14,7 @@ in the form `∀ α : Type u, Small.{v} α`.
 See the doc-string for the comparison with an alternative stronger definition.
 -/
 
-set_option autoImplicit true
+universe u v w
 
 noncomputable section
 
@@ -35,7 +35,7 @@ See also `Mathlib.CategoryTheory.UnivLE` for the statement that the stronger def
 equivalent to `EssSurj (uliftFunctor : Type v ⥤ Type max u v)`.
 -/
 @[pp_with_univ]
-abbrev UnivLE.{u, v} : Prop := ∀ α : Type u, Small.{v} α
+abbrev UnivLE : Prop := ∀ α : Type u, Small.{v} α
 
 example : UnivLE.{u, u} := inferInstance
 example : UnivLE.{u, u+1} := inferInstance
@@ -43,7 +43,7 @@ example : UnivLE.{0, u} := inferInstance
 /- This is useless as an instance due to https://github.com/leanprover/lean4/issues/2297 -/
 theorem univLE_max : UnivLE.{u, max u v} := fun α ↦ small_max.{v} α
 
-theorem Small.trans_univLE.{u, v} (α : Type w) [hα : Small.{u} α] [h : UnivLE.{u, v}] :
+theorem Small.trans_univLE (α : Type w) [hα : Small.{u} α] [h : UnivLE.{u, v}] :
     Small.{v} α :=
   let ⟨β, ⟨f⟩⟩ := hα.equiv_small
   let ⟨_, ⟨g⟩⟩ := (h β).equiv_small
@@ -53,7 +53,7 @@ theorem UnivLE.trans [UnivLE.{u, v}] [UnivLE.{v, w}] : UnivLE.{u, w} :=
   fun α ↦ Small.trans_univLE α
 
 /- This is the crucial instance that subsumes `univLE_max`. -/
-instance univLE_of_max [UnivLE.{max u v, v}] : UnivLE.{u, v} := @UnivLE.trans univLE_max.{v,u} ‹_›
+instance univLE_of_max [UnivLE.{max u v, v}] : UnivLE.{u, v} := @UnivLE.trans univLE_max ‹_›
 
 /- When `small_Pi` from `Mathlib.Logic.Small.Basic` is imported, we have : -/
 -- example (α : Type u) (β : Type v) [UnivLE.{u, v}] : Small.{v} (α → β) := inferInstance
diff --git a/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean b/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
index c7a28a5184fbf..b7ac515ab27d0 100644
--- a/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
+++ b/Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
@@ -55,14 +55,6 @@ theorem borel_anti : Antitone (@borel α) := fun _ _ h =>
 theorem borel_eq_top_of_discrete [TopologicalSpace α] [DiscreteTopology α] : borel α = ⊤ :=
   top_le_iff.1 fun s _ => GenerateMeasurable.basic s (isOpen_discrete s)
 
-theorem borel_eq_top_of_countable [TopologicalSpace α] [T1Space α] [Countable α] : borel α = ⊤ := by
-  refine top_le_iff.1 fun s _ => biUnion_of_singleton s ▸ ?_
-  apply MeasurableSet.biUnion s.to_countable
-  intro x _
-  apply MeasurableSet.of_compl
-  apply GenerateMeasurable.basic
-  exact isClosed_singleton.isOpen_compl
-
 theorem borel_eq_generateFrom_of_subbasis {s : Set (Set α)} [t : TopologicalSpace α]
     [SecondCountableTopology α] (hs : t = .generateFrom s) : borel α = .generateFrom s :=
   le_antisymm
@@ -267,6 +259,9 @@ theorem IsClosed.nullMeasurableSet {μ} (h : IsClosed s) : NullMeasurableSet s 
 theorem IsCompact.measurableSet [T2Space α] (h : IsCompact s) : MeasurableSet s :=
   h.isClosed.measurableSet
 
+theorem IsCompact.nullMeasurableSet [T2Space α] {μ} (h : IsCompact s) : NullMeasurableSet s μ :=
+  h.isClosed.nullMeasurableSet
+
 /-- If two points are topologically inseparable,
 then they can't be separated by a Borel measurable set. -/
 theorem Inseparable.mem_measurableSet_iff {x y : γ} (h : Inseparable x y) {s : Set γ}
@@ -286,7 +281,7 @@ the measure of the closure of a compact set `K` is equal to the measure of `K`.
 
 See also `MeasureTheory.Measure.OuterRegular.measure_closure_eq_of_isCompact`
 for a version that assumes `μ` to be outer regular
-but does not assume the `σ`-algebra to be Borel.  -/
+but does not assume the `σ`-algebra to be Borel. -/
 theorem IsCompact.measure_closure [R1Space γ] {K : Set γ} (hK : IsCompact K) (μ : Measure γ) :
     μ (closure K) = μ K := by
   refine le_antisymm ?_ (measure_mono subset_closure)
@@ -339,6 +334,12 @@ instance (priority := 100) OpensMeasurableSpace.separatesPoints [T0Space α] :
   rw [inseparable_iff_forall_open]
   exact fun s hs => hxy _ hs.measurableSet
 
+theorem borel_eq_top_of_countable {α : Type*} [TopologicalSpace α] [T0Space α] [Countable α] :
+    borel α = ⊤ := by
+  refine top_unique fun s _ ↦ ?_
+  borelize α
+  exact .of_discrete
+
 -- see Note [lower instance priority]
 instance (priority := 100) OpensMeasurableSpace.toMeasurableSingletonClass [T1Space α] :
     MeasurableSingletonClass α :=
@@ -501,9 +502,8 @@ instance (priority := 100) ContinuousSub.measurableSub [Sub γ] [ContinuousSub 
   measurable_sub_const _ := (continuous_id.sub continuous_const).measurable
 
 @[to_additive]
-instance (priority := 100) TopologicalGroup.measurableInv [Group γ] [TopologicalGroup γ] :
-    MeasurableInv γ :=
-  ⟨continuous_inv.measurable⟩
+instance (priority := 100) ContinuousInv.measurableInv [Inv γ] [ContinuousInv γ] :
+    MeasurableInv γ := ⟨continuous_inv.measurable⟩
 
 @[to_additive]
 instance (priority := 100) ContinuousSMul.measurableSMul {M α} [TopologicalSpace M]
@@ -624,7 +624,7 @@ instance _root_.ULift.instBorelSpace : BorelSpace (ULift α) :=
 
 instance DiscreteMeasurableSpace.toBorelSpace {α : Type*} [TopologicalSpace α] [DiscreteTopology α]
     [MeasurableSpace α] [DiscreteMeasurableSpace α] : BorelSpace α := by
-  constructor; ext; simp [MeasurableSpace.measurableSet_generateFrom, measurableSet_discrete]
+  constructor; ext; simp [MeasurableSpace.measurableSet_generateFrom, MeasurableSet.of_discrete]
 
 protected theorem Embedding.measurableEmbedding {f : α → β} (h₁ : Embedding f)
     (h₂ : MeasurableSet (range f)) : MeasurableEmbedding f :=
diff --git a/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean b/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
index 8abe04e25e044..2a7e7bc08abc6 100644
--- a/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
+++ b/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
@@ -128,7 +128,7 @@ theorem tendsto_measure_cthickening {μ : Measure α} {s : Set α}
   have A : Tendsto (fun r => μ (cthickening r s)) (𝓝[Ioi 0] 0) (𝓝 (μ (closure s))) := by
     rw [closure_eq_iInter_cthickening]
     exact
-      tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.measurableSet)
+      tendsto_measure_biInter_gt (fun r _ => isClosed_cthickening.nullMeasurableSet)
         (fun i j _ ij => cthickening_mono ij _) hs
   have B : Tendsto (fun r => μ (cthickening r s)) (𝓝[Iic 0] 0) (𝓝 (μ (closure s))) := by
     apply Tendsto.congr' _ tendsto_const_nhds
@@ -151,7 +151,7 @@ theorem tendsto_measure_thickening {μ : Measure α} {s : Set α}
     (hs : ∃ R > 0, μ (thickening R s) ≠ ∞) :
     Tendsto (fun r => μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ (closure s))) := by
   rw [closure_eq_iInter_thickening]
-  exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.measurableSet)
+  exact tendsto_measure_biInter_gt (fun r _ => isOpen_thickening.nullMeasurableSet)
       (fun i j _ ij => thickening_mono ij _) hs
 
 /-- If a closed set has a thickening with finite measure, then the measure of its
diff --git a/Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean b/Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
index d848798cf7bd9..4bd14a8a98d53 100644
--- a/Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
+++ b/Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean
@@ -99,20 +99,29 @@ variable [MeasurableSpace δ]
 
 section Preorder
 
-variable [Preorder α] [OrderClosedTopology α] {a b x : α}
+variable [Preorder α] [OrderClosedTopology α] {a b x : α} {μ : Measure α}
 
 @[simp, measurability]
 theorem measurableSet_Ici : MeasurableSet (Ici a) :=
   isClosed_Ici.measurableSet
 
+theorem nullMeasurableSet_Ici : NullMeasurableSet (Ici a) μ :=
+  measurableSet_Ici.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Iic : MeasurableSet (Iic a) :=
   isClosed_Iic.measurableSet
 
+theorem nullMeasurableSet_Iic : NullMeasurableSet (Iic a) μ :=
+  measurableSet_Iic.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Icc : MeasurableSet (Icc a b) :=
   isClosed_Icc.measurableSet
 
+theorem nullMeasurableSet_Icc : NullMeasurableSet (Icc a b) μ :=
+  measurableSet_Icc.nullMeasurableSet
+
 instance nhdsWithin_Ici_isMeasurablyGenerated : (𝓝[Ici b] a).IsMeasurablyGenerated :=
   measurableSet_Ici.nhdsWithin_isMeasurablyGenerated _
 
@@ -157,7 +166,7 @@ end PartialOrder
 
 section LinearOrder
 
-variable [LinearOrder α] [OrderClosedTopology α] {a b x : α}
+variable [LinearOrder α] [OrderClosedTopology α] {a b x : α} {μ : Measure α}
 
 -- we open this locale only here to avoid issues with list being treated as intervals above
 open Interval
@@ -166,22 +175,37 @@ open Interval
 theorem measurableSet_Iio : MeasurableSet (Iio a) :=
   isOpen_Iio.measurableSet
 
+theorem nullMeasurableSet_Iio : NullMeasurableSet (Iio a) μ :=
+  measurableSet_Iio.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Ioi : MeasurableSet (Ioi a) :=
   isOpen_Ioi.measurableSet
 
+theorem nullMeasurableSet_Ioi : NullMeasurableSet (Ioi a) μ :=
+  measurableSet_Ioi.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Ioo : MeasurableSet (Ioo a b) :=
   isOpen_Ioo.measurableSet
 
+theorem nullMeasurableSet_Ioo : NullMeasurableSet (Ioo a b) μ :=
+  measurableSet_Ioo.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Ioc : MeasurableSet (Ioc a b) :=
   measurableSet_Ioi.inter measurableSet_Iic
 
+theorem nullMeasurableSet_Ioc : NullMeasurableSet (Ioc a b) μ :=
+  measurableSet_Ioc.nullMeasurableSet
+
 @[simp, measurability]
 theorem measurableSet_Ico : MeasurableSet (Ico a b) :=
   measurableSet_Ici.inter measurableSet_Iio
 
+theorem nullMeasurableSet_Ico : NullMeasurableSet (Ico a b) μ :=
+  measurableSet_Ico.nullMeasurableSet
+
 instance nhdsWithin_Ioi_isMeasurablyGenerated : (𝓝[Ioi b] a).IsMeasurablyGenerated :=
   measurableSet_Ioi.nhdsWithin_isMeasurablyGenerated _
 
@@ -281,7 +305,7 @@ theorem Dense.borel_eq_generateFrom_Ico_mem {α : Type*} [TopologicalSpace α] [
 theorem borel_eq_generateFrom_Ico (α : Type*) [TopologicalSpace α] [SecondCountableTopology α]
     [LinearOrder α] [OrderTopology α] :
     borel α = .generateFrom { S : Set α | ∃ (l u : α), l < u ∧ Ico l u = S } := by
-  simpa only [exists_prop, mem_univ, true_and_iff] using
+  simpa only [exists_prop, mem_univ, true_and] using
     (@dense_univ α _).borel_eq_generateFrom_Ico_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>
       mem_univ _
 
@@ -307,7 +331,7 @@ theorem Dense.borel_eq_generateFrom_Ioc_mem {α : Type*} [TopologicalSpace α] [
 theorem borel_eq_generateFrom_Ioc (α : Type*) [TopologicalSpace α] [SecondCountableTopology α]
     [LinearOrder α] [OrderTopology α] :
     borel α = .generateFrom { S : Set α | ∃ l u, l < u ∧ Ioc l u = S } := by
-  simpa only [exists_prop, mem_univ, true_and_iff] using
+  simpa only [exists_prop, mem_univ, true_and] using
     (@dense_univ α _).borel_eq_generateFrom_Ioc_mem_aux (fun _ _ => mem_univ _) fun _ _ _ _ =>
       mem_univ _
 
@@ -398,11 +422,11 @@ theorem ext_of_Iic {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
   · rcases exists_countable_dense_bot_top α with ⟨s, hsc, hsd, -, hst⟩
     have : DirectedOn (· ≤ ·) s := directedOn_iff_directed.2 (Subtype.mono_coe _).directed_le
     simp only [← biSup_measure_Iic hsc (hsd.exists_ge' hst) this, h]
-  rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic,
-    measure_diff (Iic_subset_Iic.2 hlt.le) measurableSet_Iic, h a, h b]
+  rw [← Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) nullMeasurableSet_Iic,
+    measure_diff (Iic_subset_Iic.2 hlt.le) nullMeasurableSet_Iic, h a, h b]
   · rw [← h a]
-    exact (measure_lt_top μ _).ne
-  · exact (measure_lt_top μ _).ne
+    exact measure_ne_top μ _
+  · exact measure_ne_top μ _
 
 /-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite
 intervals. -/
@@ -544,7 +568,8 @@ theorem Measurable.isLUB_of_mem {ι} [Countable ι] {f : ι → δ → α} {g g'
         · simp [hb, hg' hb]
       rw [this]
       exact Measurable.piecewise hs measurable_const g'_meas
-  · let f' : ι → δ → α := fun i ↦ s.piecewise (f i) g'
+  · have : Nonempty ι := ⟨i⟩
+    let f' : ι → δ → α := fun i ↦ s.piecewise (f i) g'
     suffices ∀ b, IsLUB { a | ∃ i, f' i b = a } (g b) from
       Measurable.isLUB (fun i ↦ Measurable.piecewise hs (hf i) g'_meas) this
     intro b
@@ -552,13 +577,7 @@ theorem Measurable.isLUB_of_mem {ι} [Countable ι] {f : ι → δ → α} {g g'
     · have A : ∀ i, f' i b = f i b := fun i ↦ by simp [f', hb]
       simpa [A] using hg b hb
     · have A : ∀ i, f' i b = g' b := fun i ↦ by simp [f', hb]
-      have : {a | ∃ (_i : ι), g' b = a} = {g' b} := by
-        apply Subset.antisymm
-        · rintro - ⟨_j, rfl⟩
-          simp only [mem_singleton_iff]
-        · rintro - rfl
-          exact ⟨i, rfl⟩
-      simp [A, this, hg' hb, isLUB_singleton]
+      simp [A, hg' hb, isLUB_singleton]
 
 theorem AEMeasurable.isLUB {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α} {g : δ → α}
     (hf : ∀ i, AEMeasurable (f i) μ) (hg : ∀ᵐ b ∂μ, IsLUB { a | ∃ i, f i b = a } (g b)) :
@@ -924,8 +943,8 @@ theorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow {α : Type*} [Meas
       ext x
       simp only [mem_singleton_iff, mem_union, mem_Ioo, mem_Ioi, mem_preimage]
       obtain (H | H) : f x = ∞ ∨ f x < ∞ := eq_or_lt_of_le le_top
-      · simp only [H, eq_self_iff_true, or_false_iff, ENNReal.zero_lt_top, not_top_lt, and_false]
-      · simp only [H, H.ne, and_true_iff, false_or_iff]
+      · simp only [H, eq_self_iff_true, or_false, ENNReal.zero_lt_top, not_top_lt, and_false]
+      · simp only [H, H.ne, and_true, false_or]
     · refine disjoint_left.2 fun x hx h'x => ?_
       have : f x < ∞ := h'x.2.2
       exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm))
diff --git a/Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean b/Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
index 9e03eae8ea344..f15add556e5f0 100644
--- a/Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
+++ b/Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean
@@ -480,7 +480,7 @@ theorem exists_spanning_measurableSet_le {α : Type*} {m : MeasurableSpace α} {
   let norm_sets := fun n : ℕ => { x | f x ≤ n }
   have norm_sets_spanning : ⋃ n, norm_sets n = Set.univ := by
     ext1 x
-    simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
+    simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true]
     exact exists_nat_ge (f x)
   let sets n := sigma_finite_sets n ∩ norm_sets n
   have h_meas : ∀ n, MeasurableSet (sets n) := by
@@ -505,7 +505,7 @@ variable (μ : Measure ℝ) [IsFiniteMeasureOnCompacts μ]
 lemma tendsto_measure_Icc_nhdsWithin_right' (b : ℝ) :
     Tendsto (fun δ ↦ μ (Icc (b - δ) (b + δ))) (𝓝[>] (0 : ℝ)) (𝓝 (μ {b})) := by
   rw [Real.singleton_eq_inter_Icc]
-  apply tendsto_measure_biInter_gt (fun r hr ↦ measurableSet_Icc)
+  apply tendsto_measure_biInter_gt (fun r hr ↦ nullMeasurableSet_Icc)
   · intro r s _rpos hrs
     exact Icc_subset_Icc (by linarith) (by linarith)
   · exact ⟨1, zero_lt_one, isCompact_Icc.measure_ne_top⟩
diff --git a/Mathlib/MeasureTheory/Constructions/Cylinders.lean b/Mathlib/MeasureTheory/Constructions/Cylinders.lean
index dcdfe7b8f1a19..3ef19d096a73e 100644
--- a/Mathlib/MeasureTheory/Constructions/Cylinders.lean
+++ b/Mathlib/MeasureTheory/Constructions/Cylinders.lean
@@ -150,11 +150,11 @@ section cylinder
 /-- Given a finite set `s` of indices, a cylinder is the preimage of a set `S` of `∀ i : s, α i` by
 the projection from `∀ i, α i` to `∀ i : s, α i`. -/
 def cylinder (s : Finset ι) (S : Set (∀ i : s, α i)) : Set (∀ i, α i) :=
-  (fun (f : ∀ i, α i) (i : s) ↦ f i) ⁻¹' S
+  s.restrict ⁻¹' S
 
 @[simp]
 theorem mem_cylinder (s : Finset ι) (S : Set (∀ i : s, α i)) (f : ∀ i, α i) :
-    f ∈ cylinder s S ↔ (fun i : s ↦ f i) ∈ S :=
+    f ∈ cylinder s S ↔ s.restrict f ∈ S :=
   mem_preimage
 
 @[simp]
@@ -178,7 +178,7 @@ theorem cylinder_eq_empty_iff [h_nonempty : Nonempty (∀ i, α i)] (s : Finset
   let f' : ∀ i, α i := fun i ↦ if hi : i ∈ s then f ⟨i, hi⟩ else h_nonempty.some i
   have hf' : f' ∈ cylinder s S := by
     rw [mem_cylinder]
-    simpa only [f', Finset.coe_mem, dif_pos]
+    simpa only [Finset.restrict_def, Finset.coe_mem, dif_pos, f']
   rw [h] at hf'
   exact not_mem_empty _ hf'
 
@@ -186,8 +186,8 @@ theorem inter_cylinder (s₁ s₂ : Finset ι) (S₁ : Set (∀ i : s₁, α i))
     [DecidableEq ι] :
     cylinder s₁ S₁ ∩ cylinder s₂ S₂ =
       cylinder (s₁ ∪ s₂)
-        ((fun f ↦ fun j : s₁ ↦ f ⟨j, Finset.mem_union_left s₂ j.prop⟩) ⁻¹' S₁ ∩
-          (fun f ↦ fun j : s₂ ↦ f ⟨j, Finset.mem_union_right s₁ j.prop⟩) ⁻¹' S₂) := by
+        (Finset.restrict₂ Finset.subset_union_left ⁻¹' S₁ ∩
+          Finset.restrict₂ Finset.subset_union_right ⁻¹' S₂) := by
   ext1 f; simp only [mem_inter_iff, mem_cylinder, mem_setOf_eq]; rfl
 
 theorem inter_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
@@ -198,8 +198,8 @@ theorem union_cylinder (s₁ s₂ : Finset ι) (S₁ : Set (∀ i : s₁, α i))
     [DecidableEq ι] :
     cylinder s₁ S₁ ∪ cylinder s₂ S₂ =
       cylinder (s₁ ∪ s₂)
-        ((fun f ↦ fun j : s₁ ↦ f ⟨j, Finset.mem_union_left s₂ j.prop⟩) ⁻¹' S₁ ∪
-          (fun f ↦ fun j : s₂ ↦ f ⟨j, Finset.mem_union_right s₁ j.prop⟩) ⁻¹' S₂) := by
+        (Finset.restrict₂ Finset.subset_union_left ⁻¹' S₁ ∪
+          Finset.restrict₂ Finset.subset_union_right ⁻¹' S₂) := by
   ext1 f; simp only [mem_union, mem_cylinder, mem_setOf_eq]; rfl
 
 theorem union_cylinder_same (s : Finset ι) (S₁ : Set (∀ i : s, α i)) (S₂ : Set (∀ i : s, α i)) :
@@ -217,7 +217,7 @@ theorem diff_cylinder_same (s : Finset ι) (S T : Set (∀ i : s, α i)) :
 theorem eq_of_cylinder_eq_of_subset [h_nonempty : Nonempty (∀ i, α i)] {I J : Finset ι}
     {S : Set (∀ i : I, α i)} {T : Set (∀ i : J, α i)} (h_eq : cylinder I S = cylinder J T)
     (hJI : J ⊆ I) :
-    S = (fun f : ∀ i : I, α i ↦ fun j : J ↦ f ⟨j, hJI j.prop⟩) ⁻¹' T := by
+    S = Finset.restrict₂ hJI ⁻¹' T := by
   rw [Set.ext_iff] at h_eq
   simp only [mem_cylinder] at h_eq
   ext1 f
@@ -225,22 +225,20 @@ theorem eq_of_cylinder_eq_of_subset [h_nonempty : Nonempty (∀ i, α i)] {I J :
   classical
   specialize h_eq fun i ↦ if hi : i ∈ I then f ⟨i, hi⟩ else h_nonempty.some i
   have h_mem : ∀ j : J, ↑j ∈ I := fun j ↦ hJI j.prop
-  simp only [Finset.coe_mem, dite_true, h_mem] at h_eq
-  exact h_eq
+  simpa only [Finset.restrict_def, Finset.coe_mem, dite_true, h_mem] using h_eq
 
 theorem cylinder_eq_cylinder_union [DecidableEq ι] (I : Finset ι) (S : Set (∀ i : I, α i))
     (J : Finset ι) :
     cylinder I S =
-      cylinder (I ∪ J) ((fun f ↦ fun j : I ↦ f ⟨j, Finset.mem_union_left J j.prop⟩) ⁻¹' S) := by
-  ext1 f; simp only [mem_cylinder, mem_preimage]
+      cylinder (I ∪ J) (Finset.restrict₂ Finset.subset_union_left ⁻¹' S) := by
+  ext1 f; simp only [mem_cylinder, Finset.restrict_def, Finset.restrict₂_def, mem_preimage]
 
 theorem disjoint_cylinder_iff [Nonempty (∀ i, α i)] {s t : Finset ι} {S : Set (∀ i : s, α i)}
     {T : Set (∀ i : t, α i)} [DecidableEq ι] :
     Disjoint (cylinder s S) (cylinder t T) ↔
       Disjoint
-        ((fun f : ∀ i : (s ∪ t : Finset ι), α i
-          ↦ fun j : s ↦ f ⟨j, Finset.mem_union_left t j.prop⟩) ⁻¹' S)
-        ((fun f ↦ fun j : t ↦ f ⟨j, Finset.mem_union_right s j.prop⟩) ⁻¹' T) := by
+        (Finset.restrict₂ Finset.subset_union_left ⁻¹' S)
+        (Finset.restrict₂ Finset.subset_union_right ⁻¹' T) := by
   simp_rw [Set.disjoint_iff, subset_empty_iff, inter_cylinder, cylinder_eq_empty_iff]
 
 theorem IsClosed.cylinder [∀ i, TopologicalSpace (α i)] (s : Finset ι) {S : Set (∀ i : s, α i)}
@@ -274,6 +272,12 @@ theorem mem_measurableCylinders (t : Set (∀ i, α i)) :
     t ∈ measurableCylinders α ↔ ∃ s S, MeasurableSet S ∧ t = cylinder s S := by
   simp_rw [measurableCylinders, mem_iUnion, exists_prop, mem_singleton_iff]
 
+@[measurability]
+theorem _root_.MeasurableSet.of_mem_measurableCylinders {s : Set (Π i, α i)}
+    (hs : s ∈ measurableCylinders α) : MeasurableSet s := by
+  obtain ⟨I, t, mt, rfl⟩ := (mem_measurableCylinders s).1 hs
+  exact mt.cylinder
+
 /-- A finset `s` such that `t = cylinder s S`. `S` is given by `measurableCylinders.set`. -/
 noncomputable def measurableCylinders.finset (ht : t ∈ measurableCylinders α) : Finset ι :=
   ((mem_measurableCylinders t).mp ht).choose
@@ -304,8 +308,8 @@ theorem inter_mem_measurableCylinders (hs : s ∈ measurableCylinders α)
   obtain ⟨s₂, S₂, hS₂, rfl⟩ := ht
   classical
   refine ⟨s₁ ∪ s₂,
-    (fun f ↦ (fun i ↦ f ⟨i, Finset.mem_union_left s₂ i.prop⟩ : ∀ i : s₁, α i)) ⁻¹' S₁ ∩
-      {f | (fun i ↦ f ⟨i, Finset.mem_union_right s₁ i.prop⟩ : ∀ i : s₂, α i) ∈ S₂}, ?_, ?_⟩
+    Finset.restrict₂ Finset.subset_union_left ⁻¹' S₁ ∩
+      {f | Finset.restrict₂ Finset.subset_union_right f ∈ S₂}, ?_, ?_⟩
   · refine MeasurableSet.inter ?_ ?_
     · exact measurable_pi_lambda _ (fun _ ↦ measurable_pi_apply _) hS₁
     · exact measurable_pi_lambda _ (fun _ ↦ measurable_pi_apply _) hS₂
@@ -353,7 +357,7 @@ theorem generateFrom_measurableCylinders :
     rintro t ht rfl
     refine ⟨{i}, {f | f ⟨i, Finset.mem_singleton_self i⟩ ∈ t i}, measurable_pi_apply _ (ht i), ?_⟩
     ext1 x
-    simp only [singleton_pi, Function.eval, mem_preimage, mem_cylinder, mem_setOf_eq]
+    simp only [mem_preimage, Function.eval, mem_cylinder, mem_setOf_eq, Finset.restrict]
 
 end cylinders
 
diff --git a/Mathlib/MeasureTheory/Constructions/HaarToSphere.lean b/Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
index 9326504897129..2b64a512ceb6e 100644
--- a/Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
+++ b/Mathlib/MeasureTheory/Constructions/HaarToSphere.lean
@@ -3,7 +3,7 @@ Copyright (c) 2023 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Algebra.Order.Pointwise
+import Mathlib.Algebra.Order.Field.Pointwise
 import Mathlib.Analysis.NormedSpace.SphereNormEquiv
 import Mathlib.Analysis.SpecialFunctions.Integrals
 import Mathlib.MeasureTheory.Constructions.Prod.Integral
@@ -27,7 +27,7 @@ for a general nontrivial normed space.
 open Set Function Metric MeasurableSpace intervalIntegral
 open scoped Pointwise ENNReal NNReal
 
-local notation "dim" => FiniteDimensional.finrank ℝ
+local notation "dim" => Module.finrank ℝ
 
 noncomputable section
 namespace MeasureTheory
@@ -39,7 +39,7 @@ namespace Measure
 
 /-- If `μ` is an additive Haar measure on a normed space `E`,
 then `μ.toSphere` is the measure on the unit sphere in `E`
-such that `μ.toSphere s = FiniteDimensional.finrank ℝ E • μ (Set.Ioo (0 : ℝ) 1 • s)`. -/
+such that `μ.toSphere s = Module.finrank ℝ E • μ (Set.Ioo (0 : ℝ) 1 • s)`. -/
 def toSphere (μ : Measure E) : Measure (sphere (0 : E) 1) :=
   dim E • ((μ.comap (Subtype.val ∘ (homeomorphUnitSphereProd E).symm)).restrict
     (univ ×ˢ Iio ⟨1, mem_Ioi.2 one_pos⟩)).fst
@@ -106,7 +106,7 @@ instance (n : ℕ) : SigmaFinite (volumeIoiPow n) :=
 
 /-- The homeomorphism `homeomorphUnitSphereProd E` sends an additive Haar measure `μ`
 to the product of `μ.toSphere` and `MeasureTheory.Measure.volumeIoiPow (dim E - 1)`,
-where `dim E = FiniteDimensional.finrank ℝ E` is the dimension of `E`. -/
+where `dim E = Module.finrank ℝ E` is the dimension of `E`. -/
 theorem measurePreserving_homeomorphUnitSphereProd :
     MeasurePreserving (homeomorphUnitSphereProd E) (μ.comap (↑))
       (μ.toSphere.prod (volumeIoiPow (dim E - 1))) := by
@@ -119,7 +119,7 @@ theorem measurePreserving_homeomorphUnitSphereProd :
     fun s hs ↦ forall_mem_range.2 fun r ↦ ?_
   have : Ioo (0 : ℝ) r = r.1 • Ioo (0 : ℝ) 1 := by
     rw [LinearOrderedField.smul_Ioo r.2.out, smul_zero, smul_eq_mul, mul_one]
-  have hpos : 0 < dim E := FiniteDimensional.finrank_pos
+  have hpos : 0 < dim E := Module.finrank_pos
   rw [(Homeomorph.measurableEmbedding _).map_apply, toSphere_apply' _ hs, volumeIoiPow_apply_Iio,
     comap_subtype_coe_apply (measurableSet_singleton _).compl, toSphere_apply_aux, this,
     smul_assoc, μ.addHaar_smul_of_nonneg r.2.out.le, Nat.sub_add_cancel hpos, Nat.cast_pred hpos,
diff --git a/Mathlib/MeasureTheory/Constructions/Pi.lean b/Mathlib/MeasureTheory/Constructions/Pi.lean
index 12c48b54b7654..300e10ce1b53a 100644
--- a/Mathlib/MeasureTheory/Constructions/Pi.lean
+++ b/Mathlib/MeasureTheory/Constructions/Pi.lean
@@ -211,8 +211,7 @@ variable {δ : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
 protected def tprod (l : List δ) (μ : ∀ i, Measure (π i)) : Measure (TProd π l) := by
   induction' l with i l ih
   · exact dirac PUnit.unit
-  · have := (μ i).prod (α := π i) ih
-    exact this
+  · exact (μ i).prod (α := π i) ih
 
 @[simp]
 theorem tprod_nil (μ : ∀ i, Measure (π i)) : Measure.tprod [] μ = dirac PUnit.unit :=
diff --git a/Mathlib/MeasureTheory/Constructions/Polish/Basic.lean b/Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
index f1677d8c92db5..89db09097d3cb 100644
--- a/Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
+++ b/Mathlib/MeasureTheory/Constructions/Polish/Basic.lean
@@ -343,7 +343,7 @@ protected lemma AnalyticSet.preimage {X Y : Type*} [TopologicalSpace X] [Topolog
     [PolishSpace X] [T2Space Y] {s : Set Y} (hs : AnalyticSet s) {f : X → Y} (hf : Continuous f) :
     AnalyticSet (f ⁻¹' s) := by
   rcases analyticSet_iff_exists_polishSpace_range.1 hs with ⟨Z, _, _, g, hg, rfl⟩
-  have : IsClosed {x : X × Z | f x.1 = g x.2} := isClosed_diagonal.preimage (hf.prod_map hg)
+  have : IsClosed {x : X × Z | f x.1 = g x.2} := isClosed_eq hf.fst' hg.snd'
   convert this.analyticSet.image_of_continuous continuous_fst
   ext x
   simp [eq_comm]
diff --git a/Mathlib/MeasureTheory/Constructions/Polish/EmbeddingReal.lean b/Mathlib/MeasureTheory/Constructions/Polish/EmbeddingReal.lean
index f6436e70180fc..67a330bbb5bb7 100644
--- a/Mathlib/MeasureTheory/Constructions/Polish/EmbeddingReal.lean
+++ b/Mathlib/MeasureTheory/Constructions/Polish/EmbeddingReal.lean
@@ -63,4 +63,9 @@ lemma measurableEmbedding_embeddingReal (Ω : Type*) [MeasurableSpace Ω] [Stand
     MeasurableEmbedding (embeddingReal Ω) :=
   (exists_measurableEmbedding_real Ω).choose_spec
 
+@[fun_prop]
+lemma measurable_embeddingReal (Ω : Type*) [MeasurableSpace Ω] [StandardBorelSpace Ω] :
+    Measurable (embeddingReal Ω) :=
+  (measurableEmbedding_embeddingReal Ω).measurable
+
 end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean b/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
index 7aa58edc685c0..a8e405fc234e1 100644
--- a/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
+++ b/Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
@@ -1012,6 +1012,9 @@ theorem fst_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α}
     (hY : Measurable Y) : (μ.map fun a => (X a, Y a)).fst = μ.map X :=
   fst_map_prod_mk₀ hY.aemeasurable
 
+@[gcongr]
+theorem fst_mono {μ : Measure (α × β)} (h : ρ ≤ μ) : ρ.fst ≤ μ.fst := map_mono h measurable_fst
+
 /-- Marginal measure on `β` obtained from a measure on `ρ` `α × β`, defined by `ρ.map Prod.snd`. -/
 noncomputable def snd (ρ : Measure (α × β)) : Measure β :=
   ρ.map Prod.snd
@@ -1056,6 +1059,9 @@ theorem snd_map_prod_mk {X : α → β} {Y : α → γ} {μ : Measure α} (hX :
     (μ.map fun a => (X a, Y a)).snd = μ.map Y :=
   snd_map_prod_mk₀ hX.aemeasurable
 
+@[gcongr]
+theorem snd_mono {μ : Measure (α × β)} (h : ρ ≤ μ) : ρ.snd ≤ μ.snd := map_mono h measurable_snd
+
 @[simp] lemma fst_map_swap : (ρ.map Prod.swap).fst = ρ.snd := by
   rw [Measure.fst, Measure.map_map measurable_fst measurable_swap]
   rfl
diff --git a/Mathlib/MeasureTheory/Constructions/Prod/Integral.lean b/Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
index fd2189d76b79a..25d335405cc0f 100644
--- a/Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
+++ b/Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
@@ -61,7 +61,7 @@ along one of the variables (using either the Lebesgue or Bochner integral) is me
 
 theorem measurableSet_integrable [SFinite ν] ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} := by
-  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
+  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
   exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const
 
 section
@@ -422,7 +422,7 @@ theorem integral_prod (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
   revert f
   apply Integrable.induction
   · intro c s hs h2s
-    simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp,
+    simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp_def,
       integral_indicator (measurable_prod_mk_left hs), setIntegral_const, integral_smul_const,
       integral_toReal (measurable_measure_prod_mk_left hs).aemeasurable
         (ae_measure_lt_top hs h2s.ne)]
diff --git a/Mathlib/MeasureTheory/Constructions/Projective.lean b/Mathlib/MeasureTheory/Constructions/Projective.lean
index 1b5505b2a390d..37ef05c547f45 100644
--- a/Mathlib/MeasureTheory/Constructions/Projective.lean
+++ b/Mathlib/MeasureTheory/Constructions/Projective.lean
@@ -40,18 +40,28 @@ variable {ι : Type*} {α : ι → Type*} [∀ i, MeasurableSpace (α i)]
 the projection from `∀ i : I, α i` to `∀ i : J, α i` maps `P I` to `P J`. -/
 def IsProjectiveMeasureFamily (P : ∀ J : Finset ι, Measure (∀ j : J, α j)) : Prop :=
   ∀ (I J : Finset ι) (hJI : J ⊆ I),
-    P J = (P I).map (fun (x : ∀ i : I, α i) (j : J) ↦ x ⟨j, hJI j.2⟩)
+    P J = (P I).map (Finset.restrict₂ hJI)
 
 namespace IsProjectiveMeasureFamily
 
 variable {I J : Finset ι}
 
+lemma eq_zero_of_isEmpty [h : IsEmpty (Π i, α i)]
+    (hP : IsProjectiveMeasureFamily P) (I : Finset ι) :
+    P I = 0 := by
+  classical
+  obtain ⟨i, hi⟩ := isEmpty_pi.mp h
+  rw [hP (insert i I) I (I.subset_insert i)]
+  have : IsEmpty (Π j : ↑(insert i I), α j) := by simp [hi]
+  rw [(P (insert i I)).eq_zero_of_isEmpty]
+  simp
+
 /-- Auxiliary lemma for `measure_univ_eq`. -/
 lemma measure_univ_eq_of_subset (hP : IsProjectiveMeasureFamily P) (hJI : J ⊆ I) :
     P I univ = P J univ := by
   classical
   have : (univ : Set (∀ i : I, α i)) =
-      (fun x : ∀ i : I, α i ↦ fun i : J ↦ x ⟨i, hJI i.2⟩) ⁻¹' (univ : Set (∀ i : J, α i)) := by
+      Finset.restrict₂ hJI ⁻¹' (univ : Set (∀ i : J, α i)) := by
     rw [preimage_univ]
   rw [this, ← Measure.map_apply _ MeasurableSet.univ]
   · rw [hP I J hJI]
@@ -79,7 +89,7 @@ lemma congr_cylinder_of_subset (hP : IsProjectiveMeasureFamily P)
     have : (univ : Set ((j : {x // x ∈ ({i} : Finset ι)}) → α j)) = ∅ := by simp [hi_empty]
     simp [this]
   | inr h =>
-    have : S = (fun f : ∀ i : I, α i ↦ fun j : J ↦ f ⟨j, hJI j.prop⟩) ⁻¹' T :=
+    have : S = Finset.restrict₂ hJI ⁻¹' T :=
       eq_of_cylinder_eq_of_subset h_eq hJI
     rw [hP I J hJI, Measure.map_apply _ hT, this]
     exact measurable_pi_lambda _ (fun _ ↦ measurable_pi_apply _)
@@ -89,9 +99,8 @@ lemma congr_cylinder (hP : IsProjectiveMeasureFamily P)
     (h_eq : cylinder I S = cylinder J T) :
     P I S = P J T := by
   classical
-  let U := (fun f : ∀ i : (I ∪ J : Finset ι), α i
-        ↦ fun j : I ↦ f ⟨j, Finset.mem_union_left J j.prop⟩) ⁻¹' S ∩
-      (fun f ↦ fun j : J ↦ f ⟨j, Finset.mem_union_right I j.prop⟩) ⁻¹' T
+  let U := Finset.restrict₂ Finset.subset_union_left ⁻¹' S ∩
+      Finset.restrict₂ Finset.subset_union_right ⁻¹' T
   suffices P (I ∪ J) U = P I S ∧ P (I ∪ J) U = P J T from this.1.symm.trans this.2
   constructor
   · have h_eq_union : cylinder I S = cylinder (I ∪ J) U := by
@@ -107,7 +116,7 @@ end IsProjectiveMeasureFamily
 for all `I : Finset ι`, the projection from `∀ i, α i` to `∀ i : I, α i` maps `μ` to `P I`. -/
 def IsProjectiveLimit (μ : Measure (∀ i, α i))
     (P : ∀ J : Finset ι, Measure (∀ j : J, α j)) : Prop :=
-  ∀ I : Finset ι, (μ.map fun x : ∀ i, α i ↦ fun i : I ↦ x i) = P I
+  ∀ I : Finset ι, (μ.map I.restrict) = P I
 
 namespace IsProjectiveLimit
 
diff --git a/Mathlib/MeasureTheory/Constructions/UnitInterval.lean b/Mathlib/MeasureTheory/Constructions/UnitInterval.lean
index 631b218cfe854..d3ea8002c5a93 100644
--- a/Mathlib/MeasureTheory/Constructions/UnitInterval.lean
+++ b/Mathlib/MeasureTheory/Constructions/UnitInterval.lean
@@ -25,7 +25,7 @@ theorem volume_def : (volume : Measure I) = volume.comap Subtype.val := rfl
 
 instance : IsProbabilityMeasure (volume : Measure I) where
   measure_univ := by
-    rw [Measure.Subtype.volume_univ measurableSet_Icc.nullMeasurableSet, Real.volume_Icc, sub_zero,
+    rw [Measure.Subtype.volume_univ nullMeasurableSet_Icc, Real.volume_Icc, sub_zero,
       ENNReal.ofReal_one]
 
 @[measurability]
diff --git a/Mathlib/MeasureTheory/Covering/Besicovitch.lean b/Mathlib/MeasureTheory/Covering/Besicovitch.lean
index 24f265572ac91..91cc3db5e2be0 100644
--- a/Mathlib/MeasureTheory/Covering/Besicovitch.lean
+++ b/Mathlib/MeasureTheory/Covering/Besicovitch.lean
@@ -309,7 +309,7 @@ theorem mem_iUnionUpTo_lastStep (x : β) : p.c x ∈ p.iUnionUpTo p.lastStep :=
     apply lt_trans (mul_pos (_root_.zero_lt_one.trans p.one_lt_tau) (p.rpos _)) H
   have B : p.τ⁻¹ * p.R p.lastStep < p.R p.lastStep := by
     conv_rhs => rw [← one_mul (p.R p.lastStep)]
-    exact mul_lt_mul (inv_lt_one p.one_lt_tau) le_rfl Rpos zero_le_one
+    exact mul_lt_mul (inv_lt_one_of_one_lt₀ p.one_lt_tau) le_rfl Rpos zero_le_one
   obtain ⟨y, hy1, hy2⟩ : ∃ y, p.c y ∉ p.iUnionUpTo p.lastStep ∧ p.τ⁻¹ * p.R p.lastStep < p.r y := by
     have := exists_lt_of_lt_csSup ?_ B
     · simpa only [exists_prop, mem_range, exists_exists_and_eq_and, Subtype.exists,
@@ -340,7 +340,7 @@ theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
   have color_i : p.color i = sInf (univ \ A) := by rw [color]
   rw [color_i]
   have N_mem : N ∈ univ \ A := by
-    simp only [A, not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff,
+    simp only [A, not_exists, true_and, exists_prop, mem_iUnion, mem_singleton_iff,
       mem_closedBall, not_and, mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]
     intro j ji _
     exact (IH j ji (ji.trans hi)).ne'
@@ -356,7 +356,7 @@ theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
     intro k hk
     rw [← Inf_eq_N] at hk
     have : k ∈ A := by
-      simpa only [true_and_iff, mem_univ, Classical.not_not, mem_diff] using
+      simpa only [true_and, mem_univ, Classical.not_not, mem_diff] using
         Nat.not_mem_of_lt_sInf hk
     simp only [mem_iUnion, mem_singleton_iff, exists_prop, Subtype.exists, exists_and_right,
       and_assoc] at this
@@ -491,15 +491,15 @@ theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
       rw [color_j]
       apply csInf_mem
       refine ⟨N, ?_⟩
-      simp only [A, not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
+      simp only [A, not_exists, true_and, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
         mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk]
       intro k hk _
       exact (p.color_lt (hk.trans jy_lt) hN).ne'
-    simp only [A, not_exists, true_and_iff, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
+    simp only [A, not_exists, true_and, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
       mem_univ, mem_diff, Subtype.exists, Subtype.coe_mk] at h
     specialize h jx jxy
     contrapose! h
-    simpa only [jxi, jyi, and_true_iff, eq_self_iff_true, ← not_disjoint_iff_nonempty_inter] using h
+    simpa only [jxi, jyi, and_true, eq_self_iff_true, ← not_disjoint_iff_nonempty_inter] using h
   · -- show that the balls of color at most `N` cover every center.
     refine range_subset_iff.2 fun b => ?_
     obtain ⟨a, ha⟩ :
@@ -621,9 +621,9 @@ theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMea
     rw [← diff_inter_self_eq_diff,
       measure_diff_le_iff_le_add _ inter_subset_right (measure_lt_top μ _).ne]
     swap
-    · apply MeasurableSet.inter _ omeas
-      haveI : Encodable (u i) := (u_count i).toEncodable
-      exact MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => measurableSet_closedBall
+    · exact .inter
+        (w.nullMeasurableSet_biUnion fun _ _ ↦ measurableSet_closedBall.nullMeasurableSet)
+        omeas.nullMeasurableSet
     calc
       μ o = 1 / (N + 1) * μ s + N / (N + 1) * μ s := by
         rw [μo, ← add_mul, ENNReal.div_add_div_same, add_comm, ENNReal.div_self, one_mul] <;> simp
@@ -708,7 +708,7 @@ theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measur
       exist_finset_disjoint_balls_large_measure μ hτ hN s' r (fun x hx => (rI x hx).1) fun x hx =>
         (rI x hx).2.le
     refine ⟨t ∪ Finset.image (fun x => (x, r x)) v, Finset.subset_union_left, ⟨?_, ?_, ?_⟩, ?_⟩
-    · simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and_iff, Finset.coe_image]
+    · simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and, Finset.coe_image]
       constructor
       · intro p hp q hq hpq
         rcases (mem_image _ _ _).1 hp with ⟨p', p'v, rfl⟩
@@ -781,8 +781,8 @@ theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measur
       rw [ENNReal.div_lt_iff, one_mul]
       · conv_lhs => rw [← add_zero (N : ℝ≥0∞)]
         exact ENNReal.add_lt_add_left (ENNReal.natCast_ne_top N) zero_lt_one
-      · simp only [true_or_iff, add_eq_zero, Ne, not_false_iff, one_ne_zero, and_false_iff]
-      · simp only [ENNReal.natCast_ne_top, Ne, not_false_iff, or_true_iff]
+      · simp only [true_or, add_eq_zero, Ne, not_false_iff, one_ne_zero, and_false]
+      · simp only [ENNReal.natCast_ne_top, Ne, not_false_iff, or_true]
     rw [zero_mul] at C
     apply le_bot_iff.1
     exact le_of_tendsto_of_tendsto' tendsto_const_nhds C fun n => (A n).trans (B n)
@@ -921,7 +921,7 @@ theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ
   refine ⟨t0 ∪ ⋃ i : Fin N, ((↑) : s' → α) '' S i, r, ?_, ?_, ?_, ?_, ?_⟩
   -- it remains to check that they have the desired properties
   · exact t0_count.union (countable_iUnion fun i => (S_count i).image _)
-  · simp only [t0s, true_and_iff, union_subset_iff, image_subset_iff, iUnion_subset_iff]
+  · simp only [t0s, true_and, union_subset_iff, image_subset_iff, iUnion_subset_iff]
     intro i x _
     exact s's x.2
   · intro x hx
diff --git a/Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean b/Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
index 97a0b59d00769..2c3bc40927bd3 100644
--- a/Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
+++ b/Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
@@ -43,7 +43,7 @@ In particular, this number is bounded by `5 ^ dim` by a straightforward measure
 
 universe u
 
-open Metric Set FiniteDimensional MeasureTheory Filter Fin
+open Metric Set Module MeasureTheory Filter Fin
 
 open scoped ENNReal Topology
 
@@ -461,13 +461,13 @@ theorem exists_normalized {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) (las
   · simp_rw [c', Hj, hij.trans Hj, if_true]
     exact exists_normalized_aux1 a lastr hτ δ hδ1 hδ2 i j inej
   -- case `2 < ‖c j‖`
-  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hj
+  · have H'j : ‖a.c j‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false] using Hj
     rcases le_or_lt ‖a.c i‖ 2 with (Hi | Hi)
     · -- case `‖c i‖ ≤ 2`
       simp_rw [c', Hi, if_true, H'j, if_false]
       exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj
     · -- case `2 < ‖c i‖`
-      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false_iff] using Hi
+      have H'i : ‖a.c i‖ ≤ 2 ↔ False := by simpa only [not_le, iff_false] using Hi
       simp_rw [c', H'i, if_false, H'j, if_false]
       exact exists_normalized_aux3 a lastc lastr hτ δ hδ1 i j inej Hi hij
 
diff --git a/Mathlib/MeasureTheory/Covering/DensityTheorem.lean b/Mathlib/MeasureTheory/Covering/DensityTheorem.lean
index 35d1a28b7705d..62c58768b78a6 100644
--- a/Mathlib/MeasureTheory/Covering/DensityTheorem.lean
+++ b/Mathlib/MeasureTheory/Covering/DensityTheorem.lean
@@ -69,7 +69,7 @@ theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ
     (rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x := by
   let R := scalingScaleOf μ (max (4 * K + 3) 3)
   simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
-    isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
+    isClosed_ball, true_and, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
     measurableSet_closedBall]
   /- The measure is doubling on scales smaller than `R`. Therefore, we treat differently small
     and large balls. For large balls, this follows directly from the enlargement we used in the
@@ -123,7 +123,7 @@ theorem tendsto_closedBall_filterAt {K : ℝ} {x : α} {ι : Type*} {l : Filter
     apply (((Metric.tendsto_nhds.mp δlim _ (div_pos hε hK)).and δpos).and xmem).mono
     rintro j ⟨⟨hjε, hj₀ : 0 < δ j⟩, hx⟩ y hy
     replace hjε : (K + 1) * δ j < ε := by
-      simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff' hK).mp hjε
+      simpa [abs_eq_self.mpr hj₀.le] using (lt_div_iff₀' hK).mp hjε
     simp only [mem_closedBall] at hx hy ⊢
     linarith [dist_triangle_right y x (w j)]
 
diff --git a/Mathlib/MeasureTheory/Covering/Differentiation.lean b/Mathlib/MeasureTheory/Covering/Differentiation.lean
index abe8892caee43..803d605ca1b7f 100644
--- a/Mathlib/MeasureTheory/Covering/Differentiation.lean
+++ b/Mathlib/MeasureTheory/Covering/Differentiation.lean
@@ -245,7 +245,7 @@ theorem ae_tendsto_div : ∀ᵐ x ∂μ, ∃ c, Tendsto (fun a => ρ a / μ a) (
       intro x h1x _
       apply h1x.mono fun a ha => ?_
       refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
-      simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]
+      simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
     · simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
         mem_setOf_eq, mem_compl_iff, not_forall]
       intro x _ h2x
@@ -324,7 +324,7 @@ theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
           (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
         toMeasurable μ sᶜ ∪
           ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) := by
-      simp only [inter_union_distrib_left, union_inter_distrib_right, true_and_iff,
+      simp only [inter_union_distrib_left, union_inter_distrib_right, true_and,
         subset_union_left, union_subset_iff, inter_self]
       refine ⟨?_, ?_, ?_⟩
       · exact inter_subset_right.trans subset_union_left
@@ -368,7 +368,7 @@ theorem exists_measurable_supersets_limRatio {p q : ℝ≥0} (hpq : p < q) :
         apply I.frequently.mono fun a ha => ?_
         rw [coe_nnreal_smul_apply]
         refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
-        simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]
+        simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
       _ = p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by
         simp only [coe_nnreal_smul_apply,
           measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I]
@@ -448,7 +448,7 @@ theorem measure_le_mul_of_subset_limRatioMeas_lt {p : ℝ≥0} {s : Set α}
   apply I.frequently.mono fun a ha => ?_
   rw [coe_nnreal_smul_apply]
   refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
-  simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]
+  simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
 
 /-- If, for all `x` in a set `s`, one has frequently `q < ρ a / μ a`, then `q * μ s ≤ ρ s`, as
 proved in `measure_le_of_frequently_le`. Since `ρ a / μ a` tends almost everywhere to
@@ -485,7 +485,7 @@ theorem measure_limRatioMeas_top : μ {x | v.limRatioMeas hρ x = ∞} = 0 := by
       intro y hy
       have : v.limRatioMeas hρ y = ∞ := hy.1
       simp only [this, ENNReal.coe_lt_top, mem_setOf_eq]
-    · simp only [(zero_lt_one.trans_le hq).ne', true_or_iff, ENNReal.coe_eq_zero, Ne,
+    · simp only [(zero_lt_one.trans_le hq).ne', true_or, ENNReal.coe_eq_zero, Ne,
         not_false_iff]
   have B : Tendsto (fun q : ℝ≥0 => (q : ℝ≥0∞)⁻¹ * ρ s) atTop (𝓝 (∞⁻¹ * ρ s)) := by
     apply ENNReal.Tendsto.mul_const _ (Or.inr ρs)
@@ -573,7 +573,7 @@ theorem withDensity_le_mul {s : Set α} (hs : MeasurableSet s) {t : ℝ≥0} (ht
         conv_rhs => rw [← mul_one (t ^ n)]
         gcongr
         rw [zpow_neg_one]
-        exact inv_lt_one ht
+        exact inv_lt_one_of_one_lt₀ ht
   calc
     ν s =
       ν (s ∩ f ⁻¹' {0}) + ν (s ∩ f ⁻¹' {∞}) +
@@ -649,8 +649,8 @@ theorem withDensity_limRatioMeas_eq : μ.withDensity (v.limRatioMeas hρ) = ρ :
         ((t : ℝ≥0∞) ^ 2 * ρ s : ℝ≥0∞)) (𝓝[>] 1) (𝓝 ((1 : ℝ≥0∞) ^ 2 * ρ s)) := by
       refine ENNReal.Tendsto.mul ?_ ?_ tendsto_const_nhds ?_
       · exact ENNReal.Tendsto.pow (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds)
-      · simp only [one_pow, ENNReal.coe_one, true_or_iff, Ne, not_false_iff, one_ne_zero]
-      · simp only [one_pow, ENNReal.coe_one, Ne, or_true_iff, ENNReal.one_ne_top, not_false_iff]
+      · simp only [one_pow, ENNReal.coe_one, true_or, Ne, not_false_iff, one_ne_zero]
+      · simp only [one_pow, ENNReal.coe_one, Ne, or_true, ENNReal.one_ne_top, not_false_iff]
     simp only [one_pow, one_mul, ENNReal.coe_one] at this
     refine ge_of_tendsto this ?_
     filter_upwards [self_mem_nhdsWithin] with _ ht
@@ -659,7 +659,7 @@ theorem withDensity_limRatioMeas_eq : μ.withDensity (v.limRatioMeas hρ) = ρ :
       Tendsto (fun t : ℝ≥0 => (t : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s) (𝓝[>] 1)
         (𝓝 ((1 : ℝ≥0∞) * μ.withDensity (v.limRatioMeas hρ) s)) := by
       refine ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) ?_
-      simp only [ENNReal.coe_one, true_or_iff, Ne, not_false_iff, one_ne_zero]
+      simp only [ENNReal.coe_one, true_or, Ne, not_false_iff, one_ne_zero]
     simp only [one_mul, ENNReal.coe_one] at this
     refine ge_of_tendsto this ?_
     filter_upwards [self_mem_nhdsWithin] with _ ht
@@ -809,7 +809,7 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf :
         _ < ∞ + ∞ :=
           haveI I : Integrable ((A.set n).indicator fun _ : α => c) μ := by
             simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),
-              integrableOn_const, A.finite n, or_true_iff]
+              integrableOn_const, A.finite n, or_true]
           ENNReal.add_lt_add hf.2 I.2
   filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x
   have M :
diff --git a/Mathlib/MeasureTheory/Covering/Vitali.lean b/Mathlib/MeasureTheory/Covering/Vitali.lean
index 5928bb689a9c9..364031a842a49 100644
--- a/Mathlib/MeasureTheory/Covering/Vitali.lean
+++ b/Mathlib/MeasureTheory/Covering/Vitali.lean
@@ -112,7 +112,7 @@ theorem exists_disjoint_subfamily_covering_enlargment (B : ι → Set α) (t : S
     · refine ⟨a, ⟨hat, a_disj⟩, ?_⟩
       simpa only [← mzero, zero_div] using δnonneg a hat
     · have I : m / τ < m := by
-        rw [div_lt_iff (zero_lt_one.trans hτ)]
+        rw [div_lt_iff₀ (zero_lt_one.trans hτ)]
         conv_lhs => rw [← mul_one m]
         exact (mul_lt_mul_left mpos).2 hτ
       rcases exists_lt_of_lt_csSup (Anonempty.image _) I with ⟨x, xA, hx⟩
diff --git a/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean b/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
index aa62a9d723094..48bb69b5fadb2 100644
--- a/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
+++ b/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
@@ -390,6 +390,12 @@ lemma integral_toReal_rnDeriv [SigmaFinite μ] [SigmaFinite ν] (hμν : μ ≪
     ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal := by
   rw [← integral_univ, setIntegral_toReal_rnDeriv hμν Set.univ]
 
+lemma integral_toReal_rnDeriv' [IsFiniteMeasure μ] [SigmaFinite ν] :
+    ∫ x, (μ.rnDeriv ν x).toReal ∂ν = (μ Set.univ).toReal - (μ.singularPart ν Set.univ).toReal := by
+  rw [← ENNReal.toReal_sub_of_le (μ.singularPart_le ν Set.univ) (measure_ne_top _ _),
+    ← Measure.sub_apply .univ (Measure.singularPart_le μ ν), Measure.measure_sub_singularPart,
+    ← Measure.setIntegral_toReal_rnDeriv_eq_withDensity, integral_univ]
+
 end integral
 
 lemma rnDeriv_mul_rnDeriv {κ : Measure α} [SigmaFinite μ] [SigmaFinite ν] [SigmaFinite κ]
@@ -563,6 +569,11 @@ theorem integral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪
     contrapose! hf
     exact (integrable_rnDeriv_smul_iff hμν).mp hf
 
+lemma setIntegral_rnDeriv_smul [HaveLebesgueDecomposition μ ν] (hμν : μ ≪ ν)
+    [SigmaFinite μ] {f : α → E} {s : Set α} (hs : MeasurableSet s) :
+    ∫ x in s, (μ.rnDeriv ν x).toReal • f x ∂ν = ∫ x in s, f x ∂μ := by
+  simp_rw [← integral_indicator hs, Set.indicator_smul, integral_rnDeriv_smul hμν]
+
 end IntegralRNDerivMul
 
 end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean b/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
index 054d7829fab2f..77cd7b8a79759 100644
--- a/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
+++ b/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
@@ -81,7 +81,7 @@ To implement this in Lean, we define several auxiliary definitions.
   where
   `restrictNonposSeq s i 0 = someExistsOneDivLT s (i \ ∅)` and
   `restrictNonposSeq s i (n + 1) = someExistsOneDivLT s (i \ ⋃ k ≤ n, restrictNonposSeq k)`.
-  This definition represents the sequence $(A_n)$ in the proof as described above.
+  This definition represents the sequence $(A_n)$ in the proof as described above.
 
 With these definitions, we are able consider the case where the sequence terminates separately,
 allowing us to prove `exists_subset_restrict_nonpos`.
@@ -229,7 +229,7 @@ private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂
   have hmeas : MeasurableSet (⋃ (l : ℕ) (_ : l < k), restrictNonposSeq s i l) :=
     MeasurableSet.iUnion fun _ => MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
   refine ⟨i \ ⋃ l < k, restrictNonposSeq s i l, hi₁.diff hmeas, Set.diff_subset, hk₂, ?_⟩
-  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion_nat]
+  rw [of_diff hmeas hi₁, s.of_disjoint_iUnion]
   · have h₁ : ∀ l < k, 0 ≤ s (restrictNonposSeq s i l) := by
       intro l hl
       refine le_of_lt (measure_of_restrictNonposSeq h _ ?_)
@@ -247,7 +247,7 @@ private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂
       rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
       exact fun _ => h
     · convert le_of_eq s.empty.symm
-      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false_iff]
+      ext; simp only [exists_prop, Set.mem_empty_iff_false, Set.mem_iUnion, not_and, iff_false]
       exact fun h' => False.elim (h h')
   · intro; exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
   · intro a b hab
@@ -276,7 +276,7 @@ theorem exists_subset_restrict_nonpos (hi : s i < 0) :
         simp only [exists_prop, Set.mem_iUnion, and_congr_left_iff]
         exact fun _ => Nat.lt_succ_iff.symm
   have h₁ : s i = s A + ∑' l, s (restrictNonposSeq s i l) := by
-    rw [hA, ← s.of_disjoint_iUnion_nat, add_comm, of_add_of_diff]
+    rw [hA, ← s.of_disjoint_iUnion, add_comm, of_add_of_diff]
     · exact MeasurableSet.iUnion fun _ => restrictNonposSeq_measurableSet _
     exacts [hi₁, Set.iUnion_subset fun _ => restrictNonposSeq_subset _, fun _ =>
       restrictNonposSeq_measurableSet _, restrictNonposSeq_disjoint]
@@ -312,7 +312,7 @@ theorem exists_subset_restrict_nonpos (hi : s i < 0) :
     · have : 1 / s E < bdd k := by
         linarith only [le_of_max_le_left (hk k le_rfl)]
       rw [one_div] at this ⊢
-      rwa [inv_lt (lt_trans (inv_pos.2 hE₃) this) hE₃]
+      exact inv_lt_of_inv_lt₀ hE₃ this
   obtain ⟨k, hk₁, hk₂⟩ := this
   have hA' : A ⊆ i \ ⋃ l ≤ k, restrictNonposSeq s i l := by
     apply Set.diff_subset_diff_right
diff --git a/Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean b/Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
index f6b3615d79356..1e85f61f04295 100644
--- a/Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
+++ b/Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean
@@ -30,9 +30,8 @@ variable {α : Type*} [MeasurableSpace α] {μ ν : Measure α}
 
 /-- **Hahn decomposition theorem** -/
 theorem hahn_decomposition [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
-    ∃ s,
-      MeasurableSet s ∧
-        (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t := by
+    ∃ s, MeasurableSet s ∧ (∀ t, MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧
+      ∀ t, MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t := by
   let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal
   let c : Set ℝ := d '' { s | MeasurableSet s }
   let γ : ℝ := sSup c
@@ -50,15 +49,15 @@ theorem hahn_decomposition [IsFiniteMeasure μ] [IsFiniteMeasure ν] :
   have d_Union (s : ℕ → Set α) (hm : Monotone s) :
     Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) := by
     refine Tendsto.sub ?_ ?_ <;>
-      refine NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal ?_).comp <| tendsto_measure_iUnion hm
+      refine NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal ?_).comp <|
+        tendsto_measure_iUnion_atTop hm
     · exact hμ _
     · exact hν _
   have d_Inter (s : ℕ → Set α) (hs : ∀ n, MeasurableSet (s n)) (hm : ∀ n m, n ≤ m → s m ⊆ s n) :
         Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) := by
     refine Tendsto.sub ?_ ?_ <;>
-      refine
-        NNReal.tendsto_coe.2 <|
-          (ENNReal.tendsto_toNNReal <| ?_).comp <| tendsto_measure_iInter hs hm ?_
+      refine NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal <| ?_).comp <|
+        tendsto_measure_iInter (fun n ↦ (hs n).nullMeasurableSet) hm ?_
     exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]
   have bdd_c : BddAbove c := by
     use (μ univ).toNNReal
diff --git a/Mathlib/MeasureTheory/Function/AEEqFun.lean b/Mathlib/MeasureTheory/Function/AEEqFun.lean
index cb6030efa880d..000e0416082a7 100644
--- a/Mathlib/MeasureTheory/Function/AEEqFun.lean
+++ b/Mathlib/MeasureTheory/Function/AEEqFun.lean
@@ -5,7 +5,7 @@ Authors: Johannes Hölzl, Zhouhang Zhou
 -/
 import Mathlib.MeasureTheory.Integral.Lebesgue
 import Mathlib.Order.Filter.Germ.Basic
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
 
 /-!
@@ -93,7 +93,7 @@ variable (α)
 
 /-- The space of equivalence classes of almost everywhere strongly measurable functions, where two
     strongly measurable functions are equivalent if they agree almost everywhere, i.e.,
-    they differ on a set of measure `0`.  -/
+    they differ on a set of measure `0`. -/
 def AEEqFun (μ : Measure α) : Type _ :=
   Quotient (μ.aeEqSetoid β)
 
@@ -235,7 +235,7 @@ variable [TopologicalSpace γ] [MeasurableSpace β] {ν : MeasureTheory.Measure
 /-- Composition of an almost everywhere equal function and a quasi measure preserving function.
 
 This is an important special case of `AEEqFun.compQuasiMeasurePreserving`. We use a separate
-definition so that lemmas that need `f` to be measure preserving can be `@[simp]` lemmas.  -/
+definition so that lemmas that need `f` to be measure preserving can be `@[simp]` lemmas. -/
 def compMeasurePreserving (g : β →ₘ[ν] γ) (f : α → β) (hf : MeasurePreserving f μ ν) : α →ₘ[μ] γ :=
   g.compQuasiMeasurePreserving f hf.quasiMeasurePreserving
 
@@ -576,7 +576,7 @@ theorem coeFn_const (b : β) : (const α b : α →ₘ[μ] β) =ᵐ[μ] Function
 @[simp]
 theorem coeFn_const_eq [NeZero μ] (b : β) (x : α) : (const α b : α →ₘ[μ] β) x = b := by
   simp only [cast]
-  split_ifs with h; swap; exact h.elim ⟨b, rfl⟩
+  split_ifs with h; swap; · exact h.elim ⟨b, rfl⟩
   have := Classical.choose_spec h
   set b' := Classical.choose h
   simp_rw [const, mk_eq_mk, EventuallyEq, ← const_def, eventually_const] at this
diff --git a/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean b/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
index bb4121b69171b..37c95e1795cb8 100644
--- a/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
+++ b/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
@@ -180,7 +180,7 @@ theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
     have B : (∫⁻ x in s, g x ∂μ) ≠ ∞ :=
       (setLIntegral_lt_top_of_le_nnreal s_lt_top.ne ⟨N, fun _ h ↦ h.1.2⟩).ne
     have : (ε : ℝ≥0∞) * μ s ≤ 0 := ENNReal.le_of_add_le_add_left B A
-    simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or_iff]
+    simpa only [ENNReal.coe_eq_zero, nonpos_iff_eq_zero, mul_eq_zero, εpos.ne', false_or]
   obtain ⟨u, _, u_pos, u_lim⟩ :
     ∃ u : ℕ → ℝ≥0, StrictAnti u ∧ (∀ n, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
     exists_seq_strictAnti_tendsto (0 : ℝ≥0)
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
index 04cb417cd938d..daab2ab684f35 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
@@ -109,7 +109,7 @@ theorem condexp_of_sigmaFinite (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)]
         else aestronglyMeasurable'_condexpL1.mk (condexpL1 hm μ f)
       else 0 := by
   rw [condexp, dif_pos hm]
-  simp only [hμm, Ne, true_and_iff]
+  simp only [hμm, Ne, true_and]
   by_cases hf : Integrable f μ
   · rw [dif_pos hf, if_pos hf]
   · rw [dif_neg hf, if_neg hf]
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
index 779e0ed7f4ba6..a0b70b96c1884 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
@@ -176,11 +176,11 @@ theorem condexpIndL1_of_measurableSet_of_measure_ne_top (hs : MeasurableSet s) (
 
 theorem condexpIndL1_of_measure_eq_top (hμs : μ s = ∞) (x : G) : condexpIndL1 hm μ s x = 0 := by
   simp only [condexpIndL1, hμs, eq_self_iff_true, not_true, Ne, dif_neg, not_false_iff,
-    and_false_iff]
+    and_false]
 
 theorem condexpIndL1_of_not_measurableSet (hs : ¬MeasurableSet s) (x : G) :
     condexpIndL1 hm μ s x = 0 := by
-  simp only [condexpIndL1, hs, dif_neg, not_false_iff, false_and_iff]
+  simp only [condexpIndL1, hs, dif_neg, not_false_iff, false_and]
 
 theorem condexpIndL1_add (x y : G) :
     condexpIndL1 hm μ s (x + y) = condexpIndL1 hm μ s x + condexpIndL1 hm μ s y := by
diff --git a/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean b/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
index d16ec22981a70..633e66b2ab991 100644
--- a/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
+++ b/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
@@ -120,7 +120,7 @@ theorem exists_continuous_eLpNorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠
     have : eLpNorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ := by
       refine (eLpNorm_indicator_const_le _ _).trans_lt ?_
       simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne,
-        ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and_iff, or_false_iff, not_and, not_lt,
+        ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and, or_false, not_and, not_lt,
         ENNReal.toReal_nonneg, imp_true_iff]
     refine (eLpNorm_mono fun x => ?_).trans_lt this
     by_cases hx : x ∈ v
@@ -366,7 +366,7 @@ namespace ContinuousMap
 
 /-- Continuous functions are dense in `MeasureTheory.Lp`, `1 ≤ p < ∞`. This theorem assumes that
 the domain is a compact space because otherwise `ContinuousMap.toLp` is undefined. Use
-`BoundedContinuousFunction.toLp_denseRange` if the domain is not a compact space.  -/
+`BoundedContinuousFunction.toLp_denseRange` if the domain is not a compact space. -/
 theorem toLp_denseRange [CompactSpace α] [μ.WeaklyRegular] [IsFiniteMeasure μ] (hp : p ≠ ∞) :
     DenseRange (toLp p μ 𝕜 : C(α, E) →L[𝕜] Lp E p μ) := by
   refine (BoundedContinuousFunction.toLp_denseRange _ _ 𝕜 hp).mono ?_
diff --git a/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean b/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
index 1d97cdeb360fd..f9a7daf12fa48 100644
--- a/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
+++ b/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
@@ -191,8 +191,8 @@ theorem TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTo
     fun k => ExistsSeqTendstoAe.seqTendstoAeSeq_spec hfg k (ns k) le_rfl
   set s := Filter.atTop.limsup S with hs
   have hμs : μ s = 0 := by
-    refine measure_limsup_eq_zero (ne_of_lt <| lt_of_le_of_lt (ENNReal.tsum_le_tsum hμS_le) ?_)
-    simp only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, ENNReal.two_lt_top, inv_inv]
+    refine measure_limsup_atTop_eq_zero (ne_top_of_le_ne_top ?_ (ENNReal.tsum_le_tsum hμS_le))
+    simpa only [ENNReal.tsum_geometric, ENNReal.one_sub_inv_two, inv_inv] using ENNReal.two_ne_top
   have h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x)) := by
     refine fun x hx => Metric.tendsto_atTop.mpr fun ε hε => ?_
     rw [hs, limsup_eq_iInf_iSup_of_nat] at hx
@@ -203,12 +203,13 @@ theorem TendstoInMeasure.exists_seq_tendsto_ae (hfg : TendstoInMeasure μ f atTo
     refine ⟨max N (k - 1), fun n hn_ge => lt_of_le_of_lt ?_ hk_lt_ε⟩
     specialize hNx n ((le_max_left _ _).trans hn_ge)
     have h_inv_n_le_k : (2 : ℝ)⁻¹ ^ n ≤ 2 * (2 : ℝ)⁻¹ ^ k := by
-      rw [mul_comm, ← inv_mul_le_iff' (zero_lt_two' ℝ)]
+      rw [mul_comm, ← inv_mul_le_iff₀' (zero_lt_two' ℝ)]
       conv_lhs =>
         congr
         rw [← pow_one (2 : ℝ)⁻¹]
       rw [← pow_add, add_comm]
-      exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le) (inv_le_one one_le_two)
+      exact pow_le_pow_of_le_one (one_div (2 : ℝ) ▸ one_half_pos.le)
+        (inv_le_one_of_one_le₀ one_le_two)
         ((le_tsub_add.trans (add_le_add_right (le_max_right _ _) 1)).trans
           (add_le_add_right hn_ge 1))
     exact le_trans hNx.le h_inv_n_le_k
@@ -249,7 +250,7 @@ section TendstoInMeasureOf
 variable [NormedAddCommGroup E] {p : ℝ≥0∞}
 variable {f : ι → α → E} {g : α → E}
 
-/-- This lemma is superceded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
+/-- This lemma is superseded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
 allow `p = ∞` and only require `AEStronglyMeasurable`. -/
 theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) (hf : ∀ n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
@@ -278,7 +279,7 @@ theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero :
 alias tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable :=
   tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable
 
-/-- This lemma is superceded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
+/-- This lemma is superseded by `MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm` where we
 allow `p = ∞`. -/
 theorem tendstoInMeasure_of_tendsto_eLpNorm_of_ne_top (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ) {l : Filter ι}
@@ -306,7 +307,7 @@ theorem tendstoInMeasure_of_tendsto_eLpNorm_top {E} [NormedAddCommGroup E] {f :
   specialize hfg (ENNReal.ofReal δ / 2)
       (ENNReal.div_pos_iff.2 ⟨(ENNReal.ofReal_pos.2 hδ).ne.symm, ENNReal.two_ne_top⟩)
   refine hfg.mono fun n hn => ?_
-  simp only [true_and_iff, gt_iff_lt, zero_tsub, zero_le, zero_add, Set.mem_Icc,
+  simp only [gt_iff_lt, zero_tsub, zero_le, zero_add, Set.mem_Icc,
     Pi.sub_apply] at *
   have : essSup (fun x : α => (‖f n x - g x‖₊ : ℝ≥0∞)) μ < ENNReal.ofReal δ :=
     lt_of_le_of_lt hn
diff --git a/Mathlib/MeasureTheory/Function/Egorov.lean b/Mathlib/MeasureTheory/Function/Egorov.lean
index 402e64e727ca0..932a6fefb1d33 100644
--- a/Mathlib/MeasureTheory/Function/Egorov.lean
+++ b/Mathlib/MeasureTheory/Function/Egorov.lean
@@ -81,7 +81,8 @@ theorem measure_notConvergentSeq_tendsto_zero [SemilatticeSup ι] [Countable ι]
     rw [this]
     exact tendsto_const_nhds
   rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]
-  refine tendsto_measure_iInter (fun n => hsm.inter <| notConvergentSeq_measurableSet hf hg)
+  refine tendsto_measure_iInter
+    (fun n ↦ (hsm.inter <| notConvergentSeq_measurableSet hf hg).nullMeasurableSet)
     (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)
     ⟨h.some, ne_top_of_le_ne_top hs (measure_mono Set.inter_subset_left)⟩
 
diff --git a/Mathlib/MeasureTheory/Function/EssSup.lean b/Mathlib/MeasureTheory/Function/EssSup.lean
index 999cef7eab633..503c49f68e8df 100644
--- a/Mathlib/MeasureTheory/Function/EssSup.lean
+++ b/Mathlib/MeasureTheory/Function/EssSup.lean
@@ -36,7 +36,7 @@ variable {α β : Type*} {m : MeasurableSpace α} {μ ν : Measure α}
 
 section ConditionallyCompleteLattice
 
-variable [ConditionallyCompleteLattice β]
+variable [ConditionallyCompleteLattice β] {f : α → β}
 
 /-- Essential supremum of `f` with respect to measure `μ`: the smallest `c : β` such that
 `f x ≤ c` a.e. -/
@@ -68,6 +68,32 @@ theorem essSup_const (c : β) (hμ : μ ≠ 0) : essSup (fun _ : α => c) μ = c
 theorem essInf_const (c : β) (hμ : μ ≠ 0) : essInf (fun _ : α => c) μ = c :=
   have := NeZero.mk hμ; essInf_const' _
 
+section SMul
+variable {R : Type*} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
+  [NoZeroSMulDivisors R ℝ≥0∞] {c : R}
+
+@[simp]
+lemma essSup_smul_measure (hc : c ≠ 0) (f : α → β) : essSup f (c • μ) = essSup f μ := by
+  simp_rw [essSup, Measure.ae_smul_measure_eq hc]
+
+end SMul
+
+variable [Nonempty α]
+
+lemma essSup_eq_ciSup (hμ : ∀ a, μ {a} ≠ 0) (hf : BddAbove (Set.range f)) :
+    essSup f μ = ⨆ a, f a := by rw [essSup, ae_eq_top.2 hμ, limsup_top_eq_ciSup hf]
+
+lemma essInf_eq_ciInf (hμ : ∀ a, μ {a} ≠ 0) (hf : BddBelow (Set.range f)) :
+    essInf f μ = ⨅ a, f a := by rw [essInf, ae_eq_top.2 hμ, liminf_top_eq_ciInf hf]
+
+variable [MeasurableSingletonClass α]
+
+@[simp] lemma essSup_count_eq_ciSup (hf : BddAbove (Set.range f)) :
+    essSup f .count = ⨆ a, f a := essSup_eq_ciSup (by simp) hf
+
+@[simp] lemma essInf_count_eq_ciInf (hf : BddBelow (Set.range f)) :
+    essInf f .count = ⨅ a, f a := essInf_eq_ciInf (by simp) hf
+
 end ConditionallyCompleteLattice
 
 section ConditionallyCompleteLinearOrder
@@ -172,15 +198,11 @@ theorem essInf_antitone_measure {f : α → β} (hμν : μ ≪ ν) : essInf f 
   refine liminf_le_liminf_of_le (Measure.ae_le_iff_absolutelyContinuous.mpr hμν) ?_ ?_
   all_goals isBoundedDefault
 
-theorem essSup_smul_measure {f : α → β} {c : ℝ≥0∞} (hc : c ≠ 0) :
-    essSup f (c • μ) = essSup f μ := by
-  simp_rw [essSup, Measure.ae_smul_measure_eq hc]
-
 lemma essSup_eq_iSup (hμ : ∀ a, μ {a} ≠ 0) (f : α → β) : essSup f μ = ⨆ i, f i := by
-  rw [essSup, ae_eq_top.2 hμ, limsup_top]
+  rw [essSup, ae_eq_top.2 hμ, limsup_top_eq_iSup]
 
 lemma essInf_eq_iInf (hμ : ∀ a, μ {a} ≠ 0) (f : α → β) : essInf f μ = ⨅ i, f i := by
-  rw [essInf, ae_eq_top.2 hμ, liminf_top]
+  rw [essInf, ae_eq_top.2 hμ, liminf_top_eq_iInf]
 
 @[simp] lemma essSup_count [MeasurableSingletonClass α] (f : α → β) : essSup f .count = ⨆ i, f i :=
   essSup_eq_iSup (by simp) _
diff --git a/Mathlib/MeasureTheory/Function/Intersectivity.lean b/Mathlib/MeasureTheory/Function/Intersectivity.lean
index 5ea828b2fcf5b..c571061c05018 100644
--- a/Mathlib/MeasureTheory/Function/Intersectivity.lean
+++ b/Mathlib/MeasureTheory/Function/Intersectivity.lean
@@ -61,8 +61,7 @@ lemma bergelson' {s : ℕ → Set α} (hs : ∀ n, MeasurableSet (s n)) (hr₀ :
   have hfapp : ∀ n a, f n a = (↑(n + 1))⁻¹ * ∑ k in Finset.range (n + 1), (s k).indicator 1 a := by
     simp only [f, Pi.natCast_def, Pi.smul_apply, Pi.inv_apply, Finset.sum_apply, eq_self_iff_true,
     forall_const, imp_true_iff, smul_eq_mul]
-  have hf n : Measurable (f n) := Measurable.mul' (@measurable_const ℝ≥0∞ _ _ _ (↑(n + 1))⁻¹)
-      (Finset.measurable_sum' _ fun i _ ↦ measurable_one.indicator <| hs i)
+  have hf n : Measurable (f n) := by fun_prop (disch := exact hs _)
   have hf₁ n : f n ≤ 1 := by
     rintro a
     rw [hfapp, ← ENNReal.div_eq_inv_mul]
diff --git a/Mathlib/MeasureTheory/Function/Jacobian.lean b/Mathlib/MeasureTheory/Function/Jacobian.lean
index 38f7a8882415c..ce6116b747cb5 100644
--- a/Mathlib/MeasureTheory/Function/Jacobian.lean
+++ b/Mathlib/MeasureTheory/Function/Jacobian.lean
@@ -87,7 +87,7 @@ Change of variables in integrals
 [Fremlin, *Measure Theory* (volume 2)][fremlin_vol2]
 -/
 
-open MeasureTheory MeasureTheory.Measure Metric Filter Set FiniteDimensional Asymptotics
+open MeasureTheory MeasureTheory.Measure Metric Filter Set Module Asymptotics
   TopologicalSpace
 
 open scoped NNReal ENNReal Topology Pointwise
@@ -377,7 +377,7 @@ theorem addHaar_image_le_mul_of_det_lt (A : E →L[ℝ] E) {m : ℝ≥0}
   have L : Tendsto (fun a => (m : ℝ≥0∞) * (μ s + a)) (𝓝[>] 0) (𝓝 (m * (μ s + 0))) := by
     apply Tendsto.mono_left _ nhdsWithin_le_nhds
     apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)
-    simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]
+    simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
   rw [add_zero] at L
   exact ge_of_tendsto L J
 
@@ -420,10 +420,10 @@ theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
   -- record smallness conditions for `δ` that will be needed to apply `hδ₀` below.
   have L1 : ∀ᶠ δ in 𝓝 (0 : ℝ≥0), Subsingleton E ∨ δ < ‖(B.symm : E →L[ℝ] E)‖₊⁻¹ := by
     by_cases h : Subsingleton E
-    · simp only [h, true_or_iff, eventually_const]
-    simp only [h, false_or_iff]
+    · simp only [h, true_or, eventually_const]
+    simp only [h, false_or]
     apply Iio_mem_nhds
-    simpa only [h, false_or_iff, inv_pos] using B.subsingleton_or_nnnorm_symm_pos
+    simpa only [h, false_or, inv_pos] using B.subsingleton_or_nnnorm_symm_pos
   have L2 :
     ∀ᶠ δ in 𝓝 (0 : ℝ≥0), ‖(B.symm : E →L[ℝ] E)‖₊ * (‖(B.symm : E →L[ℝ] E)‖₊⁻¹ - δ)⁻¹ * δ < δ₀ := by
     have :
@@ -448,7 +448,7 @@ theorem mul_le_addHaar_image_of_lt_det (A : E →L[ℝ] E) {m : ℝ≥0}
       mul_comm, ← ENNReal.coe_inv mpos.ne']
     · apply Or.inl
       simpa only [ENNReal.coe_eq_zero, Ne] using mpos.ne'
-    · simp only [ENNReal.coe_ne_top, true_or_iff, Ne, not_false_iff]
+    · simp only [ENNReal.coe_ne_top, true_or, Ne, not_false_iff]
   -- as `f⁻¹` is well approximated by `B⁻¹`, the conclusion follows from `hδ₀`
   -- and our choice of `δ`.
   exact hδ₀ _ _ ((hf'.to_inv h1δ).mono_num h2δ.le)
@@ -806,7 +806,7 @@ theorem addHaar_image_le_lintegral_abs_det_fderiv_aux1 (hs : MeasurableSet s)
       rw [dist_eq_norm]
       calc
         ‖B - A‖ ≤ (min δ δ'' : ℝ≥0) := hB
-        _ ≤ δ'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]
+        _ ≤ δ'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true]
         _ < δ' := half_lt_self δ'pos
     · intro t g htg
       exact h t g (htg.mono_num (min_le_left _ _))
@@ -941,7 +941,7 @@ theorem lintegral_abs_det_fderiv_le_addHaar_image_aux1 (hs : MeasurableSet s)
     refine ⟨min δ δ'', lt_min δpos (half_pos δ'pos), ?_, ?_⟩
     · intro B hB
       apply I'' _ (hB.trans _)
-      simp only [le_refl, NNReal.coe_min, min_le_iff, or_true_iff]
+      simp only [le_refl, NNReal.coe_min, min_le_iff, or_true]
     · intro t g htg
       rcases eq_or_ne (μ t) ∞ with (ht | ht)
       · simp only [ht, εpos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne,
@@ -1155,7 +1155,7 @@ theorem integrableOn_image_iff_integrableOn_abs_det_fderiv_smul (hs : Measurable
     IntegrableOn g (f '' s) μ ↔ IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ := by
   rw [IntegrableOn, ← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,
     (measurableEmbedding_of_fderivWithin hs hf' hf).integrable_map_iff]
-  simp only [Set.restrict_eq, ← Function.comp.assoc, ENNReal.ofReal]
+  simp only [Set.restrict_eq, ← Function.comp_assoc, ENNReal.ofReal]
   rw [← (MeasurableEmbedding.subtype_coe hs).integrable_map_iff, map_comap_subtype_coe hs,
     restrict_withDensity hs, integrable_withDensity_iff_integrable_coe_smul₀]
   · simp_rw [IntegrableOn, Real.coe_toNNReal _ (abs_nonneg _), Function.comp_apply]
@@ -1185,7 +1185,7 @@ theorem det_one_smulRight {𝕜 : Type*} [NormedField 𝕜] (v : 𝕜) :
       Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]
   rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_smul,
     ContinuousLinearMap.one_def, ContinuousLinearMap.coe_id, LinearMap.det_smul,
-    FiniteDimensional.finrank_self, LinearMap.det_id, pow_one, mul_one]
+    Module.finrank_self, LinearMap.det_id, pow_one, mul_one]
 
 /-- Integrability in the change of variable formula for differentiable functions (one-variable
 version): if a function `f` is injective and differentiable on a measurable set `s ⊆ ℝ`, then a
diff --git a/Mathlib/MeasureTheory/Function/L1Space.lean b/Mathlib/MeasureTheory/Function/L1Space.lean
index 8951025e6bfb6..9994ae4d6e8b1 100644
--- a/Mathlib/MeasureTheory/Function/L1Space.lean
+++ b/Mathlib/MeasureTheory/Function/L1Space.lean
@@ -438,9 +438,13 @@ theorem integrable_const [IsFiniteMeasure μ] (c : β) : Integrable (fun _ : α
   integrable_const_iff.2 <| Or.inr <| measure_lt_top _ _
 
 @[simp]
-theorem Integrable.of_finite [Finite α] [MeasurableSpace α] [MeasurableSingletonClass α]
-    (μ : Measure α) [IsFiniteMeasure μ] (f : α → β) : Integrable (fun a ↦ f a) μ :=
-  ⟨(StronglyMeasurable.of_finite f).aestronglyMeasurable, .of_finite⟩
+lemma Integrable.of_finite [Finite α] [MeasurableSingletonClass α] [IsFiniteMeasure μ] {f : α → β} :
+    Integrable f μ := ⟨.of_finite, .of_finite⟩
+
+/-- This lemma is a special case of `Integrable.of_finite`. -/
+-- Eternal deprecation for discoverability, don't remove
+@[deprecated Integrable.of_finite, nolint deprecatedNoSince]
+lemma Integrable.of_isEmpty [IsEmpty α] {f : α → β} : Integrable f μ := .of_finite
 
 @[deprecated (since := "2024-02-05")] alias integrable_of_fintype := Integrable.of_finite
 
@@ -878,8 +882,7 @@ theorem integrable_withDensity_iff_integrable_coe_smul {f : α → ℝ≥0} (hf
     {g : α → E} :
     Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) μ := by
   by_cases H : AEStronglyMeasurable (fun x : α => (f x : ℝ) • g x) μ
-  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H,
-      true_and_iff]
+  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, HasFiniteIntegral, H]
     rw [lintegral_withDensity_eq_lintegral_mul₀' hf.coe_nnreal_ennreal.aemeasurable]
     · rw [iff_iff_eq]
       congr
@@ -889,7 +892,7 @@ theorem integrable_withDensity_iff_integrable_coe_smul {f : α → ℝ≥0} (hf
       convert H.ennnorm using 1
       ext1 x
       simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul]
-  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and_iff]
+  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and]
 
 theorem integrable_withDensity_iff_integrable_smul {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
     Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => f x • g x) μ :=
diff --git a/Mathlib/MeasureTheory/Function/LocallyIntegrable.lean b/Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
index 2acd297ad0fd6..0d8110dc76977 100644
--- a/Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
+++ b/Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
@@ -127,24 +127,16 @@ theorem LocallyIntegrableOn.aestronglyMeasurable [SecondCountableTopology X]
   rw [this, aestronglyMeasurable_iUnion_iff]
   exact fun i : ℕ => (hu i).aestronglyMeasurable
 
-/-- If `s` is either open, or closed, then `f` is locally integrable on `s` iff it is integrable on
-every compact subset contained in `s`. -/
-theorem locallyIntegrableOn_iff [LocallyCompactSpace X] [T2Space X] (hs : IsClosed s ∨ IsOpen s) :
-    LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → (IsCompact k → IntegrableOn f k μ) := by
-  -- The correct condition is that `s` be *locally closed*, i.e. for every `x ∈ s` there is some
-  -- `U ∈ 𝓝 x` such that `U ∩ s` is closed. But mathlib doesn't have locally closed sets yet.
-  refine ⟨fun hf k hk => hf.integrableOn_compact_subset hk, fun hf x hx => ?_⟩
-  cases hs with
-  | inl hs =>
-    exact
-      let ⟨K, hK, h2K⟩ := exists_compact_mem_nhds x
-      ⟨_, inter_mem_nhdsWithin s h2K,
-        hf _ inter_subset_left
-          (hK.of_isClosed_subset (hs.inter hK.isClosed) inter_subset_right)⟩
-  | inr hs =>
-    obtain ⟨K, hK, h2K, h3K⟩ := exists_compact_subset hs hx
-    refine ⟨K, ?_, hf K h3K hK⟩
-    simpa only [IsOpen.nhdsWithin_eq hs hx, interior_eq_nhds'] using h2K
+/-- If `s` is locally closed (e.g. open or closed), then `f` is locally integrable on `s` iff it is
+integrable on every compact subset contained in `s`. -/
+theorem locallyIntegrableOn_iff [LocallyCompactSpace X] (hs : IsLocallyClosed s) :
+    LocallyIntegrableOn f s μ ↔ ∀ (k : Set X), k ⊆ s → IsCompact k → IntegrableOn f k μ := by
+  refine ⟨fun hf k hk ↦ hf.integrableOn_compact_subset hk, fun hf x hx ↦ ?_⟩
+  rcases hs with ⟨U, Z, hU, hZ, rfl⟩
+  rcases exists_compact_subset hU hx.1 with ⟨K, hK, hxK, hKU⟩
+  rw [nhdsWithin_inter_of_mem (nhdsWithin_le_nhds <| hU.mem_nhds hx.1)]
+  refine ⟨Z ∩ K, inter_mem_nhdsWithin _ (mem_interior_iff_mem_nhds.1 hxK), ?_⟩
+  exact hf (Z ∩ K) (fun y hy ↦ ⟨hKU hy.2, hy.1⟩) (.inter_left hK hZ)
 
 protected theorem LocallyIntegrableOn.add
     (hf : LocallyIntegrableOn f s μ) (hg : LocallyIntegrableOn g s μ) :
@@ -506,7 +498,7 @@ theorem MonotoneOn.integrableOn_of_measure_ne_top (hmono : MonotoneOn f s) {a b
   have : IsBounded (f '' s) := Metric.isBounded_of_bddAbove_of_bddBelow habove hbelow
   rcases isBounded_iff_forall_norm_le.mp this with ⟨C, hC⟩
   have A : IntegrableOn (fun _ => C) s μ := by
-    simp only [hs.lt_top, integrableOn_const, or_true_iff]
+    simp only [hs.lt_top, integrableOn_const, or_true]
   exact
     Integrable.mono' A (aemeasurable_restrict_of_monotoneOn h's hmono).aestronglyMeasurable
       ((ae_restrict_iff' h's).mpr <| ae_of_all _ fun y hy => hC (f y) (mem_image_of_mem f hy))
@@ -619,28 +611,30 @@ namespace LocallyIntegrableOn
 
 theorem continuousOn_mul [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
-    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => g x * f x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
+    (hg : ContinuousOn g s) (hs : IsLocallyClosed s) :
+    LocallyIntegrableOn (fun x => g x * f x) s μ := by
+  rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_mul (hg.mono hk_sub) hk_c
 
 theorem mul_continuousOn [LocallyCompactSpace X] [T2Space X] [NormedRing R]
     [SecondCountableTopologyEither X R] {f g : X → R} {s : Set X} (hf : LocallyIntegrableOn f s μ)
-    (hg : ContinuousOn g s) (hs : IsOpen s) : LocallyIntegrableOn (fun x => f x * g x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
+    (hg : ContinuousOn g s) (hs : IsLocallyClosed s) :
+    LocallyIntegrableOn (fun x => f x * g x) s μ := by
+  rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).mul_continuousOn (hg.mono hk_sub) hk_c
 
 theorem continuousOn_smul [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
     [SecondCountableTopologyEither X 𝕜] [NormedSpace 𝕜 E] {f : X → E} {g : X → 𝕜} {s : Set X}
-    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
+    (hs : IsLocallyClosed s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => g x • f x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).continuousOn_smul (hg.mono hk_sub) hk_c
 
 theorem smul_continuousOn [LocallyCompactSpace X] [T2Space X] {𝕜 : Type*} [NormedField 𝕜]
     [SecondCountableTopologyEither X E] [NormedSpace 𝕜 E] {f : X → 𝕜} {g : X → E} {s : Set X}
-    (hs : IsOpen s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
+    (hs : IsLocallyClosed s) (hf : LocallyIntegrableOn f s μ) (hg : ContinuousOn g s) :
     LocallyIntegrableOn (fun x => f x • g x) s μ := by
-  rw [MeasureTheory.locallyIntegrableOn_iff (Or.inr hs)] at hf ⊢
+  rw [MeasureTheory.locallyIntegrableOn_iff hs] at hf ⊢
   exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c
 
 end LocallyIntegrableOn
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
index 021f82da8c982..fdf5f19ca96be 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
@@ -404,18 +404,18 @@ theorem eLpNorm_const_lt_top_iff {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0)
     eLpNorm (fun _ : α => c) p μ < ∞ ↔ c = 0 ∨ μ Set.univ < ∞ := by
   have hp : 0 < p.toReal := ENNReal.toReal_pos hp_ne_zero hp_ne_top
   by_cases hμ : μ = 0
-  · simp only [hμ, Measure.coe_zero, Pi.zero_apply, or_true_iff, ENNReal.zero_lt_top,
+  · simp only [hμ, Measure.coe_zero, Pi.zero_apply, or_true, ENNReal.zero_lt_top,
       eLpNorm_measure_zero]
   by_cases hc : c = 0
-  · simp only [hc, true_or_iff, eq_self_iff_true, ENNReal.zero_lt_top, eLpNorm_zero']
+  · simp only [hc, true_or, eq_self_iff_true, ENNReal.zero_lt_top, eLpNorm_zero']
   rw [eLpNorm_const' c hp_ne_zero hp_ne_top]
   by_cases hμ_top : μ Set.univ = ∞
   · simp [hc, hμ_top, hp]
   rw [ENNReal.mul_lt_top_iff]
-  simp only [true_and_iff, one_div, ENNReal.rpow_eq_zero_iff, hμ, false_or_iff, or_false_iff,
+  simp only [true_and, one_div, ENNReal.rpow_eq_zero_iff, hμ, false_or, or_false,
     ENNReal.coe_lt_top, nnnorm_eq_zero, ENNReal.coe_eq_zero,
-    MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false_iff, false_and_iff,
-    inv_pos, or_self_iff, hμ_top, Ne.lt_top hμ_top, iff_true_iff]
+    MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, false_and,
+    inv_pos, or_self_iff, hμ_top, Ne.lt_top hμ_top, iff_true]
   exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) hμ_top
 
 @[deprecated (since := "2024-07-27")]
@@ -661,7 +661,7 @@ theorem eLpNorm_norm_rpow (f : α → F) (hq_pos : 0 < q) :
       congr
       ext1 x
       conv_rhs => rw [← nnnorm_norm]
-      rw [ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_inj]
+      rw [← ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_inj]
       ext
       push_cast
       rw [Real.norm_rpow_of_nonneg (norm_nonneg _)]
@@ -794,10 +794,16 @@ theorem eLpNorm'_smul_measure {p : ℝ} (hp : 0 ≤ p) {f : α → F} (c : ℝ
 @[deprecated (since := "2024-07-27")]
 alias snorm'_smul_measure := eLpNorm'_smul_measure
 
-theorem eLpNormEssSup_smul_measure {f : α → F} {c : ℝ≥0∞} (hc : c ≠ 0) :
+section SMul
+variable {R : Type*} [Zero R] [SMulWithZero R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞]
+  [NoZeroSMulDivisors R ℝ≥0∞] {c : R}
+
+@[simp] lemma eLpNormEssSup_smul_measure (hc : c ≠ 0) (f : α → F) :
     eLpNormEssSup f (c • μ) = eLpNormEssSup f μ := by
   simp_rw [eLpNormEssSup]
-  exact essSup_smul_measure hc
+  exact essSup_smul_measure hc _
+
+end SMul
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_smul_measure := eLpNormEssSup_smul_measure
@@ -1094,7 +1100,7 @@ theorem eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul {f : α → F} {g : α →
   rw [← ENNReal.rpow_le_rpow_iff hp, ENNReal.smul_def, smul_eq_mul,
     ENNReal.mul_rpow_of_nonneg _ _ hp.le]
   simp_rw [← ENNReal.rpow_mul, one_div, inv_mul_cancel₀ hp.ne.symm, ENNReal.rpow_one,
-    ENNReal.coe_rpow_of_nonneg _ hp.le, ← lintegral_const_mul' _ _ ENNReal.coe_ne_top, ←
+    ← ENNReal.coe_rpow_of_nonneg _ hp.le, ← lintegral_const_mul' _ _ ENNReal.coe_ne_top, ←
     ENNReal.coe_mul]
   apply lintegral_mono_ae
   simp_rw [ENNReal.coe_le_coe, ← NNReal.mul_rpow, NNReal.rpow_le_rpow_iff hp]
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
index 32599fde917d0..6248e4a2c5925 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
@@ -34,7 +34,7 @@ theorem mul_meas_ge_le_pow_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞
   have : 1 / p.toReal * p.toReal = 1 := by
     refine one_div_mul_cancel ?_
     rw [Ne, ENNReal.toReal_eq_zero_iff]
-    exact not_or_of_not hp_ne_zero hp_ne_top
+    exact not_or_intro hp_ne_zero hp_ne_top
   rw [← ENNReal.rpow_one (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }), ← this, ENNReal.rpow_mul]
   gcongr
   exact pow_mul_meas_ge_le_eLpNorm μ hp_ne_zero hp_ne_top hf ε
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
index dcafb2aeecbec..e7723bfa145de 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
@@ -126,7 +126,7 @@ theorem eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le {p q : ℝ} [IsFiniteM
     _ < ∞ := by
       rw [ENNReal.mul_lt_top_iff]
       refine Or.inl ⟨hfq_lt_top, ENNReal.rpow_lt_top_of_nonneg ?_ (measure_ne_top μ Set.univ)⟩
-      rwa [le_sub_comm, sub_zero, one_div, one_div, inv_le_inv hq_pos hp_pos]
+      rwa [le_sub_comm, sub_zero, one_div, one_div, inv_le_inv₀ hq_pos hp_pos]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_lt_top_of_snorm'_lt_top_of_exponent_le :=
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
index 42f9edb4bd2b4..cc56d954aa492 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
@@ -88,9 +88,8 @@ theorem LpAddConst_lt_top (p : ℝ≥0∞) : LpAddConst p < ∞ := by
   rw [LpAddConst]
   split_ifs with h
   · apply ENNReal.rpow_lt_top_of_nonneg _ ENNReal.two_ne_top
-    simp only [one_div, sub_nonneg]
-    apply one_le_inv (ENNReal.toReal_pos h.1.ne' (h.2.trans ENNReal.one_lt_top).ne)
-    simpa using ENNReal.toReal_mono ENNReal.one_ne_top h.2.le
+    rw [one_div, sub_nonneg, ← ENNReal.toReal_inv, ← ENNReal.one_toReal]
+    exact ENNReal.toReal_mono (by simpa using h.1.ne') (ENNReal.one_le_inv.2 h.2.le)
   · exact ENNReal.one_lt_top
 
 theorem eLpNorm_add_le' (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ)
diff --git a/Mathlib/MeasureTheory/Function/LpSpace.lean b/Mathlib/MeasureTheory/Function/LpSpace.lean
index cfd54d45970a3..5410247494af6 100644
--- a/Mathlib/MeasureTheory/Function/LpSpace.lean
+++ b/Mathlib/MeasureTheory/Function/LpSpace.lean
@@ -13,7 +13,7 @@ import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality
 import Mathlib.MeasureTheory.Measure.OpenPos
 import Mathlib.MeasureTheory.Measure.Typeclasses
 import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.Compact
 import Mathlib.Order.Filter.IndicatorFunction
 
 /-!
@@ -387,7 +387,7 @@ theorem nnnorm_le_of_ae_bound [IsFiniteMeasure μ] {f : Lp E p μ} {C : ℝ≥0}
     · simp [hμ, nnnorm_def, Real.zero_rpow hp]
   rw [← ENNReal.coe_le_coe, nnnorm_def, ENNReal.coe_toNNReal (eLpNorm_ne_top _)]
   refine (eLpNorm_le_of_ae_nnnorm_bound hfC).trans_eq ?_
-  rw [← coe_measureUnivNNReal μ, ENNReal.coe_rpow_of_ne_zero (measureUnivNNReal_pos hμ).ne',
+  rw [← coe_measureUnivNNReal μ, ← ENNReal.coe_rpow_of_ne_zero (measureUnivNNReal_pos hμ).ne',
     ENNReal.coe_mul, mul_comm, ENNReal.smul_def, smul_eq_mul]
 
 theorem norm_le_of_ae_bound [IsFiniteMeasure μ] {f : Lp E p μ} {C : ℝ} (hC : 0 ≤ C)
@@ -677,7 +677,7 @@ theorem exists_eLpNorm_indicator_le (hp : p ≠ ∞) (c : E) {ε : ℝ≥0∞} (
       NNReal.nhds_zero_basis.eventually_iff.mp (eventually_le_of_tendsto_lt hε' this)
     obtain ⟨η, hη, hηδ⟩ := exists_between hδ
     refine ⟨η, hη, ?_⟩
-    rw [ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul]
+    rw [← ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul]
     exact hδε' hηδ
   refine ⟨η, hη_pos, fun s hs => ?_⟩
   refine (eLpNorm_indicator_const_le _ _).trans (le_trans ?_ hη_le)
@@ -802,7 +802,7 @@ theorem nnnorm_indicatorConstLp_le :
 theorem ennnorm_indicatorConstLp_le :
     (‖indicatorConstLp p hs hμs c‖₊ : ℝ≥0∞) ≤ ‖c‖₊ * (μ s) ^ (1 / p.toReal) := by
   refine (ENNReal.coe_le_coe.mpr nnnorm_indicatorConstLp_le).trans_eq ?_
-  simp [← ENNReal.coe_rpow_of_nonneg, ENNReal.coe_toNNReal hμs]
+  simp [ENNReal.coe_rpow_of_nonneg, ENNReal.coe_toNNReal hμs]
 
 theorem edist_indicatorConstLp_eq_nnnorm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
     edist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) =
@@ -1064,7 +1064,7 @@ theorem MeasureTheory.Memℒp.of_comp_antilipschitzWith {α E F} {K'} [Measurabl
     rw [← dist_zero_right, ← dist_zero_right, ← g0]
     apply hg'.le_mul_dist
   have B : AEStronglyMeasurable f μ :=
-    (hg'.uniformEmbedding hg).embedding.aestronglyMeasurable_comp_iff.1 hL.1
+    (hg'.isUniformEmbedding hg).embedding.aestronglyMeasurable_comp_iff.1 hL.1
   exact hL.of_le_mul B (Filter.Eventually.of_forall A)
 
 namespace LipschitzWith
@@ -1300,7 +1300,6 @@ end Composition
 We show that `L^p` is a complete space for `1 ≤ p`.
 -/
 
-
 section CompleteSpace
 
 namespace MeasureTheory
@@ -1317,7 +1316,7 @@ theorem eLpNorm'_lim_eq_lintegral_liminf {ι} [Nonempty ι] [LinearOrder ι] {f
   refine lintegral_congr_ae (h_lim.mono fun a ha => ?_)
   dsimp only
   rw [Tendsto.liminf_eq]
-  simp_rw [ENNReal.coe_rpow_of_nonneg _ hp_nonneg, ENNReal.tendsto_coe]
+  simp_rw [← ENNReal.coe_rpow_of_nonneg _ hp_nonneg, ENNReal.tendsto_coe]
   refine ((NNReal.continuous_rpow_const hp_nonneg).tendsto ‖f_lim a‖₊).comp ?_
   exact (continuous_nnnorm.tendsto (f_lim a)).comp ha
 
@@ -1877,7 +1876,7 @@ theorem toLp_inj {f g : C(α, E)} [μ.IsOpenPosMeasure] [NormedField 𝕜] [Norm
 variable {μ}
 
 /-- If a sum of continuous functions `g n` is convergent, and the same sum converges in `Lᵖ` to `h`,
-then in fact `g n` converges uniformly to `h`.  -/
+then in fact `g n` converges uniformly to `h`. -/
 theorem hasSum_of_hasSum_Lp {β : Type*} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
     {g : β → C(α, E)} {f : C(α, E)} (hg : Summable g)
     (hg2 : HasSum (toLp (E := E) p μ 𝕜 ∘ g) (toLp (E := E) p μ 𝕜 f)) : HasSum g f := by
@@ -1929,3 +1928,5 @@ theorem meas_ge_le_mul_pow_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_to
 end Lp
 
 end MeasureTheory
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/MeasureTheory/Function/SimpleFunc.lean b/Mathlib/MeasureTheory/Function/SimpleFunc.lean
index 12cb38f0a28bf..0935ab9a20775 100644
--- a/Mathlib/MeasureTheory/Function/SimpleFunc.lean
+++ b/Mathlib/MeasureTheory/Function/SimpleFunc.lean
@@ -161,7 +161,7 @@ theorem measurableSet_preimage (f : α →ₛ β) (s) : MeasurableSet (f ⁻¹'
   measurableSet_cut (fun _ b => b ∈ s) f fun b => MeasurableSet.const (b ∈ s)
 
 /-- A simple function is measurable -/
-@[measurability]
+@[measurability, fun_prop]
 protected theorem measurable [MeasurableSpace β] (f : α →ₛ β) : Measurable f := fun s _ =>
   measurableSet_preimage f s
 
@@ -353,6 +353,10 @@ theorem pair_preimage_singleton (f : α →ₛ β) (g : α →ₛ γ) (b : β) (
   rw [← singleton_prod_singleton]
   exact pair_preimage _ _ _ _
 
+@[simp] theorem map_fst_pair (f : α →ₛ β) (g : α →ₛ γ) : (f.pair g).map Prod.fst = f := rfl
+@[simp] theorem map_snd_pair (f : α →ₛ β) (g : α →ₛ γ) : (f.pair g).map Prod.snd = g := rfl
+
+@[simp]
 theorem bind_const (f : α →ₛ β) : f.bind (const α) = f := by ext; simp
 
 @[to_additive]
@@ -812,7 +816,7 @@ theorem lintegral_eq_of_subset' (f : α →ₛ ℝ≥0∞) {s : Finset ℝ≥0
   f.lintegral_eq_of_subset fun x hfx _ =>
     hs <| Finset.mem_sdiff.2 ⟨f.mem_range_self x, mt Finset.mem_singleton.1 hfx⟩
 
-/-- Calculate the integral of `(g ∘ f)`, where `g : β → ℝ≥0∞` and `f : α →ₛ β`.  -/
+/-- Calculate the integral of `(g ∘ f)`, where `g : β → ℝ≥0∞` and `f : α →ₛ β`. -/
 theorem map_lintegral (g : β → ℝ≥0∞) (f : α →ₛ β) :
     (f.map g).lintegral μ = ∑ x ∈ f.range, g x * μ (f ⁻¹' {x}) := by
   simp only [lintegral, range_map]
@@ -924,27 +928,30 @@ theorem restrict_const_lintegral (c : ℝ≥0∞) {s : Set α} (hs : MeasurableS
     ((const α c).restrict s).lintegral μ = c * μ s := by
   rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict]
 
-theorem le_sup_lintegral (f g : α →ₛ ℝ≥0∞) : f.lintegral μ ⊔ g.lintegral μ ≤ (f ⊔ g).lintegral μ :=
+@[gcongr]
+theorem lintegral_mono_fun {f g : α →ₛ ℝ≥0∞} (h : f ≤ g) : f.lintegral μ ≤ g.lintegral μ := by
+  refine Monotone.of_left_le_map_sup (f := (lintegral · μ)) (fun f g ↦ ?_) h
   calc
-    f.lintegral μ ⊔ g.lintegral μ =
-        ((pair f g).map Prod.fst).lintegral μ ⊔ ((pair f g).map Prod.snd).lintegral μ :=
-      rfl
-    _ ≤ ∑ x ∈ (pair f g).range, (x.1 ⊔ x.2) * μ (pair f g ⁻¹' {x}) := by
-      rw [map_lintegral, map_lintegral]
-      refine sup_le ?_ ?_ <;> refine Finset.sum_le_sum fun a _ => mul_le_mul_right' ?_ _
-      · exact le_sup_left
-      · exact le_sup_right
-    _ = (f ⊔ g).lintegral μ := by rw [sup_eq_map₂, map_lintegral]
+    f.lintegral μ = ((pair f g).map Prod.fst).lintegral μ := by rw [map_fst_pair]
+    _ ≤ ((pair f g).map fun p ↦ p.1 ⊔ p.2).lintegral μ := by
+      simp only [map_lintegral]
+      gcongr
+      exact le_sup_left
+
+theorem le_sup_lintegral (f g : α →ₛ ℝ≥0∞) : f.lintegral μ ⊔ g.lintegral μ ≤ (f ⊔ g).lintegral μ :=
+  Monotone.le_map_sup (fun _ _ ↦ lintegral_mono_fun) f g
+
+@[gcongr]
+theorem lintegral_mono_measure {f : α →ₛ ℝ≥0∞} (h : μ ≤ ν) : f.lintegral μ ≤ f.lintegral ν := by
+  simp only [lintegral]
+  gcongr
+  apply h
 
 /-- `SimpleFunc.lintegral` is monotone both in function and in measure. -/
-@[mono]
+@[mono, gcongr]
 theorem lintegral_mono {f g : α →ₛ ℝ≥0∞} (hfg : f ≤ g) (hμν : μ ≤ ν) :
     f.lintegral μ ≤ g.lintegral ν :=
-  calc
-    f.lintegral μ ≤ f.lintegral μ ⊔ g.lintegral μ := le_sup_left
-    _ ≤ (f ⊔ g).lintegral μ := le_sup_lintegral _ _
-    _ = g.lintegral μ := by rw [sup_of_le_right hfg]
-    _ ≤ g.lintegral ν := Finset.sum_le_sum fun y _ => ENNReal.mul_left_mono <| hμν _
+  (lintegral_mono_fun hfg).trans (lintegral_mono_measure hμν)
 
 /-- `SimpleFunc.lintegral` depends only on the measures of `f ⁻¹' {y}`. -/
 theorem lintegral_eq_of_measure_preimage [MeasurableSpace β] {f : α →ₛ ℝ≥0∞} {g : β →ₛ ℝ≥0∞}
@@ -980,7 +987,7 @@ open Finset Function
 theorem support_eq [MeasurableSpace α] [Zero β] (f : α →ₛ β) :
     support f = ⋃ y ∈ f.range.filter fun y => y ≠ 0, f ⁻¹' {y} :=
   Set.ext fun x => by
-    simp only [mem_support, Set.mem_preimage, mem_filter, mem_range_self, true_and_iff, exists_prop,
+    simp only [mem_support, Set.mem_preimage, mem_filter, mem_range_self, true_and, exists_prop,
       mem_iUnion, Set.mem_range, mem_singleton_iff, exists_eq_right']
 
 variable {m : MeasurableSpace α} [Zero β] [Zero γ] {μ : Measure α} {f : α →ₛ β}
diff --git a/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean b/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
index 679c905a01e48..e017f36bbf06c 100644
--- a/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
+++ b/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
@@ -23,7 +23,7 @@ by a sequence of simple functions.
   measurable and `Memℒp` (for `p < ∞`), then the simple functions
   `SimpleFunc.approxOn f hf s 0 h₀ n` may be considered as elements of `Lp E p μ`, and they tend
   in Lᵖ to `f`.
-* `Lp.simpleFunc.denseEmbedding`: the embedding `coeToLp` of the `Lp` simple functions into
+* `Lp.simpleFunc.isDenseEmbedding`: the embedding `coeToLp` of the `Lp` simple functions into
   `Lp` is dense.
 * `Lp.simpleFunc.induction`, `Lp.induction`, `Memℒp.induction`, `Integrable.induction`: to prove
   a predicate for all elements of one of these classes of functions, it suffices to check that it
@@ -299,7 +299,7 @@ theorem measure_preimage_lt_top_of_memℒp (hp_pos : p ≠ 0) (hp_ne_top : p ≠
   · suffices h_empty : f ⁻¹' {y} = ∅ by
       rw [h_empty, measure_empty]; exact ENNReal.coe_lt_top
     ext1 x
-    rw [Set.mem_preimage, Set.mem_singleton_iff, mem_empty_iff_false, iff_false_iff]
+    rw [Set.mem_preimage, Set.mem_singleton_iff, mem_empty_iff_false, iff_false]
     refine fun hxy => hyf ?_
     rw [mem_range, Set.mem_range]
     exact ⟨x, hxy⟩
@@ -679,16 +679,21 @@ variable [Fact (1 ≤ p)]
 protected theorem uniformContinuous : UniformContinuous ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
   uniformContinuous_comap
 
-protected theorem uniformEmbedding : UniformEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
-  uniformEmbedding_comap Subtype.val_injective
+lemma isUniformEmbedding : IsUniformEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
+  isUniformEmbedding_comap Subtype.val_injective
 
-protected theorem uniformInducing : UniformInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
-  simpleFunc.uniformEmbedding.toUniformInducing
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
 
-protected theorem denseEmbedding (hp_ne_top : p ≠ ∞) :
-    DenseEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) := by
+theorem isUniformInducing : IsUniformInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
+  simpleFunc.isUniformEmbedding.isUniformInducing
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing := isUniformInducing
+
+lemma isDenseEmbedding (hp_ne_top : p ≠ ∞) :
+    IsDenseEmbedding ((↑) : Lp.simpleFunc E p μ → Lp E p μ) := by
   borelize E
-  apply simpleFunc.uniformEmbedding.denseEmbedding
+  apply simpleFunc.isUniformEmbedding.isDenseEmbedding
   intro f
   rw [mem_closure_iff_seq_limit]
   have hfi' : Memℒp f p μ := Lp.memℒp f
@@ -703,13 +708,16 @@ protected theorem denseEmbedding (hp_ne_top : p ≠ ∞) :
   convert SimpleFunc.tendsto_approxOn_range_Lp hp_ne_top (Lp.stronglyMeasurable f).measurable hfi'
   rw [toLp_coeFn f (Lp.memℒp f)]
 
-protected theorem denseInducing (hp_ne_top : p ≠ ∞) :
-    DenseInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
-  (simpleFunc.denseEmbedding hp_ne_top).toDenseInducing
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding := isDenseEmbedding
+
+protected theorem isDenseInducing (hp_ne_top : p ≠ ∞) :
+    IsDenseInducing ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
+  (simpleFunc.isDenseEmbedding hp_ne_top).toIsDenseInducing
 
 protected theorem denseRange (hp_ne_top : p ≠ ∞) :
     DenseRange ((↑) : Lp.simpleFunc E p μ → Lp E p μ) :=
-  (simpleFunc.denseInducing hp_ne_top).dense
+  (simpleFunc.isDenseInducing hp_ne_top).dense
 
 protected theorem dense (hp_ne_top : p ≠ ∞) : Dense (Lp.simpleFunc E p μ : Set (Lp E p μ)) := by
   simpa only [denseRange_subtype_val] using simpleFunc.denseRange (E := E) (μ := μ) hp_ne_top
@@ -773,7 +781,7 @@ theorem denseRange_coeSimpleFuncNonnegToLpNonneg [hp : Fact (1 ≤ p)] (hp_ne_to
   rw [mem_closure_iff_seq_limit]
   have hg_memℒp : Memℒp (g : α → G) p μ := Lp.memℒp (g : Lp G p μ)
   have zero_mem : (0 : G) ∈ (range (g : α → G) ∪ {0} : Set G) ∩ { y | 0 ≤ y } := by
-    simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff,
+    simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or,
       mem_setOf_eq, le_refl, and_self_iff]
   have : SeparableSpace ((range (g : α → G) ∪ {0}) ∩ { y | 0 ≤ y } : Set G) := by
     apply IsSeparable.separableSpace
diff --git a/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean b/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
index 1825b6c071019..b9644215bcca2 100644
--- a/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
@@ -127,7 +127,7 @@ theorem SimpleFunc.stronglyMeasurable {α β} {_ : MeasurableSpace α} [Topologi
 @[nontriviality]
 theorem StronglyMeasurable.of_finite [Finite α] {_ : MeasurableSpace α}
     [MeasurableSingletonClass α] [TopologicalSpace β]
-    (f : α → β) : StronglyMeasurable f :=
+    {f : α → β} : StronglyMeasurable f :=
   ⟨fun _ => SimpleFunc.ofFinite f, fun _ => tendsto_const_nhds⟩
 
 @[deprecated (since := "2024-02-05")]
@@ -136,7 +136,7 @@ alias stronglyMeasurable_of_fintype := StronglyMeasurable.of_finite
 @[deprecated StronglyMeasurable.of_finite (since := "2024-02-06")]
 theorem stronglyMeasurable_of_isEmpty [IsEmpty α] {_ : MeasurableSpace α} [TopologicalSpace β]
     (f : α → β) : StronglyMeasurable f :=
-  .of_finite f
+  .of_finite
 
 theorem stronglyMeasurable_const {α β} {_ : MeasurableSpace α} [TopologicalSpace β] {b : β} :
     StronglyMeasurable fun _ : α => b :=
@@ -887,7 +887,7 @@ theorem stronglyMeasurable_of_measurableSpace_le_on {α E} {m m₂ : MeasurableS
             exact MeasurableSet.empty
           ext1 y
           simp only [mem_inter_iff, mem_preimage, mem_singleton_iff, mem_compl_iff,
-            mem_empty_iff_false, iff_false_iff, not_and, not_not_mem]
+            mem_empty_iff_false, iff_false, not_and, not_not_mem]
           refine Function.mtr fun hys => ?_
           rw [hg_seq_zero y hys n]
           exact Ne.symm hx
@@ -1108,7 +1108,7 @@ variable {m : MeasurableSpace α} {μ ν : Measure α} [TopologicalSpace β] [To
   {f g : α → β}
 
 lemma of_finite [DiscreteMeasurableSpace α] [Finite α] : AEStronglyMeasurable f μ :=
-  ⟨_, .of_finite _, ae_eq_rfl⟩
+  ⟨_, .of_finite, ae_eq_rfl⟩
 
 section Mk
 
@@ -1880,3 +1880,5 @@ theorem stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable {α β ι
   exact ((t_sf n).measurable.comp measurable_fst).subtype_mk
 
 end MeasureTheory
+
+set_option linter.style.longFile 2000
diff --git a/Mathlib/MeasureTheory/Function/UnifTight.lean b/Mathlib/MeasureTheory/Function/UnifTight.lean
index cc50e8c1de965..deaea6b197602 100644
--- a/Mathlib/MeasureTheory/Function/UnifTight.lean
+++ b/Mathlib/MeasureTheory/Function/UnifTight.lean
@@ -76,7 +76,8 @@ namespace UnifTight
 
 theorem eventually_cofinite_indicator (hf : UnifTight f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∀ᶠ s in μ.cofinite.smallSets, ∀ i, eLpNorm (s.indicator (f i)) p μ ≤ ε := by
-  by_cases hε_top : ε = ∞; subst hε_top; simp
+  by_cases hε_top : ε = ∞
+  · subst hε_top; simp
   rcases hf (pos_iff_ne_zero.2 (toNNReal_ne_zero.mpr ⟨hε,hε_top⟩)) with ⟨s, hμs, hfs⟩
   refine (eventually_smallSets' ?_).2 ⟨sᶜ, ?_, fun i ↦ (coe_toNNReal hε_top) ▸ hfs i⟩
   · intro s t hst ht i
diff --git a/Mathlib/MeasureTheory/Function/UniformIntegrable.lean b/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
index 383456a232000..52d0280c4dbba 100644
--- a/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
+++ b/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
@@ -213,14 +213,14 @@ theorem Memℒp.integral_indicator_norm_ge_le (hf : Memℒp f 1 μ) (hmeas : Str
       · assumption
   rw [ENNReal.tendsto_atTop_zero] at this
   obtain ⟨M, hM⟩ := this (ENNReal.ofReal ε) (ENNReal.ofReal_pos.2 hε)
-  simp only [true_and_iff, zero_tsub, zero_le, sub_zero, zero_add, coe_nnnorm,
+  simp only [zero_tsub, zero_le, sub_zero, zero_add, coe_nnnorm,
     Set.mem_Icc] at hM
   refine ⟨M, ?_⟩
   convert hM M le_rfl
   simp only [coe_nnnorm, ENNReal.ofReal_eq_coe_nnreal (norm_nonneg _)]
   rfl
 
-/-- This lemma is superceded by `MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le`
+/-- This lemma is superseded by `MeasureTheory.Memℒp.integral_indicator_norm_ge_nonneg_le`
 which does not require measurability. -/
 theorem Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas (hf : Memℒp f 1 μ)
     (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
@@ -288,7 +288,7 @@ theorem Memℒp.eLpNorm_indicator_norm_ge_le (hf : Memℒp f p μ) (hmeas : Stro
     ENNReal.ofReal_rpow_of_pos hε]
   convert hM
   rename_i x
-  rw [ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,
+  rw [← ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,
     nnnorm_indicator_eq_indicator_nnnorm]
   have hiff : M ^ (1 / p.toReal) ≤ ‖f x‖₊ ↔ M ≤ ‖‖f x‖ ^ p.toReal‖₊ := by
     rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,
@@ -393,7 +393,7 @@ theorem Memℒp.eLpNorm_indicator_le' (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (h
 @[deprecated (since := "2024-07-27")]
 alias Memℒp.snorm_indicator_le' := Memℒp.eLpNorm_indicator_le'
 
-/-- This lemma is superceded by `MeasureTheory.Memℒp.eLpNorm_indicator_le` which does not require
+/-- This lemma is superseded by `MeasureTheory.Memℒp.eLpNorm_indicator_le` which does not require
 measurability on `f`. -/
 theorem Memℒp.eLpNorm_indicator_le_of_meas (hp_one : 1 ≤ p) (hp_top : p ≠ ∞) (hf : Memℒp f p μ)
     (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
@@ -641,7 +641,7 @@ theorem tendstoInMeasure_iff_tendsto_Lp_finite [IsFiniteMeasure μ] (hp : 1 ≤
         (fun n => (hf n).aestronglyMeasurable) hg.aestronglyMeasurable h,
       unifIntegrable_of_tendsto_Lp hp hp' hf hg h⟩⟩
 
-/-- This lemma is superceded by `unifIntegrable_of` which do not require `C` to be positive. -/
+/-- This lemma is superseded by `unifIntegrable_of` which do not require `C` to be positive. -/
 theorem unifIntegrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
     (hf : ∀ i, StronglyMeasurable (f i))
     (h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
@@ -790,7 +790,7 @@ theorem uniformIntegrable_const {g : α → β} (hp : 1 ≤ p) (hp_ne_top : p 
   ⟨fun _ => hg.1, unifIntegrable_const hp hp_ne_top hg,
     ⟨(eLpNorm g p μ).toNNReal, fun _ => le_of_eq (ENNReal.coe_toNNReal hg.2.ne).symm⟩⟩
 
-/-- This lemma is superceded by `uniformIntegrable_of` which only requires
+/-- This lemma is superseded by `uniformIntegrable_of` which only requires
 `AEStronglyMeasurable`. -/
 theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
     (hf : ∀ i, StronglyMeasurable (f i))
@@ -851,7 +851,7 @@ theorem uniformIntegrable_of [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ 
   · rw [Set.indicator_of_not_mem hfx, Set.indicator_of_not_mem]
     rwa [Set.mem_setOf, hx] at hfx
 
-/-- This lemma is superceded by `UniformIntegrable.spec` which does not require measurability. -/
+/-- This lemma is superseded by `UniformIntegrable.spec` which does not require measurability. -/
 theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, StronglyMeasurable (f i))
     (hfu : UniformIntegrable f p μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ C : ℝ≥0, ∀ i, eLpNorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε := by
@@ -876,7 +876,7 @@ theorem UniformIntegrable.spec' (hp : p ≠ 0) (hp' : p ≠ ∞) (hf : ∀ i, St
           rwa [nnnorm_indicator_eq_indicator_nnnorm, Set.indicator_of_mem hx]
         _ ≤ eLpNorm (f (ℐ C)) p μ := eLpNorm_indicator_le _
     specialize this (2 * max M 1 * δ⁻¹ ^ (1 / p.toReal))
-    rw [ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), ← ENNReal.coe_smul,
+    rw [← ENNReal.coe_rpow_of_nonneg _ (one_div_nonneg.2 ENNReal.toReal_nonneg), ← ENNReal.coe_smul,
       smul_eq_mul, mul_assoc, NNReal.inv_rpow,
       inv_mul_cancel₀ (NNReal.rpow_pos (NNReal.coe_pos.1 hδpos)).ne.symm, mul_one, ENNReal.coe_mul,
       ← NNReal.inv_rpow] at this
diff --git a/Mathlib/MeasureTheory/Group/Action.lean b/Mathlib/MeasureTheory/Group/Action.lean
index 159a5bff0f4ff..04f432bce67b9 100644
--- a/Mathlib/MeasureTheory/Group/Action.lean
+++ b/Mathlib/MeasureTheory/Group/Action.lean
@@ -171,7 +171,7 @@ theorem smulInvariantMeasure_iterateMulAct
 section SMulHomClass
 
 universe uM uN uα uβ
-variable {M : Type uM} {N : Type uN}  {α : Type uα} {β : Type uβ}
+variable {M : Type uM} {N : Type uN} {α : Type uα} {β : Type uβ}
   [MeasurableSpace M] [MeasurableSpace N] [MeasurableSpace α] [MeasurableSpace β]
 
 @[to_additive]
@@ -226,23 +226,17 @@ theorem smulInvariantMeasure_tfae :
         ∀ (c : G) (s), μ (c • s) = μ s,
         ∀ c : G, Measure.map (c • ·) μ = μ,
         ∀ c : G, MeasurePreserving (c • ·) μ μ] := by
-  tfae_have 1 ↔ 2
-  · exact ⟨fun h => h.1, fun h => ⟨h⟩⟩
-  tfae_have 1 → 6
-  · intro h c
-    exact (measurePreserving_smul c μ).map_eq
-  tfae_have 6 → 7
-  · exact fun H c => ⟨measurable_const_smul c, H c⟩
-  tfae_have 7 → 4
-  · exact fun H c => (H c).measure_preimage_emb (measurableEmbedding_const_smul c)
+  tfae_have 1 ↔ 2 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
+  tfae_have 1 → 6 := fun h c => (measurePreserving_smul c μ).map_eq
+  tfae_have 6 → 7 := fun H c => ⟨measurable_const_smul c, H c⟩
+  tfae_have 7 → 4 := fun H c => (H c).measure_preimage_emb (measurableEmbedding_const_smul c)
   tfae_have 4 → 5
-  · exact fun H c s => by
-      rw [← preimage_smul_inv]
-      apply H
-  tfae_have 5 → 3
-  · exact fun H c s _ => H c s
+  | H, c, s => by
+    rw [← preimage_smul_inv]
+    apply H
+  tfae_have 5 → 3 := fun H c s _ => H c s
   tfae_have 3 → 2
-  · intro H c s hs
+  | H, c, s, hs => by
     rw [preimage_smul]
     exact H c⁻¹ s hs
   tfae_finish
diff --git a/Mathlib/MeasureTheory/Group/AddCircle.lean b/Mathlib/MeasureTheory/Group/AddCircle.lean
index a48b34277aca2..38e54d654f75e 100644
--- a/Mathlib/MeasureTheory/Group/AddCircle.lean
+++ b/Mathlib/MeasureTheory/Group/AddCircle.lean
@@ -35,9 +35,9 @@ theorem closedBall_ae_eq_ball {x : AddCircle T} {ε : ℝ} : closedBall x ε =
   · rw [ball_eq_empty.mpr hε, ae_eq_empty, volume_closedBall,
       min_eq_right (by linarith [hT.out] : 2 * ε ≤ T), ENNReal.ofReal_eq_zero]
     exact mul_nonpos_of_nonneg_of_nonpos zero_le_two hε
-  · suffices volume (closedBall x ε) ≤ volume (ball x ε) by
-      exact (ae_eq_of_subset_of_measure_ge ball_subset_closedBall this measurableSet_ball
-        (measure_ne_top _ _)).symm
+  · suffices volume (closedBall x ε) ≤ volume (ball x ε) from
+      (ae_eq_of_subset_of_measure_ge ball_subset_closedBall this
+        measurableSet_ball.nullMeasurableSet (measure_ne_top _ _)).symm
     have : Tendsto (fun δ => volume (closedBall x δ)) (𝓝[<] ε) (𝓝 <| volume (closedBall x ε)) := by
       simp_rw [volume_closedBall]
       refine ENNReal.tendsto_ofReal (Tendsto.min tendsto_const_nhds <| Tendsto.const_mul _ ?_)
diff --git a/Mathlib/MeasureTheory/Group/Arithmetic.lean b/Mathlib/MeasureTheory/Group/Arithmetic.lean
index d58f14fb0b7a4..a265438e4ebd7 100644
--- a/Mathlib/MeasureTheory/Group/Arithmetic.lean
+++ b/Mathlib/MeasureTheory/Group/Arithmetic.lean
@@ -87,8 +87,8 @@ export MeasurableMul₂ (measurable_mul)
 
 section Mul
 
-variable {M α : Type*} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α} {f g : α → M}
-  {μ : Measure α}
+variable {M α β : Type*} [MeasurableSpace M] [Mul M] {m : MeasurableSpace α}
+  {mβ : MeasurableSpace β} {f g : α → M} {μ : Measure α}
 
 @[to_additive (attr := fun_prop, measurability)]
 theorem Measurable.const_mul [MeasurableMul M] (hf : Measurable f) (c : M) :
@@ -110,17 +110,19 @@ theorem AEMeasurable.mul_const [MeasurableMul M] (hf : AEMeasurable f μ) (c : M
     AEMeasurable (fun x => f x * c) μ :=
   (measurable_mul_const c).comp_aemeasurable hf
 
-@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
-theorem Measurable.mul' [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
-    Measurable (f * g) :=
-  measurable_mul.comp (hf.prod_mk hg)
-
 @[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
 theorem Measurable.mul [MeasurableMul₂ M] (hf : Measurable f) (hg : Measurable g) :
     Measurable fun a => f a * g a :=
   measurable_mul.comp (hf.prod_mk hg)
 
-@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
+/-- Compositional version of `Measurable.mul` for use by `fun_prop`. -/
+@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))
+"Compositional version of `Measurable.add` for use by `fun_prop`."]
+lemma Measurable.mul' [MeasurableMul₂ M] {f g : α → β → M} {h : α → β} (hf : Measurable ↿f)
+    (hg : Measurable ↿g) (hh : Measurable h) : Measurable fun a ↦ (f a * g a) (h a) := by
+  simp; fun_prop
+
+@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
 theorem AEMeasurable.mul' [MeasurableMul₂ M] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
     AEMeasurable (f * g) μ :=
   measurable_mul.comp_aemeasurable (hf.prod_mk hg)
@@ -238,8 +240,8 @@ export MeasurableDiv₂ (measurable_div)
 
 section Div
 
-variable {G α : Type*} [MeasurableSpace G] [Div G] {m : MeasurableSpace α} {f g : α → G}
-  {μ : Measure α}
+variable {G α β : Type*} [MeasurableSpace G] [Div G] {m : MeasurableSpace α}
+  {mβ : MeasurableSpace β} {f g : α → G} {μ : Measure α}
 
 @[to_additive (attr := measurability)]
 theorem Measurable.const_div [MeasurableDiv G] (hf : Measurable f) (c : G) :
@@ -261,17 +263,17 @@ theorem AEMeasurable.div_const [MeasurableDiv G] (hf : AEMeasurable f μ) (c : G
     AEMeasurable (fun x => f x / c) μ :=
   (MeasurableDiv.measurable_div_const c).comp_aemeasurable hf
 
-@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
-theorem Measurable.div' [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
-    Measurable (f / g) :=
-  measurable_div.comp (hf.prod_mk hg)
-
 @[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
 theorem Measurable.div [MeasurableDiv₂ G] (hf : Measurable f) (hg : Measurable g) :
     Measurable fun a => f a / g a :=
   measurable_div.comp (hf.prod_mk hg)
 
-@[to_additive (attr := aesop safe 20 apply (rule_sets := [Measurable]))]
+@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
+lemma Measurable.div' [MeasurableDiv₂ G] {f g : α → β → G} {h : α → β} (hf : Measurable ↿f)
+    (hg : Measurable ↿g) (hh : Measurable h) : Measurable fun a ↦ (f a / g a) (h a) := by
+  simp; fun_prop
+
+@[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
 theorem AEMeasurable.div' [MeasurableDiv₂ G] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
     AEMeasurable (f / g) μ :=
   measurable_div.comp_aemeasurable (hf.prod_mk hg)
@@ -528,42 +530,52 @@ instance Subgroup.measurableSMul {G α} [MeasurableSpace G] [MeasurableSpace α]
 
 section SMul
 
-variable {M β α : Type*} [MeasurableSpace M] [MeasurableSpace β] [_root_.SMul M β]
-  {m : MeasurableSpace α} {f : α → M} {g : α → β}
+variable {M X α β : Type*} [MeasurableSpace M] [MeasurableSpace X] [SMul M X]
+  {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → M} {g : α → X}
 
 @[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
-theorem Measurable.smul [MeasurableSMul₂ M β] (hf : Measurable f) (hg : Measurable g) :
+theorem Measurable.smul [MeasurableSMul₂ M X] (hf : Measurable f) (hg : Measurable g) :
     Measurable fun x => f x • g x :=
   measurable_smul.comp (hf.prod_mk hg)
 
+/-- Compositional version of `Measurable.smul` for use by `fun_prop`. -/
+@[to_additive (attr := fun_prop)
+"Compositional version of `Measurable.vadd` for use by `fun_prop`."]
+lemma Measurable.smul' [MeasurableSMul₂ M X] {f : α → β → M} {g : α → β → X} {h : α → β}
+    (hf : Measurable ↿f) (hg : Measurable ↿g) (hh : Measurable h) :
+    Measurable fun a ↦ (f a • g a) (h a) := by simp; fun_prop
+
 @[to_additive (attr := fun_prop, aesop safe 20 apply (rule_sets := [Measurable]))]
-theorem AEMeasurable.smul [MeasurableSMul₂ M β] {μ : Measure α} (hf : AEMeasurable f μ)
+theorem AEMeasurable.smul [MeasurableSMul₂ M X] {μ : Measure α} (hf : AEMeasurable f μ)
     (hg : AEMeasurable g μ) : AEMeasurable (fun x => f x • g x) μ :=
   MeasurableSMul₂.measurable_smul.comp_aemeasurable (hf.prod_mk hg)
 
 @[to_additive]
-instance (priority := 100) MeasurableSMul₂.toMeasurableSMul [MeasurableSMul₂ M β] :
-    MeasurableSMul M β :=
+instance (priority := 100) MeasurableSMul₂.toMeasurableSMul [MeasurableSMul₂ M X] :
+    MeasurableSMul M X :=
   ⟨fun _ => measurable_const.smul measurable_id, fun _ => measurable_id.smul measurable_const⟩
 
-variable [MeasurableSMul M β] {μ : Measure α}
+variable [MeasurableSMul M X] {μ : Measure α}
 
 @[to_additive (attr := measurability)]
-theorem Measurable.smul_const (hf : Measurable f) (y : β) : Measurable fun x => f x • y :=
+theorem Measurable.smul_const (hf : Measurable f) (y : X) : Measurable fun x => f x • y :=
   (MeasurableSMul.measurable_smul_const y).comp hf
 
 @[to_additive (attr := measurability)]
-theorem AEMeasurable.smul_const (hf : AEMeasurable f μ) (y : β) :
+theorem AEMeasurable.smul_const (hf : AEMeasurable f μ) (y : X) :
     AEMeasurable (fun x => f x • y) μ :=
   (MeasurableSMul.measurable_smul_const y).comp_aemeasurable hf
 
-@[to_additive (attr := measurability)]
-theorem Measurable.const_smul' (hg : Measurable g) (c : M) : Measurable fun x => c • g x :=
+@[to_additive (attr := fun_prop, measurability)]
+theorem Measurable.const_smul (hg : Measurable g) (c : M) : Measurable (c • g) :=
   (MeasurableSMul.measurable_const_smul c).comp hg
 
-@[to_additive (attr := measurability)]
-theorem Measurable.const_smul (hg : Measurable g) (c : M) : Measurable (c • g) :=
-  hg.const_smul' c
+/-- Compositional version of `Measurable.const_smul` for use by `fun_prop`. -/
+@[to_additive (attr := fun_prop)
+"Compositional version of `Measurable.const_vadd` for use by `fun_prop`."]
+lemma Measurable.const_smul' {g : α → β → X} {h : α → β} (hg : Measurable ↿g) (hh : Measurable h)
+    (c : M) : Measurable fun a ↦ (c • g a) (h a) :=
+  (hg.comp <| measurable_id.prod_mk hh).const_smul _
 
 @[to_additive (attr := measurability)]
 theorem AEMeasurable.const_smul' (hg : AEMeasurable g μ) (c : M) :
@@ -638,12 +650,12 @@ variable {G : Type*} [Group G] [MeasurableSpace G] [MulAction G β] [MeasurableS
 
 @[to_additive]
 theorem measurable_const_smul_iff (c : G) : (Measurable fun x => c • f x) ↔ Measurable f :=
-  ⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
+  ⟨fun h => by simpa [inv_smul_smul, Pi.smul_def] using h.const_smul c⁻¹, fun h => h.const_smul c⟩
 
 @[to_additive]
 theorem aemeasurable_const_smul_iff (c : G) :
     AEMeasurable (fun x => c • f x) μ ↔ AEMeasurable f μ :=
-  ⟨fun h => by simpa only [inv_smul_smul] using h.const_smul' c⁻¹, fun h => h.const_smul c⟩
+  ⟨fun h => by simpa [inv_smul_smul, Pi.smul_def] using h.const_smul c⁻¹, fun h => h.const_smul c⟩
 
 @[to_additive]
 instance Units.instMeasurableSpace : MeasurableSpace Mˣ := MeasurableSpace.comap ((↑) : Mˣ → M) ‹_›
@@ -782,8 +794,8 @@ end Monoid
 
 section CommMonoid
 
-variable {M ι α : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M]
-  {m : MeasurableSpace α} {μ : Measure α} {f : ι → α → M}
+variable {M ι α β : Type*} [CommMonoid M] [MeasurableSpace M] [MeasurableMul₂ M]
+  {m : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : Measure α} {f : ι → α → M}
 
 @[to_additive (attr := measurability)]
 theorem Multiset.measurable_prod' (l : Multiset (α → M)) (hl : ∀ f ∈ l, Measurable f) :
@@ -807,15 +819,18 @@ theorem Multiset.aemeasurable_prod (s : Multiset (α → M)) (hs : ∀ f ∈ s,
     AEMeasurable (fun x => (s.map fun f : α → M => f x).prod) μ := by
   simpa only [← Pi.multiset_prod_apply] using s.aemeasurable_prod' hs
 
-@[to_additive (attr := measurability)]
-theorem Finset.measurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
-    Measurable (∏ i ∈ s, f i) :=
-  Finset.prod_induction _ _ (fun _ _ => Measurable.mul) (@measurable_one M _ _ _ _) hf
-
-@[to_additive (attr := measurability)]
+@[to_additive (attr := fun_prop, measurability)]
 theorem Finset.measurable_prod (s : Finset ι) (hf : ∀ i ∈ s, Measurable (f i)) :
-    Measurable fun a => ∏ i ∈ s, f i a := by
-  simpa only [← Finset.prod_apply] using s.measurable_prod' hf
+    Measurable fun a ↦ ∏ i ∈ s, f i a := by
+  simp_rw [← Finset.prod_apply]
+  exact Finset.prod_induction _ _ (fun _ _ => Measurable.mul) (@measurable_one M _ _ _ _) hf
+
+/-- Compositional version of `Finset.measurable_prod` for use by `fun_prop`. -/
+@[to_additive (attr := measurability, fun_prop)
+"Compositional version of `Finset.measurable_sum` for use by `fun_prop`."]
+lemma Finset.measurable_prod' {f : ι → α → β → M} {g : α → β} {s : Finset ι}
+    (hf : ∀ i, Measurable ↿(f i)) (hg : Measurable g) :
+    Measurable fun a ↦ (∏ i ∈ s, f i a) (g a) := by simp; fun_prop
 
 @[to_additive (attr := measurability)]
 theorem Finset.aemeasurable_prod' (s : Finset ι) (hf : ∀ i ∈ s, AEMeasurable (f i) μ) :
@@ -836,23 +851,23 @@ variable [MeasurableSpace α] [Mul α] [Div α] [Inv α]
 @[to_additive] -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableMul [DiscreteMeasurableSpace α] :
     MeasurableMul α where
-  measurable_const_mul _ := measurable_discrete _
-  measurable_mul_const _ := measurable_discrete _
+  measurable_const_mul _ := .of_discrete
+  measurable_mul_const _ := .of_discrete
 
 @[to_additive DiscreteMeasurableSpace.toMeasurableAdd₂] -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableMul₂
-    [DiscreteMeasurableSpace (α × α)] : MeasurableMul₂ α := ⟨measurable_discrete _⟩
+    [DiscreteMeasurableSpace (α × α)] : MeasurableMul₂ α := ⟨.of_discrete⟩
 
 @[to_additive] -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableInv [DiscreteMeasurableSpace α] :
-    MeasurableInv α := ⟨measurable_discrete _⟩
+    MeasurableInv α := ⟨.of_discrete⟩
 
 @[to_additive] -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableDiv [DiscreteMeasurableSpace α] :
     MeasurableDiv α where
-  measurable_const_div _ := measurable_discrete _
-  measurable_div_const _ := measurable_discrete _
+  measurable_const_div _ := .of_discrete
+  measurable_div_const _ := .of_discrete
 
 @[to_additive DiscreteMeasurableSpace.toMeasurableSub₂] -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableDiv₂
-    [DiscreteMeasurableSpace (α × α)] : MeasurableDiv₂ α := ⟨measurable_discrete _⟩
+    [DiscreteMeasurableSpace (α × α)] : MeasurableDiv₂ α := ⟨.of_discrete⟩
diff --git a/Mathlib/MeasureTheory/Group/Defs.lean b/Mathlib/MeasureTheory/Group/Defs.lean
index 35caa44a94cae..c68e87b9ad78f 100644
--- a/Mathlib/MeasureTheory/Group/Defs.lean
+++ b/Mathlib/MeasureTheory/Group/Defs.lean
@@ -82,15 +82,20 @@ class IsAddRightInvariant [Add G] (μ : Measure G) : Prop where
 class IsMulRightInvariant [Mul G] (μ : Measure G) : Prop where
   map_mul_right_eq_self : ∀ g : G, map (· * g) μ = μ
 
-variable [Mul G] {μ : Measure G}
+variable {μ : Measure G}
 
 @[to_additive]
-instance IsMulLeftInvariant.smulInvariantMeasure [IsMulLeftInvariant μ] :
+instance IsMulLeftInvariant.smulInvariantMeasure  [Mul G] [IsMulLeftInvariant μ] :
     SMulInvariantMeasure G G μ :=
   ⟨fun _x _s hs => measure_preimage_of_map_eq_self (map_mul_left_eq_self _) hs.nullMeasurableSet⟩
 
 @[to_additive]
-instance IsMulRightInvariant.toSMulInvariantMeasure_op [μ.IsMulRightInvariant] :
+instance [Monoid G] (s : Submonoid G) [IsMulLeftInvariant μ] :
+    SMulInvariantMeasure {x // x ∈ s} G μ :=
+  ⟨fun ⟨x, _⟩ _ h ↦ IsMulLeftInvariant.smulInvariantMeasure.1 x h⟩
+
+@[to_additive]
+instance IsMulRightInvariant.toSMulInvariantMeasure_op  [Mul G] [μ.IsMulRightInvariant] :
     SMulInvariantMeasure Gᵐᵒᵖ G μ :=
   ⟨fun _x _s hs => measure_preimage_of_map_eq_self (map_mul_right_eq_self _) hs.nullMeasurableSet⟩
 
diff --git a/Mathlib/MeasureTheory/Group/FundamentalDomain.lean b/Mathlib/MeasureTheory/Group/FundamentalDomain.lean
index 561111dd29111..b7e676a68b71b 100644
--- a/Mathlib/MeasureTheory/Group/FundamentalDomain.lean
+++ b/Mathlib/MeasureTheory/Group/FundamentalDomain.lean
@@ -346,7 +346,7 @@ protected theorem aEStronglyMeasurable_on_iff {β : Type*} [TopologicalSpace β]
       have he : MeasurableEmbedding (g⁻¹ • · : α → α) := measurableEmbedding_const_smul _
       rw [← image_smul, ← ((measurePreserving_smul g⁻¹ μ).restrict_image_emb he
         _).aestronglyMeasurable_comp_iff he]
-      simp only [(· ∘ ·), hf]
+      simp only [Function.comp_def, hf]
     _ ↔ AEStronglyMeasurable f (μ.restrict t) := by
       simp only [← aestronglyMeasurable_sum_measure_iff, ← hs.restrict_restrict,
         hs.sum_restrict_of_ac restrict_le_self.absolutelyContinuous]
diff --git a/Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean b/Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean
index 9fd4d3c6d24d8..4f7644744e85c 100644
--- a/Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean
+++ b/Mathlib/MeasureTheory/Group/GeometryOfNumbers.lean
@@ -36,7 +36,7 @@ Hermann Minkowski.
 
 namespace MeasureTheory
 
-open ENNReal FiniteDimensional MeasureTheory MeasureTheory.Measure Set Filter
+open ENNReal Module MeasureTheory MeasureTheory.Measure Set Filter
 
 open scoped Pointwise NNReal
 
@@ -57,7 +57,7 @@ theorem exists_pair_mem_lattice_not_disjoint_vadd [AddCommGroup L] [Countable L]
 
 /-- The **Minkowski Convex Body Theorem**. If `s` is a convex symmetric domain of `E` whose volume
 is large enough compared to the covolume of a lattice `L` of `E`, then it contains a non-zero
-lattice point of `L`.  -/
+lattice point of `L`. -/
 theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure [NormedAddCommGroup E]
     [NormedSpace ℝ E] [BorelSpace E] [FiniteDimensional ℝ E] [IsAddHaarMeasure μ]
     {L : AddSubgroup E} [Countable L] (fund : IsAddFundamentalDomain L F μ)
@@ -134,7 +134,7 @@ theorem exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure [NormedAddC
     rw [show μ s < _ ↔ 1 * μ s < _ by rw [one_mul]]
     refine (mul_lt_mul_right h_mes (ne_of_lt h_cpt.measure_lt_top)).mpr ?_
     rw [ofReal_pow (NNReal.coe_nonneg _)]
-    refine one_lt_pow ?_ (ne_of_gt finrank_pos)
+    refine one_lt_pow₀ ?_ (ne_of_gt finrank_pos)
     simp [(exists_seq_strictAnti_tendsto (0 : ℝ≥0)).choose_spec.2.1 n]
 
 end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Group/Measure.lean b/Mathlib/MeasureTheory/Group/Measure.lean
index ea0b382479f51..3e36cb114970b 100644
--- a/Mathlib/MeasureTheory/Group/Measure.lean
+++ b/Mathlib/MeasureTheory/Group/Measure.lean
@@ -3,15 +3,9 @@ Copyright (c) 2020 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
-import Mathlib.Dynamics.Ergodic.MeasurePreserving
-import Mathlib.GroupTheory.GroupAction.Hom
 import Mathlib.MeasureTheory.Constructions.Prod.Basic
 import Mathlib.MeasureTheory.Group.Action
-import Mathlib.MeasureTheory.Group.MeasurableEquiv
-import Mathlib.MeasureTheory.Measure.OpenPos
-import Mathlib.MeasureTheory.Measure.Regular
-import Mathlib.Topology.ContinuousFunction.CocompactMap
-import Mathlib.Topology.Homeomorph
+import Mathlib.Topology.ContinuousMap.CocompactMap
 
 /-!
 # Measures on Groups
@@ -359,7 +353,7 @@ instance inv.instIsMulRightInvariant [IsMulLeftInvariant μ] : IsMulRightInvaria
   intro g
   conv_rhs => rw [← map_mul_left_eq_self μ g⁻¹]
   simp_rw [Measure.inv, map_map (measurable_mul_const g) measurable_inv,
-    map_map measurable_inv (measurable_const_mul g⁻¹), Function.comp, mul_inv_rev, inv_inv]
+    map_map measurable_inv (measurable_const_mul g⁻¹), Function.comp_def, mul_inv_rev, inv_inv]
 
 @[to_additive]
 instance inv.instIsMulLeftInvariant [IsMulRightInvariant μ] : IsMulLeftInvariant μ.inv := by
@@ -367,7 +361,7 @@ instance inv.instIsMulLeftInvariant [IsMulRightInvariant μ] : IsMulLeftInvarian
   intro g
   conv_rhs => rw [← map_mul_right_eq_self μ g⁻¹]
   simp_rw [Measure.inv, map_map (measurable_const_mul g) measurable_inv,
-    map_map measurable_inv (measurable_mul_const g⁻¹), Function.comp, mul_inv_rev, inv_inv]
+    map_map measurable_inv (measurable_mul_const g⁻¹), Function.comp_def, mul_inv_rev, inv_inv]
 
 @[to_additive]
 theorem measurePreserving_div_left (μ : Measure G) [IsInvInvariant μ] [IsMulLeftInvariant μ]
@@ -470,7 +464,7 @@ lemma eventually_nhds_one_measure_smul_diff_lt [LocallyCompactSpace G]
     μ (g • k \ k) ≤ μ (U \ k) := by
       gcongr
       exact (smul_set_subset_smul hg).trans hVkU
-    _ < ε := measure_diff_lt_of_lt_add h'k.measurableSet hUk hk.measure_lt_top.ne hμUk
+    _ < ε := measure_diff_lt_of_lt_add h'k.nullMeasurableSet hUk hk.measure_lt_top.ne hμUk
 
 /-- Continuity of the measure of translates of a compact set:
 Given a closed compact set `k` in a topological group,
@@ -524,7 +518,7 @@ theorem null_iff_of_isMulLeftInvariant [Regular μ] {s : Set G} (hs : IsOpen s)
     μ s = 0 ↔ s = ∅ ∨ μ = 0 := by
   rcases eq_zero_or_neZero μ with rfl|hμ
   · simp
-  · simp only [or_false_iff, hs.measure_eq_zero_iff μ, NeZero.ne μ]
+  · simp only [or_false, hs.measure_eq_zero_iff μ, NeZero.ne μ]
 
 @[to_additive]
 theorem measure_ne_zero_iff_nonempty_of_isMulLeftInvariant [Regular μ] (hμ : μ ≠ 0) {s : Set G}
@@ -728,14 +722,15 @@ a Haar measure. See also `MulEquiv.isHaarMeasure_map`. -/
 "The image of an additive Haar measure under a continuous surjective proper additive group
 homomorphism is again an additive Haar measure. See also `AddEquiv.isAddHaarMeasure_map`."]
 theorem isHaarMeasure_map [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
-    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [T2Space H] [TopologicalGroup H]
+    [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H]
     (f : G →* H) (hf : Continuous f) (h_surj : Surjective f)
     (h_prop : Tendsto f (cocompact G) (cocompact H)) : IsHaarMeasure (Measure.map f μ) :=
   { toIsMulLeftInvariant := isMulLeftInvariant_map f.toMulHom hf.measurable h_surj
     lt_top_of_isCompact := by
       intro K hK
-      rw [map_apply hf.measurable hK.measurableSet]
-      exact IsCompact.measure_lt_top ((⟨⟨f, hf⟩, h_prop⟩ : CocompactMap G H).isCompact_preimage hK)
+      rw [← hK.measure_closure, map_apply hf.measurable isClosed_closure.measurableSet]
+      set g : CocompactMap G H := ⟨⟨f, hf⟩, h_prop⟩
+      exact IsCompact.measure_lt_top (g.isCompact_preimage_of_isClosed hK.closure isClosed_closure)
     toIsOpenPosMeasure := hf.isOpenPosMeasure_map h_surj }
 
 /-- The image of a finite Haar measure under a continuous surjective group homomorphism is again
@@ -747,10 +742,9 @@ theorem isHaarMeasure_map_of_isFiniteMeasure
     [BorelSpace G] [TopologicalGroup G] {H : Type*} [Group H]
     [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H] [TopologicalGroup H] [IsFiniteMeasure μ]
     (f : G →* H) (hf : Continuous f) (h_surj : Surjective f) :
-    IsHaarMeasure (Measure.map f μ) :=
-  { toIsMulLeftInvariant := isMulLeftInvariant_map f.toMulHom hf.measurable h_surj
-    lt_top_of_isCompact := fun _K hK ↦ hK.measure_lt_top
-    toIsOpenPosMeasure := hf.isOpenPosMeasure_map h_surj }
+    IsHaarMeasure (Measure.map f μ) where
+  toIsMulLeftInvariant := isMulLeftInvariant_map f.toMulHom hf.measurable h_surj
+  toIsOpenPosMeasure := hf.isOpenPosMeasure_map h_surj
 
 /-- The image of a Haar measure under map of a left action is again a Haar measure. -/
 @[to_additive
@@ -780,20 +774,11 @@ nonrec theorem _root_.MulEquiv.isHaarMeasure_map [BorelSpace G] [TopologicalGrou
     [Group H] [TopologicalSpace H] [MeasurableSpace H] [BorelSpace H]
     [TopologicalGroup H] (e : G ≃* H) (he : Continuous e) (hesymm : Continuous e.symm) :
     IsHaarMeasure (Measure.map e μ) :=
-  { toIsMulLeftInvariant := isMulLeftInvariant_map e.toMulHom he.measurable e.surjective
-    lt_top_of_isCompact := by
-      intro K hK
-      let F : G ≃ₜ H := {
-        e.toEquiv with
-        continuous_toFun := he
-        continuous_invFun := hesymm }
-      change map F.toMeasurableEquiv μ K < ∞
-      rw [F.toMeasurableEquiv.map_apply K]
-      exact (F.isCompact_preimage.mpr hK).measure_lt_top
-    toIsOpenPosMeasure := he.isOpenPosMeasure_map e.surjective }
+  let f : G ≃ₜ H := .mk e
+  isHaarMeasure_map μ e he e.surjective f.closedEmbedding.tendsto_cocompact
 
 /-- A convenience wrapper for MeasureTheory.Measure.isAddHaarMeasure_map`. -/
-theorem _root_.ContinuousLinearEquiv.isAddHaarMeasure_map
+instance _root_.ContinuousLinearEquiv.isAddHaarMeasure_map
     {E F R S : Type*} [Semiring R] [Semiring S]
     [AddCommGroup E] [Module R E] [AddCommGroup F] [Module S F]
     [TopologicalSpace E] [TopologicalAddGroup E] [TopologicalSpace F]
diff --git a/Mathlib/MeasureTheory/Group/Prod.lean b/Mathlib/MeasureTheory/Group/Prod.lean
index bfa11106c1272..38b8691d5bc0d 100644
--- a/Mathlib/MeasureTheory/Group/Prod.lean
+++ b/Mathlib/MeasureTheory/Group/Prod.lean
@@ -243,7 +243,7 @@ theorem absolutelyContinuous_of_isMulLeftInvariant [IsMulLeftInvariant ν] (hν
   refine AbsolutelyContinuous.mk fun s sm hνs => ?_
   have h1 := measure_mul_lintegral_eq μ ν sm 1 measurable_one
   simp_rw [Pi.one_apply, lintegral_one, mul_one, (measure_mul_right_null ν _).mpr hνs,
-    lintegral_zero, mul_eq_zero (M₀ := ℝ≥0∞), measure_univ_eq_zero.not.mpr hν, or_false_iff] at h1
+    lintegral_zero, mul_eq_zero (M₀ := ℝ≥0∞), measure_univ_eq_zero.not.mpr hν, or_false] at h1
   exact h1
 
 section SigmaFinite
@@ -455,7 +455,7 @@ theorem quasiMeasurePreserving_mul_left [IsMulRightInvariant μ] (g : G) :
   have :=
     (quasiMeasurePreserving_inv_of_right_invariant μ).comp
       (this.comp (quasiMeasurePreserving_inv_of_right_invariant μ))
-  simp_rw [Function.comp, mul_inv_rev, inv_inv] at this
+  simp_rw [Function.comp_def, mul_inv_rev, inv_inv] at this
   exact this
 
 end QuasiMeasurePreserving
diff --git a/Mathlib/MeasureTheory/Integral/Asymptotics.lean b/Mathlib/MeasureTheory/Integral/Asymptotics.lean
index 17d15d5d198ac..f80409c54ea35 100644
--- a/Mathlib/MeasureTheory/Integral/Asymptotics.lean
+++ b/Mathlib/MeasureTheory/Integral/Asymptotics.lean
@@ -5,6 +5,7 @@ Authors: Lawrence Wu
 -/
 import Mathlib.MeasureTheory.Group.Measure
 import Mathlib.MeasureTheory.Integral.IntegrableOn
+import Mathlib.MeasureTheory.Integral.SetIntegral
 import Mathlib.MeasureTheory.Function.LocallyIntegrable
 
 /-!
@@ -28,8 +29,13 @@ We establish integrability of `f` from `f = O(g)`.
 
 open Asymptotics MeasureTheory Set Filter
 
-variable {α E F : Type*} [MeasurableSpace α] [NormedAddCommGroup E] [NormedAddCommGroup F]
-  {f : α → E} {g : α → F} {a b : α} {μ : Measure α} {l : Filter α}
+variable {α E F : Type*} [NormedAddCommGroup E] {f : α → E} {g : α → F} {a b : α} {l : Filter α}
+
+namespace Asymptotics
+
+section Basic
+
+variable [MeasurableSpace α] [NormedAddCommGroup F] {μ : Measure α}
 
 /-- If `f = O[l] g` on measurably generated `l`, `f` is strongly measurable at `l`,
 and `g` is integrable at `l`, then `f` is integrable at `l`. -/
@@ -49,7 +55,53 @@ theorem _root_.Asymptotics.IsBigO.integrable (hfm : AEStronglyMeasurable f μ)
   rewrite [← integrableAtFilter_top] at *
   exact hf.integrableAtFilter ⟨univ, univ_mem, hfm.restrict⟩ hg
 
-variable [TopologicalSpace α] [SecondCountableTopology α]
+end Basic
+
+variable {ι : Type*} [MeasurableSpace ι] {f : ι × α → E} {s : Set ι} {μ : Measure ι}
+
+/-- Let `f : X x Y → Z`. If as `y` tends to `l`, `f(x, y) = O(g(y))` uniformly on `s : Set X`
+of finite measure, then f is eventually (as `y` tends to `l`) integrable along `s`. -/
+theorem IsBigO.eventually_integrableOn [Norm F]
+    (hf : f =O[𝓟 s ×ˢ l] (g ∘ Prod.snd))
+    (hfm : ∀ᶠ x in l, AEStronglyMeasurable (fun i ↦ f (i, x)) (μ.restrict s))
+    (hs : MeasurableSet s) (hμ : μ s < ⊤) :
+    ∀ᶠ x in l, IntegrableOn (fun i ↦ f (i, x)) s μ := by
+  obtain ⟨C, hC⟩ := hf.bound
+  obtain ⟨t, htl, ht⟩ := hC.exists_mem
+  obtain ⟨u, hu, v, hv, huv⟩ := Filter.mem_prod_iff.mp htl
+  obtain ⟨w, hwl, hw⟩ := hfm.exists_mem
+  refine eventually_iff_exists_mem.mpr ⟨w ∩ v, inter_mem hwl hv, fun x hx ↦ ?_⟩
+  haveI : IsFiniteMeasure (μ.restrict s) := ⟨Measure.restrict_apply_univ s ▸ hμ⟩
+  refine Integrable.mono' (integrable_const (C * ‖g x‖)) (hw x hx.1) ?_
+  filter_upwards [MeasureTheory.self_mem_ae_restrict hs]
+  intro y hy
+  exact ht (y, x) <| huv ⟨hu hy, hx.2⟩
+
+variable [NormedSpace ℝ E] [NormedAddCommGroup F]
+
+/-- Let `f : X x Y → Z`. If as `y` tends to `l`, `f(x, y) = O(g(y))` uniformly on `s : Set X`
+of finite measure, then the integral of `f` along `s` is `O(g(y))`. -/
+theorem IsBigO.set_integral_isBigO
+    (hf : f =O[𝓟 s ×ˢ l] (g ∘ Prod.snd)) (hs : MeasurableSet s) (hμ : μ s < ⊤)  :
+    (fun x ↦ ∫ i in s, f (i, x) ∂μ) =O[l] g := by
+  obtain ⟨C, hC⟩ := hf.bound
+  obtain ⟨t, htl, ht⟩ := hC.exists_mem
+  obtain ⟨u, hu, v, hv, huv⟩ := Filter.mem_prod_iff.mp htl
+  refine isBigO_iff.mpr ⟨C * (μ s).toReal, eventually_iff_exists_mem.mpr ⟨v, hv, fun x hx ↦ ?_⟩⟩
+  rw [mul_assoc, ← smul_eq_mul (a' := ‖g x‖), ← MeasureTheory.Measure.restrict_apply_univ,
+    ← integral_const, mul_comm, ← smul_eq_mul, ← integral_smul_const]
+  haveI : IsFiniteMeasure (μ.restrict s) := ⟨by rw [Measure.restrict_apply_univ s]; exact hμ⟩
+  refine (norm_integral_le_integral_norm _).trans <|
+    integral_mono_of_nonneg (univ_mem' fun _ ↦ norm_nonneg _) (integrable_const _) ?_
+  filter_upwards [MeasureTheory.self_mem_ae_restrict hs]
+  intro y hy
+  rw [smul_eq_mul, mul_comm]
+  exact ht (y, x) <| huv ⟨hu hy, hx⟩
+
+end Asymptotics
+
+variable [TopologicalSpace α] [SecondCountableTopology α] [MeasurableSpace α] {μ : Measure α}
+  [NormedAddCommGroup F]
 
 namespace MeasureTheory
 
diff --git a/Mathlib/MeasureTheory/Integral/Bochner.lean b/Mathlib/MeasureTheory/Integral/Bochner.lean
index ab1f1a86e1f93..411b231ffbadc 100644
--- a/Mathlib/MeasureTheory/Integral/Bochner.lean
+++ b/Mathlib/MeasureTheory/Integral/Bochner.lean
@@ -433,7 +433,7 @@ theorem norm_eq_integral (f : α →₁ₛ[μ] E) : ‖f‖ = ((toSimpleFunc f).
 
 section PosPart
 
-/-- Positive part of a simple function in L1 space.  -/
+/-- Positive part of a simple function in L1 space. -/
 nonrec def posPart (f : α →₁ₛ[μ] ℝ) : α →₁ₛ[μ] ℝ :=
   ⟨Lp.posPart (f : α →₁[μ] ℝ), by
     rcases f with ⟨f, s, hsf⟩
@@ -441,7 +441,7 @@ nonrec def posPart (f : α →₁ₛ[μ] ℝ) : α →₁ₛ[μ] ℝ :=
     simp only [Subtype.coe_mk, Lp.coe_posPart, ← hsf, AEEqFun.posPart_mk,
       SimpleFunc.coe_map, mk_eq_mk]
     -- Porting note: added
-    simp [SimpleFunc.posPart, Function.comp, EventuallyEq.rfl] ⟩
+    simp [SimpleFunc.posPart, Function.comp_def, EventuallyEq.rfl] ⟩
 
 /-- Negative part of a simple function in L1 space. -/
 def negPart (f : α →₁ₛ[μ] ℝ) : α →₁ₛ[μ] ℝ :=
@@ -588,7 +588,7 @@ variable (𝕜)
 /-- The Bochner integral in L1 space as a continuous linear map. -/
 nonrec def integralCLM' : (α →₁[μ] E) →L[𝕜] E :=
   (integralCLM' α E 𝕜 μ).extend (coeToLp α E 𝕜) (simpleFunc.denseRange one_ne_top)
-    simpleFunc.uniformInducing
+    simpleFunc.isUniformInducing
 
 variable {𝕜}
 
@@ -1000,7 +1000,7 @@ theorem continuous_of_dominated {F : X → α → G} {bound : α → ℝ}
   · simp [integral, hG, continuous_const]
 
 /-- The Bochner integral of a real-valued function `f : α → ℝ` is the difference between the
-  integral of the positive part of `f` and the integral of the negative part of `f`.  -/
+  integral of the positive part of `f` and the integral of the negative part of `f`. -/
 theorem integral_eq_lintegral_pos_part_sub_lintegral_neg_part {f : α → ℝ} (hf : Integrable f μ) :
     ∫ a, f a ∂μ =
       ENNReal.toReal (∫⁻ a, .ofReal (f a) ∂μ) - ENNReal.toReal (∫⁻ a, .ofReal (-f a) ∂μ) := by
@@ -1044,7 +1044,7 @@ theorem integral_eq_lintegral_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f
       · exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg
     rw [h_min, zero_toReal, _root_.sub_zero]
   · rw [integral_undef hfi]
-    simp_rw [Integrable, hfm, hasFiniteIntegral_iff_norm, lt_top_iff_ne_top, Ne, true_and_iff,
+    simp_rw [Integrable, hfm, hasFiniteIntegral_iff_norm, lt_top_iff_ne_top, Ne, true_and,
       Classical.not_not] at hfi
     have : ∫⁻ a : α, ENNReal.ofReal (f a) ∂μ = ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ := by
       refine lintegral_congr_ae (hf.mono fun a h => ?_)
@@ -1123,7 +1123,7 @@ theorem integral_nonpos {f : α → ℝ} (hf : f ≤ 0) : ∫ a, f a ∂μ ≤ 0
 theorem integral_eq_zero_iff_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f) (hfi : Integrable f μ) :
     ∫ x, f x ∂μ = 0 ↔ f =ᵐ[μ] 0 := by
   simp_rw [integral_eq_lintegral_of_nonneg_ae hf hfi.1, ENNReal.toReal_eq_zero_iff,
-    ← ENNReal.not_lt_top, ← hasFiniteIntegral_iff_ofReal hf, hfi.2, not_true_eq_false, or_false_iff]
+    ← ENNReal.not_lt_top, ← hasFiniteIntegral_iff_ofReal hf, hfi.2, not_true_eq_false, or_false]
   -- Porting note: split into parts, to make `rw` and `simp` work
   rw [lintegral_eq_zero_iff']
   · rw [← hf.le_iff_eq, Filter.EventuallyEq, Filter.EventuallyLE]
@@ -1468,6 +1468,13 @@ theorem integral_zero_measure {m : MeasurableSpace α} (f : α → G) :
     exact setToFun_measure_zero (dominatedFinMeasAdditive_weightedSMul _) rfl
   · simp [integral, hG]
 
+@[simp]
+theorem setIntegral_zero_measure (f : α → G) {μ : Measure α} {s : Set α} (hs : μ s = 0) :
+    ∫ x in s, f x ∂μ = 0 := Measure.restrict_eq_zero.mpr hs ▸ integral_zero_measure f
+
+lemma integral_of_isEmpty [IsEmpty α] {f : α → G} : ∫ x, f x ∂μ = 0 :=
+    μ.eq_zero_of_isEmpty ▸ integral_zero_measure _
+
 theorem integral_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
     {s : Finset ι} (hf : ∀ i ∈ s, Integrable f (μ i)) :
     ∫ a, f a ∂(∑ i ∈ s, μ i) = ∑ i ∈ s, ∫ a, f a ∂μ i := by
@@ -1958,3 +1965,5 @@ def evalIntegral : PositivityExt where eval {u α} zα pα e := do
   | _ => throwError "not MeasureTheory.integral"
 
 end Mathlib.Meta.Positivity
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean b/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
index 9fccb8bf103ac..2280f163883e0 100644
--- a/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
+++ b/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
@@ -350,7 +350,7 @@ theorem integral_divergence_of_hasFDerivWithinAt_off_countable_of_equiv {F : Typ
         simpa only [Set.mem_preimage, eL.apply_symm_apply, ← pi_univ_Icc,
           interior_pi_set (@finite_univ (Fin _) _), interior_Icc] using hx.1
       · rw [← he_vol.integrableOn_comp_preimage he_emb, hIcc]
-        simp [← hDF, (· ∘ ·), Hi]
+        simp [← hDF, Function.comp_def, Hi]
 
 end
 
diff --git a/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean b/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
index 0aa5977e8f9e6..7dc0c8c165627 100644
--- a/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
+++ b/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
@@ -255,8 +255,8 @@ variable {X : Type*} [TopologicalSpace X] [FirstCountableTopology X]
 
 /-- Continuity of interval integral with respect to a parameter, at a point within a set.
   Given `F : X → ℝ → E`, assume `F x` is ae-measurable on `[a, b]` for `x` in a
-  neighborhood of `x₀` within `s` and at `x₀`, and assume it is bounded by a function integrable
-  on `[a, b]` independent of `x` in a neighborhood of `x₀` within `s`. If `(fun x ↦ F x t)`
+  neighborhood of `x₀` within `s` and at `x₀`, and assume it is bounded by a function integrable
+  on `[a, b]` independent of `x` in a neighborhood of `x₀` within `s`. If `(fun x ↦ F x t)`
   is continuous at `x₀` within `s` for almost every `t` in `[a, b]`
   then the same holds for `(fun x ↦ ∫ t in a..b, F x t ∂μ) s x₀`. -/
 theorem continuousWithinAt_of_dominated_interval {F : X → ℝ → E} {x₀ : X} {bound : ℝ → ℝ} {a b : ℝ}
diff --git a/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean b/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
index 0ab09a7e17c02..4a55c25968c90 100644
--- a/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
+++ b/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
@@ -591,7 +591,7 @@ theorem integral_hasStrictFDerivAt_of_tendsto_ae (hf : IntervalIntegrable f volu
       (continuous_snd.snd.tendsto ((a, b), (a, b)))
       (continuous_fst.snd.tendsto ((a, b), (a, b)))
   refine (this.congr_left ?_).trans_isBigO ?_
-  · intro x; simp [sub_smul]; abel
+  · intro x; simp [sub_smul]
   · exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left
 
 /-- **Fundamental theorem of calculus-1**, strict differentiability in both endpoints.
@@ -794,7 +794,7 @@ theorem integral_hasFDerivWithinAt_of_tendsto_ae (hf : IntervalIntegrable f volu
       (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[s] a)) tendsto_fst
       (tendsto_const_pure.mono_right FTCFilter.pure_le : Tendsto _ _ (𝓝[t] b)) tendsto_snd
   refine .of_isLittleO <| (this.congr_left ?_).trans_isBigO ?_
-  · intro x; simp [sub_smul]; abel
+  · intro x; simp [sub_smul]
   · exact isBigO_fst_prod.norm_left.add isBigO_snd_prod.norm_left
 
 /-- Let `f` be a measurable function integrable on `a..b`. The function `(u, v) ↦ ∫ x in u..v, f x`
@@ -1002,7 +1002,7 @@ theorem sub_le_integral_of_hasDeriv_right_of_le_Ico (hab : a ≤ b)
         _ ≤ ∫ w in t..u, (G' w).toReal := by
           rw [intervalIntegral.integral_of_le hu.1.le, ← integral_Icc_eq_integral_Ioc]
           apply setIntegral_mono_ae_restrict
-          · simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true_iff]
+          · simp only [integrableOn_const, Real.volume_Icc, ENNReal.ofReal_lt_top, or_true]
           · exact IntegrableOn.mono_set G'int I
           · have C1 : ∀ᵐ x : ℝ ∂volume.restrict (Icc t u), G' x < ∞ :=
               ae_mono (Measure.restrict_mono I le_rfl) G'lt_top
@@ -1252,10 +1252,10 @@ theorem intervalIntegrable_deriv_of_nonneg (hcont : ContinuousOn g (uIcc a b))
     (hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) : IntervalIntegrable g' volume a b := by
   rcases le_total a b with hab | hab
   · simp only [uIcc_of_le, min_eq_left, max_eq_right, hab, IntervalIntegrable, hab,
-      Ioc_eq_empty_of_le, integrableOn_empty, and_true_iff] at hcont hderiv hpos ⊢
+      Ioc_eq_empty_of_le, integrableOn_empty, and_true] at hcont hderiv hpos ⊢
     exact integrableOn_deriv_of_nonneg hcont hderiv hpos
   · simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
-      integrableOn_empty, true_and_iff] at hcont hderiv hpos ⊢
+      integrableOn_empty, true_and] at hcont hderiv hpos ⊢
     exact integrableOn_deriv_of_nonneg hcont hderiv hpos
 
 /-!
@@ -1559,3 +1559,5 @@ theorem integral_deriv_comp_mul_deriv {f f' g g' : ℝ → ℝ}
 end Mul
 
 end intervalIntegral
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/MeasureTheory/Integral/Indicator.lean b/Mathlib/MeasureTheory/Integral/Indicator.lean
index fb1889b30f770..071281c1dc5e1 100644
--- a/Mathlib/MeasureTheory/Integral/Indicator.lean
+++ b/Mathlib/MeasureTheory/Integral/Indicator.lean
@@ -39,21 +39,25 @@ variable {ι : Type*} (L : Filter ι) [IsCountablyGenerated L] {As : ι → Set
 /-- If the indicators of measurable sets `Aᵢ` tend pointwise to the indicator of a set `A`
 and we eventually have `Aᵢ ⊆ B` for some set `B` of finite measure, then the measures of `Aᵢ`
 tend to the measure of `A`. -/
-lemma tendsto_measure_of_tendsto_indicator [NeBot L] {μ : Measure α}
+lemma tendsto_measure_of_tendsto_indicator {μ : Measure α}
     (As_mble : ∀ i, MeasurableSet (As i)) {B : Set α} (B_mble : MeasurableSet B)
     (B_finmeas : μ B ≠ ∞) (As_le_B : ∀ᶠ i in L, As i ⊆ B)
     (h_lim : ∀ x, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
     Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) := by
+  rcases L.eq_or_neBot with rfl | _
+  · exact tendsto_bot
   apply tendsto_measure_of_ae_tendsto_indicator L ?_ As_mble B_mble B_finmeas As_le_B
         (ae_of_all μ h_lim)
   exact measurableSet_of_tendsto_indicator L As_mble h_lim
 
 /-- If `μ` is a finite measure and the indicators of measurable sets `Aᵢ` tend pointwise to
 the indicator of a set `A`, then the measures `μ Aᵢ` tend to the measure `μ A`. -/
-lemma tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure [NeBot L]
+lemma tendsto_measure_of_tendsto_indicator_of_isFiniteMeasure
     (μ : Measure α) [IsFiniteMeasure μ] (As_mble : ∀ i, MeasurableSet (As i))
     (h_lim : ∀ x, ∀ᶠ i in L, x ∈ As i ↔ x ∈ A) :
     Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A)) := by
+  rcases L.eq_or_neBot with rfl | _
+  · exact tendsto_bot
   apply tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure L ?_ As_mble (ae_of_all μ h_lim)
   exact measurableSet_of_tendsto_indicator L As_mble h_lim
 
diff --git a/Mathlib/MeasureTheory/Integral/IntegrableOn.lean b/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
index 20ef2efcbbe01..ae1b813c974c1 100644
--- a/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
+++ b/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
@@ -622,7 +622,7 @@ theorem integrableOn_Icc_iff_integrableOn_Ioc' (ha : μ {a} ≠ ∞) :
     IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioc a b) μ := by
   by_cases hab : a ≤ b
   · rw [← Ioc_union_left hab, integrableOn_union,
-      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
+      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true]
   · rw [Icc_eq_empty hab, Ioc_eq_empty]
     contrapose! hab
     exact hab.le
@@ -631,7 +631,7 @@ theorem integrableOn_Icc_iff_integrableOn_Ico' (hb : μ {b} ≠ ∞) :
     IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ico a b) μ := by
   by_cases hab : a ≤ b
   · rw [← Ico_union_right hab, integrableOn_union,
-      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
+      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true]
   · rw [Icc_eq_empty hab, Ico_eq_empty]
     contrapose! hab
     exact hab.le
@@ -640,14 +640,14 @@ theorem integrableOn_Ico_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) :
     IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
   by_cases hab : a < b
   · rw [← Ioo_union_left hab, integrableOn_union,
-      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true_iff]
+      eq_true (integrableOn_singleton_iff.mpr <| Or.inr ha.lt_top), and_true]
   · rw [Ioo_eq_empty hab, Ico_eq_empty hab]
 
 theorem integrableOn_Ioc_iff_integrableOn_Ioo' (hb : μ {b} ≠ ∞) :
     IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ := by
   by_cases hab : a < b
   · rw [← Ioo_union_right hab, integrableOn_union,
-      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
+      eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true]
   · rw [Ioo_eq_empty hab, Ioc_eq_empty hab]
 
 theorem integrableOn_Icc_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) (hb : μ {b} ≠ ∞) :
@@ -657,12 +657,12 @@ theorem integrableOn_Icc_iff_integrableOn_Ioo' (ha : μ {a} ≠ ∞) (hb : μ {b
 theorem integrableOn_Ici_iff_integrableOn_Ioi' (hb : μ {b} ≠ ∞) :
     IntegrableOn f (Ici b) μ ↔ IntegrableOn f (Ioi b) μ := by
   rw [← Ioi_union_left, integrableOn_union,
-    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
+    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true]
 
 theorem integrableOn_Iic_iff_integrableOn_Iio' (hb : μ {b} ≠ ∞) :
     IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ := by
   rw [← Iio_union_right, integrableOn_union,
-    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]
+    eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true]
 
 variable [NoAtoms μ]
 
diff --git a/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean b/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
index 4b9143054ef0c..737247f5c8918 100644
--- a/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
+++ b/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
@@ -712,7 +712,7 @@ theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi
     rw [← top_le_iff, ← volume_Ici (a := b)]
     exact measure_mono hb
   rwa [B, ← Embedding.tendsto_nhds_iff] at A
-  exact (Completion.uniformEmbedding_coe E).embedding
+  exact (Completion.isUniformEmbedding_coe E).embedding
 
 variable [CompleteSpace E]
 
@@ -909,7 +909,7 @@ theorem tendsto_zero_of_hasDerivAt_of_integrableOn_Iic
     rw [← volume_Iic (a := b)]
     exact measure_mono hb
   rwa [B, ← Embedding.tendsto_nhds_iff] at A
-  exact (Completion.uniformEmbedding_coe E).embedding
+  exact (Completion.isUniformEmbedding_coe E).embedding
 
 variable [CompleteSpace E]
 
@@ -1075,7 +1075,7 @@ theorem integral_comp_rpow_Ioi (g : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
     rcases lt_or_gt_of_ne hp with (h | h)
     · apply StrictAntiOn.injOn
       intro x hx y hy hxy
-      rw [← inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx),
+      rw [← inv_lt_inv₀ (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx),
         ← rpow_neg (le_of_lt hy)]
       exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)
     exact StrictMonoOn.injOn fun x hx y _ hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h
@@ -1129,7 +1129,7 @@ theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p :
     rcases lt_or_gt_of_ne hp with (h | h)
     · apply StrictAntiOn.injOn
       intro x hx y hy hxy
-      rw [← inv_lt_inv (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx), ←
+      rw [← inv_lt_inv₀ (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx), ←
         rpow_neg (le_of_lt hy)]
       exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h)
     exact StrictMonoOn.injOn fun x hx y _hy hxy => rpow_lt_rpow (mem_Ioi.mp hx).le hxy h
@@ -1241,7 +1241,7 @@ theorem integral_mul_deriv_eq_deriv_mul [CompleteSpace A]
     (huv' : Integrable (u * v')) (hu'v : Integrable (u' * v))
     (h_bot : Tendsto (u * v) atBot (𝓝 a')) (h_top : Tendsto (u * v) atTop (𝓝 b')) :
     ∫ (x : ℝ), u x * v' x = b' - a' - ∫ (x : ℝ), u' x * v x :=
-  integral_bilinear_hasDerivAt_right_eq_sub  (L := ContinuousLinearMap.mul ℝ A)
+  integral_bilinear_hasDerivAt_right_eq_sub (L := ContinuousLinearMap.mul ℝ A)
     hu hv huv' hu'v h_bot h_top
 
 /-- **Integration by parts on (-∞, ∞).**
diff --git a/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean b/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
index 0acaf9efc4688..d7473d6e2d8df 100644
--- a/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
+++ b/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
@@ -878,7 +878,7 @@ theorem integral_const_of_cdf [CompleteSpace E] [IsFiniteMeasure μ] (c : E) :
     ∫ _ in a..b, c ∂μ = ((μ (Iic b)).toReal - (μ (Iic a)).toReal) • c := by
   simp only [sub_smul, ← setIntegral_const]
   refine (integral_Iic_sub_Iic ?_ ?_).symm <;>
-    simp only [integrableOn_const, measure_lt_top, or_true_iff]
+    simp only [integrableOn_const, measure_lt_top, or_true]
 
 theorem integral_eq_integral_of_support_subset {a b} (h : support f ⊆ Ioc a b) :
     ∫ x in a..b, f x ∂μ = ∫ x, f x ∂μ := by
@@ -936,9 +936,9 @@ theorem integral_pos_iff_support_of_nonneg_ae' (hf : 0 ≤ᵐ[μ.restrict (Ι a
     (0 < ∫ x in a..b, f x ∂μ) ↔ a < b ∧ 0 < μ (support f ∩ Ioc a b) := by
   cases' lt_or_le a b with hab hba
   · rw [uIoc_of_le hab.le] at hf
-    simp only [hab, true_and_iff, integral_of_le hab.le,
+    simp only [hab, true_and, integral_of_le hab.le,
       setIntegral_pos_iff_support_of_nonneg_ae hf hfi.1]
-  · suffices (∫ x in a..b, f x ∂μ) ≤ 0 by simp only [this.not_lt, hba.not_lt, false_and_iff]
+  · suffices (∫ x in a..b, f x ∂μ) ≤ 0 by simp only [this.not_lt, hba.not_lt, false_and]
     rw [integral_of_ge hba, neg_nonpos]
     rw [uIoc_comm, uIoc_of_le hba] at hf
     exact integral_nonneg_of_ae hf
@@ -1021,6 +1021,12 @@ theorem abs_integral_le_integral_abs (hab : a ≤ b) :
     |∫ x in a..b, f x ∂μ| ≤ ∫ x in a..b, |f x| ∂μ := by
   simpa only [← Real.norm_eq_abs] using norm_integral_le_integral_norm hab
 
+lemma integral_pos (hab : a < b)
+    (hfc : ContinuousOn f (Icc a b)) (hle : ∀ x ∈ Ioc a b, 0 ≤ f x) (hlt : ∃ c ∈ Icc a b, 0 < f c) :
+    0 < ∫ x in a..b, f x :=
+  (integral_lt_integral_of_continuousOn_of_le_of_exists_lt hab
+    continuousOn_const hfc hle hlt).trans_eq' (by simp)
+
 section Mono
 
 theorem integral_mono_interval {c d} (hca : c ≤ a) (hab : a ≤ b) (hbd : b ≤ d)
diff --git a/Mathlib/MeasureTheory/Integral/Layercake.lean b/Mathlib/MeasureTheory/Integral/Layercake.lean
index e0771f0991c48..ccdbcf47487ab 100644
--- a/Mathlib/MeasureTheory/Integral/Layercake.lean
+++ b/Mathlib/MeasureTheory/Integral/Layercake.lean
@@ -141,7 +141,7 @@ theorem lintegral_comp_eq_lintegral_meas_le_mul_of_measurable_of_sigmaFinite
         · simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le, mul_zero]
         · have : s ∉ Ioi (0 : ℝ) := h'
           simp only [this, h', indicator_of_not_mem, not_false_iff, mul_zero,
-            zero_mul, mem_Ioc, false_and_iff]
+            zero_mul, mem_Ioc, false_and]
     simp_rw [aux₁]
     rw [lintegral_const_mul']
     swap
@@ -548,7 +548,7 @@ lemma Integrable.integral_eq_integral_Ioc_meas_le {f : α → ℝ} {M : ℝ}
     ∫ ω, f ω ∂μ = ∫ t in Ioc 0 M, ENNReal.toReal (μ {a : α | t ≤ f a}) := by
   rw [f_intble.integral_eq_integral_meas_le f_nn]
   rw [setIntegral_eq_of_subset_of_ae_diff_eq_zero
-      measurableSet_Ioi.nullMeasurableSet Ioc_subset_Ioi_self ?_]
+      nullMeasurableSet_Ioi Ioc_subset_Ioi_self ?_]
   apply Eventually.of_forall (fun t ht ↦ ?_)
   have htM : M < t := by simp_all only [mem_diff, mem_Ioi, mem_Ioc, not_and, not_le]
   have obs : μ {a | M < f a} = 0 := by
diff --git a/Mathlib/MeasureTheory/Integral/Lebesgue.lean b/Mathlib/MeasureTheory/Integral/Lebesgue.lean
index 3cad10c95109e..675449008740c 100644
--- a/Mathlib/MeasureTheory/Integral/Lebesgue.lean
+++ b/Mathlib/MeasureTheory/Integral/Lebesgue.lean
@@ -358,7 +358,7 @@ theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (
   have eq : ∀ p, rs.map c ⁻¹' {p} = ⋃ n, rs.map c ⁻¹' {p} ∩ { a | p ≤ f n a } := by
     intro p
     rw [← inter_iUnion]; nth_rw 1 [← inter_univ (map c rs ⁻¹' {p})]
-    refine Set.ext fun x => and_congr_right fun hx => true_iff_iff.2 ?_
+    refine Set.ext fun x => and_congr_right fun hx => (iff_of_eq (true_iff _)).2 ?_
     by_cases p_eq : p = 0
     · simp [p_eq]
     simp only [coe_map, mem_preimage, Function.comp_apply, mem_singleton_iff] at hx
@@ -385,8 +385,7 @@ theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (
     _ = ∑ r ∈ (rs.map c).range, r * μ (⋃ n, rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
       simp only [(eq _).symm]
     _ = ∑ r ∈ (rs.map c).range, ⨆ n, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) :=
-      (Finset.sum_congr rfl fun x _ => by
-        rw [measure_iUnion_eq_iSup (mono x).directed_le, ENNReal.mul_iSup])
+      Finset.sum_congr rfl fun x _ => by rw [(mono x).measure_iUnion, ENNReal.mul_iSup]
     _ = ⨆ n, ∑ r ∈ (rs.map c).range, r * μ (rs.map c ⁻¹' {r} ∩ { a | r ≤ f n a }) := by
       refine ENNReal.finsetSum_iSup_of_monotone fun p i j h ↦ ?_
       gcongr _ * μ ?_
@@ -476,7 +475,7 @@ theorem exists_pos_setLIntegral_lt_of_measure_lt {f : α → ℝ≥0∞} (h : 
       exact SimpleFunc.lintegral_mono le_rfl Measure.restrict_le_self
     _ ≤ (SimpleFunc.const α (C : ℝ≥0∞)).lintegral (μ.restrict s) + ε₁ := by
       gcongr
-      exact SimpleFunc.lintegral_mono (fun x ↦ ENNReal.coe_le_coe.2 (hC x)) le_rfl
+      exact fun x ↦ ENNReal.coe_le_coe.2 (hC x)
     _ = C * μ s + ε₁ := by
       simp only [← SimpleFunc.lintegral_eq_lintegral, coe_const, lintegral_const,
         Measure.restrict_apply, MeasurableSet.univ, univ_inter, Function.const]
@@ -531,7 +530,7 @@ theorem lintegral_add_aux {f g : α → ℝ≥0∞} (hf : Measurable f) (hg : Me
         funext n
         rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral]
         simp only [Pi.add_apply, SimpleFunc.coe_add]
-      · measurability
+      · fun_prop
       · intro i j h a
         dsimp
         gcongr <;> exact monotone_eapprox _ h _
@@ -918,6 +917,15 @@ theorem lintegral_eq_zero_iff' {f : α → ℝ≥0∞} (hf : AEMeasurable f μ)
 theorem lintegral_eq_zero_iff {f : α → ℝ≥0∞} (hf : Measurable f) : ∫⁻ a, f a ∂μ = 0 ↔ f =ᵐ[μ] 0 :=
   lintegral_eq_zero_iff' hf.aemeasurable
 
+theorem setLIntegral_eq_zero_iff' {s : Set α} (hs : MeasurableSet s)
+    {f : α → ℝ≥0∞} (hf : AEMeasurable f (μ.restrict s)) :
+    ∫⁻ a in s, f a ∂μ = 0 ↔ ∀ᵐ x ∂μ, x ∈ s → f x = 0 :=
+  (lintegral_eq_zero_iff' hf).trans (ae_restrict_iff' hs)
+
+theorem setLIntegral_eq_zero_iff {s : Set α} (hs : MeasurableSet s) {f : α → ℝ≥0∞}
+    (hf : Measurable f) : ∫⁻ a in s, f a ∂μ = 0 ↔ ∀ᵐ x ∂μ, x ∈ s → f x = 0 :=
+  setLIntegral_eq_zero_iff' hs hf.aemeasurable
+
 theorem lintegral_pos_iff_support {f : α → ℝ≥0∞} (hf : Measurable f) :
     (0 < ∫⁻ a, f a ∂μ) ↔ 0 < μ (Function.support f) := by
   simp [pos_iff_ne_zero, hf, Filter.EventuallyEq, ae_iff, Function.support]
@@ -926,7 +934,7 @@ theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Se
     0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) := by
   rw [lintegral_pos_iff_support hf, Measure.restrict_apply (measurableSet_support hf)]
 
-/-- Weaker version of the monotone convergence theorem-/
+/-- Weaker version of the monotone convergence theorem -/
 theorem lintegral_iSup_ae {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n))
     (h_mono : ∀ n, ∀ᵐ a ∂μ, f n a ≤ f n.succ a) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by
   classical
@@ -1958,8 +1966,8 @@ theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : Measurabl
       simp only [hc, ENNReal.coe_zero, zero_mul, not_lt_zero] at hL
     have : L / c < μ s := by
       rwa [ENNReal.div_lt_iff, mul_comm]
-      · simp only [c_ne_zero, Ne, ENNReal.coe_eq_zero, not_false_iff, true_or_iff]
-      · simp only [Ne, coe_ne_top, not_false_iff, true_or_iff]
+      · simp only [c_ne_zero, Ne, ENNReal.coe_eq_zero, not_false_iff, true_or]
+      · simp only [Ne, coe_ne_top, not_false_iff, true_or]
     obtain ⟨t, ht, ts, mlt, t_top⟩ :
       ∃ t : Set α, MeasurableSet t ∧ t ⊆ s ∧ L / ↑c < μ t ∧ μ t < ∞ :=
       Measure.exists_subset_measure_lt_top hs this
@@ -1973,8 +1981,8 @@ theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : Measurabl
         piecewise_eq_indicator, ENNReal.coe_indicator, Function.const_apply, lintegral_indicator,
         lintegral_const, Measure.restrict_apply', univ_inter]
       rwa [mul_comm, ← ENNReal.div_lt_iff]
-      · simp only [c_ne_zero, Ne, ENNReal.coe_eq_zero, not_false_iff, true_or_iff]
-      · simp only [Ne, coe_ne_top, not_false_iff, true_or_iff]
+      · simp only [c_ne_zero, Ne, ENNReal.coe_eq_zero, not_false_iff, true_or]
+      · simp only [Ne, coe_ne_top, not_false_iff, true_or]
   · replace hL : L < ∫⁻ x, f₁ x ∂μ + ∫⁻ x, f₂ x ∂μ := by
       rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]
     by_cases hf₁ : ∫⁻ x, f₁ x ∂μ = 0
@@ -1987,8 +1995,7 @@ theorem SimpleFunc.exists_lt_lintegral_simpleFunc_of_lt_lintegral {m : Measurabl
       rcases h₁ hL with ⟨g, g_le, g_top, gL⟩
       refine ⟨g, fun x => (g_le x).trans ?_, g_top, gL⟩
       simp only [SimpleFunc.coe_add, Pi.add_apply, le_add_iff_nonneg_right, zero_le']
-    obtain ⟨L₁, L₂, hL₁, hL₂, hL⟩ :
-      ∃ L₁ L₂ : ℝ≥0∞, (L₁ < ∫⁻ x, f₁ x ∂μ) ∧ (L₂ < ∫⁻ x, f₂ x ∂μ) ∧ L < L₁ + L₂ :=
+    obtain ⟨L₁, hL₁, L₂, hL₂, hL⟩ : ∃ L₁ < ∫⁻ x, f₁ x ∂μ, ∃ L₂ < ∫⁻ x, f₂ x ∂μ, L < L₁ + L₂ :=
       ENNReal.exists_lt_add_of_lt_add hL hf₁ hf₂
     rcases h₁ hL₁ with ⟨g₁, g₁_le, g₁_top, hg₁⟩
     rcases h₂ hL₂ with ⟨g₂, g₂_le, g₂_top, hg₂⟩
@@ -2050,3 +2057,5 @@ lemma tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure
 end TendstoIndicator -- section
 
 end MeasureTheory
+
+set_option linter.style.longFile 2200
diff --git a/Mathlib/MeasureTheory/Integral/MeanInequalities.lean b/Mathlib/MeasureTheory/Integral/MeanInequalities.lean
index 39a15c80e968f..424952fb5cb92 100644
--- a/Mathlib/MeasureTheory/Integral/MeanInequalities.lean
+++ b/Mathlib/MeasureTheory/Integral/MeanInequalities.lean
@@ -180,8 +180,7 @@ theorem lintegral_mul_norm_pow_le {α} [MeasurableSpace α] {μ : Measure α}
   · rw [add_zero] at hpq
     simp [hpq]
   have h2p : 1 < 1 / p := by
-    rw [one_div]
-    apply one_lt_inv hp
+    rw [one_div, one_lt_inv₀ hp]
     linarith
   have h2pq : (1 / p)⁻¹ + (1 / q)⁻¹ = 1 := by simp [hp.ne', hq.ne', hpq]
   have := ENNReal.lintegral_mul_le_Lp_mul_Lq μ ⟨h2p, h2pq⟩ (hf.pow_const p) (hg.pow_const q)
@@ -312,7 +311,7 @@ theorem lintegral_Lp_mul_le_Lq_mul_Lr {α} [MeasurableSpace α] {p q r : ℝ} (h
   let p2 := q / p
   let q2 := p2.conjExponent
   have hp2q2 : p2.IsConjExponent q2 :=
-    .conjExponent (by simp [p2, q2, _root_.lt_div_iff, hpq, hp0_lt])
+    .conjExponent (by simp [p2, q2, _root_.lt_div_iff₀, hpq, hp0_lt])
   calc
     (∫⁻ a : α, (f * g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ a : α, f a ^ p * g a ^ p ∂μ) ^ (1 / p) := by
       simp_rw [Pi.mul_apply, ENNReal.mul_rpow_of_nonneg _ _ hp0]
diff --git a/Mathlib/MeasureTheory/Integral/PeakFunction.lean b/Mathlib/MeasureTheory/Integral/PeakFunction.lean
index 8db2aa64138ec..845644d23d8fb 100644
--- a/Mathlib/MeasureTheory/Integral/PeakFunction.lean
+++ b/Mathlib/MeasureTheory/Integral/PeakFunction.lean
@@ -392,7 +392,7 @@ theorem tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuou
 ### Peak functions of the form `x ↦ c ^ dim * φ (c x)`
 -/
 
-open FiniteDimensional Bornology
+open Module Bornology
 
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F]
   [MeasurableSpace F] [BorelSpace F] {μ : Measure F} [IsAddHaarMeasure μ]
@@ -438,7 +438,7 @@ theorem tendsto_integral_comp_smul_smul_of_integrable
         simp [norm_smul, abs_of_pos cpos, mul_pow]; ring
       _ < δ ^ finrank ℝ F * ε := by
         apply hM
-        rw [div_lt_iff δpos] at hc
+        rw [div_lt_iff₀ δpos] at hc
         simp only [mem_compl_iff, mem_closedBall, dist_zero_right, norm_smul, Real.norm_eq_abs,
           abs_of_nonneg cpos.le, not_le, gt_iff_lt]
         exact hc.trans_le (by gcongr)
diff --git a/Mathlib/MeasureTheory/Integral/Periodic.lean b/Mathlib/MeasureTheory/Integral/Periodic.lean
index 892fa83114568..c86458b3ee841 100644
--- a/Mathlib/MeasureTheory/Integral/Periodic.lean
+++ b/Mathlib/MeasureTheory/Integral/Periodic.lean
@@ -35,17 +35,17 @@ protected theorem AddCircle.measurable_mk' {a : ℝ} :
 theorem isAddFundamentalDomain_Ioc {T : ℝ} (hT : 0 < T) (t : ℝ)
     (μ : Measure ℝ := by volume_tac) :
     IsAddFundamentalDomain (AddSubgroup.zmultiples T) (Ioc t (t + T)) μ := by
-  refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
+  refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_
   have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
-    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strictMono_left hT).injective).bijective
+    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_left_strictMono hT).injective).bijective
   refine this.existsUnique_iff.2 ?_
   simpa only [add_comm x] using existsUnique_add_zsmul_mem_Ioc hT x t
 
 theorem isAddFundamentalDomain_Ioc' {T : ℝ} (hT : 0 < T) (t : ℝ) (μ : Measure ℝ := by volume_tac) :
     IsAddFundamentalDomain (AddSubgroup.op <| .zmultiples T) (Ioc t (t + T)) μ := by
-  refine IsAddFundamentalDomain.mk' measurableSet_Ioc.nullMeasurableSet fun x => ?_
+  refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_
   have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
-    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_strictMono_left hT).injective).bijective
+    (Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_left_strictMono hT).injective).bijective
   refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2 ?_
   simpa using existsUnique_add_zsmul_mem_Ioc hT x t
 
diff --git a/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean b/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean
index 20a484f42774c..b6a9c18db7857 100644
--- a/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean
+++ b/Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Jesse Reimann. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jesse Reimann, Kalle Kytölä
 -/
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 import Mathlib.Topology.Sets.Compacts
 
 /-!
diff --git a/Mathlib/MeasureTheory/Integral/SetIntegral.lean b/Mathlib/MeasureTheory/Integral/SetIntegral.lean
index 9982a8ab04974..be099131dfc9b 100644
--- a/Mathlib/MeasureTheory/Integral/SetIntegral.lean
+++ b/Mathlib/MeasureTheory/Integral/SetIntegral.lean
@@ -7,7 +7,7 @@ import Mathlib.MeasureTheory.Integral.IntegrableOn
 import Mathlib.MeasureTheory.Integral.Bochner
 import Mathlib.MeasureTheory.Function.LocallyIntegrable
 import Mathlib.Topology.MetricSpace.ThickenedIndicator
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.ContinuousMapZero
 import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDual
 
 /-!
@@ -228,12 +228,12 @@ theorem tendsto_setIntegral_of_monotone {ι : Type*} [Countable ι] [Semilattice
   refine Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun ε ε0 => ?_
   lift ε to ℝ≥0 using ε0.le
   have : ∀ᶠ i in atTop, ν (s i) ∈ Icc (ν S - ε) (ν S + ε) :=
-    tendsto_measure_iUnion h_mono (ENNReal.Icc_mem_nhds hfi'.ne (ENNReal.coe_pos.2 ε0).ne')
+    tendsto_measure_iUnion_atTop h_mono (ENNReal.Icc_mem_nhds hfi'.ne (ENNReal.coe_pos.2 ε0).ne')
   filter_upwards [this] with i hi
   rw [mem_closedBall_iff_norm', ← integral_diff (hsm i) hfi hsub, ← coe_nnnorm, NNReal.coe_le_coe, ←
     ENNReal.coe_le_coe]
   refine (ennnorm_integral_le_lintegral_ennnorm _).trans ?_
-  rw [← withDensity_apply _ (hSm.diff (hsm _)), ← hν, measure_diff hsub (hsm _)]
+  rw [← withDensity_apply _ (hSm.diff (hsm _)), ← hν, measure_diff hsub (hsm _).nullMeasurableSet]
   exacts [tsub_le_iff_tsub_le.mp hi.1,
     (hi.2.trans_lt <| ENNReal.add_lt_top.2 ⟨hfi', ENNReal.coe_lt_top⟩).ne]
 
@@ -255,14 +255,15 @@ theorem tendsto_setIntegral_of_antitone {ι : Type*} [Countable ι] [Semilattice
     simpa [hsm i₀, ν, ENNReal.ofReal, norm_toNNReal] using hi₀.norm.lintegral_lt_top.ne
   have νS : ν S ≠ ∞ := ((measure_mono (hsub i₀)).trans_lt νi₀.lt_top).ne
   have : ∀ᶠ i in atTop, ν (s i) ∈ Icc (ν S - ε) (ν S + ε) := by
-    apply tendsto_measure_iInter hsm h_anti ⟨i₀, νi₀⟩
+    apply tendsto_measure_iInter (fun i ↦ (hsm i).nullMeasurableSet) h_anti ⟨i₀, νi₀⟩
     apply ENNReal.Icc_mem_nhds νS (ENNReal.coe_pos.2 ε0).ne'
   filter_upwards [this, Ici_mem_atTop i₀] with i hi h'i
   rw [mem_closedBall_iff_norm, ← integral_diff hSm (hi₀.mono_set (h_anti h'i)) (hsub i),
     ← coe_nnnorm, NNReal.coe_le_coe, ← ENNReal.coe_le_coe]
   refine (ennnorm_integral_le_lintegral_ennnorm _).trans ?_
-  rw [← withDensity_apply _ ((hsm _).diff hSm), ← hν, measure_diff (hsub i) hSm νS]
-  exact tsub_le_iff_left.2 hi.2
+  rw [← withDensity_apply _ ((hsm _).diff hSm), ← hν,
+    measure_diff_le_iff_le_add hSm.nullMeasurableSet (hsub i) νS]
+  exact hi.2
 
 @[deprecated (since := "2024-04-17")]
 alias tendsto_set_integral_of_antitone := tendsto_setIntegral_of_antitone
@@ -377,7 +378,7 @@ theorem setIntegral_eq_of_subset_of_ae_diff_eq_zero_aux (hts : s ⊆ t)
       intro h'x
       by_cases xs : x ∈ s
       · simp only [xs, hts xs]
-      · simp only [xs, iff_false_iff]
+      · simp only [xs, iff_false]
         intro xt
         exact h'x (hx ⟨xt, xs⟩)
     _ = ∫ x in s ∩ k, f x ∂μ + ∫ x in s \ k, f x ∂μ := by
@@ -1259,23 +1260,34 @@ variable [NormedSpace ℝ F] [NormedSpace ℝ E]
 
 theorem integral_comp_comm (L : E ≃L[𝕜] F) (φ : X → E) : ∫ x, L (φ x) ∂μ = L (∫ x, φ x ∂μ) := by
   have : CompleteSpace E ↔ CompleteSpace F :=
-    completeSpace_congr (e := L.toEquiv) L.uniformEmbedding
+    completeSpace_congr (e := L.toEquiv) L.isUniformEmbedding
   obtain ⟨_, _⟩|⟨_, _⟩ := iff_iff_and_or_not_and_not.mp this
   · exact L.toContinuousLinearMap.integral_comp_comm' L.antilipschitz _
   · simp [integral, *]
 
 end ContinuousLinearEquiv
 
-namespace ContinuousMap
+section ContinuousMap
 
-lemma integral_apply [TopologicalSpace Y] [CompactSpace Y] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : X → C(Y, E)} (hf : Integrable f μ) (y : Y) :
-    (∫ x, f x ∂μ) y = ∫ x, f x y ∂μ := by
+variable [TopologicalSpace Y] [CompactSpace Y]
+
+lemma ContinuousMap.integral_apply [NormedSpace ℝ E] [CompleteSpace E] {f : X → C(Y, E)}
+    (hf : Integrable f μ) (y : Y) : (∫ x, f x ∂μ) y = ∫ x, f x y ∂μ := by
   calc (∫ x, f x ∂μ) y = ContinuousMap.evalCLM ℝ y (∫ x, f x ∂μ) := rfl
     _ = ∫ x, ContinuousMap.evalCLM ℝ y (f x) ∂μ :=
           (ContinuousLinearMap.integral_comp_comm _ hf).symm
     _ = _ := rfl
 
+open scoped ContinuousMapZero in
+theorem ContinuousMapZero.integral_apply {R : Type*} [NormedCommRing R] [Zero Y]
+    [NormedAlgebra ℝ R] [CompleteSpace R] {f : X → C(Y, R)₀}
+    (hf : MeasureTheory.Integrable f μ) (y : Y) :
+    (∫ (x : X), f x ∂μ) y = ∫ (x : X), (f x) y ∂μ := by
+  calc (∫ x, f x ∂μ) y = ContinuousMapZero.evalCLM ℝ y (∫ x, f x ∂μ) := rfl
+    _ = ∫ x, ContinuousMapZero.evalCLM ℝ y (f x) ∂μ :=
+          (ContinuousLinearMap.integral_comp_comm _ hf).symm
+    _ = _ := rfl
+
 end ContinuousMap
 
 @[norm_cast]
@@ -1604,3 +1616,5 @@ lemma continuousOn_integral_of_compact_support
     hk hf hfs (integrableOn_const.2 (Or.inr hk.measure_lt_top)) (μ := μ) (g := fun _ ↦ 1)
 
 end ParametricIntegral
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/MeasureTheory/Integral/SetToL1.lean b/Mathlib/MeasureTheory/Integral/SetToL1.lean
index 218126470c835..b1c2e325add44 100644
--- a/Mathlib/MeasureTheory/Integral/SetToL1.lean
+++ b/Mathlib/MeasureTheory/Integral/SetToL1.lean
@@ -116,14 +116,14 @@ theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAd
     FinMeasAdditive μ T := by
   refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
   rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
-  simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
+  simp only [hc_ne_top, or_false, Ne, false_and] at hμs
   exact hμs.2
 
 theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
     FinMeasAdditive (c • μ) T := by
   refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
   rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top]
-  simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff]
+  simp only [hc_ne_zero, true_and, Ne, not_false_iff]
   exact Or.inl hμs
 
 theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) :
@@ -231,8 +231,8 @@ theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : Dominated
     DominatedFinMeasAdditive μ T (c.toReal * C) := by
   have h : ∀ s, MeasurableSet s → c • μ s = ∞ → μ s = ∞ := by
     intro s _ hcμs
-    simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false_iff, Ne,
-      false_and_iff] at hcμs
+    simp only [hc_ne_top, Algebra.id.smul_eq_mul, ENNReal.mul_eq_top, or_false, Ne,
+      false_and] at hcμs
     exact hcμs.2
   refine ⟨hT.1.of_eq_top_imp_eq_top (μ := c • μ) h, fun s hs hμs => ?_⟩
   have hcμs : c • μ s ≠ ∞ := mt (h s hs) hμs.ne
@@ -920,18 +920,18 @@ variable (𝕜) [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace
 def setToL1' (hT : DominatedFinMeasAdditive μ T C)
     (h_smul : ∀ c : 𝕜, ∀ s x, T s (c • x) = c • T s x) : (α →₁[μ] E) →L[𝕜] F :=
   (setToL1SCLM' α E 𝕜 μ hT h_smul).extend (coeToLp α E 𝕜) (simpleFunc.denseRange one_ne_top)
-    simpleFunc.uniformInducing
+    simpleFunc.isUniformInducing
 
 variable {𝕜}
 
 /-- Extend `Set α → E →L[ℝ] F` to `(α →₁[μ] E) →L[ℝ] F`. -/
 def setToL1 (hT : DominatedFinMeasAdditive μ T C) : (α →₁[μ] E) →L[ℝ] F :=
   (setToL1SCLM α E μ hT).extend (coeToLp α E ℝ) (simpleFunc.denseRange one_ne_top)
-    simpleFunc.uniformInducing
+    simpleFunc.isUniformInducing
 
 theorem setToL1_eq_setToL1SCLM (hT : DominatedFinMeasAdditive μ T C) (f : α →₁ₛ[μ] E) :
     setToL1 hT f = setToL1SCLM α E μ hT f :=
-  uniformly_extend_of_ind simpleFunc.uniformInducing (simpleFunc.denseRange one_ne_top)
+  uniformly_extend_of_ind simpleFunc.isUniformInducing (simpleFunc.denseRange one_ne_top)
     (setToL1SCLM α E μ hT).uniformContinuous _
 
 theorem setToL1_eq_setToL1' (hT : DominatedFinMeasAdditive μ T C)
@@ -1502,7 +1502,7 @@ theorem setToFun_top_smul_measure (hT : DominatedFinMeasAdditive (∞ • μ) T
     setToFun (∞ • μ) T hT f = 0 := by
   refine setToFun_measure_zero' hT fun s _ hμs => ?_
   rw [lt_top_iff_ne_top] at hμs
-  simp only [true_and_iff, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true,
+  simp only [true_and, Measure.smul_apply, ENNReal.mul_eq_top, eq_self_iff_true,
     top_ne_zero, Ne, not_false_iff, not_or, Classical.not_not, smul_eq_mul] at hμs
   simp only [hμs.right, Measure.smul_apply, mul_zero, smul_eq_mul]
 
@@ -1645,3 +1645,5 @@ theorem continuous_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C)
 end Function
 
 end MeasureTheory
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean b/Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
index 15a4b3437ddd3..2aa0a2e287af9 100644
--- a/Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
+++ b/Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
@@ -110,11 +110,11 @@ theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ
       · simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
     have ne_top : μ s ≠ ⊤ := by
       classical
-      simpa [f, hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
+      simpa [f, hs, hc, lt_top_iff_ne_top, SimpleFunc.coe_const,
         Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
         ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
-        or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
-        SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,
+        lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
+        SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,
         restrict_apply] using h
     have : μ s < μ s + ε / c := by
       have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
@@ -326,12 +326,12 @@ theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ
           Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]
     have μs_lt_top : μ s < ∞ := by
       classical
-      simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,
+      simpa only [hs, hc, lt_top_iff_ne_top, true_and, SimpleFunc.coe_const, or_false,
         lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,
         Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
         Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
         SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,
-        false_and_iff] using int_f
+        false_and] using int_f
     have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
     obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧ IsClosed F ∧ μ s < μ F + ε / c :=
       hs.exists_isClosed_lt_add μs_lt_top.ne this.ne'
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Basic.lean b/Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
index 72f27d9b68033..61c55dee0d648 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
@@ -231,7 +231,7 @@ theorem Subsingleton.measurable [Subsingleton α] : Measurable f := fun _ _ =>
 theorem measurable_of_subsingleton_codomain [Subsingleton β] (f : α → β) : Measurable f :=
   fun s _ => Subsingleton.set_cases MeasurableSet.empty MeasurableSet.univ s
 
-@[to_additive (attr := measurability)]
+@[to_additive (attr := measurability, fun_prop)]
 theorem measurable_one [One α] : Measurable (1 : β → α) :=
   @measurable_const _ _ _ _ 1
 
@@ -351,6 +351,16 @@ theorem measurable_to_countable' [MeasurableSpace α] [Countable α] [Measurable
     (h : ∀ x, MeasurableSet (f ⁻¹' {x})) : Measurable f :=
   measurable_to_countable fun y => h (f y)
 
+theorem ENat.measurable_iff {α : Type*} [MeasurableSpace α] {f : α → ℕ∞} :
+    Measurable f ↔ ∀ n : ℕ, MeasurableSet (f ⁻¹' {↑n}) := by
+  refine ⟨fun hf n ↦ hf <| measurableSet_singleton _, fun h ↦ measurable_to_countable' fun n ↦ ?_⟩
+  cases n with
+  | top =>
+    rw [← WithTop.none_eq_top, ← compl_range_some, preimage_compl, ← iUnion_singleton_eq_range,
+      preimage_iUnion]
+    exact .compl <| .iUnion h
+  | coe n => exact h n
+
 @[measurability]
 theorem measurable_unit [MeasurableSpace α] (f : Unit → α) : Measurable f :=
   measurable_from_top
@@ -508,7 +518,7 @@ alias Measurable.subtype_val := Measurable.subtype_coe
 @[measurability]
 theorem Measurable.subtype_mk {p : β → Prop} {f : α → β} (hf : Measurable f) {h : ∀ x, p (f x)} :
     Measurable fun x => (⟨f x, h x⟩ : Subtype p) := fun t ⟨s, hs⟩ =>
-  hs.2 ▸ by simp only [← preimage_comp, (· ∘ ·), Subtype.coe_mk, hf hs.1]
+  hs.2 ▸ by simp only [← preimage_comp, Function.comp_def, Subtype.coe_mk, hf hs.1]
 
 @[measurability]
 protected theorem Measurable.rangeFactorization {f : α → β} (hf : Measurable f) :
@@ -661,6 +671,7 @@ theorem Measurable.prod_mk {β γ} {_ : MeasurableSpace β} {_ : MeasurableSpace
     {g : α → γ} (hf : Measurable f) (hg : Measurable g) : Measurable fun a : α => (f a, g a) :=
   Measurable.prod hf hg
 
+@[fun_prop]
 theorem Measurable.prod_map [MeasurableSpace δ] {f : α → β} {g : γ → δ} (hf : Measurable f)
     (hg : Measurable g) : Measurable (Prod.map f g) :=
   (hf.comp measurable_fst).prod_mk (hg.comp measurable_snd)
@@ -805,7 +816,7 @@ variable [∀ a, MeasurableSpace (π a)] [MeasurableSpace γ]
 
 theorem measurable_pi_iff {g : α → ∀ a, π a} : Measurable g ↔ ∀ a, Measurable fun x => g x a := by
   simp_rw [measurable_iff_comap_le, MeasurableSpace.pi, MeasurableSpace.comap_iSup,
-    MeasurableSpace.comap_comp, Function.comp, iSup_le_iff]
+    MeasurableSpace.comap_comp, Function.comp_def, iSup_le_iff]
 
 @[fun_prop, aesop safe 100 apply (rule_sets := [Measurable])]
 theorem measurable_pi_apply (a : δ) : Measurable fun f : ∀ a, π a => f a :=
@@ -854,6 +865,42 @@ theorem measurable_update_left {a : δ} [DecidableEq δ] {x : π a} :
     Measurable (update · a x) :=
   measurable_update'.comp measurable_prod_mk_right
 
+@[measurability, fun_prop]
+theorem Set.measurable_restrict (s : Set δ) : Measurable (s.restrict (π := π)) :=
+  measurable_pi_lambda _ fun _ ↦ measurable_pi_apply _
+
+@[measurability, fun_prop]
+theorem Set.measurable_restrict₂ {s t : Set δ} (hst : s ⊆ t) :
+    Measurable (restrict₂ (π := π) hst) :=
+  measurable_pi_lambda _ fun _ ↦ measurable_pi_apply _
+
+@[measurability, fun_prop]
+theorem Finset.measurable_restrict (s : Finset δ) : Measurable (s.restrict (π := π)) :=
+  measurable_pi_lambda _ fun _ ↦ measurable_pi_apply _
+
+@[measurability, fun_prop]
+theorem Finset.measurable_restrict₂ {s t : Finset δ} (hst : s ⊆ t) :
+    Measurable (Finset.restrict₂ (π := π) hst) :=
+  measurable_pi_lambda _ fun _ ↦ measurable_pi_apply _
+
+@[measurability, fun_prop]
+theorem Set.measurable_restrict_apply (s : Set α) {f : α → γ} (hf : Measurable f) :
+    Measurable (s.restrict f) := hf.comp measurable_subtype_coe
+
+@[measurability, fun_prop]
+theorem Set.measurable_restrict₂_apply {s t : Set α} (hst : s ⊆ t)
+    {f : t → γ} (hf : Measurable f) :
+    Measurable (restrict₂ (π := fun _ ↦ γ) hst f) := hf.comp (measurable_inclusion hst)
+
+@[measurability, fun_prop]
+theorem Finset.measurable_restrict_apply (s : Finset α) {f : α → γ} (hf : Measurable f) :
+    Measurable (s.restrict f) := hf.comp measurable_subtype_coe
+
+@[measurability, fun_prop]
+theorem Finset.measurable_restrict₂_apply {s t : Finset α} (hst : s ⊆ t)
+    {f : t → γ} (hf : Measurable f) :
+    Measurable (restrict₂ (π := fun _ ↦ γ) hst f) := hf.comp (measurable_inclusion hst)
+
 variable (π) in
 theorem measurable_eq_mp {i i' : δ} (h : i = i') : Measurable (congr_arg π h).mp := by
   cases h
@@ -1081,7 +1128,7 @@ lemma measurable_set_mem (a : α) : Measurable fun s : Set α ↦ a ∈ s := mea
 
 @[aesop safe 100 apply (rule_sets := [Measurable])]
 lemma measurable_set_not_mem (a : α) : Measurable fun s : Set α ↦ a ∉ s :=
-  (measurable_discrete Not).comp <| measurable_set_mem a
+  (Measurable.of_discrete (f := Not)).comp <| measurable_set_mem a
 
 @[aesop safe 100 apply (rule_sets := [Measurable])]
 lemma measurableSet_mem (a : α) : MeasurableSet {s : Set α | a ∈ s} :=
@@ -1094,6 +1141,20 @@ lemma measurableSet_not_mem (a : α) : MeasurableSet {s : Set α | a ∉ s} :=
 lemma measurable_compl : Measurable ((·ᶜ) : Set α → Set α) :=
   measurable_set_iff.2 fun _ ↦ measurable_set_not_mem _
 
+lemma MeasurableSet.setOf_finite [Countable α] : MeasurableSet {s : Set α | s.Finite} :=
+  Countable.setOf_finite.measurableSet
+
+lemma MeasurableSet.setOf_infinite [Countable α] : MeasurableSet {s : Set α | s.Infinite} :=
+  .setOf_finite |> .compl
+
+lemma MeasurableSet.sep_finite [Countable α] {S : Set (Set α)} (hS : MeasurableSet S) :
+    MeasurableSet {s ∈ S | s.Finite} :=
+  hS.inter .setOf_finite
+
+lemma MeasurableSet.sep_infinite [Countable α] {S : Set (Set α)} (hS : MeasurableSet S) :
+    MeasurableSet {s ∈ S | s.Infinite} :=
+  hS.inter .setOf_infinite
+
 end Set
 end Constructions
 
@@ -1204,7 +1265,7 @@ namespace MeasurableSet
 variable [MeasurableSpace α]
 
 instance Subtype.instMembership : Membership α (Subtype (MeasurableSet : Set α → Prop)) :=
-  ⟨fun a s => a ∈ (s : Set α)⟩
+  ⟨fun s a => a ∈ (s : Set α)⟩
 
 @[simp]
 theorem mem_coe (a : α) (s : Subtype (MeasurableSet : Set α → Prop)) : a ∈ (s : Set α) ↔ a ∈ s :=
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Card.lean b/Mathlib/MeasureTheory/MeasurableSpace/Card.lean
index 1416dc2d3f354..9da03d77e896e 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Card.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Card.lean
@@ -108,13 +108,15 @@ theorem generateMeasurable_eq_rec (s : Set (Set α)) :
         ⋃ (i : (aleph 1).ord.toType), generateMeasurableRec s i := by
   ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
   · inhabit ω₁
-    induction' ht with u hu u _ IH f _ IH
-    · exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
-    · exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
-    · rcases mem_iUnion.1 IH with ⟨i, hi⟩
+    induction ht with
+    | basic u hu => exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
+    | empty => exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
+    | compl _ _ IH =>
+      rcases mem_iUnion.1 IH with ⟨i, hi⟩
       obtain ⟨j, hj⟩ := exists_gt i
       exact mem_iUnion.2 ⟨j, compl_mem_generateMeasurableRec hj hi⟩
-    · have : ∀ n, ∃ i, f n ∈ generateMeasurableRec s i := fun n => by simpa using IH n
+    | iUnion f _ IH =>
+      have : ∀ n, ∃ i, f n ∈ generateMeasurableRec s i := fun n => by simpa using IH n
       choose I hI using this
       refine mem_iUnion.2 ⟨Ordinal.enum (α := ω₁) (· < ·)
         ⟨Ordinal.lsub fun n => Ordinal.typein.{u} (α := ω₁) (· < ·) (I n), ?_⟩,
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean b/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
index 00c2d47af5032..9ff18dc3605be 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean
@@ -208,6 +208,17 @@ instance (priority := 100) separatesPoints_of_measurableSingletonClass [Measurab
   simp_rw [mem_singleton_iff, forall_true_left] at h
   exact h.symm
 
+instance (priority := 50) MeasurableSingletonClass.of_separatesPoints [MeasurableSpace α]
+    [Countable α] [SeparatesPoints α] : MeasurableSingletonClass α where
+  measurableSet_singleton x := by
+    choose s hsm hxs hys using fun y (h : x ≠ y) ↦ exists_measurableSet_of_ne h
+    convert MeasurableSet.iInter fun y ↦ .iInter fun h ↦ hsm y h
+    ext y
+    rcases eq_or_ne x y with rfl | h
+    · simpa
+    · simp only [mem_singleton_iff, h.symm, false_iff, mem_iInter, not_forall]
+      exact ⟨y, h, hys y h⟩
+
 instance hasCountableSeparatingOn_of_countablySeparated_subtype
     [MeasurableSpace α] {s : Set α} [h : CountablySeparated s] :
     HasCountableSeparatingOn _ MeasurableSet s := CountablySeparated.subtype_iff.mp h
@@ -505,7 +516,8 @@ variable [MeasurableSpace β]
 
 /-- A class registering that either `α` is countable or `β` is a countably generated
 measurable space. -/
-class CountableOrCountablyGenerated (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] : Prop :=
+class CountableOrCountablyGenerated (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
+    Prop where
   countableOrCountablyGenerated : Countable α ∨ MeasurableSpace.CountablyGenerated β
 
 instance instCountableOrCountablyGeneratedOfCountable [h1 : Countable α] :
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean b/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean
index 24be3674a76da..5c9f7d9d195dc 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Defs.lean
@@ -517,7 +517,7 @@ end MeasurableFunctions
 
 /-- A typeclass mixin for `MeasurableSpace`s such that all sets are measurable. -/
 class DiscreteMeasurableSpace (α : Type*) [MeasurableSpace α] : Prop where
-  /-- Do not use this. Use `measurableSet_discrete` instead. -/
+  /-- Do not use this. Use `MeasurableSet.of_discrete` instead. -/
   forall_measurableSet : ∀ s : Set α, MeasurableSet s
 
 instance : @DiscreteMeasurableSpace α ⊤ :=
@@ -529,19 +529,24 @@ instance (priority := 100) MeasurableSingletonClass.toDiscreteMeasurableSpace [M
   forall_measurableSet _ := (Set.to_countable _).measurableSet
 
 section DiscreteMeasurableSpace
-variable [MeasurableSpace α] [MeasurableSpace β] [DiscreteMeasurableSpace α]
+variable [MeasurableSpace α] [MeasurableSpace β] [DiscreteMeasurableSpace α] {s : Set α} {f : α → β}
 
-@[measurability] lemma measurableSet_discrete (s : Set α) : MeasurableSet s :=
+@[measurability] lemma MeasurableSet.of_discrete : MeasurableSet s :=
   DiscreteMeasurableSpace.forall_measurableSet _
 
-@[measurability]
-lemma measurable_discrete (f : α → β) : Measurable f := fun _ _ ↦ measurableSet_discrete _
+@[measurability, fun_prop] lemma Measurable.of_discrete : Measurable f := fun _ _ ↦ .of_discrete
+
+@[deprecated MeasurableSet.of_discrete (since := "2024-08-25")]
+lemma measurableSet_discrete (s : Set α) : MeasurableSet s := .of_discrete
+
+@[deprecated Measurable.of_discrete (since := "2024-08-25")]
+lemma measurable_discrete (f : α → β) : Measurable f := .of_discrete
 
 /-- Warning: Creates a typeclass loop with `MeasurableSingletonClass.toDiscreteMeasurableSpace`.
 To be monitored. -/
 -- See note [lower instance priority]
 instance (priority := 100) DiscreteMeasurableSpace.toMeasurableSingletonClass :
     MeasurableSingletonClass α where
-  measurableSet_singleton _ := measurableSet_discrete _
+  measurableSet_singleton _ := .of_discrete
 
 end DiscreteMeasurableSpace
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean b/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
index 539df69ae463e..5cb861c7dd95a 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
@@ -60,6 +60,8 @@ structure MeasurableEmbedding [MeasurableSpace α] [MeasurableSpace β] (f : α
   /-- The image of a measurable set under a measurable embedding is a measurable set. -/
   protected measurableSet_image' : ∀ ⦃s⦄, MeasurableSet s → MeasurableSet (f '' s)
 
+attribute [fun_prop] MeasurableEmbedding.measurable
+
 namespace MeasurableEmbedding
 
 variable {mα : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ] {f : α → β} {g : β → γ}
@@ -155,7 +157,7 @@ instance instEquivLike : EquivLike (α ≃ᵐ β) α β where
 theorem coe_toEquiv (e : α ≃ᵐ β) : (e.toEquiv : α → β) = e :=
   rfl
 
-@[measurability]
+@[measurability, fun_prop]
 protected theorem measurable (e : α ≃ᵐ β) : Measurable (e : α → β) :=
   e.measurable_toFun
 
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Instances.lean b/Mathlib/MeasureTheory/MeasurableSpace/Instances.lean
index 2de7f4465974c..a760b97654d0d 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Instances.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Instances.lean
@@ -25,6 +25,8 @@ instance Prop.instMeasurableSpace : MeasurableSpace Prop := ⊤
 
 instance Nat.instMeasurableSpace : MeasurableSpace ℕ := ⊤
 
+instance ENat.instMeasurableSpace : MeasurableSpace ℕ∞ := ⊤
+
 instance Fin.instMeasurableSpace (n : ℕ) : MeasurableSpace (Fin n) := ⊤
 
 instance ZMod.instMeasurableSpace (n : ℕ) : MeasurableSpace (ZMod n) := ⊤
@@ -53,6 +55,10 @@ instance Prop.instMeasurableSingletonClass : MeasurableSingletonClass Prop := 
 
 instance Nat.instMeasurableSingletonClass : MeasurableSingletonClass ℕ := ⟨fun _ => trivial⟩
 
+instance ENat.instDiscreteMeasurableSpace : DiscreteMeasurableSpace ℕ∞ := ⟨fun _ ↦ trivial⟩
+
+instance ENat.instMeasurableSingletonClass : MeasurableSingletonClass ℕ∞ := inferInstance
+
 instance Fin.instMeasurableSingletonClass (n : ℕ) : MeasurableSingletonClass (Fin n) :=
   ⟨fun _ => trivial⟩
 
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/NCard.lean b/Mathlib/MeasureTheory/MeasurableSpace/NCard.lean
new file mode 100644
index 0000000000000..4385de7626560
--- /dev/null
+++ b/Mathlib/MeasureTheory/MeasurableSpace/NCard.lean
@@ -0,0 +1,27 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.MeasureTheory.MeasurableSpace.Basic
+import Mathlib.Data.Set.Card
+
+/-!
+# Measurability of `Set.encard` and `Set.ncard`
+
+In this file we prove that `Set.encard` and `Set.ncard` are measurable functions,
+provided that the ambient space is countable.
+-/
+
+open Set
+
+variable {α : Type*} [Countable α]
+
+@[measurability]
+theorem measurable_encard : Measurable (Set.encard : Set α → ℕ∞) :=
+  ENat.measurable_iff.2 fun _n ↦ Countable.measurableSet <| Countable.setOf_finite.mono fun _s hs ↦
+    finite_of_encard_eq_coe hs
+
+@[measurability]
+theorem measurable_ncard : Measurable (Set.ncard : Set α → ℕ) :=
+  Measurable.of_discrete.comp measurable_encard
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/PreorderRestrict.lean b/Mathlib/MeasureTheory/MeasurableSpace/PreorderRestrict.lean
new file mode 100644
index 0000000000000..2432d3f8d2ae9
--- /dev/null
+++ b/Mathlib/MeasureTheory/MeasurableSpace/PreorderRestrict.lean
@@ -0,0 +1,40 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.Order.Restriction
+import Mathlib.MeasureTheory.MeasurableSpace.Basic
+
+/-!
+# Measurability of the restriction function for functions indexed by a preorder
+
+We prove that the map which restricts a function `f : (i : α) → X i` to elements `≤ a` is
+measurable.
+-/
+
+open MeasureTheory
+
+namespace Preorder
+
+variable {α : Type*} [Preorder α] {X : α → Type*} [∀ a, MeasurableSpace (X a)]
+
+@[measurability, fun_prop]
+theorem measurable_restrictLe (a : α) : Measurable (restrictLe (π := X) a) :=
+    Set.measurable_restrict _
+
+@[measurability, fun_prop]
+theorem measurable_restrictLe₂ {a b : α} (hab : a ≤ b) : Measurable (restrictLe₂ (π := X) hab) :=
+  Set.measurable_restrict₂ _
+
+variable [LocallyFiniteOrderBot α]
+
+@[measurability, fun_prop]
+theorem measurable_frestrictLe (a : α) : Measurable (frestrictLe (π := X) a) :=
+  Finset.measurable_restrict _
+
+@[measurability, fun_prop]
+theorem measurable_frestrictLe₂ {a b : α} (hab : a ≤ b) : Measurable (frestrictLe₂ (π := X) hab) :=
+  Finset.measurable_restrict₂ _
+
+end Preorder
diff --git a/Mathlib/MeasureTheory/Measure/AEMeasurable.lean b/Mathlib/MeasureTheory/Measure/AEMeasurable.lean
index f9ad11c6fbc66..7ea4ac9be4b94 100644
--- a/Mathlib/MeasureTheory/Measure/AEMeasurable.lean
+++ b/Mathlib/MeasureTheory/Measure/AEMeasurable.lean
@@ -197,7 +197,7 @@ theorem exists_ae_eq_range_subset (H : AEMeasurable f μ) {t : Set β} (ht : ∀
     simp only [g, hx, piecewise_eq_of_not_mem, not_false_iff]
     contrapose! hx
     apply subset_toMeasurable
-    simp only [hx, mem_compl_iff, mem_setOf_eq, false_and_iff, not_false_iff]
+    simp only [hx, mem_compl_iff, mem_setOf_eq, false_and, not_false_iff]
 
 theorem exists_measurable_nonneg {β} [Preorder β] [Zero β] {mβ : MeasurableSpace β} {f : α → β}
     (hf : AEMeasurable f μ) (f_nn : ∀ᵐ t ∂μ, 0 ≤ f t) : ∃ g, Measurable g ∧ 0 ≤ g ∧ f =ᵐ[μ] g := by
diff --git a/Mathlib/MeasureTheory/Measure/AddContent.lean b/Mathlib/MeasureTheory/Measure/AddContent.lean
index 362cd5b634339..e07a40a78c24f 100644
--- a/Mathlib/MeasureTheory/Measure/AddContent.lean
+++ b/Mathlib/MeasureTheory/Measure/AddContent.lean
@@ -63,9 +63,8 @@ instance : Inhabited (AddContent C) :=
     sUnion' := by simp }⟩
 
 instance : DFunLike (AddContent C) (Set α) (fun _ ↦ ℝ≥0∞) where
-  coe := fun m s ↦ m.toFun s
-  coe_injective' := by
-    intro m m' h
+  coe m s := m.toFun s
+  coe_injective' m m' _ := by
     cases m
     cases m'
     congr
@@ -91,7 +90,7 @@ lemma addContent_union' (hs : s ∈ C) (ht : t ∈ C) (hst : s ∪ t ∈ C) (h_d
   rotate_left
   · simp only [coe_pair, Set.insert_subset_iff, hs, ht, Set.singleton_subset_iff, and_self_iff]
   · simp only [coe_pair, Set.pairwiseDisjoint_insert, pairwiseDisjoint_singleton,
-      mem_singleton_iff, Ne, id, forall_eq, true_and_iff]
+      mem_singleton_iff, Ne, id, forall_eq, true_and]
     exact fun _ => h_dis
   · simp only [coe_pair, sUnion_insert, sUnion_singleton]
     exact hst
diff --git a/Mathlib/MeasureTheory/Measure/Complex.lean b/Mathlib/MeasureTheory/Measure/Complex.lean
index ccb2008885c00..7ce4fcfe8f699 100644
--- a/Mathlib/MeasureTheory/Measure/Complex.lean
+++ b/Mathlib/MeasureTheory/Measure/Complex.lean
@@ -4,17 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 -/
 import Mathlib.MeasureTheory.Measure.VectorMeasure
+import Mathlib.Analysis.Complex.Basic
 
 /-!
 # Complex measure
 
-This file proves some elementary results about complex measures. In particular, we prove that
+This file defines a complex measure to be a vector measure with codomain `ℂ`.
+Then we prove some elementary results about complex measures. In particular, we prove that
 a complex measure is always in the form `s + it` where `s` and `t` are signed measures.
 
-The complex measure is defined to be vector measure over `ℂ`, this definition can be found
-in `Mathlib/MeasureTheory/Measure/VectorMeasure.lean` and is known as
-`MeasureTheory.ComplexMeasure`.
-
 ## Main definitions
 
 * `MeasureTheory.ComplexMeasure.re`: obtains a signed measure `s` from a complex measure `c`
@@ -42,6 +40,10 @@ namespace MeasureTheory
 
 open VectorMeasure
 
+/-- A `ComplexMeasure` is a `ℂ`-vector measure. -/
+abbrev ComplexMeasure (α : Type*) [MeasurableSpace α] :=
+  VectorMeasure α ℂ
+
 namespace ComplexMeasure
 
 /-- The real part of a complex measure is a signed measure. -/
@@ -54,7 +56,7 @@ def re : ComplexMeasure α →ₗ[ℝ] SignedMeasure α :=
 def im : ComplexMeasure α →ₗ[ℝ] SignedMeasure α :=
   mapRangeₗ Complex.imCLM Complex.continuous_im
 
-/-- Given `s` and `t` signed measures, `s + it` is a complex measure-/
+/-- Given `s` and `t` signed measures, `s + it` is a complex measure -/
 @[simps!]
 def _root_.MeasureTheory.SignedMeasure.toComplexMeasure (s t : SignedMeasure α) :
     ComplexMeasure α where
diff --git a/Mathlib/MeasureTheory/Measure/Content.lean b/Mathlib/MeasureTheory/Measure/Content.lean
index 5306eb0281091..b07a09b9ad57e 100644
--- a/Mathlib/MeasureTheory/Measure/Content.lean
+++ b/Mathlib/MeasureTheory/Measure/Content.lean
@@ -310,7 +310,7 @@ variable [S : MeasurableSpace G] [BorelSpace G]
 
 /-- For the outer measure coming from a content, all Borel sets are measurable. -/
 theorem borel_le_caratheodory : S ≤ μ.outerMeasure.caratheodory := by
-  rw [@BorelSpace.measurable_eq G _ _]
+  rw [BorelSpace.measurable_eq (α := G)]
   refine MeasurableSpace.generateFrom_le ?_
   intro U hU
   rw [μ.outerMeasure_caratheodory]
diff --git a/Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean b/Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean
index ca19e7aa85cca..b98c2b2b39069 100644
--- a/Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean
+++ b/Mathlib/MeasureTheory/Measure/ContinuousPreimage.lean
@@ -79,7 +79,7 @@ theorem tendsto_measure_symmDiff_preimage_nhds_zero
   obtain ⟨K, hKg, hKco, hKcl, hKμ⟩ :
       ∃ K, MapsTo g K s ∧ IsCompact K ∧ IsClosed K ∧ μ (g ⁻¹' s \ K) < ε / 2 :=
     (hg.measurable hso.measurableSet).exists_isCompact_isClosed_diff_lt hνs' <| by simp [hε.ne']
-  have hKm : MeasurableSet K := hKcl.measurableSet
+  have hKm : NullMeasurableSet K μ := hKcl.nullMeasurableSet
   -- Take `a` such that `f a` is measure preserving and maps `K` to `s`.
   -- This is possible, because `K` is a compact set and `s` is an open set.
   filter_upwards [hf, ContinuousMap.tendsto_nhds_compactOpen.mp hfg K hKco s hso hKg] with a hfa ha
diff --git a/Mathlib/MeasureTheory/Measure/EverywherePos.lean b/Mathlib/MeasureTheory/Measure/EverywherePos.lean
index c3975a2f9c5e2..1d8c5d6773ce4 100644
--- a/Mathlib/MeasureTheory/Measure/EverywherePos.lean
+++ b/Mathlib/MeasureTheory/Measure/EverywherePos.lean
@@ -243,7 +243,7 @@ lemma IsEverywherePos.IsGdelta_of_isMulLeftInvariant
     apply le_of_lt (hW _ _ ?_)
     have : W n * {z} ∈ 𝓝 z := (IsOpen.mul_right (W_open n)).mem_nhds (by simp [mem_W])
     obtain ⟨i, hi, ni⟩ : ∃ i, y i ∈ W n * {z} ∧ n < i :=
-      (((mapClusterPt_iff _ _ _).1 hz _ this).and_eventually (eventually_gt_atTop n)).exists
+      ((mapClusterPt_iff.1 hz _ this).and_eventually (eventually_gt_atTop n)).exists
     refine ⟨x * (y i) ⁻¹, ?_, y i * z⁻¹, by simpa using hi, by group⟩
     have I : V i ⊆ W n := iInter₂_subset n (by simp [ni])
     have J : x * (y i) ⁻¹ ∈ V i := by simpa [← hvy i] using hv i
diff --git a/Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean b/Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
index 5ec34e572fd68..97ffeece7c11c 100644
--- a/Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
+++ b/Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean
@@ -71,7 +71,7 @@ lemma prod_prod (s : Set α) (t : Set β) : μ.prod ν (s ×ˢ t) = μ s * ν t
 @[simp] lemma map_snd_prod : (μ.prod ν).map Prod.snd = μ univ • ν := by ext; simp
 
 lemma map_prod_map {α' : Type*} [MeasurableSpace α'] {β' : Type*} [MeasurableSpace β']
-    {f : α → α'} {g : β → β'}  (f_mble : Measurable f) (g_mble : Measurable g) :
+    {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) :
     (μ.map f).prod (ν.map g) = (μ.prod ν).map (Prod.map f g) := by
   apply Subtype.ext
   simp only [val_eq_toMeasure, toMeasure_prod, toMeasure_map]
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Basic.lean b/Mathlib/MeasureTheory/Measure/Haar/Basic.lean
index b8bf558a35d4e..c09a144500014 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Basic.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Basic.lean
@@ -199,7 +199,7 @@ theorem index_union_le (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).N
   apply Nat.sInf_le; refine ⟨_, ?_, rfl⟩; rw [mem_setOf_eq]
   apply union_subset <;> refine Subset.trans (by assumption) ?_ <;>
     apply biUnion_subset_biUnion_left <;> intro g hg <;> simp only [mem_def] at hg <;>
-    simp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true_iff, true_or_iff]
+    simp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true, true_or]
 
 @[to_additive addIndex_union_eq]
 theorem index_union_eq (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).Nonempty)
@@ -216,7 +216,7 @@ theorem index_union_eq (K₁ K₂ : Compacts G) {V : Set G} (hV : (interior V).N
     intro g hg; rcases hK hg with ⟨_, ⟨g₀, rfl⟩, _, ⟨h1g₀, rfl⟩, h2g₀⟩
     simp only [mem_preimage] at h2g₀
     simp only [mem_iUnion]; use g₀; constructor; swap
-    · simp only [Finset.mem_filter, h1g₀, true_and_iff]; use g
+    · simp only [Finset.mem_filter, h1g₀, true_and]; use g
       simp only [hg, h2g₀, mem_inter_iff, mem_preimage, and_self_iff]
     exact h2g₀
   refine
@@ -432,7 +432,7 @@ theorem chaar_sup_eq {K₀ : PositiveCompacts G}
     mem_of_subset_of_mem _
       (chaar_mem_clPrehaar K₀
         ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
-          simp only [V, mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
+          simp only [V, mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and]⟩)
   unfold clPrehaar; rw [IsClosed.closure_subset_iff]
   · rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
     simp only [eval, mem_preimage, sub_eq_zero, mem_singleton_iff]; rw [eq_comm]
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean b/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
index b499fde66b0b5..c3c47f6abc70a 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean
@@ -36,6 +36,10 @@ variable {𝕜 E F : Type*}
 variable [LocallyCompactSpace E]
 variable (L μ ν)
 
+-- Instances with keys using `Submodule`
+instance (T : Submodule 𝕜 E) : BorelSpace T := Subtype.borelSpace _
+instance (T : Submodule 𝕜 E) : OpensMeasurableSpace T := Subtype.opensMeasurableSpace _
+
 /-- The image of an additive Haar measure under a surjective linear map is proportional to a given
 additive Haar measure. The proportionality factor will be infinite if the linear map has a
 nontrivial kernel. -/
@@ -46,7 +50,6 @@ theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L
   is also true for linear equivalences, as they map Haar measure to Haar measure. The general case
   follows from these two and linear algebra, as `L` can be interpreted as the composition of the
   projection `P` on a complement `T` to its kernel `S`, together with a linear equivalence. -/
-  have : ProperSpace E := .of_locallyCompactSpace 𝕜
   have : FiniteDimensional 𝕜 E := .of_locallyCompactSpace 𝕜
   have : ProperSpace F := by
     rcases subsingleton_or_nontrivial E with hE|hE
diff --git a/Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean b/Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean
index cc3daa531971b..71eec8c794d88 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean
@@ -16,7 +16,7 @@ measure `1` to the parallelepiped spanned by any orthonormal basis, and that it
 the canonical `volume` from the `MeasureSpace` instance.
 -/
 
-open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
+open Module MeasureTheory MeasureTheory.Measure Set
 
 variable {ι E F : Type*}
 
diff --git a/Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean b/Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
index 23979e07b5cf8..a4030ed00f9d7 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
@@ -18,7 +18,7 @@ open scoped NNReal ENNReal Pointwise Topology
 
 open Inv Set Function MeasureTheory.Measure Filter
 
-open FiniteDimensional
+open Module
 
 namespace MeasureTheory
 
@@ -30,23 +30,19 @@ example {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [Nontrivial E] [F
     [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [IsAddHaarMeasure μ] : NoAtoms μ := by
   infer_instance
 
-section ContinuousLinearEquiv
+section LinearEquiv
 
 variable {𝕜 G H : Type*} [MeasurableSpace G] [MeasurableSpace H] [NontriviallyNormedField 𝕜]
   [TopologicalSpace G] [TopologicalSpace H] [AddCommGroup G] [AddCommGroup H]
   [TopologicalAddGroup G] [TopologicalAddGroup H] [Module 𝕜 G] [Module 𝕜 H] (μ : Measure G)
-  [IsAddHaarMeasure μ] [BorelSpace G] [BorelSpace H] [T2Space H]
-
-instance MapContinuousLinearEquiv.isAddHaarMeasure (e : G ≃L[𝕜] H) : IsAddHaarMeasure (μ.map e) :=
-  e.toAddEquiv.isAddHaarMeasure_map _ e.continuous e.symm.continuous
-
-variable [CompleteSpace 𝕜] [T2Space G] [FiniteDimensional 𝕜 G] [ContinuousSMul 𝕜 G]
-  [ContinuousSMul 𝕜 H]
+  [IsAddHaarMeasure μ] [BorelSpace G] [BorelSpace H]
+  [CompleteSpace 𝕜] [T2Space G] [FiniteDimensional 𝕜 G] [ContinuousSMul 𝕜 G]
+  [ContinuousSMul 𝕜 H] [T2Space H]
 
 instance MapLinearEquiv.isAddHaarMeasure (e : G ≃ₗ[𝕜] H) : IsAddHaarMeasure (μ.map e) :=
-  MapContinuousLinearEquiv.isAddHaarMeasure _ e.toContinuousLinearEquiv
+  e.toContinuousLinearEquiv.isAddHaarMeasure_map _
 
-end ContinuousLinearEquiv
+end LinearEquiv
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]
   [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ] {F : Type*} [NormedAddCommGroup F]
@@ -126,11 +122,11 @@ alias set_integral_comp_smul_of_pos := setIntegral_comp_smul_of_pos
 
 theorem integral_comp_mul_left (g : ℝ → F) (a : ℝ) :
     (∫ x : ℝ, g (a * x)) = |a⁻¹| • ∫ y : ℝ, g y := by
-  simp_rw [← smul_eq_mul, Measure.integral_comp_smul, FiniteDimensional.finrank_self, pow_one]
+  simp_rw [← smul_eq_mul, Measure.integral_comp_smul, Module.finrank_self, pow_one]
 
 theorem integral_comp_inv_mul_left (g : ℝ → F) (a : ℝ) :
     (∫ x : ℝ, g (a⁻¹ * x)) = |a| • ∫ y : ℝ, g y := by
-  simp_rw [← smul_eq_mul, Measure.integral_comp_inv_smul, FiniteDimensional.finrank_self, pow_one]
+  simp_rw [← smul_eq_mul, Measure.integral_comp_inv_smul, Module.finrank_self, pow_one]
 
 theorem integral_comp_mul_right (g : ℝ → F) (a : ℝ) :
     (∫ x : ℝ, g (x * a)) = |a⁻¹| • ∫ y : ℝ, g y := by
diff --git a/Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean b/Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
index 5c2649473b452..853e3faeab186 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean
@@ -26,7 +26,7 @@ of the basis).
 -/
 
 
-open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
+open Set TopologicalSpace MeasureTheory MeasureTheory.Measure Module
 
 open scoped Pointwise
 
@@ -162,7 +162,7 @@ theorem parallelepiped_single [DecidableEq ι] (a : ι → ℝ) :
       · rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
         exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩
       · rw [sup_eq_right.mpr hai, inf_eq_left.mpr hai] at h
-        exact ⟨div_nonneg h.1 hai, div_le_one_of_le h.2 hai⟩
+        exact ⟨div_nonneg h.1 hai, div_le_one_of_le₀ h.2 hai⟩
     · specialize h i
       simp only [smul_eq_mul, Pi.mul_apply]
       rcases eq_or_ne (a i) 0 with hai | hai
@@ -290,7 +290,18 @@ end Fintype
 /-- A finite dimensional inner product space has a canonical measure, the Lebesgue measure giving
 volume `1` to the parallelepiped spanned by any orthonormal basis. We define the measure using
 some arbitrary choice of orthonormal basis. The fact that it works with any orthonormal basis
-is proved in `orthonormalBasis.volume_parallelepiped`. -/
+is proved in `orthonormalBasis.volume_parallelepiped`.
+
+This instance creates:
+
+- a potential non-defeq diamond with the natural instance for `MeasureSpace (ULift E)`,
+  which does not exist in Mathlib at the moment;
+
+- a diamond with the existing instance `MeasureTheory.Measure.instMeasureSpacePUnit`.
+
+However, we've decided not to refactor until one of these diamonds starts creating issues, see
+https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Hausdorff.20measure.20normalisation
+-/
 instance (priority := 100) measureSpaceOfInnerProductSpace [NormedAddCommGroup E]
     [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] :
     MeasureSpace E where volume := (stdOrthonormalBasis ℝ E).toBasis.addHaar
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean b/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
index 4882bfa945404..6cb8d92d7d95f 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
@@ -223,7 +223,7 @@ theorem MeasureTheory.QuotientMeasureEqMeasurePreimage.haarMeasure_quotient [Loc
     [IsFiniteMeasure μ] : IsHaarMeasure μ := by
   obtain ⟨K⟩ := PositiveCompacts.nonempty' (α := G)
   let K' : PositiveCompacts (G ⧸ Γ) :=
-    K.map π continuous_coinduced_rng (QuotientGroup.isOpenMap_coe Γ)
+    K.map π QuotientGroup.continuous_mk QuotientGroup.isOpenMap_coe
   haveI : IsMulLeftInvariant μ :=
     MeasureTheory.QuotientMeasureEqMeasurePreimage.mulInvariantMeasure_quotient ν
   rw [haarMeasure_unique μ K']
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Unique.lean b/Mathlib/MeasureTheory/Measure/Haar/Unique.lean
index e2880770c5856..0b75436b65433 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Unique.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Unique.lean
@@ -573,7 +573,8 @@ lemma measure_isMulInvariant_eq_smul_of_isCompact_closure_of_innerRegularCompact
     exact t_comp.closure_of_subset diff_subset
   have B : μ' t = ν t :=
     measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport _ _ f_cont f_comp
-  rwa [measure_diff st hs, measure_diff st hs, ← B, ENNReal.sub_le_sub_iff_left] at A
+  rwa [measure_diff st hs.nullMeasurableSet, measure_diff st hs.nullMeasurableSet, ← B,
+    ENNReal.sub_le_sub_iff_left] at A
   · exact measure_mono st
   · exact t_comp.measure_lt_top.ne
   · exact ((measure_mono st).trans_lt t_comp.measure_lt_top).ne
@@ -908,6 +909,32 @@ theorem absolutelyContinuous_isHaarMeasure [LocallyCompactSpace G]
   rw [haarMeasure_unique μ K, h, smul_smul]
   exact AbsolutelyContinuous.smul (Eq.absolutelyContinuous rfl) _
 
+/-- A continuous surjective monoid homomorphism of topological groups with compact codomain
+is measure preserving, provided that the Haar measures on the domain and on the codomain
+have the same total mass.
+-/
+@[to_additive
+  "A continuous surjective additive monoid homomorphism of topological groups with compact codomain
+is measure preserving, provided that the Haar measures on the domain and on the codomain
+have the same total mass."]
+theorem _root_.MonoidHom.measurePreserving
+    {H : Type*} [Group H] [TopologicalSpace H] [TopologicalGroup H] [CompactSpace H]
+    [MeasurableSpace H] [BorelSpace H]
+    {μ : Measure G} [IsHaarMeasure μ] {ν : Measure H} [IsHaarMeasure ν]
+    {f : G →* H} (hcont : Continuous f) (hsurj : Surjective f) (huniv : μ univ = ν univ) :
+    MeasurePreserving f μ ν where
+  measurable := hcont.measurable
+  map_eq := by
+    have : IsFiniteMeasure μ := ⟨by rw [huniv]; apply measure_lt_top⟩
+    have : (μ.map f).IsHaarMeasure := isHaarMeasure_map_of_isFiniteMeasure μ f hcont hsurj
+    set C : ℝ≥0 := haarScalarFactor (μ.map f) ν
+    have hC : μ.map f = C • ν := isMulLeftInvariant_eq_smul_of_innerRegular _ _
+    suffices C = 1 by rwa [this, one_smul] at hC
+    have : C * ν univ = 1 * ν univ := by
+      rw [one_mul, ← smul_eq_mul, ← ENNReal.smul_def, ← smul_apply, ← hC,
+        map_apply hcont.measurable .univ, preimage_univ, huniv]
+    rwa [ENNReal.mul_eq_mul_right (NeZero.ne _) (measure_ne_top _ _), ENNReal.coe_eq_one] at this
+
 end Group
 
 section CommGroup
@@ -968,22 +995,8 @@ instance (priority := 100) IsHaarMeasure.isInvInvariant_of_innerRegular
 @[to_additive]
 theorem measurePreserving_zpow [CompactSpace G] [RootableBy G ℤ] {n : ℤ} (hn : n ≠ 0) :
     MeasurePreserving (fun g : G => g ^ n) μ μ :=
-  { measurable := (continuous_zpow n).measurable
-    map_eq := by
-      let f := @zpowGroupHom G _ n
-      have hf : Continuous f := continuous_zpow n
-      have : (μ.map f).IsHaarMeasure :=
-        isHaarMeasure_map_of_isFiniteMeasure μ f hf (RootableBy.surjective_pow G ℤ hn)
-      let C : ℝ≥0∞ := haarScalarFactor (μ.map f) μ
-      have hC : μ.map f = C • μ := isMulLeftInvariant_eq_smul_of_innerRegular _ _
-      suffices C = 1 by rwa [this, one_smul] at hC
-      have h_univ : (μ.map f) univ = μ univ := by
-        rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ,
-          preimage_univ]
-      have hμ₀ : μ univ ≠ 0 := IsOpenPosMeasure.open_pos univ isOpen_univ univ_nonempty
-      have hμ₁ : μ univ ≠ ∞ := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne
-      rwa [hC, smul_apply, Algebra.id.smul_eq_mul, mul_comm, ← ENNReal.eq_div_iff hμ₀ hμ₁,
-        ENNReal.div_self hμ₀ hμ₁] at h_univ }
+  (zpowGroupHom n).measurePreserving (continuous_zpow n)
+    (RootableBy.surjective_pow G ℤ hn) rfl
 
 @[to_additive]
 theorem MeasurePreserving.zpow [CompactSpace G] [RootableBy G ℤ]
diff --git a/Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean b/Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
index ed741f37cbf85..d824d0aecc8ad 100644
--- a/Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
+++ b/Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean
@@ -210,7 +210,7 @@ theorem measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure {Ω : Type
     have whole := h 1
     simp only [BoundedContinuousFunction.coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_const,
       one_mul] at whole
-    simpa [← whole] using IsFiniteMeasure.measure_univ_lt_top
+    simp [← whole]
   have obs_μ := HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed μ
   have obs_ν := HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed ν
   simp_rw [h] at obs_μ
diff --git a/Mathlib/MeasureTheory/Measure/Hausdorff.lean b/Mathlib/MeasureTheory/Measure/Hausdorff.lean
index 1eb27758afc84..497ba69408ba9 100644
--- a/Mathlib/MeasureTheory/Measure/Hausdorff.lean
+++ b/Mathlib/MeasureTheory/Measure/Hausdorff.lean
@@ -109,7 +109,7 @@ Hausdorff measure, measure, metric measure
 
 open scoped NNReal ENNReal Topology
 
-open EMetric Set Function Filter Encodable FiniteDimensional TopologicalSpace
+open EMetric Set Function Filter Encodable Module TopologicalSpace
 
 noncomputable section
 
@@ -183,7 +183,7 @@ theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory :
   suffices μ (⋃ n, S n) ≤ ⨆ n, μ (S n) by calc
     μ (s ∩ t) + μ (s \ t) = μ (s ∩ t) + μ (⋃ n, S n) := by rw [iUnion_S]
     _ ≤ μ (s ∩ t) + ⨆ n, μ (S n) := by gcongr
-    _ = ⨆ n, μ (s ∩ t) + μ (S n) := ENNReal.add_iSup
+    _ = ⨆ n, μ (s ∩ t) + μ (S n) := ENNReal.add_iSup ..
     _ ≤ μ s := iSup_le hSs
   /- It suffices to show that `∑' k, μ (S (k + 1) \ S k) ≠ ∞`. Indeed, if we have this,
     then for all `N` we have `μ (⋃ n, S n) ≤ μ (S N) + ∑' k, m (S (N + k + 1) \ S (N + k))`
@@ -574,7 +574,7 @@ theorem hausdorffMeasure_zero_or_top {d₁ d₂ : ℝ} (h : d₁ < d₂) (s : Se
   intro c hc
   refine le_iff'.1 (mkMetric_mono_smul ENNReal.coe_ne_top (mod_cast hc) ?_) s
   have : 0 < ((c : ℝ≥0∞) ^ (d₂ - d₁)⁻¹) := by
-    rw [ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne, ENNReal.coe_eq_zero,
+    rw [← ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne, ENNReal.coe_eq_zero,
       NNReal.rpow_eq_zero_iff]
     exact mt And.left hc
   filter_upwards [Ico_mem_nhdsWithin_Ici ⟨le_rfl, this⟩]
@@ -741,17 +741,16 @@ open scoped Pointwise
 theorem MeasureTheory.Measure.hausdorffMeasure_smul₀ {𝕜 E : Type*} [NormedAddCommGroup E]
     [NormedField 𝕜] [NormedSpace 𝕜 E] [MeasurableSpace E] [BorelSpace E] {d : ℝ} (hd : 0 ≤ d)
     {r : 𝕜} (hr : r ≠ 0) (s : Set E) : μH[d] (r • s) = ‖r‖₊ ^ d • μH[d] s := by
-  suffices ∀ {r : 𝕜}, r ≠ 0 → ∀ s : Set E, μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s by
-    refine le_antisymm (this hr s) ?_
-    rw [← le_inv_smul_iff_of_pos]
-    · dsimp
-      rw [← NNReal.inv_rpow, ← nnnorm_inv]
-      · refine Eq.trans_le ?_ (this (inv_ne_zero hr) (r • s))
-        rw [inv_smul_smul₀ hr]
-    · simp [pos_iff_ne_zero, hr]
-  intro r _ s
-  simp only [NNReal.rpow_eq_pow, ENNReal.smul_def, ← ENNReal.coe_rpow_of_nonneg _ hd, smul_eq_mul]
-  exact (lipschitzWith_smul (β := E) r).hausdorffMeasure_image_le hd s
+  have {r : 𝕜} (s : Set E) : μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s := by
+    simpa [ENNReal.coe_rpow_of_nonneg, hd]
+      using (lipschitzWith_smul r).hausdorffMeasure_image_le hd s
+  refine le_antisymm (this s) ?_
+  rw [← le_inv_smul_iff_of_pos]
+  · dsimp
+    rw [← NNReal.inv_rpow, ← nnnorm_inv]
+    · refine Eq.trans_le ?_ (this (r • s))
+      rw [inv_smul_smul₀ hr]
+  · simp [pos_iff_ne_zero, hr]
 
 /-!
 ### Antilipschitz maps do not decrease Hausdorff measures and dimension
@@ -958,6 +957,30 @@ theorem hausdorffMeasure_pi_real {ι : Type*} [Fintype ι] :
           ENNReal.ofReal_natCast]
       · simp only [ENNReal.ofReal_ne_top, Ne, not_false_iff]
 
+instance isAddHaarMeasure_hausdorffMeasure {E : Type*}
+    [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
+    [MeasurableSpace E] [BorelSpace E] :
+    IsAddHaarMeasure (G := E) μH[finrank ℝ E] where
+  lt_top_of_isCompact K hK := by
+    set e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq (by simp)
+    suffices μH[finrank ℝ E] (e '' K) < ⊤ by
+      rw [← e.symm_image_image K]
+      apply lt_of_le_of_lt <| e.symm.lipschitz.hausdorffMeasure_image_le (by simp) (e '' K)
+      rw [ENNReal.rpow_natCast]
+      exact ENNReal.mul_lt_top (ENNReal.pow_lt_top ENNReal.coe_lt_top _) this
+    conv_lhs => congr; congr; rw [← Fintype.card_fin (finrank ℝ E)]
+    rw [hausdorffMeasure_pi_real]
+    exact (hK.image e.continuous).measure_lt_top
+  open_pos U hU hU' := by
+    set e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq (by simp)
+    suffices 0 < μH[finrank ℝ E] (e '' U) from
+      (ENNReal.mul_pos_iff.mp (lt_of_lt_of_le this <|
+        e.lipschitz.hausdorffMeasure_image_le (by simp) _)).2.ne'
+    conv_rhs => congr; congr; rw [← Fintype.card_fin (finrank ℝ E)]
+    rw [hausdorffMeasure_pi_real]
+    apply (e.isOpenMap U hU).measure_pos (μ := volume)
+    simpa
+
 variable (ι X)
 
 theorem hausdorffMeasure_measurePreserving_funUnique [Unique ι]
diff --git a/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean b/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
index 7411012c4e1c6..9ede8fdf04384 100644
--- a/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
+++ b/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
@@ -100,7 +100,7 @@ theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [
 
 namespace MeasureTheory
 
-open Measure TopologicalSpace.PositiveCompacts FiniteDimensional
+open Measure TopologicalSpace.PositiveCompacts Module
 
 /-!
 ### The Lebesgue measure is a Haar measure on `ℝ` and on `ℝ^ι`.
@@ -492,7 +492,8 @@ theorem addHaar_sphere_of_ne_zero (x : E) {r : ℝ} (hr : r ≠ 0) : μ (sphere
   rcases hr.lt_or_lt with (h | h)
   · simp only [empty_diff, measure_empty, ← closedBall_diff_ball, closedBall_eq_empty.2 h]
   · rw [← closedBall_diff_ball,
-      measure_diff ball_subset_closedBall measurableSet_ball measure_ball_lt_top.ne,
+      measure_diff ball_subset_closedBall measurableSet_ball.nullMeasurableSet
+        measure_ball_lt_top.ne,
       addHaar_ball_of_pos μ _ h, addHaar_closedBall μ _ h.le, tsub_self]
 
 theorem addHaar_sphere [Nontrivial E] (x : E) (r : ℝ) : μ (sphere x r) = 0 := by
@@ -511,8 +512,8 @@ theorem addHaar_singleton_add_smul_div_singleton_add_smul {r : ℝ} (hr : r ≠
           (μ s * (μ t)⁻¹) := by
       rw [ENNReal.mul_inv]
       · ring
-      · simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne, true_or_iff]
-      · simp only [ENNReal.ofReal_ne_top, true_or_iff, Ne, not_false_iff]
+      · simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne, true_or]
+      · simp only [ENNReal.ofReal_ne_top, true_or, Ne, not_false_iff]
     _ = μ s / μ t := by
       rw [ENNReal.mul_inv_cancel, one_mul, div_eq_mul_inv]
       · simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne]
@@ -606,7 +607,6 @@ theorem tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 (s : Set E) (x : E)
     rintro r (rpos : 0 < r)
     rw [← affinity_unitClosedBall rpos.le, singleton_add, ← image_vadd]
     gcongr
-    exact smul_set_mono t_bound
   have B :
     Tendsto (fun r : ℝ => μ (closedBall x r) / μ ({x} + r • u)) (𝓝[>] 0)
       (𝓝 (μ (closedBall x 1) / μ ({x} + u))) := by
@@ -623,7 +623,7 @@ theorem tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 (s : Set E) (x : E)
       (𝓝[>] 0) (𝓝 (0 * (μ (closedBall x 1) / μ ({x} + u)))) := by
     apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)
     simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,
-      ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]
+      ENNReal.div_eq_top, h'u, not_and, and_false]
     intro aux
     exact (measure_closedBall_lt_top.ne aux).elim
     -- Porting note: it used to be enough to pass `measure_closedBall_lt_top.ne` to `simp`
@@ -696,7 +696,8 @@ theorem tendsto_addHaar_inter_smul_zero_of_density_zero (s : Set E) (x : E)
         (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
       have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
         ⟨0, ((measure_mono diff_subset).trans_lt h''t.lt_top).ne⟩
-      refine tendsto_measure_iInter (fun n ↦ ht.diff measurableSet_closedBall) (fun m n hmn ↦ ?_) N
+      refine tendsto_measure_iInter (fun n ↦ (ht.diff measurableSet_closedBall).nullMeasurableSet)
+        (fun m n hmn ↦ ?_) N
       exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))
     have : ⋂ n : ℕ, t \ closedBall 0 n = ∅ := by
       simp_rw [diff_eq, ← inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,
@@ -744,10 +745,8 @@ theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : Meas
     rw [← ENNReal.sub_mul]; swap
     · simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne, not_false_iff]
     congr 1
-    apply
-      ENNReal.sub_eq_of_add_eq (ne_top_of_le_ne_top utop (measure_mono inter_subset_right))
-    rw [inter_comm _ u, inter_comm _ u]
-    exact measure_inter_add_diff u vmeas
+    rw [inter_comm _ u, inter_comm _ u, eq_comm]
+    exact ENNReal.eq_sub_of_add_eq' utop (measure_inter_add_diff u vmeas)
   have L : Tendsto (fun r => μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0) := by
     have A : Tendsto (fun r => μ (closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1) := by
       apply tendsto_const_nhds.congr' _
@@ -778,11 +777,10 @@ theorem tendsto_addHaar_inter_smul_one_of_density_one_aux (s : Set E) (hs : Meas
   rintro r (rpos : 0 < r)
   refine I ({x} + r • t) s ?_ ?_ hs
   · simp only [h't, abs_of_nonneg rpos.le, pow_pos rpos, addHaar_smul, image_add_left,
-      ENNReal.ofReal_eq_zero, not_le, or_false_iff, Ne, measure_preimage_add, abs_pow,
+      ENNReal.ofReal_eq_zero, not_le, or_false, Ne, measure_preimage_add, abs_pow,
       singleton_add, mul_eq_zero]
   · simp [h''t, ENNReal.ofReal_ne_top, addHaar_smul, image_add_left, ENNReal.mul_eq_top,
-      Ne, not_false_iff, measure_preimage_add, singleton_add, and_false_iff, false_and_iff,
-      or_self_iff]
+      Ne, not_false_iff, measure_preimage_add, singleton_add, or_self_iff]
 
 /-- Consider a point `x` at which a set `s` has density one, with respect to closed balls (i.e.,
 a Lebesgue density point of `s`). Then `s` has also density one at `x` with respect to any
diff --git a/Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean b/Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean
index f4d39f02991d6..15ee5ceb0977a 100644
--- a/Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean
+++ b/Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean
@@ -102,7 +102,7 @@ theorem integral_comp_abs {f : ℝ → ℝ} :
       rw [← Measure.map_neg_eq_self (volume : Measure ℝ)]
       let m : MeasurableEmbedding fun x : ℝ => -x := (Homeomorph.neg ℝ).measurableEmbedding
       rw [m.integrableOn_map_iff]
-      simp_rw [Function.comp, abs_neg, neg_preimage, preimage_neg_Iic, neg_zero]
+      simp_rw [Function.comp_def, abs_neg, neg_preimage, preimage_neg_Iic, neg_zero]
       exact integrableOn_Ici_iff_integrableOn_Ioi.mpr hf
     calc
       _ = (∫ x in Iic 0, f |x|) + ∫ x in Ioi 0, f |x| := by
diff --git a/Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean b/Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
index a43d6a7133501..73abc051da088 100644
--- a/Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
+++ b/Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean
@@ -40,7 +40,7 @@ Using these formulas, we compute the volume of the unit balls in several cases.
 
 section general_case
 
-open MeasureTheory MeasureTheory.Measure FiniteDimensional ENNReal
+open MeasureTheory MeasureTheory.Measure Module ENNReal
 
 theorem MeasureTheory.measure_unitBall_eq_integral_div_gamma {E : Type*} {p : ℝ}
     [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E]
@@ -160,7 +160,7 @@ end general_case
 
 section LpSpace
 
-open Real Fintype ENNReal FiniteDimensional MeasureTheory MeasureTheory.Measure
+open Real Fintype ENNReal Module MeasureTheory MeasureTheory.Measure
 
 variable (ι : Type*) [Fintype ι] {p : ℝ}
 
@@ -212,7 +212,7 @@ theorem MeasureTheory.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p)
     simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_setOf_eq, Pi.smul_apply,
       smul_eq_mul, abs_mul, mul_rpow (abs_nonneg _) (abs_nonneg _), abs_inv,
       inv_rpow (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr),
-      inv_mul_lt_iff (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
+      inv_mul_lt_iff₀ (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff
       (rpow_nonneg (h₁ _) _) (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul
       (h₁ _), div_mul_cancel₀ _ (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]
 
@@ -284,7 +284,7 @@ theorem Complex.volume_sum_rpow_lt [Nonempty ι] {p : ℝ} (hp : 1 ≤ p) (r : 
     convert addHaar_smul_of_nonneg volume (le_of_lt hr) {x : ι → ℂ |  ∑ i, ‖x i‖ ^ p < 1} using 2
     · simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hr), Set.preimage_setOf_eq, Pi.smul_apply,
         norm_smul, mul_rpow (norm_nonneg _) (norm_nonneg _), Real.norm_eq_abs, abs_inv, inv_rpow
-        (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr), inv_mul_lt_iff
+        (abs_nonneg _), ← Finset.mul_sum, abs_eq_self.mpr (le_of_lt hr), inv_mul_lt_iff₀
         (rpow_pos_of_pos hr _), mul_one, ← rpow_lt_rpow_iff (rpow_nonneg (h₁ _) _)
         (le_of_lt hr) (by linarith : 0 < p), ← rpow_mul (h₁ _), div_mul_cancel₀ _
         (ne_of_gt (by linarith) : p ≠ 0), Real.rpow_one]
@@ -350,7 +350,7 @@ end EuclideanSpace
 
 section InnerProductSpace
 
-open MeasureTheory MeasureTheory.Measure ENNReal Real FiniteDimensional
+open MeasureTheory MeasureTheory.Measure ENNReal Real Module
 
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
   [MeasurableSpace E] [BorelSpace E] [Nontrivial E]
diff --git a/Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean b/Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
index a2b8a9ec9bdc4..fd684e18cafc6 100644
--- a/Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
+++ b/Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
@@ -28,10 +28,6 @@ import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
   probability measures on a separable space coincides with the topology of convergence in
   distribution, and in particular convergence in distribution is then pseudometrizable.
 
-## TODO
-
-* Show that in Borel spaces, the Lévy-Prokhorov distance is a metric; not just a pseudometric.
-
 ## Tags
 
 finite measure, probability measure, weak convergence, convergence in distribution, metrizability
@@ -198,6 +194,9 @@ lemma levyProkhorovDist_triangle [OpensMeasurableSpace Ω] (μ ν κ : Measure 
 when they are to be equipped with the Lévy-Prokhorov distance. -/
 def LevyProkhorov (α : Type*) := α
 
+/-- The "identity" equivalence between the type synonym `LevyProkhorov α` and `α`. -/
+def LevyProkhorov.equiv (α : Type*) : LevyProkhorov α ≃ α := Equiv.refl _
+
 variable [OpensMeasurableSpace Ω]
 
 /-- The Lévy-Prokhorov distance `levyProkhorovEDist` makes `Measure Ω` a pseudoemetric
@@ -219,9 +218,43 @@ noncomputable instance levyProkhorovDist_pseudoMetricSpace_finiteMeasure :
   dist_triangle μ ν κ := levyProkhorovDist_triangle _ _ _
   edist_dist μ ν := by simp [← ENNReal.ofReal_coe_nnreal]
 
-/-- The Lévy-Prokhorov distance `levyProkhorovDist` makes `ProbabilityMeasure Ω` a pseudoemetric
+lemma measure_le_measure_closure_of_levyProkhorovEDist_eq_zero {μ ν : Measure Ω}
+    (hLP : levyProkhorovEDist μ ν = 0) {s : Set Ω} (s_mble : MeasurableSet s)
+    (h_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞) :
+    μ s ≤ ν (closure s) := by
+  have key : Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] (0 : ℝ≥0∞)) (𝓝 (ν (closure s))) := by
+    have aux : Tendsto ENNReal.toReal (𝓝[>] 0) (𝓝[>] 0) := by
+      apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within (s := Ioi 0) ENNReal.toReal
+      · exact tendsto_nhdsWithin_of_tendsto_nhds (continuousAt_toReal zero_ne_top).tendsto
+      · filter_upwards [Ioo_mem_nhdsWithin_Ioi ⟨le_rfl, zero_lt_one⟩] with x hx
+        exact toReal_pos hx.1.ne.symm <| ne_top_of_lt hx.2
+    exact (tendsto_measure_thickening h_finite).comp aux
+  have obs := Tendsto.add key (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
+  simp only [id_eq, add_zero] at obs
+  apply ge_of_tendsto (b := μ s) obs
+  filter_upwards [self_mem_nhdsWithin] with ε ε_pos
+  exact left_measure_le_of_levyProkhorovEDist_lt (B_mble := s_mble) (hLP ▸ ε_pos)
+
+/-- Two measures at vanishing Lévy-Prokhorov distance from each other assign the same values to all
+closed sets. -/
+lemma measure_eq_measure_of_levyProkhorovEDist_eq_zero_of_isClosed {μ ν : Measure Ω}
+    (hLP : levyProkhorovEDist μ ν = 0) {s : Set Ω} (s_closed : IsClosed s)
+    (hμs : ∃ δ > 0, μ (thickening δ s) ≠ ∞) (hνs : ∃ δ > 0, ν (thickening δ s) ≠ ∞) :
+    μ s = ν s := by
+  apply le_antisymm
+  · exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
+      hLP s_closed.measurableSet hνs |>.trans <|
+      le_of_eq (congr_arg _ s_closed.closure_eq)
+  · exact measure_le_measure_closure_of_levyProkhorovEDist_eq_zero
+      (levyProkhorovEDist_comm μ ν ▸ hLP) s_closed.measurableSet hμs |>.trans <|
+      le_of_eq (congr_arg _ s_closed.closure_eq)
+
+/-- The Lévy-Prokhorov distance `levyProkhorovDist` makes `ProbabilityMeasure Ω` a pseudometric
 space. The instance is recorded on the type synonym
-`LevyProkhorov (ProbabilityMeasure Ω) := ProbabilityMeasure Ω`. -/
+`LevyProkhorov (ProbabilityMeasure Ω) := ProbabilityMeasure Ω`.
+
+Note: For this pseudometric to give the topology of convergence in distribution, one must
+furthermore assume that `Ω` is separable. -/
 noncomputable instance levyProkhorovDist_pseudoMetricSpace_probabilityMeasure :
     PseudoMetricSpace (LevyProkhorov (ProbabilityMeasure Ω)) where
   dist μ ν := levyProkhorovDist μ.toMeasure ν.toMeasure
@@ -233,6 +266,29 @@ noncomputable instance levyProkhorovDist_pseudoMetricSpace_probabilityMeasure :
 lemma LevyProkhorov.dist_def (μ ν : LevyProkhorov (ProbabilityMeasure Ω)) :
     dist μ ν = levyProkhorovDist μ.toMeasure ν.toMeasure := rfl
 
+/-- If `Ω` is a Borel space, then the Lévy-Prokhorov distance `levyProkhorovDist` makes
+`ProbabilityMeasure Ω` a metric space. The instance is recorded on the type synonym
+`LevyProkhorov (ProbabilityMeasure Ω) := ProbabilityMeasure Ω`.
+
+Note: For this metric to give the topology of convergence in distribution, one must
+furthermore assume that `Ω` is separable. -/
+noncomputable instance levyProkhorovDist_metricSpace_probabilityMeasure [BorelSpace Ω] :
+    MetricSpace (LevyProkhorov (ProbabilityMeasure Ω)) where
+  eq_of_dist_eq_zero := by
+    intro μ ν h
+    apply (LevyProkhorov.equiv _).injective
+    apply ProbabilityMeasure.toMeasure_injective
+    apply ext_of_generate_finite _ ?_ isPiSystem_isClosed ?_ (by simp)
+    · rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed]
+    · intro A A_closed
+      apply measure_eq_measure_of_levyProkhorovEDist_eq_zero_of_isClosed
+      · simpa only [levyProkhorovEDist_ne_top μ.toMeasure ν.toMeasure, mem_setOf_eq,
+                    or_false, ne_eq, zero_ne_top, not_false_eq_true, zero_toReal]
+          using (toReal_eq_zero_iff _).mp h
+      · exact A_closed
+      · exact ⟨1, Real.zero_lt_one, measure_ne_top _ _⟩
+      · exact ⟨1, Real.zero_lt_one, measure_ne_top _ _⟩
+
 /-- A simple sufficient condition for bounding `levyProkhorovEDist` between probability measures
 from above. The condition involves only one of two natural bounds, the other bound is for free. -/
 lemma levyProkhorovEDist_le_of_forall_le
@@ -279,20 +335,6 @@ open BoundedContinuousFunction
 
 variable {ι : Type*} {Ω : Type*} [MeasurableSpace Ω]
 
-/-- Coercion from the type synonym `LevyProkhorov (ProbabilityMeasure Ω)`
-to `ProbabilityMeasure Ω`. -/
-def LevyProkhorov.toProbabilityMeasure (μ : LevyProkhorov (ProbabilityMeasure Ω)) :
-    ProbabilityMeasure Ω := μ
-
-/-- Coercion to the type synonym `LevyProkhorov (ProbabilityMeasure Ω)`
-from `ProbabilityMeasure Ω`. -/
-def ProbabilityMeasure.toLevyProkhorov (μ : ProbabilityMeasure Ω) :
-    LevyProkhorov (ProbabilityMeasure Ω) := μ
-
-/-- Coercion from the type synonym `LevyProkhorov (FiniteMeasure Ω)` to `FiniteMeasure Ω`. -/
-def LevyProkhorov.finiteMeasure (μ : LevyProkhorov (FiniteMeasure Ω)) :
-    FiniteMeasure Ω := μ
-
 variable [PseudoMetricSpace Ω] [OpensMeasurableSpace Ω]
 
 /-- A version of the layer cake formula for bounded continuous functions which have finite integral:
@@ -384,13 +426,13 @@ lemma tendsto_integral_meas_thickening_le (f : Ω →ᵇ ℝ)
       · exact isClosed_le continuous_const f.continuous
     · exact measure_ne_top _ _
 
-/-- The coercion `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
-lemma LevyProkhorov.continuous_toProbabilityMeasure :
-    Continuous (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) := by
+/-- The identity map `LevyProkhorov (ProbabilityMeasure Ω) → ProbabilityMeasure Ω` is continuous. -/
+lemma LevyProkhorov.continuous_equiv_probabilityMeasure :
+    Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)) := by
   refine SeqContinuous.continuous ?_
   intro μs ν hμs
-  set P := ν.toProbabilityMeasure -- more palatable notation
-  set Ps := fun n ↦ (μs n).toProbabilityMeasure -- more palatable notation
+  set P := LevyProkhorov.equiv _ ν -- more palatable notation
+  set Ps := fun n ↦ LevyProkhorov.equiv _ (μs n) -- more palatable notation
   rw [ProbabilityMeasure.tendsto_iff_forall_integral_tendsto]
   refine fun f ↦ tendsto_integral_of_forall_limsup_integral_le_integral ?_ f
   intro f f_nn
@@ -433,9 +475,8 @@ lemma LevyProkhorov.continuous_toProbabilityMeasure :
     · rw [ENNReal.ofReal_add (by positivity) (by positivity), ← add_zero (levyProkhorovEDist _ _)]
       apply ENNReal.add_lt_add_of_le_of_lt (levyProkhorovEDist_ne_top _ _)
             (le_of_eq ?_) (ofReal_pos.mpr εs_pos)
-      rw [LevyProkhorov.dist_def, levyProkhorovDist,
-          ofReal_toReal (levyProkhorovEDist_ne_top _ _)]
-      simp only [Ps, P, LevyProkhorov.toProbabilityMeasure]
+      rw [LevyProkhorov.dist_def, levyProkhorovDist, ofReal_toReal (levyProkhorovEDist_ne_top _ _)]
+      rfl
     · exact Eventually.of_forall f_nn
   · simp only [IsCoboundedUnder, IsCobounded, eventually_map, eventually_atTop,
                forall_exists_index]
@@ -444,9 +485,9 @@ lemma LevyProkhorov.continuous_toProbabilityMeasure :
 /-- The topology of the Lévy-Prokhorov metric is at least as fine as the topology of convergence in
 distribution. -/
 theorem levyProkhorov_le_convergenceInDistribution :
-    TopologicalSpace.coinduced (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) inferInstance
+    TopologicalSpace.coinduced (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)) inferInstance
       ≤ (inferInstance : TopologicalSpace (ProbabilityMeasure Ω)) :=
-  (LevyProkhorov.continuous_toProbabilityMeasure).coinduced_le
+  (LevyProkhorov.continuous_equiv_probabilityMeasure).coinduced_le
 
 end Levy_Prokhorov_is_finer
 
@@ -456,13 +497,34 @@ section Levy_Prokhorov_metrizes_convergence_in_distribution
 
 open BoundedContinuousFunction TopologicalSpace
 
-variable {ι : Type*} (Ω : Type*) [PseudoMetricSpace Ω]
+variable {ι : Type*} {Ω : Type*} [PseudoMetricSpace Ω]
 variable [MeasurableSpace Ω] [OpensMeasurableSpace Ω]
 
+lemma ProbabilityMeasure.toMeasure_add_pos_gt_mem_nhds (P : ProbabilityMeasure Ω)
+    {G : Set Ω} (G_open : IsOpen G) {ε : ℝ≥0∞} (ε_pos : 0 < ε) :
+    {Q | P.toMeasure G < Q.toMeasure G + ε} ∈ 𝓝 P := by
+  by_cases easy : P.toMeasure G < ε
+  · exact Eventually.of_forall (fun _ ↦ lt_of_lt_of_le easy le_add_self)
+  by_cases ε_top : ε = ∞
+  · simp [ε_top, measure_lt_top]
+  simp only [not_lt] at easy
+  have aux : P.toMeasure G - ε < liminf (fun Q ↦ Q.toMeasure G) (𝓝 P) := by
+    apply lt_of_lt_of_le (ENNReal.sub_lt_self (measure_lt_top _ _).ne _ _)
+        <| ProbabilityMeasure.le_liminf_measure_open_of_tendsto tendsto_id G_open
+    · exact (lt_of_lt_of_le ε_pos easy).ne.symm
+    · exact ε_pos.ne.symm
+  filter_upwards [gt_mem_sets_of_limsInf_gt (α := ℝ≥0∞) isBounded_ge_of_bot
+      (show P.toMeasure G - ε < limsInf ((𝓝 P).map (fun Q ↦ Q.toMeasure G)) from aux)] with Q hQ
+  simp only [preimage_setOf_eq, mem_setOf_eq] at hQ
+  convert ENNReal.add_lt_add_right ε_top hQ
+  exact (tsub_add_cancel_of_le easy).symm
+
+variable [SeparableSpace Ω]
+
+variable (Ω) in
 /-- In a separable pseudometric space, for any ε > 0 there exists a countable collection of
 disjoint Borel measurable subsets of diameter at most ε that cover the whole space. -/
-lemma SeparableSpace.exists_measurable_partition_diam_le [SeparableSpace Ω]
-    {ε : ℝ} (ε_pos : 0 < ε) :
+lemma SeparableSpace.exists_measurable_partition_diam_le {ε : ℝ} (ε_pos : 0 < ε) :
     ∃ (As : ℕ → Set Ω), (∀ n, MeasurableSet (As n)) ∧ (∀ n, Bornology.IsBounded (As n)) ∧
         (∀ n, diam (As n) ≤ ε) ∧ (⋃ n, As n = univ) ∧
         (Pairwise (fun (n m : ℕ) ↦ Disjoint (As n) (As m))) := by
@@ -492,29 +554,8 @@ lemma SeparableSpace.exists_measurable_partition_diam_le [SeparableSpace Ω]
     simpa only [← aux] using iUnion_disjointed
   · exact disjoint_disjointed Bs
 
-variable {Ω}
-
-lemma ProbabilityMeasure.toMeasure_add_pos_gt_mem_nhds (P : ProbabilityMeasure Ω)
-    {G : Set Ω} (G_open : IsOpen G) {ε : ℝ≥0∞} (ε_pos : 0 < ε) :
-    {Q | P.toMeasure G < Q.toMeasure G + ε} ∈ 𝓝 P := by
-  by_cases easy : P.toMeasure G < ε
-  · exact Eventually.of_forall (fun _ ↦ lt_of_lt_of_le easy le_add_self)
-  by_cases ε_top : ε = ∞
-  · simp [ε_top, measure_lt_top]
-  simp only [not_lt] at easy
-  have aux : P.toMeasure G - ε < liminf (fun Q ↦ Q.toMeasure G) (𝓝 P) := by
-    apply lt_of_lt_of_le (ENNReal.sub_lt_self (measure_lt_top _ _).ne _ _)
-        <| ProbabilityMeasure.le_liminf_measure_open_of_tendsto tendsto_id G_open
-    · exact (lt_of_lt_of_le ε_pos easy).ne.symm
-    · exact ε_pos.ne.symm
-  filter_upwards [gt_mem_sets_of_limsInf_gt (α := ℝ≥0∞) isBounded_ge_of_bot
-      (show P.toMeasure G - ε < limsInf ((𝓝 P).map (fun Q ↦ Q.toMeasure G)) from aux)] with Q hQ
-  simp only [preimage_setOf_eq, mem_setOf_eq] at hQ
-  convert ENNReal.add_lt_add_right ε_top hQ
-  exact (tsub_add_cancel_of_le easy).symm
-
-lemma ProbabilityMeasure.continuous_toLevyProkhorov [SeparableSpace Ω] :
-    Continuous (ProbabilityMeasure.toLevyProkhorov (Ω := Ω)) := by
+lemma LevyProkhorov.continuous_equiv_symm_probabilityMeasure :
+    Continuous (LevyProkhorov.equiv (α := ProbabilityMeasure Ω)).symm := by
   -- We check continuity of `id : ProbabilityMeasure Ω → LevyProkhorov (ProbabilityMeasure Ω)` at
   -- each point `P : ProbabilityMeasure Ω`.
   rw [continuous_iff_continuousAt]
@@ -533,7 +574,7 @@ lemma ProbabilityMeasure.continuous_toLevyProkhorov [SeparableSpace Ω] :
   -- `P (⋃ n < N, Es n)ᶜ < ε/3`.
   obtain ⟨N, hN⟩ : ∃ N, P.toMeasure (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε/3) := by
     have exhaust := @tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint Ω _ P.toMeasure _
-                    Es Es_mble Es_disjoint
+                    Es (fun n ↦ (Es_mble n).nullMeasurableSet) Es_disjoint
     simp only [tendsto_atTop_nhds, Function.comp_apply] at exhaust
     obtain ⟨N, hN⟩ := exhaust (Iio (ENNReal.ofReal (ε / 3))) third_ε_pos' isOpen_Iio
     refine ⟨N, ?_⟩
@@ -612,22 +653,22 @@ lemma ProbabilityMeasure.continuous_toLevyProkhorov [SeparableSpace Ω] :
 
 /-- The topology of the Lévy-Prokhorov metric on probability measures on a separable space
 coincides with the topology of convergence in distribution. -/
-theorem levyProkhorov_eq_convergenceInDistribution [SeparableSpace Ω] :
+theorem levyProkhorov_eq_convergenceInDistribution :
     (inferInstance : TopologicalSpace (ProbabilityMeasure Ω))
-      = TopologicalSpace.coinduced (LevyProkhorov.toProbabilityMeasure (Ω := Ω)) inferInstance :=
-  le_antisymm (ProbabilityMeasure.continuous_toLevyProkhorov (Ω := Ω)).coinduced_le
+      = TopologicalSpace.coinduced (LevyProkhorov.equiv _) inferInstance :=
+  le_antisymm (LevyProkhorov.continuous_equiv_symm_probabilityMeasure (Ω := Ω)).coinduced_le
               levyProkhorov_le_convergenceInDistribution
 
 /-- The identity map is a homeomorphism from `ProbabilityMeasure Ω` with the topology of
 convergence in distribution to `ProbabilityMeasure Ω` with the Lévy-Prokhorov (pseudo)metric. -/
-def homeomorph_probabilityMeasure_levyProkhorov [SeparableSpace Ω] :
+def homeomorph_probabilityMeasure_levyProkhorov :
     ProbabilityMeasure Ω ≃ₜ LevyProkhorov (ProbabilityMeasure Ω) where
-  toFun := ProbabilityMeasure.toLevyProkhorov (Ω := Ω)
-  invFun := LevyProkhorov.toProbabilityMeasure (Ω := Ω)
+  toFun := LevyProkhorov.equiv _
+  invFun := (LevyProkhorov.equiv _).symm
   left_inv := congrFun rfl
   right_inv := congrFun rfl
-  continuous_toFun := ProbabilityMeasure.continuous_toLevyProkhorov
-  continuous_invFun := LevyProkhorov.continuous_toProbabilityMeasure
+  continuous_toFun := LevyProkhorov.continuous_equiv_symm_probabilityMeasure
+  continuous_invFun := LevyProkhorov.continuous_equiv_probabilityMeasure
 
 /-- The topology of convergence in distribution on a separable space is pseudo-metrizable. -/
 instance (X : Type*) [TopologicalSpace X] [PseudoMetrizableSpace X] [SeparableSpace X]
@@ -636,6 +677,13 @@ instance (X : Type*) [TopologicalSpace X] [PseudoMetrizableSpace X] [SeparableSp
   letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
   (homeomorph_probabilityMeasure_levyProkhorov (Ω := X)).inducing.pseudoMetrizableSpace
 
+/-- The topology of convergence in distribution on a separable Borel space is metrizable. -/
+instance instMetrizableSpaceProbabilityMeasure (X : Type*) [TopologicalSpace X]
+    [PseudoMetrizableSpace X] [SeparableSpace X] [MeasurableSpace X] [BorelSpace X] :
+    MetrizableSpace (ProbabilityMeasure X) := by
+  letI : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
+  exact homeomorph_probabilityMeasure_levyProkhorov.embedding.metrizableSpace
+
 end Levy_Prokhorov_metrizes_convergence_in_distribution
 
 end MeasureTheory -- namespace
diff --git a/Mathlib/MeasureTheory/Measure/MeasureSpace.lean b/Mathlib/MeasureTheory/Measure/MeasureSpace.lean
index 88bbbe61b968b..2c1821c1225d6 100644
--- a/Mathlib/MeasureTheory/Measure/MeasureSpace.lean
+++ b/Mathlib/MeasureTheory/Measure/MeasureSpace.lean
@@ -5,7 +5,9 @@ Authors: Johannes Hölzl, Mario Carneiro
 -/
 import Mathlib.MeasureTheory.Measure.NullMeasurable
 import Mathlib.MeasureTheory.MeasurableSpace.Embedding
+import Mathlib.MeasureTheory.OuterMeasure.BorelCantelli
 import Mathlib.Topology.Algebra.Order.LiminfLimsup
+import Mathlib.Order.Interval.Set.Monotone
 
 /-!
 # Measure spaces
@@ -126,9 +128,10 @@ theorem measure_union_add_inter' (hs : MeasurableSet s) (t : Set α) :
     μ (s ∪ t) + μ (s ∩ t) = μ s + μ t := by
   rw [union_comm, inter_comm, measure_union_add_inter t hs, add_comm]
 
-lemma measure_symmDiff_eq (hs : MeasurableSet s) (ht : MeasurableSet t) :
+lemma measure_symmDiff_eq (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
     μ (s ∆ t) = μ (s \ t) + μ (t \ s) := by
-  simpa only [symmDiff_def, sup_eq_union] using measure_union disjoint_sdiff_sdiff (ht.diff hs)
+  simpa only [symmDiff_def, sup_eq_union]
+    using measure_union₀ (ht.diff hs) disjoint_sdiff_sdiff.aedisjoint
 
 lemma measure_symmDiff_le (s t u : Set α) :
     μ (s ∆ u) ≤ μ (s ∆ t) + μ (t ∆ u) :=
@@ -205,14 +208,15 @@ theorem sum_measure_preimage_singleton (s : Finset β) {f : α → β}
 theorem measure_diff_null' (h : μ (s₁ ∩ s₂) = 0) : μ (s₁ \ s₂) = μ s₁ :=
   measure_congr <| diff_ae_eq_self.2 h
 
-theorem measure_add_diff (hs : MeasurableSet s) (t : Set α) : μ s + μ (t \ s) = μ (s ∪ t) := by
-  rw [← measure_union' disjoint_sdiff_right hs, union_diff_self]
+theorem measure_add_diff (hs : NullMeasurableSet s μ) (t : Set α) :
+    μ s + μ (t \ s) = μ (s ∪ t) := by
+  rw [← measure_union₀' hs disjoint_sdiff_right.aedisjoint, union_diff_self]
 
-theorem measure_diff' (s : Set α) (hm : MeasurableSet t) (h_fin : μ t ≠ ∞) :
+theorem measure_diff' (s : Set α) (hm : NullMeasurableSet t μ) (h_fin : μ t ≠ ∞) :
     μ (s \ t) = μ (s ∪ t) - μ t :=
-  Eq.symm <| ENNReal.sub_eq_of_add_eq h_fin <| by rw [add_comm, measure_add_diff hm, union_comm]
+  ENNReal.eq_sub_of_add_eq h_fin <| by rw [add_comm, measure_add_diff hm, union_comm]
 
-theorem measure_diff (h : s₂ ⊆ s₁) (h₂ : MeasurableSet s₂) (h_fin : μ s₂ ≠ ∞) :
+theorem measure_diff (h : s₂ ⊆ s₁) (h₂ : NullMeasurableSet s₂ μ) (h_fin : μ s₂ ≠ ∞) :
     μ (s₁ \ s₂) = μ s₁ - μ s₂ := by rw [measure_diff' _ h₂ h_fin, union_eq_self_of_subset_right h]
 
 theorem le_measure_diff : μ s₁ - μ s₂ ≤ μ (s₁ \ s₂) :=
@@ -238,13 +242,14 @@ then one has finite measure if and only if the other one does. -/
 theorem measure_ne_top_iff_of_symmDiff (hμst : μ (s ∆ t) ≠ ∞) : μ s ≠ ∞ ↔ μ t ≠ ∞ :=
     (measure_eq_top_iff_of_symmDiff hμst).ne
 
-theorem measure_diff_lt_of_lt_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞}
-    (h : μ t < μ s + ε) : μ (t \ s) < ε := by
+theorem measure_diff_lt_of_lt_add (hs : NullMeasurableSet s μ) (hst : s ⊆ t) (hs' : μ s ≠ ∞)
+    {ε : ℝ≥0∞} (h : μ t < μ s + ε) : μ (t \ s) < ε := by
   rw [measure_diff hst hs hs']; rw [add_comm] at h
   exact ENNReal.sub_lt_of_lt_add (measure_mono hst) h
 
-theorem measure_diff_le_iff_le_add (hs : MeasurableSet s) (hst : s ⊆ t) (hs' : μ s ≠ ∞) {ε : ℝ≥0∞} :
-    μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε := by rw [measure_diff hst hs hs', tsub_le_iff_left]
+theorem measure_diff_le_iff_le_add (hs : NullMeasurableSet s μ) (hst : s ⊆ t) (hs' : μ s ≠ ∞)
+    {ε : ℝ≥0∞} : μ (t \ s) ≤ ε ↔ μ t ≤ μ s + ε := by
+  rw [measure_diff hst hs hs', tsub_le_iff_left]
 
 theorem measure_eq_measure_of_null_diff {s t : Set α} (hst : s ⊆ t) (h_nulldiff : μ (t \ s) = 0) :
     μ s = μ t := measure_congr <|
@@ -295,26 +300,27 @@ theorem union_ae_eq_left_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] s ↔ t ≤
 theorem union_ae_eq_right_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] t ↔ s ≤ᵐ[μ] t := by
   rw [union_comm, union_ae_eq_left_iff_ae_subset]
 
-theorem ae_eq_of_ae_subset_of_measure_ge (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s)
-    (ht : μ t ≠ ∞) : s =ᵐ[μ] t := by
+theorem ae_eq_of_ae_subset_of_measure_ge (h₁ : s ≤ᵐ[μ] t) (h₂ : μ t ≤ μ s)
+    (hsm : NullMeasurableSet s μ) (ht : μ t ≠ ∞) : s =ᵐ[μ] t := by
   refine eventuallyLE_antisymm_iff.mpr ⟨h₁, ae_le_set.mpr ?_⟩
   replace h₂ : μ t = μ s := h₂.antisymm (measure_mono_ae h₁)
   replace ht : μ s ≠ ∞ := h₂ ▸ ht
   rw [measure_diff' t hsm ht, measure_congr (union_ae_eq_left_iff_ae_subset.mpr h₁), h₂, tsub_self]
 
 /-- If `s ⊆ t`, `μ t ≤ μ s`, `μ t ≠ ∞`, and `s` is measurable, then `s =ᵐ[μ] t`. -/
-theorem ae_eq_of_subset_of_measure_ge (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : MeasurableSet s)
+theorem ae_eq_of_subset_of_measure_ge (h₁ : s ⊆ t) (h₂ : μ t ≤ μ s) (hsm : NullMeasurableSet s μ)
     (ht : μ t ≠ ∞) : s =ᵐ[μ] t :=
   ae_eq_of_ae_subset_of_measure_ge (HasSubset.Subset.eventuallyLE h₁) h₂ hsm ht
 
-theorem measure_iUnion_congr_of_subset [Countable β] {s : β → Set α} {t : β → Set α}
-    (hsub : ∀ b, s b ⊆ t b) (h_le : ∀ b, μ (t b) ≤ μ (s b)) : μ (⋃ b, s b) = μ (⋃ b, t b) := by
-  rcases Classical.em (∃ b, μ (t b) = ∞) with (⟨b, hb⟩ | htop)
+theorem measure_iUnion_congr_of_subset {ι : Sort*} [Countable ι] {s : ι → Set α} {t : ι → Set α}
+    (hsub : ∀ i, s i ⊆ t i) (h_le : ∀ i, μ (t i) ≤ μ (s i)) : μ (⋃ i, s i) = μ (⋃ i, t i) := by
+  refine le_antisymm (by gcongr; apply hsub) ?_
+  rcases Classical.em (∃ i, μ (t i) = ∞) with (⟨i, hi⟩ | htop)
   · calc
-      μ (⋃ b, s b) = ∞ := top_unique (hb ▸ (h_le b).trans <| measure_mono <| subset_iUnion _ _)
-      _ = μ (⋃ b, t b) := Eq.symm <| top_unique <| hb ▸ measure_mono (subset_iUnion _ _)
+      μ (⋃ i, t i) ≤ ∞ := le_top
+      _ ≤ μ (s i) := hi ▸ h_le i
+      _ ≤ μ (⋃ i, s i) := measure_mono <| subset_iUnion _ _
   push_neg at htop
-  refine le_antisymm (measure_mono (iUnion_mono hsub)) ?_
   set M := toMeasurable μ
   have H : ∀ b, (M (t b) ∩ M (⋃ b, s b) : Set α) =ᵐ[μ] M (t b) := by
     refine fun b => ae_eq_of_subset_of_measure_ge inter_subset_left ?_ ?_ ?_
@@ -325,7 +331,7 @@ theorem measure_iUnion_congr_of_subset [Countable β] {s : β → Set α} {t : 
           measure_mono <|
             subset_inter ((hsub b).trans <| subset_toMeasurable _ _)
               ((subset_iUnion _ _).trans <| subset_toMeasurable _ _)
-    · exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _)
+    · measurability
     · rw [measure_toMeasurable]
       exact htop b
   calc
@@ -340,11 +346,10 @@ theorem measure_union_congr_of_subset {t₁ t₂ : Set α} (hs : s₁ ⊆ s₂)
   exact measure_iUnion_congr_of_subset (Bool.forall_bool.2 ⟨ht, hs⟩) (Bool.forall_bool.2 ⟨htμ, hsμ⟩)
 
 @[simp]
-theorem measure_iUnion_toMeasurable [Countable β] (s : β → Set α) :
-    μ (⋃ b, toMeasurable μ (s b)) = μ (⋃ b, s b) :=
-  Eq.symm <|
-    measure_iUnion_congr_of_subset (fun _b => subset_toMeasurable _ _) fun _b =>
-      (measure_toMeasurable _).le
+theorem measure_iUnion_toMeasurable {ι : Sort*} [Countable ι] (s : ι → Set α) :
+    μ (⋃ i, toMeasurable μ (s i)) = μ (⋃ i, s i) :=
+  Eq.symm <| measure_iUnion_congr_of_subset (fun _i => subset_toMeasurable _ _) fun _i ↦
+    (measure_toMeasurable _).le
 
 theorem measure_biUnion_toMeasurable {I : Set β} (hc : I.Countable) (s : β → Set α) :
     μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b) := by
@@ -416,30 +421,28 @@ theorem nonempty_inter_of_measure_lt_add' {m : MeasurableSpace α} (μ : Measure
   rw [inter_comm]
   exact nonempty_inter_of_measure_lt_add μ hs h't h's h
 
-/-- Continuity from below: the measure of the union of a directed sequence of (not necessarily
--measurable) sets is the supremum of the measures. -/
-theorem measure_iUnion_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s) :
+/-- Continuity from below:
+the measure of the union of a directed sequence of (not necessarily measurable) sets
+is the supremum of the measures. -/
+theorem _root_.Directed.measure_iUnion [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s) :
     μ (⋃ i, s i) = ⨆ i, μ (s i) := by
-  cases nonempty_encodable ι
   -- WLOG, `ι = ℕ`
-  generalize ht : Function.extend Encodable.encode s ⊥ = t
-  replace hd : Directed (· ⊆ ·) t := ht ▸ hd.extend_bot Encodable.encode_injective
+  rcases Countable.exists_injective_nat ι with ⟨e, he⟩
+  generalize ht : Function.extend e s ⊥ = t
+  replace hd : Directed (· ⊆ ·) t := ht ▸ hd.extend_bot he
   suffices μ (⋃ n, t n) = ⨆ n, μ (t n) by
-    simp only [← ht, Function.apply_extend μ, ← iSup_eq_iUnion,
-      iSup_extend_bot Encodable.encode_injective, (· ∘ ·), Pi.bot_apply, bot_eq_empty,
-      measure_empty] at this
-    exact this.trans (iSup_extend_bot Encodable.encode_injective _)
+    simp only [← ht, Function.apply_extend μ, ← iSup_eq_iUnion, iSup_extend_bot he,
+      Function.comp_def, Pi.bot_apply, bot_eq_empty, measure_empty] at this
+    exact this.trans (iSup_extend_bot he _)
   clear! ι
   -- The `≥` inequality is trivial
-  refine le_antisymm ?_ (iSup_le fun i => measure_mono <| subset_iUnion _ _)
+  refine le_antisymm ?_ (iSup_le fun i ↦ measure_mono <| subset_iUnion _ _)
   -- Choose `T n ⊇ t n` of the same measure, put `Td n = disjointed T`
   set T : ℕ → Set α := fun n => toMeasurable μ (t n)
   set Td : ℕ → Set α := disjointed T
-  have hm : ∀ n, MeasurableSet (Td n) :=
-    MeasurableSet.disjointed fun n => measurableSet_toMeasurable _ _
+  have hm : ∀ n, MeasurableSet (Td n) := .disjointed fun n ↦ measurableSet_toMeasurable _ _
   calc
-    μ (⋃ n, t n) ≤ μ (⋃ n, T n) := measure_mono (iUnion_mono fun i => subset_toMeasurable _ _)
-    _ = μ (⋃ n, Td n) := by rw [iUnion_disjointed]
+    μ (⋃ n, t n) = μ (⋃ n, Td n) := by rw [iUnion_disjointed, measure_iUnion_toMeasurable]
     _ ≤ ∑' n, μ (Td n) := measure_iUnion_le _
     _ = ⨆ I : Finset ℕ, ∑ n ∈ I, μ (Td n) := ENNReal.tsum_eq_iSup_sum
     _ ≤ ⨆ n, μ (t n) := iSup_le fun I => by
@@ -452,51 +455,72 @@ theorem measure_iUnion_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed
         _ ≤ μ (t N) := measure_mono (iUnion₂_subset hN)
         _ ≤ ⨆ n, μ (t n) := le_iSup (μ ∘ t) N
 
+@[deprecated (since := "2024-09-01")] alias measure_iUnion_eq_iSup := Directed.measure_iUnion
+
+/-- Continuity from below:
+the measure of the union of a monotone family of sets is equal to the supremum of their measures.
+The theorem assumes that the `atTop` filter on the index set is countably generated,
+so it works for a family indexed by a countable type, as well as `ℝ`.  -/
+theorem _root_.Monotone.measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)]
+    [(atTop : Filter ι).IsCountablyGenerated] {s : ι → Set α} (hs : Monotone s) :
+    μ (⋃ i, s i) = ⨆ i, μ (s i) := by
+  refine le_antisymm ?_ (iSup_le fun i ↦ measure_mono <| subset_iUnion _ _)
+  cases isEmpty_or_nonempty ι with
+  | inl _ => simp
+  | inr _ =>
+    rcases exists_seq_monotone_tendsto_atTop_atTop ι with ⟨x, hxm, hx⟩
+    calc
+      μ (⋃ i, s i) ≤ μ (⋃ n, s (x n)) := by
+        refine measure_mono <| iUnion_mono' fun i ↦ ?_
+        rcases (hx.eventually_ge_atTop i).exists with ⟨n, hn⟩
+        exact ⟨n, hs hn⟩
+      _ = ⨆ n, μ (s (x n)) := (hs.comp hxm).directed_le.measure_iUnion
+      _ ≤ ⨆ i, μ (s i) := iSup_comp_le (μ ∘ s) x
+
+theorem _root_.Antitone.measure_iUnion [Preorder ι] [IsDirected ι (· ≥ ·)]
+    [(atBot : Filter ι).IsCountablyGenerated] {s : ι → Set α} (hs : Antitone s) :
+    μ (⋃ i, s i) = ⨆ i, μ (s i) :=
+  hs.dual_left.measure_iUnion
+
 /-- Continuity from below: the measure of the union of a sequence of
 (not necessarily measurable) sets is the supremum of the measures of the partial unions. -/
-theorem measure_iUnion_eq_iSup' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
-    [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
-    {f : ι → Set α} : μ (⋃ i, f i) = ⨆ i, μ (Accumulate f i) := by
-  have hd : Directed (· ⊆ ·) (Accumulate f) := by
-    intro i j
-    rcases directed_of (· ≤ ·) i j with ⟨k, rik, rjk⟩
-    exact ⟨k, biUnion_subset_biUnion_left fun l rli ↦ le_trans rli rik,
-      biUnion_subset_biUnion_left fun l rlj ↦ le_trans rlj rjk⟩
+theorem measure_iUnion_eq_iSup_accumulate [Preorder ι] [IsDirected ι (· ≤ ·)]
+    [(atTop : Filter ι).IsCountablyGenerated] {f : ι → Set α} :
+    μ (⋃ i, f i) = ⨆ i, μ (Accumulate f i) := by
   rw [← iUnion_accumulate]
-  exact measure_iUnion_eq_iSup hd
+  exact monotone_accumulate.measure_iUnion
+
+@[deprecated (since := "2024-09-01")]
+alias measure_iUnion_eq_iSup' := measure_iUnion_eq_iSup_accumulate
 
 theorem measure_biUnion_eq_iSup {s : ι → Set α} {t : Set ι} (ht : t.Countable)
     (hd : DirectedOn ((· ⊆ ·) on s) t) : μ (⋃ i ∈ t, s i) = ⨆ i ∈ t, μ (s i) := by
-  haveI := ht.toEncodable
-  rw [biUnion_eq_iUnion, measure_iUnion_eq_iSup hd.directed_val, ← iSup_subtype'']
+  haveI := ht.to_subtype
+  rw [biUnion_eq_iUnion, hd.directed_val.measure_iUnion, ← iSup_subtype'']
 
 /-- Continuity from above: the measure of the intersection of a decreasing sequence of measurable
 sets is the infimum of the measures. -/
-theorem measure_iInter_eq_iInf [Countable ι] {s : ι → Set α} (h : ∀ i, MeasurableSet (s i))
+theorem measure_iInter_eq_iInf [Countable ι] {s : ι → Set α} (h : ∀ i, NullMeasurableSet (s i) μ)
     (hd : Directed (· ⊇ ·) s) (hfin : ∃ i, μ (s i) ≠ ∞) : μ (⋂ i, s i) = ⨅ i, μ (s i) := by
   rcases hfin with ⟨k, hk⟩
   have : ∀ t ⊆ s k, μ t ≠ ∞ := fun t ht => ne_top_of_le_ne_top hk (measure_mono ht)
   rw [← ENNReal.sub_sub_cancel hk (iInf_le _ k), ENNReal.sub_iInf, ←
     ENNReal.sub_sub_cancel hk (measure_mono (iInter_subset _ k)), ←
-    measure_diff (iInter_subset _ k) (MeasurableSet.iInter h) (this _ (iInter_subset _ k)),
-    diff_iInter, measure_iUnion_eq_iSup]
+    measure_diff (iInter_subset _ k) (.iInter h) (this _ (iInter_subset _ k)),
+    diff_iInter, Directed.measure_iUnion]
   · congr 1
-    refine le_antisymm (iSup_mono' fun i => ?_) (iSup_mono fun i => ?_)
-    · rcases hd i k with ⟨j, hji, hjk⟩
-      use j
-      rw [← measure_diff hjk (h _) (this _ hjk)]
-      gcongr
-    · rw [tsub_le_iff_right, ← measure_union, Set.union_comm]
-      · exact measure_mono (diff_subset_iff.1 Subset.rfl)
-      · apply disjoint_sdiff_left
-      · apply h i
+    refine le_antisymm (iSup_mono' fun i => ?_) (iSup_mono fun i => le_measure_diff)
+    rcases hd i k with ⟨j, hji, hjk⟩
+    use j
+    rw [← measure_diff hjk (h _) (this _ hjk)]
+    gcongr
   · exact hd.mono_comp _ fun _ _ => diff_subset_diff_right
 
 /-- Continuity from above: the measure of the intersection of a sequence of
 measurable sets is the infimum of the measures of the partial intersections. -/
 theorem measure_iInter_eq_iInf' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
     [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
-    {f : ι → Set α} (h : ∀ i, MeasurableSet (f i)) (hfin : ∃ i, μ (f i) ≠ ∞) :
+    {f : ι → Set α} (h : ∀ i, NullMeasurableSet (f i) μ) (hfin : ∃ i, μ (f i) ≠ ∞) :
     μ (⋂ i, f i) = ⨅ i, μ (⋂ j ≤ i, f j) := by
   let s := fun i ↦ ⋂ j ≤ i, f j
   have iInter_eq : ⋂ i, f i = ⋂ i, s i := by
@@ -505,8 +529,7 @@ theorem measure_iInter_eq_iInf' {α ι : Type*} [MeasurableSpace α] {μ : Measu
     · intro h i
       rcases directed_of (· ≤ ·) i i with ⟨j, rij, -⟩
       exact h j i rij
-  have ms : ∀ i, MeasurableSet (s i) :=
-    fun i ↦ MeasurableSet.biInter (countable_univ.mono <| subset_univ _) fun i _ ↦ h i
+  have ms (i) : NullMeasurableSet (s i) μ := .biInter (to_countable _) fun i _ ↦ h i
   have hd : Directed (· ⊇ ·) s := by
     intro i j
     rcases directed_of (· ≤ ·) i j with ⟨k, rik, rjk⟩
@@ -520,31 +543,47 @@ theorem measure_iInter_eq_iInf' {α ι : Type*} [MeasurableSpace α] {μ : Measu
 
 /-- Continuity from below: the measure of the union of an increasing sequence of (not necessarily
 measurable) sets is the limit of the measures. -/
-theorem tendsto_measure_iUnion [Preorder ι] [IsDirected ι (· ≤ ·)] [Countable ι]
+theorem tendsto_measure_iUnion_atTop [Preorder ι] [IsCountablyGenerated (atTop : Filter ι)]
     {s : ι → Set α} (hm : Monotone s) : Tendsto (μ ∘ s) atTop (𝓝 (μ (⋃ n, s n))) := by
-  rw [measure_iUnion_eq_iSup hm.directed_le]
+  refine .of_neBot_imp fun h ↦ ?_
+  have := (atTop_neBot_iff.1 h).2
+  rw [hm.measure_iUnion]
   exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm
 
+@[deprecated (since := "2024-09-01")] alias tendsto_measure_iUnion := tendsto_measure_iUnion_atTop
+
+theorem tendsto_measure_iUnion_atBot [Preorder ι] [IsCountablyGenerated (atBot : Filter ι)]
+    {s : ι → Set α} (hm : Antitone s) : Tendsto (μ ∘ s) atBot (𝓝 (μ (⋃ n, s n))) :=
+  tendsto_measure_iUnion_atTop (ι := ιᵒᵈ) hm.dual_left
+
 /-- Continuity from below: the measure of the union of a sequence of (not necessarily measurable)
 sets is the limit of the measures of the partial unions. -/
-theorem tendsto_measure_iUnion' {α ι : Type*} [MeasurableSpace α] {μ : Measure α} [Countable ι]
-    [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} :
+theorem tendsto_measure_iUnion_accumulate {α ι : Type*}
+    [Preorder ι] [IsCountablyGenerated (atTop : Filter ι)]
+    [MeasurableSpace α] {μ : Measure α} {f : ι → Set α} :
     Tendsto (fun i ↦ μ (Accumulate f i)) atTop (𝓝 (μ (⋃ i, f i))) := by
-  rw [measure_iUnion_eq_iSup']
+  refine .of_neBot_imp fun h ↦ ?_
+  have := (atTop_neBot_iff.1 h).2
+  rw [measure_iUnion_eq_iSup_accumulate]
   exact tendsto_atTop_iSup fun i j hij ↦ by gcongr
 
+@[deprecated (since := "2024-09-01")]
+alias tendsto_measure_iUnion' := tendsto_measure_iUnion_accumulate
+
 /-- Continuity from above: the measure of the intersection of a decreasing sequence of measurable
 sets is the limit of the measures. -/
-theorem tendsto_measure_iInter [Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)] {s : ι → Set α}
-    (hs : ∀ n, MeasurableSet (s n)) (hm : Antitone s) (hf : ∃ i, μ (s i) ≠ ∞) :
+theorem tendsto_measure_iInter [Countable ι] [Preorder ι] {s : ι → Set α}
+    (hs : ∀ n, NullMeasurableSet (s n) μ) (hm : Antitone s) (hf : ∃ i, μ (s i) ≠ ∞) :
     Tendsto (μ ∘ s) atTop (𝓝 (μ (⋂ n, s n))) := by
+  refine .of_neBot_imp fun h ↦ ?_
+  have := (atTop_neBot_iff.1 h).2
   rw [measure_iInter_eq_iInf hs hm.directed_ge hf]
   exact tendsto_atTop_iInf fun n m hnm => measure_mono <| hm hnm
 
 /-- Continuity from above: the measure of the intersection of a sequence of measurable
 sets such that one has finite measure is the limit of the measures of the partial intersections. -/
 theorem tendsto_measure_iInter' {α ι : Type*} [MeasurableSpace α] {μ : Measure α} [Countable ι]
-    [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} (hm : ∀ i, MeasurableSet (f i))
+    [Preorder ι] [IsDirected ι (· ≤ ·)] {f : ι → Set α} (hm : ∀ i, NullMeasurableSet (f i) μ)
     (hf : ∃ i, μ (f i) ≠ ∞) :
     Tendsto (fun i ↦ μ (⋂ j ∈ {j | j ≤ i}, f j)) atTop (𝓝 (μ (⋂ i, f i))) := by
   rw [measure_iInter_eq_iInf' hm hf]
@@ -555,7 +594,7 @@ theorem tendsto_measure_iInter' {α ι : Type*} [MeasurableSpace α] {μ : Measu
 sets indexed by a linear order with first countable topology is the limit of the measures. -/
 theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpace ι]
     [OrderTopology ι] [DenselyOrdered ι] [FirstCountableTopology ι] {s : ι → Set α}
-    {a : ι} (hs : ∀ r > a, MeasurableSet (s r)) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
+    {a : ι} (hs : ∀ r > a, NullMeasurableSet (s r) μ) (hm : ∀ i j, a < i → i ≤ j → s i ⊆ s j)
     (hf : ∃ r > a, μ (s r) ≠ ∞) : Tendsto (μ ∘ s) (𝓝[Ioi a] a) (𝓝 (μ (⋂ r > a, s r))) := by
   refine tendsto_order.2 ⟨fun l hl => ?_, fun L hL => ?_⟩
   · filter_upwards [self_mem_nhdsWithin (s := Ioi a)] with r hr using hl.trans_le
@@ -588,66 +627,6 @@ theorem tendsto_measure_biInter_gt {ι : Type*} [LinearOrder ι] [TopologicalSpa
   have : Ioc a (u n) ∈ 𝓝[>] a := Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, u_pos n⟩
   filter_upwards [this] with r hr using lt_of_le_of_lt (measure_mono (hm _ _ hr.1 hr.2)) hn
 
-/-- One direction of the **Borel-Cantelli lemma** (sometimes called the "*first* Borel-Cantelli
-lemma"): if (sᵢ) is a sequence of sets such that `∑ μ sᵢ` is finite, then the limit superior of the
-`sᵢ` is a null set.
-
-Note: for the *second* Borel-Cantelli lemma (applying to independent sets in a probability space),
-see `ProbabilityTheory.measure_limsup_eq_one`. -/
-theorem measure_limsup_eq_zero {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) :
-    μ (limsup s atTop) = 0 := by
-  -- First we replace the sequence `sₙ` with a sequence of measurable sets `tₙ ⊇ sₙ` of the same
-  -- measure.
-  set t : ℕ → Set α := fun n => toMeasurable μ (s n)
-  have ht : (∑' i, μ (t i)) ≠ ∞ := by simpa only [t, measure_toMeasurable] using hs
-  suffices μ (limsup t atTop) = 0 by
-    have A : s ≤ t := fun n => subset_toMeasurable μ (s n)
-    -- TODO default args fail
-    exact measure_mono_null (limsup_le_limsup (Eventually.of_forall (Pi.le_def.mp A))) this
-  -- Next we unfold `limsup` for sets and replace equality with an inequality
-  simp only [limsup_eq_iInf_iSup_of_nat', Set.iInf_eq_iInter, Set.iSup_eq_iUnion, ←
-    nonpos_iff_eq_zero]
-  -- Finally, we estimate `μ (⋃ i, t (i + n))` by `∑ i', μ (t (i + n))`
-  refine
-    le_of_tendsto_of_tendsto'
-      (tendsto_measure_iInter
-        (fun i => MeasurableSet.iUnion fun b => measurableSet_toMeasurable _ _) ?_
-        ⟨0, ne_top_of_le_ne_top ht (measure_iUnion_le t)⟩)
-      (ENNReal.tendsto_sum_nat_add (μ ∘ t) ht) fun n => measure_iUnion_le _
-  intro n m hnm x
-  simp only [Set.mem_iUnion]
-  exact fun ⟨i, hi⟩ => ⟨i + (m - n), by simpa only [add_assoc, tsub_add_cancel_of_le hnm] using hi⟩
-
-theorem measure_liminf_eq_zero {s : ℕ → Set α} (h : (∑' i, μ (s i)) ≠ ∞) :
-    μ (liminf s atTop) = 0 := by
-  rw [← le_zero_iff]
-  have : liminf s atTop ≤ limsup s atTop := liminf_le_limsup
-  exact (μ.mono this).trans (by simp [measure_limsup_eq_zero h])
-
--- Need to specify `α := Set α` below because of diamond; see #19041
-theorem limsup_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
-    (h : ∀ n, s n =ᵐ[μ] t) : limsup (α := Set α) s atTop =ᵐ[μ] t := by
-  simp_rw [ae_eq_set] at h ⊢
-  constructor
-  · rw [atTop.limsup_sdiff s t]
-    apply measure_limsup_eq_zero
-    simp [h]
-  · rw [atTop.sdiff_limsup s t]
-    apply measure_liminf_eq_zero
-    simp [h]
-
--- Need to specify `α := Set α` above because of diamond; see #19041
-theorem liminf_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
-    (h : ∀ n, s n =ᵐ[μ] t) : liminf (α := Set α) s atTop =ᵐ[μ] t := by
-  simp_rw [ae_eq_set] at h ⊢
-  constructor
-  · rw [atTop.liminf_sdiff s t]
-    apply measure_liminf_eq_zero
-    simp [h]
-  · rw [atTop.sdiff_liminf s t]
-    apply measure_limsup_eq_zero
-    simp [h]
-
 theorem measure_if {x : β} {t : Set β} {s : Set α} [Decidable (x ∈ t)] :
     μ (if x ∈ t then s else ∅) = indicator t (fun _ => μ s) x := by split_ifs with h <;> simp [h]
 
@@ -750,7 +729,7 @@ theorem coe_zero {_m : MeasurableSpace α} : ⇑(0 : Measure α) = 0 :=
   rfl
 
 @[simp] lemma _root_.MeasureTheory.OuterMeasure.toMeasure_zero
-    [ms : MeasurableSpace α](h : ms ≤ (0 : OuterMeasure α).caratheodory) :
+    [ms : MeasurableSpace α] (h : ms ≤ (0 : OuterMeasure α).caratheodory) :
     (0 : OuterMeasure α).toMeasure h = 0 := by
   ext s hs
   simp [hs]
@@ -1855,61 +1834,26 @@ theorem biSup_measure_Iic [Preorder α] {s : Set α} (hsc : s.Countable)
     exact iUnion₂_eq_univ_iff.2 hst
   · exact directedOn_iff_directed.2 (hdir.directed_val.mono_comp _ fun x y => Iic_subset_Iic.2)
 
-theorem tendsto_measure_Ico_atTop [SemilatticeSup α] [NoMaxOrder α]
+theorem tendsto_measure_Ico_atTop [Preorder α] [NoMaxOrder α]
     [(atTop : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) :
     Tendsto (fun x => μ (Ico a x)) atTop (𝓝 (μ (Ici a))) := by
-  haveI : Nonempty α := ⟨a⟩
-  have h_mono : Monotone fun x => μ (Ico a x) := fun i j hij => by simp only; gcongr
-  convert tendsto_atTop_iSup h_mono
-  obtain ⟨xs, hxs_mono, hxs_tendsto⟩ := exists_seq_monotone_tendsto_atTop_atTop α
-  have h_Ici : Ici a = ⋃ n, Ico a (xs n) := by
-    ext1 x
-    simp only [mem_Ici, mem_iUnion, mem_Ico, exists_and_left, iff_self_and]
-    intro
-    obtain ⟨y, hxy⟩ := NoMaxOrder.exists_gt x
-    obtain ⟨n, hn⟩ := tendsto_atTop_atTop.mp hxs_tendsto y
-    exact ⟨n, hxy.trans_le (hn n le_rfl)⟩
-  rw [h_Ici, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]
-  exact Monotone.directed_le fun i j hij => Ico_subset_Ico_right (hxs_mono hij)
-
-theorem tendsto_measure_Ioc_atBot [SemilatticeInf α] [NoMinOrder α]
+  rw [← iUnion_Ico_right]
+  exact tendsto_measure_iUnion_atTop (antitone_const.Ico monotone_id)
+
+theorem tendsto_measure_Ioc_atBot [Preorder α] [NoMinOrder α]
     [(atBot : Filter α).IsCountablyGenerated] (μ : Measure α) (a : α) :
     Tendsto (fun x => μ (Ioc x a)) atBot (𝓝 (μ (Iic a))) := by
-  haveI : Nonempty α := ⟨a⟩
-  have h_mono : Antitone fun x => μ (Ioc x a) := fun i j hij => by simp only; gcongr
-  convert tendsto_atBot_iSup h_mono
-  obtain ⟨xs, hxs_mono, hxs_tendsto⟩ := exists_seq_antitone_tendsto_atTop_atBot α
-  have h_Iic : Iic a = ⋃ n, Ioc (xs n) a := by
-    ext1 x
-    simp only [mem_Iic, mem_iUnion, mem_Ioc, exists_and_right, iff_and_self]
-    intro
-    obtain ⟨y, hxy⟩ := NoMinOrder.exists_lt x
-    obtain ⟨n, hn⟩ := tendsto_atTop_atBot.mp hxs_tendsto y
-    exact ⟨n, (hn n le_rfl).trans_lt hxy⟩
-  rw [h_Iic, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_antitone h_mono hxs_tendsto]
-  exact Monotone.directed_le fun i j hij => Ioc_subset_Ioc_left (hxs_mono hij)
-
-theorem tendsto_measure_Iic_atTop [SemilatticeSup α] [(atTop : Filter α).IsCountablyGenerated]
+  rw [← iUnion_Ioc_left]
+  exact tendsto_measure_iUnion_atBot (monotone_id.Ioc antitone_const)
+
+theorem tendsto_measure_Iic_atTop [Preorder α] [(atTop : Filter α).IsCountablyGenerated]
     (μ : Measure α) : Tendsto (fun x => μ (Iic x)) atTop (𝓝 (μ univ)) := by
-  cases isEmpty_or_nonempty α
-  · have h1 : ∀ x : α, Iic x = ∅ := fun x => Subsingleton.elim _ _
-    have h2 : (univ : Set α) = ∅ := Subsingleton.elim _ _
-    simp_rw [h1, h2]
-    exact tendsto_const_nhds
-  have h_mono : Monotone fun x => μ (Iic x) := fun i j hij => by simp only; gcongr
-  convert tendsto_atTop_iSup h_mono
-  obtain ⟨xs, hxs_mono, hxs_tendsto⟩ := exists_seq_monotone_tendsto_atTop_atTop α
-  have h_univ : (univ : Set α) = ⋃ n, Iic (xs n) := by
-    ext1 x
-    simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]
-    obtain ⟨n, hn⟩ := tendsto_atTop_atTop.mp hxs_tendsto x
-    exact ⟨n, hn n le_rfl⟩
-  rw [h_univ, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto]
-  exact Monotone.directed_le fun i j hij => Iic_subset_Iic.mpr (hxs_mono hij)
+  rw [← iUnion_Iic]
+  exact tendsto_measure_iUnion_atTop monotone_Iic
 
-theorem tendsto_measure_Ici_atBot [SemilatticeInf α] [h : (atBot : Filter α).IsCountablyGenerated]
+theorem tendsto_measure_Ici_atBot [Preorder α] [(atBot : Filter α).IsCountablyGenerated]
     (μ : Measure α) : Tendsto (fun x => μ (Ici x)) atBot (𝓝 (μ univ)) :=
-  @tendsto_measure_Iic_atTop αᵒᵈ _ _ h μ
+  tendsto_measure_Iic_atTop (α := αᵒᵈ) μ
 
 variable [PartialOrder α] {a b : α}
 
@@ -2028,3 +1972,5 @@ theorem quasiMeasurePreserving_symm (μ : Measure α) (e : α ≃ᵐ β) :
 end MeasurableEquiv
 
 end
+
+set_option linter.style.longFile 2000
diff --git a/Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean b/Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
index 86b668b5e3a15..5dcc0f77d0243 100644
--- a/Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
+++ b/Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
@@ -378,6 +378,9 @@ theorem Measurable.aemeasurable (h : Measurable f) : AEMeasurable f μ :=
 
 namespace AEMeasurable
 
+lemma of_discrete [DiscreteMeasurableSpace α] : AEMeasurable f μ :=
+  Measurable.of_discrete.aemeasurable
+
 /-- Given an almost everywhere measurable function `f`, associate to it a measurable function
 that coincides with it almost everywhere. `f` is explicit in the definition to make sure that
 it shows in pretty-printing. -/
diff --git a/Mathlib/MeasureTheory/Measure/NullMeasurable.lean b/Mathlib/MeasureTheory/Measure/NullMeasurable.lean
index f1d91454da789..09e4f3a49eccb 100644
--- a/Mathlib/MeasureTheory/Measure/NullMeasurable.lean
+++ b/Mathlib/MeasureTheory/Measure/NullMeasurable.lean
@@ -122,6 +122,7 @@ theorem of_subsingleton [Subsingleton α] : NullMeasurableSet s μ :=
 protected theorem congr (hs : NullMeasurableSet s μ) (h : s =ᵐ[μ] t) : NullMeasurableSet t μ :=
   EventuallyMeasurableSet.congr hs h.symm
 
+@[measurability]
 protected theorem iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
     (h : ∀ i, NullMeasurableSet (s i) μ) : NullMeasurableSet (⋃ i, s i) μ :=
   MeasurableSet.iUnion h
@@ -135,6 +136,7 @@ protected theorem sUnion {s : Set (Set α)} (hs : s.Countable) (h : ∀ t ∈ s,
   rw [sUnion_eq_biUnion]
   exact MeasurableSet.biUnion hs h
 
+@[measurability]
 protected theorem iInter {ι : Sort*} [Countable ι] {f : ι → Set α}
     (h : ∀ i, NullMeasurableSet (f i) μ) : NullMeasurableSet (⋂ i, f i) μ :=
   MeasurableSet.iInter h
diff --git a/Mathlib/MeasureTheory/Measure/OpenPos.lean b/Mathlib/MeasureTheory/Measure/OpenPos.lean
index 611930fcce591..086d4961c2a0e 100644
--- a/Mathlib/MeasureTheory/Measure/OpenPos.lean
+++ b/Mathlib/MeasureTheory/Measure/OpenPos.lean
@@ -100,6 +100,11 @@ theorem _root_.IsClosed.measure_eq_one_iff_eq_univ [OpensMeasurableSpace X] [IsP
 theorem interior_eq_empty_of_null (hs : μ s = 0) : interior s = ∅ :=
   isOpen_interior.eq_empty_of_measure_zero <| measure_mono_null interior_subset hs
 
+/-- A property satisfied almost everywhere is satisfied on a dense subset. -/
+theorem dense_of_ae {p : X → Prop} (hp : ∀ᵐ x ∂μ, p x) : Dense {x | p x} := by
+  rw [dense_iff_closure_eq, closure_eq_compl_interior_compl, compl_univ_iff]
+  exact μ.interior_eq_empty_of_null hp
+
 /-- If two functions are a.e. equal on an open set and are continuous on this set, then they are
 equal on this set. -/
 theorem eqOn_open_of_ae_eq {f g : X → Y} (h : f =ᵐ[μ.restrict U] g) (hU : IsOpen U)
diff --git a/Mathlib/MeasureTheory/Measure/Portmanteau.lean b/Mathlib/MeasureTheory/Measure/Portmanteau.lean
index 37d3edc5f8472..4d82e897d1771 100644
--- a/Mathlib/MeasureTheory/Measure/Portmanteau.lean
+++ b/Mathlib/MeasureTheory/Measure/Portmanteau.lean
@@ -399,13 +399,14 @@ Assuming that for all Borel sets E whose boundary ∂E carries no probability ma
 candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
 then for all closed sets F we have the limsup condition limsup μsᵢ(F) ≤ μ(F). -/
 lemma limsup_measure_closed_le_of_forall_tendsto_measure
-    {Ω ι : Type*} {L : Filter ι} [NeBot L]
-    [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
+    {Ω ι : Type*} {L : Filter ι} [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
     {μ : Measure Ω} [IsFiniteMeasure μ] {μs : ι → Measure Ω}
     (h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 →
             Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
     (F : Set Ω) (F_closed : IsClosed F) :
     L.limsup (fun i ↦ μs i F) ≤ μ F := by
+  rcases L.eq_or_neBot with rfl | _
+  · simp only [limsup_bot, bot_eq_zero', zero_le]
   have ex := exists_null_frontiers_thickening μ F
   let rs := Classical.choose ex
   have rs_lim : Tendsto rs atTop (𝓝 0) := (Classical.choose_spec ex).1
@@ -435,7 +436,7 @@ Assuming that for all Borel sets E whose boundary ∂E carries no probability ma
 candidate limit probability measure μ we have convergence of the measures μsᵢ(E) to μ(E),
 then for all open sets G we have the limsup condition μ(G) ≤ liminf μsᵢ(G). -/
 lemma le_liminf_measure_open_of_forall_tendsto_measure
-    {Ω ι : Type*} {L : Filter ι} [NeBot L]
+    {Ω ι : Type*} {L : Filter ι}
     [MeasurableSpace Ω] [PseudoEMetricSpace Ω] [OpensMeasurableSpace Ω]
     {μ : Measure Ω} [IsProbabilityMeasure μ] {μs : ι → Measure Ω} [∀ i, IsProbabilityMeasure (μs i)]
     (h : ∀ {E}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i ↦ μs i E) L (𝓝 (μ E)))
diff --git a/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean b/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
index 62aa6c8957d30..465298ca8e9fc 100644
--- a/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
+++ b/Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
@@ -504,8 +504,8 @@ lemma map_apply' (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMea
 lemma map_apply_of_aemeasurable (ν : ProbabilityMeasure Ω) {f : Ω → Ω'}
     (f_aemble : AEMeasurable f ν) {A : Set Ω'} (A_mble : MeasurableSet A) :
     (ν.map f_aemble) A = ν (f ⁻¹' A) := by
-  have := ν.map_apply' f_aemble A_mble
-  exact (ENNReal.toNNReal_eq_toNNReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mpr this
+  exact (ENNReal.toNNReal_eq_toNNReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mpr <|
+    ν.map_apply' f_aemble A_mble
 
 lemma map_apply (ν : ProbabilityMeasure Ω) {f : Ω → Ω'} (f_aemble : AEMeasurable f ν)
     {A : Set Ω'} (A_mble : MeasurableSet A) :
diff --git a/Mathlib/MeasureTheory/Measure/Regular.lean b/Mathlib/MeasureTheory/Measure/Regular.lean
index f592f69d8eb29..362b5a971e2a4 100644
--- a/Mathlib/MeasureTheory/Measure/Regular.lean
+++ b/Mathlib/MeasureTheory/Measure/Regular.lean
@@ -301,7 +301,7 @@ class WeaklyRegular (μ : Measure α) extends OuterRegular μ : Prop where
 /-- A measure `μ` is inner regular if, for any measurable set `s`, then
 `μ(s) = sup {μ(K) | K ⊆ s compact}`. -/
 class InnerRegular (μ : Measure α) : Prop where
-  protected innerRegular : InnerRegularWRT μ IsCompact (fun s ↦ MeasurableSet s)
+  protected innerRegular : InnerRegularWRT μ IsCompact MeasurableSet
 
 /-- A measure `μ` is inner regular for finite measure sets with respect to compact sets:
 for any measurable set `s` with finite measure, then `μ(s) = sup {μ(K) | K ⊆ s compact}`.
@@ -361,7 +361,7 @@ theorem _root_.MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α
     ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε := by
   rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
   use U, hAU, hUo, hU.trans_le le_top
-  exact measure_diff_lt_of_lt_add hA hAU hA' hU
+  exact measure_diff_lt_of_lt_add hA.nullMeasurableSet hAU hA' hU
 
 protected theorem map [OpensMeasurableSpace α] [MeasurableSpace β] [TopologicalSpace β]
     [BorelSpace β] (f : α ≃ₜ β) (μ : Measure α) [OuterRegular μ] :
@@ -457,8 +457,7 @@ lemma of_restrict {μ : Measure α} {s : ℕ → Set α}
   intro F hF r hr
   have hBU : ⋃ n, F ∩ s n = F := by rw [← inter_iUnion, univ_subset_iff.mp hs, inter_univ]
   have : μ F = ⨆ n, μ (F ∩ s n) := by
-    rw [← measure_iUnion_eq_iSup, hBU]
-    exact Monotone.directed_le fun m n h ↦ inter_subset_inter_right _ (hmono h)
+    rw [← (monotone_const.inter hmono).measure_iUnion, hBU]
   rw [this] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   rw [← restrict_apply hF] at hn
@@ -509,8 +508,7 @@ lemma of_sigmaFinite [SigmaFinite μ] :
   set B : ℕ → Set α := spanningSets μ
   have hBU : ⋃ n, s ∩ B n = s := by rw [← inter_iUnion, iUnion_spanningSets, inter_univ]
   have : μ s = ⨆ n, μ (s ∩ B n) := by
-    rw [← measure_iUnion_eq_iSup, hBU]
-    exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (monotone_spanningSets μ h)
+    rw [← (monotone_const.inter (monotone_spanningSets μ)).measure_iUnion, hBU]
   rw [this] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   refine ⟨s ∩ B n, inter_subset_left, ⟨hs.inter (measurable_spanningSets μ n), ?_⟩, hn⟩
@@ -574,7 +572,7 @@ theorem weaklyRegular_of_finite [BorelSpace α] (μ : Measure α) [IsFiniteMeasu
     refine
       ⟨Uᶜ, compl_subset_compl.2 hsU, Fᶜ, compl_subset_compl.2 hFs, hUo.isClosed_compl,
         hFc.isOpen_compl, ?_⟩
-    simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and_iff]
+    simp only [measure_compl_le_add_iff, *, hUo.measurableSet, hFc.measurableSet, true_and]
   -- check for disjoint unions
   · intro s hsd hsm H ε ε0
     have ε0' : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'
@@ -617,7 +615,7 @@ theorem of_pseudoMetrizableSpace {X : Type*} [TopologicalSpace X] [PseudoMetriza
   let A : PseudoMetricSpace X := TopologicalSpace.pseudoMetrizableSpacePseudoMetric X
   intro U hU r hr
   rcases hU.exists_iUnion_isClosed with ⟨F, F_closed, -, rfl, F_mono⟩
-  rw [measure_iUnion_eq_iSup F_mono.directed_le] at hr
+  rw [F_mono.measure_iUnion] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨F n, subset_iUnion _ _, F_closed n, hn⟩
 
@@ -629,8 +627,8 @@ theorem isCompact_isClosed {X : Type*} [TopologicalSpace X] [SigmaCompactSpace X
   have hBc : ∀ n, IsCompact (F ∩ B n) := fun n => (isCompact_compactCovering X n).inter_left hF
   have hBU : ⋃ n, F ∩ B n = F := by rw [← inter_iUnion, iUnion_compactCovering, Set.inter_univ]
   have : μ F = ⨆ n, μ (F ∩ B n) := by
-    rw [← measure_iUnion_eq_iSup, hBU]
-    exact Monotone.directed_le fun m n h => inter_subset_inter_right _ (compactCovering_subset _ h)
+    rw [← Monotone.measure_iUnion, hBU]
+    exact monotone_const.inter monotone_accumulate
   rw [this] at hr
   rcases lt_iSup_iff.1 hr with ⟨n, hn⟩
   exact ⟨_, inter_subset_left, hBc n, hn⟩
@@ -667,7 +665,7 @@ lemma innerRegularWRT_isClosed_isOpen [R1Space α] [OpensMeasurableSpace α] [h
 
 theorem exists_compact_not_null [InnerRegular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne, ← measure_univ_eq_zero, MeasurableSet.univ.measure_eq_iSup_isCompact,
-    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and]
 
 /-- If `μ` is inner regular, then any measurable set can be approximated by a compact subset.
 See also `MeasurableSet.exists_isCompact_lt_add_of_ne_top`. -/
@@ -730,7 +728,7 @@ theorem _root_.MeasurableSet.exists_isCompact_diff_lt [OpensMeasurableSpace α]
     {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ K, K ⊆ A ∧ IsCompact K ∧ μ (A \ K) < ε := by
   rcases hA.exists_isCompact_lt_add h'A hε with ⟨K, hKA, hKc, hK⟩
-  exact ⟨K, hKA, hKc, measure_diff_lt_of_lt_add hKc.measurableSet hKA
+  exact ⟨K, hKA, hKc, measure_diff_lt_of_lt_add hKc.nullMeasurableSet hKA
     (ne_top_of_le_ne_top h'A <| measure_mono hKA) hK⟩
 
 /-- If `μ` is inner regular for finite measure sets with respect to compact sets,
@@ -742,7 +740,7 @@ theorem _root_.MeasurableSet.exists_isCompact_isClosed_diff_lt [BorelSpace α] [
     {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ K, K ⊆ A ∧ IsCompact K ∧ IsClosed K ∧ μ (A \ K) < ε := by
   rcases hA.exists_isCompact_isClosed_lt_add h'A hε with ⟨K, hKA, hKco, hKcl, hK⟩
-  exact ⟨K, hKA, hKco, hKcl, measure_diff_lt_of_lt_add hKcl.measurableSet hKA
+  exact ⟨K, hKA, hKco, hKcl, measure_diff_lt_of_lt_add hKcl.nullMeasurableSet hKA
     (ne_top_of_le_ne_top h'A <| measure_mono hKA) hK⟩
 
 /-- If `μ` is inner regular for finite measure sets with respect to compact sets,
@@ -826,12 +824,12 @@ protected theorem _root_.MeasurableSet.exists_isOpen_symmDiff_lt [InnerRegularCo
   rcases hs.exists_isCompact_isClosed_diff_lt hμs this with ⟨K, hKs, hKco, hKcl, hμK⟩
   rcases hKco.exists_isOpen_lt_add (μ := μ) this with ⟨U, hKU, hUo, hμU⟩
   refine ⟨U, hUo, hμU.trans_le le_top, ?_⟩
-  rw [← ENNReal.add_halves ε, measure_symmDiff_eq hUo.measurableSet hs]
+  rw [← ENNReal.add_halves ε, measure_symmDiff_eq hUo.nullMeasurableSet hs.nullMeasurableSet]
   gcongr
   · calc
       μ (U \ s) ≤ μ (U \ K) := by gcongr
       _ < ε / 2 := by
-        apply measure_diff_lt_of_lt_add hKcl.measurableSet hKU _ hμU
+        apply measure_diff_lt_of_lt_add hKcl.nullMeasurableSet hKU _ hμU
         exact ne_top_of_le_ne_top hμs (by gcongr)
   · exact lt_of_le_of_lt (by gcongr) hμK
 
@@ -918,7 +916,7 @@ theorem _root_.MeasurableSet.exists_isClosed_diff_lt [OpensMeasurableSpace α] [
     ⦃A : Set α⦄ (hA : MeasurableSet A) (h'A : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ F, F ⊆ A ∧ IsClosed F ∧ μ (A \ F) < ε := by
   rcases hA.exists_isClosed_lt_add h'A hε with ⟨F, hFA, hFc, hF⟩
-  exact ⟨F, hFA, hFc, measure_diff_lt_of_lt_add hFc.measurableSet hFA
+  exact ⟨F, hFA, hFc, measure_diff_lt_of_lt_add hFc.nullMeasurableSet hFA
     (ne_top_of_le_ne_top h'A <| measure_mono hFA) hF⟩
 
 /-- Given a weakly regular measure, any measurable set of finite mass can be approximated from
@@ -993,7 +991,7 @@ theorem _root_.IsOpen.measure_eq_iSup_isCompact ⦃U : Set α⦄ (hU : IsOpen U)
 
 theorem exists_compact_not_null [Regular μ] : (∃ K, IsCompact K ∧ μ K ≠ 0) ↔ μ ≠ 0 := by
   simp_rw [Ne, ← measure_univ_eq_zero, isOpen_univ.measure_eq_iSup_isCompact,
-    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and_iff]
+    ENNReal.iSup_eq_zero, not_forall, exists_prop, subset_univ, true_and]
 
 /-- If `μ` is a regular measure, then any measurable set of finite measure can be approximated by a
 compact subset. See also `MeasurableSet.exists_isCompact_lt_add` and
@@ -1051,8 +1049,8 @@ instance (priority := 100) {X : Type*}
     [TopologicalSpace X] [PseudoMetrizableSpace X] [SigmaCompactSpace X] [MeasurableSpace X]
     [BorelSpace X] (μ : Measure X) [SigmaFinite μ] : InnerRegular μ := by
   refine ⟨(InnerRegularWRT.isCompact_isClosed μ).trans ?_⟩
-  refine InnerRegularWRT.of_restrict (fun n ↦ ?_)
-    (univ_subset_iff.2 (iUnion_spanningSets μ)) (monotone_spanningSets μ)
+  refine InnerRegularWRT.of_restrict (fun n ↦ ?_) (iUnion_spanningSets μ).superset
+    (monotone_spanningSets μ)
   have : Fact (μ (spanningSets μ n) < ∞) := ⟨measure_spanningSets_lt_top μ n⟩
   exact WeaklyRegular.innerRegular_measurable.trans InnerRegularWRT.of_sigmaFinite
 
diff --git a/Mathlib/MeasureTheory/Measure/Restrict.lean b/Mathlib/MeasureTheory/Measure/Restrict.lean
index c38f9cc1c8cef..5e3e7702c9a7e 100644
--- a/Mathlib/MeasureTheory/Measure/Restrict.lean
+++ b/Mathlib/MeasureTheory/Measure/Restrict.lean
@@ -76,11 +76,21 @@ theorem restrict_mono' {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ ⦃μ ν :
     _ = ν.restrict s' t := (restrict_apply ht).symm
 
 /-- Restriction of a measure to a subset is monotone both in set and in measure. -/
-@[mono]
+@[mono, gcongr]
 theorem restrict_mono {_m0 : MeasurableSpace α} ⦃s s' : Set α⦄ (hs : s ⊆ s') ⦃μ ν : Measure α⦄
     (hμν : μ ≤ ν) : μ.restrict s ≤ ν.restrict s' :=
   restrict_mono' (ae_of_all _ hs) hμν
 
+@[gcongr]
+theorem restrict_mono_measure {_ : MeasurableSpace α} {μ ν : Measure α} (h : μ ≤ ν) (s : Set α) :
+    μ.restrict s ≤ ν.restrict s :=
+  restrict_mono subset_rfl h
+
+@[gcongr]
+theorem restrict_mono_set {_ : MeasurableSpace α} (μ : Measure α) {s t : Set α} (h : s ⊆ t) :
+    μ.restrict s ≤ μ.restrict t :=
+  restrict_mono h le_rfl
+
 theorem restrict_mono_ae (h : s ≤ᵐ[μ] t) : μ.restrict s ≤ μ.restrict t :=
   restrict_mono' h (le_refl μ)
 
@@ -271,7 +281,7 @@ theorem restrict_iUnion_apply [Countable ι] {s : ι → Set α} (hd : Pairwise
 theorem restrict_iUnion_apply_eq_iSup [Countable ι] {s : ι → Set α} (hd : Directed (· ⊆ ·) s)
     {t : Set α} (ht : MeasurableSet t) : μ.restrict (⋃ i, s i) t = ⨆ i, μ.restrict (s i) t := by
   simp only [restrict_apply ht, inter_iUnion]
-  rw [measure_iUnion_eq_iSup]
+  rw [Directed.measure_iUnion]
   exacts [hd.mono_comp _ fun s₁ s₂ => inter_subset_inter_right _]
 
 /-- The restriction of the pushforward measure is the pushforward of the restriction. For a version
@@ -307,11 +317,8 @@ measurable subsets of `s ∪ t`. -/
 theorem restrict_union_congr :
     μ.restrict (s ∪ t) = ν.restrict (s ∪ t) ↔
       μ.restrict s = ν.restrict s ∧ μ.restrict t = ν.restrict t := by
-  refine
-    ⟨fun h =>
-      ⟨restrict_congr_mono subset_union_left h,
-        restrict_congr_mono subset_union_right h⟩,
-      ?_⟩
+  refine ⟨fun h ↦ ⟨restrict_congr_mono subset_union_left h,
+    restrict_congr_mono subset_union_right h⟩, ?_⟩
   rintro ⟨hs, ht⟩
   ext1 u hu
   simp only [restrict_apply hu, inter_union_distrib_left]
@@ -319,14 +326,14 @@ theorem restrict_union_congr :
   calc
     μ (u ∩ s ∪ u ∩ t) = μ (US ∪ u ∩ t) :=
       measure_union_congr_of_subset hsub hμ.le Subset.rfl le_rfl
-    _ = μ US + μ ((u ∩ t) \ US) := (measure_add_diff hm _).symm
+    _ = μ US + μ ((u ∩ t) \ US) := (measure_add_diff hm.nullMeasurableSet _).symm
     _ = restrict μ s u + restrict μ t (u \ US) := by
       simp only [restrict_apply, hu, hu.diff hm, hμ, ← inter_comm t, inter_diff_assoc]
     _ = restrict ν s u + restrict ν t (u \ US) := by rw [hs, ht]
     _ = ν US + ν ((u ∩ t) \ US) := by
       simp only [restrict_apply, hu, hu.diff hm, hν, ← inter_comm t, inter_diff_assoc]
-    _ = ν (US ∪ u ∩ t) := measure_add_diff hm _
-    _ = ν (u ∩ s ∪ u ∩ t) := Eq.symm <| measure_union_congr_of_subset hsub hν.le Subset.rfl le_rfl
+    _ = ν (US ∪ u ∩ t) := measure_add_diff hm.nullMeasurableSet _
+    _ = ν (u ∩ s ∪ u ∩ t) := .symm <| measure_union_congr_of_subset hsub hν.le Subset.rfl le_rfl
 
 theorem restrict_finset_biUnion_congr {s : Finset ι} {t : ι → Set α} :
     μ.restrict (⋃ i ∈ s, t i) = ν.restrict (⋃ i ∈ s, t i) ↔
@@ -666,20 +673,6 @@ theorem ae_restrict_congr_set {s t} (hst : s =ᵐ[μ] t) {p : α → Prop} :
     (∀ᵐ x ∂μ.restrict s, p x) ↔ ∀ᵐ x ∂μ.restrict t, p x :=
   ⟨ae_restrict_of_ae_eq_of_ae_restrict hst, ae_restrict_of_ae_eq_of_ae_restrict hst.symm⟩
 
-/-- A version of the **Borel-Cantelli lemma**: if `pᵢ` is a sequence of predicates such that
-`∑ μ {x | pᵢ x}` is finite, then the measure of `x` such that `pᵢ x` holds frequently as `i → ∞` (or
-equivalently, `pᵢ x` holds for infinitely many `i`) is equal to zero. -/
-theorem measure_setOf_frequently_eq_zero {p : ℕ → α → Prop} (hp : ∑' i, μ { x | p i x } ≠ ∞) :
-    μ { x | ∃ᶠ n in atTop, p n x } = 0 := by
-  simpa only [limsup_eq_iInf_iSup_of_nat, frequently_atTop, ← bex_def, setOf_forall,
-    setOf_exists] using measure_limsup_eq_zero hp
-
-/-- A version of the **Borel-Cantelli lemma**: if `sᵢ` is a sequence of sets such that
-`∑ μ sᵢ` exists, then for almost all `x`, `x` does not belong to almost all `sᵢ`. -/
-theorem ae_eventually_not_mem {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) :
-    ∀ᵐ x ∂μ, ∀ᶠ n in atTop, x ∉ s n :=
-  measure_setOf_frequently_eq_zero hs
-
 lemma NullMeasurable.measure_preimage_eq_measure_restrict_preimage_of_ae_compl_eq_const
     {β : Type*} [MeasurableSpace β] {b : β} {f : α → β} {s : Set α}
     (f_mble : NullMeasurable f (μ.restrict s)) (hs : f =ᵐ[Measure.restrict μ sᶜ] (fun _ ↦ b))
diff --git a/Mathlib/MeasureTheory/Measure/SeparableMeasure.lean b/Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
new file mode 100644
index 0000000000000..f40b6cba3186b
--- /dev/null
+++ b/Mathlib/MeasureTheory/Measure/SeparableMeasure.lean
@@ -0,0 +1,510 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
+import Mathlib.MeasureTheory.SetAlgebra
+
+/-!
+# Separable measure
+
+The goal of this file is to give a sufficient condition on the measure space `(X, μ)` and the
+`NormedAddCommGroup E` for the space `MeasureTheory.Lp E p μ` to have `SecondCountableTopology` when
+`1 ≤ p < ∞`. To do so we define the notion of a `MeasureTheory.MeasureDense` family and a
+separable measure (`MeasureTheory.IsSeparable`).
+We prove that if `X` is `MeasurableSpace.CountablyGenerated` and `μ` is s-finite, then `μ`
+is separable. We then prove that if `μ` is separable and `E` is second-countable,
+then `Lp E p μ` is second-countable.
+
+A family `𝒜` of subsets of `X` is said to be **measure-dense** if it contains only measurable sets
+and can approximate any measurable set with finite measure, in the sense that
+for any measurable set `s` such that `μ s ≠ ∞`, `μ (s ∆ t)` can be made
+arbitrarily small when `t ∈ 𝒜`. We show below that such a family can be chosen to contain only
+sets with finite measure.
+The term "measure-dense" is justified by the fact that the approximating condition translates
+to the usual notion of density in the metric space made by constant indicators of measurable sets
+equipped with the `Lᵖ` norm.
+
+A measure `μ` is **separable** if it admits a countable and measure-dense family of sets.
+The term "separable" is justified by the fact that the definition translates to the usual notion
+of separability in the metric space made by constant indicators equipped with the `Lᵖ` norm.
+
+## Main definitions
+
+* `MeasureTheory.Measure.MeasureDense μ 𝒜`: `𝒜` is a measure-dense family if it only contains
+  measurable sets and if the following condition is satisfied: if `s` is measurable with finite
+  measure, then for any `ε > 0` there exists `t ∈ 𝒜` such that `μ (s ∆ t) < ε `.
+* `MeasureTheory.IsSeparable`: A measure is separable if there exists a countable and
+  measure-dense family.
+
+## Main statements
+
+* `MeasureTheory.instSecondCountableLp`: If `μ` is separable, `E` is second-countable and
+  `1 ≤ p < ∞` then `Lp E p μ` is second-countable. This is in particular true if `X` is countably
+  generated and `μ` is s-finite.
+
+## Implementation notes
+
+Through the whole file we consider a measurable space `X` equipped with a measure `μ`, and also
+a normed commutative group `E`. We also consider an extended non-negative real `p` such that
+`1 ≤ p < ∞`. This is registered as instances via `one_le_p : Fact (1 ≤ p)` and
+`p_ne_top : Fact (p ≠ ∞)`, so the properties are accessible via `one_le_p.elim` and `p_ne_top.elim`.
+
+Through the whole file, when we write that an extended non-negative real is finite, it is always
+written `≠ ∞` rather than `< ∞`. See `Ne.lt_top` and `ne_of_lt` to switch from one to the other.
+
+## References
+
+* [D. L. Cohn, *Measure Theory*][cohn2013measure]
+
+## Tags
+
+separable measure, measure-dense, Lp space, second-countable
+-/
+
+open MeasurableSpace Set ENNReal TopologicalSpace BigOperators symmDiff Filter Real
+
+namespace MeasureTheory
+
+variable {X E : Type*} [m : MeasurableSpace X] [NormedAddCommGroup E] {μ : Measure X}
+variable {p : ℝ≥0∞} [one_le_p : Fact (1 ≤ p)] [p_ne_top : Fact (p ≠ ∞)] {𝒜 : Set (Set X)}
+
+section MeasureDense
+
+/-! ### Definition of a measure-dense family, basic properties and sufficient conditions -/
+
+/-- A family `𝒜` of sets of a measure space is said to be measure-dense if it contains only
+measurable sets and can approximate any measurable set with finite measure, in the sense that
+for any measurable set `s` with finite measure the symmetric difference `s ∆ t` can be made
+arbitrarily small when `t ∈ 𝒜`. We show below that such a family can be chosen to contain only
+sets with finite measures.
+
+The term "measure-dense" is justified by the fact that the approximating condition translates
+to the usual notion of density in the metric space made by constant indicators of measurable sets
+equipped with the `Lᵖ` norm. -/
+structure Measure.MeasureDense (μ : Measure X) (𝒜 : Set (Set X)) : Prop where
+  /-- Each set has to be measurable. -/
+  measurable : ∀ s ∈ 𝒜, MeasurableSet s
+  /-- Any measurable set can be approximated by sets in the family. -/
+  approx : ∀ s, MeasurableSet s → μ s ≠ ∞ → ∀ ε : ℝ, 0 < ε → ∃ t ∈ 𝒜, μ (s ∆ t) < ENNReal.ofReal ε
+
+theorem Measure.MeasureDense.nonempty (h𝒜 : μ.MeasureDense 𝒜) : 𝒜.Nonempty := by
+  rcases h𝒜.approx ∅ MeasurableSet.empty (by simp) 1 (by norm_num) with ⟨t, ht, -⟩
+  exact ⟨t, ht⟩
+
+theorem Measure.MeasureDense.nonempty' (h𝒜 : μ.MeasureDense 𝒜) :
+    {s | s ∈ 𝒜 ∧ μ s ≠ ∞}.Nonempty := by
+  rcases h𝒜.approx ∅ MeasurableSet.empty (by simp) 1 (by norm_num) with ⟨t, ht, hμt⟩
+  refine ⟨t, ht, ?_⟩
+  convert ne_top_of_lt hμt
+  rw [← bot_eq_empty, bot_symmDiff]
+
+/-- The set of measurable sets is measure-dense. -/
+theorem measureDense_measurableSet : μ.MeasureDense {s | MeasurableSet s} where
+  measurable _ h := h
+  approx s hs _ ε ε_pos := ⟨s, hs, by simpa⟩
+
+/-- If a family of sets `𝒜` is measure-dense in `X`, then any measurable set with finite measure
+can be approximated by sets in `𝒜` with finite measure. -/
+lemma Measure.MeasureDense.fin_meas_approx (h𝒜 : μ.MeasureDense 𝒜) {s : Set X}
+    (ms : MeasurableSet s) (hμs : μ s ≠ ∞) (ε : ℝ) (ε_pos : 0 < ε) :
+    ∃ t ∈ 𝒜, μ t ≠ ∞ ∧ μ (s ∆ t) < ENNReal.ofReal ε := by
+  rcases h𝒜.approx s ms hμs ε ε_pos with ⟨t, t_mem, ht⟩
+  exact ⟨t, t_mem, (measure_ne_top_iff_of_symmDiff <| ne_top_of_lt ht).1 hμs, ht⟩
+
+variable (p) in
+/-- If `𝒜` is a measure-dense family of sets and `c : E`, then the set of constant indicators
+with constant `c` whose underlying set is in `𝒜` is dense in the set of constant indicators
+which are in `Lp E p μ` when `1 ≤ p < ∞`. -/
+theorem Measure.MeasureDense.indicatorConstLp_subset_closure (h𝒜 : μ.MeasureDense 𝒜) (c : E) :
+    {indicatorConstLp p hs hμs c | (s : Set X) (hs : MeasurableSet s) (hμs : μ s ≠ ∞)} ⊆
+    closure {indicatorConstLp p (h𝒜.measurable s hs) hμs c |
+      (s : Set X) (hs : s ∈ 𝒜) (hμs : μ s ≠ ∞)} := by
+  obtain rfl | hc := eq_or_ne c 0
+  · refine Subset.trans ?_ subset_closure
+    rintro - ⟨s, ms, hμs, rfl⟩
+    obtain ⟨t, ht, hμt⟩ := h𝒜.nonempty'
+    refine ⟨t, ht, hμt, ?_⟩
+    simp_rw [indicatorConstLp]
+    congr
+    simp
+  · have p_pos : 0 < p := lt_of_lt_of_le (by norm_num) one_le_p.elim
+    rintro - ⟨s, ms, hμs, rfl⟩
+    refine Metric.mem_closure_iff.2 fun ε hε ↦ ?_
+    have aux : 0 < (ε / ‖c‖) ^ p.toReal := rpow_pos_of_pos (div_pos hε (norm_pos_iff.2 hc)) _
+    obtain ⟨t, ht, hμt, hμst⟩ := h𝒜.fin_meas_approx ms hμs ((ε / ‖c‖) ^ p.toReal) aux
+    refine ⟨indicatorConstLp p (h𝒜.measurable t ht) hμt c,
+      ⟨t, ht, hμt, rfl⟩, ?_⟩
+    rw [dist_indicatorConstLp_eq_norm, norm_indicatorConstLp p_pos.ne.symm p_ne_top.elim]
+    calc
+      ‖c‖ * (μ (s ∆ t)).toReal ^ (1 / p.toReal)
+        < ‖c‖ * (ENNReal.ofReal ((ε / ‖c‖) ^ p.toReal)).toReal ^ (1 / p.toReal) := by
+          rw [_root_.mul_lt_mul_left (norm_pos_iff.2 hc)]
+          refine Real.rpow_lt_rpow (by simp) ?_
+            (one_div_pos.2 <| toReal_pos p_pos.ne.symm p_ne_top.elim)
+          rwa [toReal_lt_toReal (measure_symmDiff_ne_top hμs hμt) ofReal_ne_top]
+      _ = ε := by
+        rw [toReal_ofReal (rpow_nonneg (div_nonneg hε.le (norm_nonneg _)) _),
+          one_div, Real.rpow_rpow_inv (div_nonneg hε.le (norm_nonneg _))
+            (toReal_pos p_pos.ne.symm p_ne_top.elim).ne.symm,
+          mul_div_cancel₀ _ (norm_ne_zero_iff.2 hc)]
+
+/-- If a family of sets `𝒜` is measure-dense in `X`, then it is also the case for the sets in `𝒜`
+with finite measure. -/
+theorem Measure.MeasureDense.fin_meas (h𝒜 : μ.MeasureDense 𝒜) :
+    μ.MeasureDense {s | s ∈ 𝒜 ∧ μ s ≠ ∞} where
+  measurable s h := h𝒜.measurable s h.1
+  approx s ms hμs ε ε_pos := by
+    rcases Measure.MeasureDense.fin_meas_approx h𝒜 ms hμs ε ε_pos with ⟨t, t_mem, hμt, hμst⟩
+    exact ⟨t, ⟨t_mem, hμt⟩, hμst⟩
+
+/-- If a measurable space equipped with a finite measure is generated by an algebra of sets, then
+this algebra of sets is measure-dense. -/
+theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite [IsFiniteMeasure μ]
+    (h𝒜 : IsSetAlgebra 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) : μ.MeasureDense 𝒜 where
+  measurable s hs := hgen ▸ measurableSet_generateFrom hs
+  approx s ms := by
+    -- We want to show that any measurable set can be approximated by sets in `𝒜`. To do so, it is
+    -- enough to show that such sets constitute a `σ`-algebra containing `𝒜`. This is contained in
+    -- the theorem `generateFrom_induction`.
+    have : MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (s ∆ t)).toReal < ε := by
+      apply generateFrom_induction
+        (p := fun s ↦ MeasurableSet s ∧ ∀ (ε : ℝ), 0 < ε → ∃ t ∈ 𝒜, (μ (s ∆ t)).toReal < ε)
+        (C := 𝒜) (hs := hgen ▸ ms)
+      · -- If `t ∈ 𝒜`, then `μ (t ∆ t) = 0` which is less than any `ε > 0`.
+        exact fun t t_mem ↦ ⟨hgen ▸ measurableSet_generateFrom t_mem,
+          fun ε ε_pos ↦ ⟨t, t_mem, by simpa⟩⟩
+      · -- `∅ ∈ 𝒜` and `μ (∅ ∆ ∅) = 0` which is less than any `ε > 0`.
+        exact ⟨MeasurableSet.empty, fun ε ε_pos ↦ ⟨∅, h𝒜.empty_mem, by simpa⟩⟩
+      · -- If `s` is measurable and `t ∈ 𝒜` such that `μ (s ∆ t) < ε`, then `tᶜ ∈ 𝒜` and
+        -- `μ (sᶜ ∆ tᶜ) = μ (s ∆ t) < ε` so `sᶜ` can be approximated.
+        refine fun t ⟨mt, ht⟩ ↦ ⟨mt.compl, fun ε ε_pos ↦ ?_⟩
+        rcases ht ε ε_pos with ⟨u, u_mem, hμtcu⟩
+        exact ⟨uᶜ, h𝒜.compl_mem u_mem, by rwa [compl_symmDiff_compl]⟩
+      · -- Let `(fₙ)` be a sequence of measurable sets and `ε > 0`.
+        refine fun f hf ↦ ⟨MeasurableSet.iUnion (fun n ↦ (hf n).1), fun ε ε_pos ↦ ?_⟩
+        -- We have  `μ (⋃ n ≤ N, fₙ) ⟶ μ (⋃ n, fₙ)`.
+        have := tendsto_measure_iUnion_accumulate (μ := μ) (f := f)
+        rw [← tendsto_toReal_iff (fun _ ↦ measure_ne_top _ _) (measure_ne_top _ _)] at this
+        -- So there exists `N` such that `μ (⋃ n, fₙ) - μ (⋃ n ≤ N, fₙ) < ε/2`.
+        rcases Metric.tendsto_atTop.1 this (ε / 2) (by linarith [ε_pos]) with ⟨N, hN⟩
+        -- For any `n ≤ N` there exists `gₙ ∈ 𝒜` such that `μ (fₙ ∆ gₙ) < ε/(2*(N+1))`.
+        choose g g_mem hg using fun n ↦ (hf n).2 (ε / (2 * (N + 1))) (div_pos ε_pos (by linarith))
+        -- Therefore we have
+        -- `μ ((⋃ n, fₙ) ∆ (⋃ n ≤ N, gₙ))`
+        --   `≤ μ ((⋃ n, fₙ) ∆ (⋃ n ≤ N, fₙ)) + μ ((⋃ n ≤ N, fₙ) ∆ (⋃ n ≤ N, gₙ))`
+        --   `< ε/2 + ∑ n ≤ N, μ (fₙ ∆ gₙ)` (see `biSup_symmDiff_biSup_le`)
+        --   `< ε/2 + (N+1)*ε/(2*(N+1)) = ε/2`.
+        refine ⟨⋃ n ∈ Finset.range (N + 1), g n, h𝒜.biUnion_mem _ (fun i _ ↦ g_mem i), ?_⟩
+        calc
+          (μ ((⋃ n, f n) ∆ (⋃ n ∈ (Finset.range (N + 1)), g n))).toReal
+            ≤ (μ ((⋃ n, f n) \ ((⋃ n ∈ (Finset.range (N + 1)), f n)) ∪
+              ((⋃ n ∈ (Finset.range (N + 1)), f n) ∆
+              (⋃ n ∈ (Finset.range (N + 1)), g ↑n)))).toReal :=
+                toReal_mono (measure_ne_top _ _)
+                  (measure_mono <| symmDiff_of_ge (iUnion_subset <|
+                  fun i ↦ iUnion_subset (fun _ ↦ subset_iUnion f i)) ▸ symmDiff_triangle ..)
+          _ ≤ (μ ((⋃ n, f n) \
+              ((⋃ n ∈ (Finset.range (N + 1)), f n)))).toReal +
+              (μ ((⋃ n ∈ (Finset.range (N + 1)), f n) ∆
+              (⋃ n ∈ (Finset.range (N + 1)), g ↑n))).toReal := by
+                rw [← toReal_add (measure_ne_top _ _) (measure_ne_top _ _)]
+                exact toReal_mono (add_ne_top.2 ⟨measure_ne_top _ _, measure_ne_top _ _⟩) <|
+                  measure_union_le _ _
+          _ < ε := by
+                rw [← add_halves ε]
+                apply _root_.add_lt_add
+                · rw [measure_diff (h_fin := measure_ne_top _ _),
+                    toReal_sub_of_le (ha := measure_ne_top _ _)]
+                  · apply lt_of_le_of_lt (sub_le_dist ..)
+                    simp only [Finset.mem_range, Nat.lt_add_one_iff]
+                    exact (dist_comm (α := ℝ) .. ▸ hN N (le_refl N))
+                  · exact measure_mono <| iUnion_subset <|
+                      fun i ↦ iUnion_subset fun _ ↦ subset_iUnion f i
+                  · exact iUnion_subset <| fun i ↦ iUnion_subset (fun _ ↦ subset_iUnion f i)
+                  · exact MeasurableSet.biUnion (countable_coe_iff.1 inferInstance)
+                      (fun n _ ↦ (hf n).1.nullMeasurableSet)
+                · calc
+                    (μ ((⋃ n ∈ (Finset.range (N + 1)), f n) ∆
+                    (⋃ n ∈ (Finset.range (N + 1)), g ↑n))).toReal
+                      ≤ (μ (⋃ n ∈ (Finset.range (N + 1)), f n ∆ g n)).toReal :=
+                          toReal_mono (measure_ne_top _ _) (measure_mono biSup_symmDiff_biSup_le)
+                    _ ≤ ∑ n in (Finset.range (N + 1)), (μ (f n ∆ g n)).toReal := by
+                          rw [← toReal_sum (fun _ _ ↦ measure_ne_top _ _)]
+                          exact toReal_mono (ne_of_lt <| sum_lt_top.2 fun _ _ ↦ measure_lt_top μ _)
+                            (measure_biUnion_finset_le _ _)
+                    _ < ∑ n in (Finset.range (N + 1)), (ε / (2 * (N + 1))) :=
+                          Finset.sum_lt_sum (fun i _ ↦ le_of_lt (hg i)) ⟨0, by simp, hg 0⟩
+                    _ ≤ ε / 2 := by
+                          simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul,
+                            Nat.cast_add, Nat.cast_one]
+                          rw [div_mul_eq_div_mul_one_div, ← mul_assoc, mul_comm ((N : ℝ) + 1),
+                            mul_assoc]
+                          exact mul_le_of_le_one_right (by linarith [ε_pos]) <|
+                            le_of_eq <| mul_one_div_cancel <| Nat.cast_add_one_ne_zero _
+    rintro - ε ε_pos
+    rcases this.2 ε ε_pos with ⟨t, t_mem, hμst⟩
+    exact ⟨t, t_mem, (lt_ofReal_iff_toReal_lt (measure_ne_top _ _)).2 hμst⟩
+
+/-- If a measure space `X` is generated by an algebra of sets which contains a monotone countable
+family of sets with finite measure spanning `X` (thus the measure is `σ`-finite), then this algebra
+of sets is measure-dense. -/
+theorem Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite (h𝒜 : IsSetAlgebra 𝒜)
+    (S : μ.FiniteSpanningSetsIn 𝒜) (hgen : m = MeasurableSpace.generateFrom 𝒜) :
+    μ.MeasureDense 𝒜 where
+  measurable s hs := hgen ▸ measurableSet_generateFrom hs
+  approx s ms hμs ε ε_pos := by
+    -- We use partial unions of (Sₙ) to get a monotone family spanning `X`.
+    let T := Accumulate S.set
+    have T_mem (n) : T n ∈ 𝒜 := by
+      simpa using h𝒜.biUnion_mem {k | k ≤ n}.toFinset (fun k _ ↦ S.set_mem k)
+    have T_finite (n) : μ (T n) < ∞ := by
+      simpa using measure_biUnion_lt_top {k | k ≤ n}.toFinset.finite_toSet
+        (fun k _ ↦ S.finite k)
+    have T_spanning : ⋃ n, T n = univ := S.spanning ▸ iUnion_accumulate
+    -- We use the fact that we already know this is true for finite measures. As `⋃ n, T n = X`,
+    -- we have that `μ ((T n) ∩ s) ⟶ μ s`.
+    have mono : Monotone (fun n ↦ (T n) ∩ s) := fun m n hmn ↦ inter_subset_inter_left s
+        (biUnion_subset_biUnion_left fun k hkm ↦ Nat.le_trans hkm hmn)
+    have := tendsto_measure_iUnion_atTop (μ := μ) mono
+    rw [← tendsto_toReal_iff] at this
+    · -- We can therefore choose `N` such that `μ s - μ ((S N) ∩ s) < ε/2`.
+      rcases Metric.tendsto_atTop.1 this (ε / 2) (by linarith [ε_pos]) with ⟨N, hN⟩
+      have : Fact (μ (T N) < ∞) := Fact.mk <| T_finite N
+      -- Then we can apply the previous result to the measure `μ ((S N) ∩ •)`.
+      -- There exists `t ∈ 𝒜` such that `μ ((S N) ∩ (s ∆ t)) < ε/2`.
+      rcases (Measure.MeasureDense.of_generateFrom_isSetAlgebra_finite
+        (μ := μ.restrict (T N)) h𝒜 hgen).approx s ms
+        (ne_of_lt (lt_of_le_of_lt (μ.restrict_apply_le _ s) hμs.lt_top))
+        (ε / 2) (by linarith [ε_pos])
+        with ⟨t, t_mem, ht⟩
+      -- We can then use `t ∩ (S N)`, because `S N ∈ 𝒜` by hypothesis.
+      -- `μ (s ∆ (t ∩ S N))`
+      --   `≤ μ (s ∆ (s ∩ S N)) + μ ((s ∩ S N) ∆ (t ∩ S N))`
+      --   `= μ s - μ (s ∩ S N) + μ (s ∆ t) ∩ S N) < ε`.
+      refine ⟨t ∩ T N, h𝒜.inter_mem t_mem (T_mem N), ?_⟩
+      calc
+        μ (s ∆ (t ∩ T N))
+          ≤ μ (s \ (s ∩ T N)) + μ ((s ∆ t) ∩ T N) := by
+              rw [← symmDiff_of_le (inter_subset_left ..), symmDiff_comm _ s,
+                inter_symmDiff_distrib_right]
+              exact measure_symmDiff_le _ _ _
+        _ < ENNReal.ofReal (ε / 2) + ENNReal.ofReal (ε / 2) := by
+              apply ENNReal.add_lt_add
+              · rw [measure_diff
+                    (inter_subset_left ..)
+                    (ms.inter (hgen ▸ measurableSet_generateFrom (T_mem N))).nullMeasurableSet
+                    (ne_top_of_le_ne_top hμs (measure_mono (inter_subset_left ..))),
+                  lt_ofReal_iff_toReal_lt (sub_ne_top hμs),
+                  toReal_sub_of_le (measure_mono (inter_subset_left ..)) hμs]
+                apply lt_of_le_of_lt (sub_le_dist ..)
+                nth_rw 1 [← univ_inter s]
+                rw [inter_comm s, dist_comm, ← T_spanning, iUnion_inter _ T]
+                apply hN N (le_refl _)
+              · rwa [← μ.restrict_apply' (hgen ▸ measurableSet_generateFrom (T_mem N))]
+        _ = ENNReal.ofReal ε := by
+              rw [← ofReal_add (by linarith [ε_pos]) (by linarith [ε_pos]), add_halves]
+    · exact fun n ↦ ne_top_of_le_ne_top hμs (measure_mono (inter_subset_right ..))
+    · exact ne_top_of_le_ne_top hμs
+        (measure_mono (iUnion_subset (fun i ↦ inter_subset_right ..)))
+
+end MeasureDense
+
+section IsSeparable
+
+/-! ### Definition of a separable measure space, sufficient condition -/
+
+/-- A measure `μ` is separable if there exists a countable and measure-dense family of sets.
+
+The term "separable" is justified by the fact that the definition translates to the usual notion
+of separability in the metric space made by constant indicators equipped with the `Lᵖ` norm. -/
+class IsSeparable (μ : Measure X) : Prop where
+  exists_countable_measureDense : ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜
+
+/-- By definition, a separable measure admits a countable and measure-dense family of sets. -/
+theorem exists_countable_measureDense [IsSeparable μ] :
+    ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜 :=
+  IsSeparable.exists_countable_measureDense
+
+/-- If a measurable space is countably generated and equipped with a `σ`-finite measure, then the
+measure is separable. This is not an instance because it is used below to prove the more
+general case where `μ` is s-finite. -/
+theorem isSeparable_of_sigmaFinite [CountablyGenerated X] [SigmaFinite μ] :
+    IsSeparable μ where
+  exists_countable_measureDense := by
+    have h := countable_countableGeneratingSet (α := X)
+    have hgen := generateFrom_countableGeneratingSet (α := X)
+    let 𝒜 := (countableGeneratingSet X) ∪ {μ.toFiniteSpanningSetsIn.set n | n : ℕ}
+    have count_𝒜 : 𝒜.Countable :=
+      countable_union.2 ⟨h, countable_iff_exists_subset_range.2
+        ⟨μ.toFiniteSpanningSetsIn.set, fun _ hx ↦ hx⟩⟩
+    refine ⟨generateSetAlgebra 𝒜, countable_generateSetAlgebra count_𝒜,
+      Measure.MeasureDense.of_generateFrom_isSetAlgebra_sigmaFinite isSetAlgebra_generateSetAlgebra
+      { set := μ.toFiniteSpanningSetsIn.set
+        set_mem := fun n ↦ self_subset_generateSetAlgebra (𝒜 := 𝒜) <| Or.inr ⟨n, rfl⟩
+        finite := μ.toFiniteSpanningSetsIn.finite
+        spanning := μ.toFiniteSpanningSetsIn.spanning }
+      (le_antisymm ?_ (generateFrom_le (fun s hs ↦ ?_)))⟩
+    · rw [← hgen]
+      exact generateFrom_mono <| le_trans self_subset_generateSetAlgebra <|
+        generateSetAlgebra_mono <| subset_union_left ..
+    · induction hs with
+      | base t t_mem =>
+        rcases t_mem with t_mem | ⟨n, rfl⟩
+        · exact hgen ▸ measurableSet_generateFrom t_mem
+        · exact μ.toFiniteSpanningSetsIn.set_mem n
+      | empty => exact MeasurableSet.empty
+      | compl t _ t_mem => exact MeasurableSet.compl t_mem
+      | union t u _ _ t_mem u_mem => exact MeasurableSet.union t_mem u_mem
+
+/-- If a measurable space is countably generated and equipped with an s-finite measure, then the
+measure is separable. -/
+instance [CountablyGenerated X] [SFinite μ] : IsSeparable μ where
+  exists_countable_measureDense := by
+    have := isSeparable_of_sigmaFinite (μ := μ.restrict μ.sigmaFiniteSet)
+    rcases exists_countable_measureDense (μ := μ.restrict μ.sigmaFiniteSet) with ⟨𝒜, count_𝒜, h𝒜⟩
+    let ℬ := {s ∩ μ.sigmaFiniteSet | s ∈ 𝒜}
+    refine ⟨ℬ, count_𝒜.image (fun s ↦ s ∩ μ.sigmaFiniteSet), ?_, ?_⟩
+    · rintro - ⟨s, s_mem, rfl⟩
+      exact (h𝒜.measurable s s_mem).inter measurableSet_sigmaFiniteSet
+    · intro s ms hμs ε ε_pos
+      rcases restrict_compl_sigmaFiniteSet_eq_zero_or_top μ s with hs | hs
+      · have : (μ.restrict μ.sigmaFiniteSet) s ≠ ∞ :=
+          ne_top_of_le_ne_top hμs <| μ.restrict_le_self _
+        rcases h𝒜.approx s ms this ε ε_pos with ⟨t, t_mem, ht⟩
+        refine ⟨t ∩ μ.sigmaFiniteSet, ⟨t, t_mem, rfl⟩, ?_⟩
+        have : μ (s ∆ (t ∩ μ.sigmaFiniteSet) \ μ.sigmaFiniteSet) = 0 := by
+          rw [diff_eq_compl_inter, inter_symmDiff_distrib_left, ← ENNReal.bot_eq_zero, eq_bot_iff]
+          calc
+            μ ((μ.sigmaFiniteSetᶜ ∩ s) ∆ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet)))
+              ≤ μ ((μ.sigmaFiniteSetᶜ ∩ s) ∪ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet))) :=
+                measure_mono symmDiff_subset_union
+            _ ≤ μ (μ.sigmaFiniteSetᶜ ∩ s) + μ (μ.sigmaFiniteSetᶜ ∩ (t ∩ μ.sigmaFiniteSet)) :=
+                measure_union_le _ _
+            _ = 0 := by
+                rw [inter_comm, ← μ.restrict_apply ms, hs, ← inter_assoc, inter_comm,
+                  ← inter_assoc, inter_compl_self, empty_inter, measure_empty, zero_add]
+        rwa [← measure_inter_add_diff _ measurableSet_sigmaFiniteSet, this, add_zero,
+          inter_symmDiff_distrib_right, inter_assoc, inter_self, ← inter_symmDiff_distrib_right,
+          ← μ.restrict_apply' measurableSet_sigmaFiniteSet]
+      · refine False.elim <| hμs ?_
+        rw [eq_top_iff, ← hs]
+        exact μ.restrict_le_self _
+
+end IsSeparable
+
+section SecondCountableLp
+
+/-! ### A sufficient condition for $L^p$ spaces to be second-countable -/
+
+/-- If the measure `μ` is separable (in particular if `X` is countably generated and `μ` is
+`s`-finite), if `E` is a second-countable `NormedAddCommGroup`, and if `1 ≤ p < +∞`,
+then the associated `Lᵖ` space is second-countable. -/
+instance Lp.SecondCountableTopology [IsSeparable μ] [TopologicalSpace.SeparableSpace E] :
+    SecondCountableTopology (Lp E p μ) := by
+  -- It is enough to show that the space is separable, i.e. admits a countable and dense susbet.
+  refine @UniformSpace.secondCountable_of_separable _ _ _ ?_
+  -- There exists a countable and measure-dense family, and we can keep only the sets with finite
+  -- measure while preserving the two properties. This family is denoted `𝒜₀`.
+  rcases exists_countable_measureDense (μ := μ) with ⟨𝒜, count_𝒜, h𝒜⟩
+  have h𝒜₀ := Measure.MeasureDense.fin_meas h𝒜
+  set 𝒜₀ := {s | s ∈ 𝒜 ∧ μ s ≠ ∞}
+  have count_𝒜₀ : 𝒜₀.Countable := count_𝒜.mono fun _ ⟨h, _⟩ ↦ h
+  -- `1 ≤ p` so `p ≠ 0`, we prove it now as it is often needed.
+  have p_ne_zero : p ≠ 0 := ne_of_gt <| lt_of_lt_of_le (by norm_num) one_le_p.elim
+  -- `E` is second-countable, therefore separable and admits a countable and dense subset `u`.
+  rcases exists_countable_dense E with ⟨u, countable_u, dense_u⟩
+  -- The countable and dense subset of `Lᵖ` we are going to build is the set of finite sums of
+  -- constant indicators with support in `𝒜₀`, and is denoted `D`. To make manipulations easier,
+  -- we define the function `key` which given an integer `n` and two families of `n` elements
+  -- in `u` and `𝒜₀` associates the corresponding sum.
+  let key (n : ℕ) (d : Fin n → u) (s : Fin n → 𝒜₀) : (Lp E p μ) :=
+    ∑ i, indicatorConstLp p (h𝒜₀.measurable (s i) (Subtype.mem (s i))) (s i).2.2 (d i : E)
+  let D := {s : Lp E p μ | ∃ n d t, s = key n d t}
+  refine ⟨D, ?_, ?_⟩
+  · -- Countability directly follows from countability of `u` and `𝒜₀`. The function `f` below
+    -- is the uncurryfied version of `key`, which is easier to manipulate as countability of the
+    -- domain is automatically infered.
+    let f (nds : Σ n : ℕ, (Fin n → u) × (Fin n → 𝒜₀)) : Lp E p μ := key nds.1 nds.2.1 nds.2.2
+    have := count_𝒜₀.to_subtype
+    have := countable_u.to_subtype
+    have : D ⊆ range f := by
+      rintro - ⟨n, d, s, rfl⟩
+      use ⟨n, (d, s)⟩
+    exact (countable_range f).mono this
+  · -- Let's turn to the density. Thanks to the density of simple functions in `Lᵖ`, it is enough
+    -- to show that the closure of `D` contains constant indicators which are in `Lᵖ` (i. e. the
+    -- set has finite measure), is closed by sum and closed.
+    -- This is given by `Lp.induction`.
+    refine Lp.induction p_ne_top.elim (P := fun f ↦ f ∈ closure D) ?_ ?_ isClosed_closure
+    · intro a s ms hμs
+      -- We want to approximate `a • 𝟙ₛ`.
+      apply ne_of_lt at hμs
+      rw [SeminormedAddCommGroup.mem_closure_iff]
+      intro ε ε_pos
+      have μs_pow_nonneg : 0 ≤ (μ s).toReal ^ (1 / p.toReal) :=
+        Real.rpow_nonneg ENNReal.toReal_nonneg _
+      -- To do so, we first pick `b ∈ u` such that `‖a - b‖ < ε / (3 * (1 + (μ s)^(1/p)))`.
+      have approx_a_pos : 0 < ε / (3 * (1 + (μ s).toReal ^ (1 / p.toReal))) :=
+        div_pos ε_pos (by linarith [μs_pow_nonneg])
+      have ⟨b, b_mem, hb⟩ := SeminormedAddCommGroup.mem_closure_iff.1 (dense_u a) _ approx_a_pos
+      -- Then we pick `t ∈ 𝒜₀` such that `‖b • 𝟙ₛ - b • 𝟙ₜ‖ < ε / 3`.
+      rcases SeminormedAddCommGroup.mem_closure_iff.1
+        (h𝒜₀.indicatorConstLp_subset_closure p b ⟨s, ms, hμs, rfl⟩)
+          (ε / 3) (by linarith [ε_pos]) with ⟨-, ⟨t, ht, hμt, rfl⟩, hst⟩
+      have mt := h𝒜₀.measurable t ht
+      -- We now show that `‖a • 𝟙ₛ - b • 𝟙ₜ‖ₚ < ε`, as follows:
+      -- `‖a • 𝟙ₛ - b • 𝟙ₜ‖ₚ`
+      --   `= ‖a • 𝟙ₛ - b • 𝟙ₛ + b • 𝟙ₛ - b • 𝟙ₜ‖ₚ`
+      --   `≤ ‖a - b‖ * ‖𝟙ₛ‖ₚ + ε / 3`
+      --   `= ‖a - b‖ * (μ s)^(1/p) + ε / 3`
+      --   `< ε * (μ s)^(1/p) / (3 * (1 + (μ s)^(1/p))) + ε / 3`
+      --   `≤ ε / 3 + ε / 3 < ε`.
+      refine ⟨indicatorConstLp p mt hμt b,
+        ⟨1, fun _ ↦ ⟨b, b_mem⟩, fun _ ↦ ⟨t, ht⟩, by simp [key]⟩, ?_⟩
+      rw [Lp.simpleFunc.coe_indicatorConst,
+        ← sub_add_sub_cancel _ (indicatorConstLp p ms hμs b), ← add_halves ε]
+      refine lt_of_le_of_lt (b := ε / 3 + ε / 3) (norm_add_le_of_le ?_ hst.le) (by linarith [ε_pos])
+      rw [indicatorConstLp_sub, norm_indicatorConstLp p_ne_zero p_ne_top.elim]
+      calc
+        ‖a - b‖ * (μ s).toReal ^ (1 / p.toReal)
+          ≤ (ε / (3 * (1 + (μ s).toReal ^ (1 / p.toReal)))) * (μ s).toReal ^ (1 / p.toReal) :=
+              mul_le_mul_of_nonneg_right (le_of_lt hb) μs_pow_nonneg
+        _ ≤ ε / 3 := by
+            rw [← mul_one (ε / 3), div_mul_eq_div_mul_one_div, mul_assoc, one_div_mul_eq_div]
+            exact mul_le_mul_of_nonneg_left
+              ((div_le_one (by linarith [μs_pow_nonneg])).2 (by linarith))
+              (by linarith [ε_pos])
+    · -- Now we have to show that the closure of `D` is closed by sum. Let `f` and `g` be two
+      -- functions in `Lᵖ` which are also in the closure of `D`.
+      rintro f g hf hg - f_mem g_mem
+      rw [SeminormedAddCommGroup.mem_closure_iff] at *
+      intro ε ε_pos
+      -- For `ε > 0`, there exists `bf, bg ∈ D` such that `‖f - bf‖ₚ < ε/2` and `‖g - bg‖ₚ < ε/2`.
+      rcases f_mem (ε / 2) (by linarith [ε_pos]) with ⟨bf, ⟨nf, df, sf, bf_eq⟩, hbf⟩
+      rcases g_mem (ε / 2) (by linarith [ε_pos]) with ⟨bg, ⟨ng, dg, sg, bg_eq⟩, hbg⟩
+      -- It is obvious that `D` is closed by sum, it suffices to concatenate the family of
+      -- elements of `u` and the family of elements of `𝒜₀`.
+      let d := fun i : Fin (nf + ng) ↦ if h : i < nf
+        then df (Fin.castLT i h)
+        else dg (Fin.subNat nf (Fin.cast (Nat.add_comm ..) i) (le_of_not_gt h))
+      let s := fun i : Fin (nf + ng) ↦ if h : i < nf
+        then sf (Fin.castLT i h)
+        else sg (Fin.subNat nf (Fin.cast (Nat.add_comm ..) i) (le_of_not_gt h))
+      -- So we can use `bf + bg`.
+      refine ⟨bf + bg, ⟨nf + ng, d, s, ?_⟩, ?_⟩
+      · simp [key, d, s, Fin.sum_univ_add, bf_eq, bg_eq]
+      · -- We have
+        -- `‖f + g - (bf + bg)‖ₚ`
+        --   `≤ ‖f - bf‖ₚ + ‖g - bg‖ₚ`
+        --   `< ε/2 + ε/2 = ε`.
+        calc
+          ‖Memℒp.toLp f hf + Memℒp.toLp g hg - (bf + bg)‖
+            = ‖(Memℒp.toLp f hf) - bf + ((Memℒp.toLp g hg) - bg)‖ := by congr; abel
+          _ ≤ ‖(Memℒp.toLp f hf) - bf‖ + ‖(Memℒp.toLp g hg) - bg‖ := norm_add_le ..
+          _ < ε := by linarith [hbf, hbg]
+
+end SecondCountableLp
+
+end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Measure/Stieltjes.lean b/Mathlib/MeasureTheory/Measure/Stieltjes.lean
index 7399a3bc6bab9..841b773d2cd9d 100644
--- a/Mathlib/MeasureTheory/Measure/Stieltjes.lean
+++ b/Mathlib/MeasureTheory/Measure/Stieltjes.lean
@@ -211,7 +211,7 @@ theorem length_subadditive_Icc_Ioo {a b : ℝ} {c d : ℕ → ℝ} (ss : Icc a b
       ⟨s, _, hf, hs⟩
     have e : ⋃ i ∈ (hf.toFinset : Set ℕ), Ioo (c i) (d i) = ⋃ i ∈ s, Ioo (c i) (d i) := by
       simp only [Set.ext_iff, exists_prop, Finset.set_biUnion_coe, mem_iUnion, forall_const,
-        iff_self_iff, Finite.mem_toFinset]
+        Finite.mem_toFinset]
     rw [ENNReal.tsum_eq_iSup_sum]
     refine le_trans ?_ (le_iSup _ hf.toFinset)
     exact this hf.toFinset _ (by simpa only [e] )
@@ -312,7 +312,7 @@ theorem measurableSet_Ioi {c : ℝ} : MeasurableSet[f.outer.caratheodory] (Ioi c
       sub_add_sub_cancel, le_refl,
       max_eq_right]
   · simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_diff_Ioi, f.length_empty,
-      zero_add, or_true_iff, le_sup_iff, f.length_Ioc, not_lt]
+      zero_add, or_true, le_sup_iff, f.length_Ioc, not_lt]
   · simp only [hac, hbc, Ioc_inter_Ioi, Ioc_diff_Ioi, f.length_Ioc, min_eq_right, _root_.sup_eq_max,
       le_refl, Ioc_eq_empty, add_zero, max_eq_left, f.length_empty, not_lt]
 
@@ -373,8 +373,9 @@ theorem measure_singleton (a : ℝ) : f.measure {a} = ofReal (f a - leftLim f a)
     simp [le_antisymm this (hx 0).2]
   have L1 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a})) := by
     rw [A]
-    refine tendsto_measure_iInter (fun n => measurableSet_Ioc) (fun m n hmn => ?_) ?_
-    · exact Ioc_subset_Ioc (u_mono.monotone hmn) le_rfl
+    refine tendsto_measure_iInter (fun n => nullMeasurableSet_Ioc)
+      (fun m n hmn => ?_) ?_
+    · exact Ioc_subset_Ioc_left (u_mono.monotone hmn)
     · exact ⟨0, by simpa only [measure_Ioc] using ENNReal.ofReal_ne_top⟩
   have L2 :
       Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (ofReal (f a - leftLim f a))) := by
diff --git a/Mathlib/MeasureTheory/Measure/Typeclasses.lean b/Mathlib/MeasureTheory/Measure/Typeclasses.lean
index c6eb904a1c082..21a35d7e02dd3 100644
--- a/Mathlib/MeasureTheory/Measure/Typeclasses.lean
+++ b/Mathlib/MeasureTheory/Measure/Typeclasses.lean
@@ -46,13 +46,14 @@ instance Restrict.isFiniteMeasure (μ : Measure α) [hs : Fact (μ s < ∞)] :
     IsFiniteMeasure (μ.restrict s) :=
   ⟨by simpa using hs.elim⟩
 
+@[simp]
 theorem measure_lt_top (μ : Measure α) [IsFiniteMeasure μ] (s : Set α) : μ s < ∞ :=
   (measure_mono (subset_univ s)).trans_lt IsFiniteMeasure.measure_univ_lt_top
 
 instance isFiniteMeasureRestrict (μ : Measure α) (s : Set α) [h : IsFiniteMeasure μ] :
-    IsFiniteMeasure (μ.restrict s) :=
-  ⟨by simpa using measure_lt_top μ s⟩
+    IsFiniteMeasure (μ.restrict s) := ⟨by simp⟩
 
+@[simp]
 theorem measure_ne_top (μ : Measure α) [IsFiniteMeasure μ] (s : Set α) : μ s ≠ ∞ :=
   ne_of_lt (measure_lt_top μ s)
 
@@ -143,13 +144,9 @@ theorem summable_measure_toReal [hμ : IsFiniteMeasure μ] {f : ℕ → Set α}
   exact ne_of_lt (measure_lt_top _ _)
 
 theorem ae_eq_univ_iff_measure_eq [IsFiniteMeasure μ] (hs : NullMeasurableSet s μ) :
-    s =ᵐ[μ] univ ↔ μ s = μ univ := by
-  refine ⟨measure_congr, fun h => ?_⟩
-  obtain ⟨t, -, ht₁, ht₂⟩ := hs.exists_measurable_subset_ae_eq
-  exact
-    ht₂.symm.trans
-      (ae_eq_of_subset_of_measure_ge (subset_univ t) (Eq.le ((measure_congr ht₂).trans h).symm) ht₁
-        (measure_ne_top μ univ))
+    s =ᵐ[μ] univ ↔ μ s = μ univ :=
+  ⟨measure_congr, fun h ↦
+    ae_eq_of_subset_of_measure_ge (subset_univ _) h.ge hs (measure_ne_top _ _)⟩
 
 theorem ae_iff_measure_eq [IsFiniteMeasure μ] {p : α → Prop}
     (hp : NullMeasurableSet { a | p a } μ) : (∀ᵐ a ∂μ, p a) ↔ μ { a | p a } = μ univ := by
@@ -161,7 +158,7 @@ theorem ae_mem_iff_measure_eq [IsFiniteMeasure μ] {s : Set α} (hs : NullMeasur
 
 lemma tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint
     {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ]
-    {Es : ℕ → Set X} (Es_mble : ∀ i, MeasurableSet (Es i))
+    {Es : ℕ → Set X} (Es_mble : ∀ i, NullMeasurableSet (Es i) μ)
     (Es_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)) :
     Tendsto (μ ∘ fun n ↦ ⋃ i ≥ n, Es i) atTop (𝓝 0) := by
   have decr : Antitone fun n ↦ ⋃ i ≥ n, Es i :=
@@ -174,15 +171,16 @@ lemma tendsto_measure_biUnion_Ici_zero_of_pairwise_disjoint
     obtain ⟨k, k_gt_j, x_in_Es_k⟩ := hx (j+1)
     have oops := (Es_disj (Nat.ne_of_lt k_gt_j)).ne_of_mem x_in_Es_j x_in_Es_k
     contradiction
-  have key :=
-    tendsto_measure_iInter (μ := μ) (fun n ↦ by measurability) decr ⟨0, measure_ne_top _ _⟩
+  -- TODO: `by measurability` fails
+  have key := tendsto_measure_iInter (μ := μ) (fun n ↦ .iUnion fun _ ↦ .iUnion fun _ ↦ Es_mble _)
+    decr ⟨0, measure_ne_top _ _⟩
   simp only [nothing, measure_empty] at key
   convert key
 
 open scoped symmDiff
 
 theorem abs_toReal_measure_sub_le_measure_symmDiff'
-    (hs : MeasurableSet s) (ht : MeasurableSet t) (hs' : μ s ≠ ∞) (ht' : μ t ≠ ∞) :
+    (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) (hs' : μ s ≠ ∞) (ht' : μ t ≠ ∞) :
     |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal := by
   have hst : μ (s \ t) ≠ ∞ := (measure_lt_top_of_subset diff_subset hs').ne
   have hts : μ (t \ s) ≠ ∞ := (measure_lt_top_of_subset diff_subset ht').ne
@@ -196,15 +194,18 @@ theorem abs_toReal_measure_sub_le_measure_symmDiff'
   abel
 
 theorem abs_toReal_measure_sub_le_measure_symmDiff [IsFiniteMeasure μ]
-    (hs : MeasurableSet s) (ht : MeasurableSet t) :
+    (hs : NullMeasurableSet s μ) (ht : NullMeasurableSet t μ) :
     |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal :=
   abs_toReal_measure_sub_le_measure_symmDiff' hs ht (measure_ne_top μ s) (measure_ne_top μ t)
 
 instance {s : Finset ι} {μ : ι → Measure α} [∀ i, IsFiniteMeasure (μ i)] :
     IsFiniteMeasure (∑ i ∈ s, μ i) where measure_univ_lt_top := by simp [measure_lt_top]
 
-instance [Fintype ι] {μ : ι → Measure α} [∀ i, IsFiniteMeasure (μ i)] :
-    IsFiniteMeasure (.sum μ) where measure_univ_lt_top := by simp [measure_lt_top]
+instance [Finite ι] {μ : ι → Measure α} [∀ i, IsFiniteMeasure (μ i)] :
+    IsFiniteMeasure (.sum μ) where
+  measure_univ_lt_top := by
+    cases nonempty_fintype ι
+    simp [measure_lt_top]
 
 end IsFiniteMeasure
 
@@ -1104,13 +1105,11 @@ instance SMul.sigmaFinite {μ : Measure α} [SigmaFinite μ] (c : ℝ≥0) :
 instance [SigmaFinite (μ.restrict s)] [SigmaFinite (μ.restrict t)] :
     SigmaFinite (μ.restrict (s ∪ t)) := sigmaFinite_of_le _ (restrict_union_le _ _)
 
-instance [h : SigmaFinite (μ.restrict s)] : SigmaFinite (μ.restrict (s ∩ t)) := by
-  convert sigmaFinite_of_le _ (restrict_mono_ae (ae_of_all _ Set.inter_subset_left))
-  exact h
+instance [SigmaFinite (μ.restrict s)] : SigmaFinite (μ.restrict (s ∩ t)) :=
+  sigmaFinite_of_le (μ.restrict s) (restrict_mono_ae (ae_of_all _ Set.inter_subset_left))
 
-instance [h : SigmaFinite (μ.restrict t)] : SigmaFinite (μ.restrict (s ∩ t)) := by
-  convert sigmaFinite_of_le _ (restrict_mono_ae (ae_of_all _ Set.inter_subset_right))
-  exact h
+instance [SigmaFinite (μ.restrict t)] : SigmaFinite (μ.restrict (s ∩ t)) :=
+  sigmaFinite_of_le (μ.restrict t) (restrict_mono_ae (ae_of_all _ Set.inter_subset_right))
 
 theorem SigmaFinite.of_map (μ : Measure α) {f : α → β} (hf : AEMeasurable f μ)
     (h : SigmaFinite (μ.map f)) : SigmaFinite μ :=
@@ -1542,3 +1541,5 @@ theorem measure_Ioo_lt_top : μ (Ioo a b) < ∞ :=
   (measure_mono Ioo_subset_Icc_self).trans_lt measure_Icc_lt_top
 
 end MeasureIxx
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/MeasureTheory/Measure/VectorMeasure.lean b/Mathlib/MeasureTheory/Measure/VectorMeasure.lean
index d9b3f84d8bdb6..50bf902e8d3cf 100644
--- a/Mathlib/MeasureTheory/Measure/VectorMeasure.lean
+++ b/Mathlib/MeasureTheory/Measure/VectorMeasure.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 -/
 import Mathlib.MeasureTheory.Measure.Typeclasses
-import Mathlib.Analysis.Complex.Basic
+import Mathlib.Topology.Algebra.InfiniteSum.Module
 
 /-!
 
@@ -13,7 +13,8 @@ import Mathlib.Analysis.Complex.Basic
 This file defines vector valued measures, which are σ-additive functions from a set to an add monoid
 `M` such that it maps the empty set and non-measurable sets to zero. In the case
 that `M = ℝ`, we called the vector measure a signed measure and write `SignedMeasure α`.
-Similarly, when `M = ℂ`, we call the measure a complex measure and write `ComplexMeasure α`.
+Similarly, when `M = ℂ`, we call the measure a complex measure and write `ComplexMeasure α`
+(defined in `MeasureTheory/Measure/Complex`).
 
 ## Main definitions
 
@@ -63,10 +64,6 @@ structure VectorMeasure (α : Type*) [MeasurableSpace α] (M : Type*) [AddCommMo
 abbrev SignedMeasure (α : Type*) [MeasurableSpace α] :=
   VectorMeasure α ℝ
 
-/-- A `ComplexMeasure` is a `ℂ`-vector measure. -/
-abbrev ComplexMeasure (α : Type*) [MeasurableSpace α] :=
-  VectorMeasure α ℂ
-
 open Set MeasureTheory
 
 namespace VectorMeasure
@@ -93,11 +90,6 @@ theorem m_iUnion (v : VectorMeasure α M) {f : ℕ → Set α} (hf₁ : ∀ i, M
     (hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) :=
   v.m_iUnion' hf₁ hf₂
 
-theorem of_disjoint_iUnion_nat [T2Space M] (v : VectorMeasure α M) {f : ℕ → Set α}
-    (hf₁ : ∀ i, MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) :
-    v (⋃ i, f i) = ∑' i, v (f i) :=
-  (v.m_iUnion hf₁ hf₂).tsum_eq.symm
-
 theorem coe_injective : @Function.Injective (VectorMeasure α M) (Set α → M) (⇑) := fun v w h => by
   cases v
   cases w
@@ -120,45 +112,29 @@ theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, Measurabl
 theorem ext {s t : VectorMeasure α M} (h : ∀ i : Set α, MeasurableSet i → s i = t i) : s = t :=
   (ext_iff s t).2 h
 
-variable [T2Space M] {v : VectorMeasure α M} {f : ℕ → Set α}
+variable [Countable β] {v : VectorMeasure α M} {f : β → Set α}
 
-theorem hasSum_of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
-    (hf₂ : Pairwise (Disjoint on f)) : HasSum (fun i => v (f i)) (v (⋃ i, f i)) := by
-  cases nonempty_encodable β
-  set g := fun i : ℕ => ⋃ (b : β) (_ : b ∈ Encodable.decode₂ β i), f b with hg
-  have hg₁ : ∀ i, MeasurableSet (g i) :=
-    fun _ => MeasurableSet.iUnion fun b => MeasurableSet.iUnion fun _ => hf₁ b
-  have hg₂ : Pairwise (Disjoint on g) := Encodable.iUnion_decode₂_disjoint_on hf₂
-  have := v.of_disjoint_iUnion_nat hg₁ hg₂
-  rw [hg, Encodable.iUnion_decode₂] at this
-  have hg₃ : (fun i : β => v (f i)) = fun i => v (g (Encodable.encode i)) := by
-    ext x
-    rw [hg]
-    simp only
-    congr
-    ext y
-    simp only [exists_prop, Set.mem_iUnion, Option.mem_def]
-    constructor
-    · intro hy
-      exact ⟨x, (Encodable.decode₂_is_partial_inv _ _).2 rfl, hy⟩
-    · rintro ⟨b, hb₁, hb₂⟩
-      rw [Encodable.decode₂_is_partial_inv _ _] at hb₁
-      rwa [← Encodable.encode_injective hb₁]
-  rw [Summable.hasSum_iff, this, ← tsum_iUnion_decode₂]
-  · exact v.empty
-  · rw [hg₃]
-    change Summable ((fun i => v (g i)) ∘ Encodable.encode)
-    rw [Function.Injective.summable_iff Encodable.encode_injective]
-    · exact (v.m_iUnion hg₁ hg₂).summable
-    · intro x hx
-      convert v.empty
-      simp only [g, Set.iUnion_eq_empty, Option.mem_def, not_exists, Set.mem_range] at hx ⊢
-      intro i hi
-      exact False.elim ((hx i) ((Encodable.decode₂_is_partial_inv _ _).1 hi))
-
-theorem of_disjoint_iUnion [Countable β] {f : β → Set α} (hf₁ : ∀ i, MeasurableSet (f i))
-    (hf₂ : Pairwise (Disjoint on f)) : v (⋃ i, f i) = ∑' i, v (f i) :=
-  (hasSum_of_disjoint_iUnion hf₁ hf₂).tsum_eq.symm
+theorem hasSum_of_disjoint_iUnion (hm : ∀ i, MeasurableSet (f i)) (hd : Pairwise (Disjoint on f)) :
+    HasSum (fun i => v (f i)) (v (⋃ i, f i)) := by
+  rcases Countable.exists_injective_nat β with ⟨e, he⟩
+  rw [← hasSum_extend_zero he]
+  convert m_iUnion v (f := Function.extend e f fun _ ↦ ∅) _ _
+  · simp only [Pi.zero_def, Function.apply_extend v, Function.comp_def, empty]
+  · exact (iSup_extend_bot he _).symm
+  · simp [Function.apply_extend MeasurableSet, Function.comp_def, hm]
+  · exact hd.disjoint_extend_bot (he.factorsThrough _)
+
+variable [T2Space M]
+
+theorem of_disjoint_iUnion (hm : ∀ i, MeasurableSet (f i)) (hd : Pairwise (Disjoint on f)) :
+    v (⋃ i, f i) = ∑' i, v (f i) :=
+  (hasSum_of_disjoint_iUnion hm hd).tsum_eq.symm
+
+@[deprecated of_disjoint_iUnion (since := "2024-09-15")]
+theorem of_disjoint_iUnion_nat (v : VectorMeasure α M) {f : ℕ → Set α}
+    (hf₁ : ∀ i, MeasurableSet (f i)) (hf₂ : Pairwise (Disjoint on f)) :
+    v (⋃ i, f i) = ∑' i, v (f i) :=
+  of_disjoint_iUnion hf₁ hf₂
 
 theorem of_union {A B : Set α} (h : Disjoint A B) (hA : MeasurableSet A) (hB : MeasurableSet B) :
     v (A ∪ B) = v A + v B := by
@@ -195,12 +171,12 @@ theorem of_diff_of_diff_eq_zero {A B : Set α} (hA : MeasurableSet A) (hB : Meas
 theorem of_iUnion_nonneg {M : Type*} [TopologicalSpace M] [OrderedAddCommMonoid M]
     [OrderClosedTopology M] {v : VectorMeasure α M} (hf₁ : ∀ i, MeasurableSet (f i))
     (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ i, 0 ≤ v (f i)) : 0 ≤ v (⋃ i, f i) :=
-  (v.of_disjoint_iUnion_nat hf₁ hf₂).symm ▸ tsum_nonneg hf₃
+  (v.of_disjoint_iUnion hf₁ hf₂).symm ▸ tsum_nonneg hf₃
 
 theorem of_iUnion_nonpos {M : Type*} [TopologicalSpace M] [OrderedAddCommMonoid M]
     [OrderClosedTopology M] {v : VectorMeasure α M} (hf₁ : ∀ i, MeasurableSet (f i))
     (hf₂ : Pairwise (Disjoint on f)) (hf₃ : ∀ i, v (f i) ≤ 0) : v (⋃ i, f i) ≤ 0 :=
-  (v.of_disjoint_iUnion_nat hf₁ hf₂).symm ▸ tsum_nonpos hf₃
+  (v.of_disjoint_iUnion hf₁ hf₂).symm ▸ tsum_nonpos hf₃
 
 theorem of_nonneg_disjoint_union_eq_zero {s : SignedMeasure α} {A B : Set α} (h : Disjoint A B)
     (hA₁ : MeasurableSet A) (hB₁ : MeasurableSet B) (hA₂ : 0 ≤ s A) (hB₂ : 0 ≤ s B)
@@ -445,7 +421,7 @@ section
 
 /-- A vector measure over `ℝ≥0∞` is a measure. -/
 def ennrealToMeasure {_ : MeasurableSpace α} (v : VectorMeasure α ℝ≥0∞) : Measure α :=
-  ofMeasurable (fun s _ => v s) v.empty fun _ hf₁ hf₂ => v.of_disjoint_iUnion_nat hf₁ hf₂
+  ofMeasurable (fun s _ => v s) v.empty fun _ hf₁ hf₂ => v.of_disjoint_iUnion hf₁ hf₂
 
 theorem ennrealToMeasure_apply {m : MeasurableSpace α} {v : VectorMeasure α ℝ≥0∞} {s : Set α}
     (hs : MeasurableSet s) : ennrealToMeasure v s = v s := by
@@ -815,7 +791,7 @@ theorem restrict_le_restrict_iUnion {f : ℕ → Set α} (hf₁ : ∀ n, Measura
     rwa [← Set.inter_iUnion, iUnion_disjointed, Set.inter_eq_left]
   have ha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n) :=
     (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right
-  rw [← ha₃, v.of_disjoint_iUnion_nat _ ha₄, w.of_disjoint_iUnion_nat _ ha₄]
+  rw [← ha₃, v.of_disjoint_iUnion _ ha₄, w.of_disjoint_iUnion _ ha₄]
   · refine tsum_le_tsum (fun n => (restrict_le_restrict_iff v w (hf₁ n)).1 (hf₂ n) ?_ ?_) ?_ ?_
     · exact ha₁.inter (MeasurableSet.disjointed hf₁ n)
     · exact Set.Subset.trans Set.inter_subset_right (disjointed_subset _ _)
@@ -1162,7 +1138,7 @@ def toMeasureOfZeroLE (s : SignedMeasure α) (i : Set α) (hi₁ : MeasurableSet
       intro n m hnm
       exact ((hf₂ hnm).inf_left' i).inf_right' i
     simp only [toMeasureOfZeroLE', s.restrict_apply hi₁ (MeasurableSet.iUnion hf₁), Set.inter_comm,
-      Set.inter_iUnion, s.of_disjoint_iUnion_nat h₁ h₂, ENNReal.some_eq_coe, id]
+      Set.inter_iUnion, s.of_disjoint_iUnion h₁ h₂, ENNReal.some_eq_coe, id]
     have h : ∀ n, 0 ≤ s (i ∩ f n) := fun n =>
       s.nonneg_of_zero_le_restrict (s.zero_le_restrict_subset hi₁ Set.inter_subset_left hi₂)
     rw [NNReal.coe_tsum_of_nonneg h, ENNReal.coe_tsum]
diff --git a/Mathlib/MeasureTheory/Measure/WithDensity.lean b/Mathlib/MeasureTheory/Measure/WithDensity.lean
index 2822135c290aa..4af7da3eaff23 100644
--- a/Mathlib/MeasureTheory/Measure/WithDensity.lean
+++ b/Mathlib/MeasureTheory/Measure/WithDensity.lean
@@ -235,14 +235,14 @@ theorem withDensity_apply_eq_zero' {f : α → ℝ≥0∞} {s : Set α} (hf : AE
     simp only [Pi.zero_apply, mem_setOf_eq, Filter.mem_mk] at A
     convert A using 2
     ext x
-    simp only [and_comm, exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq,
+    simp only [and_comm, exists_prop, mem_inter_iff, mem_setOf_eq,
       mem_compl_iff, not_forall]
   · intro hs
     let t := toMeasurable μ ({ x | f x ≠ 0 } ∩ s)
     have A : s ⊆ t ∪ { x | f x = 0 } := by
       intro x hx
       rcases eq_or_ne (f x) 0 with (fx | fx)
-      · simp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true_iff]
+      · simp only [fx, mem_union, mem_setOf_eq, eq_self_iff_true, or_true]
       · left
         apply subset_toMeasurable _ _
         exact ⟨fx, hx⟩
@@ -271,7 +271,7 @@ theorem ae_withDensity_iff' {p : α → Prop} {f : α → ℝ≥0∞} (hf : AEMe
   rw [ae_iff, ae_iff, withDensity_apply_eq_zero' hf, iff_iff_eq]
   congr
   ext x
-  simp only [exists_prop, mem_inter_iff, iff_self_iff, mem_setOf_eq, not_forall]
+  simp only [exists_prop, mem_inter_iff, mem_setOf_eq, not_forall]
 
 theorem ae_withDensity_iff {p : α → Prop} {f : α → ℝ≥0∞} (hf : Measurable f) :
     (∀ᵐ x ∂μ.withDensity f, p x) ↔ ∀ᵐ x ∂μ, f x ≠ 0 → p x :=
diff --git a/Mathlib/MeasureTheory/Order/UpperLower.lean b/Mathlib/MeasureTheory/Order/UpperLower.lean
index 5e56bec648560..6e04d112e9d58 100644
--- a/Mathlib/MeasureTheory/Order/UpperLower.lean
+++ b/Mathlib/MeasureTheory/Order/UpperLower.lean
@@ -116,7 +116,7 @@ private lemma aux₁
       mul_div_mul_left _ _ (two_ne_zero' ℝ), div_right_comm, div_self, one_div]
   all_goals try positivity
   · simp_all
-  · measurability
+  · exact measurableSet_closedBall.nullMeasurableSet
 
 theorem IsUpperSet.null_frontier (hs : IsUpperSet s) : volume (frontier s) = 0 := by
   refine measure_mono_null (fun x hx ↦ ?_)
diff --git a/Mathlib/MeasureTheory/OuterMeasure/BorelCantelli.lean b/Mathlib/MeasureTheory/OuterMeasure/BorelCantelli.lean
new file mode 100644
index 0000000000000..e3e8075a43882
--- /dev/null
+++ b/Mathlib/MeasureTheory/OuterMeasure/BorelCantelli.lean
@@ -0,0 +1,113 @@
+/-
+Copyright (c) 2020 Markus Himmel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Markus Himmel, Yury Kudryashov
+-/
+import Mathlib.MeasureTheory.OuterMeasure.AE
+
+/-!
+# Borel-Cantelli lemma, part 1
+
+In this file we show one implication of the **Borel-Cantelli lemma**:
+if `s i` is a countable family of sets such that `∑' i, μ (s i)` is finite,
+then a.e. all points belong to finitely many sets of the family.
+
+We prove several versions of this lemma:
+
+- `MeasureTheory.ae_finite_setOf_mem`: as stated above;
+- `MeasureTheory.measure_limsup_cofinite_eq_zero`:
+  in terms of `Filter.limsup` along `Filter.cofinite`;
+- `MeasureTheory.measure_limsup_atTop_eq_zero`:
+  in terms of `Filter.limsup` along `(Filter.atTop : Filter ℕ)`.
+
+For the *second* Borel-Cantelli lemma (applying to independent sets in a probability space),
+see `ProbabilityTheory.measure_limsup_eq_one`.
+-/
+
+open Filter Set
+open scoped ENNReal Topology
+
+namespace MeasureTheory
+
+variable {α ι F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] [Countable ι] {μ : F}
+
+/-- One direction of the **Borel-Cantelli lemma**
+(sometimes called the "*first* Borel-Cantelli lemma"):
+if `(s i)` is a countable family of sets such that `∑' i, μ (s i)` is finite,
+then the limit superior of the `s i` along the cofinite filter is a null set.
+
+Note: for the *second* Borel-Cantelli lemma (applying to independent sets in a probability space),
+see `ProbabilityTheory.measure_limsup_eq_one`. -/
+theorem measure_limsup_cofinite_eq_zero {s : ι → Set α} (hs : ∑' i, μ (s i) ≠ ∞) :
+    μ (limsup s cofinite) = 0 := by
+  refine bot_unique <| ge_of_tendsto' (ENNReal.tendsto_tsum_compl_atTop_zero hs) fun t ↦ ?_
+  calc
+    μ (limsup s cofinite) ≤ μ (⋃ i : {i // i ∉ t}, s i) := by
+      gcongr
+      rw [hasBasis_cofinite.limsup_eq_iInf_iSup, iUnion_subtype]
+      exact iInter₂_subset _ t.finite_toSet
+    _ ≤ ∑' i : {i // i ∉ t}, μ (s i) := measure_iUnion_le _
+
+/-- One direction of the **Borel-Cantelli lemma**
+(sometimes called the "*first* Borel-Cantelli lemma"):
+if `(s i)` is a sequence of sets such that `∑' i, μ (s i)` is finite,
+then the limit superior of the `s i` along the `atTop` filter is a null set.
+
+Note: for the *second* Borel-Cantelli lemma (applying to independent sets in a probability space),
+see `ProbabilityTheory.measure_limsup_eq_one`. -/
+theorem measure_limsup_atTop_eq_zero {s : ℕ → Set α} (hs : ∑' i, μ (s i) ≠ ∞) :
+    μ (limsup s atTop) = 0 := by
+  rw [← Nat.cofinite_eq_atTop, measure_limsup_cofinite_eq_zero hs]
+
+@[deprecated (since := "2024-09-01")]
+alias measure_limsup_eq_zero := measure_limsup_atTop_eq_zero
+
+/-- One direction of the **Borel-Cantelli lemma**
+(sometimes called the "*first* Borel-Cantelli lemma"):
+if `(s i)` is a countable family of sets such that `∑' i, μ (s i)` is finite,
+then a.e. all points belong to finitely sets of the family. -/
+theorem ae_finite_setOf_mem {s : ι → Set α} (h : ∑' i, μ (s i) ≠ ∞) :
+    ∀ᵐ x ∂μ, {i | x ∈ s i}.Finite := by
+  rw [ae_iff, ← measure_limsup_cofinite_eq_zero h]
+  congr 1 with x
+  simp [mem_limsup_iff_frequently_mem, Filter.Frequently]
+
+/-- A version of the **Borel-Cantelli lemma**: if `pᵢ` is a sequence of predicates such that
+`∑' i, μ {x | pᵢ x}` is finite, then the measure of `x` such that `pᵢ x` holds frequently as `i → ∞`
+(or equivalently, `pᵢ x` holds for infinitely many `i`) is equal to zero. -/
+theorem measure_setOf_frequently_eq_zero {p : ℕ → α → Prop} (hp : ∑' i, μ { x | p i x } ≠ ∞) :
+    μ { x | ∃ᶠ n in atTop, p n x } = 0 := by
+  simpa only [limsup_eq_iInf_iSup_of_nat, frequently_atTop, ← bex_def, setOf_forall,
+    setOf_exists] using measure_limsup_atTop_eq_zero hp
+
+/-- A version of the **Borel-Cantelli lemma**: if `sᵢ` is a sequence of sets such that
+`∑' i, μ sᵢ` is finite, then for almost all `x`, `x` does not belong to `sᵢ` for large `i`. -/
+theorem ae_eventually_not_mem {s : ℕ → Set α} (hs : (∑' i, μ (s i)) ≠ ∞) :
+    ∀ᵐ x ∂μ, ∀ᶠ n in atTop, x ∉ s n :=
+  measure_setOf_frequently_eq_zero hs
+
+theorem measure_liminf_cofinite_eq_zero [Infinite ι]  {s : ι → Set α} (h : ∑' i, μ (s i) ≠ ∞) :
+    μ (liminf s cofinite) = 0 := by
+  rw [← le_zero_iff, ← measure_limsup_cofinite_eq_zero h]
+  exact measure_mono liminf_le_limsup
+
+theorem measure_liminf_atTop_eq_zero {s : ℕ → Set α} (h : (∑' i, μ (s i)) ≠ ∞) :
+    μ (liminf s atTop) = 0 := by
+  rw [← Nat.cofinite_eq_atTop, measure_liminf_cofinite_eq_zero h]
+
+-- TODO: the next 2 lemmas are true for any filter with countable intersections, not only `ae`.
+-- Need to specify `α := Set α` below because of diamond; see #19041
+theorem limsup_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
+    (h : ∀ n, s n =ᵐ[μ] t) : limsup (α := Set α) s atTop =ᵐ[μ] t := by
+  simp only [eventuallyEq_set, ← eventually_countable_forall] at h
+  refine eventuallyEq_set.2 <| h.mono fun x hx ↦ ?_
+  simp [mem_limsup_iff_frequently_mem, hx]
+
+-- Need to specify `α := Set α` above because of diamond; see #19041
+theorem liminf_ae_eq_of_forall_ae_eq (s : ℕ → Set α) {t : Set α}
+    (h : ∀ n, s n =ᵐ[μ] t) : liminf (α := Set α) s atTop =ᵐ[μ] t := by
+  simp only [eventuallyEq_set, ← eventually_countable_forall] at h
+  refine eventuallyEq_set.2 <| h.mono fun x hx ↦ ?_
+  simp only [mem_liminf_iff_eventually_mem, hx, eventually_const]
+
+end MeasureTheory
diff --git a/Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean b/Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
index 04c80d5a09c16..167ea1437b67c 100644
--- a/Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
+++ b/Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean
@@ -148,7 +148,7 @@ theorem f_iUnion {s : ℕ → Set α} (h : ∀ i, IsCaratheodory m (s i)) (hd :
   simp only [inter_comm, inter_univ, univ_inter] at this; simp only [this]
   exact m.mono (iUnion₂_subset fun i _ => subset_iUnion _ i)
 
-/-- The Carathéodory-measurable sets for an outer measure `m` form a Dynkin system.  -/
+/-- The Carathéodory-measurable sets for an outer measure `m` form a Dynkin system. -/
 def caratheodoryDynkin : MeasurableSpace.DynkinSystem α where
   Has := IsCaratheodory m
   has_empty := isCaratheodory_empty m
diff --git a/Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean b/Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
index 0e6784ff063d2..7e859de1147fc 100644
--- a/Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
+++ b/Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean
@@ -88,10 +88,33 @@ protected def ofFunction (m : Set α → ℝ≥0∞) (m_empty : m ∅ = 0) : Out
 
 variable (m : Set α → ℝ≥0∞) (m_empty : m ∅ = 0)
 
+/-- `ofFunction` of a set `s` is the infimum of `∑ᵢ, m (tᵢ)` for all collections of sets
+`tᵢ` that cover `s`. -/
 theorem ofFunction_apply (s : Set α) :
     OuterMeasure.ofFunction m m_empty s = ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' n, m (t n) :=
   rfl
 
+/-- `ofFunction` of a set `s` is the infimum of `∑ᵢ, m (tᵢ)` for all collections of sets
+`tᵢ` that cover `s`, with all `tᵢ` satisfying a predicate `P` such that `m` is infinite for sets
+that don't satisfy `P`.
+This is similar to `ofFunction_apply`, except that the sets `tᵢ` satisfy `P`.
+The hypothesis `m_top` applies in particular to a function of the form `extend m'`. -/
+theorem ofFunction_eq_iInf_mem {P : Set α → Prop} (m_top : ∀ s, ¬ P s → m s = ∞) (s : Set α) :
+    OuterMeasure.ofFunction m m_empty s =
+      ⨅ (t : ℕ → Set α) (_ : ∀ i, P (t i)) (_ : s ⊆ ⋃ i, t i), ∑' i, m (t i) := by
+  rw [OuterMeasure.ofFunction_apply]
+  apply le_antisymm
+  · exact le_iInf fun t ↦ le_iInf fun _ ↦ le_iInf fun h ↦ iInf₂_le _ (by exact h)
+  · simp_rw [le_iInf_iff]
+    refine fun t ht_subset ↦ iInf_le_of_le t ?_
+    by_cases ht : ∀ i, P (t i)
+    · exact iInf_le_of_le ht (iInf_le_of_le ht_subset le_rfl)
+    · simp only [ht, not_false_eq_true, iInf_neg, top_le_iff]
+      push_neg at ht
+      obtain ⟨i, hti_not_mem⟩ := ht
+      have hfi_top : m (t i) = ∞ := m_top _ hti_not_mem
+      exact ENNReal.tsum_eq_top_of_eq_top ⟨i, hfi_top⟩
+
 variable {m m_empty}
 
 theorem ofFunction_le (s : Set α) : OuterMeasure.ofFunction m m_empty s ≤ m s :=
@@ -128,7 +151,7 @@ theorem ofFunction_eq_sSup : OuterMeasure.ofFunction m m_empty = sSup { μ | ∀
 
 E.g., if `α` is an (e)metric space and `m u = ∞` on any set of diameter `≥ r`, then this lemma
 implies that `μ (s ∪ t) = μ s + μ t` on any two sets such that `r ≤ edist x y` for all `x ∈ s`
-and `y ∈ t`.  -/
+and `y ∈ t`. -/
 theorem ofFunction_union_of_top_of_nonempty_inter {s t : Set α}
     (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
     OuterMeasure.ofFunction m m_empty (s ∪ t) =
@@ -209,7 +232,7 @@ theorem smul_ofFunction {c : ℝ≥0∞} (hc : c ≠ ∞) : c • OuterMeasure.o
   haveI : Nonempty { t : ℕ → Set α // s ⊆ ⋃ i, t i } := ⟨⟨fun _ => s, subset_iUnion (fun _ => s) 0⟩⟩
   simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,
   iInf_subtype']
-  rw [ENNReal.iInf_mul_left fun h => (hc h).elim]
+  rw [ENNReal.mul_iInf fun h => (hc h).elim]
 
 end OfFunction
 
@@ -291,7 +314,7 @@ theorem comap_boundedBy {β} (f : β → α)
 
 E.g., if `α` is an (e)metric space and `m u = ∞` on any set of diameter `≥ r`, then this lemma
 implies that `μ (s ∪ t) = μ s + μ t` on any two sets such that `r ≤ edist x y` for all `x ∈ s`
-and `y ∈ t`.  -/
+and `y ∈ t`. -/
 theorem boundedBy_union_of_top_of_nonempty_inter {s t : Set α}
     (h : ∀ u, (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ∞) :
     boundedBy m (s ∪ t) = boundedBy m s + boundedBy m t :=
diff --git a/Mathlib/MeasureTheory/PiSystem.lean b/Mathlib/MeasureTheory/PiSystem.lean
index f00e51c4094e7..aed108095fc15 100644
--- a/Mathlib/MeasureTheory/PiSystem.lean
+++ b/Mathlib/MeasureTheory/PiSystem.lean
@@ -207,25 +207,25 @@ theorem subset_generatePiSystem_self {α} (S : Set (Set α)) : S ⊆ generatePiS
 
 theorem generatePiSystem_subset_self {α} {S : Set (Set α)} (h_S : IsPiSystem S) :
     generatePiSystem S ⊆ S := fun x h => by
-  induction' h with _ h_s s u _ _ h_nonempty h_s h_u
-  · exact h_s
-  · exact h_S _ h_s _ h_u h_nonempty
+  induction h with
+  | base h_s => exact h_s
+  | inter _ _ h_nonempty h_s h_u => exact h_S _ h_s _ h_u h_nonempty
 
 theorem generatePiSystem_eq {α} {S : Set (Set α)} (h_pi : IsPiSystem S) : generatePiSystem S = S :=
   Set.Subset.antisymm (generatePiSystem_subset_self h_pi) (subset_generatePiSystem_self S)
 
 theorem generatePiSystem_mono {α} {S T : Set (Set α)} (hST : S ⊆ T) :
     generatePiSystem S ⊆ generatePiSystem T := fun t ht => by
-  induction' ht with s h_s s u _ _ h_nonempty h_s h_u
-  · exact generatePiSystem.base (Set.mem_of_subset_of_mem hST h_s)
-  · exact isPiSystem_generatePiSystem T _ h_s _ h_u h_nonempty
+  induction ht with
+  | base h_s => exact generatePiSystem.base (Set.mem_of_subset_of_mem hST h_s)
+  | inter _ _ h_nonempty h_s h_u => exact isPiSystem_generatePiSystem T _ h_s _ h_u h_nonempty
 
 theorem generatePiSystem_measurableSet {α} [M : MeasurableSpace α] {S : Set (Set α)}
     (h_meas_S : ∀ s ∈ S, MeasurableSet s) (t : Set α) (h_in_pi : t ∈ generatePiSystem S) :
     MeasurableSet t := by
-  induction' h_in_pi with s h_s s u _ _ _ h_s h_u
-  · apply h_meas_S _ h_s
-  · apply MeasurableSet.inter h_s h_u
+  induction h_in_pi with
+  | base h_s => apply h_meas_S _ h_s
+  | inter _ _ _ h_s h_u => apply MeasurableSet.inter h_s h_u
 
 theorem generateFrom_measurableSet_of_generatePiSystem {α} {g : Set (Set α)} (t : Set α)
     (ht : t ∈ generatePiSystem g) : MeasurableSet[generateFrom g] t :=
@@ -260,8 +260,7 @@ theorem mem_generatePiSystem_iUnion_elim {α β} {g : β → Set (Set α)} (h_pi
       rw [← forall_and]
       constructor <;> intro h1 b <;> by_cases hbs : b ∈ T_s <;> by_cases hbt : b ∈ T_t' <;>
           specialize h1 b <;>
-        simp only [hbs, hbt, if_true, if_false, true_imp_iff, and_self_iff, false_imp_iff,
-          and_true_iff, true_and_iff] at h1 ⊢
+        simp only [hbs, hbt, if_true, if_false, true_imp_iff, and_self_iff, false_imp_iff] at h1 ⊢
       all_goals exact h1
     intro b h_b
     split_ifs with hbs hbt hbt
@@ -338,11 +337,10 @@ theorem piiUnionInter_singleton (π : ι → Set (Set α)) (i : ι) :
       exact Or.inl (hfπ i hi)
     · have ht_empty : t = ∅ := by
         ext1 x
-        simp only [Finset.not_mem_empty, iff_false_iff]
+        simp only [Finset.not_mem_empty, iff_false]
         exact fun hx => hi (hti x hx ▸ hx)
       -- Porting note: `Finset.not_mem_empty` required
-      simp [ht_empty, Finset.not_mem_empty, iInter_false, iInter_univ, Set.mem_singleton univ,
-        or_true_iff]
+      simp [ht_empty, Finset.not_mem_empty, iInter_false, iInter_univ, Set.mem_singleton univ]
   · cases' h with hs hs
     · refine ⟨{i}, ?_, fun _ => s, ⟨fun x hx => ?_, ?_⟩⟩
       · rw [Finset.coe_singleton]
@@ -351,7 +349,7 @@ theorem piiUnionInter_singleton (π : ι → Set (Set α)) (i : ι) :
       · simp only [Finset.mem_singleton, iInter_iInter_eq_left]
     · refine ⟨∅, ?_⟩
       simpa only [Finset.coe_empty, subset_singleton_iff, mem_empty_iff_false, IsEmpty.forall_iff,
-        imp_true_iff, Finset.not_mem_empty, iInter_false, iInter_univ, true_and_iff,
+        imp_true_iff, Finset.not_mem_empty, iInter_false, iInter_univ, true_and,
         exists_const] using hs
 
 theorem piiUnionInter_singleton_left (s : ι → Set α) (S : Set ι) :
diff --git a/Mathlib/MeasureTheory/SetSemiring.lean b/Mathlib/MeasureTheory/SetSemiring.lean
index c17719e996a4f..0727d10d8f8f5 100644
--- a/Mathlib/MeasureTheory/SetSemiring.lean
+++ b/Mathlib/MeasureTheory/SetSemiring.lean
@@ -72,7 +72,7 @@ lemma empty_not_mem_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈
     ∅ ∉ hC.diffFinset hs ht := by
   classical
   simp only [diffFinset, mem_sdiff, Finset.mem_singleton, eq_self_iff_true, not_true,
-    and_false_iff, not_false_iff]
+    and_false, not_false_iff]
 
 lemma diffFinset_subset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
     ↑(hC.diffFinset hs ht) ⊆ C := by
@@ -209,7 +209,7 @@ lemma empty_not_mem_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : 
     ∅ ∉ hC.diffFinset₀ hs hI := by
   classical
   simp only [diffFinset₀, mem_sdiff, Finset.mem_singleton, eq_self_iff_true, not_true,
-    and_false_iff, not_false_iff]
+    and_false, not_false_iff]
 
 lemma diffFinset₀_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
     ↑(hC.diffFinset₀ hs hI) ⊆ C := by
@@ -313,9 +313,10 @@ lemma biInter_mem {ι : Type*} (hC : IsSetRing C) {s : ι → Set α}
     (S : Finset ι) (hS : S.Nonempty) (hs : ∀ n ∈ S, s n ∈ C) :
     ⋂ i ∈ S, s i ∈ C := by
   classical
-  induction' hS using Finset.Nonempty.cons_induction with _ i S hiS _ h hs
-  · simpa using hs
-  · simp_rw [← Finset.mem_coe, Finset.coe_cons, Set.biInter_insert]
+  induction hS using Finset.Nonempty.cons_induction with
+  | singleton => simpa using hs
+  | cons i S hiS _ h =>
+    simp_rw [← Finset.mem_coe, Finset.coe_cons, Set.biInter_insert]
     simp only [cons_eq_insert, Finset.mem_insert, forall_eq_or_imp] at hs
     refine hC.inter_mem hs.1 ?_
     exact h (fun n hnS ↦ hs.2 n hnS)
diff --git a/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean b/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean
new file mode 100644
index 0000000000000..2d7efbda8ea35
--- /dev/null
+++ b/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean
@@ -0,0 +1,252 @@
+/-
+Copyright (c) 2023 Chris Hughes. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Hughes
+-/
+
+import Mathlib.Data.Nat.PrimeFin
+import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
+import Mathlib.FieldTheory.IsAlgClosed.Classification
+import Mathlib.ModelTheory.Algebra.Field.CharP
+import Mathlib.ModelTheory.Satisfiability
+
+/-!
+
+# The First Order Theory of Algebraically Closed Fields
+
+This file defines the theory of algebraically closed fields of characteristic `p`, as well
+as proving completeness of the theory and the Lefschetz Principle.
+
+## Main definitions
+
+* `FirstOrder.Language.Theory.ACF p` : the theory of algebraically closed fields of characteristic
+  `p` as a theory over the language of rings.
+* `FirstOrder.Field.ACF_isComplete` : the theory of algebraically closed fields of characteristic
+  `p` is complete whenever `p` is prime or zero.
+* `FirstOrder.Field.ACF_zero_realize_iff_infinite_ACF_prime_realize` : the Lefschetz principle.
+
+## Implementation details
+
+To apply a theorem about the model theory of algebraically closed fields to a specific
+algebraically closed field `K` which does not have a `Language.ring.Structure` instance,
+you must introduce the local instance `compatibleRingOfRing K`. Theorems whose statement requires
+both a `Language.ring.Structure` instance and a `Field` instance will all be stated with the
+assumption `Field K`, `CharP K p`, `IsAlgClosed K` and `CompatibleRing K` and there are instances
+defined saying that these assumptions imply `Theory.field.Model K` and `(Theory.ACF p).Model K`
+
+## References
+
+The first order theory of algebraically closed fields, along with the Lefschetz Principle and
+the Ax-Grothendieck Theorem were first formalized in Lean 3 by Joseph Hua
+[here](https://github.com/Jlh18/ModelTheoryInLean8) with the master's thesis
+[here](https://github.com/Jlh18/ModelTheory8Report)
+
+-/
+
+variable {K : Type*}
+
+namespace FirstOrder
+
+namespace Field
+
+open Ring FreeCommRing BigOperators Polynomial Language
+
+/-- A generic monic polynomial of degree `n` as an element of the
+free commutative ring in `n+1` variables, with a variable for each
+of the `n` non-leading coefficients of the polynomial and one variable (`Fin.last n`)
+for `X`.  -/
+def genericMonicPoly (n : ℕ) : FreeCommRing (Fin (n + 1)) :=
+  of (Fin.last _) ^ n + ∑ i : Fin n, of i.castSucc * of (Fin.last _) ^ (i : ℕ)
+
+theorem lift_genericMonicPoly [CommRing K] [Nontrivial K] {n : ℕ} (v : Fin (n+1) → K) :
+    FreeCommRing.lift v (genericMonicPoly n) =
+    (((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv).symm (v ∘ Fin.castSucc)).1.eval
+      (v (Fin.last _)) := by
+  simp only [genericMonicPoly, map_add, map_pow, lift_of, map_sum, map_mul, monicEquivDegreeLT,
+    degreeLTEquiv, Equiv.symm_trans_apply, LinearEquiv.coe_toEquiv_symm, EquivLike.coe_coe,
+    LinearEquiv.coe_symm_mk, Function.comp_apply, Equiv.coe_fn_symm_mk, eval_add, eval_pow, eval_X,
+    eval_finset_sum, eval_monomial]
+
+/-- A sentence saying every monic polynomial of degree `n` has a root. -/
+noncomputable def genericMonicPolyHasRoot (n : ℕ) : Language.ring.Sentence :=
+  (∃' ((termOfFreeCommRing (genericMonicPoly n)).relabel Sum.inr =' 0)).alls
+
+theorem realize_genericMonicPolyHasRoot [Field K] [CompatibleRing K] (n : ℕ) :
+    K ⊨ genericMonicPolyHasRoot n ↔
+      ∀ p : { p : K[X] // p.Monic ∧ p.natDegree = n }, ∃ x, p.1.eval x = 0 := by
+  let _ := Classical.decEq K
+  rw [Equiv.forall_congr_left ((monicEquivDegreeLT n).trans (degreeLTEquiv K n).toEquiv)]
+  simp [Sentence.Realize, genericMonicPolyHasRoot, lift_genericMonicPoly]
+
+/-- The theory of algebraically closed fields of characteristic `p` as a theory over
+the language of rings -/
+def _root_.FirstOrder.Language.Theory.ACF (p : ℕ) : Theory .ring :=
+  Theory.fieldOfChar p ∪ genericMonicPolyHasRoot '' {n | 0 < n}
+
+instance [Language.ring.Structure K] (p : ℕ) [h : (Theory.ACF p).Model K] :
+    (Theory.fieldOfChar p).Model K :=
+  Theory.Model.mono h Set.subset_union_left
+
+instance [Field K] [CompatibleRing K] {p : ℕ} [CharP K p] [IsAlgClosed K] :
+    (Theory.ACF p).Model K := by
+  refine Theory.model_union_iff.2 ⟨inferInstance, ?_⟩
+  simp only [Theory.model_iff, Set.mem_image, Set.mem_singleton_iff,
+    exists_prop, forall_exists_index, and_imp]
+  rintro _ n hn0 rfl
+  simp only [realize_genericMonicPolyHasRoot]
+  rintro ⟨p, _, rfl⟩
+  exact IsAlgClosed.exists_root p (ne_of_gt
+    (natDegree_pos_iff_degree_pos.1 hn0))
+
+theorem modelField_of_modelACF (p : ℕ) (K : Type*) [Language.ring.Structure K]
+    [h : (Theory.ACF p).Model K] : Theory.field.Model K :=
+  Theory.Model.mono h (Set.subset_union_of_subset_left Set.subset_union_left _)
+
+/-- A model for the Theory of algebraically closed fields is a Field. After introducing
+this as a local instance on a particular Type, you should usually also introduce
+`modelField_of_modelACF p M`, `compatibleRingOfModelField` and `isAlgClosed_of_model_ACF` -/
+@[reducible]
+noncomputable def fieldOfModelACF (p : ℕ) (K : Type*)
+    [Language.ring.Structure K]
+    [h : (Theory.ACF p).Model K] : Field K := by
+  have := modelField_of_modelACF p K
+  exact fieldOfModelField K
+
+theorem isAlgClosed_of_model_ACF (p : ℕ) (K : Type*)
+    [Field K] [CompatibleRing K] [h : (Theory.ACF p).Model K] :
+    IsAlgClosed K := by
+  refine IsAlgClosed.of_exists_root _ ?_
+  intro p hpm hpi
+  have h : K ⊨ genericMonicPolyHasRoot '' {n | 0 < n} :=
+    Theory.Model.mono h (by simp [Theory.ACF])
+  simp only [Theory.model_iff, Set.mem_image, Set.mem_singleton_iff,
+    exists_prop, forall_exists_index, and_imp] at h
+  have := h _ p.natDegree (natDegree_pos_iff_degree_pos.2
+    (degree_pos_of_irreducible hpi)) rfl
+  rw [realize_genericMonicPolyHasRoot] at this
+  exact this ⟨_, hpm, rfl⟩
+
+theorem ACF_isSatisfiable {p : ℕ} (hp : p.Prime ∨ p = 0) :
+    (Theory.ACF p).IsSatisfiable := by
+  cases hp with
+  | inl hp =>
+    have : Fact p.Prime := ⟨hp⟩
+    let _ := compatibleRingOfRing (AlgebraicClosure (ZMod p))
+    have : CharP (AlgebraicClosure (ZMod p)) p :=
+      charP_of_injective_algebraMap
+        (RingHom.injective (algebraMap (ZMod p) (AlgebraicClosure (ZMod p)))) p
+    exact ⟨⟨AlgebraicClosure (ZMod p)⟩⟩
+  | inr hp =>
+    subst hp
+    let _ := compatibleRingOfRing (AlgebraicClosure ℚ)
+    have : CharP (AlgebraicClosure ℚ) 0 :=
+      charP_of_injective_algebraMap
+        (RingHom.injective (algebraMap ℚ (AlgebraicClosure ℚ))) 0
+    exact ⟨⟨AlgebraicClosure ℚ⟩⟩
+
+open Cardinal
+
+/-- The Theory `Theory.ACF p` is `κ`-categorical whenever `κ` is an uncountable cardinal.
+At the moment this is not as universe polymorphic as it could be,
+it currently requires `κ : Cardinal.{0}`, but it is true for any universe. -/
+theorem ACF_categorical {p : ℕ} (κ : Cardinal.{0}) (hκ : ℵ₀ < κ) :
+    Categorical κ (Theory.ACF p) := by
+  rintro ⟨M⟩ ⟨N⟩ hM hN
+  let _ := fieldOfModelACF p M
+  have := modelField_of_modelACF p M
+  let _ := compatibleRingOfModelField M
+  have := isAlgClosed_of_model_ACF p M
+  have := charP_of_model_fieldOfChar p M
+  let _ := fieldOfModelACF p N
+  have := modelField_of_modelACF p N
+  let _ := compatibleRingOfModelField N
+  have := isAlgClosed_of_model_ACF p N
+  have := charP_of_model_fieldOfChar p N
+  constructor
+  refine languageEquivEquivRingEquiv.symm ?_
+  apply Classical.choice
+  refine IsAlgClosed.ringEquivOfCardinalEqOfCharEq p ?_ ?_
+  · rw [hM]; exact hκ
+  · rw [hM, hN]
+
+theorem ACF_isComplete {p : ℕ} (hp : p.Prime ∨ p = 0) :
+    (Theory.ACF p).IsComplete := by
+  apply Categorical.isComplete.{0, 0, 0} (Order.succ ℵ₀) _
+    (ACF_categorical _ (Order.lt_succ _))
+    (Order.le_succ ℵ₀)
+  · simp only [card_ring, lift_id']
+    exact le_trans (le_of_lt (lt_aleph0_of_finite _)) (Order.le_succ _)
+  · exact ACF_isSatisfiable hp
+  · rintro ⟨M⟩
+    let _ := fieldOfModelACF p M
+    have := modelField_of_modelACF p M
+    let _ := compatibleRingOfModelField M
+    have := isAlgClosed_of_model_ACF p M
+    infer_instance
+
+theorem finite_ACF_prime_not_realize_of_ACF_zero_realize
+    (φ : Language.ring.Sentence) (h : Theory.ACF 0 ⊨ᵇ φ) :
+    Set.Finite { p : Nat.Primes | ¬ Theory.ACF p ⊨ᵇ φ } := by
+  rw [Theory.models_iff_finset_models] at h
+  rcases h with ⟨T0, hT0, h⟩
+  have f : ∀ ψ ∈ Theory.ACF 0,
+      { s : Finset Nat.Primes // ∀ q : Nat.Primes, q ∉ s → Theory.ACF q ⊨ᵇ ψ } := by
+    intro ψ hψ
+    rw [Theory.ACF, Theory.fieldOfChar, Set.union_right_comm, Set.mem_union, if_pos rfl,
+      Set.mem_image] at hψ
+    apply Classical.choice
+    rcases hψ with h | ⟨p, hp, rfl⟩
+    · refine ⟨⟨∅, ?_⟩⟩
+      intro q _
+      exact Theory.models_sentence_of_mem
+        (by rw [Theory.ACF, Theory.fieldOfChar, Set.union_right_comm];
+            exact Set.mem_union_left _ h)
+    · refine ⟨⟨{⟨p, hp⟩}, ?_⟩⟩
+      rintro ⟨q, _⟩ hq ⟨K⟩ _ _
+      have hqp : q ≠ p := by simpa [← Nat.Primes.coe_nat_inj] using hq
+      let _ := fieldOfModelACF q K
+      have := modelField_of_modelACF q K
+      let _ := compatibleRingOfModelField K
+      have := charP_of_model_fieldOfChar q K
+      simp only [eqZero, Term.equal, Term.relabel, BoundedFormula.realize_not,
+        BoundedFormula.realize_bdEqual, Term.realize_relabel, Sum.elim_comp_inl,
+        realize_termOfFreeCommRing, map_natCast, Term.realize_func, CompatibleRing.funMap_zero,
+        ne_eq, ← CharP.charP_iff_prime_eq_zero hp]
+      intro _
+      exact hqp <| CharP.eq K inferInstance inferInstance
+  let s : Finset Nat.Primes := T0.attach.biUnion (fun φ => f φ.1 (hT0 φ.2))
+  have hs : ∀ (p : Nat.Primes) ψ, ψ ∈ T0 → p ∉ s → Theory.ACF p ⊨ᵇ ψ := by
+    intro p ψ hψ hpψ
+    simp only [s, Finset.mem_biUnion, Finset.mem_attach, true_and,
+      Subtype.exists, not_exists] at hpψ
+    exact (f ψ (hT0 hψ)).2 p (hpψ _ hψ)
+  refine Set.Finite.subset (Finset.finite_toSet s) (Set.compl_subset_comm.2 ?_)
+  intro p hp
+  exact Theory.models_of_models_theory (fun ψ hψ => hs p ψ hψ hp) h
+
+/-- The **Lefschetz principle**. A first order sentence is modeled by the theory
+of algebraically closed fields of characteristic zero if and only if it is modeled by
+the theory of algebraically closed fields of characteristic `p` for infinitely many `p`. -/
+theorem ACF_zero_realize_iff_infinite_ACF_prime_realize {φ : Language.ring.Sentence} :
+    Theory.ACF 0 ⊨ᵇ φ ↔ Set.Infinite { p : Nat.Primes | Theory.ACF p ⊨ᵇ φ } := by
+  refine ⟨fun h => Set.infinite_of_finite_compl
+      (finite_ACF_prime_not_realize_of_ACF_zero_realize φ h),
+    not_imp_not.1 ?_⟩
+  simpa [(ACF_isComplete (Or.inr rfl)).models_not_iff,
+      fun p : Nat.Primes => (ACF_isComplete (Or.inl p.2)).models_not_iff] using
+    finite_ACF_prime_not_realize_of_ACF_zero_realize φ.not
+
+/-- Another statement of the **Lefschetz principle**. A first order sentence is modeled by the
+theory of algebraically closed fields of characteristic zero if and only if it is modeled by the
+theory of algebraically closed fields of characteristic `p` for all but finitely many primes `p`.
+-/
+theorem ACF_zero_realize_iff_finite_ACF_prime_not_realize {φ : Language.ring.Sentence} :
+    Theory.ACF 0 ⊨ᵇ φ ↔ Set.Finite { p : Nat.Primes | Theory.ACF p ⊨ᵇ φ }ᶜ :=
+  ⟨fun h => finite_ACF_prime_not_realize_of_ACF_zero_realize φ h,
+    fun h => ACF_zero_realize_iff_infinite_ACF_prime_realize.2
+      (Set.infinite_of_finite_compl h)⟩
+
+
+end Field
+
+end FirstOrder
diff --git a/Mathlib/ModelTheory/Algebra/Ring/Basic.lean b/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
index 34291af0a1546..c2c75e666e4b2 100644
--- a/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
+++ b/Mathlib/ModelTheory/Algebra/Ring/Basic.lean
@@ -59,6 +59,7 @@ inductive ringFunc : ℕ → Type
 def Language.ring : Language :=
   { Functions := ringFunc
     Relations := fun _ => Empty }
+  deriving IsAlgebraic
 
 namespace Ring
 
@@ -139,7 +140,7 @@ theorem card_ring : card Language.ring = 5 := by
   have : Fintype.card Language.ring.Symbols = 5 := rfl
   simp [Language.card, this]
 
-open Language ring Structure
+open Language Structure
 
 /-- A Type `R` is a `CompatibleRing` if it is a structure for the language of rings and this
 structure is the same as the structure already given on `R` by the classes `Add`, `Mul` etc.
@@ -225,7 +226,6 @@ def compatibleRingOfRing (R : Type*) [Add R] [Mul R] [Neg R] [One R] [Zero R] :
       | _, .neg => fun x => -x 0
       | _, .zero => fun _ => 0
       | _, .one => fun _ => 1
-    RelMap := Empty.elim,
     funMap_add := fun _ => rfl,
     funMap_mul := fun _ => rfl,
     funMap_neg := fun _ => rfl,
diff --git a/Mathlib/ModelTheory/Basic.lean b/Mathlib/ModelTheory/Basic.lean
index 04ea4484a6189..baa0ada8d4271 100644
--- a/Mathlib/ModelTheory/Basic.lean
+++ b/Mathlib/ModelTheory/Basic.lean
@@ -58,160 +58,61 @@ structure Language where
   /-- For every arity, a `Type*` of relations of that arity -/
   Relations : ℕ → Type v
 
-/-- Used to define `FirstOrder.Language₂`. -/
---@[simp]
-def Sequence₂ (a₀ a₁ a₂ : Type u) : ℕ → Type u
-  | 0 => a₀
-  | 1 => a₁
-  | 2 => a₂
-  | _ => PEmpty
-
-namespace Sequence₂
-
-variable (a₀ a₁ a₂ : Type u)
-
-instance inhabited₀ [h : Inhabited a₀] : Inhabited (Sequence₂ a₀ a₁ a₂ 0) :=
-  h
-
-instance inhabited₁ [h : Inhabited a₁] : Inhabited (Sequence₂ a₀ a₁ a₂ 1) :=
-  h
-
-instance inhabited₂ [h : Inhabited a₂] : Inhabited (Sequence₂ a₀ a₁ a₂ 2) :=
-  h
-
-instance {n : ℕ} : IsEmpty (Sequence₂ a₀ a₁ a₂ (n + 3)) := inferInstanceAs (IsEmpty PEmpty)
-
-instance [DecidableEq a₀] [DecidableEq a₁] [DecidableEq a₂] {n : ℕ} :
-    DecidableEq (Sequence₂ a₀ a₁ a₂ n) :=
-  match n with
-  | 0 | 1 | 2 => ‹_›
-  | _ + 3 => inferInstance
-
-@[simp]
-theorem lift_mk {i : ℕ} :
-    Cardinal.lift.{v,u} #(Sequence₂ a₀ a₁ a₂ i)
-      = #(Sequence₂ (ULift.{v,u} a₀) (ULift.{v,u} a₁) (ULift.{v,u} a₂) i) := by
-  rcases i with (_ | _ | _ | i) <;>
-    simp only [Sequence₂, mk_uLift, Nat.succ_ne_zero, IsEmpty.forall_iff, Nat.succ.injEq,
-      add_eq_zero, OfNat.ofNat_ne_zero, and_false, one_ne_zero, mk_eq_zero, lift_zero]
-
-@[simp]
-theorem sum_card : Cardinal.sum (fun i => #(Sequence₂ a₀ a₁ a₂ i)) = #a₀ + #a₁ + #a₂ := by
-  rw [sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ, sum_nat_eq_add_sum_succ]
-  simp [add_assoc, Sequence₂]
+namespace Language
 
-end Sequence₂
+variable (L : Language.{u, v})
 
-namespace Language
+/-- A language is relational when it has no function symbols. -/
+abbrev IsRelational : Prop := ∀ n, IsEmpty (L.Functions n)
 
-/-- A constructor for languages with only constants, unary and binary functions, and
-unary and binary relations. -/
-@[simps]
-protected def mk₂ (c f₁ f₂ : Type u) (r₁ r₂ : Type v) : Language :=
-  ⟨Sequence₂ c f₁ f₂, Sequence₂ PEmpty r₁ r₂⟩
+/-- A language is algebraic when it has no relation symbols. -/
+abbrev IsAlgebraic : Prop := ∀ n, IsEmpty (L.Relations n)
 
 /-- The empty language has no symbols. -/
 protected def empty : Language :=
   ⟨fun _ => Empty, fun _ => Empty⟩
+  deriving IsAlgebraic, IsRelational
 
 instance : Inhabited Language :=
   ⟨Language.empty⟩
 
 /-- The sum of two languages consists of the disjoint union of their symbols. -/
-protected def sum (L : Language.{u, v}) (L' : Language.{u', v'}) : Language :=
+protected def sum (L' : Language.{u', v'}) : Language :=
   ⟨fun n => L.Functions n ⊕ L'.Functions n, fun n => L.Relations n ⊕ L'.Relations n⟩
 
-variable (L : Language.{u, v})
-
 /-- The type of constants in a given language. -/
 -- Porting note(#5171): this linter isn't ported yet.
 -- @[nolint has_nonempty_instance]
-protected def Constants :=
+protected abbrev Constants :=
   L.Functions 0
 
-@[simp]
-theorem constants_mk₂ (c f₁ f₂ : Type u) (r₁ r₂ : Type v) :
-    (Language.mk₂ c f₁ f₂ r₁ r₂).Constants = c :=
-  rfl
-
 /-- The type of symbols in a given language. -/
 -- Porting note(#5171): this linter isn't ported yet.
 -- @[nolint has_nonempty_instance]
-def Symbols :=
+abbrev Symbols :=
   (Σ l, L.Functions l) ⊕ (Σ l, L.Relations l)
 
 /-- The cardinality of a language is the cardinality of its type of symbols. -/
 def card : Cardinal :=
   #L.Symbols
 
-/-- A language is relational when it has no function symbols. -/
-class IsRelational : Prop where
-  /-- There are no function symbols in the language. -/
-  empty_functions : ∀ n, IsEmpty (L.Functions n)
-
-/-- A language is algebraic when it has no relation symbols. -/
-class IsAlgebraic : Prop where
-  /-- There are no relation symbols in the language. -/
-  empty_relations : ∀ n, IsEmpty (L.Relations n)
-
 variable {L} {L' : Language.{u', v'}}
 
 theorem card_eq_card_functions_add_card_relations :
     L.card =
       (Cardinal.sum fun l => Cardinal.lift.{v} #(L.Functions l)) +
         Cardinal.sum fun l => Cardinal.lift.{u} #(L.Relations l) := by
-  simp [card, Symbols]
-
-instance [L.IsRelational] {n : ℕ} : IsEmpty (L.Functions n) :=
-  IsRelational.empty_functions n
-
-instance [L.IsAlgebraic] {n : ℕ} : IsEmpty (L.Relations n) :=
-  IsAlgebraic.empty_relations n
-
-instance isRelational_of_empty_functions {symb : ℕ → Type*} :
-    IsRelational ⟨fun _ => Empty, symb⟩ :=
-  ⟨fun _ => instIsEmptyEmpty⟩
-
-instance isAlgebraic_of_empty_relations {symb : ℕ → Type*} : IsAlgebraic ⟨symb, fun _ => Empty⟩ :=
-  ⟨fun _ => instIsEmptyEmpty⟩
-
-instance isRelational_empty : IsRelational Language.empty :=
-  Language.isRelational_of_empty_functions
-
-instance isAlgebraic_empty : IsAlgebraic Language.empty :=
-  Language.isAlgebraic_of_empty_relations
+  simp only [card, mk_sum, mk_sigma, lift_sum]
 
 instance isRelational_sum [L.IsRelational] [L'.IsRelational] : IsRelational (L.sum L') :=
-  ⟨fun _ => instIsEmptySum⟩
+  fun _ => instIsEmptySum
 
 instance isAlgebraic_sum [L.IsAlgebraic] [L'.IsAlgebraic] : IsAlgebraic (L.sum L') :=
-  ⟨fun _ => instIsEmptySum⟩
-
-instance isRelational_mk₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} [h0 : IsEmpty c] [h1 : IsEmpty f₁]
-    [h2 : IsEmpty f₂] : IsRelational (Language.mk₂ c f₁ f₂ r₁ r₂) :=
-  ⟨fun n =>
-    Nat.casesOn n h0 fun n => Nat.casesOn n h1 fun n => Nat.casesOn n h2 fun _ =>
-      inferInstanceAs (IsEmpty PEmpty)⟩
-
-instance isAlgebraic_mk₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} [h1 : IsEmpty r₁] [h2 : IsEmpty r₂] :
-    IsAlgebraic (Language.mk₂ c f₁ f₂ r₁ r₂) :=
-  ⟨fun n =>
-    Nat.casesOn n (inferInstanceAs (IsEmpty PEmpty)) fun n =>
-      Nat.casesOn n h1 fun n => Nat.casesOn n h2 fun _ => inferInstanceAs (IsEmpty PEmpty)⟩
-
-instance subsingleton_mk₂_functions {c f₁ f₂ : Type u} {r₁ r₂ : Type v} [h0 : Subsingleton c]
-    [h1 : Subsingleton f₁] [h2 : Subsingleton f₂] {n : ℕ} :
-    Subsingleton ((Language.mk₂ c f₁ f₂ r₁ r₂).Functions n) :=
-  Nat.casesOn n h0 fun n =>
-    Nat.casesOn n h1 fun n => Nat.casesOn n h2 fun _ => ⟨fun x => PEmpty.elim x⟩
-
-instance subsingleton_mk₂_relations {c f₁ f₂ : Type u} {r₁ r₂ : Type v} [h1 : Subsingleton r₁]
-    [h2 : Subsingleton r₂] {n : ℕ} : Subsingleton ((Language.mk₂ c f₁ f₂ r₁ r₂).Relations n) :=
-  Nat.casesOn n ⟨fun x => PEmpty.elim x⟩ fun n =>
-    Nat.casesOn n h1 fun n => Nat.casesOn n h2 fun _ => ⟨fun x => PEmpty.elim x⟩
+  fun _ => instIsEmptySum
 
 @[simp]
-theorem empty_card : Language.empty.card = 0 := by simp [card_eq_card_functions_add_card_relations]
+theorem empty_card : Language.empty.card = 0 := by simp only [card, mk_sum, mk_sigma, mk_eq_zero,
+  sum_const, mk_eq_aleph0, lift_id', mul_zero, add_zero]
 
 instance isEmpty_empty : IsEmpty Language.empty.Symbols := by
   simp only [Language.Symbols, isEmpty_sum, isEmpty_sigma]
@@ -232,29 +133,21 @@ theorem card_relations_sum (i : ℕ) :
       Cardinal.lift.{v'} #(L.Relations i) + Cardinal.lift.{v} #(L'.Relations i) := by
   simp [Language.sum]
 
-@[simp]
 theorem card_sum :
     (L.sum L').card = Cardinal.lift.{max u' v'} L.card + Cardinal.lift.{max u v} L'.card := by
-  simp only [card_eq_card_functions_add_card_relations, card_functions_sum, card_relations_sum,
-    sum_add_distrib', lift_add, lift_sum, lift_lift]
-  simp only [add_assoc, add_comm (Cardinal.sum fun i => (#(L'.Functions i)).lift)]
-
-@[simp]
-theorem card_mk₂ (c f₁ f₂ : Type u) (r₁ r₂ : Type v) :
-    (Language.mk₂ c f₁ f₂ r₁ r₂).card =
-      Cardinal.lift.{v} #c + Cardinal.lift.{v} #f₁ + Cardinal.lift.{v} #f₂ +
-          Cardinal.lift.{u} #r₁ + Cardinal.lift.{u} #r₂ := by
-  simp [card_eq_card_functions_add_card_relations, add_assoc]
+  simp only [card, mk_sum, mk_sigma, card_functions_sum, sum_add_distrib', lift_add, lift_sum,
+    lift_lift, card_relations_sum, add_assoc,
+    add_comm (Cardinal.sum fun i => (#(L'.Functions i)).lift)]
 
 /-- Passes a `DecidableEq` instance on a type of function symbols through the  `Language`
 constructor. Despite the fact that this is proven by `inferInstance`, it is still needed -
-see the `example`s in `ModelTheory/Ring/Basic`.  -/
+see the `example`s in `ModelTheory/Ring/Basic`. -/
 instance instDecidableEqFunctions {f : ℕ → Type*} {R : ℕ → Type*} (n : ℕ) [DecidableEq (f n)] :
     DecidableEq ((⟨f, R⟩ : Language).Functions n) := inferInstance
 
 /-- Passes a `DecidableEq` instance on a type of relation symbols through the  `Language`
 constructor. Despite the fact that this is proven by `inferInstance`, it is still needed -
-see the `example`s in `ModelTheory/Ring/Basic`.  -/
+see the `example`s in `ModelTheory/Ring/Basic`. -/
 instance instDecidableEqRelations {f : ℕ → Type*} {R : ℕ → Type*} (n : ℕ) [DecidableEq (R n)] :
     DecidableEq ((⟨f, R⟩ : Language).Relations n) := inferInstance
 
@@ -267,9 +160,11 @@ variable (L) (M : Type w)
 @[ext]
 class Structure where
   /-- Interpretation of the function symbols -/
-  funMap : ∀ {n}, L.Functions n → (Fin n → M) → M
+  funMap : ∀ {n}, L.Functions n → (Fin n → M) → M := by
+    exact fun {n} => isEmptyElim
   /-- Interpretation of the relation symbols -/
-  RelMap : ∀ {n}, L.Relations n → (Fin n → M) → Prop
+  RelMap : ∀ {n}, L.Relations n → (Fin n → M) → Prop := by
+    exact fun {n} => isEmptyElim
 
 variable (N : Type w') [L.Structure M] [L.Structure N]
 
@@ -350,71 +245,16 @@ theorem funMap_eq_coe_constants {c : L.Constants} {x : Fin 0 → M} : funMap c x
 theorem nonempty_of_nonempty_constants [h : Nonempty L.Constants] : Nonempty M :=
   h.map (↑)
 
-/-- The function map for `FirstOrder.Language.Structure₂`. -/
-def funMap₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} (c' : c → M) (f₁' : f₁ → M → M)
-    (f₂' : f₂ → M → M → M) : ∀ {n}, (Language.mk₂ c f₁ f₂ r₁ r₂).Functions n → (Fin n → M) → M
-  | 0, f, _ => c' f
-  | 1, f, x => f₁' f (x 0)
-  | 2, f, x => f₂' f (x 0) (x 1)
-  | _ + 3, f, _ => PEmpty.elim f
-
-/-- The relation map for `FirstOrder.Language.Structure₂`. -/
-def RelMap₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} (r₁' : r₁ → Set M) (r₂' : r₂ → M → M → Prop) :
-    ∀ {n}, (Language.mk₂ c f₁ f₂ r₁ r₂).Relations n → (Fin n → M) → Prop
-  | 0, r, _ => PEmpty.elim r
-  | 1, r, x => x 0 ∈ r₁' r
-  | 2, r, x => r₂' r (x 0) (x 1)
-  | _ + 3, r, _ => PEmpty.elim r
-
-/-- A structure constructor to match `FirstOrder.Language₂`. -/
-protected def Structure.mk₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} (c' : c → M) (f₁' : f₁ → M → M)
-    (f₂' : f₂ → M → M → M) (r₁' : r₁ → Set M) (r₂' : r₂ → M → M → Prop) :
-    (Language.mk₂ c f₁ f₂ r₁ r₂).Structure M :=
-  ⟨funMap₂ c' f₁' f₂', RelMap₂ r₁' r₂'⟩
-
-namespace Structure
-
-variable {c f₁ f₂ : Type u} {r₁ r₂ : Type v}
-variable {c' : c → M} {f₁' : f₁ → M → M} {f₂' : f₂ → M → M → M}
-variable {r₁' : r₁ → Set M} {r₂' : r₂ → M → M → Prop}
-
-@[simp]
-theorem funMap_apply₀ (c₀ : c) {x : Fin 0 → M} :
-    @Structure.funMap _ M (Structure.mk₂ c' f₁' f₂' r₁' r₂') 0 c₀ x = c' c₀ :=
-  rfl
-
-@[simp]
-theorem funMap_apply₁ (f : f₁) (x : M) :
-    @Structure.funMap _ M (Structure.mk₂ c' f₁' f₂' r₁' r₂') 1 f ![x] = f₁' f x :=
-  rfl
-
-@[simp]
-theorem funMap_apply₂ (f : f₂) (x y : M) :
-    @Structure.funMap _ M (Structure.mk₂ c' f₁' f₂' r₁' r₂') 2 f ![x, y] = f₂' f x y :=
-  rfl
-
-@[simp]
-theorem relMap_apply₁ (r : r₁) (x : M) :
-    @Structure.RelMap _ M (Structure.mk₂ c' f₁' f₂' r₁' r₂') 1 r ![x] = (x ∈ r₁' r) :=
-  rfl
-
-@[simp]
-theorem relMap_apply₂ (r : r₂) (x y : M) :
-    @Structure.RelMap _ M (Structure.mk₂ c' f₁' f₂' r₁' r₂') 2 r ![x, y] = r₂' r x y :=
-  rfl
-
-end Structure
-
 /-- `HomClass L F M N` states that `F` is a type of `L`-homomorphisms. You should extend this
   typeclass when you extend `FirstOrder.Language.Hom`. -/
-class HomClass (L : outParam Language) (F M N : Type*)
+class HomClass (L : outParam Language) (F : Type*) (M N : outParam Type*)
   [FunLike F M N] [L.Structure M] [L.Structure N] : Prop where
   map_fun : ∀ (φ : F) {n} (f : L.Functions n) (x), φ (funMap f x) = funMap f (φ ∘ x)
   map_rel : ∀ (φ : F) {n} (r : L.Relations n) (x), RelMap r x → RelMap r (φ ∘ x)
 
 /-- `StrongHomClass L F M N` states that `F` is a type of `L`-homomorphisms which preserve
   relations in both directions. -/
-class StrongHomClass (L : outParam Language) (F M N : Type*)
+class StrongHomClass (L : outParam Language) (F : Type*) (M N : outParam Type*)
   [FunLike F M N] [L.Structure M] [L.Structure N] : Prop where
   map_fun : ∀ (φ : F) {n} (f : L.Functions n) (x), φ (funMap f x) = funMap f (φ ∘ x)
   map_rel : ∀ (φ : F) {n} (r : L.Relations n) (x), RelMap r (φ ∘ x) ↔ RelMap r x
@@ -429,7 +269,7 @@ instance (priority := 100) StrongHomClass.homClass {F : Type*} [L.Structure M]
 theorem HomClass.strongHomClassOfIsAlgebraic [L.IsAlgebraic] {F M N} [L.Structure M] [L.Structure N]
     [FunLike F M N] [HomClass L F M N] : StrongHomClass L F M N where
   map_fun := HomClass.map_fun
-  map_rel _ n R _ := (IsAlgebraic.empty_relations n).elim R
+  map_rel _ _ := isEmptyElim
 
 theorem HomClass.map_constants {F M N} [L.Structure M] [L.Structure N] [FunLike F M N]
     [HomClass L F M N] (φ : F) (c : L.Constants) : φ c = c :=
@@ -526,7 +366,7 @@ theorem id_comp (f : M →[L] N) : (id L N).comp f = f :=
 end Hom
 
 /-- Any element of a `HomClass` can be realized as a first_order homomorphism. -/
-def HomClass.toHom {F M N} [L.Structure M] [L.Structure N] [FunLike F M N]
+@[simps] def HomClass.toHom {F M N} [L.Structure M] [L.Structure N] [FunLike F M N]
     [HomClass L F M N] : F → M →[L] N := fun φ =>
   ⟨φ, HomClass.map_fun φ, HomClass.map_rel φ⟩
 
@@ -634,7 +474,7 @@ def comp (hnp : N ↪[L] P) (hmn : M ↪[L] N) : M ↪[L] P where
   -- Porting note: should be done by autoparam?
   map_fun' := by intros; simp only [Function.comp_apply, map_fun]; trivial
   -- Porting note: should be done by autoparam?
-  map_rel' := by intros; rw [Function.comp.assoc, map_rel, map_rel]
+  map_rel' := by intros; rw [Function.comp_assoc, map_rel, map_rel]
 
 @[simp]
 theorem comp_apply (g : N ↪[L] P) (f : M ↪[L] N) (x : M) : g.comp f x = g (f x) :=
@@ -681,7 +521,7 @@ theorem refl_toHom : (refl L M).toHom = Hom.id L M :=
 end Embedding
 
 /-- Any element of an injective `StrongHomClass` can be realized as a first_order embedding. -/
-def StrongHomClass.toEmbedding {F M N} [L.Structure M] [L.Structure N] [FunLike F M N]
+@[simps] def StrongHomClass.toEmbedding {F M N} [L.Structure M] [L.Structure N] [FunLike F M N]
     [EmbeddingLike F M N] [StrongHomClass L F M N] : F → M ↪[L] N := fun φ =>
   ⟨⟨φ, EmbeddingLike.injective φ⟩, StrongHomClass.map_fun φ, StrongHomClass.map_rel φ⟩
 
@@ -711,11 +551,11 @@ def symm (f : M ≃[L] N) : N ≃[L] M :=
       simp only [Equiv.toFun_as_coe]
       rw [Equiv.symm_apply_eq]
       refine Eq.trans ?_ (f.map_fun' f' (f.toEquiv.symm ∘ x)).symm
-      rw [← Function.comp.assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.id_comp]
+      rw [← Function.comp_assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.id_comp]
     map_rel' := fun n r {x} => by
       simp only [Equiv.toFun_as_coe]
       refine (f.map_rel' r (f.toEquiv.symm ∘ x)).symm.trans ?_
-      rw [← Function.comp.assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.id_comp] }
+      rw [← Function.comp_assoc, Equiv.toFun_as_coe, Equiv.self_comp_symm, Function.id_comp] }
 
 instance hasCoeToFun : CoeFun (M ≃[L] N) fun _ => M → N :=
   DFunLike.hasCoeToFun
@@ -808,7 +648,7 @@ def comp (hnp : N ≃[L] P) (hmn : M ≃[L] N) : M ≃[L] P :=
     -- Porting note: should be done by autoparam?
     map_fun' := by intros; simp only [Function.comp_apply, map_fun]; trivial
     -- Porting note: should be done by autoparam?
-    map_rel' := by intros; rw [Function.comp.assoc, map_rel, map_rel] }
+    map_rel' := by intros; rw [Function.comp_assoc, map_rel, map_rel] }
 
 @[simp]
 theorem comp_apply (g : N ≃[L] P) (f : M ≃[L] N) (x : M) : g.comp f x = g (f x) :=
@@ -895,7 +735,7 @@ theorem comp_right_inj (h : M ≃[L] N) (f g : N ≃[L] P) : f.comp h = g.comp h
 end Equiv
 
 /-- Any element of a bijective `StrongHomClass` can be realized as a first_order isomorphism. -/
-def StrongHomClass.toEquiv {F M N} [L.Structure M] [L.Structure N] [EquivLike F M N]
+@[simps] def StrongHomClass.toEquiv {F M N} [L.Structure M] [L.Structure N] [EquivLike F M N]
     [StrongHomClass L F M N] : F → M ≃[L] N := fun φ =>
   ⟨⟨φ, EquivLike.inv φ, EquivLike.left_inv φ, EquivLike.right_inv φ⟩, StrongHomClass.map_fun φ,
     StrongHomClass.map_rel φ⟩
@@ -935,8 +775,7 @@ end SumStructure
 section Empty
 
 /-- Any type can be made uniquely into a structure over the empty language. -/
-def emptyStructure : Language.empty.Structure M :=
-  ⟨Empty.elim, Empty.elim⟩
+def emptyStructure : Language.empty.Structure M where
 
 instance : Unique (Language.empty.Structure M) :=
   ⟨⟨Language.emptyStructure⟩, fun a => by
@@ -991,8 +830,8 @@ def inducedStructureEquiv (e : M ≃ N) : @Language.Equiv L M N _ (inducedStruct
   letI : L.Structure N := inducedStructure e
   exact
   { e with
-    map_fun' := @fun n f x => by simp [← Function.comp.assoc e.symm e x]
-    map_rel' := @fun n r x => by simp [← Function.comp.assoc e.symm e x] }
+    map_fun' := @fun n f x => by simp [← Function.comp_assoc e.symm e x]
+    map_rel' := @fun n r x => by simp [← Function.comp_assoc e.symm e x] }
 
 @[simp]
 theorem toEquiv_inducedStructureEquiv (e : M ≃ N) :
diff --git a/Mathlib/ModelTheory/Complexity.lean b/Mathlib/ModelTheory/Complexity.lean
index 3414496216c54..e4d9bfc7ef31f 100644
--- a/Mathlib/ModelTheory/Complexity.lean
+++ b/Mathlib/ModelTheory/Complexity.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
 -/
-import Mathlib.ModelTheory.Satisfiability
+import Mathlib.ModelTheory.Equivalence
 
 /-!
 # Quantifier Complexity
@@ -169,11 +169,10 @@ theorem IsQF.toPrenexImpRight {φ : L.BoundedFormula α n} :
 
 theorem isPrenex_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ) (hψ : IsPrenex ψ) :
     IsPrenex (φ.toPrenexImpRight ψ) := by
-  induction' hψ with _ _ hψ _ _ _ ih1 _ _ _ ih2
-  · rw [hψ.toPrenexImpRight]
-    exact (hφ.imp hψ).isPrenex
-  · exact (ih1 hφ.liftAt).all
-  · exact (ih2 hφ.liftAt).ex
+  induction hψ with
+  | of_isQF hψ => rw [hψ.toPrenexImpRight]; exact (hφ.imp hψ).isPrenex
+  | all _ ih1 => exact (ih1 hφ.liftAt).all
+  | ex _ ih2 => exact (ih2 hφ.liftAt).ex
 
 -- Porting note: universes in different order
 /-- An auxiliary operation to `FirstOrder.Language.BoundedFormula.toPrenex`.
@@ -196,11 +195,10 @@ theorem IsQF.toPrenexImp :
 
 theorem isPrenex_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ) (hψ : IsPrenex ψ) :
     IsPrenex (φ.toPrenexImp ψ) := by
-  induction' hφ with _ _ hφ _ _ _ ih1 _ _ _ ih2
-  · rw [hφ.toPrenexImp]
-    exact isPrenex_toPrenexImpRight hφ hψ
-  · exact (ih1 hψ.liftAt).ex
-  · exact (ih2 hψ.liftAt).all
+  induction hφ with
+  | of_isQF hφ => rw [hφ.toPrenexImp]; exact isPrenex_toPrenexImpRight hφ hψ
+  | all _ ih1 => exact (ih1 hψ.liftAt).ex
+  | ex _ ih2 => exact (ih2 hψ.liftAt).all
 
 -- Porting note: universes in different order
 /-- For any bounded formula `φ`, `φ.toPrenex` is a semantically-equivalent formula in prenex normal
@@ -222,12 +220,14 @@ variable [Nonempty M]
 theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ) (hψ : IsPrenex ψ)
     {v : α → M} {xs : Fin n → M} :
     (φ.toPrenexImpRight ψ).Realize v xs ↔ (φ.imp ψ).Realize v xs := by
-  induction' hψ with _ _ hψ _ _ _hψ ih _ _ _hψ ih
-  · rw [hψ.toPrenexImpRight]
-  · refine _root_.trans (forall_congr' fun _ => ih hφ.liftAt) ?_
+  induction hψ with
+  | of_isQF hψ => rw [hψ.toPrenexImpRight]
+  | all _ ih =>
+    refine _root_.trans (forall_congr' fun _ => ih hφ.liftAt) ?_
     simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
     exact ⟨fun h1 a h2 => h1 h2 a, fun h1 h2 a => h1 a h2⟩
-  · unfold toPrenexImpRight
+  | ex _ ih =>
+    unfold toPrenexImpRight
     rw [realize_ex]
     refine _root_.trans (exists_congr fun _ => ih hφ.liftAt) ?_
     simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_ex]
@@ -243,10 +243,14 @@ theorem realize_toPrenexImpRight {φ ψ : L.BoundedFormula α n} (hφ : IsQF φ)
 theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ) (hψ : IsPrenex ψ)
     {v : α → M} {xs : Fin n → M} : (φ.toPrenexImp ψ).Realize v xs ↔ (φ.imp ψ).Realize v xs := by
   revert ψ
-  induction' hφ with _ _ hφ _ _ _hφ ih _ _ _hφ ih <;> intro ψ hψ
-  · rw [hφ.toPrenexImp]
+  induction hφ with
+  | of_isQF hφ =>
+    intro ψ hψ
+    rw [hφ.toPrenexImp]
     exact realize_toPrenexImpRight hφ hψ
-  · unfold toPrenexImp
+  | all _ ih =>
+    intro ψ hψ
+    unfold toPrenexImp
     rw [realize_ex]
     refine _root_.trans (exists_congr fun _ => ih hψ.liftAt) ?_
     simp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]
@@ -258,20 +262,24 @@ theorem realize_toPrenexImp {φ ψ : L.BoundedFormula α n} (hφ : IsPrenex φ)
         exact ⟨default, fun _h'' => h⟩
       · obtain ⟨a, ha⟩ := not_forall.1 (h ∘ h')
         exact ⟨a, fun h => (ha h).elim⟩
-  · refine _root_.trans (forall_congr' fun _ => ih hψ.liftAt) ?_
+  | ex _ ih =>
+    intro ψ hψ
+    refine _root_.trans (forall_congr' fun _ => ih hψ.liftAt) ?_
     simp
 
 @[simp]
 theorem realize_toPrenex (φ : L.BoundedFormula α n) {v : α → M} :
     ∀ {xs : Fin n → M}, φ.toPrenex.Realize v xs ↔ φ.Realize v xs := by
-  induction' φ with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h
-  · exact Iff.rfl
-  · exact Iff.rfl
-  · exact Iff.rfl
-  · intros
+  induction φ with
+  | falsum => exact Iff.rfl
+  | equal => exact Iff.rfl
+  | rel => exact Iff.rfl
+  | imp f1 f2 h1 h2 =>
+    intros
     rw [toPrenex, realize_toPrenexImp f1.toPrenex_isPrenex f2.toPrenex_isPrenex, realize_imp,
       realize_imp, h1, h2]
-  · intros
+  | all _ h =>
+    intros
     rw [realize_all, toPrenex, realize_all]
     exact forall_congr' fun a => h
 
@@ -280,25 +288,25 @@ theorem IsQF.induction_on_sup_not {P : L.BoundedFormula α n → Prop} {φ : L.B
     (ha : ∀ ψ : L.BoundedFormula α n, IsAtomic ψ → P ψ)
     (hsup : ∀ {φ₁ φ₂}, P φ₁ → P φ₂ → P (φ₁ ⊔ φ₂)) (hnot : ∀ {φ}, P φ → P φ.not)
     (hse :
-      ∀ {φ₁ φ₂ : L.BoundedFormula α n}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)) :
+      ∀ {φ₁ φ₂ : L.BoundedFormula α n}, (φ₁ ⇔[∅] φ₂) → (P φ₁ ↔ P φ₂)) :
     P φ :=
   IsQF.recOn h hf @(ha) fun {φ₁ φ₂} _ _ h1 h2 =>
-    (hse (φ₁.imp_semanticallyEquivalent_not_sup φ₂)).2 (hsup (hnot h1) h2)
+    (hse (φ₁.imp_iff_not_sup φ₂)).2 (hsup (hnot h1) h2)
 
 theorem IsQF.induction_on_inf_not {P : L.BoundedFormula α n → Prop} {φ : L.BoundedFormula α n}
     (h : IsQF φ) (hf : P (⊥ : L.BoundedFormula α n))
     (ha : ∀ ψ : L.BoundedFormula α n, IsAtomic ψ → P ψ)
     (hinf : ∀ {φ₁ φ₂}, P φ₁ → P φ₂ → P (φ₁ ⊓ φ₂)) (hnot : ∀ {φ}, P φ → P φ.not)
     (hse :
-      ∀ {φ₁ φ₂ : L.BoundedFormula α n}, Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)) :
+      ∀ {φ₁ φ₂ : L.BoundedFormula α n}, (φ₁ ⇔[∅] φ₂) → (P φ₁ ↔ P φ₂)) :
     P φ :=
   h.induction_on_sup_not hf ha
     (fun {φ₁ φ₂} h1 h2 =>
-      (hse (φ₁.sup_semanticallyEquivalent_not_inf_not φ₂)).2 (hnot (hinf (hnot h1) (hnot h2))))
+      (hse (φ₁.sup_iff_not_inf_not φ₂)).2 (hnot (hinf (hnot h1) (hnot h2))))
     (fun {_} => hnot) fun {_ _} => hse
 
-theorem semanticallyEquivalent_toPrenex (φ : L.BoundedFormula α n) :
-    (∅ : L.Theory).SemanticallyEquivalent φ φ.toPrenex := fun M v xs => by
+theorem iff_toPrenex (φ : L.BoundedFormula α n) :
+    φ ⇔[∅] φ.toPrenex := fun M v xs => by
   rw [realize_iff, realize_toPrenex]
 
 theorem induction_on_all_ex {P : ∀ {m}, L.BoundedFormula α m → Prop} (φ : L.BoundedFormula α n)
@@ -306,25 +314,25 @@ theorem induction_on_all_ex {P : ∀ {m}, L.BoundedFormula α m → Prop} (φ :
     (hall : ∀ {m} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P ψ.all)
     (hex : ∀ {m} {φ : L.BoundedFormula α (m + 1)}, P φ → P φ.ex)
     (hse : ∀ {m} {φ₁ φ₂ : L.BoundedFormula α m},
-      Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)) :
+      (φ₁ ⇔[∅] φ₂) → (P φ₁ ↔ P φ₂)) :
     P φ := by
   suffices h' : ∀ {m} {φ : L.BoundedFormula α m}, φ.IsPrenex → P φ from
-    (hse φ.semanticallyEquivalent_toPrenex).2 (h' φ.toPrenex_isPrenex)
+    (hse φ.iff_toPrenex).2 (h' φ.toPrenex_isPrenex)
   intro m φ hφ
-  induction' hφ with _ _ hφ _ _ _ hφ _ _ _ hφ
-  · exact hqf hφ
-  · exact hall hφ
-  · exact hex hφ
+  induction hφ with
+  | of_isQF hφ => exact hqf hφ
+  | all _ hφ => exact hall hφ
+  | ex _ hφ => exact hex hφ
 
 theorem induction_on_exists_not {P : ∀ {m}, L.BoundedFormula α m → Prop} (φ : L.BoundedFormula α n)
     (hqf : ∀ {m} {ψ : L.BoundedFormula α m}, IsQF ψ → P ψ)
     (hnot : ∀ {m} {φ : L.BoundedFormula α m}, P φ → P φ.not)
     (hex : ∀ {m} {φ : L.BoundedFormula α (m + 1)}, P φ → P φ.ex)
     (hse : ∀ {m} {φ₁ φ₂ : L.BoundedFormula α m},
-      Theory.SemanticallyEquivalent ∅ φ₁ φ₂ → (P φ₁ ↔ P φ₂)) :
+      (φ₁ ⇔[∅] φ₂) → (P φ₁ ↔ P φ₂)) :
     P φ :=
   φ.induction_on_all_ex (fun {_ _} => hqf)
-    (fun {_ φ} hφ => (hse φ.all_semanticallyEquivalent_not_ex_not).2 (hnot (hex (hnot hφ))))
+    (fun {_ φ} hφ => (hse φ.all_iff_not_ex_not).2 (hnot (hex (hnot hφ))))
     (fun {_ _} => hex) fun {_ _ _} => hse
 
 /-- A universal formula is a formula defined by applying only universal quantifiers to a
@@ -415,7 +423,7 @@ end BoundedFormula
 /-- A theory is universal when it is comprised only of universal sentences - these theories apply
 also to substructures. -/
 class Theory.IsUniversal (T : L.Theory) : Prop where
-  isUniversal_of_mem : ∀ {φ}, φ ∈ T → φ.IsUniversal
+  isUniversal_of_mem : ∀ ⦃φ⦄, φ ∈ T → φ.IsUniversal
 
 lemma Theory.IsUniversal.models_of_embedding {T : L.Theory} [hT : T.IsUniversal]
     {N : Type*} [L.Structure N] [N ⊨ T] (f : M ↪[L] N) : M ⊨ T := by
@@ -428,6 +436,12 @@ instance Substructure.models_of_isUniversal
     (S : L.Substructure M) (T : L.Theory) [T.IsUniversal] [M ⊨ T] : S ⊨ T :=
   Theory.IsUniversal.models_of_embedding (Substructure.subtype S)
 
+lemma Theory.IsUniversal.insert
+    {T : L.Theory} [hT : T.IsUniversal] {φ : L.Sentence} (hφ : φ.IsUniversal) :
+    (insert φ T).IsUniversal := ⟨by
+  simp only [Set.mem_insert_iff, forall_eq_or_imp, hφ, true_and]
+  exact hT.isUniversal_of_mem⟩
+
 namespace Relations
 
 open BoundedFormula
diff --git a/Mathlib/ModelTheory/Definability.lean b/Mathlib/ModelTheory/Definability.lean
index 771c189b32504..641713521c6c4 100644
--- a/Mathlib/ModelTheory/Definability.lean
+++ b/Mathlib/ModelTheory/Definability.lean
@@ -61,8 +61,8 @@ theorem definable_iff_exists_formula_sum :
   rw [Definable, Equiv.exists_congr_left (BoundedFormula.constantsVarsEquiv)]
   refine exists_congr (fun φ => iff_iff_eq.2 (congr_arg (s = ·) ?_))
   ext
-  simp only [Formula.Realize, BoundedFormula.constantsVarsEquiv, constantsOn, mk₂_Relations,
-    BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq]
+  simp only [BoundedFormula.constantsVarsEquiv, constantsOn,
+    BoundedFormula.mapTermRelEquiv_symm_apply, mem_setOf_eq, Formula.Realize]
   refine BoundedFormula.realize_mapTermRel_id ?_ (fun _ _ _ => rfl)
   intros
   simp only [Term.constantsVarsEquivLeft_symm_apply, Term.realize_varsToConstants,
@@ -74,7 +74,7 @@ theorem definable_iff_exists_formula_sum :
 theorem empty_definable_iff :
     (∅ : Set M).Definable L s ↔ ∃ φ : L.Formula α, s = setOf φ.Realize := by
   rw [Definable, Equiv.exists_congr_left (LEquiv.addEmptyConstants L (∅ : Set M)).onFormula]
-  simp [-constantsOn]
+  simp
 
 theorem definable_iff_empty_definable_with_params :
     A.Definable L s ↔ (∅ : Set M).Definable (L[[A]]) s :=
diff --git a/Mathlib/ModelTheory/DirectLimit.lean b/Mathlib/ModelTheory/DirectLimit.lean
index a9c1fe39191e1..3a74358878660 100644
--- a/Mathlib/ModelTheory/DirectLimit.lean
+++ b/Mathlib/ModelTheory/DirectLimit.lean
@@ -353,13 +353,13 @@ def lift (g : ∀ i, G i ↪[L] P) (Hg : ∀ i j hij x, g j (f i j hij x) = g i
   map_fun' F x := by
     obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x
     change _ = funMap F (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y)
-    rw [funMap_quotient_mk'_sigma_mk', ← Function.comp.assoc, Quotient.lift_comp_mk]
+    rw [funMap_quotient_mk'_sigma_mk', ← Function.comp_assoc, Quotient.lift_comp_mk]
     simp only [Quotient.lift_mk, Embedding.map_fun]
     rfl
   map_rel' R x := by
     obtain ⟨i, y, rfl⟩ := exists_quotient_mk'_sigma_mk'_eq G f x
     change RelMap R (Quotient.lift _ _ ∘ Quotient.mk _ ∘ Structure.Sigma.mk f i ∘ y) ↔ _
-    rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp.assoc,
+    rw [relMap_quotient_mk'_sigma_mk' G f, ← (g i).map_rel R y, ← Function.comp_assoc,
       Quotient.lift_comp_mk]
     rfl
 
diff --git a/Mathlib/ModelTheory/ElementaryMaps.lean b/Mathlib/ModelTheory/ElementaryMaps.lean
index 178809e7de1d5..54428842a22a5 100644
--- a/Mathlib/ModelTheory/ElementaryMaps.lean
+++ b/Mathlib/ModelTheory/ElementaryMaps.lean
@@ -79,12 +79,12 @@ theorem map_boundedFormula (f : M ↪ₑ[L] N) {α : Type*} {n : ℕ} (φ : L.Bo
       f.map_formula' ((φ.restrictFreeVar id).toFormula.relabel (Fintype.equivFin _))
         (Sum.elim (v ∘ (↑)) xs ∘ (Fintype.equivFin _).symm)
     simp only [Formula.realize_relabel, BoundedFormula.realize_toFormula, iff_eq_eq] at h
-    rw [← Function.comp.assoc _ _ (Fintype.equivFin _).symm,
-      Function.comp.assoc _ (Fintype.equivFin _).symm (Fintype.equivFin _),
-      _root_.Equiv.symm_comp_self, Function.comp_id, Function.comp.assoc, Sum.elim_comp_inl,
-      Function.comp.assoc _ _ Sum.inr, Sum.elim_comp_inr, ← Function.comp.assoc] at h
+    rw [← Function.comp_assoc _ _ (Fintype.equivFin _).symm,
+      Function.comp_assoc _ (Fintype.equivFin _).symm (Fintype.equivFin _),
+      _root_.Equiv.symm_comp_self, Function.comp_id, Function.comp_assoc, Sum.elim_comp_inl,
+      Function.comp_assoc _ _ Sum.inr, Sum.elim_comp_inr, ← Function.comp_assoc] at h
     refine h.trans ?_
-    erw [Function.comp.assoc _ _ (Fintype.equivFin _), _root_.Equiv.symm_comp_self,
+    erw [Function.comp_assoc _ _ (Fintype.equivFin _), _root_.Equiv.symm_comp_self,
       Function.comp_id, Sum.elim_comp_inl, Sum.elim_comp_inr (v ∘ Subtype.val) xs,
       ← Set.inclusion_eq_id (s := (BoundedFormula.freeVarFinset φ : Set α)) Set.Subset.rfl,
       BoundedFormula.realize_restrictFreeVar Set.Subset.rfl]
@@ -224,7 +224,7 @@ def ElementaryEmbedding.ofModelsElementaryDiagram (N : Type*) [L.Structure N] [L
                   (Constants.term ∘ Sum.inr ∘ x)).alls).trans
           ?_)
     · simp_rw [Sentence.Realize, BoundedFormula.realize_alls, BoundedFormula.realize_subst,
-        LHom.realize_onBoundedFormula, Formula.Realize, Unique.forall_iff, Function.comp,
+        LHom.realize_onBoundedFormula, Formula.Realize, Unique.forall_iff, Function.comp_def,
         Term.realize_constants]
     · simp_rw [Sentence.Realize, BoundedFormula.realize_alls, BoundedFormula.realize_subst,
         LHom.realize_onBoundedFormula, Formula.Realize, Unique.forall_iff]
diff --git a/Mathlib/ModelTheory/Encoding.lean b/Mathlib/ModelTheory/Encoding.lean
index 4c7fe629d2236..a9157aba577e8 100644
--- a/Mathlib/ModelTheory/Encoding.lean
+++ b/Mathlib/ModelTheory/Encoding.lean
@@ -83,7 +83,7 @@ theorem listDecode_encode_list (l : List (L.Term α)) :
     simp only [h, length_append, length_map, length_finRange, le_add_iff_nonneg_right,
       _root_.zero_le, ↓reduceDIte, getElem_fin, cons.injEq, func.injEq, heq_eq_eq, true_and]
     refine ⟨funext (fun i => ?_), ?_⟩
-    · rw [List.getElem_append, List.getElem_map, List.getElem_finRange]
+    · rw [List.getElem_append_left, List.getElem_map, List.getElem_finRange]
       simp only [length_map, length_finRange, i.2]
     · simp only [length_map, length_finRange, drop_left']
 
@@ -132,8 +132,8 @@ theorem card_sigma : #(Σn, L.Term (α ⊕ (Fin n))) = max ℵ₀ #(α ⊕ (Σi,
       · simp only [Sum.elim_inl, Sigma.mk.inj_iff, heq_eq_eq, var.injEq, Sum.inl.injEq, true_and]
           at h
         rw [h]
-      · simp only [Sum.elim_inl, Sum.elim_inr, Sigma.mk.inj_iff, false_and] at h
-      · simp only [Sum.elim_inr, Sum.elim_inl, Sigma.mk.inj_iff, false_and] at h
+      · simp only [Sum.elim_inl, Sum.elim_inr, Sigma.mk.inj_iff, false_and, reduceCtorEq] at h
+      · simp only [Sum.elim_inr, Sum.elim_inl, Sigma.mk.inj_iff, false_and, reduceCtorEq] at h
       · simp only [Sum.elim_inr, Sigma.mk.inj_iff, heq_eq_eq, func.injEq, true_and] at h
         rw [Sigma.ext_iff.2 ⟨h.1, h.2.1⟩]
 
@@ -146,7 +146,7 @@ instance [Encodable α] [Encodable (Σi, L.Functions i)] : Encodable (L.Term α)
 
 instance [h1 : Countable α] [h2 : Countable (Σl, L.Functions l)] : Countable (L.Term α) := by
   refine mk_le_aleph0_iff.1 (card_le.trans (max_le_iff.2 ?_))
-  simp only [le_refl, mk_sum, add_le_aleph0, lift_le_aleph0, true_and_iff]
+  simp only [le_refl, mk_sum, add_le_aleph0, lift_le_aleph0, true_and]
   exact ⟨Cardinal.mk_le_aleph0, Cardinal.mk_le_aleph0⟩
 
 instance small [Small.{u} α] : Small.{u} (L.Term α) :=
@@ -227,12 +227,16 @@ theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
       simp [listDecode]
     · rw [bind_cons, h φ _, ih]
   rintro ⟨n, φ⟩
-  induction' φ with _ _ _ _ φ_n φ_l φ_R ts _ _ _ ih1 ih2 _ _ ih <;> intro l
-  · rw [listEncode, singleton_append, listDecode]
-  · rw [listEncode, cons_append, cons_append, listDecode, dif_pos]
+  induction φ with
+  | falsum => intro l; rw [listEncode, singleton_append, listDecode]
+  | equal =>
+    intro l
+    rw [listEncode, cons_append, cons_append, listDecode, dif_pos]
     · simp only [eq_mp_eq_cast, cast_eq, eq_self_iff_true, heq_iff_eq, and_self_iff, nil_append]
     · simp only [eq_self_iff_true, heq_iff_eq, and_self_iff]
-  · rw [listEncode, cons_append, cons_append, singleton_append, cons_append, listDecode]
+  | @rel φ_n φ_l φ_R ts =>
+    intro l
+    rw [listEncode, cons_append, cons_append, singleton_append, cons_append, listDecode]
     have h : ∀ i : Fin φ_l, ((List.map Sum.getLeft? (List.map (fun i : Fin φ_l =>
       Sum.inl (⟨(⟨φ_n, rel φ_R ts⟩ : Σn, L.BoundedFormula α n).fst, ts i⟩ :
         Σn, L.Term (α ⊕ (Fin n)))) (finRange φ_l) ++ l)).get? ↑i).join = some ⟨_, ts i⟩ := by
@@ -240,7 +244,7 @@ theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
       simp only [Option.join, map_append, map_map, Option.bind_eq_some, id, exists_eq_right,
         get?_eq_some, length_append, length_map, length_finRange]
       refine ⟨lt_of_lt_of_le i.2 le_self_add, ?_⟩
-      rw [get_eq_getElem, getElem_append, getElem_map]
+      rw [get_eq_getElem, getElem_append_left, getElem_map]
       · simp only [getElem_finRange, Fin.eta, Function.comp_apply, Sum.getLeft?]
       · simp only [length_map, length_finRange, is_lt]
     rw [dif_pos]
@@ -260,11 +264,15 @@ theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
     rw [List.drop_append_eq_append_drop, length_map, length_finRange, Nat.sub_self, drop,
       drop_eq_nil_of_le, nil_append]
     rw [length_map, length_finRange]
-  · simp only [] at *
+  | imp _ _ ih1 ih2 =>
+    intro l
+    simp only [] at *
     rw [listEncode, List.append_assoc, cons_append, listDecode]
     simp only [ih1, ih2, length_cons, le_add_iff_nonneg_left, _root_.zero_le, ↓reduceDIte,
       getElem_cons_zero, getElem_cons_succ, sigmaImp_apply, drop_succ_cons, drop_zero]
-  · simp only [] at *
+  | all _ ih =>
+    intro l
+    simp only [] at *
     rw [listEncode, cons_append, listDecode]
     simp only [ih, length_cons, le_add_iff_nonneg_left, _root_.zero_le, ↓reduceDIte,
       getElem_cons_zero, sigmaAll_apply, drop_succ_cons, drop_zero]
diff --git a/Mathlib/ModelTheory/Equivalence.lean b/Mathlib/ModelTheory/Equivalence.lean
new file mode 100644
index 0000000000000..42f8aff5f8d70
--- /dev/null
+++ b/Mathlib/ModelTheory/Equivalence.lean
@@ -0,0 +1,262 @@
+/-
+Copyright (c) 2021 Aaron Anderson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Aaron Anderson
+-/
+import Mathlib.ModelTheory.Satisfiability
+
+/-!
+# Equivalence of Formulas
+
+## Main Definitions
+- `FirstOrder.Language.Theory.Imp`: `φ ⟹[T] ψ` indicates that `φ` implies `ψ` in models of `T`.
+- `FirstOrder.Language.Theory.Iff`: `φ ⇔[T] ψ` indicates that `φ` and `ψ` are equivalent formulas or
+  sentences in models of `T`.
+
+## TODO
+- Define the quotient of `L.Formula α` modulo `⇔[T]` and its Boolean Algebra structure.
+
+-/
+
+universe u v w w'
+
+open Cardinal CategoryTheory
+
+open Cardinal FirstOrder
+
+namespace FirstOrder
+
+namespace Language
+
+variable {L : Language.{u, v}} {T : L.Theory} {α : Type w} {n : ℕ}
+variable {M : Type*} [Nonempty M] [L.Structure M] [M ⊨ T]
+
+namespace Theory
+
+/-- `φ ⟹[T] ψ` indicates that `φ` implies `ψ` in models of `T`. -/
+protected def Imp (T : L.Theory) (φ ψ : L.BoundedFormula α n) : Prop :=
+  T ⊨ᵇ φ.imp ψ
+
+@[inherit_doc FirstOrder.Language.Theory.Imp]
+scoped[FirstOrder] notation:51 φ:50 " ⟹[" T "] " ψ:51 => Language.Theory.Imp T φ ψ
+
+namespace Imp
+
+@[refl]
+protected theorem refl (φ : L.BoundedFormula α n) : φ ⟹[T] φ := fun _ _ _ => id
+
+instance : IsRefl (L.BoundedFormula α n) T.Imp := ⟨Imp.refl⟩
+
+@[trans]
+protected theorem trans {φ ψ θ : L.BoundedFormula α n} (h1 : φ ⟹[T] ψ) (h2 : ψ ⟹[T] θ) :
+    φ ⟹[T] θ := fun M v xs => (h2 M v xs) ∘ (h1 M v xs)
+
+instance : IsTrans (L.BoundedFormula α n) T.Imp := ⟨fun _ _ _ => Imp.trans⟩
+
+end Imp
+
+section Imp
+
+lemma bot_imp (φ : L.BoundedFormula α n) : ⊥ ⟹[T] φ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_bot, false_implies]
+
+lemma imp_top (φ : L.BoundedFormula α n) : φ ⟹[T] ⊤ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_top, implies_true]
+
+lemma imp_sup_left (φ ψ : L.BoundedFormula α n) : φ ⟹[T] φ ⊔ ψ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_sup]
+  exact Or.inl
+
+lemma imp_sup_right (φ ψ : L.BoundedFormula α n) : ψ ⟹[T] φ ⊔ ψ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_sup]
+  exact Or.inr
+
+lemma sup_imp {φ ψ θ : L.BoundedFormula α n} (h₁ : φ ⟹[T] θ) (h₂ : ψ ⟹[T] θ) :
+    φ ⊔ ψ ⟹[T] θ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_sup]
+  exact fun h => h.elim (h₁ M v xs) (h₂ M v xs)
+
+lemma sup_imp_iff {φ ψ θ : L.BoundedFormula α n} :
+    (φ ⊔ ψ ⟹[T] θ) ↔ (φ ⟹[T] θ) ∧ (ψ ⟹[T] θ) :=
+  ⟨fun h => ⟨(imp_sup_left _ _).trans h, (imp_sup_right _ _).trans h⟩,
+    fun ⟨h₁, h₂⟩ => sup_imp h₁ h₂⟩
+
+lemma inf_imp_left (φ ψ : L.BoundedFormula α n) : φ ⊓ ψ ⟹[T] φ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
+  exact And.left
+
+lemma inf_imp_right (φ ψ : L.BoundedFormula α n) : φ ⊓ ψ ⟹[T] ψ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
+  exact And.right
+
+lemma imp_inf {φ ψ θ : L.BoundedFormula α n} (h₁ : φ ⟹[T] ψ) (h₂ : φ ⟹[T] θ) :
+    φ ⟹[T] ψ ⊓ θ := fun M v xs => by
+  simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
+  exact fun h => ⟨h₁ M v xs h, h₂ M v xs h⟩
+
+lemma imp_inf_iff {φ ψ θ : L.BoundedFormula α n} :
+    (φ ⟹[T] ψ ⊓ θ) ↔ (φ ⟹[T] ψ) ∧ (φ ⟹[T] θ) :=
+  ⟨fun h => ⟨h.trans (inf_imp_left _ _), h.trans (inf_imp_right _ _)⟩,
+    fun ⟨h₁, h₂⟩ => imp_inf h₁ h₂⟩
+
+end Imp
+
+/-- Two (bounded) formulas are semantically equivalent over a theory `T` when they have the same
+interpretation in every model of `T`. (This is also known as logical equivalence, which also has a
+proof-theoretic definition.) -/
+protected def Iff (T : L.Theory) (φ ψ : L.BoundedFormula α n) : Prop :=
+  T ⊨ᵇ φ.iff ψ
+
+@[inherit_doc FirstOrder.Language.Theory.Iff]
+scoped[FirstOrder]
+notation:51 φ:50 " ⇔[" T "] " ψ:51 => Language.Theory.Iff T φ ψ
+
+theorem iff_iff_imp_and_imp {φ ψ : L.BoundedFormula α n} :
+    (φ ⇔[T] ψ) ↔ (φ ⟹[T] ψ) ∧ (ψ ⟹[T] φ) := by
+  simp only [Theory.Imp, ModelsBoundedFormula, BoundedFormula.realize_imp, ← forall_and,
+    Theory.Iff, BoundedFormula.realize_iff, iff_iff_implies_and_implies]
+
+theorem imp_antisymm {φ ψ : L.BoundedFormula α n} (h₁ : φ ⟹[T] ψ) (h₂ : ψ ⟹[T] φ) :
+    φ ⇔[T] ψ :=
+  iff_iff_imp_and_imp.2 ⟨h₁, h₂⟩
+
+namespace Iff
+
+protected theorem mp {φ ψ : L.BoundedFormula α n} (h : φ ⇔[T] ψ) :
+    φ ⟹[T] ψ := (iff_iff_imp_and_imp.1 h).1
+
+protected theorem mpr {φ ψ : L.BoundedFormula α n} (h : φ ⇔[T] ψ) :
+    ψ ⟹[T] φ := (iff_iff_imp_and_imp.1 h).2
+
+@[refl]
+protected theorem refl (φ : L.BoundedFormula α n) : φ ⇔[T] φ :=
+  fun M v xs => by rw [BoundedFormula.realize_iff]
+
+instance : IsRefl (L.BoundedFormula α n) T.Iff :=
+  ⟨Iff.refl⟩
+
+@[symm]
+protected theorem symm {φ ψ : L.BoundedFormula α n}
+    (h : φ ⇔[T] ψ) : ψ ⇔[T] φ := fun M v xs => by
+  rw [BoundedFormula.realize_iff, Iff.comm, ← BoundedFormula.realize_iff]
+  exact h M v xs
+
+instance : IsSymm (L.BoundedFormula α n) T.Iff :=
+  ⟨fun _ _ => Iff.symm⟩
+
+@[trans]
+protected theorem trans {φ ψ θ : L.BoundedFormula α n}
+    (h1 : φ ⇔[T] ψ) (h2 : ψ ⇔[T] θ) :
+    φ ⇔[T] θ := fun M v xs => by
+  have h1' := h1 M v xs
+  have h2' := h2 M v xs
+  rw [BoundedFormula.realize_iff] at *
+  exact ⟨h2'.1 ∘ h1'.1, h1'.2 ∘ h2'.2⟩
+
+instance : IsTrans (L.BoundedFormula α n) T.Iff :=
+  ⟨fun _ _ _ => Iff.trans⟩
+
+theorem realize_bd_iff {φ ψ : L.BoundedFormula α n} (h : φ ⇔[T] ψ)
+    {v : α → M} {xs : Fin n → M} : φ.Realize v xs ↔ ψ.Realize v xs :=
+  BoundedFormula.realize_iff.1 (h.realize_boundedFormula M)
+
+theorem realize_iff {φ ψ : L.Formula α} {M : Type*} [Nonempty M]
+    [L.Structure M] [M ⊨ T] (h : φ ⇔[T] ψ) {v : α → M} :
+    φ.Realize v ↔ ψ.Realize v :=
+  h.realize_bd_iff
+
+theorem models_sentence_iff {φ ψ : L.Sentence} {M : Type*} [Nonempty M]
+    [L.Structure M] [M ⊨ T] (h : φ ⇔[T] ψ) :
+    M ⊨ φ ↔ M ⊨ ψ :=
+  h.realize_iff
+
+protected theorem all {φ ψ : L.BoundedFormula α (n + 1)}
+    (h : φ ⇔[T] ψ) : φ.all ⇔[T] ψ.all := by
+  simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
+    BoundedFormula.realize_all]
+  exact fun M v xs => forall_congr' fun a => h.realize_bd_iff
+
+protected theorem ex {φ ψ : L.BoundedFormula α (n + 1)} (h : φ ⇔[T] ψ) :
+    φ.ex ⇔[T] ψ.ex := by
+  simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
+    BoundedFormula.realize_ex]
+  exact fun M v xs => exists_congr fun a => h.realize_bd_iff
+
+protected theorem not {φ ψ : L.BoundedFormula α n} (h : φ ⇔[T] ψ) :
+    φ.not ⇔[T] ψ.not := by
+  simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
+    BoundedFormula.realize_not]
+  exact fun M v xs => not_congr h.realize_bd_iff
+
+protected theorem imp {φ ψ φ' ψ' : L.BoundedFormula α n} (h : φ ⇔[T] ψ) (h' : φ' ⇔[T] ψ') :
+    (φ.imp φ') ⇔[T] (ψ.imp ψ') := by
+  simp_rw [Theory.Iff, ModelsBoundedFormula, BoundedFormula.realize_iff,
+    BoundedFormula.realize_imp]
+  exact fun M v xs => imp_congr h.realize_bd_iff h'.realize_bd_iff
+
+end Iff
+
+/-- Semantic equivalence forms an equivalence relation on formulas. -/
+def iffSetoid (T : L.Theory) : Setoid (L.BoundedFormula α n) where
+  r := T.Iff
+  iseqv := ⟨fun _ => refl _, fun {_ _} h => h.symm, fun {_ _ _} h1 h2 => h1.trans h2⟩
+
+end Theory
+
+namespace BoundedFormula
+
+variable (φ ψ : L.BoundedFormula α n)
+
+theorem iff_not_not : φ ⇔[T] φ.not.not := fun M v xs => by
+  simp
+
+theorem imp_iff_not_sup : (φ.imp ψ) ⇔[T] (φ.not ⊔ ψ) :=
+  fun M v xs => by simp [imp_iff_not_or]
+
+theorem sup_iff_not_inf_not : (φ ⊔ ψ) ⇔[T] (φ.not ⊓ ψ.not).not :=
+  fun M v xs => by simp [imp_iff_not_or]
+
+theorem inf_iff_not_sup_not : (φ ⊓ ψ) ⇔[T] (φ.not ⊔ ψ.not).not :=
+  fun M v xs => by simp
+
+theorem all_iff_not_ex_not (φ : L.BoundedFormula α (n + 1)) :
+    φ.all ⇔[T] φ.not.ex.not := fun M v xs => by simp
+
+theorem ex_iff_not_all_not (φ : L.BoundedFormula α (n + 1)) :
+    φ.ex ⇔[T] φ.not.all.not := fun M v xs => by simp
+
+theorem iff_all_liftAt : φ ⇔[T] (φ.liftAt 1 n).all :=
+  fun M v xs => by
+  rw [realize_iff, realize_all_liftAt_one_self]
+
+lemma inf_not_iff_bot :
+    φ ⊓ ∼φ ⇔[T] ⊥ := fun M v xs => by
+  simp only [realize_iff, realize_inf, realize_not, and_not_self, realize_bot]
+
+lemma sup_not_iff_top :
+    φ ⊔ ∼φ ⇔[T] ⊤ := fun M v xs => by
+  simp only [realize_iff, realize_sup, realize_not, realize_top, iff_true, or_not]
+
+end BoundedFormula
+
+namespace Formula
+
+variable (φ ψ : L.Formula α)
+
+theorem iff_not_not : φ ⇔[T] φ.not.not :=
+  BoundedFormula.iff_not_not φ
+
+theorem imp_iff_not_sup : (φ.imp ψ) ⇔[T] (φ.not ⊔ ψ) :=
+  BoundedFormula.imp_iff_not_sup φ ψ
+
+theorem sup_iff_not_inf_not : (φ ⊔ ψ) ⇔[T] (φ.not ⊓ ψ.not).not :=
+  BoundedFormula.sup_iff_not_inf_not φ ψ
+
+theorem inf_iff_not_sup_not : (φ ⊓ ψ) ⇔[T] (φ.not ⊔ ψ.not).not :=
+  BoundedFormula.inf_iff_not_sup_not φ ψ
+
+end Formula
+
+end Language
+
+end FirstOrder
diff --git a/Mathlib/ModelTheory/FinitelyGenerated.lean b/Mathlib/ModelTheory/FinitelyGenerated.lean
index 634921b6fb8b5..1ac2ffb332707 100644
--- a/Mathlib/ModelTheory/FinitelyGenerated.lean
+++ b/Mathlib/ModelTheory/FinitelyGenerated.lean
@@ -98,7 +98,7 @@ theorem FG.of_finite {s : L.Substructure M} [h : Finite s] : s.FG :=
 theorem FG.finite [L.IsRelational] {S : L.Substructure M} (h : S.FG) : Finite S := by
   obtain ⟨s, rfl⟩ := h
   have hs := s.finite_toSet
-  rw [← ((closure L).mem_closed_iff _).1 (mem_closed_of_isRelational L (↑s : Set M))] at hs
+  rw [← closure_eq_of_isRelational L (s : Set M)] at hs
   exact hs
 
 theorem fg_iff_finite [L.IsRelational] {S : L.Substructure M} : S.FG ↔ Finite S :=
diff --git a/Mathlib/ModelTheory/Fraisse.lean b/Mathlib/ModelTheory/Fraisse.lean
index 6107ba6f5fdea..49c2b90badb5d 100644
--- a/Mathlib/ModelTheory/Fraisse.lean
+++ b/Mathlib/ModelTheory/Fraisse.lean
@@ -42,8 +42,12 @@ Fraïssé limit - the countable ultrahomogeneous structure with that age.
   essentially countable.
 - `FirstOrder.Language.exists_countable_is_age_of_iff` gives necessary and sufficient conditions
   for a class to be the age of a countable structure in a language with countably many functions.
-- `FirstOrder.Language.IsFraisseLimit.nonempty_equiv` shows that any class which is Fraïsse has
-  at most one Fraïsse limit up to equivalence.
+- `FirstOrder.Language.IsFraisseLimit.nonempty_equiv` shows that any class which is Fraïssé has
+  at most one Fraïssé limit up to equivalence.
+- `FirstOrder.Language.empty.isFraisseLimit_of_countable_infinite` shows that any countably infinite
+  structure in the empty language is a Fraïssé limit of the class of finite structures.
+- `FirstOrder.Language.empty.isFraisse_finite` shows that the class of finite structures in the
+  empty language is Fraïssé.
 
 ## Implementation Notes
 
@@ -58,7 +62,7 @@ Fraïssé limit - the countable ultrahomogeneous structure with that age.
 
 ## TODO
 
-- Show existence and uniqueness of Fraïssé limits
+- Show existence of Fraïssé limits
 
 -/
 
@@ -77,7 +81,7 @@ open Structure Substructure
 
 variable (L : Language.{u, v})
 
-/-! ### The Age of a Structure and Fraïssé Classes-/
+/-! ### The Age of a Structure and Fraïssé Classes -/
 
 
 /-- The age of a structure `M` is the class of finitely-generated structures that embed into it. -/
@@ -88,7 +92,7 @@ variable {L}
 variable (K : Set (Bundled.{w} L.Structure))
 
 /-- A class `K` has the hereditary property when all finitely-generated structures that embed into
-  structures in `K` are also in `K`.  -/
+  structures in `K` are also in `K`. -/
 def Hereditary : Prop :=
   ∀ M : Bundled.{w} L.Structure, M ∈ K → L.age M ⊆ K
 
@@ -403,6 +407,35 @@ theorem nonempty_equiv : Nonempty (M ≃[L] N) := by
 
 end IsFraisseLimit
 
+namespace empty
+
+/-- Any countable infinite structure in the empty language is a Fraïssé limit of the class of finite
+structures. -/
+theorem isFraisseLimit_of_countable_infinite
+    (M : Type*) [Countable M] [Infinite M] [Language.empty.Structure M] :
+    IsFraisseLimit { S : Bundled Language.empty.Structure | Finite S } M where
+  age := by
+    ext S
+    simp only [age, Structure.fg_iff_finite, mem_setOf_eq, and_iff_left_iff_imp]
+    intro hS
+    simp
+  ultrahomogeneous S hS f := by
+    classical
+    have : Finite S := hS.finite
+    have : Infinite { x // x ∉ S } := ((Set.toFinite _).infinite_compl).to_subtype
+    have : Finite f.toHom.range := (((Substructure.fg_iff_structure_fg S).1 hS).range _).finite
+    have : Infinite { x // x ∉ f.toHom.range } := ((Set.toFinite _).infinite_compl ).to_subtype
+    refine ⟨StrongHomClass.toEquiv (f.equivRange.subtypeCongr nonempty_equiv_of_countable.some), ?_⟩
+    ext x
+    simp [Equiv.subtypeCongr]
+
+/-- The class of finite structures in the empty language is Fraïssé. -/
+theorem isFraisse_finite : IsFraisse { S : Bundled.{w} Language.empty.Structure | Finite S } := by
+  have : Language.empty.Structure (ULift ℕ : Type w) := emptyStructure
+  exact (isFraisseLimit_of_countable_infinite (ULift ℕ)).isFraisse
+
+end empty
+
 end Language
 
 end FirstOrder
diff --git a/Mathlib/ModelTheory/Graph.lean b/Mathlib/ModelTheory/Graph.lean
index 06519b49d2e04..c4ce433290e9c 100644
--- a/Mathlib/ModelTheory/Graph.lean
+++ b/Mathlib/ModelTheory/Graph.lean
@@ -21,9 +21,7 @@ This file defines first-order languages, structures, and theories in graph theor
   of the theory of simple graphs.
 -/
 
-
-
-universe u v w w'
+universe u v
 
 namespace FirstOrder
 
@@ -33,30 +31,31 @@ open FirstOrder
 
 open Structure
 
-variable {L : Language.{u, v}} {α : Type w} {V : Type w'} {n : ℕ}
+variable {α : Type u} {V : Type v} {n : ℕ}
 
 /-! ### Simple Graphs -/
 
+/-- The type of relations for the language of graphs, consisting of a single binary relation `adj`.
+-/
+inductive graphRel : ℕ → Type
+  | adj : graphRel 2
+  deriving DecidableEq
 
 /-- The language consisting of a single relation representing adjacency. -/
-protected def graph : Language :=
-  Language.mk₂ Empty Empty Empty Empty Unit
+protected def graph : Language := ⟨fun _ => Empty, graphRel⟩
+  deriving IsRelational
 
 /-- The symbol representing the adjacency relation. -/
-def adj : Language.graph.Relations 2 :=
-  Unit.unit
+abbrev adj : Language.graph.Relations 2 := .adj
 
 /-- Any simple graph can be thought of as a structure in the language of graphs. -/
-def _root_.SimpleGraph.structure (G : SimpleGraph V) : Language.graph.Structure V :=
-  Structure.mk₂ Empty.elim Empty.elim Empty.elim Empty.elim fun _ => G.Adj
+def _root_.SimpleGraph.structure (G : SimpleGraph V) : Language.graph.Structure V where
+  RelMap | .adj => (fun x => G.Adj (x 0) (x 1))
 
 namespace graph
 
-instance instIsRelational : IsRelational Language.graph :=
-  Language.isRelational_mk₂
-
 instance instSubsingleton : Subsingleton (Language.graph.Relations n) :=
-  Language.subsingleton_mk₂_relations
+  ⟨by rintro ⟨⟩ ⟨⟩; rfl⟩
 
 end graph
 
@@ -73,7 +72,8 @@ theorem Theory.simpleGraph_model_iff [Language.graph.Structure V] :
 
 instance simpleGraph_model (G : SimpleGraph V) :
     @Theory.Model _ V G.structure Theory.simpleGraph := by
-  simp only [@Theory.simpleGraph_model_iff _ G.structure, relMap_apply₂]
+  letI := G.structure
+  rw [Theory.simpleGraph_model_iff]
   exact ⟨G.loopless, G.symm⟩
 
 variable (V)
@@ -104,19 +104,14 @@ theorem structure_simpleGraphOfStructure [S : Language.graph.Structure V] [V ⊨
     (simpleGraphOfStructure V).structure = S := by
   ext
   case funMap n f xs =>
-    exact (IsRelational.empty_functions n).elim f
+    exact isEmptyElim f
   case RelMap n r xs =>
-    rw [iff_eq_eq]
-    cases' n with n
-    · exact r.elim
-    · cases' n with n
-      · exact r.elim
-      · cases' n with n
-        · cases r
-          change RelMap adj ![xs 0, xs 1] = _
-          refine congr rfl (funext ?_)
-          simp [Fin.forall_fin_two]
-        · exact r.elim
+    match n, r with
+    | 2, .adj =>
+      rw [iff_eq_eq]
+      change RelMap adj ![xs 0, xs 1] = _
+      refine congr rfl (funext ?_)
+      simp [Fin.forall_fin_two]
 
 theorem Theory.simpleGraph_isSatisfiable : Theory.IsSatisfiable Theory.simpleGraph :=
   ⟨@Theory.ModelType.of _ _ Unit (SimpleGraph.structure ⊥) _ _⟩
diff --git a/Mathlib/ModelTheory/LanguageMap.lean b/Mathlib/ModelTheory/LanguageMap.lean
index a70065ee3ba7e..e7a57200c37cb 100644
--- a/Mathlib/ModelTheory/LanguageMap.lean
+++ b/Mathlib/ModelTheory/LanguageMap.lean
@@ -47,8 +47,10 @@ variable (L : Language.{u, v}) (L' : Language.{u', v'}) {M : Type w} [L.Structur
 
 /-- A language homomorphism maps the symbols of one language to symbols of another. -/
 structure LHom where
-  onFunction : ∀ ⦃n⦄, L.Functions n → L'.Functions n
-  onRelation : ∀ ⦃n⦄, L.Relations n → L'.Relations n
+  onFunction : ∀ ⦃n⦄, L.Functions n → L'.Functions n := by
+    exact fun {n} => isEmptyElim
+  onRelation : ∀ ⦃n⦄, L.Relations n → L'.Relations n :=by
+    exact fun {n} => isEmptyElim
 
 @[inherit_doc FirstOrder.Language.LHom]
 infixl:10 " →ᴸ " => LHom
@@ -58,16 +60,6 @@ variable {L L'}
 
 namespace LHom
 
-/-- Defines a map between languages defined with `Language.mk₂`. -/
-protected def mk₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} (φ₀ : c → L'.Constants)
-    (φ₁ : f₁ → L'.Functions 1) (φ₂ : f₂ → L'.Functions 2) (φ₁' : r₁ → L'.Relations 1)
-    (φ₂' : r₂ → L'.Relations 2) : Language.mk₂ c f₁ f₂ r₁ r₂ →ᴸ L' :=
-  ⟨fun n =>
-    Nat.casesOn n φ₀ fun n => Nat.casesOn n φ₁ fun n => Nat.casesOn n φ₂ fun _ => PEmpty.elim,
-    fun n =>
-    Nat.casesOn n PEmpty.elim fun n =>
-      Nat.casesOn n φ₁' fun n => Nat.casesOn n φ₂' fun _ => PEmpty.elim⟩
-
 variable (ϕ : L →ᴸ L')
 
 /-- Pulls a structure back along a language map. -/
@@ -97,8 +89,7 @@ variable (L L')
 
 /-- The inclusion of an empty language into any other language. -/
 @[simps]
-protected def ofIsEmpty [L.IsAlgebraic] [L.IsRelational] : L →ᴸ L' :=
-  ⟨fun n => (IsRelational.empty_functions n).elim, fun n => (IsAlgebraic.empty_relations n).elim⟩
+protected def ofIsEmpty [L.IsAlgebraic] [L.IsRelational] : L →ᴸ L' where
 
 variable {L L'} {L'' : Language}
 
@@ -113,23 +104,6 @@ protected theorem funext {F G : L →ᴸ L'} (h_fun : F.onFunction = G.onFunctio
 instance [L.IsAlgebraic] [L.IsRelational] : Unique (L →ᴸ L') :=
   ⟨⟨LHom.ofIsEmpty L L'⟩, fun _ => LHom.funext (Subsingleton.elim _ _) (Subsingleton.elim _ _)⟩
 
-theorem mk₂_funext {c f₁ f₂ : Type u} {r₁ r₂ : Type v} {F G : Language.mk₂ c f₁ f₂ r₁ r₂ →ᴸ L'}
-    (h0 : ∀ c : (Language.mk₂ c f₁ f₂ r₁ r₂).Constants, F.onFunction c = G.onFunction c)
-    (h1 : ∀ f : (Language.mk₂ c f₁ f₂ r₁ r₂).Functions 1, F.onFunction f = G.onFunction f)
-    (h2 : ∀ f : (Language.mk₂ c f₁ f₂ r₁ r₂).Functions 2, F.onFunction f = G.onFunction f)
-    (h1' : ∀ r : (Language.mk₂ c f₁ f₂ r₁ r₂).Relations 1, F.onRelation r = G.onRelation r)
-    (h2' : ∀ r : (Language.mk₂ c f₁ f₂ r₁ r₂).Relations 2, F.onRelation r = G.onRelation r) :
-    F = G :=
-  LHom.funext
-    (funext fun n =>
-      Nat.casesOn n (funext h0) fun n =>
-        Nat.casesOn n (funext h1) fun n =>
-          Nat.casesOn n (funext h2) fun _n => funext fun f => PEmpty.elim f)
-    (funext fun n =>
-      Nat.casesOn n (funext fun r => PEmpty.elim r) fun n =>
-        Nat.casesOn n (funext h1') fun n =>
-          Nat.casesOn n (funext h2') fun _n => funext fun r => PEmpty.elim r)
-
 /-- The composition of two language homomorphisms. -/
 @[simps]
 def comp (g : L' →ᴸ L'') (f : L →ᴸ L') : L →ᴸ L'' :=
@@ -219,9 +193,12 @@ noncomputable def defaultExpansion (ϕ : L →ᴸ L')
 /-- A language homomorphism is an expansion on a structure if it commutes with the interpretation of
 all symbols on that structure. -/
 class IsExpansionOn (M : Type*) [L.Structure M] [L'.Structure M] : Prop where
-  map_onFunction : ∀ {n} (f : L.Functions n) (x : Fin n → M), funMap (ϕ.onFunction f) x = funMap f x
-  map_onRelation : ∀ {n} (R : L.Relations n) (x : Fin n → M),
-    RelMap (ϕ.onRelation R) x = RelMap R x
+  map_onFunction :
+    ∀ {n} (f : L.Functions n) (x : Fin n → M), funMap (ϕ.onFunction f) x = funMap f x := by
+      exact fun {n} => isEmptyElim
+  map_onRelation :
+    ∀ {n} (R : L.Relations n) (x : Fin n → M), RelMap (ϕ.onRelation R) x = RelMap R x := by
+      exact fun {n} => isEmptyElim
 
 @[simp]
 theorem map_onFunction {M : Type*} [L.Structure M] [L'.Structure M] [ϕ.IsExpansionOn M] {n}
@@ -237,9 +214,7 @@ instance id_isExpansionOn (M : Type*) [L.Structure M] : IsExpansionOn (LHom.id L
   ⟨fun _ _ => rfl, fun _ _ => rfl⟩
 
 instance ofIsEmpty_isExpansionOn (M : Type*) [L.Structure M] [L'.Structure M] [L.IsAlgebraic]
-    [L.IsRelational] : IsExpansionOn (LHom.ofIsEmpty L L') M :=
-  ⟨fun {n} => (IsRelational.empty_functions n).elim,
-   fun {n} => (IsAlgebraic.empty_relations n).elim⟩
+    [L.IsRelational] : IsExpansionOn (LHom.ofIsEmpty L L') M where
 
 instance sumElim_isExpansionOn {L'' : Language} (ψ : L'' →ᴸ L') (M : Type*) [L.Structure M]
     [L'.Structure M] [L''.Structure M] [ϕ.IsExpansionOn M] [ψ.IsExpansionOn M] :
@@ -340,38 +315,47 @@ section ConstantsOn
 
 variable (α : Type u')
 
-/-- A language with constants indexed by a type. -/
+/-- The type of functions for a language consisting only of constant symbols. -/
 @[simp]
-def constantsOn : Language.{u', 0} :=
-  Language.mk₂ α PEmpty PEmpty PEmpty PEmpty
+def constantsOnFunc : ℕ → Type u'
+  | 0 => α
+  | (_ + 1) => PEmpty
+
+/-- A language with constants indexed by a type. -/
+@[simps]
+def constantsOn : Language.{u', 0} := ⟨constantsOnFunc α, fun _ => Empty⟩
 
 variable {α}
 
 theorem constantsOn_constants : (constantsOn α).Constants = α :=
   rfl
 
-instance isAlgebraic_constantsOn : IsAlgebraic (constantsOn α) :=
-  Language.isAlgebraic_mk₂
-
-instance isRelational_constantsOn [_ie : IsEmpty α] : IsRelational (constantsOn α) :=
-  Language.isRelational_mk₂
+instance isAlgebraic_constantsOn : IsAlgebraic (constantsOn α) := by
+  unfold constantsOn
+  infer_instance
 
 instance isEmpty_functions_constantsOn_succ {n : ℕ} : IsEmpty ((constantsOn α).Functions (n + 1)) :=
-  Nat.casesOn n (inferInstanceAs (IsEmpty PEmpty))
-    fun n => Nat.casesOn n (inferInstanceAs (IsEmpty PEmpty))
-    fun _ => (inferInstanceAs (IsEmpty PEmpty))
+  inferInstanceAs (IsEmpty PEmpty)
+
+instance isRelational_constantsOn [_ie : IsEmpty α] : IsRelational (constantsOn α) :=
+  fun n => Nat.casesOn n _ie inferInstance
 
-theorem card_constantsOn : (constantsOn α).card = #α := by simp
+theorem card_constantsOn : (constantsOn α).card = #α := by
+  simp [card_eq_card_functions_add_card_relations, sum_nat_eq_add_sum_succ]
 
 /-- Gives a `constantsOn α` structure to a type by assigning each constant a value. -/
-def constantsOn.structure (f : α → M) : (constantsOn α).Structure M :=
-  Structure.mk₂ f PEmpty.elim PEmpty.elim PEmpty.elim PEmpty.elim
+def constantsOn.structure (f : α → M) : (constantsOn α).Structure M where
+  funMap := fun {n} c _ =>
+    match n, c with
+    | 0, c => f c
 
 variable {β : Type v'}
 
 /-- A map between index types induces a map between constant languages. -/
-def LHom.constantsOnMap (f : α → β) : constantsOn α →ᴸ constantsOn β :=
-  LHom.mk₂ f PEmpty.elim PEmpty.elim PEmpty.elim PEmpty.elim
+def LHom.constantsOnMap (f : α → β) : constantsOn α →ᴸ constantsOn β where
+  onFunction := fun {n} c =>
+    match n, c with
+    | 0, c => f c
 
 theorem constantsOnMap_isExpansionOn {f : α → β} {fα : α → M} {fβ : β → M} (h : fβ ∘ f = fα) :
     @LHom.IsExpansionOn _ _ (LHom.constantsOnMap f) M (constantsOn.structure fα)
@@ -404,7 +388,7 @@ theorem card_withConstants :
     L[[α]].card = Cardinal.lift.{w'} L.card + Cardinal.lift.{max u v} #α := by
   rw [withConstants, card_sum, card_constantsOn]
 
-/-- The language map adding constants.  -/
+/-- The language map adding constants. -/
 @[simps!] -- Porting note: add `!` to `simps`
 def lhomWithConstants : L →ᴸ L[[α]] :=
   LHom.sumInl
@@ -420,7 +404,7 @@ protected def con (a : α) : L[[α]].Constants :=
 
 variable {L} (α)
 
-/-- Adds constants to a language map.  -/
+/-- Adds constants to a language map. -/
 def LHom.addConstants {L' : Language} (φ : L →ᴸ L') : L[[α]] →ᴸ L'[[α]] :=
   φ.sumMap (LHom.id _)
 
@@ -429,7 +413,7 @@ instance paramsStructure (A : Set α) : (constantsOn A).Structure α :=
 
 variable (L)
 
-/-- The language map removing an empty constant set.  -/
+/-- The language map removing an empty constant set. -/
 @[simps]
 def LEquiv.addEmptyConstants [ie : IsEmpty α] : L ≃ᴸ L[[α]] where
   toLHom := lhomWithConstants L α
@@ -451,7 +435,7 @@ theorem withConstants_relMap_sum_inl [L[[α]].Structure M] [(lhomWithConstants L
     {n} {R : L.Relations n} {x : Fin n → M} : @RelMap (L[[α]]) M _ n (Sum.inl R) x = RelMap R x :=
   (lhomWithConstants L α).map_onRelation R x
 
-/-- The language map extending the constant set.  -/
+/-- The language map extending the constant set. -/
 def lhomWithConstantsMap (f : α → β) : L[[α]] →ᴸ L[[β]] :=
   LHom.sumMap (LHom.id L) (LHom.constantsOnMap f)
 
diff --git a/Mathlib/ModelTheory/Order.lean b/Mathlib/ModelTheory/Order.lean
index 4e28ffbdd0c0e..af11c905aa569 100644
--- a/Mathlib/ModelTheory/Order.lean
+++ b/Mathlib/ModelTheory/Order.lean
@@ -3,7 +3,10 @@ Copyright (c) 2022 Aaron Anderson. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Aaron Anderson
 -/
+import Mathlib.Data.Rat.Denumerable
 import Mathlib.ModelTheory.Complexity
+import Mathlib.ModelTheory.Fraisse
+import Mathlib.Order.CountableDenseLinearOrder
 
 /-!
 # Ordered First-Ordered Structures
@@ -13,8 +16,6 @@ This file defines ordered first-order languages and structures, as well as their
 ## Main Definitions
 
 - `FirstOrder.Language.order` is the language consisting of a single relation representing `≤`.
-- `FirstOrder.Language.orderStructure` is the structure on an ordered type, assigning the symbol
-  representing `≤` to the actual relation `≤`.
 - `FirstOrder.Language.IsOrdered` points out a specific symbol in a language as representing `≤`.
 - `FirstOrder.Language.OrderedStructure` indicates that the `≤` symbol in an ordered language
   is interpreted as the actual relation `≤` in a particular structure.
@@ -22,11 +23,29 @@ This file defines ordered first-order languages and structures, as well as their
   partial orders, and linear orders.
 - `FirstOrder.Language.dlo` defines the theory of dense linear orders without endpoints, a
   particularly useful example in model theory.
+- `FirstOrder.Language.orderStructure` is the structure on an ordered type, assigning the symbol
+  representing `≤` to the actual relation `≤`.
+- Conversely, `FirstOrder.Language.LEOfStructure`, `FirstOrder.Language.preorderOfModels`,
+  `FirstOrder.Language.partialOrderOfModels`, and `FirstOrder.Language.linearOrderOfModels`
+  are the orders induced by first-order structures modelling the relevant theory.
 
 ## Main Results
 
 - `PartialOrder`s model the theory of partial orders, `LinearOrder`s model the theory of
   linear orders, and dense linear orders without endpoints model `Language.dlo`.
+- Under `L.orderedStructure` assumptions, elements of any `L.HomClass M N` are monotone, and
+  strictly monotone if injective.
+- Under `Language.order.orderedStructure` assumptions, any `OrderHomClass` has an instance of
+  `L.HomClass M N`, while `M ↪o N` and any `OrderIsoClass` have an instance of
+  `L.StrongHomClass M N`.
+- `FirstOrder.Language.isFraisseLimit_of_countable_nonempty_dlo` shows that any countable nonempty
+  model of the theory of linear orders is a Fraïssé limit of the class of finite models of the
+  theory of linear orders.
+- `FirstOrder.Language.isFraisse_finite_linear_order` shows that the class of finite models of the
+  theory of linear orders is Fraïssé.
+- `FirstOrder.Language.aleph0_categorical_dlo` shows that the theory of dense linear orders is
+  `ℵ₀`-categorical, and thus complete.
+
 -/
 
 
@@ -36,34 +55,61 @@ namespace FirstOrder
 
 namespace Language
 
-
 open FirstOrder Structure
 
 variable {L : Language.{u, v}} {α : Type w} {M : Type w'} {n : ℕ}
 
-/-- The language consisting of a single relation representing `≤`. -/
-protected def order : Language :=
-  Language.mk₂ Empty Empty Empty Empty Unit
+/-- The type of relations for the language of orders, consisting of a single binary relation `le`.
+-/
+inductive orderRel : ℕ → Type
+  | le : orderRel 2
+  deriving DecidableEq
 
-instance orderStructure [LE M] : Language.order.Structure M :=
-  Structure.mk₂ Empty.elim Empty.elim Empty.elim Empty.elim fun _ => (· ≤ ·)
+/-- The relational language consisting of a single relation representing `≤`. -/
+protected def order : Language := ⟨fun _ => Empty, orderRel⟩
+  deriving IsRelational
 
-namespace Order
+namespace order
 
-instance Language.instIsRelational : IsRelational Language.order :=
-  Language.isRelational_mk₂
+@[simp]
+lemma forall_relations {P : ∀ (n) (_ : Language.order.Relations n), Prop} :
+    (∀ {n} (R), P n R) ↔ P 2 .le := ⟨fun h => h _, fun h n R =>
+      match n, R with
+      | 2, .le => h⟩
 
-instance Language.instSubsingleton : Subsingleton (Language.order.Relations n) :=
-  Language.subsingleton_mk₂_relations
+instance instSubsingleton : Subsingleton (Language.order.Relations n) :=
+  ⟨by rintro ⟨⟩ ⟨⟩; rfl⟩
 
-end Order
+instance : IsEmpty (Language.order.Relations 0) := ⟨fun x => by cases x⟩
+
+instance : Unique (Σ n, Language.order.Relations n) :=
+  ⟨⟨⟨2, .le⟩⟩, fun ⟨n, R⟩ =>
+      match n, R with
+      | 2, .le => rfl⟩
+
+instance : Unique Language.order.Symbols := ⟨⟨Sum.inr default⟩, by
+  have : IsEmpty (Σ n, Language.order.Functions n) := isEmpty_sigma.2 inferInstance
+  simp only [Symbols, Sum.forall, reduceCtorEq, Sum.inr.injEq, IsEmpty.forall_iff, true_and]
+  exact Unique.eq_default⟩
+
+@[simp]
+lemma card_eq_one : Language.order.card = 1 := by simp [card]
+
+end order
 
 /-- A language is ordered if it has a symbol representing `≤`. -/
 class IsOrdered (L : Language.{u, v}) where
+  /-- The relation symbol representing `≤`. -/
   leSymb : L.Relations 2
 
 export IsOrdered (leSymb)
 
+instance : IsOrdered Language.order :=
+  ⟨.le⟩
+
+lemma order.relation_eq_leSymb : (R : Language.order.Relations 2) → R = leSymb
+  | .le => rfl
+
 section IsOrdered
 
 variable [IsOrdered L]
@@ -80,16 +126,11 @@ variable (L)
 
 /-- The language homomorphism sending the unique symbol `≤` of `Language.order` to `≤` in an ordered
  language. -/
-def orderLHom : Language.order →ᴸ L :=
-  LHom.mk₂ Empty.elim Empty.elim Empty.elim Empty.elim fun _ => leSymb
-
-end IsOrdered
-
-instance : IsOrdered Language.order :=
-  ⟨Unit.unit⟩
+@[simps] def orderLHom : Language.order →ᴸ L where
+  onRelation | _, .le => leSymb
 
 @[simp]
-theorem orderLHom_leSymb [L.IsOrdered] :
+theorem orderLHom_leSymb :
     (orderLHom L).onRelation leSymb = (leSymb : L.Relations 2) :=
   rfl
 
@@ -97,13 +138,6 @@ theorem orderLHom_leSymb [L.IsOrdered] :
 theorem orderLHom_order : orderLHom Language.order = LHom.id Language.order :=
   LHom.funext (Subsingleton.elim _ _) (Subsingleton.elim _ _)
 
-instance sum.instIsOrdered : IsOrdered (L.sum Language.order) :=
-  ⟨Sum.inr IsOrdered.leSymb⟩
-
-section
-
-variable (L) [IsOrdered L]
-
 /-- The theory of preorders. -/
 def preorderTheory : L.Theory :=
   {leSymb.reflexive, leSymb.transitive}
@@ -114,23 +148,17 @@ instance : Theory.IsUniversal L.preorderTheory := ⟨by
 
 /-- The theory of partial orders. -/
 def partialOrderTheory : L.Theory :=
-  {leSymb.reflexive, leSymb.antisymmetric, leSymb.transitive}
+  insert leSymb.antisymmetric L.preorderTheory
 
-instance : Theory.IsUniversal L.partialOrderTheory := ⟨by
-  simp only [partialOrderTheory,
-    Set.mem_insert_iff, Set.mem_singleton_iff, forall_eq_or_imp, forall_eq]
-  exact ⟨leSymb.isUniversal_reflexive, leSymb.isUniversal_antisymmetric,
-    leSymb.isUniversal_transitive⟩⟩
+instance : Theory.IsUniversal L.partialOrderTheory :=
+  Theory.IsUniversal.insert leSymb.isUniversal_antisymmetric
 
 /-- The theory of linear orders. -/
 def linearOrderTheory : L.Theory :=
-  {leSymb.reflexive, leSymb.antisymmetric, leSymb.transitive, leSymb.total}
+  insert leSymb.total L.partialOrderTheory
 
-instance : Theory.IsUniversal L.linearOrderTheory := ⟨by
-  simp only [linearOrderTheory,
-    Set.mem_insert_iff, Set.mem_singleton_iff, forall_eq_or_imp, forall_eq]
-  exact ⟨leSymb.isUniversal_reflexive, leSymb.isUniversal_antisymmetric,
-    leSymb.isUniversal_transitive, leSymb.isUniversal_total⟩⟩
+instance : Theory.IsUniversal L.linearOrderTheory :=
+  Theory.IsUniversal.insert leSymb.isUniversal_total
 
 example [L.Structure M] [M ⊨ L.linearOrderTheory] (S : L.Substructure M) :
     S ⊨ L.linearOrderTheory := inferInstance
@@ -154,72 +182,64 @@ def denselyOrderedSentence : L.Sentence :=
 def dlo : L.Theory :=
   L.linearOrderTheory ∪ {L.noTopOrderSentence, L.noBotOrderSentence, L.denselyOrderedSentence}
 
-end
+variable [L.Structure M]
+
+instance [h : M ⊨ L.dlo] : M ⊨ L.linearOrderTheory := h.mono Set.subset_union_left
+
+instance [h : M ⊨ L.linearOrderTheory] : M ⊨ L.partialOrderTheory := h.mono (Set.subset_insert _ _)
+
+instance [h : M ⊨ L.partialOrderTheory] : M ⊨ L.preorderTheory := h.mono (Set.subset_insert _ _)
+
+end IsOrdered
+
+instance sum.instIsOrdered : IsOrdered (L.sum Language.order) :=
+  ⟨Sum.inr IsOrdered.leSymb⟩
 
 variable (L M)
 
-/-- A structure is ordered if its language has a `≤` symbol whose interpretation is -/
-abbrev OrderedStructure [IsOrdered L] [LE M] [L.Structure M] : Prop :=
-  LHom.IsExpansionOn (orderLHom L) M
+/-- Any linearly-ordered type is naturally a structure in the language `Language.order`.
+This is not an instance, because sometimes the `Language.order.Structure` is defined first. -/
+def orderStructure [LE M] : Language.order.Structure M where
+  RelMap | .le => (fun x => x 0 ≤ x 1)
+
+/-- A structure is ordered if its language has a `≤` symbol whose interpretation is `≤`. -/
+class OrderedStructure [L.IsOrdered] [LE M] [L.Structure M] : Prop where
+  relMap_leSymb : ∀ (x : Fin 2 → M), RelMap (leSymb : L.Relations 2) x ↔ (x 0 ≤ x 1)
+
+export OrderedStructure (relMap_leSymb)
+
+attribute [simp] relMap_leSymb
 
 variable {L M}
 
-@[simp]
-theorem orderedStructure_iff [IsOrdered L] [LE M] [L.Structure M] :
-    L.OrderedStructure M ↔ LHom.IsExpansionOn (orderLHom L) M :=
-  Iff.rfl
+section order_to_structure
 
-instance orderedStructure_LE [LE M] : OrderedStructure Language.order M := by
-  rw [orderedStructure_iff, orderLHom_order]
-  exact LHom.id_isExpansionOn M
+variable [IsOrdered L] [L.Structure M]
 
-instance model_preorder [Preorder M] : M ⊨ Language.order.preorderTheory := by
-  simp only [preorderTheory, Theory.model_iff, Set.mem_insert_iff, Set.mem_singleton_iff,
-    forall_eq_or_imp, Relations.realize_reflexive, relMap_apply₂, forall_eq,
-    Relations.realize_transitive]
-  exact ⟨le_refl, fun _ _ _ => le_trans⟩
+section LE
 
-instance model_partialOrder [PartialOrder M] : M ⊨ Language.order.partialOrderTheory := by
-  simp only [partialOrderTheory, Theory.model_iff, Set.mem_insert_iff, Set.mem_singleton_iff,
-    forall_eq_or_imp, Relations.realize_reflexive, relMap_apply₂, Relations.realize_antisymmetric,
-    forall_eq, Relations.realize_transitive]
-  exact ⟨le_refl, fun _ _ => le_antisymm, fun _ _ _ => le_trans⟩
+variable [LE M]
 
-instance model_linearOrder [LinearOrder M] : M ⊨ Language.order.linearOrderTheory := by
-  simp only [linearOrderTheory, Theory.model_iff, Set.mem_insert_iff, Set.mem_singleton_iff,
-    forall_eq_or_imp, Relations.realize_reflexive, relMap_apply₂, Relations.realize_antisymmetric,
-    Relations.realize_transitive, forall_eq, Relations.realize_total]
-  exact ⟨le_refl, fun _ _ => le_antisymm, fun _ _ _ => le_trans, le_total⟩
+instance [Language.order.Structure M] [Language.order.OrderedStructure M]
+    [(orderLHom L).IsExpansionOn M] : L.OrderedStructure M where
+  relMap_leSymb x := by
+    rw [← orderLHom_leSymb L, LHom.IsExpansionOn.map_onRelation, relMap_leSymb]
 
-section OrderedStructure
+variable [L.OrderedStructure M]
 
-variable [IsOrdered L] [L.Structure M]
+instance [Language.order.Structure M] [Language.order.OrderedStructure M] :
+    LHom.IsExpansionOn (orderLHom L) M where
+  map_onRelation := by simp [order.relation_eq_leSymb]
 
-@[simp]
-theorem relMap_leSymb [LE M] [L.OrderedStructure M] {a b : M} :
-    RelMap (leSymb : L.Relations 2) ![a, b] ↔ a ≤ b := by
-  rw [← orderLHom_leSymb, LHom.map_onRelation]
-  rfl
+instance (S : L.Substructure M) : L.OrderedStructure S := ⟨fun x => relMap_leSymb (S.subtype ∘ x)⟩
 
 @[simp]
-theorem Term.realize_le [LE M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ (Fin n))} {v : α → M}
+theorem Term.realize_le {t₁ t₂ : L.Term (α ⊕ (Fin n))} {v : α → M}
     {xs : Fin n → M} :
     (t₁.le t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) ≤ t₂.realize (Sum.elim v xs) := by
   simp [Term.le]
 
-@[simp]
-theorem Term.realize_lt [Preorder M] [L.OrderedStructure M] {t₁ t₂ : L.Term (α ⊕ (Fin n))}
-    {v : α → M} {xs : Fin n → M} :
-    (t₁.lt t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) < t₂.realize (Sum.elim v xs) := by
-  simp [Term.lt, lt_iff_le_not_le]
-
-end OrderedStructure
-
-section LE
-
-variable [LE M]
-
-theorem realize_noTopOrder_iff : M ⊨ Language.order.noTopOrderSentence ↔ NoTopOrder M := by
+theorem realize_noTopOrder_iff : M ⊨ L.noTopOrderSentence ↔ NoTopOrder M := by
   simp only [noTopOrderSentence, Sentence.Realize, Formula.Realize, BoundedFormula.realize_all,
     BoundedFormula.realize_ex, BoundedFormula.realize_not, Term.realize, Term.realize_le,
     Sum.elim_inr]
@@ -227,11 +247,7 @@ theorem realize_noTopOrder_iff : M ⊨ Language.order.noTopOrderSentence ↔ NoT
   intro h a
   exact exists_not_le a
 
-@[simp]
-theorem realize_noTopOrder [h : NoTopOrder M] : M ⊨ Language.order.noTopOrderSentence :=
-  realize_noTopOrder_iff.2 h
-
-theorem realize_noBotOrder_iff : M ⊨ Language.order.noBotOrderSentence ↔ NoBotOrder M := by
+theorem realize_noBotOrder_iff : M ⊨ L.noBotOrderSentence ↔ NoBotOrder M := by
   simp only [noBotOrderSentence, Sentence.Realize, Formula.Realize, BoundedFormula.realize_all,
     BoundedFormula.realize_ex, BoundedFormula.realize_not, Term.realize, Term.realize_le,
     Sum.elim_inr]
@@ -239,14 +255,49 @@ theorem realize_noBotOrder_iff : M ⊨ Language.order.noBotOrderSentence ↔ NoB
   intro h a
   exact exists_not_ge a
 
+variable (L M)
+
 @[simp]
-theorem realize_noBotOrder [h : NoBotOrder M] : M ⊨ Language.order.noBotOrderSentence :=
+theorem realize_noTopOrder [h : NoTopOrder M] : M ⊨ L.noTopOrderSentence :=
+  realize_noTopOrder_iff.2 h
+
+@[simp]
+theorem realize_noBotOrder [h : NoBotOrder M] : M ⊨ L.noBotOrderSentence :=
   realize_noBotOrder_iff.2 h
 
+theorem noTopOrder_of_dlo [M ⊨ L.dlo] : NoTopOrder M :=
+  realize_noTopOrder_iff.1 (L.dlo.realize_sentence_of_mem (by
+    simp only [dlo, Set.union_insert, Set.union_singleton, Set.mem_insert_iff, true_or]))
+
+theorem noBotOrder_of_dlo [M ⊨ L.dlo] : NoBotOrder M :=
+  realize_noBotOrder_iff.1 (L.dlo.realize_sentence_of_mem (by
+    simp only [dlo, Set.union_insert, Set.union_singleton, Set.mem_insert_iff, true_or, or_true]))
+
 end LE
 
-theorem realize_denselyOrdered_iff [Preorder M] :
-    M ⊨ Language.order.denselyOrderedSentence ↔ DenselyOrdered M := by
+@[simp]
+theorem orderedStructure_iff
+    [LE M] [Language.order.Structure M] [Language.order.OrderedStructure M] :
+    L.OrderedStructure M ↔ LHom.IsExpansionOn (orderLHom L) M :=
+  ⟨fun _ => inferInstance, fun _ => inferInstance⟩
+
+section Preorder
+
+variable [Preorder M] [L.OrderedStructure M]
+
+instance model_preorder : M ⊨ L.preorderTheory := by
+  simp only [preorderTheory, Theory.model_insert_iff, Relations.realize_reflexive, relMap_leSymb,
+    Theory.model_singleton_iff, Relations.realize_transitive]
+  exact ⟨le_refl, fun _ _ _ => le_trans⟩
+
+@[simp]
+theorem Term.realize_lt {t₁ t₂ : L.Term (α ⊕ (Fin n))}
+    {v : α → M} {xs : Fin n → M} :
+    (t₁.lt t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) < t₂.realize (Sum.elim v xs) := by
+  simp [Term.lt, lt_iff_le_not_le]
+
+theorem realize_denselyOrdered_iff :
+    M ⊨ L.denselyOrderedSentence ↔ DenselyOrdered M := by
   simp only [denselyOrderedSentence, Sentence.Realize, Formula.Realize,
     BoundedFormula.realize_imp, BoundedFormula.realize_all, Term.realize, Term.realize_lt,
     Sum.elim_inr, BoundedFormula.realize_ex, BoundedFormula.realize_inf]
@@ -255,18 +306,255 @@ theorem realize_denselyOrdered_iff [Preorder M] :
   exact exists_between ab
 
 @[simp]
-theorem realize_denselyOrdered [Preorder M] [h : DenselyOrdered M] :
-    M ⊨ Language.order.denselyOrderedSentence :=
+theorem realize_denselyOrdered [h : DenselyOrdered M] :
+    M ⊨ L.denselyOrderedSentence :=
   realize_denselyOrdered_iff.2 h
 
-instance model_dlo [LinearOrder M] [DenselyOrdered M] [NoTopOrder M] [NoBotOrder M] :
-    M ⊨ Language.order.dlo := by
-  simp only [dlo, Set.union_insert, Set.union_singleton, Theory.model_iff, Set.mem_insert_iff,
-    forall_eq_or_imp, realize_noTopOrder, realize_noBotOrder, realize_denselyOrdered,
-    true_and_iff]
-  rw [← Theory.model_iff]
-  infer_instance
+variable (L) (M)
+
+theorem denselyOrdered_of_dlo [h: M ⊨ L.dlo] : DenselyOrdered M :=
+  realize_denselyOrdered_iff.1 (L.dlo.realize_sentence_of_mem (by
+    simp only [dlo, Set.union_insert, Set.union_singleton, Set.mem_insert_iff, true_or, or_true]))
+
+end Preorder
+
+instance model_partialOrder [PartialOrder M] [L.OrderedStructure M] :
+    M ⊨ L.partialOrderTheory := by
+  simp only [partialOrderTheory, Theory.model_insert_iff, Relations.realize_antisymmetric,
+    relMap_leSymb, Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,
+    model_preorder, and_true]
+  exact fun _ _ => le_antisymm
+
+section LinearOrder
+
+variable [LinearOrder M] [L.OrderedStructure M]
+
+instance model_linearOrder : M ⊨ L.linearOrderTheory := by
+  simp only [linearOrderTheory, Theory.model_insert_iff, Relations.realize_total, relMap_leSymb,
+    Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons, model_partialOrder,
+    and_true]
+  exact le_total
+
+instance model_dlo [DenselyOrdered M] [NoTopOrder M] [NoBotOrder M] :
+    M ⊨ L.dlo := by
+  simp [dlo, model_linearOrder, Theory.model_insert_iff]
+
+end LinearOrder
+
+end order_to_structure
+
+section structure_to_order
+
+variable (L) [IsOrdered L] (M) [L.Structure M]
+
+/-- Any structure in an ordered language can be ordered correspondingly. -/
+def leOfStructure : LE M where
+  le a b := Structure.RelMap (leSymb : L.Relations 2) ![a,b]
+
+instance : @OrderedStructure L M _ (L.leOfStructure M) _ := by
+  letI := L.leOfStructure M
+  constructor
+  simp only [Fin.forall_fin_succ_pi, Fin.cons_zero, Fin.forall_fin_zero_pi]
+  intros
+  rfl
+
+/-- The order structure on an ordered language is decidable. -/
+-- This should not be a global instance,
+-- because it will match with any `LE` typeclass search
+@[local instance]
+def decidableLEOfStructure
+    [h : DecidableRel (fun (a b : M) => Structure.RelMap (leSymb : L.Relations 2) ![a,b])] :
+    letI := L.leOfStructure M
+    DecidableRel ((· : M) ≤ ·) := h
+
+/-- Any model of a theory of preorders is a preorder. -/
+def preorderOfModels [h : M ⊨ L.preorderTheory] : Preorder M where
+  __ := L.leOfStructure M
+  le_refl := Relations.realize_reflexive.1 ((Theory.model_iff _).1 h _
+    (by simp only [preorderTheory, Set.mem_insert_iff, Set.mem_singleton_iff, true_or]))
+  le_trans := Relations.realize_transitive.1 ((Theory.model_iff _).1 h _
+    (by simp only [preorderTheory, Set.mem_insert_iff, Set.mem_singleton_iff, or_true]))
+
+/-- Any model of a theory of partial orders is a partial order. -/
+def partialOrderOfModels [h : M ⊨ L.partialOrderTheory] : PartialOrder M where
+  __ := L.preorderOfModels M
+  le_antisymm := Relations.realize_antisymmetric.1 ((Theory.model_iff _).1 h _
+    (by simp only [partialOrderTheory, Set.mem_insert_iff, Set.mem_singleton_iff, true_or]))
+
+/-- Any model of a theory of linear orders is a linear order. -/
+def linearOrderOfModels [h : M ⊨ L.linearOrderTheory]
+    [DecidableRel (fun (a b : M) => Structure.RelMap (leSymb : L.Relations 2) ![a,b])] :
+    LinearOrder M where
+  __ := L.partialOrderOfModels M
+  le_total := Relations.realize_total.1 ((Theory.model_iff _).1 h _
+    (by simp only [linearOrderTheory, Set.mem_insert_iff, Set.mem_singleton_iff, true_or]))
+  decidableLE := inferInstance
+
+end structure_to_order
+
+namespace order
+
+variable [Language.order.Structure M] [LE M] [Language.order.OrderedStructure M]
+  {N : Type*} [Language.order.Structure N] [LE N] [Language.order.OrderedStructure N]
+  {F : Type*}
+
+instance [FunLike F M N] [OrderHomClass F M N] : Language.order.HomClass F M N :=
+  ⟨fun _ => isEmptyElim, by
+    simp only [forall_relations, relation_eq_leSymb, relMap_leSymb, Fin.isValue,
+      Function.comp_apply]
+    exact fun φ x => map_rel φ⟩
+
+-- If `OrderEmbeddingClass` or `RelEmbeddingClass` is defined, this should be generalized.
+instance : Language.order.StrongHomClass (M ↪o N) M N :=
+  ⟨fun _ => isEmptyElim,
+    by simp only [order.forall_relations, order.relation_eq_leSymb, relMap_leSymb, Fin.isValue,
+    Function.comp_apply, RelEmbedding.map_rel_iff, implies_true]⟩
+
+instance [EquivLike F M N] [OrderIsoClass F M N] : Language.order.StrongHomClass F M N :=
+  ⟨fun _ => isEmptyElim,
+    by simp only [order.forall_relations, order.relation_eq_leSymb, relMap_leSymb, Fin.isValue,
+      Function.comp_apply, map_le_map_iff, implies_true]⟩
+
+end order
+
+namespace HomClass
+
+variable [L.IsOrdered] [L.Structure M] {N : Type*} [L.Structure N]
+  {F : Type*} [FunLike F M N] [L.HomClass F M N]
+
+lemma monotone [Preorder M] [L.OrderedStructure M] [Preorder N] [L.OrderedStructure N] (f : F) :
+    Monotone f := fun a b => by
+  have h := HomClass.map_rel f leSymb ![a,b]
+  simp only [relMap_leSymb, Fin.isValue, Matrix.cons_val_zero, Matrix.cons_val_one,
+    Matrix.head_cons, Function.comp_apply] at h
+  exact h
+
+lemma strictMono [EmbeddingLike F M N] [PartialOrder M] [L.OrderedStructure M]
+    [PartialOrder N] [L.OrderedStructure N] (f : F) :
+    StrictMono f :=
+  (HomClass.monotone f).strictMono_of_injective (EmbeddingLike.injective f)
+
+end HomClass
+
+/-- This is not an instance because it would form a loop with
+  `FirstOrder.Language.order.instStrongHomClassOfOrderIsoClass`.
+  As both types are `Prop`s, it would only cause a slowdown.  -/
+lemma StrongHomClass.toOrderIsoClass
+    (L : Language) [L.IsOrdered] (M : Type*) [L.Structure M] [LE M] [L.OrderedStructure M]
+    (N : Type*) [L.Structure N] [LE N] [L.OrderedStructure N]
+    (F : Type*) [EquivLike F M N] [L.StrongHomClass F M N] :
+    OrderIsoClass F M N where
+  map_le_map_iff f a b := by
+    have h := StrongHomClass.map_rel f leSymb ![a,b]
+    simp only [relMap_leSymb, Fin.isValue, Function.comp_apply, Matrix.cons_val_zero,
+      Matrix.cons_val_one, Matrix.head_cons] at h
+    exact h
+
+section Fraisse
+
+variable (M)
+
+lemma dlo_isExtensionPair
+    (M : Type w) [Language.order.Structure M] [M ⊨ Language.order.linearOrderTheory]
+    (N : Type w') [Language.order.Structure N] [N ⊨ Language.order.dlo] [Nonempty N] :
+    Language.order.IsExtensionPair M N := by
+  classical
+  rw [isExtensionPair_iff_exists_embedding_closure_singleton_sup]
+  intro S S_fg f m
+  letI := Language.order.linearOrderOfModels M
+  letI := Language.order.linearOrderOfModels N
+  have := Language.order.denselyOrdered_of_dlo N
+  have := Language.order.noBotOrder_of_dlo N
+  have := Language.order.noTopOrder_of_dlo N
+  have := NoBotOrder.to_noMinOrder N
+  have := NoTopOrder.to_noMaxOrder N
+  have hS : Set.Finite (S : Set M) := (S.fg_iff_structure_fg.1 S_fg).finite
+  obtain ⟨g, hg⟩ := Order.exists_orderEmbedding_insert hS.toFinset
+    ((OrderIso.setCongr hS.toFinset (S : Set M) hS.coe_toFinset).toOrderEmbedding.trans
+      (OrderEmbedding.ofStrictMono f (HomClass.strictMono f))) m
+  let g' :
+    ((Substructure.closure Language.order).toFun {m} ⊔ S : Language.order.Substructure M) ↪o N :=
+    ((OrderIso.setCongr _ _ (by
+      convert LowerAdjoint.closure_eq_self_of_mem_closed _
+        (Substructure.mem_closed_of_isRelational Language.order
+        ((insert m hS.toFinset : Finset M) : Set M))
+      simp only [Finset.coe_insert, Set.Finite.coe_toFinset, Substructure.closure_insert,
+        Substructure.closure_eq])).toOrderEmbedding.trans g)
+  use StrongHomClass.toEmbedding g'
+  ext ⟨x, xS⟩
+  refine ((funext_iff.1 hg) ⟨x, ?_⟩).symm
+  simp only [Set.Finite.coe_toFinset, SetLike.mem_coe, xS]
+
+instance (M : Type w) [Language.order.Structure M] [M ⊨ Language.order.dlo] [Nonempty M] :
+    Infinite M := by
+  letI := orderStructure ℚ
+  obtain ⟨f, _⟩ := embedding_from_cg cg_of_countable default (dlo_isExtensionPair ℚ M)
+  exact Infinite.of_injective f f.injective
+
+lemma dlo_age [Language.order.Structure M] [Mdlo : M ⊨ Language.order.dlo] [Nonempty M] :
+    Language.order.age M = {M : CategoryTheory.Bundled.{w'} Language.order.Structure |
+      Finite M ∧ M ⊨ Language.order.linearOrderTheory} := by
+  classical
+  rw [age]
+  ext N
+  refine ⟨fun ⟨hF, h⟩ => ⟨hF.finite, Theory.IsUniversal.models_of_embedding h.some⟩,
+    fun ⟨hF, h⟩ => ⟨FG.of_finite, ?_⟩⟩
+  letI := Language.order.linearOrderOfModels M
+  letI := Language.order.linearOrderOfModels N
+  exact ⟨StrongHomClass.toEmbedding (nonempty_orderEmbedding_of_finite_infinite N M).some⟩
+
+/-- Any countable nonempty model of the theory of dense linear orders is a Fraïssé limit of the
+class of finite models of the theory of linear orders. -/
+theorem isFraisseLimit_of_countable_nonempty_dlo (M : Type w)
+    [Language.order.Structure M] [Countable M] [Nonempty M] [M ⊨ Language.order.dlo] :
+    IsFraisseLimit {M : CategoryTheory.Bundled.{w} Language.order.Structure |
+      Finite M ∧ M ⊨ Language.order.linearOrderTheory} M :=
+  ⟨(isUltrahomogeneous_iff_IsExtensionPair cg_of_countable).2 (dlo_isExtensionPair M M), dlo_age M⟩
+
+/-- The class of finite models of the theory of linear orders is Fraïssé. -/
+theorem isFraisse_finite_linear_order :
+    IsFraisse {M : CategoryTheory.Bundled.{0} Language.order.Structure |
+      Finite M ∧ M ⊨ Language.order.linearOrderTheory} := by
+  letI : Language.order.Structure ℚ := orderStructure _
+  exact (isFraisseLimit_of_countable_nonempty_dlo ℚ).isFraisse
+
+open Cardinal
+
+/-- The theory of dense linear orders is `ℵ₀`-categorical. -/
+theorem aleph0_categorical_dlo : (ℵ₀).Categorical Language.order.dlo := fun M₁ M₂ h₁ h₂ => by
+  obtain ⟨_⟩ := denumerable_iff.2 h₁
+  obtain ⟨_⟩ := denumerable_iff.2 h₂
+  exact (isFraisseLimit_of_countable_nonempty_dlo M₁).nonempty_equiv
+    (isFraisseLimit_of_countable_nonempty_dlo M₂)
+
+/-- The theory of dense linear orders is `ℵ₀`-complete. -/
+theorem dlo_isComplete : Language.order.dlo.IsComplete :=
+  aleph0_categorical_dlo.{0}.isComplete ℵ₀ _ le_rfl (by simp [one_le_aleph0])
+    ⟨by
+      letI : Language.order.Structure ℚ := orderStructure ℚ
+      exact Theory.ModelType.of _ ℚ⟩
+    fun M => inferInstance
+
+end Fraisse
 
 end Language
 
 end FirstOrder
+
+namespace Order
+
+open FirstOrder FirstOrder.Language
+
+/-- A model-theoretic adaptation of the proof of `Order.iso_of_countable_dense`: two countable,
+  dense, nonempty linear orders without endpoints are order isomorphic. -/
+example (α β : Type w') [LinearOrder α] [LinearOrder β]
+    [Countable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α]
+    [Nonempty α] [Countable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] :
+    Nonempty (α ≃o β) := by
+  letI := orderStructure α
+  letI := orderStructure β
+  letI := StrongHomClass.toOrderIsoClass Language.order α β (α ≃[Language.order] β)
+  exact ⟨(IsFraisseLimit.nonempty_equiv (isFraisseLimit_of_countable_nonempty_dlo α)
+    (isFraisseLimit_of_countable_nonempty_dlo β)).some⟩
+
+end Order
diff --git a/Mathlib/ModelTheory/PartialEquiv.lean b/Mathlib/ModelTheory/PartialEquiv.lean
index ba0f9f88df242..3db2f33308397 100644
--- a/Mathlib/ModelTheory/PartialEquiv.lean
+++ b/Mathlib/ModelTheory/PartialEquiv.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Gabin Kolly. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Aaron Anderson, Gabin Kolly
+Authors: Aaron Anderson, Gabin Kolly, David Wärn
 -/
 import Mathlib.ModelTheory.DirectLimit
 import Mathlib.Order.Ideal
@@ -12,7 +12,26 @@ This file defines partial isomorphisms between first-order structures.
 
 ## Main Definitions
 - `FirstOrder.Language.PartialEquiv` is defined so that `L.PartialEquiv M N`, annotated
-  `M ≃ₚ[L] N`, is the type of equivalences between substructures of `M` and `N`.
+  `M ≃ₚ[L] N`, is the type of equivalences between substructures of `M` and `N`. These can be
+  ordered, with an order that is defined here in terms of a commutative square, but could also be
+  defined as the order on the graphs of the partial equivalences under inclusion as subsets of
+  `M × N`.
+- `FirstOrder.Language.FGEquiv` is the type of partial equivalences `M ≃ₚ[L] N` with
+  finitely-generated domain (or equivalently, codomain).
+- `FirstOrder.Language.IsExtensionPair` is defined so that `L.IsExtensionPair M N` indicates that
+  any finitely-generated partial equivalence from `M` to `N` can be extended to include an arbitrary
+  element `m : M` in its domain.
+
+## Main Results
+- `FirstOrder.Language.embedding_from_cg` shows that if structures `M` and `N` form an equivalence
+  pair with `M` countably-generated, then any finite-generated partial equivalence between them
+  can be extended to an embedding `M ↪[L] N`.
+- `FirstOrder.Language.equiv_from_cg` shows that if countably-generated structures `M` and `N` form
+  an equivalence pair in both directions, then any finite-generated partial equivalence between them
+  can be extended to an isomorphism `M ↪[L] N`.
+- The proofs of these results are adapted in part from David Wärn's approach to countable dense
+  linear orders, a special case of this phenomenon in the case where `L = Language.order`.
+
 -/
 
 universe u v w w'
@@ -328,8 +347,10 @@ lemma partialEquivLimit_comp_inclusion {i : ι} :
 
 theorem le_partialEquivLimit (i : ι) : S i ≤ partialEquivLimit S :=
   ⟨le_iSup (f := fun i ↦ (S i).dom) _, by
-    simp only [cod_partialEquivLimit, dom_partialEquivLimit, partialEquivLimit_comp_inclusion,
-      ← Embedding.comp_assoc, subtype_comp_inclusion]⟩
+    #adaptation_note /-- After lean4#5020, these two `simp` calls cannot be combined. -/
+    simp only [partialEquivLimit_comp_inclusion]
+    simp only [cod_partialEquivLimit, dom_partialEquivLimit, ← Embedding.comp_assoc,
+      subtype_comp_inclusion]⟩
 
 end DirectLimit
 
@@ -368,20 +389,63 @@ theorem countable_self_fgequiv_of_countable [Countable M] :
 instance inhabited_self_FGEquiv : Inhabited (L.FGEquiv M M) :=
   ⟨⟨⟨⊥, ⊥, Equiv.refl L (⊥ : L.Substructure M)⟩, fg_bot⟩⟩
 
+instance inhabited_FGEquiv_of_IsEmpty_Constants_and_Relations
+    [IsEmpty L.Constants] [IsEmpty (L.Relations 0)] [L.Structure N] :
+    Inhabited (L.FGEquiv M N) :=
+  ⟨⟨⟨⊥, ⊥, {
+      toFun := isEmptyElim
+      invFun := isEmptyElim
+      left_inv := isEmptyElim
+      right_inv := isEmptyElim
+      map_fun' := fun {n} f x => by
+        cases n
+        · exact isEmptyElim f
+        · exact isEmptyElim (x 0)
+      map_rel' := fun {n} r x => by
+        cases n
+        · exact isEmptyElim r
+        · exact isEmptyElim (x 0)
+    }⟩, fg_bot⟩⟩
+
 /-- Maps to the symmetric finitely-generated partial equivalence. -/
 @[simps]
 def FGEquiv.symm (f : L.FGEquiv M N) : L.FGEquiv N M := ⟨f.1.symm, f.1.dom_fg_iff_cod_fg.1 f.2⟩
 
-lemma IsExtensionPair_iff_cod : L.IsExtensionPair M N ↔
+lemma isExtensionPair_iff_cod : L.IsExtensionPair M N ↔
     ∀ (f : L.FGEquiv N M) (m : M), ∃ g, m ∈ g.1.cod ∧ f ≤ g := by
   refine Iff.intro ?_ ?_ <;>
   · intro h f m
     obtain ⟨g, h1, h2⟩ := h f.symm m
     exact ⟨g.symm, h1, monotone_symm h2⟩
 
+/-- An alternate characterization of an extension pair is that every finitely generated partial
+isomorphism can be extended to include any particular element of the domain. -/
+theorem isExtensionPair_iff_exists_embedding_closure_singleton_sup :
+    L.IsExtensionPair M N ↔
+    ∀ (S : L.Substructure M) (_ : S.FG) (f : S ↪[L] N) (m : M),
+      ∃ g : (closure L {m} ⊔ S : L.Substructure M) ↪[L] N, f =
+        g.comp (Substructure.inclusion le_sup_right) := by
+  refine ⟨fun h S S_FG f m => ?_, fun h ⟨f, f_FG⟩ m => ?_⟩
+  · obtain ⟨⟨f', hf'⟩, mf', ff'1, ff'2⟩ := h ⟨⟨S, _, f.equivRange⟩, S_FG⟩ m
+    refine ⟨f'.toEmbedding.comp (Substructure.inclusion ?_), ?_⟩
+    · simp only [sup_le_iff, ff'1, closure_le, singleton_subset_iff, SetLike.mem_coe, mf',
+        and_self]
+    · ext ⟨x, hx⟩
+      rw [Embedding.subtype_equivRange] at ff'2
+      simp only [← ff'2, Embedding.comp_apply, Substructure.coe_inclusion, inclusion_mk,
+        Equiv.coe_toEmbedding, coeSubtype, PartialEquiv.toEmbedding_apply]
+  · obtain ⟨f', eq_f'⟩ := h f.dom f_FG f.toEmbedding m
+    refine ⟨⟨⟨closure L {m} ⊔ f.dom, f'.toHom.range, f'.equivRange⟩,
+      (fg_closure_singleton _).sup f_FG⟩,
+      subset_closure.trans (le_sup_left : (closure L) {m} ≤ _) (mem_singleton m),
+      ⟨le_sup_right, Embedding.ext (fun _ => ?_)⟩⟩
+    rw [PartialEquiv.toEmbedding] at eq_f'
+    simp only [Embedding.comp_apply, Substructure.coe_inclusion, Equiv.coe_toEmbedding, coeSubtype,
+      Embedding.equivRange_apply, eq_f']
+
 namespace IsExtensionPair
 
-protected alias ⟨cod, _⟩ := IsExtensionPair_iff_cod
+protected alias ⟨cod, _⟩ := isExtensionPair_iff_cod
 
 /-- The cofinal set of finite equivalences with a given element in their domain. -/
 def definedAtLeft
diff --git a/Mathlib/ModelTheory/Quotients.lean b/Mathlib/ModelTheory/Quotients.lean
index 798514c512fb1..fef412e512ae8 100644
--- a/Mathlib/ModelTheory/Quotients.lean
+++ b/Mathlib/ModelTheory/Quotients.lean
@@ -64,9 +64,9 @@ theorem relMap_quotient_mk' {n : ℕ} (r : L.Relations n) (x : Fin n → M) :
 
 theorem Term.realize_quotient_mk' {β : Type*} (t : L.Term β) (x : β → M) :
     (t.realize fun i => (⟦x i⟧ : Quotient s)) = ⟦@Term.realize _ _ ps.toStructure _ x t⟧ := by
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp only [ih, funMap_quotient_mk', Term.realize]
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp only [ih, funMap_quotient_mk', Term.realize]
 
 end Language
 
diff --git a/Mathlib/ModelTheory/Satisfiability.lean b/Mathlib/ModelTheory/Satisfiability.lean
index 5adfbfe169f28..55fdc417813b7 100644
--- a/Mathlib/ModelTheory/Satisfiability.lean
+++ b/Mathlib/ModelTheory/Satisfiability.lean
@@ -20,8 +20,6 @@ This file deals with the satisfiability of first-order theories, as well as equi
   every finite subset of `T` is satisfiable.
 - `FirstOrder.Language.Theory.IsComplete`: `T.IsComplete` indicates that `T` is satisfiable and
   models each sentence or its negation.
-- `FirstOrder.Language.Theory.SemanticallyEquivalent`: `T.SemanticallyEquivalent φ ψ` indicates
-  that `φ` and `ψ` are equivalent formulas or sentences in models of `T`.
 - `Cardinal.Categorical`: A theory is `κ`-categorical if all models of size `κ` are isomorphic.
 
 ## Main Results
@@ -281,7 +279,7 @@ variable (T)
 /-- A theory models a (bounded) formula when any of its nonempty models realizes that formula on all
   inputs. -/
 def ModelsBoundedFormula (φ : L.BoundedFormula α n) : Prop :=
-  ∀ (M : ModelType.{u, v, max u v} T) (v : α → M) (xs : Fin n → M), φ.Realize v xs
+  ∀ (M : ModelType.{u, v, max u v w} T) (v : α → M) (xs : Fin n → M), φ.Realize v xs
 
 -- Porting note: In Lean3 it was `⊨` but ambiguous.
 @[inherit_doc FirstOrder.Language.Theory.ModelsBoundedFormula]
@@ -290,7 +288,7 @@ infixl:51 " ⊨ᵇ " => ModelsBoundedFormula -- input using \|= or \vDash, but n
 variable {T}
 
 theorem models_formula_iff {φ : L.Formula α} :
-    T ⊨ᵇ φ ↔ ∀ (M : ModelType.{u, v, max u v} T) (v : α → M), φ.Realize v :=
+    T ⊨ᵇ φ ↔ ∀ (M : ModelType.{u, v, max u v w} T) (v : α → M), φ.Realize v :=
   forall_congr' fun _ => forall_congr' fun _ => Unique.forall_iff
 
 theorem models_sentence_iff {φ : L.Sentence} : T ⊨ᵇ φ ↔ ∀ M : ModelType.{u, v, max u v} T, M ⊨ φ :=
@@ -327,12 +325,47 @@ theorem ModelsBoundedFormula.realize_sentence {φ : L.Sentence} (h : T ⊨ᵇ φ
     exact ⟨h, inferInstance⟩
   exact Model.isSatisfiable M
 
+theorem models_formula_iff_onTheory_models_equivSentence {φ : L.Formula α} :
+    T ⊨ᵇ φ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ Formula.equivSentence φ := by
+  refine ⟨fun h => models_sentence_iff.2 (fun M => ?_),
+    fun h => models_formula_iff.2 (fun M v => ?_)⟩
+  · letI := (L.lhomWithConstants α).reduct M
+    have : (L.lhomWithConstants α).IsExpansionOn M := LHom.isExpansionOn_reduct _ _
+      -- why doesn't that instance just work?
+    rw [Formula.realize_equivSentence]
+    have : M ⊨ T := (LHom.onTheory_model _ _).1 M.is_model -- why isn't M.is_model inferInstance?
+    let M' := Theory.ModelType.of T M
+    exact h M' (fun a => (L.con a : M)) _
+  · letI : (constantsOn α).Structure M := constantsOn.structure v
+    have : M ⊨ (L.lhomWithConstants α).onTheory T := (LHom.onTheory_model _ _).2 inferInstance
+    exact (Formula.realize_equivSentence _ _).1 (h.realize_sentence M)
+
+theorem ModelsBoundedFormula.realize_formula {φ : L.Formula α} (h : T ⊨ᵇ φ) (M : Type*)
+    [L.Structure M] [M ⊨ T] [Nonempty M] {v : α → M} : φ.Realize v := by
+  rw [models_formula_iff_onTheory_models_equivSentence] at h
+  letI : (constantsOn α).Structure M := constantsOn.structure v
+  have : M ⊨ (L.lhomWithConstants α).onTheory T := (LHom.onTheory_model _ _).2 inferInstance
+  exact (Formula.realize_equivSentence _ _).1 (h.realize_sentence M)
+
+theorem models_toFormula_iff {φ : L.BoundedFormula α n} : T ⊨ᵇ φ.toFormula ↔ T ⊨ᵇ φ := by
+  refine ⟨fun h M v xs => ?_, ?_⟩
+  · have h' : φ.toFormula.Realize (Sum.elim v xs) := h.realize_formula M
+    simp only [BoundedFormula.realize_toFormula, Sum.elim_comp_inl, Sum.elim_comp_inr] at h'
+    exact h'
+  · simp only [models_formula_iff, BoundedFormula.realize_toFormula]
+    exact fun h M v => h M _ _
+
+theorem ModelsBoundedFormula.realize_boundedFormula
+    {φ : L.BoundedFormula α n} (h : T ⊨ᵇ φ) (M : Type*)
+    [L.Structure M] [M ⊨ T] [Nonempty M] {v : α → M} {xs : Fin n → M} : φ.Realize v xs := by
+  have h' : φ.toFormula.Realize (Sum.elim v xs) := (models_toFormula_iff.2 h).realize_formula M
+  simp only [BoundedFormula.realize_toFormula, Sum.elim_comp_inl, Sum.elim_comp_inr] at h'
+  exact h'
+
 theorem models_of_models_theory {T' : L.Theory}
     (h : ∀ φ : L.Sentence, φ ∈ T' → T ⊨ᵇ φ)
-    {φ : L.Formula α} (hφ : T' ⊨ᵇ φ) : T ⊨ᵇ φ := by
-  simp only [models_sentence_iff] at h
-  intro M
-  have hM : M ⊨ T' := T'.model_iff.2 (fun ψ hψ => h ψ hψ M)
+    {φ : L.Formula α} (hφ : T' ⊨ᵇ φ) : T ⊨ᵇ φ := fun M => by
+  have hM : M ⊨ T' := T'.model_iff.2 (fun ψ hψ => (h ψ hψ).realize_sentence M)
   let M' : ModelType T' := ⟨M⟩
   exact hφ M'
 
@@ -362,12 +395,12 @@ namespace IsComplete
 
 theorem models_not_iff (h : T.IsComplete) (φ : L.Sentence) : T ⊨ᵇ φ.not ↔ ¬T ⊨ᵇ φ := by
   cases' h.2 φ with hφ hφn
-  · simp only [hφ, not_true, iff_false_iff]
+  · simp only [hφ, not_true, iff_false]
     rw [models_sentence_iff, not_forall]
     refine ⟨h.1.some, ?_⟩
     simp only [Sentence.realize_not, Classical.not_not]
     exact models_sentence_iff.1 hφ _
-  · simp only [hφn, true_iff_iff]
+  · simp only [hφn, true_iff]
     intro hφ
     rw [models_sentence_iff] at *
     exact hφn h.1.some (hφ _)
@@ -401,74 +434,6 @@ theorem IsMaximal.mem_of_models (h : T.IsMaximal) {φ : L.Sentence} (hφ : T ⊨
 theorem IsMaximal.mem_iff_models (h : T.IsMaximal) (φ : L.Sentence) : φ ∈ T ↔ T ⊨ᵇ φ :=
   ⟨models_sentence_of_mem, h.mem_of_models⟩
 
-/-- Two (bounded) formulas are semantically equivalent over a theory `T` when they have the same
-interpretation in every model of `T`. (This is also known as logical equivalence, which also has a
-proof-theoretic definition.) -/
-def SemanticallyEquivalent (T : L.Theory) (φ ψ : L.BoundedFormula α n) : Prop :=
-  T ⊨ᵇ φ.iff ψ
-
-@[refl]
-theorem SemanticallyEquivalent.refl (φ : L.BoundedFormula α n) : T.SemanticallyEquivalent φ φ :=
-  fun M v xs => by rw [BoundedFormula.realize_iff]
-
-instance : IsRefl (L.BoundedFormula α n) T.SemanticallyEquivalent :=
-  ⟨SemanticallyEquivalent.refl⟩
-
-@[symm]
-theorem SemanticallyEquivalent.symm {φ ψ : L.BoundedFormula α n}
-    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent ψ φ := fun M v xs => by
-  rw [BoundedFormula.realize_iff, Iff.comm, ← BoundedFormula.realize_iff]
-  exact h M v xs
-
-@[trans]
-theorem SemanticallyEquivalent.trans {φ ψ θ : L.BoundedFormula α n}
-    (h1 : T.SemanticallyEquivalent φ ψ) (h2 : T.SemanticallyEquivalent ψ θ) :
-    T.SemanticallyEquivalent φ θ := fun M v xs => by
-  have h1' := h1 M v xs
-  have h2' := h2 M v xs
-  rw [BoundedFormula.realize_iff] at *
-  exact ⟨h2'.1 ∘ h1'.1, h1'.2 ∘ h2'.2⟩
-
-theorem SemanticallyEquivalent.realize_bd_iff {φ ψ : L.BoundedFormula α n} {M : Type max u v}
-    [Nonempty M] [L.Structure M] [T.Model M] (h : T.SemanticallyEquivalent φ ψ)
-    {v : α → M} {xs : Fin n → M} : φ.Realize v xs ↔ ψ.Realize v xs :=
-  BoundedFormula.realize_iff.1 (h (ModelType.of T M) v xs)
-
-theorem SemanticallyEquivalent.realize_iff {φ ψ : L.Formula α} {M : Type max u v} [Nonempty M]
-    [L.Structure M] (_hM : T.Model M) (h : T.SemanticallyEquivalent φ ψ) {v : α → M} :
-    φ.Realize v ↔ ψ.Realize v :=
-  h.realize_bd_iff
-
-/-- Semantic equivalence forms an equivalence relation on formulas. -/
-def semanticallyEquivalentSetoid (T : L.Theory) : Setoid (L.BoundedFormula α n) where
-  r := SemanticallyEquivalent T
-  iseqv := ⟨fun _ => refl _, fun {_ _} h => h.symm, fun {_ _ _} h1 h2 => h1.trans h2⟩
-
-protected theorem SemanticallyEquivalent.all {φ ψ : L.BoundedFormula α (n + 1)}
-    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.all ψ.all := by
-  simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,
-    BoundedFormula.realize_all]
-  exact fun M v xs => forall_congr' fun a => h.realize_bd_iff
-
-protected theorem SemanticallyEquivalent.ex {φ ψ : L.BoundedFormula α (n + 1)}
-    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.ex ψ.ex := by
-  simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,
-    BoundedFormula.realize_ex]
-  exact fun M v xs => exists_congr fun a => h.realize_bd_iff
-
-protected theorem SemanticallyEquivalent.not {φ ψ : L.BoundedFormula α n}
-    (h : T.SemanticallyEquivalent φ ψ) : T.SemanticallyEquivalent φ.not ψ.not := by
-  simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,
-    BoundedFormula.realize_not]
-  exact fun M v xs => not_congr h.realize_bd_iff
-
-protected theorem SemanticallyEquivalent.imp {φ ψ φ' ψ' : L.BoundedFormula α n}
-    (h : T.SemanticallyEquivalent φ ψ) (h' : T.SemanticallyEquivalent φ' ψ') :
-    T.SemanticallyEquivalent (φ.imp φ') (ψ.imp ψ') := by
-  simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,
-    BoundedFormula.realize_imp]
-  exact fun M v xs => imp_congr h.realize_bd_iff h'.realize_bd_iff
-
 end Theory
 
 namespace completeTheory
@@ -490,55 +455,6 @@ theorem isComplete [Nonempty M] : (L.completeTheory M).IsComplete :=
 
 end completeTheory
 
-namespace BoundedFormula
-
-variable (φ ψ : L.BoundedFormula α n)
-
-theorem semanticallyEquivalent_not_not : T.SemanticallyEquivalent φ φ.not.not := fun M v xs => by
-  simp
-
-theorem imp_semanticallyEquivalent_not_sup : T.SemanticallyEquivalent (φ.imp ψ) (φ.not ⊔ ψ) :=
-  fun M v xs => by simp [imp_iff_not_or]
-
-theorem sup_semanticallyEquivalent_not_inf_not :
-    T.SemanticallyEquivalent (φ ⊔ ψ) (φ.not ⊓ ψ.not).not := fun M v xs => by simp [imp_iff_not_or]
-
-theorem inf_semanticallyEquivalent_not_sup_not :
-    T.SemanticallyEquivalent (φ ⊓ ψ) (φ.not ⊔ ψ.not).not := fun M v xs => by
-  simp
-
-theorem all_semanticallyEquivalent_not_ex_not (φ : L.BoundedFormula α (n + 1)) :
-    T.SemanticallyEquivalent φ.all φ.not.ex.not := fun M v xs => by simp
-
-theorem ex_semanticallyEquivalent_not_all_not (φ : L.BoundedFormula α (n + 1)) :
-    T.SemanticallyEquivalent φ.ex φ.not.all.not := fun M v xs => by simp
-
-theorem semanticallyEquivalent_all_liftAt : T.SemanticallyEquivalent φ (φ.liftAt 1 n).all :=
-  fun M v xs => by
-  rw [realize_iff, realize_all_liftAt_one_self]
-
-end BoundedFormula
-
-namespace Formula
-
-variable (φ ψ : L.Formula α)
-
-theorem semanticallyEquivalent_not_not : T.SemanticallyEquivalent φ φ.not.not :=
-  BoundedFormula.semanticallyEquivalent_not_not φ
-
-theorem imp_semanticallyEquivalent_not_sup : T.SemanticallyEquivalent (φ.imp ψ) (φ.not ⊔ ψ) :=
-  BoundedFormula.imp_semanticallyEquivalent_not_sup φ ψ
-
-theorem sup_semanticallyEquivalent_not_inf_not :
-    T.SemanticallyEquivalent (φ ⊔ ψ) (φ.not ⊓ ψ.not).not :=
-  BoundedFormula.sup_semanticallyEquivalent_not_inf_not φ ψ
-
-theorem inf_semanticallyEquivalent_not_sup_not :
-    T.SemanticallyEquivalent (φ ⊓ ψ) (φ.not ⊔ ψ.not).not :=
-  BoundedFormula.inf_semanticallyEquivalent_not_sup_not φ ψ
-
-end Formula
-
 end Language
 
 end FirstOrder
diff --git a/Mathlib/ModelTheory/Semantics.lean b/Mathlib/ModelTheory/Semantics.lean
index b25b48b668084..3da6c40283d18 100644
--- a/Mathlib/ModelTheory/Semantics.lean
+++ b/Mathlib/ModelTheory/Semantics.lean
@@ -122,16 +122,17 @@ theorem realize_con {A : Set M} {a : A} {v : α → M} : (L.con a).term.realize
 @[simp]
 theorem realize_subst {t : L.Term α} {tf : α → L.Term β} {v : β → M} :
     (t.subst tf).realize v = t.realize fun a => (tf a).realize v := by
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp [ih]
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp [ih]
 
 @[simp]
 theorem realize_restrictVar [DecidableEq α] {t : L.Term α} {s : Set α} (h : ↑t.varFinset ⊆ s)
     {v : α → M} : (t.restrictVar (Set.inclusion h)).realize (v ∘ (↑)) = t.realize v := by
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp_rw [varFinset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
+  induction t with
+  | var => rfl
+  | func _ _ ih =>
+    simp_rw [varFinset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 
 @[simp]
@@ -139,9 +140,10 @@ theorem realize_restrictVarLeft [DecidableEq α] {γ : Type*} {t : L.Term (α 
     (h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
     (t.restrictVarLeft (Set.inclusion h)).realize (Sum.elim (v ∘ (↑)) xs) =
       t.realize (Sum.elim v xs) := by
-  induction' t with a _ _ _ ih
-  · cases a <;> rfl
-  · simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
+  induction t with
+  | var a => cases a <;> rfl
+  | func _ _ ih =>
+    simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
     exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
 
 @[simp]
@@ -152,13 +154,13 @@ theorem realize_constantsToVars [L[[α]].Structure M] [(lhomWithConstants L α).
   · simp
   · cases n
     · cases f
-      · simp only [realize, ih, constantsOn, mk₂_Functions]
+      · simp only [realize, ih, constantsOn, constantsOnFunc]
         -- Porting note: below lemma does not work with simp for some reason
         rw [withConstants_funMap_sum_inl]
       · simp only [realize, constantsToVars, Sum.elim_inl, funMap_eq_coe_constants]
         rfl
     · cases' f with _ f
-      · simp only [realize, ih, constantsOn, mk₂_Functions]
+      · simp only [realize, ih, constantsOn, constantsOnFunc]
         -- Porting note: below lemma does not work with simp for some reason
         rw [withConstants_funMap_sum_inl]
       · exact isEmptyElim f
@@ -172,7 +174,7 @@ theorem realize_varsToConstants [L[[α]].Structure M] [(lhomWithConstants L α).
     -- Porting note: both cases were `simp [Language.con]`
     · simp [Language.con, realize, funMap_eq_coe_constants]
     · simp [realize, constantMap]
-  · simp only [realize, constantsOn, mk₂_Functions, ih]
+  · simp only [realize, constantsOn, constantsOnFunc, ih]
     -- Porting note: below lemma does not work with simp for some reason
     rw [withConstants_funMap_sum_inl]
 
@@ -321,12 +323,12 @@ theorem realize_mapTermRel_id [L'.Structure M]
         (ft n t).realize (Sum.elim v' xs) = t.realize (Sum.elim v xs))
     (h2 : ∀ (n) (R : L.Relations n) (x : Fin n → M), RelMap (fr n R) x = RelMap R x) :
     (φ.mapTermRel ft fr fun _ => id).Realize v' xs ↔ φ.Realize v xs := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
-  · rfl
-  · simp [mapTermRel, Realize, h1]
-  · simp [mapTermRel, Realize, h1, h2]
-  · simp [mapTermRel, Realize, ih1, ih2]
-  · simp only [mapTermRel, Realize, ih, id]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [mapTermRel, Realize, h1]
+  | rel => simp [mapTermRel, Realize, h1, h2]
+  | imp _ _ ih1 ih2 => simp [mapTermRel, Realize, ih1, ih2]
+  | all _ ih => simp only [mapTermRel, Realize, ih, id]
 
 theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     {ft : ∀ n, L.Term (α ⊕ (Fin n)) → L'.Term (β ⊕ (Fin (k + n)))}
@@ -339,31 +341,32 @@ theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
     (hv : ∀ (n) (xs : Fin (k + n) → M) (x : M), @v (n + 1) (snoc xs x : Fin _ → M) = v xs) :
     (φ.mapTermRel ft fr fun n => castLE (add_assoc _ _ _).symm.le).Realize v' xs ↔
       φ.Realize (v xs) (xs ∘ Fin.natAdd _) := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
-  · rfl
-  · simp [mapTermRel, Realize, h1]
-  · simp [mapTermRel, Realize, h1, h2]
-  · simp [mapTermRel, Realize, ih1, ih2]
-  · simp [mapTermRel, Realize, ih, hv]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [mapTermRel, Realize, h1]
+  | rel => simp [mapTermRel, Realize, h1, h2]
+  | imp _ _ ih1 ih2 => simp [mapTermRel, Realize, ih1, ih2]
+  | all _ ih => simp [mapTermRel, Realize, ih, hv]
 
 @[simp]
 theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → β ⊕ (Fin m)} {v : β → M}
     {xs : Fin (m + n) → M} :
     (φ.relabel g).Realize v xs ↔
       φ.Realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAdd m) := by
-  rw [relabel, realize_mapTermRel_add_castLe] <;> intros <;> simp
+  apply realize_mapTermRel_add_castLe <;> simp
 
 theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
     (hmn : m + n' ≤ n + 1) :
     (φ.liftAt n' m).Realize v xs ↔
       φ.Realize v (xs ∘ fun i => if ↑i < m then Fin.castAdd n' i else Fin.addNat i n') := by
   rw [liftAt]
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 k _ ih3
-  · simp [mapTermRel, Realize]
-  · simp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]
-  · simp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]
-  · simp only [mapTermRel, Realize, ih1 hmn, ih2 hmn]
-  · have h : k + 1 + n' = k + n' + 1 := by rw [add_assoc, add_comm 1 n', ← add_assoc]
+  induction φ with
+  | falsum => simp [mapTermRel, Realize]
+  | equal => simp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]
+  | rel => simp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]
+  | imp _ _ ih1 ih2 => simp only [mapTermRel, Realize, ih1 hmn, ih2 hmn]
+  | @all k _ ih3 =>
+    have h : k + 1 + n' = k + n' + 1 := by rw [add_assoc, add_comm 1 n', ← add_assoc]
     simp only [mapTermRel, Realize, realize_castLE_of_eq h, ih3 (hmn.trans k.succ.le_succ)]
     refine forall_congr' fun x => iff_eq_eq.mpr (congr rfl (funext (Fin.lastCases ?_ fun i => ?_)))
     · simp only [Function.comp_apply, val_last, snoc_last]
@@ -411,12 +414,22 @@ theorem realize_subst {φ : L.BoundedFormula α n} {tf : α → L.Term β} {v :
 theorem realize_restrictFreeVar [DecidableEq α] {n : ℕ} {φ : L.BoundedFormula α n} {s : Set α}
     (h : ↑φ.freeVarFinset ⊆ s) {v : α → M} {xs : Fin n → M} :
     (φ.restrictFreeVar (Set.inclusion h)).Realize (v ∘ (↑)) xs ↔ φ.Realize v xs := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp [restrictFreeVar, Realize]
-  · simp [restrictFreeVar, Realize]
-  · simp [restrictFreeVar, Realize, ih1, ih2]
-  · simp [restrictFreeVar, Realize, ih3]
+  induction φ with
+  | falsum => rfl
+  | equal =>
+    simp only [Realize, freeVarFinset.eq_2]
+    rw [Set.inclusion_comp_inclusion, Set.inclusion_comp_inclusion]
+    simp
+  | rel =>
+    simp only [Realize, freeVarFinset.eq_3, Finset.biUnion_val]
+    congr!
+    erw [Set.inclusion_comp_inclusion _ h]
+    simp
+  | imp _ _ ih1 ih2 =>
+    simp only [Realize, freeVarFinset.eq_4]
+    rw [Set.inclusion_comp_inclusion, Set.inclusion_comp_inclusion]
+    simp [ih1, ih2]
+  | all _ ih3 => simp [restrictFreeVar, Realize, ih3]
 
 theorem realize_constantsVarsEquiv [L[[α]].Structure M] [(lhomWithConstants L α).IsExpansionOn M]
     {n} {φ : L[[α]].BoundedFormula β n} {v : β → M} {xs : Fin n → M} :
@@ -467,15 +480,15 @@ open BoundedFormula
 theorem realize_onBoundedFormula [L'.Structure M] (φ : L →ᴸ L') [φ.IsExpansionOn M] {n : ℕ}
     (ψ : L.BoundedFormula α n) {v : α → M} {xs : Fin n → M} :
     (φ.onBoundedFormula ψ).Realize v xs ↔ ψ.Realize v xs := by
-  induction' ψ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp only [onBoundedFormula, realize_bdEqual, realize_onTerm]
-    rfl
-  · simp only [onBoundedFormula, realize_rel, LHom.map_onRelation,
+  induction ψ with
+  | falsum => rfl
+  | equal => simp only [onBoundedFormula, realize_bdEqual, realize_onTerm]; rfl
+  | rel =>
+    simp only [onBoundedFormula, realize_rel, LHom.map_onRelation,
       Function.comp_apply, realize_onTerm]
     rfl
-  · simp only [onBoundedFormula, ih1, ih2, realize_imp]
-  · simp only [onBoundedFormula, ih3, realize_all]
+  | imp _ _ ih1 ih2 => simp only [onBoundedFormula, ih1, ih2, realize_imp]
+  | all _ ih3 => simp only [onBoundedFormula, ih3, realize_all]
 
 end LHom
 
@@ -689,8 +702,12 @@ theorem model_union_iff {T' : L.Theory} : M ⊨ T ∪ T' ↔ M ⊨ T ∧ M ⊨ T
   ⟨fun h => ⟨h.mono Set.subset_union_left, h.mono Set.subset_union_right⟩, fun h =>
     h.1.union h.2⟩
 
+@[simp]
 theorem model_singleton_iff {φ : L.Sentence} : M ⊨ ({φ} : L.Theory) ↔ M ⊨ φ := by simp
 
+theorem model_insert_iff {φ : L.Sentence} : M ⊨ insert φ T ↔ M ⊨ φ ∧ M ⊨ T := by
+  rw [Set.insert_eq, model_union_iff, model_singleton_iff]
+
 theorem model_iff_subset_completeTheory : M ⊨ T ↔ T ⊆ L.completeTheory M :=
   T.model_iff
 
@@ -743,12 +760,12 @@ theorem _root_.FirstOrder.Language.Formula.realize_iAlls
       ∀ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
   let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin γ))
   rw [Formula.iAlls]
-  simp only [Nat.add_zero, realize_alls, realize_relabel, Function.comp,
+  simp only [Nat.add_zero, realize_alls, realize_relabel, Function.comp_def,
     castAdd_zero, finCongr_refl, OrderIso.refl_apply, Sum.elim_map, id_eq]
   refine Equiv.forall_congr ?_ ?_
   · exact ⟨fun v => v ∘ e, fun v => v ∘ e.symm,
-      fun _ => by simp [Function.comp],
-      fun _ => by simp [Function.comp]⟩
+      fun _ => by simp [Function.comp_def],
+      fun _ => by simp [Function.comp_def]⟩
   · intro x
     rw [Formula.Realize, iff_iff_eq]
     congr
@@ -769,13 +786,13 @@ theorem _root_.FirstOrder.Language.Formula.realize_iExs
       ∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by
   let e := Classical.choice (Classical.choose_spec (Finite.exists_equiv_fin γ))
   rw [Formula.iExs]
-  simp only [Nat.add_zero, realize_exs, realize_relabel, Function.comp,
+  simp only [Nat.add_zero, realize_exs, realize_relabel, Function.comp_def,
     castAdd_zero, finCongr_refl, OrderIso.refl_apply, Sum.elim_map, id_eq]
   rw [← not_iff_not, not_exists, not_exists]
   refine Equiv.forall_congr ?_ ?_
   · exact ⟨fun v => v ∘ e, fun v => v ∘ e.symm,
-      fun _ => by simp [Function.comp],
-      fun _ => by simp [Function.comp]⟩
+      fun _ => by simp [Function.comp_def],
+      fun _ => by simp [Function.comp_def]⟩
   · intro x
     rw [Formula.Realize, iff_iff_eq]
     congr
@@ -792,18 +809,20 @@ theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}
 @[simp]
 theorem realize_toFormula (φ : L.BoundedFormula α n) (v : α ⊕ (Fin n) → M) :
     φ.toFormula.Realize v ↔ φ.Realize (v ∘ Sum.inl) (v ∘ Sum.inr) := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 a8 a9 a0
-  · rfl
-  · simp [BoundedFormula.Realize]
-  · simp [BoundedFormula.Realize]
-  · rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,
+  induction φ with
+  | falsum => rfl
+  | equal => simp [BoundedFormula.Realize]
+  | rel => simp [BoundedFormula.Realize]
+  | imp _ _ ih1 ih2 =>
+    rw [toFormula, Formula.Realize, realize_imp, ← Formula.Realize, ih1, ← Formula.Realize, ih2,
       realize_imp]
-  · rw [toFormula, Formula.Realize, realize_all, realize_all]
+  | all _ ih3 =>
+    rw [toFormula, Formula.Realize, realize_all, realize_all]
     refine forall_congr' fun a => ?_
     have h := ih3 (Sum.elim (v ∘ Sum.inl) (snoc (v ∘ Sum.inr) a))
     simp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h
     rw [← h, realize_relabel, Formula.Realize, iff_iff_eq]
-    simp only [Function.comp]
+    simp only [Function.comp_def]
     congr with x
     · cases' x with _ x
       · simp
@@ -840,14 +859,17 @@ variable {F : Type*} [EquivLike F M N] [StrongHomClass L F M N] (g : F)
 @[simp]
 theorem realize_boundedFormula (φ : L.BoundedFormula α n) {v : α → M}
     {xs : Fin n → M} : φ.Realize (g ∘ v) (g ∘ xs) ↔ φ.Realize v xs := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp only [BoundedFormula.Realize, ← Sum.comp_elim, HomClass.realize_term,
+  induction φ with
+  | falsum => rfl
+  | equal =>
+    simp only [BoundedFormula.Realize, ← Sum.comp_elim, HomClass.realize_term,
       EmbeddingLike.apply_eq_iff_eq g]
-  · simp only [BoundedFormula.Realize, ← Sum.comp_elim, HomClass.realize_term]
+  | rel =>
+    simp only [BoundedFormula.Realize, ← Sum.comp_elim, HomClass.realize_term]
     exact StrongHomClass.map_rel g _ _
-  · rw [BoundedFormula.Realize, ih1, ih2, BoundedFormula.Realize]
-  · rw [BoundedFormula.Realize, BoundedFormula.Realize]
+  | imp _ _ ih1 ih2 => rw [BoundedFormula.Realize, ih1, ih2, BoundedFormula.Realize]
+  | all _ ih3 =>
+    rw [BoundedFormula.Realize, BoundedFormula.Realize]
     constructor
     · intro h a
       have h' := h (g a)
@@ -924,7 +946,7 @@ theorem Sentence.realize_cardGe (n) : M ⊨ Sentence.cardGe L n ↔ ↑n ≤ #M
     BoundedFormula.realize_exs]
   simp_rw [BoundedFormula.realize_foldr_inf]
   simp only [Function.comp_apply, List.mem_map, Prod.exists, Ne, List.mem_product,
-    List.mem_finRange, forall_exists_index, and_imp, List.mem_filter, true_and_iff]
+    List.mem_finRange, forall_exists_index, and_imp, List.mem_filter, true_and]
   refine ⟨?_, fun xs => ⟨xs.some, ?_⟩⟩
   · rintro ⟨xs, h⟩
     refine ⟨⟨xs, fun i j ij => ?_⟩⟩
diff --git a/Mathlib/ModelTheory/Skolem.lean b/Mathlib/ModelTheory/Skolem.lean
index eefcbc6d811c1..0ff7aa3c3c2b8 100644
--- a/Mathlib/ModelTheory/Skolem.lean
+++ b/Mathlib/ModelTheory/Skolem.lean
@@ -82,11 +82,11 @@ theorem skolem₁_reduct_isElementary (S : (L.sum L.skolem₁).Substructure M) :
   apply (LHom.sumInl.substructureReduct S).isElementary_of_exists
   intro n φ x a h
   let φ' : (L.sum L.skolem₁).Functions n := LHom.sumInr.onFunction φ
-  exact
-    ⟨⟨funMap φ' ((↑) ∘ x), S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by
-      exact fun i => (x i).2)⟩,
-      by exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default
-          (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩⟩
+  use ⟨funMap φ' ((↑) ∘ x), ?_⟩
+  · exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default
+          (Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩
+  · exact S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by
+      exact fun i => (x i).2)
 
 /-- Any `L.sum L.skolem₁`-substructure is an elementary `L`-substructure. -/
 noncomputable def elementarySkolem₁Reduct (S : (L.sum L.skolem₁).Substructure M) :
@@ -116,7 +116,7 @@ variable {M}
 /-- The **Downward Löwenheim–Skolem theorem** :
   If `s` is a set in an `L`-structure `M` and `κ` an infinite cardinal such that
   `max (#s, L.card) ≤ κ` and `κ ≤ # M`, then `M` has an elementary substructure containing `s` of
-  cardinality `κ`.  -/
+  cardinality `κ`. -/
 theorem exists_elementarySubstructure_card_eq (s : Set M) (κ : Cardinal.{w'}) (h1 : ℵ₀ ≤ κ)
     (h2 : Cardinal.lift.{w'} #s ≤ Cardinal.lift.{w} κ)
     (h3 : Cardinal.lift.{w'} L.card ≤ Cardinal.lift.{max u v} κ)
diff --git a/Mathlib/ModelTheory/Substructures.lean b/Mathlib/ModelTheory/Substructures.lean
index bd4d79e0cea0f..0fc0993999295 100644
--- a/Mathlib/ModelTheory/Substructures.lean
+++ b/Mathlib/ModelTheory/Substructures.lean
@@ -299,9 +299,17 @@ lemma mem_closed_iff (s : Set M) :
 
 variable (L)
 
-@[simp]
 lemma mem_closed_of_isRelational [L.IsRelational] (s : Set M) : s ∈ (closure L).closed :=
-  (mem_closed_iff s).2 (IsRelational.empty_functions _).elim
+  (mem_closed_iff s).2 isEmptyElim
+
+@[simp]
+lemma closure_eq_of_isRelational [L.IsRelational] (s : Set M) : closure L s = s :=
+  LowerAdjoint.closure_eq_self_of_mem_closed _ (mem_closed_of_isRelational L s)
+
+@[simp]
+lemma mem_closure_iff_of_isRelational [L.IsRelational] (s : Set M) (m : M) :
+    m ∈ closure L s ↔ m ∈ s := by
+  rw [← SetLike.mem_coe, closure_eq_of_isRelational]
 
 theorem _root_.Set.Countable.substructure_closure
     [Countable (Σl, L.Functions l)] (h : s.Countable) : Countable.{w + 1} (closure L s) := by
@@ -357,6 +365,9 @@ theorem closure_union (s t : Set M) : closure L (s ∪ t) = closure L s ⊔ clos
 theorem closure_iUnion {ι} (s : ι → Set M) : closure L (⋃ i, s i) = ⨆ i, closure L (s i) :=
   (Substructure.gi L M).gc.l_iSup
 
+theorem closure_insert (s : Set M) (m : M) : closure L (insert m s) = closure L {m} ⊔ closure L s :=
+  closure_union {m} s
+
 instance small_bot : Small.{u} (⊥ : L.Substructure M) := by
   rw [← closure_empty]
   haveI : Small.{u} (∅ : Set M) := small_subsingleton _
@@ -384,6 +395,22 @@ theorem mem_sSup_of_directedOn {S : Set (L.Substructure M)} (Sne : S.Nonempty)
   haveI : Nonempty S := Sne.to_subtype
   simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, Subtype.exists, exists_prop]
 
+variable (L) (M)
+
+instance [IsEmpty L.Constants] : IsEmpty (⊥ : L.Substructure M) := by
+  refine (isEmpty_subtype _).2 (fun x => ?_)
+  have h : (∅ : Set M) ∈ (closure L).closed := by
+    rw [mem_closed_iff]
+    intro n f
+    cases n
+    · exact isEmptyElim f
+    · intro x hx
+      simp only [mem_empty_iff_false, forall_const] at hx
+  rw [← closure_empty, ← SetLike.mem_coe, h]
+  exact Set.not_mem_empty _
+
+variable {L} {M}
+
 /-!
 ### `comap` and `map`
  -/
@@ -712,7 +739,7 @@ theorem closure_withConstants_eq :
   refine closure_eq_of_le ((A.subset_union_right).trans subset_closure) ?_
   rw [← (L.lhomWithConstants A).substructureReduct.le_iff_le]
   simp only [subset_closure, reduct_withConstants, closure_le, LHom.coe_substructureReduct,
-    Set.union_subset_iff, and_true_iff]
+    Set.union_subset_iff, and_true]
   exact subset_closure_withConstants
 
 end Substructure
@@ -876,7 +903,7 @@ theorem subtype_substructureEquivMap (f : M ↪[L] N) (s : L.Substructure M) :
   ext; rfl
 
 /-- The equivalence between the domain and the range of an embedding `f`. -/
-noncomputable def equivRange (f : M ↪[L] N) : M ≃[L] f.toHom.range where
+@[simps toEquiv_apply] noncomputable def equivRange (f : M ↪[L] N) : M ≃[L] f.toHom.range where
   toFun := codRestrict f.toHom.range f f.toHom.mem_range_self
   invFun n := Classical.choose n.2
   left_inv m :=
@@ -899,7 +926,7 @@ namespace Equiv
 
 theorem toHom_range (f : M ≃[L] N) : f.toHom.range = ⊤ := by
   ext n
-  simp only [Hom.mem_range, coe_toHom, Substructure.mem_top, iff_true_iff]
+  simp only [Hom.mem_range, coe_toHom, Substructure.mem_top, iff_true]
   exact ⟨f.symm n, apply_symm_apply _ _⟩
 
 end Equiv
diff --git a/Mathlib/ModelTheory/Syntax.lean b/Mathlib/ModelTheory/Syntax.lean
index 66286f0d43c57..7fa52ce76edb5 100644
--- a/Mathlib/ModelTheory/Syntax.lean
+++ b/Mathlib/ModelTheory/Syntax.lean
@@ -114,9 +114,9 @@ def relabel (g : α → β) : L.Term α → L.Term β
   | func f ts => func f fun {i} => (ts i).relabel g
 
 theorem relabel_id (t : L.Term α) : t.relabel id = t := by
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp [ih]
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp [ih]
 
 @[simp]
 theorem relabel_id_eq_id : (Term.relabel id : L.Term α → L.Term α) = id :=
@@ -125,9 +125,9 @@ theorem relabel_id_eq_id : (Term.relabel id : L.Term α → L.Term α) = id :=
 @[simp]
 theorem relabel_relabel (f : α → β) (g : β → γ) (t : L.Term α) :
     (t.relabel f).relabel g = t.relabel (g ∘ f) := by
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp [ih]
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp [ih]
 
 @[simp]
 theorem relabel_comp_relabel (f : α → β) (g : β → γ) :
@@ -197,9 +197,10 @@ def varsToConstants : L.Term (γ ⊕ α) → L[[γ]].Term α
 def constantsVarsEquiv : L[[γ]].Term α ≃ L.Term (γ ⊕ α) :=
   ⟨constantsToVars, varsToConstants, by
     intro t
-    induction' t with _ n f _ ih
-    · rfl
-    · cases n
+    induction t with
+    | var => rfl
+    | @func n f _ ih =>
+      cases n
       · cases f
         · simp [constantsToVars, varsToConstants, ih]
         · simp [constantsToVars, varsToConstants, Constants.term, eq_iff_true_of_subsingleton]
@@ -260,19 +261,17 @@ def onTerm (φ : L →ᴸ L') : L.Term α → L'.Term α
 @[simp]
 theorem id_onTerm : ((LHom.id L).onTerm : L.Term α → L.Term α) = id := by
   ext t
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp_rw [onTerm, ih]
-    rfl
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp_rw [onTerm, ih]; rfl
 
 @[simp]
 theorem comp_onTerm {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
     ((φ.comp ψ).onTerm : L.Term α → L''.Term α) = φ.onTerm ∘ ψ.onTerm := by
   ext t
-  induction' t with _ _ _ _ ih
-  · rfl
-  · simp_rw [onTerm, ih]
-    rfl
+  induction t with
+  | var => rfl
+  | func _ _ ih => simp_rw [onTerm, ih]; rfl
 
 end LHom
 
@@ -402,25 +401,27 @@ def castLE : ∀ {m n : ℕ} (_h : m ≤ n), L.BoundedFormula α m → L.Bounded
 
 @[simp]
 theorem castLE_rfl {n} (h : n ≤ n) (φ : L.BoundedFormula α n) : φ.castLE h = φ := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp [Fin.castLE_of_eq]
-  · simp [Fin.castLE_of_eq]
-  · simp [Fin.castLE_of_eq, ih1, ih2]
-  · simp [Fin.castLE_of_eq, ih3]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [Fin.castLE_of_eq]
+  | rel => simp [Fin.castLE_of_eq]
+  | imp _ _ ih1 ih2 => simp [Fin.castLE_of_eq, ih1, ih2]
+  | all _ ih3 => simp [Fin.castLE_of_eq, ih3]
 
 @[simp]
 theorem castLE_castLE {k m n} (km : k ≤ m) (mn : m ≤ n) (φ : L.BoundedFormula α k) :
     (φ.castLE km).castLE mn = φ.castLE (km.trans mn) := by
   revert m n
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 <;> intro m n km mn
-  · rfl
-  · simp
-  · simp only [castLE, eq_self_iff_true, heq_iff_eq, true_and_iff]
-    rw [← Function.comp.assoc, Term.relabel_comp_relabel]
+  induction φ with
+  | falsum => intros; rfl
+  | equal => simp
+  | rel =>
+    intros
+    simp only [castLE, eq_self_iff_true, heq_iff_eq]
+    rw [← Function.comp_assoc, Term.relabel_comp_relabel]
     simp
-  · simp [ih1, ih2]
-  · simp only [castLE, ih3]
+  | imp _ _ ih1 ih2 => simp [ih1, ih2]
+  | all _ ih3 => intros; simp only [castLE, ih3]
 
 @[simp]
 theorem castLE_comp_castLE {k m n} (km : k ≤ m) (mn : m ≤ n) :
@@ -483,22 +484,22 @@ theorem mapTermRel_mapTermRel {L'' : Language}
     (fr' : ∀ n, L'.Relations n → L''.Relations n) {n} (φ : L.BoundedFormula α n) :
     ((φ.mapTermRel ft fr fun _ => id).mapTermRel ft' fr' fun _ => id) =
       φ.mapTermRel (fun _ => ft' _ ∘ ft _) (fun _ => fr' _ ∘ fr _) fun _ => id := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp [mapTermRel]
-  · simp [mapTermRel]
-  · simp [mapTermRel, ih1, ih2]
-  · simp [mapTermRel, ih3]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [mapTermRel]
+  | rel => simp [mapTermRel]
+  | imp _ _ ih1 ih2 => simp [mapTermRel, ih1, ih2]
+  | all _ ih3 => simp [mapTermRel, ih3]
 
 @[simp]
 theorem mapTermRel_id_id_id {n} (φ : L.BoundedFormula α n) :
     (φ.mapTermRel (fun _ => id) (fun _ => id) fun _ => id) = φ := by
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp [mapTermRel]
-  · simp [mapTermRel]
-  · simp [mapTermRel, ih1, ih2]
-  · simp [mapTermRel, ih3]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [mapTermRel]
+  | rel => simp [mapTermRel]
+  | imp _ _ ih1 ih2 => simp [mapTermRel, ih1, ih2]
+  | all _ ih3 => simp [mapTermRel, ih3]
 
 /-- An equivalence of bounded formulas given by an equivalence of terms and an equivalence of
 relations. -/
@@ -571,12 +572,12 @@ theorem relabel_ex (g : α → β ⊕ (Fin n)) {k} (φ : L.BoundedFormula α (k
 theorem relabel_sum_inl (φ : L.BoundedFormula α n) :
     (φ.relabel Sum.inl : L.BoundedFormula α (0 + n)) = φ.castLE (ge_of_eq (zero_add n)) := by
   simp only [relabel, relabelAux_sum_inl]
-  induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]
-  · simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]; rfl
-  · simp [mapTermRel, ih1, ih2]
-  · simp [mapTermRel, ih3, castLE]
+  induction φ with
+  | falsum => rfl
+  | equal => simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]
+  | rel => simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]; rfl
+  | imp _ _ ih1 ih2 => simp_all [mapTermRel]
+  | all _ ih3 => simp_all [mapTermRel]
 
 /-- Substitutes the variables in a given formula with terms. -/
 def subst {n : ℕ} (φ : L.BoundedFormula α n) (f : α → L.Term β) : L.BoundedFormula β n :=
@@ -627,26 +628,25 @@ def onBoundedFormula (g : L →ᴸ L') : ∀ {k : ℕ}, L.BoundedFormula α k 
 theorem id_onBoundedFormula :
     ((LHom.id L).onBoundedFormula : L.BoundedFormula α n → L.BoundedFormula α n) = id := by
   ext f
-  induction' f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · rw [onBoundedFormula, LHom.id_onTerm, id, id, id, Term.bdEqual]
-  · rw [onBoundedFormula, LHom.id_onTerm]
-    rfl
-  · rw [onBoundedFormula, ih1, ih2, id, id, id]
-  · rw [onBoundedFormula, ih3, id, id]
+  induction f with
+  | falsum => rfl
+  | equal => rw [onBoundedFormula, LHom.id_onTerm, id, id, id, Term.bdEqual]
+  | rel => rw [onBoundedFormula, LHom.id_onTerm]; rfl
+  | imp _ _ ih1 ih2 => rw [onBoundedFormula, ih1, ih2, id, id, id]
+  | all _ ih3 => rw [onBoundedFormula, ih3, id, id]
 
 @[simp]
 theorem comp_onBoundedFormula {L'' : Language} (φ : L' →ᴸ L'') (ψ : L →ᴸ L') :
     ((φ.comp ψ).onBoundedFormula : L.BoundedFormula α n → L''.BoundedFormula α n) =
       φ.onBoundedFormula ∘ ψ.onBoundedFormula := by
   ext f
-  induction' f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
-  · rfl
-  · simp only [onBoundedFormula, comp_onTerm, Function.comp_apply]
-  · simp only [onBoundedFormula, comp_onRelation, comp_onTerm, Function.comp_apply]
-    rfl
-  · simp only [onBoundedFormula, Function.comp_apply, ih1, ih2, eq_self_iff_true, and_self_iff]
-  · simp only [ih3, onBoundedFormula, Function.comp_apply]
+  induction f with
+  | falsum => rfl
+  | equal => simp only [onBoundedFormula, comp_onTerm, Function.comp_apply]
+  | rel => simp only [onBoundedFormula, comp_onRelation, comp_onTerm, Function.comp_apply]; rfl
+  | imp _ _ ih1 ih2 =>
+    simp only [onBoundedFormula, Function.comp_apply, ih1, ih2, eq_self_iff_true, and_self_iff]
+  | all _ ih3 => simp only [ih3, onBoundedFormula, Function.comp_apply]
 
 /-- Maps a formula's symbols along a language map. -/
 def onFormula (g : L →ᴸ L') : L.Formula α → L'.Formula α :=
diff --git a/Mathlib/ModelTheory/Ultraproducts.lean b/Mathlib/ModelTheory/Ultraproducts.lean
index efe11fdadc8cd..c7c3922d198a4 100644
--- a/Mathlib/ModelTheory/Ultraproducts.lean
+++ b/Mathlib/ModelTheory/Ultraproducts.lean
@@ -94,23 +94,27 @@ theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormul
         (fun i => (v i : (u : Filter α).Product M))) ↔
       ∀ᶠ a : α in u, φ.Realize (fun i : β => x i a) fun i => v i a := by
   letI := (u : Filter α).productSetoid M
-  induction' φ with _ _ _ _ _ _ _ _ m _ _ ih ih' k φ ih
-  · simp only [BoundedFormula.Realize, eventually_const]
-  · have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
+  induction φ with
+  | falsum => simp only [BoundedFormula.Realize, eventually_const]
+  | equal =>
+    have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
       fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun i => rfl
     simp only [BoundedFormula.Realize, h2, term_realize_cast]
     erw [(Sum.comp_elim ((↑) : (∀ a, M a) → (u : Filter α).Product M) x v).symm,
       term_realize_cast, term_realize_cast]
     exact Quotient.eq''
-  · have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
+  | rel =>
+    have h2 : ∀ a : α, (Sum.elim (fun i : β => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=
       fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun i => rfl
     simp only [BoundedFormula.Realize, h2]
     erw [(Sum.comp_elim ((↑) : (∀ a, M a) → (u : Filter α).Product M) x v).symm]
     conv_lhs => enter [2, i]; erw [term_realize_cast]
     apply relMap_quotient_mk'
-  · simp only [BoundedFormula.Realize, ih v, ih' v]
+  | imp _ _ ih ih' =>
+    simp only [BoundedFormula.Realize, ih v, ih' v]
     rw [Ultrafilter.eventually_imp]
-  · simp only [BoundedFormula.Realize]
+  | @all k φ ih =>
+    simp only [BoundedFormula.Realize]
     apply Iff.trans (b := ∀ m : ∀ a : α, M a,
       φ.Realize (fun i : β => (x i : (u : Filter α).Product M))
         (Fin.snoc (((↑) : (∀ a, M a) → (u : Filter α).Product M) ∘ v)
@@ -124,7 +128,7 @@ theorem boundedFormula_realize_cast {β : Type*} {n : ℕ} (φ : L.BoundedFormul
       · simp only [Fin.snoc_last]
       · simp only [Fin.snoc_castSucc]
     simp only [← Fin.comp_snoc]
-    simp only [Function.comp, ih, h']
+    simp only [Function.comp_def, ih, h']
     refine ⟨fun h => ?_, fun h m => ?_⟩
     · contrapose! h
       simp_rw [← Ultrafilter.eventually_not, not_forall] at h
diff --git a/Mathlib/NumberTheory/ADEInequality.lean b/Mathlib/NumberTheory/ADEInequality.lean
index 4c41bee0d6a33..f6cf4a0e5c35e 100644
--- a/Mathlib/NumberTheory/ADEInequality.lean
+++ b/Mathlib/NumberTheory/ADEInequality.lean
@@ -3,7 +3,6 @@ Copyright (c) 2021 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Algebra.Order.Field.Basic
 import Mathlib.Algebra.Order.Ring.Rat
 import Mathlib.Data.Multiset.Sort
 import Mathlib.Data.PNat.Basic
@@ -148,7 +147,7 @@ theorem Admissible.one_lt_sumInv {pqr : Multiset ℕ+} : Admissible pqr → 1 <
   all_goals
     rw [← H, E', sumInv_pqr]
     conv_rhs => simp only [OfNat.ofNat, PNat.mk_coe]
-    rfl
+    norm_num
 
 theorem lt_three {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r) (H : 1 < sumInv {p, q, r}) : p < 3 := by
   have h3 : (0 : ℚ) < 3 := by norm_num
@@ -157,15 +156,15 @@ theorem lt_three {p q r : ℕ+} (hpq : p ≤ q) (hqr : q ≤ r) (H : 1 < sumInv
   have h3q := H.trans hpq
   have h3r := h3q.trans hqr
   have hp : (p : ℚ)⁻¹ ≤ 3⁻¹ := by
-    rw [inv_le_inv _ h3]
+    rw [inv_le_inv₀ _ h3]
     · assumption_mod_cast
     · norm_num
   have hq : (q : ℚ)⁻¹ ≤ 3⁻¹ := by
-    rw [inv_le_inv _ h3]
+    rw [inv_le_inv₀ _ h3]
     · assumption_mod_cast
     · norm_num
   have hr : (r : ℚ)⁻¹ ≤ 3⁻¹ := by
-    rw [inv_le_inv _ h3]
+    rw [inv_le_inv₀ _ h3]
     · assumption_mod_cast
     · norm_num
   calc
@@ -178,11 +177,11 @@ theorem lt_four {q r : ℕ+} (hqr : q ≤ r) (H : 1 < sumInv {2, q, r}) : q < 4
   rw [sumInv_pqr]
   have h4r := H.trans hqr
   have hq : (q : ℚ)⁻¹ ≤ 4⁻¹ := by
-    rw [inv_le_inv _ h4]
+    rw [inv_le_inv₀ _ h4]
     · assumption_mod_cast
     · norm_num
   have hr : (r : ℚ)⁻¹ ≤ 4⁻¹ := by
-    rw [inv_le_inv _ h4]
+    rw [inv_le_inv₀ _ h4]
     · assumption_mod_cast
     · norm_num
   calc
@@ -194,7 +193,7 @@ theorem lt_six {r : ℕ+} (H : 1 < sumInv {2, 3, r}) : r < 6 := by
   contrapose! H
   rw [sumInv_pqr]
   have hr : (r : ℚ)⁻¹ ≤ 6⁻¹ := by
-    rw [inv_le_inv _ h6]
+    rw [inv_le_inv₀ _ h6]
     · assumption_mod_cast
     · norm_num
   calc
diff --git a/Mathlib/NumberTheory/ArithmeticFunction.lean b/Mathlib/NumberTheory/ArithmeticFunction.lean
index ffd2563801e40..9e6e2e67edaef 100644
--- a/Mathlib/NumberTheory/ArithmeticFunction.lean
+++ b/Mathlib/NumberTheory/ArithmeticFunction.lean
@@ -390,7 +390,7 @@ instance {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] :
 
 section Zeta
 
-/-- `ζ 0 = 0`, otherwise `ζ x = 1`. The Dirichlet Series is the Riemann `ζ`.  -/
+/-- `ζ 0 = 0`, otherwise `ζ x = 1`. The Dirichlet Series is the Riemann `ζ`. -/
 def zeta : ArithmeticFunction ℕ :=
   ⟨fun x => ite (x = 0) 0 1, rfl⟩
 
@@ -625,7 +625,7 @@ theorem mul [CommSemiring R] {f g : ArithmeticFunction R} (hf : f.IsMultiplicati
     constructor
     · ring
     rw [Nat.mul_eq_zero] at *
-    apply not_or_of_not ha hb
+    apply not_or_intro ha hb
   · simp only [Set.InjOn, mem_coe, mem_divisorsAntidiagonal, Ne, mem_product, Prod.mk.inj_iff]
     rintro ⟨⟨a1, a2⟩, ⟨b1, b2⟩⟩ ⟨⟨rfl, ha⟩, ⟨rfl, hb⟩⟩ ⟨⟨c1, c2⟩, ⟨d1, d2⟩⟩ hcd h
     simp only [Prod.mk.inj_iff] at h
@@ -776,7 +776,7 @@ end IsMultiplicative
 
 section SpecialFunctions
 
-/-- The identity on `ℕ` as an `ArithmeticFunction`.  -/
+/-- The identity on `ℕ` as an `ArithmeticFunction`. -/
 nonrec  -- Porting note (#11445): added
 def id : ArithmeticFunction ℕ :=
   ⟨id, rfl⟩
diff --git a/Mathlib/NumberTheory/Bernoulli.lean b/Mathlib/NumberTheory/Bernoulli.lean
index fbd3725afff82..bda018aa1e2ea 100644
--- a/Mathlib/NumberTheory/Bernoulli.lean
+++ b/Mathlib/NumberTheory/Bernoulli.lean
@@ -3,12 +3,8 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Kevin Buzzard
 -/
-import Mathlib.Algebra.BigOperators.NatAntidiagonal
-import Mathlib.Algebra.GeomSum
-import Mathlib.Data.Fintype.BigOperators
 import Mathlib.RingTheory.PowerSeries.Inverse
 import Mathlib.RingTheory.PowerSeries.WellKnown
-import Mathlib.Tactic.FieldSimp
 
 /-!
 # Bernoulli numbers
@@ -257,7 +253,7 @@ theorem bernoulliPowerSeries_mul_exp_sub_one : bernoulliPowerSeries A * (exp A -
   simp only [bernoulliPowerSeries, coeff_mul, coeff_X, sum_antidiagonal_succ', one_div, coeff_mk,
     coeff_one, coeff_exp, LinearMap.map_sub, factorial, if_pos, cast_succ, cast_one, cast_mul,
     sub_zero, RingHom.map_one, add_eq_zero, if_false, _root_.inv_one, zero_add, one_ne_zero,
-    mul_zero, and_false_iff, sub_self, ← RingHom.map_mul, ← map_sum]
+    mul_zero, and_false, sub_self, ← RingHom.map_mul, ← map_sum]
   cases' n with n
   · simp
   rw [if_neg n.succ_succ_ne_one]
diff --git a/Mathlib/NumberTheory/BernoulliPolynomials.lean b/Mathlib/NumberTheory/BernoulliPolynomials.lean
index 4c666ceb7e29f..f333c35443ce6 100644
--- a/Mathlib/NumberTheory/BernoulliPolynomials.lean
+++ b/Mathlib/NumberTheory/BernoulliPolynomials.lean
@@ -175,8 +175,7 @@ theorem bernoulli_succ_eval (n p : ℕ) : (bernoulli p.succ).eval (n : ℚ) =
 theorem bernoulli_eval_one_add (n : ℕ) (x : ℚ) :
     (bernoulli n).eval (1 + x) = (bernoulli n).eval x + n * x ^ (n - 1) := by
   refine Nat.strong_induction_on n fun d hd => ?_
-  have nz : ((d.succ : ℕ) : ℚ) ≠ 0 := by
-    norm_cast
+  have nz : ((d.succ : ℕ) : ℚ) ≠ 0 := by norm_cast
   apply (mul_right_inj' nz).1
   rw [← smul_eq_mul, ← eval_smul, bernoulli_eq_sub_sum, mul_add, ← smul_eq_mul, ← eval_smul,
     bernoulli_eq_sub_sum, eval_sub, eval_finset_sum]
diff --git a/Mathlib/NumberTheory/Bertrand.lean b/Mathlib/NumberTheory/Bertrand.lean
index 772c5d12f2dff..f35408c70ebe0 100644
--- a/Mathlib/NumberTheory/Bertrand.lean
+++ b/Mathlib/NumberTheory/Bertrand.lean
@@ -165,13 +165,13 @@ theorem centralBinom_le_of_no_bertrand_prime (n : ℕ) (n_large : 2 < n)
     · exact pow_factorization_choose_le (mul_pos two_pos n_pos)
     have : (Finset.Icc 1 (sqrt (2 * n))).card = sqrt (2 * n) := by rw [card_Icc, Nat.add_sub_cancel]
     rw [Finset.prod_const]
-    refine pow_le_pow_right n2_pos ((Finset.card_le_card fun x hx => ?_).trans this.le)
+    refine pow_right_mono₀ n2_pos ((Finset.card_le_card fun x hx => ?_).trans this.le)
     obtain ⟨h1, h2⟩ := Finset.mem_filter.1 hx
     exact Finset.mem_Icc.mpr ⟨(Finset.mem_filter.1 h1).2.one_lt.le, h2⟩
   · refine le_trans ?_ (primorial_le_4_pow (2 * n / 3))
     refine (Finset.prod_le_prod' fun p hp => (?_ : f p ≤ p)).trans ?_
     · obtain ⟨h1, h2⟩ := Finset.mem_filter.1 hp
-      refine (pow_le_pow_right (Finset.mem_filter.1 h1).2.one_lt.le ?_).trans (pow_one p).le
+      refine (pow_right_mono₀ (Finset.mem_filter.1 h1).2.one_lt.le ?_).trans (pow_one p).le
       exact Nat.factorization_choose_le_one (sqrt_lt'.mp <| not_le.1 h2)
     refine Finset.prod_le_prod_of_subset_of_one_le' (Finset.filter_subset _ _) ?_
     exact fun p hp _ => (Finset.mem_filter.1 hp).2.one_lt.le
diff --git a/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean b/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean
index dd9d6a2408815..694594f1a0e40 100644
--- a/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean
+++ b/Mathlib/NumberTheory/ClassNumber/AdmissibleAbs.lean
@@ -38,7 +38,7 @@ theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb :
   refine ⟨fun i ↦ ⟨natAbs (floor ((A i % b : ℤ) / abs b • ε : ℝ)), ?_⟩, ?_⟩
   · rw [← ofNat_lt, natAbs_of_nonneg (hfloor i), floor_lt]
     apply lt_of_lt_of_le _ (Nat.le_ceil _)
-    rw [Algebra.smul_def, eq_intCast, ← div_div, div_lt_div_right hε, div_lt_iff hb', one_mul,
+    rw [Algebra.smul_def, eq_intCast, ← div_div, div_lt_div_right hε, div_lt_iff₀ hb', one_mul,
       cast_lt]
     exact Int.emod_lt _ hb
   intro i₀ i₁ hi
@@ -46,7 +46,7 @@ theorem exists_partition_int (n : ℕ) {ε : ℝ} (hε : 0 < ε) {b : ℤ} (hb :
     congr_arg ((↑) : ℕ → ℤ) (Fin.mk_eq_mk.mp hi)
   rw [natAbs_of_nonneg (hfloor i₀), natAbs_of_nonneg (hfloor i₁)] at hi
   have hi := abs_sub_lt_one_of_floor_eq_floor hi
-  rw [abs_sub_comm, ← sub_div, abs_div, abs_of_nonneg hbε.le, div_lt_iff hbε, one_mul] at hi
+  rw [abs_sub_comm, ← sub_div, abs_div, abs_of_nonneg hbε.le, div_lt_iff₀ hbε, one_mul] at hi
   rwa [Int.cast_abs, Int.cast_sub]
 
 /-- `abs : ℤ → ℤ` is an admissible absolute value. -/
diff --git a/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean b/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean
index 7b6a30b0c651c..fabb26655f44b 100644
--- a/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean
+++ b/Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean
@@ -131,7 +131,7 @@ theorem exists_approx_polynomial {b : Fq[X]} (hb : b ≠ 0) {ε : ℝ} (hε : 0
       cardPowDegree_nonzero _ h', cardPowDegree_nonzero _ hb, Algebra.smul_def, eq_intCast,
       Int.cast_pow, Int.cast_natCast, Int.cast_pow, Int.cast_natCast,
       log_mul (pow_ne_zero _ q_pos'.ne') hε.ne', ← rpow_natCast, ← rpow_natCast, log_rpow q_pos',
-      log_rpow q_pos', ← lt_div_iff (log_pos one_lt_q'), add_div,
+      log_rpow q_pos', ← lt_div_iff₀ (log_pos one_lt_q'), add_div,
       mul_div_cancel_right₀ _ (log_pos one_lt_q').ne']
   -- And that result follows from manipulating the result from `exists_approx_polynomial_aux`
   -- to turn the `-⌈-stuff⌉₊` into `+ stuff`.
@@ -160,7 +160,7 @@ theorem cardPowDegree_anti_archimedean {x y z : Fq[X]} {a : ℤ} (hxy : cardPowD
     cardPowDegree_nonzero _ hyz']
   have : (1 : ℤ) ≤ Fintype.card Fq := mod_cast (@Fintype.one_lt_card Fq _ _).le
   simp only [Int.cast_pow, Int.cast_natCast, le_max_iff]
-  refine Or.imp (pow_le_pow_right this) (pow_le_pow_right this) ?_
+  refine Or.imp (pow_le_pow_right₀ this) (pow_le_pow_right₀ this) ?_
   rw [natDegree_le_iff_degree_le, natDegree_le_iff_degree_le, ← le_max_iff, ←
     degree_eq_natDegree hxy', ← degree_eq_natDegree hyz']
   convert degree_add_le (x - y) (y - z) using 2
diff --git a/Mathlib/NumberTheory/Cyclotomic/Basic.lean b/Mathlib/NumberTheory/Cyclotomic/Basic.lean
index 891a8f3be52b6..3c95d15cc13ee 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Basic.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Basic.lean
@@ -5,7 +5,7 @@ Authors: Riccardo Brasca
 -/
 import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
 import Mathlib.NumberTheory.NumberField.Basic
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 
 /-!
 # Cyclotomic extensions
@@ -58,7 +58,7 @@ included in the `Cyclotomic` locale.
 -/
 
 
-open Polynomial Algebra FiniteDimensional Set
+open Polynomial Algebra Module Set
 
 universe u v w z
 
diff --git a/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean b/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean
index 1f7a73c505206..fad2bbc131b52 100644
--- a/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean
@@ -105,7 +105,8 @@ theorem toFun_spec (g : L ≃+* L) {n : ℕ+} (t : rootsOfUnity n L) :
     g (t : Lˣ) = (t ^ (χ₀ n g).val : Lˣ) := by
   rw [ModularCyclotomicCharacter_aux_spec g n t, ← zpow_natCast, ModularCyclotomicCharacter.toFun,
     ZMod.val_intCast, ← Subgroup.coe_zpow]
-  exact Units.ext_iff.1 <| SetCoe.ext_iff.2 <| zpow_eq_zpow_emod _ pow_card_eq_one
+  exact Units.ext_iff.1 <| SetCoe.ext_iff.2 <|
+    zpow_eq_zpow_emod _ pow_card_eq_one (G := rootsOfUnity n L)
 
 theorem toFun_spec' (g : L ≃+* L) {n : ℕ+} {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :
     g t = t ^ (χ₀ n g).val :=
@@ -195,7 +196,7 @@ lemma spec (g : L ≃+* L) {t : Lˣ} (ht : t ∈ rootsOfUnity n L) :
   congr 1
   exact (ZMod.ringEquivCongr_val _ _).symm
 
-lemma unique (g : L ≃+* L) {c : ZMod n}  (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :
+lemma unique (g : L ≃+* L) {c : ZMod n} (hc : ∀ t ∈ rootsOfUnity n L, g t = t ^ c.val) :
     c = ModularCyclotomicCharacter L hn g := by
   change c = (ZMod.ringEquivCongr hn) (toFun n g)
   rw [← toFun_unique' n g (ZMod.ringEquivCongr hn.symm c)
diff --git a/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean b/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean
index e526d1422118e..c4b31c98a929a 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Discriminant.lean
@@ -168,7 +168,7 @@ theorem discr_prime_pow [hcycl : IsCyclotomicExtension {p ^ k} K L] [hp : Fact (
       convert_to (discr K fun i : Fin 1 ↦ (algebraMap K L) (-1) ^ ↑i) = _
       · congr
         ext i
-        simp only [map_neg, map_one, Function.comp_apply, Fin.coe_fin_one, _root_.pow_zero]
+        simp only [map_neg, map_one, Function.comp_apply, Fin.val_eq_zero, _root_.pow_zero]
         suffices (e.symm i : ℕ) = 0 by simp [this]
         rw [← Nat.lt_one_iff]
         convert (e.symm i).2
diff --git a/Mathlib/NumberTheory/Cyclotomic/Embeddings.lean b/Mathlib/NumberTheory/Cyclotomic/Embeddings.lean
index b3d2a56cf11a2..642d6933fad59 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Embeddings.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Embeddings.lean
@@ -21,7 +21,7 @@ universe u
 
 namespace IsCyclotomicExtension.Rat
 
-open NumberField InfinitePlace FiniteDimensional Complex Nat Polynomial
+open NumberField InfinitePlace Module Complex Nat Polynomial
 
 variable {n : ℕ+} (K : Type u) [Field K] [CharZero K]
 
diff --git a/Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean b/Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
index fbbb83aa89763..e4a82b6a04edb 100644
--- a/Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean
@@ -61,7 +61,7 @@ and only at the "final step", when we need to provide an "explicit" primitive ro
 -/
 
 
-open Polynomial Algebra Finset FiniteDimensional IsCyclotomicExtension Nat PNat Set
+open Polynomial Algebra Finset Module IsCyclotomicExtension Nat PNat Set
 open scoped IntermediateField
 
 universe u v w z
@@ -113,9 +113,10 @@ variable {C}
 /-- The `PowerBasis` given by a primitive root `η`. -/
 @[simps!]
 protected noncomputable def powerBasis : PowerBasis K L :=
-  PowerBasis.map (Algebra.adjoin.powerBasis <| (integral {n} K L).isIntegral ζ) <|
-    (Subalgebra.equivOfEq _ _ (IsCyclotomicExtension.adjoin_primitive_root_eq_top hζ)).trans
-      Subalgebra.topEquiv
+  -- this is purely an optimization
+  letI pb := Algebra.adjoin.powerBasis <| (integral {n} K L).isIntegral ζ
+  pb.map <| (Subalgebra.equivOfEq _ _ (IsCyclotomicExtension.adjoin_primitive_root_eq_top hζ)).trans
+    Subalgebra.topEquiv
 
 theorem powerBasis_gen_mem_adjoin_zeta_sub_one :
     (hζ.powerBasis K).gen ∈ adjoin K ({ζ - 1} : Set L) := by
@@ -182,7 +183,7 @@ least `(lcm p q).totient`. -/
 theorem _root_.IsPrimitiveRoot.lcm_totient_le_finrank [FiniteDimensional K L] {p q : ℕ} {x y : L}
     (hx : IsPrimitiveRoot x p) (hy : IsPrimitiveRoot y q)
     (hirr : Irreducible (cyclotomic (Nat.lcm p q) K)) :
-    (Nat.lcm p q).totient ≤ FiniteDimensional.finrank K L := by
+    (Nat.lcm p q).totient ≤ Module.finrank K L := by
   rcases Nat.eq_zero_or_pos p with (rfl | hppos)
   · simp
   rcases Nat.eq_zero_or_pos q with (rfl | hqpos)
@@ -340,8 +341,8 @@ theorem sub_one_norm_eq_eval_cyclotomic [IsCyclotomicExtension {n} K L] (h : 2 <
     rfl
     ext
     rw [← neg_sub, map_neg, map_sub, map_one, neg_eq_neg_one_mul]
-  rw [prod_mul_distrib, prod_const, card_univ, AlgHom.card, IsCyclotomicExtension.finrank L hirr,
-    (totient_even h).neg_one_pow, one_mul]
+  rw [prod_mul_distrib, prod_const, Finset.card_univ, AlgHom.card,
+    IsCyclotomicExtension.finrank L hirr, (totient_even h).neg_one_pow, one_mul]
   have Hprod : (Finset.univ.prod fun σ : L →ₐ[K] E => 1 - σ ζ) = eval 1 (cyclotomic' n E) := by
     rw [cyclotomic', eval_prod, ← @Finset.prod_attach E E, ← univ_eq_attach]
     refine Fintype.prod_equiv (hζ.embeddingsEquivPrimitiveRoots E hirr) _ _ fun σ => ?_
@@ -379,7 +380,7 @@ theorem minpoly_sub_one_eq_cyclotomic_comp [Algebra K A] [IsDomain A] {ζ : A}
     minpoly K (ζ - 1) = (cyclotomic n K).comp (X + 1) := by
   haveI := IsCyclotomicExtension.neZero' n K A
   rw [show ζ - 1 = ζ + algebraMap K A (-1) by simp [sub_eq_add_neg],
-    minpoly.add_algebraMap ((integral {n} K A).isIntegral ζ),
+    minpoly.add_algebraMap ζ,
     hζ.minpoly_eq_cyclotomic_of_irreducible h]
   simp
 
@@ -392,9 +393,8 @@ theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot
     [hpri : Fact (p : ℕ).Prime] [IsCyclotomicExtension {p ^ (k + 1)} K L]
     (hirr : Irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (hs : s ≤ k)
     (htwo : p ^ (k - s + 1) ≠ 2) : norm K (ζ ^ (p : ℕ) ^ s - 1) = (p : K) ^ (p : ℕ) ^ s := by
--- Porting note: `by simp` was `by linarith` that now fails.
   have hirr₁ : Irreducible (cyclotomic ((p : ℕ) ^ (k - s + 1)) K) :=
-    cyclotomic_irreducible_pow_of_irreducible_pow hpri.1 (by simp) hirr
+    cyclotomic_irreducible_pow_of_irreducible_pow hpri.1 (by omega) hirr
   rw [← PNat.pow_coe] at hirr₁
   set η := ζ ^ (p : ℕ) ^ s - 1
   let η₁ : K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η
@@ -403,22 +403,17 @@ theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot
     refine IsPrimitiveRoot.pow (p ^ (k + 1)).pos hζ ?_
     rw [PNat.pow_coe, ← pow_add, add_comm s, Nat.sub_add_cancel (le_trans hs (Nat.le_succ k))]
   have : IsCyclotomicExtension {p ^ (k - s + 1)} K K⟮η⟯ := by
-    suffices IsCyclotomicExtension {p ^ (k - s + 1)} K K⟮η + 1⟯.toSubalgebra by
-      have H : K⟮η + 1⟯.toSubalgebra = K⟮η⟯.toSubalgebra := by
-        simp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral
-            ((integral {p ^ (k + 1)} K L).isIntegral _)]
-        refine Subalgebra.ext fun x => ⟨fun hx => adjoin_le ?_ hx, fun hx => adjoin_le ?_ hx⟩
-        · simp only [Set.singleton_subset_iff, SetLike.mem_coe]
-          exact Subalgebra.add_mem _ (subset_adjoin (mem_singleton η)) (Subalgebra.one_mem _)
-        · simp only [Set.singleton_subset_iff, SetLike.mem_coe]
-          nth_rw 1 [← add_sub_cancel_right η 1]
-          exact Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _)) (Subalgebra.one_mem _)
--- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.
-      exact IsCyclotomicExtension.equiv _ _ _ (Subalgebra.equivOfEq _ _ H)
--- Porting note: the next `refine` was `rw [H]`, abusing defeq, and it now fails.
+    have HKη : K⟮η⟯ = K⟮η + 1⟯ := by
+      refine le_antisymm ?_ ?_
+      all_goals rw [IntermediateField.adjoin_simple_le_iff]
+      · nth_rw 2 [← add_sub_cancel_right η 1]
+        exact sub_mem (IntermediateField.mem_adjoin_simple_self K (η + 1)) (one_mem _)
+      · exact add_mem (IntermediateField.mem_adjoin_simple_self K η) (one_mem _)
+    rw [HKη]
     have H := IntermediateField.adjoin_simple_toSubalgebra_of_integral
-        ((integral {p ^ (k + 1)} K L).isIntegral (η + 1))
-    refine @IsCyclotomicExtension.equiv _ _ _ _ _ _ _ _ _ ?_ (Subalgebra.equivOfEq _ _ H).symm
+      ((integral {p ^ (k + 1)} K L).isIntegral (η + 1))
+    refine IsCyclotomicExtension.equiv _ _ _ (h := ?_) (.refl : K⟮η + 1⟯.toSubalgebra ≃ₐ[K] _)
+    rw [H]
     have hη' : IsPrimitiveRoot (η + 1) ↑(p ^ (k + 1 - s)) := by simpa using hη
 -- Porting note: `using 1` was not needed.
     convert hη'.adjoin_isCyclotomicExtension K using 1
@@ -427,10 +422,10 @@ theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot
     apply coe_submonoidClass_iff.1
     convert hη using 1
     rw [Nat.sub_add_comm hs, pow_coe]
--- Porting note: the following `haveI` were not needed because the locale `cyclotomic` set them
+-- Porting note: the following `have` were not needed because the locale `cyclotomic` set them
 -- as instances.
-  haveI := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L
-  haveI := IsCyclotomicExtension.isGalois (p ^ (k + 1)) K L
+  have := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L
+  have := IsCyclotomicExtension.isGalois (p ^ (k + 1)) K L
   rw [norm_eq_norm_adjoin K]
   have H := hη.sub_one_norm_isPrimePow ?_ hirr₁ htwo
   swap; · rw [PNat.pow_coe]; exact hpri.1.isPrimePow.pow (Nat.succ_ne_zero _)
@@ -439,7 +434,7 @@ theorem norm_pow_sub_one_of_prime_pow_ne_two {k s : ℕ} (hζ : IsPrimitiveRoot
   congr
   · rw [PNat.pow_coe, Nat.pow_minFac, hpri.1.minFac_eq]
     exact Nat.succ_ne_zero _
-  have := FiniteDimensional.finrank_mul_finrank K K⟮η⟯ L
+  have := Module.finrank_mul_finrank K K⟮η⟯ L
   rw [IsCyclotomicExtension.finrank L hirr, IsCyclotomicExtension.finrank K⟮η⟯ hirr₁,
     PNat.pow_coe, PNat.pow_coe, Nat.totient_prime_pow hpri.out (k - s).succ_pos,
     Nat.totient_prime_pow hpri.out k.succ_pos, mul_comm _ ((p : ℕ) - 1), mul_assoc,
diff --git a/Mathlib/NumberTheory/Cyclotomic/Rat.lean b/Mathlib/NumberTheory/Cyclotomic/Rat.lean
index 2378d74c707e4..39cdf61a38935 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Rat.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Rat.lean
@@ -261,9 +261,7 @@ noncomputable def subOneIntegralPowerBasis [IsCyclotomicExtension {p ^ k} ℚ K]
       simp only [integralPowerBasis_gen, toInteger]
       convert Subalgebra.add_mem _ (self_mem_adjoin_singleton ℤ (⟨ζ - 1, _⟩ : 𝓞 K))
         (Subalgebra.one_mem _)
--- Porting note: `simp` was able to finish the proof.
-      · simp only [Subsemiring.coe_add, Subalgebra.coe_toSubsemiring,
-          OneMemClass.coe_one, sub_add_cancel]
+      · simp
       · exact Subalgebra.sub_mem _ (hζ.isIntegral (by simp)) (Subalgebra.one_mem _))
 
 @[simp]
@@ -295,7 +293,7 @@ theorem zeta_sub_one_prime_of_ne_two [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
     Prime (hζ.toInteger - 1) := by
   letI := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
   refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_
-  · apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow hp.out.one_lt (by simp))
+  · apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow₀ hp.out.one_lt (by simp))
     rw [sub_eq_zero] at h
     simpa using congrArg (algebraMap _ K) h
   rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
@@ -314,7 +312,7 @@ theorem zeta_sub_one_prime_of_two_pow [IsCyclotomicExtension {(2 : ℕ+) ^ (k +
     Prime (hζ.toInteger - 1) := by
   letI := IsCyclotomicExtension.numberField {(2 : ℕ+) ^ (k + 1)} ℚ K
   refine Ideal.prime_of_irreducible_absNorm_span (fun h ↦ ?_) ?_
-  · apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow (by decide) (by simp))
+  · apply hζ.pow_ne_one_of_pos_of_lt zero_lt_one (one_lt_pow₀ (by decide) (by simp))
     rw [sub_eq_zero] at h
     simpa using congrArg (algebraMap _ K) h
   rw [Nat.irreducible_iff_prime, Ideal.absNorm_span_singleton, ← Nat.prime_iff,
@@ -455,7 +453,7 @@ theorem not_exists_int_prime_dvd_sub_of_prime_pow_ne_two
     · simp only [hk, zero_add, pow_one, pow_zero, one_mul, Nat.lt_sub_iff_add_lt,
         Nat.reduceAdd] at htwo ⊢
       exact htwo.symm.lt_of_le hp.1.two_le
-    · exact one_lt_mul_of_lt_of_le (one_lt_pow hp.1.one_lt hk)
+    · exact one_lt_mul_of_lt_of_le (one_lt_pow₀ hp.1.one_lt hk)
         (have := Nat.Prime.two_le hp.out; by omega)
   rw [sub_eq_iff_eq_add] at h
   -- We are assuming that `ζ = n + p * x` for some integer `n` and `x : 𝓞 K`. Looking at the
@@ -500,7 +498,8 @@ theorem finite_quotient_span_sub_one [hcycl : IsCyclotomicExtension {p ^ (k + 1)
   have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
   refine Fintype.finite <| Ideal.fintypeQuotientOfFreeOfNeBot _ (fun h ↦ ?_)
   simp only [Ideal.span_singleton_eq_bot, sub_eq_zero, ← Subtype.coe_inj] at h
-  exact hζ.ne_one (one_lt_pow hp.1.one_lt (Nat.zero_ne_add_one k).symm) (RingOfIntegers.ext_iff.1 h)
+  exact hζ.ne_one (one_lt_pow₀ hp.1.one_lt (Nat.zero_ne_add_one k).symm)
+    (RingOfIntegers.ext_iff.1 h)
 
 theorem finite_quotient_span_sub_one' [hcycl : IsCyclotomicExtension {p} ℚ K]
     (hζ : IsPrimitiveRoot ζ ↑p) :
diff --git a/Mathlib/NumberTheory/Dioph.lean b/Mathlib/NumberTheory/Dioph.lean
index 674ecffb62b68..721b81b5271ba 100644
--- a/Mathlib/NumberTheory/Dioph.lean
+++ b/Mathlib/NumberTheory/Dioph.lean
@@ -439,14 +439,15 @@ open Vector3
 open scoped Vector3
 
 -- Porting note: Fails because declaration is in an imported module
--- attribute [local reducible] Vector3
+set_option allowUnsafeReducibility true in
+attribute [local reducible] Vector3
 
 theorem diophFn_vec_comp1 {S : Set (Vector3 ℕ (succ n))} (d : Dioph S) {f : Vector3 ℕ n → ℕ}
     (df : DiophFn f) : Dioph {v : Vector3 ℕ n | (f v::v) ∈ S} :=
   Dioph.ext (diophFn_comp1 (reindex_dioph _ (none::some) d) df) (fun v => by
     dsimp
     -- Porting note: `congr` use to be enough here
-    refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
+    suffices ((f v ::ₒ v) ∘ none :: some) = f v :: v by rw [this]; rfl
     ext x; cases x <;> rfl)
 
 theorem vec_ex1_dioph (n) {S : Set (Vector3 ℕ (succ n))} (d : Dioph S) :
@@ -470,7 +471,7 @@ theorem diophFn_compn :
     ext (reindex_dioph _ (id ⊗ Fin2.elim0) d) fun v => by
       dsimp
       -- Porting note: `congr` use to be enough here
-      refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
+      suffices v ∘ (id ⊗ elim0) = v ⊗ fun i ↦ f i v by rw [this]
       ext x; obtain _ | _ | _ := x; rfl
   | succ n, S, d, f =>
     f.consElim fun f fl => by
@@ -482,14 +483,15 @@ theorem diophFn_compn :
               fun v => by
                 dsimp
                 -- Porting note: `congr` use to be enough here
-                refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
+                suffices (f (v ∘ inl) ::ₒ v) ∘ (some ∘ inl ⊗ none :: some ∘ inr) =
+                    v ∘ inl ⊗ f (v ∘ inl) :: v ∘ inr by rw [this]
                 ext x; obtain _ | _ | _ := x <;> rfl
           have : Dioph {v | (v ⊗ f v::fun i : Fin2 n => fl i v) ∈ S} :=
             @diophFn_compn n (fun v => S (v ∘ inl ⊗ f (v ∘ inl)::v ∘ inr)) this _ dfl
           ext this fun v => by
             dsimp
             -- Porting note: `congr` use to be enough here
-            refine iff_of_eq (congrFun (congrArg Membership.mem ?_) S)
+            suffices (v ⊗ f v :: fun i ↦ fl i v) = v ⊗ fun i ↦ (f :: fl) i v by rw [this]
             ext x; obtain _ | _ | _ := x <;> rfl
 
 theorem dioph_comp {S : Set (Vector3 ℕ n)} (d : Dioph S) (f : Vector3 ((α → ℕ) → ℕ) n)
diff --git a/Mathlib/NumberTheory/DiophantineApproximation.lean b/Mathlib/NumberTheory/DiophantineApproximation.lean
index 77d109b645420..1d052c7a9f81e 100644
--- a/Mathlib/NumberTheory/DiophantineApproximation.lean
+++ b/Mathlib/NumberTheory/DiophantineApproximation.lean
@@ -235,7 +235,7 @@ theorem den_le_and_le_num_le_of_sub_lt_one_div_den_sq {ξ q : ℚ}
     · exact le_rfl
     · have hξ₀ : (0 : ℚ) < ξ.den := Nat.cast_pos.mpr ξ.pos
       rw [← Rat.num_div_den ξ, div_mul_eq_mul_div, div_sub' _ _ _ hξ₀.ne', abs_div, abs_of_pos hξ₀,
-        div_lt_iff hξ₀, div_mul_comm, mul_one] at h
+        div_lt_iff₀ hξ₀, div_mul_comm, mul_one] at h
       refine Nat.cast_le.mp ((one_lt_div hq₀).mp <| lt_of_le_of_lt ?_ h).le
       norm_cast
       rw [mul_comm _ q.num]
@@ -406,8 +406,7 @@ private theorem aux₁ : 0 < fract ξ := by
   refine fract_pos.mpr fun hf => ?_
   rw [hf] at h
   have H : (2 * v - 1 : ℝ) < 1 := by
-    refine
-      (mul_lt_iff_lt_one_right hv₀).mp ((inv_lt_inv hv₀ (mul_pos hv₁ hv₂)).mp (lt_of_le_of_lt ?_ h))
+    refine (mul_lt_iff_lt_one_right hv₀).1 ((inv_lt_inv₀ hv₀ (mul_pos hv₁ hv₂)).1 (h.trans_le' ?_))
     have h' : (⌊ξ⌋ : ℝ) - u / v = (⌊ξ⌋ * v - u) / v := by field_simp
     rw [h', abs_div, abs_of_pos hv₀, ← one_div, div_le_div_right hv₀]
     norm_cast
@@ -422,9 +421,9 @@ private theorem aux₂ : 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v := by
   obtain ⟨hcop, _, h⟩ := h
   obtain ⟨hv₀, hv₀'⟩ := aux₀ (zero_lt_two.trans_le hv)
   have hv₁ : 0 < 2 * v - 1 := by linarith only [hv]
-  rw [← one_div, lt_div_iff (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul, sub_mul,
-    ← mul_assoc, ← mul_assoc, div_mul_cancel₀ _ hv₀.ne', abs_sub_comm, abs_lt, lt_sub_iff_add_lt,
-    sub_lt_iff_lt_add, mul_assoc] at h
+  rw [← one_div, lt_div_iff₀ (mul_pos hv₀ hv₀'), ← abs_of_pos (mul_pos hv₀ hv₀'), ← abs_mul,
+    sub_mul, ← mul_assoc, ← mul_assoc, div_mul_cancel₀ _ hv₀.ne', abs_sub_comm, abs_lt,
+    lt_sub_iff_add_lt, sub_lt_iff_lt_add, mul_assoc] at h
   have hu₀ : 0 ≤ u - ⌊ξ⌋ * v := by
     -- Porting note: this abused the definitional equality `-1 + 1 = 0`
     -- refine' (mul_nonneg_iff_of_pos_right hv₁).mp ((lt_iff_add_one_le (-1 : ℤ) _).mp _)
@@ -490,7 +489,7 @@ private theorem aux₃ :
     _ < ((v : ℝ) * (2 * v - 1))⁻¹ * (v / u' / fract ξ) := (mul_lt_mul_right H₁).mpr h'
     _ = (u' * (2 * v - 1) * fract ξ)⁻¹ := help₂ hξ₀.ne' Hv.ne' Hv'.ne' Hu.ne'
     _ ≤ ((u' : ℝ) * (2 * u' - 1))⁻¹ := by
-      rwa [inv_le_inv (mul_pos (mul_pos Hu Hv') hξ₀) <| mul_pos Hu Hu', mul_assoc,
+      rwa [inv_le_inv₀ (mul_pos (mul_pos Hu Hv') hξ₀) <| mul_pos Hu Hu', mul_assoc,
         mul_le_mul_left Hu]
 
 -- The conditions `ass ξ u v` persist in the inductive step.
@@ -506,7 +505,7 @@ private theorem invariant : ContfracLegendre.Ass (fract ξ)⁻¹ v (u - ⌊ξ⌋
     have h' := (abs_sub_lt_iff.mp h.2.2).1
     rw [Huv, ← sub_sub, sub_lt_iff_lt_add, self_sub_floor, Hv] at h'
     rwa [lt_sub_iff_add_lt', (by ring : (v : ℝ) + -(1 / 2) = (2 * v - 1) / 2),
-      lt_inv (div_pos hv₀' zero_lt_two) (aux₁ hv h), inv_div]
+      lt_inv_comm₀ (div_pos hv₀' zero_lt_two) (aux₁ hv h), inv_div]
 
 end
 
@@ -538,8 +537,8 @@ theorem exists_rat_eq_convergent' {v : ℕ} (h : ContfracLegendre.Ass ξ u v) :
       · rw [Hξ, hξ₁, cast_sub, cast_one, ← sub_eq_add_neg, sub_lt_sub_iff_left] at h₁
         exact False.elim (lt_irrefl _ <| h₁.trans one_half_lt_one)
       · have hξ₂ : ⌊(fract ξ)⁻¹⌋ = 1 := by
-          rw [floor_eq_iff, cast_one, le_inv zero_lt_one (fract_pos.mpr Hξ), inv_one,
-            one_add_one_eq_two, inv_lt (fract_pos.mpr Hξ) zero_lt_two]
+          rw [floor_eq_iff, cast_one, le_inv_comm₀ zero_lt_one (fract_pos.mpr Hξ), inv_one,
+            one_add_one_eq_two, inv_lt_comm₀ (fract_pos.mpr Hξ) zero_lt_two]
           refine ⟨(fract_lt_one ξ).le, ?_⟩
           rw [fract, hξ₁, cast_sub, cast_one, lt_sub_iff_add_lt', sub_add]
           convert h₁ using 1
diff --git a/Mathlib/NumberTheory/DirichletCharacter/Basic.lean b/Mathlib/NumberTheory/DirichletCharacter/Basic.lean
index 8146354beb0aa..9fbae12e56bcd 100644
--- a/Mathlib/NumberTheory/DirichletCharacter/Basic.lean
+++ b/Mathlib/NumberTheory/DirichletCharacter/Basic.lean
@@ -161,7 +161,7 @@ noncomputable instance : Unique (DirichletCharacter R 1) := Unique.mk' (Dirichle
 
 lemma changeLevel_one {d : ℕ} (h : d ∣ n) :
     changeLevel h (1 : DirichletCharacter R d) = 1 := by
-  simp [changeLevel]
+  simp
 
 lemma factorsThrough_one_iff : FactorsThrough χ 1 ↔ χ = 1 := by
   refine ⟨fun ⟨_, χ₀, hχ₀⟩ ↦ ?_,
diff --git a/Mathlib/NumberTheory/Divisors.lean b/Mathlib/NumberTheory/Divisors.lean
index 78221e801b0c2..0c98cefad6ecd 100644
--- a/Mathlib/NumberTheory/Divisors.lean
+++ b/Mathlib/NumberTheory/Divisors.lean
@@ -82,7 +82,7 @@ theorem cons_self_properDivisors (h : n ≠ 0) :
 @[simp]
 theorem mem_divisors {m : ℕ} : n ∈ divisors m ↔ n ∣ m ∧ m ≠ 0 := by
   rcases eq_or_ne m 0 with (rfl | hm); · simp [divisors]
-  simp only [hm, Ne, not_false_iff, and_true_iff, ← filter_dvd_eq_divisors hm, mem_filter,
+  simp only [hm, Ne, not_false_iff, and_true, ← filter_dvd_eq_divisors hm, mem_filter,
     mem_range, and_iff_right_iff_imp, Nat.lt_succ_iff]
   exact le_of_dvd hm.bot_lt
 
@@ -109,7 +109,7 @@ theorem mem_divisorsAntidiagonal {x : ℕ × ℕ} :
   · rw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]
     rw [mul_eq_zero, not_or] at h
     simp only [succ_le_of_lt (Nat.pos_of_ne_zero h.1), succ_le_of_lt (Nat.pos_of_ne_zero h.2),
-      true_and_iff]
+      true_and]
     exact
       ⟨Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero h.2),
         Nat.le_mul_of_pos_left _ (Nat.pos_of_ne_zero h.1)⟩
@@ -130,7 +130,7 @@ lemma right_ne_zero_of_mem_divisorsAntidiagonal {p : ℕ × ℕ} (hp : p ∈ n.d
 theorem divisor_le {m : ℕ} : n ∈ divisors m → n ≤ m := by
   cases' m with m
   · simp
-  · simp only [mem_divisors, Nat.succ_ne_zero m, and_true_iff, Ne, not_false_iff]
+  · simp only [mem_divisors, Nat.succ_ne_zero m, and_true, Ne, not_false_iff]
     exact Nat.le_of_dvd (Nat.succ_pos m)
 
 theorem divisors_subset_of_dvd {m : ℕ} (hzero : n ≠ 0) (h : m ∣ n) : divisors m ⊆ divisors n :=
diff --git a/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean b/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
index 82d366d734cee..ba72ec28fdfcd 100644
--- a/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
+++ b/Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean
@@ -77,12 +77,8 @@ lemma summable_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs :
 open scoped LSeries.notation in
 lemma tsum_dirichletSummand {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :
     ∑' (n : ℕ), dirichletSummandHom χ (ne_zero_of_one_lt_re hs) n = L ↗χ s := by
-  simp only [LSeries, LSeries.term, dirichletSummandHom]
-  refine tsum_congr (fun n ↦ ?_)
-  rcases eq_or_ne n 0 with rfl | hn
-  · simp only [map_zero, ↓reduceIte]
-  · simp only [cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, hn, ↓reduceIte,
-      Field.div_eq_mul_inv]
+  simp only [dirichletSummandHom, cpow_neg, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, LSeries,
+    LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), div_eq_mul_inv]
 
 open Filter Nat Topology EulerProduct
 
diff --git a/Mathlib/NumberTheory/FLT/Basic.lean b/Mathlib/NumberTheory/FLT/Basic.lean
index 49253dbdcc2fd..3630dbe1368d9 100644
--- a/Mathlib/NumberTheory/FLT/Basic.lean
+++ b/Mathlib/NumberTheory/FLT/Basic.lean
@@ -1,18 +1,12 @@
 /-
 Copyright (c) 2023 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kevin Buzzard, Yaël Dillies
+Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
 -/
 import Mathlib.Algebra.EuclideanDomain.Int
 import Mathlib.Algebra.GCDMonoid.Finset
 import Mathlib.Algebra.GCDMonoid.Nat
-import Mathlib.Algebra.GroupWithZero.Divisibility
-import Mathlib.Algebra.Order.Ring.Abs
-import Mathlib.Data.Rat.Defs
 import Mathlib.RingTheory.PrincipalIdealDomain
-import Mathlib.Tactic.NormNum
-import Mathlib.Tactic.Positivity.Basic
-import Mathlib.Tactic.TFAE
 
 /-!
 # Statement of Fermat's Last Theorem
@@ -89,7 +83,7 @@ lemma FermatLastTheoremFor.mono (hmn : m ∣ n) (hm : FermatLastTheoremFor m) :
 lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
     TFAE [FermatLastTheoremWith ℕ n, FermatLastTheoremWith ℤ n, FermatLastTheoremWith ℚ n] := by
   tfae_have 1 → 2
-  · rintro h a b c ha hb hc habc
+  | h, a, b, c, ha, hb, hc, habc => by
     obtain hn | hn := n.even_or_odd
     · refine h a.natAbs b.natAbs c.natAbs (by positivity) (by positivity) (by positivity)
         (Int.natCast_inj.1 ?_)
@@ -129,7 +123,7 @@ lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
       push_cast
       simp only [abs_of_pos, habc, *]
   tfae_have 2 → 3
-  · rintro h a b c ha hb hc habc
+  | h, a, b, c, ha, hb, hc, habc => by
     rw [← Rat.num_ne_zero] at ha hb hc
     refine h (a.num * b.den * c.den) (a.den * b.num * c.den) (a.den * b.den * c.num)
       (by positivity) (by positivity) (by positivity) ?_
@@ -140,8 +134,7 @@ lemma fermatLastTheoremWith_nat_int_rat_tfae (n : ℕ) :
       div_self (by positivity : (b.den : ℚ) ≠ 0), div_self (by positivity : (c.den : ℚ) ≠ 0),
       one_mul, mul_one, Rat.num_div_den, habc]
   tfae_have 3 → 1
-  · rintro h a b c
-    exact mod_cast h a b c
+  | h, a, b, c => mod_cast h a b c
   tfae_finish
 
 lemma fermatLastTheoremFor_iff_nat {n : ℕ} : FermatLastTheoremFor n ↔ FermatLastTheoremWith ℕ n :=
@@ -153,6 +146,67 @@ lemma fermatLastTheoremFor_iff_int {n : ℕ} : FermatLastTheoremFor n ↔ Fermat
 lemma fermatLastTheoremFor_iff_rat {n : ℕ} : FermatLastTheoremFor n ↔ FermatLastTheoremWith ℚ n :=
   (fermatLastTheoremWith_nat_int_rat_tfae n).out 0 2
 
+/--
+A relaxed variant of Fermat's Last Theorem over a given commutative semiring with a specific
+exponent, allowing nonzero solutions of units and their common multiples.
+
+1. The variant `FermatLastTheoremWith' α` is weaker than `FermatLastTheoremWith α` in general.
+   In particular, it holds trivially for `[Field α]`.
+2. This variant is equivalent to the original `FermatLastTheoremWith α` for `α = ℕ` or `ℤ`.
+   In general, they are equivalent if there is no solutions of units to the Fermat equation.
+3. For a polynomial ring `α = k[X]`, the original `FermatLastTheoremWith α` is false but the weaker
+   variant `FermatLastTheoremWith' α` is true. This polynomial variant of Fermat's Last Theorem
+   can be shown elementarily using Mason--Stothers theorem.
+-/
+def FermatLastTheoremWith' (α : Type*) [CommSemiring α] (n : ℕ) : Prop :=
+  ∀ a b c : α, a ≠ 0 → b ≠ 0 → c ≠ 0 → a ^ n + b ^ n = c ^ n →
+    ∃ d a' b' c', (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ (IsUnit a' ∧ IsUnit b' ∧ IsUnit c')
+
+lemma FermatLastTheoremWith.fermatLastTheoremWith' {α : Type*} [CommSemiring α] {n : ℕ}
+    (h : FermatLastTheoremWith α n) : FermatLastTheoremWith' α n :=
+  fun a b c _ _ _ _ ↦ by exfalso; apply h a b c <;> assumption
+
+lemma fermatLastTheoremWith'_of_field (α : Type*) [Field α] (n : ℕ) : FermatLastTheoremWith' α n :=
+  fun a b c ha hb hc _ ↦
+    ⟨1, a, b, c,
+     ⟨(mul_one a).symm, (mul_one b).symm, (mul_one c).symm⟩,
+     ⟨ha.isUnit, hb.isUnit, hc.isUnit⟩⟩
+
+lemma FermatLastTheoremWith'.fermatLastTheoremWith {α : Type*} [CommSemiring α] [IsDomain α]
+    {n : ℕ} (h : FermatLastTheoremWith' α n)
+    (hn : ∀ a b c : α, IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n) :
+    FermatLastTheoremWith α n := by
+  intro a b c ha hb hc heq
+  rcases h a b c ha hb hc heq with ⟨d, a', b', c', ⟨rfl, rfl, rfl⟩, ⟨ua, ub, uc⟩⟩
+  rw [mul_pow, mul_pow, mul_pow, ← add_mul] at heq
+  exact hn _ _ _ ua ub uc <| mul_right_cancel₀ (pow_ne_zero _ (right_ne_zero_of_mul ha)) heq
+
+lemma fermatLastTheoremWith'_iff_fermatLastTheoremWith {α : Type*} [CommSemiring α] [IsDomain α]
+    {n : ℕ} (hn : ∀ a b c : α, IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n) :
+    FermatLastTheoremWith' α n ↔ FermatLastTheoremWith α n :=
+  Iff.intro (fun h ↦ h.fermatLastTheoremWith hn) (fun h ↦ h.fermatLastTheoremWith')
+
+lemma fermatLastTheoremWith'_nat_int_tfae (n : ℕ) :
+    TFAE [FermatLastTheoremFor n, FermatLastTheoremWith' ℕ n, FermatLastTheoremWith' ℤ n] := by
+  tfae_have 2 ↔ 1 := by
+    apply fermatLastTheoremWith'_iff_fermatLastTheoremWith
+    simp only [Nat.isUnit_iff]
+    intro _ _ _ ha hb hc
+    rw [ha, hb, hc]
+    simp only [one_pow, Nat.reduceAdd, ne_eq, OfNat.ofNat_ne_one, not_false_eq_true]
+  tfae_have 3 ↔ 1 := by
+    rw [fermatLastTheoremFor_iff_int]
+    apply fermatLastTheoremWith'_iff_fermatLastTheoremWith
+    intro a b c ha hb hc
+    by_cases hn : n = 0
+    · subst hn
+      simp only [pow_zero, Int.reduceAdd, ne_eq, OfNat.ofNat_ne_one, not_false_eq_true]
+    · rw [← isUnit_pow_iff hn, Int.isUnit_iff] at ha hb hc
+      -- case division
+      rcases ha with ha | ha <;> rcases hb with hb | hb <;> rcases hc with hc | hc <;>
+        rw [ha, hb, hc] <;> decide
+  tfae_finish
+
 open Finset in
 /-- To prove Fermat Last Theorem in any semiring that is a `NormalizedGCDMonoid` one can assume
 that the `gcd` of `{a, b, c}` is `1`. -/
@@ -175,7 +229,7 @@ lemma fermatLastTheoremWith_of_fermatLastTheoremWith_coprime {n : ℕ} {R : Type
   rw [← Finset.gcd_mul_left, gcd_eq_gcd_image, image_insert, image_insert, image_singleton,
       id_eq, id_eq, id_eq, ← hA, ← hB, ← hC]
 
-lemma dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT {n : ℕ} {p : ℤ} (hp : Prime p)  {a b c : ℤ}
+lemma dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT {n : ℕ} {p : ℤ} (hp : Prime p) {a b c : ℤ}
     (hpa : p ∣ a) (hpb : p ∣ b) (HF : a ^ n + b ^ n + c ^ n = 0) : p ∣ c := by
   rcases eq_or_ne n 0 with rfl | hn
   · simp at HF
diff --git a/Mathlib/NumberTheory/FLT/Three.lean b/Mathlib/NumberTheory/FLT/Three.lean
index 328eb0d18caae..555bad3224e64 100644
--- a/Mathlib/NumberTheory/FLT/Three.lean
+++ b/Mathlib/NumberTheory/FLT/Three.lean
@@ -241,7 +241,7 @@ lemma Solution.exists_minimal : ∃ (S₁ : Solution hζ), S₁.isMinimal := by
 end DecidableRel
 
 /-- Given `S' : Solution'`, then `S'.a` and `S'.b` are both congruent to `1` modulo `λ ^ 4` or are
-both congruent to `-1`.  -/
+both congruent to `-1`. -/
 lemma a_cube_b_cube_congr_one_or_neg_one :
     λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1 := by
   obtain ⟨z, hz⟩ := S'.hcdvd
diff --git a/Mathlib/NumberTheory/FactorisationProperties.lean b/Mathlib/NumberTheory/FactorisationProperties.lean
new file mode 100644
index 0000000000000..28cabcc7fd958
--- /dev/null
+++ b/Mathlib/NumberTheory/FactorisationProperties.lean
@@ -0,0 +1,198 @@
+/-
+Copyright (c) 2024 Colin Jones. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Colin Jones
+-/
+import Mathlib.Algebra.GeomSum
+import Mathlib.Algebra.IsPrimePow
+import Mathlib.NumberTheory.Divisors
+import Mathlib.Tactic.FinCases
+import Mathlib.Tactic.NormNum.Prime
+
+/-!
+# Factorisation properties of natural numbers
+
+This file defines abundant, pseudoperfect, deficient, and weird numbers and formalizes their
+relations with prime and perfect numbers.
+
+## Main Definitions
+
+* `Nat.Abundant`: a natural number `n` is _abundant_ if the sum of its proper divisors is greater
+  than `n`
+* `Nat.Pseudoperfect`: a natural number `n` is _pseudoperfect_ if the sum of a subset of its proper
+  divisors equals `n`
+* `Nat.Deficient`: a natural number `n` is _deficient_ if the sum of its proper divisors is less
+  than `n`
+* `Nat.Weird`: a natural number is _weird_ if it is abundant but not pseudoperfect
+
+## Main Results
+
+* `Nat.deficient_or_perfect_or_abundant`: A positive natural number is either deficient,
+  perfect, or abundant.
+* `Nat.Prime.deficient`: All prime natural numbers are deficient.
+* `Nat.infinite_deficient`: There are infinitely many deficient numbers.
+* `Nat.Prime.deficient_pow`: Any natural number power of a prime is deficient.
+
+## Implementation Notes
+* Zero is not included in any of the definitions and these definitions only apply to natural
+  numbers greater than zero.
+
+## References
+* [R. W. Prielipp, *PERFECT NUMBERS, ABUNDANT NUMBERS, AND DEFICIENT NUMBERS*][Prielipp1970]
+
+## Tags
+
+abundant, deficient, weird, pseudoperfect
+-/
+
+open Finset
+
+namespace Nat
+
+variable {n m p : ℕ}
+
+/-- `n : ℕ` is _abundant_ if the sum of the proper divisors of `n` is greater than `n`. -/
+def Abundant (n : ℕ) : Prop := n < ∑ i ∈ properDivisors n, i
+
+/-- `n : ℕ` is _deficient_ if the sum of the proper divisors of `n` is less than `n`. -/
+def Deficient (n : ℕ) : Prop := ∑ i ∈ properDivisors n, i < n
+
+/-- A positive natural number `n` is _pseudoperfect_ if there exists a subset of the proper
+  divisors of `n` such that the sum of that subset is equal to `n`. -/
+def Pseudoperfect (n : ℕ) : Prop :=
+  0 < n ∧ ∃ s ⊆ properDivisors n, ∑ i ∈ s, i = n
+
+/-- `n : ℕ` is a _weird_ number if and only if it is abundant but not pseudoperfect. -/
+def Weird (n : ℕ) : Prop := Abundant n ∧ ¬ Pseudoperfect n
+
+theorem not_pseudoperfect_iff_forall :
+    ¬ Pseudoperfect n ↔ n = 0 ∨ ∀ s ⊆ properDivisors n, ∑ i ∈ s, i ≠ n := by
+  rw [Pseudoperfect, not_and_or]
+  simp only [not_lt, nonpos_iff_eq_zero, mem_powerset, not_exists, not_and, ne_eq]
+
+theorem deficient_one : Deficient 1 := zero_lt_one
+theorem deficient_two : Deficient 2 := one_lt_two
+theorem deficient_three : Deficient 3 := by norm_num [Deficient]
+
+theorem abundant_twelve : Abundant 12 := by
+  rw [Abundant, show properDivisors 12 = {1,2,3,4,6} by rfl]
+  norm_num
+
+set_option maxRecDepth 1730 in
+theorem weird_seventy : Weird 70 := by
+  rw [Weird, Abundant, not_pseudoperfect_iff_forall]
+  have h : properDivisors 70 = {1, 2, 5, 7, 10, 14, 35} := by rfl
+  constructor
+  · rw [h]
+    repeat norm_num
+  · rw [h]
+    right
+    intro s hs
+    have hs' := mem_powerset.mpr hs
+    fin_cases hs' <;> decide
+
+lemma deficient_iff_not_abundant_and_not_perfect (hn : n ≠ 0) :
+    Deficient n ↔ ¬ Abundant n ∧ ¬ Perfect n := by
+  dsimp only [Perfect, Abundant, Deficient]
+  omega
+
+lemma perfect_iff_not_abundant_and_not_deficient (hn : 0 ≠ n) :
+    Perfect n ↔ ¬ Abundant n ∧ ¬ Deficient n := by
+  dsimp only [Perfect, Abundant, Deficient]
+  omega
+
+lemma abundant_iff_not_perfect_and_not_deficient (hn : 0 ≠ n) :
+    Abundant n ↔ ¬ Perfect n ∧ ¬ Deficient n := by
+  dsimp only [Perfect, Abundant, Deficient]
+  omega
+
+/-- A positive natural number is either deficient, perfect, or abundant -/
+theorem deficient_or_perfect_or_abundant (hn : 0 ≠ n) :
+    Deficient n ∨ Abundant n ∨ Perfect n := by
+  dsimp only [Perfect, Abundant, Deficient]
+  omega
+
+theorem Perfect.pseudoperfect (h : Perfect n) : Pseudoperfect n :=
+  ⟨h.2, ⟨properDivisors n, ⟨fun ⦃_⦄ a ↦ a, h.1⟩⟩⟩
+
+theorem Prime.not_abundant (h : Prime n) : ¬ Abundant n :=
+  fun h1 ↦ (h.one_lt.trans h1).ne' (sum_properDivisors_eq_one_iff_prime.mpr h)
+
+theorem Prime.not_weird (h : Prime n) : ¬ Weird n := by
+  simp only [Nat.Weird, not_and_or]
+  left
+  exact h.not_abundant
+
+theorem Prime.not_pseudoperfect (h : Prime p) : ¬ Pseudoperfect p := by
+  simp_rw [not_pseudoperfect_iff_forall, ← mem_powerset,
+    show p.properDivisors.powerset = {∅, {1}} by rw [Prime.properDivisors h]; rfl]
+  refine Or.inr (fun s hs ↦ ?_)
+  fin_cases hs <;>
+  simp only [sum_empty, sum_singleton] <;>
+  linarith [Prime.one_lt h]
+
+theorem Prime.not_perfect (h : Prime p) : ¬ Perfect p := by
+  have h1 := Prime.not_pseudoperfect h
+  revert h1
+  exact not_imp_not.mpr (Perfect.pseudoperfect)
+
+/-- Any natural number power of a prime is deficient -/
+theorem Prime.deficient_pow  (h : Prime n) : Deficient (n ^ m) := by
+  rcases Nat.eq_zero_or_pos m with (rfl | _)
+  · simpa using deficient_one
+  · have h1 : (n ^ m).properDivisors = image (n ^ ·) (range m) := by
+      apply subset_antisymm <;> intro a
+      · simp only [mem_properDivisors, mem_image, mem_range, dvd_prime_pow h]
+        rintro ⟨⟨t, ht, rfl⟩, ha'⟩
+        exact ⟨t, lt_of_le_of_ne ht (fun ht' ↦ lt_irrefl _ (ht' ▸ ha')), rfl⟩
+      · simp only [mem_image, mem_range, mem_properDivisors, forall_exists_index, and_imp]
+        intro x hx hy
+        constructor
+        · rw [← hy, dvd_prime_pow h]
+          exact ⟨x, Nat.le_of_succ_le hx, rfl⟩
+        · rw [← hy]
+          exact (Nat.pow_lt_pow_iff_right (Prime.two_le h)).mpr hx
+    have h2 : ∑ i in image (fun x => n ^ x) (range m), i = ∑ i in range m, n^i := by
+      rw [Finset.sum_image]
+      rintro x _ y _
+      apply pow_injective_of_not_isUnit h.not_unit <| Prime.ne_zero h
+    rw [Deficient, h1, h2]
+    calc
+      ∑ i ∈ range m, n ^ i
+        = (n ^ m - 1) / (n - 1) := (Nat.geomSum_eq (Prime.two_le h) _)
+      _ ≤ (n ^ m - 1) := Nat.div_le_self (n ^ m - 1) (n - 1)
+      _ < n ^ m := sub_lt (pow_pos (Prime.pos h) m) (Nat.one_pos)
+
+theorem _root_.IsPrimePow.deficient (h : IsPrimePow n) : Deficient n := by
+  obtain ⟨p, k, hp, -, rfl⟩ := h
+  exact hp.nat_prime.deficient_pow
+
+theorem Prime.deficient (h : Prime n) : Deficient n := by
+  rw [← pow_one n]
+  exact h.deficient_pow
+
+/-- There exists infinitely many deficient numbers -/
+theorem infinite_deficient : {n : ℕ | n.Deficient}.Infinite := by
+  rw [Set.infinite_iff_exists_gt]
+  intro a
+  obtain ⟨b, h1, h2⟩ := exists_infinite_primes a.succ
+  exact ⟨b, h2.deficient, h1⟩
+
+theorem infinite_even_deficient : {n : ℕ | Even n ∧ n.Deficient}.Infinite := by
+  rw [Set.infinite_iff_exists_gt]
+  intro n
+  use 2 ^ (n + 1)
+  constructor
+  · exact ⟨⟨2 ^ n, by ring⟩, prime_two.deficient_pow⟩
+  · calc
+      n ≤ 2 ^ n := Nat.le_of_lt (lt_two_pow n)
+      _ < 2 ^ (n + 1) := (Nat.pow_lt_pow_iff_right (Nat.one_lt_two)).mpr (lt_add_one n)
+
+theorem infinite_odd_deficient : {n : ℕ | Odd n ∧ n.Deficient}.Infinite := by
+  rw [Set.infinite_iff_exists_gt]
+  intro n
+  obtain ⟨p, ⟨_, h2⟩⟩ := exists_infinite_primes (max (n + 1) 3)
+  exact ⟨p, Set.mem_setOf.mpr ⟨Prime.odd_of_ne_two h2 (Ne.symm (ne_of_lt (by omega))),
+    Prime.deficient h2⟩, by omega⟩
+
+end Nat
diff --git a/Mathlib/NumberTheory/FermatPsp.lean b/Mathlib/NumberTheory/FermatPsp.lean
index 87661a60098ce..113b74cc7f237 100644
--- a/Mathlib/NumberTheory/FermatPsp.lean
+++ b/Mathlib/NumberTheory/FermatPsp.lean
@@ -61,7 +61,7 @@ instance decidableProbablePrime (n b : ℕ) : Decidable (ProbablePrime n b) :=
   Nat.decidable_dvd _ _
 
 instance decidablePsp (n b : ℕ) : Decidable (FermatPsp n b) :=
-  And.decidable
+  inferInstanceAs (Decidable (_ ∧ _))
 
 /-- If `n` passes the Fermat primality test to base `b`, then `n` is coprime with `b`, assuming that
 `n` and `b` are both positive.
@@ -94,7 +94,7 @@ theorem coprime_of_probablePrime {n b : ℕ} (h : ProbablePrime n b) (h₁ : 1 
 
 theorem probablePrime_iff_modEq (n : ℕ) {b : ℕ} (h : 1 ≤ b) :
     ProbablePrime n b ↔ b ^ (n - 1) ≡ 1 [MOD n] := by
-  have : 1 ≤ b ^ (n - 1) := one_le_pow_of_one_le h (n - 1)
+  have : 1 ≤ b ^ (n - 1) := one_le_pow₀ h
   -- For exact mod_cast
   rw [Nat.ModEq.comm]
   constructor
@@ -135,7 +135,7 @@ private theorem b_id_helper {a b : ℕ} (ha : 2 ≤ a) (hb : 2 < b) : 2 ≤ (a ^
   calc
     2 * a + 1 ≤ a ^ 2 * a := by nlinarith
     _ = a ^ 3 := by rw [Nat.pow_succ a 2]
-    _ ≤ a ^ b := pow_le_pow_right (Nat.le_of_succ_le ha) hb
+    _ ≤ a ^ b := pow_right_mono₀ (Nat.le_of_succ_le ha) hb
 
 private theorem AB_id_helper (b p : ℕ) (_ : 2 ≤ b) (hp : Odd p) :
     (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) := by
diff --git a/Mathlib/NumberTheory/FunctionField.lean b/Mathlib/NumberTheory/FunctionField.lean
index 94de3820f0adb..425189bfa3c7b 100644
--- a/Mathlib/NumberTheory/FunctionField.lean
+++ b/Mathlib/NumberTheory/FunctionField.lean
@@ -42,7 +42,7 @@ function field, ring of integers
 
 noncomputable section
 
-open scoped nonZeroDivisors Polynomial DiscreteValuation
+open scoped nonZeroDivisors Polynomial Multiplicative
 
 variable (Fq F : Type) [Field Fq] [Field F]
 
@@ -69,9 +69,9 @@ theorem functionField_iff (Fqt : Type*) [Field Fqt] [Algebra Fq[X] Fqt]
     refine IsLocalization.ext (nonZeroDivisors Fq[X]) _ _ ?_ ?_ ?_ ?_ ?_ <;> intros <;>
       simp only [map_one, map_mul, AlgEquiv.commutes, ← IsScalarTower.algebraMap_apply]
   constructor <;> intro h
-  · let b := FiniteDimensional.finBasis (RatFunc Fq) F
+  · let b := Module.finBasis (RatFunc Fq) F
     exact FiniteDimensional.of_fintype_basis (b.mapCoeffs e this)
-  · let b := FiniteDimensional.finBasis Fqt F
+  · let b := Module.finBasis Fqt F
     refine FiniteDimensional.of_fintype_basis (b.mapCoeffs e.symm ?_)
     intro c x; convert (this (e.symm c) x).symm; simp only [e.apply_symm_apply]
 
@@ -213,7 +213,7 @@ theorem inftyValuation.polynomial {p : Fq[X]} (hp : p ≠ 0) :
     inftyValuationDef Fq (algebraMap Fq[X] (RatFunc Fq) p) =
       Multiplicative.ofAdd (p.natDegree : ℤ) := by
   have hp' : algebraMap Fq[X] (RatFunc Fq) p ≠ 0 := by
-    rw [Ne, RatFunc.algebraMap_eq_zero_iff]; exact hp
+    rw [Ne, NoZeroSMulDivisors.algebraMap_eq_zero_iff]; exact hp
   rw [inftyValuationDef, if_neg hp', RatFunc.intDegree_polynomial]
 
 /-- The valued field `Fq(t)` with the valuation at infinity. -/
diff --git a/Mathlib/NumberTheory/GaussSum.lean b/Mathlib/NumberTheory/GaussSum.lean
index 5140078baa06e..6af4f9545d5ff 100644
--- a/Mathlib/NumberTheory/GaussSum.lean
+++ b/Mathlib/NumberTheory/GaussSum.lean
@@ -97,8 +97,15 @@ lemma gaussSum_mul {R : Type u} [CommRing R] [Fintype R] {R' : Type v} [CommRing
     · exact fun a _ ↦ by rw [add_sub_cancel_right, add_comm]
   rw [sum_congr rfl fun x _ ↦ sum_eq x, sum_comm]
 
--- In the following, we need `R` to be a finite field and `R'` to be a domain.
-variable {R : Type u} [Field R] [Fintype R] {R' : Type v} [CommRing R'] [IsDomain R']
+-- In the following, we need `R` to be a finite field.
+variable {R : Type u} [Field R] [Fintype R] {R' : Type v} [CommRing R']
+
+lemma mul_gaussSum_inv_eq_gaussSum (χ : MulChar R R') (ψ : AddChar R R') :
+    χ (-1) * gaussSum χ ψ⁻¹ = gaussSum χ ψ := by
+  rw [ψ.inv_mulShift, ← Units.coe_neg_one]
+  exact gaussSum_mulShift χ ψ (-1)
+
+variable [IsDomain R'] --  From now on, `R'` needs to be a domain.
 
 -- A helper lemma for `gaussSum_mul_gaussSum_eq_card` below
 -- Is this useful enough in other contexts to be public?
@@ -130,6 +137,17 @@ theorem gaussSum_mul_gaussSum_eq_card {χ : MulChar R R'} (hχ : χ ≠ 1) {ψ :
   rw [Finset.sum_ite_eq' Finset.univ (1 : R)]
   simp only [Finset.mem_univ, map_one, one_mul, if_true]
 
+/-- If `χ` is a multiplicative character of order `n` on a finite field `F`,
+then `g(χ) * g(χ^(n-1)) = χ(-1)*#F` -/
+lemma gaussSum_mul_gaussSum_pow_orderOf_sub_one {χ : MulChar R R'} {ψ : AddChar R R'}
+    (hχ : χ ≠ 1) (hψ : ψ.IsPrimitive) :
+    gaussSum χ ψ * gaussSum (χ ^ (orderOf χ - 1)) ψ = χ (-1) * Fintype.card R := by
+  have h : χ ^ (orderOf χ - 1) = χ⁻¹ := by
+    refine (inv_eq_of_mul_eq_one_right ?_).symm
+    rw [← pow_succ', Nat.sub_one_add_one_eq_of_pos χ.orderOf_pos, pow_orderOf_eq_one]
+  rw [h, ← mul_gaussSum_inv_eq_gaussSum χ⁻¹, mul_left_comm, gaussSum_mul_gaussSum_eq_card hχ hψ,
+    MulChar.inv_apply', inv_neg_one]
+
 /-- The Gauss sum of a nontrivial character on a finite field does not vanish. -/
 lemma gaussSum_ne_zero_of_nontrivial (h : (Fintype.card R : R') ≠ 0) {χ : MulChar R R'}
     (hχ : χ ≠ 1) {ψ : AddChar R R'} (hψ : ψ.IsPrimitive) :
diff --git a/Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean b/Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
index 92a72fe1377c1..e61b195e807ac 100644
--- a/Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
+++ b/Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
@@ -84,7 +84,7 @@ lemma strictAnti_eulerMascheroniSeq' : StrictAnti eulerMascheroniSeq' := by
   refine strictAnti_nat_of_succ_lt (fun n ↦ ?_)
   rcases Nat.eq_zero_or_pos n with rfl | hn
   · simp [eulerMascheroniSeq']
-  simp_rw [eulerMascheroniSeq', eq_false_intro hn.ne', if_false]
+  simp_rw [eulerMascheroniSeq', eq_false_intro hn.ne', reduceCtorEq, if_false]
   rw [← sub_pos, sub_sub_sub_comm,
     harmonic_succ, Rat.cast_add, ← sub_sub, sub_self, zero_sub, sub_eq_add_neg, neg_sub,
     ← sub_eq_neg_add, sub_pos, ← log_div (by positivity) (by positivity), ← neg_lt_neg_iff,
diff --git a/Mathlib/NumberTheory/Harmonic/GammaDeriv.lean b/Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
index a66d5d7a47b91..90e05ffd8ff51 100644
--- a/Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
+++ b/Mathlib/NumberTheory/Harmonic/GammaDeriv.lean
@@ -4,12 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
 
-import Mathlib.NumberTheory.Harmonic.EulerMascheroni
+import Mathlib.Analysis.Convex.Deriv
 import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
 import Mathlib.Data.Nat.Factorial.Basic
+import Mathlib.NumberTheory.Harmonic.EulerMascheroni
 
 /-!
-# Derivative of Γ at positive integers
+# Derivative of Γ at positive integers
 
 We prove the formula for the derivative of `Real.Gamma` at a positive integer:
 
diff --git a/Mathlib/NumberTheory/Harmonic/Int.lean b/Mathlib/NumberTheory/Harmonic/Int.lean
index cf5d3e429630e..822c1dac331cb 100644
--- a/Mathlib/NumberTheory/Harmonic/Int.lean
+++ b/Mathlib/NumberTheory/Harmonic/Int.lean
@@ -43,4 +43,4 @@ theorem harmonic_not_int {n : ℕ} (h : 2 ≤ n) : ¬ (harmonic n).isInt := by
   apply padicNorm.not_int_of_not_padic_int 2
   rw [padicNorm.eq_zpow_of_nonzero (harmonic_pos (ne_zero_of_lt h)).ne',
       padicValRat_two_harmonic, neg_neg, zpow_natCast]
-  exact one_lt_pow one_lt_two (Nat.log_pos one_lt_two h).ne'
+  exact one_lt_pow₀ one_lt_two (Nat.log_pos one_lt_two h).ne'
diff --git a/Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean b/Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
index c5c6b9529953e..99e63c94ec211 100644
--- a/Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
+++ b/Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean
@@ -37,7 +37,7 @@ namespace ZetaAsymptotics
 -- since the intermediate lemmas are of little interest in themselves we put them in a namespace
 
 /-!
-## Definitions
+## Definitions
 -/
 
 /-- Auxiliary function used in studying zeta-function asymptotics. -/
@@ -128,7 +128,7 @@ lemma term_tsum_one : HasSum (fun n ↦ term (n + 1) 1) (1 - γ) := by
   refine Tendsto.add ?_ tendsto_const_nhds
   have := (tendsto_eulerMascheroniSeq'.comp (tendsto_add_atTop_nat 1)).neg
   refine this.congr' (Eventually.of_forall (fun n ↦ ?_))
-  simp_rw [Function.comp_apply, eulerMascheroniSeq', if_false]
+  simp_rw [Function.comp_apply, eulerMascheroniSeq', reduceCtorEq, if_false]
   push_cast
   abel
 
diff --git a/Mathlib/NumberTheory/JacobiSum/Basic.lean b/Mathlib/NumberTheory/JacobiSum/Basic.lean
index 6c989fdfbe4bb..9088bfbe869f8 100644
--- a/Mathlib/NumberTheory/JacobiSum/Basic.lean
+++ b/Mathlib/NumberTheory/JacobiSum/Basic.lean
@@ -3,8 +3,9 @@ Copyright (c) 2024 Michael Stoll. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Stoll
 -/
-import Mathlib.NumberTheory.MulChar.Basic
-import Mathlib.Algebra.Module.BigOperators
+import Mathlib.NumberTheory.GaussSum
+import Mathlib.NumberTheory.MulChar.Lemmas
+import Mathlib.RingTheory.RootsOfUnity.Lemmas
 
 /-!
 # Jacobi Sums
@@ -59,18 +60,19 @@ end Def
 ### Jacobi sums over finite fields
 -/
 
-section FiniteField
+section CommRing
 
-variable {F R : Type*} [Field F] [Fintype F] [DecidableEq F] [CommRing R]
+variable {F R : Type*} [CommRing F] [Nontrivial F] [Fintype F] [DecidableEq F] [CommRing R]
 
-/-- The Jacobi sum of two multiplicative characters on a finite field `F` can be written
-as a sum over `F \ {0,1}`. -/
+/-- The Jacobi sum of two multiplicative characters on a nontrivial finite commutative ring `F`
+can be written as a sum over `F \ {0,1}`. -/
 lemma jacobiSum_eq_sum_sdiff (χ ψ : MulChar F R) :
     jacobiSum χ ψ = ∑ x ∈ univ \ {0,1}, χ x * ψ (1 - x) := by
-  simp only [jacobiSum, subset_univ, sum_sdiff_eq_sub, mem_singleton, zero_ne_one,
-    not_false_eq_true, sum_insert, isUnit_iff_ne_zero, ne_eq, not_true_eq_false,
-    MulCharClass.map_nonunit, sub_zero, map_one, mul_one, sum_singleton, sub_self, mul_zero,
-    add_zero]
+  simp only [jacobiSum, subset_univ, sum_sdiff_eq_sub, sub_eq_add_neg, self_eq_add_right,
+    neg_eq_zero]
+  apply sum_eq_zero
+  simp only [mem_insert, mem_singleton, forall_eq_or_imp, χ.map_zero, neg_zero, add_zero, map_one,
+    mul_one, forall_eq, add_neg_cancel, ψ.map_zero, mul_zero, and_self]
 
 private lemma jacobiSum_eq_aux (χ ψ : MulChar F R) :
     jacobiSum χ ψ = ∑ x : F, χ x + ∑ x : F, ψ x - Fintype.card F +
@@ -86,10 +88,17 @@ private lemma jacobiSum_eq_aux (χ ψ : MulChar F R) :
   congr
   rw [sum_pair zero_ne_one, sub_zero, ψ.map_one, χ.map_one, sub_self, mul_zero, zero_mul, add_zero]
 
+end CommRing
+
+section FiniteField
+
+variable {F R : Type*} [Field F] [Fintype F] [CommRing R]
+
 /-- The Jacobi sum of twice the trivial multiplicative character on a finite field `F`
 equals `#F-2`. -/
 theorem jacobiSum_trivial_trivial :
     jacobiSum (MulChar.trivial F R) (MulChar.trivial F R) = Fintype.card F - 2 := by
+  classical
   rw [jacobiSum_eq_sum_sdiff]
   have : ∀ x ∈ univ \ {0, 1}, (MulChar.trivial F R) x * (MulChar.trivial F R) (1 - x) = 1 := by
     intros x hx
@@ -107,4 +116,221 @@ theorem jacobiSum_trivial_trivial :
 theorem jacobiSum_one_one : jacobiSum (1 : MulChar F R) 1 = Fintype.card F - 2 :=
   jacobiSum_trivial_trivial
 
+variable [IsDomain R] -- needed for `MulChar.sum_eq_zero_of_ne_one`
+
+/-- If `χ` is a nontrivial multiplicative character on a finite field `F`, then `J(1,χ) = -1`. -/
+theorem jacobiSum_one_nontrivial {χ : MulChar F R} (hχ : χ ≠ 1) : jacobiSum 1 χ = -1 := by
+  classical
+  have : ∑ x ∈ univ \ {0, 1}, ((1 : MulChar F R) x - 1) * (χ (1 - x) - 1) = 0 := by
+    apply Finset.sum_eq_zero
+    simp (config := { contextual := true }) only [mem_sdiff, mem_univ, mem_insert, mem_singleton,
+      not_or, ← isUnit_iff_ne_zero, true_and, MulChar.one_apply, sub_self, zero_mul, and_imp,
+      implies_true]
+  simp only [jacobiSum_eq_aux, MulChar.sum_one_eq_card_units, MulChar.sum_eq_zero_of_ne_one hχ,
+    add_zero, Fintype.card_eq_card_units_add_one (α := F), Nat.cast_add, Nat.cast_one,
+    sub_add_cancel_left, this]
+
+/-- If `χ` is a nontrivial multiplicative character on a finite field `F`,
+then `J(χ,χ⁻¹) = -χ(-1)`. -/
+theorem jacobiSum_nontrivial_inv {χ : MulChar F R} (hχ : χ ≠ 1) : jacobiSum χ χ⁻¹ = -χ (-1) := by
+  classical
+  rw [jacobiSum]
+  conv => enter [1, 2, x]; rw [MulChar.inv_apply', ← map_mul, ← div_eq_mul_inv]
+  rw [sum_eq_sum_diff_singleton_add (mem_univ (1 : F)), sub_self, div_zero, χ.map_zero, add_zero]
+  have : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x := by
+    refine sum_bij' (fun a _ ↦ a / (1 - a)) (fun b _ ↦ b / (1 + b)) (fun x hx ↦ ?_)
+      (fun y hy ↦ ?_) (fun x hx ↦ ?_) (fun y hy ↦ ?_) (fun _ _ ↦ rfl)
+    · simp only [mem_sdiff, mem_univ, mem_singleton, true_and] at hx ⊢
+      rw [div_eq_iff <| sub_ne_zero.mpr ((ne_eq ..).symm ▸ hx).symm, mul_sub, mul_one,
+        neg_one_mul, sub_neg_eq_add, self_eq_add_left, neg_eq_zero]
+      exact one_ne_zero
+    · simp only [mem_sdiff, mem_univ, mem_singleton, true_and] at hy ⊢
+      rw [div_eq_iff fun h ↦ hy <| eq_neg_of_add_eq_zero_right h, one_mul, self_eq_add_left]
+      exact one_ne_zero
+    · simp only [mem_sdiff, mem_univ, mem_singleton, true_and] at hx
+      rw [eq_comm, ← sub_eq_zero] at hx
+      field_simp
+    · simp only [mem_sdiff, mem_univ, mem_singleton, true_and] at hy
+      rw [eq_comm, neg_eq_iff_eq_neg, ← sub_eq_zero, sub_neg_eq_add] at hy
+      field_simp
+  rw [this, ← add_eq_zero_iff_eq_neg, ← sum_eq_sum_diff_singleton_add (mem_univ (-1 : F))]
+  exact MulChar.sum_eq_zero_of_ne_one hχ
+
+/-- If `χ` and `φ` are multiplicative characters on a finite field `F` such that
+`χφ` is nontrivial, then `g(χφ) * J(χ,φ) = g(χ) * g(φ)`. -/
+theorem jacobiSum_mul_nontrivial {χ φ : MulChar F R} (h : χ * φ ≠ 1) (ψ : AddChar F R) :
+    gaussSum (χ * φ) ψ * jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ := by
+  classical
+  rw [gaussSum_mul _ _ ψ, sum_eq_sum_diff_singleton_add (mem_univ (0 : F))]
+  conv =>
+    enter [2, 2, 2, x]
+    rw [zero_sub, neg_eq_neg_one_mul x, map_mul, mul_left_comm (χ x) (φ (-1)),
+      ← MulChar.mul_apply, ψ.map_zero_eq_one, mul_one]
+  rw [← mul_sum _ _ (φ (-1)), MulChar.sum_eq_zero_of_ne_one h, mul_zero, add_zero]
+  have sum_eq : ∀ t ∈ univ \ {0}, (∑ x : F, χ x * φ (t - x)) * ψ t =
+      (∑ y : F, χ (t * y) * φ (t - (t * y))) * ψ t := by
+    intro t ht
+    simp only [mem_sdiff, mem_univ, mem_singleton, true_and] at ht
+    exact congrArg (· * ψ t) (Equiv.sum_comp (Equiv.mulLeft₀ t ht) _).symm
+  simp_rw [← sum_mul, sum_congr rfl sum_eq, ← mul_one_sub, map_mul, mul_assoc]
+  conv => enter [2, 2, t, 1, 2, x, 2]; rw [← mul_assoc, mul_comm (χ x) (φ t)]
+  simp_rw [← mul_assoc, ← MulChar.mul_apply, mul_assoc, ← mul_sum, mul_right_comm]
+  rw [← jacobiSum, ← sum_mul, gaussSum, sum_eq_sum_diff_singleton_add (mem_univ (0 : F)),
+    (χ * φ).map_zero, zero_mul, add_zero]
+
 end FiniteField
+
+section field_field
+
+variable {F F' : Type*} [Fintype F] [Field F] [Field F']
+
+/-- If `χ` and `φ` are multiplicative characters on a finite field `F` with values
+in another field `F'` and such that `χφ` is nontrivial, then `J(χ,φ) = g(χ) * g(φ) / g(χφ)`. -/
+theorem jacobiSum_eq_gaussSum_mul_gaussSum_div_gaussSum (h : (Fintype.card F : F') ≠ 0)
+    {χ φ : MulChar F F'} (hχφ : χ * φ ≠ 1) {ψ : AddChar F F'} (hψ : ψ.IsPrimitive) :
+    jacobiSum χ φ = gaussSum χ ψ * gaussSum φ ψ / gaussSum (χ * φ) ψ := by
+  rw [eq_div_iff <| gaussSum_ne_zero_of_nontrivial h hχφ hψ, mul_comm]
+  exact jacobiSum_mul_nontrivial hχφ ψ
+
+open AddChar MulChar in
+/-- If `χ` and `φ` are multiplicative characters on a finite field `F` with values in another
+field `F'` such that `χ`, `φ` and `χφ` are all nontrivial and `char F' ≠ char F`, then
+`J(χ,φ) * J(χ⁻¹,φ⁻¹) = #F` (in `F'`). -/
+lemma jacobiSum_mul_jacobiSum_inv (h : ringChar F' ≠ ringChar F) {χ φ : MulChar F F'} (hχ : χ ≠ 1)
+    (hφ : φ ≠ 1) (hχφ : χ * φ ≠ 1) :
+    jacobiSum χ φ * jacobiSum χ⁻¹ φ⁻¹ = Fintype.card F := by
+  obtain ⟨n, hp, hc⟩ := FiniteField.card F (ringChar F)
+  let ψ := FiniteField.primitiveChar F F' h   -- obtain primitive additive character `ψ : F → FF'`
+  let FF' := CyclotomicField ψ.n F'           -- the target field of `ψ`
+  let χ' := χ.ringHomComp (algebraMap F' FF') -- consider `χ` and `φ` as characters `F → FF'`
+  let φ' := φ.ringHomComp (algebraMap F' FF')
+  have hinj := (algebraMap F' FF').injective
+  apply hinj
+  rw [map_mul, ← jacobiSum_ringHomComp, ← jacobiSum_ringHomComp]
+  have Hχφ : χ' * φ' ≠ 1 := by
+    rw [← ringHomComp_mul]
+    exact (MulChar.ringHomComp_ne_one_iff hinj).mpr hχφ
+  have Hχφ' : χ'⁻¹ * φ'⁻¹ ≠ 1 := by
+    rwa [← mul_inv, inv_ne_one]
+  have Hχ : χ' ≠ 1 := (MulChar.ringHomComp_ne_one_iff hinj).mpr hχ
+  have Hφ : φ' ≠ 1 := (MulChar.ringHomComp_ne_one_iff hinj).mpr hφ
+  have Hcard : (Fintype.card F : FF') ≠ 0 := by
+    intro H
+    simp only [hc, Nat.cast_pow, ne_eq, PNat.ne_zero, not_false_eq_true, pow_eq_zero_iff] at H
+    exact h <| (Algebra.ringChar_eq F' FF').trans <| CharP.ringChar_of_prime_eq_zero hp H
+  have H := (gaussSum_mul_gaussSum_eq_card Hχφ ψ.prim).trans_ne Hcard
+  apply_fun (gaussSum (χ' * φ') ψ.char * gaussSum (χ' * φ')⁻¹ ψ.char⁻¹ * ·)
+    using mul_right_injective₀ H
+  simp only
+  rw [mul_mul_mul_comm, jacobiSum_mul_nontrivial Hχφ, mul_inv, ← ringHomComp_inv,
+    ← ringHomComp_inv, jacobiSum_mul_nontrivial Hχφ', map_natCast, ← mul_mul_mul_comm,
+    gaussSum_mul_gaussSum_eq_card Hχ ψ.prim, gaussSum_mul_gaussSum_eq_card Hφ ψ.prim,
+    ← mul_inv, gaussSum_mul_gaussSum_eq_card Hχφ ψ.prim]
+
+end field_field
+
+section image
+
+variable {F R : Type*} [Fintype F] [Field F] [CommRing R] [IsDomain R]
+
+/-- If `χ` and `φ` are multiplicative characters on a finite field `F` satisfying `χ^n = φ^n = 1`
+and with values in an integral domain `R`, and `μ` is a primitive `n`th root of unity in `R`,
+then the Jacobi sum `J(χ,φ)` is in `ℤ[μ] ⊆ R`. -/
+lemma jacobiSum_mem_algebraAdjoin_of_pow_eq_one {n : ℕ} (hn : n ≠ 0) {χ φ : MulChar F R}
+    (hχ : χ ^ n = 1) (hφ : φ ^ n = 1) {μ : R} (hμ : IsPrimitiveRoot μ n) :
+    jacobiSum χ φ ∈ Algebra.adjoin ℤ {μ} :=
+  Subalgebra.sum_mem _ fun _ _ ↦ Subalgebra.mul_mem _
+    (MulChar.apply_mem_algebraAdjoin_of_pow_eq_one hn hχ hμ _)
+    (MulChar.apply_mem_algebraAdjoin_of_pow_eq_one hn hφ hμ _)
+
+open Algebra in
+private
+lemma MulChar.exists_apply_sub_one_eq_mul_sub_one {n : ℕ} (hn : n ≠ 0) {χ : MulChar F R} {μ : R}
+    (hχ : χ ^ n = 1) (hμ : IsPrimitiveRoot μ n) {x : F} (hx : x ≠ 0) :
+    ∃ z ∈ Algebra.adjoin ℤ {μ}, χ x - 1 = z * (μ - 1) := by
+  obtain ⟨k, _, hk⟩ := exists_apply_eq_pow hn hχ hμ hx
+  refine hk ▸ ⟨(Finset.range k).sum (μ ^ ·), ?_, (geom_sum_mul μ k).symm⟩
+  exact Subalgebra.sum_mem _ fun m _ ↦ Subalgebra.pow_mem _ (self_mem_adjoin_singleton _ μ) _
+
+private
+lemma MulChar.exists_apply_sub_one_mul_apply_sub_one {n : ℕ} (hn : n ≠ 0) {χ ψ : MulChar F R}
+    {μ : R} (hχ : χ ^ n = 1) (hψ : ψ ^ n = 1) (hμ : IsPrimitiveRoot μ n) (x : F) :
+    ∃ z ∈ Algebra.adjoin ℤ {μ}, (χ x - 1) * (ψ (1 - x) - 1) = z * (μ - 1) ^ 2 := by
+  rcases eq_or_ne x 0 with rfl | hx₀
+  · exact ⟨0, Subalgebra.zero_mem _, by rw [sub_zero, ψ.map_one, sub_self, mul_zero, zero_mul]⟩
+  rcases eq_or_ne x 1 with rfl | hx₁
+  · exact ⟨0, Subalgebra.zero_mem _, by rw [χ.map_one, sub_self, zero_mul, zero_mul]⟩
+  obtain ⟨z₁, hz₁, Hz₁⟩ := MulChar.exists_apply_sub_one_eq_mul_sub_one hn hχ hμ hx₀
+  obtain ⟨z₂, hz₂, Hz₂⟩ :=
+    MulChar.exists_apply_sub_one_eq_mul_sub_one hn hψ hμ (sub_ne_zero_of_ne hx₁.symm)
+  rewrite [Hz₁, Hz₂, sq]
+  exact ⟨z₁ * z₂, Subalgebra.mul_mem _ hz₁ hz₂, mul_mul_mul_comm ..⟩
+
+/-- If `χ` and `ψ` are multiplicative characters of order dividing `n` on a finite field `F`
+with values in an integral domain `R` and `μ` is a primitive `n`th root of unity in `R`,
+then `J(χ,ψ) = -1 + z*(μ - 1)^2` for some `z ∈ ℤ[μ] ⊆ R`. (We assume that `#F ≡ 1 mod n`.)
+Note that we do not state this as a divisbility in `R`, as this would give a weaker statement. -/
+lemma exists_jacobiSum_eq_neg_one_add {n : ℕ} (hn : 2 < n) {χ ψ : MulChar F R}
+    {μ : R} (hχ : χ ^ n = 1) (hψ : ψ ^ n = 1) (hn' : n ∣ Fintype.card F - 1)
+    (hμ : IsPrimitiveRoot μ n) :
+    ∃ z ∈ Algebra.adjoin ℤ {μ}, jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2 := by
+  obtain ⟨q, hq⟩ := hn'
+  rw [Nat.sub_eq_iff_eq_add NeZero.one_le] at hq
+  obtain ⟨z₁, hz₁, Hz₁⟩ := hμ.self_sub_one_pow_dvd_order hn
+  by_cases hχ₀ : χ = 1 <;> by_cases hψ₀ : ψ = 1
+  · rw [hχ₀, hψ₀, jacobiSum_one_one]
+    refine ⟨q * z₁, Subalgebra.mul_mem _ (Subalgebra.natCast_mem _ q) hz₁, ?_⟩
+    rw [hq, Nat.cast_add, Nat.cast_mul, Hz₁]
+    ring
+  · refine ⟨0, Subalgebra.zero_mem _, ?_⟩
+    rw [hχ₀, jacobiSum_one_nontrivial hψ₀, zero_mul, add_zero]
+  · refine ⟨0, Subalgebra.zero_mem _, ?_⟩
+    rw [jacobiSum_comm, hψ₀, jacobiSum_one_nontrivial hχ₀, zero_mul, add_zero]
+  · classical
+    rw [jacobiSum_eq_aux, MulChar.sum_eq_zero_of_ne_one hχ₀, MulChar.sum_eq_zero_of_ne_one hψ₀, hq]
+    have H := MulChar.exists_apply_sub_one_mul_apply_sub_one (by omega) hχ hψ hμ
+    have Hcs x := (H x).choose_spec
+    refine ⟨-q * z₁ + ∑ x ∈ (univ \ {0, 1} : Finset F), (H x).choose, ?_, ?_⟩
+    · refine Subalgebra.add_mem _ (Subalgebra.mul_mem _ (Subalgebra.neg_mem _ ?_) hz₁) ?_
+      · exact Subalgebra.natCast_mem ..
+      · exact Subalgebra.sum_mem _ fun x _ ↦ (Hcs x).1
+    · conv => enter [1, 2, 2, x]; rw [(Hcs x).2]
+      rw [← Finset.sum_mul, Nat.cast_add, Nat.cast_mul, Hz₁]
+      ring
+
+end image
+
+section GaussSum
+
+variable {F R : Type*} [Fintype F] [Field F] [CommRing R] [IsDomain R]
+
+lemma gaussSum_pow_eq_prod_jacobiSum_aux (χ : MulChar F R) (ψ : AddChar F R) {n : ℕ}
+    (hn₁ : 0 < n) (hn₂ : n < orderOf χ) :
+    gaussSum χ ψ ^ n = gaussSum (χ ^ n) ψ * ∏ j ∈ Ico 1 n, jacobiSum χ (χ ^ j) := by
+  induction n, hn₁ using Nat.le_induction with
+  | base => simp only [pow_one, le_refl, Ico_eq_empty_of_le, prod_empty, mul_one]
+  | succ n hn ih =>
+      specialize ih <| lt_trans (Nat.lt_succ_self n) hn₂
+      have gauss_rw : gaussSum (χ ^ n) ψ * gaussSum χ ψ =
+            jacobiSum χ (χ ^ n) * gaussSum (χ ^ (n + 1)) ψ := by
+        have hχn : χ * (χ ^ n) ≠ 1 :=
+          pow_succ' χ n ▸ pow_ne_one_of_lt_orderOf n.add_one_ne_zero hn₂
+        rw [mul_comm, ← jacobiSum_mul_nontrivial hχn, mul_comm, ← pow_succ']
+      apply_fun (· * gaussSum χ ψ) at ih
+      rw [mul_right_comm, ← pow_succ, gauss_rw] at ih
+      rw [ih, Finset.prod_Ico_succ_top hn, mul_rotate, mul_assoc]
+
+/-- If `χ` is a multiplicative character of order `n ≥ 2` on a finite field `F`,
+then `g(χ)^n = χ(-1) * #F * J(χ,χ) * J(χ,χ²) * ... * J(χ,χⁿ⁻²)`. -/
+theorem gaussSum_pow_eq_prod_jacobiSum {χ : MulChar F R} {ψ : AddChar F R} (hχ : 2 ≤ orderOf χ)
+    (hψ : ψ.IsPrimitive) :
+    gaussSum χ ψ ^ orderOf χ =
+      χ (-1) * Fintype.card F * ∏ i ∈ Ico 1 (orderOf χ - 1), jacobiSum χ (χ ^ i) := by
+  have := gaussSum_pow_eq_prod_jacobiSum_aux χ ψ (n := orderOf χ - 1) (by omega) (by omega)
+  apply_fun (gaussSum χ ψ * ·) at this
+  rw [← pow_succ', Nat.sub_one_add_one_eq_of_pos (by omega)] at this
+  have hχ₁ : χ ≠ 1 :=
+    fun h ↦ ((orderOf_one (G := MulChar F R) ▸ h ▸ hχ).trans_lt Nat.one_lt_two).false
+  rw [this, ← mul_assoc, gaussSum_mul_gaussSum_pow_orderOf_sub_one hχ₁ hψ]
+
+end GaussSum
diff --git a/Mathlib/NumberTheory/KummerDedekind.lean b/Mathlib/NumberTheory/KummerDedekind.lean
index 747f0b3cbec26..b93799f8bfa38 100644
--- a/Mathlib/NumberTheory/KummerDedekind.lean
+++ b/Mathlib/NumberTheory/KummerDedekind.lean
@@ -111,9 +111,9 @@ variable {I : Ideal R}
 theorem prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}
     (hp : p ∈ Ideal.comap (algebraMap R S) (conductor R x)) (hz' : z ∈ I.map (algebraMap R S)) :
     algebraMap R S p * z ∈ algebraMap R<x> S '' ↑(I.map (algebraMap R R<x>)) := by
-  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_total] at hz'
+  rw [Ideal.map, Ideal.span, Finsupp.mem_span_image_iff_linearCombination] at hz'
   obtain ⟨l, H, H'⟩ := hz'
-  rw [Finsupp.total_apply] at H'
+  rw [Finsupp.linearCombination_apply] at H'
   rw [← H', mul_comm, Finsupp.sum_mul]
   have lem : ∀ {a : R}, a ∈ I → l a • algebraMap R S a * algebraMap R S p ∈
       algebraMap R<x> S '' I.map (algebraMap R R<x>) := by
@@ -168,7 +168,8 @@ theorem comap_map_eq_map_adjoin_of_coprime_conductor
     obtain ⟨x₁, hx₁, hx₂⟩ := (Set.mem_image _ _ _).mp h
     have : x₁ = ⟨z, hz⟩ := by
       apply h_alg
-      simp [hx₂, algebraMap_eq_smul_one]
+      simp only [hx₂, algebraMap_eq_smul_one]
+      rw [Submonoid.mk_smul, smul_eq_mul, mul_one]
     rwa [← this]
   · -- The converse inclusion is trivial
     have : algebraMap R S = (algebraMap _ S).comp (algebraMap R R<x>) := by ext; rfl
@@ -282,8 +283,8 @@ theorem normalizedFactors_ideal_map_eq_normalizedFactors_min_poly_mk_map (hI : I
   by_cases hJ : J ∈ normalizedFactors (I.map (algebraMap R S))
   swap
   · rw [Multiset.count_eq_zero.mpr hJ, eq_comm, Multiset.count_eq_zero, Multiset.mem_map]
-    simp only [Multiset.mem_attach, true_and_iff, not_exists]
-    rintro J' rfl
+    simp only [not_exists]
+    rintro J' ⟨_, rfl⟩
     exact
       hJ ((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx').symm J').prop
   -- Then we just have to compare the multiplicities, which we already proved are equal.
diff --git a/Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean b/Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
index 9395e10f0cc6f..64d09847b3f9e 100644
--- a/Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
+++ b/Mathlib/NumberTheory/LSeries/AbstractFuncEq.lean
@@ -167,7 +167,7 @@ lemma hf_zero' (P : WeakFEPair E) :
     filter_upwards [eventually_le_nhds zero_lt_one] with x hx' (hx : 0 < x)
     apply le_mul_of_one_le_right (norm_nonneg _)
     rw [norm_of_nonneg (rpow_pos_of_pos hx _).le, rpow_neg hx.le]
-    exact one_le_inv (rpow_pos_of_pos hx _) (rpow_le_one hx.le hx' P.hk.le)
+    exact (one_le_inv₀ (rpow_pos_of_pos hx _)).2 (rpow_le_one hx.le hx' P.hk.le)
 
 end WeakFEPair
 
@@ -397,7 +397,7 @@ theorem differentiableAt_Λ {s : ℂ} (hs : s ≠ 0 ∨ P.f₀ = 0) (hs' : s ≠
     DifferentiableAt ℂ P.Λ s := by
   refine ((P.differentiable_Λ₀ s).sub ?_).sub ?_
   · rcases hs with hs | hs
-    · simpa only [one_div] using (differentiableAt_inv' hs).smul_const P.f₀
+    · simpa only [one_div] using (differentiableAt_inv hs).smul_const P.f₀
     · simpa only [hs, smul_zero] using differentiableAt_const (0 : E)
   · rcases hs' with hs' | hs'
     · apply DifferentiableAt.smul_const
@@ -427,10 +427,8 @@ theorem functional_equation₀ (s : ℂ) : P.Λ₀ (P.k - s) = P.ε • P.symm.
 /-- Functional equation formulated for `Λ`. -/
 theorem functional_equation (s : ℂ) :
     P.Λ (P.k - s) = P.ε • P.symm.Λ s := by
-  have := P.functional_equation₀ s
-  rw [P.Λ₀_eq, P.symm_Λ₀_eq, sub_sub_cancel] at this
-  rwa [smul_add, smul_add, ← mul_smul, mul_one_div, ← mul_smul, ← mul_div_assoc,
-    mul_inv_cancel₀ P.hε, add_assoc, add_comm (_ • _), add_assoc, add_left_inj] at this
+  linear_combination (norm := module) P.functional_equation₀ s - P.Λ₀_eq (P.k - s)
+    + congr(P.ε • $(P.symm_Λ₀_eq s)) + congr(($(mul_inv_cancel₀ P.hε) / ((P.k:ℂ) - s)) • P.f₀)
 
 /-- The residue of `Λ` at `s = k` is equal to `ε • g₀`. -/
 theorem Λ_residue_k :
@@ -444,8 +442,7 @@ theorem Λ_residue_k :
     exact continuousAt_const.div continuousAt_id (ofReal_ne_zero.mpr P.hk.ne')
   · refine (tendsto_const_nhds.mono_left nhdsWithin_le_nhds).congr' ?_
     refine eventually_nhdsWithin_of_forall (fun s (hs : s ≠ P.k) ↦ ?_)
-    simp_rw [← mul_smul]
-    congr 1
+    match_scalars
     field_simp [sub_ne_zero.mpr hs.symm]
     ring
 
@@ -457,7 +454,7 @@ theorem Λ_residue_zero :
   · exact (continuous_id.smul P.differentiable_Λ₀.continuous).tendsto _
   · refine (tendsto_const_nhds.mono_left nhdsWithin_le_nhds).congr' ?_
     refine eventually_nhdsWithin_of_forall (fun s (hs : s ≠ 0) ↦ ?_)
-    simp_rw [← mul_smul]
+    match_scalars
     field_simp [sub_ne_zero.mpr hs.symm]
   · rw [show 𝓝 0 = 𝓝 ((0 : ℂ) • (P.ε / (P.k - 0 : ℂ)) • P.g₀) by rw [zero_smul]]
     exact (continuousAt_id.smul ((continuousAt_const.div ((continuous_sub_left _).continuousAt)
diff --git a/Mathlib/NumberTheory/LSeries/Basic.lean b/Mathlib/NumberTheory/LSeries/Basic.lean
index 027a30959131b..7221b06b0360f 100644
--- a/Mathlib/NumberTheory/LSeries/Basic.lean
+++ b/Mathlib/NumberTheory/LSeries/Basic.lean
@@ -77,6 +77,16 @@ lemma term_of_ne_zero {n : ℕ} (hn : n ≠ 0) (f : ℕ → ℂ) (s : ℂ) :
     term f s n = f n / n ^ s :=
   if_neg hn
 
+/--
+If `s ≠ 0`, then the `if .. then .. else` construction in `LSeries.term` isn't needed, since
+`0 ^ s = 0`.
+-/
+lemma term_of_ne_zero' {s : ℂ} (hs : s ≠ 0) (f : ℕ → ℂ) (n : ℕ) :
+    term f s n = f n / n ^ s := by
+  rcases eq_or_ne n 0 with rfl | hn
+  · rw [term_zero, Nat.cast_zero, zero_cpow hs, div_zero]
+  · rw [term_of_ne_zero hn]
+
 lemma term_congr {f g : ℕ → ℂ} (h : ∀ {n}, n ≠ 0 → f n = g n) (s : ℂ) (n : ℕ) :
     term f s n = term g s n := by
   rcases eq_or_ne n 0 with hn | hn <;> simp [hn, h]
diff --git a/Mathlib/NumberTheory/LSeries/Deriv.lean b/Mathlib/NumberTheory/LSeries/Deriv.lean
index d9a355faf2a9f..f8f5726a088aa 100644
--- a/Mathlib/NumberTheory/LSeries/Deriv.lean
+++ b/Mathlib/NumberTheory/LSeries/Deriv.lean
@@ -20,7 +20,7 @@ import Mathlib.Analysis.Complex.HalfPlane
 * We prove similar results for iterated derivatives (`LSeries.iteratedDeriv`).
 
 * We use this to show that `LSeries f` is holomorphic on the right half-plane of
-  absolute convergence (`LSeries.analyticOn`).
+  absolute convergence (`LSeries.analyticOnNhd`).
 
 ## Implementation notes
 
@@ -151,6 +151,10 @@ lemma LSeries_differentiableOn (f : ℕ → ℂ) :
   fun _ hz ↦ (LSeries_hasDerivAt hz).differentiableAt.differentiableWithinAt
 
 /-- The L-series of `f` is holomorphic on its open half-plane of absolute convergence. -/
+lemma LSeries_analyticOnNhd (f : ℕ → ℂ) :
+    AnalyticOnNhd ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} :=
+  (LSeries_differentiableOn f).analyticOnNhd <| isOpen_re_gt_EReal _
+
 lemma LSeries_analyticOn (f : ℕ → ℂ) :
     AnalyticOn ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} :=
-  (LSeries_differentiableOn f).analyticOn <| isOpen_re_gt_EReal _
+  (LSeries_analyticOnNhd f).analyticOn
diff --git a/Mathlib/NumberTheory/LSeries/Dirichlet.lean b/Mathlib/NumberTheory/LSeries/Dirichlet.lean
index 2ee009cb3d46a..6e6fed2c20936 100644
--- a/Mathlib/NumberTheory/LSeries/Dirichlet.lean
+++ b/Mathlib/NumberTheory/LSeries/Dirichlet.lean
@@ -137,7 +137,7 @@ lemma convolution_mul_moebius {n : ℕ} (χ : DirichletCharacter ℂ n) : ↗χ
 lemma modZero_eq_delta {χ : DirichletCharacter ℂ 0} : ↗χ = δ := by
   ext n
   rcases eq_or_ne n 0 with rfl | hn
-  · simp_rw [cast_zero, χ.map_nonunit not_isUnit_zero, delta, if_false]
+  · simp_rw [cast_zero, χ.map_nonunit not_isUnit_zero, delta, reduceCtorEq, if_false]
   rcases eq_or_ne n 1 with rfl | hn'
   · simp only [cast_one, map_one, delta, ↓reduceIte]
   have : ¬ IsUnit (n : ZMod 0) := fun h ↦ hn' <| ZMod.eq_one_of_isUnit_natCast h
diff --git a/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean b/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
new file mode 100644
index 0000000000000..dd4b132bc5130
--- /dev/null
+++ b/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
@@ -0,0 +1,70 @@
+/-
+Copyright (c) 2024 David Loeffler. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Loeffler
+-/
+import Mathlib.NumberTheory.LSeries.ZMod
+import Mathlib.NumberTheory.DirichletCharacter.Basic
+
+/-!
+# Analytic continuation of Dirichlet L-functions
+
+We show that if `χ` is a Dirichlet character `ZMod N → ℂ`, for a positive integer `N`, then the
+L-series of `χ` has analytic continuation (away from a pole at `s = 1` if `χ` is trivial).
+
+All definitions and theorems are in the `DirichletCharacter` namespace.
+
+## Main definitions
+
+* `LFunction χ s`: the L-function, defined as a linear combination of Hurwitz zeta functions.
+
+## Main theorems
+
+* `LFunction_eq_LSeries`: if `1 < re s` then the `LFunction` coincides with the naive `LSeries`.
+* `differentiable_LFunction`: if `χ` is nontrivial then `LFunction χ s` is differentiable
+  everywhere.
+-/
+
+open Complex
+
+namespace DirichletCharacter
+
+variable {N : ℕ} [NeZero N]
+
+/--
+The unique meromorphic function `ℂ → ℂ` which agrees with `∑' n : ℕ, χ n / n ^ s` wherever the
+latter is convergent. This is constructed as a linear combination of Hurwitz zeta functions.
+
+Note that this is not the same as `LSeries χ`: they agree in the convergence range, but
+`LSeries χ s` is defined to be `0` if `re s ≤ 1`.
+ -/
+noncomputable def LFunction (χ : DirichletCharacter ℂ N) (s : ℂ) : ℂ := ZMod.LFunction χ s
+
+/-- The L-function of the (unique) Dirichlet character mod 1 is the Riemann zeta function.
+(Compare `DirichletCharacter.LSeries_modOne_eq`.) -/
+@[simp] lemma LFunction_modOne_eq {χ : DirichletCharacter ℂ 1} :
+    LFunction χ = riemannZeta := by
+  ext1; rw [LFunction, ZMod.LFunction_modOne_eq, (by rfl : (0 : ZMod 1) = 1), map_one, one_mul]
+
+/--
+For `1 < re s` the L-function of a Dirichlet character agrees with the sum of the naive Dirichlet
+series.
+-/
+lemma LFunction_eq_LSeries (χ : DirichletCharacter ℂ N) {s : ℂ} (hs : 1 < re s) :
+    LFunction χ s = LSeries (χ ·) s :=
+  ZMod.LFunction_eq_LSeries χ hs
+
+/--
+The L-function of a Dirichlet character is differentiable, except at `s = 1` if the character is
+trivial.
+-/
+lemma differentiableAt_LFunction (χ : DirichletCharacter ℂ N) (s : ℂ) (hs : s ≠ 1 ∨ χ ≠ 1) :
+    DifferentiableAt ℂ (LFunction χ) s :=
+  ZMod.differentiableAt_LFunction χ s (hs.imp_right χ.sum_eq_zero_of_ne_one)
+
+/-- The L-function of a non-trivial Dirichlet character is differentiable everywhere. -/
+lemma differentiable_LFunction {χ : DirichletCharacter ℂ N} (hχ : χ ≠ 1) :
+    Differentiable ℂ (LFunction χ) :=
+  (differentiableAt_LFunction _ · <| Or.inr hχ)
+
+end DirichletCharacter
diff --git a/Mathlib/NumberTheory/LSeries/HurwitzZeta.lean b/Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
index 79f23403685fc..69091d6315bba 100644
--- a/Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
+++ b/Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
@@ -142,11 +142,9 @@ lemma differentiable_expZeta_of_ne_zero {a : UnitAddCircle} (ha : a ≠ 0) :
 
 /-- Reformulation of `hasSum_expZeta_of_one_lt_re` using `LSeriesHasSum`. -/
 lemma LSeriesHasSum_exp (a : ℝ) {s : ℂ} (hs : 1 < re s) :
-    LSeriesHasSum (cexp <| 2 * π * I * a * ·) s (expZeta a s) := by
-  refine (hasSum_expZeta_of_one_lt_re a hs).congr_fun (fun n ↦ ?_)
-  rcases eq_or_ne n 0 with rfl | hn
-  · rw [LSeries.term_zero, Nat.cast_zero, zero_cpow (ne_zero_of_one_lt_re hs), div_zero]
-  · apply LSeries.term_of_ne_zero hn
+    LSeriesHasSum (cexp <| 2 * π * I * a * ·) s (expZeta a s) :=
+  (hasSum_expZeta_of_one_lt_re a hs).congr_fun
+    (LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs) _)
 
 /-!
 ## The functional equation
diff --git a/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean b/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
index 48fb9b87f7ce9..ac307e2324e77 100644
--- a/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
+++ b/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
@@ -768,12 +768,9 @@ lemma hasSum_nat_cosZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
 
 /-- Reformulation of `hasSum_nat_cosZeta` using `LSeriesHasSum`. -/
 lemma LSeriesHasSum_cos (a : ℝ) {s : ℂ} (hs : 1 < re s) :
-    LSeriesHasSum (Real.cos <| 2 * π * a * ·) s (cosZeta a s) := by
-  refine (hasSum_nat_cosZeta a hs).congr_fun (fun n ↦ ?_)
-  rcases eq_or_ne n 0 with rfl | hn
-  · rw [LSeries.term_zero, Nat.cast_zero, Nat.cast_zero, zero_cpow (ne_zero_of_one_lt_re hs),
-      div_zero]
-  · apply LSeries.term_of_ne_zero hn
+    LSeriesHasSum (Real.cos <| 2 * π * a * ·) s (cosZeta a s) :=
+  (hasSum_nat_cosZeta a hs).congr_fun
+    (LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs) _)
 
 /-!
 ## Functional equations for the un-completed zetas
diff --git a/Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean b/Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
index e14ca16ba1108..cf70f0a830ee7 100644
--- a/Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
+++ b/Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
@@ -547,11 +547,9 @@ lemma hasSum_nat_sinZeta (a : ℝ) {s : ℂ} (hs : 1 < re s) :
 
 /-- Reformulation of `hasSum_nat_sinZeta` using `LSeriesHasSum`. -/
 lemma LSeriesHasSum_sin (a : ℝ) {s : ℂ} (hs : 1 < re s) :
-    LSeriesHasSum (Real.sin <| 2 * π * a * ·) s (sinZeta a s) := by
-  refine (hasSum_nat_sinZeta a hs).congr_fun (fun n ↦ ?_)
-  rcases eq_or_ne n 0 with rfl | hn
-  · rw [LSeries.term_zero, Nat.cast_zero, mul_zero, Real.sin_zero, ofReal_zero, zero_div]
-  · apply LSeries.term_of_ne_zero hn
+    LSeriesHasSum (Real.sin <| 2 * π * a * ·) s (sinZeta a s) :=
+  (hasSum_nat_sinZeta a hs).congr_fun
+    (LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs) _)
 
 /-- The trivial zeroes of the odd Hurwitz zeta function. -/
 theorem hurwitzZetaOdd_neg_two_mul_nat_sub_one (a : UnitAddCircle) (n : ℕ) :
diff --git a/Mathlib/NumberTheory/LSeries/RiemannZeta.lean b/Mathlib/NumberTheory/LSeries/RiemannZeta.lean
index e62b2028ff8b0..bcc35496bf6f6 100644
--- a/Mathlib/NumberTheory/LSeries/RiemannZeta.lean
+++ b/Mathlib/NumberTheory/LSeries/RiemannZeta.lean
@@ -158,7 +158,7 @@ def RiemannHypothesis : Prop :=
   ∀ (s : ℂ) (_ : riemannZeta s = 0) (_ : ¬∃ n : ℕ, s = -2 * (n + 1)) (_ : s ≠ 1), s.re = 1 / 2
 
 /-!
-## Relating the Mellin transform to the Dirichlet series
+## Relating the Mellin transform to the Dirichlet series
 -/
 
 theorem completedZeta_eq_tsum_of_one_lt_re {s : ℂ} (hs : 1 < re s) :
@@ -182,7 +182,7 @@ theorem zeta_eq_tsum_one_div_nat_cpow {s : ℂ} (hs : 1 < re s) :
     ofReal_one] using (hasSum_nat_cosZeta 0 hs).tsum_eq.symm
 
 /-- Alternate formulation of `zeta_eq_tsum_one_div_nat_cpow` with a `+ 1` (to avoid relying
-on mathlib's conventions for `0 ^ s`).  -/
+on mathlib's conventions for `0 ^ s`). -/
 theorem zeta_eq_tsum_one_div_nat_add_one_cpow {s : ℂ} (hs : 1 < re s) :
     riemannZeta s = ∑' n : ℕ, 1 / (n + 1 : ℂ) ^ s := by
   have := zeta_eq_tsum_one_div_nat_cpow hs
diff --git a/Mathlib/NumberTheory/LSeries/ZMod.lean b/Mathlib/NumberTheory/LSeries/ZMod.lean
new file mode 100644
index 0000000000000..2ef6c910b132d
--- /dev/null
+++ b/Mathlib/NumberTheory/LSeries/ZMod.lean
@@ -0,0 +1,522 @@
+/-
+Copyright (c) 2024 David Loeffler. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Loeffler
+-/
+
+import Mathlib.Analysis.Fourier.ZMod
+import Mathlib.Analysis.NormedSpace.Connected
+import Mathlib.NumberTheory.LSeries.RiemannZeta
+
+/-!
+# L-series of functions on `ZMod N`
+
+We show that if `N` is a positive integer and `Φ : ZMod N → ℂ`, then the L-series of `Φ` has
+analytic continuation (away from a pole at `s = 1` if `∑ j, Φ j ≠ 0`) and satisfies a functional
+equation. We also define completed L-functions (given by multiplying the naive L-function by a
+Gamma-factor), and prove analytic continuation and functional equations for these too, assuming `Φ`
+is either even or odd.
+
+The most familiar case is when `Φ` is a Dirichlet character, but the results here are valid
+for general functions; for the specific case of Dirichlet characters see
+`Mathlib.NumberTheory.LSeries.DirichletContinuation`.
+
+## Main definitions
+
+* `ZMod.LFunction Φ s`: the meromorphic continuation of the function `∑ n : ℕ, Φ n * n ^ (-s)`.
+* `ZMod.completedLFunction Φ s`: the completed L-function, which for *almost* all `s` is equal to
+  `LFunction Φ s` multiplied by an Archimedean Gamma-factor.
+
+Note that `ZMod.completedLFunction Φ s` is only mathematically well-defined if `Φ` is either even
+or odd. Here we extend it to all functions `Φ` by linearity  (but the functional equation only holds
+if `Φ` is either even or odd).
+
+## Main theorems
+
+Results for non-completed L-functions:
+
+* `ZMod.LFunction_eq_LSeries`: if `1 < re s` then the `LFunction` coincides with the naive
+  `LSeries`.
+* `ZMod.differentiableAt_LFunction`: `ZMod.LFunction Φ` is differentiable at `s ∈ ℂ` if either
+  `s ≠ 1` or `∑ j, Φ j = 0`.
+* `ZMod.LFunction_one_sub`: the functional equation relating `LFunction Φ (1 - s)` to
+  `LFunction (𝓕 Φ) s`, where `𝓕` is the Fourier transform.
+
+Results for completed L-functions:
+
+* `ZMod.LFunction_eq_completed_div_gammaFactor_even` and
+  `LFunction_eq_completed_div_gammaFactor_odd`: we have
+  `LFunction Φ s = completedLFunction Φ s / Gammaℝ s` for `Φ` even, and
+  `LFunction Φ s = completedLFunction Φ s / Gammaℝ (s + 1)` for `Φ` odd. (We formulate it this way
+  around so it is still valid at the poles of the Gamma factor.)
+* `ZMod.differentiableAt_completedLFunction`: `ZMod.completedLFunction Φ` is differentiable at
+  `s ∈ ℂ`, unless `s = 1` and `∑ j, Φ j ≠ 0`, or `s = 0` and `Φ 0 ≠ 0`.
+* `ZMod.completedLFunction_one_sub_even` and `ZMod.completedLFunction_one_sub_odd`:
+  the functional equation relating `completedLFunction Φ (1 - s)` to `completedLFunction (𝓕 Φ) s`.
+-/
+
+open HurwitzZeta Complex ZMod Finset Classical Topology Filter Set
+
+open scoped Real
+
+namespace ZMod
+
+variable {N : ℕ} [NeZero N]
+
+/-- If `Φ` is a periodic function, then the L-series of `Φ` converges for `1 < re s`. -/
+lemma LSeriesSummable_of_one_lt_re (Φ : ZMod N → ℂ) {s : ℂ} (hs : 1 < re s) :
+    LSeriesSummable (Φ ·) s := by
+  let c := max' _ <| univ_nonempty.image (Complex.abs ∘ Φ)
+  refine LSeriesSummable_of_bounded_of_one_lt_re (fun n _ ↦ le_max' _ _ ?_) (m := c) hs
+  exact mem_image_of_mem _ (mem_univ _)
+
+/--
+The unique meromorphic function `ℂ → ℂ` which agrees with `∑' n : ℕ, Φ n / n ^ s` wherever the
+latter is convergent. This is constructed as a linear combination of Hurwitz zeta functions.
+
+Note that this is not the same as `LSeries Φ`: they agree in the convergence range, but
+`LSeries Φ s` is defined to be `0` if `re s ≤ 1`.
+ -/
+noncomputable def LFunction (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
+  N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZeta (toAddCircle j) s
+
+/-- The L-function of a function on `ZMod 1` is a scalar multiple of the Riemann zeta function. -/
+lemma LFunction_modOne_eq (Φ : ZMod 1 → ℂ) (s : ℂ) :
+    LFunction Φ s = Φ 0 * riemannZeta s := by
+  simp only [LFunction, Nat.cast_one, one_cpow, ← singleton_eq_univ (0 : ZMod 1), sum_singleton,
+    map_zero, hurwitzZeta_zero, one_mul]
+
+/-- For `1 < re s` the congruence L-function agrees with the sum of the Dirichlet series. -/
+lemma LFunction_eq_LSeries (Φ : ZMod N → ℂ) {s : ℂ} (hs : 1 < re s) :
+    LFunction Φ s = LSeries (Φ ·) s := by
+  rw [LFunction, LSeries, mul_sum, Nat.sumByResidueClasses (LSeriesSummable_of_one_lt_re Φ hs) N]
+  congr 1 with j
+  have : (j.val / N : ℝ) ∈ Set.Icc 0 1 := mem_Icc.mpr ⟨by positivity,
+    (div_le_one (Nat.cast_pos.mpr <| NeZero.pos _)).mpr <| Nat.cast_le.mpr (val_lt j).le⟩
+  rw [toAddCircle_apply, ← (hasSum_hurwitzZeta_of_one_lt_re this hs).tsum_eq, ← mul_assoc,
+    ← tsum_mul_left]
+  congr 1 with m
+  -- The following manipulation is slightly delicate because `(x * y) ^ s = x ^ s * y ^ s` is
+  -- false for general complex `x`, `y`, but it is true if `x` and `y` are non-negative reals, so
+  -- we have to carefully juggle coercions `ℕ → ℝ → ℂ`.
+  calc N ^ (-s) * Φ j * (1 / (m + (j.val / N : ℝ)) ^ s)
+  _ = Φ j * (N ^ (-s) * (1 / (m + (j.val / N : ℝ)) ^ s)) := by
+    rw [← mul_assoc, mul_comm _ (Φ _)]
+  _ = Φ j * (1 / (N : ℝ) ^ s * (1 / ((j.val + N * m) / N : ℝ) ^ s)) := by
+    simp only [cpow_neg, ← one_div, ofReal_div, ofReal_natCast, add_comm, add_div, ofReal_add,
+      ofReal_mul, mul_div_cancel_left₀ (m : ℂ) (Nat.cast_ne_zero.mpr (NeZero.ne N))]
+  _ = Φ j / ((N : ℝ) * ((j.val + N * m) / N : ℝ)) ^ s := by -- this is the delicate step!
+    rw [one_div_mul_one_div, mul_one_div, mul_cpow_ofReal_nonneg] <;> positivity
+  _ = Φ j / (N * (j.val + N * m) / N) ^ s := by
+    simp only [ofReal_natCast, ofReal_div, ofReal_add, ofReal_mul, mul_div_assoc]
+  _ = Φ j / (j.val + N * m) ^ s := by
+    rw [mul_div_cancel_left₀ _ (Nat.cast_ne_zero.mpr (NeZero.ne N))]
+  _ = Φ ↑(j.val + N * m) / (↑(j.val + N * m)) ^ s := by
+    simp only [Nat.cast_add, Nat.cast_mul, natCast_zmod_val, natCast_self, zero_mul, add_zero]
+  _ = LSeries.term (Φ ·) s (j.val + N * m) := by
+    rw [LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs)]
+
+lemma differentiableAt_LFunction (Φ : ZMod N → ℂ) (s : ℂ) (hs : s ≠ 1 ∨ ∑ j, Φ j = 0) :
+    DifferentiableAt ℂ (LFunction Φ) s := by
+  apply (differentiable_neg.const_cpow (Or.inl <| NeZero.ne _) s).mul
+  rcases ne_or_eq s 1 with hs' | rfl
+  · exact .sum fun j _ ↦ (differentiableAt_hurwitzZeta _ hs').const_mul _
+  · have := DifferentiableAt.sum (u := univ) fun j _ ↦
+      (differentiableAt_hurwitzZeta_sub_one_div (toAddCircle j)).const_mul (Φ j)
+    simpa only [mul_sub, sum_sub_distrib, ← sum_mul, hs.neg_resolve_left rfl, zero_mul, sub_zero]
+
+lemma differentiable_LFunction_of_sum_zero {Φ : ZMod N → ℂ} (hΦ : ∑ j, Φ j = 0) :
+    Differentiable ℂ (LFunction Φ) :=
+  fun s ↦ differentiableAt_LFunction Φ s (Or.inr hΦ)
+
+/-- The L-function of `Φ` has a residue at `s = 1` equal to the average value of `Φ`. -/
+lemma LFunction_residue_one (Φ : ZMod N → ℂ) :
+    Tendsto (fun s ↦ (s - 1) * LFunction Φ s) (𝓝[≠] 1) (𝓝 (∑ j, Φ j / N)) := by
+  simp only [sum_div, LFunction, mul_sum]
+  refine tendsto_finset_sum _ fun j _ ↦ ?_
+  rw [(by ring : Φ j / N = Φ j * (1 / N * 1)), one_div, ← cpow_neg_one]
+  simp only [show ∀ a b c d : ℂ, a * (b * (c * d)) = c * (b * (a * d)) by intros; ring]
+  refine tendsto_const_nhds.mul (.mul ?_ <| hurwitzZeta_residue_one _)
+  exact ((continuous_neg.const_cpow (Or.inl <| NeZero.ne _)).tendsto _).mono_left
+    nhdsWithin_le_nhds
+
+local notation "𝕖" => stdAddChar
+
+/--
+The `LFunction` of the function `x ↦ e (j * x)`, where `e : ZMod N → ℂ` is the standard additive
+character, agrees with `expZeta (j / N)` on `1 < re s`. Private since it is a stepping-stone to
+the more general result `LFunction_stdAddChar_eq_expZeta` below.
+-/
+private lemma LFunction_stdAddChar_eq_expZeta_of_one_lt_re (j : ZMod N) {s : ℂ} (hs : 1 < s.re) :
+    LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (ZMod.toAddCircle j) s := by
+  rw [toAddCircle_apply, ← (hasSum_expZeta_of_one_lt_re (j.val / N) hs).tsum_eq,
+    LFunction_eq_LSeries _ hs, LSeries]
+  congr 1 with n
+  rw [LSeries.term_of_ne_zero' (ne_zero_of_one_lt_re hs), ofReal_div, ofReal_natCast,
+    ofReal_natCast, mul_assoc, div_mul_eq_mul_div, stdAddChar_apply]
+  have := ZMod.toCircle_intCast (N := N) (j.val * n)
+  conv_rhs at this => rw [Int.cast_mul, Int.cast_natCast, Int.cast_natCast, mul_div_assoc]
+  rw [← this, Int.cast_mul, Int.cast_natCast, Int.cast_natCast, natCast_zmod_val]
+
+/--
+The `LFunction` of the function `x ↦ e (j * x)`, where `e : ZMod N → ℂ` is the standard additive
+character, is `expZeta (j / N)`.
+
+Note this is not at all obvious from the definitions, and we prove it by analytic continuation
+from the convergence range.
+-/
+lemma LFunction_stdAddChar_eq_expZeta (j : ZMod N) (s : ℂ) (hjs : j ≠ 0 ∨ s ≠ 1) :
+    LFunction (fun k ↦ 𝕖 (j * k)) s = expZeta (ZMod.toAddCircle j) s := by
+  let U := if j = 0 then {z : ℂ | z ≠ 1} else univ -- region of analyticity of both functions
+  let V := {z : ℂ | 1 < re z} -- convergence region
+  have hUo : IsOpen U := by
+    by_cases h : j = 0
+    · simpa only [h, ↓reduceIte, U] using isOpen_compl_singleton
+    · simp only [h, ↓reduceIte, isOpen_univ, U]
+  let f := LFunction (fun k ↦ stdAddChar (j * k))
+  let g := expZeta (toAddCircle j)
+  have hU {u} : u ∈ U ↔ u ≠ 1 ∨ j ≠ 0 := by simp only [mem_ite_univ_right, U]; tauto
+  -- hypotheses for uniqueness of analytic continuation
+  have hf : AnalyticOnNhd ℂ f U := by
+    refine DifferentiableOn.analyticOnNhd (fun u hu ↦ ?_) hUo
+    refine (differentiableAt_LFunction _ _ ((hU.mp hu).imp_right fun h ↦ ?_)).differentiableWithinAt
+    simp only [mul_comm j, AddChar.sum_mulShift _ (isPrimitive_stdAddChar _), h,
+      ↓reduceIte, CharP.cast_eq_zero, or_true]
+  have hg : AnalyticOnNhd ℂ g U := by
+    refine DifferentiableOn.analyticOnNhd (fun u hu ↦ ?_) hUo
+    refine (differentiableAt_expZeta _ _ ((hU.mp hu).imp_right fun h ↦ ?_)).differentiableWithinAt
+    rwa [ne_eq, toAddCircle_eq_zero]
+  have hUc : IsPreconnected U := by
+    by_cases h : j = 0
+    · simpa only [h, ↓reduceIte, U] using
+        (isConnected_compl_singleton_of_one_lt_rank (by simp) _).isPreconnected
+    · simpa only [h, ↓reduceIte, U] using isPreconnected_univ
+  have hV : V ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
+  have hUmem : 2 ∈ U := by simp [U]
+  have hUmem' : s ∈ U := hU.mpr hjs.symm
+  -- apply uniqueness result
+  refine hf.eqOn_of_preconnected_of_eventuallyEq hg hUc hUmem ?_ hUmem'
+  -- now remains to prove equality on `1 < re s`
+  filter_upwards [hV] with z using LFunction_stdAddChar_eq_expZeta_of_one_lt_re _
+
+/-- Explicit formula for the L-function of `𝓕 Φ`, where `𝓕` is the discrete Fourier transform. -/
+lemma LFunction_dft (Φ : ZMod N → ℂ) {s : ℂ} (hs : Φ 0 = 0 ∨ s ≠ 1) :
+    LFunction (𝓕 Φ) s = ∑ j : ZMod N, Φ j * expZeta (toAddCircle (-j)) s := by
+  have (j : ZMod N) : Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s =
+      Φ j * expZeta (toAddCircle (-j)) s := by
+    by_cases h : -j ≠ 0 ∨ s ≠ 1
+    · rw [LFunction_stdAddChar_eq_expZeta _ _ h]
+    · simp only [neg_ne_zero, not_or, not_not] at h
+      rw [h.1, show Φ 0 = 0 by tauto, zero_mul, zero_mul]
+  simp only [LFunction, ← this, mul_sum]
+  rw [dft_def, sum_comm]
+  simp only [sum_mul, mul_sum, Circle.smul_def, smul_eq_mul, stdAddChar_apply, ← mul_assoc]
+  congr 1 with j
+  congr 1 with k
+  rw [mul_assoc (Φ _), mul_comm (Φ _), neg_mul]
+
+/-- Functional equation for `ZMod` L-functions, in terms of discrete Fourier transform. -/
+theorem LFunction_one_sub (Φ : ZMod N → ℂ) {s : ℂ}
+    (hs : ∀ (n : ℕ), s ≠ -n) (hs' : Φ 0 = 0 ∨ s ≠ 1) :
+    LFunction Φ (1 - s) = N ^ (s - 1) * (2 * π) ^ (-s) * Gamma s *
+      (cexp (π * I * s / 2) * LFunction (𝓕 Φ) s
+       + cexp (-π * I * s / 2) * LFunction (𝓕 fun x ↦ Φ (-x)) s) := by
+  rw [LFunction]
+  have (j : ZMod N) :  Φ j * hurwitzZeta (toAddCircle j) (1 - s) = Φ j *
+      ((2 * π) ^ (-s) * Gamma s * (cexp (-π * I * s / 2) *
+      expZeta (toAddCircle j) s + cexp (π * I * s / 2) * expZeta (-toAddCircle j) s)) := by
+    rcases eq_or_ne j 0 with rfl | hj
+    · rcases hs' with hΦ | hs'
+      · simp only [hΦ, zero_mul]
+      · rw [hurwitzZeta_one_sub _ hs (Or.inr hs')]
+    · rw [hurwitzZeta_one_sub _ hs (Or.inl <| toAddCircle_eq_zero.not.mpr hj)]
+  simp only [this, mul_assoc _ _ (Gamma s)]
+  -- get rid of Gamma terms and power of N
+  generalize (2 * π) ^ (-s) * Gamma s = C
+  simp_rw [← mul_assoc, mul_comm _ C, mul_assoc, ← mul_sum, ← mul_assoc, mul_comm _ C, mul_assoc,
+    neg_sub]
+  congr 2
+  -- now gather sum terms
+  rw [LFunction_dft _ hs', LFunction_dft _ (hs'.imp_left <| by simp only [neg_zero, imp_self])]
+  conv_rhs => enter [2, 2]; rw [← (Equiv.neg _).sum_comp _ _ (by simp), Equiv.neg_apply]
+  simp_rw [neg_neg, mul_sum, ← sum_add_distrib, ← mul_assoc, mul_comm _ (Φ _), mul_assoc,
+    ← mul_add, map_neg, add_comm]
+
+section signed
+
+variable {Φ : ZMod N → ℂ}
+
+lemma LFunction_def_even (hΦ : Φ.Even) (s : ℂ) :
+    LFunction Φ s = N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZetaEven (toAddCircle j) s := by
+  simp only [LFunction, hurwitzZeta, mul_add (Φ _), sum_add_distrib]
+  congr 1
+  simp only [add_right_eq_self, ← neg_eq_self ℂ, ← sum_neg_distrib]
+  refine Fintype.sum_equiv (.neg _) _ _ fun i ↦ ?_
+  simp only [Equiv.neg_apply, hΦ i, map_neg, hurwitzZetaOdd_neg, mul_neg]
+
+lemma LFunction_def_odd (hΦ : Φ.Odd) (s : ℂ) :
+    LFunction Φ s = N ^ (-s) * ∑ j : ZMod N, Φ j * hurwitzZetaOdd (toAddCircle j) s := by
+  simp only [LFunction, hurwitzZeta, mul_add (Φ _), sum_add_distrib]
+  congr 1
+  simp only [add_left_eq_self, ← neg_eq_self ℂ, ← sum_neg_distrib]
+  refine Fintype.sum_equiv (.neg _) _ _ fun i ↦ ?_
+  simp only [Equiv.neg_apply, hΦ i, map_neg, hurwitzZetaEven_neg, neg_mul]
+
+/-- Explicit formula for `LFunction Φ 0` when `Φ` is even. -/
+@[simp] lemma LFunction_apply_zero_of_even (hΦ : Φ.Even) :
+    LFunction Φ 0 = -Φ 0 / 2 := by
+  simp only [LFunction_def_even hΦ, neg_zero, cpow_zero, hurwitzZetaEven_apply_zero,
+    toAddCircle_eq_zero, mul_ite, mul_div, mul_neg_one, mul_zero, sum_ite_eq', Finset.mem_univ,
+    ↓reduceIte, one_mul]
+
+/-- The L-function of an even function vanishes at negative even integers. -/
+@[simp] lemma LFunction_neg_two_mul_nat_add_one (hΦ : Φ.Even) (n : ℕ) :
+    LFunction Φ (-(2 * (n + 1))) = 0 := by
+  simp only [LFunction_def_even hΦ, hurwitzZetaEven_neg_two_mul_nat_add_one, mul_zero,
+    sum_const_zero, ← neg_mul]
+
+/-- The L-function of an odd function vanishes at negative odd integers. -/
+@[simp] lemma LFunction_neg_two_mul_nat_sub_one (hΦ : Φ.Odd) (n : ℕ) :
+    LFunction Φ (-(2 * n) - 1) = 0 := by
+  simp only [LFunction_def_odd hΦ, hurwitzZetaOdd_neg_two_mul_nat_sub_one, mul_zero, ← neg_mul,
+    sum_const_zero]
+
+/--
+The completed L-function of a function `Φ : ZMod N → ℂ`.
+
+This is only mathematically meaningful if `Φ` is either even, or odd; here we extend this to all `Φ`
+by linearity.
+-/
+noncomputable def completedLFunction (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
+  N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s
+  + N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s
+
+@[simp] lemma completedLFunction_zero (s : ℂ) : completedLFunction (0 : ZMod N → ℂ) s = 0 := by
+  simp only [completedLFunction, Pi.zero_apply, zero_mul, sum_const_zero, mul_zero, zero_add]
+
+lemma completedLFunction_def_even (hΦ : Φ.Even) (s : ℂ) :
+    completedLFunction Φ s = N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s := by
+  suffices ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0 by
+    rw [completedLFunction, this, mul_zero, add_zero]
+  refine (hΦ.mul_odd fun j ↦ ?_).sum_eq_zero
+  rw [map_neg, completedHurwitzZetaOdd_neg]
+
+lemma completedLFunction_def_odd (hΦ : Φ.Odd) (s : ℂ) :
+    completedLFunction Φ s = N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s := by
+  suffices ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s = 0 by
+    rw [completedLFunction, this, mul_zero, zero_add]
+  refine (hΦ.mul_even fun j ↦ ?_).sum_eq_zero
+  rw [map_neg, completedHurwitzZetaEven_neg]
+
+/--
+The completed L-function of a function `ZMod 1 → ℂ` is a scalar multiple of the completed Riemann
+zeta function.
+-/
+lemma completedLFunction_modOne_eq (Φ : ZMod 1 → ℂ) (s : ℂ) :
+    completedLFunction Φ s = Φ 1 * completedRiemannZeta s := by
+  rw [completedLFunction_def_even (show Φ.Even from fun _ ↦ congr_arg Φ (Subsingleton.elim ..)),
+    Nat.cast_one, one_cpow, one_mul, ← singleton_eq_univ 0, sum_singleton, map_zero,
+    completedHurwitzZetaEven_zero, Subsingleton.elim 0 1]
+
+/--
+The completed L-function of a function `ZMod N → ℂ`, modified by adding multiples of `N ^ (-s) / s`
+and `N ^ (-s) / (1 - s)` to make it entire.
+-/
+noncomputable def completedLFunction₀ (Φ : ZMod N → ℂ) (s : ℂ) : ℂ :=
+  N ^ (-s) * ∑ j : ZMod N, Φ j * completedHurwitzZetaEven₀ (toAddCircle j) s
+  + N ^ (-s) * ∑ j : ZMod N, Φ j * completedHurwitzZetaOdd (toAddCircle j) s
+
+/-- The function `completedLFunction₀ Φ` is differentiable. -/
+lemma differentiable_completedLFunction₀ (Φ : ZMod N → ℂ) :
+    Differentiable ℂ (completedLFunction₀ Φ) := by
+  refine .add ?_ ?_ <;>
+  refine (differentiable_neg.const_cpow <| .inl <| NeZero.ne _).mul (.sum fun i _ ↦ .const_mul ?_ _)
+  exacts [differentiable_completedHurwitzZetaEven₀ _, differentiable_completedHurwitzZetaOdd _]
+
+lemma completedLFunction_eq (Φ : ZMod N → ℂ) (s : ℂ) :
+    completedLFunction Φ s =
+      completedLFunction₀ Φ s - N ^ (-s) * Φ 0 / s - N ^ (-s) * (∑ j, Φ j) / (1 - s) := by
+  simp only [completedLFunction, completedHurwitzZetaEven_eq, toAddCircle_eq_zero, div_eq_mul_inv,
+    ite_mul, one_mul, zero_mul, mul_sub, mul_ite, mul_zero, sum_sub_distrib, Fintype.sum_ite_eq',
+    ← sum_mul, completedLFunction₀, mul_assoc]
+  abel
+
+/--
+The completed L-function of a function `ZMod N → ℂ` is differentiable, with the following
+exceptions: at `s = 1` if `∑ j, Φ j ≠ 0`; and at `s = 0` if `Φ 0 ≠ 0`.
+-/
+lemma differentiableAt_completedLFunction (Φ : ZMod N → ℂ) (s : ℂ) (hs₀ : s ≠ 0 ∨ Φ 0 = 0)
+    (hs₁ : s ≠ 1 ∨ ∑ j, Φ j = 0) : DifferentiableAt ℂ (completedLFunction Φ) s := by
+  simp only [funext (completedLFunction_eq Φ), mul_div_assoc]
+  -- We know `completedLFunction₀` is differentiable everywhere, so it suffices to show that the
+  -- correction terms from `completedLFunction_eq` are differentiable at `s`.
+  refine ((differentiable_completedLFunction₀ _ _).sub ?_).sub ?_
+  · -- term with `1 / s`
+    refine ((differentiable_neg.const_cpow <| .inl <| NeZero.ne _) s).mul (hs₀.elim ?_ ?_)
+    · exact fun h ↦ (differentiableAt_const _).div differentiableAt_id h
+    · exact fun h ↦ by simp only [h, funext zero_div, differentiableAt_const]
+  · -- term with `1 / (1 - s)`
+    refine ((differentiable_neg.const_cpow <| .inl <| NeZero.ne _) s).mul (hs₁.elim ?_ ?_)
+    · exact fun h ↦ (differentiableAt_const _).div (by fun_prop) (by rwa [sub_ne_zero, ne_comm])
+    · exact fun h ↦ by simp only [h, zero_div, differentiableAt_const]
+
+/--
+Special case of `differentiableAt_completedLFunction` asserting differentiability everywhere
+under suitable hypotheses.
+-/
+lemma differentiable_completedLFunction (hΦ₂ : Φ 0 = 0) (hΦ₃ : ∑ j, Φ j = 0) :
+    Differentiable ℂ (completedLFunction Φ) :=
+  fun s ↦ differentiableAt_completedLFunction Φ s (.inr hΦ₂) (.inr hΦ₃)
+
+/--
+Relation between the completed L-function and the usual one (even case).
+We state it this way around so it holds at the poles of the gamma factor as well
+(except at `s = 0`, where it is genuinely false if `N > 1` and `Φ 0 ≠ 0`).
+-/
+lemma LFunction_eq_completed_div_gammaFactor_even (hΦ : Φ.Even) (s : ℂ) (hs : s ≠ 0 ∨ Φ 0 = 0) :
+    LFunction Φ s = completedLFunction Φ s / Gammaℝ s := by
+  simp only [completedLFunction_def_even hΦ, LFunction_def_even hΦ, mul_div_assoc, sum_div]
+  congr 2 with i
+  rcases ne_or_eq i 0 with hi | rfl
+  · rw [hurwitzZetaEven_def_of_ne_or_ne (.inl (hi ∘ toAddCircle_eq_zero.mp))]
+  · rcases hs with hs | hΦ'
+    · rw [hurwitzZetaEven_def_of_ne_or_ne (.inr hs)]
+    · simp only [hΦ', map_zero, zero_mul]
+
+/--
+Relation between the completed L-function and the usual one (odd case).
+We state it this way around so it holds at the poles of the gamma factor as well.
+-/
+lemma LFunction_eq_completed_div_gammaFactor_odd (hΦ : Φ.Odd) (s : ℂ) :
+    LFunction Φ s = completedLFunction Φ s / Gammaℝ (s + 1) := by
+  simp only [LFunction_def_odd hΦ, completedLFunction_def_odd hΦ, hurwitzZetaOdd, mul_div_assoc,
+    sum_div]
+
+/--
+First form of functional equation for completed L-functions (even case).
+
+Private because it is superseded by `completedLFunction_one_sub_even` below, which is valid for a
+much wider range of `s`.
+-/
+private lemma completedLFunction_one_sub_of_one_lt_even (hΦ : Φ.Even) {s : ℂ} (hs : 1 < re s) :
+    completedLFunction Φ (1 - s) = N ^ (s - 1) * completedLFunction (𝓕 Φ) s := by
+  have hs₀ : s ≠ 0 := ne_zero_of_one_lt_re hs
+  have hs₁ : s ≠ 1 := (lt_irrefl _ <| one_re ▸ · ▸ hs)
+  -- strip down to the key equality:
+  suffices ∑ x, Φ x * completedCosZeta (toAddCircle x) s = completedLFunction (𝓕 Φ) s by
+    simp only [completedLFunction_def_even hΦ, neg_sub, completedHurwitzZetaEven_one_sub, this]
+  -- reduce to equality with un-completed L-functions:
+  suffices ∑ x, Φ x * cosZeta (toAddCircle x) s = LFunction (𝓕 Φ) s by
+    simpa only [cosZeta, Function.update_noteq hs₀, ← mul_div_assoc, ← sum_div,
+      LFunction_eq_completed_div_gammaFactor_even (dft_even_iff.mpr hΦ) _ (.inl hs₀),
+      div_left_inj' (Gammaℝ_ne_zero_of_re_pos (zero_lt_one.trans hs))]
+  -- expand out `LFunction (𝓕 Φ)` and use parity:
+  simp only [cosZeta_eq, ← mul_div_assoc _ _ (2 : ℂ), mul_add, ← sum_div, sum_add_distrib,
+    LFunction_dft Φ (.inr hs₁), map_neg, div_eq_iff (two_ne_zero' ℂ), mul_two, add_left_inj]
+  exact Fintype.sum_equiv (.neg _) _ _ (by simp [hΦ _])
+
+/--
+First form of functional equation for completed L-functions (odd case).
+
+Private because it is superseded by `completedLFunction_one_sub_odd` below, which is valid for a
+much wider range of `s`.
+-/
+private lemma completedLFunction_one_sub_of_one_lt_odd (hΦ : Φ.Odd) {s : ℂ} (hs : 1 < re s) :
+    completedLFunction Φ (1 - s) = N ^ (s - 1) * I * completedLFunction (𝓕 Φ) s := by
+  -- strip down to the key equality:
+  suffices ∑ x, Φ x * completedSinZeta (toAddCircle x) s = I * completedLFunction (𝓕 Φ) s by
+    simp only [completedLFunction_def_odd hΦ, neg_sub, completedHurwitzZetaOdd_one_sub, this,
+      mul_assoc]
+  -- reduce to equality with un-completed L-functions:
+  suffices ∑ x, Φ x * sinZeta (toAddCircle x) s = I * LFunction (𝓕 Φ) s by
+    have hs' : 0 < re (s + 1) := by simp only [add_re, one_re]; linarith
+    simpa only [sinZeta, ← mul_div_assoc, ← sum_div, div_left_inj' (Gammaℝ_ne_zero_of_re_pos hs'),
+      LFunction_eq_completed_div_gammaFactor_odd (dft_odd_iff.mpr hΦ)]
+  -- now calculate:
+  calc ∑ x, Φ x * sinZeta (toAddCircle x) s
+  _ = (∑ x, Φ x * expZeta (toAddCircle x) s) / (2 * I)
+      - (∑ x, Φ x * expZeta (toAddCircle (-x)) s) / (2 * I) := by
+    simp only [sinZeta_eq, ← mul_div_assoc, mul_sub, sub_div, sum_sub_distrib, sum_div, map_neg]
+  _ = (∑ x, Φ (-x) * expZeta (toAddCircle (-x)) s) / (_) - (_) := by
+    congrm ?_ / _ - _
+    exact (Fintype.sum_equiv (.neg _) _ _ fun x ↦ by rfl).symm
+  _ = -I⁻¹ * LFunction (𝓕 Φ) s := by
+    simp only [hΦ _, neg_mul, sum_neg_distrib, LFunction_dft Φ (.inl hΦ.map_zero)]
+    ring
+  _ = I * LFunction (𝓕 Φ) s := by rw [inv_I, neg_neg]
+
+/--
+Functional equation for completed L-functions (even case), valid at all points of differentiability.
+-/
+theorem completedLFunction_one_sub_even (hΦ : Φ.Even) (s : ℂ)
+    (hs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0) (hs₁ : s ≠ 1 ∨ Φ 0 = 0) :
+    completedLFunction Φ (1 - s) = N ^ (s - 1) * completedLFunction (𝓕 Φ) s := by
+  -- We prove this using `AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq`, so we need to
+  -- gather up the ingredients for this big theorem.
+  -- First set up some notations:
+  let F (t) := completedLFunction Φ (1 - t)
+  let G (t) := ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
+  -- Set on which F, G are analytic:
+  let U := {t : ℂ | (t ≠ 0 ∨ ∑ j, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
+  -- Properties of U:
+  have hsU : s ∈ U := ⟨hs₀, hs₁⟩
+  have h2U : 2 ∈ U := ⟨.inl two_ne_zero, .inl (OfNat.ofNat_ne_one _)⟩
+  have hUo : IsOpen U := (isOpen_compl_singleton.union isOpen_const).inter
+    (isOpen_compl_singleton.union isOpen_const)
+  have hUp : IsPreconnected U := by
+    -- need to write `U` as the complement of an obviously countable set
+    let Uc : Set ℂ := (if ∑ j, Φ j = 0 then ∅ else {0}) ∪ (if Φ 0 = 0 then ∅ else {1})
+    have : Uc.Countable := by
+      apply Countable.union <;>
+      split_ifs <;>
+      simp only [countable_singleton, countable_empty]
+    convert (this.isConnected_compl_of_one_lt_rank ?_).isPreconnected using 1
+    · ext x
+      by_cases h : Φ 0 = 0 <;>
+      by_cases h' : ∑ j, Φ j = 0 <;>
+      simp [U, Uc, h, h', and_comm]
+    · simp only [rank_real_complex, Nat.one_lt_ofNat]
+  -- Analyticity on U:
+  have hF : AnalyticOnNhd ℂ F U := by
+    refine DifferentiableOn.analyticOnNhd
+      (fun t ht ↦ DifferentiableAt.differentiableWithinAt ?_) hUo
+    refine (differentiableAt_completedLFunction Φ _ ?_ ?_).comp t (differentiableAt_id.const_sub 1)
+    exacts [ht.2.imp_left (sub_ne_zero.mpr ∘ Ne.symm), ht.1.imp_left sub_eq_self.not.mpr]
+  have hG : AnalyticOnNhd ℂ G U := by
+    refine DifferentiableOn.analyticOnNhd
+      (fun t ht ↦ DifferentiableAt.differentiableWithinAt ?_) hUo
+    apply ((differentiableAt_id.sub_const 1).const_cpow (.inl (NeZero.ne _))).mul
+    apply differentiableAt_completedLFunction _ _ (ht.1.imp_right fun h ↦ dft_apply_zero Φ ▸ h)
+    exact ht.2.imp_right (fun h ↦ by simp only [← dft_apply_zero, dft_dft, neg_zero, h, smul_zero])
+  -- set where we know equality
+  have hV : {z | 1 < re z} ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
+  have hFG : F =ᶠ[𝓝 2] G := eventually_of_mem hV <| fun t ht ↦ by
+    simpa only [F, G, pow_zero, mul_one] using completedLFunction_one_sub_of_one_lt_even hΦ ht
+  -- now apply the big hammer to finish
+  exact hF.eqOn_of_preconnected_of_eventuallyEq hG hUp h2U hFG hsU
+
+/-- Functional equation for completed L-functions (odd case), valid for all `s`. -/
+theorem completedLFunction_one_sub_odd (hΦ : Φ.Odd) (s : ℂ) :
+    completedLFunction Φ (1 - s) = N ^ (s - 1) * I * completedLFunction (𝓕 Φ) s := by
+  -- This is much easier than the even case since both functions are entire.
+  -- First set up some notations:
+  let F (t) := completedLFunction Φ (1 - t)
+  let G (t) := ↑N ^ (t - 1) * I * completedLFunction (𝓕 Φ) t
+  -- check F, G globally differentiable
+  have hF : Differentiable ℂ F := (differentiable_completedLFunction hΦ.map_zero
+    hΦ.sum_eq_zero).comp (differentiable_id.const_sub 1)
+  have hG : Differentiable ℂ G := by
+    apply (((differentiable_id.sub_const 1).const_cpow (.inl (NeZero.ne _))).mul_const _).mul
+    rw [← dft_odd_iff] at hΦ
+    exact differentiable_completedLFunction hΦ.map_zero hΦ.sum_eq_zero
+  -- set where we know equality
+  have : {z | 1 < re z} ∈ 𝓝 2 := (continuous_re.isOpen_preimage _ isOpen_Ioi).mem_nhds (by simp)
+  have hFG : F =ᶠ[𝓝 2] G := by filter_upwards [this] with t ht
+    using completedLFunction_one_sub_of_one_lt_odd hΦ ht
+  -- now apply the big hammer to finish
+  rw [← analyticOnNhd_univ_iff_differentiable] at hF hG
+  exact congr_fun (hF.eq_of_eventuallyEq hG hFG) s
+
+end signed
+
+end ZMod
diff --git a/Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean b/Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
index 32524bd59090a..392736e26084d 100644
--- a/Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
+++ b/Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
 import Mathlib.NumberTheory.LegendreSymbol.Basic
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 
 /-!
 # Lemmas of Gauss and Eisenstein
@@ -34,7 +34,7 @@ theorem Ico_map_valMinAbs_natAbs_eq_Ico_map_id (p : ℕ) [hp : Fact p.Prime] (a
     lt_of_le_of_lt (he hx).2 (Nat.div_lt_self hp.1.pos (by decide))
   have hpe : ∀ {x}, x ∈ Ico 1 (p / 2).succ → ¬p ∣ x := fun hx hpx =>
     not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero (he hx).1) hpx) (hep hx)
-  have hmem : ∀ (x : ℕ) (hx : x ∈ Ico 1 (p / 2).succ),
+  have hmem : ∀ (x : ℕ) (_ : x ∈ Ico 1 (p / 2).succ),
       (a * x : ZMod p).valMinAbs.natAbs ∈ Ico 1 (p / 2).succ := by
     intro x hx
     simp [hap, CharP.cast_eq_zero_iff (ZMod p) p, hpe hx, Nat.lt_succ_iff, succ_le_iff,
@@ -205,7 +205,7 @@ theorem sum_mul_div_add_sum_mul_div_eq_mul (p q : ℕ) [hp : Fact p.Prime] (hq0
       simpa [hq0] using congr_arg ((↑) : ℕ → ZMod p) (le_antisymm hpq hqp)
     apply_fun ZMod.val at this
     rw [val_cast_of_lt hxp, val_zero] at this
-    simp only [this, nonpos_iff_eq_zero, mem_Ico, one_ne_zero, false_and_iff, mem_product] at hx
+    simp only [this, nonpos_iff_eq_zero, mem_Ico, one_ne_zero, false_and, mem_product] at hx
   have hunion :
       (((Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter fun x : ℕ × ℕ => x.2 * p ≤ x.1 * q) ∪
         (Ico 1 (p / 2).succ ×ˢ Ico 1 (q / 2).succ).filter fun x : ℕ × ℕ => x.1 * q ≤ x.2 * p) =
diff --git a/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean b/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
index 7002957721377..c4b313daf7c59 100644
--- a/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
+++ b/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
@@ -156,6 +156,10 @@ theorem mul_left (a₁ a₂ : ℤ) (b : ℕ) : J(a₁ * a₂ | b) = J(a₁ | b)
     (f := fun x ↦ @legendreSym x {out := prime_of_mem_primeFactorsList x.2} a₁)
     (g := fun x ↦ @legendreSym x {out := prime_of_mem_primeFactorsList x.2} a₂)
 
+#adaptation_note
+/--
+After nightly-2024-09-06 we can remove the `_root_` prefixes below.
+-/
 /-- The symbol `J(a | b)` vanishes iff `a` and `b` are not coprime (assuming `b ≠ 0`). -/
 theorem eq_zero_iff_not_coprime {a : ℤ} {b : ℕ} [NeZero b] : J(a | b) = 0 ↔ a.gcd b ≠ 1 :=
   List.prod_eq_zero_iff.trans
@@ -165,7 +169,7 @@ theorem eq_zero_iff_not_coprime {a : ℤ} {b : ℕ} [NeZero b] : J(a | b) = 0 
       -- been deprecated so we replace them with `and_assoc` and `and_comm`
       simp_rw [legendreSym.eq_zero_iff _ _, intCast_zmod_eq_zero_iff_dvd,
         mem_primeFactorsList (NeZero.ne b), ← Int.natCast_dvd, Int.natCast_dvd_natCast, exists_prop,
-        and_assoc, and_comm])
+        _root_.and_assoc, _root_.and_comm])
 
 /-- The symbol `J(a | b)` is nonzero when `a` and `b` are coprime. -/
 protected theorem ne_zero {a : ℤ} {b : ℕ} (h : a.gcd b = 1) : J(a | b) ≠ 0 := by
@@ -214,7 +218,7 @@ theorem sq_one' {a : ℤ} {b : ℕ} (h : a.gcd b = 1) : J(a ^ 2 | b) = 1 := by r
 /-- The symbol `J(a | b)` depends only on `a` mod `b`. -/
 theorem mod_left (a : ℤ) (b : ℕ) : J(a | b) = J(a % b | b) :=
   congr_arg List.prod <|
-    List.pmap_congr _
+    List.pmap_congr_left _
       (by
         -- Porting note: Lean does not synthesize the instance [Fact (Nat.Prime p)] automatically
         -- (it is needed for `legendreSym.mod` on line 227). Thus, we name the hypothesis
@@ -309,7 +313,7 @@ theorem value_at (a : ℤ) {R : Type*} [CommSemiring R] (χ : R →* ℤ)
   conv_rhs => rw [← prod_primeFactorsList hb.pos.ne', cast_list_prod, map_list_prod χ]
   rw [jacobiSym, List.map_map, ← List.pmap_eq_map Nat.Prime _ _
     fun _ => prime_of_mem_primeFactorsList]
-  congr 1; apply List.pmap_congr
+  congr 1; apply List.pmap_congr_left
   exact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| dvd_of_mem_primeFactorsList h)
 
 /-- If `b` is odd, then `J(-1 | b)` is given by `χ₄ b`. -/
@@ -529,7 +533,7 @@ decreasing_by
 private theorem fastJacobiSymAux.eq_jacobiSym {a b : ℕ} {flip : Bool} {ha0 : a > 0}
     (hb2 : b % 2 = 1) (hb1 : b > 1) :
     fastJacobiSymAux a b flip ha0 = if flip then -J(a | b) else J(a | b) := by
-  induction' a using Nat.strongInductionOn with a IH generalizing b flip
+  induction' a using Nat.strongRecOn with a IH generalizing b flip
   unfold fastJacobiSymAux
   split <;> rename_i ha4
   · rw [IH (a / 4) (a.div_lt_self ha0 (by decide)) hb2 hb1]
@@ -569,7 +573,7 @@ private def fastJacobiSym (a : ℤ) (b : ℕ) : ℤ :=
 
 @[csimp] private theorem fastJacobiSym.eq : jacobiSym = fastJacobiSym := by
   ext a b
-  induction' b using Nat.strongInductionOn with b IH
+  induction' b using Nat.strongRecOn with b IH
   unfold fastJacobiSym
   split_ifs with hb0 hb2 ha2 hb1 hab
   · rw [hb0, zero_right]
diff --git a/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean b/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
index b9a2c36a497c9..9ba104ca9690a 100644
--- a/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
+++ b/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean
@@ -64,7 +64,7 @@ theorem quadraticCharFun_eq_zero_iff {a : F} : quadraticCharFun F a = 0 ↔ a =
   simp only [quadraticCharFun]
   by_cases ha : a = 0
   · simp only [ha, if_true]
-  · simp only [ha, if_false, iff_false_iff]
+  · simp only [ha, if_false]
     split_ifs <;> simp only [neg_eq_zero, one_ne_zero, not_false_iff]
 
 @[simp]
@@ -137,7 +137,7 @@ theorem quadraticChar_zero : quadraticChar F 0 = 0 := by
 theorem quadraticChar_one_iff_isSquare {a : F} (ha : a ≠ 0) :
     quadraticChar F a = 1 ↔ IsSquare a := by
   simp only [quadraticChar_apply, quadraticCharFun, ha, if_false, ite_eq_left_iff,
-    (by omega : (-1 : ℤ) ≠ 1), imp_false, not_not]
+    (by omega : (-1 : ℤ) ≠ 1), imp_false, not_not, reduceCtorEq]
 
 /-- The quadratic character takes the value `1` on nonzero squares. -/
 theorem quadraticChar_sq_one' {a : F} (ha : a ≠ 0) : quadraticChar F (a ^ 2) = 1 := by
@@ -213,7 +213,7 @@ when the domain has odd characteristic. -/
 theorem quadraticChar_ne_one (hF : ringChar F ≠ 2) : quadraticChar F ≠ 1 := by
   rcases quadraticChar_exists_neg_one' hF with ⟨a, ha⟩
   intro hχ
-  simp only [hχ, one_apply a.isUnit, one_ne_zero] at ha
+  simp only [hχ, one_apply a.isUnit, one_ne_zero, reduceCtorEq] at ha
 
 set_option linter.deprecated false in
 @[deprecated quadraticChar_ne_one (since := "2024-06-16")]
@@ -247,7 +247,7 @@ theorem quadraticChar_card_sqrts (hF : ringChar F ≠ 2) (a : F) :
       ext1
       -- Porting note(https://github.com/leanprover-community/mathlib4/issues/5026):
       -- added (Set.mem_toFinset), Set.mem_setOf
-      simp only [(Set.mem_toFinset), Set.mem_setOf, not_mem_empty, iff_false_iff]
+      simp only [(Set.mem_toFinset), Set.mem_setOf, not_mem_empty, iff_false]
       rw [isSquare_iff_exists_sq] at h
       exact fun h' ↦ h ⟨_, h'.symm⟩
 
@@ -284,7 +284,7 @@ theorem quadraticChar_neg_one [DecidableEq F] (hF : ringChar F ≠ 2) :
 theorem FiniteField.isSquare_neg_one_iff : IsSquare (-1 : F) ↔ Fintype.card F % 4 ≠ 3 := by
   classical -- suggested by the linter (instead of `[DecidableEq F]`)
   by_cases hF : ringChar F = 2
-  · simp only [FiniteField.isSquare_of_char_two hF, Ne, true_iff_iff]
+  · simp only [FiniteField.isSquare_of_char_two hF, Ne, true_iff]
     exact fun hf ↦
       one_ne_zero <|
         (Nat.odd_of_mod_four_eq_three hf).symm.trans <| FiniteField.even_card_of_char_two hF
@@ -292,7 +292,7 @@ theorem FiniteField.isSquare_neg_one_iff : IsSquare (-1 : F) ↔ Fintype.card F
     rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (one_ne_zero' F)),
       quadraticChar_neg_one hF, χ₄_nat_eq_if_mod_four, h₁]
     simp only [Nat.one_ne_zero, if_false, ite_eq_left_iff, (by omega : (-1 : ℤ) ≠ 1), imp_false,
-      not_not, Ne]
+      not_not, Ne, reduceCtorEq]
     exact ⟨fun h ↦ ne_of_eq_of_ne h (by omega), (Nat.odd_mod_four_iff.mp h₁).resolve_right⟩
 
 end SpecialValues
diff --git a/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean b/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
index 15a271ae8ab8f..b10a0d651815b 100644
--- a/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
+++ b/Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean
@@ -41,7 +41,7 @@ theorem FiniteField.isSquare_two_iff :
   classical
   by_cases hF : ringChar F = 2
   · have h := FiniteField.even_card_of_char_two hF
-    simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]
+    simp only [FiniteField.isSquare_of_char_two hF, true_iff]
     omega
   · have h := FiniteField.odd_card_of_char_ne_two hF
     rw [← quadraticChar_one_iff_isSquare (Ring.two_ne_zero hF), quadraticChar_two hF,
@@ -60,7 +60,7 @@ theorem FiniteField.isSquare_neg_two_iff :
   classical
   by_cases hF : ringChar F = 2
   · have h := FiniteField.even_card_of_char_two hF
-    simp only [FiniteField.isSquare_of_char_two hF, true_iff_iff]
+    simp only [FiniteField.isSquare_of_char_two hF, true_iff]
     omega
   · have h := FiniteField.odd_card_of_char_ne_two hF
     rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)),
@@ -102,7 +102,7 @@ theorem FiniteField.isSquare_odd_prime_iff (hF : ringChar F ≠ 2) {p : ℕ} [Fa
   classical
   by_cases hFp : ringChar F = p
   · rw [show (p : F) = 0 by rw [← hFp]; exact ringChar.Nat.cast_ringChar]
-    simp only [isSquare_zero, Ne, true_iff_iff, map_mul]
+    simp only [isSquare_zero, Ne, true_iff, map_mul]
     obtain ⟨n, _, hc⟩ := FiniteField.card F (ringChar F)
     have hchar : ringChar F = ringChar (ZMod p) := by rw [hFp]; exact (ringChar_zmod_n p).symm
     conv => enter [1, 1, 2]; rw [hc, Nat.cast_pow, map_pow, hchar, map_ringChar]
diff --git a/Mathlib/NumberTheory/Liouville/Basic.lean b/Mathlib/NumberTheory/Liouville/Basic.lean
index a9e820e8d2cad..04403810047a7 100644
--- a/Mathlib/NumberTheory/Liouville/Basic.lean
+++ b/Mathlib/NumberTheory/Liouville/Basic.lean
@@ -143,7 +143,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ |fR.derivative.eval xm| ?_ z0
       (fun y hy => ?_) fun z a hq => ?_
   -- 1: the denominators are positive -- essentially by definition;
-  · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _
+  · exact fun a => one_le_pow₀ ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg)
   -- 2: the polynomial `fR` is Lipschitz at `α` -- as its derivative continuous;
   · rw [mul_comm]
     rw [Real.closedBall_eq_Icc] at hy
@@ -191,11 +191,11 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   -- recall, this is a proof by contradiction!
   refine lt_irrefl ((b : ℝ) ^ f.natDegree * |x - ↑a / ↑b|) ?_
   -- clear denominators at `a1`
-  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
+  rw [lt_div_iff₀' (pow_pos b0 _), pow_add, mul_assoc] at a1
   -- split the inequality via `1 / A`.
   refine (?_ : (b : ℝ) ^ f.natDegree * |x - a / b| < 1 / A).trans_le ?_
   -- This branch of the proof uses the Liouville condition and the Archimedean property
-  · refine (lt_div_iff' hA).mpr ?_
+  · refine (lt_div_iff₀' hA).mpr ?_
     refine lt_of_le_of_lt ?_ a1
     gcongr
     refine hn.le.trans ?_
diff --git a/Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean b/Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean
index 2a8dad8fad8f5..49a8325713112 100644
--- a/Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean
+++ b/Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean
@@ -145,7 +145,7 @@ theorem aux_calc (n : ℕ) {m : ℝ} (hm : 2 ≤ m) :
       any_goals exact pow_pos (zero_lt_two.trans_le hm) _
       -- `2 ≤ m ^ n!` is a consequence of monotonicity of exponentiation at `2 ≤ m`.
       exact _root_.trans (_root_.trans hm (pow_one _).symm.le)
-        (pow_right_mono (one_le_two.trans hm) n.factorial_pos)
+        (pow_right_mono₀ (one_le_two.trans hm) n.factorial_pos)
     _ = 1 / (m ^ n !) ^ n := congr_arg (1 / ·) (pow_mul m n ! n)
 
 /-- An upper estimate on the remainder. This estimate works with `m ∈ ℝ` satisfying `2 ≤ m` and is
@@ -183,7 +183,7 @@ theorem liouville_liouvilleNumber {m : ℕ} (hm : 2 ≤ m) : Liouville (liouvill
   intro n
   -- the first `n` terms sum to `p / m ^ k!`
   rcases partialSum_eq_rat (zero_lt_two.trans_le hm) n with ⟨p, hp⟩
-  refine ⟨p, m ^ n !, one_lt_pow mZ1 n.factorial_ne_zero, ?_⟩
+  refine ⟨p, m ^ n !, one_lt_pow₀ mZ1 n.factorial_ne_zero, ?_⟩
   push_cast
   rw [Nat.cast_pow] at hp
   -- separate out the sum of the first `n` terms and the rest
diff --git a/Mathlib/NumberTheory/Liouville/LiouvilleWith.lean b/Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
index ac2e9e02f07f7..26c83fc23c3a5 100644
--- a/Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
+++ b/Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
@@ -53,7 +53,7 @@ theorem liouvilleWith_one (x : ℝ) : LiouvilleWith 1 x := by
   refine ((eventually_gt_atTop 0).mono fun n hn => ?_).frequently
   have hn' : (0 : ℝ) < n := by simpa
   have : x < ↑(⌊x * ↑n⌋ + 1) / ↑n := by
-    rw [lt_div_iff hn', Int.cast_add, Int.cast_one]
+    rw [lt_div_iff₀ hn', Int.cast_add, Int.cast_one]
     exact Int.lt_floor_add_one _
   refine ⟨⌊x * n⌋ + 1, this.ne, ?_⟩
   rw [abs_sub_comm, abs_of_pos (sub_pos.2 this), rpow_one, sub_lt_iff_lt_add',
@@ -100,10 +100,10 @@ theorem frequently_lt_rpow_neg (h : LiouvilleWith p x) (hlt : q < p) :
   refine (this.and_frequently hC).mono ?_
   rintro n ⟨hnC, hn, m, hne, hlt⟩
   replace hn : (0 : ℝ) < n := Nat.cast_pos.2 hn
-  refine ⟨m, hne, hlt.trans <| (div_lt_iff <| rpow_pos_of_pos hn _).2 ?_⟩
+  refine ⟨m, hne, hlt.trans <| (div_lt_iff₀ <| rpow_pos_of_pos hn _).2 ?_⟩
   rwa [mul_comm, ← rpow_add hn, ← sub_eq_add_neg]
 
-/-- The product of a Liouville number and a nonzero rational number is again a Liouville number.  -/
+/-- The product of a Liouville number and a nonzero rational number is again a Liouville number. -/
 theorem mul_rat (h : LiouvilleWith p x) (hr : r ≠ 0) : LiouvilleWith p (x * r) := by
   rcases h.exists_pos with ⟨C, _hC₀, hC⟩
   refine ⟨r.den ^ p * (|r| * C), (tendsto_id.nsmul_atTop r.pos).frequently (hC.mono ?_)⟩
diff --git a/Mathlib/NumberTheory/Liouville/Measure.lean b/Mathlib/NumberTheory/Liouville/Measure.lean
index c6bd2d0a641c4..618b456b25505 100644
--- a/Mathlib/NumberTheory/Liouville/Measure.lean
+++ b/Mathlib/NumberTheory/Liouville/Measure.lean
@@ -97,7 +97,7 @@ theorem volume_iUnion_setOf_liouvilleWith :
       _ = _ := by
         have : 1 - r ≠ 0 := by linarith
         rw [ENNReal.coe_inj]
-        simp [add_mul, div_eq_mul_inv, NNReal.rpow_neg, NNReal.rpow_sub' _ this, mul_add,
+        simp [add_mul, div_eq_mul_inv, NNReal.rpow_neg, NNReal.rpow_sub' this, mul_add,
           mul_left_comm]
   refine ne_top_of_le_ne_top (ENNReal.tsum_coe_ne_top_iff_summable.2 ?_) (ENNReal.tsum_le_tsum this)
   refine (Summable.add ?_ ?_).mul_left _ <;> simp only [NNReal.summable_rpow] <;> linarith
diff --git a/Mathlib/NumberTheory/Liouville/Residual.lean b/Mathlib/NumberTheory/Liouville/Residual.lean
index 4d72a848d850a..38feb5335e927 100644
--- a/Mathlib/NumberTheory/Liouville/Residual.lean
+++ b/Mathlib/NumberTheory/Liouville/Residual.lean
@@ -53,7 +53,7 @@ theorem setOf_liouville_eq_irrational_inter_iInter_iUnion :
 theorem eventually_residual_liouville : ∀ᶠ x in residual ℝ, Liouville x := by
   rw [Filter.Eventually, setOf_liouville_eq_irrational_inter_iInter_iUnion]
   refine eventually_residual_irrational.and ?_
-  refine residual_of_dense_Gδ ?_ (Rat.denseEmbedding_coe_real.dense.mono ?_)
+  refine residual_of_dense_Gδ ?_ (Rat.isDenseEmbedding_coe_real.dense.mono ?_)
   · exact .iInter fun n => IsOpen.isGδ <|
           isOpen_iUnion fun a => isOpen_iUnion fun b => isOpen_iUnion fun _hb => isOpen_ball
   · rintro _ ⟨r, rfl⟩
diff --git a/Mathlib/NumberTheory/LucasLehmer.lean b/Mathlib/NumberTheory/LucasLehmer.lean
index b6ad0fed7b57e..fd4fbbd8441a4 100644
--- a/Mathlib/NumberTheory/LucasLehmer.lean
+++ b/Mathlib/NumberTheory/LucasLehmer.lean
@@ -1,16 +1,9 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Scott Morrison, Ainsley Pahljina
+Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
 -/
-import Mathlib.Algebra.Order.Ring.Abs
-import Mathlib.Algebra.Order.Ring.Basic
-import Mathlib.Algebra.Ring.Nat
-import Mathlib.Data.ZMod.Basic
-import Mathlib.GroupTheory.OrderOfElement
 import Mathlib.RingTheory.Fintype
-import Mathlib.Tactic.IntervalCases
-import Mathlib.Tactic.Zify
 
 /-!
 # The Lucas-Lehmer test for Mersenne primes.
@@ -31,7 +24,7 @@ primes using `lucas_lehmer_sufficiency`.
 ## History
 
 This development began as a student project by Ainsley Pahljina,
-and was then cleaned up for mathlib by Scott Morrison.
+and was then cleaned up for mathlib by Kim Morrison.
 The tactic for certified computation of Lucas-Lehmer residues was provided by Mario Carneiro.
 This tactic was ported by Thomas Murrills to Lean 4, and then it was converted to a `norm_num`
 extension and made to use kernel reductions by Kyle Miller.
@@ -62,7 +55,7 @@ theorem mersenne_le_mersenne {p q : ℕ} : mersenne p ≤ mersenne q ↔ p ≤ q
 @[simp] lemma mersenne_odd : ∀ {p : ℕ}, Odd (mersenne p) ↔ p ≠ 0
   | 0 => by simp
   | p + 1 => by
-    simpa using Nat.Even.sub_odd (one_le_pow_of_one_le one_le_two _)
+    simpa using Nat.Even.sub_odd (one_le_pow₀ one_le_two)
       (even_two.pow_of_ne_zero p.succ_ne_zero) odd_one
 
 @[simp] theorem mersenne_pos {p : ℕ} : 0 < mersenne p ↔ 0 < p := mersenne_lt_mersenne (p := 0)
@@ -94,7 +87,7 @@ theorem one_lt_mersenne {p : ℕ} : 1 < mersenne p ↔ 1 < p :=
 @[simp]
 theorem succ_mersenne (k : ℕ) : mersenne k + 1 = 2 ^ k := by
   rw [mersenne, tsub_add_cancel_of_le]
-  exact one_le_pow_of_one_le (by norm_num) k
+  exact one_le_pow₀ (by norm_num)
 
 namespace LucasLehmer
 
@@ -456,7 +449,8 @@ theorem order_ω (p' : ℕ) (h : lucasLehmerResidue (p' + 2) = 0) :
     have ω_pow := orderOf_dvd_iff_pow_eq_one.1 o
     replace ω_pow :=
       congr_arg (Units.coeHom (X (q (p' + 2))) : Units (X (q (p' + 2))) → X (q (p' + 2))) ω_pow
-    simp? at ω_pow says simp only [map_pow, Units.coeHom_apply, ωUnit_coe, map_one] at ω_pow
+    simp? at ω_pow says
+      simp only [Units.coeHom_apply, Units.val_pow_eq_pow_val, ωUnit_coe, Units.val_one] at ω_pow
     have h : (1 : ZMod (q (p' + 2))) = -1 :=
       congr_arg Prod.fst (ω_pow.symm.trans (ω_pow_eq_neg_one p' h))
     haveI : Fact (2 < (q (p' + 2) : ℕ)) := ⟨two_lt_q _⟩
diff --git a/Mathlib/NumberTheory/LucasPrimality.lean b/Mathlib/NumberTheory/LucasPrimality.lean
index ac590aca60046..2f2f71e0b2bae 100644
--- a/Mathlib/NumberTheory/LucasPrimality.lean
+++ b/Mathlib/NumberTheory/LucasPrimality.lean
@@ -7,9 +7,10 @@ import Mathlib.Data.Fintype.Basic
 import Mathlib.GroupTheory.OrderOfElement
 import Mathlib.Tactic.Zify
 import Mathlib.Data.Nat.Totient
+import Mathlib.RingTheory.IntegralDomain
 
 /-!
-# The Lucas test for primes.
+# The Lucas test for primes
 
 This file implements the Lucas test for primes (not to be confused with the Lucas-Lehmer test for
 Mersenne primes). A number `a` witnesses that `n` is prime if `a` has order `n-1` in the
@@ -18,16 +19,13 @@ and `a^d ≠ 1 (mod n)` for any divisor `d | n - 1`. This test is the basis of t
 certificate.
 
 ## TODO
-
-- Bonus: Show the reverse implication i.e. if a number is prime then it has a Lucas witness.
-  Use `Units.IsCyclic` from `RingTheory/IntegralDomain` to show the group is cyclic.
 - Write a tactic that uses this theorem to generate Pratt primality certificates
 - Integrate Pratt primality certificates into the norm_num primality verifier
 
 ## Implementation notes
 
 Note that the proof for `lucas_primality` relies on analyzing the multiplicative group
-modulo `p`. Despite this, the theorem still holds vacuously for `p = 0` and `p = 1`: In these
+modulo `p`. Despite this, the theorem still holds vacuously for `p = 0` and `p = 1`. In these
 cases, we can take `q` to be any prime and see that `hd` does not hold, since `a^((p-1)/q)` reduces
 to `1`.
 -/
@@ -39,23 +37,37 @@ group must itself have order `p-1`, which only happens when `p` is prime.
 -/
 theorem lucas_primality (p : ℕ) (a : ZMod p) (ha : a ^ (p - 1) = 1)
     (hd : ∀ q : ℕ, q.Prime → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1) : p.Prime := by
-  have h0 : p ≠ 0 := by
-    rintro ⟨⟩
-    exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
-  have h1 : p ≠ 1 := by
-    rintro ⟨⟩
-    exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
-  have hp1 : 1 < p := lt_of_le_of_ne h0.bot_lt h1.symm
-  have order_of_a : orderOf a = p - 1 := by
-    apply orderOf_eq_of_pow_and_pow_div_prime _ ha hd
-    exact tsub_pos_of_lt hp1
-  haveI : NeZero p := ⟨h0⟩
+  have h : p ≠ 0 ∧ p ≠ 1 := by
+    constructor <;> rintro rfl <;> exact hd 2 Nat.prime_two (dvd_zero _) (pow_zero _)
+  have hp1 : 1 < p := Nat.one_lt_iff_ne_zero_and_ne_one.2 h
+  have : NeZero p := ⟨h.1⟩
   rw [Nat.prime_iff_card_units]
-  -- Prove cardinality of `Units` of `ZMod p` is both `≤ p-1` and `≥ p-1`
-  refine le_antisymm (Nat.card_units_zmod_lt_sub_one hp1) ?_
-  have hp' : p - 2 + 1 = p - 1 := tsub_add_eq_add_tsub hp1
-  let a' : (ZMod p)ˣ := Units.mkOfMulEqOne a (a ^ (p - 2)) (by rw [← pow_succ', hp', ha])
-  calc
-    p - 1 = orderOf a := order_of_a.symm
-    _ = orderOf a' := (orderOf_injective (Units.coeHom (ZMod p)) Units.ext a')
+  apply (Nat.card_units_zmod_lt_sub_one hp1).antisymm
+  let a' : (ZMod p)ˣ := Units.mkOfMulEqOne a _ (by rwa [← pow_succ', tsub_add_eq_add_tsub hp1])
+  calc p - 1 = orderOf a := (orderOf_eq_of_pow_and_pow_div_prime (tsub_pos_of_lt hp1) ha hd).symm
+    _ = orderOf a' := orderOf_injective (Units.coeHom _) Units.ext a'
     _ ≤ Fintype.card (ZMod p)ˣ := orderOf_le_card_univ
+
+/-- If `p` is prime, then there exists an `a` such that `a^(p-1) = 1 mod p`
+and `a^((p-1)/q) ≠ 1 mod p` for all prime factors `q` of `p-1`.
+The multiplicative group mod `p` is cyclic, so `a` can be any generator of the group
+(which must have order `p-1`).
+-/
+theorem reverse_lucas_primality (p : ℕ) (hP : p.Prime) :
+    ∃ a : ZMod p, a ^ (p - 1) = 1 ∧ ∀ q : ℕ, q.Prime → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1 := by
+  have : Fact p.Prime := ⟨hP⟩
+  obtain ⟨g, hg⟩ := IsCyclic.exists_generator (α := (ZMod p)ˣ)
+  have h1 : orderOf g = p - 1 := by
+    rwa [orderOf_eq_card_of_forall_mem_zpowers hg, ← Nat.prime_iff_card_units]
+  have h2 := tsub_pos_iff_lt.2 hP.one_lt
+  rw [← orderOf_injective (Units.coeHom _) Units.ext _, orderOf_eq_iff h2] at h1
+  refine ⟨g, h1.1, fun q hq hqd ↦ ?_⟩
+  replace hq := hq.one_lt
+  exact h1.2 _ (Nat.div_lt_self h2 hq) (Nat.div_pos (Nat.le_of_dvd h2 hqd) (zero_lt_one.trans hq))
+
+/-- A number `p` is prime if and only if there exists an `a` such that
+`a^(p-1) = 1 mod p` and `a^((p-1)/q) ≠ 1 mod p` for all prime factors `q` of `p-1`.
+-/
+theorem lucas_primality_iff (p : ℕ) : p.Prime ↔
+    ∃ a : ZMod p, a ^ (p - 1) = 1 ∧ ∀ q : ℕ, q.Prime → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1 :=
+  ⟨reverse_lucas_primality p, fun ⟨a, ⟨ha, hb⟩⟩ ↦ lucas_primality p a ha hb⟩
diff --git a/Mathlib/NumberTheory/Modular.lean b/Mathlib/NumberTheory/Modular.lean
index e14a58682b9f1..d27a687df5826 100644
--- a/Mathlib/NumberTheory/Modular.lean
+++ b/Mathlib/NumberTheory/Modular.lean
@@ -289,7 +289,7 @@ theorem exists_max_im : ∃ g : SL(2, ℤ), ∀ g' : SL(2, ℤ), (g' • z).im 
   · exact normSq_denom_pos g z
 
 /-- Given `z : ℍ` and a bottom row `(c,d)`, among the `g : SL(2,ℤ)` with this bottom row, minimize
-  `|(g•z).re|`.  -/
+  `|(g•z).re|`. -/
 theorem exists_row_one_eq_and_min_re {cd : Fin 2 → ℤ} (hcd : IsCoprime (cd 0) (cd 1)) :
     ∃ g : SL(2, ℤ), (↑ₘg) 1 = cd ∧ ∀ g' : SL(2, ℤ), (↑ₘg) 1 = (↑ₘg') 1 →
       |(g • z).re| ≤ |(g' • z).re| := by
@@ -351,13 +351,13 @@ theorem g_eq_of_c_eq_one (hc : (↑ₘg) 1 0 = 1) : g = T ^ (↑ₘg) 0 0 * S *
 
 /-- If `1 < |z|`, then `|S • z| < 1`. -/
 theorem normSq_S_smul_lt_one (h : 1 < normSq z) : normSq ↑(S • z) < 1 := by
-  simpa [coe_S, num, denom] using (inv_lt_inv z.normSq_pos zero_lt_one).mpr h
+  simpa [coe_S, num, denom] using (inv_lt_inv₀ z.normSq_pos zero_lt_one).mpr h
 
 /-- If `|z| < 1`, then applying `S` strictly decreases `im`. -/
 theorem im_lt_im_S_smul (h : normSq z < 1) : z.im < (S • z).im := by
   have : z.im < z.im / normSq (z : ℂ) := by
     have imz : 0 < z.im := im_pos z
-    apply (lt_div_iff z.normSq_pos).mpr
+    apply (lt_div_iff₀ z.normSq_pos).mpr
     nlinarith
   convert this
   simp only [ModularGroup.im_smul_eq_div_normSq]
@@ -380,7 +380,7 @@ scoped[Modular] notation "𝒟ᵒ" => ModularGroup.fdo
 open scoped Modular
 
 theorem abs_two_mul_re_lt_one_of_mem_fdo (h : z ∈ 𝒟ᵒ) : |2 * z.re| < 1 := by
-  rw [abs_mul, abs_two, ← lt_div_iff' (zero_lt_two' ℝ)]
+  rw [abs_mul, abs_two, ← lt_div_iff₀' (zero_lt_two' ℝ)]
   exact h.2
 
 theorem three_lt_four_mul_im_sq_of_mem_fdo (h : z ∈ 𝒟ᵒ) : 3 < 4 * z.im ^ 2 := by
@@ -465,7 +465,7 @@ theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |(↑ₘg
         (by linarith) (by linarith))
       hc
   have h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1 :=
-    div_le_one_of_le
+    div_le_one_of_le₀
       (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))
       (sq_nonneg _)
   let nsq := normSq (denom g z)
diff --git a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean
index 91bae3555b311..ce4577936eb32 100644
--- a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean
+++ b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/Defs.lean
@@ -64,7 +64,7 @@ lemma vecMul_SL2_mem_gammaSet {v : Fin 2 → ℤ} (hv : v ∈ gammaSet N a) (γ
   refine ⟨?_, hv.2.vecMulSL γ⟩
   have := RingHom.map_vecMul (m := Fin 2) (n := Fin 2) (Int.castRingHom (ZMod N)) γ v
   simp only [eq_intCast, Int.coe_castRingHom] at this
-  simp_rw [Function.comp, this, hv.1]
+  simp_rw [Function.comp_def, this, hv.1]
   simp
 
 variable (a) in
diff --git a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean
index 0b6f47e0894a7..7268d68dd4275 100644
--- a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean
+++ b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/UniformConvergence.lean
@@ -56,7 +56,7 @@ lemma r1_pos : 0 < r1 z := by
   dsimp only [r1]
   positivity
 
-/-- For `c, d ∈ ℝ` with `1 ≤ d ^ 2`, we have `r1 z ≤ |c * z + d| ^ 2`.  -/
+/-- For `c, d ∈ ℝ` with `1 ≤ d ^ 2`, we have `r1 z ≤ |c * z + d| ^ 2`. -/
 lemma r1_aux_bound (c : ℝ) {d : ℝ} (hd : 1 ≤ d ^ 2) :
     r1 z ≤ (c * z.re + d) ^ 2 + (c * z.im) ^ 2 := by
   have H1 : (c * z.re + d) ^ 2 + (c * z.im) ^ 2 =
@@ -78,7 +78,7 @@ lemma r_lower_bound_on_verticalStrip {A B : ℝ} (h : 0 < B) (hz : z ∈ vertica
   apply min_le_min hz.2
   rw [Real.sqrt_le_sqrt_iff (by apply (r1_pos z).le)]
   simp only [r1_eq, div_pow, one_div]
-  rw [inv_le_inv (by positivity) (by positivity), add_le_add_iff_right]
+  rw [inv_le_inv₀ (by positivity) (by positivity), add_le_add_iff_right]
   apply div_le_div (sq_nonneg _) _ (by positivity) (pow_le_pow_left h.le hz.2 2)
   simpa only [even_two.pow_abs] using pow_le_pow_left (abs_nonneg _) hz.1 2
 
diff --git a/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean b/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
index f1931f2f1bb78..5af4baf5ed34a 100644
--- a/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
+++ b/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
@@ -7,7 +7,7 @@ Authors: David Loeffler
 import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
 
 /-!
-# Asymptotic bounds for Jacobi theta functions
+# Asymptotic bounds for Jacobi theta functions
 
 The goal of this file is to establish some technical lemmas about the asymptotics of the sums
 
diff --git a/Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean b/Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
index 1740046e80dae..fb58da50e6f50 100644
--- a/Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
+++ b/Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
@@ -285,7 +285,7 @@ lemma jacobiTheta₂'_undef (z : ℂ) {τ : ℂ} (hτ : im τ ≤ 0) : jacobiThe
   exact not_lt.mpr hτ
 
 /-!
-## Derivatives and continuity
+## Derivatives and continuity
 -/
 
 lemma hasFDerivAt_jacobiTheta₂ (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
@@ -373,7 +373,7 @@ lemma continuousAt_jacobiTheta₂' (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
   · exact norm_jacobiTheta₂'_term_le hT (le_of_lt hz') (le_of_lt hτ') n
 
 /-!
-## Periodicity and conjugation
+## Periodicity and conjugation
 -/
 
 /-- The two-variable Jacobi theta function is periodic in `τ` with period 2. -/
diff --git a/Mathlib/NumberTheory/MulChar/Basic.lean b/Mathlib/NumberTheory/MulChar/Basic.lean
index f9a5760d1bcda..e4ed0932f9876 100644
--- a/Mathlib/NumberTheory/MulChar/Basic.lean
+++ b/Mathlib/NumberTheory/MulChar/Basic.lean
@@ -159,14 +159,14 @@ noncomputable def ofUnitHom (f : Rˣ →* R'ˣ) : MulChar R R' where
     classical
       intro x y
       by_cases hx : IsUnit x
-      · simp only [hx, IsUnit.mul_iff, true_and_iff, dif_pos]
+      · simp only [hx, IsUnit.mul_iff, true_and, dif_pos]
         by_cases hy : IsUnit y
         · simp only [hy, dif_pos]
           have hm : (IsUnit.mul_iff.mpr ⟨hx, hy⟩).unit = hx.unit * hy.unit := Units.eq_iff.mp rfl
           rw [hm, map_mul]
           norm_cast
         · simp only [hy, not_false_iff, dif_neg, mul_zero]
-      · simp only [hx, IsUnit.mul_iff, false_and_iff, not_false_iff, dif_neg, zero_mul]
+      · simp only [hx, IsUnit.mul_iff, false_and, not_false_iff, dif_neg, zero_mul]
   map_nonunit' := by
     intro a ha
     simp only [ha, not_false_iff, dif_neg]
@@ -454,7 +454,7 @@ lemma injective_ringHomComp {f : R' →+* R''} (hf : Function.Injective f) :
 
 lemma ringHomComp_eq_one_iff {f : R' →+* R''} (hf : Function.Injective f) {χ : MulChar R R'} :
     χ.ringHomComp f = 1 ↔ χ = 1 := by
-  conv_lhs => rw [← (show  (1 : MulChar R R').ringHomComp f = 1 by ext; simp)]
+  conv_lhs => rw [← (show (1 : MulChar R R').ringHomComp f = 1 by ext; simp)]
   exact (injective_ringHomComp hf).eq_iff
 
 lemma ringHomComp_ne_one_iff {f : R' →+* R''} (hf : Function.Injective f) {χ : MulChar R R'} :
@@ -577,7 +577,7 @@ theorem sum_one_eq_card_units [DecidableEq R] :
     · exact map_nonunit _ h
   · congr
     ext a
-    simp only [Finset.mem_filter, Finset.mem_univ, true_and_iff, Finset.mem_map,
+    simp only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_map,
       Function.Embedding.coeFn_mk, exists_true_left, IsUnit]
 
 end sum
diff --git a/Mathlib/NumberTheory/MulChar/Lemmas.lean b/Mathlib/NumberTheory/MulChar/Lemmas.lean
index b0e6da9769998..d4ac05321aeb8 100644
--- a/Mathlib/NumberTheory/MulChar/Lemmas.lean
+++ b/Mathlib/NumberTheory/MulChar/Lemmas.lean
@@ -111,7 +111,7 @@ noncomputable def equiv_rootsOfUnity [inst_cyc : IsCyclic Mˣ] :
     ext
     simp only [toUnitHom_eq, coe_equivToUnitHom, ofRootOfUnity_spec]
   map_mul' x y := by
-    simp only [toUnitHom_eq, equivToUnitHom_mul_apply, Submonoid.mk_mul_mk]
+    simp only [toUnitHom_eq, equivToUnitHom_mul_apply, MulMemClass.mk_mul_mk]
 
 end IsCyclic
 
diff --git a/Mathlib/NumberTheory/Multiplicity.lean b/Mathlib/NumberTheory/Multiplicity.lean
index dd1a44061a628..43de4a819a527 100644
--- a/Mathlib/NumberTheory/Multiplicity.lean
+++ b/Mathlib/NumberTheory/Multiplicity.lean
@@ -389,7 +389,7 @@ theorem pow_add_pow (hxy : p ∣ x + y) (hx : ¬p ∣ x) {n : ℕ} (hn : Odd n)
   iterate 3 rw [padicValNat_def, PartENat.natCast_get]
   · exact multiplicity.Nat.pow_add_pow hp.out hp1 hxy hx hn
   · exact Odd.pos hn
-  · simp only [add_pos_iff, Nat.succ_pos', or_true_iff]
+  · simp only [add_pos_iff, Nat.succ_pos', or_true]
   · exact Nat.lt_add_left _ (pow_pos y.succ_pos _)
 
 end padicValNat
diff --git a/Mathlib/NumberTheory/NumberField/Basic.lean b/Mathlib/NumberTheory/NumberField/Basic.lean
index e7031b91989cc..cd142c881c370 100644
--- a/Mathlib/NumberTheory/NumberField/Basic.lean
+++ b/Mathlib/NumberTheory/NumberField/Basic.lean
@@ -134,22 +134,55 @@ lemma mk_eq_mk (x y : K) (hx hy) : (⟨x, hx⟩ : 𝓞 K) = ⟨y, hy⟩ ↔ x =
 @[simp] lemma neg_mk (x : K) (hx) : (-⟨x, hx⟩ : 𝓞 K) = ⟨-x, neg_mem hx⟩ :=
   rfl
 
+/-- The ring homomorphism `(𝓞 K) →+* (𝓞 L)` given by restricting a ring homomorphism
+  `f : K →+* L` to `𝓞 K`. -/
+def mapRingHom {K L F : Type*} [Field K] [Field L] [FunLike F K L]
+    [RingHomClass F K L] (f : F) : (𝓞 K) →+* (𝓞 L) where
+  toFun k := ⟨f k.val, map_isIntegral_int f k.2⟩
+  map_zero' := by ext; simp only [map_mk, map_zero]
+  map_one' := by ext; simp only [map_mk, map_one]
+  map_add' x y:= by ext; simp only [map_mk, map_add]
+  map_mul' x y := by ext; simp only [map_mk, map_mul]
+
+/-- The ring isomorphsim `(𝓞 K) ≃+* (𝓞 L)` given by restricting
+  a ring isomorphsim `e : K ≃+* L` to `𝓞 K`. -/
+def mapRingEquiv {K L E : Type*} [Field K] [Field L] [EquivLike E K L]
+    [RingEquivClass E K L] (e : E) : (𝓞 K) ≃+* (𝓞 L) :=
+  RingEquiv.ofRingHom (mapRingHom e) (mapRingHom (e : K ≃+* L).symm)
+    (RingHom.ext fun x => ext (EquivLike.right_inv e x.1))
+      (RingHom.ext fun x => ext (EquivLike.left_inv e x.1))
+
 end RingOfIntegers
 
 /-- Given an algebra between two fields, create an algebra between their two rings of integers. -/
 instance inst_ringOfIntegersAlgebra [Algebra K L] : Algebra (𝓞 K) (𝓞 L) :=
-  RingHom.toAlgebra
-    { toFun := fun k => ⟨algebraMap K L (algebraMap _ K k), IsIntegral.algebraMap k.2⟩
-      map_zero' := by ext; simp only [RingOfIntegers.map_mk, map_zero]
-      map_one' := by ext; simp only [RingOfIntegers.map_mk, map_one]
-      map_add' := fun x y => by ext; simp only [RingOfIntegers.map_mk, map_add]
-      map_mul' := fun x y => by ext; simp only [RingOfIntegers.map_mk, map_mul] }
+  (RingOfIntegers.mapRingHom (algebraMap K L)).toAlgebra
 
 -- diamond at `reducible_and_instances` #10906
 example : Algebra.id (𝓞 K) = inst_ringOfIntegersAlgebra K K := rfl
 
 namespace RingOfIntegers
 
+/-- The algebra homomorphism `(𝓞 K) →ₐ[𝓞 k] (𝓞 L)` given by restricting a algebra homomorphism
+  `f : K →ₐ[k] L` to `𝓞 K`. -/
+def mapAlgHom {k K L F : Type*} [Field k] [Field K] [Field L] [Algebra k K]
+    [Algebra k L] [FunLike F K L] [AlgHomClass F k K L] (f : F) : (𝓞 K) →ₐ[𝓞 k] (𝓞 L) where
+  toRingHom := mapRingHom f
+  commutes' x := SetCoe.ext (AlgHomClass.commutes ((f : K →ₐ[k] L).restrictScalars (𝓞 k)) x)
+
+/-- The isomorphism of algebras `(𝓞 K) ≃ₐ[𝓞 k] (𝓞 L)` given by restricting
+  an isomorphism of algebras `e : K ≃ₐ[k] L` to `𝓞 K`. -/
+def mapAlgEquiv {k K L E : Type*} [Field k] [Field K] [Field L] [Algebra k K]
+    [Algebra k L] [EquivLike E K L] [AlgEquivClass E k K L] (e : E) : (𝓞 K) ≃ₐ[𝓞 k] (𝓞 L) :=
+  AlgEquiv.ofAlgHom (mapAlgHom e) (mapAlgHom (e : K ≃ₐ[k] L).symm)
+    (AlgHom.ext fun x => ext (EquivLike.right_inv e x.1))
+      (AlgHom.ext fun x => ext (EquivLike.left_inv e x.1))
+
+instance inst_isScalarTower (k K L : Type*) [Field k] [Field K] [Field L]
+    [Algebra k K] [Algebra k L] [Algebra K L] [IsScalarTower k K L] :
+    IsScalarTower (𝓞 k) (𝓞 K) (𝓞 L) :=
+  IsScalarTower.of_algHom (mapAlgHom (IsScalarTower.toAlgHom k K L))
+
 variable {K}
 
 /-- The canonical map from `𝓞 K` to `K` is injective.
@@ -164,7 +197,7 @@ lemma coe_injective : Function.Injective (algebraMap (𝓞 K) K) :=
 This is a convenient abbreviation for `map_eq_zero_iff` applied to
 `NoZeroSMulDivisors.algebraMap_injective`.
 -/
-@[simp] lemma coe_eq_zero_iff {x : 𝓞 K} : algebraMap _ K x = 0 ↔ x = 0 :=
+lemma coe_eq_zero_iff {x : 𝓞 K} : algebraMap _ K x = 0 ↔ x = 0 :=
   map_eq_zero_iff _ coe_injective
 
 /-- The canonical map from `𝓞 K` to `K` is injective.
@@ -274,7 +307,7 @@ theorem mem_span_integralBasis {x : K} :
   rw [integralBasis, Basis.localizationLocalization_span, LinearMap.mem_range,
       IsScalarTower.coe_toAlgHom', RingHom.mem_range]
 
-theorem RingOfIntegers.rank : FiniteDimensional.finrank ℤ (𝓞 K) = FiniteDimensional.finrank ℚ K :=
+theorem RingOfIntegers.rank : Module.finrank ℤ (𝓞 K) = Module.finrank ℚ K :=
   IsIntegralClosure.rank ℤ ℚ K (𝓞 K)
 
 end NumberField
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
index 5e0cd9a7ebdb7..c62e465bc30d5 100644
--- a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
@@ -3,9 +3,9 @@ Copyright (c) 2022 Xavier Roblot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Xavier Roblot
 -/
-import Mathlib.Algebra.Module.Zlattice.Basic
-import Mathlib.NumberTheory.NumberField.Embeddings
+import Mathlib.Algebra.Module.ZLattice.Basic
 import Mathlib.NumberTheory.NumberField.FractionalIdeal
+import Mathlib.NumberTheory.NumberField.Units.Basic
 
 /-!
 # Canonical embedding of a number field
@@ -24,11 +24,11 @@ sending `x : K` to the vector `(φ x)` indexed by `φ : K →+* ℂ`.
 image of the ring of integers by the canonical embedding and any ball centered at `0` of finite
 radius is finite.
 
-* `NumberField.mixedEmbedding`: the ring homomorphism from `K →+* ({ w // IsReal w } → ℝ) ×
-({ w // IsComplex w } → ℂ)` that sends `x ∈ K` to `(φ_w x)_w` where `φ_w` is the embedding
-associated to the infinite place `w`. In particular, if `w` is real then `φ_w : K →+* ℝ` and, if
-`w` is complex, `φ_w` is an arbitrary choice between the two complex embeddings defining the place
-`w`.
+* `NumberField.mixedEmbedding`: the ring homomorphism from `K` to the mixed space
+`K →+* ({ w // IsReal w } → ℝ) × ({ w // IsComplex w } → ℂ)` that sends `x ∈ K` to `(φ_w x)_w`
+where `φ_w` is the embedding associated to the infinite place `w`. In particular, if `w` is real
+then `φ_w : K →+* ℝ` and, if `w` is complex, `φ_w` is an arbitrary choice between the two complex
+embeddings defining the place `w`.
 
 ## Tags
 
@@ -39,8 +39,6 @@ variable (K : Type*) [Field K]
 
 namespace NumberField.canonicalEmbedding
 
---open NumberField
-
 /-- The canonical embedding of a number field `K` of degree `n` into `ℂ^n`. -/
 def _root_.NumberField.canonicalEmbedding : K →+* ((K →+* ℂ) → ℂ) := Pi.ringHom fun φ => φ
 
@@ -102,7 +100,7 @@ theorem integerLattice.inter_ball_finite [NumberField K] (r : ℝ) :
     · rintro ⟨x, ⟨hx1, hx2⟩, rfl⟩
       exact ⟨⟨x, ⟨⟨x, hx1⟩, rfl⟩, rfl⟩, (heq x).mpr hx2⟩
 
-open Module Fintype FiniteDimensional
+open Module Fintype Module
 
 /-- A `ℂ`-basis of `ℂ^n` that is also a `ℤ`-basis of the `integerLattice`. -/
 noncomputable def latticeBasis [NumberField K] :
@@ -178,18 +176,28 @@ end NumberField.canonicalEmbedding
 
 namespace NumberField.mixedEmbedding
 
-open NumberField.InfinitePlace FiniteDimensional Finset
+open NumberField.InfinitePlace Module Finset
 
-/-- The space `ℝ^r₁ × ℂ^r₂` with `(r₁, r₂)` the signature of `K`. -/
-local notation "E" K =>
+/-- The mixed space `ℝ^r₁ × ℂ^r₂` with `(r₁, r₂)` the signature of `K`. -/
+abbrev mixedSpace :=
   ({w : InfinitePlace K // IsReal w} → ℝ) × ({w : InfinitePlace K // IsComplex w} → ℂ)
 
-/-- The mixed embedding of a number field `K` of signature `(r₁, r₂)` into `ℝ^r₁ × ℂ^r₂`. -/
-noncomputable def _root_.NumberField.mixedEmbedding : K →+* (E K) :=
+/-- The mixed embedding of a number field `K` into the mixed space of `K`. -/
+noncomputable def _root_.NumberField.mixedEmbedding : K →+* (mixedSpace K) :=
   RingHom.prod (Pi.ringHom fun w => embedding_of_isReal w.prop)
     (Pi.ringHom fun w => w.val.embedding)
 
-instance [NumberField K] : Nontrivial (E K) := by
+@[simp]
+theorem mixedEmbedding_apply_ofIsReal (x : K) (w : {w // IsReal w}) :
+    (mixedEmbedding K x).1 w = embedding_of_isReal w.prop x := by
+  simp_rw [mixedEmbedding, RingHom.prod_apply, Pi.ringHom_apply]
+
+@[simp]
+theorem mixedEmbedding_apply_ofIsComplex (x : K) (w : {w // IsComplex w}) :
+    (mixedEmbedding K x).2 w = w.val.embedding x := by
+  simp_rw [mixedEmbedding, RingHom.prod_apply, Pi.ringHom_apply]
+
+instance [NumberField K] : Nontrivial (mixedSpace K) := by
   obtain ⟨w⟩ := (inferInstance : Nonempty (InfinitePlace K))
   obtain hw | hw := w.isReal_or_isComplex
   · have : Nonempty {w : InfinitePlace K // IsReal w} := ⟨⟨w, hw⟩⟩
@@ -197,7 +205,7 @@ instance [NumberField K] : Nontrivial (E K) := by
   · have : Nonempty {w : InfinitePlace K // IsComplex w} := ⟨⟨w, hw⟩⟩
     exact nontrivial_prod_right
 
-protected theorem finrank [NumberField K] : finrank ℝ (E K) = finrank ℚ K := by
+protected theorem finrank [NumberField K] : finrank ℝ (mixedSpace K) = finrank ℚ K := by
   classical
   rw [finrank_prod, finrank_pi, finrank_pi_fintype, Complex.finrank_real_complex, sum_const,
     card_univ, ← NrRealPlaces, ← NrComplexPlaces, ← card_real_embeddings, Algebra.id.smul_eq_mul,
@@ -212,7 +220,7 @@ section commMap
 
 /-- The linear map that makes `canonicalEmbedding` and `mixedEmbedding` commute, see
 `commMap_canonical_eq_mixed`. -/
-noncomputable def commMap : ((K →+* ℂ) → ℂ) →ₗ[ℝ] (E K) where
+noncomputable def commMap : ((K →+* ℂ) → ℂ) →ₗ[ℝ] (mixedSpace K) where
   toFun := fun x => ⟨fun w => (x w.val.embedding).re, fun w => x w.val.embedding⟩
   map_add' := by
     simp only [Pi.add_apply, Complex.add_re, Prod.mk_add_mk, Prod.mk.injEq]
@@ -236,7 +244,7 @@ theorem commMap_canonical_eq_mixed (x : K) :
   exact ⟨rfl, rfl⟩
 
 /-- This is a technical result to ensure that the image of the `ℂ`-basis of `ℂ^n` defined in
-`canonicalEmbedding.latticeBasis` is a `ℝ`-basis of `ℝ^r₁ × ℂ^r₂`,
+`canonicalEmbedding.latticeBasis` is a `ℝ`-basis of the mixed space `ℝ^r₁ × ℂ^r₂`,
 see `mixedEmbedding.latticeBasis`. -/
 theorem disjoint_span_commMap_ker [NumberField K] :
     Disjoint (Submodule.span ℝ (Set.range (canonicalEmbedding.latticeBasis K)))
@@ -270,30 +278,29 @@ open scoped Classical
 
 variable {K}
 
-/-- The norm at the infinite place `w` of an element of
-`({w // IsReal w} → ℝ) × ({ w // IsComplex w } → ℂ)`. -/
-def normAtPlace (w : InfinitePlace K) : (E K) →*₀ ℝ where
+/-- The norm at the infinite place `w` of an element of the mixed space. --/
+def normAtPlace (w : InfinitePlace K) : (mixedSpace K) →*₀ ℝ where
   toFun x := if hw : IsReal w then ‖x.1 ⟨w, hw⟩‖ else ‖x.2 ⟨w, not_isReal_iff_isComplex.mp hw⟩‖
   map_zero' := by simp
   map_one' := by simp
   map_mul' x y := by split_ifs <;> simp
 
-theorem normAtPlace_nonneg  (w : InfinitePlace K) (x : E K) :
+theorem normAtPlace_nonneg (w : InfinitePlace K) (x : mixedSpace K) :
     0 ≤ normAtPlace w x := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
   split_ifs <;> exact norm_nonneg _
 
-theorem normAtPlace_neg (w : InfinitePlace K) (x : E K)  :
+theorem normAtPlace_neg (w : InfinitePlace K) (x : mixedSpace K)  :
     normAtPlace w (- x) = normAtPlace w x := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
   split_ifs <;> simp
 
-theorem normAtPlace_add_le (w : InfinitePlace K) (x y : E K) :
+theorem normAtPlace_add_le (w : InfinitePlace K) (x y : mixedSpace K) :
     normAtPlace w (x + y) ≤ normAtPlace w x + normAtPlace w y := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
   split_ifs <;> exact norm_add_le _ _
 
-theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : ℝ) :
+theorem normAtPlace_smul (w : InfinitePlace K) (x : mixedSpace K) (c : ℝ) :
     normAtPlace w (c • x) = |c| * normAtPlace w x := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
   split_ifs
@@ -301,15 +308,15 @@ theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : ℝ) :
   · rw [Prod.smul_snd, Pi.smul_apply, norm_smul, Real.norm_eq_abs, Complex.norm_eq_abs]
 
 theorem normAtPlace_real (w : InfinitePlace K) (c : ℝ) :
-    normAtPlace w ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| := by
-  rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, normAtPlace_smul, map_one,
-    mul_one]
+    normAtPlace w ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = |c| := by
+  rw [show ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = c • 1 by ext <;> simp, normAtPlace_smul,
+    map_one, mul_one]
 
-theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K) :
+theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : mixedSpace K) :
     normAtPlace w x = ‖x.1 ⟨w, hw⟩‖ := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
 
-theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : E K) :
+theorem normAtPlace_apply_isComplex {w : InfinitePlace K} (hw : IsComplex w) (x : mixedSpace K) :
     normAtPlace w x = ‖x.2 ⟨w, hw⟩‖ := by
   rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk,
     dif_neg (not_isReal_iff_isComplex.mpr hw)]
@@ -321,7 +328,7 @@ theorem normAtPlace_apply (w : InfinitePlace K) (x : K) :
     RingHom.prod_apply, Pi.ringHom_apply, norm_embedding_of_isReal, norm_embedding_eq, dite_eq_ite,
     ite_id]
 
-theorem normAtPlace_eq_zero {x : E K} :
+theorem forall_normAtPlace_eq_zero_iff {x : mixedSpace K} :
     (∀ w, normAtPlace w x = 0) ↔ x = 0 := by
   refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
   · ext w
@@ -329,9 +336,16 @@ theorem normAtPlace_eq_zero {x : E K} :
     · exact norm_eq_zero'.mp (normAtPlace_apply_isComplex w.prop _ ▸ h w.1)
   · simp_rw [h, map_zero, implies_true]
 
+@[deprecated (since := "2024-09-13")] alias normAtPlace_eq_zero := forall_normAtPlace_eq_zero_iff
+
+@[simp]
+theorem exists_normAtPlace_ne_zero_iff {x : mixedSpace K} :
+    (∃ w, normAtPlace w x ≠ 0) ↔ x ≠ 0 := by
+  rw [ne_eq, ← forall_normAtPlace_eq_zero_iff, not_forall]
+
 variable [NumberField K]
 
-theorem nnnorm_eq_sup_normAtPlace (x : E K) :
+theorem nnnorm_eq_sup_normAtPlace (x : mixedSpace K) :
     ‖x‖₊ = univ.sup fun w ↦ ⟨normAtPlace w x, normAtPlace_nonneg w x⟩ := by
   have :
       (univ : Finset (InfinitePlace K)) =
@@ -346,7 +360,7 @@ theorem nnnorm_eq_sup_normAtPlace (x : E K) :
   · ext w
     simp [normAtPlace_apply_isComplex w.prop]
 
-theorem norm_eq_sup'_normAtPlace (x : E K) :
+theorem norm_eq_sup'_normAtPlace (x : mixedSpace K) :
     ‖x‖ = univ.sup' univ_nonempty fun w ↦ normAtPlace w x := by
   rw [← coe_nnnorm, nnnorm_eq_sup_normAtPlace, ← sup'_eq_sup univ_nonempty, ← NNReal.val_eq_coe,
     ← OrderHom.Subtype.val_coe, map_finset_sup', OrderHom.Subtype.val_coe]
@@ -355,43 +369,53 @@ theorem norm_eq_sup'_normAtPlace (x : E K) :
 /-- The norm of `x` is `∏ w, (normAtPlace x) ^ mult w`. It is defined such that the norm of
 `mixedEmbedding K a` for `a : K` is equal to the absolute value of the norm of `a` over `ℚ`,
 see `norm_eq_norm`. -/
-protected def norm : (E K) →*₀ ℝ where
+protected def norm : (mixedSpace K) →*₀ ℝ where
   toFun x := ∏ w, (normAtPlace w x) ^ (mult w)
   map_one' := by simp only [map_one, one_pow, prod_const_one]
   map_zero' := by simp [mult]
   map_mul' _ _ := by simp only [map_mul, mul_pow, prod_mul_distrib]
 
-protected theorem norm_apply (x : E K) :
+protected theorem norm_apply (x : mixedSpace K) :
     mixedEmbedding.norm x = ∏ w, (normAtPlace w x) ^ (mult w) := rfl
 
-protected theorem norm_nonneg (x : E K) :
+protected theorem norm_nonneg (x : mixedSpace K) :
     0 ≤ mixedEmbedding.norm x := univ.prod_nonneg fun _ _ ↦ pow_nonneg (normAtPlace_nonneg _ _) _
 
-protected theorem norm_eq_zero_iff {x : E K} :
+protected theorem norm_eq_zero_iff {x : mixedSpace K} :
     mixedEmbedding.norm x = 0 ↔ ∃ w, normAtPlace w x = 0 := by
   simp_rw [mixedEmbedding.norm, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, prod_eq_zero_iff,
     mem_univ, true_and, pow_eq_zero_iff mult_ne_zero]
 
-protected theorem norm_ne_zero_iff {x : E K} :
+protected theorem norm_ne_zero_iff {x : mixedSpace K} :
     mixedEmbedding.norm x ≠ 0 ↔ ∀ w, normAtPlace w x ≠ 0 := by
   rw [← not_iff_not]
   simp_rw [ne_eq, mixedEmbedding.norm_eq_zero_iff, not_not, not_forall, not_not]
 
-theorem norm_smul (c : ℝ) (x : E K) :
+theorem norm_eq_of_normAtPlace_eq {x y : mixedSpace K}
+    (h : ∀ w, normAtPlace w x = normAtPlace w y) :
+    mixedEmbedding.norm x = mixedEmbedding.norm y := by
+  simp_rw [mixedEmbedding.norm_apply, h]
+
+theorem norm_smul (c : ℝ) (x : mixedSpace K) :
     mixedEmbedding.norm (c • x) = |c| ^ finrank ℚ K * (mixedEmbedding.norm x) := by
   simp_rw [mixedEmbedding.norm_apply, normAtPlace_smul, mul_pow, prod_mul_distrib,
     prod_pow_eq_pow_sum, sum_mult_eq]
 
 theorem norm_real (c : ℝ) :
-    mixedEmbedding.norm ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| ^ finrank ℚ K := by
-  rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, norm_smul, map_one, mul_one]
+    mixedEmbedding.norm ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = |c| ^ finrank ℚ K := by
+  rw [show ((fun _ ↦ c, fun _ ↦ c) : (mixedSpace K)) = c • 1 by ext <;> simp, norm_smul, map_one,
+    mul_one]
 
 @[simp]
 theorem norm_eq_norm (x : K) :
     mixedEmbedding.norm (mixedEmbedding K x) = |Algebra.norm ℚ x| := by
   simp_rw [mixedEmbedding.norm_apply, normAtPlace_apply, prod_eq_abs_norm]
 
-theorem norm_eq_zero_iff' {x : E K} (hx : x ∈ Set.range (mixedEmbedding K)) :
+theorem norm_unit (u : (𝓞 K)ˣ) :
+    mixedEmbedding.norm (mixedEmbedding K u) = 1 := by
+  rw [norm_eq_norm, Units.norm, Rat.cast_one]
+
+theorem norm_eq_zero_iff' {x : mixedSpace K} (hx : x ∈ Set.range (mixedEmbedding K)) :
     mixedEmbedding.norm x = 0 ↔ x = 0 := by
   obtain ⟨a, rfl⟩ := hx
   rw [norm_eq_norm, Rat.cast_abs, abs_eq_zero, Rat.cast_eq_zero, Algebra.norm_eq_zero_iff,
@@ -403,32 +427,34 @@ noncomputable section stdBasis
 
 open scoped Classical
 
-open Complex MeasureTheory MeasureTheory.Measure Zspan Matrix ComplexConjugate
+open Complex MeasureTheory MeasureTheory.Measure ZSpan Matrix ComplexConjugate
 
 variable [NumberField K]
 
 /-- The type indexing the basis `stdBasis`. -/
 abbrev index := {w : InfinitePlace K // IsReal w} ⊕ ({w : InfinitePlace K // IsComplex w}) × (Fin 2)
 
-/-- The `ℝ`-basis of `({w // IsReal w} → ℝ) × ({ w // IsComplex w } → ℂ)` formed by the vector
-equal to `1` at `w` and `0` elsewhere for `IsReal w` and by the couple of vectors equal to `1`
-(resp. `I`) at `w` and `0` elsewhere for `IsComplex w`. -/
-def stdBasis : Basis (index K) ℝ (E K) :=
+/-- The `ℝ`-basis of the mixed space of `K` formed by the vector equal to `1` at `w` and `0`
+elsewhere for `IsReal w` and by the couple of vectors equal to `1` (resp. `I`) at `w` and `0`
+elsewhere for `IsComplex w`. -/
+def stdBasis : Basis (index K) ℝ (mixedSpace K) :=
   Basis.prod (Pi.basisFun ℝ _)
     (Basis.reindex (Pi.basis fun _ => basisOneI) (Equiv.sigmaEquivProd _ _))
 
 variable {K}
 
 @[simp]
-theorem stdBasis_apply_ofIsReal (x : E K) (w : {w : InfinitePlace K // IsReal w}) :
+theorem stdBasis_apply_ofIsReal (x : mixedSpace K) (w : {w : InfinitePlace K // IsReal w}) :
     (stdBasis K).repr x (Sum.inl w) = x.1 w := rfl
 
 @[simp]
-theorem stdBasis_apply_ofIsComplex_fst (x : E K) (w : {w : InfinitePlace K // IsComplex w}) :
+theorem stdBasis_apply_ofIsComplex_fst (x : mixedSpace K)
+    (w : {w : InfinitePlace K // IsComplex w}) :
     (stdBasis K).repr x (Sum.inr ⟨w, 0⟩) = (x.2 w).re := rfl
 
 @[simp]
-theorem stdBasis_apply_ofIsComplex_snd (x : E K) (w : {w : InfinitePlace K // IsComplex w}) :
+theorem stdBasis_apply_ofIsComplex_snd (x : mixedSpace K)
+    (w : {w : InfinitePlace K // IsComplex w}) :
     (stdBasis K).repr x (Sum.inr ⟨w, 1⟩) = (x.2 w).im := rfl
 
 variable (K)
@@ -465,7 +491,7 @@ def indexEquiv : (index K) ≃ (K →+* ℂ) := by
       · exact ⟨Sum.inr ⟨InfinitePlace.mkComplex ⟨φ, hφ⟩, 1⟩,
           by simp [(embedding_mk_eq φ).resolve_left hw]⟩
   · rw [Embeddings.card, ← mixedEmbedding.finrank K,
-      ← FiniteDimensional.finrank_eq_card_basis (stdBasis K)]
+      ← Module.finrank_eq_card_basis (stdBasis K)]
 
 variable {K}
 
@@ -547,9 +573,13 @@ open Module.Free
 
 open scoped nonZeroDivisors
 
-/-- A `ℝ`-basis of `ℝ^r₁ × ℂ^r₂` that is also a `ℤ`-basis of the image of `𝓞 K`. -/
+/-- The image of the ring of integers of `K` in the mixed space. -/
+protected abbrev integerLattice : Submodule ℤ (mixedSpace K) :=
+  LinearMap.range ((mixedEmbedding K).comp (algebraMap (𝓞 K) K)).toIntAlgHom.toLinearMap
+
+/-- A `ℝ`-basis of the mixed space that is also a `ℤ`-basis of the image of `𝓞 K`. -/
 def latticeBasis :
-    Basis (ChooseBasisIndex ℤ (𝓞 K)) ℝ (E K) := by
+    Basis (ChooseBasisIndex ℤ (𝓞 K)) ℝ (mixedSpace K) := by
   classical
     -- We construct an `ℝ`-linear independent family from the image of
     -- `canonicalEmbedding.lattice_basis` by `commMap`
@@ -571,7 +601,7 @@ theorem latticeBasis_apply (i : ChooseBasisIndex ℤ (𝓞 K)) :
   simp only [latticeBasis, coe_basisOfLinearIndependentOfCardEqFinrank, Function.comp_apply,
     canonicalEmbedding.latticeBasis_apply, integralBasis_apply, commMap_canonical_eq_mixed]
 
-theorem mem_span_latticeBasis (x : (E K)) :
+theorem mem_span_latticeBasis (x : (mixedSpace K)) :
     x ∈ Submodule.span ℤ (Set.range (latticeBasis K)) ↔
       x ∈ ((mixedEmbedding K).comp (algebraMap (𝓞 K) K)).range := by
   rw [show Set.range (latticeBasis K) =
@@ -582,6 +612,20 @@ theorem mem_span_latticeBasis (x : (E K)) :
     RingHom.mem_range, exists_exists_eq_and]
   rfl
 
+theorem span_latticeBasis :
+    Submodule.span ℤ (Set.range (latticeBasis K)) = mixedEmbedding.integerLattice K :=
+  Submodule.ext_iff.mpr (mem_span_latticeBasis K)
+
+instance : DiscreteTopology (mixedEmbedding.integerLattice K) := by
+  classical
+  rw [← span_latticeBasis]
+  infer_instance
+
+open Classical in
+instance : IsZLattice ℝ (mixedEmbedding.integerLattice K) := by
+  simp_rw [← span_latticeBasis]
+  exact ZSpan.isZLattice (latticeBasis K)
+
 theorem mem_rat_span_latticeBasis (x : K) :
     mixedEmbedding K x ∈ Submodule.span ℚ (Set.range (latticeBasis K)) := by
   rw [← Basis.sum_repr (integralBasis K) x, map_sum]
@@ -611,8 +655,8 @@ variable (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
 /-- The generalized index of the lattice generated by `I` in the lattice generated by
 `𝓞 K` is equal to the norm of the ideal `I`. The result is stated in terms of base change
 determinant and is the translation of `NumberField.det_basisOfFractionalIdeal_eq_absNorm` in
-`ℝ^r₁ × ℂ^r₂`. This is useful, in particular, to prove that the family obtained from
-the `ℤ`-basis of `I` is actually an `ℝ`-basis of `ℝ^r₁ × ℂ^r₂`, see
+the mixed space. This is useful, in particular, to prove that the family obtained from
+the `ℤ`-basis of `I` is actually an `ℝ`-basis of the mixed space, see
 `fractionalIdealLatticeBasis`. -/
 theorem det_basisOfFractionalIdeal_eq_norm
     (e : (ChooseBasisIndex ℤ (𝓞 K)) ≃ (ChooseBasisIndex ℤ I)) :
@@ -628,10 +672,10 @@ theorem det_basisOfFractionalIdeal_eq_norm
   simp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, Basis.toMatrix_apply, Function.comp_apply]
   exact latticeBasis_repr_apply K _ i
 
-/-- A `ℝ`-basis of `ℝ^r₁ × ℂ^r₂` that is also a `ℤ`-basis of the image of the fractional
+/-- A `ℝ`-basis of the mixed space of `K` that is also a `ℤ`-basis of the image of the fractional
 ideal `I`. -/
 def fractionalIdealLatticeBasis :
-    Basis (ChooseBasisIndex ℤ I) ℝ (E K) := by
+    Basis (ChooseBasisIndex ℤ I) ℝ (mixedSpace K) := by
   let e : (ChooseBasisIndex ℤ (𝓞 K)) ≃ (ChooseBasisIndex ℤ I) := by
     refine Fintype.equivOfCardEq ?_
     rw [← finrank_eq_card_chooseBasisIndex, ← finrank_eq_card_chooseBasisIndex,
@@ -650,7 +694,7 @@ theorem fractionalIdealLatticeBasis_apply (i : ChooseBasisIndex ℤ I) :
   simp only [fractionalIdealLatticeBasis, Basis.coe_reindex, Basis.coe_mk, Function.comp_apply,
     Equiv.apply_symm_apply]
 
-theorem mem_span_fractionalIdealLatticeBasis (x : (E K)) :
+theorem mem_span_fractionalIdealLatticeBasis (x : (mixedSpace K)) :
     x ∈ Submodule.span ℤ (Set.range (fractionalIdealLatticeBasis K I)) ↔
       x ∈ mixedEmbedding K '' I := by
   rw [show Set.range (fractionalIdealLatticeBasis K I) =
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
index 6bf5d0e285fa8..36a06aabe738b 100644
--- a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
@@ -10,7 +10,7 @@ import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
 /-!
 # Convex Bodies
 
-The file contains the definitions of several convex bodies lying in the space `ℝ^r₁ × ℂ^r₂`
+The file contains the definitions of several convex bodies lying in the mixed space `ℝ^r₁ × ℂ^r₂`
 associated to a number field of signature `K` and proves several existence theorems by applying
 *Minkowski Convex Body Theorem* to those.
 
@@ -40,11 +40,7 @@ variable (K : Type*) [Field K]
 
 namespace NumberField.mixedEmbedding
 
-open NumberField NumberField.InfinitePlace FiniteDimensional
-
-/-- The space `ℝ^r₁ × ℂ^r₂` with `(r₁, r₂)` the signature of `K`. -/
-local notation "E" K =>
-  ({w : InfinitePlace K // IsReal w} → ℝ) × ({w : InfinitePlace K // IsComplex w} → ℂ)
+open NumberField NumberField.InfinitePlace Module
 
 section convexBodyLT
 
@@ -54,7 +50,7 @@ variable (f : InfinitePlace K → ℝ≥0)
 
 /-- The convex body defined by `f`: the set of points `x : E` such that `‖x w‖ < f w` for all
 infinite places `w`. -/
-abbrev convexBodyLT : Set (E K) :=
+abbrev convexBodyLT : Set (mixedSpace K) :=
   (Set.univ.pi (fun w : { w : InfinitePlace K // IsReal w } => ball 0 (f w))) ×ˢ
   (Set.univ.pi (fun w : { w : InfinitePlace K // IsComplex w } => ball 0 (f w)))
 
@@ -65,7 +61,7 @@ theorem convexBodyLT_mem {x : K} :
     embedding_of_isReal_apply, Subtype.forall, ← forall₂_or_left, ← not_isReal_iff_isComplex, em,
     forall_true_left, norm_embedding_eq]
 
-theorem convexBodyLT_neg_mem (x : E K) (hx : x ∈ (convexBodyLT K f)) :
+theorem convexBodyLT_neg_mem (x : mixedSpace K) (hx : x ∈ (convexBodyLT K f)) :
     -x ∈ (convexBodyLT K f) := by
   simp only [Set.mem_prod, Prod.fst_neg, Set.mem_pi, Set.mem_univ, Pi.neg_apply,
     mem_ball_zero_iff, norm_neg, Real.norm_eq_abs, forall_true_left, Subtype.forall,
@@ -81,9 +77,9 @@ open scoped Classical
 
 variable [NumberField K]
 
-instance : IsAddHaarMeasure (volume : Measure (E K)) := prod.instIsAddHaarMeasure volume volume
+instance : IsAddHaarMeasure (volume : Measure (mixedSpace K)) := prod.instIsAddHaarMeasure _ _
 
-instance : NoAtoms (volume : Measure (E K)) := by
+instance : NoAtoms (volume : Measure (mixedSpace K)) := by
   obtain ⟨w⟩ := (inferInstance : Nonempty (InfinitePlace K))
   by_cases hw : IsReal w
   · exact @prod.instNoAtoms_fst _ _ _ _ volume volume _ (pi_noAtoms ⟨w, hw⟩)
@@ -98,8 +94,7 @@ theorem convexBodyLTFactor_ne_zero : convexBodyLTFactor K ≠ 0 :=
   mul_ne_zero (pow_ne_zero _ two_ne_zero) (pow_ne_zero _ pi_ne_zero)
 
 theorem one_le_convexBodyLTFactor : 1 ≤ convexBodyLTFactor K :=
-  one_le_mul (one_le_pow_of_one_le one_le_two _)
-    (one_le_pow_of_one_le (le_trans one_le_two Real.two_le_pi) _)
+  one_le_mul (one_le_pow₀ one_le_two) (one_le_pow₀ (one_le_two.trans Real.two_le_pi))
 
 /-- The volume of `(ConvexBodyLt K f)` where `convexBodyLT K f` is the set of points `x`
 such that `‖x w‖ < f w` for all infinite places `w`. -/
@@ -160,7 +155,7 @@ variable (f : InfinitePlace K → ℝ≥0) (w₀ : {w : InfinitePlace K // IsCom
 needed to ensure the element constructed is not real, see for example
 `exists_primitive_element_lt_of_isComplex`.
 -/
-abbrev convexBodyLT' : Set (E K) :=
+abbrev convexBodyLT' : Set (mixedSpace K) :=
   (Set.univ.pi (fun w : { w : InfinitePlace K // IsReal w } ↦ ball 0 (f w))) ×ˢ
   (Set.univ.pi (fun w : { w : InfinitePlace K // IsComplex w } ↦
     if w = w₀ then {x | |x.re| < 1 ∧ |x.im| < (f w : ℝ) ^ 2} else ball 0 (f w)))
@@ -170,21 +165,24 @@ theorem convexBodyLT'_mem {x : K} :
       (∀ w : InfinitePlace K, w ≠ w₀ → w x < f w) ∧
       |(w₀.val.embedding x).re| < 1 ∧ |(w₀.val.embedding x).im| < (f w₀ : ℝ) ^ 2 := by
   simp_rw [mixedEmbedding, RingHom.prod_apply, Set.mem_prod, Set.mem_pi, Set.mem_univ,
-    forall_true_left, Pi.ringHom_apply, apply_ite, mem_ball_zero_iff, ← Complex.norm_real,
-    embedding_of_isReal_apply, norm_embedding_eq, Subtype.forall, Set.mem_setOf_eq]
+    forall_true_left, Pi.ringHom_apply, mem_ball_zero_iff, ← Complex.norm_real,
+    embedding_of_isReal_apply, norm_embedding_eq, Subtype.forall]
   refine ⟨fun ⟨h₁, h₂⟩ ↦ ⟨fun w h_ne ↦ ?_, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ⟨fun w hw ↦ ?_, fun w hw ↦ ?_⟩⟩
   · by_cases hw : IsReal w
     · exact norm_embedding_eq w _ ▸ h₁ w hw
     · specialize h₂ w (not_isReal_iff_isComplex.mp hw)
+      rw [apply_ite (w.embedding x ∈ ·), Set.mem_setOf_eq,
+        mem_ball_zero_iff, norm_embedding_eq] at h₂
       rwa [if_neg (by exact Subtype.coe_ne_coe.1 h_ne)] at h₂
   · simpa [if_true] using h₂ w₀.val w₀.prop
   · exact h₁ w (ne_of_isReal_isComplex hw w₀.prop)
   · by_cases h_ne : w = w₀
     · simpa [h_ne]
     · rw [if_neg (by exact Subtype.coe_ne_coe.1 h_ne)]
+      rw [mem_ball_zero_iff, norm_embedding_eq]
       exact h₁ w h_ne
 
-theorem convexBodyLT'_neg_mem (x : E K) (hx : x ∈ convexBodyLT' K f w₀) :
+theorem convexBodyLT'_neg_mem (x : mixedSpace K) (hx : x ∈ convexBodyLT' K f w₀) :
     -x ∈ convexBodyLT' K f w₀ := by
   simp only [Set.mem_prod, Set.mem_pi, Set.mem_univ, mem_ball, dist_zero_right, Real.norm_eq_abs,
     true_implies, Subtype.forall, Prod.fst_neg, Pi.neg_apply, norm_neg, Prod.snd_neg] at hx ⊢
@@ -195,8 +193,8 @@ theorem convexBodyLT'_convex : Convex ℝ (convexBodyLT' K f w₀) := by
   refine Convex.prod (convex_pi (fun _ _ => convex_ball _ _)) (convex_pi (fun _ _ => ?_))
   split_ifs
   · simp_rw [abs_lt]
-    refine Convex.inter ((convex_halfspace_re_gt _).inter  (convex_halfspace_re_lt _))
-      ((convex_halfspace_im_gt _).inter  (convex_halfspace_im_lt _))
+    refine Convex.inter ((convex_halfspace_re_gt _).inter (convex_halfspace_re_lt _))
+      ((convex_halfspace_im_gt _).inter (convex_halfspace_im_lt _))
   · exact convex_ball _ _
 
 open MeasureTheory MeasureTheory.Measure
@@ -213,8 +211,7 @@ theorem convexBodyLT'Factor_ne_zero : convexBodyLT'Factor K ≠ 0 :=
   mul_ne_zero (pow_ne_zero _ two_ne_zero) (pow_ne_zero _ pi_ne_zero)
 
 theorem one_le_convexBodyLT'Factor : 1 ≤ convexBodyLT'Factor K :=
-  one_le_mul (one_le_pow_of_one_le one_le_two _)
-    (one_le_pow_of_one_le (le_trans one_le_two Real.two_le_pi) _)
+  one_le_mul (one_le_pow₀ one_le_two) (one_le_pow₀ (one_le_two.trans Real.two_le_pi))
 
 theorem convexBodyLT'_volume :
     volume (convexBodyLT' K f w₀) = convexBodyLT'Factor K * ∏ w, (f w) ^ (mult w) := by
@@ -274,14 +271,14 @@ open scoped Real Classical NNReal
 variable [NumberField K] (B : ℝ)
 variable {K}
 
-/-- The function that sends `x : ({w // IsReal w} → ℝ) × ({w // IsComplex w} → ℂ)` to
-  `∑ w, ‖x.1 w‖ + 2 * ∑ w, ‖x.2 w‖`. It defines a norm and it used to define `convexBodySum`. -/
-noncomputable abbrev convexBodySumFun (x : E K) : ℝ := ∑ w, mult w * normAtPlace w x
+/-- The function that sends `x : mixedSpace K` to `∑ w, ‖x.1 w‖ + 2 * ∑ w, ‖x.2 w‖`. It defines a
+norm and it used to define `convexBodySum`. -/
+noncomputable abbrev convexBodySumFun (x : mixedSpace K) : ℝ := ∑ w, mult w * normAtPlace w x
 
-theorem convexBodySumFun_apply (x : E K) :
+theorem convexBodySumFun_apply (x : mixedSpace K) :
     convexBodySumFun x = ∑ w,  mult w * normAtPlace w x := rfl
 
-theorem convexBodySumFun_apply' (x : E K) :
+theorem convexBodySumFun_apply' (x : mixedSpace K) :
     convexBodySumFun x = ∑ w, ‖x.1 w‖ + 2 * ∑ w, ‖x.2 w‖ := by
   simp_rw [convexBodySumFun_apply, ← Finset.sum_add_sum_compl {w | IsReal w}.toFinset,
     Set.toFinset_setOf, Finset.compl_filter, not_isReal_iff_isComplex, ← Finset.subtype_univ,
@@ -293,35 +290,35 @@ theorem convexBodySumFun_apply' (x : E K) :
     rw [mult, if_neg (not_isReal_iff_isComplex.mpr w.prop), normAtPlace_apply_isComplex,
       Nat.cast_ofNat]
 
-theorem convexBodySumFun_nonneg (x : E K) :
+theorem convexBodySumFun_nonneg (x : mixedSpace K) :
     0 ≤ convexBodySumFun x :=
   Finset.sum_nonneg (fun _ _ => mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _))
 
-theorem convexBodySumFun_neg (x : E K) :
+theorem convexBodySumFun_neg (x : mixedSpace K) :
     convexBodySumFun (- x) = convexBodySumFun x := by
   simp_rw [convexBodySumFun, normAtPlace_neg]
 
-theorem convexBodySumFun_add_le (x y : E K) :
+theorem convexBodySumFun_add_le (x y : mixedSpace K) :
     convexBodySumFun (x + y) ≤ convexBodySumFun x + convexBodySumFun y := by
   simp_rw [convexBodySumFun, ← Finset.sum_add_distrib, ← mul_add]
   exact Finset.sum_le_sum
     fun _ _ ↦ mul_le_mul_of_nonneg_left (normAtPlace_add_le _ x y) (Nat.cast_pos.mpr mult_pos).le
 
-theorem convexBodySumFun_smul (c : ℝ) (x : E K) :
+theorem convexBodySumFun_smul (c : ℝ) (x : mixedSpace K) :
     convexBodySumFun (c • x) = |c| * convexBodySumFun x := by
   simp_rw [convexBodySumFun, normAtPlace_smul, ← mul_assoc, mul_comm, Finset.mul_sum, mul_assoc]
 
-theorem convexBodySumFun_eq_zero_iff (x : E K) :
+theorem convexBodySumFun_eq_zero_iff (x : mixedSpace K) :
     convexBodySumFun x = 0 ↔ x = 0 := by
-  rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>
-    mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
+  rw [← forall_normAtPlace_eq_zero_iff, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg
+    fun _ _ ↦ mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
   conv =>
     enter [1, w, hw]
     rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
       (mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_zero)]
   simp_rw [Finset.mem_univ, true_implies]
 
-theorem norm_le_convexBodySumFun (x : E K) : ‖x‖ ≤ convexBodySumFun x := by
+theorem norm_le_convexBodySumFun (x : mixedSpace K) : ‖x‖ ≤ convexBodySumFun x := by
   rw [norm_eq_sup'_normAtPlace]
   refine (Finset.sup'_le_iff _ _).mpr fun w _ ↦ ?_
   rw [convexBodySumFun_apply, ← Finset.univ.add_sum_erase _ (Finset.mem_univ w)]
@@ -333,16 +330,16 @@ theorem norm_le_convexBodySumFun (x : E K) : ‖x‖ ≤ convexBodySumFun x := b
 variable (K)
 
 theorem convexBodySumFun_continuous :
-    Continuous (convexBodySumFun : (E K) → ℝ) := by
+    Continuous (convexBodySumFun : mixedSpace K → ℝ) := by
   refine continuous_finset_sum Finset.univ fun w ↦ ?_
   obtain hw | hw := isReal_or_isComplex w
   all_goals
   · simp only [normAtPlace_apply_isReal, normAtPlace_apply_isComplex, hw]
     fun_prop
 
-/-- The convex body equal to the set of points `x : E` such that
+/-- The convex body equal to the set of points `x : mixedSpace K` such that
   `∑ w real, ‖x w‖ + 2 * ∑ w complex, ‖x w‖ ≤ B`. -/
-abbrev convexBodySum : Set (E K)  := { x | convexBodySumFun x ≤ B }
+abbrev convexBodySum : Set (mixedSpace K)  := { x | convexBodySumFun x ≤ B }
 
 theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
     volume (convexBodySum K B) = 0 := by
@@ -363,7 +360,7 @@ theorem convexBodySum_mem {x : K} :
   simp_rw [Set.mem_setOf_eq, convexBodySumFun, normAtPlace_apply]
   rfl
 
-theorem convexBodySum_neg_mem {x : E K} (hx : x ∈ (convexBodySum K B)) :
+theorem convexBodySum_neg_mem {x : mixedSpace K} (hx : x ∈ (convexBodySum K B)) :
     -x ∈ (convexBodySum K B) := by
   rw [Set.mem_setOf, convexBodySumFun_neg]
   exact hx
@@ -406,7 +403,7 @@ theorem convexBodySum_volume :
       convert addHaar_smul volume B (convexBodySum K 1)
       · simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hB), Set.preimage_setOf_eq, convexBodySumFun,
         normAtPlace_smul, abs_inv, abs_eq_self.mpr (le_of_lt hB), ← mul_assoc, mul_comm, mul_assoc,
-        ← Finset.mul_sum, inv_mul_le_iff hB, mul_one]
+        ← Finset.mul_sum, inv_mul_le_iff₀ hB, mul_one]
       · rw [abs_pow, ofReal_pow (abs_nonneg _), abs_eq_self.mpr (le_of_lt hB),
           mixedEmbedding.finrank]
       · exact this.symm
@@ -414,13 +411,13 @@ theorem convexBodySum_volume :
       convexBodySumFun_neg convexBodySumFun_add_le
       (fun hx => (convexBodySumFun_eq_zero_iff _).mp hx)
       (fun r x => le_of_eq (convexBodySumFun_smul r x))]
-    rw [measure_lt_one_eq_integral_div_gamma (g := fun x : (E K) => convexBodySumFun x)
+    rw [measure_lt_one_eq_integral_div_gamma (g := fun x : (mixedSpace K) => convexBodySumFun x)
       volume ((convexBodySumFun_eq_zero_iff 0).mpr rfl) convexBodySumFun_neg convexBodySumFun_add_le
       (fun hx => (convexBodySumFun_eq_zero_iff _).mp hx)
       (fun r x => le_of_eq (convexBodySumFun_smul r x)) zero_lt_one]
     simp_rw [mixedEmbedding.finrank, div_one, Gamma_nat_eq_factorial, ofReal_div_of_pos
       (Nat.cast_pos.mpr (Nat.factorial_pos _)), Real.rpow_one, ofReal_natCast]
-    suffices ∫ x : E K, exp (-convexBodySumFun x) =
+    suffices ∫ x : mixedSpace K, exp (-convexBodySumFun x) =
         (2 : ℝ) ^ NrRealPlaces K * (π / 2) ^ NrComplexPlaces K by
       rw [this, convexBodySumFactor, ofReal_mul (by positivity), ofReal_pow zero_le_two,
         ofReal_pow (by positivity), ofReal_div_of_pos zero_lt_two, ofReal_ofNat,
@@ -453,7 +450,7 @@ end convexBodySum
 section minkowski
 
 open scoped Classical
-open MeasureTheory MeasureTheory.Measure FiniteDimensional Zspan Real Submodule
+open MeasureTheory MeasureTheory.Measure Module ZSpan Real Submodule
 
 open scoped ENNReal NNReal nonZeroDivisors IntermediateField
 
@@ -465,7 +462,8 @@ variable [NumberField K] (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
 `NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis` for the computation of
 `volume (fundamentalDomain (idealLatticeBasis K))`. -/
 noncomputable def minkowskiBound : ℝ≥0∞ :=
-  volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) * (2 : ℝ≥0∞) ^ (finrank ℝ (E K))
+  volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) *
+    (2 : ℝ≥0∞) ^ (finrank ℝ (mixedSpace K))
 
 theorem volume_fundamentalDomain_fractionalIdealLatticeBasis :
     volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) =
@@ -488,7 +486,7 @@ theorem minkowskiBound_lt_top : minkowskiBound K I < ⊤ := by
 
 theorem minkowskiBound_pos : 0 < minkowskiBound K I := by
   refine zero_lt_iff.mpr (mul_ne_zero ?_ ?_)
-  · exact Zspan.measure_fundamentalDomain_ne_zero _
+  · exact ZSpan.measure_fundamentalDomain_ne_zero _
   · exact ENNReal.pow_ne_zero two_ne_zero _
 
 variable {f : InfinitePlace K → ℝ≥0} (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
@@ -500,9 +498,9 @@ the computation of this volume), then there exists a nonzero algebraic number `a
 that `w a < f w` for all infinite places `w`. -/
 theorem exists_ne_zero_mem_ideal_lt (h : minkowskiBound K I < volume (convexBodyLT K f)) :
     ∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧ ∀ w : InfinitePlace K, w a < f w := by
-  have h_fund := Zspan.isAddFundamentalDomain (fractionalIdealLatticeBasis K I) volume
+  have h_fund := ZSpan.isAddFundamentalDomain' (fractionalIdealLatticeBasis K I) volume
   have : Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I))).toAddSubgroup := by
-    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)) : Set (E K))
+    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)))
     infer_instance
   obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
     h_fund (convexBodyLT_neg_mem K f) (convexBodyLT_convex K f) h
@@ -516,9 +514,9 @@ theorem exists_ne_zero_mem_ideal_lt' (w₀ : {w : InfinitePlace K // IsComplex w
     (h : minkowskiBound K I < volume (convexBodyLT' K f w₀)) :
     ∃ a ∈ (I : FractionalIdeal (𝓞 K)⁰ K), a ≠ 0 ∧ (∀ w : InfinitePlace K, w ≠ w₀ → w a < f w) ∧
       |(w₀.val.embedding a).re| < 1 ∧ |(w₀.val.embedding a).im| < (f w₀ : ℝ) ^ 2 := by
-  have h_fund := Zspan.isAddFundamentalDomain (fractionalIdealLatticeBasis K I) volume
+  have h_fund := ZSpan.isAddFundamentalDomain' (fractionalIdealLatticeBasis K I) volume
   have : Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I))).toAddSubgroup := by
-    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)) : Set (E K))
+    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)))
     infer_instance
   obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure
     h_fund (convexBodyLT'_neg_mem K f w₀) (convexBodyLT'_convex K f w₀) h
@@ -604,9 +602,9 @@ theorem exists_ne_zero_mem_ideal_of_norm_le {B : ℝ}
   -- Some inequalities that will be useful later on
   have h1 : 0 < (finrank ℚ K : ℝ)⁻¹ := inv_pos.mpr (Nat.cast_pos.mpr finrank_pos)
   have h2 : 0 ≤ B / (finrank ℚ K) := div_nonneg hB (Nat.cast_nonneg _)
-  have h_fund := Zspan.isAddFundamentalDomain (fractionalIdealLatticeBasis K I) volume
+  have h_fund := ZSpan.isAddFundamentalDomain' (fractionalIdealLatticeBasis K I) volume
   have : Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I))).toAddSubgroup := by
-    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)) : Set (E K))
+    change Countable (span ℤ (Set.range (fractionalIdealLatticeBasis K I)))
     infer_instance
   obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
       h_fund (fun _ ↦ convexBodySum_neg_mem K B) (convexBodySum_convex K B)
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean
new file mode 100644
index 0000000000000..419b0f6e623c0
--- /dev/null
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/FundamentalCone.lean
@@ -0,0 +1,502 @@
+/-
+Copyright (c) 2024 Xavier Roblot. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Xavier Roblot
+-/
+import Mathlib.RingTheory.Ideal.IsPrincipal
+import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
+
+/-!
+# Fundamental Cone
+
+Let `K` be a number field of signature `(r₁, r₂)`. We define an action of the units `(𝓞 K)ˣ` on
+the mixed space `ℝ^r₁ × ℂ^r₂` via the `mixedEmbedding`. The fundamental cone is a cone in the
+mixed space that is a fundamental domain for the action of `(𝓞 K)ˣ` modulo torsion.
+
+## Main definitions and results
+
+* `NumberField.mixedEmbedding.unitSMul`: the action of `(𝓞 K)ˣ` on the mixed space defined, for
+`u : (𝓞 K)ˣ`, by multiplication component by component with `mixedEmbedding K u`.
+
+* `NumberField.mixedEmbedding.fundamentalCone`: a cone in the mixed space, ie. a subset stable
+by multiplication by a nonzero real number, see `smul_mem_of_mem`, that is also a fundamental
+domain for the action of `(𝓞 K)ˣ` modulo torsion, see `exists_unit_smul_mem` and
+`torsion_unit_smul_mem_of_mem`.
+
+* `NumberField.mixedEmbedding.fundamentalCone.integralPoint`: the subset of elements of the
+fundamental cone that are images of algebraic integers of `K`.
+
+* `NumberField.mixedEmbedding.fundamentalCone.integralPointEquiv`: the equivalence between
+`fundamentalCone.integralPoint K` and the principal nonzero ideals of `𝓞 K` times the
+torsion of `K`.
+
+* `NumberField.mixedEmbedding.fundamentalCone.card_isPrincipal_norm_eq_mul_torsion`: the number of
+principal nonzero ideals in `𝓞 K` of norm `n` multiplied by the order of the torsion of `K` is
+equal to the number of `fundamentalCone.integralPoint K` of norm `n`.
+
+## Tags
+
+number field, canonical embedding, units, principal ideals
+-/
+
+variable (K : Type*) [Field K]
+
+namespace NumberField.mixedEmbedding
+
+open NumberField NumberField.InfinitePlace
+
+noncomputable section UnitSMul
+
+/-- The action of `(𝓞 K)ˣ` on the mixed space `ℝ^r₁ × ℂ^r₂` defined, for `u : (𝓞 K)ˣ`, by
+multiplication component by component with `mixedEmbedding K u`. -/
+@[simps]
+instance unitSMul : SMul (𝓞 K)ˣ (mixedSpace K) where
+  smul u x := mixedEmbedding K u * x
+
+instance : MulAction (𝓞 K)ˣ (mixedSpace K) where
+  one_smul := fun _ ↦ by simp_rw [unitSMul_smul, Units.coe_one, map_one, one_mul]
+  mul_smul := fun _ _ _ ↦ by simp_rw [unitSMul_smul, Units.coe_mul, map_mul, mul_assoc]
+
+instance : SMulZeroClass (𝓞 K)ˣ (mixedSpace K) where
+  smul_zero := fun _ ↦ by simp_rw [unitSMul_smul, mul_zero]
+
+variable {K}
+
+theorem unit_smul_eq_zero (u : (𝓞 K)ˣ) (x : mixedSpace K) :
+    u • x = 0 ↔ x = 0 := by
+  refine ⟨fun h ↦ ?_, fun h ↦ by rw [h, smul_zero]⟩
+  contrapose! h
+  obtain ⟨w, h⟩ := exists_normAtPlace_ne_zero_iff.mpr h
+  refine exists_normAtPlace_ne_zero_iff.mp ⟨w, ?_⟩
+  rw [unitSMul_smul, map_mul]
+  exact mul_ne_zero (by simp) h
+
+variable [NumberField K]
+
+theorem unit_smul_eq_iff_mul_eq {x y : 𝓞 K} {u : (𝓞 K)ˣ} :
+    u • mixedEmbedding K x = mixedEmbedding K y ↔ u * x = y := by
+  rw [unitSMul_smul, ← map_mul, Function.Injective.eq_iff, ← RingOfIntegers.coe_eq_algebraMap,
+    ← map_mul, ← RingOfIntegers.ext_iff]
+  exact mixedEmbedding_injective K
+
+theorem norm_unit_smul (u : (𝓞 K)ˣ) (x : mixedSpace K) :
+    mixedEmbedding.norm (u • x) = mixedEmbedding.norm x := by
+  rw [unitSMul_smul, map_mul, norm_unit, one_mul]
+
+end UnitSMul
+
+noncomputable section logMap
+
+open NumberField.Units NumberField.Units.dirichletUnitTheorem Module
+
+variable [NumberField K] {K}
+
+open Classical in
+/-- The map from the mixed space to `{w : InfinitePlace K // w ≠ w₀} → ℝ` (with `w₀` the fixed
+place from the proof of Dirichlet Unit Theorem) defined in such way that: 1) it factors the map
+`logEmbedding`, see `logMap_eq_logEmbedding`; 2) it is constant on the sets
+`{c • x | c ∈ ℝ, c ≠ 0}` if `norm x ≠ 0`, see `logMap_real_smul`. -/
+def logMap (x : mixedSpace K) : {w : InfinitePlace K // w ≠ w₀} → ℝ := fun w ↦
+  mult w.val * (Real.log (normAtPlace w.val x) -
+    Real.log (mixedEmbedding.norm x) * (finrank ℚ K : ℝ)⁻¹)
+
+@[simp]
+theorem logMap_apply (x : mixedSpace K) (w : {w : InfinitePlace K // w ≠ w₀}) :
+    logMap x w = mult w.val * (Real.log (normAtPlace w.val x) -
+      Real.log (mixedEmbedding.norm x) * (finrank ℚ K : ℝ)⁻¹) := rfl
+
+@[simp]
+theorem logMap_zero : logMap (0 : mixedSpace K) = 0 := by
+  ext; simp
+
+@[simp]
+theorem logMap_one : logMap (1 : mixedSpace K) = 0 := by
+  ext; simp
+
+variable {x y : mixedSpace K}
+
+theorem logMap_mul (hx : mixedEmbedding.norm x ≠ 0) (hy : mixedEmbedding.norm y ≠ 0) :
+    logMap (x * y) = logMap x + logMap y := by
+  ext w
+  simp_rw [Pi.add_apply, logMap_apply]
+  rw [map_mul, map_mul, Real.log_mul, Real.log_mul hx hy, add_mul]
+  · ring
+  · exact mixedEmbedding.norm_ne_zero_iff.mp hx w
+  · exact mixedEmbedding.norm_ne_zero_iff.mp hy w
+
+theorem logMap_apply_of_norm_one (hx : mixedEmbedding.norm x = 1)
+    (w : {w : InfinitePlace K // w ≠ w₀}) :
+    logMap x w = mult w.val * Real.log (normAtPlace w x) := by
+  rw [logMap_apply, hx, Real.log_one, zero_mul, sub_zero]
+
+@[simp]
+theorem logMap_eq_logEmbedding (u : (𝓞 K)ˣ) :
+    logMap (mixedEmbedding K u) = logEmbedding K (Additive.ofMul u) := by
+  ext; simp
+
+theorem logMap_unit_smul (u : (𝓞 K)ˣ) (hx : mixedEmbedding.norm x ≠ 0) :
+    logMap (u • x) = logEmbedding K (Additive.ofMul u) + logMap x := by
+  rw [unitSMul_smul, logMap_mul (by rw [norm_unit]; norm_num) hx, logMap_eq_logEmbedding]
+
+variable (x) in
+theorem logMap_torsion_smul {ζ : (𝓞 K)ˣ} (hζ : ζ ∈ torsion K) :
+    logMap (ζ • x) = logMap x := by
+  ext
+  simp_rw [logMap_apply, unitSMul_smul, map_mul, norm_eq_norm, Units.norm, Rat.cast_one, one_mul,
+    normAtPlace_apply, (mem_torsion K).mp hζ, one_mul]
+
+theorem logMap_real (c : ℝ) :
+    logMap (c • (1 : mixedSpace K)) = 0 := by
+  ext
+  rw [logMap_apply, normAtPlace_smul, norm_smul, map_one, map_one, mul_one, mul_one, Real.log_pow,
+    mul_comm (finrank ℚ K : ℝ) _, mul_assoc, mul_inv_cancel₀ (Nat.cast_ne_zero.mpr finrank_pos.ne'),
+    mul_one, sub_self, mul_zero, Pi.zero_apply]
+
+theorem logMap_real_smul (hx : mixedEmbedding.norm x ≠ 0) {c : ℝ} (hc : c ≠ 0) :
+    logMap (c • x) = logMap x := by
+  have : mixedEmbedding.norm (c • (1 : mixedSpace K)) ≠ 0 := by
+    rw [norm_smul, map_one, mul_one]
+    exact pow_ne_zero _ (abs_ne_zero.mpr hc)
+  rw [← smul_one_mul, logMap_mul this hx, logMap_real, zero_add]
+
+theorem logMap_eq_of_normAtPlace_eq (h : ∀ w, normAtPlace w x = normAtPlace w y) :
+    logMap x = logMap y := by
+  ext
+  simp_rw [logMap_apply, h, norm_eq_of_normAtPlace_eq h]
+
+end logMap
+
+noncomputable section
+
+open NumberField.Units NumberField.Units.dirichletUnitTheorem
+
+variable [NumberField K]
+
+open Classical in
+/-- The fundamental cone is a cone in the mixed space, ie. a subset fixed by multiplication by
+a nonzero real number, see `smul_mem_of_mem`, that is also a fundamental domain for the action
+of `(𝓞 K)ˣ` modulo torsion, see `exists_unit_smul_mem` and `torsion_smul_mem_of_mem`. -/
+def fundamentalCone : Set (mixedSpace K) :=
+  logMap⁻¹' (ZSpan.fundamentalDomain ((basisUnitLattice K).ofZLatticeBasis ℝ _)) \
+      {x | mixedEmbedding.norm x = 0}
+
+namespace fundamentalCone
+
+variable {K} {x y : mixedSpace K} {c : ℝ}
+
+theorem norm_pos_of_mem (hx : x ∈ fundamentalCone K) :
+    0 < mixedEmbedding.norm x :=
+  lt_of_le_of_ne (mixedEmbedding.norm_nonneg _) (Ne.symm hx.2)
+
+theorem normAtPlace_pos_of_mem (hx : x ∈ fundamentalCone K) (w : InfinitePlace K) :
+    0 < normAtPlace w x :=
+  lt_of_le_of_ne (normAtPlace_nonneg _ _)
+    (mixedEmbedding.norm_ne_zero_iff.mp (norm_pos_of_mem hx).ne' w).symm
+
+theorem mem_of_normAtPlace_eq (hx : x ∈ fundamentalCone K)
+    (hy : ∀ w, normAtPlace w y = normAtPlace w x) :
+    y ∈ fundamentalCone K := by
+  refine ⟨?_, by simpa [norm_eq_of_normAtPlace_eq hy] using hx.2⟩
+  rw [Set.mem_preimage, logMap_eq_of_normAtPlace_eq hy]
+  exact hx.1
+
+theorem smul_mem_of_mem (hx : x ∈ fundamentalCone K) (hc : c ≠ 0) :
+    c • x ∈ fundamentalCone K := by
+  refine ⟨?_, ?_⟩
+  · rw [Set.mem_preimage, logMap_real_smul hx.2 hc]
+    exact hx.1
+  · rw [Set.mem_setOf_eq, mixedEmbedding.norm_smul, mul_eq_zero, not_or]
+    exact ⟨pow_ne_zero _ (abs_ne_zero.mpr hc), hx.2⟩
+
+theorem smul_mem_iff_mem (hc : c ≠ 0) :
+    c • x ∈ fundamentalCone K ↔ x ∈ fundamentalCone K := by
+  refine ⟨fun h ↦ ?_, fun h ↦ smul_mem_of_mem h hc⟩
+  convert smul_mem_of_mem h (inv_ne_zero hc)
+  rw [eq_inv_smul_iff₀ hc]
+
+theorem exists_unit_smul_mem (hx : mixedEmbedding.norm x ≠ 0) :
+    ∃ u : (𝓞 K)ˣ, u • x ∈ fundamentalCone K := by
+  classical
+  let B := (basisUnitLattice K).ofZLatticeBasis ℝ
+  rsuffices ⟨⟨_, ⟨u, _, rfl⟩⟩, hu⟩ : ∃ e : unitLattice K, e + logMap x ∈ ZSpan.fundamentalDomain B
+  · exact ⟨u, by rwa [Set.mem_preimage, logMap_unit_smul u hx], by simp [hx]⟩
+  · obtain ⟨⟨e, h₁⟩, h₂, -⟩ := ZSpan.exist_unique_vadd_mem_fundamentalDomain B (logMap x)
+    exact ⟨⟨e, by rwa [← Basis.ofZLatticeBasis_span ℝ (unitLattice K)]⟩, h₂⟩
+
+theorem torsion_smul_mem_of_mem (hx : x ∈ fundamentalCone K) {ζ : (𝓞 K)ˣ} (hζ : ζ ∈ torsion K) :
+    ζ • x ∈ fundamentalCone K := by
+  constructor
+  · rw [Set.mem_preimage, logMap_torsion_smul _ hζ]
+    exact hx.1
+  · rw [Set.mem_setOf_eq, unitSMul_smul, map_mul, norm_unit, one_mul]
+    exact hx.2
+
+theorem unit_smul_mem_iff_mem_torsion (hx : x ∈ fundamentalCone K) (u : (𝓞 K)ˣ) :
+    u • x ∈ fundamentalCone K ↔ u ∈ torsion K := by
+  classical
+  refine ⟨fun h ↦ ?_, fun h ↦ torsion_smul_mem_of_mem hx h⟩
+  rw [← logEmbedding_eq_zero_iff]
+  let B := (basisUnitLattice K).ofZLatticeBasis ℝ
+  refine (Subtype.mk_eq_mk (h := ?_) (h' := Submodule.zero_mem _)).mp <|
+    (ZSpan.exist_unique_vadd_mem_fundamentalDomain B (logMap x)).unique ?_ ?_
+  · rw [Basis.ofZLatticeBasis_span ℝ (unitLattice K)]
+    exact ⟨u, trivial, rfl⟩
+  · rw [AddSubmonoid.mk_vadd, vadd_eq_add, ← logMap_unit_smul _ hx.2]
+    exact h.1
+  · rw [AddSubmonoid.mk_vadd, vadd_eq_add, zero_add]
+    exact hx.1
+
+variable (K) in
+/-- The set of images by `mixedEmbedding` of algebraic integers of `K` contained in the
+fundamental cone. -/
+def integralPoint : Set (mixedSpace K) :=
+  fundamentalCone K ∩ mixedEmbedding.integerLattice K
+
+theorem mem_integralPoint {a : mixedSpace K} :
+    a ∈ integralPoint K ↔ a ∈ fundamentalCone K ∧ ∃ x : 𝓞 K, mixedEmbedding K x = a := by
+  simp only [integralPoint, Set.mem_inter_iff, SetLike.mem_coe, LinearMap.mem_range,
+    AlgHom.toLinearMap_apply, RingHom.toIntAlgHom_coe, RingHom.coe_comp, Function.comp_apply]
+
+/-- If `a` is an integral point, then there is a *unique* algebraic integer in `𝓞 K` such
+that `mixedEmbedding K x = a`. -/
+theorem exists_unique_preimage_of_integralPoint {a : mixedSpace K} (ha : a ∈ integralPoint K) :
+    ∃! x : 𝓞 K, mixedEmbedding K x = a := by
+  obtain ⟨_, ⟨x, rfl⟩⟩ := mem_integralPoint.mp ha
+  refine Function.Injective.existsUnique_of_mem_range ?_ (Set.mem_range_self x)
+  exact (mixedEmbedding_injective K).comp RingOfIntegers.coe_injective
+
+theorem integralPoint_ne_zero (a : integralPoint K) : (a : mixedSpace K) ≠ 0 := by
+  by_contra!
+  exact a.prop.1.2 (this.symm ▸ mixedEmbedding.norm.map_zero')
+
+open scoped nonZeroDivisors
+
+/-- For `a : fundamentalCone K`, the unique nonzero algebraic integer `x` such that its image by
+`mixedEmbedding` is equal to `a`. Note that we state the fact that `x ≠ 0` by saying that `x` is
+a nonzero divisors since we will use later on the isomorphism
+`Ideal.associatesNonZeroDivisorsEquivIsPrincipal`, see `integralPointEquiv`. -/
+def preimageOfIntegralPoint (a : integralPoint K) : (𝓞 K)⁰ :=
+  ⟨(mem_integralPoint.mp a.prop).2.choose, mem_nonZeroDivisors_of_ne_zero (by
+  simp_rw [ne_eq, ← RingOfIntegers.coe_injective.eq_iff, ← (mixedEmbedding_injective K).eq_iff,
+    map_zero, (mem_integralPoint.mp a.prop).2.choose_spec, integralPoint_ne_zero,
+    not_false_eq_true])⟩
+
+@[simp]
+theorem mixedEmbedding_preimageOfIntegralPoint (a : integralPoint K) :
+    mixedEmbedding K (preimageOfIntegralPoint a : 𝓞 K) = (a : mixedSpace K) := by
+  rw [preimageOfIntegralPoint, (mem_integralPoint.mp a.prop).2.choose_spec]
+
+theorem preimageOfIntegralPoint_mixedEmbedding {x : (𝓞 K)⁰}
+    (hx : mixedEmbedding K (x : 𝓞 K) ∈ integralPoint K) :
+    preimageOfIntegralPoint (⟨mixedEmbedding K (x : 𝓞 K), hx⟩) = x := by
+  simp_rw [Subtype.ext_iff, RingOfIntegers.ext_iff, ← (mixedEmbedding_injective K).eq_iff,
+    mixedEmbedding_preimageOfIntegralPoint]
+
+/-- If `x : mixedSpace K` is nonzero and the image of an algebraic integer, then there exists a
+unit such that `u • x ∈ integralPoint K`. -/
+theorem exists_unitSMul_mem_integralPoint {x : mixedSpace K} (hx : x ≠ 0)
+    (hx' : x ∈ mixedEmbedding K '' (Set.range (algebraMap (𝓞 K) K))) :
+    ∃ u : (𝓞 K)ˣ, u • x ∈ integralPoint K := by
+  replace hx : mixedEmbedding.norm x ≠ 0 :=
+      (norm_eq_zero_iff' (Set.mem_range_of_mem_image (mixedEmbedding K) _ hx')).not.mpr hx
+  obtain ⟨u, hu⟩ := exists_unit_smul_mem hx
+  obtain ⟨_, ⟨x, rfl⟩, _, rfl⟩ := hx'
+  exact ⟨u, mem_integralPoint.mpr ⟨hu, u * x, by simp_rw [unitSMul_smul, ← map_mul]⟩⟩
+
+/-- The set `integralPoint K` is stable under the action of the torsion. -/
+theorem torsion_unitSMul_mem_integralPoint {x : mixedSpace K} {ζ : (𝓞 K)ˣ} (hζ : ζ ∈ torsion K)
+    (hx : x ∈ integralPoint K) : ζ • x ∈ integralPoint K := by
+  obtain ⟨a, ⟨_, rfl⟩, rfl⟩ := (mem_integralPoint.mp hx).2
+  refine mem_integralPoint.mpr ⟨torsion_smul_mem_of_mem hx.1 hζ, ⟨ζ * a, by simp⟩⟩
+
+/-- The action of `torsion K` on `integralPoint K`. -/
+@[simps]
+instance integralPoint_torsionSMul: SMul (torsion K) (integralPoint K) where
+  smul := fun ⟨ζ, hζ⟩ ⟨x, hx⟩ ↦ ⟨ζ • x, torsion_unitSMul_mem_integralPoint hζ hx⟩
+
+instance : MulAction (torsion K) (integralPoint K) where
+  one_smul := fun _ ↦ by
+    rw [Subtype.mk_eq_mk, integralPoint_torsionSMul_smul_coe, OneMemClass.coe_one, one_smul]
+  mul_smul := fun _ _ _ ↦ by
+    rw [Subtype.mk_eq_mk]
+    simp_rw [integralPoint_torsionSMul_smul_coe, Subgroup.coe_mul, mul_smul]
+
+/-- The `mixedEmbedding.norm` of `a : integralPoint K` as a natural number, see also
+`intNorm_coe`. -/
+def intNorm (a : integralPoint K) : ℕ := (Algebra.norm ℤ (preimageOfIntegralPoint a : 𝓞 K)).natAbs
+
+@[simp]
+theorem intNorm_coe (a : integralPoint K) :
+    (intNorm a : ℝ) = mixedEmbedding.norm (a : mixedSpace K) := by
+  rw [intNorm, Int.cast_natAbs, ← Rat.cast_intCast, Int.cast_abs, Algebra.coe_norm_int,
+    ← norm_eq_norm, mixedEmbedding_preimageOfIntegralPoint]
+
+/-- The norm `intNorm` lifts to a function on `integralPoint K` modulo `torsion K`. -/
+def quotIntNorm :
+    Quotient (MulAction.orbitRel (torsion K) (integralPoint K)) → ℕ :=
+  Quotient.lift (fun x ↦ intNorm x) fun a b ⟨u, hu⟩ ↦ by
+    rw [← Nat.cast_inj (R := ℝ), intNorm_coe, intNorm_coe, ← hu, integralPoint_torsionSMul_smul_coe,
+      norm_unit_smul]
+
+@[simp]
+theorem quotIntNorm_apply (a : integralPoint K) : quotIntNorm ⟦a⟧ = intNorm a := rfl
+
+variable (K) in
+/-- The map that sends an element of `a : integralPoint K` to the associates class
+of its preimage in `(𝓞 K)⁰`. By quotienting by the kernel of the map, which is equal to the
+subgroup of torsion, we get the equivalence `integralPointQuotEquivAssociates`. -/
+def integralPointToAssociates (a : integralPoint K) : Associates (𝓞 K)⁰ :=
+  ⟦preimageOfIntegralPoint a⟧
+
+@[simp]
+theorem integralPointToAssociates_apply (a : integralPoint K) :
+    integralPointToAssociates K a = ⟦preimageOfIntegralPoint a⟧ := rfl
+
+variable (K) in
+theorem integralPointToAssociates_surjective :
+    Function.Surjective (integralPointToAssociates K) := by
+  rintro ⟨x⟩
+  obtain ⟨u, hu⟩ : ∃ u : (𝓞 K)ˣ, u • mixedEmbedding K (x : 𝓞 K) ∈ integralPoint K := by
+    refine exists_unitSMul_mem_integralPoint ?_ ⟨(x : 𝓞 K), Set.mem_range_self _, rfl⟩
+    exact (map_ne_zero _).mpr <| RingOfIntegers.coe_ne_zero_iff.mpr (nonZeroDivisors.coe_ne_zero _)
+  refine ⟨⟨u • mixedEmbedding K (x : 𝓞 K), hu⟩,
+    Quotient.sound ⟨unitsNonZeroDivisorsEquiv.symm u⁻¹, ?_⟩⟩
+  simp_rw [Subtype.ext_iff, RingOfIntegers.ext_iff, ← (mixedEmbedding_injective K).eq_iff,
+    Submonoid.coe_mul, map_mul, mixedEmbedding_preimageOfIntegralPoint,
+    unitSMul_smul, ← map_mul, mul_comm, map_inv, val_inv_unitsNonZeroDivisorsEquiv_symm_apply_coe,
+    Units.mul_inv_cancel_right]
+
+theorem integralPointToAssociates_eq_iff (a b : integralPoint K) :
+    integralPointToAssociates K a = integralPointToAssociates K b ↔
+      ∃ ζ : torsion K, ζ • a = b := by
+  simp_rw [integralPointToAssociates_apply, Associates.quotient_mk_eq_mk,
+    Associates.mk_eq_mk_iff_associated, Associated, mul_comm, Subtype.ext_iff,
+    RingOfIntegers.ext_iff, ← (mixedEmbedding_injective K).eq_iff, Submonoid.coe_mul, map_mul,
+    mixedEmbedding_preimageOfIntegralPoint, integralPoint_torsionSMul_smul_coe]
+  refine ⟨fun ⟨u, h⟩ ↦  ⟨⟨unitsNonZeroDivisorsEquiv u, ?_⟩, by simpa using h⟩,
+    fun ⟨⟨u, _⟩, h⟩ ↦ ⟨unitsNonZeroDivisorsEquiv.symm u, by simpa using h⟩⟩
+  exact (unit_smul_mem_iff_mem_torsion a.prop.1 _).mp (by simpa [h] using b.prop.1)
+
+variable (K) in
+/-- The equivalence between `integralPoint K` modulo `torsion K` and `Associates (𝓞 K)⁰`. -/
+def integralPointQuotEquivAssociates :
+    Quotient (MulAction.orbitRel (torsion K) (integralPoint K)) ≃ Associates (𝓞 K)⁰ :=
+  Equiv.ofBijective
+    (Quotient.lift (integralPointToAssociates K)
+      fun _ _ h ↦ ((integralPointToAssociates_eq_iff _ _).mpr h).symm)
+    ⟨by convert Setoid.ker_lift_injective (integralPointToAssociates K)
+        all_goals
+        · ext a b
+          rw [Setoid.ker_def, eq_comm, integralPointToAssociates_eq_iff b a,
+            MulAction.orbitRel_apply, MulAction.mem_orbit_iff],
+        (Quot.surjective_lift _).mpr (integralPointToAssociates_surjective K)⟩
+
+@[simp]
+theorem integralPointQuotEquivAssociates_apply (a : integralPoint K) :
+    integralPointQuotEquivAssociates K ⟦a⟧ = ⟦preimageOfIntegralPoint a⟧ := rfl
+
+theorem integralPoint_torsionSMul_stabilizer (a : integralPoint K) :
+    MulAction.stabilizer (torsion K) a = ⊥ := by
+  refine (Subgroup.eq_bot_iff_forall _).mpr fun ζ hζ ↦ ?_
+  rwa [MulAction.mem_stabilizer_iff, Subtype.ext_iff, integralPoint_torsionSMul_smul_coe,
+    unitSMul_smul, ← mixedEmbedding_preimageOfIntegralPoint, ← map_mul,
+    (mixedEmbedding_injective K).eq_iff, ← map_mul, ← RingOfIntegers.ext_iff, mul_eq_right₀,
+    Units.val_eq_one, OneMemClass.coe_eq_one] at hζ
+  exact nonZeroDivisors.coe_ne_zero _
+
+open Submodule Ideal
+
+variable (K) in
+/-- The equivalence between `integralPoint K` and the product of the set of nonzero principal
+ideals of `K` and the torsion of `K`. -/
+def integralPointEquiv :
+    integralPoint K ≃ {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.val} × torsion K :=
+  (MulAction.selfEquivSigmaOrbitsQuotientStabilizer (torsion K) (integralPoint K)).trans
+    ((Equiv.sigmaEquivProdOfEquiv (by
+        intro _
+        simp_rw [integralPoint_torsionSMul_stabilizer]
+        exact QuotientGroup.quotientBot.toEquiv)).trans
+      (Equiv.prodCongrLeft (fun _ ↦ (integralPointQuotEquivAssociates K).trans
+        (Ideal.associatesNonZeroDivisorsEquivIsPrincipal (𝓞 K)))))
+
+@[simp]
+theorem integralPointEquiv_apply_fst (a : integralPoint K) :
+    ((integralPointEquiv K a).1 : Ideal (𝓞 K)) = span {(preimageOfIntegralPoint a : 𝓞 K)} := rfl
+
+variable (K) in
+/-- For an integer `n`, The equivalence between the `integralPoint K` of norm `n` and the product
+of the set of nonzero principal ideals of `K` of norm `n` and the torsion of `K`. -/
+def integralPointEquivNorm (n : ℕ) :
+    {a : integralPoint K // intNorm a = n} ≃
+      {I : (Ideal (𝓞 K))⁰ // IsPrincipal (I : Ideal (𝓞 K)) ∧
+        absNorm (I : Ideal (𝓞 K)) = n} × (torsion K) :=
+  calc {a // intNorm a = n}
+      ≃ {I : {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1} × torsion K //
+          absNorm (I.1 : Ideal (𝓞 K)) = n} :=
+      (Equiv.subtypeEquiv (integralPointEquiv K) fun _ ↦ by simp [intNorm, absNorm_span_singleton])
+    _ ≃ {I : {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1} // absNorm (I.1 : Ideal (𝓞 K)) = n} ×
+          torsion K :=
+      Equiv.prodSubtypeFstEquivSubtypeProd (p := fun I : {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1} ↦
+        absNorm (I : Ideal (𝓞 K)) = n)
+    _ ≃ {I : (Ideal (𝓞 K))⁰ // IsPrincipal (I : Ideal (𝓞 K)) ∧
+          absNorm (I : Ideal (𝓞 K)) = n} × (torsion K) :=
+      Equiv.prodCongrLeft fun _ ↦ (Equiv.subtypeSubtypeEquivSubtypeInter
+        (fun I : (Ideal (𝓞 K))⁰ ↦ IsPrincipal I.1) (fun I ↦ absNorm I.1 = n))
+
+@[simp]
+theorem integralPointEquivNorm_apply_fst {n : ℕ} {a : integralPoint K} (ha : intNorm a = n) :
+    ((integralPointEquivNorm K n ⟨a, ha⟩).1 : Ideal (𝓞 K)) =
+      span {(preimageOfIntegralPoint a : 𝓞 K)} := by
+  simp_rw [integralPointEquivNorm, Equiv.prodSubtypeFstEquivSubtypeProd, Equiv.instTrans_trans,
+    Equiv.prodCongrLeft, Equiv.trans_apply, Equiv.subtypeEquiv_apply, Equiv.coe_fn_mk,
+    Equiv.subtypeSubtypeEquivSubtypeInter_apply_coe, integralPointEquiv_apply_fst]
+
+variable (K)
+
+/-- For `n` positive, the number of principal ideals in `𝓞 K` of norm `n` multiplied by the order
+of the torsion of `K` is equal to the number of `integralPoint K` of norm `n`. -/
+theorem card_isPrincipal_norm_eq_mul_torsion (n : ℕ) :
+    Nat.card {I : (Ideal (𝓞 K))⁰ | IsPrincipal (I : Ideal (𝓞 K)) ∧
+      absNorm (I : Ideal (𝓞 K)) = n} * torsionOrder K =
+        Nat.card {a : integralPoint K | intNorm a = n} := by
+  rw [torsionOrder, PNat.mk_coe, ← Nat.card_eq_fintype_card, ← Nat.card_prod]
+  exact Nat.card_congr (integralPointEquivNorm K n).symm
+
+/-- For `s : ℝ`, the number of principal nonzero ideals in `𝓞 K` of norm `≤ s` multiplied by the
+order of the torsion of `K` is equal to the number of `integralPoint K` of norm `≤ s`. -/
+theorem card_isPrincipal_norm_le_mul_torsion (s : ℝ) :
+    Nat.card {I : (Ideal (𝓞 K))⁰ | IsPrincipal (I : Ideal (𝓞 K)) ∧
+      absNorm (I : Ideal (𝓞 K)) ≤ s} * torsionOrder K =
+        Nat.card {a : integralPoint K | intNorm a ≤ s} := by
+  obtain hs | hs := le_or_gt 0 s
+  · simp_rw [← Nat.le_floor_iff hs]
+    rw [torsionOrder, PNat.mk_coe, ← Nat.card_eq_fintype_card, ← Nat.card_prod]
+    refine Nat.card_congr <| @Equiv.ofFiberEquiv _ (γ := Finset.Iic _) _
+      (fun I ↦ ⟨absNorm (I.1 : Ideal (𝓞 K)), Finset.mem_Iic.mpr I.1.2.2⟩)
+      (fun a ↦ ⟨intNorm a.1, Finset.mem_Iic.mpr a.2⟩) fun ⟨i, hi⟩ ↦ ?_
+    simp_rw [Subtype.mk.injEq]
+    calc
+    _ ≃ {I : {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1 ∧ absNorm I.1 ≤ _} // absNorm I.1.1 = i}
+          × torsion K := Equiv.prodSubtypeFstEquivSubtypeProd
+    _ ≃ {I : (Ideal (𝓞 K))⁰ // (IsPrincipal I.1 ∧ absNorm I.1 ≤ _) ∧ absNorm I.1 = i}
+          × torsion K :=
+      Equiv.prodCongrLeft fun _ ↦ (Equiv.subtypeSubtypeEquivSubtypeInter
+        (p := fun I : (Ideal (𝓞 K))⁰ ↦ IsPrincipal I.1 ∧ absNorm I.1 ≤ _)
+        (q := fun I ↦ absNorm I.1 = i))
+    _ ≃ {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1 ∧ absNorm I.1 = i ∧ absNorm I.1 ≤ _}
+          × torsion K :=
+      Equiv.prodCongrLeft fun _ ↦ (Equiv.subtypeEquivRight fun _ ↦ by aesop)
+    _ ≃ {I : (Ideal (𝓞 K))⁰ // IsPrincipal I.1 ∧ absNorm I.1 = i} × torsion K :=
+      Equiv.prodCongrLeft fun _ ↦ (Equiv.subtypeEquivRight fun _ ↦ by
+      rw [and_iff_left_of_imp (a := absNorm _ = _) fun h ↦ Finset.mem_Iic.mp (h ▸ hi)])
+    _ ≃ {a : integralPoint K // intNorm a = i} := (integralPointEquivNorm K i).symm
+    _ ≃ {a : {a : integralPoint K // intNorm a ≤ _} // intNorm a.1 = i} :=
+      (Equiv.subtypeSubtypeEquivSubtype fun h ↦ Finset.mem_Iic.mp (h ▸ hi)).symm
+  · simp_rw [lt_iff_not_le.mp (lt_of_lt_of_le hs (Nat.cast_nonneg _)), and_false, Set.setOf_false,
+      Nat.card_eq_fintype_card, Fintype.card_ofIsEmpty, zero_mul]
+
+end fundamentalCone
+
+end
+
+end NumberField.mixedEmbedding
diff --git a/Mathlib/NumberTheory/NumberField/ClassNumber.lean b/Mathlib/NumberTheory/NumberField/ClassNumber.lean
index c6726b41347b6..06ed53e2b5316 100644
--- a/Mathlib/NumberTheory/NumberField/ClassNumber.lean
+++ b/Mathlib/NumberTheory/NumberField/ClassNumber.lean
@@ -41,7 +41,7 @@ variable {K}
 theorem classNumber_eq_one_iff : classNumber K = 1 ↔ IsPrincipalIdealRing (𝓞 K) :=
   card_classGroup_eq_one_iff
 
-open FiniteDimensional NumberField.InfinitePlace
+open Module NumberField.InfinitePlace
 
 open scoped nonZeroDivisors Real
 
@@ -75,7 +75,7 @@ theorem _root_.RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt
       ((finrank ℚ K) ^ (finrank ℚ K) / (finrank ℚ K).factorial)) ^ 2) :
     IsPrincipalIdealRing (𝓞 K) := by
   have : 0 < finrank ℚ K := finrank_pos -- Lean needs to know that for positivity to succeed
-  rw [← Real.sqrt_lt (by positivity) (by positivity), mul_assoc, ← inv_mul_lt_iff' (by positivity),
+  rw [← Real.sqrt_lt (by positivity) (by positivity), mul_assoc, ← inv_mul_lt_iff₀' (by positivity),
     mul_inv, ← inv_pow, inv_div, inv_div, mul_assoc, Int.cast_abs] at h
   rw [← classNumber_eq_one_iff, classNumber, Fintype.card_eq_one_iff]
   refine ⟨1, fun C ↦ ?_⟩
diff --git a/Mathlib/NumberTheory/NumberField/Discriminant.lean b/Mathlib/NumberTheory/NumberField/Discriminant.lean
index d48472626310e..52a08d5551c4a 100644
--- a/Mathlib/NumberTheory/NumberField/Discriminant.lean
+++ b/Mathlib/NumberTheory/NumberField/Discriminant.lean
@@ -32,7 +32,7 @@ number field, discriminant
 
 namespace NumberField
 
-open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
+open Module NumberField NumberField.InfinitePlace Matrix
 
 open scoped Classical Real nonZeroDivisors
 
@@ -66,7 +66,7 @@ theorem discr_eq_discr_of_algEquiv {L : Type*} [Field L] [NumberField L] (f : K
     Basis.map_apply]
   rfl
 
-open MeasureTheory MeasureTheory.Measure Zspan NumberField.mixedEmbedding
+open MeasureTheory MeasureTheory.Measure ZSpan NumberField.mixedEmbedding
   NumberField.InfinitePlace ENNReal NNReal Complex
 
 theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
@@ -95,7 +95,7 @@ theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis
           coe_inv two_ne_zero, coe_ofNat, nnnorm_pow, NNReal.sqrt_sq]
       _ = (2 : ℝ≥0∞)⁻¹ ^ Fintype.card { w // IsComplex w } * NNReal.sqrt ‖discr K‖₊ := by
         rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
-          ← coe_discr, map_intCast, ← Complex.nnnorm_int]
+          ← coe_discr, map_intCast, ← Complex.nnnorm_intCast]
   ext : 2
   dsimp only [M]
   rw [Matrix.map_apply, Basis.toMatrix_apply, Basis.coe_reindex, Function.comp_apply,
@@ -173,7 +173,7 @@ theorem abs_discr_ge (h : 1 < finrank ℚ K) :
     rw [← Algebra.coe_norm_int, ← Int.cast_one, ← Int.cast_abs, Rat.cast_intCast, Int.cast_le]
     exact Int.one_le_abs (Algebra.norm_ne_zero_iff.mpr h_nz)
   replace h_bd := le_trans h_nm h_bd
-  rw [← inv_mul_le_iff (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
+  rw [← inv_mul_le_iff₀ (by positivity), inv_div, mul_one, Real.le_sqrt (by positivity)
     (by positivity), ← Int.cast_abs, div_pow, mul_pow, ← pow_mul, ← pow_mul] at h_bd
   refine le_trans ?_ h_bd
   -- The sequence `a n` is a lower bound for `|discr K|`. We prove below by induction an uniform
@@ -277,7 +277,7 @@ theorem rank_le_rankOfDiscrBdd :
     refine fun h ↦ discr_ne_zero K ?_
     rwa [h, Nat.cast_zero, abs_nonpos_iff] at hK
   have h₂ : 1 < 3 * π / 4 := by
-    rw [_root_.lt_div_iff (by positivity), ← _root_.div_lt_iff' (by positivity), one_mul]
+    rw [_root_.lt_div_iff₀ (by positivity), ← _root_.div_lt_iff₀' (by positivity), one_mul]
     linarith [Real.pi_gt_three]
   obtain h | h := lt_or_le 1 (finrank ℚ K)
   · apply le_max_of_le_right
@@ -307,7 +307,7 @@ theorem minkowskiBound_lt_boundOfDiscBdd : minkowskiBound K ↑1 < boundOfDiscBd
     ENNReal.ofReal_one, one_mul, mixedEmbedding.finrank, volume_fundamentalDomain_latticeBasis,
     coe_mul, ENNReal.coe_pow, coe_ofNat, show sqrt N = (1 : ℝ≥0∞) * sqrt N by rw [one_mul]]
   gcongr
-  · exact pow_le_one _ (by positivity) (by norm_num)
+  · exact pow_le_one₀ (by positivity) (by norm_num)
   · rwa [sqrt_le_sqrt, ← NNReal.coe_le_coe, coe_nnnorm, Int.norm_eq_abs, ← Int.cast_abs,
       NNReal.coe_natCast, ← Int.cast_natCast, Int.cast_le]
   · exact one_le_two
@@ -337,6 +337,8 @@ theorem finite_of_discr_bdd_of_isReal :
       (Set.finite_Icc (-C : ℤ) C)) (fun ⟨K, hK₀⟩ ⟨hK₁, hK₂⟩ ↦ ?_)
   -- We now need to prove that each field is generated by an element of the union of the rootset
   simp_rw [Set.mem_iUnion]
+  -- this is purely an optimization
+  have : CharZero K := SubsemiringClass.instCharZero K
   haveI : NumberField K := @NumberField.mk _ _ inferInstance hK₀
   obtain ⟨w₀, hw₀⟩ := hK₁
   suffices minkowskiBound K ↑1 < (convexBodyLTFactor K) * B by
@@ -360,12 +362,15 @@ theorem finite_of_discr_bdd_of_isReal :
     · refine mem_rootSet.mpr ⟨minpoly.ne_zero hx, ?_⟩
       exact (aeval_algebraMap_eq_zero_iff _ _ _).mpr (minpoly.aeval ℤ (x : K))
     · rw [← (IntermediateField.lift_injective _).eq_iff, eq_comm] at hx₁
-      convert hx₁ <;> simp
+      convert hx₁
+      · simp only [IntermediateField.lift_top]
+      · simp only [IntermediateField.lift_adjoin, Set.image_singleton]
   have := one_le_convexBodyLTFactor K
   convert lt_of_le_of_lt (mul_right_mono (coe_le_coe.mpr this))
     (ENNReal.mul_lt_mul_left' (by positivity) coe_ne_top (minkowskiBound_lt_boundOfDiscBdd hK₂))
   simp_rw [ENNReal.coe_one, one_mul]
 
+
 theorem finite_of_discr_bdd_of_isComplex :
     {K : { F : IntermediateField ℚ A // FiniteDimensional ℚ F} |
       haveI :  NumberField K := @NumberField.mk _ _ inferInstance K.prop
@@ -380,6 +385,8 @@ theorem finite_of_discr_bdd_of_isComplex :
       (Set.finite_Icc (-C : ℤ) C)) (fun ⟨K, hK₀⟩ ⟨hK₁, hK₂⟩ ↦ ?_)
   -- We now need to prove that each field is generated by an element of the union of the rootset
   simp_rw [Set.mem_iUnion]
+  -- this is purely an optimization
+  have : CharZero K := SubsemiringClass.instCharZero K
   haveI : NumberField K := @NumberField.mk _ _ inferInstance hK₀
   obtain ⟨w₀, hw₀⟩ := hK₁
   suffices minkowskiBound K ↑1 < (convexBodyLT'Factor K) * boundOfDiscBdd N by
@@ -404,7 +411,9 @@ theorem finite_of_discr_bdd_of_isComplex :
     · refine mem_rootSet.mpr ⟨minpoly.ne_zero hx, ?_⟩
       exact (aeval_algebraMap_eq_zero_iff _ _ _).mpr (minpoly.aeval ℤ (x : K))
     · rw [← (IntermediateField.lift_injective _).eq_iff, eq_comm] at hx₁
-      convert hx₁ <;> simp
+      convert hx₁
+      · simp only [IntermediateField.lift_top]
+      · simp only [IntermediateField.lift_adjoin, Set.image_singleton]
   have := one_le_convexBodyLT'Factor K
   convert lt_of_le_of_lt (mul_right_mono (coe_le_coe.mpr this))
     (ENNReal.mul_lt_mul_left' (by positivity) coe_ne_top (minkowskiBound_lt_boundOfDiscBdd hK₂))
@@ -419,6 +428,8 @@ theorem _root_.NumberField.finite_of_discr_bdd :
   refine Set.Finite.subset (Set.Finite.union (finite_of_discr_bdd_of_isReal A N)
     (finite_of_discr_bdd_of_isComplex A N)) ?_
   rintro ⟨K, hK₀⟩ hK₁
+  -- this is purely an optimization
+  have : CharZero K := SubsemiringClass.instCharZero K
   haveI : NumberField K := @NumberField.mk _ _ inferInstance hK₀
   obtain ⟨w₀⟩ := (inferInstance : Nonempty (InfinitePlace K))
   by_cases hw₀ : IsReal w₀
diff --git a/Mathlib/NumberTheory/NumberField/Embeddings.lean b/Mathlib/NumberTheory/NumberField/Embeddings.lean
index d3f1dd0fe2cb7..9e1f2fc310559 100644
--- a/Mathlib/NumberTheory/NumberField/Embeddings.lean
+++ b/Mathlib/NumberTheory/NumberField/Embeddings.lean
@@ -38,7 +38,7 @@ namespace NumberField.Embeddings
 
 section Fintype
 
-open FiniteDimensional
+open Module
 
 variable (K : Type*) [Field K] [NumberField K]
 variable (A : Type*) [Field A] [CharZero A]
@@ -55,7 +55,7 @@ theorem card : Fintype.card (K →+* A) = finrank ℚ K := by
 
 instance : Nonempty (K →+* A) := by
   rw [← Fintype.card_pos_iff, NumberField.Embeddings.card K A]
-  exact FiniteDimensional.finrank_pos
+  exact Module.finrank_pos
 
 end Fintype
 
@@ -78,7 +78,7 @@ end Roots
 
 section Bounded
 
-open FiniteDimensional Polynomial Set
+open Module Polynomial Set
 
 variable {K : Type*} [Field K] [NumberField K]
 variable {A : Type*} [NormedField A] [IsAlgClosed A] [NormedAlgebra ℚ A]
@@ -259,7 +259,7 @@ open NumberField
 
 instance {K : Type*} [Field K] : FunLike (InfinitePlace K) K ℝ where
   coe w x := w.1 x
-  coe_injective' := fun _ _ h => Subtype.eq (AbsoluteValue.ext fun x => congr_fun h x)
+  coe_injective' _ _ h := Subtype.eq (AbsoluteValue.ext fun x => congr_fun h x)
 
 instance : MonoidWithZeroHomClass (InfinitePlace K) K ℝ where
   map_mul w _ _ := w.1.map_mul _ _
@@ -450,7 +450,7 @@ noncomputable instance NumberField.InfinitePlace.fintype [NumberField K] :
     Fintype (InfinitePlace K) := Set.fintypeRange _
 
 theorem sum_mult_eq [NumberField K] :
-    ∑ w : InfinitePlace K, mult w = FiniteDimensional.finrank ℚ K := by
+    ∑ w : InfinitePlace K, mult w = Module.finrank ℚ K := by
   rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
     (fun φ => InfinitePlace.mk φ)]
   exact Finset.sum_congr rfl
@@ -483,7 +483,7 @@ section NumberField
 variable [NumberField K]
 
 /-- The infinite part of the product formula : for `x ∈ K`, we have `Π_w ‖x‖_w = |norm(x)|` where
-`‖·‖_w` is the normalized absolute value for `w`.  -/
+`‖·‖_w` is the normalized absolute value for `w`. -/
 theorem prod_eq_abs_norm (x : K) :
     ∏ w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm ℚ x) := by
   convert (congr_arg Complex.abs (@Algebra.norm_eq_prod_embeddings ℚ _ _ _ _ ℂ _ _ _ _ _ x)).symm
@@ -505,7 +505,7 @@ theorem one_le_of_lt_one {w : InfinitePlace K} {a : (𝓞 K)} (ha : a ≠ 0)
     rw [← InfinitePlace.prod_eq_abs_norm, ← Finset.prod_const_one]
     refine Finset.prod_lt_prod_of_nonempty (fun _ _ ↦ ?_) (fun z _ ↦ ?_) Finset.univ_nonempty
     · exact pow_pos (pos_iff.mpr ((Subalgebra.coe_eq_zero _).not.mpr ha)) _
-    · refine pow_lt_one (apply_nonneg _ _) ?_ (by rw [mult]; split_ifs <;> norm_num)
+    · refine pow_lt_one₀ (apply_nonneg _ _) ?_ (by rw [mult]; split_ifs <;> norm_num)
       by_cases hz : z = w
       · rwa [hz]
       · exact h hz
@@ -546,7 +546,7 @@ theorem _root_.NumberField.adjoin_eq_top_of_infinitePlace_lt {x : 𝓞 K} {w : I
 
 end NumberField
 
-open Fintype FiniteDimensional
+open Fintype Module
 
 variable (K)
 
@@ -1024,12 +1024,12 @@ lemma IsUnramifiedAtInfinitePlaces_of_odd_card_aut [IsGalois k K] [FiniteDimensi
   ⟨fun _ ↦ not_not.mp (Nat.not_even_iff_odd.2 h ∘ InfinitePlace.even_card_aut_of_not_isUnramified)⟩
 
 lemma IsUnramifiedAtInfinitePlaces_of_odd_finrank [IsGalois k K]
-    (h : Odd (FiniteDimensional.finrank k K)) : IsUnramifiedAtInfinitePlaces k K :=
+    (h : Odd (Module.finrank k K)) : IsUnramifiedAtInfinitePlaces k K :=
   ⟨fun _ ↦ not_not.mp (Nat.not_even_iff_odd.2 h ∘ InfinitePlace.even_finrank_of_not_isUnramified)⟩
 
 variable (k K)
 
-open FiniteDimensional in
+open Module in
 lemma IsUnramifiedAtInfinitePlaces.card_infinitePlace [NumberField k] [NumberField K]
     [IsGalois k K] [IsUnramifiedAtInfinitePlaces k K] :
     Fintype.card (InfinitePlace K) = Fintype.card (InfinitePlace k) * finrank k K := by
diff --git a/Mathlib/NumberTheory/NumberField/EquivReindex.lean b/Mathlib/NumberTheory/NumberField/EquivReindex.lean
index 67fc4926a8307..0226ce9f8e5d5 100644
--- a/Mathlib/NumberTheory/NumberField/EquivReindex.lean
+++ b/Mathlib/NumberTheory/NumberField/EquivReindex.lean
@@ -21,7 +21,7 @@ namespace NumberField
 
 noncomputable section
 
-open Module.Free FiniteDimensional canonicalEmbedding Matrix Finset
+open Module.Free Module canonicalEmbedding Matrix Finset
 
 /-- An equivalence between the set of embeddings of `K` into `ℂ` and the
   index set of the chosen basis of the ring of integers of `K`. -/
diff --git a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
index 371f141943b5f..1444de4b7e903 100644
--- a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
+++ b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean
@@ -61,7 +61,7 @@ instance (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
     · refine Submonoid.mul_mem _ hd (mem_nonZeroDivisors_of_ne_zero ?_)
       rw [Nat.cast_ne_zero, ne_eq, Ideal.absNorm_eq_zero_iff]
       exact FractionalIdeal.num_eq_zero_iff.not.mpr <| Units.ne_zero I
-    · simp_rw [LinearMap.coe_restrictScalars, Submodule.coeSubtype] at h ⊢
+    · simp_rw [LinearMap.coe_restrictScalars, Submodule.coe_subtype] at h ⊢
       rw [← h]
       simp only [Submonoid.mk_smul, zsmul_eq_mul, Int.cast_mul, Int.cast_natCast, algebraMap_int_eq,
         eq_intCast, map_intCast]
@@ -89,7 +89,7 @@ theorem mem_span_basisOfFractionalIdeal {I : (FractionalIdeal (𝓞 K)⁰ K)ˣ}
   rw [basisOfFractionalIdeal, (fractionalIdealBasis K I.1).ofIsLocalizedModule_span ℚ ℤ⁰ _]
   simp
 
-open FiniteDimensional in
+open Module in
 theorem fractionalIdeal_rank (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ) :
     finrank ℤ I = finrank ℤ (𝓞 K) := by
   rw [finrank_eq_card_chooseBasisIndex, RingOfIntegers.rank,
@@ -103,7 +103,7 @@ open Module
 
 /-- The absolute value of the determinant of the base change from `integralBasis` to
 `basisOfFractionalIdeal I` is equal to the norm of `I`. -/
-theorem det_basisOfFractionalIdeal_eq_absNorm  (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
+theorem det_basisOfFractionalIdeal_eq_absNorm (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
     (e : (Free.ChooseBasisIndex ℤ (𝓞 K)) ≃ (Free.ChooseBasisIndex ℤ I)) :
     |(integralBasis K).det ((basisOfFractionalIdeal K I).reindex e.symm)| =
       FractionalIdeal.absNorm I.1 := by
diff --git a/Mathlib/NumberTheory/NumberField/House.lean b/Mathlib/NumberTheory/NumberField/House.lean
index 5d0cc0062fc60..d56eb4d18c668 100644
--- a/Mathlib/NumberTheory/NumberField/House.lean
+++ b/Mathlib/NumberTheory/NumberField/House.lean
@@ -27,7 +27,7 @@ namespace NumberField
 
 noncomputable section
 
-open Module.Free FiniteDimensional canonicalEmbedding Matrix Finset
+open Module.Free Module canonicalEmbedding Matrix Finset
 
 attribute [local instance] Matrix.seminormedAddCommGroup
 
@@ -62,7 +62,7 @@ noncomputable section
 
 variable (K)
 
-open Module.Free FiniteDimensional canonicalEmbedding Matrix Finset
+open Module.Free Module canonicalEmbedding Matrix Finset
 
 attribute [local instance] Matrix.seminormedAddCommGroup
 
diff --git a/Mathlib/NumberTheory/NumberField/Norm.lean b/Mathlib/NumberTheory/NumberField/Norm.lean
index 2b529a1d239bb..e8964a687a66b 100644
--- a/Mathlib/NumberTheory/NumberField/Norm.lean
+++ b/Mathlib/NumberTheory/NumberField/Norm.lean
@@ -22,7 +22,7 @@ rings of integers.
 
 open scoped NumberField
 
-open Finset NumberField Algebra FiniteDimensional
+open Finset NumberField Algebra Module
 
 section Rat
 
diff --git a/Mathlib/NumberTheory/NumberField/Units/Basic.lean b/Mathlib/NumberTheory/NumberField/Units/Basic.lean
index eaabbc19a44f8..7907293a0c950 100644
--- a/Mathlib/NumberTheory/NumberField/Units/Basic.lean
+++ b/Mathlib/NumberTheory/NumberField/Units/Basic.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Xavier Roblot
 -/
 import Mathlib.NumberTheory.NumberField.Embeddings
+import Mathlib.RingTheory.LocalRing.RingHom.Basic
 
 /-!
 # Units of a number field
diff --git a/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean b/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
index 33f35ed619614..12b15e1ddd7d7 100644
--- a/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
+++ b/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
@@ -84,10 +84,11 @@ variable {K}
 
 @[simp]
 theorem logEmbedding_component (x : (𝓞 K)ˣ) (w : {w : InfinitePlace K // w ≠ w₀}) :
-    (logEmbedding K x) w = mult w.val * Real.log (w.val x) := rfl
+    (logEmbedding K (Additive.ofMul x)) w = mult w.val * Real.log (w.val x) := rfl
 
 theorem sum_logEmbedding_component (x : (𝓞 K)ˣ) :
-    ∑ w, logEmbedding K x w = - mult (w₀ : InfinitePlace K) * Real.log (w₀ (x : K)) := by
+    ∑ w, logEmbedding K (Additive.ofMul x) w =
+      - mult (w₀ : InfinitePlace K) * Real.log (w₀ (x : K)) := by
   have h := congr_arg Real.log (prod_eq_abs_norm (x : K))
   rw [Units.norm, Rat.cast_one, Real.log_one, Real.log_prod] at h
   · simp_rw [Real.log_pow] at h
@@ -112,7 +113,7 @@ theorem mult_log_place_eq_zero {x : (𝓞 K)ˣ} {w : InfinitePlace K} :
 variable [NumberField K]
 
 theorem logEmbedding_eq_zero_iff {x : (𝓞 K)ˣ} :
-    logEmbedding K x = 0 ↔ x ∈ torsion K := by
+    logEmbedding K (Additive.ofMul x) = 0 ↔ x ∈ torsion K := by
   rw [mem_torsion]
   refine ⟨fun h w => ?_, fun h => ?_⟩
   · by_cases hw : w = w₀
@@ -126,13 +127,14 @@ theorem logEmbedding_eq_zero_iff {x : (𝓞 K)ˣ} :
     rw [logEmbedding_component, h w.val, Real.log_one, mul_zero, Pi.zero_apply]
 
 theorem logEmbedding_component_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
-    (w : {w : InfinitePlace K // w ≠ w₀}) : |logEmbedding K x w| ≤ r := by
+    (w : {w : InfinitePlace K // w ≠ w₀}) : |logEmbedding K (Additive.ofMul x) w| ≤ r := by
   lift r to NNReal using hr
   simp_rw [Pi.norm_def, NNReal.coe_le_coe, Finset.sup_le_iff, ← NNReal.coe_le_coe] at h
   exact h w (mem_univ _)
 
-theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r) (h : ‖logEmbedding K x‖ ≤ r)
-    (w : InfinitePlace K) : |Real.log (w x)| ≤ (Fintype.card (InfinitePlace K)) * r := by
+theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r)
+    (h : ‖logEmbedding K (Additive.ofMul x)‖ ≤ r) (w : InfinitePlace K) :
+    |Real.log (w x)| ≤ (Fintype.card (InfinitePlace K)) * r := by
   have tool : ∀ x : ℝ, 0 ≤ x → x ≤ mult w * x := fun x hx => by
     nth_rw 1 [← one_mul x]
     refine mul_le_mul ?_ le_rfl hx ?_
@@ -160,8 +162,8 @@ variable (K)
 
 /-- The lattice formed by the image of the logarithmic embedding. -/
 noncomputable def _root_.NumberField.Units.unitLattice :
-    AddSubgroup ({w : InfinitePlace K // w ≠ w₀} → ℝ) :=
-  AddSubgroup.map (logEmbedding K) ⊤
+    Submodule ℤ ({w : InfinitePlace K // w ≠ w₀} → ℝ) :=
+  Submodule.map (logEmbedding K).toIntLinearMap ⊤
 
 theorem unitLattice_inter_ball_finite (r : ℝ) :
     ((unitLattice K : Set ({ w : InfinitePlace K // w ≠ w₀} → ℝ)) ∩
@@ -215,7 +217,7 @@ theorem seq_next {x : 𝓞 K} (hx : x ≠ 0) :
     fun w => ⟨(w x) / 2, div_nonneg (AbsoluteValue.nonneg _ _) (by norm_num)⟩
   suffices ∀ w, w ≠ w₁ → f w ≠ 0 by
     obtain ⟨g, h_geqf, h_gprod⟩ := adjust_f K B this
-    obtain ⟨y, h_ynz, h_yle⟩ := exists_ne_zero_mem_ringOfIntegers_lt (f := g)
+    obtain ⟨y, h_ynz, h_yle⟩ := exists_ne_zero_mem_ringOfIntegers_lt K (f := g)
       (by rw [convexBodyLT_volume]; convert hB; exact congr_arg ((↑) : NNReal → ENNReal) h_gprod)
     refine ⟨y, h_ynz, fun w hw => (h_geqf w hw ▸ h_yle w).trans ?_, ?_⟩
     · rw [← Rat.cast_le (K := ℝ), Rat.cast_natCast]
@@ -302,7 +304,7 @@ theorem exists_unit (w₁ : InfinitePlace K) :
         _ = w (algebraMap (𝓞 K) K (seq K w₁ hB m)) * w (algebraMap (𝓞 K) K (seq K w₁ hB n))⁻¹ :=
           _root_.map_mul _ _ _
         _ < 1 := by
-          rw [map_inv₀, mul_inv_lt_iff (pos_iff.mpr (seq_ne_zero K w₁ hB n)), mul_one]
+          rw [map_inv₀, mul_inv_lt_iff₀ (pos_iff.mpr (seq_ne_zero K w₁ hB n)), one_mul]
           exact seq_decreasing K w₁ hB hnm w hw
   refine Set.Finite.exists_lt_map_eq_of_forall_mem
     (t := { I : Ideal (𝓞 K) | 1 ≤ Ideal.absNorm I ∧ Ideal.absNorm I ≤ B })
@@ -320,13 +322,13 @@ theorem unitLattice_span_eq_top :
   -- The standard basis
   let B := Pi.basisFun ℝ {w : InfinitePlace K // w ≠ w₀}
   -- The image by log_embedding of the family of units constructed above
-  let v := fun w : { w : InfinitePlace K // w ≠ w₀ } => logEmbedding K (exists_unit K w).choose
+  let v := fun w : { w : InfinitePlace K // w ≠ w₀ } =>
+    logEmbedding K (Additive.ofMul (exists_unit K w).choose)
   -- To prove the result, it is enough to prove that the family `v` is linearly independent
   suffices B.det v ≠ 0 by
     rw [← isUnit_iff_ne_zero, ← is_basis_iff_det] at this
     rw [← this.2]
-    exact Submodule.span_monotone (fun _ ⟨w, hw⟩ =>
-      ⟨(exists_unit K w).choose, trivial, by rw [← hw]⟩)
+    refine  Submodule.span_monotone fun _ ⟨w, hw⟩ ↦ ⟨(exists_unit K w).choose, trivial, hw⟩
   rw [Basis.det_apply]
   -- We use a specific lemma to prove that this determinant is nonzero
   refine det_ne_zero_of_sum_col_lt_diag (fun w => ?_)
@@ -352,7 +354,7 @@ section statements
 variable [NumberField K]
 
 open scoped Classical
-open dirichletUnitTheorem FiniteDimensional
+open dirichletUnitTheorem Module
 
 /-- The unit rank of the number field `K`, it is equal to `card (InfinitePlace K) - 1`. -/
 def rank : ℕ := Fintype.card (InfinitePlace K) - 1
@@ -368,7 +370,7 @@ instance instDiscrete_unitLattice : DiscreteTopology (unitLattice K) := by
     rintro ⟨x, hx, rfl⟩
     exact ⟨Subtype.mem x, hx⟩
 
-instance instZlattice_unitLattice : IsZlattice ℝ (unitLattice K) where
+instance instZLattice_unitLattice : IsZLattice ℝ (unitLattice K) where
   span_top := unitLattice_span_eq_top K
 
 protected theorem finrank_eq_rank :
@@ -379,7 +381,7 @@ protected theorem finrank_eq_rank :
 @[simp]
 theorem unitLattice_rank :
     finrank ℤ (unitLattice K) = Units.rank K := by
-  rw [← Units.finrank_eq_rank, Zlattice.rank ℝ]
+  rw [← Units.finrank_eq_rank, ZLattice.rank ℝ]
 
 /-- The map obtained by quotienting by the kernel of `logEmbedding`. -/
 def logEmbeddingQuot :
@@ -389,12 +391,12 @@ def logEmbeddingQuot :
       (QuotientGroup.quotientMulEquivOfEq (by
         ext
         rw [MonoidHom.mem_ker, AddMonoidHom.toMultiplicative'_apply_apply, ofAdd_eq_one,
-          ← logEmbedding_eq_zero_iff]
-        rfl)).toMonoidHom
+          ← logEmbedding_eq_zero_iff])).toMonoidHom
 
 @[simp]
 theorem logEmbeddingQuot_apply (x : (𝓞 K)ˣ) :
-    logEmbeddingQuot K ⟦x⟧ = logEmbedding K x := rfl
+    logEmbeddingQuot K (Additive.ofMul (QuotientGroup.mk x)) =
+      logEmbedding K (Additive.ofMul x) := rfl
 
 theorem logEmbeddingQuot_injective :
     Function.Injective (logEmbeddingQuot K) := by
@@ -420,15 +422,18 @@ set_option maxSynthPendingDepth 2 -- Note this is active for the remainder of th
 `unitLattice` . -/
 def logEmbeddingEquiv :
     Additive ((𝓞 K)ˣ ⧸ (torsion K)) ≃ₗ[ℤ] (unitLattice K) :=
-  (AddEquiv.ofBijective (AddMonoidHom.codRestrict (logEmbeddingQuot K) _
-  (Quotient.ind fun x ↦ logEmbeddingQuot_apply K _ ▸ AddSubgroup.mem_map_of_mem _ trivial))
-  ⟨fun _ _ ↦ by
-    rw [AddMonoidHom.codRestrict_apply, AddMonoidHom.codRestrict_apply, Subtype.mk.injEq]
-    apply logEmbeddingQuot_injective K, fun ⟨a, ⟨b, _, ha⟩⟩ ↦ ⟨⟦b⟧, by simp [ha]⟩⟩).toIntLinearEquiv
+  LinearEquiv.ofBijective ((logEmbeddingQuot K).codRestrict (unitLattice K)
+    (Quotient.ind fun x ↦ logEmbeddingQuot_apply K _ ▸
+      Submodule.mem_map_of_mem trivial)).toIntLinearMap
+    ⟨fun _ _ ↦ by
+      rw [AddMonoidHom.coe_toIntLinearMap, AddMonoidHom.codRestrict_apply,
+        AddMonoidHom.codRestrict_apply, Subtype.mk.injEq]
+      apply logEmbeddingQuot_injective K, fun ⟨a, ⟨b, _, ha⟩⟩ ↦ ⟨⟦b⟧, by simpa using ha⟩⟩
 
 @[simp]
 theorem logEmbeddingEquiv_apply (x : (𝓞 K)ˣ) :
-    logEmbeddingEquiv K ⟦x⟧ = logEmbedding K x := rfl
+    logEmbeddingEquiv K (Additive.ofMul (QuotientGroup.mk x)) =
+      logEmbedding K (Additive.ofMul x) := rfl
 
 instance : Module.Free ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) :=
   Module.Free.of_equiv (logEmbeddingEquiv K).symm
@@ -457,13 +462,17 @@ instance : Monoid.FG (𝓞 K)ˣ := by
   infer_instance
 
 theorem rank_modTorsion :
-    FiniteDimensional.finrank ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) = rank K := by
+    Module.finrank ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) = rank K := by
   rw [← LinearEquiv.finrank_eq (logEmbeddingEquiv K).symm, unitLattice_rank]
 
 /-- A basis of the quotient `(𝓞 K)ˣ ⧸ (torsion K)` seen as an additive ℤ-module. -/
 def basisModTorsion : Basis (Fin (rank K)) ℤ (Additive ((𝓞 K)ˣ ⧸ (torsion K))) :=
   Basis.reindex (Module.Free.chooseBasis ℤ _) (Fintype.equivOfCardEq <| by
-    rw [← FiniteDimensional.finrank_eq_card_chooseBasisIndex, rank_modTorsion, Fintype.card_fin])
+    rw [← Module.finrank_eq_card_chooseBasisIndex, rank_modTorsion, Fintype.card_fin])
+
+/-- The basis of the `unitLattice` obtained by mapping `basisModTorsion` via `logEmbedding`. -/
+def basisUnitLattice : Basis (Fin (rank K)) ℤ (unitLattice K) :=
+  (basisModTorsion K).map (logEmbeddingEquiv K)
 
 /-- A fundamental system of units of `K`. The units of `fundSystem` are arbitrary lifts of the
 units in `basisModTorsion`. -/
@@ -472,10 +481,12 @@ def fundSystem : Fin (rank K) → (𝓞 K)ˣ :=
   fun i => Quotient.out' (Additive.toMul (basisModTorsion K i):)
 
 theorem fundSystem_mk (i : Fin (rank K)) :
-    Additive.ofMul ⟦fundSystem K i⟧ = (basisModTorsion K i) := by
-  rw [fundSystem, Equiv.apply_eq_iff_eq_symm_apply, @Quotient.mk_eq_iff_out,
-    Quotient.out', Quotient.out_equiv_out]
-  rfl
+    Additive.ofMul (QuotientGroup.mk (fundSystem K i)) = (basisModTorsion K i) := by
+  simp_rw [fundSystem, Equiv.apply_eq_iff_eq_symm_apply, Additive.ofMul_symm_eq, Quotient.out_eq']
+
+theorem logEmbedding_fundSystem (i : Fin (rank K)) :
+    logEmbedding K (Additive.ofMul (fundSystem K i)) = basisUnitLattice K i := by
+  rw [basisUnitLattice, Basis.map_apply, ← fundSystem_mk, logEmbeddingEquiv_apply]
 
 /-- The exponents that appear in the unique decomposition of a unit as the product of
 a root of unity and powers of the units of the fundamental system `fundSystem` (see
@@ -513,5 +524,4 @@ theorem exist_unique_eq_mul_prod (x : (𝓞 K)ˣ) : ∃! ζe : torsion K × (Fin
 
 end statements
 
-
 end NumberField.Units
diff --git a/Mathlib/NumberTheory/NumberField/Units/Regulator.lean b/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
index f25c3dd48cda3..34d7ce2339491 100644
--- a/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
+++ b/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Xavier Roblot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Xavier Roblot
 -/
-import Mathlib.Algebra.Module.Zlattice.Covolume
+import Mathlib.Algebra.Module.ZLattice.Covolume
 import Mathlib.LinearAlgebra.Matrix.Determinant.Misc
 import Mathlib.NumberTheory.NumberField.Units.DirichletTheorem
 
@@ -38,11 +38,11 @@ open MeasureTheory Classical BigOperators NumberField.InfinitePlace
 variable [NumberField K]
 
 /-- The regulator of a number field `K`. -/
-def regulator : ℝ := Zlattice.covolume (unitLattice K)
+def regulator : ℝ := ZLattice.covolume (unitLattice K)
 
-theorem regulator_ne_zero : regulator K ≠ 0 := Zlattice.covolume_ne_zero (unitLattice K) volume
+theorem regulator_ne_zero : regulator K ≠ 0 := ZLattice.covolume_ne_zero (unitLattice K) volume
 
-theorem regulator_pos : 0 < regulator K := Zlattice.covolume_pos (unitLattice K) volume
+theorem regulator_pos : 0 < regulator K := ZLattice.covolume_pos (unitLattice K) volume
 
 #adaptation_note
 /--
@@ -57,11 +57,11 @@ local instance : CommGroup (𝓞 K)ˣ := inferInstance
 set_option maxSynthPendingDepth 2 -- Note this is active for the remainder of the file.
 
 theorem regulator_eq_det' (e : {w : InfinitePlace K // w ≠ w₀} ≃ Fin (rank K)) :
-    regulator K = |(Matrix.of fun i ↦ (logEmbedding K) (fundSystem K (e i))).det| := by
-  simp_rw [regulator, Zlattice.covolume_eq_det _
+    regulator K = |(Matrix.of fun i ↦
+      logEmbedding K (Additive.ofMul (fundSystem K (e i)))).det| := by
+  simp_rw [regulator, ZLattice.covolume_eq_det _
     (((basisModTorsion K).map (logEmbeddingEquiv K)).reindex e.symm), Basis.coe_reindex,
-    Function.comp, Basis.map_apply, ← fundSystem_mk, Equiv.symm_symm]
-  rfl
+    Function.comp_def, Basis.map_apply, ← fundSystem_mk, Equiv.symm_symm, logEmbeddingEquiv_apply]
 
 /-- Let `u : Fin (rank K) → (𝓞 K)ˣ` be a family of units and let `w₁` and `w₂` be two infinite
 places. Then, the two square matrices with entries `(mult w * log w (u i))_i, {w ≠ w_i}`, `i = 1,2`,
diff --git a/Mathlib/NumberTheory/Ostrowski.lean b/Mathlib/NumberTheory/Ostrowski.lean
index c41547159ecef..de32a03798da5 100644
--- a/Mathlib/NumberTheory/Ostrowski.lean
+++ b/Mathlib/NumberTheory/Ostrowski.lean
@@ -6,7 +6,7 @@ María Inés de Frutos-Fernández, Sam van Gool, Silvain Rideau-Kikuchi, Amos Tu
 Francesco Veneziano
 -/
 
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Analysis.SpecialFunctions.Log.Base
 import Mathlib.Analysis.SpecialFunctions.Pow.Real
 import Mathlib.Analysis.Normed.Ring.Seminorm
diff --git a/Mathlib/NumberTheory/Padics/Hensel.lean b/Mathlib/NumberTheory/Padics/Hensel.lean
index d32fe3c1f17a2..05f97027ce268 100644
--- a/Mathlib/NumberTheory/Padics/Hensel.lean
+++ b/Mathlib/NumberTheory/Padics/Hensel.lean
@@ -131,7 +131,7 @@ private theorem T_lt_one : T < 1 := by
   have h := (div_lt_one (deriv_sq_norm_pos hnorm)).2 hnorm
   rw [T_def]; exact h
 
-private theorem T_pow {n : ℕ} (hn : n ≠ 0) : T ^ n < 1 := pow_lt_one T_nonneg (T_lt_one hnorm) hn
+private theorem T_pow {n : ℕ} (hn : n ≠ 0) : T ^ n < 1 := pow_lt_one₀ T_nonneg (T_lt_one hnorm) hn
 
 private theorem T_pow' (n : ℕ) : T ^ 2 ^ n < 1 := T_pow hnorm (pow_ne_zero _ two_ne_zero)
 
@@ -156,7 +156,7 @@ private theorem calc_norm_le_one {n : ℕ} {z : ℤ_[p]} (hz : ih n z) :
       gcongr
       apply hz.2
     _ = ‖F.derivative.eval a‖ * T ^ 2 ^ n := div_sq_cancel _ _
-    _ ≤ 1 := mul_le_one (PadicInt.norm_le_one _) (T_pow_nonneg _) (le_of_lt (T_pow' hnorm _))
+    _ ≤ 1 := mul_le_one₀ (PadicInt.norm_le_one _) (T_pow_nonneg _) (le_of_lt (T_pow' hnorm _))
 
 
 private theorem calc_deriv_dist {z z' z1 : ℤ_[p]} (hz' : z' = z - z1)
@@ -183,7 +183,7 @@ private def calc_eval_z' {z z' z1 : ℤ_[p]} (hz' : z' = z - z1) {n} (hz : ih n
   obtain ⟨q, hq⟩ := F.binomExpansion z (-z1)
   have : ‖(↑(F.derivative.eval z) * (↑(F.eval z) / ↑(F.derivative.eval z)) : ℚ_[p])‖ ≤ 1 := by
     rw [padicNormE.mul]
-    exact mul_le_one (PadicInt.norm_le_one _) (norm_nonneg _) h1
+    exact mul_le_one₀ (PadicInt.norm_le_one _) (norm_nonneg _) h1
   have : F.derivative.eval z * -z1 = -F.eval z := by
     calc
       F.derivative.eval z * -z1 =
@@ -275,7 +275,7 @@ private theorem newton_seq_dist_aux (n : ℕ) :
   | 0 => by simp [T_pow_nonneg, mul_nonneg]
   | k + 1 =>
     have : 2 ^ n ≤ 2 ^ (n + k) := by
-      apply pow_le_pow_right
+      apply pow_right_mono₀
       · norm_num
       · apply Nat.le_add_right
     calc
@@ -356,7 +356,7 @@ private theorem T_pos : T > 0 := by
 private theorem newton_seq_succ_dist_weak (n : ℕ) :
     ‖newton_seq (n + 2) - newton_seq (n + 1)‖ < ‖F.eval a‖ / ‖F.derivative.eval a‖ :=
   have : 2 ≤ 2 ^ (n + 1) := by
-    have := pow_le_pow_right (by norm_num : 1 ≤ 2) (Nat.le_add_left _ _ : 1 ≤ n + 1)
+    have := pow_right_mono₀ (by norm_num : 1 ≤ 2) (Nat.le_add_left _ _ : 1 ≤ n + 1)
     simpa using this
   calc
     ‖newton_seq (n + 2) - newton_seq (n + 1)‖ ≤ ‖F.derivative.eval a‖ * T ^ 2 ^ (n + 1) :=
@@ -401,7 +401,7 @@ private theorem soln_dist_to_a : ‖soln - a‖ = ‖F.eval a‖ / ‖F.derivati
   tendsto_nhds_unique (newton_seq_dist_tendsto' hnorm) (newton_seq_dist_tendsto hnorm hnsol)
 
 private theorem soln_dist_to_a_lt_deriv : ‖soln - a‖ < ‖F.derivative.eval a‖ := by
-  rw [soln_dist_to_a, div_lt_iff (deriv_norm_pos _), ← sq] <;> assumption
+  rw [soln_dist_to_a, div_lt_iff₀ (deriv_norm_pos _), ← sq] <;> assumption
 
 private theorem soln_unique (z : ℤ_[p]) (hev : F.eval z = 0)
     (hnlt : ‖z - a‖ < ‖F.derivative.eval a‖) : z = soln :=
diff --git a/Mathlib/NumberTheory/Padics/PadicIntegers.lean b/Mathlib/NumberTheory/Padics/PadicIntegers.lean
index b7a318c97639b..a7add4280a7e7 100644
--- a/Mathlib/NumberTheory/Padics/PadicIntegers.lean
+++ b/Mathlib/NumberTheory/Padics/PadicIntegers.lean
@@ -78,7 +78,7 @@ def subring : Subring ℚ_[p] where
   zero_mem' := by norm_num
   one_mem' := by norm_num
   add_mem' hx hy := (padicNormE.nonarchimedean _ _).trans <| max_le_iff.2 ⟨hx, hy⟩
-  mul_mem' hx hy := (padicNormE.mul _ _).trans_le <| mul_le_one hx (norm_nonneg _) hy
+  mul_mem' hx hy := (padicNormE.mul _ _).trans_le <| mul_le_one₀ hx (norm_nonneg _) hy
   neg_mem' hx := (norm_neg _).trans_le hx
 
 @[simp]
@@ -304,7 +304,7 @@ variable (p : ℕ) [hp : Fact p.Prime]
 theorem exists_pow_neg_lt {ε : ℝ} (hε : 0 < ε) : ∃ k : ℕ, (p : ℝ) ^ (-(k : ℤ)) < ε := by
   obtain ⟨k, hk⟩ := exists_nat_gt ε⁻¹
   use k
-  rw [← inv_lt_inv hε (_root_.zpow_pos_of_pos _ _)]
+  rw [← inv_lt_inv₀ hε (_root_.zpow_pos_of_pos _ _)]
   · rw [zpow_neg, inv_inv, zpow_natCast]
     apply lt_of_lt_of_le hk
     norm_cast
@@ -334,7 +334,7 @@ theorem norm_int_le_pow_iff_dvd {k : ℤ} {n : ℕ} :
 /-! ### Valuation on `ℤ_[p]` -/
 
 
-/-- `PadicInt.valuation` lifts the `p`-adic valuation on `ℚ` to `ℤ_[p]`.  -/
+/-- `PadicInt.valuation` lifts the `p`-adic valuation on `ℚ` to `ℤ_[p]`. -/
 def valuation (x : ℤ_[p]) :=
   Padic.valuation (x : ℚ_[p])
 
@@ -468,7 +468,7 @@ theorem norm_le_pow_iff_le_valuation (x : ℤ_[p]) (hx : x ≠ 0) (n : ℕ) :
     intro m
     refine pow_pos ?_ m
     exact mod_cast hp.1.pos
-  rw [inv_le_inv (aux _) (aux _)]
+  rw [inv_le_inv₀ (aux _) (aux _)]
   have : p ^ n ≤ p ^ k ↔ n ≤ k := (pow_right_strictMono hp.1.one_lt).le_iff_le
   rw [← this]
   norm_cast
@@ -494,7 +494,7 @@ theorem norm_le_pow_iff_mem_span_pow (x : ℤ_[p]) (n : ℕ) :
     ‖x‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ x ∈ (Ideal.span {(p : ℤ_[p]) ^ n} : Ideal ℤ_[p]) := by
   by_cases hx : x = 0
   · subst hx
-    simp only [norm_zero, zpow_neg, zpow_natCast, inv_nonneg, iff_true_iff, Submodule.zero_mem]
+    simp only [norm_zero, zpow_neg, zpow_natCast, inv_nonneg, iff_true, Submodule.zero_mem]
     exact mod_cast Nat.zero_le _
   rw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx]
 
@@ -528,7 +528,7 @@ instance : LocalRing ℤ_[p] :=
   LocalRing.of_nonunits_add <| by simp only [mem_nonunits]; exact fun x y => norm_lt_one_add
 
 theorem p_nonnunit : (p : ℤ_[p]) ∈ nonunits ℤ_[p] := by
-  have : (p : ℝ)⁻¹ < 1 := inv_lt_one <| mod_cast hp.1.one_lt
+  have : (p : ℝ)⁻¹ < 1 := inv_lt_one_of_one_lt₀ <| mod_cast hp.1.one_lt
   rwa [← norm_p, ← mem_nonunits] at this
 
 theorem maximalIdeal_eq_span_p : maximalIdeal ℤ_[p] = Ideal.span {(p : ℤ_[p])} := by
diff --git a/Mathlib/NumberTheory/Padics/PadicNorm.lean b/Mathlib/NumberTheory/Padics/PadicNorm.lean
index 164a53abcb173..c61676a7816b8 100644
--- a/Mathlib/NumberTheory/Padics/PadicNorm.lean
+++ b/Mathlib/NumberTheory/Padics/PadicNorm.lean
@@ -92,7 +92,7 @@ theorem padicNorm_of_prime_of_ne {q : ℕ} [p_prime : Fact p.Prime] [q_prime : F
 
 See also `padicNorm.padicNorm_p_lt_one_of_prime` for a version assuming `p` is prime. -/
 theorem padicNorm_p_lt_one (hp : 1 < p) : padicNorm p p < 1 := by
-  rw [padicNorm_p hp, inv_lt_one_iff]
+  rw [padicNorm_p hp, inv_lt_one_iff₀]
   exact mod_cast Or.inr hp
 
 /-- The `p`-adic norm of `p` is less than `1` if `p` is prime.
@@ -246,7 +246,7 @@ theorem int_eq_one_iff (m : ℤ) : padicNorm p m = 1 ↔ ¬(p : ℤ) ∣ m := by
   simp only [dvd_iff_norm_le, Int.cast_natCast, Nat.cast_one, zpow_neg, zpow_one, not_le]
   constructor
   · intro h
-    rw [h, inv_lt_one_iff_of_pos] <;> norm_cast
+    rw [h, inv_lt_one₀] <;> norm_cast
     · exact Nat.Prime.one_lt Fact.out
     · exact Nat.Prime.pos Fact.out
   · simp only [padicNorm]
@@ -262,7 +262,7 @@ theorem int_eq_one_iff (m : ℤ) : padicNorm p m = 1 ↔ ¬(p : ℤ) ∣ m := by
 
 theorem int_lt_one_iff (m : ℤ) : padicNorm p m < 1 ↔ (p : ℤ) ∣ m := by
   rw [← not_iff_not, ← int_eq_one_iff, eq_iff_le_not_lt]
-  simp only [padicNorm.of_int, true_and_iff]
+  simp only [padicNorm.of_int, true_and]
 
 theorem of_nat (m : ℕ) : padicNorm p m ≤ 1 :=
   padicNorm.of_int (m : ℤ)
diff --git a/Mathlib/NumberTheory/Padics/PadicNumbers.lean b/Mathlib/NumberTheory/Padics/PadicNumbers.lean
index 58bc95a1f23b1..ff2f78e73d60b 100644
--- a/Mathlib/NumberTheory/Padics/PadicNumbers.lean
+++ b/Mathlib/NumberTheory/Padics/PadicNumbers.lean
@@ -5,7 +5,7 @@ Authors: Robert Y. Lewis
 -/
 import Mathlib.RingTheory.Valuation.Basic
 import Mathlib.NumberTheory.Padics.PadicNorm
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.Tactic.Peel
 import Mathlib.Topology.MetricSpace.Ultra.Basic
 
@@ -779,8 +779,7 @@ theorem norm_p : ‖(p : ℚ_[p])‖ = (p : ℝ)⁻¹ := by
 
 theorem norm_p_lt_one : ‖(p : ℚ_[p])‖ < 1 := by
   rw [norm_p]
-  apply inv_lt_one
-  exact mod_cast hp.1.one_lt
+  exact inv_lt_one_of_one_lt₀ <| mod_cast hp.1.one_lt
 
 -- Porting note: Linter thinks this is a duplicate simp lemma, so `priority` is assigned
 @[simp (high)]
@@ -833,7 +832,7 @@ theorem norm_rat_le_one : ∀ {q : ℚ} (_ : ¬p ∣ q.den), ‖(q : ℚ_[p])‖
       -- Porting note: `Nat.cast_zero` instead of another `norm_cast` call
       rw [padicNorm.eq_zpow_of_nonzero hnz', padicValRat, neg_sub,
         padicValNat.eq_zero_of_not_dvd hq, Nat.cast_zero, zero_sub, zpow_neg, zpow_natCast]
-      apply inv_le_one
+      apply inv_le_one_of_one_le₀
       norm_cast
       apply one_le_pow
       exact hp.1.pos
@@ -860,8 +859,7 @@ theorem norm_int_lt_one_iff_dvd (k : ℤ) : ‖(k : ℚ_[p])‖ < 1 ↔ ↑p ∣
         mul_le_mul le_rfl (by simpa using norm_int_le_one _) (norm_nonneg _) (norm_nonneg _)
       _ < 1 := by
         rw [mul_one, padicNormE.norm_p]
-        apply inv_lt_one
-        exact mod_cast hp.1.one_lt
+        exact inv_lt_one_of_one_lt₀ <| mod_cast hp.1.one_lt
 
 theorem norm_int_le_pow_iff_dvd (k : ℤ) (n : ℕ) :
     ‖(k : ℚ_[p])‖ ≤ (p : ℝ) ^ (-n : ℤ) ↔ (p ^ n : ℤ) ∣ k := by
diff --git a/Mathlib/NumberTheory/Padics/ProperSpace.lean b/Mathlib/NumberTheory/Padics/ProperSpace.lean
new file mode 100644
index 0000000000000..5aeae0f7642ee
--- /dev/null
+++ b/Mathlib/NumberTheory/Padics/ProperSpace.lean
@@ -0,0 +1,67 @@
+/-
+Copyright (c) 2024 Jou Glasheen. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jou Glasheen, Kevin Buzzard
+-/
+
+import Mathlib.Analysis.Normed.Field.ProperSpace
+import Mathlib.NumberTheory.Padics.RingHoms
+
+/-!
+# Properness of the p-adic numbers
+
+In this file, we prove that `ℤ_[p]` is totally bounded and compact,
+and that `ℚ_[p]` is proper.
+
+## Main results
+
+- `PadicInt.totallyBounded_univ` : The set of p-adic integers `ℤ_[p]` is totally bounded.
+- `PadicInt.compactSpace` : The set of p-adic integers `ℤ_[p]` is a compact topological space.
+- `Padic.instProperSpace` : The field of p-adic numbers `ℚ_[p]` is a proper metric space.
+
+## Notation
+
+ - `p` : Is a natural prime.
+
+## References
+
+Gouvêa, F. Q. (2020) p-adic Numbers An Introduction. 3rd edition.
+  Cham, Springer International Publishing
+-/
+
+assert_not_exists FiniteDimensional
+
+open Metric Topology
+
+variable (p : ℕ) [Fact (Nat.Prime p)]
+
+namespace PadicInt
+
+/-- The set of p-adic integers `ℤ_[p]` is totally bounded. -/
+theorem totallyBounded_univ : TotallyBounded (Set.univ : Set ℤ_[p]) := by
+  refine Metric.totallyBounded_iff.mpr (fun ε hε ↦ ?_)
+  obtain ⟨k, hk⟩ := exists_pow_neg_lt p hε
+  refine ⟨Nat.cast '' Finset.range (p ^ k), Set.toFinite _, fun z _ ↦ ?_⟩
+  simp only [PadicInt, Set.mem_iUnion, Metric.mem_ball, exists_prop, Set.exists_mem_image]
+  refine ⟨z.appr k, ?_, ?_⟩
+  · simpa only [Finset.mem_coe, Finset.mem_range] using z.appr_lt k
+  · exact (((z - z.appr k).norm_le_pow_iff_mem_span_pow k).mpr (z.appr_spec k)).trans_lt hk
+
+/-- The set of p-adic integers `ℤ_[p]` is a compact topological space. -/
+instance compactSpace : CompactSpace ℤ_[p] := by
+  rw [← isCompact_univ_iff, isCompact_iff_totallyBounded_isComplete]
+  exact ⟨totallyBounded_univ p, complete_univ⟩
+
+end PadicInt
+
+namespace Padic
+
+/-- The field of p-adic numbers `ℚ_[p]` is a proper metric space. -/
+instance : ProperSpace ℚ_[p] := by
+  suffices LocallyCompactSpace ℚ_[p] from .of_nontriviallyNormedField_of_weaklyLocallyCompactSpace _
+  have : closedBall 0 1 ∈ 𝓝 (0 : ℚ_[p]) := closedBall_mem_nhds _ zero_lt_one
+  simp only [closedBall, dist_eq_norm_sub, sub_zero] at this
+  refine IsCompact.locallyCompactSpace_of_mem_nhds_of_addGroup ?_ this
+  simpa only [isCompact_iff_compactSpace] using PadicInt.compactSpace p
+
+end Padic
diff --git a/Mathlib/NumberTheory/Padics/RingHoms.lean b/Mathlib/NumberTheory/Padics/RingHoms.lean
index 31ba8f2ff384a..8f65a9cbc788e 100644
--- a/Mathlib/NumberTheory/Padics/RingHoms.lean
+++ b/Mathlib/NumberTheory/Padics/RingHoms.lean
@@ -256,7 +256,7 @@ theorem toZMod_spec : x - (ZMod.cast (toZMod x) : ℤ_[p]) ∈ maximalIdeal ℤ_
   dsimp [toZMod, toZModHom]
   rcases Nat.exists_eq_add_of_lt hp_prime.1.pos with ⟨p', rfl⟩
   change ↑((_ : ZMod (0 + p' + 1)).val) = (_ : ℤ_[0 + p' + 1])
-  simp only [ZMod.val_natCast, add_zero, add_def, Nat.cast_inj, zero_add]
+  rw [Nat.cast_inj]
   apply mod_eq_of_lt
   simpa only [zero_add] using zmodRepr_lt_p x
 
diff --git a/Mathlib/NumberTheory/Pell.lean b/Mathlib/NumberTheory/Pell.lean
index 4ffe309128a95..8cdd56d82e73b 100644
--- a/Mathlib/NumberTheory/Pell.lean
+++ b/Mathlib/NumberTheory/Pell.lean
@@ -206,7 +206,7 @@ theorem y_ne_zero_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : a.y ≠ 0 :=
 theorem d_pos_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : 0 < d := by
   refine pos_of_mul_pos_left ?_ (sq_nonneg a.y)
   rw [a.prop_y, sub_pos]
-  exact one_lt_pow ha two_ne_zero
+  exact one_lt_pow₀ ha two_ne_zero
 
 /-- If a solution has `x > 1`, then `d` is not a square. -/
 theorem d_nonsquare_of_one_lt_x {a : Solution₁ d} (ha : 1 < a.x) : ¬IsSquare d := by
@@ -305,8 +305,8 @@ theorem exists_pos_variant (h₀ : 0 < d) (a : Solution₁ d) :
         (lt_or_gt_of_ne (a.x_ne_zero h₀.le)).elim
           ((le_total 0 a.y).elim (fun hy hx => ⟨-a⁻¹, ?_, ?_, ?_⟩) fun hy hx => ⟨-a, ?_, ?_, ?_⟩)
           ((le_total 0 a.y).elim (fun hy hx => ⟨a, hx, hy, ?_⟩) fun hy hx => ⟨a⁻¹, hx, ?_, ?_⟩) <;>
-      simp only [neg_neg, inv_inv, neg_inv, Set.mem_insert_iff, Set.mem_singleton_iff, true_or_iff,
-        eq_self_iff_true, x_neg, x_inv, y_neg, y_inv, neg_pos, neg_nonneg, or_true_iff] <;>
+      simp only [neg_neg, inv_inv, neg_inv, Set.mem_insert_iff, Set.mem_singleton_iff, true_or,
+        eq_self_iff_true, x_neg, x_inv, y_neg, y_inv, neg_pos, neg_nonneg, or_true] <;>
     assumption
 
 end Solution₁
diff --git a/Mathlib/NumberTheory/PrimeCounting.lean b/Mathlib/NumberTheory/PrimeCounting.lean
index 92dca84e7b65e..3005d0cb7165a 100644
--- a/Mathlib/NumberTheory/PrimeCounting.lean
+++ b/Mathlib/NumberTheory/PrimeCounting.lean
@@ -3,8 +3,8 @@ Copyright (c) 2021 Bolton Bailey. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Bolton Bailey, Ralf Stephan
 -/
-import Mathlib.Data.Nat.Totient
 import Mathlib.Data.Nat.Nth
+import Mathlib.Data.Nat.Totient
 import Mathlib.NumberTheory.SmoothNumbers
 
 /-!
@@ -56,6 +56,10 @@ def primeCounting (n : ℕ) : ℕ :=
 
 open scoped Nat.Prime
 
+@[simp]
+theorem primeCounting_sub_one (n : ℕ) : π (n - 1) = π' n := by
+  cases n <;> rfl
+
 theorem monotone_primeCounting' : Monotone primeCounting' :=
   count_monotone Prime
 
@@ -66,13 +70,38 @@ theorem monotone_primeCounting : Monotone primeCounting :=
 theorem primeCounting'_nth_eq (n : ℕ) : π' (nth Prime n) = n :=
   count_nth_of_infinite infinite_setOf_prime _
 
+@[simp]
+theorem zeroth_prime_eq_two : nth Prime 0 = 2 := nth_count prime_two
+
+/-- The `n`th prime is greater or equal to `n + 2`. -/
+theorem add_two_le_nth_prime (n : ℕ) : n + 2 ≤ nth Prime n :=
+  zeroth_prime_eq_two ▸ (nth_strictMono infinite_setOf_prime).add_le_nat n 0
+
+theorem surjective_primeCounting' : Function.Surjective π' :=
+  Nat.surjective_count_of_infinite_setOf infinite_setOf_prime
+
+theorem surjective_primeCounting : Function.Surjective π := by
+  suffices Function.Surjective (π ∘ fun n => n - 1) from this.of_comp
+  convert surjective_primeCounting'
+  ext
+  exact primeCounting_sub_one _
+
+open Filter
+
+theorem tendsto_primeCounting' : Tendsto π' atTop atTop := by
+  apply tendsto_atTop_atTop_of_monotone' monotone_primeCounting'
+  simp [Set.range_iff_surjective.mpr surjective_primeCounting']
+
+theorem tensto_primeCounting : Tendsto π atTop atTop :=
+  (tendsto_add_atTop_iff_nat 1).mpr tendsto_primeCounting'
+
 @[simp]
 theorem prime_nth_prime (n : ℕ) : Prime (nth Prime n) :=
   nth_mem_of_infinite infinite_setOf_prime _
 
 /-- The cardinality of the finset `primesBelow n` equals the counting function
 `primeCounting'` at `n`. -/
-lemma primesBelow_card_eq_primeCounting' (n : ℕ) : n.primesBelow.card = primeCounting' n := by
+theorem primesBelow_card_eq_primeCounting' (n : ℕ) : n.primesBelow.card = primeCounting' n := by
   simp only [primesBelow, primeCounting']
   exact (count_eq_card_filter_range Prime n).symm
 
@@ -98,11 +127,4 @@ theorem primeCounting'_add_le {a k : ℕ} (h0 : 0 < a) (h1 : a < k) (n : ℕ) :
       rw [add_le_add_iff_left]
       exact Ico_filter_coprime_le k n h0
 
-@[simp]
-theorem zeroth_prime_eq_two : nth Prime 0 = 2 := nth_count prime_two
-
-/-- The `n`th prime is greater or equal to `n + 2`. -/
-lemma add_two_le_nth_prime (n : ℕ) : n + 2 ≤ nth Prime n :=
-  zeroth_prime_eq_two ▸ (nth_strictMono infinite_setOf_prime).add_le_nat n 0
-
 end Nat
diff --git a/Mathlib/NumberTheory/PythagoreanTriples.lean b/Mathlib/NumberTheory/PythagoreanTriples.lean
index b2482ab6382cc..57ca7ff109082 100644
--- a/Mathlib/NumberTheory/PythagoreanTriples.lean
+++ b/Mathlib/NumberTheory/PythagoreanTriples.lean
@@ -179,7 +179,7 @@ theorem normalize : PythagoreanTriple (x / Int.gcd x y) (y / Int.gcd x y) (z / I
     have hz : z = 0 := by
       simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,
         or_self_iff] using h
-    simp only [hx, hy, hz, Int.zero_div]
+    simp only [hx, hy, hz]
     exact zero
   rcases h.gcd_dvd with ⟨z0, rfl⟩
   obtain ⟨k, x0, y0, k0, h2, rfl, rfl⟩ :
@@ -462,8 +462,7 @@ theorem isPrimitiveClassified_of_coprime_of_odd_of_pos (hc : Int.gcd x y = 1) (h
   let q := (circleEquivGen hQ).symm ⟨⟨v, w⟩, hp⟩
   have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
     apply Prod.mk.inj
-    have := ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
-    exact congr_arg Subtype.val this
+    exact congr_arg Subtype.val ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm
   let m := (q.den : ℤ)
   let n := q.num
   have hm0 : m ≠ 0 := by
diff --git a/Mathlib/NumberTheory/RamificationInertia.lean b/Mathlib/NumberTheory/RamificationInertia.lean
index 08845015bb89a..10d1ba5cf00ef 100644
--- a/Mathlib/NumberTheory/RamificationInertia.lean
+++ b/Mathlib/NumberTheory/RamificationInertia.lean
@@ -47,7 +47,7 @@ variable {R : Type u} [CommRing R]
 variable {S : Type v} [CommRing S] (f : R →+* S)
 variable (p : Ideal R) (P : Ideal S)
 
-open FiniteDimensional
+open Module
 
 open UniqueFactorizationMonoid
 
@@ -699,7 +699,7 @@ instance Factors.finiteDimensional_quotient [IsNoetherian R S] [p.IsMaximal]
 
 theorem Factors.inertiaDeg_ne_zero [IsNoetherian R S] [p.IsMaximal]
     (P : (factors (map (algebraMap R S) p)).toFinset) : inertiaDeg (algebraMap R S) p P ≠ 0 := by
-  rw [inertiaDeg_algebraMap]; exact (FiniteDimensional.finrank_pos_iff.mpr inferInstance).ne'
+  rw [inertiaDeg_algebraMap]; exact (Module.finrank_pos_iff.mpr inferInstance).ne'
 
 instance Factors.finiteDimensional_quotient_pow [IsNoetherian R S] [p.IsMaximal]
     (P : (factors (map (algebraMap R S) p)).toFinset) :
@@ -793,4 +793,69 @@ theorem sum_ramification_inertia (K L : Type*) [Field K] [Field L] [IsDedekindDo
 
 end FactorsMap
 
+section tower
+
+variable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
+
+theorem ramificationIdx_tower [IsDedekindDomain S] [IsDedekindDomain T] {f : R →+* S} {g : S →+* T}
+    {p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
+    (hg0 : map g P ≠ ⊥) (hfg : map (g.comp f) p ≠ ⊥) (hg : map g P ≤ Q) :
+    ramificationIdx (g.comp f) p Q = ramificationIdx f p P * ramificationIdx g P Q := by
+  classical
+  have hf0 : map f p ≠ ⊥ :=
+    ne_bot_of_map_ne_bot (Eq.mp (congrArg (fun I ↦ I ≠ ⊥) (map_map f g).symm) hfg)
+  have hp0 : P ≠ ⊥ := ne_bot_of_map_ne_bot hg0
+  have hq0 : Q ≠ ⊥ := ne_bot_of_le_ne_bot hg0 hg
+  letI : P.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp0 hpm
+  rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hf0 hpm hp0,
+    IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hg0 hqm hq0,
+    IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hfg hqm hq0, ← map_map]
+  rcases eq_prime_pow_mul_coprime hf0 P with ⟨I, hcp, heq⟩
+  have hcp : ⊤ = map g P ⊔ map g I := by rw [← map_sup, hcp, map_top g]
+  have hntq : ¬ ⊤ ≤ Q := fun ht ↦ IsPrime.ne_top hqm (Iff.mpr (eq_top_iff_one Q) (ht trivial))
+  nth_rw 1 [heq, map_mul, Ideal.map_pow, normalizedFactors_mul (pow_ne_zero _ hg0) <| by
+    by_contra h
+    simp only [h, Submodule.zero_eq_bot, bot_le, sup_of_le_left] at hcp
+    exact hntq (hcp.trans_le hg), Multiset.count_add, normalizedFactors_pow, Multiset.count_nsmul]
+  exact add_right_eq_self.mpr <| Decidable.byContradiction fun h ↦ hntq <| hcp.trans_le <|
+    sup_le hg <| le_of_dvd <| dvd_of_mem_normalizedFactors <| Multiset.count_ne_zero.mp h
+
+attribute [local instance] Quotient.field in
+theorem inertiaDeg_tower {f : R →+* S} {g : S →+* T} {p : Ideal R} {P : Ideal S} {I : Ideal T}
+    [p.IsMaximal] [P.IsMaximal] (hp : p = comap f P) (hP : P = comap g I) :
+    inertiaDeg (g.comp f) p I = inertiaDeg f p P * inertiaDeg g P I := by
+  have h : comap (g.comp f) I = p := by rw [hp, hP, comap_comap]
+  simp only [inertiaDeg, dif_pos hp.symm, dif_pos hP.symm, dif_pos h]
+  letI : Algebra (R ⧸ p) (S ⧸ P) := Ideal.Quotient.algebraQuotientOfLEComap (le_of_eq hp)
+  letI : Algebra (S ⧸ P) (T ⧸ I) := Ideal.Quotient.algebraQuotientOfLEComap (le_of_eq hP)
+  letI : Algebra (R ⧸ p) (T ⧸ I) := Ideal.Quotient.algebraQuotientOfLEComap (le_of_eq h.symm)
+  letI : IsScalarTower (R ⧸ p) (S ⧸ P) (T ⧸ I) := IsScalarTower.of_algebraMap_eq (by rintro ⟨⟩; rfl)
+  exact (finrank_mul_finrank (R ⧸ p) (S ⧸ P) (T ⧸ I)).symm
+
+variable [Algebra R S] [Algebra S T] [Algebra R T] [IsScalarTower R S T]
+
+/-- Let `T / S / R` be a tower of algebras, `p, P, Q` be ideals in `R, S, T` respectively,
+  and `P` and `Q` are prime. If `P = Q ∩ S`, then `e (Q | p) = e (P | p) * e (Q | P)`. -/
+theorem ramificationIdx_algebra_tower [IsDedekindDomain S] [IsDedekindDomain T]
+    {p : Ideal R} {P : Ideal S} {Q : Ideal T} [hpm : P.IsPrime] [hqm : Q.IsPrime]
+    (hg0 : map (algebraMap S T) P ≠ ⊥)
+    (hfg : map (algebraMap R T) p ≠ ⊥) (hg : map (algebraMap S T) P ≤ Q) :
+    ramificationIdx (algebraMap R T) p Q =
+    ramificationIdx (algebraMap R S) p P * ramificationIdx (algebraMap S T) P Q := by
+  classical
+  rw [IsScalarTower.algebraMap_eq R S T] at hfg ⊢
+  exact ramificationIdx_tower hg0 hfg hg
+
+/-- Let `T / S / R` be a tower of algebras, `p, P, I` be ideals in `R, S, T`, respectively,
+  and `p` and `P` are maximal. If `p = P ∩ S` and `P = Q ∩ S`,
+  then `f (Q | p) = f (P | p) * f (Q | P)`. -/
+theorem inertiaDeg_algebra_tower {p : Ideal R} {P : Ideal S} {I : Ideal T} [p.IsMaximal]
+    [P.IsMaximal] (hp : p = comap (algebraMap R S) P) (hP : P = comap (algebraMap S T) I) :
+    inertiaDeg (algebraMap R T) p I =
+    inertiaDeg (algebraMap R S) p P * inertiaDeg (algebraMap S T) P I := by
+  rw [IsScalarTower.algebraMap_eq R S T]
+  exact inertiaDeg_tower hp hP
+
+end tower
+
 end Ideal
diff --git a/Mathlib/NumberTheory/Rayleigh.lean b/Mathlib/NumberTheory/Rayleigh.lean
index d7a2e5f205472..0e29615583c61 100644
--- a/Mathlib/NumberTheory/Rayleigh.lean
+++ b/Mathlib/NumberTheory/Rayleigh.lean
@@ -63,9 +63,9 @@ private theorem no_collision (hrs : r.IsConjExponent s) :
     Disjoint {beattySeq r k | k} {beattySeq' s k | k} := by
   rw [Set.disjoint_left]
   intro j ⟨k, h₁⟩ ⟨m, h₂⟩
-  rw [beattySeq, Int.floor_eq_iff, ← div_le_iff₀ hrs.pos, ← lt_div_iff hrs.pos] at h₁
+  rw [beattySeq, Int.floor_eq_iff, ← div_le_iff₀ hrs.pos, ← lt_div_iff₀ hrs.pos] at h₁
   rw [beattySeq', sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one,
-    add_sub_cancel_right, ← div_lt_iff hrs.symm.pos, ← le_div_iff₀ hrs.symm.pos] at h₂
+    add_sub_cancel_right, ← div_lt_iff₀ hrs.symm.pos, ← le_div_iff₀ hrs.symm.pos] at h₂
   have h₃ := add_lt_add_of_le_of_lt h₁.1 h₂.1
   have h₄ := add_lt_add_of_lt_of_le h₁.2 h₂.2
   simp_rw [div_eq_inv_mul, ← right_distrib, hrs.inv_add_inv_conj, one_mul] at h₃ h₄
@@ -91,10 +91,10 @@ private theorem hit_or_miss (h : r > 0) :
   -- for both cases, the candidate is `k = ⌈(j + 1) / r⌉ - 1`
   cases lt_or_ge ((⌈(j + 1) / r⌉ - 1) * r) j
   · refine Or.inr ⟨⌈(j + 1) / r⌉ - 1, ?_⟩
-    rw [Int.cast_sub, Int.cast_one, lt_div_iff h, sub_add_cancel]
+    rw [Int.cast_sub, Int.cast_one, lt_div_iff₀ h, sub_add_cancel]
     exact ⟨‹_›, Int.le_ceil _⟩
   · refine Or.inl ⟨⌈(j + 1) / r⌉ - 1, ?_⟩
-    rw [beattySeq, Int.floor_eq_iff, Int.cast_sub, Int.cast_one, ← lt_div_iff h, sub_lt_iff_lt_add]
+    rw [beattySeq, Int.floor_eq_iff, Int.cast_sub, Int.cast_one, ← lt_div_iff₀ h, sub_lt_iff_lt_add]
     exact ⟨‹_›, Int.ceil_lt_add_one _⟩
 
 /-- Let `0 < r ∈ ℝ` and `j ∈ ℤ`. Then either `j ∈ B'_r` or `B'_r` jumps over `j`. -/
diff --git a/Mathlib/NumberTheory/SmoothNumbers.lean b/Mathlib/NumberTheory/SmoothNumbers.lean
index 18f25f9900202..a130280fc2731 100644
--- a/Mathlib/NumberTheory/SmoothNumbers.lean
+++ b/Mathlib/NumberTheory/SmoothNumbers.lean
@@ -233,7 +233,7 @@ def equivProdNatFactoredNumbers {s : Finset ℕ} {p : ℕ} (hp : p.Prime) (hs :
       refine prod_eq <|
         (filter _ <| perm_primeFactorsList_mul (pow_ne_zero e hp.ne_zero) hm₀).trans ?_
       rw [filter_append, hp.primeFactorsList_pow,
-          filter_eq_nil.mpr fun q hq ↦ by rw [mem_replicate] at hq; simp [hq.2, hs],
+          filter_eq_nil_iff.mpr fun q hq ↦ by rw [mem_replicate] at hq; simp [hq.2, hs],
           nil_append, filter_eq_self.mpr fun q hq ↦ by simp only [hm q hq, decide_True]]
   right_inv := by
     rintro ⟨m, hm₀, hm⟩
diff --git a/Mathlib/NumberTheory/WellApproximable.lean b/Mathlib/NumberTheory/WellApproximable.lean
index 46e2e5d3371ff..7b72f49ff60e8 100644
--- a/Mathlib/NumberTheory/WellApproximable.lean
+++ b/Mathlib/NumberTheory/WellApproximable.lean
@@ -248,7 +248,7 @@ theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto 
     let f : 𝕊 → 𝕊 := fun y => (p : ℕ) • y
     suffices
       f '' A p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (p * δ n)) atTop fun n => 0 < n ∧ p∤n by
-      apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA₀ p)
+      apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA₀ p).nullMeasurableSet
       apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl
       exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∤n) (fun n => {y | addOrderOf y = n})
         (Nat.cast_pos.mpr hp.pos) _ hδ
@@ -261,7 +261,8 @@ theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto 
     let f : 𝕊 → 𝕊 := fun y => p • y + x
     suffices
       f '' B p ⊆ blimsup (fun n => approxAddOrderOf 𝕊 n (p * δ n)) atTop fun n => 0 < n ∧ p∣∣n by
-      apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le (hB₀ p)
+      apply (ergodic_nsmul_add x hp.one_lt).ae_empty_or_univ_of_image_ae_le
+        (hB₀ p).nullMeasurableSet
       apply (HasSubset.Subset.eventuallyLE this).congr EventuallyEq.rfl
       exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∣∣n) (fun n => {y | addOrderOf y = n})
         (Nat.cast_pos.mpr hp.pos) _ hδ
diff --git a/Mathlib/NumberTheory/Zsqrtd/Basic.lean b/Mathlib/NumberTheory/Zsqrtd/Basic.lean
index acbc027998af2..c73a74bb7a6b6 100644
--- a/Mathlib/NumberTheory/Zsqrtd/Basic.lean
+++ b/Mathlib/NumberTheory/Zsqrtd/Basic.lean
@@ -893,8 +893,8 @@ theorem norm_eq_zero {d : ℤ} (h_nonsquare : ∀ n : ℤ, d ≠ n * n) (a : ℤ
   · push_neg at h
     suffices a.re * a.re = 0 by
       rw [eq_zero_of_mul_self_eq_zero this] at ha ⊢
-      simpa only [true_and_iff, or_self_right, zero_re, zero_im, eq_self_iff_true, zero_eq_mul,
-        mul_zero, mul_eq_zero, h.ne, false_or_iff, or_self_iff] using ha
+      simpa only [true_and, or_self_right, zero_re, zero_im, eq_self_iff_true, zero_eq_mul,
+        mul_zero, mul_eq_zero, h.ne, false_or, or_self_iff] using ha
     apply _root_.le_antisymm _ (mul_self_nonneg _)
     rw [ha, mul_assoc]
     exact mul_nonpos_of_nonpos_of_nonneg h.le (mul_self_nonneg _)
diff --git a/Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean b/Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean
index a22d6696a31a2..413c023dba9e7 100644
--- a/Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean
+++ b/Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean
@@ -5,7 +5,7 @@ Authors: Chris Hughes
 -/
 import Mathlib.NumberTheory.Zsqrtd.GaussianInt
 import Mathlib.NumberTheory.LegendreSymbol.Basic
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 
 /-!
 # Facts about the gaussian integers relying on quadratic reciprocity.
diff --git a/Mathlib/Order/Atoms/Finite.lean b/Mathlib/Order/Atoms/Finite.lean
index c17782a493d32..93cca39127a7d 100644
--- a/Mathlib/Order/Atoms/Finite.lean
+++ b/Mathlib/Order/Atoms/Finite.lean
@@ -22,23 +22,26 @@ variable {α β : Type*}
 
 namespace IsSimpleOrder
 
+variable [LE α] [BoundedOrder α] [IsSimpleOrder α]
+
 section DecidableEq
 
 /- It is important that `IsSimpleOrder` is the last type-class argument of this instance,
 so that type-class inference fails quickly if it doesn't apply. -/
-instance (priority := 200) {α} [DecidableEq α] [LE α] [BoundedOrder α] [IsSimpleOrder α] :
-    Fintype α :=
+instance (priority := 200) [DecidableEq α] : Fintype α :=
   Fintype.ofEquiv Bool equivBool.symm
 
 end DecidableEq
 
+instance (priority := 200) : Finite α := by classical infer_instance
+
 end IsSimpleOrder
 
 namespace Fintype
 
 namespace IsSimpleOrder
 
-variable [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]
+variable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]
 
 theorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by
   change Finset.map _ (Finset.univ : Finset Bool) = _
diff --git a/Mathlib/Order/Basic.lean b/Mathlib/Order/Basic.lean
index 663f2c7b91967..08b21b2de4986 100644
--- a/Mathlib/Order/Basic.lean
+++ b/Mathlib/Order/Basic.lean
@@ -186,7 +186,7 @@ end
 
 namespace Eq
 
-variable [Preorder α] {x y z : α}
+variable [Preorder α] {x y : α}
 
 /-- If `x = y` then `y ≤ x`. Note: this lemma uses `y ≤ x` instead of `x ≥ y`, because `le` is used
 almost exclusively in mathlib. -/
@@ -308,7 +308,7 @@ alias LE.le.eq_iff_not_lt := eq_iff_not_lt_of_le
 protected theorem Decidable.eq_iff_le_not_lt [@DecidableRel α (· ≤ ·)] :
     a = b ↔ a ≤ b ∧ ¬a < b :=
   ⟨fun h ↦ ⟨h.le, h ▸ lt_irrefl _⟩, fun ⟨h₁, h₂⟩ ↦
-    h₁.antisymm <| Decidable.by_contradiction fun h₃ ↦ h₂ (h₁.lt_of_not_le h₃)⟩
+    h₁.antisymm <| Decidable.byContradiction fun h₃ ↦ h₂ (h₁.lt_of_not_le h₃)⟩
 
 theorem eq_iff_le_not_lt : a = b ↔ a ≤ b ∧ ¬a < b :=
   haveI := Classical.dec
@@ -442,7 +442,7 @@ theorem eq_of_forall_lt_iff [LinearOrder α] {a b : α} (h : ∀ c, c < a ↔ c
 theorem eq_of_forall_gt_iff [LinearOrder α] {a b : α} (h : ∀ c, a < c ↔ b < c) : a = b :=
   (le_of_forall_lt' fun _ ↦ (h _).2).antisymm <| le_of_forall_lt' fun _ ↦ (h _).1
 
-/-- A symmetric relation implies two values are equal, when it implies they're less-equal.  -/
+/-- A symmetric relation implies two values are equal, when it implies they're less-equal. -/
 theorem rel_imp_eq_of_rel_imp_le [PartialOrder β] (r : α → α → Prop) [IsSymm α r] {f : α → β}
     (h : ∀ a b, r a b → f a ≤ f b) {a b : α} : r a b → f a = f b := fun hab ↦
   le_antisymm (h a b hab) (h b a <| symm hab)
@@ -463,11 +463,12 @@ theorem commutative_of_le {f : β → β → α} (comm : ∀ a b, f a b ≤ f b
 
 /-- To prove associativity of a commutative binary operation `○`, we only to check
 `(a ○ b) ○ c ≤ a ○ (b ○ c)` for all `a`, `b`, `c`. -/
-theorem associative_of_commutative_of_le {f : α → α → α} (comm : Commutative f)
-    (assoc : ∀ a b c, f (f a b) c ≤ f a (f b c)) : Associative f := fun a b c ↦
-  le_antisymm (assoc _ _ _) <| by
-    rw [comm, comm b, comm _ c, comm a]
-    exact assoc _ _ _
+theorem associative_of_commutative_of_le {f : α → α → α} (comm : Std.Commutative f)
+    (assoc : ∀ a b c, f (f a b) c ≤ f a (f b c)) : Std.Associative f where
+  assoc a b c :=
+    le_antisymm (assoc _ _ _) <| by
+      rw [comm.comm, comm.comm b, comm.comm _ c, comm.comm a]
+      exact assoc _ _ _
 
 end PartialOrder
 
@@ -710,6 +711,10 @@ instance instLinearOrder (α : Type*) [LinearOrder α] : LinearOrder αᵒᵈ wh
   decidableLE := (inferInstance : DecidableRel (fun a b : α ↦ b ≤ a))
   decidableLT := (inferInstance : DecidableRel (fun a b : α ↦ b < a))
 
+/-- The opposite linear order to a given linear order -/
+def _root_.LinearOrder.swap (α : Type*) (_ : LinearOrder α) : LinearOrder α :=
+  inferInstanceAs <| LinearOrder (OrderDual α)
+
 instance : ∀ [Inhabited α], Inhabited αᵒᵈ := fun [x : Inhabited α] => x
 
 theorem Preorder.dual_dual (α : Type*) [H : Preorder α] : OrderDual.instPreorder αᵒᵈ = H :=
@@ -1100,7 +1105,8 @@ instance (α β : Type*) [LE α] [LE β] : LE (α × β) :=
 
 -- Porting note (#10754): new instance
 instance instDecidableLE (α β : Type*) [LE α] [LE β] (x y : α × β)
-    [Decidable (x.1 ≤ y.1)] [Decidable (x.2 ≤ y.2)] : Decidable (x ≤ y) := And.decidable
+    [Decidable (x.1 ≤ y.1)] [Decidable (x.2 ≤ y.2)] : Decidable (x ≤ y) :=
+  inferInstanceAs (Decidable (x.1 ≤ y.1 ∧ x.2 ≤ y.2))
 
 theorem le_def [LE α] [LE β] {x y : α × β} : x ≤ y ↔ x.1 ≤ y.1 ∧ x.2 ≤ y.2 :=
   Iff.rfl
@@ -1190,6 +1196,12 @@ instance OrderDual.denselyOrdered (α : Type*) [LT α] [h : DenselyOrdered α] :
 theorem denselyOrdered_orderDual [LT α] : DenselyOrdered αᵒᵈ ↔ DenselyOrdered α :=
   ⟨by convert @OrderDual.denselyOrdered αᵒᵈ _, @OrderDual.denselyOrdered α _⟩
 
+/-- Any ordered subsingleton is densely ordered. Not an instance to avoid a heavy subsingleton
+typeclass search. -/
+lemma Subsingleton.instDenselyOrdered {X : Type*} [Subsingleton X] [Preorder X] :
+    DenselyOrdered X :=
+  ⟨fun _ _ h ↦ (not_lt_of_subsingleton h).elim⟩
+
 instance [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] : DenselyOrdered (α × β) :=
   ⟨fun a b ↦ by
     simp_rw [Prod.lt_iff]
diff --git a/Mathlib/Order/Booleanisation.lean b/Mathlib/Order/Booleanisation.lean
index dd3aa408d0c38..aa7f7370581b4 100644
--- a/Mathlib/Order/Booleanisation.lean
+++ b/Mathlib/Order/Booleanisation.lean
@@ -58,7 +58,7 @@ instance instCompl : HasCompl (Booleanisation α) where
 @[simp] lemma compl_lift (a : α) : (lift a)ᶜ = comp a := rfl
 @[simp] lemma compl_comp (a : α) : (comp a)ᶜ = lift a := rfl
 
-variable [GeneralizedBooleanAlgebra α] {x y : Booleanisation α} {a b : α}
+variable [GeneralizedBooleanAlgebra α] {a b : α}
 
 /-- The order on `Booleanisation α` is as follows: For `a b : α`,
 * `a ≤ b` iff `a ≤ b` in `α`
diff --git a/Mathlib/Order/BoundedOrder.lean b/Mathlib/Order/BoundedOrder.lean
index d4039a093f12d..a28c66b0685a2 100644
--- a/Mathlib/Order/BoundedOrder.lean
+++ b/Mathlib/Order/BoundedOrder.lean
@@ -33,7 +33,7 @@ open Function OrderDual
 
 universe u v
 
-variable {α : Type u} {β : Type v} {γ δ : Type*}
+variable {α : Type u} {β : Type v}
 
 /-! ### Top, bottom element -/
 
@@ -84,6 +84,11 @@ theorem ne_top_of_lt (h : a < b) : a ≠ ⊤ :=
 
 alias LT.lt.ne_top := ne_top_of_lt
 
+theorem lt_top_of_lt (h : a < b) : a < ⊤ :=
+  lt_of_lt_of_le h le_top
+
+alias LT.lt.lt_top := lt_top_of_lt
+
 end Preorder
 
 variable [PartialOrder α] [OrderTop α] [Preorder β] {f : α → β} {a b : α}
@@ -144,6 +149,9 @@ theorem StrictMono.apply_eq_top_iff (hf : StrictMono f) : f a = f ⊤ ↔ a = 
 theorem StrictAnti.apply_eq_top_iff (hf : StrictAnti f) : f a = f ⊤ ↔ a = ⊤ :=
   ⟨fun h => not_lt_top_iff.1 fun ha => (hf ha).ne' h, congr_arg _⟩
 
+lemma top_not_mem_iff {s : Set α} : ⊤ ∉ s ↔ ∀ x ∈ s, x < ⊤ :=
+  ⟨fun h x hx ↦ Ne.lt_top (fun hx' : x = ⊤ ↦ h (hx' ▸ hx)), fun h h₀ ↦ (h ⊤ h₀).false⟩
+
 variable [Nontrivial α]
 
 theorem not_isMin_top : ¬IsMin (⊤ : α) := fun h =>
@@ -250,6 +258,11 @@ theorem ne_bot_of_gt (h : a < b) : b ≠ ⊥ :=
 
 alias LT.lt.ne_bot := ne_bot_of_gt
 
+theorem bot_lt_of_lt (h : a < b) : ⊥ < b :=
+  lt_of_le_of_lt bot_le h
+
+alias LT.lt.bot_lt := bot_lt_of_lt
+
 end Preorder
 
 variable [PartialOrder α] [OrderBot α] [Preorder β] {f : α → β} {a b : α}
@@ -313,6 +326,9 @@ theorem StrictMono.apply_eq_bot_iff (hf : StrictMono f) : f a = f ⊥ ↔ a = 
 theorem StrictAnti.apply_eq_bot_iff (hf : StrictAnti f) : f a = f ⊥ ↔ a = ⊥ :=
   hf.dual.apply_eq_top_iff
 
+lemma bot_not_mem_iff {s : Set α} : ⊥ ∉ s ↔ ∀ x ∈ s, ⊥ < x :=
+  top_not_mem_iff (α := αᵒᵈ)
+
 variable [Nontrivial α]
 
 theorem not_isMax_bot : ¬IsMax (⊥ : α) :=
@@ -337,7 +353,7 @@ theorem OrderBot.ext_bot {α} {hA : PartialOrder α} (A : OrderBot α) {hB : Par
 
 section SemilatticeSupTop
 
-variable [SemilatticeSup α] [OrderTop α] {a : α}
+variable [SemilatticeSup α] [OrderTop α]
 
 -- Porting note: Not simp because simp can prove it
 theorem top_sup_eq (a : α) : ⊤ ⊔ a = ⊤ :=
@@ -384,7 +400,7 @@ end SemilatticeInfTop
 
 section SemilatticeInfBot
 
-variable [SemilatticeInf α] [OrderBot α] {a : α}
+variable [SemilatticeInf α] [OrderBot α]
 
 -- Porting note: Not simp because simp can prove it
 lemma bot_inf_eq (a : α) : ⊥ ⊓ a = ⊥ := inf_of_le_left bot_le
diff --git a/Mathlib/Order/Bounds/Basic.lean b/Mathlib/Order/Bounds/Basic.lean
index b8c81cbad1d9d..86984514ba3e7 100644
--- a/Mathlib/Order/Bounds/Basic.lean
+++ b/Mathlib/Order/Bounds/Basic.lean
@@ -759,7 +759,7 @@ theorem nonempty_of_not_bddBelow [Nonempty α] (h : ¬BddBelow s) : s.Nonempty :
 @[simp]
 theorem bddAbove_insert [IsDirected α (· ≤ ·)] {s : Set α} {a : α} :
     BddAbove (insert a s) ↔ BddAbove s := by
-  simp only [insert_eq, bddAbove_union, bddAbove_singleton, true_and_iff]
+  simp only [insert_eq, bddAbove_union, bddAbove_singleton, true_and]
 
 protected theorem BddAbove.insert [IsDirected α (· ≤ ·)] {s : Set α} (a : α) :
     BddAbove s → BddAbove (insert a s) :=
@@ -769,7 +769,7 @@ protected theorem BddAbove.insert [IsDirected α (· ≤ ·)] {s : Set α} (a :
 @[simp]
 theorem bddBelow_insert [IsDirected α (· ≥ ·)] {s : Set α} {a : α} :
     BddBelow (insert a s) ↔ BddBelow s := by
-  simp only [insert_eq, bddBelow_union, bddBelow_singleton, true_and_iff]
+  simp only [insert_eq, bddBelow_union, bddBelow_singleton, true_and]
 
 protected theorem BddBelow.insert [IsDirected α (· ≥ ·)] {s : Set α} (a : α) :
     BddBelow s → BddBelow (insert a s) :=
@@ -1430,31 +1430,3 @@ lemma BddBelow.range_comp {γ : Type*} [Preorder β] [Preorder γ] {f : α → 
     (hf : BddBelow (range f)) (hg : Monotone g) : BddBelow (range (fun x => g (f x))) := by
   change BddBelow (range (g ∘ f))
   simpa only [Set.range_comp] using hg.map_bddBelow hf
-
-section ScottContinuous
-variable [Preorder α] [Preorder β] {f : α → β} {a : α}
-
-/-- A function between preorders is said to be Scott continuous if it preserves `IsLUB` on directed
-sets. It can be shown that a function is Scott continuous if and only if it is continuous wrt the
-Scott topology.
-
-The dual notion
-
-```lean
-∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≥ ·) d → ∀ ⦃a⦄, IsGLB d a → IsGLB (f '' d) (f a)
-```
-
-does not appear to play a significant role in the literature, so is omitted here.
--/
-def ScottContinuous (f : α → β) : Prop :=
-  ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → IsLUB (f '' d) (f a)
-
-protected theorem ScottContinuous.monotone (h : ScottContinuous f) : Monotone f := by
-  refine fun a b hab =>
-    (h (insert_nonempty _ _) (directedOn_pair le_refl hab) ?_).1
-      (mem_image_of_mem _ <| mem_insert _ _)
-  rw [IsLUB, upperBounds_insert, upperBounds_singleton,
-    inter_eq_self_of_subset_right (Ici_subset_Ici.2 hab)]
-  exact isLeast_Ici
-
-end ScottContinuous
diff --git a/Mathlib/Order/Category/OmegaCompletePartialOrder.lean b/Mathlib/Order/Category/OmegaCompletePartialOrder.lean
index d5b3f6975277c..c92cacfbb73ae 100644
--- a/Mathlib/Order/Category/OmegaCompletePartialOrder.lean
+++ b/Mathlib/Order/Category/OmegaCompletePartialOrder.lean
@@ -120,7 +120,8 @@ def isEqualizer {X Y : ωCPO.{v}} (f g : X ⟶ Y) : IsLimit (equalizer f g) :=
     -- Porting note: Changed `s.ι x` to `s.ι.toFun x`
     ⟨{  toFun := fun x => ⟨s.ι.toFun x, by apply ContinuousHom.congr_fun s.condition⟩
         monotone' := fun x y h => s.ι.monotone h
-        cont := fun x => Subtype.ext (s.ι.continuous x) }, by ext; rfl, fun hm => by
+        map_ωSup' := fun x => Subtype.ext (s.ι.continuous x)
+      }, by ext; rfl, fun hm => by
       apply ContinuousHom.ext _ _ fun x => Subtype.ext ?_ -- Porting note: Originally `ext`
       apply ContinuousHom.congr_fun hm⟩
 
diff --git a/Mathlib/Order/CompactlyGenerated/Basic.lean b/Mathlib/Order/CompactlyGenerated/Basic.lean
index 5a2890bd05342..6f7a8bd1d8782 100644
--- a/Mathlib/Order/CompactlyGenerated/Basic.lean
+++ b/Mathlib/Order/CompactlyGenerated/Basic.lean
@@ -129,7 +129,7 @@ theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
         apply sSup_le_sSup
         intro x hx
         use {x}
-        simpa only [and_true_iff, id, Finset.coe_singleton, eq_self_iff_true,
+        simpa only [and_true, id, Finset.coe_singleton, eq_self_iff_true,
           Finset.sup_singleton, Set.singleton_subset_iff]
       have Sne : S.Nonempty := by
         suffices ⊥ ∈ S from Set.nonempty_of_mem this
@@ -189,10 +189,10 @@ theorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β
     specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)
     simpa only [exists_prop]
 
-theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
+theorem WellFoundedGT.isSupFiniteCompact [WellFoundedGT α] :
     IsSupFiniteCompact α := fun s => by
   let S := { x | ∃ t : Finset α, ↑t ⊆ s ∧ t.sup id = x }
-  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := h.has_min S ⟨⊥, ∅, by simp⟩
+  obtain ⟨m, ⟨t, ⟨ht₁, rfl⟩⟩, hm⟩ := wellFounded_gt.has_min S ⟨⊥, ∅, by simp⟩
   refine ⟨t, ht₁, (sSup_le _ _ fun y hy => ?_).antisymm ?_⟩
   · classical
     rw [eq_of_le_of_not_lt (Finset.sup_mono (t.subset_insert y))
@@ -212,25 +212,26 @@ theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
   · rw [ht₂]
     exact hsc.finsetSup_mem h ht₁
 
-theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
-    WellFounded ((· > ·) : α → α → Prop) := by
-  refine RelEmbedding.wellFounded_iff_no_descending_seq.mpr ⟨fun a => ?_⟩
-  suffices sSup (Set.range a) ∈ Set.range a by
-    obtain ⟨n, hn⟩ := Set.mem_range.mp this
-    have h' : sSup (Set.range a) < a (n + 1) := by
-      change _ > _
-      simp [← hn, a.map_rel_iff]
-    apply lt_irrefl (a (n + 1))
-    apply lt_of_le_of_lt _ h'
-    apply le_sSup
-    apply Set.mem_range_self
-  apply h (Set.range a)
-  · use a 37
-    apply Set.mem_range_self
-  · rintro x ⟨m, hm⟩ y ⟨n, hn⟩
-    use m ⊔ n
-    rw [← hm, ← hn]
-    apply RelHomClass.map_sup a
+theorem IsSupClosedCompact.wellFoundedGT (h : IsSupClosedCompact α) :
+    WellFoundedGT α where
+  wf := by
+    refine RelEmbedding.wellFounded_iff_no_descending_seq.mpr ⟨fun a => ?_⟩
+    suffices sSup (Set.range a) ∈ Set.range a by
+      obtain ⟨n, hn⟩ := Set.mem_range.mp this
+      have h' : sSup (Set.range a) < a (n + 1) := by
+        change _ > _
+        simp [← hn, a.map_rel_iff]
+      apply lt_irrefl (a (n + 1))
+      apply lt_of_le_of_lt _ h'
+      apply le_sSup
+      apply Set.mem_range_self
+    apply h (Set.range a)
+    · use a 37
+      apply Set.mem_range_self
+    · rintro x ⟨m, hm⟩ y ⟨n, hn⟩
+      use m ⊔ n
+      rw [← hm, ← hn]
+      apply RelHomClass.map_sup a
 
 theorem isSupFiniteCompact_iff_all_elements_compact :
     IsSupFiniteCompact α ↔ ∀ k : α, IsCompactElement k := by
@@ -247,43 +248,38 @@ theorem isSupFiniteCompact_iff_all_elements_compact :
     exact ⟨t, hts, this⟩
 
 open List in
-theorem wellFounded_characterisations : List.TFAE
-    [WellFounded ((· > ·) : α → α → Prop),
-      IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] := by
-  tfae_have 1 → 2
-  · exact WellFounded.isSupFiniteCompact α
-  tfae_have 2 → 3
-  · exact IsSupFiniteCompact.isSupClosedCompact α
-  tfae_have 3 → 1
-  · exact IsSupClosedCompact.wellFounded α
-  tfae_have 2 ↔ 4
-  · exact isSupFiniteCompact_iff_all_elements_compact α
+theorem wellFoundedGT_characterisations : List.TFAE
+    [WellFoundedGT α, IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] := by
+  tfae_have 1 → 2 := @WellFoundedGT.isSupFiniteCompact α _
+  tfae_have 2 → 3 := IsSupFiniteCompact.isSupClosedCompact α
+  tfae_have 3 → 1 := IsSupClosedCompact.wellFoundedGT α
+  tfae_have 2 ↔ 4 := isSupFiniteCompact_iff_all_elements_compact α
   tfae_finish
 
-theorem wellFounded_iff_isSupFiniteCompact :
-    WellFounded ((· > ·) : α → α → Prop) ↔ IsSupFiniteCompact α :=
-  (wellFounded_characterisations α).out 0 1
+theorem wellFoundedGT_iff_isSupFiniteCompact :
+    WellFoundedGT α ↔ IsSupFiniteCompact α :=
+  (wellFoundedGT_characterisations α).out 0 1
 
 theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ IsSupClosedCompact α :=
-  (wellFounded_characterisations α).out 1 2
+  (wellFoundedGT_characterisations α).out 1 2
 
-theorem isSupClosedCompact_iff_wellFounded :
-    IsSupClosedCompact α ↔ WellFounded ((· > ·) : α → α → Prop) :=
-  (wellFounded_characterisations α).out 2 0
+theorem isSupClosedCompact_iff_wellFoundedGT :
+    IsSupClosedCompact α ↔ WellFoundedGT α :=
+  (wellFoundedGT_characterisations α).out 2 0
 
-alias ⟨_, IsSupFiniteCompact.wellFounded⟩ := wellFounded_iff_isSupFiniteCompact
+alias ⟨_, IsSupFiniteCompact.wellFoundedGT⟩ := wellFoundedGT_iff_isSupFiniteCompact
 
 alias ⟨_, IsSupClosedCompact.isSupFiniteCompact⟩ := isSupFiniteCompact_iff_isSupClosedCompact
 
-alias ⟨_, _root_.WellFounded.isSupClosedCompact⟩ := isSupClosedCompact_iff_wellFounded
+alias ⟨_, WellFoundedGT.isSupClosedCompact⟩ := isSupClosedCompact_iff_wellFoundedGT
 
 variable {α}
 
-theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
+theorem WellFoundedGT.finite_of_setIndependent [WellFoundedGT α] {s : Set α}
     (hs : SetIndependent s) : s.Finite := by
   classical
     refine Set.not_infinite.mp fun contra => ?_
-    obtain ⟨t, ht₁, ht₂⟩ := WellFounded.isSupFiniteCompact α h s
+    obtain ⟨t, ht₁, ht₂⟩ := WellFoundedGT.isSupFiniteCompact α s
     replace contra : ∃ x : α, x ∈ s ∧ x ≠ ⊥ ∧ x ∉ t := by
       have : (s \ (insert ⊥ t : Finset α)).Infinite := contra.diff (Finset.finite_toSet _)
       obtain ⟨x, hx₁, hx₂⟩ := this.nonempty
@@ -296,14 +292,36 @@ theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α 
     rw [← hs, eq_comm, inf_eq_left]
     exact le_sSup _ _ hx₀
 
-theorem WellFounded.finite_ne_bot_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop))
+theorem WellFoundedGT.finite_ne_bot_of_independent [WellFoundedGT α]
     {ι : Type*} {t : ι → α} (ht : Independent t) : Set.Finite {i | t i ≠ ⊥} := by
   refine Finite.of_finite_image (Finite.subset ?_ (image_subset_range t _)) ht.injOn
-  exact WellFounded.finite_of_setIndependent hwf ht.setIndependent_range
+  exact WellFoundedGT.finite_of_setIndependent ht.setIndependent_range
 
-theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type*}
+theorem WellFoundedGT.finite_of_independent [WellFoundedGT α] {ι : Type*}
     {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
-  haveI := (WellFounded.finite_of_setIndependent hwf ht.setIndependent_range).to_subtype
+  haveI := (WellFoundedGT.finite_of_setIndependent ht.setIndependent_range).to_subtype
+  Finite.of_injective_finite_range (ht.injective h_ne_bot)
+
+theorem WellFoundedLT.finite_of_setIndependent [WellFoundedLT α] {s : Set α}
+    (hs : SetIndependent s) : s.Finite := by
+  by_contra inf
+  let e := (Infinite.diff inf <| finite_singleton ⊥).to_subtype.natEmbedding
+  let a n := ⨆ i ≥ n, (e i).1
+  have sup_le n : (e n).1 ⊔ a (n + 1) ≤ a n := sup_le_iff.mpr ⟨le_iSup₂_of_le n le_rfl le_rfl,
+    iSup₂_le fun i hi ↦ le_iSup₂_of_le i (n.le_succ.trans hi) le_rfl⟩
+  have lt n : a (n + 1) < a n := (Disjoint.right_lt_sup_of_left_ne_bot
+    ((hs (e n).2.1).mono_right <| iSup₂_le fun i hi ↦ le_sSup _ _ ?_) (e n).2.2).trans_le (sup_le n)
+  · exact (RelEmbedding.natGT a lt).not_wellFounded_of_decreasing_seq wellFounded_lt
+  exact ⟨(e i).2.1, fun h ↦ n.lt_succ_self.not_le <| hi.trans_eq <| e.2 <| Subtype.val_injective h⟩
+
+theorem WellFoundedLT.finite_ne_bot_of_independent [WellFoundedLT α]
+    {ι : Type*} {t : ι → α} (ht : Independent t) : Set.Finite {i | t i ≠ ⊥} := by
+  refine Finite.of_finite_image (Finite.subset ?_ (image_subset_range t _)) ht.injOn
+  exact WellFoundedLT.finite_of_setIndependent ht.setIndependent_range
+
+theorem WellFoundedLT.finite_of_independent [WellFoundedLT α] {ι : Type*}
+    {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι :=
+  haveI := (WellFoundedLT.finite_of_setIndependent ht.setIndependent_range).to_subtype
   Finite.of_injective_finite_range (ht.injective h_ne_bot)
 
 end CompleteLattice
@@ -456,12 +474,35 @@ end
 
 namespace CompleteLattice
 
-theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
+theorem isCompactlyGenerated_of_wellFoundedGT [h : WellFoundedGT α] :
     IsCompactlyGenerated α := by
-  rw [wellFounded_iff_isSupFiniteCompact, isSupFiniteCompact_iff_all_elements_compact] at h
+  rw [wellFoundedGT_iff_isSupFiniteCompact, isSupFiniteCompact_iff_all_elements_compact] at h
   -- x is the join of the set of compact elements {x}
   exact ⟨fun x => ⟨{x}, ⟨fun x _ => h x, sSup_singleton⟩⟩⟩
 
+@[deprecated (since := "2024-10-07")]
+alias WellFounded.isSupFiniteCompact := WellFoundedGT.isSupFiniteCompact
+@[deprecated (since := "2024-10-07")]
+alias IsSupClosedCompact.wellFounded := IsSupClosedCompact.wellFoundedGT
+@[deprecated (since := "2024-10-07")]
+alias wellFounded_characterisations := wellFoundedGT_characterisations
+@[deprecated (since := "2024-10-07")]
+alias wellFounded_iff_isSupFiniteCompact := wellFoundedGT_iff_isSupFiniteCompact
+@[deprecated (since := "2024-10-07")]
+alias isSupClosedCompact_iff_wellFounded := isSupClosedCompact_iff_wellFoundedGT
+@[deprecated (since := "2024-10-07")]
+alias IsSupFiniteCompact.wellFounded := IsSupFiniteCompact.wellFoundedGT
+@[deprecated (since := "2024-10-07")]
+alias _root_.WellFounded.isSupClosedCompact := WellFoundedGT.isSupClosedCompact
+@[deprecated (since := "2024-10-07")]
+alias WellFounded.finite_of_setIndependent := WellFoundedGT.finite_of_setIndependent
+@[deprecated (since := "2024-10-07")]
+alias WellFounded.finite_ne_bot_of_independent := WellFoundedGT.finite_ne_bot_of_independent
+@[deprecated (since := "2024-10-07")]
+alias WellFounded.finite_of_independent := WellFoundedGT.finite_of_independent
+@[deprecated (since := "2024-10-07")]
+alias isCompactlyGenerated_of_wellFounded := isCompactlyGenerated_of_wellFoundedGT
+
 /-- A compact element `k` has the property that any `b < k` lies below a "maximal element below
 `k`", which is to say `[⊥, k]` is coatomic. -/
 theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) :
diff --git a/Mathlib/Order/Compare.lean b/Mathlib/Order/Compare.lean
index 4bd82d405b971..0fc1605e23a0d 100644
--- a/Mathlib/Order/Compare.lean
+++ b/Mathlib/Order/Compare.lean
@@ -31,33 +31,15 @@ def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
 theorem cmpLE_swap {α} [LE α] [IsTotal α (· ≤ ·)] [@DecidableRel α (· ≤ ·)] (x y : α) :
     (cmpLE x y).swap = cmpLE y x := by
   by_cases xy : x ≤ y <;> by_cases yx : y ≤ x <;> simp [cmpLE, *, Ordering.swap]
-  cases not_or_of_not xy yx (total_of _ _ _)
+  cases not_or_intro xy yx (total_of _ _ _)
 
 theorem cmpLE_eq_cmp {α} [Preorder α] [IsTotal α (· ≤ ·)] [@DecidableRel α (· ≤ ·)]
     [@DecidableRel α (· < ·)] (x y : α) : cmpLE x y = cmp x y := by
   by_cases xy : x ≤ y <;> by_cases yx : y ≤ x <;> simp [cmpLE, lt_iff_le_not_le, *, cmp, cmpUsing]
-  cases not_or_of_not xy yx (total_of _ _ _)
+  cases not_or_intro xy yx (total_of _ _ _)
 
 namespace Ordering
 
-/-- `Compares o a b` means that `a` and `b` have the ordering relation `o` between them, assuming
-that the relation `a < b` is defined. -/
--- Porting note: we have removed `@[simp]` here in favour of separate simp lemmas,
--- otherwise this definition will unfold to a match.
-def Compares [LT α] : Ordering → α → α → Prop
-  | lt, a, b => a < b
-  | eq, a, b => a = b
-  | gt, a, b => a > b
-
-@[simp]
-lemma compares_lt [LT α] (a b : α) : Compares lt a b = (a < b) := rfl
-
-@[simp]
-lemma compares_eq [LT α] (a b : α) : Compares eq a b = (a = b) := rfl
-
-@[simp]
-lemma compares_gt [LT α] (a b : α) : Compares gt a b = (a > b) := rfl
-
 theorem compares_swap [LT α] {a b : α} {o : Ordering} : o.swap.Compares a b ↔ o.Compares b a := by
   cases o
   · exact Iff.rfl
@@ -118,11 +100,13 @@ theorem compares_iff_of_compares_impl [LinearOrder α] [Preorder β] {a b : α}
   · have hab : Compares Ordering.gt a b := hab
     rwa [ho.inj (h hab)]
 
-theorem swap_orElse (o₁ o₂) : (orElse o₁ o₂).swap = orElse o₁.swap o₂.swap := by
-  cases o₁ <;> rfl
+set_option linter.deprecated false in
+@[deprecated swap_then (since := "2024-09-13")]
+theorem swap_orElse (o₁ o₂) : (orElse o₁ o₂).swap = orElse o₁.swap o₂.swap := swap_then ..
 
-theorem orElse_eq_lt (o₁ o₂) : orElse o₁ o₂ = lt ↔ o₁ = lt ∨ o₁ = eq ∧ o₂ = lt := by
-  cases o₁ <;> cases o₂ <;> decide
+set_option linter.deprecated false in
+@[deprecated then_eq_lt (since := "2024-09-13")]
+theorem orElse_eq_lt (o₁ o₂) : orElse o₁ o₂ = lt ↔ o₁ = lt ∨ o₁ = eq ∧ o₂ = lt := then_eq_lt ..
 
 end Ordering
 
diff --git a/Mathlib/Order/CompleteLattice.lean b/Mathlib/Order/CompleteLattice.lean
index 9e27855231d95..9e2df5ddb29f1 100644
--- a/Mathlib/Order/CompleteLattice.lean
+++ b/Mathlib/Order/CompleteLattice.lean
@@ -326,40 +326,44 @@ open OrderDual
 
 section
 
-variable [CompleteLattice α] {s t : Set α} {a b : α}
+section OrderDual
 
 @[simp]
-theorem toDual_sSup (s : Set α) : toDual (sSup s) = sInf (ofDual ⁻¹' s) :=
+theorem toDual_sSup [SupSet α] (s : Set α) : toDual (sSup s) = sInf (ofDual ⁻¹' s) :=
   rfl
 
 @[simp]
-theorem toDual_sInf (s : Set α) : toDual (sInf s) = sSup (ofDual ⁻¹' s) :=
+theorem toDual_sInf [InfSet α] (s : Set α) : toDual (sInf s) = sSup (ofDual ⁻¹' s) :=
   rfl
 
 @[simp]
-theorem ofDual_sSup (s : Set αᵒᵈ) : ofDual (sSup s) = sInf (toDual ⁻¹' s) :=
+theorem ofDual_sSup [InfSet α] (s : Set αᵒᵈ) : ofDual (sSup s) = sInf (toDual ⁻¹' s) :=
   rfl
 
 @[simp]
-theorem ofDual_sInf (s : Set αᵒᵈ) : ofDual (sInf s) = sSup (toDual ⁻¹' s) :=
+theorem ofDual_sInf [SupSet α] (s : Set αᵒᵈ) : ofDual (sInf s) = sSup (toDual ⁻¹' s) :=
   rfl
 
 @[simp]
-theorem toDual_iSup (f : ι → α) : toDual (⨆ i, f i) = ⨅ i, toDual (f i) :=
+theorem toDual_iSup [SupSet α] (f : ι → α) : toDual (⨆ i, f i) = ⨅ i, toDual (f i) :=
   rfl
 
 @[simp]
-theorem toDual_iInf (f : ι → α) : toDual (⨅ i, f i) = ⨆ i, toDual (f i) :=
+theorem toDual_iInf [InfSet α] (f : ι → α) : toDual (⨅ i, f i) = ⨆ i, toDual (f i) :=
   rfl
 
 @[simp]
-theorem ofDual_iSup (f : ι → αᵒᵈ) : ofDual (⨆ i, f i) = ⨅ i, ofDual (f i) :=
+theorem ofDual_iSup [InfSet α] (f : ι → αᵒᵈ) : ofDual (⨆ i, f i) = ⨅ i, ofDual (f i) :=
   rfl
 
 @[simp]
-theorem ofDual_iInf (f : ι → αᵒᵈ) : ofDual (⨅ i, f i) = ⨆ i, ofDual (f i) :=
+theorem ofDual_iInf [SupSet α] (f : ι → αᵒᵈ) : ofDual (⨅ i, f i) = ⨆ i, ofDual (f i) :=
   rfl
 
+end OrderDual
+
+variable [CompleteLattice α] {s t : Set α} {a b : α}
+
 theorem sInf_le_sSup (hs : s.Nonempty) : sInf s ≤ sSup s :=
   isGLB_le_isLUB (isGLB_sInf s) (isLUB_sSup s) hs
 
@@ -753,6 +757,12 @@ theorem sSup_eq_iSup {s : Set α} : sSup s = ⨆ a ∈ s, a :=
 theorem sInf_eq_iInf {s : Set α} : sInf s = ⨅ a ∈ s, a :=
   @sSup_eq_iSup αᵒᵈ _ _
 
+lemma sSup_lowerBounds_eq_sInf (s : Set α) : sSup (lowerBounds s) = sInf s :=
+  (isLUB_sSup _).unique (isGLB_sInf _).isLUB
+
+lemma sInf_upperBounds_eq_csSup (s : Set α) : sInf (upperBounds s) = sSup s :=
+  (isGLB_sInf _).unique (isLUB_sSup _).isGLB
+
 theorem Monotone.le_map_iSup [CompleteLattice β] {f : α → β} (hf : Monotone f) :
     ⨆ i, f (s i) ≤ f (iSup s) :=
   iSup_le fun _ => hf <| le_iSup _ _
@@ -866,6 +876,9 @@ theorem iSup_eq_bot : iSup s = ⊥ ↔ ∀ i, s i = ⊥ :=
 theorem iInf_eq_top : iInf s = ⊤ ↔ ∀ i, s i = ⊤ :=
   sInf_eq_top.trans forall_mem_range
 
+@[simp] lemma bot_lt_iSup : ⊥ < ⨆ i, s i ↔ ∃ i, ⊥ < s i := by simp [bot_lt_iff_ne_bot]
+@[simp] lemma iInf_lt_top : ⨅ i, s i < ⊤ ↔ ∃ i, s i < ⊤ := by simp [lt_top_iff_ne_top]
+
 theorem iSup₂_eq_bot {f : ∀ i, κ i → α} : ⨆ (i) (j), f i j = ⊥ ↔ ∀ i j, f i j = ⊥ := by
   simp
 
@@ -1273,12 +1286,24 @@ theorem iInf_sigma {p : β → Type*} {f : Sigma p → α} : ⨅ x, f x = ⨅ (i
   @iSup_sigma αᵒᵈ _ _ _ _
 
 lemma iSup_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :
-    (⨆ i, ⨆ j, f i j) = ⨆ x : Σ i, κ i, f x.1 x.2 :=
-(iSup_sigma (f := fun x ↦ f x.1 x.2)).symm
+    (⨆ i, ⨆ j, f i j) = ⨆ x : Σ i, κ i, f x.1 x.2 := (iSup_sigma (f := fun x ↦ f x.1 x.2)).symm
 
 lemma iInf_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :
-    (⨅ i, ⨅ j, f i j) = ⨅ x : Σ i, κ i, f x.1 x.2 :=
-(iInf_sigma (f := fun x ↦ f x.1 x.2)).symm
+    (⨅ i, ⨅ j, f i j) = ⨅ x : Σ i, κ i, f x.1 x.2 := (iInf_sigma (f := fun x ↦ f x.1 x.2)).symm
+
+lemma iSup_psigma {ι : Sort*} {κ : ι → Sort*} (f : (Σ' i, κ i) → α) :
+    ⨆ ij, f ij = ⨆ i, ⨆ j, f ⟨i, j⟩ :=
+  eq_of_forall_ge_iff fun c ↦ by simp only [iSup_le_iff, PSigma.forall]
+
+lemma iInf_psigma {ι : Sort*} {κ : ι → Sort*} (f : (Σ' i, κ i) → α) :
+    ⨅ ij, f ij = ⨅ i, ⨅ j, f ⟨i, j⟩ :=
+  eq_of_forall_le_iff fun c ↦ by simp only [le_iInf_iff, PSigma.forall]
+
+lemma iSup_psigma' {ι : Sort*} {κ : ι → Sort*} (f : ∀ i, κ i → α) :
+    (⨆ i, ⨆ j, f i j) = ⨆ ij : Σ' i, κ i, f ij.1 ij.2 := (iSup_psigma fun x ↦ f x.1 x.2).symm
+
+lemma iInf_psigma' {ι : Sort*} {κ : ι → Sort*} (f : ∀ i, κ i → α) :
+    (⨅ i, ⨅ j, f i j) = ⨅ ij : Σ' i, κ i, f ij.1 ij.2 := (iInf_psigma fun x ↦ f x.1 x.2).symm
 
 theorem iSup_prod {f : β × γ → α} : ⨆ x, f x = ⨆ (i) (j), f (i, j) :=
   eq_of_forall_ge_iff fun c => by simp only [iSup_le_iff, Prod.forall]
@@ -1412,6 +1437,12 @@ theorem iSup_eq_top (f : ι → α) : iSup f = ⊤ ↔ ∀ b < ⊤, ∃ i, b < f
 theorem iInf_eq_bot (f : ι → α) : iInf f = ⊥ ↔ ∀ b > ⊥, ∃ i, f i < b := by
   simp only [← sInf_range, sInf_eq_bot, Set.exists_range_iff]
 
+lemma iSup₂_eq_top (f : ∀ i, κ i → α) : ⨆ i, ⨆ j, f i j = ⊤ ↔ ∀ b < ⊤, ∃ i j, b < f i j := by
+  simp_rw [iSup_psigma', iSup_eq_top, PSigma.exists]
+
+lemma iInf₂_eq_bot (f : ∀ i, κ i → α) : ⨅ i, ⨅ j, f i j = ⊥ ↔ ∀ b > ⊥, ∃ i j, f i j < b := by
+  simp_rw [iInf_psigma', iInf_eq_bot, PSigma.exists]
+
 end CompleteLinearOrder
 
 /-!
@@ -1575,7 +1606,7 @@ theorem snd_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).snd =
   congr_arg sInf (range_comp _ _).symm
 
 theorem swap_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).swap = ⨅ i, (f i).swap := by
-  simp_rw [iInf, swap_sInf, ← range_comp, Function.comp]  -- Porting note: need to unfold `∘`
+  simp_rw [iInf, swap_sInf, ← range_comp, comp_def]  -- Porting note: need to unfold `∘`
 
 theorem iInf_mk [InfSet α] [InfSet β] (f : ι → α) (g : ι → β) :
     ⨅ i, (f i, g i) = (⨅ i, f i, ⨅ i, g i) :=
@@ -1588,7 +1619,7 @@ theorem snd_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).snd =
   congr_arg sSup (range_comp _ _).symm
 
 theorem swap_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).swap = ⨆ i, (f i).swap := by
-  simp_rw [iSup, swap_sSup, ← range_comp, Function.comp]  -- Porting note: need to unfold `∘`
+  simp_rw [iSup, swap_sSup, ← range_comp, comp_def]  -- Porting note: need to unfold `∘`
 
 theorem iSup_mk [SupSet α] [SupSet β] (f : ι → α) (g : ι → β) :
     ⨆ i, (f i, g i) = (⨆ i, f i, ⨆ i, g i) :=
@@ -1719,3 +1750,5 @@ instance instCompleteLinearOrder : CompleteLinearOrder PUnit where
   top_sdiff := by intros; trivial
 
 end PUnit
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/Order/CompleteLattice/Finset.lean b/Mathlib/Order/CompleteLattice/Finset.lean
new file mode 100644
index 0000000000000..b96b173756c62
--- /dev/null
+++ b/Mathlib/Order/CompleteLattice/Finset.lean
@@ -0,0 +1,247 @@
+/-
+Copyright (c) 2018 Mario Carneiro. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro
+-/
+import Mathlib.Data.Finset.Option
+import Mathlib.Order.Minimal
+
+/-!
+# Lattice operations on finsets
+
+This file is concerned with how big lattice or set operations behave when indexed by a finset.
+
+See also Lattice.lean, which is concerned with folding binary lattice operations over a finset.
+-/
+
+assert_not_exists OrderedCommMonoid
+assert_not_exists MonoidWithZero
+
+open Function Multiset OrderDual
+
+variable {F α β γ ι κ : Type*}
+
+section Lattice
+
+variable {ι' : Sort*} [CompleteLattice α]
+
+/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : Finset ι` of suprema
+`⨆ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `iSup_eq_iSup_finset'` for a version
+that works for `ι : Sort*`. -/
+theorem iSup_eq_iSup_finset (s : ι → α) : ⨆ i, s i = ⨆ t : Finset ι, ⨆ i ∈ t, s i := by
+  classical
+  refine le_antisymm ?_ ?_
+  · exact iSup_le fun b => le_iSup_of_le {b} <| le_iSup_of_le b <| le_iSup_of_le (by simp) <| le_rfl
+  · exact iSup_le fun t => iSup_le fun b => iSup_le fun _ => le_iSup _ _
+
+/-- Supremum of `s i`, `i : ι`, is equal to the supremum over `t : Finset ι` of suprema
+`⨆ i ∈ t, s i`. This version works for `ι : Sort*`. See `iSup_eq_iSup_finset` for a version
+that assumes `ι : Type*` but has no `PLift`s. -/
+theorem iSup_eq_iSup_finset' (s : ι' → α) :
+    ⨆ i, s i = ⨆ t : Finset (PLift ι'), ⨆ i ∈ t, s (PLift.down i) := by
+  rw [← iSup_eq_iSup_finset, ← Equiv.plift.surjective.iSup_comp]; rfl
+
+/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : Finset ι` of infima
+`⨅ i ∈ t, s i`. This version assumes `ι` is a `Type*`. See `iInf_eq_iInf_finset'` for a version
+that works for `ι : Sort*`. -/
+theorem iInf_eq_iInf_finset (s : ι → α) : ⨅ i, s i = ⨅ (t : Finset ι) (i ∈ t), s i :=
+  @iSup_eq_iSup_finset αᵒᵈ _ _ _
+
+/-- Infimum of `s i`, `i : ι`, is equal to the infimum over `t : Finset ι` of infima
+`⨅ i ∈ t, s i`. This version works for `ι : Sort*`. See `iInf_eq_iInf_finset` for a version
+that assumes `ι : Type*` but has no `PLift`s. -/
+theorem iInf_eq_iInf_finset' (s : ι' → α) :
+    ⨅ i, s i = ⨅ t : Finset (PLift ι'), ⨅ i ∈ t, s (PLift.down i) :=
+  @iSup_eq_iSup_finset' αᵒᵈ _ _ _
+
+end Lattice
+
+namespace Set
+
+variable {ι' : Sort*}
+
+/-- Union of an indexed family of sets `s : ι → Set α` is equal to the union of the unions
+of finite subfamilies. This version assumes `ι : Type*`. See also `iUnion_eq_iUnion_finset'` for
+a version that works for `ι : Sort*`. -/
+theorem iUnion_eq_iUnion_finset (s : ι → Set α) : ⋃ i, s i = ⋃ t : Finset ι, ⋃ i ∈ t, s i :=
+  iSup_eq_iSup_finset s
+
+/-- Union of an indexed family of sets `s : ι → Set α` is equal to the union of the unions
+of finite subfamilies. This version works for `ι : Sort*`. See also `iUnion_eq_iUnion_finset` for
+a version that assumes `ι : Type*` but avoids `PLift`s in the right hand side. -/
+theorem iUnion_eq_iUnion_finset' (s : ι' → Set α) :
+    ⋃ i, s i = ⋃ t : Finset (PLift ι'), ⋃ i ∈ t, s (PLift.down i) :=
+  iSup_eq_iSup_finset' s
+
+/-- Intersection of an indexed family of sets `s : ι → Set α` is equal to the intersection of the
+intersections of finite subfamilies. This version assumes `ι : Type*`. See also
+`iInter_eq_iInter_finset'` for a version that works for `ι : Sort*`. -/
+theorem iInter_eq_iInter_finset (s : ι → Set α) : ⋂ i, s i = ⋂ t : Finset ι, ⋂ i ∈ t, s i :=
+  iInf_eq_iInf_finset s
+
+/-- Intersection of an indexed family of sets `s : ι → Set α` is equal to the intersection of the
+intersections of finite subfamilies. This version works for `ι : Sort*`. See also
+`iInter_eq_iInter_finset` for a version that assumes `ι : Type*` but avoids `PLift`s in the right
+hand side. -/
+theorem iInter_eq_iInter_finset' (s : ι' → Set α) :
+    ⋂ i, s i = ⋂ t : Finset (PLift ι'), ⋂ i ∈ t, s (PLift.down i) :=
+  iInf_eq_iInf_finset' s
+
+end Set
+
+namespace Finset
+
+section minimal
+
+variable [DecidableEq α] {P : Finset α → Prop} {s : Finset α}
+
+theorem maximal_iff_forall_insert (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
+    Maximal P s ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s) := by
+  simp only [Maximal, and_congr_right_iff]
+  exact fun _ ↦ ⟨fun h x hxs hx ↦ hxs <| h hx (subset_insert _ _) (mem_insert_self x s),
+    fun h t ht hst x hxt ↦ by_contra fun hxs ↦ h x hxs (hP ht (insert_subset hxt hst))⟩
+
+theorem minimal_iff_forall_diff_singleton (hP : ∀ ⦃s t⦄, P t → t ⊆ s → P s) :
+    Minimal P s ↔ P s ∧ ∀ x ∈ s, ¬ P (s.erase x) where
+  mp h := ⟨h.prop, fun x hxs hx ↦ by simpa using h.le_of_le hx (erase_subset _ _) hxs⟩
+  mpr h := ⟨h.1, fun t ht hts x hxs ↦ by_contra fun hxt ↦
+    h.2 x hxs <| hP ht (subset_erase.2 ⟨hts, hxt⟩)⟩
+
+end minimal
+
+/-! ### Interaction with big lattice/set operations -/
+
+section Lattice
+
+theorem iSup_coe [SupSet β] (f : α → β) (s : Finset α) : ⨆ x ∈ (↑s : Set α), f x = ⨆ x ∈ s, f x :=
+  rfl
+
+theorem iInf_coe [InfSet β] (f : α → β) (s : Finset α) : ⨅ x ∈ (↑s : Set α), f x = ⨅ x ∈ s, f x :=
+  rfl
+
+variable [CompleteLattice β]
+
+theorem iSup_singleton (a : α) (s : α → β) : ⨆ x ∈ ({a} : Finset α), s x = s a := by simp
+
+theorem iInf_singleton (a : α) (s : α → β) : ⨅ x ∈ ({a} : Finset α), s x = s a := by simp
+
+theorem iSup_option_toFinset (o : Option α) (f : α → β) : ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x := by
+  simp
+
+theorem iInf_option_toFinset (o : Option α) (f : α → β) : ⨅ x ∈ o.toFinset, f x = ⨅ x ∈ o, f x :=
+  @iSup_option_toFinset _ βᵒᵈ _ _ _
+
+variable [DecidableEq α]
+
+theorem iSup_union {f : α → β} {s t : Finset α} :
+    ⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x := by simp [iSup_or, iSup_sup_eq]
+
+theorem iInf_union {f : α → β} {s t : Finset α} :
+    ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x :=
+  @iSup_union α βᵒᵈ _ _ _ _ _
+
+theorem iSup_insert (a : α) (s : Finset α) (t : α → β) :
+    ⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x := by
+  rw [insert_eq]
+  simp only [iSup_union, Finset.iSup_singleton]
+
+theorem iInf_insert (a : α) (s : Finset α) (t : α → β) :
+    ⨅ x ∈ insert a s, t x = t a ⊓ ⨅ x ∈ s, t x :=
+  @iSup_insert α βᵒᵈ _ _ _ _ _
+
+theorem iSup_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
+    ⨆ x ∈ s.image f, g x = ⨆ y ∈ s, g (f y) := by rw [← iSup_coe, coe_image, iSup_image, iSup_coe]
+
+theorem iInf_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
+    ⨅ x ∈ s.image f, g x = ⨅ y ∈ s, g (f y) := by rw [← iInf_coe, coe_image, iInf_image, iInf_coe]
+
+theorem iSup_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
+    ⨆ i ∈ insert x t, Function.update f x s i = s ⊔ ⨆ i ∈ t, f i := by
+  simp only [Finset.iSup_insert, update_same]
+  rcongr (i hi); apply update_noteq; rintro rfl; exact hx hi
+
+theorem iInf_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
+    ⨅ i ∈ insert x t, update f x s i = s ⊓ ⨅ i ∈ t, f i :=
+  @iSup_insert_update α βᵒᵈ _ _ _ _ f _ hx
+
+theorem iSup_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
+    ⨆ y ∈ s.biUnion t, f y = ⨆ (x ∈ s) (y ∈ t x), f y := by simp [@iSup_comm _ α, iSup_and]
+
+theorem iInf_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
+    ⨅ y ∈ s.biUnion t, f y = ⨅ (x ∈ s) (y ∈ t x), f y :=
+  @iSup_biUnion _ βᵒᵈ _ _ _ _ _ _
+
+end Lattice
+
+theorem set_biUnion_coe (s : Finset α) (t : α → Set β) : ⋃ x ∈ (↑s : Set α), t x = ⋃ x ∈ s, t x :=
+  rfl
+
+theorem set_biInter_coe (s : Finset α) (t : α → Set β) : ⋂ x ∈ (↑s : Set α), t x = ⋂ x ∈ s, t x :=
+  rfl
+
+theorem set_biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Finset α), s x = s a :=
+  iSup_singleton a s
+
+theorem set_biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Finset α), s x = s a :=
+  iInf_singleton a s
+
+@[simp]
+theorem set_biUnion_preimage_singleton (f : α → β) (s : Finset β) :
+    ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s :=
+  Set.biUnion_preimage_singleton f s
+
+theorem set_biUnion_option_toFinset (o : Option α) (f : α → Set β) :
+    ⋃ x ∈ o.toFinset, f x = ⋃ x ∈ o, f x :=
+  iSup_option_toFinset o f
+
+theorem set_biInter_option_toFinset (o : Option α) (f : α → Set β) :
+    ⋂ x ∈ o.toFinset, f x = ⋂ x ∈ o, f x :=
+  iInf_option_toFinset o f
+
+theorem subset_set_biUnion_of_mem {s : Finset α} {f : α → Set β} {x : α} (h : x ∈ s) :
+    f x ⊆ ⋃ y ∈ s, f y :=
+  show f x ≤ ⨆ y ∈ s, f y from le_iSup_of_le x <| by simp only [h, iSup_pos, le_refl]
+
+variable [DecidableEq α]
+
+theorem set_biUnion_union (s t : Finset α) (u : α → Set β) :
+    ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x :=
+  iSup_union
+
+theorem set_biInter_inter (s t : Finset α) (u : α → Set β) :
+    ⋂ x ∈ s ∪ t, u x = (⋂ x ∈ s, u x) ∩ ⋂ x ∈ t, u x :=
+  iInf_union
+
+theorem set_biUnion_insert (a : α) (s : Finset α) (t : α → Set β) :
+    ⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x :=
+  iSup_insert a s t
+
+theorem set_biInter_insert (a : α) (s : Finset α) (t : α → Set β) :
+    ⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x :=
+  iInf_insert a s t
+
+theorem set_biUnion_finset_image {f : γ → α} {g : α → Set β} {s : Finset γ} :
+    ⋃ x ∈ s.image f, g x = ⋃ y ∈ s, g (f y) :=
+  iSup_finset_image
+
+theorem set_biInter_finset_image {f : γ → α} {g : α → Set β} {s : Finset γ} :
+    ⋂ x ∈ s.image f, g x = ⋂ y ∈ s, g (f y) :=
+  iInf_finset_image
+
+theorem set_biUnion_insert_update {x : α} {t : Finset α} (f : α → Set β) {s : Set β} (hx : x ∉ t) :
+    ⋃ i ∈ insert x t, @update _ _ _ f x s i = s ∪ ⋃ i ∈ t, f i :=
+  iSup_insert_update f hx
+
+theorem set_biInter_insert_update {x : α} {t : Finset α} (f : α → Set β) {s : Set β} (hx : x ∉ t) :
+    ⋂ i ∈ insert x t, @update _ _ _ f x s i = s ∩ ⋂ i ∈ t, f i :=
+  iInf_insert_update f hx
+
+theorem set_biUnion_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → Set β) :
+    ⋃ y ∈ s.biUnion t, f y = ⋃ (x ∈ s) (y ∈ t x), f y :=
+  iSup_biUnion s t f
+
+theorem set_biInter_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → Set β) :
+    ⋂ y ∈ s.biUnion t, f y = ⋂ (x ∈ s) (y ∈ t x), f y :=
+  iInf_biUnion s t f
+
+end Finset
diff --git a/Mathlib/Order/CompleteLatticeIntervals.lean b/Mathlib/Order/CompleteLatticeIntervals.lean
index 44fa32da464fc..149df0ed5225d 100644
--- a/Mathlib/Order/CompleteLatticeIntervals.lean
+++ b/Mathlib/Order/CompleteLatticeIntervals.lean
@@ -52,7 +52,7 @@ theorem subset_sSup_def [Inhabited s] :
   rfl
 
 theorem subset_sSup_of_within [Inhabited s] {t : Set s}
-    (h' : t.Nonempty) (h'' : BddAbove t)  (h : sSup ((↑) '' t : Set α) ∈ s) :
+    (h' : t.Nonempty) (h'' : BddAbove t) (h : sSup ((↑) '' t : Set α) ∈ s) :
     sSup ((↑) '' t : Set α) = (@sSup s _ t : α) := by simp [dif_pos, h, h', h'']
 
 theorem subset_sSup_emptyset [Inhabited s] :
diff --git a/Mathlib/Order/CompleteSublattice.lean b/Mathlib/Order/CompleteSublattice.lean
index 908ecea1281e8..cabb0e732606e 100644
--- a/Mathlib/Order/CompleteSublattice.lean
+++ b/Mathlib/Order/CompleteSublattice.lean
@@ -66,10 +66,10 @@ instance instTop : Top L where
   top := ⟨⊤, by simpa using L.sInfClosed' <| empty_subset _⟩
 
 instance instSupSet : SupSet L where
-  sSup s := ⟨sSup s, L.sSupClosed' image_val_subset⟩
+  sSup s := ⟨sSup <| (↑) '' s, L.sSupClosed' image_val_subset⟩
 
 instance instInfSet : InfSet L where
-  sInf s := ⟨sInf s, L.sInfClosed' image_val_subset⟩
+  sInf s := ⟨sInf <| (↑) '' s, L.sInfClosed' image_val_subset⟩
 
 theorem sSupClosed {s : Set α} (h : s ⊆ L) : sSup s ∈ L := L.sSupClosed' h
 
@@ -89,6 +89,10 @@ theorem coe_sSup' (S : Set L) : (↑(sSup S) : α) = ⨆ N ∈ S, (N : α) := by
 theorem coe_sInf' (S : Set L) : (↑(sInf S) : α) = ⨅ N ∈ S, (N : α) := by
   rw [coe_sInf, ← Set.image, sInf_image]
 
+-- Redeclaring to get proper keys for these instances
+instance : Sup {x // x ∈ L} := Sublattice.instSupCoe
+instance : Inf {x // x ∈ L} := Sublattice.instInfCoe
+
 instance instCompleteLattice : CompleteLattice L :=
   Subtype.coe_injective.completeLattice _
     Sublattice.coe_sup Sublattice.coe_inf coe_sSup' coe_sInf' coe_top coe_bot
diff --git a/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean b/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
index 206708c8c5347..9f80770bcdc85 100644
--- a/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
+++ b/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
@@ -155,7 +155,7 @@ every nonempty subset which is bounded below has an infimum.
 Typical examples are real numbers or natural numbers.
 
 To differentiate the statements from the corresponding statements in (unconditional)
-complete lattices, we prefix sInf and subₛ by a c everywhere. The same statements should
+complete lattices, we prefix `sInf` and `sSup` by a `c` everywhere. The same statements should
 hold in both worlds, sometimes with additional assumptions of nonemptiness or
 boundedness. -/
 class ConditionallyCompleteLattice (α : Type*) extends Lattice α, SupSet α, InfSet α where
@@ -175,7 +175,7 @@ every nonempty subset which is bounded below has an infimum.
 Typical examples are real numbers or natural numbers.
 
 To differentiate the statements from the corresponding statements in (unconditional)
-complete linear orders, we prefix sInf and sSup by a c everywhere. The same statements should
+complete linear orders, we prefix `sInf` and `sSup` by a `c` everywhere. The same statements should
 hold in both worlds, sometimes with additional assumptions of nonemptiness or
 boundedness. -/
 class ConditionallyCompleteLinearOrder (α : Type*) extends ConditionallyCompleteLattice α where
@@ -215,7 +215,7 @@ every nonempty subset which is bounded above has a supremum, and every nonempty
 bounded below) has an infimum.  A typical example is the natural numbers.
 
 To differentiate the statements from the corresponding statements in (unconditional)
-complete linear orders, we prefix `sInf` and `sSup` by a c everywhere. The same statements should
+complete linear orders, we prefix `sInf` and `sSup` by a `c` everywhere. The same statements should
 hold in both worlds, sometimes with additional assumptions of nonemptiness or
 boundedness. -/
 class ConditionallyCompleteLinearOrderBot (α : Type*) extends ConditionallyCompleteLinearOrder α,
@@ -539,14 +539,20 @@ theorem csSup_le_iff (hb : BddAbove s) (hs : s.Nonempty) : sSup s ≤ a ↔ ∀
 theorem le_csInf_iff (hb : BddBelow s) (hs : s.Nonempty) : a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b :=
   le_isGLB_iff (isGLB_csInf hs hb)
 
-theorem csSup_lower_bounds_eq_csInf {s : Set α} (h : BddBelow s) (hs : s.Nonempty) :
+theorem csSup_lowerBounds_eq_csInf {s : Set α} (h : BddBelow s) (hs : s.Nonempty) :
     sSup (lowerBounds s) = sInf s :=
   (isLUB_csSup h <| hs.mono fun _ hx _ hy => hy hx).unique (isGLB_csInf hs h).isLUB
 
-theorem csInf_upper_bounds_eq_csSup {s : Set α} (h : BddAbove s) (hs : s.Nonempty) :
+theorem csInf_upperBounds_eq_csSup {s : Set α} (h : BddAbove s) (hs : s.Nonempty) :
     sInf (upperBounds s) = sSup s :=
   (isGLB_csInf h <| hs.mono fun _ hx _ hy => hy hx).unique (isLUB_csSup hs h).isGLB
 
+@[deprecated (since := "2024-08-25")]
+alias csSup_lower_bounds_eq_csInf := csSup_lowerBounds_eq_csInf
+
+@[deprecated (since := "2024-08-25")]
+alias csInf_upper_bounds_eq_csSup := csInf_upperBounds_eq_csSup
+
 theorem not_mem_of_lt_csInf {x : α} {s : Set α} (h : x < sInf s) (hs : BddBelow s) : x ∉ s :=
   fun hx => lt_irrefl _ (h.trans_le (csInf_le hs hx))
 
@@ -591,12 +597,12 @@ theorem exists_between_of_forall_le (sne : s.Nonempty) (tne : t.Nonempty)
     (hst : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) : (upperBounds s ∩ lowerBounds t).Nonempty :=
   ⟨sInf t, fun x hx => le_csInf tne <| hst x hx, fun _ hy => csInf_le (sne.mono hst) hy⟩
 
-/-- The supremum of a singleton is the element of the singleton-/
+/-- The supremum of a singleton is the element of the singleton -/
 @[simp]
 theorem csSup_singleton (a : α) : sSup {a} = a :=
   isGreatest_singleton.csSup_eq
 
-/-- The infimum of a singleton is the element of the singleton-/
+/-- The infimum of a singleton is the element of the singleton -/
 @[simp]
 theorem csInf_singleton (a : α) : sInf {a} = a :=
   isLeast_singleton.csInf_eq
@@ -698,18 +704,18 @@ theorem csSup_Ioc (h : a < b) : sSup (Ioc a b) = b :=
 theorem csSup_Ioo [DenselyOrdered α] (h : a < b) : sSup (Ioo a b) = b :=
   (isLUB_Ioo h).csSup_eq (nonempty_Ioo.2 h)
 
-/-- The indexed supremum of a function is bounded above by a uniform bound-/
+/-- The indexed supremum of a function is bounded above by a uniform bound -/
 theorem ciSup_le [Nonempty ι] {f : ι → α} {c : α} (H : ∀ x, f x ≤ c) : iSup f ≤ c :=
   csSup_le (range_nonempty f) (by rwa [forall_mem_range])
 
-/-- The indexed supremum of a function is bounded below by the value taken at one point-/
+/-- The indexed supremum of a function is bounded below by the value taken at one point -/
 theorem le_ciSup {f : ι → α} (H : BddAbove (range f)) (c : ι) : f c ≤ iSup f :=
   le_csSup H (mem_range_self _)
 
 theorem le_ciSup_of_le {f : ι → α} (H : BddAbove (range f)) (c : ι) (h : a ≤ f c) : a ≤ iSup f :=
   le_trans h (le_ciSup H c)
 
-/-- The indexed supremum of two functions are comparable if the functions are pointwise comparable-/
+/-- The indexed suprema of two functions are comparable if the functions are pointwise comparable -/
 theorem ciSup_mono {f g : ι → α} (B : BddAbove (range g)) (H : ∀ x, f x ≤ g x) :
     iSup f ≤ iSup g := by
   cases isEmpty_or_nonempty ι
@@ -720,15 +726,15 @@ theorem le_ciSup_set {f : β → α} {s : Set β} (H : BddAbove (f '' s)) {c : 
     f c ≤ ⨆ i : s, f i :=
   (le_csSup H <| mem_image_of_mem f hc).trans_eq sSup_image'
 
-/-- The indexed infimum of two functions are comparable if the functions are pointwise comparable-/
+/-- The indexed infimum of two functions are comparable if the functions are pointwise comparable -/
 theorem ciInf_mono {f g : ι → α} (B : BddBelow (range f)) (H : ∀ x, f x ≤ g x) : iInf f ≤ iInf g :=
   ciSup_mono (α := αᵒᵈ) B H
 
-/-- The indexed minimum of a function is bounded below by a uniform lower bound-/
+/-- The indexed minimum of a function is bounded below by a uniform lower bound -/
 theorem le_ciInf [Nonempty ι] {f : ι → α} {c : α} (H : ∀ x, c ≤ f x) : c ≤ iInf f :=
   ciSup_le (α := αᵒᵈ) H
 
-/-- The indexed infimum of a function is bounded above by the value taken at one point-/
+/-- The indexed infimum of a function is bounded above by the value taken at one point -/
 theorem ciInf_le {f : ι → α} (H : BddBelow (range f)) (c : ι) : iInf f ≤ f c :=
   le_ciSup (α := αᵒᵈ) H c
 
@@ -1206,6 +1212,9 @@ theorem exists_lt_of_lt_ciSup' {f : ι → α} {a : α} (h : a < ⨆ i, f i) : 
   contrapose! h
   exact ciSup_le' h
 
+theorem not_mem_of_lt_csInf' {x : α} {s : Set α} (h : x < sInf s) : x ∉ s :=
+  not_mem_of_lt_csInf h (OrderBot.bddBelow s)
+
 theorem ciSup_mono' {ι'} {f : ι → α} {g : ι' → α} (hg : BddAbove (range g))
     (h : ∀ i, ∃ i', f i ≤ g i') : iSup f ≤ iSup g :=
   ciSup_le' fun i => Exists.elim (h i) (le_ciSup_of_le hg)
@@ -1213,6 +1222,12 @@ theorem ciSup_mono' {ι'} {f : ι → α} {g : ι' → α} (hg : BddAbove (range
 theorem csInf_le_csInf' {s t : Set α} (h₁ : t.Nonempty) (h₂ : t ⊆ s) : sInf s ≤ sInf t :=
   csInf_le_csInf (OrderBot.bddBelow s) h₁ h₂
 
+theorem csSup_le_csSup' {s t : Set α} (h₁ : BddAbove t) (h₂ : s ⊆ t) : sSup s ≤ sSup t := by
+  rcases eq_empty_or_nonempty s with rfl | h
+  · rw [csSup_empty]
+    exact bot_le
+  · exact csSup_le_csSup h₁ h h₂
+
 lemma ciSup_or' (p q : Prop) (f : p ∨ q → α) :
     ⨆ (h : p ∨ q), f h = (⨆ h : p, f (.inl h)) ⊔ ⨆ h : q, f (.inr h) := by
   by_cases hp : p <;> by_cases hq : q
@@ -1639,3 +1654,5 @@ lemma iInf_coe_lt_top : ⨅ i, (f i : WithTop α) < ⊤ ↔ Nonempty ι := by
 
 end WithTop
 end WithTopBot
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Order/ConditionallyCompleteLattice/Finset.lean b/Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
index 18c36b633f629..c493a2324fc8c 100644
--- a/Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
+++ b/Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
@@ -46,12 +46,143 @@ theorem Set.Finite.csSup_lt_iff (hs : s.Finite) (h : s.Nonempty) : sSup s < a 
 theorem Set.Finite.lt_csInf_iff (hs : s.Finite) (h : s.Nonempty) : a < sInf s ↔ ∀ x ∈ s, a < x :=
   @Set.Finite.csSup_lt_iff αᵒᵈ _ _ _ hs h
 
+variable (f : ι → α)
+
+theorem Finset.ciSup_eq_max'_image {s : Finset ι} (h : ∃ x ∈ s, sSup ∅ ≤ f x)
+    (h' : (s.image f).Nonempty := by classical exact image_nonempty.mpr (h.imp fun _ ↦ And.left)) :
+    ⨆ i ∈ s, f i = (s.image f).max' h' := by
+  classical
+  rw [iSup, ← h'.csSup_eq_max', coe_image]
+  refine csSup_eq_csSup_of_forall_exists_le ?_ ?_
+  · simp only [ciSup_eq_ite, dite_eq_ite, Set.mem_range, Set.mem_image, mem_coe,
+      exists_exists_and_eq_and, forall_exists_index, forall_apply_eq_imp_iff]
+    intro i
+    split_ifs
+    · exact ⟨_, by assumption, le_rfl⟩
+    · obtain ⟨a, ha, ha'⟩ := h
+      exact ⟨a, ha, ha'⟩
+  · simp only [Set.mem_image, mem_coe, ciSup_eq_ite, dite_eq_ite, Set.mem_range,
+      exists_exists_eq_and, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+    intro i hi
+    refine ⟨i, ?_⟩
+    simp [hi]
+
+theorem Finset.ciInf_eq_min'_image {s : Finset ι} (h : ∃ x ∈ s, f x ≤ sInf ∅)
+    (h' : (s.image f).Nonempty := by classical exact image_nonempty.mpr (h.imp fun _ ↦ And.left)) :
+    ⨅ i ∈ s, f i = (s.image f).min' h' := by
+  classical
+  rw [← OrderDual.toDual_inj, toDual_min', toDual_iInf]
+  simp only [Function.comp_apply, toDual_iInf]
+  rw [ciSup_eq_max'_image _ h]
+  simp only [image_image]
+  congr
+
+theorem Finset.ciSup_mem_image {s : Finset ι} (h : ∃ x ∈ s, sSup ∅ ≤ f x) :
+    ⨆ i ∈ s, f i ∈ s.image f := by
+  rw [ciSup_eq_max'_image _ h]
+  exact max'_mem (image f s) _
+
+theorem Finset.ciInf_mem_image {s : Finset ι} (h : ∃ x ∈ s, f x ≤ sInf ∅) :
+    ⨅ i ∈ s, f i ∈ s.image f := by
+  rw [ciInf_eq_min'_image _ h]
+  exact min'_mem (image f s) _
+
+theorem Set.Finite.ciSup_mem_image {s : Set ι} (hs : s.Finite) (h : ∃ x ∈ s, sSup ∅ ≤ f x) :
+    ⨆ i ∈ s, f i ∈ f '' s := by
+  lift s to Finset ι using hs
+  simp only [Finset.mem_coe] at h
+  simpa using Finset.ciSup_mem_image f h
+
+theorem Set.Finite.ciInf_mem_image {s : Set ι} (hs : s.Finite) (h : ∃ x ∈ s, f x ≤ sInf ∅) :
+    ⨅ i ∈ s, f i ∈ f '' s := by
+  lift s to Finset ι using hs
+  simp only [Finset.mem_coe] at h
+  simpa using Finset.ciInf_mem_image f h
+
+theorem Set.Finite.ciSup_lt_iff {s : Set ι} {f : ι → α} (hs : s.Finite)
+    (h : ∃ x ∈ s, sSup ∅ ≤ f x) :
+    ⨆ i ∈ s, f i < a ↔ ∀ x ∈ s, f x < a := by
+  constructor
+  · intro h x hx
+    refine h.trans_le' (le_csSup ?_ ?_)
+    · classical
+      refine (((hs.image f).union (finite_singleton (sSup ∅))).subset ?_).bddAbove
+      intro
+      simp only [ciSup_eq_ite, dite_eq_ite, mem_range, union_singleton, mem_insert_iff, mem_image,
+        forall_exists_index]
+      intro x hx
+      split_ifs at hx
+      · exact Or.inr ⟨_, by assumption, hx⟩
+      · simp_all
+    · simp only [mem_range]
+      refine ⟨x, ?_⟩
+      simp [hx]
+  · intro H
+    have := hs.ciSup_mem_image _ h
+    simp only [mem_image] at this
+    obtain ⟨_, hmem, hx⟩ := this
+    rw [← hx]
+    exact H _ hmem
+
+theorem Set.Finite.lt_ciInf_iff {s : Set ι} {f : ι → α} (hs : s.Finite)
+    (h : ∃ x ∈ s, f x ≤ sInf ∅) :
+    a < ⨅ i ∈ s, f i ↔ ∀ x ∈ s, a < f x := by
+  constructor
+  · intro h x hx
+    refine h.trans_le (csInf_le ?_ ?_)
+    · classical
+      refine (((hs.image f).union (finite_singleton (sInf ∅))).subset ?_).bddBelow
+      intro
+      simp only [ciInf_eq_ite, dite_eq_ite, mem_range, union_singleton, mem_insert_iff, mem_image,
+        forall_exists_index]
+      intro x hx
+      split_ifs at hx
+      · exact Or.inr ⟨_, by assumption, hx⟩
+      · simp_all
+    · simp only [mem_range]
+      refine ⟨x, ?_⟩
+      simp [hx]
+  · intro H
+    have := hs.ciInf_mem_image _ h
+    simp only [mem_image] at this
+    obtain ⟨_, hmem, hx⟩ := this
+    rw [← hx]
+    exact H _ hmem
+
+section ListMultiset
+
+lemma List.iSup_mem_map_of_exists_sSup_empty_le {l : List ι} (f : ι → α)
+    (h : ∃ x ∈ l, sSup ∅ ≤ f x) :
+    ⨆ x ∈ l, f x ∈ l.map f := by
+  classical
+  simpa using l.toFinset.ciSup_mem_image f (by simpa using h)
+
+lemma List.iInf_mem_map_of_exists_le_sInf_empty {l : List ι} (f : ι → α)
+    (h : ∃ x ∈ l, f x ≤ sInf ∅) :
+    ⨅ x ∈ l, f x ∈ l.map f := by
+  classical
+  simpa using l.toFinset.ciInf_mem_image f (by simpa using h)
+
+lemma Multiset.iSup_mem_map_of_exists_sSup_empty_le {s : Multiset ι} (f : ι → α)
+    (h : ∃ x ∈ s, sSup ∅ ≤ f x) :
+    ⨆ x ∈ s, f x ∈ s.map f := by
+  classical
+  simpa using s.toFinset.ciSup_mem_image f (by simpa using h)
+
+lemma Multiset.iInf_mem_map_of_exists_le_sInf_empty {s : Multiset ι} (f : ι → α)
+    (h : ∃ x ∈ s, f x ≤ sInf ∅) :
+    ⨅ x ∈ s, f x ∈ s.map f := by
+  classical
+  simpa using s.toFinset.ciInf_mem_image f (by simpa using h)
+
 theorem exists_eq_ciSup_of_finite [Nonempty ι] [Finite ι] {f : ι → α} : ∃ i, f i = ⨆ i, f i :=
   Nonempty.csSup_mem (range_nonempty f) (finite_range f)
 
 theorem exists_eq_ciInf_of_finite [Nonempty ι] [Finite ι] {f : ι → α} : ∃ i, f i = ⨅ i, f i :=
   Nonempty.csInf_mem (range_nonempty f) (finite_range f)
 
+end ListMultiset
+
 end ConditionallyCompleteLinearOrder
 
 /-!
@@ -102,3 +233,38 @@ lemma sup_univ_eq_ciSup [Fintype ι] (f : ι → α) : univ.sup f = ⨆ i, f i :
 end ConditionallyCompleteLinearOrderBot
 
 end Finset
+
+section ConditionallyCompleteLinearOrderBot
+
+variable [ConditionallyCompleteLinearOrderBot α] (f : ι → α)
+
+theorem Finset.Nonempty.ciSup_eq_max'_image {s : Finset ι} (h : s.Nonempty)
+    (h' : (s.image f).Nonempty := h.image f) :
+    ⨆ i ∈ s, f i = (s.image f).max' h' :=
+  s.ciSup_eq_max'_image _ (h.imp (by simp)) _
+
+theorem Finset.Nonempty.ciSup_mem_image {s : Finset ι} (h : s.Nonempty) :
+    ⨆ i ∈ s, f i ∈ s.image f :=
+  s.ciSup_mem_image _ (h.imp (by simp))
+
+theorem Set.Nonempty.ciSup_mem_image {s : Set ι} (h : s.Nonempty) (hs : s.Finite) :
+    ⨆ i ∈ s, f i ∈ f '' s :=
+  hs.ciSup_mem_image _ (h.imp (by simp))
+
+theorem Set.Nonempty.ciSup_lt_iff {s : Set ι} {a : α} {f : ι → α} (h : s.Nonempty) (hs : s.Finite) :
+    ⨆ i ∈ s, f i < a ↔ ∀ x ∈ s, f x < a :=
+  hs.ciSup_lt_iff (h.imp (by simp))
+
+section ListMultiset
+
+lemma List.iSup_mem_map_of_ne_nil {l : List ι} (f : ι → α) (h : l ≠ []) :
+    ⨆ x ∈ l, f x ∈ l.map f :=
+  l.iSup_mem_map_of_exists_sSup_empty_le _ (by simpa using exists_mem_of_ne_nil _ h)
+
+lemma Multiset.iSup_mem_map_of_ne_zero {s : Multiset ι} (f : ι → α) (h : s ≠ 0) :
+    ⨆ x ∈ s, f x ∈ s.map f :=
+  s.iSup_mem_map_of_exists_sSup_empty_le _ (by simpa using exists_mem_of_ne_zero h)
+
+end ListMultiset
+
+end ConditionallyCompleteLinearOrderBot
diff --git a/Mathlib/Order/CountableDenseLinearOrder.lean b/Mathlib/Order/CountableDenseLinearOrder.lean
index 0e43df278296d..1ab3a1faebbc2 100644
--- a/Mathlib/Order/CountableDenseLinearOrder.lean
+++ b/Mathlib/Order/CountableDenseLinearOrder.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Wärn
 -/
 import Mathlib.Order.Ideal
-import Mathlib.Data.Finset.Lattice
+import Mathlib.Data.Finset.Max
 
 /-!
 # The back and forth method and countable dense linear orders
@@ -33,11 +33,13 @@ noncomputable section
 
 namespace Order
 
+variable {α β : Type*} [LinearOrder α] [LinearOrder β]
+
 /-- Suppose `α` is a nonempty dense linear order without endpoints, and
     suppose `lo`, `hi`, are finite subsets with all of `lo` strictly
     before `hi`. Then there is an element of `α` strictly between `lo`
     and `hi`. -/
-theorem exists_between_finsets {α : Type*} [LinearOrder α] [DenselyOrdered α] [NoMinOrder α]
+theorem exists_between_finsets [DenselyOrdered α] [NoMinOrder α]
     [NoMaxOrder α] [nonem : Nonempty α] (lo hi : Finset α) (lo_lt_hi : ∀ x ∈ lo, ∀ y ∈ hi, x < y) :
     ∃ m : α, (∀ x ∈ lo, x < m) ∧ ∀ y ∈ hi, m < y :=
   if nlo : lo.Nonempty then
@@ -61,7 +63,40 @@ theorem exists_between_finsets {α : Type*} [LinearOrder α] [DenselyOrdered α]
           nonem.elim
         fun m ↦ ⟨m, fun x hx ↦ (nlo ⟨x, hx⟩).elim, fun y hy ↦ (nhi ⟨y, hy⟩).elim⟩
 
-variable (α β : Type*) [LinearOrder α] [LinearOrder β]
+lemma exists_orderEmbedding_insert [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β]
+    [nonem : Nonempty β]  (S : Finset α) (f : S ↪o β) (a : α) :
+    ∃ (g : (insert a S : Finset α) ↪o β),
+      g ∘ (Set.inclusion ((S.subset_insert a) : ↑S ⊆ ↑(insert a S))) = f := by
+  let Slt := (S.attach.filter (fun (x : S) => x < a)).image f
+  let Sgt := (S.attach.filter (fun (x : S) => a < x)).image f
+  obtain ⟨b, hb, hb'⟩ := Order.exists_between_finsets Slt Sgt (fun x hx y hy => by
+    simp only [Finset.mem_image, Finset.mem_filter, Finset.mem_attach, true_and, Subtype.exists,
+      exists_and_left, Slt, Sgt] at hx hy
+    obtain ⟨_, hx, _, rfl⟩ := hx
+    obtain ⟨_, hy, _, rfl⟩ := hy
+    exact f.strictMono (hx.trans hy))
+  refine ⟨OrderEmbedding.ofStrictMono
+    (fun (x : (insert a S : Finset α)) => if hx : x.1 ∈ S then f ⟨x.1, hx⟩ else b) ?_, ?_⟩
+  · rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
+    if hxS : x ∈ S
+    then if hyS : y ∈ S
+      then simpa only [hxS, hyS, ↓reduceDIte, OrderEmbedding.lt_iff_lt, Subtype.mk_lt_mk]
+      else
+        obtain rfl := Finset.eq_of_not_mem_of_mem_insert hy hyS
+        simp only [hxS, hyS, ↓reduceDIte]
+        exact hb _ (Finset.mem_image_of_mem _ (Finset.mem_filter.2 ⟨Finset.mem_attach _ _, hxy⟩))
+    else
+      obtain rfl := Finset.eq_of_not_mem_of_mem_insert hx hxS
+      if hyS : y ∈ S
+      then
+        simp only [hxS, hyS, ↓reduceDIte]
+        exact hb' _ (Finset.mem_image_of_mem _ (Finset.mem_filter.2 ⟨Finset.mem_attach _ _, hxy⟩))
+      else simp only [Finset.eq_of_not_mem_of_mem_insert hy hyS, lt_self_iff_false] at hxy
+  · ext x
+    simp only [Finset.coe_sort_coe, OrderEmbedding.coe_ofStrictMono, Finset.insert_val,
+      Function.comp_apply, Finset.coe_mem, ↓reduceDIte, Subtype.coe_eta]
+
+variable (α β)
 
 -- Porting note: Mathport warning: expanding binder collection (p q «expr ∈ » f)
 /-- The type of partial order isomorphisms between `α` and `β` defined on finite subsets.
diff --git a/Mathlib/Order/Cover.lean b/Mathlib/Order/Cover.lean
index 860d217d8fddb..cf13cbab95f7f 100644
--- a/Mathlib/Order/Cover.lean
+++ b/Mathlib/Order/Cover.lean
@@ -80,7 +80,7 @@ theorem wcovBy_congr_right (hab : AntisymmRel (· ≤ ·) a b) : c ⩿ a ↔ c 
 
 /-- If `a ≤ b`, then `b` does not cover `a` iff there's an element in between. -/
 theorem not_wcovBy_iff (h : a ≤ b) : ¬a ⩿ b ↔ ∃ c, a < c ∧ c < b := by
-  simp_rw [WCovBy, h, true_and_iff, not_forall, exists_prop, not_not]
+  simp_rw [WCovBy, h, true_and, not_forall, exists_prop, not_not]
 
 instance WCovBy.isRefl : IsRefl α (· ⩿ ·) :=
   ⟨WCovBy.refl⟩
@@ -127,6 +127,19 @@ alias ⟨_, WCovBy.toDual⟩ := toDual_wcovBy_toDual_iff
 
 alias ⟨_, WCovBy.ofDual⟩ := ofDual_wcovBy_ofDual_iff
 
+theorem OrderEmbedding.wcovBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
+    (f : α ↪o β) {x y : α} (h : f x ⩿ f y) : x ⩿ y := by
+  use f.le_iff_le.1 h.1
+  intro a
+  rw [← f.lt_iff_lt, ← f.lt_iff_lt]
+  apply h.2
+
+theorem OrderIso.map_wcovBy {α β : Type*} [Preorder α] [Preorder β]
+    (f : α ≃o β) {x y : α} : f x ⩿ f y ↔ x ⩿ y := by
+  use f.toOrderEmbedding.wcovBy_of_apply
+  conv_lhs => rw [← f.symm_apply_apply x, ← f.symm_apply_apply y]
+  exact f.symm.toOrderEmbedding.wcovBy_of_apply
+
 end Preorder
 
 section PartialOrder
@@ -195,7 +208,7 @@ theorem CovBy.lt (h : a ⋖ b) : a < b :=
 
 /-- If `a < b`, then `b` does not cover `a` iff there's an element in between. -/
 theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by
-  simp_rw [CovBy, h, true_and_iff, not_forall, exists_prop, not_not]
+  simp_rw [CovBy, h, true_and, not_forall, exists_prop, not_not]
 
 alias ⟨exists_lt_lt_of_not_covBy, _⟩ := not_covBy_iff
 
@@ -313,6 +326,19 @@ theorem apply_covBy_apply_iff {E : Type*} [EquivLike E α β] [OrderIsoClass E 
 theorem covBy_of_eq_or_eq (hab : a < b) (h : ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b) : a ⋖ b :=
   ⟨hab, fun c ha hb => (h c ha.le hb.le).elim ha.ne' hb.ne⟩
 
+theorem OrderEmbedding.covBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
+    (f : α ↪o β) {x y : α} (h : f x ⋖ f y) : x ⋖ y := by
+  use f.lt_iff_lt.1 h.1
+  intro a
+  rw [← f.lt_iff_lt, ← f.lt_iff_lt]
+  apply h.2
+
+theorem OrderIso.map_covBy {α β : Type*} [Preorder α] [Preorder β]
+    (f : α ≃o β) {x y : α} : f x ⋖ f y ↔ x ⋖ y := by
+  use f.toOrderEmbedding.covBy_of_apply
+  conv_lhs => rw [← f.symm_apply_apply x, ← f.symm_apply_apply y]
+  exact f.symm.toOrderEmbedding.covBy_of_apply
+
 end Preorder
 
 section PartialOrder
@@ -565,3 +591,19 @@ variable [Preorder α] {a b : α}
   simp only [wcovBy_iff_Ioo_eq, ← image_coe_Iio, bot_le, image_eq_empty, true_and, Iio_eq_empty_iff]
 
 end WithBot
+
+section WellFounded
+
+variable [Preorder α]
+
+lemma exists_covBy_of_wellFoundedLT [wf : WellFoundedLT α] ⦃a : α⦄ (h : ¬ IsMax a) :
+    ∃ a', a ⋖ a' := by
+  rw [not_isMax_iff] at h
+  exact ⟨_, wellFounded_lt.min_mem _ h, fun a' ↦ wf.wf.not_lt_min _ h⟩
+
+lemma exists_covBy_of_wellFoundedGT [wf : WellFoundedGT α] ⦃a : α⦄ (h : ¬ IsMin a) :
+    ∃ a', a' ⋖ a := by
+  rw [not_isMin_iff] at h
+  exact ⟨_, wf.wf.min_mem _ h, fun a' h₁ h₂ ↦ wf.wf.not_lt_min _ h h₂ h₁⟩
+
+end WellFounded
diff --git a/Mathlib/Order/Defs.lean b/Mathlib/Order/Defs.lean
index 9b7259b58ef99..a6a6ffa250fe6 100644
--- a/Mathlib/Order/Defs.lean
+++ b/Mathlib/Order/Defs.lean
@@ -4,9 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 import Batteries.Classes.Order
-import Mathlib.Init.Logic
 import Mathlib.Data.Ordering.Basic
+import Mathlib.Tactic.Lemma
+import Mathlib.Tactic.Relation.Trans
 import Mathlib.Tactic.SplitIfs
+import Mathlib.Tactic.TypeStar
 
 /-!
 # Orders
@@ -17,8 +19,8 @@ and proves some basic lemmas about them.
 
 /-! ### Unbundled classes -/
 
-universe u
-variable {α : Type u}
+/-- An empty relation does not relate any elements. -/
+@[nolint unusedArguments] def EmptyRelation {α : Sort*} := fun _ _ : α ↦ False
 
 /-- `IsIrrefl X r` means the binary relation `r` on `X` is irreflexive (that is, `r x x` never
 holds). -/
@@ -71,7 +73,7 @@ class IsPartialOrder (α : Sort*) (r : α → α → Prop) extends IsPreorder α
 
 /-- `IsLinearOrder X r` means that the binary relation `r` on `X` is a linear order, that is,
 `IsPartialOrder X r` and `IsTotal X r`. -/
-class IsLinearOrder (α : Sort u) (r : α → α → Prop) extends IsPartialOrder α r, IsTotal α r : Prop
+class IsLinearOrder (α : Sort*) (r : α → α → Prop) extends IsPartialOrder α r, IsTotal α r : Prop
 
 /-- `IsEquiv X r` means that the binary relation `r` on `X` is an equivalence relation, that
 is, `IsPreorder X r` and `IsSymm X r`. -/
@@ -83,7 +85,7 @@ class IsStrictOrder (α : Sort*) (r : α → α → Prop) extends IsIrrefl α r,
 
 /-- `IsStrictWeakOrder X lt` means that the binary relation `lt` on `X` is a strict weak order,
 that is, `IsStrictOrder X lt` and `¬lt a b ∧ ¬lt b a → ¬lt b c ∧ ¬lt c b → ¬lt a c ∧ ¬lt c a`. -/
-class IsStrictWeakOrder (α : Sort u) (lt : α → α → Prop) extends IsStrictOrder α lt : Prop where
+class IsStrictWeakOrder (α : Sort*) (lt : α → α → Prop) extends IsStrictOrder α lt : Prop where
   incomp_trans : ∀ a b c, ¬lt a b ∧ ¬lt b a → ¬lt b c ∧ ¬lt c b → ¬lt a c ∧ ¬lt c a
 
 /-- `IsTrichotomous X lt` means that the binary relation `lt` on `X` is trichotomous, that is,
@@ -106,6 +108,7 @@ section
 
 variable {α : Sort*} {r : α → α → Prop} {a b c : α}
 
+/-- Local notation for an arbitrary binary relation `r`. -/
 local infixl:50 " ≺ " => r
 
 lemma irrefl [IsIrrefl α r] (a : α) : ¬a ≺ a := IsIrrefl.irrefl a
@@ -136,10 +139,55 @@ lemma total_of [IsTotal α r] (a b : α) : a ≺ b ∨ b ≺ a := IsTotal.total
 @[elab_without_expected_type]
 lemma trichotomous_of [IsTrichotomous α r] : ∀ a b : α, a ≺ b ∨ a = b ∨ b ≺ a := trichotomous
 
+section
+
+/-- `IsRefl` as a definition, suitable for use in proofs. -/
+def Reflexive := ∀ x, x ≺ x
+
+/-- `IsSymm` as a definition, suitable for use in proofs. -/
+def Symmetric := ∀ ⦃x y⦄, x ≺ y → y ≺ x
+
+/-- `IsTrans` as a definition, suitable for use in proofs. -/
+def Transitive := ∀ ⦃x y z⦄, x ≺ y → y ≺ z → x ≺ z
+
+/-- `IsIrrefl` as a definition, suitable for use in proofs. -/
+def Irreflexive := ∀ x, ¬x ≺ x
+
+/-- `IsAntisymm` as a definition, suitable for use in proofs. -/
+def AntiSymmetric := ∀ ⦃x y⦄, x ≺ y → y ≺ x → x = y
+
+/-- `IsTotal` as a definition, suitable for use in proofs. -/
+def Total := ∀ x y, x ≺ y ∨ y ≺ x
+
+@[deprecated Equivalence.refl (since := "2024-09-13")]
+theorem Equivalence.reflexive (h : Equivalence r) : Reflexive r := h.refl
+
+@[deprecated Equivalence.symm (since := "2024-09-13")]
+theorem Equivalence.symmetric (h : Equivalence r) : Symmetric r :=
+  fun _ _ ↦ h.symm
+
+@[deprecated Equivalence.trans (since := "2024-09-13")]
+theorem Equivalence.transitive (h : Equivalence r) : Transitive r :=
+  fun _ _ _ ↦ h.trans
+
+variable {β : Sort*} (r : β → β → Prop) (f : α → β)
+
+@[deprecated (since := "2024-09-13")]
+theorem InvImage.trans (h : Transitive r) : Transitive (InvImage r f) :=
+  fun (a₁ a₂ a₃ : α) (h₁ : InvImage r f a₁ a₂) (h₂ : InvImage r f a₂ a₃) ↦ h h₁ h₂
+
+@[deprecated (since := "2024-09-13")]
+theorem InvImage.irreflexive (h : Irreflexive r) : Irreflexive (InvImage r f) :=
+  fun (a : α) (h₁ : InvImage r f a a) ↦ h (f a) h₁
+
+end
+
 end
 
 /-! ### Bundled classes -/
 
+variable {α : Type*}
+
 section Preorder
 
 /-!
@@ -147,7 +195,7 @@ section Preorder
 -/
 
 /-- A preorder is a reflexive, transitive relation `≤` with `a < b` defined in the obvious way. -/
-class Preorder (α : Type u) extends LE α, LT α where
+class Preorder (α : Type*) extends LE α, LT α where
   le_refl : ∀ a : α, a ≤ a
   le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c
   lt := fun a b => a ≤ b ∧ ¬b ≤ a
@@ -235,7 +283,7 @@ section PartialOrder
 -/
 
 /-- A partial order is a reflexive, transitive, antisymmetric relation `≤`. -/
-class PartialOrder (α : Type u) extends Preorder α where
+class PartialOrder (α : Type*) extends Preorder α where
   le_antisymm : ∀ a b : α, a ≤ b → b ≤ a → a = b
 
 variable [PartialOrder α] {a b : α}
@@ -286,11 +334,11 @@ section LinearOrder
 -/
 
 /-- Default definition of `max`. -/
-def maxDefault {α : Type u} [LE α] [DecidableRel ((· ≤ ·) : α → α → Prop)] (a b : α) :=
+def maxDefault [LE α] [DecidableRel ((· ≤ ·) : α → α → Prop)] (a b : α) :=
   if a ≤ b then b else a
 
 /-- Default definition of `min`. -/
-def minDefault {α : Type u} [LE α] [DecidableRel ((· ≤ ·) : α → α → Prop)] (a b : α) :=
+def minDefault [LE α] [DecidableRel ((· ≤ ·) : α → α → Prop)] (a b : α) :=
   if a ≤ b then a else b
 
 /-- This attempts to prove that a given instance of `compare` is equal to `compareOfLessAndEq` by
@@ -309,7 +357,7 @@ macro "compareOfLessAndEq_rfl" : tactic =>
 
 /-- A linear order is reflexive, transitive, antisymmetric and total relation `≤`.
 We assume that every linear ordered type has decidable `(≤)`, `(<)`, and `(=)`. -/
-class LinearOrder (α : Type u) extends PartialOrder α, Min α, Max α, Ord α :=
+class LinearOrder (α : Type*) extends PartialOrder α, Min α, Max α, Ord α where
   /-- A linear order is total. -/
   le_total (a b : α) : a ≤ b ∨ b ≤ a
   /-- In a linearly ordered type, we assume the order relations are all decidable. -/
@@ -390,6 +438,22 @@ def ltByCases (x y : α) {P : Sort*} (h₁ : x < y → P) (h₂ : x = y → P) (
   if h : x < y then h₁ h
   else if h' : y < x then h₃ h' else h₂ (le_antisymm (le_of_not_gt h') (le_of_not_gt h))
 
+namespace Nat
+
+/-! Deprecated properties of inequality on `Nat` -/
+
+@[deprecated (since := "2024-08-23")]
+protected def ltGeByCases {a b : Nat} {C : Sort*} (h₁ : a < b → C) (h₂ : b ≤ a → C) : C :=
+  Decidable.byCases h₁ fun h => h₂ (Or.elim (Nat.lt_or_ge a b) (fun a => absurd a h) fun a => a)
+
+set_option linter.deprecated false in
+@[deprecated ltByCases (since := "2024-08-23")]
+protected def ltByCases {a b : Nat} {C : Sort*} (h₁ : a < b → C) (h₂ : a = b → C)
+    (h₃ : b < a → C) : C :=
+  Nat.ltGeByCases h₁ fun h₁ => Nat.ltGeByCases h₃ fun h => h₂ (Nat.le_antisymm h h₁)
+
+end Nat
+
 theorem le_imp_le_of_lt_imp_lt {α β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}
     (H : d < c → b < a) (h : a ≤ b) : c ≤ d :=
   le_of_not_lt fun h' => not_le_of_gt (H h') h
@@ -437,13 +501,16 @@ lemma min_comm (a b : α) : min a b = min b a :=
 
 lemma min_assoc (a b c : α) : min (min a b) c = min a (min b c) := by
   apply eq_min
-  · apply le_trans; apply min_le_left; apply min_le_left
-  · apply le_min; apply le_trans; apply min_le_left; apply min_le_right; apply min_le_right
-  · intro d h₁ h₂; apply le_min; apply le_min h₁; apply le_trans h₂; apply min_le_left
-    apply le_trans h₂; apply min_le_right
+  · apply le_trans (min_le_left ..); apply min_le_left
+  · apply le_min
+    · apply le_trans (min_le_left ..); apply min_le_right
+    · apply min_le_right
+  · intro d h₁ h₂; apply le_min
+    · apply le_min h₁; apply le_trans h₂; apply min_le_left
+    · apply le_trans h₂; apply min_le_right
 
-lemma min_left_comm : ∀ a b c : α, min a (min b c) = min b (min a c) :=
-  left_comm (@min α _) (@min_comm α _) (@min_assoc α _)
+lemma min_left_comm (a b c : α) : min a (min b c) = min b (min a c) := by
+  rw [← min_assoc, min_comm a, min_assoc]
 
 @[simp] lemma min_self (a : α) : min a a = a := by simp [min_def]
 
@@ -461,13 +528,16 @@ lemma max_comm (a b : α) : max a b = max b a :=
 
 lemma max_assoc (a b c : α) : max (max a b) c = max a (max b c) := by
   apply eq_max
-  · apply le_trans; apply le_max_left a b; apply le_max_left
-  · apply max_le; apply le_trans; apply le_max_right a b; apply le_max_left; apply le_max_right
-  · intro d h₁ h₂; apply max_le; apply max_le h₁; apply le_trans (le_max_left _ _) h₂
-    apply le_trans (le_max_right _ _) h₂
+  · apply le_trans (le_max_left a b); apply le_max_left
+  · apply max_le
+    · apply le_trans (le_max_right a b); apply le_max_left
+    · apply le_max_right
+  · intro d h₁ h₂; apply max_le
+    · apply max_le h₁; apply le_trans (le_max_left _ _) h₂
+    · apply le_trans (le_max_right _ _) h₂
 
-lemma max_left_comm : ∀ a b c : α, max a (max b c) = max b (max a c) :=
-  left_comm (@max α _) (@max_comm α _) (@max_assoc α _)
+lemma max_left_comm (a b c : α) : max a (max b c) = max b (max a c) := by
+  rw [← max_assoc, max_comm a, max_assoc]
 
 @[simp] lemma max_self (a : α) : max a a = a := by simp [max_def]
 
@@ -488,25 +558,24 @@ lemma max_lt (h₁ : a < c) (h₂ : b < c) : max a b < c := by
   cases le_total a b <;> simp [max_eq_left, max_eq_right, *]
 
 section Ord
-variable {o : Ordering}
 
 lemma compare_lt_iff_lt : compare a b = .lt ↔ a < b := by
   rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
-  split_ifs <;> simp only [*, lt_irrefl]
+  split_ifs <;> simp only [*, lt_irrefl, reduceCtorEq]
 
 lemma compare_gt_iff_gt : compare a b = .gt ↔ a > b := by
   rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
-  split_ifs <;> simp only [*, lt_irrefl, not_lt_of_gt]
+  split_ifs <;> simp only [*, lt_irrefl, not_lt_of_gt, reduceCtorEq]
   case _ h₁ h₂ =>
     have h : b < a := lt_trichotomy a b |>.resolve_left h₁ |>.resolve_left h₂
-    exact true_iff_iff.2 h
+    rwa [true_iff]
 
 lemma compare_eq_iff_eq : compare a b = .eq ↔ a = b := by
   rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq]
-  split_ifs <;> try simp only
-  case _ h   => exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h
-  case _ _ h => exact true_iff_iff.2 h
-  case _ _ h => exact false_iff_iff.2 h
+  split_ifs <;> try simp only [reduceCtorEq]
+  case _ h   => rw [false_iff]; exact ne_iff_lt_or_gt.2 <| .inl h
+  case _ _ h => rwa [true_iff]
+  case _ _ h => rwa [false_iff]
 
 lemma compare_le_iff_le : compare a b ≠ .gt ↔ a ≤ b := by
   cases h : compare a b <;> simp
@@ -520,13 +589,23 @@ lemma compare_ge_iff_ge : compare a b ≠ .lt ↔ a ≥ b := by
   · exact le_of_eq <| (·.symm) <| compare_eq_iff_eq.1 h
   · exact le_of_lt <| compare_gt_iff_gt.1 h
 
-lemma compare_iff (a b : α) : compare a b = o ↔ o.toRel a b := by
-  cases o <;> simp only [Ordering.toRel]
+lemma compare_iff (a b : α) {o : Ordering} : compare a b = o ↔ o.Compares a b := by
+  cases o <;> simp only [Ordering.Compares]
   · exact compare_lt_iff_lt
   · exact compare_eq_iff_eq
   · exact compare_gt_iff_gt
 
-instance : Batteries.TransCmp (compare (α := α)) where
+theorem cmp_eq_compare (a b : α) : cmp a b = compare a b := by
+  refine ((compare_iff ..).2 ?_).symm
+  unfold cmp cmpUsing; split_ifs with h1 h2
+  · exact h1
+  · exact h2
+  · exact le_antisymm (not_lt.1 h2) (not_lt.1 h1)
+
+theorem cmp_eq_compareOfLessAndEq (a b : α) : cmp a b = compareOfLessAndEq a b :=
+  (cmp_eq_compare ..).trans (LinearOrder.compare_eq_compareOfLessAndEq ..)
+
+instance : Batteries.LawfulCmp (compare (α := α)) where
   symm a b := by
     cases h : compare a b <;>
     simp only [Ordering.swap] <;> symm
@@ -535,6 +614,9 @@ instance : Batteries.TransCmp (compare (α := α)) where
     · exact compare_lt_iff_lt.2 <| compare_gt_iff_gt.1 h
   le_trans := fun h₁ h₂ ↦
     compare_le_iff_le.2 <| le_trans (compare_le_iff_le.1 h₁) (compare_le_iff_le.1 h₂)
+  cmp_iff_beq := by simp [compare_eq_iff_eq]
+  cmp_iff_lt := by simp [compare_lt_iff_lt]
+  cmp_iff_le := by simp [compare_le_iff_le]
 
 end Ord
 
diff --git a/Mathlib/Order/Directed.lean b/Mathlib/Order/Directed.lean
index 8ab4e604c1f95..29285fca18124 100644
--- a/Mathlib/Order/Directed.lean
+++ b/Mathlib/Order/Directed.lean
@@ -17,8 +17,6 @@ directed iff each pair of elements has a shared upper bound.
 * `DirectedOn r s`: Predicate stating that the set `s` is `r`-directed.
 * `IsDirected α r`: Prop-valued mixin stating that `α` is `r`-directed. Follows the style of the
   unbundled relation classes such as `IsTotal`.
-* `ScottContinuous`: Predicate stating that a function between preorders preserves `IsLUB` on
-  directed sets.
 
 ## TODO
 
@@ -40,7 +38,7 @@ variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
 local infixl:50 " ≼ " => r
 
 /-- A family of elements of α is directed (with respect to a relation `≼` on α)
-  if there is a member of the family `≼`-above any pair in the family.  -/
+  if there is a member of the family `≼`-above any pair in the family. -/
 def Directed (f : ι → α) :=
   ∀ x y, ∃ z, f x ≼ f z ∧ f y ≼ f z
 
@@ -110,7 +108,7 @@ theorem Directed.extend_bot [Preorder α] [OrderBot α] {e : ι → β} {f : ι
     simp [Function.extend_apply' _ _ _ hb]
   rcases hf i j with ⟨k, hi, hj⟩
   use e k
-  simp only [he.extend_apply, *, true_and_iff]
+  simp only [he.extend_apply, *, true_and]
 
 /-- A set stable by infimum is `≥`-directed. -/
 theorem directedOn_of_inf_mem [SemilatticeInf α] {S : Set α}
@@ -169,7 +167,7 @@ instance OrderDual.isDirected_le [LE α] [IsDirected α (· ≥ ·)] : IsDirecte
 /-- A monotone function on an upwards-directed type is directed. -/
 theorem directed_of_isDirected_le [LE α] [IsDirected α (· ≤ ·)] {f : α → β} {r : β → β → Prop}
     (H : ∀ ⦃i j⦄, i ≤ j → r (f i) (f j)) : Directed r f :=
-  directed_id.mono_comp H
+  directed_id.mono_comp _ H
 
 theorem Monotone.directed_le [Preorder α] [IsDirected α (· ≤ ·)] [Preorder β] {f : α → β} :
     Monotone f → Directed (· ≤ ·) f :=
@@ -250,18 +248,48 @@ theorem isBot_iff_isMin [IsDirected α (· ≥ ·)] : IsBot a ↔ IsMin a :=
 theorem isTop_iff_isMax [IsDirected α (· ≤ ·)] : IsTop a ↔ IsMax a :=
   ⟨IsTop.isMax, IsMax.isTop⟩
 
-variable (β) [PartialOrder β]
+end Preorder
+
+section PartialOrder
+
+variable [PartialOrder β]
 
-theorem exists_lt_of_directed_ge [IsDirected β (· ≥ ·)] [Nontrivial β] : ∃ a b : β, a < b := by
+section Nontrivial
+
+variable [Nontrivial β]
+
+variable (β) in
+theorem exists_lt_of_directed_ge [IsDirected β (· ≥ ·)] :
+    ∃ a b : β, a < b := by
   rcases exists_pair_ne β with ⟨a, b, hne⟩
   rcases isBot_or_exists_lt a with (ha | ⟨c, hc⟩)
   exacts [⟨a, b, (ha b).lt_of_ne hne⟩, ⟨_, _, hc⟩]
 
-theorem exists_lt_of_directed_le [IsDirected β (· ≤ ·)] [Nontrivial β] : ∃ a b : β, a < b :=
+variable (β) in
+theorem exists_lt_of_directed_le [IsDirected β (· ≤ ·)] :
+    ∃ a b : β, a < b :=
   let ⟨a, b, h⟩ := exists_lt_of_directed_ge βᵒᵈ
   ⟨b, a, h⟩
 
-variable {f : α → β} {s : Set α}
+protected theorem IsMin.not_isMax [IsDirected β (· ≥ ·)] {b : β} (hb : IsMin b) : ¬ IsMax b := by
+  intro hb'
+  obtain ⟨a, c, hac⟩ := exists_lt_of_directed_ge β
+  have := hb.isBot a
+  obtain rfl := (hb' <| this).antisymm this
+  exact hb'.not_lt hac
+
+protected theorem IsMin.not_isMax' [IsDirected β (· ≤ ·)] {b : β} (hb : IsMin b) : ¬ IsMax b :=
+  fun hb' ↦ hb'.toDual.not_isMax hb.toDual
+
+protected theorem IsMax.not_isMin [IsDirected β (· ≤ ·)] {b : β} (hb : IsMax b) : ¬ IsMin b :=
+  fun hb' ↦ hb.toDual.not_isMax hb'.toDual
+
+protected theorem IsMax.not_isMin' [IsDirected β (· ≥ ·)] {b : β} (hb : IsMax b) : ¬ IsMin b :=
+  fun hb' ↦ hb'.toDual.not_isMin hb.toDual
+
+end Nontrivial
+
+variable [Preorder α] {f : α → β} {s : Set α}
 
 -- TODO: Generalise the following two lemmas to connected orders
 
@@ -278,7 +306,7 @@ lemma constant_of_monotoneOn_antitoneOn (hf : MonotoneOn f s) (hf' : AntitoneOn
   obtain ⟨c, hc, hac, hbc⟩ := hs _ ha _ hb
   exact le_antisymm ((hf ha hc hac).trans <| hf' hb hc hbc) ((hf hb hc hbc).trans <| hf' ha hc hac)
 
-end Preorder
+end PartialOrder
 
 -- see Note [lower instance priority]
 instance (priority := 100) SemilatticeSup.to_isDirected_le [SemilatticeSup α] :
diff --git a/Mathlib/Order/Disjoint.lean b/Mathlib/Order/Disjoint.lean
index 29678782f984f..959e072e1b1b6 100644
--- a/Mathlib/Order/Disjoint.lean
+++ b/Mathlib/Order/Disjoint.lean
@@ -108,7 +108,7 @@ end PartialBoundedOrder
 
 section SemilatticeInfBot
 
-variable [SemilatticeInf α] [OrderBot α] {a b c d : α}
+variable [SemilatticeInf α] [OrderBot α] {a b c : α}
 
 theorem disjoint_iff_inf_le : Disjoint a b ↔ a ⊓ b ≤ ⊥ :=
   ⟨fun hd ↦ hd inf_le_left inf_le_right, fun h _ ha hb ↦ (le_inf ha hb).trans h⟩
@@ -155,6 +155,10 @@ theorem Disjoint.of_disjoint_inf_of_le' (h : Disjoint (a ⊓ b) c) (hle : b ≤
 
 end SemilatticeInfBot
 
+theorem Disjoint.right_lt_sup_of_left_ne_bot [SemilatticeSup α] [OrderBot α] {a b : α}
+    (h : Disjoint a b) (ha : a ≠ ⊥) : b < a ⊔ b :=
+  le_sup_right.lt_of_ne fun eq ↦ ha (le_bot_iff.mp <| h le_rfl <| sup_eq_right.mp eq.symm)
+
 section DistribLatticeBot
 
 variable [DistribLattice α] [OrderBot α] {a b c : α}
@@ -267,7 +271,7 @@ end PartialBoundedOrder
 
 section SemilatticeSupTop
 
-variable [SemilatticeSup α] [OrderTop α] {a b c d : α}
+variable [SemilatticeSup α] [OrderTop α] {a b c : α}
 
 theorem codisjoint_iff_le_sup : Codisjoint a b ↔ ⊤ ≤ a ⊔ b :=
   @disjoint_iff_inf_le αᵒᵈ _ _ _ _
@@ -401,7 +405,7 @@ namespace IsCompl
 
 section BoundedPartialOrder
 
-variable [PartialOrder α] [BoundedOrder α] {x y z : α}
+variable [PartialOrder α] [BoundedOrder α] {x y : α}
 
 @[symm]
 protected theorem symm (h : IsCompl x y) : IsCompl y x :=
@@ -419,7 +423,7 @@ end BoundedPartialOrder
 
 section BoundedLattice
 
-variable [Lattice α] [BoundedOrder α] {x y z : α}
+variable [Lattice α] [BoundedOrder α] {x y : α}
 
 theorem of_le (h₁ : x ⊓ y ≤ ⊥) (h₂ : ⊤ ≤ x ⊔ y) : IsCompl x y :=
   ⟨disjoint_iff_inf_le.mpr h₁, codisjoint_iff_le_sup.mpr h₂⟩
diff --git a/Mathlib/Order/Estimator.lean b/Mathlib/Order/Estimator.lean
index 6cbb66c675781..29676a5330f47 100644
--- a/Mathlib/Order/Estimator.lean
+++ b/Mathlib/Order/Estimator.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Set.Operations
 import Mathlib.Order.Heyting.Basic
diff --git a/Mathlib/Order/Filter/AtTopBot.lean b/Mathlib/Order/Filter/AtTopBot.lean
index b108dbc04048d..f393265bd856c 100644
--- a/Mathlib/Order/Filter/AtTopBot.lean
+++ b/Mathlib/Order/Filter/AtTopBot.lean
@@ -3,17 +3,11 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
 -/
-import Mathlib.Algebra.BigOperators.Group.Finset
-import Mathlib.Algebra.Order.Field.Defs
-import Mathlib.Algebra.Order.Group.Instances
-import Mathlib.Algebra.Order.Group.MinMax
-import Mathlib.Algebra.Order.Ring.Basic
 import Mathlib.Data.Finset.Preimage
-import Mathlib.Order.Interval.Set.Disjoint
-import Mathlib.Order.Interval.Set.OrderIso
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 import Mathlib.Order.Filter.Bases
-import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Order.Filter.Prod
+import Mathlib.Order.Interval.Set.Disjoint
 
 /-!
 # `Filter.atTop` and `Filter.atBot` filters on preorders, monoids and groups.
@@ -184,6 +178,12 @@ instance (priority := 200) atBot.isCountablyGenerated [Preorder α] [Countable 
     (atBot : Filter <| α).IsCountablyGenerated :=
   isCountablyGenerated_seq _
 
+instance _root_.OrderDual.instIsCountablyGeneratedAtTop [Preorder α]
+    [IsCountablyGenerated (atBot : Filter α)] : IsCountablyGenerated (atTop : Filter αᵒᵈ) := ‹_›
+
+instance _root_.OrderDual.instIsCountablyGeneratedAtBot [Preorder α]
+    [IsCountablyGenerated (atTop : Filter α)] : IsCountablyGenerated (atBot : Filter αᵒᵈ) := ‹_›
+
 theorem _root_.IsTop.atTop_eq [Preorder α] {a : α} (ha : IsTop a) : atTop = 𝓟 (Ici a) :=
   (iInf_le _ _).antisymm <| le_iInf fun b ↦ principal_mono.2 <| Ici_subset_Ici.2 <| ha b
 
@@ -214,6 +214,19 @@ theorem tendsto_atBot_pure [PartialOrder α] [OrderBot α] (f : α → β) :
     Tendsto f atBot (pure <| f ⊥) :=
   @tendsto_atTop_pure αᵒᵈ _ _ _ _
 
+theorem atTop_eq_generate_Ici [Preorder α] : atTop = generate (range (Ici (α := α))) := by
+  simp only [generate_eq_biInf, atTop, iInf_range]
+
+theorem Frequently.forall_exists_of_atTop [Preorder α] {p : α → Prop}
+    (h : ∃ᶠ x in atTop, p x) (a : α) : ∃ b ≥ a, p b := by
+  rw [Filter.Frequently] at h
+  contrapose! h
+  exact (eventually_ge_atTop a).mono h
+
+theorem Frequently.forall_exists_of_atBot [Preorder α] {p : α → Prop}
+    (h : ∃ᶠ x in atBot, p x) (a : α) : ∃ b ≤ a, p b :=
+  Frequently.forall_exists_of_atTop (α := αᵒᵈ) h _
+
 section IsDirected
 variable [Preorder α] [IsDirected α (· ≤ ·)] {p : α → Prop}
 
@@ -223,11 +236,6 @@ theorem hasAntitoneBasis_atTop [Nonempty α] : (@atTop α _).HasAntitoneBasis Ic
 theorem atTop_basis [Nonempty α] : (@atTop α _).HasBasis (fun _ => True) Ici :=
   hasAntitoneBasis_atTop.1
 
-theorem atTop_eq_generate_Ici : atTop = generate (range (Ici (α := α))) := by
-  rcases isEmpty_or_nonempty α with hα|hα
-  · simp only [eq_iff_true_of_subsingleton]
-  · simp [(atTop_basis (α := α)).eq_generate, range]
-
 lemma atTop_basis_Ioi [Nonempty α] [NoMaxOrder α] : (@atTop α _).HasBasis (fun _ => True) Ioi :=
   atTop_basis.to_hasBasis (fun a ha => ⟨a, ha, Ioi_subset_Ici_self⟩) fun a ha =>
     (exists_gt a).imp fun _b hb => ⟨ha, Ici_subset_Ioi.2 hb⟩
@@ -239,7 +247,7 @@ lemma atTop_basis_Ioi' [NoMaxOrder α] (a : α) : atTop.HasBasis (a < ·) Ioi :=
   obtain ⟨d, hcd⟩ := exists_gt c
   exact ⟨d, hac.trans_lt hcd, Ioi_subset_Ioi (hbc.trans hcd.le)⟩
 
-theorem atTop_basis' (a : α) : (@atTop α _).HasBasis (fun x => a ≤ x) Ici := by
+theorem atTop_basis' (a : α) : atTop.HasBasis (a ≤ ·) Ici := by
   have : Nonempty α := ⟨a⟩
   refine atTop_basis.to_hasBasis (fun b _ ↦ ?_) fun b _ ↦ ⟨b, trivial, Subset.rfl⟩
   obtain ⟨c, hac, hbc⟩ := exists_ge_ge a b
@@ -250,8 +258,17 @@ variable [Nonempty α]
 @[instance]
 lemma atTop_neBot : NeBot (atTop : Filter α) := atTop_basis.neBot_iff.2 fun _ => nonempty_Ici
 
+theorem atTop_neBot_iff {α : Type*} [Preorder α] :
+    (atTop : Filter α).NeBot ↔ Nonempty α ∧ IsDirected α (· ≤ ·) := by
+  refine ⟨fun h ↦ ⟨nonempty_of_neBot atTop, ⟨fun x y ↦ ?_⟩⟩, fun ⟨h₁, h₂⟩ ↦ atTop_neBot⟩
+  exact ((eventually_ge_atTop x).and (eventually_ge_atTop y)).exists
+
+theorem atBot_neBot_iff {α : Type*} [Preorder α] :
+    (atBot : Filter α).NeBot ↔ Nonempty α ∧ IsDirected α (· ≥ ·) :=
+  atTop_neBot_iff (α := αᵒᵈ)
+
 @[simp] lemma mem_atTop_sets {s : Set α} : s ∈ (atTop : Filter α) ↔ ∃ a : α, ∀ b ≥ a, b ∈ s :=
-  atTop_basis.mem_iff.trans <| exists_congr fun _ => true_and_iff _
+  atTop_basis.mem_iff.trans <| exists_congr fun _ => iff_of_eq (true_and _)
 
 @[simp] lemma eventually_atTop : (∀ᶠ x in atTop, p x) ↔ ∃ a, ∀ b ≥ a, p b := mem_atTop_sets
 
@@ -259,7 +276,6 @@ theorem frequently_atTop : (∃ᶠ x in atTop, p x) ↔ ∀ a, ∃ b ≥ a, p b
   atTop_basis.frequently_iff.trans <| by simp
 
 alias ⟨Eventually.exists_forall_of_atTop, _⟩ := eventually_atTop
-alias ⟨Frequently.forall_exists_of_atTop, _⟩ := frequently_atTop
 
 lemma exists_eventually_atTop {r : α → β → Prop} :
     (∃ b, ∀ᶠ a in atTop, r a b) ↔ ∀ᶠ a₀ in atTop, ∃ b, ∀ a ≥ a₀, r a b := by
@@ -304,13 +320,10 @@ lemma atBot_basis : (@atBot α _).HasBasis (fun _ => True) Iic := atTop_basis (
 theorem frequently_atBot : (∃ᶠ x in atBot, p x) ↔ ∀ a, ∃ b ≤ a, p b := frequently_atTop (α := αᵒᵈ)
 
 alias ⟨Eventually.exists_forall_of_atBot, _⟩ := eventually_atBot
-alias ⟨Frequently.forall_exists_of_atBot, _⟩ := frequently_atBot
 
 lemma exists_eventually_atBot {r : α → β → Prop} :
-    (∃ b, ∀ᶠ a in atBot, r a b) ↔ ∀ᶠ a₀ in atBot, ∃ b, ∀ a ≤ a₀, r a b := by
-  simp_rw [eventually_atBot, ← exists_swap (α := α)]
-  exact exists_congr fun a ↦ .symm <| forall_le_iff <| Antitone.exists fun _ _ _ hb H n hn ↦
-    H n (hn.trans hb)
+    (∃ b, ∀ᶠ a in atBot, r a b) ↔ ∀ᶠ a₀ in atBot, ∃ b, ∀ a ≤ a₀, r a b :=
+  exists_eventually_atTop (α := αᵒᵈ)
 
 theorem map_atBot_eq {f : α → β} : atBot.map f = ⨅ a, 𝓟 (f '' { a' | a' ≤ a }) :=
   map_atTop_eq (α := αᵒᵈ)
@@ -562,616 +575,6 @@ theorem strictMono_subseq_of_id_le {u : ℕ → ℕ} (hu : ∀ n, n ≤ u n) :
 theorem _root_.StrictMono.tendsto_atTop {φ : ℕ → ℕ} (h : StrictMono φ) : Tendsto φ atTop atTop :=
   tendsto_atTop_mono h.id_le tendsto_id
 
-section OrderedAddCommMonoid
-
-variable [OrderedAddCommMonoid β] {l : Filter α} {f g : α → β}
-
-theorem tendsto_atTop_add_nonneg_left' (hf : ∀ᶠ x in l, 0 ≤ f x) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_mono' l (hf.mono fun _ => le_add_of_nonneg_left) hg
-
-theorem tendsto_atBot_add_nonpos_left' (hf : ∀ᶠ x in l, f x ≤ 0) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_nonneg_left' _ βᵒᵈ _ _ _ _ hf hg
-
-theorem tendsto_atTop_add_nonneg_left (hf : ∀ x, 0 ≤ f x) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_add_nonneg_left' (Eventually.of_forall hf) hg
-
-theorem tendsto_atBot_add_nonpos_left (hf : ∀ x, f x ≤ 0) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_nonneg_left _ βᵒᵈ _ _ _ _ hf hg
-
-theorem tendsto_atTop_add_nonneg_right' (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, 0 ≤ g x) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_mono' l (monotone_mem (fun _ => le_add_of_nonneg_right) hg) hf
-
-theorem tendsto_atBot_add_nonpos_right' (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ 0) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_nonneg_right' _ βᵒᵈ _ _ _ _ hf hg
-
-theorem tendsto_atTop_add_nonneg_right (hf : Tendsto f l atTop) (hg : ∀ x, 0 ≤ g x) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_add_nonneg_right' hf (Eventually.of_forall hg)
-
-theorem tendsto_atBot_add_nonpos_right (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ 0) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_nonneg_right _ βᵒᵈ _ _ _ _ hf hg
-
-theorem tendsto_atTop_add (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_add_nonneg_left' (tendsto_atTop.mp hf 0) hg
-
-theorem tendsto_atBot_add (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add _ βᵒᵈ _ _ _ _ hf hg
-
-theorem Tendsto.nsmul_atTop (hf : Tendsto f l atTop) {n : ℕ} (hn : 0 < n) :
-    Tendsto (fun x => n • f x) l atTop :=
-  tendsto_atTop.2 fun y =>
-    (tendsto_atTop.1 hf y).mp <|
-      (tendsto_atTop.1 hf 0).mono fun x h₀ hy =>
-        calc
-          y ≤ f x := hy
-          _ = 1 • f x := (one_nsmul _).symm
-          _ ≤ n • f x := nsmul_le_nsmul_left h₀ hn
-
-theorem Tendsto.nsmul_atBot (hf : Tendsto f l atBot) {n : ℕ} (hn : 0 < n) :
-    Tendsto (fun x => n • f x) l atBot :=
-  @Tendsto.nsmul_atTop α βᵒᵈ _ l f hf n hn
-
-end OrderedAddCommMonoid
-
-section OrderedCancelAddCommMonoid
-
-variable [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β}
-
-theorem tendsto_atTop_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atTop) :
-    Tendsto f l atTop :=
-  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (C + b)).mono fun _ => le_of_add_le_add_left
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atBot) :
-    Tendsto f l atBot :=
-  tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (C + b)).mono fun _ => le_of_add_le_add_left
-
-theorem tendsto_atTop_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atTop) :
-    Tendsto f l atTop :=
-  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b + C)).mono fun _ => le_of_add_le_add_right
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atBot) :
-    Tendsto f l atBot :=
-  tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (b + C)).mono fun _ => le_of_add_le_add_right
-
-theorem tendsto_atTop_of_add_bdd_above_left' (C) (hC : ∀ᶠ x in l, f x ≤ C)
-    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto g l atTop :=
-  tendsto_atTop_of_add_const_left C
-    (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h)
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_bdd_below_left' (C) (hC : ∀ᶠ x in l, C ≤ f x)
-    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto g l atBot :=
-  tendsto_atBot_of_add_const_left C
-    (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h)
-
-theorem tendsto_atTop_of_add_bdd_above_left (C) (hC : ∀ x, f x ≤ C) :
-    Tendsto (fun x => f x + g x) l atTop → Tendsto g l atTop :=
-  tendsto_atTop_of_add_bdd_above_left' C (univ_mem' hC)
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_bdd_below_left (C) (hC : ∀ x, C ≤ f x) :
-    Tendsto (fun x => f x + g x) l atBot → Tendsto g l atBot :=
-  tendsto_atBot_of_add_bdd_below_left' C (univ_mem' hC)
-
-theorem tendsto_atTop_of_add_bdd_above_right' (C) (hC : ∀ᶠ x in l, g x ≤ C)
-    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto f l atTop :=
-  tendsto_atTop_of_add_const_right C
-    (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h)
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_bdd_below_right' (C) (hC : ∀ᶠ x in l, C ≤ g x)
-    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto f l atBot :=
-  tendsto_atBot_of_add_const_right C
-    (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h)
-
-theorem tendsto_atTop_of_add_bdd_above_right (C) (hC : ∀ x, g x ≤ C) :
-    Tendsto (fun x => f x + g x) l atTop → Tendsto f l atTop :=
-  tendsto_atTop_of_add_bdd_above_right' C (univ_mem' hC)
-
--- Porting note: the "order dual" trick timeouts
-theorem tendsto_atBot_of_add_bdd_below_right (C) (hC : ∀ x, C ≤ g x) :
-    Tendsto (fun x => f x + g x) l atBot → Tendsto f l atBot :=
-  tendsto_atBot_of_add_bdd_below_right' C (univ_mem' hC)
-
-end OrderedCancelAddCommMonoid
-
-section OrderedGroup
-
-variable [OrderedAddCommGroup β] (l : Filter α) {f g : α → β}
-
-theorem tendsto_atTop_add_left_of_le' (C : β) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  @tendsto_atTop_of_add_bdd_above_left' _ _ _ l (fun x => -f x) (fun x => f x + g x) (-C) (by simpa)
-    (by simpa)
-
-theorem tendsto_atBot_add_left_of_ge' (C : β) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_left_of_le' _ βᵒᵈ _ _ _ _ C hf hg
-
-theorem tendsto_atTop_add_left_of_le (C : β) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_add_left_of_le' l C (univ_mem' hf) hg
-
-theorem tendsto_atBot_add_left_of_ge (C : β) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_left_of_le _ βᵒᵈ _ _ _ _ C hf hg
-
-theorem tendsto_atTop_add_right_of_le' (C : β) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  @tendsto_atTop_of_add_bdd_above_right' _ _ _ l (fun x => f x + g x) (fun x => -g x) (-C)
-    (by simp [hg]) (by simp [hf])
-
-theorem tendsto_atBot_add_right_of_ge' (C : β) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_right_of_le' _ βᵒᵈ _ _ _ _ C hf hg
-
-theorem tendsto_atTop_add_right_of_le (C : β) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) :
-    Tendsto (fun x => f x + g x) l atTop :=
-  tendsto_atTop_add_right_of_le' l C hf (univ_mem' hg)
-
-theorem tendsto_atBot_add_right_of_ge (C : β) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) :
-    Tendsto (fun x => f x + g x) l atBot :=
-  @tendsto_atTop_add_right_of_le _ βᵒᵈ _ _ _ _ C hf hg
-
-theorem tendsto_atTop_add_const_left (C : β) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => C + f x) l atTop :=
-  tendsto_atTop_add_left_of_le' l C (univ_mem' fun _ => le_refl C) hf
-
-theorem tendsto_atBot_add_const_left (C : β) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => C + f x) l atBot :=
-  @tendsto_atTop_add_const_left _ βᵒᵈ _ _ _ C hf
-
-theorem tendsto_atTop_add_const_right (C : β) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => f x + C) l atTop :=
-  tendsto_atTop_add_right_of_le' l C hf (univ_mem' fun _ => le_refl C)
-
-theorem tendsto_atBot_add_const_right (C : β) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => f x + C) l atBot :=
-  @tendsto_atTop_add_const_right _ βᵒᵈ _ _ _ C hf
-
-theorem map_neg_atBot : map (Neg.neg : β → β) atBot = atTop :=
-  (OrderIso.neg β).map_atBot
-
-theorem map_neg_atTop : map (Neg.neg : β → β) atTop = atBot :=
-  (OrderIso.neg β).map_atTop
-
-theorem comap_neg_atBot : comap (Neg.neg : β → β) atBot = atTop :=
-  (OrderIso.neg β).comap_atTop
-
-theorem comap_neg_atTop : comap (Neg.neg : β → β) atTop = atBot :=
-  (OrderIso.neg β).comap_atBot
-
-theorem tendsto_neg_atTop_atBot : Tendsto (Neg.neg : β → β) atTop atBot :=
-  (OrderIso.neg β).tendsto_atTop
-
-theorem tendsto_neg_atBot_atTop : Tendsto (Neg.neg : β → β) atBot atTop :=
-  @tendsto_neg_atTop_atBot βᵒᵈ _
-
-variable {l}
-
-@[simp]
-theorem tendsto_neg_atTop_iff : Tendsto (fun x => -f x) l atTop ↔ Tendsto f l atBot :=
-  (OrderIso.neg β).tendsto_atBot_iff
-
-@[simp]
-theorem tendsto_neg_atBot_iff : Tendsto (fun x => -f x) l atBot ↔ Tendsto f l atTop :=
-  (OrderIso.neg β).tendsto_atTop_iff
-
-end OrderedGroup
-
-section OrderedSemiring
-
-variable [OrderedSemiring α] {l : Filter β} {f g : β → α}
-
-theorem Tendsto.atTop_mul_atTop (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x * g x) l atTop := by
-  refine tendsto_atTop_mono' _ ?_ hg
-  filter_upwards [hg.eventually (eventually_ge_atTop 0),
-    hf.eventually (eventually_ge_atTop 1)] with _ using le_mul_of_one_le_left
-
-theorem tendsto_mul_self_atTop : Tendsto (fun x : α => x * x) atTop atTop :=
-  tendsto_id.atTop_mul_atTop tendsto_id
-
-/-- The monomial function `x^n` tends to `+∞` at `+∞` for any positive natural `n`.
-A version for positive real powers exists as `tendsto_rpow_atTop`. -/
-theorem tendsto_pow_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (fun x : α => x ^ n) atTop atTop :=
-  tendsto_atTop_mono' _ ((eventually_ge_atTop 1).mono fun _x hx => le_self_pow hx hn) tendsto_id
-
-end OrderedSemiring
-
-theorem zero_pow_eventuallyEq [MonoidWithZero α] :
-    (fun n : ℕ => (0 : α) ^ n) =ᶠ[atTop] fun _ => 0 :=
-  eventually_atTop.2 ⟨1, fun _n hn ↦ zero_pow <| Nat.one_le_iff_ne_zero.1 hn⟩
-
-section OrderedRing
-
-variable [OrderedRing α] {l : Filter β} {f g : β → α}
-
-theorem Tendsto.atTop_mul_atBot (hf : Tendsto f l atTop) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x * g x) l atBot := by
-  have := hf.atTop_mul_atTop <| tendsto_neg_atBot_atTop.comp hg
-  simpa only [(· ∘ ·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this
-
-theorem Tendsto.atBot_mul_atTop (hf : Tendsto f l atBot) (hg : Tendsto g l atTop) :
-    Tendsto (fun x => f x * g x) l atBot := by
-  have : Tendsto (fun x => -f x * g x) l atTop :=
-    (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop hg
-  simpa only [(· ∘ ·), neg_mul_eq_neg_mul, neg_neg] using tendsto_neg_atTop_atBot.comp this
-
-theorem Tendsto.atBot_mul_atBot (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) :
-    Tendsto (fun x => f x * g x) l atTop := by
-  have : Tendsto (fun x => -f x * -g x) l atTop :=
-    (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop (tendsto_neg_atBot_atTop.comp hg)
-  simpa only [neg_mul_neg] using this
-
-end OrderedRing
-
-section LinearOrderedAddCommGroup
-
-variable [LinearOrderedAddCommGroup α]
-
-/-- $\lim_{x\to+\infty}|x|=+\infty$ -/
-theorem tendsto_abs_atTop_atTop : Tendsto (abs : α → α) atTop atTop :=
-  tendsto_atTop_mono le_abs_self tendsto_id
-
-/-- $\lim_{x\to-\infty}|x|=+\infty$ -/
-theorem tendsto_abs_atBot_atTop : Tendsto (abs : α → α) atBot atTop :=
-  tendsto_atTop_mono neg_le_abs tendsto_neg_atBot_atTop
-
-@[simp]
-theorem comap_abs_atTop : comap (abs : α → α) atTop = atBot ⊔ atTop := by
-  refine
-    le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 ?_)
-      (sup_le tendsto_abs_atBot_atTop.le_comap tendsto_abs_atTop_atTop.le_comap)
-  rintro ⟨a, b⟩ -
-  refine ⟨max (-a) b, trivial, fun x hx => ?_⟩
-  rw [mem_preimage, mem_Ici, le_abs', max_le_iff, ← min_neg_neg, le_min_iff, neg_neg] at hx
-  exact hx.imp And.left And.right
-
-end LinearOrderedAddCommGroup
-
-section LinearOrderedSemiring
-
-variable [LinearOrderedSemiring α] {l : Filter β} {f : β → α}
-
-theorem Tendsto.atTop_of_const_mul {c : α} (hc : 0 < c) (hf : Tendsto (fun x => c * f x) l atTop) :
-    Tendsto f l atTop :=
-  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (c * b)).mono
-    fun _x hx => le_of_mul_le_mul_left hx hc
-
-theorem Tendsto.atTop_of_mul_const {c : α} (hc : 0 < c) (hf : Tendsto (fun x => f x * c) l atTop) :
-    Tendsto f l atTop :=
-  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b * c)).mono
-    fun _x hx => le_of_mul_le_mul_right hx hc
-
-@[simp]
-theorem tendsto_pow_atTop_iff {n : ℕ} : Tendsto (fun x : α => x ^ n) atTop atTop ↔ n ≠ 0 :=
-  ⟨fun h hn => by simp only [hn, pow_zero, not_tendsto_const_atTop] at h, tendsto_pow_atTop⟩
-
-end LinearOrderedSemiring
-
-theorem not_tendsto_pow_atTop_atBot [LinearOrderedRing α] :
-    ∀ {n : ℕ}, ¬Tendsto (fun x : α => x ^ n) atTop atBot
-  | 0 => by simp [not_tendsto_const_atBot]
-  | n + 1 => (tendsto_pow_atTop n.succ_ne_zero).not_tendsto disjoint_atTop_atBot
-
-section LinearOrderedSemifield
-
-variable [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r c : α} {n : ℕ}
-
-/-!
-### Multiplication by constant: iff lemmas
--/
-
-
-/-- If `r` is a positive constant, `fun x ↦ r * f x` tends to infinity along a filter
-if and only if `f` tends to infinity along the same filter. -/
-theorem tendsto_const_mul_atTop_of_pos (hr : 0 < r) :
-    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atTop :=
-  ⟨fun h => h.atTop_of_const_mul hr, fun h =>
-    Tendsto.atTop_of_const_mul (inv_pos.2 hr) <| by simpa only [inv_mul_cancel_left₀ hr.ne'] ⟩
-
-/-- If `r` is a positive constant, `fun x ↦ f x * r` tends to infinity along a filter
-if and only if `f` tends to infinity along the same filter. -/
-theorem tendsto_mul_const_atTop_of_pos (hr : 0 < r) :
-    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atTop := by
-  simpa only [mul_comm] using tendsto_const_mul_atTop_of_pos hr
-
-/-- If `r` is a positive constant, `x ↦ f x / r` tends to infinity along a filter
-if and only if `f` tends to infinity along the same filter. -/
-lemma tendsto_div_const_atTop_of_pos (hr : 0 < r) :
-    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atTop := by
-  simpa only [div_eq_mul_inv] using tendsto_mul_const_atTop_of_pos (inv_pos.2 hr)
-
-/-- If `f` tends to infinity along a nontrivial filter `l`, then
-`fun x ↦ r * f x` tends to infinity if and only if `0 < r. `-/
-theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x => r * f x) l atTop ↔ 0 < r := by
-  refine ⟨fun hrf => not_le.mp fun hr => ?_, fun hr => (tendsto_const_mul_atTop_of_pos hr).mpr h⟩
-  rcases ((h.eventually_ge_atTop 0).and (hrf.eventually_gt_atTop 0)).exists with ⟨x, hx, hrx⟩
-  exact (mul_nonpos_of_nonpos_of_nonneg hr hx).not_lt hrx
-
-/-- If `f` tends to infinity along a nontrivial filter `l`, then
-`fun x ↦ f x * r` tends to infinity if and only if `0 < r. `-/
-theorem tendsto_mul_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x => f x * r) l atTop ↔ 0 < r := by
-  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_pos h]
-
-/-- If `f` tends to infinity along a nontrivial filter `l`, then
-`x ↦ f x * r` tends to infinity if and only if `0 < r. `-/
-lemma tendsto_div_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r := by
-  simp only [div_eq_mul_inv, tendsto_mul_const_atTop_iff_pos h, inv_pos]
-
-/-- If `f` tends to infinity along a filter, then `f` multiplied by a positive
-constant (on the left) also tends to infinity. For a version working in `ℕ` or `ℤ`, use
-`Filter.Tendsto.const_mul_atTop'` instead. -/
-theorem Tendsto.const_mul_atTop (hr : 0 < r) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => r * f x) l atTop :=
-  (tendsto_const_mul_atTop_of_pos hr).2 hf
-
-/-- If a function `f` tends to infinity along a filter, then `f` multiplied by a positive
-constant (on the right) also tends to infinity. For a version working in `ℕ` or `ℤ`, use
-`Filter.Tendsto.atTop_mul_const'` instead. -/
-theorem Tendsto.atTop_mul_const (hr : 0 < r) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => f x * r) l atTop :=
-  (tendsto_mul_const_atTop_of_pos hr).2 hf
-
-/-- If a function `f` tends to infinity along a filter, then `f` divided by a positive
-constant also tends to infinity. -/
-theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => f x / r) l atTop := by
-  simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)
-
-theorem tendsto_const_mul_pow_atTop (hn : n ≠ 0) (hc : 0 < c) :
-    Tendsto (fun x => c * x ^ n) atTop atTop :=
-  Tendsto.const_mul_atTop hc (tendsto_pow_atTop hn)
-
-theorem tendsto_const_mul_pow_atTop_iff :
-    Tendsto (fun x => c * x ^ n) atTop atTop ↔ n ≠ 0 ∧ 0 < c := by
-  refine ⟨fun h => ⟨?_, ?_⟩, fun h => tendsto_const_mul_pow_atTop h.1 h.2⟩
-  · rintro rfl
-    simp only [pow_zero, not_tendsto_const_atTop] at h
-  · rcases ((h.eventually_gt_atTop 0).and (eventually_ge_atTop 0)).exists with ⟨k, hck, hk⟩
-    exact pos_of_mul_pos_left hck (pow_nonneg hk _)
-
-lemma tendsto_zpow_atTop_atTop {n : ℤ} (hn : 0 < n) : Tendsto (fun x : α ↦ x ^ n) atTop atTop := by
-  lift n to ℕ+ using hn; simp
-
-end LinearOrderedSemifield
-
-section LinearOrderedField
-
-variable [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α}
-
-/-- If `r` is a positive constant, `fun x ↦ r * f x` tends to negative infinity along a filter
-if and only if `f` tends to negative infinity along the same filter. -/
-theorem tendsto_const_mul_atBot_of_pos (hr : 0 < r) :
-    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atBot := by
-  simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr
-
-/-- If `r` is a positive constant, `fun x ↦ f x * r` tends to negative infinity along a filter
-if and only if `f` tends to negative infinity along the same filter. -/
-theorem tendsto_mul_const_atBot_of_pos (hr : 0 < r) :
-    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot := by
-  simpa only [mul_comm] using tendsto_const_mul_atBot_of_pos hr
-
-/-- If `r` is a positive constant, `fun x ↦ f x / r` tends to negative infinity along a filter
-if and only if `f` tends to negative infinity along the same filter. -/
-lemma tendsto_div_const_atBot_of_pos (hr : 0 < r) :
-    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atBot := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_pos, hr]
-
-/-- If `r` is a negative constant, `fun x ↦ r * f x` tends to infinity along a filter `l`
-if and only if `f` tends to negative infinity along `l`. -/
-theorem tendsto_const_mul_atTop_of_neg (hr : r < 0) :
-    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atBot := by
-  simpa only [neg_mul, tendsto_neg_atBot_iff] using tendsto_const_mul_atBot_of_pos (neg_pos.2 hr)
-
-/-- If `r` is a negative constant, `fun x ↦ f x * r` tends to infinity along a filter `l`
-if and only if `f` tends to negative infinity along `l`. -/
-theorem tendsto_mul_const_atTop_of_neg (hr : r < 0) :
-    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atBot := by
-  simpa only [mul_comm] using tendsto_const_mul_atTop_of_neg hr
-
-/-- If `r` is a negative constant, `fun x ↦ f x / r` tends to infinity along a filter `l`
-if and only if `f` tends to negative infinity along `l`. -/
-lemma tendsto_div_const_atTop_of_neg (hr : r < 0) :
-    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atBot := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atTop_of_neg, hr]
-
-/-- If `r` is a negative constant, `fun x ↦ r * f x` tends to negative infinity along a filter `l`
-if and only if `f` tends to infinity along `l`. -/
-theorem tendsto_const_mul_atBot_of_neg (hr : r < 0) :
-    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atTop := by
-  simpa only [neg_mul, tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos (neg_pos.2 hr)
-
-/-- If `r` is a negative constant, `fun x ↦ f x * r` tends to negative infinity along a filter `l`
-if and only if `f` tends to infinity along `l`. -/
-theorem tendsto_mul_const_atBot_of_neg (hr : r < 0) :
-    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atTop := by
-  simpa only [mul_comm] using tendsto_const_mul_atBot_of_neg hr
-
-/-- If `r` is a negative constant, `fun x ↦ f x / r` tends to negative infinity along a filter `l`
-if and only if `f` tends to infinity along `l`. -/
-lemma tendsto_div_const_atBot_of_neg (hr : r < 0) :
-    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atTop := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_neg, hr]
-
-/-- The function `fun x ↦ r * f x` tends to infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
-theorem tendsto_const_mul_atTop_iff [NeBot l] :
-    Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
-  rcases lt_trichotomy r 0 with (hr | rfl | hr)
-  · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_neg]
-  · simp [not_tendsto_const_atTop]
-  · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_pos]
-
-/-- The function `fun x ↦ f x * r` tends to infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
-theorem tendsto_mul_const_atTop_iff [NeBot l] :
-    Tendsto (fun x => f x * r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
-  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff]
-
-/-- The function `fun x ↦ f x / r` tends to infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
-lemma tendsto_div_const_atTop_iff [NeBot l] :
-    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff]
-
-/-- The function `fun x ↦ r * f x` tends to negative infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
-theorem tendsto_const_mul_atBot_iff [NeBot l] :
-    Tendsto (fun x => r * f x) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
-  simp only [← tendsto_neg_atTop_iff, ← mul_neg, tendsto_const_mul_atTop_iff, neg_neg]
-
-/-- The function `fun x ↦ f x * r` tends to negative infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
-theorem tendsto_mul_const_atBot_iff [NeBot l] :
-    Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
-  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff]
-
-/-- The function `fun x ↦ f x / r` tends to negative infinity along a nontrivial filter
-if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
-lemma tendsto_div_const_atBot_iff [NeBot l] :
-    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`,
-then `fun x ↦ r * f x` tends to infinity if and only if `r < 0. `-/
-theorem tendsto_const_mul_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x => r * f x) l atTop ↔ r < 0 := by
-  simp [tendsto_const_mul_atTop_iff, h, h.not_tendsto disjoint_atBot_atTop]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`,
-then `fun x ↦ f x * r` tends to infinity if and only if `r < 0. `-/
-theorem tendsto_mul_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x => f x * r) l atTop ↔ r < 0 := by
-  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_neg h]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`,
-then `fun x ↦ f x / r` tends to infinity if and only if `r < 0. `-/
-lemma tendsto_div_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0 := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff_neg h]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
-`fun x ↦ r * f x` tends to negative infinity if and only if `0 < r. `-/
-theorem tendsto_const_mul_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x => r * f x) l atBot ↔ 0 < r := by
-  simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atBot_atTop]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
-`fun x ↦ f x * r` tends to negative infinity if and only if `0 < r. `-/
-theorem tendsto_mul_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x => f x * r) l atBot ↔ 0 < r := by
-  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff_pos h]
-
-/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
-`fun x ↦ f x / r` tends to negative infinity if and only if `0 < r. `-/
-lemma tendsto_div_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
-    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff_pos h]
-
-/-- If `f` tends to infinity along a nontrivial filter,
-`fun x ↦ r * f x` tends to negative infinity if and only if `r < 0. `-/
-theorem tendsto_const_mul_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x => r * f x) l atBot ↔ r < 0 := by
-  simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atTop_atBot]
-
-/-- If `f` tends to infinity along a nontrivial filter,
-`fun x ↦ f x * r` tends to negative infinity if and only if `r < 0. `-/
-theorem tendsto_mul_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x => f x * r) l atBot ↔ r < 0 := by
-  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff_neg h]
-
-/-- If `f` tends to infinity along a nontrivial filter,
-`fun x ↦ f x / r` tends to negative infinity if and only if `r < 0. `-/
-lemma tendsto_div_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
-    Tendsto (fun x ↦ f x / r) l atBot ↔ r < 0 := by
-  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff_neg h]
-
-/-- If a function `f` tends to infinity along a filter,
-then `f` multiplied by a negative constant (on the left) tends to negative infinity. -/
-theorem Tendsto.const_mul_atTop_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => r * f x) l atBot :=
-  (tendsto_const_mul_atBot_of_neg hr).2 hf
-
-/-- If a function `f` tends to infinity along a filter,
-then `f` multiplied by a negative constant (on the right) tends to negative infinity. -/
-theorem Tendsto.atTop_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
-    Tendsto (fun x => f x * r) l atBot :=
-  (tendsto_mul_const_atBot_of_neg hr).2 hf
-
-/-- If a function `f` tends to infinity along a filter,
-then `f` divided by a negative constant tends to negative infinity. -/
-lemma Tendsto.atTop_div_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
-    Tendsto (fun x ↦ f x / r) l atBot := (tendsto_div_const_atBot_of_neg hr).2 hf
-
-/-- If a function `f` tends to negative infinity along a filter, then `f` multiplied by
-a positive constant (on the left) also tends to negative infinity. -/
-theorem Tendsto.const_mul_atBot (hr : 0 < r) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => r * f x) l atBot :=
-  (tendsto_const_mul_atBot_of_pos hr).2 hf
-
-/-- If a function `f` tends to negative infinity along a filter, then `f` multiplied by
-a positive constant (on the right) also tends to negative infinity. -/
-theorem Tendsto.atBot_mul_const (hr : 0 < r) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => f x * r) l atBot :=
-  (tendsto_mul_const_atBot_of_pos hr).2 hf
-
-/-- If a function `f` tends to negative infinity along a filter, then `f` divided by
-a positive constant also tends to negative infinity. -/
-theorem Tendsto.atBot_div_const (hr : 0 < r) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => f x / r) l atBot := (tendsto_div_const_atBot_of_pos hr).2 hf
-
-/-- If a function `f` tends to negative infinity along a filter,
-then `f` multiplied by a negative constant (on the left) tends to positive infinity. -/
-theorem Tendsto.const_mul_atBot_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => r * f x) l atTop :=
-  (tendsto_const_mul_atTop_of_neg hr).2 hf
-
-/-- If a function tends to negative infinity along a filter,
-then `f` multiplied by a negative constant (on the right) tends to positive infinity. -/
-theorem Tendsto.atBot_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
-    Tendsto (fun x => f x * r) l atTop :=
-  (tendsto_mul_const_atTop_of_neg hr).2 hf
-
-theorem tendsto_neg_const_mul_pow_atTop {c : α} {n : ℕ} (hn : n ≠ 0) (hc : c < 0) :
-    Tendsto (fun x => c * x ^ n) atTop atBot :=
-  (tendsto_pow_atTop hn).const_mul_atTop_of_neg hc
-
-theorem tendsto_const_mul_pow_atBot_iff {c : α} {n : ℕ} :
-    Tendsto (fun x => c * x ^ n) atTop atBot ↔ n ≠ 0 ∧ c < 0 := by
-  simp only [← tendsto_neg_atTop_iff, ← neg_mul, tendsto_const_mul_pow_atTop_iff, neg_pos]
-
-@[deprecated (since := "2024-05-06")]
-alias Tendsto.neg_const_mul_atTop := Tendsto.const_mul_atTop_of_neg
-
-@[deprecated (since := "2024-05-06")]
-alias Tendsto.atTop_mul_neg_const := Tendsto.atTop_mul_const_of_neg
-
-@[deprecated (since := "2024-05-06")]
-alias Tendsto.neg_const_mul_atBot := Tendsto.const_mul_atBot_of_neg
-
-@[deprecated (since := "2024-05-06")]
-alias Tendsto.atBot_mul_neg_const := Tendsto.atBot_mul_const_of_neg
-
-end LinearOrderedField
-
-open Filter
-
 theorem tendsto_atTop_atTop_of_monotone [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
     (h : ∀ b, ∃ a, b ≤ f a) : Tendsto f atTop atTop :=
   tendsto_iInf.2 fun b =>
@@ -1321,10 +724,32 @@ theorem prod_atTop_atTop_eq [Preorder α] [Preorder β] :
   · subsingleton
   simpa [atTop, prod_iInf_left, prod_iInf_right, iInf_prod] using iInf_comm
 
+instance instIsCountablyGeneratedAtTopProd [Preorder α] [IsCountablyGenerated (atTop : Filter α)]
+    [Preorder β] [IsCountablyGenerated (atTop : Filter β)] :
+    IsCountablyGenerated (atTop : Filter (α × β)) := by
+  rw [← prod_atTop_atTop_eq]
+  infer_instance
+
+lemma tendsto_finset_prod_atTop :
+    Tendsto (fun (p : Finset ι × Finset ι') ↦ p.1 ×ˢ p.2) atTop atTop := by
+  classical
+  apply Monotone.tendsto_atTop_atTop
+  · intro p q hpq
+    simpa using Finset.product_subset_product hpq.1 hpq.2
+  · intro b
+    use (Finset.image Prod.fst b, Finset.image Prod.snd b)
+    exact Finset.subset_product
+
 theorem prod_atBot_atBot_eq [Preorder α] [Preorder β] :
     (atBot : Filter α) ×ˢ (atBot : Filter β) = (atBot : Filter (α × β)) :=
   @prod_atTop_atTop_eq αᵒᵈ βᵒᵈ _ _
 
+instance instIsCountablyGeneratedAtBotProd [Preorder α] [IsCountablyGenerated (atBot : Filter α)]
+    [Preorder β] [IsCountablyGenerated (atBot : Filter β)] :
+    IsCountablyGenerated (atBot : Filter (α × β)) := by
+  rw [← prod_atBot_atBot_eq]
+  infer_instance
+
 theorem prod_map_atTop_eq {α₁ α₂ β₁ β₂ : Type*} [Preorder β₁] [Preorder β₂]
     (u₁ : β₁ → α₁) (u₂ : β₂ → α₂) : map u₁ atTop ×ˢ map u₂ atTop = map (Prod.map u₁ u₂) atTop := by
   rw [prod_map_map_eq, prod_atTop_atTop_eq, Prod.map_def]
@@ -1397,132 +822,167 @@ theorem eventually_atBot_curry [Preorder α] [Preorder β] {p : α × β → Pro
 
 /-- A function `f` maps upwards closed sets (atTop sets) to upwards closed sets when it is a
 Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an
-insertion and a connection above `b'`. -/
-theorem map_atTop_eq_of_gc [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (g : β → α) (b' : β)
-    (hf : Monotone f) (gc : ∀ a, ∀ b ≥ b', f a ≤ b ↔ a ≤ g b) (hgi : ∀ b ≥ b', b ≤ f (g b)) :
-    map f atTop = atTop := by
-  refine
-    le_antisymm
-      (hf.tendsto_atTop_atTop fun b => ⟨g (b ⊔ b'), le_sup_left.trans <| hgi _ le_sup_right⟩) ?_
-  have : Nonempty α := ⟨g b'⟩
-  rw [map_atTop_eq]
-  refine le_iInf fun a => iInf_le_of_le (f a ⊔ b') <| principal_mono.2 fun b hb => ?_
-  rw [mem_Ici, sup_le_iff] at hb
-  exact ⟨g b, (gc _ _ hb.2).1 hb.1, le_antisymm ((gc _ _ hb.2).2 le_rfl) (hgi _ hb.2)⟩
-
-theorem map_atBot_eq_of_gc [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (g : β → α) (b' : β)
+insertion and a connection above `b`. -/
+theorem map_atTop_eq_of_gc_preorder
+    [Preorder α] [IsDirected α (· ≤ ·)] [Preorder β] [IsDirected β (· ≤ ·)] {f : α → β}
+    (hf : Monotone f) (b : β)
+    (hgi : ∀ c ≥ b, ∃ x, f x = c ∧ ∀ a, f a ≤ c ↔ a ≤ x) : map f atTop = atTop := by
+  have : Nonempty α := (hgi b le_rfl).nonempty
+  choose! g hfg hgle using hgi
+  refine le_antisymm (hf.tendsto_atTop_atTop fun c ↦ ?_) ?_
+  · rcases exists_ge_ge c b with ⟨d, hcd, hbd⟩
+    exact ⟨g d, hcd.trans (hfg d hbd).ge⟩
+  · have : Nonempty α := ⟨g b⟩
+    rw [(atTop_basis.map f).ge_iff]
+    intro a _
+    filter_upwards [eventually_ge_atTop (f a), eventually_ge_atTop b] with c hac hbc
+    exact ⟨g c, (hgle _ hbc _).1 hac, hfg _ hbc⟩
+
+
+/-- A function `f` maps upwards closed sets (atTop sets) to upwards closed sets when it is a
+Galois insertion. The Galois "insertion" and "connection" is weakened to only require it to be an
+insertion and a connection above `b`. -/
+theorem map_atTop_eq_of_gc
+    [Preorder α] [IsDirected α (· ≤ ·)] [PartialOrder β] [IsDirected β (· ≤ ·)]
+    {f : α → β} (g : β → α) (b : β) (hf : Monotone f)
+    (gc : ∀ a, ∀ c ≥ b, f a ≤ c ↔ a ≤ g c) (hgi : ∀ c ≥ b, c ≤ f (g c)) :
+    map f atTop = atTop :=
+  map_atTop_eq_of_gc_preorder hf b fun c hc ↦
+    ⟨g c, le_antisymm ((gc _ _ hc).2 le_rfl) (hgi c hc), (gc · c hc)⟩
+
+theorem map_atBot_eq_of_gc_preorder
+    [Preorder α] [IsDirected α (· ≥ ·)] [Preorder β] [IsDirected β (· ≥ ·)] {f : α → β}
+    (hf : Monotone f) (b : β)
+    (hgi : ∀ c ≤ b, ∃ x, f x = c ∧ ∀ a, c ≤ f a ↔ x ≤ a) : map f atBot = atBot :=
+  map_atTop_eq_of_gc_preorder (α := αᵒᵈ) (β := βᵒᵈ) hf.dual _ hgi
+
+theorem map_atBot_eq_of_gc [Preorder α] [IsDirected α (· ≥ ·)]
+    [PartialOrder β] [IsDirected β (· ≥ ·)] {f : α → β} (g : β → α) (b' : β)
     (hf : Monotone f) (gc : ∀ a, ∀ b ≤ b', b ≤ f a ↔ g b ≤ a) (hgi : ∀ b ≤ b', f (g b) ≤ b) :
     map f atBot = atBot :=
-  @map_atTop_eq_of_gc αᵒᵈ βᵒᵈ _ _ _ _ _ hf.dual gc hgi
-
-theorem map_val_atTop_of_Ici_subset [SemilatticeSup α] {a : α} {s : Set α} (h : Ici a ⊆ s) :
-    map ((↑) : s → α) atTop = atTop := by
-  haveI : Nonempty s := ⟨⟨a, h le_rfl⟩⟩
-  have : Directed (· ≥ ·) fun x : s => 𝓟 (Ici x) := fun x y ↦ by
-    use ⟨x ⊔ y ⊔ a, h le_sup_right⟩
-    simp only [principal_mono, Ici_subset_Ici, ← Subtype.coe_le_coe, Subtype.coe_mk]
-    exact ⟨le_sup_left.trans le_sup_left, le_sup_right.trans le_sup_left⟩
-  simp only [le_antisymm_iff, atTop, le_iInf_iff, le_principal_iff, mem_map, mem_setOf_eq,
-    map_iInf_eq this, map_principal]
-  constructor
-  · intro x
-    refine mem_of_superset (mem_iInf_of_mem ⟨x ⊔ a, h le_sup_right⟩ (mem_principal_self _)) ?_
-    rintro _ ⟨y, hy, rfl⟩
-    exact le_trans le_sup_left (Subtype.coe_le_coe.2 hy)
-  · intro x
-    filter_upwards [mem_atTop (↑x ⊔ a)] with b hb
-    exact ⟨⟨b, h <| le_sup_right.trans hb⟩, Subtype.coe_le_coe.1 (le_sup_left.trans hb), rfl⟩
+  map_atTop_eq_of_gc (α := αᵒᵈ) (β := βᵒᵈ) _ _ hf.dual gc hgi
+
+theorem map_val_atTop_of_Ici_subset [Preorder α] [IsDirected α (· ≤ ·)] {a : α} {s : Set α}
+    (h : Ici a ⊆ s) : map ((↑) : s → α) atTop = atTop := by
+  choose f hl hr using exists_ge_ge (α := α)
+  have : DirectedOn (· ≤ ·) s := fun x _ y _ ↦
+    ⟨f a (f x y), h <| hl _ _, (hl x y).trans (hr _ _), (hr x y).trans (hr _ _)⟩
+  have : IsDirected s (· ≤ ·) := by
+    rw [directedOn_iff_directed] at this
+    rwa [← directed_id_iff]
+  refine map_atTop_eq_of_gc_preorder (Subtype.mono_coe _) a fun c hc ↦ ?_
+  exact ⟨⟨c, h hc⟩, rfl, fun _ ↦ .rfl⟩
+
+@[simp]
+theorem _root_.Nat.map_cast_int_atTop : map ((↑) : ℕ → ℤ) atTop = atTop := by
+  refine map_atTop_eq_of_gc_preorder (fun _ _ ↦ Int.ofNat_le.2) 0 fun n hn ↦ ?_
+  lift n to ℕ using hn
+  exact ⟨n, rfl, fun _ ↦ Int.ofNat_le⟩
 
 /-- The image of the filter `atTop` on `Ici a` under the coercion equals `atTop`. -/
 @[simp]
-theorem map_val_Ici_atTop [SemilatticeSup α] (a : α) : map ((↑) : Ici a → α) atTop = atTop :=
-  map_val_atTop_of_Ici_subset (Subset.refl _)
+theorem map_val_Ici_atTop [Preorder α] [IsDirected α (· ≤ ·)] (a : α) :
+    map ((↑) : Ici a → α) atTop = atTop :=
+  map_val_atTop_of_Ici_subset Subset.rfl
 
 /-- The image of the filter `atTop` on `Ioi a` under the coercion equals `atTop`. -/
 @[simp]
-theorem map_val_Ioi_atTop [SemilatticeSup α] [NoMaxOrder α] (a : α) :
+theorem map_val_Ioi_atTop [Preorder α] [IsDirected α (· ≤ ·)] [NoMaxOrder α] (a : α) :
     map ((↑) : Ioi a → α) atTop = atTop :=
   let ⟨_b, hb⟩ := exists_gt a
   map_val_atTop_of_Ici_subset <| Ici_subset_Ioi.2 hb
 
 /-- The `atTop` filter for an open interval `Ioi a` comes from the `atTop` filter in the ambient
 order. -/
-theorem atTop_Ioi_eq [SemilatticeSup α] (a : α) : atTop = comap ((↑) : Ioi a → α) atTop := by
+theorem atTop_Ioi_eq [Preorder α] [IsDirected α (· ≤ ·)] (a : α) :
+    atTop = comap ((↑) : Ioi a → α) atTop := by
   rcases isEmpty_or_nonempty (Ioi a) with h|⟨⟨b, hb⟩⟩
   · subsingleton
   · rw [← map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective]
 
 /-- The `atTop` filter for an open interval `Ici a` comes from the `atTop` filter in the ambient
 order. -/
-theorem atTop_Ici_eq [SemilatticeSup α] (a : α) : atTop = comap ((↑) : Ici a → α) atTop := by
+theorem atTop_Ici_eq [Preorder α] [IsDirected α (· ≤ ·)] (a : α) :
+    atTop = comap ((↑) : Ici a → α) atTop := by
   rw [← map_val_Ici_atTop a, comap_map Subtype.coe_injective]
 
 /-- The `atBot` filter for an open interval `Iio a` comes from the `atBot` filter in the ambient
 order. -/
 @[simp]
-theorem map_val_Iio_atBot [SemilatticeInf α] [NoMinOrder α] (a : α) :
+theorem map_val_Iio_atBot [Preorder α] [IsDirected α (· ≥ ·)] [NoMinOrder α] (a : α) :
     map ((↑) : Iio a → α) atBot = atBot :=
-  @map_val_Ioi_atTop αᵒᵈ _ _ _
+  map_val_Ioi_atTop (OrderDual.toDual a)
 
 /-- The `atBot` filter for an open interval `Iio a` comes from the `atBot` filter in the ambient
 order. -/
-theorem atBot_Iio_eq [SemilatticeInf α] (a : α) : atBot = comap ((↑) : Iio a → α) atBot :=
-  @atTop_Ioi_eq αᵒᵈ _ _
+theorem atBot_Iio_eq [Preorder α] [IsDirected α (· ≥ ·)] (a : α) :
+    atBot = comap ((↑) : Iio a → α) atBot :=
+  atTop_Ioi_eq (OrderDual.toDual a)
 
 /-- The `atBot` filter for an open interval `Iic a` comes from the `atBot` filter in the ambient
 order. -/
 @[simp]
-theorem map_val_Iic_atBot [SemilatticeInf α] (a : α) : map ((↑) : Iic a → α) atBot = atBot :=
-  @map_val_Ici_atTop αᵒᵈ _ _
+theorem map_val_Iic_atBot [Preorder α] [IsDirected α (· ≥ ·)] (a : α) :
+    map ((↑) : Iic a → α) atBot = atBot :=
+  map_val_Ici_atTop (OrderDual.toDual a)
 
 /-- The `atBot` filter for an open interval `Iic a` comes from the `atBot` filter in the ambient
 order. -/
-theorem atBot_Iic_eq [SemilatticeInf α] (a : α) : atBot = comap ((↑) : Iic a → α) atBot :=
-  @atTop_Ici_eq αᵒᵈ _ _
+theorem atBot_Iic_eq [Preorder α] [IsDirected α (· ≥ ·)] (a : α) :
+    atBot = comap ((↑) : Iic a → α) atBot :=
+  atTop_Ici_eq (OrderDual.toDual a)
 
-theorem tendsto_Ioi_atTop [SemilatticeSup α] {a : α} {f : β → Ioi a} {l : Filter β} :
+theorem tendsto_Ioi_atTop [Preorder α] [IsDirected α (· ≤ ·)]
+    {a : α} {f : β → Ioi a} {l : Filter β} :
     Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l atTop := by
   rw [atTop_Ioi_eq, tendsto_comap_iff, Function.comp_def]
 
-theorem tendsto_Iio_atBot [SemilatticeInf α] {a : α} {f : β → Iio a} {l : Filter β} :
-    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l atBot := by
-  rw [atBot_Iio_eq, tendsto_comap_iff, Function.comp_def]
+theorem tendsto_Iio_atBot [Preorder α] [IsDirected α (· ≥ ·)]
+    {a : α} {f : β → Iio a} {l : Filter β} :
+    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l atBot :=
+  tendsto_Ioi_atTop (α := αᵒᵈ)
 
-theorem tendsto_Ici_atTop [SemilatticeSup α] {a : α} {f : β → Ici a} {l : Filter β} :
+theorem tendsto_Ici_atTop [Preorder α] [IsDirected α (· ≤ ·)]
+    {a : α} {f : β → Ici a} {l : Filter β} :
     Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l atTop := by
   rw [atTop_Ici_eq, tendsto_comap_iff, Function.comp_def]
 
-theorem tendsto_Iic_atBot [SemilatticeInf α] {a : α} {f : β → Iic a} {l : Filter β} :
-    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l atBot := by
-  rw [atBot_Iic_eq, tendsto_comap_iff, Function.comp_def]
+theorem tendsto_Iic_atBot [Preorder α] [IsDirected α (· ≥ ·)]
+    {a : α} {f : β → Iic a} {l : Filter β} :
+    Tendsto f l atBot ↔ Tendsto (fun x => (f x : α)) l atBot :=
+  tendsto_Ici_atTop (α := αᵒᵈ)
 
-@[simp, nolint simpNF] -- Porting note: linter claims that LHS doesn't simplify. It does.
-theorem tendsto_comp_val_Ioi_atTop [SemilatticeSup α] [NoMaxOrder α] {a : α} {f : α → β}
-    {l : Filter β} : Tendsto (fun x : Ioi a => f x) atTop l ↔ Tendsto f atTop l := by
+@[simp]
+theorem tendsto_comp_val_Ioi_atTop [Preorder α] [IsDirected α (· ≤ ·)] [NoMaxOrder α]
+    {a : α} {f : α → β} {l : Filter β} :
+    Tendsto (fun x : Ioi a => f x) atTop l ↔ Tendsto f atTop l := by
   rw [← map_val_Ioi_atTop a, tendsto_map'_iff, Function.comp_def]
 
-@[simp, nolint simpNF] -- Porting note: linter claims that LHS doesn't simplify. It does.
-theorem tendsto_comp_val_Ici_atTop [SemilatticeSup α] {a : α} {f : α → β} {l : Filter β} :
+@[simp]
+theorem tendsto_comp_val_Ici_atTop [Preorder α] [IsDirected α (· ≤ ·)]
+    {a : α} {f : α → β} {l : Filter β} :
     Tendsto (fun x : Ici a => f x) atTop l ↔ Tendsto f atTop l := by
   rw [← map_val_Ici_atTop a, tendsto_map'_iff, Function.comp_def]
 
-@[simp, nolint simpNF] -- Porting note: linter claims that LHS doesn't simplify. It does.
-theorem tendsto_comp_val_Iio_atBot [SemilatticeInf α] [NoMinOrder α] {a : α} {f : α → β}
-    {l : Filter β} : Tendsto (fun x : Iio a => f x) atBot l ↔ Tendsto f atBot l := by
-  rw [← map_val_Iio_atBot a, tendsto_map'_iff, Function.comp_def]
+@[simp]
+theorem tendsto_comp_val_Iio_atBot [Preorder α] [IsDirected α (· ≥ ·)] [NoMinOrder α]
+    {a : α} {f : α → β} {l : Filter β} :
+    Tendsto (fun x : Iio a => f x) atBot l ↔ Tendsto f atBot l :=
+  tendsto_comp_val_Ioi_atTop (α := αᵒᵈ)
 
-@[simp, nolint simpNF] -- Porting note: linter claims that LHS doesn't simplify. It does.
-theorem tendsto_comp_val_Iic_atBot [SemilatticeInf α] {a : α} {f : α → β} {l : Filter β} :
-    Tendsto (fun x : Iic a => f x) atBot l ↔ Tendsto f atBot l := by
-  rw [← map_val_Iic_atBot a, tendsto_map'_iff, Function.comp_def]
+@[simp]
+theorem tendsto_comp_val_Iic_atBot [Preorder α] [IsDirected α (· ≥ ·)]
+    {a : α} {f : α → β} {l : Filter β} :
+    Tendsto (fun x : Iic a => f x) atBot l ↔ Tendsto f atBot l :=
+  tendsto_comp_val_Ici_atTop (α := αᵒᵈ)
 
 theorem map_add_atTop_eq_nat (k : ℕ) : map (fun a => a + k) atTop = atTop :=
-  map_atTop_eq_of_gc (fun a => a - k) k (fun a b h => add_le_add_right h k)
-    (fun a b h => (le_tsub_iff_right h).symm) fun a h => by rw [tsub_add_cancel_of_le h]
+  map_atTop_eq_of_gc (· - k) k (fun a b h => Nat.add_le_add_right h k)
+    (fun a b h => (Nat.le_sub_iff_add_le h).symm) fun a h => by rw [Nat.sub_add_cancel h]
 
 theorem map_sub_atTop_eq_nat (k : ℕ) : map (fun a => a - k) atTop = atTop :=
-  map_atTop_eq_of_gc (fun a => a + k) 0 (fun a b h => tsub_le_tsub_right h _)
-    (fun a b _ => tsub_le_iff_right) fun b _ => by rw [add_tsub_cancel_right]
+  map_atTop_eq_of_gc (· + k) 0 (fun a b h => Nat.sub_le_sub_right h _)
+    (fun a b _ => Nat.sub_le_iff_le_add) fun b _ => by rw [Nat.add_sub_cancel_right]
 
 theorem tendsto_add_atTop_nat (k : ℕ) : Tendsto (fun a => a + k) atTop atTop :=
   le_of_eq (map_add_atTop_eq_nat k)
@@ -1536,14 +996,10 @@ theorem tendsto_add_atTop_iff_nat {f : ℕ → α} {l : Filter α} (k : ℕ) :
     rw [← tendsto_map'_iff, map_add_atTop_eq_nat]
 
 theorem map_div_atTop_eq_nat (k : ℕ) (hk : 0 < k) : map (fun a => a / k) atTop = atTop :=
-  map_atTop_eq_of_gc (fun b => b * k + (k - 1)) 1 (fun a b h => Nat.div_le_div_right h)
+  map_atTop_eq_of_gc (fun b => k * b + (k - 1)) 1 (fun a b h => Nat.div_le_div_right h)
     -- Porting note: there was a parse error in `calc`, use `simp` instead
-    (fun a b _ => by simp only [← Nat.lt_succ_iff, Nat.div_lt_iff_lt_mul hk, Nat.succ_eq_add_one,
-      add_assoc, tsub_add_cancel_of_le (Nat.one_le_iff_ne_zero.2 hk.ne'), add_mul, one_mul])
-    fun b _ =>
-    calc
-      b = b * k / k := by rw [Nat.mul_div_cancel b hk]
-      _ ≤ (b * k + (k - 1)) / k := Nat.div_le_div_right <| Nat.le_add_right _ _
+    (fun a b _ => by rw [Nat.div_le_iff_le_mul_add_pred hk])
+    fun b _ => by rw [Nat.mul_add_div hk, Nat.div_eq_of_lt, add_zero]; omega
 
 /-- If `u` is a monotone function with linear ordered codomain and the range of `u` is not bounded
 above, then `Tendsto u atTop atTop`. -/
@@ -1610,25 +1066,6 @@ theorem tendsto_atBot_of_monotone_of_subseq [Preorder ι] [Preorder α] {u : ι
     Tendsto u atBot atBot :=
   tendsto_atBot_of_monotone_of_filter h (tendsto_map' H)
 
-/-- Let `f` and `g` be two maps to the same commutative monoid. This lemma gives a sufficient
-condition for comparison of the filter `atTop.map (fun s ↦ ∏ b ∈ s, f b)` with
-`atTop.map (fun s ↦ ∏ b ∈ s, g b)`. This is useful to compare the set of limit points of
-`Π b in s, f b` as `s → atTop` with the similar set for `g`. -/
-@[to_additive "Let `f` and `g` be two maps to the same commutative additive monoid. This lemma gives
-a sufficient condition for comparison of the filter `atTop.map (fun s ↦ ∑ b ∈ s, f b)` with
-`atTop.map (fun s ↦ ∑ b ∈ s, g b)`. This is useful to compare the set of limit points of
-`∑ b ∈ s, f b` as `s → atTop` with the similar set for `g`."]
-theorem map_atTop_finset_prod_le_of_prod_eq [CommMonoid α] {f : β → α} {g : γ → α}
-    (h_eq : ∀ u : Finset γ,
-      ∃ v : Finset β, ∀ v', v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b) :
-    (atTop.map fun s : Finset β => ∏ b ∈ s, f b) ≤
-      atTop.map fun s : Finset γ => ∏ x ∈ s, g x := by
-  classical
-    refine ((atTop_basis.map _).le_basis_iff (atTop_basis.map _)).2 fun b _ => ?_
-    let ⟨v, hv⟩ := h_eq b
-    refine ⟨v, trivial, ?_⟩
-    simpa [image_subset_iff] using hv
-
 theorem HasAntitoneBasis.eventually_subset [Preorder ι] {l : Filter α} {s : ι → Set α}
     (hl : l.HasAntitoneBasis s) {t : Set α} (ht : t ∈ l) : ∀ᶠ i in atTop, s i ⊆ t :=
   let ⟨i, _, hi⟩ := hl.1.mem_iff.1 ht
@@ -1673,21 +1110,25 @@ theorem exists_seq_tendsto (f : Filter α) [IsCountablyGenerated f] [NeBot f] :
   choose x hx using fun n => Filter.nonempty_of_mem (h.mem n)
   exact ⟨x, h.tendsto hx⟩
 
-theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [SemilatticeSup α] [Nonempty α]
-    [(atTop : Filter α).IsCountablyGenerated] :
+theorem exists_seq_monotone_tendsto_atTop_atTop (α : Type*) [Preorder α] [Nonempty α]
+    [IsDirected α (· ≤ ·)] [(atTop : Filter α).IsCountablyGenerated] :
     ∃ xs : ℕ → α, Monotone xs ∧ Tendsto xs atTop atTop := by
   obtain ⟨ys, h⟩ := exists_seq_tendsto (atTop : Filter α)
-  let xs : ℕ → α := fun n => Finset.sup' (Finset.range (n + 1)) Finset.nonempty_range_succ ys
-  have h_mono : Monotone xs := fun i j hij ↦ by
-    simp only [xs] -- Need to unfold `xs` and do alpha reduction, otherwise `gcongr` fails
-    gcongr
-  refine ⟨xs, h_mono, tendsto_atTop_mono (fun n ↦ Finset.le_sup' _ ?_) h⟩
-  simp
-
-theorem exists_seq_antitone_tendsto_atTop_atBot (α : Type*) [SemilatticeInf α] [Nonempty α]
-    [h2 : (atBot : Filter α).IsCountablyGenerated] :
+  choose c hleft hright using exists_ge_ge (α := α)
+  set xs : ℕ → α := fun n => (List.range n).foldl (fun x n ↦ c x (ys n)) (ys 0)
+  have hsucc (n : ℕ) : xs (n + 1) = c (xs n) (ys n) := by simp [xs, List.range_succ]
+  refine ⟨xs, ?_, ?_⟩
+  · refine monotone_nat_of_le_succ fun n ↦ ?_
+    rw [hsucc]
+    apply hleft
+  · refine (tendsto_add_atTop_iff_nat 1).1 <| tendsto_atTop_mono (fun n ↦ ?_) h
+    rw [hsucc]
+    apply hright
+
+theorem exists_seq_antitone_tendsto_atTop_atBot (α : Type*) [Preorder α] [Nonempty α]
+    [IsDirected α (· ≥ ·)] [(atBot : Filter α).IsCountablyGenerated] :
     ∃ xs : ℕ → α, Antitone xs ∧ Tendsto xs atTop atBot :=
-  @exists_seq_monotone_tendsto_atTop_atTop αᵒᵈ _ _ h2
+  exists_seq_monotone_tendsto_atTop_atTop αᵒᵈ
 
 /-- An abstract version of continuity of sequentially continuous functions on metric spaces:
 if a filter `k` is countably generated then `Tendsto f k l` iff for every sequence `u`
@@ -1765,21 +1206,6 @@ end Filter
 
 open Filter Finset
 
-section
-
-variable {R : Type*} [LinearOrderedSemiring R]
-
-theorem exists_lt_mul_self (a : R) : ∃ x ≥ 0, a < x * x :=
-  let ⟨x, hxa, hx0⟩ :=
-    ((tendsto_mul_self_atTop.eventually (eventually_gt_atTop a)).and (eventually_ge_atTop 0)).exists
-  ⟨x, hx0, hxa⟩
-
-theorem exists_le_mul_self (a : R) : ∃ x ≥ 0, a ≤ x * x :=
-  let ⟨x, hx0, hxa⟩ := exists_lt_mul_self a
-  ⟨x, hx0, hxa.le⟩
-
-end
-
 theorem Monotone.piecewise_eventually_eq_iUnion {β : α → Type*} [Preorder ι] {s : ι → Set α}
     [∀ i, DecidablePred (· ∈ s i)] [DecidablePred (· ∈ ⋃ i, s i)]
     (hs : Monotone s) (f g : (a : α) → β a) (a : α) :
@@ -1800,34 +1226,44 @@ theorem Antitone.piecewise_eventually_eq_iInter {β : α → Type*} [Preorder ι
   · convert congr_fun (Set.piecewise_compl (s _) g f) a
   · simp only [(· ∘ ·), ← compl_iInter, Set.piecewise_compl]
 
-/-- Let `g : γ → β` be an injective function and `f : β → α` be a function from the codomain of `g`
-to a commutative monoid. Suppose that `f x = 1` outside of the range of `g`. Then the filters
-`atTop.map (fun s ↦ ∏ i ∈ s, f (g i))` and `atTop.map (fun s ↦ ∏ i ∈ s, f i)` coincide.
-
-The additive version of this lemma is used to prove the equality `∑' x, f (g x) = ∑' y, f y` under
-the same assumptions. -/
-@[to_additive]
-theorem Function.Injective.map_atTop_finset_prod_eq [CommMonoid α] {g : γ → β}
-    (hg : Function.Injective g) {f : β → α} (hf : ∀ x, x ∉ Set.range g → f x = 1) :
-    map (fun s => ∏ i ∈ s, f (g i)) atTop = map (fun s => ∏ i ∈ s, f i) atTop := by
-  haveI := Classical.decEq β
-  apply le_antisymm <;> refine map_atTop_finset_prod_le_of_prod_eq fun s => ?_
-  · refine ⟨s.preimage g hg.injOn, fun t ht => ?_⟩
-    refine ⟨t.image g ∪ s, Finset.subset_union_right, ?_⟩
-    rw [← Finset.prod_image hg.injOn]
-    refine (prod_subset subset_union_left ?_).symm
-    simp only [Finset.mem_union, Finset.mem_image]
-    refine fun y hy hyt => hf y (mt ?_ hyt)
-    rintro ⟨x, rfl⟩
-    exact ⟨x, ht (Finset.mem_preimage.2 <| hy.resolve_left hyt), rfl⟩
-  · refine ⟨s.image g, fun t ht => ?_⟩
-    simp only [← prod_preimage _ _ hg.injOn _ fun x _ => hf x]
-    exact ⟨_, (image_subset_iff_subset_preimage _).1 ht, rfl⟩
-
-/-- Let `g : γ → β` be an injective function and `f : β → α` be a function from the codomain of `g`
-to an additive commutative monoid. Suppose that `f x = 0` outside of the range of `g`. Then the
-filters `atTop.map (fun s ↦ ∑ i ∈ s, f (g i))` and `atTop.map (fun s ↦ ∑ i ∈ s, f i)` coincide.
-
-This lemma is used to prove the equality `∑' x, f (g x) = ∑' y, f y` under
-the same assumptions. -/
-add_decl_doc Function.Injective.map_atTop_finset_sum_eq
+namespace Nat
+
+theorem eventually_pow_lt_factorial_sub (c d : ℕ) : ∀ᶠ n in atTop, c ^ n < (n - d)! := by
+  rw [eventually_atTop]
+  refine ⟨2 * (c ^ 2 + d + 1), ?_⟩
+  intro n hn
+  obtain ⟨d', rfl⟩ := Nat.exists_eq_add_of_le hn
+  obtain (rfl | c0) := c.eq_zero_or_pos
+  · simp [Nat.two_mul, ← Nat.add_assoc, Nat.add_right_comm _ 1, Nat.factorial_pos]
+  refine (Nat.le_mul_of_pos_right _ (Nat.pow_pos (n := d') c0)).trans_lt ?_
+  convert_to (c ^ 2) ^ (c ^ 2 + d' + d + 1) < (c ^ 2 + (c ^ 2 + d' + d + 1) + 1)!
+  · rw [← pow_mul, ← pow_add]
+    congr 1
+    omega
+  · congr 1
+    omega
+  refine (lt_of_lt_of_le ?_ Nat.factorial_mul_pow_le_factorial).trans_le <|
+    (factorial_le (Nat.le_succ _))
+  rw [← one_mul (_ ^ _ : ℕ)]
+  apply Nat.mul_lt_mul_of_le_of_lt
+  · exact Nat.one_le_of_lt (Nat.factorial_pos _)
+  · exact Nat.pow_lt_pow_left (Nat.lt_succ_self _) (Nat.succ_ne_zero _)
+  · exact (Nat.factorial_pos _)
+
+theorem eventually_mul_pow_lt_factorial_sub (a c d : ℕ) :
+    ∀ᶠ n in atTop, a * c ^ n < (n - d)! := by
+  filter_upwards [Nat.eventually_pow_lt_factorial_sub (a * c) d, Filter.eventually_gt_atTop 0]
+    with n hn hn0
+  rw [mul_pow] at hn
+  exact (Nat.mul_le_mul_right _ (Nat.le_self_pow hn0.ne' _)).trans_lt hn
+
+@[deprecated eventually_pow_lt_factorial_sub (since := "2024-09-25")]
+theorem exists_pow_lt_factorial (c : ℕ) : ∃ n0 > 1, ∀ n ≥ n0, c ^ n < (n - 1)! :=
+  let ⟨n0, h⟩ := (eventually_pow_lt_factorial_sub c 1).exists_forall_of_atTop
+  ⟨max n0 2, by omega, fun n hn ↦ h n (by omega)⟩
+
+@[deprecated eventually_mul_pow_lt_factorial_sub (since := "2024-09-25")]
+theorem exists_mul_pow_lt_factorial (a : ℕ) (c : ℕ) : ∃ n0, ∀ n ≥ n0, a * c ^ n < (n - 1)! :=
+  (eventually_mul_pow_lt_factorial_sub a c 1).exists_forall_of_atTop
+
+end Nat
diff --git a/Mathlib/Order/Filter/Archimedean.lean b/Mathlib/Order/Filter/AtTopBot/Archimedean.lean
similarity index 99%
rename from Mathlib/Order/Filter/Archimedean.lean
rename to Mathlib/Order/Filter/AtTopBot/Archimedean.lean
index 7b5d1fe77631c..143e3e2e610f0 100644
--- a/Mathlib/Order/Filter/Archimedean.lean
+++ b/Mathlib/Order/Filter/AtTopBot/Archimedean.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Yury Kudryashov
 -/
 import Mathlib.Algebra.Order.Archimedean.Basic
-import Mathlib.Order.Filter.AtTopBot
+import Mathlib.Order.Filter.AtTopBot.Group
 import Mathlib.Tactic.GCongr
 
 /-!
diff --git a/Mathlib/Order/Filter/AtTopBot/BigOperators.lean b/Mathlib/Order/Filter/AtTopBot/BigOperators.lean
new file mode 100644
index 0000000000000..bbc99acf7d7b1
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/BigOperators.lean
@@ -0,0 +1,70 @@
+/-
+Copyright (c) 2020 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Order.Filter.AtTopBot
+
+/-!
+# Two lemmas about limit of `Π b ∈ s, f b` along
+
+In this file we prove two auxiliary lemmas
+about `Filter.atTop : Filter (Finset _)` and `∏ b ∈ s, f b`.
+These lemmas are useful to build the theory of absolutely convergent series.
+-/
+
+open Filter Finset
+
+variable {α β M : Type*} [CommMonoid M]
+
+/-- Let `f` and `g` be two maps to the same commutative monoid. This lemma gives a sufficient
+condition for comparison of the filter `atTop.map (fun s ↦ ∏ b ∈ s, f b)` with
+`atTop.map (fun s ↦ ∏ b ∈ s, g b)`. This is useful to compare the set of limit points of
+`Π b in s, f b` as `s → atTop` with the similar set for `g`. -/
+@[to_additive "Let `f` and `g` be two maps to the same commutative additive monoid. This lemma gives
+a sufficient condition for comparison of the filter `atTop.map (fun s ↦ ∑ b ∈ s, f b)` with
+`atTop.map (fun s ↦ ∑ b ∈ s, g b)`. This is useful to compare the set of limit points of
+`∑ b ∈ s, f b` as `s → atTop` with the similar set for `g`."]
+theorem Filter.map_atTop_finset_prod_le_of_prod_eq {f : α → M} {g : β → M}
+    (h_eq : ∀ u : Finset β,
+      ∃ v : Finset α, ∀ v', v ⊆ v' → ∃ u', u ⊆ u' ∧ ∏ x ∈ u', g x = ∏ b ∈ v', f b) :
+    (atTop.map fun s : Finset α => ∏ b ∈ s, f b) ≤
+      atTop.map fun s : Finset β => ∏ x ∈ s, g x := by
+  classical
+    refine ((atTop_basis.map _).le_basis_iff (atTop_basis.map _)).2 fun b _ => ?_
+    let ⟨v, hv⟩ := h_eq b
+    refine ⟨v, trivial, ?_⟩
+    simpa [Finset.image_subset_iff] using hv
+
+/-- Let `g : γ → β` be an injective function and `f : β → α` be a function from the codomain of `g`
+to a commutative monoid. Suppose that `f x = 1` outside of the range of `g`. Then the filters
+`atTop.map (fun s ↦ ∏ i ∈ s, f (g i))` and `atTop.map (fun s ↦ ∏ i ∈ s, f i)` coincide.
+
+The additive version of this lemma is used to prove the equality `∑' x, f (g x) = ∑' y, f y` under
+the same assumptions. -/
+@[to_additive]
+theorem Function.Injective.map_atTop_finset_prod_eq {g : α → β}
+    (hg : Function.Injective g) {f : β → M} (hf : ∀ x, x ∉ Set.range g → f x = 1) :
+    map (fun s => ∏ i ∈ s, f (g i)) atTop = map (fun s => ∏ i ∈ s, f i) atTop := by
+  haveI := Classical.decEq β
+  apply le_antisymm <;> refine map_atTop_finset_prod_le_of_prod_eq fun s => ?_
+  · refine ⟨s.preimage g hg.injOn, fun t ht => ?_⟩
+    refine ⟨t.image g ∪ s, Finset.subset_union_right, ?_⟩
+    rw [← Finset.prod_image hg.injOn]
+    refine (prod_subset subset_union_left ?_).symm
+    simp only [Finset.mem_union, Finset.mem_image]
+    refine fun y hy hyt => hf y (mt ?_ hyt)
+    rintro ⟨x, rfl⟩
+    exact ⟨x, ht (Finset.mem_preimage.2 <| hy.resolve_left hyt), rfl⟩
+  · refine ⟨s.image g, fun t ht => ?_⟩
+    simp only [← prod_preimage _ _ hg.injOn _ fun x _ => hf x]
+    exact ⟨_, (image_subset_iff_subset_preimage _).1 ht, rfl⟩
+
+/-- Let `g : γ → β` be an injective function and `f : β → α` be a function from the codomain of `g`
+to an additive commutative monoid. Suppose that `f x = 0` outside of the range of `g`. Then the
+filters `atTop.map (fun s ↦ ∑ i ∈ s, f (g i))` and `atTop.map (fun s ↦ ∑ i ∈ s, f i)` coincide.
+
+This lemma is used to prove the equality `∑' x, f (g x) = ∑' y, f y` under
+the same assumptions. -/
+add_decl_doc Function.Injective.map_atTop_finset_sum_eq
diff --git a/Mathlib/Order/Filter/AtTopBot/Field.lean b/Mathlib/Order/Filter/AtTopBot/Field.lean
new file mode 100644
index 0000000000000..519009d383724
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/Field.lean
@@ -0,0 +1,320 @@
+/-
+Copyright (c) 2019 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Order.Field.Defs
+import Mathlib.Order.Filter.AtTopBot.Ring
+
+/-!
+# Convergence to ±infinity in linear ordered (semi)fields
+-/
+
+namespace Filter
+
+variable {α β : Type*}
+
+section LinearOrderedSemifield
+
+variable [LinearOrderedSemifield α] {l : Filter β} {f : β → α} {r c : α} {n : ℕ}
+
+/-!
+### Multiplication by constant: iff lemmas
+-/
+
+/-- If `r` is a positive constant, `fun x ↦ r * f x` tends to infinity along a filter
+if and only if `f` tends to infinity along the same filter. -/
+theorem tendsto_const_mul_atTop_of_pos (hr : 0 < r) :
+    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atTop :=
+  ⟨fun h => h.atTop_of_const_mul hr, fun h =>
+    Tendsto.atTop_of_const_mul (inv_pos.2 hr) <| by simpa only [inv_mul_cancel_left₀ hr.ne'] ⟩
+
+/-- If `r` is a positive constant, `fun x ↦ f x * r` tends to infinity along a filter
+if and only if `f` tends to infinity along the same filter. -/
+theorem tendsto_mul_const_atTop_of_pos (hr : 0 < r) :
+    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atTop := by
+  simpa only [mul_comm] using tendsto_const_mul_atTop_of_pos hr
+
+/-- If `r` is a positive constant, `x ↦ f x / r` tends to infinity along a filter
+if and only if `f` tends to infinity along the same filter. -/
+lemma tendsto_div_const_atTop_of_pos (hr : 0 < r) :
+    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atTop := by
+  simpa only [div_eq_mul_inv] using tendsto_mul_const_atTop_of_pos (inv_pos.2 hr)
+
+/-- If `f` tends to infinity along a nontrivial filter `l`, then
+`fun x ↦ r * f x` tends to infinity if and only if `0 < r. `-/
+theorem tendsto_const_mul_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x => r * f x) l atTop ↔ 0 < r := by
+  refine ⟨fun hrf => not_le.mp fun hr => ?_, fun hr => (tendsto_const_mul_atTop_of_pos hr).mpr h⟩
+  rcases ((h.eventually_ge_atTop 0).and (hrf.eventually_gt_atTop 0)).exists with ⟨x, hx, hrx⟩
+  exact (mul_nonpos_of_nonpos_of_nonneg hr hx).not_lt hrx
+
+/-- If `f` tends to infinity along a nontrivial filter `l`, then
+`fun x ↦ f x * r` tends to infinity if and only if `0 < r. `-/
+theorem tendsto_mul_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x => f x * r) l atTop ↔ 0 < r := by
+  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_pos h]
+
+/-- If `f` tends to infinity along a nontrivial filter `l`, then
+`x ↦ f x * r` tends to infinity if and only if `0 < r. `-/
+lemma tendsto_div_const_atTop_iff_pos [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r := by
+  simp only [div_eq_mul_inv, tendsto_mul_const_atTop_iff_pos h, inv_pos]
+
+/-- If `f` tends to infinity along a filter, then `f` multiplied by a positive
+constant (on the left) also tends to infinity. For a version working in `ℕ` or `ℤ`, use
+`Filter.Tendsto.const_mul_atTop'` instead. -/
+theorem Tendsto.const_mul_atTop (hr : 0 < r) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => r * f x) l atTop :=
+  (tendsto_const_mul_atTop_of_pos hr).2 hf
+
+/-- If a function `f` tends to infinity along a filter, then `f` multiplied by a positive
+constant (on the right) also tends to infinity. For a version working in `ℕ` or `ℤ`, use
+`Filter.Tendsto.atTop_mul_const'` instead. -/
+theorem Tendsto.atTop_mul_const (hr : 0 < r) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => f x * r) l atTop :=
+  (tendsto_mul_const_atTop_of_pos hr).2 hf
+
+/-- If a function `f` tends to infinity along a filter, then `f` divided by a positive
+constant also tends to infinity. -/
+theorem Tendsto.atTop_div_const (hr : 0 < r) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => f x / r) l atTop := by
+  simpa only [div_eq_mul_inv] using hf.atTop_mul_const (inv_pos.2 hr)
+
+theorem tendsto_const_mul_pow_atTop (hn : n ≠ 0) (hc : 0 < c) :
+    Tendsto (fun x => c * x ^ n) atTop atTop :=
+  Tendsto.const_mul_atTop hc (tendsto_pow_atTop hn)
+
+theorem tendsto_const_mul_pow_atTop_iff :
+    Tendsto (fun x => c * x ^ n) atTop atTop ↔ n ≠ 0 ∧ 0 < c := by
+  refine ⟨fun h => ⟨?_, ?_⟩, fun h => tendsto_const_mul_pow_atTop h.1 h.2⟩
+  · rintro rfl
+    simp only [pow_zero, not_tendsto_const_atTop] at h
+  · rcases ((h.eventually_gt_atTop 0).and (eventually_ge_atTop 0)).exists with ⟨k, hck, hk⟩
+    exact pos_of_mul_pos_left hck (pow_nonneg hk _)
+
+lemma tendsto_zpow_atTop_atTop {n : ℤ} (hn : 0 < n) : Tendsto (fun x : α ↦ x ^ n) atTop atTop := by
+  lift n to ℕ+ using hn; simp
+
+end LinearOrderedSemifield
+
+
+section LinearOrderedField
+
+variable [LinearOrderedField α] {l : Filter β} {f : β → α} {r : α}
+
+/-- If `r` is a positive constant, `fun x ↦ r * f x` tends to negative infinity along a filter
+if and only if `f` tends to negative infinity along the same filter. -/
+theorem tendsto_const_mul_atBot_of_pos (hr : 0 < r) :
+    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atBot := by
+  simpa only [← mul_neg, ← tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos hr
+
+/-- If `r` is a positive constant, `fun x ↦ f x * r` tends to negative infinity along a filter
+if and only if `f` tends to negative infinity along the same filter. -/
+theorem tendsto_mul_const_atBot_of_pos (hr : 0 < r) :
+    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot := by
+  simpa only [mul_comm] using tendsto_const_mul_atBot_of_pos hr
+
+/-- If `r` is a positive constant, `fun x ↦ f x / r` tends to negative infinity along a filter
+if and only if `f` tends to negative infinity along the same filter. -/
+lemma tendsto_div_const_atBot_of_pos (hr : 0 < r) :
+    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atBot := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_pos, hr]
+
+/-- If `r` is a negative constant, `fun x ↦ r * f x` tends to infinity along a filter `l`
+if and only if `f` tends to negative infinity along `l`. -/
+theorem tendsto_const_mul_atTop_of_neg (hr : r < 0) :
+    Tendsto (fun x => r * f x) l atTop ↔ Tendsto f l atBot := by
+  simpa only [neg_mul, tendsto_neg_atBot_iff] using tendsto_const_mul_atBot_of_pos (neg_pos.2 hr)
+
+/-- If `r` is a negative constant, `fun x ↦ f x * r` tends to infinity along a filter `l`
+if and only if `f` tends to negative infinity along `l`. -/
+theorem tendsto_mul_const_atTop_of_neg (hr : r < 0) :
+    Tendsto (fun x => f x * r) l atTop ↔ Tendsto f l atBot := by
+  simpa only [mul_comm] using tendsto_const_mul_atTop_of_neg hr
+
+/-- If `r` is a negative constant, `fun x ↦ f x / r` tends to infinity along a filter `l`
+if and only if `f` tends to negative infinity along `l`. -/
+lemma tendsto_div_const_atTop_of_neg (hr : r < 0) :
+    Tendsto (fun x ↦ f x / r) l atTop ↔ Tendsto f l atBot := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atTop_of_neg, hr]
+
+/-- If `r` is a negative constant, `fun x ↦ r * f x` tends to negative infinity along a filter `l`
+if and only if `f` tends to infinity along `l`. -/
+theorem tendsto_const_mul_atBot_of_neg (hr : r < 0) :
+    Tendsto (fun x => r * f x) l atBot ↔ Tendsto f l atTop := by
+  simpa only [neg_mul, tendsto_neg_atTop_iff] using tendsto_const_mul_atTop_of_pos (neg_pos.2 hr)
+
+/-- If `r` is a negative constant, `fun x ↦ f x * r` tends to negative infinity along a filter `l`
+if and only if `f` tends to infinity along `l`. -/
+theorem tendsto_mul_const_atBot_of_neg (hr : r < 0) :
+    Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atTop := by
+  simpa only [mul_comm] using tendsto_const_mul_atBot_of_neg hr
+
+/-- If `r` is a negative constant, `fun x ↦ f x / r` tends to negative infinity along a filter `l`
+if and only if `f` tends to infinity along `l`. -/
+lemma tendsto_div_const_atBot_of_neg (hr : r < 0) :
+    Tendsto (fun x ↦ f x / r) l atBot ↔ Tendsto f l atTop := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atBot_of_neg, hr]
+
+/-- The function `fun x ↦ r * f x` tends to infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
+theorem tendsto_const_mul_atTop_iff [NeBot l] :
+    Tendsto (fun x => r * f x) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
+  rcases lt_trichotomy r 0 with (hr | rfl | hr)
+  · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_neg]
+  · simp [not_tendsto_const_atTop]
+  · simp [hr, hr.not_lt, tendsto_const_mul_atTop_of_pos]
+
+/-- The function `fun x ↦ f x * r` tends to infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
+theorem tendsto_mul_const_atTop_iff [NeBot l] :
+    Tendsto (fun x => f x * r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
+  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff]
+
+/-- The function `fun x ↦ f x / r` tends to infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to infinity or `r < 0` and `f` tends to negative infinity. -/
+lemma tendsto_div_const_atTop_iff [NeBot l] :
+    Tendsto (fun x ↦ f x / r) l atTop ↔ 0 < r ∧ Tendsto f l atTop ∨ r < 0 ∧ Tendsto f l atBot := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff]
+
+/-- The function `fun x ↦ r * f x` tends to negative infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
+theorem tendsto_const_mul_atBot_iff [NeBot l] :
+    Tendsto (fun x => r * f x) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
+  simp only [← tendsto_neg_atTop_iff, ← mul_neg, tendsto_const_mul_atTop_iff, neg_neg]
+
+/-- The function `fun x ↦ f x * r` tends to negative infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
+theorem tendsto_mul_const_atBot_iff [NeBot l] :
+    Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
+  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff]
+
+/-- The function `fun x ↦ f x / r` tends to negative infinity along a nontrivial filter
+if and only if `r > 0` and `f` tends to negative infinity or `r < 0` and `f` tends to infinity. -/
+lemma tendsto_div_const_atBot_iff [NeBot l] :
+    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`,
+then `fun x ↦ r * f x` tends to infinity if and only if `r < 0. `-/
+theorem tendsto_const_mul_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x => r * f x) l atTop ↔ r < 0 := by
+  simp [tendsto_const_mul_atTop_iff, h, h.not_tendsto disjoint_atBot_atTop]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`,
+then `fun x ↦ f x * r` tends to infinity if and only if `r < 0. `-/
+theorem tendsto_mul_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x => f x * r) l atTop ↔ r < 0 := by
+  simp only [mul_comm _ r, tendsto_const_mul_atTop_iff_neg h]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`,
+then `fun x ↦ f x / r` tends to infinity if and only if `r < 0. `-/
+lemma tendsto_div_const_atTop_iff_neg [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x ↦ f x / r) l atTop ↔ r < 0 := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atTop_iff_neg h]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
+`fun x ↦ r * f x` tends to negative infinity if and only if `0 < r. `-/
+theorem tendsto_const_mul_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x => r * f x) l atBot ↔ 0 < r := by
+  simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atBot_atTop]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
+`fun x ↦ f x * r` tends to negative infinity if and only if `0 < r. `-/
+theorem tendsto_mul_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x => f x * r) l atBot ↔ 0 < r := by
+  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff_pos h]
+
+/-- If `f` tends to negative infinity along a nontrivial filter `l`, then
+`fun x ↦ f x / r` tends to negative infinity if and only if `0 < r. `-/
+lemma tendsto_div_const_atBot_iff_pos [NeBot l] (h : Tendsto f l atBot) :
+    Tendsto (fun x ↦ f x / r) l atBot ↔ 0 < r := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff_pos h]
+
+/-- If `f` tends to infinity along a nontrivial filter,
+`fun x ↦ r * f x` tends to negative infinity if and only if `r < 0. `-/
+theorem tendsto_const_mul_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x => r * f x) l atBot ↔ r < 0 := by
+  simp [tendsto_const_mul_atBot_iff, h, h.not_tendsto disjoint_atTop_atBot]
+
+/-- If `f` tends to infinity along a nontrivial filter,
+`fun x ↦ f x * r` tends to negative infinity if and only if `r < 0. `-/
+theorem tendsto_mul_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x => f x * r) l atBot ↔ r < 0 := by
+  simp only [mul_comm _ r, tendsto_const_mul_atBot_iff_neg h]
+
+/-- If `f` tends to infinity along a nontrivial filter,
+`fun x ↦ f x / r` tends to negative infinity if and only if `r < 0. `-/
+lemma tendsto_div_const_atBot_iff_neg [NeBot l] (h : Tendsto f l atTop) :
+    Tendsto (fun x ↦ f x / r) l atBot ↔ r < 0 := by
+  simp [div_eq_mul_inv, tendsto_mul_const_atBot_iff_neg h]
+
+/-- If a function `f` tends to infinity along a filter,
+then `f` multiplied by a negative constant (on the left) tends to negative infinity. -/
+theorem Tendsto.const_mul_atTop_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => r * f x) l atBot :=
+  (tendsto_const_mul_atBot_of_neg hr).2 hf
+
+/-- If a function `f` tends to infinity along a filter,
+then `f` multiplied by a negative constant (on the right) tends to negative infinity. -/
+theorem Tendsto.atTop_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => f x * r) l atBot :=
+  (tendsto_mul_const_atBot_of_neg hr).2 hf
+
+/-- If a function `f` tends to infinity along a filter,
+then `f` divided by a negative constant tends to negative infinity. -/
+lemma Tendsto.atTop_div_const_of_neg (hr : r < 0) (hf : Tendsto f l atTop) :
+    Tendsto (fun x ↦ f x / r) l atBot := (tendsto_div_const_atBot_of_neg hr).2 hf
+
+/-- If a function `f` tends to negative infinity along a filter, then `f` multiplied by
+a positive constant (on the left) also tends to negative infinity. -/
+theorem Tendsto.const_mul_atBot (hr : 0 < r) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => r * f x) l atBot :=
+  (tendsto_const_mul_atBot_of_pos hr).2 hf
+
+/-- If a function `f` tends to negative infinity along a filter, then `f` multiplied by
+a positive constant (on the right) also tends to negative infinity. -/
+theorem Tendsto.atBot_mul_const (hr : 0 < r) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => f x * r) l atBot :=
+  (tendsto_mul_const_atBot_of_pos hr).2 hf
+
+/-- If a function `f` tends to negative infinity along a filter, then `f` divided by
+a positive constant also tends to negative infinity. -/
+theorem Tendsto.atBot_div_const (hr : 0 < r) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => f x / r) l atBot := (tendsto_div_const_atBot_of_pos hr).2 hf
+
+/-- If a function `f` tends to negative infinity along a filter,
+then `f` multiplied by a negative constant (on the left) tends to positive infinity. -/
+theorem Tendsto.const_mul_atBot_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => r * f x) l atTop :=
+  (tendsto_const_mul_atTop_of_neg hr).2 hf
+
+/-- If a function tends to negative infinity along a filter,
+then `f` multiplied by a negative constant (on the right) tends to positive infinity. -/
+theorem Tendsto.atBot_mul_const_of_neg (hr : r < 0) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => f x * r) l atTop :=
+  (tendsto_mul_const_atTop_of_neg hr).2 hf
+
+theorem tendsto_neg_const_mul_pow_atTop {c : α} {n : ℕ} (hn : n ≠ 0) (hc : c < 0) :
+    Tendsto (fun x => c * x ^ n) atTop atBot :=
+  (tendsto_pow_atTop hn).const_mul_atTop_of_neg hc
+
+theorem tendsto_const_mul_pow_atBot_iff {c : α} {n : ℕ} :
+    Tendsto (fun x => c * x ^ n) atTop atBot ↔ n ≠ 0 ∧ c < 0 := by
+  simp only [← tendsto_neg_atTop_iff, ← neg_mul, tendsto_const_mul_pow_atTop_iff, neg_pos]
+
+@[deprecated (since := "2024-05-06")]
+alias Tendsto.neg_const_mul_atTop := Tendsto.const_mul_atTop_of_neg
+
+@[deprecated (since := "2024-05-06")]
+alias Tendsto.atTop_mul_neg_const := Tendsto.atTop_mul_const_of_neg
+
+@[deprecated (since := "2024-05-06")]
+alias Tendsto.neg_const_mul_atBot := Tendsto.const_mul_atBot_of_neg
+
+@[deprecated (since := "2024-05-06")]
+alias Tendsto.atBot_mul_neg_const := Tendsto.atBot_mul_const_of_neg
+
+end LinearOrderedField
+end Filter
diff --git a/Mathlib/Order/Filter/AtTopBot/Floor.lean b/Mathlib/Order/Filter/AtTopBot/Floor.lean
new file mode 100644
index 0000000000000..32e3e3aad7c86
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/Floor.lean
@@ -0,0 +1,28 @@
+/-
+Copyright (c) 2022 Yuyang Zhao. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuyang Zhao
+-/
+import Mathlib.Algebra.Order.Floor
+import Mathlib.Order.Filter.AtTopBot
+
+/-!
+# `a * c ^ n < (n - d)!` holds true for sufficiently large `n`.
+-/
+
+open Filter
+open scoped Nat
+
+variable {K : Type*} [LinearOrderedRing K] [FloorSemiring K]
+
+theorem FloorSemiring.eventually_mul_pow_lt_factorial_sub (a c : K) (d : ℕ) :
+    ∀ᶠ n in atTop, a * c ^ n < (n - d)! := by
+  filter_upwards [Nat.eventually_mul_pow_lt_factorial_sub ⌈|a|⌉₊ ⌈|c|⌉₊ d] with n h
+  calc a * c ^ n
+    _ ≤ |a * c ^ n| := le_abs_self _
+    _ ≤ ⌈|a|⌉₊ * (⌈|c|⌉₊ : K) ^ n := ?_
+    _ = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n) := ?_
+    _ < (n - d)! := Nat.cast_lt.mpr h
+  · rw [abs_mul, abs_pow]
+    gcongr <;> try first | positivity | apply Nat.le_ceil
+  · simp_rw [Nat.cast_mul, Nat.cast_pow]
diff --git a/Mathlib/Order/Filter/AtTopBot/Group.lean b/Mathlib/Order/Filter/AtTopBot/Group.lean
new file mode 100644
index 0000000000000..4c4b84aeb895d
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/Group.lean
@@ -0,0 +1,127 @@
+/-
+Copyright (c) 2019 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Order.Group.Instances
+import Mathlib.Algebra.Order.Group.MinMax
+import Mathlib.Order.Filter.AtTopBot.Monoid
+
+/-!
+# Convergence to ±infinity in ordered commutative groups
+-/
+
+variable {α β : Type*}
+open Set
+
+namespace Filter
+
+section OrderedGroup
+
+variable [OrderedAddCommGroup β] (l : Filter α) {f g : α → β}
+
+theorem tendsto_atTop_add_left_of_le' (C : β) (hf : ∀ᶠ x in l, C ≤ f x) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  @tendsto_atTop_of_add_bdd_above_left' _ _ _ l (fun x => -f x) (fun x => f x + g x) (-C) (by simpa)
+    (by simpa)
+
+theorem tendsto_atBot_add_left_of_ge' (C : β) (hf : ∀ᶠ x in l, f x ≤ C) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_left_of_le' _ βᵒᵈ _ _ _ _ C hf hg
+
+theorem tendsto_atTop_add_left_of_le (C : β) (hf : ∀ x, C ≤ f x) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_add_left_of_le' l C (univ_mem' hf) hg
+
+theorem tendsto_atBot_add_left_of_ge (C : β) (hf : ∀ x, f x ≤ C) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_left_of_le _ βᵒᵈ _ _ _ _ C hf hg
+
+theorem tendsto_atTop_add_right_of_le' (C : β) (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, C ≤ g x) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  @tendsto_atTop_of_add_bdd_above_right' _ _ _ l (fun x => f x + g x) (fun x => -g x) (-C)
+    (by simp [hg]) (by simp [hf])
+
+theorem tendsto_atBot_add_right_of_ge' (C : β) (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ C) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_right_of_le' _ βᵒᵈ _ _ _ _ C hf hg
+
+theorem tendsto_atTop_add_right_of_le (C : β) (hf : Tendsto f l atTop) (hg : ∀ x, C ≤ g x) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_add_right_of_le' l C hf (univ_mem' hg)
+
+theorem tendsto_atBot_add_right_of_ge (C : β) (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ C) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_right_of_le _ βᵒᵈ _ _ _ _ C hf hg
+
+theorem tendsto_atTop_add_const_left (C : β) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => C + f x) l atTop :=
+  tendsto_atTop_add_left_of_le' l C (univ_mem' fun _ => le_refl C) hf
+
+theorem tendsto_atBot_add_const_left (C : β) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => C + f x) l atBot :=
+  @tendsto_atTop_add_const_left _ βᵒᵈ _ _ _ C hf
+
+theorem tendsto_atTop_add_const_right (C : β) (hf : Tendsto f l atTop) :
+    Tendsto (fun x => f x + C) l atTop :=
+  tendsto_atTop_add_right_of_le' l C hf (univ_mem' fun _ => le_refl C)
+
+theorem tendsto_atBot_add_const_right (C : β) (hf : Tendsto f l atBot) :
+    Tendsto (fun x => f x + C) l atBot :=
+  @tendsto_atTop_add_const_right _ βᵒᵈ _ _ _ C hf
+
+theorem map_neg_atBot : map (Neg.neg : β → β) atBot = atTop :=
+  (OrderIso.neg β).map_atBot
+
+theorem map_neg_atTop : map (Neg.neg : β → β) atTop = atBot :=
+  (OrderIso.neg β).map_atTop
+
+theorem comap_neg_atBot : comap (Neg.neg : β → β) atBot = atTop :=
+  (OrderIso.neg β).comap_atTop
+
+theorem comap_neg_atTop : comap (Neg.neg : β → β) atTop = atBot :=
+  (OrderIso.neg β).comap_atBot
+
+theorem tendsto_neg_atTop_atBot : Tendsto (Neg.neg : β → β) atTop atBot :=
+  (OrderIso.neg β).tendsto_atTop
+
+theorem tendsto_neg_atBot_atTop : Tendsto (Neg.neg : β → β) atBot atTop :=
+  @tendsto_neg_atTop_atBot βᵒᵈ _
+
+variable {l}
+
+@[simp]
+theorem tendsto_neg_atTop_iff : Tendsto (fun x => -f x) l atTop ↔ Tendsto f l atBot :=
+  (OrderIso.neg β).tendsto_atBot_iff
+
+@[simp]
+theorem tendsto_neg_atBot_iff : Tendsto (fun x => -f x) l atBot ↔ Tendsto f l atTop :=
+  (OrderIso.neg β).tendsto_atTop_iff
+
+end OrderedGroup
+
+section LinearOrderedAddCommGroup
+
+variable [LinearOrderedAddCommGroup α]
+
+/-- $\lim_{x\to+\infty}|x|=+\infty$ -/
+theorem tendsto_abs_atTop_atTop : Tendsto (abs : α → α) atTop atTop :=
+  tendsto_atTop_mono le_abs_self tendsto_id
+
+/-- $\lim_{x\to-\infty}|x|=+\infty$ -/
+theorem tendsto_abs_atBot_atTop : Tendsto (abs : α → α) atBot atTop :=
+  tendsto_atTop_mono neg_le_abs tendsto_neg_atBot_atTop
+
+@[simp]
+theorem comap_abs_atTop : comap (abs : α → α) atTop = atBot ⊔ atTop := by
+  refine
+    le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 ?_)
+      (sup_le tendsto_abs_atBot_atTop.le_comap tendsto_abs_atTop_atTop.le_comap)
+  rintro ⟨a, b⟩ -
+  refine ⟨max (-a) b, trivial, fun x hx => ?_⟩
+  rw [mem_preimage, mem_Ici, le_abs', max_le_iff, ← min_neg_neg, le_min_iff, neg_neg] at hx
+  exact hx.imp And.left And.right
+
+end LinearOrderedAddCommGroup
+
+end Filter
diff --git a/Mathlib/Order/Filter/ModEq.lean b/Mathlib/Order/Filter/AtTopBot/ModEq.lean
similarity index 86%
rename from Mathlib/Order/Filter/ModEq.lean
rename to Mathlib/Order/Filter/AtTopBot/ModEq.lean
index 0b9c0b89ea5f7..d45c17cc0d74b 100644
--- a/Mathlib/Order/Filter/ModEq.lean
+++ b/Mathlib/Order/Filter/AtTopBot/ModEq.lean
@@ -3,9 +3,12 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Algebra.Order.Ring.Abs
+import Mathlib.Algebra.Order.Ring.Basic
+import Mathlib.Algebra.Order.Ring.Nat
+import Mathlib.Algebra.Ring.Divisibility.Basic
+import Mathlib.Algebra.Ring.Int
 import Mathlib.Data.Nat.ModEq
-import Mathlib.Order.Filter.AtTopBot
+import Mathlib.Order.Filter.AtTopBot.Monoid
 
 /-!
 # Numbers are frequently ModEq to fixed numbers
diff --git a/Mathlib/Order/Filter/AtTopBot/Monoid.lean b/Mathlib/Order/Filter/AtTopBot/Monoid.lean
new file mode 100644
index 0000000000000..0cfa275d9a426
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/Monoid.lean
@@ -0,0 +1,141 @@
+/-
+Copyright (c) 2019 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Order.Monoid.OrderDual
+import Mathlib.Order.Filter.AtTopBot
+
+/-!
+# Convergence to ±infinity in ordered commutative monoids
+-/
+
+variable {α β : Type*}
+
+namespace Filter
+
+section OrderedAddCommMonoid
+
+variable [OrderedAddCommMonoid β] {l : Filter α} {f g : α → β}
+
+theorem tendsto_atTop_add_nonneg_left' (hf : ∀ᶠ x in l, 0 ≤ f x) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_mono' l (hf.mono fun _ => le_add_of_nonneg_left) hg
+
+theorem tendsto_atBot_add_nonpos_left' (hf : ∀ᶠ x in l, f x ≤ 0) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_nonneg_left' _ βᵒᵈ _ _ _ _ hf hg
+
+theorem tendsto_atTop_add_nonneg_left (hf : ∀ x, 0 ≤ f x) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_add_nonneg_left' (Eventually.of_forall hf) hg
+
+theorem tendsto_atBot_add_nonpos_left (hf : ∀ x, f x ≤ 0) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_nonneg_left _ βᵒᵈ _ _ _ _ hf hg
+
+theorem tendsto_atTop_add_nonneg_right' (hf : Tendsto f l atTop) (hg : ∀ᶠ x in l, 0 ≤ g x) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_mono' l (monotone_mem (fun _ => le_add_of_nonneg_right) hg) hf
+
+theorem tendsto_atBot_add_nonpos_right' (hf : Tendsto f l atBot) (hg : ∀ᶠ x in l, g x ≤ 0) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_nonneg_right' _ βᵒᵈ _ _ _ _ hf hg
+
+theorem tendsto_atTop_add_nonneg_right (hf : Tendsto f l atTop) (hg : ∀ x, 0 ≤ g x) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_add_nonneg_right' hf (Eventually.of_forall hg)
+
+theorem tendsto_atBot_add_nonpos_right (hf : Tendsto f l atBot) (hg : ∀ x, g x ≤ 0) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add_nonneg_right _ βᵒᵈ _ _ _ _ hf hg
+
+theorem tendsto_atTop_add (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x + g x) l atTop :=
+  tendsto_atTop_add_nonneg_left' (tendsto_atTop.mp hf 0) hg
+
+theorem tendsto_atBot_add (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x + g x) l atBot :=
+  @tendsto_atTop_add _ βᵒᵈ _ _ _ _ hf hg
+
+theorem Tendsto.nsmul_atTop (hf : Tendsto f l atTop) {n : ℕ} (hn : 0 < n) :
+    Tendsto (fun x => n • f x) l atTop :=
+  tendsto_atTop.2 fun y =>
+    (tendsto_atTop.1 hf y).mp <|
+      (tendsto_atTop.1 hf 0).mono fun x h₀ hy =>
+        calc
+          y ≤ f x := hy
+          _ = 1 • f x := (one_nsmul _).symm
+          _ ≤ n • f x := nsmul_le_nsmul_left h₀ hn
+
+theorem Tendsto.nsmul_atBot (hf : Tendsto f l atBot) {n : ℕ} (hn : 0 < n) :
+    Tendsto (fun x => n • f x) l atBot :=
+  @Tendsto.nsmul_atTop α βᵒᵈ _ l f hf n hn
+
+end OrderedAddCommMonoid
+
+section OrderedCancelAddCommMonoid
+
+variable [OrderedCancelAddCommMonoid β] {l : Filter α} {f g : α → β}
+
+theorem tendsto_atTop_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atTop) :
+    Tendsto f l atTop :=
+  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (C + b)).mono fun _ => le_of_add_le_add_left
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_const_left (C : β) (hf : Tendsto (fun x => C + f x) l atBot) :
+    Tendsto f l atBot :=
+  tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (C + b)).mono fun _ => le_of_add_le_add_left
+
+theorem tendsto_atTop_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atTop) :
+    Tendsto f l atTop :=
+  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b + C)).mono fun _ => le_of_add_le_add_right
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_const_right (C : β) (hf : Tendsto (fun x => f x + C) l atBot) :
+    Tendsto f l atBot :=
+  tendsto_atBot.2 fun b => (tendsto_atBot.1 hf (b + C)).mono fun _ => le_of_add_le_add_right
+
+theorem tendsto_atTop_of_add_bdd_above_left' (C) (hC : ∀ᶠ x in l, f x ≤ C)
+    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto g l atTop :=
+  tendsto_atTop_of_add_const_left C
+    (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h)
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_bdd_below_left' (C) (hC : ∀ᶠ x in l, C ≤ f x)
+    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto g l atBot :=
+  tendsto_atBot_of_add_const_left C
+    (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_right hx (g x)) h)
+
+theorem tendsto_atTop_of_add_bdd_above_left (C) (hC : ∀ x, f x ≤ C) :
+    Tendsto (fun x => f x + g x) l atTop → Tendsto g l atTop :=
+  tendsto_atTop_of_add_bdd_above_left' C (univ_mem' hC)
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_bdd_below_left (C) (hC : ∀ x, C ≤ f x) :
+    Tendsto (fun x => f x + g x) l atBot → Tendsto g l atBot :=
+  tendsto_atBot_of_add_bdd_below_left' C (univ_mem' hC)
+
+theorem tendsto_atTop_of_add_bdd_above_right' (C) (hC : ∀ᶠ x in l, g x ≤ C)
+    (h : Tendsto (fun x => f x + g x) l atTop) : Tendsto f l atTop :=
+  tendsto_atTop_of_add_const_right C
+    (tendsto_atTop_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h)
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_bdd_below_right' (C) (hC : ∀ᶠ x in l, C ≤ g x)
+    (h : Tendsto (fun x => f x + g x) l atBot) : Tendsto f l atBot :=
+  tendsto_atBot_of_add_const_right C
+    (tendsto_atBot_mono' l (hC.mono fun x hx => add_le_add_left hx (f x)) h)
+
+theorem tendsto_atTop_of_add_bdd_above_right (C) (hC : ∀ x, g x ≤ C) :
+    Tendsto (fun x => f x + g x) l atTop → Tendsto f l atTop :=
+  tendsto_atTop_of_add_bdd_above_right' C (univ_mem' hC)
+
+-- Porting note: the "order dual" trick timeouts
+theorem tendsto_atBot_of_add_bdd_below_right (C) (hC : ∀ x, C ≤ g x) :
+    Tendsto (fun x => f x + g x) l atBot → Tendsto f l atBot :=
+  tendsto_atBot_of_add_bdd_below_right' C (univ_mem' hC)
+
+end OrderedCancelAddCommMonoid
+
+end Filter
diff --git a/Mathlib/Order/Filter/AtTopBot/Ring.lean b/Mathlib/Order/Filter/AtTopBot/Ring.lean
new file mode 100644
index 0000000000000..74207b0e55f14
--- /dev/null
+++ b/Mathlib/Order/Filter/AtTopBot/Ring.lean
@@ -0,0 +1,104 @@
+/-
+Copyright (c) 2019 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Algebra.Order.Ring.Defs
+import Mathlib.Order.Filter.AtTopBot.Group
+
+/-!
+# Convergence to ±infinity in ordered rings
+-/
+
+variable {α β : Type*}
+
+namespace Filter
+
+section OrderedSemiring
+
+variable [OrderedSemiring α] {l : Filter β} {f g : β → α}
+
+theorem Tendsto.atTop_mul_atTop (hf : Tendsto f l atTop) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x * g x) l atTop := by
+  refine tendsto_atTop_mono' _ ?_ hg
+  filter_upwards [hg.eventually (eventually_ge_atTop 0),
+    hf.eventually (eventually_ge_atTop 1)] with _ using le_mul_of_one_le_left
+
+theorem tendsto_mul_self_atTop : Tendsto (fun x : α => x * x) atTop atTop :=
+  tendsto_id.atTop_mul_atTop tendsto_id
+
+/-- The monomial function `x^n` tends to `+∞` at `+∞` for any positive natural `n`.
+A version for positive real powers exists as `tendsto_rpow_atTop`. -/
+theorem tendsto_pow_atTop {n : ℕ} (hn : n ≠ 0) : Tendsto (fun x : α => x ^ n) atTop atTop :=
+  tendsto_atTop_mono' _ ((eventually_ge_atTop 1).mono fun _x hx => le_self_pow₀ hx hn) tendsto_id
+
+end OrderedSemiring
+
+theorem zero_pow_eventuallyEq [MonoidWithZero α] :
+    (fun n : ℕ => (0 : α) ^ n) =ᶠ[atTop] fun _ => 0 :=
+  eventually_atTop.2 ⟨1, fun _n hn ↦ zero_pow <| Nat.one_le_iff_ne_zero.1 hn⟩
+
+section OrderedRing
+
+variable [OrderedRing α] {l : Filter β} {f g : β → α}
+
+theorem Tendsto.atTop_mul_atBot (hf : Tendsto f l atTop) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x * g x) l atBot := by
+  have := hf.atTop_mul_atTop <| tendsto_neg_atBot_atTop.comp hg
+  simpa only [Function.comp_def, neg_mul_eq_mul_neg, neg_neg] using
+    tendsto_neg_atTop_atBot.comp this
+
+theorem Tendsto.atBot_mul_atTop (hf : Tendsto f l atBot) (hg : Tendsto g l atTop) :
+    Tendsto (fun x => f x * g x) l atBot := by
+  have : Tendsto (fun x => -f x * g x) l atTop :=
+    (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop hg
+  simpa only [Function.comp_def, neg_mul_eq_neg_mul, neg_neg] using
+    tendsto_neg_atTop_atBot.comp this
+
+theorem Tendsto.atBot_mul_atBot (hf : Tendsto f l atBot) (hg : Tendsto g l atBot) :
+    Tendsto (fun x => f x * g x) l atTop := by
+  have : Tendsto (fun x => -f x * -g x) l atTop :=
+    (tendsto_neg_atBot_atTop.comp hf).atTop_mul_atTop (tendsto_neg_atBot_atTop.comp hg)
+  simpa only [neg_mul_neg] using this
+
+end OrderedRing
+
+section LinearOrderedSemiring
+
+variable [LinearOrderedSemiring α] {l : Filter β} {f : β → α}
+
+theorem Tendsto.atTop_of_const_mul {c : α} (hc : 0 < c) (hf : Tendsto (fun x => c * f x) l atTop) :
+    Tendsto f l atTop :=
+  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (c * b)).mono
+    fun _x hx => le_of_mul_le_mul_left hx hc
+
+theorem Tendsto.atTop_of_mul_const {c : α} (hc : 0 < c) (hf : Tendsto (fun x => f x * c) l atTop) :
+    Tendsto f l atTop :=
+  tendsto_atTop.2 fun b => (tendsto_atTop.1 hf (b * c)).mono
+    fun _x hx => le_of_mul_le_mul_right hx hc
+
+@[simp]
+theorem tendsto_pow_atTop_iff {n : ℕ} : Tendsto (fun x : α => x ^ n) atTop atTop ↔ n ≠ 0 :=
+  ⟨fun h hn => by simp only [hn, pow_zero, not_tendsto_const_atTop] at h, tendsto_pow_atTop⟩
+
+end LinearOrderedSemiring
+
+theorem not_tendsto_pow_atTop_atBot [LinearOrderedRing α] :
+    ∀ {n : ℕ}, ¬Tendsto (fun x : α => x ^ n) atTop atBot
+  | 0 => by simp [not_tendsto_const_atBot]
+  | n + 1 => (tendsto_pow_atTop n.succ_ne_zero).not_tendsto disjoint_atTop_atBot
+
+end Filter
+
+open Filter
+
+variable {R : Type*} [LinearOrderedSemiring R]
+
+theorem exists_lt_mul_self (a : R) : ∃ x ≥ 0, a < x * x :=
+  let ⟨x, hxa, hx0⟩ :=
+    ((tendsto_mul_self_atTop.eventually (eventually_gt_atTop a)).and (eventually_ge_atTop 0)).exists
+  ⟨x, hx0, hxa⟩
+
+theorem exists_le_mul_self (a : R) : ∃ x ≥ 0, a ≤ x * x :=
+  let ⟨x, hx0, hxa⟩ := exists_lt_mul_self a
+  ⟨x, hx0, hxa.le⟩
diff --git a/Mathlib/Order/Filter/Bases.lean b/Mathlib/Order/Filter/Bases.lean
index dee2d71c9f043..ffd5fe2a6c9e4 100644
--- a/Mathlib/Order/Filter/Bases.lean
+++ b/Mathlib/Order/Filter/Bases.lean
@@ -5,8 +5,7 @@ Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
 import Mathlib.Data.Prod.PProd
 import Mathlib.Data.Set.Countable
-import Mathlib.Order.Filter.Prod
-import Mathlib.Order.Filter.Ker
+import Mathlib.Order.Filter.Basic
 
 /-!
 # Filter bases
@@ -100,7 +99,7 @@ instance FilterBasis.nonempty_sets (B : FilterBasis α) : Nonempty B.sets :=
 /-- If `B` is a filter basis on `α`, and `U` a subset of `α` then we can write `U ∈ B` as
 on paper. -/
 instance {α : Type*} : Membership (Set α) (FilterBasis α) :=
-  ⟨fun U B => U ∈ B.sets⟩
+  ⟨fun B U => U ∈ B.sets⟩
 
 @[simp] theorem FilterBasis.mem_sets {s : Set α} {B : FilterBasis α} : s ∈ B.sets ↔ s ∈ B := Iff.rfl
 
@@ -117,7 +116,7 @@ def Filter.asBasis (f : Filter α) : FilterBasis α :=
   ⟨f.sets, ⟨univ, univ_mem⟩, fun {x y} hx hy => ⟨x ∩ y, inter_mem hx hy, subset_rfl⟩⟩
 
 -- Porting note: was `protected` in Lean 3 but `protected` didn't work; removed
-/-- `is_basis p s` means the image of `s` bounded by `p` is a filter basis. -/
+/-- `IsBasis p s` means the image of `s` bounded by `p` is a filter basis. -/
 structure Filter.IsBasis (p : ι → Prop) (s : ι → Set α) : Prop where
   /-- There exists at least one `i` that satisfies `p`. -/
   nonempty : ∃ i, p i
@@ -543,7 +542,7 @@ theorem hasBasis_iSup {ι : Sort*} {ι' : ι → Type*} {l : ι → Filter α} {
 theorem HasBasis.sup_principal (hl : l.HasBasis p s) (t : Set α) :
     (l ⊔ 𝓟 t).HasBasis p fun i => s i ∪ t :=
   ⟨fun u => by
-    simp only [(hl.sup' (hasBasis_principal t)).mem_iff, PProd.exists, exists_prop, and_true_iff,
+    simp only [(hl.sup' (hasBasis_principal t)).mem_iff, PProd.exists, exists_prop, and_true,
       Unique.exists_iff]⟩
 
 theorem HasBasis.sup_pure (hl : l.HasBasis p s) (x : α) :
@@ -632,10 +631,6 @@ alias ⟨_, _root_.Disjoint.filter_principal⟩ := disjoint_principal_principal
 theorem disjoint_pure_pure {x y : α} : Disjoint (pure x : Filter α) (pure y) ↔ x ≠ y := by
   simp only [← principal_singleton, disjoint_principal_principal, disjoint_singleton]
 
-@[simp]
-theorem compl_diagonal_mem_prod {l₁ l₂ : Filter α} : (diagonal α)ᶜ ∈ l₁ ×ˢ l₂ ↔ Disjoint l₁ l₂ := by
-  simp only [mem_prod_iff, Filter.disjoint_iff, prod_subset_compl_diagonal_iff_disjoint]
-
 -- Porting note: use `∃ i, p i ∧ _` instead of `∃ i (hi : p i), _`.
 theorem HasBasis.disjoint_iff_left (h : l.HasBasis p s) :
     Disjoint l l' ↔ ∃ i, p i ∧ (s i)ᶜ ∈ l' := by
@@ -668,10 +663,10 @@ theorem HasBasis.eq_iInf (h : l.HasBasis (fun _ => True) s) : l = ⨅ i, 𝓟 (s
 theorem hasBasis_iInf_principal {s : ι → Set α} (h : Directed (· ≥ ·) s) [Nonempty ι] :
     (⨅ i, 𝓟 (s i)).HasBasis (fun _ => True) s :=
   ⟨fun t => by
-    simpa only [true_and] using mem_iInf_of_directed (h.mono_comp monotone_principal.dual) t⟩
+    simpa only [true_and] using mem_iInf_of_directed (h.mono_comp _ monotone_principal.dual) t⟩
 
 /-- If `s : ι → Set α` is an indexed family of sets, then finite intersections of `s i` form a basis
-of `⨅ i, 𝓟 (s i)`.  -/
+of `⨅ i, 𝓟 (s i)`. -/
 theorem hasBasis_iInf_principal_finite {ι : Type*} (s : ι → Set α) :
     (⨅ i, 𝓟 (s i)).HasBasis (fun t : Set ι => t.Finite) fun t => ⋂ i ∈ t, s i := by
   refine ⟨fun U => (mem_iInf_finite _).trans ?_⟩
@@ -683,7 +678,7 @@ theorem hasBasis_biInf_principal {s : β → Set α} {S : Set β} (h : DirectedO
   ⟨fun t => by
     refine mem_biInf_of_directed ?_ ne
     rw [directedOn_iff_directed, ← directed_comp] at h ⊢
-    refine h.mono_comp ?_
+    refine h.mono_comp _ ?_
     exact fun _ _ => principal_mono.2⟩
 
 theorem hasBasis_biInf_principal' {ι : Type*} {p : ι → Prop} {s : ι → Set α}
@@ -722,7 +717,7 @@ protected theorem HasBasis.biInter_mem {f : Set α → Set β} (h : HasBasis l p
   h.biInf_mem hf
 
 protected theorem HasBasis.ker (h : HasBasis l p s) : l.ker = ⋂ (i) (_ : p i), s i :=
-  l.ker_def.trans <| h.biInter_mem monotone_id
+  sInter_eq_biInter.trans <| h.biInter_mem monotone_id
 
 variable {ι'' : Type*} [Preorder ι''] (l) (s'' : ι'' → Set α)
 
diff --git a/Mathlib/Order/Filter/Basic.lean b/Mathlib/Order/Filter/Basic.lean
index effd009d22e4c..85fed348f7035 100644
--- a/Mathlib/Order/Filter/Basic.lean
+++ b/Mathlib/Order/Filter/Basic.lean
@@ -4,24 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Jeremy Avigad
 -/
 import Mathlib.Data.Set.Finite
+import Mathlib.Order.Filter.Defs
 
 /-!
 # Theory of filters on sets
 
-## Main definitions
-
-* `Filter` : filters on a set;
-* `Filter.principal` : filter of all sets containing a given set;
-* `Filter.map`, `Filter.comap` : operations on filters;
-* `Filter.Tendsto` : limit with respect to filters;
-* `Filter.Eventually` : `f.eventually p` means `{x | p x} ∈ f`;
-* `Filter.Frequently` : `f.frequently p` means `{x | ¬p x} ∉ f`;
-* `filter_upwards [h₁, ..., hₙ]` :
-  a tactic that takes a list of proofs `hᵢ : sᵢ ∈ f`,
-  and replaces a goal `s ∈ f` with `∀ x, x ∈ s₁ → ... → x ∈ sₙ → x ∈ s`;
-* `Filter.NeBot f` : a utility class stating that `f` is a non-trivial filter.
-
-Filters on a type `X` are sets of sets of `X` satisfying three conditions. They are mostly used to
+A *filter* on a type `α` is a collection of sets of `α` which contains the whole `α`,
+is upwards-closed, and is stable under intersection. They are mostly used to
 abstract two related kinds of ideas:
 * *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions
   at a point or at infinity, etc...
@@ -30,8 +19,10 @@ abstract two related kinds of ideas:
   sense of measure theory. Dually, filters can also express the idea of *things happening often*:
   for arbitrarily large `n`, or at a point in any neighborhood of given a point etc...
 
-In this file, we define the type `Filter X` of filters on `X`, and endow it with a complete lattice
-structure. This structure is lifted from the lattice structure on `Set (Set X)` using the Galois
+## Main definitions
+
+In this file, we endow `Filter α` it with a complete lattice structure.
+This structure is lifted from the lattice structure on `Set (Set X)` using the Galois
 insertion which maps a filter to its elements in one direction, and an arbitrary set of sets to
 the smallest filter containing it in the other direction.
 We also prove `Filter` is a monadic functor, with a push-forward operation
@@ -82,65 +73,26 @@ open scoped symmDiff
 
 universe u v w x y
 
-/-- A filter `F` on a type `α` is a collection of sets of `α` which contains the whole `α`,
-is upwards-closed, and is stable under intersection. We do not forbid this collection to be
-all sets of `α`. -/
-structure Filter (α : Type*) where
-  /-- The set of sets that belong to the filter. -/
-  sets : Set (Set α)
-  /-- The set `Set.univ` belongs to any filter. -/
-  univ_sets : Set.univ ∈ sets
-  /-- If a set belongs to a filter, then its superset belongs to the filter as well. -/
-  sets_of_superset {x y} : x ∈ sets → x ⊆ y → y ∈ sets
-  /-- If two sets belong to a filter, then their intersection belongs to the filter as well. -/
-  inter_sets {x y} : x ∈ sets → y ∈ sets → x ∩ y ∈ sets
-
-/-- If `F` is a filter on `α`, and `U` a subset of `α` then we can write `U ∈ F` as on paper. -/
-instance {α : Type*} : Membership (Set α) (Filter α) :=
-  ⟨fun U F => U ∈ F.sets⟩
-
 namespace Filter
 
 variable {α : Type u} {f g : Filter α} {s t : Set α}
 
-@[simp]
-protected theorem mem_mk {t : Set (Set α)} {h₁ h₂ h₃} : s ∈ mk t h₁ h₂ h₃ ↔ s ∈ t :=
-  Iff.rfl
-
-@[simp]
-protected theorem mem_sets : s ∈ f.sets ↔ s ∈ f :=
-  Iff.rfl
-
 instance inhabitedMem : Inhabited { s : Set α // s ∈ f } :=
   ⟨⟨univ, f.univ_sets⟩⟩
 
-theorem filter_eq : ∀ {f g : Filter α}, f.sets = g.sets → f = g
-  | ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl
-
 theorem filter_eq_iff : f = g ↔ f.sets = g.sets :=
   ⟨congr_arg _, filter_eq⟩
 
-@[ext]
-protected theorem ext (h : ∀ s, s ∈ f ↔ s ∈ g) : f = g := by
-  simpa [filter_eq_iff, Set.ext_iff, Filter.mem_sets]
-
 /-- An extensionality lemma that is useful for filters with good lemmas about `sᶜ ∈ f` (e.g.,
 `Filter.comap`, `Filter.coprod`, `Filter.Coprod`, `Filter.cofinite`). -/
 protected theorem coext (h : ∀ s, sᶜ ∈ f ↔ sᶜ ∈ g) : f = g :=
   Filter.ext <| compl_surjective.forall.2 h
 
-@[simp]
-theorem univ_mem : univ ∈ f :=
-  f.univ_sets
-
-theorem mem_of_superset {x y : Set α} (hx : x ∈ f) (hxy : x ⊆ y) : y ∈ f :=
-  f.sets_of_superset hx hxy
-
 instance : Trans (· ⊇ ·) ((· ∈ ·) : Set α → Filter α → Prop) (· ∈ ·) where
   trans h₁ h₂ := mem_of_superset h₂ h₁
 
-theorem inter_mem {s t : Set α} (hs : s ∈ f) (ht : t ∈ f) : s ∩ t ∈ f :=
-  f.inter_sets hs ht
+instance : Trans Membership.mem (· ⊆ ·) (Membership.mem : Filter α → Set α → Prop) where
+  trans h₁ h₂ := mem_of_superset h₁ h₂
 
 @[simp]
 theorem inter_mem_iff {s t : Set α} : s ∩ t ∈ f ↔ s ∈ f ∧ t ∈ f :=
@@ -150,27 +102,12 @@ theorem inter_mem_iff {s t : Set α} : s ∩ t ∈ f ↔ s ∈ f ∧ t ∈ f :=
 theorem diff_mem {s t : Set α} (hs : s ∈ f) (ht : tᶜ ∈ f) : s \ t ∈ f :=
   inter_mem hs ht
 
-theorem univ_mem' (h : ∀ a, a ∈ s) : s ∈ f :=
-  mem_of_superset univ_mem fun x _ => h x
-
-theorem mp_mem (hs : s ∈ f) (h : { x | x ∈ s → x ∈ t } ∈ f) : t ∈ f :=
-  mem_of_superset (inter_mem hs h) fun _ ⟨h₁, h₂⟩ => h₂ h₁
-
 theorem congr_sets (h : { x | x ∈ s ↔ x ∈ t } ∈ f) : s ∈ f ↔ t ∈ f :=
   ⟨fun hs => mp_mem hs (mem_of_superset h fun _ => Iff.mp), fun hs =>
     mp_mem hs (mem_of_superset h fun _ => Iff.mpr)⟩
 
-/-- Override `sets` field of a filter to provide better definitional equality. -/
-protected def copy (f : Filter α) (S : Set (Set α)) (hmem : ∀ s, s ∈ S ↔ s ∈ f) : Filter α where
-  sets := S
-  univ_sets := (hmem _).2 univ_mem
-  sets_of_superset h hsub := (hmem _).2 <| mem_of_superset ((hmem _).1 h) hsub
-  inter_sets h₁ h₂ := (hmem _).2 <| inter_mem ((hmem _).1 h₁) ((hmem _).1 h₂)
-
 lemma copy_eq {S} (hmem : ∀ s, s ∈ S ↔ s ∈ f) : f.copy S hmem = f := Filter.ext hmem
 
-@[simp] lemma mem_copy {S hmem} : s ∈ f.copy S hmem ↔ s ∈ S := Iff.rfl
-
 @[simp]
 theorem biInter_mem {β : Type v} {s : β → Set α} {is : Set β} (hf : is.Finite) :
     (⋂ i ∈ is, s i) ∈ f ↔ ∀ i ∈ is, s i ∈ f :=
@@ -214,100 +151,17 @@ theorem forall_in_swap {β : Type*} {p : Set α → β → Prop} :
 
 end Filter
 
-namespace Mathlib.Tactic
-
-open Lean Meta Elab Tactic
-
-/--
-`filter_upwards [h₁, ⋯, hₙ]` replaces a goal of the form `s ∈ f` and terms
-`h₁ : t₁ ∈ f, ⋯, hₙ : tₙ ∈ f` with `∀ x, x ∈ t₁ → ⋯ → x ∈ tₙ → x ∈ s`.
-The list is an optional parameter, `[]` being its default value.
-
-`filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ` is a short form for
-`{ filter_upwards [h₁, ⋯, hₙ], intros a₁ a₂ ⋯ aₖ }`.
-
-`filter_upwards [h₁, ⋯, hₙ] using e` is a short form for
-`{ filter_upwards [h1, ⋯, hn], exact e }`.
-
-Combining both shortcuts is done by writing `filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ using e`.
-Note that in this case, the `aᵢ` terms can be used in `e`.
--/
-syntax (name := filterUpwards) "filter_upwards" (" [" term,* "]")?
-  (" with" (ppSpace colGt term:max)*)? (" using " term)? : tactic
-
-elab_rules : tactic
-| `(tactic| filter_upwards $[[$[$args],*]]? $[with $wth*]? $[using $usingArg]?) => do
-  let config : ApplyConfig := {newGoals := ApplyNewGoals.nonDependentOnly}
-  for e in args.getD #[] |>.reverse do
-    let goal ← getMainGoal
-    replaceMainGoal <| ← goal.withContext <| runTermElab do
-      let m ← mkFreshExprMVar none
-      let lem ← Term.elabTermEnsuringType
-        (← ``(Filter.mp_mem $e $(← Term.exprToSyntax m))) (← goal.getType)
-      goal.assign lem
-      return [m.mvarId!]
-  liftMetaTactic fun goal => do
-    goal.apply (← mkConstWithFreshMVarLevels ``Filter.univ_mem') config
-  evalTactic <|← `(tactic| dsimp (config := {zeta := false}) only [Set.mem_setOf_eq])
-  if let some l := wth then
-    evalTactic <|← `(tactic| intro $[$l]*)
-  if let some e := usingArg then
-    evalTactic <|← `(tactic| exact $e)
-
-end Mathlib.Tactic
 
 namespace Filter
 
 variable {α : Type u} {β : Type v} {γ : Type w} {δ : Type*} {ι : Sort x}
 
-section Principal
-
-/-- The principal filter of `s` is the collection of all supersets of `s`. -/
-def principal (s : Set α) : Filter α where
-  sets := { t | s ⊆ t }
-  univ_sets := subset_univ s
-  sets_of_superset hx := Subset.trans hx
-  inter_sets := subset_inter
-
-@[inherit_doc]
-scoped notation "𝓟" => Filter.principal
-
-@[simp] theorem mem_principal {s t : Set α} : s ∈ 𝓟 t ↔ t ⊆ s := Iff.rfl
-
 theorem mem_principal_self (s : Set α) : s ∈ 𝓟 s := Subset.rfl
 
-end Principal
-
-open Filter
-
-section Join
-
-/-- The join of a filter of filters is defined by the relation `s ∈ join f ↔ {t | s ∈ t} ∈ f`. -/
-def join (f : Filter (Filter α)) : Filter α where
-  sets := { s | { t : Filter α | s ∈ t } ∈ f }
-  univ_sets := by simp only [mem_setOf_eq, univ_sets, ← Filter.mem_sets, setOf_true]
-  sets_of_superset hx xy := mem_of_superset hx fun f h => mem_of_superset h xy
-  inter_sets hx hy := mem_of_superset (inter_mem hx hy) fun f ⟨h₁, h₂⟩ => inter_mem h₁ h₂
-
-@[simp]
-theorem mem_join {s : Set α} {f : Filter (Filter α)} : s ∈ join f ↔ { t | s ∈ t } ∈ f :=
-  Iff.rfl
-
-end Join
-
 section Lattice
 
 variable {f g : Filter α} {s t : Set α}
 
-instance : PartialOrder (Filter α) where
-  le f g := ∀ ⦃U : Set α⦄, U ∈ g → U ∈ f
-  le_antisymm a b h₁ h₂ := filter_eq <| Subset.antisymm h₂ h₁
-  le_refl a := Subset.rfl
-  le_trans a b c h₁ h₂ := Subset.trans h₂ h₁
-
-theorem le_def : f ≤ g ↔ ∀ x ∈ g, x ∈ f :=
-  Iff.rfl
-
 protected theorem not_le : ¬f ≤ g ↔ ∃ s ∈ g, s ∉ f := by simp_rw [le_def, not_forall, exists_prop]
 
 /-- `GenerateSets g s`: `s` is in the filter closure of `g`. -/
@@ -375,23 +229,6 @@ def giGenerate (α : Type*) :
   choice s hs := Filter.mkOfClosure s (le_antisymm hs <| le_generate_iff.1 <| le_rfl)
   choice_eq _ _ := mkOfClosure_sets
 
-/-- The infimum of filters is the filter generated by intersections
-  of elements of the two filters. -/
-instance : Inf (Filter α) :=
-  ⟨fun f g : Filter α =>
-    { sets := { s | ∃ a ∈ f, ∃ b ∈ g, s = a ∩ b }
-      univ_sets := ⟨_, univ_mem, _, univ_mem, by simp⟩
-      sets_of_superset := by
-        rintro x y ⟨a, ha, b, hb, rfl⟩ xy
-        refine
-          ⟨a ∪ y, mem_of_superset ha subset_union_left, b ∪ y,
-            mem_of_superset hb subset_union_left, ?_⟩
-        rw [← inter_union_distrib_right, union_eq_self_of_subset_left xy]
-      inter_sets := by
-        rintro x y ⟨a, ha, b, hb, rfl⟩ ⟨c, hc, d, hd, rfl⟩
-        refine ⟨a ∩ c, inter_mem ha hc, b ∩ d, inter_mem hb hd, ?_⟩
-        ac_rfl }⟩
-
 theorem mem_inf_iff {f g : Filter α} {s : Set α} : s ∈ f ⊓ g ↔ ∃ t₁ ∈ f, ∃ t₂ ∈ g, s = t₁ ∩ t₂ :=
   Iff.rfl
 
@@ -414,19 +251,6 @@ theorem mem_inf_iff_superset {f g : Filter α} {s : Set α} :
   ⟨fun ⟨t₁, h₁, t₂, h₂, Eq⟩ => ⟨t₁, h₁, t₂, h₂, Eq ▸ Subset.rfl⟩, fun ⟨_, h₁, _, h₂, sub⟩ =>
     mem_inf_of_inter h₁ h₂ sub⟩
 
-instance : Top (Filter α) :=
-  ⟨{  sets := { s | ∀ x, x ∈ s }
-      univ_sets := fun x => mem_univ x
-      sets_of_superset := fun hx hxy a => hxy (hx a)
-      inter_sets := fun hx hy _ => mem_inter (hx _) (hy _) }⟩
-
-theorem mem_top_iff_forall {s : Set α} : s ∈ (⊤ : Filter α) ↔ ∀ x, x ∈ s :=
-  Iff.rfl
-
-@[simp]
-theorem mem_top {s : Set α} : s ∈ (⊤ : Filter α) ↔ s = univ := by
-  rw [mem_top_iff_forall, eq_univ_iff_forall]
-
 section CompleteLattice
 
 /- We lift the complete lattice along the Galois connection `generate` / `sets`. Unfortunately,
@@ -449,16 +273,6 @@ instance : Inhabited (Filter α) := ⟨⊥⟩
 
 end CompleteLattice
 
-/-- A filter is `NeBot` if it is not equal to `⊥`, or equivalently the empty set does not belong to
-the filter. Bourbaki include this assumption in the definition of a filter but we prefer to have a
-`CompleteLattice` structure on `Filter _`, so we use a typeclass argument in lemmas instead. -/
-class NeBot (f : Filter α) : Prop where
-  /-- The filter is nontrivial: `f ≠ ⊥` or equivalently, `∅ ∉ f`. -/
-  ne' : f ≠ ⊥
-
-theorem neBot_iff {f : Filter α} : NeBot f ↔ f ≠ ⊥ :=
-  ⟨fun h => h.1, fun h => ⟨h⟩⟩
-
 theorem NeBot.ne {f : Filter α} (hf : NeBot f) : f ≠ ⊥ := hf.ne'
 
 @[simp] theorem not_neBot {f : Filter α} : ¬f.NeBot ↔ f = ⊥ := neBot_iff.not_left
@@ -503,10 +317,6 @@ theorem generate_iUnion {s : ι → Set (Set α)} :
     Filter.generate (⋃ i, s i) = ⨅ i, Filter.generate (s i) :=
   (giGenerate α).gc.l_iSup
 
-@[simp]
-theorem mem_bot {s : Set α} : s ∈ (⊥ : Filter α) :=
-  trivial
-
 @[simp]
 theorem mem_sup {f g : Filter α} {s : Set α} : s ∈ f ⊔ g ↔ s ∈ f ∧ s ∈ g :=
   Iff.rfl
@@ -520,7 +330,7 @@ theorem mem_sSup {x : Set α} {s : Set (Filter α)} : x ∈ sSup s ↔ ∀ f ∈
 
 @[simp]
 theorem mem_iSup {x : Set α} {f : ι → Filter α} : x ∈ iSup f ↔ ∀ i, x ∈ f i := by
-  simp only [← Filter.mem_sets, iSup_sets_eq, iff_self_iff, mem_iInter]
+  simp only [← Filter.mem_sets, iSup_sets_eq, mem_iInter]
 
 @[simp]
 theorem iSup_neBot {f : ι → Filter α} : (⨆ i, f i).NeBot ↔ ∃ i, (f i).NeBot := by
@@ -533,20 +343,21 @@ theorem mem_iInf_of_mem {f : ι → Filter α} (i : ι) {s} (hs : s ∈ f i) : s
   iInf_le f i hs
 
 theorem mem_iInf_of_iInter {ι} {s : ι → Filter α} {U : Set α} {I : Set ι} (I_fin : I.Finite)
-    {V : I → Set α} (hV : ∀ i, V i ∈ s i) (hU : ⋂ i, V i ⊆ U) : U ∈ ⨅ i, s i := by
+    {V : I → Set α} (hV : ∀ (i : I), V i ∈ s i) (hU : ⋂ i, V i ⊆ U) : U ∈ ⨅ i, s i := by
   haveI := I_fin.fintype
   refine mem_of_superset (iInter_mem.2 fun i => ?_) hU
   exact mem_iInf_of_mem (i : ι) (hV _)
 
 theorem mem_iInf {ι} {s : ι → Filter α} {U : Set α} :
-    (U ∈ ⨅ i, s i) ↔ ∃ I : Set ι, I.Finite ∧ ∃ V : I → Set α, (∀ i, V i ∈ s i) ∧ U = ⋂ i, V i := by
+    (U ∈ ⨅ i, s i) ↔
+      ∃ I : Set ι, I.Finite ∧ ∃ V : I → Set α, (∀ (i : I), V i ∈ s i) ∧ U = ⋂ i, V i := by
   constructor
   · rw [iInf_eq_generate, mem_generate_iff]
     rintro ⟨t, tsub, tfin, tinter⟩
     rcases eq_finite_iUnion_of_finite_subset_iUnion tfin tsub with ⟨I, Ifin, σ, σfin, σsub, rfl⟩
     rw [sInter_iUnion] at tinter
     set V := fun i => U ∪ ⋂₀ σ i with hV
-    have V_in : ∀ i, V i ∈ s i := by
+    have V_in : ∀ (i : I), V i ∈ s i := by
       rintro i
       have : ⋂₀ σ i ∈ s i := by
         rw [sInter_mem (σfin _)]
@@ -568,10 +379,10 @@ theorem mem_iInf' {ι} {s : ι → Filter α} {U : Set α} :
   refine ⟨I, If, fun i => if hi : i ∈ I then V ⟨i, hi⟩ else univ, fun i => ?_, fun i hi => ?_, ?_⟩
   · dsimp only
     split_ifs
-    exacts [hV _, univ_mem]
+    exacts [hV ⟨i,_⟩, univ_mem]
   · exact dif_neg hi
   · simp only [iInter_dite, biInter_eq_iInter, dif_pos (Subtype.coe_prop _), Subtype.coe_eta,
-      iInter_univ, inter_univ, eq_self_iff_true, true_and_iff]
+      iInter_univ, inter_univ, eq_self_iff_true, true_and]
 
 theorem exists_iInter_of_mem_iInf {ι : Type*} {α : Type*} {f : ι → Filter α} {s}
     (hs : s ∈ ⨅ i, f i) : ∃ t : ι → Set α, (∀ i, t i ∈ f i) ∧ s = ⋂ i, t i :=
@@ -591,7 +402,7 @@ theorem Iic_principal (s : Set α) : Iic (𝓟 s) = { l | s ∈ l } :=
   Set.ext fun _ => le_principal_iff
 
 theorem principal_mono {s t : Set α} : 𝓟 s ≤ 𝓟 t ↔ s ⊆ t := by
-  simp only [le_principal_iff, iff_self_iff, mem_principal]
+  simp only [le_principal_iff, mem_principal]
 
 @[gcongr] alias ⟨_, _root_.GCongr.filter_principal_mono⟩ := principal_mono
 
@@ -707,7 +518,7 @@ theorem eq_sInf_of_mem_iff_exists_mem {S : Set (Filter α)} {l : Filter α}
 
 theorem eq_iInf_of_mem_iff_exists_mem {f : ι → Filter α} {l : Filter α}
     (h : ∀ {s}, s ∈ l ↔ ∃ i, s ∈ f i) : l = iInf f :=
-  eq_sInf_of_mem_iff_exists_mem <| h.trans exists_range_iff.symm
+  eq_sInf_of_mem_iff_exists_mem <| h.trans (exists_range_iff (p := (_ ∈ ·))).symm
 
 theorem eq_biInf_of_mem_iff_exists_mem {f : ι → Filter α} {p : ι → Prop} {l : Filter α}
     (h : ∀ {s}, s ∈ l ↔ ∃ i, p i ∧ s ∈ f i) : l = ⨅ (i) (_ : p i), f i := by
@@ -781,7 +592,8 @@ instance : DistribLattice (Filter α) :=
           x.sets_of_superset hs inter_subset_right, ht₂, rfl⟩ }
 
 /-- The dual version does not hold! `Filter α` is not a `CompleteDistribLattice`. -/
-def coframeMinimalAxioms : Coframe.MinimalAxioms (Filter α) :=
+-- See note [reducible non-instances]
+abbrev coframeMinimalAxioms : Coframe.MinimalAxioms (Filter α) :=
   { Filter.instCompleteLatticeFilter with
     iInf_sup_le_sup_sInf := fun f s t ⟨h₁, h₂⟩ => by
       classical
@@ -945,14 +757,6 @@ theorem join_mono {f₁ f₂ : Filter (Filter α)} (h : f₁ ≤ f₂) : join f
 
 /-! ### Eventually -/
 
-/-- `f.Eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in atTop, p x`
-means that `p` holds true for sufficiently large `x`. -/
-protected def Eventually (p : α → Prop) (f : Filter α) : Prop :=
-  { x | p x } ∈ f
-
-@[inherit_doc Filter.Eventually]
-notation3 "∀ᶠ "(...)" in "f", "r:(scoped p => Filter.Eventually p f) => r
-
 theorem eventually_iff {f : Filter α} {P : α → Prop} : (∀ᶠ x in f, P x) ↔ { x | P x } ∈ f :=
   Iff.rfl
 
@@ -1101,14 +905,6 @@ theorem eventually_inf_principal {f : Filter α} {p : α → Prop} {s : Set α}
 
 /-! ### Frequently -/
 
-/-- `f.Frequently p` or `∃ᶠ x in f, p x` mean that `{x | ¬p x} ∉ f`. E.g., `∃ᶠ x in atTop, p x`
-means that there exist arbitrarily large `x` for which `p` holds true. -/
-protected def Frequently (p : α → Prop) (f : Filter α) : Prop :=
-  ¬∀ᶠ x in f, ¬p x
-
-@[inherit_doc Filter.Frequently]
-notation3 "∃ᶠ "(...)" in "f", "r:(scoped p => Filter.Frequently p f) => r
-
 theorem Eventually.frequently {f : Filter α} [NeBot f] {p : α → Prop} (h : ∀ᶠ x in f, p x) :
     ∃ᶠ x in f, p x :=
   compl_not_mem h
@@ -1261,17 +1057,12 @@ theorem Eventually.choice {r : α → β → Prop} {l : Filter α} [l.NeBot] (h
 ### Relation “eventually equal”
 -/
 
-/-- Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
-`f x = g x` belongs to `l`. -/
-def EventuallyEq (l : Filter α) (f g : α → β) : Prop :=
-  ∀ᶠ x in l, f x = g x
+section EventuallyEq
+variable {l : Filter α} {f g : α → β}
 
-@[inherit_doc]
-notation:50 f " =ᶠ[" l:50 "] " g:50 => EventuallyEq l f g
+theorem EventuallyEq.eventually (h : f =ᶠ[l] g) : ∀ᶠ x in l, f x = g x := h
 
-theorem EventuallyEq.eventually {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) :
-    ∀ᶠ x in l, f x = g x :=
-  h
+@[simp] lemma eventuallyEq_top : f =ᶠ[⊤] g ↔ f = g := by simp [EventuallyEq, funext_iff]
 
 theorem EventuallyEq.rw {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop)
     (hf : ∀ᶠ x in l, p x (f x)) : ∀ᶠ x in l, p x (g x) :=
@@ -1309,6 +1100,9 @@ theorem EventuallyEq.refl (l : Filter α) (f : α → β) : f =ᶠ[l] f :=
 protected theorem EventuallyEq.rfl {l : Filter α} {f : α → β} : f =ᶠ[l] f :=
   EventuallyEq.refl l f
 
+theorem EventuallyEq.of_eq {l : Filter α} {f g : α → β} (h : f = g) : f =ᶠ[l] g := h ▸ .rfl
+alias _root_.Eq.eventuallyEq := EventuallyEq.of_eq
+
 @[symm]
 theorem EventuallyEq.symm {f g : α → β} {l : Filter α} (H : f =ᶠ[l] g) : g =ᶠ[l] f :=
   H.mono fun _ => Eq.symm
@@ -1438,13 +1232,6 @@ section LE
 
 variable [LE β] {l : Filter α}
 
-/-- A function `f` is eventually less than or equal to a function `g` at a filter `l`. -/
-def EventuallyLE (l : Filter α) (f g : α → β) : Prop :=
-  ∀ᶠ x in l, f x ≤ g x
-
-@[inherit_doc]
-notation:50 f " ≤ᶠ[" l:50 "] " g:50 => EventuallyLE l f g
-
 theorem EventuallyLE.congr {f f' g g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
     f' ≤ᶠ[l] g' :=
   H.mp <| hg.mp <| hf.mono fun x hf hg H => by rwa [hf, hg] at H
@@ -1612,7 +1399,7 @@ theorem set_eventuallyLE_iff_mem_inf_principal {s t : Set α} {l : Filter α} :
 theorem set_eventuallyLE_iff_inf_principal_le {s t : Set α} {l : Filter α} :
     s ≤ᶠ[l] t ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t :=
   set_eventuallyLE_iff_mem_inf_principal.trans <| by
-    simp only [le_inf_iff, inf_le_left, true_and_iff, le_principal_iff]
+    simp only [le_inf_iff, inf_le_left, true_and, le_principal_iff]
 
 theorem set_eventuallyEq_iff_inf_principal {s t : Set α} {l : Filter α} :
     s =ᶠ[l] t ↔ l ⊓ 𝓟 s = l ⊓ 𝓟 t := by
@@ -1637,17 +1424,12 @@ theorem EventuallyLE.le_sup_of_le_right [SemilatticeSup β] {l : Filter α} {f g
 theorem join_le {f : Filter (Filter α)} {l : Filter α} (h : ∀ᶠ m in f, m ≤ l) : join f ≤ l :=
   fun _ hs => h.mono fun _ hm => hm hs
 
+end EventuallyEq
+
 /-! ### Push-forwards, pull-backs, and the monad structure -/
 
 section Map
 
-/-- The forward map of a filter -/
-def map (m : α → β) (f : Filter α) : Filter β where
-  sets := preimage m ⁻¹' f.sets
-  univ_sets := univ_mem
-  sets_of_superset hs st := mem_of_superset hs <| preimage_mono st
-  inter_sets hs ht := inter_mem hs ht
-
 @[simp]
 theorem map_principal {s : Set α} {f : α → β} : map f (𝓟 s) = 𝓟 (Set.image f s) :=
   Filter.ext fun _ => image_subset_iff.symm
@@ -1712,20 +1494,6 @@ end Map
 
 section Comap
 
-/-- The inverse map of a filter. A set `s` belongs to `Filter.comap m f` if either of the following
-equivalent conditions hold.
-
-1. There exists a set `t ∈ f` such that `m ⁻¹' t ⊆ s`. This is used as a definition.
-2. The set `kernImage m s = {y | ∀ x, m x = y → x ∈ s}` belongs to `f`, see `Filter.mem_comap'`.
-3. The set `(m '' sᶜ)ᶜ` belongs to `f`, see `Filter.mem_comap_iff_compl` and
-`Filter.compl_mem_comap`. -/
-def comap (m : α → β) (f : Filter β) : Filter α where
-  sets := { s | ∃ t ∈ f, m ⁻¹' t ⊆ s }
-  univ_sets := ⟨univ, univ_mem, by simp only [subset_univ, preimage_univ]⟩
-  sets_of_superset := fun ⟨a', ha', ma'a⟩ ab => ⟨a', ha', ma'a.trans ab⟩
-  inter_sets := fun ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩ =>
-    ⟨a' ∩ b', inter_mem ha₁ hb₁, inter_subset_inter ha₂ hb₂⟩
-
 variable {f : α → β} {l : Filter β} {p : α → Prop} {s : Set α}
 
 theorem mem_comap' : s ∈ comap f l ↔ { y | ∀ ⦃x⦄, f x = y → x ∈ s } ∈ l :=
@@ -1796,37 +1564,6 @@ theorem compl_mem_kernMap {s : Set β} : sᶜ ∈ kernMap m f ↔ ∃ t, tᶜ 
 
 end KernMap
 
-/-- The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
-
-Unfortunately, this `bind` does not result in the expected applicative. See `Filter.seq` for the
-applicative instance. -/
-def bind (f : Filter α) (m : α → Filter β) : Filter β :=
-  join (map m f)
-
-/-- The applicative sequentiation operation. This is not induced by the bind operation. -/
-def seq (f : Filter (α → β)) (g : Filter α) : Filter β where
-  sets := { s | ∃ u ∈ f, ∃ t ∈ g, ∀ m ∈ u, ∀ x ∈ t, (m : α → β) x ∈ s }
-  univ_sets := ⟨univ, univ_mem, univ, univ_mem, fun _ _ _ _ => trivial⟩
-  sets_of_superset := fun ⟨t₀, t₁, h₀, h₁, h⟩ hst =>
-    ⟨t₀, t₁, h₀, h₁, fun _ hx _ hy => hst <| h _ hx _ hy⟩
-  inter_sets := fun ⟨t₀, ht₀, t₁, ht₁, ht⟩ ⟨u₀, hu₀, u₁, hu₁, hu⟩ =>
-    ⟨t₀ ∩ u₀, inter_mem ht₀ hu₀, t₁ ∩ u₁, inter_mem ht₁ hu₁, fun _ ⟨hx₀, hx₁⟩ _ ⟨hy₀, hy₁⟩ =>
-      ⟨ht _ hx₀ _ hy₀, hu _ hx₁ _ hy₁⟩⟩
-
-/-- `pure x` is the set of sets that contain `x`. It is equal to `𝓟 {x}` but
-with this definition we have `s ∈ pure a` defeq `a ∈ s`. -/
-instance : Pure Filter :=
-  ⟨fun x =>
-    { sets := { s | x ∈ s }
-      inter_sets := And.intro
-      sets_of_superset := fun hs hst => hst hs
-      univ_sets := trivial }⟩
-
-instance : Bind Filter :=
-  ⟨@Filter.bind⟩
-
-instance : Functor Filter where map := @Filter.map
-
 instance : LawfulFunctor (Filter : Type u → Type u) where
   id_map _ := map_id
   comp_map _ _ _ := map_map.symm
@@ -2443,7 +2180,7 @@ theorem mem_seq_def {f : Filter (α → β)} {g : Filter α} {s : Set β} :
 
 theorem mem_seq_iff {f : Filter (α → β)} {g : Filter α} {s : Set β} :
     s ∈ f.seq g ↔ ∃ u ∈ f, ∃ t ∈ g, Set.seq u t ⊆ s := by
-  simp only [mem_seq_def, seq_subset, exists_prop, iff_self_iff]
+  simp only [mem_seq_def, seq_subset, exists_prop]
 
 theorem mem_map_seq_iff {f : Filter α} {g : Filter β} {m : α → β → γ} {s : Set γ} :
     s ∈ (f.map m).seq g ↔ ∃ t u, t ∈ g ∧ u ∈ f ∧ ∀ x ∈ u, ∀ y ∈ t, m x y ∈ s :=
@@ -2583,12 +2320,6 @@ end Bind
 
 /-! ### Limits -/
 
-/-- `Filter.Tendsto` is the generic "limit of a function" predicate.
-  `Tendsto f l₁ l₂` asserts that for every `l₂` neighborhood `a`,
-  the `f`-preimage of `a` is an `l₁` neighborhood. -/
-def Tendsto (f : α → β) (l₁ : Filter α) (l₂ : Filter β) :=
-  l₁.map f ≤ l₂
-
 theorem tendsto_def {f : α → β} {l₁ : Filter α} {l₂ : Filter β} :
     Tendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f ⁻¹' s ∈ l₁ :=
   Iff.rfl
@@ -2625,6 +2356,12 @@ theorem Tendsto.frequently_map {l₁ : Filter α} {l₂ : Filter β} {p : α →
 @[simp]
 theorem tendsto_bot {f : α → β} {l : Filter β} : Tendsto f ⊥ l := by simp [Tendsto]
 
+theorem Tendsto.of_neBot_imp {f : α → β} {la : Filter α} {lb : Filter β}
+    (h : NeBot la → Tendsto f la lb) : Tendsto f la lb := by
+  rcases eq_or_neBot la with rfl | hla
+  · exact tendsto_bot
+  · exact h hla
+
 @[simp] theorem tendsto_top {f : α → β} {l : Filter α} : Tendsto f l ⊤ := le_top
 
 theorem le_map_of_right_inverse {mab : α → β} {mba : β → α} {f : Filter α} {g : Filter β}
@@ -2733,7 +2470,7 @@ theorem map_eq_of_inverse {f : Filter α} {g : Filter β} {φ : α → β} (ψ :
 
 theorem tendsto_inf {f : α → β} {x : Filter α} {y₁ y₂ : Filter β} :
     Tendsto f x (y₁ ⊓ y₂) ↔ Tendsto f x y₁ ∧ Tendsto f x y₂ := by
-  simp only [Tendsto, le_inf_iff, iff_self_iff]
+  simp only [Tendsto, le_inf_iff]
 
 theorem tendsto_inf_left {f : α → β} {x₁ x₂ : Filter α} {y : Filter β} (h : Tendsto f x₁ y) :
     Tendsto f (x₁ ⊓ x₂) y :=
@@ -2750,7 +2487,7 @@ theorem Tendsto.inf {f : α → β} {x₁ x₂ : Filter α} {y₁ y₂ : Filter
 @[simp]
 theorem tendsto_iInf {f : α → β} {x : Filter α} {y : ι → Filter β} :
     Tendsto f x (⨅ i, y i) ↔ ∀ i, Tendsto f x (y i) := by
-  simp only [Tendsto, iff_self_iff, le_iInf_iff]
+  simp only [Tendsto, le_iInf_iff]
 
 theorem tendsto_iInf' {f : α → β} {x : ι → Filter α} {y : Filter β} (i : ι)
     (hi : Tendsto f (x i) y) : Tendsto f (⨅ i, x i) y :=
@@ -2902,23 +2639,10 @@ alias ⟨_, Set.InjOn.filter_map_Iic⟩ := Filter.filter_injOn_Iic_iff_injOn
 
 namespace Filter
 
-/-- Construct a filter from a property that is stable under finite unions.
-A set `s` belongs to `Filter.comk p _ _ _` iff its complement satisfies the predicate `p`.
-This constructor is useful to define filters like `Filter.cofinite`. -/
-def comk (p : Set α → Prop) (he : p ∅) (hmono : ∀ t, p t → ∀ s ⊆ t, p s)
-    (hunion : ∀ s, p s → ∀ t, p t → p (s ∪ t)) : Filter α where
-  sets := {t | p tᶜ}
-  univ_sets := by simpa
-  sets_of_superset := fun ht₁ ht => hmono _ ht₁ _ (compl_subset_compl.2 ht)
-  inter_sets := fun ht₁ ht₂ => by simp [compl_inter, hunion _ ht₁ _ ht₂]
-
-@[simp]
-lemma mem_comk {p : Set α → Prop} {he hmono hunion s} :
-    s ∈ comk p he hmono hunion ↔ p sᶜ :=
-  .rfl
-
 lemma compl_mem_comk {p : Set α → Prop} {he hmono hunion s} :
     sᶜ ∈ comk p he hmono hunion ↔ p s := by
   simp
 
 end Filter
+
+set_option linter.style.longFile 3000
diff --git a/Mathlib/Order/Filter/CardinalInter.lean b/Mathlib/Order/Filter/CardinalInter.lean
index 7462de92c6dd8..df6c37ffccbb7 100644
--- a/Mathlib/Order/Filter/CardinalInter.lean
+++ b/Mathlib/Order/Filter/CardinalInter.lean
@@ -300,16 +300,19 @@ theorem mem_cardinaleGenerate_iff {s : Set α} {hreg : c.IsRegular} :
     s ∈ cardinalGenerate g (IsRegular.nat_lt hreg 2) ↔
     ∃ S : Set (Set α), S ⊆ g ∧ (#S < c) ∧ ⋂₀ S ⊆ s := by
   constructor <;> intro h
-  · induction' h with s hs s t _ st ih S Sct _ ih
-    · refine ⟨{s}, singleton_subset_iff.mpr hs, ?_⟩
+  · induction h with
+    | @basic s hs =>
+      refine ⟨{s}, singleton_subset_iff.mpr hs, ?_⟩
       simpa [subset_refl] using IsRegular.nat_lt hreg 1
-    · exact ⟨∅, ⟨empty_subset g, mk_eq_zero (∅ : Set <| Set α) ▸ IsRegular.nat_lt hreg 0, by simp⟩⟩
-    · exact Exists.imp (by tauto) ih
-    choose T Tg Tct hT using ih
-    refine ⟨⋃ (s) (H : s ∈ S), T s H, by simpa,
-      (Cardinal.card_biUnion_lt_iff_forall_of_isRegular hreg Sct).2 Tct, ?_⟩
-    apply subset_sInter
-    apply fun s H => subset_trans (sInter_subset_sInter (subset_iUnion₂ s H)) (hT s H)
+    | univ =>
+      exact ⟨∅, ⟨empty_subset g, mk_eq_zero (∅ : Set <| Set α) ▸ IsRegular.nat_lt hreg 0, by simp⟩⟩
+    | superset _ _ ih => exact Exists.imp (by tauto) ih
+    | @sInter S Sct _ ih =>
+      choose T Tg Tct hT using ih
+      refine ⟨⋃ (s) (H : s ∈ S), T s H, by simpa,
+        (Cardinal.card_biUnion_lt_iff_forall_of_isRegular hreg Sct).2 Tct, ?_⟩
+      apply subset_sInter
+      apply fun s H => subset_trans (sInter_subset_sInter (subset_iUnion₂ s H)) (hT s H)
   rcases h with ⟨S, Sg, Sct, hS⟩
   have : CardinalInterFilter (cardinalGenerate g (IsRegular.nat_lt hreg 2)) c :=
     cardinalInter_ofCardinalGenerate _ _
diff --git a/Mathlib/Order/Filter/Cofinite.lean b/Mathlib/Order/Filter/Cofinite.lean
index 03cc4bdf98814..94c269c447ffa 100644
--- a/Mathlib/Order/Filter/Cofinite.lean
+++ b/Mathlib/Order/Filter/Cofinite.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov
 -/
 import Mathlib.Order.Filter.AtTopBot
+import Mathlib.Order.Filter.Ker
 import Mathlib.Order.Filter.Pi
 
 /-!
diff --git a/Mathlib/Order/Filter/CountableInter.lean b/Mathlib/Order/Filter/CountableInter.lean
index ea7172ae2ea14..b9d5d1ef60f8d 100644
--- a/Mathlib/Order/Filter/CountableInter.lean
+++ b/Mathlib/Order/Filter/CountableInter.lean
@@ -242,16 +242,16 @@ it contains a countable intersection of elements of `g`. -/
 theorem mem_countableGenerate_iff {s : Set α} :
     s ∈ countableGenerate g ↔ ∃ S : Set (Set α), S ⊆ g ∧ S.Countable ∧ ⋂₀ S ⊆ s := by
   constructor <;> intro h
-  · induction' h with s hs s t _ st ih S Sct _ ih
-    · exact ⟨{s}, by simp [hs, subset_refl]⟩
-    · exact ⟨∅, by simp⟩
-    · refine Exists.imp (fun S => ?_) ih
-      tauto
-    choose T Tg Tct hT using ih
-    refine ⟨⋃ (s) (H : s ∈ S), T s H, by simpa, Sct.biUnion Tct, ?_⟩
-    apply subset_sInter
-    intro s H
-    exact subset_trans (sInter_subset_sInter (subset_iUnion₂ s H)) (hT s H)
+  · induction h with
+    | @basic s hs => exact ⟨{s}, by simp [hs, subset_refl]⟩
+    | univ => exact ⟨∅, by simp⟩
+    | superset _ _ ih => refine Exists.imp (fun S => ?_) ih; tauto
+    | @sInter S Sct _ ih =>
+      choose T Tg Tct hT using ih
+      refine ⟨⋃ (s) (H : s ∈ S), T s H, by simpa, Sct.biUnion Tct, ?_⟩
+      apply subset_sInter
+      intro s H
+      exact subset_trans (sInter_subset_sInter (subset_iUnion₂ s H)) (hT s H)
   rcases h with ⟨S, Sg, Sct, hS⟩
   refine mem_of_superset ((countable_sInter_mem Sct).mpr ?_) hS
   intro s H
@@ -262,11 +262,11 @@ theorem le_countableGenerate_iff_of_countableInterFilter {f : Filter α} [Counta
   constructor <;> intro h
   · exact subset_trans (fun s => CountableGenerateSets.basic) h
   intro s hs
-  induction' hs with s hs s t _ st ih S Sct _ ih
-  · exact h hs
-  · exact univ_mem
-  · exact mem_of_superset ih st
-  exact (countable_sInter_mem Sct).mpr ih
+  induction hs with
+  | basic hs => exact h hs
+  | univ => exact univ_mem
+  | superset _ st ih => exact mem_of_superset ih st
+  | sInter Sct _ ih => exact (countable_sInter_mem Sct).mpr ih
 
 variable (g)
 
diff --git a/Mathlib/Order/Filter/Curry.lean b/Mathlib/Order/Filter/Curry.lean
index 561f48bd033b3..7f91d3cc76ef9 100644
--- a/Mathlib/Order/Filter/Curry.lean
+++ b/Mathlib/Order/Filter/Curry.lean
@@ -49,13 +49,6 @@ namespace Filter
 
 variable {α β γ : Type*}
 
-/-- This filter is characterized by `Filter.eventually_curry_iff`:
-`(∀ᶠ (x : α × β) in f.curry g, p x) ↔ ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in g, p (x, y)`. Useful
-in adding quantifiers to the middle of `Tendsto`s. See
-`hasFDerivAt_of_tendstoUniformlyOnFilter`. -/
-def curry (f : Filter α) (g : Filter β) : Filter (α × β) :=
-  bind f fun a ↦ map (a, ·) g
-
 theorem eventually_curry_iff {f : Filter α} {g : Filter β} {p : α × β → Prop} :
     (∀ᶠ x : α × β in f.curry g, p x) ↔ ∀ᶠ x : α in f, ∀ᶠ y : β in g, p (x, y) :=
   Iff.rfl
diff --git a/Mathlib/Order/Filter/Defs.lean b/Mathlib/Order/Filter/Defs.lean
new file mode 100644
index 0000000000000..cd23371bb3f2f
--- /dev/null
+++ b/Mathlib/Order/Filter/Defs.lean
@@ -0,0 +1,378 @@
+/-
+Copyright (c) 2017 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl, Jeremy Avigad
+-/
+import Mathlib.Data.Set.Basic
+import Mathlib.Order.SetNotation
+
+/-!
+# Definitions about filters
+
+A *filter* on a type `α` is a collection of sets of `α` which contains the whole `α`,
+is upwards-closed, and is stable under intersection. Filters are mostly used to
+abstract two related kinds of ideas:
+* *limits*, including finite or infinite limits of sequences, finite or infinite limits of functions
+  at a point or at infinity, etc...
+* *things happening eventually*, including things happening for large enough `n : ℕ`, or near enough
+  a point `x`, or for close enough pairs of points, or things happening almost everywhere in the
+  sense of measure theory. Dually, filters can also express the idea of *things happening often*:
+  for arbitrarily large `n`, or at a point in any neighborhood of given a point etc...
+
+## Main definitions
+
+* `Filter` : filters on a set;
+* `Filter.principal`, `𝓟 s` : filter of all sets containing a given set;
+* `Filter.map`, `Filter.comap` : operations on filters;
+* `Filter.Tendsto` : limit with respect to filters;
+* `Filter.Eventually` : `f.Eventually p` means `{x | p x} ∈ f`;
+* `Filter.Frequently` : `f.Frequently p` means `{x | ¬p x} ∉ f`;
+* `filter_upwards [h₁, ..., hₙ]` :
+  a tactic that takes a list of proofs `hᵢ : sᵢ ∈ f`,
+  and replaces a goal `s ∈ f` with `∀ x, x ∈ s₁ → ... → x ∈ sₙ → x ∈ s`;
+* `Filter.NeBot f` : a utility class stating that `f` is a non-trivial filter.
+
+## Notations
+
+* `∀ᶠ x in f, p x` : `f.Eventually p`;
+* `∃ᶠ x in f, p x` : `f.Frequently p`;
+* `f =ᶠ[l] g` : `∀ᶠ x in l, f x = g x`;
+* `f ≤ᶠ[l] g` : `∀ᶠ x in l, f x ≤ g x`;
+* `𝓟 s` : `Filter.Principal s`, localized in `Filter`.
+
+## Implementation Notes
+
+Important note: Bourbaki requires that a filter on `X` cannot contain all sets of `X`,
+which we do *not* require.
+This gives `Filter X` better formal properties,
+in particular a bottom element `⊥` for its lattice structure,
+at the cost of including the assumption `[NeBot f]` in a number of lemmas and definitions.
+
+## References
+
+*  [N. Bourbaki, *General Topology*][bourbaki1966]
+-/
+
+open Set
+
+/-- A filter `F` on a type `α` is a collection of sets of `α` which contains the whole `α`,
+is upwards-closed, and is stable under intersection. We do not forbid this collection to be
+all sets of `α`. -/
+structure Filter (α : Type*) where
+  /-- The set of sets that belong to the filter. -/
+  sets : Set (Set α)
+  /-- The set `Set.univ` belongs to any filter. -/
+  univ_sets : Set.univ ∈ sets
+  /-- If a set belongs to a filter, then its superset belongs to the filter as well. -/
+  sets_of_superset {x y} : x ∈ sets → x ⊆ y → y ∈ sets
+  /-- If two sets belong to a filter, then their intersection belongs to the filter as well. -/
+  inter_sets {x y} : x ∈ sets → y ∈ sets → x ∩ y ∈ sets
+
+namespace Filter
+
+variable {α β : Type*} {f g : Filter α} {s t : Set α}
+
+theorem filter_eq : ∀ {f g : Filter α}, f.sets = g.sets → f = g
+  | ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl
+
+/-- If `F` is a filter on `α`, and `U` a subset of `α` then we can write `U ∈ F` as on paper. -/
+instance instMembership : Membership (Set α) (Filter α) := ⟨fun F U => U ∈ F.sets⟩
+
+@[ext]
+protected theorem ext (h : ∀ s, s ∈ f ↔ s ∈ g) : f = g := filter_eq <| Set.ext h
+
+@[simp]
+protected theorem mem_mk {t : Set (Set α)} {h₁ h₂ h₃} : s ∈ mk t h₁ h₂ h₃ ↔ s ∈ t :=
+  Iff.rfl
+
+@[simp]
+protected theorem mem_sets : s ∈ f.sets ↔ s ∈ f :=
+  Iff.rfl
+
+@[simp]
+theorem univ_mem : univ ∈ f :=
+  f.univ_sets
+
+theorem mem_of_superset {x y : Set α} (hx : x ∈ f) (hxy : x ⊆ y) : y ∈ f :=
+  f.sets_of_superset hx hxy
+
+theorem univ_mem' (h : ∀ a, a ∈ s) : s ∈ f :=
+  mem_of_superset univ_mem fun x _ => h x
+
+theorem inter_mem (hs : s ∈ f) (ht : t ∈ f) : s ∩ t ∈ f :=
+  f.inter_sets hs ht
+
+theorem mp_mem (hs : s ∈ f) (h : { x | x ∈ s → x ∈ t } ∈ f) : t ∈ f :=
+  mem_of_superset (inter_mem hs h) fun _ ⟨h₁, h₂⟩ => h₂ h₁
+
+/-- Override `sets` field of a filter to provide better definitional equality. -/
+protected def copy (f : Filter α) (S : Set (Set α)) (hmem : ∀ s, s ∈ S ↔ s ∈ f) : Filter α where
+  sets := S
+  univ_sets := (hmem _).2 univ_mem
+  sets_of_superset h hsub := (hmem _).2 <| mem_of_superset ((hmem _).1 h) hsub
+  inter_sets h₁ h₂ := (hmem _).2 <| inter_mem ((hmem _).1 h₁) ((hmem _).1 h₂)
+
+@[simp] theorem mem_copy {S hmem} : s ∈ f.copy S hmem ↔ s ∈ S := Iff.rfl
+
+/-- Construct a filter from a property that is stable under finite unions.
+A set `s` belongs to `Filter.comk p _ _ _` iff its complement satisfies the predicate `p`.
+This constructor is useful to define filters like `Filter.cofinite`. -/
+def comk (p : Set α → Prop) (he : p ∅) (hmono : ∀ t, p t → ∀ s ⊆ t, p s)
+    (hunion : ∀ s, p s → ∀ t, p t → p (s ∪ t)) : Filter α where
+  sets := {t | p tᶜ}
+  univ_sets := by simpa
+  sets_of_superset := fun ht₁ ht => hmono _ ht₁ _ (compl_subset_compl.2 ht)
+  inter_sets := fun ht₁ ht₂ => by simp [compl_inter, hunion _ ht₁ _ ht₂]
+
+@[simp]
+lemma mem_comk {p : Set α → Prop} {he hmono hunion s} :
+    s ∈ comk p he hmono hunion ↔ p sᶜ :=
+  .rfl
+
+/-- The principal filter of `s` is the collection of all supersets of `s`. -/
+def principal (s : Set α) : Filter α where
+  sets := { t | s ⊆ t }
+  univ_sets := subset_univ s
+  sets_of_superset hx := Subset.trans hx
+  inter_sets := subset_inter
+
+@[inherit_doc]
+scoped notation "𝓟" => Filter.principal
+
+@[simp] theorem mem_principal : s ∈ 𝓟 t ↔ t ⊆ s := Iff.rfl
+
+/-- The *kernel* of a filter is the intersection of all its sets. -/
+def ker (f : Filter α) : Set α := ⋂₀ f.sets
+
+/-- The join of a filter of filters is defined by the relation `s ∈ join f ↔ {t | s ∈ t} ∈ f`. -/
+def join (f : Filter (Filter α)) : Filter α where
+  sets := { s | { t : Filter α | s ∈ t } ∈ f }
+  univ_sets := by simp only [mem_setOf_eq, univ_mem, setOf_true]
+  sets_of_superset hx xy := mem_of_superset hx fun f h => mem_of_superset h xy
+  inter_sets hx hy := mem_of_superset (inter_mem hx hy) fun f ⟨h₁, h₂⟩ => inter_mem h₁ h₂
+
+@[simp]
+theorem mem_join {s : Set α} {f : Filter (Filter α)} : s ∈ join f ↔ { t | s ∈ t } ∈ f :=
+  Iff.rfl
+
+instance : PartialOrder (Filter α) where
+  le f g := ∀ ⦃U : Set α⦄, U ∈ g → U ∈ f
+  le_antisymm a b h₁ h₂ := filter_eq <| Subset.antisymm h₂ h₁
+  le_refl a := Subset.rfl
+  le_trans a b c h₁ h₂ := Subset.trans h₂ h₁
+
+theorem le_def : f ≤ g ↔ ∀ x ∈ g, x ∈ f :=
+  Iff.rfl
+
+instance : Top (Filter α) :=
+  ⟨{  sets := { s | ∀ x, x ∈ s }
+      univ_sets := fun x => mem_univ x
+      sets_of_superset := fun hx hxy a => hxy (hx a)
+      inter_sets := fun hx hy _ => mem_inter (hx _) (hy _) }⟩
+
+theorem mem_top_iff_forall {s : Set α} : s ∈ (⊤ : Filter α) ↔ ∀ x, x ∈ s :=
+  Iff.rfl
+
+@[simp]
+theorem mem_top {s : Set α} : s ∈ (⊤ : Filter α) ↔ s = univ := by
+  rw [mem_top_iff_forall, eq_univ_iff_forall]
+
+instance : Bot (Filter α) where
+  bot :=
+    { sets := univ
+      univ_sets := trivial
+      sets_of_superset := fun _ _ ↦ trivial
+      inter_sets := fun _ _ ↦ trivial }
+
+@[simp]
+theorem mem_bot {s : Set α} : s ∈ (⊥ : Filter α) :=
+  trivial
+
+/-- The infimum of filters is the filter generated by intersections
+  of elements of the two filters. -/
+instance : Inf (Filter α) :=
+  ⟨fun f g : Filter α =>
+    { sets := { s | ∃ a ∈ f, ∃ b ∈ g, s = a ∩ b }
+      univ_sets := ⟨_, univ_mem, _, univ_mem, by simp⟩
+      sets_of_superset := by
+        rintro x y ⟨a, ha, b, hb, rfl⟩ xy
+        refine ⟨a ∪ y, mem_of_superset ha subset_union_left, b ∪ y,
+          mem_of_superset hb subset_union_left, ?_⟩
+        rw [← inter_union_distrib_right, union_eq_self_of_subset_left xy]
+      inter_sets := by
+        rintro x y ⟨a, ha, b, hb, rfl⟩ ⟨c, hc, d, hd, rfl⟩
+        refine ⟨a ∩ c, inter_mem ha hc, b ∩ d, inter_mem hb hd, ?_⟩
+        ac_rfl }⟩
+
+/-- The supremum of two filters is the filter that contains sets that belong to both filters. -/
+instance : Sup (Filter α) where
+  sup f g :=
+    { sets := {s | s ∈ f ∧ s ∈ g}
+      univ_sets := ⟨univ_mem, univ_mem⟩
+      sets_of_superset := fun h₁ h₂ ↦ ⟨mem_of_superset h₁.1 h₂, mem_of_superset h₁.2 h₂⟩
+      inter_sets := fun h₁ h₂ ↦ ⟨inter_mem h₁.1 h₂.1, inter_mem h₁.2 h₂.2⟩ }
+
+/-- A filter is `NeBot` if it is not equal to `⊥`, or equivalently the empty set does not belong to
+the filter. Bourbaki include this assumption in the definition of a filter but we prefer to have a
+`CompleteLattice` structure on `Filter _`, so we use a typeclass argument in lemmas instead. -/
+class NeBot (f : Filter α) : Prop where
+  /-- The filter is nontrivial: `f ≠ ⊥` or equivalently, `∅ ∉ f`. -/
+  ne' : f ≠ ⊥
+
+theorem neBot_iff {f : Filter α} : NeBot f ↔ f ≠ ⊥ :=
+  ⟨fun h => h.1, fun h => ⟨h⟩⟩
+
+/-- `f.Eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in atTop, p x`
+means that `p` holds true for sufficiently large `x`. -/
+protected def Eventually (p : α → Prop) (f : Filter α) : Prop :=
+  { x | p x } ∈ f
+
+@[inherit_doc Filter.Eventually]
+notation3 "∀ᶠ "(...)" in "f", "r:(scoped p => Filter.Eventually p f) => r
+
+/-- `f.Frequently p` or `∃ᶠ x in f, p x` mean that `{x | ¬p x} ∉ f`. E.g., `∃ᶠ x in atTop, p x`
+means that there exist arbitrarily large `x` for which `p` holds true. -/
+protected def Frequently (p : α → Prop) (f : Filter α) : Prop :=
+  ¬∀ᶠ x in f, ¬p x
+
+@[inherit_doc Filter.Frequently]
+notation3 "∃ᶠ "(...)" in "f", "r:(scoped p => Filter.Frequently p f) => r
+
+/-- Two functions `f` and `g` are *eventually equal* along a filter `l` if the set of `x` such that
+`f x = g x` belongs to `l`. -/
+def EventuallyEq (l : Filter α) (f g : α → β) : Prop :=
+  ∀ᶠ x in l, f x = g x
+
+@[inherit_doc]
+notation:50 f " =ᶠ[" l:50 "] " g:50 => EventuallyEq l f g
+
+/-- A function `f` is eventually less than or equal to a function `g` at a filter `l`. -/
+def EventuallyLE [LE β] (l : Filter α) (f g : α → β) : Prop :=
+  ∀ᶠ x in l, f x ≤ g x
+
+@[inherit_doc]
+notation:50 f " ≤ᶠ[" l:50 "] " g:50 => EventuallyLE l f g
+
+/-- The forward map of a filter -/
+def map (m : α → β) (f : Filter α) : Filter β where
+  sets := preimage m ⁻¹' f.sets
+  univ_sets := univ_mem
+  sets_of_superset hs st := mem_of_superset hs fun _x hx ↦ st hx
+  inter_sets hs ht := inter_mem hs ht
+
+/-- `Filter.Tendsto` is the generic "limit of a function" predicate.
+  `Tendsto f l₁ l₂` asserts that for every `l₂` neighborhood `a`,
+  the `f`-preimage of `a` is an `l₁` neighborhood. -/
+def Tendsto (f : α → β) (l₁ : Filter α) (l₂ : Filter β) :=
+  l₁.map f ≤ l₂
+
+/-- The inverse map of a filter. A set `s` belongs to `Filter.comap m f` if either of the following
+equivalent conditions hold.
+
+1. There exists a set `t ∈ f` such that `m ⁻¹' t ⊆ s`. This is used as a definition.
+2. The set `kernImage m s = {y | ∀ x, m x = y → x ∈ s}` belongs to `f`, see `Filter.mem_comap'`.
+3. The set `(m '' sᶜ)ᶜ` belongs to `f`, see `Filter.mem_comap_iff_compl` and
+`Filter.compl_mem_comap`. -/
+def comap (m : α → β) (f : Filter β) : Filter α where
+  sets := { s | ∃ t ∈ f, m ⁻¹' t ⊆ s }
+  univ_sets := ⟨univ, univ_mem, subset_univ _⟩
+  sets_of_superset := fun ⟨a', ha', ma'a⟩ ab => ⟨a', ha', ma'a.trans ab⟩
+  inter_sets := fun ⟨a', ha₁, ha₂⟩ ⟨b', hb₁, hb₂⟩ =>
+    ⟨a' ∩ b', inter_mem ha₁ hb₁, inter_subset_inter ha₂ hb₂⟩
+
+/-- Product of filters. This is the filter generated by cartesian products
+of elements of the component filters. -/
+protected def prod (f : Filter α) (g : Filter β) : Filter (α × β) :=
+  f.comap Prod.fst ⊓ g.comap Prod.snd
+
+/-- Coproduct of filters. -/
+protected def coprod (f : Filter α) (g : Filter β) : Filter (α × β) :=
+  f.comap Prod.fst ⊔ g.comap Prod.snd
+
+instance instSProd : SProd (Filter α) (Filter β) (Filter (α × β)) where
+  sprod := Filter.prod
+
+theorem prod_eq_inf (f : Filter α) (g : Filter β) : f ×ˢ g = f.comap Prod.fst ⊓ g.comap Prod.snd :=
+  rfl
+
+/-- The monadic bind operation on filter is defined the usual way in terms of `map` and `join`.
+
+Unfortunately, this `bind` does not result in the expected applicative. See `Filter.seq` for the
+applicative instance. -/
+def bind (f : Filter α) (m : α → Filter β) : Filter β :=
+  join (map m f)
+
+/-- The applicative sequentiation operation. This is not induced by the bind operation. -/
+def seq (f : Filter (α → β)) (g : Filter α) : Filter β where
+  sets := { s | ∃ u ∈ f, ∃ t ∈ g, ∀ m ∈ u, ∀ x ∈ t, (m : α → β) x ∈ s }
+  univ_sets := ⟨univ, univ_mem, univ, univ_mem, fun _ _ _ _ => trivial⟩
+  sets_of_superset := fun ⟨t₀, t₁, h₀, h₁, h⟩ hst =>
+    ⟨t₀, t₁, h₀, h₁, fun _ hx _ hy => hst <| h _ hx _ hy⟩
+  inter_sets := fun ⟨t₀, ht₀, t₁, ht₁, ht⟩ ⟨u₀, hu₀, u₁, hu₁, hu⟩ =>
+    ⟨t₀ ∩ u₀, inter_mem ht₀ hu₀, t₁ ∩ u₁, inter_mem ht₁ hu₁, fun _ ⟨hx₀, hx₁⟩ _ ⟨hy₀, hy₁⟩ =>
+      ⟨ht _ hx₀ _ hy₀, hu _ hx₁ _ hy₁⟩⟩
+
+/-- This filter is characterized by `Filter.eventually_curry_iff`:
+`(∀ᶠ (x : α × β) in f.curry g, p x) ↔ ∀ᶠ (x : α) in f, ∀ᶠ (y : β) in g, p (x, y)`. Useful
+in adding quantifiers to the middle of `Tendsto`s. See
+`hasFDerivAt_of_tendstoUniformlyOnFilter`. -/
+def curry (f : Filter α) (g : Filter β) : Filter (α × β) :=
+  bind f fun a ↦ map (a, ·) g
+
+/-- `pure x` is the set of sets that contain `x`. It is equal to `𝓟 {x}` but
+with this definition we have `s ∈ pure a` defeq `a ∈ s`. -/
+instance : Pure Filter :=
+  ⟨fun x =>
+    { sets := { s | x ∈ s }
+      inter_sets := And.intro
+      sets_of_superset := fun hs hst => hst hs
+      univ_sets := trivial }⟩
+
+instance : Bind Filter :=
+  ⟨@Filter.bind⟩
+
+instance : Functor Filter where map := @Filter.map
+
+end Filter
+
+namespace Mathlib.Tactic
+
+open Lean Meta Elab Tactic
+
+/--
+`filter_upwards [h₁, ⋯, hₙ]` replaces a goal of the form `s ∈ f` and terms
+`h₁ : t₁ ∈ f, ⋯, hₙ : tₙ ∈ f` with `∀ x, x ∈ t₁ → ⋯ → x ∈ tₙ → x ∈ s`.
+The list is an optional parameter, `[]` being its default value.
+
+`filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ` is a short form for
+`{ filter_upwards [h₁, ⋯, hₙ], intros a₁ a₂ ⋯ aₖ }`.
+
+`filter_upwards [h₁, ⋯, hₙ] using e` is a short form for
+`{ filter_upwards [h1, ⋯, hn], exact e }`.
+
+Combining both shortcuts is done by writing `filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ using e`.
+Note that in this case, the `aᵢ` terms can be used in `e`.
+-/
+syntax (name := filterUpwards) "filter_upwards" (" [" term,* "]")?
+  (" with" (ppSpace colGt term:max)*)? (" using " term)? : tactic
+
+elab_rules : tactic
+| `(tactic| filter_upwards $[[$[$args],*]]? $[with $wth*]? $[using $usingArg]?) => do
+  let config : ApplyConfig := {newGoals := ApplyNewGoals.nonDependentOnly}
+  for e in args.getD #[] |>.reverse do
+    let goal ← getMainGoal
+    replaceMainGoal <| ← goal.withContext <| runTermElab do
+      let m ← mkFreshExprMVar none
+      let lem ← Term.elabTermEnsuringType
+        (← ``(Filter.mp_mem $e $(← Term.exprToSyntax m))) (← goal.getType)
+      goal.assign lem
+      return [m.mvarId!]
+  liftMetaTactic fun goal => do
+    goal.apply (← mkConstWithFreshMVarLevels ``Filter.univ_mem') config
+  evalTactic <|← `(tactic| dsimp (config := {zeta := false}) only [Set.mem_setOf_eq])
+  if let some l := wth then
+    evalTactic <|← `(tactic| intro $[$l]*)
+  if let some e := usingArg then
+    evalTactic <|← `(tactic| exact $e)
+
+end Mathlib.Tactic
diff --git a/Mathlib/Order/Filter/EventuallyConst.lean b/Mathlib/Order/Filter/EventuallyConst.lean
index 5ab7e4b3e49cd..be981a5e72142 100644
--- a/Mathlib/Order/Filter/EventuallyConst.lean
+++ b/Mathlib/Order/Filter/EventuallyConst.lean
@@ -3,6 +3,7 @@ Copyright (c) 2023 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov, Floris van Doorn
 -/
+import Mathlib.Algebra.Group.Indicator
 import Mathlib.Order.Filter.AtTopBot
 import Mathlib.Order.Filter.Subsingleton
 /-!
@@ -133,7 +134,7 @@ variable [One β] {s : Set α} {c : β}
 @[to_additive]
 lemma of_mulIndicator_const (h : EventuallyConst (s.mulIndicator fun _ ↦ c) l) (hc : c ≠ 1) :
     EventuallyConst s l := by
-  simpa [(· ∘ ·), hc, imp_false] using h.comp (· = c)
+  simpa [Function.comp_def, hc, imp_false] using h.comp (· = c)
 
 @[to_additive]
 theorem mulIndicator_const (h : EventuallyConst s l) (c : β) :
diff --git a/Mathlib/Order/Filter/Extr.lean b/Mathlib/Order/Filter/Extr.lean
index b46294e1e6f5c..aa4829ffbd4a4 100644
--- a/Mathlib/Order/Filter/Extr.lean
+++ b/Mathlib/Order/Filter/Extr.lean
@@ -577,3 +577,34 @@ theorem IsMinOn.iInf_eq (hx₀ : x₀ ∈ s) (h : IsMinOn f s x₀) : ⨅ x : s,
   @IsMaxOn.iSup_eq αᵒᵈ β _ _ _ _ hx₀ h
 
 end ConditionallyCompleteLinearOrder
+
+/-! ### Value of `Finset.sup` / `Finset.inf` -/
+
+section SemilatticeSup
+
+variable [SemilatticeSup β] [OrderBot β] {D : α → β} {s : Finset α}
+
+theorem sup_eq_of_isMaxOn {a : α} (hmem : a ∈ s) (hmax : IsMaxOn D s a) : s.sup D = D a :=
+  (Finset.sup_le hmax).antisymm (Finset.le_sup hmem)
+
+theorem sup_eq_of_max [Nonempty α] {b : β} (hb : b ∈ Set.range D) (hmem : D.invFun b ∈ s)
+    (hmax : ∀ a ∈ s, D a ≤ b) : s.sup D = b := by
+  obtain ⟨a, rfl⟩ := hb
+  rw [← Function.apply_invFun_apply (f := D)]
+  apply sup_eq_of_isMaxOn hmem; intro
+  rw [Function.apply_invFun_apply (f := D)]; apply hmax
+
+end SemilatticeSup
+
+section SemilatticeInf
+
+variable [SemilatticeInf β] [OrderTop β] {D : α → β} {s : Finset α}
+
+theorem inf_eq_of_isMinOn {a : α} (hmem : a ∈ s) (hmax : IsMinOn D s a) : s.inf D = D a :=
+  sup_eq_of_isMaxOn (α := αᵒᵈ) (β := βᵒᵈ) hmem hmax.dual
+
+theorem inf_eq_of_min [Nonempty α] {b : β} (hb : b ∈ Set.range D) (hmem : D.invFun b ∈ s)
+    (hmin : ∀ a ∈ s, b ≤ D a) : s.inf D = b :=
+  sup_eq_of_max (α := αᵒᵈ) (β := βᵒᵈ) hb hmem hmin
+
+end SemilatticeInf
diff --git a/Mathlib/Order/Filter/FilterProduct.lean b/Mathlib/Order/Filter/FilterProduct.lean
index 99c01de709197..4e69166e08aea 100644
--- a/Mathlib/Order/Filter/FilterProduct.lean
+++ b/Mathlib/Order/Filter/FilterProduct.lean
@@ -3,8 +3,10 @@ Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Abhimanyu Pallavi Sudhir, Yury Kudryashov
 -/
-import Mathlib.Order.Filter.Ultrafilter
+import Mathlib.Algebra.Order.Field.Defs
+import Mathlib.Algebra.Order.Group.Unbundled.Abs
 import Mathlib.Order.Filter.Ring
+import Mathlib.Order.Filter.Ultrafilter
 
 /-!
 # Ultraproducts
@@ -38,7 +40,7 @@ instance instGroupWithZero [GroupWithZero β] : GroupWithZero β* where
   __ := instMonoidWithZero
   mul_inv_cancel f := inductionOn f fun f hf ↦ coe_eq.2 <| (φ.em fun y ↦ f y = 0).elim
     (fun H ↦ (hf <| coe_eq.2 H).elim) fun H ↦ H.mono fun x ↦ mul_inv_cancel₀
-  inv_zero := coe_eq.2 <| by simp only [Function.comp, inv_zero, EventuallyEq.rfl]
+  inv_zero := coe_eq.2 <| by simp only [Function.comp_def, inv_zero, EventuallyEq.rfl]
 
 instance instDivisionSemiring [DivisionSemiring β] : DivisionSemiring β* where
   toSemiring := instSemiring
diff --git a/Mathlib/Order/Filter/Germ/Basic.lean b/Mathlib/Order/Filter/Germ/Basic.lean
index dd4039e19d250..dee43bb2b8c71 100644
--- a/Mathlib/Order/Filter/Germ/Basic.lean
+++ b/Mathlib/Order/Filter/Germ/Basic.lean
@@ -512,7 +512,7 @@ instance instAddGroupWithOne [AddGroupWithOne G] : AddGroupWithOne (Germ l G) wh
   __ := instAddMonoidWithOne
   __ := instAddGroup
   intCast_ofNat _ := congrArg ofFun <| by simp
-  intCast_negSucc _ := congrArg ofFun <| by simp [Function.comp]; rfl
+  intCast_negSucc _ := congrArg ofFun <| by simp [Function.comp_def]; rfl
 
 end Monoid
 
diff --git a/Mathlib/Order/Filter/IndicatorFunction.lean b/Mathlib/Order/Filter/IndicatorFunction.lean
index d7338ff33d231..2b818c2fa7218 100644
--- a/Mathlib/Order/Filter/IndicatorFunction.lean
+++ b/Mathlib/Order/Filter/IndicatorFunction.lean
@@ -3,6 +3,7 @@ Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou, Yury Kudryashov
 -/
+import Mathlib.Algebra.Group.Indicator
 import Mathlib.Order.Filter.AtTopBot
 
 /-!
diff --git a/Mathlib/Order/Filter/Interval.lean b/Mathlib/Order/Filter/Interval.lean
index e5b7087c92c4f..02a5a679d0685 100644
--- a/Mathlib/Order/Filter/Interval.lean
+++ b/Mathlib/Order/Filter/Interval.lean
@@ -18,7 +18,7 @@ from the left or from the right, or it could also be infinity, and "around this
 to these directed neighborhoods. Therefore, the above theorem has many variants. Instead of stating
 all these variants, one can look for the common abstraction and have a single version. One has to
 be careful: if one considers convergence along a sequence, then the function may tend to infinity
-but have a derivative which is small along the sequence (with big jumps inbetween), so in the end
+but have a derivative which is small along the sequence (with big jumps in between), so in the end
 the derivative may be integrable on a neighborhood of the sequence. What really matters for such
 calculus issues in terms of derivatives is that whole intervals are included in the sets we
 consider.
diff --git a/Mathlib/Order/Filter/Ker.lean b/Mathlib/Order/Filter/Ker.lean
index 64eecf4a76499..84ab8db20e0cf 100644
--- a/Mathlib/Order/Filter/Ker.lean
+++ b/Mathlib/Order/Filter/Ker.lean
@@ -21,9 +21,6 @@ namespace Filter
 
 variable {ι : Sort*} {α β : Type*} {f g : Filter α} {s : Set α} {a : α}
 
-/-- The *kernel* of a filter is the intersection of all its sets. -/
-def ker (f : Filter α) : Set α := ⋂₀ f.sets
-
 lemma ker_def (f : Filter α) : f.ker = ⋂ s ∈ f, s := sInter_eq_biInter
 
 @[simp] lemma mem_ker : a ∈ f.ker ↔ ∀ s ∈ f, a ∈ s := mem_sInter
@@ -54,4 +51,21 @@ lemma ker_surjective : Surjective (ker : Filter α → Set α) := gi_principal_k
   simp only [mem_ker, mem_comap, forall_exists_index, and_imp, @forall_swap (Set α), mem_preimage]
   exact forall₂_congr fun s _ ↦ ⟨fun h ↦ h _ Subset.rfl, fun ha t ht ↦ ht ha⟩
 
+@[simp]
+theorem ker_iSup (f : ι → Filter α) : ker (⨆ i, f i) = ⋃ i, ker (f i) := by
+  refine subset_antisymm (fun x hx ↦ ?_) ker_mono.le_map_iSup
+  simp only [mem_iUnion, mem_ker] at hx ⊢
+  contrapose! hx
+  choose s hsf hxs using hx
+  refine ⟨⋃ i, s i, ?_, by simpa⟩
+  exact mem_iSup.2 fun i ↦ mem_of_superset (hsf i) (subset_iUnion s i)
+
+@[simp]
+theorem ker_sSup (S : Set (Filter α)) : ker (sSup S) = ⋃ f ∈ S, ker f := by
+  simp [sSup_eq_iSup]
+
+@[simp]
+theorem ker_sup (f g : Filter α) : ker (f ⊔ g) = ker f ∪ ker g := by
+  rw [← sSup_pair, ker_sSup, biUnion_pair]
+
 end Filter
diff --git a/Mathlib/Order/Filter/Lift.lean b/Mathlib/Order/Filter/Lift.lean
index 6a193e98904ff..95e8d4708a1be 100644
--- a/Mathlib/Order/Filter/Lift.lean
+++ b/Mathlib/Order/Filter/Lift.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
 import Mathlib.Order.Filter.Bases
+import Mathlib.Order.Filter.Prod
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 
 /-!
@@ -249,7 +250,7 @@ theorem lift'_cong (hh : ∀ s ∈ f, h₁ s = h₂ s) : f.lift' h₁ = f.lift'
 theorem map_lift'_eq {m : β → γ} (hh : Monotone h) : map m (f.lift' h) = f.lift' (image m ∘ h) :=
   calc
     map m (f.lift' h) = f.lift (map m ∘ 𝓟 ∘ h) := map_lift_eq <| monotone_principal.comp hh
-    _ = f.lift' (image m ∘ h) := by simp only [comp, Filter.lift', map_principal]
+    _ = f.lift' (image m ∘ h) := by simp only [comp_def, Filter.lift', map_principal]
 
 theorem lift'_map_le {g : Set β → Set γ} {m : α → β} : (map m f).lift' g ≤ f.lift' (g ∘ image m) :=
   lift_map_le
@@ -259,7 +260,7 @@ theorem map_lift'_eq2 {g : Set β → Set γ} {m : α → β} (hg : Monotone g)
   map_lift_eq2 <| monotone_principal.comp hg
 
 theorem comap_lift'_eq {m : γ → β} : comap m (f.lift' h) = f.lift' (preimage m ∘ h) := by
-  simp only [Filter.lift', comap_lift_eq, (· ∘ ·), comap_principal]
+  simp only [Filter.lift', comap_lift_eq, comp_def, comap_principal]
 
 theorem comap_lift'_eq2 {m : β → α} {g : Set β → Set γ} (hg : Monotone g) :
     (comap m f).lift' g = f.lift' (g ∘ preimage m) :=
@@ -362,7 +363,7 @@ theorem prod_same_eq : f ×ˢ f = f.lift' fun t : Set α => t ×ˢ t :=
 
 theorem tendsto_prod_self_iff {f : α × α → β} {x : Filter α} {y : Filter β} :
     Filter.Tendsto f (x ×ˢ x) y ↔ ∀ W ∈ y, ∃ U ∈ x, ∀ x x' : α, x ∈ U → x' ∈ U → f (x, x') ∈ W := by
-  simp only [tendsto_def, mem_prod_same_iff, prod_sub_preimage_iff, exists_prop, iff_self_iff]
+  simp only [tendsto_def, mem_prod_same_iff, prod_sub_preimage_iff, exists_prop]
 
 variable {α₁ : Type*} {α₂ : Type*} {β₁ : Type*} {β₂ : Type*}
 
diff --git a/Mathlib/Order/Filter/NAry.lean b/Mathlib/Order/Filter/NAry.lean
index dfc9e8e2826f9..80b0db70b8043 100644
--- a/Mathlib/Order/Filter/NAry.lean
+++ b/Mathlib/Order/Filter/NAry.lean
@@ -51,7 +51,7 @@ theorem map_prod_eq_map₂ (m : α → β → γ) (f : Filter α) (g : Filter β
 
 theorem map_prod_eq_map₂' (m : α × β → γ) (f : Filter α) (g : Filter β) :
     Filter.map m (f ×ˢ g) = map₂ (fun a b => m (a, b)) f g :=
-  map_prod_eq_map₂ (curry m) f g
+  map_prod_eq_map₂ m.curry f g
 
 @[simp]
 theorem map₂_mk_eq_prod (f : Filter α) (g : Filter β) : map₂ Prod.mk f g = f ×ˢ g := by
@@ -145,7 +145,7 @@ theorem map₂_map_right (m : α → γ → δ) (n : β → γ) :
 
 @[simp]
 theorem map₂_curry (m : α × β → γ) (f : Filter α) (g : Filter β) :
-    map₂ (curry m) f g = (f ×ˢ g).map m :=
+    map₂ m.curry f g = (f ×ˢ g).map m :=
   (map_prod_eq_map₂' _ _ _).symm
 
 @[simp]
@@ -168,7 +168,7 @@ theorem map₂_assoc {m : δ → γ → ε} {n : α → β → δ} {m' : α →
     map₂ m (map₂ n f g) h = map₂ m' f (map₂ n' g h) := by
   rw [← map_prod_eq_map₂ n, ← map_prod_eq_map₂ n', map₂_map_left, map₂_map_right,
     ← map_prod_eq_map₂, ← map_prod_eq_map₂, ← prod_assoc, map_map]
-  simp only [h_assoc, Function.comp, Equiv.prodAssoc_apply]
+  simp only [h_assoc, Function.comp_def, Equiv.prodAssoc_apply]
 
 theorem map₂_comm {n : β → α → γ} (h_comm : ∀ a b, m a b = n b a) : map₂ m f g = map₂ n g f :=
   (map₂_swap _ _ _).trans <| by simp_rw [h_comm]
diff --git a/Mathlib/Order/Filter/Pointwise.lean b/Mathlib/Order/Filter/Pointwise.lean
index 86f157845a039..a1c46ad3a9f3f 100644
--- a/Mathlib/Order/Filter/Pointwise.lean
+++ b/Mathlib/Order/Filter/Pointwise.lean
@@ -3,6 +3,8 @@ Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou, Yaël Dillies
 -/
+import Mathlib.Algebra.Order.Group.Defs
+import Mathlib.Algebra.Order.Group.OrderIso
 import Mathlib.Data.Set.Pointwise.SMul
 import Mathlib.Order.Filter.NAry
 import Mathlib.Order.Filter.Ultrafilter
@@ -549,7 +551,7 @@ theorem pow_mem_pow (hs : s ∈ f) : ∀ n : ℕ, s ^ n ∈ f ^ n
 
 @[to_additive (attr := simp) nsmul_bot]
 theorem bot_pow {n : ℕ} (hn : n ≠ 0) : (⊥ : Filter α) ^ n = ⊥ := by
-  rw [← tsub_add_cancel_of_le (Nat.succ_le_of_lt <| Nat.pos_of_ne_zero hn), pow_succ', bot_mul]
+  rw [← Nat.sub_one_add_one hn, pow_succ', bot_mul]
 
 @[to_additive]
 theorem mul_top_of_one_le (hf : 1 ≤ f) : f * ⊤ = ⊤ := by
@@ -595,7 +597,7 @@ variable [DivisionMonoid α] {f g : Filter α}
 @[to_additive]
 protected theorem mul_eq_one_iff : f * g = 1 ↔ ∃ a b, f = pure a ∧ g = pure b ∧ a * b = 1 := by
   refine ⟨fun hfg => ?_, ?_⟩
-  · obtain ⟨t₁, h₁, t₂, h₂, h⟩ : (1 : Set α) ∈ f * g := hfg.symm.subst one_mem_one
+  · obtain ⟨t₁, h₁, t₂, h₂, h⟩ : (1 : Set α) ∈ f * g := hfg.symm ▸ one_mem_one
     have hfg : (f * g).NeBot := hfg.symm.subst one_neBot
     rw [(hfg.nonempty_of_mem <| mul_mem_mul h₁ h₂).subset_one_iff, Set.mul_eq_one_iff] at h
     obtain ⟨a, b, rfl, rfl, h⟩ := h
@@ -713,7 +715,7 @@ theorem isUnit_pure (a : α) : IsUnit (pure a : Filter α) :=
 
 @[simp]
 theorem isUnit_iff_singleton : IsUnit f ↔ ∃ a, f = pure a := by
-  simp only [isUnit_iff, Group.isUnit, and_true_iff]
+  simp only [isUnit_iff, Group.isUnit, and_true]
 
 @[to_additive]
 theorem map_inv' : f⁻¹.map m = (f.map m)⁻¹ :=
@@ -1048,7 +1050,7 @@ protected def mulAction [Monoid α] [MulAction α β] : MulAction (Filter α) (F
 @[to_additive "An additive action of an additive monoid on a type `β` gives an additive action on
  `Filter β`."]
 protected def mulActionFilter [Monoid α] [MulAction α β] : MulAction α (Filter β) where
-  mul_smul a b f := by simp only [← Filter.map_smul, map_map, Function.comp, ← mul_smul]
+  mul_smul a b f := by simp only [← Filter.map_smul, map_map, Function.comp_def, ← mul_smul]
   one_smul f := by simp only [← Filter.map_smul, one_smul, map_id']
 
 scoped[Pointwise] attribute [instance] Filter.mulAction Filter.addAction Filter.mulActionFilter
diff --git a/Mathlib/Order/Filter/Prod.lean b/Mathlib/Order/Filter/Prod.lean
index 0202f4d85b27e..b2af14e42295d 100644
--- a/Mathlib/Order/Filter/Prod.lean
+++ b/Mathlib/Order/Filter/Prod.lean
@@ -46,14 +46,6 @@ section Prod
 
 variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
 
-/-- Product of filters. This is the filter generated by cartesian products
-of elements of the component filters. -/
-protected def prod (f : Filter α) (g : Filter β) : Filter (α × β) :=
-  f.comap Prod.fst ⊓ g.comap Prod.snd
-
-instance instSProd : SProd (Filter α) (Filter β) (Filter (α × β)) where
-  sprod := Filter.prod
-
 theorem prod_mem_prod (hs : s ∈ f) (ht : t ∈ g) : s ×ˢ t ∈ f ×ˢ g :=
   inter_mem_inf (preimage_mem_comap hs) (preimage_mem_comap ht)
 
@@ -66,6 +58,10 @@ theorem mem_prod_iff {s : Set (α × β)} {f : Filter α} {g : Filter β} :
   · rintro ⟨t₁, ht₁, t₂, ht₂, h⟩
     exact mem_inf_of_inter (preimage_mem_comap ht₁) (preimage_mem_comap ht₂) h
 
+@[simp]
+theorem compl_diagonal_mem_prod {l₁ l₂ : Filter α} : (diagonal α)ᶜ ∈ l₁ ×ˢ l₂ ↔ Disjoint l₁ l₂ := by
+  simp only [mem_prod_iff, Filter.disjoint_iff, prod_subset_compl_diagonal_iff_disjoint]
+
 @[simp]
 theorem prod_mem_prod_iff [f.NeBot] [g.NeBot] : s ×ˢ t ∈ f ×ˢ g ↔ s ∈ f ∧ t ∈ g :=
   ⟨fun h =>
@@ -99,6 +95,10 @@ theorem comap_prod (f : α → β × γ) (b : Filter β) (c : Filter γ) :
     comap f (b ×ˢ c) = comap (Prod.fst ∘ f) b ⊓ comap (Prod.snd ∘ f) c := by
   erw [comap_inf, Filter.comap_comap, Filter.comap_comap]
 
+theorem comap_prodMap_prod (f : α → β) (g : γ → δ) (lb : Filter β) (ld : Filter δ) :
+    comap (Prod.map f g) (lb ×ˢ ld) = comap f lb ×ˢ comap g ld := by
+  simp [prod_eq_inf, comap_comap, Function.comp_def]
+
 theorem prod_top : f ×ˢ (⊤ : Filter β) = f.comap Prod.fst := by
   dsimp only [SProd.sprod]
   rw [Filter.prod, comap_top, inf_top_eq]
@@ -229,10 +229,11 @@ theorem prod_mono_right (f : Filter α) {g₁ g₂ : Filter β} (hf : g₁ ≤ g
 theorem prod_comap_comap_eq.{u, v, w, x} {α₁ : Type u} {α₂ : Type v} {β₁ : Type w} {β₂ : Type x}
     {f₁ : Filter α₁} {f₂ : Filter α₂} {m₁ : β₁ → α₁} {m₂ : β₂ → α₂} :
     comap m₁ f₁ ×ˢ comap m₂ f₂ = comap (fun p : β₁ × β₂ => (m₁ p.1, m₂ p.2)) (f₁ ×ˢ f₂) := by
-  simp only [SProd.sprod, Filter.prod, comap_comap, comap_inf, (· ∘ ·)]
+  simp only [SProd.sprod, Filter.prod, comap_comap, comap_inf, Function.comp_def]
 
 theorem prod_comm' : f ×ˢ g = comap Prod.swap (g ×ˢ f) := by
-  simp only [SProd.sprod, Filter.prod, comap_comap, (· ∘ ·), inf_comm, Prod.swap, comap_inf]
+  simp only [SProd.sprod, Filter.prod, comap_comap, Function.comp_def, inf_comm, Prod.swap,
+    comap_inf]
 
 theorem prod_comm : f ×ˢ g = map (fun p : β × α => (p.2, p.1)) (g ×ˢ f) := by
   rw [prod_comm', ← map_swap_eq_comap_swap]
@@ -279,13 +280,13 @@ theorem eventually_swap_iff {p : α × β → Prop} :
 
 theorem prod_assoc (f : Filter α) (g : Filter β) (h : Filter γ) :
     map (Equiv.prodAssoc α β γ) ((f ×ˢ g) ×ˢ h) = f ×ˢ (g ×ˢ h) := by
-  simp_rw [← comap_equiv_symm, SProd.sprod, Filter.prod, comap_inf, comap_comap, inf_assoc, (· ∘ ·),
-    Equiv.prodAssoc_symm_apply]
+  simp_rw [← comap_equiv_symm, SProd.sprod, Filter.prod, comap_inf, comap_comap, inf_assoc,
+    Function.comp_def, Equiv.prodAssoc_symm_apply]
 
 theorem prod_assoc_symm (f : Filter α) (g : Filter β) (h : Filter γ) :
     map (Equiv.prodAssoc α β γ).symm (f ×ˢ (g ×ˢ h)) = (f ×ˢ g) ×ˢ h := by
   simp_rw [map_equiv_symm, SProd.sprod, Filter.prod, comap_inf, comap_comap, inf_assoc,
-    Function.comp, Equiv.prodAssoc_apply]
+    Function.comp_def, Equiv.prodAssoc_apply]
 
 theorem tendsto_prodAssoc {h : Filter γ} :
     Tendsto (Equiv.prodAssoc α β γ) ((f ×ˢ g) ×ˢ h) (f ×ˢ (g ×ˢ h)) :=
@@ -408,13 +409,13 @@ theorem frequently_prod_and {p : α → Prop} {q : β → Prop} :
 
 theorem tendsto_prod_iff {f : α × β → γ} {x : Filter α} {y : Filter β} {z : Filter γ} :
     Tendsto f (x ×ˢ y) z ↔ ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W := by
-  simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop, iff_self_iff]
+  simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop]
 
 theorem tendsto_prod_iff' {g' : Filter γ} {s : α → β × γ} :
     Tendsto s f (g ×ˢ g') ↔ Tendsto (fun n => (s n).1) f g ∧ Tendsto (fun n => (s n).2) f g' := by
   dsimp only [SProd.sprod]
   unfold Filter.prod
-  simp only [tendsto_inf, tendsto_comap_iff, (· ∘ ·)]
+  simp only [tendsto_inf, tendsto_comap_iff, Function.comp_def]
 
 theorem le_prod {f : Filter (α × β)} {g : Filter α} {g' : Filter β} :
     (f ≤ g ×ˢ g') ↔ Tendsto Prod.fst f g ∧ Tendsto Prod.snd f g' :=
@@ -428,10 +429,6 @@ section Coprod
 
 variable {f : Filter α} {g : Filter β}
 
-/-- Coproduct of filters. -/
-protected def coprod (f : Filter α) (g : Filter β) : Filter (α × β) :=
-  f.comap Prod.fst ⊔ g.comap Prod.snd
-
 theorem coprod_eq_prod_top_sup_top_prod (f : Filter α) (g : Filter β) :
     Filter.coprod f g = f ×ˢ ⊤ ⊔ ⊤ ×ˢ g := by
   rw [prod_top, top_prod]
diff --git a/Mathlib/Order/Filter/SmallSets.lean b/Mathlib/Order/Filter/SmallSets.lean
index fba0431ca4a38..fe284b82bd719 100644
--- a/Mathlib/Order/Filter/SmallSets.lean
+++ b/Mathlib/Order/Filter/SmallSets.lean
@@ -109,7 +109,7 @@ theorem smallSets_principal (s : Set α) : (𝓟 s).smallSets = 𝓟 (𝒫 s) :=
 theorem smallSets_comap_eq_comap_image (l : Filter β) (f : α → β) :
     (comap f l).smallSets = comap (image f) l.smallSets := by
   refine (gc_map_comap _).u_comm_of_l_comm (gc_map_comap _) bind_smallSets_gc bind_smallSets_gc ?_
-  simp [Function.comp, map_bind, bind_map]
+  simp [Function.comp_def, map_bind, bind_map]
 
 theorem smallSets_comap (l : Filter β) (f : α → β) :
     (comap f l).smallSets = l.lift' (powerset ∘ preimage f) :=
diff --git a/Mathlib/Order/Filter/Ultrafilter.lean b/Mathlib/Order/Filter/Ultrafilter.lean
index d73a8c74c3df8..6f238409f1d35 100644
--- a/Mathlib/Order/Filter/Ultrafilter.lean
+++ b/Mathlib/Order/Filter/Ultrafilter.lean
@@ -50,7 +50,7 @@ instance : CoeTC (Ultrafilter α) (Filter α) :=
   ⟨Ultrafilter.toFilter⟩
 
 instance : Membership (Set α) (Ultrafilter α) :=
-  ⟨fun s f => s ∈ (f : Filter α)⟩
+  ⟨fun f s => s ∈ (f : Filter α)⟩
 
 theorem unique (f : Ultrafilter α) {g : Filter α} (h : g ≤ f) (hne : NeBot g := by infer_instance) :
     g = f :=
@@ -115,7 +115,7 @@ theorem diff_mem_iff (f : Ultrafilter α) : s \ t ∈ f ↔ s ∈ f ∧ t ∉ f
   inter_mem_iff.trans <| and_congr Iff.rfl compl_mem_iff_not_mem
 
 /-- If `sᶜ ∉ f ↔ s ∈ f`, then `f` is an ultrafilter. The other implication is given by
-`Ultrafilter.compl_not_mem_iff`.  -/
+`Ultrafilter.compl_not_mem_iff`. -/
 def ofComplNotMemIff (f : Filter α) (h : ∀ s, sᶜ ∉ f ↔ s ∈ f) : Ultrafilter α where
   toFilter := f
   neBot' := ⟨fun hf => by simp [hf] at h⟩
@@ -465,4 +465,8 @@ theorem ofComapInfPrincipal_eq_of_map (h : m '' s ∈ g) : (ofComapInfPrincipal
     _ ≤ ↑g ⊓ (𝓟 <| m '' s) := inf_le_inf_right _ map_comap_le
     _ = ↑g := inf_of_le_left (le_principal_iff.mpr h)
 
+theorem eq_of_le_pure {X : Type _} {α : Filter X} (hα : α.NeBot) {x y : X}
+    (hx : α ≤ pure x) (hy : α ≤ pure y) : x = y :=
+  Filter.pure_injective (hα.le_pure_iff.mp hx ▸ hα.le_pure_iff.mp hy)
+
 end Ultrafilter
diff --git a/Mathlib/Order/Fin/Basic.lean b/Mathlib/Order/Fin/Basic.lean
index 3477d9965e7f1..10983f1527c03 100644
--- a/Mathlib/Order/Fin/Basic.lean
+++ b/Mathlib/Order/Fin/Basic.lean
@@ -5,7 +5,6 @@ Authors: Robert Y. Lewis, Keeley Hoek
 -/
 import Mathlib.Data.Fin.Basic
 import Mathlib.Order.Hom.Set
-import Mathlib.Init.Data.Nat.Lemmas
 
 /-!
 # `Fin n` forms a bounded linear order
@@ -123,6 +122,7 @@ end FromFin
 /-! #### Monotonicity -/
 
 lemma val_strictMono : StrictMono (val : Fin n → ℕ) := fun _ _ ↦ id
+lemma cast_strictMono {k l : ℕ} (h : k = l) : StrictMono (cast h) := fun {_ _} h ↦ h
 
 lemma strictMono_succ : StrictMono (succ : Fin n → Fin (n + 1)) := fun _ _ ↦ succ_lt_succ
 lemma strictMono_castLE (h : n ≤ m) : StrictMono (castLE h : Fin n → Fin m) := fun _ _ ↦ id
@@ -279,25 +279,16 @@ map. In this lemma we state that for each `i : Fin n` we have `(e i : ℕ) = (i
     simpa using h _ this (e.symm _).is_lt
   · rwa [← h j hj (hj.trans hi), ← lt_iff_val_lt_val, e.lt_iff_lt]
 
-instance orderIso_subsingleton : Subsingleton (Fin n ≃o α) :=
-  ⟨fun e e' => by
-    ext i
-    rw [← e.symm.apply_eq_iff_eq, e.symm_apply_apply, ← e'.trans_apply, Fin.ext_iff,
-      coe_orderIso_apply]⟩
-
-instance orderIso_subsingleton' : Subsingleton (α ≃o Fin n) := OrderIso.symm_injective.subsingleton
-
-instance orderIsoUnique : Unique (Fin n ≃o Fin n) := Unique.mk' _
-
 /-- Two strictly monotone functions from `Fin n` are equal provided that their ranges
 are equal. -/
+@[deprecated StrictMono.range_inj (since := "2024-09-17")]
 lemma strictMono_unique {f g : Fin n → α} (hf : StrictMono f) (hg : StrictMono g)
     (h : range f = range g) : f = g :=
-  have : (hf.orderIso f).trans (OrderIso.setCongr _ _ h) = hg.orderIso g := Subsingleton.elim _ _
-  congr_arg (Function.comp (Subtype.val : range g → α)) (funext <| RelIso.ext_iff.1 this)
+  (hf.range_inj hg).1 h
 
 /-- Two order embeddings of `Fin n` are equal provided that their ranges are equal. -/
+@[deprecated OrderEmbedding.range_inj (since := "2024-09-17")]
 lemma orderEmbedding_eq {f g : Fin n ↪o α} (h : range f = range g) : f = g :=
-  RelEmbedding.ext <| funext_iff.1 <| strictMono_unique f.strictMono g.strictMono h
+  OrderEmbedding.range_inj.1 h
 
 end Fin
diff --git a/Mathlib/Order/Fin/Tuple.lean b/Mathlib/Order/Fin/Tuple.lean
index 428d79cc5cd09..1f92421e88291 100644
--- a/Mathlib/Order/Fin/Tuple.lean
+++ b/Mathlib/Order/Fin/Tuple.lean
@@ -39,12 +39,12 @@ lemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove
 lemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}
     (hx : x ∈ Icc (q₁ i) (q₂ i)) :
     i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) :=
-  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and_iff]
+  Set.ext fun p ↦ by simp only [mem_preimage, insertNth_mem_Icc, hx, true_and]
 
 lemma preimage_insertNth_Icc_of_not_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}
     (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ :=
   Set.ext fun p ↦ by
-    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and_iff, mem_empty_iff_false]
+    simp only [mem_preimage, insertNth_mem_Icc, hx, false_and, mem_empty_iff_false]
 
 end Fin
 
diff --git a/Mathlib/Order/FixedPoints.lean b/Mathlib/Order/FixedPoints.lean
index 9d6da8b67abcb..e6f40735b5ddc 100644
--- a/Mathlib/Order/FixedPoints.lean
+++ b/Mathlib/Order/FixedPoints.lean
@@ -280,16 +280,17 @@ open OmegaCompletePartialOrder fixedPoints
 
 /-- **Kleene's fixed point Theorem**: The least fixed point in a complete lattice is
 the supremum of iterating a function on bottom arbitrary often. -/
-theorem lfp_eq_sSup_iterate (h : Continuous f) :
+theorem lfp_eq_sSup_iterate (h : ωScottContinuous f) :
     lfp f = ⨆ n, f^[n] ⊥ := by
   apply le_antisymm
   · apply lfp_le_fixed
-    exact Function.mem_fixedPoints.mp (ωSup_iterate_mem_fixedPoint ⟨f, h⟩ ⊥ bot_le)
+    exact Function.mem_fixedPoints.mp (ωSup_iterate_mem_fixedPoint
+      ⟨f, h.map_ωSup_of_orderHom⟩ ⊥ bot_le)
   · apply le_lfp
     intro a h_a
-    exact ωSup_iterate_le_prefixedPoint ⟨f, h⟩ ⊥ bot_le h_a bot_le
+    exact ωSup_iterate_le_prefixedPoint ⟨f, h.map_ωSup_of_orderHom⟩ ⊥ bot_le h_a bot_le
 
-theorem gfp_eq_sInf_iterate (h : Continuous (OrderHom.dual f)) :
+theorem gfp_eq_sInf_iterate (h : ωScottContinuous (OrderHom.dual f)) :
     gfp f = ⨅ n, f^[n] ⊤ :=
   lfp_eq_sSup_iterate (OrderHom.dual f) h
 
diff --git a/Mathlib/Order/Height.lean b/Mathlib/Order/Height.lean
index 531157c49d046..22b9697c5dd05 100644
--- a/Mathlib/Order/Height.lean
+++ b/Mathlib/Order/Height.lean
@@ -100,9 +100,9 @@ theorem exists_chain_of_le_chainHeight {n : ℕ} (hn : ↑n ≤ s.chainHeight) :
 
 theorem le_chainHeight_TFAE (n : ℕ) :
     TFAE [↑n ≤ s.chainHeight, ∃ l ∈ s.subchain, length l = n, ∃ l ∈ s.subchain, n ≤ length l] := by
-  tfae_have 1 → 2; · exact s.exists_chain_of_le_chainHeight
-  tfae_have 2 → 3; · rintro ⟨l, hls, he⟩; exact ⟨l, hls, he.ge⟩
-  tfae_have 3 → 1; · rintro ⟨l, hs, hn⟩; exact le_iSup₂_of_le l hs (WithTop.coe_le_coe.2 hn)
+  tfae_have 1 → 2 := s.exists_chain_of_le_chainHeight
+  tfae_have 2 → 3 := fun ⟨l, hls, he⟩ ↦ ⟨l, hls, he.ge⟩
+  tfae_have 3 → 1 := fun ⟨l, hs, hn⟩ ↦ le_iSup₂_of_le l hs (WithTop.coe_le_coe.2 hn)
   tfae_finish
 
 variable {s t}
@@ -169,10 +169,10 @@ theorem chainHeight_add_le_chainHeight_add (s : Set α) (t : Set β) (n m : ℕ)
 theorem chainHeight_le_chainHeight_TFAE (s : Set α) (t : Set β) :
     TFAE [s.chainHeight ≤ t.chainHeight, ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l = length l',
       ∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l ≤ length l'] := by
-  tfae_have 1 ↔ 3
-  · convert ← chainHeight_add_le_chainHeight_add s t 0 0 <;> apply add_zero
-  tfae_have 2 ↔ 3
-  · refine forall₂_congr fun l hl ↦ ?_
+  tfae_have 1 ↔ 3 := by
+    convert ← chainHeight_add_le_chainHeight_add s t 0 0 <;> apply add_zero
+  tfae_have 2 ↔ 3 := by
+    refine forall₂_congr fun l _ ↦ ?_
     simp_rw [← (le_chainHeight_TFAE t l.length).out 1 2, eq_comm]
   tfae_finish
 
diff --git a/Mathlib/Order/Heyting/Basic.lean b/Mathlib/Order/Heyting/Basic.lean
index 671d477c88c61..f12c306e19f10 100644
--- a/Mathlib/Order/Heyting/Basic.lean
+++ b/Mathlib/Order/Heyting/Basic.lean
@@ -124,36 +124,36 @@ theorem hnot_apply [∀ i, HNot (π i)] (a : ∀ i, π i) (i : ι) : (￢a) i =
 end Pi
 
 /-- A generalized Heyting algebra is a lattice with an additional binary operation `⇨` called
-Heyting implication such that `a ⇨` is right adjoint to `a ⊓`.
+Heyting implication such that `(a ⇨ ·)` is right adjoint to `(a ⊓ ·)`.
 
  This generalizes `HeytingAlgebra` by not requiring a bottom element. -/
 class GeneralizedHeytingAlgebra (α : Type*) extends Lattice α, OrderTop α, HImp α where
-  /-- `a ⇨` is right adjoint to `a ⊓` -/
+  /-- `(a ⇨ ·)` is right adjoint to `(a ⊓ ·)` -/
   le_himp_iff (a b c : α) : a ≤ b ⇨ c ↔ a ⊓ b ≤ c
 
 /-- A generalized co-Heyting algebra is a lattice with an additional binary
-difference operation `\` such that `\ a` is right adjoint to `⊔ a`.
+difference operation `\` such that `(· \ a)` is right adjoint to `(· ⊔ a)`.
 
 This generalizes `CoheytingAlgebra` by not requiring a top element. -/
 class GeneralizedCoheytingAlgebra (α : Type*) extends Lattice α, OrderBot α, SDiff α where
-  /-- `\ a` is right adjoint to `⊔ a` -/
+  /-- `(· \ a)` is right adjoint to `(· ⊔ a)` -/
   sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c
 
 /-- A Heyting algebra is a bounded lattice with an additional binary operation `⇨` called Heyting
-implication such that `a ⇨` is right adjoint to `a ⊓`. -/
+implication such that `(a ⇨ ·)` is right adjoint to `(a ⊓ ·)`. -/
 class HeytingAlgebra (α : Type*) extends GeneralizedHeytingAlgebra α, OrderBot α, HasCompl α where
-  /-- `a ⇨` is right adjoint to `a ⊓` -/
+  /-- `aᶜ` is defined as `a ⇨ ⊥` -/
   himp_bot (a : α) : a ⇨ ⊥ = aᶜ
 
 /-- A co-Heyting algebra is a bounded lattice with an additional binary difference operation `\`
-such that `\ a` is right adjoint to `⊔ a`. -/
+such that `(· \ a)` is right adjoint to `(· ⊔ a)`. -/
 class CoheytingAlgebra (α : Type*) extends GeneralizedCoheytingAlgebra α, OrderTop α, HNot α where
   /-- `⊤ \ a` is `￢a` -/
   top_sdiff (a : α) : ⊤ \ a = ￢a
 
 /-- A bi-Heyting algebra is a Heyting algebra that is also a co-Heyting algebra. -/
 class BiheytingAlgebra (α : Type*) extends HeytingAlgebra α, SDiff α, HNot α where
-  /-- `\ a` is right adjoint to `⊔ a` -/
+  /-- `(· \ a)` is right adjoint to `(· ⊔ a)` -/
   sdiff_le_iff (a b c : α) : a \ b ≤ c ↔ a ≤ b ⊔ c
   /-- `⊤ \ a` is `￢a` -/
   top_sdiff (a : α) : ⊤ \ a = ￢a
@@ -589,7 +589,7 @@ end GeneralizedCoheytingAlgebra
 
 section HeytingAlgebra
 
-variable [HeytingAlgebra α] {a b c : α}
+variable [HeytingAlgebra α] {a b : α}
 
 @[simp]
 theorem himp_bot (a : α) : a ⇨ ⊥ = aᶜ :=
@@ -760,7 +760,7 @@ end HeytingAlgebra
 
 section CoheytingAlgebra
 
-variable [CoheytingAlgebra α] {a b c : α}
+variable [CoheytingAlgebra α] {a b : α}
 
 @[simp]
 theorem top_sdiff' (a : α) : ⊤ \ a = ￢a :=
diff --git a/Mathlib/Order/Heyting/Hom.lean b/Mathlib/Order/Heyting/Hom.lean
index 358a7b6215806..6b0940817f6d3 100644
--- a/Mathlib/Order/Heyting/Hom.lean
+++ b/Mathlib/Order/Heyting/Hom.lean
@@ -190,7 +190,6 @@ theorem map_compl (a : α) : f aᶜ = (f a)ᶜ := by rw [← himp_bot, ← himp_
 @[simp]
 theorem map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b := by simp_rw [bihimp, map_inf, map_himp]
 
--- TODO: `map_bihimp`
 end HeytingAlgebra
 
 section CoheytingAlgebra
diff --git a/Mathlib/Order/Hom/Basic.lean b/Mathlib/Order/Hom/Basic.lean
index 107ce7003d6f8..96b44c3f6a264 100644
--- a/Mathlib/Order/Hom/Basic.lean
+++ b/Mathlib/Order/Hom/Basic.lean
@@ -111,7 +111,8 @@ abbrev OrderHomClass (F : Type*) (α β : outParam Type*) [LE α] [LE β] [FunLi
 /-- `OrderIsoClass F α β` states that `F` is a type of order isomorphisms.
 
 You should extend this class when you extend `OrderIso`. -/
-class OrderIsoClass (F α β : Type*) [LE α] [LE β] [EquivLike F α β] : Prop where
+class OrderIsoClass (F : Type*) (α β : outParam Type*) [LE α] [LE β] [EquivLike F α β] :
+    Prop where
   /-- An order isomorphism respects `≤`. -/
   map_le_map_iff (f : F) {a b : α} : f a ≤ f b ↔ a ≤ b
 
@@ -173,6 +174,10 @@ theorem map_inv_le_iff (f : F) {a : α} {b : β} : EquivLike.inv f b ≤ a ↔ b
   convert (map_le_map_iff f (a := EquivLike.inv f b) (b := a)).symm
   exact (EquivLike.right_inv f _).symm
 
+theorem map_inv_le_map_inv_iff (f : F) {a b : β} :
+    EquivLike.inv f b ≤ EquivLike.inv f a ↔ b ≤ a := by
+  simp
+
 -- Porting note: needed to add explicit arguments to map_le_map_iff
 @[simp]
 theorem le_map_inv_iff (f : F) {a : α} {b : β} : a ≤ EquivLike.inv f b ↔ f a ≤ b := by
@@ -191,6 +196,10 @@ theorem map_inv_lt_iff (f : F) {a : α} {b : β} : EquivLike.inv f b < a ↔ b <
   rw [← map_lt_map_iff f]
   simp only [EquivLike.apply_inv_apply]
 
+theorem map_inv_lt_map_inv_iff (f : F) {a b : β} :
+    EquivLike.inv f b < EquivLike.inv f a ↔ b < a := by
+  simp
+
 @[simp]
 theorem lt_map_inv_iff (f : F) {a : α} {b : β} : a < EquivLike.inv f b ↔ f a < b := by
   rw [← map_lt_map_iff f]
@@ -426,7 +435,7 @@ def coeFnHom : (α →o β) →o α → β where
   monotone' _ _ h := h
 
 /-- Function application `fun f => f a` (for fixed `a`) is a monotone function from the
-monotone function space `α →o β` to `β`. See also `Pi.evalOrderHom`.  -/
+monotone function space `α →o β` to `β`. See also `Pi.evalOrderHom`. -/
 @[simps! (config := .asFn)]
 def apply (x : α) : (α →o β) →o β :=
   (Pi.evalOrderHom x).comp coeFnHom
@@ -447,7 +456,7 @@ def piIso : (α →o ∀ i, π i) ≃o ∀ i, α →o π i where
   right_inv _ := rfl
   map_rel_iff' := forall_swap
 
-/-- `Subtype.val` as a bundled monotone function.  -/
+/-- `Subtype.val` as a bundled monotone function. -/
 @[simps (config := .asFn)]
 def Subtype.val (p : α → Prop) : Subtype p →o α :=
   ⟨_root_.Subtype.val, fun _ _ h => h⟩
@@ -906,7 +915,7 @@ open Set
 
 section LE
 
-variable [LE α] [LE β] [LE γ]
+variable [LE α] [LE β]
 
 --@[simp] Porting note (#10618): simp can prove it
 theorem le_iff_le (e : α ≃o β) {x y : α} : e x ≤ e y ↔ x ≤ y :=
@@ -920,7 +929,7 @@ theorem symm_apply_le (e : α ≃o β) {x : α} {y : β} : e.symm y ≤ x ↔ y
 
 end LE
 
-variable [Preorder α] [Preorder β] [Preorder γ]
+variable [Preorder α] [Preorder β]
 
 protected theorem monotone (e : α ≃o β) : Monotone e :=
   e.toOrderEmbedding.monotone
@@ -1044,7 +1053,7 @@ end Equiv
 namespace StrictMono
 
 variable [LinearOrder α] [Preorder β]
-variable (f : α → β) (h_mono : StrictMono f) (h_surj : Function.Surjective f)
+variable (f : α → β) (h_mono : StrictMono f)
 
 /-- A strictly monotone function with a right inverse is an order isomorphism. -/
 @[simps (config := .asFn)]
@@ -1097,6 +1106,18 @@ theorem OrderIso.map_sup [SemilatticeSup α] [SemilatticeSup β] (f : α ≃o β
     f (x ⊔ y) = f x ⊔ f y :=
   f.dual.map_inf x y
 
+theorem OrderIso.isMax_apply {α β : Type*} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} :
+    IsMax (f x) ↔ IsMax x := by
+  refine ⟨f.strictMono.isMax_of_apply, ?_⟩
+  conv_lhs => rw [← f.symm_apply_apply x]
+  exact f.symm.strictMono.isMax_of_apply
+
+theorem OrderIso.isMin_apply {α β : Type*} [Preorder α] [Preorder β] (f : α ≃o β) {x : α} :
+    IsMin (f x) ↔ IsMin x := by
+  refine ⟨f.strictMono.isMin_of_apply, ?_⟩
+  conv_lhs => rw [← f.symm_apply_apply x]
+  exact f.symm.strictMono.isMin_of_apply
+
 /-- Note that this goal could also be stated `(Disjoint on f) a b` -/
 theorem Disjoint.map_orderIso [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β]
     {a b : α} (f : α ≃o β) (ha : Disjoint a b) : Disjoint (f a) (f b) := by
@@ -1151,6 +1172,13 @@ theorem coe_toDualTopEquiv_eq [LE α] :
     (WithBot.toDualTopEquiv : WithBot αᵒᵈ → (WithTop α)ᵒᵈ) = toDual ∘ WithBot.ofDual :=
   funext fun _ => rfl
 
+/-- The coercion `α → WithBot α` bundled as monotone map. -/
+@[simps]
+def coeOrderHom {α : Type*} [Preorder α] : α ↪o WithBot α where
+  toFun := (↑)
+  inj' := WithBot.coe_injective
+  map_rel_iff' := WithBot.coe_le_coe
+
 end WithBot
 
 namespace WithTop
@@ -1182,6 +1210,13 @@ theorem coe_toDualBotEquiv [LE α] :
     (WithTop.toDualBotEquiv : WithTop αᵒᵈ → (WithBot α)ᵒᵈ) = toDual ∘ WithTop.ofDual :=
   funext fun _ => rfl
 
+/-- The coercion `α → WithTop α` bundled as monotone map. -/
+@[simps]
+def coeOrderHom {α : Type*} [Preorder α] : α ↪o WithTop α where
+  toFun := (↑)
+  inj' := WithTop.coe_injective
+  map_rel_iff' := WithTop.coe_le_coe
+
 end WithTop
 
 namespace OrderIso
@@ -1251,3 +1286,20 @@ theorem OrderIso.complementedLattice_iff (f : α ≃o β) :
 end BoundedOrder
 
 end LatticeIsos
+
+section DenselyOrdered
+
+lemma denselyOrdered_iff_of_orderIsoClass {X Y F : Type*} [Preorder X] [Preorder Y]
+    [EquivLike F X Y] [OrderIsoClass F X Y] (f : F) :
+    DenselyOrdered X ↔ DenselyOrdered Y := by
+  constructor
+  · intro H
+    refine ⟨fun a b h ↦ ?_⟩
+    obtain ⟨c, hc⟩ := exists_between ((map_inv_lt_map_inv_iff f).mpr h)
+    exact ⟨f c, by simpa using hc⟩
+  · intro H
+    refine ⟨fun a b h ↦ ?_⟩
+    obtain ⟨c, hc⟩ := exists_between ((map_lt_map_iff f).mpr h)
+    exact ⟨EquivLike.inv f c, by simpa using hc⟩
+
+end DenselyOrdered
diff --git a/Mathlib/Order/Hom/Bounded.lean b/Mathlib/Order/Hom/Bounded.lean
index 971df11a1b4ff..92fa0698730b3 100644
--- a/Mathlib/Order/Hom/Bounded.lean
+++ b/Mathlib/Order/Hom/Bounded.lean
@@ -59,14 +59,16 @@ section
 /-- `TopHomClass F α β` states that `F` is a type of `⊤`-preserving morphisms.
 
 You should extend this class when you extend `TopHom`. -/
-class TopHomClass (F α β : Type*) [Top α] [Top β] [FunLike F α β] : Prop where
+class TopHomClass (F : Type*) (α β : outParam Type*) [Top α] [Top β] [FunLike F α β] :
+    Prop where
   /-- A `TopHomClass` morphism preserves the top element. -/
   map_top (f : F) : f ⊤ = ⊤
 
 /-- `BotHomClass F α β` states that `F` is a type of `⊥`-preserving morphisms.
 
 You should extend this class when you extend `BotHom`. -/
-class BotHomClass (F α β : Type*) [Bot α] [Bot β] [FunLike F α β] : Prop where
+class BotHomClass (F : Type*) (α β : outParam Type*) [Bot α] [Bot β] [FunLike F α β] :
+    Prop where
   /-- A `BotHomClass` morphism preserves the bottom element. -/
   map_bot (f : F) : f ⊥ = ⊥
 
diff --git a/Mathlib/Order/Hom/CompleteLattice.lean b/Mathlib/Order/Hom/CompleteLattice.lean
index 004c9a77b9bef..8fecfd2875b9b 100644
--- a/Mathlib/Order/Hom/CompleteLattice.lean
+++ b/Mathlib/Order/Hom/CompleteLattice.lean
@@ -118,13 +118,13 @@ section Hom
 variable [FunLike F α β]
 
 @[simp] theorem map_iSup [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ι → α) :
-    f (⨆ i, g i) = ⨆ i, f (g i) := by simp [iSup, ← Set.range_comp, Function.comp]
+    f (⨆ i, g i) = ⨆ i, f (g i) := by simp [iSup, ← Set.range_comp, Function.comp_def]
 
 theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
     f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) := by simp_rw [map_iSup]
 
 @[simp] theorem map_iInf [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ι → α) :
-    f (⨅ i, g i) = ⨅ i, f (g i) := by simp [iInf, ← Set.range_comp, Function.comp]
+    f (⨅ i, g i) = ⨅ i, f (g i) := by simp [iInf, ← Set.range_comp, Function.comp_def]
 
 theorem map_iInf₂ [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
     f (⨅ (i) (j), g i j) = ⨅ (i) (j), f (g i j) := by simp_rw [map_iInf]
diff --git a/Mathlib/Order/Hom/Lattice.lean b/Mathlib/Order/Hom/Lattice.lean
index a6513d866da90..94af3f7566504 100644
--- a/Mathlib/Order/Hom/Lattice.lean
+++ b/Mathlib/Order/Hom/Lattice.lean
@@ -474,6 +474,24 @@ theorem bot_apply [Bot β] (a : α) : (⊥ : SupHom α β) a = ⊥ :=
 theorem top_apply [Top β] (a : α) : (⊤ : SupHom α β) a = ⊤ :=
   rfl
 
+/-- `Subtype.val` as a `SupHom`. -/
+def subtypeVal {P : β → Prop}
+    (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) :
+    letI := Subtype.semilatticeSup Psup
+    SupHom {x : β // P x} β :=
+  letI := Subtype.semilatticeSup Psup
+  .mk Subtype.val (by simp)
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) (x : {x : β // P x}) :
+    subtypeVal Psup x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop}
+    (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) :
+    ⇑(subtypeVal Psup) = Subtype.val := rfl
+
 end SupHom
 
 /-! ### Infimum homomorphisms -/
@@ -632,6 +650,24 @@ theorem bot_apply [Bot β] (a : α) : (⊥ : InfHom α β) a = ⊥ :=
 theorem top_apply [Top β] (a : α) : (⊤ : InfHom α β) a = ⊤ :=
   rfl
 
+/-- `Subtype.val` as an `InfHom`. -/
+def subtypeVal {P : β → Prop}
+    (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) :
+    letI := Subtype.semilatticeInf Pinf
+    InfHom {x : β // P x} β :=
+  letI := Subtype.semilatticeInf Pinf
+  .mk Subtype.val (by simp)
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) (x : {x : β // P x}) :
+    subtypeVal Pinf x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop}
+    (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) :
+    ⇑(subtypeVal Pinf) = Subtype.val := rfl
+
 end InfHom
 
 /-! ### Finitary supremum homomorphisms -/
@@ -762,6 +798,26 @@ theorem sup_apply (f g : SupBotHom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a :
 theorem bot_apply (a : α) : (⊥ : SupBotHom α β) a = ⊥ :=
   rfl
 
+/-- `Subtype.val` as a `SupBotHom`. -/
+def subtypeVal {P : β → Prop}
+    (Pbot : P ⊥) (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) :
+    letI := Subtype.orderBot Pbot
+    letI := Subtype.semilatticeSup Psup
+    SupBotHom {x : β // P x} β :=
+  letI := Subtype.orderBot Pbot
+  letI := Subtype.semilatticeSup Psup
+  .mk (SupHom.subtypeVal Psup) (by simp [Subtype.coe_bot Pbot])
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Pbot : P ⊥) (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) (x : {x : β // P x}) :
+    subtypeVal Pbot Psup x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop}
+    (Pbot : P ⊥) (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y)) :
+    ⇑(subtypeVal Pbot Psup) = Subtype.val := rfl
+
 end SupBotHom
 
 /-! ### Finitary infimum homomorphisms -/
@@ -893,6 +949,26 @@ theorem inf_apply (f g : InfTopHom α β) (a : α) : (f ⊓ g) a = f a ⊓ g a :
 theorem top_apply (a : α) : (⊤ : InfTopHom α β) a = ⊤ :=
   rfl
 
+/-- `Subtype.val` as an `InfTopHom`. -/
+def subtypeVal {P : β → Prop}
+    (Ptop : P ⊤) (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) :
+    letI := Subtype.orderTop Ptop
+    letI := Subtype.semilatticeInf Pinf
+    InfTopHom {x : β // P x} β :=
+  letI := Subtype.orderTop Ptop
+  letI := Subtype.semilatticeInf Pinf
+  .mk (InfHom.subtypeVal Pinf) (by simp [Subtype.coe_top Ptop])
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Ptop : P ⊤) (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) (x : {x : β // P x}) :
+    subtypeVal Ptop Pinf x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop}
+    (Ptop : P ⊤) (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) :
+    ⇑(subtypeVal Ptop Pinf) = Subtype.val := rfl
+
 end InfTopHom
 
 /-! ### Lattice homomorphisms -/
@@ -1013,6 +1089,25 @@ theorem cancel_left {g : LatticeHom β γ} {f₁ f₂ : LatticeHom α β} (hg :
   ⟨fun h => LatticeHom.ext fun a => hg <| by rw [← LatticeHom.comp_apply, h, LatticeHom.comp_apply],
     congr_arg _⟩
 
+/-- `Subtype.val` as a `LatticeHom`. -/
+def subtypeVal {P : β → Prop}
+    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
+    letI := Subtype.lattice Psup Pinf
+    LatticeHom {x : β // P x} β :=
+  letI := Subtype.lattice Psup Pinf
+  .mk (SupHom.subtypeVal Psup) (by simp)
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y))
+    (x : {x : β // P x}) :
+    subtypeVal Psup Pinf x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop}
+    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
+    ⇑(subtypeVal Psup Pinf) = Subtype.val := rfl
+
 end LatticeHom
 
 namespace OrderHomClass
@@ -1184,6 +1279,27 @@ theorem cancel_left {g : BoundedLatticeHom β γ} {f₁ f₂ : BoundedLatticeHom
     g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ :=
   ⟨fun h => ext fun a => hg <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩
 
+/-- `Subtype.val` as a `BoundedLatticeHom`. -/
+def subtypeVal {P : β → Prop} (Pbot : P ⊥) (Ptop : P ⊤)
+    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
+    letI := Subtype.lattice Psup Pinf
+    letI := Subtype.boundedOrder Pbot Ptop
+    BoundedLatticeHom {x : β // P x} β :=
+  letI := Subtype.lattice Psup Pinf
+  letI := Subtype.boundedOrder Pbot Ptop
+  .mk (.subtypeVal Psup Pinf) (by simp [Subtype.coe_top Ptop]) (by simp [Subtype.coe_bot Pbot])
+
+@[simp]
+lemma subtypeVal_apply {P : β → Prop}
+    (Pbot : P ⊥) (Ptop : P ⊤) (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y))
+    (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x : {x : β // P x}) :
+    subtypeVal Pbot Ptop Psup Pinf x = x := rfl
+
+@[simp]
+lemma subtypeVal_coe {P : β → Prop} (Pbot : P ⊥) (Ptop : P ⊤)
+    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
+    ⇑(subtypeVal Pbot Ptop Psup Pinf) = Subtype.val := rfl
+
 end BoundedLatticeHom
 
 /-! ### Dual homs -/
@@ -1660,3 +1776,5 @@ def withTopWithBot' [BoundedOrder β] (f : LatticeHom α β) :
   map_bot' := rfl
 
 end LatticeHom
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Order/Hom/Set.lean b/Mathlib/Order/Hom/Set.lean
index 6838a14b4bba9..382a2354e4f5d 100644
--- a/Mathlib/Order/Hom/Set.lean
+++ b/Mathlib/Order/Hom/Set.lean
@@ -5,7 +5,9 @@ Authors: Johan Commelin
 -/
 import Mathlib.Order.Hom.Basic
 import Mathlib.Logic.Equiv.Set
+import Mathlib.Data.Set.Monotone
 import Mathlib.Data.Set.Image
+import Mathlib.Order.WellFounded
 
 /-!
 # Order homomorphisms and sets
@@ -14,13 +16,13 @@ import Mathlib.Data.Set.Image
 
 open OrderDual
 
-variable {F α β γ δ : Type*}
+variable {α β : Type*}
 
 namespace OrderIso
 
 section LE
 
-variable [LE α] [LE β] [LE γ]
+variable [LE α] [LE β]
 
 theorem range_eq (e : α ≃o β) : Set.range e = Set.univ :=
   e.surjective.range_eq
@@ -56,7 +58,7 @@ end LE
 
 open Set
 
-variable [Preorder α] [Preorder β] [Preorder γ]
+variable [Preorder α]
 
 /-- Order isomorphism between two equal sets. -/
 def setCongr (s t : Set α) (h : s = t) :
@@ -119,6 +121,46 @@ theorem orderIsoOfSurjective_self_symm_apply (b : β) :
 
 end StrictMono
 
+/-- Two order embeddings on a well-order are equal provided that their ranges are equal. -/
+lemma OrderEmbedding.range_inj [LinearOrder α] [WellFoundedLT α] [Preorder β] {f g : α ↪o β} :
+    Set.range f = Set.range g ↔ f = g := by
+  rw [f.strictMono.range_inj g.strictMono, DFunLike.coe_fn_eq]
+
+namespace OrderIso
+
+-- These results are also true whenever β is well-founded instead of α.
+-- You can use `RelEmbedding.isWellFounded` to transfer the instance over.
+
+instance subsingleton_of_wellFoundedLT [LinearOrder α] [WellFoundedLT α] [Preorder β] :
+    Subsingleton (α ≃o β) := by
+  refine ⟨fun f g ↦ ?_⟩
+  rw [OrderIso.ext_iff, ← coe_toOrderEmbedding, ← coe_toOrderEmbedding, DFunLike.coe_fn_eq,
+    ← OrderEmbedding.range_inj, coe_toOrderEmbedding, coe_toOrderEmbedding, range_eq, range_eq]
+
+instance subsingleton_of_wellFoundedLT' [LinearOrder β] [WellFoundedLT β] [Preorder α] :
+    Subsingleton (α ≃o β) := by
+  refine ⟨fun f g ↦ ?_⟩
+  change f.symm.symm = g.symm.symm
+  rw [Subsingleton.elim f.symm]
+
+instance unique_of_wellFoundedLT [LinearOrder α] [WellFoundedLT α] : Unique (α ≃o α) := Unique.mk' _
+
+instance subsingleton_of_wellFoundedGT [LinearOrder α] [WellFoundedGT α] [Preorder β] :
+    Subsingleton (α ≃o β) := by
+  refine ⟨fun f g ↦ ?_⟩
+  change f.dual.dual = g.dual.dual
+  rw [Subsingleton.elim f.dual]
+
+instance subsingleton_of_wellFoundedGT' [LinearOrder β] [WellFoundedGT β] [Preorder α] :
+    Subsingleton (α ≃o β) := by
+  refine ⟨fun f g ↦ ?_⟩
+  change f.dual.dual = g.dual.dual
+  rw [Subsingleton.elim f.dual]
+
+instance unique_of_wellFoundedGT [LinearOrder α] [WellFoundedGT α] : Unique (α ≃o α) := Unique.mk' _
+
+end OrderIso
+
 section BooleanAlgebra
 
 variable (α) [BooleanAlgebra α]
diff --git a/Mathlib/Order/Ideal.lean b/Mathlib/Order/Ideal.lean
index f10e971e09067..3e6b195981b8f 100644
--- a/Mathlib/Order/Ideal.lean
+++ b/Mathlib/Order/Ideal.lean
@@ -476,7 +476,7 @@ instance : Inhabited (Cofinal P) :=
       mem_gt := fun x ↦ ⟨x, trivial, le_rfl⟩ }⟩
 
 instance : Membership P (Cofinal P) :=
-  ⟨fun x D ↦ x ∈ D.carrier⟩
+  ⟨fun D x ↦ x ∈ D.carrier⟩
 
 variable (D : Cofinal P) (x : P)
 
diff --git a/Mathlib/Order/InitialSeg.lean b/Mathlib/Order/InitialSeg.lean
index 8ab6a88adb7ee..f2e7f48acdac1 100644
--- a/Mathlib/Order/InitialSeg.lean
+++ b/Mathlib/Order/InitialSeg.lean
@@ -6,6 +6,7 @@ Authors: Mario Carneiro, Floris van Doorn
 import Mathlib.Logic.Equiv.Set
 import Mathlib.Order.RelIso.Set
 import Mathlib.Order.WellFounded
+import Mathlib.Data.Sum.Order
 /-!
 # Initial and principal segments
 
@@ -47,7 +48,7 @@ embedding whose range is an initial segment. That is, whenever `b < f a` in `β`
 range of `f`. -/
 structure InitialSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where
   /-- The order embedding is an initial segment -/
-  init' : ∀ a b, s b (toRelEmbedding a) → ∃ a', toRelEmbedding a' = b
+  mem_range_of_rel' : ∀ a b, s b (toRelEmbedding a) → b ∈ Set.range toRelEmbedding
 
 -- Porting note: Deleted `scoped[InitialSeg]`
 /-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≼i s` is an order
@@ -77,18 +78,24 @@ instance : EmbeddingLike (r ≼i s) α β where
 theorem coe_coe_fn (f : r ≼i s) : ((f : r ↪r s) : α → β) = f :=
   rfl
 
-theorem init (f : r ≼i s) {a : α} {b : β} : s b (f a) → ∃ a', f a' = b :=
-  f.init' _ _
+theorem mem_range_of_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) → b ∈ Set.range f :=
+  f.mem_range_of_rel' _ _
+
+@[deprecated mem_range_of_rel (since := "2024-09-21")]
+alias init := mem_range_of_rel
 
 theorem map_rel_iff {a b : α} (f : r ≼i s) : s (f a) (f b) ↔ r a b :=
   f.map_rel_iff'
 
-theorem init_iff (f : r ≼i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
+theorem exists_eq_iff_rel (f : r ≼i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
   ⟨fun h => by
-    rcases f.init h with ⟨a', rfl⟩
+    rcases f.mem_range_of_rel h with ⟨a', rfl⟩
     exact ⟨a', rfl, f.map_rel_iff.1 h⟩,
     fun ⟨a', e, h⟩ => e ▸ f.map_rel_iff.2 h⟩
 
+@[deprecated exists_eq_iff_rel (since := "2024-09-21")]
+alias init_iff := exists_eq_iff_rel
+
 /-- An order isomorphism is an initial segment -/
 def ofIso (f : r ≃r s) : r ≼i s :=
   ⟨f, fun _ b _ => ⟨f.symm b, RelIso.apply_symm_apply f _⟩⟩
@@ -123,7 +130,7 @@ instance subsingleton_of_trichotomous_of_irrefl [IsTrichotomous β s] [IsIrrefl
     ext a
     refine IsWellFounded.induction r a fun b IH =>
       extensional_of_trichotomous_of_irrefl s fun x => ?_
-    rw [f.init_iff, g.init_iff]
+    rw [f.exists_eq_iff_rel, g.exists_eq_iff_rel]
     exact exists_congr fun x => and_congr_left fun hx => IH _ hx ▸ Iff.rfl⟩
 
 instance [IsWellOrder β s] : Subsingleton (r ≼i s) :=
@@ -151,21 +158,24 @@ theorem antisymm_symm [IsWellOrder α r] [IsWellOrder β s] (f : r ≼i s) (g :
   RelIso.coe_fn_injective rfl
 
 theorem eq_or_principal [IsWellOrder β s] (f : r ≼i s) :
-    Surjective f ∨ ∃ b, ∀ x, s x b ↔ ∃ y, f y = x :=
-  or_iff_not_imp_right.2 fun h b =>
-    Acc.recOn (IsWellFounded.wf.apply b : Acc s b) fun x _ IH =>
-      not_forall_not.1 fun hn =>
-        h
-          ⟨x, fun y =>
-            ⟨IH _, fun ⟨a, e⟩ => by
-              rw [← e]
-              exact (trichotomous _ _).resolve_right
-                (not_or_of_not (hn a) fun hl => not_exists.2 hn (f.init hl))⟩⟩
+    Surjective f ∨ ∃ b, ∀ x, x ∈ Set.range f ↔ s x b := by
+  apply or_iff_not_imp_right.2
+  intro h b
+  push_neg at h
+  apply IsWellFounded.induction s b
+  intro x IH
+  obtain ⟨y, ⟨hy, hs⟩ | ⟨hy, hs⟩⟩ := h x
+  · obtain (rfl | h) := (trichotomous y x).resolve_left hs
+    · exact hy
+    · obtain ⟨z, rfl⟩ := hy
+      exact f.mem_range_of_rel h
+  · obtain ⟨z, rfl⟩ := IH y hs
+    cases hy (Set.mem_range_self z)
 
 /-- Restrict the codomain of an initial segment -/
 def codRestrict (p : Set β) (f : r ≼i s) (H : ∀ a, f a ∈ p) : r ≼i Subrel s p :=
   ⟨RelEmbedding.codRestrict p f H, fun a ⟨b, m⟩ h =>
-    let ⟨a', e⟩ := f.init h
+    let ⟨a', e⟩ := f.mem_range_of_rel h
     ⟨a', by subst e; rfl⟩⟩
 
 @[simp]
@@ -188,7 +198,7 @@ theorem leAdd_apply (r : α → α → Prop) (s : β → β → Prop) (a) : leAd
 protected theorem acc (f : r ≼i s) (a : α) : Acc r a ↔ Acc s (f a) :=
   ⟨by
     refine fun h => Acc.recOn h fun a _ ha => Acc.intro _ fun b hb => ?_
-    obtain ⟨a', rfl⟩ := f.init hb
+    obtain ⟨a', rfl⟩ := f.mem_range_of_rel hb
     exact ha _ (f.map_rel_iff.mp hb), f.toRelEmbedding.acc a⟩
 
 end InitialSeg
@@ -209,8 +219,8 @@ embeddings are called principal segments -/
 structure PrincipalSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where
   /-- The supremum of the principal segment -/
   top : β
-  /-- The image of the order embedding is the set of elements `b` such that `s b top` -/
-  down' : ∀ b, s b top ↔ ∃ a, toRelEmbedding a = b
+  /-- The range of the order embedding is the set of elements `b` such that `s b top` -/
+  mem_range_iff_rel' : ∀ b, b ∈ Set.range toRelEmbedding ↔ s b top
 
 -- Porting note: deleted `scoped[InitialSeg]`
 /-- If `r` is a relation on `α` and `s` in a relation on `β`, then `f : r ≺i s` is an order
@@ -230,24 +240,49 @@ instance : CoeFun (r ≺i s) fun _ => α → β :=
 theorem coe_fn_mk (f : r ↪r s) (t o) : (@PrincipalSeg.mk _ _ r s f t o : α → β) = f :=
   rfl
 
+theorem mem_range_iff_rel (f : r ≺i s) : ∀ {b : β}, b ∈ Set.range f ↔ s b f.top :=
+  f.mem_range_iff_rel' _
+
+@[deprecated mem_range_iff_rel (since := "2024-10-07")]
 theorem down (f : r ≺i s) : ∀ {b : β}, s b f.top ↔ ∃ a, f a = b :=
-  f.down' _
+  f.mem_range_iff_rel.symm
 
 theorem lt_top (f : r ≺i s) (a : α) : s (f a) f.top :=
-  f.down.2 ⟨_, rfl⟩
+  f.mem_range_iff_rel.1 ⟨_, rfl⟩
+
+theorem mem_range_of_rel_top (f : r ≺i s) {b : β} (h : s b f.top) : b ∈ Set.range f :=
+  f.mem_range_iff_rel.2 h
 
-theorem init [IsTrans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f a)) : ∃ a', f a' = b :=
-  f.down.1 <| _root_.trans h <| f.lt_top _
+theorem mem_range_of_rel [IsTrans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f a)) :
+    b ∈ Set.range f :=
+  f.mem_range_of_rel_top <| _root_.trans h <| f.lt_top _
+
+@[deprecated mem_range_of_rel (since := "2024-09-21")]
+alias init := mem_range_of_rel
 
 /-- A principal segment is in particular an initial segment. -/
 instance hasCoeInitialSeg [IsTrans β s] : Coe (r ≺i s) (r ≼i s) :=
-  ⟨fun f => ⟨f.toRelEmbedding, fun _ _ => f.init⟩⟩
+  ⟨fun f => ⟨f.toRelEmbedding, fun _ _ => f.mem_range_of_rel⟩⟩
 
 theorem coe_coe_fn' [IsTrans β s] (f : r ≺i s) : ((f : r ≼i s) : α → β) = f :=
   rfl
 
-theorem init_iff [IsTrans β s] (f : r ≺i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
-  @InitialSeg.init_iff α β r s f a b
+theorem exists_eq_iff_rel [IsTrans β s] (f : r ≺i s) {a : α} {b : β} :
+    s b (f a) ↔ ∃ a', f a' = b ∧ r a' a :=
+  @InitialSeg.exists_eq_iff_rel α β r s f a b
+
+@[deprecated exists_eq_iff_rel (since := "2024-09-21")]
+alias init_iff := exists_eq_iff_rel
+
+/-- A principal segment is the same as a non-surjective initial segment. -/
+noncomputable def _root_.InitialSeg.toPrincipalSeg [IsWellOrder β s] (f : r ≼i s)
+    (hf : ¬ Surjective f) : r ≺i s :=
+  ⟨f, _, Classical.choose_spec (f.eq_or_principal.resolve_left hf)⟩
+
+@[simp]
+theorem _root_.InitialSeg.toPrincipalSeg_apply [IsWellOrder β s] (f : r ≼i s)
+    (hf : ¬ Surjective f) (x : α) : f.toPrincipalSeg hf x = f x :=
+  rfl
 
 theorem irrefl {r : α → α → Prop} [IsWellOrder α r] (f : r ≺i r) : False := by
   have h := f.lt_top f.top
@@ -260,8 +295,7 @@ instance (r : α → α → Prop) [IsWellOrder α r] : IsEmpty (r ≺i r) :=
 /-- Composition of a principal segment with an initial segment, as a principal segment -/
 def ltLe (f : r ≺i s) (g : s ≼i t) : r ≺i t :=
   ⟨@RelEmbedding.trans _ _ _ r s t f g, g f.top, fun a => by
-    simp only [g.init_iff, PrincipalSeg.down, exists_and_left.symm, exists_swap,
-        RelEmbedding.trans_apply, exists_eq_right', InitialSeg.coe_coe_fn]⟩
+    simp [g.exists_eq_iff_rel, ← PrincipalSeg.mem_range_iff_rel, exists_swap, ← exists_and_left]⟩
 
 @[simp]
 theorem lt_le_apply (f : r ≺i s) (g : s ≼i t) (a : α) : (f.ltLe g) a = g (f a) :=
@@ -287,7 +321,7 @@ theorem trans_top [IsTrans γ t] (f : r ≺i s) (g : s ≺i t) : (f.trans g).top
 /-- Composition of an order isomorphism with a principal segment, as a principal segment -/
 def equivLT (f : r ≃r s) (g : s ≺i t) : r ≺i t :=
   ⟨@RelEmbedding.trans _ _ _ r s t f g, g.top, fun c =>
-    suffices (∃ a : β, g a = c) ↔ ∃ a : α, g (f a) = c by simpa [PrincipalSeg.down]
+    suffices (∃ a, g a = c) ↔ ∃ a, g (f a) = c by simp [← PrincipalSeg.mem_range_iff_rel]
     ⟨fun ⟨b, h⟩ => ⟨f.symm b, by simp only [h, RelIso.apply_symm_apply]⟩,
       fun ⟨a, h⟩ => ⟨f a, h⟩⟩⟩
 
@@ -296,8 +330,8 @@ def ltEquiv {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ →
     (g : s ≃r t) : PrincipalSeg r t :=
   ⟨@RelEmbedding.trans _ _ _ r s t f g, g f.top, by
     intro x
-    rw [← g.apply_symm_apply x, g.map_rel_iff, f.down', exists_congr]
-    intro y; exact ⟨congr_arg g, fun h => g.toEquiv.bijective.1 h⟩⟩
+    rw [← g.apply_symm_apply x, g.map_rel_iff, ← f.mem_range_iff_rel]
+    exact exists_congr <| fun _ ↦ ⟨fun h => g.toEquiv.bijective.1 h, congr_arg g⟩⟩
 
 @[simp]
 theorem equivLT_apply (f : r ≃r s) (g : s ≺i t) (a : α) : (equivLT f g) a = g (f a) :=
@@ -315,7 +349,7 @@ instance [IsWellOrder β s] : Subsingleton (r ≺i s) :=
       rw [@Subsingleton.elim _ _ (f : r ≼i s) g]
     have et : f.top = g.top := by
       refine extensional_of_trichotomous_of_irrefl s fun x => ?_
-      simp only [PrincipalSeg.down, ef]
+      simp only [← PrincipalSeg.mem_range_iff_rel, ef]
     cases f
     cases g
     have := RelEmbedding.coe_fn_injective ef; congr ⟩
@@ -330,7 +364,7 @@ theorem topLTTop {r : α → α → Prop} {s : β → β → Prop} {t : γ → 
 
 /-- Any element of a well order yields a principal segment -/
 def ofElement {α : Type*} (r : α → α → Prop) (a : α) : Subrel r { b | r b a } ≺i r :=
-  ⟨Subrel.relEmbedding _ _, a, fun _ => ⟨fun h => ⟨⟨_, h⟩, rfl⟩, fun ⟨⟨_, h⟩, rfl⟩ => h⟩⟩
+  ⟨Subrel.relEmbedding _ _, a, fun _ => ⟨fun ⟨⟨_, h⟩, rfl⟩ => h, fun h => ⟨⟨_, h⟩, rfl⟩⟩⟩
 
 -- This lemma was always bad, but the linter only noticed after lean4#2644
 @[simp, nolint simpNF]
@@ -346,7 +380,7 @@ theorem ofElement_top {α : Type*} (r : α → α → Prop) (a : α) : (ofElemen
 noncomputable def subrelIso (f : r ≺i s) : Subrel s {b | s b f.top} ≃r r :=
   RelIso.symm
   { toEquiv := ((Equiv.ofInjective f f.injective).trans (Equiv.setCongr
-      (funext fun _ ↦ propext f.down.symm))),
+      (funext fun _ ↦ propext f.mem_range_iff_rel))),
     map_rel_iff' := f.map_rel_iff }
 
 -- This lemma was always bad, but the linter only noticed after lean4#2644
@@ -360,15 +394,12 @@ theorem apply_subrelIso (f : r ≺i s) (b : {b | s b f.top}) :
 
 -- This lemma was always bad, but the linter only noticed after lean4#2644
 @[simp, nolint simpNF]
-theorem subrelIso_apply (f : r ≺i s) (a : α) :
-    f.subrelIso ⟨f a, f.down.mpr ⟨a, rfl⟩⟩ = a :=
+theorem subrelIso_apply (f : r ≺i s) (a : α) : f.subrelIso ⟨f a, f.lt_top a⟩ = a :=
   Equiv.ofInjective_symm_apply f.injective _
 
 /-- Restrict the codomain of a principal segment -/
 def codRestrict (p : Set β) (f : r ≺i s) (H : ∀ a, f a ∈ p) (H₂ : f.top ∈ p) : r ≺i Subrel s p :=
-  ⟨RelEmbedding.codRestrict p f H, ⟨f.top, H₂⟩, fun ⟨_, _⟩ =>
-    f.down.trans <|
-      exists_congr fun a => show (⟨f a, H a⟩ : p).1 = _ ↔ _ from ⟨Subtype.eq, congr_arg _⟩⟩
+  ⟨RelEmbedding.codRestrict p f H, ⟨f.top, H₂⟩, fun ⟨_, _⟩ => by simp [← f.mem_range_iff_rel]⟩
 
 @[simp]
 theorem codRestrict_apply (p) (f : r ≺i s) (H H₂ a) : codRestrict p f H H₂ a = ⟨f a, H a⟩ :=
@@ -382,7 +413,7 @@ theorem codRestrict_top (p) (f : r ≺i s) (H H₂) : (codRestrict p f H H₂).t
 def ofIsEmpty (r : α → α → Prop) [IsEmpty α] {b : β} (H : ∀ b', ¬s b' b) : r ≺i s :=
   { RelEmbedding.ofIsEmpty r s with
     top := b
-    down' := by simp [H] }
+    mem_range_iff_rel' := by simp [H] }
 
 @[simp]
 theorem ofIsEmpty_top (r : α → α → Prop) [IsEmpty α] {b : β} (H : ∀ b', ¬s b' b) :
@@ -409,7 +440,7 @@ theorem wellFounded_iff_wellFounded_subrel {β : Type*} {s : β → β → Prop}
     ⟨fun wf b => ⟨fun b' => ((PrincipalSeg.ofElement _ b).acc b').mpr (wf.apply b')⟩, fun wf =>
       ⟨fun b => Acc.intro _ fun b' hb' => ?_⟩⟩
   let f := PrincipalSeg.ofElement s b
-  obtain ⟨b', rfl⟩ := f.down.mp ((PrincipalSeg.ofElement_top s b).symm ▸ hb' : s b' f.top)
+  obtain ⟨b', rfl⟩ := f.mem_range_of_rel_top ((PrincipalSeg.ofElement_top s b).symm ▸ hb')
   exact (f.acc b').mp ((wf b).apply b')
 
 theorem wellFounded_iff_principalSeg.{u} {β : Type u} {s : β → β → Prop} [IsTrans β s] :
@@ -420,37 +451,40 @@ theorem wellFounded_iff_principalSeg.{u} {β : Type u} {s : β → β → Prop}
 /-! ### Properties of initial and principal segments -/
 
 
+namespace InitialSeg
+
 /-- To an initial segment taking values in a well order, one can associate either a principal
 segment (if the range is not everything, hence one can take as top the minimum of the complement
 of the range) or an order isomorphism (if the range is everything). -/
-noncomputable def InitialSeg.ltOrEq [IsWellOrder β s] (f : r ≼i s) : (r ≺i s) ⊕ (r ≃r s) := by
+noncomputable def ltOrEq [IsWellOrder β s] (f : r ≼i s) : (r ≺i s) ⊕ (r ≃r s) := by
   by_cases h : Surjective f
   · exact Sum.inr (RelIso.ofSurjective f h)
-  · have h' : _ := (InitialSeg.eq_or_principal f).resolve_left h
-    exact Sum.inl ⟨f, Classical.choose h', Classical.choose_spec h'⟩
+  · exact Sum.inl (f.toPrincipalSeg h)
 
-theorem InitialSeg.ltOrEq_apply_left [IsWellOrder β s] (f : r ≼i s) (g : r ≺i s) (a : α) :
+theorem ltOrEq_apply_left [IsWellOrder β s] (f : r ≼i s) (g : r ≺i s) (a : α) :
     g a = f a :=
   @InitialSeg.eq α β r s _ g f a
 
-theorem InitialSeg.ltOrEq_apply_right [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α) :
+theorem ltOrEq_apply_right [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α) :
     g a = f a :=
   InitialSeg.eq (InitialSeg.ofIso g) f a
 
 /-- Composition of an initial segment taking values in a well order and a principal segment. -/
-noncomputable def InitialSeg.leLT [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) :
+noncomputable def leLT [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) :
     r ≺i t :=
   match f.ltOrEq with
   | Sum.inl f' => f'.trans g
   | Sum.inr f' => PrincipalSeg.equivLT f' g
 
 @[simp]
-theorem InitialSeg.leLT_apply [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) (a : α) :
+theorem leLT_apply [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) (a : α) :
     (f.leLT g) a = g (f a) := by
   delta InitialSeg.leLT; cases' f.ltOrEq with f' f'
   · simp only [PrincipalSeg.trans_apply, f.ltOrEq_apply_left]
   · simp only [PrincipalSeg.equivLT_apply, f.ltOrEq_apply_right]
 
+end InitialSeg
+
 namespace RelEmbedding
 
 /-- Given an order embedding into a well order, collapse the order embedding by filling the
@@ -498,5 +532,23 @@ theorem collapse_apply [IsWellOrder β s] (f : r ↪r s) (a) : collapse f a = (c
   rfl
 
 end RelEmbedding
+
+/-- For any two well orders, one is an initial segment of the other. -/
+noncomputable def InitialSeg.total (r s) [IsWellOrder α r] [IsWellOrder β s] :
+    (r ≼i s) ⊕ (s ≼i r) :=
+  match (leAdd r s).ltOrEq, (RelEmbedding.sumLexInr r s).collapse.ltOrEq with
+  | Sum.inl f, Sum.inr g => Sum.inl <| f.ltEquiv g.symm
+  | Sum.inr f, Sum.inl g => Sum.inr <| g.ltEquiv f.symm
+  | Sum.inr f, Sum.inr g => Sum.inl <| InitialSeg.ofIso (f.trans g.symm)
+  | Sum.inl f, Sum.inl g => Classical.choice <| by
+      obtain h | h | h := trichotomous_of (Sum.Lex r s) f.top g.top
+      · exact ⟨Sum.inl <| (f.codRestrict {x | Sum.Lex r s x g.top}
+          (fun a => _root_.trans (f.lt_top a) h) h).ltEquiv g.subrelIso⟩
+      · let f := f.subrelIso
+        rw [h] at f
+        exact ⟨Sum.inl <| InitialSeg.ofIso (f.symm.trans g.subrelIso)⟩
+      · exact ⟨Sum.inr <| (g.codRestrict {x | Sum.Lex r s x f.top}
+          (fun a => _root_.trans (g.lt_top a) h) h).ltEquiv f.subrelIso⟩
+
 attribute [nolint simpNF] PrincipalSeg.ofElement_apply PrincipalSeg.subrelIso_symm_apply
   PrincipalSeg.apply_subrelIso PrincipalSeg.subrelIso_apply
diff --git a/Mathlib/Order/Interval/Basic.lean b/Mathlib/Order/Interval/Basic.lean
index bb31af6cef48d..a8d64a2279fcd 100644
--- a/Mathlib/Order/Interval/Basic.lean
+++ b/Mathlib/Order/Interval/Basic.lean
@@ -101,7 +101,7 @@ instance : Coe (NonemptyInterval α) (Set α) :=
   ⟨fun s => Icc s.fst s.snd⟩
 
 instance (priority := 100) : Membership α (NonemptyInterval α) :=
-  ⟨fun a s => a ∈ (s : Set α)⟩
+  ⟨fun s a => a ∈ (s : Set α)⟩
 
 @[simp]
 theorem mem_mk {hx : x.1 ≤ x.2} : a ∈ mk x hx ↔ x.1 ≤ a ∧ a ≤ x.2 :=
diff --git a/Mathlib/Order/Interval/Finset/Basic.lean b/Mathlib/Order/Interval/Finset/Basic.lean
index cd0f3a4b760b2..53ccb40f79265 100644
--- a/Mathlib/Order/Interval/Finset/Basic.lean
+++ b/Mathlib/Order/Interval/Finset/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Yaël Dillies
+Authors: Kim Morrison, Yaël Dillies
 -/
 import Mathlib.Order.Cover
 import Mathlib.Order.Interval.Finset.Defs
@@ -17,7 +17,7 @@ respectively, `⩿` and `⋖`, which then leads to a characterization of monoton
 functions whose domain is a locally finite order. In particular, this file proves:
 
 * `le_iff_transGen_wcovBy`: `≤` is the transitive closure of `⩿`
-* `lt_iff_transGen_covBy`: `≤` is the transitive closure of `⩿`
+* `lt_iff_transGen_covBy`: `<` is the transitive closure of `⋖`
 * `monotone_iff_forall_wcovBy`: Characterization of monotone functions
 * `strictMono_iff_forall_covBy`: Characterization of strictly monotone functions
 
@@ -51,18 +51,27 @@ section LocallyFiniteOrder
 
 variable [LocallyFiniteOrder α]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b := by
   rw [← coe_nonempty, coe_Icc, Set.nonempty_Icc]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.nonempty_Icc_of_le⟩ := nonempty_Icc
+
+@[simp]
 theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b := by
   rw [← coe_nonempty, coe_Ico, Set.nonempty_Ico]
 
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.nonempty_Ico_of_lt⟩ := nonempty_Ico
+
+@[simp]
 theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b := by
   rw [← coe_nonempty, coe_Ioc, Set.nonempty_Ioc]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.nonempty_Ioc_of_lt⟩ := nonempty_Ioc
+
 -- TODO: This is nonsense. A locally finite order is never densely ordered
 @[simp]
 theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by
@@ -113,19 +122,19 @@ theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ :=
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
-theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, true_and_iff, le_rfl]
+theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, true_and, le_rfl]
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
-theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp only [mem_Ico, true_and_iff, le_refl]
+theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by simp only [mem_Ico, true_and, le_refl]
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
-theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, and_true_iff, le_rfl]
+theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b := by simp only [mem_Icc, and_true, le_rfl]
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
-theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp only [mem_Ioc, and_true_iff, le_rfl]
+theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b := by simp only [mem_Ioc, and_true, le_rfl]
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
@@ -331,9 +340,18 @@ variable [LocallyFiniteOrderTop α]
 
 @[simp, aesop safe apply (rule_sets := [finsetNonempty])]
 lemma nonempty_Ici : (Ici a).Nonempty := ⟨a, mem_Ici.2 le_rfl⟩
-@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+@[simp]
 lemma nonempty_Ioi : (Ioi a).Nonempty ↔ ¬ IsMax a := by simp [Finset.Nonempty]
 
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.nonempty_Ioi_of_not_isMax⟩ := nonempty_Ioi
+
+theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a := by
+  simpa [← coe_subset] using Set.Ici_subset_Ici
+
+theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a := by
+  simpa [← coe_subset] using Set.Ioi_subset_Ioi h
+
 variable [LocallyFiniteOrder α]
 
 theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a := by
@@ -360,8 +378,19 @@ section LocallyFiniteOrderBot
 
 variable [LocallyFiniteOrderBot α]
 
-@[simp] lemma nonempty_Iic : (Iic a).Nonempty := ⟨a, mem_Iic.2 le_rfl⟩
-@[simp] lemma nonempty_Iio : (Iio a).Nonempty ↔ ¬ IsMin a := by simp [Finset.Nonempty]
+@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
+lemma nonempty_Iic : (Iic a).Nonempty := ⟨a, mem_Iic.2 le_rfl⟩
+@[simp]
+lemma nonempty_Iio : (Iio a).Nonempty ↔ ¬ IsMin a := by simp [Finset.Nonempty]
+
+@[aesop safe apply (rule_sets := [finsetNonempty])]
+alias ⟨_, Aesop.nonempty_Iio_of_not_isMin⟩ := nonempty_Iio
+
+theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b := by
+  simpa [← coe_subset] using Set.Iic_subset_Iic
+
+theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b := by
+  simpa [← coe_subset] using Set.Iio_subset_Iio h
 
 variable [LocallyFiniteOrder α]
 
@@ -654,7 +683,7 @@ variable [LinearOrder α]
 
 section LocallyFiniteOrder
 
-variable [LocallyFiniteOrder α] {a b : α}
+variable [LocallyFiniteOrder α]
 
 theorem Ico_subset_Ico_iff {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) :
     Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ := by
@@ -761,7 +790,7 @@ end LinearOrder
 
 section Lattice
 
-variable [Lattice α] [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ c x : α}
+variable [Lattice α] [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ x : α}
 
 theorem uIcc_toDual (a b : α) : [[toDual a, toDual b]] = [[a, b]].map toDual.toEmbedding :=
   Icc_toDual _ _
@@ -833,7 +862,7 @@ end Lattice
 
 section DistribLattice
 
-variable [DistribLattice α] [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ c x : α}
+variable [DistribLattice α] [LocallyFiniteOrder α] {a b c : α}
 
 theorem eq_of_mem_uIcc_of_mem_uIcc : a ∈ [[b, c]] → b ∈ [[a, c]] → a = b := by
   simp_rw [mem_uIcc]
@@ -854,7 +883,7 @@ end DistribLattice
 
 section LinearOrder
 
-variable [LinearOrder α] [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ c x : α}
+variable [LinearOrder α] [LocallyFiniteOrder α] {a a₁ a₂ b b₁ b₂ c : α}
 
 theorem Icc_min_max : Icc (min a b) (max a b) = [[a, b]] :=
   rfl
diff --git a/Mathlib/Order/Interval/Finset/Defs.lean b/Mathlib/Order/Interval/Finset/Defs.lean
index 8216c8ad29540..d81c2dec0066c 100644
--- a/Mathlib/Order/Interval/Finset/Defs.lean
+++ b/Mathlib/Order/Interval/Finset/Defs.lean
@@ -271,23 +271,23 @@ section LocallyFiniteOrder
 
 variable [LocallyFiniteOrder α] {a b x : α}
 
-/-- The finset of elements `x` such that `a ≤ x` and `x ≤ b`. Basically `Set.Icc a b` as a finset.
--/
+/-- The finset $[a, b]$ of elements `x` such that `a ≤ x` and `x ≤ b`. Basically `Set.Icc a b` as a
+finset. -/
 def Icc (a b : α) : Finset α :=
   LocallyFiniteOrder.finsetIcc a b
 
-/-- The finset of elements `x` such that `a ≤ x` and `x < b`. Basically `Set.Ico a b` as a finset.
--/
+/-- The finset $[a, b)$ of elements `x` such that `a ≤ x` and `x < b`. Basically `Set.Ico a b` as a
+finset. -/
 def Ico (a b : α) : Finset α :=
   LocallyFiniteOrder.finsetIco a b
 
-/-- The finset of elements `x` such that `a < x` and `x ≤ b`. Basically `Set.Ioc a b` as a finset.
--/
+/-- The finset $(a, b]$ of elements `x` such that `a < x` and `x ≤ b`. Basically `Set.Ioc a b` as a
+finset. -/
 def Ioc (a b : α) : Finset α :=
   LocallyFiniteOrder.finsetIoc a b
 
-/-- The finset of elements `x` such that `a < x` and `x < b`. Basically `Set.Ioo a b` as a finset.
--/
+/-- The finset $(a, b)$ of elements `x` such that `a < x` and `x < b`. Basically `Set.Ioo a b` as a
+finset. -/
 def Ioo (a b : α) : Finset α :=
   LocallyFiniteOrder.finsetIoo a b
 
@@ -329,11 +329,11 @@ section LocallyFiniteOrderTop
 
 variable [LocallyFiniteOrderTop α] {a x : α}
 
-/-- The finset of elements `x` such that `a ≤ x`. Basically `Set.Ici a` as a finset. -/
+/-- The finset $[a, ∞)$ of elements `x` such that `a ≤ x`. Basically `Set.Ici a` as a finset. -/
 def Ici (a : α) : Finset α :=
   LocallyFiniteOrderTop.finsetIci a
 
-/-- The finset of elements `x` such that `a < x`. Basically `Set.Ioi a` as a finset. -/
+/-- The finset $(a, ∞)$ of elements `x` such that `a < x`. Basically `Set.Ioi a` as a finset. -/
 def Ioi (a : α) : Finset α :=
   LocallyFiniteOrderTop.finsetIoi a
 
@@ -359,13 +359,13 @@ section LocallyFiniteOrderBot
 
 variable [LocallyFiniteOrderBot α] {a x : α}
 
-/-- The finset of elements `x` such that `a ≤ x`. Basically `Set.Iic a` as a finset. -/
-def Iic (a : α) : Finset α :=
-  LocallyFiniteOrderBot.finsetIic a
+/-- The finset $(-∞, b]$ of elements `x` such that `x ≤ b`. Basically `Set.Iic b` as a finset. -/
+def Iic (b : α) : Finset α :=
+  LocallyFiniteOrderBot.finsetIic b
 
-/-- The finset of elements `x` such that `a < x`. Basically `Set.Iio a` as a finset. -/
-def Iio (a : α) : Finset α :=
-  LocallyFiniteOrderBot.finsetIio a
+/-- The finset $(-∞, b)$ of elements `x` such that `x < b`. Basically `Set.Iio b` as a finset. -/
+def Iio (b : α) : Finset α :=
+  LocallyFiniteOrderBot.finsetIio b
 
 @[simp]
 theorem mem_Iic : x ∈ Iic a ↔ x ≤ a :=
@@ -884,7 +884,7 @@ instance locallyFiniteOrder : LocallyFiniteOrder (WithTop α) where
         rw [← some_eq_coe, some_mem_insertNone, mem_Ici]
     | (a : α), (b : α), ⊤ => by
         simp only [Embedding.some, mem_map, mem_Icc, and_false, exists_const, some, le_top,
-          top_le_iff]
+          top_le_iff, reduceCtorEq]
     | (a : α), (b : α), (x : α) => by
         simp only [le_eq_subset, Embedding.some, mem_map, mem_Icc, Embedding.coeFn_mk, coe_le_coe]
         -- This used to be in the above `simp` before leanprover/lean4#2644
diff --git a/Mathlib/Order/Interval/Finset/Nat.lean b/Mathlib/Order/Interval/Finset/Nat.lean
index 133d9233ac466..8be56194d6481 100644
--- a/Mathlib/Order/Interval/Finset/Nat.lean
+++ b/Mathlib/Order/Interval/Finset/Nat.lean
@@ -187,8 +187,8 @@ theorem Ico_image_const_sub_eq_Ico (hac : a ≤ c) :
 
 theorem Ico_succ_left_eq_erase_Ico : Ico a.succ b = erase (Ico a b) a := by
   ext x
-  rw [Ico_succ_left, mem_erase, mem_Ico, mem_Ioo, ← and_assoc, ne_comm, @and_comm (a ≠ x),
-    lt_iff_le_and_ne]
+  rw [Ico_succ_left, mem_erase, mem_Ico, mem_Ioo, ← and_assoc, ne_comm,
+    and_comm (a := a ≠ x), lt_iff_le_and_ne]
 
 theorem mod_injOn_Ico (n a : ℕ) : Set.InjOn (· % a) (Finset.Ico n (n + a)) := by
   induction' n with n ih
diff --git a/Mathlib/Order/Interval/Set/Basic.lean b/Mathlib/Order/Interval/Set/Basic.lean
index a8522056697d5..0be1e96eff231 100644
--- a/Mathlib/Order/Interval/Set/Basic.lean
+++ b/Mathlib/Order/Interval/Set/Basic.lean
@@ -38,35 +38,35 @@ section Preorder
 
 variable [Preorder α] {a a₁ a₂ b b₁ b₂ c x : α}
 
-/-- Left-open right-open interval -/
+/-- `Ioo a b` is the left-open right-open interval $(a, b)$. -/
 def Ioo (a b : α) :=
   { x | a < x ∧ x < b }
 
-/-- Left-closed right-open interval -/
+/-- `Ico a b` is the left-closed right-open interval $[a, b)$. -/
 def Ico (a b : α) :=
   { x | a ≤ x ∧ x < b }
 
-/-- Left-infinite right-open interval -/
-def Iio (a : α) :=
-  { x | x < a }
+/-- `Iio b` is the left-infinite right-open interval $(-∞, b)$. -/
+def Iio (b : α) :=
+  { x | x < b }
 
-/-- Left-closed right-closed interval -/
+/-- `Icc a b` is the left-closed right-closed interval $[a, b]$. -/
 def Icc (a b : α) :=
   { x | a ≤ x ∧ x ≤ b }
 
-/-- Left-infinite right-closed interval -/
+/-- `Iic b` is the left-infinite right-closed interval $(-∞, b]$. -/
 def Iic (b : α) :=
   { x | x ≤ b }
 
-/-- Left-open right-closed interval -/
+/-- `Ioc a b` is the left-open right-closed interval $(a, b]$. -/
 def Ioc (a b : α) :=
   { x | a < x ∧ x ≤ b }
 
-/-- Left-closed right-infinite interval -/
+/-- `Ici a` is the left-closed right-infinite interval $[a, ∞)$. -/
 def Ici (a : α) :=
   { x | a ≤ x }
 
-/-- Left-open right-infinite interval -/
+/-- `Ioi a` is the left-open right-infinite interval $(a, ∞)$. -/
 def Ioi (a : α) :=
   { x | a < x }
 
@@ -620,8 +620,8 @@ theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by
   refine ⟨fun h => ?_, ?_⟩
   · have hab : a ≤ b := nonempty_Icc.1 (h.symm.subst <| singleton_nonempty c)
     exact
-      ⟨eq_of_mem_singleton <| h.subst <| left_mem_Icc.2 hab,
-        eq_of_mem_singleton <| h.subst <| right_mem_Icc.2 hab⟩
+      ⟨eq_of_mem_singleton <| h ▸ left_mem_Icc.2 hab,
+        eq_of_mem_singleton <| h ▸ right_mem_Icc.2 hab⟩
   · rintro ⟨rfl, rfl⟩
     exact Icc_self _
 
@@ -1566,16 +1566,11 @@ theorem Ioc_union_Ioc_symm : Ioc a b ∪ Ioc b a = Ioc (min a b) (max a b) := by
 @[simp]
 theorem Ioc_union_Ioc_union_Ioc_cycle :
     Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc (min a (min b c)) (max a (max b c)) := by
-  rw [Ioc_union_Ioc, Ioc_union_Ioc] <;>
-  -- Porting note: mathlib3 proof finished from here as follows:
-  -- (It can probably be restored after https://github.com/leanprover-community/mathlib4/pull/856)
-  -- ac_rfl
-  -- all_goals
-  --   solve_by_elim (config := { max_depth := 5 }) [min_le_of_left_le, min_le_of_right_le,
-  --     le_max_of_le_left, le_max_of_le_right, le_refl]
-  simp [min_le_of_left_le, min_le_of_right_le, le_max_of_le_left, le_max_of_le_right, le_refl,
-    min_assoc, max_comm]
-
+  rw [Ioc_union_Ioc, Ioc_union_Ioc]
+  · ac_rfl
+  all_goals
+  solve_by_elim (config := { maxDepth := 5 }) [min_le_of_left_le, min_le_of_right_le,
+       le_max_of_le_left, le_max_of_le_right, le_refl]
 end LinearOrder
 
 /-!
@@ -1650,3 +1645,22 @@ instance : NoMaxOrder (Set.Iio x) :=
     exact ⟨⟨b, hb₂⟩, hb₁⟩⟩
 
 end Dense
+
+/-!
+### Intervals in `Prop`
+-/
+
+namespace Set
+
+@[simp] lemma Iic_False : Iic False = {False} := by aesop
+@[simp] lemma Iic_True : Iic True = univ := by aesop
+@[simp] lemma Ici_False : Ici False = univ := by aesop
+@[simp] lemma Ici_True : Ici True = {True} := by aesop
+@[simp] lemma Iio_False : Iio False = ∅ := by aesop
+@[simp] lemma Iio_True : Iio True = {False} := by aesop (add simp [Ioi, lt_iff_le_not_le])
+@[simp] lemma Ioi_False : Ioi False = {True} := by aesop (add simp [Ioi, lt_iff_le_not_le])
+@[simp] lemma Ioi_True : Ioi True = ∅ := by aesop
+
+end Set
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Order/Interval/Set/Image.lean b/Mathlib/Order/Interval/Set/Image.lean
index 0b194b81b848d..61ca9a56a17ad 100644
--- a/Mathlib/Order/Interval/Set/Image.lean
+++ b/Mathlib/Order/Interval/Set/Image.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger, Yaël Dillies
+Authors: Kim Morrison, Yaël Dillies
 -/
 import Mathlib.Order.Interval.Set.Basic
 import Mathlib.Data.Set.Function
diff --git a/Mathlib/Order/Interval/Set/Monotone.lean b/Mathlib/Order/Interval/Set/Monotone.lean
index 4b95aaa3c724e..6b8430931f6c7 100644
--- a/Mathlib/Order/Interval/Set/Monotone.lean
+++ b/Mathlib/Order/Interval/Set/Monotone.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.Order.Interval.Set.Disjoint
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 
 /-!
 # Monotonicity on intervals
@@ -188,18 +188,34 @@ theorem strictMonoOn_Iic_of_lt_succ [SuccOrder α] [IsSuccArchimedean α] {n : 
   refine hψ _ (lt_of_lt_of_le ?_ hy)
   rwa [Function.iterate_succ', Function.comp_apply, lt_succ_iff_not_isMax]
 
+theorem strictMono_of_lt_succ [SuccOrder α] [IsSuccArchimedean α]
+    (hψ : ∀ m, ψ m < ψ (succ m)) : StrictMono ψ := fun _ _ h ↦
+  (strictMonoOn_Iic_of_lt_succ fun m _ ↦ hψ m) h.le le_rfl h
+
 theorem strictAntiOn_Iic_of_succ_lt [SuccOrder α] [IsSuccArchimedean α] {n : α}
     (hψ : ∀ m, m < n → ψ (succ m) < ψ m) : StrictAntiOn ψ (Set.Iic n) := fun i hi j hj hij =>
   @strictMonoOn_Iic_of_lt_succ α βᵒᵈ _ _ ψ _ _ n hψ i hi j hj hij
 
+theorem strictAnti_of_succ_lt [SuccOrder α] [IsSuccArchimedean α]
+    (hψ : ∀ m, ψ (succ m) < ψ m) : StrictAnti ψ := fun _ _ h ↦
+  (strictAntiOn_Iic_of_succ_lt fun m _ ↦ hψ m) h.le le_rfl h
+
 theorem strictMonoOn_Ici_of_pred_lt [PredOrder α] [IsPredArchimedean α] {n : α}
     (hψ : ∀ m, n < m → ψ (pred m) < ψ m) : StrictMonoOn ψ (Set.Ici n) := fun i hi j hj hij =>
   @strictMonoOn_Iic_of_lt_succ αᵒᵈ βᵒᵈ _ _ ψ _ _ n hψ j hj i hi hij
 
+theorem strictMono_of_pred_lt [PredOrder α] [IsPredArchimedean α]
+    (hψ : ∀ m, ψ (pred m) < ψ m) : StrictMono ψ := fun _ _ h ↦
+  (strictMonoOn_Ici_of_pred_lt fun m _ ↦ hψ m) le_rfl h.le h
+
 theorem strictAntiOn_Ici_of_lt_pred [PredOrder α] [IsPredArchimedean α] {n : α}
     (hψ : ∀ m, n < m → ψ m < ψ (pred m)) : StrictAntiOn ψ (Set.Ici n) := fun i hi j hj hij =>
   @strictAntiOn_Iic_of_succ_lt αᵒᵈ βᵒᵈ _ _ ψ _ _ n hψ j hj i hi hij
 
+theorem strictAnti_of_lt_pred [PredOrder α] [IsPredArchimedean α]
+    (hψ : ∀ m, ψ m < ψ (pred m)) : StrictAnti ψ := fun _ _ h ↦
+  (strictAntiOn_Ici_of_lt_pred fun m _ ↦ hψ m) le_rfl h.le h
+
 end SuccOrder
 
 section LinearOrder
diff --git a/Mathlib/Order/Interval/Set/SurjOn.lean b/Mathlib/Order/Interval/Set/SurjOn.lean
index fd36a0e6fb432..8ebe61b8fbd13 100644
--- a/Mathlib/Order/Interval/Set/SurjOn.lean
+++ b/Mathlib/Order/Interval/Set/SurjOn.lean
@@ -35,8 +35,8 @@ theorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Functi
   · intro p hp
     rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')
     · exact mem_image_of_mem f (left_mem_Ico.mpr hab)
-    · have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
-      exact image_subset f Ioo_subset_Ico_self this
+    · exact image_subset f Ioo_subset_Ico_self <|
+        surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
   · rw [Ico_eq_empty (h_mono hab).not_lt]
     exact surjOn_empty f _
 
@@ -51,8 +51,8 @@ theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Functi
   rcases eq_endpoints_or_mem_Ioo_of_mem_Icc hp with (rfl | rfl | hp')
   · exact ⟨a, left_mem_Icc.mpr hab, rfl⟩
   · exact ⟨b, right_mem_Icc.mpr hab, rfl⟩
-  · have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
-    exact image_subset f Ioo_subset_Icc_self this
+  · exact image_subset f Ioo_subset_Icc_self <|
+      surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'
 
 theorem surjOn_Ioi_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
     (a : α) : SurjOn f (Ioi a) (Ioi (f a)) := by
diff --git a/Mathlib/Order/Interval/Set/UnorderedInterval.lean b/Mathlib/Order/Interval/Set/UnorderedInterval.lean
index a0556c3932ff6..ea241a7fbf3dc 100644
--- a/Mathlib/Order/Interval/Set/UnorderedInterval.lean
+++ b/Mathlib/Order/Interval/Set/UnorderedInterval.lean
@@ -44,7 +44,7 @@ namespace Set
 
 section Lattice
 
-variable [Lattice α] [Lattice β] {a a₁ a₂ b b₁ b₂ c x : α}
+variable [Lattice α] [Lattice β] {a a₁ a₂ b b₁ b₂ x : α}
 
 /-- `uIcc a b` is the set of elements lying between `a` and `b`, with `a` and `b` included.
 Note that we define it more generally in a lattice as `Set.Icc (a ⊓ b) (a ⊔ b)`. In a product type,
@@ -134,7 +134,7 @@ open Interval
 
 section DistribLattice
 
-variable [DistribLattice α] {a a₁ a₂ b b₁ b₂ c x : α}
+variable [DistribLattice α] {a b c : α}
 
 lemma eq_of_mem_uIcc_of_mem_uIcc (ha : a ∈ [[b, c]]) (hb : b ∈ [[a, c]]) : a = b :=
   eq_of_inf_eq_sup_eq (inf_congr_right ha.1 hb.1) <| sup_congr_right ha.2 hb.2
@@ -155,7 +155,7 @@ section LinearOrder
 variable [LinearOrder α]
 
 section Lattice
-variable [Lattice β] {f : α → β} {s : Set α} {a b : α}
+variable [Lattice β] {f : α → β} {a b : α}
 
 lemma _root_.MonotoneOn.mapsTo_uIcc (hf : MonotoneOn f (uIcc a b)) :
     MapsTo f (uIcc a b) (uIcc (f a) (f b)) := by
@@ -187,7 +187,7 @@ lemma _root_.Antitone.image_uIcc_subset (hf : Antitone f) : f '' uIcc a b ⊆ uI
 
 end Lattice
 
-variable [LinearOrder β] {f : α → β} {s : Set α} {a a₁ a₂ b b₁ b₂ c d x : α}
+variable [LinearOrder β] {f : α → β} {s : Set α} {a a₁ a₂ b b₁ b₂ c : α}
 
 theorem Icc_min_max : Icc (min a b) (max a b) = [[a, b]] :=
   rfl
@@ -294,8 +294,8 @@ lemma uIoc_injective_right (a : α) : Injective fun b => Ι b a := by
   rw [Set.ext_iff] at h
   obtain ha | ha := le_or_lt b a
   · have hb := (h b).not
-    simp only [ha, left_mem_uIoc, not_lt, true_iff_iff, not_mem_uIoc, ← not_le,
-      and_true_iff, not_true, false_and_iff, not_false_iff, true_iff_iff, or_false_iff] at hb
+    simp only [ha, left_mem_uIoc, not_lt, true_iff, not_mem_uIoc, ← not_le,
+      and_true, not_true, false_and, not_false_iff, or_false] at hb
     refine hb.eq_of_not_lt fun hc => ?_
     simpa [ha, and_iff_right hc, ← @not_le _ _ _ a, iff_not_self, -not_le] using h c
   · refine
diff --git a/Mathlib/Order/IsWellOrderLimitElement.lean b/Mathlib/Order/IsWellOrderLimitElement.lean
index 1628b90dac2f6..2d6bde7b9282b 100644
--- a/Mathlib/Order/IsWellOrderLimitElement.lean
+++ b/Mathlib/Order/IsWellOrderLimitElement.lean
@@ -25,7 +25,7 @@ section
 variable [IsWellOrder α (· < ·)]
 
 /-- Given an element `a : α` in a well ordered set, this is the successor of `a`,
-i.e. the smallest element stricly greater than `a` if it exists (or `a` itself otherwise). -/
+i.e. the smallest element strictly greater than `a` if it exists (or `a` itself otherwise). -/
 noncomputable def wellOrderSucc (a : α) : α :=
   (IsWellFounded.wf (r := (· < ·))).succ a
 
diff --git a/Mathlib/Order/KonigLemma.lean b/Mathlib/Order/KonigLemma.lean
index 64ff1a4d6bd67..b38fa275198d5 100644
--- a/Mathlib/Order/KonigLemma.lean
+++ b/Mathlib/Order/KonigLemma.lean
@@ -69,7 +69,7 @@ theorem exists_orderEmbedding_covby_of_forall_covby_finite (hfin : ∀ (a : α),
   obtain ⟨f, hf⟩ := exists_seq_covby_of_forall_covby_finite hfin hb
   exact ⟨OrderEmbedding.ofStrictMono f (strictMono_nat_of_lt_succ (fun i ↦ (hf.2 i).lt)), hf⟩
 
-/-- A version of Kőnig's lemma where the sequence starts at the minimum of an infinite order.  -/
+/-- A version of Kőnig's lemma where the sequence starts at the minimum of an infinite order. -/
 theorem exists_orderEmbedding_covby_of_forall_covby_finite_of_bot [OrderBot α] [Infinite α]
     (hfin : ∀ (a : α), {x | a ⋖ x}.Finite) : ∃ f : ℕ ↪o α, f 0 = ⊥ ∧ ∀ i, f i ⋖ f (i+1) :=
   exists_orderEmbedding_covby_of_forall_covby_finite hfin (by simpa using infinite_univ)
@@ -79,8 +79,9 @@ theorem GradeMinOrder.exists_nat_orderEmbedding_of_forall_covby_finite
     ∃ f : ℕ ↪o α, f 0 = ⊥ ∧ (∀ i, f i ⋖ f (i+1)) ∧ ∀ i, grade ℕ (f i) = i := by
   obtain ⟨f, h0, hf⟩ := exists_orderEmbedding_covby_of_forall_covby_finite_of_bot hfin
   refine ⟨f, h0, hf, fun i ↦ ?_⟩
-  induction' i with i ih; simp [h0]
-  simpa [Nat.covBy_iff_succ_eq, ih, eq_comm] using CovBy.grade ℕ <| hf i
+  induction' i with i ih
+  · simp [h0]
+  · simpa [Order.covBy_iff_add_one_eq, ih, eq_comm] using CovBy.grade ℕ <| hf i
 
 end Sequence
 
diff --git a/Mathlib/Order/KrullDimension.lean b/Mathlib/Order/KrullDimension.lean
index 6847b5762d6a8..4631e6250c4ba 100644
--- a/Mathlib/Order/KrullDimension.lean
+++ b/Mathlib/Order/KrullDimension.lean
@@ -1,26 +1,36 @@
 /-
 Copyright (c) 2023 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jujian Zhang, Fangming Li
+Authors: Jujian Zhang, Fangming Li, Joachim Breitner
 -/
 
 import Mathlib.Order.RelSeries
-import Mathlib.Order.WithBot
-import Mathlib.Data.Nat.Lattice
+import Mathlib.Data.ENat.Lattice
 
 /-!
-# Krull dimension of a preordered set
+# Krull dimension of a preordered set and height of an element
+
+If `α` is a preordered set, then `krullDim α : WithBot ℕ∞` is defined to be
+`sup {n | a₀ < a₁ < ... < aₙ}`.
 
-If `α` is a preordered set, then `krullDim α` is defined to be `sup {n | a₀ < a₁ < ... < aₙ}`.
 In case that `α` is empty, then its Krull dimension is defined to be negative infinity; if the
-length of all series `a₀ < a₁ < ... < aₙ` is unbounded, then its Krull dimension is defined to
-be positive infinity.
+length of all series `a₀ < a₁ < ... < aₙ` is unbounded, then its Krull dimension is defined to be
+positive infinity.
+
+For `a : α`, its height (in `ℕ∞`) is defined to be `sup {n | a₀ < a₁ < ... < aₙ ≤ a}` while its
+coheight is defined to be `sup {n | a ≤ a₀ < a₁ < ... < aₙ}` .
+
+## Main results
+
+* The Krull dimension is the same as that of the dual order (`krullDim_orderDual`).
 
-For `a : α`, its height is defined to be the krull dimension of the subset `(-∞, a]` while its
-coheight is defined to be the krull dimension of `[a, +∞)`.
+* The Krull dimension is the supremum of the heights of the elements (`krullDim_eq_iSup_height`).
 
-## Implementation notes
-Krull dimensions are defined to take value in `WithBot (WithTop ℕ)` so that `(-∞) + (+∞)` is
+* The height is monotone.
+
+## Design notes
+
+Krull dimensions are defined to take value in `WithBot ℕ∞` so that `(-∞) + (+∞)` is
 also negative infinity. This is because we want Krull dimensions to be additive with respect
 to product of varieties so that `-∞` being the Krull dimension of empty variety is equal to
 sum of `-∞` and the Krull dimension of any other varieties.
@@ -30,40 +40,152 @@ in this file would generalize as well. But we don't think it would be useful, so
 Krull dimension of a preorder.
 -/
 
-section definitions
+namespace Order
 
-variable (α : Type*) [Preorder α]
+section definitions
 
 /--
 The **Krull dimension** of a preorder `α` is the supremum of the rightmost index of all relation
-series of `α` order by `<`. If there is no series `a₀ < a₁ < ... < aₙ` in `α`, then its Krull
+series of `α` ordered by `<`. If there is no series `a₀ < a₁ < ... < aₙ` in `α`, then its Krull
 dimension is defined to be negative infinity; if the length of all series `a₀ < a₁ < ... < aₙ` is
 unbounded, its Krull dimension is defined to be positive infinity.
 -/
-noncomputable def krullDim : WithBot (WithTop ℕ) :=
+noncomputable def krullDim (α : Type*) [Preorder α] : WithBot ℕ∞ :=
   ⨆ (p : LTSeries α), p.length
 
 /--
-Height of an element `a` of a preordered set `α` is the Krull dimension of the subset `(-∞, a]`
+The **height** of an element `a` in a preorder `α` is the supremum of the rightmost index of all
+relation series of `α` ordered by `<` and ending below or at `a`.
 -/
-noncomputable def height (a : α) : WithBot (WithTop ℕ) := krullDim (Set.Iic a)
+noncomputable def height {α : Type*} [Preorder α] (a : α) : ℕ∞ :=
+  ⨆ (p : LTSeries α) (_ : p.last ≤ a), p.length
 
 /--
-Coheight of an element `a` of a pre-ordered set `α` is the Krull dimension of the subset `[a, +∞)`
+The **coheight** of an element `a` in a preorder `α` is the supremum of the rightmost index of all
+relation series of `α` ordered by `<` and beginning with `a`.
+
+The definition of `coheight` is via the `height` in the dual order, in order to easily transfer
+theorems between `height` and `coheight`.
 -/
-noncomputable def coheight (a : α) : WithBot (WithTop ℕ) := krullDim (Set.Ici a)
+noncomputable def coheight {α : Type*} [Preorder α] (a : α) : ℕ∞ := height (α := αᵒᵈ) a
 
 end definitions
 
+/-!
+## Height
+-/
+
+section height
+
+variable {α β : Type*}
+
+variable [Preorder α] [Preorder β]
+
+lemma height_le_iff {a : α} {n : ℕ∞} :
+    height a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, p.last ≤ a → p.length ≤ n := by
+ rw [height, iSup₂_le_iff]
+
+lemma height_le {a : α} {n : ℕ∞} (h : ∀ (p : LTSeries α), p.last = a → p.length ≤ n) :
+    height a ≤ n := by
+  apply height_le_iff.mpr
+  intro p hlast
+  wlog hlenpos : p.length ≠ 0
+  · simp_all
+  -- We replace the last element in the series with `a`
+  let p' := p.eraseLast.snoc a (lt_of_lt_of_le (p.eraseLast_last_rel_last (by simp_all)) hlast)
+  rw [show p.length = p'.length by simp [p']; omega]
+  apply h
+  simp [p']
+
+lemma height_le_iff' {a : α} {n : ℕ∞} :
+    height a ≤ n ↔ ∀ ⦃p : LTSeries α⦄, p.last = a → p.length ≤ n := by
+ constructor
+ · rw [height_le_iff]
+   exact fun h p hlast => h (le_of_eq hlast)
+ · exact height_le
+
+/--
+Alternative definition of height, with the supremum ranging only over those series that end at `a`.
+-/
+lemma height_eq_iSup_last_eq (a : α) :
+    height a = ⨆ (p : LTSeries α) (_ : p.last = a), ↑(p.length) := by
+  apply eq_of_forall_ge_iff
+  intro n
+  rw [height_le_iff', iSup₂_le_iff]
+
+lemma length_le_height {p : LTSeries α} {x : α} (hlast : p.last ≤ x) :
+    p.length ≤ height x := by
+  by_cases hlen0 : p.length ≠ 0
+  · let p' := p.eraseLast.snoc x (by
+      apply lt_of_lt_of_le
+      · apply p.step ⟨p.length - 1, by omega⟩
+      · convert hlast
+        simp only [Fin.succ_mk, Nat.succ_eq_add_one, RelSeries.last, Fin.last]
+        congr; omega)
+    suffices p'.length ≤ height x by
+      simp [p'] at this
+      convert this
+      norm_cast
+      omega
+    refine le_iSup₂_of_le p' ?_ le_rfl
+    simp [p']
+  · simp_all
+
+/--
+The height of the last element in a series is larger or equal to the length of the series.
+-/
+lemma length_le_height_last {p : LTSeries α} : p.length ≤ height p.last :=
+  length_le_height le_rfl
+
+/--
+The height of an element in a series is larger or equal to its index in the series.
+-/
+lemma index_le_height (p : LTSeries α) (i : Fin (p.length + 1)) : i ≤ height (p i) :=
+  length_le_height_last (p := p.take i)
+
+/--
+In a maximally long series, i.e one as long as the height of the last element, the height of each
+element is its index in the series.
+-/
+lemma height_eq_index_of_length_eq_height_last {p : LTSeries α} (h : p.length = height p.last)
+    (i : Fin (p.length + 1)) : height (p i) = i := by
+  refine le_antisymm (height_le ?_) (index_le_height p i)
+  intro p' hp'
+  have hp'' := length_le_height_last (p := p'.smash (p.drop i) (by simpa))
+  simp [← h] at hp''; clear h
+  norm_cast at *
+  omega
+
+lemma height_mono : Monotone (α := α) height :=
+  fun _ _ hab ↦ biSup_mono (fun _ hla => hla.trans hab)
+
+@[gcongr] protected lemma _root_.GCongr.height_le_height (a b : α) (hab : a ≤ b) :
+    height a ≤ height b := height_mono hab
+
+end height
+
+/-!
+## Krull dimension
+-/
+
 section krullDim
 
 variable {α β : Type*}
 
 variable [Preorder α] [Preorder β]
 
+lemma LTSeries.length_le_krullDim (p : LTSeries α) : p.length ≤ krullDim α := le_sSup ⟨_, rfl⟩
+
 lemma krullDim_nonneg_of_nonempty [Nonempty α] : 0 ≤ krullDim α :=
   le_sSup ⟨⟨0, fun _ ↦ @Nonempty.some α inferInstance, fun f ↦ f.elim0⟩, rfl⟩
 
+/-- A definition of krullDim for nonempty `α` that avoids `WithBot` -/
+lemma krullDim_eq_iSup_length [Nonempty α] :
+    krullDim α = ⨆ (p : LTSeries α), (p.length : ℕ∞) := by
+  unfold krullDim
+  rw [WithBot.coe_iSup (OrderTop.bddAbove _)]
+  rfl
+
 lemma krullDim_eq_bot_of_isEmpty [IsEmpty α] : krullDim α = ⊥ := WithBot.ciSup_empty _
 
 lemma krullDim_eq_top_of_infiniteDimensionalOrder [InfiniteDimensionalOrder α] :
@@ -81,9 +203,6 @@ lemma krullDim_eq_top_of_infiniteDimensionalOrder [InfiniteDimensionalOrder α]
 lemma krullDim_le_of_strictMono (f : α → β) (hf : StrictMono f) : krullDim α ≤ krullDim β :=
   iSup_le <| fun p ↦ le_sSup ⟨p.map f hf, rfl⟩
 
-lemma height_mono {a b : α} (h : a ≤ b) : height α a ≤ height α b :=
-  krullDim_le_of_strictMono (fun x ↦ ⟨x, le_trans x.2 h⟩) <| fun _ _ ↦ id
-
 lemma krullDim_eq_length_of_finiteDimensionalOrder [FiniteDimensionalOrder α] :
     krullDim α = (LTSeries.longestOf α).length :=
   le_antisymm
@@ -97,6 +216,14 @@ lemma krullDim_eq_zero_of_unique [Unique α] : krullDim α = 0 := by
   by_contra r
   exact ne_of_lt (q.step ⟨0, not_le.mp r⟩) <| Subsingleton.elim _ _
 
+lemma krullDim_nonpos_of_subsingleton [Subsingleton α] : krullDim α ≤ 0 := by
+  by_cases hα : Nonempty α
+  · have := uniqueOfSubsingleton (Classical.choice hα)
+    exact le_of_eq krullDim_eq_zero_of_unique
+  · have := not_nonempty_iff.mp hα
+    exact le_of_lt <| lt_of_eq_of_lt krullDim_eq_bot_of_isEmpty <|
+      Batteries.compareOfLessAndEq_eq_lt.mp rfl
+
 lemma krullDim_le_of_strictComono_and_surj
     (f : α → β) (hf : ∀ ⦃a b⦄, f a < f b → a < b) (hf' : Function.Surjective f) :
     krullDim β ≤ krullDim α :=
@@ -106,15 +233,29 @@ lemma krullDim_eq_of_orderIso (f : α ≃o β) : krullDim α = krullDim β :=
   le_antisymm (krullDim_le_of_strictMono _ f.strictMono) <|
     krullDim_le_of_strictMono _ f.symm.strictMono
 
-lemma krullDim_eq_iSup_height : krullDim α = ⨆ (a : α), height α a :=
-  le_antisymm
-    (iSup_le fun i ↦ le_iSup_of_le (i ⟨i.length, lt_add_one _⟩) <|
-    le_sSup ⟨⟨_, fun m ↦ ⟨i m, i.strictMono.monotone <| show m.1 ≤ i.length by omega⟩,
-      i.step⟩, rfl⟩) <|
-    iSup_le fun a ↦ krullDim_le_of_strictMono Subtype.val fun _ _ h ↦ h
-
 @[simp] lemma krullDim_orderDual : krullDim αᵒᵈ = krullDim α :=
   le_antisymm (iSup_le fun i ↦ le_sSup ⟨i.reverse, rfl⟩) <|
     iSup_le fun i ↦ le_sSup ⟨i.reverse, rfl⟩
 
+/--
+The Krull dimension is the supremum of the elements' heights.
+-/
+lemma krullDim_eq_iSup_height : krullDim α = ⨆ (a : α), ↑(height a) := by
+  cases isEmpty_or_nonempty α with
+  | inl h => simp [krullDim_eq_bot_of_isEmpty]
+  | inr h =>
+    rw [← WithBot.coe_iSup (OrderTop.bddAbove _)]
+    apply le_antisymm
+    · apply iSup_le
+      intro p
+      suffices p.length ≤ ⨆ (a : α), height a by
+        exact (WithBot.unbot'_le_iff fun _ => this).mp this
+      apply le_iSup_of_le p.last (length_le_height_last (p := p))
+    · rw [krullDim_eq_iSup_length]
+      simp only [WithBot.coe_le_coe, iSup_le_iff]
+      intro x
+      exact height_le fun p _ ↦ le_iSup_of_le p le_rfl
+
 end krullDim
+
+end Order
diff --git a/Mathlib/Order/Lattice.lean b/Mathlib/Order/Lattice.lean
index 26522df88f179..8383d090d9c12 100644
--- a/Mathlib/Order/Lattice.lean
+++ b/Mathlib/Order/Lattice.lean
@@ -6,7 +6,7 @@ Authors: Johannes Hölzl
 import Mathlib.Data.Bool.Basic
 import Mathlib.Order.Monotone.Basic
 import Mathlib.Order.ULift
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 
 /-!
 # (Semi-)lattices
@@ -196,7 +196,7 @@ instance : Std.Commutative (α := α) (· ⊔ ·) := ⟨sup_comm⟩
 theorem sup_assoc (a b c : α) : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) :=
   eq_of_forall_ge_iff fun x => by simp only [sup_le_iff]; rw [and_assoc]
 
-instance : Std.Associative (α := α)  (· ⊔ ·) := ⟨sup_assoc⟩
+instance : Std.Associative (α := α) (· ⊔ ·) := ⟨sup_assoc⟩
 
 theorem sup_left_right_swap (a b c : α) : a ⊔ b ⊔ c = c ⊔ b ⊔ a := by
   rw [sup_comm, sup_comm a, sup_assoc]
@@ -528,7 +528,7 @@ def Lattice.mk' {α : Type*} [Sup α] [Inf α] (sup_comm : ∀ a b : α, a ⊔ b
 
 section Lattice
 
-variable [Lattice α] {a b c d : α}
+variable [Lattice α] {a b c : α}
 
 theorem inf_le_sup : a ⊓ b ≤ a ⊔ b :=
   inf_le_left.trans le_sup_left
@@ -901,9 +901,27 @@ theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h :
     f (x ⊓ y) ≤ f x ⊓ f y :=
   le_inf (h inf_le_left) (h inf_le_right)
 
+theorem of_map_inf_le_left [SemilatticeInf α] [Preorder β] {f : α → β}
+    (h : ∀ x y, f (x ⊓ y) ≤ f x) : Monotone f := by
+  intro x y hxy
+  rw [← inf_eq_right.2 hxy]
+  apply h
+
+theorem of_map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
+    (h : ∀ x y, f (x ⊓ y) ≤ f x ⊓ f y) : Monotone f :=
+  of_map_inf_le_left fun x y ↦ (h x y).trans inf_le_left
+
 theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
     (h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f :=
-  fun x y hxy => inf_eq_left.1 <| by rw [← h, inf_eq_left.2 hxy]
+  of_map_inf_le fun x y ↦ (h x y).le
+
+theorem of_left_le_map_sup [SemilatticeSup α] [Preorder β] {f : α → β}
+    (h : ∀ x y, f x ≤ f (x ⊔ y)) : Monotone f :=
+  monotone_dual_iff.1 <| of_map_inf_le_left h
+
+theorem of_le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
+    (h : ∀ x y, f x ⊔ f y ≤ f (x ⊔ y)) : Monotone f :=
+  monotone_dual_iff.mp <| of_map_inf_le h
 
 theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
     (h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f :=
diff --git a/Mathlib/Order/LiminfLimsup.lean b/Mathlib/Order/LiminfLimsup.lean
index 2661fb80c53a6..0d6faea4d46f6 100644
--- a/Mathlib/Order/LiminfLimsup.lean
+++ b/Mathlib/Order/LiminfLimsup.lean
@@ -3,6 +3,9 @@ Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
 -/
+import Mathlib.Algebra.BigOperators.Group.Finset
+import Mathlib.Algebra.Order.Group.Defs
+import Mathlib.Algebra.Order.Group.Unbundled.Abs
 import Mathlib.Order.Filter.Cofinite
 import Mathlib.Order.Hom.CompleteLattice
 
@@ -123,9 +126,17 @@ lemma isBoundedUnder_iff_eventually_bddBelow :
 lemma _root_.BddAbove.isBoundedUnder (hs : s ∈ f) (hu : BddAbove (u '' s)) :
     f.IsBoundedUnder (· ≤ ·) u := isBoundedUnder_iff_eventually_bddAbove.2 ⟨_, hu, hs⟩
 
+/-- A bounded above function `u` is in particular eventually bounded above. -/
+lemma _root_.BddAbove.isBoundedUnder_of_range (hu : BddAbove (Set.range u)) :
+    f.IsBoundedUnder (· ≤ ·) u := BddAbove.isBoundedUnder (s := univ) f.univ_mem (by simpa)
+
 lemma _root_.BddBelow.isBoundedUnder (hs : s ∈ f) (hu : BddBelow (u '' s)) :
     f.IsBoundedUnder (· ≥ ·) u := isBoundedUnder_iff_eventually_bddBelow.2 ⟨_, hu, hs⟩
 
+/-- A bounded below function `u` is in particular eventually bounded below. -/
+lemma _root_.BddBelow.isBoundedUnder_of_range (hu : BddBelow (Set.range u)) :
+    f.IsBoundedUnder (· ≥ ·) u := BddBelow.isBoundedUnder (s := univ) f.univ_mem (by simpa)
+
 end Preorder
 
 theorem _root_.Monotone.isBoundedUnder_le_comp [Preorder α] [Preorder β] {l : Filter γ} {u : γ → α}
@@ -443,7 +454,7 @@ macro "isBoundedDefault" : tactic =>
 
 section ConditionallyCompleteLattice
 
-variable [ConditionallyCompleteLattice α]
+variable [ConditionallyCompleteLattice α] {s : Set α} {u : β → α}
 
 -- Porting note: Renamed from Limsup and Liminf to limsSup and limsInf
 /-- The `limsSup` of a filter `f` is the infimum of the `a` such that, eventually for `f`,
@@ -632,12 +643,17 @@ theorem liminf_le_liminf_of_le {α β} [ConditionallyCompleteLattice β] {f g :
     liminf u f ≤ liminf u g :=
   limsInf_le_limsInf_of_le (map_mono h) hf hg
 
-theorem limsSup_principal {s : Set α} (h : BddAbove s) (hs : s.Nonempty) :
-    limsSup (𝓟 s) = sSup s := by
-  simp only [limsSup, eventually_principal]; exact csInf_upper_bounds_eq_csSup h hs
+lemma limsSup_principal_eq_csSup (h : BddAbove s) (hs : s.Nonempty) : limsSup (𝓟 s) = sSup s := by
+  simp only [limsSup, eventually_principal]; exact csInf_upperBounds_eq_csSup h hs
+
+lemma limsInf_principal_eq_csSup (h : BddBelow s) (hs : s.Nonempty) : limsInf (𝓟 s) = sInf s :=
+  limsSup_principal_eq_csSup (α := αᵒᵈ) h hs
 
-theorem limsInf_principal {s : Set α} (h : BddBelow s) (hs : s.Nonempty) : limsInf (𝓟 s) = sInf s :=
-  limsSup_principal (α := αᵒᵈ) h hs
+lemma limsup_top_eq_ciSup [Nonempty β] (hu : BddAbove (range u)) : limsup u ⊤ = ⨆ i, u i := by
+  rw [limsup, map_top, limsSup_principal_eq_csSup hu (range_nonempty _), sSup_range]
+
+lemma liminf_top_eq_ciInf [Nonempty β] (hu : BddBelow (range u)) : liminf u ⊤ = ⨅ i, u i := by
+  rw [liminf, map_top, limsInf_principal_eq_csSup hu (range_nonempty _), sInf_range]
 
 theorem limsup_congr {α : Type*} [ConditionallyCompleteLattice β] {f : Filter α} {u v : α → β}
     (h : ∀ᶠ a in f, u a = v a) : limsup u f = limsup v f := by
@@ -786,7 +802,22 @@ theorem HasBasis.limsup_eq_iInf_iSup {p : ι → Prop} {s : ι → Set β} {f :
     (h : f.HasBasis p s) : limsup u f = ⨅ (i) (_ : p i), ⨆ a ∈ s i, u a :=
   (h.map u).limsSup_eq_iInf_sSup.trans <| by simp only [sSup_image, id]
 
-@[simp] lemma limsup_top (u : β → α) : limsup u ⊤ = ⨆ i, u i := by simp [limsup_eq_iInf_iSup]
+lemma limsSup_principal_eq_sSup (s : Set α) : limsSup (𝓟 s) = sSup s := by
+  simpa only [limsSup, eventually_principal] using sInf_upperBounds_eq_csSup s
+
+lemma limsInf_principal_eq_sInf (s : Set α) : limsInf (𝓟 s) = sInf s := by
+  simpa only [limsInf, eventually_principal] using sSup_lowerBounds_eq_sInf s
+
+@[simp] lemma limsup_top_eq_iSup (u : β → α) : limsup u ⊤ = ⨆ i, u i := by
+  rw [limsup, map_top, limsSup_principal_eq_sSup, sSup_range]
+
+@[simp] lemma liminf_top_eq_iInf (u : β → α) : liminf u ⊤ = ⨅ i, u i := by
+  rw [liminf, map_top, limsInf_principal_eq_sInf, sInf_range]
+
+@[deprecated (since := "2024-08-27")] alias limsSup_principal := limsSup_principal_eq_sSup
+@[deprecated (since := "2024-08-27")] alias limsInf_principal := limsInf_principal_eq_sInf
+@[deprecated (since := "2024-08-27")] alias limsup_top := limsup_top_eq_iSup
+@[deprecated (since := "2024-08-27")] alias liminf_top := liminf_top_eq_iInf
 
 theorem blimsup_congr' {f : Filter β} {p q : β → Prop} {u : β → α}
     (h : ∀ᶠ x in f, u x ≠ ⊥ → (p x ↔ q x)) : blimsup u f p = blimsup u f q := by
@@ -825,8 +856,6 @@ theorem liminf_eq_iSup_iInf_of_nat {u : ℕ → α} : liminf u atTop = ⨆ n : 
 theorem liminf_eq_iSup_iInf_of_nat' {u : ℕ → α} : liminf u atTop = ⨆ n : ℕ, ⨅ i : ℕ, u (i + n) :=
   @limsup_eq_iInf_iSup_of_nat' αᵒᵈ _ _
 
-@[simp] lemma liminf_top (u : β → α) : liminf u ⊤ = ⨅ i, u i := by simp [liminf_eq_iSup_iInf]
-
 theorem HasBasis.liminf_eq_iSup_iInf {p : ι → Prop} {s : ι → Set β} {f : Filter β} {u : β → α}
     (h : f.HasBasis p s) : liminf u f = ⨆ (i) (_ : p i), ⨅ a ∈ s i, u a :=
   HasBasis.limsup_eq_iInf_iSup (α := αᵒᵈ) h
@@ -1092,11 +1121,11 @@ theorem liminf_sdiff [NeBot f] (a : α) : liminf u f \ a = liminf (fun b => u b
 
 theorem sdiff_limsup [NeBot f] (a : α) : a \ limsup u f = liminf (fun b => a \ u b) f := by
   rw [← compl_inj_iff]
-  simp only [sdiff_eq, liminf_compl, (· ∘ ·), compl_inf, compl_compl, sup_limsup]
+  simp only [sdiff_eq, liminf_compl, comp_def, compl_inf, compl_compl, sup_limsup]
 
 theorem sdiff_liminf (a : α) : a \ liminf u f = limsup (fun b => a \ u b) f := by
   rw [← compl_inj_iff]
-  simp only [sdiff_eq, limsup_compl, (· ∘ ·), compl_inf, compl_compl, sup_liminf]
+  simp only [sdiff_eq, limsup_compl, comp_def, compl_inf, compl_compl, sup_liminf]
 
 end CompleteBooleanAlgebra
 
@@ -1130,12 +1159,12 @@ theorem cofinite.bliminf_set_eq : bliminf s cofinite p = { x | { n | p n ∧ x 
 /-- In other words, `limsup cofinite s` is the set of elements lying inside the family `s`
 infinitely often. -/
 theorem cofinite.limsup_set_eq : limsup s cofinite = { x | { n | x ∈ s n }.Infinite } := by
-  simp only [← cofinite.blimsup_true s, cofinite.blimsup_set_eq, true_and_iff]
+  simp only [← cofinite.blimsup_true s, cofinite.blimsup_set_eq, true_and]
 
 /-- In other words, `liminf cofinite s` is the set of elements lying outside the family `s`
 finitely often. -/
 theorem cofinite.liminf_set_eq : liminf s cofinite = { x | { n | x ∉ s n }.Finite } := by
-  simp only [← cofinite.bliminf_true s, cofinite.bliminf_set_eq, true_and_iff]
+  simp only [← cofinite.bliminf_true s, cofinite.bliminf_set_eq, true_and]
 
 theorem exists_forall_mem_of_hasBasis_mem_blimsup {l : Filter β} {b : ι → Set β} {q : ι → Prop}
     (hl : l.HasBasis q b) {u : β → Set α} {p : β → Prop} {x : α} (hx : x ∈ blimsup u l p) :
@@ -1707,3 +1736,5 @@ lemma Antitone.isCoboundedUnder_ge_of_isCobounded {f : R → S} (f_decr : Antito
   Monotone.isCoboundedUnder_le_of_isCobounded (S := Sᵒᵈ) f_decr cobdd
 
 end frequently_bounded
+
+set_option linter.style.longFile 1800
diff --git a/Mathlib/Order/Max.lean b/Mathlib/Order/Max.lean
index 8938fb68ba6e2..36518a3cb4aa5 100644
--- a/Mathlib/Order/Max.lean
+++ b/Mathlib/Order/Max.lean
@@ -158,7 +158,7 @@ theorem NoMaxOrder.not_acc [LT α] [NoMaxOrder α] (a : α) : ¬Acc (· > ·) a
 
 section LE
 
-variable [LE α] {a b : α}
+variable [LE α] {a : α}
 
 /-- `a : α` is a bottom element of `α` if it is less than or equal to any other element of `α`.
 This predicate is roughly an unbundled version of `OrderBot`, except that a preorder may have
@@ -332,11 +332,31 @@ protected theorem IsMax.eq_of_le (ha : IsMax a) (h : a ≤ b) : a = b :=
 protected theorem IsMax.eq_of_ge (ha : IsMax a) (h : a ≤ b) : b = a :=
   h.antisymm' <| ha h
 
+protected theorem IsBot.lt_of_ne (ha : IsBot a) (h : a ≠ b) : a < b :=
+  (ha b).lt_of_ne h
+
+protected theorem IsTop.lt_of_ne (ha : IsTop a) (h : b ≠ a) : b < a :=
+  (ha b).lt_of_ne h
+
+protected theorem IsBot.not_isMax [Nontrivial α] (ha : IsBot a) : ¬ IsMax a := by
+  intro ha'
+  obtain ⟨b, hb⟩ := exists_ne a
+  exact hb <| ha'.eq_of_ge (ha.lt_of_ne hb.symm).le
+
+protected theorem IsTop.not_isMin [Nontrivial α] (ha : IsTop a) : ¬ IsMin a :=
+  ha.toDual.not_isMax
+
+protected theorem IsBot.not_isTop [Nontrivial α] (ha : IsBot a) : ¬ IsTop a :=
+  mt IsTop.isMax ha.not_isMax
+
+protected theorem IsTop.not_isBot [Nontrivial α] (ha : IsTop a) : ¬ IsBot a :=
+  ha.toDual.not_isTop
+
 end PartialOrder
 
 section Prod
 
-variable [Preorder α] [Preorder β] {a a₁ a₂ : α} {b b₁ b₂ : β} {x y : α × β}
+variable [Preorder α] [Preorder β] {a : α} {b : β} {x : α × β}
 
 theorem IsBot.prod_mk (ha : IsBot a) (hb : IsBot b) : IsBot (a, b) := fun _ => ⟨ha _, hb _⟩
 
diff --git a/Mathlib/Order/MinMax.lean b/Mathlib/Order/MinMax.lean
index cd79b7297eee5..f815d61f44816 100644
--- a/Mathlib/Order/MinMax.lean
+++ b/Mathlib/Order/MinMax.lean
@@ -3,6 +3,8 @@ Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
+import Mathlib.Init.Logic
+import Mathlib.Logic.OpClass
 import Mathlib.Order.Lattice
 
 /-!
@@ -188,14 +190,14 @@ theorem max_lt_max (h₁ : a < c) (h₂ : b < d) : max a b < max c d :=
 theorem min_lt_min (h₁ : a < c) (h₂ : b < d) : min a b < min c d :=
   @max_lt_max αᵒᵈ _ _ _ _ _ h₁ h₂
 
-theorem min_right_comm (a b c : α) : min (min a b) c = min (min a c) b :=
-  right_comm min min_comm min_assoc a b c
+theorem min_right_comm (a b c : α) : min (min a b) c = min (min a c) b := by
+  rw [min_assoc, min_comm b, min_assoc]
 
-theorem Max.left_comm (a b c : α) : max a (max b c) = max b (max a c) :=
-  _root_.left_comm max max_comm max_assoc a b c
+theorem Max.left_comm (a b c : α) : max a (max b c) = max b (max a c) := by
+  rw [← max_assoc, max_comm a, max_assoc]
 
-theorem Max.right_comm (a b c : α) : max (max a b) c = max (max a c) b :=
-  _root_.right_comm max max_comm max_assoc a b c
+theorem Max.right_comm (a b c : α) : max (max a b) c = max (max a c) b := by
+  rw [max_assoc, max_comm b, max_assoc]
 
 theorem MonotoneOn.map_max (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (max a b) =
     max (f a) (f b) := by
@@ -234,34 +236,26 @@ theorem le_of_max_le_left {a b c : α} (h : max a b ≤ c) : a ≤ c :=
 theorem le_of_max_le_right {a b c : α} (h : max a b ≤ c) : b ≤ c :=
   le_trans (le_max_right _ _) h
 
-theorem max_commutative : Commutative (max : α → α → α) :=
-  max_comm
+instance instCommutativeMax : Std.Commutative (α := α) max where comm := max_comm
+instance instAssociativeMax : Std.Associative (α := α) max where assoc := max_assoc
+instance instCommutativeMin : Std.Commutative (α := α) min where comm := min_comm
+instance instAssociativeMin : Std.Associative (α := α) min where assoc := min_assoc
 
-theorem max_associative : Associative (max : α → α → α) :=
-  max_assoc
+theorem max_left_commutative : LeftCommutative (max : α → α → α) := ⟨max_left_comm⟩
+theorem min_left_commutative : LeftCommutative (min : α → α → α) := ⟨min_left_comm⟩
 
-instance : Std.Commutative (α := α) max where
-  comm := max_comm
+section deprecated
+set_option linter.deprecated false
 
-instance : Std.Associative (α := α) max where
-  assoc := max_assoc
+@[deprecated instCommutativeMax (since := "2024-09-12")]
+theorem max_commutative : Commutative (α := α) max := max_comm
+@[deprecated instAssociativeMax (since := "2024-09-12")]
+theorem max_associative : Associative (α := α) max := max_assoc
+@[deprecated instCommutativeMin (since := "2024-09-12")]
+theorem min_commutative : Commutative (α := α) min := min_comm
+@[deprecated instAssociativeMin (since := "2024-09-12")]
+theorem min_associative : Associative (α := α) min := min_assoc
 
-theorem max_left_commutative : LeftCommutative (max : α → α → α) :=
-  max_left_comm
-
-theorem min_commutative : Commutative (min : α → α → α) :=
-  min_comm
-
-theorem min_associative : Associative (α := α) min :=
-  min_assoc
-
-instance : Std.Commutative (α := α) min where
-  comm := min_comm
-
-instance : Std.Associative (α := α) min where
-  assoc := min_assoc
-
-theorem min_left_commutative : LeftCommutative (min : α → α → α) :=
-  min_left_comm
+end deprecated
 
 end
diff --git a/Mathlib/Order/Minimal.lean b/Mathlib/Order/Minimal.lean
index eec9b8aea31ce..728087a7dbdf4 100644
--- a/Mathlib/Order/Minimal.lean
+++ b/Mathlib/Order/Minimal.lean
@@ -381,7 +381,7 @@ variable [Preorder α]
 theorem setOf_minimal_subset (s : Set α) : {x | Minimal (· ∈ s) x} ⊆ s :=
   sep_subset ..
 
-theorem setOf_maximal_subset (s : Set α) : {x | Minimal (· ∈ s) x} ⊆ s :=
+theorem setOf_maximal_subset (s : Set α) : {x | Maximal (· ∈ s) x} ⊆ s :=
   sep_subset ..
 
 theorem Set.Subsingleton.maximal_mem_iff (h : s.Subsingleton) : Maximal (· ∈ s) x ↔ x ∈ s := by
diff --git a/Mathlib/Order/ModularLattice.lean b/Mathlib/Order/ModularLattice.lean
index 5c8d4fa9e256e..7b2b7f77ae7f8 100644
--- a/Mathlib/Order/ModularLattice.lean
+++ b/Mathlib/Order/ModularLattice.lean
@@ -218,6 +218,10 @@ theorem sup_lt_sup_of_lt_of_inf_le_inf (hxy : x < y) (hinf : y ⊓ z ≤ x ⊓ z
 theorem inf_lt_inf_of_lt_of_sup_le_sup (hxy : x < y) (hinf : y ⊔ z ≤ x ⊔ z) : x ⊓ z < y ⊓ z :=
   sup_lt_sup_of_lt_of_inf_le_inf (α := αᵒᵈ) hxy hinf
 
+theorem strictMono_inf_prod_sup : StrictMono fun x ↦ (x ⊓ z, x ⊔ z) := fun _x _y hxy ↦
+  ⟨⟨inf_le_inf_right _ hxy.le, sup_le_sup_right hxy.le _⟩,
+    fun ⟨inf_le, sup_le⟩ ↦ (sup_lt_sup_of_lt_of_inf_le_inf hxy inf_le).not_le sup_le⟩
+
 /-- A generalization of the theorem that if `N` is a submodule of `M` and
   `N` and `M / N` are both Artinian, then `M` is Artinian. -/
 theorem wellFounded_lt_exact_sequence {β γ : Type*} [PartialOrder β] [Preorder γ]
@@ -225,19 +229,12 @@ theorem wellFounded_lt_exact_sequence {β γ : Type*} [PartialOrder β] [Preorde
     (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂)
     (gi : GaloisInsertion g₂ g₁) (hf : ∀ a, f₁ (f₂ a) = a ⊓ K) (hg : ∀ a, g₁ (g₂ a) = a ⊔ K) :
     WellFoundedLT α :=
-  ⟨Subrelation.wf
-    (@fun A B hAB =>
-      show Prod.Lex (· < ·) (· < ·) (f₂ A, g₂ A) (f₂ B, g₂ B) by
-        simp only [Prod.lex_def, lt_iff_le_not_le, ← gci.l_le_l_iff, ← gi.u_le_u_iff, hf, hg,
-          le_antisymm_iff]
-        simp only [gci.l_le_l_iff, gi.u_le_u_iff, ← lt_iff_le_not_le, ← le_antisymm_iff]
-        rcases lt_or_eq_of_le (inf_le_inf_right K (le_of_lt hAB)) with h | h
-        · exact Or.inl h
-        · exact Or.inr ⟨h, sup_lt_sup_of_lt_of_inf_le_inf hAB (le_of_eq h.symm)⟩)
-    (InvImage.wf _ (h₁.wf.prod_lex h₂.wf))⟩
+  StrictMono.wellFoundedLT (f := fun A ↦ (f₂ A, g₂ A)) fun A B hAB ↦ by
+    simp only [Prod.le_def, lt_iff_le_not_le, ← gci.l_le_l_iff, ← gi.u_le_u_iff, hf, hg]
+    exact strictMono_inf_prod_sup hAB
 
 /-- A generalization of the theorem that if `N` is a submodule of `M` and
-  `N` and `M / N` are both Noetherian, then `M` is Noetherian.  -/
+  `N` and `M / N` are both Noetherian, then `M` is Noetherian. -/
 theorem wellFounded_gt_exact_sequence {β γ : Type*} [Preorder β] [PartialOrder γ]
     [WellFoundedGT β] [WellFoundedGT γ] (K : α)
     (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂)
@@ -261,8 +258,8 @@ def infIccOrderIsoIccSup (a b : α) : Set.Icc (a ⊓ b) a ≃o Set.Icc b (a ⊔
       (by
         change a ⊓ ↑x ⊔ b = ↑x
         rw [inf_comm, inf_sup_assoc_of_le _ x.prop.1, inf_eq_left.2 x.prop.2])
-  map_rel_iff' := @fun x y => by
-    simp only [Subtype.mk_le_mk, Equiv.coe_fn_mk, and_true_iff, le_sup_right]
+  map_rel_iff' {x y} := by
+    simp only [Subtype.mk_le_mk, Equiv.coe_fn_mk, le_sup_right]
     rw [← Subtype.coe_le_coe]
     refine ⟨fun h => ?_, fun h => sup_le_sup_right h _⟩
     rw [← sup_eq_right.2 x.prop.1, inf_sup_assoc_of_le _ x.prop.2, sup_comm, ←
diff --git a/Mathlib/Order/Monotone/Basic.lean b/Mathlib/Order/Monotone/Basic.lean
index c1a6c68abafef..bafac4b6ac367 100644
--- a/Mathlib/Order/Monotone/Basic.lean
+++ b/Mathlib/Order/Monotone/Basic.lean
@@ -66,7 +66,6 @@ open Function OrderDual
 universe u v w
 
 variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {δ : Type*} {π : ι → Type*}
-  {r : α → α → Prop}
 
 section MonotoneDef
 
@@ -1014,9 +1013,6 @@ theorem Antitone.ne_of_lt_of_lt_int {f : ℤ → α} (hf : Antitone f) (n : ℤ)
   rintro rfl
   exact (hf.reflect_lt h2).not_le (Int.le_of_lt_add_one <| hf.reflect_lt h1)
 
-theorem StrictMono.id_le {φ : ℕ → ℕ} (h : StrictMono φ) : ∀ n, n ≤ φ n := fun n ↦
-  Nat.recOn n (Nat.zero_le _) fun n hn ↦ Nat.succ_le_of_lt (hn.trans_lt <| h <| Nat.lt_succ_self n)
-
 end Preorder
 
 theorem Subtype.mono_coe [Preorder α] (t : Set α) : Monotone ((↑) : Subtype t → α) :=
@@ -1028,7 +1024,7 @@ theorem Subtype.strictMono_coe [Preorder α] (t : Set α) :
 
 section Preorder
 
-variable [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] {f : α → γ} {g : β → δ} {a b : α}
+variable [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] {f : α → γ} {g : β → δ}
 
 theorem monotone_fst : Monotone (@Prod.fst α β) := fun _ _ ↦ And.left
 
@@ -1077,7 +1073,7 @@ theorem const_strictMono [Nonempty β] : StrictMono (const β : α → β → α
 end Function
 
 section apply
-variable {ι α : Type*} {β : ι → Type*} [∀ i, Preorder (β i)] [Preorder α] {f : α → ∀ i, β i}
+variable {β : ι → Type*} [∀ i, Preorder (β i)] [Preorder α] {f : α → ∀ i, β i}
 
 lemma monotone_iff_apply₂ : Monotone f ↔ ∀ i, Monotone (f · i) := by
   simp [Monotone, Pi.le_def, @forall_swap ι]
diff --git a/Mathlib/Order/Monotone/Monovary.lean b/Mathlib/Order/Monotone/Monovary.lean
index 8b2766ecaefea..9dcfecb90a91f 100644
--- a/Mathlib/Order/Monotone/Monovary.lean
+++ b/Mathlib/Order/Monotone/Monovary.lean
@@ -31,7 +31,7 @@ variable {ι ι' α β γ : Type*}
 
 section Preorder
 
-variable [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {g' : β → γ}
+variable [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β}
   {s t : Set ι}
 
 /-- `f` monovaries with `g` if `g i < g j` implies `f i ≤ f j`. -/
@@ -249,7 +249,7 @@ end PartialOrder
 variable [LinearOrder ι]
 
 /- Porting note: Due to a bug in `alias`, many of the below lemmas have dot notation removed in the
-proof-/
+proof -/
 
 protected theorem Monotone.monovary (hf : Monotone f) (hg : Monotone g) : Monovary f g :=
   fun _ _ hij => hf (hg.reflect_lt hij).le
@@ -282,7 +282,7 @@ end Preorder
 
 section LinearOrder
 
-variable [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {g' : β → γ}
+variable [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {g : ι → β} {g' : β → γ}
   {s : Set ι}
 
 theorem MonovaryOn.comp_monotoneOn_right (h : MonovaryOn f g s) (hg : MonotoneOn g' (g '' s)) :
diff --git a/Mathlib/Order/Nat.lean b/Mathlib/Order/Nat.lean
index 9ecce65c098b4..fb4ba2eb92da7 100644
--- a/Mathlib/Order/Nat.lean
+++ b/Mathlib/Order/Nat.lean
@@ -23,6 +23,9 @@ instance instOrderBot : OrderBot ℕ where
   bot := 0
   bot_le := zero_le
 
+instance instNoMaxOrder : NoMaxOrder ℕ where
+  exists_gt n := ⟨n + 1, n.lt_succ_self⟩
+
 /-! ### Miscellaneous lemmas -/
 
 -- We want to use this lemma earlier than the lemma simp can prove it with
diff --git a/Mathlib/Order/OmegaCompletePartialOrder.lean b/Mathlib/Order/OmegaCompletePartialOrder.lean
index 91826bf421721..911cd139607e1 100644
--- a/Mathlib/Order/OmegaCompletePartialOrder.lean
+++ b/Mathlib/Order/OmegaCompletePartialOrder.lean
@@ -9,6 +9,7 @@ import Mathlib.Order.Chain
 import Mathlib.Order.Hom.Order
 import Mathlib.Order.Iterate
 import Mathlib.Order.Part
+import Mathlib.Order.ScottContinuity
 
 /-!
 # Omega Complete Partial Orders
@@ -56,6 +57,7 @@ supremum helps define the meaning of recursive procedures.
 assert_not_exists OrderedCommMonoid
 
 universe u v
+variable {ι : Sort*} {α β γ δ : Type*}
 
 namespace OmegaCompletePartialOrder
 
@@ -66,8 +68,6 @@ def Chain (α : Type u) [Preorder α] :=
   ℕ →o α
 
 namespace Chain
-
-variable {α : Type u} {β : Type v} {γ : Type*}
 variable [Preorder α] [Preorder β] [Preorder γ]
 
 instance : FunLike (Chain α) ℕ α := inferInstanceAs <| FunLike (ℕ →o α) ℕ α
@@ -78,7 +78,7 @@ instance [Inhabited α] : Inhabited (Chain α) :=
   ⟨⟨default, fun _ _ _ => le_rfl⟩⟩
 
 instance : Membership α (Chain α) :=
-  ⟨fun a (c : ℕ →o α) => ∃ i, a = c i⟩
+  ⟨fun (c : ℕ →o α) a => ∃ i, a = c i⟩
 
 variable (c c' : Chain α)
 variable (f : α →o β)
@@ -133,6 +133,23 @@ def zip (c₀ : Chain α) (c₁ : Chain β) : Chain (α × β) :=
 
 @[simp] theorem zip_coe (c₀ : Chain α) (c₁ : Chain β) (n : ℕ) : c₀.zip c₁ n = (c₀ n, c₁ n) := rfl
 
+/-- An example of a `Chain` constructed from an ordered pair. -/
+def pair (a b : α) (hab : a ≤ b) : Chain α where
+  toFun n := match n with
+    | 0 => a
+    | _ => b
+  monotone' _ _ _ := by aesop
+
+@[simp] lemma pair_zero (a b : α) (hab) : pair a b hab 0 = a := rfl
+@[simp] lemma pair_succ (a b : α) (hab) (n : ℕ) : pair a b hab (n + 1) = b := rfl
+
+@[simp] lemma range_pair (a b : α) (hab) : Set.range (pair a b hab) = {a, b} := by
+  ext; exact Nat.or_exists_add_one.symm.trans (by aesop)
+
+@[simp] lemma pair_zip_pair (a₁ a₂ : α) (b₁ b₂ : β) (ha hb) :
+    (pair a₁ a₂ ha).zip (pair b₁ b₂ hb) = pair (a₁, b₁) (a₂, b₂) (Prod.le_def.2 ⟨ha, hb⟩) := by
+  unfold Chain; ext n : 2; cases n <;> rfl
+
 end Chain
 
 end OmegaCompletePartialOrder
@@ -156,8 +173,6 @@ class OmegaCompletePartialOrder (α : Type*) extends PartialOrder α where
   ωSup_le : ∀ (c : Chain α) (x), (∀ i, c i ≤ x) → ωSup c ≤ x
 
 namespace OmegaCompletePartialOrder
-
-variable {α : Type u} {β : Type v} {γ : Type*}
 variable [OmegaCompletePartialOrder α]
 
 /-- Transfer an `OmegaCompletePartialOrder` on `β` to an `OmegaCompletePartialOrder` on `α`
@@ -188,7 +203,7 @@ theorem ωSup_le_ωSup_of_le {c₀ c₁ : Chain α} (h : c₀ ≤ c₁) : ωSup
     obtain ⟨_, h⟩ := h i
     exact le_trans h (le_ωSup _ _)
 
-theorem ωSup_le_iff (c : Chain α) (x : α) : ωSup c ≤ x ↔ ∀ i, c i ≤ x := by
+@[simp] theorem ωSup_le_iff {c : Chain α} {x : α} : ωSup c ≤ x ↔ ∀ i, c i ≤ x := by
   constructor <;> intros
   · trans ωSup c
     · exact le_ωSup _ _
@@ -228,6 +243,61 @@ open Chain
 
 variable [OmegaCompletePartialOrder β]
 variable [OmegaCompletePartialOrder γ]
+variable {f : α → β} {g : β → γ}
+
+/-- A function `f` between `ω`-complete partial orders is `ωScottContinuous` if it is
+Scott continuous over chains. -/
+def ωScottContinuous (f : α → β) : Prop :=
+    ScottContinuousOn (Set.range fun c : Chain α => Set.range c) f
+
+lemma _root_.ScottContinuous.ωScottContinuous (hf : ScottContinuous f) : ωScottContinuous f :=
+  hf.scottContinuousOn
+
+lemma ωScottContinuous.monotone (h : ωScottContinuous f) : Monotone f :=
+  ScottContinuousOn.monotone _ (fun a b hab => by
+    use pair a b hab; exact range_pair a b hab) h
+
+lemma ωScottContinuous.isLUB {c : Chain α} (hf : ωScottContinuous f) :
+    IsLUB (Set.range (c.map ⟨f, hf.monotone⟩)) (f (ωSup c)) := by
+  simpa [map_coe, OrderHom.coe_mk, Set.range_comp]
+    using hf (by simp) (Set.range_nonempty _) (isChain_range c).directedOn (isLUB_range_ωSup c)
+
+lemma ωScottContinuous.id : ωScottContinuous (id : α → α) := ScottContinuousOn.id
+
+lemma ωScottContinuous.map_ωSup (hf : ωScottContinuous f) (c : Chain α) :
+    f (ωSup c) = ωSup (c.map ⟨f, hf.monotone⟩) := ωSup_eq_of_isLUB hf.isLUB
+
+/-- `ωScottContinuous f` asserts that `f` is both monotone and distributes over ωSup. -/
+lemma ωScottContinuous_iff_monotone_map_ωSup :
+    ωScottContinuous f ↔ ∃ hf : Monotone f, ∀ c : Chain α, f (ωSup c) = ωSup (c.map ⟨f, hf⟩) := by
+  refine ⟨fun hf ↦ ⟨hf.monotone, hf.map_ωSup⟩, ?_⟩
+  intro hf _ ⟨c, hc⟩ _ _ _ hda
+  convert isLUB_range_ωSup (c.map { toFun := f, monotone' := hf.1 })
+  · rw [map_coe, OrderHom.coe_mk, ← hc, ← (Set.range_comp f ⇑c)]
+  · rw [← hc] at hda
+    rw [← hf.2 c, ωSup_eq_of_isLUB hda]
+
+alias ⟨ωScottContinuous.monotone_map_ωSup, ωScottContinuous.of_monotone_map_ωSup⟩ :=
+  ωScottContinuous_iff_monotone_map_ωSup
+
+/- A monotone function `f : α →o β` is ωScott continuous if and only if it distributes over ωSup. -/
+lemma ωScottContinuous_iff_map_ωSup_of_orderHom {f : α →o β} :
+    ωScottContinuous f ↔ ∀ c : Chain α, f (ωSup c) = ωSup (c.map f) := by
+  rw [ωScottContinuous_iff_monotone_map_ωSup]
+  exact exists_prop_of_true f.monotone'
+
+alias ⟨ωScottContinuous.map_ωSup_of_orderHom, ωScottContinuous.of_map_ωSup_of_orderHom⟩ :=
+  ωScottContinuous_iff_map_ωSup_of_orderHom
+
+lemma ωScottContinuous.comp (hg : ωScottContinuous g) (hf : ωScottContinuous f) :
+    ωScottContinuous (g.comp f) :=
+  ωScottContinuous.of_monotone_map_ωSup
+    ⟨hg.monotone.comp hf.monotone, by simp [hf.map_ωSup, hg.map_ωSup, map_comp]⟩
+
+lemma ωScottContinuous.const {x : β} : ωScottContinuous (Function.const α x) := by
+  simp [ωScottContinuous, ScottContinuousOn, Set.range_nonempty]
+
+set_option linter.deprecated false
 
 /-- A monotone function `f : α →o β` is continuous if it distributes over ωSup.
 
@@ -235,56 +305,67 @@ In order to distinguish it from the (more commonly used) continuity from topolog
 (see `Mathlib/Topology/Basic.lean`), the present definition is often referred to as
 "Scott-continuity" (referring to Dana Scott). It corresponds to continuity
 in Scott topological spaces (not defined here). -/
+@[deprecated ωScottContinuous (since := "2024-05-29")]
 def Continuous (f : α →o β) : Prop :=
   ∀ c : Chain α, f (ωSup c) = ωSup (c.map f)
 
 /-- `Continuous' f` asserts that `f` is both monotone and continuous. -/
+@[deprecated ωScottContinuous (since := "2024-05-29")]
 def Continuous' (f : α → β) : Prop :=
   ∃ hf : Monotone f, Continuous ⟨f, hf⟩
 
+@[deprecated ωScottContinuous.isLUB (since := "2024-05-29")]
 lemma isLUB_of_scottContinuous {c : Chain α} {f : α → β} (hf : ScottContinuous f) :
-    IsLUB (Set.range (Chain.map c ⟨f, (ScottContinuous.monotone hf)⟩)) (f (ωSup c)) := by
-  simp only [map_coe, OrderHom.coe_mk]
-  rw [(Set.range_comp f ↑c)]
-  exact hf (Set.range_nonempty ↑c) (IsChain.directedOn (isChain_range c)) (isLUB_range_ωSup c)
+    IsLUB (Set.range (Chain.map c ⟨f, (ScottContinuous.monotone hf)⟩)) (f (ωSup c)) :=
+  ωScottContinuous.isLUB hf.scottContinuousOn
 
+@[deprecated ScottContinuous.ωScottContinuous (since := "2024-05-29")]
 lemma ScottContinuous.continuous' {f : α → β} (hf : ScottContinuous f) : Continuous' f := by
   constructor
   · intro c
     rw [← (ωSup_eq_of_isLUB (isLUB_of_scottContinuous hf))]
     simp only [OrderHom.coe_mk]
 
+@[deprecated ωScottContinuous.monotone (since := "2024-05-29")]
 theorem Continuous'.to_monotone {f : α → β} (hf : Continuous' f) : Monotone f :=
   hf.fst
 
+@[deprecated ωScottContinuous.of_monotone_map_ωSup (since := "2024-05-29")]
 theorem Continuous.of_bundled (f : α → β) (hf : Monotone f) (hf' : Continuous ⟨f, hf⟩) :
     Continuous' f :=
   ⟨hf, hf'⟩
 
+@[deprecated ωScottContinuous.of_monotone_map_ωSup (since := "2024-05-29")]
 theorem Continuous.of_bundled' (f : α →o β) (hf' : Continuous f) : Continuous' f :=
   ⟨f.mono, hf'⟩
 
+@[deprecated ωScottContinuous_iff_monotone_map_ωSup (since := "2024-05-29")]
 theorem Continuous'.to_bundled (f : α → β) (hf : Continuous' f) : Continuous ⟨f, hf.to_monotone⟩ :=
   hf.snd
 
-@[simp, norm_cast]
+@[simp, norm_cast, deprecated ωScottContinuous_iff_monotone_map_ωSup (since := "2024-05-29")]
 theorem continuous'_coe : ∀ {f : α →o β}, Continuous' f ↔ Continuous f
   | ⟨_, hf⟩ => ⟨fun ⟨_, hc⟩ => hc, fun hc => ⟨hf, hc⟩⟩
 
 variable (f : α →o β) (g : β →o γ)
 
+@[deprecated ωScottContinuous.id (since := "2024-05-29")]
 theorem continuous_id : Continuous (@OrderHom.id α _) := by intro c; rw [c.map_id]; rfl
 
+@[deprecated ωScottContinuous.comp (since := "2024-05-29")]
 theorem continuous_comp (hfc : Continuous f) (hgc : Continuous g) : Continuous (g.comp f) := by
   dsimp [Continuous] at *; intro
   rw [hfc, hgc, Chain.map_comp]
 
+@[deprecated ωScottContinuous.id (since := "2024-05-29")]
 theorem id_continuous' : Continuous' (@id α) :=
   continuous_id.of_bundled' _
 
+@[deprecated ωScottContinuous.const (since := "2024-05-29")]
 theorem continuous_const (x : β) : Continuous (OrderHom.const α x) := fun c =>
   eq_of_forall_ge_iff fun z => by rw [ωSup_le_iff, Chain.map_coe, OrderHom.const_coe_coe]; simp
 
+@[deprecated ωScottContinuous.const (since := "2024-05-29")]
 theorem const_continuous' (x : β) : Continuous' (Function.const α x) :=
   Continuous.of_bundled' (OrderHom.const α x) (continuous_const x)
 
@@ -294,8 +375,6 @@ end OmegaCompletePartialOrder
 
 namespace Part
 
-variable {α : Type u} {β : Type v} {γ : Type*}
-
 open OmegaCompletePartialOrder
 
 theorem eq_of_chain {c : Chain (Part α)} {a b : α} (ha : some a ∈ c) (hb : some b ∈ c) : a = b := by
@@ -374,9 +453,9 @@ end Inst
 
 end Part
 
-namespace Pi
+section Pi
 
-variable {α : Type*} {β : α → Type*} {γ : Type*}
+variable {β : α → Type*}
 
 open OmegaCompletePartialOrder OmegaCompletePartialOrder.Chain
 
@@ -393,11 +472,27 @@ namespace OmegaCompletePartialOrder
 
 variable [∀ x, OmegaCompletePartialOrder <| β x]
 variable [OmegaCompletePartialOrder γ]
+variable {f : γ → ∀ x, β x}
+
+lemma ωScottContinuous.apply₂ (hf : ωScottContinuous f) (a : α) : ωScottContinuous (f · a) :=
+  ωScottContinuous.of_monotone_map_ωSup
+    ⟨fun _ _ h ↦ hf.monotone h a, fun c ↦ congr_fun (hf.map_ωSup c) a⟩
+
+lemma ωScottContinuous.of_apply₂ (hf : ∀ a, ωScottContinuous (f · a)) : ωScottContinuous f :=
+  ωScottContinuous.of_monotone_map_ωSup
+    ⟨fun x y h a ↦ (hf a).monotone h, fun c ↦ by ext a; apply (hf a).map_ωSup c⟩
 
+lemma ωScottContinuous_iff_apply₂ : ωScottContinuous f ↔ ∀ a, ωScottContinuous (f · a) :=
+  ⟨ωScottContinuous.apply₂, ωScottContinuous.of_apply₂⟩
+
+set_option linter.deprecated false
+
+@[deprecated ωScottContinuous.apply₂ (since := "2024-05-29")]
 theorem flip₁_continuous' (f : ∀ x : α, γ → β x) (a : α) (hf : Continuous' fun x y => f y x) :
     Continuous' (f a) :=
   Continuous.of_bundled _ (fun _ _ h => hf.to_monotone h a) fun c => congr_fun (hf.to_bundled _ c) a
 
+@[deprecated ωScottContinuous.of_apply₂ (since := "2024-05-29")]
 theorem flip₂_continuous' (f : γ → ∀ x, β x) (hf : ∀ x, Continuous' fun g => f g x) :
     Continuous' f :=
   Continuous.of_bundled _ (fun x y h a => (hf a).to_monotone h)
@@ -411,7 +506,6 @@ namespace Prod
 
 open OmegaCompletePartialOrder
 
-variable {α : Type*} {β : Type*} {γ : Type*}
 variable [OmegaCompletePartialOrder α]
 variable [OmegaCompletePartialOrder β]
 variable [OmegaCompletePartialOrder γ]
@@ -437,8 +531,6 @@ open OmegaCompletePartialOrder
 
 namespace CompleteLattice
 
-variable (α : Type u)
-
 -- see Note [lower instance priority]
 /-- Any complete lattice has an `ω`-CPO structure where the countable supremum is a special case
 of arbitrary suprema. -/
@@ -448,20 +540,55 @@ instance (priority := 100) [CompleteLattice α] : OmegaCompletePartialOrder α w
     simp only [iSup_le_iff, OrderHom.coe_mk] at hs ⊢; intro i; apply hs i
   le_ωSup := fun ⟨c, _⟩ i => by simp only [OrderHom.coe_mk]; apply le_iSup_of_le i; rfl
 
-variable {α} {β : Type v} [OmegaCompletePartialOrder α] [CompleteLattice β]
+variable [OmegaCompletePartialOrder α] [CompleteLattice β] {f g : α → β}
+
+-- TODO Prove this result for `ScottContinuousOn` and deduce this as a special case
+-- https://github.com/leanprover-community/mathlib4/pull/15412
+open Chain in
+lemma ωScottContinuous.prodMk (hf : ωScottContinuous f) (hg : ωScottContinuous g) :
+    ωScottContinuous fun x => (f x, g x) := ScottContinuousOn.prodMk (fun a b hab => by
+  use pair a b hab; exact range_pair a b hab) hf hg
+
+lemma ωScottContinuous.iSup {f : ι → α → β} (hf : ∀ i, ωScottContinuous (f i)) :
+    ωScottContinuous (⨆ i, f i) := by
+  refine ωScottContinuous.of_monotone_map_ωSup
+    ⟨Monotone.iSup fun i ↦ (hf i).monotone, fun c ↦ eq_of_forall_ge_iff fun a ↦ ?_⟩
+  simp (config := { contextual := true }) [ωSup_le_iff, (hf _).map_ωSup, @forall_swap ι]
+
+lemma ωScottContinuous.sSup {s : Set (α → β)} (hs : ∀ f ∈ s, ωScottContinuous f) :
+    ωScottContinuous (sSup s) := by
+  rw [sSup_eq_iSup]; exact ωScottContinuous.iSup fun f ↦ ωScottContinuous.iSup <| hs f
+
+lemma ωScottContinuous.sup (hf : ωScottContinuous f) (hg : ωScottContinuous g) :
+    ωScottContinuous (f ⊔ g) := by
+  rw [← sSup_pair]
+  apply ωScottContinuous.sSup
+  rintro f (rfl | rfl | _) <;> assumption
+
+lemma ωScottContinuous.top : ωScottContinuous (⊤ : α → β) :=
+  ωScottContinuous.of_monotone_map_ωSup
+    ⟨monotone_const, fun c ↦ eq_of_forall_ge_iff fun a ↦ by simp⟩
+
+lemma ωScottContinuous.bot : ωScottContinuous (⊥ : α → β) := by
+  rw [← sSup_empty]; exact ωScottContinuous.sSup (by simp)
 
+set_option linter.deprecated false
+
+@[deprecated ωScottContinuous.sSup (since := "2024-05-29")]
 theorem sSup_continuous (s : Set <| α →o β) (hs : ∀ f ∈ s, Continuous f) : Continuous (sSup s) := by
   intro c
   apply eq_of_forall_ge_iff
   intro z
-  suffices (∀ f ∈ s, ∀ (n), (f : _) (c n) ≤ z) ↔ ∀ (n), ∀ f ∈ s, (f : _) (c n) ≤ z by
+  suffices (∀ f ∈ s, ∀ n, f (c n) ≤ z) ↔ ∀ n, ∀ f ∈ s, f (c n) ≤ z by
     simpa (config := { contextual := true }) [ωSup_le_iff, hs _ _ _] using this
   exact ⟨fun H n f hf => H f hf n, fun H f hf n => H n f hf⟩
 
+@[deprecated ωScottContinuous.iSup (since := "2024-05-29")]
 theorem iSup_continuous {ι : Sort*} {f : ι → α →o β} (h : ∀ i, Continuous (f i)) :
     Continuous (⨆ i, f i) :=
   sSup_continuous _ <| Set.forall_mem_range.2 h
 
+@[deprecated ωScottContinuous.sSup (since := "2024-05-29")]
 theorem sSup_continuous' (s : Set (α → β)) (hc : ∀ f ∈ s, Continuous' f) :
     Continuous' (sSup s) := by
   lift s to Set (α →o β) using fun f hf => (hc f hf).to_monotone
@@ -470,16 +597,19 @@ theorem sSup_continuous' (s : Set (α → β)) (hc : ∀ f ∈ s, Continuous' f)
   norm_cast
   exact iSup_continuous fun f ↦ iSup_continuous fun hf ↦ hc hf
 
+@[deprecated ωScottContinuous.sup (since := "2024-05-29")]
 theorem sup_continuous {f g : α →o β} (hf : Continuous f) (hg : Continuous g) :
     Continuous (f ⊔ g) := by
   rw [← sSup_pair]; apply sSup_continuous
   rintro f (rfl | rfl | _) <;> assumption
 
+@[deprecated ωScottContinuous.top (since := "2024-05-29")]
 theorem top_continuous : Continuous (⊤ : α →o β) := by
   intro c; apply eq_of_forall_ge_iff; intro z
   simp only [OrderHom.instTopOrderHom_top, OrderHom.const_coe_coe, Function.const, top_le_iff,
     ωSup_le_iff, Chain.map_coe, Function.comp, forall_const]
 
+@[deprecated ωScottContinuous.bot (since := "2024-05-29")]
 theorem bot_continuous : Continuous (⊥ : α →o β) := by
   rw [← sSup_empty]
   exact sSup_continuous _ fun f hf => hf.elim
@@ -488,8 +618,27 @@ end CompleteLattice
 
 namespace CompleteLattice
 
-variable {α β : Type*} [OmegaCompletePartialOrder α] [CompleteLinearOrder β]
+variable [OmegaCompletePartialOrder α] [CompleteLinearOrder β] {f g : α → β}
+
+-- TODO Prove this result for `ScottContinuousOn` and deduce this as a special case
+-- Also consider if it holds in greater generality (e.g. finite sets)
+-- N.B. The Scott Topology coincides with the Upper Topology on a Complete Linear Order
+-- `Topology.IsScott.scott_eq_upper_of_completeLinearOrder`
+-- We have that the product topology coincides with the upper topology
+-- https://github.com/leanprover-community/mathlib4/pull/12133
+lemma ωScottContinuous.inf (hf : ωScottContinuous f) (hg : ωScottContinuous g) :
+    ωScottContinuous (f ⊓ g) := by
+  refine ωScottContinuous.of_monotone_map_ωSup
+    ⟨hf.monotone.inf hg.monotone, fun c ↦ eq_of_forall_ge_iff fun a ↦ ?_⟩
+  simp only [Pi.inf_apply, hf.map_ωSup c, hg.map_ωSup c, inf_le_iff, ωSup_le_iff, Chain.map_coe,
+    Function.comp, OrderHom.coe_mk, ← forall_or_left, ← forall_or_right]
+  exact ⟨fun h _ ↦ h _ _, fun h i j ↦
+    (h (max j i)).imp (le_trans <| hf.monotone <| c.mono <| le_max_left _ _)
+      (le_trans <| hg.monotone <| c.mono <| le_max_right _ _)⟩
+
+set_option linter.deprecated false
 
+@[deprecated ωScottContinuous.inf (since := "2024-05-29")]
 theorem inf_continuous (f g : α →o β) (hf : Continuous f) (hg : Continuous g) :
     Continuous (f ⊓ g) := by
   refine fun c => eq_of_forall_ge_iff fun z => ?_
@@ -499,6 +648,7 @@ theorem inf_continuous (f g : α →o β) (hf : Continuous f) (hg : Continuous g
     (h (max j i)).imp (le_trans <| f.mono <| c.mono <| le_max_left _ _)
       (le_trans <| g.mono <| c.mono <| le_max_right _ _)⟩
 
+@[deprecated ωScottContinuous.inf (since := "2024-05-29")]
 theorem inf_continuous' {f g : α → β} (hf : Continuous' f) (hg : Continuous' g) :
     Continuous' (f ⊓ g) :=
   ⟨_, inf_continuous _ _ hf.snd hg.snd⟩
@@ -506,11 +656,8 @@ theorem inf_continuous' {f g : α → β} (hf : Continuous' f) (hg : Continuous'
 end CompleteLattice
 
 namespace OmegaCompletePartialOrder
-
-variable {α : Type u} {α' : Type*} {β : Type v} {β' : Type*} {γ : Type*} {φ : Type*}
 variable [OmegaCompletePartialOrder α] [OmegaCompletePartialOrder β]
-variable [OmegaCompletePartialOrder γ] [OmegaCompletePartialOrder φ]
-variable [OmegaCompletePartialOrder α'] [OmegaCompletePartialOrder β']
+variable [OmegaCompletePartialOrder γ] [OmegaCompletePartialOrder δ]
 
 namespace OrderHom
 
@@ -526,16 +673,13 @@ instance omegaCompletePartialOrder : OmegaCompletePartialOrder (α →o β) :=
 
 end OrderHom
 
-section
-
-variable (α β)
-
+variable (α β) in
 /-- A monotone function on `ω`-continuous partial orders is said to be continuous
 if for every chain `c : chain α`, `f (⊔ i, c i) = ⊔ i, f (c i)`.
 This is just the bundled version of `OrderHom.continuous`. -/
 structure ContinuousHom extends OrderHom α β where
   /-- The underlying function of a `ContinuousHom` is continuous, i.e. it preserves `ωSup` -/
-  cont : Continuous toOrderHom
+  protected map_ωSup' (c : Chain α) : toFun (ωSup c) = ωSup (c.map toOrderHom)
 
 attribute [nolint docBlame] ContinuousHom.toOrderHom
 
@@ -554,14 +698,16 @@ instance : OrderHomClass (α →𝒄 β) α β where
 instance : PartialOrder (α →𝒄 β) :=
   (PartialOrder.lift fun f => f.toOrderHom.toFun) <| by rintro ⟨⟨⟩⟩ ⟨⟨⟩⟩ h; congr
 
-end
-
 namespace ContinuousHom
 
+protected lemma ωScottContinuous (f : α →𝒄 β) : ωScottContinuous f :=
+  ωScottContinuous.of_map_ωSup_of_orderHom f.map_ωSup'
+
 -- Not a `simp` lemma because in many cases projection is simpler than a generic coercion
 theorem toOrderHom_eq_coe (f : α →𝒄 β) : f.1 = f := rfl
 
-@[simp] theorem coe_mk (f : α →o β) (hf : Continuous f) : ⇑(mk f hf) = f := rfl
+@[simp] theorem coe_mk (f : α →o β) (hf) : ⇑(mk f hf) = f := rfl
+
 @[simp] theorem coe_toOrderHom (f : α →𝒄 β) : ⇑f.1 = f := rfl
 
 /-- See Note [custom simps projection]. We specify this explicitly because we don't have a DFunLike
@@ -585,6 +731,8 @@ protected theorem monotone (f : α →𝒄 β) : Monotone f :=
 theorem apply_mono {f g : α →𝒄 β} {x y : α} (h₁ : f ≤ g) (h₂ : x ≤ y) : f x ≤ g y :=
   OrderHom.apply_mono (show (f : α →o β) ≤ g from h₁) h₂
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-07-27")]
 theorem ite_continuous' {p : Prop} [hp : Decidable p] (f g : α → β) (hf : Continuous' f)
     (hg : Continuous' g) : Continuous' fun x => if p then f x else g x := by
   split_ifs <;> simp [*]
@@ -617,26 +765,48 @@ theorem ωSup_bind {β γ : Type v} (c : Chain α) (f : α →o Part β) (g : α
     · apply le_ωSup (c.map g) _ _ _ hb₁
     · apply le_ωSup (c.map f) i _ hb₀
 
+-- TODO: We should move `ωScottContinuous` to the root namespace
+lemma ωScottContinuous.bind {β γ} {f : α → Part β} {g : α → β → Part γ} (hf : ωScottContinuous f)
+    (hg : ωScottContinuous g) : ωScottContinuous fun x ↦ f x >>= g x :=
+  ωScottContinuous.of_monotone_map_ωSup
+    ⟨hf.monotone.partBind hg.monotone, fun c ↦ by rw [hf.map_ωSup, hg.map_ωSup, ← ωSup_bind]; rfl⟩
+
+lemma ωScottContinuous.map {β γ} {f : β → γ} {g : α → Part β} (hg : ωScottContinuous g) :
+    ωScottContinuous fun x ↦ f <$> g x := by
+  simpa only [map_eq_bind_pure_comp] using ωScottContinuous.bind hg ωScottContinuous.const
+
+lemma ωScottContinuous.seq {β γ} {f : α → Part (β → γ)} {g : α → Part β} (hf : ωScottContinuous f)
+    (hg : ωScottContinuous g) : ωScottContinuous fun x ↦ f x <*> g x := by
+  simp only [seq_eq_bind_map]
+  exact ωScottContinuous.bind hf <| ωScottContinuous.of_apply₂ fun _ ↦ ωScottContinuous.map hg
+
+set_option linter.deprecated false
+
+@[deprecated ωScottContinuous.bind (since := "2024-05-29")]
 theorem bind_continuous' {β γ : Type v} (f : α → Part β) (g : α → β → Part γ) :
     Continuous' f → Continuous' g → Continuous' fun x => f x >>= g x
   | ⟨hf, hf'⟩, ⟨hg, hg'⟩ =>
     Continuous.of_bundled' (OrderHom.partBind ⟨f, hf⟩ ⟨g, hg⟩)
       (by intro c; rw [ωSup_bind, ← hf', ← hg']; rfl)
 
+@[deprecated ωScottContinuous.map (since := "2024-05-29")]
 theorem map_continuous' {β γ : Type v} (f : β → γ) (g : α → Part β) (hg : Continuous' g) :
     Continuous' fun x => f <$> g x := by
   simp only [map_eq_bind_pure_comp]; apply bind_continuous' _ _ hg; apply const_continuous'
 
+@[deprecated ωScottContinuous.seq (since := "2024-05-29")]
 theorem seq_continuous' {β γ : Type v} (f : α → Part (β → γ)) (g : α → Part β) (hf : Continuous' f)
     (hg : Continuous' g) : Continuous' fun x => f x <*> g x := by
   simp only [seq_eq_bind_map]
   apply bind_continuous' _ _ hf
-  apply Pi.OmegaCompletePartialOrder.flip₂_continuous'
+  apply OmegaCompletePartialOrder.flip₂_continuous'
   intro
   apply map_continuous' _ _ hg
 
 theorem continuous (F : α →𝒄 β) (C : Chain α) : F (ωSup C) = ωSup (C.map F) :=
-  ContinuousHom.cont _ _
+  F.ωScottContinuous.map_ωSup _
+
+set_option linter.deprecated true
 
 /-- Construct a continuous function from a bare function, a continuous function, and a proof that
 they are equal. -/
@@ -644,17 +814,18 @@ they are equal. -/
 @[simps!]
 def copy (f : α → β) (g : α →𝒄 β) (h : f = g) : α →𝒄 β where
   toOrderHom := g.1.copy f h
-  cont := by rw [OrderHom.copy_eq]; exact g.cont
+  map_ωSup' := by rw [OrderHom.copy_eq]; exact g.map_ωSup'
 
 -- Porting note: `of_mono` now defeq `mk`
 
 /-- The identity as a continuous function. -/
 @[simps!]
-def id : α →𝒄 α := ⟨OrderHom.id, continuous_id⟩
+def id : α →𝒄 α := ⟨OrderHom.id, ωScottContinuous.id.map_ωSup⟩
 
 /-- The composition of continuous functions. -/
 @[simps!]
-def comp (f : β →𝒄 γ) (g : α →𝒄 β) : α →𝒄 γ := ⟨.comp f.1 g.1, continuous_comp _ _ g.cont f.cont⟩
+def comp (f : β →𝒄 γ) (g : α →𝒄 β) : α →𝒄 γ :=
+  ⟨.comp f.1 g.1, (f.ωScottContinuous.comp g.ωScottContinuous).map_ωSup⟩
 
 @[ext]
 protected theorem ext (f g : α →𝒄 β) (h : ∀ x, f x = g x) : f = g := DFunLike.ext f g h
@@ -669,7 +840,7 @@ theorem comp_id (f : β →𝒄 γ) : f.comp id = f := rfl
 theorem id_comp (f : β →𝒄 γ) : id.comp f = f := rfl
 
 @[simp]
-theorem comp_assoc (f : γ →𝒄 φ) (g : β →𝒄 γ) (h : α →𝒄 β) : f.comp (g.comp h) = (f.comp g).comp h :=
+theorem comp_assoc (f : γ →𝒄 δ) (g : β →𝒄 γ) (h : α →𝒄 β) : f.comp (g.comp h) = (f.comp g).comp h :=
   rfl
 
 @[simp]
@@ -678,7 +849,7 @@ theorem coe_apply (a : α) (f : α →𝒄 β) : (f : α →o β) a = f a :=
 
 /-- `Function.const` is a continuous function. -/
 @[simps!]
-def const (x : β) : α →𝒄 β := ⟨.const _ x, continuous_const x⟩
+def const (x : β) : α →𝒄 β := ⟨.const _ x, ωScottContinuous.const.map_ωSup⟩
 
 instance [Inhabited β] : Inhabited (α →𝒄 β) :=
   ⟨const default⟩
@@ -715,12 +886,9 @@ theorem forall_forall_merge' (c₀ : Chain (α →𝒄 β)) (c₁ : Chain α) (z
 /-- The `ωSup` operator for continuous functions, which takes the pointwise countable supremum
 of the functions in the `ω`-chain. -/
 @[simps!]
-protected def ωSup (c : Chain (α →𝒄 β)) : α →𝒄 β :=
-  .mk (ωSup <| c.map toMono) fun c' ↦ by
-    apply eq_of_forall_ge_iff; intro z
-    simp only [ωSup_le_iff, (c _).continuous, Chain.map_coe, OrderHom.apply_coe, toMono_coe,
-      OrderHom.omegaCompletePartialOrder_ωSup_coe, forall_forall_merge, OrderHomClass.coe_coe,
-      forall_forall_merge', (· ∘ ·), Function.eval]
+protected def ωSup (c : Chain (α →𝒄 β)) : α →𝒄 β where
+  toOrderHom := ωSup <| c.map toMono
+  map_ωSup' c' := eq_of_forall_ge_iff fun a ↦ by simp [(c _).ωScottContinuous.map_ωSup]
 
 @[simps ωSup]
 instance : OmegaCompletePartialOrder (α →𝒄 β) :=
@@ -729,15 +897,14 @@ instance : OmegaCompletePartialOrder (α →𝒄 β) :=
 
 namespace Prod
 
-/-- The application of continuous functions as a continuous function.  -/
+/-- The application of continuous functions as a continuous function. -/
 @[simps]
 def apply : (α →𝒄 β) × α →𝒄 β where
   toFun f := f.1 f.2
   monotone' x y h := by
     dsimp
     trans y.fst x.snd <;> [apply h.1; apply y.1.monotone h.2]
-  cont := by
-    intro c
+  map_ωSup' c := by
     apply le_antisymm
     · apply ωSup_le
       intro i
@@ -770,7 +937,7 @@ theorem ωSup_apply_ωSup (c₀ : Chain (α →𝒄 β)) (c₁ : Chain α) :
 def flip {α : Type*} (f : α → β →𝒄 γ) : β →𝒄 α → γ where
   toFun x y := f y x
   monotone' x y h a := (f a).monotone h
-  cont := by intro _; ext x; change f _ _ = _; rw [(f _).continuous]; rfl
+  map_ωSup' _ := by ext x; change f _ _ = _; rw [(f _).continuous]; rfl
 
 /-- `Part.bind` as a continuous function. -/
 @[simps! apply] -- Porting note: removed `(config := { rhsMd := reducible })`
diff --git a/Mathlib/Order/OrderIsoNat.lean b/Mathlib/Order/OrderIsoNat.lean
index 5eddb07567f64..da0fabb452ad3 100644
--- a/Mathlib/Order/OrderIsoNat.lean
+++ b/Mathlib/Order/OrderIsoNat.lean
@@ -232,3 +232,20 @@ theorem WellFounded.iSup_eq_monotonicSequenceLimit [CompleteLattice α]
   · cases' WellFounded.monotone_chain_condition'.1 h a with n hn
     have : n ∈ {n | ∀ m, n ≤ m → a n = a m} := fun k hk => (a.mono hk).eq_of_not_lt (hn k hk)
     exact (Nat.sInf_mem ⟨n, this⟩ m hm.le).ge
+
+theorem exists_covBy_seq_of_wellFoundedLT_wellFoundedGT (α) [Preorder α]
+    [Nonempty α] [wfl : WellFoundedLT α] [wfg : WellFoundedGT α] :
+    ∃ a : ℕ → α, IsMin (a 0) ∧ ∃ n, IsMax (a n) ∧ ∀ i < n, a i ⋖ a (i + 1) := by
+  choose next hnext using exists_covBy_of_wellFoundedLT (α := α)
+  have hα := Set.nonempty_iff_univ_nonempty.mp ‹_›
+  classical
+  let a : ℕ → α := Nat.rec (wfl.wf.min _ hα) fun _n a ↦ if ha : IsMax a then a else next ha
+  refine ⟨a, isMin_iff_forall_not_lt.mpr fun _ ↦ wfl.wf.not_lt_min _ hα trivial, ?_⟩
+  have cov n (hn : ¬ IsMax (a n)) : a n ⋖ a (n + 1) := by
+    change a n ⋖ if ha : IsMax (a n) then a n else _
+    rw [dif_neg hn]
+    exact hnext hn
+  have H : ∃ n, IsMax (a n) := by
+    by_contra!
+    exact (RelEmbedding.natGT a fun n ↦ (cov n (this n)).1).not_wellFounded_of_decreasing_seq wfg.wf
+  exact ⟨_, wellFounded_lt.min_mem _ H, fun i h ↦ cov _ fun h' ↦ wellFounded_lt.not_lt_min _ H h' h⟩
diff --git a/Mathlib/Order/PartialSups.lean b/Mathlib/Order/PartialSups.lean
index cfb8d50271716..5fbe0f73945d5 100644
--- a/Mathlib/Order/PartialSups.lean
+++ b/Mathlib/Order/PartialSups.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Finset.Lattice
 import Mathlib.Order.Hom.Basic
diff --git a/Mathlib/Order/Partition/Finpartition.lean b/Mathlib/Order/Partition/Finpartition.lean
index 13a431112afd8..91dbd15b3b57b 100644
--- a/Mathlib/Order/Partition/Finpartition.lean
+++ b/Mathlib/Order/Partition/Finpartition.lean
@@ -67,7 +67,7 @@ structure Finpartition [Lattice α] [OrderBot α] (a : α) where
   supIndep : parts.SupIndep id
   /-- The supremum of the partition is `a` -/
   sup_parts : parts.sup id = a
-  /-- No element of the partition is bottom-/
+  /-- No element of the partition is bottom -/
   not_bot_mem : ⊥ ∉ parts
   deriving DecidableEq
 
@@ -257,6 +257,15 @@ theorem parts_top_subsingleton (a : α) [Decidable (a = ⊥)] :
     ((⊤ : Finpartition a).parts : Set α).Subsingleton :=
   Set.subsingleton_of_subset_singleton fun _ hb ↦ mem_singleton.1 <| parts_top_subset _ hb
 
+-- TODO: this instance takes double-exponential time to generate all partitions, find a faster way
+instance [DecidableEq α] {s : Finset α} : Fintype (Finpartition s) where
+  elems := s.powerset.powerset.image
+    fun ps ↦ if h : ps.sup id = s ∧ ⊥ ∉ ps ∧ ps.SupIndep id then ⟨ps, h.2.2, h.1, h.2.1⟩ else ⊤
+  complete P := by
+    refine mem_image.mpr ⟨P.parts, ?_, ?_⟩
+    · rw [mem_powerset]; intro p hp; rw [mem_powerset]; exact P.le hp
+    · simp only [P.supIndep, P.sup_parts, P.not_bot_mem]; rfl
+
 end Order
 
 end Lattice
@@ -465,7 +474,8 @@ def part (a : α) : Finset α := if ha : a ∈ s then choose (hp := P.existsUniq
 
 lemma part_mem (ha : a ∈ s) : P.part a ∈ P.parts := by simp [part, ha, choose_mem]
 
-lemma mem_part (ha : a ∈ s) : a ∈ P.part a := by simp [part, ha, choose_property]
+lemma mem_part (ha : a ∈ s) : a ∈ P.part a := by
+  simp [part, ha, choose_property (p := fun s => a ∈ s) P.parts (P.existsUnique_mem ha)]
 
 lemma part_eq_of_mem (ht : t ∈ P.parts) (hat : a ∈ t) : P.part a = t := by
   apply P.eq_of_mem_parts (P.part_mem _) ht (P.mem_part _) hat <;> exact mem_of_subset (P.le ht) hat
@@ -547,6 +557,8 @@ lemma card_mod_card_parts_le : s.card % P.parts.card ≤ P.parts.card := by
     rw [h, h']
   · exact (Nat.mod_lt _ h).le
 
+section Setoid
+
 variable [Fintype α]
 
 /-- A setoid over a finite type induces a finpartition of the type's elements,
@@ -586,6 +598,8 @@ theorem mem_part_ofSetoid_iff_rel {s : Setoid α} [DecidableRel s.r] {b : α} :
   simp only [← hc, mem_univ, mem_filter, true_and] at this ⊢
   exact ⟨s.trans (s.symm this), s.trans this⟩
 
+end Setoid
+
 section Atomise
 
 /-- Cuts `s` along the finsets in `F`: Two elements of `s` will be in the same part if they are
diff --git a/Mathlib/Order/RelClasses.lean b/Mathlib/Order/RelClasses.lean
index 1981aaa5b754d..be726cd3e66f4 100644
--- a/Mathlib/Order/RelClasses.lean
+++ b/Mathlib/Order/RelClasses.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
 -/
 import Mathlib.Data.Nat.Defs
+import Mathlib.Init.Algebra.Classes
 import Mathlib.Logic.IsEmpty
 import Mathlib.Order.Basic
 import Mathlib.Tactic.MkIffOfInductiveProp
@@ -41,21 +42,21 @@ theorem antisymm_iff [IsRefl α r] [IsAntisymm α r] {a b : α} : r a b ∧ r b
 
 /-- A version of `antisymm` with `r` explicit.
 
-This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there.  -/
+This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there. -/
 @[elab_without_expected_type]
 theorem antisymm_of (r : α → α → Prop) [IsAntisymm α r] {a b : α} : r a b → r b a → a = b :=
   antisymm
 
 /-- A version of `antisymm'` with `r` explicit.
 
-This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there.  -/
+This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there. -/
 @[elab_without_expected_type]
 theorem antisymm_of' (r : α → α → Prop) [IsAntisymm α r] {a b : α} : r a b → r b a → b = a :=
   antisymm'
 
 /-- A version of `comm` with `r` explicit.
 
-This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there.  -/
+This lemma matches the lemmas from lean core in `Init.Algebra.Classes`, but is missing there. -/
 theorem comm_of (r : α → α → Prop) [IsSymm α r] {a b : α} : r a b ↔ r b a :=
   comm
 
@@ -170,7 +171,7 @@ abbrev partialOrderOfSO (r) [IsStrictOrder α r] : PartialOrder α where
     | _, _, Or.inl rfl => rfl
     | _, Or.inr h₁, Or.inr h₂ => (asymm h₁ h₂).elim
   lt_iff_le_not_le x y :=
-    ⟨fun h => ⟨Or.inr h, not_or_of_not (fun e => by rw [e] at h; exact irrefl _ h) (asymm h)⟩,
+    ⟨fun h => ⟨Or.inr h, not_or_intro (fun e => by rw [e] at h; exact irrefl _ h) (asymm h)⟩,
       fun ⟨h₁, h₂⟩ => h₁.resolve_left fun e => h₂ <| e ▸ Or.inl rfl⟩
 
 /-- Construct a linear order from an `IsStrictTotalOrder` relation.
@@ -199,7 +200,7 @@ theorem IsStrictTotalOrder.swap (r) [IsStrictTotalOrder α r] : IsStrictTotalOrd
 /-- A connected order is one satisfying the condition `a < c → a < b ∨ b < c`.
   This is recognizable as an intuitionistic substitute for `a ≤ b ∨ b ≤ a` on
   the constructive reals, and is also known as negative transitivity,
-  since the contrapositive asserts transitivity of the relation `¬ a < b`.  -/
+  since the contrapositive asserts transitivity of the relation `¬ a < b`. -/
 class IsOrderConnected (α : Type u) (lt : α → α → Prop) : Prop where
   /-- A connected order is one satisfying the condition `a < c → a < b ∨ b < c`. -/
   conn : ∀ a b c, lt a c → lt a b ∨ lt b c
diff --git a/Mathlib/Order/RelIso/Basic.lean b/Mathlib/Order/RelIso/Basic.lean
index bfbc41e545f91..4c5eb975fb749 100644
--- a/Mathlib/Order/RelIso/Basic.lean
+++ b/Mathlib/Order/RelIso/Basic.lean
@@ -3,7 +3,6 @@ Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
-import Mathlib.Init.Algebra.Classes
 import Mathlib.Data.FunLike.Basic
 import Mathlib.Logic.Embedding.Basic
 import Mathlib.Order.RelClasses
@@ -59,7 +58,7 @@ satisfy `r a b → s (f a) (f b)`.
 The relations `r` and `s` are `outParam`s since figuring them out from a goal is a higher-order
 matching problem that Lean usually can't do unaided.
 -/
-class RelHomClass (F : Type*) {α β : Type*} (r : outParam <| α → α → Prop)
+class RelHomClass (F : Type*) {α β : outParam Type*} (r : outParam <| α → α → Prop)
   (s : outParam <| β → β → Prop) [FunLike F α β] : Prop where
   /-- A `RelHomClass` sends related elements to related elements -/
   map_rel : ∀ (f : F) {a b}, r a b → s (f a) (f b)
@@ -202,7 +201,7 @@ instance : Coe (r ↪r s) (r →r s) :=
 
 -- TODO: define and instantiate a `RelEmbeddingClass` when `EmbeddingLike` is defined
 instance : FunLike (r ↪r s) α β where
-  coe := fun x => x.toFun
+  coe x := x.toFun
   coe_injective' f g h := by
     rcases f with ⟨⟨⟩⟩
     rcases g with ⟨⟨⟩⟩
@@ -537,7 +536,7 @@ instance : CoeOut (r ≃r s) (r ↪r s) :=
 
 -- TODO: define and instantiate a `RelIsoClass` when `EquivLike` is defined
 instance : FunLike (r ≃r s) α β where
-  coe := fun x => x
+  coe x := x
   coe_injective' := Equiv.coe_fn_injective.comp toEquiv_injective
 
 -- TODO: define and instantiate a `RelIsoClass` when `EquivLike` is defined
diff --git a/Mathlib/Order/RelSeries.lean b/Mathlib/Order/RelSeries.lean
index e3396ccf701b7..d01a0e54fc8ee 100644
--- a/Mathlib/Order/RelSeries.lean
+++ b/Mathlib/Order/RelSeries.lean
@@ -100,7 +100,7 @@ lemma toList_chain' (x : RelSeries r) : x.toList.Chain' r := by
 lemma toList_ne_nil (x : RelSeries r) : x.toList ≠ [] := fun m =>
   List.eq_nil_iff_forall_not_mem.mp m (x 0) <| (List.mem_ofFn _ _).mpr ⟨_, rfl⟩
 
-/-- Every nonempty list satisfying the chain condition gives a relation series-/
+/-- Every nonempty list satisfying the chain condition gives a relation series -/
 @[simps]
 def fromListChain' (x : List α) (x_ne_nil : x ≠ []) (hx : x.Chain' r) : RelSeries r where
   length := x.length - 1
@@ -169,7 +169,7 @@ lemma nonempty_of_infiniteDimensional [r.InfiniteDimensional] : Nonempty α :=
   ⟨RelSeries.withLength r 0 0⟩
 
 instance membership : Membership α (RelSeries r) :=
-  ⟨(· ∈ Set.range ·)⟩
+  ⟨Function.swap (· ∈ Set.range ·)⟩
 
 theorem mem_def : x ∈ s ↔ x ∈ Set.range s := Iff.rfl
 
@@ -734,6 +734,66 @@ noncomputable def comap (p : LTSeries β) (f : α → β)
   LTSeries α := mk p.length (fun i ↦ (surjective (p i)).choose)
     (fun i j h ↦ comap (by simpa only [(surjective _).choose_spec] using p.strictMono h))
 
+/-- The strict series `0 < … < n` in `ℕ`. -/
+def range (n : ℕ) : LTSeries ℕ where
+  length := n
+  toFun := fun i => i
+  step i := Nat.lt_add_one i
+
+@[simp] lemma length_range (n : ℕ) : (range n).length = n := rfl
+
+@[simp] lemma range_apply (n : ℕ) (i : Fin (n+1)) : (range n) i = i := rfl
+
+@[simp] lemma head_range (n : ℕ) : (range n).head = 0 := rfl
+
+@[simp] lemma last_range (n : ℕ) : (range n).last = n := rfl
+
+/--
+In ℕ, two entries in an `LTSeries` differ by at least the difference of their indices.
+(Expressed in a way that avoids subtraction).
+ -/
+lemma apply_add_index_le_apply_add_index_nat (p : LTSeries ℕ) (i j : Fin (p.length + 1))
+    (hij : i ≤ j) : p i + j ≤ p j + i := by
+  have ⟨i, hi⟩ := i
+  have ⟨j, hj⟩ := j
+  simp only [Fin.mk_le_mk] at hij
+  simp only at *
+  induction j, hij using Nat.le_induction with
+  | base => simp
+  | succ j _hij ih =>
+    specialize ih (Nat.lt_of_succ_lt hj)
+    have step : p ⟨j, _⟩ < p ⟨j + 1, _⟩ := p.step ⟨j, by omega⟩
+    norm_cast at *; omega
+
+/--
+In ℤ, two entries in an `LTSeries` differ by at least the difference of their indices.
+(Expressed in a way that avoids subtraction).
+-/
+lemma apply_add_index_le_apply_add_index_int (p : LTSeries ℤ) (i j : Fin (p.length + 1))
+    (hij : i ≤ j) : p i + j ≤ p j + i := by
+  -- The proof is identical to `LTSeries.apply_add_index_le_apply_add_index_nat`, but seemed easier
+  -- to copy rather than to abstract
+  have ⟨i, hi⟩ := i
+  have ⟨j, hj⟩ := j
+  simp only [Fin.mk_le_mk] at hij
+  simp only at *
+  induction j, hij using Nat.le_induction with
+  | base => simp
+  | succ j _hij ih =>
+    specialize ih (Nat.lt_of_succ_lt hj)
+    have step : p ⟨j, _⟩ < p ⟨j + 1, _⟩:= p.step ⟨j, by omega⟩
+    norm_cast at *; omega
+
+/-- In ℕ, the head and tail of an `LTSeries` differ at least by the length of the series -/
+lemma head_add_length_le_nat (p : LTSeries ℕ) : p.head + p.length ≤ p.last :=
+  LTSeries.apply_add_index_le_apply_add_index_nat _ _ (Fin.last _) (Fin.zero_le _)
+
+/-- In ℤ, the head and tail of an `LTSeries` differ at least by the length of the series -/
+lemma head_add_length_le_int (p : LTSeries ℤ) : p.head + p.length ≤ p.last := by
+  simpa using LTSeries.apply_add_index_le_apply_add_index_int _ _ (Fin.last _) (Fin.zero_le _)
+
+section Fintype
+
 variable [Fintype α]
 
 lemma length_lt_card (s : LTSeries α) : s.length < Fintype.card α := by
@@ -751,6 +811,8 @@ instance [DecidableRel ((· < ·) : α → α → Prop)] : Fintype (LTSeries α)
     simp_rw [Finset.mem_map, Finset.mem_univ, true_and, Subtype.exists]
     use ⟨⟨l, bl⟩, f⟩, Fin.strictMono_iff_lt_succ.mpr mf; rfl
 
+end Fintype
+
 end LTSeries
 
 end LTSeries
diff --git a/Mathlib/Order/Restriction.lean b/Mathlib/Order/Restriction.lean
new file mode 100644
index 0000000000000..53d992f41e60b
--- /dev/null
+++ b/Mathlib/Order/Restriction.lean
@@ -0,0 +1,84 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.Order.Interval.Set.Basic
+import Mathlib.Order.Interval.Finset.Basic
+
+/-!
+# Restriction of a function indexed by a preorder
+
+Given a preorder `α` and dependent function `f : (i : α) → π i` and `a : α`, one might want
+to consider the restriction of `f` to elements `≤ a`.
+This is defined in this file as `Preorder.restrictLe a f`.
+Similarly, if we have `a b : α`, `hab : a ≤ b` and `f : (i : ↑(Set.Iic b)) → π ↑i`,
+one might want to restrict it to elements `≤ a`.
+This is defined in this file as `Preorder.restrictLe₂ hab f`.
+
+We also provide versions where the intervals are seen as finite sets, see `Preorder.frestrictLe`
+and `Preorder.frestrictLe₂`.
+
+## Main definitions
+* `Preorder.restrictLe a f`: Restricts the function `f` to the variables indexed by elements `≤ a`.
+-/
+
+namespace Preorder
+
+variable {α : Type*} [Preorder α] {π : α → Type*}
+
+section Set
+
+open Set
+
+/-- Restrict domain of a function `f` indexed by `α` to elements `≤ a`. -/
+def restrictLe (a : α) := (Iic a).restrict (π := π)
+
+@[simp]
+lemma restrictLe_apply (a : α) (f : (a : α) → π a) (i : Iic a) : restrictLe a f i = f i := rfl
+
+/-- If a function `f` indexed by `α` is restricted to elements `≤ π`, and `a ≤ b`,
+this is the restriction to elements `≤ a`. -/
+def restrictLe₂ {a b : α} (hab : a ≤ b) := Set.restrict₂ (π := π) (Iic_subset_Iic.2 hab)
+
+@[simp]
+lemma restrictLe₂_apply {a b : α} (hab : a ≤ b) (f : (i : Iic b) → π i) (i : Iic a) :
+    restrictLe₂ hab f i = f ⟨i.1, Iic_subset_Iic.2 hab i.2⟩ := rfl
+
+theorem restrictLe₂_comp_restrictLe {a b : α} (hab : a ≤ b) :
+    (restrictLe₂ (π := π) hab) ∘ (restrictLe b) = restrictLe a := rfl
+
+theorem restrictLe₂_comp_restrictLe₂ {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
+    (restrictLe₂ (π := π) hab) ∘ (restrictLe₂ hbc) = restrictLe₂ (hab.trans hbc) := rfl
+
+end Set
+
+section Finset
+
+variable [LocallyFiniteOrderBot α]
+
+open Finset
+
+/-- Restrict domain of a function `f` indexed by `α` to elements `≤ α`, seen as a finite set. -/
+def frestrictLe (a : α) := (Iic a).restrict (π := π)
+
+@[simp]
+lemma frestrictLe_apply (a : α) (f : (a : α) → π a) (i : Iic a) : frestrictLe a f i = f i := rfl
+
+/-- If a function `f` indexed by `α` is restricted to elements `≤ b`, and `a ≤ b`,
+this is the restriction to elements `≤ b`. Intervals are seen as finite sets. -/
+def frestrictLe₂ {a b : α} (hab : a ≤ b) := Finset.restrict₂ (π := π) (Iic_subset_Iic.2 hab)
+
+@[simp]
+lemma frestrictLe₂_apply {a b : α} (hab : a ≤ b) (f : (i : Iic b) → π i) (i : Iic a) :
+    frestrictLe₂ hab f i = f ⟨i.1, Iic_subset_Iic.2 hab i.2⟩ := rfl
+
+theorem frestrictLe₂_comp_frestrictLe {a b : α} (hab : a ≤ b) :
+    (frestrictLe₂ (π := π) hab) ∘ (frestrictLe b) = frestrictLe a := rfl
+
+theorem frestrictLe₂_comp_frestrictLe₂ {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
+    (frestrictLe₂ (π := π) hab) ∘ (frestrictLe₂ hbc) = frestrictLe₂ (hab.trans hbc) := rfl
+
+end Finset
+
+end Preorder
diff --git a/Mathlib/Order/ScottContinuity.lean b/Mathlib/Order/ScottContinuity.lean
new file mode 100644
index 0000000000000..511092cd0a000
--- /dev/null
+++ b/Mathlib/Order/ScottContinuity.lean
@@ -0,0 +1,136 @@
+/-
+Copyright (c) 2022 Christopher Hoskin. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christopher Hoskin
+-/
+import Mathlib.Order.Bounds.Basic
+
+/-!
+# Scott continuity
+
+A function `f : α → β` between preorders is Scott continuous (referring to Dana Scott) if it
+distributes over `IsLUB`. Scott continuity corresponds to continuity in Scott topological spaces
+(defined in `Mathlib/Topology/Order/ScottTopology.lean`). It is distinct from the (more commonly
+used) continuity from topology (see `Mathlib/Topology/Basic.lean`).
+
+## Implementation notes
+
+Given a set `D` of directed sets, we define say `f` is `ScottContinuousOn D` if it distributes over
+`IsLUB` for all elements of `D`. This allows us to consider Scott Continuity on all directed sets
+in this file, and ωScott Continuity on chains later in
+`Mathlib/Order/OmegaCompletePartialOrder.lean`.
+
+## References
+
+* [Abramsky and Jung, *Domain Theory*][abramsky_gabbay_maibaum_1994]
+* [Gierz et al, *A Compendium of Continuous Lattices*][GierzEtAl1980]
+
+-/
+
+open Set
+
+variable {α β : Type*}
+
+section ScottContinuous
+variable [Preorder α] [Preorder β] {D D₁ D₂ : Set (Set α)} {E : Set (Set β)}
+  {f : α → β} {a : α}
+
+/-- A function between preorders is said to be Scott continuous on a set `D` of directed sets if it
+preserves `IsLUB` on elements of `D`.
+
+The dual notion
+
+```lean
+∀ ⦃d : Set α⦄, d ∈ D →  d.Nonempty → DirectedOn (· ≥ ·) d → ∀ ⦃a⦄, IsGLB d a → IsGLB (f '' d) (f a)
+```
+
+does not appear to play a significant role in the literature, so is omitted here.
+-/
+def ScottContinuousOn (D : Set (Set α)) (f : α → β) : Prop :=
+  ∀ ⦃d : Set α⦄, d ∈ D → d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → IsLUB (f '' d) (f a)
+
+lemma ScottContinuousOn.mono (hD : D₁ ⊆ D₂) (hf : ScottContinuousOn D₂ f) :
+    ScottContinuousOn D₁ f := fun _  hdD₁ hd₁ hd₂ _ hda => hf (hD hdD₁) hd₁ hd₂ hda
+
+protected theorem ScottContinuousOn.monotone (D : Set (Set α)) (hD : ∀ a b : α, a ≤ b → {a, b} ∈ D)
+    (h : ScottContinuousOn D f) : Monotone f := by
+  refine fun a b hab =>
+    (h (hD a b hab) (insert_nonempty _ _) (directedOn_pair le_refl hab) ?_).1
+      (mem_image_of_mem _ <| mem_insert _ _)
+  rw [IsLUB, upperBounds_insert, upperBounds_singleton,
+    inter_eq_self_of_subset_right (Ici_subset_Ici.2 hab)]
+  exact isLeast_Ici
+
+@[simp] lemma ScottContinuousOn.id : ScottContinuousOn D (id : α → α) := by simp [ScottContinuousOn]
+
+variable {g : α → β}
+
+lemma ScottContinuousOn.prodMk (hD : ∀ a b : α, a ≤ b → {a, b} ∈ D)
+    (hf : ScottContinuousOn D f) (hg : ScottContinuousOn D g) :
+    ScottContinuousOn D fun x => (f x, g x) := fun d hd₁ hd₂ hd₃ a hda => by
+  rw [IsLUB, IsLeast, upperBounds]
+  constructor
+  · simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf_eq,
+      Prod.mk_le_mk]
+    intro b hb
+    exact ⟨hf.monotone D hD (hda.1 hb), hg.monotone D hD (hda.1 hb)⟩
+  · intro ⟨p₁, p₂⟩ hp
+    simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf_eq,
+      Prod.mk_le_mk] at hp
+    constructor
+    · rw [isLUB_le_iff (hf hd₁ hd₂ hd₃ hda), upperBounds]
+      simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf_eq]
+      intro _ hb
+      exact (hp _ hb).1
+    · rw [isLUB_le_iff (hg hd₁ hd₂ hd₃ hda), upperBounds]
+      simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂, mem_setOf_eq]
+      intro _ hb
+      exact (hp _ hb).2
+
+/-- A function between preorders is said to be Scott continuous if it preserves `IsLUB` on directed
+sets. It can be shown that a function is Scott continuous if and only if it is continuous wrt the
+Scott topology.
+-/
+def ScottContinuous (f : α → β) : Prop :=
+  ∀ ⦃d : Set α⦄, d.Nonempty → DirectedOn (· ≤ ·) d → ∀ ⦃a⦄, IsLUB d a → IsLUB (f '' d) (f a)
+
+@[simp] lemma scottContinuousOn_univ : ScottContinuousOn univ f ↔ ScottContinuous f := by
+  simp [ScottContinuousOn, ScottContinuous]
+
+lemma ScottContinuous.scottContinuousOn {D : Set (Set α)} :
+    ScottContinuous f → ScottContinuousOn D f := fun h _ _ d₂ d₃ _ hda => h d₂ d₃ hda
+
+protected theorem ScottContinuous.monotone (h : ScottContinuous f) : Monotone f :=
+  h.scottContinuousOn.monotone univ (fun _ _ _ ↦ mem_univ _)
+
+@[simp] lemma ScottContinuous.id : ScottContinuous (id : α → α) := by simp [ScottContinuous]
+
+end ScottContinuous
+
+section SemilatticeSup
+
+variable [Preorder α] [SemilatticeSup β]
+
+lemma ScottContinuousOn.sup₂ {D : Set (Set (β × β))} :
+    ScottContinuousOn D fun (a, b) => (a ⊔ b : β) := by
+  simp only
+  intro d _ _ _ ⟨p₁, p₂⟩ hdp
+  rw [IsLUB, IsLeast, upperBounds] at hdp
+  simp only [Prod.forall, mem_setOf_eq, Prod.mk_le_mk] at hdp
+  rw [IsLUB, IsLeast, upperBounds]
+  constructor
+  · simp only [mem_image, Prod.exists, forall_exists_index, and_imp, mem_setOf_eq]
+    intro a b₁ b₂ hbd hba
+    rw [← hba]
+    exact sup_le_sup (hdp.1 _ _ hbd).1 (hdp.1 _ _ hbd).2
+  · simp only [mem_image, Prod.exists, forall_exists_index, and_imp]
+    intro b hb
+    simp only [sup_le_iff]
+    have e1 : (p₁, p₂) ∈ lowerBounds {x | ∀ (b₁ b₂ : β), (b₁, b₂) ∈ d → (b₁, b₂) ≤ x} := hdp.2
+    rw [lowerBounds] at e1
+    simp only [mem_setOf_eq, Prod.forall, Prod.mk_le_mk] at e1
+    apply e1
+    intro b₁ b₂ hb'
+    exact sup_le_iff.mp (hb b₁ b₂ hb' rfl)
+
+end SemilatticeSup
diff --git a/Mathlib/Order/SemiconjSup.lean b/Mathlib/Order/SemiconjSup.lean
index 835ea7b923fb9..ebadc1795ef66 100644
--- a/Mathlib/Order/SemiconjSup.lean
+++ b/Mathlib/Order/SemiconjSup.lean
@@ -96,7 +96,7 @@ theorem semiconj_of_isLUB [PartialOrder α] [Group G] (f₁ f₂ : G →* α ≃
   refine fun y => (H _).unique ?_
   have := (f₁ g).leftOrdContinuous (H y)
   rw [← range_comp, ← (Equiv.mulRight g).surjective.range_comp _] at this
-  simpa [(· ∘ ·)] using this
+  simpa [comp_def] using this
 
 /-- Consider two actions `f₁ f₂ : G → α → α` of a group on a complete lattice by order
 isomorphisms. Then the map `x ↦ ⨆ g : G, (f₁ g)⁻¹ (f₂ g x)` semiconjugates each `f₁ g'` to `f₂ g'`.
diff --git a/Mathlib/Order/SuccPred/Archimedean.lean b/Mathlib/Order/SuccPred/Archimedean.lean
new file mode 100644
index 0000000000000..0db5ba2b12320
--- /dev/null
+++ b/Mathlib/Order/SuccPred/Archimedean.lean
@@ -0,0 +1,300 @@
+/-
+Copyright (c) 2021 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies
+-/
+import Mathlib.Order.SuccPred.Basic
+
+/-!
+# Archimedean successor and predecessor
+
+* `IsSuccArchimedean`: `SuccOrder` where `succ` iterated to an element gives all the greater
+  ones.
+* `IsPredArchimedean`: `PredOrder` where `pred` iterated to an element gives all the smaller
+  ones.
+-/
+
+variable {α β : Type*}
+
+open Order Function
+
+/-- A `SuccOrder` is succ-archimedean if one can go from any two comparable elements by iterating
+`succ` -/
+class IsSuccArchimedean (α : Type*) [Preorder α] [SuccOrder α] : Prop where
+  /-- If `a ≤ b` then one can get to `a` from `b` by iterating `succ` -/
+  exists_succ_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, succ^[n] a = b
+
+/-- A `PredOrder` is pred-archimedean if one can go from any two comparable elements by iterating
+`pred` -/
+class IsPredArchimedean (α : Type*) [Preorder α] [PredOrder α] : Prop where
+  /-- If `a ≤ b` then one can get to `b` from `a` by iterating `pred` -/
+  exists_pred_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, pred^[n] b = a
+
+export IsSuccArchimedean (exists_succ_iterate_of_le)
+
+export IsPredArchimedean (exists_pred_iterate_of_le)
+
+section Preorder
+
+variable [Preorder α]
+
+section SuccOrder
+
+variable [SuccOrder α] [IsSuccArchimedean α] {a b : α}
+
+instance : IsPredArchimedean αᵒᵈ :=
+  ⟨fun {a b} h => by convert exists_succ_iterate_of_le h.ofDual⟩
+
+theorem LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] a = b :=
+  exists_succ_iterate_of_le h
+
+theorem exists_succ_iterate_iff_le : (∃ n, succ^[n] a = b) ↔ a ≤ b := by
+  refine ⟨?_, exists_succ_iterate_of_le⟩
+  rintro ⟨n, rfl⟩
+  exact id_le_iterate_of_id_le le_succ n a
+
+/-- Induction principle on a type with a `SuccOrder` for all elements above a given element `m`. -/
+@[elab_as_elim]
+theorem Succ.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, m ≤ n → P n → P (succ n)) ⦃n : α⦄
+    (hmn : m ≤ n) : P n := by
+  obtain ⟨n, rfl⟩ := hmn.exists_succ_iterate; clear hmn
+  induction' n with n ih
+  · exact h0
+  · rw [Function.iterate_succ_apply']
+    exact h1 _ (id_le_iterate_of_id_le le_succ n m) ih
+
+theorem Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :
+    p a ↔ p b := by
+  obtain ⟨n, rfl⟩ := h.exists_succ_iterate
+  exact Iterate.rec (fun b => p a ↔ p b) (fun c hc => hc.trans (hsucc _)) Iff.rfl n
+
+end SuccOrder
+
+section PredOrder
+
+variable [PredOrder α] [IsPredArchimedean α] {a b : α}
+
+instance : IsSuccArchimedean αᵒᵈ :=
+  ⟨fun {a b} h => by convert exists_pred_iterate_of_le h.ofDual⟩
+
+theorem LE.le.exists_pred_iterate (h : a ≤ b) : ∃ n, pred^[n] b = a :=
+  exists_pred_iterate_of_le h
+
+theorem exists_pred_iterate_iff_le : (∃ n, pred^[n] b = a) ↔ a ≤ b :=
+  exists_succ_iterate_iff_le (α := αᵒᵈ)
+
+/-- Induction principle on a type with a `PredOrder` for all elements below a given element `m`. -/
+@[elab_as_elim]
+theorem Pred.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, n ≤ m → P n → P (pred n)) ⦃n : α⦄
+    (hmn : n ≤ m) : P n :=
+  Succ.rec (α := αᵒᵈ) (P := P) h0 h1 hmn
+
+theorem Pred.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) {a b : α} (h : a ≤ b) :
+    p a ↔ p b :=
+  (Succ.rec_iff (α := αᵒᵈ) hsucc h).symm
+
+end PredOrder
+
+end Preorder
+
+section LinearOrder
+
+variable [LinearOrder α]
+
+section SuccOrder
+variable [SuccOrder α]
+
+lemma succ_max (a b : α) : succ (max a b) = max (succ a) (succ b) := succ_mono.map_max
+lemma succ_min (a b : α) : succ (min a b) = min (succ a) (succ b) := succ_mono.map_min
+
+variable [IsSuccArchimedean α] {a b : α}
+
+theorem exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a :=
+  (le_total a b).imp exists_succ_iterate_of_le exists_succ_iterate_of_le
+
+theorem Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p b :=
+  (le_total a b).elim (Succ.rec_iff hsucc) fun h => (Succ.rec_iff hsucc h).symm
+
+end SuccOrder
+
+section PredOrder
+variable [PredOrder α]
+
+lemma pred_max (a b : α) : pred (max a b) = max (pred a) (pred b) := pred_mono.map_max
+lemma pred_min (a b : α) : pred (min a b) = min (pred a) (pred b) := pred_mono.map_min
+
+variable [IsPredArchimedean α] {a b : α}
+
+theorem exists_pred_iterate_or : (∃ n, pred^[n] b = a) ∨ ∃ n, pred^[n] a = b :=
+  (le_total a b).imp exists_pred_iterate_of_le exists_pred_iterate_of_le
+
+theorem Pred.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) (a b : α) : p a ↔ p b :=
+  (le_total a b).elim (Pred.rec_iff hsucc) fun h => (Pred.rec_iff hsucc h).symm
+
+end PredOrder
+
+end LinearOrder
+
+section bdd_range
+variable [Preorder α] [Nonempty α] [Preorder β] {f : α → β}
+
+lemma StrictMono.not_bddAbove_range_of_isSuccArchimedean [NoMaxOrder α] [SuccOrder β]
+    [IsSuccArchimedean β] (hf : StrictMono f) : ¬ BddAbove (Set.range f) := by
+  rintro ⟨m, hm⟩
+  have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _
+  obtain ⟨a₀⟩ := ‹Nonempty α›
+  suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by
+    obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)
+    exact ha.not_le (hm' a)
+  have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)
+  apply Succ.rec
+  · exact h a₀
+  rintro b _ ⟨a, hba⟩
+  exact (h a).imp (fun a' ↦ (succ_le_of_lt hba).trans_lt)
+
+@[deprecated StrictMono.not_bddAbove_range_of_isSuccArchimedean (since := "2024-09-21")]
+alias StrictMono.not_bddAbove_range := StrictMono.not_bddAbove_range_of_isSuccArchimedean
+
+lemma StrictMono.not_bddBelow_range_of_isPredArchimedean [NoMinOrder α] [PredOrder β]
+    [IsPredArchimedean β] (hf : StrictMono f) : ¬ BddBelow (Set.range f) :=
+  hf.dual.not_bddAbove_range_of_isSuccArchimedean
+
+@[deprecated StrictMono.not_bddBelow_range_of_isPredArchimedean (since := "2024-09-21")]
+alias StrictMono.not_bddBelow_range := StrictMono.not_bddBelow_range_of_isPredArchimedean
+
+lemma StrictAnti.not_bddBelow_range_of_isSuccArchimedean [NoMinOrder α] [SuccOrder β]
+    [IsSuccArchimedean β] (hf : StrictAnti f) : ¬ BddAbove (Set.range f) :=
+  hf.dual_right.not_bddBelow_range_of_isPredArchimedean
+
+@[deprecated StrictAnti.not_bddBelow_range_of_isSuccArchimedean (since := "2024-09-21")]
+alias StrictAnti.not_bddAbove_range := StrictAnti.not_bddBelow_range_of_isSuccArchimedean
+
+lemma StrictAnti.not_bddBelow_range_of_isPredArchimedean [NoMaxOrder α] [PredOrder β]
+    [IsPredArchimedean β] (hf : StrictAnti f) : ¬ BddBelow (Set.range f) :=
+  hf.dual_right.not_bddAbove_range_of_isSuccArchimedean
+
+@[deprecated StrictAnti.not_bddBelow_range_of_isPredArchimedean (since := "2024-09-21")]
+alias StrictAnti.not_bddBelow_range := StrictAnti.not_bddBelow_range_of_isPredArchimedean
+
+end bdd_range
+
+section IsWellFounded
+
+variable [PartialOrder α]
+
+instance (priority := 100) WellFoundedLT.toIsPredArchimedean [h : WellFoundedLT α]
+    [PredOrder α] : IsPredArchimedean α :=
+  ⟨fun {a b} => by
+    refine WellFounded.fix (C := fun b => a ≤ b → ∃ n, Nat.iterate pred n b = a)
+      h.wf ?_ b
+    intros b ih hab
+    replace hab := eq_or_lt_of_le hab
+    rcases hab with (rfl | hab)
+    · exact ⟨0, rfl⟩
+    rcases eq_or_lt_of_le (pred_le b) with hb | hb
+    · cases (min_of_le_pred hb.ge).not_lt hab
+    dsimp at ih
+    obtain ⟨k, hk⟩ := ih (pred b) hb (le_pred_of_lt hab)
+    refine ⟨k + 1, ?_⟩
+    rw [iterate_add_apply, iterate_one, hk]⟩
+
+instance (priority := 100) WellFoundedGT.toIsSuccArchimedean [h : WellFoundedGT α]
+    [SuccOrder α] : IsSuccArchimedean α :=
+  let h : IsPredArchimedean αᵒᵈ := by infer_instance
+  ⟨h.1⟩
+
+end IsWellFounded
+
+section OrderBot
+
+variable [Preorder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
+
+theorem Succ.rec_bot (p : α → Prop) (hbot : p ⊥) (hsucc : ∀ a, p a → p (succ a)) (a : α) : p a :=
+  Succ.rec hbot (fun x _ h => hsucc x h) (bot_le : ⊥ ≤ a)
+
+end OrderBot
+
+section OrderTop
+
+variable [Preorder α] [OrderTop α] [PredOrder α] [IsPredArchimedean α]
+
+theorem Pred.rec_top (p : α → Prop) (htop : p ⊤) (hpred : ∀ a, p a → p (pred a)) (a : α) : p a :=
+  Pred.rec htop (fun x _ h => hpred x h) (le_top : a ≤ ⊤)
+
+end OrderTop
+
+lemma SuccOrder.forall_ne_bot_iff
+    [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
+    (P : α → Prop) :
+    (∀ i, i ≠ ⊥ → P i) ↔ (∀ i, P (SuccOrder.succ i)) := by
+  refine ⟨fun h i ↦ h _ (Order.succ_ne_bot i), fun h i hi ↦ ?_⟩
+  obtain ⟨j, rfl⟩ := exists_succ_iterate_of_le (bot_le : ⊥ ≤ i)
+  have hj : 0 < j := by apply Nat.pos_of_ne_zero; contrapose! hi; simp [hi]
+  rw [← Nat.succ_pred_eq_of_pos hj]
+  simp only [Function.iterate_succ', Function.comp_apply]
+  apply h
+
+section IsLeast
+
+-- TODO: generalize to PartialOrder and `DirectedOn` after #16272
+lemma BddAbove.exists_isGreatest_of_nonempty {X : Type*} [LinearOrder X] [SuccOrder X]
+    [IsSuccArchimedean X] {S : Set X} (hS : BddAbove S) (hS' : S.Nonempty) :
+    ∃ x, IsGreatest S x := by
+  obtain ⟨m, hm⟩ := hS
+  obtain ⟨n, hn⟩ := hS'
+  by_cases hm' : m ∈ S
+  · exact ⟨_, hm', hm⟩
+  have hn' := hm hn
+  revert hn hm hm'
+  refine Succ.rec ?_ ?_ hn'
+  · simp (config := {contextual := true})
+  intro m _ IH hm hn hm'
+  rw [mem_upperBounds] at IH hm
+  simp_rw [Order.le_succ_iff_eq_or_le] at hm
+  replace hm : ∀ x ∈ S, x ≤ m := by
+    intro x hx
+    refine (hm x hx).resolve_left ?_
+    rintro rfl
+    exact hm' hx
+  by_cases hmS : m ∈ S
+  · exact ⟨m, hmS, hm⟩
+  · exact IH hm hn hmS
+
+lemma BddBelow.exists_isLeast_of_nonempty {X : Type*} [LinearOrder X] [PredOrder X]
+    [IsPredArchimedean X] {S : Set X} (hS : BddBelow S) (hS' : S.Nonempty) :
+    ∃ x, IsLeast S x :=
+  hS.dual.exists_isGreatest_of_nonempty hS'
+
+end IsLeast
+
+section OrderIso
+
+variable {X Y : Type*} [PartialOrder X] [PartialOrder Y]
+
+/-- `IsSuccArchimedean` transfers across equivalences between `SuccOrder`s. -/
+protected lemma IsSuccArchimedean.of_orderIso [SuccOrder X] [IsSuccArchimedean X] [SuccOrder Y]
+    (f : X ≃o Y) : IsSuccArchimedean Y where
+  exists_succ_iterate_of_le {a b} h := by
+    refine (exists_succ_iterate_of_le ((map_inv_le_map_inv_iff f).mpr h)).imp ?_
+    intro n
+    rw [← f.apply_eq_iff_eq, EquivLike.apply_inv_apply]
+    rintro rfl
+    clear h
+    induction n generalizing a with
+    | zero => simp
+    | succ n IH => simp only [Function.iterate_succ', Function.comp_apply, IH, f.map_succ]
+
+/-- `IsPredArchimedean` transfers across equivalences between `PredOrder`s. -/
+protected lemma IsPredArchimedean.of_orderIso [PredOrder X] [IsPredArchimedean X] [PredOrder Y]
+    (f : X ≃o Y) : IsPredArchimedean Y where
+  exists_pred_iterate_of_le {a b} h := by
+    refine (exists_pred_iterate_of_le ((map_inv_le_map_inv_iff f).mpr h)).imp ?_
+    intro n
+    rw [← f.apply_eq_iff_eq, EquivLike.apply_inv_apply]
+    rintro rfl
+    clear h
+    induction n generalizing b with
+    | zero => simp
+    | succ n IH => simp only [Function.iterate_succ', Function.comp_apply, IH, f.map_pred]
+
+end OrderIso
diff --git a/Mathlib/Order/SuccPred/Basic.lean b/Mathlib/Order/SuccPred/Basic.lean
index 98516d9942e27..43142cc9da386 100644
--- a/Mathlib/Order/SuccPred/Basic.lean
+++ b/Mathlib/Order/SuccPred/Basic.lean
@@ -5,9 +5,7 @@ Authors: Yaël Dillies
 -/
 import Mathlib.Order.ConditionallyCompleteLattice.Basic
 import Mathlib.Order.Cover
-import Mathlib.Order.GaloisConnection
 import Mathlib.Order.Iterate
-import Mathlib.Order.WellFounded
 
 /-!
 # Successor and predecessor
@@ -21,16 +19,12 @@ order...
 
 * `SuccOrder`: Order equipped with a sensible successor function.
 * `PredOrder`: Order equipped with a sensible predecessor function.
-* `IsSuccArchimedean`: `SuccOrder` where `succ` iterated to an element gives all the greater
-  ones.
-* `IsPredArchimedean`: `PredOrder` where `pred` iterated to an element gives all the smaller
-  ones.
 
 ## Implementation notes
 
 Maximal elements don't have a sensible successor. Thus the naïve typeclass
 ```lean
-class NaiveSuccOrder (α : Type*) [Preorder α] :=
+class NaiveSuccOrder (α : Type*) [Preorder α] where
   (succ : α → α)
   (succ_le_iff : ∀ {a b}, succ a ≤ b ↔ a < b)
   (lt_succ_iff : ∀ {a b}, a < succ b ↔ a ≤ b)
@@ -51,11 +45,11 @@ variable {α β : Type*}
 /-- Order equipped with a sensible successor function. -/
 @[ext]
 class SuccOrder (α : Type*) [Preorder α] where
-  /-- Successor function-/
+  /-- Successor function -/
   succ : α → α
   /-- Proof of basic ordering with respect to `succ`-/
   le_succ : ∀ a, a ≤ succ a
-  /-- Proof of interaction between `succ` and maximal element-/
+  /-- Proof of interaction between `succ` and maximal element -/
   max_of_succ_le {a} : succ a ≤ a → IsMax a
   /-- Proof that `succ a` is the least element greater than `a`-/
   succ_le_of_lt {a b} : a < b → succ a ≤ b
@@ -63,11 +57,11 @@ class SuccOrder (α : Type*) [Preorder α] where
 /-- Order equipped with a sensible predecessor function. -/
 @[ext]
 class PredOrder (α : Type*) [Preorder α] where
-  /-- Predecessor function-/
+  /-- Predecessor function -/
   pred : α → α
   /-- Proof of basic ordering with respect to `pred`-/
   pred_le : ∀ a, pred a ≤ a
-  /-- Proof of interaction between `pred` and minimal element-/
+  /-- Proof of interaction between `pred` and minimal element -/
   min_of_le_pred {a} : a ≤ pred a → IsMin a
   /-- Proof that `pred b` is the greatest element less than `b`-/
   le_pred_of_lt {a b} : a < b → a ≤ pred b
@@ -229,15 +223,13 @@ lemma le_succ_of_wcovBy (h : a ⩿ b) : b ≤ succ a := by
 
 alias _root_.WCovBy.le_succ := le_succ_of_wcovBy
 
-theorem le_succ_iterate (k : ℕ) (x : α) : x ≤ succ^[k] x := by
-  conv_lhs => rw [(by simp only [Function.iterate_id, id] : x = id^[k] x)]
-  exact Monotone.le_iterate_of_le succ_mono le_succ k x
+theorem le_succ_iterate (k : ℕ) (x : α) : x ≤ succ^[k] x :=
+  id_le_iterate_of_id_le le_succ _ _
 
 theorem isMax_iterate_succ_of_eq_of_lt {n m : ℕ} (h_eq : succ^[n] a = succ^[m] a)
     (h_lt : n < m) : IsMax (succ^[n] a) := by
   refine max_of_succ_le (le_trans ?_ h_eq.symm.le)
-  have : succ (succ^[n] a) = succ^[n + 1] a := by rw [Function.iterate_succ', comp]
-  rw [this]
+  rw [← iterate_succ_apply' succ]
   have h_le : n + 1 ≤ m := Nat.succ_le_of_lt h_lt
   exact Monotone.monotone_iterate_of_le_map succ_mono (le_succ a) h_le
 
@@ -347,6 +339,12 @@ lemma succ_eq_of_covBy (h : a ⋖ b) : succ a = b := (succ_le_of_lt h.lt).antisy
 
 alias _root_.CovBy.succ_eq := succ_eq_of_covBy
 
+theorem _root_.OrderIso.map_succ {β : Type*} [PartialOrder β] [SuccOrder β] (f : α ≃o β) (a : α) :
+    f (succ a) = succ (f a) := by
+  by_cases h : IsMax a
+  · rw [h.succ_eq, (f.isMax_apply.2 h).succ_eq]
+  · exact (f.map_covBy.2 <| covBy_succ_of_not_isMax h).succ_eq.symm
+
 section NoMaxOrder
 
 variable [NoMaxOrder α]
@@ -427,6 +425,11 @@ theorem le_succ_iff_eq_or_le : a ≤ succ b ↔ a = succ b ∨ a ≤ b := by
 theorem lt_succ_iff_eq_or_lt_of_not_isMax (hb : ¬IsMax b) : a < succ b ↔ a = b ∨ a < b :=
   (lt_succ_iff_of_not_isMax hb).trans le_iff_eq_or_lt
 
+theorem not_isMin_succ [Nontrivial α] (a : α) : ¬ IsMin (succ a) := by
+  obtain ha | ha := (le_succ a).eq_or_lt
+  · exact (ha ▸ succ_eq_iff_isMax.1 ha.symm).not_isMin
+  · exact not_isMin_of_lt ha
+
 theorem Iic_succ (a : α) : Iic (succ a) = insert (succ a) (Iic a) :=
   ext fun _ => le_succ_iff_eq_or_le
 
@@ -716,6 +719,10 @@ lemma pred_eq_of_covBy (h : a ⋖ b) : pred b = a := h.wcovBy.pred_le.antisymm (
 
 alias _root_.CovBy.pred_eq := pred_eq_of_covBy
 
+theorem _root_.OrderIso.map_pred {β : Type*} [PartialOrder β] [PredOrder β] (f : α ≃o β) (a : α) :
+    f (pred a) = pred (f a) :=
+  f.dual.map_succ a
+
 section NoMinOrder
 
 variable [NoMinOrder α]
@@ -798,6 +805,9 @@ theorem pred_le_iff_eq_or_le : pred a ≤ b ↔ b = pred a ∨ a ≤ b := by
 theorem pred_lt_iff_eq_or_lt_of_not_isMin (ha : ¬IsMin a) : pred a < b ↔ a = b ∨ a < b :=
   (pred_lt_iff_of_not_isMin ha).trans le_iff_eq_or_lt
 
+theorem not_isMax_pred [Nontrivial α] (a : α) : ¬ IsMax (pred a) :=
+  not_isMin_succ (α := αᵒᵈ) a
+
 theorem Ici_pred (a : α) : Ici (pred a) = insert (pred a) (Ici a) :=
   ext fun _ => pred_le_iff_eq_or_le
 
@@ -917,7 +927,7 @@ lemma gc_pred_succ : GaloisConnection (pred : α → α) succ := fun _ _ ↦ pre
 
 end Preorder
 
-variable [PartialOrder α] [SuccOrder α] [PredOrder α] {a b : α}
+variable [PartialOrder α] [SuccOrder α] [PredOrder α] {a : α}
 
 @[simp]
 theorem succ_pred_of_not_isMin (h : ¬IsMin a) : succ (pred a) = a :=
@@ -1199,205 +1209,31 @@ end Succ
 
 end WithBot
 
-/-! ### Archimedeanness -/
-
-/-- A `SuccOrder` is succ-archimedean if one can go from any two comparable elements by iterating
-`succ` -/
-class IsSuccArchimedean (α : Type*) [Preorder α] [SuccOrder α] : Prop where
-  /-- If `a ≤ b` then one can get to `a` from `b` by iterating `succ` -/
-  exists_succ_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, succ^[n] a = b
-
-/-- A `PredOrder` is pred-archimedean if one can go from any two comparable elements by iterating
-`pred` -/
-class IsPredArchimedean (α : Type*) [Preorder α] [PredOrder α] : Prop where
-  /-- If `a ≤ b` then one can get to `b` from `a` by iterating `pred` -/
-  exists_pred_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, pred^[n] b = a
-
-export IsSuccArchimedean (exists_succ_iterate_of_le)
-
-export IsPredArchimedean (exists_pred_iterate_of_le)
-
-section Preorder
-
-variable [Preorder α]
-
-section SuccOrder
-
-variable [SuccOrder α] [IsSuccArchimedean α] {a b : α}
-
-instance : IsPredArchimedean αᵒᵈ :=
-  ⟨fun {a b} h => by convert exists_succ_iterate_of_le h.ofDual⟩
-
-theorem LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] a = b :=
-  exists_succ_iterate_of_le h
-
-theorem exists_succ_iterate_iff_le : (∃ n, succ^[n] a = b) ↔ a ≤ b := by
-  refine ⟨?_, exists_succ_iterate_of_le⟩
-  rintro ⟨n, rfl⟩
-  exact id_le_iterate_of_id_le le_succ n a
-
-/-- Induction principle on a type with a `SuccOrder` for all elements above a given element `m`. -/
-@[elab_as_elim]
-theorem Succ.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, m ≤ n → P n → P (succ n)) ⦃n : α⦄
-    (hmn : m ≤ n) : P n := by
-  obtain ⟨n, rfl⟩ := hmn.exists_succ_iterate; clear hmn
-  induction' n with n ih
-  · exact h0
-  · rw [Function.iterate_succ_apply']
-    exact h1 _ (id_le_iterate_of_id_le le_succ n m) ih
-
-theorem Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :
-    p a ↔ p b := by
-  obtain ⟨n, rfl⟩ := h.exists_succ_iterate
-  exact Iterate.rec (fun b => p a ↔ p b) (fun c hc => hc.trans (hsucc _)) Iff.rfl n
-
-end SuccOrder
-
-section PredOrder
-
-variable [PredOrder α] [IsPredArchimedean α] {a b : α}
-
-instance : IsSuccArchimedean αᵒᵈ :=
-  ⟨fun {a b} h => by convert exists_pred_iterate_of_le h.ofDual⟩
-
-theorem LE.le.exists_pred_iterate (h : a ≤ b) : ∃ n, pred^[n] b = a :=
-  exists_pred_iterate_of_le h
-
-theorem exists_pred_iterate_iff_le : (∃ n, pred^[n] b = a) ↔ a ≤ b :=
-  exists_succ_iterate_iff_le (α := αᵒᵈ)
-
-/-- Induction principle on a type with a `PredOrder` for all elements below a given element `m`. -/
-@[elab_as_elim]
-theorem Pred.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, n ≤ m → P n → P (pred n)) ⦃n : α⦄
-    (hmn : n ≤ m) : P n :=
-  Succ.rec (α := αᵒᵈ) (P := P) h0 h1 hmn
-
-theorem Pred.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) {a b : α} (h : a ≤ b) :
-    p a ↔ p b :=
-  (Succ.rec_iff (α := αᵒᵈ) hsucc h).symm
-
-end PredOrder
-
-end Preorder
-
-section LinearOrder
-
-variable [LinearOrder α]
-
-section SuccOrder
-variable [SuccOrder α]
-
-lemma succ_max (a b : α) : succ (max a b) = max (succ a) (succ b) := succ_mono.map_max
-lemma succ_min (a b : α) : succ (min a b) = min (succ a) (succ b) := succ_mono.map_min
-
-variable [IsSuccArchimedean α] {a b : α}
-
-theorem exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a :=
-  (le_total a b).imp exists_succ_iterate_of_le exists_succ_iterate_of_le
-
-theorem Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p b :=
-  (le_total a b).elim (Succ.rec_iff hsucc) fun h => (Succ.rec_iff hsucc h).symm
-
-end SuccOrder
-
-section PredOrder
-variable [PredOrder α]
-
-lemma pred_max (a b : α) : pred (max a b) = max (pred a) (pred b) := pred_mono.map_max
-lemma pred_min (a b : α) : pred (min a b) = min (pred a) (pred b) := pred_mono.map_min
-
-variable [IsPredArchimedean α] {a b : α}
-
-theorem exists_pred_iterate_or : (∃ n, pred^[n] b = a) ∨ ∃ n, pred^[n] a = b :=
-  (le_total a b).imp exists_pred_iterate_of_le exists_pred_iterate_of_le
-
-theorem Pred.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) (a b : α) : p a ↔ p b :=
-  (le_total a b).elim (Pred.rec_iff hsucc) fun h => (Pred.rec_iff hsucc h).symm
-
-end PredOrder
-
-end LinearOrder
-
-section bdd_range
-variable [Preorder α] [Nonempty α] [Preorder β] {f : α → β}
-
-lemma StrictMono.not_bddAbove_range [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β]
-    (hf : StrictMono f) : ¬ BddAbove (Set.range f) := by
-  rintro ⟨m, hm⟩
-  have hm' : ∀ a, f a ≤ m := fun a ↦ hm <| Set.mem_range_self _
-  obtain ⟨a₀⟩ := ‹Nonempty α›
-  suffices ∀ b, f a₀ ≤ b → ∃ a, b < f a by
-    obtain ⟨a, ha⟩ : ∃ a, m < f a := this m (hm' a₀)
-    exact ha.not_le (hm' a)
-  have h : ∀ a, ∃ a', f a < f a' := fun a ↦ (exists_gt a).imp (fun a' h ↦ hf h)
-  apply Succ.rec
-  · exact h a₀
-  rintro b _ ⟨a, hba⟩
-  exact (h a).imp (fun a' ↦ (succ_le_of_lt hba).trans_lt)
-
-lemma StrictMono.not_bddBelow_range [NoMinOrder α] [PredOrder β] [IsPredArchimedean β]
-    (hf : StrictMono f) : ¬ BddBelow (Set.range f) := hf.dual.not_bddAbove_range
-
-lemma StrictAnti.not_bddAbove_range [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β]
-    (hf : StrictAnti f) : ¬ BddAbove (Set.range f) := hf.dual_right.not_bddBelow_range
-
-lemma StrictAnti.not_bddBelow_range [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β]
-    (hf : StrictAnti f) : ¬ BddBelow (Set.range f) := hf.dual_right.not_bddAbove_range
-
-end bdd_range
-
-section IsWellOrder
-
-variable [LinearOrder α]
-
-instance (priority := 100) IsWellOrder.toIsPredArchimedean [h : IsWellOrder α (· < ·)]
-    [PredOrder α] : IsPredArchimedean α :=
-  ⟨fun {a b} => by
-    refine WellFounded.fix (C := fun b => a ≤ b → ∃ n, Nat.iterate pred n b = a)
-      h.wf ?_ b
-    intros b ih hab
-    replace hab := eq_or_lt_of_le hab
-    rcases hab with (rfl | hab)
-    · exact ⟨0, rfl⟩
-    rcases le_or_lt b (pred b) with hb | hb
-    · cases (min_of_le_pred hb).not_lt hab
-    dsimp at ih
-    obtain ⟨k, hk⟩ := ih (pred b) hb (le_pred_of_lt hab)
-    refine ⟨k + 1, ?_⟩
-    rw [iterate_add_apply, iterate_one, hk]⟩
-
-instance (priority := 100) IsWellOrder.toIsSuccArchimedean [h : IsWellOrder α (· > ·)]
-    [SuccOrder α] : IsSuccArchimedean α :=
-  let h : IsPredArchimedean αᵒᵈ := by infer_instance
-  ⟨h.1⟩
-
-end IsWellOrder
-
-section OrderBot
-
-variable [Preorder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
-
-theorem Succ.rec_bot (p : α → Prop) (hbot : p ⊥) (hsucc : ∀ a, p a → p (succ a)) (a : α) : p a :=
-  Succ.rec hbot (fun x _ h => hsucc x h) (bot_le : ⊥ ≤ a)
-
-end OrderBot
-
-section OrderTop
-
-variable [Preorder α] [OrderTop α] [PredOrder α] [IsPredArchimedean α]
-
-theorem Pred.rec_top (p : α → Prop) (htop : p ⊤) (hpred : ∀ a, p a → p (pred a)) (a : α) : p a :=
-  Pred.rec htop (fun x _ h => hpred x h) (le_top : a ≤ ⊤)
-
-end OrderTop
-
-lemma SuccOrder.forall_ne_bot_iff
-    [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
-    (P : α → Prop) :
-    (∀ i, i ≠ ⊥ → P i) ↔ (∀ i, P (SuccOrder.succ i)) := by
-  refine ⟨fun h i ↦ h _ (Order.succ_ne_bot i), fun h i hi ↦ ?_⟩
-  obtain ⟨j, rfl⟩ := exists_succ_iterate_of_le (bot_le : ⊥ ≤ i)
-  have hj : 0 < j := by apply Nat.pos_of_ne_zero; contrapose! hi; simp [hi]
-  rw [← Nat.succ_pred_eq_of_pos hj]
-  simp only [Function.iterate_succ', Function.comp_apply]
-  apply h
+section OrderIso
+
+variable {X Y : Type*} [Preorder X] [Preorder Y]
+
+-- See note [reducible non instances]
+/-- `SuccOrder` transfers across equivalences between orders. -/
+protected abbrev SuccOrder.ofOrderIso [SuccOrder X] (f : X ≃o Y) : SuccOrder Y where
+  succ y := f (succ (f.symm y))
+  le_succ y := by rw [← map_inv_le_iff f]; exact le_succ (f.symm y)
+  max_of_succ_le h := by
+    rw [← f.symm.isMax_apply]
+    refine max_of_succ_le ?_
+    simp [f.le_symm_apply, h]
+  succ_le_of_lt h := by rw [← le_map_inv_iff]; exact succ_le_of_lt (by simp [h])
+
+-- See note [reducible non instances]
+/-- `PredOrder` transfers across equivalences between orders. -/
+protected abbrev PredOrder.ofOrderIso [PredOrder X] (f : X ≃o Y) :
+    PredOrder Y where
+  pred y := f (pred (f.symm y))
+  pred_le y := by rw [← le_map_inv_iff f]; exact pred_le (f.symm y)
+  min_of_le_pred h := by
+    rw [← f.symm.isMin_apply]
+    refine min_of_le_pred ?_
+    simp [f.symm_apply_le, h]
+  le_pred_of_lt h := by rw [← map_inv_le_iff]; exact le_pred_of_lt (by simp [h])
+
+end OrderIso
diff --git a/Mathlib/Order/SuccPred/CompleteLinearOrder.lean b/Mathlib/Order/SuccPred/CompleteLinearOrder.lean
index 6e68665cc40b0..6b8b1de8ce9f3 100644
--- a/Mathlib/Order/SuccPred/CompleteLinearOrder.lean
+++ b/Mathlib/Order/SuccPred/CompleteLinearOrder.lean
@@ -8,7 +8,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
 
 /-!
 
-# Relation between `IsSuccLimit` and `iSup` in (conditionally) complete linear orders.
+# Relation between `IsSuccPrelimit` and `iSup` in (conditionally) complete linear orders.
 
 -/
 
@@ -19,45 +19,69 @@ variable {ι α : Type*}
 section ConditionallyCompleteLinearOrder
 variable [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} {s : Set α} {x : α}
 
-lemma csSup_mem_of_not_isSuccLimit
-    (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
+lemma csSup_mem_of_not_isSuccPrelimit
+    (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) :
     sSup s ∈ s := by
   obtain ⟨y, hy⟩ := not_forall_not.mp hlim
   obtain ⟨i, his, hi⟩ := exists_lt_of_lt_csSup hne hy.lt
   exact eq_of_le_of_not_lt (le_csSup hbdd his) (hy.2 hi) ▸ his
 
-lemma csInf_mem_of_not_isPredLimit
-    (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬ IsPredLimit (sInf s)) :
+@[deprecated csSup_mem_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias csSup_mem_of_not_isSuccLimit := csSup_mem_of_not_isSuccPrelimit
+
+lemma csInf_mem_of_not_isPredPrelimit
+    (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬ IsPredPrelimit (sInf s)) :
     sInf s ∈ s := by
   obtain ⟨y, hy⟩ := not_forall_not.mp hlim
   obtain ⟨i, his, hi⟩ := exists_lt_of_csInf_lt hne hy.lt
   exact eq_of_le_of_not_lt (csInf_le hbdd his) (hy.2 · hi) ▸ his
 
-lemma exists_eq_ciSup_of_not_isSuccLimit
-    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
+@[deprecated csInf_mem_of_not_isPredPrelimit (since := "2024-09-05")]
+alias csInf_mem_of_not_isPredLimit := csInf_mem_of_not_isPredPrelimit
+
+lemma exists_eq_ciSup_of_not_isSuccPrelimit
+    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccPrelimit (⨆ i, f i)) :
     ∃ i, f i = ⨆ i, f i :=
-  csSup_mem_of_not_isSuccLimit (Set.range_nonempty f) hf hf'
+  csSup_mem_of_not_isSuccPrelimit (Set.range_nonempty f) hf hf'
+
+@[deprecated exists_eq_ciSup_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias exists_eq_ciSup_of_not_isSuccLimit := exists_eq_ciSup_of_not_isSuccPrelimit
 
-lemma exists_eq_ciInf_of_not_isPredLimit
-    (hf : BddBelow (Set.range f)) (hf' : ¬ IsPredLimit (⨅ i, f i)) :
+lemma exists_eq_ciInf_of_not_isPredPrelimit
+    (hf : BddBelow (Set.range f)) (hf' : ¬ IsPredPrelimit (⨅ i, f i)) :
     ∃ i, f i = ⨅ i, f i :=
-  csInf_mem_of_not_isPredLimit (Set.range_nonempty f) hf hf'
+  csInf_mem_of_not_isPredPrelimit (Set.range_nonempty f) hf hf'
+
+@[deprecated exists_eq_ciInf_of_not_isPredPrelimit (since := "2024-09-05")]
+alias exists_eq_ciInf_of_not_isPredLimit := exists_eq_ciInf_of_not_isPredPrelimit
+
+lemma IsLUB.mem_of_nonempty_of_not_isSuccPrelimit
+    (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬ IsSuccPrelimit x) : x ∈ s :=
+  hs.csSup_eq hne ▸ csSup_mem_of_not_isSuccPrelimit hne hs.bddAbove (hs.csSup_eq hne ▸ hx)
 
-lemma IsLUB.mem_of_nonempty_of_not_isSuccLimit
-    (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬ IsSuccLimit x) : x ∈ s :=
-  hs.csSup_eq hne ▸ csSup_mem_of_not_isSuccLimit hne hs.bddAbove (hs.csSup_eq hne ▸ hx)
+@[deprecated IsLUB.mem_of_nonempty_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias IsLUB.mem_of_nonempty_of_not_isSuccLimit := IsLUB.mem_of_nonempty_of_not_isSuccPrelimit
 
-lemma IsGLB.mem_of_nonempty_of_not_isPredLimit
-    (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬ IsPredLimit x) : x ∈ s :=
-  hs.csInf_eq hne ▸ csInf_mem_of_not_isPredLimit hne hs.bddBelow (hs.csInf_eq hne ▸ hx)
+lemma IsGLB.mem_of_nonempty_of_not_isPredPrelimit
+    (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬ IsPredPrelimit x) : x ∈ s :=
+  hs.csInf_eq hne ▸ csInf_mem_of_not_isPredPrelimit hne hs.bddBelow (hs.csInf_eq hne ▸ hx)
 
-lemma IsLUB.exists_of_nonempty_of_not_isSuccLimit
-    (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
-    ∃ i, f i = x := hf.mem_of_nonempty_of_not_isSuccLimit (Set.range_nonempty f) hx
+@[deprecated IsGLB.mem_of_nonempty_of_not_isPredPrelimit (since := "2024-09-05")]
+alias IsGLB.mem_of_nonempty_of_not_isPredLimit := IsGLB.mem_of_nonempty_of_not_isPredPrelimit
 
-lemma IsGLB.exists_of_nonempty_of_not_isPredLimit
-    (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
-    ∃ i, f i = x := hf.mem_of_nonempty_of_not_isPredLimit (Set.range_nonempty f) hx
+lemma IsLUB.exists_of_nonempty_of_not_isSuccPrelimit
+    (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccPrelimit x) :
+    ∃ i, f i = x := hf.mem_of_nonempty_of_not_isSuccPrelimit (Set.range_nonempty f) hx
+
+@[deprecated IsLUB.exists_of_nonempty_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias IsLUB.exists_of_nonempty_of_not_isSuccLimit := IsLUB.exists_of_nonempty_of_not_isSuccPrelimit
+
+lemma IsGLB.exists_of_nonempty_of_not_isPredPrelimit
+    (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredPrelimit x) :
+    ∃ i, f i = x := hf.mem_of_nonempty_of_not_isPredPrelimit (Set.range_nonempty f) hx
+
+@[deprecated IsGLB.exists_of_nonempty_of_not_isPredPrelimit (since := "2024-09-05")]
+alias IsGLB.exists_of_nonempty_of_not_isPredLimit := IsGLB.exists_of_nonempty_of_not_isPredPrelimit
 
 open Classical in
 /-- Every conditionally complete linear order with well-founded `<` is a successor order, by setting
@@ -85,60 +109,90 @@ end ConditionallyCompleteLinearOrder
 section ConditionallyCompleteLinearOrderBot
 variable [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {s : Set α} {x : α}
 
-/-- See `csSup_mem_of_not_isSuccLimit` for the `ConditionallyCompleteLinearOrder` version. -/
-lemma csSup_mem_of_not_isSuccLimit'
-    (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
+/-- See `csSup_mem_of_not_isSuccPrelimit` for the `ConditionallyCompleteLinearOrder` version. -/
+lemma csSup_mem_of_not_isSuccPrelimit'
+    (hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) :
     sSup s ∈ s := by
   obtain (rfl|hs) := s.eq_empty_or_nonempty
-  · simp [isSuccLimit_bot] at hlim
-  · exact csSup_mem_of_not_isSuccLimit hs hbdd hlim
+  · simp [isSuccPrelimit_bot] at hlim
+  · exact csSup_mem_of_not_isSuccPrelimit hs hbdd hlim
+
+@[deprecated csSup_mem_of_not_isSuccPrelimit' (since := "2024-09-05")]
+alias csSup_mem_of_not_isSuccLimit' := csSup_mem_of_not_isSuccPrelimit'
 
-/-- See `exists_eq_ciSup_of_not_isSuccLimit` for the
+/-- See `exists_eq_ciSup_of_not_isSuccPrelimit` for the
 `ConditionallyCompleteLinearOrder` version. -/
-lemma exists_eq_ciSup_of_not_isSuccLimit'
-    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
+lemma exists_eq_ciSup_of_not_isSuccPrelimit'
+    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccPrelimit (⨆ i, f i)) :
     ∃ i, f i = ⨆ i, f i :=
-  csSup_mem_of_not_isSuccLimit' hf hf'
+  csSup_mem_of_not_isSuccPrelimit' hf hf'
 
-lemma IsLUB.mem_of_not_isSuccLimit (hs : IsLUB s x) (hx : ¬ IsSuccLimit x) :
+@[deprecated exists_eq_ciSup_of_not_isSuccPrelimit' (since := "2024-09-05")]
+alias exists_eq_ciSup_of_not_isSuccLimit' := exists_eq_ciSup_of_not_isSuccPrelimit'
+
+lemma IsLUB.mem_of_not_isSuccPrelimit (hs : IsLUB s x) (hx : ¬ IsSuccPrelimit x) :
     x ∈ s := by
   obtain (rfl|hs') := s.eq_empty_or_nonempty
-  · simp [show x = ⊥ by simpa using hs, isSuccLimit_bot] at hx
-  · exact hs.mem_of_nonempty_of_not_isSuccLimit hs' hx
+  · simp [show x = ⊥ by simpa using hs, isSuccPrelimit_bot] at hx
+  · exact hs.mem_of_nonempty_of_not_isSuccPrelimit hs' hx
+
+@[deprecated IsLUB.mem_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias IsLUB.mem_of_not_isSuccLimit := IsLUB.mem_of_not_isSuccPrelimit
+
+lemma IsLUB.exists_of_not_isSuccPrelimit (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccPrelimit x) :
+    ∃ i, f i = x := hf.mem_of_not_isSuccPrelimit hx
 
-lemma IsLUB.exists_of_not_isSuccLimit (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
-    ∃ i, f i = x := hf.mem_of_not_isSuccLimit hx
+@[deprecated IsLUB.exists_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias IsLUB.exists_of_not_isSuccLimit := IsLUB.exists_of_not_isSuccPrelimit
 
 end ConditionallyCompleteLinearOrderBot
 
 section CompleteLinearOrder
 variable [CompleteLinearOrder α] {s : Set α} {f : ι → α} {x : α}
 
-lemma sSup_mem_of_not_isSuccLimit (hlim : ¬ IsSuccLimit (sSup s)) :
+lemma sSup_mem_of_not_isSuccPrelimit (hlim : ¬ IsSuccPrelimit (sSup s)) :
     sSup s ∈ s := by
   obtain ⟨y, hy⟩ := not_forall_not.mp hlim
   obtain ⟨i, his, hi⟩ := lt_sSup_iff.mp hy.lt
   exact eq_of_le_of_not_lt (le_sSup his) (hy.2 hi) ▸ his
 
-lemma sInf_mem_of_not_isPredLimit (hlim : ¬ IsPredLimit (sInf s)) :
+@[deprecated sSup_mem_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias sSup_mem_of_not_isSuccLimit := sSup_mem_of_not_isSuccPrelimit
+
+lemma sInf_mem_of_not_isPredPrelimit (hlim : ¬ IsPredPrelimit (sInf s)) :
     sInf s ∈ s := by
   obtain ⟨y, hy⟩ := not_forall_not.mp hlim
   obtain ⟨i, his, hi⟩ := sInf_lt_iff.mp hy.lt
   exact eq_of_le_of_not_lt (sInf_le his) (hy.2 · hi) ▸ his
 
-lemma exists_eq_iSup_of_not_isSuccLimit (hf : ¬ IsSuccLimit (⨆ i, f i)) :
+@[deprecated sInf_mem_of_not_isPredPrelimit (since := "2024-09-05")]
+alias sInf_mem_of_not_isPredLimit := sInf_mem_of_not_isPredPrelimit
+
+lemma exists_eq_iSup_of_not_isSuccPrelimit (hf : ¬ IsSuccPrelimit (⨆ i, f i)) :
     ∃ i, f i = ⨆ i, f i :=
-  sSup_mem_of_not_isSuccLimit hf
+  sSup_mem_of_not_isSuccPrelimit hf
 
-lemma exists_eq_iInf_of_not_isPredLimit (hf : ¬ IsPredLimit (⨅ i, f i)) :
+@[deprecated exists_eq_iSup_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias exists_eq_iSup_of_not_isSuccLimit := exists_eq_iSup_of_not_isSuccPrelimit
+
+lemma exists_eq_iInf_of_not_isPredPrelimit (hf : ¬ IsPredPrelimit (⨅ i, f i)) :
     ∃ i, f i = ⨅ i, f i :=
-  sInf_mem_of_not_isPredLimit hf
+  sInf_mem_of_not_isPredPrelimit hf
+
+@[deprecated exists_eq_iInf_of_not_isPredPrelimit (since := "2024-09-05")]
+alias exists_eq_iInf_of_not_isPredLimit := exists_eq_iInf_of_not_isPredPrelimit
 
-lemma IsGLB.mem_of_not_isPredLimit (hs : IsGLB s x) (hx : ¬ IsPredLimit x) :
+lemma IsGLB.mem_of_not_isPredPrelimit (hs : IsGLB s x) (hx : ¬ IsPredPrelimit x) :
     x ∈ s :=
-  hs.sInf_eq ▸ sInf_mem_of_not_isPredLimit (hs.sInf_eq ▸ hx)
+  hs.sInf_eq ▸ sInf_mem_of_not_isPredPrelimit (hs.sInf_eq ▸ hx)
+
+@[deprecated IsGLB.mem_of_not_isPredPrelimit (since := "2024-09-05")]
+alias IsGLB.mem_of_not_isPredLimit := IsGLB.mem_of_not_isPredPrelimit
+
+lemma IsGLB.exists_of_not_isPredPrelimit (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredPrelimit x) :
+    ∃ i, f i = x := hf.mem_of_not_isPredPrelimit hx
 
-lemma IsGLB.exists_of_not_isPredLimit (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
-    ∃ i, f i = x := hf.mem_of_not_isPredLimit hx
+@[deprecated IsGLB.exists_of_not_isPredPrelimit (since := "2024-09-05")]
+alias IsGLB.exists_of_not_isPredLimit := IsGLB.exists_of_not_isPredPrelimit
 
 end CompleteLinearOrder
diff --git a/Mathlib/Order/SuccPred/IntervalSucc.lean b/Mathlib/Order/SuccPred/IntervalSucc.lean
index 96bbcd2e0b85a..4e11847ea9a33 100644
--- a/Mathlib/Order/SuccPred/IntervalSucc.lean
+++ b/Mathlib/Order/SuccPred/IntervalSucc.lean
@@ -5,7 +5,7 @@ Authors: Yury Kudryashov
 -/
 import Mathlib.Data.Set.Pairwise.Basic
 import Mathlib.Data.Set.Lattice
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 
 /-!
 # Intervals `Ixx (f x) (f (Order.succ x))`
diff --git a/Mathlib/Order/SuccPred/Limit.lean b/Mathlib/Order/SuccPred/Limit.lean
index dc95eb49132a8..45933dc60feb8 100644
--- a/Mathlib/Order/SuccPred/Limit.lean
+++ b/Mathlib/Order/SuccPred/Limit.lean
@@ -3,15 +3,19 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 import Mathlib.Order.BoundedOrder
 
 /-!
 # Successor and predecessor limits
 
-We define the predicate `Order.IsSuccLimit` for "successor limits", values that don't cover any
-others. They are so named since they can't be the successors of anything smaller. We define
-`Order.IsPredLimit` analogously, and prove basic results.
+We define the predicate `Order.IsSuccPrelimit` for "successor pre-limits", values that don't cover
+any others. They are so named since they can't be the successors of anything smaller. We define
+`Order.IsPredPrelimit` analogously, and prove basic results.
+
+For some applications, it is desirable to exclude minimal elements from being successor limits, or
+maximal elements from being predecessor limits. As such, we also provide `Order.IsSuccLimit` and
+`Order.IsPredLimit`, which exclude these cases.
 
 ## TODO
 
@@ -20,7 +24,7 @@ predicate `Order.IsSuccLimit`.
 -/
 
 
-variable {α : Type*}
+variable {α : Type*} {a b : α}
 
 namespace Order
 
@@ -33,49 +37,113 @@ section LT
 
 variable [LT α]
 
-/-- A successor limit is a value that doesn't cover any other.
+/-- A successor pre-limit is a value that doesn't cover any other.
 
-It's so named because in a successor order, a successor limit can't be the successor of anything
-smaller. -/
-def IsSuccLimit (a : α) : Prop :=
+It's so named because in a successor order, a successor pre-limit can't be the successor of anything
+smaller.
+
+Use `IsSuccLimit` if you want to exclude the case of a minimal element. -/
+def IsSuccPrelimit (a : α) : Prop :=
   ∀ b, ¬b ⋖ a
 
-theorem not_isSuccLimit_iff_exists_covBy (a : α) : ¬IsSuccLimit a ↔ ∃ b, b ⋖ a := by
-  simp [IsSuccLimit]
+theorem not_isSuccPrelimit_iff_exists_covBy (a : α) : ¬IsSuccPrelimit a ↔ ∃ b, b ⋖ a := by
+  simp [IsSuccPrelimit]
+
+@[deprecated not_isSuccPrelimit_iff_exists_covBy (since := "2024-09-05")]
+alias not_isSuccLimit_iff_exists_covBy := not_isSuccPrelimit_iff_exists_covBy
 
 @[simp]
-theorem isSuccLimit_of_dense [DenselyOrdered α] (a : α) : IsSuccLimit a := fun _ => not_covBy
+theorem isSuccPrelimit_of_dense [DenselyOrdered α] (a : α) : IsSuccPrelimit a := fun _ => not_covBy
+
+@[deprecated isSuccPrelimit_of_dense (since := "2024-09-05")]
+alias isSuccLimit_of_dense := isSuccPrelimit_of_dense
 
 end LT
 
 section Preorder
 
-variable [Preorder α] {a : α}
+variable [Preorder α]
+
+/-- A successor limit is a value that isn't minimal and doesn't cover any other.
+
+It's so named because in a successor order, a successor limit can't be the successor of anything
+smaller.
+
+This previously allowed the element to be minimal. This usage is now covered by `IsSuccPrelimit`. -/
+def IsSuccLimit (a : α) : Prop :=
+  ¬ IsMin a ∧ IsSuccPrelimit a
+
+protected theorem IsSuccLimit.not_isMin (h : IsSuccLimit a) : ¬ IsMin a := h.1
+protected theorem IsSuccLimit.isSuccPrelimit (h : IsSuccLimit a) : IsSuccPrelimit a := h.2
+
+theorem IsSuccPrelimit.isSuccLimit_of_not_isMin (h : IsSuccPrelimit a) (ha : ¬ IsMin a) :
+    IsSuccLimit a :=
+  ⟨ha, h⟩
+
+theorem IsSuccPrelimit.isSuccLimit [NoMinOrder α] (h : IsSuccPrelimit a) : IsSuccLimit a :=
+  h.isSuccLimit_of_not_isMin (not_isMin a)
 
-protected theorem _root_.IsMin.isSuccLimit : IsMin a → IsSuccLimit a := fun h _ hab =>
+theorem isSuccPrelimit_iff_isSuccLimit_of_not_isMin (h : ¬ IsMin a) :
+    IsSuccPrelimit a ↔ IsSuccLimit a :=
+  ⟨fun ha ↦ ha.isSuccLimit_of_not_isMin h, IsSuccLimit.isSuccPrelimit⟩
+
+theorem isSuccPrelimit_iff_isSuccLimit [NoMinOrder α] : IsSuccPrelimit a ↔ IsSuccLimit a :=
+  isSuccPrelimit_iff_isSuccLimit_of_not_isMin (not_isMin a)
+
+protected theorem _root_.IsMin.not_isSuccLimit (h : IsMin a) : ¬ IsSuccLimit a :=
+  fun ha ↦ ha.not_isMin h
+
+protected theorem _root_.IsMin.isSuccPrelimit : IsMin a → IsSuccPrelimit a := fun h _ hab =>
   not_isMin_of_lt hab.lt h
 
-theorem isSuccLimit_bot [OrderBot α] : IsSuccLimit (⊥ : α) :=
-  IsMin.isSuccLimit isMin_bot
+@[deprecated _root_.IsMin.isSuccPrelimit (since := "2024-09-05")]
+alias _root_.IsMin.isSuccLimit := _root_.IsMin.isSuccPrelimit
+
+theorem isSuccPrelimit_bot [OrderBot α] : IsSuccPrelimit (⊥ : α) :=
+  isMin_bot.isSuccPrelimit
+
+theorem not_isSuccLimit_bot [OrderBot α] : ¬ IsSuccLimit (⊥ : α) :=
+  isMin_bot.not_isSuccLimit
+
+theorem IsSuccLimit.ne_bot [OrderBot α] (h : IsSuccLimit a) : a ≠ ⊥ := by
+  rintro rfl
+  exact not_isSuccLimit_bot h
+
+@[deprecated isSuccPrelimit_bot (since := "2024-09-05")]
+alias isSuccLimit_bot := isSuccPrelimit_bot
+
+theorem not_isSuccLimit_iff : ¬ IsSuccLimit a ↔ IsMin a ∨ ¬ IsSuccPrelimit a := by
+  rw [IsSuccLimit, not_and_or, not_not]
 
 variable [SuccOrder α]
 
-protected theorem IsSuccLimit.isMax (h : IsSuccLimit (succ a)) : IsMax a := by
+protected theorem IsSuccPrelimit.isMax (h : IsSuccPrelimit (succ a)) : IsMax a := by
   by_contra H
   exact h a (covBy_succ_of_not_isMax H)
 
-theorem not_isSuccLimit_succ_of_not_isMax (ha : ¬IsMax a) : ¬IsSuccLimit (succ a) := by
-  contrapose! ha
-  exact ha.isMax
+protected theorem IsSuccLimit.isMax (h : IsSuccLimit (succ a)) : IsMax a :=
+  h.isSuccPrelimit.isMax
+
+theorem not_isSuccPrelimit_succ_of_not_isMax (ha : ¬ IsMax a) : ¬ IsSuccPrelimit (succ a) :=
+  mt IsSuccPrelimit.isMax ha
+
+theorem not_isSuccLimit_succ_of_not_isMax (ha : ¬ IsMax a) : ¬ IsSuccLimit (succ a) :=
+  mt IsSuccLimit.isMax ha
 
 section NoMaxOrder
 
 variable [NoMaxOrder α]
 
-theorem IsSuccLimit.succ_ne (h : IsSuccLimit a) (b : α) : succ b ≠ a := by
+theorem IsSuccPrelimit.succ_ne (h : IsSuccPrelimit a) (b : α) : succ b ≠ a := by
   rintro rfl
   exact not_isMax _ h.isMax
 
+theorem IsSuccLimit.succ_ne (h : IsSuccLimit a) (b : α) : succ b ≠ a :=
+  h.isSuccPrelimit.succ_ne b
+
+@[simp]
+theorem not_isSuccPrelimit_succ (a : α) : ¬IsSuccPrelimit (succ a) := fun h => h.succ_ne _ rfl
+
 @[simp]
 theorem not_isSuccLimit_succ (a : α) : ¬IsSuccLimit (succ a) := fun h => h.succ_ne _ rfl
 
@@ -83,19 +151,30 @@ end NoMaxOrder
 
 section IsSuccArchimedean
 
-variable [IsSuccArchimedean α]
+variable [IsSuccArchimedean α] [NoMaxOrder α]
 
-theorem IsSuccLimit.isMin_of_noMax [NoMaxOrder α] (h : IsSuccLimit a) : IsMin a := fun b hb => by
+theorem IsSuccPrelimit.isMin_of_noMax (h : IsSuccPrelimit a) : IsMin a := by
+  intro b hb
   rcases hb.exists_succ_iterate with ⟨_ | n, rfl⟩
   · exact le_rfl
   · rw [iterate_succ_apply'] at h
-    exact (not_isSuccLimit_succ _ h).elim
+    exact (not_isSuccPrelimit_succ _ h).elim
+
+@[deprecated IsSuccPrelimit.isMin_of_noMax (since := "2024-09-05")]
+alias IsSuccLimit.isMin_of_noMax := IsSuccPrelimit.isMin_of_noMax
 
 @[simp]
-theorem isSuccLimit_iff_of_noMax [NoMaxOrder α] : IsSuccLimit a ↔ IsMin a :=
-  ⟨IsSuccLimit.isMin_of_noMax, IsMin.isSuccLimit⟩
+theorem isSuccPrelimit_iff_of_noMax : IsSuccPrelimit a ↔ IsMin a :=
+  ⟨IsSuccPrelimit.isMin_of_noMax, IsMin.isSuccPrelimit⟩
 
-theorem not_isSuccLimit_of_noMax [NoMinOrder α] [NoMaxOrder α] : ¬IsSuccLimit a := by simp
+@[deprecated isSuccPrelimit_iff_of_noMax (since := "2024-09-05")]
+alias isSuccLimit_iff_of_noMax := isSuccPrelimit_iff_of_noMax
+
+@[simp]
+theorem not_isSuccLimit_of_noMax : ¬ IsSuccLimit a :=
+  fun h ↦ h.not_isMin h.isSuccPrelimit.isMin_of_noMax
+
+theorem not_isSuccPrelimit_of_noMax [NoMinOrder α] : ¬ IsSuccPrelimit a := by simp
 
 end IsSuccArchimedean
 
@@ -103,30 +182,48 @@ end Preorder
 
 section PartialOrder
 
-variable [PartialOrder α] [SuccOrder α] {a b : α} {C : α → Sort*}
+variable [PartialOrder α]
+
+theorem isSuccLimit_iff [OrderBot α] : IsSuccLimit a ↔ a ≠ ⊥ ∧ IsSuccPrelimit a := by
+  rw [IsSuccLimit, isMin_iff_eq_bot]
+
+variable [SuccOrder α]
 
-theorem isSuccLimit_of_succ_ne (h : ∀ b, succ b ≠ a) : IsSuccLimit a := fun b hba =>
+theorem isSuccPrelimit_of_succ_ne (h : ∀ b, succ b ≠ a) : IsSuccPrelimit a := fun b hba =>
   h b (CovBy.succ_eq hba)
 
-theorem not_isSuccLimit_iff : ¬IsSuccLimit a ↔ ∃ b, ¬IsMax b ∧ succ b = a := by
-  rw [not_isSuccLimit_iff_exists_covBy]
+@[deprecated isSuccPrelimit_of_succ_ne (since := "2024-09-05")]
+alias isSuccLimit_of_succ_ne := isSuccPrelimit_of_succ_ne
+
+theorem not_isSuccPrelimit_iff : ¬ IsSuccPrelimit a ↔ ∃ b, ¬ IsMax b ∧ succ b = a := by
+  rw [not_isSuccPrelimit_iff_exists_covBy]
   refine exists_congr fun b => ⟨fun hba => ⟨hba.lt.not_isMax, (CovBy.succ_eq hba)⟩, ?_⟩
   rintro ⟨h, rfl⟩
   exact covBy_succ_of_not_isMax h
 
-/-- See `not_isSuccLimit_iff` for a version that states that `a` is a successor of a value other
+/-- See `not_isSuccPrelimit_iff` for a version that states that `a` is a successor of a value other
 than itself. -/
-theorem mem_range_succ_of_not_isSuccLimit (h : ¬IsSuccLimit a) : a ∈ range (@succ α _ _) := by
-  cases' not_isSuccLimit_iff.1 h with b hb
+theorem mem_range_succ_of_not_isSuccPrelimit (h : ¬ IsSuccPrelimit a) :
+    a ∈ range (succ : α → α) := by
+  obtain ⟨b, hb⟩ := not_isSuccPrelimit_iff.1 h
   exact ⟨b, hb.2⟩
 
-theorem mem_range_succ_or_isSuccLimit (a) : a ∈ range (@succ α _ _) ∨ IsSuccLimit a :=
-  or_iff_not_imp_right.2 <| mem_range_succ_of_not_isSuccLimit
+@[deprecated mem_range_succ_of_not_isSuccPrelimit (since := "2024-09-05")]
+alias mem_range_succ_of_not_isSuccLimit := mem_range_succ_of_not_isSuccPrelimit
+
+theorem mem_range_succ_or_isSuccPrelimit (a) : a ∈ range (succ : α → α) ∨ IsSuccPrelimit a :=
+  or_iff_not_imp_right.2 <| mem_range_succ_of_not_isSuccPrelimit
+
+@[deprecated mem_range_succ_or_isSuccPrelimit (since := "2024-09-05")]
+alias mem_range_succ_or_isSuccLimit := mem_range_succ_or_isSuccPrelimit
 
-theorem isSuccLimit_of_succ_lt (H : ∀ a < b, succ a < b) : IsSuccLimit b := fun a hab =>
+theorem isSuccPrelimit_of_succ_lt (H : ∀ a < b, succ a < b) : IsSuccPrelimit b := fun a hab =>
   (H a hab.lt).ne (CovBy.succ_eq hab)
 
-theorem IsSuccLimit.succ_lt (hb : IsSuccLimit b) (ha : a < b) : succ a < b := by
+@[deprecated isSuccPrelimit_of_succ_lt (since := "2024-09-05")]
+alias isSuccLimit_of_succ_lt := isSuccPrelimit_of_succ_lt
+
+theorem IsSuccPrelimit.succ_lt (hb : IsSuccPrelimit b) (ha : a < b) : succ a < b := by
   by_cases h : IsMax a
   · rwa [h.succ_eq]
   · rw [lt_iff_le_and_ne, succ_le_iff_of_not_isMax h]
@@ -134,60 +231,36 @@ theorem IsSuccLimit.succ_lt (hb : IsSuccLimit b) (ha : a < b) : succ a < b := by
     subst hab
     exact (h hb.isMax).elim
 
-theorem IsSuccLimit.succ_lt_iff (hb : IsSuccLimit b) : succ a < b ↔ a < b :=
-  ⟨fun h => (le_succ a).trans_lt h, hb.succ_lt⟩
-
-theorem isSuccLimit_iff_succ_lt : IsSuccLimit b ↔ ∀ a < b, succ a < b :=
-  ⟨fun hb _ => hb.succ_lt, isSuccLimit_of_succ_lt⟩
-
-/-- A value can be built by building it on successors and successor limits. -/
-@[elab_as_elim]
-noncomputable def isSuccLimitRecOn (b : α) (hs : ∀ a, ¬IsMax a → C (succ a))
-    (hl : ∀ a, IsSuccLimit a → C a) : C b := by
-  by_cases hb : IsSuccLimit b
-  · exact hl b hb
-  · have H := Classical.choose_spec (not_isSuccLimit_iff.1 hb)
-    rw [← H.2]
-    exact hs _ H.1
-
-theorem isSuccLimitRecOn_limit (hs : ∀ a, ¬IsMax a → C (succ a)) (hl : ∀ a, IsSuccLimit a → C a)
-    (hb : IsSuccLimit b) : @isSuccLimitRecOn α _ _ C b hs hl = hl b hb := by
-  classical exact dif_pos hb
-
-section limitRecOn
+theorem IsSuccLimit.succ_lt (hb : IsSuccLimit b) (ha : a < b) : succ a < b :=
+  hb.isSuccPrelimit.succ_lt ha
 
-variable [WellFoundedLT α]
-  (H_succ : ∀ a, ¬IsMax a → C a → C (succ a))
-  (H_lim : ∀ a, IsSuccLimit a → (∀ b < a, C b) → C a)
+theorem IsSuccPrelimit.succ_lt_iff (hb : IsSuccPrelimit b) : succ a < b ↔ a < b :=
+  ⟨fun h => (le_succ a).trans_lt h, hb.succ_lt⟩
 
-open scoped Classical in
-variable (a) in
-/-- Recursion principle on a well-founded partial `SuccOrder`. -/
-@[elab_as_elim] noncomputable def _root_.SuccOrder.limitRecOn : C a :=
-  wellFounded_lt.fix
-    (fun a IH ↦ if h : IsSuccLimit a then H_lim a h IH else
-      let x := Classical.indefiniteDescription _ (not_isSuccLimit_iff.mp h)
-      x.2.2 ▸ H_succ x x.2.1 (IH x <| x.2.2.subst <| lt_succ_of_not_isMax x.2.1))
-    a
+theorem IsSuccLimit.succ_lt_iff (hb : IsSuccLimit b) : succ a < b ↔ a < b :=
+  hb.isSuccPrelimit.succ_lt_iff
 
-@[simp]
-theorem _root_.SuccOrder.limitRecOn_limit (ha : IsSuccLimit a) :
-    SuccOrder.limitRecOn a H_succ H_lim
-      = H_lim a ha fun x _ ↦ SuccOrder.limitRecOn x H_succ H_lim := by
-  rw [SuccOrder.limitRecOn, WellFounded.fix_eq, dif_pos ha]; rfl
+theorem isSuccPrelimit_iff_succ_lt : IsSuccPrelimit b ↔ ∀ a < b, succ a < b :=
+  ⟨fun hb _ => hb.succ_lt, isSuccPrelimit_of_succ_lt⟩
 
-end limitRecOn
+@[deprecated isSuccPrelimit_iff_succ_lt (since := "2024-09-05")]
+alias isSuccLimit_iff_succ_lt := isSuccPrelimit_iff_succ_lt
 
 section NoMaxOrder
 
 variable [NoMaxOrder α]
 
-theorem isSuccLimit_iff_succ_ne : IsSuccLimit a ↔ ∀ b, succ b ≠ a :=
-  ⟨IsSuccLimit.succ_ne, isSuccLimit_of_succ_ne⟩
+theorem isSuccPrelimit_iff_succ_ne : IsSuccPrelimit a ↔ ∀ b, succ b ≠ a :=
+  ⟨IsSuccPrelimit.succ_ne, isSuccPrelimit_of_succ_ne⟩
+
+@[deprecated isSuccPrelimit_iff_succ_ne (since := "2024-09-05")]
+alias isSuccLimit_iff_succ_ne := isSuccPrelimit_iff_succ_ne
 
-theorem not_isSuccLimit_iff' : ¬IsSuccLimit a ↔ a ∈ range (@succ α _ _) := by
-  simp_rw [isSuccLimit_iff_succ_ne, not_forall, not_ne_iff]
-  rfl
+theorem not_isSuccPrelimit_iff' : ¬ IsSuccPrelimit a ↔ a ∈ range (succ : α → α) := by
+  simp_rw [isSuccPrelimit_iff_succ_ne, not_forall, not_ne_iff, mem_range]
+
+@[deprecated not_isSuccPrelimit_iff' (since := "2024-09-05")]
+alias not_isSuccLimit_iff' := not_isSuccPrelimit_iff'
 
 end NoMaxOrder
 
@@ -195,7 +268,7 @@ section IsSuccArchimedean
 
 variable [IsSuccArchimedean α]
 
-protected theorem IsSuccLimit.isMin (h : IsSuccLimit a) : IsMin a := fun b hb => by
+protected theorem IsSuccPrelimit.isMin (h : IsSuccPrelimit a) : IsMin a := fun b hb => by
   revert h
   refine Succ.rec (fun _ => le_rfl) (fun c _ H hc => ?_) hb
   have := hc.isMax.succ_eq
@@ -203,33 +276,94 @@ protected theorem IsSuccLimit.isMin (h : IsSuccLimit a) : IsMin a := fun b hb =>
   exact H hc
 
 @[simp]
-theorem isSuccLimit_iff : IsSuccLimit a ↔ IsMin a :=
-  ⟨IsSuccLimit.isMin, IsMin.isSuccLimit⟩
+theorem isSuccPrelimit_iff : IsSuccPrelimit a ↔ IsMin a :=
+  ⟨IsSuccPrelimit.isMin, IsMin.isSuccPrelimit⟩
+
+@[simp]
+theorem not_isSuccLimit : ¬ IsSuccLimit a :=
+  fun h ↦ h.not_isMin <| h.isSuccPrelimit.isMin
 
-theorem not_isSuccLimit [NoMinOrder α] : ¬IsSuccLimit a := by simp
+theorem not_isSuccPrelimit [NoMinOrder α] : ¬ IsSuccPrelimit a := by simp
 
 end IsSuccArchimedean
 
 end PartialOrder
 
+section LinearOrder
+
+variable [LinearOrder α]
+
+theorem IsSuccPrelimit.le_iff_forall_le (h : IsSuccPrelimit a) : a ≤ b ↔ ∀ c < a, c ≤ b := by
+  use fun ha c hc ↦ hc.le.trans ha
+  intro H
+  by_contra! ha
+  exact h b ⟨ha, fun c hb hc ↦ (H c hc).not_lt hb⟩
+
+theorem IsSuccPrelimit.lt_iff_exists_lt (h : IsSuccPrelimit b) : a < b ↔ ∃ c < b, a < c := by
+  rw [← not_iff_not]
+  simp [h.le_iff_forall_le]
+
+end LinearOrder
+
 /-! ### Predecessor limits -/
 
 
 section LT
 
-variable [LT α] {a : α}
+variable [LT α]
+
+/-- A predecessor pre-limit is a value that isn't covered by any other.
+
+It's so named because in a predecessor order, a predecessor pre-limit can't be the predecessor of
+anything smaller.
+
+Use `IsPredLimit` to exclude the case of a maximal element. -/
+def IsPredPrelimit (a : α) : Prop :=
+  ∀ b, ¬ a ⋖ b
 
-/-- A predecessor limit is a value that isn't covered by any other.
+theorem not_isPredPrelimit_iff_exists_covBy (a : α) : ¬IsPredPrelimit a ↔ ∃ b, a ⋖ b := by
+  simp [IsPredPrelimit]
+
+@[deprecated not_isPredPrelimit_iff_exists_covBy (since := "2024-09-05")]
+alias not_isPredLimit_iff_exists_covBy := not_isPredPrelimit_iff_exists_covBy
+
+theorem isPredPrelimit_of_dense [DenselyOrdered α] (a : α) : IsPredPrelimit a := fun _ => not_covBy
+
+@[deprecated isPredPrelimit_of_dense (since := "2024-09-05")]
+alias isPredLimit_of_dense := isPredPrelimit_of_dense
+
+@[simp]
+theorem isSuccPrelimit_toDual_iff : IsSuccPrelimit (toDual a) ↔ IsPredPrelimit a := by
+  simp [IsSuccPrelimit, IsPredPrelimit]
+
+@[simp]
+theorem isPredPrelimit_toDual_iff : IsPredPrelimit (toDual a) ↔ IsSuccPrelimit a := by
+  simp [IsSuccPrelimit, IsPredPrelimit]
+
+alias ⟨_, IsPredPrelimit.dual⟩ := isSuccPrelimit_toDual_iff
+alias ⟨_, IsSuccPrelimit.dual⟩ := isPredPrelimit_toDual_iff
+@[deprecated IsPredPrelimit.dual (since := "2024-09-05")]
+alias isPredLimit.dual := IsPredPrelimit.dual
+@[deprecated IsSuccPrelimit.dual (since := "2024-09-05")]
+alias isSuccLimit.dual := IsSuccPrelimit.dual
+
+end LT
+
+section Preorder
+
+variable [Preorder α]
+
+/-- A predecessor limit is a value that isn't maximal and doesn't cover any other.
 
 It's so named because in a predecessor order, a predecessor limit can't be the predecessor of
-anything greater. -/
-def IsPredLimit (a : α) : Prop :=
-  ∀ b, ¬a ⋖ b
+anything larger.
 
-theorem not_isPredLimit_iff_exists_covBy (a : α) : ¬IsPredLimit a ↔ ∃ b, a ⋖ b := by
-  simp [IsPredLimit]
+This previously allowed the element to be maximal. This usage is now covered by `IsPredPreLimit`. -/
+def IsPredLimit (a : α) : Prop :=
+  ¬ IsMax a ∧ IsPredPrelimit a
 
-theorem isPredLimit_of_dense [DenselyOrdered α] (a : α) : IsPredLimit a := fun _ => not_covBy
+protected theorem IsPredLimit.not_isMax (h : IsPredLimit a) : ¬ IsMax a := h.1
+protected theorem IsPredLimit.isPredPrelimit (h : IsPredLimit a) : IsPredPrelimit a := h.2
 
 @[simp]
 theorem isSuccLimit_toDual_iff : IsSuccLimit (toDual a) ↔ IsPredLimit a := by
@@ -239,57 +373,104 @@ theorem isSuccLimit_toDual_iff : IsSuccLimit (toDual a) ↔ IsPredLimit a := by
 theorem isPredLimit_toDual_iff : IsPredLimit (toDual a) ↔ IsSuccLimit a := by
   simp [IsSuccLimit, IsPredLimit]
 
-alias ⟨_, isPredLimit.dual⟩ := isSuccLimit_toDual_iff
+alias ⟨_, IsPredLimit.dual⟩ := isSuccLimit_toDual_iff
+alias ⟨_, IsSuccLimit.dual⟩ := isPredLimit_toDual_iff
 
-alias ⟨_, isSuccLimit.dual⟩ := isPredLimit_toDual_iff
+theorem IsPredPrelimit.isPredLimit_of_not_isMax (h : IsPredPrelimit a) (ha : ¬ IsMax a) :
+    IsPredLimit a :=
+  ⟨ha, h⟩
 
-end LT
+theorem IsPredPrelimit.isPredLimit [NoMaxOrder α] (h : IsPredPrelimit a) : IsPredLimit a :=
+  h.isPredLimit_of_not_isMax (not_isMax a)
 
-section Preorder
+theorem isPredPrelimit_iff_isPredLimit_of_not_isMax (h : ¬ IsMax a) :
+    IsPredPrelimit a ↔ IsPredLimit a :=
+  ⟨fun ha ↦ ha.isPredLimit_of_not_isMax h, IsPredLimit.isPredPrelimit⟩
 
-variable [Preorder α] {a : α}
+theorem isPredPrelimit_iff_isPredLimit [NoMaxOrder α] : IsPredPrelimit a ↔ IsPredLimit a :=
+  isPredPrelimit_iff_isPredLimit_of_not_isMax (not_isMax a)
 
-protected theorem _root_.IsMax.isPredLimit : IsMax a → IsPredLimit a := fun h _ hab =>
+protected theorem _root_.IsMax.not_isPredLimit (h : IsMax a) : ¬ IsPredLimit a :=
+  fun ha ↦ ha.not_isMax h
+
+protected theorem _root_.IsMax.isPredPrelimit : IsMax a → IsPredPrelimit a := fun h _ hab =>
   not_isMax_of_lt hab.lt h
 
-theorem isPredLimit_top [OrderTop α] : IsPredLimit (⊤ : α) :=
-   IsMax.isPredLimit isMax_top
+@[deprecated _root_.IsMax.isPredPrelimit (since := "2024-09-05")]
+alias _root_.IsMax.isPredLimit := _root_.IsMax.isPredPrelimit
+
+theorem isPredPrelimit_top [OrderTop α] : IsPredPrelimit (⊤ : α) :=
+  isMax_top.isPredPrelimit
+
+@[deprecated isPredPrelimit_top (since := "2024-09-05")]
+alias isPredLimit_top := isPredPrelimit_top
+
+theorem not_isPredLimit_top [OrderTop α] : ¬ IsPredLimit (⊤ : α) :=
+  isMax_top.not_isPredLimit
+
+theorem IsPredLimit.ne_top [OrderTop α] (h : IsPredLimit a) : a ≠ ⊤ :=
+  h.dual.ne_bot
+
+theorem not_isPredLimit_iff : ¬ IsPredLimit a ↔ IsMax a ∨ ¬ IsPredPrelimit a := by
+  rw [IsPredLimit, not_and_or, not_not]
+
+theorem not_isPredLimit_of_not_isPredPrelimit (h : ¬ IsPredPrelimit a) : ¬ IsPredLimit a :=
+  not_isPredLimit_iff.2 (Or.inr h)
 
 variable [PredOrder α]
 
-protected theorem IsPredLimit.isMin (h : IsPredLimit (pred a)) : IsMin a := by
-  by_contra H
-  exact h a (pred_covBy_of_not_isMin H)
+protected theorem IsPredPrelimit.isMin (h : IsPredPrelimit (pred a)) : IsMin a :=
+  h.dual.isMax
+
+protected theorem IsPredLimit.isMin (h : IsPredLimit (pred a)) : IsMin a :=
+  h.dual.isMax
 
-theorem not_isPredLimit_pred_of_not_isMin (ha : ¬IsMin a) : ¬IsPredLimit (pred a) := by
-  contrapose! ha
-  exact ha.isMin
+theorem not_isPredPrelimit_pred_of_not_isMin (ha : ¬ IsMin a) : ¬ IsPredPrelimit (pred a) :=
+  mt IsPredPrelimit.isMin ha
+
+theorem not_isPredLimit_pred_of_not_isMin (ha : ¬ IsMin a) : ¬ IsPredLimit (pred a) :=
+  mt IsPredLimit.isMin ha
 
 section NoMinOrder
 
 variable [NoMinOrder α]
 
-theorem IsPredLimit.pred_ne (h : IsPredLimit a) (b : α) : pred b ≠ a := by
-  rintro rfl
-  exact not_isMin _ h.isMin
+theorem IsPredPrelimit.pred_ne (h : IsPredPrelimit a) (b : α) : pred b ≠ a :=
+  h.dual.succ_ne b
+
+theorem IsPredLimit.pred_ne (h : IsPredLimit a) (b : α) : pred b ≠ a :=
+  h.isPredPrelimit.pred_ne b
+
+@[simp]
+theorem not_isPredPrelimit_pred (a : α) : ¬ IsPredPrelimit (pred a) := fun h => h.pred_ne _ rfl
 
 @[simp]
-theorem not_isPredLimit_pred (a : α) : ¬IsPredLimit (pred a) := fun h => h.pred_ne _ rfl
+theorem not_isPredLimit_pred (a : α) : ¬ IsPredLimit (pred a) := fun h => h.pred_ne _ rfl
 
 end NoMinOrder
 
 section IsPredArchimedean
 
-variable [IsPredArchimedean α]
+variable [IsPredArchimedean α] [NoMinOrder α]
+
+theorem IsPredPrelimit.isMax_of_noMin (h : IsPredPrelimit a) : IsMax a :=
+  h.dual.isMin_of_noMax
 
-protected theorem IsPredLimit.isMax_of_noMin [NoMinOrder α] (h : IsPredLimit a) : IsMax a :=
-  (isPredLimit.dual h).isMin_of_noMax
+@[deprecated IsPredPrelimit.isMax_of_noMin (since := "2024-09-05")]
+alias IsPredLimit.isMax_of_noMin := IsPredPrelimit.isMax_of_noMin
 
 @[simp]
-theorem isPredLimit_iff_of_noMin [NoMinOrder α] : IsPredLimit a ↔ IsMax a :=
-  isSuccLimit_toDual_iff.symm.trans isSuccLimit_iff_of_noMax
+theorem isPredPrelimit_iff_of_noMin : IsPredPrelimit a ↔ IsMax a :=
+  ⟨IsPredPrelimit.isMax_of_noMin, IsMax.isPredPrelimit⟩
 
-theorem not_isPredLimit_of_noMin [NoMinOrder α] [NoMaxOrder α] : ¬IsPredLimit a := by simp
+@[deprecated isPredPrelimit_iff_of_noMin (since := "2024-09-05")]
+alias isPredLimit_iff_of_noMin := isPredPrelimit_iff_of_noMin
+
+theorem not_isPredPrelimit_of_noMin [NoMaxOrder α] : ¬ IsPredPrelimit a := by simp
+
+@[simp]
+theorem not_isPredLimit_of_noMin : ¬ IsPredLimit a :=
+  fun h ↦ h.not_isMax h.isPredPrelimit.isMax_of_noMin
 
 end IsPredArchimedean
 
@@ -297,88 +478,94 @@ end Preorder
 
 section PartialOrder
 
-variable [PartialOrder α] [PredOrder α] {a b : α} {C : α → Sort*}
+variable [PartialOrder α]
+
+theorem isPredLimit_iff [OrderTop α] : IsPredLimit a ↔ a ≠ ⊤ ∧ IsPredPrelimit a := by
+  rw [IsPredLimit, isMax_iff_eq_top]
+
+variable [PredOrder α]
 
-theorem isPredLimit_of_pred_ne (h : ∀ b, pred b ≠ a) : IsPredLimit a := fun b hba =>
+theorem isPredPrelimit_of_pred_ne (h : ∀ b, pred b ≠ a) : IsPredPrelimit a := fun b hba =>
   h b (CovBy.pred_eq hba)
 
-theorem not_isPredLimit_iff : ¬IsPredLimit a ↔ ∃ b, ¬IsMin b ∧ pred b = a := by
-  rw [← isSuccLimit_toDual_iff]
-  exact not_isSuccLimit_iff
+@[deprecated isPredPrelimit_of_pred_ne (since := "2024-09-05")]
+alias isPredLimit_of_pred_ne := isPredPrelimit_of_pred_ne
 
-/-- See `not_isPredLimit_iff` for a version that states that `a` is a successor of a value other
+theorem not_isPredPrelimit_iff : ¬ IsPredPrelimit a ↔ ∃ b, ¬ IsMin b ∧ pred b = a := by
+  rw [← isSuccPrelimit_toDual_iff]
+  exact not_isSuccPrelimit_iff
+
+/-- See `not_isPredPrelimit_iff` for a version that states that `a` is a successor of a value other
 than itself. -/
-theorem mem_range_pred_of_not_isPredLimit (h : ¬IsPredLimit a) : a ∈ range (@pred α _ _) := by
-  cases' not_isPredLimit_iff.1 h with b hb
+theorem mem_range_pred_of_not_isPredPrelimit (h : ¬ IsPredPrelimit a) :
+    a ∈ range (pred : α → α) := by
+  obtain ⟨b, hb⟩ := not_isPredPrelimit_iff.1 h
   exact ⟨b, hb.2⟩
 
-theorem mem_range_pred_or_isPredLimit (a) : a ∈ range (@pred α _ _) ∨ IsPredLimit a :=
-  or_iff_not_imp_right.2 <| mem_range_pred_of_not_isPredLimit
+@[deprecated mem_range_pred_of_not_isPredPrelimit (since := "2024-09-05")]
+alias mem_range_pred_of_not_isPredLimit := mem_range_pred_of_not_isPredPrelimit
 
-theorem isPredLimit_of_pred_lt (H : ∀ a > b, pred a < b) : IsPredLimit b := fun a hab =>
-  (H a hab.lt).ne (CovBy.pred_eq hab)
+theorem mem_range_pred_or_isPredPrelimit (a) : a ∈ range (pred : α → α) ∨ IsPredPrelimit a :=
+  or_iff_not_imp_right.2 <| mem_range_pred_of_not_isPredPrelimit
 
-theorem IsPredLimit.lt_pred (h : IsPredLimit a) : a < b → a < pred b :=
-  (isPredLimit.dual h).succ_lt
+@[deprecated mem_range_pred_or_isPredPrelimit (since := "2024-09-05")]
+alias mem_range_pred_or_isPredLimit := mem_range_pred_or_isPredPrelimit
 
-theorem IsPredLimit.lt_pred_iff (h : IsPredLimit a) : a < pred b ↔ a < b :=
-  (isPredLimit.dual h).succ_lt_iff
+theorem isPredPrelimit_of_pred_lt (H : ∀ b > a, a < pred b) : IsPredPrelimit a := fun a hab =>
+  (H a hab.lt).ne (CovBy.pred_eq hab).symm
 
-theorem isPredLimit_iff_lt_pred : IsPredLimit a ↔ ∀ ⦃b⦄, a < b → a < pred b :=
-  isSuccLimit_toDual_iff.symm.trans isSuccLimit_iff_succ_lt
+@[deprecated isPredPrelimit_of_pred_lt (since := "2024-09-05")]
+alias isPredLimit_of_pred_lt := isPredPrelimit_of_pred_lt
 
-/-- A value can be built by building it on predecessors and predecessor limits. -/
-@[elab_as_elim]
-noncomputable def isPredLimitRecOn (b : α) (hs : ∀ a, ¬IsMin a → C (pred a))
-    (hl : ∀ a, IsPredLimit a → C a) : C b :=
-  @isSuccLimitRecOn αᵒᵈ _ _ _ _ hs fun _ ha => hl _ (isSuccLimit.dual ha)
+theorem IsPredPrelimit.lt_pred (ha : IsPredPrelimit a) (hb : a < b) : a < pred b :=
+  ha.dual.succ_lt hb
 
-theorem isPredLimitRecOn_limit (hs : ∀ a, ¬IsMin a → C (pred a)) (hl : ∀ a, IsPredLimit a → C a)
-    (hb : IsPredLimit b) : @isPredLimitRecOn α _ _ C b hs hl = hl b hb :=
-  isSuccLimitRecOn_limit _ _ (isPredLimit.dual hb)
+theorem IsPredLimit.lt_pred (ha : IsPredLimit a) (hb : a < b) : a < pred b :=
+  ha.isPredPrelimit.lt_pred hb
 
-section limitRecOn
+theorem IsPredPrelimit.lt_pred_iff (ha : IsPredPrelimit a) : a < pred b ↔ a < b :=
+  ha.dual.succ_lt_iff
 
-variable [WellFoundedGT α]
-  (H_pred : ∀ a, ¬IsMin a → C a → C (pred a))
-  (H_lim : ∀ a, IsPredLimit a → (∀ b > a, C b) → C a)
+theorem IsPredLimit.lt_pred_iff (ha : IsPredLimit a) : a < pred b ↔ a < b :=
+  ha.dual.succ_lt_iff
 
-open scoped Classical in
-variable (a) in
-/-- Recursion principle on a well-founded partial `PredOrder`. -/
-@[elab_as_elim] noncomputable def _root_.PredOrder.limitRecOn : C a :=
-  wellFounded_gt.fix
-    (fun a IH ↦ if h : IsPredLimit a then H_lim a h IH else
-      let x := Classical.indefiniteDescription _ (not_isPredLimit_iff.mp h)
-      x.2.2 ▸ H_pred x x.2.1 (IH x <| x.2.2.subst <| pred_lt_of_not_isMin x.2.1))
-    a
+theorem isPredPrelimit_iff_lt_pred : IsPredPrelimit a ↔ ∀ b > a, a < pred b :=
+  ⟨fun hb _ => hb.lt_pred, isPredPrelimit_of_pred_lt⟩
 
-@[simp]
-theorem _root_.PredOrder.limitRecOn_limit (ha : IsPredLimit a) :
-    PredOrder.limitRecOn a H_pred H_lim
-      = H_lim a ha fun x _ ↦ PredOrder.limitRecOn x H_pred H_lim := by
-  rw [PredOrder.limitRecOn, WellFounded.fix_eq, dif_pos ha]; rfl
-
-end limitRecOn
+@[deprecated isPredPrelimit_iff_lt_pred (since := "2024-09-05")]
+alias isPredLimit_iff_lt_pred := isPredPrelimit_iff_lt_pred
 
 section NoMinOrder
 
 variable [NoMinOrder α]
 
+theorem isPredPrelimit_iff_pred_ne : IsPredPrelimit a ↔ ∀ b, pred b ≠ a :=
+  ⟨IsPredPrelimit.pred_ne, isPredPrelimit_of_pred_ne⟩
+
+theorem not_isPredPrelimit_iff' : ¬ IsPredPrelimit a ↔ a ∈ range (pred : α → α) := by
+  simp_rw [isPredPrelimit_iff_pred_ne, not_forall, not_ne_iff, mem_range]
+
 end NoMinOrder
 
 section IsPredArchimedean
 
 variable [IsPredArchimedean α]
 
-protected theorem IsPredLimit.isMax (h : IsPredLimit a) : IsMax a :=
-  (isPredLimit.dual h).isMin
+protected theorem IsPredPrelimit.isMax (h : IsPredPrelimit a) : IsMax a :=
+  h.dual.isMin
+
+@[deprecated IsPredPrelimit.isMax (since := "2024-09-05")]
+alias IsPredLimit.isMax := IsPredPrelimit.isMax
+
+@[simp]
+theorem isPredPrelimit_iff : IsPredPrelimit a ↔ IsMax a :=
+  ⟨IsPredPrelimit.isMax, IsMax.isPredPrelimit⟩
 
 @[simp]
-theorem isPredLimit_iff : IsPredLimit a ↔ IsMax a :=
-  isSuccLimit_toDual_iff.symm.trans isSuccLimit_iff
+theorem not_isPredLimit : ¬ IsPredLimit a :=
+  fun h ↦ h.not_isMax <| h.isPredPrelimit.isMax
 
-theorem not_isPredLimit [NoMaxOrder α] : ¬IsPredLimit a := by simp
+theorem not_isPredPrelimit [NoMaxOrder α] : ¬ IsPredPrelimit a := by simp
 
 end IsPredArchimedean
 
@@ -386,70 +573,411 @@ end PartialOrder
 
 section LinearOrder
 
-section Succ
+variable [LinearOrder α]
+
+theorem IsPredPrelimit.le_iff_forall_le (h : IsPredPrelimit a) : b ≤ a ↔ ∀ ⦃c⦄, a < c → b ≤ c :=
+  h.dual.le_iff_forall_le
+
+theorem IsPredPrelimit.lt_iff_exists_lt (h : IsPredPrelimit b) : b < a ↔ ∃ c, b < c ∧ c < a :=
+  h.dual.lt_iff_exists_lt
+
+end LinearOrder
+
+end Order
+
+/-! ### Induction principles -/
+
+
+variable {C : α → Sort*}
+
+namespace Order
+
+section isSuccPrelimitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [SuccOrder α]
+  (hs : ∀ a, ¬ IsMax a → C (succ a)) (hl : ∀ a, IsSuccPrelimit a → C a)
+
+variable (b) in
+open Classical in
+/-- A value can be built by building it on successors and successor pre-limits. -/
+@[elab_as_elim]
+noncomputable def isSuccPrelimitRecOn : C b :=
+  if hb : IsSuccPrelimit b then hl b hb else
+    haveI H := Classical.choose_spec (not_isSuccPrelimit_iff.1 hb)
+    cast (congr_arg C H.2) (hs _ H.1)
+
+theorem isSuccPrelimitRecOn_of_isSuccPrelimit (hb : IsSuccPrelimit b) :
+    isSuccPrelimitRecOn b hs hl = hl b hb :=
+  dif_pos hb
+
+@[deprecated isSuccPrelimitRecOn_of_isSuccPrelimit (since := "2024-09-05")]
+alias isSuccLimitRecOn_limit := isSuccPrelimitRecOn_of_isSuccPrelimit
+@[deprecated isSuccPrelimitRecOn_of_isSuccPrelimit (since := "2024-09-14")]
+alias isSuccPrelimitRecOn_limit := isSuccPrelimitRecOn_of_isSuccPrelimit
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [SuccOrder α]
+  (hs : ∀ a, ¬ IsMax a → C (succ a)) (hl : ∀ a, IsSuccPrelimit a → C a)
+
+theorem isSuccPrelimitRecOn_succ_of_not_isMax (hb : ¬ IsMax b) :
+    isSuccPrelimitRecOn (succ b) hs hl = hs b hb := by
+  have hb' := not_isSuccPrelimit_succ_of_not_isMax hb
+  have H := Classical.choose_spec (not_isSuccPrelimit_iff.1 hb')
+  rw [isSuccPrelimitRecOn, dif_neg hb', cast_eq_iff_heq]
+  congr
+  exacts [(succ_eq_succ_iff_of_not_isMax H.1 hb).1 H.2, proof_irrel_heq _ _]
+
+@[deprecated isSuccPrelimitRecOn_succ_of_not_isMax (since := "2024-09-05")]
+alias isSuccLimitRecOn_succ' := isSuccPrelimitRecOn_succ_of_not_isMax
+@[deprecated isSuccPrelimitRecOn_succ_of_not_isMax (since := "2024-09-14")]
+alias isSuccPrelimitRecOn_succ' := isSuccPrelimitRecOn_succ_of_not_isMax
+
+@[simp]
+theorem isSuccPrelimitRecOn_succ [NoMaxOrder α] (b : α) :
+    isSuccPrelimitRecOn (succ b) hs hl = hs b (not_isMax b) :=
+  isSuccPrelimitRecOn_succ_of_not_isMax _ _ _
+
+end LinearOrder
+
+end isSuccPrelimitRecOn
+
+section isPredPrelimitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [PredOrder α]
+  (hs : ∀ a, ¬ IsMin a → C (pred a)) (hl : ∀ a, IsPredPrelimit a → C a)
+
+variable (b) in
+/-- A value can be built by building it on predecessors and predecessor pre-limits. -/
+@[elab_as_elim]
+noncomputable def isPredPrelimitRecOn : C b :=
+  isSuccPrelimitRecOn (α := αᵒᵈ) b hs (fun a ha ↦ hl a ha.dual)
+
+theorem isPredPrelimitRecOn_of_isPredPrelimit (hb : IsPredPrelimit b) :
+    isPredPrelimitRecOn b hs hl = hl b hb :=
+  isSuccPrelimitRecOn_of_isSuccPrelimit _ _ hb.dual
+
+@[deprecated isPredPrelimitRecOn_of_isPredPrelimit (since := "2024-09-05")]
+alias isPredLimitRecOn_limit := isPredPrelimitRecOn_of_isPredPrelimit
+@[deprecated isPredPrelimitRecOn_of_isPredPrelimit (since := "2024-09-14")]
+alias isPredPrelimitRecOn_limit := isPredPrelimitRecOn_of_isPredPrelimit
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [PredOrder α]
+  (hs : ∀ a, ¬ IsMin a → C (pred a)) (hl : ∀ a, IsPredPrelimit a → C a)
+
+theorem isPredPrelimitRecOn_pred_of_not_isMin (hb : ¬ IsMin b) :
+    isPredPrelimitRecOn (pred b) hs hl = hs b hb :=
+  isSuccPrelimitRecOn_succ_of_not_isMax (α := αᵒᵈ) _ _ _
+
+@[deprecated isPredPrelimitRecOn_pred_of_not_isMin (since := "2024-09-05")]
+alias isPredLimitRecOn_pred' := isPredPrelimitRecOn_pred_of_not_isMin
+@[deprecated isPredPrelimitRecOn_pred_of_not_isMin (since := "2024-09-14")]
+alias isPredPrelimitRecOn_pred' := isPredPrelimitRecOn_pred_of_not_isMin
+
+@[simp]
+theorem isPredPrelimitRecOn_pred [NoMinOrder α] (b : α) :
+    isPredPrelimitRecOn (pred b) hs hl = hs b (not_isMin b) :=
+  isPredPrelimitRecOn_pred_of_not_isMin _ _ _
+
+end LinearOrder
+
+end isPredPrelimitRecOn
+
+section isSuccLimitRecOn
 
-variable [LinearOrder α] [SuccOrder α] {a b : α} {C : α → Sort*}
+section PartialOrder
 
-theorem isSuccLimitRecOn_succ' (hs : ∀ a, ¬IsMax a → C (succ a)) (hl : ∀ a, IsSuccLimit a → C a)
-    {b : α} (hb : ¬IsMax b) : @isSuccLimitRecOn α _ _ C (succ b) hs hl = hs b hb := by
-  have hb' := not_isSuccLimit_succ_of_not_isMax hb
-  have H := Classical.choose_spec (not_isSuccLimit_iff.1 hb')
-  rw [isSuccLimitRecOn]
-  simp only [cast_eq_iff_heq, hb', not_false_iff, eq_mpr_eq_cast, dif_neg]
-  congr 1 <;> first |
-    exact (succ_eq_succ_iff_of_not_isMax H.left hb).mp H.right |
-    exact proof_irrel_heq H.left hb
+variable [PartialOrder α] [SuccOrder α]
+  (hm : ∀ a, IsMin a → C a) (hs : ∀ a, ¬ IsMax a → C (succ a)) (hl : ∀ a, IsSuccLimit a → C a)
+
+variable (b) in
+open Classical in
+/-- A value can be built by building it on minimal elements, successors, and successor limits. -/
+@[elab_as_elim]
+noncomputable def isSuccLimitRecOn : C b :=
+  isSuccPrelimitRecOn b hs fun a ha ↦
+    if h : IsMin a then hm a h else hl a (ha.isSuccLimit_of_not_isMin h)
 
 @[simp]
-theorem _root_.SuccOrder.limitRecOn_succ [WellFoundedLT α]
-    (H_succ : ∀ a, ¬IsMax a → C a → C (succ a)) (H_lim : ∀ a, IsSuccLimit a → (∀ b < a, C b) → C a)
-    (ha : ¬ IsMax a) :
-    SuccOrder.limitRecOn (succ a) H_succ H_lim
-      = H_succ a ha (SuccOrder.limitRecOn a H_succ H_lim) := by
-  have h := not_isSuccLimit_succ_of_not_isMax ha
-  rw [SuccOrder.limitRecOn, WellFounded.fix_eq, dif_neg h]
-  have {b c hb hc} {x : ∀ a, C a} (h : b = c) :
-    congr_arg succ h ▸ H_succ b hb (x b) = H_succ c hc (x c) := by subst h; rfl
-  let x := Classical.indefiniteDescription _ (not_isSuccLimit_iff.mp h)
-  exact this ((succ_eq_succ_iff_of_not_isMax x.2.1 ha).mp x.2.2)
+theorem isSuccLimitRecOn_of_isSuccLimit (hb : IsSuccLimit b) :
+    isSuccLimitRecOn b hm hs hl = hl b hb := by
+  rw [isSuccLimitRecOn, isSuccPrelimitRecOn_of_isSuccPrelimit _ _ hb.isSuccPrelimit,
+    dif_neg hb.not_isMin]
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [SuccOrder α]
+  (hm : ∀ a, IsMin a → C a) (hs : ∀ a, ¬ IsMax a → C (succ a)) (hl : ∀ a, IsSuccLimit a → C a)
+
+theorem isSuccLimitRecOn_succ_of_not_isMax (hb : ¬ IsMax b) :
+    isSuccLimitRecOn (succ b) hm hs hl = hs b hb := by
+  rw [isSuccLimitRecOn, isSuccPrelimitRecOn_succ_of_not_isMax]
 
 @[simp]
-theorem isSuccLimitRecOn_succ [NoMaxOrder α] (hs : ∀ a, ¬IsMax a → C (succ a))
-    (hl : ∀ a, IsSuccLimit a → C a) (b : α) :
-    @isSuccLimitRecOn α _ _ C (succ b) hs hl = hs b (not_isMax b) :=
-  isSuccLimitRecOn_succ' _ _ _
+theorem isSuccLimitRecOn_succ [NoMaxOrder α] (b : α) :
+    isSuccLimitRecOn (succ b) hm hs hl = hs b (not_isMax b) :=
+  isSuccLimitRecOn_succ_of_not_isMax hm hs hl _
 
-end Succ
+theorem isSuccLimitRecOn_of_isMin (hb : IsMin b) : isSuccLimitRecOn b hm hs hl = hm b hb := by
+  rw [isSuccLimitRecOn, isSuccPrelimitRecOn_of_isSuccPrelimit _ _ hb.isSuccPrelimit, dif_pos hb]
 
-section Pred
+end LinearOrder
 
-variable [LinearOrder α] [PredOrder α] {a b : α} {C : α → Sort*}
+end isSuccLimitRecOn
 
-theorem isPredLimitRecOn_pred' (hs : ∀ a, ¬IsMin a → C (pred a)) (hl : ∀ a, IsPredLimit a → C a)
-    {b : α} (hb : ¬IsMin b) : @isPredLimitRecOn α _ _ C (pred b) hs hl = hs b hb :=
-  isSuccLimitRecOn_succ' _ _ _
+section isPredLimitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [PredOrder α]
+  (hm : ∀ a, IsMax a → C a) (hs : ∀ a, ¬ IsMin a → C (pred a)) (hl : ∀ a, IsPredLimit a → C a)
+
+variable (b) in
+/-- A value can be built by building it on maximal elements, predecessors,
+and predecessor limits. -/
+@[elab_as_elim]
+noncomputable def isPredLimitRecOn : C b :=
+  isSuccLimitRecOn (α := αᵒᵈ) b hm hs (fun a ha => hl a ha.dual)
 
 @[simp]
-theorem _root_.PredOrder.limitRecOn_pred [WellFoundedGT α]
-    (H_pred : ∀ a, ¬IsMin a → C a → C (pred a)) (H_lim : ∀ a, IsPredLimit a → (∀ b > a, C b) → C a)
-    (ha : ¬ IsMin a) :
-    PredOrder.limitRecOn (pred a) H_pred H_lim
-      = H_pred a ha (PredOrder.limitRecOn a H_pred H_lim) := by
-  have h := not_isPredLimit_pred_of_not_isMin ha
-  rw [PredOrder.limitRecOn, WellFounded.fix_eq, dif_neg h]
-  have {b c hb hc} {x : ∀ a, C a} (h : b = c) :
-    congr_arg pred h ▸ H_pred b hb (x b) = H_pred c hc (x c) := by subst h; rfl
-  let x := Classical.indefiniteDescription _ (not_isPredLimit_iff.mp h)
-  exact this ((pred_eq_pred_iff_of_not_isMin x.2.1 ha).mp x.2.2)
+theorem isPredLimitRecOn_of_isPredLimit (hb : IsPredLimit b) :
+    isPredLimitRecOn b hm hs hl = hl b hb :=
+  isSuccLimitRecOn_of_isSuccLimit (α := αᵒᵈ) hm hs _ hb.dual
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [PredOrder α]
+  (hm : ∀ a, IsMax a → C a) (hs : ∀ a, ¬ IsMin a → C (pred a)) (hl : ∀ a, IsPredLimit a → C a)
+
+theorem isPredLimitRecOn_pred_of_not_isMin (hb : ¬ IsMin b) :
+    isPredLimitRecOn (pred b) hm hs hl = hs b hb :=
+  isSuccLimitRecOn_succ_of_not_isMax (α := αᵒᵈ) hm hs _ hb
 
 @[simp]
-theorem isPredLimitRecOn_pred [NoMinOrder α] (hs : ∀ a, ¬IsMin a → C (pred a))
-    (hl : ∀ a, IsPredLimit a → C a) (b : α) :
-    @isPredLimitRecOn α _ _ C (pred b) hs hl = hs b (not_isMin b) :=
-  isSuccLimitRecOn_succ _ _ _
+theorem isPredLimitRecOn_pred [NoMinOrder α] :
+    isPredLimitRecOn (pred b) hm hs hl = hs b (not_isMin b) :=
+  isSuccLimitRecOn_succ (α := αᵒᵈ) hm hs _ b
 
-end Pred
+theorem isPredLimitRecOn_of_isMax (hb : IsMax b) : isPredLimitRecOn b hm hs hl = hm b hb :=
+  isSuccLimitRecOn_of_isMin (α := αᵒᵈ) hm hs _ hb
 
 end LinearOrder
 
+end isPredLimitRecOn
+
 end Order
+
+open Order
+
+namespace SuccOrder
+
+section prelimitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [SuccOrder α] [WellFoundedLT α]
+  (hs : ∀ a, ¬ IsMax a → C a → C (Order.succ a)) (hl : ∀ a, IsSuccPrelimit a → (∀ b < a, C b) → C a)
+
+variable (b) in
+open Classical in
+/-- Recursion principle on a well-founded partial `SuccOrder`. -/
+@[elab_as_elim] noncomputable def prelimitRecOn : C b :=
+  wellFounded_lt.fix
+    (fun a IH ↦ if h : IsSuccPrelimit a then hl a h IH else
+      haveI H := Classical.choose_spec (not_isSuccPrelimit_iff.1 h)
+      cast (congr_arg C H.2) (hs _ H.1 <| IH _ <| H.2.subst <| lt_succ_of_not_isMax H.1))
+    b
+
+@[simp]
+theorem prelimitRecOn_of_isSuccPrelimit (hb : IsSuccPrelimit b) :
+    prelimitRecOn b hs hl = hl b hb fun x _ ↦ SuccOrder.prelimitRecOn x hs hl := by
+  rw [prelimitRecOn, WellFounded.fix_eq, dif_pos hb]; rfl
+
+@[deprecated prelimitRecOn_of_isSuccPrelimit (since := "2024-09-05")]
+alias limitRecOn_limit := prelimitRecOn_of_isSuccPrelimit
+@[deprecated prelimitRecOn_of_isSuccPrelimit (since := "2024-09-14")]
+alias prelimitRecOn_limit := prelimitRecOn_of_isSuccPrelimit
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [SuccOrder α] [WellFoundedLT α]
+  (hs : ∀ a, ¬ IsMax a → C a → C (Order.succ a)) (hl : ∀ a, IsSuccPrelimit a → (∀ b < a, C b) → C a)
+
+theorem prelimitRecOn_succ_of_not_isMax (hb : ¬ IsMax b) :
+    prelimitRecOn (Order.succ b) hs hl = hs b hb (prelimitRecOn b hs hl) := by
+  have h := not_isSuccPrelimit_succ_of_not_isMax hb
+  have H := Classical.choose_spec (not_isSuccPrelimit_iff.1 h)
+  rw [prelimitRecOn, WellFounded.fix_eq, dif_neg h]
+  have {a c : α} {ha hc} {x : ∀ a, C a} (h : a = c) :
+    cast (congr_arg (C ∘ succ) h) (hs a ha (x a)) = hs c hc (x c) := by subst h; rfl
+  exact this <| (succ_eq_succ_iff_of_not_isMax H.1 hb).1 H.2
+
+@[deprecated prelimitRecOn_succ_of_not_isMax (since := "2024-09-05")]
+alias limitRecOn_succ' := prelimitRecOn_succ_of_not_isMax
+@[deprecated prelimitRecOn_succ_of_not_isMax (since := "2024-09-14")]
+alias prelimitRecOn_succ' := prelimitRecOn_succ_of_not_isMax
+
+@[simp]
+theorem prelimitRecOn_succ [NoMaxOrder α] (b : α) :
+    prelimitRecOn (Order.succ b) hs hl = hs b (not_isMax b) (prelimitRecOn b hs hl) :=
+  prelimitRecOn_succ_of_not_isMax _ _ _
+
+end LinearOrder
+
+end prelimitRecOn
+
+section limitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [SuccOrder α] [WellFoundedLT α] (hm : ∀ a, IsMin a → C a)
+  (hs : ∀ a, ¬ IsMax a → C a → C (Order.succ a)) (hl : ∀ a, IsSuccLimit a → (∀ b < a, C b) → C a)
+
+variable (b) in
+open Classical in
+/-- Recursion principle on a well-founded partial `SuccOrder`, separating out the case of a
+minimal element. -/
+@[elab_as_elim] noncomputable def limitRecOn : C b :=
+  prelimitRecOn b hs fun a ha IH ↦
+    if h : IsMin a then hm a h else hl a (ha.isSuccLimit_of_not_isMin h) IH
+
+@[simp]
+theorem limitRecOn_isMin (hb : IsMin b) : limitRecOn b hm hs hl = hm b hb := by
+  rw [limitRecOn, prelimitRecOn_of_isSuccPrelimit _ _ hb.isSuccPrelimit, dif_pos hb]
+
+@[simp]
+theorem limitRecOn_of_isSuccLimit (hb : IsSuccLimit b) :
+    limitRecOn b hm hs hl = hl b hb fun x _ ↦ limitRecOn x hm hs hl := by
+  rw [limitRecOn, prelimitRecOn_of_isSuccPrelimit _ _ hb.isSuccPrelimit, dif_neg hb.not_isMin]; rfl
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [SuccOrder α] [WellFoundedLT α] (hm : ∀ a, IsMin a → C a)
+  (hs : ∀ a, ¬ IsMax a → C a → C (Order.succ a)) (hl : ∀ a, IsSuccLimit a → (∀ b < a, C b) → C a)
+
+theorem limitRecOn_succ_of_not_isMax (hb : ¬ IsMax b) :
+    limitRecOn (Order.succ b) hm hs hl = hs b hb (limitRecOn b hm hs hl) := by
+  rw [limitRecOn, prelimitRecOn_succ_of_not_isMax]; rfl
+
+@[simp]
+theorem limitRecOn_succ [NoMaxOrder α] (b : α) :
+    limitRecOn (Order.succ b) hm hs hl = hs b (not_isMax b) (limitRecOn b hm hs hl) :=
+  limitRecOn_succ_of_not_isMax hm hs hl _
+
+end LinearOrder
+
+end limitRecOn
+
+end SuccOrder
+
+namespace PredOrder
+
+section prelimitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [PredOrder α] [WellFoundedGT α]
+  (hp : ∀ a, ¬ IsMin a → C a → C (Order.pred a)) (hl : ∀ a, IsPredPrelimit a → (∀ b > a, C b) → C a)
+
+variable (b) in
+/-- Recursion principle on a well-founded partial `PredOrder`. -/
+@[elab_as_elim] noncomputable def prelimitRecOn : C b :=
+  SuccOrder.prelimitRecOn (α := αᵒᵈ) b hp (fun a ha => hl a ha.dual)
+
+@[simp]
+theorem prelimitRecOn_of_isPredPrelimit (hb : IsPredPrelimit b) :
+    prelimitRecOn b hp hl = hl b hb fun x _ ↦ prelimitRecOn x hp hl :=
+  SuccOrder.prelimitRecOn_of_isSuccPrelimit _ _ hb.dual
+
+@[deprecated prelimitRecOn_of_isPredPrelimit (since := "2024-09-05")]
+alias limitRecOn_limit := prelimitRecOn_of_isPredPrelimit
+@[deprecated prelimitRecOn_of_isPredPrelimit (since := "2024-09-14")]
+alias prelimitRecOn_limit := prelimitRecOn_of_isPredPrelimit
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [PredOrder α] [WellFoundedGT α]
+  (hp : ∀ a, ¬ IsMin a → C a → C (Order.pred a)) (hl : ∀ a, IsPredPrelimit a → (∀ b > a, C b) → C a)
+
+theorem prelimitRecOn_pred_of_not_isMin (hb : ¬ IsMin b) :
+    prelimitRecOn (Order.pred b) hp hl = hp b hb (prelimitRecOn b hp hl) :=
+  SuccOrder.prelimitRecOn_succ_of_not_isMax _ _ _
+
+@[deprecated prelimitRecOn_pred_of_not_isMin (since := "2024-09-05")]
+alias limitRecOn_pred' := prelimitRecOn_pred_of_not_isMin
+@[deprecated prelimitRecOn_pred_of_not_isMin (since := "2024-09-14")]
+alias prelimitRecOn_pred' := prelimitRecOn_pred_of_not_isMin
+
+@[simp]
+theorem prelimitRecOn_pred [NoMinOrder α] (b : α) :
+    prelimitRecOn (Order.pred b) hp hl = hp b (not_isMin b) (prelimitRecOn b hp hl) :=
+  prelimitRecOn_pred_of_not_isMin _ _ _
+
+end LinearOrder
+
+end prelimitRecOn
+
+section limitRecOn
+
+section PartialOrder
+
+variable [PartialOrder α] [PredOrder α] [WellFoundedGT α] (hm : ∀ a, IsMax a → C a)
+  (hs : ∀ a, ¬ IsMin a → C a → C (Order.pred a)) (hl : ∀ a, IsPredLimit a → (∀ b > a, C b) → C a)
+
+variable (b) in
+open Classical in
+/-- Recursion principle on a well-founded partial `PredOrder`, separating out the case of a
+maximal element. -/
+@[elab_as_elim] noncomputable def limitRecOn : C b :=
+  SuccOrder.limitRecOn (α := αᵒᵈ) b hm hs (fun a ha => hl a ha.dual)
+
+@[simp]
+theorem limitRecOn_isMax (hb : IsMax b) : limitRecOn b hm hs hl = hm b hb :=
+  SuccOrder.limitRecOn_isMin (α := αᵒᵈ) hm hs _ hb
+
+@[simp]
+theorem limitRecOn_of_isPredLimit (hb : IsPredLimit b) :
+    limitRecOn b hm hs hl = hl b hb fun x _ ↦ limitRecOn x hm hs hl :=
+  SuccOrder.limitRecOn_of_isSuccLimit (α := αᵒᵈ) hm hs _ hb.dual
+
+end PartialOrder
+
+section LinearOrder
+
+variable [LinearOrder α] [PredOrder α] [WellFoundedGT α] (hm : ∀ a, IsMax a → C a)
+  (hs : ∀ a, ¬ IsMin a → C a → C (Order.pred a)) (hl : ∀ a, IsPredLimit a → (∀ b > a, C b) → C a)
+
+theorem limitRecOn_pred_of_not_isMin (hb : ¬ IsMin b) :
+    limitRecOn (Order.pred b) hm hs hl = hs b hb (limitRecOn b hm hs hl) :=
+  SuccOrder.limitRecOn_succ_of_not_isMax (α := αᵒᵈ) hm hs _ hb
+
+@[simp]
+theorem limitRecOn_pred [NoMinOrder α] (b : α) :
+    limitRecOn (Order.pred b) hm hs hl = hs b (not_isMin b) (limitRecOn b hm hs hl) :=
+  SuccOrder.limitRecOn_succ (α := αᵒᵈ) hm hs _ b
+
+end LinearOrder
+
+end limitRecOn
+
+end PredOrder
diff --git a/Mathlib/Order/SuccPred/LinearLocallyFinite.lean b/Mathlib/Order/SuccPred/LinearLocallyFinite.lean
index 3e5665723a687..e758ae8c0f2f4 100644
--- a/Mathlib/Order/SuccPred/LinearLocallyFinite.lean
+++ b/Mathlib/Order/SuccPred/LinearLocallyFinite.lean
@@ -5,7 +5,7 @@ Authors: Rémy Degenne
 -/
 import Mathlib.Data.Countable.Basic
 import Mathlib.Logic.Encodable.Basic
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 import Mathlib.Order.Interval.Finset.Defs
 import Mathlib.Algebra.Order.Ring.Nat
 
@@ -56,6 +56,41 @@ open Order
 
 variable {ι : Type*} [LinearOrder ι]
 
+namespace LinearOrder
+
+variable [SuccOrder ι] [PredOrder ι]
+
+instance (priority := 100) isPredArchimedean_of_isSuccArchimedean [IsSuccArchimedean ι] :
+    IsPredArchimedean ι where
+  exists_pred_iterate_of_le {i j} hij := by
+    have h_exists := exists_succ_iterate_of_le hij
+    obtain ⟨n, hn_eq, hn_lt_ne⟩ : ∃ n, succ^[n] i = j ∧ ∀ m < n, succ^[m] i ≠ j :=
+      ⟨Nat.find h_exists, Nat.find_spec h_exists, fun m hmn ↦ Nat.find_min h_exists hmn⟩
+    refine ⟨n, ?_⟩
+    rw [← hn_eq]
+    cases n with
+    | zero => simp only [Function.iterate_zero, id]
+    | succ n =>
+      rw [pred_succ_iterate_of_not_isMax]
+      rw [Nat.succ_sub_succ_eq_sub, tsub_zero]
+      suffices succ^[n] i < succ^[n.succ] i from not_isMax_of_lt this
+      refine lt_of_le_of_ne ?_ ?_
+      · rw [Function.iterate_succ_apply']
+        exact le_succ _
+      · rw [hn_eq]
+        exact hn_lt_ne _ (Nat.lt_succ_self n)
+
+instance isSuccArchimedean_of_isPredArchimedean [IsPredArchimedean ι] : IsSuccArchimedean ι :=
+  inferInstanceAs (IsSuccArchimedean ιᵒᵈᵒᵈ)
+
+/-- In a linear `SuccOrder` that's also a `PredOrder`, `IsSuccArchimedean` and `IsPredArchimedean`
+are equivalent. -/
+theorem isSuccArchimedean_iff_isPredArchimedean : IsSuccArchimedean ι ↔ IsPredArchimedean ι where
+  mp _ := isPredArchimedean_of_isSuccArchimedean
+  mpr _ := isSuccArchimedean_of_isPredArchimedean
+
+end LinearOrder
+
 namespace LinearLocallyFiniteOrder
 
 /-- Successor in a linear order. This defines a true successor only when `i` is isolated from above,
@@ -111,12 +146,9 @@ noncomputable instance (priority := 100) [LocallyFiniteOrder ι] : SuccOrder ι
   succ_le_of_lt h := succFn_le_of_lt _ _ h
 
 noncomputable instance (priority := 100) [LocallyFiniteOrder ι] : PredOrder ι :=
-  (inferInstance : PredOrder (OrderDual ιᵒᵈ))
+  inferInstanceAs (PredOrder ιᵒᵈᵒᵈ)
 
-end LinearLocallyFiniteOrder
-
-instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFiniteOrder ι] :
-    IsSuccArchimedean ι where
+instance (priority := 100) [LocallyFiniteOrder ι] : IsSuccArchimedean ι where
   exists_succ_iterate_of_le := by
     intro i j hij
     rw [le_iff_lt_or_eq] at hij
@@ -145,28 +177,14 @@ instance (priority := 100) LinearLocallyFiniteOrder.isSuccArchimedean [LocallyFi
     have h_max : IsMax (succ^[n] i) := isMax_iterate_succ_of_eq_of_ne h_eq hnm.ne
     exact not_le.mpr (h_lt n) (h_max (h_lt n).le)
 
-instance (priority := 100) LinearOrder.isPredArchimedean_of_isSuccArchimedean [SuccOrder ι]
-    [PredOrder ι] [IsSuccArchimedean ι] : IsPredArchimedean ι where
-  exists_pred_iterate_of_le := by
-    intro i j hij
-    have h_exists := exists_succ_iterate_of_le hij
-    obtain ⟨n, hn_eq, hn_lt_ne⟩ : ∃ n, succ^[n] i = j ∧ ∀ m < n, succ^[m] i ≠ j :=
-      ⟨Nat.find h_exists, Nat.find_spec h_exists, fun m hmn ↦ Nat.find_min h_exists hmn⟩
-    refine ⟨n, ?_⟩
-    rw [← hn_eq]
-    induction' n with n
-    · simp only [Function.iterate_zero, id]
-    · rw [pred_succ_iterate_of_not_isMax]
-      rw [Nat.succ_sub_succ_eq_sub, tsub_zero]
-      suffices succ^[n] i < succ^[n.succ] i from not_isMax_of_lt this
-      refine lt_of_le_of_ne ?_ ?_
-      · rw [Function.iterate_succ']
-        exact le_succ _
-      · rw [hn_eq]
-        exact hn_lt_ne _ (Nat.lt_succ_self n)
+instance (priority := 100) [LocallyFiniteOrder ι] : IsPredArchimedean ι :=
+  inferInstance
+
+end LinearLocallyFiniteOrder
 
 section toZ
 
+-- Requiring either of `IsSuccArchimedean` or `IsPredArchimedean` is equivalent.
 variable [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι}
 
 -- For "to_Z"
@@ -176,12 +194,13 @@ variable [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι}
 the range of `toZ`. -/
 def toZ (i0 i : ι) : ℤ :=
   dite (i0 ≤ i) (fun hi ↦ Nat.find (exists_succ_iterate_of_le hi)) fun hi ↦
-    -Nat.find (exists_pred_iterate_of_le (not_le.mp hi).le)
+    -Nat.find (exists_pred_iterate_of_le (α := ι) (not_le.mp hi).le)
 
 theorem toZ_of_ge (hi : i0 ≤ i) : toZ i0 i = Nat.find (exists_succ_iterate_of_le hi) :=
   dif_pos hi
 
-theorem toZ_of_lt (hi : i < i0) : toZ i0 i = -Nat.find (exists_pred_iterate_of_le hi.le) :=
+theorem toZ_of_lt (hi : i < i0) :
+    toZ i0 i = -Nat.find (exists_pred_iterate_of_le (α := ι) hi.le) :=
   dif_neg (not_le.mpr hi)
 
 @[simp]
@@ -292,8 +311,8 @@ theorem toZ_mono {i j : ι} (h_le : i ≤ j) : toZ i0 i ≤ toZ i0 j := by
     · exact le_of_not_le h
   · exact absurd h_le (not_le.mpr (hj.trans_le hi))
   · exact (toZ_neg hi).le.trans (toZ_nonneg hj)
-  · let m := Nat.find (exists_pred_iterate_of_le h_le)
-    have hm : pred^[m] j = i := Nat.find_spec (exists_pred_iterate_of_le h_le)
+  · let m := Nat.find (exists_pred_iterate_of_le (α := ι) h_le)
+    have hm : pred^[m] j = i := Nat.find_spec (exists_pred_iterate_of_le (α := ι) h_le)
     have hj_eq : i = pred^[(-toZ i0 j).toNat + m] i0 := by
       rw [← hm, add_comm]
       nth_rw 1 [← iterate_pred_toZ j hj]
diff --git a/Mathlib/Order/SuccPred/Relation.lean b/Mathlib/Order/SuccPred/Relation.lean
index 3b9c6848d52a5..b54a6dd23a17f 100644
--- a/Mathlib/Order/SuccPred/Relation.lean
+++ b/Mathlib/Order/SuccPred/Relation.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
-import Mathlib.Order.SuccPred.Basic
+import Mathlib.Order.SuccPred.Archimedean
 
 /-!
 # Relations on types with a `SuccOrder`
diff --git a/Mathlib/Order/SupIndep.lean b/Mathlib/Order/SupIndep.lean
index e3a967e7b0c55..cb3c3c12cac6a 100644
--- a/Mathlib/Order/SupIndep.lean
+++ b/Mathlib/Order/SupIndep.lean
@@ -131,14 +131,14 @@ theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
       have : ({i, k} : Finset ι).erase k = {i} := by
         ext
         rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
-          not_and_self_iff, or_false_iff, and_iff_right_of_imp]
+          not_and_self_iff, or_false, and_iff_right_of_imp]
         rintro rfl
         exact hij
       rw [this, Finset.sup_singleton]⟩
 
 theorem supIndep_univ_bool (f : Bool → α) :
     (Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) :=
-  haveI : true ≠ false := by simp only [Ne, not_false_iff]
+  haveI : true ≠ false := by simp only [Ne, not_false_iff, reduceCtorEq]
   (supIndep_pair this).trans disjoint_comm
 
 @[simp]
@@ -174,7 +174,7 @@ theorem supIndep_attach : (s.attach.SupIndep fun a => f a) ↔ s.SupIndep f := b
   convert h (filter_subset (fun (i : { x // x ∈ s }) => (i : ι) ∈ t) _) (mem_attach _ ⟨i, ‹_›⟩)
     fun hi => hit <| by simpa using hi using 1
   refine eq_of_forall_ge_iff ?_
-  simp only [Finset.sup_le_iff, mem_filter, mem_attach, true_and_iff, Function.comp_apply,
+  simp only [Finset.sup_le_iff, mem_filter, mem_attach, true_and, Function.comp_apply,
     Subtype.forall, Subtype.coe_mk]
   exact fun a => forall_congr' fun j => ⟨fun h _ => h, fun h hj => h (ht hj) hj⟩
 
diff --git a/Mathlib/Order/SymmDiff.lean b/Mathlib/Order/SymmDiff.lean
index f16a508003350..fa8d23d3aa081 100644
--- a/Mathlib/Order/SymmDiff.lean
+++ b/Mathlib/Order/SymmDiff.lean
@@ -82,7 +82,7 @@ theorem symmDiff_eq_Xor' (p q : Prop) : p ∆ q = Xor' p q :=
 
 @[simp]
 theorem bihimp_iff_iff {p q : Prop} : p ⇔ q ↔ (p ↔ q) :=
-  (iff_iff_implies_and_implies _ _).symm.trans Iff.comm
+  iff_iff_implies_and_implies.symm.trans Iff.comm
 
 @[simp]
 theorem Bool.symmDiff_eq_xor : ∀ p q : Bool, p ∆ q = xor p q := by decide
@@ -397,7 +397,7 @@ theorem symmDiff_symmDiff_right :
 theorem symmDiff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) := by
   rw [symmDiff_symmDiff_left, symmDiff_symmDiff_right]
 
-instance symmDiff_isAssociative : Std.Associative (α := α)  (· ∆ ·) :=
+instance symmDiff_isAssociative : Std.Associative (α := α) (· ∆ ·) :=
   ⟨symmDiff_assoc⟩
 
 theorem symmDiff_left_comm : a ∆ (b ∆ c) = b ∆ (a ∆ c) := by
diff --git a/Mathlib/Order/Synonym.lean b/Mathlib/Order/Synonym.lean
index 70558d627478e..35decae0e1c87 100644
--- a/Mathlib/Order/Synonym.lean
+++ b/Mathlib/Order/Synonym.lean
@@ -34,7 +34,7 @@ This file is similar to `Algebra.Group.TypeTags`.
 -/
 
 
-variable {α β γ : Type*}
+variable {α : Type*}
 
 /-! ### Order dual -/
 
@@ -44,11 +44,11 @@ namespace OrderDual
 instance [h : Nontrivial α] : Nontrivial αᵒᵈ :=
   h
 
-/-- `toDual` is the identity function to the `OrderDual` of a linear order.  -/
+/-- `toDual` is the identity function to the `OrderDual` of a linear order. -/
 def toDual : α ≃ αᵒᵈ :=
   Equiv.refl _
 
-/-- `ofDual` is the identity function from the `OrderDual` of a linear order.  -/
+/-- `ofDual` is the identity function from the `OrderDual` of a linear order. -/
 def ofDual : αᵒᵈ ≃ α :=
   Equiv.refl _
 
@@ -134,12 +134,12 @@ end OrderDual
 def Lex (α : Type*) :=
   α
 
-/-- `toLex` is the identity function to the `Lex` of a type.  -/
+/-- `toLex` is the identity function to the `Lex` of a type. -/
 @[match_pattern]
 def toLex : α ≃ Lex α :=
   Equiv.refl _
 
-/-- `ofLex` is the identity function from the `Lex` of a type.  -/
+/-- `ofLex` is the identity function from the `Lex` of a type. -/
 @[match_pattern]
 def ofLex : Lex α ≃ α :=
   Equiv.refl _
@@ -170,6 +170,18 @@ theorem toLex_inj {a b : α} : toLex a = toLex b ↔ a = b :=
 theorem ofLex_inj {a b : Lex α} : ofLex a = ofLex b ↔ a = b :=
   Iff.rfl
 
+instance (α : Type*) [BEq α] : BEq (Lex α) where
+  beq a b := ofLex a == ofLex b
+
+instance (α : Type*) [BEq α] [LawfulBEq α] : LawfulBEq (Lex α) :=
+  inferInstanceAs (LawfulBEq α)
+
+instance (α : Type*) [DecidableEq α] : DecidableEq (Lex α) :=
+  inferInstanceAs (DecidableEq α)
+
+instance (α : Type*) [Inhabited α] : Inhabited (Lex α) :=
+  inferInstanceAs (Inhabited α)
+
 /-- A recursor for `Lex`. Use as `induction x`. -/
 @[elab_as_elim, induction_eliminator, cases_eliminator]
 protected def Lex.rec {β : Lex α → Sort*} (h : ∀ a, β (toLex a)) : ∀ a, β a := fun a => h (ofLex a)
diff --git a/Mathlib/Order/ULift.lean b/Mathlib/Order/ULift.lean
index b3fa9d2fcbc4c..05f7ece9dd38b 100644
--- a/Mathlib/Order/ULift.lean
+++ b/Mathlib/Order/ULift.lean
@@ -14,6 +14,8 @@ the corresponding `Prod` instances.
 
 namespace ULift
 
+open Batteries
+
 universe v u
 
 variable {α : Type u}
@@ -28,6 +30,11 @@ instance [LT α] : LT (ULift.{v} α) where lt x y := x.down < y.down
 @[simp] theorem up_lt [LT α] {a b : α} : up a < up b ↔ a < b := Iff.rfl
 @[simp] theorem down_lt [LT α] {a b : ULift α} : down a < down b ↔ a < b := Iff.rfl
 
+instance [BEq α] : BEq (ULift.{v} α) where beq x y := x.down == y.down
+
+@[simp] theorem up_beq [BEq α] (a b : α) : (up a == up b) = (a == b) := rfl
+@[simp] theorem down_beq [BEq α] (a b : ULift α) : (down a == down b) = (a == b) := rfl
+
 instance [Ord α] : Ord (ULift.{v} α) where compare x y := compare x.down y.down
 
 @[simp] theorem up_compare [Ord α] (a b : α) : compare (up a) (up b) = compare a b := rfl
@@ -54,6 +61,25 @@ instance [HasCompl α] : HasCompl (ULift.{v} α) where compl x := up <| x.down
 @[simp] theorem up_compl [HasCompl α] (a : α) : up (aᶜ) = (up a)ᶜ := rfl
 @[simp] theorem down_compl [HasCompl α] (a : ULift α) : down aᶜ = (down a)ᶜ := rfl
 
+instance [Ord α] [inst : OrientedOrd α] : OrientedOrd (ULift.{v} α) where
+  symm _ _ := inst.symm ..
+
+instance [Ord α] [inst : TransOrd α] : TransOrd (ULift.{v} α) where
+  le_trans := inst.le_trans
+
+instance [BEq α] [Ord α] [inst : BEqOrd α] : BEqOrd (ULift.{v} α) where
+  cmp_iff_beq := inst.cmp_iff_beq
+
+instance [LT α] [Ord α] [inst : LTOrd α] : LTOrd (ULift.{v} α) where
+  cmp_iff_lt := inst.cmp_iff_lt
+
+instance [LE α] [Ord α] [inst : LEOrd α] : LEOrd (ULift.{v} α) where
+  cmp_iff_le := inst.cmp_iff_le
+
+instance [LE α] [LT α] [BEq α] [Ord α] [inst : LawfulOrd α] : LawfulOrd (ULift.{v} α) where
+  cmp_iff_lt := inst.cmp_iff_lt
+  cmp_iff_le := inst.cmp_iff_le
+
 instance [Preorder α] : Preorder (ULift.{v} α) :=
   Preorder.lift ULift.down
 
diff --git a/Mathlib/Order/UpperLower/Basic.lean b/Mathlib/Order/UpperLower/Basic.lean
index efe287062c81e..1c2dbcf53ed26 100644
--- a/Mathlib/Order/UpperLower/Basic.lean
+++ b/Mathlib/Order/UpperLower/Basic.lean
@@ -1802,3 +1802,5 @@ theorem lowerClosure_prod (s : Set α) (t : Set β) :
   simp [Prod.le_def, @and_and_and_comm _ (_ ∈ t)]
 
 end Preorder
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/Order/WellFounded.lean b/Mathlib/Order/WellFounded.lean
index cacf754c0c287..613d28890f078 100644
--- a/Mathlib/Order/WellFounded.lean
+++ b/Mathlib/Order/WellFounded.lean
@@ -3,7 +3,8 @@ Copyright (c) 2020 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Mario Carneiro
 -/
-import Mathlib.Data.Set.Basic
+import Mathlib.Data.Set.Function
+import Mathlib.Order.Bounds.Basic
 
 /-!
 # Well-founded relations
@@ -119,47 +120,86 @@ protected theorem lt_succ_iff {r : α → α → Prop} [wo : IsWellOrder α r] {
     exact hy
   rintro (hy | rfl); (· exact _root_.trans hy (wo.wf.lt_succ h)); exact wo.wf.lt_succ h
 
+end WellFounded
+
 section LinearOrder
 
-variable [LinearOrder β] [PartialOrder γ]
+variable [LinearOrder β] [Preorder γ]
 
-theorem min_le (h : WellFounded ((· < ·) : β → β → Prop)) {x : β} {s : Set β} (hx : x ∈ s)
-    (hne : s.Nonempty := ⟨x, hx⟩) : h.min s hne ≤ x :=
+theorem WellFounded.min_le (h : WellFounded ((· < ·) : β → β → Prop))
+    {x : β} {s : Set β} (hx : x ∈ s) (hne : s.Nonempty := ⟨x, hx⟩) : h.min s hne ≤ x :=
   not_lt.1 <| h.not_lt_min _ _ hx
 
-private theorem eq_strictMono_iff_eq_range_aux {f g : β → γ} (hf : StrictMono f)
-    (hg : StrictMono g) (hfg : Set.range f = Set.range g) {b : β} (H : ∀ a < b, f a = g a) :
-    f b ≤ g b := by
-  obtain ⟨c, hc⟩ : g b ∈ Set.range f := by
-    rw [hfg]
-    exact Set.mem_range_self b
-  rcases lt_or_le c b with hcb | hbc
-  · rw [H c hcb] at hc
-    rw [hg.injective hc] at hcb
-    exact hcb.false.elim
-  · rw [← hc]
-    exact hf.monotone hbc
-
-theorem eq_strictMono_iff_eq_range (h : WellFounded ((· < ·) : β → β → Prop))
+theorem Set.range_injOn_strictMono [WellFoundedLT β] :
+    Set.InjOn Set.range { f : β → γ | StrictMono f } := by
+  intro f hf g hg hfg
+  ext a
+  apply WellFoundedLT.induction a
+  intro a IH
+  obtain ⟨b, hb⟩ := hfg ▸ mem_range_self a
+  obtain h | rfl | h := lt_trichotomy b a
+  · rw [← IH b h] at hb
+    cases (hf.injective hb).not_lt h
+  · rw [hb]
+  · obtain ⟨c, hc⟩ := hfg.symm ▸ mem_range_self a
+    have := hg h
+    rw [hb, ← hc, hf.lt_iff_lt] at this
+    rw [IH c this] at hc
+    cases (hg.injective hc).not_lt this
+
+theorem Set.range_injOn_strictAnti [WellFoundedGT β] :
+    Set.InjOn Set.range { f : β → γ | StrictAnti f } :=
+  fun _ hf _ hg ↦ Set.range_injOn_strictMono (β := βᵒᵈ) hf.dual hg.dual
+
+theorem StrictMono.range_inj [WellFoundedLT β] {f g : β → γ}
+    (hf : StrictMono f) (hg : StrictMono g) : Set.range f = Set.range g ↔ f = g :=
+  Set.range_injOn_strictMono.eq_iff hf hg
+
+theorem StrictAnti.range_inj [WellFoundedGT β] {f g : β → γ}
+    (hf : StrictAnti f) (hg : StrictAnti g) : Set.range f = Set.range g ↔ f = g :=
+  Set.range_injOn_strictAnti.eq_iff hf hg
+
+@[deprecated StrictMono.range_inj (since := "2024-09-11")]
+theorem WellFounded.eq_strictMono_iff_eq_range (h : WellFounded ((· < ·) : β → β → Prop))
     {f g : β → γ} (hf : StrictMono f) (hg : StrictMono g) :
     Set.range f = Set.range g ↔ f = g :=
-  ⟨fun hfg => by
-    funext a
-    apply h.induction a
-    exact fun b H =>
-      le_antisymm (eq_strictMono_iff_eq_range_aux hf hg hfg H)
-        (eq_strictMono_iff_eq_range_aux hg hf hfg.symm fun a hab => (H a hab).symm),
-    congr_arg _⟩
-
-theorem self_le_of_strictMono (h : WellFounded ((· < ·) : β → β → Prop))
+  @StrictMono.range_inj β γ _ _ ⟨h⟩ f g hf hg
+
+/-- A strictly monotone function `f` on a well-order satisfies `x ≤ f x` for all `x`. -/
+theorem StrictMono.id_le [WellFoundedLT β] {f : β → β} (hf : StrictMono f) : id ≤ f := by
+  rw [Pi.le_def]
+  by_contra! H
+  obtain ⟨m, hm, hm'⟩ := wellFounded_lt.has_min _ H
+  exact hm' _ (hf hm) hm
+
+theorem StrictMono.le_apply [WellFoundedLT β] {f : β → β} (hf : StrictMono f) {x} : x ≤ f x :=
+  hf.id_le x
+
+/-- A strictly monotone function `f` on a cowell-order satisfies `f x ≤ x` for all `x`. -/
+theorem StrictMono.le_id [WellFoundedGT β] {f : β → β} (hf : StrictMono f) : f ≤ id :=
+  StrictMono.id_le (β := βᵒᵈ) hf.dual
+
+theorem StrictMono.apply_le [WellFoundedGT β] {f : β → β} (hf : StrictMono f) {x} : f x ≤ x :=
+  StrictMono.le_apply (β := βᵒᵈ) hf.dual
+
+@[deprecated StrictMono.le_apply (since := "2024-09-11")]
+theorem WellFounded.self_le_of_strictMono (h : WellFounded ((· < ·) : β → β → Prop))
     {f : β → β} (hf : StrictMono f) : ∀ n, n ≤ f n := by
   by_contra! h₁
   have h₂ := h.min_mem _ h₁
   exact h.not_lt_min _ h₁ (hf h₂) h₂
 
-end LinearOrder
+theorem StrictMono.not_bddAbove_range_of_wellFoundedLT {f : β → β} [WellFoundedLT β] [NoMaxOrder β]
+    (hf : StrictMono f) : ¬ BddAbove (Set.range f) := by
+  rintro ⟨a, ha⟩
+  obtain ⟨b, hb⟩ := exists_gt a
+  exact ((hf.le_apply.trans_lt (hf hb)).trans_le <| ha (Set.mem_range_self _)).false
 
-end WellFounded
+theorem StrictMono.not_bddBelow_range_of_wellFoundedGT {f : β → β} [WellFoundedGT β] [NoMinOrder β]
+    (hf : StrictMono f) : ¬ BddBelow (Set.range f) :=
+  hf.dual.not_bddAbove_range_of_wellFoundedLT
+
+end LinearOrder
 
 namespace Function
 
diff --git a/Mathlib/Order/WellFoundedSet.lean b/Mathlib/Order/WellFoundedSet.lean
index 84f0ca38e5c7b..5c7e684d6af00 100644
--- a/Mathlib/Order/WellFoundedSet.lean
+++ b/Mathlib/Order/WellFoundedSet.lean
@@ -134,16 +134,15 @@ theorem acc_iff_wellFoundedOn {α} {r : α → α → Prop} {a : α} :
     TFAE [Acc r a,
       WellFoundedOn { b | ReflTransGen r b a } r,
       WellFoundedOn { b | TransGen r b a } r] := by
-  tfae_have 1 → 2
-  · refine fun h => ⟨fun b => InvImage.accessible _ ?_⟩
+  tfae_have 1 → 2 := by
+    refine fun h => ⟨fun b => InvImage.accessible _ ?_⟩
     rw [← acc_transGen_iff] at h ⊢
     obtain h' | h' := reflTransGen_iff_eq_or_transGen.1 b.2
     · rwa [h'] at h
     · exact h.inv h'
-  tfae_have 2 → 3
-  · exact fun h => h.subset fun _ => TransGen.to_reflTransGen
-  tfae_have 3 → 1
-  · refine fun h => Acc.intro _ (fun b hb => (h.apply ⟨b, .single hb⟩).of_fibration Subtype.val ?_)
+  tfae_have 2 → 3 := fun h => h.subset fun _ => TransGen.to_reflTransGen
+  tfae_have 3 → 1 := by
+    refine fun h => Acc.intro _ (fun b hb => (h.apply ⟨b, .single hb⟩).of_fibration Subtype.val ?_)
     exact fun ⟨c, hc⟩ d h => ⟨⟨d, .head h hc⟩, h, rfl⟩
   tfae_finish
 
@@ -167,10 +166,10 @@ theorem wellFoundedOn_iff_no_descending_seq :
   · rintro ⟨⟨f, hf⟩⟩
     have H : ∀ n, f n ∈ s := fun n => (hf.2 n.lt_succ_self).2.2
     refine ⟨⟨f, ?_⟩, H⟩
-    simpa only [H, and_true_iff] using @hf
+    simpa only [H, and_true] using @hf
   · rintro ⟨⟨f, hf⟩, hfs : ∀ n, f n ∈ s⟩
     refine ⟨⟨f, ?_⟩⟩
-    simpa only [hfs, and_true_iff] using @hf
+    simpa only [hfs, and_true] using @hf
 
 theorem WellFoundedOn.union (hs : s.WellFoundedOn r) (ht : t.WellFoundedOn r) :
     (s ∪ t).WellFoundedOn r := by
@@ -310,7 +309,7 @@ theorem Subsingleton.partiallyWellOrderedOn (hs : s.Subsingleton) : PartiallyWel
 theorem partiallyWellOrderedOn_insert :
     PartiallyWellOrderedOn (insert a s) r ↔ PartiallyWellOrderedOn s r := by
   simp only [← singleton_union, partiallyWellOrderedOn_union,
-    partiallyWellOrderedOn_singleton, true_and_iff]
+    partiallyWellOrderedOn_singleton, true_and]
 
 protected theorem PartiallyWellOrderedOn.insert (h : PartiallyWellOrderedOn s r) (a : α) :
     PartiallyWellOrderedOn (insert a s) r :=
@@ -431,7 +430,7 @@ protected theorem Subsingleton.isPWO (hs : s.Subsingleton) : IsPWO s := hs.finit
 
 @[simp]
 theorem isPWO_insert {a} : IsPWO (insert a s) ↔ IsPWO s := by
-  simp only [← singleton_union, isPWO_union, isPWO_singleton, true_and_iff]
+  simp only [← singleton_union, isPWO_union, isPWO_singleton, true_and]
 
 protected theorem IsPWO.insert (h : IsPWO s) (a : α) : IsPWO (insert a s) :=
   isPWO_insert.2 h
@@ -444,7 +443,7 @@ protected theorem Subsingleton.isWF (hs : s.Subsingleton) : IsWF s := hs.isPWO.i
 
 @[simp]
 theorem isWF_insert {a} : IsWF (insert a s) ↔ IsWF s := by
-  simp only [← singleton_union, isWF_union, isWF_singleton, true_and_iff]
+  simp only [← singleton_union, isWF_union, isWF_singleton, true_and]
 
 protected theorem IsWF.insert (h : IsWF s) (a : α) : IsWF (insert a s) :=
   isWF_insert.2 h
@@ -468,7 +467,7 @@ protected theorem Subsingleton.wellFoundedOn (hs : s.Subsingleton) : s.WellFound
 
 @[simp]
 theorem wellFoundedOn_insert : WellFoundedOn (insert a s) r ↔ WellFoundedOn s r := by
-  simp only [← singleton_union, wellFoundedOn_union, wellFoundedOn_singleton, true_and_iff]
+  simp only [← singleton_union, wellFoundedOn_union, wellFoundedOn_singleton, true_and]
 
 protected theorem WellFoundedOn.insert (h : WellFoundedOn s r) (a : α) :
     WellFoundedOn (insert a s) r :=
@@ -695,7 +694,7 @@ theorem iff_not_exists_isMinBadSeq (rk : α → ℕ) {s : Set α} :
 /-- Higman's Lemma, which states that for any reflexive, transitive relation `r` which is
   partially well-ordered on a set `s`, the relation `List.SublistForall₂ r` is partially
   well-ordered on the set of lists of elements of `s`. That relation is defined so that
-  `List.SublistForall₂ r l₁ l₂` whenever `l₁` related pointwise by `r` to a sublist of `l₂`.  -/
+  `List.SublistForall₂ r l₁ l₂` whenever `l₁` related pointwise by `r` to a sublist of `l₂`. -/
 theorem partiallyWellOrderedOn_sublistForall₂ (r : α → α → Prop) [IsRefl α r] [IsTrans α r]
     {s : Set α} (h : s.PartiallyWellOrderedOn r) :
     { l : List α | ∀ x, x ∈ l → x ∈ s }.PartiallyWellOrderedOn (List.SublistForall₂ r) := by
diff --git a/Mathlib/Order/WithBot.lean b/Mathlib/Order/WithBot.lean
index 02c7ec3c5e101..81f3b53ee346a 100644
--- a/Mathlib/Order/WithBot.lean
+++ b/Mathlib/Order/WithBot.lean
@@ -3,11 +3,11 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Mathlib.Init.Algebra.Classes
 import Mathlib.Logic.Nontrivial.Basic
 import Mathlib.Order.BoundedOrder
 import Mathlib.Order.TypeTags
 import Mathlib.Data.Option.NAry
+import Mathlib.Tactic.Contrapose
 import Mathlib.Tactic.Lift
 import Mathlib.Data.Option.Basic
 
@@ -903,11 +903,37 @@ theorem ofDual_map (f : αᵒᵈ → βᵒᵈ) (a : WithBot αᵒᵈ) :
 
 lemma forall_lt_iff_eq_bot [Preorder α] {x : WithBot α} :
     (∀ y : α, x < y) ↔ x = ⊥ :=
-  ⟨fun h ↦ forall_ne_iff_eq_bot.mp (fun x ↦ (h x).ne'), fun h ↦ h ▸ fun y ↦ bot_lt_coe y⟩
+  ⟨fun h ↦ forall_ne_iff_eq_bot.mp (fun x ↦ (h x).ne'), fun h y ↦ h ▸ bot_lt_coe y⟩
+
+lemma forall_le_iff_eq_bot [Preorder α] [NoMinOrder α] {x : WithBot α} :
+    (∀ y : α, x ≤ y) ↔ x = ⊥ := by
+  refine ⟨fun h ↦ forall_lt_iff_eq_bot.1 fun y ↦ ?_, fun h _ ↦ h ▸ bot_le⟩
+  obtain ⟨w, hw⟩ := exists_lt y
+  exact (h w).trans_lt (coe_lt_coe.2 hw)
+
+lemma le_of_forall_lt_iff_le [LinearOrder α] [DenselyOrdered α] [NoMinOrder α]
+    {x y : WithBot α} : (∀ z : α, x < z → y ≤ z) ↔ y ≤ x := by
+  refine ⟨fun h ↦ ?_, fun h z x_z ↦ h.trans x_z.le⟩
+  induction x with
+  | bot => exact le_of_eq <| forall_le_iff_eq_bot.1 fun z ↦ h z (bot_lt_coe z)
+  | coe x =>
+    rw [le_coe_iff]
+    rintro y rfl
+    exact le_of_forall_le_of_dense (by exact_mod_cast h)
+
+lemma ge_of_forall_gt_iff_ge [LinearOrder α] [DenselyOrdered α] [NoMinOrder α]
+    {x y : WithBot α} : (∀ z : α, z < x → z ≤ y) ↔ x ≤ y := by
+  apply Iff.intro _ (fun h _ x_z ↦ x_z.le.trans h)
+  induction y with
+  | bot => simpa using forall_le_iff_eq_bot.1
+  | coe y =>
+    rw [le_coe_iff]
+    rintro h x rfl
+    exact le_of_forall_ge_of_dense (by exact_mod_cast h)
 
 section LE
 
-variable [LE α] {a b : α}
+variable [LE α]
 
 theorem toDual_le_iff {a : WithBot α} {b : WithTop αᵒᵈ} :
     WithBot.toDual a ≤ b ↔ WithTop.ofDual b ≤ a :=
@@ -937,7 +963,7 @@ end LE
 
 section LT
 
-variable [LT α] {a b : α}
+variable [LT α]
 
 theorem toDual_lt_iff {a : WithBot α} {b : WithTop αᵒᵈ} :
     WithBot.toDual a < b ↔ WithTop.ofDual b < a :=
@@ -1072,11 +1098,22 @@ theorem coe_untop'_le (a : WithTop α) (b : α) : a.untop' b ≤ a :=
 theorem coe_top_lt [OrderTop α] {x : WithTop α} : (⊤ : α) < x ↔ x = ⊤ :=
   WithBot.lt_coe_bot (α := αᵒᵈ)
 
-lemma forall_lt_iff_eq_top {x : WithTop α} : (∀ y : α, y < x) ↔ x = ⊤ :=
-  ⟨fun h ↦ forall_ne_iff_eq_top.mp (fun x ↦ (h x).ne), fun h ↦ h ▸ fun y ↦ coe_lt_top y⟩
+lemma forall_gt_iff_eq_top {x : WithTop α} : (∀ y : α, y < x) ↔ x = ⊤ :=
+  WithBot.forall_lt_iff_eq_bot (α := αᵒᵈ)
+
+lemma forall_ge_iff_eq_top [NoMaxOrder α] {x : WithTop α} : (∀ y : α, y ≤ x) ↔ x = ⊤ :=
+  WithBot.forall_le_iff_eq_bot (α := αᵒᵈ)
 
 end Preorder
 
+lemma le_of_forall_lt_iff_le [LinearOrder α] [DenselyOrdered α] [NoMaxOrder α]
+    {x y : WithTop α} : (∀ z : α, x < z → y ≤ z) ↔ y ≤ x :=
+  WithBot.ge_of_forall_gt_iff_ge (α := αᵒᵈ)
+
+lemma ge_of_forall_gt_iff_ge [LinearOrder α] [DenselyOrdered α] [NoMaxOrder α]
+    {x y : WithTop α} : (∀ z : α, z < x → z ≤ y) ↔ x ≤ y :=
+  WithBot.le_of_forall_lt_iff_le (α := αᵒᵈ)
+
 instance semilatticeInf [SemilatticeInf α] : SemilatticeInf (WithTop α) :=
   { WithTop.partialOrder with
     inf := Option.liftOrGet (· ⊓ ·),
diff --git a/Mathlib/Probability/BorelCantelli.lean b/Mathlib/Probability/BorelCantelli.lean
index 0e50407c65298..be4a2299f904e 100644
--- a/Mathlib/Probability/BorelCantelli.lean
+++ b/Mathlib/Probability/BorelCantelli.lean
@@ -29,7 +29,7 @@ open MeasureTheory
 
 namespace ProbabilityTheory
 
-variable {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω} [IsProbabilityMeasure μ]
+variable {Ω : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω}
 
 section BorelCantelli
 
@@ -45,8 +45,9 @@ theorem iIndepFun.indep_comap_natural_of_lt (hf : ∀ i, StronglyMeasurable (f i
 
 theorem iIndepFun.condexp_natural_ae_eq_of_lt [SecondCountableTopology β] [CompleteSpace β]
     [NormedSpace ℝ β] (hf : ∀ i, StronglyMeasurable (f i)) (hfi : iIndepFun (fun _ => mβ) f μ)
-    (hij : i < j) : μ[f j|Filtration.natural f hf i] =ᵐ[μ] fun _ => μ[f j] :=
-  condexp_indep_eq (hf j).measurable.comap_le (Filtration.le _ _)
+    (hij : i < j) : μ[f j|Filtration.natural f hf i] =ᵐ[μ] fun _ => μ[f j] := by
+  have : IsProbabilityMeasure μ := hfi.isProbabilityMeasure
+  exact condexp_indep_eq (hf j).measurable.comap_le (Filtration.le _ _)
     (comap_measurable <| f j).stronglyMeasurable (hfi.indep_comap_natural_of_lt hf hij)
 
 theorem iIndepSet.condexp_indicator_filtrationOfSet_ae_eq (hsm : ∀ n, MeasurableSet (s n))
@@ -63,6 +64,7 @@ open Filter
 `∑ n, μ sₙ = ∞`, `limsup sₙ` has measure 1. -/
 theorem measure_limsup_eq_one {s : ℕ → Set Ω} (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ)
     (hs' : (∑' n, μ (s n)) = ∞) : μ (limsup s atTop) = 1 := by
+  have : IsProbabilityMeasure μ := hs.isProbabilityMeasure
   rw [measure_congr (eventuallyEq_set.2 (ae_mem_limsup_atTop_iff μ <|
     measurableSet_filtrationOfSet' hsm) : (limsup s atTop : Set Ω) =ᵐ[μ]
       {ω | Tendsto (fun n => ∑ k ∈ Finset.range n,
diff --git a/Mathlib/Probability/ConditionalExpectation.lean b/Mathlib/Probability/ConditionalExpectation.lean
index 66719c1935974..18a5ab72ec167 100644
--- a/Mathlib/Probability/ConditionalExpectation.lean
+++ b/Mathlib/Probability/ConditionalExpectation.lean
@@ -58,11 +58,11 @@ theorem condexp_indep_eq (hle₁ : m₁ ≤ m) (hle₂ : m₂ ≤ m) [SigmaFinit
   · have heq₁ : (fun f : lpMeas E ℝ m₁ 1 μ => ∫ x, (f : Ω → E) x ∂μ) =
         (fun f : Lp E 1 μ => ∫ x, f x ∂μ) ∘ Submodule.subtypeL _ := by
       refine funext fun f => integral_congr_ae ?_
-      simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]; norm_cast
+      simp_rw [Submodule.coe_subtypeL', Submodule.coe_subtype]; norm_cast
     have heq₂ : (fun f : lpMeas E ℝ m₁ 1 μ => ∫ x in s, (f : Ω → E) x ∂μ) =
         (fun f : Lp E 1 μ => ∫ x in s, f x ∂μ) ∘ Submodule.subtypeL _ := by
       refine funext fun f => integral_congr_ae (ae_restrict_of_ae ?_)
-      simp_rw [Submodule.coe_subtypeL', Submodule.coeSubtype]
+      simp_rw [Submodule.coe_subtypeL', Submodule.coe_subtype]
       exact Eventually.of_forall fun _ => (by trivial)
     refine isClosed_eq (Continuous.const_smul ?_ _) ?_
     · rw [heq₁]
diff --git a/Mathlib/Probability/ConditionalProbability.lean b/Mathlib/Probability/ConditionalProbability.lean
index dc3fcd9562f40..aa9148f363412 100644
--- a/Mathlib/Probability/ConditionalProbability.lean
+++ b/Mathlib/Probability/ConditionalProbability.lean
@@ -144,6 +144,9 @@ theorem cond_apply (hms : MeasurableSet s) (t : Set Ω) : μ[t|s] = (μ s)⁻¹
 theorem cond_apply' {t : Set Ω} (hA : MeasurableSet t) : μ[t|s] = (μ s)⁻¹ * μ (s ∩ t) := by
   rw [cond, Measure.smul_apply, Measure.restrict_apply hA, Set.inter_comm, smul_eq_mul]
 
+@[simp] lemma cond_apply_self (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : μ[s|s] = 1 := by
+  simpa [cond] using ENNReal.inv_mul_cancel hs₀ hs
+
 theorem cond_inter_self (hms : MeasurableSet s) (t : Set Ω) : μ[s ∩ t|s] = μ[t|s] := by
   rw [cond_apply _ hms, ← Set.inter_assoc, Set.inter_self, ← cond_apply _ hms]
 
diff --git a/Mathlib/Probability/Distributions/Exponential.lean b/Mathlib/Probability/Distributions/Exponential.lean
index fb141c97472bc..2a2c503ce5f51 100644
--- a/Mathlib/Probability/Distributions/Exponential.lean
+++ b/Mathlib/Probability/Distributions/Exponential.lean
@@ -73,7 +73,7 @@ lemma measurable_exponentialPDFReal (r : ℝ) : Measurable (exponentialPDFReal r
 lemma exponentialPDFReal_pos {x r : ℝ} (hr : 0 < r) (hx : 0 < x) :
     0 < exponentialPDFReal r x := gammaPDFReal_pos zero_lt_one hr hx
 
-/-- The exponential pdf is nonnegative-/
+/-- The exponential pdf is nonnegative -/
 lemma exponentialPDFReal_nonneg {r : ℝ} (hr : 0 < r) (x : ℝ) :
     0 ≤ exponentialPDFReal r x := gammaPDFReal_nonneg zero_lt_one hr x
 
diff --git a/Mathlib/Probability/Distributions/Gamma.lean b/Mathlib/Probability/Distributions/Gamma.lean
index 0548d04bfd3c0..562c9b3c3b6f9 100644
--- a/Mathlib/Probability/Distributions/Gamma.lean
+++ b/Mathlib/Probability/Distributions/Gamma.lean
@@ -30,8 +30,8 @@ open MeasureTheory Real Set Filter Topology
 lemma lintegral_Iic_eq_lintegral_Iio_add_Icc {y z : ℝ} (f : ℝ → ℝ≥0∞) (hzy : z ≤ y) :
     ∫⁻ x in Iic y, f x = (∫⁻ x in Iio z, f x) + ∫⁻ x in Icc z y, f x := by
   rw [← Iio_union_Icc_eq_Iic hzy, lintegral_union measurableSet_Icc]
-  rw [Set.disjoint_iff]
-  rintro x ⟨h1 : x < _, h2, _⟩
+  simp_rw [Set.disjoint_iff_forall_ne, mem_Iio, mem_Icc]
+  intros
   linarith
 
 namespace ProbabilityTheory
@@ -49,8 +49,9 @@ def gammaPDF (a r x : ℝ) : ℝ≥0∞ :=
   ENNReal.ofReal (gammaPDFReal a r x)
 
 lemma gammaPDF_eq (a r x : ℝ) :
-    gammaPDF a r x = ENNReal.ofReal (if 0 ≤ x then
-    r ^ a / (Gamma a) * x ^ (a-1) * exp (-(r * x)) else 0) := rfl
+    gammaPDF a r x =
+      ENNReal.ofReal (if 0 ≤ x then r ^ a / (Gamma a) * x ^ (a-1) * exp (-(r * x)) else 0) :=
+  rfl
 
 lemma gammaPDF_of_neg {a r x : ℝ} (hx : x < 0) : gammaPDF a r x = 0 := by
   simp only [gammaPDF_eq, if_neg (not_le.mpr hx), ENNReal.ofReal_zero]
diff --git a/Mathlib/Probability/Distributions/Gaussian.lean b/Mathlib/Probability/Distributions/Gaussian.lean
index b31dba6a5ddf3..10a5c010266d0 100644
--- a/Mathlib/Probability/Distributions/Gaussian.lean
+++ b/Mathlib/Probability/Distributions/Gaussian.lean
@@ -93,7 +93,7 @@ lemma integrable_gaussianPDFReal (μ : ℝ) (v : ℝ≥0) :
     field_simp
   exact Integrable.comp_sub_right hg μ
 
-/-- The gaussian distribution pdf integrates to 1 when the variance is not zero.  -/
+/-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
 lemma lintegral_gaussianPDFReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
     ∫⁻ x, ENNReal.ofReal (gaussianPDFReal μ v x) = 1 := by
   rw [← ENNReal.toReal_eq_one_iff]
@@ -112,7 +112,7 @@ lemma lintegral_gaussianPDFReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
     ring
   · positivity
 
-/-- The gaussian distribution pdf integrates to 1 when the variance is not zero.  -/
+/-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
 lemma integral_gaussianPDFReal_eq_one (μ : ℝ) {v : ℝ≥0} (hv : v ≠ 0) :
     ∫ x, gaussianPDFReal μ v x = 1 := by
   have h := lintegral_gaussianPDFReal_eq_one μ hv
@@ -283,7 +283,7 @@ lemma gaussianReal_map_const_mul (c : ℝ) :
     rw [Measure.map_const]
     simp only [ne_eq, measure_univ, one_smul, mul_eq_zero]
     convert (gaussianReal_zero_var 0).symm
-    simp only [ne_eq, zero_pow, mul_eq_zero, hv, or_false, not_false_eq_true]
+    simp only [ne_eq, zero_pow, mul_eq_zero, hv, or_false, not_false_eq_true, reduceCtorEq]
     rfl
   let e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv
   have he' : ∀ x, HasDerivAt e ((fun _ ↦ c⁻¹) x) x := by
diff --git a/Mathlib/Probability/Distributions/Pareto.lean b/Mathlib/Probability/Distributions/Pareto.lean
new file mode 100644
index 0000000000000..abc5e6e63621a
--- /dev/null
+++ b/Mathlib/Probability/Distributions/Pareto.lean
@@ -0,0 +1,147 @@
+/-
+Copyright (c) 2024 Alvan Caleb Arulandu. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Alvan Caleb Arulandu
+-/
+import Mathlib.Probability.Notation
+import Mathlib.Probability.CDF
+import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
+
+/-! # Pareto distributions over ℝ
+
+Define the Pareto measure over the reals.
+
+## Main definitions
+* `paretoPDFReal`: the function `t r x ↦ r * t ^ r * x ^ -(r + 1)`
+  for `t ≤ x` or `0` else, which is the probability density function of a Pareto distribution with
+  scale `t` and shape `r` (when `ht : 0 < t` and `hr : 0 < r`).
+* `paretoPDF`: `ℝ≥0∞`-valued pdf,
+  `paretoPDF t r = ENNReal.ofReal (paretoPDFReal t r)`.
+* `paretoMeasure`: a Pareto measure on `ℝ`, parametrized by its scale `t` and shape `r`.
+* `paretoCDFReal`: the CDF given by the definition of CDF in `ProbabilityTheory.CDF` applied to the
+  Pareto measure.
+-/
+
+open scoped ENNReal NNReal
+
+open MeasureTheory Real Set Filter Topology
+
+namespace ProbabilityTheory
+variable {t r x : ℝ}
+
+section ParetoPDF
+
+/-- The pdf of the Pareto distribution depending on its scale `t` and rate `r`. -/
+noncomputable def paretoPDFReal (t r x : ℝ) : ℝ :=
+  if t ≤ x then r * t ^ r * x ^ (-(r + 1)) else 0
+
+/-- The pdf of the Pareto distribution, as a function valued in `ℝ≥0∞`. -/
+noncomputable def paretoPDF (t r x : ℝ) : ℝ≥0∞ :=
+  ENNReal.ofReal (paretoPDFReal t r x)
+
+lemma paretoPDF_eq (t r x : ℝ) :
+    paretoPDF t r x = ENNReal.ofReal (if t ≤ x then r * t ^ r * x ^ (-(r + 1)) else 0) := rfl
+
+lemma paretoPDF_of_lt (hx : x < t) : paretoPDF t r x = 0 := by
+  simp only [paretoPDF_eq, if_neg (not_le.mpr hx), ENNReal.ofReal_zero]
+
+lemma paretoPDF_of_le (hx : t ≤ x) :
+    paretoPDF t r x = ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1))) := by
+  simp only [paretoPDF_eq, if_pos hx]
+
+/-- The Lebesgue integral of the Pareto pdf over reals `≤ t` equals `0`. -/
+lemma lintegral_paretoPDF_of_le (hx : x ≤ t) :
+    ∫⁻ y in Iio x, paretoPDF t r y = 0 := by
+  rw [setLIntegral_congr_fun (g := fun _ ↦ 0) measurableSet_Iio]
+  · rw [lintegral_zero, ← ENNReal.ofReal_zero]
+  · simp only [paretoPDF_eq, ge_iff_le, ENNReal.ofReal_eq_zero]
+    filter_upwards with a (_ : a < _)
+    rw [if_neg (by linarith)]
+
+/-- The Pareto pdf is measurable. -/
+@[measurability, fun_prop]
+lemma measurable_paretoPDFReal (t r : ℝ) : Measurable (paretoPDFReal t r) :=
+  Measurable.ite measurableSet_Ici ((measurable_id.pow_const _).const_mul _) measurable_const
+
+/-- The Pareto pdf is strongly measurable. -/
+@[measurability]
+lemma stronglyMeasurable_paretoPDFReal (t r : ℝ) :
+    StronglyMeasurable (paretoPDFReal t r) :=
+  (measurable_paretoPDFReal t r).stronglyMeasurable
+
+/-- The Pareto pdf is positive for all reals `>= t`. -/
+lemma paretoPDFReal_pos (ht : 0 < t) (hr : 0 < r) (hx : t ≤ x) :
+    0 < paretoPDFReal t r x := by
+  rw [paretoPDFReal, if_pos hx]
+  have _ : 0 < x := by linarith
+  positivity
+
+/-- The Pareto pdf is nonnegative. -/
+lemma paretoPDFReal_nonneg (ht : 0 ≤ t) (hr : 0 ≤ r) (x : ℝ) :
+    0 ≤ paretoPDFReal t r x := by
+  unfold paretoPDFReal
+  split_ifs with h
+  · cases le_iff_eq_or_lt.1 ht with
+    | inl ht0 =>
+      rw [← ht0] at h
+      positivity
+    | inr htp =>
+      have := lt_of_lt_of_le htp h
+      positivity
+  · positivity
+
+open Measure
+
+/-- The pdf of the Pareto distribution integrates to `1`. -/
+@[simp]
+lemma lintegral_paretoPDF_eq_one (ht : 0 < t) (hr : 0 < r) :
+    ∫⁻ x, paretoPDF t r x = 1 := by
+  have leftSide : ∫⁻ x in Iio t, paretoPDF t r x = 0 := lintegral_paretoPDF_of_le (le_refl t)
+  have rightSide : ∫⁻ x in Ici t, paretoPDF t r x =
+      ∫⁻ x in Ici t, ENNReal.ofReal (r * t ^ r * x ^ (-(r + 1))) :=
+    setLIntegral_congr_fun measurableSet_Ici (ae_of_all _ (fun _ ↦ paretoPDF_of_le))
+  rw [← ENNReal.toReal_eq_one_iff, ← lintegral_add_compl _ measurableSet_Ici, compl_Ici,
+    leftSide, rightSide, add_zero, ← integral_eq_lintegral_of_nonneg_ae]
+  · rw [integral_Ici_eq_integral_Ioi, integral_mul_left, integral_Ioi_rpow_of_lt _ ht]
+    · field_simp [hr]
+      rw [mul_assoc, ← rpow_add ht]
+      simp
+    linarith
+  · rw [EventuallyLE, ae_restrict_iff' measurableSet_Ici]
+    refine ae_of_all _ fun x (hx : t ≤ x) ↦ ?_
+    have := lt_of_lt_of_le ht hx
+    positivity
+  · apply (measurable_paretoPDFReal t r).aestronglyMeasurable.congr
+    refine (ae_restrict_iff' measurableSet_Ici).mpr <| ae_of_all _ fun x (hx : t ≤ x) ↦ ?_
+    simp_rw [paretoPDFReal, eq_true_intro hx, ite_true]
+
+end ParetoPDF
+
+open MeasureTheory
+
+/-- Measure defined by the Pareto distribution. -/
+noncomputable def paretoMeasure (t r : ℝ) : Measure ℝ :=
+  volume.withDensity (paretoPDF t r)
+
+lemma isProbabilityMeasure_paretoMeasure (ht : 0 < t) (hr : 0 < r) :
+    IsProbabilityMeasure (paretoMeasure t r) where
+  measure_univ := by simp [paretoMeasure, lintegral_paretoPDF_eq_one ht hr]
+
+section ParetoCDF
+
+/-- CDF of the Pareto distribution equals the integral of the PDF. -/
+lemma paretoCDFReal_eq_integral (ht : 0 < t) (hr : 0 < r) (x : ℝ) :
+    cdf (paretoMeasure t r) x = ∫ x in Iic x, paretoPDFReal t r x := by
+  have : IsProbabilityMeasure (paretoMeasure t r) := isProbabilityMeasure_paretoMeasure ht hr
+  rw [cdf_eq_toReal, paretoMeasure, withDensity_apply _ measurableSet_Iic]
+  refine (integral_eq_lintegral_of_nonneg_ae ?_ ?_).symm
+  · exact ae_of_all _ fun _ ↦ by simp only [Pi.zero_apply, paretoPDFReal_nonneg ht.le hr.le]
+  · exact (measurable_paretoPDFReal t r).aestronglyMeasurable.restrict
+
+lemma paretoCDFReal_eq_lintegral (ht : 0 < t) (hr : 0 < r) (x : ℝ) :
+    cdf (paretoMeasure t r) x = ENNReal.toReal (∫⁻ x in Iic x, paretoPDF t r x) := by
+  have : IsProbabilityMeasure (paretoMeasure t r) := isProbabilityMeasure_paretoMeasure ht hr
+  rw [cdf_eq_toReal, paretoMeasure, withDensity_apply _ measurableSet_Iic]
+
+end ParetoCDF
+end ProbabilityTheory
diff --git a/Mathlib/Probability/Distributions/Uniform.lean b/Mathlib/Probability/Distributions/Uniform.lean
index 67f4746b80fcf..c639f53703f5b 100644
--- a/Mathlib/Probability/Distributions/Uniform.lean
+++ b/Mathlib/Probability/Distributions/Uniform.lean
@@ -346,7 +346,7 @@ theorem mem_support_ofMultiset_iff (a : α) : a ∈ (ofMultiset s hs).support 
 
 theorem ofMultiset_apply_of_not_mem {a : α} (ha : a ∉ s) : ofMultiset s hs a = 0 := by
   simpa only [ofMultiset_apply, ENNReal.div_eq_zero_iff, Nat.cast_eq_zero, Multiset.count_eq_zero,
-    ENNReal.natCast_ne_top, or_false_iff] using ha
+    ENNReal.natCast_ne_top, or_false] using ha
 
 section Measure
 
diff --git a/Mathlib/Probability/Independence/Basic.lean b/Mathlib/Probability/Independence/Basic.lean
index 37bd8bdc8c405..cf6f2aebad8a9 100644
--- a/Mathlib/Probability/Independence/Basic.lean
+++ b/Mathlib/Probability/Independence/Basic.lean
@@ -64,6 +64,8 @@ when defining `μ` in the example above, the measurable space used is the last o
 Part A, Chapter 4.
 -/
 
+assert_not_exists MeasureTheory.Integrable
+
 open MeasureTheory MeasurableSpace Set
 
 open scoped MeasureTheory ENNReal
@@ -144,6 +146,9 @@ lemma iIndepSets.meas_biInter (h : iIndepSets π μ) (s : Finset ι) {f : ι →
     (hf : ∀ i, i ∈ s → f i ∈ π i) : μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) :=
   (iIndepSets_iff _ _).1 h s hf
 
+lemma iIndepSets.isProbabilityMeasure (h : iIndepSets π μ) : IsProbabilityMeasure μ :=
+  ⟨by simpa using h ∅ (f := fun _ ↦ univ)⟩
+
 lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π μ) (hs : ∀ i, s i ∈ π i) :
     μ (⋂ i, s i) = ∏ i, μ (s i) := by simp [← h.meas_biInter _ fun _i _ ↦ hs _]
 
@@ -159,6 +164,9 @@ lemma iIndep.iIndepSets' {m : ι → MeasurableSpace Ω}
     {_ : MeasurableSpace Ω} {μ : Measure Ω} (hμ : iIndep m μ) :
     iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) μ := (iIndep_iff_iIndepSets _ _).1 hμ
 
+lemma iIndep.isProbabilityMeasure (h : iIndep m μ) : IsProbabilityMeasure μ :=
+  h.iIndepSets'.isProbabilityMeasure
+
 lemma iIndep_iff (m : ι → MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω} (μ : Measure Ω) :
     iIndep m μ ↔ ∀ (s : Finset ι) {f : ι → Set Ω} (_H : ∀ i, i ∈ s → MeasurableSet[m i] (f i)),
       μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) := by
@@ -183,6 +191,9 @@ lemma iIndepSet_iff_iIndep (s : ι → Set Ω) (μ : Measure Ω) :
     iIndepSet s μ ↔ iIndep (fun i ↦ generateFrom {s i}) μ := by
   simp only [iIndepSet, iIndep, Kernel.iIndepSet]
 
+lemma iIndepSet.isProbabilityMeasure (h : iIndepSet s μ) : IsProbabilityMeasure μ :=
+  ((iIndepSet_iff_iIndep _ _).1 h).isProbabilityMeasure
+
 lemma iIndepSet_iff (s : ι → Set Ω) (μ : Measure Ω) :
     iIndepSet s μ ↔ ∀ (s' : Finset ι) {f : ι → Set Ω}
       (_H : ∀ i, i ∈ s' → MeasurableSet[generateFrom {s i}] (f i)),
@@ -253,15 +264,15 @@ theorem IndepSets.symm {s₁ s₂ : Set (Set Ω)} (h : IndepSets s₁ s₂ μ) :
 @[symm]
 theorem Indep.symm (h : Indep m₁ m₂ μ) : Indep m₂ m₁ μ := IndepSets.symm h
 
-theorem indep_bot_right [IsProbabilityMeasure μ] : Indep m' ⊥ μ :=
+theorem indep_bot_right [IsZeroOrProbabilityMeasure μ] : Indep m' ⊥ μ :=
   Kernel.indep_bot_right m'
 
-theorem indep_bot_left [IsProbabilityMeasure μ] : Indep ⊥ m' μ := (indep_bot_right m').symm
+theorem indep_bot_left [IsZeroOrProbabilityMeasure μ] : Indep ⊥ m' μ := (indep_bot_right m').symm
 
-theorem indepSet_empty_right [IsProbabilityMeasure μ] (s : Set Ω) : IndepSet s ∅ μ :=
+theorem indepSet_empty_right [IsZeroOrProbabilityMeasure μ] (s : Set Ω) : IndepSet s ∅ μ :=
   Kernel.indepSet_empty_right s
 
-theorem indepSet_empty_left [IsProbabilityMeasure μ] (s : Set Ω) : IndepSet ∅ s μ :=
+theorem indepSet_empty_left [IsZeroOrProbabilityMeasure μ] (s : Set Ω) : IndepSet ∅ s μ :=
   Kernel.indepSet_empty_left s
 
 theorem indepSets_of_indepSets_of_le_left {s₁ s₂ s₃ : Set (Set Ω)}
@@ -327,7 +338,7 @@ end Indep
 
 section FromIndepToIndep
 
-variable {m : ι → MeasurableSpace Ω}  {_mΩ : MeasurableSpace Ω} {μ : Measure Ω}
+variable {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} {μ : Measure Ω}
 
 theorem iIndepSets.indepSets {s : ι → Set (Set Ω)}
     (h_indep : iIndepSets s μ) {i j : ι} (hij : i ≠ j) : IndepSets (s i) (s j) μ :=
@@ -376,61 +387,62 @@ variable {m : ι → MeasurableSpace Ω} {m1 m2 _mΩ : MeasurableSpace Ω} {μ :
 
 /-! ### Independence of generating π-systems implies independence of measurable space structures -/
 
-theorem IndepSets.indep [IsProbabilityMeasure μ]
+theorem IndepSets.indep [IsZeroOrProbabilityMeasure μ]
     {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ _mΩ) (h2 : m2 ≤ _mΩ) (hp1 : IsPiSystem p1)
     (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
     (hyp : IndepSets p1 p2 μ) :
     Indep m1 m2 μ :=
   Kernel.IndepSets.indep h1 h2 hp1 hp2 hpm1 hpm2 hyp
 
-theorem IndepSets.indep' [IsProbabilityMeasure μ]
+theorem IndepSets.indep' [IsZeroOrProbabilityMeasure μ]
     {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
     (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 μ) :
     Indep (generateFrom p1) (generateFrom p2) μ :=
   Kernel.IndepSets.indep' hp1m hp2m hp1 hp2 hyp
 
-theorem indepSets_piiUnionInter_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set (Set Ω)}
+theorem indepSets_piiUnionInter_of_disjoint {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : iIndepSets s μ) (hST : Disjoint S T) :
     IndepSets (piiUnionInter s S) (piiUnionInter s T) μ :=
   Kernel.indepSets_piiUnionInter_of_disjoint h_indep hST
 
-theorem iIndepSet.indep_generateFrom_of_disjoint [IsProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_of_disjoint {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (S T : Set ι) (hST : Disjoint S T) :
     Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) μ :=
   Kernel.iIndepSet.indep_generateFrom_of_disjoint hsm hs S T hST
 
-theorem indep_iSup_of_disjoint [IsProbabilityMeasure μ]
+theorem indep_iSup_of_disjoint
     (h_le : ∀ i, m i ≤ _mΩ) (h_indep : iIndep m μ) {S T : Set ι} (hST : Disjoint S T) :
     Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) μ :=
   Kernel.indep_iSup_of_disjoint h_le h_indep hST
 
 theorem indep_iSup_of_directed_le
-    [IsProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m1 μ)
+    [IsZeroOrProbabilityMeasure μ] (h_indep : ∀ i, Indep (m i) m1 μ)
     (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Directed (· ≤ ·) m) :
     Indep (⨆ i, m i) m1 μ :=
   Kernel.indep_iSup_of_directed_le h_indep h_le h_le' hm
 
-theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_lt [Preorder ι] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) :
     Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) μ :=
   Kernel.iIndepSet.indep_generateFrom_lt hsm hs i
 
-theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsProbabilityMeasure μ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_le [LinearOrder ι]
+    {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (i : ι) {k : ι} (hk : i < k) :
     Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) μ :=
   Kernel.iIndepSet.indep_generateFrom_le hsm hs i hk
 
-theorem iIndepSet.indep_generateFrom_le_nat [IsProbabilityMeasure μ] {s : ℕ → Set Ω}
+theorem iIndepSet.indep_generateFrom_le_nat {s : ℕ → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s μ) (n : ℕ) :
     Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) μ :=
   Kernel.iIndepSet.indep_generateFrom_le_nat hsm hs n
 
-theorem indep_iSup_of_monotone [SemilatticeSup ι] [IsProbabilityMeasure μ]
+theorem indep_iSup_of_monotone [SemilatticeSup ι] [IsZeroOrProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m1 μ) (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Monotone m) :
     Indep (⨆ i, m i) m1 μ :=
   Kernel.indep_iSup_of_monotone h_indep h_le h_le' hm
 
-theorem indep_iSup_of_antitone [SemilatticeInf ι] [IsProbabilityMeasure μ]
+theorem indep_iSup_of_antitone [SemilatticeInf ι] [IsZeroOrProbabilityMeasure μ]
     (h_indep : ∀ i, Indep (m i) m1 μ) (h_le : ∀ i, m i ≤ _mΩ) (h_le' : m1 ≤ _mΩ) (hm : Antitone m) :
     Indep (⨆ i, m i) m1 μ :=
   Kernel.indep_iSup_of_antitone h_indep h_le h_le' hm
@@ -441,7 +453,7 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
   Kernel.iIndepSets.piiUnionInter_of_not_mem hp_ind haS
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
-theorem iIndepSets.iIndep [IsProbabilityMeasure μ]
+theorem iIndepSets.iIndep
     (h_le : ∀ i, m i ≤ _mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π μ) :
     iIndep m μ :=
@@ -463,12 +475,12 @@ variable {m₁ m₂ _mΩ : MeasurableSpace Ω} {μ : Measure Ω} {s t : Set Ω}
 
 theorem indepSet_iff_indepSets_singleton (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
-    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
+    [IsZeroOrProbabilityMeasure μ] : IndepSet s t μ ↔ IndepSets {s} {t} μ :=
   Kernel.indepSet_iff_indepSets_singleton hs_meas ht_meas _ _
 
 theorem indepSet_iff_measure_inter_eq_mul (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (μ : Measure Ω := by volume_tac)
-    [IsProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
+    [IsZeroOrProbabilityMeasure μ] : IndepSet s t μ ↔ μ (s ∩ t) = μ s * μ t :=
   (indepSet_iff_indepSets_singleton hs_meas ht_meas μ).trans indepSets_singleton_iff
 
 lemma IndepSet.measure_inter_eq_mul {μ : Measure Ω} (h : IndepSet s t μ) :
@@ -477,7 +489,7 @@ lemma IndepSet.measure_inter_eq_mul {μ : Measure Ω} (h : IndepSet s t μ) :
 
 theorem IndepSets.indepSet_of_mem (hs : s ∈ S) (ht : t ∈ T)
     (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
-    (μ : Measure Ω := by volume_tac) [IsProbabilityMeasure μ]
+    (μ : Measure Ω := by volume_tac) [IsZeroOrProbabilityMeasure μ]
     (h_indep : IndepSets S T μ) :
     IndepSet s t μ :=
   Kernel.IndepSets.indepSet_of_mem _ _ hs ht hs_meas ht_meas _ _ h_indep
@@ -506,8 +518,6 @@ theorem iIndepSet.meas_biInter {f : ι → Set Ω} (h : iIndepSet f μ) (s : Fin
     μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i) := by
   simpa using Kernel.iIndepSet.meas_biInter h s
 
-variable [IsProbabilityMeasure μ]
-
 theorem iIndepSet_iff_iIndepSets_singleton {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
     iIndepSet f μ ↔ iIndepSets (fun i ↦ {f i}) μ :=
   Kernel.iIndepSet_iff_iIndepSets_singleton hf
@@ -554,7 +564,7 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Ty
 alias ⟨iIndepFun.measure_inter_preimage_eq_mul, _⟩ := iIndepFun_iff_measure_inter_preimage_eq_mul
 
 theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    [IsProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
+    [IsZeroOrProbabilityMeasure μ] (hf : Measurable f) (hg : Measurable g) :
     IndepFun f g μ ↔
       ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) μ := by
   simp only [IndepFun, IndepSet, Kernel.indepFun_iff_indepSet_preimage hf hg, ae_dirac_eq,
@@ -616,13 +626,11 @@ tuple `(f i)_i` for `i ∈ T`. -/
 lemma iIndepFun.indepFun_finset (S T : Finset ι) (hST : Disjoint S T) (hf_Indep : iIndepFun m f μ)
     (hf_meas : ∀ i, Measurable (f i)) :
     IndepFun (fun a (i : S) ↦ f i a) (fun a (i : T) ↦ f i a) μ :=
-  have := hf_Indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_finset S T hST hf_Indep hf_meas
 
 lemma iIndepFun.indepFun_prod_mk (hf_Indep : iIndepFun m f μ) (hf_meas : ∀ i, Measurable (f i))
     (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) μ :=
-  have := hf_Indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_prod_mk hf_Indep hf_meas i j k hik hjk
 
 open Finset in
@@ -644,14 +652,12 @@ variable {β : Type*} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f
 lemma iIndepFun.indepFun_mul_left (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (f i * f j) (f k) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_mul_left hf_indep hf_meas i j k hik hjk
 
 @[to_additive]
 lemma iIndepFun.indepFun_mul_right (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
     IndepFun (f i) (f j * f k) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_mul_right hf_indep hf_meas i j k hij hik
 
 @[to_additive]
@@ -659,7 +665,6 @@ lemma iIndepFun.indepFun_mul_mul (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i))
     (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
     IndepFun (f i * f j) (f k * f l) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_mul_mul hf_indep hf_meas i j k l hik hil hjk hjl
 
 end Mul
@@ -671,14 +676,12 @@ variable {β : Type*} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f
 lemma iIndepFun.indepFun_div_left (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (f i / f j) (f k) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_div_left hf_indep hf_meas i j k hik hjk
 
 @[to_additive]
 lemma iIndepFun.indepFun_div_right (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hij : i ≠ j) (hik : i ≠ k) :
     IndepFun (f i) (f j / f k) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_div_right hf_indep hf_meas i j k hij hik
 
 @[to_additive]
@@ -686,7 +689,6 @@ lemma iIndepFun.indepFun_div_div (hf_indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i))
     (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
     IndepFun (f i / f j) (f k / f l) μ :=
-  have := hf_indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_div_div hf_indep hf_meas i j k l hik hil hjk hjl
 
 end Div
@@ -698,13 +700,11 @@ variable {β : Type*} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂
 lemma iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) {s : Finset ι} {i : ι} (hi : i ∉ s) :
     IndepFun (∏ j ∈ s, f j) (f i) μ :=
-  have := hf_Indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_finset_prod_of_not_mem hf_Indep hf_meas hi
 
 @[to_additive]
 lemma iIndepFun.indepFun_prod_range_succ {f : ℕ → Ω → β} (hf_Indep : iIndepFun (fun _ ↦ m) f μ)
     (hf_meas : ∀ i, Measurable (f i)) (n : ℕ) : IndepFun (∏ j ∈ Finset.range n, f j) (f n) μ :=
-  have := hf_Indep.isProbabilityMeasure
   Kernel.iIndepFun.indepFun_prod_range_succ hf_Indep hf_meas n
 
 end CommMonoid
@@ -717,44 +717,3 @@ theorem iIndepSet.iIndepFun_indicator [Zero β] [One β] {m : MeasurableSpace β
 end IndepFun
 
 end ProbabilityTheory
-
-namespace MeasureTheory
-
-open Filter ProbabilityTheory
-open scoped NNReal Topology
-
-/-- If a nonzero function belongs to `ℒ^p` and is independent of another function, then
-the space is a probability space. -/
-lemma Memℒp.isProbabilityMeasure_of_indepFun
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
-    {F : Type*} [MeasurableSpace F]
-    (f : Ω → E) (g : Ω → F) {p : ℝ≥0∞} (hp : p ≠ 0) (hp' : p ≠ ∞)
-    (hℒp : Memℒp f p μ) (h'f : ¬(∀ᵐ ω ∂μ, f ω = 0)) (hindep : IndepFun f g μ) :
-    IsProbabilityMeasure μ := by
-  obtain ⟨c, c_pos, hc⟩ : ∃ (c : ℝ≥0), 0 < c ∧ 0 < μ {ω | c ≤ ‖f ω‖₊} := by
-    contrapose! h'f
-    have A (c : ℝ≥0) (hc : 0 < c) : ∀ᵐ ω ∂μ, ‖f ω‖₊ < c := by simpa [ae_iff] using h'f c hc
-    obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), 0 < u n)
-      ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto (0 : ℝ≥0)
-    filter_upwards [ae_all_iff.2 (fun n ↦ A (u n) (u_pos n))] with ω hω
-    simpa using ge_of_tendsto' u_lim (fun i ↦ (hω i).le)
-  have h'c : μ {ω | c ≤ ‖f ω‖₊} < ∞ := hℒp.meas_ge_lt_top hp hp' c_pos.ne'
-  have := hindep.measure_inter_preimage_eq_mul {x | c ≤ ‖x‖₊} Set.univ
-    (isClosed_le continuous_const continuous_nnnorm).measurableSet MeasurableSet.univ
-  simp only [Set.preimage_setOf_eq, Set.preimage_univ, Set.inter_univ] at this
-  exact ⟨(ENNReal.mul_eq_left hc.ne' h'c.ne).1 this.symm⟩
-
-/-- If a nonzero function is integrable and is independent of another function, then
-the space is a probability space. -/
-lemma Integrable.isProbabilityMeasure_of_indepFun
-    {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
-    {E : Type*} [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
-    {F : Type*} [MeasurableSpace F]
-    (f : Ω → E) (g : Ω → F)
-    (hf : Integrable f μ) (h'f : ¬(∀ᵐ ω ∂μ, f ω = 0)) (hindep : IndepFun f g μ) :
-    IsProbabilityMeasure μ :=
-  Memℒp.isProbabilityMeasure_of_indepFun f g one_ne_zero ENNReal.one_ne_top
-    (memℒp_one_iff_integrable.mpr hf) h'f hindep
-
-end MeasureTheory
diff --git a/Mathlib/Probability/Independence/Integrable.lean b/Mathlib/Probability/Independence/Integrable.lean
new file mode 100644
index 0000000000000..c33b372d816a2
--- /dev/null
+++ b/Mathlib/Probability/Independence/Integrable.lean
@@ -0,0 +1,54 @@
+/-
+Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sébastien Gouëzel
+-/
+import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.Probability.Independence.Basic
+
+/-!
+# Independence of functions implies that the measure is a probability measure
+
+If a nonzero function belongs to `ℒ^p` (in particular if it is integrable) and is independent
+of another function, then the space is a probability space.
+
+-/
+
+open Filter ProbabilityTheory
+
+open scoped ENNReal NNReal Topology
+
+namespace MeasureTheory
+
+variable {Ω E F: Type*} [MeasurableSpace Ω] {μ : Measure Ω}
+  [NormedAddCommGroup E] [MeasurableSpace E] [BorelSpace E]
+  [MeasurableSpace F]
+
+/-- If a nonzero function belongs to `ℒ^p` and is independent of another function, then
+the space is a probability space. -/
+lemma Memℒp.isProbabilityMeasure_of_indepFun
+    (f : Ω → E) (g : Ω → F) {p : ℝ≥0∞} (hp : p ≠ 0) (hp' : p ≠ ∞)
+    (hℒp : Memℒp f p μ) (h'f : ¬ (∀ᵐ ω ∂μ, f ω = 0)) (hindep : IndepFun f g μ) :
+    IsProbabilityMeasure μ := by
+  obtain ⟨c, c_pos, hc⟩ : ∃ (c : ℝ≥0), 0 < c ∧ 0 < μ {ω | c ≤ ‖f ω‖₊} := by
+    contrapose! h'f
+    have A (c : ℝ≥0) (hc : 0 < c) : ∀ᵐ ω ∂μ, ‖f ω‖₊ < c := by simpa [ae_iff] using h'f c hc
+    obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), 0 < u n)
+      ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto (0 : ℝ≥0)
+    filter_upwards [ae_all_iff.2 (fun n ↦ A (u n) (u_pos n))] with ω hω
+    simpa using ge_of_tendsto' u_lim (fun i ↦ (hω i).le)
+  have h'c : μ {ω | c ≤ ‖f ω‖₊} < ∞ := hℒp.meas_ge_lt_top hp hp' c_pos.ne'
+  have := hindep.measure_inter_preimage_eq_mul {x | c ≤ ‖x‖₊} Set.univ
+    (isClosed_le continuous_const continuous_nnnorm).measurableSet MeasurableSet.univ
+  simp only [Set.preimage_setOf_eq, Set.preimage_univ, Set.inter_univ] at this
+  exact ⟨(ENNReal.mul_eq_left hc.ne' h'c.ne).1 this.symm⟩
+
+/-- If a nonzero function is integrable and is independent of another function, then
+the space is a probability space. -/
+lemma Integrable.isProbabilityMeasure_of_indepFun (f : Ω → E) (g : Ω → F)
+    (hf : Integrable f μ) (h'f : ¬ (∀ᵐ ω ∂μ, f ω = 0)) (hindep : IndepFun f g μ) :
+    IsProbabilityMeasure μ :=
+  Memℒp.isProbabilityMeasure_of_indepFun f g one_ne_zero ENNReal.one_ne_top
+    (memℒp_one_iff_integrable.mpr hf) h'f hindep
+
+end MeasureTheory
diff --git a/Mathlib/Probability/Independence/Kernel.lean b/Mathlib/Probability/Independence/Kernel.lean
index c07e53b9ec329..146667d5bfbdb 100644
--- a/Mathlib/Probability/Independence/Kernel.lean
+++ b/Mathlib/Probability/Independence/Kernel.lean
@@ -5,6 +5,7 @@ Authors: Rémy Degenne
 -/
 import Mathlib.MeasureTheory.Constructions.Pi
 import Mathlib.Probability.Kernel.Basic
+import Mathlib.Tactic.Peel
 
 /-!
 # Independence with respect to a kernel and a measure
@@ -118,13 +119,98 @@ section ByDefinition
 
 variable {β : ι → Type*} {mβ : ∀ i, MeasurableSpace (β i)}
   {_mα : MeasurableSpace α} {m : ι → MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
-  {κ : Kernel α Ω} {μ : Measure α}
+  {κ η : Kernel α Ω} {μ : Measure α}
   {π : ι → Set (Set Ω)} {s : ι → Set Ω} {S : Finset ι} {f : ∀ x : ι, Ω → β x}
+  {s1 s2 : Set (Set Ω)}
+
+@[simp] lemma iIndepSets_zero_right : iIndepSets π κ 0 := by simp [iIndepSets]
+
+@[simp] lemma indepSets_zero_right : IndepSets s1 s2 κ 0 := by simp [IndepSets]
+
+@[simp] lemma indepSets_zero_left : IndepSets s1 s2 (0 : Kernel α Ω) μ := by simp [IndepSets]
+
+@[simp] lemma iIndep_zero_right : iIndep m κ 0 := by simp [iIndep]
+
+@[simp] lemma indep_zero_right {m₁ m₂ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
+    {κ : Kernel α Ω} : Indep m₁ m₂ κ 0 := by simp [Indep]
+
+@[simp] lemma indep_zero_left {m₁ m₂ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω} :
+    Indep m₁ m₂ (0 : Kernel α Ω) μ  := by simp [Indep]
+
+@[simp] lemma iIndepSet_zero_right : iIndepSet s κ 0 := by simp [iIndepSet]
+
+@[simp] lemma indepSet_zero_right {s t : Set Ω} : IndepSet s t κ 0 := by simp [IndepSet]
+
+@[simp] lemma indepSet_zero_left {s t : Set Ω} : IndepSet s t (0 : Kernel α Ω) μ := by
+  simp [IndepSet]
+
+@[simp] lemma iIndepFun_zero_right {β : ι → Type*} {m : ∀ x : ι, MeasurableSpace (β x)}
+    {f : ∀ x : ι, Ω → β x} : iIndepFun m f κ 0 := by simp [iIndepFun]
+
+@[simp] lemma indepFun_zero_right {β γ} [MeasurableSpace β] [MeasurableSpace γ]
+    {f : Ω → β} {g : Ω → γ} : IndepFun f g κ 0 := by simp [IndepFun]
+
+@[simp] lemma indepFun_zero_left {β γ} [MeasurableSpace β] [MeasurableSpace γ]
+    {f : Ω → β} {g : Ω → γ} : IndepFun f g (0 : Kernel α Ω) μ := by simp [IndepFun]
+
+lemma iIndepSets_congr (h : κ =ᵐ[μ] η) : iIndepSets π κ μ ↔ iIndepSets π η μ := by
+  peel 3
+  refine ⟨fun h' ↦ ?_, fun h' ↦ ?_⟩ <;>
+  · filter_upwards [h, h'] with a ha h'a
+    simpa [ha] using h'a
+
+alias ⟨iIndepSets.congr, _⟩ := iIndepSets_congr
+
+lemma indepSets_congr (h : κ =ᵐ[μ] η) : IndepSets s1 s2 κ μ ↔ IndepSets s1 s2 η μ := by
+  peel 4
+  refine ⟨fun h' ↦ ?_, fun h' ↦ ?_⟩ <;>
+  · filter_upwards [h, h'] with a ha h'a
+    simpa [ha] using h'a
+
+alias ⟨IndepSets.congr, _⟩ := indepSets_congr
+
+lemma iIndep_congr (h : κ =ᵐ[μ] η) : iIndep m κ μ ↔ iIndep m η μ :=
+  iIndepSets_congr h
+
+alias ⟨iIndep.congr, _⟩ := iIndep_congr
+
+lemma indep_congr {m₁ m₂ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
+    {κ η : Kernel α Ω} (h : κ =ᵐ[μ] η) : Indep m₁ m₂ κ μ ↔ Indep m₁ m₂ η μ :=
+  indepSets_congr h
+
+alias ⟨Indep.congr, _⟩ := indep_congr
+
+lemma iIndepSet_congr (h : κ =ᵐ[μ] η) : iIndepSet s κ μ ↔ iIndepSet s η μ :=
+  iIndep_congr h
+
+alias ⟨iIndepSet.congr, _⟩ := iIndepSet_congr
+
+lemma indepSet_congr {s t : Set Ω} (h : κ =ᵐ[μ] η) : IndepSet s t κ μ ↔ IndepSet s t η μ :=
+  indep_congr h
+
+alias ⟨indepSet.congr, _⟩ := indepSet_congr
+
+lemma iIndepFun_congr {β : ι → Type*} {m : ∀ x : ι, MeasurableSpace (β x)}
+    {f : ∀ x : ι, Ω → β x} (h : κ =ᵐ[μ] η) : iIndepFun m f κ μ ↔ iIndepFun m f η μ :=
+  iIndep_congr h
+
+alias ⟨iIndepFun.congr, _⟩ := iIndepFun_congr
+
+lemma indepFun_congr {β γ} [MeasurableSpace β] [MeasurableSpace γ]
+    {f : Ω → β} {g : Ω → γ} (h : κ =ᵐ[μ] η) : IndepFun f g κ μ ↔ IndepFun f g η μ :=
+  indep_congr h
+
+alias ⟨IndepFun.congr, _⟩ := indepFun_congr
 
 lemma iIndepSets.meas_biInter (h : iIndepSets π κ μ) (s : Finset ι)
     {f : ι → Set Ω} (hf : ∀ i, i ∈ s → f i ∈ π i) :
     ∀ᵐ a ∂μ, κ a (⋂ i ∈ s, f i) = ∏ i ∈ s, κ a (f i) := h s hf
 
+lemma iIndepSets.ae_isProbabilityMeasure (h : iIndepSets π κ μ) :
+    ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a) := by
+  filter_upwards [h.meas_biInter ∅ (f := fun _ ↦ Set.univ) (by simp)] with a ha
+  exact ⟨by simpa using ha⟩
+
 lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π κ μ) (hs : ∀ i, s i ∈ π i) :
     ∀ᵐ a ∂μ, κ a (⋂ i, s i) = ∏ i, κ a (s i) := by
   filter_upwards [h.meas_biInter Finset.univ (fun _i _ ↦ hs _)] with a ha using by simp [← ha]
@@ -132,6 +218,10 @@ lemma iIndepSets.meas_iInter [Fintype ι] (h : iIndepSets π κ μ) (hs : ∀ i,
 lemma iIndep.iIndepSets' (hμ : iIndep m κ μ) :
     iIndepSets (fun x ↦ {s | MeasurableSet[m x] s}) κ μ := hμ
 
+lemma iIndep.ae_isProbabilityMeasure (h : iIndep m κ μ) :
+    ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a) :=
+  h.iIndepSets'.ae_isProbabilityMeasure
+
 lemma iIndep.meas_biInter (hμ : iIndep m κ μ) (hs : ∀ i, i ∈ S → MeasurableSet[m i] (s i)) :
     ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, s i) = ∏ i ∈ S, κ a (s i) := hμ _ hs
 
@@ -143,6 +233,10 @@ lemma iIndep.meas_iInter [Fintype ι] (h : iIndep m κ μ) (hs : ∀ i, Measurab
 protected lemma iIndepFun.iIndep (hf : iIndepFun mβ f κ μ) :
     iIndep (fun x ↦ (mβ x).comap (f x)) κ μ := hf
 
+lemma iIndepFun.ae_isProbabilityMeasure (h : iIndepFun mβ f κ μ) :
+    ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a) :=
+  h.iIndep.ae_isProbabilityMeasure
+
 lemma iIndepFun.meas_biInter (hf : iIndepFun mβ f κ μ)
     (hs : ∀ i, i ∈ S → MeasurableSet[(mβ i).comap (f i)] (s i)) :
     ∀ᵐ a ∂μ, κ a (⋂ i ∈ S, s i) = ∏ i ∈ S, κ a (s i) := hf.iIndep.meas_biInter hs
@@ -177,27 +271,29 @@ theorem Indep.symm {m₁ m₂ : MeasurableSpace Ω} {_mΩ : MeasurableSpace Ω}
   IndepSets.symm h
 
 theorem indep_bot_right (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] :
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] :
     Indep m' ⊥ κ μ := by
   intros s t _ ht
   rw [Set.mem_setOf_eq, MeasurableSpace.measurableSet_bot_iff] at ht
+  rcases eq_zero_or_isMarkovKernel κ with rfl| h
+  · simp
   refine Filter.Eventually.of_forall (fun a ↦ ?_)
   cases' ht with ht ht
   · rw [ht, Set.inter_empty, measure_empty, mul_zero]
   · rw [ht, Set.inter_univ, measure_univ, mul_one]
 
 theorem indep_bot_left (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] :
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] :
     Indep ⊥ m' κ μ := (indep_bot_right m').symm
 
 theorem indepSet_empty_right {_mΩ : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] (s : Set Ω) :
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] (s : Set Ω) :
     IndepSet s ∅ κ μ := by
   simp only [IndepSet, generateFrom_singleton_empty]
   exact indep_bot_right _
 
 theorem indepSet_empty_left {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
-    {μ : Measure α} [IsMarkovKernel κ] (s : Set Ω) :
+    {μ : Measure α} [IsZeroOrMarkovKernel κ] (s : Set Ω) :
     IndepSet ∅ s κ μ :=
   (indepSet_empty_right s).symm
 
@@ -376,10 +472,12 @@ section FromPiSystemsToMeasurableSpaces
 variable {_mα : MeasurableSpace α}
 
 theorem IndepSets.indep_aux {m₂ m : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h2 : m₂ ≤ m)
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h2 : m₂ ≤ m)
     (hp2 : IsPiSystem p2) (hpm2 : m₂ = generateFrom p2) (hyp : IndepSets p1 p2 κ μ) {t1 t2 : Set Ω}
     (ht1 : t1 ∈ p1) (ht1m : MeasurableSet[m] t1) (ht2m : MeasurableSet[m₂] t2) :
     ∀ᵐ a ∂μ, κ a (t1 ∩ t2) = κ a t1 * κ a t2 := by
+  rcases eq_zero_or_isMarkovKernel κ with rfl | h
+  · simp
   refine @induction_on_inter _ (fun t ↦ ∀ᵐ a ∂μ, κ a (t1 ∩ t) = κ a t1 * κ a t) _
     m₂ hpm2 hp2 ?_ ?_ ?_ ?_ t2 ht2m
   · simp only [Set.inter_empty, measure_empty, mul_zero, eq_self_iff_true,
@@ -390,7 +488,8 @@ theorem IndepSets.indep_aux {m₂ m : MeasurableSpace Ω}
     have : t1 ∩ tᶜ = t1 \ (t1 ∩ t) := by
       rw [Set.diff_self_inter, Set.diff_eq_compl_inter, Set.inter_comm]
     rw [this,
-      measure_diff Set.inter_subset_left (ht1m.inter (h2 _ ht)) (measure_ne_top (κ a) _),
+      measure_diff Set.inter_subset_left (ht1m.inter (h2 _ ht)).nullMeasurableSet
+        (measure_ne_top (κ a) _),
       measure_compl (h2 _ ht) (measure_ne_top (κ a) t), measure_univ,
       ENNReal.mul_sub (fun _ _ ↦ measure_ne_top (κ a) _), mul_one, ha]
   · intros f hf_disj hf_meas h
@@ -408,10 +507,12 @@ theorem IndepSets.indep_aux {m₂ m : MeasurableSpace Ω}
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
 theorem IndepSets.indep {m1 m2 m : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α}
-    [IsMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
+    [IsZeroOrMarkovKernel κ] {p1 p2 : Set (Set Ω)} (h1 : m1 ≤ m) (h2 : m2 ≤ m) (hp1 : IsPiSystem p1)
     (hp2 : IsPiSystem p2) (hpm1 : m1 = generateFrom p1) (hpm2 : m2 = generateFrom p2)
     (hyp : IndepSets p1 p2 κ μ) :
     Indep m1 m2 κ μ := by
+  rcases eq_zero_or_isMarkovKernel κ with rfl | h
+  · simp
   intros t1 t2 ht1 ht2
   refine @induction_on_inter _ (fun t ↦ ∀ᵐ (a : α) ∂μ, κ a (t ∩ t2) = κ a t * κ a t2) _ m1 hpm1 hp1
     ?_ ?_ ?_ ?_ _ ht1
@@ -428,7 +529,7 @@ theorem IndepSets.indep {m1 m2 m : MeasurableSpace Ω} {κ : Kernel α Ω} {μ :
     have : tᶜ ∩ t2 = t2 \ (t ∩ t2) := by
       rw [Set.inter_comm t, Set.diff_self_inter, Set.diff_eq_compl_inter]
     rw [this, Set.inter_comm t t2,
-      measure_diff Set.inter_subset_left ((h2 _ ht2).inter (h1 _ ht))
+      measure_diff Set.inter_subset_left ((h2 _ ht2).inter (h1 _ ht)).nullMeasurableSet
         (measure_ne_top (κ a) _),
       Set.inter_comm, ha, measure_compl (h1 _ ht) (measure_ne_top (κ a) t), measure_univ,
       mul_comm (1 - κ a t), ENNReal.mul_sub (fun _ _ ↦ measure_ne_top (κ a) _), mul_one, mul_comm]
@@ -446,7 +547,7 @@ theorem IndepSets.indep {m1 m2 m : MeasurableSpace Ω} {κ : Kernel α Ω} {μ :
     · exact fun i ↦ (h2 _ ht2).inter (h1 _ (hf_meas i))
 
 theorem IndepSets.indep' {_mΩ : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ]
     {p1 p2 : Set (Set Ω)} (hp1m : ∀ s ∈ p1, MeasurableSet s) (hp2m : ∀ s ∈ p2, MeasurableSet s)
     (hp1 : IsPiSystem p1) (hp2 : IsPiSystem p2) (hyp : IndepSets p1 p2 κ μ) :
     Indep (generateFrom p1) (generateFrom p2) κ μ :=
@@ -454,7 +555,7 @@ theorem IndepSets.indep' {_mΩ : MeasurableSpace Ω}
 
 variable {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α}
 
-theorem indepSets_piiUnionInter_of_disjoint [IsMarkovKernel κ] {s : ι → Set (Set Ω)}
+theorem indepSets_piiUnionInter_of_disjoint {s : ι → Set (Set Ω)}
     {S T : Set ι} (h_indep : iIndepSets s κ μ) (hST : Disjoint S T) :
     IndepSets (piiUnionInter s S) (piiUnionInter s T) κ μ := by
   rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩
@@ -481,7 +582,8 @@ theorem indepSets_piiUnionInter_of_disjoint [IsMarkovKernel κ] {s : ι → Set
           ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩
     filter_upwards [h_indep _ hgm] with a ha
     rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← ha]
-  filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m] with a h_P_inter ha1 ha2
+  filter_upwards [h_P_inter, h_indep p1 ht1_m, h_indep p2 ht2_m, h_indep.ae_isProbabilityMeasure]
+    with a h_P_inter ha1 ha2 h'
   have h_μg : ∀ n, κ a (g n) = (ite (n ∈ p1) (κ a (f1 n)) 1) * (ite (n ∈ p2) (κ a (f2 n)) 1) := by
     intro n
     dsimp only [g]
@@ -493,9 +595,15 @@ theorem indepSets_piiUnionInter_of_disjoint [IsMarkovKernel κ] {s : ι → Set
     Finset.prod_ite_mem (p1 ∪ p2) p2 (fun x => κ a (f2 x)), Finset.union_inter_cancel_right, ht1_eq,
       ← ha1, ht2_eq, ← ha2]
 
-theorem iIndepSet.indep_generateFrom_of_disjoint [IsMarkovKernel κ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_of_disjoint {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (S T : Set ι) (hST : Disjoint S T) :
     Indep (generateFrom { t | ∃ n ∈ S, s n = t }) (generateFrom { t | ∃ k ∈ T, s k = t }) κ μ := by
+  classical
+  rcases eq_or_ne μ 0 with rfl | hμ
+  · simp
+  obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
+    exists_ae_eq_isMarkovKernel hs.ae_isProbabilityMeasure hμ
+  apply Indep.congr (Filter.EventuallyEq.symm η_eq)
   rw [← generateFrom_piiUnionInter_singleton_left, ← generateFrom_piiUnionInter_singleton_left]
   refine
     IndepSets.indep'
@@ -505,21 +613,27 @@ theorem iIndepSet.indep_generateFrom_of_disjoint [IsMarkovKernel κ] {s : ι →
   · exact fun k => generateFrom_le fun t ht => (Set.mem_singleton_iff.1 ht).symm ▸ hsm k
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
   · exact isPiSystem_piiUnionInter _ (fun k => IsPiSystem.singleton _) _
-  · classical exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) hs) hST
+  · exact indepSets_piiUnionInter_of_disjoint (iIndep.iIndepSets (fun n => rfl) (hs.congr η_eq)) hST
 
-theorem indep_iSup_of_disjoint [IsMarkovKernel κ] {m : ι → MeasurableSpace Ω}
+theorem indep_iSup_of_disjoint {m : ι → MeasurableSpace Ω}
     (h_le : ∀ i, m i ≤ _mΩ) (h_indep : iIndep m κ μ) {S T : Set ι} (hST : Disjoint S T) :
     Indep (⨆ i ∈ S, m i) (⨆ i ∈ T, m i) κ μ := by
+  classical
+  rcases eq_or_ne μ 0 with rfl | hμ
+  · simp
+  obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
+    exists_ae_eq_isMarkovKernel h_indep.ae_isProbabilityMeasure hμ
+  apply Indep.congr (Filter.EventuallyEq.symm η_eq)
   refine
     IndepSets.indep (iSup₂_le fun i _ => h_le i) (iSup₂_le fun i _ => h_le i) ?_ ?_
       (generateFrom_piiUnionInter_measurableSet m S).symm
       (generateFrom_piiUnionInter_measurableSet m T).symm ?_
   · exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _
   · exact isPiSystem_piiUnionInter _ (fun n => @isPiSystem_measurableSet Ω (m n)) _
-  · classical exact indepSets_piiUnionInter_of_disjoint h_indep hST
+  · exact indepSets_piiUnionInter_of_disjoint (h_indep.congr η_eq) hST
 
 theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 : MeasurableSpace Ω}
-    {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] (h_indep : ∀ i, Indep (m i) m' κ μ)
+    {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] (h_indep : ∀ i, Indep (m i) m' κ μ)
     (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0) (hm : Directed (· ≤ ·) m) :
     Indep (⨆ i, m i) m' κ μ := by
   let p : ι → Set (Set Ω) := fun n => { t | MeasurableSet[m n] t }
@@ -541,34 +655,34 @@ theorem indep_iSup_of_directed_le {Ω} {m : ι → MeasurableSpace Ω} {m' m0 :
   refine IndepSets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi ?_ h_gen' h_pi_system_indep
   exact (generateFrom_iUnion_measurableSet _).symm
 
-theorem iIndepSet.indep_generateFrom_lt [Preorder ι] [IsMarkovKernel κ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_lt [Preorder ι] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (i : ι) :
     Indep (generateFrom {s i}) (generateFrom { t | ∃ j < i, s j = t }) κ μ := by
   convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {i} { j | j < i }
-    (Set.disjoint_singleton_left.mpr (lt_irrefl _))
+    (Set.disjoint_singleton_left.mpr (lt_irrefl _)) using 1
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
 
-theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] [IsMarkovKernel κ] {s : ι → Set Ω}
+theorem iIndepSet.indep_generateFrom_le [LinearOrder ι] {s : ι → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (i : ι) {k : ι} (hk : i < k) :
     Indep (generateFrom {s k}) (generateFrom { t | ∃ j ≤ i, s j = t }) κ μ := by
   convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j | j ≤ i }
-      (Set.disjoint_singleton_left.mpr hk.not_le)
+      (Set.disjoint_singleton_left.mpr hk.not_le) using 1
   simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']
 
-theorem iIndepSet.indep_generateFrom_le_nat [IsMarkovKernel κ] {s : ℕ → Set Ω}
+theorem iIndepSet.indep_generateFrom_le_nat {s : ℕ → Set Ω}
     (hsm : ∀ n, MeasurableSet (s n)) (hs : iIndepSet s κ μ) (n : ℕ) :
     Indep (generateFrom {s (n + 1)}) (generateFrom { t | ∃ k ≤ n, s k = t }) κ μ :=
   iIndepSet.indep_generateFrom_le hsm hs _ n.lt_succ_self
 
 theorem indep_iSup_of_monotone [SemilatticeSup ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
+    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ]
     (h_indep : ∀ i, Indep (m i) m' κ μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Monotone m) :
     Indep (⨆ i, m i) m' κ μ :=
   indep_iSup_of_directed_le h_indep h_le h_le' (Monotone.directed_le hm)
 
 theorem indep_iSup_of_antitone [SemilatticeInf ι] {Ω} {m : ι → MeasurableSpace Ω}
-    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsMarkovKernel κ]
+    {m' m0 : MeasurableSpace Ω} {κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ]
     (h_indep : ∀ i, Indep (m i) m' κ μ) (h_le : ∀ i, m i ≤ m0) (h_le' : m' ≤ m0)
     (hm : Antitone m) :
     Indep (⨆ i, m i) m' κ μ :=
@@ -612,11 +726,16 @@ theorem iIndepSets.piiUnionInter_of_not_mem {π : ι → Set (Set Ω)} {a : ι}
   rw [ha2, Finset.prod_insert has, h_t2, mul_comm, ha1]
 
 /-- The measurable space structures generated by independent pi-systems are independent. -/
-theorem iIndepSets.iIndep [IsMarkovKernel κ] (m : ι → MeasurableSpace Ω)
+theorem iIndepSets.iIndep (m : ι → MeasurableSpace Ω)
     (h_le : ∀ i, m i ≤ _mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
     (h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π κ μ) :
     iIndep m κ μ := by
   classical
+  rcases eq_or_ne μ 0 with rfl | hμ
+  · simp
+  obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
+    exists_ae_eq_isMarkovKernel h_ind.ae_isProbabilityMeasure hμ
+  apply iIndep.congr (Filter.EventuallyEq.symm η_eq)
   intro s f
   refine Finset.induction ?_ ?_ s
   · simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,
@@ -626,12 +745,12 @@ theorem iIndepSets.iIndep [IsMarkovKernel κ] (m : ι → MeasurableSpace Ω)
     have hf_m_S : ∀ x ∈ S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])
     let p := piiUnionInter π S
     set m_p := generateFrom p with hS_eq_generate
-    have h_indep : Indep m_p (m a) κ μ := by
+    have h_indep : Indep m_p (m a) η μ := by
       have hp : IsPiSystem p := isPiSystem_piiUnionInter π h_pi S
       have h_le' : ∀ i, generateFrom (π i) ≤ _mΩ := fun i ↦ (h_generate i).symm.trans_le (h_le i)
       have hm_p : m_p ≤ _mΩ := generateFrom_piiUnionInter_le π h_le' S
       exact IndepSets.indep hm_p (h_le a) hp (h_pi a) hS_eq_generate (h_generate a)
-        (iIndepSets.piiUnionInter_of_not_mem h_ind ha_notin_S)
+        (iIndepSets.piiUnionInter_of_not_mem (h_ind.congr η_eq) ha_notin_S)
     have h := h_indep.symm (f a) (⋂ n ∈ S, f n) (hf_m a (Finset.mem_insert_self a S)) ?_
     · filter_upwards [h_rec hf_m_S, h] with a' ha' h'
       rwa [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← ha']
@@ -657,8 +776,7 @@ We prove the following equivalences on `IndepSet`, for measurable sets `s, t`.
 variable {_mα : MeasurableSpace α}
 
 theorem iIndepSet_iff_iIndepSets_singleton {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
-    [IsMarkovKernel κ] {μ : Measure α} {f : ι → Set Ω}
-    (hf : ∀ i, MeasurableSet (f i)) :
+    {μ : Measure α} {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
     iIndepSet f κ μ ↔ iIndepSets (fun i ↦ {f i}) κ μ :=
   ⟨iIndep.iIndepSets fun _ ↦ rfl,
     iIndepSets.iIndep _ (fun i ↦ generateFrom_le <| by rintro t (rfl : t = _); exact hf _) _
@@ -670,14 +788,13 @@ theorem iIndepSet.meas_biInter {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
   iIndep.iIndepSets (fun _ ↦ rfl) h _ (by simp)
 
 theorem iIndepSet_iff_meas_biInter {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
-    [IsMarkovKernel κ] {μ : Measure α} {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
+    {μ : Measure α} {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
     iIndepSet f κ μ ↔ ∀ s, ∀ᵐ a ∂μ, κ a (⋂ i ∈ s, f i) = ∏ i ∈ s, κ a (f i) :=
   (iIndepSet_iff_iIndepSets_singleton hf).trans iIndepSets_singleton_iff
 
 theorem iIndepSets.iIndepSet_of_mem {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
-    [IsMarkovKernel κ] {μ : Measure α} {π : ι → Set (Set Ω)} {f : ι → Set Ω}
-    (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i))
-    (hπ : iIndepSets π κ μ) :
+    {μ : Measure α} {π : ι → Set (Set Ω)} {f : ι → Set Ω}
+    (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i)) (hπ : iIndepSets π κ μ) :
     iIndepSet f κ μ :=
   (iIndepSet_iff_meas_biInter hf).2 fun _t ↦ hπ.meas_biInter _ fun _i _ ↦ hfπ _
 
@@ -685,7 +802,7 @@ variable {s t : Set Ω} (S T : Set (Set Ω))
 
 theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (κ : Kernel α Ω) (μ : Measure α)
-    [IsMarkovKernel κ] :
+    [IsZeroOrMarkovKernel κ] :
     IndepSet s t κ μ ↔ IndepSets {s} {t} κ μ :=
   ⟨Indep.indepSets, fun h =>
     IndepSets.indep
@@ -695,7 +812,7 @@ theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : Me
 
 theorem indepSet_iff_measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (hs_meas : MeasurableSet s)
     (ht_meas : MeasurableSet t) (κ : Kernel α Ω) (μ : Measure α)
-    [IsMarkovKernel κ] :
+    [IsZeroOrMarkovKernel κ] :
     IndepSet s t κ μ ↔ ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t :=
   (indepSet_iff_indepSets_singleton hs_meas ht_meas κ μ).trans indepSets_singleton_iff
 
@@ -705,7 +822,7 @@ theorem IndepSet.measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (κ : Kernel α
 
 theorem IndepSets.indepSet_of_mem {_m0 : MeasurableSpace Ω} (hs : s ∈ S) (ht : t ∈ T)
     (hs_meas : MeasurableSet s) (ht_meas : MeasurableSet t)
-    (κ : Kernel α Ω) (μ : Measure α) [IsMarkovKernel κ]
+    (κ : Kernel α Ω) (μ : Measure α) [IsZeroOrMarkovKernel κ]
     (h_indep : IndepSets S T κ μ) :
     IndepSet s t κ μ :=
   (indepSet_iff_measure_inter_eq_mul hs_meas ht_meas κ μ).mpr (h_indep s t hs ht)
@@ -791,7 +908,7 @@ theorem iIndepFun_iff_measure_inter_preimage_eq_mul {ι : Type*} {β : ι → Ty
 alias ⟨iIndepFun.measure_inter_preimage_eq_mul, _⟩ := iIndepFun_iff_measure_inter_preimage_eq_mul
 
 theorem indepFun_iff_indepSet_preimage {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'}
-    [IsMarkovKernel κ] (hf : Measurable f) (hg : Measurable g) :
+    [IsZeroOrMarkovKernel κ] (hf : Measurable f) (hg : Measurable g) :
     IndepFun f g κ μ ↔
       ∀ s t, MeasurableSet s → MeasurableSet t → IndepSet (f ⁻¹' s) (g ⁻¹' t) κ μ := by
   refine indepFun_iff_measure_inter_preimage_eq_mul.trans ?_
@@ -841,16 +958,17 @@ lemma iIndepFun.of_subsingleton [IsMarkovKernel κ] [Subsingleton ι] : iIndepFu
   · have : s = {x} := by ext y; simp [Subsingleton.elim y x, hx]
     simp [this]
 
-lemma iIndepFun.ae_isProbabilityMeasure (h : iIndepFun m f κ μ) :
-    ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a) := by
-  simpa [isProbabilityMeasure_iff] using h.meas_biInter (S := ∅) (s := fun _ ↦ Set.univ)
-
 /-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
 two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
 tuple `(f i)_i` for `i ∈ T`. -/
-theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : Disjoint S T)
+theorem iIndepFun.indepFun_finset (S T : Finset ι) (hST : Disjoint S T)
     (hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
     IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ := by
+  rcases eq_or_ne μ 0 with rfl | hμ
+  · simp
+  obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μ] η ∧ IsMarkovKernel η :=
+    exists_ae_eq_isMarkovKernel hf_Indep.ae_isProbabilityMeasure hμ
+  apply IndepFun.congr (Filter.EventuallyEq.symm η_eq)
   -- We introduce π-systems, built from the π-system of boxes which generates `MeasurableSpace.pi`.
   let πSβ := Set.pi (Set.univ : Set S) ''
     Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | MeasurableSet[m i] s }
@@ -902,6 +1020,7 @@ theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : D
     constructor <;> intro h
     · intro i hi; simp_rw [sets_t', dif_pos hi]; exact h ⟨i, hi⟩
     · rintro ⟨i, hi⟩; specialize h i hi; simp_rw [sets_t', dif_pos hi] at h; exact h
+  replace hf_Indep := hf_Indep.congr η_eq
   rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf_Indep
   have h_Inter_inter :
     ((⋂ i ∈ S, f i ⁻¹' sets_s' i) ∩ ⋂ i ∈ T, f i ⁻¹' sets_t' i) =
@@ -938,7 +1057,7 @@ theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : D
   · refine Finset.prod_congr rfl fun i hi => ?_
     rw [h_sets_s'_univ hi, Set.univ_inter]
 
-theorem iIndepFun.indepFun_prod_mk [IsMarkovKernel κ] (hf_Indep : iIndepFun m f κ μ)
+theorem iIndepFun.indepFun_prod_mk (hf_Indep : iIndepFun m f κ μ)
     (hf_meas : ∀ i, Measurable (f i)) (i j k : ι) (hik : i ≠ k) (hjk : j ≠ k) :
     IndepFun (fun a => (f i a, f j a)) (f k) κ μ := by
   classical
@@ -965,7 +1084,7 @@ theorem iIndepFun.indepFun_prod_mk [IsMarkovKernel κ] (hf_Indep : iIndepFun m f
   exact ⟨hik.symm, hjk.symm⟩
 
 open Finset in
-lemma iIndepFun.indepFun_prod_mk_prod_mk [IsMarkovKernel κ] (hf_indep : iIndepFun m f κ μ)
+lemma iIndepFun.indepFun_prod_mk_prod_mk (hf_indep : iIndepFun m f κ μ)
     (hf_meas : ∀ i, Measurable (f i))
     (i j k l : ι) (hik : i ≠ k) (hil : i ≠ l) (hjk : j ≠ k) (hjl : j ≠ l) :
     IndepFun (fun a ↦ (f i a, f j a)) (fun a ↦ (f k a, f l a)) κ μ := by
@@ -979,7 +1098,6 @@ end iIndepFun
 
 section Mul
 variable {β : Type*} {m : MeasurableSpace β} [Mul β] [MeasurableMul₂ β] {f : ι → Ω → β}
-  [IsMarkovKernel κ]
 
 @[to_additive]
 lemma iIndepFun.indepFun_mul_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
@@ -1007,7 +1125,6 @@ end Mul
 
 section Div
 variable {β : Type*} {m : MeasurableSpace β} [Div β] [MeasurableDiv₂ β] {f : ι → Ω → β}
-  [IsMarkovKernel κ]
 
 @[to_additive]
 lemma iIndepFun.indepFun_div_left (hf_indep : iIndepFun (fun _ ↦ m) f κ μ)
@@ -1035,7 +1152,6 @@ end Div
 
 section CommMonoid
 variable {β : Type*} {m : MeasurableSpace β} [CommMonoid β] [MeasurableMul₂ β] {f : ι → Ω → β}
-  [IsMarkovKernel κ]
 
 @[to_additive]
 theorem iIndepFun.indepFun_finset_prod_of_not_mem (hf_Indep : iIndepFun (fun _ ↦ m) f κ μ)
diff --git a/Mathlib/Probability/Independence/ZeroOne.lean b/Mathlib/Probability/Independence/ZeroOne.lean
index 417ff18e8e79b..06403152bcee2 100644
--- a/Mathlib/Probability/Independence/ZeroOne.lean
+++ b/Mathlib/Probability/Independence/ZeroOne.lean
@@ -46,12 +46,17 @@ theorem measure_eq_zero_or_one_or_top_of_indepSet_self {t : Set Ω}
   simpa only [ae_dirac_eq, Filter.eventually_pure]
     using Kernel.measure_eq_zero_or_one_or_top_of_indepSet_self h_indep
 
-theorem Kernel.measure_eq_zero_or_one_of_indepSet_self [∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
-    (h_indep : IndepSet t t κ μα) :
+theorem Kernel.measure_eq_zero_or_one_of_indepSet_self' (h : ∀ᵐ a ∂μα, IsFiniteMeasure (κ a))
+    {t : Set Ω} (h_indep : IndepSet t t κ μα) :
     ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by
-  filter_upwards [measure_eq_zero_or_one_or_top_of_indepSet_self h_indep] with a h_0_1_top
+  filter_upwards [measure_eq_zero_or_one_or_top_of_indepSet_self h_indep, h] with a h_0_1_top h'
   simpa only [measure_ne_top (κ a), or_false] using h_0_1_top
 
+theorem Kernel.measure_eq_zero_or_one_of_indepSet_self [h : ∀ a, IsFiniteMeasure (κ a)] {t : Set Ω}
+    (h_indep : IndepSet t t κ μα) :
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
+  Kernel.measure_eq_zero_or_one_of_indepSet_self' (ae_of_all μα h) h_indep
+
 theorem measure_eq_zero_or_one_of_indepSet_self [IsFiniteMeasure μ] {t : Set Ω}
     (h_indep : IndepSet t t μ) : μ t = 0 ∨ μ t = 1 := by
   simpa only [ae_dirac_eq, Filter.eventually_pure]
@@ -71,16 +76,13 @@ theorem condexp_eq_zero_or_one_of_condIndepSet_self
   | inl h => exact Or.inl (Or.inl h)
   | inr h => exact Or.inr h
 
-variable [IsMarkovKernel κ]
-
 open Filter
 
 theorem Kernel.indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα) (t : Set ι) :
     Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) κ μα :=
   indep_iSup_of_disjoint h_le h_indep disjoint_compl_right
 
-theorem indep_biSup_compl [IsProbabilityMeasure μ]
-    (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
+theorem indep_biSup_compl (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (t : Set ι) :
     Indep (⨆ n ∈ t, s n) (⨆ n ∈ tᶜ, s n) μ :=
   Kernel.indep_biSup_compl h_le h_indep t
 
@@ -92,7 +94,7 @@ theorem condIndep_biSup_compl [StandardBorelSpace Ω]
 
 section Abstract
 
-variable {α : Type*} {p : Set ι → Prop} {f : Filter ι} {ns : α → Set ι}
+variable {β : Type*} {p : Set ι → Prop} {f : Filter ι} {ns : β → Set ι}
 
 /-! We prove a version of Kolmogorov's 0-1 law for the σ-algebra `limsup s f` where `f` is a filter
 for which we can define the following two functions:
@@ -112,7 +114,7 @@ theorem Kernel.indep_biSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s
   simp only [Set.mem_compl_iff, eventually_map]
   exact eventually_of_mem (hf t ht) le_iSup₂
 
-theorem indep_biSup_limsup [IsProbabilityMeasure μ]
+theorem indep_biSup_limsup
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     {t : Set ι} (ht : p t) :
     Indep (⨆ n ∈ t, s n) (limsup s f) μ :=
@@ -128,6 +130,12 @@ theorem condIndep_biSup_limsup [StandardBorelSpace Ω]
 theorem Kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s κ μα)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
     Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) κ μα := by
+  rcases eq_or_ne μα 0 with rfl | hμ
+  · simp
+  obtain ⟨η, η_eq, hη⟩ : ∃ (η : Kernel α Ω), κ =ᵐ[μα] η ∧ IsMarkovKernel η :=
+    exists_ae_eq_isMarkovKernel h_indep.ae_isProbabilityMeasure hμ
+  replace h_indep := h_indep.congr η_eq
+  apply Indep.congr (Filter.EventuallyEq.symm η_eq)
   apply indep_iSup_of_directed_le
   · exact fun a => indep_biSup_limsup h_le h_indep hf (hnsp a)
   · exact fun a => iSup₂_le fun n _ => h_le n
@@ -138,7 +146,7 @@ theorem Kernel.indep_iSup_directed_limsup (h_le : ∀ n, s n ≤ m0) (h_indep :
     · exact hc.1 hn
     · exact hc.2 hn
 
-theorem indep_iSup_directed_limsup [IsProbabilityMeasure μ]
+theorem indep_iSup_directed_limsup
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) :
     Indep (⨆ a, ⨆ n ∈ ns a, s n) (limsup s f) μ :=
@@ -160,11 +168,11 @@ theorem Kernel.indep_iSup_limsup (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s
     exact indep_iSup_directed_limsup h_le h_indep hf hns hnsp
   rw [iSup_comm]
   refine iSup_congr fun n => ?_
-  have h : ⨆ (i : α) (_ : n ∈ ns i), s n = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
-  haveI : Nonempty (∃ i : α, n ∈ ns i) := ⟨hns_univ n⟩
+  have h : ⨆ (i : β) (_ : n ∈ ns i), s n = ⨆ _ : ∃ i, n ∈ ns i, s n := by rw [iSup_exists]
+  haveI : Nonempty (∃ i : β, n ∈ ns i) := ⟨hns_univ n⟩
   rw [h, iSup_const]
 
-theorem indep_iSup_limsup [IsProbabilityMeasure μ]
+theorem indep_iSup_limsup
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
     Indep (⨆ n, s n) (limsup s f) μ :=
@@ -183,7 +191,7 @@ theorem Kernel.indep_limsup_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s
     Indep (limsup s f) (limsup s f) κ μα :=
   indep_of_indep_of_le_left (indep_iSup_limsup h_le h_indep hf hns hnsp hns_univ) limsup_le_iSup
 
-theorem indep_limsup_self [IsProbabilityMeasure μ]
+theorem indep_limsup_self
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) (hf : ∀ t, p t → tᶜ ∈ f)
     (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a)) (hns_univ : ∀ n, ∃ a, n ∈ ns a) :
     Indep (limsup s f) (limsup s f) μ :=
@@ -200,12 +208,13 @@ theorem Kernel.measure_zero_or_one_of_measurableSet_limsup (h_le : ∀ n, s n 
     (h_indep : iIndep s κ μα)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
     (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
-    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
-  measure_eq_zero_or_one_of_indepSet_self
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by
+  apply measure_eq_zero_or_one_of_indepSet_self' ?_
     ((indep_limsup_self h_le h_indep hf hns hnsp hns_univ).indepSet_of_measurableSet ht_tail
       ht_tail)
+  filter_upwards [h_indep.ae_isProbabilityMeasure] with a ha using by infer_instance
 
-theorem measure_zero_or_one_of_measurableSet_limsup [IsProbabilityMeasure μ]
+theorem measure_zero_or_one_of_measurableSet_limsup
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ)
     (hf : ∀ t, p t → tᶜ ∈ f) (hns : Directed (· ≤ ·) ns) (hnsp : ∀ a, p (ns a))
     (hns_univ : ∀ n, ∃ a, n ∈ ns a) {t : Set Ω} (ht_tail : MeasurableSet[limsup s f] t) :
@@ -249,8 +258,7 @@ theorem Kernel.indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIn
   · exact Monotone.directed_le fun i j hij k hki => le_trans hki hij
   · exact fun n => ⟨n, le_rfl⟩
 
-theorem indep_limsup_atTop_self [IsProbabilityMeasure μ]
-    (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
+theorem indep_limsup_atTop_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
     Indep (limsup s atTop) (limsup s atTop) μ :=
   Kernel.indep_limsup_atTop_self h_le h_indep
 
@@ -262,14 +270,15 @@ theorem condIndep_limsup_atTop_self [StandardBorelSpace Ω]
 
 theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atTop (h_le : ∀ n, s n ≤ m0)
     (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
-    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
-  measure_eq_zero_or_one_of_indepSet_self
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by
+  apply measure_eq_zero_or_one_of_indepSet_self' ?_
     ((indep_limsup_atTop_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+  filter_upwards [h_indep.ae_isProbabilityMeasure] with a ha using by infer_instance
 
 /-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
 sub-σ-algebras has probability 0 or 1.
 The tail σ-algebra `limsup s atTop` is the same as `⋂ n, ⋃ i ≥ n, s i`. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atTop [IsProbabilityMeasure μ]
+theorem measure_zero_or_one_of_measurableSet_limsup_atTop
     (h_le : ∀ n, s n ≤ m0)
     (h_indep : iIndep s μ) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atTop] t) :
     μ t = 0 ∨ μ t = 1 := by
@@ -303,7 +312,7 @@ theorem Kernel.indep_limsup_atBot_self (h_le : ∀ n, s n ≤ m0) (h_indep : iIn
   · exact Antitone.directed_le fun _ _ ↦ Set.Ici_subset_Ici.2
   · exact fun n => ⟨n, le_rfl⟩
 
-theorem indep_limsup_atBot_self [IsProbabilityMeasure μ]
+theorem indep_limsup_atBot_self
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) :
     Indep (limsup s atBot) (limsup s atBot) μ :=
   Kernel.indep_limsup_atBot_self h_le h_indep
@@ -318,13 +327,14 @@ theorem condIndep_limsup_atBot_self [StandardBorelSpace Ω]
 sequence of sub-σ-algebras has probability 0 or 1 almost surely. -/
 theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
     (h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[limsup s atBot] t) :
-    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 :=
-  measure_eq_zero_or_one_of_indepSet_self
+    ∀ᵐ a ∂μα, κ a t = 0 ∨ κ a t = 1 := by
+  apply measure_eq_zero_or_one_of_indepSet_self' ?_
     ((indep_limsup_atBot_self h_le h_indep).indepSet_of_measurableSet ht_tail ht_tail)
+  filter_upwards [h_indep.ae_isProbabilityMeasure] with a ha using by infer_instance
 
 /-- **Kolmogorov's 0-1 law** : any event in the tail σ-algebra of an independent sequence of
 sub-σ-algebras has probability 0 or 1. -/
-theorem measure_zero_or_one_of_measurableSet_limsup_atBot [IsProbabilityMeasure μ]
+theorem measure_zero_or_one_of_measurableSet_limsup_atBot
     (h_le : ∀ n, s n ≤ m0) (h_indep : iIndep s μ) {t : Set Ω}
     (ht_tail : MeasurableSet[limsup s atBot] t) :
     μ t = 0 ∨ μ t = 1 := by
diff --git a/Mathlib/Probability/Kernel/Basic.lean b/Mathlib/Probability/Kernel/Basic.lean
index 2ba8229ef510e..3fc42e7dc57d2 100644
--- a/Mathlib/Probability/Kernel/Basic.lean
+++ b/Mathlib/Probability/Kernel/Basic.lean
@@ -3,361 +3,44 @@ Copyright (c) 2022 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathlib.MeasureTheory.Integral.Bochner
-import Mathlib.MeasureTheory.Measure.GiryMonad
+import Mathlib.Probability.Kernel.Defs
 
 /-!
-# Markov Kernels
+# Basic kernels
 
-A kernel from a measurable space `α` to another measurable space `β` is a measurable map
-`α → MeasureTheory.Measure β`, where the measurable space instance on `measure β` is the one defined
-in `MeasureTheory.Measure.instMeasurableSpace`. That is, a kernel `κ` verifies that for all
-measurable sets `s` of `β`, `a ↦ κ a s` is measurable.
+This file contains basic results about kernels in general and definitions of some particular
+kernels.
 
 ## Main definitions
 
-Classes of kernels:
-* `ProbabilityTheory.Kernel α β`: kernels from `α` to `β`.
-* `ProbabilityTheory.IsMarkovKernel κ`: a kernel from `α` to `β` is said to be a Markov kernel
-  if for all `a : α`, `k a` is a probability measure.
-* `ProbabilityTheory.IsFiniteKernel κ`: a kernel from `α` to `β` is said to be finite if there
-  exists `C : ℝ≥0∞` such that `C < ∞` and for all `a : α`, `κ a univ ≤ C`. This implies in
-  particular that all measures in the image of `κ` are finite, but is stronger since it requires a
-  uniform bound. This stronger condition is necessary to ensure that the composition of two finite
-  kernels is finite.
-* `ProbabilityTheory.IsSFiniteKernel κ`: a kernel is called s-finite if it is a countable
-  sum of finite kernels.
-
-Particular kernels:
 * `ProbabilityTheory.Kernel.deterministic (f : α → β) (hf : Measurable f)`:
   kernel `a ↦ Measure.dirac (f a)`.
 * `ProbabilityTheory.Kernel.const α (μβ : measure β)`: constant kernel `a ↦ μβ`.
 * `ProbabilityTheory.Kernel.restrict κ (hs : MeasurableSet s)`: kernel for which the image of
   `a : α` is `(κ a).restrict s`.
   Integral: `∫⁻ b, f b ∂(κ.restrict hs a) = ∫⁻ b in s, f b ∂(κ a)`
+* `ProbabilityTheory.Kernel.comapRight`: Kernel with value `(κ a).comap f`,
+  for a measurable embedding `f`. That is, for a measurable set `t : Set β`,
+  `ProbabilityTheory.Kernel.comapRight κ hf a t = κ a (f '' t)`
+* `ProbabilityTheory.Kernel.piecewise (hs : MeasurableSet s) κ η`: the kernel equal to `κ`
+  on the measurable set `s` and to `η` on its complement.
 
 ## Main statements
 
-* `ProbabilityTheory.Kernel.ext_fun`: if `∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a)` for all measurable
-  functions `f` and all `a`, then the two kernels `κ` and `η` are equal.
-
 -/
 
+assert_not_exists MeasureTheory.integral
 
 open MeasureTheory
 
-open scoped MeasureTheory ENNReal NNReal
+open scoped ENNReal
 
 namespace ProbabilityTheory
 
-/-- A kernel from a measurable space `α` to another measurable space `β` is a measurable function
-`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
-`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
-iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`. -/
-structure Kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] where
-  /-- The underlying function of a kernel.
-
-  Do not use this function directly. Instead use the coercion coming from the `DFunLike`
-  instance. -/
-  toFun : α → Measure β
-  /-- A kernel is a measurable map.
-
-  Do not use this lemma directly. Use `Kernel.measurable` instead. -/
-  measurable' : Measurable toFun
-
-@[deprecated (since := "2024-07-22")] alias kernel := Kernel
-
-/-- Notation for `Kernel` with respect to a non-standard σ-algebra in the domain. -/
-scoped notation "Kernel[" mα "]" α:arg β:arg => @Kernel α β mα _
-
-/-- Notation for `Kernel` with respect to a non-standard σ-algebra in the domain and codomain. -/
-scoped notation "Kernel[" mα ", " mβ "]" α:arg β:arg => @Kernel α β mα mβ
-
-initialize_simps_projections Kernel (toFun → apply)
-
-variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
-
-namespace Kernel
-
-instance instFunLike : FunLike (Kernel α β) α (Measure β) where
-  coe := toFun
-  coe_injective' f g h := by cases f; cases g; congr
-
-lemma measurable (κ : Kernel α β) : Measurable κ := κ.measurable'
-
-instance instZero : Zero (Kernel α β) where zero := ⟨0, measurable_zero⟩
-noncomputable instance instAdd : Add (Kernel α β) where add κ η := ⟨κ + η, κ.2.add η.2⟩
-noncomputable instance instSMulNat : SMul ℕ (Kernel α β) where
-  smul n κ := ⟨n • κ, (measurable_const (a := n)).smul κ.2⟩
-
-@[simp, norm_cast] lemma coe_zero : ⇑(0 : Kernel α β) = 0 := rfl
-@[simp, norm_cast] lemma coe_add (κ η : Kernel α β) : ⇑(κ + η) = κ + η := rfl
-@[simp, norm_cast] lemma coe_nsmul (n : ℕ) (κ : Kernel α β) : ⇑(n • κ) = n • κ := rfl
-
-@[simp] lemma zero_apply (a : α) : (0 : Kernel α β) a = 0 := rfl
-@[simp] lemma add_apply (κ η : Kernel α β) (a : α) : (κ + η) a = κ a + η a := rfl
-@[simp] lemma nsmul_apply (n : ℕ) (κ : Kernel α β) (a : α) : (n • κ) a = n • κ a := rfl
-
-noncomputable instance instAddCommMonoid : AddCommMonoid (Kernel α β) :=
-  DFunLike.coe_injective.addCommMonoid _ coe_zero coe_add (by intros; rfl)
-
-instance instPartialOrder : PartialOrder (Kernel α β) := .lift _ DFunLike.coe_injective
-
-instance instCovariantAddLE {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
-    CovariantClass (Kernel α β) (Kernel α β) (· + ·) (· ≤ ·) :=
-  ⟨fun _ _ _ hμ a ↦ add_le_add_left (hμ a) _⟩
-
-noncomputable
-instance instOrderBot {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
-    OrderBot (Kernel α β) where
-  bot := 0
-  bot_le κ a := by simp only [coe_zero, Pi.zero_apply, Measure.zero_le]
-
-/-- Coercion to a function as an additive monoid homomorphism. -/
-def coeAddHom (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
-    Kernel α β →+ α → Measure β where
-  toFun := (⇑)
-  map_zero' := coe_zero
-  map_add' := coe_add
-
-@[simp]
-theorem coe_finset_sum (I : Finset ι) (κ : ι → Kernel α β) : ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i) :=
-  map_sum (coeAddHom α β) _ _
-
-theorem finset_sum_apply (I : Finset ι) (κ : ι → Kernel α β) (a : α) :
-    (∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by rw [coe_finset_sum, Finset.sum_apply]
-
-theorem finset_sum_apply' (I : Finset ι) (κ : ι → Kernel α β) (a : α) (s : Set β) :
-    (∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by rw [finset_sum_apply, Measure.finset_sum_apply]
-
-end Kernel
-
-/-- A kernel is a Markov kernel if every measure in its image is a probability measure. -/
-class IsMarkovKernel (κ : Kernel α β) : Prop where
-  isProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
-
-/-- A kernel is finite if every measure in its image is finite, with a uniform bound. -/
-class IsFiniteKernel (κ : Kernel α β) : Prop where
-  exists_univ_le : ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ a, κ a Set.univ ≤ C
-
-/-- A constant `C : ℝ≥0∞` such that `C < ∞` (`ProbabilityTheory.IsFiniteKernel.bound_lt_top κ`) and
-for all `a : α` and `s : Set β`, `κ a s ≤ C` (`ProbabilityTheory.Kernel.measure_le_bound κ a s`).
-
-Porting note (#11215): TODO: does it make sense to
--- make `ProbabilityTheory.IsFiniteKernel.bound` the least possible bound?
--- Should it be an `NNReal` number? -/
-noncomputable def IsFiniteKernel.bound (κ : Kernel α β) [h : IsFiniteKernel κ] : ℝ≥0∞ :=
-  h.exists_univ_le.choose
-
-theorem IsFiniteKernel.bound_lt_top (κ : Kernel α β) [h : IsFiniteKernel κ] :
-    IsFiniteKernel.bound κ < ∞ :=
-  h.exists_univ_le.choose_spec.1
-
-theorem IsFiniteKernel.bound_ne_top (κ : Kernel α β) [IsFiniteKernel κ] :
-    IsFiniteKernel.bound κ ≠ ∞ :=
-  (IsFiniteKernel.bound_lt_top κ).ne
-
-theorem Kernel.measure_le_bound (κ : Kernel α β) [h : IsFiniteKernel κ] (a : α) (s : Set β) :
-    κ a s ≤ IsFiniteKernel.bound κ :=
-  (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
-
-instance isFiniteKernel_zero (α β : Type*) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
-    IsFiniteKernel (0 : Kernel α β) :=
-  ⟨⟨0, ENNReal.coe_lt_top, fun _ => by
-      simp only [Kernel.zero_apply, Measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
-
-instance IsFiniteKernel.add (κ η : Kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :
-    IsFiniteKernel (κ + η) := by
-  refine ⟨⟨IsFiniteKernel.bound κ + IsFiniteKernel.bound η,
-    ENNReal.add_lt_top.mpr ⟨IsFiniteKernel.bound_lt_top κ, IsFiniteKernel.bound_lt_top η⟩,
-    fun a => ?_⟩⟩
-  exact add_le_add (Kernel.measure_le_bound _ _ _) (Kernel.measure_le_bound _ _ _)
-
-lemma isFiniteKernel_of_le {κ ν : Kernel α β} [hν : IsFiniteKernel ν] (hκν : κ ≤ ν) :
-    IsFiniteKernel κ := by
-  refine ⟨hν.bound, hν.bound_lt_top, fun a ↦ (hκν _ _).trans (Kernel.measure_le_bound ν a Set.univ)⟩
-
-variable {κ : Kernel α β}
-
-instance IsMarkovKernel.is_probability_measure' [IsMarkovKernel κ] (a : α) :
-    IsProbabilityMeasure (κ a) :=
-  IsMarkovKernel.isProbabilityMeasure a
-
-instance IsFiniteKernel.isFiniteMeasure [IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
-  ⟨(Kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
-
-instance (priority := 100) IsMarkovKernel.isFiniteKernel [IsMarkovKernel κ] :
-    IsFiniteKernel κ :=
-  ⟨⟨1, ENNReal.one_lt_top, fun _ => prob_le_one⟩⟩
+variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β}
 
 namespace Kernel
 
-@[ext]
-theorem ext {η : Kernel α β} (h : ∀ a, κ a = η a) : κ = η := DFunLike.ext _ _ h
-
-theorem ext_iff' {η : Kernel α β} :
-    κ = η ↔ ∀ a s, MeasurableSet s → κ a s = η a s := by
-  simp_rw [Kernel.ext_iff, Measure.ext_iff]
-
-theorem ext_fun {η : Kernel α β} (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
-    κ = η := by
-  ext a s hs
-  specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
-  simp_rw [lintegral_indicator_const hs, one_mul] at h
-  rw [h]
-
-theorem ext_fun_iff {η : Kernel α β} :
-    κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a :=
-  ⟨fun h a f _ => by rw [h], ext_fun⟩
-
-protected theorem measurable_coe (κ : Kernel α β) {s : Set β} (hs : MeasurableSet s) :
-    Measurable fun a => κ a s :=
-  (Measure.measurable_coe hs).comp κ.measurable
-
-lemma apply_congr_of_mem_measurableAtom (κ : Kernel α β) {y' y : α} (hy' : y' ∈ measurableAtom y) :
-    κ y' = κ y := by
-  ext s hs
-  exact mem_of_mem_measurableAtom hy' (κ.measurable_coe hs (measurableSet_singleton (κ y s))) rfl
-
-lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
-    (κ : Kernel α β) [IsFiniteKernel κ] {s : Set β} (hs : MeasurableSet s) :
-    Integrable (fun x => (κ x s).toReal) μ := by
-  refine Integrable.mono' (integrable_const (IsFiniteKernel.bound κ).toReal)
-    ((κ.measurable_coe hs).ennreal_toReal.aestronglyMeasurable)
-    (ae_of_all μ fun x => ?_)
-  rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg,
-    ENNReal.toReal_le_toReal (measure_ne_top _ _) (IsFiniteKernel.bound_ne_top _)]
-  exact Kernel.measure_le_bound _ _ _
-
-lemma IsMarkovKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
-    (κ : Kernel α β) [IsMarkovKernel κ] {s : Set β} (hs : MeasurableSet s) :
-    Integrable (fun x => (κ x s).toReal) μ :=
-  IsFiniteKernel.integrable μ κ hs
-
-lemma integral_congr_ae₂ {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f g : α → β → E}
-    {μ : Measure α} (h : ∀ᵐ a ∂μ, f a =ᵐ[κ a] g a) :
-    ∫ a, ∫ b, f a b ∂(κ a) ∂μ = ∫ a, ∫ b, g a b ∂(κ a) ∂μ := by
-  apply integral_congr_ae
-  filter_upwards [h] with _ ha
-  apply integral_congr_ae
-  filter_upwards [ha] with _ hb using hb
-
-section Sum
-
-/-- Sum of an indexed family of kernels. -/
-protected noncomputable def sum [Countable ι] (κ : ι → Kernel α β) : Kernel α β where
-  toFun a := Measure.sum fun n => κ n a
-  measurable' := by
-    refine Measure.measurable_of_measurable_coe _ fun s hs => ?_
-    simp_rw [Measure.sum_apply _ hs]
-    exact Measurable.ennreal_tsum fun n => Kernel.measurable_coe (κ n) hs
-
-theorem sum_apply [Countable ι] (κ : ι → Kernel α β) (a : α) :
-    Kernel.sum κ a = Measure.sum fun n => κ n a :=
-  rfl
-
-theorem sum_apply' [Countable ι] (κ : ι → Kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s) :
-    Kernel.sum κ a s = ∑' n, κ n a s := by rw [sum_apply κ a, Measure.sum_apply _ hs]
-
-@[simp]
-theorem sum_zero [Countable ι] : (Kernel.sum fun _ : ι => (0 : Kernel α β)) = 0 := by
-  ext a s hs
-  rw [sum_apply' _ a hs]
-  simp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero]
-
-theorem sum_comm [Countable ι] (κ : ι → ι → Kernel α β) :
-    (Kernel.sum fun n => Kernel.sum (κ n)) = Kernel.sum fun m => Kernel.sum fun n => κ n m := by
-  ext a s; simp_rw [sum_apply]; rw [Measure.sum_comm]
-
-@[simp]
-theorem sum_fintype [Fintype ι] (κ : ι → Kernel α β) : Kernel.sum κ = ∑ i, κ i := by
-  ext a s hs
-  simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
-
-theorem sum_add [Countable ι] (κ η : ι → Kernel α β) :
-    (Kernel.sum fun n => κ n + η n) = Kernel.sum κ + Kernel.sum η := by
-  ext a s hs
-  simp only [coe_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply,
-    Measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
-
-end Sum
-
-section SFinite
-
-/-- A kernel is s-finite if it can be written as the sum of countably many finite kernels. -/
-class _root_.ProbabilityTheory.IsSFiniteKernel (κ : Kernel α β) : Prop where
-  tsum_finite : ∃ κs : ℕ → Kernel α β, (∀ n, IsFiniteKernel (κs n)) ∧ κ = Kernel.sum κs
-
-instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
-    IsSFiniteKernel κ :=
-  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by
-      simp only; split_ifs
-      · exact h
-      · infer_instance, by
-      ext a s hs
-      rw [Kernel.sum_apply' _ _ hs]
-      have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by
-        ext1 i; split_ifs <;> rfl
-      rw [this, tsum_ite_eq]⟩⟩
-
-/-- A sequence of finite kernels such that `κ = ProbabilityTheory.Kernel.sum (seq κ)`. See
-`ProbabilityTheory.Kernel.isFiniteKernel_seq` and `ProbabilityTheory.Kernel.kernel_sum_seq`. -/
-noncomputable def seq (κ : Kernel α β) [h : IsSFiniteKernel κ] : ℕ → Kernel α β :=
-  h.tsum_finite.choose
-
-theorem kernel_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] : Kernel.sum (seq κ) = κ :=
-  h.tsum_finite.choose_spec.2.symm
-
-theorem measure_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] (a : α) :
-    (Measure.sum fun n => seq κ n a) = κ a := by rw [← Kernel.sum_apply, kernel_sum_seq κ]
-
-instance isFiniteKernel_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :
-    IsFiniteKernel (Kernel.seq κ n) :=
-  h.tsum_finite.choose_spec.1 n
-
-instance _root_.ProbabilityTheory.IsSFiniteKernel.sFinite [IsSFiniteKernel κ] (a : α) :
-    SFinite (κ a) :=
-  ⟨⟨fun n ↦ seq κ n a, inferInstance, (measure_sum_seq κ a).symm⟩⟩
-
-instance IsSFiniteKernel.add (κ η : Kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
-    IsSFiniteKernel (κ + η) := by
-  refine ⟨⟨fun n => seq κ n + seq η n, fun n => inferInstance, ?_⟩⟩
-  rw [sum_add, kernel_sum_seq κ, kernel_sum_seq η]
-
-theorem IsSFiniteKernel.finset_sum {κs : ι → Kernel α β} (I : Finset ι)
-    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i ∈ I, κs i) := by
-  classical
-  induction' I using Finset.induction with i I hi_nmem_I h_ind h
-  · rw [Finset.sum_empty]; infer_instance
-  · rw [Finset.sum_insert hi_nmem_I]
-    haveI : IsSFiniteKernel (κs i) := h i (Finset.mem_insert_self _ _)
-    have : IsSFiniteKernel (∑ x ∈ I, κs x) :=
-      h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
-    exact IsSFiniteKernel.add _ _
-
-theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → Kernel α β}
-    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) := by
-  let e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm
-  refine ⟨⟨fun n => seq (κs (e n).1) (e n).2, inferInstance, ?_⟩⟩
-  have hκ_eq : Kernel.sum κs = Kernel.sum fun n => Kernel.sum (seq (κs n)) := by
-    simp_rw [kernel_sum_seq]
-  ext a s hs
-  rw [hκ_eq]
-  simp_rw [Kernel.sum_apply' _ _ hs]
-  change (∑' i, ∑' m, seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
-  rw [e.tsum_eq (fun im : ι × ℕ => seq (κs im.fst) im.snd a s),
-    tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
-
-theorem isSFiniteKernel_sum [Countable ι] {κs : ι → Kernel α β}
-    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) := by
-  cases fintypeOrInfinite ι
-  · rw [sum_fintype]
-    exact IsSFiniteKernel.finset_sum Finset.univ fun i _ => hκs i
-  cases nonempty_denumerable ι
-  exact isSFiniteKernel_sum_of_denumerable hκs
-
-end SFinite
-
 section Deterministic
 
 /-- Kernel which to `a` associates the dirac measure at `f a`. This is a Markov kernel. -/
@@ -408,36 +91,6 @@ theorem setLIntegral_deterministic {f : β → ℝ≥0∞} {g : α → β} {a :
 @[deprecated (since := "2024-06-29")]
 alias set_lintegral_deterministic := setLIntegral_deterministic
 
-theorem integral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : StronglyMeasurable f) : ∫ x, f x ∂deterministic g hg a = f (g a) := by
-  rw [deterministic_apply, integral_dirac' _ _ hf]
-
-@[simp]
-theorem integral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] : ∫ x, f x ∂deterministic g hg a = f (g a) := by
-  rw [deterministic_apply, integral_dirac _ (g a)]
-
-theorem setIntegral_deterministic' {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
-    ∫ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
-  rw [deterministic_apply, setIntegral_dirac' hf _ hs]
-
-@[deprecated (since := "2024-04-17")]
-alias set_integral_deterministic' := setIntegral_deterministic'
-
-@[simp]
-theorem setIntegral_deterministic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    [CompleteSpace E] {f : β → E} {g : α → β} {a : α} (hg : Measurable g)
-    [MeasurableSingletonClass β] (s : Set β) [Decidable (g a ∈ s)] :
-    ∫ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
-  rw [deterministic_apply, setIntegral_dirac f _ s]
-
-@[deprecated (since := "2024-04-17")]
-alias set_integral_deterministic := setIntegral_deterministic
-
 end Deterministic
 
 section Const
@@ -477,6 +130,13 @@ instance const.instIsMarkovKernel {μβ : Measure β} [hμβ : IsProbabilityMeas
     IsMarkovKernel (const α μβ) :=
   ⟨fun _ => hμβ⟩
 
+instance const.instIsZeroOrMarkovKernel {μβ : Measure β} [hμβ : IsZeroOrProbabilityMeasure μβ] :
+    IsZeroOrMarkovKernel (const α μβ) := by
+  rcases eq_zero_or_isProbabilityMeasure μβ with rfl | h
+  · simp only [const_zero]
+    infer_instance
+  · infer_instance
+
 lemma isSFiniteKernel_const [Nonempty α] {μβ : Measure β} :
     IsSFiniteKernel (const α μβ) ↔ SFinite μβ :=
   ⟨fun h ↦ h.sFinite (Classical.arbitrary α), fun _ ↦ inferInstance⟩
@@ -492,19 +152,6 @@ theorem setLIntegral_const {f : β → ℝ≥0∞} {μ : Measure β} {a : α} {s
 @[deprecated (since := "2024-06-29")]
 alias set_lintegral_const := setLIntegral_const
 
-@[simp]
-theorem integral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {f : β → E} {μ : Measure β} {a : α} : ∫ x, f x ∂const α μ a = ∫ x, f x ∂μ := by
-  rw [const_apply]
-
-@[simp]
-theorem setIntegral_const {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {f : β → E} {μ : Measure β} {a : α} {s : Set β} :
-    ∫ x in s, f x ∂const α μ a = ∫ x in s, f x ∂μ := by rw [const_apply]
-
-@[deprecated (since := "2024-04-17")]
-alias set_integral_const := setIntegral_const
-
 end Const
 
 /-- In a countable space with measurable singletons, every function `α → MeasureTheory.Measure β`
@@ -551,14 +198,6 @@ theorem setLIntegral_restrict (κ : Kernel α β) (hs : MeasurableSet s) (a : α
 @[deprecated (since := "2024-06-29")]
 alias set_lintegral_restrict := setLIntegral_restrict
 
-@[simp]
-theorem setIntegral_restrict {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    {f : β → E} {a : α} (hs : MeasurableSet s) (t : Set β) :
-    ∫ x in t, f x ∂κ.restrict hs a = ∫ x in t ∩ s, f x ∂κ a := by
-  rw [restrict_apply, Measure.restrict_restrict' hs]
-
-@[deprecated (since := "2024-04-17")]
-alias set_integral_restrict := setIntegral_restrict
 
 instance IsFiniteKernel.restrict (κ : Kernel α β) [IsFiniteKernel κ] (hs : MeasurableSet s) :
     IsFiniteKernel (κ.restrict hs) := by
@@ -683,20 +322,28 @@ theorem setLIntegral_piecewise (a : α) (g : β → ℝ≥0∞) (t : Set β) :
 @[deprecated (since := "2024-06-29")]
 alias set_lintegral_piecewise := setLIntegral_piecewise
 
-theorem integral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (a : α) (g : β → E) :
-    ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
-  simp_rw [piecewise_apply]; split_ifs <;> rfl
-
-theorem setIntegral_piecewise {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
-    (a : α) (g : β → E) (t : Set β) :
-    ∫ b in t, g b ∂piecewise hs κ η a =
-      if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a := by
-  simp_rw [piecewise_apply]; split_ifs <;> rfl
+end Piecewise
 
-@[deprecated (since := "2024-04-17")]
-alias set_integral_piecewise := setIntegral_piecewise
+lemma exists_ae_eq_isMarkovKernel {μ : Measure α}
+    (h : ∀ᵐ a ∂μ, IsProbabilityMeasure (κ a)) (h' : μ ≠ 0) :
+    ∃ (η : Kernel α β), (κ =ᵐ[μ] η) ∧ IsMarkovKernel η := by
+  classical
+  obtain ⟨s, s_meas, μs, hs⟩ : ∃ s, MeasurableSet s ∧ μ s = 0
+      ∧ ∀ a ∉ s, IsProbabilityMeasure (κ a) := by
+    refine ⟨toMeasurable μ {a | ¬ IsProbabilityMeasure (κ a)}, measurableSet_toMeasurable _ _,
+      by simpa [measure_toMeasurable] using h, ?_⟩
+    intro a ha
+    contrapose! ha
+    exact subset_toMeasurable _ _ ha
+  obtain ⟨a, ha⟩ : sᶜ.Nonempty := by
+    contrapose! h'; simpa [μs, h'] using measure_univ_le_add_compl s (μ := μ)
+  refine ⟨Kernel.piecewise s_meas (Kernel.const _ (κ a)) κ, ?_, ?_⟩
+  · filter_upwards [measure_zero_iff_ae_nmem.1 μs] with b hb
+    simp [hb, piecewise]
+  · refine ⟨fun b ↦ ?_⟩
+    by_cases hb : b ∈ s
+    · simpa [hb, piecewise] using hs _ ha
+    · simpa [hb, piecewise] using hs _ hb
 
-end Piecewise
 end Kernel
 end ProbabilityTheory
diff --git a/Mathlib/Probability/Kernel/Composition.lean b/Mathlib/Probability/Kernel/Composition.lean
index 39106448b1afa..513325918f94f 100644
--- a/Mathlib/Probability/Kernel/Composition.lean
+++ b/Mathlib/Probability/Kernel/Composition.lean
@@ -29,8 +29,8 @@ Kernels built from other kernels:
 * `compProd (κ : Kernel α β) (η : Kernel (α × β) γ) : Kernel α (β × γ)`: composition-product of 2
   s-finite kernels. We define a notation `κ ⊗ₖ η = compProd κ η`.
   `∫⁻ bc, f bc ∂((κ ⊗ₖ η) a) = ∫⁻ b, ∫⁻ c, f (b, c) ∂(η (a, b)) ∂(κ a)`
-* `map (κ : Kernel α β) (f : β → γ) (hf : Measurable f) : Kernel α γ`
-  `∫⁻ c, g c ∂(map κ f hf a) = ∫⁻ b, g (f b) ∂(κ a)`
+* `map (κ : Kernel α β) (f : β → γ) : Kernel α γ`
+  `∫⁻ c, g c ∂(map κ f a) = ∫⁻ b, g (f b) ∂(κ a)`
 * `comap (κ : Kernel α β) (f : γ → α) (hf : Measurable f) : Kernel γ β`
   `∫⁻ b, g b ∂(comap κ f hf c) = ∫⁻ b, g b ∂(κ (f c))`
 * `comp (η : Kernel β γ) (κ : Kernel α β) : Kernel α γ`: composition of 2 kernels.
@@ -96,16 +96,14 @@ theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSF
     (f : ℕ → Set (β × γ)) (hf_meas : ∀ i, MeasurableSet (f i))
     (hf_disj : Pairwise (Disjoint on f)) :
     compProdFun κ η a (⋃ i, f i) = ∑' i, compProdFun κ η a (f i) := by
-  have h_Union :
-    (fun b => η (a, b) {c : γ | (b, c) ∈ ⋃ i, f i}) = fun b =>
-      η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i}) := by
+  have h_Union : (fun b ↦ η (a, b) {c : γ | (b, c) ∈ ⋃ i, f i})
+      = fun b ↦ η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i}) := by
     ext1 b
     congr with c
     simp only [Set.mem_iUnion, Set.iSup_eq_iUnion, Set.mem_setOf_eq]
   rw [compProdFun, h_Union]
-  have h_tsum :
-    (fun b => η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i})) = fun b =>
-      ∑' i, η (a, b) {c : γ | (b, c) ∈ f i} := by
+  have h_tsum : (fun b ↦ η (a, b) (⋃ i, {c : γ | (b, c) ∈ f i}))
+      = fun b ↦ ∑' i, η (a, b) {c : γ | (b, c) ∈ f i} := by
     ext1 b
     rw [measure_iUnion]
     · intro i j hij s hsi hsj c hcs
@@ -113,11 +111,9 @@ theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSF
       have hbcj : {(b, c)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hcs
       simpa only [Set.bot_eq_empty, Set.le_eq_subset, Set.singleton_subset_iff,
         Set.mem_empty_iff_false] using hf_disj hij hbci hbcj
-    · -- Porting note: behavior of `@` changed relative to lean 3, was
-      -- exact fun i => (@measurable_prod_mk_left β γ _ _ b) _ (hf_meas i)
-      exact fun i => (@measurable_prod_mk_left β γ _ _ b) (hf_meas i)
+    · exact fun i ↦ measurable_prod_mk_left (hf_meas i)
   rw [h_tsum, lintegral_tsum]
-  · rfl
+  · simp [compProdFun]
   · intro i
     have hm : MeasurableSet {p : (α × β) × γ | (p.1.2, p.2) ∈ f i} :=
       measurable_fst.snd.prod_mk measurable_snd (hf_meas i)
@@ -126,15 +122,13 @@ theorem compProdFun_iUnion (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSF
 theorem compProdFun_tsum_right (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel η] (a : α)
     (hs : MeasurableSet s) : compProdFun κ η a s = ∑' n, compProdFun κ (seq η n) a s := by
   simp_rw [compProdFun, (measure_sum_seq η _).symm]
-  have :
-    ∫⁻ b, Measure.sum (fun n => seq η n (a, b)) {c : γ | (b, c) ∈ s} ∂κ a =
-      ∫⁻ b, ∑' n, seq η n (a, b) {c : γ | (b, c) ∈ s} ∂κ a := by
-    congr
-    ext1 b
+  have : ∫⁻ b, Measure.sum (fun n => seq η n (a, b)) {c : γ | (b, c) ∈ s} ∂κ a
+      = ∫⁻ b, ∑' n, seq η n (a, b) {c : γ | (b, c) ∈ s} ∂κ a := by
+    congr with b
     rw [Measure.sum_apply]
     exact measurable_prod_mk_left hs
   rw [this, lintegral_tsum]
-  exact fun n => ((measurable_kernel_prod_mk_left (κ := (seq η n))
+  exact fun n ↦ ((measurable_kernel_prod_mk_left (κ := (seq η n))
     ((measurable_fst.snd.prod_mk measurable_snd) hs)).comp measurable_prod_mk_left).aemeasurable
 
 theorem compProdFun_tsum_left (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel κ] (a : α)
@@ -146,34 +140,21 @@ theorem compProdFun_eq_tsum (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kerne
     compProdFun κ η a s = ∑' (n) (m), compProdFun (seq κ n) (seq η m) a s := by
   simp_rw [compProdFun_tsum_left κ η a s, compProdFun_tsum_right _ η a hs]
 
-/-- Auxiliary lemma for `measurable_compProdFun`. -/
-theorem measurable_compProdFun_of_finite (κ : Kernel α β) [IsFiniteKernel κ] (η : Kernel (α × β) γ)
-    [IsFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s := by
+theorem measurable_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
+    [IsSFiniteKernel η] (hs : MeasurableSet s) :
+    Measurable fun a ↦ compProdFun κ η a s := by
   simp only [compProdFun]
   have h_meas : Measurable (Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s}) := by
-    have :
-      (Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
-        η p {c : γ | (p.2, c) ∈ s} := by
+    have : (Function.uncurry fun a b => η (a, b) {c : γ | (b, c) ∈ s})
+        = fun p ↦ η p {c : γ | (p.2, c) ∈ s} := by
       ext1 p
       rw [Function.uncurry_apply_pair]
     rw [this]
     exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs)
   exact h_meas.lintegral_kernel_prod_right
 
-theorem measurable_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
-    [IsSFiniteKernel η] (hs : MeasurableSet s) : Measurable fun a => compProdFun κ η a s := by
-  simp_rw [compProdFun_tsum_right κ η _ hs]
-  refine Measurable.ennreal_tsum fun n => ?_
-  simp only [compProdFun]
-  have h_meas : Measurable (Function.uncurry fun a b => seq η n (a, b) {c : γ | (b, c) ∈ s}) := by
-    have :
-      (Function.uncurry fun a b => seq η n (a, b) {c : γ | (b, c) ∈ s}) = fun p =>
-        seq η n p {c : γ | (p.2, c) ∈ s} := by
-      ext1 p
-      rw [Function.uncurry_apply_pair]
-    rw [this]
-    exact measurable_kernel_prod_mk_left (measurable_fst.snd.prod_mk measurable_snd hs)
-  exact h_meas.lintegral_kernel_prod_right
+@[deprecated (since := "2024-08-30")]
+alias measurable_compProdFun_of_finite := measurable_compProdFun
 
 open scoped Classical
 
@@ -182,24 +163,24 @@ open scoped Classical
 (see `ProbabilityTheory.Kernel.lintegral_compProd`).
 If either of the kernels is not s-finite, `compProd` is given the junk value 0. -/
 noncomputable def compProd (κ : Kernel α β) (η : Kernel (α × β) γ) : Kernel α (β × γ) :=
-if h : IsSFiniteKernel κ ∧ IsSFiniteKernel η then
-{ toFun := fun a ↦
-    Measure.ofMeasurable (fun s _ => compProdFun κ η a s) (compProdFun_empty κ η a)
-      (@compProdFun_iUnion _ _ _ _ _ _ κ η h.2 a)
-  measurable' := by
-    have : IsSFiniteKernel κ := h.1
-    have : IsSFiniteKernel η := h.2
-    refine Measure.measurable_of_measurable_coe _ fun s hs => ?_
-    have :
-      (fun a =>
-          Measure.ofMeasurable (fun s _ => compProdFun κ η a s) (compProdFun_empty κ η a)
-            (compProdFun_iUnion κ η a) s) =
-        fun a => compProdFun κ η a s := by
-      ext1 a; rwa [Measure.ofMeasurable_apply]
-    rw [this]
-    exact measurable_compProdFun κ η hs }
-else 0
-
+  if h : IsSFiniteKernel κ ∧ IsSFiniteKernel η then
+  { toFun := fun a ↦
+      have : IsSFiniteKernel η := h.2
+      Measure.ofMeasurable (fun s _ ↦ compProdFun κ η a s) (compProdFun_empty κ η a)
+        (compProdFun_iUnion κ η a)
+    measurable' := by
+      have : IsSFiniteKernel κ := h.1
+      have : IsSFiniteKernel η := h.2
+      refine Measure.measurable_of_measurable_coe _ fun s hs ↦ ?_
+      have : (fun a ↦ Measure.ofMeasurable (fun s _ ↦ compProdFun κ η a s) (compProdFun_empty κ η a)
+              (compProdFun_iUnion κ η a) s)
+          = fun a ↦ compProdFun κ η a s := by
+        ext1 a; rwa [Measure.ofMeasurable_apply]
+      rw [this]
+      exact measurable_compProdFun κ η hs }
+  else 0
+
+@[inherit_doc]
 scoped[ProbabilityTheory] infixl:100 " ⊗ₖ " => ProbabilityTheory.Kernel.compProd
 
 theorem compProd_apply_eq_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
@@ -253,7 +234,7 @@ lemma compProd_zero_left (κ : Kernel (α × β) γ) :
   · rw [Kernel.compProd_of_not_isSFiniteKernel_right _ _ h]
 
 @[simp]
-lemma compProd_zero_right (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
+lemma compProd_zero_right (κ : Kernel α β) (γ : Type*) {mγ : MeasurableSpace γ} :
     κ ⊗ₖ (0 : Kernel (α × β) γ) = 0 := by
   by_cases h : IsSFiniteKernel κ
   · ext a s hs
@@ -339,9 +320,8 @@ theorem compProd_restrict {s : Set β} {t : Set γ} (hs : MeasurableSet s) (ht :
     classical
     rw [Set.indicator_apply]
     split_ifs with h
-    · simp only [h, true_and_iff]
-      rfl
-    · simp only [h, false_and_iff, and_false_iff, Set.setOf_false, measure_empty]
+    · simp only [h, true_and, Set.inter_def, Set.mem_setOf]
+    · simp only [h, false_and, and_false, Set.setOf_false, measure_empty]
   simp_rw [this]
   rw [lintegral_indicator _ hs]
 
@@ -588,60 +568,92 @@ section MapComap
 
 variable {γ δ : Type*} {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ} {f : β → γ} {g : γ → α}
 
-/-- The pushforward of a kernel along a measurable function.
-We include measurability in the assumptions instead of using junk values
-to make sure that typeclass inference can infer that the `map` of a Markov kernel
-is again a Markov kernel. -/
-noncomputable def map (κ : Kernel α β) (f : β → γ) (hf : Measurable f) : Kernel α γ where
+/-- The pushforward of a kernel along a measurable function. This is an implementation detail,
+use `map κ f` instead. -/
+noncomputable def mapOfMeasurable (κ : Kernel α β) (f : β → γ) (hf : Measurable f) :
+    Kernel α γ where
   toFun a := (κ a).map f
   measurable' := (Measure.measurable_map _ hf).comp (Kernel.measurable κ)
 
-theorem map_apply (κ : Kernel α β) (hf : Measurable f) (a : α) : map κ f hf a = (κ a).map f :=
-  rfl
+open Classical in
+/-- The pushforward of a kernel along a function.
+If the function is not measurable, we use zero instead. This choice of junk
+value ensures that typeclass inference can infer that the `map` of a kernel
+satisfying `IsZeroOrMarkovKernel` again satisfies this property. -/
+noncomputable def map [MeasurableSpace γ] (κ : Kernel α β) (f : β → γ) : Kernel α γ :=
+  if hf : Measurable f then mapOfMeasurable κ f hf else 0
+
+theorem map_of_not_measurable (κ : Kernel α β) {f : β → γ} (hf : ¬(Measurable f)) :
+    map κ f = 0 := by
+  simp [map, hf]
+
+@[simp] theorem mapOfMeasurable_eq_map (κ : Kernel α β) {f : β → γ} (hf : Measurable f) :
+    mapOfMeasurable κ f hf = map κ f := by
+  simp [map, hf]
+
+theorem map_apply (κ : Kernel α β) (hf : Measurable f) (a : α) : map κ f a = (κ a).map f := by
+  simp only [map, hf, ↓reduceDIte, mapOfMeasurable, coe_mk]
 
 theorem map_apply' (κ : Kernel α β) (hf : Measurable f) (a : α) {s : Set γ} (hs : MeasurableSet s) :
-    map κ f hf a s = κ a (f ⁻¹' s) := by rw [map_apply, Measure.map_apply hf hs]
+    map κ f a s = κ a (f ⁻¹' s) := by rw [map_apply _ hf, Measure.map_apply hf hs]
 
 @[simp]
-lemma map_zero (hf : Measurable f) : Kernel.map (0 : Kernel α β) f hf = 0 := by
-  ext; rw [Kernel.map_apply]; simp
+lemma map_zero : Kernel.map (0 : Kernel α β) f = 0 := by
+  ext
+  by_cases hf : Measurable f
+  · simp [map_apply, hf]
+  · simp [map_of_not_measurable _ hf, map_apply]
 
 @[simp]
-lemma map_id (κ : Kernel α β) : map κ id measurable_id = κ := by ext a; rw [map_apply]; simp
+lemma map_id (κ : Kernel α β) : map κ id = κ := by
+  ext a
+  simp [map_apply, measurable_id]
 
 @[simp]
-lemma map_id' (κ : Kernel α β) : map κ (fun a ↦ a) measurable_id = κ := map_id κ
+lemma map_id' (κ : Kernel α β) : map κ (fun a ↦ a) = κ := map_id κ
 
 nonrec theorem lintegral_map (κ : Kernel α β) (hf : Measurable f) (a : α) {g' : γ → ℝ≥0∞}
-    (hg : Measurable g') : ∫⁻ b, g' b ∂map κ f hf a = ∫⁻ a, g' (f a) ∂κ a := by
+    (hg : Measurable g') : ∫⁻ b, g' b ∂map κ f a = ∫⁻ a, g' (f a) ∂κ a := by
   rw [map_apply _ hf, lintegral_map hg hf]
 
-theorem sum_map_seq (κ : Kernel α β) [IsSFiniteKernel κ] (hf : Measurable f) :
-    (Kernel.sum fun n => map (seq κ n) f hf) = map κ f hf := by
-  ext a s hs
-  rw [Kernel.sum_apply, map_apply' κ hf a hs, Measure.sum_apply _ hs, ← measure_sum_seq κ,
-    Measure.sum_apply _ (hf hs)]
-  simp_rw [map_apply' _ hf _ hs]
+theorem sum_map_seq (κ : Kernel α β) [IsSFiniteKernel κ] (f : β → γ) :
+    (Kernel.sum fun n => map (seq κ n) f) = map κ f := by
+  by_cases hf : Measurable f
+  · ext a s hs
+    rw [Kernel.sum_apply, map_apply' κ hf a hs, Measure.sum_apply _ hs, ← measure_sum_seq κ,
+      Measure.sum_apply _ (hf hs)]
+    simp_rw [map_apply' _ hf _ hs]
+  · simp [map_of_not_measurable _ hf]
 
-instance IsMarkovKernel.map (κ : Kernel α β) [IsMarkovKernel κ] (hf : Measurable f) :
-    IsMarkovKernel (map κ f hf) :=
+lemma IsMarkovKernel.map (κ : Kernel α β) [IsMarkovKernel κ] (hf : Measurable f) :
+    IsMarkovKernel (map κ f) :=
   ⟨fun a => ⟨by rw [map_apply' κ hf a MeasurableSet.univ, Set.preimage_univ, measure_univ]⟩⟩
 
-instance IsFiniteKernel.map (κ : Kernel α β) [IsFiniteKernel κ] (hf : Measurable f) :
-    IsFiniteKernel (map κ f hf) := by
+instance IsZeroOrMarkovKernel.map (κ : Kernel α β) [IsZeroOrMarkovKernel κ] (f : β → γ) :
+    IsZeroOrMarkovKernel (map κ f) := by
+  by_cases hf : Measurable f
+  · rcases eq_zero_or_isMarkovKernel κ with rfl | h
+    · simp only [map_zero]; infer_instance
+    · have := IsMarkovKernel.map κ hf; infer_instance
+  · simp only [map_of_not_measurable _ hf]; infer_instance
+
+instance IsFiniteKernel.map (κ : Kernel α β) [IsFiniteKernel κ] (f : β → γ) :
+    IsFiniteKernel (map κ f) := by
   refine ⟨⟨IsFiniteKernel.bound κ, IsFiniteKernel.bound_lt_top κ, fun a => ?_⟩⟩
-  rw [map_apply' κ hf a MeasurableSet.univ]
-  exact measure_le_bound κ a _
+  by_cases hf : Measurable f
+  · rw [map_apply' κ hf a MeasurableSet.univ]
+    exact measure_le_bound κ a _
+  · simp [map_of_not_measurable _ hf]
 
-instance IsSFiniteKernel.map (κ : Kernel α β) [IsSFiniteKernel κ] (hf : Measurable f) :
-    IsSFiniteKernel (map κ f hf) :=
-  ⟨⟨fun n => Kernel.map (seq κ n) f hf, inferInstance, (sum_map_seq κ hf).symm⟩⟩
+instance IsSFiniteKernel.map (κ : Kernel α β) [IsSFiniteKernel κ] (f : β → γ) :
+    IsSFiniteKernel (map κ f) :=
+  ⟨⟨fun n => Kernel.map (seq κ n) f, inferInstance, (sum_map_seq κ f).symm⟩⟩
 
 @[simp]
 lemma map_const (μ : Measure α) {f : α → β} (hf : Measurable f) :
-    map (const γ μ) f hf = const γ (μ.map f) := by
+    map (const γ μ) f = const γ (μ.map f) := by
   ext x s hs
-  rw [map_apply' _ _ _ hs, const_apply, const_apply, Measure.map_apply hf hs]
+  rw [map_apply' _ hf _ hs, const_apply, const_apply, Measure.map_apply hf hs]
 
 /-- Pullback of a kernel, such that for each set s `comap κ g hg c s = κ (g c) s`.
 We include measurability in the assumptions instead of using junk values
@@ -695,9 +707,9 @@ instance IsSFiniteKernel.comap (κ : Kernel α β) [IsSFiniteKernel κ] (hg : Me
 
 lemma comap_map_comm (κ : Kernel β γ) {f : α → β} {g : γ → δ}
     (hf : Measurable f) (hg : Measurable g) :
-    comap (map κ g hg) f hf = map (comap κ f hf) g hg := by
+    comap (map κ g) f hf = map (comap κ f hf) g := by
   ext x s _
-  rw [comap_apply, map_apply, map_apply, comap_apply]
+  rw [comap_apply, map_apply _ hg, map_apply _ hg, comap_apply]
 
 end MapComap
 
@@ -783,13 +795,19 @@ lemma isSFiniteKernel_prodMkRight_unit {κ : Kernel α β} :
   change IsSFiniteKernel ((prodMkRight Unit κ).comap (fun a ↦ (a, ())) (by fun_prop))
   infer_instance
 
-lemma map_prodMkLeft (γ : Type*) [MeasurableSpace γ] (κ : Kernel α β)
-    {f : β → δ} (hf : Measurable f) :
-    map (prodMkLeft γ κ) f hf = prodMkLeft γ (map κ f hf) := rfl
+lemma map_prodMkLeft (γ : Type*) [MeasurableSpace γ] (κ : Kernel α β) (f : β → δ) :
+    map (prodMkLeft γ κ) f = prodMkLeft γ (map κ f) := by
+  by_cases hf : Measurable f
+  · simp only [map, hf, ↓reduceDIte]
+    rfl
+  · simp [map_of_not_measurable _ hf]
 
-lemma map_prodMkRight (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ]
-    {f : β → δ} (hf : Measurable f) :
-    map (prodMkRight γ κ) f hf = prodMkRight γ (map κ f hf) := rfl
+lemma map_prodMkRight (κ : Kernel α β) (γ : Type*) {mγ : MeasurableSpace γ} (f : β → δ) :
+    map (prodMkRight γ κ) f = prodMkRight γ (map κ f) := by
+  by_cases hf : Measurable f
+  · simp only [map, hf, ↓reduceDIte]
+    rfl
+  · simp [map_of_not_measurable _ hf]
 
 /-- Define a `Kernel (β × α) γ` from a `Kernel (α × β) γ` by taking the comap of `Prod.swap`. -/
 def swapLeft (κ : Kernel (α × β) γ) : Kernel (β × α) γ :=
@@ -814,15 +832,19 @@ instance IsFiniteKernel.swapLeft (κ : Kernel (α × β) γ) [IsFiniteKernel κ]
 instance IsSFiniteKernel.swapLeft (κ : Kernel (α × β) γ) [IsSFiniteKernel κ] :
     IsSFiniteKernel (swapLeft κ) := by rw [Kernel.swapLeft]; infer_instance
 
-@[simp] lemma swapLeft_prodMkLeft (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
+@[simp] lemma swapLeft_prodMkLeft (κ : Kernel α β) (γ : Type*) {_ : MeasurableSpace γ} :
     swapLeft (prodMkLeft γ κ) = prodMkRight γ κ := rfl
 
-@[simp] lemma swapLeft_prodMkRight (κ : Kernel α β) (γ : Type*) [MeasurableSpace γ] :
+@[simp] lemma swapLeft_prodMkRight (κ : Kernel α β) (γ : Type*) {_ : MeasurableSpace γ} :
     swapLeft (prodMkRight γ κ) = prodMkLeft γ κ := rfl
 
-/-- Define a `Kernel α (γ × β)` from a `Kernel α (β × γ)` by taking the map of `Prod.swap`. -/
+/-- Define a `Kernel α (γ × β)` from a `Kernel α (β × γ)` by taking the map of `Prod.swap`.
+We use `mapOfMeasurable` in the definition for better defeqs. -/
 noncomputable def swapRight (κ : Kernel α (β × γ)) : Kernel α (γ × β) :=
-  map κ Prod.swap measurable_swap
+  mapOfMeasurable κ Prod.swap measurable_swap
+
+lemma swapRight_eq (κ : Kernel α (β × γ)) : swapRight κ = map κ Prod.swap := by
+  simp [swapRight]
 
 theorem swapRight_apply (κ : Kernel α (β × γ)) (a : α) : swapRight κ a = (κ a).map Prod.swap :=
   rfl
@@ -833,20 +855,27 @@ theorem swapRight_apply' (κ : Kernel α (β × γ)) (a : α) {s : Set (γ × β
 
 theorem lintegral_swapRight (κ : Kernel α (β × γ)) (a : α) {g : γ × β → ℝ≥0∞} (hg : Measurable g) :
     ∫⁻ c, g c ∂swapRight κ a = ∫⁻ bc : β × γ, g bc.swap ∂κ a := by
-  rw [swapRight, lintegral_map _ measurable_swap a hg]
+  rw [swapRight_eq, lintegral_map _ measurable_swap a hg]
 
 instance IsMarkovKernel.swapRight (κ : Kernel α (β × γ)) [IsMarkovKernel κ] :
-    IsMarkovKernel (swapRight κ) := by rw [Kernel.swapRight]; infer_instance
+    IsMarkovKernel (swapRight κ) := by
+  rw [Kernel.swapRight_eq]; exact IsMarkovKernel.map _ measurable_swap
+
+instance IsZeroOrMarkovKernel.swapRight (κ : Kernel α (β × γ)) [IsZeroOrMarkovKernel κ] :
+    IsZeroOrMarkovKernel (swapRight κ) := by rw [Kernel.swapRight_eq]; infer_instance
 
 instance IsFiniteKernel.swapRight (κ : Kernel α (β × γ)) [IsFiniteKernel κ] :
-    IsFiniteKernel (swapRight κ) := by rw [Kernel.swapRight]; infer_instance
+    IsFiniteKernel (swapRight κ) := by rw [Kernel.swapRight_eq]; infer_instance
 
 instance IsSFiniteKernel.swapRight (κ : Kernel α (β × γ)) [IsSFiniteKernel κ] :
-    IsSFiniteKernel (swapRight κ) := by rw [Kernel.swapRight]; infer_instance
+    IsSFiniteKernel (swapRight κ) := by rw [Kernel.swapRight_eq]; infer_instance
 
-/-- Define a `Kernel α β` from a `Kernel α (β × γ)` by taking the map of the first projection. -/
+/-- Define a `Kernel α β` from a `Kernel α (β × γ)` by taking the map of the first projection.
+We use `mapOfMeasurable` for better defeqs. -/
 noncomputable def fst (κ : Kernel α (β × γ)) : Kernel α β :=
-  map κ Prod.fst measurable_fst
+  mapOfMeasurable κ Prod.fst measurable_fst
+
+theorem fst_eq (κ : Kernel α (β × γ)) : fst κ = map κ Prod.fst := by simp [fst]
 
 theorem fst_apply (κ : Kernel α (β × γ)) (a : α) : fst κ a = (κ a).map Prod.fst :=
   rfl
@@ -859,16 +888,20 @@ lemma fst_zero : fst (0 : Kernel α (β × γ)) = 0 := by simp [fst]
 
 theorem lintegral_fst (κ : Kernel α (β × γ)) (a : α) {g : β → ℝ≥0∞} (hg : Measurable g) :
     ∫⁻ c, g c ∂fst κ a = ∫⁻ bc : β × γ, g bc.fst ∂κ a := by
-  rw [fst, lintegral_map _ measurable_fst a hg]
+  rw [fst_eq, lintegral_map _ measurable_fst a hg]
 
 instance IsMarkovKernel.fst (κ : Kernel α (β × γ)) [IsMarkovKernel κ] : IsMarkovKernel (fst κ) := by
-  rw [Kernel.fst]; infer_instance
+  rw [Kernel.fst_eq]; exact IsMarkovKernel.map _ measurable_fst
+
+instance IsZeroOrMarkovKernel.fst (κ : Kernel α (β × γ)) [IsZeroOrMarkovKernel κ] :
+    IsZeroOrMarkovKernel (fst κ) := by
+  rw [Kernel.fst_eq]; infer_instance
 
 instance IsFiniteKernel.fst (κ : Kernel α (β × γ)) [IsFiniteKernel κ] : IsFiniteKernel (fst κ) := by
-  rw [Kernel.fst]; infer_instance
+  rw [Kernel.fst_eq]; infer_instance
 
 instance IsSFiniteKernel.fst (κ : Kernel α (β × γ)) [IsSFiniteKernel κ] :
-    IsSFiniteKernel (fst κ) := by rw [Kernel.fst]; infer_instance
+    IsSFiniteKernel (fst κ) := by rw [Kernel.fst_eq]; infer_instance
 
 instance (priority := 100) isFiniteKernel_of_isFiniteKernel_fst {κ : Kernel α (β × γ)}
     [h : IsFiniteKernel (fst κ)] :
@@ -877,18 +910,21 @@ instance (priority := 100) isFiniteKernel_of_isFiniteKernel_fst {κ : Kernel α
   rw [fst_apply' _ _ MeasurableSet.univ]
   simp
 
-lemma fst_map_prod (κ : Kernel α β) {f : β → γ} {g : β → δ}
-    (hf : Measurable f) (hg : Measurable g) :
-    fst (map κ (fun x ↦ (f x, g x)) (hf.prod_mk hg)) = map κ f hf := by
-  ext x s hs
-  rw [fst_apply' _ _ hs, map_apply', map_apply' _ _ _ hs]
-  · rfl
-  · exact measurable_fst hs
+lemma fst_map_prod (κ : Kernel α β) {f : β → γ} {g : β → δ} (hg : Measurable g) :
+    fst (map κ (fun x ↦ (f x, g x))) = map κ f := by
+  by_cases hf : Measurable f
+  · ext x s hs
+    rw [fst_apply' _ _ hs, map_apply' _ (hf.prod hg) _, map_apply' _ hf _ hs]
+    · simp only [Set.preimage, Set.mem_setOf]
+    · exact measurable_fst hs
+  · have : ¬ Measurable (fun x ↦ (f x, g x)) := by
+      contrapose! hf; exact hf.fst
+    simp [map_of_not_measurable _ hf, map_of_not_measurable _ this]
 
 lemma fst_map_id_prod (κ : Kernel α β) {γ : Type*} {mγ : MeasurableSpace γ}
     {f : β → γ} (hf : Measurable f) :
-    fst (map κ (fun a ↦ (a, f a)) (measurable_id.prod_mk hf)) = κ := by
-  rw [fst_map_prod _ measurable_id' hf, Kernel.map_id']
+    fst (map κ (fun a ↦ (a, f a))) = κ := by
+  rw [fst_map_prod _ hf, Kernel.map_id']
 
 @[simp]
 lemma fst_compProd (κ : Kernel α β) (η : Kernel (α × β) γ) [IsSFiniteKernel κ] [IsMarkovKernel η] :
@@ -910,9 +946,12 @@ lemma fst_prodMkLeft (δ : Type*) [MeasurableSpace δ] (κ : Kernel α (β × γ
 lemma fst_prodMkRight (κ : Kernel α (β × γ)) (δ : Type*) [MeasurableSpace δ] :
     fst (prodMkRight δ κ) = prodMkRight δ (fst κ) := rfl
 
-/-- Define a `Kernel α γ` from a `Kernel α (β × γ)` by taking the map of the second projection. -/
+/-- Define a `Kernel α γ` from a `Kernel α (β × γ)` by taking the map of the second projection.
+We use `mapOfMeasurable` for better defeqs. -/
 noncomputable def snd (κ : Kernel α (β × γ)) : Kernel α γ :=
-  map κ Prod.snd measurable_snd
+  mapOfMeasurable κ Prod.snd measurable_snd
+
+theorem snd_eq (κ : Kernel α (β × γ)) : snd κ = map κ Prod.snd := by simp [snd]
 
 theorem snd_apply (κ : Kernel α (β × γ)) (a : α) : snd κ a = (κ a).map Prod.snd :=
   rfl
@@ -925,16 +964,20 @@ lemma snd_zero : snd (0 : Kernel α (β × γ)) = 0 := by simp [snd]
 
 theorem lintegral_snd (κ : Kernel α (β × γ)) (a : α) {g : γ → ℝ≥0∞} (hg : Measurable g) :
     ∫⁻ c, g c ∂snd κ a = ∫⁻ bc : β × γ, g bc.snd ∂κ a := by
-  rw [snd, lintegral_map _ measurable_snd a hg]
+  rw [snd_eq, lintegral_map _ measurable_snd a hg]
 
 instance IsMarkovKernel.snd (κ : Kernel α (β × γ)) [IsMarkovKernel κ] : IsMarkovKernel (snd κ) := by
-  rw [Kernel.snd]; infer_instance
+  rw [Kernel.snd_eq]; exact IsMarkovKernel.map _ measurable_snd
+
+instance IsZeroOrMarkovKernel.snd (κ : Kernel α (β × γ)) [IsZeroOrMarkovKernel κ] :
+    IsZeroOrMarkovKernel (snd κ) := by
+  rw [Kernel.snd_eq]; infer_instance
 
 instance IsFiniteKernel.snd (κ : Kernel α (β × γ)) [IsFiniteKernel κ] : IsFiniteKernel (snd κ) := by
-  rw [Kernel.snd]; infer_instance
+  rw [Kernel.snd_eq]; infer_instance
 
 instance IsSFiniteKernel.snd (κ : Kernel α (β × γ)) [IsSFiniteKernel κ] :
-    IsSFiniteKernel (snd κ) := by rw [Kernel.snd]; infer_instance
+    IsSFiniteKernel (snd κ) := by rw [Kernel.snd_eq]; infer_instance
 
 instance (priority := 100) isFiniteKernel_of_isFiniteKernel_snd {κ : Kernel α (β × γ)}
     [h : IsFiniteKernel (snd κ)] :
@@ -943,18 +986,21 @@ instance (priority := 100) isFiniteKernel_of_isFiniteKernel_snd {κ : Kernel α
   rw [snd_apply' _ _ MeasurableSet.univ]
   simp
 
-lemma snd_map_prod (κ : Kernel α β) {f : β → γ} {g : β → δ}
-    (hf : Measurable f) (hg : Measurable g) :
-    snd (map κ (fun x ↦ (f x, g x)) (hf.prod_mk hg)) = map κ g hg := by
-  ext x s hs
-  rw [snd_apply' _ _ hs, map_apply', map_apply' _ _ _ hs]
-  · rfl
-  · exact measurable_snd hs
+lemma snd_map_prod (κ : Kernel α β) {f : β → γ} {g : β → δ} (hf : Measurable f) :
+    snd (map κ (fun x ↦ (f x, g x))) = map κ g := by
+  by_cases hg : Measurable g
+  · ext x s hs
+    rw [snd_apply' _ _ hs, map_apply' _ (hf.prod hg), map_apply' _ hg _ hs]
+    · simp only [Set.preimage, Set.mem_setOf]
+    · exact measurable_snd hs
+  · have : ¬ Measurable (fun x ↦ (f x, g x)) := by
+      contrapose! hg; exact hg.snd
+    simp [map_of_not_measurable _ hg, map_of_not_measurable _ this]
 
 lemma snd_map_prod_id (κ : Kernel α β) {γ : Type*} {mγ : MeasurableSpace γ}
     {f : β → γ} (hf : Measurable f) :
-    snd (map κ (fun a ↦ (f a, a)) (hf.prod_mk measurable_id)) = κ := by
-  rw [snd_map_prod _ hf measurable_id', Kernel.map_id']
+    snd (map κ (fun a ↦ (f a, a))) = κ := by
+  rw [snd_map_prod _ hf, Kernel.map_id']
 
 lemma snd_prodMkLeft (δ : Type*) [MeasurableSpace δ] (κ : Kernel α (β × γ)) :
     snd (prodMkLeft δ κ) = prodMkLeft δ (snd κ) := rfl
@@ -990,6 +1036,7 @@ noncomputable def comp (η : Kernel β γ) (κ : Kernel α β) : Kernel α γ wh
   toFun a := (κ a).bind η
   measurable' := (Measure.measurable_bind' η.measurable).comp κ.measurable
 
+@[inherit_doc]
 scoped[ProbabilityTheory] infixl:100 " ∘ₖ " => ProbabilityTheory.Kernel.comp
 
 theorem comp_apply (η : Kernel β γ) (κ : Kernel α β) (a : α) : (η ∘ₖ κ) a = (κ a).bind η :=
@@ -1027,9 +1074,9 @@ theorem comp_assoc {δ : Type*} {mδ : MeasurableSpace δ} (ξ : Kernel γ δ) [
   simp_rw [lintegral_comp _ _ _ hf, lintegral_comp _ _ _ hf.lintegral_kernel]
 
 theorem deterministic_comp_eq_map (hf : Measurable f) (κ : Kernel α β) :
-    deterministic f hf ∘ₖ κ = map κ f hf := by
+    deterministic f hf ∘ₖ κ = map κ f := by
   ext a s hs
-  simp_rw [map_apply' _ _ _ hs, comp_apply' _ _ _ hs, deterministic_apply' hf _ hs,
+  simp_rw [map_apply' _ hf _ hs, comp_apply' _ _ _ hs, deterministic_apply' hf _ hs,
     lintegral_indicator_const_comp hf hs, one_mul]
 
 theorem comp_deterministic_eq_comap (κ : Kernel α β) (hg : Measurable g) :
@@ -1089,6 +1136,14 @@ theorem lintegral_prod (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel α
 instance IsMarkovKernel.prod (κ : Kernel α β) [IsMarkovKernel κ] (η : Kernel α γ)
     [IsMarkovKernel η] : IsMarkovKernel (κ ×ₖ η) := by rw [Kernel.prod]; infer_instance
 
+nonrec instance IsZeroOrMarkovKernel.prod (κ : Kernel α β) [h : IsZeroOrMarkovKernel κ]
+    (η : Kernel α γ) [IsZeroOrMarkovKernel η] : IsZeroOrMarkovKernel (κ ×ₖ η) := by
+  rcases eq_zero_or_isMarkovKernel κ with rfl | h
+  · simp only [prod, swapLeft_prodMkLeft, compProd_zero_left]; infer_instance
+  rcases eq_zero_or_isMarkovKernel η with rfl | h'
+  · simp only [prod, swapLeft, prodMkLeft_zero, comap_zero, compProd_zero_right]; infer_instance
+  infer_instance
+
 instance IsFiniteKernel.prod (κ : Kernel α β) [IsFiniteKernel κ] (η : Kernel α γ)
     [IsFiniteKernel η] : IsFiniteKernel (κ ×ₖ η) := by rw [Kernel.prod]; infer_instance
 
diff --git a/Mathlib/Probability/Kernel/Defs.lean b/Mathlib/Probability/Kernel/Defs.lean
new file mode 100644
index 0000000000000..707e45278f5bc
--- /dev/null
+++ b/Mathlib/Probability/Kernel/Defs.lean
@@ -0,0 +1,354 @@
+/-
+Copyright (c) 2022 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne
+-/
+import Mathlib.MeasureTheory.Measure.GiryMonad
+
+/-!
+# Markov Kernels
+
+A kernel from a measurable space `α` to another measurable space `β` is a measurable map
+`α → MeasureTheory.Measure β`, where the measurable space instance on `measure β` is the one defined
+in `MeasureTheory.Measure.instMeasurableSpace`. That is, a kernel `κ` verifies that for all
+measurable sets `s` of `β`, `a ↦ κ a s` is measurable.
+
+## Main definitions
+
+Classes of kernels:
+* `ProbabilityTheory.Kernel α β`: kernels from `α` to `β`.
+* `ProbabilityTheory.IsMarkovKernel κ`: a kernel from `α` to `β` is said to be a Markov kernel
+  if for all `a : α`, `k a` is a probability measure.
+* `ProbabilityTheory.IsZeroOrMarkovKernel κ`: a kernel from `α` to `β` which is zero or
+  a Markov kernel.
+* `ProbabilityTheory.IsFiniteKernel κ`: a kernel from `α` to `β` is said to be finite if there
+  exists `C : ℝ≥0∞` such that `C < ∞` and for all `a : α`, `κ a univ ≤ C`. This implies in
+  particular that all measures in the image of `κ` are finite, but is stronger since it requires a
+  uniform bound. This stronger condition is necessary to ensure that the composition of two finite
+  kernels is finite.
+* `ProbabilityTheory.IsSFiniteKernel κ`: a kernel is called s-finite if it is a countable
+  sum of finite kernels.
+
+## Main statements
+
+* `ProbabilityTheory.Kernel.ext_fun`: if `∫⁻ b, f b ∂(κ a) = ∫⁻ b, f b ∂(η a)` for all measurable
+  functions `f` and all `a`, then the two kernels `κ` and `η` are equal.
+
+-/
+
+assert_not_exists MeasureTheory.integral
+
+open MeasureTheory
+
+open scoped ENNReal
+
+namespace ProbabilityTheory
+
+/-- A kernel from a measurable space `α` to another measurable space `β` is a measurable function
+`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
+`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
+iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`. -/
+structure Kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] where
+  /-- The underlying function of a kernel.
+
+  Do not use this function directly. Instead use the coercion coming from the `DFunLike`
+  instance. -/
+  toFun : α → Measure β
+  /-- A kernel is a measurable map.
+
+  Do not use this lemma directly. Use `Kernel.measurable` instead. -/
+  measurable' : Measurable toFun
+
+@[deprecated (since := "2024-07-22")] alias kernel := Kernel
+
+/-- Notation for `Kernel` with respect to a non-standard σ-algebra in the domain. -/
+scoped notation "Kernel[" mα "]" α:arg β:arg => @Kernel α β mα _
+
+/-- Notation for `Kernel` with respect to a non-standard σ-algebra in the domain and codomain. -/
+scoped notation "Kernel[" mα ", " mβ "]" α:arg β:arg => @Kernel α β mα mβ
+
+variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+
+namespace Kernel
+
+instance instFunLike : FunLike (Kernel α β) α (Measure β) where
+  coe := toFun
+  coe_injective' f g h := by cases f; cases g; congr
+
+lemma measurable (κ : Kernel α β) : Measurable κ := κ.measurable'
+@[simp, norm_cast] lemma coe_mk (f : α → Measure β) (hf) : mk f hf = f := rfl
+
+initialize_simps_projections Kernel (toFun → apply)
+
+instance instZero : Zero (Kernel α β) where zero := ⟨0, measurable_zero⟩
+noncomputable instance instAdd : Add (Kernel α β) where add κ η := ⟨κ + η, κ.2.add η.2⟩
+noncomputable instance instSMulNat : SMul ℕ (Kernel α β) where
+  smul n κ := ⟨n • κ, (measurable_const (a := n)).smul κ.2⟩
+
+@[simp, norm_cast] lemma coe_zero : ⇑(0 : Kernel α β) = 0 := rfl
+@[simp, norm_cast] lemma coe_add (κ η : Kernel α β) : ⇑(κ + η) = κ + η := rfl
+@[simp, norm_cast] lemma coe_nsmul (n : ℕ) (κ : Kernel α β) : ⇑(n • κ) = n • κ := rfl
+
+@[simp] lemma zero_apply (a : α) : (0 : Kernel α β) a = 0 := rfl
+@[simp] lemma add_apply (κ η : Kernel α β) (a : α) : (κ + η) a = κ a + η a := rfl
+@[simp] lemma nsmul_apply (n : ℕ) (κ : Kernel α β) (a : α) : (n • κ) a = n • κ a := rfl
+
+noncomputable instance instAddCommMonoid : AddCommMonoid (Kernel α β) :=
+  DFunLike.coe_injective.addCommMonoid _ coe_zero coe_add (by intros; rfl)
+
+instance instPartialOrder : PartialOrder (Kernel α β) := .lift _ DFunLike.coe_injective
+
+instance instCovariantAddLE {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
+    CovariantClass (Kernel α β) (Kernel α β) (· + ·) (· ≤ ·) :=
+  ⟨fun _ _ _ hμ a ↦ add_le_add_left (hμ a) _⟩
+
+noncomputable
+instance instOrderBot {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
+    OrderBot (Kernel α β) where
+  bot := 0
+  bot_le κ a := by simp only [coe_zero, Pi.zero_apply, Measure.zero_le]
+
+/-- Coercion to a function as an additive monoid homomorphism. -/
+def coeAddHom (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
+    Kernel α β →+ α → Measure β where
+  toFun := (⇑)
+  map_zero' := coe_zero
+  map_add' := coe_add
+
+@[simp]
+theorem coe_finset_sum (I : Finset ι) (κ : ι → Kernel α β) : ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i) :=
+  map_sum (coeAddHom α β) _ _
+
+theorem finset_sum_apply (I : Finset ι) (κ : ι → Kernel α β) (a : α) :
+    (∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by rw [coe_finset_sum, Finset.sum_apply]
+
+theorem finset_sum_apply' (I : Finset ι) (κ : ι → Kernel α β) (a : α) (s : Set β) :
+    (∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by rw [finset_sum_apply, Measure.finset_sum_apply]
+
+end Kernel
+
+/-- A kernel is a Markov kernel if every measure in its image is a probability measure. -/
+class IsMarkovKernel (κ : Kernel α β) : Prop where
+  isProbabilityMeasure : ∀ a, IsProbabilityMeasure (κ a)
+
+/-- A class for kernels which are zero or a Markov kernel. -/
+class IsZeroOrMarkovKernel (κ : Kernel α β) : Prop where
+  eq_zero_or_isMarkovKernel' : κ = 0 ∨ IsMarkovKernel κ
+
+/-- A kernel is finite if every measure in its image is finite, with a uniform bound. -/
+class IsFiniteKernel (κ : Kernel α β) : Prop where
+  exists_univ_le : ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ a, κ a Set.univ ≤ C
+
+theorem eq_zero_or_isMarkovKernel
+    (κ : Kernel α β) [h : IsZeroOrMarkovKernel κ] :
+    κ = 0 ∨ IsMarkovKernel κ :=
+  h.eq_zero_or_isMarkovKernel'
+
+/-- A constant `C : ℝ≥0∞` such that `C < ∞` (`ProbabilityTheory.IsFiniteKernel.bound_lt_top κ`) and
+for all `a : α` and `s : Set β`, `κ a s ≤ C` (`ProbabilityTheory.Kernel.measure_le_bound κ a s`).
+
+Porting note (#11215): TODO: does it make sense to
+-- make `ProbabilityTheory.IsFiniteKernel.bound` the least possible bound?
+-- Should it be an `NNReal` number? -/
+noncomputable def IsFiniteKernel.bound (κ : Kernel α β) [h : IsFiniteKernel κ] : ℝ≥0∞ :=
+  h.exists_univ_le.choose
+
+theorem IsFiniteKernel.bound_lt_top (κ : Kernel α β) [h : IsFiniteKernel κ] :
+    IsFiniteKernel.bound κ < ∞ :=
+  h.exists_univ_le.choose_spec.1
+
+theorem IsFiniteKernel.bound_ne_top (κ : Kernel α β) [IsFiniteKernel κ] :
+    IsFiniteKernel.bound κ ≠ ∞ :=
+  (IsFiniteKernel.bound_lt_top κ).ne
+
+theorem Kernel.measure_le_bound (κ : Kernel α β) [h : IsFiniteKernel κ] (a : α) (s : Set β) :
+    κ a s ≤ IsFiniteKernel.bound κ :=
+  (measure_mono (Set.subset_univ s)).trans (h.exists_univ_le.choose_spec.2 a)
+
+instance isFiniteKernel_zero (α β : Type*) {mα : MeasurableSpace α} {mβ : MeasurableSpace β} :
+    IsFiniteKernel (0 : Kernel α β) :=
+  ⟨⟨0, ENNReal.coe_lt_top, fun _ => by
+      simp only [Kernel.zero_apply, Measure.coe_zero, Pi.zero_apply, le_zero_iff]⟩⟩
+
+instance IsFiniteKernel.add (κ η : Kernel α β) [IsFiniteKernel κ] [IsFiniteKernel η] :
+    IsFiniteKernel (κ + η) := by
+  refine ⟨⟨IsFiniteKernel.bound κ + IsFiniteKernel.bound η,
+    ENNReal.add_lt_top.mpr ⟨IsFiniteKernel.bound_lt_top κ, IsFiniteKernel.bound_lt_top η⟩,
+    fun a => ?_⟩⟩
+  exact add_le_add (Kernel.measure_le_bound _ _ _) (Kernel.measure_le_bound _ _ _)
+
+lemma isFiniteKernel_of_le {κ ν : Kernel α β} [hν : IsFiniteKernel ν] (hκν : κ ≤ ν) :
+    IsFiniteKernel κ := by
+  refine ⟨hν.bound, hν.bound_lt_top, fun a ↦ (hκν _ _).trans (Kernel.measure_le_bound ν a Set.univ)⟩
+
+variable {κ η : Kernel α β}
+
+instance IsMarkovKernel.is_probability_measure' [IsMarkovKernel κ] (a : α) :
+    IsProbabilityMeasure (κ a) :=
+  IsMarkovKernel.isProbabilityMeasure a
+
+instance : IsZeroOrMarkovKernel (0 : Kernel α β) := ⟨Or.inl rfl⟩
+
+instance (priority := 100) IsMarkovKernel.IsZeroOrMarkovKernel [h : IsMarkovKernel κ] :
+    IsZeroOrMarkovKernel κ := ⟨Or.inr h⟩
+
+instance (priority := 100) IsZeroOrMarkovKernel.isZeroOrProbabilityMeasure
+    [IsZeroOrMarkovKernel κ] (a : α) : IsZeroOrProbabilityMeasure (κ a) := by
+  rcases eq_zero_or_isMarkovKernel κ with rfl | h'
+  · simp only [Kernel.zero_apply]
+    infer_instance
+  · infer_instance
+
+instance IsFiniteKernel.isFiniteMeasure [IsFiniteKernel κ] (a : α) : IsFiniteMeasure (κ a) :=
+  ⟨(Kernel.measure_le_bound κ a Set.univ).trans_lt (IsFiniteKernel.bound_lt_top κ)⟩
+
+instance (priority := 100) IsZeroOrMarkovKernel.isFiniteKernel [h : IsZeroOrMarkovKernel κ] :
+    IsFiniteKernel κ := by
+  rcases eq_zero_or_isMarkovKernel κ with rfl | _h'
+  · infer_instance
+  · exact ⟨⟨1, ENNReal.one_lt_top, fun _ => prob_le_one⟩⟩
+
+namespace Kernel
+
+@[ext]
+theorem ext (h : ∀ a, κ a = η a) : κ = η := DFunLike.ext _ _ h
+
+theorem ext_iff' : κ = η ↔ ∀ a s, MeasurableSet s → κ a s = η a s := by
+  simp_rw [Kernel.ext_iff, Measure.ext_iff]
+
+theorem ext_fun (h : ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a) :
+    κ = η := by
+  ext a s hs
+  specialize h a (s.indicator fun _ => 1) (Measurable.indicator measurable_const hs)
+  simp_rw [lintegral_indicator_const hs, one_mul] at h
+  rw [h]
+
+theorem ext_fun_iff : κ = η ↔ ∀ a f, Measurable f → ∫⁻ b, f b ∂κ a = ∫⁻ b, f b ∂η a :=
+  ⟨fun h a f _ => by rw [h], ext_fun⟩
+
+protected theorem measurable_coe (κ : Kernel α β) {s : Set β} (hs : MeasurableSet s) :
+    Measurable fun a => κ a s :=
+  (Measure.measurable_coe hs).comp κ.measurable
+
+lemma apply_congr_of_mem_measurableAtom (κ : Kernel α β) {y' y : α} (hy' : y' ∈ measurableAtom y) :
+    κ y' = κ y := by
+  ext s hs
+  exact mem_of_mem_measurableAtom hy' (κ.measurable_coe hs (measurableSet_singleton (κ y s))) rfl
+
+section Sum
+
+/-- Sum of an indexed family of kernels. -/
+protected noncomputable def sum [Countable ι] (κ : ι → Kernel α β) : Kernel α β where
+  toFun a := Measure.sum fun n => κ n a
+  measurable' := by
+    refine Measure.measurable_of_measurable_coe _ fun s hs => ?_
+    simp_rw [Measure.sum_apply _ hs]
+    exact Measurable.ennreal_tsum fun n => Kernel.measurable_coe (κ n) hs
+
+theorem sum_apply [Countable ι] (κ : ι → Kernel α β) (a : α) :
+    Kernel.sum κ a = Measure.sum fun n => κ n a :=
+  rfl
+
+theorem sum_apply' [Countable ι] (κ : ι → Kernel α β) (a : α) {s : Set β} (hs : MeasurableSet s) :
+    Kernel.sum κ a s = ∑' n, κ n a s := by rw [sum_apply κ a, Measure.sum_apply _ hs]
+
+@[simp]
+theorem sum_zero [Countable ι] : (Kernel.sum fun _ : ι => (0 : Kernel α β)) = 0 := by
+  ext a s hs
+  rw [sum_apply' _ a hs]
+  simp only [zero_apply, Measure.coe_zero, Pi.zero_apply, tsum_zero]
+
+theorem sum_comm [Countable ι] (κ : ι → ι → Kernel α β) :
+    (Kernel.sum fun n => Kernel.sum (κ n)) = Kernel.sum fun m => Kernel.sum fun n => κ n m := by
+  ext a s; simp_rw [sum_apply]; rw [Measure.sum_comm]
+
+@[simp]
+theorem sum_fintype [Fintype ι] (κ : ι → Kernel α β) : Kernel.sum κ = ∑ i, κ i := by
+  ext a s hs
+  simp only [sum_apply' κ a hs, finset_sum_apply' _ κ a s, tsum_fintype]
+
+theorem sum_add [Countable ι] (κ η : ι → Kernel α β) :
+    (Kernel.sum fun n => κ n + η n) = Kernel.sum κ + Kernel.sum η := by
+  ext a s hs
+  simp only [coe_add, Pi.add_apply, sum_apply, Measure.sum_apply _ hs, Pi.add_apply,
+    Measure.coe_add, tsum_add ENNReal.summable ENNReal.summable]
+
+end Sum
+
+section SFinite
+
+/-- A kernel is s-finite if it can be written as the sum of countably many finite kernels. -/
+class _root_.ProbabilityTheory.IsSFiniteKernel (κ : Kernel α β) : Prop where
+  tsum_finite : ∃ κs : ℕ → Kernel α β, (∀ n, IsFiniteKernel (κs n)) ∧ κ = Kernel.sum κs
+
+instance (priority := 100) IsFiniteKernel.isSFiniteKernel [h : IsFiniteKernel κ] :
+    IsSFiniteKernel κ :=
+  ⟨⟨fun n => if n = 0 then κ else 0, fun n => by
+      simp only; split_ifs
+      · exact h
+      · infer_instance, by
+      ext a s hs
+      rw [Kernel.sum_apply' _ _ hs]
+      have : (fun i => ((ite (i = 0) κ 0) a) s) = fun i => ite (i = 0) (κ a s) 0 := by
+        ext1 i; split_ifs <;> rfl
+      rw [this, tsum_ite_eq]⟩⟩
+
+/-- A sequence of finite kernels such that `κ = ProbabilityTheory.Kernel.sum (seq κ)`. See
+`ProbabilityTheory.Kernel.isFiniteKernel_seq` and `ProbabilityTheory.Kernel.kernel_sum_seq`. -/
+noncomputable def seq (κ : Kernel α β) [h : IsSFiniteKernel κ] : ℕ → Kernel α β :=
+  h.tsum_finite.choose
+
+theorem kernel_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] : Kernel.sum (seq κ) = κ :=
+  h.tsum_finite.choose_spec.2.symm
+
+theorem measure_sum_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] (a : α) :
+    (Measure.sum fun n => seq κ n a) = κ a := by rw [← Kernel.sum_apply, kernel_sum_seq κ]
+
+instance isFiniteKernel_seq (κ : Kernel α β) [h : IsSFiniteKernel κ] (n : ℕ) :
+    IsFiniteKernel (Kernel.seq κ n) :=
+  h.tsum_finite.choose_spec.1 n
+
+instance _root_.ProbabilityTheory.IsSFiniteKernel.sFinite [IsSFiniteKernel κ] (a : α) :
+    SFinite (κ a) :=
+  ⟨⟨fun n ↦ seq κ n a, inferInstance, (measure_sum_seq κ a).symm⟩⟩
+
+instance IsSFiniteKernel.add (κ η : Kernel α β) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    IsSFiniteKernel (κ + η) := by
+  refine ⟨⟨fun n => seq κ n + seq η n, fun n => inferInstance, ?_⟩⟩
+  rw [sum_add, kernel_sum_seq κ, kernel_sum_seq η]
+
+theorem IsSFiniteKernel.finset_sum {κs : ι → Kernel α β} (I : Finset ι)
+    (h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i ∈ I, κs i) := by
+  classical
+  induction' I using Finset.induction with i I hi_nmem_I h_ind h
+  · rw [Finset.sum_empty]; infer_instance
+  · rw [Finset.sum_insert hi_nmem_I]
+    haveI : IsSFiniteKernel (κs i) := h i (Finset.mem_insert_self _ _)
+    have : IsSFiniteKernel (∑ x ∈ I, κs x) :=
+      h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)
+    exact IsSFiniteKernel.add _ _
+
+theorem isSFiniteKernel_sum_of_denumerable [Denumerable ι] {κs : ι → Kernel α β}
+    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) := by
+  let e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm
+  refine ⟨⟨fun n => seq (κs (e n).1) (e n).2, inferInstance, ?_⟩⟩
+  have hκ_eq : Kernel.sum κs = Kernel.sum fun n => Kernel.sum (seq (κs n)) := by
+    simp_rw [kernel_sum_seq]
+  ext a s hs
+  rw [hκ_eq]
+  simp_rw [Kernel.sum_apply' _ _ hs]
+  change (∑' i, ∑' m, seq (κs i) m a s) = ∑' n, (fun im : ι × ℕ => seq (κs im.fst) im.snd a s) (e n)
+  rw [e.tsum_eq (fun im : ι × ℕ => seq (κs im.fst) im.snd a s),
+    tsum_prod' ENNReal.summable fun _ => ENNReal.summable]
+
+theorem isSFiniteKernel_sum [Countable ι] {κs : ι → Kernel α β}
+    (hκs : ∀ n, IsSFiniteKernel (κs n)) : IsSFiniteKernel (Kernel.sum κs) := by
+  cases fintypeOrInfinite ι
+  · rw [sum_fintype]
+    exact IsSFiniteKernel.finset_sum Finset.univ fun i _ => hκs i
+  cases nonempty_denumerable ι
+  exact isSFiniteKernel_sum_of_denumerable hκs
+
+end SFinite
+end Kernel
+end ProbabilityTheory
diff --git a/Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean b/Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean
index a2a9b77161b60..cd5425670e316 100644
--- a/Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/CDFToKernel.lean
@@ -63,7 +63,8 @@ to `ν` if is measurable, if `fun b ↦ f (a, b) x` is `(ν a)`-integrable for a
 and for all measurable sets `s : Set β`, `∫ b in s, f (a, b) x ∂(ν a) = (κ a (s ×ˢ Iic x)).toReal`.
 Also the `ℚ → ℝ` function `f (a, b)` should satisfy the properties of a Sieltjes function for
 `(ν a)`-almost all `b : β`. -/
-structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) : Prop :=
+structure IsRatCondKernelCDF (f : α × β → ℚ → ℝ) (κ : Kernel α (β × ℝ)) (ν : Kernel α β) :
+    Prop where
   measurable : Measurable f
   isRatStieltjesPoint_ae (a : α) : ∀ᵐ b ∂(ν a), IsRatStieltjesPoint f (a, b)
   integrable (a : α) (q : ℚ) : Integrable (fun b ↦ f (a, b) q) (ν a)
@@ -147,7 +148,7 @@ lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondK
       · exact mod_cast ha.le
       · refine le_of_forall_lt_rat_imp_le fun q hq ↦ h q ?_
         exact mod_cast hq
-    · exact fun _ ↦ measurableSet_Iic
+    · exact fun _ ↦ nullMeasurableSet_Iic
     · refine Monotone.directed_ge fun r r' hrr' ↦ Iic_subset_Iic.mpr ?_
       exact mod_cast hrr'
     · obtain ⟨q, hq⟩ := exists_rat_gt x
@@ -172,7 +173,7 @@ lemma setLIntegral_stieltjesOfMeasurableRat [IsFiniteKernel κ] (hf : IsRatCondK
     congr with y
     simp only [mem_iInter, mem_Iic, Subtype.forall, Subtype.coe_mk]
     exact ⟨le_of_forall_lt_rat_imp_le, fun hyx q hq ↦ hyx.trans hq.le⟩
-  · exact fun i ↦ hs.prod measurableSet_Iic
+  · exact fun i ↦ (hs.prod measurableSet_Iic).nullMeasurableSet
   · refine Monotone.directed_ge fun i j hij ↦ ?_
     refine prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩)
     exact mod_cast hij
@@ -355,7 +356,7 @@ lemma _root_.MeasureTheory.Measure.iInf_rat_gt_prod_Iic {ρ : Measure (α × ℝ
     · refine le_of_forall_lt_rat_imp_le fun q htq ↦ h q ?_
       exact mod_cast htq
     · exact mod_cast hta.le
-  · exact fun _ => hs.prod measurableSet_Iic
+  · exact fun _ => (hs.prod measurableSet_Iic).nullMeasurableSet
   · refine Monotone.directed_ge fun r r' hrr' ↦ prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, ?_⟩)
     refine Iic_subset_Iic.mpr ?_
     exact mod_cast hrr'
@@ -550,7 +551,7 @@ lemma setLIntegral_toKernel_univ [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ
     refine Monotone.directed_le fun i j hij ↦ ?_
     refine prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩)
     exact mod_cast hij
-  simp_rw [measure_iUnion_eq_iSup h_dir, measure_iUnion_eq_iSup h_dir_prod]
+  simp_rw [h_dir.measure_iUnion, h_dir_prod.measure_iUnion]
   rw [lintegral_iSup_directed]
   · simp_rw [setLIntegral_toKernel_Iic hf _ _ hs]
   · refine fun q ↦ Measurable.aemeasurable ?_
@@ -589,7 +590,7 @@ lemma setLIntegral_toKernel_prod [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ
     _ = κ a (s ×ˢ univ) - κ a (s ×ˢ t) := by
         rw [setLIntegral_toKernel_univ hf a hs, ht_lintegral]
     _ = κ a (s ×ˢ tᶜ) := by
-        rw [← measure_diff _ (hs.prod ht) (measure_ne_top _ _)]
+        rw [← measure_diff _ (hs.prod ht).nullMeasurableSet (measure_ne_top _ _)]
         · rw [prod_diff_prod, compl_eq_univ_diff]
           simp only [diff_self, empty_prod, union_empty]
         · rw [prod_subset_prod_iff]
@@ -621,7 +622,7 @@ lemma lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
     simp only [mem_setOf_eq] at ht₁ ht₂
     have h_prod_eq_snd : ∀ a ∈ t₁, {x : ℝ | (a, x) ∈ t₁ ×ˢ t₂} = t₂ := by
       intro a ha
-      simp only [ha, prod_mk_mem_set_prod_eq, true_and_iff, setOf_mem_eq]
+      simp only [ha, prod_mk_mem_set_prod_eq, true_and, setOf_mem_eq]
     rw [← lintegral_add_compl _ ht₁]
     have h_eq1 : ∫⁻ x in t₁, hf.toKernel f (a, x) {y : ℝ | (x, y) ∈ t₁ ×ˢ t₂} ∂(ν a)
         = ∫⁻ x in t₁, hf.toKernel f (a, x) t₂ ∂(ν a) := by
@@ -635,7 +636,7 @@ lemma lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
         simp only [lintegral_const, zero_mul]
       intro a hat₁
       rw [mem_compl_iff] at hat₁
-      simp only [hat₁, prod_mk_mem_set_prod_eq, false_and_iff, setOf_false, measure_empty]
+      simp only [hat₁, prod_mk_mem_set_prod_eq, false_and, setOf_false, measure_empty]
     rw [h_eq1, h_eq2, add_zero]
     exact setLIntegral_toKernel_prod hf a ht₁ ht₂
   · intro t ht ht_eq
@@ -667,7 +668,7 @@ lemma lintegral_toKernel_mem [IsFiniteKernel κ] (hf : IsCondKernelCDF f κ ν)
       have h_disj := hf_disj hij
       rw [Function.onFun, disjoint_iff_inter_eq_empty] at h_disj ⊢
       ext1 x
-      simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false_iff]
+      simp only [mem_inter_iff, mem_setOf_eq, mem_empty_iff_false, iff_false]
       intro h_mem_both
       suffices (a, x) ∈ ∅ by rwa [mem_empty_iff_false] at this
       rwa [← h_disj, mem_inter_iff]
diff --git a/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean b/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
index 0659092ade23a..22bfd49d0e410 100644
--- a/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
@@ -50,23 +50,18 @@ noncomputable def IicSnd (r : ℝ) : Measure α :=
 
 theorem IicSnd_apply (r : ℝ) {s : Set α} (hs : MeasurableSet s) :
     ρ.IicSnd r s = ρ (s ×ˢ Iic r) := by
-  rw [IicSnd, fst_apply hs,
-    restrict_apply' (MeasurableSet.univ.prod (measurableSet_Iic : MeasurableSet (Iic r))), ←
-    prod_univ, prod_inter_prod, inter_univ, univ_inter]
+  rw [IicSnd, fst_apply hs, restrict_apply' (MeasurableSet.univ.prod measurableSet_Iic),
+    univ_prod, Set.prod_eq]
 
 theorem IicSnd_univ (r : ℝ) : ρ.IicSnd r univ = ρ (univ ×ˢ Iic r) :=
   IicSnd_apply ρ r MeasurableSet.univ
 
+@[gcongr]
 theorem IicSnd_mono {r r' : ℝ} (h_le : r ≤ r') : ρ.IicSnd r ≤ ρ.IicSnd r' := by
-  refine Measure.le_iff.2 fun s hs ↦ ?_
-  simp_rw [IicSnd_apply ρ _ hs]
-  refine measure_mono (prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, Iic_subset_Iic.mpr ?_⟩))
-  exact mod_cast h_le
+  unfold IicSnd; gcongr
 
-theorem IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst := by
-  refine Measure.le_iff.2 fun s hs ↦ ?_
-  simp_rw [fst_apply hs, IicSnd_apply ρ r hs]
-  exact measure_mono (prod_subset_preimage_fst _ _)
+theorem IicSnd_le_fst (r : ℝ) : ρ.IicSnd r ≤ ρ.fst :=
+  fst_mono restrict_le_self
 
 theorem IicSnd_ac_fst (r : ℝ) : ρ.IicSnd r ≪ ρ.fst :=
   Measure.absolutelyContinuous_of_le (IicSnd_le_fst ρ r)
@@ -83,10 +78,8 @@ theorem tendsto_IicSnd_atTop {s : Set α} (hs : MeasurableSet s) :
     Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atTop (𝓝 (ρ.fst s)) := by
   simp_rw [ρ.IicSnd_apply _ hs, fst_apply hs, ← prod_univ]
   rw [← Real.iUnion_Iic_rat, prod_iUnion]
-  refine tendsto_measure_iUnion fun r q hr_le_q x ↦ ?_
-  simp only [mem_prod, mem_Iic, and_imp]
-  refine fun hxs hxr ↦ ⟨hxs, hxr.trans ?_⟩
-  exact mod_cast hr_le_q
+  apply tendsto_measure_iUnion_atTop
+  exact monotone_const.set_prod Rat.cast_mono.Iic
 
 theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableSet s) :
     Tendsto (fun r : ℚ ↦ ρ.IicSnd r s) atBot (𝓝 0) := by
@@ -106,8 +99,9 @@ theorem tendsto_IicSnd_atBot [IsFiniteMeasure ρ] {s : Set α} (hs : MeasurableS
       simp_rw [neg_neg]
     rw [h_fun_eq]
     exact h_neg.comp tendsto_neg_atBot_atTop
-  refine tendsto_measure_iInter (fun q ↦ hs.prod measurableSet_Iic) ?_ ⟨0, measure_ne_top ρ _⟩
-  refine fun q r hqr ↦ prod_subset_prod_iff.mpr (Or.inl ⟨subset_rfl, fun x hx ↦ ?_⟩)
+  refine tendsto_measure_iInter (fun q ↦ (hs.prod measurableSet_Iic).nullMeasurableSet)
+    ?_ ⟨0, measure_ne_top ρ _⟩
+  refine fun q r hqr ↦ Set.prod_mono subset_rfl fun x hx ↦ ?_
   simp only [Rat.cast_neg, mem_Iic] at hx ⊢
   refine hx.trans (neg_le_neg ?_)
   exact mod_cast hqr
@@ -126,7 +120,7 @@ attribute [local instance] MeasureTheory.Measure.IsFiniteMeasure.IicSnd
 
 We build towards the definition of `ProbabilityTheory.condCDF`. We first define
 `ProbabilityTheory.preCDF`, a function defined on `α × ℚ` with the properties of a cdf almost
-everywhere.  -/
+everywhere. -/
 
 /-- `preCDF` is the Radon-Nikodym derivative of `ρ.IicSnd` with respect to `ρ.fst` at each
 `r : ℚ`. This function `ℚ → α → ℝ≥0∞` is such that for almost all `a : α`, the function `ℚ → ℝ≥0∞`
diff --git a/Mathlib/Probability/Kernel/Disintegration/Density.lean b/Mathlib/Probability/Kernel/Disintegration/Density.lean
index 8b2f314be1286..1e1d4fe17ac12 100644
--- a/Mathlib/Probability/Kernel/Disintegration/Density.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/Density.lean
@@ -385,7 +385,7 @@ lemma tendsto_densityProcess_atTop_empty_of_antitone (κ : Kernel α (γ × β))
       ?_ ?_ ?_
     · convert h
       rw [← prod_iInter, hseq_iInter]
-    · exact fun m ↦ MeasurableSet.prod (measurableSet_countablePartitionSet _ _) (hseq_meas m)
+    · exact fun m ↦ ((measurableSet_countablePartitionSet _ _).prod (hseq_meas m)).nullMeasurableSet
     · intro m m' hmm'
       simp only [le_eq_subset, prod_subset_prod_iff, subset_rfl, true_and]
       exact Or.inl <| hseq hmm'
@@ -599,7 +599,7 @@ lemma setIntegral_density (hκν : fst κ ≤ ν) [IsFiniteKernel ν]
     have : Aᶜ ×ˢ s = univ ×ˢ s \ A ×ˢ s := by
       rw [prod_diff_prod, compl_eq_univ_diff]
       simp
-    rw [this, measure_diff (by intro x; simp) (hA.prod hs) (measure_ne_top (κ a) _),
+    rw [this, measure_diff (by intro; simp) (hA.prod hs).nullMeasurableSet (measure_ne_top (κ a) _),
       ENNReal.toReal_sub_of_le (measure_mono (by intro x; simp)) (measure_ne_top _ _)]
     rw [eq_tsub_iff_add_eq_of_le, add_comm]
     · exact h
@@ -656,11 +656,8 @@ lemma tendsto_integral_density_of_monotone (hκν : fst κ ≤ ν) [IsFiniteKern
     · simp only [mem_Iio]
       exact ENNReal.lt_add_right (measure_ne_top _ _) one_ne_zero
   refine h_cont.tendsto.comp ?_
-  have h := tendsto_measure_iUnion (s := fun m ↦ univ ×ˢ seq m) (μ := κ a) ?_
-  swap; · intro n m hnm x; simp only [mem_prod, mem_univ, true_and]; exact fun h ↦ hseq hnm h
-  convert h
-  rw [← prod_iUnion, hseq_iUnion]
-  simp only [univ_prod_univ, measure_univ]
+  convert tendsto_measure_iUnion_atTop (monotone_const.set_prod hseq)
+  rw [← prod_iUnion, hseq_iUnion, univ_prod_univ]
 
 lemma tendsto_integral_density_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKernel ν] (a : α)
     (seq : ℕ → Set β) (hseq : Antitone seq) (hseq_iInter : ⋂ i, seq i = ∅)
@@ -677,7 +674,7 @@ lemma tendsto_integral_density_of_antitone (hκν : fst κ ≤ ν) [IsFiniteKern
     · simp
   refine h_cont.tendsto.comp ?_
   have h := tendsto_measure_iInter (s := fun m ↦ univ ×ˢ seq m) (μ := κ a)
-    (fun m ↦ MeasurableSet.univ.prod (hseq_meas m)) ?_ ?_
+    (fun m ↦ (MeasurableSet.univ.prod (hseq_meas m)).nullMeasurableSet) ?_ ?_
   rotate_left
   · intro n m hnm x; simp only [mem_prod, mem_univ, true_and]; exact fun h ↦ hseq hnm h
   · refine ⟨0, measure_ne_top _ _⟩
@@ -776,13 +773,7 @@ lemma tendsto_densityProcess_fst_atTop_univ_of_monotone (κ : Kernel α (γ × 
     simp only [mem_prod, mem_setOf_eq, and_imp]
     exact fun h _ ↦ h
   refine ENNReal.Tendsto.div_const ?_ ?_
-  · have h := tendsto_measure_iUnion (μ := κ a)
-      (s := fun m ↦ countablePartitionSet n x ×ˢ seq m) ?_
-    swap
-    · intro m m' hmm'
-      simp only [le_eq_subset, prod_subset_prod_iff, subset_rfl, true_and]
-      exact Or.inl <| hseq hmm'
-    convert h
+  · convert tendsto_measure_iUnion_atTop (monotone_const.set_prod hseq)
     rw [← prod_iUnion, hseq_iUnion]
   · exact Or.inr h0
 
diff --git a/Mathlib/Probability/Kernel/Disintegration/Integral.lean b/Mathlib/Probability/Kernel/Disintegration/Integral.lean
index f712bc83a22b8..026257adcaf92 100644
--- a/Mathlib/Probability/Kernel/Disintegration/Integral.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/Integral.lean
@@ -146,12 +146,12 @@ end ProbabilityTheory
 
 namespace MeasureTheory.Measure
 
-variable {α β Ω : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+variable {β Ω : Type*} {mβ : MeasurableSpace β}
   [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω]
 
 section Lintegral
 
-variable [CountableOrCountablyGenerated α β] {ρ : Measure (β × Ω)} [IsFiniteMeasure ρ]
+variable {ρ : Measure (β × Ω)} [IsFiniteMeasure ρ]
   {f : β × Ω → ℝ≥0∞}
 
 lemma lintegral_condKernel_mem {s : Set (β × Ω)} (hs : MeasurableSet s) :
diff --git a/Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean b/Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
index 1659fdaec44bc..c9a1f8966a14e 100644
--- a/Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/StandardBorel.lean
@@ -234,22 +234,18 @@ instance instIsMarkovKernelBorelMarkovFromReal (η : Kernel α ℝ) [IsMarkovKer
   · rw [deterministic_apply]
     simp [(range_nonempty (embeddingReal Ω)).choose_spec]
 
-/-- For `κ' := map κ (Prod.map (id : β → β) e) (measurable_id.prod_map he.measurable)`, the
-hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`. The conclusion of the lemma is
-`fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) _`. -/
+/-- For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
+The conclusion of the lemma is `fst κ ⊗ₖ borelMarkovFromReal Ω η = comapRight (fst κ' ⊗ₖ η) _`. -/
 lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
     (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ] (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
-    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
-        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable))) ⊗ₖ η
-      = map κ (Prod.map (id : β → β) (embeddingReal Ω))
-        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) :
+    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω)))) ⊗ₖ η
+      = map κ (Prod.map (id : β → β) (embeddingReal Ω))) :
     fst κ ⊗ₖ borelMarkovFromReal Ω η
-      = comapRight (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
-          (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) ⊗ₖ η)
+      = comapRight (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))) ⊗ₖ η)
         (MeasurableEmbedding.id.prod_mk (measurableEmbedding_embeddingReal Ω)) := by
   let e := embeddingReal Ω
   let he := measurableEmbedding_embeddingReal Ω
-  let κ' := map κ (Prod.map (id : β → β) e) (measurable_id.prod_map he.measurable)
+  let κ' := map κ (Prod.map (id : β → β) e)
   have hη' : fst κ' ⊗ₖ η = κ' := hη
   have h_prod_embed : MeasurableEmbedding (Prod.map (id : β → β) e) :=
     MeasurableEmbedding.id.prod_mk he
@@ -258,7 +254,8 @@ lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
   have h_fst : fst κ' = fst κ := by
     ext a u
     unfold_let κ'
-    rw [fst_apply, map_apply, Measure.map_map measurable_fst h_prod_embed.measurable, fst_apply]
+    rw [fst_apply, map_apply _ (by fun_prop),
+      Measure.map_map measurable_fst h_prod_embed.measurable, fst_apply]
     congr
   rw [h_fst]
   ext a t ht : 2
@@ -268,7 +265,7 @@ lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
     rw [← h_fst]
     have h_compProd : κ' a (univ ×ˢ range e)ᶜ = 0 := by
       unfold_let κ'
-      rw [map_apply']
+      rw [map_apply' _ (by fun_prop)]
       swap; · exact (MeasurableSet.univ.prod he.measurableSet_range).compl
       suffices Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ = ∅ by rw [this]; simp
       ext x
@@ -286,26 +283,23 @@ lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
   rw [piecewise_apply, if_pos]
   exact ha
 
-/-- For `κ' := map κ (Prod.map (id : β → β) e) (measurable_id.prod_map he.measurable)`, the
-hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`. With that hypothesis,
-`fst κ ⊗ₖ borelMarkovFromReal κ η = κ`.-/
+/-- For `κ' := map κ (Prod.map (id : β → β) e)`, the hypothesis `hη` is `fst κ' ⊗ₖ η = κ'`.
+With that hypothesis, `fst κ ⊗ₖ borelMarkovFromReal κ η = κ`.-/
 lemma compProd_fst_borelMarkovFromReal (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ]
     (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
-    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
-        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable))) ⊗ₖ η
-      = map κ (Prod.map (id : β → β) (embeddingReal Ω))
-        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) :
+    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω)))) ⊗ₖ η
+      = map κ (Prod.map (id : β → β) (embeddingReal Ω))) :
     fst κ ⊗ₖ borelMarkovFromReal Ω η = κ := by
   let e := embeddingReal Ω
   let he := measurableEmbedding_embeddingReal Ω
-  let κ' := map κ (Prod.map (id : β → β) e) (measurable_id.prod_map he.measurable)
+  let κ' := map κ (Prod.map (id : β → β) e)
   have hη' : fst κ' ⊗ₖ η = κ' := hη
   have h_prod_embed : MeasurableEmbedding (Prod.map (id : β → β) e) :=
     MeasurableEmbedding.id.prod_mk he
   have : κ = comapRight κ' h_prod_embed := by
     ext c t : 2
     unfold_let κ'
-    rw [comapRight_apply, map_apply, h_prod_embed.comap_map]
+    rw [comapRight_apply, map_apply _ (by fun_prop), h_prod_embed.comap_map]
   conv_rhs => rw [this, ← hη']
   exact compProd_fst_borelMarkovFromReal_eq_comapRight_compProd κ η hη
 
@@ -320,9 +314,7 @@ A conditional kernel for `κ : Kernel α (γ × Ω)` where `γ` is countably gen
 standard Borel. -/
 noncomputable
 def condKernelBorel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] : Kernel (α × γ) Ω :=
-  let e := embeddingReal Ω
-  let he := measurableEmbedding_embeddingReal Ω
-  let κ' := map κ (Prod.map (id : γ → γ) e) (measurable_id.prod_map he.measurable)
+  let κ' := map κ (Prod.map (id : γ → γ) (embeddingReal Ω))
   borelMarkovFromReal Ω (condKernelReal κ')
 
 instance instIsMarkovKernelCondKernelBorel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] :
@@ -349,9 +341,7 @@ variable (κ : Kernel Unit (α × Ω)) [IsFiniteKernel κ]
 A conditional kernel for `κ : Kernel Unit (α × Ω)` where `Ω` is standard Borel. -/
 noncomputable
 def condKernelUnitBorel : Kernel (Unit × α) Ω :=
-  let e := embeddingReal Ω
-  let he := measurableEmbedding_embeddingReal Ω
-  let κ' := map κ (Prod.map (id : α → α) e) (measurable_id.prod_map he.measurable)
+  let κ' := map κ (Prod.map (id : α → α) (embeddingReal Ω))
   borelMarkovFromReal Ω (condKernelUnitReal κ')
 
 instance instIsMarkovKernelCondKernelUnitBorel : IsMarkovKernel κ.condKernelUnitBorel := by
diff --git a/Mathlib/Probability/Kernel/Disintegration/Unique.lean b/Mathlib/Probability/Kernel/Disintegration/Unique.lean
index a55d8224d6a44..e1b79719abfe1 100644
--- a/Mathlib/Probability/Kernel/Disintegration/Unique.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/Unique.lean
@@ -91,35 +91,35 @@ theorem eq_condKernel_of_measure_eq_compProd (κ : Kernel α Ω) [IsFiniteKernel
     rw [hρ'def, Measure.fst_apply, Measure.fst_apply, Measure.map_apply]
     exacts [rfl, Measurable.prod measurable_fst <| hf.measurable.comp measurable_snd,
       measurable_fst hs, hs, hs]
-  have hρ'' : ∀ᵐ x ∂ρ.fst, Kernel.map κ f hf.measurable x = ρ'.condKernel x := by
+  have hρ'' : ∀ᵐ x ∂ρ.fst, Kernel.map κ f x = ρ'.condKernel x := by
     rw [← hρ']
-    refine eq_condKernel_of_measure_eq_compProd_real (Kernel.map κ f hf.measurable) ?_
+    refine eq_condKernel_of_measure_eq_compProd_real (Kernel.map κ f) ?_
     ext s hs
     conv_lhs => rw [hρ'def, hκ]
     rw [Measure.map_apply (measurable_id.prod_map hf.measurable) hs, hρ',
       Measure.compProd_apply hs, Measure.compProd_apply (measurable_id.prod_map hf.measurable hs)]
     congr with a
-    rw [Kernel.map_apply']
+    rw [Kernel.map_apply' _ hf.measurable]
     exacts [rfl, measurable_prod_mk_left hs]
   suffices ∀ᵐ x ∂ρ.fst, ∀ s, MeasurableSet s → ρ'.condKernel x s = ρ.condKernel x (f ⁻¹' s) by
     filter_upwards [hρ'', this] with x hx h
-    rw [Kernel.map_apply] at hx
+    rw [Kernel.map_apply _ hf.measurable] at hx
     ext s hs
     rw [← Set.preimage_image_eq s hf.injective,
       ← Measure.map_apply hf.measurable <| hf.measurableSet_image.2 hs, hx,
       h _ <| hf.measurableSet_image.2 hs]
-  suffices ρ.map (Prod.map id f) = (ρ.fst ⊗ₘ (Kernel.map ρ.condKernel f hf.measurable)) by
+  suffices ρ.map (Prod.map id f) = (ρ.fst ⊗ₘ (Kernel.map ρ.condKernel f)) by
     rw [← hρ'] at this
     have heq := eq_condKernel_of_measure_eq_compProd_real _ this
     rw [hρ'] at heq
     filter_upwards [heq] with x hx s hs
-    rw [← hx, Kernel.map_apply, Measure.map_apply hf.measurable hs]
+    rw [← hx, Kernel.map_apply _ hf.measurable, Measure.map_apply hf.measurable hs]
   ext s hs
   conv_lhs => rw [← ρ.disintegrate ρ.condKernel]
   rw [Measure.compProd_apply hs, Measure.map_apply (measurable_id.prod_map hf.measurable) hs,
     Measure.compProd_apply]
   · congr with a
-    rw [Kernel.map_apply']
+    rw [Kernel.map_apply' _ hf.measurable]
     exacts [rfl, measurable_prod_mk_left hs]
   · exact measurable_id.prod_map hf.measurable hs
 
diff --git a/Mathlib/Probability/Kernel/Integral.lean b/Mathlib/Probability/Kernel/Integral.lean
new file mode 100644
index 0000000000000..f259203acbe03
--- /dev/null
+++ b/Mathlib/Probability/Kernel/Integral.lean
@@ -0,0 +1,130 @@
+/-
+Copyright (c) 2022 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne
+-/
+import Mathlib.MeasureTheory.Integral.Bochner
+import Mathlib.Probability.Kernel.Basic
+
+/-!
+# Bochner integrals of kernels
+
+-/
+
+open MeasureTheory
+
+namespace ProbabilityTheory
+
+variable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : Kernel α β}
+  {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : β → E} {a : α}
+
+namespace Kernel
+
+lemma IsFiniteKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
+    (κ : Kernel α β) [IsFiniteKernel κ] {s : Set β} (hs : MeasurableSet s) :
+    Integrable (fun x ↦ (κ x s).toReal) μ := by
+  refine Integrable.mono' (integrable_const (IsFiniteKernel.bound κ).toReal)
+    ((κ.measurable_coe hs).ennreal_toReal.aestronglyMeasurable)
+    (ae_of_all μ fun x ↦ ?_)
+  rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg,
+    ENNReal.toReal_le_toReal (measure_ne_top _ _) (IsFiniteKernel.bound_ne_top _)]
+  exact Kernel.measure_le_bound _ _ _
+
+lemma IsMarkovKernel.integrable (μ : Measure α) [IsFiniteMeasure μ]
+    (κ : Kernel α β) [IsMarkovKernel κ] {s : Set β} (hs : MeasurableSet s) :
+    Integrable (fun x => (κ x s).toReal) μ :=
+  IsFiniteKernel.integrable μ κ hs
+
+lemma integral_congr_ae₂ {f g : α → β → E} {μ : Measure α} (h : ∀ᵐ a ∂μ, f a =ᵐ[κ a] g a) :
+    ∫ a, ∫ b, f a b ∂(κ a) ∂μ = ∫ a, ∫ b, g a b ∂(κ a) ∂μ := by
+  apply integral_congr_ae
+  filter_upwards [h] with _ ha
+  apply integral_congr_ae
+  filter_upwards [ha] with _ hb using hb
+
+section Deterministic
+
+variable [CompleteSpace E] {g : α → β}
+
+theorem integral_deterministic' (hg : Measurable g) (hf : StronglyMeasurable f) :
+    ∫ x, f x ∂deterministic g hg a = f (g a) := by
+  rw [deterministic_apply, integral_dirac' _ _ hf]
+
+@[simp]
+theorem integral_deterministic [MeasurableSingletonClass β] (hg : Measurable g) :
+    ∫ x, f x ∂deterministic g hg a = f (g a) := by
+  rw [deterministic_apply, integral_dirac _ (g a)]
+
+theorem setIntegral_deterministic' (hg : Measurable g)
+    (hf : StronglyMeasurable f) {s : Set β} (hs : MeasurableSet s) [Decidable (g a ∈ s)] :
+    ∫ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
+  rw [deterministic_apply, setIntegral_dirac' hf _ hs]
+
+@[deprecated (since := "2024-04-17")]
+alias set_integral_deterministic' := setIntegral_deterministic'
+
+@[simp]
+theorem setIntegral_deterministic [MeasurableSingletonClass β] (hg : Measurable g)
+    (s : Set β) [Decidable (g a ∈ s)] :
+    ∫ x in s, f x ∂deterministic g hg a = if g a ∈ s then f (g a) else 0 := by
+  rw [deterministic_apply, setIntegral_dirac f _ s]
+
+@[deprecated (since := "2024-04-17")]
+alias set_integral_deterministic := setIntegral_deterministic
+
+end Deterministic
+
+section Const
+
+@[simp]
+theorem integral_const {μ : Measure β} : ∫ x, f x ∂const α μ a = ∫ x, f x ∂μ := by
+  rw [const_apply]
+
+@[simp]
+theorem setIntegral_const {μ : Measure β} {s : Set β} :
+    ∫ x in s, f x ∂const α μ a = ∫ x in s, f x ∂μ := by rw [const_apply]
+
+@[deprecated (since := "2024-04-17")]
+alias set_integral_const := setIntegral_const
+
+end Const
+
+section Restrict
+
+variable {s t : Set β}
+
+@[simp]
+theorem integral_restrict (hs : MeasurableSet s) :
+    ∫ x, f x ∂κ.restrict hs a = ∫ x in s, f x ∂κ a := by
+  rw [restrict_apply]
+
+@[simp]
+theorem setIntegral_restrict (hs : MeasurableSet s) (t : Set β) :
+    ∫ x in t, f x ∂κ.restrict hs a = ∫ x in t ∩ s, f x ∂κ a := by
+  rw [restrict_apply, Measure.restrict_restrict' hs]
+
+@[deprecated (since := "2024-04-17")]
+alias set_integral_restrict := setIntegral_restrict
+
+end Restrict
+
+section Piecewise
+
+variable {η : Kernel α β} {s : Set α} {hs : MeasurableSet s} [DecidablePred (· ∈ s)]
+
+theorem integral_piecewise (a : α) (g : β → E) :
+    ∫ b, g b ∂piecewise hs κ η a = if a ∈ s then ∫ b, g b ∂κ a else ∫ b, g b ∂η a := by
+  simp_rw [piecewise_apply]; split_ifs <;> rfl
+
+theorem setIntegral_piecewise (a : α) (g : β → E) (t : Set β) :
+    ∫ b in t, g b ∂piecewise hs κ η a =
+      if a ∈ s then ∫ b in t, g b ∂κ a else ∫ b in t, g b ∂η a := by
+  simp_rw [piecewise_apply]; split_ifs <;> rfl
+
+@[deprecated (since := "2024-04-17")]
+alias set_integral_piecewise := setIntegral_piecewise
+
+end Piecewise
+
+end Kernel
+end ProbabilityTheory
diff --git a/Mathlib/Probability/Kernel/IntegralCompProd.lean b/Mathlib/Probability/Kernel/IntegralCompProd.lean
index 6e080b423d194..a8ad7e64ece6c 100644
--- a/Mathlib/Probability/Kernel/IntegralCompProd.lean
+++ b/Mathlib/Probability/Kernel/IntegralCompProd.lean
@@ -116,7 +116,7 @@ theorem integrable_compProd_iff ⦃f : β × γ → E⦄ (hf : AEStronglyMeasura
       (∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x))) ∧
         Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
   simp only [Integrable, hasFiniteIntegral_compProd_iff' hf, hf.norm.integral_kernel_compProd,
-    hf, hf.compProd_mk_left, eventually_and, true_and_iff]
+    hf, hf.compProd_mk_left, eventually_and, true_and]
 
 theorem _root_.MeasureTheory.Integrable.compProd_mk_left_ae ⦃f : β × γ → E⦄
     (hf : Integrable f ((κ ⊗ₖ η) a)) : ∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x)) :=
@@ -226,7 +226,7 @@ theorem integral_compProd :
   · simp [integral, hE]
   apply Integrable.induction
   · intro c s hs h2s
-    simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp,
+    simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp_def,
       integral_indicator (measurable_prod_mk_left hs), MeasureTheory.setIntegral_const,
       integral_smul_const]
     congr 1
@@ -249,7 +249,7 @@ theorem integral_compProd :
 theorem setIntegral_compProd {f : β × γ → E} {s : Set β} {t : Set γ} (hs : MeasurableSet s)
     (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)) :
     ∫ z in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y in t, f (x, y) ∂η (a, x) ∂κ a := by
-  -- Porting note: `compProd_restrict` needed some explicit argumnts
+  -- Porting note: `compProd_restrict` needed some explicit arguments
   rw [← Kernel.restrict_apply (κ ⊗ₖ η) (hs.prod ht), ← compProd_restrict hs ht, integral_compProd]
   · simp_rw [Kernel.restrict_apply]
   · rw [compProd_restrict, Kernel.restrict_apply]; exact hf
diff --git a/Mathlib/Probability/Kernel/MeasurableIntegral.lean b/Mathlib/Probability/Kernel/MeasurableIntegral.lean
index 6790cc25384d2..dc0acc26cac82 100644
--- a/Mathlib/Probability/Kernel/MeasurableIntegral.lean
+++ b/Mathlib/Probability/Kernel/MeasurableIntegral.lean
@@ -3,9 +3,8 @@ Copyright (c) 2023 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathlib.Probability.Kernel.Basic
-import Mathlib.MeasureTheory.Constructions.Prod.Basic
 import Mathlib.MeasureTheory.Integral.DominatedConvergence
+import Mathlib.Probability.Kernel.Basic
 
 /-!
 # Measurability of the integral against a kernel
@@ -64,14 +63,14 @@ theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : Meas
     have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' := by
       intro a
       ext1 b
-      simp only [mem_compl_iff, mem_preimage, mem_diff, mem_univ, true_and_iff]
+      simp only [mem_compl_iff, mem_preimage, mem_diff, mem_univ, true_and]
     simp_rw [h_eq_sdiff]
     have :
       (fun a => κ a (Set.univ \ Prod.mk a ⁻¹' t')) = fun a =>
         κ a Set.univ - κ a (Prod.mk a ⁻¹' t') := by
       ext1 a
       rw [← Set.diff_inter_self_eq_diff, Set.inter_univ, measure_diff (Set.subset_univ _)]
-      · exact (@measurable_prod_mk_left α β _ _ a) ht'
+      · exact (measurable_prod_mk_left ht').nullMeasurableSet
       · exact measure_ne_top _ _
     rw [this]
     exact Measurable.sub (Kernel.measurable_coe κ MeasurableSet.univ) h_meas
@@ -236,7 +235,7 @@ variable {E : Type*} [NormedAddCommGroup E] [IsSFiniteKernel κ] [IsSFiniteKerne
 
 theorem measurableSet_kernel_integrable ⦃f : α → β → E⦄ (hf : StronglyMeasurable (uncurry f)) :
     MeasurableSet {x | Integrable (f x) (κ x)} := by
-  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and_iff]
+  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
   exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
 
 end ProbabilityTheory
diff --git a/Mathlib/Probability/Kernel/RadonNikodym.lean b/Mathlib/Probability/Kernel/RadonNikodym.lean
index 7f5426267ce77..d99f2e27d910e 100644
--- a/Mathlib/Probability/Kernel/RadonNikodym.lean
+++ b/Mathlib/Probability/Kernel/RadonNikodym.lean
@@ -69,9 +69,9 @@ Theorem 1.28 in [O. Kallenberg, Random Measures, Theory and Applications][kallen
 
 -/
 
-open MeasureTheory Set Filter
+open MeasureTheory Set Filter ENNReal
 
-open scoped NNReal ENNReal MeasureTheory Topology ProbabilityTheory
+open scoped NNReal MeasureTheory Topology ProbabilityTheory
 
 namespace ProbabilityTheory.Kernel
 
@@ -88,8 +88,7 @@ noncomputable
 def rnDerivAux (κ η : Kernel α γ) (a : α) (x : γ) : ℝ :=
   if hα : Countable α then ((κ a).rnDeriv (η a) x).toReal
   else haveI := hαγ.countableOrCountablyGenerated.resolve_left hα
-    density (map κ (fun a ↦ (a, ()))
-      (@measurable_prod_mk_right γ Unit _ inferInstance _)) η a x univ
+    density (map κ (fun a ↦ (a, ()))) η a x univ
 
 lemma rnDerivAux_nonneg (hκη : κ ≤ η) {a : α} {x : γ} : 0 ≤ rnDerivAux κ η a x := by
   rw [rnDerivAux]
@@ -144,7 +143,7 @@ lemma setLIntegral_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFini
     rw [ENNReal.ofReal_toReal hx_lt.ne]
   · have := hαγ.countableOrCountablyGenerated.resolve_left hα
     rw [setLIntegral_density ((fst_map_id_prod _ measurable_const).trans_le h_le) _
-      MeasurableSet.univ hs, map_apply' _ _ _ (hs.prod MeasurableSet.univ)]
+      MeasurableSet.univ hs, map_apply' _ (by fun_prop) _ (hs.prod MeasurableSet.univ)]
     congr with x
     simp
 
@@ -157,8 +156,7 @@ lemma withDensity_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFinit
   rw [Kernel.withDensity_apply']
   swap
   · exact (measurable_rnDerivAux _ _).ennreal_ofReal
-  have : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
-  simp_rw [this]
+  simp_rw [ofNNReal_toNNReal]
   exact setLIntegral_rnDerivAux κ η a hs
 
 lemma withDensity_one_sub_rnDerivAux (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
@@ -217,11 +215,10 @@ lemma measurableSet_mutuallySingularSetSlice (κ η : Kernel α γ) (a : α) :
 lemma measure_mutuallySingularSetSlice (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η]
     (a : α) :
     η a (mutuallySingularSetSlice κ η a) = 0 := by
-  have h_coe : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
   suffices withDensity (κ + η) (fun a x ↦ Real.toNNReal
       (1 - rnDerivAux κ (κ + η) a x)) a {x | 1 ≤ rnDerivAux κ (κ + η) a x} = 0 by
     rwa [withDensity_one_sub_rnDerivAux κ η] at this
-  simp_rw [h_coe]
+  simp_rw [ofNNReal_toNNReal]
   rw [Kernel.withDensity_apply', lintegral_eq_zero_iff, EventuallyEq, ae_restrict_iff]
   rotate_left
   · exact (measurable_const.sub
@@ -289,10 +286,9 @@ lemma measurable_singularPart_fun_right (κ η : Kernel α γ) (a : α) :
 lemma singularPart_compl_mutuallySingularSetSlice (κ η : Kernel α γ) [IsSFiniteKernel κ]
     [IsSFiniteKernel η] (a : α) :
     singularPart κ η a (mutuallySingularSetSlice κ η a)ᶜ = 0 := by
-  have h_coe : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
   rw [singularPart, Kernel.withDensity_apply', lintegral_eq_zero_iff, EventuallyEq,
     ae_restrict_iff]
-  all_goals simp_rw [h_coe]
+  all_goals simp_rw [ofNNReal_toNNReal]
   rotate_left
   · exact measurableSet_preimage (measurable_singularPart_fun_right κ η a)
       (measurableSet_singleton _)
@@ -351,7 +347,6 @@ lemma withDensity_rnDeriv_of_subset_compl_mutuallySingularSetSlice
     [IsFiniteKernel κ] [IsFiniteKernel η] {a : α} {s : Set γ} (hsm : MeasurableSet s)
     (hs : s ⊆ (mutuallySingularSetSlice κ η a)ᶜ) :
     withDensity η (rnDeriv κ η) a s = κ a s := by
-  have h_coe : ∀ b, (Real.toNNReal b : ℝ≥0∞) = ENNReal.ofReal b := fun _ ↦ rfl
   have : withDensity η (rnDeriv κ η)
       = withDensity (withDensity (κ + η)
         (fun a x ↦ Real.toNNReal (1 - rnDerivAux κ (κ + η) a x))) (rnDeriv κ η) := by
@@ -374,8 +369,8 @@ lemma withDensity_rnDeriv_of_subset_compl_mutuallySingularSetSlice
         ENNReal.ofReal (1 - rnDerivAux κ (κ + η) a x)) ∂(κ + η) a
   _ = ∫⁻ x in s, ENNReal.ofReal (rnDerivAux κ (κ + η) a x) ∂(κ + η) a := by
       refine setLIntegral_congr_fun hsm (ae_of_all _ fun x hx ↦ ?_)
-      rw [h_coe, ← ENNReal.ofReal_div_of_pos, div_eq_inv_mul, ← ENNReal.ofReal_mul, ← mul_assoc,
-        mul_inv_cancel₀, one_mul]
+      rw [ofNNReal_toNNReal, ← ENNReal.ofReal_div_of_pos, div_eq_inv_mul, ← ENNReal.ofReal_mul,
+        ← mul_assoc, mul_inv_cancel₀, one_mul]
       · rw [ne_eq, sub_eq_zero]
         exact (hs' x hx).ne'
       · simp [(hs' x hx).le]
@@ -391,7 +386,7 @@ lemma mutuallySingular_singularPart (κ η : Kernel α γ) [IsFiniteKernel κ] [
     measure_mutuallySingularSetSlice κ η a, singularPart_compl_mutuallySingularSetSlice κ η a⟩
 
 /-- Lebesgue decomposition of a finite kernel `κ` with respect to another one `η`.
-`κ` is the sum of an abolutely continuous part `withDensity η (rnDeriv κ η)` and a singular part
+`κ` is the sum of an absolutely continuous part `withDensity η (rnDeriv κ η)` and a singular part
 `singularPart κ η`. -/
 lemma rnDeriv_add_singularPart (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
     withDensity η (rnDeriv κ η) + singularPart κ η = κ := by
diff --git a/Mathlib/Probability/Martingale/BorelCantelli.lean b/Mathlib/Probability/Martingale/BorelCantelli.lean
index bf606fa145ff1..fffa57f7af53e 100644
--- a/Mathlib/Probability/Martingale/BorelCantelli.lean
+++ b/Mathlib/Probability/Martingale/BorelCantelli.lean
@@ -61,7 +61,7 @@ theorem leastGE_le {i : ℕ} {r : ℝ} (ω : Ω) : leastGE f r i ω ≤ i :=
 
 -- The following four lemmas shows `leastGE` behaves like a stopped process. Ideally we should
 -- define `leastGE` as a stopping time and take its stopped process. However, we can't do that
--- with our current definition since a stopping time takes only finite indicies. An upcomming
+-- with our current definition since a stopping time takes only finite indices. An upcoming
 -- refactor should hopefully make it possible to have stopping times taking infinity as a value
 theorem leastGE_mono {n m : ℕ} (hnm : n ≤ m) (r : ℝ) (ω : Ω) : leastGE f r n ω ≤ leastGE f r m ω :=
   hitting_mono hnm
diff --git a/Mathlib/Probability/Martingale/Convergence.lean b/Mathlib/Probability/Martingale/Convergence.lean
index a9308ef0c2aab..5cbec74af7d1e 100644
--- a/Mathlib/Probability/Martingale/Convergence.lean
+++ b/Mathlib/Probability/Martingale/Convergence.lean
@@ -173,8 +173,8 @@ theorem Submartingale.upcrossings_ae_lt_top' [IsFiniteMeasure μ] (hf : Submarti
           exact norm_nonneg _
       · simp only [Ne, ENNReal.coe_ne_top, not_false_iff]
     · simp only [hab, Ne, ENNReal.ofReal_eq_zero, sub_nonpos, not_le]
-  · simp only [hab, Ne, ENNReal.ofReal_eq_zero, sub_nonpos, not_le, true_or_iff]
-  · simp only [Ne, ENNReal.ofReal_ne_top, not_false_iff, true_or_iff]
+  · simp only [hab, Ne, ENNReal.ofReal_eq_zero, sub_nonpos, not_le, true_or]
+  · simp only [Ne, ENNReal.ofReal_ne_top, not_false_iff, true_or]
 
 theorem Submartingale.upcrossings_ae_lt_top [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
     (hbdd : ∀ n, eLpNorm (f n) 1 μ ≤ R) : ∀ᵐ ω ∂μ, ∀ a b : ℚ, a < b → upcrossings a b f ω < ∞ := by
@@ -269,12 +269,12 @@ $(f_n)_n$ is a martingale by the tower property for conditional expectations. Fu
 $(f_n)_n$ is uniformly integrable in the probability sense. Indeed, as a single function is
 uniformly integrable in the measure theory sense, for all $\epsilon > 0$, there exists some
 $\delta > 0$ such that for all measurable set $A$ with $\mu(A) < δ$, we have
-$\mathbb{E}|h|\mathbf{1}_A < \epsilon$. So, since for sufficently large $\lambda$, by the Markov
+$\mathbb{E}|h|\mathbf{1}_A < \epsilon$. So, since for sufficiently large $\lambda$, by the Markov
 inequality, we have for all $n$,
 $$
   \mu(|f_n| \ge \lambda) \le \lambda^{-1}\mathbb{E}|f_n| \le \lambda^{-1}\mathbb|g| < \delta,
 $$
-we have for sufficently large $\lambda$, for all $n$,
+we have for sufficiently large $\lambda$, for all $n$,
 $$
   \mathbb{E}|f_n|\mathbf{1}_{|f_n| \ge \lambda} \le
     \mathbb|g|\mathbf{1}_{|f_n| \ge \lambda} < \epsilon,
diff --git a/Mathlib/Probability/Martingale/OptionalSampling.lean b/Mathlib/Probability/Martingale/OptionalSampling.lean
index 3132d43cb03f9..535963b759aba 100644
--- a/Mathlib/Probability/Martingale/OptionalSampling.lean
+++ b/Mathlib/Probability/Martingale/OptionalSampling.lean
@@ -66,7 +66,7 @@ theorem condexp_stopping_time_ae_eq_restrict_eq_const_of_le_const (h : Martingal
     rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff]
   · suffices {x : Ω | τ x = i} = ∅ by simp [this]; norm_cast
     ext1 x
-    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
+    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]
     rintro rfl
     exact hin (hτ_le x)
 
@@ -88,7 +88,7 @@ theorem stoppedValue_ae_eq_condexp_of_le_const_of_countable_range (h : Martingal
     stoppedValue f τ =ᵐ[μ] μ[f n|hτ.measurableSpace] := by
   have : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i} := by
     ext1 x
-    simp only [Set.mem_univ, Set.mem_range, true_and_iff, Set.iUnion_exists, Set.iUnion_iUnion_eq',
+    simp only [Set.mem_univ, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',
       Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']
   nth_rw 1 [← @Measure.restrict_univ Ω _ μ]
   rw [this, ae_eq_restrict_biUnion_iff _ h_countable_range]
diff --git a/Mathlib/Probability/Martingale/OptionalStopping.lean b/Mathlib/Probability/Martingale/OptionalStopping.lean
index 9ccbef6e43662..fee8f4ea1c1a3 100644
--- a/Mathlib/Probability/Martingale/OptionalStopping.lean
+++ b/Mathlib/Probability/Martingale/OptionalStopping.lean
@@ -130,7 +130,7 @@ theorem smul_le_stoppedValue_hitting [IsFiniteMeasure μ] (hsub : Submartingale
 we have `ε • μ {ε ≤ f* n} ≤ ∫ ω in {ε ≤ f* n}, f n` where `f* n ω = max_{k ≤ n}, f k ω`.
 
 In some literature, the Doob's maximal inequality refers to what we call Doob's Lp inequality
-(which is a corollary of this lemma and will be proved in an upcomming PR). -/
+(which is a corollary of this lemma and will be proved in an upcoming PR). -/
 theorem maximal_ineq [IsFiniteMeasure μ] (hsub : Submartingale f 𝒢 μ) (hnonneg : 0 ≤ f) {ε : ℝ≥0}
     (n : ℕ) : ε • μ {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω} ≤
     ENNReal.ofReal (∫ ω in {ω | (ε : ℝ) ≤ (range (n + 1)).sup' nonempty_range_succ fun k => f k ω},
diff --git a/Mathlib/Probability/Martingale/Upcrossing.lean b/Mathlib/Probability/Martingale/Upcrossing.lean
index 54131389f0654..a1694af83142d 100644
--- a/Mathlib/Probability/Martingale/Upcrossing.lean
+++ b/Mathlib/Probability/Martingale/Upcrossing.lean
@@ -430,7 +430,7 @@ theorem upperCrossingTime_lt_of_le_upcrossingsBefore (hN : 0 < N) (hab : a < b)
 theorem upperCrossingTime_eq_of_upcrossingsBefore_lt (hab : a < b)
     (hn : upcrossingsBefore a b f N ω < n) : upperCrossingTime a b f N n ω = N := by
   refine le_antisymm upperCrossingTime_le (not_lt.1 ?_)
-  convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab)
+  convert not_mem_of_csSup_lt hn (upperCrossingTime_lt_bddAbove hab) using 1
 
 theorem upcrossingsBefore_le (f : ℕ → Ω → ℝ) (ω : Ω) (hab : a < b) :
     upcrossingsBefore a b f N ω ≤ N := by
@@ -449,7 +449,7 @@ theorem crossing_eq_crossing_of_lowerCrossingTime_lt {M : ℕ} (hNM : N ≤ M)
     lt_of_le_of_lt upperCrossingTime_le_lowerCrossingTime h
   induction' n with k ih
   · simp only [upperCrossingTime_zero, bot_eq_zero', eq_self_iff_true,
-      lowerCrossingTime_zero, true_and_iff, eq_comm]
+      lowerCrossingTime_zero, true_and, eq_comm]
     refine hitting_eq_hitting_of_exists hNM ?_
     rw [lowerCrossingTime, hitting_lt_iff] at h
     · obtain ⟨j, hj₁, hj₂⟩ := h
diff --git a/Mathlib/Probability/Moments.lean b/Mathlib/Probability/Moments.lean
index 9f39ce4a332a9..3ad21345e302f 100644
--- a/Mathlib/Probability/Moments.lean
+++ b/Mathlib/Probability/Moments.lean
@@ -166,7 +166,7 @@ theorem mgf_pos' (hμ : μ ≠ 0) (h_int_X : Integrable (fun ω => exp (t * X ω
   rw [this, setIntegral_pos_iff_support_of_nonneg_ae _ _]
   · have h_eq_univ : (Function.support fun x : Ω => exp (t * X x)) = Set.univ := by
       ext1 x
-      simp only [Function.mem_support, Set.mem_univ, iff_true_iff]
+      simp only [Function.mem_support, Set.mem_univ, iff_true]
       exact (exp_pos _).ne'
     rw [h_eq_univ, Set.inter_univ _]
     refine Ne.bot_lt ?_
diff --git a/Mathlib/Probability/ProbabilityMassFunction/Basic.lean b/Mathlib/Probability/ProbabilityMassFunction/Basic.lean
index ec56f812e2f47..877ecc41e7f9d 100644
--- a/Mathlib/Probability/ProbabilityMassFunction/Basic.lean
+++ b/Mathlib/Probability/ProbabilityMassFunction/Basic.lean
@@ -176,7 +176,8 @@ theorem toOuterMeasure_apply_eq_one_iff : p.toOuterMeasure s = 1 ↔ p.support 
       (fun x => Set.indicator_apply_le fun _ => le_rfl) hsa
   · suffices ∀ (x) (_ : x ∉ s), p x = 0 from
       _root_.trans (tsum_congr
-        fun a => (Set.indicator_apply s p a).trans (ite_eq_left_iff.2 <| symm ∘ this a)) p.tsum_coe
+        fun a => (Set.indicator_apply s p a).trans
+          (ite_eq_left_iff.2 <| symm ∘ this a)) p.tsum_coe
     exact fun a ha => (p.apply_eq_zero_iff a).2 <| Set.not_mem_subset h ha
 
 @[simp]
diff --git a/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean b/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
index 9b85c12e07cf7..9ae6765fb79bd 100644
--- a/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
+++ b/Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
@@ -58,7 +58,7 @@ theorem bind_pure_comp : bind p (pure ∘ f) = map f p := rfl
 theorem map_id : map id p = p :=
   bind_pure _
 
-theorem map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) := by simp [map, Function.comp]
+theorem map_comp (g : β → γ) : (p.map f).map g = p.map (g ∘ f) := by simp [map, Function.comp_def]
 
 theorem pure_map (a : α) : (pure a).map f = pure (f a) :=
   pure_bind _ _
@@ -72,7 +72,7 @@ theorem bind_map (p : PMF α) (f : α → β) (q : β → PMF γ) : (p.map f).bi
 
 @[simp]
 theorem map_const : p.map (Function.const α b) = pure b := by
-  simp only [map, Function.comp, bind_const, Function.const]
+  simp only [map, Function.comp_def, bind_const, Function.const]
 
 section Measure
 
diff --git a/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean b/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean
index 0803f711af0ab..befbc17ee812f 100644
--- a/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean
+++ b/Mathlib/Probability/ProbabilityMassFunction/Integrals.lean
@@ -42,7 +42,7 @@ theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeas
 
 theorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :
     ∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a := by
-  rw [integral_fintype _ (.of_finite _ f)]
+  rw [integral_fintype _ .of_finite]
   congr with x; congr 2
   exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
 
diff --git a/Mathlib/Probability/ProbabilityMassFunction/Monad.lean b/Mathlib/Probability/ProbabilityMassFunction/Monad.lean
index d137f3c66d02f..31a9cbfa903d1 100644
--- a/Mathlib/Probability/ProbabilityMassFunction/Monad.lean
+++ b/Mathlib/Probability/ProbabilityMassFunction/Monad.lean
@@ -67,9 +67,11 @@ theorem toOuterMeasure_pure_apply : (pure a).toOuterMeasure s = if a ∈ s then
   refine (toOuterMeasure_apply (pure a) s).trans ?_
   split_ifs with ha
   · refine (tsum_congr fun b => ?_).trans (tsum_ite_eq a 1)
-    exact ite_eq_left_iff.2 fun hb => symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)
+    exact ite_eq_left_iff.2 fun hb =>
+      symm (ite_eq_right_iff.2 fun h => (hb <| h.symm ▸ ha).elim)
   · refine (tsum_congr fun b => ?_).trans tsum_zero
-    exact ite_eq_right_iff.2 fun hb => ite_eq_right_iff.2 fun h => (ha <| h ▸ hb).elim
+    exact ite_eq_right_iff.2 fun hb =>
+      ite_eq_right_iff.2 fun h => (ha <| h ▸ hb).elim
 
 variable [MeasurableSpace α]
 
diff --git a/Mathlib/Probability/Process/HittingTime.lean b/Mathlib/Probability/Process/HittingTime.lean
index 28eb2728601cf..161d19118807f 100644
--- a/Mathlib/Probability/Process/HittingTime.lean
+++ b/Mathlib/Probability/Process/HittingTime.lean
@@ -164,7 +164,7 @@ theorem hitting_le_iff_of_lt [IsWellOrder ι (· < ·)] {m : ι} (i : ι) (hi :
   · rw [hitting_le_iff_of_exists h_exists]
   · simp_rw [hitting, if_neg h_exists]
     push_neg at h_exists
-    simp only [not_le.mpr hi, Set.mem_Icc, false_iff_iff, not_exists, not_and, and_imp]
+    simp only [not_le.mpr hi, Set.mem_Icc, false_iff, not_exists, not_and, and_imp]
     exact fun k hkn hki => h_exists k ⟨hkn, hki.trans hi.le⟩
 
 theorem hitting_lt_iff [IsWellOrder ι (· < ·)] {m : ι} (i : ι) (hi : i ≤ m) :
@@ -251,7 +251,7 @@ theorem isStoppingTime_hitting_isStoppingTime [ConditionallyCompleteLinearOrder
   have h₂ : ⋃ i > n, {x | τ x = i} ∩ {x | hitting u s i N x ≤ n} = ∅ := by
     ext x
     simp only [gt_iff_lt, Set.mem_iUnion, Set.mem_inter_iff, Set.mem_setOf_eq, exists_prop,
-      Set.mem_empty_iff_false, iff_false_iff, not_exists, not_and, not_le]
+      Set.mem_empty_iff_false, iff_false, not_exists, not_and, not_le]
     rintro m hm rfl
     exact lt_of_lt_of_le hm (le_hitting (hτbdd _) _)
   rw [h₁, h₂, Set.union_empty]
@@ -283,7 +283,7 @@ theorem hitting_bot_le_iff {i n : ι} {ω : Ω} (hx : ∃ j, j ≤ n ∧ u j ω
   cases' lt_or_le i n with hi hi
   · rw [hitting_le_iff_of_lt _ hi]
     simp
-  · simp only [(hitting_le ω).trans hi, true_iff_iff]
+  · simp only [(hitting_le ω).trans hi, true_iff]
     obtain ⟨j, hj₁, hj₂⟩ := hx
     exact ⟨j, hj₁.trans hi, hj₂⟩
 
diff --git a/Mathlib/Probability/Process/Stopping.lean b/Mathlib/Probability/Process/Stopping.lean
index 23d8793910896..0b6419fc37ae6 100644
--- a/Mathlib/Probability/Process/Stopping.lean
+++ b/Mathlib/Probability/Process/Stopping.lean
@@ -69,7 +69,7 @@ theorem IsStoppingTime.measurableSet_lt_of_pred [PredOrder ι] (hτ : IsStopping
   by_cases hi_min : IsMin i
   · suffices {ω : Ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)
     ext1 ω
-    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
+    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]
     rw [isMin_iff_forall_not_lt] at hi_min
     exact hi_min (τ ω)
   have : {ω : Ω | τ ω < i} = τ ⁻¹' Set.Iic (pred i) := by ext; simp [Iic_pred_of_not_isMin hi_min]
@@ -152,7 +152,7 @@ theorem IsStoppingTime.measurableSet_lt_of_isLUB (hτ : IsStoppingTime f τ) (i
   by_cases hi_min : IsMin i
   · suffices {ω | τ ω < i} = ∅ by rw [this]; exact @MeasurableSet.empty _ (f i)
     ext1 ω
-    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false_iff]
+    simp only [Set.mem_setOf_eq, Set.mem_empty_iff_false, iff_false]
     exact isMin_iff_forall_not_lt.mp hi_min (τ ω)
   obtain ⟨seq, -, -, h_tendsto, h_bound⟩ :
       ∃ seq : ℕ → ι, Monotone seq ∧ (∀ j, seq j ≤ i) ∧ Tendsto seq atTop (𝓝 i) ∧ ∀ j, seq j < i :=
@@ -261,7 +261,7 @@ theorem add_const_nat {f : Filtration ℕ m} {τ : Ω → ℕ} (hτ : IsStopping
   · rw [not_le] at hij
     convert @MeasurableSet.empty _ (f.1 j)
     ext ω
-    simp only [Set.mem_empty_iff_false, iff_false_iff, Set.mem_setOf]
+    simp only [Set.mem_empty_iff_false, iff_false, Set.mem_setOf]
     omega
 
 -- generalize to certain countable type?
@@ -567,18 +567,17 @@ theorem measurableSet_inter_le [TopologicalSpace ι] [SecondCountableTopology ι
       s ∩ {ω | τ ω ≤ i} ∩ {ω | min (τ ω) (π ω) ≤ i} ∩
         {ω | min (τ ω) i ≤ min (min (τ ω) (π ω)) i} := by
     ext1 ω
-    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and_iff,
-      and_true_iff, true_or_iff, or_true_iff]
+    simp only [min_le_iff, Set.mem_inter_iff, Set.mem_setOf_eq, le_min_iff, le_refl, true_and,
+      true_or]
     by_cases hτi : τ ω ≤ i
-    · simp only [hτi, true_or_iff, and_true_iff, and_congr_right_iff]
+    · simp only [hτi, true_or, and_true, and_congr_right_iff]
       intro
       constructor <;> intro h
       · exact Or.inl h
       · cases' h with h h
         · exact h
         · exact hτi.trans h
-    simp only [hτi, false_or_iff, and_false_iff, false_and_iff, iff_false_iff, not_and, not_le,
-      and_imp]
+    simp only [hτi, false_or, and_false, false_and, iff_false, not_and, not_le, and_imp]
     refine fun _ hτ_le_π => lt_of_lt_of_le ?_ hτ_le_π
     rw [← not_le]
     exact hτi
@@ -617,10 +616,10 @@ theorem measurableSet_le_stopping_time [TopologicalSpace ι] [SecondCountableTop
   intro j
   have : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j} := by
     ext1 ω
-    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl, and_true_iff,
+    simp only [Set.mem_inter_iff, Set.mem_setOf_eq, min_le_iff, le_min_iff, le_refl,
       and_congr_left_iff]
     intro h
-    simp only [h, or_self_iff, and_true_iff]
+    simp only [h, or_self_iff, and_true]
     rw [Iff.comm, or_iff_left_iff_imp]
     exact h.trans
   rw [this]
@@ -704,7 +703,7 @@ variable [LinearOrder ι]
 /-- Given a map `u : ι → Ω → E`, the stopped process with respect to `τ` is `u i ω` if
 `i ≤ τ ω`, and `u (τ ω) ω` otherwise.
 
-Intuitively, the stopped process stops evolving once the stopping time has occured. -/
+Intuitively, the stopped process stops evolving once the stopping time has occurred. -/
 def stoppedProcess (u : ι → Ω → β) (τ : Ω → ι) : ι → Ω → β := fun i ω => u (min i (τ ω)) ω
 
 theorem stoppedProcess_eq_stoppedValue {u : ι → Ω → β} {τ : Ω → ι} :
diff --git a/Mathlib/Probability/StrongLaw.lean b/Mathlib/Probability/StrongLaw.lean
index 17244a35ade7a..34d9233a9272f 100644
--- a/Mathlib/Probability/StrongLaw.lean
+++ b/Mathlib/Probability/StrongLaw.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
 import Mathlib.Probability.IdentDistrib
+import Mathlib.Probability.Independence.Integrable
 import Mathlib.MeasureTheory.Integral.DominatedConvergence
 import Mathlib.Analysis.SpecificLimits.FloorPow
 import Mathlib.Analysis.PSeries
@@ -107,11 +108,11 @@ theorem truncation_eq_of_nonneg {f : α → ℝ} {A : ℝ} (h : ∀ x, 0 ≤ f x
     truncation f A = indicator (Set.Ioc 0 A) id ∘ f := by
   ext x
   rcases (h x).lt_or_eq with (hx | hx)
-  · simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply, true_and_iff]
+  · simp only [truncation, indicator, hx, Set.mem_Ioc, id, Function.comp_apply]
     by_cases h'x : f x ≤ A
     · have : -A < f x := by linarith [h x]
-      simp only [this, true_and_iff]
-    · simp only [h'x, and_false_iff]
+      simp only [this, true_and]
+    · simp only [h'x, and_false]
   · simp only [truncation, indicator, hx, id, Function.comp_apply, ite_self]
 
 theorem truncation_nonneg {f : α → ℝ} (A : ℝ) {x : α} (h : 0 ≤ f x) : 0 ≤ truncation f A x :=
@@ -298,7 +299,7 @@ theorem tsum_prob_mem_Ioi_lt_top {X : Ω → ℝ} (hint : Integrable X) (hnonneg
       · simp (config := {contextual := true}) only [Set.mem_Ioc, Set.mem_Ioi,
           Set.iUnion_subset_iff, Set.setOf_subset_setOf, imp_true_iff]
     rw [this]
-    apply tendsto_measure_iUnion
+    apply tendsto_measure_iUnion_atTop
     intro m n hmn x hx
     exact ⟨hx.1, hx.2.trans (Nat.cast_le.2 hmn)⟩
   apply le_of_tendsto_of_tendsto A tendsto_const_nhds
@@ -380,11 +381,11 @@ theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) :
     ε * ⌊c ^ n⌋₊ := by
   /- Let `S n = ∑ i ∈ range n, Y i` where `Y i = truncation (X i) i`. We should show that
     `|S k - 𝔼[S k]| / k ≤ ε` along the sequence of powers of `c`. For this, we apply Borel-Cantelli:
-    it suffices to show that the converse probabilites are summable. From Chebyshev inequality, this
-    will follow from a variance control `∑' Var[S (c^i)] / (c^i)^2 < ∞`. This is checked in `I2`
-    using pairwise independence to expand the variance of the sum as the sum of the variances,
+    it suffices to show that the converse probabilities are summable. From Chebyshev inequality,
+    this will follow from a variance control `∑' Var[S (c^i)] / (c^i)^2 < ∞`. This is checked in
+    `I2` using pairwise independence to expand the variance of the sum as the sum of the variances,
     and then a straightforward but tedious computation (essentially boiling down to the fact that
-    the sum of `1/(c ^ i)^2` beyong a threshold `j` is comparable to `1/j^2`).
+    the sum of `1/(c ^ i)^2` beyond a threshold `j` is comparable to `1/j^2`).
     Note that we have written `c^i` in the above proof sketch, but rigorously one should put integer
     parts everywhere, making things more painful. We write `u i = ⌊c^i⌋₊` for brevity. -/
   have c_pos : 0 < c := zero_lt_one.trans c_one
@@ -395,7 +396,7 @@ theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) :
   set Y := fun n : ℕ => truncation (X n) n
   set S := fun n => ∑ i ∈ range n, Y i with hS
   let u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
-  have u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_le_pow_right c_one.le hij)
+  have u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_right_mono₀ c_one.le hij)
   have I1 : ∀ K, ∑ j ∈ range K, ((j : ℝ) ^ 2)⁻¹ * Var[Y j] ≤ 2 * 𝔼[X 0] := by
     intro K
     calc
@@ -456,7 +457,7 @@ theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) :
           refine zero_lt_one.trans_le ?_
           apply Nat.le_floor
           rw [Nat.cast_one]
-          apply one_le_pow_of_one_le c_one.le
+          apply one_le_pow₀ c_one.le
       _ = ENNReal.ofReal (∑ i ∈ range N, Var[S (u i)] / (u i * ε) ^ 2) := by
         rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]
         exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)
@@ -546,7 +547,7 @@ theorem strong_law_aux5 :
     · have : -(n : ℝ) < X n ω := by
         apply lt_of_lt_of_le _ (hnonneg n ω)
         simpa only [Right.neg_neg_iff, Nat.cast_pos] using npos
-      simp only [this, true_and_iff, not_le] at h
+      simp only [this, true_and, not_le] at h
       exact (hn h).elim
   filter_upwards [B] with ω hω
   convert isLittleO_sum_range_of_tendsto_zero hω using 1
@@ -562,7 +563,7 @@ theorem strong_law_aux6 {c : ℝ} (c_one : 1 < c) :
   have H : ∀ n : ℕ, (0 : ℝ) < ⌊c ^ n⌋₊ := by
     intro n
     refine zero_lt_one.trans_le ?_
-    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le c_one.le n]
+    simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow₀ c_one.le]
   filter_upwards [strong_law_aux4 X hint hindep hident hnonneg c_one,
     strong_law_aux5 X hint hident hnonneg] with ω hω h'ω
   rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ]
diff --git a/Mathlib/Probability/Variance.lean b/Mathlib/Probability/Variance.lean
index d2036c459f53a..9b0a9bcb2fe5c 100644
--- a/Mathlib/Probability/Variance.lean
+++ b/Mathlib/Probability/Variance.lean
@@ -228,7 +228,7 @@ theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : A
     rw [Memℒp, not_and] at hℒ
     specialize hℒ hX
     simp only [eLpNorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
-      ENNReal.toReal_ofNat, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true_iff,
+      ENNReal.toReal_ofNat, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true,
       or_iff_not_imp_left, not_and_or, zero_lt_two] at hℒ
     exact mod_cast hℒ fun _ => zero_le_two
 
@@ -255,7 +255,7 @@ theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ}
     (hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) := by
   rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero hc.ne.symm), hX.ofReal_variance_eq]
   convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1
-  · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
+  · simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true]
   · rw [ENNReal.ofReal_pow hc.le]
     rfl
 
diff --git a/Mathlib/RepresentationTheory/Action/Basic.lean b/Mathlib/RepresentationTheory/Action/Basic.lean
index f52d5604cf170..e09c00af54f4e 100644
--- a/Mathlib/RepresentationTheory/Action/Basic.lean
+++ b/Mathlib/RepresentationTheory/Action/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Grp.Basic
 import Mathlib.CategoryTheory.SingleObj
@@ -125,6 +125,16 @@ theorem comp_hom {M N K : Action V G} (f : M ⟶ N) (g : N ⟶ K) :
     (f ≫ g : Hom M K).hom = f.hom ≫ g.hom :=
   rfl
 
+@[simp]
+theorem hom_inv_hom {M N : Action V G} (f : M ≅ N) :
+    f.hom.hom ≫ f.inv.hom = 𝟙 M.V := by
+  rw [← comp_hom, Iso.hom_inv_id, id_hom]
+
+@[simp]
+theorem inv_hom_hom {M N : Action V G} (f : M ≅ N) :
+    f.inv.hom ≫ f.hom.hom = 𝟙 N.V := by
+  rw [← comp_hom, Iso.inv_hom_id, id_hom]
+
 /-- Construct an isomorphism of `G` actions/representations
 from an isomorphism of the underlying objects,
 where the forward direction commutes with the group action. -/
@@ -145,6 +155,12 @@ instance isIso_hom_mk {M N : Action V G} (f : M.V ⟶ N.V) [IsIso f] (w) :
     @IsIso _ _ M N (Hom.mk f w) :=
   (mkIso (asIso f) w).isIso_hom
 
+instance {M N : Action V G} (f : M ≅ N) : IsIso f.hom.hom where
+  out := ⟨f.inv.hom, by simp⟩
+
+instance {M N : Action V G} (f : M ≅ N) : IsIso f.inv.hom where
+  out := ⟨f.hom.hom, by simp⟩
+
 namespace FunctorCategoryEquivalence
 
 /-- Auxiliary definition for `functorCategoryEquivalence`. -/
diff --git a/Mathlib/RepresentationTheory/Action/Concrete.lean b/Mathlib/RepresentationTheory/Action/Concrete.lean
index 1ae786bb201e9..e2ebc3a42dff7 100644
--- a/Mathlib/RepresentationTheory/Action/Concrete.lean
+++ b/Mathlib/RepresentationTheory/Action/Concrete.lean
@@ -1,10 +1,12 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Group.Action.Pi
 import Mathlib.CategoryTheory.FintypeCat
+import Mathlib.GroupTheory.GroupAction.Quotient
+import Mathlib.GroupTheory.QuotientGroup.Basic
 import Mathlib.RepresentationTheory.Action.Basic
 
 /-!
@@ -68,6 +70,12 @@ def diagonalOneIsoLeftRegular (G : Type u) [Monoid G] : diagonal G 1 ≅ leftReg
 
 namespace FintypeCat
 
+/-- If `X` is a type with `[Fintype X]` and `G` acts on `X`, then `G` also acts on
+`FintypeCat.of X`. -/
+instance (G : Type*) (X : Type*) [Monoid G] [MulAction G X] [Fintype X] :
+    MulAction G (FintypeCat.of X) :=
+  inferInstanceAs <| MulAction G X
+
 /-- Bundles a finite type `H` with a multiplicative action of `G` as an `Action`. -/
 def ofMulAction (G : Type u) (H : FintypeCat.{u}) [Monoid G] [MulAction G H] :
     Action FintypeCat (MonCat.of G) where
@@ -79,6 +87,82 @@ theorem ofMulAction_apply {G : Type u} {H : FintypeCat.{u}} [Monoid G] [MulActio
     (g : G) (x : H) : (FintypeCat.ofMulAction G H).ρ g x = (g • x : H) :=
   rfl
 
+section
+
+/-- Shorthand notation for the quotient of `G` by `H` as a finite `G`-set. -/
+notation:10 G:10 " ⧸ₐ " H:10 => Action.FintypeCat.ofMulAction G (FintypeCat.of <| G ⧸ H)
+
+variable {G : Type*} [Group G] (H N : Subgroup G) [Fintype (G ⧸ N)]
+
+/-- If `N` is a normal subgroup of `G`, then this is the group homomorphism
+sending an element `g` of `G` to the `G`-endomorphism of `G ⧸ₐ N` given by
+multiplication with `g⁻¹` on the right. -/
+def toEndHom [N.Normal] : G →* End (G ⧸ₐ N) where
+  toFun v := {
+    hom := Quotient.lift (fun σ ↦ ⟦σ * v⁻¹⟧) <| fun a b h ↦ Quotient.sound <| by
+      apply (QuotientGroup.leftRel_apply).mpr
+      simp only [mul_inv_rev, inv_inv]
+      convert_to v * (a⁻¹ * b) * v⁻¹ ∈ N
+      · group
+      · exact Subgroup.Normal.conj_mem ‹_› _ (QuotientGroup.leftRel_apply.mp h) _
+    comm := fun (g : G) ↦ by
+      ext (x : G ⧸ N)
+      induction' x using Quotient.inductionOn with x
+      simp only [FintypeCat.comp_apply, Action.FintypeCat.ofMulAction_apply, Quotient.lift_mk]
+      show Quotient.lift (fun σ ↦ ⟦σ * v⁻¹⟧) _ (⟦g • x⟧) = _
+      simp only [smul_eq_mul, Quotient.lift_mk, mul_assoc]
+      rfl
+  }
+  map_one' := by
+    apply Action.hom_ext
+    ext (x : G ⧸ N)
+    induction' x using Quotient.inductionOn with x
+    simp
+  map_mul' σ τ := by
+    apply Action.hom_ext
+    ext (x : G ⧸ N)
+    induction' x using Quotient.inductionOn with x
+    show ⟦x * (σ * τ)⁻¹⟧ = ⟦x * τ⁻¹ * σ⁻¹⟧
+    rw [mul_inv_rev, mul_assoc]
+
+@[simp]
+lemma toEndHom_apply [N.Normal] (g h : G) : (toEndHom N g).hom ⟦h⟧ = ⟦h * g⁻¹⟧ := rfl
+
+variable {N} in
+lemma toEndHom_trivial_of_mem [N.Normal] {n : G} (hn : n ∈ N) : toEndHom N n = 𝟙 (G ⧸ₐ N) := by
+  apply Action.hom_ext
+  ext (x : G ⧸ N)
+  induction' x using Quotient.inductionOn with μ
+  exact Quotient.sound ((QuotientGroup.leftRel_apply).mpr <| by simpa)
+
+/-- If `H` and `N` are subgroups of a group `G` with `N` normal, there is a canonical
+group homomorphism `H ⧸ N ⊓ H` to the `G`-endomorphisms of `G ⧸ N`. -/
+def quotientToEndHom [N.Normal] : H ⧸ Subgroup.subgroupOf N H →* End (G ⧸ₐ N) :=
+  QuotientGroup.lift (Subgroup.subgroupOf N H) ((toEndHom N).comp H.subtype) <| fun _ uinU' ↦
+    toEndHom_trivial_of_mem uinU'
+
+@[simp]
+lemma quotientToEndHom_mk [N.Normal] (x : H) (g : G) :
+    (quotientToEndHom H N ⟦x⟧).hom ⟦g⟧ = ⟦g * x⁻¹⟧ :=
+  rfl
+
+/-- If `N` and `H` are subgroups of a group `G` with `N ≤ H`, this is the canonical
+`G`-morphism `G ⧸ N ⟶ G ⧸ H`. -/
+def quotientToQuotientOfLE [Fintype (G ⧸ H)] (h : N ≤ H) : (G ⧸ₐ N) ⟶ (G ⧸ₐ H) where
+  hom := Quotient.lift _ <| fun a b hab ↦ Quotient.sound <|
+    (QuotientGroup.leftRel_apply).mpr (h <| (QuotientGroup.leftRel_apply).mp hab)
+  comm g := by
+    ext (x : G ⧸ N)
+    induction' x using Quotient.inductionOn with μ
+    rfl
+
+@[simp]
+lemma quotientToQuotientOfLE_hom_mk [Fintype (G ⧸ H)] (h : N ≤ H) (x : G) :
+    (quotientToQuotientOfLE H N h).hom ⟦x⟧ = ⟦x⟧ :=
+  rfl
+
+end
+
 end FintypeCat
 
 section ToMulAction
diff --git a/Mathlib/RepresentationTheory/Action/Limits.lean b/Mathlib/RepresentationTheory/Action/Limits.lean
index 60d9a97e9d74f..3978ae5ce16ce 100644
--- a/Mathlib/RepresentationTheory/Action/Limits.lean
+++ b/Mathlib/RepresentationTheory/Action/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Abelian.FunctorCategory
 import Mathlib.CategoryTheory.Abelian.Transfer
diff --git a/Mathlib/RepresentationTheory/Action/Monoidal.lean b/Mathlib/RepresentationTheory/Action/Monoidal.lean
index 022783ffbf8d2..e9ebdc2e143b1 100644
--- a/Mathlib/RepresentationTheory/Action/Monoidal.lean
+++ b/Mathlib/RepresentationTheory/Action/Monoidal.lean
@@ -1,17 +1,15 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
-import Mathlib.RepresentationTheory.Action.Limits
-import Mathlib.RepresentationTheory.Action.Concrete
-import Mathlib.CategoryTheory.Monoidal.FunctorCategory
-import Mathlib.CategoryTheory.Monoidal.Transport
-import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
-import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory
 import Mathlib.CategoryTheory.Monoidal.Linear
-import Mathlib.CategoryTheory.Monoidal.Braided.Basic
+import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCategory
+import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
+import Mathlib.CategoryTheory.Monoidal.Transport
 import Mathlib.CategoryTheory.Monoidal.Types.Basic
+import Mathlib.RepresentationTheory.Action.Concrete
+import Mathlib.RepresentationTheory.Action.Limits
 
 /-!
 # Induced monoidal structure on `Action V G`
@@ -35,77 +33,36 @@ open MonoidalCategory
 
 variable [MonoidalCategory V]
 
+@[simps! tensorUnit_V tensorObj_V tensorHom_hom whiskerLeft_hom whiskerRight_hom
+  associator_hom_hom associator_inv_hom leftUnitor_hom_hom leftUnitor_inv_hom
+  rightUnitor_hom_hom rightUnitor_inv_hom]
 instance instMonoidalCategory : MonoidalCategory (Action V G) :=
   Monoidal.transport (Action.functorCategoryEquivalence _ _).symm
 
+/- Adding this solves `simpNF` linter report at `tensorUnit_ρ` -/
 @[simp]
-theorem tensorUnit_v : (𝟙_ (Action V G)).V = 𝟙_ V :=
-  rfl
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem tensorUnit_rho {g : G} : (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) :=
+theorem tensorUnit_ρ' {g : G} :
+    @DFunLike.coe (G →* MonCat.of (End (𝟙_ V))) _ _ _ (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) := by
   rfl
 
 @[simp]
-theorem tensor_v {X Y : Action V G} : (X ⊗ Y).V = X.V ⊗ Y.V :=
-  rfl
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem tensor_rho {X Y : Action V G} {g : G} : (X ⊗ Y).ρ g = X.ρ g ⊗ Y.ρ g :=
+theorem tensorUnit_ρ {g : G} : (𝟙_ (Action V G)).ρ g = 𝟙 (𝟙_ V) :=
   rfl
 
+/- Adding this solves `simpNF` linter report at `tensor_ρ` -/
 @[simp]
-theorem tensor_hom {W X Y Z : Action V G} (f : W ⟶ X) (g : Y ⟶ Z) : (f ⊗ g).hom = f.hom ⊗ g.hom :=
+theorem tensor_ρ' {X Y : Action V G} {g : G} :
+    @DFunLike.coe (G →* MonCat.of (End (X.V ⊗ Y.V))) _ _ _ (X ⊗ Y).ρ g = X.ρ g ⊗ Y.ρ g :=
   rfl
 
 @[simp]
-theorem whiskerLeft_hom (X : Action V G) {Y Z : Action V G} (f : Y ⟶ Z) :
-    (X ◁ f).hom = X.V ◁ f.hom :=
+theorem tensor_ρ {X Y : Action V G} {g : G} : (X ⊗ Y).ρ g = X.ρ g ⊗ Y.ρ g :=
   rfl
 
-@[simp]
-theorem whiskerRight_hom {X Y : Action V G} (f : X ⟶ Y) (Z : Action V G) :
-    (f ▷ Z).hom = f.hom ▷ Z.V :=
-  rfl
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem associator_hom_hom {X Y Z : Action V G} :
-    Hom.hom (α_ X Y Z).hom = (α_ X.V Y.V Z.V).hom := by
-  dsimp
-  simp
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem associator_inv_hom {X Y Z : Action V G} :
-    Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by
-  dsimp
-  simp
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem leftUnitor_hom_hom {X : Action V G} : Hom.hom (λ_ X).hom = (λ_ X.V).hom := by
-  dsimp
-  simp
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem leftUnitor_inv_hom {X : Action V G} : Hom.hom (λ_ X).inv = (λ_ X.V).inv := by
-  dsimp
-  simp
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem rightUnitor_hom_hom {X : Action V G} : Hom.hom (ρ_ X).hom = (ρ_ X.V).hom := by
-  dsimp
-  simp
-
--- Porting note: removed @[simp] as the simpNF linter complains
-theorem rightUnitor_inv_hom {X : Action V G} : Hom.hom (ρ_ X).inv = (ρ_ X.V).inv := by
-  dsimp
-  simp
-
 /-- Given an object `X` isomorphic to the tensor unit of `V`, `X` equipped with the trivial action
 is isomorphic to the tensor unit of `Action V G`. -/
 def tensorUnitIso {X : V} (f : 𝟙_ V ≅ X) : 𝟙_ (Action V G) ≅ Action.mk X 1 :=
-  Action.mkIso f fun _ => by
-    simp only [MonoidHom.one_apply, End.one_def, Category.id_comp f.hom, tensorUnit_rho,
-      MonCat.oneHom_apply, MonCat.one_of, Category.comp_id]
+  Action.mkIso f
 
 variable (V G)
 
@@ -284,7 +241,7 @@ noncomputable def leftRegularTensorIso (G : Type u) [Group G] (X : Action (Type
       comm := fun (g : G) => by
         funext ⟨(x₁ : G), (x₂ : X.V)⟩
         refine Prod.ext rfl ?_
-        erw [tensor_rho, tensor_rho]
+        rw [tensor_ρ, tensor_ρ]
         dsimp
         -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
         erw [leftRegular_ρ_apply]
diff --git a/Mathlib/RepresentationTheory/Basic.lean b/Mathlib/RepresentationTheory/Basic.lean
index 6f61040dc4ac8..979151b7da507 100644
--- a/Mathlib/RepresentationTheory/Basic.lean
+++ b/Mathlib/RepresentationTheory/Basic.lean
@@ -3,12 +3,7 @@ Copyright (c) 2022 Antoine Labelle. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Labelle
 -/
-import Mathlib.Algebra.Group.Equiv.TypeTags
-import Mathlib.Algebra.Module.Defs
-import Mathlib.Algebra.MonoidAlgebra.Basic
-import Mathlib.LinearAlgebra.Dual
 import Mathlib.LinearAlgebra.Contraction
-import Mathlib.RingTheory.TensorProduct.Basic
 
 /-!
 # Monoid representations
diff --git a/Mathlib/RepresentationTheory/Character.lean b/Mathlib/RepresentationTheory/Character.lean
index a0566328c0f4a..b629a2dc92e8b 100644
--- a/Mathlib/RepresentationTheory/Character.lean
+++ b/Mathlib/RepresentationTheory/Character.lean
@@ -32,7 +32,7 @@ noncomputable section
 
 universe u
 
-open CategoryTheory LinearMap CategoryTheory.MonoidalCategory Representation FiniteDimensional
+open CategoryTheory LinearMap CategoryTheory.MonoidalCategory Representation Module
 
 variable {k : Type u} [Field k]
 
@@ -51,21 +51,14 @@ theorem char_mul_comm (V : FDRep k G) (g : G) (h : G) :
     V.character (h * g) = V.character (g * h) := by simp only [trace_mul_comm, character, map_mul]
 
 @[simp]
-theorem char_one (V : FDRep k G) : V.character 1 = FiniteDimensional.finrank k V := by
+theorem char_one (V : FDRep k G) : V.character 1 = Module.finrank k V := by
   simp only [character, map_one, trace_one]
 
 /-- The character is multiplicative under the tensor product. -/
+@[simp]
 theorem char_tensor (V W : FDRep k G) : (V ⊗ W).character = V.character * W.character := by
   ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g)
 
--- Porting note: adding variant of `char_tensor` to make the simp-set confluent
-@[simp]
-theorem char_tensor' (V W : FDRep k G) :
-    character (Action.FunctorCategoryEquivalence.inverse.obj
-    (Action.FunctorCategoryEquivalence.functor.obj V ⊗
-     Action.FunctorCategoryEquivalence.functor.obj W)) = V.character * W.character := by
-  simp [← char_tensor]
-
 /-- The character of isomorphic representations is the same. -/
 theorem char_iso {V W : FDRep k G} (i : V ≅ W) : V.character = W.character := by
   ext g
diff --git a/Mathlib/RepresentationTheory/FDRep.lean b/Mathlib/RepresentationTheory/FDRep.lean
index 8cc68e7568dae..1f3f8535f0114 100644
--- a/Mathlib/RepresentationTheory/FDRep.lean
+++ b/Mathlib/RepresentationTheory/FDRep.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.FGModuleCat.Limits
 import Mathlib.CategoryTheory.Monoidal.Rigid.Braided
@@ -114,11 +114,11 @@ example : MonoidalPreadditive (FDRep k G) := by infer_instance
 
 example : MonoidalLinear k (FDRep k G) := by infer_instance
 
-open FiniteDimensional
+open Module
 
 open scoped Classical
 
--- We need to provide this instance explicitely as otherwise `finrank_hom_simple_simple` gives a
+-- We need to provide this instance explicitly as otherwise `finrank_hom_simple_simple` gives a
 -- deterministic timeout.
 instance : HasKernels (FDRep k G) := by infer_instance
 
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean b/Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean
index 79fadf7a25697..b666cf6409176 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/Hilbert90.lean
@@ -55,7 +55,7 @@ variable {K L : Type*} [Field K] [Field L] [Algebra K L] [FiniteDimensional K L]
 
 /-- Given `f : Aut_K(L) → Lˣ`, the sum `∑ f(φ) • φ` for `φ ∈ Aut_K(L)`, as a function `L → L`. -/
 noncomputable def aux (f : (L ≃ₐ[K] L) → Lˣ) : L → L :=
-  Finsupp.total L (fun φ : L ≃ₐ[K] L ↦ (φ : L → L))
+  Finsupp.linearCombination L (fun φ : L ≃ₐ[K] L ↦ (φ : L → L))
     (Finsupp.equivFunOnFinite.symm (fun φ => (f φ : L)))
 
 theorem aux_ne_zero (f : (L ≃ₐ[K] L) → Lˣ) : aux f ≠ 0 :=
@@ -83,7 +83,7 @@ theorem isMulOneCoboundary_of_isMulOneCocycle_of_aut_to_units
 /- Let `z : L` be such that `∑ f(h) * h(z) ≠ 0`, for `h ∈ Aut_K(L)` -/
   obtain ⟨z, hz⟩ : ∃ z, aux f z ≠ 0 :=
     not_forall.1 (fun H => aux_ne_zero f <| funext <| fun x => H x)
-  have : aux f z = ∑ h, f h * h z := by simp [aux, Finsupp.total, Finsupp.sum_fintype]
+  have : aux f z = ∑ h, f h * h z := by simp [aux, Finsupp.linearCombination, Finsupp.sum_fintype]
 /- Let `β = (∑ f(h) * h(z))⁻¹.` -/
   use (Units.mk0 (aux f z) hz)⁻¹
   intro g
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean b/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
index fbe67ec9e3438..7cd53f1275514 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
@@ -146,7 +146,7 @@ theorem dZero_comp_eq : dZero A ∘ₗ (zeroCochainsLequiv A) =
     oneCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 0 1 := by
   ext x y
   show A.ρ y (x default) - x default = _ + ({0} : Finset _).sum _
-  simp_rw [Fin.coe_fin_one, zero_add, pow_one, neg_smul, one_smul,
+  simp_rw [Fin.val_eq_zero, zero_add, pow_one, neg_smul, one_smul,
     Finset.sum_singleton, sub_eq_add_neg]
   rcongr i <;> exact Fin.elim0 i
 
@@ -202,9 +202,9 @@ theorem dOne_comp_dZero : dOne A ∘ₗ dZero A = 0 := by
   rfl
 
 theorem dTwo_comp_dOne : dTwo A ∘ₗ dOne A = 0 := by
-  show ModuleCat.ofHom (dOne A) ≫ ModuleCat.ofHom (dTwo A) = _
-  have h1 : _ ≫ ModuleCat.ofHom (dOne A) = _ ≫ _ := congr_arg ModuleCat.ofHom (dOne_comp_eq A)
-  have h2 : _ ≫ ModuleCat.ofHom (dTwo A) = _ ≫ _ := congr_arg ModuleCat.ofHom (dTwo_comp_eq A)
+  show ModuleCat.asHom (dOne A) ≫ ModuleCat.asHom (dTwo A) = _
+  have h1 : _ ≫ ModuleCat.asHom (dOne A) = _ ≫ _ := congr_arg ModuleCat.asHom (dOne_comp_eq A)
+  have h2 : _ ≫ ModuleCat.asHom (dTwo A) = _ ≫ _ := congr_arg ModuleCat.asHom (dTwo_comp_eq A)
   simp only [← LinearEquiv.toModuleIso_hom] at h1 h2
   simp only [(Iso.eq_inv_comp _).2 h2, (Iso.eq_inv_comp _).2 h1,
     Category.assoc, Iso.hom_inv_id_assoc, HomologicalComplex.d_comp_d_assoc, zero_comp, comp_zero]
@@ -716,7 +716,7 @@ lemma shortComplexH0_exact : (shortComplexH0 A).Exact := by
 `(inhomogeneousCochains A).d 0 1` of the complex of inhomogeneous cochains of `A`. -/
 @[simps! hom_left hom_right inv_left inv_right]
 def dZeroArrowIso : Arrow.mk ((inhomogeneousCochains A).d 0 1) ≅
-    Arrow.mk (ModuleCat.ofHom (dZero A)) :=
+    Arrow.mk (ModuleCat.asHom (dZero A)) :=
   Arrow.isoMk (zeroCochainsLequiv A).toModuleIso
     (oneCochainsLequiv A).toModuleIso (dZero_comp_eq A)
 
@@ -764,7 +764,7 @@ def isoOneCocycles : cocycles A 1 ≅ ModuleCat.of k (oneCocycles A) :=
     cyclesMapIso (shortComplexH1Iso A) ≪≫ (shortComplexH1 A).moduleCatCyclesIso
 
 lemma isoOneCocycles_hom_comp_subtype :
-    (isoOneCocycles A).hom ≫ ModuleCat.ofHom (oneCocycles A).subtype =
+    (isoOneCocycles A).hom ≫ ModuleCat.asHom (oneCocycles A).subtype =
       iCocycles A 1 ≫ (oneCochainsLequiv A).toModuleIso.hom := by
   dsimp [isoOneCocycles]
   rw [Category.assoc, Category.assoc]
@@ -774,7 +774,7 @@ lemma isoOneCocycles_hom_comp_subtype :
 lemma toCocycles_comp_isoOneCocycles_hom :
     toCocycles A 0 1 ≫ (isoOneCocycles A).hom =
       (zeroCochainsLequiv A).toModuleIso.hom ≫
-        ModuleCat.ofHom (shortComplexH1 A).moduleCatToCycles := by
+        ModuleCat.asHom (shortComplexH1 A).moduleCatToCycles := by
   simp [isoOneCocycles]
   rfl
 
@@ -812,7 +812,7 @@ def isoTwoCocycles : cocycles A 2 ≅ ModuleCat.of k (twoCocycles A) :=
     cyclesMapIso (shortComplexH2Iso A) ≪≫ (shortComplexH2 A).moduleCatCyclesIso
 
 lemma isoTwoCocycles_hom_comp_subtype :
-    (isoTwoCocycles A).hom ≫ ModuleCat.ofHom (twoCocycles A).subtype =
+    (isoTwoCocycles A).hom ≫ ModuleCat.asHom (twoCocycles A).subtype =
       iCocycles A 2 ≫ (twoCochainsLequiv A).toModuleIso.hom := by
   dsimp [isoTwoCocycles]
   rw [Category.assoc, Category.assoc]
@@ -822,7 +822,7 @@ lemma isoTwoCocycles_hom_comp_subtype :
 lemma toCocycles_comp_isoTwoCocycles_hom :
     toCocycles A 1 2 ≫ (isoTwoCocycles A).hom =
       (oneCochainsLequiv A).toModuleIso.hom ≫
-        ModuleCat.ofHom (shortComplexH2 A).moduleCatToCycles := by
+        ModuleCat.asHom (shortComplexH2 A).moduleCatToCycles := by
   simp [isoTwoCocycles]
   rfl
 
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean b/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
index 612922d39d32a..b7e92987380e6 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
@@ -68,7 +68,7 @@ universe u v w
 
 variable {k G : Type u} [CommRing k] {n : ℕ}
 
-open CategoryTheory
+open CategoryTheory Finsupp
 
 local notation "Gⁿ" => Fin n → G
 
@@ -136,7 +136,7 @@ theorem actionDiagonalSucc_inv_apply {G : Type u} [Group G] {n : ℕ} (g : G) (f
     dsimp only [actionDiagonalSucc]
     simp only [Iso.trans_inv, comp_hom, hn, diagonalSucc_inv_hom, types_comp_apply, tensorIso_inv,
       Iso.refl_inv, Action.tensorHom, id_hom, tensor_apply, types_id_apply,
-      leftRegularTensorIso_inv_hom, tensor_rho, leftRegular_ρ_apply, Pi.smul_apply, smul_eq_mul]
+      leftRegularTensorIso_inv_hom, tensor_ρ, leftRegular_ρ_apply, Pi.smul_apply, smul_eq_mul]
     refine' Fin.cases _ _ x
     · simp only [Fin.cons_zero, Fin.partialProd_zero, mul_one]
     · intro i
@@ -565,12 +565,12 @@ def forget₂ToModuleCatHomotopyEquiv :
 
 /-- The hom of `k`-linear `G`-representations `k[G¹] → k` sending `∑ nᵢgᵢ ↦ ∑ nᵢ`. -/
 def ε : Rep.ofMulAction k G (Fin 1 → G) ⟶ Rep.trivial k G k where
-  hom := Finsupp.total _ fun _ => (1 : k)
+  hom := Finsupp.linearCombination _ fun _ => (1 : k)
   comm g := Finsupp.lhom_ext' fun _ => LinearMap.ext_ring (by
     show
-      Finsupp.total k (fun _ => (1 : k)) (Finsupp.mapDomain _ (Finsupp.single _ _)) =
-        Finsupp.total k (fun _ => (1 : k)) (Finsupp.single _ _)
-    simp only [Finsupp.mapDomain_single, Finsupp.total_single])
+      Finsupp.linearCombination k (fun _ => (1 : k)) (Finsupp.mapDomain _ (Finsupp.single _ _)) =
+        Finsupp.linearCombination k (fun _ => (1 : k)) (Finsupp.single _ _)
+    simp only [Finsupp.mapDomain_single, Finsupp.linearCombination_single])
 
 /-- The homotopy equivalence of complexes of `k`-modules between the standard resolution of `k` as
 a trivial `G`-representation, and the complex which is `k` at 0 and 0 everywhere else, acts as
@@ -583,9 +583,9 @@ theorem forget₂ToModuleCatHomotopyEquiv_f_0_eq :
   convert Category.id_comp (X := (forget₂ToModuleCat k G).X 0) _
   · dsimp only [HomotopyEquiv.ofIso, compForgetAugmentedIso]
     simp only [Iso.symm_hom, eqToIso.inv, HomologicalComplex.eqToHom_f, eqToHom_refl]
-  trans (Finsupp.total _ fun _ => (1 : k)).comp ((ModuleCat.free k).map (terminal.from _))
+  trans (linearCombination _ fun _ => (1 : k)).comp ((ModuleCat.free k).map (terminal.from _))
   · dsimp
-    erw [Finsupp.lmapDomain_total (α := Fin 1 → G) (R := k) (α' := ⊤_ Type u)
+    erw [Finsupp.lmapDomain_linearCombination (α := Fin 1 → G) (R := k) (α' := ⊤_ Type u)
         (v := fun _ => (1 : k)) (v' := fun _ => (1 : k))
         (terminal.from
           ((classifyingSpaceUniversalCover G).obj (Opposite.op (SimplexCategory.mk 0))).V)
@@ -596,7 +596,8 @@ theorem forget₂ToModuleCatHomotopyEquiv_f_0_eq :
     · ext x
       dsimp (config := { unfoldPartialApp := true }) [HomotopyEquiv.ofIso,
         Finsupp.LinearEquiv.finsuppUnique]
-      rw [Finsupp.total_single, one_smul, @Unique.eq_default _ Types.terminalIso.toEquiv.unique x,
+      rw [linearCombination_single, one_smul,
+        @Unique.eq_default _ Types.terminalIso.toEquiv.unique x,
         ChainComplex.single₀_map_f_zero, LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply,
         Finsupp.equivFunOnFinite_apply, Finsupp.single_eq_same]
     · exact @Subsingleton.elim _ (@Unique.instSubsingleton _ (Limits.uniqueToTerminal _)) _ _
diff --git a/Mathlib/RepresentationTheory/Maschke.lean b/Mathlib/RepresentationTheory/Maschke.lean
index f35efe285d18d..7eaec76c225f9 100644
--- a/Mathlib/RepresentationTheory/Maschke.lean
+++ b/Mathlib/RepresentationTheory/Maschke.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.MonoidAlgebra.Basic
 import Mathlib.LinearAlgebra.Basis.VectorSpace
diff --git a/Mathlib/RepresentationTheory/Rep.lean b/Mathlib/RepresentationTheory/Rep.lean
index 8cf8b61925ff3..fc63a62f92cea 100644
--- a/Mathlib/RepresentationTheory/Rep.lean
+++ b/Mathlib/RepresentationTheory/Rep.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.ModuleCat.Adjunctions
 import Mathlib.Algebra.Category.ModuleCat.Limits
@@ -332,7 +332,7 @@ variable [Group G] (A B C : Rep k G)
 protected def ihom (A : Rep k G) : Rep k G ⥤ Rep k G where
   obj B := Rep.of (Representation.linHom A.ρ B.ρ)
   map := fun {X} {Y} f =>
-    { hom := ModuleCat.ofHom (LinearMap.llcomp k _ _ _ f.hom)
+    { hom := ModuleCat.asHom (LinearMap.llcomp k _ _ _ f.hom)
       comm := fun g => LinearMap.ext fun x => LinearMap.ext fun y => by
         show f.hom (X.ρ g _) = _
         simp only [hom_comm_apply]; rfl }
@@ -361,7 +361,7 @@ def homEquiv (A B C : Rep k G) : (A ⊗ B ⟶ C) ≃ (B ⟶ (Rep.ihom A).obj C)
       comm := fun g => TensorProduct.ext' fun x y => by
 /- Porting note: rest of broken proof was
         dsimp only [MonoidalCategory.tensorLeft_obj, ModuleCat.comp_def, LinearMap.comp_apply,
-          tensor_rho, ModuleCat.MonoidalCategory.hom_apply, TensorProduct.map_tmul]
+          tensor_ρ, ModuleCat.MonoidalCategory.hom_apply, TensorProduct.map_tmul]
         simp only [TensorProduct.uncurry_apply f.hom.flip, LinearMap.flip_apply, Action_ρ_eq_ρ,
           hom_comm_apply f g y, Rep.ihom_obj_ρ_apply, LinearMap.comp_apply, ρ_inv_self_apply] -/
         change TensorProduct.uncurry k _ _ _ f.hom.flip (A.ρ g x ⊗ₜ[k] B.ρ g y) =
@@ -404,7 +404,7 @@ theorem ihom_obj_ρ_def (A B : Rep k G) : ((ihom A).obj B).ρ = ((Rep.ihom A).ob
 
 @[simp]
 theorem homEquiv_def (A B C : Rep k G) : (ihom.adjunction A).homEquiv B C = Rep.homEquiv A B C :=
-  rfl
+  congrFun (congrFun (Adjunction.mkOfHomEquiv_homEquiv _) _) _
 
 @[simp]
 theorem ihom_ev_app_hom (A B : Rep k G) :
@@ -445,7 +445,9 @@ theorem MonoidalClosed.linearHomEquivComm_hom (f : A ⊗ B ⟶ C) :
   rfl
 
 theorem MonoidalClosed.linearHomEquiv_symm_hom (f : B ⟶ A ⟶[Rep k G] C) :
-    ((MonoidalClosed.linearHomEquiv A B C).symm f).hom = TensorProduct.uncurry k A B C f.hom.flip :=
+    ((MonoidalClosed.linearHomEquiv A B C).symm f).hom =
+      TensorProduct.uncurry k A B C f.hom.flip := by
+  simp [linearHomEquiv]
   rfl
 
 theorem MonoidalClosed.linearHomEquivComm_symm_hom (f : A ⟶ B ⟶[Rep k G] C) :
diff --git a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
index afc24e101812b..1c4dfed5cf15a 100644
--- a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
+++ b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean
@@ -241,17 +241,17 @@ private instance : AddCommGroup (AdicCompletion I R ⊗[R] (LinearMap.ker f)) :=
 
 private def firstRow : ComposableArrows (ModuleCat (AdicCompletion I R)) 4 :=
   ComposableArrows.mk₄
-    (ModuleCat.ofHom <| lTensorKerIncl I M f)
-    (ModuleCat.ofHom <| lTensorf I M f)
-    (ModuleCat.ofHom (0 : AdicCompletion I R ⊗[R] M →ₗ[AdicCompletion I R] PUnit))
-    (ModuleCat.ofHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
+    (ModuleCat.asHom <| lTensorKerIncl I M f)
+    (ModuleCat.asHom <| lTensorf I M f)
+    (ModuleCat.asHom (0 : AdicCompletion I R ⊗[R] M →ₗ[AdicCompletion I R] PUnit))
+    (ModuleCat.asHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
 
 private def secondRow : ComposableArrows (ModuleCat (AdicCompletion I R)) 4 :=
   ComposableArrows.mk₄
-    (ModuleCat.ofHom (map I <| (LinearMap.ker f).subtype))
-    (ModuleCat.ofHom (map I f))
-    (ModuleCat.ofHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
-    (ModuleCat.ofHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
+    (ModuleCat.asHom (map I <| (LinearMap.ker f).subtype))
+    (ModuleCat.asHom (map I f))
+    (ModuleCat.asHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
+    (ModuleCat.asHom (0 : _ →ₗ[AdicCompletion I R] PUnit))
 
 include hf
 
@@ -282,25 +282,25 @@ private lemma secondRow_exact [Fintype ι] [IsNoetherianRing R] : (secondRow I M
 /- The compatible vertical maps between the first and the second row. -/
 private def firstRowToSecondRow : firstRow I M f ⟶ secondRow I M f :=
   ComposableArrows.homMk₄
-    (ModuleCat.ofHom (ofTensorProduct I (LinearMap.ker f)))
-    (ModuleCat.ofHom (ofTensorProduct I (ι → R)))
-    (ModuleCat.ofHom (ofTensorProduct I M))
-    (ModuleCat.ofHom 0)
-    (ModuleCat.ofHom 0)
+    (ModuleCat.asHom (ofTensorProduct I (LinearMap.ker f)))
+    (ModuleCat.asHom (ofTensorProduct I (ι → R)))
+    (ModuleCat.asHom (ofTensorProduct I M))
+    (ModuleCat.asHom 0)
+    (ModuleCat.asHom 0)
     (ofTensorProduct_naturality I <| (LinearMap.ker f).subtype).symm
     (ofTensorProduct_naturality I f).symm
     rfl
     rfl
 
 private lemma ofTensorProduct_iso [Fintype ι] [IsNoetherianRing R] :
-    IsIso (ModuleCat.ofHom (ofTensorProduct I M)) := by
+    IsIso (ModuleCat.asHom (ofTensorProduct I M)) := by
   refine Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
     (firstRow_exact I M f hf) (secondRow_exact I M f hf) (firstRowToSecondRow I M f) ?_ ?_ ?_ ?_
   · apply ConcreteCategory.epi_of_surjective
     exact ofTensorProduct_surjective_of_finite I (LinearMap.ker f)
   · apply (ConcreteCategory.isIso_iff_bijective _).mpr
     exact ofTensorProduct_bijective_of_pi_of_fintype I ι
-  · show IsIso (ModuleCat.ofHom 0)
+  · show IsIso (ModuleCat.asHom 0)
     apply Limits.isIso_of_isTerminal
       <;> exact Limits.IsZero.isTerminal (ModuleCat.isZero_of_subsingleton _)
   · apply ConcreteCategory.mono_of_injective
@@ -310,9 +310,9 @@ private lemma ofTensorProduct_iso [Fintype ι] [IsNoetherianRing R] :
 private
 lemma ofTensorProduct_bijective_of_map_from_fin [Fintype ι] [IsNoetherianRing R] :
     Function.Bijective (ofTensorProduct I M) := by
-  have : IsIso (ModuleCat.ofHom (ofTensorProduct I M)) :=
+  have : IsIso (ModuleCat.asHom (ofTensorProduct I M)) :=
     ofTensorProduct_iso I M f hf
-  exact ConcreteCategory.bijective_of_isIso (ModuleCat.ofHom (ofTensorProduct I M))
+  exact ConcreteCategory.bijective_of_isIso (ModuleCat.asHom (ofTensorProduct I M))
 
 end
 
diff --git a/Mathlib/RingTheory/AdicCompletion/Basic.lean b/Mathlib/RingTheory/AdicCompletion/Basic.lean
index b1a5ab9bf4f6c..a680f5898fdf6 100644
--- a/Mathlib/RingTheory/AdicCompletion/Basic.lean
+++ b/Mathlib/RingTheory/AdicCompletion/Basic.lean
@@ -555,7 +555,7 @@ theorem le_jacobson_bot [IsAdicComplete I R] : I ≤ (⊥ : Ideal R).jacobson :=
     ring
   cases n
   · simp only [Ideal.one_eq_top, pow_zero, mem_top]
-  · rw [← neg_sub _ (1 : R), neg_mul, mul_geom_sum, neg_sub, sub_sub, add_comm, ← sub_sub,
+  · rw [← neg_sub _ (1 : R), neg_mul, mul_geom_sum, neg_sub, sub_sub, add_comm (_ ^ _), ← sub_sub,
       sub_self, zero_sub, @neg_mem_iff, mul_pow]
     exact Ideal.mul_mem_right _ (I ^ _) (Ideal.pow_mem_pow hx _)
 
diff --git a/Mathlib/RingTheory/AdicCompletion/Exactness.lean b/Mathlib/RingTheory/AdicCompletion/Exactness.lean
index e8545415bd9fd..c184632614480 100644
--- a/Mathlib/RingTheory/AdicCompletion/Exactness.lean
+++ b/Mathlib/RingTheory/AdicCompletion/Exactness.lean
@@ -98,7 +98,7 @@ theorem map_injective {f : M →ₗ[R] N} (hf : Function.Injective f) :
   rw [← Submodule.comap_map_eq_of_injective hf (I ^ n • ⊤ : Submodule R M),
     Submodule.map_smul'', Submodule.map_top]
   apply (smul_mono_right _ inf_le_right : I ^ n • (I ^ k • ⊤ ⊓ (range f)) ≤ _)
-  nth_rw 2 [show n = n + k - k by omega]
+  nth_rw 1 [show n = n + k - k by omega]
   rw [← hk (n + k) (show n + k ≥ k by omega)]
   exact ⟨by simpa using congrArg (fun x ↦ x.val (n + k)) hx, ⟨a (n + k), rfl⟩⟩
 
diff --git a/Mathlib/RingTheory/Adjoin/Basic.lean b/Mathlib/RingTheory/Adjoin/Basic.lean
index 2e497804ccf8a..83e58942dce2b 100644
--- a/Mathlib/RingTheory/Adjoin/Basic.lean
+++ b/Mathlib/RingTheory/Adjoin/Basic.lean
@@ -251,7 +251,7 @@ theorem adjoin_inl_union_inr_eq_prod (s) (t) :
     simpa [P] using Subalgebra.add_mem _ Ha Hb
 
 /-- If all elements of `s : Set A` commute pairwise, then `adjoin R s` is a commutative
-semiring.  -/
+semiring. -/
 def adjoinCommSemiringOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
     CommSemiring (adjoin R s) :=
   { (adjoin R s).toSemiring with
@@ -355,7 +355,7 @@ theorem pow_smul_mem_of_smul_subset_of_mem_adjoin [CommSemiring B] [Algebra R B]
     [IsScalarTower R A B] (r : A) (s : Set B) (B' : Subalgebra R B) (hs : r • s ⊆ B') {x : B}
     (hx : x ∈ adjoin R s) (hr : algebraMap A B r ∈ B') : ∃ n₀ : ℕ, ∀ n ≥ n₀, r ^ n • x ∈ B' := by
   change x ∈ Subalgebra.toSubmodule (adjoin R s) at hx
-  rw [adjoin_eq_span, Finsupp.mem_span_iff_total] at hx
+  rw [adjoin_eq_span, Finsupp.mem_span_iff_linearCombination] at hx
   rcases hx with ⟨l, rfl : (l.sum fun (i : Submonoid.closure s) (c : R) => c • (i : B)) = x⟩
   choose n₁ n₂ using fun x : Submonoid.closure s => Submonoid.pow_smul_mem_closure_smul r s x.prop
   use l.support.sup n₁
@@ -403,7 +403,7 @@ theorem adjoin_eq_ring_closure (s : Set A) :
 variable (R)
 
 /-- If all elements of `s : Set A` commute pairwise, then `adjoin R s` is a commutative
-ring.  -/
+ring. -/
 def adjoinCommRingOfComm {s : Set A} (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) :
     CommRing (adjoin R s) :=
   { (adjoin R s).toRing, adjoinCommSemiringOfComm R hcomm with }
diff --git a/Mathlib/RingTheory/Adjoin/FG.lean b/Mathlib/RingTheory/Adjoin/FG.lean
index eae1117ae1555..1256cc4e9a973 100644
--- a/Mathlib/RingTheory/Adjoin/FG.lean
+++ b/Mathlib/RingTheory/Adjoin/FG.lean
@@ -58,17 +58,17 @@ theorem fg_trans (h1 : (adjoin R s).toSubmodule.FG) (h2 : (adjoin (adjoin R s) t
     change r ∈ adjoin R (s ∪ t) at hr
     rw [adjoin_union_eq_adjoin_adjoin] at hr
     change r ∈ Subalgebra.toSubmodule (adjoin (adjoin R s) t) at hr
-    rw [← hq', ← Set.image_id q, Finsupp.mem_span_image_iff_total (adjoin R s)] at hr
+    rw [← hq', ← Set.image_id q, Finsupp.mem_span_image_iff_linearCombination (adjoin R s)] at hr
     rcases hr with ⟨l, hlq, rfl⟩
-    have := @Finsupp.total_apply A A (adjoin R s)
+    have := @Finsupp.linearCombination_apply A A (adjoin R s)
     rw [this, Finsupp.sum]
     refine sum_mem ?_
     intro z hz
     change (l z).1 * _ ∈ _
     have : (l z).1 ∈ Subalgebra.toSubmodule (adjoin R s) := (l z).2
-    rw [← hp', ← Set.image_id p, Finsupp.mem_span_image_iff_total R] at this
+    rw [← hp', ← Set.image_id p, Finsupp.mem_span_image_iff_linearCombination R] at this
     rcases this with ⟨l2, hlp, hl⟩
-    have := @Finsupp.total_apply A A R
+    have := @Finsupp.linearCombination_apply A A R
     rw [this] at hl
     rw [← hl, Finsupp.sum_mul]
     refine sum_mem ?_
diff --git a/Mathlib/RingTheory/AdjoinRoot.lean b/Mathlib/RingTheory/AdjoinRoot.lean
index 75dc495a1ff01..bba7f1e3e7457 100644
--- a/Mathlib/RingTheory/AdjoinRoot.lean
+++ b/Mathlib/RingTheory/AdjoinRoot.lean
@@ -353,11 +353,11 @@ noncomputable instance instField [Fact (Irreducible f)] : Field (AdjoinRoot f) w
   ratCast_def q := by
     rw [← map_natCast (of f), ← map_intCast (of f), ← map_div₀, ← Rat.cast_def]; rfl
   nnqsmul_def q x :=
-    AdjoinRoot.induction_on (C := fun y ↦ q • y = (of f) q * y) x fun p ↦ by
+    AdjoinRoot.induction_on f (C := fun y ↦ q • y = (of f) q * y) x fun p ↦ by
       simp only [smul_mk, of, RingHom.comp_apply, ← (mk f).map_mul, Polynomial.nnqsmul_eq_C_mul]
   qsmul_def q x :=
     -- Porting note: I gave the explicit motive and changed `rw` to `simp`.
-    AdjoinRoot.induction_on (C := fun y ↦ q • y = (of f) q * y) x fun p ↦ by
+    AdjoinRoot.induction_on f (C := fun y ↦ q • y = (of f) q * y) x fun p ↦ by
       simp only [smul_mk, of, RingHom.comp_apply, ← (mk f).map_mul, Polynomial.qsmul_eq_C_mul]
 
 theorem coe_injective (h : degree f ≠ 0) : Function.Injective ((↑) : K → AdjoinRoot f) :=
@@ -736,7 +736,7 @@ theorem quotAdjoinRootEquivQuotPolynomialQuot_symm_mk_mk (p : R[X]) :
     quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_symm_quotQuotMk,
     quotMapOfEquivQuotMapCMapSpanMk_symm_mk]
 
-/-- Promote `AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot` to an alg_equiv.  -/
+/-- Promote `AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot` to an alg_equiv. -/
 @[simps!]
 noncomputable def quotEquivQuotMap (f : R[X]) (I : Ideal R) :
     (AdjoinRoot f ⧸ Ideal.map (of f) I) ≃ₐ[R]
diff --git a/Mathlib/RingTheory/Algebraic.lean b/Mathlib/RingTheory/Algebraic.lean
index 0326b17fa9001..67c966cf31c26 100644
--- a/Mathlib/RingTheory/Algebraic.lean
+++ b/Mathlib/RingTheory/Algebraic.lean
@@ -5,6 +5,7 @@ Authors: Johan Commelin
 -/
 import Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic
 import Mathlib.RingTheory.Polynomial.IntegralNormalization
+import Mathlib.RingTheory.LocalRing.Basic
 
 /-!
 # Algebraic elements and algebraic extensions
@@ -79,7 +80,7 @@ theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
 /-- An algebra is algebraic if and only if it is algebraic as a subalgebra. -/
 theorem Algebra.isAlgebraic_iff : Algebra.IsAlgebraic R A ↔ (⊤ : Subalgebra R A).IsAlgebraic := by
   delta Subalgebra.IsAlgebraic
-  simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true, iff_self_iff]
+  simp only [Algebra.isAlgebraic_def, Algebra.mem_top, forall_prop_of_true]
 
 theorem isAlgebraic_iff_not_injective {x : A} :
     IsAlgebraic R x ↔ ¬Function.Injective (Polynomial.aeval x : R[X] →ₐ[R] A) := by
diff --git a/Mathlib/RingTheory/AlgebraicIndependent.lean b/Mathlib/RingTheory/AlgebraicIndependent.lean
index 7f5a9e71866e3..0e358adaa9418 100644
--- a/Mathlib/RingTheory/AlgebraicIndependent.lean
+++ b/Mathlib/RingTheory/AlgebraicIndependent.lean
@@ -82,11 +82,7 @@ theorem algebraicIndependent_iff_injective_aeval :
 @[simp]
 theorem algebraicIndependent_empty_type_iff [IsEmpty ι] :
     AlgebraicIndependent R x ↔ Injective (algebraMap R A) := by
-  have : aeval x = (Algebra.ofId R A).comp (@isEmptyAlgEquiv R ι _ _).toAlgHom := by
-    ext i
-    exact IsEmpty.elim' ‹IsEmpty ι› i
-  rw [AlgebraicIndependent, this, ← Injective.of_comp_iff' _ (@isEmptyAlgEquiv R ι _ _).bijective]
-  rfl
+  rw [algebraicIndependent_iff_injective_aeval, MvPolynomial.aeval_injective_iff_of_isEmpty]
 
 namespace AlgebraicIndependent
 
@@ -100,19 +96,19 @@ variable (hx : AlgebraicIndependent R x)
 include hx
 
 theorem algebraMap_injective : Injective (algebraMap R A) := by
-  simpa [Function.comp] using
+  simpa [Function.comp_def] using
     (Injective.of_comp_iff (algebraicIndependent_iff_injective_aeval.1 hx) MvPolynomial.C).2
       (MvPolynomial.C_injective _ _)
 
 theorem linearIndependent : LinearIndependent R x := by
-  rw [linearIndependent_iff_injective_total]
-  have : Finsupp.total R x =
-      (MvPolynomial.aeval x).toLinearMap.comp (Finsupp.total R X) := by
+  rw [linearIndependent_iff_injective_linearCombination]
+  have : Finsupp.linearCombination R x =
+      (MvPolynomial.aeval x).toLinearMap.comp (Finsupp.linearCombination R X) := by
     ext
     simp
   rw [this]
   refine hx.comp ?_
-  rw [← linearIndependent_iff_injective_total]
+  rw [← linearIndependent_iff_injective_linearCombination]
   exact linearIndependent_X _ _
 
 protected theorem injective [Nontrivial R] : Injective x :=
@@ -135,7 +131,7 @@ theorem map {f : A →ₐ[R] A'} (hf_inj : Set.InjOn f (adjoin R (range x))) :
     intro p
     rw [AlgHom.mem_range]
     refine ⟨MvPolynomial.rename (codRestrict x (range x) mem_range_self) p, ?_⟩
-    simp [Function.comp, aeval_rename]
+    simp [Function.comp_def, aeval_rename]
   intro x y hxy
   rw [this] at hxy
   rw [adjoin_eq_range] at hf_inj
diff --git a/Mathlib/RingTheory/Artinian.lean b/Mathlib/RingTheory/Artinian.lean
index 1dfc6d4350316..9e43f7cbc62b9 100644
--- a/Mathlib/RingTheory/Artinian.lean
+++ b/Mathlib/RingTheory/Artinian.lean
@@ -91,22 +91,33 @@ theorem isArtinian_of_surjective (f : M →ₗ[R] P) (hf : Function.Surjective f
       show A.comap f < B.comap f from Submodule.comap_strictMono_of_surjective hf hAB)
     (InvImage.wf (Submodule.comap f) IsWellFounded.wf)⟩
 
+instance isArtinian_of_quotient_of_artinian
+    (N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
+  isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N)
+
 variable {M}
 
 theorem isArtinian_of_linearEquiv (f : M ≃ₗ[R] P) [IsArtinian R M] : IsArtinian R P :=
   isArtinian_of_surjective _ f.toLinearMap f.toEquiv.surjective
 
 theorem isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P)
-    (hf : Function.Injective f) (hg : Function.Surjective g)
     (h : LinearMap.range f = LinearMap.ker g) : IsArtinian R N :=
-  wellFounded_lt_exact_sequence (LinearMap.range f) (Submodule.map f)
-    (Submodule.comap f) (Submodule.comap g) (Submodule.map g) (Submodule.gciMapComap hf)
-    (Submodule.giMapComap hg)
-    (by simp [Submodule.map_comap_eq, inf_comm]) (by simp [Submodule.comap_map_eq, h])
+  wellFounded_lt_exact_sequence (LinearMap.range f) (Submodule.map (f.ker.liftQ f le_rfl))
+    (Submodule.comap (f.ker.liftQ f le_rfl))
+    (Submodule.comap g.rangeRestrict) (Submodule.map g.rangeRestrict)
+    (Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <| Submodule.ker_liftQ_eq_bot _ _ _ le_rfl)
+    (Submodule.giMapComap g.surjective_rangeRestrict)
+    (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
+    (by simp [Submodule.comap_map_eq, h])
+
+theorem isArtinian_iff_submodule_quotient (S : Submodule R P) :
+    IsArtinian R P ↔ IsArtinian R S ∧ IsArtinian R (P ⧸ S) := by
+  refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
+  apply isArtinian_of_range_eq_ker S.subtype S.mkQ
+  rw [Submodule.ker_mkQ, Submodule.range_subtype]
 
 instance isArtinian_prod [IsArtinian R M] [IsArtinian R P] : IsArtinian R (M × P) :=
-  isArtinian_of_range_eq_ker (LinearMap.inl R M P) (LinearMap.snd R M P) LinearMap.inl_injective
-    LinearMap.snd_surjective (LinearMap.range_inl R M P)
+  isArtinian_of_range_eq_ker (LinearMap.inl R M P) (LinearMap.snd R M P) (LinearMap.range_inl R M P)
 
 instance (priority := 100) isArtinian_of_finite [Finite M] : IsArtinian R M :=
   ⟨Finite.wellFounded_of_trans_of_irrefl _⟩
@@ -115,17 +126,12 @@ instance (priority := 100) isArtinian_of_finite [Finite M] : IsArtinian R M :=
 -- attribute [local elab_as_elim] Finite.induction_empty_option
 
 instance isArtinian_pi {R ι : Type*} [Finite ι] :
-    ∀ {M : ι → Type*} [Ring R] [∀ i, AddCommGroup (M i)],
-      ∀ [∀ i, Module R (M i)], ∀ [∀ i, IsArtinian R (M i)], IsArtinian R (∀ i, M i) := by
+    ∀ {M : ι → Type*} [Ring R] [∀ i, AddCommGroup (M i)]
+      [∀ i, Module R (M i)] [∀ i, IsArtinian R (M i)], IsArtinian R (∀ i, M i) := by
   apply Finite.induction_empty_option _ _ _ ι
-  · intro α β e hα M _ _ _ _
-    have := @hα
-    exact isArtinian_of_linearEquiv (LinearEquiv.piCongrLeft R M e)
-  · intro M _ _ _ _
-    infer_instance
-  · intro α _ ih M _ _ _ _
-    have := @ih
-    exact isArtinian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm
+  · exact fun e h ↦ isArtinian_of_linearEquiv (LinearEquiv.piCongrLeft R _ e)
+  · infer_instance
+  · exact fun ih ↦ isArtinian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm
 
 /-- A version of `isArtinian_pi` for non-dependent functions. We need this instance because
 sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to
@@ -336,10 +342,6 @@ theorem Ring.isArtinian_of_zero_eq_one {R} [Ring R] (h01 : (0 : R) = 1) : IsArti
 theorem isArtinian_of_submodule_of_artinian (R M) [Ring R] [AddCommGroup M] [Module R M]
     (N : Submodule R M) (_ : IsArtinian R M) : IsArtinian R N := inferInstance
 
-instance isArtinian_of_quotient_of_artinian (R) [Ring R] (M) [AddCommGroup M] [Module R M]
-    (N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
-  isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N)
-
 /-- If `M / S / R` is a scalar tower, and `M / R` is Artinian, then `M / S` is also Artinian. -/
 theorem isArtinian_of_tower (R) {S M} [CommRing R] [Ring S] [AddCommGroup M] [Algebra R S]
     [Module S M] [Module R M] [IsScalarTower R S M] (h : IsArtinian R M) : IsArtinian S M :=
@@ -355,7 +357,7 @@ theorem isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M
   haveI := Classical.decEq R
   have : ∀ x ∈ s, x ∈ N := fun x hx => hs ▸ Submodule.subset_span hx
   refine @isArtinian_of_surjective _ ((↑s : Set M) →₀ R) N _ _ _ _ _ ?_ ?_ isArtinian_finsupp
-  · exact Finsupp.total R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
+  · exact Finsupp.linearCombination R (fun i => ⟨i, hs ▸ subset_span i.2⟩)
   · rw [← LinearMap.range_eq_top, eq_top_iff,
        ← map_le_map_iff_of_injective (show Injective (Submodule.subtype N)
          from Subtype.val_injective), Submodule.map_top, range_subtype,
@@ -520,7 +522,7 @@ instance [IsReduced R] : DecompositionMonoid (Polynomial R) :=
 theorem isSemisimpleRing_of_isReduced [IsReduced R] : IsSemisimpleRing R :=
   (equivPi R).symm.isSemisimpleRing
 
-proof_wanted IsSemisimpleRing.isArtinianRing (R : Type*) [CommRing R] [IsSemisimpleRing R] :
+proof_wanted IsSemisimpleRing.isArtinianRing (R : Type*) [Ring R] [IsSemisimpleRing R] :
     IsArtinianRing R
 
 end IsArtinianRing
diff --git a/Mathlib/RingTheory/Bezout.lean b/Mathlib/RingTheory/Bezout.lean
index 4ef7731887a6d..37aef0df9ae66 100644
--- a/Mathlib/RingTheory/Bezout.lean
+++ b/Mathlib/RingTheory/Bezout.lean
@@ -51,13 +51,13 @@ theorem TFAE [IsBezout R] [IsDomain R] :
     [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R] := by
   classical
     tfae_have 1 → 2
-    · intro H; exact ⟨fun I => isPrincipal_of_FG _ (IsNoetherian.noetherian _)⟩
+    | H => ⟨fun I => isPrincipal_of_FG _ (IsNoetherian.noetherian _)⟩
     tfae_have 2 → 3
-    · intro; infer_instance
+    | _ => inferInstance
     tfae_have 3 → 4
-    · intro; infer_instance
+    | _ => inferInstance
     tfae_have 4 → 1
-    · rintro ⟨h⟩
+    | ⟨h⟩ => by
       rw [isNoetherianRing_iff, isNoetherian_iff_fg_wellFounded]
       refine ⟨RelEmbedding.wellFounded ?_ h⟩
       have : ∀ I : { J : Ideal R // J.FG }, ∃ x : R, (I : Ideal R) = Ideal.span {x} :=
diff --git a/Mathlib/RingTheory/Bialgebra/Equiv.lean b/Mathlib/RingTheory/Bialgebra/Equiv.lean
index a94379991aff9..c0160257534d0 100644
--- a/Mathlib/RingTheory/Bialgebra/Equiv.lean
+++ b/Mathlib/RingTheory/Bialgebra/Equiv.lean
@@ -39,7 +39,7 @@ attribute [nolint docBlame] BialgEquiv.toCoalgEquiv
 notation:50 A " ≃ₐc[" R "] " B => BialgEquiv R A B
 
 /-- `BialgEquivClass F R A B` asserts `F` is a type of bundled bialgebra equivalences
-from `A` to `B`.  -/
+from `A` to `B`. -/
 class BialgEquivClass (F : Type*) (R A B : outParam Type*) [CommSemiring R]
     [Semiring A] [Semiring B] [Algebra R A] [Algebra R B]
     [CoalgebraStruct R A] [CoalgebraStruct R B] [EquivLike F A B]
diff --git a/Mathlib/RingTheory/Bialgebra/Hom.lean b/Mathlib/RingTheory/Bialgebra/Hom.lean
index fe171794c8b66..f1b2923cd334c 100644
--- a/Mathlib/RingTheory/Bialgebra/Hom.lean
+++ b/Mathlib/RingTheory/Bialgebra/Hom.lean
@@ -45,7 +45,7 @@ infixr:25 " →ₐc " => BialgHom _
 notation:25 A " →ₐc[" R "] " B => BialgHom R A B
 
 /-- `BialgHomClass F R A B` asserts `F` is a type of bundled bialgebra homomorphisms
-from `A` to `B`.  -/
+from `A` to `B`. -/
 class BialgHomClass (F : Type*) (R A B : outParam Type*)
     [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
     [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]
diff --git a/Mathlib/RingTheory/Binomial.lean b/Mathlib/RingTheory/Binomial.lean
index 31d44ff0e4ac7..45bbba6501dad 100644
--- a/Mathlib/RingTheory/Binomial.lean
+++ b/Mathlib/RingTheory/Binomial.lean
@@ -241,7 +241,7 @@ instance Int.instBinomialRing : BinomialRing ℤ where
   nsmul_right_injective n hn r s hrs := Int.eq_of_mul_eq_mul_left (Int.ofNat_ne_zero.mpr hn) hrs
   multichoose := Int.multichoose
   factorial_nsmul_multichoose r k := by
-    rw [Int.multichoose, nsmul_eq_mul]
+    rw [Int.multichoose.eq_def, nsmul_eq_mul]
     cases r with
     | ofNat n =>
       simp only [multichoose, nsmul_eq_mul, Int.ofNat_eq_coe, Int.ofNat_mul_out]
@@ -452,7 +452,7 @@ theorem choose_smul_choose [NatPowAssoc R] (r : R) (n k : ℕ) (hkn : k ≤ n) :
   refine nsmul_right_injective (Nat.factorial n) (Nat.factorial_ne_zero n) ?_
   simp only
   rw [nsmul_left_comm, ← descPochhammer_eq_factorial_smul_choose,
-    ← Nat.choose_mul_factorial_mul_factorial hkn, ← smul_mul_smul,
+    ← Nat.choose_mul_factorial_mul_factorial hkn, ← smul_mul_smul_comm,
     ← descPochhammer_eq_factorial_smul_choose, mul_nsmul',
     ← descPochhammer_eq_factorial_smul_choose, smul_mul_assoc]
   nth_rw 2 [← Nat.sub_add_cancel hkn]
diff --git a/Mathlib/RingTheory/ChainOfDivisors.lean b/Mathlib/RingTheory/ChainOfDivisors.lean
index ed816bb3683ea..684edca6086c9 100644
--- a/Mathlib/RingTheory/ChainOfDivisors.lean
+++ b/Mathlib/RingTheory/ChainOfDivisors.lean
@@ -160,7 +160,7 @@ theorem element_of_chain_eq_pow_second_of_chain {q r : Associates M} {n : ℕ} (
       rw [Finset.card_image_iff]
       refine Set.injOn_of_injective (fun m m' h => Fin.ext ?_)
       refine
-        pow_injective_of_not_unit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
+        pow_injective_of_not_isUnit (element_of_chain_not_isUnit_of_index_ne_zero (by simp) h₁) ?_ h
       exact Irreducible.ne_zero (second_of_chain_is_irreducible hn h₁ (@h₂) hq)
     suffices H' : ∀ r ∈ Finset.univ.image fun m : Fin (i + 1) => c 1 ^ (m : ℕ), r ≤ q by
       simp only [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← this]
@@ -332,7 +332,7 @@ theorem multiplicity_prime_eq_multiplicity_image_by_factor_orderIso {m p : Assoc
 
 end
 
-variable [Unique Mˣ] [Unique Nˣ]
+variable [Subsingleton Mˣ] [Subsingleton Nˣ]
 
 /-- The order isomorphism between the factors of `mk m` and the factors of `mk n` induced by a
   bijection between the factors of `m` and the factors of `n` that preserves `∣`. -/
diff --git a/Mathlib/RingTheory/Coalgebra/Basic.lean b/Mathlib/RingTheory/Coalgebra/Basic.lean
index 03811b4c741c4..643086a03c0e1 100644
--- a/Mathlib/RingTheory/Coalgebra/Basic.lean
+++ b/Mathlib/RingTheory/Coalgebra/Basic.lean
@@ -62,6 +62,9 @@ def Coalgebra.Repr.arbitrary (R : Type u) {A : Type v}
   index := TensorProduct.exists_finset (R := R) (CoalgebraStruct.comul a) |>.choose
   eq := TensorProduct.exists_finset (R := R) (CoalgebraStruct.comul a) |>.choose_spec.symm
 
+@[inherit_doc Coalgebra.Repr.arbitrary]
+scoped[Coalgebra] notation "ℛ" => Coalgebra.Repr.arbitrary
+
 namespace Coalgebra
 export CoalgebraStruct (comul counit)
 end Coalgebra
@@ -115,6 +118,44 @@ lemma sum_tmul_counit_eq {a : A} (repr : Coalgebra.Repr R a) :
     ∑ i ∈ repr.index, (repr.left i) ⊗ₜ counit (R := R) (repr.right i) = a ⊗ₜ[R] 1 := by
   simpa [← repr.eq, map_sum] using congr($(lTensor_counit_comp_comul (R := R) (A := A)) a)
 
+@[simp]
+lemma sum_tmul_tmul_eq {a : A} (repr : Repr R a)
+    (a₁ : (i : repr.ι) → Repr R (repr.left i)) (a₂ : (i : repr.ι) → Repr R (repr.right i)) :
+    ∑ i in repr.index, ∑ j in (a₁ i).index,
+      (a₁ i).left j ⊗ₜ[R] (a₁ i).right j ⊗ₜ[R] repr.right i
+      = ∑ i in repr.index, ∑ j in (a₂ i).index,
+      repr.left i ⊗ₜ[R] (a₂ i).left j ⊗ₜ[R] (a₂ i).right j := by
+  simpa [(a₂ _).eq, ← (a₁ _).eq, ← TensorProduct.tmul_sum,
+    TensorProduct.sum_tmul, ← repr.eq] using congr($(coassoc (R := R)) a)
+
+@[simp]
+theorem sum_counit_tmul_map_eq {B : Type*} [AddCommMonoid B] [Module R B]
+    {F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Repr R a} :
+    ∑ i in repr.index, counit (R := R) (repr.left i) ⊗ₜ f (repr.right i) = 1 ⊗ₜ[R] f a := by
+  have := sum_counit_tmul_eq repr
+  apply_fun LinearMap.lTensor R (f : A →ₗ[R] B) at this
+  simp_all only [map_sum, LinearMap.lTensor_tmul, LinearMap.coe_coe]
+
+@[simp]
+theorem sum_map_tmul_counit_eq {B : Type*} [AddCommMonoid B] [Module R B]
+    {F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f : F) (a : A) {repr : Repr R a} :
+    ∑ i in repr.index, f (repr.left i) ⊗ₜ counit (R := R) (repr.right i) = f a ⊗ₜ[R] 1 := by
+  have := sum_tmul_counit_eq repr
+  apply_fun LinearMap.rTensor R (f : A →ₗ[R] B) at this
+  simp_all only [map_sum, LinearMap.rTensor_tmul, LinearMap.coe_coe]
+
+@[simp]
+theorem sum_map_tmul_tmul_eq {B : Type*} [AddCommMonoid B] [Module R B]
+    {F : Type*} [FunLike F A B] [LinearMapClass F R A B] (f g h : F) (a : A) {repr : Repr R a}
+    {a₁ : (i : repr.ι) → Repr R (repr.left i)} {a₂ : (i : repr.ι) → Repr R (repr.right i)} :
+    ∑ i in repr.index, ∑ j in (a₂ i).index,
+      f (repr.left i) ⊗ₜ (g ((a₂ i).left j) ⊗ₜ h ((a₂ i).right j)) =
+    ∑ i in repr.index, ∑ j in (a₁ i).index,
+      f ((a₁ i).left j) ⊗ₜ[R] (g ((a₁ i).right j) ⊗ₜ[R] h (repr.right i)) := by
+  have := sum_tmul_tmul_eq repr a₁ a₂
+  apply_fun TensorProduct.map (f : A →ₗ[R] B)
+    (TensorProduct.map (g : A →ₗ[R] B) (h : A →ₗ[R] B)) at this
+  simp_all only [map_sum, TensorProduct.map_tmul, LinearMap.coe_coe]
 
 end Coalgebra
 
@@ -284,8 +325,7 @@ open Coalgebra
 variable {R A B : Type*} [CommSemiring R] [AddCommMonoid A] [AddCommMonoid B]
   [Module R A] [Module R B] [CoalgebraStruct R A] [CoalgebraStruct R B]
 
-/-- The coalgebra instance will be defined in #11975, in
-`Mathlib.RingTheory.Coalgebra.TensorProduct`. -/
+/-- See `Mathlib.RingTheory.Coalgebra.TensorProduct` for the `Coalgebra` instance. -/
 @[simps] instance instCoalgebraStruct :
     CoalgebraStruct R (A ⊗[R] B) where
   comul := TensorProduct.tensorTensorTensorComm R A A B B ∘ₗ TensorProduct.map comul comul
diff --git a/Mathlib/RingTheory/Coalgebra/Equiv.lean b/Mathlib/RingTheory/Coalgebra/Equiv.lean
index 5b15a25b51a3b..843220c0bfa2f 100644
--- a/Mathlib/RingTheory/Coalgebra/Equiv.lean
+++ b/Mathlib/RingTheory/Coalgebra/Equiv.lean
@@ -38,7 +38,7 @@ attribute [nolint docBlame] CoalgEquiv.toLinearEquiv
 notation:50 A " ≃ₗc[" R "] " B => CoalgEquiv R A B
 
 /-- `CoalgEquivClass F R A B` asserts `F` is a type of bundled coalgebra equivalences
-from `A` to `B`.  -/
+from `A` to `B`. -/
 class CoalgEquivClass (F : Type*) (R A B : outParam Type*) [CommSemiring R]
     [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B]
     [CoalgebraStruct R A] [CoalgebraStruct R B] [EquivLike F A B]
@@ -200,10 +200,23 @@ def symm (e : A ≃ₗc[R] B) : B ≃ₗc[R] A :=
 theorem symm_toLinearEquiv (e : A ≃ₗc[R] B) :
     e.symm = (e : A ≃ₗ[R] B).symm := rfl
 
+theorem coe_symm_toLinearEquiv (e : A ≃ₗc[R] B) :
+    ⇑(e : A ≃ₗ[R] B).symm = e.symm := rfl
+
 @[simp]
 theorem symm_toCoalgHom (e : A ≃ₗc[R] B) :
     ((e.symm : B →ₗc[R] A) : B →ₗ[R] A) = (e : A ≃ₗ[R] B).symm := rfl
 
+@[simp]
+theorem symm_apply_apply (e : A ≃ₗc[R] B) (x) :
+    e.symm (e x) = x :=
+  LinearEquiv.symm_apply_apply (e : A ≃ₗ[R] B) x
+
+@[simp]
+theorem apply_symm_apply (e : A ≃ₗc[R] B) (x) :
+    e (e.symm x) = x :=
+  LinearEquiv.apply_symm_apply (e : A ≃ₗ[R] B) x
+
 /-- See Note [custom simps projection] -/
 def Simps.symm_apply {R : Type*} [CommSemiring R]
     {A : Type*} {B : Type*} [AddCommMonoid A] [AddCommMonoid B] [Module R A] [Module R B]
@@ -240,4 +253,35 @@ theorem coe_toEquiv_trans : (e₁₂ : A ≃ B).trans e₂₃ = (e₁₂.trans e
   rfl
 
 end
+variable [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A]
+  [AddCommMonoid B] [Module R B] [CoalgebraStruct R B]
+
+/-- Let `A` be an `R`-coalgebra and let `B` be an `R`-module with a `CoalgebraStruct`.
+A linear equivalence `A ≃ₗ[R] B` that respects the `CoalgebraStruct`s defines an `R`-coalgebra
+structure on `B`. -/
+@[reducible] def toCoalgebra (f : A ≃ₗc[R] B) :
+    Coalgebra R B where
+  coassoc := by
+    simp only [← ((f : A ≃ₗ[R] B).comp_toLinearMap_symm_eq _ _).2 f.map_comp_comul,
+      ← LinearMap.comp_assoc]
+    congr 1
+    ext x
+    simpa only [toCoalgHom_eq_coe, CoalgHom.toLinearMap_eq_coe, LinearMap.coe_comp,
+      LinearEquiv.coe_coe, Function.comp_apply, ← (ℛ R _).eq, map_sum, TensorProduct.map_tmul,
+      LinearMap.coe_coe, CoalgHom.coe_coe, LinearMap.rTensor_tmul, coe_symm_toLinearEquiv,
+      symm_apply_apply, LinearMap.lTensor_comp_map, TensorProduct.sum_tmul,
+      TensorProduct.assoc_tmul, TensorProduct.tmul_sum] using (sum_map_tmul_tmul_eq f f f x).symm
+  rTensor_counit_comp_comul := by
+    simp_rw [(f.toLinearEquiv.eq_comp_toLinearMap_symm _ _).2 f.counit_comp,
+      ← (f.toLinearEquiv.comp_toLinearMap_symm_eq _ _).2 f.map_comp_comul, ← LinearMap.comp_assoc,
+      f.toLinearEquiv.comp_toLinearMap_symm_eq]
+    ext x
+    simp [← (ℛ R _).eq, coe_symm_toLinearEquiv]
+  lTensor_counit_comp_comul := by
+    simp_rw [(f.toLinearEquiv.eq_comp_toLinearMap_symm _ _).2 f.counit_comp,
+      ← (f.toLinearEquiv.comp_toLinearMap_symm_eq _ _).2 f.map_comp_comul, ← LinearMap.comp_assoc,
+      f.toLinearEquiv.comp_toLinearMap_symm_eq]
+    ext x
+    simp [← (ℛ R _).eq, coe_symm_toLinearEquiv]
+
 end CoalgEquiv
diff --git a/Mathlib/RingTheory/Coalgebra/Hom.lean b/Mathlib/RingTheory/Coalgebra/Hom.lean
index 75d7d5fd60d0e..c6c26e0bf65b9 100644
--- a/Mathlib/RingTheory/Coalgebra/Hom.lean
+++ b/Mathlib/RingTheory/Coalgebra/Hom.lean
@@ -43,7 +43,7 @@ infixr:25 " →ₗc " => CoalgHom _
 notation:25 A " →ₗc[" R "] " B => CoalgHom R A B
 
 /-- `CoalgHomClass F R A B` asserts `F` is a type of bundled coalgebra homomorphisms
-from `A` to `B`.  -/
+from `A` to `B`. -/
 class CoalgHomClass (F : Type*) (R A B : outParam Type*)
     [CommSemiring R] [AddCommMonoid A] [Module R A] [AddCommMonoid B] [Module R B]
     [CoalgebraStruct R A] [CoalgebraStruct R B] [FunLike F A B]
@@ -308,16 +308,4 @@ def Repr.induced {a : A} (repr : Repr R a)
   eq := (congr($((CoalgHomClass.map_comp_comul φ).symm) a).trans <|
       by rw [LinearMap.comp_apply, ← repr.eq, map_sum]; rfl).symm
 
-@[simp]
-lemma sum_tmul_counit_apply_eq
-    {F : Type*} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) {a : A} (repr : Repr R a) :
-    ∑ i ∈ repr.index, counit (R := R) (repr.left i) ⊗ₜ φ (repr.right i) = 1 ⊗ₜ[R] φ a := by
-  simp [← sum_counit_tmul_eq  (repr.induced φ)]
-
-@[simp]
-lemma sum_tmul_apply_counit_eq
-    {F : Type*} [FunLike F A B] [CoalgHomClass F R A B] (φ : F) {a : A} (repr : Repr R a) :
-    ∑ i ∈ repr.index, φ (repr.left i) ⊗ₜ counit (R := R) (repr.right i) = φ a ⊗ₜ[R] 1 := by
-  simp [← sum_tmul_counit_eq (repr.induced φ)]
-
 end Coalgebra
diff --git a/Mathlib/RingTheory/Coalgebra/TensorProduct.lean b/Mathlib/RingTheory/Coalgebra/TensorProduct.lean
new file mode 100644
index 0000000000000..5226e76ef5d0e
--- /dev/null
+++ b/Mathlib/RingTheory/Coalgebra/TensorProduct.lean
@@ -0,0 +1,201 @@
+/-
+Copyright (c) 2024 Amelia Livingston. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Amelia Livingston
+-/
+import Mathlib.Algebra.Category.CoalgebraCat.ComonEquivalence
+
+/-!
+# Tensor products of coalgebras
+
+Given two `R`-coalgebras `M, N`, we can define a natural comultiplication map
+`Δ : M ⊗[R] N → (M ⊗[R] N) ⊗[R] (M ⊗[R] N)` and counit map `ε : M ⊗[R] N → R` induced by
+the comultiplication and counit maps of `M` and `N`. These `Δ, ε` are declared as linear maps
+in `TensorProduct.instCoalgebraStruct` in `Mathlib.RingTheory.Coalgebra.Basic`.
+
+In this file we show that `Δ, ε` satisfy the axioms of a coalgebra, and also define other data
+in the monoidal structure on `R`-coalgebras, like the tensor product of two coalgebra morphisms
+as a coalgebra morphism.
+
+## Implementation notes
+
+We keep the linear maps underlying `Δ, ε` and other definitions in this file syntactically equal
+to the corresponding definitions for tensor products of modules in the hope that this makes
+life easier. However, to fill in prop fields, we use the API in
+`Mathlib.Algebra.Category.CoalgebraCat.ComonEquivalence`. That file defines the monoidal category
+structure on coalgebras induced by an equivalence with comonoid objects in the category of modules,
+`CoalgebraCat.instMonoidalCategoryAux`, but we do not declare this as an instance - we just use it
+to prove things. Then, we use the definitions in this file to make a monoidal category instance on
+`CoalgebraCat R` in `Mathlib.Algebra.Category.CoalgebraCat.Monoidal` that has simpler data.
+
+However, this approach forces our coalgebras to be in the same universe as the base ring `R`,
+since it relies on the monoidal category structure on `ModuleCat R`, which is currently
+universe monomorphic. Any contribution that achieves universe polymorphism would be welcome. For
+instance, the tensor product of coalgebras in the
+[FLT repo](https://github.com/ImperialCollegeLondon/FLT/blob/eef74b4538c8852363936dfaad23e6ffba72eca5/FLT/mathlibExperiments/Coalgebra/TensorProduct.lean)
+is already universe polymorphic since it does not go via category theory.
+
+-/
+
+universe v u
+
+open CategoryTheory
+open scoped TensorProduct
+
+section
+
+variable {R M N P Q : Type u} [CommRing R]
+  [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] [Coalgebra R M] [Coalgebra R N]
+
+open MonoidalCategory in
+noncomputable instance TensorProduct.instCoalgebra : Coalgebra R (M ⊗[R] N) :=
+  let I := Monoidal.transport ((CoalgebraCat.comonEquivalence R).symm)
+  CoalgEquiv.toCoalgebra
+    (A := (CoalgebraCat.of R M ⊗ CoalgebraCat.of R N : CoalgebraCat R))
+    { LinearEquiv.refl R _ with
+      counit_comp := rfl
+      map_comp_comul := by
+        rw [CoalgebraCat.ofComonObjCoalgebraStruct_comul]
+        simp [-Mon_.monMonoidalStruct_tensorObj_X,
+          ModuleCat.MonoidalCategory.instMonoidalCategoryStruct_tensorHom,
+          ModuleCat.comp_def, ModuleCat.of, ModuleCat.asHom,
+          ModuleCat.MonoidalCategory.tensor_μ_eq_tensorTensorTensorComm] }
+
+end
+
+namespace Coalgebra
+namespace TensorProduct
+
+open CoalgebraCat.MonoidalCategoryAux MonoidalCategory
+
+variable {R M N P Q : Type u} [CommRing R]
+  [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [AddCommGroup Q] [Module R M] [Module R N]
+  [Module R P] [Module R Q] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P] [Coalgebra R Q]
+
+attribute [local instance] CoalgebraCat.instMonoidalCategoryAux in
+section
+
+/-- The tensor product of two coalgebra morphisms as a coalgebra morphism. -/
+noncomputable def map (f : M →ₗc[R] N) (g : P →ₗc[R] Q) :
+    M ⊗[R] P →ₗc[R] N ⊗[R] Q where
+  toLinearMap := _root_.TensorProduct.map f.toLinearMap g.toLinearMap
+  counit_comp := by
+    simp_rw [← tensorHom_toLinearMap]
+    apply (CoalgebraCat.ofHom f ⊗ CoalgebraCat.ofHom g).1.counit_comp
+  map_comp_comul := by
+    simp_rw [← tensorHom_toLinearMap, ← comul_tensorObj]
+    apply (CoalgebraCat.ofHom f ⊗ CoalgebraCat.ofHom g).1.map_comp_comul
+
+@[simp]
+theorem map_tmul (f : M →ₗc[R] N) (g : P →ₗc[R] Q) (x : M) (y : P) :
+    map f g (x ⊗ₜ y) = f x ⊗ₜ g y :=
+  rfl
+
+@[simp]
+theorem map_toLinearMap (f : M →ₗc[R] N) (g : P →ₗc[R] Q) :
+    map f g = _root_.TensorProduct.map (f : M →ₗ[R] N) (g : P →ₗ[R] Q) := rfl
+
+variable (R M N P)
+
+/-- The associator for tensor products of R-coalgebras, as a coalgebra equivalence. -/
+protected noncomputable def assoc :
+    (M ⊗[R] N) ⊗[R] P ≃ₗc[R] M ⊗[R] (N ⊗[R] P) :=
+  { _root_.TensorProduct.assoc R M N P with
+    counit_comp := by
+      simp_rw [← associator_hom_toLinearMap, ← counit_tensorObj_tensorObj_right,
+        ← counit_tensorObj_tensorObj_left]
+      apply CoalgHom.counit_comp (α_ (CoalgebraCat.of R M) (CoalgebraCat.of R N)
+        (CoalgebraCat.of R P)).hom.1
+    map_comp_comul := by
+      simp_rw [← associator_hom_toLinearMap, ← comul_tensorObj_tensorObj_left,
+        ← comul_tensorObj_tensorObj_right]
+      exact CoalgHom.map_comp_comul (α_ (CoalgebraCat.of R M)
+        (CoalgebraCat.of R N) (CoalgebraCat.of R P)).hom.1 }
+
+variable {R M N P}
+
+@[simp]
+theorem assoc_tmul (x : M) (y : N) (z : P) :
+    Coalgebra.TensorProduct.assoc R M N P ((x ⊗ₜ y) ⊗ₜ z) = x ⊗ₜ (y ⊗ₜ z) :=
+  rfl
+
+@[simp]
+theorem assoc_symm_tmul (x : M) (y : N) (z : P) :
+    (Coalgebra.TensorProduct.assoc R M N P).symm (x ⊗ₜ (y ⊗ₜ z)) = (x ⊗ₜ y) ⊗ₜ z :=
+  rfl
+
+@[simp]
+theorem assoc_toLinearEquiv :
+    Coalgebra.TensorProduct.assoc R M N P = _root_.TensorProduct.assoc R M N P := rfl
+
+variable (R M)
+
+/-- The base ring is a left identity for the tensor product of coalgebras, up to
+coalgebra equivalence. -/
+protected noncomputable def lid : R ⊗[R] M ≃ₗc[R] M :=
+  { _root_.TensorProduct.lid R M with
+    counit_comp := by
+      simp only [← leftUnitor_hom_toLinearMap]
+      apply CoalgHom.counit_comp (λ_ (CoalgebraCat.of R M)).hom.1
+    map_comp_comul := by
+      simp_rw [← leftUnitor_hom_toLinearMap, ← comul_tensorObj]
+      apply CoalgHom.map_comp_comul (λ_ (CoalgebraCat.of R M)).hom.1 }
+
+variable {R M}
+
+@[simp]
+theorem lid_toLinearEquiv :
+    (Coalgebra.TensorProduct.lid R M) = _root_.TensorProduct.lid R M := rfl
+
+@[simp]
+theorem lid_tmul (r : R) (a : M) : Coalgebra.TensorProduct.lid R M (r ⊗ₜ a) = r • a := rfl
+
+@[simp]
+theorem lid_symm_apply (a : M) : (Coalgebra.TensorProduct.lid R M).symm a = 1 ⊗ₜ a := rfl
+
+variable (R M)
+
+/-- The base ring is a right identity for the tensor product of coalgebras, up to
+coalgebra equivalence. -/
+protected noncomputable def rid : M ⊗[R] R ≃ₗc[R] M :=
+  { _root_.TensorProduct.rid R M with
+    counit_comp := by
+      simp only [← rightUnitor_hom_toLinearMap]
+      apply CoalgHom.counit_comp (ρ_ (CoalgebraCat.of R M)).hom.1
+    map_comp_comul := by
+      simp_rw [← rightUnitor_hom_toLinearMap, ← comul_tensorObj]
+      apply CoalgHom.map_comp_comul (ρ_ (CoalgebraCat.of R M)).hom.1 }
+
+variable {R M}
+
+@[simp]
+theorem rid_toLinearEquiv :
+    (Coalgebra.TensorProduct.rid R M) = _root_.TensorProduct.rid R M := rfl
+
+@[simp]
+theorem rid_tmul (r : R) (a : M) : Coalgebra.TensorProduct.rid R M (a ⊗ₜ r) = r • a := rfl
+
+@[simp]
+theorem rid_symm_apply (a : M) : (Coalgebra.TensorProduct.rid R M).symm a = a ⊗ₜ 1 := rfl
+
+end
+
+end TensorProduct
+end Coalgebra
+namespace CoalgHom
+
+variable {R M N P : Type u} [CommRing R]
+  [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N]
+  [Module R P] [Coalgebra R M] [Coalgebra R N] [Coalgebra R P]
+
+variable (M)
+
+/-- `lTensor M f : M ⊗ N →ₗc M ⊗ P` is the natural coalgebra morphism induced by `f : N →ₗc P`. -/
+noncomputable abbrev lTensor (f : N →ₗc[R] P) : M ⊗[R] N →ₗc[R] M ⊗[R] P :=
+  Coalgebra.TensorProduct.map (CoalgHom.id R M) f
+
+/-- `rTensor M f : N ⊗ M →ₗc P ⊗ M` is the natural coalgebra morphism induced by `f : N →ₗc P`. -/
+noncomputable abbrev rTensor (f : N →ₗc[R] P) : N ⊗[R] M →ₗc[R] P ⊗[R] M :=
+  Coalgebra.TensorProduct.map f (CoalgHom.id R M)
+
+end CoalgHom
diff --git a/Mathlib/RingTheory/Congruence/Basic.lean b/Mathlib/RingTheory/Congruence/Basic.lean
index 30337d5e5beaf..131575077c37b 100644
--- a/Mathlib/RingTheory/Congruence/Basic.lean
+++ b/Mathlib/RingTheory/Congruence/Basic.lean
@@ -67,16 +67,15 @@ section Basic
 
 variable [Add R] [Mul R] (c : RingCon R)
 
--- Porting note: upgrade to `FunLike`
 /-- A coercion from a congruence relation to its underlying binary relation. -/
-instance : FunLike (RingCon R) R (R → Prop) :=
-  { coe := fun c => c.r,
-    coe_injective' := fun x y h => by
-      rcases x with ⟨⟨x, _⟩, _⟩
-      rcases y with ⟨⟨y, _⟩, _⟩
-      congr!
-      rw [Setoid.ext_iff,(show x.Rel = y.Rel from h)]
-      simp}
+instance : FunLike (RingCon R) R (R → Prop) where
+  coe c := c.r
+  coe_injective' x y h := by
+    rcases x with ⟨⟨x, _⟩, _⟩
+    rcases y with ⟨⟨y, _⟩, _⟩
+    congr!
+    rw [Setoid.ext_iff, (show x.Rel = y.Rel from h)]
+    simp
 
 theorem rel_eq_coe : c.r = c :=
   rfl
@@ -126,6 +125,16 @@ theorem ext' {c d : RingCon R} (H : ⇑c = ⇑d) : c = d := DFunLike.coe_injecti
 theorem ext {c d : RingCon R} (H : ∀ x y, c x y ↔ d x y) : c = d :=
   ext' <| by ext; apply H
 
+/--
+Pulling back a `RingCon` across a ring homomorphism.
+-/
+def comap {R R' F : Type*} [Add R] [Add R']
+    [FunLike F R R'] [AddHomClass F R R'] [Mul R] [Mul R'] [MulHomClass F R R']
+    (J : RingCon R') (f : F) :
+    RingCon R where
+  __ := J.toCon.comap f (map_mul f)
+  __ := J.toAddCon.comap f (map_add f)
+
 end Basic
 
 section Quotient
@@ -409,7 +418,6 @@ end Quotient
 The API in this section is copied from `Mathlib/GroupTheory/Congruence.lean`
 -/
 
-
 section Lattice
 
 variable [Add R] [Mul R]
@@ -448,7 +456,7 @@ theorem coe_sInf (S : Set (RingCon R)) : ⇑(sInf S) = sInf ((⇑) '' S) := by
 
 @[simp, norm_cast]
 theorem coe_iInf {ι : Sort*} (f : ι → RingCon R) : ⇑(iInf f) = ⨅ i, ⇑(f i) := by
-  rw [iInf, coe_sInf, ← Set.range_comp, sInf_range, Function.comp]
+  rw [iInf, coe_sInf, ← Set.range_comp, sInf_range, Function.comp_def]
 
 instance : PartialOrder (RingCon R) where
   le_refl _c _ _ := id
diff --git a/Mathlib/RingTheory/Coprime/Basic.lean b/Mathlib/RingTheory/Coprime/Basic.lean
index ea3e3dce7eff2..851c87915e7a1 100644
--- a/Mathlib/RingTheory/Coprime/Basic.lean
+++ b/Mathlib/RingTheory/Coprime/Basic.lean
@@ -232,7 +232,7 @@ variable {R G : Type*} [CommSemiring R] [Group G] [MulAction G R] [SMulCommClass
 
 theorem isCoprime_group_smul_left : IsCoprime (x • y) z ↔ IsCoprime y z :=
   ⟨fun ⟨a, b, h⟩ => ⟨x • a, b, by rwa [smul_mul_assoc, ← mul_smul_comm]⟩, fun ⟨a, b, h⟩ =>
-    ⟨x⁻¹ • a, b, by rwa [smul_mul_smul, inv_mul_cancel, one_smul]⟩⟩
+    ⟨x⁻¹ • a, b, by rwa [smul_mul_smul_comm, inv_mul_cancel, one_smul]⟩⟩
 
 theorem isCoprime_group_smul_right : IsCoprime y (x • z) ↔ IsCoprime y z :=
   isCoprime_comm.trans <| (isCoprime_group_smul_left x z y).trans isCoprime_comm
diff --git a/Mathlib/RingTheory/Coprime/Ideal.lean b/Mathlib/RingTheory/Coprime/Ideal.lean
index 3a76aa94c02e4..3f7176c9ae4c7 100644
--- a/Mathlib/RingTheory/Coprime/Ideal.lean
+++ b/Mathlib/RingTheory/Coprime/Ideal.lean
@@ -32,10 +32,10 @@ theorem iSup_iInf_eq_top_iff_pairwise {t : Finset ι} (h : t.Nonempty) (I : ι 
   haveI : DecidableEq ι := Classical.decEq ι
   rw [eq_top_iff_one, Submodule.mem_iSup_finset_iff_exists_sum]
   refine h.cons_induction ?_ ?_ <;> clear t h
-  · simp only [Finset.sum_singleton, Finset.coe_singleton, Set.pairwise_singleton, iff_true_iff]
+  · simp only [Finset.sum_singleton, Finset.coe_singleton, Set.pairwise_singleton, iff_true]
     refine fun a => ⟨fun i => if h : i = a then ⟨1, ?_⟩ else 0, ?_⟩
     · simp [h]
-    · simp only [dif_pos, dif_ctx_congr, Submodule.coe_mk, eq_self_iff_true]
+    · simp only [dif_pos, Submodule.coe_mk, eq_self_iff_true]
   intro a t hat h ih
   rw [Finset.coe_cons,
     Set.pairwise_insert_of_symmetric fun i j (h : I i ⊔ I j = ⊤) ↦ (sup_comm _ _).trans h]
diff --git a/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean b/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
index 9659b2099784c..4163653269e60 100644
--- a/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
+++ b/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
@@ -58,7 +58,7 @@ dedekind domain, dedekind ring, adic valuation
 
 noncomputable section
 
-open scoped Classical DiscreteValuation
+open scoped Classical Multiplicative
 
 open Multiplicative IsDedekindDomain
 
@@ -458,11 +458,11 @@ instance : Algebra R (v.adicCompletionIntegers K) where
   map_one' := by simp only [map_one]; rfl
   map_mul' x y := by
     ext
-    simp_rw [RingHom.map_mul, Subring.coe_mul, UniformSpace.Completion.coe_mul]
+    simp only [map_mul, UniformSpace.Completion.coe_mul, MulMemClass.mk_mul_mk]
   map_zero' := by simp only [map_zero]; rfl
   map_add' x y := by
     ext
-    simp_rw [RingHom.map_add, Subring.coe_add, UniformSpace.Completion.coe_add]
+    simp only [map_add, UniformSpace.Completion.coe_add, AddMemClass.mk_add_mk]
   commutes' r x := by
     rw [mul_comm]
   smul_def' r x := by
diff --git a/Mathlib/RingTheory/DedekindDomain/Basic.lean b/Mathlib/RingTheory/DedekindDomain/Basic.lean
index 29a6df2a644da..aae1db626fae1 100644
--- a/Mathlib/RingTheory/DedekindDomain/Basic.lean
+++ b/Mathlib/RingTheory/DedekindDomain/Basic.lean
@@ -49,7 +49,7 @@ variable (R A K : Type*) [CommRing R] [CommRing A] [Field K]
 open scoped nonZeroDivisors Polynomial
 
 /-- A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. -/
-class Ring.DimensionLEOne : Prop :=
+class Ring.DimensionLEOne : Prop where
   (maximalOfPrime : ∀ {p : Ideal R}, p ≠ ⊥ → p.IsPrime → p.IsMaximal)
 
 open Ideal Ring
@@ -122,7 +122,6 @@ use `isDedekindDomain_iff` to prove `IsDedekindDomain` for a given `fraction_map
 
 This is the default implementation, but there are equivalent definitions,
 `IsDedekindDomainDvr` and `IsDedekindDomainInv`.
-TODO: Prove that these are actually equivalent definitions.
 -/
 class IsDedekindDomain
   extends IsDomain A, IsDedekindRing A : Prop
diff --git a/Mathlib/RingTheory/DedekindDomain/Dvr.lean b/Mathlib/RingTheory/DedekindDomain/Dvr.lean
index 04b1f0a1bc959..2edc66bfc7bfa 100644
--- a/Mathlib/RingTheory/DedekindDomain/Dvr.lean
+++ b/Mathlib/RingTheory/DedekindDomain/Dvr.lean
@@ -1,18 +1,16 @@
 /-
 Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
+Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio, Yongle Hu
 -/
-import Mathlib.RingTheory.Localization.LocalizationLocalization
-import Mathlib.RingTheory.Localization.Submodule
 import Mathlib.RingTheory.DiscreteValuationRing.TFAE
+import Mathlib.RingTheory.LocalProperties.IntegrallyClosed
 
 /-!
 # Dedekind domains
 
 This file defines an equivalent notion of a Dedekind domain (or Dedekind ring),
-namely a Noetherian integral domain where the localization at all nonzero prime ideals is a DVR
-(TODO: and shows that implies the main definition).
+namely a Noetherian integral domain where the localization at all nonzero prime ideals is a DVR.
 
 ## Main definitions
 
@@ -53,10 +51,8 @@ open scoped nonZeroDivisors Polynomial
 localization at every nonzero prime is a discrete valuation ring.
 
 This is equivalent to `IsDedekindDomain`.
-TODO: prove the equivalence.
 -/
-structure IsDedekindDomainDvr : Prop where
-  isNoetherianRing : IsNoetherianRing A
+class IsDedekindDomainDvr extends IsNoetherian A A : Prop where
   is_dvr_at_nonzero_prime : ∀ P ≠ (⊥ : Ideal A), ∀ _ : P.IsPrime,
     DiscreteValuationRing (Localization.AtPrime P)
 
@@ -94,7 +90,7 @@ theorem IsLocalization.isDedekindDomain [IsDedekindDomain A] {M : Submonoid A} (
     IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M _ _
   refine (isDedekindDomain_iff _ (FractionRing A)).mpr ⟨?_, ?_, ?_, ?_⟩
   · infer_instance
-  · exact IsLocalization.isNoetherianRing M _ (by infer_instance)
+  · exact IsLocalization.isNoetherianRing M _ inferInstance
   · exact Ring.DimensionLEOne.localization Aₘ hM
   · intro x hx
     obtain ⟨⟨y, y_mem⟩, hy⟩ := hx.exists_multiple_integral_of_isLocalization M _
@@ -109,8 +105,12 @@ theorem IsLocalization.AtPrime.isDedekindDomain [IsDedekindDomain A] (P : Ideal
     IsDedekindDomain Aₘ :=
   IsLocalization.isDedekindDomain A P.primeCompl_le_nonZeroDivisors Aₘ
 
+instance Localization.AtPrime.isDedekindDomain [IsDedekindDomain A] (P : Ideal A) [P.IsPrime] :
+    IsDedekindDomain (Localization.AtPrime P) :=
+  IsLocalization.AtPrime.isDedekindDomain A P _
+
 theorem IsLocalization.AtPrime.not_isField {P : Ideal A} (hP : P ≠ ⊥) [pP : P.IsPrime] (Aₘ : Type*)
-    [CommRing Aₘ] [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] : ¬IsField Aₘ := by
+    [CommRing Aₘ] [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] : ¬ IsField Aₘ := by
   intro h
   letI := h.toField
   obtain ⟨x, x_mem, x_ne⟩ := P.ne_bot_iff.mp hP
@@ -139,7 +139,36 @@ theorem IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain [IsDedek
 /-- Dedekind domains, in the sense of Noetherian integrally closed domains of Krull dimension ≤ 1,
 are also Dedekind domains in the sense of Noetherian domains where the localization at every
 nonzero prime ideal is a DVR. -/
-theorem IsDedekindDomain.isDedekindDomainDvr [IsDedekindDomain A] : IsDedekindDomainDvr A :=
-  { isNoetherianRing := IsDedekindRing.toIsNoetherian
-    is_dvr_at_nonzero_prime := fun _ hP _ =>
-      IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain A hP _ }
+instance IsDedekindDomain.isDedekindDomainDvr [IsDedekindDomain A] : IsDedekindDomainDvr A where
+  is_dvr_at_nonzero_prime := fun _ hP _ =>
+    IsLocalization.AtPrime.discreteValuationRing_of_dedekind_domain A hP _
+
+instance IsDedekindDomainDvr.ring_dimensionLEOne [h : IsDedekindDomainDvr A] :
+    Ring.DimensionLEOne A where
+  maximalOfPrime := by
+    intro p hp hpp
+    rcases p.exists_le_maximal (Ideal.IsPrime.ne_top hpp) with ⟨q, hq, hpq⟩
+    let f := (IsLocalization.orderIsoOfPrime q.primeCompl (Localization.AtPrime q)).symm
+    let P := f ⟨p, hpp, hpq.disjoint_compl_left⟩
+    let Q := f ⟨q, hq.isPrime, Set.disjoint_left.mpr fun _ a => a⟩
+    have hinj : Function.Injective (algebraMap A (Localization.AtPrime q)) :=
+      IsLocalization.injective (Localization.AtPrime q) q.primeCompl_le_nonZeroDivisors
+    have hp1 : P.1 ≠ ⊥ := fun x => hp ((p.map_eq_bot_iff_of_injective hinj).mp x)
+    have hq1 : Q.1 ≠ ⊥ :=
+      fun x => (ne_bot_of_le_ne_bot hp hpq) ((q.map_eq_bot_iff_of_injective hinj).mp x)
+    rcases (DiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime q)).mp
+      (h.is_dvr_at_nonzero_prime q (ne_bot_of_le_ne_bot hp hpq) hq.isPrime) with ⟨_, huq⟩
+    rw [show p = q from Subtype.val_inj.mpr <| f.injective <|
+      Subtype.val_inj.mp (huq.unique ⟨hp1, P.2⟩ ⟨hq1, Q.2⟩)]
+    exact hq
+
+instance IsDedekindDomainDvr.isIntegrallyClosed [h : IsDedekindDomainDvr A] :
+    IsIntegrallyClosed A :=
+  IsIntegrallyClosed.of_localization_maximal <| fun p hp0 hpm =>
+    let ⟨_, _⟩ := (DiscreteValuationRing.iff_pid_with_one_nonzero_prime (Localization.AtPrime p)).mp
+      (h.is_dvr_at_nonzero_prime p hp0 hpm.isPrime)
+    inferInstance
+
+/-- If an integral domain is Noetherian, and the localization at every nonzero prime is
+a discrete valuation ring, then it is a Dedekind domain. -/
+instance IsDedekindDomainDvr.isDedekindDomain [IsDedekindDomainDvr A] : IsDedekindDomain A where
diff --git a/Mathlib/RingTheory/DedekindDomain/Factorization.lean b/Mathlib/RingTheory/DedekindDomain/Factorization.lean
index 21265293630ad..9c8f207b3ee90 100644
--- a/Mathlib/RingTheory/DedekindDomain/Factorization.lean
+++ b/Mathlib/RingTheory/DedekindDomain/Factorization.lean
@@ -406,7 +406,7 @@ theorem count_neg_zpow (n : ℤ) (I : FractionalIdeal R⁰ K) :
       ← zpow_add₀ hI, neg_add_cancel, zpow_zero]
     exact count_one K v
 
-theorem count_inv  (I : FractionalIdeal R⁰ K) :
+theorem count_inv (I : FractionalIdeal R⁰ K) :
     count K v (I⁻¹) = - count K v I := by
   rw [← zpow_neg_one, count_neg_zpow K v (1 : ℤ) I, zpow_one]
 
diff --git a/Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean b/Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
index fc15e6527ca9e..9cd04bf371ef8 100644
--- a/Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
+++ b/Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
@@ -54,6 +54,9 @@ instance : CommRing (FiniteIntegralAdeles R K) :=
 instance : TopologicalSpace (FiniteIntegralAdeles R K) :=
   inferInstanceAs (TopologicalSpace (∀ v : HeightOneSpectrum R, v.adicCompletionIntegers K))
 
+instance (v : HeightOneSpectrum R) : TopologicalRing (v.adicCompletionIntegers K) :=
+  Subring.instTopologicalRing ..
+
 instance : TopologicalRing (FiniteIntegralAdeles R K) :=
   inferInstanceAs (TopologicalRing (∀ v : HeightOneSpectrum R, v.adicCompletionIntegers K))
 
@@ -212,7 +215,7 @@ theorem zero : (0 : K_hat R K).IsFiniteAdele := by
   rw [IsFiniteAdele, Filter.eventually_cofinite]
   have h_empty :
     {v : HeightOneSpectrum R | ¬(0 : v.adicCompletion K) ∈ v.adicCompletionIntegers K} = ∅ := by
-    ext v; rw [mem_empty_iff_false, iff_false_iff]; intro hv
+    ext v; rw [mem_empty_iff_false, iff_false]; intro hv
     rw [mem_setOf] at hv; apply hv; rw [mem_adicCompletionIntegers]
     have h_zero : (Valued.v (0 : v.adicCompletion K) : WithZero (Multiplicative ℤ)) = 0 :=
       Valued.v.map_zero'
@@ -256,13 +259,13 @@ theorem one : (1 : K_hat R K).IsFiniteAdele := by
   rw [IsFiniteAdele, Filter.eventually_cofinite]
   have h_empty :
     {v : HeightOneSpectrum R | ¬(1 : v.adicCompletion K) ∈ v.adicCompletionIntegers K} = ∅ := by
-    ext v; rw [mem_empty_iff_false, iff_false_iff]; intro hv
+    ext v; rw [mem_empty_iff_false, iff_false]; intro hv
     rw [mem_setOf] at hv; apply hv; rw [mem_adicCompletionIntegers]
     exact le_of_eq Valued.v.map_one'
   -- Porting note: was `exact`, but `OfNat` got in the way.
   convert finite_empty
 
-open scoped DiscreteValuation
+open scoped Multiplicative
 
 theorem algebraMap' (k : K) : (_root_.algebraMap K (K_hat R K) k).IsFiniteAdele := by
   rw [IsFiniteAdele, Filter.eventually_cofinite]
@@ -334,7 +337,7 @@ instance : IsScalarTower R K (FiniteAdeleRing R K) :=
   IsScalarTower.of_algebraMap_eq' rfl
 
 instance : Coe (FiniteAdeleRing R K) (K_hat R K) where
-  coe := fun x ↦ x.1
+  coe x := x.1
 
 @[ext]
 lemma ext {a₁ a₂ : FiniteAdeleRing R K} (h : (a₁ : K_hat R K) = a₂) : a₁ = a₂ :=
@@ -368,7 +371,7 @@ lemma exists_finiteIntegralAdele_iff (a : FiniteAdeleRing R K) : (∃ c : R_hat
 section Topology
 
 open nonZeroDivisors
-open scoped DiscreteValuation
+open scoped Multiplicative
 
 variable {R K} in
 lemma mul_nonZeroDivisor_mem_finiteIntegralAdeles (a : FiniteAdeleRing R K) :
diff --git a/Mathlib/RingTheory/DedekindDomain/Ideal.lean b/Mathlib/RingTheory/DedekindDomain/Ideal.lean
index 5c53166947b18..426d939612987 100644
--- a/Mathlib/RingTheory/DedekindDomain/Ideal.lean
+++ b/Mathlib/RingTheory/DedekindDomain/Ideal.lean
@@ -108,7 +108,7 @@ theorem right_inverse_eq (I J : FractionalIdeal R₁⁰ K) (h : I * J = 1) : J =
   apply (le_div_iff_of_nonzero hI).mpr _
   intro y hy x hx
   rw [mul_comm]
-  exact mul_mem_mul hx hy
+  exact mul_mem_mul hy hx
 
 theorem mul_inv_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * I⁻¹ = 1 ↔ ∃ J, I * J = 1 :=
   ⟨fun h => ⟨I⁻¹, h⟩, fun ⟨J, hJ⟩ => by rwa [← right_inverse_eq K I J hJ]⟩
@@ -609,7 +609,7 @@ theorem Ideal.dvdNotUnit_iff_lt {I J : Ideal A} : DvdNotUnit I J ↔ J < I :=
       (mt Ideal.dvd_iff_le.mp (not_le_of_lt h))⟩
 
 instance : WfDvdMonoid (Ideal A) where
-  wellFounded_dvdNotUnit := by
+  wf := by
     have : WellFoundedGT (Ideal A) := inferInstance
     convert this.wf
     ext
@@ -630,7 +630,7 @@ instance Ideal.uniqueFactorizationMonoid : UniqueFactorizationMonoid (Ideal A) :
         rintro ⟨⟨x, x_mem, x_not_mem⟩, ⟨y, y_mem, y_not_mem⟩⟩
         exact
           ⟨x * y, Ideal.mul_mem_mul x_mem y_mem,
-            mt this.isPrime.mem_or_mem (not_or_of_not x_not_mem y_not_mem)⟩⟩, Prime.irreducible⟩ }
+            mt this.isPrime.mem_or_mem (not_or_intro x_not_mem y_not_mem)⟩⟩, Prime.irreducible⟩ }
 
 instance Ideal.normalizationMonoid : NormalizationMonoid (Ideal A) :=
   normalizationMonoidOfUniqueUnits
@@ -720,7 +720,7 @@ theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime
 theorem Ideal.pow_succ_lt_pow {P : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥) (i : ℕ) :
     P ^ (i + 1) < P ^ i :=
   lt_of_le_of_ne (Ideal.pow_le_pow_right (Nat.le_succ _))
-    (mt (pow_eq_pow_iff hP (mt Ideal.isUnit_iff.mp P_prime.ne_top)).mp i.succ_ne_self)
+    (mt (pow_inj_of_not_isUnit (mt Ideal.isUnit_iff.mp P_prime.ne_top) hP).mp i.succ_ne_self)
 
 theorem Associates.le_singleton_iff (x : A) (n : ℕ) (I : Ideal A) :
     Associates.mk I ^ n ≤ Associates.mk (Ideal.span {x}) ↔ x ∈ I ^ n := by
@@ -892,7 +892,7 @@ theorem sup_eq_prod_inf_factors [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : J
     · exact ne_bot_of_le_ne_bot hI le_sup_left
     · exact this
 
-theorem irreducible_pow_sup [DecidableEq (Ideal T)](hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ) :
+theorem irreducible_pow_sup [DecidableEq (Ideal T)] (hI : I ≠ ⊥) (hJ : Irreducible J) (n : ℕ) :
     J ^ n ⊔ I = J ^ min ((normalizedFactors I).count J) n := by
   rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
     normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
@@ -916,6 +916,17 @@ theorem irreducible_pow_sup_of_ge [DecidableRel fun (x : Ideal T) x_1 ↦ x ∣
       multiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J]
   · rwa [multiplicity_eq_count_normalizedFactors hJ hI, PartENat.coe_le_coe, normalize_eq J] at hn
 
+theorem Ideal.eq_prime_pow_mul_coprime [DecidableEq (Ideal T)] {I : Ideal T} (hI : I ≠ ⊥)
+    (P : Ideal T) [hpm : P.IsMaximal] :
+    ∃ Q : Ideal T, P ⊔ Q = ⊤ ∧ I = P ^ (Multiset.count P (normalizedFactors I)) * Q := by
+  use (filter (¬ P = ·) (normalizedFactors I)).prod
+  constructor
+  · refine P.sup_multiset_prod_eq_top (fun p hpi ↦ ?_)
+    have hp : Prime p := prime_of_normalized_factor p (filter_subset _ (normalizedFactors I) hpi)
+    exact hpm.coprime_of_ne ((isPrime_of_prime hp).isMaximal hp.ne_zero) (of_mem_filter hpi)
+  · nth_rw 1 [← prod_normalizedFactors_eq_self hI, ← filter_add_not (P = ·) (normalizedFactors I)]
+    rw [prod_add, pow_count]
+
 end IsDedekindDomain
 
 /-!
@@ -1180,26 +1191,6 @@ theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
     (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)
   exact fun hPJ hPK => mt Ideal.isPrime_of_prime (coprime _ hPJ hPK)
 
-theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
-    I ≤ P := by
-  by_cases hP0 : P = ⊥
-  · simp only [hP0, le_bot_iff] at h ⊢
-    exact pow_eq_zero h
-  rw [← Ideal.dvd_iff_le] at h ⊢
-  exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h
-
-theorem Ideal.pow_le_prime_iff {I P : Ideal R} [_hP : P.IsPrime] {n : ℕ} (hn : n ≠ 0) :
-    I ^ n ≤ P ↔ I ≤ P :=
-  ⟨Ideal.le_of_pow_le_prime, fun h => _root_.trans (Ideal.pow_le_self hn) h⟩
-
-theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
-    [hP : P.IsPrime] : ∏ i ∈ s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
-  by_cases hP0 : P = ⊥
-  · simp only [hP0, le_bot_iff]
-    rw [← Ideal.zero_eq_bot, Finset.prod_eq_zero_iff]
-  simp only [← Ideal.dvd_iff_le]
-  exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_finset_prod_iff _
-
 /-- The intersection of distinct prime powers in a Dedekind domain is the product of these
 prime powers. -/
 theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
@@ -1217,15 +1208,13 @@ theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f :
   rw [Finset.inf_insert, Finset.prod_insert ha, ih]
   refine le_antisymm (Ideal.le_mul_of_no_prime_factors ?_ inf_le_left inf_le_right) Ideal.mul_le_inf
   intro P hPa hPs hPp
-  obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
+  obtain ⟨b, hb, hPb⟩ := hPp.prod_le.mp hPs
   haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
   haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
   refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
   · exact (ne_of_mem_of_not_mem hb ha).symm
-  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
-      (Ideal.le_of_pow_le_prime hPa)).trans
-      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
-      (Ideal.le_of_pow_le_prime hPb)).symm
+  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp (hPp.le_of_pow_le hPa)).trans
+      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp (hPp.le_of_pow_le hPb)).symm
     · exact (prime a (Finset.mem_insert_self a s)).ne_zero
     · exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
 
@@ -1240,17 +1229,13 @@ noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype
       simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
         ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i)
         (coprime.set_pairwise _)])).trans <|
-    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
+    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge <| by
       intro P hPi hPj hPp
       haveI := Ideal.isPrime_of_prime (prime i)
       haveI := Ideal.isPrime_of_prime (prime j)
-      refine coprime hij ?_
-      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
-        (Ideal.le_of_pow_le_prime hPi)).trans
-        ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
-          (Ideal.le_of_pow_le_prime hPj)).symm
-      · exact (prime i).ne_zero
-      · exact (prime j).ne_zero)
+      exact coprime hij <| ((Ring.DimensionLeOne.prime_le_prime_iff_eq (prime i).ne_zero).mp
+        (hPp.le_of_pow_le hPi)).trans <| Eq.symm <|
+          (Ring.DimensionLeOne.prime_le_prime_iff_eq (prime j).ne_zero).mp (hPp.le_of_pow_le hPj)
 
 open scoped Classical
 
@@ -1267,7 +1252,7 @@ noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I
         (factors I).toFinset.prod_coe_sort fun P => P ^ (factors I).count P
       _ = ((factors I).map fun P => P).prod := (Finset.prod_multiset_map_count (factors I) id).symm
       _ = (factors I).prod := by rw [Multiset.map_id']
-      _ = I := (@associated_iff_eq (Ideal R) _ Ideal.uniqueUnits _ _).mp (factors_prod hI)
+      _ = I := associated_iff_eq.mp (factors_prod hI)
       )
 
 @[simp]
diff --git a/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean b/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
index 8f37a34bfe33b..86796148fbcf3 100644
--- a/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
+++ b/Mathlib/RingTheory/DedekindDomain/IntegralClosure.lean
@@ -156,7 +156,7 @@ theorem FiniteDimensional.exists_is_basis_integral :
   · intro x; simp only [mul_inv_cancel_left₀ hy']
   · rintro ⟨x', hx'⟩
     simp only [Algebra.smul_def, Finset.mem_image, exists_prop, Finset.mem_univ,
-      true_and_iff] at his'
+      true_and] at his'
     simp only [Basis.map_apply, LinearEquiv.coe_mk]
     exact his' _ ⟨_, rfl⟩
 
@@ -198,13 +198,13 @@ theorem IsIntegralClosure.module_free [NoZeroSMulDivisors A L] [IsPrincipalIdeal
 and `L` has no zero smul divisors by `A`, the `A`-rank of the integral closure `C` of `A` in `L`
 is equal to the `K`-rank of `L`. -/
 theorem IsIntegralClosure.rank [IsPrincipalIdealRing A] [NoZeroSMulDivisors A L] :
-    FiniteDimensional.finrank A C = FiniteDimensional.finrank K L := by
+    Module.finrank A C = Module.finrank K L := by
   haveI : Module.Free A C := IsIntegralClosure.module_free A K L C
   haveI : IsNoetherian A C := IsIntegralClosure.isNoetherian A K L C
   haveI : IsLocalization (Algebra.algebraMapSubmonoid C A⁰) L :=
     IsIntegralClosure.isLocalization A K L C
   let b := Basis.localizationLocalization K A⁰ L (Module.Free.chooseBasis A C)
-  rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex, FiniteDimensional.finrank_eq_card_basis b]
+  rw [Module.finrank_eq_card_chooseBasisIndex, Module.finrank_eq_card_basis b]
 
 variable {A K}
 
diff --git a/Mathlib/RingTheory/DedekindDomain/PID.lean b/Mathlib/RingTheory/DedekindDomain/PID.lean
index a476970611575..258435cca7f9a 100644
--- a/Mathlib/RingTheory/DedekindDomain/PID.lean
+++ b/Mathlib/RingTheory/DedekindDomain/PID.lean
@@ -96,7 +96,7 @@ theorem FractionalIdeal.isPrincipal_of_unit_of_comap_mul_span_singleton_eq_top {
   · conv_rhs => rw [← hinv, mul_comm]
     apply FractionalIdeal.mul_le_mul_left (FractionalIdeal.spanSingleton_le_iff_mem.mpr hw)
   · rw [FractionalIdeal.one_le, ← hvw, mul_comm]
-    exact FractionalIdeal.mul_mem_mul hv (FractionalIdeal.mem_spanSingleton_self _ _)
+    exact FractionalIdeal.mul_mem_mul (FractionalIdeal.mem_spanSingleton_self _ _) hv
 
 /--
 An invertible fractional ideal of a commutative ring with finitely many maximal ideals is principal.
diff --git a/Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean b/Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean
index 6ea9dbd5a87bc..3ac49f7143a12 100644
--- a/Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean
+++ b/Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean
@@ -67,7 +67,7 @@ namespace IsDedekindDomain
 
 noncomputable section
 
-open scoped DiscreteValuation nonZeroDivisors
+open scoped Multiplicative nonZeroDivisors
 
 universe u v
 
@@ -173,7 +173,7 @@ theorem monotone (hS : S ≤ S') : K⟮S,n⟯ ≤ K⟮S',n⟯ := fun _ hx v => h
 def valuation : K⟮S,n⟯ →* S → Multiplicative (ZMod n) where
   toFun x v := (v : HeightOneSpectrum R).valuationOfNeZeroMod n (x : K/n)
   map_one' := funext fun v => map_one _
-  map_mul' x y := by simp only [Submonoid.coe_mul, Subgroup.coe_toSubmonoid, map_mul]; rfl
+  map_mul' x y := by simp only [Subgroup.coe_mul, map_mul]; rfl
 
 theorem valuation_ker_eq :
     valuation.ker = K⟮(∅ : Set <| HeightOneSpectrum R),n⟯.subgroupOf (K⟮S,n⟯) := by
@@ -191,8 +191,8 @@ def fromUnit {n : ℕ} : Rˣ →* K⟮(∅ : Set <| HeightOneSpectrum R),n⟯ wh
     ⟨QuotientGroup.mk <| Units.map (algebraMap R K).toMonoidHom x, fun v _ =>
       v.valuation_of_unit_mod_eq n x⟩
   map_one' := by simp only [map_one, QuotientGroup.mk_one, Subgroup.mk_eq_one]
-  map_mul' _ _ := by simp only [RingHom.toMonoidHom_eq_coe, map_mul, MonoidHom.mem_range,
-    powMonoidHom_apply, QuotientGroup.mk_mul, Submonoid.mk_mul_mk]
+  map_mul' _ _ := by simp only [RingHom.toMonoidHom_eq_coe, map_mul, QuotientGroup.mk_mul,
+    MulMemClass.mk_mul_mk]
 
 theorem fromUnit_ker [hn : Fact <| 0 < n] :
     (@fromUnit R _ _ K _ _ _ n).ker = (powMonoidHom n : Rˣ →* Rˣ).range := by
diff --git a/Mathlib/RingTheory/Derivation/MapCoeffs.lean b/Mathlib/RingTheory/Derivation/MapCoeffs.lean
new file mode 100644
index 0000000000000..01ab5a9134a4a
--- /dev/null
+++ b/Mathlib/RingTheory/Derivation/MapCoeffs.lean
@@ -0,0 +1,101 @@
+/-
+Copyright (c) 2024 Daniel Weber. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Daniel Weber
+-/
+import Mathlib.RingTheory.Derivation.Basic
+import Mathlib.Algebra.Polynomial.Module.Basic
+
+/-!
+# Coefficient-wise derivation on polynomials
+
+In this file we define applying a derivation on the coefficients of a polynomial,
+show this forms a derivation, and prove `apply_eval_eq`, which shows that for a derivation `D`,
+`D(p(x)) = (D.mapCoeffs p)(x) + D(x) * p'(x)`. `apply_aeval_eq` and `apply_aeval_eq'`
+are generalizations of that for algebras.
+-/
+
+noncomputable section
+
+open Polynomial Module
+
+namespace Derivation
+
+variable {R A M : Type*} [CommRing R] [CommRing A] [Algebra R A] [AddCommGroup M]
+  [Module A M] [Module R M] (d : Derivation R A M) (a : A)
+
+/--
+The `R`-derivation from `A[X]` to `M[X]` which applies the derivative to each
+of the coefficients.
+-/
+def mapCoeffs : Derivation R A[X] (PolynomialModule A M) where
+  __ := (PolynomialModule.map A d.toLinearMap).comp
+    PolynomialModule.equivPolynomial.symm.toLinearMap
+  map_one_eq_zero' := show (Finsupp.single 0 1).mapRange (d : A → M) d.map_zero = 0 by simp
+  leibniz' p q := by
+    dsimp
+    induction p using Polynomial.induction_on' with
+    | h_add => simp only [add_mul, map_add, add_smul, smul_add, add_add_add_comm, *]
+    | h_monomial n a =>
+      induction q using Polynomial.induction_on' with
+      | h_add => simp only [mul_add, map_add, add_smul, smul_add, add_add_add_comm, *]
+      | h_monomial m b =>
+        refine Finsupp.ext fun i ↦ ?_
+        dsimp [PolynomialModule.equivPolynomial, PolynomialModule.map]
+        simp only [toFinsupp_mul, toFinsupp_monomial, AddMonoidAlgebra.single_mul_single]
+        show d _ = _ + _
+        erw [Finsupp.mapRange.linearMap_apply, Finsupp.mapRange.linearMap_apply]
+        rw [Finsupp.mapRange_single, Finsupp.mapRange_single]
+        erw [PolynomialModule.monomial_smul_single, PolynomialModule.monomial_smul_single]
+        simp only [AddMonoidAlgebra.single_apply, apply_ite d, leibniz, map_zero, coeFn_coe,
+          PolynomialModule.single_apply, ite_add_zero, add_comm m n]
+
+@[simp]
+lemma mapCoeffs_apply (p : A[X]) (i) :
+    d.mapCoeffs p i = d (coeff p i) := rfl
+
+@[simp]
+lemma mapCoeffs_monomial (n : ℕ) (x : A) :
+    d.mapCoeffs (monomial n x) = .single A n (d x) := Finsupp.ext fun _ ↦ by
+  simp [coeff_monomial, apply_ite d, PolynomialModule.single_apply]
+
+@[simp]
+lemma mapCoeffs_X :
+    d.mapCoeffs (X : A[X]) = 0 := by simp [← monomial_one_one_eq_X]
+
+@[simp]
+lemma mapCoeffs_C (x : A) :
+    d.mapCoeffs (C x) = .single A 0 (d x) := by simp [← monomial_zero_left]
+
+variable {B M' : Type*} [CommRing B] [Algebra R B] [Algebra A B]
+    [AddCommGroup M'] [Module B M'] [Module R M'] [Module A M']
+
+theorem apply_aeval_eq' (d' : Derivation R B M') (f : M →ₗ[A] M')
+    (h : ∀ a, f (d a) = d' (algebraMap A B a)) (x : B) (p : A[X]) :
+    d' (aeval x p) = PolynomialModule.eval x (PolynomialModule.map B f (d.mapCoeffs p)) +
+      aeval x (derivative p) • d' x := by
+  induction p using Polynomial.induction_on' with
+  | h_add => simp_all only [eval_add, map_add, add_smul]; abel
+  | h_monomial =>
+    simp only [aeval_monomial, leibniz, leibniz_pow, mapCoeffs_monomial,
+      PolynomialModule.map_single, PolynomialModule.eval_single, derivative_monomial, map_mul,
+      _root_.map_natCast, h]
+    rw [add_comm, ← smul_smul, ← smul_smul, Nat.cast_smul_eq_nsmul]
+
+
+theorem apply_aeval_eq [IsScalarTower R A B] [IsScalarTower A B M'] (d : Derivation R B M')
+    (x : B) (p : A[X]) :
+    d (aeval x p) = PolynomialModule.eval x ((d.compAlgebraMap A).mapCoeffs p) +
+      aeval x (derivative p) • d x := by
+  convert apply_aeval_eq' (d.compAlgebraMap A) d LinearMap.id _ x p
+  · apply Finsupp.ext
+    intro x
+    rfl
+  · intro a
+    rfl
+
+theorem apply_eval_eq (x : A) (p : A[X]) :
+    d (eval x p) = PolynomialModule.eval x (d.mapCoeffs p) + eval x (derivative p) • d x :=
+  apply_aeval_eq d x p
+
+end Derivation
diff --git a/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean b/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
index 518c5943e0624..214213dcd11a6 100644
--- a/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
+++ b/Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
@@ -3,10 +3,10 @@ Copyright (c) 2020 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kevin Buzzard
 -/
-import Mathlib.RingTheory.PrincipalIdealDomain
-import Mathlib.RingTheory.Valuation.PrimeMultiplicity
 import Mathlib.RingTheory.AdicCompletion.Basic
 import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
+import Mathlib.RingTheory.Valuation.PrimeMultiplicity
+import Mathlib.RingTheory.Valuation.ValuationRing
 
 /-!
 # Discrete valuation rings
@@ -193,7 +193,7 @@ theorem toUniqueFactorizationMonoid (hR : HasUnitMulPowIrreducibleFactorization
       intro a b h
       by_cases ha : a = 0
       · rw [ha]
-        simp only [true_or_iff, dvd_zero]
+        simp only [true_or, dvd_zero]
       obtain ⟨m, u, rfl⟩ := spec.2 ha
       rw [mul_assoc, mul_left_comm, Units.dvd_mul_left] at h
       rw [Units.dvd_mul_right]
@@ -222,8 +222,7 @@ theorem of_ufd_of_unique_irreducible [UniqueFactorizationMonoid R] (h₁ : ∃ p
   congr 1
   symm
   rw [Multiset.eq_replicate]
-  simp only [true_and_iff, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map,
-    exists_imp]
+  simp only [true_and, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map, exists_imp]
   rintro _ q hq rfl
   rw [Associates.mk_eq_mk_iff_associated]
   apply h₂ (hfx.1 _ hq) hp
@@ -311,8 +310,7 @@ theorem associated_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irred
   rw [← H, ← Associates.prod_mk, Associates.mk_pow, ← Multiset.prod_replicate]
   congr 1
   rw [Multiset.eq_replicate]
-  simp only [true_and_iff, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map,
-    exists_imp]
+  simp only [true_and, and_imp, Multiset.card_map, eq_self_iff_true, Multiset.mem_map, exists_imp]
   rintro _ _ _ rfl
   rw [Associates.mk_eq_mk_iff_associated]
   refine associated_of_irreducible _ ?_ hirr
@@ -452,3 +450,31 @@ instance (R : Type*) [CommRing R] [IsDomain R] [DiscreteValuationRing R] :
     rwa [← addVal_eq_top_iff, PartENat.eq_top_iff_forall_le]
 
 end DiscreteValuationRing
+
+
+section
+
+variable (A : Type u) [CommRing A] [IsDomain A] [DiscreteValuationRing A]
+
+/-- A DVR is a valuation ring. -/
+instance (priority := 100) of_discreteValuationRing : ValuationRing A := by
+  constructor
+  intro a b
+  by_cases ha : a = 0; · use 0; right; simp [ha]
+  by_cases hb : b = 0; · use 0; left; simp [hb]
+  obtain ⟨ϖ, hϖ⟩ := DiscreteValuationRing.exists_irreducible A
+  obtain ⟨m, u, rfl⟩ := DiscreteValuationRing.eq_unit_mul_pow_irreducible ha hϖ
+  obtain ⟨n, v, rfl⟩ := DiscreteValuationRing.eq_unit_mul_pow_irreducible hb hϖ
+  rcases le_total m n with h | h
+  · use (u⁻¹ * v : Aˣ) * ϖ ^ (n - m); left
+    simp_rw [mul_comm (u : A), Units.val_mul, ← mul_assoc, mul_assoc _ (u : A)]
+    simp only [Units.mul_inv, mul_one, mul_comm _ (v : A), mul_assoc, ← pow_add]
+    congr 2
+    exact Nat.add_sub_of_le h
+  · use (v⁻¹ * u : Aˣ) * ϖ ^ (m - n); right
+    simp_rw [mul_comm (v : A), Units.val_mul, ← mul_assoc, mul_assoc _ (v : A)]
+    simp only [Units.mul_inv, mul_one, mul_comm _ (u : A), mul_assoc, ← pow_add]
+    congr 2
+    exact Nat.add_sub_of_le h
+
+end
diff --git a/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean b/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
index 96d349b2cab5b..5957f090f0b89 100644
--- a/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
+++ b/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean
@@ -3,10 +3,9 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathlib.RingTheory.Ideal.Cotangent
 import Mathlib.RingTheory.DedekindDomain.Basic
-import Mathlib.RingTheory.Valuation.ValuationRing
-import Mathlib.RingTheory.Nakayama
+import Mathlib.RingTheory.DiscreteValuationRing.Basic
+import Mathlib.RingTheory.Ideal.Cotangent
 
 /-!
 
@@ -28,9 +27,9 @@ Also see `tfae_of_isNoetherianRing_of_localRing_of_isDomain` for a version witho
 
 variable (R : Type*) [CommRing R] (K : Type*) [Field K] [Algebra R K] [IsFractionRing R K]
 
-open scoped DiscreteValuation
+open scoped Multiplicative
 
-open LocalRing FiniteDimensional
+open LocalRing Module
 
 theorem exists_maximalIdeal_pow_eq_of_principal [IsNoetherianRing R] [LocalRing R] [IsDomain R]
     (h' : (maximalIdeal R).IsPrincipal) (I : Ideal R) (hI : I ≠ ⊥) :
@@ -167,23 +166,17 @@ theorem tfae_of_isNoetherianRing_of_localRing_of_isDomain
         (maximalIdeal R).IsPrincipal,
         finrank (ResidueField R) (CotangentSpace R) ≤ 1,
         ∀ (I) (_ : I ≠ ⊥), ∃ n : ℕ, I = maximalIdeal R ^ n] := by
-  tfae_have 1 → 2
-  · exact fun _ ↦ inferInstance
-  tfae_have 2 → 1
-  · exact fun _ ↦ ((IsBezout.TFAE (R := R)).out 0 1).mp ‹_›
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 2 → 1 := fun _ ↦ ((IsBezout.TFAE (R := R)).out 0 1).mp ‹_›
   tfae_have 1 → 4
-  · intro H
-    exact ⟨inferInstance, fun P hP hP' ↦ eq_maximalIdeal (hP'.isMaximal hP)⟩
-  tfae_have 4 → 3
-  · exact fun ⟨h₁, h₂⟩ ↦ { h₁ with maximalOfPrime := (h₂ _ · · ▸ maximalIdeal.isMaximal R) }
-  tfae_have 3 → 5
-  · exact fun h ↦ maximalIdeal_isPrincipal_of_isDedekindDomain R
-  tfae_have 6 ↔ 5
-  · exact finrank_cotangentSpace_le_one_iff
-  tfae_have 5 → 7
-  · exact exists_maximalIdeal_pow_eq_of_principal R
-  tfae_have 7 → 2
-  · rw [ValuationRing.iff_ideal_total]
+  | H => ⟨inferInstance, fun P hP hP' ↦ eq_maximalIdeal (hP'.isMaximal hP)⟩
+  tfae_have 4 → 3 :=
+    fun ⟨h₁, h₂⟩ ↦ { h₁ with maximalOfPrime := (h₂ _ · · ▸ maximalIdeal.isMaximal R) }
+  tfae_have 3 → 5 := fun h ↦ maximalIdeal_isPrincipal_of_isDedekindDomain R
+  tfae_have 6 ↔ 5 := finrank_cotangentSpace_le_one_iff
+  tfae_have 5 → 7 := exists_maximalIdeal_pow_eq_of_principal R
+  tfae_have 7 → 2 := by
+    rw [ValuationRing.iff_ideal_total]
     intro H
     constructor
     intro I J
diff --git a/Mathlib/RingTheory/Discriminant.lean b/Mathlib/RingTheory/Discriminant.lean
index f6285f4b7aa27..f3faffa03b668 100644
--- a/Mathlib/RingTheory/Discriminant.lean
+++ b/Mathlib/RingTheory/Discriminant.lean
@@ -50,7 +50,7 @@ universe u v w z
 
 open scoped Matrix
 
-open Matrix FiniteDimensional Fintype Polynomial Finset IntermediateField
+open Matrix Module Fintype Polynomial Finset IntermediateField
 
 namespace Algebra
 
@@ -190,7 +190,7 @@ theorem discr_powerBasis_eq_prod'' [Algebra.IsSeparable K L] (e : Fin pb.dim ≃
   have h₂ : 2 ∣ pb.dim * (pb.dim - 1) := pb.dim.even_mul_pred_self.two_dvd
   have hne : ((2 : ℕ) : ℚ) ≠ 0 := by simp
   have hle : 1 ≤ pb.dim := by
-    rw [← hn, Nat.one_le_iff_ne_zero, ← zero_lt_iff, FiniteDimensional.finrank_pos_iff]
+    rw [← hn, Nat.one_le_iff_ne_zero, ← zero_lt_iff, Module.finrank_pos_iff]
     infer_instance
   rw [hn, Nat.cast_div h₂ hne, Nat.cast_mul, Nat.cast_sub hle]
   field_simp
diff --git a/Mathlib/RingTheory/DualNumber.lean b/Mathlib/RingTheory/DualNumber.lean
new file mode 100644
index 0000000000000..cf0ac64836b99
--- /dev/null
+++ b/Mathlib/RingTheory/DualNumber.lean
@@ -0,0 +1,168 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky
+-/
+import Mathlib.Algebra.DualNumber
+import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
+
+/-!
+# Algebraic properties of dual numbers
+
+## Main results
+
+* `DualNumber.instLocalRing`: The dual numbers over a field `K` form a local ring.
+* `DualNumber.instPrincipalIdealRing`: The dual numbers over a field `K` form a principal ideal
+  ring.
+
+-/
+
+namespace TrivSqZeroExt
+
+variable {R M : Type*}
+
+section Semiring
+variable [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
+
+lemma isNilpotent_iff_isNilpotent_fst {x : TrivSqZeroExt R M} :
+    IsNilpotent x ↔ IsNilpotent x.fst := by
+  constructor <;> rintro ⟨n, hn⟩
+  · refine ⟨n, ?_⟩
+    rw [← fst_pow, hn, fst_zero]
+  · refine ⟨n * 2, ?_⟩
+    rw [pow_mul]
+    ext
+    · rw [fst_pow, fst_pow, hn, zero_pow two_ne_zero, fst_zero]
+    · rw [pow_two, snd_mul, fst_pow, hn, MulOpposite.op_zero, zero_smul, zero_smul, zero_add,
+        snd_zero]
+
+@[simp]
+lemma isNilpotent_inl_iff (r : R) : IsNilpotent (.inl r : TrivSqZeroExt R M) ↔ IsNilpotent r := by
+  rw [isNilpotent_iff_isNilpotent_fst, fst_inl]
+
+@[simp]
+lemma isNilpotent_inr (x : M) : IsNilpotent (.inr x : TrivSqZeroExt R M) := by
+  refine ⟨2, by simp [pow_two]⟩
+
+end Semiring
+
+lemma isUnit_or_isNilpotent_of_isMaximal_isNilpotent [CommSemiring R] [AddCommGroup M]
+    [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
+    (h : ∀ I : Ideal R, I.IsMaximal → IsNilpotent I)
+    (a : TrivSqZeroExt R M) :
+    IsUnit a ∨ IsNilpotent a := by
+  rw [isUnit_iff_isUnit_fst, isNilpotent_iff_isNilpotent_fst]
+  refine (em _).imp_right fun ha ↦ ?_
+  obtain ⟨I, hI, haI⟩ := exists_max_ideal_of_mem_nonunits (mem_nonunits_iff.mpr ha)
+  refine (h _ hI).imp fun n hn ↦ ?_
+  exact hn.le (Ideal.pow_mem_pow haI _)
+
+lemma isUnit_or_isNilpotent [DivisionSemiring R] [AddCommGroup M]
+    [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
+    (a : TrivSqZeroExt R M) :
+    IsUnit a ∨ IsNilpotent a := by
+  simp [isUnit_iff_isUnit_fst, isNilpotent_iff_isNilpotent_fst, em']
+
+end TrivSqZeroExt
+
+namespace DualNumber
+variable {R : Type*}
+
+lemma fst_eq_zero_iff_eps_dvd [Semiring R] {x : R[ε]} :
+    x.fst = 0 ↔ ε ∣ x := by
+  simp_rw [dvd_def, TrivSqZeroExt.ext_iff, TrivSqZeroExt.fst_mul, TrivSqZeroExt.snd_mul,
+    fst_eps, snd_eps, zero_mul, zero_smul, zero_add, MulOpposite.smul_eq_mul_unop,
+    MulOpposite.unop_op, one_mul, exists_and_left, iff_self_and]
+  intro
+  exact ⟨.inl x.snd, rfl⟩
+
+lemma isNilpotent_eps [Semiring R] :
+    IsNilpotent (ε : R[ε]) :=
+  TrivSqZeroExt.isNilpotent_inr 1
+
+open TrivSqZeroExt
+
+lemma isNilpotent_iff_eps_dvd [DivisionSemiring R] {x : R[ε]} :
+    IsNilpotent x ↔ ε ∣ x := by
+  simp only [isNilpotent_iff_isNilpotent_fst, isNilpotent_iff_eq_zero, fst_eq_zero_iff_eps_dvd]
+
+section Field
+
+variable {K : Type*}
+
+instance [DivisionRing K] : LocalRing K[ε] where
+  isUnit_or_isUnit_of_add_one {a b} h := by
+    rw [add_comm, ← eq_sub_iff_add_eq] at h
+    rcases eq_or_ne (fst a) 0 with ha|ha <;>
+    simp [isUnit_iff_isUnit_fst, h, ha]
+
+lemma ideal_trichotomy [DivisionRing K] (I : Ideal K[ε]) :
+    I = ⊥ ∨ I = .span {ε} ∨ I = ⊤ := by
+  refine (eq_or_ne I ⊥).imp_right fun hb ↦ ?_
+  refine (eq_or_ne I ⊤).symm.imp_left fun ht ↦ ?_
+  have hd : ∀ x ∈ I, ε ∣ x := by
+    intro x hxI
+    rcases isUnit_or_isNilpotent x with hx|hx
+    · exact absurd (Ideal.eq_top_of_isUnit_mem _ hxI hx) ht
+    · rwa [← isNilpotent_iff_eps_dvd]
+  have hd' : ∀ x ∈ I, x ≠ 0 → ∃ r, ε = r * x := by
+    intro x hxI hx0
+    obtain ⟨r, rfl⟩ := hd _ hxI
+    have : ε * r = (fst r) • ε := by ext <;> simp
+    rw [this] at hxI hx0 ⊢
+    have hr : fst r ≠ 0 := by
+      contrapose! hx0
+      simp [hx0]
+    refine ⟨r⁻¹, ?_⟩
+    simp [TrivSqZeroExt.ext_iff, inv_mul_cancel₀ hr]
+  refine le_antisymm ?_ ?_ <;> intro x <;>
+    simp_rw [Ideal.mem_span_singleton', (commute_eps_right _).eq, eq_comm, ← dvd_def]
+  · intro hx
+    simp_rw [hd _ hx]
+  · intro hx
+    obtain ⟨p, rfl⟩ := hx
+    obtain ⟨y, hyI, hy0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hb
+    obtain ⟨r, hr⟩ := hd' _ hyI hy0
+    rw [(commute_eps_left _).eq, hr, ← mul_assoc]
+    exact Ideal.mul_mem_left _ _ hyI
+
+lemma isMaximal_span_singleton_eps [DivisionRing K] :
+    (Ideal.span {ε} : Ideal K[ε]).IsMaximal := by
+  refine ⟨?_, fun I hI ↦ ?_⟩
+  · simp [ne_eq, Ideal.eq_top_iff_one, Ideal.mem_span_singleton', TrivSqZeroExt.ext_iff]
+  · rcases ideal_trichotomy I with rfl|rfl|rfl <;>
+    first | simp at hI | simp
+
+lemma maximalIdeal_eq_span_singleton_eps [Field K] :
+    LocalRing.maximalIdeal K[ε] = Ideal.span {ε} :=
+  (LocalRing.eq_maximalIdeal isMaximal_span_singleton_eps).symm
+
+instance [DivisionRing K] : IsPrincipalIdealRing K[ε] where
+  principal I := by
+    rcases ideal_trichotomy I with rfl|rfl|rfl
+    · exact bot_isPrincipal
+    · exact ⟨_, rfl⟩
+    · exact top_isPrincipal
+
+lemma exists_mul_left_or_mul_right [DivisionRing K] (a b : K[ε]) :
+    ∃ c, a * c = b ∨ b * c = a := by
+  rcases isUnit_or_isNilpotent a with ha|ha
+  · lift a to K[ε]ˣ using ha
+    exact ⟨a⁻¹ * b, by simp⟩
+  rcases isUnit_or_isNilpotent b with hb|hb
+  · lift b to K[ε]ˣ using hb
+    exact ⟨b⁻¹ * a, by simp⟩
+  rw [isNilpotent_iff_eps_dvd] at ha hb
+  obtain ⟨x, rfl⟩ := ha
+  obtain ⟨y, rfl⟩ := hb
+  suffices ∃ c, fst x * fst c = fst y ∨ fst y * fst c = fst x by
+    simpa [TrivSqZeroExt.ext_iff] using this
+  rcases eq_or_ne (fst x) 0 with hx|hx
+  · refine ⟨ε, Or.inr ?_⟩
+    simp [hx]
+  refine ⟨inl ((fst x)⁻¹ * fst y), ?_⟩
+  simp [← inl_mul, ← mul_assoc, mul_inv_cancel₀ hx]
+
+end Field
+
+end DualNumber
diff --git a/Mathlib/RingTheory/EssentialFiniteness.lean b/Mathlib/RingTheory/EssentialFiniteness.lean
index 65b40d237017c..c23b4328ea0fb 100644
--- a/Mathlib/RingTheory/EssentialFiniteness.lean
+++ b/Mathlib/RingTheory/EssentialFiniteness.lean
@@ -34,7 +34,7 @@ class EssFiniteType : Prop where
     IsLocalization ((IsUnit.submonoid S).comap (algebraMap (adjoin R (s : Set S)) S)) S
 
 /-- Let `S` be an `R`-algebra essentially of finite type, this is a choice of a finset `s ⊆ S`
-such that `S` is the localization of `R[s]`.  -/
+such that `S` is the localization of `R[s]`. -/
 noncomputable
 def EssFiniteType.finset [h : EssFiniteType R S] : Finset S := h.cond.choose
 
diff --git a/Mathlib/RingTheory/Etale/Basic.lean b/Mathlib/RingTheory/Etale/Basic.lean
index 0a5d43bc3a05a..171f57f41c31c 100644
--- a/Mathlib/RingTheory/Etale/Basic.lean
+++ b/Mathlib/RingTheory/Etale/Basic.lean
@@ -11,7 +11,7 @@ import Mathlib.RingTheory.Unramified.Basic
 
 # Etale morphisms
 
-An `R`-algebra `A` is formally étale if for every `R`-algebra,
+An `R`-algebra `A` is formally étale if for every `R`-algebra `B`,
 every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
 exactly one lift `A →ₐ[R] B`.
 It is étale if it is formally étale and of finite presentation.
diff --git a/Mathlib/RingTheory/Filtration.lean b/Mathlib/RingTheory/Filtration.lean
index a73354fbc33c0..8ce8b760670a3 100644
--- a/Mathlib/RingTheory/Filtration.lean
+++ b/Mathlib/RingTheory/Filtration.lean
@@ -294,11 +294,11 @@ theorem submodule_eq_span_le_iff_stable_ge (n₀ : ℕ) :
   · intro H n hn
     refine (F.smul_le n).antisymm ?_
     intro x hx
-    obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total _ _ _).mp (H _ ⟨x, hx, rfl⟩)
+    obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination _ _ _).mp (H _ ⟨x, hx, rfl⟩)
     replace hl := congr_arg (fun f : ℕ →₀ M => f (n + 1)) hl
     dsimp only at hl
     erw [Finsupp.single_eq_same] at hl
-    rw [← hl, Finsupp.total_apply, Finsupp.sum_apply]
+    rw [← hl, Finsupp.linearCombination_apply, Finsupp.sum_apply]
     apply Submodule.sum_mem _ _
     rintro ⟨_, _, ⟨n', rfl⟩, _, ⟨hn', rfl⟩, m, hm, rfl⟩ -
     dsimp only [Subtype.coe_mk]
diff --git a/Mathlib/RingTheory/FiniteLength.lean b/Mathlib/RingTheory/FiniteLength.lean
new file mode 100644
index 0000000000000..d0305806bd193
--- /dev/null
+++ b/Mathlib/RingTheory/FiniteLength.lean
@@ -0,0 +1,78 @@
+/-
+Copyright (c) 2024 Junyan Xu. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Junyan Xu
+-/
+import Mathlib.Order.Atoms.Finite
+import Mathlib.RingTheory.Artinian
+
+/-!
+# Modules of finite length
+
+We define modules of finite length (`IsFiniteLength`) to be finite iterated extensions of
+simple modules, and show that a module is of finite length iff it is both Noetherian and Artinian,
+iff it admits a composition series.
+We do not make `IsFiniteLength` a class, instead we use `[IsNoetherian R M] [IsArtinian R M]`.
+
+## Tag
+
+Finite length, Composition series
+-/
+
+universe u
+
+variable (R : Type*) [Ring R]
+
+/-- A module of finite length is either trivial or a simple extension of a module known
+to be of finite length. -/
+inductive IsFiniteLength : ∀ (M : Type u) [AddCommGroup M] [Module R M], Prop
+  | of_subsingleton {M} [AddCommGroup M] [Module R M] [Subsingleton M] : IsFiniteLength M
+  | of_simple_quotient {M} [AddCommGroup M] [Module R M] {N : Submodule R M}
+      [IsSimpleModule R (M ⧸ N)] : IsFiniteLength N → IsFiniteLength M
+
+variable {R} {M N : Type*} [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
+
+theorem LinearEquiv.isFiniteLength (e : M ≃ₗ[R] N)
+    (h : IsFiniteLength R M) : IsFiniteLength R N := by
+  induction' h with M _ _ _ M _ _ S _ _ ih generalizing N
+  · have := e.symm.toEquiv.subsingleton; exact .of_subsingleton
+  · have : IsSimpleModule R (N ⧸ Submodule.map (e : M →ₗ[R] N) S) :=
+      IsSimpleModule.congr (Submodule.Quotient.equiv S _ e rfl).symm
+    exact .of_simple_quotient (ih <| e.submoduleMap S)
+
+variable (R M) in
+theorem exists_compositionSeries_of_isNoetherian_isArtinian [IsNoetherian R M] [IsArtinian R M] :
+    ∃ s : CompositionSeries (Submodule R M), s.head = ⊥ ∧ s.last = ⊤ := by
+  obtain ⟨f, f0, n, hn⟩ := exists_covBy_seq_of_wellFoundedLT_wellFoundedGT (Submodule R M)
+  exact ⟨⟨n, fun i ↦ f i, fun i ↦ hn.2 i i.2⟩, f0.eq_bot, hn.1.eq_top⟩
+
+theorem isFiniteLength_of_exists_compositionSeries
+    (h : ∃ s : CompositionSeries (Submodule R M), s.head = ⊥ ∧ s.last = ⊤) :
+    IsFiniteLength R M :=
+  Submodule.topEquiv.isFiniteLength <| by
+    obtain ⟨s, s_head, s_last⟩ := h
+    rw [← s_last]
+    suffices ∀ i, IsFiniteLength R (s i) from this (Fin.last _)
+    intro i
+    induction' i using Fin.induction with i ih
+    · change IsFiniteLength R s.head; rw [s_head]; exact .of_subsingleton
+    let cov := s.step i
+    have := (covBy_iff_quot_is_simple cov.le).mp cov
+    have := ((s i.castSucc).comap (s i.succ).subtype).equivMapOfInjective
+      _ (Submodule.injective_subtype _)
+    rw [Submodule.map_comap_subtype, inf_of_le_right cov.le] at this
+    exact .of_simple_quotient (this.symm.isFiniteLength ih)
+
+theorem isFiniteLength_iff_isNoetherian_isArtinian :
+    IsFiniteLength R M ↔ IsNoetherian R M ∧ IsArtinian R M :=
+  ⟨fun h ↦ h.rec (fun {M} _ _ _ ↦ ⟨inferInstance, inferInstance⟩) fun M _ _ {N} _ _ ⟨_, _⟩ ↦
+    ⟨(isNoetherian_iff_submodule_quotient N).mpr ⟨‹_›, isNoetherian_iff'.mpr inferInstance⟩,
+      (isArtinian_iff_submodule_quotient N).mpr ⟨‹_›, inferInstance⟩⟩,
+    fun ⟨_, _⟩ ↦ isFiniteLength_of_exists_compositionSeries
+      (exists_compositionSeries_of_isNoetherian_isArtinian R M)⟩
+
+theorem isFiniteLength_iff_exists_compositionSeries :
+    IsFiniteLength R M ↔ ∃ s : CompositionSeries (Submodule R M), s.head = ⊥ ∧ s.last = ⊤ :=
+  ⟨fun h ↦ have ⟨_, _⟩ := isFiniteLength_iff_isNoetherian_isArtinian.mp h
+    exists_compositionSeries_of_isNoetherian_isArtinian R M,
+    isFiniteLength_of_exists_compositionSeries⟩
diff --git a/Mathlib/RingTheory/FinitePresentation.lean b/Mathlib/RingTheory/FinitePresentation.lean
index a2955e2776f05..b5f8a9e4728c8 100644
--- a/Mathlib/RingTheory/FinitePresentation.lean
+++ b/Mathlib/RingTheory/FinitePresentation.lean
@@ -69,8 +69,7 @@ theorem of_finiteType [IsNoetherianRing R] : FiniteType R A ↔ FinitePresentati
   -- Porting note: rewrote code to help typeclass inference
   rw [isNoetherianRing_iff] at hnoet
   letI : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule
-  have := hnoet.noetherian (RingHom.ker f.toRingHom)
-  convert this
+  convert hnoet.noetherian (RingHom.ker f.toRingHom)
 
 /-- If `e : A ≃ₐ[R] B` and `A` is finitely presented, then so is `B`. -/
 theorem equiv [FinitePresentation R A] (e : A ≃ₐ[R] B) : FinitePresentation R B := by
@@ -395,6 +394,7 @@ variable {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
 
 /-- A ring morphism `A →+* B` is of `RingHom.FinitePresentation` if `B` is finitely presented as
 `A`-algebra. -/
+@[algebraize]
 def FinitePresentation (f : A →+* B) : Prop :=
   @Algebra.FinitePresentation A B _ _ f.toAlgebra
 
@@ -415,11 +415,9 @@ theorem id : FinitePresentation (RingHom.id A) :=
 variable {A}
 
 theorem comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.FinitePresentation) (hg : Surjective g)
-    (hker : g.ker.FG) : (g.comp f).FinitePresentation :=
-  letI := f.toAlgebra
-  letI := (g.comp f).toAlgebra
-  letI : Algebra.FinitePresentation A B := hf
-  Algebra.FinitePresentation.of_surjective
+    (hker : g.ker.FG) : (g.comp f).FinitePresentation := by
+  algebraize [f, g.comp f]
+  exact Algebra.FinitePresentation.of_surjective
     (f :=
       { g with
         toFun := g
@@ -435,28 +433,16 @@ theorem of_finiteType [IsNoetherianRing A] {f : A →+* B} : f.FiniteType ↔ f.
   @Algebra.FinitePresentation.of_finiteType A B _ _ f.toAlgebra _
 
 theorem comp {g : B →+* C} {f : A →+* B} (hg : g.FinitePresentation) (hf : f.FinitePresentation) :
-    (g.comp f).FinitePresentation :=
+    (g.comp f).FinitePresentation := by
   -- Porting note: specify `Algebra` instances to get `SMul`
-  letI ins1 := RingHom.toAlgebra f
-  letI ins2 := RingHom.toAlgebra g
-  letI ins3 := RingHom.toAlgebra (g.comp f)
-  letI ins4 : IsScalarTower A B C :=
-    { smul_assoc := fun a b c => by simp [Algebra.smul_def, mul_assoc]; rfl }
-  letI : Algebra.FinitePresentation A B := hf
-  letI : Algebra.FinitePresentation B C := hg
-  Algebra.FinitePresentation.trans A B C
+  algebraize [f, g, g.comp f]
+  exact Algebra.FinitePresentation.trans A B C
 
 theorem of_comp_finiteType (f : A →+* B) {g : B →+* C} (hg : (g.comp f).FinitePresentation)
-    (hf : f.FiniteType) : g.FinitePresentation :=
+    (hf : f.FiniteType) : g.FinitePresentation := by
   -- Porting note: need to specify some instances
-  letI ins1 := RingHom.toAlgebra f
-  letI ins2 := RingHom.toAlgebra g
-  letI ins3 := RingHom.toAlgebra (g.comp f)
-  letI ins4 : IsScalarTower A B C :=
-    { smul_assoc := fun a b c => by simp [Algebra.smul_def, mul_assoc]; rfl }
-  letI : Algebra.FinitePresentation A C := hg
-  letI : Algebra.FiniteType A B := hf
-  Algebra.FinitePresentation.of_restrict_scalars_finitePresentation A B C
+  algebraize [f, g, g.comp f]
+  exact Algebra.FinitePresentation.of_restrict_scalars_finitePresentation A B C
 
 end FinitePresentation
 
diff --git a/Mathlib/RingTheory/FiniteType.lean b/Mathlib/RingTheory/FiniteType.lean
index 6272fa5f1de4d..ed78e1a416de7 100644
--- a/Mathlib/RingTheory/FiniteType.lean
+++ b/Mathlib/RingTheory/FiniteType.lean
@@ -45,7 +45,7 @@ section Algebra
 
 -- see Note [lower instance priority]
 instance (priority := 100) finiteType {R : Type*} (A : Type*) [CommSemiring R] [Semiring A]
-    [Algebra R A] [hRA : Finite R A] : Algebra.FiniteType R A :=
+    [Algebra R A] [hRA : Module.Finite R A] : Algebra.FiniteType R A :=
   ⟨Subalgebra.fg_of_submodule_fg hRA.1⟩
 
 end Algebra
@@ -122,7 +122,8 @@ theorem iff_quotient_freeAlgebra :
     intro x
     have hrw : (↑s : Set A) = fun x : A => x ∈ s.val := rfl
     rw [← Set.mem_range, ← AlgHom.coe_range]
-    erw [← adjoin_eq_range_freeAlgebra_lift, ← hrw, hs]
+    erw [← adjoin_eq_range_freeAlgebra_lift]
+    simp_rw [← hrw, hs]
     exact Set.mem_univ x
   · rintro ⟨s, ⟨f, hsur⟩⟩
     exact FiniteType.of_surjective (FiniteType.freeAlgebra R s) f hsur
@@ -137,7 +138,8 @@ theorem iff_quotient_mvPolynomial :
     use s, MvPolynomial.aeval (↑)
     intro x
     have hrw : (↑s : Set S) = fun x : S => x ∈ s.val := rfl
-    rw [← Set.mem_range, ← AlgHom.coe_range, ← adjoin_eq_range, ← hrw, hs]
+    rw [← Set.mem_range, ← AlgHom.coe_range, ← adjoin_eq_range]
+    simp_rw [← hrw, hs]
     exact Set.mem_univ x
   · rintro ⟨s, ⟨f, hsur⟩⟩
     exact FiniteType.of_surjective (FiniteType.mvPolynomial R { x // x ∈ s }) f hsur
@@ -205,6 +207,7 @@ namespace RingHom
 variable {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
 
 /-- A ring morphism `A →+* B` is of `FiniteType` if `B` is finitely generated as `A`-algebra. -/
+@[algebraize]
 def FiniteType (f : A →+* B) : Prop :=
   @Algebra.FiniteType A B _ _ f.toAlgebra
 
@@ -226,8 +229,7 @@ variable {A}
 
 theorem comp_surjective {f : A →+* B} {g : B →+* C} (hf : f.FiniteType) (hg : Surjective g) :
     (g.comp f).FiniteType := by
-  let _ : Algebra A B := f.toAlgebra
-  let _ : Algebra A C := (g.comp f).toAlgebra
+  algebraize_only [f, g.comp f]
   exact Algebra.FiniteType.of_surjective hf
     { g with
       toFun := g
@@ -240,15 +242,8 @@ theorem of_surjective (f : A →+* B) (hf : Surjective f) : f.FiniteType := by
 
 theorem comp {g : B →+* C} {f : A →+* B} (hg : g.FiniteType) (hf : f.FiniteType) :
     (g.comp f).FiniteType := by
-  let _ : Algebra A B := f.toAlgebra
-  let _ : Algebra A C := (g.comp f).toAlgebra
-  let _ : Algebra B C := g.toAlgebra
-  exact @Algebra.FiniteType.trans A B C _ _ _ f.toAlgebra (g.comp f).toAlgebra g.toAlgebra
-    ⟨by
-      intro a b c
-      simp [Algebra.smul_def, RingHom.map_mul, mul_assoc]
-      rfl⟩
-    hf hg
+  algebraize_only [f, g, g.comp f]
+  exact Algebra.FiniteType.trans hf hg
 
 theorem of_finite {f : A →+* B} (hf : f.Finite) : f.FiniteType :=
   @Module.Finite.finiteType _ _ _ _ f.toAlgebra hf
@@ -257,11 +252,7 @@ alias _root_.RingHom.Finite.to_finiteType := of_finite
 
 theorem of_comp_finiteType {f : A →+* B} {g : B →+* C} (h : (g.comp f).FiniteType) :
     g.FiniteType := by
-  let _ := f.toAlgebra
-  let _ := g.toAlgebra
-  let _ := (g.comp f).toAlgebra
-  let _ : IsScalarTower A B C := RestrictScalars.isScalarTower A B C
-  let _ : Algebra.FiniteType A C := h
+  algebraize [f, g, g.comp f]
   exact Algebra.FiniteType.of_restrictScalars_finiteType A B C
 
 end FiniteType
@@ -744,7 +735,7 @@ This is similar to `IsNoetherian.injective_of_surjective_endomorphism` but only
 commutative case, but does not use a Noetherian hypothesis. -/
 @[deprecated OrzechProperty.injective_of_surjective_endomorphism (since := "2024-05-30")]
 theorem Module.Finite.injective_of_surjective_endomorphism {R : Type*} [CommRing R] {M : Type*}
-    [AddCommGroup M] [Module R M] [Finite R M] (f : M →ₗ[R] M)
+    [AddCommGroup M] [Module R M] [Module.Finite R M] (f : M →ₗ[R] M)
     (f_surj : Function.Surjective f) : Function.Injective f :=
   OrzechProperty.injective_of_surjective_endomorphism f f_surj
 
diff --git a/Mathlib/RingTheory/Finiteness.lean b/Mathlib/RingTheory/Finiteness.lean
index 01080ae164e3c..0039e1437e417 100644
--- a/Mathlib/RingTheory/Finiteness.lean
+++ b/Mathlib/RingTheory/Finiteness.lean
@@ -11,6 +11,7 @@ import Mathlib.GroupTheory.Finiteness
 import Mathlib.RingTheory.Ideal.Maps
 import Mathlib.RingTheory.Nilpotent.Defs
 import Mathlib.LinearAlgebra.Basis.Cardinality
+import Mathlib.Tactic.Algebraize
 
 /-!
 # Finiteness conditions in commutative algebra
@@ -35,6 +36,7 @@ In this file we define a notion of finiteness that is common in commutative alge
 
 
 open Function (Surjective)
+open Finsupp
 
 namespace Submodule
 
@@ -274,14 +276,14 @@ theorem fg_of_fg_map_of_fg_inf_ker {R M P : Type*} [Ring R] [AddCommGroup M] [Mo
   have : f x ∈ s.map f := by
     rw [mem_map]
     exact ⟨x, hx, rfl⟩
-  rw [← ht1, ← Set.image_id (t1 : Set P), Finsupp.mem_span_image_iff_total] at this
+  rw [← ht1, ← Set.image_id (t1 : Set P), Finsupp.mem_span_image_iff_linearCombination] at this
   rcases this with ⟨l, hl1, hl2⟩
   refine
     mem_sup.2
-      ⟨(Finsupp.total R id).toFun ((Finsupp.lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
-        x - Finsupp.total R id ((Finsupp.lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
+      ⟨(linearCombination R id).toFun ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
+        x - linearCombination R id ((lmapDomain R R g : (P →₀ R) → M →₀ R) l), ?_,
         add_sub_cancel _ _⟩
-  · rw [← Set.image_id (g '' ↑t1), Finsupp.mem_span_image_iff_total]
+  · rw [← Set.image_id (g '' ↑t1), Finsupp.mem_span_image_iff_linearCombination]
     refine ⟨_, ?_, rfl⟩
     haveI : Inhabited P := ⟨0⟩
     rw [← Finsupp.lmapDomain_supported _ _ g, mem_map]
@@ -290,14 +292,14 @@ theorem fg_of_fg_map_of_fg_inf_ker {R M P : Type*} [Ring R] [AddCommGroup M] [Mo
   rw [ht2, mem_inf]
   constructor
   · apply s.sub_mem hx
-    rw [Finsupp.total_apply, Finsupp.lmapDomain_apply, Finsupp.sum_mapDomain_index]
+    rw [Finsupp.linearCombination_apply, Finsupp.lmapDomain_apply, Finsupp.sum_mapDomain_index]
     · refine s.sum_mem ?_
       intro y hy
       exact s.smul_mem _ (hg y (hl1 hy)).1
     · exact zero_smul _
     · exact fun _ _ _ => add_smul _ _ _
   · rw [LinearMap.mem_ker, f.map_sub, ← hl2]
-    rw [Finsupp.total_apply, Finsupp.total_apply, Finsupp.lmapDomain_apply]
+    rw [Finsupp.linearCombination_apply, Finsupp.linearCombination_apply, Finsupp.lmapDomain_apply]
     rw [Finsupp.sum_mapDomain_index, Finsupp.sum, Finsupp.sum, map_sum]
     · rw [sub_eq_zero]
       refine Finset.sum_congr rfl fun y hy => ?_
@@ -311,11 +313,13 @@ theorem fg_induction (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
     (h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : N.FG) : P N := by
   classical
     obtain ⟨s, rfl⟩ := hN
-    induction s using Finset.induction
-    · rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton]
-      apply h₁
-    · rw [Finset.coe_insert, Submodule.span_insert]
-      apply h₂ <;> apply_assumption
+    induction s using Finset.induction with
+    | empty =>
+      rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton]
+      exact h₁ _
+    | insert _ ih =>
+      rw [Finset.coe_insert, Submodule.span_insert]
+      exact h₂ _ _ (h₁ _) ih
 
 /-- The kernel of the composition of two linear maps is finitely generated if both kernels are and
 the first morphism is surjective. -/
@@ -377,9 +381,7 @@ theorem fg_iff_compact (s : Submodule R M) : s.FG ↔ CompleteLattice.IsCompactE
         suffices u.sup id ≤ s from le_antisymm husup this
         rw [sSup', Finset.sup_id_eq_sSup]
         exact sSup_le_sSup huspan
-      -- Porting note: had to split this out of the `obtain`
-      have := Finset.subset_image_iff.mp huspan
-      obtain ⟨t, ⟨-, rfl⟩⟩ := this
+      obtain ⟨t, -, rfl⟩ := Finset.subset_set_image_iff.mp huspan
       rw [Finset.sup_image, Function.id_comp, Finset.sup_eq_iSup, supr_rw, ←
         span_eq_iSup_of_singleton_spans, eq_comm] at ssup
       exact ⟨t, ssup⟩
@@ -498,7 +500,7 @@ section ModuleAndAlgebra
 variable (R A B M N : Type*)
 
 /-- A module over a semiring is `Finite` if it is finitely generated as a module. -/
-class Module.Finite [Semiring R] [AddCommMonoid M] [Module R M] : Prop where
+protected class Module.Finite [Semiring R] [AddCommMonoid M] [Module R M] : Prop where
   out : (⊤ : Submodule R M).FG
 
 attribute [inherit_doc Module.Finite] Module.Finite.out
@@ -508,7 +510,7 @@ namespace Module
 variable [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N]
 
 theorem finite_def {R M} [Semiring R] [AddCommMonoid M] [Module R M] :
-    Finite R M ↔ (⊤ : Submodule R M).FG :=
+    Module.Finite R M ↔ (⊤ : Submodule R M).FG :=
   ⟨fun h => h.1, fun h => ⟨h⟩⟩
 
 namespace Finite
@@ -526,46 +528,47 @@ theorem iff_addGroup_fg {G : Type*} [AddCommGroup G] : Module.Finite ℤ G ↔ A
 variable {R M N}
 
 /-- See also `Module.Finite.exists_fin'`. -/
-theorem exists_fin [Finite R M] : ∃ (n : ℕ) (s : Fin n → M), Submodule.span R (range s) = ⊤ :=
+lemma exists_fin [Module.Finite R M] : ∃ (n : ℕ) (s : Fin n → M), Submodule.span R (range s) = ⊤ :=
   Submodule.fg_iff_exists_fin_generating_family.mp out
 
 variable (R M) in
-lemma exists_fin' [Finite R M] : ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), Surjective f := by
+lemma exists_fin' [Module.Finite R M] : ∃ (n : ℕ) (f : (Fin n → R) →ₗ[R] M), Surjective f := by
   have ⟨n, s, hs⟩ := exists_fin (R := R) (M := M)
   refine ⟨n, Basis.constr (Pi.basisFun R _) ℕ s, ?_⟩
   rw [← LinearMap.range_eq_top, Basis.constr_range, hs]
 
-theorem of_surjective [hM : Finite R M] (f : M →ₗ[R] N) (hf : Surjective f) : Finite R N :=
+theorem of_surjective [hM : Module.Finite R M] (f : M →ₗ[R] N) (hf : Surjective f) :
+    Module.Finite R N :=
   ⟨by
     rw [← LinearMap.range_eq_top.2 hf, ← Submodule.map_top]
     exact hM.1.map f⟩
 
 instance quotient (R) {A M} [Semiring R] [AddCommGroup M] [Ring A] [Module A M] [Module R M]
-    [SMul R A] [IsScalarTower R A M] [Finite R M]
-    (N : Submodule A M) : Finite R (M ⧸ N) :=
+    [SMul R A] [IsScalarTower R A M] [Module.Finite R M]
+    (N : Submodule A M) : Module.Finite R (M ⧸ N) :=
   Module.Finite.of_surjective (N.mkQ.restrictScalars R) N.mkQ_surjective
 
 /-- The range of a linear map from a finite module is finite. -/
-instance range {F : Type*} [FunLike F M N] [SemilinearMapClass F (RingHom.id R) M N] [Finite R M]
-    (f : F) : Finite R (LinearMap.range f) :=
+instance range {F : Type*} [FunLike F M N] [SemilinearMapClass F (RingHom.id R) M N]
+    [Module.Finite R M] (f : F) : Module.Finite R (LinearMap.range f) :=
   of_surjective (SemilinearMapClass.semilinearMap f).rangeRestrict
     fun ⟨_, y, hy⟩ => ⟨y, Subtype.ext hy⟩
 
 /-- Pushforwards of finite submodules are finite. -/
-instance map (p : Submodule R M) [Finite R p] (f : M →ₗ[R] N) : Finite R (p.map f) :=
+instance map (p : Submodule R M) [Module.Finite R p] (f : M →ₗ[R] N) : Module.Finite R (p.map f) :=
   of_surjective (f.restrict fun _ => Submodule.mem_map_of_mem) fun ⟨_, _, hy, hy'⟩ =>
     ⟨⟨_, hy⟩, Subtype.ext hy'⟩
 
 variable (R)
 
-instance self : Finite R R :=
+instance self : Module.Finite R R :=
   ⟨⟨{1}, by simpa only [Finset.coe_singleton] using Ideal.span_singleton_one⟩⟩
 
 variable (M)
 
 theorem of_restrictScalars_finite (R A M : Type*) [CommSemiring R] [Semiring A] [AddCommMonoid M]
-    [Module R M] [Module A M] [Algebra R A] [IsScalarTower R A M] [hM : Finite R M] :
-    Finite A M := by
+    [Module R M] [Module A M] [Algebra R A] [IsScalarTower R A M] [hM : Module.Finite R M] :
+    Module.Finite A M := by
   rw [finite_def, Submodule.fg_def] at hM ⊢
   obtain ⟨S, hSfin, hSgen⟩ := hM
   refine ⟨S, hSfin, eq_top_iff.2 ?_⟩
@@ -575,24 +578,24 @@ theorem of_restrictScalars_finite (R A M : Type*) [CommSemiring R] [Semiring A]
 
 variable {R M}
 
-instance prod [hM : Finite R M] [hN : Finite R N] : Finite R (M × N) :=
+instance prod [hM : Module.Finite R M] [hN : Module.Finite R N] : Module.Finite R (M × N) :=
   ⟨by
     rw [← Submodule.prod_top]
     exact hM.1.prod hN.1⟩
 
 instance pi {ι : Type*} {M : ι → Type*} [_root_.Finite ι] [∀ i, AddCommMonoid (M i)]
-    [∀ i, Module R (M i)] [h : ∀ i, Finite R (M i)] : Finite R (∀ i, M i) :=
+    [∀ i, Module R (M i)] [h : ∀ i, Module.Finite R (M i)] : Module.Finite R (∀ i, M i) :=
   ⟨by
     rw [← Submodule.pi_top]
     exact Submodule.fg_pi fun i => (h i).1⟩
 
-theorem equiv [Finite R M] (e : M ≃ₗ[R] N) : Finite R N :=
+theorem equiv [Module.Finite R M] (e : M ≃ₗ[R] N) : Module.Finite R N :=
   of_surjective (e : M →ₗ[R] N) e.surjective
 
-theorem equiv_iff (e : M ≃ₗ[R] N) : Finite R M ↔ Finite R N :=
+theorem equiv_iff (e : M ≃ₗ[R] N) : Module.Finite R M ↔ Module.Finite R N :=
   ⟨fun _ ↦ equiv e, fun _ ↦ equiv e.symm⟩
 
-instance ulift [Finite R M] : Finite R (ULift M) := equiv ULift.moduleEquiv.symm
+instance ulift [Module.Finite R M] : Module.Finite R (ULift M) := equiv ULift.moduleEquiv.symm
 
 theorem iff_fg {N : Submodule R M} : Module.Finite R N ↔ N.FG := Module.finite_def.trans (fg_top _)
 
@@ -600,7 +603,7 @@ variable (R M)
 
 instance bot : Module.Finite R (⊥ : Submodule R M) := iff_fg.mpr fg_bot
 
-instance top [Finite R M] : Module.Finite R (⊤ : Submodule R M) := iff_fg.mpr out
+instance top [Module.Finite R M] : Module.Finite R (⊤ : Submodule R M) := iff_fg.mpr out
 
 variable {M}
 
@@ -639,7 +642,7 @@ section Algebra
 
 theorem trans {R : Type*} (A M : Type*) [Semiring R] [Semiring A] [Module R A]
     [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] :
-    ∀ [Finite R A] [Finite A M], Finite R M
+    ∀ [Module.Finite R A] [Module.Finite A M], Module.Finite R M
   | ⟨⟨s, hs⟩⟩, ⟨⟨t, ht⟩⟩ =>
     ⟨Submodule.fg_def.2
         ⟨Set.image2 (· • ·) (↑s : Set A) (↑t : Set M),
@@ -699,7 +702,7 @@ instance Module.Finite.tensorProduct [CommSemiring R] [AddCommMonoid M] [Module
   out := (TensorProduct.map₂_mk_top_top_eq_top R M N).subst (hM.out.map₂ _ hN.out)
 
 /-- If a free module is finite, then any arbitrary basis is finite. -/
-lemma Module.Finite.finite_basis {R M} [Ring R] [Nontrivial R] [AddCommGroup M] [Module R M]
+lemma Module.Finite.finite_basis {R M} [Semiring R] [Nontrivial R] [AddCommGroup M] [Module R M]
     {ι} [Module.Finite R M] (b : Basis ι R M) :
     _root_.Finite ι :=
   let ⟨s, hs⟩ := ‹Module.Finite R M›
@@ -757,6 +760,7 @@ namespace RingHom
 variable {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
 
 /-- A ring morphism `A →+* B` is `Finite` if `B` is finitely generated as `A`-module. -/
+@[algebraize Module.Finite]
 def Finite (f : A →+* B) : Prop :=
   letI : Algebra A B := f.toAlgebra
   Module.Finite A B
@@ -775,20 +779,11 @@ theorem of_surjective (f : A →+* B) (hf : Surjective f) : f.Finite :=
   Module.Finite.of_surjective (Algebra.linearMap A B) hf
 
 theorem comp {g : B →+* C} {f : A →+* B} (hg : g.Finite) (hf : f.Finite) : (g.comp f).Finite := by
-  letI := f.toAlgebra
-  letI := g.toAlgebra
-  letI := (g.comp f).toAlgebra
-  letI : IsScalarTower A B C := RestrictScalars.isScalarTower A B C
-  letI : Module.Finite A B := hf
-  letI : Module.Finite B C := hg
+  algebraize [f, g, g.comp f]
   exact Module.Finite.trans B C
 
 theorem of_comp_finite {f : A →+* B} {g : B →+* C} (h : (g.comp f).Finite) : g.Finite := by
-  letI := f.toAlgebra
-  letI := g.toAlgebra
-  letI := (g.comp f).toAlgebra
-  letI : IsScalarTower A B C := RestrictScalars.isScalarTower A B C
-  letI : Module.Finite A C := h
+  algebraize [f, g, g.comp f]
   exact Module.Finite.of_restrictScalars_finite A B C
 
 end Finite
diff --git a/Mathlib/RingTheory/Fintype.lean b/Mathlib/RingTheory/Fintype.lean
index ea2719912c171..9ee3bc2a315f2 100644
--- a/Mathlib/RingTheory/Fintype.lean
+++ b/Mathlib/RingTheory/Fintype.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Data.Fintype.Units
 import Mathlib.Data.ZMod.Basic
diff --git a/Mathlib/RingTheory/Flat/Algebra.lean b/Mathlib/RingTheory/Flat/Algebra.lean
index 0aa4d1079abf0..53bc49e2510e5 100644
--- a/Mathlib/RingTheory/Flat/Algebra.lean
+++ b/Mathlib/RingTheory/Flat/Algebra.lean
@@ -60,6 +60,7 @@ theorem isBaseChange [Algebra R S] (R' : Type w) (S' : Type t) [CommRing R'] [Co
 end Algebra.Flat
 
 /-- A ring homomorphism `f : R →+* S` is flat if `S` is flat as an `R` algebra. -/
+@[algebraize RingHom.Flat.out]
 class RingHom.Flat {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) : Prop where
   out : f.toAlgebra.Flat := by infer_instance
 
@@ -75,11 +76,8 @@ variable {R : Type u} {S : Type v} {T : Type w} [CommRing R] [CommRing S] [CommR
 
 /-- Composition of flat ring homomorphisms is flat. -/
 instance comp [RingHom.Flat f] [RingHom.Flat g] : RingHom.Flat (g.comp f) where
-  out :=
-    letI : Algebra R S := f.toAlgebra
-    letI : Algebra S T := g.toAlgebra
-    letI : Algebra R T := (g.comp f).toAlgebra
-    letI : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq (congrFun rfl)
-    Algebra.Flat.comp R S T
+  out := by
+    algebraize_only [f, g, g.comp f]
+    exact Algebra.Flat.comp R S T
 
 end RingHom.Flat
diff --git a/Mathlib/RingTheory/Flat/Basic.lean b/Mathlib/RingTheory/Flat/Basic.lean
index 6398e62b89386..813b32f84d0ac 100644
--- a/Mathlib/RingTheory/Flat/Basic.lean
+++ b/Mathlib/RingTheory/Flat/Basic.lean
@@ -3,17 +3,16 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin, Jujian Zhang
 -/
-import Mathlib.RingTheory.Noetherian
-import Mathlib.Algebra.DirectSum.Module
 import Mathlib.Algebra.DirectSum.Finsupp
-import Mathlib.Algebra.Module.Projective
-import Mathlib.Algebra.Module.Injective
+import Mathlib.Algebra.DirectSum.Module
+import Mathlib.Algebra.Exact
 import Mathlib.Algebra.Module.CharacterModule
+import Mathlib.Algebra.Module.Injective
+import Mathlib.Algebra.Module.Projective
 import Mathlib.LinearAlgebra.DirectSum.TensorProduct
 import Mathlib.LinearAlgebra.FreeModule.Basic
-import Mathlib.Algebra.Module.Projective
 import Mathlib.LinearAlgebra.TensorProduct.RightExactness
-import Mathlib.Algebra.Exact
+import Mathlib.RingTheory.Noetherian
 
 /-!
 # Flat modules
@@ -173,7 +172,7 @@ instance directSum (ι : Type v) (M : ι → Type w) [(i : ι) → AddCommGroup
     apply TensorProduct.ext'
     intro a m
     simp only [ρ, ψ, φ, η, η₁, coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
-      directSumRight_symm_lof_tmul, rTensor_tmul, Submodule.coeSubtype, lid_tmul, map_smul]
+      directSumRight_symm_lof_tmul, rTensor_tmul, Submodule.coe_subtype, lid_tmul, map_smul]
     rw [DirectSum.component.of, DirectSum.component.of]
     by_cases h₂ : j = i
     · subst j; simp
diff --git a/Mathlib/RingTheory/Flat/CategoryTheory.lean b/Mathlib/RingTheory/Flat/CategoryTheory.lean
index ea7bc816d2f57..ea6d1442dad38 100644
--- a/Mathlib/RingTheory/Flat/CategoryTheory.lean
+++ b/Mathlib/RingTheory/Flat/CategoryTheory.lean
@@ -52,7 +52,7 @@ lemma iff_lTensor_preserves_shortComplex_exact :
   ⟨fun _ _ ↦ lTensor_shortComplex_exact _ _, fun H ↦ iff_lTensor_exact.2 <|
     fun _ _ _ _ _ _ _ _ _ f g h ↦
       moduleCat_exact_iff_function_exact _ |>.1 <|
-      H (.mk (ModuleCat.ofHom f) (ModuleCat.ofHom g)
+      H (.mk (ModuleCat.asHom f) (ModuleCat.asHom g)
         (DFunLike.ext _ _ h.apply_apply_eq_zero))
           (moduleCat_exact_iff_function_exact _ |>.2 h)⟩
 
@@ -62,7 +62,7 @@ lemma iff_rTensor_preserves_shortComplex_exact :
   ⟨fun _ _ ↦ rTensor_shortComplex_exact _ _, fun H ↦ iff_rTensor_exact.2 <|
     fun _ _ _ _ _ _ _ _ _ f g h ↦
       moduleCat_exact_iff_function_exact _ |>.1 <|
-      H (.mk (ModuleCat.ofHom f) (ModuleCat.ofHom g)
+      H (.mk (ModuleCat.asHom f) (ModuleCat.asHom g)
         (DFunLike.ext _ _ h.apply_apply_eq_zero))
           (moduleCat_exact_iff_function_exact _ |>.2 h)⟩
 
diff --git a/Mathlib/RingTheory/Flat/EquationalCriterion.lean b/Mathlib/RingTheory/Flat/EquationalCriterion.lean
index a0cd9e88c3f4a..763d9dc2c83db 100644
--- a/Mathlib/RingTheory/Flat/EquationalCriterion.lean
+++ b/Mathlib/RingTheory/Flat/EquationalCriterion.lean
@@ -120,19 +120,17 @@ theorem tfae_equational_criterion : List.TFAE [
     ∀ {ι : Type u} [Fintype ι] {f : ι →₀ R} {x : (ι →₀ R) →ₗ[R] M}, x f = 0 →
       ∃ (κ : Type u) (_ : Fintype κ) (a : (ι →₀ R) →ₗ[R] (κ →₀ R)) (y : (κ →₀ R) →ₗ[R] M),
         x = y ∘ₗ a ∧ a f = 0,
-    ∀ {N : Type u} [AddCommGroup N] [Module R N] [Free R N] [Finite R N] {f : N} {x : N →ₗ[R] M},
-      x f = 0 →
+    ∀ {N : Type u} [AddCommGroup N] [Module R N] [Free R N] [Module.Finite R N] {f : N}
+      {x : N →ₗ[R] M}, x f = 0 →
         ∃ (κ : Type u) (_ : Fintype κ) (a : N →ₗ[R] (κ →₀ R)) (y : (κ →₀ R) →ₗ[R] M),
           x = y ∘ₗ a ∧ a f = 0] := by
   classical
-  tfae_have 1 ↔ 2
-  · exact iff_rTensor_injective' R M
-  tfae_have 3 ↔ 2
-  · exact forall_vanishesTrivially_iff_forall_rTensor_injective R
-  tfae_have 3 ↔ 4
-  · simp [(TensorProduct.lid R M).injective.eq_iff.symm, isTrivialRelation_iff_vanishesTrivially]
+  tfae_have 1 ↔ 2 := iff_rTensor_injective' R M
+  tfae_have 3 ↔ 2 := forall_vanishesTrivially_iff_forall_rTensor_injective R
+  tfae_have 3 ↔ 4 := by
+    simp [(TensorProduct.lid R M).injective.eq_iff.symm, isTrivialRelation_iff_vanishesTrivially]
   tfae_have 4 → 5
-  · intro h₄ ι hι f x hfx
+  | h₄, ι, hι, f, x, hfx => by
     let f' : ι → R := f
     let x' : ι → M := fun i ↦ x (single i 1)
     have := calc
@@ -144,20 +142,20 @@ theorem tfae_equational_criterion : List.TFAE [
       _ = 0                                 := hfx
     obtain ⟨κ, hκ, a', y', ⟨ha'y', ha'⟩⟩ := h₄ this
     use κ, hκ
-    use Finsupp.total R (fun i ↦ equivFunOnFinite.symm (a' i))
-    use Finsupp.total R y'
+    use Finsupp.linearCombination R (fun i ↦ equivFunOnFinite.symm (a' i))
+    use Finsupp.linearCombination R y'
     constructor
     · apply Finsupp.basisSingleOne.ext
       intro i
-      simpa [total_apply, sum_fintype, Finsupp.single_apply] using ha'y' i
+      simpa [linearCombination_apply, sum_fintype, Finsupp.single_apply] using ha'y' i
     · ext j
-      simp only [total_apply, zero_smul, implies_true, sum_fintype, finset_sum_apply]
+      simp only [linearCombination_apply, zero_smul, implies_true, sum_fintype, finset_sum_apply]
       exact ha' j
   tfae_have 5 → 4
-  · intro h₅ ι hi f x hfx
+  | h₅, ι, hi, f, x, hfx => by
     let f' : ι →₀ R := equivFunOnFinite.symm f
-    let x' : (ι →₀ R) →ₗ[R] M := Finsupp.total R x
-    have : x' f' = 0 := by simpa [x', f', total_apply, sum_fintype] using hfx
+    let x' : (ι →₀ R) →ₗ[R] M := Finsupp.linearCombination R x
+    have : x' f' = 0 := by simpa [x', f', linearCombination_apply, sum_fintype] using hfx
     obtain ⟨κ, hκ, a', y', ha'y', ha'⟩ := h₅ this
     refine ⟨κ, hκ, fun i ↦ a' (single i 1), fun j ↦ y' (single j 1), fun i ↦ ?_, fun j ↦ ?_⟩
     · simpa [x', ← map_smul, ← map_sum, smul_single] using
@@ -167,7 +165,7 @@ theorem tfae_equational_criterion : List.TFAE [
         ← (fun _ ↦ equivFunOnFinite_symm_apply_toFun _ _ : ∀ x, f' x = f x), univ_sum_single]
       simpa using DFunLike.congr_fun ha' j
   tfae_have 5 → 6
-  · intro h₅ N _ _ _ _ f x hfx
+  | h₅, N, _, _, _, _, f, x, hfx => by
     have ϕ := Module.Free.repr R N
     have : (x ∘ₗ ϕ.symm) (ϕ f) = 0 := by simpa
     obtain ⟨κ, hκ, a', y, ha'y, ha'⟩ := h₅ this
@@ -175,8 +173,7 @@ theorem tfae_equational_criterion : List.TFAE [
     · simpa [LinearMap.comp_assoc] using congrArg (fun g ↦ (g ∘ₗ ϕ : N →ₗ[R] M)) ha'y
     · simpa using ha'
   tfae_have 6 → 5
-  · intro h₆ _ _ _ _ hfx
-    exact h₆ hfx
+  | h₆, _, _, _, _, hfx => h₆ hfx
   tfae_finish
 
 /-- **Equational criterion for flatness** [Stacks 00HK](https://stacks.math.columbia.edu/tag/00HK).
@@ -246,8 +243,8 @@ Let $M$ be a flat module over a commutative ring $R$. Let $N$ be a finite free m
 let $f \in N$, and let $x \colon N \to M$ be a homomorphism such that $x(f) = 0$. Then there exist a
 finite index type $\kappa$ and module homomorphisms $a \colon N \to R^{\kappa}$ and
 $y \colon R^{\kappa} \to M$ such that $x = y \circ a$ and $a(f) = 0$. -/
-theorem exists_factorization_of_apply_eq_zero_of_free [Flat R M] {N : Type u}
-    [AddCommGroup N] [Module R N] [Free R N] [Finite R N] {f : N} {x : N →ₗ[R] M} (h : x f = 0) :
+theorem exists_factorization_of_apply_eq_zero_of_free [Flat R M] {N : Type u} [AddCommGroup N]
+    [Module R N] [Free R N] [Module.Finite R N] {f : N} {x : N →ₗ[R] M} (h : x f = 0) :
     ∃ (κ : Type u) (_ : Fintype κ) (a : N →ₗ[R] (κ →₀ R)) (y : (κ →₀ R) →ₗ[R] M),
       x = y ∘ₗ a ∧ a f = 0 := by
   exact ((tfae_equational_criterion R M).out 0 5 rfl rfl).mp ‹Flat R M› h
@@ -257,8 +254,8 @@ free, and let $f \colon K \to N$ and $x \colon N \to M$ be homomorphisms such th
 $x \circ f = 0$. Then there exist a finite index type $\kappa$ and module homomorphisms
 $a \colon N \to R^{\kappa}$ and $y \colon R^{\kappa} \to M$ such that $x = y \circ a$ and
 $a \circ f = 0$. -/
-theorem exists_factorization_of_comp_eq_zero_of_free [Flat R M] {K N : Type u}
-    [AddCommGroup K] [Module R K] [Finite R K] [AddCommGroup N] [Module R N] [Free R N] [Finite R N]
+theorem exists_factorization_of_comp_eq_zero_of_free [Flat R M] {K N : Type u} [AddCommGroup K]
+    [Module R K] [Module.Finite R K] [AddCommGroup N] [Module R N] [Free R N] [Module.Finite R N]
     {f : K →ₗ[R] N} {x : N →ₗ[R] M} (h : x ∘ₗ f = 0) :
     ∃ (κ : Type u) (_ : Fintype κ) (a : N →ₗ[R] (κ →₀ R)) (y : (κ →₀ R) →ₗ[R] M),
       x = y ∘ₗ a ∧ a ∘ₗ f = 0 := by
@@ -286,7 +283,7 @@ theorem exists_factorization_of_isFinitelyPresented [Flat R M] {P : Type u} [Add
       ∃ (κ : Type u) (_ : Fintype κ) (h₂ : P →ₗ[R] (κ →₀ R)) (h₃ : (κ →₀ R) →ₗ[R] M),
         h₁ = h₃ ∘ₗ h₂ := by
   obtain ⟨L, _, _, K, ϕ, _, _, hK⟩ := FinitePresentation.equiv_quotient R P
-  haveI : Finite R ↥K := Module.Finite.iff_fg.mpr hK
+  haveI : Module.Finite R ↥K := Module.Finite.iff_fg.mpr hK
   have : (h₁ ∘ₗ ϕ.symm ∘ₗ K.mkQ) ∘ₗ K.subtype = 0 := by
     simp_rw [comp_assoc, (LinearMap.exact_subtype_mkQ K).linearMap_comp_eq_zero, comp_zero]
   obtain ⟨κ, hκ, a, y, hay, ha⟩ := exists_factorization_of_comp_eq_zero_of_free this
diff --git a/Mathlib/RingTheory/FractionalIdeal/Operations.lean b/Mathlib/RingTheory/FractionalIdeal/Operations.lean
index 56982ee5c698d..d6ab280bcde60 100644
--- a/Mathlib/RingTheory/FractionalIdeal/Operations.lean
+++ b/Mathlib/RingTheory/FractionalIdeal/Operations.lean
@@ -5,6 +5,7 @@ Authors: Anne Baanen, Filippo A. E. Nuccio
 -/
 import Mathlib.RingTheory.FractionalIdeal.Basic
 import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic
+import Mathlib.RingTheory.LocalRing.Basic
 
 /-!
 # More operations on fractional ideals
@@ -226,7 +227,7 @@ theorem canonicalEquiv_symm : (canonicalEquiv S P P').symm = canonicalEquiv S P'
       exact ⟨fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩, fun ⟨y, mem, Eq⟩ => ⟨y, mem, Eq⟩⟩
 
 theorem canonicalEquiv_flip (I) : canonicalEquiv S P P' (canonicalEquiv S P' P I) = I := by
-  rw [← canonicalEquiv_symm]; erw [RingEquiv.apply_symm_apply]
+  rw [← canonicalEquiv_symm, RingEquiv.symm_apply_apply]
 
 @[simp]
 theorem canonicalEquiv_canonicalEquiv (P'' : Type*) [CommRing P''] [Algebra R P'']
@@ -454,7 +455,7 @@ theorem eq_one_div_of_mul_eq_one_right (I J : FractionalIdeal R₁⁰ K) (h : I
   apply (le_div_iff_of_nonzero hI).mpr _
   intro y hy x hx
   rw [mul_comm]
-  exact mul_mem_mul hx hy
+  exact mul_mem_mul hy hx
 
 theorem mul_div_self_cancel_iff {I : FractionalIdeal R₁⁰ K} : I * (1 / I) = 1 ↔ ∃ J, I * J = 1 :=
   ⟨fun h => ⟨1 / I, h⟩, fun ⟨J, hJ⟩ => by rwa [← eq_one_div_of_mul_eq_one_right I J hJ]⟩
diff --git a/Mathlib/RingTheory/Generators.lean b/Mathlib/RingTheory/Generators.lean
index 147df9df37990..3ded91b21c690 100644
--- a/Mathlib/RingTheory/Generators.lean
+++ b/Mathlib/RingTheory/Generators.lean
@@ -45,7 +45,7 @@ variable (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S]
 2. `val : vars → S`: The assignment of each variable to a value in `S`.
 3. `σ`: A section of `R[X] → S`. -/
 structure Algebra.Generators where
-  /-- The type of variables.  -/
+  /-- The type of variables. -/
   vars : Type w
   /-- The assignment of each variable to a value in `S`. -/
   val : vars → S
@@ -106,6 +106,18 @@ def ofSurjective {vars} (val : vars → S) (h : Function.Surjective (aeval (R :=
   σ' x := (h x).choose
   aeval_val_σ' x := (h x).choose_spec
 
+/-- If `algebraMap R S` is surjective, the empty type generates `S`. -/
+noncomputable def ofSurjectiveAlgebraMap (h : Function.Surjective (algebraMap R S)) :
+    Generators.{w} R S :=
+  ofSurjective PEmpty.elim <| fun s ↦ by
+    use C (h s).choose
+    simp [(h s).choose_spec]
+
+/-- The canonical generators for `R` as an `R`-algebra. -/
+noncomputable def id : Generators.{w} R R := ofSurjectiveAlgebraMap <| by
+  rw [id.map_eq_id]
+  exact RingHomSurjective.is_surjective
+
 /-- Construct `Generators` from an assignment `I → S` such that `R[X] → S` is surjective. -/
 noncomputable
 def ofAlgHom {I} (f : MvPolynomial I R →ₐ[R] S) (h : Function.Surjective f) :
@@ -166,7 +178,7 @@ def comp (Q : Generators S T) (P : Generators R S) : Generators R T where
   σ' x := (Q.σ x).sum (fun n r ↦ rename Sum.inr (P.σ r) * monomial (n.mapDomain Sum.inl) 1)
   aeval_val_σ' s := by
     have (x : P.Ring) : aeval (algebraMap S T ∘ P.val) x = algebraMap S T (aeval P.val x) := by
-      rw [map_aeval, aeval_def, coe_eval₂Hom, ← IsScalarTower.algebraMap_eq, Function.comp]
+      rw [map_aeval, aeval_def, coe_eval₂Hom, ← IsScalarTower.algebraMap_eq, Function.comp_def]
     conv_rhs => rw [← Q.aeval_val_σ s, ← (Q.σ s).sum_single]
     simp only [map_finsupp_sum, map_mul, aeval_rename, Sum.elim_comp_inr, this, aeval_val_σ,
       aeval_monomial, map_one, Finsupp.prod_mapDomain_index_inj Sum.inl_injective, Sum.elim_inl,
@@ -381,10 +393,10 @@ instance : AddCommGroup P.Cotangent := inferInstanceAs (AddCommGroup P.ker.Cotan
 
 variable {P}
 
-/-- The identity map `P.ker.Cotangent → P.Cotangent` into the type synonym.  -/
+/-- The identity map `P.ker.Cotangent → P.Cotangent` into the type synonym. -/
 def Cotangent.of (x : P.ker.Cotangent) : P.Cotangent := x
 
-/-- The identity map `P.Cotangent → P.ker.Cotangent` from the type synonym.  -/
+/-- The identity map `P.Cotangent → P.ker.Cotangent` from the type synonym. -/
 def Cotangent.val (x : P.Cotangent) : P.ker.Cotangent := x
 
 @[ext]
diff --git a/Mathlib/RingTheory/GradedAlgebra/Basic.lean b/Mathlib/RingTheory/GradedAlgebra/Basic.lean
index 458f9d5da4ef5..0d6693692b34f 100644
--- a/Mathlib/RingTheory/GradedAlgebra/Basic.lean
+++ b/Mathlib/RingTheory/GradedAlgebra/Basic.lean
@@ -207,7 +207,7 @@ end DirectSum
 
 open DirectSum
 
-/-- The projection maps of graded algebra-/
+/-- The projection maps of graded algebra -/
 def GradedAlgebra.proj (𝒜 : ι → Submodule R A) [GradedAlgebra 𝒜] (i : ι) : A →ₗ[R] A :=
   (𝒜 i).subtype.comp <| (DFinsupp.lapply i).comp <| (decomposeAlgEquiv 𝒜).toAlgHom.toLinearMap
 
diff --git a/Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean b/Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
index f19c6b57a7077..dc2ec96afc47a 100644
--- a/Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
+++ b/Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
@@ -154,10 +154,10 @@ theorem Ideal.mul_homogeneous_element_mem_of_mem {I : Ideal A} (r x : A) (hx₁
 theorem Ideal.homogeneous_span (s : Set A) (h : ∀ x ∈ s, Homogeneous 𝒜 x) :
     (Ideal.span s).IsHomogeneous 𝒜 := by
   rintro i r hr
-  rw [Ideal.span, Finsupp.span_eq_range_total] at hr
+  rw [Ideal.span, Finsupp.span_eq_range_linearCombination] at hr
   rw [LinearMap.mem_range] at hr
   obtain ⟨s, rfl⟩ := hr
-  rw [Finsupp.total_apply, Finsupp.sum, decompose_sum, DFinsupp.finset_sum_apply,
+  rw [Finsupp.linearCombination_apply, Finsupp.sum, decompose_sum, DFinsupp.finset_sum_apply,
     AddSubmonoidClass.coe_finset_sum]
   refine Ideal.sum_mem _ ?_
   rintro z hz1
diff --git a/Mathlib/RingTheory/HahnSeries/Multiplication.lean b/Mathlib/RingTheory/HahnSeries/Multiplication.lean
index 148770477311f..153ac24993c5d 100644
--- a/Mathlib/RingTheory/HahnSeries/Multiplication.lean
+++ b/Mathlib/RingTheory/HahnSeries/Multiplication.lean
@@ -6,6 +6,7 @@ Authors: Aaron Anderson, Scott Carnahan
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Data.Finset.MulAntidiagonal
 import Mathlib.Data.Finset.SMulAntidiagonal
+import Mathlib.GroupTheory.GroupAction.Ring
 import Mathlib.RingTheory.HahnSeries.Addition
 
 /-!
@@ -228,10 +229,10 @@ theorem add_smul [AddCommMonoid R] [SMulWithZero R V] {x y : HahnSeries Γ R}
   ext a
   have hwf := x.isPWO_support.union y.isPWO_support
   rw [smul_coeff_left hwf, HahnSeries.add_coeff', of_symm_add]
-  simp_all only [Pi.add_apply, HahnSeries.add_coeff']
-  rw [smul_coeff_left hwf Set.subset_union_right,
-    smul_coeff_left hwf Set.subset_union_left]
-  · simp only [HahnSeries.add_coeff, h, sum_add_distrib]
+  · simp_all only [Pi.add_apply, HahnSeries.add_coeff']
+    rw [smul_coeff_left hwf Set.subset_union_right,
+      smul_coeff_left hwf Set.subset_union_left]
+    simp only [HahnSeries.add_coeff, h, sum_add_distrib]
   · intro b
     simp_all only [Set.isPWO_union, HahnSeries.isPWO_support, and_self, HahnSeries.mem_support,
       HahnSeries.add_coeff, ne_eq, Set.mem_union, Set.mem_setOf_eq, mem_support]
@@ -252,7 +253,7 @@ theorem single_smul_coeff_add [MulZeroClass R] [SMulWithZero R V] {r : R} {x : H
     rw [sum_congr _ fun _ _ => rfl, sum_empty]
     ext ⟨a1, a2⟩
     simp only [not_mem_empty, not_and, Set.mem_singleton_iff, Classical.not_not,
-      mem_vaddAntidiagonal, Set.mem_setOf_eq, iff_false_iff]
+      mem_vaddAntidiagonal, Set.mem_setOf_eq, iff_false]
     rintro rfl h2 h1
     rw [IsCancelVAdd.left_cancel a1 a2 a h1] at h2
     exact h2 hx
@@ -386,7 +387,7 @@ theorem mul_single_coeff_add [NonUnitalNonAssocSemiring R] {r : R} {x : HahnSeri
     rw [sum_congr _ fun _ _ => rfl, sum_empty]
     ext ⟨a1, a2⟩
     simp only [not_mem_empty, not_and, Set.mem_singleton_iff, Classical.not_not,
-      mem_addAntidiagonal, Set.mem_setOf_eq, iff_false_iff]
+      mem_addAntidiagonal, Set.mem_setOf_eq, iff_false]
     rintro h2 rfl h1
     rw [← add_right_cancel h1] at hx
     exact h2 hx
@@ -753,7 +754,7 @@ instance [Nontrivial Γ] [Nontrivial R] : Nontrivial (Subalgebra R (HahnSeries 
       rw [Ne, SetLike.ext_iff, not_forall]
       obtain ⟨a, ha⟩ := exists_ne (0 : Γ)
       refine ⟨single a 1, ?_⟩
-      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true_iff, Algebra.mem_top]
+      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true, Algebra.mem_top]
       intro x
       rw [HahnSeries.ext_iff, Function.funext_iff, not_forall]
       refine ⟨a, ?_⟩
diff --git a/Mathlib/RingTheory/HahnSeries/PowerSeries.lean b/Mathlib/RingTheory/HahnSeries/PowerSeries.lean
index e37029a7b20db..094a71acddd64 100644
--- a/Mathlib/RingTheory/HahnSeries/PowerSeries.lean
+++ b/Mathlib/RingTheory/HahnSeries/PowerSeries.lean
@@ -5,6 +5,7 @@ Authors: Aaron Anderson
 -/
 import Mathlib.RingTheory.HahnSeries.Multiplication
 import Mathlib.RingTheory.PowerSeries.Basic
+import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
 import Mathlib.Data.Finsupp.PWO
 
 /-!
@@ -18,6 +19,11 @@ we get the more familiar semiring of formal power series with coefficients in `R
   * `toPowerSeries` the isomorphism from `HahnSeries ℕ R` to `PowerSeries R`.
   * `ofPowerSeries` the inverse, casting a `PowerSeries R` to a `HahnSeries ℕ R`.
 
+## Instances
+  * For `Finite σ`, the instance `NoZeroDivisors (HahnSeries (σ →₀ ℕ) R)`,
+  deduced from the case of `MvPowerSeries`
+  The case of `HahnSeries ℕ R` is taken care of by `instNoZeroDivisors`.
+
 ## TODO
   * Build an API for the variable `X` (defined to be `single 1 1 : HahnSeries Γ R`) in analogy to
     `X : R[X]` and `X : PowerSeries R`
@@ -87,7 +93,7 @@ theorem ofPowerSeries_injective : Function.Injective (ofPowerSeries Γ R) :=
   embDomain_injective.comp toPowerSeries.symm.injective
 
 /-@[simp] Porting note: removing simp. RHS is more complicated and it makes linter
-failures elsewhere-/
+failures elsewhere -/
 theorem ofPowerSeries_apply (x : PowerSeries R) :
     ofPowerSeries Γ R x =
       HahnSeries.embDomain
@@ -166,6 +172,12 @@ def toMvPowerSeries {σ : Type*} [Finite σ] : HahnSeries (σ →₀ ℕ) R ≃+
 
 variable {σ : Type*} [Finite σ]
 
+-- TODO : generalize to all (?) rings of Hahn Series
+/-- If R has no zero divisors and `σ` is finite,
+then `HahnSeries (σ →₀ ℕ) R` has no zero divisors -/
+instance [NoZeroDivisors R] : NoZeroDivisors (HahnSeries (σ →₀ ℕ) R) :=
+  toMvPowerSeries.toMulEquiv.noZeroDivisors (A := HahnSeries (σ →₀ ℕ) R) (MvPowerSeries σ R)
+
 theorem coeff_toMvPowerSeries {f : HahnSeries (σ →₀ ℕ) R} {n : σ →₀ ℕ} :
     MvPowerSeries.coeff R n (toMvPowerSeries f) = f.coeff n :=
   rfl
diff --git a/Mathlib/RingTheory/HahnSeries/Summable.lean b/Mathlib/RingTheory/HahnSeries/Summable.lean
index 517632b481a87..871e001eae93b 100644
--- a/Mathlib/RingTheory/HahnSeries/Summable.lean
+++ b/Mathlib/RingTheory/HahnSeries/Summable.lean
@@ -418,8 +418,8 @@ def powers (x : HahnSeries Γ R) (hx : 0 < x.orderTop) : SummableFamily Γ R ℕ
       · exact Set.mem_union_right _ (Set.mem_singleton 0)
       · obtain ⟨i, hi, j, hj, rfl⟩ := support_mul_subset_add_support hn
         refine Set.mem_union_left _ ⟨n, Set.mem_iUnion.2 ⟨⟨j, i⟩, Set.mem_iUnion.2 ⟨?_, hi⟩⟩, rfl⟩
-        simp only [and_true_iff, Set.mem_iUnion, mem_addAntidiagonal, mem_coe, eq_self_iff_true,
-          Ne, mem_support, Set.mem_setOf_eq]
+        simp only [Set.mem_iUnion, mem_addAntidiagonal, mem_coe, eq_self_iff_true, Ne, mem_support,
+          Set.mem_setOf_eq]
         exact ⟨hj, ⟨n, hi⟩, add_comm j i⟩
 
 variable {x : HahnSeries Γ R} (hx : 0 < x.orderTop)
diff --git a/Mathlib/RingTheory/HahnSeries/Valuation.lean b/Mathlib/RingTheory/HahnSeries/Valuation.lean
index 86b6d16e4a9a0..95c72f7dfe144 100644
--- a/Mathlib/RingTheory/HahnSeries/Valuation.lean
+++ b/Mathlib/RingTheory/HahnSeries/Valuation.lean
@@ -35,7 +35,7 @@ section Valuation
 variable (Γ R) [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R]
 
 /-- The additive valuation on `HahnSeries Γ R`, returning the smallest index at which
-  a Hahn Series has a nonzero coefficient, or `⊤` for the 0 series.  -/
+  a Hahn Series has a nonzero coefficient, or `⊤` for the 0 series. -/
 def addVal : AddValuation (HahnSeries Γ R) (WithTop Γ) :=
   AddValuation.of orderTop orderTop_zero (orderTop_one) (fun x y => min_orderTop_le_orderTop_add)
   fun x y => by
diff --git a/Mathlib/RingTheory/Henselian.lean b/Mathlib/RingTheory/Henselian.lean
index 18e8607dbc93a..aa9b54b1479e5 100644
--- a/Mathlib/RingTheory/Henselian.lean
+++ b/Mathlib/RingTheory/Henselian.lean
@@ -121,10 +121,9 @@ theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [LocalRing R] :
           ∀ (φ : R →+* K), Surjective φ → ∀ f : R[X], f.Monic → ∀ a₀ : K,
             f.eval₂ φ a₀ = 0 → f.derivative.eval₂ φ a₀ ≠ 0 → ∃ a : R, f.IsRoot a ∧ φ a = a₀] := by
   tfae_have 3 → 2
-  · intro H
-    exact H (residue R) Ideal.Quotient.mk_surjective
+  | H => H (residue R) Ideal.Quotient.mk_surjective
   tfae_have 2 → 1
-  · intro H
+  | H => by
     constructor
     intro f hf a₀ h₁ h₂
     specialize H f hf (residue R a₀)
@@ -136,10 +135,10 @@ theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [LocalRing R] :
     rw [← Ideal.Quotient.eq_zero_iff_mem]
     rwa [← sub_eq_zero, ← RingHom.map_sub] at ha₂
   tfae_have 1 → 3
-  · intro hR K _K φ hφ f hf a₀ h₁ h₂
+  | hR, K, _K, φ, hφ, f, hf, a₀, h₁, h₂ => by
     obtain ⟨a₀, rfl⟩ := hφ a₀
     have H := HenselianLocalRing.is_henselian f hf a₀
-    simp only [← ker_eq_maximalIdeal φ hφ, eval₂_at_apply, RingHom.mem_ker φ] at H h₁ h₂
+    simp only [← ker_eq_maximalIdeal φ hφ, eval₂_at_apply, RingHom.mem_ker] at H h₁ h₂
     obtain ⟨a, ha₁, ha₂⟩ := H h₁ (by
       contrapose! h₂
       rwa [← mem_nonunits_iff, ← LocalRing.mem_maximalIdeal, ← LocalRing.ker_eq_maximalIdeal φ hφ,
@@ -195,7 +194,7 @@ instance (priority := 100) IsAdicComplete.henselianRing (R : Type*) [CommRing R]
       have hf'c : ∀ n, IsUnit (f'.eval (c n)) := by
         intro n
         haveI := isLocalRingHom_of_le_jacobson_bot I (IsAdicComplete.le_jacobson_bot I)
-        apply isUnit_of_map_unit (Ideal.Quotient.mk I)
+        apply IsUnit.of_map (Ideal.Quotient.mk I)
         convert h₂ using 1
         exact SModEq.def.mp ((hc_mod n).eval _)
       have hfcI : ∀ n, f.eval (c n) ∈ I ^ (n + 1) := by
diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime.lean
index e37e8ce6c85bf..b4ddec4e36127 100644
--- a/Mathlib/RingTheory/Ideal/AssociatedPrime.lean
+++ b/Mathlib/RingTheory/Ideal/AssociatedPrime.lean
@@ -104,7 +104,7 @@ theorem LinearEquiv.AssociatedPrimes.eq (l : M ≃ₗ[R] M') :
     (associatedPrimes.subset_of_injective l.symm l.symm.injective)
 
 theorem associatedPrimes.eq_empty_of_subsingleton [Subsingleton M] : associatedPrimes R M = ∅ := by
-  ext; simp only [Set.mem_empty_iff_false, iff_false_iff]
+  ext; simp only [Set.mem_empty_iff_false, iff_false]
   apply not_isAssociatedPrime_of_subsingleton
 
 variable (R M)
diff --git a/Mathlib/RingTheory/Ideal/Basic.lean b/Mathlib/RingTheory/Ideal/Basic.lean
index bb631626fa74b..69a42212b475e 100644
--- a/Mathlib/RingTheory/Ideal/Basic.lean
+++ b/Mathlib/RingTheory/Ideal/Basic.lean
@@ -28,14 +28,14 @@ Support right ideals, and two-sided ideals over non-commutative rings.
 
 universe u v w
 
-variable {α : Type u} {β : Type v}
+variable {α : Type u} {β : Type v} {F : Type w}
 
 open Set Function
 
 open Pointwise
 
 /-- A (left) ideal in a semiring `R` is an additive submonoid `s` such that
-`a * b ∈ s` whenever `b ∈ s`. If `R` is a ring, then `s` is an additive subgroup.  -/
+`a * b ∈ s` whenever `b ∈ s`. If `R` is a ring, then `s` is an additive subgroup. -/
 abbrev Ideal (R : Type u) [Semiring R] :=
   Submodule R R
 
@@ -76,9 +76,9 @@ theorem sum_mem (I : Ideal α) {ι : Type*} {t : Finset ι} {f : ι → α} :
 theorem eq_top_of_unit_mem (x y : α) (hx : x ∈ I) (h : y * x = 1) : I = ⊤ :=
   eq_top_iff.2 fun z _ =>
     calc
-      z = z * (y * x) := by simp [h]
-      _ = z * y * x := Eq.symm <| mul_assoc z y x
-      _ ∈ I := I.mul_mem_left _ hx
+      z * y * x ∈ I := I.mul_mem_left _ hx
+      _ = z * (y * x) := mul_assoc z y x
+      _ = z := by rw [h, mul_one]
 
 theorem eq_top_of_isUnit_mem {x} (hx : x ∈ I) (h : IsUnit x) : I = ⊤ :=
   let ⟨y, hy⟩ := h.exists_left_inv
@@ -149,6 +149,9 @@ theorem mem_span_insert {s : Set α} {x y} :
 theorem mem_span_singleton' {x y : α} : x ∈ span ({y} : Set α) ↔ ∃ a, a * y = x :=
   Submodule.mem_span_singleton
 
+theorem mem_span_singleton_self (x : α) : x ∈ span ({x} : Set α) :=
+  Submodule.mem_span_singleton_self x
+
 theorem span_singleton_le_iff_mem {x : α} : span {x} ≤ I ↔ x ∈ I :=
   Submodule.span_singleton_le_iff_mem _ _
 
@@ -184,8 +187,8 @@ theorem span_eq_top_iff_finite (s : Set α) :
   simp_rw [eq_top_iff_one]
   exact ⟨Submodule.mem_span_finite_of_mem_span, fun ⟨s', h₁, h₂⟩ => span_mono h₁ h₂⟩
 
-theorem mem_span_singleton_sup {S : Type*} [CommSemiring S] {x y : S} {I : Ideal S} :
-    x ∈ Ideal.span {y} ⊔ I ↔ ∃ a : S, ∃ b ∈ I, a * y + b = x := by
+theorem mem_span_singleton_sup {x y : α} {I : Ideal α} :
+    x ∈ Ideal.span {y} ⊔ I ↔ ∃ a : α, ∃ b ∈ I, a * y + b = x := by
   rw [Submodule.mem_sup]
   constructor
   · rintro ⟨ya, hya, b, hb, rfl⟩
@@ -431,9 +434,6 @@ theorem mul_unit_mem_iff_mem {x y : α} (hy : IsUnit y) : x * y ∈ I ↔ x ∈
 theorem mem_span_singleton {x y : α} : x ∈ span ({y} : Set α) ↔ y ∣ x :=
   mem_span_singleton'.trans <| exists_congr fun _ => by rw [eq_comm, mul_comm]
 
-theorem mem_span_singleton_self (x : α) : x ∈ span ({x} : Set α) :=
-  mem_span_singleton.mpr dvd_rfl
-
 theorem span_singleton_le_span_singleton {x y : α} :
     span ({x} : Set α) ≤ span ({y} : Set α) ↔ y ∣ x :=
   span_le.trans <| singleton_subset_iff.trans mem_span_singleton
@@ -490,6 +490,9 @@ theorem mul_mem_right (h : a ∈ I) : a * b ∈ I :=
 
 variable {b}
 
+lemma mem_of_dvd (hab : a ∣ b) (ha : a ∈ I) : b ∈ I := by
+  obtain ⟨c, rfl⟩ := hab; exact I.mul_mem_right _ ha
+
 theorem pow_mem_of_mem (ha : a ∈ I) (n : ℕ) (hn : 0 < n) : a ^ n ∈ I :=
   Nat.casesOn n (Not.elim (by decide))
     (fun m _hm => (pow_succ a m).symm ▸ I.mul_mem_left (a ^ m) ha) hn
@@ -570,7 +573,7 @@ theorem span_pow_eq_top (s : Set α) (hs : span s = ⊤) (n : ℕ) :
     · rw [Set.image_empty, hs]
       trivial
     · exact subset_span ⟨_, hx, pow_zero _⟩
-  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_total] at hs
+  rw [eq_top_iff_one, span, Finsupp.mem_span_iff_linearCombination] at hs
   rcases hs with ⟨f, hf⟩
   have hf : (f.support.sum fun a => f a * a) = 1 := hf -- Porting note: was `change ... at hf`
   have := sum_pow_mem_span_pow f.support (fun a => f a * a) n
@@ -802,6 +805,12 @@ theorem zero_mem_nonunits [Semiring α] : 0 ∈ nonunits α ↔ (0 : α) ≠ 1 :
 theorem one_not_mem_nonunits [Monoid α] : (1 : α) ∉ nonunits α :=
   not_not_intro isUnit_one
 
+-- Porting note : as this can be proved by other `simp` lemmas, this is marked as high priority.
+@[simp (high)]
+theorem map_mem_nonunits_iff [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F)
+    [IsLocalRingHom f] (a) : f a ∈ nonunits β ↔ a ∈ nonunits α :=
+  ⟨fun h ha => h <| ha.map f, fun h ha => h <| ha.of_map⟩
+
 theorem coe_subset_nonunits [Semiring α] {I : Ideal α} (h : I ≠ ⊤) : (I : Set α) ⊆ nonunits α :=
   fun _x hx hu => h <| I.eq_top_of_isUnit_mem hx hu
 
diff --git a/Mathlib/RingTheory/Ideal/Cotangent.lean b/Mathlib/RingTheory/Ideal/Cotangent.lean
index 3156cab853769..9ce01fea0e26b 100644
--- a/Mathlib/RingTheory/Ideal/Cotangent.lean
+++ b/Mathlib/RingTheory/Ideal/Cotangent.lean
@@ -149,8 +149,9 @@ theorem cotangentEquivIdeal_apply (x : I.Cotangent) :
 
 theorem cotangentEquivIdeal_symm_apply (x : R) (hx : x ∈ I) :
     -- Note: #8386 had to specify `(R₂ := R)` because `I.toCotangent` suggested `R ⧸ I^2` instead
-    I.cotangentEquivIdeal.symm ⟨(I ^ 2).mkQ x, Submodule.mem_map_of_mem (R₂ := R) hx⟩ =
-      I.toCotangent ⟨x, hx⟩ := by
+    I.cotangentEquivIdeal.symm ⟨(I ^ 2).mkQ x,
+      Submodule.mem_map_of_mem (F := R →ₗ[R] R ⧸ I ^ 2) (f := (I ^ 2).mkQ) hx⟩ =
+      I.toCotangent (R := R) ⟨x, hx⟩ := by
   apply I.cotangentEquivIdeal.injective
   rw [I.cotangentEquivIdeal.apply_symm_apply]
   ext
@@ -250,7 +251,7 @@ lemma CotangentSpace.span_image_eq_top_iff [IsNoetherianRing R] {s : Set (maxima
   · simp only [Ideal.toCotangent_apply, Submodule.restrictScalars_top, Submodule.map_span]
   · exact Ideal.Quotient.mk_surjective
 
-open FiniteDimensional
+open Module
 
 lemma finrank_cotangentSpace_eq_zero_iff [IsNoetherianRing R] :
     finrank (ResidueField R) (CotangentSpace R) = 0 ↔ IsField R := by
diff --git a/Mathlib/RingTheory/Ideal/Maps.lean b/Mathlib/RingTheory/Ideal/Maps.lean
index 8b0482cf861a6..5a46dd702e132 100644
--- a/Mathlib/RingTheory/Ideal/Maps.lean
+++ b/Mathlib/RingTheory/Ideal/Maps.lean
@@ -125,11 +125,21 @@ theorem comap_comap {T : Type*} [Semiring T] {I : Ideal T} (f : R →+* S) (g :
     (I.comap g).comap f = I.comap (g.comp f) :=
   rfl
 
+lemma comap_comapₐ {R A B C : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B]
+    [Algebra R B] [Semiring C] [Algebra R C] {I : Ideal C} (f : A →ₐ[R] B) (g : B →ₐ[R] C) :
+    (I.comap g).comap f = I.comap (g.comp f) :=
+  I.comap_comap f.toRingHom g.toRingHom
+
 theorem map_map {T : Type*} [Semiring T] {I : Ideal R} (f : R →+* S) (g : S →+* T) :
     (I.map f).map g = I.map (g.comp f) :=
   ((gc_map_comap f).compose (gc_map_comap g)).l_unique (gc_map_comap (g.comp f)) fun _ =>
     comap_comap _ _
 
+lemma map_mapₐ {R A B C : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B]
+    [Algebra R B] [Semiring C] [Algebra R C] {I : Ideal A} (f : A →ₐ[R] B) (g : B →ₐ[R] C) :
+    (I.map f).map g = I.map (g.comp f) :=
+  I.map_map f.toRingHom g.toRingHom
+
 theorem map_span (f : F) (s : Set R) : map f (span s) = span (f '' s) := by
   refine (Submodule.span_eq_of_le _ ?_ ?_).symm
   · rintro _ ⟨x, hx, rfl⟩; exact mem_map_of_mem f (subset_span hx)
@@ -163,6 +173,9 @@ theorem comap_eq_top_iff {I : Ideal S} : I.comap f = ⊤ ↔ I = ⊤ :=
 theorem map_bot : (⊥ : Ideal R).map f = ⊥ :=
   (gc_map_comap f).l_bot
 
+theorem ne_bot_of_map_ne_bot (hI : map f I ≠ ⊥) : I ≠ ⊥ :=
+  fun h => hI (Eq.mpr (congrArg (fun I ↦ map f I = ⊥) h) map_bot)
+
 variable (f I J K L)
 
 @[simp]
@@ -433,6 +446,10 @@ theorem comap_le_iff_le_map (hf : Function.Bijective f) {I : Ideal R} {K : Ideal
   ⟨fun h => le_map_of_comap_le_of_surjective f hf.right h, fun h =>
     (relIsoOfBijective f hf).right_inv I ▸ comap_mono h⟩
 
+lemma comap_map_of_bijective (hf : Function.Bijective f) {I : Ideal R} :
+    (I.map f).comap f = I :=
+  le_antisymm ((comap_le_iff_le_map f hf).mpr fun _ ↦ id) le_comap_map
+
 theorem map.isMaximal (hf : Function.Bijective f) {I : Ideal R} (H : IsMaximal I) :
     IsMaximal (map f I) := by
   refine
@@ -529,8 +546,9 @@ variable (f : F) (g : G)
 def ker : Ideal R :=
   Ideal.comap f ⊥
 
+variable {f} in
 /-- An element is in the kernel if and only if it maps to zero. -/
-theorem mem_ker {r} : r ∈ ker f ↔ f r = 0 := by rw [ker, Ideal.mem_comap, Submodule.mem_bot]
+@[simp] theorem mem_ker {r} : r ∈ ker f ↔ f r = 0 := by rw [ker, Ideal.mem_comap, Submodule.mem_bot]
 
 theorem ker_eq : (ker f : Set R) = Set.preimage f {0} :=
   rfl
@@ -609,13 +627,13 @@ theorem ker_isMaximal_of_surjective {R K F : Type*} [Ring R] [Field K]
     (hf : Function.Surjective f) : (ker f).IsMaximal := by
   refine
     Ideal.isMaximal_iff.mpr
-      ⟨fun h1 => one_ne_zero' K <| map_one f ▸ (mem_ker f).mp h1, fun J x hJ hxf hxJ => ?_⟩
+      ⟨fun h1 => one_ne_zero' K <| map_one f ▸ mem_ker.mp h1, fun J x hJ hxf hxJ => ?_⟩
   obtain ⟨y, hy⟩ := hf (f x)⁻¹
   have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm
   rw [H]
   refine J.sub_mem (J.mul_mem_left _ hxJ) (hJ ?_)
   rw [mem_ker]
-  simp only [hy, map_sub, map_one, map_mul, inv_mul_cancel₀ (mt (mem_ker f).mpr hxf), sub_self]
+  simp only [hy, map_sub, map_one, map_mul, inv_mul_cancel₀ (mt mem_ker.mpr hxf :), sub_self]
 
 end RingHom
 
@@ -631,7 +649,7 @@ theorem map_eq_bot_iff_le_ker {I : Ideal R} (f : F) : I.map f = ⊥ ↔ I ≤ Ri
   rw [RingHom.ker, eq_bot_iff, map_le_iff_le_comap]
 
 theorem ker_le_comap {K : Ideal S} (f : F) : RingHom.ker f ≤ comap f K := fun _ hx =>
-  mem_comap.2 (((RingHom.mem_ker f).1 hx).symm ▸ K.zero_mem)
+  mem_comap.2 (RingHom.mem_ker.1 hx ▸ K.zero_mem)
 
 theorem map_isPrime_of_equiv {F' : Type*} [EquivLike F' R S] [RingEquivClass F' R S]
     (f : F') {I : Ideal R} [IsPrime I] : IsPrime (map f I) := by
@@ -645,6 +663,10 @@ section Ring
 
 variable [Ring R] [Ring S] [FunLike F R S] [rc : RingHomClass F R S]
 
+lemma comap_map_of_surjective' (f : F) (hf : Function.Surjective f) (I : Ideal R) :
+    (I.map f).comap f = I ⊔ RingHom.ker f :=
+  comap_map_of_surjective f hf I
+
 theorem map_sInf {A : Set (Ideal R)} {f : F} (hf : Function.Surjective f) :
     (∀ J ∈ A, RingHom.ker f ≤ J) → map f (sInf A) = sInf (map f '' A) := by
   refine fun h => le_antisymm (le_sInf ?_) ?_
@@ -726,15 +748,15 @@ def liftOfRightInverseAux (hf : Function.RightInverse f_inv f) (g : A →+* C)
   { AddMonoidHom.liftOfRightInverse f.toAddMonoidHom f_inv hf ⟨g.toAddMonoidHom, hg⟩ with
     toFun := fun b => g (f_inv b)
     map_one' := by
-      rw [← map_one g, ← sub_eq_zero, ← map_sub g, ← mem_ker g]
+      rw [← map_one g, ← sub_eq_zero, ← map_sub g, ← mem_ker]
       apply hg
-      rw [mem_ker f, map_sub f, sub_eq_zero, map_one f]
+      rw [mem_ker, map_sub f, sub_eq_zero, map_one f]
       exact hf 1
     map_mul' := by
       intro x y
-      rw [← map_mul g, ← sub_eq_zero, ← map_sub g, ← mem_ker g]
+      rw [← map_mul g, ← sub_eq_zero, ← map_sub g, ← mem_ker]
       apply hg
-      rw [mem_ker f, map_sub f, sub_eq_zero, map_mul f]
+      rw [mem_ker, map_sub f, sub_eq_zero, map_mul f]
       simp only [hf _] }
 
 @[simp]
@@ -764,7 +786,7 @@ See `RingHom.eq_liftOfRightInverse` for the uniqueness lemma.
 def liftOfRightInverse (hf : Function.RightInverse f_inv f) :
     { g : A →+* C // RingHom.ker f ≤ RingHom.ker g } ≃ (B →+* C) where
   toFun g := f.liftOfRightInverseAux f_inv hf g.1 g.2
-  invFun φ := ⟨φ.comp f, fun x hx => (mem_ker _).mpr <| by simp [(mem_ker _).mp hx]⟩
+  invFun φ := ⟨φ.comp f, fun x hx => mem_ker.mpr <| by simp [mem_ker.mp hx]⟩
   left_inv g := by
     ext
     simp only [comp_apply, liftOfRightInverseAux_comp_apply, Subtype.coe_mk]
@@ -807,6 +829,10 @@ lemma coe_ker : RingHom.ker f = RingHom.ker (f : A →+* B) := rfl
 lemma coe_ideal_map (I : Ideal A) :
     Ideal.map f I = Ideal.map (f : A →+* B) I := rfl
 
+lemma comap_ker {C : Type*} [Semiring C] [Algebra R C] (f : B →ₐ[R] C) (g : A →ₐ[R] B) :
+    (RingHom.ker f).comap g = RingHom.ker (f.comp g) :=
+  RingHom.comap_ker f.toRingHom g.toRingHom
+
 end AlgHom
 
 namespace Algebra
diff --git a/Mathlib/RingTheory/Ideal/Norm.lean b/Mathlib/RingTheory/Ideal/Norm.lean
index b30d0820ffb94..6e1a0aa1de64f 100644
--- a/Mathlib/RingTheory/Ideal/Norm.lean
+++ b/Mathlib/RingTheory/Ideal/Norm.lean
@@ -12,7 +12,7 @@ import Mathlib.LinearAlgebra.FreeModule.Determinant
 import Mathlib.LinearAlgebra.FreeModule.IdealQuotient
 import Mathlib.RingTheory.DedekindDomain.PID
 import Mathlib.RingTheory.Ideal.Basis
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.LocalProperties.Basic
 import Mathlib.RingTheory.Localization.NormTrace
 
 /-!
diff --git a/Mathlib/RingTheory/Ideal/Operations.lean b/Mathlib/RingTheory/Ideal/Operations.lean
index 28847bdce17f9..952cf0b1f19a1 100644
--- a/Mathlib/RingTheory/Ideal/Operations.lean
+++ b/Mathlib/RingTheory/Ideal/Operations.lean
@@ -6,6 +6,8 @@ Authors: Kenny Lau
 import Mathlib.Algebra.Algebra.Operations
 import Mathlib.Data.Fintype.Lattice
 import Mathlib.RingTheory.Coprime.Lemmas
+import Mathlib.RingTheory.NonUnitalSubring.Basic
+import Mathlib.RingTheory.NonUnitalSubsemiring.Basic
 
 /-!
 # More operations on modules and ideals
@@ -606,6 +608,11 @@ theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s
     (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK)
     (by simp_rw [one_eq_top, sup_top_eq]) h
 
+theorem sup_multiset_prod_eq_top {s : Multiset (Ideal R)} (h : ∀  p ∈ s, I ⊔ p = ⊤) :
+    I ⊔ Multiset.prod s = ⊤ :=
+  Multiset.prod_induction (I ⊔ · = ⊤) s (fun _ _ hp hq ↦ (sup_mul_eq_of_coprime_left hp).trans hq)
+    (by simp only [one_eq_top, ge_iff_le, top_le_iff, le_top, sup_of_le_right]) h
+
 theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) :
     (I ⊔ ⨅ i ∈ s, J i) = ⊤ :=
   eq_top_iff.mpr <|
@@ -973,10 +980,37 @@ theorem IsPrime.multiset_prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime)
     s.prod ∈ I ↔ ∃ p ∈ s, p ∈ I := by
   simpa [span_singleton_le_iff_mem] using (hI.multiset_prod_map_le (span {·}))
 
+theorem IsPrime.pow_le_iff {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (hn : n ≠ 0) :
+    I ^ n ≤ P ↔ I ≤ P := by
+  have h : (Multiset.replicate n I).prod ≤ P ↔ _ := hP.multiset_prod_le
+  simp_rw [Multiset.prod_replicate, Multiset.mem_replicate, ne_eq, hn, not_false_eq_true,
+    true_and, exists_eq_left] at h
+  exact h
+
+@[deprecated (since := "2024-10-06")] alias pow_le_prime_iff := IsPrime.pow_le_iff
+
+theorem IsPrime.le_of_pow_le {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
+    I ≤ P := by
+  by_cases hn : n = 0
+  · rw [hn, pow_zero, one_eq_top] at h
+    exact fun ⦃_⦄ _ ↦ h Submodule.mem_top
+  · exact (pow_le_iff hn).mp h
+
+@[deprecated (since := "2024-10-06")] alias le_of_pow_le_prime := IsPrime.le_of_pow_le
+
 theorem IsPrime.prod_le {s : Finset ι} {f : ι → Ideal R} {P : Ideal R} (hp : IsPrime P) :
     s.prod f ≤ P ↔ ∃ i ∈ s, f i ≤ P :=
   hp.multiset_prod_map_le f
 
+@[deprecated (since := "2024-10-06")] alias prod_le_prime := IsPrime.prod_le
+
+/-- The product of a finite number of elements in the commutative semiring `R` lies in the
+  prime ideal `p` if and only if at least one of those elements is in `p`. -/
+theorem IsPrime.prod_mem_iff {s : Finset ι} {x : ι → R} {p : Ideal R} [hp : p.IsPrime] :
+    ∏ i in s, x i ∈ p ↔ ∃ i ∈ s, x i ∈ p := by
+  simp_rw [← span_singleton_le_iff_mem, ← prod_span_singleton]
+  exact hp.prod_le
+
 theorem IsPrime.prod_mem_iff_exists_mem {I : Ideal R} (hI : I.IsPrime) (s : Finset R) :
     s.prod (fun x ↦ x) ∈ I ↔ ∃ p ∈ s, p ∈ I := by
   rw [Finset.prod_eq_multiset_prod, Multiset.map_id']
@@ -1015,7 +1049,7 @@ theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι 
     rw [Finset.card_eq_zero] at hn
     subst hn
     rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h
-    simpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]
+    simpa only [exists_prop, Finset.not_mem_empty, false_and, exists_false, or_false]
   classical
     replace hn : ∃ (i : ι) (t : Finset ι), i ∉ t ∧ insert i t = s ∧ t.card = n :=
       Finset.card_eq_succ.1 hn
@@ -1185,16 +1219,16 @@ variable (ι : Type*)
 variable (M : Type*) [AddCommGroup M] {R : Type*} [CommRing R] [Module R M] (I : Ideal R)
 variable (v : ι → M) (hv : Submodule.span R (Set.range v) = ⊤)
 
-/-- A variant of `Finsupp.total` that takes in vectors valued in `I`. -/
+/-- A variant of `Finsupp.linearCombination` that takes in vectors valued in `I`. -/
 noncomputable def finsuppTotal : (ι →₀ I) →ₗ[R] M :=
-  (Finsupp.total R v).comp (Finsupp.mapRange.linearMap I.subtype)
+  (Finsupp.linearCombination R v).comp (Finsupp.mapRange.linearMap I.subtype)
 
 variable {ι M v}
 
 theorem finsuppTotal_apply (f : ι →₀ I) :
     finsuppTotal ι M I v f = f.sum fun i x => (x : R) • v i := by
   dsimp [finsuppTotal]
-  rw [Finsupp.total_apply, Finsupp.sum_mapRange_index]
+  rw [Finsupp.linearCombination_apply, Finsupp.sum_mapRange_index]
   exact fun _ => zero_smul _ _
 
 theorem finsuppTotal_apply_eq_of_fintype [Fintype ι] (f : ι →₀ I) :
@@ -1209,7 +1243,8 @@ theorem range_finsuppTotal :
   refine ⟨fun ⟨f, h⟩ => ⟨Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h⟩, ?_⟩
   rintro ⟨a, ha, rfl⟩
   classical
-    refine ⟨a.mapRange (fun r => if h : r ∈ I then ⟨r, h⟩ else 0) (by simp), ?_⟩
+    refine ⟨a.mapRange (fun r => if h : r ∈ I then ⟨r, h⟩ else 0)
+      (by simp only [Submodule.zero_mem, ↓reduceDIte]; rfl), ?_⟩
     rw [finsuppTotal_apply, Finsupp.sum_mapRange_index]
     · apply Finsupp.sum_congr
       intro i _
@@ -1273,3 +1308,7 @@ theorem set_smul_top_eq_span (s : Set R) :
     Eq.trans (smul_eq_mul (Ideal R)) (Ideal.mul_top (.span s))
 
 end Submodule
+
+instance {R} [Semiring R] : NonUnitalSubsemiringClass (Ideal R) R where
+  mul_mem _ hb := Ideal.mul_mem_left _ _ hb
+instance {R} [Ring R] : NonUnitalSubringClass (Ideal R) R where
diff --git a/Mathlib/RingTheory/Ideal/QuotientOperations.lean b/Mathlib/RingTheory/Ideal/QuotientOperations.lean
index 7abf1424f8ab5..5fb624f6f2143 100644
--- a/Mathlib/RingTheory/Ideal/QuotientOperations.lean
+++ b/Mathlib/RingTheory/Ideal/QuotientOperations.lean
@@ -36,7 +36,7 @@ This is an isomorphism if `f` has a right inverse (`quotientKerEquivOfRightInver
 is surjective (`quotientKerEquivOfSurjective`).
 -/
 def kerLift : R ⧸ ker f →+* S :=
-  Ideal.Quotient.lift _ f fun _ => f.mem_ker.mp
+  Ideal.Quotient.lift _ f fun _ => mem_ker.mp
 
 @[simp]
 theorem kerLift_mk (r : R) : kerLift f (Ideal.Quotient.mk (ker f) r) = f r :=
@@ -49,7 +49,7 @@ theorem lift_injective_of_ker_le_ideal (I : Ideal R) {f : R →+* S} (H : ∀ a
   obtain ⟨v, rfl⟩ := Ideal.Quotient.mk_surjective u
   rw [Ideal.Quotient.lift_mk] at hu
   rw [Ideal.Quotient.eq_zero_iff_mem]
-  exact hI ((RingHom.mem_ker f).mpr hu)
+  exact hI (RingHom.mem_ker.mpr hu)
 
 /-- The induced map from the quotient by the kernel is injective. -/
 theorem kerLift_injective : Function.Injective (kerLift f) :=
diff --git a/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean b/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
index c84cfb1553f9f..37343ddf2c611 100644
--- a/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
@@ -33,23 +33,28 @@ section
 variable {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
 variable (f : A →ₐ[R] B) (hf : Function.Injective f)
 
+theorem Algebra.IsIntegral.of_injective (hf : Function.Injective f) [Algebra.IsIntegral R B] :
+    Algebra.IsIntegral R A :=
+  ⟨fun _ ↦ (isIntegral_algHom_iff f hf).mp (isIntegral _)⟩
+
 end
 
 instance Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   ⟨LinearMap.exists_monic_and_aeval_eq_zero R⟩
 
-variable (R)
+variable (R) in
 theorem IsIntegral.of_finite [Module.Finite R B] (x : B) : IsIntegral R x :=
   (isIntegral_algHom_iff (Algebra.lmul R B) Algebra.lmul_injective).mp
     (Algebra.IsIntegral.isIntegral _)
 
-variable (B)
+theorem isIntegral_of_noetherian (_ : IsNoetherian R B) (x : B) : IsIntegral R x :=
+  .of_finite R x
+
+variable (R B) in
 instance Algebra.IsIntegral.of_finite [Module.Finite R B] : Algebra.IsIntegral R B :=
   ⟨.of_finite R⟩
 
-variable {R B}
-
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
 theorem IsIntegral.of_mem_of_fg {A} [Ring A] [Algebra R A] (S : Subalgebra R A)
@@ -57,6 +62,61 @@ theorem IsIntegral.of_mem_of_fg {A} [Ring A] [Algebra R A] (S : Subalgebra R A)
   have : Module.Finite R S := ⟨(fg_top _).mpr HS⟩
   (isIntegral_algHom_iff S.val Subtype.val_injective).mpr (.of_finite R (⟨x, hx⟩ : S))
 
+theorem isIntegral_of_submodule_noetherian (S : Subalgebra R B)
+    (H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x :=
+  .of_mem_of_fg _ ((fg_top _).mp <| H.noetherian _) _ hx
+
+/-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
+and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
+theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
+    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
+    (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x := by
+  let A' : Subalgebra R A :=
+    { carrier := { x | ∀ n ∈ N, x • n ∈ N }
+      mul_mem' := fun {a b} ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
+      one_mem' := fun n hn => (one_smul A n).symm ▸ hn
+      add_mem' := fun {a b} ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
+      zero_mem' := fun n _hn => (zero_smul A n).symm ▸ N.zero_mem
+      algebraMap_mem' := fun r n hn => (algebraMap_smul A r n).symm ▸ N.smul_mem r hn }
+  let f : A' →ₐ[R] Module.End R N :=
+    AlgHom.ofLinearMap
+      { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop
+        -- Porting note: was
+                -- `fun x y => LinearMap.ext fun n => Subtype.ext <| add_smul x y n`
+        map_add' := by intros x y; ext; exact add_smul _ _ _
+        -- Porting note: was
+                --  `fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n`
+        map_smul' := by intros r s; ext; apply smul_assoc }
+      -- Porting note: the next two lines were
+      --`(LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>`
+      --`LinearMap.ext fun n => Subtype.ext <| mul_smul x y n`
+      (by ext; apply one_smul)
+      (by intros x y; ext; apply mul_smul)
+  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
+    by_contra! h'
+    apply hN
+    rwa [eq_bot_iff]
+  have : Function.Injective f := by
+    show Function.Injective f.toLinearMap
+    rw [← LinearMap.ker_eq_bot, eq_bot_iff]
+    intro s hs
+    have : s.1 • a = 0 := congr_arg Subtype.val (LinearMap.congr_fun hs ⟨a, ha₁⟩)
+    exact Subtype.ext ((eq_zero_or_eq_zero_of_smul_eq_zero this).resolve_right ha₂)
+  show IsIntegral R (A'.val ⟨x, hx⟩)
+  rw [isIntegral_algHom_iff A'.val Subtype.val_injective, ← isIntegral_algHom_iff f this]
+  haveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top]
+  apply Algebra.IsIntegral.isIntegral
+
+variable {f}
+
+theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
+  letI := f.toAlgebra
+  fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
+
+alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
+
+variable (f)
+
 theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z := by
   letI : Algebra R S := f.toAlgebra
@@ -88,9 +148,23 @@ variable (f : R →+* S)
 theorem RingHom.IsIntegralElem.neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
   hx.of_mem_closure f hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
 
+theorem RingHom.IsIntegralElem.of_neg {x : S} (h : f.IsIntegralElem (-x)) : f.IsIntegralElem x :=
+  neg_neg x ▸ h.neg
+
+@[simp]
+theorem RingHom.IsIntegralElem.neg_iff {x : S} : f.IsIntegralElem (-x) ↔ f.IsIntegralElem x :=
+  ⟨fun h => h.of_neg, fun h => h.neg⟩
+
 theorem IsIntegral.neg {x : B} (hx : IsIntegral R x) : IsIntegral R (-x) :=
   .of_mem_of_fg _ hx.fg_adjoin_singleton _ (Subalgebra.neg_mem _ <| Algebra.subset_adjoin rfl)
 
+theorem IsIntegral.of_neg {x : B} (hx : IsIntegral R (-x)) : IsIntegral R x :=
+  neg_neg x ▸ hx.neg
+
+@[simp]
+theorem IsIntegral.neg_iff {x : B} : IsIntegral R (-x) ↔ IsIntegral R x :=
+  ⟨IsIntegral.of_neg, IsIntegral.neg⟩
+
 theorem RingHom.IsIntegralElem.sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using hx.add f (hy.neg f)
@@ -125,3 +199,7 @@ def integralClosure : Subalgebra R A where
   algebraMap_mem' _ := isIntegral_algebraMap
 
 end
+
+theorem mem_integralClosure_iff (R A : Type*) [CommRing R] [CommRing A] [Algebra R A] {a : A} :
+    a ∈ integralClosure R A ↔ IsIntegral R a :=
+  Iff.rfl
diff --git a/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean b/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
index f0f556d8c02f6..5253fd9dff111 100644
--- a/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
@@ -3,11 +3,11 @@ Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
-import Mathlib.RingTheory.LocalProperties
-import Mathlib.RingTheory.Localization.NormTrace
-import Mathlib.RingTheory.Localization.LocalizationLocalization
 import Mathlib.RingTheory.DedekindDomain.IntegralClosure
+import Mathlib.RingTheory.RingHom.Finite
+import Mathlib.RingTheory.Localization.LocalizationLocalization
+import Mathlib.RingTheory.Localization.NormTrace
+
 /-!
 # Restriction of various maps between fields to integrally closed subrings.
 
diff --git a/Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean b/Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean
index aa5706c755a4c..c987ccc2c77c5 100644
--- a/Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IntegrallyClosed.lean
@@ -3,7 +3,6 @@ Copyright (c) 2021 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
-import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
 import Mathlib.RingTheory.Localization.Integral
 
 /-!
@@ -275,3 +274,10 @@ theorem isIntegrallyClosedOfFiniteExtension [IsDomain R] [FiniteDimensional K L]
   (integralClosure_eq_bot_iff L).mp integralClosure_idem
 
 end integralClosure
+
+/-- Any field is integral closed. -/
+/- Although `infer_instance` can find this if you import Mathlib, in this file they have not been
+  proven yet. However, it is used to prove a fundamental property of `IsIntegrallyClosed`,
+  and it is not desirable to involve more content from other files. -/
+instance Field.instIsIntegrallyClosed (K : Type*) [Field K] : IsIntegrallyClosed K :=
+  (isIntegrallyClosed_iff K).mpr fun {x} _ ↦ ⟨x, rfl⟩
diff --git a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
index 4950bcffb0f99..ab277015154a5 100644
--- a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
@@ -104,3 +104,97 @@ theorem IsIntegral.of_pow [Algebra R B] {x : B} {n : ℕ} (hn : 0 < n) (hx : IsI
   exact ⟨expand R n p, hmonic.expand hn, by rwa [← aeval_def, expand_aeval]⟩
 
 end
+
+section
+
+variable {R A B S : Type*}
+variable [CommRing R] [CommRing A] [Ring B] [CommRing S]
+variable [Algebra R A] [Algebra R B] (f : R →+* S)
+
+theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [Ring S]
+    [CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
+    (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
+    IsIntegral T (ψ a) :=
+  let ⟨p, hp⟩ := ha
+  ⟨p.map φ, hp.1.map _, by
+    rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.2, ψ.map_zero]⟩
+
+@[simp]
+theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
+    (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
+  ⟨fun h ↦ by simpa using h.map f.symm, IsIntegral.map f⟩
+
+/-- If `R → A → B` is an algebra tower,
+then if the entire tower is an integral extension so is `A → B`. -/
+theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
+    (hx : IsIntegral R x) : IsIntegral A x :=
+  let ⟨p, hp, hpx⟩ := hx
+  ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]⟩
+
+/- If `R` and `T` are isomorphic commutative rings and `S` is an `R`-algebra and a `T`-algebra in
+  a compatible way, then an element `a ∈ S` is integral over `R` if and only if it is integral
+  over `T`.-/
+theorem RingEquiv.isIntegral_iff {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
+    [Algebra R S] [Algebra T S] (φ : R ≃+* T)
+    (h : (algebraMap T S).comp φ.toRingHom = algebraMap R S) (a : S) :
+    IsIntegral R a ↔ IsIntegral T a := by
+  constructor <;> intro ha
+  · letI : Algebra R T := φ.toRingHom.toAlgebra
+    letI : IsScalarTower R T S :=
+      ⟨fun r t s ↦ by simp only [Algebra.smul_def, map_mul, ← h, mul_assoc]; rfl⟩
+    exact IsIntegral.tower_top ha
+  · have h' : (algebraMap T S) = (algebraMap R S).comp φ.symm.toRingHom := by
+      simp only [← h, RingHom.comp_assoc, RingEquiv.toRingHom_eq_coe, RingEquiv.comp_symm,
+        RingHomCompTriple.comp_eq]
+    letI : Algebra T R := φ.symm.toRingHom.toAlgebra
+    letI : IsScalarTower T R S :=
+      ⟨fun r t s ↦ by simp only [Algebra.smul_def, map_mul, h', mul_assoc]; rfl⟩
+    exact IsIntegral.tower_top ha
+
+theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B}
+    [FunLike F B C] [RingHomClass F B C] (f : F)
+    (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
+  hb.map (f : B →+* C).toIntAlgHom
+
+theorem IsIntegral.of_subring {x : B} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
+  hx.tower_top
+
+protected theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
+    (h : IsIntegral R x) : IsIntegral R (algebraMap A B x) := by
+  rcases h with ⟨f, hf, hx⟩
+  use f, hf
+  rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
+
+theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
+    (hAB : Function.Injective (algebraMap A B)) :
+    IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
+  isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
+
+theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
+    IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r := by
+  constructor <;> intro hr
+  · rcases hr with ⟨p, hmp, hpr⟩
+    refine ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, ?_⟩
+    rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
+  rcases hr with ⟨s, _, hsr⟩
+  exact hsr.of_subring _
+
+theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
+    (Algebra.adjoin R s).toSubmodule.FG :=
+  Set.Finite.induction_on hfs
+    (fun _ =>
+      ⟨{1},
+        Submodule.ext fun x => by
+          rw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, Algebra.toSubmodule_bot]⟩)
+    (fun {a s} _ _ ih his => by
+      rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
+      exact
+        FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
+          (his a <| Set.mem_insert a s).fg_adjoin_singleton)
+    his
+
+theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
+    (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) :=
+  isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
+
+end
diff --git a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean
index 891b51eb8aa16..a145ced7ef9ab 100644
--- a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Defs.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
 import Mathlib.Algebra.Polynomial.Eval
+import Mathlib.Tactic.Algebraize
 
 /-!
 # Integral closure of a subring.
@@ -35,6 +36,7 @@ def RingHom.IsIntegralElem (f : R →+* A) (x : A) :=
 
 /-- A ring homomorphism `f : R →+* A` is said to be integral
 if every element `A` is integral with respect to the map `f` -/
+@[algebraize Algebra.IsIntegral.mk]
 def RingHom.IsIntegral (f : R →+* A) :=
   ∀ x : A, f.IsIntegralElem x
 
diff --git a/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean b/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
index b1fb223b3f0ba..146fc3d8f12a9 100644
--- a/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
@@ -7,6 +7,7 @@ import Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Defs
 import Mathlib.RingTheory.IntegralClosure.Algebra.Basic
 import Mathlib.RingTheory.FiniteType
 import Mathlib.RingTheory.Polynomial.ScaleRoots
+import Mathlib.LinearAlgebra.FiniteDimensional.Defs
 
 /-!
 # # Integral closure as a characteristic predicate
@@ -22,140 +23,7 @@ section
 
 variable {R A B S : Type*}
 variable [CommRing R] [CommRing A] [Ring B] [CommRing S]
-variable [Algebra R A] [Algebra R B] (f : R →+* S)
-
-theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [Ring S]
-    [CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
-    (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
-    IsIntegral T (ψ a) :=
-  let ⟨p, hp⟩ := ha
-  ⟨p.map φ, hp.1.map _, by
-    rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.2, ψ.map_zero]⟩
-
-section
-
-variable {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
-variable (f : A →ₐ[R] B)
-
-theorem Algebra.IsIntegral.of_injective (hf : Function.Injective f) [Algebra.IsIntegral R B] :
-    Algebra.IsIntegral R A :=
-  ⟨fun _ ↦ (isIntegral_algHom_iff f hf).mp (isIntegral _)⟩
-
-end
-
-@[simp]
-theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
-    (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
-  ⟨fun h ↦ by simpa using h.map f.symm, IsIntegral.map f⟩
-
-/-- If `R → A → B` is an algebra tower,
-then if the entire tower is an integral extension so is `A → B`. -/
-theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
-    (hx : IsIntegral R x) : IsIntegral A x :=
-  let ⟨p, hp, hpx⟩ := hx
-  ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]⟩
-
-theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B}
-    [FunLike F B C] [RingHomClass F B C] (f : F)
-    (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
-  hb.map (f : B →+* C).toIntAlgHom
-
-theorem IsIntegral.of_subring {x : B} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
-  hx.tower_top
-
-protected theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
-    (h : IsIntegral R x) : IsIntegral R (algebraMap A B x) := by
-  rcases h with ⟨f, hf, hx⟩
-  use f, hf
-  rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
-
-theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
-    (hAB : Function.Injective (algebraMap A B)) :
-    IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
-  isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
-
-theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
-    IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r := by
-  constructor <;> intro hr
-  · rcases hr with ⟨p, hmp, hpr⟩
-    refine ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, ?_⟩
-    rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
-  rcases hr with ⟨s, _, hsr⟩
-  exact hsr.of_subring _
-
-theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
-    (Algebra.adjoin R s).toSubmodule.FG :=
-  Set.Finite.induction_on hfs
-    (fun _ =>
-      ⟨{1},
-        Submodule.ext fun x => by
-          rw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, Algebra.toSubmodule_bot]⟩)
-    (fun {a s} _ _ ih his => by
-      rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
-      exact
-        FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-          (his a <| Set.mem_insert a s).fg_adjoin_singleton)
-    his
-
-theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
-    (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) :=
-  isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
-
-theorem isIntegral_of_noetherian (_ : IsNoetherian R B) (x : B) : IsIntegral R x :=
-  .of_finite R x
-
-theorem isIntegral_of_submodule_noetherian (S : Subalgebra R B)
-    (H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x :=
-  .of_mem_of_fg _ ((fg_top _).mp <| H.noetherian _) _ hx
-
-/-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
-and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
-theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
-    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
-    (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x := by
-  let A' : Subalgebra R A :=
-    { carrier := { x | ∀ n ∈ N, x • n ∈ N }
-      mul_mem' := fun {a b} ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
-      one_mem' := fun n hn => (one_smul A n).symm ▸ hn
-      add_mem' := fun {a b} ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
-      zero_mem' := fun n _hn => (zero_smul A n).symm ▸ N.zero_mem
-      algebraMap_mem' := fun r n hn => (algebraMap_smul A r n).symm ▸ N.smul_mem r hn }
-  let f : A' →ₐ[R] Module.End R N :=
-    AlgHom.ofLinearMap
-      { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop
-        -- Porting note: was
-                -- `fun x y => LinearMap.ext fun n => Subtype.ext <| add_smul x y n`
-        map_add' := by intros x y; ext; exact add_smul _ _ _
-        -- Porting note: was
-                --  `fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n`
-        map_smul' := by intros r s; ext; apply smul_assoc }
-      -- Porting note: the next two lines were
-      --`(LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>`
-      --`LinearMap.ext fun n => Subtype.ext <| mul_smul x y n`
-      (by ext; apply one_smul)
-      (by intros x y; ext; apply mul_smul)
-  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
-    by_contra! h'
-    apply hN
-    rwa [eq_bot_iff]
-  have : Function.Injective f := by
-    show Function.Injective f.toLinearMap
-    rw [← LinearMap.ker_eq_bot, eq_bot_iff]
-    intro s hs
-    have : s.1 • a = 0 := congr_arg Subtype.val (LinearMap.congr_fun hs ⟨a, ha₁⟩)
-    exact Subtype.ext ((eq_zero_or_eq_zero_of_smul_eq_zero this).resolve_right ha₂)
-  show IsIntegral R (A'.val ⟨x, hx⟩)
-  rw [isIntegral_algHom_iff A'.val Subtype.val_injective, ← isIntegral_algHom_iff f this]
-  haveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top]
-  apply Algebra.IsIntegral.isIntegral
-
-variable {f}
-
-theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
-  letI := f.toAlgebra
-  fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
-
-alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
+variable [Algebra R A] [Algebra R B] {f : R →+* S}
 
 /-- The [Kurosh problem](https://en.wikipedia.org/wiki/Kurosh_problem) asks to show that
   this is still true when `A` is not necessarily commutative and `R` is a field, but it has
@@ -527,14 +395,14 @@ variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A'
 /-- Integral closures are all isomorphic to each other. -/
 noncomputable def equiv : A ≃ₐ[R] A' :=
   AlgEquiv.ofAlgHom
-    (lift _ B (isIntegral := isIntegral_algebra R B))
-    (lift _ B (isIntegral := isIntegral_algebra R B))
+    (lift R A' B (isIntegral := isIntegral_algebra R B))
+    (lift R A B (isIntegral := isIntegral_algebra R B))
     (by ext x; apply algebraMap_injective A' R B; simp)
     (by ext x; apply algebraMap_injective A R B; simp)
 
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
-  algebraMap_lift A' B (isIntegral := isIntegral_algebra R B) x
+  algebraMap_lift R A' B (isIntegral := isIntegral_algebra R B) x
 
 end Equiv
 
@@ -693,7 +561,7 @@ theorem Algebra.IsIntegral.isField_iff_isField {R S : Type*} [CommRing R]
 end Algebra
 
 theorem integralClosure_idem {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
-    integralClosure (integralClosure R A : Set A) A = ⊥ :=
+    integralClosure (integralClosure R A) A = ⊥ :=
   letI := (integralClosure R A).algebra
   eq_bot_iff.2 fun x hx ↦ Algebra.mem_bot.2
     ⟨⟨x, isIntegral_trans (A := integralClosure R A) x hx⟩, rfl⟩
diff --git a/Mathlib/RingTheory/IntegralDomain.lean b/Mathlib/RingTheory/IntegralDomain.lean
index 544f816a2f202..2d5e9ee2a2c51 100644
--- a/Mathlib/RingTheory/IntegralDomain.lean
+++ b/Mathlib/RingTheory/IntegralDomain.lean
@@ -54,7 +54,7 @@ def Fintype.groupWithZeroOfCancel (M : Type*) [CancelMonoidWithZero M] [Decidabl
     inv_zero := by simp [Inv.inv, dif_pos rfl] }
 
 theorem exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [CommSemiring R] [IsDomain R]
-    [GCDMonoid R] [Unique Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n) :
+    [GCDMonoid R] [Subsingleton Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n) :
     ∃ d : R, a = d ^ n := by
   refine exists_eq_pow_of_mul_eq_pow (isUnit_of_dvd_one ?_) h
   obtain ⟨x, y, hxy⟩ := cp
@@ -65,7 +65,7 @@ theorem exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [CommSemiring R] [IsD
 
 nonrec
 theorem Finset.exists_eq_pow_of_mul_eq_pow_of_coprime {ι R : Type*} [CommSemiring R] [IsDomain R]
-    [GCDMonoid R] [Unique Rˣ] {n : ℕ} {c : R} {s : Finset ι} {f : ι → R}
+    [GCDMonoid R] [Subsingleton Rˣ] {n : ℕ} {c : R} {s : Finset ι} {f : ι → R}
     (h : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → IsCoprime (f i) (f j))
     (hprod : ∏ i ∈ s, f i = c ^ n) : ∀ i ∈ s, ∃ d : R, f i = d ^ n := by
   classical
diff --git a/Mathlib/RingTheory/IsAdjoinRoot.lean b/Mathlib/RingTheory/IsAdjoinRoot.lean
index 4057eec806f6f..1b153b9e33498 100644
--- a/Mathlib/RingTheory/IsAdjoinRoot.lean
+++ b/Mathlib/RingTheory/IsAdjoinRoot.lean
@@ -121,10 +121,10 @@ theorem subsingleton (h : IsAdjoinRoot S f) [Subsingleton R] : Subsingleton S :=
 theorem algebraMap_apply (h : IsAdjoinRoot S f) (x : R) :
     algebraMap R S x = h.map (Polynomial.C x) := by rw [h.algebraMap_eq, RingHom.comp_apply]
 
-@[simp]
 theorem mem_ker_map (h : IsAdjoinRoot S f) {p} : p ∈ RingHom.ker h.map ↔ f ∣ p := by
   rw [h.ker_map, Ideal.mem_span_singleton]
 
+@[simp]
 theorem map_eq_zero_iff (h : IsAdjoinRoot S f) {p} : h.map p = 0 ↔ f ∣ p := by
   rw [← h.mem_ker_map, RingHom.mem_ker]
 
@@ -393,12 +393,10 @@ def basis (h : IsAdjoinRootMonic S f) : Basis (Fin (natDegree f)) R S :=
         ext i
         simp only [h.modByMonicHom_map, Finsupp.comapDomain_apply, Polynomial.toFinsupp_apply]
         rw [(Polynomial.modByMonic_eq_self_iff h.Monic).mpr, Polynomial.coeff]
-        · dsimp only -- Porting note (#10752): added `dsimp only`
-          rw [Finsupp.mapDomain_apply Fin.val_injective]
+        · rw [Finsupp.mapDomain_apply Fin.val_injective]
         rw [degree_eq_natDegree h.Monic.ne_zero, degree_lt_iff_coeff_zero]
         intro m hm
         rw [Polynomial.coeff]
-        dsimp only -- Porting note (#10752): added `dsimp only`
         rw [Finsupp.mapDomain_notin_range]
         rw [Set.mem_range, not_exists]
         rintro i rfl
diff --git a/Mathlib/RingTheory/IsTensorProduct.lean b/Mathlib/RingTheory/IsTensorProduct.lean
index 942b0ab45ddf4..b02662937d43f 100644
--- a/Mathlib/RingTheory/IsTensorProduct.lean
+++ b/Mathlib/RingTheory/IsTensorProduct.lean
@@ -318,11 +318,40 @@ theorem IsBaseChange.comp {f : M →ₗ[R] N} (hf : IsBaseChange S f) {g : N →
   ext
   rfl
 
+/-- If `N` is the base change of `M` to `S` and `O` the base change of `M` to `T`, then
+`O` is the base change of `N` to `T`. -/
+lemma IsBaseChange.of_comp {f : M →ₗ[R] N} (hf : IsBaseChange S f) {h : N →ₗ[S] O}
+    (hc : IsBaseChange T ((h : N →ₗ[R] O) ∘ₗ f)) :
+    IsBaseChange T h := by
+  apply IsBaseChange.of_lift_unique
+  intro Q _ _ _ _ r
+  letI : Module R Q := inferInstanceAs (Module R (RestrictScalars R S Q))
+  haveI : IsScalarTower R S Q := IsScalarTower.of_algebraMap_smul fun r ↦ congrFun rfl
+  haveI : IsScalarTower R T Q := IsScalarTower.of_algebraMap_smul fun r x ↦ by
+    simp [IsScalarTower.algebraMap_apply R S T]
+  let r' : M →ₗ[R] Q := r ∘ₗ f
+  let q : O →ₗ[T] Q := hc.lift r'
+  refine ⟨q, ?_, ?_⟩
+  · apply hf.algHom_ext'
+    simp [LinearMap.comp_assoc, hc.lift_comp]
+  · intro q' hq'
+    apply hc.algHom_ext'
+    apply_fun LinearMap.restrictScalars R at hq'
+    rw [← LinearMap.comp_assoc]
+    rw [show q'.restrictScalars R ∘ₗ h.restrictScalars R = _ from hq', hc.lift_comp]
+
+/-- If `N` is the base change `M` to `S`, then `O` is the base change of `M` to `T` if and
+only if `O` is the base change of `N` to `T`. -/
+lemma IsBaseChange.comp_iff {f : M →ₗ[R] N} (hf : IsBaseChange S f) {h : N →ₗ[S] O} :
+    IsBaseChange T ((h : N →ₗ[R] O) ∘ₗ f) ↔ IsBaseChange T h :=
+  ⟨fun hc ↦ IsBaseChange.of_comp hf hc, fun hh ↦ IsBaseChange.comp hf hh⟩
+
 variable {R' S' : Type*} [CommSemiring R'] [CommSemiring S']
 variable [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S']
 variable [IsScalarTower R R' S'] [IsScalarTower R S S']
 
 open IsScalarTower (toAlgHom)
+open IsScalarTower (algebraMap_apply)
 
 variable (R S R' S')
 
@@ -424,7 +453,7 @@ noncomputable def Algebra.pushoutDesc [H : Algebra.IsPushout R S R' S'] {A : Typ
       rw [mul_add, map_add, map_add, mul_add, e₁, e₂]
 
 @[simp]
-theorem Algebra.pushoutDesc_left [H : Algebra.IsPushout R S R' S'] {A : Type*} [Semiring A]
+theorem Algebra.pushoutDesc_left [Algebra.IsPushout R S R' S'] {A : Type*} [Semiring A]
     [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) (x : S) :
     Algebra.pushoutDesc S' f g H (algebraMap S S' x) = f x := by
   letI := Module.compHom A f.toRingHom
@@ -442,7 +471,7 @@ theorem Algebra.lift_algHom_comp_left [Algebra.IsPushout R S R' S'] {A : Type*}
   AlgHom.ext fun x => (Algebra.pushoutDesc_left S' f g H x : _)
 
 @[simp]
-theorem Algebra.pushoutDesc_right [H : Algebra.IsPushout R S R' S'] {A : Type*} [Semiring A]
+theorem Algebra.pushoutDesc_right [Algebra.IsPushout R S R' S'] {A : Type*} [Semiring A]
     [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H) (x : R') :
     Algebra.pushoutDesc S' f g H (algebraMap R' S' x) = g x :=
   letI := Module.compHom A f.toRingHom
@@ -471,4 +500,33 @@ theorem Algebra.IsPushout.algHom_ext [H : Algebra.IsPushout R S R' S'] {A : Type
   · intro s₁ s₂ e₁ e₂
     rw [map_add, map_add, e₁, e₂]
 
+/--
+Let the following be a commutative diagram of rings
+```
+  R  →  S  →  T
+  ↓     ↓     ↓
+  R' →  S' →  T'
+```
+where the left-hand square is a pushout. Then the following are equivalent:
+- the big rectangle is a pushout.
+- the right-hand square is a pushout.
+
+Note that this is essentially the isomorphism `T ⊗[S] (S ⊗[R] R') ≃ₐ[T] T ⊗[R] R'`.
+-/
+lemma Algebra.IsPushout.comp_iff {T' : Type*} [CommRing T'] [Algebra R T']
+    [Algebra S' T'] [Algebra S T'] [Algebra T T'] [Algebra R' T']
+    [IsScalarTower R T T'] [IsScalarTower S T T'] [IsScalarTower S S' T']
+    [IsScalarTower R R' T'] [IsScalarTower R S' T'] [IsScalarTower R' S' T']
+    [Algebra.IsPushout R S R' S'] :
+    Algebra.IsPushout R T R' T' ↔ Algebra.IsPushout S T S' T' := by
+  let f : R' →ₗ[R] S' := (IsScalarTower.toAlgHom R R' S').toLinearMap
+  haveI : IsScalarTower R S T' := IsScalarTower.of_algebraMap_eq <| fun x ↦ by
+    rw [algebraMap_apply R S' T', algebraMap_apply R S S', ← algebraMap_apply S S' T']
+  have heq : (toAlgHom S S' T').toLinearMap.restrictScalars R ∘ₗ f =
+      (toAlgHom R R' T').toLinearMap := by
+    ext x
+    simp [f, ← IsScalarTower.algebraMap_apply]
+  rw [isPushout_iff, isPushout_iff, ← heq, IsBaseChange.comp_iff]
+  exact Algebra.IsPushout.out
+
 end IsBaseChange
diff --git a/Mathlib/RingTheory/Jacobson.lean b/Mathlib/RingTheory/Jacobson.lean
index f9ca45cdd5b0b..29cd5339dde32 100644
--- a/Mathlib/RingTheory/Jacobson.lean
+++ b/Mathlib/RingTheory/Jacobson.lean
@@ -127,7 +127,7 @@ theorem isJacobson_of_isIntegral [Algebra R S] [Algebra.IsIntegral R S] (hR : Is
       ((isJacobson_iff_prime_eq.1 hR) (comap (algebraMap R S) P) (comap_isPrime _ _)),
       comap_jacobson]
     refine sInf_le_sInf fun J hJ => ?_
-    simp only [true_and_iff, Set.mem_image, bot_le, Set.mem_setOf_eq]
+    simp only [true_and, Set.mem_image, bot_le, Set.mem_setOf_eq]
     have : J.IsMaximal := by simpa using hJ
     exact exists_ideal_over_maximal_of_isIntegral J
       (comap_bot_le_of_injective _ algebraMap_quotient_injective)
diff --git a/Mathlib/RingTheory/JacobsonIdeal.lean b/Mathlib/RingTheory/JacobsonIdeal.lean
index c51eef801b35e..5f513287fa395 100644
--- a/Mathlib/RingTheory/JacobsonIdeal.lean
+++ b/Mathlib/RingTheory/JacobsonIdeal.lean
@@ -1,11 +1,12 @@
 /-
 Copyright (c) 2020 Devon Tuma. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kenny Lau, Devon Tuma
+Authors: Kenny Lau, Devon Tuma, Wojciech Nawrocki
 -/
 import Mathlib.RingTheory.Ideal.IsPrimary
 import Mathlib.RingTheory.Ideal.Quotient
 import Mathlib.RingTheory.Polynomial.Quotient
+import Mathlib.RingTheory.TwoSidedIdeal.Operations
 
 /-!
 # Jacobson radical
@@ -13,20 +14,24 @@ import Mathlib.RingTheory.Polynomial.Quotient
 The Jacobson radical of a ring `R` is defined to be the intersection of all maximal ideals of `R`.
 This is similar to how the nilradical is equal to the intersection of all prime ideals of `R`.
 
-We can extend the idea of the nilradical to ideals of `R`,
-by letting the radical of an ideal `I` be the intersection of prime ideals containing `I`.
+We can extend the idea of the nilradical of `R` to ideals of `R`,
+by letting the nilradical of an ideal `I` be the intersection of prime ideals containing `I`.
 Under this extension, the original nilradical is the radical of the zero ideal `⊥`.
 Here we define the Jacobson radical of an ideal `I` in a similar way,
 as the intersection of maximal ideals containing `I`.
 
 ## Main definitions
 
-Let `R` be a commutative ring, and `I` be an ideal of `R`
+Let `R` be a ring, and `I` be a left ideal of `R`
 
-* `Ideal.jacobson I` is the jacobson radical, i.e. the infimum of all maximal ideals containing I.
+* `Ideal.jacobson I` is the Jacobson radical, i.e. the infimum of all maximal ideals containing `I`.
 
 * `Ideal.IsLocal I` is the proposition that the jacobson radical of `I` is itself a maximal ideal
 
+Furthermore when `I` is a two-sided ideal of `R`
+
+* `TwoSidedIdeal.jacobson I` is the Jacobson radical as a two-sided ideal
+
 ## Main statements
 
 * `mem_jacobson_iff` gives a characterization of members of the jacobson of I
@@ -111,11 +116,17 @@ theorem mem_jacobson_iff {x : R} : x ∈ jacobson I ↔ ∀ y, ∃ z, z * y * x
             sub_add_cancel]
           exact M.mul_mem_left _ hi) <| him hz⟩
 
-theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
-    ∃ s, s * r - 1 ∈ I := by
+theorem exists_mul_add_sub_mem_of_mem_jacobson {I : Ideal R} (r : R) (h : r ∈ jacobson I) :
+    ∃ s, s * (r + 1) - 1 ∈ I := by
   cases' mem_jacobson_iff.1 h 1 with s hs
   use s
-  simpa [mul_sub] using hs
+  rw [mul_add, mul_one]
+  simpa using hs
+
+theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
+    ∃ s, s * r - 1 ∈ I := by
+  convert exists_mul_add_sub_mem_of_mem_jacobson _ h
+  simp
 
 /-- An ideal equals its Jacobson radical iff it is the intersection of a set of maximal ideals.
 Allowing the set to include ⊤ is equivalent, and is included only to simplify some proofs. -/
@@ -214,6 +225,44 @@ theorem jacobson_mono {I J : Ideal R} : I ≤ J → I.jacobson ≤ J.jacobson :=
   erw [mem_sInf] at hx ⊢
   exact fun K ⟨hK, hK_max⟩ => hx ⟨Trans.trans h hK, hK_max⟩
 
+/-- The Jacobson radical of a two-sided ideal is two-sided.
+
+It is preferable to use `TwoSidedIdeal.jacobson` instead of this lemma. -/
+theorem jacobson_mul_mem_right {I : Ideal R}
+    (mul_mem_right : ∀ {x y}, x ∈ I → x * y ∈ I) :
+    ∀ {x y}, x ∈ I.jacobson → x * y ∈ I.jacobson := by
+  -- Proof generalized from
+  -- https://ysharifi.wordpress.com/2022/08/16/the-jacobson-radical-definition-and-basic-results/
+  intro x r xJ
+  apply mem_sInf.mpr
+  intro 𝔪 𝔪_mem
+  by_cases r𝔪 : r ∈ 𝔪
+  · apply 𝔪.smul_mem _ r𝔪
+  -- 𝔪₀ := { a : R | a*r ∈ 𝔪 }
+  let 𝔪₀ : Ideal R := Submodule.comap (DistribMulAction.toLinearMap R (S := Rᵐᵒᵖ) R (.op r)) 𝔪
+  suffices x ∈ 𝔪₀ by simpa [𝔪₀] using this
+  have I𝔪₀ : I ≤ 𝔪₀ := fun i iI =>
+    𝔪_mem.left (mul_mem_right iI)
+  have 𝔪₀_maximal : IsMaximal 𝔪₀ := by
+    refine isMaximal_iff.mpr ⟨
+      fun h => r𝔪 (by simpa [𝔪₀] using h),
+      fun J b 𝔪₀J b𝔪₀ bJ => ?_⟩
+    let K : Ideal R := Ideal.span {b*r} ⊔ 𝔪
+    have ⟨s, y, y𝔪, sbyr⟩ :=
+      mem_span_singleton_sup.mp <|
+        mul_mem_left _ r <|
+          (isMaximal_iff.mp 𝔪_mem.right).right K (b*r)
+          le_sup_right b𝔪₀
+          (mem_sup_left <| mem_span_singleton_self _)
+    have : 1 - s*b ∈ 𝔪₀ := by
+      rw [mul_one, add_comm, ← eq_sub_iff_add_eq] at sbyr
+      rw [sbyr, ← mul_assoc] at y𝔪
+      simp [𝔪₀, sub_mul, y𝔪]
+    have : 1 - s*b + s*b ∈ J := by
+      apply add_mem (𝔪₀J this) (J.mul_mem_left _ bJ)
+    simpa using this
+  exact mem_sInf.mp xJ ⟨I𝔪₀, 𝔪₀_maximal⟩
+
 end Ring
 
 section CommRing
@@ -375,3 +424,16 @@ theorem isPrimary_of_isMaximal_radical [CommRing R] {I : Ideal R} (hi : IsMaxima
       (this ▸ id)⟩
 
 end Ideal
+
+namespace TwoSidedIdeal
+
+variable {R : Type u} [Ring R]
+
+/-- The Jacobson radical of `I` is the infimum of all maximal (left) ideals containing `I`. -/
+def jacobson (I : TwoSidedIdeal R) : TwoSidedIdeal R :=
+  (asIdeal I).jacobson.toTwoSided (Ideal.jacobson_mul_mem_right <| I.mul_mem_right _ _)
+
+lemma asIdeal_jacobson (I : TwoSidedIdeal R) : asIdeal I.jacobson = (asIdeal I).jacobson := by
+  ext; simp [jacobson]
+
+end TwoSidedIdeal
diff --git a/Mathlib/RingTheory/Kaehler/Basic.lean b/Mathlib/RingTheory/Kaehler/Basic.lean
index 7ebe02604993f..09bc7c6dc0c45 100644
--- a/Mathlib/RingTheory/Kaehler/Basic.lean
+++ b/Mathlib/RingTheory/Kaehler/Basic.lean
@@ -49,7 +49,7 @@ suppress_compilation
 section KaehlerDifferential
 
 open scoped TensorProduct
-open Algebra
+open Algebra Finsupp
 
 universe u v
 
@@ -320,7 +320,7 @@ theorem KaehlerDifferential.tensorProductTo_surjective :
   exact ⟨x, KaehlerDifferential.D_tensorProductTo x⟩
 
 /-- The `S`-linear maps from `Ω[S⁄R]` to `M` are (`S`-linearly) equivalent to `R`-derivations
-from `S` to `M`.  -/
+from `S` to `M`. -/
 @[simps! symm_apply apply_apply]
 def KaehlerDifferential.linearMapEquivDerivation : (Ω[S⁄R] →ₗ[S] M) ≃ₗ[S] Derivation R S M :=
   { Derivation.llcomp.flip <| KaehlerDifferential.D R S with
@@ -499,8 +499,9 @@ the relations:
 3. `dr = 0` for `r ∈ R`
 where `db` is the unit in the copy of `S` with index `b`.
 
-This is the kernel of the surjection `Finsupp.total S Ω[S⁄R] S (KaehlerDifferential.D R S)`.
-See `KaehlerDifferential.kerTotal_eq` and `KaehlerDifferential.total_surjective`.
+This is the kernel of the surjection
+`Finsupp.linearCombination S Ω[S⁄R] S (KaehlerDifferential.D R S)`.
+See `KaehlerDifferential.kerTotal_eq` and `KaehlerDifferential.linearCombination_surjective`.
 -/
 noncomputable def KaehlerDifferential.kerTotal : Submodule S (S →₀ S) :=
   Submodule.span S
@@ -555,46 +556,55 @@ theorem KaehlerDifferential.derivationQuotKerTotal_apply (x) :
     KaehlerDifferential.derivationQuotKerTotal R S x = 1𝖣x :=
   rfl
 
-theorem KaehlerDifferential.derivationQuotKerTotal_lift_comp_total :
+theorem KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination :
     (KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential.comp
-        (Finsupp.total S (KaehlerDifferential.D R S)) =
+        (Finsupp.linearCombination S (KaehlerDifferential.D R S)) =
       Submodule.mkQ _ := by
   apply Finsupp.lhom_ext
   intro a b
   conv_rhs => rw [← Finsupp.smul_single_one a b, LinearMap.map_smul]
   simp [KaehlerDifferential.derivationQuotKerTotal_apply]
 
+@[deprecated (since := "2024-08-29")] alias
+  KaehlerDifferential.derivationQuotKerTotal_lift_comp_total :=
+  KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination
+
 theorem KaehlerDifferential.kerTotal_eq :
-    LinearMap.ker (Finsupp.total S (KaehlerDifferential.D R S)) =
+    LinearMap.ker (Finsupp.linearCombination S (KaehlerDifferential.D R S)) =
       KaehlerDifferential.kerTotal R S := by
   apply le_antisymm
   · conv_rhs => rw [← (KaehlerDifferential.kerTotal R S).ker_mkQ]
-    rw [← KaehlerDifferential.derivationQuotKerTotal_lift_comp_total]
+    rw [← KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination]
     exact LinearMap.ker_le_ker_comp _ _
   · rw [KaehlerDifferential.kerTotal, Submodule.span_le]
     rintro _ ((⟨⟨x, y⟩, rfl⟩ | ⟨⟨x, y⟩, rfl⟩) | ⟨x, rfl⟩) <;> dsimp <;> simp [LinearMap.mem_ker]
 
-theorem KaehlerDifferential.total_surjective :
-    Function.Surjective (Finsupp.total S (KaehlerDifferential.D R S)) := by
-  rw [← LinearMap.range_eq_top, Finsupp.range_total, KaehlerDifferential.span_range_derivation]
+theorem KaehlerDifferential.linearCombination_surjective :
+    Function.Surjective (Finsupp.linearCombination S (KaehlerDifferential.D R S)) := by
+  rw [← LinearMap.range_eq_top, range_linearCombination, KaehlerDifferential.span_range_derivation]
+
+@[deprecated (since := "2024-08-29")] alias KaehlerDifferential.total_surjective :=
+  KaehlerDifferential.linearCombination_surjective
 
 /-- `Ω[S⁄R]` is isomorphic to `S` copies of `S` with kernel `KaehlerDifferential.kerTotal`. -/
 @[simps!]
 noncomputable def KaehlerDifferential.quotKerTotalEquiv :
     ((S →₀ S) ⧸ KaehlerDifferential.kerTotal R S) ≃ₗ[S] Ω[S⁄R] :=
   { (KaehlerDifferential.kerTotal R S).liftQ
-      (Finsupp.total S (KaehlerDifferential.D R S))
+      (Finsupp.linearCombination S (KaehlerDifferential.D R S))
       (KaehlerDifferential.kerTotal_eq R S).ge with
     invFun := (KaehlerDifferential.derivationQuotKerTotal R S).liftKaehlerDifferential
     left_inv := by
       intro x
       obtain ⟨x, rfl⟩ := Submodule.mkQ_surjective _ x
       exact
-        LinearMap.congr_fun (KaehlerDifferential.derivationQuotKerTotal_lift_comp_total R S : _) x
+        LinearMap.congr_fun
+          (KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination R S : _) x
     right_inv := by
       intro x
-      obtain ⟨x, rfl⟩ := KaehlerDifferential.total_surjective R S x
-      have := LinearMap.congr_fun (KaehlerDifferential.derivationQuotKerTotal_lift_comp_total R S) x
+      obtain ⟨x, rfl⟩ := KaehlerDifferential.linearCombination_surjective R S x
+      have := LinearMap.congr_fun
+        (KaehlerDifferential.derivationQuotKerTotal_lift_comp_linearCombination R S) x
       rw [LinearMap.comp_apply] at this
       rw [this]
       rfl }
@@ -704,13 +714,13 @@ theorem KaehlerDifferential.map_D (x : A) :
 theorem KaehlerDifferential.ker_map :
     LinearMap.ker (KaehlerDifferential.map R S A B) =
       (((kerTotal S B).restrictScalars A).comap finsupp_map).map
-        (Finsupp.total (M := Ω[A⁄R]) A (D R A)) := by
-  rw [← Submodule.map_comap_eq_of_surjective (total_surjective R A) (LinearMap.ker _)]
+        (Finsupp.linearCombination (M := Ω[A⁄R]) A (D R A)) := by
+  rw [← Submodule.map_comap_eq_of_surjective (linearCombination_surjective R A) (LinearMap.ker _)]
   congr 1
   ext x
-  simp only [Submodule.mem_comap, LinearMap.mem_ker, Finsupp.apply_total, ← kerTotal_eq,
+  simp only [Submodule.mem_comap, LinearMap.mem_ker, Finsupp.apply_linearCombination, ← kerTotal_eq,
     Submodule.restrictScalars_mem]
-  simp only [Finsupp.total_apply, Function.comp_apply, LinearMap.coe_comp, Finsupp.lmapDomain_apply,
+  simp only [linearCombination_apply, Function.comp_apply, LinearMap.coe_comp, lmapDomain_apply,
     Finsupp.mapRange.linearMap_apply]
   rw [Finsupp.sum_mapRange_index, Finsupp.sum_mapDomain_index]
   · simp [ofId]
@@ -720,10 +730,10 @@ theorem KaehlerDifferential.ker_map :
 
 lemma KaehlerDifferential.ker_map_of_surjective (h : Function.Surjective (algebraMap A B)) :
     LinearMap.ker (map R R A B) =
-      (LinearMap.ker finsupp_map).map (Finsupp.total A (D R A)) := by
+      (LinearMap.ker finsupp_map).map (Finsupp.linearCombination A (D R A)) := by
   rw [ker_map, ← kerTotal_map' R A B h, Submodule.comap_map_eq, Submodule.map_sup,
     Submodule.map_sup, ← kerTotal_eq, ← Submodule.comap_bot,
-    Submodule.map_comap_eq_of_surjective (total_surjective _ _),
+    Submodule.map_comap_eq_of_surjective (linearCombination_surjective _ _),
     bot_sup_eq, Submodule.map_span, ← Set.range_comp]
   convert bot_sup_eq _
   rw [Submodule.span_eq_bot]; simp
@@ -763,14 +773,14 @@ lemma KaehlerDifferential.range_mapBaseChange :
   · rintro _ ⟨x, rfl⟩
     induction' x with r s
     · simp
-    · obtain ⟨x, rfl⟩ := total_surjective _ _ s
+    · obtain ⟨x, rfl⟩ := linearCombination_surjective _ _ s
       simp only [mapBaseChange_tmul, LinearMap.mem_ker, map_smul]
       induction x using Finsupp.induction_linear
       · simp
       · simp [smul_add, *]
       · simp
     · rw [map_add]; exact add_mem ‹_› ‹_›
-  · convert_to (kerTotal A B).map (Finsupp.total B (D R B)) ≤ _
+  · convert_to (kerTotal A B).map (Finsupp.linearCombination B (D R B)) ≤ _
     · rw [KaehlerDifferential.ker_map]
       congr 1
       convert Submodule.comap_id _
@@ -795,11 +805,11 @@ noncomputable
 def KaehlerDifferential.kerToTensor :
     RingHom.ker (algebraMap A B) →ₗ[A] B ⊗[A] Ω[A⁄R] where
   toFun x := 1 ⊗ₜ D R A x
-  map_add' x y := by simp only [AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, map_add,
-    TensorProduct.tmul_add]
-  map_smul' r x := by simp only [SetLike.val_smul, Derivation.leibniz, RingHom.id_apply,
-    TensorProduct.tmul_smul, TensorProduct.smul_tmul', add_zero, ← Algebra.algebraMap_eq_smul_one,
-    TensorProduct.zero_tmul, smul_eq_mul, TensorProduct.tmul_add, (RingHom.mem_ker _).mp x.prop]
+  map_add' x y := by simp only [Submodule.coe_add, map_add, TensorProduct.tmul_add]
+  map_smul' r x := by simp only [SetLike.val_smul, smul_eq_mul, Derivation.leibniz,
+    TensorProduct.tmul_add, TensorProduct.tmul_smul, TensorProduct.smul_tmul', ←
+    algebraMap_eq_smul_one, RingHom.mem_ker.mp x.prop, TensorProduct.zero_tmul, add_zero,
+    RingHom.id_apply]
 
 /-- The map `I/I² → B ⊗[A] B ⊗[A] Ω[A⁄R]` where `I = ker(A → B)`. -/
 noncomputable
@@ -810,7 +820,7 @@ def KaehlerDifferential.kerCotangentToTensor :
     rintro x hx y -
     simp only [Submodule.mem_comap, LinearMap.lsmul_apply, LinearMap.mem_ker, map_smul,
       kerToTensor_apply, TensorProduct.smul_tmul', ← algebraMap_eq_smul_one,
-      (RingHom.mem_ker _).mp hx, TensorProduct.zero_tmul]))
+      RingHom.mem_ker.mp hx, TensorProduct.zero_tmul]))
 
 @[simp]
 lemma KaehlerDifferential.kerCotangentToTensor_toCotangent (x) :
@@ -827,7 +837,7 @@ theorem KaehlerDifferential.range_kerCotangentToTensor
   constructor
   · rintro ⟨x, rfl⟩
     obtain ⟨x, rfl⟩ := Ideal.toCotangent_surjective _ x
-    simp [kerCotangentToTensor_toCotangent, (RingHom.mem_ker _).mp x.2]
+    simp [kerCotangentToTensor_toCotangent, RingHom.mem_ker.mp x.2]
   · intro hx
     obtain ⟨x, rfl⟩ := LinearMap.rTensor_surjective (Ω[A⁄R]) (g := Algebra.linearMap A B) h x
     obtain ⟨x, rfl⟩ := (TensorProduct.lid _ _).symm.surjective x
@@ -844,21 +854,23 @@ theorem KaehlerDifferential.range_kerCotangentToTensor
     intro c _
     simp only [Finset.filter_congr_decidable, TensorProduct.lid_symm_apply, LinearMap.rTensor_tmul,
       AlgHom.toLinearMap_apply, map_one, LinearMap.mem_range]
-    simp only [map_sum, Finsupp.total_single]
+    simp only [map_sum, Finsupp.linearCombination_single]
     have : (x.support.filter (algebraMap A B · = c)).sum x ∈ RingHom.ker (algebraMap A B) := by
       simpa [Finsupp.mapDomain, Finsupp.sum, Finsupp.finset_sum_apply, RingHom.mem_ker,
         Finsupp.single_apply, ← Finset.sum_filter] using DFunLike.congr_fun hx c
     obtain ⟨a, ha⟩ := h c
     use (x.support.filter (algebraMap A B · = c)).attach.sum
         fun i ↦ x i • Ideal.toCotangent _ ⟨i - a, ?_⟩; swap
-    · have : x i ≠ 0 ∧ algebraMap A B i = c := by simpa using i.prop
+    · have : x i ≠ 0 ∧ algebraMap A B i = c := by
+        convert i.prop
+        simp_rw [Finset.mem_filter, Finsupp.mem_support_iff]
       simp [RingHom.mem_ker, ha, this.2]
     · simp only [map_sum, LinearMapClass.map_smul, kerCotangentToTensor_toCotangent, map_sub]
       simp_rw [← TensorProduct.tmul_smul]
       simp only [smul_sub, TensorProduct.tmul_sub, Finset.sum_sub_distrib, ← TensorProduct.tmul_sum,
         ← Finset.sum_smul, Finset.sum_attach, sub_eq_self,
         Finset.sum_attach (f := fun i ↦ x i • KaehlerDifferential.D R A i)]
-      rw [← TensorProduct.smul_tmul, ← Algebra.algebraMap_eq_smul_one, (RingHom.mem_ker _).mp this,
+      rw [← TensorProduct.smul_tmul, ← Algebra.algebraMap_eq_smul_one, RingHom.mem_ker.mp this,
         TensorProduct.zero_tmul]
 
 theorem KaehlerDifferential.exact_kerCotangentToTensor_mapBaseChange
diff --git a/Mathlib/RingTheory/Kaehler/Polynomial.lean b/Mathlib/RingTheory/Kaehler/Polynomial.lean
index 11d31caf3a25a..c5e94eaca9de6 100644
--- a/Mathlib/RingTheory/Kaehler/Polynomial.lean
+++ b/Mathlib/RingTheory/Kaehler/Polynomial.lean
@@ -27,20 +27,23 @@ section MvPolynomial
 def KaehlerDifferential.mvPolynomialEquiv (σ : Type*) :
     Ω[MvPolynomial σ R⁄R] ≃ₗ[MvPolynomial σ R] σ →₀ MvPolynomial σ R where
   __ := (MvPolynomial.mkDerivation _ (Finsupp.single · 1)).liftKaehlerDifferential
-  invFun := Finsupp.total (α := σ) _ (fun x ↦ D _ _ (MvPolynomial.X x))
+  invFun := Finsupp.linearCombination (α := σ) _ (fun x ↦ D _ _ (MvPolynomial.X x))
   right_inv := by
     intro x
-    induction' x using Finsupp.induction_linear with _ _ _ _ a b
-    · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]; rw [map_zero, map_zero]
-    · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, map_add] at *; simp only [*]
-    · simp [LinearMap.map_smul, -map_smul]
+    induction x using Finsupp.induction_linear with
+    | h0 => simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]; rw [map_zero, map_zero]
+    | hadd => simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, map_add] at *; simp only [*]
+    | hsingle a b => simp [LinearMap.map_smul, -map_smul]
   left_inv := by
     intro x
-    obtain ⟨x, rfl⟩ := total_surjective _ _ x
-    induction' x using Finsupp.induction_linear with _ _ _ _ a b
-    · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]; rw [map_zero, map_zero, map_zero]
-    · simp only [map_add, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom] at *; simp only [*]
-    · simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Finsupp.total_single,
+    obtain ⟨x, rfl⟩ := linearCombination_surjective _ _ x
+    induction x using Finsupp.induction_linear with
+    | h0 =>
+      simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom]
+      rw [map_zero, map_zero, map_zero]
+    | hadd => simp only [map_add, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom] at *; simp only [*]
+    | hsingle a b =>
+      simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, Finsupp.linearCombination_single,
         LinearMap.map_smul, Derivation.liftKaehlerDifferential_comp_D]
       congr 1
       induction a using MvPolynomial.induction_on
@@ -109,7 +112,7 @@ def KaehlerDifferential.polynomialEquiv : Ω[R[X]⁄R] ≃ₗ[R[X]] R[X] where
   invFun := (Algebra.lsmul R R _).toLinearMap.flip (D R R[X] X)
   left_inv := by
     intro x
-    obtain ⟨x, rfl⟩ := total_surjective _ _ x
+    obtain ⟨x, rfl⟩ := linearCombination_surjective _ _ x
     induction' x using Finsupp.induction_linear with x y hx hy x y
     · simp
     · simp only [map_add, AddHom.toFun_eq_coe, LinearMap.coe_toAddHom, LinearMap.flip_apply,
diff --git a/Mathlib/RingTheory/KrullDimension/Basic.lean b/Mathlib/RingTheory/KrullDimension/Basic.lean
new file mode 100644
index 0000000000000..cca9f211a88e5
--- /dev/null
+++ b/Mathlib/RingTheory/KrullDimension/Basic.lean
@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Fangming Li, Jujian Zhang
+-/
+import Mathlib.Algebra.MvPolynomial.CommRing
+import Mathlib.Algebra.Polynomial.Basic
+import Mathlib.RingTheory.PrimeSpectrum
+import Mathlib.Order.KrullDimension
+
+/-!
+# Krull dimensions of (commutative) rings
+
+Given a commutative ring, its ring theoretic Krull dimension is the order theoretic Krull dimension
+of its prime spectrum. Unfolding this definition, it is the length of the longest sequence(s) of
+prime ideals ordered by strict inclusion.
+-/
+
+open Order
+
+/--
+The ring theoretic Krull dimension is the Krull dimension of its spectrum ordered by inclusion.
+-/
+noncomputable def ringKrullDim (R : Type*) [CommRing R] : WithBot (WithTop ℕ) :=
+  krullDim (PrimeSpectrum R)
+
+variable {R S : Type*} [CommRing R] [CommRing S]
+
+@[nontriviality]
+lemma ringKrullDim_eq_bot_of_subsingleton [Subsingleton R] :
+    ringKrullDim R = ⊥ :=
+  krullDim_eq_bot_of_isEmpty
+
+lemma ringKrullDim_nonneg_of_nontrivial [Nontrivial R] :
+    0 ≤ ringKrullDim R :=
+  krullDim_nonneg_of_nonempty
+
+/-- If `f : R →+* S` is surjective, then `ringKrullDim S ≤ ringKrullDim R`. -/
+theorem ringKrullDim_le_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
+    ringKrullDim S ≤ ringKrullDim R :=
+  krullDim_le_of_strictMono (fun I ↦ ⟨Ideal.comap f I.asIdeal, inferInstance⟩)
+    (Monotone.strictMono_of_injective (fun _ _ hab ↦ Ideal.comap_mono hab)
+      (fun _ _ h => PrimeSpectrum.ext_iff.mpr <| Ideal.comap_injective_of_surjective f hf <| by
+        simpa using h))
+
+/-- If `I` is an ideal of `R`, then `ringKrullDim (R ⧸ I) ≤ ringKrullDim R`. -/
+theorem ringKrullDim_quotient_le (I : Ideal R) :
+    ringKrullDim (R ⧸ I) ≤ ringKrullDim R :=
+  ringKrullDim_le_of_surjective _ Ideal.Quotient.mk_surjective
+
+/-- If `R` and `S` are isomorphic, then `ringKrullDim R = ringKrullDim S`. -/
+theorem ringKrullDim_eq_of_ringEquiv (e : R ≃+* S) :
+    ringKrullDim R = ringKrullDim S :=
+  le_antisymm (ringKrullDim_le_of_surjective e.symm e.symm.surjective)
+    (ringKrullDim_le_of_surjective e e.surjective)
+
+alias RingEquiv.ringKrullDim := ringKrullDim_eq_of_ringEquiv
+
+proof_wanted Polynomial.ringKrullDim_le :
+    ringKrullDim (Polynomial R) ≤ 2 * (ringKrullDim R) + 1
+
+proof_wanted MvPolynomial.fin_ringKrullDim_eq_add_of_isNoetherianRing
+    [IsNoetherianRing R] (n : ℕ) :
+    ringKrullDim (MvPolynomial (Fin n) R) = ringKrullDim R + n
diff --git a/Mathlib/RingTheory/KrullDimension/Field.lean b/Mathlib/RingTheory/KrullDimension/Field.lean
new file mode 100644
index 0000000000000..bc8ebdf09a387
--- /dev/null
+++ b/Mathlib/RingTheory/KrullDimension/Field.lean
@@ -0,0 +1,22 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Fangming Li, Jujian Zhang
+-/
+import Mathlib.RingTheory.KrullDimension.Basic
+
+/-!
+# The Krull dimension of a field
+
+This file proves that the Krull dimension of a field is zero.
+-/
+
+open Order
+
+@[simp]
+theorem ringKrullDim_eq_zero_of_field (F : Type*) [Field F] : ringKrullDim F = 0 :=
+  krullDim_eq_zero_of_unique
+
+theorem ringKrullDim_eq_zero_of_isField {F : Type*} [CommRing F] (hF : IsField F) :
+    ringKrullDim F = 0 :=
+  @krullDim_eq_zero_of_unique _ _ <| @PrimeSpectrum.instUnique _ hF.toField
diff --git a/Mathlib/RingTheory/LaurentSeries.lean b/Mathlib/RingTheory/LaurentSeries.lean
index 7ea03ca4e499d..eea4d59465c7d 100644
--- a/Mathlib/RingTheory/LaurentSeries.lean
+++ b/Mathlib/RingTheory/LaurentSeries.lean
@@ -11,6 +11,7 @@ import Mathlib.RingTheory.HahnSeries.Summable
 import Mathlib.RingTheory.PowerSeries.Inverse
 import Mathlib.FieldTheory.RatFunc.AsPolynomial
 import Mathlib.RingTheory.Localization.FractionRing
+import Mathlib.Topology.UniformSpace.Cauchy
 
 /-!
 # Laurent Series
@@ -26,16 +27,23 @@ import Mathlib.RingTheory.Localization.FractionRing
 * Embedding of rational functions into Laurent series, provided as a coercion, utilizing
 the underlying `RatFunc.coeAlgHom`.
 * Study of the `X`-Adic valuation on the ring of Laurent series over a field
+* In `LaurentSeries.uniformContinuous_coeff` we show that sending a Laurent series to its `d`th
+coefficient is uniformly continuous, ensuring that it sends a Cauchy filter `ℱ` in `LaurentSeries K`
+to a Cauchy filter in `K`: since this latter is given the discrete topology, this provides an
+element `LaurentSeries.Cauchy.coeff ℱ d` in `K` that serves as `d`th coefficient of the Laurent
+series to which the filter `ℱ` converges.
 
 ## Main Results
 * Basic properties of Hasse derivatives
 ### About the `X`-Adic valuation:
 * The (integral) valuation of a power series is the order of the first non-zero coefficient, see
-`intValuation_le_iff_coeff_lt_eq_zero`.
+`LaurentSeries.intValuation_le_iff_coeff_lt_eq_zero`.
 * The valuation of a Laurent series is the order of the first non-zero coefficient, see
-`valuation_le_iff_coeff_lt_eq_zero`.
+`LaurentSeries.valuation_le_iff_coeff_lt_eq_zero`.
 * Every Laurent series of valuation less than `(1 : ℤₘ₀)` comes from a power series, see
-`val_le_one_iff_eq_coe`.
+`LaurentSeries.val_le_one_iff_eq_coe`.
+* The uniform space of `LaurentSeries` over a field is complete, formalized in the instance
+`instLaurentSeriesComplete`.
 
 ## Implementation details
 * Since `LaurentSeries` is just an abbreviation of `HahnSeries ℤ _`, the definition of the
@@ -101,7 +109,7 @@ theorem coeff_coe_powerSeries (x : PowerSeries R) (n : ℕ) :
 
 /-- This is a power series that can be multiplied by an integer power of `X` to give our
   Laurent series. If the Laurent series is nonzero, `powerSeriesPart` has a nonzero
-  constant term.  -/
+  constant term. -/
 def powerSeriesPart (x : LaurentSeries R) : PowerSeries R :=
   PowerSeries.mk fun n => x.coeff (x.order + n)
 
@@ -233,7 +241,7 @@ theorem coeff_coe (i : ℤ) :
   · rw [ofPowerSeries_apply, embDomain_notin_image_support, if_pos (Int.negSucc_lt_zero _)]
     simp only [not_exists, RelEmbedding.coe_mk, Set.mem_image, not_and, Function.Embedding.coeFn_mk,
       Ne, toPowerSeries_symm_apply_coeff, mem_support, imp_true_iff,
-      not_false_iff]
+      not_false_iff, reduceCtorEq]
 
 -- Porting note (#10618): simp can prove this
 -- Porting note: removed norm_cast attribute
@@ -396,7 +404,7 @@ instance : IsScalarTower F[X] (RatFunc F) (LaurentSeries F) :=
 end RatFunc
 section AdicValuation
 
-open scoped DiscreteValuation
+open scoped Multiplicative
 
 variable (K : Type*) [Field K]
 namespace PowerSeries
@@ -464,6 +472,7 @@ end RatFunc
 
 namespace LaurentSeries
 
+
 open IsDedekindDomain.HeightOneSpectrum PowerSeries RatFunc
 
 instance : Valued (LaurentSeries K) ℤₘ₀ := Valued.mk' (PowerSeries.idealX K).valuation
@@ -601,7 +610,7 @@ theorem val_le_one_iff_eq_coe (f : LaurentSeries K) : Valued.v f ≤ (1 : ℤₘ
   all_goals
     apply HahnSeries.embDomain_notin_range
     simp only [Nat.coe_castAddMonoidHom, RelEmbedding.coe_mk, Function.Embedding.coeFn_mk,
-      Set.mem_range, not_exists, Int.negSucc_lt_zero,]
+      Set.mem_range, not_exists, Int.negSucc_lt_zero, reduceCtorEq]
     intro
   · simp only [not_false_eq_true]
   · linarith
@@ -609,3 +618,169 @@ theorem val_le_one_iff_eq_coe (f : LaurentSeries K) : Valued.v f ≤ (1 : ℤₘ
 end LaurentSeries
 
 end AdicValuation
+namespace LaurentSeries
+section Complete
+
+open Filter
+
+open scoped Multiplicative
+
+variable {K : Type*} [Field K]
+
+/- Sending a Laurent series to its `d`-th coefficient is uniformly continuous (independently of the
+ uniformity with which `K` is endowed). -/
+theorem uniformContinuous_coeff {uK : UniformSpace K} (d : ℤ) :
+    UniformContinuous fun f : LaurentSeries K ↦ f.coeff d := by
+  refine uniformContinuous_iff_eventually.mpr fun S hS ↦ eventually_iff_exists_mem.mpr ?_
+  let γ : ℤₘ₀ˣ := Units.mk0 (↑(Multiplicative.ofAdd (-(d + 1)))) WithZero.coe_ne_zero
+  use {P | Valued.v (P.snd - P.fst) < ↑γ}
+  refine ⟨(Valued.hasBasis_uniformity (LaurentSeries K) ℤₘ₀).mem_of_mem (by tauto), fun P hP ↦ ?_⟩
+  rw [eq_coeff_of_valuation_sub_lt K (le_of_lt hP) (lt_add_one _)]
+  exact mem_uniformity_of_eq hS rfl
+
+/-- Since extracting coefficients is uniformly continuous, every Cauchy filter in
+`laurentSeries K` gives rise to a Cauchy filter in `K` for every `d : ℤ`, and such Cauchy filter
+in `K` converges to a principal filter -/
+def Cauchy.coeff {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) : ℤ → K :=
+  let _ : UniformSpace K := ⊥
+  fun d ↦ UniformSpace.DiscreteUnif.cauchyConst rfl <| hℱ.map (uniformContinuous_coeff d)
+
+theorem Cauchy.coeff_tendsto {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) (D : ℤ) :
+    Tendsto (fun f : LaurentSeries K ↦ f.coeff D) ℱ (𝓟 {coeff hℱ D}) :=
+  let _ : UniformSpace K := ⊥
+  le_of_eq <| UniformSpace.DiscreteUnif.eq_const_of_cauchy (by rfl)
+    (hℱ.map (uniformContinuous_coeff D)) ▸ (principal_singleton _).symm
+
+/- For every Cauchy filter of Laurent series, there is a `N` such that the `n`-th coefficient
+vanishes for all `n ≤ N` and almost all series in the filter. This is an auxiliary lemma used
+to construct the limit of the Cauchy filter as a Laurent series, ensuring that the support of the
+limit is `PWO`.
+The result is true also for more general Hahn Series indexed over a partially ordered group `Γ`
+beyond the special case `Γ = ℤ`, that corresponds to Laurent Series: nevertheless the proof below
+does not generalise, as it relies on the study of the `X`-adic valuation attached to the height-one
+prime `X`, and this is peculiar to the one-variable setting. In the future we should prove this
+result in full generality and deduce the case `Γ = ℤ` from that one.-/
+lemma Cauchy.exists_lb_eventual_support {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) :
+    ∃ N, ∀ᶠ f : LaurentSeries K in ℱ, ∀ n < N, f.coeff n = (0 : K) := by
+  let entourage : Set (LaurentSeries K × LaurentSeries K) :=
+    {P : LaurentSeries K × LaurentSeries K |
+      Valued.v (P.snd - P.fst) < ((Multiplicative.ofAdd 0 : Multiplicative ℤ) : ℤₘ₀)}
+  let ζ := Units.mk0 (G₀ := ℤₘ₀) _ (WithZero.coe_ne_zero (a := (Multiplicative.ofAdd 0)))
+  obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := mem_prod_iff.mp <| Filter.le_def.mp hℱ.2 entourage
+    <| (Valued.hasBasis_uniformity (LaurentSeries K) ℤₘ₀).mem_of_mem (i := ζ) (by tauto)
+  obtain ⟨f, hf⟩ := forall_mem_nonempty_iff_neBot.mpr hℱ.1 (S ∩ T) (inter_mem_iff.mpr ⟨hS, hT⟩)
+  obtain ⟨N, hN⟩ :  ∃ N : ℤ, ∀ g : LaurentSeries K,
+    Valued.v (g - f) ≤ ↑(Multiplicative.ofAdd (0 : ℤ)) → ∀ n < N, g.coeff n = 0 := by
+    by_cases hf : f = 0
+    · refine ⟨0, fun x hg ↦ ?_⟩
+      rw [hf, sub_zero] at hg
+      exact (valuation_le_iff_coeff_lt_eq_zero K).mp hg
+    · refine ⟨min (f.2.isWF.min (HahnSeries.support_nonempty_iff.mpr hf)) 0 - 1, fun _ hg n hn ↦ ?_⟩
+      rw [eq_coeff_of_valuation_sub_lt K hg (d := 0)]
+      · exact Function.nmem_support.mp fun h ↦
+        f.2.isWF.not_lt_min (HahnSeries.support_nonempty_iff.mpr hf) h
+        <| lt_trans hn <| Int.sub_one_lt_iff.mpr <| min_le_left _ _
+      exact lt_of_lt_of_le hn <| le_of_lt (Int.sub_one_lt_of_le <| min_le_right _ _)
+  use N
+  apply mem_of_superset (inter_mem hS hT)
+  intro g hg
+  have h_prod : (f, g) ∈ entourage := Set.prod_mono (Set.inter_subset_left (t := T))
+    (Set.inter_subset_right (s := S)) |>.trans H <| Set.mem_prod.mpr ⟨hf, hg⟩
+  exact hN g (le_of_lt h_prod)
+
+/- The support of `Cauchy.coeff` has a lower bound. -/
+theorem Cauchy.exists_lb_support {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) :
+    ∃ N, ∀ n, n < N → coeff hℱ n = 0 := by
+  let _ : UniformSpace K := ⊥
+  obtain ⟨N, hN⟩ := exists_lb_eventual_support hℱ
+  refine ⟨N, fun n hn ↦ Ultrafilter.eq_of_le_pure (hℱ.map (uniformContinuous_coeff n)).1
+      ((principal_singleton _).symm ▸ coeff_tendsto _ _) ?_⟩
+  simp only [pure_zero, nonpos_iff]
+  apply Filter.mem_of_superset hN (fun _ ha ↦ ha _ hn)
+
+/- The support of `Cauchy.coeff` is bounded below -/
+theorem Cauchy.coeff_support_bddBelow {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) :
+    BddBelow (coeff hℱ).support := by
+  refine ⟨(exists_lb_support hℱ).choose, fun d hd ↦ ?_⟩
+  by_contra hNd
+  exact hd ((exists_lb_support hℱ).choose_spec d (not_le.mp hNd))
+
+/-- To any Cauchy filter ℱ of `LaurentSeries K`, we can attach a laurent series that is the limit
+of the filter. Its `d`-th coefficient is defined as the limit of `Cauchy.coeff hℱ d`, which is
+again Cauchy but valued in the discrete space `K`. That sufficiently negative coefficients vanish
+follows from `Cauchy.coeff_support_bddBelow` -/
+def Cauchy.limit {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) : LaurentSeries K :=
+  HahnSeries.mk (coeff hℱ) <| Set.IsWF.isPWO (coeff_support_bddBelow _).wellFoundedOn_lt
+
+/- The following lemma shows that for every `d` smaller than the minimum between the integers
+produced in `Cauchy.exists_lb_eventual_support` and `Cauchy.exists_lb_support`, for almost all
+series in `ℱ` the `d`th coefficient coincides with the `d`th coefficient of `Cauchy.coeff hℱ`. -/
+theorem Cauchy.exists_lb_coeff_ne {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) :
+    ∃ N, ∀ᶠ f : LaurentSeries K in ℱ, ∀ d < N, coeff hℱ d = f.coeff d := by
+  obtain ⟨⟨N₁, hN₁⟩, ⟨N₂, hN₂⟩⟩ := exists_lb_eventual_support hℱ, exists_lb_support hℱ
+  refine ⟨min N₁ N₂, ℱ.3 hN₁ fun _ hf d hd ↦ ?_⟩
+  rw [hf d (lt_of_lt_of_le hd (min_le_left _ _)), hN₂ d (lt_of_lt_of_le hd (min_le_right _ _))]
+
+/- Given a Cauchy filter `ℱ` in the Laurent Series and a bound `D`, for almost all series in the
+filter the coefficients below `D` coincide with `Caucy.coeff hℱ`-/
+theorem Cauchy.coeff_eventually_equal {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ) {D : ℤ} :
+    ∀ᶠ f : LaurentSeries K in ℱ, ∀ d, d < D → coeff hℱ d = f.coeff d := by
+  -- `φ` sends `d` to the set of Laurent Series having `d`th coefficient equal to `ℱ.coeff`.
+  let φ : ℤ → Set (LaurentSeries K) := fun d ↦ {f | coeff hℱ d = f.coeff d}
+  have intersec₁ :
+    (⋂ n ∈ Set.Iio D, φ n) ⊆ {x : LaurentSeries K | ∀ d : ℤ, d < D → coeff hℱ d = x.coeff d} := by
+    intro _ hf
+    simpa only [Set.mem_iInter] using hf
+  -- The goal is now to show that the intersection of all `φ d` (for `d < D`) is in `ℱ`.
+  let ℓ := (exists_lb_coeff_ne hℱ).choose
+  let N := max ℓ D
+  have intersec₂ : ⋂ n ∈ Set.Iio D, φ n ⊇ (⋂ n ∈ Set.Iio ℓ, φ n) ∩ (⋂ n ∈ Set.Icc ℓ N, φ n) := by
+    simp only [Set.mem_Iio, Set.mem_Icc, Set.subset_iInter_iff]
+    intro i hi x hx
+    simp only [Set.mem_inter_iff, Set.mem_iInter, and_imp] at hx
+    by_cases H : i < ℓ
+    exacts [hx.1 _ H, hx.2 _ (le_of_not_lt H) <| le_of_lt <| lt_max_of_lt_right hi]
+  suffices (⋂ n ∈ Set.Iio ℓ, φ n) ∩ (⋂ n ∈ Set.Icc ℓ N, φ n) ∈ ℱ by
+    exact ℱ.sets_of_superset this <| intersec₂.trans intersec₁
+  /- To show that the intersection we have in sight is in `ℱ`, we use that it contains a double
+  intersection (an infinite and a finite one): by general properties of filters, we are reduced
+  to show that both terms are in `ℱ`, which is easy in light of their definition. -/
+  · simp only [Set.mem_Iio, Set.mem_Ico, inter_mem_iff]
+    constructor
+    · have := (exists_lb_coeff_ne hℱ).choose_spec
+      rw [Filter.eventually_iff] at this
+      convert this
+      ext
+      simp only [Set.mem_iInter, Set.mem_setOf_eq]; rfl
+    · rw [biInter_mem (Set.finite_Icc ℓ N)]
+      intro _ _
+      apply coeff_tendsto hℱ
+      simp only [principal_singleton, mem_pure]; rfl
+
+
+open scoped Topology
+
+/- The main result showing that the Cauchy filter tends to the `Cauchy.limit`-/
+theorem Cauchy.eventually_mem_nhds {ℱ : Filter (LaurentSeries K)} (hℱ : Cauchy ℱ)
+    {U : Set (LaurentSeries K)} (hU : U ∈ 𝓝 (Cauchy.limit hℱ)) : ∀ᶠ f in ℱ, f ∈ U := by
+  obtain ⟨γ, hU₁⟩ := Valued.mem_nhds.mp hU
+  suffices ∀ᶠ f in ℱ, f ∈ {y : LaurentSeries K | Valued.v (y - limit hℱ) < ↑γ} by
+    apply this.mono fun _ hf ↦ hU₁ hf
+  set D := -(Multiplicative.toAdd (WithZero.unzero γ.ne_zero) - 1) with hD₀
+  have hD : ((Multiplicative.ofAdd (-D) : Multiplicative ℤ) : ℤₘ₀) < γ := by
+    rw [← WithZero.coe_unzero γ.ne_zero, WithZero.coe_lt_coe, hD₀, neg_neg, ofAdd_sub,
+      ofAdd_toAdd, div_lt_comm, div_self', ← ofAdd_zero, Multiplicative.ofAdd_lt]
+    exact zero_lt_one
+  apply coeff_eventually_equal (D := D) hℱ |>.mono
+  intro _ hf
+  apply lt_of_le_of_lt (valuation_le_iff_coeff_lt_eq_zero K |>.mpr _) hD
+  intro n hn
+  rw [HahnSeries.sub_coeff, sub_eq_zero, hf n hn |>.symm]; rfl
+
+/- Laurent Series with coefficients in a field are complete w.r.t. the `X`-adic valuation -/
+instance instLaurentSeriesComplete : CompleteSpace (LaurentSeries K) :=
+  ⟨fun hℱ ↦ ⟨Cauchy.limit hℱ, fun _ hS ↦ Cauchy.eventually_mem_nhds hℱ hS⟩⟩
+
+end Complete
+
+end LaurentSeries
diff --git a/Mathlib/RingTheory/LittleWedderburn.lean b/Mathlib/RingTheory/LittleWedderburn.lean
index bb054fb0dc9a4..42f44f6d9f529 100644
--- a/Mathlib/RingTheory/LittleWedderburn.lean
+++ b/Mathlib/RingTheory/LittleWedderburn.lean
@@ -48,7 +48,7 @@ private def InductionHyp : Prop :=
 
 namespace InductionHyp
 
-open FiniteDimensional Polynomial
+open Module Polynomial
 
 variable {D}
 
@@ -96,7 +96,7 @@ private theorem center_eq_top [Finite D] (hD : InductionHyp D) : Subring.center
       refine not_le_of_lt hZ.lt_top (fun y _ ↦ Subring.mem_center_iff.mpr fun z ↦ ?_)
       obtain ⟨r, rfl⟩ := hx y
       obtain ⟨s, rfl⟩ := hx z
-      rw [smul_mul_smul, smul_mul_smul, mul_comm]
+      rw [smul_mul_smul_comm, smul_mul_smul_comm, mul_comm]
   rw [Nat.cast_sum]
   apply Finset.dvd_sum
   rintro ⟨x⟩ hx
@@ -149,7 +149,7 @@ private theorem center_eq_top [Finite D] : Subring.center D = ⊤ := by
   rw [IH (Fintype.card R) _ R inferInstance rfl]
   · trivial
   rw [← hn, ← Subring.card_top D]
-  exact Set.card_lt_card hR
+  convert Set.card_lt_card hR
 
 end LittleWedderburn
 
diff --git a/Mathlib/RingTheory/LocalProperties.lean b/Mathlib/RingTheory/LocalProperties.lean
deleted file mode 100644
index 95ef9fd34099e..0000000000000
--- a/Mathlib/RingTheory/LocalProperties.lean
+++ /dev/null
@@ -1,665 +0,0 @@
-/-
-Copyright (c) 2021 Andrew Yang. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Andrew Yang
--/
-import Mathlib.RingTheory.FiniteType
-import Mathlib.RingTheory.Localization.AtPrime
-import Mathlib.RingTheory.Localization.Away.Basic
-import Mathlib.RingTheory.Localization.Integer
-import Mathlib.RingTheory.Localization.Submodule
-import Mathlib.RingTheory.Nilpotent.Lemmas
-import Mathlib.RingTheory.RingHomProperties
-import Mathlib.Data.Set.Subsingleton
-
-/-!
-# Local properties of commutative rings
-
-In this file, we provide the proofs of various local properties.
-
-## Naming Conventions
-
-* `localization_P` : `P` holds for `S⁻¹R` if `P` holds for `R`.
-* `P_of_localization_maximal` : `P` holds for `R` if `P` holds for `Rₘ` for all maximal `m`.
-* `P_of_localization_prime` : `P` holds for `R` if `P` holds for `Rₘ` for all prime `m`.
-* `P_ofLocalizationSpan` : `P` holds for `R` if given a spanning set `{fᵢ}`, `P` holds for all
-  `R_{fᵢ}`.
-
-## Main results
-
-The following properties are covered:
-
-* The triviality of an ideal or an element:
-  `ideal_eq_bot_of_localization`, `eq_zero_of_localization`
-* `IsReduced` : `localization_isReduced`, `isReduced_of_localization_maximal`.
-* `RingHom.finite`: `localization_finite`, `finite_ofLocalizationSpan`
-* `RingHom.finiteType`: `localization_finiteType`, `finiteType_ofLocalizationSpan`
-
--/
-
-open scoped Pointwise Classical
-
-universe u
-
-variable {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R)
-variable (N : Submonoid S) (R' S' : Type u) [CommRing R'] [CommRing S'] (f : R →+* S)
-variable [Algebra R R'] [Algebra S S']
-
-section Properties
-
-section CommRing
-
-variable (P : ∀ (R : Type u) [CommRing R], Prop)
-
-/-- A property `P` of comm rings is said to be preserved by localization
-  if `P` holds for `M⁻¹R` whenever `P` holds for `R`. -/
-def LocalizationPreserves : Prop :=
-  ∀ {R : Type u} [hR : CommRing R] (M : Submonoid R) (S : Type u) [hS : CommRing S] [Algebra R S]
-    [IsLocalization M S], @P R hR → @P S hS
-
-/-- A property `P` of comm rings satisfies `OfLocalizationMaximal`
-  if `P` holds for `R` whenever `P` holds for `Rₘ` for all maximal ideal `m`. -/
-def OfLocalizationMaximal : Prop :=
-  ∀ (R : Type u) [CommRing R],
-    (∀ (J : Ideal R) (_ : J.IsMaximal), P (Localization.AtPrime J)) → P R
-
-end CommRing
-
-section RingHom
-
-variable (P : ∀ {R S : Type u} [CommRing R] [CommRing S] (_ : R →+* S), Prop)
-
-/-- A property `P` of ring homs is said to be preserved by localization
- if `P` holds for `M⁻¹R →+* M⁻¹S` whenever `P` holds for `R →+* S`. -/
-def RingHom.LocalizationPreserves :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (M : Submonoid R) (R' S' : Type u)
-    [CommRing R'] [CommRing S'] [Algebra R R'] [Algebra S S'] [IsLocalization M R']
-    [IsLocalization (M.map f) S'],
-    P f → P (IsLocalization.map S' f (Submonoid.le_comap_map M) : R' →+* S')
-
-/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpan`
-if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `R` such that
-`P` holds for `Rᵣ →+* Sᵣ`.
-
-Note that this is equivalent to `RingHom.OfLocalizationSpan` via
-`RingHom.ofLocalizationSpan_iff_finite`, but this is easier to prove. -/
-def RingHom.OfLocalizationFiniteSpan :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Finset R)
-    (_ : Ideal.span (s : Set R) = ⊤) (_ : ∀ r : s, P (Localization.awayMap f r)), P f
-
-/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpan`
-if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `R` such that
-`P` holds for `Rᵣ →+* Sᵣ`.
-
-Note that this is equivalent to `RingHom.OfLocalizationFiniteSpan` via
-`RingHom.ofLocalizationSpan_iff_finite`, but this has less restrictions when applying. -/
-def RingHom.OfLocalizationSpan :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Set R) (_ : Ideal.span s = ⊤)
-    (_ : ∀ r : s, P (Localization.awayMap f r)), P f
-
-/-- A property `P` of ring homs satisfies `RingHom.HoldsForLocalizationAway`
- if `P` holds for each localization map `R →+* Rᵣ`. -/
-def RingHom.HoldsForLocalizationAway : Prop :=
-  ∀ ⦃R : Type u⦄ (S : Type u) [CommRing R] [CommRing S] [Algebra R S] (r : R)
-    [IsLocalization.Away r S], P (algebraMap R S)
-
-/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpanTarget`
-if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `S` such that
-`P` holds for `R →+* Sᵣ`.
-
-Note that this is equivalent to `RingHom.OfLocalizationSpanTarget` via
-`RingHom.ofLocalizationSpanTarget_iff_finite`, but this is easier to prove. -/
-def RingHom.OfLocalizationFiniteSpanTarget : Prop :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Finset S)
-    (_ : Ideal.span (s : Set S) = ⊤)
-    (_ : ∀ r : s, P ((algebraMap S (Localization.Away (r : S))).comp f)), P f
-
-/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationSpanTarget`
-if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `S` such that
-`P` holds for `R →+* Sᵣ`.
-
-Note that this is equivalent to `RingHom.OfLocalizationFiniteSpanTarget` via
-`RingHom.ofLocalizationSpanTarget_iff_finite`, but this has less restrictions when applying. -/
-def RingHom.OfLocalizationSpanTarget : Prop :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (_ : Ideal.span s = ⊤)
-    (_ : ∀ r : s, P ((algebraMap S (Localization.Away (r : S))).comp f)), P f
-
-/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationPrime`
-if `P` holds for `R` whenever `P` holds for `Rₘ` for all prime ideals `p`. -/
-def RingHom.OfLocalizationPrime : Prop :=
-  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S),
-    (∀ (J : Ideal S) (_ : J.IsPrime), P (Localization.localRingHom _ J f rfl)) → P f
-
-/-- A property of ring homs is local if it is preserved by localizations and compositions, and for
-each `{ r }` that spans `S`, we have `P (R →+* S) ↔ ∀ r, P (R →+* Sᵣ)`. -/
-structure RingHom.PropertyIsLocal : Prop where
-  LocalizationPreserves : RingHom.LocalizationPreserves @P
-  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
-  StableUnderComposition : RingHom.StableUnderComposition @P
-  HoldsForLocalizationAway : RingHom.HoldsForLocalizationAway @P
-
-theorem RingHom.ofLocalizationSpan_iff_finite :
-    RingHom.OfLocalizationSpan @P ↔ RingHom.OfLocalizationFiniteSpan @P := by
-  delta RingHom.OfLocalizationSpan RingHom.OfLocalizationFiniteSpan
-  apply forall₅_congr
-  -- TODO: Using `refine` here breaks `resetI`.
-  intros
-  constructor
-  · intro h s; exact h s
-  · intro h s hs hs'
-    obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
-    exact h s' h₂ fun x => hs' ⟨_, h₁ x.prop⟩
-
-theorem RingHom.ofLocalizationSpanTarget_iff_finite :
-    RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P := by
-  delta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget
-  apply forall₅_congr
-  -- TODO: Using `refine` here breaks `resetI`.
-  intros
-  constructor
-  · intro h s; exact h s
-  · intro h s hs hs'
-    obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
-    exact h s' h₂ fun x => hs' ⟨_, h₁ x.prop⟩
-
-theorem RingHom.HoldsForLocalizationAway.of_bijective
-    (H : RingHom.HoldsForLocalizationAway P) (hf : Function.Bijective f) :
-    P f := by
-  letI := f.toAlgebra
-  have := IsLocalization.at_units (.powers (1 : R)) (by simp)
-  have := IsLocalization.isLocalization_of_algEquiv (.powers (1 : R))
-    (AlgEquiv.ofBijective (Algebra.ofId R S) hf)
-  exact H _ 1
-
-variable {P f R' S'}
-
-theorem RingHom.PropertyIsLocal.respectsIso (hP : RingHom.PropertyIsLocal @P) :
-    RingHom.RespectsIso @P := by
-  apply hP.StableUnderComposition.respectsIso
-  introv
-  letI := e.toRingHom.toAlgebra
-  -- Porting note: was `apply_with hP.holds_for_localization_away { instances := ff }`
-  have : IsLocalization.Away (1 : R) S := by
-    apply IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
-  exact RingHom.PropertyIsLocal.HoldsForLocalizationAway hP S (1 : R)
-
--- Almost all arguments are implicit since this is not intended to use mid-proof.
-theorem RingHom.LocalizationPreserves.away (H : RingHom.LocalizationPreserves @P) (r : R)
-    [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f) :
-    P (IsLocalization.Away.map R' S' f r) := by
-  have : IsLocalization ((Submonoid.powers r).map f) S' := by rw [Submonoid.map_powers]; assumption
-  exact H f (Submonoid.powers r) R' S' hf
-
-theorem RingHom.PropertyIsLocal.ofLocalizationSpan (hP : RingHom.PropertyIsLocal @P) :
-    RingHom.OfLocalizationSpan @P := by
-  introv R hs hs'
-  apply_fun Ideal.map f at hs
-  rw [Ideal.map_span, Ideal.map_top] at hs
-  apply hP.OfLocalizationSpanTarget _ _ hs
-  rintro ⟨_, r, hr, rfl⟩
-  convert hP.StableUnderComposition
-    _ _ (hP.HoldsForLocalizationAway (Localization.Away r) r) (hs' ⟨r, hr⟩) using 1
-  exact (IsLocalization.map_comp _).symm
-
-lemma RingHom.OfLocalizationSpanTarget.ofIsLocalization
-    (hP : RingHom.OfLocalizationSpanTarget P) (hP' : RingHom.RespectsIso P)
-    {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (hs : Ideal.span s = ⊤)
-    (hT : ∀ r : s, ∃ (T : Type u) (_ : CommRing T) (_ : Algebra S T)
-      (_ : IsLocalization.Away (r : S) T), P ((algebraMap S T).comp f)) : P f := by
-  apply hP _ s hs
-  intros r
-  obtain ⟨T, _, _, _, hT⟩ := hT r
-  convert hP'.1 _
-    (Localization.algEquiv (R := S) (Submonoid.powers (r : S)) T).symm.toRingEquiv hT
-  rw [← RingHom.comp_assoc, RingEquiv.toRingHom_eq_coe, AlgEquiv.toRingEquiv_eq_coe,
-    AlgEquiv.toRingEquiv_toRingHom, Localization.coe_algEquiv_symm, IsLocalization.map_comp,
-    RingHom.comp_id]
-
-end RingHom
-
-end Properties
-
-section Ideal
-
-open scoped nonZeroDivisors
-
-/-- Let `I J : Ideal R`. If the localization of `I` at each maximal ideal `P` is included in
-the localization of `J` at `P`, then `I ≤ J`. -/
-theorem Ideal.le_of_localization_maximal {I J : Ideal R}
-    (h : ∀ (P : Ideal R) (hP : P.IsMaximal),
-      Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤
-        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
-    I ≤ J := by
-  intro x hx
-  suffices J.colon (Ideal.span {x}) = ⊤ by
-    simpa using Submodule.mem_colon.mp
-      (show (1 : R) ∈ J.colon (Ideal.span {x}) from this.symm ▸ Submodule.mem_top) x
-      (Ideal.mem_span_singleton_self x)
-  refine Not.imp_symm (J.colon (Ideal.span {x})).exists_le_maximal ?_
-  push_neg
-  intro P hP le
-  obtain ⟨⟨⟨a, ha⟩, ⟨s, hs⟩⟩, eq⟩ :=
-    (IsLocalization.mem_map_algebraMap_iff P.primeCompl _).mp (h P hP (Ideal.mem_map_of_mem _ hx))
-  rw [← _root_.map_mul, ← sub_eq_zero, ← map_sub] at eq
-  obtain ⟨⟨m, hm⟩, eq⟩ := (IsLocalization.map_eq_zero_iff P.primeCompl _ _).mp eq
-  refine hs ((hP.isPrime.mem_or_mem (le (Ideal.mem_colon_singleton.mpr ?_))).resolve_right hm)
-  simp only [Subtype.coe_mk, mul_sub, sub_eq_zero, mul_comm x s, mul_left_comm] at eq
-  simpa only [mul_assoc, eq] using J.mul_mem_left m ha
-
-/-- Let `I J : Ideal R`. If the localization of `I` at each maximal ideal `P` is equal to
-the localization of `J` at `P`, then `I = J`. -/
-theorem Ideal.eq_of_localization_maximal {I J : Ideal R}
-    (h : ∀ (P : Ideal R) (_ : P.IsMaximal),
-      Ideal.map (algebraMap R (Localization.AtPrime P)) I =
-        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
-    I = J :=
-  le_antisymm (Ideal.le_of_localization_maximal fun P hP => (h P hP).le)
-    (Ideal.le_of_localization_maximal fun P hP => (h P hP).ge)
-
-/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/
-theorem ideal_eq_bot_of_localization' (I : Ideal R)
-    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
-      Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥) :
-    I = ⊥ :=
-  Ideal.eq_of_localization_maximal fun P hP => by simpa using h P hP
-
--- TODO: This proof should work for all modules, once we have enough material on submodules of
--- localized modules.
-/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/
-theorem ideal_eq_bot_of_localization (I : Ideal R)
-    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
-      IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥) :
-    I = ⊥ :=
-  ideal_eq_bot_of_localization' _ fun P hP =>
-    (Ideal.map_eq_bot_iff_le_ker _).mpr fun x hx => by
-      rw [RingHom.mem_ker, ← Submodule.mem_bot R, ← h P hP, IsLocalization.mem_coeSubmodule]
-      exact ⟨x, hx, rfl⟩
-
-theorem eq_zero_of_localization (r : R)
-    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
-    r = 0 := by
-  rw [← Ideal.span_singleton_eq_bot]
-  apply ideal_eq_bot_of_localization
-  intro J hJ
-  delta IsLocalization.coeSubmodule
-  erw [Submodule.map_span, Submodule.span_eq_bot]
-  rintro _ ⟨_, h', rfl⟩
-  cases Set.mem_singleton_iff.mpr h'
-  exact h J hJ
-
-end Ideal
-
-section Reduced
-
-theorem localization_isReduced : LocalizationPreserves fun R hR => IsReduced R := by
-  introv R _ _
-  constructor
-  rintro x ⟨_ | n, e⟩
-  · simpa using congr_arg (· * x) e
-  obtain ⟨⟨y, m⟩, hx⟩ := IsLocalization.surj M x
-  dsimp only at hx
-  let hx' := congr_arg (· ^ n.succ) hx
-  simp only [mul_pow, e, zero_mul, ← RingHom.map_pow] at hx'
-  rw [← (algebraMap R S).map_zero] at hx'
-  obtain ⟨m', hm'⟩ := (IsLocalization.eq_iff_exists M S).mp hx'
-  apply_fun (· * (m' : R) ^ n) at hm'
-  simp only [mul_assoc, zero_mul, mul_zero] at hm'
-  rw [← mul_left_comm, ← pow_succ', ← mul_pow] at hm'
-  replace hm' := IsNilpotent.eq_zero ⟨_, hm'.symm⟩
-  rw [← (IsLocalization.map_units S m).mul_left_inj, hx, zero_mul,
-    IsLocalization.map_eq_zero_iff M]
-  exact ⟨m', by rw [← hm', mul_comm]⟩
-
-instance [IsReduced R] : IsReduced (Localization M) :=
-  localization_isReduced M _ inferInstance
-
-theorem isReduced_ofLocalizationMaximal : OfLocalizationMaximal fun R hR => IsReduced R := by
-  introv R h
-  constructor
-  intro x hx
-  apply eq_zero_of_localization
-  intro J hJ
-  specialize h J hJ
-  exact (hx.map <| algebraMap R <| Localization.AtPrime J).eq_zero
-
-end Reduced
-
-section Surjective
-
-theorem localizationPreserves_surjective :
-    RingHom.LocalizationPreserves fun {R S} _ _ f => Function.Surjective f := by
-  introv R H x
-  obtain ⟨x, ⟨_, s, hs, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (M.map f) x
-  obtain ⟨y, rfl⟩ := H x
-  use IsLocalization.mk' R' y ⟨s, hs⟩
-  rw [IsLocalization.map_mk']
-
-theorem surjective_ofLocalizationSpan :
-    RingHom.OfLocalizationSpan fun {R S} _ _ f => Function.Surjective f := by
-  introv R e H
-  rw [← Set.range_iff_surjective, Set.eq_univ_iff_forall]
-  letI := f.toAlgebra
-  intro x
-  apply Submodule.mem_of_span_eq_top_of_smul_pow_mem
-    (LinearMap.range (Algebra.linearMap R S)) s e
-  intro r
-  obtain ⟨a, e'⟩ := H r (algebraMap _ _ x)
-  obtain ⟨b, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers (r : R)) a
-  erw [IsLocalization.map_mk'] at e'
-  rw [eq_comm, IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, Subtype.coe_mk, ← map_mul] at e'
-  obtain ⟨⟨_, n', rfl⟩, e''⟩ := (IsLocalization.eq_iff_exists (Submonoid.powers (f r)) _).mp e'
-  dsimp only at e''
-  rw [mul_comm x, ← mul_assoc, ← map_pow, ← map_mul, ← map_mul, ← pow_add] at e''
-  exact ⟨n' + n, _, e''.symm⟩
-
-/-- A surjective ring homomorphism `R →+* S` induces a surjective homomorphism `R_{f⁻¹(P)} →+* S_P`
-for every prime ideal `P` of `S`. -/
-theorem surjective_localRingHom_of_surjective (h : Function.Surjective f) (P : Ideal S)
-    [P.IsPrime] : Function.Surjective (Localization.localRingHom (P.comap f) P f rfl) :=
-  have : IsLocalization (Submonoid.map f (Ideal.comap f P).primeCompl) (Localization.AtPrime P) :=
-    (Submonoid.map_comap_eq_of_surjective h P.primeCompl).symm ▸ Localization.isLocalization
-  localizationPreserves_surjective _ _ _ _ h
-
-lemma surjective_respectsIso : RingHom.RespectsIso (fun f ↦ Function.Surjective f) := by
-  apply RingHom.StableUnderComposition.respectsIso
-  · intro R S T _ _ _ f g hf hg
-    simp only [RingHom.coe_comp]
-    exact Function.Surjective.comp hg hf
-  · intro R S _ _ e
-    exact EquivLike.surjective e
-
-end Surjective
-
-section Finite
-
-lemma Module.Finite_of_isLocalization (R S Rₚ Sₚ) [CommSemiring R] [CommRing S] [CommRing Rₚ]
-    [CommRing Sₚ] [Algebra R S] [Algebra R Rₚ] [Algebra R Sₚ] [Algebra S Sₚ] [Algebra Rₚ Sₚ]
-    [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] (M : Submonoid R) [IsLocalization M Rₚ]
-    [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ] [hRS : Module.Finite R S] :
-    Module.Finite Rₚ Sₚ := by
-  classical
-  have : algebraMap Rₚ Sₚ = IsLocalization.map (T := Algebra.algebraMapSubmonoid S M) Sₚ
-      (algebraMap R S) (Submonoid.le_comap_map M) := by
-    apply IsLocalization.ringHom_ext M
-    simp only [IsLocalization.map_comp, ← IsScalarTower.algebraMap_eq]
-  -- We claim that if `S` is generated by `T` as an `R`-module,
-  -- then `S'` is generated by `T` as an `R'`-module.
-  obtain ⟨T, hT⟩ := hRS
-  use T.image (algebraMap S Sₚ)
-  rw [eq_top_iff]
-  rintro x -
-  -- By the hypotheses, for each `x : S'`, we have `x = y / (f r)` for some `y : S` and `r : M`.
-  -- Since `S` is generated by `T`, the image of `y` should fall in the span of the image of `T`.
-  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ :=
-    IsLocalization.mk'_surjective (Algebra.algebraMapSubmonoid S M) x
-  rw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]
-  have hy : y ∈ Submodule.span R ↑T := by rw [hT]; trivial
-  replace hy : algebraMap S Sₚ y ∈ Submodule.map (IsScalarTower.toAlgHom R S Sₚ).toLinearMap
-    (Submodule.span R (T : Set S)) := Submodule.mem_map_of_mem
---     -- Note: #8386 had to specify the value of `f` below
-      (f := (IsScalarTower.toAlgHom R S Sₚ).toLinearMap) hy
-  rw [Submodule.map_span (IsScalarTower.toAlgHom R S Sₚ).toLinearMap T] at hy
-  have H : Submodule.span R (algebraMap S Sₚ '' T) ≤
-      (Submodule.span Rₚ (algebraMap S Sₚ '' T)).restrictScalars R := by
-    rw [Submodule.span_le]; exact Submodule.subset_span
-  -- Now, since `y ∈ span T`, and `(f r)⁻¹ ∈ R'`, `x / (f r)` is in `span T` as well.
-  convert (Submodule.span Rₚ (algebraMap S Sₚ '' T)).smul_mem
-    (IsLocalization.mk' Rₚ (1 : R) ⟨r, hr⟩) (H hy) using 1
-  rw [Algebra.smul_def, this, IsLocalization.map_mk', map_one]
-
-/-- If `S` is a finite `R`-algebra, then `S' = M⁻¹S` is a finite `R' = M⁻¹R`-algebra. -/
-theorem localization_finite : RingHom.LocalizationPreserves @RingHom.Finite := by
-  introv R hf
-  letI := f.toAlgebra
-  letI := ((algebraMap S S').comp f).toAlgebra
-  let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
-  letI := f'.toAlgebra
-  have : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq'
-    (IsLocalization.map_comp M.le_comap_map).symm
-  have : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' rfl
-  have : IsLocalization (Algebra.algebraMapSubmonoid S M) S' := by
-    rwa [Algebra.algebraMapSubmonoid, RingHom.algebraMap_toAlgebra]
-  have : Module.Finite R S := hf
-  apply Module.Finite_of_isLocalization R S R' S' M
-
-theorem localization_away_map_finite (r : R) [IsLocalization.Away r R']
-    [IsLocalization.Away (f r) S'] (hf : f.Finite) : (IsLocalization.Away.map R' S' f r).Finite :=
-  localization_finite.away r hf
-
-/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
-If the image of some `x : S` falls in the span of some finite `s ⊆ S'` over `R`,
-then there exists some `m : M` such that `m • x` falls in the
-span of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
--/
-theorem IsLocalization.smul_mem_finsetIntegerMultiple_span [Algebra R S] [Algebra R S']
-    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
-    (hx : algebraMap S S' x ∈ Submodule.span R (s : Set S')) :
-    ∃ m : M, m • x ∈
-      Submodule.span R
-        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) := by
-  let g : S →ₐ[R] S' :=
-    AlgHom.mk' (algebraMap S S') fun c x => by simp [Algebra.algebraMap_eq_smul_one]
-  -- We first obtain the `y' ∈ M` such that `s' = y' • s` is falls in the image of `S` in `S'`.
-  let y := IsLocalization.commonDenomOfFinset (M.map (algebraMap R S)) s
-  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
-    (IsLocalization.finsetIntegerMultiple_image _ s).symm
-  obtain ⟨y', hy', e : algebraMap R S y' = y⟩ := y.prop
-  have : algebraMap R S y' • (s : Set S') = y' • (s : Set S') := by
-    simp_rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
-  rw [← e, this] at hx₁
-  replace hx₁ := congr_arg (Submodule.span R) hx₁
-  rw [Submodule.span_smul] at hx₁
-  replace hx : _ ∈ y' • Submodule.span R (s : Set S') := Set.smul_mem_smul_set hx
-  rw [hx₁] at hx
-  erw [← _root_.map_smul g, ← Submodule.map_span (g : S →ₗ[R] S')] at hx
-  -- Since `x` falls in the span of `s` in `S'`, `y' • x : S` falls in the span of `s'` in `S'`.
-  -- That is, there exists some `x' : S` in the span of `s'` in `S` and `x' = y' • x` in `S'`.
-  -- Thus `a • (y' • x) = a • x' ∈ span s'` in `S` for some `a ∈ M`.
-  obtain ⟨x', hx', hx'' : algebraMap _ _ _ = _⟩ := hx
-  obtain ⟨⟨_, a, ha₁, rfl⟩, ha₂⟩ :=
-    (IsLocalization.eq_iff_exists (M.map (algebraMap R S)) S').mp hx''
-  use (⟨a, ha₁⟩ : M) * (⟨y', hy'⟩ : M)
-  convert (Submodule.span R
-    (IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s : Set S)).smul_mem
-      a hx' using 1
-  convert ha₂.symm using 1
-  · rw [Subtype.coe_mk, Submonoid.smul_def, Submonoid.coe_mul, ← smul_smul]
-    exact Algebra.smul_def _ _
-  · exact Algebra.smul_def _ _
-
-/-- If `M` is an `R' = S⁻¹R` module, and `x ∈ span R' s`,
-then `t • x ∈ span R s` for some `t : S`. -/
-theorem multiple_mem_span_of_mem_localization_span
-    {N : Type*} [AddCommMonoid N] [Module R N] [Module R' N]
-    [IsScalarTower R R' N] [IsLocalization M R'] (s : Set N) (x : N)
-    (hx : x ∈ Submodule.span R' s) : ∃ (t : M), t • x ∈ Submodule.span R s := by
-  classical
-  obtain ⟨s', hss', hs'⟩ := Submodule.mem_span_finite_of_mem_span hx
-  rsuffices ⟨t, ht⟩ : ∃ t : M, t • x ∈ Submodule.span R (s' : Set N)
-  · exact ⟨t, Submodule.span_mono hss' ht⟩
-  clear hx hss' s
-  induction s' using Finset.induction_on generalizing x
-  · use 1; simpa using hs'
-  rename_i a s _ hs
-  simp only [Finset.coe_insert, Finset.image_insert, Finset.coe_image, Subtype.coe_mk,
-    Submodule.mem_span_insert] at hs' ⊢
-  rcases hs' with ⟨y, z, hz, rfl⟩
-  rcases IsLocalization.surj M y with ⟨⟨y', s'⟩, e⟩
-  apply congrArg (fun x ↦ x • a) at e
-  simp only [algebraMap_smul] at e
-  rcases hs _ hz with ⟨t, ht⟩
-  refine ⟨t * s', t * y', _, (Submodule.span R (s : Set N)).smul_mem s' ht, ?_⟩
-  rw [smul_add, ← smul_smul, mul_comm, ← smul_smul, ← smul_smul, ← e, mul_comm, ← Algebra.smul_def]
-  simp
-  rfl
-
-/-- If `S` is an `R' = M⁻¹R` algebra, and `x ∈ adjoin R' s`,
-then `t • x ∈ adjoin R s` for some `t : M`. -/
-theorem multiple_mem_adjoin_of_mem_localization_adjoin [Algebra R' S] [Algebra R S]
-    [IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S)
-    (hx : x ∈ Algebra.adjoin R' s) : ∃ t : M, t • x ∈ Algebra.adjoin R s := by
-  change ∃ t : M, t • x ∈ Subalgebra.toSubmodule (Algebra.adjoin R s)
-  change x ∈ Subalgebra.toSubmodule (Algebra.adjoin R' s) at hx
-  simp_rw [Algebra.adjoin_eq_span] at hx ⊢
-  exact multiple_mem_span_of_mem_localization_span M R' _ _ hx
-
-theorem finite_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.Finite := by
-  rw [RingHom.ofLocalizationSpan_iff_finite]
-  introv R hs H
-  -- We first setup the instances
-  letI := f.toAlgebra
-  letI := fun r : s => (Localization.awayMap f r).toAlgebra
-  have : ∀ r : s,
-      IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
-    by intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization
-  haveI : ∀ r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) :=
-    fun r => IsScalarTower.of_algebraMap_eq'
-      (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm
-  -- By the hypothesis, we may find a finite generating set for each `Sᵣ`. This set can then be
-  -- lifted into `R` by multiplying a sufficiently large power of `r`. I claim that the union of
-  -- these generates `S`.
-  constructor
-  replace H := fun r => (H r).1
-  choose s₁ s₂ using H
-  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s₁ x)
-  use s.attach.biUnion sf
-  rw [Submodule.span_attach_biUnion, eq_top_iff]
-  -- It suffices to show that `r ^ n • x ∈ span T` for each `r : s`, since `{ r ^ n }` spans `R`.
-  -- This then follows from the fact that each `x : R` is a linear combination of the generating set
-  -- of `Sᵣ`. By multiplying a sufficiently large power of `r`, we can cancel out the `r`s in the
-  -- denominators of both the generating set and the coefficients.
-  rintro x -
-  apply Submodule.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set R) hs _ _
-  intro r
-  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ :=
-    multiple_mem_span_of_mem_localization_span (Submonoid.powers (r : R))
-      (Localization.Away (r : R)) (s₁ r : Set (Localization.Away (f r))) (algebraMap S _ x)
-      (by rw [s₂ r]; trivial)
-  dsimp only at hn₁
-  rw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, ← map_mul] at hn₁
-  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ :=
-    IsLocalization.smul_mem_finsetIntegerMultiple_span (Submonoid.powers (r : R))
-      (Localization.Away (f r)) _ (s₁ r) hn₁
-  rw [Submonoid.smul_def, ← Algebra.smul_def, smul_smul, Subtype.coe_mk, ← pow_add] at hn₂
-  simp_rw [Submonoid.map_powers] at hn₂
-  use n₂ + n₁
-  exact le_iSup (fun x : s => Submodule.span R (sf x : Set S)) r hn₂
-
-end Finite
-
-section FiniteType
-
-theorem localization_finiteType : RingHom.LocalizationPreserves @RingHom.FiniteType := by
-  introv R hf
-  -- mirrors the proof of `localization_map_finite`
-  letI := f.toAlgebra
-  letI := ((algebraMap S S').comp f).toAlgebra
-  let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
-  letI := f'.toAlgebra
-  haveI : IsScalarTower R R' S' :=
-    IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm
-  let fₐ : S →ₐ[R] S' := AlgHom.mk' (algebraMap S S') fun c x => RingHom.map_mul _ _ _
-  obtain ⟨T, hT⟩ := id hf
-  use T.image (algebraMap S S')
-  rw [eq_top_iff]
-  rintro x -
-  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ := IsLocalization.mk'_surjective (M.map f) x
-  rw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]
-  have hy : y ∈ Algebra.adjoin R (T : Set S) := by rw [hT]; trivial
-  replace hy : algebraMap S S' y ∈ (Algebra.adjoin R (T : Set S)).map fₐ :=
-    Subalgebra.mem_map.mpr ⟨_, hy, rfl⟩
-  rw [fₐ.map_adjoin T] at hy
-  have H : Algebra.adjoin R (algebraMap S S' '' T) ≤
-      (Algebra.adjoin R' (algebraMap S S' '' T)).restrictScalars R := by
-    rw [Algebra.adjoin_le_iff]; exact Algebra.subset_adjoin
-  convert (Algebra.adjoin R' (algebraMap S S' '' T)).smul_mem (H hy)
-    (IsLocalization.mk' R' (1 : R) ⟨r, hr⟩) using 1
-  rw [Algebra.smul_def]
-  erw [IsLocalization.map_mk' M.le_comap_map]
-  rw [map_one]
-
-theorem localization_away_map_finiteType (r : R) [IsLocalization.Away r R']
-    [IsLocalization.Away (f r) S'] (hf : f.FiniteType) :
-    (IsLocalization.Away.map R' S' f r).FiniteType :=
-  localization_finiteType.away r hf
-
-variable {S'}
-
-/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
-Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
-and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
-Then, there exists some `m : M` such that `m • x` falls in `A`.
--/
-theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
-    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
-    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
-    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
-    ∃ m : M, m • x ∈ A := by
-  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
-  let y := IsLocalization.commonDenomOfFinset M s
-  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
-    (IsLocalization.finsetIntegerMultiple_image _ s).symm
-  obtain ⟨n, hn⟩ :=
-    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
-      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
-  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
-  rw [Algebra.smul_def, ← _root_.map_mul] at hx''
-  obtain ⟨a, ha₂⟩ := (IsLocalization.eq_iff_exists M S').mp hx''
-  use a * y ^ n
-  convert A.mul_mem hx' (hA₂ a.prop) using 1
-  rw [Submonoid.smul_def, smul_eq_mul, Submonoid.coe_mul, SubmonoidClass.coe_pow, mul_assoc, ← ha₂,
-    mul_comm]
-
-/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
-If the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
-then there exists some `m : M` such that `m • x` falls in the
-adjoin of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
--/
-theorem IsLocalization.lift_mem_adjoin_finsetIntegerMultiple [Algebra R S] [Algebra R S']
-    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
-    (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
-    ∃ m : M, m • x ∈
-      Algebra.adjoin R
-        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) := by
-  obtain ⟨⟨_, a, ha, rfl⟩, e⟩ :=
-    IsLocalization.exists_smul_mem_of_mem_adjoin (M.map (algebraMap R S)) x s (Algebra.adjoin R _)
-      Algebra.subset_adjoin (by rintro _ ⟨a, _, rfl⟩; exact Subalgebra.algebraMap_mem _ a) hx
-  refine ⟨⟨a, ha⟩, ?_⟩
-  simpa only [Submonoid.smul_def, algebraMap_smul] using e
-
-theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType := by
-  rw [RingHom.ofLocalizationSpan_iff_finite]
-  introv R hs H
-  -- mirrors the proof of `finite_ofLocalizationSpan`
-  letI := f.toAlgebra
-  letI := fun r : s => (Localization.awayMap f r).toAlgebra
-  have : ∀ r : s,
-      IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
-    by intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization
-  haveI : ∀ r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) :=
-    fun r => IsScalarTower.of_algebraMap_eq'
-      (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm
-  constructor
-  replace H := fun r => (H r).1
-  choose s₁ s₂ using H
-  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s₁ x)
-  use s.attach.biUnion sf
-  convert (Algebra.adjoin_attach_biUnion (R := R) sf).trans _
-  rw [eq_top_iff]
-  rintro x -
-  apply (⨆ x : s, Algebra.adjoin R (sf x : Set S)).toSubmodule.mem_of_span_eq_top_of_smul_pow_mem
-    _ hs _ _
-  intro r
-  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ :=
-    multiple_mem_adjoin_of_mem_localization_adjoin (Submonoid.powers (r : R))
-      (Localization.Away (r : R)) (s₁ r : Set (Localization.Away (f r)))
-      (algebraMap S (Localization.Away (f r)) x) (by rw [s₂ r]; trivial)
-  rw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, ← map_mul] at hn₁
-  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ :=
-    IsLocalization.lift_mem_adjoin_finsetIntegerMultiple (Submonoid.powers (r : R)) _ (s₁ r) hn₁
-  rw [Submonoid.smul_def, ← Algebra.smul_def, smul_smul, Subtype.coe_mk, ← pow_add] at hn₂
-  simp_rw [Submonoid.map_powers] at hn₂
-  use n₂ + n₁
-  exact le_iSup (fun x : s => Algebra.adjoin R (sf x : Set S)) r hn₂
-
-end FiniteType
diff --git a/Mathlib/RingTheory/LocalProperties/Basic.lean b/Mathlib/RingTheory/LocalProperties/Basic.lean
new file mode 100644
index 0000000000000..a3ece3c9a9726
--- /dev/null
+++ b/Mathlib/RingTheory/LocalProperties/Basic.lean
@@ -0,0 +1,332 @@
+/-
+Copyright (c) 2021 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+-/
+import Mathlib.RingTheory.Localization.AtPrime
+import Mathlib.RingTheory.Localization.Submodule
+import Mathlib.RingTheory.RingHomProperties
+
+/-!
+# Local properties of commutative rings
+
+In this file, we define local properties in general.
+
+## Naming Conventions
+
+* `localization_P` : `P` holds for `S⁻¹R` if `P` holds for `R`.
+* `P_of_localization_maximal` : `P` holds for `R` if `P` holds for `Rₘ` for all maximal `m`.
+* `P_of_localization_prime` : `P` holds for `R` if `P` holds for `Rₘ` for all prime `m`.
+* `P_ofLocalizationSpan` : `P` holds for `R` if given a spanning set `{fᵢ}`, `P` holds for all
+  `R_{fᵢ}`.
+
+## Main definitions
+
+* `LocalizationPreserves` : A property `P` of comm rings is said to be preserved by localization
+  if `P` holds for `M⁻¹R` whenever `P` holds for `R`.
+* `OfLocalizationMaximal` : A property `P` of comm rings satisfies `OfLocalizationMaximal`
+  if `P` holds for `R` whenever `P` holds for `Rₘ` for all maximal ideal `m`.
+* `RingHom.LocalizationPreserves` : A property `P` of ring homs is said to be preserved by
+  localization if `P` holds for `M⁻¹R →+* M⁻¹S` whenever `P` holds for `R →+* S`.
+* `RingHom.OfLocalizationSpan` : A property `P` of ring homs satisfies
+  `RingHom.OfLocalizationSpan` if `P` holds for `R →+* S` whenever there exists a
+  set `{ r }` that spans `R` such that `P` holds for `Rᵣ →+* Sᵣ`.
+
+## Main results
+
+* The triviality of an ideal or an element:
+  `ideal_eq_bot_of_localization`, `eq_zero_of_localization`
+
+-/
+
+open scoped Pointwise Classical
+
+universe u
+
+variable {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) (f : R →+* S)
+variable (N : Submonoid S) (R' S' : Type u) [CommRing R'] [CommRing S']
+variable [Algebra R R'] [Algebra S S']
+
+section Properties
+
+section CommRing
+
+variable (P : ∀ (R : Type u) [CommRing R], Prop)
+
+/-- A property `P` of comm rings is said to be preserved by localization
+  if `P` holds for `M⁻¹R` whenever `P` holds for `R`. -/
+def LocalizationPreserves : Prop :=
+  ∀ {R : Type u} [hR : CommRing R] (M : Submonoid R) (S : Type u) [hS : CommRing S] [Algebra R S]
+    [IsLocalization M S], @P R hR → @P S hS
+
+/-- A property `P` of comm rings satisfies `OfLocalizationMaximal`
+  if `P` holds for `R` whenever `P` holds for `Rₘ` for all maximal ideal `m`. -/
+def OfLocalizationMaximal : Prop :=
+  ∀ (R : Type u) [CommRing R],
+    (∀ (J : Ideal R) (_ : J.IsMaximal), P (Localization.AtPrime J)) → P R
+
+end CommRing
+
+section RingHom
+
+variable (P : ∀ {R S : Type u} [CommRing R] [CommRing S] (_ : R →+* S), Prop)
+
+/-- A property `P` of ring homs is said to contain identities if `P` holds
+for the identity homomorphism of every ring. -/
+def RingHom.ContainsIdentities := ∀ (R : Type u) [CommRing R], P (RingHom.id R)
+
+/-- A property `P` of ring homs is said to be preserved by localization
+ if `P` holds for `M⁻¹R →+* M⁻¹S` whenever `P` holds for `R →+* S`. -/
+def RingHom.LocalizationPreserves :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (M : Submonoid R) (R' S' : Type u)
+    [CommRing R'] [CommRing S'] [Algebra R R'] [Algebra S S'] [IsLocalization M R']
+    [IsLocalization (M.map f) S'],
+    P f → P (IsLocalization.map S' f (Submonoid.le_comap_map M) : R' →+* S')
+
+/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpan`
+if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `R` such that
+`P` holds for `Rᵣ →+* Sᵣ`.
+
+Note that this is equivalent to `RingHom.OfLocalizationSpan` via
+`RingHom.ofLocalizationSpan_iff_finite`, but this is easier to prove. -/
+def RingHom.OfLocalizationFiniteSpan :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Finset R)
+    (_ : Ideal.span (s : Set R) = ⊤) (_ : ∀ r : s, P (Localization.awayMap f r)), P f
+
+/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpan`
+if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `R` such that
+`P` holds for `Rᵣ →+* Sᵣ`.
+
+Note that this is equivalent to `RingHom.OfLocalizationFiniteSpan` via
+`RingHom.ofLocalizationSpan_iff_finite`, but this has less restrictions when applying. -/
+def RingHom.OfLocalizationSpan :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Set R) (_ : Ideal.span s = ⊤)
+    (_ : ∀ r : s, P (Localization.awayMap f r)), P f
+
+/-- A property `P` of ring homs satisfies `RingHom.HoldsForLocalizationAway`
+ if `P` holds for each localization map `R →+* Rᵣ`. -/
+def RingHom.HoldsForLocalizationAway : Prop :=
+  ∀ ⦃R : Type u⦄ (S : Type u) [CommRing R] [CommRing S] [Algebra R S] (r : R)
+    [IsLocalization.Away r S], P (algebraMap R S)
+
+/-- A property `P` of ring homs satisfies `RingHom.StableUnderCompositionWithLocalizationAway`
+if whenever `P` holds for `f` it also holds for the composition with
+localization maps on the left and on the right. -/
+def RingHom.StableUnderCompositionWithLocalizationAway : Prop :=
+  (∀ ⦃R S : Type u⦄ (T : Type u) [CommRing R] [CommRing S] [CommRing T] [Algebra S T] (s : S)
+    [IsLocalization.Away s T] (f : R →+* S), P f → P ((algebraMap S T).comp f)) ∧
+    ∀ ⦃R : Type u⦄ (S : Type u) ⦃T : Type u⦄ [CommRing R] [CommRing S] [CommRing T] [Algebra R S]
+      (r : R) [IsLocalization.Away r S] (f : S →+* T), P f → P (f.comp (algebraMap R S))
+
+/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationFiniteSpanTarget`
+if `P` holds for `R →+* S` whenever there exists a finite set `{ r }` that spans `S` such that
+`P` holds for `R →+* Sᵣ`.
+
+Note that this is equivalent to `RingHom.OfLocalizationSpanTarget` via
+`RingHom.ofLocalizationSpanTarget_iff_finite`, but this is easier to prove. -/
+def RingHom.OfLocalizationFiniteSpanTarget : Prop :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Finset S)
+    (_ : Ideal.span (s : Set S) = ⊤)
+    (_ : ∀ r : s, P ((algebraMap S (Localization.Away (r : S))).comp f)), P f
+
+/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationSpanTarget`
+if `P` holds for `R →+* S` whenever there exists a set `{ r }` that spans `S` such that
+`P` holds for `R →+* Sᵣ`.
+
+Note that this is equivalent to `RingHom.OfLocalizationFiniteSpanTarget` via
+`RingHom.ofLocalizationSpanTarget_iff_finite`, but this has less restrictions when applying. -/
+def RingHom.OfLocalizationSpanTarget : Prop :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (_ : Ideal.span s = ⊤)
+    (_ : ∀ r : s, P ((algebraMap S (Localization.Away (r : S))).comp f)), P f
+
+/-- A property `P` of ring homs satisfies `RingHom.OfLocalizationPrime`
+if `P` holds for `R` whenever `P` holds for `Rₘ` for all prime ideals `p`. -/
+def RingHom.OfLocalizationPrime : Prop :=
+  ∀ ⦃R S : Type u⦄ [CommRing R] [CommRing S] (f : R →+* S),
+    (∀ (J : Ideal S) (_ : J.IsPrime), P (Localization.localRingHom _ J f rfl)) → P f
+
+/-- A property of ring homs is local if it is preserved by localizations and compositions, and for
+each `{ r }` that spans `S`, we have `P (R →+* S) ↔ ∀ r, P (R →+* Sᵣ)`. -/
+structure RingHom.PropertyIsLocal : Prop where
+  LocalizationPreserves : RingHom.LocalizationPreserves @P
+  OfLocalizationSpanTarget : RingHom.OfLocalizationSpanTarget @P
+  StableUnderCompositionWithLocalizationAway : RingHom.StableUnderCompositionWithLocalizationAway @P
+
+theorem RingHom.ofLocalizationSpan_iff_finite :
+    RingHom.OfLocalizationSpan @P ↔ RingHom.OfLocalizationFiniteSpan @P := by
+  delta RingHom.OfLocalizationSpan RingHom.OfLocalizationFiniteSpan
+  apply forall₅_congr
+  -- TODO: Using `refine` here breaks `resetI`.
+  intros
+  constructor
+  · intro h s; exact h s
+  · intro h s hs hs'
+    obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
+    exact h s' h₂ fun x => hs' ⟨_, h₁ x.prop⟩
+
+theorem RingHom.ofLocalizationSpanTarget_iff_finite :
+    RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P := by
+  delta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget
+  apply forall₅_congr
+  -- TODO: Using `refine` here breaks `resetI`.
+  intros
+  constructor
+  · intro h s; exact h s
+  · intro h s hs hs'
+    obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
+    exact h s' h₂ fun x => hs' ⟨_, h₁ x.prop⟩
+
+theorem RingHom.HoldsForLocalizationAway.of_bijective
+    (H : RingHom.HoldsForLocalizationAway P) (hf : Function.Bijective f) :
+    P f := by
+  letI := f.toAlgebra
+  have := IsLocalization.at_units (.powers (1 : R)) (by simp)
+  have := IsLocalization.isLocalization_of_algEquiv (.powers (1 : R))
+    (AlgEquiv.ofBijective (Algebra.ofId R S) hf)
+  exact H _ 1
+
+variable {P f R' S'}
+
+lemma RingHom.StableUnderComposition.stableUnderCompositionWithLocalizationAway
+    (hPc : RingHom.StableUnderComposition P) (hPl : HoldsForLocalizationAway P) :
+    StableUnderCompositionWithLocalizationAway P := by
+  constructor
+  · introv _ hf
+    exact hPc _ _ hf (hPl T s)
+  · introv _ hf
+    exact hPc _ _ (hPl S r) hf
+
+lemma RingHom.HoldsForLocalizationAway.containsIdentities (hPl : HoldsForLocalizationAway P) :
+    ContainsIdentities P := by
+  introv R
+  exact hPl.of_bijective _ _ Function.bijective_id
+
+theorem RingHom.PropertyIsLocal.respectsIso (hP : RingHom.PropertyIsLocal @P) :
+    RingHom.RespectsIso @P := by
+  constructor
+  · intro R S T _ _ _ f e hf
+    letI := e.toRingHom.toAlgebra
+    have : IsLocalization.Away (1 : S) T :=
+      IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
+    exact hP.StableUnderCompositionWithLocalizationAway.left T (1 : S) f hf
+  · intro R S T _ _ _ f e hf
+    letI := e.toRingHom.toAlgebra
+    have : IsLocalization.Away (1 : R) S :=
+      IsLocalization.away_of_isUnit_of_bijective _ isUnit_one e.bijective
+    exact hP.StableUnderCompositionWithLocalizationAway.right S (1 : R) f hf
+
+-- Almost all arguments are implicit since this is not intended to use mid-proof.
+theorem RingHom.LocalizationPreserves.away (H : RingHom.LocalizationPreserves @P) (r : R)
+    [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] (hf : P f) :
+    P (IsLocalization.Away.map R' S' f r) := by
+  have : IsLocalization ((Submonoid.powers r).map f) S' := by rw [Submonoid.map_powers]; assumption
+  exact H f (Submonoid.powers r) R' S' hf
+
+lemma RingHom.PropertyIsLocal.HoldsForLocalizationAway (hP : RingHom.PropertyIsLocal @P)
+    (hPi : ContainsIdentities P) :
+    RingHom.HoldsForLocalizationAway @P := by
+  introv R _
+  have : algebraMap R S = (algebraMap R S).comp (RingHom.id R) := by simp
+  rw [this]
+  apply (hP.StableUnderCompositionWithLocalizationAway).left S r
+  apply hPi
+
+theorem RingHom.PropertyIsLocal.ofLocalizationSpan (hP : RingHom.PropertyIsLocal @P) :
+    RingHom.OfLocalizationSpan @P := by
+  introv R hs hs'
+  apply_fun Ideal.map f at hs
+  rw [Ideal.map_span, Ideal.map_top] at hs
+  apply hP.OfLocalizationSpanTarget _ _ hs
+  rintro ⟨_, r, hr, rfl⟩
+  rw [← IsLocalization.map_comp (M := Submonoid.powers r) (S := Localization.Away r)
+    (T := Submonoid.powers (f r))]
+  · apply hP.StableUnderCompositionWithLocalizationAway.right _ r
+    exact hs' ⟨r, hr⟩
+
+lemma RingHom.OfLocalizationSpanTarget.ofIsLocalization
+    (hP : RingHom.OfLocalizationSpanTarget P) (hP' : RingHom.RespectsIso P)
+    {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) (s : Set S) (hs : Ideal.span s = ⊤)
+    (hT : ∀ r : s, ∃ (T : Type u) (_ : CommRing T) (_ : Algebra S T)
+      (_ : IsLocalization.Away (r : S) T), P ((algebraMap S T).comp f)) : P f := by
+  apply hP _ s hs
+  intros r
+  obtain ⟨T, _, _, _, hT⟩ := hT r
+  convert hP'.1 _
+    (Localization.algEquiv (R := S) (Submonoid.powers (r : S)) T).symm.toRingEquiv hT
+  rw [← RingHom.comp_assoc, RingEquiv.toRingHom_eq_coe, AlgEquiv.toRingEquiv_eq_coe,
+    AlgEquiv.toRingEquiv_toRingHom, Localization.coe_algEquiv_symm, IsLocalization.map_comp,
+    RingHom.comp_id]
+
+end RingHom
+
+end Properties
+
+section Ideal
+
+open scoped nonZeroDivisors
+
+/-- Let `I J : Ideal R`. If the localization of `I` at each maximal ideal `P` is included in
+the localization of `J` at `P`, then `I ≤ J`. -/
+theorem Ideal.le_of_localization_maximal {I J : Ideal R}
+    (h : ∀ (P : Ideal R) (hP : P.IsMaximal),
+      Ideal.map (algebraMap R (Localization.AtPrime P)) I ≤
+        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
+    I ≤ J := by
+  intro x hx
+  suffices J.colon (Ideal.span {x}) = ⊤ by
+    simpa using Submodule.mem_colon.mp
+      (show (1 : R) ∈ J.colon (Ideal.span {x}) from this.symm ▸ Submodule.mem_top) x
+      (Ideal.mem_span_singleton_self x)
+  refine Not.imp_symm (J.colon (Ideal.span {x})).exists_le_maximal ?_
+  push_neg
+  intro P hP le
+  obtain ⟨⟨⟨a, ha⟩, ⟨s, hs⟩⟩, eq⟩ :=
+    (IsLocalization.mem_map_algebraMap_iff P.primeCompl _).mp (h P hP (Ideal.mem_map_of_mem _ hx))
+  rw [← _root_.map_mul, ← sub_eq_zero, ← map_sub] at eq
+  obtain ⟨⟨m, hm⟩, eq⟩ := (IsLocalization.map_eq_zero_iff P.primeCompl _ _).mp eq
+  refine hs ((hP.isPrime.mem_or_mem (le (Ideal.mem_colon_singleton.mpr ?_))).resolve_right hm)
+  simp only [Subtype.coe_mk, mul_sub, sub_eq_zero, mul_comm x s, mul_left_comm] at eq
+  simpa only [mul_assoc, eq] using J.mul_mem_left m ha
+
+/-- Let `I J : Ideal R`. If the localization of `I` at each maximal ideal `P` is equal to
+the localization of `J` at `P`, then `I = J`. -/
+theorem Ideal.eq_of_localization_maximal {I J : Ideal R}
+    (h : ∀ (P : Ideal R) (_ : P.IsMaximal),
+      Ideal.map (algebraMap R (Localization.AtPrime P)) I =
+        Ideal.map (algebraMap R (Localization.AtPrime P)) J) :
+    I = J :=
+  le_antisymm (Ideal.le_of_localization_maximal fun P hP => (h P hP).le)
+    (Ideal.le_of_localization_maximal fun P hP => (h P hP).ge)
+
+/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/
+theorem ideal_eq_bot_of_localization' (I : Ideal R)
+    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
+      Ideal.map (algebraMap R (Localization.AtPrime J)) I = ⊥) :
+    I = ⊥ :=
+  Ideal.eq_of_localization_maximal fun P hP => by simpa using h P hP
+
+-- TODO: This proof should work for all modules, once we have enough material on submodules of
+-- localized modules.
+/-- An ideal is trivial if its localization at every maximal ideal is trivial. -/
+theorem ideal_eq_bot_of_localization (I : Ideal R)
+    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal),
+      IsLocalization.coeSubmodule (Localization.AtPrime J) I = ⊥) :
+    I = ⊥ :=
+  ideal_eq_bot_of_localization' _ fun P hP =>
+    (Ideal.map_eq_bot_iff_le_ker _).mpr fun x hx => by
+      rw [RingHom.mem_ker, ← Submodule.mem_bot R, ← h P hP, IsLocalization.mem_coeSubmodule]
+      exact ⟨x, hx, rfl⟩
+
+theorem eq_zero_of_localization (r : R)
+    (h : ∀ (J : Ideal R) (hJ : J.IsMaximal), algebraMap R (Localization.AtPrime J) r = 0) :
+    r = 0 := by
+  rw [← Ideal.span_singleton_eq_bot]
+  apply ideal_eq_bot_of_localization
+  intro J hJ
+  delta IsLocalization.coeSubmodule
+  erw [Submodule.map_span, Submodule.span_eq_bot]
+  rintro _ ⟨_, h', rfl⟩
+  cases Set.mem_singleton_iff.mpr h'
+  exact h J hJ
+
+end Ideal
diff --git a/Mathlib/RingTheory/LocalProperties/IntegrallyClosed.lean b/Mathlib/RingTheory/LocalProperties/IntegrallyClosed.lean
new file mode 100644
index 0000000000000..3753f3c70b5be
--- /dev/null
+++ b/Mathlib/RingTheory/LocalProperties/IntegrallyClosed.lean
@@ -0,0 +1,68 @@
+/-
+Copyright (c) 2024 Yongle Hu. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yongle Hu
+-/
+import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
+import Mathlib.RingTheory.LocalProperties.Basic
+import Mathlib.RingTheory.Localization.LocalizationLocalization
+
+/-!
+# `IsIntegrallyClosed` is a local property
+
+In this file, we prove that `IsIntegrallyClosed` is a local property.
+
+## Main results
+
+* `IsIntegrallyClosed.of_localization_maximal` : An integral domain `R` is integral closed
+  if `Rₘ` is integral closed for any maximal ideal `m` of `R`.
+
+## TODO
+
+Prove that `IsIntegrallyClosed` is preserved by localization
+
+-/
+
+open scoped nonZeroDivisors
+
+open Localization Ideal IsLocalization
+
+/-- An integral domain `R` is integral closed if `Rₘ` is integral closed
+  for any maximal ideal `m` of `R`. -/
+theorem IsIntegrallyClosed.of_localization_maximal {R : Type*} [CommRing R] [IsDomain R]
+    (h : ∀ p : Ideal R, p ≠ ⊥ → [p.IsMaximal] → IsIntegrallyClosed (Localization.AtPrime p)) :
+    IsIntegrallyClosed R := by
+  by_cases hf : IsField R
+  · exact hf.toField.instIsIntegrallyClosed
+  apply (isIntegrallyClosed_iff (FractionRing R)).mpr
+  rintro ⟨x⟩ hx
+  let I : Ideal R := span {x.2.1} / span {x.1}
+  have h1 : 1 ∈ I := by
+    apply I.eq_top_iff_one.mp
+    by_contra hn
+    rcases I.exists_le_maximal hn with ⟨p, hpm, hpi⟩
+    have hic := h p (Ring.ne_bot_of_isMaximal_of_not_isField hpm hf)
+    have hxp : IsIntegral (Localization.AtPrime p) (mk x.1 x.2) := hx.tower_top
+    /- `x.1 / x.2.1 ∈ Rₚ` since it is integral over `Rₚ` and `Rₚ` is integrally closed.
+      More precisely, `x.1 / x.2.1 = y.1 / y.2.1` where `y.1, y.2.1 ∈ R` and `y.2.1 ∉ p`. -/
+    rcases (isIntegrallyClosed_iff (FractionRing R)).mp hic hxp with ⟨⟨y⟩, hy⟩
+    /- `y.2.1 ∈ I` since for all `a ∈ Ideal.span {x.1}`, say `a = b * x.1`,
+      we have `y.2 * a = b * x.1 * y.2 = b * y.1 * x.2.1 ∈ Ideal.span {x.2.1}`. -/
+    have hyi : y.2.1 ∈ I := by
+      intro a ha
+      rcases mem_span_singleton'.mp ha with ⟨b, hb⟩
+      apply mem_span_singleton'.mpr ⟨b * y.1, _⟩
+      rw [← hb, ← mul_assoc, mul_comm y.2.1 b, mul_assoc, mul_assoc]
+      exact congrArg (HMul.hMul b) <| (mul_comm y.1 x.2.1).trans <|
+        NoZeroSMulDivisors.algebraMap_injective R (Localization R⁰) <| mk'_eq_iff_eq.mp <|
+          (mk'_eq_algebraMap_mk'_of_submonoid_le _ _ p.primeCompl_le_nonZeroDivisors y.1 y.2).trans
+            <| show algebraMap (Localization.AtPrime p) _ (mk' _ y.1 y.2) = mk' _ x.1 x.2
+              by simpa only [← mk_eq_mk', ← hy] using by rfl
+    -- `y.2.1 ∈ I` implies `y.2.1 ∈ p` since `I ⊆ p`, which contradicts to the choice of `y`.
+    exact y.2.2 (hpi hyi)
+  rcases mem_span_singleton'.mp (h1 x.1 (mem_span_singleton_self x.1)) with ⟨y, hy⟩
+  exact ⟨y, (eq_mk'_of_mul_eq (hy.trans (one_mul x.1))).trans (mk_eq_mk'_apply x.1 x.2).symm⟩
+
+theorem isIntegrallyClosed_ofLocalizationMaximal :
+    OfLocalizationMaximal fun R _ => ([IsDomain R] → IsIntegrallyClosed R) :=
+  fun _ _ h _ ↦ IsIntegrallyClosed.of_localization_maximal fun p _ hpm ↦ h p hpm
diff --git a/Mathlib/RingTheory/LocalProperties/Reduced.lean b/Mathlib/RingTheory/LocalProperties/Reduced.lean
new file mode 100644
index 0000000000000..65df178848887
--- /dev/null
+++ b/Mathlib/RingTheory/LocalProperties/Reduced.lean
@@ -0,0 +1,53 @@
+/-
+Copyright (c) 2021 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+-/
+import Mathlib.RingTheory.LocalProperties.Basic
+
+/-!
+# `IsReduced` is a local property
+
+In this file, we prove that `IsReduced` is a local property.
+
+## Main results
+
+Let `R` be a commutative ring, `M` be a submonoid of `R`.
+
+* `isReduced_localizationPreserves` :  `M⁻¹R` is reduced if `R` is reduced.
+* `isReduced_ofLocalizationMaximal` : `R` is reduced if `Rₘ` is reduced for all maximal ideal `m`.
+
+-/
+
+/-- `M⁻¹R` is reduced if `R` is reduced. -/
+theorem isReduced_localizationPreserves : LocalizationPreserves fun R hR => IsReduced R := by
+  introv R _ _
+  constructor
+  rintro x ⟨_ | n, e⟩
+  · simpa using congr_arg (· * x) e
+  obtain ⟨⟨y, m⟩, hx⟩ := IsLocalization.surj M x
+  dsimp only at hx
+  let hx' := congr_arg (· ^ n.succ) hx
+  simp only [mul_pow, e, zero_mul, ← RingHom.map_pow] at hx'
+  rw [← (algebraMap R S).map_zero] at hx'
+  obtain ⟨m', hm'⟩ := (IsLocalization.eq_iff_exists M S).mp hx'
+  apply_fun (· * (m' : R) ^ n) at hm'
+  simp only [mul_assoc, zero_mul, mul_zero] at hm'
+  rw [← mul_left_comm, ← pow_succ', ← mul_pow] at hm'
+  replace hm' := IsNilpotent.eq_zero ⟨_, hm'.symm⟩
+  rw [← (IsLocalization.map_units S m).mul_left_inj, hx, zero_mul,
+    IsLocalization.map_eq_zero_iff M]
+  exact ⟨m', by rw [← hm', mul_comm]⟩
+
+instance {R : Type*} [CommRing R] (M : Submonoid R) [IsReduced R] : IsReduced (Localization M) :=
+  isReduced_localizationPreserves M _ inferInstance
+
+/-- `R` is reduced if `Rₘ` is reduced for all maximal ideal `m`. -/
+theorem isReduced_ofLocalizationMaximal : OfLocalizationMaximal fun R hR => IsReduced R := by
+  introv R h
+  constructor
+  intro x hx
+  apply eq_zero_of_localization
+  intro J hJ
+  specialize h J hJ
+  exact (hx.map <| algebraMap R <| Localization.AtPrime J).eq_zero
diff --git a/Mathlib/RingTheory/LocalRing/Basic.lean b/Mathlib/RingTheory/LocalRing/Basic.lean
index 8e057c3e33536..a7bb4b615fa7b 100644
--- a/Mathlib/RingTheory/LocalRing/Basic.lean
+++ b/Mathlib/RingTheory/LocalRing/Basic.lean
@@ -63,7 +63,7 @@ theorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a
   apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right
 
 theorem nonunits_add {a b : R} (ha : a ∈ nonunits R) (hb : b ∈ nonunits R) : a + b ∈ nonunits R :=
-  fun H => not_or_of_not ha hb (isUnit_or_isUnit_of_isUnit_add H)
+  fun H => not_or_intro ha hb (isUnit_or_isUnit_of_isUnit_add H)
 
 end LocalRing
 
diff --git a/Mathlib/RingTheory/LocalRing/Module.lean b/Mathlib/RingTheory/LocalRing/Module.lean
index 0d777447494d0..332bccee7c8f5 100644
--- a/Mathlib/RingTheory/LocalRing/Module.lean
+++ b/Mathlib/RingTheory/LocalRing/Module.lean
@@ -93,7 +93,7 @@ theorem span_eq_top_of_tmul_eq_basis [Module.Finite R M] {ι}
   rw [← map_tensorProduct_mk_eq_top, Submodule.map_span, ← Submodule.restrictScalars_span R k
     Ideal.Quotient.mk_surjective, Submodule.restrictScalars_eq_top_iff,
     ← b.span_eq, ← Set.range_comp]
-  simp only [Function.comp, mk_apply, hb, Basis.span_eq]
+  simp only [Function.comp_def, mk_apply, hb, Basis.span_eq]
 
 end LocalRing
 
@@ -152,9 +152,9 @@ theorem free_of_maximalIdeal_rTensor_injective [Module.FinitePresentation R M]
     isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (I →₀ R))
   choose f hf using TensorProduct.mk_surjective R M k Ideal.Quotient.mk_surjective
   -- By choosing an arbitrary lift of `b` to `I → M`, we get a surjection `i : Rᴵ → M`.
-  let i := Finsupp.total R (f ∘ b)
+  let i := Finsupp.linearCombination R (f ∘ b)
   have hi : Surjective i := by
-    rw [← LinearMap.range_eq_top, Finsupp.range_total]
+    rw [← LinearMap.range_eq_top, Finsupp.range_linearCombination]
     exact LocalRing.span_eq_top_of_tmul_eq_basis (R := R) (f := f ∘ b) b (fun _ ↦ hf _)
   have : Module.Finite R (LinearMap.ker i) := by
     constructor
@@ -171,26 +171,28 @@ theorem free_of_maximalIdeal_rTensor_injective [Module.FinitePresentation R M]
     refine ⟨?_, this⟩
     rw [← LinearMap.ker_eq_bot (M := k ⊗[R] (I →₀ R)) (f := i.baseChange k),
       ← Submodule.finrank_eq_zero (R := k) (M := k ⊗[R] (I →₀ R)),
-      ← Nat.add_right_inj (n := FiniteDimensional.finrank k (LinearMap.range <| i.baseChange k)),
+      ← Nat.add_right_inj (n := Module.finrank k (LinearMap.range <| i.baseChange k)),
       LinearMap.finrank_range_add_finrank_ker (V := k ⊗[R] (I →₀ R)),
       LinearMap.range_eq_top.mpr this, finrank_top]
-    simp only [FiniteDimensional.finrank_tensorProduct, FiniteDimensional.finrank_self,
-      FiniteDimensional.finrank_finsupp_self, one_mul, add_zero]
-    rw [FiniteDimensional.finrank_eq_card_chooseBasisIndex]
+    simp only [Module.finrank_tensorProduct, Module.finrank_self,
+      Module.finrank_finsupp_self, one_mul, add_zero]
+    rw [Module.finrank_eq_card_chooseBasisIndex]
   -- On the other hand, `m ⊗ M → M` injective => `Tor₁(k, M) = 0` => `k ⊗ ker(i) → kᴵ` injective.
-  have := @lTensor_injective_of_exact_of_exact_of_rTensor_injective
-    (N₁ := LinearMap.ker i) (N₂ := I →₀ R) (N₃ := M)
-    (f₁ := (𝔪).subtype) (f₂ := Submodule.mkQ 𝔪) inferInstance inferInstance inferInstance
-    inferInstance inferInstance inferInstance
   intro x
-  apply @this (LinearMap.ker i).subtype i (LinearMap.exact_subtype_mkQ 𝔪)
-    (Submodule.mkQ_surjective _) (LinearMap.exact_subtype_ker_map i) hi H
-    (Module.Flat.lTensor_preserves_injective_linearMap _ Subtype.val_injective)
-  apply hi'.injective
-  rw [LinearMap.baseChange_eq_ltensor]
-  erw [← LinearMap.comp_apply (i.lTensor k), ← LinearMap.lTensor_comp]
-  rw [(LinearMap.exact_subtype_ker_map i).linearMap_comp_eq_zero]
-  simp only [LinearMap.lTensor_zero, LinearMap.zero_apply, map_zero]
+  refine lTensor_injective_of_exact_of_exact_of_rTensor_injective
+    (N₁ := LinearMap.ker i) (N₂ := I →₀ R) (N₃ := M)
+    (f₁ := (𝔪).subtype) (f₂ := Submodule.mkQ 𝔪)
+    (g₁ := (LinearMap.ker i).subtype) (g₂ := i) (LinearMap.exact_subtype_mkQ 𝔪)
+    (Submodule.mkQ_surjective _) (LinearMap.exact_subtype_ker_map i) hi H ?_ ?_
+  · apply Module.Flat.lTensor_preserves_injective_linearMap
+      (N := LinearMap.ker i) (N' := I →₀ R)
+      (L := (LinearMap.ker i).subtype)
+    exact Subtype.val_injective
+  · apply hi'.injective
+    rw [LinearMap.baseChange_eq_ltensor]
+    erw [← LinearMap.comp_apply (i.lTensor k), ← LinearMap.lTensor_comp]
+    rw [(LinearMap.exact_subtype_ker_map i).linearMap_comp_eq_zero]
+    simp only [LinearMap.lTensor_zero, LinearMap.zero_apply, map_zero]
 
 -- TODO: Generalise this to finite free modules.
 theorem free_of_flat_of_localRing [Module.FinitePresentation R P] [Module.Flat R P] :
diff --git a/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean b/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean
index 25391bf0ccb40..885d727ff53f4 100644
--- a/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean
+++ b/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean
@@ -3,9 +3,9 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Chris Hughes, Mario Carneiro
 -/
+import Mathlib.LinearAlgebra.FiniteDimensional.Defs
 import Mathlib.RingTheory.LocalRing.ResidueField.Defs
 import Mathlib.RingTheory.LocalRing.RingHom.Basic
-import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
 
 /-!
 
@@ -145,6 +145,34 @@ theorem residue_smul (g : G) (r : R) : residue R (g • r) = g • residue R r :
 
 end MulSemiringAction
 
+section FiniteDimensional
+
+variable [Algebra R S] [IsLocalRingHom (algebraMap R S)]
+
+noncomputable instance : Algebra (ResidueField R) (ResidueField S) :=
+  (ResidueField.map (algebraMap R S)).toAlgebra
+
+noncomputable instance : Algebra R (ResidueField S) :=
+  ((ResidueField.map <| algebraMap R S).comp <| residue R).toAlgebra
+
+instance : IsScalarTower R (ResidueField R) (ResidueField S) :=
+  IsScalarTower.of_algebraMap_eq (congrFun rfl)
+
+instance finiteDimensional_of_noetherian [IsNoetherian R S] :
+    FiniteDimensional (ResidueField R) (ResidueField S) := by
+  apply IsNoetherian.iff_fg.mp <|
+    isNoetherian_of_tower R (S := ResidueField R) (M := ResidueField S) _
+  convert isNoetherian_of_surjective S (Ideal.Quotient.mkₐ R (maximalIdeal S)).toLinearMap
+    (LinearMap.range_eq_top.mpr Ideal.Quotient.mk_surjective)
+  exact Algebra.algebra_ext _ _ (fun r => rfl)
+
+lemma finite_of_finite [IsNoetherian R S] (hfin : Finite (ResidueField R)) :
+    Finite (ResidueField S) := by
+  have := @finiteDimensional_of_noetherian R S
+  exact FiniteDimensional.finite_of_finite (ResidueField R) (ResidueField S)
+
+end FiniteDimensional
+
 end ResidueField
 
 theorem isLocalRingHom_residue : IsLocalRingHom (LocalRing.residue R) := by
diff --git a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
index fc9ea8c3bc6a3..4b44e62f3ee66 100644
--- a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
+++ b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
@@ -3,7 +3,7 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Chris Hughes, Mario Carneiro
 -/
-import Mathlib.RingTheory.LocalRing.RingHom.Defs
+import Mathlib.Algebra.Group.Units.Hom
 import Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs
 import Mathlib.RingTheory.Ideal.Maps
 import Mathlib.Logic.Equiv.TransferInstance
@@ -24,34 +24,14 @@ variable [Semiring R] [Semiring S] [Semiring T]
 instance isLocalRingHom_id (R : Type*) [Semiring R] : IsLocalRingHom (RingHom.id R) where
   map_nonunit _ := id
 
-@[simp]
-theorem isUnit_map_iff (f : R →+* S) [IsLocalRingHom f] (a) : IsUnit (f a) ↔ IsUnit a :=
-  ⟨IsLocalRingHom.map_nonunit a, f.isUnit_map⟩
+-- see note [lower instance priority]
+instance (priority := 100)  isLocalRingHom_toRingHom {F : Type*} [FunLike F R S]
+   [RingHomClass F R S] (f : F) [IsLocalRingHom f] : IsLocalRingHom (f : R →+* S) :=
+  ⟨IsLocalRingHom.map_nonunit (f := f)⟩
 
--- Porting note: as this can be proved by other `simp` lemmas, this is marked as high priority.
-@[simp (high)]
-theorem map_mem_nonunits_iff (f : R →+* S) [IsLocalRingHom f] (a) :
-    f a ∈ nonunits S ↔ a ∈ nonunits R :=
-  ⟨fun h ha => h <| (isUnit_map_iff f a).mpr ha, fun h ha => h <| (isUnit_map_iff f a).mp ha⟩
-
-instance isLocalRingHom_comp (g : S →+* T) (f : R →+* S) [IsLocalRingHom g] [IsLocalRingHom f] :
-    IsLocalRingHom (g.comp f) where
-  map_nonunit a := IsLocalRingHom.map_nonunit a ∘ IsLocalRingHom.map_nonunit (f a)
-
-instance isLocalRingHom_equiv (f : R ≃+* S) : IsLocalRingHom (f : R →+* S) where
-  map_nonunit a ha := by
-    convert RingHom.isUnit_map (f.symm : S →+* R) ha
-    exact (RingEquiv.symm_apply_apply f a).symm
-
-@[simp]
-theorem isUnit_of_map_unit (f : R →+* S) [IsLocalRingHom f] (a) (h : IsUnit (f a)) : IsUnit a :=
-  IsLocalRingHom.map_nonunit a h
-
-theorem of_irreducible_map (f : R →+* S) [h : IsLocalRingHom f] {x} (hfx : Irreducible (f x)) :
-    Irreducible x :=
-  ⟨fun h => hfx.not_unit <| IsUnit.map f h, fun p q hx =>
-    let ⟨H⟩ := h
-    Or.imp (H p) (H q) <| hfx.isUnit_or_isUnit <| f.map_mul p q ▸ congr_arg f hx⟩
+instance RingHom.isLocalRingHom_comp (g : S →+* T) (f : R →+* S) [IsLocalRingHom g]
+    [IsLocalRingHom f] : IsLocalRingHom (g.comp f) where
+  map_nonunit a := IsLocalRingHom.map_nonunit a ∘ IsLocalRingHom.map_nonunit (f := g) (f a)
 
 theorem isLocalRingHom_of_comp (f : R →+* S) (g : S →+* T) [IsLocalRingHom (g.comp f)] :
     IsLocalRingHom f :=
@@ -97,22 +77,13 @@ theorem local_hom_TFAE (f : R →+* S) :
         (maximalIdeal R).map f ≤ maximalIdeal S, maximalIdeal R ≤ (maximalIdeal S).comap f,
         (maximalIdeal S).comap f = maximalIdeal R] := by
   tfae_have 1 → 2
-  · rintro _ _ ⟨a, ha, rfl⟩
-    exact map_nonunit f a ha
-  tfae_have 2 → 4
-  · exact Set.image_subset_iff.1
-  tfae_have 3 ↔ 4
-  · exact Ideal.map_le_iff_le_comap
-  tfae_have 4 → 1
-  · intro h
-    constructor
-    exact fun x => not_imp_not.1 (@h x)
+  | _, _, ⟨a, ha, rfl⟩ => map_nonunit f a ha
+  tfae_have 2 → 4 := Set.image_subset_iff.1
+  tfae_have 3 ↔ 4 := Ideal.map_le_iff_le_comap
+  tfae_have 4 → 1 := fun h ↦ ⟨fun x => not_imp_not.1 (@h x)⟩
   tfae_have 1 → 5
-  · intro
-    ext
-    exact not_iff_not.2 (isUnit_map_iff f _)
-  tfae_have 5 → 4
-  · exact fun h => le_of_eq h.symm
+  | _ => by ext; exact not_iff_not.2 (isUnit_map_iff f _)
+  tfae_have 5 → 4 := fun h ↦ le_of_eq h.symm
   tfae_finish
 
 end
diff --git a/Mathlib/RingTheory/LocalRing/RingHom/Defs.lean b/Mathlib/RingTheory/LocalRing/RingHom/Defs.lean
deleted file mode 100644
index 5949c6dcab0ff..0000000000000
--- a/Mathlib/RingTheory/LocalRing/RingHom/Defs.lean
+++ /dev/null
@@ -1,28 +0,0 @@
-/-
-Copyright (c) 2018 Kenny Lau. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kenny Lau, Chris Hughes, Mario Carneiro
--/
-import Mathlib.Algebra.Group.Units
-import Mathlib.Algebra.Ring.Hom.Defs
-
-/-!
-
-# Local rings homomorphisms
-
-We Define local ring homomorhpisms.
-
-## Main definitions
-
-* `IsLocalRingHom`: A predicate on semiring homomorphisms, requiring that it maps nonunits
-  to nonunits. For local rings, this means that the image of the unique maximal ideal is again
-  contained in the unique maximal ideal.
-
--/
-
-/-- A local ring homomorphism is a homomorphism `f` between local rings such that `a` in the domain
-  is a unit if `f a` is a unit for any `a`. See `LocalRing.local_hom_TFAE` for other equivalent
-  definitions. -/
-class IsLocalRingHom {R S : Type*} [Semiring R] [Semiring S] (f : R →+* S) : Prop where
-  /-- A local ring homomorphism `f : R ⟶ S` will send nonunits of `R` to nonunits of `S`. -/
-  map_nonunit : ∀ a, IsUnit (f a) → IsUnit a
diff --git a/Mathlib/RingTheory/Localization/Algebra.lean b/Mathlib/RingTheory/Localization/Algebra.lean
index bf2fa84c5d38c..794c17c9aea4a 100644
--- a/Mathlib/RingTheory/Localization/Algebra.lean
+++ b/Mathlib/RingTheory/Localization/Algebra.lean
@@ -56,12 +56,12 @@ variable (S) in
 /-- The canonical linear map from the kernel of `g` to the kernel of its localization. -/
 def RingHom.toKerIsLocalization (hy : M ≤ Submonoid.comap g T) :
     RingHom.ker g →ₗ[R] RingHom.ker (IsLocalization.map Q g hy : S →+* Q) where
-  toFun x := ⟨algebraMap R S x, by simp [RingHom.mem_ker, (RingHom.mem_ker g).mp x.property]⟩
+  toFun x := ⟨algebraMap R S x, by simp [RingHom.mem_ker, RingHom.mem_ker.mp x.property]⟩
   map_add' x y := by
-    simp only [AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, map_add, AddSubmonoid.mk_add_mk]
+    simp only [Submodule.coe_add, map_add, AddMemClass.mk_add_mk]
   map_smul' a x := by
-    simp only [SetLike.val_smul, smul_eq_mul, map_mul, RingHom.id_apply,
-      SetLike.mk_smul_of_tower_mk, Algebra.smul_def]
+    simp only [SetLike.val_smul, smul_eq_mul, map_mul, id_apply, SetLike.mk_smul_of_tower_mk,
+      Algebra.smul_def]
 
 @[simp]
 lemma RingHom.toKerIsLocalization_apply (hy : M ≤ Submonoid.comap g T) (r : RingHom.ker g) :
diff --git a/Mathlib/RingTheory/Localization/AtPrime.lean b/Mathlib/RingTheory/Localization/AtPrime.lean
index 6ff9e226beb8a..5b122fb2d696c 100644
--- a/Mathlib/RingTheory/Localization/AtPrime.lean
+++ b/Mathlib/RingTheory/Localization/AtPrime.lean
@@ -5,7 +5,7 @@ Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baan
 -/
 import Mathlib.RingTheory.Localization.Ideal
 import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic
-import Mathlib.RingTheory.LocalRing.RingHom.Defs
+import Mathlib.Algebra.Group.Units.Hom
 
 /-!
 # Localizations of commutative rings at the complement of a prime ideal
@@ -89,7 +89,7 @@ theorem AtPrime.localRing [IsLocalization.AtPrime S P] : LocalRing S :=
       obtain ⟨t, ht⟩ := IsLocalization.eq.1 hxyz
       simp only [mul_one, one_mul, Submonoid.coe_mul, Subtype.coe_mk] at ht
       suffices (t : R) * (sx * sy * sz) ∈ P from
-        not_or_of_not (mt hp.mem_or_mem <| not_or_of_not sx.2 sy.2) sz.2
+        not_or_intro (mt hp.mem_or_mem <| not_or_intro sx.2 sy.2) sz.2
           (hp.mem_or_mem <| (hp.mem_or_mem this).resolve_left t.2)
       rw [← ht]
       exact
diff --git a/Mathlib/RingTheory/Localization/Away/Basic.lean b/Mathlib/RingTheory/Localization/Away/Basic.lean
index c41d650a75ef4..eaaebda203892 100644
--- a/Mathlib/RingTheory/Localization/Away/Basic.lean
+++ b/Mathlib/RingTheory/Localization/Away/Basic.lean
@@ -38,7 +38,8 @@ section Away
 variable (x : R)
 
 /-- Given `x : R`, the typeclass `IsLocalization.Away x S` states that `S` is
-isomorphic to the localization of `R` at the submonoid generated by `x`. -/
+isomorphic to the localization of `R` at the submonoid generated by `x`.
+See `IsLocalization.Away.mk` for a specialized constructor. -/
 abbrev Away (S : Type*) [CommSemiring S] [Algebra R S] :=
   IsLocalization (Submonoid.powers x) S
 
@@ -68,6 +69,58 @@ lemma sec_spec (s : S) : s * (algebraMap R S) (x ^ (IsLocalization.Away.sec x s)
   congr
   exact (IsLocalization.sec (Submonoid.powers x) s).2.property.choose_spec
 
+lemma algebraMap_pow_isUnit (n : ℕ) : IsUnit (algebraMap R S x ^ n) :=
+  IsUnit.pow _ <| IsLocalization.map_units _ (⟨x, 1, by simp⟩ : Submonoid.powers x)
+
+lemma algebraMap_isUnit : IsUnit (algebraMap R S x) :=
+  IsLocalization.map_units _ (⟨x, 1, by simp⟩ : Submonoid.powers x)
+
+lemma surj (z : S) : ∃ (n : ℕ) (a : R), z * algebraMap R S x ^ n = algebraMap R S a := by
+  obtain ⟨⟨a, ⟨-, n, rfl⟩⟩, h⟩ := IsLocalization.surj (Submonoid.powers x) z
+  use n, a
+  simpa using h
+
+lemma exists_of_eq {a b : R} (h : algebraMap R S a = algebraMap R S b) :
+    ∃ (n : ℕ), x ^ n * a = x ^ n * b := by
+  obtain ⟨⟨-, n, rfl⟩, hx⟩ := IsLocalization.exists_of_eq (M := Submonoid.powers x) h
+  use n
+
+/-- Specialized constructor for `IsLocalization.Away`. -/
+lemma mk (r : R) (map_unit : IsUnit (algebraMap R S r))
+    (surj : ∀ s, ∃ (n : ℕ) (a : R), s * algebraMap R S r ^ n = algebraMap R S a)
+    (exists_of_eq : ∀ a b, algebraMap R S a = algebraMap R S b → ∃ (n : ℕ), r ^ n * a = r ^ n * b) :
+    IsLocalization.Away r S where
+  map_units' := by
+    rintro ⟨-, n, rfl⟩
+    simp only [map_pow]
+    exact IsUnit.pow _ map_unit
+  surj' z := by
+    obtain ⟨n, a, hn⟩ := surj z
+    use ⟨a, ⟨r ^ n, n, rfl⟩⟩
+    simpa using hn
+  exists_of_eq {x y} h := by
+    obtain ⟨n, hn⟩ := exists_of_eq x y h
+    use ⟨r ^ n, n, rfl⟩
+
+lemma of_associated {r r' : R} (h : Associated r r') [IsLocalization.Away r S] :
+    IsLocalization.Away r' S := by
+  obtain ⟨u, rfl⟩ := h
+  refine mk _ ?_ (fun s ↦ ?_) (fun a b hab ↦ ?_)
+  · simp [algebraMap_isUnit r, IsUnit.map _ u.isUnit]
+  · obtain ⟨n, a, hn⟩ := surj r s
+    use n, a * u ^ n
+    simp [mul_pow, ← mul_assoc, hn]
+  · obtain ⟨n, hn⟩ := exists_of_eq r hab
+    use n
+    rw [mul_pow, mul_comm (r ^ n), mul_assoc, mul_assoc, hn]
+
+/-- If `r` and `r'` are associated elements of `R`, an `R`-algebra `S`
+is the localization of `R` away from `r` if and only of it is the localization of `R` away from
+`r'`. -/
+lemma iff_of_associated {r r' : R} (h : Associated r r') :
+    IsLocalization.Away r S ↔ IsLocalization.Away r' S :=
+  ⟨fun _ ↦ IsLocalization.Away.of_associated h, fun _ ↦ IsLocalization.Away.of_associated h.symm⟩
+
 variable {g : R →+* P}
 
 /-- Given `x : R`, a localization map `F : R →+* S` away from `x`, and a map of `CommSemiring`s
@@ -146,6 +199,53 @@ lemma mapₐ_surjective_of_surjective {f : A →ₐ[R] B} (a : A) [Away a Aₚ]
 
 end Algebra
 
+/-- Localizing the localization of `R` at `x` at the image of `y` is the same as localizing
+`R` at `y * x`. See `IsLocalization.Away.mul'` for the `x * y` version. -/
+lemma mul (T : Type*) [CommSemiring T] [Algebra S T]
+    [Algebra R T] [IsScalarTower R S T] (x y : R)
+    [IsLocalization.Away x S] [IsLocalization.Away (algebraMap R S y) T] :
+    IsLocalization.Away (y * x) T := by
+  refine mk _ ?_ (fun z ↦ ?_) (fun a b h ↦ ?_)
+  · simp only [map_mul, IsUnit.mul_iff, IsScalarTower.algebraMap_apply R S T]
+    exact ⟨algebraMap_isUnit _, IsUnit.map _ (algebraMap_isUnit x)⟩
+  · obtain ⟨m, p, hpq⟩ := surj (algebraMap R S y) z
+    obtain ⟨n, a, hab⟩ := surj x p
+    use m + n, a * x ^ m * y ^ n
+    simp only [mul_pow, pow_add, map_pow, map_mul, ← mul_assoc, hpq,
+      IsScalarTower.algebraMap_apply R S T, ← hab]
+    ring
+  · repeat rw [IsScalarTower.algebraMap_apply R S T] at h
+    obtain ⟨n, hn⟩ := exists_of_eq (algebraMap R S y) h
+    simp only [← map_pow, ← map_mul, ← map_mul] at hn
+    obtain ⟨m, hm⟩ := exists_of_eq x hn
+    use n + m
+    convert_to y ^ m * x ^ n * (x ^ m * (y ^ n * a)) = y ^ m * x ^ n * (x ^ m * (y ^ n * b))
+    · ring
+    · ring
+    · rw [hm]
+
+/-- Localizing the localization of `R` at `x` at the image of `y` is the same as localizing
+`R` at `x * y`. See `IsLocalization.Away.mul` for the `y * x` version. -/
+lemma mul' (T : Type*) [CommSemiring T] [Algebra S T] [Algebra R T] [IsScalarTower R S T] (x y : R)
+    [IsLocalization.Away x S] [IsLocalization.Away (algebraMap R S y) T] :
+    IsLocalization.Away (x * y) T :=
+  mul_comm x y ▸ mul S T x y
+
+/-- If `S₁` is the localization of `R` away from `f` and `S₂` is the localization away from `g`,
+then any localization `T` of `S₂` away from `f` is also a localization of `S₁` away from `g`. -/
+lemma commutes {R : Type*} [CommSemiring R] (S₁ S₂ T : Type*) [CommSemiring S₁]
+    [CommSemiring S₂] [CommSemiring T] [Algebra R S₁] [Algebra R S₂] [Algebra R T] [Algebra S₁ T]
+    [Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T] (x y : R)
+    [IsLocalization.Away x S₁] [IsLocalization.Away y S₂]
+    [IsLocalization.Away (algebraMap R S₂ x) T] :
+    IsLocalization.Away (algebraMap R S₁ y) T := by
+  haveI : IsLocalization (Algebra.algebraMapSubmonoid S₂ (Submonoid.powers x)) T := by
+    simp only [Algebra.algebraMapSubmonoid, Submonoid.map_powers]
+    infer_instance
+  convert IsLocalization.commutes S₁ S₂ T (Submonoid.powers x) (Submonoid.powers y)
+  ext x
+  simp [Algebra.algebraMapSubmonoid]
+
 end Away
 
 end Away
diff --git a/Mathlib/RingTheory/Localization/Away/Lemmas.lean b/Mathlib/RingTheory/Localization/Away/Lemmas.lean
new file mode 100644
index 0000000000000..640763c874cbe
--- /dev/null
+++ b/Mathlib/RingTheory/Localization/Away/Lemmas.lean
@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.RingTheory.Localization.Away.Basic
+import Mathlib.RingTheory.Localization.Submodule
+
+/-!
+# More lemmas on localization away
+
+This file contains lemmas on localization away from an element requiring more imports.
+
+-/
+
+variable {R : Type*} [CommRing R]
+
+namespace IsLocalization
+
+namespace Away
+
+/-- Given a set `s` in a ring `R` and for every `t : s` a set `p t` of fractions in
+a localization of `R` at `t`, this is the function sending a pair `(t, y)`, with
+`t : s` and `y : t a`, to `t` multiplied with a numerator of `y`. The range
+of this function spans the unit ideal, if `s` and every `p t` do. -/
+noncomputable def mulNumerator (s : Set R)
+    {Rₜ : s → Type*} [∀ t, CommRing (Rₜ t)] [∀ t, Algebra R (Rₜ t)]
+    [∀ t, IsLocalization.Away t.val (Rₜ t)]
+    (p : (t : s) → Set (Rₜ t)) (x : (t : s) × p t) : R :=
+  x.1 * (IsLocalization.Away.sec x.1.1 x.2.1).1
+
+lemma span_range_mulNumerator_eq_top {s : Set R}
+    (hsone : Ideal.span s = ⊤) {Rₜ : s → Type*} [∀ t, CommRing (Rₜ t)] [∀ t, Algebra R (Rₜ t)]
+    [∀ t, IsLocalization.Away t.val (Rₜ t)]
+    {p : (t : s) → Set (Rₜ t)} (htone : ∀ (r : s), Ideal.span (p r) = ⊤) :
+    Ideal.span (Set.range (IsLocalization.Away.mulNumerator s p)) = ⊤ := by
+  rw [← Ideal.radical_eq_top, eq_top_iff, ← hsone, Ideal.span_le]
+  intro a ha
+  haveI : IsLocalization (Submonoid.powers a) (Rₜ ⟨a, ha⟩) :=
+    inferInstanceAs <| IsLocalization.Away (⟨a, ha⟩ : s).val (Rₜ ⟨a, ha⟩)
+  have h₁ : Ideal.span (p ⟨a, ha⟩) ≤ Ideal.span
+      (algebraMap R (Rₜ ⟨a, ha⟩) '' Set.range (IsLocalization.Away.mulNumerator s p)) := by
+    rw [Ideal.span_le]
+    intro x hx
+    rw [SetLike.mem_coe, IsLocalization.mem_span_map (Submonoid.powers a)]
+    refine ⟨a * (IsLocalization.Away.sec a x).1, Ideal.subset_span ⟨⟨⟨a, ha⟩, ⟨x, hx⟩⟩, rfl⟩, ?_⟩
+    use ⟨a ^ ((IsLocalization.Away.sec a x).2 + 1), _, rfl⟩
+    rw [IsLocalization.eq_mk'_iff_mul_eq, map_pow, map_mul, ← map_pow, pow_add, map_mul,
+      ← mul_assoc, IsLocalization.Away.sec_spec a x, mul_comm, pow_one]
+  have h₂ : IsLocalization.mk' (Rₜ ⟨a, ha⟩) 1 (1 : Submonoid.powers a) ∈ Ideal.span
+      (algebraMap R (Rₜ ⟨a, ha⟩) ''
+        (Set.range <| IsLocalization.Away.mulNumerator s p)) := by
+    rw [IsLocalization.mk'_one]
+    apply h₁
+    simp [htone]
+  rw [IsLocalization.mem_span_map (Submonoid.powers a)] at h₂
+  obtain ⟨y, hy, ⟨-, m, rfl⟩, hyz⟩ := h₂
+  rw [IsLocalization.eq] at hyz
+  obtain ⟨⟨-, n, rfl⟩, hc⟩ := hyz
+  simp only [← mul_assoc, OneMemClass.coe_one, one_mul, mul_one] at hc
+  use n + m
+  simpa [pow_add, hc] using Ideal.mul_mem_left _ _ hy
+
+end Away
+
+end IsLocalization
diff --git a/Mathlib/RingTheory/Localization/Basic.lean b/Mathlib/RingTheory/Localization/Basic.lean
index ec5333ad7be07..ee3d1fbf1a3d4 100644
--- a/Mathlib/RingTheory/Localization/Basic.lean
+++ b/Mathlib/RingTheory/Localization/Basic.lean
@@ -262,6 +262,9 @@ theorem eq_mk'_iff_mul_eq {x} {y : M} {z} :
     z = mk' S x y ↔ z * algebraMap R S y = algebraMap R S x :=
   (toLocalizationMap M S).eq_mk'_iff_mul_eq
 
+theorem eq_mk'_of_mul_eq {x : R} {y : M} {z : R} (h : z * y = x) : (algebraMap R S) z = mk' S x y :=
+  eq_mk'_iff_mul_eq.mpr (by rw [← h, map_mul])
+
 theorem mk'_eq_iff_eq_mul {x} {y : M} {z} :
     mk' S x y = z ↔ algebraMap R S x = z * algebraMap R S y :=
   (toLocalizationMap M S).mk'_eq_iff_eq_mul
@@ -825,6 +828,48 @@ theorem map_nonZeroDivisors_le [IsLocalization M S] :
     (nonZeroDivisors R).map (algebraMap R S) ≤ nonZeroDivisors S :=
   Submonoid.map_le_iff_le_comap.mpr (nonZeroDivisors_le_comap M S)
 
+/-- If `S₁` is the localization of `R` at `M₁` and `S₂` is the localization of
+`R` at `M₂`, then every localization `T` of `S₂` at `M₁` is also a localization of
+`S₁` at `M₂`, in other words `M₁⁻¹M₂⁻¹R` can be identified with `M₂⁻¹M₁⁻¹R`. -/
+lemma commutes (S₁ S₂ T : Type*) [CommSemiring S₁]
+    [CommSemiring S₂] [CommSemiring T] [Algebra R S₁] [Algebra R S₂] [Algebra R T] [Algebra S₁ T]
+    [Algebra S₂ T] [IsScalarTower R S₁ T] [IsScalarTower R S₂ T] (M₁ M₂ : Submonoid R)
+    [IsLocalization M₁ S₁] [IsLocalization M₂ S₂]
+    [IsLocalization (Algebra.algebraMapSubmonoid S₂ M₁) T] :
+    IsLocalization (Algebra.algebraMapSubmonoid S₁ M₂) T where
+  map_units' := by
+    rintro ⟨m, ⟨a, ha, rfl⟩⟩
+    rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R S₂ T]
+    exact IsUnit.map _ (IsLocalization.map_units' ⟨a, ha⟩)
+  surj' a := by
+    obtain ⟨⟨y, -, m, hm, rfl⟩, hy⟩ := surj (M := Algebra.algebraMapSubmonoid S₂ M₁) a
+    rw [← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R S₁ T] at hy
+    obtain ⟨⟨z, n, hn⟩, hz⟩ := IsLocalization.surj (M := M₂) y
+    have hunit : IsUnit (algebraMap R S₁ m) := map_units' ⟨m, hm⟩
+    use ⟨algebraMap R S₁ z * hunit.unit⁻¹, ⟨algebraMap R S₁ n, n, hn, rfl⟩⟩
+    rw [map_mul, ← IsScalarTower.algebraMap_apply, IsScalarTower.algebraMap_apply R S₂ T]
+    conv_rhs => rw [← IsScalarTower.algebraMap_apply]
+    rw [IsScalarTower.algebraMap_apply R S₂ T, ← hz, map_mul, ← hy]
+    convert_to _ = a * (algebraMap S₂ T) ((algebraMap R S₂) n) *
+        (algebraMap S₁ T) (((algebraMap R S₁) m) * hunit.unit⁻¹.val)
+    · rw [map_mul]
+      ring
+    simp
+  exists_of_eq {x y} hxy := by
+    obtain ⟨r, s, d, hr, hs⟩ := IsLocalization.surj₂ M₁ S₁ x y
+    apply_fun (· * algebraMap S₁ T (algebraMap R S₁ d)) at hxy
+    simp_rw [← map_mul, hr, hs, ← IsScalarTower.algebraMap_apply,
+      IsScalarTower.algebraMap_apply R S₂ T] at hxy
+    obtain ⟨⟨-, c, hmc, rfl⟩, hc⟩ := exists_of_eq (M := Algebra.algebraMapSubmonoid S₂ M₁) hxy
+    simp_rw [← map_mul] at hc
+    obtain ⟨a, ha⟩ := IsLocalization.exists_of_eq (M := M₂) hc
+    use ⟨algebraMap R S₁ a, a, a.property, rfl⟩
+    apply (map_units S₁ d).mul_right_cancel
+    rw [mul_assoc, hr, mul_assoc, hs]
+    apply (map_units S₁ ⟨c, hmc⟩).mul_right_cancel
+    rw [← map_mul, ← map_mul, mul_assoc, mul_comm _ c, ha, map_mul, map_mul]
+    ring
+
 end IsLocalization
 
 namespace Localization
diff --git a/Mathlib/RingTheory/Localization/Finiteness.lean b/Mathlib/RingTheory/Localization/Finiteness.lean
index 10abd059206ce..ab340bfec9330 100644
--- a/Mathlib/RingTheory/Localization/Finiteness.lean
+++ b/Mathlib/RingTheory/Localization/Finiteness.lean
@@ -5,7 +5,7 @@ Authors: Christian Merten
 -/
 import Mathlib.Algebra.Module.LocalizedModuleIntegers
 import Mathlib.RingTheory.Localization.Algebra
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.RingHom.Finite
 
 /-!
 
@@ -53,10 +53,9 @@ lemma of_isLocalizedModule [Module.Finite R M] : Module.Finite Rₚ Mₚ := by
       (Submodule.span Rₚ (f '' T)).restrictScalars R := by
     rw [Submodule.span_le]; exact Submodule.subset_span
   convert (Submodule.span Rₚ (f '' T)).smul_mem
-    (IsLocalization.mk' Rₚ (1 : R) m) (H this) using 1
+    (IsLocalization.mk' Rₚ (1 : R) m) (H this) using 0
   · rw [← hyx, ← IsLocalizedModule.mk'_one S, IsLocalizedModule.mk'_smul_mk']
     simp
-  · simp
 
 end
 
diff --git a/Mathlib/RingTheory/Localization/FractionRing.lean b/Mathlib/RingTheory/Localization/FractionRing.lean
index 1960bbcd7e01d..f4b231a725789 100644
--- a/Mathlib/RingTheory/Localization/FractionRing.lean
+++ b/Mathlib/RingTheory/Localization/FractionRing.lean
@@ -132,7 +132,7 @@ noncomputable abbrev toField : Field K where
 
 lemma surjective_iff_isField [IsDomain R] : Function.Surjective (algebraMap R K) ↔ IsField R where
   mp h := (RingEquiv.ofBijective (algebraMap R K)
-      ⟨IsFractionRing.injective R K, h⟩).toMulEquiv.isField (IsFractionRing.toField R).toIsField
+      ⟨IsFractionRing.injective R K, h⟩).toMulEquiv.isField _ (IsFractionRing.toField R).toIsField
   mpr h :=
     letI := h.toField
     (IsLocalization.atUnits R _ (S := K)
diff --git a/Mathlib/RingTheory/Localization/LocalizationLocalization.lean b/Mathlib/RingTheory/Localization/LocalizationLocalization.lean
index bc2b5cc8d572a..a8796591c0f6c 100644
--- a/Mathlib/RingTheory/Localization/LocalizationLocalization.lean
+++ b/Mathlib/RingTheory/Localization/LocalizationLocalization.lean
@@ -183,6 +183,11 @@ noncomputable instance (x : Ideal R) [H : x.IsPrime] [IsDomain R] :
       rw [mem_nonZeroDivisors_iff_ne_zero]
       exact fun h => ha (h.symm ▸ x.zero_mem))
 
+instance {R : Type*} [CommRing R] [IsDomain R] (p : Ideal R) [p.IsPrime] :
+    IsScalarTower R (Localization.AtPrime p) (FractionRing R) :=
+  localization_isScalarTower_of_submonoid_le (Localization.AtPrime p) (FractionRing R)
+    p.primeCompl (nonZeroDivisors R) p.primeCompl_le_nonZeroDivisors
+
 /-- If `M ≤ N` are submonoids of `R`, then `N⁻¹S` is also the localization of `M⁻¹S` at `N`. -/
 theorem isLocalization_of_submonoid_le (M N : Submonoid R) (h : M ≤ N) [IsLocalization M S]
     [IsLocalization N T] [Algebra S T] [IsScalarTower R S T] :
@@ -235,6 +240,11 @@ theorem isLocalization_of_is_exists_mul_mem (M N : Submonoid R) [IsLocalization
       rw [IsLocalization.eq_iff_exists M]
       exact fun ⟨x, hx⟩ => ⟨⟨_, h x.prop⟩, hx⟩ }
 
+theorem mk'_eq_algebraMap_mk'_of_submonoid_le {M N : Submonoid R} (h : M ≤ N) [IsLocalization M S]
+    [IsLocalization N T] [Algebra S T] [IsScalarTower R S T] (x : R) (y : {a : R // a ∈ M}) :
+    mk' T x ⟨y.1, h y.2⟩ = algebraMap S T (mk' S x y) :=
+  mk'_eq_iff_eq_mul.mpr (by simp only [IsScalarTower.algebraMap_apply R S T, ← map_mul, mk'_spec])
+
 end LocalizationLocalization
 
 end IsLocalization
@@ -276,4 +286,9 @@ theorem isFractionRing_of_isDomain_of_isLocalization [IsDomain R] (S T : Type*)
   rw [← (algebraMap R S).map_one, ← @mk'_one R _ M, @comm _ Eq, mk'_eq_zero_iff]
   exact ⟨⟨x, hx⟩, by simp [hx']⟩
 
+instance {R : Type*} [CommRing R] [IsDomain R] (p : Ideal R) [p.IsPrime] :
+    IsFractionRing (Localization.AtPrime p) (FractionRing R) :=
+  IsFractionRing.isFractionRing_of_isDomain_of_isLocalization p.primeCompl
+    (Localization.AtPrime p) (FractionRing R)
+
 end IsFractionRing
diff --git a/Mathlib/RingTheory/MatrixAlgebra.lean b/Mathlib/RingTheory/MatrixAlgebra.lean
index 8525a367e4fcf..4a71967b89d79 100644
--- a/Mathlib/RingTheory/MatrixAlgebra.lean
+++ b/Mathlib/RingTheory/MatrixAlgebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Eric Wieser
+Authors: Kim Morrison, Eric Wieser
 -/
 import Mathlib.Data.Matrix.Basis
 import Mathlib.RingTheory.TensorProduct.Basic
@@ -96,13 +96,13 @@ theorem invFun_algebraMap (M : Matrix n n R) : invFun R A n (M.map (algebraMap R
   simp only [Algebra.algebraMap_eq_smul_one, smul_tmul, ← tmul_sum, mul_boole]
   congr
   conv_rhs => rw [matrix_eq_sum_stdBasisMatrix M]
-  convert Finset.sum_product (β := Matrix n n R); simp
+  convert Finset.sum_product (β := Matrix n n R) ..; simp
 
 theorem right_inv (M : Matrix n n A) : (toFunAlgHom R A n) (invFun R A n M) = M := by
   simp only [invFun, map_sum, stdBasisMatrix, apply_ite ↑(algebraMap R A), smul_eq_mul,
     mul_boole, toFunAlgHom_apply, RingHom.map_zero, RingHom.map_one, Matrix.map_apply,
     Pi.smul_def]
-  convert Finset.sum_product (β := Matrix n n A)
+  convert Finset.sum_product (β := Matrix n n A) ..
   conv_lhs => rw [matrix_eq_sum_stdBasisMatrix M]
   refine Finset.sum_congr rfl fun i _ => Finset.sum_congr rfl fun j _ => Matrix.ext fun a b => ?_
   simp only [stdBasisMatrix, smul_apply, Matrix.map_apply]
diff --git a/Mathlib/RingTheory/Multiplicity.lean b/Mathlib/RingTheory/Multiplicity.lean
index effafa7779249..457a6e35fcef7 100644
--- a/Mathlib/RingTheory/Multiplicity.lean
+++ b/Mathlib/RingTheory/Multiplicity.lean
@@ -506,7 +506,7 @@ protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b))
             1) ∣
         a * b :=
     fun h =>
-    not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
+    not_or_intro (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
       (_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
   rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
 
diff --git a/Mathlib/RingTheory/MvPolynomial/Homogeneous.lean b/Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
index f9a0ed2868cf3..4772bb3e6107b 100644
--- a/Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
+++ b/Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
@@ -49,7 +49,7 @@ variable [CommSemiring R]
 theorem weightedTotalDegree_one (φ : MvPolynomial σ R) :
     weightedTotalDegree (1 : σ → ℕ) φ = φ.totalDegree := by
   simp only [totalDegree, weightedTotalDegree, weight, LinearMap.toAddMonoidHom_coe,
-    Finsupp.total, Pi.one_apply, Finsupp.coe_lsum, LinearMap.coe_smulRight, LinearMap.id_coe,
+    linearCombination, Pi.one_apply, Finsupp.coe_lsum, LinearMap.coe_smulRight, LinearMap.id_coe,
     id, Algebra.id.smul_eq_mul, mul_one]
 
 variable (σ R)
diff --git a/Mathlib/RingTheory/MvPolynomial/Ideal.lean b/Mathlib/RingTheory/MvPolynomial/Ideal.lean
index 25562372ca086..de669fd310c46 100644
--- a/Mathlib/RingTheory/MvPolynomial/Ideal.lean
+++ b/Mathlib/RingTheory/MvPolynomial/Ideal.lean
@@ -37,7 +37,7 @@ theorem mem_ideal_span_monomial_image_iff_dvd {x : MvPolynomial σ R} {s : Set (
     x ∈ Ideal.span ((fun s => monomial s (1 : R)) '' s) ↔
       ∀ xi ∈ x.support, ∃ si ∈ s, monomial si 1 ∣ monomial xi (x.coeff xi) := by
   refine mem_ideal_span_monomial_image.trans (forall₂_congr fun xi hxi => ?_)
-  simp_rw [monomial_dvd_monomial, one_dvd, and_true_iff, mem_support_iff.mp hxi, false_or_iff]
+  simp_rw [monomial_dvd_monomial, one_dvd, and_true, mem_support_iff.mp hxi, false_or]
 
 /-- `x` is in a monomial ideal generated by variables `X` iff every element of its support
 has a component in `s`. -/
diff --git a/Mathlib/RingTheory/MvPolynomial/Symmetric.lean b/Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean
similarity index 100%
rename from Mathlib/RingTheory/MvPolynomial/Symmetric.lean
rename to Mathlib/RingTheory/MvPolynomial/Symmetric/Defs.lean
diff --git a/Mathlib/RingTheory/MvPolynomial/Symmetric/FundamentalTheorem.lean b/Mathlib/RingTheory/MvPolynomial/Symmetric/FundamentalTheorem.lean
new file mode 100644
index 0000000000000..b003422369bc5
--- /dev/null
+++ b/Mathlib/RingTheory/MvPolynomial/Symmetric/FundamentalTheorem.lean
@@ -0,0 +1,342 @@
+/-
+Copyright (c) 2023 Junyan Xu. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Junyan Xu
+-/
+import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
+import Mathlib.RingTheory.MvPolynomial.Tower
+import Mathlib.Data.Finsupp.Notation
+import Mathlib.Data.Finsupp.WellFounded
+
+/-!
+# The Fundamental Theorem of Symmetric Polynomials
+
+In a polynomial ring in `n` variables over a commutative ring, the subalgebra of symmetric
+polynomials is freely generated by the first `n` elementary symmetric polynomials (excluding
+the 0th, which is simply 1). This is expressed as an isomorphism
+`MvPolynomial.esymmAlgEquiv` between `MvPolynomial (Fin n) R` and
+the symmetric subalgebra of any polynomial ring `MvPolynomial σ R` with `#σ = n`.
+The forward map is called `MvPolynomial.esymmAlgHom`.
+
+## Proof strategy
+
+We follow the alternative proof on the Wikipedia page
+https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial#Alternative_proof
+It suffices to show `esymmAlgHom` is both injective and surjective.
+
+Endow the Fintype `σ` with a linear order and endow the (monic) monomials in the polynomial ring
+`MvPolynomial σ R` with the lexicographic order on `σ →₀ ℕ`, which is a well order.
+Then any nonzero polynomial `p : MvPolynomial σ R` has a largest nonzero monomial
+(`AddMonoidAlgebra.supDegree toLex p`) and the corresponding coefficient is
+`AddMonoidAlgebra.leadingCoeff toLex p`. If `p` is symmetric, any permutation of a nonzero monomial
+in `p` must also be a nonzero monomial in `p`, so the largest nonzero monomial must be antitone
+as a function `σ → ℕ` (`MvPolynomial.IsSymmetric.antitone_supDegree`). We can then construct a
+monomial in `MvPolynomial (Fin n) R` whose image under `esymmAlgHom` has the same `supDegree` and
+`leadingCoeff` as `p`: `MvPolynomial.supDegree_esymmAlgHomMonomial` says that the `supDegree` of
+the image is given by `Fin.accumulate`, and `Fin.accumulate_invAccumulate` says that
+`Fin.invAccumulate` is inverse to `Fin.accumulate` for antitone monomials.
+If we subtract the image from `p`, we are left with a symmetric polynomial of
+lower `supDegree`, which we may assume to be in the image by induction,
+thanks to the well-orderedness of `Lex (σ →₀ ℕ)`; the surjectivity of `esymmAlgHom`
+follows. For injectivity, just notice that the images of different monic monomials in
+`MvPolynomial (Fin n) R` have different `supDegree` (`Fin.accumulate_injective`), so if there is
+at least one nonzero monomial, the images cannot all cancel out
+(`AddMonoidAlgebra.sum_ne_zero_of_injOn_supDegree`).
+
+We actually only define `Fin.accumulate` in the case `σ := Fin m` rather than an arbitrary Fintype
+with a linear order; we show that `esymmAlgHom` is in fact surjective whenever `m ≤ n` and
+injective whenever `n ≤ m`, and then transfer the results to any Fintype `σ`. See
+`MvPolynomial.injective_esymmAlgHom` and `MvPolynomial.esymmAlgHom_surjective`.
+
+-/
+
+variable {σ τ R : Type*} {n m k : ℕ}
+
+open AddMonoidAlgebra Finset
+
+namespace Fin
+
+section accumulate
+
+/-- The `j`th entry of `accumulate n m t` is the sum of `t i` over all `i ≥ j`. -/
+@[simps] def accumulate (n m : ℕ) : (Fin n → ℕ) →+ (Fin m → ℕ) where
+  toFun t j := ∑ i in univ.filter (fun i : Fin n ↦ (j : ℕ) ≤ i), t i
+  map_zero' := funext <| fun j ↦ sum_eq_zero <| fun h _ ↦ rfl
+  map_add' t₁ t₂ := funext <| fun j ↦ by dsimp only; exact sum_add_distrib
+
+/-- The `i`th entry of `invAccumulate n m s` is `s i - s (i+1)`, where `s j = 0` if `j ≥ m`. -/
+def invAccumulate (n m : ℕ) (s : Fin m → ℕ) (i : Fin n) : ℕ :=
+  (if hi : i < m then s ⟨i, hi⟩ else 0) - (if hi : i + 1 < m then s ⟨i + 1, hi⟩ else 0)
+
+lemma accumulate_rec {i n m : ℕ} (hin : i < n) (him : i + 1 < m) (t : Fin n → ℕ) :
+    accumulate n m t ⟨i, Nat.lt_of_succ_lt him⟩ = t ⟨i, hin⟩ + accumulate n m t ⟨i + 1, him⟩ := by
+  simp_rw [accumulate_apply]
+  convert (add_sum_erase _ _ _).symm
+  · ext
+    rw [mem_erase]
+    simp_rw [mem_filter, mem_univ, true_and, i.succ_le_iff, lt_iff_le_and_ne]
+    rw [and_comm, ne_comm, ← Fin.val_ne_iff]
+  · exact mem_filter.2 ⟨mem_univ _, le_rfl⟩
+
+lemma accumulate_last {i n m : ℕ} (hin : i < n) (hmi : m = i + 1) (t : Fin n → ℕ)
+    (ht : ∀ j : Fin n, m ≤ j → t j = 0) :
+    accumulate n m t ⟨i, i.lt_succ_self.trans_eq hmi.symm⟩ = t ⟨i, hin⟩ := by
+  rw [accumulate_apply]
+  apply sum_eq_single_of_mem
+  · rw [mem_filter]; exact ⟨mem_univ _, le_rfl⟩
+  refine fun j hij hji ↦ ht j ?_
+  simp_rw [mem_filter, mem_univ, true_and] at hij
+  exact hmi.trans_le (hij.lt_of_ne (Fin.val_ne_iff.2 hji).symm).nat_succ_le
+
+lemma accumulate_injective {n m} (hnm : n ≤ m) : Function.Injective (accumulate n m) := by
+  refine fun t s he ↦ funext fun i ↦ ?_
+  obtain h|h := lt_or_le (i.1 + 1) m
+  · have := accumulate_rec i.2 h s
+    rwa [← he, accumulate_rec i.2 h t, add_right_cancel_iff] at this
+  · have := h.antisymm (i.2.nat_succ_le.trans hnm)
+    rw [← accumulate_last i.2 this t, ← accumulate_last i.2 this s, he]
+    iterate 2 { intro j hj; exact ((j.2.trans_le hnm).not_le hj).elim }
+
+lemma accumulate_invAccumulate {n m} (hmn : m ≤ n) {s : Fin m → ℕ} (hs : Antitone s) :
+    accumulate n m (invAccumulate n m s) = s := funext <| fun ⟨i, hi⟩ ↦ by
+  have := Nat.le_sub_one_of_lt hi
+  revert hi
+  refine Nat.decreasingInduction' (fun i hi _ ih him ↦ ?_) this fun hm ↦ ?_
+  · rw [← Nat.pred_eq_sub_one, Nat.lt_pred_iff, Nat.succ_eq_add_one] at hi
+    rw [accumulate_rec (him.trans_le hmn) hi, ih hi, invAccumulate, dif_pos him, dif_pos hi]
+    simp only
+    exact Nat.sub_add_cancel (hs i.le_succ)
+  · have := (Nat.sub_one_add_one <| Nat.not_eq_zero_of_lt hm).symm
+    rw [accumulate_last (hm.trans_le hmn) this, invAccumulate, dif_pos hm, dif_neg this.not_gt,
+      Nat.sub_zero]
+    intro j hj
+    rw [invAccumulate, dif_neg hj.not_lt, Nat.zero_sub]
+
+end accumulate
+
+end Fin
+
+namespace MvPolynomial
+
+open Fin
+
+section CommSemiring
+
+variable [CommSemiring R] [Fintype σ] [Fintype τ]
+
+variable (σ R n) in
+/-- The `R`-algebra homomorphism from $R[x_1,\dots,x_n]$ to the symmetric subalgebra of
+  $R[\{x_i \mid i ∈ σ\}]$ sending $x_i$ to the $i$-th elementary symmetric polynomial. -/
+noncomputable def esymmAlgHom :
+    MvPolynomial (Fin n) R →ₐ[R] symmetricSubalgebra σ R :=
+  aeval (fun i ↦ ⟨esymm σ R (i + 1), esymm_isSymmetric σ R _⟩)
+
+lemma esymmAlgHom_apply (p : MvPolynomial (Fin n) R) :
+    (esymmAlgHom σ R n p).val = aeval (fun i : Fin n ↦ esymm σ R (i + 1)) p :=
+  (Subalgebra.mvPolynomial_aeval_coe _ _ _).symm
+
+lemma rename_esymmAlgHom (e : σ ≃ τ) :
+    (renameSymmetricSubalgebra e).toAlgHom.comp (esymmAlgHom σ R n) = esymmAlgHom τ R n := by
+  ext i : 2
+  simp_rw [AlgHom.comp_apply, esymmAlgHom, aeval_X, AlgEquiv.toAlgHom_eq_coe, AlgHom.coe_coe,
+    renameSymmetricSubalgebra_apply_coe, rename_esymm]
+
+variable (σ) in
+/-- The image of a monomial under `esymmAlgHom`. -/
+noncomputable def esymmAlgHomMonomial (t : Fin n →₀ ℕ) (r : R) :
+    MvPolynomial σ R := (esymmAlgHom σ R n <| monomial t r).val
+
+variable {i : Fin n} {j : Fin m} {r : R}
+
+lemma isSymmetric_esymmAlgHomMonomial (t : Fin n →₀ ℕ) (r : R) :
+    (esymmAlgHomMonomial σ t r).IsSymmetric := (esymmAlgHom _ _ _ _).2
+
+lemma esymmAlgHomMonomial_single :
+    esymmAlgHomMonomial σ (Finsupp.single i k) r = C r * esymm σ R (i + 1) ^ k := by
+  rw [esymmAlgHomMonomial, esymmAlgHom_apply, aeval_monomial, algebraMap_eq,
+    Finsupp.prod_single_index]
+  exact pow_zero _
+
+lemma esymmAlgHomMonomial_single_one :
+    esymmAlgHomMonomial σ (Finsupp.single i k) 1 = esymm σ R (i + 1) ^ k := by
+  rw [esymmAlgHomMonomial_single, map_one, one_mul]
+
+lemma esymmAlgHomMonomial_add {t s : Fin n →₀ ℕ} :
+    esymmAlgHomMonomial σ (t + s) r = esymmAlgHomMonomial σ t r * esymmAlgHomMonomial σ s 1 := by
+  simp_rw [esymmAlgHomMonomial, esymmAlgHom_apply, ← map_mul, monomial_mul, mul_one]
+
+lemma esymmAlgHom_zero : esymmAlgHomMonomial σ (0 : Fin n →₀ ℕ) r = C r := by
+  rw [esymmAlgHomMonomial, monomial_zero', esymmAlgHom_apply, aeval_C, algebraMap_eq]
+
+private lemma supDegree_monic_esymm [Nontrivial R] {i : ℕ} (him : i < m) :
+    supDegree toLex (esymm (Fin m) R (i + 1)) =
+      toLex (Finsupp.indicator (Iic ⟨i, him⟩) fun _ _ ↦ 1) ∧
+    Monic toLex (esymm (Fin m) R (i + 1)) := by
+  have := supDegree_leadingCoeff_sum_eq (D := toLex) (s := univ.powersetCard (i + 1))
+    (i := Iic (⟨i, him⟩ : Fin m)) ?_ (f := fun s ↦ monomial (∑ j in s, fun₀ | j => 1) (1 : R)) ?_
+  · rwa [← esymm_eq_sum_monomial, ← Finsupp.indicator_eq_sum_single, ← single_eq_monomial,
+      supDegree_single_ne_zero _ one_ne_zero, leadingCoeff_single toLex.injective] at this
+  · exact mem_powersetCard.2 ⟨subset_univ _, Fin.card_Iic _⟩
+  intro t ht hne
+  have ht' : t.card = (Iic (⟨i, him⟩ : Fin m)).card := by
+    rw [(mem_powersetCard.1 ht).2, Fin.card_Iic]
+  simp_rw [← single_eq_monomial, supDegree_single_ne_zero _ one_ne_zero,
+    ← Finsupp.indicator_eq_sum_single]
+  rw [ne_comm, Ne, ← subset_iff_eq_of_card_le ht'.le, not_subset] at hne
+  simp_rw [← mem_sdiff] at hne
+  have hkm := mem_sdiff.1 (min'_mem _ hne)
+  refine ⟨min' _ hne, fun k hk ↦ ?_, ?_⟩
+  all_goals simp only [Pi.toLex_apply, ofLex_toLex, Finsupp.indicator_apply]
+  · have hki := mem_Iic.2 (hk.le.trans <| mem_Iic.1 hkm.1)
+    rw [dif_pos hki, dif_pos]
+    by_contra h
+    exact lt_irrefl k <| ((lt_min'_iff _ _).1 hk) _ <| mem_sdiff.2 ⟨hki, h⟩
+  · rw [dif_neg hkm.2, dif_pos hkm.1]; exact Nat.zero_lt_one
+
+lemma supDegree_esymm [Nontrivial R] (him : i < m) :
+    ofLex (supDegree toLex <| esymm (Fin m) R (i + 1)) = accumulate n m (Finsupp.single i 1) := by
+  rw [(supDegree_monic_esymm him).1, ofLex_toLex]
+  ext j
+  simp_rw [Finsupp.indicator_apply, dite_eq_ite, mem_Iic, accumulate_apply, Finsupp.single_apply,
+    sum_ite_eq, mem_filter, mem_univ, true_and, Fin.le_def]
+
+lemma monic_esymm {i : ℕ} (him : i ≤ m) : Monic toLex (esymm (Fin m) R i) := by
+  cases i with
+  | zero =>
+    rw [esymm_zero]
+    exact monic_one toLex.injective
+  | succ i =>
+    nontriviality R
+    exact (supDegree_monic_esymm him).2
+
+lemma leadingCoeff_esymmAlgHomMonomial (t : Fin n →₀ ℕ) (hnm : n ≤ m) :
+    leadingCoeff toLex (esymmAlgHomMonomial (Fin m) t r) = r := by
+  induction t using Finsupp.induction₂ with
+  | h0 => rw [esymmAlgHom_zero, leadingCoeff_toLex_C]
+  | ha i _ _ _ _ ih =>
+    rw [esymmAlgHomMonomial_add, esymmAlgHomMonomial_single_one,
+        ((monic_esymm <| i.2.trans_le hnm).pow toLex_add toLex.injective).leadingCoeff_mul_eq_left,
+        ih]
+    exacts [toLex.injective, toLex_add]
+
+lemma supDegree_esymmAlgHomMonomial (hr : r ≠ 0) (t : Fin n →₀ ℕ) (hnm : n ≤ m) :
+    ofLex (supDegree toLex <| esymmAlgHomMonomial (Fin m) t r) = accumulate n m t := by
+  nontriviality R
+  induction t using Finsupp.induction₂ with
+  | h0 => simp_rw [esymmAlgHom_zero, supDegree_toLex_C, ofLex_zero, Finsupp.coe_zero, map_zero]
+  | ha  i _ _ _ _ ih =>
+    have := i.2.trans_le hnm
+    rw [esymmAlgHomMonomial_add, esymmAlgHomMonomial_single_one,
+        Monic.supDegree_mul_of_ne_zero_left toLex.injective toLex_add, ofLex_add, Finsupp.coe_add,
+        ih, Finsupp.coe_add, map_add, Monic.supDegree_pow rfl toLex_add toLex.injective, ofLex_smul,
+        Finsupp.coe_smul, supDegree_esymm this, ← map_nsmul, ← Finsupp.coe_smul,
+        Finsupp.smul_single, nsmul_one, Nat.cast_id]
+    · exact monic_esymm this
+    · exact (monic_esymm this).pow toLex_add toLex.injective
+    · rwa [Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_esymmAlgHomMonomial _ hnm]
+
+omit [Fintype σ] in
+lemma IsSymmetric.antitone_supDegree [LinearOrder σ] {p : MvPolynomial σ R} (hp : p.IsSymmetric) :
+    Antitone ↑(ofLex <| p.supDegree toLex) := by
+  obtain rfl | h0 := eq_or_ne p 0
+  · rw [supDegree_zero, Finsupp.bot_eq_zero]
+    exact Pi.zero_mono
+  rw [Antitone]
+  by_contra! h
+  obtain ⟨i, j, hle, hlt⟩ := h
+  apply (le_sup (s := p.support) (f := toLex) _).not_lt
+  pick_goal 3
+  · rw [← hp (Equiv.swap i j), mem_support_iff, coeff_rename_mapDomain _ (Equiv.injective _)]
+    rw [Ne, ← leadingCoeff_eq_zero toLex.injective, leadingCoeff_toLex] at h0
+    assumption
+  refine ⟨i, fun k hk ↦ ?_, ?_⟩
+  all_goals dsimp only [Pi.toLex_apply, ofLex_toLex]
+  · conv_rhs => rw [← Equiv.swap_apply_of_ne_of_ne hk.ne (hk.trans_le hle).ne]
+    rw [Finsupp.mapDomain_apply (Equiv.injective _), supDegree]; rfl
+  · apply hlt.trans_eq
+    simp_rw [Finsupp.mapDomain_equiv_apply, Equiv.symm_swap, Equiv.swap_apply_left]
+
+end CommSemiring
+
+section CommRing
+
+variable (R)
+variable [Fintype σ] [CommRing R]
+
+open AddMonoidAlgebra
+
+/- Also holds for a cancellative CommSemiring. -/
+lemma esymmAlgHom_fin_injective (h : n ≤ m) :
+    Function.Injective (esymmAlgHom (Fin m) R n) := by
+  rw [injective_iff_map_eq_zero]
+  refine fun p ↦ (fun hp ↦ ?_).mtr
+  rw [p.as_sum, map_sum (esymmAlgHom (Fin m) R n), ← Subalgebra.coe_eq_zero,
+    AddSubmonoidClass.coe_finset_sum]
+  refine sum_ne_zero_of_injOn_supDegree (D := toLex) (support_eq_empty.not.2 hp) (fun t ht ↦ ?_)
+    (fun t ht s hs he ↦ DFunLike.ext' <| accumulate_injective h ?_)
+  · rw [← esymmAlgHomMonomial, Ne, ← leadingCoeff_eq_zero toLex.injective,
+      leadingCoeff_esymmAlgHomMonomial t h]
+    rwa [mem_support_iff] at ht
+  rw [mem_coe, mem_support_iff] at ht hs
+  dsimp only [Function.comp] at he
+  rwa [← esymmAlgHomMonomial, ← esymmAlgHomMonomial, ← ofLex_inj, DFunLike.ext'_iff,
+       supDegree_esymmAlgHomMonomial ht t h, supDegree_esymmAlgHomMonomial hs s h] at he
+
+lemma esymmAlgHom_injective (hn : n ≤ Fintype.card σ) :
+    Function.Injective (esymmAlgHom σ R n) := by
+  rw [← rename_esymmAlgHom (Fintype.equivFin σ).symm, AlgHom.coe_comp]
+  exact (AlgEquiv.injective _).comp (esymmAlgHom_fin_injective R hn)
+
+lemma esymmAlgHom_fin_bijective (n : ℕ) :
+    Function.Bijective (esymmAlgHom (Fin n) R n) := by
+  use esymmAlgHom_fin_injective R le_rfl
+  rintro ⟨p, hp⟩
+  rw [← AlgHom.mem_range]
+  obtain rfl | h0 := eq_or_ne p 0
+  · exact Subalgebra.zero_mem _
+  induction' he : p.supDegree toLex using WellFoundedLT.induction with t ih generalizing p; subst he
+  let t := Finsupp.equivFunOnFinite.symm (invAccumulate n n <| ↑(ofLex <| p.supDegree toLex))
+  have hd :
+      (esymmAlgHomMonomial _ t <| p.leadingCoeff toLex).supDegree toLex = p.supDegree toLex := by
+    rw [← ofLex_inj, DFunLike.ext'_iff, supDegree_esymmAlgHomMonomial _ _ le_rfl]
+    · exact accumulate_invAccumulate le_rfl hp.antitone_supDegree
+    · rwa [Ne, leadingCoeff_eq_zero toLex.injective]
+  obtain he | hne := eq_or_ne p (esymmAlgHomMonomial _ t <| p.leadingCoeff toLex)
+  · convert AlgHom.mem_range_self _ (monomial t <| p.leadingCoeff toLex)
+  have := (supDegree_sub_lt_of_leadingCoeff_eq toLex.injective hd.symm ?_).resolve_right hne
+  · specialize ih _ this _ (Subalgebra.sub_mem _ hp <| isSymmetric_esymmAlgHomMonomial _ _) _ rfl
+    · rwa [sub_ne_zero]
+    convert ← Subalgebra.add_mem _ ih ⟨monomial t (p.leadingCoeff toLex), rfl⟩
+    apply sub_add_cancel p
+  · rw [leadingCoeff_esymmAlgHomMonomial t le_rfl]
+
+lemma esymmAlgHom_fin_surjective (h : m ≤ n) :
+    Function.Surjective (esymmAlgHom (Fin m) R n) := by
+  intro p
+  obtain ⟨q, rfl⟩ := (esymmAlgHom_fin_bijective R m).2 p
+  rw [← AlgHom.mem_range]
+  induction q using MvPolynomial.induction_on with
+  | h_C r => rw [← algebraMap_eq, AlgHom.commutes]; apply Subalgebra.algebraMap_mem
+  | h_add p q hp hq => rw [map_add]; exact Subalgebra.add_mem _ hp hq
+  | h_X p i hp =>
+    rw [map_mul]
+    apply Subalgebra.mul_mem _ hp
+    rw [AlgHom.mem_range]
+    refine ⟨X ⟨i, i.2.trans_le h⟩, ?_⟩
+    simp_rw [esymmAlgHom, aeval_X]
+
+lemma esymmAlgHom_surjective (hn : Fintype.card σ ≤ n) :
+    Function.Surjective (esymmAlgHom σ R n) := by
+  rw [← rename_esymmAlgHom (Fintype.equivFin σ).symm, AlgHom.coe_comp]
+  exact (AlgEquiv.surjective _).comp (esymmAlgHom_fin_surjective R hn)
+
+/-- If the cardinality of `σ` is `n`, then `esymmAlgHom σ R n` is an isomorphism. -/
+@[simps! apply]
+noncomputable def esymmAlgEquiv (hn : Fintype.card σ = n) :
+    MvPolynomial (Fin n) R ≃ₐ[R] symmetricSubalgebra σ R :=
+  AlgEquiv.ofBijective (esymmAlgHom σ R n)
+    ⟨esymmAlgHom_injective R hn.ge, esymmAlgHom_surjective R hn.le⟩
+
+end CommRing
+
+end MvPolynomial
diff --git a/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean b/Mathlib/RingTheory/MvPolynomial/Symmetric/NewtonIdentities.lean
similarity index 99%
rename from Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
rename to Mathlib/RingTheory/MvPolynomial/Symmetric/NewtonIdentities.lean
index 5d1767881cad0..3aa09ead7b4ab 100644
--- a/Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean
+++ b/Mathlib/RingTheory/MvPolynomial/Symmetric/NewtonIdentities.lean
@@ -8,7 +8,7 @@ import Mathlib.Algebra.MvPolynomial.CommRing
 import Mathlib.Algebra.MvPolynomial.Rename
 import Mathlib.Data.Finset.Card
 import Mathlib.Data.Fintype.Basic
-import Mathlib.RingTheory.MvPolynomial.Symmetric
+import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
 
 /-!
 # Newton's Identities
diff --git a/Mathlib/RingTheory/MvPolynomial/Tower.lean b/Mathlib/RingTheory/MvPolynomial/Tower.lean
index 691f027ab1158..77228738a64aa 100644
--- a/Mathlib/RingTheory/MvPolynomial/Tower.lean
+++ b/Mathlib/RingTheory/MvPolynomial/Tower.lean
@@ -47,12 +47,12 @@ theorem aeval_algebraMap_apply (x : σ → A) (p : MvPolynomial σ R) :
   rw [aeval_def, aeval_def, ← coe_eval₂Hom, ← coe_eval₂Hom, map_eval₂Hom, ←
     IsScalarTower.algebraMap_eq]
   -- Porting note: added
-  simp only [Function.comp]
+  simp only [Function.comp_def]
 
 theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : σ → A)
     (p : MvPolynomial σ R) : aeval (algebraMap A B ∘ x) p = 0 ↔ aeval x p = 0 := by
   rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
-    iff_false_intro (one_ne_zero' B), or_false_iff]
+    iff_false_intro (one_ne_zero' B), or_false]
 
 theorem aeval_algebraMap_eq_zero_iff_of_injective {x : σ → A} {p : MvPolynomial σ R}
     (h : Function.Injective (algebraMap A B)) :
diff --git a/Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean b/Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
index 2f0b07d5dd242..a5d6459f7d56f 100644
--- a/Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
+++ b/Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
@@ -214,7 +214,7 @@ theorem isWeightedHomogeneous_one (w : σ → M) : IsWeightedHomogeneous w (1 :
 theorem isWeightedHomogeneous_X (w : σ → M) (i : σ) :
     IsWeightedHomogeneous w (X i : MvPolynomial σ R) (w i) := by
   apply isWeightedHomogeneous_monomial
-  simp only [weight, LinearMap.toAddMonoidHom_coe, total_single, one_nsmul]
+  simp only [weight, LinearMap.toAddMonoidHom_coe, linearCombination_single, one_nsmul]
 
 namespace IsWeightedHomogeneous
 
@@ -506,7 +506,7 @@ theorem weightedHomogeneousComponent_zero [NoZeroSMulDivisors ℕ M] (hw : ∀ i
   rcases Classical.em (d = 0) with (rfl | hd)
   · simp only [coeff_weightedHomogeneousComponent, if_pos, map_zero, coeff_zero_C]
   · rw [coeff_weightedHomogeneousComponent, if_neg, coeff_C, if_neg (Ne.symm hd)]
-    simp only [weight, LinearMap.toAddMonoidHom_coe, Finsupp.total_apply, Finsupp.sum,
+    simp only [weight, LinearMap.toAddMonoidHom_coe, Finsupp.linearCombination_apply, Finsupp.sum,
       sum_eq_zero_iff, Finsupp.mem_support_iff, Ne, smul_eq_zero, not_forall, not_or,
       and_self_left, exists_prop]
     simp only [DFunLike.ext_iff, Finsupp.coe_zero, Pi.zero_apply, not_forall] at hd
@@ -531,7 +531,7 @@ variable [CanonicallyLinearOrderedAddCommMonoid M] {w : σ → M} (φ : MvPolyno
   has weighted degree zero if and only if `∀ x : σ, m x = 0`. -/
 theorem weightedDegree_eq_zero_iff (hw : NonTorsionWeight w) {m : σ →₀ ℕ} :
     weight w m = 0 ↔ ∀ x : σ, m x = 0 := by
-  simp only [weight, Finsupp.total, LinearMap.toAddMonoidHom_coe, coe_lsum,
+  simp only [weight, Finsupp.linearCombination, LinearMap.toAddMonoidHom_coe, coe_lsum,
     LinearMap.coe_smulRight, LinearMap.id_coe, id_eq]
   rw [Finsupp.sum, Finset.sum_eq_zero_iff]
   apply forall_congr'
@@ -582,7 +582,7 @@ theorem weightedHomogeneousComponent_eq_zero_of_not_mem [DecidableEq M]
 
 variable (R)
 
-/-- The `decompose'` argument of `weightedDecomposition`.  -/
+/-- The `decompose'` argument of `weightedDecomposition`. -/
 def decompose' [DecidableEq M] := fun φ : MvPolynomial σ R =>
   DirectSum.mk (fun i : M => ↥(weightedHomogeneousSubmodule R w i))
     (Finset.image (weight w) φ.support) fun m =>
diff --git a/Mathlib/RingTheory/MvPowerSeries/Basic.lean b/Mathlib/RingTheory/MvPowerSeries/Basic.lean
index 8e2b9f0fb7e79..3e649d881d0c8 100644
--- a/Mathlib/RingTheory/MvPowerSeries/Basic.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/Basic.lean
@@ -730,7 +730,7 @@ instance [Nonempty σ] [Nontrivial R] : Nontrivial (Subalgebra R (MvPowerSeries
       rw [Ne, SetLike.ext_iff, not_forall]
       inhabit σ
       refine ⟨X default, ?_⟩
-      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true_iff, Algebra.mem_top]
+      simp only [Algebra.mem_bot, not_exists, Set.mem_range, iff_true, Algebra.mem_top]
       intro x
       rw [MvPowerSeries.ext_iff, not_forall]
       refine ⟨Finsupp.single default 1, ?_⟩
diff --git a/Mathlib/RingTheory/MvPowerSeries/Inverse.lean b/Mathlib/RingTheory/MvPowerSeries/Inverse.lean
index a7e076eb389ea..d53136ff960c2 100644
--- a/Mathlib/RingTheory/MvPowerSeries/Inverse.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/Inverse.lean
@@ -7,7 +7,7 @@ Authors: Johan Commelin, Kenny Lau
 import Mathlib.Algebra.Group.Units
 import Mathlib.RingTheory.MvPowerSeries.Basic
 import Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
-import Mathlib.RingTheory.LocalRing.RingHom.Basic
+import Mathlib.RingTheory.LocalRing.Basic
 
 /-!
 # Formal (multivariate) power series - Inverses
diff --git a/Mathlib/RingTheory/MvPowerSeries/LexOrder.lean b/Mathlib/RingTheory/MvPowerSeries/LexOrder.lean
new file mode 100644
index 0000000000000..13ecad6b47dce
--- /dev/null
+++ b/Mathlib/RingTheory/MvPowerSeries/LexOrder.lean
@@ -0,0 +1,186 @@
+/-
+Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Antoine Chambert-Loir
+-/
+
+import Mathlib.RingTheory.MvPowerSeries.Basic
+import Mathlib.Data.Finsupp.WellFounded
+
+/-! LexOrder of multivariate power series
+
+Given an ordering of `σ` such that `WellOrderGT σ`,
+the lexicographic order on `σ →₀ ℕ` is a well ordering,
+which can be used to define a natural valuation `lexOrder` on the ring `MvPowerSeries σ R`:
+the smallest exponent in the support.
+
+-/
+
+namespace MvPowerSeries
+
+variable {σ R : Type*}
+variable [Semiring R]
+
+section LexOrder
+
+open Finsupp
+variable [LinearOrder σ] [WellFoundedGT σ]
+
+/-- The lex order on multivariate power series.  -/
+noncomputable def lexOrder (φ : MvPowerSeries σ R) : (WithTop (Lex (σ →₀ ℕ))) := by
+  classical
+  exact if h : φ = 0 then ⊤ else by
+    have ne : Set.Nonempty (toLex '' φ.support) := by
+      simp only [Set.image_nonempty, Function.support_nonempty_iff, ne_eq, h, not_false_eq_true]
+    apply WithTop.some
+    apply WellFounded.min _ (toLex '' φ.support) ne
+    · exact Finsupp.instLTLex.lt
+    · exact wellFounded_lt
+
+theorem lexOrder_def_of_ne_zero {φ : MvPowerSeries σ R} (hφ : φ ≠ 0) :
+    ∃ (ne : Set.Nonempty (toLex '' φ.support)),
+      lexOrder φ = WithTop.some ((@wellFounded_lt (Lex (σ →₀ ℕ))
+        (instLTLex) (Lex.wellFoundedLT)).min (toLex '' φ.support) ne) := by
+  suffices ne : Set.Nonempty (toLex '' φ.support) by
+    use ne
+    unfold lexOrder
+    simp only [dif_neg hφ]
+  simp only [Set.image_nonempty, Function.support_nonempty_iff, ne_eq, hφ, not_false_eq_true]
+
+@[simp]
+theorem lexOrder_eq_top_iff_eq_zero (φ : MvPowerSeries σ R) :
+    lexOrder φ = ⊤ ↔ φ = 0 := by
+  unfold lexOrder
+  split_ifs with h
+  · simp only [h]
+  · simp only [h, WithTop.coe_ne_top]
+
+theorem lexOrder_zero : lexOrder (0 : MvPowerSeries σ R) = ⊤ := by
+  unfold lexOrder
+  rw [dif_pos rfl]
+
+theorem exists_finsupp_eq_lexOrder_of_ne_zero {φ : MvPowerSeries σ R} (hφ : φ ≠ 0) :
+    ∃ (d : σ →₀ ℕ), lexOrder φ = toLex d := by
+  simp only [ne_eq, ← lexOrder_eq_top_iff_eq_zero, WithTop.ne_top_iff_exists] at hφ
+  obtain ⟨p, hp⟩ := hφ
+  exact ⟨ofLex p, by simp only [toLex_ofLex, hp]⟩
+
+theorem coeff_ne_zero_of_lexOrder {φ : MvPowerSeries σ R} {d : σ →₀ ℕ}
+    (h : toLex d = lexOrder φ) : coeff R d φ ≠ 0 := by
+  have hφ : φ ≠ 0 := by
+    simp only [ne_eq, ← lexOrder_eq_top_iff_eq_zero, ← h, WithTop.coe_ne_top, not_false_eq_true]
+  have hφ' := lexOrder_def_of_ne_zero hφ
+  rcases hφ' with ⟨ne, hφ'⟩
+  simp only [← h, WithTop.coe_eq_coe] at hφ'
+  suffices toLex d ∈ toLex '' φ.support by
+    simp only [Set.mem_image_equiv, toLex_symm_eq, ofLex_toLex, Function.mem_support, ne_eq] at this
+    apply this
+  rw [hφ']
+  apply WellFounded.min_mem
+
+theorem coeff_eq_zero_of_lt_lexOrder {φ : MvPowerSeries σ R} {d : σ →₀ ℕ}
+    (h : toLex d < lexOrder φ) : coeff R d φ = 0 := by
+  by_cases hφ : φ = 0
+  · simp only [hφ, map_zero]
+  · rcases lexOrder_def_of_ne_zero hφ with ⟨ne, hφ'⟩
+    rw [hφ', WithTop.coe_lt_coe] at h
+    by_contra h'
+    exact WellFounded.not_lt_min _ (toLex '' φ.support) ne (Set.mem_image_equiv.mpr h') h
+
+theorem lexOrder_le_of_coeff_ne_zero {φ : MvPowerSeries σ R} {d : σ →₀ ℕ}
+    (h : coeff R d φ ≠ 0) : lexOrder φ ≤ toLex d := by
+  rw [← not_lt]
+  intro h'
+  exact h (coeff_eq_zero_of_lt_lexOrder h')
+
+theorem le_lexOrder_iff {φ : MvPowerSeries σ R} {w : WithTop (Lex (σ →₀ ℕ))} :
+    w ≤ lexOrder φ ↔ (∀ (d : σ →₀ ℕ) (_ : toLex d < w), coeff R d φ = 0) := by
+  constructor
+  · intro h d hd
+    apply coeff_eq_zero_of_lt_lexOrder
+    exact lt_of_lt_of_le hd h
+  · intro h
+    rw [← not_lt]
+    intro h'
+    have hφ : φ ≠ 0 := by
+      rw [ne_eq, ← lexOrder_eq_top_iff_eq_zero]
+      exact ne_top_of_lt h'
+    obtain ⟨d, hd⟩ := exists_finsupp_eq_lexOrder_of_ne_zero hφ
+    refine coeff_ne_zero_of_lexOrder hd.symm (h d ?_)
+    rwa [← hd]
+
+theorem min_lexOrder_le {φ ψ : MvPowerSeries σ R} :
+    min (lexOrder φ) (lexOrder ψ) ≤ lexOrder (φ + ψ)  := by
+  rw [le_lexOrder_iff]
+  intro d hd
+  simp only [lt_min_iff] at hd
+  rw [map_add, coeff_eq_zero_of_lt_lexOrder hd.1, coeff_eq_zero_of_lt_lexOrder hd.2, add_zero]
+
+theorem coeff_mul_of_add_lexOrder {φ ψ : MvPowerSeries σ R}
+    {p q : σ →₀ ℕ} (hp : lexOrder φ = toLex p) (hq : lexOrder ψ = toLex q) :
+    coeff R (p + q) (φ * ψ) = coeff R p φ * coeff R q ψ := by
+  rw [coeff_mul]
+  apply Finset.sum_eq_single (⟨p, q⟩ : (σ →₀ ℕ) × (σ →₀ ℕ))
+  · rintro ⟨u, v⟩ h h'
+    simp only [Finset.mem_antidiagonal] at h
+    simp only
+    by_cases hu : toLex u < toLex p
+    · rw [coeff_eq_zero_of_lt_lexOrder (R := R) (d := u), zero_mul]
+      simp only [hp, WithTop.coe_lt_coe, hu]
+    · rw [coeff_eq_zero_of_lt_lexOrder (d := v), mul_zero]
+      simp only [hq, WithTop.coe_lt_coe, ← not_le]
+      simp only [not_lt] at hu
+      intro hv
+      simp only [WithTop.coe_le_coe] at hv
+      apply h'
+      simp only [Prod.mk.injEq]
+      constructor
+      · apply toLex.injective
+        apply Or.resolve_right (eq_or_gt_of_le hu)
+        intro hu'
+        exact not_le.mpr (add_lt_add_of_lt_of_le hu' hv) (le_of_eq h)
+      · apply toLex.injective
+        apply Or.resolve_right (eq_or_gt_of_le hv)
+        intro hv'
+        exact not_le.mpr (add_lt_add_of_le_of_lt hu hv') (le_of_eq h)
+  · intro h
+    simp only [Finset.mem_antidiagonal, not_true_eq_false] at h
+
+theorem le_lexOrder_mul (φ ψ : MvPowerSeries σ R) :
+    lexOrder φ + lexOrder ψ ≤ lexOrder (φ * ψ) := by
+  rw [le_lexOrder_iff]
+  intro d hd
+  rw [coeff_mul]
+  apply Finset.sum_eq_zero
+  rintro ⟨u, v⟩ h
+  simp only [Finset.mem_antidiagonal] at h
+  simp only
+  suffices toLex u < lexOrder φ ∨ toLex v < lexOrder ψ by
+    rcases this with (hu | hv)
+    · rw [coeff_eq_zero_of_lt_lexOrder hu, zero_mul]
+    · rw [coeff_eq_zero_of_lt_lexOrder hv, mul_zero]
+  rw [or_iff_not_imp_left, not_lt, ← not_le]
+  intro hu hv
+  rw [← not_le] at hd
+  apply hd
+  simp only [← h, toLex_add, WithTop.coe_add, add_le_add hu hv]
+
+alias lexOrder_mul_ge := le_lexOrder_mul
+
+theorem lexOrder_mul [NoZeroDivisors R] (φ ψ : MvPowerSeries σ R) :
+    lexOrder (φ * ψ) = lexOrder φ + lexOrder ψ := by
+  by_cases hφ : φ = 0
+  · simp only [hφ, zero_mul, lexOrder_zero, top_add]
+  by_cases hψ : ψ = 0
+  · simp only [hψ, mul_zero, lexOrder_zero, add_top]
+  rcases exists_finsupp_eq_lexOrder_of_ne_zero hφ with ⟨p, hp⟩
+  rcases exists_finsupp_eq_lexOrder_of_ne_zero hψ with ⟨q, hq⟩
+  apply le_antisymm _ (lexOrder_mul_ge φ ψ)
+  rw [hp, hq]
+  apply lexOrder_le_of_coeff_ne_zero (d := p + q)
+  rw [coeff_mul_of_add_lexOrder hp hq, mul_ne_zero_iff]
+  exact ⟨coeff_ne_zero_of_lexOrder hp.symm, coeff_ne_zero_of_lexOrder hq.symm⟩
+
+end LexOrder
+
+end MvPowerSeries
diff --git a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
index 76197b816b090..fb9d75cb2281a 100644
--- a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
+++ b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir
 -/
 
-import Mathlib.RingTheory.MvPowerSeries.Basic
 import Mathlib.Data.Finsupp.WellFounded
+import Mathlib.RingTheory.MvPowerSeries.LexOrder
 
 /-! # ZeroDivisors in a MvPowerSeries ring
 
@@ -65,6 +65,15 @@ theorem mem_nonZeroDivisors_of_constantCoeff {φ : MvPowerSeries σ R}
 
 end Semiring
 
+instance {σ R : Type*} [Semiring R] [NoZeroDivisors R] :
+    NoZeroDivisors (MvPowerSeries σ R) where
+  eq_zero_or_eq_zero_of_mul_eq_zero {φ ψ} h := by
+    letI : LinearOrder σ := LinearOrder.swap σ WellOrderingRel.isWellOrder.linearOrder
+    letI : WellFoundedGT σ := by
+      change IsWellFounded σ fun x y ↦ WellOrderingRel x y
+      exact IsWellOrder.toIsWellFounded
+    simpa only [← lexOrder_eq_top_iff_eq_zero, lexOrder_mul, WithTop.add_eq_top] using h
+
 end MvPowerSeries
 
 end
diff --git a/Mathlib/RingTheory/Nakayama.lean b/Mathlib/RingTheory/Nakayama.lean
index 3c19bfdae99d5..2cf0c3850d107 100644
--- a/Mathlib/RingTheory/Nakayama.lean
+++ b/Mathlib/RingTheory/Nakayama.lean
@@ -46,7 +46,7 @@ namespace Submodule
 
 /-- **Nakayama's Lemma** - A slightly more general version of (2) in
 [Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
-See also `eq_bot_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`.  -/
+See also `eq_bot_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`. -/
 theorem eq_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N : Submodule R M} (hN : N.FG)
     (hIN : N ≤ I • N) (hIjac : I ≤ jacobson J) : N = J • N := by
   refine le_antisymm ?_ (Submodule.smul_le.2 fun _ _ _ => Submodule.smul_mem _ _)
@@ -123,7 +123,7 @@ theorem sup_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submod
 
 /-- **Nakayama's Lemma** - A slightly more general version of (4) in
 [Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
-See also `smul_le_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`.  -/
+See also `smul_le_of_le_smul_of_le_jacobson_bot` for the special case when `J = ⊥`. -/
 theorem sup_smul_eq_sup_smul_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
     (hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N' ≤ N ⊔ I • N') : N ⊔ I • N' = N ⊔ J • N' :=
   ((sup_le_sup_left smul_le_right _).antisymm (sup_le le_sup_left hNN)).trans
diff --git a/Mathlib/RingTheory/Nilpotent/Basic.lean b/Mathlib/RingTheory/Nilpotent/Basic.lean
index 0080e768a4300..71e196c394db9 100644
--- a/Mathlib/RingTheory/Nilpotent/Basic.lean
+++ b/Mathlib/RingTheory/Nilpotent/Basic.lean
@@ -82,6 +82,15 @@ theorem IsNilpotent.isUnit_add_right_of_commute [Ring R] {r u : R}
     IsUnit (r + u) :=
   add_comm r u ▸ hnil.isUnit_add_left_of_commute hu h_comm
 
+lemma IsUnit.not_isNilpotent [Ring R] [Nontrivial R] {x : R} (hx : IsUnit x) :
+    ¬ IsNilpotent x := by
+  intro H
+  simpa using H.isUnit_add_right_of_commute hx.neg (by simp)
+
+lemma IsNilpotent.not_isUnit [Ring R] [Nontrivial R] {x : R} (hx : IsNilpotent x) :
+    ¬ IsUnit x :=
+  mt IsUnit.not_isNilpotent (by simpa only [not_not] using hx)
+
 instance [Zero R] [Pow R ℕ] [Zero S] [Pow S ℕ] [IsReduced R] [IsReduced S] : IsReduced (R × S) where
   eq_zero _ := fun ⟨n, hn⟩ ↦ have hn := Prod.ext_iff.1 hn
     Prod.ext (IsReduced.eq_zero _ ⟨n, hn.1⟩) (IsReduced.eq_zero _ ⟨n, hn.2⟩)
diff --git a/Mathlib/RingTheory/Nilpotent/Defs.lean b/Mathlib/RingTheory/Nilpotent/Defs.lean
index 0404b768832fa..cbdde6aab35bc 100644
--- a/Mathlib/RingTheory/Nilpotent/Defs.lean
+++ b/Mathlib/RingTheory/Nilpotent/Defs.lean
@@ -168,6 +168,29 @@ class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where
   /-- A reduced structure has no nonzero nilpotent elements. -/
   eq_zero : ∀ x : R, IsNilpotent x → x = 0
 
+namespace IsReduced
+
+theorem pow_eq_zero [Zero R] [Pow R ℕ] [IsReduced R] {n : ℕ} (h : x ^ n = 0) :
+    x = 0 := IsReduced.eq_zero x ⟨n, h⟩
+
+@[simp]
+theorem pow_eq_zero_iff [MonoidWithZero R] [IsReduced R] {n : ℕ} (hn : n ≠ 0) :
+    x ^ n = 0 ↔ x = 0 := ⟨pow_eq_zero, fun h ↦ h.symm ▸ zero_pow hn⟩
+
+theorem pow_ne_zero_iff [MonoidWithZero R] [IsReduced R] {n : ℕ} (hn : n ≠ 0) :
+    x ^ n ≠ 0 ↔ x ≠ 0 := not_congr (pow_eq_zero_iff hn)
+
+theorem pow_ne_zero [Zero R] [Pow R ℕ] [IsReduced R] (n : ℕ) (h : x ≠ 0) :
+    x ^ n ≠ 0 := fun H ↦ h (pow_eq_zero H)
+
+/-- A variant of `IsReduced.pow_eq_zero_iff` assuming `R` is not trivial. -/
+@[simp]
+theorem pow_eq_zero_iff' [MonoidWithZero R] [IsReduced R] [Nontrivial R] {n : ℕ} :
+    x ^ n = 0 ↔ x = 0 ∧ n ≠ 0 := by
+  cases n <;> simp
+
+end IsReduced
+
 instance (priority := 900) isReduced_of_noZeroDivisors [MonoidWithZero R] [NoZeroDivisors R] :
     IsReduced R :=
   ⟨fun _ ⟨_, hn⟩ => pow_eq_zero hn⟩
diff --git a/Mathlib/RingTheory/Nilpotent/Lemmas.lean b/Mathlib/RingTheory/Nilpotent/Lemmas.lean
index 36d5f8aa60445..fdbae3b577172 100644
--- a/Mathlib/RingTheory/Nilpotent/Lemmas.lean
+++ b/Mathlib/RingTheory/Nilpotent/Lemmas.lean
@@ -31,16 +31,6 @@ theorem isRadical_iff_span_singleton [CommSemiring R] :
   simp_rw [IsRadical, ← Ideal.mem_span_singleton]
   exact forall_swap.trans (forall_congr' fun r => exists_imp.symm)
 
-namespace Commute
-
-section Semiring
-
-variable [Semiring R] (h_comm : Commute x y)
-
-end Semiring
-
-end Commute
-
 section CommSemiring
 
 variable [CommSemiring R] {x y : R}
@@ -99,11 +89,30 @@ end LinearMap
 
 namespace Module.End
 
-lemma isNilpotent.restrict {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
+section
+
+variable {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
+
+lemma isNilpotent_restrict_of_le {f : End R M} {p q : Submodule R M}
+    {hp : MapsTo f p p} {hq : MapsTo f q q} (h : p ≤ q) (hf : IsNilpotent (f.restrict hq)) :
+    IsNilpotent (f.restrict hp) := by
+  obtain ⟨n, hn⟩ := hf
+  use n
+  ext ⟨x, hx⟩
+  replace hn := DFunLike.congr_fun hn ⟨x, h hx⟩
+  simp_rw [LinearMap.zero_apply, ZeroMemClass.coe_zero, ZeroMemClass.coe_eq_zero] at hn ⊢
+  rw [LinearMap.pow_restrict, LinearMap.restrict_apply] at hn ⊢
+  ext
+  exact (congr_arg Subtype.val hn : _)
+
+lemma isNilpotent.restrict
     {f : M →ₗ[R] M} {p : Submodule R M} (hf : MapsTo f p p) (hnil : IsNilpotent f) :
     IsNilpotent (f.restrict hf) := by
   obtain ⟨n, hn⟩ := hnil
-  exact ⟨n, LinearMap.ext fun m ↦ by simp [LinearMap.pow_restrict n, LinearMap.restrict_apply, hn]⟩
+  exact ⟨n, LinearMap.ext fun m ↦ by simp only [LinearMap.pow_restrict n, hn,
+    LinearMap.restrict_apply, LinearMap.zero_apply]; rfl⟩
+
+end
 
 variable {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 variable {f : Module.End R M} {p : Submodule R M} (hp : p ≤ p.comap f)
diff --git a/Mathlib/RingTheory/Noetherian.lean b/Mathlib/RingTheory/Noetherian.lean
index 59a7d5f1a5552..903c8862e2b78 100644
--- a/Mathlib/RingTheory/Noetherian.lean
+++ b/Mathlib/RingTheory/Noetherian.lean
@@ -145,17 +145,21 @@ variable [Semiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R
 variable (R M)
 
 -- see Note [lower instance priority]
-instance (priority := 100) IsNoetherian.finite [IsNoetherian R M] : Finite R M :=
+instance (priority := 80) _root_.isNoetherian_of_finite [Finite M] : IsNoetherian R M :=
+  ⟨fun s => ⟨(s : Set M).toFinite.toFinset, by rw [Set.Finite.coe_toFinset, Submodule.span_eq]⟩⟩
+
+-- see Note [lower instance priority]
+instance (priority := 100) IsNoetherian.finite [IsNoetherian R M] : Module.Finite R M :=
   ⟨IsNoetherian.noetherian ⊤⟩
 
 instance {R₁ S : Type*} [CommSemiring R₁] [Semiring S] [Algebra R₁ S]
-    [IsNoetherian R₁ S] (I : Ideal S) : Finite R₁ I :=
+    [IsNoetherian R₁ S] (I : Ideal S) : Module.Finite R₁ I :=
   IsNoetherian.finite R₁ ((I : Submodule S S).restrictScalars R₁)
 
 variable {R M}
 
 theorem Finite.of_injective [IsNoetherian R N] (f : M →ₗ[R] N) (hf : Function.Injective f) :
-    Finite R M :=
+    Module.Finite R M :=
   ⟨fg_of_injective f hf⟩
 
 end Module
@@ -184,77 +188,13 @@ instance isNoetherian_prod [IsNoetherian R M] [IsNoetherian R P] : IsNoetherian
         fun x ⟨_, hx2⟩ => ⟨x.1, Prod.ext rfl <| Eq.symm <| LinearMap.mem_ker.1 hx2⟩
       Submodule.map_comap_eq_self this ▸ (noetherian _).map _⟩
 
-instance isNoetherian_pi {R ι : Type*} {M : ι → Type*}
-    [Ring R] [∀ i, AddCommGroup (M i)] [∀ i, Module R (M i)] [Finite ι]
-    [∀ i, IsNoetherian R (M i)] : IsNoetherian R (∀ i, M i) := by
-  cases nonempty_fintype ι
-  haveI := Classical.decEq ι
-  suffices on_finset : ∀ s : Finset ι, IsNoetherian R (∀ i : s, M i) by
-    let coe_e := Equiv.subtypeUnivEquiv <| @Finset.mem_univ ι _
-    letI : IsNoetherian R (∀ i : Finset.univ, M (coe_e i)) := on_finset Finset.univ
-    exact isNoetherian_of_linearEquiv (LinearEquiv.piCongrLeft R M coe_e)
-  intro s
-  induction' s using Finset.induction with a s has ih
-  · exact ⟨fun s => by
-      have : s = ⊥ := by simp only [eq_iff_true_of_subsingleton]
-      rw [this]
-      apply Submodule.fg_bot⟩
-  refine
-    @isNoetherian_of_linearEquiv R (M a × ((i : s) → M i)) _ _ _ _ _ _ ?_ <|
-      @isNoetherian_prod R (M a) _ _ _ _ _ _ _ ih
-  refine
-  { toFun := fun f i =>
-      (Finset.mem_insert.1 i.2).by_cases
-        (fun h : i.1 = a => show M i.1 from Eq.recOn h.symm f.1)
-        (fun h : i.1 ∈ s => show M i.1 from f.2 ⟨i.1, h⟩),
-    invFun := fun f =>
-      (f ⟨a, Finset.mem_insert_self _ _⟩, fun i => f ⟨i.1, Finset.mem_insert_of_mem i.2⟩),
-    map_add' := ?_,
-    map_smul' := ?_
-    left_inv := ?_,
-    right_inv := ?_ }
-  · intro f g
-    ext i
-    unfold Or.by_cases
-    cases' i with i hi
-    rcases Finset.mem_insert.1 hi with (rfl | h)
-    · change _ = _ + _
-      simp only [dif_pos]
-      rfl
-    · change _ = _ + _
-      have : ¬i = a := by
-        rintro rfl
-        exact has h
-      simp only [dif_neg this, dif_pos h]
-      rfl
-  · intro c f
-    ext i
-    unfold Or.by_cases
-    cases' i with i hi
-    rcases Finset.mem_insert.1 hi with (rfl | h)
-    · dsimp
-      simp only [dif_pos]
-    · dsimp
-      have : ¬i = a := by
-        rintro rfl
-        exact has h
-      simp only [dif_neg this, dif_pos h]
-  · intro f
-    apply Prod.ext
-    · simp only [Or.by_cases, dif_pos]
-    · ext ⟨i, his⟩
-      have : ¬i = a := by
-        rintro rfl
-        exact has his
-      simp only [Or.by_cases, this, not_false_iff, dif_neg]
-  · intro f
-    ext ⟨i, hi⟩
-    rcases Finset.mem_insert.1 hi with (rfl | h)
-    · simp only [Or.by_cases, dif_pos]
-    · have : ¬i = a := by
-        rintro rfl
-        exact has h
-      simp only [Or.by_cases, dif_neg this, dif_pos h]
+instance isNoetherian_pi {R ι : Type*} [Finite ι] :
+    ∀ {M : ι → Type*} [Ring R] [∀ i, AddCommGroup (M i)]
+      [∀ i, Module R (M i)] [∀ i, IsNoetherian R (M i)], IsNoetherian R (∀ i, M i) := by
+  apply Finite.induction_empty_option _ _ _ ι
+  · exact fun e h ↦ isNoetherian_of_linearEquiv (LinearEquiv.piCongrLeft R _ e)
+  · infer_instance
+  · exact fun ih ↦ isNoetherian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm
 
 /-- A version of `isNoetherian_pi` for non-dependent functions. We need this instance because
 sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to
@@ -289,19 +229,19 @@ universe w
 variable {R M P : Type*} {N : Type w} [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N]
   [Module R N] [AddCommMonoid P] [Module R P]
 
-theorem isNoetherian_iff :
-    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by
-  have := (CompleteLattice.wellFounded_characterisations <| Submodule R M).out 0 3
+theorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by
+  have := (CompleteLattice.wellFoundedGT_characterisations <| Submodule R M).out 0 3
   -- Porting note: inlining this makes rw complain about it being a metavariable
   rw [this]
   exact
     ⟨fun ⟨h⟩ => fun k => (fg_iff_compact k).mp (h k), fun h =>
       ⟨fun k => (fg_iff_compact k).mpr (h k)⟩⟩
 
-alias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff
+theorem isNoetherian_iff :
+    IsNoetherian R M ↔ WellFounded ((· > ·) : Submodule R M → Submodule R M → Prop) := by
+  rw [isNoetherian_iff', ← isWellFounded_iff]
 
-theorem isNoetherian_iff' : IsNoetherian R M ↔ WellFoundedGT (Submodule R M) := by
-  rw [isNoetherian_iff, ← isWellFounded_iff]
+alias ⟨IsNoetherian.wf, _⟩ := isNoetherian_iff
 
 alias ⟨IsNoetherian.wellFoundedGT, isNoetherian_mk⟩ := isNoetherian_iff'
 
@@ -363,14 +303,13 @@ variable {R M P : Type*} {N : Type w} [Ring R] [AddCommGroup M] [Module R M] [Ad
 lemma Submodule.finite_ne_bot_of_independent {ι : Type*} {N : ι → Submodule R M}
     (h : CompleteLattice.Independent N) :
     Set.Finite {i | N i ≠ ⊥} :=
-  CompleteLattice.WellFounded.finite_ne_bot_of_independent
-    (IsWellFounded.wf) h
+  CompleteLattice.WellFoundedGT.finite_ne_bot_of_independent h
 
 /-- A linearly-independent family of vectors in a module over a non-trivial ring must be finite if
 the module is Noetherian. -/
 theorem LinearIndependent.finite_of_isNoetherian [Nontrivial R] {ι} {v : ι → M}
     (hv : LinearIndependent R v) : Finite ι := by
-  refine CompleteLattice.WellFounded.finite_of_independent IsWellFounded.wf
+  refine CompleteLattice.WellFoundedGT.finite_of_independent
     hv.independent_span_singleton
     fun i contra => ?_
   apply hv.ne_zero i
@@ -381,9 +320,6 @@ theorem LinearIndependent.set_finite_of_isNoetherian [Nontrivial R] {s : Set M}
     (hi : LinearIndependent R ((↑) : s → M)) : s.Finite :=
   @Set.toFinite _ _ hi.finite_of_isNoetherian
 
-@[deprecated (since := "2023-12-30")]
-alias finite_of_linearIndependent := LinearIndependent.set_finite_of_isNoetherian
-
 /-- If the first and final modules in an exact sequence are Noetherian,
   then the middle module is also Noetherian. -/
 theorem isNoetherian_of_range_eq_ker [IsNoetherian R P]
@@ -392,15 +328,20 @@ theorem isNoetherian_of_range_eq_ker [IsNoetherian R P]
   isNoetherian_mk <|
     wellFounded_gt_exact_sequence
       (LinearMap.range f)
-      (Submodule.map (f.ker.liftQ f <| le_rfl))
-      (Submodule.comap (f.ker.liftQ f <| le_rfl))
+      (Submodule.map (f.ker.liftQ f le_rfl))
+      (Submodule.comap (f.ker.liftQ f le_rfl))
       (Submodule.comap g.rangeRestrict) (Submodule.map g.rangeRestrict)
-      (Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <|
-        Submodule.ker_liftQ_eq_bot _ _ _ (le_refl _))
+      (Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <| Submodule.ker_liftQ_eq_bot _ _ _ le_rfl)
       (Submodule.giMapComap g.surjective_rangeRestrict)
       (by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
       (by simp [Submodule.comap_map_eq, h])
 
+theorem isNoetherian_iff_submodule_quotient (S : Submodule R P) :
+    IsNoetherian R P ↔ IsNoetherian R S ∧ IsNoetherian R (P ⧸ S) := by
+  refine ⟨fun _ ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
+  apply isNoetherian_of_range_eq_ker S.subtype S.mkQ
+  rw [Submodule.ker_mkQ, Submodule.range_subtype]
+
 /-- For an endomorphism of a Noetherian module, any sufficiently large iterate has disjoint kernel
 and range. -/
 theorem LinearMap.eventually_disjoint_ker_pow_range_pow (f : M →ₗ[R] M) :
@@ -503,11 +444,6 @@ theorem isNoetherianRing_iff_ideal_fg (R : Type*) [Semiring R] :
     IsNoetherianRing R ↔ ∀ I : Ideal R, I.FG :=
   isNoetherianRing_iff.trans isNoetherian_def
 
--- see Note [lower instance priority]
-instance (priority := 80) isNoetherian_of_finite (R M) [Finite M] [Semiring R] [AddCommMonoid M]
-    [Module R M] : IsNoetherian R M :=
-  ⟨fun s => ⟨(s : Set M).toFinite.toFinset, by rw [Set.Finite.coe_toFinset, Submodule.span_eq]⟩⟩
-
 -- see Note [lower instance priority]
 /-- Modules over the trivial ring are Noetherian. -/
 instance (priority := 100) isNoetherian_of_subsingleton (R M) [Subsingleton R] [Semiring R]
@@ -550,12 +486,12 @@ theorem isNoetherian_of_fg_of_noetherian {R M} [Ring R] [AddCommGroup M] [Module
   · rw [LinearMap.range_eq_top]
     rintro ⟨n, hn⟩
     change n ∈ N at hn
-    rw [← hs, ← Set.image_id (s : Set M), Finsupp.mem_span_image_iff_total] at hn
+    rw [← hs, ← Set.image_id (s : Set M), Finsupp.mem_span_image_iff_linearCombination] at hn
     rcases hn with ⟨l, hl1, hl2⟩
     refine ⟨fun x => l x, Subtype.ext ?_⟩
     change (∑ i ∈ s.attach, l i • (i : M)) = n
     rw [s.sum_attach fun i ↦ l i • i, ← hl2,
-      Finsupp.total_apply, Finsupp.sum, eq_comm]
+      Finsupp.linearCombination_apply, Finsupp.sum, eq_comm]
     refine Finset.sum_subset hl1 fun x _ hx => ?_
     rw [Finsupp.not_mem_support_iff.1 hx, zero_smul]
 
diff --git a/Mathlib/RingTheory/NonUnitalSubring/Basic.lean b/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
index 0a1478bc862f1..f71dbda3f4e85 100644
--- a/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
+++ b/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
@@ -71,7 +71,7 @@ universe u v w
 
 section Basic
 
-variable {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocRing R]
+variable {R : Type u} {S : Type v} [NonUnitalNonAssocRing R]
 
 section NonUnitalSubringClass
 
@@ -124,8 +124,6 @@ end NonUnitalSubringClass
 
 end NonUnitalSubringClass
 
-variable [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
-
 /-- `NonUnitalSubring R` is the type of non-unital subrings of `R`. A non-unital subring of `R`
 is a subset `s` that is a multiplicative subsemigroup and an additive subgroup. Note in particular
 that it shares the same 0 as R. -/
@@ -387,7 +385,7 @@ section Hom
 
 namespace NonUnitalSubring
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*} {SR : Type*}
+variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
   [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
   [FunLike F R S] [NonUnitalRingHomClass F R S] (s : NonUnitalSubring R)
 
@@ -503,10 +501,7 @@ namespace NonUnitalSubring
 
 section Order
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
-  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
-  [FunLike F R S] [NonUnitalRingHomClass F R S]
-  (g : S →ₙ+* T) (f : R →ₙ+* S)
+variable {R : Type u} [NonUnitalNonAssocRing R]
 
 /-! ## bot -/
 
@@ -640,10 +635,9 @@ end Center
 
 /-! ## `NonUnitalSubring` closure of a subset -/
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
-  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
+variable {F : Type w} {R : Type u} {S : Type v}
+  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
   [FunLike F R S] [NonUnitalRingHomClass F R S]
-  (g : S →ₙ+* T) (f : R →ₙ+* S)
 
 /-- The `NonUnitalSubring` generated by a set. -/
 def closure (s : Set R) : NonUnitalSubring R :=
@@ -743,7 +737,7 @@ theorem mem_closure_iff {s : Set R} {x} :
           mul_mem hx hy)
       (zero_mem _) (fun x y hx hy => add_mem hx hy) fun x hx => neg_mem hx⟩
 
-/-- If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring.  -/
+/-- If all elements of `s : Set A` commute pairwise, then `closure s` is a commutative ring. -/
 def closureNonUnitalCommRingOfComm {R : Type u} [NonUnitalRing R] {s : Set R}
     (hcomm : ∀ a ∈ s, ∀ b ∈ s, a * b = b * a) : NonUnitalCommRing (closure s) :=
   { (closure s).toNonUnitalRing with
@@ -800,6 +794,14 @@ theorem map_iSup {ι : Sort*} (f : F) (s : ι → NonUnitalSubring R) :
     (iSup s).map f = ⨆ i, (s i).map f :=
   (gc_map_comap f).l_iSup
 
+theorem map_inf (s t : NonUnitalSubring R) (f : F) (hf : Function.Injective f) :
+    (s ⊓ t).map f = s.map f ⊓ t.map f := SetLike.coe_injective (Set.image_inter hf)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
+    (s : ι → NonUnitalSubring R) : (iInf s).map f = ⨅ i, (s i).map f := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem comap_inf (s t : NonUnitalSubring S) (f : F) : (s ⊓ t).comap f = s.comap f ⊓ t.comap f :=
   (gc_map_comap f).u_inf
 
@@ -902,11 +904,8 @@ end NonUnitalSubring
 
 namespace NonUnitalRingHom
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
-  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
-  [FunLike F R S] [NonUnitalRingHomClass F R S]
-  (g : S →ₙ+* T) (f : R →ₙ+* S)
-  {s : NonUnitalSubring R}
+variable {R : Type u} {S : Type v}
+  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
 
 open NonUnitalSubring
 
@@ -972,11 +971,8 @@ end NonUnitalRingHom
 
 namespace NonUnitalSubring
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
-  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [NonUnitalNonAssocRing T]
-  [FunLike F R S] [NonUnitalRingHomClass F R S]
-  (g : S →ₙ+* T) (f : R →ₙ+* S)
-  {s : NonUnitalSubring R}
+variable {R : Type u} {S : Type v}
+  [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
 
 open NonUnitalRingHom
 
@@ -998,11 +994,7 @@ end NonUnitalSubring
 
 namespace RingEquiv
 
-variable {F : Type w} {R : Type u} {S : Type v} {T : Type*}
-  [NonUnitalRing R] [NonUnitalRing S] [NonUnitalRing T]
-  [FunLike F R S] [NonUnitalRingHomClass F R S]
-  (g : S →ₙ+* T) (f : R →ₙ+* S)
-  {s t : NonUnitalSubring R}
+variable {R : Type u} {S : Type v} [NonUnitalRing R] [NonUnitalRing S] {s t : NonUnitalSubring R}
 
 /-- Makes the identity isomorphism from a proof two `NonUnitalSubring`s of a multiplicative
     monoid are equal. -/
diff --git a/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean b/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
index 3e465645af7db..891e01b70140c 100644
--- a/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
+++ b/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
@@ -29,7 +29,7 @@ variable {R : Type u} {S : Type v} {T : Type w} [NonUnitalNonAssocSemiring R] (M
 
 /-- `NonUnitalSubsemiringClass S R` states that `S` is a type of subsets `s ⊆ R` that
 are both an additive submonoid and also a multiplicative subsemigroup. -/
-class NonUnitalSubsemiringClass (S : Type*) (R : Type u) [NonUnitalNonAssocSemiring R]
+class NonUnitalSubsemiringClass (S : Type*) (R : outParam (Type u)) [NonUnitalNonAssocSemiring R]
   [SetLike S R] extends AddSubmonoidClass S R : Prop where
   mul_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s
 
@@ -379,6 +379,15 @@ theorem coe_sInf (S : Set (NonUnitalSubsemiring R)) :
 theorem mem_sInf {S : Set (NonUnitalSubsemiring R)} {x : R} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p :=
   Set.mem_iInter₂
 
+@[simp, norm_cast]
+theorem coe_iInf {ι : Sort*} {S : ι → NonUnitalSubsemiring R} :
+    (↑(⨅ i, S i) : Set R) = ⋂ i, S i := by
+  simp only [iInf, coe_sInf, Set.biInter_range]
+
+theorem mem_iInf {ι : Sort*} {S : ι → NonUnitalSubsemiring R} {x : R} :
+    (x ∈ ⨅ i, S i) ↔ ∀ i, x ∈ S i := by
+  simp only [iInf, mem_sInf, Set.forall_mem_range]
+
 @[simp]
 theorem sInf_toSubsemigroup (s : Set (NonUnitalSubsemiring R)) :
     (sInf s).toSubsemigroup = ⨅ t ∈ s, NonUnitalSubsemiring.toSubsemigroup t :=
@@ -670,6 +679,16 @@ theorem map_iSup {ι : Sort*} (f : F) (s : ι → NonUnitalSubsemiring R) :
     (map f (iSup s) : NonUnitalSubsemiring S) = ⨆ i, map f (s i) :=
   @GaloisConnection.l_iSup _ _ _ _ _ _ _ (gc_map_comap f) s
 
+theorem map_inf (s t : NonUnitalSubsemiring R) (f : F) (hf : Function.Injective f) :
+    (map f (s ⊓ t) : NonUnitalSubsemiring S) = map f s ⊓ map f t :=
+  SetLike.coe_injective (Set.image_inter hf)
+
+theorem map_iInf {ι : Sort*} [Nonempty ι] (f : F) (hf : Function.Injective f)
+    (s : ι → NonUnitalSubsemiring R) :
+    (map f (iInf s) : NonUnitalSubsemiring S) = ⨅ i, map f (s i) := by
+  apply SetLike.coe_injective
+  simpa using (Set.injOn_of_injective hf).image_iInter_eq (s := SetLike.coe ∘ s)
+
 theorem comap_inf (s t : NonUnitalSubsemiring S) (f : F) :
     (comap f (s ⊓ t) : NonUnitalSubsemiring R) = comap f s ⊓ comap f t :=
   @GaloisConnection.u_inf _ _ s t _ _ _ _ (gc_map_comap f)
diff --git a/Mathlib/RingTheory/Norm/Basic.lean b/Mathlib/RingTheory/Norm/Basic.lean
index a051d47766d54..e58b1fc23e208 100644
--- a/Mathlib/RingTheory/Norm/Basic.lean
+++ b/Mathlib/RingTheory/Norm/Basic.lean
@@ -8,7 +8,7 @@ import Mathlib.FieldTheory.PrimitiveElement
 import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
 import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
 import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 
 /-!
 # Norm for (finite) ring extensions
@@ -46,7 +46,7 @@ variable {K L F : Type*} [Field K] [Field L] [Field F]
 variable [Algebra K L] [Algebra K F]
 variable {ι : Type w}
 
-open FiniteDimensional
+open Module
 
 open LinearMap
 
@@ -74,7 +74,7 @@ theorem PowerBasis.norm_gen_eq_prod_roots [Algebra R F] (pb : PowerBasis R S)
   have := minpoly.monic pb.isIntegral_gen
   rw [PowerBasis.norm_gen_eq_coeff_zero_minpoly, ← pb.natDegree_minpoly, RingHom.map_mul,
     ← coeff_map,
-    prod_roots_eq_coeff_zero_of_monic_of_split (this.map _) ((splits_id_iff_splits _).2 hf),
+    prod_roots_eq_coeff_zero_of_monic_of_splits (this.map _) ((splits_id_iff_splits _).2 hf),
     this.natDegree_map, map_pow, ← mul_assoc, ← mul_pow]
   simp only [map_neg, _root_.map_one, neg_mul, neg_neg, one_pow, one_mul]
 
@@ -153,7 +153,7 @@ theorem _root_.IntermediateField.AdjoinSimple.norm_gen_eq_one {x : L} (hx : ¬Is
   contrapose! hx
   obtain ⟨s, ⟨b⟩⟩ := hx
   refine .of_mem_of_fg K⟮x⟯.toSubalgebra ?_ x ?_
-  · exact (Submodule.fg_iff_finiteDimensional _).mpr (of_fintype_basis b)
+  · exact (Submodule.fg_iff_finiteDimensional _).mpr (.of_fintype_basis b)
   · exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)
 
 theorem _root_.IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots (x : L)
diff --git a/Mathlib/RingTheory/Norm/Defs.lean b/Mathlib/RingTheory/Norm/Defs.lean
index 6f1c21fae5c00..b7f2ce47f5c97 100644
--- a/Mathlib/RingTheory/Norm/Defs.lean
+++ b/Mathlib/RingTheory/Norm/Defs.lean
@@ -41,7 +41,7 @@ variable {K L F : Type*} [Field K] [Field L] [Field F]
 variable [Algebra K L] [Algebra K F]
 variable {ι : Type w}
 
-open FiniteDimensional
+open Module
 
 open LinearMap
 
diff --git a/Mathlib/RingTheory/PiTensorProduct.lean b/Mathlib/RingTheory/PiTensorProduct.lean
index ed8ee78470106..c4eeb9284a5f1 100644
--- a/Mathlib/RingTheory/PiTensorProduct.lean
+++ b/Mathlib/RingTheory/PiTensorProduct.lean
@@ -187,7 +187,7 @@ The map `Aᵢ ⟶ ⨂ᵢ Aᵢ` given by `a ↦ 1 ⊗ ... ⊗ a ⊗ 1 ⊗ ...`
 def singleAlgHom [DecidableEq ι] (i : ι) : A i →ₐ[R] ⨂[R] i, A i where
   toFun a := tprod R (MonoidHom.mulSingle _ i a)
   map_one' := by simp only [_root_.map_one]; rfl
-  map_mul' a a' := by simp
+  map_mul' a a' := by simp [_root_.map_mul]
   map_zero' := MultilinearMap.map_update_zero _ _ _
   map_add' _ _ := MultilinearMap.map_add _ _ _ _ _
   commutes' r := show tprodCoeff R _ _ = r • tprodCoeff R _ _ by
diff --git a/Mathlib/RingTheory/Polynomial/Basic.lean b/Mathlib/RingTheory/Polynomial/Basic.lean
index 9a06f73be4237..ab6d03ba20af2 100644
--- a/Mathlib/RingTheory/Polynomial/Basic.lean
+++ b/Mathlib/RingTheory/Polynomial/Basic.lean
@@ -173,6 +173,34 @@ theorem degreeLT_succ_eq_degreeLE {n : ℕ} : degreeLT R (n + 1) = degreeLE R n
   · rw [mem_degreeLT, mem_degreeLE, ← natDegree_lt_iff_degree_lt (by rwa [ne_eq]),
       ← natDegree_le_iff_degree_le, Nat.lt_succ]
 
+/-- The equivalence between monic polynomials of degree `n` and polynomials of degree less than
+`n`, formed by adding a term `X ^ n`. -/
+def monicEquivDegreeLT [Nontrivial R] (n : ℕ) :
+    { p : R[X] // p.Monic ∧ p.natDegree = n } ≃ degreeLT R n where
+  toFun p := ⟨p.1.eraseLead, by
+    rcases p with ⟨p, hp, rfl⟩
+    simp only [mem_degreeLT]
+    refine lt_of_lt_of_le ?_ degree_le_natDegree
+    exact degree_eraseLead_lt (ne_zero_of_ne_zero_of_monic one_ne_zero hp)⟩
+  invFun := fun p =>
+    ⟨X^n + p.1, monic_X_pow_add (mem_degreeLT.1 p.2), by
+        rw [natDegree_add_eq_left_of_degree_lt]
+        · simp
+        · simp [mem_degreeLT.1 p.2]⟩
+  left_inv := by
+    rintro ⟨p, hp, rfl⟩
+    ext1
+    simp only
+    conv_rhs => rw [← eraseLead_add_C_mul_X_pow p]
+    simp [Monic.def.1 hp, add_comm]
+  right_inv := by
+    rintro ⟨p, hp⟩
+    ext1
+    simp only
+    rw [eraseLead_add_of_degree_lt_left]
+    · simp
+    · simp [mem_degreeLT.1 hp]
+
 /-- For every polynomial `p` in the span of a set `s : Set R[X]`, there exists a polynomial of
   `p' ∈ s` with higher degree. See also `Polynomial.exists_degree_le_of_mem_span_of_finite`. -/
 theorem exists_degree_le_of_mem_span {s : Set R[X]} {p : R[X]}
@@ -219,7 +247,7 @@ theorem span_of_finite_le_degreeLT {s : Set R[X]} (s_fin : s.Finite) :
   exact ⟨n + 1, by rwa [degreeLT_succ_eq_degreeLE]⟩
 
 /-- If `R` is a nontrivial ring, the polynomials `R[X]` are not finite as an `R`-module. When `R` is
-a field, this is equivalent to `R[X]` being an infinite-dimensional vector space over `R`.  -/
+a field, this is equivalent to `R[X]` being an infinite-dimensional vector space over `R`. -/
 theorem not_finite [Nontrivial R] : ¬ Module.Finite R R[X] := by
   rw [Module.finite_def, Submodule.fg_def]
   push_neg
@@ -238,6 +266,7 @@ def coeffs (p : R[X]) : Finset R :=
 
 @[deprecated (since := "2024-05-17")] noncomputable alias frange := coeffs
 
+@[simp]
 theorem coeffs_zero : coeffs (0 : R[X]) = ∅ :=
   rfl
 
@@ -262,6 +291,10 @@ theorem coeff_mem_coeffs (p : R[X]) (n : ℕ) (h : p.coeff n ≠ 0) : p.coeff n
 
 @[deprecated (since := "2024-05-17")] alias coeff_mem_frange := coeff_mem_coeffs
 
+theorem coeffs_monomial (n : ℕ) {c : R} (hc : c ≠ 0) : (monomial n c).coeffs = {c} := by
+  rw [coeffs, support_monomial n hc]
+  simp
+
 theorem geom_sum_X_comp_X_add_one_eq_sum (n : ℕ) :
     (∑ i ∈ range n, (X : R[X]) ^ i).comp (X + 1) =
       (Finset.range n).sum fun i : ℕ => (n.choose (i + 1) : R[X]) * X ^ i := by
@@ -572,8 +605,7 @@ theorem _root_.Polynomial.ker_mapRingHom (f : R →+* S) :
   ext
   simp only [LinearMap.mem_ker, RingHom.toSemilinearMap_apply, coe_mapRingHom]
   rw [mem_map_C_iff, Polynomial.ext_iff]
-  simp_rw [RingHom.mem_ker f]
-  simp
+  simp [RingHom.mem_ker]
 
 variable (I : Ideal R[X])
 
@@ -724,8 +756,7 @@ theorem isPrime_map_C_iff_isPrime (P : Ideal R) :
           · rw [← not_le]
             intro hnj
             exact (add_lt_add_of_lt_of_le hmi hnj).ne hij.2.symm
-          · simp only [eq_self_iff_true, not_true, false_or_iff, add_right_inj,
-              not_and_self_iff] at hij
+          · simp only [eq_self_iff_true, not_true, false_or, add_right_inj, not_and_self_iff] at hij
         · rw [mul_comm]
           apply P.mul_mem_left
           exact Classical.not_not.1 (Nat.find_min hf hi)
@@ -840,14 +871,14 @@ namespace Polynomial
 
 instance (priority := 100) wfDvdMonoid {R : Type*} [CommRing R] [IsDomain R] [WfDvdMonoid R] :
     WfDvdMonoid R[X] where
-  wellFounded_dvdNotUnit := by
+  wf := by
     classical
       refine
         RelHomClass.wellFounded
           (⟨fun p : R[X] =>
               ((if p = 0 then ⊤ else ↑p.degree : WithTop (WithBot ℕ)), p.leadingCoeff), ?_⟩ :
             DvdNotUnit →r Prod.Lex (· < ·) DvdNotUnit)
-          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wellFounded_dvdNotUnit)
+          (wellFounded_lt.prod_lex ‹WfDvdMonoid R›.wf)
       rintro a b ⟨ane0, ⟨c, ⟨not_unit_c, rfl⟩⟩⟩
       dsimp
       rw [Polynomial.degree_mul, if_neg ane0]
@@ -968,8 +999,8 @@ theorem exists_irreducible_of_natDegree_ne_zero {R : Type u} [CommRing R] [IsDom
 theorem linearIndependent_powers_iff_aeval (f : M →ₗ[R] M) (v : M) :
     (LinearIndependent R fun n : ℕ => (f ^ n) v) ↔ ∀ p : R[X], aeval f p v = 0 → p = 0 := by
   rw [linearIndependent_iff]
-  simp only [Finsupp.total_apply, aeval_endomorphism, forall_iff_forall_finsupp, Sum, support,
-    coeff, ofFinsupp_eq_zero]
+  simp only [Finsupp.linearCombination_apply, aeval_endomorphism, forall_iff_forall_finsupp, Sum,
+    support, coeff, ofFinsupp_eq_zero]
   exact Iff.rfl
 
 attribute [-instance] Ring.toNonAssocRing
@@ -1183,6 +1214,11 @@ theorem ker_map (f : R →+* S) :
   rw [MvPolynomial.mem_map_C_iff, RingHom.mem_ker, MvPolynomial.ext_iff]
   simp_rw [coeff_map, coeff_zero, RingHom.mem_ker]
 
+lemma ker_mapAlgHom {S₁ S₂ σ : Type*} [CommRing S₁] [CommRing S₂] [Algebra R S₁]
+    [Algebra R S₂] (f : S₁ →ₐ[R] S₂) :
+    RingHom.ker (MvPolynomial.mapAlgHom (σ := σ) f) = Ideal.map MvPolynomial.C (RingHom.ker f) :=
+  MvPolynomial.ker_map (f.toRingHom : S₁ →+* S₂)
+
 end MvPolynomial
 
 section UniqueFactorizationDomain
diff --git a/Mathlib/RingTheory/Polynomial/Bernstein.lean b/Mathlib/RingTheory/Polynomial/Bernstein.lean
index d3efc2a2b5cb0..8cae9a4f38a0d 100644
--- a/Mathlib/RingTheory/Polynomial/Bernstein.lean
+++ b/Mathlib/RingTheory/Polynomial/Bernstein.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.MvPolynomial.PDeriv
 import Mathlib.Algebra.Polynomial.AlgebraMap
diff --git a/Mathlib/RingTheory/Polynomial/Content.lean b/Mathlib/RingTheory/Polynomial/Content.lean
index 273f4f4ba8827..c92ff552d810e 100644
--- a/Mathlib/RingTheory/Polynomial/Content.lean
+++ b/Mathlib/RingTheory/Polynomial/Content.lean
@@ -207,7 +207,7 @@ theorem IsPrimitive.content_eq_one {p : R[X]} (hp : p.IsPrimitive) : p.content =
 section PrimPart
 
 /-- The primitive part of a polynomial `p` is the primitive polynomial gained by dividing `p` by
-  `p.content`. If `p = 0`, then `p.primPart = 1`.  -/
+  `p.content`. If `p = 0`, then `p.primPart = 1`. -/
 noncomputable def primPart (p : R[X]) : R[X] :=
   letI := Classical.decEq R
   if p = 0 then 1 else Classical.choose (C_content_dvd p)
diff --git a/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean b/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean
index af7686e449eba..4a1ad15664ee8 100644
--- a/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean
+++ b/Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean
@@ -147,7 +147,7 @@ theorem eval_one_cyclotomic_not_prime_pow {R : Type*} [Ring R] {n : ℕ}
     rw [← Finset.prod_sdiff <| show {n} ⊆ _ from _] at this
     swap
     · simp only [singleton_subset_iff, mem_sdiff, mem_erase, Ne, mem_divisors, dvd_refl,
-        true_and_iff, mem_image, mem_range, exists_prop, not_exists, not_and]
+        true_and, mem_image, mem_range, exists_prop, not_exists, not_and]
       exact ⟨⟨hn.ne', hn'.ne'⟩, fun t _ => h hp _⟩
     rw [← Int.natAbs_ofNat p, Int.natAbs_dvd_natAbs] at hpe
     obtain ⟨t, ht⟩ := hpe
@@ -203,7 +203,7 @@ theorem sub_one_pow_totient_lt_cyclotomic_eval {n : ℕ} {q : ℝ} (hn' : 2 ≤
       Units.val_le_val, ← NNReal.coe_le_coe, Complex.abs.nonneg, hq'.le, Units.val_mk0,
       Real.coe_toNNReal', coe_nnnorm, Complex.norm_eq_abs, max_le_iff, tsub_le_iff_right]
     intro x hx
-    simpa only [and_true_iff, tsub_le_iff_right] using hfor x hx
+    simpa only [and_true, tsub_le_iff_right] using hfor x hx
   · simp only [Subtype.coe_mk, Finset.mem_attach, exists_true_left, Subtype.exists, ←
       NNReal.coe_lt_coe, ← Units.val_lt_val, Units.val_mk0 _, coe_nnnorm]
     simpa [hq'.le, Real.coe_toNNReal', max_eq_left, sub_nonneg] using hex
diff --git a/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean b/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean
index 5f2544087a31a..d66fdc3f66ab9 100644
--- a/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean
+++ b/Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean
@@ -3,8 +3,8 @@ Copyright (c) 2020 Riccardo Brasca. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Riccardo Brasca
 -/
+import Mathlib.Algebra.Algebra.ZMod
 import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
-import Mathlib.Data.ZMod.Algebra
 
 /-!
 # Cyclotomic polynomials and `expand`.
diff --git a/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean b/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
index ac741208832d2..74c4705dddc41 100644
--- a/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
+++ b/Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean
@@ -113,7 +113,7 @@ theorem cyclotomic.roots_to_finset_eq_primitiveRoots [NeZero (n : R)] :
   -- `simp [cyclotomic_ne_zero n R, isRoot_cyclotomic_iff, mem_primitiveRoots,`
   -- `  NeZero.pos_of_neZero_natCast R]`
   simp only [mem_primitiveRoots, NeZero.pos_of_neZero_natCast R]
-  convert isRoot_cyclotomic_iff (n := n) (μ := a)
+  convert isRoot_cyclotomic_iff (n := n) (μ := a) using 0
   simp [cyclotomic_ne_zero n R]
 
 theorem cyclotomic.roots_eq_primitiveRoots_val [NeZero (n : R)] :
diff --git a/Mathlib/RingTheory/Polynomial/Dickson.lean b/Mathlib/RingTheory/Polynomial/Dickson.lean
index 6f3bfc0a478bf..fdb6b3b533b34 100644
--- a/Mathlib/RingTheory/Polynomial/Dickson.lean
+++ b/Mathlib/RingTheory/Polynomial/Dickson.lean
@@ -231,7 +231,7 @@ theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p)
           mem_roots hφ, IsRoot, eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one,
           Multiset.mem_singleton]
         by_cases hy : y = 0
-        · simp only [hy, eq_self_iff_true, or_true_iff]
+        · simp only [hy, eq_self_iff_true, or_true]
         apply or_congr _ Iff.rfl
         rw [← mul_left_inj' hy, eq_comm, ← sub_eq_zero, add_mul, inv_mul_cancel₀ hy]
         apply eq_iff_eq_cancel_right.mpr
@@ -241,9 +241,9 @@ theorem dickson_one_one_zmod_p (p : ℕ) [Fact p.Prime] : dickson 1 (1 : ZMod p)
     intro x
     simp only [exists_prop, Set.mem_iUnion, Set.bind_def, Ne, Set.mem_setOf_eq]
     by_cases hx : x = 0
-    · simp only [hx, and_true_iff, eq_self_iff_true, inv_zero, or_true_iff]
+    · simp only [hx, and_true, eq_self_iff_true, inv_zero, or_true]
       exact ⟨_, 1, rfl, one_ne_zero⟩
-    · simp only [hx, or_false_iff, exists_eq_right]
+    · simp only [hx, or_false, exists_eq_right]
       exact ⟨_, rfl, hx⟩
 
 theorem dickson_one_one_charP (p : ℕ) [Fact p.Prime] [CharP R p] : dickson 1 (1 : R) p = X ^ p := by
diff --git a/Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean b/Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
index 50e6698cdb932..6365d391b724f 100644
--- a/Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
+++ b/Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean
@@ -53,6 +53,8 @@ theorem cyclotomic_comp_X_add_one_isEisensteinAt [hp : Fact p.Prime] :
       congr
       congr
       next => skip
+      congr
+      next => skip
       ext
       rw [lcoeff_apply, ← C_eq_natCast, C_mul_X_pow_eq_monomial, coeff_monomial]
     rw [natDegree_comp, show (X + 1 : ℤ[X]) = X + C 1 by simp, natDegree_X_add_C, mul_one,
@@ -135,7 +137,7 @@ theorem dvd_coeff_zero_of_aeval_eq_prime_smul_of_minpoly_isEisensteinAt {B : Pow
   letI := B.finite
   let P := minpoly R B.gen
   obtain ⟨n, hn⟩ := Nat.exists_eq_succ_of_ne_zero B.dim_pos.ne'
-  have finrank_K_L : FiniteDimensional.finrank K L = B.dim := B.finrank
+  have finrank_K_L : Module.finrank K L = B.dim := B.finrank
   have deg_K_P : (minpoly K B.gen).natDegree = B.dim := B.natDegree_minpoly
   have deg_R_P : P.natDegree = B.dim := by
     rw [← deg_K_P, minpoly.isIntegrallyClosed_eq_field_fractions' K hBint,
diff --git a/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean b/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
index 467b55e7f137e..b8383bd8e208a 100644
--- a/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
+++ b/Mathlib/RingTheory/Polynomial/IntegralNormalization.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
+Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Polynomial.AlgebraMap
 import Mathlib.Algebra.Polynomial.Degree.Lemmas
diff --git a/Mathlib/RingTheory/Polynomial/Pochhammer.lean b/Mathlib/RingTheory/Polynomial/Pochhammer.lean
index f4d2df4b5c382..a96e49ebd3495 100644
--- a/Mathlib/RingTheory/Polynomial/Pochhammer.lean
+++ b/Mathlib/RingTheory/Polynomial/Pochhammer.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Polynomial.Degree.Definitions
 import Mathlib.Algebra.Polynomial.Eval
diff --git a/Mathlib/RingTheory/Polynomial/Selmer.lean b/Mathlib/RingTheory/Polynomial/Selmer.lean
index 765741980303a..c6f4ed8e30afd 100644
--- a/Mathlib/RingTheory/Polynomial/Selmer.lean
+++ b/Mathlib/RingTheory/Polynomial/Selmer.lean
@@ -34,7 +34,7 @@ theorem X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬(z ^ n = z + 1 ∧ z ^
     rw [← Nat.mod_add_div n 3, pow_add, pow_mul, h3, one_pow, mul_one]
     have : n % 3 < 3 := Nat.mod_lt n zero_lt_three
     interval_cases n % 3 <;>
-    simp only [this, pow_zero, pow_one, eq_self_iff_true, or_true_iff, true_or_iff]
+    simp only [this, pow_zero, pow_one, eq_self_iff_true, or_true, true_or]
   have z_ne_zero : z ≠ 0 := fun h =>
     zero_ne_one ((zero_pow three_ne_zero).symm.trans (show (0 : ℂ) ^ 3 = 1 from h ▸ h3))
   rcases key with (key | key | key)
diff --git a/Mathlib/RingTheory/Polynomial/Tower.lean b/Mathlib/RingTheory/Polynomial/Tower.lean
index 2c16e8138d977..5e49599ddbda9 100644
--- a/Mathlib/RingTheory/Polynomial/Tower.lean
+++ b/Mathlib/RingTheory/Polynomial/Tower.lean
@@ -56,7 +56,7 @@ theorem aeval_algebraMap_apply (x : A) (p : R[X]) :
 theorem aeval_algebraMap_eq_zero_iff [NoZeroSMulDivisors A B] [Nontrivial B] (x : A) (p : R[X]) :
     aeval (algebraMap A B x) p = 0 ↔ aeval x p = 0 := by
   rw [aeval_algebraMap_apply, Algebra.algebraMap_eq_smul_one, smul_eq_zero,
-    iff_false_intro (one_ne_zero' B), or_false_iff]
+    iff_false_intro (one_ne_zero' B), or_false]
 
 variable {B}
 
diff --git a/Mathlib/RingTheory/Polynomial/Vieta.lean b/Mathlib/RingTheory/Polynomial/Vieta.lean
index e4bb4bc99037d..6ad52b3eb26ac 100644
--- a/Mathlib/RingTheory/Polynomial/Vieta.lean
+++ b/Mathlib/RingTheory/Polynomial/Vieta.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Hanting Zhang
 -/
 import Mathlib.Algebra.Polynomial.Splits
-import Mathlib.RingTheory.MvPolynomial.Symmetric
+import Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
 
 /-!
 # Vieta's Formula
diff --git a/Mathlib/RingTheory/PolynomialAlgebra.lean b/Mathlib/RingTheory/PolynomialAlgebra.lean
index d21058ee4247c..410b46c3aec39 100644
--- a/Mathlib/RingTheory/PolynomialAlgebra.lean
+++ b/Mathlib/RingTheory/PolynomialAlgebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Polynomial.AlgebraMap
 import Mathlib.Data.Matrix.Basis
diff --git a/Mathlib/RingTheory/PowerBasis.lean b/Mathlib/RingTheory/PowerBasis.lean
index ce69b35f56744..a9cf86b0049c1 100644
--- a/Mathlib/RingTheory/PowerBasis.lean
+++ b/Mathlib/RingTheory/PowerBasis.lean
@@ -19,7 +19,7 @@ gives a `PowerBasis` structure generated by `x`.
 * `PowerBasis R A`: a structure containing an `x` and an `n` such that
 `1, x, ..., x^n` is a basis for the `R`-algebra `A` (viewed as an `R`-module).
 
-* `finrank (hf : f ≠ 0) : FiniteDimensional.finrank K (AdjoinRoot f) = f.natDegree`,
+* `finrank (hf : f ≠ 0) : Module.finrank K (AdjoinRoot f) = f.natDegree`,
   the dimension of `AdjoinRoot f` equals the degree of `f`
 
 * `PowerBasis.lift (pb : PowerBasis R S)`: if `y : S'` satisfies the same
@@ -40,9 +40,7 @@ power basis, powerbasis
 -/
 
 
-open Polynomial
-
-open Polynomial
+open Polynomial Finsupp
 
 variable {R S T : Type*} [CommRing R] [Ring S] [Algebra R S]
 variable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B]
@@ -79,8 +77,8 @@ theorem finite (pb : PowerBasis R S) : Module.Finite R S := .of_basis pb.basis
 @[deprecated (since := "2024-03-05")] alias finiteDimensional := PowerBasis.finite
 
 theorem finrank [StrongRankCondition R] (pb : PowerBasis R S) :
-    FiniteDimensional.finrank R S = pb.dim := by
-  rw [FiniteDimensional.finrank_eq_card_basis pb.basis, Fintype.card_fin]
+    Module.finrank R S = pb.dim := by
+  rw [Module.finrank_eq_card_basis pb.basis, Fintype.card_fin]
 
 theorem mem_span_pow' {x y : S} {d : ℕ} :
     y ∈ Submodule.span R (Set.range fun i : Fin d => x ^ (i : ℕ)) ↔
@@ -89,9 +87,9 @@ theorem mem_span_pow' {x y : S} {d : ℕ} :
     ext n
     simp_rw [Set.mem_range, Set.mem_image, Finset.mem_coe, Finset.mem_range]
     exact ⟨fun ⟨⟨i, hi⟩, hy⟩ => ⟨i, hi, hy⟩, fun ⟨i, hi, hy⟩ => ⟨⟨i, hi⟩, hy⟩⟩
-  simp only [this, Finsupp.mem_span_image_iff_total, degree_lt_iff_coeff_zero, support,
+  simp only [this, mem_span_image_iff_linearCombination, degree_lt_iff_coeff_zero, Finsupp.support,
     exists_iff_exists_finsupp, coeff, aeval_def, eval₂RingHom', eval₂_eq_sum, Polynomial.sum,
-    Finsupp.mem_supported', Finsupp.total, Finsupp.sum, Algebra.smul_def, eval₂_zero, exists_prop,
+    mem_supported', linearCombination, Finsupp.sum, Algebra.smul_def, eval₂_zero, exists_prop,
     LinearMap.id_coe, eval₂_one, id, not_lt, Finsupp.coe_lsum, LinearMap.coe_smulRight,
     Finset.mem_range, AlgHom.coe_mks, Finset.mem_coe]
   simp_rw [@eq_comm _ y]
@@ -164,8 +162,8 @@ noncomputable def minpolyGen (pb : PowerBasis A S) : A[X] :=
 
 theorem aeval_minpolyGen (pb : PowerBasis A S) : aeval pb.gen (minpolyGen pb) = 0 := by
   simp_rw [minpolyGen, map_sub, map_sum, map_mul, map_pow, aeval_C, ← Algebra.smul_def, aeval_X]
-  refine sub_eq_zero.mpr ((pb.basis.total_repr (pb.gen ^ pb.dim)).symm.trans ?_)
-  rw [Finsupp.total_apply, Finsupp.sum_fintype] <;>
+  refine sub_eq_zero.mpr ((pb.basis.linearCombination_repr (pb.gen ^ pb.dim)).symm.trans ?_)
+  rw [Finsupp.linearCombination_apply, Finsupp.sum_fintype] <;>
     simp only [pb.coe_basis, zero_smul, eq_self_iff_true, imp_true_iff]
 
 theorem minpolyGen_monic (pb : PowerBasis A S) : Monic (minpolyGen pb) := by
diff --git a/Mathlib/RingTheory/PowerSeries/Basic.lean b/Mathlib/RingTheory/PowerSeries/Basic.lean
index 33cea1c3cd017..57422669ec4cd 100644
--- a/Mathlib/RingTheory/PowerSeries/Basic.lean
+++ b/Mathlib/RingTheory/PowerSeries/Basic.lean
@@ -603,14 +603,14 @@ lemma coeff_one_pow (n : ℕ) (φ : R⟦X⟧) :
             CharP.cast_eq_zero, zero_add, mul_one, not_true_eq_false] at h''
           norm_num at h''
         · rw [ih]
-          conv => lhs; arg 2; rw [mul_comm, ← mul_assoc]
-          move_mul [← (constantCoeff R) φ ^ (n' - 1)]
-          conv => enter [1, 2, 1, 1, 2]; rw [← pow_one (a := constantCoeff R φ)]
-          rw [← pow_add (a := constantCoeff R φ)]
-          conv => enter [1, 2, 1, 1]; rw [Nat.sub_add_cancel h']
-          conv => enter [1, 2, 1]; rw [mul_comm]
-          rw [mul_assoc, ← one_add_mul, add_comm, mul_assoc]
-          conv => enter [1, 2]; rw [mul_comm]
+          · conv => lhs; arg 2; rw [mul_comm, ← mul_assoc]
+            move_mul [← (constantCoeff R) φ ^ (n' - 1)]
+            conv => enter [1, 2, 1, 1, 2]; rw [← pow_one (a := constantCoeff R φ)]
+            rw [← pow_add (a := constantCoeff R φ)]
+            conv => enter [1, 2, 1, 1]; rw [Nat.sub_add_cancel h']
+            conv => enter [1, 2, 1]; rw [mul_comm]
+            rw [mul_assoc, ← one_add_mul, add_comm, mul_assoc]
+            conv => enter [1, 2]; rw [mul_comm]
           exact h'
       · decide
 
diff --git a/Mathlib/RingTheory/PowerSeries/Order.lean b/Mathlib/RingTheory/PowerSeries/Order.lean
index f73b2bb21d977..cec9d16a7d726 100644
--- a/Mathlib/RingTheory/PowerSeries/Order.lean
+++ b/Mathlib/RingTheory/PowerSeries/Order.lean
@@ -272,7 +272,7 @@ theorem order_eq_multiplicity_X {R : Type*} [Semiring R] [@DecidableRel R⟦X⟧
       exact Nat.lt_succ_self _
 
 /-- Given a non-zero power series `f`, `divided_by_X_pow_order f` is the power series obtained by
-  dividing out the largest power of X that divides `f`, that is its order-/
+  dividing out the largest power of X that divides `f`, that is its order -/
 def divided_by_X_pow_order {f : PowerSeries R} (hf : f ≠ 0) : R⟦X⟧ :=
   (exists_eq_mul_right_of_dvd (X_pow_order_dvd (order_finite_iff_ne_zero.2 hf))).choose
 
diff --git a/Mathlib/RingTheory/Presentation.lean b/Mathlib/RingTheory/Presentation.lean
index 5dfe5599b9bc1..2cdda618659b5 100644
--- a/Mathlib/RingTheory/Presentation.lean
+++ b/Mathlib/RingTheory/Presentation.lean
@@ -27,11 +27,10 @@ A presentation of an `R`-algebra `S` is a distinguished family of generators and
 - `Algebra.Presentation.dimension`: The dimension of a presentation is the number of generators
   minus the number of relations.
 
-We also give constructors for localization and base change.
+We also give constructors for localization, base change and composition.
 
 ## TODO
 
-- Define composition of presentations.
 - Define `Hom`s of presentations.
 
 ## Notes
@@ -56,7 +55,7 @@ each relation to a polynomial in the generators.
 -/
 @[nolint checkUnivs]
 structure Algebra.Presentation extends Algebra.Generators.{w} R S where
-  /-- The type of relations.  -/
+  /-- The type of relations. -/
   rels : Type t
   /-- The assignment of each relation to a polynomial in the generators. -/
   relation : rels → toGenerators.Ring
@@ -123,6 +122,39 @@ lemma finitePresentation_of_isFinite [P.IsFinite] :
 
 section Construction
 
+/-- If `algebraMap R S` is bijective, the empty generators are a presentation with no relations. -/
+noncomputable def ofBijectiveAlgebraMap (h : Function.Bijective (algebraMap R S)) :
+    Presentation.{t, w} R S where
+  __ := Generators.ofSurjectiveAlgebraMap h.surjective
+  rels := PEmpty
+  relation := PEmpty.elim
+  span_range_relation_eq_ker := by
+    simp only [Set.range_eq_empty, Ideal.span_empty]
+    symm
+    rw [← RingHom.injective_iff_ker_eq_bot]
+    show Function.Injective (aeval PEmpty.elim)
+    rw [aeval_injective_iff_of_isEmpty]
+    exact h.injective
+
+instance ofBijectiveAlgebraMap_isFinite (h : Function.Bijective (algebraMap R S)) :
+    (ofBijectiveAlgebraMap.{t, w} h).IsFinite where
+  finite_vars := inferInstanceAs (Finite PEmpty.{w + 1})
+  finite_rels := inferInstanceAs (Finite PEmpty.{t + 1})
+
+lemma ofBijectiveAlgebraMap_dimension (h : Function.Bijective (algebraMap R S)) :
+    (ofBijectiveAlgebraMap h).dimension = 0 := by
+  show Nat.card PEmpty - Nat.card PEmpty = 0
+  simp only [Nat.card_eq_fintype_card, Fintype.card_ofIsEmpty, le_refl, tsub_eq_zero_of_le]
+
+variable (R) in
+/-- The canonical `R`-presentation of `R` with no generators and no relations. -/
+noncomputable def id : Presentation.{t, w} R R := ofBijectiveAlgebraMap Function.bijective_id
+
+instance : (id R).IsFinite := ofBijectiveAlgebraMap_isFinite (R := R) Function.bijective_id
+
+lemma id_dimension : (Presentation.id R).dimension = 0 :=
+  ofBijectiveAlgebraMap_dimension (R := R) Function.bijective_id
+
 section Localization
 
 variable (r : R) [IsLocalization.Away r S]
@@ -147,6 +179,7 @@ private lemma span_range_relation_eq_ker_localizationAway :
   show Ideal.span {C r * X () - 1} = Ideal.comap _ (RingHom.ker (mvPolynomialQuotientEquiv S r))
   simp [RingHom.ker_equiv, ← RingHom.ker_eq_comap_bot]
 
+variable (S) in
 /-- If `S` is the localization of `R` away from `r`, we can construct a natural
 presentation of `S` as `R`-algebra with a single generator `X` and the relation `r * X - 1 = 0`. -/
 @[simps relation, simps (config := .lemmasOnly) rels]
@@ -158,17 +191,22 @@ noncomputable def localizationAway : Presentation R S where
     simp only [Generators.localizationAway_vars, Set.range_const]
     apply span_range_relation_eq_ker_localizationAway r
 
-instance localizationAway_isFinite : (localizationAway r (S := S)).IsFinite where
+instance localizationAway_isFinite : (localizationAway S r).IsFinite where
   finite_vars := inferInstanceAs <| Finite Unit
   finite_rels := inferInstanceAs <| Finite Unit
 
+instance : Fintype (localizationAway S r).rels :=
+  inferInstanceAs (Fintype Unit)
+
 @[simp]
-lemma localizationAway_dimension_zero : (localizationAway r (S := S)).dimension = 0 := by
+lemma localizationAway_dimension_zero : (localizationAway S r).dimension = 0 := by
   simp [Presentation.dimension, localizationAway, Generators.localizationAway_vars]
 
 end Localization
 
-variable {T} [CommRing T] [Algebra R T] (P : Presentation R S)
+section BaseChange
+
+variable (T) [CommRing T] [Algebra R T] (P : Presentation R S)
 
 private lemma span_range_relation_eq_ker_baseChange :
     Ideal.span (Set.range fun i ↦ (MvPolynomial.map (algebraMap R T)) (P.relation i)) =
@@ -227,7 +265,169 @@ def baseChange : Presentation T (T ⊗[R] S) where
   __ := Generators.baseChange P.toGenerators
   rels := P.rels
   relation i := MvPolynomial.map (algebraMap R T) (P.relation i)
-  span_range_relation_eq_ker := P.span_range_relation_eq_ker_baseChange
+  span_range_relation_eq_ker := P.span_range_relation_eq_ker_baseChange T
+
+instance baseChange_isFinite [P.IsFinite] : (P.baseChange T).IsFinite where
+  finite_vars := inferInstanceAs <| Finite (P.vars)
+  finite_rels := inferInstanceAs <| Finite (P.rels)
+
+end BaseChange
+
+section Composition
+
+/-!
+### Composition of presentations
+
+Let `S` be an `R`-algebra with presentation `P` and `T` be an `S`-algebra with
+presentation `Q`. In this section we construct a presentation of `T` as an `R`-algebra.
+
+For the underlying generators see `Algebra.Generators.comp`. The family of relations is
+indexed by `Q.rels ⊕ P.rels`.
+
+We have two canonical maps:
+`MvPolynomial P.vars R →ₐ[R] MvPolynomial (Q.vars ⊕ P.vars) R` induced by `Sum.inr`
+and `aux : MvPolynomial (Q.vars ⊕ P.vars) R →ₐ[R] MvPolynomial Q.vars S` induced by
+the evaluation `MvPolynomial P.vars R →ₐ[R] S` (see below).
+
+Now `i : P.rels` is mapped to the image of `P.relation i` under the first map and
+`j : Q.rels` is mapped to a pre-image under `aux` of `Q.relation j` (see `comp_relation_aux`
+for the construction of the pre-image and `comp_relation_aux_map` for a proof that it is indeed
+a pre-image).
+
+The evaluation map factors as:
+`MvPolynomial (Q.vars ⊕ P.vars) R →ₐ[R] MvPolynomial Q.vars S →ₐ[R] T`, where
+the first map is `aux`. The goal is to compute that the kernel of this composition
+is spanned by the relations indexed by `Q.rels ⊕ P.rels` (`span_range_relation_eq_ker_comp`).
+One easily sees that this kernel is the pre-image under `aux` of the kernel of the evaluation
+of `Q`, where the latter is by assumption spanned by the relations `Q.relation j`.
+
+Since `aux` is surjective (`aux_surjective`), the pre-image is the sum of the ideal spanned
+by the constructed pre-images of the `Q.relation j` and the kernel of `aux`. It hence
+remains to show that the kernel of `aux` is spanned by the image of the `P.relation i`
+under the canonical map `MvPolynomial P.vars R →ₐ[R] MvPolynomial (Q.vars ⊕ P.vars) R`. By
+assumption this span is the kernel of the evaluation map of `P`. For this, we use the isomorphism
+`MvPolynomial (Q.vars ⊕ P.vars) R ≃ₐ[R] MvPolynomial Q.vars (MvPolynomial P.vars R)` and
+`MvPolynomial.ker_map`.
+
+-/
+
+variable {T} [CommRing T] [Algebra S T]
+variable (Q : Presentation S T) (P : Presentation R S)
+
+/-- The evaluation map `MvPolynomial (Q.vars ⊕ P.vars) →ₐ[R] T` factors via this map. For more
+details, see the module docstring at the beginning of the section. -/
+private noncomputable def aux : MvPolynomial (Q.vars ⊕ P.vars) R →ₐ[R] MvPolynomial Q.vars S :=
+  aeval (Sum.elim X (MvPolynomial.C ∘ P.val))
+
+/-- A choice of pre-image of `Q.relation r` under `aux`. -/
+private noncomputable def comp_relation_aux (r : Q.rels) : MvPolynomial (Q.vars ⊕ P.vars) R :=
+  Finsupp.sum (Q.relation r)
+    (fun x j ↦ (MvPolynomial.rename Sum.inr <| P.σ j) * monomial (x.mapDomain Sum.inl) 1)
+
+@[simp]
+private lemma aux_X (i : Q.vars ⊕ P.vars) : (Q.aux P) (X i) = Sum.elim X (C ∘ P.val) i :=
+  aeval_X (Sum.elim X (C ∘ P.val)) i
+
+/-- The pre-images constructed in `comp_relation_aux` are indeed pre-images under `aux`. -/
+private lemma comp_relation_aux_map (r : Q.rels) :
+    (Q.aux P) (Q.comp_relation_aux P r) = Q.relation r := by
+  simp only [aux, comp_relation_aux, Generators.comp_vars, Sum.elim_inl, map_finsupp_sum]
+  simp only [_root_.map_mul, aeval_rename, aeval_monomial, Sum.elim_comp_inr]
+  conv_rhs => rw [← Finsupp.sum_single (Q.relation r)]
+  congr
+  ext u s m
+  simp only [MvPolynomial.single_eq_monomial, aeval, AlgHom.coe_mk, coe_eval₂Hom]
+  rw [monomial_eq, IsScalarTower.algebraMap_eq R S, algebraMap_eq, ← eval₂_comp_left, ← aeval_def]
+  simp [Finsupp.prod_mapDomain_index_inj (Sum.inl_injective)]
+
+private lemma aux_surjective : Function.Surjective (Q.aux P) := fun p ↦ by
+  induction' p using MvPolynomial.induction_on with a p q hp hq p i h
+  · use rename Sum.inr <| P.σ a
+    simp only [aux, aeval_rename, Sum.elim_comp_inr]
+    have (p : MvPolynomial P.vars R) :
+        aeval (C ∘ P.val) p = (C (aeval P.val p) : MvPolynomial Q.vars S) := by
+      induction' p using MvPolynomial.induction_on with a p q hp hq p i h
+      · simp
+      · simp [hp, hq]
+      · simp [h]
+    simp [this]
+  · obtain ⟨a, rfl⟩ := hp
+    obtain ⟨b, rfl⟩ := hq
+    exact ⟨a + b, map_add _ _ _⟩
+  · obtain ⟨a, rfl⟩ := h
+    exact ⟨(a * X (Sum.inl i)), by simp⟩
+
+private lemma aux_image_relation :
+    Q.aux P '' (Set.range (Algebra.Presentation.comp_relation_aux Q P)) = Set.range Q.relation := by
+  ext x
+  constructor
+  · rintro ⟨y, ⟨a, rfl⟩, rfl⟩
+    exact ⟨a, (Q.comp_relation_aux_map P a).symm⟩
+  · rintro ⟨y, rfl⟩
+    use Q.comp_relation_aux P y
+    simp only [Set.mem_range, exists_apply_eq_apply, true_and, comp_relation_aux_map]
+
+private lemma aux_eq_comp : Q.aux P =
+    (MvPolynomial.mapAlgHom (aeval P.val)).comp (sumAlgEquiv R Q.vars P.vars).toAlgHom := by
+  ext i : 1
+  cases i <;> simp
+
+private lemma aux_ker :
+    RingHom.ker (Q.aux P) = Ideal.map (rename Sum.inr) (RingHom.ker (aeval P.val)) := by
+  rw [aux_eq_comp, ← AlgHom.comap_ker, MvPolynomial.ker_mapAlgHom]
+  show Ideal.comap _ (Ideal.map (IsScalarTower.toAlgHom R (MvPolynomial P.vars R) _) _) = _
+  rw [← sumAlgEquiv_comp_rename_inr, ← Ideal.map_mapₐ, Ideal.comap_map_of_bijective]
+  simpa using AlgEquiv.bijective (sumAlgEquiv R Q.vars P.vars)
+
+variable [Algebra R T] [IsScalarTower R S T]
+
+private lemma aeval_comp_val_eq :
+    (aeval (Q.comp P.toGenerators).val) =
+      (aevalTower (IsScalarTower.toAlgHom R S T) Q.val).comp (Q.aux P) := by
+  ext i
+  simp only [AlgHom.coe_comp, Function.comp_apply]
+  erw [Q.aux_X P i]
+  cases i <;> simp
+
+private lemma span_range_relation_eq_ker_comp : Ideal.span
+    (Set.range (Sum.elim (Algebra.Presentation.comp_relation_aux Q P)
+      fun rp ↦ (rename Sum.inr) (P.relation rp))) = (Q.comp P.toGenerators).ker := by
+  rw [Generators.ker_eq_ker_aeval_val, Q.aeval_comp_val_eq, ← AlgHom.comap_ker]
+  show _ = Ideal.comap _ (Q.ker)
+  rw [← Q.span_range_relation_eq_ker, ← Q.aux_image_relation P, ← Ideal.map_span,
+    Ideal.comap_map_of_surjective' _ (Q.aux_surjective P)]
+  rw [Set.Sum.elim_range, Ideal.span_union, Q.aux_ker, ← P.ker_eq_ker_aeval_val,
+    ← P.span_range_relation_eq_ker, Ideal.map_span]
+  congr
+  ext
+  simp
+
+/-- Given presentations of `T` over `S` and of `S` over `R`,
+we may construct a presentation of `T` over `R`. -/
+@[simps rels, simps (config := .lemmasOnly) relation]
+noncomputable def comp : Presentation R T where
+  toGenerators := Q.toGenerators.comp P.toGenerators
+  rels := Q.rels ⊕ P.rels
+  relation := Sum.elim (Q.comp_relation_aux P)
+    (fun rp ↦ MvPolynomial.rename Sum.inr <| P.relation rp)
+  span_range_relation_eq_ker := Q.span_range_relation_eq_ker_comp P
+
+@[simp]
+lemma comp_relation_inr (r : P.rels) :
+    (Q.comp P).relation (Sum.inr r) = rename Sum.inr (P.relation r) :=
+  rfl
+
+lemma comp_aeval_relation_inl (r : Q.rels) :
+    aeval (Sum.elim X (MvPolynomial.C ∘ P.val)) ((Q.comp P).relation (Sum.inl r)) =
+      Q.relation r := by
+  show (Q.aux P) _ = _
+  simp [comp_relation, comp_relation_aux_map]
+
+instance comp_isFinite [P.IsFinite] [Q.IsFinite] : (Q.comp P).IsFinite where
+  finite_vars := inferInstanceAs <| Finite (Q.vars ⊕ P.vars)
+  finite_rels := inferInstanceAs <| Finite (Q.rels ⊕ P.rels)
+
+end Composition
 
 end Construction
 
diff --git a/Mathlib/RingTheory/PrimeSpectrum.lean b/Mathlib/RingTheory/PrimeSpectrum.lean
index 42e6f81ae506d..e97719ff6ca9f 100644
--- a/Mathlib/RingTheory/PrimeSpectrum.lean
+++ b/Mathlib/RingTheory/PrimeSpectrum.lean
@@ -5,6 +5,7 @@ Authors: Johan Commelin, Filippo A. E. Nuccio, Andrew Yang
 -/
 import Mathlib.LinearAlgebra.Finsupp
 import Mathlib.RingTheory.Ideal.Prod
+import Mathlib.RingTheory.Localization.Ideal
 import Mathlib.RingTheory.Nilpotent.Lemmas
 import Mathlib.RingTheory.Noetherian
 
@@ -186,7 +187,7 @@ theorem gc_set :
     @GaloisConnection (Set R) (Set (PrimeSpectrum R))ᵒᵈ _ _ (fun s => zeroLocus s) fun t =>
       vanishingIdeal t := by
   have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi R R).gc
-  simpa [zeroLocus_span, Function.comp] using ideal_gc.compose (gc R)
+  simpa [zeroLocus_span, Function.comp_def] using ideal_gc.compose (gc R)
 
 theorem subset_zeroLocus_iff_subset_vanishingIdeal (t : Set (PrimeSpectrum R)) (s : Set R) :
     t ⊆ zeroLocus s ↔ s ⊆ vanishingIdeal t :=
@@ -480,3 +481,112 @@ end Noetherian
 end CommSemiRing
 
 end PrimeSpectrum
+
+open PrimeSpectrum
+
+/-- The pullback of an element of `PrimeSpectrum S` along a ring homomorphism `f : R →+* S`.
+The bundled continuous version is `PrimeSpectrum.comap`. -/
+abbrev RingHom.specComap {R S : Type*} [CommSemiring R] [CommSemiring S] (f : R →+* S) :
+    PrimeSpectrum S → PrimeSpectrum R :=
+  fun y => ⟨Ideal.comap f y.asIdeal, inferInstance⟩
+
+namespace PrimeSpectrum
+
+open RingHom
+
+variable {R S} {S' : Type*} [CommSemiring R] [CommSemiring S] [CommSemiring S']
+
+theorem preimage_specComap_zeroLocus_aux (f : R →+* S) (s : Set R) :
+    f.specComap ⁻¹' zeroLocus s = zeroLocus (f '' s) := by
+  ext x
+  simp only [mem_zeroLocus, Set.image_subset_iff, Set.mem_preimage, mem_zeroLocus, Ideal.coe_comap]
+
+variable (f : R →+* S)
+
+@[simp]
+theorem specComap_asIdeal (y : PrimeSpectrum S) :
+    (f.specComap y).asIdeal = Ideal.comap f y.asIdeal :=
+  rfl
+
+@[simp]
+theorem specComap_id : (RingHom.id R).specComap = fun x => x :=
+  rfl
+
+@[simp]
+theorem specComap_comp (f : R →+* S) (g : S →+* S') :
+    (g.comp f).specComap = f.specComap.comp g.specComap :=
+  rfl
+
+theorem specComap_comp_apply (f : R →+* S) (g : S →+* S') (x : PrimeSpectrum S') :
+    (g.comp f).specComap x = f.specComap (g.specComap x) :=
+  rfl
+
+@[simp]
+theorem preimage_specComap_zeroLocus (s : Set R) :
+    f.specComap ⁻¹' zeroLocus s = zeroLocus (f '' s) :=
+  preimage_specComap_zeroLocus_aux f s
+
+theorem specComap_injective_of_surjective (f : R →+* S) (hf : Function.Surjective f) :
+    Function.Injective f.specComap := fun x y h =>
+  PrimeSpectrum.ext
+    (Ideal.comap_injective_of_surjective f hf
+      (congr_arg PrimeSpectrum.asIdeal h : (f.specComap x).asIdeal = (f.specComap y).asIdeal))
+
+variable (S)
+
+theorem localization_specComap_injective [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
+    Function.Injective (algebraMap R S).specComap := by
+  intro p q h
+  replace h := _root_.congr_arg (fun x : PrimeSpectrum R => Ideal.map (algebraMap R S) x.asIdeal) h
+  dsimp only [specComap] at h
+  rw [IsLocalization.map_comap M S, IsLocalization.map_comap M S] at h
+  ext1
+  exact h
+
+theorem localization_specComap_range [Algebra R S] (M : Submonoid R) [IsLocalization M S] :
+    Set.range (algebraMap R S).specComap = { p | Disjoint (M : Set R) p.asIdeal } := by
+  ext x
+  constructor
+  · simp_rw [disjoint_iff_inf_le]
+    rintro ⟨p, rfl⟩ x ⟨hx₁, hx₂⟩
+    exact (p.2.1 : ¬_) (p.asIdeal.eq_top_of_isUnit_mem hx₂ (IsLocalization.map_units S ⟨x, hx₁⟩))
+  · intro h
+    use ⟨x.asIdeal.map (algebraMap R S), IsLocalization.isPrime_of_isPrime_disjoint M S _ x.2 h⟩
+    ext1
+    exact IsLocalization.comap_map_of_isPrime_disjoint M S _ x.2 h
+
+end PrimeSpectrum
+
+section SpecOfSurjective
+
+open Function RingHom
+
+variable [CommRing R] [CommRing S]
+variable (f : R →+* S)
+variable {R}
+
+theorem image_specComap_zeroLocus_eq_zeroLocus_comap (hf : Surjective f) (I : Ideal S) :
+    f.specComap '' zeroLocus I = zeroLocus (I.comap f) := by
+  simp only [Set.ext_iff, Set.mem_image, mem_zeroLocus, SetLike.coe_subset_coe]
+  refine fun p => ⟨?_, fun h_I_p => ?_⟩
+  · rintro ⟨p, hp, rfl⟩ a ha
+    exact hp ha
+  · have hp : ker f ≤ p.asIdeal := (Ideal.comap_mono bot_le).trans h_I_p
+    refine ⟨⟨p.asIdeal.map f, Ideal.map_isPrime_of_surjective hf hp⟩, fun x hx => ?_, ?_⟩
+    · obtain ⟨x', rfl⟩ := hf x
+      exact Ideal.mem_map_of_mem f (h_I_p hx)
+    · ext x
+      rw [specComap_asIdeal, Ideal.mem_comap, Ideal.mem_map_iff_of_surjective f hf]
+      refine ⟨?_, fun hx => ⟨x, hx, rfl⟩⟩
+      rintro ⟨x', hx', heq⟩
+      rw [← sub_sub_cancel x' x]
+      refine p.asIdeal.sub_mem hx' (hp ?_)
+      rwa [mem_ker, map_sub, sub_eq_zero]
+
+theorem range_specComap_of_surjective (hf : Surjective f) :
+    Set.range f.specComap = zeroLocus (ker f) := by
+  rw [← Set.image_univ]
+  convert image_specComap_zeroLocus_eq_zeroLocus_comap _ _ hf _
+  rw [zeroLocus_bot]
+
+end SpecOfSurjective
diff --git a/Mathlib/RingTheory/PrincipalIdealDomain.lean b/Mathlib/RingTheory/PrincipalIdealDomain.lean
index 29bea641d71f3..40c180a84813b 100644
--- a/Mathlib/RingTheory/PrincipalIdealDomain.lean
+++ b/Mathlib/RingTheory/PrincipalIdealDomain.lean
@@ -92,8 +92,9 @@ theorem _root_.Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
 
 @[simp]
 theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S := by
-  conv_rhs => rw [← span_singleton_generator S]
-  exact subset_span (mem_singleton _)
+  have : generator S ∈ span R {generator S} := subset_span (mem_singleton _)
+  convert this
+  exact span_singleton_generator S |>.symm
 
 theorem mem_iff_eq_smul_generator (S : Submodule R M) [S.IsPrincipal] {x : M} :
     x ∈ S ↔ ∃ s : R, x = s • generator S := by
diff --git a/Mathlib/RingTheory/Regular/IsSMulRegular.lean b/Mathlib/RingTheory/Regular/IsSMulRegular.lean
index 34012a4084c2b..efe4d52d7cef9 100644
--- a/Mathlib/RingTheory/Regular/IsSMulRegular.lean
+++ b/Mathlib/RingTheory/Regular/IsSMulRegular.lean
@@ -96,7 +96,7 @@ lemma isSMulRegular_on_submodule_iff_mem_imp_smul_eq_zero_imp_eq_zero :
     IsSMulRegular N r ↔ ∀ x ∈ N, r • x = 0 → x = 0 :=
   Iff.trans (isSMulRegular_iff_smul_eq_zero_imp_eq_zero N r) <|
     Iff.trans Subtype.forall <| by
-      simp only [SetLike.mk_smul_mk, AddSubmonoid.mk_eq_zero]
+      simp only [SetLike.mk_smul_mk, Submodule.mk_eq_zero]
 
 lemma isSMulRegular_on_quot_iff_smul_mem_implies_mem :
     IsSMulRegular (M ⧸ N) r ↔ ∀ x : M, r • x ∈ N → x ∈ N :=
diff --git a/Mathlib/RingTheory/Regular/RegularSequence.lean b/Mathlib/RingTheory/Regular/RegularSequence.lean
index 159e21e692cb7..a1e81fdf08b75 100644
--- a/Mathlib/RingTheory/Regular/RegularSequence.lean
+++ b/Mathlib/RingTheory/Regular/RegularSequence.lean
@@ -45,7 +45,7 @@ abbrev ofList (rs : List R) := span { r | r ∈ rs }
   have : { r | r ∈ rs₁ ++ rs₂ } = _ := Set.ext (fun _ => List.mem_append)
   Eq.trans (congrArg span this) (span_union _ _)
 
-@[simp] lemma ofList_singleton (r : R) : ofList [r] = span {r} :=
+lemma ofList_singleton (r : R) : ofList [r] = span {r} :=
   congrArg span (Set.ext fun _ => List.mem_singleton)
 
 @[simp] lemma ofList_cons (r : R) (rs : List R) :
@@ -555,12 +555,14 @@ lemma map_first_exact_on_four_term_right_exact_of_isSMulRegular_last
     (h₄ : IsWeaklyRegular M₄ rs) :
     Exact (mapQ _ _ _ (smul_top_le_comap_smul_top (Ideal.ofList rs) f₁))
           (mapQ _ _ _ (smul_top_le_comap_smul_top (Ideal.ofList rs) f₂)) := by
-  induction' h₄ with _ _ _ N _ _ r rs h₄ _ ih generalizing M M₂ M₃
-  · apply (Exact.iff_of_ladder_linearEquiv ?_ ?_).mp h₁₂
+  induction h₄ generalizing M M₂ M₃ with
+  | nil =>
+    apply (Exact.iff_of_ladder_linearEquiv ?_ ?_).mp h₁₂
     any_goals exact quotEquivOfEqBot _ <|
       Eq.trans (congrArg (· • ⊤) Ideal.ofList_nil) (bot_smul ⊤)
     all_goals exact quot_hom_ext _ _ _ fun _ => rfl
-  · specialize ih
+  | cons r rs h₄ _ ih =>
+    specialize ih
       (map_first_exact_on_four_term_exact_of_isSMulRegular_last h₁₂ h₂₃ h₄)
       (map_exact r h₂₃ h₃) (map_surjective r h₃)
     have H₁ := quotOfListConsSMulTopEquivQuotSMulTopInner_naturality r rs f₁
diff --git a/Mathlib/RingTheory/RingHom/Finite.lean b/Mathlib/RingTheory/RingHom/Finite.lean
index 164f6abef7265..40a5dbe72b097 100644
--- a/Mathlib/RingTheory/RingHom/Finite.lean
+++ b/Mathlib/RingTheory/RingHom/Finite.lean
@@ -3,12 +3,22 @@ Copyright (c) 2021 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathlib.RingTheory.RingHomProperties
+import Mathlib.RingTheory.LocalProperties.Basic
+import Mathlib.RingTheory.Localization.Integer
 
 /-!
 
 # The meta properties of finite ring homomorphisms.
 
+## Main results
+
+Let `R` be a commutative ring, `S` is an `R`-algebra, `M` be a submonoid of `R`.
+
+* `finite_localizationPreserves` : If `S` is a finite `R`-algebra, then `S' = M⁻¹S` is a
+  finite `R' = M⁻¹R`-algebra.
+* `finite_ofLocalizationSpan` : `S` is a finite `R`-algebra if there exists
+  a set `{ r }` that spans `R` such that `Sᵣ` is a finite `Rᵣ`-algebra.
+
 -/
 
 
@@ -38,3 +48,186 @@ theorem finite_stableUnderBaseChange : StableUnderBaseChange @Finite := by
   exact inferInstance
 
 end RingHom
+
+open scoped Pointwise Classical
+
+universe u
+
+variable {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) (f : R →+* S)
+variable (R' S' : Type u) [CommRing R'] [CommRing S']
+variable [Algebra R R'] [Algebra S S']
+
+lemma Module.Finite_of_isLocalization (R S Rₚ Sₚ) [CommSemiring R] [CommRing S] [CommRing Rₚ]
+    [CommRing Sₚ] [Algebra R S] [Algebra R Rₚ] [Algebra R Sₚ] [Algebra S Sₚ] [Algebra Rₚ Sₚ]
+    [IsScalarTower R S Sₚ] [IsScalarTower R Rₚ Sₚ] (M : Submonoid R) [IsLocalization M Rₚ]
+    [IsLocalization (Algebra.algebraMapSubmonoid S M) Sₚ] [hRS : Module.Finite R S] :
+    Module.Finite Rₚ Sₚ := by
+  classical
+  have : algebraMap Rₚ Sₚ = IsLocalization.map (T := Algebra.algebraMapSubmonoid S M) Sₚ
+      (algebraMap R S) (Submonoid.le_comap_map M) := by
+    apply IsLocalization.ringHom_ext M
+    simp only [IsLocalization.map_comp, ← IsScalarTower.algebraMap_eq]
+  -- We claim that if `S` is generated by `T` as an `R`-module,
+  -- then `S'` is generated by `T` as an `R'`-module.
+  obtain ⟨T, hT⟩ := hRS
+  use T.image (algebraMap S Sₚ)
+  rw [eq_top_iff]
+  rintro x -
+  -- By the hypotheses, for each `x : S'`, we have `x = y / (f r)` for some `y : S` and `r : M`.
+  -- Since `S` is generated by `T`, the image of `y` should fall in the span of the image of `T`.
+  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ :=
+    IsLocalization.mk'_surjective (Algebra.algebraMapSubmonoid S M) x
+  rw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]
+  have hy : y ∈ Submodule.span R ↑T := by rw [hT]; trivial
+  replace hy : algebraMap S Sₚ y ∈ Submodule.map (IsScalarTower.toAlgHom R S Sₚ).toLinearMap
+    (Submodule.span R (T : Set S)) := Submodule.mem_map_of_mem
+--     -- Note: #8386 had to specify the value of `f` below
+      (f := (IsScalarTower.toAlgHom R S Sₚ).toLinearMap) hy
+  rw [Submodule.map_span (IsScalarTower.toAlgHom R S Sₚ).toLinearMap T] at hy
+  have H : Submodule.span R (algebraMap S Sₚ '' T) ≤
+      (Submodule.span Rₚ (algebraMap S Sₚ '' T)).restrictScalars R := by
+    rw [Submodule.span_le]; exact Submodule.subset_span
+  -- Now, since `y ∈ span T`, and `(f r)⁻¹ ∈ R'`, `x / (f r)` is in `span T` as well.
+  convert (Submodule.span Rₚ (algebraMap S Sₚ '' T)).smul_mem
+    (IsLocalization.mk' Rₚ (1 : R) ⟨r, hr⟩) (H hy) using 1
+  rw [Algebra.smul_def, this, IsLocalization.map_mk', map_one]
+
+/-- If `S` is a finite `R`-algebra, then `S' = M⁻¹S` is a finite `R' = M⁻¹R`-algebra. -/
+theorem RingHom.finite_localizationPreserves : RingHom.LocalizationPreserves @RingHom.Finite := by
+  introv R hf
+  letI := f.toAlgebra
+  letI := ((algebraMap S S').comp f).toAlgebra
+  let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
+  letI := f'.toAlgebra
+  have : IsScalarTower R R' S' := IsScalarTower.of_algebraMap_eq'
+    (IsLocalization.map_comp M.le_comap_map).symm
+  have : IsScalarTower R S S' := IsScalarTower.of_algebraMap_eq' rfl
+  have : IsLocalization (Algebra.algebraMapSubmonoid S M) S' := by
+    rwa [Algebra.algebraMapSubmonoid, RingHom.algebraMap_toAlgebra]
+  have : Module.Finite R S := hf
+  apply Module.Finite_of_isLocalization R S R' S' M
+
+theorem RingHom.localization_away_map_finite (r : R) [IsLocalization.Away r R']
+    [IsLocalization.Away (f r) S'] (hf : f.Finite) : (IsLocalization.Away.map R' S' f r).Finite :=
+  finite_localizationPreserves.away r hf
+
+/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
+If the image of some `x : S` falls in the span of some finite `s ⊆ S'` over `R`,
+then there exists some `m : M` such that `m • x` falls in the
+span of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
+-/
+theorem IsLocalization.smul_mem_finsetIntegerMultiple_span [Algebra R S] [Algebra R S']
+    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
+    (hx : algebraMap S S' x ∈ Submodule.span R (s : Set S')) :
+    ∃ m : M, m • x ∈
+      Submodule.span R
+        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) := by
+  let g : S →ₐ[R] S' :=
+    AlgHom.mk' (algebraMap S S') fun c x => by simp [Algebra.algebraMap_eq_smul_one]
+  -- We first obtain the `y' ∈ M` such that `s' = y' • s` is falls in the image of `S` in `S'`.
+  let y := IsLocalization.commonDenomOfFinset (M.map (algebraMap R S)) s
+  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
+    (IsLocalization.finsetIntegerMultiple_image _ s).symm
+  obtain ⟨y', hy', e : algebraMap R S y' = y⟩ := y.prop
+  have : algebraMap R S y' • (s : Set S') = y' • (s : Set S') := by
+    simp_rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
+  rw [← e, this] at hx₁
+  replace hx₁ := congr_arg (Submodule.span R) hx₁
+  rw [Submodule.span_smul] at hx₁
+  replace hx : _ ∈ y' • Submodule.span R (s : Set S') := Set.smul_mem_smul_set hx
+  rw [hx₁] at hx
+  erw [← _root_.map_smul g, ← Submodule.map_span (g : S →ₗ[R] S')] at hx
+  -- Since `x` falls in the span of `s` in `S'`, `y' • x : S` falls in the span of `s'` in `S'`.
+  -- That is, there exists some `x' : S` in the span of `s'` in `S` and `x' = y' • x` in `S'`.
+  -- Thus `a • (y' • x) = a • x' ∈ span s'` in `S` for some `a ∈ M`.
+  obtain ⟨x', hx', hx'' : algebraMap _ _ _ = _⟩ := hx
+  obtain ⟨⟨_, a, ha₁, rfl⟩, ha₂⟩ :=
+    (IsLocalization.eq_iff_exists (M.map (algebraMap R S)) S').mp hx''
+  use (⟨a, ha₁⟩ : M) * (⟨y', hy'⟩ : M)
+  convert (Submodule.span R
+    (IsLocalization.finsetIntegerMultiple (Submonoid.map (algebraMap R S) M) s : Set S)).smul_mem
+      a hx' using 1
+  convert ha₂.symm using 1
+  · rw [Subtype.coe_mk, Submonoid.smul_def, Submonoid.coe_mul, ← smul_smul]
+    exact Algebra.smul_def _ _
+  · exact Algebra.smul_def _ _
+
+/-- If `M` is an `R' = S⁻¹R` module, and `x ∈ span R' s`,
+then `t • x ∈ span R s` for some `t : S`. -/
+theorem multiple_mem_span_of_mem_localization_span
+    {N : Type*} [AddCommMonoid N] [Module R N] [Module R' N]
+    [IsScalarTower R R' N] [IsLocalization M R'] (s : Set N) (x : N)
+    (hx : x ∈ Submodule.span R' s) : ∃ (t : M), t • x ∈ Submodule.span R s := by
+  classical
+  obtain ⟨s', hss', hs'⟩ := Submodule.mem_span_finite_of_mem_span hx
+  rsuffices ⟨t, ht⟩ : ∃ t : M, t • x ∈ Submodule.span R (s' : Set N)
+  · exact ⟨t, Submodule.span_mono hss' ht⟩
+  clear hx hss' s
+  induction s' using Finset.induction_on generalizing x
+  · use 1; simpa using hs'
+  rename_i a s _ hs
+  simp only [Finset.coe_insert, Finset.image_insert, Finset.coe_image, Subtype.coe_mk,
+    Submodule.mem_span_insert] at hs' ⊢
+  rcases hs' with ⟨y, z, hz, rfl⟩
+  rcases IsLocalization.surj M y with ⟨⟨y', s'⟩, e⟩
+  apply congrArg (fun x ↦ x • a) at e
+  simp only [algebraMap_smul] at e
+  rcases hs _ hz with ⟨t, ht⟩
+  refine ⟨t * s', t * y', _, (Submodule.span R (s : Set N)).smul_mem s' ht, ?_⟩
+  rw [smul_add, ← smul_smul, mul_comm, ← smul_smul, ← smul_smul, ← e, mul_comm, ← Algebra.smul_def]
+  simp
+  rfl
+
+/-- If `S` is an `R' = M⁻¹R` algebra, and `x ∈ adjoin R' s`,
+then `t • x ∈ adjoin R s` for some `t : M`. -/
+theorem multiple_mem_adjoin_of_mem_localization_adjoin [Algebra R' S] [Algebra R S]
+    [IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S)
+    (hx : x ∈ Algebra.adjoin R' s) : ∃ t : M, t • x ∈ Algebra.adjoin R s := by
+  change ∃ t : M, t • x ∈ Subalgebra.toSubmodule (Algebra.adjoin R s)
+  change x ∈ Subalgebra.toSubmodule (Algebra.adjoin R' s) at hx
+  simp_rw [Algebra.adjoin_eq_span] at hx ⊢
+  exact multiple_mem_span_of_mem_localization_span M R' _ _ hx
+
+/-- `S` is a finite `R`-algebra if there exists a set `{ r }` that
+  spans `R` such that `Sᵣ` is a finite `Rᵣ`-algebra. -/
+theorem RingHom.finite_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.Finite := by
+  rw [RingHom.ofLocalizationSpan_iff_finite]
+  introv R hs H
+  -- We first setup the instances
+  letI := f.toAlgebra
+  letI := fun r : s => (Localization.awayMap f r).toAlgebra
+  have : ∀ r : s,
+      IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
+    by intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization
+  haveI : ∀ r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) :=
+    fun r => IsScalarTower.of_algebraMap_eq'
+      (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm
+  -- By the hypothesis, we may find a finite generating set for each `Sᵣ`. This set can then be
+  -- lifted into `R` by multiplying a sufficiently large power of `r`. I claim that the union of
+  -- these generates `S`.
+  constructor
+  replace H := fun r => (H r).1
+  choose s₁ s₂ using H
+  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s₁ x)
+  use s.attach.biUnion sf
+  rw [Submodule.span_attach_biUnion, eq_top_iff]
+  -- It suffices to show that `r ^ n • x ∈ span T` for each `r : s`, since `{ r ^ n }` spans `R`.
+  -- This then follows from the fact that each `x : R` is a linear combination of the generating set
+  -- of `Sᵣ`. By multiplying a sufficiently large power of `r`, we can cancel out the `r`s in the
+  -- denominators of both the generating set and the coefficients.
+  rintro x -
+  apply Submodule.mem_of_span_eq_top_of_smul_pow_mem _ (s : Set R) hs _ _
+  intro r
+  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ :=
+    multiple_mem_span_of_mem_localization_span (Submonoid.powers (r : R))
+      (Localization.Away (r : R)) (s₁ r : Set (Localization.Away (f r))) (algebraMap S _ x)
+      (by rw [s₂ r]; trivial)
+  dsimp only at hn₁
+  rw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, ← map_mul] at hn₁
+  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ :=
+    IsLocalization.smul_mem_finsetIntegerMultiple_span (Submonoid.powers (r : R))
+      (Localization.Away (f r)) _ (s₁ r) hn₁
+  rw [Submonoid.smul_def, ← Algebra.smul_def, smul_smul, ← pow_add] at hn₂
+  simp_rw [Submonoid.map_powers] at hn₂
+  use n₂ + n₁
+  exact le_iSup (fun x : s => Submodule.span R (sf x : Set S)) r hn₂
diff --git a/Mathlib/RingTheory/RingHom/FinitePresentation.lean b/Mathlib/RingTheory/RingHom/FinitePresentation.lean
index 21b4cbcffb892..ad4ae18aa6419 100644
--- a/Mathlib/RingTheory/RingHom/FinitePresentation.lean
+++ b/Mathlib/RingTheory/RingHom/FinitePresentation.lean
@@ -113,7 +113,7 @@ theorem finitePresentation_ofLocalizationSpanTarget :
     · infer_instance
   rw [RingHom.FinitePresentation]
   obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp hfintype
-  obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_total S (s : Set S) 1).mp
+  obtain ⟨l, hl⟩ := (Finsupp.mem_span_iff_linearCombination S (s : Set S) 1).mp
       (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)
   choose g' hg' using (fun g : s ↦ hf g)
   choose h' hh' using (fun g : s ↦ hf (l g))
@@ -124,7 +124,7 @@ theorem finitePresentation_ofLocalizationSpanTarget :
     simp only [Finset.univ_eq_attach, I, Ideal.mem_span_singleton] at hp
     obtain ⟨q, rfl⟩ := hp
     simp only [map_mul, map_sub, map_sum, map_one, hg', hh']
-    erw [Finsupp.total_apply_of_mem_supported S (s := s.attach)] at hl
+    erw [Finsupp.linearCombination_apply_of_mem_supported S (s := s.attach)] at hl
     · rw [← hl]
       simp only [Finset.coe_sort_coe, smul_eq_mul, mul_comm, sub_self, mul_zero, zero_mul]
     · rintro a -
@@ -149,7 +149,9 @@ theorem finitePresentation_ofLocalizationSpanTarget :
     exact ⟨{∑ g ∈ s.attach, g' g * h' g - 1}, by simp⟩
   have Ht (g : t) : Algebra.FinitePresentation R (Localization.Away (f' g)) := by
     have : ∃ (a : S) (hb : a ∈ s), (Ideal.Quotient.mk I) (g' ⟨a, hb⟩) = g.val := by
-      simpa [t] using g.property
+      obtain ⟨g, hg⟩ := g
+      convert hg
+      simp [t]
     obtain ⟨r, hr, hrr⟩ := this
     simp only [f']
     rw [← hrr, Ideal.Quotient.liftₐ_apply, Ideal.Quotient.lift_mk]
@@ -161,8 +163,9 @@ theorem finitePresentation_ofLocalizationSpanTarget :
 /-- Being finitely-presented is a local property of rings. -/
 theorem finitePresentation_isLocal : PropertyIsLocal @FinitePresentation :=
   ⟨finitePresentation_localizationPreserves,
-    finitePresentation_ofLocalizationSpanTarget, finitePresentation_stableUnderComposition,
-    finitePresentation_holdsForLocalizationAway⟩
+    finitePresentation_ofLocalizationSpanTarget,
+    finitePresentation_stableUnderComposition.stableUnderCompositionWithLocalizationAway
+      finitePresentation_holdsForLocalizationAway⟩
 
 /-- Being finitely-presented respects isomorphisms. -/
 theorem finitePresentation_respectsIso : RingHom.RespectsIso @RingHom.FinitePresentation :=
diff --git a/Mathlib/RingTheory/RingHom/FiniteType.lean b/Mathlib/RingTheory/RingHom/FiniteType.lean
index 568e607bdf1ac..1e07b805c0c4a 100644
--- a/Mathlib/RingTheory/RingHom/FiniteType.lean
+++ b/Mathlib/RingTheory/RingHom/FiniteType.lean
@@ -4,26 +4,154 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.RingTheory.FiniteStability
-import Mathlib.RingTheory.LocalProperties
 import Mathlib.RingTheory.Localization.InvSubmonoid
+import Mathlib.RingTheory.RingHom.Finite
 
 /-!
 
 # The meta properties of finite-type ring homomorphisms.
 
-The main result is `RingHom.finiteType_is_local`.
+## Main results
+
+Let `R` be a commutative ring, `S` is an `R`-algebra, `M` be a submonoid of `R`.
+
+* `finiteType_localizationPreserves` : If `S` is a finite type `R`-algebra, then `S' = M⁻¹S` is a
+  finite type `R' = M⁻¹R`-algebra.
+* `finiteType_ofLocalizationSpan` : `S` is a finite type `R`-algebra if there exists
+  a set `{ r }` that spans `R` such that `Sᵣ` is a finite type `Rᵣ`-algebra.
+*`RingHom.finiteType_is_local`: `RingHom.FiniteType` is a local property.
 
 -/
 
 
 namespace RingHom
 
-open scoped Pointwise TensorProduct
+open scoped Pointwise TensorProduct Classical
+
+universe u
+
+variable {R S : Type u} [CommRing R] [CommRing S] (M : Submonoid R) (f : R →+* S)
+variable (R' S' : Type u) [CommRing R'] [CommRing S']
+variable [Algebra R R'] [Algebra S S']
 
 theorem finiteType_stableUnderComposition : StableUnderComposition @FiniteType := by
   introv R hf hg
   exact hg.comp hf
 
+/-- If `S` is a finite type `R`-algebra, then `S' = M⁻¹S` is a finite type `R' = M⁻¹R`-algebra. -/
+theorem finiteType_localizationPreserves : RingHom.LocalizationPreserves @RingHom.FiniteType := by
+  introv R hf
+  -- mirrors the proof of `localization_map_finite`
+  letI := f.toAlgebra
+  letI := ((algebraMap S S').comp f).toAlgebra
+  let f' : R' →+* S' := IsLocalization.map S' f (Submonoid.le_comap_map M)
+  letI := f'.toAlgebra
+  haveI : IsScalarTower R R' S' :=
+    IsScalarTower.of_algebraMap_eq' (IsLocalization.map_comp M.le_comap_map).symm
+  let fₐ : S →ₐ[R] S' := AlgHom.mk' (algebraMap S S') fun c x => RingHom.map_mul _ _ _
+  obtain ⟨T, hT⟩ := hf
+  use T.image (algebraMap S S')
+  rw [eq_top_iff]
+  rintro x -
+  obtain ⟨y, ⟨_, ⟨r, hr, rfl⟩⟩, rfl⟩ := IsLocalization.mk'_surjective (M.map f) x
+  rw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, Finset.coe_image]
+  have hy : y ∈ Algebra.adjoin R (T : Set S) := by rw [hT]; trivial
+  replace hy : algebraMap S S' y ∈ (Algebra.adjoin R (T : Set S)).map fₐ :=
+    Subalgebra.mem_map.mpr ⟨_, hy, rfl⟩
+  rw [fₐ.map_adjoin T] at hy
+  have H : Algebra.adjoin R (algebraMap S S' '' T) ≤
+      (Algebra.adjoin R' (algebraMap S S' '' T)).restrictScalars R := by
+    rw [Algebra.adjoin_le_iff]; exact Algebra.subset_adjoin
+  convert (Algebra.adjoin R' (algebraMap S S' '' T)).smul_mem (H hy)
+    (IsLocalization.mk' R' (1 : R) ⟨r, hr⟩) using 1
+  rw [Algebra.smul_def]
+  erw [IsLocalization.map_mk' M.le_comap_map]
+  rw [map_one]
+
+theorem localization_away_map_finiteType (r : R) [IsLocalization.Away r R']
+    [IsLocalization.Away (f r) S'] (hf : f.FiniteType) :
+    (IsLocalization.Away.map R' S' f r).FiniteType :=
+  finiteType_localizationPreserves.away r hf
+
+variable {S'}
+
+/-- Let `S` be an `R`-algebra, `M` a submonoid of `S`, `S' = M⁻¹S`.
+Suppose the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
+and `A` is an `R`-subalgebra of `S` containing both `M` and the numerators of `s`.
+Then, there exists some `m : M` such that `m • x` falls in `A`.
+-/
+theorem IsLocalization.exists_smul_mem_of_mem_adjoin [Algebra R S] [Algebra R S']
+    [IsScalarTower R S S'] (M : Submonoid S) [IsLocalization M S'] (x : S) (s : Finset S')
+    (A : Subalgebra R S) (hA₁ : (IsLocalization.finsetIntegerMultiple M s : Set S) ⊆ A)
+    (hA₂ : M ≤ A.toSubmonoid) (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
+    ∃ m : M, m • x ∈ A := by
+  let g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'
+  let y := IsLocalization.commonDenomOfFinset M s
+  have hx₁ : (y : S) • (s : Set S') = g '' _ :=
+    (IsLocalization.finsetIntegerMultiple_image _ s).symm
+  obtain ⟨n, hn⟩ :=
+    Algebra.pow_smul_mem_of_smul_subset_of_mem_adjoin (y : S) (s : Set S') (A.map g)
+      (by rw [hx₁]; exact Set.image_subset _ hA₁) hx (Set.mem_image_of_mem _ (hA₂ y.2))
+  obtain ⟨x', hx', hx''⟩ := hn n (le_of_eq rfl)
+  rw [Algebra.smul_def, ← _root_.map_mul] at hx''
+  obtain ⟨a, ha₂⟩ := (IsLocalization.eq_iff_exists M S').mp hx''
+  use a * y ^ n
+  convert A.mul_mem hx' (hA₂ a.prop) using 1
+  rw [Submonoid.smul_def, smul_eq_mul, Submonoid.coe_mul, SubmonoidClass.coe_pow, mul_assoc, ← ha₂,
+    mul_comm]
+
+/-- Let `S` be an `R`-algebra, `M` a submonoid of `R`, and `S' = M⁻¹S`.
+If the image of some `x : S` falls in the adjoin of some finite `s ⊆ S'` over `R`,
+then there exists some `m : M` such that `m • x` falls in the
+adjoin of `IsLocalization.finsetIntegerMultiple _ s` over `R`.
+-/
+theorem IsLocalization.lift_mem_adjoin_finsetIntegerMultiple [Algebra R S] [Algebra R S']
+    [IsScalarTower R S S'] [IsLocalization (M.map (algebraMap R S)) S'] (x : S) (s : Finset S')
+    (hx : algebraMap S S' x ∈ Algebra.adjoin R (s : Set S')) :
+    ∃ m : M, m • x ∈
+      Algebra.adjoin R
+        (IsLocalization.finsetIntegerMultiple (M.map (algebraMap R S)) s : Set S) := by
+  obtain ⟨⟨_, a, ha, rfl⟩, e⟩ :=
+    IsLocalization.exists_smul_mem_of_mem_adjoin (M.map (algebraMap R S)) x s (Algebra.adjoin R _)
+      Algebra.subset_adjoin (by rintro _ ⟨a, _, rfl⟩; exact Subalgebra.algebraMap_mem _ a) hx
+  refine ⟨⟨a, ha⟩, ?_⟩
+  simpa only [Submonoid.smul_def, algebraMap_smul] using e
+
+theorem finiteType_ofLocalizationSpan : RingHom.OfLocalizationSpan @RingHom.FiniteType := by
+  rw [RingHom.ofLocalizationSpan_iff_finite]
+  introv R hs H
+  -- mirrors the proof of `finite_ofLocalizationSpan`
+  letI := f.toAlgebra
+  letI := fun r : s => (Localization.awayMap f r).toAlgebra
+  have : ∀ r : s,
+      IsLocalization ((Submonoid.powers (r : R)).map (algebraMap R S)) (Localization.Away (f r)) :=
+    by intro r; rw [Submonoid.map_powers]; exact Localization.isLocalization
+  haveI : ∀ r : s, IsScalarTower R (Localization.Away (r : R)) (Localization.Away (f r)) :=
+    fun r => IsScalarTower.of_algebraMap_eq'
+      (IsLocalization.map_comp (Submonoid.powers (r : R)).le_comap_map).symm
+  constructor
+  replace H := fun r => (H r).1
+  choose s₁ s₂ using H
+  let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (f x)) (s₁ x)
+  use s.attach.biUnion sf
+  convert (Algebra.adjoin_attach_biUnion (R := R) sf).trans _
+  rw [eq_top_iff]
+  rintro x -
+  apply (⨆ x : s, Algebra.adjoin R (sf x : Set S)).toSubmodule.mem_of_span_eq_top_of_smul_pow_mem
+    _ hs _ _
+  intro r
+  obtain ⟨⟨_, n₁, rfl⟩, hn₁⟩ :=
+    multiple_mem_adjoin_of_mem_localization_adjoin (Submonoid.powers (r : R))
+      (Localization.Away (r : R)) (s₁ r : Set (Localization.Away (f r)))
+      (algebraMap S (Localization.Away (f r)) x) (by rw [s₂ r]; trivial)
+  rw [Submonoid.smul_def, Algebra.smul_def, IsScalarTower.algebraMap_apply R S, ← map_mul] at hn₁
+  obtain ⟨⟨_, n₂, rfl⟩, hn₂⟩ :=
+    IsLocalization.lift_mem_adjoin_finsetIntegerMultiple (Submonoid.powers (r : R)) _ (s₁ r) hn₁
+  rw [Submonoid.smul_def, ← Algebra.smul_def, smul_smul, ← pow_add] at hn₂
+  simp_rw [Submonoid.map_powers] at hn₂
+  use n₂ + n₁
+  exact le_iSup (fun x : s => Algebra.adjoin R (sf x : Set S)) r hn₂
+
 theorem finiteType_holdsForLocalizationAway : HoldsForLocalizationAway @FiniteType := by
   introv R _
   suffices Algebra.FiniteType R S by
@@ -47,7 +175,7 @@ theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteTy
   -- `∑ lᵢ * sᵢ = 1`. I claim that all `s` and `l` and the numerators of `t` and generates `S`.
   choose t ht using H
   obtain ⟨l, hl⟩ :=
-    (Finsupp.mem_span_iff_total S (s : Set S) 1).mp
+    (Finsupp.mem_span_iff_linearCombination S (s : Set S) 1).mp
       (show (1 : S) ∈ Ideal.span (s : Set S) by rw [hs]; trivial)
   let sf := fun x : s => IsLocalization.finsetIntegerMultiple (Submonoid.powers (x : S)) (t x)
   use s.attach.biUnion sf ∪ s ∪ l.support.image l
@@ -88,8 +216,9 @@ theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteTy
     · rw [ht]; trivial
 
 theorem finiteType_is_local : PropertyIsLocal @FiniteType :=
-  ⟨localization_finiteType, finiteType_ofLocalizationSpanTarget, finiteType_stableUnderComposition,
-    finiteType_holdsForLocalizationAway⟩
+  ⟨finiteType_localizationPreserves, finiteType_ofLocalizationSpanTarget,
+    finiteType_stableUnderComposition.stableUnderCompositionWithLocalizationAway
+      finiteType_holdsForLocalizationAway⟩
 
 theorem finiteType_respectsIso : RingHom.RespectsIso @RingHom.FiniteType :=
   RingHom.finiteType_is_local.respectsIso
diff --git a/Mathlib/RingTheory/RingHom/Locally.lean b/Mathlib/RingTheory/RingHom/Locally.lean
new file mode 100644
index 0000000000000..fc54b147cef5b
--- /dev/null
+++ b/Mathlib/RingTheory/RingHom/Locally.lean
@@ -0,0 +1,146 @@
+/-
+Copyright (c) 2024 Christian Merten. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Christian Merten
+-/
+import Mathlib.RingTheory.LocalProperties.Basic
+import Mathlib.RingTheory.Localization.Away.Lemmas
+
+/-!
+# Target local closure of ring homomorphism properties
+
+If `P` is a property of ring homomorphisms, we call `Locally P` the closure of `P` with
+respect to standard open coverings on the (algebraic) target (i.e. geometric source). Hence
+for `f : R →+* S`, the property `Locally P` holds if it holds locally on `S`, i.e. if there exists
+a subset `{ t }` of `S` generating the unit ideal, such that `P` holds for all compositions
+`R →+* Sₜ`.
+
+Assuming without further mention that `P` is stable under composition with isomorphisms,
+`Locally P` is local on the target by construction, i.e. it satisfies
+`OfLocalizationSpanTarget`. If `P` itself is local on the target, `Locally P` coincides with `P`.
+
+The `Locally` construction preserves various properties of `P`, e.g. if `P` is stable under
+composition, base change, etc., so is `Locally P`.
+
+## Main results
+
+- `RingHom.locally_ofLocalizationSpanTarget`: `Locally P` is local on the target.
+
+-/
+
+universe u v
+
+open TensorProduct
+
+namespace RingHom
+
+variable (P : ∀ {R S : Type u} [CommRing R] [CommRing S] (_ : R →+* S), Prop)
+
+/--
+For a property of ring homomorphisms `P`, `Locally P` holds for `f : R →+* S` if
+it holds locally on `S`, i.e. if there exists a subset `{ t }` of `S` generating
+the unit ideal, such that `P` holds for all compositions `R →+* Sₜ`.
+
+We may require `s` to be finite here, for the equivalence, see `locally_iff_finite`.
+-/
+def Locally {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) : Prop :=
+  ∃ (s : Set S) (_ : Ideal.span s = ⊤),
+    ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)
+
+variable {R S : Type u} [CommRing R] [CommRing S]
+
+lemma locally_iff_finite (f : R →+* S) :
+    Locally P f ↔ ∃ (s : Finset S) (_ : Ideal.span (s : Set S) = ⊤),
+      ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f) := by
+  constructor
+  · intro ⟨s, hsone, hs⟩
+    obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hsone
+    exact ⟨s', h₂, fun t ht ↦ hs t (h₁ ht)⟩
+  · intro ⟨s, hsone, hs⟩
+    use s, hsone, hs
+
+variable {P}
+
+/-- If `P` respects isomorphisms, to check `P` holds locally for `f : R →+* S`, it suffices
+to check `P` holds on a standard open cover. -/
+lemma locally_of_exists (hP : RespectsIso P) (f : R →+* S) {ι : Type*} (s : ι → S)
+    (hsone : Ideal.span (Set.range s) = ⊤)
+    (Sₜ : ι → Type u) [∀ i, CommRing (Sₜ i)] [∀ i, Algebra S (Sₜ i)]
+    [∀ i, IsLocalization.Away (s i) (Sₜ i)] (hf : ∀ i, P ((algebraMap S (Sₜ i)).comp f)) :
+    Locally P f := by
+  use Set.range s, hsone
+  rintro - ⟨i, rfl⟩
+  let e : Localization.Away (s i) ≃+* Sₜ i :=
+    (IsLocalization.algEquiv (Submonoid.powers (s i)) _ _).toRingEquiv
+  have : algebraMap S (Localization.Away (s i)) = e.symm.toRingHom.comp (algebraMap S (Sₜ i)) :=
+    RingHom.ext (fun x ↦ (AlgEquiv.commutes (IsLocalization.algEquiv _ _ _).symm _).symm)
+  rw [this, RingHom.comp_assoc]
+  exact hP.left _ _ (hf i)
+
+/-- Equivalence variant of `locally_of_exists`. This is sometimes easier to use, if the
+`IsLocalization.Away` instance can't be automatically inferred. -/
+lemma locally_iff_exists (hP : RespectsIso P) (f : R →+* S) :
+    Locally P f ↔ ∃ (ι : Type u) (s : ι → S) (_ : Ideal.span (Set.range s) = ⊤) (Sₜ : ι → Type u)
+      (_ : (i : ι) → CommRing (Sₜ i)) (_ : (i : ι) → Algebra S (Sₜ i))
+      (_ : (i : ι) → IsLocalization.Away (s i : S) (Sₜ i)),
+      ∀ i, P ((algebraMap S (Sₜ i)).comp f) :=
+  ⟨fun ⟨s, hsone, hs⟩ ↦ ⟨s, fun t : s ↦ (t : S), by simpa, fun t ↦ Localization.Away (t : S),
+      inferInstance, inferInstance, inferInstance, fun t ↦ hs t.val t.property⟩,
+    fun ⟨ι, s, hsone, Sₜ, _, _, hislocal, hs⟩ ↦ locally_of_exists hP f s hsone Sₜ hs⟩
+
+/-- In the definition of `Locally` we may replace `Localization.Away` with an arbitrary
+algebra satisfying `IsLocalization.Away`. -/
+lemma locally_iff_isLocalization (hP : RespectsIso P) (f : R →+* S) :
+    Locally P f ↔ ∃ (s : Finset S) (_ : Ideal.span (s : Set S) = ⊤),
+      ∀ t ∈ s, ∀ (Sₜ : Type u) [CommRing Sₜ] [Algebra S Sₜ] [IsLocalization.Away t Sₜ],
+      P ((algebraMap S Sₜ).comp f) := by
+  rw [locally_iff_finite P f]
+  refine ⟨fun ⟨s, hsone, hs⟩ ↦ ⟨s, hsone, fun t ht Sₜ _ _ _ ↦ ?_⟩, fun ⟨s, hsone, hs⟩ ↦ ?_⟩
+  · let e : Localization.Away t ≃+* Sₜ :=
+      (IsLocalization.algEquiv (Submonoid.powers t) _ _).toRingEquiv
+    have : algebraMap S Sₜ = e.toRingHom.comp (algebraMap S (Localization.Away t)) :=
+      RingHom.ext (fun x ↦ (AlgEquiv.commutes (IsLocalization.algEquiv _ _ _) _).symm)
+    rw [this, RingHom.comp_assoc]
+    exact hP.left _ _ (hs t ht)
+  · exact ⟨s, hsone, fun t ht ↦ hs t ht _⟩
+
+/-- If `f` satisfies `P`, then in particular it satisfies `Locally P`. -/
+lemma locally_of (hP : RespectsIso P) (f : R →+* S) (hf : P f) : Locally P f := by
+  use {1}
+  let e : S ≃+* Localization.Away (1 : S) :=
+    (IsLocalization.atUnits S (Submonoid.powers 1) (by simp)).toRingEquiv
+  simp only [Set.mem_singleton_iff, forall_eq, Ideal.span_singleton_one, exists_const]
+  exact hP.left f e hf
+
+/-- If `P` is local on the target, then `Locally P` coincides with `P`. -/
+lemma locally_iff_of_localizationSpanTarget (hPi : RespectsIso P)
+    (hPs : OfLocalizationSpanTarget P) {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) :
+    Locally P f ↔ P f :=
+  ⟨fun ⟨s, hsone, hs⟩ ↦ hPs f s hsone (fun a ↦ hs a.val a.property), locally_of hPi f⟩
+
+section OfLocalizationSpanTarget
+
+/-- `Locally P` is local on the target. -/
+lemma locally_ofLocalizationSpanTarget (hP : RespectsIso P) :
+    OfLocalizationSpanTarget (Locally P) := by
+  intro R S _ _ f s hsone hs
+  choose t htone ht using hs
+  rw [locally_iff_exists hP]
+  refine ⟨(a : s) × t a, IsLocalization.Away.mulNumerator s t,
+      IsLocalization.Away.span_range_mulNumerator_eq_top hsone htone,
+      fun ⟨a, b⟩ ↦ Localization.Away b.val, inferInstance, inferInstance, fun ⟨a, b⟩ ↦ ?_, ?_⟩
+  · haveI : IsLocalization.Away ((algebraMap S (Localization.Away a.val))
+        (IsLocalization.Away.sec a.val b.val).1) (Localization.Away b.val) := by
+      apply IsLocalization.Away.of_associated (r := b.val)
+      rw [← IsLocalization.Away.sec_spec]
+      apply associated_mul_unit_right
+      rw [map_pow _ _]
+      exact IsUnit.pow _ (IsLocalization.Away.algebraMap_isUnit _)
+    apply IsLocalization.Away.mul' (Localization.Away a.val) (Localization.Away b.val)
+  · intro ⟨a, b⟩
+    rw [IsScalarTower.algebraMap_eq S (Localization.Away a.val) (Localization.Away b.val)]
+    apply ht _ _ b.property
+
+end OfLocalizationSpanTarget
+
+end RingHom
diff --git a/Mathlib/RingTheory/RingHom/Surjective.lean b/Mathlib/RingTheory/RingHom/Surjective.lean
index 732f8377e880b..b20eeffe62179 100644
--- a/Mathlib/RingTheory/RingHom/Surjective.lean
+++ b/Mathlib/RingTheory/RingHom/Surjective.lean
@@ -3,12 +3,22 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-import Mathlib.RingTheory.LocalProperties
+import Mathlib.RingTheory.LocalProperties.Basic
 
 /-!
 
 # The meta properties of surjective ring homomorphisms.
 
+## Main results
+
+Let `R` be a commutative ring, `M` be a submonoid of `R`.
+
+* `surjective_localizationPreserves` :  `M⁻¹R →+* M⁻¹S` is surjective if `R →+* S` is surjective.
+* `surjective_ofLocalizationSpan` : `R →+* S` is surjective if there exists a set `{ r }` that
+  spans `R` such that `Rᵣ →+* Sᵣ` is surjective.
+* `surjective_localRingHom_of_surjective` : A surjective ring homomorphism `R →+* S` induces a
+  surjective homomorphism `R_{f⁻¹(P)} →+* S_P` for every prime ideal `P` of `S`.
+
 -/
 
 
@@ -18,6 +28,8 @@ open scoped TensorProduct
 
 open TensorProduct Algebra.TensorProduct
 
+universe u
+
 local notation "surjective" => fun {X Y : Type _} [CommRing X] [CommRing Y] => fun f : X →+* Y =>
   Function.Surjective f
 
@@ -40,28 +52,41 @@ theorem surjective_stableUnderBaseChange : StableUnderBaseChange surjective := b
     rw [TensorProduct.smul_tmul, Algebra.algebraMap_eq_smul_one]
   | add x y ex ey => obtain ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ := ex, ey; exact ⟨x + y, map_add _ x y⟩
 
+/-- `M⁻¹R →+* M⁻¹S` is surjective if `R →+* S` is surjective. -/
+theorem surjective_localizationPreserves :
+    LocalizationPreserves surjective := by
+  introv R H x
+  obtain ⟨x, ⟨_, s, hs, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (M.map f) x
+  obtain ⟨y, rfl⟩ := H x
+  use IsLocalization.mk' R' y ⟨s, hs⟩
+  rw [IsLocalization.map_mk']
+
+/-- `R →+* S` is surjective if there exists a set `{ r }` that spans `R` such that
+  `Rᵣ →+* Sᵣ` is surjective. -/
 theorem surjective_ofLocalizationSpan : OfLocalizationSpan surjective := by
-  introv R hs H
+  introv R e H
+  rw [← Set.range_iff_surjective, Set.eq_univ_iff_forall]
   letI := f.toAlgebra
-  show Function.Surjective (Algebra.ofId R S)
-  rw [← Algebra.range_top_iff_surjective, eq_top_iff]
-  rintro x -
-  obtain ⟨l, hl⟩ :=
-    (Finsupp.mem_span_iff_total R s 1).mp (show _ ∈ Ideal.span s by rw [hs]; trivial)
-  fapply
-    Subalgebra.mem_of_finset_sum_eq_one_of_pow_smul_mem _ l.support (fun x : s => f x) fun x : s =>
-      f (l x)
-  · simp_rw [← _root_.map_mul, ← map_sum, ← f.map_one]; exact f.congr_arg hl
-  · exact fun _ => Set.mem_range_self _
-  · exact fun _ => Set.mem_range_self _
-  · intro r
-    obtain ⟨y, hy⟩ := H r (IsLocalization.mk' _ x (1 : Submonoid.powers (f r)))
-    obtain ⟨z, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers (r : R)) y
-    erw [IsLocalization.map_mk', IsLocalization.eq] at hy
-    obtain ⟨⟨_, m, rfl⟩, hm⟩ := hy
-    refine ⟨m + n, ?_⟩
-    dsimp at hm ⊢
-    simp_rw [_root_.one_mul, ← _root_.mul_assoc, ← map_pow, ← f.map_mul, ← pow_add, map_pow] at hm
-    exact ⟨_, hm⟩
+  intro x
+  apply Submodule.mem_of_span_eq_top_of_smul_pow_mem
+    (LinearMap.range (Algebra.linearMap R S)) s e
+  intro r
+  obtain ⟨a, e'⟩ := H r (algebraMap _ _ x)
+  obtain ⟨b, ⟨_, n, rfl⟩, rfl⟩ := IsLocalization.mk'_surjective (Submonoid.powers (r : R)) a
+  erw [IsLocalization.map_mk'] at e'
+  rw [eq_comm, IsLocalization.eq_mk'_iff_mul_eq, Subtype.coe_mk, Subtype.coe_mk, ← map_mul] at e'
+  obtain ⟨⟨_, n', rfl⟩, e''⟩ := (IsLocalization.eq_iff_exists (Submonoid.powers (f r)) _).mp e'
+  dsimp only at e''
+  rw [mul_comm x, ← mul_assoc, ← map_pow, ← map_mul, ← map_mul, ← pow_add] at e''
+  exact ⟨n' + n, _, e''.symm⟩
+
+/-- A surjective ring homomorphism `R →+* S` induces a surjective homomorphism `R_{f⁻¹(P)} →+* S_P`
+for every prime ideal `P` of `S`. -/
+theorem surjective_localRingHom_of_surjective {R S : Type u} [CommRing R] [CommRing S]
+    (f : R →+* S) (h : Function.Surjective f) (P : Ideal S) [P.IsPrime] :
+    Function.Surjective (Localization.localRingHom (P.comap f) P f rfl) :=
+  have : IsLocalization (Submonoid.map f (Ideal.comap f P).primeCompl) (Localization.AtPrime P) :=
+    (Submonoid.map_comap_eq_of_surjective h P.primeCompl).symm ▸ Localization.isLocalization
+  surjective_localizationPreserves _ _ _ _ h
 
 end RingHom
diff --git a/Mathlib/RingTheory/RingHomProperties.lean b/Mathlib/RingTheory/RingHomProperties.lean
index 280f3184fd167..7008cd0ef9a74 100644
--- a/Mathlib/RingTheory/RingHomProperties.lean
+++ b/Mathlib/RingTheory/RingHomProperties.lean
@@ -179,15 +179,17 @@ variable {P}
 
 lemma toMorphismProperty_respectsIso_iff :
     RespectsIso P ↔ (toMorphismProperty P).RespectsIso := by
-  refine ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ ⟨?_, ?_⟩⟩
+  refine ⟨fun h ↦ MorphismProperty.RespectsIso.mk _ ?_ ?_, fun h ↦ ⟨?_, ?_⟩⟩
   · intro X Y Z e f hf
     exact h.right f e.commRingCatIsoToRingEquiv hf
   · intro X Y Z e f hf
     exact h.left f e.commRingCatIsoToRingEquiv hf
+  · intro X Y Z _ _ _ f e hf
+    exact MorphismProperty.RespectsIso.postcomp (toMorphismProperty P)
+      e.toCommRingCatIso.hom (CommRingCat.ofHom f) hf
   · intro X Y Z _ _ _ f e
-    exact h.postcomp e.toCommRingCatIso (CommRingCat.ofHom f)
-  · intro X Y Z _ _ _ f e
-    exact h.precomp e.toCommRingCatIso (CommRingCat.ofHom f)
+    exact MorphismProperty.RespectsIso.precomp (toMorphismProperty P)
+      e.toCommRingCatIso.hom (CommRingCat.ofHom f)
 
 end ToMorphismProperty
 
diff --git a/Mathlib/RingTheory/RootsOfUnity/Basic.lean b/Mathlib/RingTheory/RootsOfUnity/Basic.lean
index 04f0e45847dbc..fbfd625bc62c1 100644
--- a/Mathlib/RingTheory/RootsOfUnity/Basic.lean
+++ b/Mathlib/RingTheory/RootsOfUnity/Basic.lean
@@ -442,7 +442,7 @@ then there is a `b`-th primitive root of unity in `R`. -/
 theorem pow {n : ℕ} {a b : ℕ} (hn : 0 < n) (h : IsPrimitiveRoot ζ n) (hprod : n = a * b) :
     IsPrimitiveRoot (ζ ^ a) b := by
   subst n
-  simp only [iff_def, ← pow_mul, h.pow_eq_one, eq_self_iff_true, true_and_iff]
+  simp only [iff_def, ← pow_mul, h.pow_eq_one, eq_self_iff_true, true_and]
   intro l hl
   -- Porting note: was `by rintro rfl; simpa only [Nat.not_lt_zero, zero_mul] using hn`
   have ha0 : a ≠ 0 := left_ne_zero_of_mul hn.ne'
@@ -868,14 +868,14 @@ theorem card_primitiveRoots {ζ : R} {k : ℕ} (h : IsPrimitiveRoot ζ k) :
   · simp [h0]
   symm
   refine Finset.card_bij (fun i _ ↦ ζ ^ i) ?_ ?_ ?_
-  · simp only [true_and_iff, and_imp, mem_filter, mem_range, mem_univ]
+  · simp only [and_imp, mem_filter, mem_range, mem_univ]
     rintro i - hi
     rw [mem_primitiveRoots (Nat.pos_of_ne_zero h0)]
     exact h.pow_of_coprime i hi.symm
-  · simp only [true_and_iff, and_imp, mem_filter, mem_range, mem_univ]
+  · simp only [and_imp, mem_filter, mem_range, mem_univ]
     rintro i hi - j hj - H
     exact h.pow_inj hi hj H
-  · simp only [exists_prop, true_and_iff, mem_filter, mem_range, mem_univ]
+  · simp only [exists_prop, mem_filter, mem_range, mem_univ]
     intro ξ hξ
     rw [mem_primitiveRoots (Nat.pos_of_ne_zero h0),
       h.isPrimitiveRoot_iff (Nat.pos_of_ne_zero h0)] at hξ
@@ -898,7 +898,7 @@ theorem nthRoots_one_eq_biUnion_primitiveRoots' {ζ : R} {n : ℕ+} (h : IsPrimi
   · intro x
     simp only [nthRootsFinset, ← Multiset.toFinset_eq (nthRoots_one_nodup h), exists_prop,
       Finset.mem_biUnion, Finset.mem_filter, Finset.mem_range, mem_nthRoots, Finset.mem_mk,
-      Nat.mem_divisors, and_true_iff, Ne, PNat.ne_zero, PNat.pos, not_false_iff]
+      Nat.mem_divisors, and_true, Ne, PNat.ne_zero, PNat.pos, not_false_iff]
     rintro ⟨a, ⟨d, hd⟩, ha⟩
     have hazero : 0 < a := by
       contrapose! hd with ha0
diff --git a/Mathlib/RingTheory/RootsOfUnity/Complex.lean b/Mathlib/RingTheory/RootsOfUnity/Complex.lean
index ec56155951282..a11bd30bfad0a 100644
--- a/Mathlib/RingTheory/RootsOfUnity/Complex.lean
+++ b/Mathlib/RingTheory/RootsOfUnity/Complex.lean
@@ -168,7 +168,7 @@ theorem IsPrimitiveRoot.arg {n : ℕ} {ζ : ℂ} (h : IsPrimitiveRoot ζ n) (hn
     exact mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr <| mod_cast h.le)
       (div_nonneg (by simp [Real.pi_pos.le]) <| by simp)
   rw [← mul_rotate', mul_div_assoc, neg_lt, ← mul_neg, mul_lt_iff_lt_one_right Real.pi_pos, ←
-    neg_div, ← neg_mul, neg_sub, div_lt_iff, one_mul, sub_mul, sub_lt_comm, ← mul_sub_one]
+    neg_div, ← neg_mul, neg_sub, div_lt_iff₀, one_mul, sub_mul, sub_lt_comm, ← mul_sub_one]
   · norm_num
     exact mod_cast not_le.mp h₂
   · exact Nat.cast_pos.mpr hn.bot_lt
diff --git a/Mathlib/RingTheory/RootsOfUnity/Minpoly.lean b/Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
index 18723ca9b73e4..fddcf3946281c 100644
--- a/Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
+++ b/Mathlib/RingTheory/RootsOfUnity/Minpoly.lean
@@ -199,7 +199,7 @@ theorem is_roots_of_minpoly [DecidableEq K] :
   intro x hx
   obtain ⟨m, _, hcop, rfl⟩ := (isPrimitiveRoot_iff h hpos).1 ((mem_primitiveRoots hpos).1 hx)
   simp only [Multiset.mem_toFinset, mem_roots]
-  convert pow_isRoot_minpoly h hcop
+  convert pow_isRoot_minpoly h hcop using 0
   rw [← mem_roots]
   exact map_monic_ne_zero <| minpoly.monic <| isIntegral h hpos
 
diff --git a/Mathlib/RingTheory/SimpleModule.lean b/Mathlib/RingTheory/SimpleModule.lean
index e9412b0e43039..9a27c4b72d304 100644
--- a/Mathlib/RingTheory/SimpleModule.lean
+++ b/Mathlib/RingTheory/SimpleModule.lean
@@ -163,7 +163,7 @@ theorem isSimpleModule_self_iff_isUnit :
     exact ⟨⟨x, y, left_inv_eq_right_inv hzy hyx ▸ hzy, hyx⟩, rfl⟩
 
 theorem isSimpleModule_iff_finrank_eq_one {R} [DivisionRing R] [Module R M] :
-    IsSimpleModule R M ↔ FiniteDimensional.finrank R M = 1 :=
+    IsSimpleModule R M ↔ Module.finrank R M = 1 :=
   ⟨fun h ↦ have := h.nontrivial; have ⟨v, hv⟩ := exists_ne (0 : M)
     (finrank_eq_one_iff_of_nonzero' v hv).mpr (IsSimpleModule.toSpanSingleton_surjective R hv),
   is_simple_module_of_finrank_eq_one⟩
@@ -285,7 +285,7 @@ theorem IsSemisimpleModule.sup {p q : Submodule R M}
 instance IsSemisimpleRing.isSemisimpleModule [IsSemisimpleRing R] : IsSemisimpleModule R M :=
   have : IsSemisimpleModule R (M →₀ R) := isSemisimpleModule_of_isSemisimpleModule_submodule'
     (fun _ ↦ .congr (LinearMap.quotKerEquivRange _).symm) Finsupp.iSup_lsingle_range
-  .congr (LinearMap.quotKerEquivOfSurjective _ <| Finsupp.total_id_surjective R M).symm
+  .congr (LinearMap.quotKerEquivOfSurjective _ <| Finsupp.linearCombination_id_surjective R M).symm
 
 instance IsSemisimpleRing.isCoatomic_submodule [IsSemisimpleRing R] : IsCoatomic (Submodule R M) :=
   isCoatomic_of_isAtomic_of_complementedLattice_of_isModular
@@ -338,7 +338,7 @@ variable (ι R)
 
 proof_wanted IsSemisimpleRing.mulOpposite [IsSemisimpleRing R] : IsSemisimpleRing Rᵐᵒᵖ
 
-proof_wanted IsSemisimpleRing.module_end [IsSemisimpleRing R] [Module.Finite R M] :
+proof_wanted IsSemisimpleRing.module_end [IsSemisimpleModule R M] [Module.Finite R M] :
     IsSemisimpleRing (Module.End R M)
 
 proof_wanted IsSemisimpleRing.matrix [Fintype ι] [DecidableEq ι] [IsSemisimpleRing R] :
diff --git a/Mathlib/RingTheory/SimpleRing/Basic.lean b/Mathlib/RingTheory/SimpleRing/Basic.lean
new file mode 100644
index 0000000000000..3f9711cafff02
--- /dev/null
+++ b/Mathlib/RingTheory/SimpleRing/Basic.lean
@@ -0,0 +1,87 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang
+-/
+
+import Mathlib.RingTheory.SimpleRing.Defs
+import Mathlib.Algebra.Field.Equiv
+import Mathlib.Algebra.Ring.Subring.Basic
+
+/-! # Basic Properties of Simple rings
+
+A ring `R` is **simple** if it has only two two-sided ideals, namely `⊥` and `⊤`.
+
+## Main results
+
+- `IsSimpleRing.nontrivial`: simple rings are non-trivial.
+- `DivisionRing.IsSimpleRing`: division rings are simple.
+- `IsSimpleRing.center_isField`: the center of a simple ring is a field.
+
+-/
+
+variable (R : Type*) [NonUnitalNonAssocRing R]
+
+namespace IsSimpleRing
+
+variable {R}
+
+instance [IsSimpleRing R] : IsSimpleOrder (TwoSidedIdeal R) := IsSimpleRing.simple
+
+instance [simple : IsSimpleRing R] : Nontrivial R := by
+  obtain ⟨x, hx⟩ := SetLike.exists_of_lt (bot_lt_top : (⊥ : TwoSidedIdeal R) < ⊤)
+  have h (hx : x = 0) : False := by simp_all [TwoSidedIdeal.zero_mem]
+  use x, 0, h
+
+lemma one_mem_of_ne_bot {A : Type*} [NonAssocRing A] [IsSimpleRing A] (I : TwoSidedIdeal A)
+    (hI : I ≠ ⊥) : (1 : A) ∈ I :=
+  (eq_bot_or_eq_top I).resolve_left hI ▸ ⟨⟩
+
+lemma one_mem_of_ne_zero_mem {A : Type*} [NonAssocRing A] [IsSimpleRing A] (I : TwoSidedIdeal A)
+    {x : A} (hx : x ≠ 0) (hxI : x ∈ I) : (1 : A) ∈ I :=
+  one_mem_of_ne_bot I (by rintro rfl; exact hx hxI)
+
+lemma of_eq_bot_or_eq_top [Nontrivial R] (h : ∀ I : TwoSidedIdeal R, I = ⊥ ∨ I = ⊤) :
+    IsSimpleRing R where
+  simple := { eq_bot_or_eq_top := h }
+
+instance _root_.DivisionRing.isSimpleRing (A : Type*) [DivisionRing A] : IsSimpleRing A :=
+  .of_eq_bot_or_eq_top <| fun I ↦ by
+    rw [or_iff_not_imp_left, ← I.one_mem_iff]
+    intro H
+    obtain ⟨x, hx1, hx2 : x ≠ 0⟩ := SetLike.exists_of_lt (bot_lt_iff_ne_bot.mpr H : ⊥ < I)
+    simpa [inv_mul_cancel₀ hx2] using I.mul_mem_left x⁻¹ _ hx1
+
+open TwoSidedIdeal in
+lemma isField_center (A : Type*) [Ring A] [IsSimpleRing A] : IsField (Subring.center A) where
+  exists_pair_ne := ⟨0, 1, zero_ne_one⟩
+  mul_comm := mul_comm
+  mul_inv_cancel := by
+    rintro ⟨x, hx1⟩ hx2
+    rw [Subring.mem_center_iff] at hx1
+    replace hx2 : x ≠ 0 := by simpa [Subtype.ext_iff] using hx2
+    -- Todo: golf the following `let` once `TwoSidedIdeal.span` is defined
+    let I := TwoSidedIdeal.mk' (Set.range (x * ·)) ⟨0, by simp⟩
+      (by rintro _ _ ⟨x, rfl⟩ ⟨y, rfl⟩; exact ⟨x + y, mul_add _ _ _⟩)
+      (by rintro _ ⟨x, rfl⟩; exact ⟨-x, by simp⟩)
+      (by rintro a _ ⟨c, rfl⟩; exact ⟨a * c, by dsimp; rw [← mul_assoc, ← hx1, mul_assoc]⟩)
+      (by rintro _ b ⟨a, rfl⟩; exact ⟨a * b, by dsimp; rw [← mul_assoc, ← hx1, mul_assoc]⟩)
+    have mem : 1 ∈ I := one_mem_of_ne_zero_mem I hx2 (by simpa [I, mem_mk'] using ⟨1, by simp⟩)
+    simp only [TwoSidedIdeal.mem_mk', Set.mem_range, I] at mem
+    obtain ⟨y, hy⟩ := mem
+    refine ⟨⟨y, Subring.mem_center_iff.2 fun a ↦ ?_⟩, by ext; exact hy⟩
+    calc a * y
+      _ = (x * y) * a * y := by rw [hy, one_mul]
+      _ = (y * x) * a * y := by rw [hx1]
+      _ = y * (x * a) * y := by rw [mul_assoc y x a]
+      _ = y * (a * x) * y := by rw [hx1]
+      _ = y * ((a * x) * y) := by rw [mul_assoc]
+      _ = y * (a * (x * y)) := by rw [mul_assoc a x y]
+      _ = y * a := by rw [hy, mul_one]
+
+end IsSimpleRing
+
+lemma isSimpleRing_iff_isField (A : Type*) [CommRing A] : IsSimpleRing A ↔ IsField A :=
+  ⟨fun _ ↦ Subring.topEquiv.symm.toMulEquiv.isField _ <| by
+    rw [← Subring.center_eq_top A]; exact IsSimpleRing.isField_center A,
+    fun h ↦ letI := h.toField; inferInstance⟩
diff --git a/Mathlib/RingTheory/SimpleRing/Defs.lean b/Mathlib/RingTheory/SimpleRing/Defs.lean
new file mode 100644
index 0000000000000..ce50f0bbc7460
--- /dev/null
+++ b/Mathlib/RingTheory/SimpleRing/Defs.lean
@@ -0,0 +1,25 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang
+-/
+
+import Mathlib.RingTheory.TwoSidedIdeal.Lattice
+import Mathlib.Order.Atoms
+
+/-! # Simple rings
+
+A ring `R` is **simple** if it has only two two-sided ideals, namely `⊥` and `⊤`.
+
+## Main definitions
+
+- `IsSimpleRing`: a predicate expressing that a ring is simple.
+
+-/
+
+
+/--
+A ring `R` is **simple** if it has only two two-sided ideals, namely `⊥` and `⊤`.
+-/
+class IsSimpleRing (R : Type*) [NonUnitalNonAssocRing R] : Prop where
+  simple : IsSimpleOrder (TwoSidedIdeal R)
diff --git a/Mathlib/RingTheory/Smooth/Kaehler.lean b/Mathlib/RingTheory/Smooth/Kaehler.lean
index 06b7cba8d586b..e11be1b55e317 100644
--- a/Mathlib/RingTheory/Smooth/Kaehler.lean
+++ b/Mathlib/RingTheory/Smooth/Kaehler.lean
@@ -48,14 +48,14 @@ def derivationOfSectionOfKerSqZero (f : P →ₐ[R] S) (hf' : (RingHom.ker f) ^
     (hg : f.comp g = AlgHom.id R S) : Derivation R P (RingHom.ker f) where
   toFun x := ⟨x - g (f x), by
     simpa [RingHom.mem_ker, sub_eq_zero] using AlgHom.congr_fun hg.symm (f x)⟩
-  map_add' x y := by simp only [map_add, AddSubmonoid.mk_add_mk, Subtype.mk.injEq]; ring
+  map_add' x y := by simp only [map_add, AddMemClass.mk_add_mk, Subtype.mk.injEq]; ring
   map_smul' x y := by
     ext
     simp only [Algebra.smul_def, map_mul, ← IsScalarTower.algebraMap_apply, AlgHom.commutes,
       RingHom.id_apply, Submodule.coe_smul_of_tower]
     ring
   map_one_eq_zero' := by simp only [LinearMap.coe_mk, AddHom.coe_mk, map_one, sub_self,
-    AddSubmonoid.mk_eq_zero]
+    Submodule.mk_eq_zero]
   leibniz' a b := by
     have : (a - g (f a)) * (b - g (f b)) = 0 := by
       rw [← Ideal.mem_bot, ← hf', pow_two]
@@ -66,7 +66,7 @@ def derivationOfSectionOfKerSqZero (f : P →ₐ[R] S) (hf' : (RingHom.ker f) ^
     rw [← sub_eq_zero]
     conv_rhs => rw [← neg_zero, ← this]
     simp only [LinearMap.coe_mk, AddHom.coe_mk, map_mul, SetLike.mk_smul_mk, smul_eq_mul, mul_sub,
-      AddSubmonoid.mk_add_mk, sub_mul, neg_sub]
+      AddMemClass.mk_add_mk, sub_mul, neg_sub]
     ring
 
 variable (hf' : (RingHom.ker (algebraMap P S)) ^ 2 = ⊥)
@@ -112,7 +112,7 @@ lemma retractionOfSectionOfKerSqZero_tmul_D (s : S) (t : P) :
 
 lemma retractionOfSectionOfKerSqZero_comp_kerToTensor :
     (retractionOfSectionOfKerSqZero g hf' hg).comp (kerToTensor R P S) = LinearMap.id := by
-  ext x; simp [(RingHom.mem_ker _).mp x.2]
+  ext x; simp [RingHom.mem_ker.mp x.2]
 
 end ofSection
 
@@ -149,12 +149,12 @@ def sectionOfRetractionKerToTensorAux : S →ₐ[R] P where
     have (x y) : (l x).1 * (l y).1 = 0 := by
       rw [← Ideal.mem_bot, ← hf', pow_two]; exact Ideal.mul_mem_mul (l x).2 (l y).2
     simp only [sectionOfRetractionKerToTensorAux_prop l hl (σ (a * b)) (σ a * σ b) (by simp [hσ]),
-      Derivation.leibniz, tmul_add, tmul_smul, map_add, map_smul, AddSubmonoid.coe_add, this,
-      Submodule.coe_toAddSubmonoid, SetLike.val_smul, smul_eq_mul, mul_sub, sub_mul, sub_zero]
+      Derivation.leibniz, tmul_add, tmul_smul, map_add, map_smul, Submodule.coe_add,
+      SetLike.val_smul, smul_eq_mul, mul_sub, sub_mul, this, sub_zero]
     ring
   map_add' a b := by
     simp only [sectionOfRetractionKerToTensorAux_prop l hl (σ (a + b)) (σ a + σ b) (by simp [hσ]),
-      map_add, tmul_add, AddSubmonoid.coe_add, Submodule.coe_toAddSubmonoid, add_sub_add_comm]
+      map_add, tmul_add, Submodule.coe_add, add_sub_add_comm]
   map_zero' := by simp [sectionOfRetractionKerToTensorAux_prop l hl (σ 0) 0 (by simp [hσ])]
   commutes' r := by
     simp [sectionOfRetractionKerToTensorAux_prop l hl
@@ -172,7 +172,7 @@ lemma toAlgHom_comp_sectionOfRetractionKerToTensorAux :
       (sectionOfRetractionKerToTensorAux l hl σ hσ hf') = AlgHom.id _ _ := by
   ext x
   obtain ⟨x, rfl⟩ := hf x
-  simp [sectionOfRetractionKerToTensorAux_algebraMap, (RingHom.mem_ker _).mp]
+  simp [sectionOfRetractionKerToTensorAux_algebraMap, RingHom.mem_ker.mp]
 
 /--
 Given a surjective algebra homomorphism `f : P →ₐ[R] S` with square-zero kernel `I`.
diff --git a/Mathlib/RingTheory/Smooth/StandardSmooth.lean b/Mathlib/RingTheory/Smooth/StandardSmooth.lean
index 33b0d6650afea..988a61b9c879e 100644
--- a/Mathlib/RingTheory/Smooth/StandardSmooth.lean
+++ b/Mathlib/RingTheory/Smooth/StandardSmooth.lean
@@ -62,12 +62,6 @@ Finally, for ring homomorphisms we define:
 
 ## TODO
 
-- Show that the canonical presentation for localization away from an element is standard smooth
-  of relative dimension 0.
-- Show that the base change of a submersive presentation is submersive of equal relative
-  dimension.
-- Show that the composition of submersive presentations of relative dimensions `n` and `m` is
-  submersive of relative dimension `n + m`.
 - Show that the module of Kaehler differentials of a standard smooth `R`-algebra `S` of relative
   dimension `n` is `S`-free of rank `n`. In particular this shows that the relative dimension
   is independent of the choice of the standard smooth presentation.
@@ -89,9 +83,9 @@ in June 2024.
 
 universe t t' w w' u v
 
-open TensorProduct
+open TensorProduct MvPolynomial Classical
 
-variable (n : ℕ)
+variable (n m : ℕ)
 
 namespace Algebra
 
@@ -163,10 +157,231 @@ lemma jacobian_eq_jacobiMatrix_det : P.jacobian = algebraMap P.Ring S P.jacobiMa
 
 lemma jacobiMatrix_apply (i j : P.rels) :
     P.jacobiMatrix i j = MvPolynomial.pderiv (P.map i) (P.relation j) := by
-  simp [jacobiMatrix, LinearMap.toMatrix, differential]
+  simp [jacobiMatrix, LinearMap.toMatrix, differential, basis]
 
 end Matrix
 
+section Constructions
+
+/-- If `algebraMap R S` is bijective, the empty generators are a pre-submersive
+presentation with no relations. -/
+noncomputable def ofBijectiveAlgebraMap (h : Function.Bijective (algebraMap R S)) :
+    PreSubmersivePresentation.{t, w} R S where
+  toPresentation := Presentation.ofBijectiveAlgebraMap.{t, w} h
+  map := PEmpty.elim
+  map_inj (a b : PEmpty) h := by contradiction
+  relations_finite := inferInstanceAs (Finite PEmpty.{t + 1})
+
+instance (h : Function.Bijective (algebraMap R S)) : Fintype (ofBijectiveAlgebraMap h).vars :=
+  inferInstanceAs (Fintype PEmpty)
+
+instance (h : Function.Bijective (algebraMap R S)) : Fintype (ofBijectiveAlgebraMap h).rels :=
+  inferInstanceAs (Fintype PEmpty)
+
+@[simp]
+lemma ofBijectiveAlgebraMap_jacobian (h : Function.Bijective (algebraMap R S)) :
+    (ofBijectiveAlgebraMap h).jacobian = 1 := by
+  have : (algebraMap (ofBijectiveAlgebraMap h).Ring S).mapMatrix
+      (ofBijectiveAlgebraMap h).jacobiMatrix = 1 := by
+    ext (i j : PEmpty)
+    contradiction
+  rw [jacobian_eq_jacobiMatrix_det, RingHom.map_det, this, Matrix.det_one]
+
+section Localization
+
+variable (r : R) [IsLocalization.Away r S]
+
+variable (S) in
+/-- If `S` is the localization of `R` at `r`, this is the canonical submersive presentation
+of `S` as `R`-algebra. -/
+@[simps map]
+noncomputable def localizationAway : PreSubmersivePresentation R S where
+  __ := Presentation.localizationAway S r
+  map _ := ()
+  map_inj _ _ h := h
+  relations_finite := inferInstanceAs <| Finite Unit
+
+instance : Fintype (localizationAway S r).rels :=
+  inferInstanceAs (Fintype Unit)
+
+instance : DecidableEq (localizationAway S r).rels :=
+  inferInstanceAs (DecidableEq Unit)
+
+@[simp]
+lemma localizationAway_jacobiMatrix :
+    (localizationAway S r).jacobiMatrix = Matrix.diagonal (fun () ↦ MvPolynomial.C r) := by
+  have h : (pderiv ()) (C r * X () - 1) = C r := by simp
+  ext (i : Unit) (j : Unit) : 1
+  rwa [jacobiMatrix_apply]
+
+@[simp]
+lemma localizationAway_jacobian : (localizationAway S r).jacobian = algebraMap R S r := by
+  rw [jacobian_eq_jacobiMatrix_det, localizationAway_jacobiMatrix]
+  simp [show Fintype.card (localizationAway r (S := S)).rels = 1 from rfl]
+
+end Localization
+
+section Composition
+
+variable {T} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
+variable (Q : PreSubmersivePresentation S T) (P : PreSubmersivePresentation R S)
+
+/-- Given an `R`-algebra `S` and an `S`-algebra `T` with pre-submersive presentations,
+this is the canonical pre-submersive presentation of `T` as an `R`-algebra. -/
+@[simps map]
+noncomputable def comp : PreSubmersivePresentation R T where
+  __ := Q.toPresentation.comp P.toPresentation
+  map := Sum.elim (fun rq ↦ Sum.inl <| Q.map rq) (fun rp ↦ Sum.inr <| P.map rp)
+  map_inj := Function.Injective.sum_elim ((Sum.inl_injective).comp (Q.map_inj))
+    ((Sum.inr_injective).comp (P.map_inj)) <| by simp
+  relations_finite := inferInstanceAs <| Finite (Q.rels ⊕ P.rels)
+
+/-- The dimension of the composition of two finite submersive presentations is
+the sum of the dimensions. -/
+lemma dimension_comp_eq_dimension_add_dimension [Q.IsFinite] [P.IsFinite] :
+    (Q.comp P).dimension = Q.dimension + P.dimension := by
+  simp only [Presentation.dimension]
+  erw [Presentation.comp_rels, Generators.comp_vars]
+  have : Nat.card P.rels ≤ Nat.card P.vars :=
+    card_relations_le_card_vars_of_isFinite P
+  have : Nat.card Q.rels ≤ Nat.card Q.vars :=
+    card_relations_le_card_vars_of_isFinite Q
+  simp only [Nat.card_sum]
+  omega
+
+section
+
+/-!
+### Jacobian of composition
+
+Let `S` be an `R`-algebra and `T` be an `S`-algebra with presentations `P` and `Q` respectively.
+In this section we compute the jacobian of the composition of `Q` and `P` to be
+the product of the jacobians. For this we use a block decomposition of the jacobi matrix and show
+that the upper-right block vanishes, the upper-left block has determinant jacobian of `Q` and
+the lower-right block has determinant jacobian of `P`.
+
+-/
+
+variable [Fintype (Q.comp P).rels]
+
+private lemma jacobiMatrix_comp_inl_inr (i : Q.rels) (j : P.rels) :
+    (Q.comp P).jacobiMatrix (Sum.inl i) (Sum.inr j) = 0 := by
+  rw [jacobiMatrix_apply]
+  refine MvPolynomial.pderiv_eq_zero_of_not_mem_vars (fun hmem ↦ ?_)
+  apply MvPolynomial.vars_rename at hmem
+  simp at hmem
+
+private lemma jacobiMatrix_comp_₁₂ : (Q.comp P).jacobiMatrix.toBlocks₁₂ = 0 := by
+  ext i j : 1
+  simp [Matrix.toBlocks₁₂, jacobiMatrix_comp_inl_inr]
+
+section Q
+
+variable [Fintype Q.rels]
+
+private lemma jacobiMatrix_comp_inl_inl (i j : Q.rels) :
+    aeval (Sum.elim X (MvPolynomial.C ∘ P.val))
+      ((Q.comp P).jacobiMatrix (Sum.inl j) (Sum.inl i)) = Q.jacobiMatrix j i := by
+  rw [jacobiMatrix_apply, jacobiMatrix_apply, comp_map, Sum.elim_inl,
+    ← Q.comp_aeval_relation_inl P.toPresentation]
+  apply aeval_sum_elim_pderiv_inl
+
+private lemma jacobiMatrix_comp_₁₁_det :
+    (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₁₁.det = Q.jacobian := by
+  rw [jacobian_eq_jacobiMatrix_det, AlgHom.map_det (aeval (Q.comp P).val), RingHom.map_det]
+  congr
+  ext i j : 1
+  simp only [Matrix.map_apply, RingHom.mapMatrix_apply, ← Q.jacobiMatrix_comp_inl_inl P]
+  apply aeval_sum_elim
+
+end Q
+
+section P
+
+variable [Fintype P.rels]
+
+private lemma jacobiMatrix_comp_inr_inr (i j : P.rels) :
+    (Q.comp P).jacobiMatrix (Sum.inr i) (Sum.inr j) =
+      MvPolynomial.rename Sum.inr (P.jacobiMatrix i j) := by
+  rw [jacobiMatrix_apply, jacobiMatrix_apply]
+  simp only [comp_map, Sum.elim_inr]
+  apply pderiv_rename Sum.inr_injective
+
+private lemma jacobiMatrix_comp_₂₂_det :
+    (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₂₂.det = algebraMap S T P.jacobian := by
+  rw [jacobian_eq_jacobiMatrix_det]
+  rw [AlgHom.map_det (aeval (Q.comp P).val), RingHom.map_det, RingHom.map_det]
+  congr
+  ext i j : 1
+  simp only [Matrix.toBlocks₂₂, AlgHom.mapMatrix_apply, Matrix.map_apply, Matrix.of_apply,
+    RingHom.mapMatrix_apply, Generators.algebraMap_apply, map_aeval, coe_eval₂Hom]
+  rw [jacobiMatrix_comp_inr_inr, ← IsScalarTower.algebraMap_eq]
+  simp only [aeval, AlgHom.coe_mk, coe_eval₂Hom]
+  generalize P.jacobiMatrix i j = p
+  induction' p using MvPolynomial.induction_on with a p q hp hq p i hp
+  · simp only [algHom_C, algebraMap_eq, eval₂_C]
+    erw [MvPolynomial.eval₂_C]
+  · simp [hp, hq]
+  · simp only [map_mul, rename_X, eval₂_mul, hp, eval₂_X]
+    erw [Generators.comp_val]
+    simp
+
+end P
+
+end
+
+/-- The jacobian of the composition of presentations is the product of the jacobians. -/
+@[simp]
+lemma comp_jacobian_eq_jacobian_smul_jacobian : (Q.comp P).jacobian = P.jacobian • Q.jacobian := by
+  cases nonempty_fintype Q.rels
+  cases nonempty_fintype P.rels
+  letI : Fintype (Q.comp P).rels := inferInstanceAs <| Fintype (Q.rels ⊕ P.rels)
+  rw [jacobian_eq_jacobiMatrix_det, ← Matrix.fromBlocks_toBlocks ((Q.comp P).jacobiMatrix),
+    jacobiMatrix_comp_₁₂]
+  convert_to
+    (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₁₁.det *
+    (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₂₂.det = P.jacobian • Q.jacobian
+  · simp only [Generators.algebraMap_apply, ← map_mul]
+    congr
+    convert Matrix.det_fromBlocks_zero₁₂ (Q.comp P).jacobiMatrix.toBlocks₁₁
+      (Q.comp P).jacobiMatrix.toBlocks₂₁ (Q.comp P).jacobiMatrix.toBlocks₂₂
+  · rw [jacobiMatrix_comp_₁₁_det, jacobiMatrix_comp_₂₂_det, mul_comm, Algebra.smul_def]
+
+end Composition
+
+section BaseChange
+
+variable (T) [CommRing T] [Algebra R T] (P : PreSubmersivePresentation R S)
+
+/-- If `P` is a pre-submersive presentation of `S` over `R` and `T` is an `R`-algebra, we
+obtain a natural pre-submersive presentation of `T ⊗[R] S` over `T`. -/
+noncomputable def baseChange : PreSubmersivePresentation T (T ⊗[R] S) where
+  __ := P.toPresentation.baseChange T
+  map := P.map
+  map_inj := P.map_inj
+  relations_finite := P.relations_finite
+
+@[simp]
+lemma baseChange_jacobian : (P.baseChange T).jacobian = 1 ⊗ₜ P.jacobian := by
+  classical
+  cases nonempty_fintype P.rels
+  letI : Fintype (P.baseChange T).rels := inferInstanceAs <| Fintype P.rels
+  simp_rw [jacobian_eq_jacobiMatrix_det]
+  have h : (baseChange T P).jacobiMatrix =
+      (MvPolynomial.map (algebraMap R T)).mapMatrix P.jacobiMatrix := by
+    ext i j : 1
+    simp only [baseChange, jacobiMatrix_apply, Presentation.baseChange_relation,
+      RingHom.mapMatrix_apply, Matrix.map_apply]
+    erw [MvPolynomial.pderiv_map]
+    rfl
+  rw [h]
+  erw [← RingHom.map_det, aeval_map_algebraMap]
+  apply aeval_one_tmul
+
+end BaseChange
+
+end Constructions
+
 end PreSubmersivePresentation
 
 /--
@@ -180,6 +395,75 @@ structure SubmersivePresentation extends PreSubmersivePresentation.{t, w} R S wh
 
 attribute [instance] SubmersivePresentation.isFinite
 
+namespace SubmersivePresentation
+
+open PreSubmersivePresentation
+
+section Constructions
+
+variable {R S} in
+/-- If `algebraMap R S` is bijective, the empty generators are a submersive
+presentation with no relations. -/
+noncomputable def ofBijectiveAlgebraMap (h : Function.Bijective (algebraMap R S)) :
+    SubmersivePresentation.{t, w} R S where
+  __ := PreSubmersivePresentation.ofBijectiveAlgebraMap.{t, w} h
+  jacobian_isUnit := by
+    rw [ofBijectiveAlgebraMap_jacobian]
+    exact isUnit_one
+  isFinite := Presentation.ofBijectiveAlgebraMap_isFinite h
+
+/-- The canonical submersive `R`-presentation of `R` with no generators and no relations. -/
+noncomputable def id : SubmersivePresentation.{t, w} R R :=
+  ofBijectiveAlgebraMap Function.bijective_id
+
+section Composition
+
+variable {R S T} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
+variable (Q : SubmersivePresentation S T) (P : SubmersivePresentation R S)
+
+/-- Given an `R`-algebra `S` and an `S`-algebra `T` with submersive presentations,
+this is the canonical submersive presentation of `T` as an `R`-algebra. -/
+noncomputable def comp : SubmersivePresentation R T where
+  __ := Q.toPreSubmersivePresentation.comp P.toPreSubmersivePresentation
+  jacobian_isUnit := by
+    rw [comp_jacobian_eq_jacobian_smul_jacobian, Algebra.smul_def, IsUnit.mul_iff]
+    exact ⟨RingHom.isUnit_map _ <| P.jacobian_isUnit, Q.jacobian_isUnit⟩
+  isFinite := Presentation.comp_isFinite Q.toPresentation P.toPresentation
+
+end Composition
+
+section Localization
+
+variable {R} (r : R) [IsLocalization.Away r S]
+
+/-- If `S` is the localization of `R` at `r`, this is the canonical submersive presentation
+of `S` as `R`-algebra. -/
+noncomputable def localizationAway : SubmersivePresentation R S where
+  __ := PreSubmersivePresentation.localizationAway S r
+  jacobian_isUnit := by
+    rw [localizationAway_jacobian]
+    apply IsLocalization.map_units' (⟨r, 1, by simp⟩ : Submonoid.powers r)
+  isFinite := Presentation.localizationAway_isFinite r
+
+end Localization
+
+section BaseChange
+
+variable (T) [CommRing T] [Algebra R T] (P : SubmersivePresentation R S)
+
+/-- If `P` is a submersive presentation of `S` over `R` and `T` is an `R`-algebra, we
+obtain a natural submersive presentation of `T ⊗[R] S` over `T`. -/
+noncomputable def baseChange : SubmersivePresentation T (T ⊗[R] S) where
+  toPreSubmersivePresentation := P.toPreSubmersivePresentation.baseChange T
+  jacobian_isUnit := P.baseChange_jacobian T ▸ P.jacobian_isUnit.map TensorProduct.includeRight
+  isFinite := Presentation.baseChange_isFinite T P.toPresentation
+
+end BaseChange
+
+end Constructions
+
+end SubmersivePresentation
+
 /--
 An `R`-algebra `S` is called standard smooth, if there
 exists a submersive presentation.
@@ -211,6 +495,66 @@ lemma IsStandardSmoothOfRelativeDimension.isStandardSmooth
     IsStandardSmooth.{t, w} R S :=
   ⟨‹IsStandardSmoothOfRelativeDimension n R S›.out.nonempty⟩
 
+lemma IsStandardSmoothOfRelativeDimension.of_algebraMap_bijective
+    (h : Function.Bijective (algebraMap R S)) :
+    IsStandardSmoothOfRelativeDimension.{t, w} 0 R S :=
+  ⟨SubmersivePresentation.ofBijectiveAlgebraMap h, Presentation.ofBijectiveAlgebraMap_dimension h⟩
+
+variable (R) in
+instance IsStandardSmoothOfRelativeDimension.id :
+    IsStandardSmoothOfRelativeDimension.{t, w} 0 R R :=
+  IsStandardSmoothOfRelativeDimension.of_algebraMap_bijective Function.bijective_id
+
+section Composition
+
+variable (R S T) [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T]
+
+lemma IsStandardSmooth.trans [IsStandardSmooth.{t, w} R S] [IsStandardSmooth.{t', w'} S T] :
+    IsStandardSmooth.{max t t', max w w'} R T where
+  out := by
+    obtain ⟨⟨P⟩⟩ := ‹IsStandardSmooth R S›
+    obtain ⟨⟨Q⟩⟩ := ‹IsStandardSmooth S T›
+    exact ⟨Q.comp P⟩
+
+lemma IsStandardSmoothOfRelativeDimension.trans [IsStandardSmoothOfRelativeDimension.{t, w} n R S]
+    [IsStandardSmoothOfRelativeDimension.{t', w'} m S T] :
+    IsStandardSmoothOfRelativeDimension.{max t t', max w w'} (m + n) R T where
+  out := by
+    obtain ⟨P, hP⟩ := ‹IsStandardSmoothOfRelativeDimension n R S›
+    obtain ⟨Q, hQ⟩ := ‹IsStandardSmoothOfRelativeDimension m S T›
+    refine ⟨Q.comp P, hP ▸ hQ ▸ ?_⟩
+    apply PreSubmersivePresentation.dimension_comp_eq_dimension_add_dimension
+
+end Composition
+
+lemma IsStandardSmooth.localization_away (r : R) [IsLocalization.Away r S] :
+    IsStandardSmooth.{0, 0} R S where
+  out := ⟨SubmersivePresentation.localizationAway S r⟩
+
+lemma IsStandardSmoothOfRelativeDimension.localization_away (r : R) [IsLocalization.Away r S] :
+    IsStandardSmoothOfRelativeDimension.{0, 0} 0 R S where
+  out := ⟨SubmersivePresentation.localizationAway S r,
+    Presentation.localizationAway_dimension_zero r⟩
+
+section BaseChange
+
+variable (T) [CommRing T] [Algebra R T]
+
+instance IsStandardSmooth.baseChange [IsStandardSmooth.{t, w} R S] :
+    IsStandardSmooth.{t, w} T (T ⊗[R] S) where
+  out := by
+    obtain ⟨⟨P⟩⟩ := ‹IsStandardSmooth R S›
+    exact ⟨P.baseChange R S T⟩
+
+instance IsStandardSmoothOfRelativeDimension.baseChange
+    [IsStandardSmoothOfRelativeDimension.{t, w} n R S] :
+    IsStandardSmoothOfRelativeDimension.{t, w} n T (T ⊗[R] S) where
+  out := by
+    obtain ⟨P, hP⟩ := ‹IsStandardSmoothOfRelativeDimension n R S›
+    exact ⟨P.baseChange R S T, hP⟩
+
+end BaseChange
+
 end Algebra
 
 namespace RingHom
diff --git a/Mathlib/RingTheory/SurjectiveOnStalks.lean b/Mathlib/RingTheory/SurjectiveOnStalks.lean
index 67f0451472874..94c5443047b71 100644
--- a/Mathlib/RingTheory/SurjectiveOnStalks.lean
+++ b/Mathlib/RingTheory/SurjectiveOnStalks.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
 import Mathlib.RingTheory.Localization.AtPrime
-import Mathlib.RingTheory.LocalRing.RingHom.Basic
 import Mathlib.RingTheory.TensorProduct.Basic
 
 /-!
diff --git a/Mathlib/RingTheory/TensorProduct/Basic.lean b/Mathlib/RingTheory/TensorProduct/Basic.lean
index 228b5be3855a4..3bddb995b0910 100644
--- a/Mathlib/RingTheory/TensorProduct/Basic.lean
+++ b/Mathlib/RingTheory/TensorProduct/Basic.lean
@@ -1,12 +1,13 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Johan Commelin
+Authors: Kim Morrison, Johan Commelin
 -/
-import Mathlib.LinearAlgebra.FiniteDimensional.Defs
-import Mathlib.LinearAlgebra.TensorProduct.Tower
+import Mathlib.GroupTheory.MonoidLocalization.Basic
+import Mathlib.LinearAlgebra.FreeModule.Basic
+import Mathlib.LinearAlgebra.Matrix.ToLin
 import Mathlib.RingTheory.Adjoin.Basic
-import Mathlib.LinearAlgebra.DirectSum.Finsupp
+import Mathlib.RingTheory.Finiteness
 
 /-!
 # The tensor product of R-algebras
@@ -34,6 +35,8 @@ multiplication is characterized by `(a₁ ⊗ₜ b₁) * (a₂ ⊗ₜ b₂) = (a
 
 -/
 
+assert_not_exists Equiv.Perm.cycleType
+
 suppress_compilation
 
 open scoped TensorProduct
@@ -779,7 +782,7 @@ Note that if `A` is commutative this can be instantiated with `S = A`.
 -/
 protected nonrec def rid : A ⊗[R] R ≃ₐ[S] A :=
   algEquivOfLinearEquivTensorProduct (AlgebraTensorModule.rid R S A)
-    (fun a₁ a₂ r₁ r₂ => smul_mul_smul r₁ r₂ a₁ a₂ |>.symm)
+    (fun a₁ a₂ r₁ r₂ => smul_mul_smul_comm r₁ a₁ r₂ a₂ |>.symm)
     (one_smul R _)
 
 @[simp] theorem rid_toLinearEquiv :
@@ -1213,11 +1216,6 @@ theorem Subalgebra.finite_sup {K L : Type*} [CommSemiring K] [CommSemiring L] [A
   rw [← E1.range_val, ← E2.range_val, ← Algebra.TensorProduct.productMap_range]
   exact Module.Finite.range (Algebra.TensorProduct.productMap E1.val E2.val).toLinearMap
 
-@[deprecated Subalgebra.finite_sup (since := "2024-04-11")]
-theorem Subalgebra.finiteDimensional_sup {K L : Type*} [Field K] [CommRing L] [Algebra K L]
-    (E1 E2 : Subalgebra K L) [FiniteDimensional K E1] [FiniteDimensional K E2] :
-    FiniteDimensional K (E1 ⊔ E2 : Subalgebra K L) := Subalgebra.finite_sup E1 E2
-
 namespace TensorProduct.Algebra
 
 variable {R A B M : Type*}
diff --git a/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean b/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean
index 10e82ca9edd87..5b8ae61882f73 100644
--- a/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean
+++ b/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean
@@ -229,6 +229,15 @@ lemma algebraTensorAlgEquiv_symm_monomial (m : σ →₀ ℕ) (a : A) :
     nth_rw 2 [← mul_one a]
     rw [Algebra.TensorProduct.tmul_mul_tmul]
 
+lemma aeval_one_tmul (f : σ → S) (p : MvPolynomial σ R) :
+    (aeval fun x ↦ (1 ⊗ₜ[R] f x : N ⊗[R] S)) p = 1 ⊗ₜ[R] (aeval f) p := by
+  induction' p using MvPolynomial.induction_on with a p q hp hq p i h
+  · simp only [map_C, algHom_C, Algebra.TensorProduct.algebraMap_apply,
+      RingHomCompTriple.comp_apply]
+    rw [← mul_one ((algebraMap R N) a), ← Algebra.smul_def, smul_tmul, Algebra.smul_def, mul_one]
+  · simp [hp, hq, tmul_add]
+  · simp [h]
+
 end Algebra
 
 end MvPolynomial
diff --git a/Mathlib/RingTheory/Trace/Basic.lean b/Mathlib/RingTheory/Trace/Basic.lean
index ee8a5a44c08f7..e06b0141ff923 100644
--- a/Mathlib/RingTheory/Trace/Basic.lean
+++ b/Mathlib/RingTheory/Trace/Basic.lean
@@ -5,7 +5,7 @@ Authors: Anne Baanen
 -/
 import Mathlib.RingTheory.Trace.Defs
 import Mathlib.LinearAlgebra.Determinant
-import Mathlib.FieldTheory.Galois
+import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
 import Mathlib.LinearAlgebra.Vandermonde
 import Mathlib.FieldTheory.Minpoly.MinpolyDiv
@@ -47,7 +47,7 @@ variable [Algebra R S] [Algebra R T]
 variable {K L : Type*} [Field K] [Field L] [Algebra K L]
 variable {ι κ : Type w} [Fintype ι]
 
-open FiniteDimensional
+open Module
 
 open LinearMap (BilinForm)
 open LinearMap
@@ -435,7 +435,7 @@ variable (K L)
 theorem traceForm_nondegenerate [FiniteDimensional K L] [Algebra.IsSeparable K L] :
     (traceForm K L).Nondegenerate :=
   BilinForm.nondegenerate_of_det_ne_zero (traceForm K L) _
-    (det_traceForm_ne_zero (FiniteDimensional.finBasis K L))
+    (det_traceForm_ne_zero (Module.finBasis K L))
 
 theorem Algebra.trace_ne_zero [FiniteDimensional K L] [Algebra.IsSeparable K L] :
     Algebra.trace K L ≠ 0 := by
diff --git a/Mathlib/RingTheory/Trace/Defs.lean b/Mathlib/RingTheory/Trace/Defs.lean
index 86218f0b3ecff..8d961bf7c82e6 100644
--- a/Mathlib/RingTheory/Trace/Defs.lean
+++ b/Mathlib/RingTheory/Trace/Defs.lean
@@ -48,7 +48,7 @@ variable {R S T : Type*} [CommRing R] [CommRing S] [CommRing T]
 variable [Algebra R S] [Algebra R T]
 variable {ι κ : Type w} [Fintype ι]
 
-open FiniteDimensional
+open Module
 
 open LinearMap (BilinForm)
 open LinearMap
diff --git a/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean b/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean
index 0358a0fe72767..dd7cee192929b 100644
--- a/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean
+++ b/Mathlib/RingTheory/TwoSidedIdeal/Basic.lean
@@ -45,6 +45,10 @@ section NonUnitalNonAssocRing
 
 variable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)
 
+instance [Nontrivial R] : Nontrivial (TwoSidedIdeal R) := by
+  obtain ⟨I, J, h⟩ : Nontrivial (RingCon R) := inferInstance
+  exact ⟨⟨I⟩, ⟨J⟩, by contrapose! h; aesop⟩
+
 instance setLike : SetLike (TwoSidedIdeal R) R where
   coe t := {r | t.ringCon r 0}
   coe_injective'  := by
@@ -152,17 +156,18 @@ def mk' (carrier : Set R)
         rw [show a + c - (b + d) = (a - b) + (c - d) by abel]
         exact add_mem h1 h2 }
 
-lemma mem_mk' (carrier : Set R)
-    (zero_mem : 0 ∈ carrier)
-    (add_mem : ∀ {x y}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier)
-    (neg_mem : ∀ {x}, x ∈ carrier → -x ∈ carrier)
-    (mul_mem_left : ∀ {x y}, y ∈ carrier → x * y ∈ carrier)
-    (mul_mem_right : ∀ {x y}, x ∈ carrier → x * y ∈ carrier)
-    (x : R) :
+@[simp]
+lemma mem_mk' (carrier : Set R) (zero_mem add_mem neg_mem mul_mem_left mul_mem_right) (x : R) :
     x ∈ mk' carrier zero_mem add_mem neg_mem mul_mem_left mul_mem_right ↔ x ∈ carrier := by
   rw [mem_iff]
   simp [mk']
 
+set_option linter.docPrime false in
+@[simp]
+lemma coe_mk' (carrier : Set R) (zero_mem add_mem neg_mem mul_mem_left mul_mem_right) :
+    (mk' carrier zero_mem add_mem neg_mem mul_mem_left mul_mem_right : Set R) = carrier :=
+  Set.ext <| mem_mk' carrier zero_mem add_mem neg_mem mul_mem_left mul_mem_right
+
 instance : SMulMemClass (TwoSidedIdeal R) R R where
   smul_mem _ _ h := TwoSidedIdeal.mul_mem_left _ _ _ h
 
diff --git a/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean b/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean
new file mode 100644
index 0000000000000..8ed5a222e9ed2
--- /dev/null
+++ b/Mathlib/RingTheory/TwoSidedIdeal/BigOperators.lean
@@ -0,0 +1,74 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang
+-/
+
+import Mathlib.RingTheory.Congruence.BigOperators
+import Mathlib.RingTheory.TwoSidedIdeal.Basic
+
+/-!
+# Interactions between `∑, ∏` and two sided ideals
+
+-/
+
+namespace TwoSidedIdeal
+
+section sum
+
+variable {R : Type*} [NonUnitalNonAssocRing R] (I : TwoSidedIdeal R)
+
+lemma listSum_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∀ x ∈ l, f x ∈ I) :
+    (l.map f).sum ∈ I := by
+  rw [mem_iff, ← List.sum_map_zero]
+  exact I.ringCon.listSum l hl
+
+lemma multisetSum_mem {ι : Type*} (s : Multiset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :
+    (s.map f).sum ∈ I := by
+  rw [mem_iff, ← Multiset.sum_map_zero]
+  exact I.ringCon.multisetSum s hs
+
+lemma finsetSum_mem {ι : Type*} (s : Finset ι) (f : ι → R) (hs : ∀ x ∈ s, f x ∈ I) :
+    s.sum f ∈ I := by
+  rw [mem_iff, ← Finset.sum_const_zero]
+  exact I.ringCon.finsetSum s hs
+
+end sum
+
+section prod
+
+section ring
+
+variable {R : Type*} [Ring R] (I : TwoSidedIdeal R)
+
+lemma listProd_mem {ι : Type*} (l : List ι) (f : ι → R) (hl : ∃ x ∈ l, f x ∈ I) :
+    (l.map f).prod ∈ I := by
+  induction l with
+  | nil => simp only [List.not_mem_nil, false_and, exists_false] at hl
+  | cons x l ih =>
+    simp only [List.mem_cons, exists_eq_or_imp] at hl
+    rcases hl with h | hal
+    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_right _ _ h
+    · simpa only [List.map_cons, List.prod_cons] using I.mul_mem_left _ _ <| ih hal
+
+end ring
+
+section commRing
+
+variable {R : Type*} [CommRing R] (I : TwoSidedIdeal R)
+
+lemma multiSetProd_mem {ι : Type*} (s : Multiset ι) (f : ι → R) (hs : ∃ x ∈ s, f x ∈ I) :
+    (s.map f).prod ∈ I := by
+  rcases s
+  simpa using listProd_mem (hl := hs)
+
+lemma finsetProd_mem {ι : Type*} (s : Finset ι) (f : ι → R) (hs : ∃ x ∈ s, f x ∈ I) :
+    s.prod f ∈ I := by
+  rcases s
+  simpa using multiSetProd_mem (hs := hs)
+
+end commRing
+
+end prod
+
+end TwoSidedIdeal
diff --git a/Mathlib/RingTheory/TwoSidedIdeal/Instances.lean b/Mathlib/RingTheory/TwoSidedIdeal/Instances.lean
new file mode 100644
index 0000000000000..dc62e64d97bc8
--- /dev/null
+++ b/Mathlib/RingTheory/TwoSidedIdeal/Instances.lean
@@ -0,0 +1,14 @@
+/-
+Copyright (c) 2024. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: euprunin
+-/
+import Mathlib.Algebra.Ring.Defs
+import Mathlib.RingTheory.NonUnitalSubring.Basic
+import Mathlib.RingTheory.TwoSidedIdeal.Basic
+
+/-!
+# Additional instances for two sided ideals.
+-/
+instance {R} [NonUnitalNonAssocRing R] : NonUnitalSubringClass (TwoSidedIdeal R) R where
+  mul_mem _ hb := TwoSidedIdeal.mul_mem_left _ _ _ hb
diff --git a/Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean b/Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean
index 9d596c74f10e6..55d3b2a16a8c7 100644
--- a/Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean
+++ b/Mathlib/RingTheory/TwoSidedIdeal/Lattice.lean
@@ -113,6 +113,9 @@ instance : Top (TwoSidedIdeal R) where
 
 lemma top_ringCon : (⊤ : TwoSidedIdeal R).ringCon = ⊤ := rfl
 
+@[simp]
+lemma mem_top {x : R} : x ∈ (⊤: TwoSidedIdeal R) := trivial
+
 instance : Bot (TwoSidedIdeal R) where
   bot := { ringCon := ⊥ }
 
@@ -129,4 +132,10 @@ instance : CompleteLattice (TwoSidedIdeal R) where
   le_top _ := by rw [ringCon_le_iff]; exact le_top
   bot_le _ := by rw [ringCon_le_iff]; exact bot_le
 
+lemma one_mem_iff {R : Type*} [NonAssocRing R] (I : TwoSidedIdeal R) :
+    (1 : R) ∈ I ↔ I = ⊤ :=
+  ⟨fun h => eq_top_iff.2 fun x _ => by simpa using I.mul_mem_left x _ h, fun h ↦ h.symm ▸ trivial⟩
+
+alias ⟨eq_top, one_mem⟩ := one_mem_iff
+
 end TwoSidedIdeal
diff --git a/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean b/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean
new file mode 100644
index 0000000000000..78ccd01d0a217
--- /dev/null
+++ b/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean
@@ -0,0 +1,300 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang, Jireh Loreaux
+-/
+
+import Mathlib.RingTheory.TwoSidedIdeal.Lattice
+import Mathlib.RingTheory.Congruence.Opposite
+import Mathlib.Algebra.BigOperators.Ring
+import Mathlib.Data.Fintype.BigOperators
+import Mathlib.RingTheory.Ideal.Basic
+import Mathlib.Order.GaloisConnection
+
+/-!
+# Operations on two-sided ideals
+
+This file defines operations on two-sided ideals of a ring `R`.
+
+## Main definitions and results
+
+- `TwoSidedIdeal.span`: the span of `s ⊆ R` is the smallest two-sided ideal containing the set.
+- `TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure_nonunital`: in an associative but non-unital
+  ring, an element `x` is in the two-sided ideal spanned by `s` if and only if `x` is in the closure
+  of `s ∪ {y * a | y ∈ s, a ∈ R} ∪ {a * y | y ∈ s, a ∈ R} ∪ {a * y * b | y ∈ s, a, b ∈ R}` as an
+  additive subgroup.
+- `TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure`: in a unital and associative ring, an
+  element  `x` is in the two-sided ideal spanned by `s` if and only if `x` is in the closure of
+  `{a*y*b | a, b ∈ R, y ∈ s}` as an additive subgroup.
+
+
+- `TwoSidedIdeal.comap`: pullback of a two-sided ideal; defined as the preimage of a
+  two-sided ideal.
+- `TwoSidedIdeal.map`: pushforward of a two-sided ideal; defined as the span of the image of a
+  two-sided ideal.
+- `TwoSidedIdeal.ker`: the kernel of a ring homomorphism as a two-sided ideal.
+
+- `TwoSidedIdeal.gc`: `fromIdeal` and `asIdeal` form a Galois connection where
+  `fromIdeal : Ideal R → TwoSidedIdeal R` is defined as the smallest two-sided ideal containing an
+  ideal and `asIdeal : TwoSidedIdeal R → Ideal R` the inclusion map.
+-/
+
+namespace TwoSidedIdeal
+
+section NonUnitalNonAssocRing
+
+variable {R S : Type*} [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S]
+variable {F : Type*} [FunLike F R S]
+variable (f : F)
+
+/--
+The smallest two-sided ideal containing a set.
+-/
+abbrev span (s : Set R) : TwoSidedIdeal R :=
+  { ringCon := ringConGen (fun a b ↦ a - b ∈ s) }
+
+lemma subset_span {s : Set R} : s ⊆ (span s : Set R) := by
+  intro x hx
+  rw [SetLike.mem_coe, mem_iff]
+  exact RingConGen.Rel.of _ _ (by simpa using hx)
+
+lemma mem_span_iff {s : Set R} {x} :
+    x ∈ span s ↔ ∀ (I : TwoSidedIdeal R), s ⊆ I → x ∈ I := by
+  refine ⟨?_, fun h => h _ subset_span⟩
+  delta span
+  rw [RingCon.ringConGen_eq]
+  intro h I hI
+  refine sInf_le (α := RingCon R) ?_ h
+  intro x y hxy
+  specialize hI hxy
+  rwa [SetLike.mem_coe, ← rel_iff] at hI
+
+lemma span_mono {s t : Set R} (h : s ⊆ t) : span s ≤ span t := by
+  intro x hx
+  rw [mem_span_iff] at hx ⊢
+  exact fun I hI => hx I <| h.trans hI
+
+/--
+Pushout of a two-sided ideal. Defined as the span of the image of a two-sided ideal under a ring
+homomorphism.
+-/
+def map (I : TwoSidedIdeal R) : TwoSidedIdeal S :=
+  span (f '' I)
+
+lemma map_mono {I J : TwoSidedIdeal R} (h : I ≤ J) :
+    map f I ≤ map f J :=
+  span_mono <| Set.image_mono h
+
+variable [NonUnitalRingHomClass F R S]
+
+/--
+Preimage of a two-sided ideal, as a two-sided ideal. -/
+def comap (I : TwoSidedIdeal S) : TwoSidedIdeal R :=
+{ ringCon := I.ringCon.comap f }
+
+lemma mem_comap {I : TwoSidedIdeal S} {x : R} :
+    x ∈ I.comap f ↔ f x ∈ I := by
+  simp [comap, RingCon.comap, mem_iff]
+
+/--
+The kernel of a ring homomorphism, as a two-sided ideal.
+-/
+def ker : TwoSidedIdeal R :=
+  .mk'
+    {r | f r = 0} (map_zero _) (by rintro _ _ (h1 : f _ = 0) (h2 : f _ = 0); simp [h1, h2])
+    (by rintro _ (h : f _ = 0); simp [h]) (by rintro _ _ (h : f _ = 0); simp [h])
+    (by rintro _ _ (h : f _ = 0); simp [h])
+
+lemma mem_ker {x : R} : x ∈ ker f ↔ f x = 0 := by
+  delta ker; rw [mem_mk']; rfl
+
+end NonUnitalNonAssocRing
+
+section NonUnitalRing
+
+variable {R : Type*} [NonUnitalRing R]
+
+open AddSubgroup in
+/-- If `s : Set R` is absorbing under multiplication, then its `TwoSidedIdeal.span` coincides with
+its `AddSubgroup.closure`, as sets. -/
+lemma mem_span_iff_mem_addSubgroup_closure_absorbing {s : Set R}
+    (h_left : ∀ x y, y ∈ s → x * y ∈ s) (h_right : ∀ y x, y ∈ s → y * x ∈ s) {z : R} :
+    z ∈ span s ↔ z ∈ closure s := by
+  have h_left' {x y} (hy : y ∈ closure s) : x * y ∈ closure s := by
+    have := (AddMonoidHom.mulLeft x).map_closure s ▸ mem_map_of_mem _ hy
+    refine closure_mono ?_ this
+    rintro - ⟨y, hy, rfl⟩
+    exact h_left x y hy
+  have h_right' {y x} (hy : y ∈ closure s) : y * x ∈ closure s := by
+    have := (AddMonoidHom.mulRight x).map_closure s ▸ mem_map_of_mem _ hy
+    refine closure_mono ?_ this
+    rintro - ⟨y, hy, rfl⟩
+    exact h_right y x hy
+  let I : TwoSidedIdeal R := .mk' (closure s) (AddSubgroup.zero_mem _)
+    (AddSubgroup.add_mem _) (AddSubgroup.neg_mem _) h_left' h_right'
+  suffices z ∈ span s ↔ z ∈ I by simpa only [I, mem_mk', SetLike.mem_coe]
+  rw [mem_span_iff]
+  -- Suppose that for every ideal `J` with `s ⊆ J`, then `z ∈ J`. Apply this to `I` to get `z ∈ I`.
+  refine ⟨fun h ↦ h I fun x hx ↦ ?mem_closure_of_forall, fun hz J hJ ↦ ?mem_ideal_of_subset⟩
+  case mem_closure_of_forall => simpa only [I, SetLike.mem_coe, mem_mk'] using subset_closure hx
+  /- Conversely, suppose that `z ∈ I` and that `J` is any ideal containing `s`. Then by the
+  induction principle for `AddSubgroup`, we must also have `z ∈ J`. -/
+  case mem_ideal_of_subset =>
+    simp only [I, SetLike.mem_coe, mem_mk'] at hz
+    induction hz using closure_induction' with
+    | mem x hx => exact hJ hx
+    | one => exact zero_mem _
+    | mul x _ y _ hx hy => exact J.add_mem hx hy
+    | inv x _ hx => exact J.neg_mem hx
+
+open Pointwise Set
+
+lemma set_mul_subset {s : Set R} {I : TwoSidedIdeal R} (h : s ⊆ I) (t : Set R):
+    t * s ⊆ I := by
+  rintro - ⟨r, -, x, hx, rfl⟩
+  exact mul_mem_left _ _ _ (h hx)
+
+lemma subset_mul_set {s : Set R} {I : TwoSidedIdeal R} (h : s ⊆ I) (t : Set R):
+    s * t ⊆ I := by
+  rintro - ⟨x, hx, r, -, rfl⟩
+  exact mul_mem_right _ _ _ (h hx)
+
+lemma mem_span_iff_mem_addSubgroup_closure_nonunital {s : Set R} {z : R} :
+    z ∈ span s ↔ z ∈ AddSubgroup.closure (s ∪ s * univ ∪ univ * s ∪ univ * s * univ) := by
+  trans z ∈ span (s ∪ s * univ ∪ univ * s ∪ univ * s * univ)
+  · refine ⟨(span_mono (by simp only [Set.union_assoc, Set.subset_union_left]) ·), fun h ↦ ?_⟩
+    refine mem_span_iff.mp h (span s) ?_
+    simp only [union_subset_iff, union_assoc]
+    exact ⟨subset_span, subset_mul_set subset_span _, set_mul_subset subset_span _,
+      subset_mul_set (set_mul_subset subset_span _) _⟩
+  · refine mem_span_iff_mem_addSubgroup_closure_absorbing ?_ ?_
+    · rintro x y (((hy | ⟨y, hy, r, -, rfl⟩) | ⟨r, -, y, hy, rfl⟩) |
+        ⟨-, ⟨r', -, y, hy, rfl⟩, r, -, rfl⟩)
+      · exact .inl <| .inr <| ⟨x, mem_univ _, y, hy, rfl⟩
+      · exact .inr <| ⟨x * y, ⟨x, mem_univ _, y, hy, rfl⟩, r, mem_univ _, mul_assoc ..⟩
+      · exact .inl <| .inr <| ⟨x * r, mem_univ _, y, hy, mul_assoc ..⟩
+      · refine .inr <| ⟨x * r' * y, ⟨x * r', mem_univ _, y, hy, ?_⟩, ⟨r, mem_univ _, ?_⟩⟩
+        all_goals simp [mul_assoc]
+    · rintro y x (((hy | ⟨y, hy, r, -, rfl⟩) | ⟨r, -, y, hy, rfl⟩) |
+        ⟨-, ⟨r', -, y, hy, rfl⟩, r, -, rfl⟩)
+      · exact .inl <| .inl <| .inr ⟨y, hy, x, mem_univ _, rfl⟩
+      · exact .inl <| .inl <| .inr ⟨y, hy, r * x, mem_univ _, (mul_assoc ..).symm⟩
+      · exact .inr <| ⟨r * y, ⟨r, mem_univ _, y, hy, rfl⟩, x, mem_univ _, rfl⟩
+      · refine .inr <| ⟨r' * y, ⟨r', mem_univ _, y, hy, rfl⟩, r * x, mem_univ _, ?_⟩
+        simp [mul_assoc]
+
+end NonUnitalRing
+
+section Ring
+
+variable {R : Type*} [Ring R]
+
+open Pointwise Set in
+lemma mem_span_iff_mem_addSubgroup_closure {s : Set R} {z : R} :
+    z ∈ span s ↔ z ∈ AddSubgroup.closure (univ * s * univ) := by
+  trans z ∈ span (univ * s * univ)
+  · refine ⟨(span_mono (fun x hx ↦ ?_) ·), fun hz ↦ ?_⟩
+    · exact ⟨1 * x, ⟨1, mem_univ _, x, hx, rfl⟩, 1, mem_univ _, by simp⟩
+    · exact mem_span_iff.mp hz (span s) <| subset_mul_set (set_mul_subset subset_span _) _
+  · refine mem_span_iff_mem_addSubgroup_closure_absorbing ?_ ?_
+    · intro x y hy
+      rw [mul_assoc] at hy ⊢
+      obtain ⟨r, -, y, hy, rfl⟩ := hy
+      exact ⟨x * r, mem_univ _, y, hy, mul_assoc ..⟩
+    · rintro - x ⟨y, hy, r, -, rfl⟩
+      exact ⟨y, hy, r * x, mem_univ _, (mul_assoc ..).symm⟩
+
+/-- Given an ideal `I`, `span I` is the smallest two-sided ideal containing `I`. -/
+def fromIdeal : Ideal R →o TwoSidedIdeal R where
+  toFun I := span I
+  monotone' _ _ := span_mono
+
+lemma mem_fromIdeal {I : Ideal R} {x : R} :
+    x ∈ fromIdeal I ↔ x ∈ span I := by simp [fromIdeal]
+
+/-- Every two-sided ideal is also a left ideal. -/
+def asIdeal : TwoSidedIdeal R →o Ideal R where
+  toFun I :=
+  { carrier := I
+    add_mem' := I.add_mem
+    zero_mem' := I.zero_mem
+    smul_mem' := fun r x hx => I.mul_mem_left r x hx }
+  monotone' _ _ h _ h' := h h'
+
+@[simp]
+lemma mem_asIdeal {I : TwoSidedIdeal R} {x : R} :
+    x ∈ asIdeal I ↔ x ∈ I := by simp [asIdeal]
+
+lemma gc : GaloisConnection fromIdeal (asIdeal (R := R)) :=
+  fun I J => ⟨fun h x hx ↦ h <| mem_span_iff.2 fun _ H ↦ H hx, fun h x hx ↦ by
+    simp only [fromIdeal, OrderHom.coe_mk, mem_span_iff] at hx
+    exact hx _ h⟩
+
+@[simp]
+lemma coe_asIdeal {I : TwoSidedIdeal R} : (asIdeal I : Set R) = I := rfl
+
+/-- Every two-sided ideal is also a right ideal. -/
+def asIdealOpposite : TwoSidedIdeal R →o Ideal Rᵐᵒᵖ where
+  toFun I := asIdeal ⟨I.ringCon.op⟩
+  monotone' I J h x h' := by
+    simp only [mem_asIdeal, mem_iff, RingCon.op_iff, MulOpposite.unop_zero] at h' ⊢
+    exact J.rel_iff _ _ |>.2 <| h <| I.rel_iff 0 x.unop |>.1 h'
+
+lemma mem_asIdealOpposite {I : TwoSidedIdeal R} {x : Rᵐᵒᵖ} :
+    x ∈ asIdealOpposite I ↔ x.unop ∈ I := by
+  simpa [asIdealOpposite, asIdeal, TwoSidedIdeal.mem_iff, RingCon.op_iff] using
+    ⟨I.ringCon.symm, I.ringCon.symm⟩
+
+end Ring
+
+section CommRing
+
+variable {R : Type*} [CommRing R]
+
+/--
+When the ring is commutative, two-sided ideals are exactly the same as left ideals.
+-/
+def orderIsoIdeal : TwoSidedIdeal R ≃o Ideal R where
+  toFun := asIdeal
+  invFun := fromIdeal
+  map_rel_iff' := ⟨fun h _ hx ↦ h hx, fun h ↦ asIdeal.monotone' h⟩
+  left_inv _ := SetLike.ext fun _ ↦ mem_span_iff.trans <| by aesop
+  right_inv J := SetLike.ext fun x ↦ mem_span_iff.trans
+    ⟨fun h ↦ mem_mk' _ _ _ _ _ _ _ |>.1 <| h (mk'
+      J J.zero_mem J.add_mem J.neg_mem (J.mul_mem_left _) (J.mul_mem_right _))
+      (fun x => by simp), by aesop⟩
+
+end CommRing
+
+end TwoSidedIdeal
+
+namespace Ideal
+variable {R : Type*} [Ring R]
+
+/-- Bundle an `Ideal` that is already two-sided as a `TwoSidedIdeal`. -/
+def toTwoSided (I : Ideal R) (mul_mem_right : ∀ {x y}, x ∈ I → x * y ∈ I) : TwoSidedIdeal R :=
+  TwoSidedIdeal.mk' I I.zero_mem I.add_mem I.neg_mem (I.smul_mem _) mul_mem_right
+
+@[simp]
+lemma mem_toTwoSided {I : Ideal R} {h} {x : R} :
+    x ∈ I.toTwoSided h ↔ x ∈ I := by
+  simp [toTwoSided]
+
+@[simp]
+lemma coe_toTwoSided (I : Ideal R) (h) : (I.toTwoSided h : Set R) = I := by
+  simp [toTwoSided]
+
+@[simp]
+lemma toTwoSided_asIdeal (I : TwoSidedIdeal R) (h) : (TwoSidedIdeal.asIdeal I).toTwoSided h = I :=
+  by ext; simp
+
+@[simp]
+lemma asIdeal_toTwoSided (I : Ideal R) (h) : TwoSidedIdeal.asIdeal (I.toTwoSided h) = I := by
+  ext
+  simp
+
+instance : CanLift (Ideal R) (TwoSidedIdeal R) TwoSidedIdeal.asIdeal
+    (fun I => ∀ {x y}, x ∈ I → x * y ∈ I) where
+  prf I mul_mem_right := ⟨I.toTwoSided mul_mem_right, asIdeal_toTwoSided ..⟩
+
+end Ideal
diff --git a/Mathlib/RingTheory/UniqueFactorizationDomain.lean b/Mathlib/RingTheory/UniqueFactorizationDomain.lean
index 6c8cde46f8b98..ee5e511abe687 100644
--- a/Mathlib/RingTheory/UniqueFactorizationDomain.lean
+++ b/Mathlib/RingTheory/UniqueFactorizationDomain.lean
@@ -34,10 +34,12 @@ local infixl:50 " ~ᵤ " => Associated
 condition on divisibility and to the ascending chain condition on
 principal ideals in an integral domain.
   -/
-class WfDvdMonoid (α : Type*) [CommMonoidWithZero α] : Prop where
-  wellFounded_dvdNotUnit : WellFounded (@DvdNotUnit α _)
+abbrev WfDvdMonoid (α : Type*) [CommMonoidWithZero α] : Prop :=
+  IsWellFounded α DvdNotUnit
 
-export WfDvdMonoid (wellFounded_dvdNotUnit)
+theorem wellFounded_dvdNotUnit {α : Type*} [CommMonoidWithZero α] [h : WfDvdMonoid α] :
+    WellFounded (DvdNotUnit (α := α)) :=
+  h.wf
 
 -- see Note [lower instance priority]
 instance (priority := 100) IsNoetherianRing.wfDvdMonoid [CommRing α] [IsDomain α]
@@ -61,6 +63,10 @@ variable [WfDvdMonoid α]
 instance wfDvdMonoid_associates : WfDvdMonoid (Associates α) :=
   ⟨(mk_surjective.wellFounded_iff mk_dvdNotUnit_mk_iff.symm).1 wellFounded_dvdNotUnit⟩
 
+theorem wellFoundedLT_associates : WellFoundedLT (Associates α) :=
+  ⟨Subrelation.wf dvdNotUnit_of_lt wellFounded_dvdNotUnit⟩
+
+@[deprecated wellFoundedLT_associates (since := "2024-09-02")]
 theorem wellFounded_associates : WellFounded ((· < ·) : Associates α → Associates α → Prop) :=
   Subrelation.wf dvdNotUnit_of_lt wellFounded_dvdNotUnit
 
@@ -124,6 +130,15 @@ theorem isRelPrime_of_no_irreducible_factors {x y : α} (nonzero : ¬(x = 0 ∧
 
 end WfDvdMonoid
 
+theorem WfDvdMonoid.of_wellFoundedLT_associates [CancelCommMonoidWithZero α]
+    (h : WellFoundedLT (Associates α)) : WfDvdMonoid α :=
+  WfDvdMonoid.of_wfDvdMonoid_associates
+    ⟨by
+      convert h.wf
+      ext
+      exact Associates.dvdNotUnit_iff_lt⟩
+
+@[deprecated WfDvdMonoid.of_wellFoundedLT_associates (since := "2024-09-02")]
 theorem WfDvdMonoid.of_wellFounded_associates [CancelCommMonoidWithZero α]
     (h : WellFounded ((· < ·) : Associates α → Associates α → Prop)) : WfDvdMonoid α :=
   WfDvdMonoid.of_wfDvdMonoid_associates
@@ -133,8 +148,8 @@ theorem WfDvdMonoid.of_wellFounded_associates [CancelCommMonoidWithZero α]
       exact Associates.dvdNotUnit_iff_lt⟩
 
 theorem WfDvdMonoid.iff_wellFounded_associates [CancelCommMonoidWithZero α] :
-    WfDvdMonoid α ↔ WellFounded ((· < ·) : Associates α → Associates α → Prop) :=
-  ⟨by apply WfDvdMonoid.wellFounded_associates, WfDvdMonoid.of_wellFounded_associates⟩
+    WfDvdMonoid α ↔ WellFoundedLT (Associates α) :=
+  ⟨by apply WfDvdMonoid.wellFoundedLT_associates, WfDvdMonoid.of_wellFoundedLT_associates⟩
 
 theorem WfDvdMonoid.max_power_factor' [CommMonoidWithZero α] [WfDvdMonoid α] {a₀ x : α}
     (h : a₀ ≠ 0) (hx : ¬IsUnit x) : ∃ (n : ℕ) (a : α), ¬x ∣ a ∧ a₀ = x ^ n * a := by
@@ -176,8 +191,8 @@ To define a UFD using the definition in terms of multisets
 of prime factors, use the definition `of_exists_prime_factors`
 
 -/
-class UniqueFactorizationMonoid (α : Type*) [CancelCommMonoidWithZero α] extends WfDvdMonoid α :
-  Prop where
+class UniqueFactorizationMonoid (α : Type*) [CancelCommMonoidWithZero α] extends
+    IsWellFounded α DvdNotUnit : Prop where
   protected irreducible_iff_prime : ∀ {a : α}, Irreducible a ↔ Prime a
 
 /-- Can't be an instance because it would cause a loop `ufm → WfDvdMonoid → ufm → ...`. -/
@@ -308,8 +323,7 @@ theorem WfDvdMonoid.of_exists_prime_factors : WfDvdMonoid α :=
       rw [dif_neg ane0]
       by_cases h : b = 0
       · simp [h, lt_top_iff_ne_top]
-      · rw [dif_neg h]
-        erw [WithTop.coe_lt_coe]
+      · rw [dif_neg h, Nat.cast_lt]
         have cne0 : c ≠ 0 := by
           refine mt (fun con => ?_) h
           rw [b_eq, con, mul_zero]
@@ -375,8 +389,8 @@ theorem MulEquiv.uniqueFactorizationMonoid (e : α ≃* β) (hα : UniqueFactori
         he ▸ e.prime_iff.1 (hp c hc),
         Units.map e.toMonoidHom u,
       by
-        erw [Multiset.prod_hom, ← map_mul e, h]
-        simp⟩
+        rw [Multiset.prod_hom, toMonoidHom_eq_coe, Units.coe_map, MonoidHom.coe_coe, ← map_mul e, h,
+          apply_symm_apply]⟩
 
 theorem MulEquiv.uniqueFactorizationMonoid_iff (e : α ≃* β) :
     UniqueFactorizationMonoid α ↔ UniqueFactorizationMonoid β :=
@@ -555,7 +569,7 @@ noncomputable def normalizedFactors (a : α) : Multiset α :=
 if `M` has a trivial group of units. -/
 @[simp]
 theorem factors_eq_normalizedFactors {M : Type*} [CancelCommMonoidWithZero M]
-    [UniqueFactorizationMonoid M] [Unique Mˣ] (x : M) : factors x = normalizedFactors x := by
+    [UniqueFactorizationMonoid M] [Subsingleton Mˣ] (x : M) : factors x = normalizedFactors x := by
   unfold normalizedFactors
   convert (Multiset.map_id (factors x)).symm
   ext p
@@ -728,7 +742,7 @@ theorem dvd_of_mem_normalizedFactors {a p : α} (H : p ∈ normalizedFactors a)
     exact dvd_zero p
   · exact dvd_trans (Multiset.dvd_prod H) (Associated.dvd (normalizedFactors_prod hcases))
 
-theorem mem_normalizedFactors_iff [Unique αˣ] {p x : α} (hx : x ≠ 0) :
+theorem mem_normalizedFactors_iff [Subsingleton αˣ] {p x : α} (hx : x ≠ 0) :
     p ∈ normalizedFactors x ↔ Prime p ∧ p ∣ x := by
   constructor
   · intro h
@@ -744,11 +758,16 @@ theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α}
   have := UniqueFactorizationMonoid.normalizedFactors_prod hr
   rwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this
 
-theorem normalizedFactors_prod_of_prime [Nontrivial α] [Unique αˣ] {m : Multiset α}
+theorem normalizedFactors_prod_of_prime [Subsingleton αˣ] {m : Multiset α}
     (h : ∀ p ∈ m, Prime p) : normalizedFactors m.prod = m := by
-  simpa only [← Multiset.rel_eq, ← associated_eq_eq] using
-    prime_factors_unique prime_of_normalized_factor h
-      (normalizedFactors_prod (m.prod_ne_zero_of_prime h))
+  cases subsingleton_or_nontrivial α
+  · obtain rfl : m = 0 := by
+      refine Multiset.eq_zero_of_forall_not_mem fun x hx ↦ ?_
+      simpa [Subsingleton.elim x 0] using h x hx
+    simp
+  · simpa only [← Multiset.rel_eq, ← associated_eq_eq] using
+      prime_factors_unique prime_of_normalized_factor h
+        (normalizedFactors_prod (m.prod_ne_zero_of_prime h))
 
 theorem mem_normalizedFactors_eq_of_associated {a b c : α} (ha : a ∈ normalizedFactors c)
     (hb : b ∈ normalizedFactors c) (h : Associated a b) : a = b := by
@@ -897,20 +916,8 @@ theorem exists_reduced_factors' (a b : R) (hb : b ≠ 0) :
   let ⟨b', a', c', no_factor, hb, ha⟩ := exists_reduced_factors b hb a
   ⟨a', b', c', fun _ hpb hpa => no_factor hpa hpb, ha, hb⟩
 
-theorem pow_right_injective {a : R} (ha0 : a ≠ 0) (ha1 : ¬IsUnit a) :
-    Function.Injective (a ^ · : ℕ → R) := by
-  letI := Classical.decEq R
-  intro i j hij
-  letI : Nontrivial R := ⟨⟨a, 0, ha0⟩⟩
-  letI : NormalizationMonoid R := UniqueFactorizationMonoid.normalizationMonoid
-  obtain ⟨p', hp', dvd'⟩ := WfDvdMonoid.exists_irreducible_factor ha1 ha0
-  obtain ⟨p, mem, _⟩ := exists_mem_normalizedFactors_of_dvd ha0 hp' dvd'
-  have := congr_arg (fun x => Multiset.count p (normalizedFactors x)) hij
-  simp only [normalizedFactors_pow, Multiset.count_nsmul] at this
-  exact mul_right_cancel₀ (Multiset.count_ne_zero.mpr mem) this
-
-theorem pow_eq_pow_iff {a : R} (ha0 : a ≠ 0) (ha1 : ¬IsUnit a) {i j : ℕ} : a ^ i = a ^ j ↔ i = j :=
-  (pow_right_injective ha0 ha1).eq_iff
+@[deprecated (since := "2024-09-21")] alias pow_right_injective := pow_injective_of_not_isUnit
+@[deprecated (since := "2024-09-21")] alias pow_eq_pow_iff := pow_inj_of_not_isUnit
 
 section multiplicity
 
@@ -1013,7 +1020,7 @@ theorem prime_pow_coprime_prod_of_coprime_insert [DecidableEq α] {s : Finset α
   obtain ⟨q, q_mem, rfl⟩ := Multiset.mem_map.mp q_mem'
   replace hdq := hd.dvd_of_dvd_pow hdq
   have : p ∣ q := dvd_trans (hd.irreducible.dvd_symm hp.irreducible hdp) hdq
-  convert q_mem
+  convert q_mem using 0
   rw [Finset.mem_val,
     is_coprime _ (Finset.mem_insert_self p s) _ (Finset.mem_insert_of_mem q_mem) this]
 
@@ -1201,9 +1208,9 @@ theorem prod_mono : ∀ {a b : FactorSet α}, a ≤ b → a.prod ≤ b.prod
 theorem FactorSet.prod_eq_zero_iff [Nontrivial α] (p : FactorSet α) : p.prod = 0 ↔ p = ⊤ := by
   unfold FactorSet at p
   induction p  -- TODO: `induction_eliminator` doesn't work with `abbrev`
-  · simp only [iff_self_iff, eq_self_iff_true, Associates.prod_top]
+  · simp only [eq_self_iff_true, Associates.prod_top]
   · rw [prod_coe, Multiset.prod_eq_zero_iff, Multiset.mem_map, eq_false WithTop.coe_ne_top,
-      iff_false_iff, not_exists]
+      iff_false, not_exists]
     exact fun a => not_and_of_not_right _ a.prop.ne_zero
 
 section count
@@ -1248,7 +1255,7 @@ def BfactorSetMem : { a : Associates α // Irreducible a } → FactorSet α →
 `s : FactorSet α`.
 
 If `p` is not irreducible, `p` is not a member of any `FactorSet`. -/
-def FactorSetMem (p : Associates α) (s : FactorSet α) : Prop :=
+def FactorSetMem (s : FactorSet α) (p : Associates α) : Prop :=
   letI : Decidable (Irreducible p) := Classical.dec _
   if hp : Irreducible p then BfactorSetMem ⟨p, hp⟩ s else False
 
@@ -1256,7 +1263,7 @@ instance : Membership (Associates α) (FactorSet α) :=
   ⟨FactorSetMem⟩
 
 @[simp]
-theorem factorSetMem_eq_mem (p : Associates α) (s : FactorSet α) : FactorSetMem p s = (p ∈ s) :=
+theorem factorSetMem_eq_mem (p : Associates α) (s : FactorSet α) : FactorSetMem s p = (p ∈ s) :=
   rfl
 
 theorem mem_factorSet_top {p : Associates α} {hp : Irreducible p} : p ∈ (⊤ : FactorSet α) := by
@@ -1572,21 +1579,21 @@ theorem coprime_iff_inf_one {a b : α} (ha0 : a ≠ 0) (hb0 : b ≠ 0) :
 theorem factors_self [Nontrivial α] {p : Associates α} (hp : Irreducible p) :
     p.factors = WithTop.some {⟨p, hp⟩} :=
   eq_of_prod_eq_prod
-    (by rw [factors_prod, FactorSet.prod]; dsimp; rw [prod_singleton])
+    (by rw [factors_prod, FactorSet.prod.eq_def]; dsimp; rw [prod_singleton])
 
 theorem factors_prime_pow [Nontrivial α] {p : Associates α} (hp : Irreducible p) (k : ℕ) :
     factors (p ^ k) = WithTop.some (Multiset.replicate k ⟨p, hp⟩) :=
   eq_of_prod_eq_prod
     (by
-      rw [Associates.factors_prod, FactorSet.prod]
+      rw [Associates.factors_prod, FactorSet.prod.eq_def]
       dsimp; rw [Multiset.map_replicate, Multiset.prod_replicate, Subtype.coe_mk])
 
 theorem prime_pow_le_iff_le_bcount [DecidableEq (Associates α)] {m p : Associates α}
     (h₁ : m ≠ 0) (h₂ : Irreducible p) {k : ℕ} : p ^ k ≤ m ↔ k ≤ bcount ⟨p, h₂⟩ m.factors := by
   rcases Associates.exists_non_zero_rep h₁ with ⟨m, hm, rfl⟩
   have := nontrivial_of_ne _ _ hm
-  rw [bcount, factors_mk, Multiset.le_count_iff_replicate_le, ← factors_le, factors_prime_pow,
-    factors_mk, WithTop.coe_le_coe] <;> assumption
+  rw [bcount.eq_def, factors_mk, Multiset.le_count_iff_replicate_le, ← factors_le,
+    factors_prime_pow, factors_mk, WithTop.coe_le_coe] <;> assumption
 
 @[simp]
 theorem factors_one [Nontrivial α] : factors (1 : Associates α) = 0 := by
@@ -1856,7 +1863,7 @@ noncomputable def fintypeSubtypeDvd {M : Type*} [CancelCommMonoidWithZero M]
       (((normalizedFactors y).powerset.toFinset ×ˢ (Finset.univ : Finset Mˣ)).image fun s =>
         (s.snd : M) * s.fst.prod)
       fun x => ?_
-  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true_iff,
+  simp only [exists_prop, Finset.mem_image, Finset.mem_product, Finset.mem_univ, and_true,
     Multiset.mem_toFinset, Multiset.mem_powerset, exists_eq_right, Multiset.mem_map]
   constructor
   · rintro ⟨s, hs, rfl⟩
@@ -1935,7 +1942,7 @@ theorem Ideal.IsPrime.exists_mem_prime_of_ne_bot {R : Type*} [CommSemiring R] [I
 namespace Nat
 
 instance instWfDvdMonoid : WfDvdMonoid ℕ where
-  wellFounded_dvdNotUnit := by
+  wf := by
     refine RelHomClass.wellFounded
       (⟨fun x : ℕ => if x = 0 then (⊤ : ℕ∞) else x, ?_⟩ : DvdNotUnit →r (· < ·)) wellFounded_lt
     intro a b h
@@ -1975,3 +1982,5 @@ lemma factors_multiset_prod_of_irreducible {s : Multiset ℕ} (h : ∀ x : ℕ,
   exact fun con ↦ not_irreducible_zero (h 0 con)
 
 end Nat
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/RingTheory/Unramified/Basic.lean b/Mathlib/RingTheory/Unramified/Basic.lean
index af45772e8e3eb..baf22e56e7160 100644
--- a/Mathlib/RingTheory/Unramified/Basic.lean
+++ b/Mathlib/RingTheory/Unramified/Basic.lean
@@ -104,23 +104,12 @@ theorem lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
     (g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
   FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
 
-end
-
-section OfEquiv
-
-variable {R : Type u} [CommSemiring R]
-variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
-
-theorem of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
-    FormallyUnramified R B := by
+instance : FormallyUnramified R R := by
   constructor
-  intro C _ _ I hI f₁ f₂ e'
-  rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
-  congr 1
-  refine FormallyUnramified.comp_injective I hI ?_
-  rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
+  intros B _ _ _ _ f₁ f₂ _
+  exact Subsingleton.elim _ _
 
-end OfEquiv
+end
 
 section Comp
 
@@ -155,6 +144,33 @@ theorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
 
 end Comp
 
+section of_surjective
+
+variable {R : Type u} [CommSemiring R]
+variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
+
+/-- This holds in general for epimorphisms. -/
+theorem of_surjective [FormallyUnramified R A] (f : A →ₐ[R] B) (H : Function.Surjective f) :
+    FormallyUnramified R B := by
+  constructor
+  intro Q _ _ I hI f₁ f₂ e
+  ext x
+  obtain ⟨x, rfl⟩ := H x
+  rw [← AlgHom.comp_apply, ← AlgHom.comp_apply]
+  congr 1
+  apply FormallyUnramified.comp_injective I hI
+  ext x; exact DFunLike.congr_fun e (f x)
+
+instance quotient {A} [CommRing A] [Algebra R A] [FormallyUnramified R A] (I : Ideal A) :
+    FormallyUnramified R (A ⧸ I) :=
+    FormallyUnramified.of_surjective (IsScalarTower.toAlgHom _ _ _) Ideal.Quotient.mk_surjective
+
+theorem of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
+    FormallyUnramified R B :=
+  of_surjective e.toAlgHom e.surjective
+
+end of_surjective
+
 section BaseChange
 
 open scoped TensorProduct
diff --git a/Mathlib/RingTheory/Unramified/Field.lean b/Mathlib/RingTheory/Unramified/Field.lean
new file mode 100644
index 0000000000000..3a0152d738c1c
--- /dev/null
+++ b/Mathlib/RingTheory/Unramified/Field.lean
@@ -0,0 +1,213 @@
+/-
+Copyright (c) 2024 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+-/
+import Mathlib.FieldTheory.PurelyInseparable
+import Mathlib.RingTheory.Artinian
+import Mathlib.RingTheory.LocalProperties.Basic
+import Mathlib.Algebra.Polynomial.Taylor
+import Mathlib.RingTheory.Unramified.Finite
+
+/-!
+# Unramified algebras over fields
+
+## Main results
+
+Let `K` be a field, `A` be a `K`-algebra and `L` be a field extension of `K`.
+
+- `Algebra.FormallyUnramified.bijective_of_isAlgClosed_of_localRing`:
+    If `A` is `K`-unramified and `K` is alg-closed, then `K = A`.
+- `Algebra.FormallyUnramified.isReduced_of_field`:
+    If `A` is `K`-unramified then `A` is reduced.
+- `Algebra.FormallyUnramified.iff_isSeparable`:
+    `L` is unramified over `K` iff `L` is separable over `K`.
+
+## References
+
+- [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen]
+
+-/
+
+universe u
+
+variable (K A L : Type u) [Field K] [Field L] [CommRing A] [Algebra K A] [Algebra K L]
+
+open Algebra Polynomial
+
+open scoped TensorProduct
+
+namespace Algebra.FormallyUnramified
+
+theorem of_isSeparable [Algebra.IsSeparable K L] : FormallyUnramified K L := by
+  constructor
+  intros B _ _ I hI f₁ f₂ e
+  ext x
+  have : f₁ x - f₂ x ∈ I := by
+    simpa [Ideal.Quotient.mk_eq_mk_iff_sub_mem] using AlgHom.congr_fun e x
+  have := Polynomial.eval_add_of_sq_eq_zero ((minpoly K x).map (algebraMap K B)) (f₂ x)
+    (f₁ x - f₂ x) (show (f₁ x - f₂ x) ^ 2 ∈ ⊥ from hI ▸ Ideal.pow_mem_pow this 2)
+  simp only [add_sub_cancel, eval_map_algebraMap, aeval_algHom_apply, minpoly.aeval, map_zero,
+    derivative_map, zero_add] at this
+  rwa [eq_comm, ((isUnit_iff_ne_zero.mpr
+    ((Algebra.IsSeparable.isSeparable K x).aeval_derivative_ne_zero
+      (minpoly.aeval K x))).map f₂).mul_right_eq_zero, sub_eq_zero] at this
+
+variable [FormallyUnramified K A] [EssFiniteType K A]
+variable [FormallyUnramified K L] [EssFiniteType K L]
+
+theorem bijective_of_isAlgClosed_of_localRing
+    [IsAlgClosed K] [LocalRing A] :
+    Function.Bijective (algebraMap K A) := by
+  have := finite_of_free (R := K) (S := A)
+  have : IsArtinianRing A := isArtinian_of_tower K inferInstance
+  have hA : IsNilpotent (LocalRing.maximalIdeal A) := by
+    rw [← LocalRing.jacobson_eq_maximalIdeal ⊥]
+    · exact IsArtinianRing.isNilpotent_jacobson_bot
+    · exact bot_ne_top
+  have : Function.Bijective (Algebra.ofId K (A ⧸ LocalRing.maximalIdeal A)) :=
+    ⟨RingHom.injective _, IsAlgClosed.algebraMap_surjective_of_isIntegral⟩
+  let e : K ≃ₐ[K] A ⧸ LocalRing.maximalIdeal A := {
+    __ := Algebra.ofId K (A ⧸ LocalRing.maximalIdeal A)
+    __ := Equiv.ofBijective _ this }
+  let e' : A ⊗[K] (A ⧸ LocalRing.maximalIdeal A) ≃ₐ[A] A :=
+    (Algebra.TensorProduct.congr AlgEquiv.refl e.symm).trans (Algebra.TensorProduct.rid K A A)
+  let f : A ⧸ LocalRing.maximalIdeal A →ₗ[A] A := e'.toLinearMap.comp (sec K A _)
+  have hf : (Algebra.ofId _ _).toLinearMap ∘ₗ f = LinearMap.id := by
+    dsimp [f]
+    rw [← LinearMap.comp_assoc, ← comp_sec K A]
+    congr 1
+    apply LinearMap.restrictScalars_injective K
+    apply _root_.TensorProduct.ext'
+    intros r s
+    obtain ⟨s, rfl⟩ := e.surjective s
+    suffices s • (Ideal.Quotient.mk (LocalRing.maximalIdeal A)) r = r • e s by
+      simpa [ofId, e']
+    simp [Algebra.smul_def, e, ofId, mul_comm]
+  have hf₁ : f 1 • (1 : A ⧸ LocalRing.maximalIdeal A) = 1 := by
+    rw [← algebraMap_eq_smul_one]
+    exact LinearMap.congr_fun hf 1
+  have hf₂ : 1 - f 1 ∈ LocalRing.maximalIdeal A := by
+    rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, map_one, ← Ideal.Quotient.algebraMap_eq,
+     algebraMap_eq_smul_one, hf₁, sub_self]
+  have hf₃ : IsIdempotentElem (1 - f 1) := by
+    apply IsIdempotentElem.one_sub
+    rw [IsIdempotentElem, ← smul_eq_mul, ← map_smul, hf₁]
+  have hf₄ : f 1 = 1 := by
+    obtain ⟨n, hn⟩ := hA
+    have : (1 - f 1) ^ n = 0 := by
+      rw [← Ideal.mem_bot, ← Ideal.zero_eq_bot, ← hn]
+      exact Ideal.pow_mem_pow hf₂ n
+    rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul]
+  refine Equiv.bijective ⟨algebraMap K A, ⇑e.symm ∘ ⇑(algebraMap A _), fun x ↦ by simp, fun x ↦ ?_⟩
+  have : ⇑(algebraMap K A) = ⇑f ∘ ⇑e := by
+    ext k
+    conv_rhs => rw [← mul_one k, ← smul_eq_mul, Function.comp_apply, map_smul,
+      LinearMap.map_smul_of_tower, map_one, hf₄, ← algebraMap_eq_smul_one]
+  rw [this]
+  simp only [Function.comp_apply, AlgEquiv.apply_symm_apply, algebraMap_eq_smul_one,
+    map_smul, hf₄, smul_eq_mul, mul_one]
+
+theorem isField_of_isAlgClosed_of_localRing
+    [IsAlgClosed K] [LocalRing A] : IsField A := by
+  rw [LocalRing.isField_iff_maximalIdeal_eq, eq_bot_iff]
+  intro x hx
+  obtain ⟨x, rfl⟩ := (bijective_of_isAlgClosed_of_localRing K A).surjective x
+  show _ = 0
+  rw [← (algebraMap K A).map_zero]
+  by_contra hx'
+  exact hx ((isUnit_iff_ne_zero.mpr
+    (fun e ↦ hx' ((algebraMap K A).congr_arg e))).map (algebraMap K A))
+
+include K in
+theorem isReduced_of_field :
+    IsReduced A := by
+  constructor
+  intro x hx
+  let f := (Algebra.TensorProduct.includeRight (R := K) (A := AlgebraicClosure K) (B := A))
+  have : Function.Injective f := by
+    have : ⇑f = (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap).comp
+        (Algebra.TensorProduct.lid K A).symm.toLinearMap := by
+      ext x; simp [f]
+    rw [this]
+    suffices Function.Injective
+        (LinearMap.rTensor A (Algebra.ofId K (AlgebraicClosure K)).toLinearMap) by
+      exact this.comp (Algebra.TensorProduct.lid K A).symm.injective
+    apply Module.Flat.rTensor_preserves_injective_linearMap
+    exact (algebraMap K _).injective
+  apply this
+  rw [map_zero]
+  apply eq_zero_of_localization
+  intro M hM
+  have hy := (hx.map f).map (algebraMap _ (Localization.AtPrime M))
+  generalize algebraMap _ (Localization.AtPrime M) (f x) = y at *
+  have := EssFiniteType.of_isLocalization (Localization.AtPrime M) M.primeCompl
+  have := of_isLocalization (Rₘ := Localization.AtPrime M) M.primeCompl
+  have := EssFiniteType.comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
+    (Localization.AtPrime M)
+  have := comp (AlgebraicClosure K) (AlgebraicClosure K ⊗[K] A)
+    (Localization.AtPrime M)
+  letI := (isField_of_isAlgClosed_of_localRing (AlgebraicClosure K)
+    (A := Localization.AtPrime M)).toField
+  exact hy.eq_zero
+
+theorem range_eq_top_of_isPurelyInseparable
+    [IsPurelyInseparable K L] : (algebraMap K L).range = ⊤ := by
+  classical
+  have : Nontrivial (L ⊗[K] L) := by
+    rw [← not_subsingleton_iff_nontrivial, ← rank_zero_iff (R := K), rank_tensorProduct',
+      mul_eq_zero, or_self, rank_zero_iff, not_subsingleton_iff_nontrivial]
+    infer_instance
+  rw [← top_le_iff]
+  intro x _
+  obtain ⟨n, hn⟩ := IsPurelyInseparable.pow_mem K (ringExpChar K) x
+  have : ExpChar (L ⊗[K] L) (ringExpChar K) := by
+    refine expChar_of_injective_ringHom (algebraMap K _).injective (ringExpChar K)
+  have : (1 ⊗ₜ x - x ⊗ₜ 1 : L ⊗[K] L) ^ (ringExpChar K) ^ n = 0 := by
+    rw [sub_pow_expChar_pow, TensorProduct.tmul_pow, one_pow, TensorProduct.tmul_pow, one_pow]
+    obtain ⟨r, hr⟩ := hn
+    rw [← hr, algebraMap_eq_smul_one, TensorProduct.smul_tmul, sub_self]
+  have H : (1 ⊗ₜ x : L ⊗[K] L) = x ⊗ₜ 1 := by
+    have inst : IsReduced (L ⊗[K] L) := isReduced_of_field L _
+    exact sub_eq_zero.mp (IsNilpotent.eq_zero ⟨_, this⟩)
+  by_cases h' : LinearIndependent K ![1, x]
+  · have h := h'.coe_range
+    let S := h.extend (Set.subset_univ _)
+    let a : S := ⟨1, h.subset_extend _ (by simp)⟩; have ha : Basis.extend h a = 1 := by simp
+    let b : S := ⟨x, h.subset_extend _ (by simp)⟩; have hb : Basis.extend h b = x := by simp
+    by_cases e : a = b
+    · obtain rfl : 1 = x := congr_arg Subtype.val e
+      exact ⟨1, map_one _⟩
+    have := DFunLike.congr_fun
+      (DFunLike.congr_arg ((Basis.extend h).tensorProduct (Basis.extend h)).repr H) (a, b)
+    simp only [Basis.tensorProduct_repr_tmul_apply, ← ha, ← hb, Basis.repr_self, smul_eq_mul,
+      Finsupp.single_apply, e, Ne.symm e, ↓reduceIte, mul_one, mul_zero, one_ne_zero] at this
+  · rw [LinearIndependent.pair_iff] at h'
+    simp only [not_forall, not_and, exists_prop] at h'
+    obtain ⟨a, b, e, hab⟩ := h'
+    have : IsUnit b := by
+      rw [isUnit_iff_ne_zero]
+      rintro rfl
+      rw [zero_smul, ← algebraMap_eq_smul_one, add_zero,
+        (injective_iff_map_eq_zero' _).mp (algebraMap K L).injective] at e
+      cases hab e rfl
+    use (-this.unit⁻¹ * a)
+    rw [map_mul, ← Algebra.smul_def, algebraMap_eq_smul_one, eq_neg_iff_add_eq_zero.mpr e,
+      smul_neg, neg_smul, neg_neg, smul_smul, this.val_inv_mul, one_smul]
+
+theorem isSeparable : Algebra.IsSeparable K L := by
+  have := finite_of_free (R := K) (S := L)
+  rw [← separableClosure.eq_top_iff]
+  have := of_comp K (separableClosure K L) L
+  have := EssFiniteType.of_comp K (separableClosure K L) L
+  have := separableClosure.isPurelyInseparable K L
+  ext
+  show _ ↔ _ ∈ (⊤ : Subring _)
+  rw [← range_eq_top_of_isPurelyInseparable (separableClosure K L) L]
+  simp
+
+theorem iff_isSeparable (L) [Field L] [Algebra K L] [EssFiniteType K L] :
+    FormallyUnramified K L ↔ Algebra.IsSeparable K L :=
+  ⟨fun _ ↦ isSeparable K L, fun _ ↦ of_isSeparable K L⟩
+
+end Algebra.FormallyUnramified
diff --git a/Mathlib/RingTheory/Unramified/Finite.lean b/Mathlib/RingTheory/Unramified/Finite.lean
index 90faa21a6fe0c..8652ea0e4337f 100644
--- a/Mathlib/RingTheory/Unramified/Finite.lean
+++ b/Mathlib/RingTheory/Unramified/Finite.lean
@@ -90,8 +90,9 @@ lemma finite_of_free_aux (I) [DecidableEq I] (b : Basis I R S)
   let a i := b.repr (b i * x)
   conv_lhs =>
     simp only [TensorProduct.tmul_mul_tmul, one_mul, mul_comm x (b _),
-      ← show ∀ i, Finsupp.total _ b (a i) = b i * x from fun _ ↦ b.total_repr _]
-  conv_lhs => simp only [Finsupp.total, Finsupp.coe_lsum,
+      ← show ∀ i, Finsupp.linearCombination _ b (a i) = b i * x from
+          fun _ ↦ b.linearCombination_repr _]
+  conv_lhs => simp only [Finsupp.linearCombination, Finsupp.coe_lsum,
     LinearMap.coe_smulRight, LinearMap.id_coe, id_eq, Finsupp.sum, TensorProduct.tmul_sum,
     ← TensorProduct.smul_tmul]
   have h₁ : ∀ k,
@@ -107,7 +108,7 @@ lemma finite_of_free_aux (I) [DecidableEq I] (b : Basis I R S)
     · intro; simp only [zero_smul]
     · intros; simp only [add_smul]
   have h₂ : ∀ (x : S), ((b.repr x).support.sum fun a ↦ b.repr x a • b a) = x := by
-    simpa only [Finsupp.total_apply, Finsupp.sum] using b.total_repr
+    simpa only [Finsupp.linearCombination_apply, Finsupp.sum] using b.linearCombination_repr
   simp_rw [map_finsupp_sum, map_smul, h₁, Finsupp.sum, Finset.sum_comm (t := f.support),
     TensorProduct.smul_tmul', ← TensorProduct.sum_tmul, ← Finset.smul_sum, h₂]
   apply Finset.sum_congr rfl
@@ -159,7 +160,7 @@ lemma finite_of_free [Module.Free R S] : Module.Finite R S := by
   have ⟨f, hf⟩ : ∃ (a : I →₀ S), elem R S = a.sum (fun i x ↦ x ⊗ₜ b i) := by
     let b' := ((Basis.singleton PUnit.{1} S).tensorProduct b).reindex (Equiv.punitProd I)
     use b'.repr (elem R S)
-    conv_lhs => rw [← b'.total_repr (elem R S), Finsupp.total_apply]
+    conv_lhs => rw [← b'.linearCombination_repr (elem R S), Finsupp.linearCombination_apply]
     congr! with _ i x
     simp [b', Basis.tensorProduct, TensorProduct.smul_tmul']
   constructor
@@ -197,15 +198,15 @@ lemma finite_of_free [Module.Free R S] : Module.Finite R S := by
     simp only [Basis.repr_symm_apply, Finsupp.coe_lsum, LinearMap.coe_smulRight,
       LinearMap.id_coe, id_eq, Basis.tensorProduct_apply, Finsupp.finsuppProdEquiv,
       Equiv.coe_fn_symm_mk, Finsupp.uncurry, map_finsupp_sum,
-      Finsupp.total_single, Basis.tensorProduct_apply, Finsupp.equivCongrLeft_apply,
-      Finsupp.total_equivMapDomain, Equiv.coe_prodComm]
+      Finsupp.linearCombination_single, Basis.tensorProduct_apply, Finsupp.equivCongrLeft_apply,
+      Finsupp.linearCombination_equivMapDomain, Equiv.coe_prodComm]
     rw [Finsupp.onFinset_sum, Finsupp.onFinset_sum]
     simp only [Function.comp_apply, Prod.swap_prod_mk, Basis.tensorProduct_apply]
     have : ∀ i, ((b.repr (x * f i)).sum fun j k ↦ k • b j ⊗ₜ[R] b i) = (x * f i) ⊗ₜ[R] b i := by
       intro i
       simp_rw [Finsupp.sum, TensorProduct.smul_tmul', ← TensorProduct.sum_tmul]
       congr 1
-      exact b.total_repr _
+      exact b.linearCombination_repr _
     trans (x ⊗ₜ 1) * elem R S
     · simp_rw [this, hf, Finsupp.sum, Finset.mul_sum, TensorProduct.tmul_mul_tmul, one_mul]
     · rw [← one_tmul_mul_elem, hf, finite_of_free_aux]
diff --git a/Mathlib/RingTheory/Unramified/Pi.lean b/Mathlib/RingTheory/Unramified/Pi.lean
new file mode 100644
index 0000000000000..dc1d2ec8c816b
--- /dev/null
+++ b/Mathlib/RingTheory/Unramified/Pi.lean
@@ -0,0 +1,98 @@
+/-
+Copyright (c) 2024 Andrew Yang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Andrew Yang
+-/
+import Mathlib.RingTheory.Unramified.Basic
+
+/-!
+
+# Formal-unramification of finite products of rings
+
+## Main result
+
+- `Algebra.FormallyUnramified.pi_iff`: If `I` is finite, `Π i : I, A i` is `R`-formally-smooth
+  if and only if each `A i` is `R`-formally-smooth.
+
+-/
+
+namespace Algebra.FormallyUnramified
+
+universe u v
+
+variable {R : Type max u v} {I : Type v} [Finite I] (f : I → Type max u v)
+variable [CommRing R] [∀ i, CommRing (f i)] [∀ i, Algebra R (f i)]
+
+theorem pi_iff :
+    FormallyUnramified R (∀ i, f i) ↔ ∀ i, FormallyUnramified R (f i) := by
+  classical
+  cases nonempty_fintype I
+  constructor
+  · intro _ i
+    exact FormallyUnramified.of_surjective (Pi.evalAlgHom R f i) (Function.surjective_eval i)
+  · intro H
+    constructor
+    intros B _ _ J hJ f₁ f₂ e
+    ext g
+    rw [← Finset.univ_sum_single g, map_sum, map_sum]
+    refine Finset.sum_congr rfl ?_
+    rintro x -
+    have hf : ∀ x, f₁ x - f₂ x ∈ J := by
+      intro g
+      rw [← Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero]
+      exact AlgHom.congr_fun e g
+    let e : ∀ i, f i := Pi.single x 1
+    have he : IsIdempotentElem e := by simp [IsIdempotentElem, e, ← Pi.single_mul]
+    have h₁ : (f₁ e) * (1 - f₂ e) = 0 := by
+      rw [← Ideal.mem_bot, ← hJ, ← ((he.map f₁).mul (he.map f₂).one_sub).eq, ← pow_two]
+      apply Ideal.pow_mem_pow
+      convert Ideal.mul_mem_left _ (f₁ e) (hf e) using 1
+      rw [mul_sub, mul_sub, mul_one, (he.map f₁).eq]
+    have h₂ : (f₂ e) * (1 - f₁ e) = 0 := by
+      rw [← Ideal.mem_bot, ← hJ, ← ((he.map f₂).mul (he.map f₁).one_sub).eq, ← pow_two]
+      apply Ideal.pow_mem_pow
+      convert Ideal.mul_mem_left _ (-f₂ e) (hf e) using 1
+      rw [neg_mul, mul_sub, mul_sub, mul_one, neg_sub, (he.map f₂).eq]
+    have H : f₁ e = f₂ e := by
+      trans f₁ e * f₂ e
+      · rw [← sub_eq_zero, ← h₁, mul_sub, mul_one]
+      · rw [eq_comm, ← sub_eq_zero, ← h₂, mul_sub, mul_one, mul_comm]
+    let J' := Ideal.span {1 - f₁ e}
+    let f₁' : f x →ₐ[R] B ⧸ J' := by
+      apply AlgHom.ofLinearMap
+        (((Ideal.Quotient.mkₐ R J').comp f₁).toLinearMap.comp (LinearMap.single _ _ x))
+      · simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, LinearMap.coe_single,
+          Function.comp_apply, AlgHom.toLinearMap_apply, Ideal.Quotient.mkₐ_eq_mk]
+        rw [eq_comm, ← sub_eq_zero, ← (Ideal.Quotient.mk J').map_one, ← map_sub,
+          Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
+      · intros r s; simp [Pi.single_mul]
+    let f₂' : f x →ₐ[R] B ⧸ J' := by
+      apply AlgHom.ofLinearMap
+        (((Ideal.Quotient.mkₐ R J').comp f₂).toLinearMap.comp (LinearMap.single _ _ x))
+      · simp only [AlgHom.comp_toLinearMap, LinearMap.coe_comp, LinearMap.coe_single,
+          Function.comp_apply, AlgHom.toLinearMap_apply, Ideal.Quotient.mkₐ_eq_mk]
+        rw [eq_comm, ← sub_eq_zero, ← (Ideal.Quotient.mk J').map_one, ← map_sub,
+          Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton, H]
+      · intros r s; simp [Pi.single_mul]
+    suffices f₁' = f₂' by
+      have := AlgHom.congr_fun this (g x)
+      simp only [AlgHom.comp_toLinearMap, AlgHom.ofLinearMap_apply, LinearMap.coe_comp,
+        LinearMap.coe_single, Function.comp_apply, AlgHom.toLinearMap_apply, ← map_sub,
+        Ideal.Quotient.mkₐ_eq_mk, ← sub_eq_zero (b := Ideal.Quotient.mk J' _), sub_zero, f₁', f₂',
+        Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton, J'] at this
+      obtain ⟨c, hc⟩ := this
+      apply_fun (f₁ e * ·) at hc
+      rwa [← mul_assoc, mul_sub, mul_sub, mul_one, (he.map f₁).eq, sub_self, zero_mul,
+        ← map_mul, H, ← map_mul, ← Pi.single_mul, one_mul, sub_eq_zero] at hc
+    apply FormallyUnramified.comp_injective (I := J.map (algebraMap _ _))
+    · rw [← Ideal.map_pow, hJ, Ideal.map_bot]
+    · ext r
+      rw [← sub_eq_zero]
+      simp only [Ideal.Quotient.algebraMap_eq, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
+        Function.comp_apply, ← map_sub, Ideal.Quotient.eq_zero_iff_mem, f₁', f₂',
+        AlgHom.comp_toLinearMap, AlgHom.ofLinearMap_apply, LinearMap.coe_comp,
+        LinearMap.coe_single, Function.comp_apply, AlgHom.toLinearMap_apply,
+        Ideal.Quotient.mkₐ_eq_mk]
+      exact Ideal.mem_map_of_mem (Ideal.Quotient.mk J') (hf (Pi.single x r))
+
+end Algebra.FormallyUnramified
diff --git a/Mathlib/RingTheory/Valuation/AlgebraInstances.lean b/Mathlib/RingTheory/Valuation/AlgebraInstances.lean
index 41b57feefcfbd..91ff35d526b16 100644
--- a/Mathlib/RingTheory/Valuation/AlgebraInstances.lean
+++ b/Mathlib/RingTheory/Valuation/AlgebraInstances.lean
@@ -25,14 +25,14 @@ of a field with a valuation, as well as their unit balls.
 
 open Function Valuation
 
-open scoped DiscreteValuation
+open scoped Multiplicative
 
 variable {K : Type*} [Field K] (v : Valuation K ℤₘ₀) (L : Type*) [Field L] [Algebra K L]
 
 namespace ValuationSubring
 
 -- Implementation note : this instance was automatic in Lean3
-instance smul : SMul v.valuationSubring (integralClosure v.valuationSubring L) := Algebra.toSMul
+instance : Algebra v.valuationSubring L := Algebra.ofSubring v.valuationSubring.toSubring
 
 theorem algebraMap_injective : Injective (algebraMap v.valuationSubring L) :=
   (NoZeroSMulDivisors.algebraMap_injective K L).comp (IsFractionRing.injective _ _)
@@ -50,6 +50,8 @@ theorem isIntegral_of_mem_ringOfIntegers' {x : (integralClosure v.valuationSubri
 
 variable (E : Type _) [Field E] [Algebra K E] [Algebra L E] [IsScalarTower K L E]
 
+instance : IsScalarTower v.valuationSubring L E := Subring.instIsScalarTowerSubtypeMem _
+
 /-- Given an algebra between two field extensions `L` and `E` of a field `K` with a valuation `v`,
   create an algebra between their two rings of integers. -/
 instance algebra :
diff --git a/Mathlib/RingTheory/Valuation/Basic.lean b/Mathlib/RingTheory/Valuation/Basic.lean
index e32f03f67f513..eec13a2b17f5f 100644
--- a/Mathlib/RingTheory/Valuation/Basic.lean
+++ b/Mathlib/RingTheory/Valuation/Basic.lean
@@ -207,6 +207,9 @@ theorem zero_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x = 0 
 theorem ne_zero_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : v x ≠ 0 ↔ x ≠ 0 :=
   map_ne_zero v
 
+lemma pos_iff [Nontrivial Γ₀] (v : Valuation K Γ₀) {x : K} : 0 < v x ↔ x ≠ 0 := by
+  rw [zero_lt_iff, ne_zero_iff]
+
 theorem unit_map_eq (u : Rˣ) : (Units.map (v : R →* Γ₀) u : Γ₀) = v u :=
   rfl
 
@@ -322,17 +325,16 @@ theorem map_one_sub_of_lt (h : v x < 1) : v (1 - x) = 1 := by
   simpa only [v.map_one, v.map_neg] using v.map_add_eq_of_lt_left h
 
 theorem one_lt_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 < v x ↔ v x⁻¹ < 1 := by
-  simpa using (inv_lt_inv₀ (v.ne_zero_iff.2 h) one_ne_zero).symm
+  simp [inv_lt_one₀ (v.pos_iff.2 h)]
 
 theorem one_le_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 ≤ v x ↔ v x⁻¹ ≤ 1 := by
-  convert (one_lt_val_iff v (inv_ne_zero h)).symm.not <;>
-  push_neg <;> simp only [inv_inv]
+  simp [inv_le_one₀ (v.pos_iff.2 h)]
 
 theorem val_lt_one_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : v x < 1 ↔ 1 < v x⁻¹ := by
-  simpa only [inv_inv] using (one_lt_val_iff v (inv_ne_zero h)).symm
+  simp [one_lt_inv₀ (v.pos_iff.2 h)]
 
 theorem val_le_one_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : v x ≤ 1 ↔ 1 ≤ v x⁻¹ := by
-  simpa [inv_inv] using (one_le_val_iff v (inv_ne_zero h)).symm
+  simp [one_le_inv₀ (v.pos_iff.2 h)]
 
 theorem val_eq_one_iff (v : Valuation K Γ₀) {x : K} : v x = 1 ↔ v x⁻¹ = 1 := by
   by_cases h : x = 0
@@ -412,8 +414,16 @@ theorem isEquiv_of_map_strictMono [LinearOrderedCommMonoidWithZero Γ₀]
     (H : StrictMono f) : IsEquiv (v.map f H.monotone) v := fun _x _y =>
   ⟨H.le_iff_le.mp, fun h => H.monotone h⟩
 
+theorem isEquiv_iff_val_lt_val [LinearOrderedCommGroupWithZero Γ₀]
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
+    v.IsEquiv v' ↔ ∀ {x y : K}, v x < v y ↔ v' x < v' y := by
+  simp only [IsEquiv, le_iff_le_iff_lt_iff_lt]
+  exact forall_comm
+
+alias ⟨IsEquiv.lt_iff_lt, _⟩ := isEquiv_iff_val_lt_val
+
 theorem isEquiv_of_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
-    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀)
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀}
     (h : ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1) : v.IsEquiv v' := by
   intro x y
   obtain rfl | hy := eq_or_ne y 0
@@ -422,12 +432,14 @@ theorem isEquiv_of_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
       rwa [zero_lt_iff, ne_zero_iff]
 
 theorem isEquiv_iff_val_le_one [LinearOrderedCommGroupWithZero Γ₀]
-    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
     v.IsEquiv v' ↔ ∀ {x : K}, v x ≤ 1 ↔ v' x ≤ 1 :=
-  ⟨fun h x => by simpa using h x 1, isEquiv_of_val_le_one _ _⟩
+  ⟨fun h x => by simpa using h x 1, isEquiv_of_val_le_one⟩
+
+alias ⟨IsEquiv.le_one_iff_le_one, _⟩ := isEquiv_iff_val_le_one
 
 theorem isEquiv_iff_val_eq_one [LinearOrderedCommGroupWithZero Γ₀]
-    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
     v.IsEquiv v' ↔ ∀ {x : K}, v x = 1 ↔ v' x = 1 := by
   constructor
   · intro h x
@@ -461,13 +473,15 @@ theorem isEquiv_iff_val_eq_one [LinearOrderedCommGroupWithZero Γ₀]
       · rw [← h] at hx'
         exact le_of_eq hx'
 
+alias ⟨IsEquiv.eq_one_iff_eq_one, _⟩ := isEquiv_iff_val_eq_one
+
 theorem isEquiv_iff_val_lt_one [LinearOrderedCommGroupWithZero Γ₀]
-    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
     v.IsEquiv v' ↔ ∀ {x : K}, v x < 1 ↔ v' x < 1 := by
   constructor
   · intro h x
     simp only [lt_iff_le_and_ne,
-      and_congr ((isEquiv_iff_val_le_one _ _).1 h) ((isEquiv_iff_val_eq_one _ _).1 h).not]
+      and_congr h.le_one_iff_le_one h.eq_one_iff_eq_one.not]
   · rw [isEquiv_iff_val_eq_one]
     intro h x
     by_cases hx : x = 0
@@ -488,20 +502,29 @@ theorem isEquiv_iff_val_lt_one [LinearOrderedCommGroupWithZero Γ₀]
         rw [← inv_one, ← inv_eq_iff_eq_inv, ← map_inv₀] at hh
         exact hh.not_lt (h.1 ((one_lt_val_iff v hx).1 h_2))
 
+alias ⟨IsEquiv.lt_one_iff_lt_one, _⟩ := isEquiv_iff_val_lt_one
+
 theorem isEquiv_iff_val_sub_one_lt_one [LinearOrderedCommGroupWithZero Γ₀]
-    [LinearOrderedCommGroupWithZero Γ'₀] (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
+    [LinearOrderedCommGroupWithZero Γ'₀] {v : Valuation K Γ₀} {v' : Valuation K Γ'₀} :
     v.IsEquiv v' ↔ ∀ {x : K}, v (x - 1) < 1 ↔ v' (x - 1) < 1 := by
   rw [isEquiv_iff_val_lt_one]
   exact (Equiv.subRight 1).surjective.forall
 
+alias ⟨IsEquiv.val_sub_one_lt_one_iff, _⟩ := isEquiv_iff_val_sub_one_lt_one
+
 theorem isEquiv_tfae [LinearOrderedCommGroupWithZero Γ₀] [LinearOrderedCommGroupWithZero Γ'₀]
     (v : Valuation K Γ₀) (v' : Valuation K Γ'₀) :
-    [v.IsEquiv v', ∀ {x}, v x ≤ 1 ↔ v' x ≤ 1, ∀ {x}, v x = 1 ↔ v' x = 1, ∀ {x}, v x < 1 ↔ v' x < 1,
-        ∀ {x}, v (x - 1) < 1 ↔ v' (x - 1) < 1].TFAE := by
-  tfae_have 1 ↔ 2; · apply isEquiv_iff_val_le_one
-  tfae_have 1 ↔ 3; · apply isEquiv_iff_val_eq_one
-  tfae_have 1 ↔ 4; · apply isEquiv_iff_val_lt_one
-  tfae_have 1 ↔ 5; · apply isEquiv_iff_val_sub_one_lt_one
+    [ v.IsEquiv v',
+      ∀ {x y}, v x < v y ↔ v' x < v' y,
+      ∀ {x}, v x ≤ 1 ↔ v' x ≤ 1,
+      ∀ {x}, v x = 1 ↔ v' x = 1,
+      ∀ {x}, v x < 1 ↔ v' x < 1,
+      ∀ {x}, v (x - 1) < 1 ↔ v' (x - 1) < 1 ].TFAE := by
+  tfae_have 1 ↔ 2; · apply isEquiv_iff_val_lt_val
+  tfae_have 1 ↔ 3; · apply isEquiv_iff_val_le_one
+  tfae_have 1 ↔ 4; · apply isEquiv_iff_val_eq_one
+  tfae_have 1 ↔ 5; · apply isEquiv_iff_val_lt_one
+  tfae_have 1 ↔ 6; · apply isEquiv_iff_val_sub_one_lt_one
   tfae_finish
 
 end
@@ -831,13 +854,3 @@ end Supp
 
 -- end of section
 end AddValuation
-
-section ValuationNotation
-
-/-- Notation for `WithZero (Multiplicative ℕ)` -/
-scoped[DiscreteValuation] notation "ℕₘ₀" => WithZero (Multiplicative ℕ)
-
-/-- Notation for `WithZero (Multiplicative ℤ)` -/
-scoped[DiscreteValuation] notation "ℤₘ₀" => WithZero (Multiplicative ℤ)
-
-end ValuationNotation
diff --git a/Mathlib/RingTheory/Valuation/Integers.lean b/Mathlib/RingTheory/Valuation/Integers.lean
index ae1c553270829..83132e005a57e 100644
--- a/Mathlib/RingTheory/Valuation/Integers.lean
+++ b/Mathlib/RingTheory/Valuation/Integers.lean
@@ -13,6 +13,7 @@ The elements with valuation less than or equal to 1.
 TODO: Define characteristic predicate.
 -/
 
+open Set
 
 universe u v w
 
@@ -144,6 +145,50 @@ theorem eq_algebraMap_or_inv_eq_algebraMap (hv : Integers v O) (x : F) :
   obtain ⟨a, ha⟩ := exists_of_le_one hv h
   exacts [⟨a, Or.inl ha.symm⟩, ⟨a, Or.inr ha.symm⟩]
 
+lemma isPrincipal_iff_exists_isGreatest (hv : Integers v O) {I : Ideal O} :
+    I.IsPrincipal ↔ ∃ x, IsGreatest (v ∘ algebraMap O F '' I) x := by
+  constructor <;> rintro ⟨x, hx⟩
+  · refine ⟨(v ∘ algebraMap O F) x, ?_, ?_⟩
+    · refine Set.mem_image_of_mem _ ?_
+      simp [hx, Ideal.mem_span_singleton_self]
+    · intro y hy
+      simp only [Function.comp_apply, hx, Ideal.submodule_span_eq, Set.mem_image,
+        SetLike.mem_coe, Ideal.mem_span_singleton] at hy
+      obtain ⟨y, hy, rfl⟩ := hy
+      exact le_of_dvd hv hy
+  · obtain ⟨a, ha, rfl⟩ : ∃ a ∈ I, (v ∘ algebraMap O F) a = x := by simpa using hx.left
+    refine ⟨a, ?_⟩
+    ext b
+    simp only [Ideal.submodule_span_eq, Ideal.mem_span_singleton]
+    exact ⟨fun hb ↦ dvd_of_le hv (hx.2 <| mem_image_of_mem _ hb), fun hb ↦ I.mem_of_dvd hb ha⟩
+
+lemma not_denselyOrdered_of_isPrincipalIdealRing [IsPrincipalIdealRing O] (hv : Integers v O) :
+    ¬ DenselyOrdered (range v) := by
+  intro H
+  -- nonunits as an ideal isn't defined here, nor shown to be equivalent to `v x < 1`
+  set I : Ideal O := {
+    carrier := v ∘ algebraMap O F ⁻¹' Iio (1 : Γ₀)
+    add_mem' := fun {a b} ha hb ↦ by simpa using map_add_lt v ha hb
+    zero_mem' := by simp
+    smul_mem' := by
+      intro c x
+      simp only [mem_preimage, Function.comp_apply, mem_Iio, smul_eq_mul, _root_.map_mul]
+      intro hx
+      exact Right.mul_lt_one_of_le_of_lt (hv.map_le_one c) hx
+  }
+  obtain ⟨x, hx₁, hx⟩ :
+    ∃ x, v (algebraMap O F x) < 1 ∧
+      v (algebraMap O F x) ∈ upperBounds (Iio 1 ∩ range (v ∘ algebraMap O F)) := by
+    simpa [I, IsGreatest, hv.isPrincipal_iff_exists_isGreatest, ← image_preimage_eq_inter_range]
+      using IsPrincipalIdealRing.principal I
+  obtain ⟨y, hy, hy₁⟩ : ∃ y, v (algebraMap O F x) < v y ∧ v y < 1 := by
+    simpa only [Subtype.exists, Subtype.mk_lt_mk, exists_range_iff, exists_prop]
+      using H.dense ⟨v (algebraMap O F x), mem_range_self _⟩ ⟨1, 1, v.map_one⟩ hx₁
+  obtain ⟨z, rfl⟩ := hv.exists_of_le_one hy₁.le
+  exact hy.not_le <| hx ⟨hy₁, mem_range_self _⟩
+
+-- TODO: isPrincipalIdealRing_iff_not_denselyOrdered when MulArchimedean
+
 end Integers
 
 end Field
diff --git a/Mathlib/RingTheory/Valuation/Minpoly.lean b/Mathlib/RingTheory/Valuation/Minpoly.lean
index 3448962da4e70..00094cc933c86 100644
--- a/Mathlib/RingTheory/Valuation/Minpoly.lean
+++ b/Mathlib/RingTheory/Valuation/Minpoly.lean
@@ -21,7 +21,7 @@ Let `K` be a field with a valuation `v` and let `L` be a field extension of `K`.
   is helpful for defining the valuation on `L` inducing `v`.
 -/
 
-open FiniteDimensional minpoly Polynomial
+open Module minpoly Polynomial
 
 variable {K : Type*} [Field K] {Γ₀ : Type*} [LinearOrderedCommGroupWithZero Γ₀]
   (v : Valuation K Γ₀) (L : Type*) [Field L] [Algebra K L]
diff --git a/Mathlib/RingTheory/Valuation/ValExtension.lean b/Mathlib/RingTheory/Valuation/ValExtension.lean
new file mode 100644
index 0000000000000..68c03f9397d28
--- /dev/null
+++ b/Mathlib/RingTheory/Valuation/ValExtension.lean
@@ -0,0 +1,158 @@
+/-
+Copyright (c) 2024 Jiedong Jiang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jiedong Jiang, Bichang Lei
+-/
+import Mathlib.RingTheory.Valuation.Integers
+import Mathlib.Algebra.Group.Units.Hom
+
+/-!
+# Extension of Valuation
+
+In this file, we define the typeclass for valuation extensions and prove basic facts about the
+extension of valuations. Let `A` be an `R` algebra, equipped with valuations `vA` and `vR`
+respectively. Here, the extension of a valuation means that the pullback of valuation `vA` to `R`
+is equivalent to the valuation `vR` on `R`. We only require equivalence, not equality, of
+valuations here.
+
+Note that we do not require the ring map from `R` to `A` to be injective. This holds automatically
+when `R` is a division ring and `A` is nontrivial.
+
+A motivation for choosing the more flexible `Valuation.Equiv` rather than strict equality here is
+to allow for possible normalization. As an example, consider a finite extension `K` of `ℚ_[p]`,
+which is a discretely valued field. We may choose the valuation on `K` to be either:
+
+1. the valuation where the uniformizer is mapped to one (more precisely, `-1` in `ℤₘ₀`) or
+
+2. the valuation where `p` is mapped to one.
+
+For the algebraic closure of `ℚ_[p]`, if we choose the valuation of `p` to be one, then the
+restriction of this valuation to `K` equals the second valuation, but is only equivalent to the
+first valuation. The flexibility of equivalence here allows us to develop theory for both cases
+without first determining the normalizations once and for all.
+
+## Main Definition
+
+* `IsValExtension vR vA` : The valuation `vA` on `A` is an extension of the valuation `vR` on `R`.
+
+## References
+
+* [Bourbaki, Nicolas. *Commutative algebra*] Chapter VI §3, Valuations.
+* <https://en.wikipedia.org/wiki/Valuation_(algebra)#Extension_of_valuations>
+
+## Tags
+Valuation, Extension of Valuations
+
+-/
+open Valuation
+
+variable {R A ΓR ΓA : Type*} [CommRing R] [Ring A]
+    [LinearOrderedCommMonoidWithZero ΓR] [LinearOrderedCommMonoidWithZero ΓA] [Algebra R A]
+    (vR : Valuation R ΓR) (vA : Valuation A ΓA)
+
+/--
+The class `IsValExtension R A` states that the valuation of `A` is an extension of the valuation
+on `R`. More precisely, the valuation on `R` is equivlent to the comap of the valuation on `A`.
+-/
+class IsValExtension : Prop where
+  /-- The valuation on `R` is equivalent to the comap of the valuation on `A` -/
+  val_isEquiv_comap : vR.IsEquiv <| vA.comap (algebraMap R A)
+
+namespace IsValExtension
+
+section algebraMap
+
+variable [IsValExtension vR vA]
+
+-- @[simp] does not work because `vR` cannot be inferred from `R`.
+theorem val_map_le_iff (x y : R) : vA (algebraMap R A x) ≤ vA (algebraMap R A y) ↔ vR x ≤ vR y :=
+  val_isEquiv_comap.symm x y
+
+theorem val_map_lt_iff (x y : R) : vA (algebraMap R A x) < vA (algebraMap R A y) ↔ vR x < vR y := by
+  simpa only [not_le] using ((val_map_le_iff vR vA _ _).not)
+
+theorem val_map_eq_iff (x y : R) : vA (algebraMap R A x) = vA (algebraMap R A y) ↔ vR x = vR y :=
+  (IsEquiv.val_eq val_isEquiv_comap).symm
+
+theorem val_map_le_one_iff (x : R) : vA (algebraMap R A x) ≤ 1 ↔ vR x ≤ 1 := by
+  simpa only [_root_.map_one] using val_map_le_iff vR vA x 1
+
+theorem val_map_lt_one_iff (x : R) : vA (algebraMap R A x) < 1 ↔ vR x < 1 := by
+  simpa only [_root_.map_one, not_le] using (val_map_le_iff vR vA 1 x).not
+
+theorem val_map_eq_one_iff (x : R) : vA (algebraMap R A x) = 1 ↔ vR x = 1 := by
+  simpa only [le_antisymm_iff, _root_.map_one] using
+    and_congr (val_map_le_iff vR vA x 1) (val_map_le_iff vR vA 1 x)
+
+end algebraMap
+
+instance id : IsValExtension vR vR where
+  val_isEquiv_comap := by
+    simp only [Algebra.id.map_eq_id, comap_id, IsEquiv.refl]
+
+section integer
+
+variable {K : Type*} [Field K] [Algebra K A] {ΓR ΓA ΓK: Type*}
+    [LinearOrderedCommGroupWithZero ΓR] [LinearOrderedCommGroupWithZero ΓK]
+    [LinearOrderedCommGroupWithZero ΓA] {vR : Valuation R ΓR} {vK : Valuation K ΓK}
+    {vA : Valuation A ΓA} [IsValExtension vR vA]
+
+/--
+When `K` is a field, if the preimage of the valuation integers of `A` equals to the valuation
+integers of `K`, then the valuation on `A` is an extension of the valuation on `K`.
+-/
+theorem ofComapInteger (h : vA.integer.comap (algebraMap K A) = vK.integer) :
+    IsValExtension vK vA where
+  val_isEquiv_comap := by
+    rw [isEquiv_iff_val_le_one]
+    intro x
+    simp_rw [← Valuation.mem_integer_iff, ← h, Subring.mem_comap, mem_integer_iff, comap_apply]
+
+instance instAlgebraInteger : Algebra vR.integer vA.integer where
+  smul r a := ⟨r • a,
+    Algebra.smul_def r (a : A) ▸ mul_mem ((val_map_le_one_iff vR vA _).mpr r.2) a.2⟩
+  __ := (algebraMap R A).restrict vR.integer vA.integer
+    (by simp [Valuation.mem_integer_iff, val_map_le_one_iff vR vA])
+  commutes' _ _ := Subtype.ext (Algebra.commutes _ _)
+  smul_def' _ _ := Subtype.ext (Algebra.smul_def _ _)
+
+@[simp, norm_cast]
+theorem val_smul (r : vR.integer) (a : vA.integer) : ↑(r • a : vA.integer) = (r : R) • (a : A) := by
+  rfl
+
+@[simp, norm_cast]
+theorem val_algebraMap (r : vR.integer) :
+    ((algebraMap vR.integer vA.integer) r : A) = (algebraMap R A) (r : R) := by
+  rfl
+
+instance instIsScalarTowerInteger : IsScalarTower vR.integer vA.integer A where
+  smul_assoc x y z := by
+    simp only [Algebra.smul_def]
+    exact mul_assoc _ _ _
+
+instance instNoZeroSMulDivisorsInteger [NoZeroSMulDivisors R A] :
+    NoZeroSMulDivisors vR.integer vA.integer := by
+  refine ⟨fun {x y} e ↦ ?_⟩
+  have : (x : R) • (y : A) = 0 := by simpa [Subtype.ext_iff, Algebra.smul_def] using e
+  simpa only [Subtype.ext_iff, smul_eq_zero] using this
+
+theorem algebraMap_injective [IsValExtension vK vA] [Nontrivial A] :
+    Function.Injective (algebraMap vK.integer vA.integer) := by
+  intro x y h
+  simp only [Subtype.ext_iff, val_algebraMap] at h
+  ext
+  apply RingHom.injective (algebraMap K A) h
+
+instance instIsLocalRingHomValuationInteger {S ΓS: Type*} [CommRing S]
+    [LinearOrderedCommGroupWithZero ΓS]
+    [Algebra R S] [IsLocalRingHom (algebraMap R S)] {vS : Valuation S ΓS}
+    [IsValExtension vR vS] : IsLocalRingHom (algebraMap vR.integer vS.integer) where
+  map_nonunit r hr := by
+    apply (Valuation.integer.integers (v := vR)).isUnit_of_one
+    · exact (isUnit_map_iff (algebraMap R S) _).mp (hr.map (algebraMap _ S))
+    · apply (Valuation.integer.integers (v := vS)).one_of_isUnit at hr
+      exact (val_map_eq_one_iff vR vS _).mp hr
+
+end integer
+
+end IsValExtension
diff --git a/Mathlib/RingTheory/Valuation/ValuationRing.lean b/Mathlib/RingTheory/Valuation/ValuationRing.lean
index ed5c3bf093421..864d73e5b39e6 100644
--- a/Mathlib/RingTheory/Valuation/ValuationRing.lean
+++ b/Mathlib/RingTheory/Valuation/ValuationRing.lean
@@ -3,12 +3,12 @@ Copyright (c) 2022 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Adam Topaz
 -/
-import Mathlib.RingTheory.Valuation.Integers
+import Mathlib.Algebra.EuclideanDomain.Basic
+import Mathlib.RingTheory.Bezout
+import Mathlib.RingTheory.LocalRing.Basic
 import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.RingTheory.Localization.Integer
-import Mathlib.RingTheory.DiscreteValuationRing.Basic
-import Mathlib.RingTheory.Bezout
-import Mathlib.Tactic.FieldSimp
+import Mathlib.RingTheory.Valuation.Integers
 
 /-!
 # Valuation Rings
@@ -33,6 +33,7 @@ We also show that, given a valuation `v` on a field `K`, the ring of valuation i
 valuation ring and `K` is the fraction field of this ring.
 -/
 
+assert_not_exists DiscreteValuationRing
 
 universe u v w
 
@@ -362,10 +363,10 @@ protected theorem TFAE (R : Type u) [CommRing R] [IsDomain R] :
       [ValuationRing R,
         ∀ x : FractionRing R, IsLocalization.IsInteger R x ∨ IsLocalization.IsInteger R x⁻¹,
         IsTotal R (· ∣ ·), IsTotal (Ideal R) (· ≤ ·), LocalRing R ∧ IsBezout R] := by
-  tfae_have 1 ↔ 2; · exact iff_isInteger_or_isInteger R _
-  tfae_have 1 ↔ 3; · exact iff_dvd_total
-  tfae_have 1 ↔ 4; · exact iff_ideal_total
-  tfae_have 1 ↔ 5; · exact iff_local_bezout_domain
+  tfae_have 1 ↔ 2 := iff_isInteger_or_isInteger R _
+  tfae_have 1 ↔ 3 := iff_dvd_total
+  tfae_have 1 ↔ 4 := iff_ideal_total
+  tfae_have 1 ↔ 5 := iff_local_bezout_domain
   tfae_finish
 
 end
@@ -440,31 +441,4 @@ instance (priority := 100) of_field : ValuationRing K := by
 
 end
 
-section
-
-variable (A : Type u) [CommRing A] [IsDomain A] [DiscreteValuationRing A]
-
-/-- A DVR is a valuation ring. -/
-instance (priority := 100) of_discreteValuationRing : ValuationRing A := by
-  constructor
-  intro a b
-  by_cases ha : a = 0; · use 0; right; simp [ha]
-  by_cases hb : b = 0; · use 0; left; simp [hb]
-  obtain ⟨ϖ, hϖ⟩ := DiscreteValuationRing.exists_irreducible A
-  obtain ⟨m, u, rfl⟩ := DiscreteValuationRing.eq_unit_mul_pow_irreducible ha hϖ
-  obtain ⟨n, v, rfl⟩ := DiscreteValuationRing.eq_unit_mul_pow_irreducible hb hϖ
-  rcases le_total m n with h | h
-  · use (u⁻¹ * v : Aˣ) * ϖ ^ (n - m); left
-    simp_rw [mul_comm (u : A), Units.val_mul, ← mul_assoc, mul_assoc _ (u : A)]
-    simp only [Units.mul_inv, mul_one, mul_comm _ (v : A), mul_assoc, ← pow_add]
-    congr 2
-    exact Nat.add_sub_of_le h
-  · use (v⁻¹ * u : Aˣ) * ϖ ^ (m - n); right
-    simp_rw [mul_comm (v : A), Units.val_mul, ← mul_assoc, mul_assoc _ (v : A)]
-    simp only [Units.mul_inv, mul_one, mul_comm _ (u : A), mul_assoc, ← pow_add]
-    congr 2
-    exact Nat.add_sub_of_le h
-
-end
-
 end ValuationRing
diff --git a/Mathlib/RingTheory/Valuation/ValuationSubring.lean b/Mathlib/RingTheory/Valuation/ValuationSubring.lean
index fb8a0c376dfbb..a66cc1e821359 100644
--- a/Mathlib/RingTheory/Valuation/ValuationSubring.lean
+++ b/Mathlib/RingTheory/Valuation/ValuationSubring.lean
@@ -300,9 +300,8 @@ theorem ofPrime_idealOfLE (R S : ValuationSubring K) (h : R ≤ S) :
   · rintro ⟨a, r, hr, rfl⟩; apply mul_mem; · exact h a.2
     · rw [← valuation_le_one_iff, map_inv₀, ← inv_one, inv_le_inv₀]
       · exact not_lt.1 ((not_iff_not.2 <| valuation_lt_one_iff S _).1 hr)
-      · intro hh; erw [Valuation.zero_iff, Subring.coe_eq_zero_iff] at hh
-        apply hr; rw [hh]; apply Ideal.zero_mem (R.idealOfLE S h)
-      · exact one_ne_zero
+      · simpa [Valuation.pos_iff] using fun hr₀ ↦ hr₀ ▸ hr <| Ideal.zero_mem (R.idealOfLE S h)
+      · exact zero_lt_one
   · intro hx; by_cases hr : x ∈ R; · exact R.le_ofPrime _ hr
     have : x ≠ 0 := fun h => hr (by rw [h]; exact R.zero_mem)
     replace hr := (R.mem_or_inv_mem x).resolve_left hr
@@ -532,8 +531,6 @@ theorem image_maximalIdeal : ((↑) : A → K) '' LocalRing.maximalIdeal A = A.n
   simp only [Set.mem_image, SetLike.mem_coe, mem_nonunits_iff_exists_mem_maximalIdeal]
   erw [Subtype.exists]
   simp_rw [exists_and_right, exists_eq_right]
-  -- Porting note: added
-  simp
 
 end nonunits
 
@@ -613,7 +610,7 @@ def principalUnitGroupEquiv :
       A.coe_mem_principalUnitGroup_iff.1 x.2⟩
   invFun x :=
     ⟨A.unitGroupMulEquiv.symm x, by
-      rw [A.coe_mem_principalUnitGroup_iff]; simpa using SetLike.coe_mem x⟩
+      rw [A.coe_mem_principalUnitGroup_iff]; simp⟩
   left_inv x := by simp
   right_inv x := by simp
   map_mul' x y := rfl
@@ -790,6 +787,6 @@ variable {Γ : Type*} [LinearOrderedCommGroupWithZero Γ] (v : Valuation K Γ) (
 
 -- @[simp] -- Porting note: not in simpNF
 theorem mem_unitGroup_iff : x ∈ v.valuationSubring.unitGroup ↔ v x = 1 :=
-  (Valuation.isEquiv_iff_val_eq_one _ _).mp (Valuation.isEquiv_valuation_valuationSubring _).symm
+  IsEquiv.eq_one_iff_eq_one (Valuation.isEquiv_valuation_valuationSubring _).symm
 
 end Valuation
diff --git a/Mathlib/RingTheory/WittVector/Frobenius.lean b/Mathlib/RingTheory/WittVector/Frobenius.lean
index 0492664ca9471..0d989e4ecdebe 100644
--- a/Mathlib/RingTheory/WittVector/Frobenius.lean
+++ b/Mathlib/RingTheory/WittVector/Frobenius.lean
@@ -3,11 +3,11 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
+import Mathlib.Algebra.Algebra.ZMod
 import Mathlib.Data.Nat.Multiplicity
-import Mathlib.Data.ZMod.Algebra
+import Mathlib.FieldTheory.Perfect
 import Mathlib.RingTheory.WittVector.Basic
 import Mathlib.RingTheory.WittVector.IsPoly
-import Mathlib.FieldTheory.Perfect
 
 /-!
 ## The Frobenius operator
diff --git a/Mathlib/RingTheory/WittVector/IsPoly.lean b/Mathlib/RingTheory/WittVector/IsPoly.lean
index a9cd11b9fd4c6..ef5430b614c18 100644
--- a/Mathlib/RingTheory/WittVector/IsPoly.lean
+++ b/Mathlib/RingTheory/WittVector/IsPoly.lean
@@ -116,7 +116,7 @@ theorem poly_eq_of_wittPolynomial_bind_eq' [Fact p.Prime] (f g : ℕ → MvPolyn
   rw [← Function.funext_iff] at h
   replace h :=
     congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
-  simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
+  simpa only [Function.comp_def, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
     bind₁_wittPolynomial_xInTermsOfW, bind₁_X_right] using h
 
 theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolynomial ℕ ℤ)
@@ -126,7 +126,7 @@ theorem poly_eq_of_wittPolynomial_bind_eq [Fact p.Prime] (f g : ℕ → MvPolyno
   rw [← Function.funext_iff] at h
   replace h :=
     congr_arg (fun fam => bind₁ (MvPolynomial.map (Int.castRingHom ℚ) ∘ fam) (xInTermsOfW p ℚ n)) h
-  simpa only [Function.comp, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
+  simpa only [Function.comp_def, map_bind₁, map_wittPolynomial, ← bind₁_bind₁,
     bind₁_wittPolynomial_xInTermsOfW, bind₁_X_right] using h
 
 -- Ideally, we would generalise this to n-ary functions
@@ -233,7 +233,7 @@ instance IsPoly₂.comp {h f g} [hh : IsPoly₂ p h] [hf : IsPoly p f] [hg : IsP
     uncurry]
   apply eval₂Hom_congr rfl _ rfl
   ext ⟨i, n⟩
-  fin_cases i <;> simp [aeval_eq_eval₂Hom, eval₂Hom_rename, Function.comp]
+  fin_cases i <;> simp [aeval_eq_eval₂Hom, eval₂Hom_rename, Function.comp_def]
 
 /-- The composition of a polynomial function with a binary polynomial function is polynomial. -/
 -- Porting note (#10754): made this an instance
diff --git a/Mathlib/RingTheory/WittVector/Isocrystal.lean b/Mathlib/RingTheory/WittVector/Isocrystal.lean
index 5ba7feb02f7c6..0f5d266cc6c93 100644
--- a/Mathlib/RingTheory/WittVector/Isocrystal.lean
+++ b/Mathlib/RingTheory/WittVector/Isocrystal.lean
@@ -54,7 +54,7 @@ This file introduces notation in the locale `Isocrystal`.
 
 noncomputable section
 
-open FiniteDimensional
+open Module
 
 namespace WittVector
 
@@ -181,7 +181,7 @@ admits an isomorphism to one of the standard (indexed by `m : ℤ`) one-dimensio
 theorem isocrystal_classification (k : Type*) [Field k] [IsAlgClosed k] [CharP k p] (V : Type*)
     [AddCommGroup V] [Isocrystal p k V] (h_dim : finrank K(p, k) V = 1) :
     ∃ m : ℤ, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V) := by
-  haveI : Nontrivial V := FiniteDimensional.nontrivial_of_finrank_eq_succ h_dim
+  haveI : Nontrivial V := Module.nontrivial_of_finrank_eq_succ h_dim
   obtain ⟨x, hx⟩ : ∃ x : V, x ≠ 0 := exists_ne 0
   have : Φ(p, k) x ≠ 0 := by simpa only [map_zero] using Φ(p, k).injective.ne hx
   obtain ⟨a, ha, hax⟩ : ∃ a : K(p, k), a ≠ 0 ∧ Φ(p, k) x = a • x := by
diff --git a/Mathlib/RingTheory/WittVector/MulCoeff.lean b/Mathlib/RingTheory/WittVector/MulCoeff.lean
index edb4720376f65..fa74982d2d6fa 100644
--- a/Mathlib/RingTheory/WittVector/MulCoeff.lean
+++ b/Mathlib/RingTheory/WittVector/MulCoeff.lean
@@ -118,11 +118,10 @@ theorem mul_polyOfInterest_aux1 (n : ℕ) :
     congr 1
     have hsupp : (Finsupp.single i (p ^ (n - i))).support = {i} := by
       rw [Finsupp.support_eq_singleton]
-      simp only [and_true_iff, Finsupp.single_eq_same, eq_self_iff_true, Ne]
+      simp only [and_true, Finsupp.single_eq_same, eq_self_iff_true, Ne]
       exact pow_ne_zero _ hp.out.ne_zero
     simp only [bind₁_monomial, hsupp, Int.cast_natCast, prod_singleton, eq_intCast,
-      Finsupp.single_eq_same, C_pow, mul_eq_mul_left_iff, true_or_iff, eq_self_iff_true,
-      Int.cast_pow]
+      Finsupp.single_eq_same, C_pow, mul_eq_mul_left_iff, eq_self_iff_true, Int.cast_pow]
   · simp only [map_mul, bind₁_X_right]
 
 theorem mul_polyOfInterest_aux2 (n : ℕ) :
@@ -222,7 +221,7 @@ theorem peval_polyOfInterest' (n : ℕ) (x y : 𝕎 k) :
   rw [peval_polyOfInterest]
   have : (p : k) = 0 := CharP.cast_eq_zero k p
   simp only [this, Nat.cast_pow, ne_eq, add_eq_zero, and_false, zero_pow, zero_mul, add_zero,
-    not_false_eq_true]
+    not_false_eq_true, reduceCtorEq]
   have sum_zero_pow_mul_pow_p (y : 𝕎 k) : ∑ x ∈ range (n + 1 + 1),
       (0 : k) ^ x * y.coeff x ^ p ^ (n + 1 - x) = y.coeff 0 ^ p ^ (n + 1) := by
     rw [Finset.sum_eq_single_of_mem 0] <;> simp (config := { contextual := true })
@@ -243,10 +242,15 @@ theorem nth_mul_coeff' (n : ℕ) :
     apply f₀
     rintro ⟨a, ha⟩
     apply Function.uncurry ![x, y]
-    simp_rw [product_val, this, Multiset.mem_product, mem_univ_val, true_and_iff, range_val,
-      Multiset.range_succ, Multiset.mem_cons, Multiset.mem_range] at ha
+    simp_rw [product_val, this, range_val, Multiset.range_succ] at ha
+    let S : Set (Fin 2 × ℕ) := (fun a => a.2 = n ∨ a.2 < n)
+    have ha' : a ∈ S := by
+      convert ha
+      dsimp [S]
+      congr!
+      simp
     refine ⟨a.fst, ⟨a.snd, ?_⟩⟩
-    cases' ha with ha ha <;> omega
+    cases' ha' with ha ha <;> omega
   use f
   intro x y
   dsimp [f, peval]
diff --git a/Mathlib/RingTheory/WittVector/MulP.lean b/Mathlib/RingTheory/WittVector/MulP.lean
index c5519b92ba158..6cdf37a89f03d 100644
--- a/Mathlib/RingTheory/WittVector/MulP.lean
+++ b/Mathlib/RingTheory/WittVector/MulP.lean
@@ -70,7 +70,7 @@ theorem bind₁_wittMulN_wittPolynomial (n k : ℕ) :
   · rw [wittMulN, ← bind₁_bind₁, wittAdd, wittStructureInt_prop]
     simp only [map_add, Nat.cast_succ, bind₁_X_right]
     rw [add_mul, one_mul, bind₁_rename, bind₁_rename]
-    simp only [ih, Function.uncurry, Function.comp, bind₁_X_left, AlgHom.id_apply,
+    simp only [ih, Function.uncurry, Function.comp_def, bind₁_X_left, AlgHom.id_apply,
       Matrix.cons_val_zero, Matrix.head_cons, Matrix.cons_val_one]
 
 end
diff --git a/Mathlib/RingTheory/WittVector/StructurePolynomial.lean b/Mathlib/RingTheory/WittVector/StructurePolynomial.lean
index ad4f137a05748..5a89447cb88ff 100644
--- a/Mathlib/RingTheory/WittVector/StructurePolynomial.lean
+++ b/Mathlib/RingTheory/WittVector/StructurePolynomial.lean
@@ -370,7 +370,7 @@ theorem wittStructureRat_vars [Fintype idx] (Φ : MvPolynomial idx ℚ) (n : ℕ
     (wittStructureRat p Φ n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1) := by
   rw [wittStructureRat]
   intro x hx
-  simp only [Finset.mem_product, true_and_iff, Finset.mem_univ, Finset.mem_range]
+  simp only [Finset.mem_product, true_and, Finset.mem_univ, Finset.mem_range]
   obtain ⟨k, hk, hx'⟩ := mem_vars_bind₁ _ _ hx
   obtain ⟨i, -, hx''⟩ := mem_vars_bind₁ _ _ hx'
   obtain ⟨j, hj, rfl⟩ := mem_vars_rename _ _ hx''
diff --git a/Mathlib/RingTheory/WittVector/WittPolynomial.lean b/Mathlib/RingTheory/WittVector/WittPolynomial.lean
index de01b15cb0a5c..ea728fea38133 100644
--- a/Mathlib/RingTheory/WittVector/WittPolynomial.lean
+++ b/Mathlib/RingTheory/WittVector/WittPolynomial.lean
@@ -201,8 +201,7 @@ theorem xInTermsOfW_eq [Invertible (p : R)] {n : ℕ} : xInTermsOfW p R n =
 @[simp]
 theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
     constantCoeff (xInTermsOfW p R n) = 0 := by
-  apply Nat.strongInductionOn n; clear n
-  intro n IH
+  induction n using Nat.strongRecOn with | ind n IH => ?_
   rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
     constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]
   -- Porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that
@@ -227,12 +226,11 @@ variable [hp : Fact p.Prime]
 
 theorem xInTermsOfW_vars_aux (n : ℕ) :
     n ∈ (xInTermsOfW p ℚ n).vars ∧ (xInTermsOfW p ℚ n).vars ⊆ range (n + 1) := by
-  apply Nat.strongInductionOn n; clear n
-  intro n ih
+  induction n using Nat.strongRecOn with | ind n ih => ?_
   rw [xInTermsOfW_eq, mul_comm, vars_C_mul _ (Invertible.ne_zero _),
     vars_sub_of_disjoint, vars_X, range_succ, insert_eq]
   on_goal 1 =>
-    simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]
+    simp only [true_and, true_or, eq_self_iff_true, mem_union, mem_singleton]
     intro i
     rw [mem_union, mem_union]
     apply Or.imp id
@@ -276,9 +274,7 @@ theorem bind₁_xInTermsOfW_wittPolynomial [Invertible (p : R)] (k : ℕ) :
 @[simp]
 theorem bind₁_wittPolynomial_xInTermsOfW [Invertible (p : R)] (n : ℕ) :
     bind₁ (W_ R) (xInTermsOfW p R n) = X n := by
-  apply Nat.strongInductionOn n
-  clear n
-  intro n H
+  induction n using Nat.strongRecOn with | ind n H => ?_
   rw [xInTermsOfW_eq, map_mul, map_sub, bind₁_X_right, algHom_C, map_sum,
     show X n = (X n * C ((p : R) ^ n)) * C ((⅟p : R) ^ n) by
       rw [mul_assoc, ← C_mul, ← mul_pow, mul_invOf_self, one_pow, map_one, mul_one]]
diff --git a/Mathlib/SetTheory/Cardinal/Basic.lean b/Mathlib/SetTheory/Cardinal/Basic.lean
index bf03366d250b6..170827f668647 100644
--- a/Mathlib/SetTheory/Cardinal/Basic.lean
+++ b/Mathlib/SetTheory/Cardinal/Basic.lean
@@ -83,10 +83,12 @@ open Function Set Order
 
 noncomputable section
 
-universe u v w
+universe u v w v' w'
 
 variable {α β : Type u}
 
+/-! ### Definition of cardinals -/
+
 /-- The equivalence relation on types given by equivalence (bijective correspondence) of types.
   Quotienting by this equivalence relation gives the cardinal numbers.
 -/
@@ -130,6 +132,10 @@ theorem inductionOn₃ {p : Cardinal → Cardinal → Cardinal → Prop} (c₁ :
     (c₃ : Cardinal) (h : ∀ α β γ, p #α #β #γ) : p c₁ c₂ c₃ :=
   Quotient.inductionOn₃ c₁ c₂ c₃ h
 
+theorem induction_on_pi {ι : Type u} {p : (ι → Cardinal.{v}) → Prop}
+    (f : ι → Cardinal.{v}) (h : ∀ f : ι → Type v, p fun i ↦ #(f i)) : p f :=
+  Quotient.induction_on_pi f h
+
 protected theorem eq : #α = #β ↔ Nonempty (α ≃ β) :=
   Quotient.eq'
 
@@ -164,52 +170,6 @@ def map₂ (f : Type u → Type v → Type w) (hf : ∀ α β γ δ, α ≃ β 
     Cardinal.{u} → Cardinal.{v} → Cardinal.{w} :=
   Quotient.map₂ f fun α β ⟨e₁⟩ γ δ ⟨e₂⟩ => ⟨hf α β γ δ e₁ e₂⟩
 
-/-- The universe lift operation on cardinals. You can specify the universes explicitly with
-  `lift.{u v} : Cardinal.{v} → Cardinal.{max v u}` -/
-@[pp_with_univ]
-def lift (c : Cardinal.{v}) : Cardinal.{max v u} :=
-  map ULift.{u, v} (fun _ _ e => Equiv.ulift.trans <| e.trans Equiv.ulift.symm) c
-
-@[simp]
-theorem mk_uLift (α) : #(ULift.{v, u} α) = lift.{v} #α :=
-  rfl
-
--- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
--- further down in this file
-/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/
-@[simp, nolint simpNF]
-theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
-  funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
-
--- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
--- further down in this file
-/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/
-@[simp, nolint simpNF]
-theorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=
-  lift_umax
-
--- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
--- further down in this file
-/-- A cardinal lifted to a lower or equal universe equals itself. -/
-@[simp, nolint simpNF]
-theorem lift_id' (a : Cardinal.{max u v}) : lift.{u} a = a :=
-  inductionOn a fun _ => mk_congr Equiv.ulift
-
-/-- A cardinal lifted to the same universe equals itself. -/
-@[simp]
-theorem lift_id (a : Cardinal) : lift.{u, u} a = a :=
-  lift_id'.{u, u} a
-
-/-- A cardinal lifted to the zero universe equals itself. -/
--- porting note (#10618): simp can prove this
--- @[simp]
-theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=
-  lift_id'.{0, u} a
-
-@[simp]
-theorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a :=
-  inductionOn a fun _ => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
-
 /-- We define the order on cardinal numbers by `#α ≤ #β` if and only if
   there exists an embedding (injective function) from α to β. -/
 instance : LE Cardinal.{u} :=
@@ -258,6 +218,54 @@ theorem mk_subtype_le {α : Type u} (p : α → Prop) : #(Subtype p) ≤ #α :=
 theorem mk_set_le (s : Set α) : #s ≤ #α :=
   mk_subtype_le s
 
+/-! ### Lifting cardinals to a higher universe -/
+
+/-- The universe lift operation on cardinals. You can specify the universes explicitly with
+  `lift.{u v} : Cardinal.{v} → Cardinal.{max v u}` -/
+@[pp_with_univ]
+def lift (c : Cardinal.{v}) : Cardinal.{max v u} :=
+  map ULift.{u, v} (fun _ _ e => Equiv.ulift.trans <| e.trans Equiv.ulift.symm) c
+
+@[simp]
+theorem mk_uLift (α) : #(ULift.{v, u} α) = lift.{v} #α :=
+  rfl
+
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
+-- further down in this file
+/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/
+@[simp, nolint simpNF]
+theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
+  funext fun a => inductionOn a fun _ => (Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
+
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
+-- further down in this file
+/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/
+@[simp, nolint simpNF]
+theorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=
+  lift_umax
+
+-- Porting note: simpNF is not happy with universe levels, but this is needed as simp lemma
+-- further down in this file
+/-- A cardinal lifted to a lower or equal universe equals itself. -/
+@[simp, nolint simpNF]
+theorem lift_id' (a : Cardinal.{max u v}) : lift.{u} a = a :=
+  inductionOn a fun _ => mk_congr Equiv.ulift
+
+/-- A cardinal lifted to the same universe equals itself. -/
+@[simp]
+theorem lift_id (a : Cardinal) : lift.{u, u} a = a :=
+  lift_id'.{u, u} a
+
+/-- A cardinal lifted to the zero universe equals itself. -/
+-- porting note (#10618): simp can prove this
+-- @[simp]
+theorem lift_uzero (a : Cardinal.{u}) : lift.{0} a = a :=
+  lift_id'.{0, u} a
+
+@[simp]
+theorem lift_lift.{u_1} (a : Cardinal.{u_1}) : lift.{w} (lift.{v} a) = lift.{max v w} a :=
+  inductionOn a fun _ => (Equiv.ulift.trans <| Equiv.ulift.trans Equiv.ulift.symm).cardinal_eq
+
 @[simp]
 lemma mk_preimage_down {s : Set α} : #(ULift.down.{v} ⁻¹' s) = lift.{v} (#s) := by
   rw [← mk_uLift, Cardinal.eq]
@@ -303,6 +311,32 @@ theorem lift_le {a b : Cardinal.{v}} : lift.{u, v} a ≤ lift.{u, v} b ↔ a ≤
     rw [← lift_umax]
     exact lift_mk_le.{u}
 
+-- Porting note: simpNF is not happy with universe levels.
+@[simp, nolint simpNF]
+theorem lift_mk_shrink (α : Type u) [Small.{v} α] :
+    Cardinal.lift.{max u w} #(Shrink.{v} α) = Cardinal.lift.{max v w} #α :=
+  lift_mk_eq.2 ⟨(equivShrink α).symm⟩
+
+@[simp]
+theorem lift_mk_shrink' (α : Type u) [Small.{v} α] :
+    Cardinal.lift.{u} #(Shrink.{v} α) = Cardinal.lift.{v} #α :=
+  lift_mk_shrink.{u, v, 0} α
+
+@[simp]
+theorem lift_mk_shrink'' (α : Type max u v) [Small.{v} α] :
+    Cardinal.lift.{u} #(Shrink.{v} α) = #α := by
+  rw [← lift_umax', lift_mk_shrink.{max u v, v, 0} α, ← lift_umax, lift_id]
+
+theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
+    b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b :=
+  inductionOn₂ a b fun α β => by
+    rw [← lift_id #β, ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}]
+    exact fun ⟨f⟩ =>
+      ⟨#(Set.range f),
+        Eq.symm <| lift_mk_eq.{_, _, v}.2
+          ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)
+              fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩
+
 -- Porting note: changed `simps` to `simps!` because the linter told to do so.
 /-- `Cardinal.lift` as an `OrderEmbedding`. -/
 @[simps! (config := .asFn)]
@@ -325,6 +359,36 @@ theorem lift_strictMono : StrictMono lift := fun _ _ => lift_lt.2
 theorem lift_monotone : Monotone lift :=
   lift_strictMono.monotone
 
+@[simp]
+theorem lift_min {a b : Cardinal} : lift.{u, v} (min a b) = min (lift.{u, v} a) (lift.{u, v} b) :=
+  lift_monotone.map_min
+
+@[simp]
+theorem lift_max {a b : Cardinal} : lift.{u, v} (max a b) = max (lift.{u, v} a) (lift.{u, v} b) :=
+  lift_monotone.map_max
+
+-- Porting note: simpNF is not happy with universe levels.
+@[simp, nolint simpNF]
+theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
+    lift.{max v w} a = lift.{max u w} b ↔ lift.{v} a = lift.{u} b := by
+  rw [← lift_lift.{v, w, u}, ← lift_lift.{u, w, v}, lift_inj]
+
+theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
+    b ≤ lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' ≤ a :=
+  ⟨fun h =>
+    let ⟨a', e⟩ := lift_down h
+    ⟨a', e, lift_le.1 <| e.symm ▸ h⟩,
+    fun ⟨_, e, h⟩ => e ▸ lift_le.2 h⟩
+
+theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
+    b < lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' < a :=
+  ⟨fun h =>
+    let ⟨a', e⟩ := lift_down h.le
+    ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
+    fun ⟨_, e, h⟩ => e ▸ lift_lt.2 h⟩
+
+/-! ### Basic cardinals -/
+
 instance : Zero Cardinal.{u} :=
   -- `PEmpty` might be more canonical, but this is convenient for defeq with natCast
   ⟨lift #(Fin 0)⟩
@@ -401,11 +465,6 @@ theorem mk_psum (α : Type u) (β : Type v) : #(α ⊕' β) = lift.{v} #α + lif
 theorem mk_fintype (α : Type u) [h : Fintype α] : #α = Fintype.card α :=
   mk_congr (Fintype.equivOfCardEq (by simp))
 
-protected theorem cast_succ (n : ℕ) : ((n + 1 : ℕ) : Cardinal.{u}) = n + 1 := by
-  change #(ULift.{u} (Fin (n+1))) = # (ULift.{u} (Fin n)) + 1
-  rw [← mk_option, mk_fintype, mk_fintype]
-  simp only [Fintype.card_ulift, Fintype.card_fin, Fintype.card_option]
-
 instance : Mul Cardinal.{u} :=
   ⟨map₂ Prod fun _ _ _ _ => Equiv.prodCongr⟩
 
@@ -416,9 +475,6 @@ theorem mul_def (α β : Type u) : #α * #β = #(α × β) :=
 theorem mk_prod (α : Type u) (β : Type v) : #(α × β) = lift.{v, u} #α * lift.{u, v} #β :=
   mk_congr (Equiv.ulift.symm.prodCongr Equiv.ulift.symm)
 
-private theorem mul_comm' (a b : Cardinal.{u}) : a * b = b * a :=
-  inductionOn₂ a b fun α β => mk_congr <| Equiv.prodComm α β
-
 /-- The cardinal exponential. `#α ^ #β` is the cardinal of `β → α`. -/
 instance instPowCardinal : Pow Cardinal.{u} Cardinal.{u} :=
   ⟨map₂ (fun α β => β → α) fun _ _ _ _ e₁ e₂ => e₂.arrowCongr e₁⟩
@@ -435,41 +491,45 @@ theorem lift_power (a b : Cardinal.{u}) : lift.{v} (a ^ b) = lift.{v} a ^ lift.{
     mk_congr <| Equiv.ulift.trans (Equiv.ulift.arrowCongr Equiv.ulift).symm
 
 @[simp]
-theorem power_zero {a : Cardinal} : a ^ (0 : Cardinal) = 1 :=
+theorem power_zero (a : Cardinal) : a ^ (0 : Cardinal) = 1 :=
   inductionOn a fun _ => mk_eq_one _
 
 @[simp]
-theorem power_one {a : Cardinal.{u}} : a ^ (1 : Cardinal) = a :=
+theorem power_one (a : Cardinal.{u}) : a ^ (1 : Cardinal) = a :=
   inductionOn a fun α => mk_congr (Equiv.funUnique (ULift.{u} (Fin 1)) α)
 
-theorem power_add {a b c : Cardinal} : a ^ (b + c) = a ^ b * a ^ c :=
+theorem power_add (a b c : Cardinal) : a ^ (b + c) = a ^ b * a ^ c :=
   inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumArrowEquivProdArrow β γ α
 
+private theorem cast_succ (n : ℕ) : ((n + 1 : ℕ) : Cardinal.{u}) = n + 1 := by
+  change #(ULift.{u} _) = #(ULift.{u} _) + 1
+  rw [← mk_option]
+  simp
+
 instance commSemiring : CommSemiring Cardinal.{u} where
   zero := 0
   one := 1
   add := (· + ·)
   mul := (· * ·)
-  zero_add a := inductionOn a fun α => mk_congr <| Equiv.emptySum (ULift (Fin 0)) α
-  add_zero a := inductionOn a fun α => mk_congr <| Equiv.sumEmpty α (ULift (Fin 0))
+  zero_add a := inductionOn a fun α => mk_congr <| Equiv.emptySum _ α
+  add_zero a := inductionOn a fun α => mk_congr <| Equiv.sumEmpty α _
   add_assoc a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumAssoc α β γ
   add_comm a b := inductionOn₂ a b fun α β => mk_congr <| Equiv.sumComm α β
   zero_mul a := inductionOn a fun α => mk_eq_zero _
   mul_zero a := inductionOn a fun α => mk_eq_zero _
-  one_mul a := inductionOn a fun α => mk_congr <| Equiv.uniqueProd α (ULift (Fin 1))
-  mul_one a := inductionOn a fun α => mk_congr <| Equiv.prodUnique α (ULift (Fin 1))
+  one_mul a := inductionOn a fun α => mk_congr <| Equiv.uniqueProd α _
+  mul_one a := inductionOn a fun α => mk_congr <| Equiv.prodUnique α _
   mul_assoc a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodAssoc α β γ
-  mul_comm := mul_comm'
+  mul_comm a b := inductionOn₂ a b fun α β => mk_congr <| Equiv.prodComm α β
   left_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.prodSumDistrib α β γ
   right_distrib a b c := inductionOn₃ a b c fun α β γ => mk_congr <| Equiv.sumProdDistrib α β γ
   nsmul := nsmulRec
   npow n c := c ^ (n : Cardinal)
-  npow_zero := @power_zero
-  npow_succ n c := show c ^ (↑(n + 1) : Cardinal) = c ^ (↑n : Cardinal) * c
-    by rw [Cardinal.cast_succ, power_add, power_one, mul_comm']
-  natCast := (fun n => lift.{u} #(Fin n) : ℕ → Cardinal.{u})
+  npow_zero := power_zero
+  npow_succ n c := by dsimp; rw [cast_succ, power_add, power_one]
+  natCast n := lift #(Fin n)
   natCast_zero := rfl
-  natCast_succ := Cardinal.cast_succ
+  natCast_succ n := cast_succ n
 
 @[simp]
 theorem one_power {a : Cardinal} : (1 : Cardinal) ^ a = 1 :=
@@ -538,9 +598,7 @@ theorem mk_powerset {α : Type u} (s : Set α) : #(↥(𝒫 s)) = 2 ^ #(↥s) :=
 theorem lift_two_power (a : Cardinal) : lift.{v} (2 ^ a) = 2 ^ lift.{v} a := by
   simp [← one_add_one_eq_two]
 
-section OrderProperties
-
-open Sum
+/-! ### Order properties -/
 
 protected theorem zero_le : ∀ a : Cardinal, 0 ≤ a := by
   rintro ⟨α⟩
@@ -610,7 +668,7 @@ theorem self_le_power (a : Cardinal) {b : Cardinal} (hb : 1 ≤ b) : a ≤ a ^ b
   rcases eq_or_ne a 0 with (rfl | ha)
   · exact zero_le _
   · convert power_le_power_left ha hb
-    exact power_one.symm
+    exact (power_one a).symm
 
 /-- **Cantor's theorem** -/
 theorem cantor (a : Cardinal.{u}) : a < 2 ^ a := by
@@ -639,8 +697,6 @@ theorem power_le_power_right {a b c : Cardinal} : a ≤ b → a ^ c ≤ b ^ c :=
 theorem power_pos {a : Cardinal} (b : Cardinal) (ha : 0 < a) : 0 < a ^ b :=
   (power_ne_zero _ ha.ne').bot_lt
 
-end OrderProperties
-
 protected theorem lt_wf : @WellFounded Cardinal.{u} (· < ·) :=
   ⟨fun a =>
     by_contradiction fun h => by
@@ -682,6 +738,22 @@ lemma iInf_eq_zero_iff {ι : Sort*} {f : ι → Cardinal} :
     (⨅ i, f i) = 0 ↔ IsEmpty ι ∨ ∃ i, f i = 0 := by
   simp [iInf, sInf_eq_zero_iff]
 
+/-- A variant of `ciSup_of_empty` but with `0` on the RHS for convenience -/
+protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
+  ciSup_of_empty f
+
+@[simp]
+theorem lift_sInf (s : Set Cardinal) : lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) := by
+  rcases eq_empty_or_nonempty s with (rfl | hs)
+  · simp
+  · exact lift_monotone.map_csInf hs
+
+@[simp]
+theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u, v} (iInf f) = ⨅ i, lift.{u, v} (f i) := by
+  unfold iInf
+  convert lift_sInf (range f)
+  simp_rw [← comp_apply (f := lift), range_comp]
+
 /-- Note that the successor of `c` is not the same as `c + 1` except in the case of finite `c`. -/
 instance : SuccOrder Cardinal := ConditionallyCompleteLinearOrder.toSuccOrder
 
@@ -708,38 +780,76 @@ theorem add_one_le_succ (c : Cardinal.{u}) : c + 1 ≤ succ c := by
     #γ + 1 = #(Option γ) := mk_option.symm
     _ ≤ #β := (f.optionElim b hb).cardinal_le
 
+@[simp]
+theorem lift_succ (a) : lift.{v, u} (succ a) = succ (lift.{v, u} a) :=
+  le_antisymm
+    (le_of_not_gt fun h => by
+      rcases lt_lift_iff.1 h with ⟨b, e, h⟩
+      rw [lt_succ_iff, ← lift_le, e] at h
+      exact h.not_lt (lt_succ _))
+    (succ_le_of_lt <| lift_lt.2 <| lt_succ a)
+
 /-- A cardinal is a limit if it is not zero or a successor cardinal. Note that `ℵ₀` is a limit
   cardinal by this definition, but `0` isn't.
-
-  Use `IsSuccLimit` if you want to include the `c = 0` case. -/
+Deprecated. Use `Order.IsSuccLimit` instead. -/
+@[deprecated IsSuccLimit (since := "2024-09-17")]
 def IsLimit (c : Cardinal) : Prop :=
-  c ≠ 0 ∧ IsSuccLimit c
+  c ≠ 0 ∧ IsSuccPrelimit c
+
+theorem ne_zero_of_isSuccLimit {c} (h : IsSuccLimit c) : c ≠ 0 :=
+  h.ne_bot
 
+theorem isSuccPrelimit_zero : IsSuccPrelimit (0 : Cardinal) :=
+  isSuccPrelimit_bot
+
+protected theorem isSuccLimit_iff {c : Cardinal} : IsSuccLimit c ↔ c ≠ 0 ∧ IsSuccPrelimit c :=
+  isSuccLimit_iff
+
+section deprecated
+
+set_option linter.deprecated false
+
+@[deprecated IsSuccLimit.isSuccPrelimit (since := "2024-09-17")]
+protected theorem IsLimit.isSuccPrelimit {c} (h : IsLimit c) : IsSuccPrelimit c :=
+  h.2
+
+@[deprecated ne_zero_of_isSuccLimit (since := "2024-09-17")]
 protected theorem IsLimit.ne_zero {c} (h : IsLimit c) : c ≠ 0 :=
   h.1
 
-protected theorem IsLimit.isSuccLimit {c} (h : IsLimit c) : IsSuccLimit c :=
-  h.2
+@[deprecated IsLimit.isSuccPrelimit (since := "2024-09-05")]
+alias IsLimit.isSuccLimit := IsLimit.isSuccPrelimit
 
+@[deprecated IsSuccLimit.succ_lt (since := "2024-09-17")]
 theorem IsLimit.succ_lt {x c} (h : IsLimit c) : x < c → succ x < c :=
-  h.isSuccLimit.succ_lt
+  h.isSuccPrelimit.succ_lt
 
-theorem isSuccLimit_zero : IsSuccLimit (0 : Cardinal) :=
-  isSuccLimit_bot
+@[deprecated isSuccPrelimit_zero (since := "2024-09-05")]
+alias isSuccLimit_zero := isSuccPrelimit_zero
+
+end deprecated
+
+/-! ### Indexed cardinal `sum` -/
 
 /-- The indexed sum of cardinals is the cardinality of the
   indexed disjoint union, i.e. sigma type. -/
 def sum {ι} (f : ι → Cardinal) : Cardinal :=
-  mk (Σi, (f i).out)
+  mk (Σ i, (f i).out)
 
 theorem le_sum {ι} (f : ι → Cardinal) (i) : f i ≤ sum f := by
   rw [← Quotient.out_eq (f i)]
   exact ⟨⟨fun a => ⟨i, a⟩, fun a b h => by injection h⟩⟩
 
+theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
+  ciSup_le' <| le_sum _
+
 @[simp]
 theorem mk_sigma {ι} (f : ι → Type*) : #(Σ i, f i) = sum fun i => #(f i) :=
   mk_congr <| Equiv.sigmaCongrRight fun _ => outMkEquiv.symm
 
+theorem mk_sigma_arrow {ι} (α : Type*) (f : ι → Type*) :
+    #(Sigma f → α) = #(Π i, f i → α) := mk_congr <| Equiv.piCurry fun _ _ ↦ α
+
 @[simp]
 theorem sum_const (ι : Type u) (a : Cardinal.{v}) :
     (sum fun _ : ι => a) = lift.{v} #ι * lift.{u} a :=
@@ -798,6 +908,42 @@ theorem lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le {α : Type u} {β : Type v
                 Equiv.ulift.symm)).trans_le
         (hf b)
 
+theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
+    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by
+  refine (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans ?_
+  simp only [mk_sum, mk_out, lift_id, mk_sigma]
+
+end Cardinal
+
+/-! ### Well-ordering theorem -/
+
+open Cardinal in
+theorem nonempty_embedding_to_cardinal : Nonempty (α ↪ Cardinal.{u}) :=
+  (Embedding.total _ _).resolve_left fun ⟨⟨f, hf⟩⟩ =>
+    let g : α → Cardinal.{u} := invFun f
+    let ⟨x, (hx : g x = 2 ^ sum g)⟩ := invFun_surjective hf (2 ^ sum g)
+    have : g x ≤ sum g := le_sum.{u, u} g x
+    not_le_of_gt (by rw [hx]; exact cantor _) this
+
+/-- An embedding of any type to the set of cardinals in its universe. -/
+def embeddingToCardinal : α ↪ Cardinal.{u} :=
+  Classical.choice nonempty_embedding_to_cardinal
+
+/-- Any type can be endowed with a well order, obtained by pulling back the well order over
+cardinals by some embedding. -/
+def WellOrderingRel : α → α → Prop :=
+  embeddingToCardinal ⁻¹'o (· < ·)
+
+instance WellOrderingRel.isWellOrder : IsWellOrder α WellOrderingRel :=
+  (RelEmbedding.preimage _ _).isWellOrder
+
+instance IsWellOrder.subtype_nonempty : Nonempty { r // IsWellOrder α r } :=
+  ⟨⟨WellOrderingRel, inferInstance⟩⟩
+
+/-! ### Small sets of cardinals -/
+
+namespace Cardinal
+
 /-- The range of an indexed cardinal function, whose outputs live in a higher universe than the
     inputs, is always bounded above. -/
 theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove (Set.range f) :=
@@ -805,14 +951,17 @@ theorem bddAbove_range {ι : Type u} (f : ι → Cardinal.{max u v}) : BddAbove
     rintro a ⟨i, rfl⟩
     exact le_sum f i⟩
 
-instance (a : Cardinal.{u}) : Small.{u} (Set.Iic a) := by
+instance small_Iic (a : Cardinal.{u}) : Small.{u} (Iic a) := by
   rw [← mk_out a]
   apply @small_of_surjective (Set a.out) (Iic #a.out) _ fun x => ⟨#x, mk_set_le x⟩
   rintro ⟨x, hx⟩
   simpa using le_mk_iff_exists_set.1 hx
 
-instance (a : Cardinal.{u}) : Small.{u} (Set.Iio a) :=
-  small_subset Iio_subset_Iic_self
+instance small_Iio (a : Cardinal.{u}) : Small.{u} (Iio a) := small_subset Iio_subset_Iic_self
+instance small_Icc (a b : Cardinal.{u}) : Small.{u} (Icc a b) := small_subset Icc_subset_Iic_self
+instance small_Ico (a b : Cardinal.{u}) : Small.{u} (Ico a b) := small_subset Ico_subset_Iio_self
+instance small_Ioc (a b : Cardinal.{u}) : Small.{u} (Ioc a b) := small_subset Ioc_subset_Iic_self
+instance small_Ioo (a b : Cardinal.{u}) : Small.{u} (Ioo a b) := small_subset Ioo_subset_Iio_self
 
 /-- A set of cardinals is bounded above iff it's small, i.e. it corresponds to a usual ZFC set. -/
 theorem bddAbove_iff_small {s : Set Cardinal.{u}} : BddAbove s ↔ Small.{u} s :=
@@ -840,8 +989,7 @@ theorem bddAbove_range_comp {ι : Type u} {f : ι → Cardinal.{v}} (hf : BddAbo
   rw [range_comp]
   exact bddAbove_image g hf
 
-theorem iSup_le_sum {ι} (f : ι → Cardinal) : iSup f ≤ sum f :=
-  ciSup_le' <| le_sum _
+/-! ### Bounds on suprema -/
 
 theorem sum_le_iSup_lift {ι : Type u}
     (f : ι → Cardinal.{max u v}) : sum f ≤ Cardinal.lift #ι * iSup f := by
@@ -852,60 +1000,97 @@ theorem sum_le_iSup {ι : Type u} (f : ι → Cardinal.{u}) : sum f ≤ #ι * iS
   rw [← lift_id #ι]
   exact sum_le_iSup_lift f
 
-theorem sum_nat_eq_add_sum_succ (f : ℕ → Cardinal.{u}) :
-    Cardinal.sum f = f 0 + Cardinal.sum fun i => f (i + 1) := by
-  refine (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans ?_
-  simp only [mk_sum, mk_out, lift_id, mk_sigma]
+/-- The lift of a supremum is the supremum of the lifts. -/
+theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :
+    lift.{u} (sSup s) = sSup (lift.{u} '' s) := by
+  apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
+  · intro c hc
+    by_contra h
+    obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
+    simp_rw [lift_le] at h hc
+    rw [csSup_le_iff' hs] at h
+    exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
+  · rintro i ⟨j, hj, rfl⟩
+    exact lift_le.2 (le_csSup hs hj)
 
--- Porting note: LFS is not in normal form.
--- @[simp]
-/-- A variant of `ciSup_of_empty` but with `0` on the RHS for convenience -/
-protected theorem iSup_of_empty {ι} (f : ι → Cardinal) [IsEmpty ι] : iSup f = 0 :=
-  ciSup_of_empty f
+/-- The lift of a supremum is the supremum of the lifts. -/
+theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
+    lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by
+  rw [iSup, iSup, lift_sSup hf, ← range_comp]
+  simp [Function.comp_def]
 
-lemma exists_eq_of_iSup_eq_of_not_isSuccLimit
+/-- To prove that the lift of a supremum is bounded by some cardinal `t`,
+it suffices to show that the lift of each cardinal is bounded by `t`. -/
+theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
+    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by
+  rw [lift_iSup hf]
+  exact ciSup_le' w
+
+@[simp]
+theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
+    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by
+  rw [lift_iSup hf]
+  exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _)
+
+/-- To prove an inequality between the lifts to a common universe of two different supremums,
+it suffices to show that the lift of each cardinal from the smaller supremum
+if bounded by the lift of some cardinal from the larger supremum.
+-/
+theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
+    {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
+    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (iSup f) ≤ lift.{w} (iSup f') := by
+  rw [lift_iSup hf, lift_iSup hf']
+  exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩
+
+/-- A variant of `lift_iSup_le_lift_iSup` with universes specialized via `w = v` and `w' = v'`.
+This is sometimes necessary to avoid universe unification issues. -/
+theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
+    {f' : ι' → Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : ι → ι')
+    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') :=
+  lift_iSup_le_lift_iSup hf hf' h
+
+lemma exists_eq_of_iSup_eq_of_not_isSuccPrelimit
     {ι : Type u} (f : ι → Cardinal.{v}) (ω : Cardinal.{v})
-    (hω : ¬ Order.IsSuccLimit ω)
+    (hω : ¬ IsSuccPrelimit ω)
     (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω := by
   subst h
-  refine (isLUB_csSup' ?_).exists_of_not_isSuccLimit hω
+  refine (isLUB_csSup' ?_).exists_of_not_isSuccPrelimit hω
   contrapose! hω with hf
   rw [iSup, csSup_of_not_bddAbove hf, csSup_empty]
-  exact Order.isSuccLimit_bot
+  exact isSuccPrelimit_bot
+
+lemma exists_eq_of_iSup_eq_of_not_isSuccLimit
+    {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{v}) (hf : BddAbove (range f))
+    {c : Cardinal.{v}} (hc : ¬ IsSuccLimit c)
+    (h : ⨆ i, f i = c) : ∃ i, f i = c := by
+  rw [Cardinal.isSuccLimit_iff] at hc
+  refine (not_and_or.mp hc).elim (fun e ↦ ⟨hι.some, ?_⟩)
+    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccPrelimit.{u, v} f c · h)
+  cases not_not.mp e
+  rw [← le_zero_iff] at h ⊢
+  exact (le_ciSup hf _).trans h
 
+set_option linter.deprecated false in
+@[deprecated exists_eq_of_iSup_eq_of_not_isSuccLimit (since := "2024-09-17")]
 lemma exists_eq_of_iSup_eq_of_not_isLimit
     {ι : Type u} [hι : Nonempty ι] (f : ι → Cardinal.{v}) (hf : BddAbove (range f))
     (ω : Cardinal.{v}) (hω : ¬ ω.IsLimit)
     (h : ⨆ i : ι, f i = ω) : ∃ i, f i = ω := by
   refine (not_and_or.mp hω).elim (fun e ↦ ⟨hι.some, ?_⟩)
-    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f ω · h)
+    (Cardinal.exists_eq_of_iSup_eq_of_not_isSuccPrelimit.{u, v} f ω · h)
   cases not_not.mp e
   rw [← le_zero_iff] at h ⊢
   exact (le_ciSup hf _).trans h
 
--- Porting note: simpNF is not happy with universe levels.
-@[simp, nolint simpNF]
-theorem lift_mk_shrink (α : Type u) [Small.{v} α] :
-    Cardinal.lift.{max u w} #(Shrink.{v} α) = Cardinal.lift.{max v w} #α :=
-  lift_mk_eq.2 ⟨(equivShrink α).symm⟩
-
-@[simp]
-theorem lift_mk_shrink' (α : Type u) [Small.{v} α] :
-    Cardinal.lift.{u} #(Shrink.{v} α) = Cardinal.lift.{v} #α :=
-  lift_mk_shrink.{u, v, 0} α
-
-@[simp]
-theorem lift_mk_shrink'' (α : Type max u v) [Small.{v} α] :
-    Cardinal.lift.{u} #(Shrink.{v} α) = #α := by
-  rw [← lift_umax', lift_mk_shrink.{max u v, v, 0} α, ← lift_umax, lift_id]
+/-! ### Indexed cardinal `prod` -/
 
 /-- The indexed product of cardinals is the cardinality of the Pi type
   (dependent product). -/
 def prod {ι : Type u} (f : ι → Cardinal) : Cardinal :=
-  #(∀ i, (f i).out)
+  #(Π i, (f i).out)
 
 @[simp]
-theorem mk_pi {ι : Type u} (α : ι → Type v) : #(∀ i, α i) = prod fun i => #(α i) :=
+theorem mk_pi {ι : Type u} (α : ι → Type v) : #(Π i, α i) = prod fun i => #(α i) :=
   mk_congr <| Equiv.piCongrRight fun _ => outMkEquiv.symm
 
 @[simp]
@@ -928,6 +1113,17 @@ theorem prod_eq_zero {ι} (f : ι → Cardinal.{u}) : prod f = 0 ↔ ∃ i, f i
 
 theorem prod_ne_zero {ι} (f : ι → Cardinal) : prod f ≠ 0 ↔ ∀ i, f i ≠ 0 := by simp [prod_eq_zero]
 
+theorem power_sum {ι} (a : Cardinal) (f : ι → Cardinal) :
+    a ^ sum f = prod fun i ↦ a ^ f i := by
+  induction a using Cardinal.inductionOn with | _ α =>
+  induction f using induction_on_pi with | _ f =>
+  simp_rw [prod, sum, power_def]
+  apply mk_congr
+  refine (Equiv.piCurry fun _ _ => α).trans ?_
+  refine Equiv.piCongrRight fun b => ?_
+  refine (Equiv.arrowCongr outMkEquiv (Equiv.refl α)).trans ?_
+  exact outMkEquiv.symm
+
 @[simp]
 theorem lift_prod {ι : Type u} (c : ι → Cardinal.{v}) :
     lift.{w} (prod c) = prod fun i => lift.{w} (c i) := by
@@ -950,115 +1146,26 @@ theorem prod_eq_of_fintype {α : Type u} [h : Fintype α] (f : α → Cardinal.{
         Cardinal.prod, lift_prod, Fintype.prod_option, lift_mul, ← h fun a => f (some a)]
     simp only [lift_id]
 
-@[simp]
-theorem lift_sInf (s : Set Cardinal) : lift.{u, v} (sInf s) = sInf (lift.{u, v} '' s) := by
-  rcases eq_empty_or_nonempty s with (rfl | hs)
-  · simp
-  · exact lift_monotone.map_csInf hs
-
-@[simp]
-theorem lift_iInf {ι} (f : ι → Cardinal) : lift.{u, v} (iInf f) = ⨅ i, lift.{u, v} (f i) := by
-  unfold iInf
-  convert lift_sInf (range f)
-  simp_rw [← comp_apply (f := lift), range_comp]
-
-theorem lift_down {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
-    b ≤ lift.{v,u} a → ∃ a', lift.{v,u} a' = b :=
-  inductionOn₂ a b fun α β => by
-    rw [← lift_id #β, ← lift_umax, ← lift_umax.{u, v}, lift_mk_le.{v}]
-    exact fun ⟨f⟩ =>
-      ⟨#(Set.range f),
-        Eq.symm <| lift_mk_eq.{_, _, v}.2
-          ⟨Function.Embedding.equivOfSurjective (Embedding.codRestrict _ f Set.mem_range_self)
-              fun ⟨a, ⟨b, e⟩⟩ => ⟨b, Subtype.eq e⟩⟩⟩
-
-theorem le_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
-    b ≤ lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' ≤ a :=
-  ⟨fun h =>
-    let ⟨a', e⟩ := lift_down h
-    ⟨a', e, lift_le.1 <| e.symm ▸ h⟩,
-    fun ⟨_, e, h⟩ => e ▸ lift_le.2 h⟩
-
-theorem lt_lift_iff {a : Cardinal.{u}} {b : Cardinal.{max u v}} :
-    b < lift.{v, u} a ↔ ∃ a', lift.{v, u} a' = b ∧ a' < a :=
-  ⟨fun h =>
-    let ⟨a', e⟩ := lift_down h.le
-    ⟨a', e, lift_lt.1 <| e.symm ▸ h⟩,
-    fun ⟨_, e, h⟩ => e ▸ lift_lt.2 h⟩
-
-@[simp]
-theorem lift_succ (a) : lift.{v, u} (succ a) = succ (lift.{v, u} a) :=
-  le_antisymm
-    (le_of_not_gt fun h => by
-      rcases lt_lift_iff.1 h with ⟨b, e, h⟩
-      rw [lt_succ_iff, ← lift_le, e] at h
-      exact h.not_lt (lt_succ _))
-    (succ_le_of_lt <| lift_lt.2 <| lt_succ a)
-
--- Porting note: simpNF is not happy with universe levels.
-@[simp, nolint simpNF]
-theorem lift_umax_eq {a : Cardinal.{u}} {b : Cardinal.{v}} :
-    lift.{max v w} a = lift.{max u w} b ↔ lift.{v} a = lift.{u} b := by
-  rw [← lift_lift.{v, w, u}, ← lift_lift.{u, w, v}, lift_inj]
-
-@[simp]
-theorem lift_min {a b : Cardinal} : lift.{u, v} (min a b) = min (lift.{u, v} a) (lift.{u, v} b) :=
-  lift_monotone.map_min
-
-@[simp]
-theorem lift_max {a b : Cardinal} : lift.{u, v} (max a b) = max (lift.{u, v} a) (lift.{u, v} b) :=
-  lift_monotone.map_max
-
-/-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_sSup {s : Set Cardinal} (hs : BddAbove s) :
-    lift.{u} (sSup s) = sSup (lift.{u} '' s) := by
-  apply ((le_csSup_iff' (bddAbove_image.{_,u} _ hs)).2 fun c hc => _).antisymm (csSup_le' _)
-  · intro c hc
-    by_contra h
-    obtain ⟨d, rfl⟩ := Cardinal.lift_down (not_le.1 h).le
-    simp_rw [lift_le] at h hc
-    rw [csSup_le_iff' hs] at h
-    exact h fun a ha => lift_le.1 <| hc (mem_image_of_mem _ ha)
-  · rintro i ⟨j, hj, rfl⟩
-    exact lift_le.2 (le_csSup hs hj)
-
-/-- The lift of a supremum is the supremum of the lifts. -/
-theorem lift_iSup {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f)) :
-    lift.{u} (iSup f) = ⨆ i, lift.{u} (f i) := by
-  rw [iSup, iSup, lift_sSup hf, ← range_comp]
-  simp [Function.comp]
-
-/-- To prove that the lift of a supremum is bounded by some cardinal `t`,
-it suffices to show that the lift of each cardinal is bounded by `t`. -/
-theorem lift_iSup_le {ι : Type v} {f : ι → Cardinal.{w}} {t : Cardinal} (hf : BddAbove (range f))
-    (w : ∀ i, lift.{u} (f i) ≤ t) : lift.{u} (iSup f) ≤ t := by
-  rw [lift_iSup hf]
-  exact ciSup_le' w
-
-@[simp]
-theorem lift_iSup_le_iff {ι : Type v} {f : ι → Cardinal.{w}} (hf : BddAbove (range f))
-    {t : Cardinal} : lift.{u} (iSup f) ≤ t ↔ ∀ i, lift.{u} (f i) ≤ t := by
-  rw [lift_iSup hf]
-  exact ciSup_le_iff' (bddAbove_range_comp.{_,_,u} hf _)
-
-universe v' w'
-
-/-- To prove an inequality between the lifts to a common universe of two different supremums,
-it suffices to show that the lift of each cardinal from the smaller supremum
-if bounded by the lift of some cardinal from the larger supremum.
--/
-theorem lift_iSup_le_lift_iSup {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{w}}
-    {f' : ι' → Cardinal.{w'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) {g : ι → ι'}
-    (h : ∀ i, lift.{w'} (f i) ≤ lift.{w} (f' (g i))) : lift.{w'} (iSup f) ≤ lift.{w} (iSup f') := by
-  rw [lift_iSup hf, lift_iSup hf']
-  exact ciSup_mono' (bddAbove_range_comp.{_,_,w} hf' _) fun i => ⟨_, h i⟩
+/-- **König's theorem** -/
+theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
+  lt_of_not_ge fun ⟨F⟩ => by
+    have : Inhabited (∀ i : ι, (g i).out) := by
+      refine ⟨fun i => Classical.choice <| mk_ne_zero_iff.1 ?_⟩
+      rw [mk_out]
+      exact (H i).ne_bot
+    let G := invFun F
+    have sG : Surjective G := invFun_surjective F.2
+    choose C hc using
+      show ∀ i, ∃ b, ∀ a, G ⟨i, a⟩ i ≠ b by
+        intro i
+        simp only [not_exists.symm, not_forall.symm]
+        refine fun h => (H i).not_le ?_
+        rw [← mk_out (f i), ← mk_out (g i)]
+        exact ⟨Embedding.ofSurjective _ h⟩
+    let ⟨⟨i, a⟩, h⟩ := sG C
+    exact hc i a (congr_fun h _)
 
-/-- A variant of `lift_iSup_le_lift_iSup` with universes specialized via `w = v` and `w' = v'`.
-This is sometimes necessary to avoid universe unification issues. -/
-theorem lift_iSup_le_lift_iSup' {ι : Type v} {ι' : Type v'} {f : ι → Cardinal.{v}}
-    {f' : ι' → Cardinal.{v'}} (hf : BddAbove (range f)) (hf' : BddAbove (range f')) (g : ι → ι')
-    (h : ∀ i, lift.{v'} (f i) ≤ lift.{v} (f' (g i))) : lift.{v'} (iSup f) ≤ lift.{v} (iSup f') :=
-  lift_iSup_le_lift_iSup hf hf' h
+/-! ### The first infinite cardinal `aleph0` -/
 
 /-- `ℵ₀` is the smallest infinite cardinal. -/
 def aleph0 : Cardinal.{u} :=
@@ -1097,7 +1204,6 @@ theorem lift_lt_aleph0 {c : Cardinal.{u}} : lift.{v} c < ℵ₀ ↔ c < ℵ₀ :
   rw [← lift_aleph0.{v, u}, lift_lt]
 
 /-! ### Properties about the cast from `ℕ` -/
-section castFromN
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
@@ -1264,6 +1370,9 @@ lemma two_le_iff_one_lt {c : Cardinal} : 2 ≤ c ↔ 1 < c := by
 @[simp]
 theorem succ_zero : succ (0 : Cardinal) = 1 := by norm_cast
 
+-- This works generally to prove inequalities between numeric cardinals.
+theorem one_lt_two : (1 : Cardinal) < 2 := by norm_cast
+
 theorem exists_finset_le_card (α : Type*) (n : ℕ) (h : n ≤ #α) :
     ∃ s : Finset α, n ≤ s.card := by
   obtain hα|hα := finite_or_infinite α
@@ -1292,6 +1401,8 @@ theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by
 theorem lt_one_iff_zero {c : Cardinal} : c < 1 ↔ c = 0 := by
   simpa using lt_succ_bot_iff (a := c)
 
+/-! ### Properties about `aleph0` -/
+
 theorem nat_lt_aleph0 (n : ℕ) : (n : Cardinal.{u}) < ℵ₀ :=
   succ_le_iff.1
     (by
@@ -1325,27 +1436,52 @@ theorem aleph0_le {c : Cardinal} : ℵ₀ ≤ c ↔ ∀ n : ℕ, ↑n ≤ c :=
       rcases lt_aleph0.1 hn with ⟨n, rfl⟩
       exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))⟩
 
-theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ :=
-  isSuccLimit_of_succ_lt fun a ha => by
+theorem isSuccPrelimit_aleph0 : IsSuccPrelimit ℵ₀ :=
+  isSuccPrelimit_of_succ_lt fun a ha => by
     rcases lt_aleph0.1 ha with ⟨n, rfl⟩
     rw [← nat_succ]
     apply nat_lt_aleph0
 
+theorem isSuccLimit_aleph0 : IsSuccLimit ℵ₀ := by
+  rw [Cardinal.isSuccLimit_iff]
+  exact ⟨aleph0_ne_zero, isSuccPrelimit_aleph0⟩
+
+lemma not_isSuccLimit_natCast : (n : ℕ) → ¬ IsSuccLimit (n : Cardinal.{u})
+  | 0, e => e.1 isMin_bot
+  | Nat.succ n, e => Order.not_isSuccPrelimit_succ _ (nat_succ n ▸ e.2)
+
+theorem not_isSuccLimit_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ¬ IsSuccLimit c := by
+  obtain ⟨n, rfl⟩ := lt_aleph0.1 h
+  exact not_isSuccLimit_natCast n
+
+theorem aleph0_le_of_isSuccLimit {c : Cardinal} (h : IsSuccLimit c) : ℵ₀ ≤ c := by
+  contrapose! h
+  exact not_isSuccLimit_of_lt_aleph0 h
+
+section deprecated
+
+set_option linter.deprecated false
+
+@[deprecated isSuccLimit_aleph0 (since := "2024-09-17")]
 theorem isLimit_aleph0 : IsLimit ℵ₀ :=
-  ⟨aleph0_ne_zero, isSuccLimit_aleph0⟩
+  ⟨aleph0_ne_zero, isSuccPrelimit_aleph0⟩
 
+@[deprecated not_isSuccLimit_natCast (since := "2024-09-17")]
 lemma not_isLimit_natCast : (n : ℕ) → ¬ IsLimit (n : Cardinal.{u})
   | 0, e => e.1 rfl
-  | Nat.succ n, e => Order.not_isSuccLimit_succ _ (nat_succ n ▸ e.2)
+  | Nat.succ n, e => Order.not_isSuccPrelimit_succ _ (nat_succ n ▸ e.2)
 
+@[deprecated aleph0_le_of_isSuccLimit (since := "2024-09-17")]
 theorem IsLimit.aleph0_le {c : Cardinal} (h : IsLimit c) : ℵ₀ ≤ c := by
   by_contra! h'
   rcases lt_aleph0.1 h' with ⟨n, rfl⟩
   exact not_isLimit_natCast n h
 
+end deprecated
+
 lemma exists_eq_natCast_of_iSup_eq {ι : Type u} [Nonempty ι] (f : ι → Cardinal.{v})
     (hf : BddAbove (range f)) (n : ℕ) (h : ⨆ i, f i = n) : ∃ i, f i = n :=
-  exists_eq_of_iSup_eq_of_not_isLimit.{u, v} f hf _ (not_isLimit_natCast n) h
+  exists_eq_of_iSup_eq_of_not_isSuccLimit.{u, v} f hf (not_isSuccLimit_natCast n) h
 
 @[simp]
 theorem range_natCast : range ((↑) : ℕ → Cardinal) = Iio ℵ₀ :=
@@ -1413,7 +1549,7 @@ theorem nsmul_lt_aleph0_iff {n : ℕ} {a : Cardinal} : n • a < ℵ₀ ↔ n =
   cases n with
   | zero => simpa using nat_lt_aleph0 0
   | succ n =>
-      simp only [Nat.succ_ne_zero, false_or_iff]
+      simp only [Nat.succ_ne_zero, false_or]
       induction' n with n ih
       · simp
       rw [succ_nsmul, add_lt_aleph0_iff, ih, and_self_iff]
@@ -1552,31 +1688,7 @@ theorem mk_int : #ℤ = ℵ₀ :=
 theorem mk_pNat : #ℕ+ = ℵ₀ :=
   mk_denumerable ℕ+
 
-end castFromN
-
-variable {c : Cardinal}
-
-/-- **König's theorem** -/
-theorem sum_lt_prod {ι} (f g : ι → Cardinal) (H : ∀ i, f i < g i) : sum f < prod g :=
-  lt_of_not_ge fun ⟨F⟩ => by
-    have : Inhabited (∀ i : ι, (g i).out) := by
-      refine ⟨fun i => Classical.choice <| mk_ne_zero_iff.1 ?_⟩
-      rw [mk_out]
-      exact (H i).ne_bot
-    let G := invFun F
-    have sG : Surjective G := invFun_surjective F.2
-    choose C hc using
-      show ∀ i, ∃ b, ∀ a, G ⟨i, a⟩ i ≠ b by
-        intro i
-        simp only [not_exists.symm, not_forall.symm]
-        refine fun h => (H i).not_le ?_
-        rw [← mk_out (f i), ← mk_out (g i)]
-        exact ⟨Embedding.ofSurjective _ h⟩
-    let ⟨⟨i, a⟩, h⟩ := sG C
-    exact hc i a (congr_fun h _)
-
-/-! Cardinalities of sets: cardinality of empty, finite sets, unions, subsets etc. -/
-section sets
+/-! ### Cardinalities of basic sets and types -/
 
 -- porting note (#10618): simp can prove this
 -- @[simp]
@@ -1924,9 +2036,7 @@ theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z
   have := exists_not_mem_of_length_lt [x, y] this
   simpa [not_or] using this
 
-end sets
-
-section powerlt
+/-! ### `powerlt` operation -/
 
 /-- The function `a ^< b`, defined as the supremum of `a ^ c` for `c < b`. -/
 def powerlt (a b : Cardinal.{u}) : Cardinal.{u} :=
@@ -1971,7 +2081,6 @@ theorem powerlt_zero {a : Cardinal} : a ^< 0 = 0 := by
   convert Cardinal.iSup_of_empty _
   exact Subtype.isEmpty_of_false fun x => mem_Iio.not.mpr (Cardinal.zero_le x).not_lt
 
-end powerlt
 end Cardinal
 
 -- namespace Tactic
@@ -1993,3 +2102,5 @@ end Cardinal
 --     failed
 
 -- end Tactic
+
+set_option linter.style.longFile 2200
diff --git a/Mathlib/SetTheory/Cardinal/Cofinality.lean b/Mathlib/SetTheory/Cardinal/Cofinality.lean
index 0db148a847ba8..53bfb76a4a06e 100644
--- a/Mathlib/SetTheory/Cardinal/Cofinality.lean
+++ b/Mathlib/SetTheory/Cardinal/Cofinality.lean
@@ -303,30 +303,59 @@ theorem lsub_lt_ord {ι} {f : ι → Ordinal} {c : Ordinal} (hι : #ι < c.cof)
     (∀ i, f i < c) → lsub.{u, u} f < c :=
   lsub_lt_ord_lift (by rwa [(#ι).lift_id])
 
+set_option linter.deprecated false in
+theorem cof_iSup_le_lift {ι} {f : ι → Ordinal} (H : ∀ i, f i < iSup f) :
+    cof (iSup f) ≤ Cardinal.lift.{v, u} #ι := by
+  rw [← Ordinal.sup] at *
+  rw [← sup_eq_lsub_iff_lt_sup.{u, v}] at H
+  rw [H]
+  exact cof_lsub_le_lift f
+
+set_option linter.deprecated false in
+@[deprecated cof_iSup_le_lift (since := "2024-08-27")]
 theorem cof_sup_le_lift {ι} {f : ι → Ordinal} (H : ∀ i, f i < sup.{u, v} f) :
     cof (sup.{u, v} f) ≤ Cardinal.lift.{v, u} #ι := by
   rw [← sup_eq_lsub_iff_lt_sup.{u, v}] at H
   rw [H]
   exact cof_lsub_le_lift f
 
+theorem cof_iSup_le {ι} {f : ι → Ordinal} (H : ∀ i, f i < iSup f) :
+    cof (iSup f) ≤ #ι := by
+  rw [← (#ι).lift_id]
+  exact cof_iSup_le_lift H
+
+set_option linter.deprecated false in
+@[deprecated cof_iSup_le (since := "2024-08-27")]
 theorem cof_sup_le {ι} {f : ι → Ordinal} (H : ∀ i, f i < sup.{u, u} f) :
     cof (sup.{u, u} f) ≤ #ι := by
   rw [← (#ι).lift_id]
   exact cof_sup_le_lift H
 
+theorem iSup_lt_ord_lift {ι} {f : ι → Ordinal} {c : Ordinal} (hι : Cardinal.lift.{v, u} #ι < c.cof)
+    (hf : ∀ i, f i < c) : iSup f < c :=
+  (sup_le_lsub.{u, v} f).trans_lt (lsub_lt_ord_lift hι hf)
+
+set_option linter.deprecated false in
+@[deprecated iSup_lt_ord_lift (since := "2024-08-27")]
 theorem sup_lt_ord_lift {ι} {f : ι → Ordinal} {c : Ordinal} (hι : Cardinal.lift.{v, u} #ι < c.cof)
     (hf : ∀ i, f i < c) : sup.{u, v} f < c :=
-  (sup_le_lsub.{u, v} f).trans_lt (lsub_lt_ord_lift hι hf)
+  iSup_lt_ord_lift hι hf
 
+theorem iSup_lt_ord {ι} {f : ι → Ordinal} {c : Ordinal} (hι : #ι < c.cof) :
+    (∀ i, f i < c) → iSup f < c :=
+  iSup_lt_ord_lift (by rwa [(#ι).lift_id])
+
+set_option linter.deprecated false in
+@[deprecated iSup_lt_ord (since := "2024-08-27")]
 theorem sup_lt_ord {ι} {f : ι → Ordinal} {c : Ordinal} (hι : #ι < c.cof) :
     (∀ i, f i < c) → sup.{u, u} f < c :=
   sup_lt_ord_lift (by rwa [(#ι).lift_id])
 
 theorem iSup_lt_lift {ι} {f : ι → Cardinal} {c : Cardinal}
     (hι : Cardinal.lift.{v, u} #ι < c.ord.cof)
-    (hf : ∀ i, f i < c) : iSup.{max u v + 1, u + 1} f < c := by
+    (hf : ∀ i, f i < c) : iSup f < c := by
   rw [← ord_lt_ord, iSup_ord (Cardinal.bddAbove_range.{u, v} _)]
-  refine sup_lt_ord_lift hι fun i => ?_
+  refine iSup_lt_ord_lift hι fun i => ?_
   rw [ord_lt_ord]
   apply hf
 
@@ -337,7 +366,7 @@ theorem iSup_lt {ι} {f : ι → Cardinal} {c : Cardinal} (hι : #ι < c.ord.cof
 theorem nfpFamily_lt_ord_lift {ι} {f : ι → Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
     (hc' : Cardinal.lift.{v, u} #ι < cof c) (hf : ∀ (i), ∀ b < c, f i b < c) {a} (ha : a < c) :
     nfpFamily.{u, v} f a < c := by
-  refine sup_lt_ord_lift ((Cardinal.lift_le.2 (mk_list_le_max ι)).trans_lt ?_) fun l => ?_
+  refine iSup_lt_ord_lift ((Cardinal.lift_le.2 (mk_list_le_max ι)).trans_lt ?_) fun l => ?_
   · rw [lift_max]
     apply max_lt _ hc'
     rwa [Cardinal.lift_aleph0]
@@ -651,11 +680,14 @@ theorem aleph_cof {o : Ordinal} (ho : o.IsLimit) : (aleph o).ord.cof = o.cof :=
   aleph_isNormal.cof_eq ho
 
 @[simp]
-theorem cof_omega : cof ω = ℵ₀ :=
-  (aleph0_le_cof.2 omega_isLimit).antisymm' <| by
-    rw [← card_omega]
+theorem cof_omega0 : cof ω = ℵ₀ :=
+  (aleph0_le_cof.2 omega0_isLimit).antisymm' <| by
+    rw [← card_omega0]
     apply cof_le_card
 
+@[deprecated (since := "2024-09-30")]
+alias cof_omega := cof_omega0
+
 theorem cof_eq' (r : α → α → Prop) [IsWellOrder α r] (h : IsLimit (type r)) :
     ∃ S : Set α, (∀ a, ∃ b ∈ S, r a b) ∧ #S = cof (type r) :=
   let ⟨S, H, e⟩ := cof_eq r
@@ -685,11 +717,11 @@ theorem cof_univ : cof univ.{u, v} = Cardinal.univ.{u, v} :=
       cases' Quotient.exact e with f
       have f := Equiv.ulift.symm.trans f
       let g a := (f a).1
-      let o := succ (sup.{u, u} g)
+      let o := succ (iSup g)
       rcases H o with ⟨b, h, l⟩
       refine l (lt_succ_iff.2 ?_)
       rw [← show g (f.symm ⟨b, h⟩) = b by simp [g]]
-      apply le_sup)
+      apply Ordinal.le_iSup)
 
 /-! ### Infinite pigeonhole principle -/
 
@@ -777,18 +809,28 @@ theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ :=
     rcases lt_aleph0.1 hx with ⟨n, rfl⟩
     exact mod_cast nat_lt_aleph0 (2 ^ n)⟩
 
-protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c :=
-  isSuccLimit_of_succ_lt fun x h => (succ_le_of_lt <| cantor x).trans_lt (H.two_power_lt h)
+protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c := by
+  rw [Cardinal.isSuccLimit_iff]
+  exact ⟨H.ne_zero, isSuccPrelimit_of_succ_lt fun x h =>
+    (succ_le_of_lt <| cantor x).trans_lt (H.two_power_lt h)⟩
+
+protected theorem IsStrongLimit.isSuccPrelimit {c} (H : IsStrongLimit c) : IsSuccPrelimit c :=
+  H.isSuccLimit.isSuccPrelimit
 
+theorem IsStrongLimit.aleph0_le {c} (H : IsStrongLimit c) : ℵ₀ ≤ c :=
+  aleph0_le_of_isSuccLimit H.isSuccLimit
+
+set_option linter.deprecated false in
+@[deprecated IsStrongLimit.isSuccLimit (since := "2024-09-17")]
 theorem IsStrongLimit.isLimit {c} (H : IsStrongLimit c) : IsLimit c :=
-  ⟨H.ne_zero, H.isSuccLimit⟩
+  ⟨H.ne_zero, H.isSuccPrelimit⟩
 
-theorem isStrongLimit_beth {o : Ordinal} (H : IsSuccLimit o) : IsStrongLimit (beth o) := by
+theorem isStrongLimit_beth {o : Ordinal} (H : IsSuccPrelimit o) : IsStrongLimit (beth o) := by
   rcases eq_or_ne o 0 with (rfl | h)
   · rw [beth_zero]
     exact isStrongLimit_aleph0
   · refine ⟨beth_ne_zero o, fun a ha => ?_⟩
-    rw [beth_limit ⟨h, isSuccLimit_iff_succ_lt.1 H⟩] at ha
+    rw [beth_limit ⟨h, isSuccPrelimit_iff_succ_lt.1 H⟩] at ha
     rcases exists_lt_of_lt_ciSup' ha with ⟨⟨i, hi⟩, ha⟩
     have := power_le_power_left two_ne_zero ha.le
     rw [← beth_succ] at this
@@ -804,7 +846,7 @@ theorem mk_bounded_subset {α : Type*} (h : ∀ x < #α, (2^x) < #α) {r : α 
     rintro ⟨s, hs⟩
     exact (not_unbounded_iff s).2 hs (unbounded_of_isEmpty s)
   have h' : IsStrongLimit #α := ⟨ha, h⟩
-  have ha := h'.isLimit.aleph0_le
+  have ha := h'.aleph0_le
   apply le_antisymm
   · have : { s : Set α | Bounded r s } = ⋃ i, 𝒫{ j | r j i } := setOf_exists _
     rw [← coe_setOf, this]
@@ -839,7 +881,7 @@ theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, (2^x) < #α) :
     exact lt_cof_type hs
   · refine @mk_le_of_injective α _ (fun x => Subtype.mk {x} ?_) ?_
     · rw [mk_singleton]
-      exact one_lt_aleph0.trans_le (aleph0_le_cof.2 (ord_isLimit h'.isLimit.aleph0_le))
+      exact one_lt_aleph0.trans_le (aleph0_le_cof.2 (ord_isLimit h'.aleph0_le))
     · intro a b hab
       simpa [singleton_eq_singleton_iff] using hab
 
@@ -952,13 +994,25 @@ theorem lsub_lt_ord_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c
     (∀ i, f i < c.ord) → Ordinal.lsub f < c.ord :=
   lsub_lt_ord (by rwa [hc.cof_eq])
 
+theorem iSup_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c)
+    (hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c.ord) → iSup f < c.ord :=
+  iSup_lt_ord_lift (by rwa [hc.cof_eq])
+
+set_option linter.deprecated false in
+@[deprecated iSup_lt_ord_lift_of_isRegular (since := "2024-08-27")]
 theorem sup_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c)
     (hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c.ord) → Ordinal.sup.{u, v} f < c.ord :=
-  sup_lt_ord_lift (by rwa [hc.cof_eq])
+  iSup_lt_ord_lift_of_isRegular hc hι
 
+theorem iSup_lt_ord_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c) (hι : #ι < c) :
+    (∀ i, f i < c.ord) → iSup f < c.ord :=
+  iSup_lt_ord (by rwa [hc.cof_eq])
+
+set_option linter.deprecated false in
+@[deprecated iSup_lt_ord_of_isRegular (since := "2024-08-27")]
 theorem sup_lt_ord_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c) (hι : #ι < c) :
     (∀ i, f i < c.ord) → Ordinal.sup f < c.ord :=
-  sup_lt_ord (by rwa [hc.cof_eq])
+  iSup_lt_ord_of_isRegular hc hι
 
 theorem blsub_lt_ord_lift_of_isRegular {o : Ordinal} {f : ∀ a < o, Ordinal} {c} (hc : IsRegular c)
     (ho : Cardinal.lift.{v, u} o.card < c) :
@@ -1145,6 +1199,16 @@ namespace Ordinal
 open Cardinal
 open scoped Ordinal
 
+-- TODO: generalize universes
+lemma iSup_sequence_lt_omega1 {α : Type u} [Countable α]
+    (o : α → Ordinal.{max u v}) (ho : ∀ n, o n < ω₁) :
+    iSup o < ω₁ := by
+  apply iSup_lt_ord_lift _ ho
+  rw [Cardinal.isRegular_aleph_one.cof_eq]
+  exact lt_of_le_of_lt mk_le_aleph0 aleph0_lt_aleph_one
+
+set_option linter.deprecated false in
+@[deprecated iSup_sequence_lt_omega1 (since := "2024-08-27")]
 lemma sup_sequence_lt_omega1 {α} [Countable α] (o : α → Ordinal) (ho : ∀ n, o n < ω₁) :
     sup o < ω₁ := by
   apply sup_lt_ord_lift _ ho
diff --git a/Mathlib/SetTheory/Cardinal/Continuum.lean b/Mathlib/SetTheory/Cardinal/Continuum.lean
index 8614f68343c3b..9aeb9a15becb8 100644
--- a/Mathlib/SetTheory/Cardinal/Continuum.lean
+++ b/Mathlib/SetTheory/Cardinal/Continuum.lean
@@ -166,15 +166,22 @@ theorem continuum_mul_ofNat {n : ℕ} [Nat.AtLeastTwo n] : 𝔠 * no_index (OfNa
 
 
 @[simp]
-theorem aleph0_power_aleph0 : aleph0.{u} ^ aleph0.{u} = 𝔠 :=
+theorem aleph0_power_aleph0 : ℵ₀ ^ ℵ₀ = 𝔠 :=
   power_self_eq le_rfl
 
 @[simp]
-theorem nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) : (n ^ aleph0.{u} : Cardinal.{u}) = 𝔠 :=
+theorem nat_power_aleph0 {n : ℕ} (hn : 2 ≤ n) : n ^ ℵ₀ = 𝔠 :=
   nat_power_eq le_rfl hn
 
 @[simp]
-theorem continuum_power_aleph0 : continuum.{u} ^ aleph0.{u} = 𝔠 := by
+theorem continuum_power_aleph0 : 𝔠 ^ ℵ₀ = 𝔠 := by
   rw [← two_power_aleph0, ← power_mul, mul_eq_left le_rfl le_rfl aleph0_ne_zero]
 
+theorem power_aleph0_of_le_continuum {x : Cardinal} (h₁ : 2 ≤ x) (h₂ : x ≤ 𝔠) : x ^ ℵ₀ = 𝔠 := by
+  apply le_antisymm
+  · rw [← continuum_power_aleph0]
+    exact power_le_power_right h₂
+  · rw [← two_power_aleph0]
+    exact power_le_power_right h₁
+
 end Cardinal
diff --git a/Mathlib/SetTheory/Cardinal/CountableCover.lean b/Mathlib/SetTheory/Cardinal/CountableCover.lean
index 35f9e9f8483e6..65caa5dd364f5 100644
--- a/Mathlib/SetTheory/Cardinal/CountableCover.lean
+++ b/Mathlib/SetTheory/Cardinal/CountableCover.lean
@@ -25,7 +25,7 @@ universe u v
 
 /-- If a set `t` is eventually covered by a countable family of sets, all with cardinality at
 most `a`, then the cardinality of `t` is also bounded by `a`.
-Supersed by `mk_le_of_countable_eventually_mem` which does not assume
+Superseded by `mk_le_of_countable_eventually_mem` which does not assume
 that the indexing set lives in the same universe. -/
 lemma mk_subtype_le_of_countable_eventually_mem_aux {α ι : Type u} {a : Cardinal}
     [Countable ι] {f : ι → Set α} {l : Filter ι} [NeBot l]
diff --git a/Mathlib/SetTheory/Cardinal/Divisibility.lean b/Mathlib/SetTheory/Cardinal/Divisibility.lean
index 658f7943b383d..4dac7f8531de2 100644
--- a/Mathlib/SetTheory/Cardinal/Divisibility.lean
+++ b/Mathlib/SetTheory/Cardinal/Divisibility.lean
@@ -134,7 +134,7 @@ theorem is_prime_iff {a : Cardinal} : Prime a ↔ ℵ₀ ≤ a ∨ ∃ p : ℕ,
 theorem isPrimePow_iff {a : Cardinal} : IsPrimePow a ↔ ℵ₀ ≤ a ∨ ∃ n : ℕ, a = n ∧ IsPrimePow n := by
   by_cases h : ℵ₀ ≤ a
   · simp [h, (prime_of_aleph0_le h).isPrimePow]
-  simp only [h, Nat.cast_inj, exists_eq_left', false_or_iff, isPrimePow_nat_iff]
+  simp only [h, Nat.cast_inj, exists_eq_left', false_or, isPrimePow_nat_iff]
   lift a to ℕ using not_le.mp h
   rw [isPrimePow_def]
   refine
diff --git a/Mathlib/SetTheory/Cardinal/Finite.lean b/Mathlib/SetTheory/Cardinal/Finite.lean
index 16ceb279b9710..08b850cedaa5a 100644
--- a/Mathlib/SetTheory/Cardinal/Finite.lean
+++ b/Mathlib/SetTheory/Cardinal/Finite.lean
@@ -95,15 +95,19 @@ protected theorem bijective_iff_injective_and_card [Finite β] (f : α → β) :
   rw [← and_congr_right_iff, ← Bijective,
     card_eq_fintype_card, card_eq_fintype_card, Fintype.bijective_iff_injective_and_card]
 
+#adaptation_note
+/--
+After nightly-2024-09-06 we can remove the `_root_` prefixes below.
+-/
 protected theorem bijective_iff_surjective_and_card [Finite α] (f : α → β) :
     Bijective f ↔ Surjective f ∧ Nat.card α = Nat.card β := by
   classical
-  rw [and_comm, Bijective, and_congr_left_iff]
+  rw [_root_.and_comm, Bijective, and_congr_left_iff]
   intro h
   have := Fintype.ofFinite α
   have := Fintype.ofSurjective f h
   revert h
-  rw [← and_congr_left_iff, ← Bijective, ← and_comm,
+  rw [← and_congr_left_iff, ← Bijective, ← _root_.and_comm,
     card_eq_fintype_card, card_eq_fintype_card, Fintype.bijective_iff_surjective_and_card]
 
 theorem _root_.Function.Injective.bijective_of_nat_card_le [Finite β] {f : α → β}
@@ -152,6 +156,14 @@ lemma card_preimage_of_injOn {f : α → β} {s : Set β} (hf : (f ⁻¹' s).Inj
 lemma card_preimage_of_injective {f : α → β} {s : Set β} (hf : Injective f) (hsf : s ⊆ range f) :
     Nat.card (f ⁻¹' s) = Nat.card s := card_preimage_of_injOn hf.injOn hsf
 
+@[simp] lemma card_univ : Nat.card (univ : Set α) = Nat.card α :=
+  card_congr (Equiv.Set.univ α)
+
+lemma card_range_of_injective {f : α → β} (hf : Injective f) :
+    Nat.card (range f) = Nat.card α := by
+  rw [← Nat.card_preimage_of_injective hf le_rfl]
+  simp
+
 end Set
 
 /-- If the cardinality is positive, that means it is a finite type, so there is
diff --git a/Mathlib/SetTheory/Cardinal/Ordinal.lean b/Mathlib/SetTheory/Cardinal/Ordinal.lean
index b9f6c63e10014..2126055537d6c 100644
--- a/Mathlib/SetTheory/Cardinal/Ordinal.lean
+++ b/Mathlib/SetTheory/Cardinal/Ordinal.lean
@@ -6,6 +6,7 @@ Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 import Mathlib.Order.Bounded
 import Mathlib.SetTheory.Cardinal.PartENat
 import Mathlib.SetTheory.Ordinal.Principal
+import Mathlib.SetTheory.Ordinal.Enum
 import Mathlib.Tactic.Linarith
 
 /-!
@@ -16,8 +17,7 @@ using ordinals.
 
 ## Main definitions
 
-* The function `Cardinal.aleph'` gives the cardinals listed by their ordinal
-  index, and is the inverse of `Cardinal.aleph/idx`.
+* The function `Cardinal.aleph'` gives the cardinals listed by their ordinal index.
   `aleph' n = n`, `aleph' ω = ℵ₀`, `aleph' (ω + 1) = succ ℵ₀`, etc.
   It is an order isomorphism between ordinals and cardinals.
 * The function `Cardinal.aleph` gives the infinite cardinals listed by their
@@ -29,13 +29,11 @@ using ordinals.
   `beth (succ o) = 2 ^ beth o`, and for a limit ordinal `o`, `beth o` is the supremum of `beth a`
   for `a < o`.
 
-## Main Statements
+## Main statements
 
 * `Cardinal.mul_eq_max` and `Cardinal.add_eq_max` state that the product (resp. sum) of two infinite
   cardinals is just their maximum. Several variations around this fact are also given.
 * `Cardinal.mk_list_eq_mk` : when `α` is infinite, `α` and `List α` have the same cardinality.
-* simp lemmas for inequalities between `bit0 a` and `bit1 b` are registered, making `simp`
-  able to prove inequalities about numeral cardinals.
 
 ## Tags
 
@@ -64,7 +62,7 @@ theorem ord_isLimit {c} (co : ℵ₀ ≤ c) : (ord c).IsLimit := by
     rw [← ord_le, ← le_succ_of_isLimit, ord_le]
     · exact co.trans h
     · rw [ord_aleph0]
-      exact omega_isLimit
+      exact omega0_isLimit
 
 theorem noMaxOrder {c} (h : ℵ₀ ≤ c) : NoMaxOrder c.ord.toType :=
   toType_noMax_of_succ_lt (ord_isLimit h).2
@@ -73,6 +71,26 @@ theorem noMaxOrder {c} (h : ℵ₀ ≤ c) : NoMaxOrder c.ord.toType :=
 
 section aleph
 
+/-- The `aleph'` function gives the cardinals listed by their ordinal index. `aleph' n = n`,
+`aleph' ω = ℵ₀`, `aleph' (ω + 1) = succ ℵ₀`, etc.
+
+For the more common aleph function skipping over finite cardinals, see `Cardinal.aleph`. -/
+def aleph' : Ordinal.{u} ≃o Cardinal.{u} := by
+  let f := RelEmbedding.collapse Cardinal.ord.orderEmbedding.ltEmbedding.{u}
+  refine (OrderIso.ofRelIsoLT <| RelIso.ofSurjective f ?_).symm
+  apply f.eq_or_principal.resolve_right
+  rintro ⟨o, e⟩
+  have : ∀ c, f c < o := fun c => (e _).1 ⟨_, rfl⟩
+  refine Ordinal.inductionOn o ?_ this
+  intro α r _ h
+  let s := ⨆ a, invFun f (Ordinal.typein r a)
+  apply (lt_succ s).not_le
+  have I : Injective f := f.toEmbedding.injective
+  simpa only [typein_enum, leftInverse_invFun I (succ s)] using
+    le_ciSup
+      (Cardinal.bddAbove_range.{u, u} fun a : α => invFun f (Ordinal.typein r a))
+      (Ordinal.enum r ⟨_, h (succ s)⟩)
+
 /-- The `aleph'` index function, which gives the ordinal index of a cardinal.
   (The `aleph'` part is because unlike `aleph` this counts also the
   finite stages. So `alephIdx n = n`, `alephIdx ω = ω`,
@@ -81,59 +99,59 @@ section aleph
   i.e., it is order preserving and its range is an initial segment of the ordinals.
   For the basic function version, see `alephIdx`.
   For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/
+@[deprecated aleph' (since := "2024-08-28")]
 def alephIdx.initialSeg : @InitialSeg Cardinal Ordinal (· < ·) (· < ·) :=
   @RelEmbedding.collapse Cardinal Ordinal (· < ·) (· < ·) _ Cardinal.ord.orderEmbedding.ltEmbedding
 
+/-- The `aleph'` index function, which gives the ordinal index of a cardinal.
+  (The `aleph'` part is because unlike `aleph` this counts also the
+  finite stages. So `alephIdx n = n`, `alephIdx ℵ₀ = ω`,
+  `alephIdx ℵ₁ = ω + 1` and so on.)
+  In this version, we register additionally that this function is an order isomorphism
+  between cardinals and ordinals.
+  For the basic function version, see `alephIdx`. -/
+@[deprecated aleph' (since := "2024-08-28")]
+def alephIdx.relIso : @RelIso Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) :=
+  aleph'.symm.toRelIsoLT
+
 /-- The `aleph'` index function, which gives the ordinal index of a cardinal.
   (The `aleph'` part is because unlike `aleph` this counts also the
   finite stages. So `alephIdx n = n`, `alephIdx ω = ω`,
   `alephIdx ℵ₁ = ω + 1` and so on.)
   For an upgraded version stating that the range is everything, see `AlephIdx.rel_iso`. -/
+@[deprecated aleph' (since := "2024-08-28")]
 def alephIdx : Cardinal → Ordinal :=
-  alephIdx.initialSeg
+  aleph'.symm
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx.initialSeg_coe : (alephIdx.initialSeg : Cardinal → Ordinal) = alephIdx :=
   rfl
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx_lt {a b} : alephIdx a < alephIdx b ↔ a < b :=
   alephIdx.initialSeg.toRelEmbedding.map_rel_iff
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx_le {a b} : alephIdx a ≤ alephIdx b ↔ a ≤ b := by
   rw [← not_lt, ← not_lt, alephIdx_lt]
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx.init {a b} : b < alephIdx a → ∃ c, alephIdx c = b :=
   alephIdx.initialSeg.init
 
-/-- The `aleph'` index function, which gives the ordinal index of a cardinal.
-  (The `aleph'` part is because unlike `aleph` this counts also the
-  finite stages. So `alephIdx n = n`, `alephIdx ℵ₀ = ω`,
-  `alephIdx ℵ₁ = ω + 1` and so on.)
-  In this version, we register additionally that this function is an order isomorphism
-  between cardinals and ordinals.
-  For the basic function version, see `alephIdx`. -/
-def alephIdx.relIso : @RelIso Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) :=
-  @RelIso.ofSurjective Cardinal.{u} Ordinal.{u} (· < ·) (· < ·) alephIdx.initialSeg.{u} <|
-    (InitialSeg.eq_or_principal alephIdx.initialSeg.{u}).resolve_right fun ⟨o, e⟩ => by
-      have : ∀ c, alephIdx c < o := fun c => (e _).2 ⟨_, rfl⟩
-      refine Ordinal.inductionOn o ?_ this; intro α r _ h
-      let s := ⨆ a, invFun alephIdx (Ordinal.typein r a)
-      apply (lt_succ s).not_le
-      have I : Injective.{u+2, u+2} alephIdx := alephIdx.initialSeg.toEmbedding.injective
-      simpa only [typein_enum, leftInverse_invFun I (succ s)] using
-        le_ciSup
-          (Cardinal.bddAbove_range.{u, u} fun a : α => invFun alephIdx (Ordinal.typein r a))
-          (Ordinal.enum r ⟨_, h (succ s)⟩)
-
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx.relIso_coe : (alephIdx.relIso : Cardinal → Ordinal) = alephIdx :=
   rfl
 
 @[simp]
 theorem type_cardinal : @type Cardinal (· < ·) _ = Ordinal.univ.{u, u + 1} := by
-  rw [Ordinal.univ_id]; exact Quotient.sound ⟨alephIdx.relIso⟩
+  rw [Ordinal.univ_id]
+  exact Quotient.sound ⟨aleph'.symm.toRelIsoLT⟩
 
 @[simp]
 theorem mk_cardinal : #Cardinal = univ.{u, u + 1} := by
@@ -145,45 +163,41 @@ theorem mk_cardinal : #Cardinal = univ.{u, u + 1} := by
   In this version, we register additionally that this function is an order isomorphism
   between ordinals and cardinals.
   For the basic function version, see `aleph'`. -/
+@[deprecated aleph' (since := "2024-08-28")]
 def Aleph'.relIso :=
-  Cardinal.alephIdx.relIso.symm
+  aleph'
 
-/-- The `aleph'` function gives the cardinals listed by their ordinal
-  index, and is the inverse of `aleph_idx`.
-  `aleph' n = n`, `aleph' ω = ω`, `aleph' (ω + 1) = succ ℵ₀`, etc. -/
-def aleph' : Ordinal → Cardinal :=
-  Aleph'.relIso
-
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem aleph'.relIso_coe : (Aleph'.relIso : Ordinal → Cardinal) = aleph' :=
   rfl
 
-@[simp]
 theorem aleph'_lt {o₁ o₂ : Ordinal} : aleph' o₁ < aleph' o₂ ↔ o₁ < o₂ :=
-  Aleph'.relIso.map_rel_iff
+  aleph'.lt_iff_lt
 
-@[simp]
 theorem aleph'_le {o₁ o₂ : Ordinal} : aleph' o₁ ≤ aleph' o₂ ↔ o₁ ≤ o₂ :=
-  le_iff_le_iff_lt_iff_lt.2 aleph'_lt
+  aleph'.le_iff_le
 
-@[simp]
+theorem aleph'_max (o₁ o₂ : Ordinal) : aleph' (max o₁ o₂) = max (aleph' o₁) (aleph' o₂) :=
+  aleph'.monotone.map_max
+
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem aleph'_alephIdx (c : Cardinal) : aleph' c.alephIdx = c :=
   Cardinal.alephIdx.relIso.toEquiv.symm_apply_apply c
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-28")]
 theorem alephIdx_aleph' (o : Ordinal) : (aleph' o).alephIdx = o :=
   Cardinal.alephIdx.relIso.toEquiv.apply_symm_apply o
 
 @[simp]
-theorem aleph'_zero : aleph' 0 = 0 := by
-  rw [← nonpos_iff_eq_zero, ← aleph'_alephIdx 0, aleph'_le]
-  apply Ordinal.zero_le
+theorem aleph'_zero : aleph' 0 = 0 :=
+  aleph'.map_bot
 
 @[simp]
-theorem aleph'_succ {o : Ordinal} : aleph' (succ o) = succ (aleph' o) := by
-  apply (succ_le_of_lt <| aleph'_lt.2 <| lt_succ o).antisymm' (Cardinal.alephIdx_le.1 <| _)
-  rw [alephIdx_aleph', succ_le_iff, ← aleph'_lt, aleph'_alephIdx]
-  apply lt_succ
+theorem aleph'_succ (o : Ordinal) : aleph' (succ o) = succ (aleph' o) :=
+  aleph'.map_succ o
 
 @[simp]
 theorem aleph'_nat : ∀ n : ℕ, aleph' n = n
@@ -193,9 +207,9 @@ theorem aleph'_nat : ∀ n : ℕ, aleph' n = n
 theorem aleph'_le_of_limit {o : Ordinal} (l : o.IsLimit) {c} :
     aleph' o ≤ c ↔ ∀ o' < o, aleph' o' ≤ c :=
   ⟨fun h o' h' => (aleph'_le.2 <| h'.le).trans h, fun h => by
-    rw [← aleph'_alephIdx c, aleph'_le, limit_le l]
+    rw [← aleph'.apply_symm_apply c, aleph'_le, limit_le l]
     intro x h'
-    rw [← aleph'_le, aleph'_alephIdx]
+    rw [← aleph'_le, aleph'.apply_symm_apply]
     exact h _ h'⟩
 
 theorem aleph'_limit {o : Ordinal} (ho : o.IsLimit) : aleph' o = ⨆ a : Iio o, aleph' a := by
@@ -204,59 +218,66 @@ theorem aleph'_limit {o : Ordinal} (ho : o.IsLimit) : aleph' o = ⨆ a : Iio o,
   exact fun a ha => le_ciSup (bddAbove_of_small _) (⟨a, ha⟩ : Iio o)
 
 @[simp]
-theorem aleph'_omega : aleph' ω = ℵ₀ :=
+theorem aleph'_omega0 : aleph' ω = ℵ₀ :=
   eq_of_forall_ge_iff fun c => by
-    simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le]
+    simp only [aleph'_le_of_limit omega0_isLimit, lt_omega0, exists_imp, aleph0_le]
     exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat])
 
+@[deprecated (since := "2024-09-30")]
+alias aleph'_omega := aleph'_omega0
+
+set_option linter.deprecated false in
 /-- `aleph'` and `aleph_idx` form an equivalence between `Ordinal` and `Cardinal` -/
-@[simp]
+@[deprecated aleph' (since := "2024-08-28")]
 def aleph'Equiv : Ordinal ≃ Cardinal :=
   ⟨aleph', alephIdx, alephIdx_aleph', aleph'_alephIdx⟩
 
-/-- The `aleph` function gives the infinite cardinals listed by their
-  ordinal index. `aleph 0 = ℵ₀`, `aleph 1 = succ ℵ₀` is the first
-  uncountable cardinal, and so on. -/
-def aleph (o : Ordinal) : Cardinal :=
-  aleph' (ω + o)
+/-- The `aleph` function gives the infinite cardinals listed by their ordinal index. `aleph 0 = ℵ₀`,
+`aleph 1 = succ ℵ₀` is the first uncountable cardinal, and so on.
+
+For a version including finite cardinals, see `Cardinal.aleph'`. -/
+def aleph : Ordinal ↪o Cardinal :=
+  (OrderEmbedding.addLeft ω).trans aleph'.toOrderEmbedding
+
+theorem aleph_eq_aleph' (o : Ordinal) : aleph o = aleph' (ω + o) :=
+  rfl
 
-@[simp]
 theorem aleph_lt {o₁ o₂ : Ordinal} : aleph o₁ < aleph o₂ ↔ o₁ < o₂ :=
-  aleph'_lt.trans (add_lt_add_iff_left _)
+  aleph.lt_iff_lt
 
-@[simp]
 theorem aleph_le {o₁ o₂ : Ordinal} : aleph o₁ ≤ aleph o₂ ↔ o₁ ≤ o₂ :=
-  le_iff_le_iff_lt_iff_lt.2 aleph_lt
+  aleph.le_iff_le
 
-@[simp]
-theorem max_aleph_eq (o₁ o₂ : Ordinal) : max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) := by
-  rcases le_total (aleph o₁) (aleph o₂) with h | h
-  · rw [max_eq_right h, max_eq_right (aleph_le.1 h)]
-  · rw [max_eq_left h, max_eq_left (aleph_le.1 h)]
+theorem aleph_max (o₁ o₂ : Ordinal) : aleph (max o₁ o₂) = max (aleph o₁) (aleph o₂) :=
+  aleph.monotone.map_max
+
+@[deprecated aleph_max (since := "2024-08-28")]
+theorem max_aleph_eq (o₁ o₂ : Ordinal) : max (aleph o₁) (aleph o₂) = aleph (max o₁ o₂) :=
+  (aleph_max o₁ o₂).symm
 
 @[simp]
-theorem aleph_succ {o : Ordinal} : aleph (succ o) = succ (aleph o) := by
-  rw [aleph, add_succ, aleph'_succ, aleph]
+theorem aleph_succ (o : Ordinal) : aleph (succ o) = succ (aleph o) := by
+  rw [aleph_eq_aleph', add_succ, aleph'_succ, aleph_eq_aleph']
 
 @[simp]
-theorem aleph_zero : aleph 0 = ℵ₀ := by rw [aleph, add_zero, aleph'_omega]
+theorem aleph_zero : aleph 0 = ℵ₀ := by rw [aleph_eq_aleph', add_zero, aleph'_omega0]
 
 theorem aleph_limit {o : Ordinal} (ho : o.IsLimit) : aleph o = ⨆ a : Iio o, aleph a := by
   apply le_antisymm _ (ciSup_le' _)
-  · rw [aleph, aleph'_limit (ho.add _)]
+  · rw [aleph_eq_aleph', aleph'_limit (ho.add _)]
     refine ciSup_mono' (bddAbove_of_small _) ?_
     rintro ⟨i, hi⟩
     cases' lt_or_le i ω with h h
-    · rcases lt_omega.1 h with ⟨n, rfl⟩
+    · rcases lt_omega0.1 h with ⟨n, rfl⟩
       use ⟨0, ho.pos⟩
       simpa using (nat_lt_aleph0 n).le
     · exact ⟨⟨_, (sub_lt_of_le h).2 hi⟩, aleph'_le.2 (le_add_sub _ _)⟩
   · exact fun i => aleph_le.2 (le_of_lt i.2)
 
-theorem aleph0_le_aleph' {o : Ordinal} : ℵ₀ ≤ aleph' o ↔ ω ≤ o := by rw [← aleph'_omega, aleph'_le]
+theorem aleph0_le_aleph' {o : Ordinal} : ℵ₀ ≤ aleph' o ↔ ω ≤ o := by rw [← aleph'_omega0, aleph'_le]
 
 theorem aleph0_le_aleph (o : Ordinal) : ℵ₀ ≤ aleph o := by
-  rw [aleph, aleph0_le_aleph']
+  rw [aleph_eq_aleph', aleph0_le_aleph']
   apply Ordinal.le_add_right
 
 theorem aleph'_pos {o : Ordinal} (ho : 0 < o) : 0 < aleph' o := by rwa [← aleph'_zero, aleph'_lt]
@@ -284,9 +305,9 @@ instance (o : Ordinal) : NoMaxOrder (aleph o).ord.toType :=
 
 theorem exists_aleph {c : Cardinal} : ℵ₀ ≤ c ↔ ∃ o, c = aleph o :=
   ⟨fun h =>
-    ⟨alephIdx c - ω, by
-      rw [aleph, Ordinal.add_sub_cancel_of_le, aleph'_alephIdx]
-      rwa [← aleph0_le_aleph', aleph'_alephIdx]⟩,
+    ⟨aleph'.symm c - ω, by
+      rw [aleph_eq_aleph', Ordinal.add_sub_cancel_of_le, aleph'.apply_symm_apply]
+      rwa [← aleph0_le_aleph', aleph'.apply_symm_apply]⟩,
     fun ⟨o, e⟩ => e.symm ▸ aleph0_le_aleph _⟩
 
 theorem aleph'_isNormal : IsNormal (ord ∘ aleph') :=
@@ -305,7 +326,17 @@ theorem aleph0_lt_aleph_one : ℵ₀ < aleph 1 := by
 theorem countable_iff_lt_aleph_one {α : Type*} (s : Set α) : s.Countable ↔ #s < aleph 1 := by
   rw [← succ_aleph0, lt_succ_iff, le_aleph0_iff_set_countable]
 
+section deprecated
+
+set_option linter.deprecated false
+
+-- TODO: these lemmas should be stated in terms of the `ω` function and of an `IsInitial` predicate,
+-- neither of which currently exist.
+--
+-- They should also use `¬ BddAbove` instead of `Unbounded (· < ·)`.
+
 /-- Ordinals that are cardinals are unbounded. -/
+@[deprecated (since := "2024-09-24")]
 theorem ord_card_unbounded : Unbounded (· < ·) { b : Ordinal | b.card.ord = b } :=
   unbounded_lt_iff.2 fun a =>
     ⟨_,
@@ -313,40 +344,24 @@ theorem ord_card_unbounded : Unbounded (· < ·) { b : Ordinal | b.card.ord = b
         dsimp
         rw [card_ord], (lt_ord_succ_card a).le⟩⟩
 
+@[deprecated (since := "2024-09-24")]
 theorem eq_aleph'_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) : ∃ a, (aleph' a).ord = o :=
-  ⟨Cardinal.alephIdx.relIso o.card, by simpa using ho⟩
-
-/-- `ord ∘ aleph'` enumerates the ordinals that are cardinals. -/
-theorem ord_aleph'_eq_enum_card : ord ∘ aleph' = enumOrd { b : Ordinal | b.card.ord = b } := by
-  rw [← eq_enumOrd _ ord_card_unbounded, range_eq_iff]
-  exact
-    ⟨aleph'_isNormal.strictMono,
-      ⟨fun a => by
-        dsimp
-        rw [card_ord], fun b hb => eq_aleph'_of_eq_card_ord hb⟩⟩
+  ⟨aleph'.symm o.card, by simpa using ho⟩
 
 /-- Infinite ordinals that are cardinals are unbounded. -/
+@[deprecated (since := "2024-09-24")]
 theorem ord_card_unbounded' : Unbounded (· < ·) { b : Ordinal | b.card.ord = b ∧ ω ≤ b } :=
   (unbounded_lt_inter_le ω).2 ord_card_unbounded
 
+@[deprecated (since := "2024-09-24")]
 theorem eq_aleph_of_eq_card_ord {o : Ordinal} (ho : o.card.ord = o) (ho' : ω ≤ o) :
     ∃ a, (aleph a).ord = o := by
   cases' eq_aleph'_of_eq_card_ord ho with a ha
   use a - ω
-  unfold aleph
-  rwa [Ordinal.add_sub_cancel_of_le]
+  rwa [aleph_eq_aleph', Ordinal.add_sub_cancel_of_le]
   rwa [← aleph0_le_aleph', ← ord_le_ord, ha, ord_aleph0]
 
-/-- `ord ∘ aleph` enumerates the infinite ordinals that are cardinals. -/
-theorem ord_aleph_eq_enum_card :
-    ord ∘ aleph = enumOrd { b : Ordinal | b.card.ord = b ∧ ω ≤ b } := by
-  rw [← eq_enumOrd _ ord_card_unbounded']
-  use aleph_isNormal.strictMono
-  rw [range_eq_iff]
-  refine ⟨fun a => ⟨?_, ?_⟩, fun b hb => eq_aleph_of_eq_card_ord hb.1 hb.2⟩
-  · rw [Function.comp_apply, card_ord]
-  · rw [← ord_aleph0, Function.comp_apply, ord_le_ord]
-    exact aleph0_le_aleph _
+end deprecated
 
 end aleph
 
@@ -498,7 +513,7 @@ theorem mul_mk_eq_max {α β : Type u} [Infinite α] [Infinite β] : #α * #β =
 
 @[simp]
 theorem aleph_mul_aleph (o₁ o₂ : Ordinal) : aleph o₁ * aleph o₂ = aleph (max o₁ o₂) := by
-  rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq]
+  rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), aleph_max]
 
 @[simp]
 theorem aleph0_mul_eq {a : Cardinal} (ha : ℵ₀ ≤ a) : ℵ₀ * a = a :=
@@ -767,8 +782,8 @@ protected theorem ciSup_add (hf : BddAbove (range f)) (c : Cardinal.{v}) :
   refine le_antisymm ?_ (ciSup_le' this)
   have bdd : BddAbove (range (f · + c)) := ⟨_, forall_mem_range.mpr this⟩
   obtain hs | hs := lt_or_le (⨆ i, f i) ℵ₀
-  · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isLimit
-      f hf _ (fun h ↦ hs.not_le h.aleph0_le) rfl
+  · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isSuccLimit
+      f hf (not_isSuccLimit_of_lt_aleph0 hs) rfl
     exact hi ▸ le_ciSup bdd i
   rw [add_eq_max hs, max_le_iff]
   exact ⟨ciSup_mono bdd fun i ↦ self_le_add_right _ c,
@@ -796,8 +811,8 @@ protected theorem ciSup_mul (c : Cardinal.{v}) : (⨆ i, f i) * c = ⨆ i, f i *
   refine le_antisymm ?_ (ciSup_le' this)
   have bdd : BddAbove (range (f · * c)) := ⟨_, forall_mem_range.mpr this⟩
   obtain hs | hs := lt_or_le (⨆ i, f i) ℵ₀
-  · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isLimit
-      f hf _ (fun h ↦ hs.not_le h.aleph0_le) rfl
+  · obtain ⟨i, hi⟩ := exists_eq_of_iSup_eq_of_not_isSuccLimit
+      f hf (not_isSuccLimit_of_lt_aleph0 hs) rfl
     exact hi ▸ le_ciSup bdd i
   rw [mul_eq_max_of_aleph0_le_left hs h0, max_le_iff]
   obtain ⟨i, hi⟩ := exists_lt_of_lt_ciSup' (one_lt_aleph0.trans_le hs)
@@ -815,7 +830,7 @@ end ciSup
 
 @[simp]
 theorem aleph_add_aleph (o₁ o₂ : Ordinal) : aleph o₁ + aleph o₂ = aleph (max o₁ o₂) := by
-  rw [Cardinal.add_eq_max (aleph0_le_aleph o₁), max_aleph_eq]
+  rw [Cardinal.add_eq_max (aleph0_le_aleph o₁), aleph_max]
 
 theorem principal_add_ord {c : Cardinal} (hc : ℵ₀ ≤ c) : Ordinal.Principal (· + ·) c.ord :=
   fun a b ha hb => by
@@ -836,7 +851,7 @@ theorem add_nat_inj {α β : Cardinal} (n : ℕ) : α + n = β + n ↔ α = β :
 theorem add_one_inj {α β : Cardinal} : α + 1 = β + 1 ↔ α = β :=
   add_right_inj_of_lt_aleph0 one_lt_aleph0
 
-theorem add_le_add_iff_of_lt_aleph0 {α β γ : Cardinal} (γ₀ : γ < Cardinal.aleph0) :
+theorem add_le_add_iff_of_lt_aleph0 {α β γ : Cardinal} (γ₀ : γ < ℵ₀) :
     α + γ ≤ β + γ ↔ α ≤ β := by
   refine ⟨fun h => ?_, fun h => add_le_add_right h γ⟩
   contrapose h
@@ -1206,209 +1221,6 @@ theorem extend_function_of_lt {α β : Type*} {s : Set α} (f : s ↪ β) (hs :
     rwa [mk_compl_of_infinite s hs, mk_compl_of_infinite]
     rwa [← lift_lt, mk_range_eq_of_injective f.injective, ← h, lift_lt]
 
-
--- Porting note: we no longer express literals as `bit0` and `bit1` in Lean 4, so we can't use this
--- section Bit
-
--- /-!
--- This section proves inequalities for `bit0` and `bit1`, enabling `simp` to solve inequalities
--- for numeral cardinals. The complexity of the resulting algorithm is not good, as in some cases
--- `simp` reduces an inequality to a disjunction of two situations, depending on whether a cardinal
--- is finite or infinite. Since the evaluation of the branches is not lazy, this is bad. It is good
--- enough for practical situations, though.
-
--- For specific numbers, these inequalities could also be deduced from the corresponding
--- inequalities of natural numbers using `norm_cast`:
--- ```
--- example : (37 : cardinal) < 42 :=
--- by { norm_cast, norm_num }
--- ```
--- -/
-
-
--- theorem bit0_ne_zero (a : Cardinal) : ¬bit0 a = 0 ↔ ¬a = 0 := by simp [bit0]
-
--- @[simp]
--- theorem bit1_ne_zero (a : Cardinal) : ¬bit1 a = 0 := by simp [bit1]
-
--- @[simp]
--- theorem zero_lt_bit0 (a : Cardinal) : 0 < bit0 a ↔ 0 < a := by
---   rw [← not_iff_not]
---   simp [bit0]
-
--- @[simp]
--- theorem zero_lt_bit1 (a : Cardinal) : 0 < bit1 a :=
---   zero_lt_one.trans_le (self_le_add_left _ _)
-
--- @[simp]
--- theorem one_le_bit0 (a : Cardinal) : 1 ≤ bit0 a ↔ 0 < a :=
---   ⟨fun h => (zero_lt_bit0 a).mp (zero_lt_one.trans_le h), fun h =>
---     (one_le_iff_pos.mpr h).trans (self_le_add_left a a)⟩
-
--- @[simp]
--- theorem one_le_bit1 (a : Cardinal) : 1 ≤ bit1 a :=
---   self_le_add_left _ _
-
--- theorem bit0_eq_self {c : Cardinal} (h : ℵ₀ ≤ c) : bit0 c = c :=
---   add_eq_self h
-
--- @[simp]
--- theorem bit0_lt_aleph0 {c : Cardinal} : bit0 c < ℵ₀ ↔ c < ℵ₀ :=
---   by simp [bit0, add_lt_aleph_0_iff]
-
--- @[simp]
--- theorem aleph0_le_bit0 {c : Cardinal} : ℵ₀ ≤ bit0 c ↔ ℵ₀ ≤ c := by
---   rw [← not_iff_not]
---   simp
-
--- @[simp]
--- theorem bit1_eq_self_iff {c : Cardinal} : bit1 c = c ↔ ℵ₀ ≤ c := by
---   by_cases h : ℵ₀ ≤ c
---   · simp only [bit1, bit0_eq_self h, h, eq_self_iff_true, add_one_of_aleph_0_le]
---   · refine' iff_of_false (ne_of_gt _) h
---     rcases lt_aleph_0.1 (not_le.1 h) with ⟨n, rfl⟩
---     norm_cast
---     dsimp [bit1, bit0]
---     linarith
-
--- @[simp]
--- theorem bit1_lt_aleph0 {c : Cardinal} : bit1 c < ℵ₀ ↔ c < ℵ₀ := by
---   simp [bit1, bit0, add_lt_aleph_0_iff, one_lt_aleph_0]
-
--- @[simp]
--- theorem aleph0_le_bit1 {c : Cardinal} : ℵ₀ ≤ bit1 c ↔ ℵ₀ ≤ c := by
---   rw [← not_iff_not]
---   simp
-
--- @[simp]
--- theorem bit0_le_bit0 {a b : Cardinal} : bit0 a ≤ bit0 b ↔ a ≤ b := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · rw [bit0_eq_self ha, bit0_eq_self hb]
---   · rw [bit0_eq_self ha]
---     refine' iff_of_false (fun h => _) (hb.trans_le ha).not_le
---     have A : bit0 b < ℵ₀ := by simpa using hb
---     exact lt_irrefl _ ((A.trans_le ha).trans_le h)
---   · rw [bit0_eq_self hb]
---     exact iff_of_true ((bit0_lt_aleph_0.2 ha).le.trans hb) (ha.le.trans hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     exact bit0_le_bit0
-
--- @[simp]
--- theorem bit0_le_bit1 {a b : Cardinal} : bit0 a ≤ bit1 b ↔ a ≤ b := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · rw [bit0_eq_self ha, bit1_eq_self_iff.2 hb]
---   · rw [bit0_eq_self ha]
---     refine' iff_of_false (fun h => _) (hb.trans_le ha).not_le
---     have A : bit1 b < ℵ₀ := by simpa using hb
---     exact lt_irrefl _ ((A.trans_le ha).trans_le h)
---   · rw [bit1_eq_self_iff.2 hb]
---     exact iff_of_true ((bit0_lt_aleph_0.2 ha).le.trans hb) (ha.le.trans hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     exact Nat.bit0_le_bit1_iff
-
--- @[simp]
--- theorem bit1_le_bit1 {a b : Cardinal} : bit1 a ≤ bit1 b ↔ a ≤ b :=
---   ⟨fun h => bit0_le_bit1.1 ((self_le_add_right (bit0 a) 1).trans h), fun h =>
---     (add_le_add_right (add_le_add_left h a) 1).trans (add_le_add_right (add_le_add_right h b) 1)⟩
-
--- @[simp]
--- theorem bit1_le_bit0 {a b : Cardinal} : bit1 a ≤ bit0 b ↔ a < b ∨ a ≤ b ∧ ℵ₀ ≤ a := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · simp only [bit1_eq_self_iff.mpr ha, bit0_eq_self hb, ha, and_true_iff]
---     refine' ⟨fun h => Or.inr h, fun h => _⟩
---     cases h
---     · exact le_of_lt h
---     · exact h
---   · rw [bit1_eq_self_iff.2 ha]
---     refine' iff_of_false (fun h => _) fun h => _
---     · have A : bit0 b < ℵ₀ := by simpa using hb
---       exact lt_irrefl _ ((A.trans_le ha).trans_le h)
---     · exact not_le_of_lt (hb.trans_le ha) (h.elim le_of_lt And.left)
---   · rw [bit0_eq_self hb]
---     exact iff_of_true ((bit1_lt_aleph_0.2 ha).le.trans hb) (Or.inl <| ha.trans_le hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     simp [not_le.mpr ha]
-
--- @[simp]
--- theorem bit0_lt_bit0 {a b : Cardinal} : bit0 a < bit0 b ↔ a < b := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · rw [bit0_eq_self ha, bit0_eq_self hb]
---   · rw [bit0_eq_self ha]
---     refine' iff_of_false (fun h => _) (hb.le.trans ha).not_lt
---     have A : bit0 b < ℵ₀ := by simpa using hb
---     exact lt_irrefl _ ((A.trans_le ha).trans h)
---   · rw [bit0_eq_self hb]
---     exact iff_of_true ((bit0_lt_aleph_0.2 ha).trans_le hb) (ha.trans_le hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     exact bit0_lt_bit0
-
--- @[simp]
--- theorem bit1_lt_bit0 {a b : Cardinal} : bit1 a < bit0 b ↔ a < b := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · rw [bit1_eq_self_iff.2 ha, bit0_eq_self hb]
---   · rw [bit1_eq_self_iff.2 ha]
---     refine' iff_of_false (fun h => _) (hb.le.trans ha).not_lt
---     have A : bit0 b < ℵ₀ := by simpa using hb
---     exact lt_irrefl _ ((A.trans_le ha).trans h)
---   · rw [bit0_eq_self hb]
---     exact iff_of_true ((bit1_lt_aleph_0.2 ha).trans_le hb) (ha.trans_le hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     exact Nat.bit1_lt_bit0_iff
-
--- @[simp]
--- theorem bit1_lt_bit1 {a b : Cardinal} : bit1 a < bit1 b ↔ a < b := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · rw [bit1_eq_self_iff.2 ha, bit1_eq_self_iff.2 hb]
---   · rw [bit1_eq_self_iff.2 ha]
---     refine' iff_of_false (fun h => _) (hb.le.trans ha).not_lt
---     have A : bit1 b < ℵ₀ := by simpa using hb
---     exact lt_irrefl _ ((A.trans_le ha).trans h)
---   · rw [bit1_eq_self_iff.2 hb]
---     exact iff_of_true ((bit1_lt_aleph_0.2 ha).trans_le hb) (ha.trans_le hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     exact bit1_lt_bit1
-
--- @[simp]
--- theorem bit0_lt_bit1 {a b : Cardinal} : bit0 a < bit1 b ↔ a < b ∨ a ≤ b ∧ a < ℵ₀ := by
---   rcases le_or_lt ℵ₀ a with ha | ha <;> rcases le_or_lt ℵ₀ b with hb | hb
---   · simp [bit0_eq_self ha, bit1_eq_self_iff.2 hb, not_lt.mpr ha]
---   · rw [bit0_eq_self ha]
---     refine' iff_of_false (fun h => _) fun h => _
---     · have A : bit1 b < ℵ₀ := by simpa using hb
---       exact lt_irrefl _ ((A.trans_le ha).trans h)
---     · exact (hb.trans_le ha).not_le (h.elim le_of_lt And.left)
---   · rw [bit1_eq_self_iff.2 hb]
---     exact iff_of_true ((bit0_lt_aleph_0.2 ha).trans_le hb) (Or.inl <| ha.trans_le hb)
---   · rcases lt_aleph_0.1 ha with ⟨m, rfl⟩
---     rcases lt_aleph_0.1 hb with ⟨n, rfl⟩
---     norm_cast
---     simp only [ha, and_true_iff, Nat.bit0_lt_bit1_iff, or_iff_right_of_imp le_of_lt]
-
--- theorem one_lt_two : (1 : Cardinal) < 2 := by
---   -- This strategy works generally to prove inequalities between numerals in `cardinality`.
---   norm_cast
---   norm_num
-
--- @[simp]
--- theorem one_lt_bit0 {a : Cardinal} : 1 < bit0 a ↔ 0 < a := by simp [← bit1_zero]
-
--- @[simp]
--- theorem one_lt_bit1 (a : Cardinal) : 1 < bit1 a ↔ 0 < a := by simp [← bit1_zero]
-
--- end Bit
-
 end Cardinal
 
 section Initial
@@ -1426,7 +1238,7 @@ scoped notation "ω_" o => ord <| aleph o
 -/
 scoped notation "ω₁" => ord <| aleph 1
 
-lemma omega_lt_omega1 : ω < ω₁ := ord_aleph0.symm.trans_lt (ord_lt_ord.mpr (aleph0_lt_aleph_one))
+lemma omega0_lt_omega1 : ω < ω₁ := ord_aleph0.symm.trans_lt (ord_lt_ord.mpr (aleph0_lt_aleph_one))
 
 section OrdinalIndices
 /-!
diff --git a/Mathlib/SetTheory/Game/Basic.lean b/Mathlib/SetTheory/Game/Basic.lean
index b8a4c1008298e..8c72765907e52 100644
--- a/Mathlib/SetTheory/Game/Basic.lean
+++ b/Mathlib/SetTheory/Game/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison, Apurva Nakade
+Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade
 -/
 import Mathlib.Algebra.Order.Group.Defs
 import Mathlib.Algebra.Ring.Int
@@ -78,6 +78,9 @@ instance instAddCommGroupWithOneGame : AddCommGroupWithOne Game where
 instance : Inhabited Game :=
   ⟨0⟩
 
+theorem zero_def : (0 : Game) = ⟦0⟧ :=
+  rfl
+
 instance instPartialOrderGame : PartialOrder Game where
   le := Quotient.lift₂ (· ≤ ·) fun x₁ y₁ x₂ y₂ hx hy => propext (le_congr hx hy)
   le_refl := by
@@ -134,7 +137,7 @@ end Game
 
 namespace PGame
 
--- Porting note: In a lot of places, I had to add explicitely that the quotient element was a Game.
+-- Porting note: In a lot of places, I had to add explicitly that the quotient element was a Game.
 -- In Lean4, quotients don't have the setoid as an instance argument,
 -- but as an explicit argument, see https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/confusion.20between.20equivalence.20and.20instance.20setoid/near/360822354
 theorem le_iff_game_le {x y : PGame} : x ≤ y ↔ (⟦x⟧ : Game) ≤ ⟦y⟧ :=
@@ -219,23 +222,24 @@ end Game
 
 namespace PGame
 
-@[simp]
-theorem quot_neg (a : PGame) : (⟦-a⟧ : Game) = -⟦a⟧ :=
-  rfl
+@[simp] theorem quot_zero : (⟦0⟧ : Game) = 0 := rfl
+@[simp] theorem quot_one : (⟦1⟧ : Game) = 1 := rfl
+@[simp] theorem quot_neg (a : PGame) : (⟦-a⟧ : Game) = -⟦a⟧ := rfl
+@[simp] theorem quot_add (a b : PGame) : ⟦a + b⟧ = (⟦a⟧ : Game) + ⟦b⟧ := rfl
+@[simp] theorem quot_sub (a b : PGame) : ⟦a - b⟧ = (⟦a⟧ : Game) - ⟦b⟧ := rfl
 
 @[simp]
-theorem quot_add (a b : PGame) : ⟦a + b⟧ = (⟦a⟧ : Game) + ⟦b⟧ :=
-  rfl
-
-@[simp]
-theorem quot_sub (a b : PGame) : ⟦a - b⟧ = (⟦a⟧ : Game) - ⟦b⟧ :=
-  rfl
+theorem quot_natCast : ∀ n : ℕ, ⟦(n : PGame)⟧ = (n : Game)
+  | 0 => rfl
+  | n + 1 => by
+    rw [PGame.nat_succ, quot_add, Nat.cast_add, Nat.cast_one, quot_natCast]
+    rfl
 
 theorem quot_eq_of_mk'_quot_eq {x y : PGame} (L : x.LeftMoves ≃ y.LeftMoves)
     (R : x.RightMoves ≃ y.RightMoves) (hl : ∀ i, (⟦x.moveLeft i⟧ : Game) = ⟦y.moveLeft (L i)⟧)
-    (hr : ∀ j, (⟦x.moveRight j⟧ : Game) = ⟦y.moveRight (R j)⟧) : (⟦x⟧ : Game) = ⟦y⟧ := by
-  exact Quot.sound (equiv_of_mk_equiv L R (fun _ => equiv_iff_game_eq.2 (hl _))
-                                          (fun _ => equiv_iff_game_eq.2 (hr _)))
+    (hr : ∀ j, (⟦x.moveRight j⟧ : Game) = ⟦y.moveRight (R j)⟧) : (⟦x⟧ : Game) = ⟦y⟧ :=
+  game_eq (.of_equiv L R (fun _ => equiv_iff_game_eq.2 (hl _))
+    (fun _ => equiv_iff_game_eq.2 (hr _)))
 
 /-! Multiplicative operations can be defined at the level of pre-games,
 but to prove their properties we need to use the abelian group structure of games.
@@ -246,8 +250,8 @@ Hence we define them here. -/
 `{xL*y + x*yL - xL*yL, xR*y + x*yR - xR*yR | xL*y + x*yR - xL*yR, x*yL + xR*y - xR*yL }`. -/
 instance : Mul PGame.{u} :=
   ⟨fun x y => by
-    induction' x with xl xr _ _ IHxl IHxr generalizing y
-    induction' y with yl yr yL yR IHyl IHyr
+    induction x generalizing y with | mk xl xr _ _ IHxl IHxr => _
+    induction y with | mk yl yr yL yR IHyl IHyr => _
     have y := mk yl yr yL yR
     refine ⟨(xl × yl) ⊕ (xr × yr), (xl × yr) ⊕ (xr × yl), ?_, ?_⟩ <;> rintro (⟨i, j⟩ | ⟨i, j⟩)
     · exact IHxl i y + IHyl j - IHxl i (yL j)
@@ -392,7 +396,7 @@ def mulCommRelabelling (x y : PGame.{u}) : x * y ≡r y * x :=
   termination_by (x, y)
 
 theorem quot_mul_comm (x y : PGame.{u}) : (⟦x * y⟧ : Game) = ⟦y * x⟧ :=
-  Quot.sound (mulCommRelabelling x y).equiv
+  game_eq (mulCommRelabelling x y).equiv
 
 /-- `x * y` is equivalent to `y * x`. -/
 theorem mul_comm_equiv (x y : PGame) : x * y ≈ y * x :=
@@ -421,8 +425,8 @@ theorem mul_zero_equiv (x : PGame) : x * 0 ≈ 0 :=
   (mulZeroRelabelling x).equiv
 
 @[simp]
-theorem quot_mul_zero (x : PGame) : (⟦x * 0⟧ : Game) = ⟦0⟧ :=
-  @Quotient.sound _ _ (x * 0) _ x.mul_zero_equiv
+theorem quot_mul_zero (x : PGame) : (⟦x * 0⟧ : Game) = 0 :=
+  game_eq x.mul_zero_equiv
 
 /-- `0 * x` has exactly the same moves as `0`. -/
 def zeroMulRelabelling (x : PGame) : 0 * x ≡r 0 :=
@@ -433,8 +437,8 @@ theorem zero_mul_equiv (x : PGame) : 0 * x ≈ 0 :=
   (zeroMulRelabelling x).equiv
 
 @[simp]
-theorem quot_zero_mul (x : PGame) : (⟦0 * x⟧ : Game) = ⟦0⟧ :=
-  @Quotient.sound _ _ (0 * x) _ x.zero_mul_equiv
+theorem quot_zero_mul (x : PGame) : (⟦0 * x⟧ : Game) = 0 :=
+  game_eq x.zero_mul_equiv
 
 /-- `-x * y` and `-(x * y)` have the same moves. -/
 def negMulRelabelling (x y : PGame.{u}) : -x * y ≡r -(x * y) :=
@@ -457,7 +461,7 @@ def negMulRelabelling (x y : PGame.{u}) : -x * y ≡r -(x * y) :=
 
 @[simp]
 theorem quot_neg_mul (x y : PGame) : (⟦-x * y⟧ : Game) = -⟦x * y⟧ :=
-  Quot.sound (negMulRelabelling x y).equiv
+  game_eq (negMulRelabelling x y).equiv
 
 /-- `x * -y` and `-(x * y)` have the same moves. -/
 def mulNegRelabelling (x y : PGame) : x * -y ≡r -(x * y) :=
@@ -465,7 +469,7 @@ def mulNegRelabelling (x y : PGame) : x * -y ≡r -(x * y) :=
 
 @[simp]
 theorem quot_mul_neg (x y : PGame) : ⟦x * -y⟧ = (-⟦x * y⟧ : Game) :=
-  Quot.sound (mulNegRelabelling x y).equiv
+  game_eq (mulNegRelabelling x y).equiv
 
 theorem quot_neg_mul_neg (x y : PGame) : ⟦-x * -y⟧ = (⟦x * y⟧ : Game) := by simp
 
@@ -601,13 +605,13 @@ def mulOneRelabelling : ∀ x : PGame.{u}, x * 1 ≡r x
     (try rintro (⟨i, ⟨⟩⟩ | ⟨i, ⟨⟩⟩)) <;>
     { dsimp
       apply (Relabelling.subCongr (Relabelling.refl _) (mulZeroRelabelling _)).trans
-      rw [sub_zero]
+      rw [sub_zero_eq_add_zero]
       exact (addZeroRelabelling _).trans <|
         (((mulOneRelabelling _).addCongr (mulZeroRelabelling _)).trans <| addZeroRelabelling _) }
 
 @[simp]
 theorem quot_mul_one (x : PGame) : (⟦x * 1⟧ : Game) = ⟦x⟧ :=
-  Quot.sound <| PGame.Relabelling.equiv <| mulOneRelabelling x
+  game_eq <| PGame.Relabelling.equiv <| mulOneRelabelling x
 
 /-- `x * 1` is equivalent to `x`. -/
 theorem mul_one_equiv (x : PGame) : x * 1 ≈ x :=
@@ -619,7 +623,7 @@ def oneMulRelabelling (x : PGame) : 1 * x ≡r x :=
 
 @[simp]
 theorem quot_one_mul (x : PGame) : (⟦1 * x⟧ : Game) = ⟦x⟧ :=
-  Quot.sound <| PGame.Relabelling.equiv <| oneMulRelabelling x
+  game_eq <| PGame.Relabelling.equiv <| oneMulRelabelling x
 
 /-- `1 * x` is equivalent to `x`. -/
 theorem one_mul_equiv (x : PGame) : 1 * x ≈ x :=
diff --git a/Mathlib/SetTheory/Game/Birthday.lean b/Mathlib/SetTheory/Game/Birthday.lean
index 28f8edebd14c0..016fd8366eb3a 100644
--- a/Mathlib/SetTheory/Game/Birthday.lean
+++ b/Mathlib/SetTheory/Game/Birthday.lean
@@ -3,27 +3,32 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
+import Mathlib.Algebra.Order.Group.OrderIso
 import Mathlib.SetTheory.Game.Ordinal
 import Mathlib.SetTheory.Ordinal.NaturalOps
 
 /-!
 # Birthdays of games
 
-The birthday of a game is an ordinal that represents at which "step" the game was constructed. We
-define it recursively as the least ordinal larger than the birthdays of its left and right games. We
-prove the basic properties about these.
+There are two related but distinct notions of a birthday within combinatorial game theory. One is
+the birthday of a pre-game, which represents the "step" at which it is constructed. We define it
+recursively as the least ordinal larger than the birthdays of its left and right options. On the
+other hand, the birthday of a game is the smallest birthday among all pre-games that quotient to it.
+
+The birthday of a pre-game can be understood as representing the depth of its game tree. On the
+other hand, the birthday of a game more closely matches Conway's original description. The lemma
+`SetTheory.Game.birthday_eq_pGameBirthday` links both definitions together.
 
 # Main declarations
 
 - `SetTheory.PGame.birthday`: The birthday of a pre-game.
+- `SetTheory.Game.birthday`: The birthday of a game.
 
 # Todo
 
-- Define the birthdays of `SetTheory.Game`s and `Surreal`s.
-- Characterize the birthdays of basic arithmetical operations.
+- Characterize the birthdays of other basic arithmetical operations.
 -/
 
-
 universe u
 
 open Ordinal
@@ -99,13 +104,13 @@ theorem birthday_one : birthday 1 = 1 := by rw [birthday_def]; simp
 theorem birthday_star : birthday star = 1 := by rw [birthday_def]; simp
 
 @[simp]
-theorem neg_birthday : ∀ x : PGame, (-x).birthday = x.birthday
+theorem birthday_neg : ∀ x : PGame, (-x).birthday = x.birthday
   | ⟨xl, xr, xL, xR⟩ => by
     rw [birthday_def, birthday_def, max_comm]
-    congr <;> funext <;> apply neg_birthday
+    congr <;> funext <;> apply birthday_neg
 
 @[simp]
-theorem toPGame_birthday (o : Ordinal) : o.toPGame.birthday = o := by
+theorem birthday_ordinalToPGame (o : Ordinal) : o.toPGame.birthday = o := by
   induction' o using Ordinal.induction with o IH
   rw [toPGame_def, PGame.birthday]
   simp only [lsub_empty, max_zero_right]
@@ -124,16 +129,14 @@ theorem le_birthday : ∀ x : PGame, x ≤ x.birthday.toPGame
 variable (a b x : PGame.{u})
 
 theorem neg_birthday_le : -x.birthday.toPGame ≤ x := by
-  simpa only [neg_birthday, ← neg_le_iff] using le_birthday (-x)
+  simpa only [birthday_neg, ← neg_le_iff] using le_birthday (-x)
 
 @[simp]
-theorem birthday_add : ∀ x y : PGame.{u}, (x + y).birthday = x.birthday ♯ y.birthday
+theorem birthday_add : ∀ x y : PGame, (x + y).birthday = x.birthday ♯ y.birthday
   | ⟨xl, xr, xL, xR⟩, ⟨yl, yr, yL, yR⟩ => by
-    rw [birthday_def, nadd_def]
-    -- Porting note: `simp` doesn't apply
-    erw [lsub_sum, lsub_sum]
-    simp only [lsub_sum, mk_add_moveLeft_inl, moveLeft_mk, mk_add_moveLeft_inr,
-      mk_add_moveRight_inl, moveRight_mk, mk_add_moveRight_inr]
+    rw [birthday_def, nadd_def, lsub_sum, lsub_sum]
+    simp only [mk_add_moveLeft_inl, mk_add_moveLeft_inr, mk_add_moveRight_inl, mk_add_moveRight_inr,
+      moveLeft_mk, moveRight_mk]
     -- Porting note: Originally `simp only [birthday_add]`, but this causes an error in
     -- `termination_by`. Use a workaround.
     conv_lhs => left; left; right; intro a; rw [birthday_add (xL a) ⟨yl, yr, yL, yR⟩]
@@ -156,26 +159,117 @@ theorem birthday_add : ∀ x y : PGame.{u}, (x + y).birthday = x.birthday ♯ y.
     · exact lt_max_of_lt_right ((nadd_le_nadd_left hj _).trans_lt (lt_lsub _ _))
 termination_by a b => (a, b)
 
-theorem birthday_add_zero : (a + 0).birthday = a.birthday := by simp
-
-theorem birthday_zero_add : (0 + a).birthday = a.birthday := by simp
-
-theorem birthday_add_one : (a + 1).birthday = Order.succ a.birthday := by simp
-
-theorem birthday_one_add : (1 + a).birthday = Order.succ a.birthday := by simp
+@[simp]
+theorem birthday_sub (x y : PGame) : (x - y).birthday = x.birthday ♯ y.birthday := by
+  apply (birthday_add x _).trans
+  rw [birthday_neg]
 
 @[simp]
 theorem birthday_natCast : ∀ n : ℕ, birthday n = n
   | 0 => birthday_zero
   | n + 1 => by simp [birthday_natCast]
 
-@[deprecated (since := "2024-04-17")]
-alias birthday_nat_cast := birthday_natCast
+end PGame
 
-theorem birthday_add_nat (n : ℕ) : (a + n).birthday = a.birthday + n := by simp
+namespace Game
 
-theorem birthday_nat_add (n : ℕ) : (↑n + a).birthday = a.birthday + n := by simp
+/-- The birthday of a game is defined as the least birthday among all pre-games that define it. -/
+noncomputable def birthday (x : Game.{u}) : Ordinal.{u} :=
+  sInf (PGame.birthday '' (Quotient.mk' ⁻¹' {x}))
 
-end PGame
+theorem birthday_eq_pGameBirthday (x : Game) :
+    ∃ y : PGame.{u}, ⟦y⟧ = x ∧ y.birthday = birthday x := by
+  refine csInf_mem (Set.image_nonempty.2 ?_)
+  exact ⟨_, x.out_eq⟩
+
+theorem birthday_quot_le_pGameBirthday  (x : PGame) : birthday ⟦x⟧ ≤ x.birthday :=
+  csInf_le' ⟨x, rfl, rfl⟩
+
+@[simp]
+theorem birthday_zero : birthday 0 = 0 := by
+  rw [← Ordinal.le_zero, ← PGame.birthday_zero]
+  exact birthday_quot_le_pGameBirthday  _
+
+@[simp]
+theorem birthday_eq_zero {x : Game} : birthday x = 0 ↔ x = 0 := by
+  constructor
+  · intro h
+    let ⟨y, hy₁, hy₂⟩ := birthday_eq_pGameBirthday x
+    rw [← hy₁]
+    rw [h, PGame.birthday_eq_zero] at hy₂
+    exact PGame.game_eq (@PGame.Equiv.isEmpty _ hy₂.1 hy₂.2)
+  · rintro rfl
+    exact birthday_zero
+
+@[simp]
+theorem birthday_ordinalToGame (o : Ordinal) : birthday o.toGame = o := by
+  apply le_antisymm
+  · conv_rhs => rw [← PGame.birthday_ordinalToPGame o]
+    apply birthday_quot_le_pGameBirthday
+  · let ⟨x, hx₁, hx₂⟩ := birthday_eq_pGameBirthday o.toGame
+    rw [← hx₂, ← toPGame_le_iff]
+    rw [← PGame.equiv_iff_game_eq] at hx₁
+    exact hx₁.2.trans (PGame.le_birthday x)
+
+@[simp, norm_cast]
+theorem birthday_natCast (n : ℕ) : birthday n = n := by
+  rw [← toGame_natCast]
+  exact birthday_ordinalToGame _
+
+-- See note [no_index around OfNat.ofNat]
+@[simp]
+theorem birthday_ofNat (n : ℕ) [n.AtLeastTwo] :
+    birthday (no_index (OfNat.ofNat n)) = OfNat.ofNat n :=
+  birthday_natCast n
+
+@[simp]
+theorem birthday_one : birthday 1 = 1 := by
+  rw [← Nat.cast_one, birthday_natCast, Nat.cast_one]
+
+@[simp]
+theorem birthday_star : birthday ⟦PGame.star⟧ = 1 := by
+  apply le_antisymm
+  · rw [← PGame.birthday_star]
+    exact birthday_quot_le_pGameBirthday  _
+  · rw [Ordinal.one_le_iff_ne_zero, ne_eq, birthday_eq_zero, Game.zero_def,
+      ← PGame.equiv_iff_game_eq]
+    exact PGame.star_fuzzy_zero.not_equiv
+
+private theorem birthday_neg' (x : Game) : (-x).birthday ≤ x.birthday := by
+  let ⟨y, hy₁, hy₂⟩ := birthday_eq_pGameBirthday x
+  rw [← hy₂, ← PGame.birthday_neg y]
+  conv_lhs => rw [← hy₁]
+  apply birthday_quot_le_pGameBirthday
+
+@[simp]
+theorem birthday_neg (x : Game) : (-x).birthday = x.birthday := by
+  apply le_antisymm (birthday_neg' x)
+  conv_lhs => rw [← neg_neg x]
+  exact birthday_neg' _
+
+theorem le_birthday (x : Game) : x ≤ x.birthday.toGame := by
+  let ⟨y, hy₁, hy₂⟩ := birthday_eq_pGameBirthday x
+  rw [← hy₁]
+  apply (y.le_birthday).trans
+  rw [toPGame_le_iff, hy₁, hy₂]
+
+theorem neg_birthday_le (x : Game) : -x.birthday.toGame ≤ x := by
+  rw [neg_le, ← birthday_neg]
+  exact le_birthday _
+
+theorem birthday_add_le (x y : Game) : (x + y).birthday ≤ x.birthday ♯ y.birthday := by
+  let ⟨a, ha₁, ha₂⟩ := birthday_eq_pGameBirthday x
+  let ⟨b, hb₁, hb₂⟩ := birthday_eq_pGameBirthday y
+  rw [← ha₂, ← hb₂, ← ha₁, ← hb₁, ← PGame.birthday_add]
+  exact birthday_quot_le_pGameBirthday  _
+
+theorem birthday_sub_le (x y : Game) : (x - y).birthday ≤ x.birthday ♯ y.birthday := by
+  apply (birthday_add_le x _).trans_eq
+  rw [birthday_neg]
+
+/- The bound `(x * y).birthday ≤ x.birthday ⨳ y.birthday` is currently an open problem. See
+  https://mathoverflow.net/a/476829/147705. -/
+
+end Game
 
 end SetTheory
diff --git a/Mathlib/SetTheory/Game/Domineering.lean b/Mathlib/SetTheory/Game/Domineering.lean
index e542ce9026c9b..e2301050c9961 100644
--- a/Mathlib/SetTheory/Game/Domineering.lean
+++ b/Mathlib/SetTheory/Game/Domineering.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.SetTheory.Game.State
 
diff --git a/Mathlib/SetTheory/Game/Impartial.lean b/Mathlib/SetTheory/Game/Impartial.lean
index 24c984856e4bb..fea623654ef44 100644
--- a/Mathlib/SetTheory/Game/Impartial.lean
+++ b/Mathlib/SetTheory/Game/Impartial.lean
@@ -25,13 +25,12 @@ open scoped PGame
 namespace PGame
 
 /-- The definition for an impartial game, defined using Conway induction. -/
-def ImpartialAux : PGame → Prop
-  | G => (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j)
-termination_by G => G -- Porting note: Added `termination_by`
+def ImpartialAux (G : PGame) : Prop :=
+  (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j)
+termination_by G
 
-theorem impartialAux_def {G : PGame} :
-    G.ImpartialAux ↔
-      (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j) := by
+theorem impartialAux_def {G : PGame} : G.ImpartialAux ↔
+    (G ≈ -G) ∧ (∀ i, ImpartialAux (G.moveLeft i)) ∧ ∀ j, ImpartialAux (G.moveRight j) := by
   rw [ImpartialAux]
 
 /-- A typeclass on impartial games. -/
@@ -47,18 +46,20 @@ theorem impartial_def {G : PGame} :
 
 namespace Impartial
 
-instance impartial_zero : Impartial 0 := by rw [impartial_def]; dsimp; simp
+instance impartial_zero : Impartial 0 := by
+  rw [impartial_def]
+  simp
 
 instance impartial_star : Impartial star := by
-  rw [impartial_def]; simpa using Impartial.impartial_zero
+  rw [impartial_def]
+  simpa using Impartial.impartial_zero
 
 theorem neg_equiv_self (G : PGame) [h : G.Impartial] : G ≈ -G :=
   (impartial_def.1 h).1
 
--- Porting note: Changed `-⟦G⟧` to `-(⟦G⟧ : Quotient setoid)`
 @[simp]
-theorem mk'_neg_equiv_self (G : PGame) [G.Impartial] : -(⟦G⟧ : Quotient setoid) = ⟦G⟧ :=
-  Quot.sound (Equiv.symm (neg_equiv_self G))
+theorem mk'_neg_equiv_self (G : PGame) [G.Impartial] : -(⟦G⟧ : Game) = ⟦G⟧ :=
+  game_eq (Equiv.symm (neg_equiv_self G))
 
 instance moveLeft_impartial {G : PGame} [h : G.Impartial] (i : G.LeftMoves) :
     (G.moveLeft i).Impartial :=
@@ -68,49 +69,47 @@ instance moveRight_impartial {G : PGame} [h : G.Impartial] (j : G.RightMoves) :
     (G.moveRight j).Impartial :=
   (impartial_def.1 h).2.2 j
 
-theorem impartial_congr : ∀ {G H : PGame} (_ : G ≡r H) [G.Impartial], H.Impartial
-  | G, H => fun e => by
-    intro h
-    exact impartial_def.2
-      ⟨Equiv.trans e.symm.equiv (Equiv.trans (neg_equiv_self G) (neg_equiv_neg_iff.2 e.equiv)),
-        fun i => impartial_congr (e.moveLeftSymm i), fun j => impartial_congr (e.moveRightSymm j)⟩
-termination_by G H => (G, H)
-
-instance impartial_add : ∀ (G H : PGame) [G.Impartial] [H.Impartial], (G + H).Impartial
-  | G, H, _, _ => by
-    rw [impartial_def]
-    refine ⟨Equiv.trans (add_congr (neg_equiv_self G) (neg_equiv_self _))
-        (Equiv.symm (negAddRelabelling _ _).equiv), fun k => ?_, fun k => ?_⟩
-    · apply leftMoves_add_cases k
-      all_goals
-        intro i; simp only [add_moveLeft_inl, add_moveLeft_inr]
-        apply impartial_add
-    · apply rightMoves_add_cases k
-      all_goals
-        intro i; simp only [add_moveRight_inl, add_moveRight_inr]
-        apply impartial_add
-termination_by G H => (G, H)
-
-instance impartial_neg : ∀ (G : PGame) [G.Impartial], (-G).Impartial
-  | G, _ => by
-    rw [impartial_def]
-    refine ⟨?_, fun i => ?_, fun i => ?_⟩
-    · rw [neg_neg]
-      exact Equiv.symm (neg_equiv_self G)
-    · rw [moveLeft_neg']
-      apply impartial_neg
-    · rw [moveRight_neg']
-      apply impartial_neg
-termination_by G => G
+theorem impartial_congr {G H : PGame} (e : G ≡r H) [G.Impartial] : H.Impartial :=
+  impartial_def.2
+    ⟨Equiv.trans e.symm.equiv (Equiv.trans (neg_equiv_self G) (neg_equiv_neg_iff.2 e.equiv)),
+      fun i => impartial_congr (e.moveLeftSymm i), fun j => impartial_congr (e.moveRightSymm j)⟩
+termination_by G
+
+instance impartial_add (G H : PGame) [G.Impartial] [H.Impartial] : (G + H).Impartial := by
+  rw [impartial_def]
+  refine ⟨Equiv.trans (add_congr (neg_equiv_self G) (neg_equiv_self _))
+      (Equiv.symm (negAddRelabelling _ _).equiv), fun k => ?_, fun k => ?_⟩
+  · apply leftMoves_add_cases k
+    all_goals
+      intro i; simp only [add_moveLeft_inl, add_moveLeft_inr]
+      apply impartial_add
+  · apply rightMoves_add_cases k
+    all_goals
+      intro i; simp only [add_moveRight_inl, add_moveRight_inr]
+      apply impartial_add
+termination_by (G, H)
+
+instance impartial_neg (G : PGame) [G.Impartial] : (-G).Impartial := by
+  rw [impartial_def]
+  refine ⟨?_, fun i => ?_, fun i => ?_⟩
+  · rw [neg_neg]
+    exact Equiv.symm (neg_equiv_self G)
+  · rw [moveLeft_neg']
+    exact impartial_neg _
+  · rw [moveRight_neg']
+    exact impartial_neg _
+termination_by G
 
 variable (G : PGame) [Impartial G]
 
-theorem nonpos : ¬0 < G := fun h => by
+theorem nonpos : ¬0 < G := by
+  intro h
   have h' := neg_lt_neg_iff.2 h
   rw [neg_zero, lt_congr_left (Equiv.symm (neg_equiv_self G))] at h'
   exact (h.trans h').false
 
-theorem nonneg : ¬G < 0 := fun h => by
+theorem nonneg : ¬G < 0 := by
+  intro h
   have h' := neg_lt_neg_iff.2 h
   rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h'
   exact (h.trans h').false
@@ -134,22 +133,19 @@ theorem not_fuzzy_zero_iff : ¬G ‖ 0 ↔ (G ≈ 0) :=
 theorem add_self : G + G ≈ 0 :=
   Equiv.trans (add_congr_left (neg_equiv_self G)) (neg_add_cancel_equiv G)
 
--- Porting note: Changed `⟦G⟧` to `(⟦G⟧ : Quotient setoid)`
 @[simp]
-theorem mk'_add_self : (⟦G⟧ : Quotient setoid) + ⟦G⟧ = 0 :=
-  Quot.sound (add_self G)
+theorem mk'_add_self : (⟦G⟧ : Game) + ⟦G⟧ = 0 :=
+  game_eq (add_self G)
 
 /-- This lemma doesn't require `H` to be impartial. -/
 theorem equiv_iff_add_equiv_zero (H : PGame) : (H ≈ G) ↔ (H + G ≈ 0) := by
-  rw [equiv_iff_game_eq, ← @add_right_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add,
-    equiv_iff_game_eq]
-  rfl
+  rw [equiv_iff_game_eq, ← add_right_cancel_iff (a := ⟦G⟧), mk'_add_self, ← quot_add,
+    equiv_iff_game_eq, quot_zero]
 
 /-- This lemma doesn't require `H` to be impartial. -/
 theorem equiv_iff_add_equiv_zero' (H : PGame) : (G ≈ H) ↔ (G + H ≈ 0) := by
-  rw [equiv_iff_game_eq, ← @add_left_cancel_iff _ _ _ ⟦G⟧, mk'_add_self, ← quot_add,
-    equiv_iff_game_eq]
-  exact ⟨Eq.symm, Eq.symm⟩
+  rw [equiv_iff_game_eq, ← add_left_cancel_iff, mk'_add_self, ← quot_add, equiv_iff_game_eq,
+    Eq.comm, quot_zero]
 
 theorem le_zero_iff {G : PGame} [G.Impartial] : G ≤ 0 ↔ 0 ≤ G := by
   rw [← zero_le_neg_iff, le_congr_right (neg_equiv_self G)]
diff --git a/Mathlib/SetTheory/Game/Nim.lean b/Mathlib/SetTheory/Game/Nim.lean
index 6ed82c179218b..f563fafa77d4b 100644
--- a/Mathlib/SetTheory/Game/Nim.lean
+++ b/Mathlib/SetTheory/Game/Nim.lean
@@ -65,11 +65,11 @@ theorem moveLeft_nim_hEq (o : Ordinal) :
 theorem moveRight_nim_hEq (o : Ordinal) :
     HEq (nim o).moveRight fun i : o.toType => nim ((enumIsoToType o).symm i) := by rw [nim_def]; rfl
 
-/-- Turns an ordinal less than `o` into a left move for `nim o` and viceversa. -/
+/-- Turns an ordinal less than `o` into a left move for `nim o` and vice versa. -/
 noncomputable def toLeftMovesNim {o : Ordinal} : Set.Iio o ≃ (nim o).LeftMoves :=
   (enumIsoToType o).toEquiv.trans (Equiv.cast (leftMoves_nim o).symm)
 
-/-- Turns an ordinal less than `o` into a right move for `nim o` and viceversa. -/
+/-- Turns an ordinal less than `o` into a right move for `nim o` and vice versa. -/
 noncomputable def toRightMovesNim {o : Ordinal} : Set.Iio o ≃ (nim o).RightMoves :=
   (enumIsoToType o).toEquiv.trans (Equiv.cast (rightMoves_nim o).symm)
 
@@ -206,56 +206,69 @@ theorem nim_add_fuzzy_zero_iff {o₁ o₂ : Ordinal} : nim o₁ + nim o₂ ‖ 0
 theorem nim_equiv_iff_eq {o₁ o₂ : Ordinal} : (nim o₁ ≈ nim o₂) ↔ o₁ = o₂ := by
   rw [Impartial.equiv_iff_add_equiv_zero, nim_add_equiv_zero_iff]
 
-/-- The Grundy value of an impartial game, the ordinal which corresponds to the game of nim that the
- game is equivalent to -/
-noncomputable def grundyValue : PGame.{u} → Ordinal.{u}
-  | G => Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i)
-termination_by G => G
+/-- The Grundy value of an impartial game is recursively defined as the minimum excluded value
+(the infimum of the complement) of the Grundy values of either its left or right options.
 
+This is the ordinal which corresponds to the game of nim that the game is equivalent to. -/
+noncomputable def grundyValue (G : PGame.{u}) : Ordinal.{u} :=
+  sInf (Set.range fun i => grundyValue (G.moveLeft i))ᶜ
+termination_by G
+
+theorem grundyValue_eq_sInf_moveLeft (G : PGame) :
+    grundyValue G = sInf (Set.range (grundyValue ∘ G.moveLeft))ᶜ := by
+  rw [grundyValue]; rfl
+
+set_option linter.deprecated false in
+@[deprecated grundyValue_eq_sInf_moveLeft (since := "2024-09-16")]
 theorem grundyValue_eq_mex_left (G : PGame) :
-    grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i) := by rw [grundyValue]
-
-/-- The Sprague-Grundy theorem which states that every impartial game is equivalent to a game of
- nim, namely the game of nim corresponding to the games Grundy value -/
-theorem equiv_nim_grundyValue : ∀ (G : PGame.{u}) [G.Impartial], G ≈ nim (grundyValue G)
-  | G => by
-    rw [Impartial.equiv_iff_add_equiv_zero, ← Impartial.forall_leftMoves_fuzzy_iff_equiv_zero]
-    intro i
-    apply leftMoves_add_cases i
-    · intro i₁
-      rw [add_moveLeft_inl]
-      apply
-        (fuzzy_congr_left (add_congr_left (Equiv.symm (equiv_nim_grundyValue (G.moveLeft i₁))))).1
-      rw [nim_add_fuzzy_zero_iff]
-      intro heq
-      rw [eq_comm, grundyValue_eq_mex_left G] at heq
-      -- Porting note: added universe annotation, argument
-      have h := Ordinal.ne_mex.{u, u} (fun i ↦ grundyValue (moveLeft G i))
-      rw [heq] at h
-      exact (h i₁).irrefl
-    · intro i₂
-      rw [add_moveLeft_inr, ← Impartial.exists_left_move_equiv_iff_fuzzy_zero]
-      revert i₂
-      rw [nim_def]
-      intro i₂
-      have h' :
-        ∃ i : G.LeftMoves,
-          grundyValue (G.moveLeft i) = Ordinal.typein (α := toType (grundyValue G)) (· < ·) i₂ := by
-        revert i₂
-        rw [grundyValue_eq_mex_left]
-        intro i₂
-        have hnotin : _ ∉ _ := fun hin => not_le_of_lt (Ordinal.typein_lt_self i₂) (csInf_le' hin)
-        simpa using hnotin
-      cases' h' with i hi
-      use toLeftMovesAdd (Sum.inl i)
-      rw [add_moveLeft_inl, moveLeft_mk]
-      apply Equiv.trans (add_congr_left (equiv_nim_grundyValue (G.moveLeft i)))
-      simpa only [hi] using Impartial.add_self (nim (grundyValue (G.moveLeft i)))
-termination_by G => G
-decreasing_by all_goals pgame_wf_tac
+    grundyValue G = Ordinal.mex fun i => grundyValue (G.moveLeft i) :=
+  grundyValue_eq_sInf_moveLeft G
+
+theorem grundyValue_ne_moveLeft {G : PGame} (i : G.LeftMoves) :
+    grundyValue (G.moveLeft i) ≠ grundyValue G := by
+  conv_rhs => rw [grundyValue_eq_sInf_moveLeft]
+  have := csInf_mem (nonempty_of_not_bddAbove <|
+    not_bddAbove_compl_of_small (Set.range fun i => grundyValue (G.moveLeft i)))
+  rw [Set.mem_compl_iff, Set.mem_range, not_exists] at this
+  exact this _
+
+theorem le_grundyValue_of_Iio_subset_moveLeft {G : PGame} {o : Ordinal}
+    (h : Set.Iio o ⊆ Set.range (grundyValue ∘ G.moveLeft)) : o ≤ grundyValue G := by
+  by_contra! ho
+  obtain ⟨i, hi⟩ := h ho
+  exact grundyValue_ne_moveLeft i hi
+
+theorem exists_grundyValue_moveLeft_of_lt {G : PGame} {o : Ordinal} (h : o < grundyValue G) :
+    ∃ i, grundyValue (G.moveLeft i) = o := by
+  rw [grundyValue_eq_sInf_moveLeft] at h
+  by_contra ha
+  exact h.not_le (csInf_le' ha)
+
+theorem grundyValue_le_of_forall_moveLeft {G : PGame} {o : Ordinal}
+    (h : ∀ i, grundyValue (G.moveLeft i) ≠ o) : G.grundyValue ≤ o := by
+  contrapose! h
+  exact exists_grundyValue_moveLeft_of_lt h
+
+/-- The **Sprague-Grundy theorem** states that every impartial game is equivalent to a game of nim,
+namely the game of nim corresponding to the game's Grundy value. -/
+theorem equiv_nim_grundyValue (G : PGame.{u}) [G.Impartial] : G ≈ nim (grundyValue G) := by
+  rw [Impartial.equiv_iff_add_equiv_zero, ← Impartial.forall_leftMoves_fuzzy_iff_equiv_zero]
+  intro x
+  apply leftMoves_add_cases x <;>
+  intro i
+  · rw [add_moveLeft_inl,
+      ← fuzzy_congr_left (add_congr_left (Equiv.symm (equiv_nim_grundyValue _))),
+      nim_add_fuzzy_zero_iff]
+    exact grundyValue_ne_moveLeft i
+  · rw [add_moveLeft_inr, ← Impartial.exists_left_move_equiv_iff_fuzzy_zero]
+    obtain ⟨j, hj⟩ := exists_grundyValue_moveLeft_of_lt <| toLeftMovesNim_symm_lt i
+    use toLeftMovesAdd (Sum.inl j)
+    rw [add_moveLeft_inl, moveLeft_nim']
+    exact Equiv.trans (add_congr_left (equiv_nim_grundyValue _)) (hj ▸ Impartial.add_self _)
+termination_by G
 
 theorem grundyValue_eq_iff_equiv_nim {G : PGame} [G.Impartial] {o : Ordinal} :
-    grundyValue G = o ↔ (G ≈ nim o) :=
+    grundyValue G = o ↔ G ≈ nim o :=
   ⟨by rintro rfl; exact equiv_nim_grundyValue G,
    by intro h; rw [← nim_equiv_iff_eq]; exact Equiv.trans (Equiv.symm (equiv_nim_grundyValue G)) h⟩
 
@@ -282,58 +295,77 @@ theorem grundyValue_star : grundyValue star = 1 :=
 theorem grundyValue_neg (G : PGame) [G.Impartial] : grundyValue (-G) = grundyValue G := by
   rw [grundyValue_eq_iff_equiv_nim, neg_equiv_iff, neg_nim, ← grundyValue_eq_iff_equiv_nim]
 
-theorem grundyValue_eq_mex_right :
-    ∀ (G : PGame) [G.Impartial],
-      grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i)
-   | ⟨l, r, L, R⟩, _ => by
-    rw [← grundyValue_neg, grundyValue_eq_mex_left]
-    congr
-    ext i
-    haveI : (R i).Impartial := @Impartial.moveRight_impartial ⟨l, r, L, R⟩ _ i
-    apply grundyValue_neg
-
--- Todo: this actually generalizes to all ordinals, by defining `Ordinal.lxor` as the pairwise
--- `Nat.xor` of base `ω` Cantor normal forms.
+theorem grundyValue_eq_sInf_moveRight (G : PGame) [G.Impartial] :
+    grundyValue G = sInf (Set.range (grundyValue ∘ G.moveRight))ᶜ := by
+  obtain ⟨l, r, L, R⟩ := G
+  rw [← grundyValue_neg, grundyValue_eq_sInf_moveLeft]
+  iterate 3 apply congr_arg
+  ext i
+  exact @grundyValue_neg _ (@Impartial.moveRight_impartial ⟨l, r, L, R⟩ _ _)
+
+set_option linter.deprecated false in
+@[deprecated grundyValue_eq_sInf_moveRight (since := "2024-09-16")]
+theorem grundyValue_eq_mex_right (G : PGame) [G.Impartial] :
+    grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveRight i) :=
+  grundyValue_eq_sInf_moveRight G
+
+theorem grundyValue_ne_moveRight {G : PGame} [G.Impartial] (i : G.RightMoves) :
+    grundyValue (G.moveRight i) ≠ grundyValue G := by
+  convert grundyValue_ne_moveLeft (toLeftMovesNeg i) using 1 <;> simp
+
+theorem le_grundyValue_of_Iio_subset_moveRight {G : PGame} [G.Impartial] {o : Ordinal}
+    (h : Set.Iio o ⊆ Set.range (grundyValue ∘ G.moveRight)) : o ≤ grundyValue G := by
+  by_contra! ho
+  obtain ⟨i, hi⟩ := h ho
+  exact grundyValue_ne_moveRight i hi
+
+theorem exists_grundyValue_moveRight_of_lt {G : PGame} [G.Impartial] {o : Ordinal}
+    (h : o < grundyValue G) : ∃ i, grundyValue (G.moveRight i) = o := by
+  rw [← grundyValue_neg] at h
+  obtain ⟨i, hi⟩ := exists_grundyValue_moveLeft_of_lt h
+  use toLeftMovesNeg.symm i
+  rwa [← grundyValue_neg, ← moveLeft_neg']
+
+theorem grundyValue_le_of_forall_moveRight {G : PGame} [G.Impartial] {o : Ordinal}
+    (h : ∀ i, grundyValue (G.moveRight i) ≠ o) : G.grundyValue ≤ o := by
+  contrapose! h
+  exact exists_grundyValue_moveRight_of_lt h
+
+-- Todo: redefine `grundyValue` as a nimber, and prove `grundyValue (nim a + nim b) = a + b` for all
+-- nimbers.
+
 /-- The Grundy value of the sum of two nim games with natural numbers of piles equals their bitwise
 xor. -/
 @[simp]
-theorem grundyValue_nim_add_nim (n m : ℕ) :
-    grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
-  -- We do strong induction on both variables.
-  induction' n using Nat.strong_induction_on with n hn generalizing m
-  induction' m using Nat.strong_induction_on with m hm
-  rw [grundyValue_eq_mex_left]
-  refine (Ordinal.mex_le_of_ne.{u, u} fun i => ?_).antisymm
-    (Ordinal.le_mex_of_forall fun ou hu => ?_)
-  -- The Grundy value `n ^^^ m` can't be reached by left moves.
-  · apply leftMoves_add_cases i <;>
-      · -- A left move leaves us with a Grundy value of `k ^^^ m` for `k < n`, or
-        -- `n ^^^ k` for `k < m`.
-        refine fun a => leftMovesNimRecOn a fun ok hk => ?_
-        obtain ⟨k, rfl⟩ := Ordinal.lt_omega.1 (hk.trans (Ordinal.nat_lt_omega _))
-        simp only [add_moveLeft_inl, add_moveLeft_inr, moveLeft_nim', Equiv.symm_apply_apply]
-        -- The inequality follows from injectivity.
-        rw [natCast_lt] at hk
-        first
-        | rw [hn _ hk]
-        | rw [hm _ hk]
-        refine fun h => hk.ne ?_
-        rw [Ordinal.natCast_inj] at h
-        first
-        | rwa [Nat.xor_left_inj] at h
-        | rwa [Nat.xor_right_inj] at h
-  -- Every other smaller Grundy value can be reached by left moves.
-  · -- If `u < m ^^^ n`, then either `u ^^^ n < m` or `u ^^^ m < n`.
-    obtain ⟨u, rfl⟩ := Ordinal.lt_omega.1 (hu.trans (Ordinal.nat_lt_omega _))
-    replace hu := Ordinal.natCast_lt.1 hu
-    cases' Nat.lt_xor_cases hu with h h
-    -- In the first case, reducing the `m` pile to `u ^^^ n` gives the desired Grundy value.
-    · refine ⟨toLeftMovesAdd (Sum.inl <| toLeftMovesNim ⟨_, Ordinal.natCast_lt.2 h⟩), ?_⟩
-      simp [Nat.xor_cancel_right, hn _ h]
-    -- In the second case, reducing the `n` pile to `u ^^^ m` gives the desired Grundy value.
-    · refine ⟨toLeftMovesAdd (Sum.inr <| toLeftMovesNim ⟨_, Ordinal.natCast_lt.2 h⟩), ?_⟩
-      have : n ^^^ (u ^^^ n) = u := by rw [Nat.xor_comm u, Nat.xor_cancel_left]
-      simpa [hm _ h] using this
+theorem grundyValue_nim_add_nim (n m : ℕ) : grundyValue (nim.{u} n + nim.{u} m) = n ^^^ m := by
+  apply (grundyValue_le_of_forall_moveLeft _).antisymm (le_grundyValue_of_Iio_subset_moveLeft _)
+  -- Since XOR is injective, no left moves of `nim n + nim m` will have `n ^^^ m` as a Grundy value.
+  · intro i
+    apply leftMoves_add_cases i
+    all_goals
+      intro j
+      have hj := toLeftMovesNim_symm_lt j
+      obtain ⟨k, hk⟩ := lt_omega0.1 (hj.trans (nat_lt_omega0 _))
+      rw [hk, Nat.cast_lt] at hj
+      have := hj.ne
+      have := hj -- The termination checker doesn't work without this.
+    · rwa [add_moveLeft_inl, moveLeft_nim', ne_eq, hk, grundyValue_nim_add_nim, Nat.cast_inj,
+        Nat.xor_left_inj]
+    · rwa [add_moveLeft_inr, moveLeft_nim', ne_eq, hk, grundyValue_nim_add_nim, Nat.cast_inj,
+        Nat.xor_right_inj]
+  -- For any `k < n ^^^ m`, either `nim (k ^^^ m) + nim m` or `nim n + nim (k ^^^ n)` is a left
+  -- option with Grundy value `k`.
+  · intro k hk
+    obtain ⟨k, rfl⟩ := Ordinal.lt_omega0.1 (hk.trans (Ordinal.nat_lt_omega0 _))
+    rw [Set.mem_Iio, Nat.cast_lt] at hk
+    obtain hk | hk := Nat.lt_xor_cases hk <;> rw [← natCast_lt] at hk
+    · use toLeftMovesAdd (Sum.inl (toLeftMovesNim ⟨_, hk⟩))
+      rw [Function.comp_apply, add_moveLeft_inl, moveLeft_nim', Equiv.symm_apply_apply,
+        grundyValue_nim_add_nim, Nat.xor_cancel_right]
+    · use toLeftMovesAdd (Sum.inr (toLeftMovesNim ⟨_, hk⟩))
+      rw [Function.comp_apply, add_moveLeft_inr, moveLeft_nim', Equiv.symm_apply_apply,
+        grundyValue_nim_add_nim, Nat.xor_comm, Nat.xor_cancel_right]
+termination_by (n, m)
 
 theorem nim_add_nim_equiv {n m : ℕ} : nim n + nim m ≈ nim (n ^^^ m) := by
   rw [← grundyValue_eq_iff_equiv_nim, grundyValue_nim_add_nim]
diff --git a/Mathlib/SetTheory/Game/Ordinal.lean b/Mathlib/SetTheory/Game/Ordinal.lean
index 5fdf75f21eaa4..b04e1d965c6ca 100644
--- a/Mathlib/SetTheory/Game/Ordinal.lean
+++ b/Mathlib/SetTheory/Game/Ordinal.lean
@@ -200,4 +200,15 @@ termination_by (a, b)
 theorem toGame_nmul (a b : Ordinal) : (a ⨳ b).toGame = ⟦a.toPGame * b.toPGame⟧ :=
   Quot.sound (toPGame_nmul a b)
 
+@[simp, norm_cast]
+theorem toGame_natCast : ∀ n : ℕ, toGame n = n
+  | 0 => Quot.sound (zeroToPGameRelabelling).equiv
+  | n + 1 => by
+    have : toGame 1 = 1 := Quot.sound oneToPGameRelabelling.equiv
+    rw [Nat.cast_add, ← nadd_nat, toGame_nadd, toGame_natCast, Nat.cast_one, this]
+    rfl
+
+theorem toPGame_natCast (n : ℕ) : toPGame n ≈ n := by
+  rw [PGame.equiv_iff_game_eq, ← toGame, toGame_natCast, quot_natCast]
+
 end Ordinal
diff --git a/Mathlib/SetTheory/Game/PGame.lean b/Mathlib/SetTheory/Game/PGame.lean
index 2f4a23900dd0c..55a01998f3b98 100644
--- a/Mathlib/SetTheory/Game/PGame.lean
+++ b/Mathlib/SetTheory/Game/PGame.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
+Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison
 -/
 import Mathlib.Algebra.Order.ZeroLEOne
 import Mathlib.Data.List.InsertNth
@@ -216,9 +216,9 @@ theorem wf_isOption : WellFounded IsOption :=
   ⟨fun x =>
     moveRecOn x fun x IHl IHr =>
       Acc.intro x fun y h => by
-        induction' h with _ i _ j
-        · exact IHl i
-        · exact IHr j⟩
+        induction h with
+        | moveLeft i => exact IHl i
+        | moveRight j => exact IHr j⟩
 
 /-- `Subsequent x y` says that `x` can be obtained by playing some nonempty sequence of moves from
 `y`. It is the transitive closure of `IsOption`. -/
@@ -458,7 +458,7 @@ private theorem le_trans_aux {x y z : PGame}
 instance : Preorder PGame :=
   { PGame.le with
     le_refl := fun x => by
-      induction' x with _ _ _ _ IHl IHr
+      induction x with | mk _ _ _ _ IHl IHr => _
       exact
         le_of_forall_lf (fun i => lf_of_le_moveLeft (IHl i)) fun i => lf_of_moveRight_le (IHr i)
     le_trans := by
@@ -676,7 +676,7 @@ instance : IsEquiv _ PGame.Equiv where
   trans := fun _ _ _ ⟨xy, yx⟩ ⟨yz, zy⟩ => ⟨xy.trans yz, zy.trans yx⟩
   symm _ _ := And.symm
 
--- Porting note: moved the setoid instance from Basic.lean to here
+-- Porting note: moved the setoid instance from Basic.lean to here
 
 instance setoid : Setoid PGame :=
   ⟨Equiv, refl, symm, Trans.trans⟩
@@ -822,13 +822,15 @@ theorem equiv_congr_right {x₁ x₂ : PGame} : (x₁ ≈ x₂) ↔ ∀ y₁, (x
   ⟨fun h _ => ⟨fun h' => Equiv.trans (Equiv.symm h) h', fun h' => Equiv.trans h h'⟩,
    fun h => (h x₂).2 <| equiv_rfl⟩
 
-theorem equiv_of_mk_equiv {x y : PGame} (L : x.LeftMoves ≃ y.LeftMoves)
+theorem Equiv.of_equiv {x y : PGame} (L : x.LeftMoves ≃ y.LeftMoves)
     (R : x.RightMoves ≃ y.RightMoves) (hl : ∀ i, x.moveLeft i ≈ y.moveLeft (L i))
     (hr : ∀ j, x.moveRight j ≈ y.moveRight (R j)) : x ≈ y := by
   constructor <;> rw [le_def]
   · exact ⟨fun i => Or.inl ⟨_, (hl i).1⟩, fun j => Or.inr ⟨_, by simpa using (hr (R.symm j)).1⟩⟩
   · exact ⟨fun i => Or.inl ⟨_, by simpa using (hl (L.symm i)).2⟩, fun j => Or.inr ⟨_, (hr j).2⟩⟩
 
+@[deprecated (since := "2024-09-26")] alias equiv_of_mk_equiv := Equiv.of_equiv
+
 /-- The fuzzy, confused, or incomparable relation on pre-games.
 
 If `x ‖ 0`, then the first player can always win `x`. -/
@@ -1262,8 +1264,8 @@ theorem zero_fuzzy_neg_iff {x : PGame} : 0 ‖ -x ↔ 0 ‖ x := by rw [← neg_
 /-- The sum of `x = {xL | xR}` and `y = {yL | yR}` is `{xL + y, x + yL | xR + y, x + yR}`. -/
 instance : Add PGame.{u} :=
   ⟨fun x y => by
-    induction' x with xl xr _ _ IHxl IHxr generalizing y
-    induction' y with yl yr yL yR IHyl IHyr
+    induction x generalizing y with | mk xl xr _ _ IHxl IHxr => _
+    induction y with | mk yl yr yL yR IHyl IHyr => _
     have y := mk yl yr yL yR
     refine ⟨xl ⊕ yl, xr ⊕ yr, Sum.rec ?_ ?_, Sum.rec ?_ ?_⟩
     · exact fun i => IHxl i y
@@ -1275,7 +1277,8 @@ instance : Add PGame.{u} :=
 
 Note that this is **not** the usual recursive definition `n = {0, 1, … | }`. For instance,
 `2 = 0 + 1 + 1 = {0 + 0 + 1, 0 + 1 + 0 | }` does not contain any left option equivalent to `0`. For
-an implementation of said definition, see `Ordinal.toPGame`. -/
+an implementation of said definition, see `Ordinal.toPGame`. For the proof that these games are
+equivalent, see `Ordinal.toPGame_natCast`. -/
 instance : NatCast PGame :=
   ⟨Nat.unaryCast⟩
 
@@ -1431,9 +1434,11 @@ instance : Sub PGame :=
   ⟨fun x y => x + -y⟩
 
 @[simp]
-theorem sub_zero (x : PGame) : x - 0 = x + 0 :=
+theorem sub_zero_eq_add_zero (x : PGame) : x - 0 = x + 0 :=
   show x + -0 = x + 0 by rw [neg_zero]
 
+@[deprecated (since := "2024-09-26")] alias sub_zero := sub_zero_eq_add_zero
+
 /-- If `w` has the same moves as `x` and `y` has the same moves as `z`,
 then `w - y` has the same moves as `x - z`. -/
 def Relabelling.subCongr {w x y z : PGame} (h₁ : w ≡r x) (h₂ : y ≡r z) : w - y ≡r x - z :=
@@ -1732,15 +1737,18 @@ instance uniqueStarLeftMoves : Unique star.LeftMoves :=
 instance uniqueStarRightMoves : Unique star.RightMoves :=
   PUnit.unique
 
+theorem zero_lf_star : 0 ⧏ star := by
+  rw [zero_lf]
+  use default
+  rintro ⟨⟩
+
+theorem star_lf_zero : star ⧏ 0 := by
+  rw [lf_zero]
+  use default
+  rintro ⟨⟩
+
 theorem star_fuzzy_zero : star ‖ 0 :=
-  ⟨by
-    rw [lf_zero]
-    use default
-    rintro ⟨⟩,
-   by
-    rw [zero_lf]
-    use default
-    rintro ⟨⟩⟩
+  ⟨star_lf_zero, zero_lf_star⟩
 
 @[simp]
 theorem neg_star : -star = star := by simp [star]
@@ -1759,3 +1767,5 @@ theorem zero_lf_one : (0 : PGame) ⧏ 1 :=
 end PGame
 
 end SetTheory
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/SetTheory/Game/Short.lean b/Mathlib/SetTheory/Game/Short.lean
index fa9a0a1b91291..03c8ecbda2c5b 100644
--- a/Mathlib/SetTheory/Game/Short.lean
+++ b/Mathlib/SetTheory/Game/Short.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 
 import Mathlib.Data.Fintype.Basic
@@ -136,7 +136,7 @@ def moveRightShort' {xl xr} (xL xR) [S : Short (mk xl xr xL xR)] (j : xr) : Shor
 
 attribute [local instance] moveRightShort'
 
-theorem short_birthday (x : PGame.{u}) : [Short x] → x.birthday < Ordinal.omega := by
+theorem short_birthday (x : PGame.{u}) : [Short x] → x.birthday < Ordinal.omega0 := by
   -- Porting note: Again `induction` is used instead of `pgame_wf_tac`
   induction x with
   | mk xl xr xL xR ihl ihr =>
@@ -234,21 +234,25 @@ def leLFDecidable : ∀ (x y : PGame.{u}) [Short x] [Short y], Decidable (x ≤
   | mk xl xr xL xR, mk yl yr yL yR, shortx, shorty => by
     constructor
     · refine @decidable_of_iff' _ _ mk_le_mk (id ?_)
-      apply @And.decidable _ _ ?_ ?_
-      · apply @Fintype.decidableForallFintype xl _ ?_ _
+      have : Decidable (∀ (i : xl), xL i ⧏ mk yl yr yL yR) := by
+        apply @Fintype.decidableForallFintype xl _ ?_ _
         intro i
         apply (leLFDecidable _ _).2
-      · apply @Fintype.decidableForallFintype yr _ ?_ _
+      have : Decidable (∀ (j : yr), mk xl xr xL xR ⧏ yR j) := by
+        apply @Fintype.decidableForallFintype yr _ ?_ _
         intro i
         apply (leLFDecidable _ _).2
+      exact inferInstanceAs (Decidable (_ ∧ _))
     · refine @decidable_of_iff' _ _ mk_lf_mk (id ?_)
-      apply @Or.decidable _ _ ?_ ?_
-      · apply @Fintype.decidableExistsFintype yl _ ?_ _
+      have : Decidable (∃ i, mk xl xr xL xR ≤ yL i) := by
+        apply @Fintype.decidableExistsFintype yl _ ?_ _
         intro i
         apply (leLFDecidable _ _).1
-      · apply @Fintype.decidableExistsFintype xr _ ?_ _
+      have : Decidable (∃ j, xR j ≤ mk yl yr yL yR) := by
+        apply @Fintype.decidableExistsFintype xr _ ?_ _
         intro i
         apply (leLFDecidable _ _).1
+      exact inferInstanceAs (Decidable (_ ∨ _))
 termination_by x y => (x, y)
 
 instance leDecidable (x y : PGame.{u}) [Short x] [Short y] : Decidable (x ≤ y) :=
@@ -258,10 +262,10 @@ instance lfDecidable (x y : PGame.{u}) [Short x] [Short y] : Decidable (x ⧏ y)
   (leLFDecidable x y).2
 
 instance ltDecidable (x y : PGame.{u}) [Short x] [Short y] : Decidable (x < y) :=
-  And.decidable
+  inferInstanceAs (Decidable (_ ∧ _))
 
 instance equivDecidable (x y : PGame.{u}) [Short x] [Short y] : Decidable (x ≈ y) :=
-  And.decidable
+  inferInstanceAs (Decidable (_ ∧ _))
 
 example : Short 0 := by infer_instance
 
diff --git a/Mathlib/SetTheory/Game/State.lean b/Mathlib/SetTheory/Game/State.lean
index 79423b09551f8..d7871744adc58 100644
--- a/Mathlib/SetTheory/Game/State.lean
+++ b/Mathlib/SetTheory/Game/State.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.SetTheory.Game.Short
 
diff --git a/Mathlib/SetTheory/Lists.lean b/Mathlib/SetTheory/Lists.lean
index b7bd039b50980..cd2aeac9c7722 100644
--- a/Mathlib/SetTheory/Lists.lean
+++ b/Mathlib/SetTheory/Lists.lean
@@ -129,7 +129,7 @@ instance : HasSubset (Lists' α true) :=
 /-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is
 equivalent as a ZFA list to this ZFA list. -/
 instance {b} : Membership (Lists α) (Lists' α b) :=
-  ⟨fun a l => ∃ a' ∈ l.toList, a ~ a'⟩
+  ⟨fun l a => ∃ a' ∈ l.toList, a ~ a'⟩
 
 theorem mem_def {b a} {l : Lists' α b} : a ∈ l ↔ ∃ a' ∈ l.toList, a ~ a' :=
   Iff.rfl
@@ -148,9 +148,11 @@ theorem cons_subset {a} {l₁ l₂ : Lists' α true} : Lists'.cons a l₁ ⊆ l
 
 theorem ofList_subset {l₁ l₂ : List (Lists α)} (h : l₁ ⊆ l₂) :
     Lists'.ofList l₁ ⊆ Lists'.ofList l₂ := by
-  induction' l₁ with _ _ l₁_ih; · exact Subset.nil
-  refine Subset.cons (Lists.Equiv.refl _) ?_ (l₁_ih (List.subset_of_cons_subset h))
-  simp only [List.cons_subset] at h; simp [h]
+  induction l₁ with
+  | nil => exact Subset.nil
+  | cons _ _ l₁_ih =>
+    refine Subset.cons (Lists.Equiv.refl _) ?_ (l₁_ih (List.subset_of_cons_subset h))
+    simp only [List.cons_subset] at h; simp [h]
 
 @[refl]
 theorem Subset.refl {l : Lists' α true} : l ⊆ l := by
@@ -246,8 +248,8 @@ def mem (a : Lists α) : Lists α → Prop
   | ⟨false, _⟩ => False
   | ⟨_, l⟩ => a ∈ l
 
-instance : Membership (Lists α) (Lists α) :=
-  ⟨mem⟩
+instance : Membership (Lists α) (Lists α) where
+  mem ls l := mem l ls
 
 theorem isList_of_mem {a : Lists α} : ∀ {l : Lists α}, a ∈ l → IsList l
   | ⟨_, Lists'.nil⟩, _ => rfl
diff --git a/Mathlib/SetTheory/Ordinal/Arithmetic.lean b/Mathlib/SetTheory/Ordinal/Arithmetic.lean
index bbfc46be66810..d76f66a8b12b9 100644
--- a/Mathlib/SetTheory/Ordinal/Arithmetic.lean
+++ b/Mathlib/SetTheory/Ordinal/Arithmetic.lean
@@ -104,7 +104,8 @@ instance add_contravariantClass_le : ContravariantClass Ordinal.{u} Ordinal.{u}
                   simpa only [Sum.lex_inr_inr, fr, InitialSeg.coe_coe_fn, Embedding.coeFn_mk] using
                     @RelEmbedding.map_rel_iff _ _ _ _ f.toRelEmbedding (Sum.inr a) (Sum.inr b)⟩,
               fun a b H => by
-                rcases f.init (by rw [fr] <;> exact Sum.lex_inr_inr.2 H) with ⟨a' | a', h⟩
+                rcases f.mem_range_of_rel (by rw [fr] <;> exact Sum.lex_inr_inr.2 H) with
+                  ⟨a' | a', h⟩
                 · rw [fl] at h
                   cases h
                 · rw [fr] at h
@@ -173,7 +174,7 @@ theorem pred_eq_iff_not_succ' {o} : pred o = o ↔ ∀ a, o ≠ succ a := by
   simpa using pred_eq_iff_not_succ
 
 theorem pred_lt_iff_is_succ {o} : pred o < o ↔ ∃ a, o = succ a :=
-  Iff.trans (by simp only [le_antisymm_iff, pred_le_self, true_and_iff, not_le])
+  Iff.trans (by simp only [le_antisymm_iff, pred_le_self, true_and, not_le])
     (iff_not_comm.1 pred_eq_iff_not_succ).symm
 
 @[simp]
@@ -212,16 +213,25 @@ theorem lift_pred (o : Ordinal.{v}) : lift.{u} (pred o) = pred (lift.{u} o) := b
 /-! ### Limit ordinals -/
 
 
-/-- A limit ordinal is an ordinal which is not zero and not a successor. -/
+/-- A limit ordinal is an ordinal which is not zero and not a successor.
+
+TODO: deprecate this in favor of `Order.IsSuccLimit`. -/
 def IsLimit (o : Ordinal) : Prop :=
   o ≠ 0 ∧ ∀ a < o, succ a < o
 
-theorem IsLimit.isSuccLimit {o} (h : IsLimit o) : IsSuccLimit o := isSuccLimit_iff_succ_lt.mpr h.2
+theorem IsLimit.isSuccPrelimit {o} (h : IsLimit o) : IsSuccPrelimit o :=
+  isSuccPrelimit_iff_succ_lt.mpr h.2
+
+@[deprecated IsLimit.isSuccPrelimit (since := "2024-09-05")]
+alias IsLimit.isSuccLimit := IsLimit.isSuccPrelimit
 
 theorem IsLimit.succ_lt {o a : Ordinal} (h : IsLimit o) : a < o → succ a < o :=
   h.2 a
 
-theorem isSuccLimit_zero : IsSuccLimit (0 : Ordinal) := isSuccLimit_bot
+theorem isSuccPrelimit_zero : IsSuccPrelimit (0 : Ordinal) := isSuccPrelimit_bot
+
+@[deprecated isSuccPrelimit_zero (since := "2024-09-05")]
+alias isSuccLimit_zero := isSuccPrelimit_zero
 
 theorem not_zero_isLimit : ¬IsLimit 0
   | ⟨h, _⟩ => h rfl
@@ -277,22 +287,50 @@ theorem zero_or_succ_or_limit (o : Ordinal) : o = 0 ∨ (∃ a, o = succ a) ∨
 @[elab_as_elim]
 def limitRecOn {C : Ordinal → Sort*} (o : Ordinal) (H₁ : C 0) (H₂ : ∀ o, C o → C (succ o))
     (H₃ : ∀ o, IsLimit o → (∀ o' < o, C o') → C o) : C o :=
-  SuccOrder.limitRecOn o (fun o _ ↦ H₂ o) fun o hl ↦
+  SuccOrder.prelimitRecOn o (fun o _ ↦ H₂ o) fun o hl ↦
     if h : o = 0 then fun _ ↦ h ▸ H₁ else H₃ o ⟨h, fun _ ↦ hl.succ_lt⟩
 
 @[simp]
 theorem limitRecOn_zero {C} (H₁ H₂ H₃) : @limitRecOn C 0 H₁ H₂ H₃ = H₁ := by
-  rw [limitRecOn, SuccOrder.limitRecOn_limit _ _ isSuccLimit_zero, dif_pos rfl]
+  rw [limitRecOn, SuccOrder.prelimitRecOn_of_isSuccPrelimit _ _ isSuccPrelimit_zero, dif_pos rfl]
 
 @[simp]
 theorem limitRecOn_succ {C} (o H₁ H₂ H₃) :
     @limitRecOn C (succ o) H₁ H₂ H₃ = H₂ o (@limitRecOn C o H₁ H₂ H₃) := by
-  simp_rw [limitRecOn, SuccOrder.limitRecOn_succ _ _ (not_isMax _)]
+  rw [limitRecOn, limitRecOn, SuccOrder.prelimitRecOn_succ]
 
 @[simp]
 theorem limitRecOn_limit {C} (o H₁ H₂ H₃ h) :
     @limitRecOn C o H₁ H₂ H₃ = H₃ o h fun x _h => @limitRecOn C x H₁ H₂ H₃ := by
-  simp_rw [limitRecOn, SuccOrder.limitRecOn_limit _ _ h.isSuccLimit, dif_neg h.1]
+  simp_rw [limitRecOn, SuccOrder.prelimitRecOn_of_isSuccPrelimit _ _ h.isSuccPrelimit, dif_neg h.1]
+
+/-- Bounded recursion on ordinals. Similar to `limitRecOn`, with the assumption `o < l`
+  added to all cases. The final term's domain is the ordinals below `l`. -/
+@[elab_as_elim]
+def boundedLimitRecOn {l : Ordinal} (lLim : l.IsLimit) {C : Iio l → Sort*} (o : Iio l)
+    (H₁ : C ⟨0, lLim.pos⟩) (H₂ : (o : Iio l) → C o → C ⟨succ o, lLim.succ_lt o.2⟩)
+    (H₃ : (o : Iio l) → IsLimit o → (Π o' < o, C o') → C o) : C o :=
+  limitRecOn (C := fun p ↦ (h : p < l) → C ⟨p, h⟩) o.1 (fun _ ↦ H₁)
+    (fun o ih h ↦ H₂ ⟨o, _⟩ <| ih <| (lt_succ o).trans h)
+    (fun _o ho ih _ ↦ H₃ _ ho fun _o' h ↦ ih _ h _) o.2
+
+@[simp]
+theorem boundedLimitRec_zero {l} (lLim : l.IsLimit) {C} (H₁ H₂ H₃) :
+    @boundedLimitRecOn l lLim C ⟨0, lLim.pos⟩ H₁ H₂ H₃ = H₁ := by
+  rw [boundedLimitRecOn, limitRecOn_zero]
+
+@[simp]
+theorem boundedLimitRec_succ {l} (lLim : l.IsLimit) {C} (o H₁ H₂ H₃) :
+    @boundedLimitRecOn l lLim C ⟨succ o.1, lLim.succ_lt o.2⟩ H₁ H₂ H₃ = H₂ o
+    (@boundedLimitRecOn l lLim C o H₁ H₂ H₃) := by
+  rw [boundedLimitRecOn, limitRecOn_succ]
+  rfl
+
+theorem boundedLimitRec_limit {l} (lLim : l.IsLimit) {C} (o H₁ H₂ H₃ oLim) :
+    @boundedLimitRecOn l lLim C o H₁ H₂ H₃ = H₃ o oLim (fun x _ ↦
+    @boundedLimitRecOn l lLim C x H₁ H₂ H₃) := by
+  rw [boundedLimitRecOn, limitRecOn_limit]
+  rfl
 
 instance orderTopToTypeSucc (o : Ordinal) : OrderTop (succ o).toType :=
   @OrderTop.mk _ _ (Top.mk _) le_enum_succ
@@ -323,23 +361,35 @@ theorem bounded_singleton {r : α → α → Prop} [IsWellOrder α r] (hr : (typ
   rw [@enum_lt_enum _ r, Subtype.mk_lt_mk]
   apply lt_succ
 
--- Porting note: `· < ·` requires a type ascription for an `IsWellOrder` instance.
-theorem type_subrel_lt (o : Ordinal.{u}) :
-    type (@Subrel Ordinal (· < ·) { o' : Ordinal | o' < o }) = Ordinal.lift.{u + 1} o := by
+@[simp]
+theorem typein_ordinal (o : Ordinal.{u}) :
+    @typein Ordinal (· < ·) _ o = Ordinal.lift.{u + 1} o := by
   refine Quotient.inductionOn o ?_
   rintro ⟨α, r, wo⟩; apply Quotient.sound
   constructor; refine ((RelIso.preimage Equiv.ulift r).trans (enum r).symm).symm
 
+-- Porting note: `· < ·` requires a type ascription for an `IsWellOrder` instance.
+@[deprecated typein_ordinal (since := "2024-09-19")]
+theorem type_subrel_lt (o : Ordinal.{u}) :
+    type (@Subrel Ordinal (· < ·) { o' : Ordinal | o' < o }) = Ordinal.lift.{u + 1} o :=
+  typein_ordinal o
+
+theorem mk_Iio_ordinal (o : Ordinal.{u}) :
+    #(Iio o) = Cardinal.lift.{u + 1} o.card := by
+  rw [lift_card, ← typein_ordinal]
+  rfl
+
+@[deprecated mk_Iio_ordinal (since := "2024-09-19")]
 theorem mk_initialSeg (o : Ordinal.{u}) :
-    #{ o' : Ordinal | o' < o } = Cardinal.lift.{u + 1} o.card := by
-  rw [lift_card, ← type_subrel_lt, card_type]
+    #{ o' : Ordinal | o' < o } = Cardinal.lift.{u + 1} o.card := mk_Iio_ordinal o
+
 
 /-! ### Normal ordinal functions -/
 
 
 /-- A normal ordinal function is a strictly increasing function which is
   order-continuous, i.e., the image `f o` of a limit ordinal `o` is the sup of `f a` for
-  `a < o`.  -/
+  `a < o`. -/
 def IsNormal (f : Ordinal → Ordinal) : Prop :=
   (∀ o, f o < f (succ o)) ∧ ∀ o, IsLimit o → ∀ a, f o ≤ a ↔ ∀ b < o, f b ≤ a
 
@@ -375,8 +425,18 @@ theorem IsNormal.le_iff {f} (H : IsNormal f) {a b} : f a ≤ f b ↔ a ≤ b :=
 theorem IsNormal.inj {f} (H : IsNormal f) {a b} : f a = f b ↔ a = b := by
   simp only [le_antisymm_iff, H.le_iff]
 
+theorem IsNormal.id_le {f} (H : IsNormal f) : id ≤ f :=
+  H.strictMono.id_le
+
+theorem IsNormal.le_apply {f} (H : IsNormal f) {a} : a ≤ f a :=
+  H.strictMono.le_apply
+
+@[deprecated IsNormal.le_apply (since := "2024-09-11")]
 theorem IsNormal.self_le {f} (H : IsNormal f) (a) : a ≤ f a :=
-  lt_wf.self_le_of_strictMono H.strictMono a
+  H.strictMono.le_apply
+
+theorem IsNormal.le_iff_eq {f} (H : IsNormal f) {a} : f a ≤ a ↔ f a = a :=
+  H.le_apply.le_iff_eq
 
 theorem IsNormal.le_set {f o} (H : IsNormal f) (p : Set Ordinal) (p0 : p.Nonempty) (b)
     (H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f a ≤ o :=
@@ -412,9 +472,6 @@ theorem IsNormal.isLimit {f} (H : IsNormal f) {o} (l : IsLimit o) : IsLimit (f o
     let ⟨_b, h₁, h₂⟩ := (H.limit_lt l).1 h
     (succ_le_of_lt h₂).trans_lt (H.lt_iff.2 h₁)⟩
 
-theorem IsNormal.le_iff_eq {f} (H : IsNormal f) {a} : f a ≤ a ↔ f a = a :=
-  (H.self_le a).le_iff_eq
-
 theorem add_le_of_limit {a b c : Ordinal} (h : IsLimit b) : a + b ≤ c ↔ ∀ b' < b, a + b' ≤ c :=
   ⟨fun h b' l => (add_le_add_left l.le _).trans h, fun H =>
     le_of_not_lt <| by
@@ -450,11 +507,11 @@ theorem add_isLimit (a) {b} : IsLimit b → IsLimit (a + b) :=
 
 alias IsLimit.add := add_isLimit
 
-/-! ### Subtraction on ordinals-/
+/-! ### Subtraction on ordinals -/
 
 
 /-- The set in the definition of subtraction is nonempty. -/
-theorem sub_nonempty {a b : Ordinal} : { o | a ≤ b + o }.Nonempty :=
+private theorem sub_nonempty {a b : Ordinal} : { o | a ≤ b + o }.Nonempty :=
   ⟨a, le_add_left _ _⟩
 
 /-- `a - b` is the unique ordinal satisfying `b + (a - b) = a` when `b ≤ a`. -/
@@ -522,9 +579,9 @@ theorem sub_isLimit {a b} (l : IsLimit a) (h : b < a) : IsLimit (a - b) :=
     rw [lt_sub, add_succ]; exact l.2 _ (lt_sub.1 h)⟩
 
 -- @[simp] -- Porting note (#10618): simp can prove this
-theorem one_add_omega : 1 + ω = ω := by
+theorem one_add_omega0 : 1 + ω = ω := by
   refine le_antisymm ?_ (le_add_left _ _)
-  rw [omega, ← lift_one.{0}, ← lift_add, lift_le, ← type_unit, ← type_sum_lex]
+  rw [omega0, ← lift_one.{0}, ← lift_add, lift_le, ← type_unit, ← type_sum_lex]
   refine ⟨RelEmbedding.collapse (RelEmbedding.ofMonotone ?_ ?_)⟩
   · apply Sum.rec
     · exact fun _ => 0
@@ -533,11 +590,17 @@ theorem one_add_omega : 1 + ω = ω := by
     cases a <;> cases b <;> intro H <;> cases' H with _ _ H _ _ H <;>
       [exact H.elim; exact Nat.succ_pos _; exact Nat.succ_lt_succ H]
 
+@[deprecated (since := "2024-09-30")]
+alias one_add_omega := one_add_omega0
+
 @[simp]
-theorem one_add_of_omega_le {o} (h : ω ≤ o) : 1 + o = o := by
-  rw [← Ordinal.add_sub_cancel_of_le h, ← add_assoc, one_add_omega]
+theorem one_add_of_omega0_le {o} (h : ω ≤ o) : 1 + o = o := by
+  rw [← Ordinal.add_sub_cancel_of_le h, ← add_assoc, one_add_omega0]
 
-/-! ### Multiplication of ordinals-/
+@[deprecated (since := "2024-09-30")]
+alias one_add_of_omega_le := one_add_of_omega0_le
+
+/-! ### Multiplication of ordinals -/
 
 
 /-- The multiplication of ordinals `o₁` and `o₂` is the (well founded) lexicographic order on
@@ -563,15 +626,15 @@ instance monoid : Monoid Ordinal.{u} where
       Quotient.sound
         ⟨⟨punitProd _, @fun a b => by
             rcases a with ⟨⟨⟨⟩⟩, a⟩; rcases b with ⟨⟨⟨⟩⟩, b⟩
-            simp only [Prod.lex_def, EmptyRelation, false_or_iff]
-            simp only [eq_self_iff_true, true_and_iff]
+            simp only [Prod.lex_def, EmptyRelation, false_or]
+            simp only [eq_self_iff_true, true_and]
             rfl⟩⟩
   one_mul a :=
     inductionOn a fun α r _ =>
       Quotient.sound
         ⟨⟨prodPUnit _, @fun a b => by
             rcases a with ⟨a, ⟨⟨⟩⟩⟩; rcases b with ⟨b, ⟨⟨⟩⟩⟩
-            simp only [Prod.lex_def, EmptyRelation, and_false_iff, or_false_iff]
+            simp only [Prod.lex_def, EmptyRelation, and_false, or_false]
             rfl⟩⟩
 
 @[simp]
@@ -613,11 +676,10 @@ instance leftDistribClass : LeftDistribClass Ordinal.{u} :=
       Quotient.sound
         ⟨⟨sumProdDistrib _ _ _, by
           rintro ⟨a₁ | a₁, a₂⟩ ⟨b₁ | b₁, b₂⟩ <;>
-            simp only [Prod.lex_def, Sum.lex_inl_inl, Sum.Lex.sep, Sum.lex_inr_inl,
-              Sum.lex_inr_inr, sumProdDistrib_apply_left, sumProdDistrib_apply_right] <;>
+            simp only [Prod.lex_def, Sum.lex_inl_inl, Sum.Lex.sep, Sum.lex_inr_inl, Sum.lex_inr_inr,
+              sumProdDistrib_apply_left, sumProdDistrib_apply_right, reduceCtorEq] <;>
             -- Porting note: `Sum.inr.inj_iff` is required.
-            simp only [Sum.inl.inj_iff, Sum.inr.inj_iff,
-              true_or_iff, false_and_iff, false_or_iff]⟩⟩⟩
+            simp only [Sum.inl.inj_iff, Sum.inr.inj_iff, true_or, false_and, false_or]⟩⟩⟩
 
 theorem mul_succ (a b : Ordinal) : a * succ b = a * b + a :=
   mul_add_one a b
@@ -677,11 +739,11 @@ private theorem mul_le_of_limit_aux {α β r s} [IsWellOrder α r] [IsWellOrder
     intro h
     by_cases e₁ : b = b₁ <;> by_cases e₂ : b = b₂
     · substs b₁ b₂
-      simpa only [subrel_val, Prod.lex_def, @irrefl _ s _ b, true_and_iff, false_or_iff,
+      simpa only [subrel_val, Prod.lex_def, @irrefl _ s _ b, true_and, false_or,
         eq_self_iff_true, dif_pos, Sum.lex_inr_inr] using h
     · subst b₁
       simp only [subrel_val, Prod.lex_def, e₂, Prod.lex_def, dif_pos, subrel_val, eq_self_iff_true,
-        or_false_iff, dif_neg, not_false_iff, Sum.lex_inr_inl, false_and_iff] at h ⊢
+        or_false, dif_neg, not_false_iff, Sum.lex_inr_inl, false_and] at h ⊢
       cases' h₂ with _ _ _ _ h₂_h h₂_h <;> [exact asymm h h₂_h; exact e₂ rfl]
     · simp [e₂, dif_neg e₁, show b₂ ≠ b₁ from e₂ ▸ e₁]
     · simpa only [dif_neg e₁, dif_neg e₂, Prod.lex_def, subrel_val, Subtype.mk_eq_mk,
@@ -748,20 +810,20 @@ theorem smul_eq_mul : ∀ (n : ℕ) (a : Ordinal), n • a = a * n
 
 
 /-- The set in the definition of division is nonempty. -/
-theorem div_nonempty {a b : Ordinal} (h : b ≠ 0) : { o | a < b * succ o }.Nonempty :=
+private theorem div_nonempty {a b : Ordinal} (h : b ≠ 0) : { o | a < b * succ o }.Nonempty :=
   ⟨a, (succ_le_iff (a := a) (b := b * succ a)).1 <| by
     simpa only [succ_zero, one_mul] using
       mul_le_mul_right' (succ_le_of_lt (Ordinal.pos_iff_ne_zero.2 h)) (succ a)⟩
 
 /-- `a / b` is the unique ordinal `o` satisfying `a = b * o + o'` with `o' < b`. -/
 instance div : Div Ordinal :=
-  ⟨fun a b => if _h : b = 0 then 0 else sInf { o | a < b * succ o }⟩
+  ⟨fun a b => if b = 0 then 0 else sInf { o | a < b * succ o }⟩
 
 @[simp]
 theorem div_zero (a : Ordinal) : a / 0 = 0 :=
   dif_pos rfl
 
-theorem div_def (a) {b : Ordinal} (h : b ≠ 0) : a / b = sInf { o | a < b * succ o } :=
+private theorem div_def (a) {b : Ordinal} (h : b ≠ 0) : a / b = sInf { o | a < b * succ o } :=
   dif_neg h
 
 theorem lt_mul_succ_div (a) {b : Ordinal} (h : b ≠ 0) : a < b * succ (a / b) := by
@@ -785,8 +847,7 @@ theorem le_div {a b c : Ordinal} (c0 : c ≠ 0) : a ≤ b / c ↔ c * a ≤ b :=
   | H₂ _ _ => rw [succ_le_iff, lt_div c0]
   | H₃ _ h₁ h₂ =>
     revert h₁ h₂
-    simp (config := { contextual := true }) only [mul_le_of_limit, limit_le, iff_self_iff,
-      forall_true_iff]
+    simp (config := { contextual := true }) only [mul_le_of_limit, limit_le, forall_true_iff]
 
 theorem div_lt {a b c : Ordinal} (b0 : b ≠ 0) : a / b < c ↔ a < b * c :=
   lt_iff_lt_of_le_iff_le <| le_div b0
@@ -822,6 +883,26 @@ theorem div_eq_zero_of_lt {a b : Ordinal} (h : a < b) : a / b = 0 := by
 theorem mul_div_cancel (a) {b : Ordinal} (b0 : b ≠ 0) : b * a / b = a := by
   simpa only [add_zero, zero_div] using mul_add_div a b0 0
 
+theorem mul_add_div_mul {a c : Ordinal} (hc : c < a) (b d : Ordinal) :
+    (a * b + c) / (a * d) = b / d := by
+  have ha : a ≠ 0 := ((Ordinal.zero_le c).trans_lt hc).ne'
+  obtain rfl | hd := eq_or_ne d 0
+  · rw [mul_zero, div_zero, div_zero]
+  · have H := mul_ne_zero ha hd
+    apply le_antisymm
+    · rw [← lt_succ_iff, div_lt H, mul_assoc]
+      · apply (add_lt_add_left hc _).trans_le
+        rw [← mul_succ]
+        apply mul_le_mul_left'
+        rw [succ_le_iff]
+        exact lt_mul_succ_div b hd
+    · rw [le_div H, mul_assoc]
+      exact (mul_le_mul_left' (mul_div_le b d) a).trans (le_add_right _ c)
+
+theorem mul_div_mul_cancel {a : Ordinal} (ha : a ≠ 0) (b c) : a * b / (a * c) = b / c := by
+  convert mul_add_div_mul (Ordinal.pos_iff_ne_zero.2 ha) b c using 1
+  rw [add_zero]
+
 @[simp]
 theorem div_one (a : Ordinal) : a / 1 = a := by
   simpa only [one_mul] using mul_div_cancel a Ordinal.one_ne_zero
@@ -934,6 +1015,18 @@ theorem mul_add_mod_self (x y z : Ordinal) : (x * y + z) % x = z % x := by
 theorem mul_mod (x y : Ordinal) : x * y % x = 0 := by
   simpa using mul_add_mod_self x y 0
 
+theorem mul_add_mod_mul {w x : Ordinal} (hw : w < x) (y z : Ordinal) :
+    (x * y + w) % (x * z) = x * (y % z) + w := by
+  rw [mod_def, mul_add_div_mul hw]
+  apply sub_eq_of_add_eq
+  rw [← add_assoc, mul_assoc, ← mul_add, div_add_mod]
+
+theorem mul_mod_mul (x y z : Ordinal) : (x * y) % (x * z) = x * (y % z) := by
+  obtain rfl | hx := Ordinal.eq_zero_or_pos x
+  · simp
+  · convert mul_add_mod_mul hx y z using 1 <;>
+    rw [add_zero]
+
 theorem mod_mod_of_dvd (a : Ordinal) {b c : Ordinal} (h : c ∣ b) : a % b % c = a % c := by
   nth_rw 2 [← div_add_mod a b]
   rcases h with ⟨d, rfl⟩
@@ -1058,13 +1151,14 @@ theorem comp_familyOfBFamily {o} (f : ∀ a < o, α) (g : α → β) :
 
 /-! ### Supremum of a family of ordinals -/
 
--- Porting note: Universes should be specified in `sup`s.
-
 /-- The supremum of a family of ordinals -/
+
+@[deprecated iSup (since := "2024-08-27")]
 def sup {ι : Type u} (f : ι → Ordinal.{max u v}) : Ordinal.{max u v} :=
   iSup f
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-27")]
 theorem sSup_eq_sup {ι : Type u} (f : ι → Ordinal.{max u v}) : sSup (Set.range f) = sup.{_, v} f :=
   rfl
 
@@ -1076,29 +1170,83 @@ theorem bddAbove_range {ι : Type u} (f : ι → Ordinal.{max u v}) : BddAbove (
     exact le_of_lt (Cardinal.lt_ord.2 ((lt_succ _).trans_le
       (le_ciSup (Cardinal.bddAbove_range.{_, v} _) _)))⟩
 
+/-- `le_ciSup` whenever the outputs live in a higher universe than the inputs. -/
+protected theorem le_iSup {ι : Type u} (f : ι → Ordinal.{max u v}) : ∀ i, f i ≤ iSup f :=
+  le_ciSup (bddAbove_range f)
+
+set_option linter.deprecated false in
+@[deprecated Ordinal.le_iSup (since := "2024-08-27")]
 theorem le_sup {ι : Type u} (f : ι → Ordinal.{max u v}) : ∀ i, f i ≤ sup.{_, v} f := fun i =>
-  le_csSup (bddAbove_range.{_, v} f) (mem_range_self i)
+  Ordinal.le_iSup f i
+
+/-- `ciSup_le_iff'` whenever the outputs live in a higher universe than the inputs. -/
+protected theorem iSup_le_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} :
+    iSup f ≤ a ↔ ∀ i, f i ≤ a :=
+  ciSup_le_iff' (bddAbove_range f)
 
+set_option linter.deprecated false in
+@[deprecated Ordinal.iSup_le_iff (since := "2024-08-27")]
 theorem sup_le_iff {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : sup.{_, v} f ≤ a ↔ ∀ i, f i ≤ a :=
-  (csSup_le_iff' (bddAbove_range.{_, v} f)).trans (by simp)
+  Ordinal.iSup_le_iff
 
+/-- `ciSup_le'` whenever the outputs live in a higher universe than the inputs. -/
+protected theorem iSup_le {ι : Type u} {f : ι → Ordinal.{max u v}} {a} :
+    (∀ i, f i ≤ a) → iSup f ≤ a :=
+  ciSup_le'
+
+set_option linter.deprecated false in
+@[deprecated Ordinal.iSup_le (since := "2024-08-27")]
 theorem sup_le {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : (∀ i, f i ≤ a) → sup.{_, v} f ≤ a :=
-  sup_le_iff.2
+  Ordinal.iSup_le
+
+-- TODO: generalize to conditionally complete linear orders.
+protected theorem lt_iSup {ι : Type u} {f : ι → Ordinal.{max u v}} {a} :
+    a < iSup f ↔ ∃ i, a < f i := by
+  rw [← not_iff_not]
+  simpa using Ordinal.iSup_le_iff
 
+set_option linter.deprecated false in
+@[deprecated Ordinal.lt_iSup (since := "2024-08-27")]
 theorem lt_sup {ι : Type u} {f : ι → Ordinal.{max u v}} {a} : a < sup.{_, v} f ↔ ∃ i, a < f i := by
   simpa only [not_forall, not_le] using not_congr (@sup_le_iff.{_, v} _ f a)
 
+@[deprecated (since := "2024-08-27")]
+theorem ne_iSup_iff_lt_iSup {ι : Type u} {f : ι → Ordinal.{max u v}} :
+    (∀ i, f i ≠ iSup f) ↔ ∀ i, f i < iSup f :=
+  forall_congr' fun i => (Ordinal.le_iSup f i).lt_iff_ne.symm
+
+set_option linter.deprecated false in
+@[deprecated ne_iSup_iff_lt_iSup (since := "2024-08-27")]
 theorem ne_sup_iff_lt_sup {ι : Type u} {f : ι → Ordinal.{max u v}} :
     (∀ i, f i ≠ sup.{_, v} f) ↔ ∀ i, f i < sup.{_, v} f :=
-  ⟨fun hf _ => lt_of_le_of_ne (le_sup _ _) (hf _), fun hf _ => ne_of_lt (hf _)⟩
+  ne_iSup_iff_lt_iSup
+
+-- TODO: state in terms of `IsSuccLimit`.
+theorem succ_lt_iSup_of_ne_iSup {ι : Type u} {f : ι → Ordinal.{max u v}}
+    (hf : ∀ i, f i ≠ iSup f) {a} (hao : a < iSup f) : succ a < iSup f := by
+  by_contra! hoa
+  exact hao.not_le (Ordinal.iSup_le fun i => le_of_lt_succ <|
+    (lt_of_le_of_ne (Ordinal.le_iSup _ _) (hf i)).trans_le hoa)
 
+set_option linter.deprecated false in
+@[deprecated succ_lt_iSup_of_ne_iSup (since := "2024-08-27")]
 theorem sup_not_succ_of_ne_sup {ι : Type u} {f : ι → Ordinal.{max u v}}
     (hf : ∀ i, f i ≠ sup.{_, v} f) {a} (hao : a < sup.{_, v} f) : succ a < sup.{_, v} f := by
   by_contra! hoa
   exact
     hao.not_le (sup_le fun i => le_of_lt_succ <| (lt_of_le_of_ne (le_sup _ _) (hf i)).trans_le hoa)
 
-@[simp]
+-- TODO: generalize to conditionally complete lattices.
+theorem iSup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} :
+    iSup f = 0 ↔ ∀ i, f i = 0 := by
+  refine
+    ⟨fun h i => ?_, fun h =>
+      le_antisymm (Ordinal.iSup_le fun i => Ordinal.le_zero.2 (h i)) (Ordinal.zero_le _)⟩
+  rw [← Ordinal.le_zero, ← h]
+  exact Ordinal.le_iSup f i
+
+set_option linter.deprecated false in
+@[deprecated iSup_eq_zero_iff (since := "2024-08-27")]
 theorem sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} :
     sup.{_, v} f = 0 ↔ ∀ i, f i = 0 := by
   refine
@@ -1107,35 +1255,71 @@ theorem sup_eq_zero_iff {ι : Type u} {f : ι → Ordinal.{max u v}} :
   rw [← Ordinal.le_zero, ← h]
   exact le_sup f i
 
+-- TODO: generalize universes, make sSup version.
+theorem IsNormal.map_iSup {f : Ordinal.{max u v} → Ordinal.{max u w}} (H : IsNormal f) {ι : Type u}
+    (g : ι → Ordinal.{max u v}) [Nonempty ι] : f (⨆ i, g i) = ⨆ i, f (g i) := by
+  apply eq_of_forall_ge_iff
+  intro a
+  rw [H.le_set' Set.univ Set.univ_nonempty g]
+  · rw [Ordinal.iSup_le_iff]
+    simp
+  · intro o
+    rw [Ordinal.iSup_le_iff]
+    simp
+
+set_option linter.deprecated false in
+@[deprecated IsNormal.map_iSup (since := "2024-08-27")]
 theorem IsNormal.sup {f : Ordinal.{max u v} → Ordinal.{max u w}} (H : IsNormal f) {ι : Type u}
     (g : ι → Ordinal.{max u v}) [Nonempty ι] : f (sup.{_, v} g) = sup.{_, w} (f ∘ g) :=
-  eq_of_forall_ge_iff fun a => by
-    rw [sup_le_iff]; simp only [comp]; rw [H.le_set' Set.univ Set.univ_nonempty g] <;>
-      simp [sup_le_iff]
+  H.map_iSup g
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated ciSup_of_empty (since := "2024-08-27")]
 theorem sup_empty {ι} [IsEmpty ι] (f : ι → Ordinal) : sup f = 0 :=
   ciSup_of_empty f
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated ciSup_const (since := "2024-08-27")]
 theorem sup_const {ι} [_hι : Nonempty ι] (o : Ordinal) : (sup fun _ : ι => o) = o :=
   ciSup_const
 
-@[simp]
+set_option linter.deprecated false in
+@[deprecated ciSup_unique (since := "2024-08-27")]
 theorem sup_unique {ι} [Unique ι] (f : ι → Ordinal) : sup f = f default :=
   ciSup_unique
 
+set_option linter.deprecated false in
+@[deprecated csSup_le_csSup' (since := "2024-08-27")]
 theorem sup_le_of_range_subset {ι ι'} {f : ι → Ordinal} {g : ι' → Ordinal}
     (h : Set.range f ⊆ Set.range g) : sup.{u, max v w} f ≤ sup.{v, max u w} g :=
-  sup_le fun i =>
-    match h (mem_range_self i) with
-    | ⟨_j, hj⟩ => hj ▸ le_sup _ _
+  csSup_le_csSup' (bddAbove_range.{v, max u w} _) h
 
-theorem sup_eq_of_range_eq {ι ι'} {f : ι → Ordinal} {g : ι' → Ordinal}
+-- TODO: generalize or remove
+theorem iSup_eq_of_range_eq {ι ι'} {f : ι → Ordinal} {g : ι' → Ordinal}
+    (h : Set.range f = Set.range g) : iSup f = iSup g :=
+  congr_arg _ h
+
+set_option linter.deprecated false in
+@[deprecated iSup_eq_of_range_eq (since := "2024-08-27")]
+theorem sup_eq_of_range_eq {ι : Type u} {ι' : Type v}
+    {f : ι → Ordinal.{max u v w}} {g : ι' → Ordinal.{max u v w}}
     (h : Set.range f = Set.range g) : sup.{u, max v w} f = sup.{v, max u w} g :=
-  (sup_le_of_range_subset.{u, v, w} h.le).antisymm (sup_le_of_range_subset.{v, u, w} h.ge)
+  Ordinal.iSup_eq_of_range_eq h
 
-@[simp]
+-- TODO: generalize to conditionally complete lattices
+theorem iSup_sum {α : Type u} {β : Type v} (f : α ⊕ β → Ordinal.{max u v w}) :
+    iSup f = max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b)) := by
+  apply (Ordinal.iSup_le _).antisymm (max_le _ _)
+  · rintro (i | i)
+    · exact le_max_of_le_left (Ordinal.le_iSup.{u, max u v w} _ i)
+    · exact le_max_of_le_right (Ordinal.le_iSup.{v, max u v w} _ i)
+  all_goals
+    apply csSup_le_csSup' (bddAbove_range _)
+    rintro i ⟨a, rfl⟩
+    apply mem_range_self
+
+set_option linter.deprecated false in
+@[deprecated iSup_sum (since := "2024-08-27")]
 theorem sup_sum {α : Type u} {β : Type v} (f : α ⊕ β → Ordinal) :
     sup.{max u v, w} f =
       max (sup.{u, max v w} fun a => f (Sum.inl a)) (sup.{v, max u w} fun b => f (Sum.inr b)) := by
@@ -1148,41 +1332,40 @@ theorem sup_sum {α : Type u} {β : Type v} (f : α ⊕ β → Ordinal) :
     rintro i ⟨a, rfl⟩
     apply mem_range_self
 
+theorem unbounded_range_of_le_iSup {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β → α)
+    (h : type r ≤ ⨆ i, typein r (f i)) : Unbounded r (range f) :=
+  (not_bounded_iff _).1 fun ⟨x, hx⟩ =>
+    h.not_lt <| lt_of_le_of_lt
+      (Ordinal.iSup_le fun y => ((typein_lt_typein r).2 <| hx _ <| mem_range_self y).le)
+      (typein_lt_type r x)
+
+set_option linter.deprecated false in
+@[deprecated unbounded_range_of_le_iSup (since := "2024-08-27")]
 theorem unbounded_range_of_sup_ge {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β → α)
     (h : type r ≤ sup.{u, u} (typein r ∘ f)) : Unbounded r (range f) :=
-  (not_bounded_iff _).1 fun ⟨x, hx⟩ =>
-    not_lt_of_le h <|
-      lt_of_le_of_lt
-        (sup_le fun y => le_of_lt <| (typein_lt_typein r).2 <| hx _ <| mem_range_self y)
-        (typein_lt_type r x)
+  unbounded_range_of_le_iSup r f h
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-27")]
 theorem le_sup_shrink_equiv {s : Set Ordinal.{u}} (hs : Small.{u} s) (a) (ha : a ∈ s) :
     a ≤ sup.{u, u} fun x => ((@equivShrink s hs).symm x).val := by
   convert le_sup.{u, u} (fun x => ((@equivShrink s hs).symm x).val) ((@equivShrink s hs) ⟨a, ha⟩)
   rw [symm_apply_apply]
 
-instance small_Iio (o : Ordinal.{u}) : Small.{u} (Set.Iio o) :=
-  let f : o.toType → Set.Iio o :=
-    fun x => ⟨typein (α := o.toType) (· < ·) x, typein_lt_self x⟩
-  let hf : Surjective f := fun b =>
-    ⟨enum (α := o.toType) (· < ·) ⟨b.val,
-        by
-          rw [type_lt]
-          exact b.prop⟩,
-      Subtype.ext (typein_enum _ _)⟩
-  small_of_surjective hf
-
-instance small_Iic (o : Ordinal.{u}) : Small.{u} (Set.Iic o) := by
-  rw [← Iio_succ]
-  infer_instance
+-- TODO: move this together with `bddAbove_range`.
 
-theorem bddAbove_iff_small {s : Set Ordinal.{u}} : BddAbove s ↔ Small.{u} s :=
-  ⟨fun ⟨a, h⟩ => small_subset <| show s ⊆ Iic a from fun _x hx => h hx, fun h =>
-    ⟨sup.{u, u} fun x => ((@equivShrink s h).symm x).val, le_sup_shrink_equiv h⟩⟩
+theorem bddAbove_of_small (s : Set Ordinal.{u}) [h : Small.{u} s] : BddAbove s := by
+  obtain ⟨a, ha⟩ := bddAbove_range (fun x => ((@equivShrink s h).symm x).val)
+  use a
+  intro b hb
+  simpa using ha (mem_range_self (equivShrink s ⟨b, hb⟩))
 
-theorem bddAbove_of_small (s : Set Ordinal.{u}) [h : Small.{u} s] : BddAbove s :=
-  bddAbove_iff_small.2 h
+theorem bddAbove_iff_small {s : Set Ordinal.{u}} : BddAbove s ↔ Small.{u} s :=
+  ⟨fun ⟨a, h⟩ => small_subset <| show s ⊆ Iic a from fun _ hx => h hx, fun _ =>
+    bddAbove_of_small _⟩
 
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-27")]
 theorem sup_eq_sSup {s : Set Ordinal.{u}} (hs : Small.{u} s) :
     (sup.{u, u} fun x => (@equivShrink s hs).symm x) = sSup s :=
   let hs' := bddAbove_iff_small.2 hs
@@ -1204,6 +1387,27 @@ theorem iSup_ord {ι} {f : ι → Cardinal} (hf : BddAbove (range f)) :
   conv_lhs => change range (ord ∘ f)
   rw [range_comp]
 
+theorem sInf_compl_lt_lift_ord_succ {ι : Type u} (f : ι → Ordinal.{max u v}) :
+    sInf (range f)ᶜ < lift.{v} (succ #ι).ord := by
+  by_contra! h
+  have : Iio (lift.{v} (succ #ι).ord) ⊆ range f := by
+    intro o ho
+    have := not_mem_of_lt_csInf' (ho.trans_le h)
+    rwa [not_mem_compl_iff] at this
+  have := mk_le_mk_of_subset this
+  rw [mk_Iio_ordinal, ← lift_card, Cardinal.lift_lift, card_ord, Cardinal.lift_succ,
+    succ_le_iff, ← Cardinal.lift_id'.{u, max (u + 1) (v + 1)} #_] at this
+  exact this.not_le mk_range_le_lift
+
+theorem sInf_compl_lt_ord_succ {ι : Type u} (f : ι → Ordinal.{u}) :
+    sInf (range f)ᶜ < (succ #ι).ord :=
+  lift_id (succ #ι).ord ▸ sInf_compl_lt_lift_ord_succ f
+
+-- TODO: remove `bsup` in favor of `iSup` in a future refactor.
+
+section bsup
+set_option linter.deprecated false
+
 private theorem sup_le_sup {ι ι' : Type u} (r : ι → ι → Prop) (r' : ι' → ι' → Prop)
     [IsWellOrder ι r] [IsWellOrder ι' r'] {o} (ho : type r = o) (ho' : type r' = o)
     (f : ∀ a < o, Ordinal.{max u v}) :
@@ -1343,6 +1547,13 @@ theorem bsup_eq_of_brange_eq {o o'} {f : ∀ a < o, Ordinal} {g : ∀ a < o', Or
     (h : brange o f = brange o' g) : bsup.{u, max v w} o f = bsup.{v, max u w} o' g :=
   (bsup_le_of_brange_subset.{u, v, w} h.le).antisymm (bsup_le_of_brange_subset.{v, u, w} h.ge)
 
+end bsup
+
+-- TODO: bring the lsub API in line with the sSup / iSup API, or deprecate it altogether.
+
+section lsub
+set_option linter.deprecated false
+
 /-- The least strict upper bound of a family of ordinals. -/
 def lsub {ι} (f : ι → Ordinal) : Ordinal :=
   sup (succ ∘ f)
@@ -1487,6 +1698,14 @@ theorem sup_typein_succ {o : Ordinal} :
   rw [← succ_eq_succ_iff, h]
   apply lsub_typein
 
+end lsub
+
+-- TODO: either deprecate this in favor of `lsub` when its universes are generalized, or deprecate
+-- both of them at once.
+
+section blsub
+set_option linter.deprecated false
+
 /-- The least strict upper bound of a family of ordinals indexed by the set of ordinals less than
     some `o : Ordinal.{u}`.
 
@@ -1695,12 +1914,14 @@ theorem IsNormal.eq_iff_zero_and_succ {f g : Ordinal.{u} → Ordinal.{u}} (hf :
     (hg : IsNormal g) : f = g ↔ f 0 = g 0 ∧ ∀ a, f a = g a → f (succ a) = g (succ a) :=
   ⟨fun h => by simp [h], fun ⟨h₁, h₂⟩ =>
     funext fun a => by
-      induction' a using limitRecOn with _ _ _ ho H
-      any_goals solve_by_elim
-      rw [← IsNormal.bsup_eq.{u, u} hf ho, ← IsNormal.bsup_eq.{u, u} hg ho]
-      congr
-      ext b hb
-      exact H b hb⟩
+      induction a using limitRecOn with
+      | H₁ => solve_by_elim
+      | H₂ => solve_by_elim
+      | H₃ _ ho H =>
+        rw [← IsNormal.bsup_eq.{u, u} hf ho, ← IsNormal.bsup_eq.{u, u} hg ho]
+        congr
+        ext b hb
+        exact H b hb⟩
 
 /-- A two-argument version of `Ordinal.blsub`.
 We don't develop a full API for this, since it's only used in a handful of existence results. -/
@@ -1708,6 +1929,7 @@ def blsub₂ (o₁ o₂ : Ordinal) (op : {a : Ordinal} → (a < o₁) → {b : O
     Ordinal :=
   lsub (fun x : o₁.toType × o₂.toType => op (typein_lt_self x.1) (typein_lt_self x.2))
 
+-- TODO: deprecate this, and replace the arguments using it by arguments about small sets.
 theorem lt_blsub₂ {o₁ o₂ : Ordinal}
     (op : {a : Ordinal} → (a < o₁) → {b : Ordinal} → (b < o₂) → Ordinal) {a b : Ordinal}
     (ha : a < o₁) (hb : b < o₂) : op ha hb < blsub₂ o₁ o₂ op := by
@@ -1715,34 +1937,49 @@ theorem lt_blsub₂ {o₁ o₂ : Ordinal}
     (enum (· < ·) ⟨b, by rwa [type_lt]⟩))
   simp only [typein_enum]
 
+end blsub
+
+-- TODO: deprecate in favor of `sInf sᶜ`.
+
+section mex
+set_option linter.deprecated false
+
 /-! ### Minimum excluded ordinals -/
 
 
 /-- The minimum excluded ordinal in a family of ordinals. -/
+@[deprecated "use sInf sᶜ instead" (since := "2024-09-20")]
 def mex {ι : Type u} (f : ι → Ordinal.{max u v}) : Ordinal :=
   sInf (Set.range f)ᶜ
 
+@[deprecated (since := "2024-09-20")]
 theorem mex_not_mem_range {ι : Type u} (f : ι → Ordinal.{max u v}) : mex.{_, v} f ∉ Set.range f :=
   csInf_mem (nonempty_compl_range.{_, v} f)
 
+@[deprecated (since := "2024-09-20")]
 theorem le_mex_of_forall {ι : Type u} {f : ι → Ordinal.{max u v}} {a : Ordinal}
     (H : ∀ b < a, ∃ i, f i = b) : a ≤ mex.{_, v} f := by
   by_contra! h
   exact mex_not_mem_range f (H _ h)
 
+@[deprecated (since := "2024-09-20")]
 theorem ne_mex {ι : Type u} (f : ι → Ordinal.{max u v}) : ∀ i, f i ≠ mex.{_, v} f := by
   simpa using mex_not_mem_range.{_, v} f
 
+@[deprecated (since := "2024-09-20")]
 theorem mex_le_of_ne {ι} {f : ι → Ordinal} {a} (ha : ∀ i, f i ≠ a) : mex f ≤ a :=
   csInf_le' (by simp [ha])
 
+@[deprecated (since := "2024-09-20")]
 theorem exists_of_lt_mex {ι} {f : ι → Ordinal} {a} (ha : a < mex f) : ∃ i, f i = a := by
   by_contra! ha'
   exact ha.not_le (mex_le_of_ne ha')
 
+@[deprecated (since := "2024-09-20")]
 theorem mex_le_lsub {ι : Type u} (f : ι → Ordinal.{max u v}) : mex.{_, v} f ≤ lsub.{_, v} f :=
   csInf_le' (lsub_not_mem_range f)
 
+@[deprecated (since := "2024-09-20")]
 theorem mex_monotone {α β : Type u} {f : α → Ordinal.{max u v}} {g : β → Ordinal.{max u v}}
     (h : Set.range f ⊆ Set.range g) : mex.{_, v} f ≤ mex.{_, v} g := by
   refine mex_le_of_ne fun i hi => ?_
@@ -1750,6 +1987,7 @@ theorem mex_monotone {α β : Type u} {f : α → Ordinal.{max u v}} {g : β →
   rw [← hj] at hi
   exact ne_mex g j hi
 
+@[deprecated sInf_compl_lt_ord_succ (since := "2024-09-20")]
 theorem mex_lt_ord_succ_mk {ι : Type u} (f : ι → Ordinal.{u}) :
     mex.{_, u} f < (succ #ι).ord := by
   by_contra! h
@@ -1770,18 +2008,22 @@ theorem mex_lt_ord_succ_mk {ι : Type u} (f : ι → Ordinal.{u}) :
     `familyOfBFamily`.
 
     This is to `mex` as `bsup` is to `sup`. -/
+@[deprecated "use sInf sᶜ instead" (since := "2024-09-20")]
 def bmex (o : Ordinal) (f : ∀ a < o, Ordinal) : Ordinal :=
   mex (familyOfBFamily o f)
 
+@[deprecated (since := "2024-09-20")]
 theorem bmex_not_mem_brange {o : Ordinal} (f : ∀ a < o, Ordinal) : bmex o f ∉ brange o f := by
   rw [← range_familyOfBFamily]
   apply mex_not_mem_range
 
+@[deprecated (since := "2024-09-20")]
 theorem le_bmex_of_forall {o : Ordinal} (f : ∀ a < o, Ordinal) {a : Ordinal}
     (H : ∀ b < a, ∃ i hi, f i hi = b) : a ≤ bmex o f := by
   by_contra! h
   exact bmex_not_mem_brange f (H _ h)
 
+@[deprecated (since := "2024-09-20")]
 theorem ne_bmex {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) {i} (hi) :
     f i hi ≠ bmex.{_, v} o f := by
   convert (config := {transparency := .default})
@@ -1789,29 +2031,36 @@ theorem ne_bmex {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) {i} (hi) :
   -- Porting note: `familyOfBFamily_enum` → `typein_enum`
   rw [typein_enum]
 
+@[deprecated (since := "2024-09-20")]
 theorem bmex_le_of_ne {o : Ordinal} {f : ∀ a < o, Ordinal} {a} (ha : ∀ i hi, f i hi ≠ a) :
     bmex o f ≤ a :=
   mex_le_of_ne fun _i => ha _ _
 
+@[deprecated (since := "2024-09-20")]
 theorem exists_of_lt_bmex {o : Ordinal} {f : ∀ a < o, Ordinal} {a} (ha : a < bmex o f) :
     ∃ i hi, f i hi = a := by
   cases' exists_of_lt_mex ha with i hi
   exact ⟨_, typein_lt_self i, hi⟩
 
+@[deprecated (since := "2024-09-20")]
 theorem bmex_le_blsub {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{max u v}) :
     bmex.{_, v} o f ≤ blsub.{_, v} o f :=
   mex_le_lsub _
 
+@[deprecated (since := "2024-09-20")]
 theorem bmex_monotone {o o' : Ordinal.{u}}
     {f : ∀ a < o, Ordinal.{max u v}} {g : ∀ a < o', Ordinal.{max u v}}
     (h : brange o f ⊆ brange o' g) : bmex.{_, v} o f ≤ bmex.{_, v} o' g :=
   mex_monotone (by rwa [range_familyOfBFamily, range_familyOfBFamily])
 
+@[deprecated sInf_compl_lt_ord_succ (since := "2024-09-20")]
 theorem bmex_lt_ord_succ_card {o : Ordinal.{u}} (f : ∀ a < o, Ordinal.{u}) :
     bmex.{_, u} o f < (succ o.card).ord := by
   rw [← mk_toType]
   exact mex_lt_ord_succ_mk (familyOfBFamily o f)
 
+end mex
+
 end Ordinal
 
 /-! ### Results about injectivity and surjectivity -/
@@ -1834,136 +2083,19 @@ the Burali-Forti paradox. -/
 theorem not_small_ordinal : ¬Small.{u} Ordinal.{max u v} := fun h =>
   @not_injective_of_ordinal_of_small _ h _ fun _a _b => Ordinal.lift_inj.{v, u}.1
 
-/-! ### Enumerating unbounded sets of ordinals with ordinals -/
-
-
-namespace Ordinal
-
-section
-
-/-- Enumerator function for an unbounded set of ordinals. -/
-def enumOrd (S : Set Ordinal.{u}) : Ordinal → Ordinal :=
-  lt_wf.fix fun o f => sInf (S ∩ Set.Ici (blsub.{u, u} o f))
-
-variable {S : Set Ordinal.{u}}
-
-/-- The equation that characterizes `enumOrd` definitionally. This isn't the nicest expression to
-    work with, so consider using `enumOrd_def` instead. -/
-theorem enumOrd_def' (o) :
-    enumOrd S o = sInf (S ∩ Set.Ici (blsub.{u, u} o fun a _ => enumOrd S a)) :=
-  lt_wf.fix_eq _ _
-
-/-- The set in `enumOrd_def'` is nonempty. -/
-theorem enumOrd_def'_nonempty (hS : Unbounded (· < ·) S) (a) : (S ∩ Set.Ici a).Nonempty :=
-  let ⟨b, hb, hb'⟩ := hS a
-  ⟨b, hb, le_of_not_gt hb'⟩
-
-private theorem enumOrd_mem_aux (hS : Unbounded (· < ·) S) (o) :
-    enumOrd S o ∈ S ∩ Set.Ici (blsub.{u, u} o fun c _ => enumOrd S c) := by
-  rw [enumOrd_def']
-  exact csInf_mem (enumOrd_def'_nonempty hS _)
-
-theorem enumOrd_mem (hS : Unbounded (· < ·) S) (o) : enumOrd S o ∈ S :=
-  (enumOrd_mem_aux hS o).left
-
-theorem blsub_le_enumOrd (hS : Unbounded (· < ·) S) (o) :
-    (blsub.{u, u} o fun c _ => enumOrd S c) ≤ enumOrd S o :=
-  (enumOrd_mem_aux hS o).right
-
-theorem enumOrd_strictMono (hS : Unbounded (· < ·) S) : StrictMono (enumOrd S) := fun _ _ h =>
-  (lt_blsub.{u, u} _ _ h).trans_le (blsub_le_enumOrd hS _)
-
-/-- A more workable definition for `enumOrd`. -/
-theorem enumOrd_def (o) : enumOrd S o = sInf (S ∩ { b | ∀ c, c < o → enumOrd S c < b }) := by
-  rw [enumOrd_def']
-  congr; ext
-  exact ⟨fun h a hao => (lt_blsub.{u, u} _ _ hao).trans_le h, blsub_le⟩
-
-/-- The set in `enumOrd_def` is nonempty. -/
-theorem enumOrd_def_nonempty (hS : Unbounded (· < ·) S) {o} :
-    { x | x ∈ S ∧ ∀ c, c < o → enumOrd S c < x }.Nonempty :=
-  ⟨_, enumOrd_mem hS o, fun _ b => enumOrd_strictMono hS b⟩
-
-@[simp]
-theorem enumOrd_range {f : Ordinal → Ordinal} (hf : StrictMono f) : enumOrd (range f) = f :=
-  funext fun o => by
-    apply Ordinal.induction o
-    intro a H
-    rw [enumOrd_def a]
-    have Hfa : f a ∈ range f ∩ { b | ∀ c, c < a → enumOrd (range f) c < b } :=
-      ⟨mem_range_self a, fun b hb => by
-        rw [H b hb]
-        exact hf hb⟩
-    refine (csInf_le' Hfa).antisymm ((le_csInf_iff'' ⟨_, Hfa⟩).2 ?_)
-    rintro _ ⟨⟨c, rfl⟩, hc : ∀ b < a, enumOrd (range f) b < f c⟩
-    rw [hf.le_iff_le]
-    contrapose! hc
-    exact ⟨c, hc, (H c hc).ge⟩
-
-@[simp]
-theorem enumOrd_univ : enumOrd Set.univ = id := by
-  rw [← range_id]
-  exact enumOrd_range strictMono_id
-
-@[simp]
-theorem enumOrd_zero : enumOrd S 0 = sInf S := by
-  rw [enumOrd_def]
-  simp [Ordinal.not_lt_zero]
-
-theorem enumOrd_succ_le {a b} (hS : Unbounded (· < ·) S) (ha : a ∈ S) (hb : enumOrd S b < a) :
-    enumOrd S (succ b) ≤ a := by
-  rw [enumOrd_def]
-  exact
-    csInf_le' ⟨ha, fun c hc => ((enumOrd_strictMono hS).monotone (le_of_lt_succ hc)).trans_lt hb⟩
-
-theorem enumOrd_le_of_subset {S T : Set Ordinal} (hS : Unbounded (· < ·) S) (hST : S ⊆ T) (a) :
-    enumOrd T a ≤ enumOrd S a := by
-  apply Ordinal.induction a
-  intro b H
-  rw [enumOrd_def]
-  exact csInf_le' ⟨hST (enumOrd_mem hS b), fun c h => (H c h).trans_lt (enumOrd_strictMono hS h)⟩
-
-theorem enumOrd_surjective (hS : Unbounded (· < ·) S) : ∀ s ∈ S, ∃ a, enumOrd S a = s := fun s hs =>
-  ⟨sSup { a | enumOrd S a ≤ s }, by
-    apply le_antisymm
-    · rw [enumOrd_def]
-      refine csInf_le' ⟨hs, fun a ha => ?_⟩
-      have : enumOrd S 0 ≤ s := by
-        rw [enumOrd_zero]
-        exact csInf_le' hs
-      -- Porting note: `flip` is required to infer a metavariable.
-      rcases flip exists_lt_of_lt_csSup ha ⟨0, this⟩ with ⟨b, hb, hab⟩
-      exact (enumOrd_strictMono hS hab).trans_le hb
-    · by_contra! h
-      exact
-        (le_csSup ⟨s, fun a => (lt_wf.self_le_of_strictMono (enumOrd_strictMono hS) a).trans⟩
-              (enumOrd_succ_le hS hs h)).not_lt
-          (lt_succ _)⟩
-
-/-- An order isomorphism between an unbounded set of ordinals and the ordinals. -/
-def enumOrdOrderIso (hS : Unbounded (· < ·) S) : Ordinal ≃o S :=
-  StrictMono.orderIsoOfSurjective (fun o => ⟨_, enumOrd_mem hS o⟩) (enumOrd_strictMono hS) fun s =>
-    let ⟨a, ha⟩ := enumOrd_surjective hS s s.prop
-    ⟨a, Subtype.eq ha⟩
-
-theorem range_enumOrd (hS : Unbounded (· < ·) S) : range (enumOrd S) = S := by
-  rw [range_eq_iff]
-  exact ⟨enumOrd_mem hS, enumOrd_surjective hS⟩
-
-/-- A characterization of `enumOrd`: it is the unique strict monotonic function with range `S`. -/
-theorem eq_enumOrd (f : Ordinal → Ordinal) (hS : Unbounded (· < ·) S) :
-    StrictMono f ∧ range f = S ↔ f = enumOrd S := by
-  constructor
-  · rintro ⟨h₁, h₂⟩
-    rwa [← lt_wf.eq_strictMono_iff_eq_range h₁ (enumOrd_strictMono hS), range_enumOrd hS]
-  · rintro rfl
-    exact ⟨enumOrd_strictMono hS, range_enumOrd hS⟩
-
-end
+theorem Ordinal.not_bddAbove_compl_of_small (s : Set Ordinal.{u}) [hs : Small.{u} s] :
+    ¬BddAbove sᶜ := by
+  rw [bddAbove_iff_small]
+  intro h
+  have := small_union s sᶜ
+  rw [union_compl_self, small_univ_iff] at this
+  exact not_small_ordinal this
 
 /-! ### Casting naturals into ordinals, compatibility with operations -/
 
 
+namespace Ordinal
+
 @[simp]
 theorem one_add_natCast (m : ℕ) : 1 + (m : Ordinal) = succ m := by
   rw [← Nat.cast_one, ← Nat.cast_add, add_comm]
@@ -2077,7 +2209,7 @@ theorem lift_ofNat (n : ℕ) [n.AtLeastTwo] :
 
 end Ordinal
 
-/-! ### Properties of `omega` -/
+/-! ### Properties of ω -/
 
 
 namespace Cardinal
@@ -2094,7 +2226,7 @@ theorem ord_aleph0 : ord.{u} ℵ₀ = ω :=
 
 @[simp]
 theorem add_one_of_aleph0_le {c} (h : ℵ₀ ≤ c) : c + 1 = c := by
-  rw [add_comm, ← card_ord c, ← card_one, ← card_add, one_add_of_omega_le]
+  rw [add_comm, ← card_ord c, ← card_one, ← card_add, one_add_of_omega0_le]
   rwa [← ord_aleph0, ord_le_ord]
 
 end Cardinal
@@ -2106,34 +2238,61 @@ theorem lt_add_of_limit {a b c : Ordinal.{u}} (h : IsLimit c) :
   -- Porting note: `bex_def` is required.
   rw [← IsNormal.bsup_eq.{u, u} (add_isNormal b) h, lt_bsup, bex_def]
 
-theorem lt_omega {o : Ordinal} : o < ω ↔ ∃ n : ℕ, o = n := by
+theorem lt_omega0 {o : Ordinal} : o < ω ↔ ∃ n : ℕ, o = n := by
   simp_rw [← Cardinal.ord_aleph0, Cardinal.lt_ord, lt_aleph0, card_eq_nat]
 
-theorem nat_lt_omega (n : ℕ) : ↑n < ω :=
-  lt_omega.2 ⟨_, rfl⟩
+@[deprecated (since := "2024-09-30")]
+alias lt_omega := lt_omega0
+
+theorem nat_lt_omega0 (n : ℕ) : ↑n < ω :=
+  lt_omega0.2 ⟨_, rfl⟩
 
+@[deprecated (since := "2024-09-30")]
+alias nat_lt_omega := nat_lt_omega0
+
+theorem omega0_pos : 0 < ω :=
+  nat_lt_omega0 0
+
+@[deprecated (since := "2024-09-30")]
 theorem omega_pos : 0 < ω :=
-  nat_lt_omega 0
+  nat_lt_omega0 0
+
+theorem omega0_ne_zero : ω ≠ 0 :=
+  omega0_pos.ne'
+
+@[deprecated (since := "2024-09-30")]
+alias omega_ne_zero := omega0_ne_zero
 
-theorem omega_ne_zero : ω ≠ 0 :=
-  omega_pos.ne'
+theorem one_lt_omega0 : 1 < ω := by simpa only [Nat.cast_one] using nat_lt_omega0 1
 
-theorem one_lt_omega : 1 < ω := by simpa only [Nat.cast_one] using nat_lt_omega 1
+@[deprecated (since := "2024-09-30")]
+alias one_lt_omega := one_lt_omega0
 
-theorem omega_isLimit : IsLimit ω :=
-  ⟨omega_ne_zero, fun o h => by
-    let ⟨n, e⟩ := lt_omega.1 h
-    rw [e]; exact nat_lt_omega (n + 1)⟩
+theorem omega0_isLimit : IsLimit ω :=
+  ⟨omega0_ne_zero, fun o h => by
+    let ⟨n, e⟩ := lt_omega0.1 h
+    rw [e]; exact nat_lt_omega0 (n + 1)⟩
 
-theorem omega_le {o : Ordinal} : ω ≤ o ↔ ∀ n : ℕ, ↑n ≤ o :=
-  ⟨fun h n => (nat_lt_omega _).le.trans h, fun H =>
+@[deprecated (since := "2024-09-30")]
+alias omega_isLimit := omega0_isLimit
+
+theorem omega0_le {o : Ordinal} : ω ≤ o ↔ ∀ n : ℕ, ↑n ≤ o :=
+  ⟨fun h n => (nat_lt_omega0 _).le.trans h, fun H =>
     le_of_forall_lt fun a h => by
-      let ⟨n, e⟩ := lt_omega.1 h
+      let ⟨n, e⟩ := lt_omega0.1 h
       rw [e, ← succ_le_iff]; exact H (n + 1)⟩
 
+@[deprecated (since := "2024-09-30")]
+alias omega_le := omega0_le
+
 @[simp]
+theorem iSup_natCast : iSup Nat.cast = ω :=
+  (Ordinal.iSup_le fun n => (nat_lt_omega0 n).le).antisymm <| omega0_le.2 <| Ordinal.le_iSup _
+
+set_option linter.deprecated false in
+@[deprecated iSup_natCast (since := "2024-04-17")]
 theorem sup_natCast : sup Nat.cast = ω :=
-  (sup_le fun n => (nat_lt_omega n).le).antisymm <| omega_le.2 <| le_sup _
+  iSup_natCast
 
 @[deprecated (since := "2024-04-17")]
 alias sup_nat_cast := sup_natCast
@@ -2142,24 +2301,30 @@ theorem nat_lt_limit {o} (h : IsLimit o) : ∀ n : ℕ, ↑n < o
   | 0 => lt_of_le_of_ne (Ordinal.zero_le o) h.1.symm
   | n + 1 => h.2 _ (nat_lt_limit h n)
 
-theorem omega_le_of_isLimit {o} (h : IsLimit o) : ω ≤ o :=
-  omega_le.2 fun n => le_of_lt <| nat_lt_limit h n
+theorem omega0_le_of_isLimit {o} (h : IsLimit o) : ω ≤ o :=
+  omega0_le.2 fun n => le_of_lt <| nat_lt_limit h n
+
+@[deprecated (since := "2024-09-30")]
+alias omega_le_of_isLimit := omega0_le_of_isLimit
 
-theorem isLimit_iff_omega_dvd {a : Ordinal} : IsLimit a ↔ a ≠ 0 ∧ ω ∣ a := by
+theorem isLimit_iff_omega0_dvd {a : Ordinal} : IsLimit a ↔ a ≠ 0 ∧ ω ∣ a := by
   refine ⟨fun l => ⟨l.1, ⟨a / ω, le_antisymm ?_ (mul_div_le _ _)⟩⟩, fun h => ?_⟩
   · refine (limit_le l).2 fun x hx => le_of_lt ?_
-    rw [← div_lt omega_ne_zero, ← succ_le_iff, le_div omega_ne_zero, mul_succ,
-      add_le_of_limit omega_isLimit]
+    rw [← div_lt omega0_ne_zero, ← succ_le_iff, le_div omega0_ne_zero, mul_succ,
+      add_le_of_limit omega0_isLimit]
     intro b hb
-    rcases lt_omega.1 hb with ⟨n, rfl⟩
+    rcases lt_omega0.1 hb with ⟨n, rfl⟩
     exact
       (add_le_add_right (mul_div_le _ _) _).trans
         (lt_sub.1 <| nat_lt_limit (sub_isLimit l hx) _).le
   · rcases h with ⟨a0, b, rfl⟩
-    refine mul_isLimit_left omega_isLimit (Ordinal.pos_iff_ne_zero.2 <| mt ?_ a0)
+    refine mul_isLimit_left omega0_isLimit (Ordinal.pos_iff_ne_zero.2 <| mt ?_ a0)
     intro e
     simp only [e, mul_zero]
 
+@[deprecated (since := "2024-09-30")]
+alias isLimit_iff_omega_dvd := isLimit_iff_omega0_dvd
+
 theorem add_mul_limit_aux {a b c : Ordinal} (ba : b + a = a) (l : IsLimit c)
     (IH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b) : (a + b) * c = a * c :=
   le_antisymm
@@ -2198,19 +2363,35 @@ theorem add_le_of_forall_add_lt {a b c : Ordinal} (hb : 0 < b) (h : ∀ d < b, a
   by_contra! hb
   exact (h _ hb).ne H
 
-theorem IsNormal.apply_omega {f : Ordinal.{u} → Ordinal.{u}} (hf : IsNormal f) :
-    Ordinal.sup.{0, u} (f ∘ Nat.cast) = f ω := by rw [← sup_natCast, IsNormal.sup.{0, u, u} hf]
+theorem IsNormal.apply_omega0 {f : Ordinal.{u} → Ordinal.{v}} (hf : IsNormal f) :
+    ⨆ n : ℕ, f n = f ω := by rw [← iSup_natCast, hf.map_iSup]
+
+@[deprecated (since := "2024-09-30")]
+alias IsNormal.apply_omega := IsNormal.apply_omega0
 
 @[simp]
+theorem iSup_add_nat (o : Ordinal) : ⨆ n : ℕ, o + n = o + ω :=
+  (add_isNormal o).apply_omega0
+
+set_option linter.deprecated false in
+@[deprecated iSup_add_nat (since := "2024-08-27")]
 theorem sup_add_nat (o : Ordinal) : (sup fun n : ℕ => o + n) = o + ω :=
-  (add_isNormal o).apply_omega
+  (add_isNormal o).apply_omega0
 
 @[simp]
+theorem iSup_mul_nat (o : Ordinal) : ⨆ n : ℕ, o * n = o * ω := by
+  rcases eq_zero_or_pos o with (rfl | ho)
+  · rw [zero_mul]
+    exact iSup_eq_zero_iff.2 fun n => zero_mul (n : Ordinal)
+  · exact (mul_isNormal ho).apply_omega0
+
+set_option linter.deprecated false in
+@[deprecated iSup_add_nat (since := "2024-08-27")]
 theorem sup_mul_nat (o : Ordinal) : (sup fun n : ℕ => o * n) = o * ω := by
   rcases eq_zero_or_pos o with (rfl | ho)
   · rw [zero_mul]
     exact sup_eq_zero_iff.2 fun n => zero_mul (n : Ordinal)
-  · exact (mul_isNormal ho).apply_omega
+  · exact (mul_isNormal ho).apply_omega0
 
 end Ordinal
 
@@ -2222,10 +2403,10 @@ namespace Acc
 smallest ordinal greater than the ranks of all elements below it (i.e. elements `b` such that
 `r b a`). -/
 noncomputable def rank (h : Acc r a) : Ordinal.{u} :=
-  Acc.recOn h fun a _h ih => Ordinal.sup.{u, u} fun b : { b // r b a } => Order.succ <| ih b b.2
+  Acc.recOn h fun a _h ih => ⨆ b : { b // r b a }, Order.succ (ih b b.2)
 
 theorem rank_eq (h : Acc r a) :
-    h.rank = Ordinal.sup.{u, u} fun b : { b // r b a } => Order.succ (h.inv b.2).rank := by
+    h.rank = ⨆ b : { b // r b a }, Order.succ (h.inv b.2).rank := by
   change (Acc.intro a fun _ => h.inv).rank = _
   rfl
 
@@ -2233,7 +2414,7 @@ theorem rank_eq (h : Acc r a) :
 theorem rank_lt_of_rel (hb : Acc r b) (h : r a b) : (hb.inv h).rank < hb.rank :=
   (Order.lt_succ _).trans_le <| by
     rw [hb.rank_eq]
-    refine le_trans ?_ (Ordinal.le_sup _ ⟨a, h⟩)
+    refine le_trans ?_ (Ordinal.le_iSup _ ⟨a, h⟩)
     rfl
 
 end Acc
@@ -2249,7 +2430,7 @@ noncomputable def rank (a : α) : Ordinal.{u} :=
   (hwf.apply a).rank
 
 theorem rank_eq :
-    hwf.rank a = Ordinal.sup.{u, u} fun b : { b // r b a } => Order.succ <| hwf.rank b := by
+    hwf.rank a = ⨆ b : { b // r b a }, Order.succ (hwf.rank b) := by
   rw [rank, Acc.rank_eq]
   rfl
 
@@ -2263,3 +2444,5 @@ theorem rank_strictAnti [Preorder α] [WellFoundedGT α] :
     StrictAnti (rank <| @wellFounded_gt α _ _) := fun _ _ => rank_lt_of_rel wellFounded_gt
 
 end WellFounded
+
+set_option linter.style.longFile 2700
diff --git a/Mathlib/SetTheory/Ordinal/Basic.lean b/Mathlib/SetTheory/Ordinal/Basic.lean
index c638012729c42..6f37148318d91 100644
--- a/Mathlib/SetTheory/Ordinal/Basic.lean
+++ b/Mathlib/SetTheory/Ordinal/Basic.lean
@@ -3,6 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Floris van Doorn
 -/
+import Mathlib.Algebra.Order.SuccPred
 import Mathlib.Data.Sum.Order
 import Mathlib.Order.InitialSeg
 import Mathlib.SetTheory.Cardinal.Basic
@@ -27,12 +28,13 @@ initial segment (or, equivalently, in any way). This total order is well founded
 * `Ordinal.card o`: the cardinality of an ordinal `o`.
 * `Ordinal.lift` lifts an ordinal in universe `u` to an ordinal in universe `max u v`.
   For a version registering additionally that this is an initial segment embedding, see
-  `Ordinal.lift.initialSeg`.
+  `Ordinal.liftInitialSeg`.
   For a version registering that it is a principal segment embedding if `u < v`, see
-  `Ordinal.lift.principalSeg`.
-* `Ordinal.omega` or `ω` is the order type of `ℕ`. This definition is universe polymorphic:
-  `Ordinal.omega.{u} : Ordinal.{u}` (contrast with `ℕ : Type`, which lives in a specific
-  universe). In some cases the universe level has to be given explicitly.
+  `Ordinal.liftPrincipalSeg`.
+* `Ordinal.omega0` or `ω` is the order type of `ℕ`. It is called this to match `Cardinal.aleph0`
+  and so that the omega function can be named `Ordinal.omega`. This definition is universe
+  polymorphic: `Ordinal.omega0.{u} : Ordinal.{u}` (contrast with `ℕ : Type`, which lives in
+  a specific universe). In some cases the universe level has to be given explicitly.
 
 * `o₁ + o₂` is the order on the disjoint union of `o₁` and `o₂` obtained by declaring that
   every element of `o₁` is smaller than every element of `o₂`.
@@ -66,42 +68,6 @@ universe u v w
 variable {α : Type u} {β : Type*} {γ : Type*} {r : α → α → Prop} {s : β → β → Prop}
   {t : γ → γ → Prop}
 
-/-! ### Well order on an arbitrary type -/
-
-
-section WellOrderingThm
-
--- Porting note: `parameter` does not work
--- parameter {σ : Type u}
-variable {σ : Type u}
-
-
-open Function
-
-theorem nonempty_embedding_to_cardinal : Nonempty (σ ↪ Cardinal.{u}) :=
-  (Embedding.total _ _).resolve_left fun ⟨⟨f, hf⟩⟩ =>
-    let g : σ → Cardinal.{u} := invFun f
-    let ⟨x, (hx : g x = 2 ^ sum g)⟩ := invFun_surjective hf (2 ^ sum g)
-    have : g x ≤ sum g := le_sum.{u, u} g x
-    not_le_of_gt (by rw [hx]; exact cantor _) this
-
-/-- An embedding of any type to the set of cardinals. -/
-def embeddingToCardinal : σ ↪ Cardinal.{u} :=
-  Classical.choice nonempty_embedding_to_cardinal
-
-/-- Any type can be endowed with a well order, obtained by pulling back the well order over
-cardinals by some embedding. -/
-def WellOrderingRel : σ → σ → Prop :=
-  embeddingToCardinal ⁻¹'o (· < ·)
-
-instance WellOrderingRel.isWellOrder : IsWellOrder σ WellOrderingRel :=
-  (RelEmbedding.preimage _ _).isWellOrder
-
-instance IsWellOrder.subtype_nonempty : Nonempty { r // IsWellOrder σ r } :=
-  ⟨⟨WellOrderingRel, inferInstance⟩⟩
-
-end WellOrderingThm
-
 /-! ### Definition of ordinals -/
 
 
@@ -272,16 +238,11 @@ protected theorem one_ne_zero : (1 : Ordinal) ≠ 0 :=
 instance nontrivial : Nontrivial Ordinal.{u} :=
   ⟨⟨1, 0, Ordinal.one_ne_zero⟩⟩
 
---@[simp] -- Porting note: not in simp nf, added aux lemma below
+@[simp]
 theorem type_preimage {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β ≃ α) :
     type (f ⁻¹'o r) = type r :=
   (RelIso.preimage f r).ordinal_type_eq
 
-@[simp, nolint simpNF] -- `simpNF` incorrectly complains the LHS doesn't simplify.
-theorem type_preimage_aux {α β : Type u} (r : α → α → Prop) [IsWellOrder α r] (f : β ≃ α) :
-    @type _ (fun x y => r (f x) (f y)) (inferInstanceAs (IsWellOrder β (↑f ⁻¹'o r))) = type r := by
-  convert (RelIso.preimage f r).ordinal_type_eq
-
 @[elab_as_elim]
 theorem inductionOn {C : Ordinal → Prop} (o : Ordinal)
     (H : ∀ (α r) [IsWellOrder α r], C (type r)) : C o :=
@@ -294,9 +255,11 @@ For `Ordinal`:
 
 * less-equal is defined such that well orders `r` and `s` satisfy `type r ≤ type s` if there exists
   a function embedding `r` as an *initial* segment of `s`.
-
 * less-than is defined such that well orders `r` and `s` satisfy `type r < type s` if there exists
   a function embedding `r` as a *principal* segment of `s`.
+
+Note that most of the relevant results on initial and principal segments are proved in the
+`Order.InitialSeg` file.
 -/
 instance partialOrder : PartialOrder Ordinal where
   le a b :=
@@ -317,11 +280,17 @@ instance partialOrder : PartialOrder Ordinal where
   lt_iff_le_not_le a b :=
     Quotient.inductionOn₂ a b fun _ _ =>
       ⟨fun ⟨f⟩ => ⟨⟨f⟩, fun ⟨g⟩ => (f.ltLe g).irrefl⟩, fun ⟨⟨f⟩, h⟩ =>
-        Sum.recOn f.ltOrEq (fun g => ⟨g⟩) fun g => (h ⟨InitialSeg.ofIso g.symm⟩).elim⟩
+        f.ltOrEq.recOn (fun g => ⟨g⟩) fun g => (h ⟨InitialSeg.ofIso g.symm⟩).elim⟩
   le_antisymm a b :=
     Quotient.inductionOn₂ a b fun _ _ ⟨h₁⟩ ⟨h₂⟩ =>
       Quot.sound ⟨InitialSeg.antisymm h₁ h₂⟩
 
+instance linearOrder : LinearOrder Ordinal :=
+  {inferInstanceAs (PartialOrder Ordinal) with
+    le_total := fun a b => Quotient.inductionOn₂ a b fun ⟨α, r, _⟩ ⟨β, s, _⟩ =>
+      (InitialSeg.total r s).recOn (fun f => Or.inl ⟨f⟩) fun f => Or.inr ⟨f⟩
+    decidableLE := Classical.decRel _ }
+
 theorem type_le_iff {α β} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r]
     [IsWellOrder β s] : type r ≤ type s ↔ Nonempty (r ≼i s) :=
   Iff.rfl
@@ -413,7 +382,7 @@ theorem typein_top {α β} {r : α → α → Prop} {s : β → β → Prop} [Is
   Eq.symm <|
     Quot.sound
       ⟨RelIso.ofSurjective (RelEmbedding.codRestrict _ f f.lt_top) fun ⟨a, h⟩ => by
-          rcases f.down.1 h with ⟨b, rfl⟩; exact ⟨b, rfl⟩⟩
+          rcases f.mem_range_of_rel_top h with ⟨b, rfl⟩; exact ⟨b, rfl⟩⟩
 
 @[simp]
 theorem typein_apply {α β} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r] [IsWellOrder β s]
@@ -424,7 +393,7 @@ theorem typein_apply {α β} {r : α → α → Prop} {s : β → β → Prop} [
         (RelEmbedding.codRestrict _ ((Subrel.relEmbedding _ _).trans f) fun ⟨x, h⟩ => by
           rw [RelEmbedding.trans_apply]; exact f.toRelEmbedding.map_rel_iff.2 h)
           fun ⟨y, h⟩ => by
-            rcases f.init h with ⟨a, rfl⟩
+            rcases f.mem_range_of_rel h with ⟨a, rfl⟩
             exact ⟨⟨a, f.toRelEmbedding.map_rel_iff.1 h⟩,
               Subtype.eq <| RelEmbedding.trans_apply _ _ _⟩⟩
 
@@ -457,7 +426,7 @@ principal segment. -/
 def typein.principalSeg {α : Type u} (r : α → α → Prop) [IsWellOrder α r] :
     @PrincipalSeg α Ordinal.{u} r (· < ·) :=
   ⟨⟨⟨typein r, typein_injective r⟩, typein_lt_typein r⟩, type r,
-    fun _ ↦ ⟨typein_surj r, fun ⟨a, h⟩ ↦ h ▸ typein_lt_type r a⟩⟩
+    fun _ ↦ ⟨fun ⟨a, h⟩ ↦ h ▸ typein_lt_type r a, typein_surj r⟩⟩
 
 @[simp]
 theorem typein.principalSeg_coe (r : α → α → Prop) [IsWellOrder α r] :
@@ -513,6 +482,14 @@ theorem lt_wf : @WellFounded Ordinal (· < ·) :=
 instance wellFoundedRelation : WellFoundedRelation Ordinal :=
   ⟨(· < ·), lt_wf⟩
 
+instance wellFoundedLT : WellFoundedLT Ordinal :=
+  ⟨lt_wf⟩
+
+instance isWellOrder : IsWellOrder Ordinal (· < ·) where
+
+instance : ConditionallyCompleteLinearOrderBot Ordinal :=
+  WellFoundedLT.conditionallyCompleteLinearOrderBot _
+
 /-- Reformulation of well founded induction on ordinals as a lemma that works with the
 `induction` tactic, as in `induction i using Ordinal.induction with | h i IH => ?_`. -/
 theorem induction {p : Ordinal.{u} → Prop} (i : Ordinal.{u}) (h : ∀ j, (∀ k, k < j → p k) → p j) :
@@ -531,8 +508,7 @@ def card : Ordinal → Cardinal :=
 theorem card_type (r : α → α → Prop) [IsWellOrder α r] : card (type r) = #α :=
   rfl
 
--- Porting note: nolint, simpNF linter falsely claims the lemma never applies
-@[simp, nolint simpNF]
+@[simp]
 theorem card_typein {r : α → α → Prop} [IsWellOrder α r] (x : α) :
     #{ y // r y x } = (typein r x).card :=
   rfl
@@ -551,27 +527,19 @@ theorem card_one : card 1 = 1 := mk_eq_one _
 -- Porting note: Needed to add universe hint .{u} below
 /-- The universe lift operation for ordinals, which embeds `Ordinal.{u}` as
   a proper initial segment of `Ordinal.{v}` for `v > u`. For the initial segment version,
-  see `lift.initialSeg`. -/
+  see `liftInitialSeg`. -/
 @[pp_with_univ]
 def lift (o : Ordinal.{v}) : Ordinal.{max v u} :=
   Quotient.liftOn o (fun w => type <| ULift.down.{u} ⁻¹'o w.r) fun ⟨_, r, _⟩ ⟨_, s, _⟩ ⟨f⟩ =>
     Quot.sound
       ⟨(RelIso.preimage Equiv.ulift r).trans <| f.trans (RelIso.preimage Equiv.ulift s).symm⟩
 
--- Porting note: Needed to add universe hints ULift.down.{v,u} below
--- @[simp] -- Porting note: Not in simpnf, added aux lemma below
+@[simp]
 theorem type_uLift (r : α → α → Prop) [IsWellOrder α r] :
-    type (ULift.down.{v,u} ⁻¹'o r) = lift.{v} (type r) := by
+    type (ULift.down ⁻¹'o r) = lift.{v} (type r) := by
   simp (config := { unfoldPartialApp := true })
   rfl
 
--- Porting note: simpNF linter falsely claims that this never applies
-@[simp, nolint simpNF]
-theorem type_uLift_aux (r : α → α → Prop) [IsWellOrder α r] :
-    @type.{max v u} _ (fun x y => r (ULift.down.{v,u} x) (ULift.down.{v,u} y))
-      (inferInstanceAs (IsWellOrder (ULift α) (ULift.down ⁻¹'o r))) = lift.{v} (type r) :=
-  rfl
-
 theorem _root_.RelIso.ordinal_lift_type_eq {α : Type u} {β : Type v} {r : α → α → Prop}
     {s : β → β → Prop} [IsWellOrder α r] [IsWellOrder β s] (f : r ≃r s) :
     lift.{v} (type r) = lift.{u} (type s) :=
@@ -589,20 +557,23 @@ theorem type_lift_preimage_aux {α : Type u} {β : Type v} (r : α → α → Pr
       (inferInstanceAs (IsWellOrder β (f ⁻¹'o r)))) = lift.{v} (type r) :=
   (RelIso.preimage f r).ordinal_lift_type_eq
 
-/-- `lift.{max u v, u}` equals `lift.{v, u}`. -/
--- @[simp] -- Porting note: simp lemma never applies, tested
+/-- `lift.{max u v, u}` equals `lift.{v, u}`.
+
+Unfortunately, the simp lemma doesn't seem to work. -/
 theorem lift_umax : lift.{max u v, u} = lift.{v, u} :=
   funext fun a =>
     inductionOn a fun _ r _ =>
       Quotient.sound ⟨(RelIso.preimage Equiv.ulift r).trans (RelIso.preimage Equiv.ulift r).symm⟩
 
-/-- `lift.{max v u, u}` equals `lift.{v, u}`. -/
--- @[simp] -- Porting note: simp lemma never applies, tested
+/-- `lift.{max v u, u}` equals `lift.{v, u}`.
+
+Unfortunately, the simp lemma doesn't seem to work. -/
 theorem lift_umax' : lift.{max v u, u} = lift.{v, u} :=
   lift_umax
 
-/-- An ordinal lifted to a lower or equal universe equals itself. -/
--- @[simp] -- Porting note: simp lemma never applies, tested
+/-- An ordinal lifted to a lower or equal universe equals itself.
+
+Unfortunately, the simp lemma doesn't work. -/
 theorem lift_id' (a : Ordinal) : lift a = a :=
   inductionOn a fun _ r _ => Quotient.sound ⟨RelIso.preimage Equiv.ulift r⟩
 
@@ -716,36 +687,53 @@ theorem lt_lift_iff {a : Ordinal.{u}} {b : Ordinal.{max u v}} :
 
 /-- Initial segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
   `ordinal.{v}` as an initial segment when `u ≤ v`. -/
-def lift.initialSeg : @InitialSeg Ordinal.{u} Ordinal.{max u v} (· < ·) (· < ·) :=
+def liftInitialSeg : @InitialSeg Ordinal.{u} Ordinal.{max u v} (· < ·) (· < ·) :=
   ⟨⟨⟨lift.{v}, fun _ _ => lift_inj.1⟩, lift_lt⟩, fun _ _ h => lift_down (le_of_lt h)⟩
 
+@[deprecated liftInitialSeg (since := "2024-09-21")]
+alias lift.initialSeg := liftInitialSeg
+
 @[simp]
-theorem lift.initialSeg_coe : (lift.initialSeg.{u,v} : Ordinal → Ordinal) = lift.{v,u} :=
+theorem liftInitialSeg_coe : (liftInitialSeg.{u, v} : Ordinal → Ordinal) = lift.{v, u} :=
+  rfl
+
+set_option linter.deprecated false in
+@[deprecated liftInitialSeg_coe (since := "2024-09-21")]
+theorem lift.initialSeg_coe : (lift.initialSeg.{u, v} : Ordinal → Ordinal) = lift.{v, u} :=
   rfl
 
-/-! ### The first infinite ordinal `omega` -/
+/-! ### The first infinite ordinal ω -/
 
 
 /-- `ω` is the first infinite ordinal, defined as the order type of `ℕ`. -/
-def omega : Ordinal.{u} :=
+def omega0 : Ordinal.{u} :=
   lift <| @type ℕ (· < ·) _
 
+@[deprecated Ordinal.omega0 (since := "2024-09-26")]
+alias omega := omega0
+
 @[inherit_doc]
-scoped notation "ω" => Ordinal.omega
+scoped notation "ω" => Ordinal.omega0
 
-/-- Note that the presence of this lemma makes `simp [omega]` form a loop. -/
+/-- Note that the presence of this lemma makes `simp [omega0]` form a loop. -/
 @[simp]
 theorem type_nat_lt : @type ℕ (· < ·) _ = ω :=
   (lift_id _).symm
 
 @[simp]
-theorem card_omega : card ω = ℵ₀ :=
+theorem card_omega0 : card ω = ℵ₀ :=
   rfl
 
+@[deprecated (since := "2024-09-30")]
+alias card_omega := card_omega0
+
 @[simp]
-theorem lift_omega : lift ω = ω :=
+theorem lift_omega0 : lift ω = ω :=
   lift_lift _
 
+@[deprecated (since := "2024-09-30")]
+alias lift_omega := lift_omega0
+
 /-!
 ### Definition and first properties of addition on ordinals
 
@@ -848,32 +836,6 @@ theorem le_add_right (a b : Ordinal) : a ≤ a + b := by
 theorem le_add_left (a b : Ordinal) : a ≤ b + a := by
   simpa only [zero_add] using add_le_add_right (Ordinal.zero_le b) a
 
-instance linearOrder : LinearOrder Ordinal :=
-  {inferInstanceAs (PartialOrder Ordinal) with
-    le_total := fun a b =>
-      match lt_or_eq_of_le (le_add_left b a), lt_or_eq_of_le (le_add_right a b) with
-      | Or.inr h, _ => by rw [h]; exact Or.inl (le_add_right _ _)
-      | _, Or.inr h => by rw [h]; exact Or.inr (le_add_left _ _)
-      | Or.inl h₁, Or.inl h₂ => by
-        revert h₁ h₂
-        refine inductionOn a ?_
-        intro α₁ r₁ _
-        refine inductionOn b ?_
-        intro α₂ r₂ _ ⟨f⟩ ⟨g⟩
-        rw [← typein_top f, ← typein_top g, le_iff_lt_or_eq, le_iff_lt_or_eq,
-                 typein_lt_typein, typein_lt_typein]
-        rcases trichotomous_of (Sum.Lex r₁ r₂) g.top f.top with (h | h | h) <;>
-          [exact Or.inl (Or.inl h); (left; right; rw [h]); exact Or.inr (Or.inl h)]
-    decidableLE := Classical.decRel _ }
-
-instance wellFoundedLT : WellFoundedLT Ordinal :=
-  ⟨lt_wf⟩
-
-instance isWellOrder : IsWellOrder Ordinal (· < ·) where
-
-instance : ConditionallyCompleteLinearOrderBot Ordinal :=
-  WellFoundedLT.conditionallyCompleteLinearOrderBot _
-
 theorem max_zero_left : ∀ a : Ordinal, max 0 a = a :=
   max_bot_left
 
@@ -895,8 +857,8 @@ private theorem succ_le_iff' {a b : Ordinal} : a + 1 ≤ b ↔ a < b :=
       (inductionOn a fun α r _ =>
         ⟨⟨⟨⟨fun x => Sum.inl x, fun _ _ => Sum.inl.inj⟩, Sum.lex_inl_inl⟩,
             Sum.inr PUnit.unit, fun b =>
-            Sum.recOn b (fun x => ⟨fun _ => ⟨x, rfl⟩, fun _ => Sum.Lex.sep _ _⟩) fun x =>
-              Sum.lex_inr_inr.trans ⟨False.elim, fun ⟨x, H⟩ => Sum.inl_ne_inr H⟩⟩⟩),
+            Sum.recOn b (fun x => ⟨fun _ => Sum.Lex.sep _ _, fun _ => ⟨x, rfl⟩⟩) fun x =>
+              (Sum.lex_inr_inr.trans ⟨False.elim, fun ⟨x, H⟩ => Sum.inl_ne_inr H⟩).symm⟩⟩),
     inductionOn a fun α r hr =>
       inductionOn b fun β s hs ⟨⟨f, t, hf⟩⟩ => by
         haveI := hs
@@ -904,25 +866,27 @@ private theorem succ_le_iff' {a b : Ordinal} : a + 1 ≤ b ↔ a < b :=
         · rcases a with (a | _) <;> rcases b with (b | _)
           · simpa only [Sum.lex_inl_inl] using f.map_rel_iff.2
           · intro
-            rw [hf]
+            rw [← hf]
             exact ⟨_, rfl⟩
           · exact False.elim ∘ Sum.lex_inr_inl
           · exact False.elim ∘ Sum.lex_inr_inr.1
         · rcases a with (a | _)
           · intro h
-            have := @PrincipalSeg.init _ _ _ _ _ ⟨f, t, hf⟩ _ _ h
+            have := @PrincipalSeg.mem_range_of_rel _ _ _ _ _ ⟨f, t, hf⟩ _ _ h
             cases' this with w h
             exact ⟨Sum.inl w, h⟩
           · intro h
-            cases' (hf b).1 h with w h
+            cases' (hf b).2 h with w h
             exact ⟨Sum.inl w, h⟩⟩
 
 instance noMaxOrder : NoMaxOrder Ordinal :=
   ⟨fun _ => ⟨_, succ_le_iff'.1 le_rfl⟩⟩
 
-instance succOrder : SuccOrder Ordinal.{u} :=
+instance instSuccOrder : SuccOrder Ordinal.{u} :=
   SuccOrder.ofSuccLeIff (fun o => o + 1) succ_le_iff'
 
+instance instSuccAddOrder : SuccAddOrder Ordinal := ⟨fun _ => rfl⟩
+
 @[simp]
 theorem add_one_eq_succ (o : Ordinal) : o + 1 = succ o :=
   rfl
@@ -938,10 +902,12 @@ theorem succ_one : succ (1 : Ordinal) = 2 := by congr; simp only [Nat.unaryCast,
 theorem add_succ (o₁ o₂ : Ordinal) : o₁ + succ o₂ = succ (o₁ + o₂) :=
   (add_assoc _ _ _).symm
 
-theorem one_le_iff_pos {o : Ordinal} : 1 ≤ o ↔ 0 < o := by rw [← succ_zero, succ_le_iff]
+@[deprecated Order.one_le_iff_pos (since := "2024-09-04")]
+protected theorem one_le_iff_pos {o : Ordinal} : 1 ≤ o ↔ 0 < o :=
+  Order.one_le_iff_pos
 
 theorem one_le_iff_ne_zero {o : Ordinal} : 1 ≤ o ↔ o ≠ 0 := by
-  rw [one_le_iff_pos, Ordinal.pos_iff_ne_zero]
+  rw [Order.one_le_iff_pos, Ordinal.pos_iff_ne_zero]
 
 theorem succ_pos (o : Ordinal) : 0 < succ o :=
   bot_lt_succ o
@@ -999,13 +965,12 @@ alias typein_one_out := typein_one_toType
 
 @[simp]
 theorem typein_le_typein (r : α → α → Prop) [IsWellOrder α r] {x y : α} :
-    typein r x ≤ typein r y ↔ ¬r y x := by rw [← not_lt, typein_lt_typein]
+    typein r x ≤ typein r y ↔ ¬r y x := by
+  rw [← not_lt, typein_lt_typein]
 
--- @[simp] -- Porting note (#10618): simp can prove this
 theorem typein_le_typein' (o : Ordinal) {x y : o.toType} :
     typein (α := o.toType) (· < ·) x ≤ typein (α := o.toType) (· < ·) y ↔ x ≤ y := by
-  rw [typein_le_typein]
-  exact not_lt
+  simp
 
 theorem enum_le_enum (r : α → α → Prop) [IsWellOrder α r] {o₁ o₂ : {o // o < type r}} :
     ¬r (enum r o₁) (enum r o₂) ↔ o₂ ≤ o₁ := by
@@ -1037,7 +1002,7 @@ theorem enum_inj {r : α → α → Prop} [IsWellOrder α r] {o₁ o₂ : {o //
   rw [EmbeddingLike.apply_eq_iff_eq, Subtype.mk.injEq]
 
 /-- The order isomorphism between ordinals less than `o` and `o.toType`. -/
-@[simps!]
+@[simps! (config := .lemmasOnly)]
 noncomputable def enumIsoToType (o : Ordinal) : Set.Iio o ≃o o.toType where
   toFun x :=
     enum (α := o.toType) (· < ·) ⟨x.1, by
@@ -1053,6 +1018,18 @@ noncomputable def enumIsoToType (o : Ordinal) : Set.Iio o ≃o o.toType where
 @[deprecated (since := "2024-08-26")]
 alias enumIsoOut := enumIsoToType
 
+instance small_Iio (o : Ordinal.{u}) : Small.{u} (Iio o) :=
+  ⟨_, ⟨(enumIsoToType _).toEquiv⟩⟩
+
+instance small_Iic (o : Ordinal.{u}) : Small.{u} (Iic o) := by
+  rw [← Iio_succ]
+  exact small_Iio _
+
+instance small_Ico (a b : Ordinal.{u}) : Small.{u} (Ico a b) := small_subset Ico_subset_Iio_self
+instance small_Icc (a b : Ordinal.{u}) : Small.{u} (Icc a b) := small_subset Icc_subset_Iic_self
+instance small_Ioo (a b : Ordinal.{u}) : Small.{u} (Ioo a b) := small_subset Ioo_subset_Iio_self
+instance small_Ioc (a b : Ordinal.{u}) : Small.{u} (Ioc a b) := small_subset Ioc_subset_Iic_self
+
 /-- `o.toType` is an `OrderBot` whenever `0 < o`. -/
 def toTypeOrderBotOfPos {o : Ordinal} (ho : 0 < o) : OrderBot o.toType where
   bot_le := enum_zero_le' ho
@@ -1088,10 +1065,15 @@ theorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=
 
 /-- Principal segment version of the lift operation on ordinals, embedding `ordinal.{u}` in
   `ordinal.{v}` as a principal segment when `u < v`. -/
-def lift.principalSeg : @PrincipalSeg Ordinal.{u} Ordinal.{max (u + 1) v} (· < ·) (· < ·) :=
-  ⟨↑lift.initialSeg.{u, max (u + 1) v}, univ.{u, v}, by
+def liftPrincipalSeg : @PrincipalSeg Ordinal.{u} Ordinal.{max (u + 1) v} (· < ·) (· < ·) :=
+  ⟨↑liftInitialSeg.{u, max (u + 1) v}, univ.{u, v}, by
     refine fun b => inductionOn b ?_; intro β s _
     rw [univ, ← lift_umax]; constructor <;> intro h
+    · cases' h with a e
+      rw [← e]
+      refine inductionOn a ?_
+      intro α r _
+      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein.principalSeg r⟩
     · rw [← lift_id (type s)] at h ⊢
       cases' lift_type_lt.{_,_,v}.1 h with f
       cases' f with f a hf
@@ -1100,34 +1082,46 @@ def lift.principalSeg : @PrincipalSeg Ordinal.{u} Ordinal.{max (u + 1) v} (· <
       -- Porting note: apply inductionOn does not work, refine does
       refine inductionOn a ?_
       intro α r _ hf
-      refine
-        lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2
-          ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩
-      · exact fun b => enum r ⟨f b, (hf _).2 ⟨_, rfl⟩⟩
+      refine lift_type_eq.{u, max (u + 1) v, max (u + 1) v}.2
+        ⟨(RelIso.ofSurjective (RelEmbedding.ofMonotone ?_ ?_) ?_).symm⟩
+      · exact fun b => enum r ⟨f b, (hf _).1 ⟨_, rfl⟩⟩
       · refine fun a b h => (typein_lt_typein r).1 ?_
         rw [typein_enum, typein_enum]
         exact f.map_rel_iff.2 h
       · intro a'
-        cases' (hf _).1 (typein_lt_type _ a') with b e
+        cases' (hf _).2 (typein_lt_type _ a') with b e
         exists b
         simp only [RelEmbedding.ofMonotone_coe]
-        simp [e]
-    · cases' h with a e
-      rw [← e]
-      refine inductionOn a ?_
-      intro α r _
-      exact lift_type_lt.{u, u + 1, max (u + 1) v}.2 ⟨typein.principalSeg r⟩⟩
+        simp [e]⟩
+
+@[deprecated liftPrincipalSeg (since := "2024-09-21")]
+alias lift.principalSeg := liftPrincipalSeg
 
 @[simp]
+theorem liftPrincipalSeg_coe :
+    (liftPrincipalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=
+  rfl
+
+set_option linter.deprecated false in
+@[deprecated liftPrincipalSeg_coe (since := "2024-09-21")]
 theorem lift.principalSeg_coe :
     (lift.principalSeg.{u, v} : Ordinal → Ordinal) = lift.{max (u + 1) v} :=
   rfl
 
--- Porting note: Added universe hints below
 @[simp]
-theorem lift.principalSeg_top : (lift.principalSeg.{u,v}).top = univ.{u,v} :=
+theorem liftPrincipalSeg_top : (liftPrincipalSeg.{u, v}).top = univ.{u, v} :=
   rfl
 
+set_option linter.deprecated false in
+@[deprecated liftPrincipalSeg_top (since := "2024-09-21")]
+theorem lift.principalSeg_top : (lift.principalSeg.{u, v}).top = univ.{u, v} :=
+  rfl
+
+theorem liftPrincipalSeg_top' : liftPrincipalSeg.{u, u + 1}.top = @type Ordinal (· < ·) _ := by
+  simp only [liftPrincipalSeg_top, univ_id]
+
+set_option linter.deprecated false in
+@[deprecated liftPrincipalSeg_top (since := "2024-09-21")]
 theorem lift.principalSeg_top' : lift.principalSeg.{u, u + 1}.top = @type Ordinal (· < ·) _ := by
   simp only [lift.principalSeg_top, univ_id]
 
@@ -1200,6 +1194,9 @@ theorem card_ord (c) : (ord c).card = c :=
     -- Porting note: cardinal.mk_def is now Cardinal.mk'_def, not sure why
     simp only [mk'_def, e, card_type]
 
+theorem card_surjective : Function.Surjective card :=
+  fun c ↦ ⟨_, card_ord c⟩
+
 /-- Galois coinsertion between `Cardinal.ord` and `Ordinal.card`. -/
 def gciOrdCard : GaloisCoinsertion ord card :=
   gc_ord_card.toGaloisCoinsertion fun c => c.card_ord.le
@@ -1294,6 +1291,18 @@ theorem ord_injective : Injective ord := by
   intro c c' h
   rw [← card_ord c, ← card_ord c', h]
 
+@[simp]
+theorem ord_inj {a b : Cardinal} : a.ord = b.ord ↔ a = b :=
+  ord_injective.eq_iff
+
+@[simp]
+theorem ord_eq_zero {a : Cardinal} : a.ord = 0 ↔ a = 0 :=
+  ord_injective.eq_iff' ord_zero
+
+@[simp]
+theorem ord_eq_one {a : Cardinal} : a.ord = 1 ↔ a = 1 :=
+  ord_injective.eq_iff' ord_one
+
 /-- The ordinal corresponding to a cardinal `c` is the least ordinal
   whose cardinal is `c`. This is the order-embedding version. For the regular function, see `ord`.
 -/
@@ -1324,8 +1333,8 @@ theorem univ_umax : univ.{u, max (u + 1) v} = univ.{u, v} :=
   congr_fun lift_umax _
 
 theorem lift_lt_univ (c : Cardinal) : lift.{u + 1, u} c < univ.{u, u + 1} := by
-  simpa only [lift.principalSeg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using
-    le_of_lt (lift.principalSeg.{u, u + 1}.lt_top (succ c).ord)
+  simpa only [liftPrincipalSeg_coe, lift_ord, lift_succ, ord_le, succ_le_iff] using
+    le_of_lt (liftPrincipalSeg.{u, u + 1}.lt_top (succ c).ord)
 
 theorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} := by
   have := lift_lt.{_, max (u+1) v}.2 (lift_lt_univ c)
@@ -1335,18 +1344,18 @@ theorem lift_lt_univ' (c : Cardinal) : lift.{max (u + 1) v, u} c < univ.{u, v} :
 @[simp]
 theorem ord_univ : ord univ.{u, v} = Ordinal.univ.{u, v} := by
   refine le_antisymm (ord_card_le _) <| le_of_forall_lt fun o h => lt_ord.2 ?_
-  have := lift.principalSeg.{u, v}.down.1 (by simpa only [lift.principalSeg_coe] using h)
+  have := liftPrincipalSeg.mem_range_of_rel_top (by simpa only [liftPrincipalSeg_coe] using h)
   rcases this with ⟨o, h'⟩
-  rw [← h', lift.principalSeg_coe, ← lift_card]
+  rw [← h', liftPrincipalSeg_coe, ← lift_card]
   apply lift_lt_univ'
 
 theorem lt_univ {c} : c < univ.{u, u + 1} ↔ ∃ c', c = lift.{u + 1, u} c' :=
   ⟨fun h => by
     have := ord_lt_ord.2 h
     rw [ord_univ] at this
-    cases' lift.principalSeg.{u, u + 1}.down.1 (by simpa only [lift.principalSeg_top] ) with o e
+    cases' liftPrincipalSeg.mem_range_of_rel_top (by simpa only [liftPrincipalSeg_top]) with o e
     have := card_ord c
-    rw [← e, lift.principalSeg_coe, ← lift_card] at this
+    rw [← e, liftPrincipalSeg_coe, ← lift_card] at this
     exact ⟨_, this.symm⟩, fun ⟨c', e⟩ => e.symm ▸ lift_lt_univ _⟩
 
 theorem lt_univ' {c} : c < univ.{u, v} ↔ ∃ c', c = lift.{max (u + 1) v, u} c' :=
diff --git a/Mathlib/SetTheory/Ordinal/Enum.lean b/Mathlib/SetTheory/Ordinal/Enum.lean
new file mode 100644
index 0000000000000..c5a4c3b7442d6
--- /dev/null
+++ b/Mathlib/SetTheory/Ordinal/Enum.lean
@@ -0,0 +1,125 @@
+/-
+Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+-/
+import Mathlib.SetTheory.Ordinal.Arithmetic
+
+/-!
+# Enumerating sets of ordinals by ordinals
+
+The ordinals have the peculiar property that every subset bounded above is a small type, while
+themselves not being small. As a consequence of this, every unbounded subset of `Ordinal` is order
+isomorphic to `Ordinal`.
+
+We define this correspondence as `enumOrd`, and use it to then define an order isomorphism
+`enumOrdOrderIso`.
+
+This can be thought of as an ordinal analog of `Nat.nth`.
+-/
+
+universe u
+
+open Order Set
+
+namespace Ordinal
+
+variable {o a b : Ordinal.{u}}
+
+/-- Enumerator function for an unbounded set of ordinals. -/
+noncomputable def enumOrd (s : Set Ordinal.{u}) (o : Ordinal.{u}) : Ordinal.{u} :=
+  sInf (s ∩ { b | ∀ c, c < o → enumOrd s c < b })
+termination_by o
+
+variable {s : Set Ordinal.{u}}
+
+@[deprecated (since := "2024-09-20")]
+theorem enumOrd_def (o : Ordinal.{u}) :
+    enumOrd s o = sInf (s ∩ { b | ∀ c, c < o → enumOrd s c < b }) := by
+  rw [enumOrd]
+
+theorem enumOrd_le_of_forall_lt (ha : a ∈ s) (H : ∀ b < o, enumOrd s b < a) : enumOrd s o ≤ a := by
+  rw [enumOrd]
+  exact csInf_le' ⟨ha, H⟩
+
+/-- The set in the definition of `enumOrd` is nonempty. -/
+private theorem enumOrd_nonempty (hs : ¬ BddAbove s) (o : Ordinal) :
+    (s ∩ { b | ∀ c, c < o → enumOrd s c < b }).Nonempty := by
+  rw [not_bddAbove_iff] at hs
+  obtain ⟨a, ha⟩ := bddAbove_of_small (enumOrd s '' Iio o)
+  obtain ⟨b, hb, hba⟩ := hs a
+  exact ⟨b, hb, fun c hc ↦ (ha (mem_image_of_mem _ hc)).trans_lt hba⟩
+
+private theorem enumOrd_mem_aux (hs : ¬ BddAbove s) (o : Ordinal) :
+    enumOrd s o ∈ s ∩ { b | ∀ c, c < o → enumOrd s c < b } := by
+  rw [enumOrd]
+  exact csInf_mem (enumOrd_nonempty hs o)
+
+theorem enumOrd_mem (hs : ¬ BddAbove s) (o : Ordinal) : enumOrd s o ∈ s :=
+  (enumOrd_mem_aux hs o).1
+
+theorem enumOrd_strictMono (hs : ¬ BddAbove s) : StrictMono (enumOrd s) :=
+  fun a b ↦ (enumOrd_mem_aux hs b).2 a
+
+theorem enumOrd_succ_le (hs : ¬ BddAbove s) (ha : a ∈ s) (hb : enumOrd s b < a) :
+    enumOrd s (succ b) ≤ a := by
+  apply enumOrd_le_of_forall_lt ha
+  intro c hc
+  rw [lt_succ_iff] at hc
+  exact ((enumOrd_strictMono hs).monotone hc).trans_lt hb
+
+theorem range_enumOrd (hs : ¬ BddAbove s) : range (enumOrd s) = s := by
+  ext a
+  let t := { b | a ≤ enumOrd s b }
+  constructor
+  · rintro ⟨b, rfl⟩
+    exact enumOrd_mem hs b
+  · intro ha
+    refine ⟨sInf t, (enumOrd_le_of_forall_lt ha ?_).antisymm ?_⟩
+    · intro b hb
+      by_contra! hb'
+      exact hb.not_le (csInf_le' hb')
+    · exact csInf_mem (s := t) ⟨a, (enumOrd_strictMono hs).id_le a⟩
+
+theorem enumOrd_surjective (hs : ¬ BddAbove s) {b : Ordinal} (hb : b ∈ s) :
+    ∃ a, enumOrd s a = b := by
+  rwa [← range_enumOrd hs] at hb
+
+theorem enumOrd_le_of_subset {t : Set Ordinal} (hs : ¬ BddAbove s) (hst : s ⊆ t) :
+    enumOrd t ≤ enumOrd s := by
+  intro a
+  rw [enumOrd, enumOrd]
+  apply csInf_le_csInf' (enumOrd_nonempty hs a) (inter_subset_inter hst _)
+  intro b hb c hc
+  exact (enumOrd_le_of_subset hs hst c).trans_lt <| hb c hc
+termination_by a => a
+
+/-- A characterization of `enumOrd`: it is the unique strict monotonic function with range `s`. -/
+theorem eq_enumOrd (f : Ordinal → Ordinal) (hs : ¬ BddAbove s) :
+    enumOrd s = f ↔ StrictMono f ∧ range f = s := by
+  constructor
+  · rintro rfl
+    exact ⟨enumOrd_strictMono hs, range_enumOrd hs⟩
+  · rintro ⟨h₁, h₂⟩
+    rwa [← (enumOrd_strictMono hs).range_inj h₁, range_enumOrd hs, eq_comm]
+
+theorem enumOrd_range {f : Ordinal → Ordinal} (hf : StrictMono f) : enumOrd (range f) = f :=
+  (eq_enumOrd _ hf.not_bddAbove_range_of_wellFoundedLT).2 ⟨hf, rfl⟩
+
+@[simp]
+theorem enumOrd_univ : enumOrd Set.univ = id := by
+  rw [← range_id]
+  exact enumOrd_range strictMono_id
+
+@[simp]
+theorem enumOrd_zero : enumOrd s 0 = sInf s := by
+  rw [enumOrd]
+  simp [Ordinal.not_lt_zero]
+
+/-- An order isomorphism between an unbounded set of ordinals and the ordinals. -/
+noncomputable def enumOrdOrderIso (s : Set Ordinal) (hs : ¬ BddAbove s) : Ordinal ≃o s :=
+  StrictMono.orderIsoOfSurjective (fun o => ⟨_, enumOrd_mem hs o⟩) (enumOrd_strictMono hs) fun s =>
+    let ⟨a, ha⟩ := enumOrd_surjective hs s.prop
+    ⟨a, Subtype.eq ha⟩
+
+end Ordinal
diff --git a/Mathlib/SetTheory/Ordinal/Exponential.lean b/Mathlib/SetTheory/Ordinal/Exponential.lean
index 6753d1d6c71f1..ca373660f3136 100644
--- a/Mathlib/SetTheory/Ordinal/Exponential.lean
+++ b/Mathlib/SetTheory/Ordinal/Exponential.lean
@@ -86,6 +86,14 @@ theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b := by
 theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a ≠ 0) : a ^ b ≠ 0 :=
   Ordinal.pos_iff_ne_zero.1 <| opow_pos b <| Ordinal.pos_iff_ne_zero.2 a0
 
+@[simp]
+theorem opow_eq_zero {a b : Ordinal} : a ^ b = 0 ↔ a = 0 ∧ b ≠ 0 := by
+  obtain rfl | ha := eq_or_ne a 0
+  · obtain rfl | hb := eq_or_ne b 0
+    · simp
+    · simp [hb]
+  · simp [opow_ne_zero b ha, ha]
+
 @[simp, norm_cast]
 theorem opow_natCast (a : Ordinal) (n : ℕ) : a ^ (n : Ordinal) = a ^ n := by
   induction n with
@@ -151,7 +159,7 @@ theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b := b
   rwa [opow_le_opow_iff_right a1, one_le_iff_pos]
 
 theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b ≤ a ^ b :=
-  (opow_isNormal a1).self_le _
+  (opow_isNormal a1).id_le _
 
 theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c := by
   rw [opow_succ, opow_succ]
@@ -286,17 +294,67 @@ theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b
     rwa [← succ_log_def hb hx] at this
   · rwa [one_opow, one_le_iff_ne_zero]
 
-/-- `opow b` and `log b` (almost) form a Galois connection. -/
-theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : b ^ c ≤ x ↔ c ≤ log b x :=
-  ⟨fun h =>
-    le_of_not_lt fun hn =>
-      (lt_opow_succ_log_self hb x).not_le <|
-        ((opow_le_opow_iff_right hb).2 (succ_le_of_lt hn)).trans h,
-    fun h => ((opow_le_opow_iff_right hb).2 h).trans (opow_log_le_self b hx)⟩
-
+/-- `opow b` and `log b` (almost) form a Galois connection.
+
+See `opow_le_iff_le_log'` for a variant assuming `c ≠ 0` rather than `x ≠ 0`. See also
+`le_log_of_opow_le` and `opow_le_of_le_log`, which are both separate implications under weaker
+assumptions. -/
+theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) :
+    b ^ c ≤ x ↔ c ≤ log b x := by
+  constructor <;>
+  intro h
+  · apply le_of_not_lt
+    intro hn
+    apply (lt_opow_succ_log_self hb x).not_le <|
+      ((opow_le_opow_iff_right hb).2 <| succ_le_of_lt hn).trans h
+  · exact ((opow_le_opow_iff_right hb).2 h).trans <| opow_log_le_self b hx
+
+/-- `opow b` and `log b` (almost) form a Galois connection.
+
+See `opow_le_iff_le_log` for a variant assuming `x ≠ 0` rather than `c ≠ 0`. See also
+`le_log_of_opow_le` and `opow_le_of_le_log`, which are both separate implications under weaker
+assumptions. -/
+theorem opow_le_iff_le_log' {b x c : Ordinal} (hb : 1 < b) (hc : c ≠ 0) :
+    b ^ c ≤ x ↔ c ≤ log b x := by
+  obtain rfl | hx := eq_or_ne x 0
+  · rw [log_zero_right, Ordinal.le_zero, Ordinal.le_zero, opow_eq_zero]
+    simp [hc, (zero_lt_one.trans hb).ne']
+  · exact opow_le_iff_le_log hb hx
+
+theorem le_log_of_opow_le {b x c : Ordinal} (hb : 1 < b) (h : b ^ c ≤ x) : c ≤ log b x := by
+  obtain rfl | hx := eq_or_ne x 0
+  · rw [Ordinal.le_zero, opow_eq_zero] at h
+    exact (zero_lt_one.asymm <| h.1 ▸ hb).elim
+  · exact (opow_le_iff_le_log hb hx).1 h
+
+theorem opow_le_of_le_log {b x c : Ordinal} (hc : c ≠ 0) (h : c ≤ log b x) : b ^ c ≤ x := by
+  obtain hb | hb := le_or_lt b 1
+  · rw [log_of_left_le_one hb] at h
+    exact (h.not_lt (Ordinal.pos_iff_ne_zero.2 hc)).elim
+  · rwa [opow_le_iff_le_log' hb hc]
+
+/-- `opow b` and `log b` (almost) form a Galois connection.
+
+See `lt_opow_iff_log_lt'` for a variant assuming `c ≠ 0` rather than `x ≠ 0`. See also
+`lt_opow_of_log_lt` and `lt_log_of_lt_opow`, which are both separate implications under weaker
+assumptions. -/
 theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x ≠ 0) : x < b ^ c ↔ log b x < c :=
   lt_iff_lt_of_le_iff_le (opow_le_iff_le_log hb hx)
 
+/-- `opow b` and `log b` (almost) form a Galois connection.
+
+See `lt_opow_iff_log_lt` for a variant assuming `x ≠ 0` rather than `c ≠ 0`. See also
+`lt_opow_of_log_lt` and `lt_log_of_lt_opow`, which are both separate implications under weaker
+assumptions. -/
+theorem lt_opow_iff_log_lt' {b x c : Ordinal} (hb : 1 < b) (hc : c ≠ 0) : x < b ^ c ↔ log b x < c :=
+  lt_iff_lt_of_le_iff_le (opow_le_iff_le_log' hb hc)
+
+theorem lt_opow_of_log_lt {b x c : Ordinal} (hb : 1 < b) : log b x < c → x < b ^ c :=
+  lt_imp_lt_of_le_imp_le <| le_log_of_opow_le hb
+
+theorem lt_log_of_lt_opow {b x c : Ordinal} (hc : c ≠ 0) : x < b ^ c → log b x < c :=
+  lt_imp_lt_of_le_imp_le <| opow_le_of_le_log hc
+
 theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o ≠ 0) (hbo : b ≤ o) : 0 < log b o := by
   rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one]
 
@@ -400,13 +458,21 @@ theorem natCast_opow (m : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (m : Ordinal) ^
   | n + 1 => by
     rw [pow_succ, natCast_mul, natCast_opow m n, Nat.cast_succ, add_one_eq_succ, opow_succ]
 
-@[deprecated (since := "2024-04-17")]
-alias nat_cast_opow := natCast_opow
+theorem iSup_pow {o : Ordinal} (ho : 0 < o) : ⨆ n : ℕ, o ^ n = o ^ ω := by
+  simp_rw [← opow_natCast]
+  rcases (one_le_iff_pos.2 ho).lt_or_eq with ho₁ | rfl
+  · exact (opow_isNormal ho₁).apply_omega0
+  · rw [one_opow]
+    refine le_antisymm (Ordinal.iSup_le fun n => by rw [one_opow]) ?_
+    convert Ordinal.le_iSup _ 0
+    rw [Nat.cast_zero, opow_zero]
 
+set_option linter.deprecated false in
+@[deprecated iSup_pow (since := "2024-08-27")]
 theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : ℕ => o ^ n) = o ^ ω := by
   simp_rw [← opow_natCast]
   rcases (one_le_iff_pos.2 ho).lt_or_eq with ho₁ | rfl
-  · exact (opow_isNormal ho₁).apply_omega
+  · exact (opow_isNormal ho₁).apply_omega0
   · rw [one_opow]
     refine le_antisymm (sup_le fun n => by rw [one_opow]) ?_
     convert le_sup (fun n : ℕ => 1 ^ (n : Ordinal)) 0
diff --git a/Mathlib/SetTheory/Ordinal/FixedPoint.lean b/Mathlib/SetTheory/Ordinal/FixedPoint.lean
index c7a4b2b31ea32..cedf18985846e 100644
--- a/Mathlib/SetTheory/Ordinal/FixedPoint.lean
+++ b/Mathlib/SetTheory/Ordinal/FixedPoint.lean
@@ -3,7 +3,7 @@ Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reser
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios, Mario Carneiro
 -/
-import Mathlib.SetTheory.Ordinal.Arithmetic
+import Mathlib.SetTheory.Ordinal.Enum
 import Mathlib.SetTheory.Ordinal.Exponential
 
 /-!
@@ -21,8 +21,8 @@ Moreover, we prove some lemmas about the fixed points of specific normal functio
 * `nfpFamily`, `nfpBFamily`, `nfp`: the next fixed point of a (family of) normal function(s).
 * `fp_family_unbounded`, `fp_bfamily_unbounded`, `fp_unbounded`: the (common) fixed points of a
   (family of) normal function(s) are unbounded in the ordinals.
-* `deriv_add_eq_mul_omega_add`: a characterization of the derivative of addition.
-* `deriv_mul_eq_opow_omega_mul`: a characterization of the derivative of multiplication.
+* `deriv_add_eq_mul_omega0_add`: a characterization of the derivative of addition.
+* `deriv_mul_eq_opow_omega0_mul`: a characterization of the derivative of multiplication.
 -/
 
 
@@ -48,44 +48,48 @@ finitely many functions in the family to `a`.
 
 `Ordinal.nfpFamily_fp` shows this is a fixed point, `Ordinal.le_nfpFamily` shows it's at
 least `a`, and `Ordinal.nfpFamily_le_fp` shows this is the least ordinal with these properties. -/
-def nfpFamily (f : ι → Ordinal → Ordinal) (a : Ordinal) : Ordinal :=
-  sup (List.foldr f a)
+def nfpFamily (f : ι → Ordinal.{max u v} → Ordinal.{max u v}) (a : Ordinal.{max u v}) : Ordinal :=
+  ⨆ i, List.foldr f a i
 
 theorem nfpFamily_eq_sup (f : ι → Ordinal.{max u v} → Ordinal.{max u v}) (a : Ordinal.{max u v}) :
-    nfpFamily.{u, v} f a = sup.{u, v} (List.foldr f a) :=
+    nfpFamily.{u, v} f a = ⨆ i, List.foldr f a i :=
   rfl
 
 theorem foldr_le_nfpFamily (f : ι → Ordinal → Ordinal)
     (a l) : List.foldr f a l ≤ nfpFamily.{u, v} f a :=
-  le_sup.{u, v} _ _
+  Ordinal.le_iSup _ _
 
 theorem le_nfpFamily (f : ι → Ordinal → Ordinal) (a) : a ≤ nfpFamily f a :=
-  le_sup _ []
+  Ordinal.le_iSup _ []
 
 theorem lt_nfpFamily {a b} : a < nfpFamily.{u, v} f b ↔ ∃ l, a < List.foldr f b l :=
-  lt_sup.{u, v}
+  Ordinal.lt_iSup
 
 theorem nfpFamily_le_iff {a b} : nfpFamily.{u, v} f a ≤ b ↔ ∀ l, List.foldr f a l ≤ b :=
-  sup_le_iff
+  Ordinal.iSup_le_iff
 
 theorem nfpFamily_le {a b} : (∀ l, List.foldr f a l ≤ b) → nfpFamily.{u, v} f a ≤ b :=
-  sup_le.{u, v}
+  Ordinal.iSup_le
 
-theorem nfpFamily_monotone (hf : ∀ i, Monotone (f i)) : Monotone (nfpFamily.{u, v} f) :=
-  fun _ _ h => sup_le.{u, v} fun l => (List.foldr_monotone hf l h).trans (le_sup.{u, v} _ l)
+theorem nfpFamily_monotone (hf : ∀ i, Monotone (f i)) : Monotone (nfpFamily.{u, v} f) := by
+  intro _ _ h
+  apply Ordinal.iSup_le
+  intro l
+  exact (List.foldr_monotone hf l h).trans (Ordinal.le_iSup _ l)
 
 theorem apply_lt_nfpFamily (H : ∀ i, IsNormal (f i)) {a b} (hb : b < nfpFamily.{u, v} f a) (i) :
     f i b < nfpFamily.{u, v} f a :=
   let ⟨l, hl⟩ := lt_nfpFamily.1 hb
-  lt_sup.2 ⟨i::l, (H i).strictMono hl⟩
+  Ordinal.lt_iSup.2 ⟨i::l, (H i).strictMono hl⟩
 
 theorem apply_lt_nfpFamily_iff [Nonempty ι] (H : ∀ i, IsNormal (f i)) {a b} :
-    (∀ i, f i b < nfpFamily.{u, v} f a) ↔ b < nfpFamily.{u, v} f a :=
-  ⟨fun h =>
-    lt_nfpFamily.2 <|
-      let ⟨l, hl⟩ := lt_sup.1 <| h <| Classical.arbitrary ι
-      ⟨l, ((H _).self_le b).trans_lt hl⟩,
-    apply_lt_nfpFamily H⟩
+    (∀ i, f i b < nfpFamily.{u, v} f a) ↔ b < nfpFamily.{u, v} f a := by
+  constructor
+  · intro h
+    exact lt_nfpFamily.2 <|
+      let ⟨l, hl⟩ := Ordinal.lt_iSup.1 <| h <| Classical.arbitrary ι
+      ⟨l, (H _).le_apply.trans_lt hl⟩
+  · exact apply_lt_nfpFamily H
 
 theorem nfpFamily_le_apply [Nonempty ι] (H : ∀ i, IsNormal (f i)) {a b} :
     (∃ i, nfpFamily.{u, v} f a ≤ f i b) ↔ nfpFamily.{u, v} f a ≤ b := by
@@ -94,37 +98,45 @@ theorem nfpFamily_le_apply [Nonempty ι] (H : ∀ i, IsNormal (f i)) {a b} :
   exact apply_lt_nfpFamily_iff H
 
 theorem nfpFamily_le_fp (H : ∀ i, Monotone (f i)) {a b} (ab : a ≤ b) (h : ∀ i, f i b ≤ b) :
-    nfpFamily.{u, v} f a ≤ b :=
-  sup_le fun l => by
-    by_cases hι : IsEmpty ι
-    · rwa [Unique.eq_default l]
-    · induction' l with i l IH generalizing a
-      · exact ab
-      exact (H i (IH ab)).trans (h i)
+    nfpFamily.{u, v} f a ≤ b := by
+  apply Ordinal.iSup_le
+  intro l
+  induction' l with i l IH generalizing a
+  · exact ab
+  · exact (H i (IH ab)).trans (h i)
 
 theorem nfpFamily_fp {i} (H : IsNormal (f i)) (a) :
     f i (nfpFamily.{u, v} f a) = nfpFamily.{u, v} f a := by
-  unfold nfpFamily
-  rw [@IsNormal.sup.{u, v, v} _ H _ _ ⟨[]⟩]
-  apply le_antisymm <;> refine Ordinal.sup_le fun l => ?_
-  · exact le_sup _ (i::l)
-  · exact (H.self_le _).trans (le_sup _ _)
+  rw [nfpFamily, H.map_iSup]
+  apply le_antisymm <;> refine Ordinal.iSup_le fun l => ?_
+  · exact Ordinal.le_iSup _ (i::l)
+  · exact H.le_apply.trans (Ordinal.le_iSup _ _)
 
 theorem apply_le_nfpFamily [hι : Nonempty ι] {f : ι → Ordinal → Ordinal} (H : ∀ i, IsNormal (f i))
     {a b} : (∀ i, f i b ≤ nfpFamily.{u, v} f a) ↔ b ≤ nfpFamily.{u, v} f a := by
   refine ⟨fun h => ?_, fun h i => ?_⟩
   · cases' hι with i
-    exact ((H i).self_le b).trans (h i)
+    exact (H i).le_apply.trans (h i)
   rw [← nfpFamily_fp (H i)]
   exact (H i).monotone h
 
 theorem nfpFamily_eq_self {f : ι → Ordinal → Ordinal} {a} (h : ∀ i, f i a = a) :
-    nfpFamily f a = a :=
-  le_antisymm (sup_le fun l => by rw [List.foldr_fixed' h l]) <| le_nfpFamily f a
+    nfpFamily f a = a := by
+  apply (Ordinal.iSup_le ?_).antisymm (le_nfpFamily f a)
+  intro l
+  rw [List.foldr_fixed' h l]
 
 -- Todo: This is actually a special case of the fact the intersection of club sets is a club set.
 /-- A generalization of the fixed point lemma for normal functions: any family of normal functions
     has an unbounded set of common fixed points. -/
+theorem not_bddAbove_fp_family (H : ∀ i, IsNormal (f i)) :
+    ¬ BddAbove (⋂ i, Function.fixedPoints (f i)) := by
+  rw [not_bddAbove_iff]
+  refine fun a ↦ ⟨nfpFamily f (succ a), ?_, (lt_succ a).trans_le (le_nfpFamily f _)⟩
+  rintro _ ⟨i, rfl⟩
+  exact nfpFamily_fp (H i) _
+
+@[deprecated not_bddAbove_fp_family (since := "2024-09-20")]
 theorem fp_family_unbounded (H : ∀ i, IsNormal (f i)) :
     (⋂ i, Function.fixedPoints (f i)).Unbounded (· < ·) := fun a =>
   ⟨nfpFamily.{u, v} f a, fun s ⟨i, hi⟩ => by
@@ -173,7 +185,7 @@ theorem le_iff_derivFamily (H : ∀ i, IsNormal (f i)) {a} :
     (∀ i, f i a ≤ a) ↔ ∃ o, derivFamily.{u, v} f o = a :=
   ⟨fun ha => by
     suffices ∀ (o) (_ : a ≤ derivFamily.{u, v} f o), ∃ o, derivFamily.{u, v} f o = a from
-      this a ((derivFamily_isNormal _).self_le _)
+      this a (derivFamily_isNormal _).le_apply
     intro o
     induction' o using limitRecOn with o IH o l IH
     · intro h₁
@@ -202,7 +214,7 @@ theorem fp_iff_derivFamily (H : ∀ i, IsNormal (f i)) {a} :
 /-- For a family of normal functions, `Ordinal.derivFamily` enumerates the common fixed points. -/
 theorem derivFamily_eq_enumOrd (H : ∀ i, IsNormal (f i)) :
     derivFamily.{u, v} f = enumOrd (⋂ i, Function.fixedPoints (f i)) := by
-  rw [← eq_enumOrd _ (fp_family_unbounded.{u, v} H)]
+  rw [eq_comm, eq_enumOrd _ (not_bddAbove_fp_family H)]
   use (derivFamily_isNormal f).strictMono
   rw [Set.range_eq_iff]
   refine ⟨?_, fun a ha => ?_⟩
@@ -234,23 +246,23 @@ theorem nfpBFamily_eq_nfpFamily {o : Ordinal} (f : ∀ b < o, Ordinal → Ordina
 theorem foldr_le_nfpBFamily {o : Ordinal}
     (f : ∀ b < o, Ordinal → Ordinal) (a l) :
     List.foldr (familyOfBFamily o f) a l ≤ nfpBFamily.{u, v} o f a :=
-  le_sup.{u, v} _ _
+  Ordinal.le_iSup _ _
 
 theorem le_nfpBFamily {o : Ordinal} (f : ∀ b < o, Ordinal → Ordinal) (a) :
     a ≤ nfpBFamily.{u, v} o f a :=
-  le_sup.{u, v} _ []
+  Ordinal.le_iSup _ []
 
 theorem lt_nfpBFamily {a b} :
     a < nfpBFamily.{u, v} o f b ↔ ∃ l, a < List.foldr (familyOfBFamily o f) b l :=
-  lt_sup.{u, v}
+  Ordinal.lt_iSup
 
 theorem nfpBFamily_le_iff {o : Ordinal} {f : ∀ b < o, Ordinal → Ordinal} {a b} :
     nfpBFamily.{u, v} o f a ≤ b ↔ ∀ l, List.foldr (familyOfBFamily o f) a l ≤ b :=
-  sup_le_iff.{u, v}
+  Ordinal.iSup_le_iff
 
 theorem nfpBFamily_le {o : Ordinal} {f : ∀ b < o, Ordinal → Ordinal} {a b} :
     (∀ l, List.foldr (familyOfBFamily o f) a l ≤ b) → nfpBFamily.{u, v} o f a ≤ b :=
-  sup_le.{u, v}
+  Ordinal.iSup_le.{u, v}
 
 theorem nfpBFamily_monotone (hf : ∀ i hi, Monotone (f i hi)) : Monotone (nfpBFamily.{u, v} o f) :=
   nfpFamily_monotone fun _ => hf _ _
@@ -288,7 +300,7 @@ theorem apply_le_nfpBFamily (ho : o ≠ 0) (H : ∀ i hi, IsNormal (f i hi)) {a
     (∀ i hi, f i hi b ≤ nfpBFamily.{u, v} o f a) ↔ b ≤ nfpBFamily.{u, v} o f a := by
   refine ⟨fun h => ?_, fun h i hi => ?_⟩
   · have ho' : 0 < o := Ordinal.pos_iff_ne_zero.2 ho
-    exact ((H 0 ho').self_le b).trans (h 0 ho')
+    exact (H 0 ho').le_apply.trans (h 0 ho')
   · rw [← nfpBFamily_fp (H i hi)]
     exact (H i hi).monotone h
 
@@ -297,6 +309,16 @@ theorem nfpBFamily_eq_self {a} (h : ∀ i hi, f i hi a = a) : nfpBFamily.{u, v}
 
 /-- A generalization of the fixed point lemma for normal functions: any family of normal functions
     has an unbounded set of common fixed points. -/
+theorem not_bddAbove_fp_bfamily (H : ∀ i hi, IsNormal (f i hi)) :
+    ¬ BddAbove (⋂ (i) (hi), Function.fixedPoints (f i hi)) := by
+  rw [not_bddAbove_iff]
+  refine fun a ↦ ⟨nfpBFamily _ f (succ a), ?_, (lt_succ a).trans_le (le_nfpBFamily f _)⟩
+  rw [Set.mem_iInter₂]
+  exact fun i hi ↦ nfpBFamily_fp (H i hi) _
+
+/-- A generalization of the fixed point lemma for normal functions: any family of normal functions
+    has an unbounded set of common fixed points. -/
+@[deprecated not_bddAbove_fp_bfamily (since := "2024-09-20")]
 theorem fp_bfamily_unbounded (H : ∀ i hi, IsNormal (f i hi)) :
     (⋂ (i) (hi), Function.fixedPoints (f i hi)).Unbounded (· < ·) := fun a =>
   ⟨nfpBFamily.{u, v} _ f a, by
@@ -343,7 +365,7 @@ theorem fp_iff_derivBFamily (H : ∀ i hi, IsNormal (f i hi)) {a} :
 /-- For a family of normal functions, `Ordinal.derivBFamily` enumerates the common fixed points. -/
 theorem derivBFamily_eq_enumOrd (H : ∀ i hi, IsNormal (f i hi)) :
     derivBFamily.{u, v} o f = enumOrd (⋂ (i) (hi), Function.fixedPoints (f i hi)) := by
-  rw [← eq_enumOrd _ (fp_bfamily_unbounded.{u, v} H)]
+  rw [eq_comm, eq_enumOrd _ (not_bddAbove_fp_bfamily H)]
   use (derivBFamily_isNormal f).strictMono
   rw [Set.range_eq_iff]
   refine ⟨fun a => Set.mem_iInter₂.2 fun i hi => derivBFamily_fp (H i hi) a, fun a ha => ?_⟩
@@ -368,6 +390,20 @@ def nfp (f : Ordinal → Ordinal) : Ordinal → Ordinal :=
 theorem nfp_eq_nfpFamily (f : Ordinal → Ordinal) : nfp f = nfpFamily fun _ : Unit => f :=
   rfl
 
+theorem iSup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) (a : Ordinal.{u}) :
+    ⨆ n : ℕ, f^[n] a = nfp f a := by
+  apply le_antisymm
+  · rw [Ordinal.iSup_le_iff]
+    intro n
+    rw [← List.length_replicate n Unit.unit, ← List.foldr_const f a]
+    exact Ordinal.le_iSup _ _
+  · apply Ordinal.iSup_le
+    intro l
+    rw [List.foldr_const f a l]
+    exact Ordinal.le_iSup _ _
+
+set_option linter.deprecated false in
+@[deprecated (since := "2024-08-27")]
 theorem sup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) (a : Ordinal.{u}) :
     (sup fun n : ℕ => f^[n] a) = nfp f a := by
   refine le_antisymm ?_ (sup_le fun l => ?_)
@@ -379,28 +415,28 @@ theorem sup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) (a : Ordinal.{u}) :
     exact le_sup _ _
 
 theorem iterate_le_nfp (f a n) : f^[n] a ≤ nfp f a := by
-  rw [← sup_iterate_eq_nfp]
-  exact le_sup _ n
+  rw [← iSup_iterate_eq_nfp]
+  exact Ordinal.le_iSup _ n
 
 theorem le_nfp (f a) : a ≤ nfp f a :=
   iterate_le_nfp f a 0
 
 theorem lt_nfp {a b} : a < nfp f b ↔ ∃ n, a < f^[n] b := by
-  rw [← sup_iterate_eq_nfp]
-  exact lt_sup
+  rw [← iSup_iterate_eq_nfp]
+  exact Ordinal.lt_iSup
 
 theorem nfp_le_iff {a b} : nfp f a ≤ b ↔ ∀ n, f^[n] a ≤ b := by
-  rw [← sup_iterate_eq_nfp]
-  exact sup_le_iff
+  rw [← iSup_iterate_eq_nfp]
+  exact Ordinal.iSup_le_iff
 
 theorem nfp_le {a b} : (∀ n, f^[n] a ≤ b) → nfp f a ≤ b :=
   nfp_le_iff.2
 
 @[simp]
-theorem nfp_id : nfp id = id :=
-  funext fun a => by
-    simp_rw [← sup_iterate_eq_nfp, iterate_id]
-    exact sup_const a
+theorem nfp_id : nfp id = id := by
+  ext
+  simp_rw [← iSup_iterate_eq_nfp, iterate_id]
+  exact ciSup_const
 
 theorem nfp_monotone (hf : Monotone f) : Monotone (nfp f) :=
   nfpFamily_monotone fun _ => hf
@@ -420,13 +456,21 @@ theorem IsNormal.nfp_fp {f} (H : IsNormal f) : ∀ a, f (nfp f a) = nfp f a :=
   @nfpFamily_fp Unit (fun _ => f) Unit.unit H
 
 theorem IsNormal.apply_le_nfp {f} (H : IsNormal f) {a b} : f b ≤ nfp f a ↔ b ≤ nfp f a :=
-  ⟨le_trans (H.self_le _), fun h => by simpa only [H.nfp_fp] using H.le_iff.2 h⟩
+  ⟨H.le_apply.trans, fun h => by simpa only [H.nfp_fp] using H.le_iff.2 h⟩
 
 theorem nfp_eq_self {f : Ordinal → Ordinal} {a} (h : f a = a) : nfp f a = a :=
   nfpFamily_eq_self fun _ => h
 
 /-- The fixed point lemma for normal functions: any normal function has an unbounded set of
 fixed points. -/
+theorem not_bddAbove_fp (H : IsNormal f) : ¬ BddAbove (Function.fixedPoints f) := by
+  convert not_bddAbove_fp_family fun _ : Unit => H
+  exact (Set.iInter_const _).symm
+
+set_option linter.deprecated false in
+/-- The fixed point lemma for normal functions: any normal function has an unbounded set of
+fixed points. -/
+@[deprecated not_bddAbove_fp (since := "2024-09-20")]
 theorem fp_unbounded (H : IsNormal f) : (Function.fixedPoints f).Unbounded (· < ·) := by
   convert fp_family_unbounded fun _ : Unit => H
   exact (Set.iInter_const _).symm
@@ -477,8 +521,9 @@ theorem deriv_eq_id_of_nfp_eq_id {f : Ordinal → Ordinal} (h : nfp f = id) : de
   (IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) IsNormal.refl).2 <| by simp [h]
 
 theorem nfp_zero_left (a) : nfp 0 a = a := by
-  rw [← sup_iterate_eq_nfp]
-  apply (sup_le fun n => ?_).antisymm (le_sup (fun n => 0^[n] a) 0)
+  rw [← iSup_iterate_eq_nfp]
+  apply (Ordinal.iSup_le ?_).antisymm (Ordinal.le_iSup _ 0)
+  intro n
   induction' n with n _
   · rfl
   · rw [Function.iterate_succ']
@@ -502,20 +547,23 @@ end
 /-! ### Fixed points of addition -/
 
 @[simp]
-theorem nfp_add_zero (a) : nfp (a + ·) 0 = a * omega := by
-  simp_rw [← sup_iterate_eq_nfp, ← sup_mul_nat]
+theorem nfp_add_zero (a) : nfp (a + ·) 0 = a * ω := by
+  simp_rw [← iSup_iterate_eq_nfp, ← iSup_mul_nat]
   congr; funext n
   induction' n with n hn
   · rw [Nat.cast_zero, mul_zero, iterate_zero_apply]
   · rw [iterate_succ_apply', Nat.add_comm, Nat.cast_add, Nat.cast_one, mul_one_add, hn]
 
-theorem nfp_add_eq_mul_omega {a b} (hba : b ≤ a * omega) : nfp (a + ·) b = a * omega := by
+theorem nfp_add_eq_mul_omega0 {a b} (hba : b ≤ a * ω) : nfp (a + ·) b = a * ω := by
   apply le_antisymm (nfp_le_fp (add_isNormal a).monotone hba _)
   · rw [← nfp_add_zero]
     exact nfp_monotone (add_isNormal a).monotone (Ordinal.zero_le b)
-  · dsimp; rw [← mul_one_add, one_add_omega]
+  · dsimp; rw [← mul_one_add, one_add_omega0]
+
+@[deprecated (since := "2024-09-30")]
+alias nfp_add_eq_mul_omega := nfp_add_eq_mul_omega0
 
-theorem add_eq_right_iff_mul_omega_le {a b : Ordinal} : a + b = b ↔ a * omega ≤ b := by
+theorem add_eq_right_iff_mul_omega0_le {a b : Ordinal} : a + b = b ↔ a * ω ≤ b := by
   refine ⟨fun h => ?_, fun h => ?_⟩
   · rw [← nfp_add_zero a, ← deriv_zero_right]
     cases' (add_isNormal a).fp_iff_deriv.1 h with c hc
@@ -523,26 +571,35 @@ theorem add_eq_right_iff_mul_omega_le {a b : Ordinal} : a + b = b ↔ a * omega
     exact (deriv_isNormal _).monotone (Ordinal.zero_le _)
   · have := Ordinal.add_sub_cancel_of_le h
     nth_rw 1 [← this]
-    rwa [← add_assoc, ← mul_one_add, one_add_omega]
+    rwa [← add_assoc, ← mul_one_add, one_add_omega0]
+
+@[deprecated (since := "2024-09-30")]
+alias add_eq_right_iff_mul_omega_le := add_eq_right_iff_mul_omega0_le
 
-theorem add_le_right_iff_mul_omega_le {a b : Ordinal} : a + b ≤ b ↔ a * omega ≤ b := by
-  rw [← add_eq_right_iff_mul_omega_le]
+theorem add_le_right_iff_mul_omega0_le {a b : Ordinal} : a + b ≤ b ↔ a * ω ≤ b := by
+  rw [← add_eq_right_iff_mul_omega0_le]
   exact (add_isNormal a).le_iff_eq
 
-theorem deriv_add_eq_mul_omega_add (a b : Ordinal.{u}) : deriv (a + ·) b = a * omega + b := by
+@[deprecated (since := "2024-09-30")]
+alias add_le_right_iff_mul_omega_le := add_le_right_iff_mul_omega0_le
+
+theorem deriv_add_eq_mul_omega0_add (a b : Ordinal.{u}) : deriv (a + ·) b = a * ω + b := by
   revert b
   rw [← funext_iff, IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (add_isNormal _)]
   refine ⟨?_, fun a h => ?_⟩
   · rw [deriv_zero_right, add_zero]
     exact nfp_add_zero a
   · rw [deriv_succ, h, add_succ]
-    exact nfp_eq_self (add_eq_right_iff_mul_omega_le.2 ((le_add_right _ _).trans (le_succ _)))
+    exact nfp_eq_self (add_eq_right_iff_mul_omega0_le.2 ((le_add_right _ _).trans (le_succ _)))
+
+@[deprecated (since := "2024-09-30")]
+alias deriv_add_eq_mul_omega_add := deriv_add_eq_mul_omega0_add
 
 /-! ### Fixed points of multiplication -/
 
 @[simp]
-theorem nfp_mul_one {a : Ordinal} (ha : 0 < a) : nfp (a * ·) 1 = (a^omega) := by
-  rw [← sup_iterate_eq_nfp, ← sup_opow_nat ha]
+theorem nfp_mul_one {a : Ordinal} (ha : 0 < a) : nfp (a * ·) 1 = (a ^ ω) := by
+  rw [← iSup_iterate_eq_nfp, ← iSup_pow ha]
   congr
   funext n
   induction' n with n hn
@@ -556,22 +613,25 @@ theorem nfp_mul_zero (a : Ordinal) : nfp (a * ·) 0 = 0 := by
   induction' n with n hn; · rfl
   dsimp only; rwa [iterate_succ_apply, mul_zero]
 
-theorem nfp_mul_eq_opow_omega {a b : Ordinal} (hb : 0 < b) (hba : b ≤ (a^omega)) :
-    nfp (a * ·) b = (a^omega.{u}) := by
+theorem nfp_mul_eq_opow_omega0 {a b : Ordinal} (hb : 0 < b) (hba : b ≤ (a ^ ω)) :
+    nfp (a * ·) b = (a ^ (ω : Ordinal.{u})) := by
   rcases eq_zero_or_pos a with ha | ha
-  · rw [ha, zero_opow omega_ne_zero] at hba ⊢
+  · rw [ha, zero_opow omega0_ne_zero] at hba ⊢
     simp_rw [Ordinal.le_zero.1 hba, zero_mul]
     exact nfp_zero_left 0
   apply le_antisymm
   · apply nfp_le_fp (mul_isNormal ha).monotone hba
-    rw [← opow_one_add, one_add_omega]
+    rw [← opow_one_add, one_add_omega0]
   rw [← nfp_mul_one ha]
   exact nfp_monotone (mul_isNormal ha).monotone (one_le_iff_pos.2 hb)
 
-theorem eq_zero_or_opow_omega_le_of_mul_eq_right {a b : Ordinal} (hab : a * b = b) :
-    b = 0 ∨ (a^omega.{u}) ≤ b := by
+@[deprecated (since := "2024-09-30")]
+alias nfp_mul_eq_opow_omega := nfp_mul_eq_opow_omega0
+
+theorem eq_zero_or_opow_omega0_le_of_mul_eq_right {a b : Ordinal} (hab : a * b = b) :
+    b = 0 ∨ (a ^ (ω : Ordinal.{u})) ≤ b := by
   rcases eq_zero_or_pos a with ha | ha
-  · rw [ha, zero_opow omega_ne_zero]
+  · rw [ha, zero_opow omega0_ne_zero]
     exact Or.inr (Ordinal.zero_le b)
   rw [or_iff_not_imp_left]
   intro hb
@@ -579,51 +639,66 @@ theorem eq_zero_or_opow_omega_le_of_mul_eq_right {a b : Ordinal} (hab : a * b =
   rw [← Ne, ← one_le_iff_ne_zero] at hb
   exact nfp_le_fp (mul_isNormal ha).monotone hb (le_of_eq hab)
 
-theorem mul_eq_right_iff_opow_omega_dvd {a b : Ordinal} : a * b = b ↔ (a^omega) ∣ b := by
+@[deprecated (since := "2024-09-30")]
+alias eq_zero_or_opow_omega_le_of_mul_eq_right := eq_zero_or_opow_omega0_le_of_mul_eq_right
+
+theorem mul_eq_right_iff_opow_omega0_dvd {a b : Ordinal} : a * b = b ↔ (a ^ ω) ∣ b := by
   rcases eq_zero_or_pos a with ha | ha
-  · rw [ha, zero_mul, zero_opow omega_ne_zero, zero_dvd_iff]
+  · rw [ha, zero_mul, zero_opow omega0_ne_zero, zero_dvd_iff]
     exact eq_comm
   refine ⟨fun hab => ?_, fun h => ?_⟩
   · rw [dvd_iff_mod_eq_zero]
-    rw [← div_add_mod b (a^omega), mul_add, ← mul_assoc, ← opow_one_add, one_add_omega,
+    rw [← div_add_mod b (a ^ ω), mul_add, ← mul_assoc, ← opow_one_add, one_add_omega0,
       add_left_cancel] at hab
-    cases' eq_zero_or_opow_omega_le_of_mul_eq_right hab with hab hab
+    cases' eq_zero_or_opow_omega0_le_of_mul_eq_right hab with hab hab
     · exact hab
-    refine (not_lt_of_le hab (mod_lt b (opow_ne_zero omega ?_))).elim
+    refine (not_lt_of_le hab (mod_lt b (opow_ne_zero ω ?_))).elim
     rwa [← Ordinal.pos_iff_ne_zero]
   cases' h with c hc
-  rw [hc, ← mul_assoc, ← opow_one_add, one_add_omega]
+  rw [hc, ← mul_assoc, ← opow_one_add, one_add_omega0]
+
+@[deprecated (since := "2024-09-30")]
+alias mul_eq_right_iff_opow_omega_dvd := mul_eq_right_iff_opow_omega0_dvd
 
-theorem mul_le_right_iff_opow_omega_dvd {a b : Ordinal} (ha : 0 < a) :
-    a * b ≤ b ↔ (a^omega) ∣ b := by
-  rw [← mul_eq_right_iff_opow_omega_dvd]
+theorem mul_le_right_iff_opow_omega0_dvd {a b : Ordinal} (ha : 0 < a) :
+    a * b ≤ b ↔ (a ^ ω) ∣ b := by
+  rw [← mul_eq_right_iff_opow_omega0_dvd]
   exact (mul_isNormal ha).le_iff_eq
 
-theorem nfp_mul_opow_omega_add {a c : Ordinal} (b) (ha : 0 < a) (hc : 0 < c) (hca : c ≤ (a^omega)) :
-    nfp (a * ·) ((a^omega) * b + c) = (a^omega.{u}) * succ b := by
+@[deprecated (since := "2024-09-30")]
+alias mul_le_right_iff_opow_omega_dvd := mul_le_right_iff_opow_omega0_dvd
+
+theorem nfp_mul_opow_omega0_add {a c : Ordinal} (b) (ha : 0 < a) (hc : 0 < c)
+    (hca : c ≤ a ^ ω) : nfp (a * ·) (a ^ ω * b + c) = (a ^ (ω : Ordinal.{u})) * succ b := by
   apply le_antisymm
   · apply nfp_le_fp (mul_isNormal ha).monotone
     · rw [mul_succ]
       apply add_le_add_left hca
-    · dsimp only; rw [← mul_assoc, ← opow_one_add, one_add_omega]
-  · cases' mul_eq_right_iff_opow_omega_dvd.1 ((mul_isNormal ha).nfp_fp ((a^omega) * b + c)) with
+    · dsimp only; rw [← mul_assoc, ← opow_one_add, one_add_omega0]
+  · cases' mul_eq_right_iff_opow_omega0_dvd.1 ((mul_isNormal ha).nfp_fp ((a ^ ω) * b + c)) with
       d hd
     rw [hd]
     apply mul_le_mul_left'
-    have := le_nfp (Mul.mul a) ((a^omega) * b + c)
+    have := le_nfp (a * ·) (a ^ ω * b + c)
     erw [hd] at this
-    have := (add_lt_add_left hc ((a^omega) * b)).trans_le this
-    rw [add_zero, mul_lt_mul_iff_left (opow_pos omega ha)] at this
+    have := (add_lt_add_left hc (a ^ ω * b)).trans_le this
+    rw [add_zero, mul_lt_mul_iff_left (opow_pos ω ha)] at this
     rwa [succ_le_iff]
 
-theorem deriv_mul_eq_opow_omega_mul {a : Ordinal.{u}} (ha : 0 < a) (b) :
-    deriv (a * ·) b = (a^omega) * b := by
+@[deprecated (since := "2024-09-30")]
+alias nfp_mul_opow_omega_add := nfp_mul_opow_omega0_add
+
+theorem deriv_mul_eq_opow_omega0_mul {a : Ordinal.{u}} (ha : 0 < a) (b) :
+    deriv (a * ·) b = (a ^ ω) * b := by
   revert b
   rw [← funext_iff,
-    IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (mul_isNormal (opow_pos omega ha))]
+    IsNormal.eq_iff_zero_and_succ (deriv_isNormal _) (mul_isNormal (opow_pos ω ha))]
   refine ⟨?_, fun c h => ?_⟩
   · dsimp only; rw [deriv_zero_right, nfp_mul_zero, mul_zero]
   · rw [deriv_succ, h]
-    exact nfp_mul_opow_omega_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha))
+    exact nfp_mul_opow_omega0_add c ha zero_lt_one (one_le_iff_pos.2 (opow_pos _ ha))
+
+@[deprecated (since := "2024-09-30")]
+alias deriv_mul_eq_opow_omega_mul := deriv_mul_eq_opow_omega0_mul
 
 end Ordinal
diff --git a/Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean b/Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
index 22ea84fdfe087..97272de665f4f 100644
--- a/Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
+++ b/Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean
@@ -8,7 +8,7 @@ import Mathlib.SetTheory.Ordinal.Arithmetic
 /-!
 # Ordinal Approximants for the Fixed points on complete lattices
 
-This file sets up the ordinal approximation theory of fixed points
+This file sets up the ordinal-indexed approximation theory of fixed points
 of a monotone function in a complete lattice [Cousot1979].
 The proof follows loosely the one from [Echenique2005].
 
@@ -17,15 +17,15 @@ ordinals from mathlib. It still allows an approximation scheme indexed over the
 
 ## Main definitions
 
-* `OrdinalApprox.lfpApprox`: The ordinal approximation of the least fixed point
-  greater or equal then an initial value of a bundled monotone function.
-* `OrdinalApprox.gfpApprox`: The ordinal approximation of the greatest fixed point
-  less or equal then an initial value of a bundled monotone function.
+* `OrdinalApprox.lfpApprox`: The ordinal-indexed approximation of the least fixed point
+  greater or equal than an initial value of a bundled monotone function.
+* `OrdinalApprox.gfpApprox`: The ordinal-indexed approximation of the greatest fixed point
+  less or equal than an initial value of a bundled monotone function.
 
 ## Main theorems
-* `OrdinalApprox.lfp_mem_range_lfpApprox`: The approximation of
+* `OrdinalApprox.lfp_mem_range_lfpApprox`: The ordinal-indexed approximation of
   the least fixed point eventually reaches the least fixed point
-* `OrdinalApprox.gfp_mem_range_gfpApprox`: The approximation of
+* `OrdinalApprox.gfp_mem_range_gfpApprox`: The ordinal-indexed approximation of
   the greatest fixed point eventually reaches the greatest fixed point
 
 ## References
@@ -50,12 +50,10 @@ theorem not_injective_limitation_set : ¬ InjOn g (Iio (ord <| succ #α)) := by
   have h := lift_mk_le_lift_mk_of_injective <| injOn_iff_injective.1 h_inj
   have mk_initialSeg_subtype :
       #(Iio (ord <| succ #α)) = lift.{u + 1} (succ #α) := by
-    simpa only [coe_setOf, card_typein, card_ord] using mk_initialSeg (ord <| succ #α)
+    simpa only [coe_setOf, card_typein, card_ord] using mk_Iio_ordinal (ord <| succ #α)
   rw [mk_initialSeg_subtype, lift_lift, lift_le] at h
   exact not_le_of_lt (Order.lt_succ #α) h
 
-
-
 end Cardinal
 
 namespace OrdinalApprox
@@ -67,14 +65,17 @@ variable [CompleteLattice α] (f : α →o α) (x : α)
 open Function fixedPoints Cardinal Order OrderHom
 
 set_option linter.unusedVariables false in
-/-- Ordinal approximants of the least fixed point greater then an initial value x -/
+/-- The ordinal-indexed sequence approximating the least fixed point greater than
+an initial value `x`. It is defined in such a way that we have `lfpApprox 0 x = x` and
+`lfpApprox a x = ⨆ b < a, f (lfpApprox b x)`. -/
 def lfpApprox (a : Ordinal.{u}) : α :=
   sSup ({ f (lfpApprox b) | (b : Ordinal) (h : b < a) } ∪ {x})
 termination_by a
 decreasing_by exact h
 
 theorem lfpApprox_monotone : Monotone (lfpApprox f x) := by
-  unfold Monotone; intros a b h; unfold lfpApprox
+  intros a b h
+  rw [lfpApprox, lfpApprox]
   refine sSup_le_sSup ?h
   apply sup_le_sup_right
   simp only [exists_prop, Set.le_eq_subset, Set.setOf_subset_setOf, forall_exists_index, and_imp,
@@ -84,14 +85,14 @@ theorem lfpApprox_monotone : Monotone (lfpApprox f x) := by
   exact ⟨lt_of_lt_of_le h' h, rfl⟩
 
 theorem le_lfpApprox {a : Ordinal} : x ≤ lfpApprox f x a := by
-  unfold lfpApprox
+  rw [lfpApprox]
   apply le_sSup
   simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, true_or]
 
 theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
     lfpApprox f x (a+1) = f (lfpApprox f x a) := by
   apply le_antisymm
-  · conv => left; unfold lfpApprox
+  · conv => left; rw [lfpApprox]
     apply sSup_le
     simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop, Set.union_singleton,
       Set.mem_insert_iff, Set.mem_setOf_eq, forall_eq_or_imp, forall_exists_index, and_imp,
@@ -102,7 +103,7 @@ theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
       exact le_lfpApprox f x
     · intros a' h
       apply f.2; apply lfpApprox_monotone; exact h
-  · conv => right; unfold lfpApprox
+  · conv => right; rw [lfpApprox]
     apply le_sSup
     simp only [Ordinal.add_one_eq_succ, lt_succ_iff, exists_prop]
     rw [Set.mem_union]
@@ -110,14 +111,46 @@ theorem lfpApprox_add_one (h : x ≤ f x) (a : Ordinal) :
     simp only [Set.mem_setOf_eq]
     use a
 
-/-- The ordinal approximants of the least fixed point are stabilizing
-  when reaching a fixed point of f -/
+theorem lfpApprox_mono_left : Monotone (lfpApprox : (α →o α) → _) := by
+  intro f g h x a
+  induction a using Ordinal.induction with
+  | h i ih =>
+    rw [lfpApprox, lfpApprox]
+    apply sSup_le
+    simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, sSup_insert,
+      forall_eq_or_imp, le_sup_left, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
+      true_and]
+    intro i' h_lt
+    apply le_sup_of_le_right
+    apply le_sSup_of_le
+    · use i'
+    · apply le_trans (h _)
+      simp only [OrderHom.toFun_eq_coe]
+      exact g.monotone (ih i' h_lt)
+
+theorem lfpApprox_mono_mid : Monotone (lfpApprox f) := by
+  intro x₁ x₂ h a
+  induction a using Ordinal.induction with
+  | h i ih =>
+    rw [lfpApprox, lfpApprox]
+    apply sSup_le
+    simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq, sSup_insert,
+      forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
+    constructor
+    · exact le_sup_of_le_left h
+    · intro i' h_i'
+      apply le_sup_of_le_right
+      apply le_sSup_of_le
+      · use i'
+      · exact f.monotone (ih i' h_i')
+
+/-- The approximations of the least fixed point stabilize at a fixed point of `f` -/
 theorem lfpApprox_eq_of_mem_fixedPoints {a b : Ordinal} (h_init : x ≤ f x) (h_ab : a ≤ b)
     (h : lfpApprox f x a ∈ fixedPoints f) : lfpApprox f x b = lfpApprox f x a := by
   rw [mem_fixedPoints_iff] at h
   induction b using Ordinal.induction with | h b IH =>
   apply le_antisymm
-  · conv => left; unfold lfpApprox
+  · conv => left; rw [lfpApprox]
     apply sSup_le
     simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq,
       forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
@@ -131,8 +164,8 @@ theorem lfpApprox_eq_of_mem_fixedPoints {a b : Ordinal} (h_init : x ≤ f x) (h_
     · rw [IH a' ha'b (le_of_not_lt haa), h]
   · exact lfpApprox_monotone f x h_ab
 
-/-- There are distinct ordinals smaller than the successor of the domains cardinals
-  with equal value -/
+/-- There are distinct indices smaller than the successor of the domain's cardinality
+yielding the same value -/
 theorem exists_lfpApprox_eq_lfpApprox : ∃ a < ord <| succ #α, ∃ b < ord <| succ #α,
     a ≠ b ∧ lfpApprox f x a = lfpApprox f x b := by
   have h_ninj := not_injective_limitation_set <| lfpApprox f x
@@ -144,8 +177,8 @@ theorem exists_lfpApprox_eq_lfpApprox : ∃ a < ord <| succ #α, ∃ b < ord <|
   · intro h_eq; rw [Subtype.coe_inj] at h_eq; exact h_nab h_eq
   · exact h_fab
 
-/-- If there are distinct ordinals with equal value then
-  every value succeding the smaller ordinal are fixed points -/
+/-- If the sequence of ordinal-indexed approximations takes a value twice,
+then it actually stabilised at that value. -/
 lemma lfpApprox_mem_fixedPoints_of_eq {a b c : Ordinal}
     (h_init : x ≤ f x) (h_ab : a < b) (h_ac : a ≤ c) (h_fab : lfpApprox f x a = lfpApprox f x b) :
     lfpApprox f x c ∈ fixedPoints f := by
@@ -159,7 +192,7 @@ lemma lfpApprox_mem_fixedPoints_of_eq {a b c : Ordinal}
   · exact h_ac
   · exact lfpApprox_mem_fixedPoint
 
-/-- A fixed point of f is reached after the successor of the domains cardinality -/
+/-- The approximation at the index of the successor of the domain's cardinality is a fixed point -/
 theorem lfpApprox_ord_mem_fixedPoint (h_init : x ≤ f x) :
     lfpApprox f x (ord <| succ #α) ∈ fixedPoints f := by
   let ⟨a, h_a, b, h_b, h_nab, h_fab⟩ := exists_lfpApprox_eq_lfpApprox f x
@@ -171,13 +204,13 @@ theorem lfpApprox_ord_mem_fixedPoint (h_init : x ≤ f x) :
     exact lfpApprox_mem_fixedPoints_of_eq f x h_init
       (h_nab.symm.lt_of_le h_ba) (le_of_lt h_b) (h_fab.symm)
 
-/-- Every value of the ordinal approximants are less or equal than every fixed point of f greater
-  then the initial value -/
+/-- Every value of the approximation is less or equal than every fixed point of `f`
+greater or equal than the initial value -/
 theorem lfpApprox_le_of_mem_fixedPoints {a : α}
     (h_a : a ∈ fixedPoints f) (h_le_init : x ≤ a) (i : Ordinal) : lfpApprox f x i ≤ a := by
   induction i using Ordinal.induction with
   | h i IH =>
-    unfold lfpApprox
+    rw [lfpApprox]
     apply sSup_le
     simp only [exists_prop]
     intro y h_y
@@ -186,13 +219,13 @@ theorem lfpApprox_le_of_mem_fixedPoints {a : α}
     | inl h_y =>
       let ⟨j, h_j_lt, h_j⟩ := h_y
       rw [← h_j, ← h_a]
-      apply f.monotone'
-      exact IH j h_j_lt
+      exact f.monotone' (IH j h_j_lt)
     | inr h_y =>
       rw [h_y]
       exact h_le_init
 
-/-- The least fixed point of f is reached after the successor of the domains cardinality -/
+/-- The approximation sequence converges at the successor of the domain's cardinality
+to the least fixed point if starting from `⊥` -/
 theorem lfpApprox_ord_eq_lfp : lfpApprox f ⊥ (ord <| succ #α) = lfp f := by
   apply le_antisymm
   · have h_lfp : ∃ y : fixedPoints f, lfp f = y := by use ⊥; exact rfl
@@ -204,13 +237,15 @@ theorem lfpApprox_ord_eq_lfp : lfpApprox f ⊥ (ord <| succ #α) = lfp f := by
     let ⟨x, h_x⟩ := h_fix; rw [h_x]
     exact lfp_le_fixed f x.prop
 
-/-- Some ordinal approximation of the least fixed point is the least fixed point. -/
+/-- Some approximation of the least fixed point starting from `⊥` is the least fixed point. -/
 theorem lfp_mem_range_lfpApprox : lfp f ∈ Set.range (lfpApprox f ⊥) := by
   use ord <| succ #α
   exact lfpApprox_ord_eq_lfp f
 
 set_option linter.unusedVariables false in
-/-- Ordinal approximants of the greatest fixed point -/
+/-- The ordinal-indexed sequence approximating the greatest fixed point greater than
+an initial value `x`. It is defined in such a way that we have `gfpApprox 0 x = x` and
+`gfpApprox a x = ⨅ b < a, f (lfpApprox b x)`. -/
 def gfpApprox (a : Ordinal.{u}) : α :=
   sInf ({ f (gfpApprox b) | (b : Ordinal) (h : b < a) } ∪ {x})
 termination_by a
@@ -230,34 +265,42 @@ theorem gfpApprox_add_one (h : f x ≤ x) (a : Ordinal) :
     gfpApprox f x (a+1) = f (gfpApprox f x a) :=
   lfpApprox_add_one (OrderHom.dual f) x h a
 
-/-- The ordinal approximants of the least fixed point are stabilizing
-  when reaching a fixed point of f -/
+theorem gfpApprox_mono_left : Monotone (gfpApprox : (α →o α) → _) := by
+  intro f g h
+  have : OrderHom.dual g ≤ OrderHom.dual f := h
+  exact lfpApprox_mono_left this
+
+theorem gfpApprox_mono_mid : Monotone (gfpApprox f) :=
+  fun _ _ h => lfpApprox_mono_mid (OrderHom.dual f) h
+
+/-- The approximations of the greatest fixed point stabilize at a fixed point of `f` -/
 theorem gfpApprox_eq_of_mem_fixedPoints {a b : Ordinal} (h_init : f x ≤ x) (h_ab : a ≤ b)
     (h : gfpApprox f x a ∈ fixedPoints f) : gfpApprox f x b = gfpApprox f x a :=
   lfpApprox_eq_of_mem_fixedPoints (OrderHom.dual f) x h_init h_ab h
 
-/-- There are distinct ordinals smaller than the successor of the domains cardinals with
-  equal value -/
+/-- There are distinct indices smaller than the successor of the domain's cardinality
+yielding the same value -/
 theorem exists_gfpApprox_eq_gfpApprox : ∃ a < ord <| succ #α, ∃ b < ord <| succ #α,
     a ≠ b ∧ gfpApprox f x a = gfpApprox f x b :=
   exists_lfpApprox_eq_lfpApprox (OrderHom.dual f) x
 
-/-- A fixed point of f is reached after the successor of the domains cardinality -/
+/-- The approximation at the index of the successor of the domain's cardinality is a fixed point -/
 lemma gfpApprox_ord_mem_fixedPoint (h_init : f x ≤ x) :
     gfpApprox f x (ord <| succ #α) ∈ fixedPoints f :=
   lfpApprox_ord_mem_fixedPoint (OrderHom.dual f) x h_init
 
-/-- Every value of the ordinal approximants are greater or equal than every fixed point of f
-  that is smaller then the initial value -/
+/-- Every value of the approximation is greater or equal than every fixed point of `f`
+less or equal than the initial value -/
 lemma le_gfpApprox_of_mem_fixedPoints {a : α}
     (h_a : a ∈ fixedPoints f) (h_le_init : a ≤ x) (i : Ordinal) : a ≤ gfpApprox f x i :=
   lfpApprox_le_of_mem_fixedPoints (OrderHom.dual f) x h_a h_le_init i
 
-/-- The greatest fixed point of f is reached after the successor of the domains cardinality -/
+/-- The approximation sequence converges at the successor of the domain's cardinality
+to the greatest fixed point if starting from `⊥` -/
 theorem gfpApprox_ord_eq_gfp : gfpApprox f ⊤ (ord <| succ #α) = gfp f :=
   lfpApprox_ord_eq_lfp (OrderHom.dual f)
 
-/-- Some ordinal approximation of the greatest fixed point is the greatest fixed point. -/
+/-- Some approximation of the least fixed point starting from `⊤` is the greatest fixed point. -/
 theorem gfp_mem_range_gfpApprox : gfp f ∈ Set.range (gfpApprox f ⊤) :=
   lfp_mem_range_lfpApprox (OrderHom.dual f)
 
diff --git a/Mathlib/SetTheory/Ordinal/NaturalOps.lean b/Mathlib/SetTheory/Ordinal/NaturalOps.lean
index 15a117be039a3..b901e17cfa608 100644
--- a/Mathlib/SetTheory/Ordinal/NaturalOps.lean
+++ b/Mathlib/SetTheory/Ordinal/NaturalOps.lean
@@ -27,13 +27,13 @@ The natural addition of two ordinals corresponds to adding their Cantor normal f
 polynomials in `ω`. Likewise, their natural multiplication corresponds to multiplying the Cantor
 normal forms as polynomials.
 
-# Implementation notes
+## Implementation notes
 
 Given the rich algebraic structure of these two operations, we choose to create a type synonym
 `NatOrdinal`, where we provide the appropriate instances. However, to avoid casting back and forth
 between both types, we attempt to prove and state most results on `Ordinal`.
 
-# Todo
+## Todo
 
 - Prove the characterizations of natural addition and multiplication in terms of the Cantor normal
   form.
@@ -54,8 +54,9 @@ def NatOrdinal : Type _ :=
   Ordinal deriving Zero, Inhabited, One, WellFoundedRelation
 
 instance NatOrdinal.linearOrder : LinearOrder NatOrdinal := {Ordinal.linearOrder with}
-
-instance NatOrdinal.succOrder : SuccOrder NatOrdinal := {Ordinal.succOrder with}
+instance NatOrdinal.instSuccOrder : SuccOrder NatOrdinal := {Ordinal.instSuccOrder with}
+instance NatOrdinal.orderBot : OrderBot NatOrdinal := {Ordinal.orderBot with}
+instance NatOrdinal.noMaxOrder : NoMaxOrder NatOrdinal := {Ordinal.noMaxOrder with}
 
 /-- The identity function between `Ordinal` and `NatOrdinal`. -/
 @[match_pattern]
@@ -87,7 +88,14 @@ instance : WellFoundedLT NatOrdinal :=
   Ordinal.wellFoundedLT
 
 instance : IsWellOrder NatOrdinal (· < ·) :=
-  Ordinal.isWellOrder
+  { }
+
+instance : ConditionallyCompleteLinearOrderBot NatOrdinal :=
+  WellFoundedLT.conditionallyCompleteLinearOrderBot _
+
+@[simp]
+theorem bot_eq_zero : ⊥ = 0 :=
+  rfl
 
 @[simp]
 theorem toOrdinal_zero : toOrdinal 0 = 0 :=
@@ -98,19 +106,19 @@ theorem toOrdinal_one : toOrdinal 1 = 1 :=
   rfl
 
 @[simp]
-theorem toOrdinal_eq_zero (a) : toOrdinal a = 0 ↔ a = 0 :=
+theorem toOrdinal_eq_zero {a} : toOrdinal a = 0 ↔ a = 0 :=
   Iff.rfl
 
 @[simp]
-theorem toOrdinal_eq_one (a) : toOrdinal a = 1 ↔ a = 1 :=
+theorem toOrdinal_eq_one {a} : toOrdinal a = 1 ↔ a = 1 :=
   Iff.rfl
 
 @[simp]
-theorem toOrdinal_max {a b : NatOrdinal} : toOrdinal (max a b) = max (toOrdinal a) (toOrdinal b) :=
+theorem toOrdinal_max (a b : NatOrdinal) : toOrdinal (max a b) = max (toOrdinal a) (toOrdinal b) :=
   rfl
 
 @[simp]
-theorem toOrdinal_min {a b : NatOrdinal} : toOrdinal (min a b) = min (toOrdinal a) (toOrdinal b) :=
+theorem toOrdinal_min (a b : NatOrdinal) : toOrdinal (min a b) = min (toOrdinal a) (toOrdinal b) :=
   rfl
 
 theorem succ_def (a : NatOrdinal) : succ a = toNatOrdinal (toOrdinal a + 1) :=
@@ -162,7 +170,7 @@ theorem toNatOrdinal_max (a b : Ordinal) :
 
 @[simp]
 theorem toNatOrdinal_min (a b : Ordinal) :
-    toNatOrdinal (linearOrder.min a b) = linearOrder.min (toNatOrdinal a) (toNatOrdinal b) :=
+    toNatOrdinal (min a b) = min (toNatOrdinal a) (toNatOrdinal b) :=
   rfl
 
 /-! We place the definitions of `nadd` and `nmul` before actually developing their API, as this
@@ -184,7 +192,7 @@ scoped[NaturalOps] infixl:65 " ♯ " => Ordinal.nadd
 open NaturalOps
 
 /-- Natural multiplication on ordinals `a ⨳ b`, also known as the Hessenberg product, is recursively
-defined as the least ordinal such that `a ⨳ b + a' ⨳ b'` is greater than `a' ⨳ b + a ⨳ b'` for all
+defined as the least ordinal such that `a ⨳ b ♯ a' ⨳ b'` is greater than `a' ⨳ b ♯ a ⨳ b'` for all
 `a' < a` and `b < b'`. In contrast to normal ordinal multiplication, it is commutative and
 distributive (over natural addition).
 
@@ -308,8 +316,8 @@ namespace NatOrdinal
 
 open Ordinal NaturalOps
 
-instance : Add NatOrdinal :=
-  ⟨nadd⟩
+instance : Add NatOrdinal := ⟨nadd⟩
+instance : SuccAddOrder NatOrdinal := ⟨fun x => (nadd_one x).symm⟩
 
 instance add_covariantClass_lt : CovariantClass NatOrdinal.{u} NatOrdinal.{u} (· + ·) (· < ·) :=
   ⟨fun a _ _ h => nadd_lt_nadd_left h a⟩
@@ -338,9 +346,9 @@ instance orderedCancelAddCommMonoid : OrderedCancelAddCommMonoid NatOrdinal :=
 instance addMonoidWithOne : AddMonoidWithOne NatOrdinal :=
   AddMonoidWithOne.unary
 
-@[simp]
-theorem add_one_eq_succ : ∀ a : NatOrdinal, a + 1 = succ a :=
-  nadd_one
+@[deprecated Order.succ_eq_add_one (since := "2024-09-04")]
+theorem add_one_eq_succ (a : NatOrdinal) : a + 1 = succ a :=
+  (Order.succ_eq_add_one a).symm
 
 @[simp]
 theorem toOrdinal_cast_nat (n : ℕ) : toOrdinal n = n := by
@@ -438,7 +446,7 @@ theorem nmul_def (a b : Ordinal) :
     a ⨳ b = sInf {c | ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'} := by rw [nmul]
 
 /-- The set in the definition of `nmul` is nonempty. -/
-theorem nmul_nonempty (a b : Ordinal.{u}) :
+private theorem nmul_nonempty (a b : Ordinal.{u}) :
     {c : Ordinal.{u} | ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b'}.Nonempty :=
   ⟨_, fun _ ha _ hb => (lt_blsub₂.{u, u, u} _ ha hb).trans_le le_self_nadd⟩
 
diff --git a/Mathlib/SetTheory/Ordinal/Nimber.lean b/Mathlib/SetTheory/Ordinal/Nimber.lean
new file mode 100644
index 0000000000000..538d61ec4400d
--- /dev/null
+++ b/Mathlib/SetTheory/Ordinal/Nimber.lean
@@ -0,0 +1,361 @@
+/-
+Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Violeta Hernández Palacios
+-/
+import Mathlib.SetTheory.Ordinal.Arithmetic
+
+/-!
+# Nimbers
+
+The goal of this file is to define the field of nimbers, constructed as ordinals endowed with new
+arithmetical operations. The nim sum `a + b` is recursively defined as the least ordinal not equal
+to any `a' + b` or `a + b'` for `a' < a` and `b' < b`. The nim product `a * b` is likewise
+recursively defined as the least ordinal not equal to any `a' * b + a * b' + a' * b'` for `a' < a`
+and `b' < b`.
+
+Nim addition arises within the context of impartial games. By the Sprague-Grundy theorem, each
+impartial game is equivalent to some game of nim. If `x ≈ nim o₁` and `y ≈ nim o₂`, then
+`x + y ≈ nim (o₁ + o₂)`, where the ordinals are summed together as nimbers. Unfortunately, the
+nim product admits no such characterization.
+
+## Implementation notes
+
+The nimbers inherit the order from the ordinals - this makes working with minimum excluded values
+much more convenient. However, the fact that nimbers are of characteristic 2 prevents the order from
+interacting with the arithmetic in any nice way.
+
+To reduce API duplication, we opt not to implement operations on `Nimber` on `Ordinal`. The order
+isomorphisms `Ordinal.toNimber` and `Nimber.toOrdinal` allow us to cast between them whenever
+needed.
+
+## Todo
+
+- Add a `CharP 2` instance.
+- Define nim multiplication and prove nimbers are a commutative ring.
+- Define nim division and prove nimbers are a field.
+- Show the nimbers are algebraically closed.
+-/
+
+universe u v
+
+open Function Order
+
+noncomputable section
+
+/-! ### Basic casts between `Ordinal` and `Nimber` -/
+
+/-- A type synonym for ordinals with natural addition and multiplication. -/
+def Nimber : Type _ :=
+  Ordinal deriving Zero, Inhabited, One, WellFoundedRelation
+
+instance Nimber.linearOrder : LinearOrder Nimber := {Ordinal.linearOrder with}
+instance Nimber.succOrder : SuccOrder Nimber := {Ordinal.instSuccOrder with}
+instance Nimber.orderBot : OrderBot Nimber := {Ordinal.orderBot with}
+instance Nimber.noMaxOrder : NoMaxOrder Nimber := {Ordinal.noMaxOrder with}
+instance Nimber.zeroLEOneClass : ZeroLEOneClass Nimber := {Ordinal.zeroLEOneClass with}
+
+/-- The identity function between `Ordinal` and `Nimber`. -/
+@[match_pattern]
+def Ordinal.toNimber : Ordinal ≃o Nimber :=
+  OrderIso.refl _
+
+/-- The identity function between `Nimber` and `Ordinal`. -/
+@[match_pattern]
+def Nimber.toOrdinal : Nimber ≃o Ordinal :=
+  OrderIso.refl _
+
+namespace Nimber
+
+open Ordinal
+
+@[simp]
+theorem toOrdinal_symm_eq : Nimber.toOrdinal.symm = Ordinal.toNimber :=
+  rfl
+
+@[simp]
+theorem toOrdinal_toNimber (a : Nimber) :
+    Ordinal.toNimber (Nimber.toOrdinal a) = a := rfl
+
+theorem lt_wf : @WellFounded Nimber (· < ·) :=
+  Ordinal.lt_wf
+
+instance : WellFoundedLT Nimber :=
+  Ordinal.wellFoundedLT
+
+instance : IsWellOrder Nimber (· < ·) :=
+  { }
+
+instance : ConditionallyCompleteLinearOrderBot Nimber :=
+  WellFoundedLT.conditionallyCompleteLinearOrderBot _
+
+@[simp]
+theorem bot_eq_zero : ⊥ = 0 :=
+  rfl
+
+@[simp]
+theorem toOrdinal_zero : toOrdinal 0 = 0 :=
+  rfl
+
+@[simp]
+theorem toOrdinal_one : toOrdinal 1 = 1 :=
+  rfl
+
+@[simp]
+theorem toOrdinal_eq_zero (a) : toOrdinal a = 0 ↔ a = 0 :=
+  Iff.rfl
+
+@[simp]
+theorem toOrdinal_eq_one (a) : toOrdinal a = 1 ↔ a = 1 :=
+  Iff.rfl
+
+@[simp]
+theorem toOrdinal_max {a b : Nimber} : toOrdinal (max a b) = max (toOrdinal a) (toOrdinal b) :=
+  rfl
+
+@[simp]
+theorem toOrdinal_min {a b : Nimber} : toOrdinal (min a b) = min (toOrdinal a) (toOrdinal b) :=
+  rfl
+
+theorem succ_def (a : Nimber) : succ a = toNimber (toOrdinal a + 1) :=
+  rfl
+
+/-- A recursor for `Nimber`. Use as `induction x`. -/
+@[elab_as_elim, cases_eliminator, induction_eliminator]
+protected def rec {β : Nimber → Sort*} (h : ∀ a, β (toNimber a)) : ∀ a, β a := fun a =>
+  h (toOrdinal a)
+
+/-- `Ordinal.induction` but for `Nimber`. -/
+theorem induction {p : Nimber → Prop} : ∀ (i) (_ : ∀ j, (∀ k, k < j → p k) → p j), p i :=
+  Ordinal.induction
+
+protected theorem le_zero {a : Nimber} : a ≤ 0 ↔ a = 0 :=
+  Ordinal.le_zero
+
+protected theorem not_lt_zero (a : Nimber) : ¬ a < 0 :=
+  Ordinal.not_lt_zero a
+
+protected theorem pos_iff_ne_zero {a : Nimber} : 0 < a ↔ a ≠ 0 :=
+  Ordinal.pos_iff_ne_zero
+
+instance small_Iio (a : Nimber.{u}) : Small.{u} (Set.Iio a) := Ordinal.small_Iio a
+instance small_Iic (a : Nimber.{u}) : Small.{u} (Set.Iic a) := Ordinal.small_Iic a
+instance small_Ico (a b : Nimber.{u}) : Small.{u} (Set.Ico a b) := Ordinal.small_Ico a b
+instance small_Icc (a b : Nimber.{u}) : Small.{u} (Set.Icc a b) := Ordinal.small_Icc a b
+instance small_Ioo (a b : Nimber.{u}) : Small.{u} (Set.Ioo a b) := Ordinal.small_Ioo a b
+instance small_Ioc (a b : Nimber.{u}) : Small.{u} (Set.Ioc a b) := Ordinal.small_Ioc a b
+
+end Nimber
+
+theorem not_small_nimber : ¬ Small.{u} Nimber.{max u v} :=
+  not_small_ordinal
+
+namespace Ordinal
+
+variable {a b c : Ordinal.{u}}
+
+@[simp]
+theorem toNimber_symm_eq : toNimber.symm = Nimber.toOrdinal :=
+  rfl
+
+@[simp]
+theorem toNimber_toOrdinal (a : Ordinal) :  Nimber.toOrdinal (toNimber a) = a :=
+  rfl
+
+@[simp]
+theorem toNimber_zero : toNimber 0 = 0 :=
+  rfl
+
+@[simp]
+theorem toNimber_one : toNimber 1 = 1 :=
+  rfl
+
+@[simp]
+theorem toNimber_eq_zero (a) : toNimber a = 0 ↔ a = 0 :=
+  Iff.rfl
+
+@[simp]
+theorem toNimber_eq_one (a) : toNimber a = 1 ↔ a = 1 :=
+  Iff.rfl
+
+@[simp]
+theorem toNimber_max (a b : Ordinal) :
+    toNimber (max a b) = max (toNimber a) (toNimber b) :=
+  rfl
+
+@[simp]
+theorem toNimber_min (a b : Ordinal) :
+    toNimber (min a b) = min (toNimber a) (toNimber b) :=
+  rfl
+
+end Ordinal
+
+/-! ### Nimber addition -/
+
+namespace Nimber
+
+variable {a b c : Nimber.{u}}
+
+/-- Nimber addition is recursively defined so that `a + b` is the smallest number not equal to
+`a' + b` or `a + b'` for `a' < a` and `b' < b`. -/
+-- We write the binders like this so that the termination checker works.
+protected def add (a b : Nimber.{u}) : Nimber.{u} :=
+  sInf {x | (∃ a', ∃ (_ : a' < a), Nimber.add a' b = x) ∨
+    ∃ b', ∃ (_ : b' < b), Nimber.add a b' = x}ᶜ
+termination_by (a, b)
+
+instance : Add Nimber :=
+  ⟨Nimber.add⟩
+
+theorem add_def (a b : Nimber) :
+    a + b = sInf {x | (∃ a' < a, a' + b = x) ∨ ∃ b' < b, a + b' = x}ᶜ := by
+  change Nimber.add a b = _
+  rw [Nimber.add]
+  simp_rw [exists_prop]
+  rfl
+
+/-- The set in the definition of `Nimber.add` is nonempty. -/
+private theorem add_nonempty (a b : Nimber.{u}) :
+    {x | (∃ a' < a, a' + b = x) ∨ ∃ b' < b, a + b' = x}ᶜ.Nonempty := by
+  simp_rw [Set.nonempty_compl, Set.setOf_or, ← Set.mem_Iio, ← Set.image.eq_1]
+  apply_fun (fun a : Set Nimber ↦ Small.{u} a)
+  have : Small.{u} ↑((· + b) '' Set.Iio a ∪ (a + ·) '' Set.Iio b) := inferInstance
+  simpa [this, small_congr (Equiv.Set.univ _)] using not_small_nimber
+
+theorem exists_of_lt_add (h : c < a + b) : (∃ a' < a, a' + b = c) ∨ ∃ b' < b, a + b' = c := by
+  rw [add_def] at h
+  have := not_mem_of_lt_csInf h ⟨_, bot_mem_lowerBounds _⟩
+  rwa [Set.mem_compl_iff, not_not] at this
+
+theorem add_le_of_forall_ne (h₁ : ∀ a' < a, a' + b ≠ c) (h₂ : ∀ b' < b, a + b' ≠ c) :
+    a + b ≤ c := by
+  by_contra! h
+  have := exists_of_lt_add h
+  tauto
+
+private theorem add_ne_of_lt (a b : Nimber) :
+    (∀ a' < a, a' + b ≠ a + b) ∧ ∀ b' < b, a + b' ≠ a + b := by
+  have H := csInf_mem (add_nonempty a b)
+  rw [← add_def] at H
+  simpa using H
+
+instance : IsLeftCancelAdd Nimber := by
+  constructor
+  intro a b c h
+  apply le_antisymm <;>
+  apply le_of_not_lt
+  · exact fun hc => (add_ne_of_lt a b).2 c hc h.symm
+  · exact fun hb => (add_ne_of_lt a c).2 b hb h
+
+instance : IsRightCancelAdd Nimber := by
+  constructor
+  intro a b c h
+  apply le_antisymm <;>
+  apply le_of_not_lt
+  · exact fun hc => (add_ne_of_lt a b).1 c hc h.symm
+  · exact fun ha => (add_ne_of_lt c b).1 a ha h
+
+protected theorem add_comm (a b : Nimber) : a + b = b + a := by
+  rw [add_def, add_def]
+  simp_rw [or_comm]
+  congr! 7 <;>
+  (rw [and_congr_right_iff]; intro; rw [Nimber.add_comm])
+termination_by (a, b)
+
+theorem add_eq_zero {a b : Nimber} : a + b = 0 ↔ a = b := by
+  constructor <;>
+  intro hab
+  · obtain h | rfl | h := lt_trichotomy a b
+    · have ha : a + a = 0 := add_eq_zero.2 rfl
+      rwa [← ha, add_right_inj, eq_comm] at hab
+    · rfl
+    · have hb : b + b = 0 := add_eq_zero.2 rfl
+      rwa [← hb, add_left_inj] at hab
+  · rw [← Nimber.le_zero]
+    apply add_le_of_forall_ne <;>
+    simp_rw [ne_eq] <;>
+    intro x hx
+    · rw [add_eq_zero, ← hab]
+      exact hx.ne
+    · rw [add_eq_zero, hab]
+      exact hx.ne'
+termination_by (a, b)
+
+theorem add_ne_zero_iff : a + b ≠ 0 ↔ a ≠ b :=
+  add_eq_zero.not
+
+@[simp]
+theorem add_self (a : Nimber) : a + a = 0 :=
+  add_eq_zero.2 rfl
+
+protected theorem add_assoc (a b c : Nimber) : a + b + c = a + (b + c) := by
+  apply le_antisymm <;>
+  apply add_le_of_forall_ne <;>
+  intro x hx <;>
+  try obtain ⟨y, hy, rfl⟩ | ⟨y, hy, rfl⟩ := exists_of_lt_add hx
+  on_goal 1 => rw [Nimber.add_assoc y, add_ne_add_left]
+  on_goal 2 => rw [Nimber.add_assoc _ y, add_ne_add_right, add_ne_add_left]
+  on_goal 3 => rw [Nimber.add_assoc _ _ x, add_ne_add_right, add_ne_add_right]
+  on_goal 4 => rw [← Nimber.add_assoc x, add_ne_add_left, add_ne_add_left]
+  on_goal 5 => rw [← Nimber.add_assoc _ y, add_ne_add_left, add_ne_add_right]
+  on_goal 6 => rw [← Nimber.add_assoc _ _ y, add_ne_add_right]
+  all_goals apply ne_of_lt; assumption
+termination_by (a, b, c)
+
+protected theorem add_zero (a : Nimber) : a + 0 = a := by
+  apply le_antisymm
+  · apply add_le_of_forall_ne
+    · intro a' ha
+      rw [Nimber.add_zero]
+      exact ha.ne
+    · intro _ h
+      exact (Nimber.not_lt_zero _ h).elim
+  · -- by_contra! doesn't work for whatever reason.
+    by_contra h
+    replace h := lt_of_not_le h
+    have := Nimber.add_zero (a + 0)
+    rw [add_left_inj] at this
+    exact this.not_lt h
+termination_by a
+
+protected theorem zero_add (a : Nimber) : 0 + a = a := by
+  rw [Nimber.add_comm, Nimber.add_zero]
+
+instance : Neg Nimber :=
+  ⟨id⟩
+
+@[simp]
+protected theorem neg_eq (a : Nimber) : -a = a :=
+  rfl
+
+instance : AddCommGroupWithOne Nimber where
+  add_assoc := Nimber.add_assoc
+  add_zero := Nimber.add_zero
+  zero_add := Nimber.zero_add
+  nsmul := nsmulRec
+  zsmul := zsmulRec
+  neg_add_cancel := add_self
+  add_comm := Nimber.add_comm
+
+@[simp]
+theorem add_cancel_right (a b : Nimber) : a + b + b = a := by
+  rw [add_assoc, add_self, add_zero]
+
+@[simp]
+theorem add_cancel_left (a b : Nimber) : a + (a + b) = b := by
+  rw [← add_assoc, add_self, zero_add]
+
+theorem add_trichotomy {a b c : Nimber} (h : a + b + c ≠ 0) :
+    b + c < a ∨ c + a < b ∨ a + b < c := by
+  rw [← Nimber.pos_iff_ne_zero] at h
+  obtain ⟨x, hx, hx'⟩ | ⟨x, hx, hx'⟩ := exists_of_lt_add h <;>
+  rw [add_eq_zero] at hx'
+  · obtain ⟨x, hx, hx'⟩ | ⟨x, hx, hx'⟩ := exists_of_lt_add (hx' ▸ hx)
+    · rw [← hx', add_comm, add_cancel_right]
+      exact Or.inl hx
+    · rw [← hx', add_comm a, add_cancel_right]
+      exact Or.inr <| Or.inl hx
+  · rw [← hx'] at hx
+    exact Or.inr <| Or.inr hx
+
+end Nimber
diff --git a/Mathlib/SetTheory/Ordinal/Notation.lean b/Mathlib/SetTheory/Ordinal/Notation.lean
index 270577ceb3a74..fd54a141e2131 100644
--- a/Mathlib/SetTheory/Ordinal/Notation.lean
+++ b/Mathlib/SetTheory/Ordinal/Notation.lean
@@ -139,12 +139,15 @@ theorem repr_ofNat (n : ℕ) : repr (ofNat n) = n := by cases n <;> simp
 -- @[simp] -- Porting note (#10618): simp can prove this
 theorem repr_one : repr (ofNat 1) = (1 : ℕ) := repr_ofNat 1
 
-theorem omega_le_oadd (e n a) : ω ^ repr e ≤ repr (oadd e n a) := by
+theorem omega0_le_oadd (e n a) : ω ^ repr e ≤ repr (oadd e n a) := by
   refine le_trans ?_ (le_add_right _ _)
-  simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos (repr e) omega_pos).2 (natCast_le.2 n.2)
+  simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos (repr e) omega0_pos).2 (natCast_le.2 n.2)
+
+@[deprecated (since := "2024-09-30")]
+alias omega_le_oadd := omega0_le_oadd
 
 theorem oadd_pos (e n a) : 0 < oadd e n a :=
-  @lt_of_lt_of_le _ _ _ (ω ^ repr e) _ (opow_pos (repr e) omega_pos) (omega_le_oadd e n a)
+  @lt_of_lt_of_le _ _ _ (ω ^ repr e) _ (opow_pos (repr e) omega0_pos) (omega0_le_oadd e n a)
 
 /-- Compare ordinal notations -/
 def cmp : ONote → ONote → Ordering
@@ -152,7 +155,7 @@ def cmp : ONote → ONote → Ordering
   | _, 0 => Ordering.gt
   | 0, _ => Ordering.lt
   | _o₁@(oadd e₁ n₁ a₁), _o₂@(oadd e₂ n₂ a₂) =>
-    (cmp e₁ e₂).orElse <| (_root_.cmp (n₁ : ℕ) n₂).orElse (cmp a₁ a₂)
+    (cmp e₁ e₂).then <| (_root_.cmp (n₁ : ℕ) n₂).then (cmp a₁ a₂)
 
 theorem eq_of_cmp_eq : ∀ {o₁ o₂}, cmp o₁ o₂ = Ordering.eq → o₁ = o₂
   | 0, 0, _ => rfl
@@ -230,18 +233,20 @@ theorem NF.zero_of_zero {e n a} (h : NF (ONote.oadd e n a)) (e0 : e = 0) : a = 0
   simpa [e0, NFBelow_zero] using h.snd'
 
 theorem NFBelow.repr_lt {o b} (h : NFBelow o b) : repr o < ω ^ b := by
-  induction' h with _ e n a eb b h₁ h₂ h₃ _ IH
-  · exact opow_pos _ omega_pos
-  · rw [repr]
+  induction h with
+  | zero => exact opow_pos _ omega0_pos
+  | oadd' _ _ h₃ _ IH =>
+    rw [repr]
     apply ((add_lt_add_iff_left _).2 IH).trans_le
     rw [← mul_succ]
-    apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega _)) _).trans
+    apply (mul_le_mul_left' (succ_le_of_lt (nat_lt_omega0 _)) _).trans
     rw [← opow_succ]
-    exact opow_le_opow_right omega_pos (succ_le_of_lt h₃)
+    exact opow_le_opow_right omega0_pos (succ_le_of_lt h₃)
 
 theorem NFBelow.mono {o b₁ b₂} (bb : b₁ ≤ b₂) (h : NFBelow o b₁) : NFBelow o b₂ := by
-  induction' h with _ e n a eb b h₁ h₂ h₃ _ _ <;> constructor
-  exacts [h₁, h₂, lt_of_lt_of_le h₃ bb]
+  induction h with
+  | zero => exact zero
+  | oadd' h₁ h₂ h₃ _ _ => constructor; exacts [h₁, h₂, lt_of_lt_of_le h₃ bb]
 
 theorem NF.below_of_lt {e n a b} (H : repr e < b) :
     NF (ONote.oadd e n a) → NFBelow (ONote.oadd e n a) b
@@ -251,7 +256,7 @@ theorem NF.below_of_lt' : ∀ {o b}, repr o < ω ^ b → NF o → NFBelow o b
   | 0, _, _, _ => NFBelow.zero
   | ONote.oadd _ _ _, _, H, h =>
     h.below_of_lt <|
-      (opow_lt_opow_iff_right one_lt_omega).1 <| lt_of_le_of_lt (omega_le_oadd _ _ _) H
+      (opow_lt_opow_iff_right one_lt_omega0).1 <| lt_of_le_of_lt (omega0_le_oadd _ _ _) H
 
 theorem nfBelow_ofNat : ∀ n, NFBelow (ofNat n) 1
   | 0 => NFBelow.zero
@@ -265,13 +270,13 @@ instance nf_one : NF 1 := by rw [← ofNat_one]; infer_instance
 theorem oadd_lt_oadd_1 {e₁ n₁ o₁ e₂ n₂ o₂} (h₁ : NF (oadd e₁ n₁ o₁)) (h : e₁ < e₂) :
     oadd e₁ n₁ o₁ < oadd e₂ n₂ o₂ :=
   @lt_of_lt_of_le _ _ (repr (oadd e₁ n₁ o₁)) _ _
-    (NF.below_of_lt h h₁).repr_lt (omega_le_oadd e₂ n₂ o₂)
+    (NF.below_of_lt h h₁).repr_lt (omega0_le_oadd e₂ n₂ o₂)
 
 theorem oadd_lt_oadd_2 {e o₁ o₂ : ONote} {n₁ n₂ : ℕ+} (h₁ : NF (oadd e n₁ o₁)) (h : (n₁ : ℕ) < n₂) :
     oadd e n₁ o₁ < oadd e n₂ o₂ := by
   simp only [lt_def, repr]
   refine lt_of_lt_of_le ((add_lt_add_iff_left _).2 h₁.snd'.repr_lt) (le_trans ?_ (le_add_right _ _))
-  rwa [← mul_succ,Ordinal.mul_le_mul_iff_left (opow_pos _ omega_pos), succ_le_iff, natCast_lt]
+  rwa [← mul_succ,Ordinal.mul_le_mul_iff_left (opow_pos _ omega0_pos), succ_le_iff, natCast_lt]
 
 theorem oadd_lt_oadd_3 {e n a₁ a₂} (h : a₁ < a₂) : oadd e n a₁ < oadd e n a₂ := by
   rw [lt_def]; unfold repr
@@ -323,7 +328,7 @@ theorem repr_inj {a b} [NF a] [NF b] : repr a = repr b ↔ a = b :=
     | Ordering.eq, h => h,
     congr_arg _⟩
 
-theorem NF.of_dvd_omega_opow {b e n a} (h : NF (ONote.oadd e n a))
+theorem NF.of_dvd_omega0_opow {b e n a} (h : NF (ONote.oadd e n a))
     (d : ω ^ b ∣ repr (ONote.oadd e n a)) :
     b ≤ repr e ∧ ω ^ b ∣ repr a := by
   have := mt repr_inj.1 (fun h => by injection h : ONote.oadd e n a ≠ 0)
@@ -331,9 +336,15 @@ theorem NF.of_dvd_omega_opow {b e n a} (h : NF (ONote.oadd e n a))
   simp only [repr] at d
   exact ⟨L, (dvd_add_iff <| (opow_dvd_opow _ L).mul_right _).1 d⟩
 
-theorem NF.of_dvd_omega {e n a} (h : NF (ONote.oadd e n a)) :
+@[deprecated (since := "2024-09-30")]
+alias NF.of_dvd_omega_opow := NF.of_dvd_omega0_opow
+
+theorem NF.of_dvd_omega0 {e n a} (h : NF (ONote.oadd e n a)) :
     ω ∣ repr (ONote.oadd e n a) → repr e ≠ 0 ∧ ω ∣ repr a := by
-  (rw [← opow_one ω, ← one_le_iff_ne_zero]; exact h.of_dvd_omega_opow)
+  (rw [← opow_one ω, ← one_le_iff_ne_zero]; exact h.of_dvd_omega0_opow)
+
+@[deprecated (since := "2024-09-30")]
+alias NF.of_dvd_omega := NF.of_dvd_omega0
 
 /-- `TopBelow b o` asserts that the largest exponent in `o`, if
   it exists, is less than `b`. This is an auxiliary definition
@@ -441,7 +452,7 @@ theorem repr_add : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ + o₂) = rep
         unfold repr at this
         cases he' : e' <;> simp only [he', zero_def, opow_zero, repr, gt_iff_lt] at this ⊢ <;>
         exact lt_of_le_of_lt (le_add_right _ _) this
-      · simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos (repr e') omega_pos).2
+      · simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos (repr e') omega0_pos).2
           (natCast_le.2 n'.pos)
     · rw [ee, ← add_assoc, ← mul_add]
 
@@ -501,7 +512,7 @@ theorem repr_sub : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ - o₂) = rep
         simpa using mul_le_mul_left' (natCast_le.2 <| Nat.succ_pos _) _
     · exact
         (Ordinal.sub_eq_of_add_eq <|
-            add_absorp (h₂.below_of_lt ee).repr_lt <| omega_le_oadd _ _ _).symm
+            add_absorp (h₂.below_of_lt ee).repr_lt <| omega0_le_oadd _ _ _).symm
 
 /-- Multiplication of ordinal notations (correct only for normal input) -/
 def mul : ONote → ONote → ONote
@@ -555,7 +566,7 @@ theorem repr_mul : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ * o₂) = rep
       simp [(· * ·)]
     have ao : repr a₁ + ω ^ repr e₁ * (n₁ : ℕ) = ω ^ repr e₁ * (n₁ : ℕ) := by
       apply add_absorp h₁.snd'.repr_lt
-      simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos _ omega_pos).2 (natCast_le.2 n₁.2)
+      simpa using (Ordinal.mul_le_mul_iff_left <| opow_pos _ omega0_pos).2 (natCast_le.2 n₁.2)
     by_cases e0 : e₂ = 0
     · cases' Nat.exists_eq_succ_of_ne_zero n₂.ne_zero with x xe
       simp only [e0, repr, PNat.mul_coe, natCast_mul, opow_zero, one_mul]
@@ -568,8 +579,8 @@ theorem repr_mul : ∀ (o₁ o₂) [NF o₁] [NF o₂], repr (o₁ * o₂) = rep
       rw [← mul_assoc]
       congr 2
       have := mt repr_inj.1 e0
-      rw [add_mul_limit ao (opow_isLimit_left omega_isLimit this), mul_assoc,
-        mul_omega_dvd (natCast_pos.2 n₁.pos) (nat_lt_omega _)]
+      rw [add_mul_limit ao (opow_isLimit_left omega0_isLimit this), mul_assoc,
+        mul_omega0_dvd (natCast_pos.2 n₁.pos) (nat_lt_omega0 _)]
       simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 this)
 
 /-- Calculate division and remainder of `o` mod ω.
@@ -652,9 +663,7 @@ theorem split_eq_scale_split' : ∀ {o o' m} [NF o], split' o = (o', m) → spli
         simp only [repr_add, repr, opow_zero, Nat.succPNat_coe, Nat.cast_one, mul_one, add_zero,
           repr_sub]
         have := mt repr_inj.1 e0
-        refine Ordinal.add_sub_cancel_of_le ?_
-        have := one_le_iff_ne_zero.2 this
-        exact this
+        exact Ordinal.add_sub_cancel_of_le <| one_le_iff_ne_zero.2 this
       intros
       substs o' m
       simp [scale, this]
@@ -680,7 +689,7 @@ theorem nf_repr_split' : ∀ {o o' m} [NF o], split' o = (o', m) → NF o' ∧ r
       · simp at this ⊢
         refine
           IH₁.below_of_lt'
-            ((Ordinal.mul_lt_mul_iff_left omega_pos).1 <| lt_of_le_of_lt (le_add_right _ m') ?_)
+            ((Ordinal.mul_lt_mul_iff_left omega0_pos).1 <| lt_of_le_of_lt (le_add_right _ m') ?_)
         rw [← this, ← IH₂]
         exact h.snd'.repr_lt
       · rw [this]
@@ -723,9 +732,9 @@ theorem split_dvd {o o' m} [NF o] (h : split o = (o', m)) : ω ∣ repr o' := by
 theorem split_add_lt {o e n a m} [NF o] (h : split o = (oadd e n a, m)) :
     repr a + m < ω ^ repr e := by
   cases' nf_repr_split h with h₁ h₂
-  cases' h₁.of_dvd_omega (split_dvd h) with e0 d
-  apply principal_add_omega_opow _ h₁.snd'.repr_lt (lt_of_lt_of_le (nat_lt_omega _) _)
-  simpa using opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0)
+  cases' h₁.of_dvd_omega0 (split_dvd h) with e0 d
+  apply principal_add_omega0_opow _ h₁.snd'.repr_lt (lt_of_lt_of_le (nat_lt_omega0 _) _)
+  simpa using opow_le_opow_right omega0_pos (one_le_iff_ne_zero.2 e0)
 
 @[simp]
 theorem mulNat_eq_mul (n o) : mulNat o n = o * ofNat n := by cases o <;> cases n <;> rfl
@@ -781,22 +790,22 @@ theorem repr_opow_aux₁ {e a} [Ne : NF e] [Na : NF a] {a' : Ordinal} (e0 : repr
       (ω ^ repr e) ^ (ω : Ordinal.{0}) := by
   subst aa
   have No := Ne.oadd n (Na.below_of_lt' h)
-  have := omega_le_oadd e n a
+  have := omega0_le_oadd e n a
   rw [repr] at this
   refine le_antisymm ?_ (opow_le_opow_left _ this)
-  apply (opow_le_of_limit ((opow_pos _ omega_pos).trans_le this).ne' omega_isLimit).2
+  apply (opow_le_of_limit ((opow_pos _ omega0_pos).trans_le this).ne' omega0_isLimit).2
   intro b l
   have := (No.below_of_lt (lt_succ _)).repr_lt
   rw [repr] at this
   apply (opow_le_opow_left b <| this.le).trans
   rw [← opow_mul, ← opow_mul]
-  apply opow_le_opow_right omega_pos
+  apply opow_le_opow_right omega0_pos
   rcases le_or_lt ω (repr e) with h | h
   · apply (mul_le_mul_left' (le_succ b) _).trans
-    rw [← add_one_eq_succ, add_mul_succ _ (one_add_of_omega_le h), add_one_eq_succ, succ_le_iff,
+    rw [← add_one_eq_succ, add_mul_succ _ (one_add_of_omega0_le h), add_one_eq_succ, succ_le_iff,
       Ordinal.mul_lt_mul_iff_left (Ordinal.pos_iff_ne_zero.2 e0)]
-    exact omega_isLimit.2 _ l
-  · apply (principal_mul_omega (omega_isLimit.2 _ h) l).le.trans
+    exact omega0_isLimit.2 _ l
+  · apply (principal_mul_omega0 (omega0_isLimit.2 _ h) l).le.trans
     simpa using mul_le_mul_right' (one_le_iff_ne_zero.2 e0) ω
 
 section
@@ -827,30 +836,30 @@ theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω 
     · simp only [R', ONote.repr_scale, ONote.repr, ONote.mulNat_eq_mul, ONote.opowAux,
         ONote.repr_ofNat, ONote.repr_mul, ONote.repr_add, Ordinal.opow_mul, ONote.zero_add]
   have α0 : 0 < α' := by simpa [lt_def, repr] using oadd_pos a0 n a'
-  have ω00 : 0 < ω0 ^ (k : Ordinal) := opow_pos _ (opow_pos _ omega_pos)
+  have ω00 : 0 < ω0 ^ (k : Ordinal) := opow_pos _ (opow_pos _ omega0_pos)
   have Rl : R < ω ^ (repr a0 * succ ↑k) := by
     by_cases k0 : k = 0
     · simp only [k0, Nat.cast_zero, succ_zero, mul_one, R]
-      refine lt_of_lt_of_le ?_ (opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0))
-      cases' m with m <;> simp [opowAux, omega_pos]
+      refine lt_of_lt_of_le ?_ (opow_le_opow_right omega0_pos (one_le_iff_ne_zero.2 e0))
+      cases' m with m <;> simp [opowAux, omega0_pos]
       rw [← add_one_eq_succ, ← Nat.cast_succ]
-      apply nat_lt_omega
+      apply nat_lt_omega0
     · rw [opow_mul]
       exact IH.1 k0
   refine ⟨fun _ => ?_, ?_⟩
   · rw [RR, ← opow_mul _ _ (succ k.succ)]
     have e0 := Ordinal.pos_iff_ne_zero.2 e0
     have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 (le_add_left _ _)
-    apply principal_add_omega_opow
+    apply principal_add_omega0_opow
     · simp only [Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, add_one_eq_succ,
         opow_mul, opow_succ, mul_assoc]
       rw [Ordinal.mul_lt_mul_iff_left ω00, ← Ordinal.opow_add]
       have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt
-      · exact mul_lt_omega_opow rr0 this (nat_lt_omega _)
+      · exact mul_lt_omega0_opow rr0 this (nat_lt_omega0 _)
       · simpa using (add_lt_add_iff_left (repr a0)).2 e0
     · exact
         lt_of_lt_of_le Rl
-          (opow_le_opow_right omega_pos <|
+          (opow_le_opow_right omega0_pos <|
             mul_le_mul_left' (succ_le_succ_iff.2 (natCast_le.2 (le_of_lt k.lt_succ_self))) _)
   calc
     (ω0 ^ (k.succ : Ordinal)) * α' + R'
@@ -862,10 +871,10 @@ theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω 
   · have αd : ω ∣ α' :=
       dvd_add (dvd_mul_of_dvd_left (by simpa using opow_dvd_opow ω (one_le_iff_ne_zero.2 e0)) _) d
     rw [mul_add (ω0 ^ (k : Ordinal)), add_assoc, ← mul_assoc, ← opow_succ,
-      add_mul_limit _ (isLimit_iff_omega_dvd.2 ⟨ne_of_gt α0, αd⟩), mul_assoc,
-      @mul_omega_dvd n (natCast_pos.2 n.pos) (nat_lt_omega _) _ αd]
+      add_mul_limit _ (isLimit_iff_omega0_dvd.2 ⟨ne_of_gt α0, αd⟩), mul_assoc,
+      @mul_omega0_dvd n (natCast_pos.2 n.pos) (nat_lt_omega0 _) _ αd]
     apply @add_absorp _ (repr a0 * succ ↑k)
-    · refine principal_add_omega_opow _ ?_ Rl
+    · refine principal_add_omega0_opow _ ?_ Rl
       rw [opow_mul, opow_succ, Ordinal.mul_lt_mul_iff_left ω00]
       exact No.snd'.repr_lt
     · have := mul_le_mul_left' (one_le_iff_pos.2 <| natCast_pos.2 n.pos) (ω0 ^ succ (k : Ordinal))
@@ -878,7 +887,7 @@ theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω 
       apply add_absorp Rl
       rw [opow_mul, opow_succ]
       apply mul_le_mul_left'
-      simpa [repr] using omega_le_oadd a0 n a'
+      simpa [repr] using omega0_le_oadd a0 n a'
 
 end
 
@@ -897,18 +906,18 @@ theorem repr_opow (o₁ o₂) [NF o₁] [NF o₂] : repr (o₁ ^ o₂) = repr o
       simp only [opow_def, opowAux2, opow, e₁, h, r₁, e₂, r₂, repr,
           opow_zero, Nat.succPNat_coe, Nat.cast_succ, Nat.cast_zero, _root_.zero_add, mul_one,
           add_zero, one_opow, npow_eq_pow]
-      rw [opow_add, opow_mul, opow_omega, add_one_eq_succ]
+      rw [opow_add, opow_mul, opow_omega0, add_one_eq_succ]
       · congr
         conv_lhs =>
           dsimp [(· ^ ·)]
           simp [Pow.pow, opow, Ordinal.succ_ne_zero]
         rw [opow_natCast]
       · simpa [Nat.one_le_iff_ne_zero]
-      · rw [← Nat.cast_succ, lt_omega]
+      · rw [← Nat.cast_succ, lt_omega0]
         exact ⟨_, rfl⟩
   · haveI := N₁.fst
     haveI := N₁.snd
-    cases' N₁.of_dvd_omega (split_dvd e₁) with a00 ad
+    cases' N₁.of_dvd_omega0 (split_dvd e₁) with a00 ad
     have al := split_add_lt e₁
     have aa : repr (a' + ofNat m) = repr a' + m := by
       simp only [eq_self_iff_true, ONote.repr_ofNat, ONote.repr_add]
@@ -955,13 +964,13 @@ private theorem exists_lt_add {α} [hα : Nonempty α] {o : Ordinal} {f : α →
     refine (H h).imp fun i H => ?_
     rwa [← Ordinal.add_sub_cancel_of_le h', add_lt_add_iff_left]
 
-private theorem exists_lt_mul_omega' {o : Ordinal} ⦃a⦄ (h : a < o * ω) :
+private theorem exists_lt_mul_omega0' {o : Ordinal} ⦃a⦄ (h : a < o * ω) :
     ∃ i : ℕ, a < o * ↑i + o := by
-  obtain ⟨i, hi, h'⟩ := (lt_mul_of_limit omega_isLimit).1 h
-  obtain ⟨i, rfl⟩ := lt_omega.1 hi
+  obtain ⟨i, hi, h'⟩ := (lt_mul_of_limit omega0_isLimit).1 h
+  obtain ⟨i, rfl⟩ := lt_omega0.1 hi
   exact ⟨i, h'.trans_le (le_add_right _ _)⟩
 
-private theorem exists_lt_omega_opow' {α} {o b : Ordinal} (hb : 1 < b) (ho : o.IsLimit)
+private theorem exists_lt_omega0_opow' {α} {o b : Ordinal} (hb : 1 < b) (ho : o.IsLimit)
     {f : α → Ordinal} (H : ∀ ⦃a⦄, a < o → ∃ i, a < f i) ⦃a⦄ (h : a < b ^ o) :
         ∃ i, a < b ^ f i := by
   obtain ⟨d, hd, h'⟩ := (lt_opow_of_limit (zero_lt_one.trans hb).ne' ho).1 h
@@ -1014,39 +1023,40 @@ theorem fundamentalSequence_has_prop (o) : FundamentalSequenceProp o (fundamenta
             have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm]) <;>
       (try rw [show m = (m' + 1).succPNat by
               rw [← e', ← PNat.coe_inj, Nat.succPNat_coe, ← Nat.add_one, PNat.natPred_add_one]]) <;>
-      simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega, add_lt_add_iff_left, add_zero,
-        eq_self_iff_true, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero, Nat.cast_succ,
-        Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe, succ_zero,
-        true_and_iff, _root_.zero_add, zero_def]
+      simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega0, add_lt_add_iff_left,
+        add_zero, eq_self_iff_true, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero,
+        Nat.cast_succ, Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe, succ_zero,
+        _root_.zero_add, zero_def]
     · decide
     · exact ⟨rfl, inferInstance⟩
-    · have := opow_pos (repr a') omega_pos
+    · have := opow_pos (repr a') omega0_pos
       refine
-        ⟨mul_isLimit this omega_isLimit, fun i =>
-          ⟨this, ?_, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)⟩, exists_lt_mul_omega'⟩
+        ⟨mul_isLimit this omega0_isLimit, fun i =>
+          ⟨this, ?_, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)⟩, exists_lt_mul_omega0'⟩
       rw [← mul_succ, ← natCast_succ, Ordinal.mul_lt_mul_iff_left this]
-      apply nat_lt_omega
-    · have := opow_pos (repr a') omega_pos
+      apply nat_lt_omega0
+    · have := opow_pos (repr a') omega0_pos
       refine
-        ⟨add_isLimit _ (mul_isLimit this omega_isLimit), fun i => ⟨this, ?_, ?_⟩,
-          exists_lt_add exists_lt_mul_omega'⟩
+        ⟨add_isLimit _ (mul_isLimit this omega0_isLimit), fun i => ⟨this, ?_, ?_⟩,
+          exists_lt_add exists_lt_mul_omega0'⟩
       · rw [← mul_succ, ← natCast_succ, Ordinal.mul_lt_mul_iff_left this]
-        apply nat_lt_omega
+        apply nat_lt_omega0
       · refine fun H => H.fst.oadd _ (NF.below_of_lt' ?_ (@NF.oadd_zero _ _ (iha.2 H.fst)))
         rw [repr, ← zero_def, repr, add_zero, iha.1, opow_succ, Ordinal.mul_lt_mul_iff_left this]
-        apply nat_lt_omega
+        apply nat_lt_omega0
     · rcases iha with ⟨h1, h2, h3⟩
-      refine ⟨opow_isLimit one_lt_omega h1, fun i => ?_, exists_lt_omega_opow' one_lt_omega h1 h3⟩
+      refine ⟨opow_isLimit one_lt_omega0 h1, fun i => ?_,
+        exists_lt_omega0_opow' one_lt_omega0 h1 h3⟩
       obtain ⟨h4, h5, h6⟩ := h2 i
       exact ⟨h4, h5, fun H => @NF.oadd_zero _ _ (h6 H.fst)⟩
     · rcases iha with ⟨h1, h2, h3⟩
       refine
-        ⟨add_isLimit _ (opow_isLimit one_lt_omega h1), fun i => ?_,
-          exists_lt_add (exists_lt_omega_opow' one_lt_omega h1 h3)⟩
+        ⟨add_isLimit _ (opow_isLimit one_lt_omega0 h1), fun i => ?_,
+          exists_lt_add (exists_lt_omega0_opow' one_lt_omega0 h1 h3)⟩
       obtain ⟨h4, h5, h6⟩ := h2 i
       refine ⟨h4, h5, fun H => H.fst.oadd _ (NF.below_of_lt' ?_ (@NF.oadd_zero _ _ (h6 H.fst)))⟩
       rwa [repr, ← zero_def, repr, add_zero, PNat.one_coe, Nat.cast_one, mul_one,
-        opow_lt_opow_iff_right one_lt_omega]
+        opow_lt_opow_iff_right one_lt_omega0]
   · refine ⟨by
       rw [repr, ihb.1, add_succ, repr], fun H => H.fst.oadd _ (NF.below_of_lt' ?_ (ihb.2 H.snd))⟩
     have := H.snd'.repr_lt
diff --git a/Mathlib/SetTheory/Ordinal/Principal.lean b/Mathlib/SetTheory/Ordinal/Principal.lean
index 8d13d1a443605..2835fd286b3f3 100644
--- a/Mathlib/SetTheory/Ordinal/Principal.lean
+++ b/Mathlib/SetTheory/Ordinal/Principal.lean
@@ -14,9 +14,9 @@ We define principal or indecomposable ordinals, and we prove the standard proper
 * `Principal`: A principal or indecomposable ordinal under some binary operation. We include 0 and
   any other typically excluded edge cases for simplicity.
 * `unbounded_principal`: Principal ordinals are unbounded.
-* `principal_add_iff_zero_or_omega_opow`: The main characterization theorem for additive principal
+* `principal_add_iff_zero_or_omega0_opow`: The main characterization theorem for additive principal
   ordinals.
-* `principal_mul_iff_le_two_or_omega_opow_opow`: The main characterization theorem for
+* `principal_mul_iff_le_two_or_omega0_opow_opow`: The main characterization theorem for
   multiplicative principal ordinals.
 
 ## TODO
@@ -75,7 +75,7 @@ theorem Principal.iterate_lt {a o : Ordinal} (hao : a < o) (ho : Principal op o)
 
 theorem op_eq_self_of_principal {a o : Ordinal.{u}} (hao : a < o) (H : IsNormal (op a))
     (ho : Principal op o) (ho' : IsLimit o) : op a o = o := by
-  refine le_antisymm ?_ (H.self_le _)
+  apply H.le_apply.antisymm'
   rw [← IsNormal.bsup_eq.{u, u} H ho', bsup_le_iff]
   exact fun b hbo => (ho hao hbo).le
 
@@ -163,52 +163,64 @@ theorem principal_add_iff_add_lt_ne_self {a} :
     rcases exists_lt_add_of_not_principal_add ha with ⟨b, hb, c, hc, rfl⟩
     exact (H b hb c hc).irrefl⟩
 
-theorem add_omega {a : Ordinal} (h : a < ω) : a + ω = ω := by
-  rcases lt_omega.1 h with ⟨n, rfl⟩
+theorem add_omega0 {a : Ordinal} (h : a < ω) : a + ω = ω := by
+  rcases lt_omega0.1 h with ⟨n, rfl⟩
   clear h; induction' n with n IH
   · rw [Nat.cast_zero, zero_add]
-  · rwa [Nat.cast_succ, add_assoc, one_add_of_omega_le (le_refl _)]
+  · rwa [Nat.cast_succ, add_assoc, one_add_of_omega0_le (le_refl _)]
 
-theorem principal_add_omega : Principal (· + ·) ω :=
-  principal_add_iff_add_left_eq_self.2 fun _ => add_omega
+@[deprecated (since := "2024-09-30")]
+alias add_omega := add_omega0
 
-theorem add_omega_opow {a b : Ordinal} (h : a < ω ^ b) : a + ω ^ b = ω ^ b := by
+theorem principal_add_omega0 : Principal (· + ·) ω :=
+  principal_add_iff_add_left_eq_self.2 fun _ => add_omega0
+
+@[deprecated (since := "2024-09-30")]
+alias principal_add_omega := principal_add_omega0
+
+theorem add_omega0_opow {a b : Ordinal} (h : a < ω ^ b) : a + ω ^ b = ω ^ b := by
   refine le_antisymm ?_ (le_add_left _ a)
   induction' b using limitRecOn with b _ b l IH
   · rw [opow_zero, ← succ_zero, lt_succ_iff, Ordinal.le_zero] at h
     rw [h, zero_add]
   · rw [opow_succ] at h
-    rcases (lt_mul_of_limit omega_isLimit).1 h with ⟨x, xo, ax⟩
+    rcases (lt_mul_of_limit omega0_isLimit).1 h with ⟨x, xo, ax⟩
     apply (add_le_add_right ax.le _).trans
-    rw [opow_succ, ← mul_add, add_omega xo]
-  · rcases (lt_opow_of_limit omega_ne_zero l).1 h with ⟨x, xb, ax⟩
-    apply (((add_isNormal a).trans <| opow_isNormal one_lt_omega).limit_le l).2
+    rw [opow_succ, ← mul_add, add_omega0 xo]
+  · rcases (lt_opow_of_limit omega0_ne_zero l).1 h with ⟨x, xb, ax⟩
+    apply (((add_isNormal a).trans <| opow_isNormal one_lt_omega0).limit_le l).2
     intro y yb
     calc a + ω ^ y ≤ a + ω ^ max x y :=
-      add_le_add_left (opow_le_opow_right omega_pos (le_max_right x y)) _
+      add_le_add_left (opow_le_opow_right omega0_pos (le_max_right x y)) _
     _ ≤ ω ^ max x y :=
-      IH _ (max_lt xb yb) <| ax.trans_le <| opow_le_opow_right omega_pos <| le_max_left x y
+      IH _ (max_lt xb yb) <| ax.trans_le <| opow_le_opow_right omega0_pos <| le_max_left x y
     _ ≤ ω ^ b :=
-      opow_le_opow_right omega_pos <| (max_lt xb yb).le
+      opow_le_opow_right omega0_pos <| (max_lt xb yb).le
+
+@[deprecated (since := "2024-09-30")]
+alias add_omega_opow := add_omega0_opow
+
+theorem principal_add_omega0_opow (o : Ordinal) : Principal (· + ·) (ω ^ o) :=
+  principal_add_iff_add_left_eq_self.2 fun _ => add_omega0_opow
 
-theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (ω ^ o) :=
-  principal_add_iff_add_left_eq_self.2 fun _ => add_omega_opow
+@[deprecated (since := "2024-09-30")]
+alias principal_add_omega_opow := principal_add_omega0_opow
 
 /-- The main characterization theorem for additive principal ordinals. -/
-theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :
+theorem principal_add_iff_zero_or_omega0_opow {o : Ordinal} :
     Principal (· + ·) o ↔ o = 0 ∨ o ∈ Set.range (ω ^ · : Ordinal → Ordinal) := by
   rcases eq_or_ne o 0 with (rfl | ho)
   · simp only [principal_zero, Or.inl]
   · rw [principal_add_iff_add_left_eq_self]
-    simp only [ho, false_or_iff]
+    simp only [ho, false_or]
     refine
       ⟨fun H => ⟨_, ((lt_or_eq_of_le (opow_log_le_self _ ho)).resolve_left fun h => ?_)⟩,
-        fun ⟨b, e⟩ => e.symm ▸ fun a => add_omega_opow⟩
+        fun ⟨b, e⟩ => e.symm ▸ fun a => add_omega0_opow⟩
     have := H _ h
-    have := lt_opow_succ_log_self one_lt_omega o
-    rw [opow_succ, lt_mul_of_limit omega_isLimit] at this
+    have := lt_opow_succ_log_self one_lt_omega0 o
+    rw [opow_succ, lt_mul_of_limit omega0_isLimit] at this
     rcases this with ⟨a, ao, h'⟩
-    rcases lt_omega.1 ao with ⟨n, rfl⟩
+    rcases lt_omega0.1 ao with ⟨n, rfl⟩
     clear ao
     revert h'
     apply not_lt_of_le
@@ -218,18 +230,21 @@ theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :
     · simp [Nat.cast_zero, mul_zero, zero_add]
     · simp only [Nat.cast_succ, mul_add_one, add_assoc, this, IH]
 
+@[deprecated (since := "2024-09-30")]
+alias principal_add_iff_zero_or_omega_opow := principal_add_iff_zero_or_omega0_opow
+
 theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :
     Principal (· + ·) (a ^ b) := by
-  rcases principal_add_iff_zero_or_omega_opow.1 ha with (rfl | ⟨c, rfl⟩)
+  rcases principal_add_iff_zero_or_omega0_opow.1 ha with (rfl | ⟨c, rfl⟩)
   · rcases eq_or_ne b 0 with (rfl | hb)
     · rw [opow_zero]
       exact principal_add_one
     · rwa [zero_opow hb]
   · rw [← opow_mul]
-    exact principal_add_omega_opow _
+    exact principal_add_omega0_opow _
 
 theorem add_absorp {a b c : Ordinal} (h₁ : a < ω ^ b) (h₂ : ω ^ b ≤ c) : a + c = c := by
-  rw [← Ordinal.add_sub_cancel_of_le h₂, ← add_assoc, add_omega_opow h₁]
+  rw [← Ordinal.add_sub_cancel_of_le h₂, ← add_assoc, add_omega0_opow h₁]
 
 theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)
     (hb : Principal (· + ·) b) : Principal (· + ·) (a * b) := by
@@ -302,44 +317,59 @@ theorem principal_mul_iff_mul_left_eq {o : Ordinal} :
     rw [← h a ha hao]
     exact (mul_isNormal ha).strictMono hbo
 
-theorem principal_mul_omega : Principal (· * ·) ω := fun a b ha hb =>
-  match a, b, lt_omega.1 ha, lt_omega.1 hb with
+theorem principal_mul_omega0 : Principal (· * ·) ω := fun a b ha hb =>
+  match a, b, lt_omega0.1 ha, lt_omega0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by
     dsimp only; rw [← natCast_mul]
-    apply nat_lt_omega
+    apply nat_lt_omega0
+
+@[deprecated (since := "2024-09-30")]
+alias principal_mul_omega := principal_mul_omega0
+
+theorem mul_omega0 {a : Ordinal} (a0 : 0 < a) (ha : a < ω) : a * ω = ω :=
+  principal_mul_iff_mul_left_eq.1 principal_mul_omega0 a a0 ha
 
-theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < ω) : a * ω = ω :=
-  principal_mul_iff_mul_left_eq.1 principal_mul_omega a a0 ha
+@[deprecated (since := "2024-09-30")]
+alias mul_omega := mul_omega0
 
-theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < ω ^ c) (hb : b < ω) :
+theorem mul_lt_omega0_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < ω ^ c) (hb : b < ω) :
     a * b < ω ^ c := by
   rcases zero_or_succ_or_limit c with (rfl | ⟨c, rfl⟩ | l)
   · exact (lt_irrefl _).elim c0
   · rw [opow_succ] at ha
-    rcases ((mul_isNormal <| opow_pos _ omega_pos).limit_lt omega_isLimit).1 ha with ⟨n, hn, an⟩
+    rcases ((mul_isNormal <| opow_pos _ omega0_pos).limit_lt omega0_isLimit).1 ha with ⟨n, hn, an⟩
     apply (mul_le_mul_right' (le_of_lt an) _).trans_lt
-    rw [opow_succ, mul_assoc, mul_lt_mul_iff_left (opow_pos _ omega_pos)]
-    exact principal_mul_omega hn hb
-  · rcases ((opow_isNormal one_lt_omega).limit_lt l).1 ha with ⟨x, hx, ax⟩
+    rw [opow_succ, mul_assoc, mul_lt_mul_iff_left (opow_pos _ omega0_pos)]
+    exact principal_mul_omega0 hn hb
+  · rcases ((opow_isNormal one_lt_omega0).limit_lt l).1 ha with ⟨x, hx, ax⟩
     refine (mul_le_mul' (le_of_lt ax) (le_of_lt hb)).trans_lt ?_
-    rw [← opow_succ, opow_lt_opow_iff_right one_lt_omega]
+    rw [← opow_succ, opow_lt_opow_iff_right one_lt_omega0]
     exact l.2 _ hx
 
-theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < ω ^ ω ^ b) :
+@[deprecated (since := "2024-09-30")]
+alias mul_lt_omega_opow := mul_lt_omega0_opow
+
+theorem mul_omega0_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < ω ^ ω ^ b) :
     a * ω ^ ω ^ b = ω ^ ω ^ b := by
   obtain rfl | b0 := eq_or_ne b 0
   · rw [opow_zero, opow_one] at h ⊢
-    exact mul_omega a0 h
+    exact mul_omega0 a0 h
   · apply le_antisymm
     · obtain ⟨x, xb, ax⟩ :=
-        (lt_opow_of_limit omega_ne_zero (opow_isLimit_left omega_isLimit b0)).1 h
+        (lt_opow_of_limit omega0_ne_zero (opow_isLimit_left omega0_isLimit b0)).1 h
       apply (mul_le_mul_right' (le_of_lt ax) _).trans
-      rw [← opow_add, add_omega_opow xb]
+      rw [← opow_add, add_omega0_opow xb]
     · conv_lhs => rw [← one_mul (ω ^ _)]
       exact mul_le_mul_right' (one_le_iff_pos.2 a0) _
 
-theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (ω ^ ω ^ o) :=
-  principal_mul_iff_mul_left_eq.2 fun _ => mul_omega_opow_opow
+@[deprecated (since := "2024-09-30")]
+alias mul_omega_opow_opow := mul_omega0_opow_opow
+
+theorem principal_mul_omega0_opow_opow (o : Ordinal) : Principal (· * ·) (ω ^ ω ^ o) :=
+  principal_mul_iff_mul_left_eq.2 fun _ => mul_omega0_opow_opow
+
+@[deprecated (since := "2024-09-30")]
+alias principal_mul_omega_opow_opow := principal_mul_omega0_opow_opow
 
 theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)
     (ho : Principal (· * ·) (b ^ o)) : Principal (· + ·) o := by
@@ -349,24 +379,30 @@ theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)
   rwa [← opow_add, opow_lt_opow_iff_right hb] at this
 
 /-- The main characterization theorem for multiplicative principal ordinals. -/
-theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :
+theorem principal_mul_iff_le_two_or_omega0_opow_opow {o : Ordinal} :
     Principal (· * ·) o ↔ o ≤ 2 ∨ o ∈ Set.range (ω ^ ω ^ · : Ordinal → Ordinal) := by
   refine ⟨fun ho => ?_, ?_⟩
   · rcases le_or_lt o 2 with ho₂ | ho₂
     · exact Or.inl ho₂
-    · rcases principal_add_iff_zero_or_omega_opow.1 (principal_add_of_principal_mul ho ho₂.ne') with
-      (rfl | ⟨a, rfl⟩)
+    · rcases principal_add_iff_zero_or_omega0_opow.1 (principal_add_of_principal_mul ho ho₂.ne')
+        with (rfl | ⟨a, rfl⟩)
       · exact (Ordinal.not_lt_zero 2 ho₂).elim
-      · rcases principal_add_iff_zero_or_omega_opow.1
-          (principal_add_of_principal_mul_opow one_lt_omega ho) with (rfl | ⟨b, rfl⟩)
+      · rcases principal_add_iff_zero_or_omega0_opow.1
+          (principal_add_of_principal_mul_opow one_lt_omega0 ho) with (rfl | ⟨b, rfl⟩)
         · simp
         · exact Or.inr ⟨b, rfl⟩
   · rintro (ho₂ | ⟨a, rfl⟩)
     · exact principal_mul_of_le_two ho₂
-    · exact principal_mul_omega_opow_opow a
+    · exact principal_mul_omega0_opow_opow a
+
+@[deprecated (since := "2024-09-30")]
+alias principal_mul_iff_le_two_or_omega_opow_opow := principal_mul_iff_le_two_or_omega0_opow_opow
 
-theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < ω) : ∀ {b}, ω ∣ b → a * b = b
-  | _, ⟨b, rfl⟩ => by rw [← mul_assoc, mul_omega a0 ha]
+theorem mul_omega0_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < ω) : ∀ {b}, ω ∣ b → a * b = b
+  | _, ⟨b, rfl⟩ => by rw [← mul_assoc, mul_omega0 a0 ha]
+
+@[deprecated (since := "2024-09-30")]
+alias mul_omega_dvd := mul_omega0_dvd
 
 theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)
     (hb₂ : 2 < b) : a * b = b ^ succ (log b a) := by
@@ -387,15 +423,21 @@ theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal
 /-! #### Exponential principal ordinals -/
 
 
-theorem principal_opow_omega : Principal (· ^ ·) ω := fun a b ha hb =>
-  match a, b, lt_omega.1 ha, lt_omega.1 hb with
+theorem principal_opow_omega0 : Principal (· ^ ·) ω := fun a b ha hb =>
+  match a, b, lt_omega0.1 ha, lt_omega0.1 hb with
   | _, _, ⟨m, rfl⟩, ⟨n, rfl⟩ => by
     simp_rw [← natCast_opow]
-    apply nat_lt_omega
+    apply nat_lt_omega0
+
+@[deprecated (since := "2024-09-30")]
+alias principal_opow_omega := principal_opow_omega0
 
-theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < ω) : a ^ ω = ω :=
-  ((opow_le_of_limit (one_le_iff_ne_zero.1 <| le_of_lt a1) omega_isLimit).2 fun _ hb =>
-      (principal_opow_omega h hb).le).antisymm
+theorem opow_omega0 {a : Ordinal} (a1 : 1 < a) (h : a < ω) : a ^ ω = ω :=
+  ((opow_le_of_limit (one_le_iff_ne_zero.1 <| le_of_lt a1) omega0_isLimit).2 fun _ hb =>
+      (principal_opow_omega0 h hb).le).antisymm
   (right_le_opow _ a1)
 
+@[deprecated (since := "2024-09-30")]
+alias opow_omega := opow_omega0
+
 end Ordinal
diff --git a/Mathlib/SetTheory/Ordinal/Topology.lean b/Mathlib/SetTheory/Ordinal/Topology.lean
index 427fe767bdbf7..6993d517742e4 100644
--- a/Mathlib/SetTheory/Ordinal/Topology.lean
+++ b/Mathlib/SetTheory/Ordinal/Topology.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Violeta Hernández Palacios
 -/
-import Mathlib.SetTheory.Ordinal.Arithmetic
+import Mathlib.SetTheory.Ordinal.Enum
 import Mathlib.Tactic.TFAE
 import Mathlib.Topology.Order.Monotone
 
@@ -14,7 +14,7 @@ We prove some miscellaneous results involving the order topology of ordinals.
 
 ### Main results
 
-* `Ordinal.isClosed_iff_sup` / `Ordinal.isClosed_iff_bsup`: A set of ordinals is closed iff it's
+* `Ordinal.isClosed_iff_iSup` / `Ordinal.isClosed_iff_bsup`: A set of ordinals is closed iff it's
   closed under suprema.
 * `Ordinal.isNormal_iff_strictMono_and_continuous`: A characterization of normal ordinal
   functions.
@@ -86,20 +86,20 @@ theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
       ∃ t, t ⊆ s ∧ t.Nonempty ∧ BddAbove t ∧ sSup t = a,
       ∃ (o : Ordinal.{u}), o ≠ 0 ∧ ∃ (f : ∀ x < o, Ordinal),
         (∀ x hx, f x hx ∈ s) ∧ bsup.{u, u} o f = a,
-      ∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ sup.{u, u} f = a] := by
-  tfae_have 1 → 2
-  · simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
+      ∃ (ι : Type u), Nonempty ι ∧ ∃ f : ι → Ordinal, (∀ i, f i ∈ s) ∧ ⨆ i, f i = a] := by
+  tfae_have 1 → 2 := by
+    simp only [mem_closure_iff_nhdsWithin_neBot, inter_comm s, nhdsWithin_inter', nhds_left_eq_nhds]
     exact id
   tfae_have 2 → 3
-  · intro h
+  | h => by
     rcases (s ∩ Iic a).eq_empty_or_nonempty with he | hne
     · simp [he] at h
     · refine ⟨hne, (isLUB_of_mem_closure ?_ h).csSup_eq hne⟩
       exact fun x hx => hx.2
   tfae_have 3 → 4
-  · exact fun h => ⟨_, inter_subset_left, h.1, bddAbove_Iic.mono inter_subset_right, h.2⟩
-  tfae_have 4 → 5
-  · rintro ⟨t, hts, hne, hbdd, rfl⟩
+  | h => ⟨_, inter_subset_left, h.1, bddAbove_Iic.mono inter_subset_right, h.2⟩
+  tfae_have 4 → 5 := by
+    rintro ⟨t, hts, hne, hbdd, rfl⟩
     have hlub : IsLUB t (sSup t) := isLUB_csSup hne hbdd
     let ⟨y, hyt⟩ := hne
     classical
@@ -109,37 +109,46 @@ theorem mem_closure_tfae (a : Ordinal.{u}) (s : Set Ordinal) :
       · refine le_antisymm (bsup_le fun x _ => ?_) (csSup_le hne fun x hx => ?_)
         · split_ifs <;> exact hlub.1 ‹_›
         · refine (if_pos hx).symm.trans_le (le_bsup _ _ <| (hlub.1 hx).trans_lt (lt_succ _))
-  tfae_have 5 → 6
-  · rintro ⟨o, h₀, f, hfs, rfl⟩
+  tfae_have 5 → 6 := by
+    rintro ⟨o, h₀, f, hfs, rfl⟩
     exact ⟨_, toType_nonempty_iff_ne_zero.2 h₀, familyOfBFamily o f, fun _ => hfs _ _, rfl⟩
-  tfae_have 6 → 1
-  · rintro ⟨ι, hne, f, hfs, rfl⟩
-    rw [sup, iSup]
+  tfae_have 6 → 1 := by
+    rintro ⟨ι, hne, f, hfs, rfl⟩
     exact closure_mono (range_subset_iff.2 hfs) <| csSup_mem_closure (range_nonempty f)
       (bddAbove_range.{u, u} f)
   tfae_finish
 
+theorem mem_closure_iff_iSup :
+    a ∈ closure s ↔
+      ∃ (ι : Type u) (_ : Nonempty ι) (f : ι → Ordinal), (∀ i, f i ∈ s) ∧ ⨆ i, f i = a := by
+  apply ((mem_closure_tfae a s).out 0 5).trans
+  simp_rw [exists_prop]
+
+set_option linter.deprecated false in
+@[deprecated mem_closure_iff_iSup (since := "2024-08-27")]
 theorem mem_closure_iff_sup :
     a ∈ closure s ↔
-      ∃ (ι : Type u) (_ : Nonempty ι) (f : ι → Ordinal), (∀ i, f i ∈ s) ∧ sup.{u, u} f = a :=
-  calc
-    _ ↔ (∃ (ι : Type u), Nonempty ι ∧ ∃ f, (∀ (i : ι), f i ∈ s) ∧ sup f = a) :=
-             (mem_closure_tfae a s).out 0 5
-    _ ↔ _ := by simp only [exists_prop]
+      ∃ (ι : Type u) (_ : Nonempty ι) (f : ι → Ordinal), (∀ i, f i ∈ s) ∧ sup f = a :=
+  mem_closure_iff_iSup
 
+theorem mem_iff_iSup_of_isClosed (hs : IsClosed s) :
+    a ∈ s ↔ ∃ (ι : Type u) (_hι : Nonempty ι) (f : ι → Ordinal),
+      (∀ i, f i ∈ s) ∧ ⨆ i, f i = a := by
+  rw [← mem_closure_iff_iSup, hs.closure_eq]
+
+set_option linter.deprecated false in
+@[deprecated mem_iff_iSup_of_isClosed (since := "2024-08-27")]
 theorem mem_closed_iff_sup (hs : IsClosed s) :
     a ∈ s ↔ ∃ (ι : Type u) (_hι : Nonempty ι) (f : ι → Ordinal),
-      (∀ i, f i ∈ s) ∧ sup.{u, u} f = a := by
-  rw [← mem_closure_iff_sup, hs.closure_eq]
+      (∀ i, f i ∈ s) ∧ sup f = a :=
+  mem_iff_iSup_of_isClosed hs
 
 theorem mem_closure_iff_bsup :
     a ∈ closure s ↔
       ∃ (o : Ordinal) (_ho : o ≠ 0) (f : ∀ a < o, Ordinal),
-        (∀ i hi, f i hi ∈ s) ∧ bsup.{u, u} o f = a :=
-  calc
-    _ ↔ ∃ o, o ≠ 0 ∧ ∃ f, (∀ (x : Ordinal.{u}) (hx : x < o), f x hx ∈ s) ∧ o.bsup f = a :=
-             (mem_closure_tfae a s).out 0 4
-    _ ↔ _ := by simp only [exists_prop]
+        (∀ i hi, f i hi ∈ s) ∧ bsup.{u, u} o f = a := by
+  apply ((mem_closure_tfae a s).out 0 4).trans
+  simp_rw [exists_prop]
 
 theorem mem_closed_iff_bsup (hs : IsClosed s) :
     a ∈ s ↔
@@ -147,9 +156,20 @@ theorem mem_closed_iff_bsup (hs : IsClosed s) :
         (∀ i hi, f i hi ∈ s) ∧ bsup.{u, u} o f = a := by
   rw [← mem_closure_iff_bsup, hs.closure_eq]
 
+theorem isClosed_iff_iSup :
+    IsClosed s ↔
+      ∀ {ι : Type u}, Nonempty ι → ∀ f : ι → Ordinal, (∀ i, f i ∈ s) → ⨆ i, f i ∈ s := by
+  use fun hs ι hι f hf => (mem_iff_iSup_of_isClosed hs).2 ⟨ι, hι, f, hf, rfl⟩
+  rw [← closure_subset_iff_isClosed]
+  intro h x hx
+  rcases mem_closure_iff_iSup.1 hx with ⟨ι, hι, f, hf, rfl⟩
+  exact h hι f hf
+
+set_option linter.deprecated false in
+@[deprecated mem_iff_iSup_of_isClosed (since := "2024-08-27")]
 theorem isClosed_iff_sup :
     IsClosed s ↔
-      ∀ {ι : Type u}, Nonempty ι → ∀ f : ι → Ordinal, (∀ i, f i ∈ s) → sup.{u, u} f ∈ s := by
+      ∀ {ι : Type u}, Nonempty ι → ∀ f : ι → Ordinal, (∀ i, f i ∈ s) → ⨆ i, f i ∈ s := by
   use fun hs ι hι f hf => (mem_closed_iff_sup hs).2 ⟨ι, hι, f, hf, rfl⟩
   rw [← closure_subset_iff_isClosed]
   intro h x hx
@@ -160,10 +180,10 @@ theorem isClosed_iff_bsup :
     IsClosed s ↔
       ∀ {o : Ordinal}, o ≠ 0 → ∀ f : ∀ a < o, Ordinal,
         (∀ i hi, f i hi ∈ s) → bsup.{u, u} o f ∈ s := by
-  rw [isClosed_iff_sup]
+  rw [isClosed_iff_iSup]
   refine ⟨fun H o ho f hf => H (toType_nonempty_iff_ne_zero.2 ho) _ ?_, fun H ι hι f hf => ?_⟩
   · exact fun i => hf _ _
-  · rw [← bsup_eq_sup]
+  · rw [← Ordinal.sup, ← bsup_eq_sup]
     apply H (type_ne_zero_iff_nonempty.2 hι)
     exact fun i hi => hf _
 
@@ -194,32 +214,31 @@ theorem isNormal_iff_strictMono_and_continuous (f : Ordinal.{u} → Ordinal.{u})
     rintro ⟨h, h'⟩
     refine ⟨h, fun o ho a h => ?_⟩
     suffices o ∈ f ⁻¹' Set.Iic a from Set.mem_preimage.1 this
-    rw [mem_closed_iff_sup (IsClosed.preimage h' (@isClosed_Iic _ _ _ _ a))]
+    rw [mem_iff_iSup_of_isClosed (IsClosed.preimage h' (@isClosed_Iic _ _ _ _ a))]
     exact
       ⟨_, toType_nonempty_iff_ne_zero.2 ho.1, typein (· < ·), fun i => h _ (typein_lt_self i),
         sup_typein_limit ho.2⟩
 
-theorem enumOrd_isNormal_iff_isClosed (hs : s.Unbounded (· < ·)) :
+theorem enumOrd_isNormal_iff_isClosed (hs : ¬ BddAbove s) :
     IsNormal (enumOrd s) ↔ IsClosed s := by
   have Hs := enumOrd_strictMono hs
   refine
-    ⟨fun h => isClosed_iff_sup.2 fun {ι} hι f hf => ?_, fun h =>
+    ⟨fun h => isClosed_iff_iSup.2 fun {ι} hι f hf => ?_, fun h =>
       (isNormal_iff_strictMono_limit _).2 ⟨Hs, fun a ha o H => ?_⟩⟩
-  · let g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso hs).symm ⟨_, hf i⟩
-    suffices enumOrd s (sup.{u, u} g) = sup.{u, u} f by
+  · let g : ι → Ordinal.{u} := fun i => (enumOrdOrderIso s hs).symm ⟨_, hf i⟩
+    suffices enumOrd s (⨆ i, g i) = ⨆ i, f i by
       rw [← this]
       exact enumOrd_mem hs _
-    rw [@IsNormal.sup.{u, u, u} _ h ι g hι]
+    rw [IsNormal.map_iSup h g]
     congr
     ext x
-    change ((enumOrdOrderIso hs) _).val = f x
+    change (enumOrdOrderIso s hs _).val = f x
     rw [OrderIso.apply_symm_apply]
   · rw [isClosed_iff_bsup] at h
     suffices enumOrd s a ≤ bsup.{u, u} a fun b (_ : b < a) => enumOrd s b from
       this.trans (bsup_le H)
-    cases' enumOrd_surjective hs _
-        (h ha.1 (fun b _ => enumOrd s b) fun b _ => enumOrd_mem hs b) with
-      b hb
+    obtain ⟨b, hb⟩ := enumOrd_surjective hs (h ha.1 (fun b _ => enumOrd s b)
+      fun b _ => enumOrd_mem hs b)
     rw [← hb]
     apply Hs.monotone
     by_contra! hba
diff --git a/Mathlib/SetTheory/Surreal/Basic.lean b/Mathlib/SetTheory/Surreal/Basic.lean
index bbb97f8232f9c..6b466f6a32173 100644
--- a/Mathlib/SetTheory/Surreal/Basic.lean
+++ b/Mathlib/SetTheory/Surreal/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Scott Morrison
+Authors: Mario Carneiro, Kim Morrison
 -/
 import Mathlib.Algebra.Order.Hom.Monoid
 import Mathlib.SetTheory.Game.Ordinal
diff --git a/Mathlib/SetTheory/Surreal/Multiplication.lean b/Mathlib/SetTheory/Surreal/Multiplication.lean
index 3f931a7db4fd7..581ee2b7acc94 100644
--- a/Mathlib/SetTheory/Surreal/Multiplication.lean
+++ b/Mathlib/SetTheory/Surreal/Multiplication.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Theodore Hwa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Scott Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
+Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
 -/
 import Mathlib.Logic.Hydra
 import Mathlib.SetTheory.Surreal.Basic
@@ -173,7 +173,7 @@ open Relation
 
 /-- The relation specifying when a list of (pregame) arguments is considered simpler than another:
   `ArgsRel a₁ a₂` is true if `a₁`, considered as a multiset, can be obtained from `a₂` by
-  repeatedly removing a pregame from `a₂` and adding back one or two options of the pregame.  -/
+  repeatedly removing a pregame from `a₂` and adding back one or two options of the pregame. -/
 def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset
 
 /-- `ArgsRel` is well-founded. -/
@@ -463,7 +463,7 @@ theorem Equiv.mul_congr_left (hx₁ : x₁.Numeric) (hx₂ : x₂.Numeric) (hy :
     (he : x₁ ≈ x₂) : x₁ * y ≈ x₂ * y :=
   equiv_iff_game_eq.2 <| (P24 hx₁ hx₂ hy).1 he
 
-theorem Equiv.mul_congr_right (hx : x.Numeric)  (hy₁ : y₁.Numeric) (hy₂ : y₂.Numeric)
+theorem Equiv.mul_congr_right (hx : x.Numeric) (hy₁ : y₁.Numeric) (hy₂ : y₂.Numeric)
     (he : y₁ ≈ y₂) : x * y₁ ≈ x * y₂ :=
   .trans (mul_comm_equiv _ _) <| .trans (mul_congr_left hy₁ hy₂ hx he) (mul_comm_equiv _ _)
 
diff --git a/Mathlib/SetTheory/ZFC/Basic.lean b/Mathlib/SetTheory/ZFC/Basic.lean
index c6408760cb942..137369f0640e1 100644
--- a/Mathlib/SetTheory/ZFC/Basic.lean
+++ b/Mathlib/SetTheory/ZFC/Basic.lean
@@ -193,7 +193,7 @@ theorem Subset.congr_right : ∀ {x y z : PSet}, Equiv x y → (z ⊆ x ↔ z 
       ⟨a, cb.trans (Equiv.symm ab)⟩⟩
 
 /-- `x ∈ y` as pre-sets if `x` is extensionally equivalent to a member of the family `y`. -/
-protected def Mem (x y : PSet.{u}) : Prop :=
+protected def Mem (y x : PSet.{u}) : Prop :=
   ∃ b, Equiv x (y.Func b)
 
 instance : Membership PSet PSet :=
@@ -260,10 +260,10 @@ instance : IsIrrefl PSet (· ∈ ·) :=
   mem_wf.isIrrefl
 
 theorem mem_asymm {x y : PSet} : x ∈ y → y ∉ x :=
-  asymm
+  asymm (r := (· ∈ ·))
 
 theorem mem_irrefl (x : PSet) : x ∉ x :=
-  irrefl x
+  irrefl (r := (· ∈ ·)) x
 
 /-- Convert a pre-set to a `Set` of pre-sets. -/
 def toSet (u : PSet.{u}) : Set PSet.{u} :=
@@ -617,11 +617,11 @@ theorem eval_mk {n f x} :
 
 /-- The membership relation for ZFC sets is inherited from the membership relation for pre-sets. -/
 protected def Mem : ZFSet → ZFSet → Prop :=
-  Quotient.lift₂ PSet.Mem fun _ _ _ _ hx hy =>
+  Quotient.lift₂ (· ∈ ·) fun _ _ _ _ hx hy =>
     propext ((Mem.congr_left hx).trans (Mem.congr_right hy))
 
-instance : Membership ZFSet ZFSet :=
-  ⟨ZFSet.Mem⟩
+instance : Membership ZFSet ZFSet where
+  mem t s := ZFSet.Mem s t
 
 @[simp]
 theorem mk_mem_iff {x y : PSet} : mk x ∈ mk y ↔ x ∈ y :=
@@ -834,6 +834,10 @@ theorem mem_sep {p : ZFSet.{u} → Prop} {x y : ZFSet.{u}} :
   Quotient.inductionOn₂ x y fun _ _ =>
     PSet.mem_sep (p := p ∘ mk) fun _ _ h => (Quotient.sound h).subst
 
+@[simp]
+theorem sep_empty (p : ZFSet → Prop) : (∅ : ZFSet).sep p = ∅ :=
+  (eq_empty _).mpr fun _ h ↦ not_mem_empty _ (mem_sep.mp h).1
+
 @[simp]
 theorem toSet_sep (a : ZFSet) (p : ZFSet → Prop) :
     (ZFSet.sep p a).toSet = { x ∈ a.toSet | p x } := by
@@ -886,9 +890,8 @@ def sUnion : ZFSet → ZFSet :=
 prefix:110 "⋃₀ " => ZFSet.sUnion
 
 /-- The intersection operator, the collection of elements in all of the elements of a ZFC set. We
-special-case `⋂₀ ∅ = ∅`. -/
-noncomputable def sInter (x : ZFSet) : ZFSet := by
-   classical exact if h : x.Nonempty then ZFSet.sep (fun y => ∀ z ∈ x, y ∈ z) h.some else ∅
+define `⋂₀ ∅ = ∅`. -/
+def sInter (x : ZFSet) : ZFSet := (⋃₀ x).sep (fun y => ∀ z ∈ x, y ∈ z)
 
 @[inherit_doc]
 prefix:110 "⋂₀ " => ZFSet.sInter
@@ -899,9 +902,12 @@ theorem mem_sUnion {x y : ZFSet.{u}} : y ∈ ⋃₀ x ↔ ∃ z ∈ x, y ∈ z :
     ⟨fun ⟨z, h⟩ => ⟨⟦z⟧, h⟩, fun ⟨z, h⟩ => Quotient.inductionOn z (fun z h => ⟨z, h⟩) h⟩
 
 theorem mem_sInter {x y : ZFSet} (h : x.Nonempty) : y ∈ ⋂₀ x ↔ ∀ z ∈ x, y ∈ z := by
-  rw [sInter, dif_pos h]
-  simp only [mem_toSet, mem_sep, and_iff_right_iff_imp]
-  exact fun H => H _ h.some_mem
+  unfold sInter
+  simp only [and_iff_right_iff_imp, mem_sep]
+  intro mem
+  apply mem_sUnion.mpr
+  replace ⟨s, h⟩ := h
+  exact ⟨_, h, mem _ h⟩
 
 @[simp]
 theorem sUnion_empty : ⋃₀ (∅ : ZFSet.{u}) = ∅ := by
@@ -909,7 +915,7 @@ theorem sUnion_empty : ⋃₀ (∅ : ZFSet.{u}) = ∅ := by
   simp
 
 @[simp]
-theorem sInter_empty : ⋂₀ (∅ : ZFSet) = ∅ := dif_neg <| by simp
+theorem sInter_empty : ⋂₀ (∅ : ZFSet) = ∅ := by simp [sInter]
 
 theorem mem_of_mem_sInter {x y z : ZFSet} (hy : y ∈ ⋂₀ x) (hz : z ∈ x) : y ∈ z := by
   rcases eq_empty_or_nonempty x with (rfl | hx)
@@ -1022,10 +1028,10 @@ instance : IsIrrefl ZFSet (· ∈ ·) :=
   mem_wf.isIrrefl
 
 theorem mem_asymm {x y : ZFSet} : x ∈ y → y ∉ x :=
-  asymm
+  asymm (r := (· ∈ ·))
 
 theorem mem_irrefl (x : ZFSet) : x ∉ x :=
-  irrefl x
+  irrefl (r := (· ∈ ·)) x
 
 theorem regularity (x : ZFSet.{u}) (h : x ≠ ∅) : ∃ y ∈ x, x ∩ y = ∅ :=
   by_contradiction fun ne =>
@@ -1120,7 +1126,7 @@ theorem pair_injective : Function.Injective2 pair := fun x x' y y' H => by
       rw [mem_singleton.mp m]
   have he : x = y → y = y' := by
     rintro rfl
-    cases' (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true_iff]) with xy'x xy'xx
+    cases' (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true]) with xy'x xy'xx
     · rw [eq_comm, ← mem_singleton, ← xy'x, mem_pair]
       exact Or.inr rfl
     · simpa [eq_comm] using (ZFSet.ext_iff.1 xy'xx y').1 (by simp)
@@ -1264,7 +1270,7 @@ def ToSet (B : Class.{u}) (A : Class.{u}) : Prop :=
   ∃ x : ZFSet, ↑x = A ∧ B x
 
 /-- `A ∈ B` if `A` is a ZFC set which satisfies `B` -/
-protected def Mem (A B : Class.{u}) : Prop :=
+protected def Mem (B A : Class.{u}) : Prop :=
   ToSet.{u} B A
 
 instance : Membership Class Class :=
@@ -1282,7 +1288,7 @@ theorem not_empty_hom (x : ZFSet.{u}) : ¬(∅ : Class.{u}) x :=
 
 @[simp]
 theorem mem_univ {A : Class.{u}} : A ∈ univ.{u} ↔ ∃ x : ZFSet.{u}, ↑x = A :=
-  exists_congr fun _ => and_true_iff _
+  exists_congr fun _ => iff_of_eq (and_true _)
 
 @[simp]
 theorem mem_univ_hom (x : ZFSet.{u}) : univ.{u} x :=
@@ -1315,10 +1321,10 @@ instance : IsIrrefl Class (· ∈ ·) :=
   mem_wf.isIrrefl
 
 theorem mem_asymm {x y : Class} : x ∈ y → y ∉ x :=
-  asymm
+  asymm (r := (· ∈ ·))
 
 theorem mem_irrefl (x : Class) : x ∉ x :=
-  irrefl x
+  irrefl (r := (· ∈ ·)) x
 
 /-- **There is no universal set.**
 This is stated as `univ ∉ univ`, meaning that `univ` (the class of all sets) is proper (does not
@@ -1333,7 +1339,7 @@ def congToClass (x : Set Class.{u}) : Class.{u} :=
 @[simp]
 theorem congToClass_empty : congToClass ∅ = ∅ := by
   ext z
-  simp only [congToClass, not_empty_hom, iff_false_iff]
+  simp only [congToClass, not_empty_hom, iff_false]
   exact Set.not_mem_empty z
 
 /-- Convert a class into a conglomerate (a collection of classes) -/
@@ -1391,7 +1397,7 @@ theorem coe_sep (p : Class.{u}) (x : ZFSet.{u}) :
 
 @[simp, norm_cast]
 theorem coe_empty : ↑(∅ : ZFSet.{u}) = (∅ : Class.{u}) :=
-  ext fun y => iff_false_iff.2 <| ZFSet.not_mem_empty y
+  ext fun y => iff_false _ ▸ ZFSet.not_mem_empty y
 
 @[simp, norm_cast]
 theorem coe_insert (x y : ZFSet.{u}) : ↑(insert x y) = @insert ZFSet.{u} Class.{u} _ x y :=
@@ -1564,3 +1570,5 @@ noncomputable def toSet_equiv : ZFSet.{u} ≃ {s : Set ZFSet.{u} // Small.{u, u+
   right_inv s := Subtype.coe_injective <| toSet_equiv_aux s.2
 
 end ZFSet
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/SetTheory/ZFC/Rank.lean b/Mathlib/SetTheory/ZFC/Rank.lean
new file mode 100644
index 0000000000000..2cecc412eaa35
--- /dev/null
+++ b/Mathlib/SetTheory/ZFC/Rank.lean
@@ -0,0 +1,209 @@
+/-
+Copyright (c) 2024 Dexin Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dexin Zhang
+-/
+import Mathlib.SetTheory.Ordinal.Arithmetic
+import Mathlib.SetTheory.ZFC.Basic
+
+/-!
+# Ordinal ranks of PSet and ZFSet
+
+In this file, we define the ordinal ranks of `PSet` and `ZFSet`. These ranks are the same as
+`WellFounded.rank` over `∈`, but are defined in a way that the universe levels of ranks are the
+same as the indexing types.
+
+## Definitions
+
+* `PSet.rank`: Ordinal rank of a pre-set.
+* `ZFSet.rank`: Ordinal rank of a ZFC set.
+-/
+
+universe u v
+
+open Ordinal Order
+
+namespace PSet
+
+/-- The ordinal rank of a pre-set -/
+noncomputable def rank : PSet.{u} → Ordinal.{u}
+  | ⟨_, A⟩ => lsub fun a => rank (A a)
+
+theorem rank_congr : ∀ {x y : PSet}, Equiv x y → rank x = rank y
+  | ⟨_, _⟩, ⟨_, _⟩, ⟨αβ, βα⟩ =>
+    lsub_eq_of_range_eq (by
+      ext
+      constructor <;> simp <;> intro a h
+      · obtain ⟨b, h'⟩ := αβ a
+        exists b
+        rw [← h, rank_congr h']
+      · obtain ⟨b, h'⟩ := βα a
+        exists b
+        rw [← h, rank_congr h'])
+
+theorem rank_lt_of_mem : ∀ {x y : PSet}, y ∈ x → rank y < rank x
+  | ⟨_, _⟩, _, ⟨_, h⟩ => by
+    rw [rank_congr h]
+    apply lt_lsub
+
+theorem rank_le_iff {o : Ordinal} : ∀ {x : PSet}, rank x ≤ o ↔ ∀ ⦃y⦄, y ∈ x → rank y < o
+  | ⟨_, A⟩ =>
+    ⟨fun h _ h' => (rank_lt_of_mem h').trans_le h, fun h =>
+      lsub_le fun a => h (Mem.mk A a)⟩
+
+theorem lt_rank_iff {o : Ordinal} {x : PSet} : o < rank x ↔ ∃ y ∈ x, o ≤ rank y := by
+  rw [← not_iff_not, not_lt, rank_le_iff]
+  simp
+
+variable {x y : PSet.{u}}
+
+@[gcongr] theorem rank_mono (h : x ⊆ y) : rank x ≤ rank y :=
+  rank_le_iff.2 fun _ h₁ => rank_lt_of_mem (mem_of_subset h h₁)
+
+@[simp]
+theorem rank_empty : rank ∅ = 0 := by simp [rank]
+
+@[simp]
+theorem rank_insert : rank (insert x y) = max (succ (rank x)) (rank y) := by
+  apply le_antisymm
+  · simp_rw [rank_le_iff, mem_insert_iff]
+    rintro _ (h | h)
+    · simp [rank_congr h]
+    · simp [rank_lt_of_mem h]
+  · apply max_le
+    · exact (rank_lt_of_mem (mem_insert x y)).succ_le
+    · exact rank_mono (subset_iff.2 fun z => mem_insert_of_mem x)
+
+@[simp]
+theorem rank_singleton : rank {x} = succ (rank x) :=
+  rank_insert.trans (by simp)
+
+theorem rank_pair : rank {x, y} = max (succ (rank x)) (succ (rank y)) := by
+  simp
+
+@[simp]
+theorem rank_powerset : rank (powerset x) = succ (rank x) := by
+  apply le_antisymm
+  · simp_rw [rank_le_iff, mem_powerset, lt_succ_iff]
+    intro
+    exact rank_mono
+  · rw [succ_le_iff]
+    apply rank_lt_of_mem
+    simp
+
+/-- For the rank of `⋃₀ x`, we only have `rank (⋃₀ x) ≤ rank x ≤ rank (⋃₀ x) + 1`.
+
+This inequality is split into `rank_sUnion_le` and `le_succ_rank_sUnion`. -/
+theorem rank_sUnion_le : rank (⋃₀ x) ≤ rank x := by
+  simp_rw [rank_le_iff, mem_sUnion]
+  intro _ ⟨_, _, _⟩
+  trans <;> apply rank_lt_of_mem <;> assumption
+
+theorem le_succ_rank_sUnion : rank x ≤ succ (rank (⋃₀ x)) := by
+  rw [← rank_powerset]
+  apply rank_mono
+  rw [subset_iff]
+  intro z _
+  rw [mem_powerset, subset_iff]
+  intro _ _
+  rw [mem_sUnion]
+  exists z
+
+/-- `PSet.rank` is equal to the `WellFounded.rank` over `∈`. -/
+theorem rank_eq_wfRank : lift.{u + 1, u} (rank x) = mem_wf.rank x := by
+  induction' x using mem_wf.induction with x ih
+  rw [mem_wf.rank_eq]
+  simp_rw [← fun y : { y // y ∈ x } => ih y y.2]
+  apply (le_of_forall_lt _).antisymm (Ordinal.iSup_le.{u + 1, u} _) <;> intro h
+  · rw [lt_lift_iff]
+    rintro ⟨o, rfl, h⟩
+    simpa [Ordinal.lt_iSup.{u + 1, u}] using lt_rank_iff.1 h
+  · simpa using rank_lt_of_mem h.2
+
+end PSet
+
+namespace ZFSet
+
+variable {x y : ZFSet.{u}}
+
+/-- The ordinal rank of a ZFC set -/
+noncomputable def rank : ZFSet.{u} → Ordinal.{u} :=
+  Quotient.lift _ fun _ _ => PSet.rank_congr
+
+theorem rank_lt_of_mem : y ∈ x → rank y < rank x :=
+  Quotient.inductionOn₂ x y fun _ _ => PSet.rank_lt_of_mem
+
+theorem rank_le_iff {o : Ordinal} : rank x ≤ o ↔ ∀ ⦃y⦄, y ∈ x → rank y < o :=
+  ⟨fun h _ h' => (rank_lt_of_mem h').trans_le h,
+    Quotient.inductionOn x fun _ h =>
+      PSet.rank_le_iff.2 fun y h' => @h ⟦y⟧ h'⟩
+
+theorem lt_rank_iff {o : Ordinal} : o < rank x ↔ ∃ y ∈ x, o ≤ rank y := by
+  rw [← not_iff_not, not_lt, rank_le_iff]
+  simp
+
+@[gcongr] theorem rank_mono (h : x ⊆ y) : rank x ≤ rank y :=
+  rank_le_iff.2 fun _ h₁ => rank_lt_of_mem (h h₁)
+
+@[simp]
+theorem rank_empty : rank ∅ = 0 := PSet.rank_empty
+
+@[simp]
+theorem rank_insert : rank (insert x y) = max (succ (rank x)) (rank y) :=
+  Quotient.inductionOn₂ x y fun _ _ => PSet.rank_insert
+
+@[simp]
+theorem rank_singleton : rank {x} = succ (rank x) :=
+  rank_insert.trans (by simp)
+
+theorem rank_pair : rank {x, y} = max (succ (rank x)) (succ (rank y)) := by
+  simp
+
+@[simp]
+theorem rank_union : rank (x ∪ y) = max (rank x) (rank y) := by
+  apply le_antisymm
+  · simp_rw [rank_le_iff, mem_union, lt_max_iff]
+    intro
+    apply Or.imp <;> apply rank_lt_of_mem
+  · apply max_le <;> apply rank_mono <;> intro _ h <;> simp [h]
+
+@[simp]
+theorem rank_powerset : rank (powerset x) = succ (rank x) :=
+  Quotient.inductionOn x fun _ => PSet.rank_powerset
+
+/-- For the rank of `⋃₀ x`, we only have `rank (⋃₀ x) ≤ rank x ≤ rank (⋃₀ x) + 1`.
+
+This inequality is split into `rank_sUnion_le` and `le_succ_rank_sUnion`. -/
+theorem rank_sUnion_le : rank (⋃₀ x) ≤ rank x := by
+  simp_rw [rank_le_iff, mem_sUnion]
+  intro _ ⟨_, _, _⟩
+  trans <;> apply rank_lt_of_mem <;> assumption
+
+theorem le_succ_rank_sUnion : rank x ≤ succ (rank (⋃₀ x)) := by
+  rw [← rank_powerset]
+  apply rank_mono
+  intro z _
+  rw [mem_powerset]
+  intro _ _
+  rw [mem_sUnion]
+  exists z
+
+@[simp]
+theorem rank_range {α : Type u} {f : α → ZFSet.{max u v}} :
+    rank (range f) = lsub fun i => rank (f i) := by
+  apply (lsub_le _).antisymm'
+  · simpa [rank_le_iff] using lt_lsub _
+  · simp [rank_lt_of_mem]
+
+/-- `ZFSet.rank` is equal to the `WellFounded.rank` over `∈`. -/
+theorem rank_eq_wfRank : lift.{u + 1, u} (rank x) = mem_wf.rank x := by
+  induction' x using inductionOn with x ih
+  rw [mem_wf.rank_eq]
+  simp_rw [← fun y : { y // y ∈ x } => ih y y.2]
+  apply (le_of_forall_lt _).antisymm (Ordinal.iSup_le.{u + 1, u} _) <;> intro h
+  · rw [lt_lift_iff]
+    rintro ⟨o, rfl, h⟩
+    simpa [Ordinal.lt_iSup.{u + 1, u}] using lt_rank_iff.1 h
+  · simpa using rank_lt_of_mem h.2
+
+end ZFSet
diff --git a/Mathlib/Std/Data/HashMap.lean b/Mathlib/Std/Data/HashMap.lean
new file mode 100644
index 0000000000000..19199a7f7e67b
--- /dev/null
+++ b/Mathlib/Std/Data/HashMap.lean
@@ -0,0 +1,22 @@
+/-
+Copyright (c) 2024 Lean FRO. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kim Morrison
+-/
+import Std.Data.HashMap.Basic
+
+/-!
+# Convenience functions for hash maps
+
+These will be reimplemented in the Lean standard library.
+-/
+
+namespace Std.HashMap
+
+variable {α β γ : Type _} [BEq α] [Hashable α]
+
+/-- Apply a function to the values of a hash map. -/
+def mapVal (f : α → β → γ) (m : HashMap α β) : HashMap α γ :=
+  m.fold (fun acc k v => acc.insert k (f k v)) HashMap.empty
+
+end Std.HashMap
diff --git a/Mathlib/Tactic.lean b/Mathlib/Tactic.lean
index 487a9cf1008d9..46952152d3576 100644
--- a/Mathlib/Tactic.lean
+++ b/Mathlib/Tactic.lean
@@ -1,5 +1,6 @@
 import Mathlib.Tactic.Abel
 import Mathlib.Tactic.AdaptationNote
+import Mathlib.Tactic.Algebraize
 import Mathlib.Tactic.ApplyAt
 import Mathlib.Tactic.ApplyCongr
 import Mathlib.Tactic.ApplyFun
@@ -22,13 +23,25 @@ import Mathlib.Tactic.CancelDenoms.Core
 import Mathlib.Tactic.Cases
 import Mathlib.Tactic.CasesM
 import Mathlib.Tactic.CategoryTheory.BicategoricalComp
+import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
+import Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes
+import Mathlib.Tactic.CategoryTheory.Bicategory.Normalize
+import Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence
 import Mathlib.Tactic.CategoryTheory.BicategoryCoherence
 import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.Tactic.CategoryTheory.Coherence.Basic
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+import Mathlib.Tactic.CategoryTheory.Coherence.Normalize
+import Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence
 import Mathlib.Tactic.CategoryTheory.Elementwise
-import Mathlib.Tactic.CategoryTheory.Monoidal
+import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
+import Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes
+import Mathlib.Tactic.CategoryTheory.Monoidal.Normalize
+import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
 import Mathlib.Tactic.CategoryTheory.MonoidalComp
 import Mathlib.Tactic.CategoryTheory.Reassoc
 import Mathlib.Tactic.CategoryTheory.Slice
+import Mathlib.Tactic.CategoryTheory.ToApp
 import Mathlib.Tactic.Change
 import Mathlib.Tactic.Check
 import Mathlib.Tactic.Choose
@@ -84,6 +97,7 @@ import Mathlib.Tactic.FunProp.ToBatteries
 import Mathlib.Tactic.FunProp.Types
 import Mathlib.Tactic.GCongr
 import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 import Mathlib.Tactic.GCongr.ForwardAttr
 import Mathlib.Tactic.Generalize
 import Mathlib.Tactic.GeneralizeProofs
@@ -118,14 +132,18 @@ import Mathlib.Tactic.Linarith.Preprocessing
 import Mathlib.Tactic.Linarith.Verification
 import Mathlib.Tactic.LinearCombination
 import Mathlib.Tactic.LinearCombination'
+import Mathlib.Tactic.LinearCombination.Lemmas
 import Mathlib.Tactic.Linter
 import Mathlib.Tactic.Linter.AdmitLinter
+import Mathlib.Tactic.Linter.DocPrime
+import Mathlib.Tactic.Linter.FlexibleLinter
 import Mathlib.Tactic.Linter.GlobalAttributeIn
 import Mathlib.Tactic.Linter.HashCommandLinter
 import Mathlib.Tactic.Linter.HaveLetLinter
 import Mathlib.Tactic.Linter.Lint
 import Mathlib.Tactic.Linter.MinImports
 import Mathlib.Tactic.Linter.OldObtain
+import Mathlib.Tactic.Linter.PPRoundtrip
 import Mathlib.Tactic.Linter.RefineLinter
 import Mathlib.Tactic.Linter.Style
 import Mathlib.Tactic.Linter.TextBased
@@ -135,6 +153,7 @@ import Mathlib.Tactic.Measurability.Init
 import Mathlib.Tactic.MinImports
 import Mathlib.Tactic.MkIffOfInductiveProp
 import Mathlib.Tactic.ModCases
+import Mathlib.Tactic.Module
 import Mathlib.Tactic.Monotonicity
 import Mathlib.Tactic.Monotonicity.Attr
 import Mathlib.Tactic.Monotonicity.Basic
@@ -190,6 +209,7 @@ import Mathlib.Tactic.RewriteSearch
 import Mathlib.Tactic.Rify
 import Mathlib.Tactic.Ring
 import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.Ring.Compare
 import Mathlib.Tactic.Ring.PNat
 import Mathlib.Tactic.Ring.RingNF
 import Mathlib.Tactic.Sat.FromLRAT
diff --git a/Mathlib/Tactic/Abel.lean b/Mathlib/Tactic/Abel.lean
index a0c5c9ec31aff..5ccbf7d2844fd 100644
--- a/Mathlib/Tactic/Abel.lean
+++ b/Mathlib/Tactic/Abel.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2018 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Scott Morrison
+Authors: Mario Carneiro, Kim Morrison
 -/
 import Mathlib.Tactic.NormNum.Basic
 import Mathlib.Tactic.TryThis
diff --git a/Mathlib/Tactic/Algebraize.lean b/Mathlib/Tactic/Algebraize.lean
new file mode 100644
index 0000000000000..74c3a9eaf8230
--- /dev/null
+++ b/Mathlib/Tactic/Algebraize.lean
@@ -0,0 +1,298 @@
+/-
+Copyright (c) 2024 Calle Sönne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johan Commelin, Nick Kuhn, Arend Mellendijk, Christian Merten, Calle Sönne, Adam Topaz
+-/
+
+import Mathlib.Algebra.Algebra.Tower
+
+/-!
+
+## Algebraize tactic
+
+This file defines the `algebraize` tactic. The basic functionality of this tactic is to
+automatically add `Algebra` instances given `RingHom`s. For example, `algebraize [f, g]` where
+`f : A →+* B` and `g : B →+* C` are `RingHom`s, will add the instances `Algebra A B` and
+`Algebra B C` corresponding to these `RingHom`s.
+
+## Further functionality
+
+When given a composition of `RingHom`s, e.g. `algebraize [g.comp f]`, the tactic will also try to
+add the instance `IsScalarTower A B C` if possible.
+
+After having added suitable `Algebra` and `IsScalarTower` instances, the tactic will search through
+the local context for `RingHom` properties that can be converted to properties of the corresponding
+`Algebra`. For example, given `f : A →+* B` and `hf : f.FiniteType`, then `algebraize [f]` will add
+the instance `Algebra A B` and the corresponding property `Algebra.FiniteType A B`. The tactic knows
+which `RingHom` properties have a corresponding `Algebra` property through the `algebraize`
+attribute.
+
+## Algebraize attribute
+
+The `algebraize` attribute is used to tag `RingHom` properties that can be converted to `Algebra`
+properties. It assumes that the tagged declaration has a name of the form `RingHom.Property` and
+that the corresponding `Algebra` property has the name `Algebra.Property`.
+
+If not, the `Name` of the corresponding algebra property can be provided as optional argument. The
+specified declaration should be one of the following:
+
+1. An inductive type (i.e. the `Algebra` property itself), in this case it is assumed that the
+`RingHom` and the `Algebra` property are definitionally the same, and the tactic will construct the
+`Algebra` property by giving the `RingHom` property as a term.
+2. A constructor for the `Algebra` property. In this case it is assumed that the `RingHom` property
+is the last argument of the constructor, and that no other explicit argument is needed. The tactic
+then constructs the `Algebra` property by applying the constructor to the `RingHom` property.
+
+Here are three examples of properties tagged with the `algebraize` attribute:
+```
+@[algebraize]
+def RingHom.FiniteType (f : A →+* B) : Prop :=
+  @Algebra.FiniteType A B _ _ f.toAlgebra
+```
+An example when the `Name` is provided (as the `Algebra` does not have the expected name):
+```
+@[algebraize Module.Finite]
+def RingHom.Finite (f : A →+* B) : Prop :=
+  letI : Algebra A B := f.toAlgebra
+  Module.Finite A B
+```
+An example with a constructor as parameter (as the two properties are not definitonally the same):
+```
+@[algebraize Algebra.Flat.out]
+class RingHom.Flat {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) : Prop where
+  out : f.toAlgebra.Flat := by infer_instance
+```
+
+## algebraize_only
+
+To avoid searching through the local context and adding corresponding `Algebra` properties, use
+`algebraize_only` which only adds `Algebra` and `IsScalarTower` instances.
+-/
+
+open Lean Elab Tactic Term Meta
+
+namespace Lean.Attr
+
+/-- Function that extracts the name of the corresponding `Algebra` property from a `RingHom`
+property that has been tagged with the `algebraize` attribute. This is done by either returning the
+parameter of the attribute, or by assuming that the tagged declaration has name `RingHom.Property`
+and then returning `Algebra.Property`. -/
+def algebraizeGetParam (thm : Name) (stx : Syntax) : AttrM Name := do
+  match stx with
+  | `(attr| algebraize $name:ident) => return name.getId
+  /- If no argument is provided, assume `thm` is of the form `RingHom.Property`,
+  and return `Algebra.Property` -/
+  | `(attr| algebraize) =>
+    match thm with
+    | .str `RingHom t => return .str `Algebra t
+    | _ =>
+      throwError "theorem name must be of the form `RingHom.Property` if no argument is provided"
+  | _ => throwError "unexpected algebraize argument"
+
+/-- A user attribute that is used to tag `RingHom` properties that can be converted to `Algebra`
+properties. Using an (optional) parameter, it will also generate a `Name` of a declaration which
+will help the `algebraize` tactic access the corresponding `Algebra` property.
+
+There are two cases for what declaration corresponding to this `Name` can be.
+
+1. An inductive type (i.e. the `Algebra` property itself), in this case it is assumed that the
+`RingHom` and the `Algebra` property are definitionally the same, and the tactic will construct the
+`Algebra` property by giving the `RingHom` property as a term.
+2. A constructor for the `Algebra` property. In this case it is assumed that the `RingHom` property
+is the last argument of the constructor, and that no other explicit argument is needed. The tactic
+then constructs the `Algebra` property by applying the constructor to the `RingHom` property.
+
+Finally, if no argument is provided to the `algebraize` attribute, it is assumed that the tagged
+declaration has name `RingHom.Property` and that the corresponding `Algebra` property has name
+`Algebra.Property`. The attribute then returns `Algebra.Property` (so assume case 1 above). -/
+initialize algebraizeAttr : ParametricAttribute Name ←
+  registerParametricAttribute {
+    name := `algebraize,
+    descr :=
+"Tag that lets the `algebraize` tactic know which `Algebra` property corresponds to this `RingHom`
+property.",
+    getParam := algebraizeGetParam }
+
+end Lean.Attr
+
+namespace Mathlib.Tactic
+
+namespace Algebraize
+
+/-- Given an expression `f` of type `RingHom A B` where `A` and `B` are commutative semirings,
+this function adds the instance `Algebra A B` to the context (if it does not already exist).
+
+This function also requries the type of `f`, given by the parameter `ft`. The reason this is done
+(even though `ft` can be inferred from `f`) is to avoid recomputing `ft` in the `algebraize` tactic,
+as when `algebraize` calls `addAlgebraInstanceFromRingHom` it has already computed `ft`. -/
+def addAlgebraInstanceFromRingHom (f ft : Expr) : TacticM Unit := withMainContext do
+  let (_, l) := ft.getAppFnArgs
+  -- The type of the corresponding algebra instance
+  let alg ← mkAppOptM ``Algebra #[l[0]!, l[1]!, none, none]
+  -- If the instance already exists, we do not do anything
+  unless (← synthInstance? alg).isSome do
+  liftMetaTactic fun mvarid => do
+    let nm ← mkFreshBinderNameForTactic `algInst
+    let mvar ← mvarid.define nm alg (← mkAppM ``RingHom.toAlgebra #[f])
+    let (_, mvar) ← mvar.intro1P
+    return [mvar]
+
+/-- Given an expression `g.comp f` which is the composition of two `RingHom`s, this function adds
+the instance `IsScalarTower A B C` to the context (if it does not already exist). -/
+def addIsScalarTowerInstanceFromRingHomComp (fn : Expr) : TacticM Unit := withMainContext do
+  let (_, l) := fn.getAppFnArgs
+  let tower ← mkAppOptM ``IsScalarTower #[l[0]!, l[1]!, l[2]!, none, none, none]
+  -- If the instance already exists, we do not do anything
+  unless (← synthInstance? tower).isSome do
+  liftMetaTactic fun mvarid => do
+    let nm ← mkFreshBinderNameForTactic `scalarTowerInst
+    let h ← mkFreshExprMVar (← mkAppM ``Eq #[
+      ← mkAppOptM ``algebraMap #[l[0]!, l[2]!, none, none, none],
+      ← mkAppM ``RingHom.comp #[
+        ← mkAppOptM ``algebraMap #[l[1]!, l[2]!, none, none, none],
+        ← mkAppOptM ``algebraMap #[l[0]!, l[1]!, none, none, none]]])
+    -- Note: this could fail, but then `algebraize` will just continue, and won't add this instance
+    h.mvarId!.refl
+    let val ← mkAppOptM ``IsScalarTower.of_algebraMap_eq'
+      #[l[0]!, l[1]!, l[2]!, none, none, none, none, none, none, h]
+    let mvar ← mvarid.define nm tower val
+    let (_, mvar) ← mvar.intro1P
+    return [mvar]
+
+/-- This function takes an array of expressions `t`, all of which are assumed to be `RingHom`s,
+and searches through the local context to find any additional properties of these `RingHoms`, after
+which it tries to add the corresponding `Algebra` properties to the context. It only looks for
+properties that have been tagged with the `algebraize` attribute, and uses this tag to find the
+corresponding `Algebra` property. -/
+def addProperties (t : Array Expr) : TacticM Unit := withMainContext do
+  let ctx ← getLCtx
+  ctx.forM fun decl => do
+    if decl.isImplementationDetail then return
+    let (nm, args) := decl.type.getAppFnArgs
+    -- Check if the type of the current hypothesis has been tagged with the `algebraize` attribute
+    match Attr.algebraizeAttr.getParam? (← getEnv) nm with
+    -- If it has, `p` will be the name of the corresponding `Algebra` property (or a constructor)
+    | some p =>
+      -- The last argument of the `RingHom` property is assumed to be `f`
+      let f := args[args.size - 1]!
+      -- Check that `f` appears in the list of functions given to `algebraize`
+      if ¬ (← t.anyM (Meta.isDefEq · f)) then return
+
+      let cinfo ← getConstInfo p
+      let n ← getExpectedNumArgs cinfo.type
+      let pargs := Array.mkArray n (none : Option Expr)
+      /- If the attribute points to the corresponding `Algebra` property itself, we assume that it
+      is definitionally the same as the `RingHom` property. Then, we just need to construct its type
+      and the local declaration will already give a valid term. -/
+      match cinfo with
+      | .inductInfo _ =>
+        let pargs := pargs.set! 0 args[0]!
+        let pargs := pargs.set! 1 args[1]!
+        let tp ← mkAppOptM p pargs -- This should be the type `Algebra.Property A B`
+        unless (← synthInstance? tp).isSome do
+        liftMetaTactic fun mvarid => do
+          let nm ← mkFreshBinderNameForTactic `algebraizeInst
+          let mvar ← mvarid.define nm tp decl.toExpr
+          let (_, mvar) ← mvar.intro1P
+          return [mvar]
+      /- Otherwise, the attribute points to a constructor of the `Algebra` property. In this case,
+      we assume that the `RingHom` property is the last argument of the constructor (and that
+      this is all we need to supply explicitly). -/
+      | .ctorInfo ctor =>
+        -- construct the desired value
+        let pargs := pargs.set! (n - 1) decl.toExpr
+        let val ← mkAppOptM p pargs
+
+        -- construct the expected type
+        let alg ← mkAppOptM ``Algebra #[args[0]!, args[1]!, none, none]
+        let algInst := (← synthInstance? alg)
+        let mut argsType := Array.mkArray (ctor.numParams) (none : Option Expr)
+        argsType := argsType.set! 0 args[0]!
+        argsType := argsType.set! 1 args[1]!
+        argsType := argsType.set! (ctor.numParams - 1) algInst
+        let tp := ← mkAppOptM ctor.induct argsType
+
+        unless (← synthInstance? tp).isSome do
+        liftMetaTactic fun mvarid => do
+          let nm ← mkFreshBinderNameForTactic `algebraizeInst
+          let mvar ← mvarid.define nm tp val
+          let (_, mvar) ← mvar.intro1P
+          return [mvar]
+      | _ => logError s!"bad argument to `algebraize` attribute: {p}. \
+        Only supporting inductive types or constructors."
+    | none => return
+
+/-- Configuration for `algebraize`. -/
+structure Config where
+  /-- If true (default), the tactic will search the local context for `RingHom` properties
+    that can be converted to `Algebra` properties. -/
+  properties : Bool := true
+deriving Inhabited
+
+/-- Function elaborating `Algebraize.Config`. -/
+declare_config_elab elabAlgebraizeConfig Algebraize.Config
+
+end Algebraize
+
+open Algebraize Lean.Parser.Tactic
+
+/-- A list of terms passed to `algebraize` as argument. -/
+syntax algebraizeTermSeq := " [" withoutPosition(term,*,?) "]"
+
+/-- Tactic that, given `RingHom`s, adds the corresponding `Algebra` and (if possible)
+`IsScalarTower` instances, as well as `Algebra` corresponding to `RingHom` properties available
+as hypotheses.
+
+Example: given `f : A →+* B` and `g : B →+* C`, and `hf : f.FiniteType`, `algebraize [f, g]` will
+add the instances `Algebra A B`, `Algebra B C`, and `Algebra.FiniteType A B`.
+
+See the `algebraize` tag for instructions on what properties can be added.
+
+The tactic also comes with a configuration option `properties`. If set to `true` (default), the
+tactic searches through the local context for `RingHom` properties that can be converted to
+`Algebra` properties. The macro `algebraize_only` calls
+`algebraize (config := {properties := false})`,
+so in other words it only adds `Algebra` and `IsScalarTower` instances. -/
+syntax "algebraize" (ppSpace config)? (ppSpace algebraizeTermSeq)? : tactic
+
+elab_rules : tactic
+  | `(tactic| algebraize $[$config]? $args) => do
+    let cfg ← elabAlgebraizeConfig (mkOptionalNode config)
+    let t ← match args with
+    | `(algebraizeTermSeq| [$rs,*]) => rs.getElems.mapM fun i => Term.elabTerm i none
+    | _ =>
+      throwError ""
+    if t.size == 0 then
+      logWarningAt args "`algebraize []` without arguments has no effect!"
+    -- We loop through the given terms and add algebra instances
+    for f in t do
+      let ft ← inferType f
+      match ft.getAppFn with
+      | Expr.const ``RingHom _ => addAlgebraInstanceFromRingHom f ft
+      | _ => throwError m!"{f} is not of type `RingHom`"
+    -- After having added the algebra instances we try to add scalar tower instances
+    for f in t do
+      match f.getAppFn with
+      | Expr.const ``RingHom.comp _ =>
+        try addIsScalarTowerInstanceFromRingHomComp f
+        catch _ => continue
+      | _ => continue
+
+    -- Search through the local context to find other instances of algebraize
+    if cfg.properties then
+      addProperties t
+  | `(tactic| algebraize $[$config]?) => do
+    throwError "`algebraize` expects a list of arguments: `algebraize [f]`"
+
+/-- Version of `algebraize`, which only adds `Algebra` instances and `IsScalarTower` instances,
+but does not try to add any instances about any properties tagged with
+`@[algebraize]`, like for example `Finite` or `IsIntegral`. -/
+syntax "algebraize_only" (ppSpace algebraizeTermSeq)? : tactic
+
+macro_rules
+  | `(tactic| algebraize_only $args) =>
+    `(tactic| algebraize (config := {properties := false}) $args)
+  | `(tactic| algebraize_only) =>
+    `(tactic| algebraize (config := {properties := false}))
+
+end Mathlib.Tactic
diff --git a/Mathlib/Tactic/ApplyCongr.lean b/Mathlib/Tactic/ApplyCongr.lean
index f2ae29d7ee7c3..ee79d93eba02a 100644
--- a/Mathlib/Tactic/ApplyCongr.lean
+++ b/Mathlib/Tactic/ApplyCongr.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Lucas Allen, Scott Morrison
+Authors: Lucas Allen, Kim Morrison
 -/
 
 import Mathlib.Tactic.Conv
diff --git a/Mathlib/Tactic/ApplyFun.lean b/Mathlib/Tactic/ApplyFun.lean
index e9aa791b2eb59..2b55685474bb7 100644
--- a/Mathlib/Tactic/ApplyFun.lean
+++ b/Mathlib/Tactic/ApplyFun.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Keeley Hoek, Patrick Massot, Scott Morrison
+Authors: Keeley Hoek, Patrick Massot, Kim Morrison
 -/
 import Mathlib.Lean.Expr.Basic
 import Mathlib.Order.Monotone.Basic
diff --git a/Mathlib/Tactic/Attr/Register.lean b/Mathlib/Tactic/Attr/Register.lean
index 98101a86367d7..cc80f10b4abbf 100644
--- a/Mathlib/Tactic/Attr/Register.lean
+++ b/Mathlib/Tactic/Attr/Register.lean
@@ -84,3 +84,6 @@ register_simp_attr nontriviality
 
 /-- A stub attribute for `is_poly`. -/
 register_label_attr is_poly
+
+/-- A simp set for the `fin_omega` wrapper around `omega`. -/
+register_simp_attr fin_omega
diff --git a/Mathlib/Tactic/Bound/Attribute.lean b/Mathlib/Tactic/Bound/Attribute.lean
index a33e962df9870..83685e229820e 100644
--- a/Mathlib/Tactic/Bound/Attribute.lean
+++ b/Mathlib/Tactic/Bound/Attribute.lean
@@ -6,6 +6,7 @@ Authors: Geoffrey Irving
 import Mathlib.Algebra.Group.ZeroOne
 import Mathlib.Tactic.Bound.Init
 import Qq
+import Aesop
 
 /-!
 # The `bound` attribute
diff --git a/Mathlib/Tactic/CC.lean b/Mathlib/Tactic/CC.lean
index 97367e1d058d8..e256310915550 100644
--- a/Mathlib/Tactic/CC.lean
+++ b/Mathlib/Tactic/CC.lean
@@ -125,7 +125,7 @@ def proofFor (ccs : CCState) (e : Expr) : MetaM Expr := do
 def refutationFor (ccs : CCState) (e : Expr) : MetaM Expr := do
   let (some r, _) ← CCM.run (CCM.getEqProof e (.const ``False [])) { ccs with }
     | throwError "CCState.refutationFor failed to build proof"
-  mkAppM ``not_of_eq_false #[r]
+  mkAppM ``of_eq_false #[r]
 
 /-- If the given state is inconsistent, return a proof for `False`. Otherwise fail. -/
 def proofForFalse (ccs : CCState) : MetaM Expr := do
diff --git a/Mathlib/Tactic/CC/Addition.lean b/Mathlib/Tactic/CC/Addition.lean
index 5a805f46f8ba9..51dcb46a44da0 100644
--- a/Mathlib/Tactic/CC/Addition.lean
+++ b/Mathlib/Tactic/CC/Addition.lean
@@ -192,8 +192,8 @@ def mkCCHCongrTheorem (fn : Expr) (nargs : Nat) : CCM (Option CCCongrTheorem) :=
 
   -- Check if `{ fn, nargs }` is in the cache
   let key₁ : CCCongrTheoremKey := { fn, nargs }
-  if let some it₁ := cache.findEntry? key₁ then
-    return it₁.2
+  if let some it := cache[key₁]? then
+    return it
 
   -- Try automatically generated congruence lemma with support for heterogeneous equality.
   let lemm ← mkCCHCongrWithArity fn nargs
@@ -662,7 +662,7 @@ equality to the todo list. If not, add `e` to the congruence table. -/
 def addCongruenceTable (e : Expr) : CCM Unit := do
   guard e.isApp
   let k ← mkCongruencesKey e
-  if let some es := (← get).congruences.find? k then
+  if let some es := (← get).congruences[k]? then
     for oldE in es do
       if ← isCongruent e oldE then
         -- Found new equivalence: `e ~ oldE`
@@ -687,7 +687,7 @@ def addSymmCongruenceTable (e : Expr) : CCM Unit := do
   let some (rel, lhs, rhs) ← e.relSidesIfSymm? | failure
   let k ← mkSymmCongruencesKey lhs rhs
   let newP := (e, rel)
-  if let some ps := (← get).symmCongruences.find? k then
+  if let some ps := (← get).symmCongruences[k]? then
     for p in ps do
       if ← compareSymm newP p then
         -- Found new equivalence: `e ~ p.1`
@@ -842,7 +842,7 @@ def dbgTraceACState : CCM Unit := do
 def mkACProof (e₁ e₂ : Expr) : MetaM Expr := do
   let eq ← mkEq e₁ e₂
   let .mvar m ← mkFreshExprSyntheticOpaqueMVar eq | failure
-  AC.rewriteUnnormalized m
+  AC.rewriteUnnormalizedRefl m
   let pr ← instantiateMVars (.mvar m)
   mkExpectedTypeHint pr eq
 
@@ -1470,7 +1470,8 @@ partial def propagateEqUp (e : Expr) : CCM Unit := do
     if ← isInterpretedValue ra <&&> isInterpretedValue rb <&&>
         pure (ra.int?.isNone || ra.int? != rb.int?) then
       raNeRb := some
-        (Expr.app (.proj ``Iff 0 (← mkAppM ``bne_iff_ne #[ra, rb])) (← mkEqRefl (.const ``true [])))
+        (Expr.app (.proj ``Iff 0 (← mkAppOptM ``bne_iff_ne #[none, none, none, ra, rb]))
+          (← mkEqRefl (.const ``true [])))
     else
       if let some c₁ ← isConstructorApp? ra then
       if let some c₂ ← isConstructorApp? rb then
@@ -1623,7 +1624,7 @@ def removeParents (e : Expr) (parentsToPropagate : Array Expr := #[]) : CCM (Arr
       if pocc.symmTable then
         let some (rel, lhs, rhs) ← p.relSidesIfSymm? | failure
         let k' ← mkSymmCongruencesKey lhs rhs
-        if let some lst := (← get).symmCongruences.find? k' then
+        if let some lst := (← get).symmCongruences[k']? then
           let k := (p, rel)
           let newLst ← lst.filterM fun k₂ => (!·) <$> compareSymm k k₂
           if !newLst.isEmpty then
@@ -1634,7 +1635,7 @@ def removeParents (e : Expr) (parentsToPropagate : Array Expr := #[]) : CCM (Arr
               { ccs with symmCongruences := ccs.symmCongruences.erase k' }
       else
         let k' ← mkCongruencesKey p
-        if let some es := (← get).congruences.find? k' then
+        if let some es := (← get).congruences[k']? then
           let newEs := es.erase p
           if !newEs.isEmpty then
             modify fun ccs =>
@@ -1808,7 +1809,8 @@ def propagateValueInconsistency (e₁ e₂ : Expr) : CCM Unit := do
   let some eqProof ← getEqProof e₁ e₂ | failure
   let trueEqFalse ← mkEq (.const ``True []) (.const ``False [])
   let neProof :=
-    Expr.app (.proj ``Iff 0 (← mkAppM ``bne_iff_ne #[e₁, e₂])) (← mkEqRefl (.const ``true []))
+    Expr.app (.proj ``Iff 0 (← mkAppOptM ``bne_iff_ne #[none, none, none, e₁, e₂]))
+      (← mkEqRefl (.const ``true []))
   let H ← mkAbsurd trueEqFalse eqProof neProof
   pushEq (.const ``True []) (.const ``False []) H
 
@@ -1848,7 +1850,7 @@ def propagateEqDown (e : Expr) : CCM Unit := do
 /-- Propagate equality from `¬∃ x, p x` to `∀ x, ¬p x`. -/
 def propagateExistsDown (e : Expr) : CCM Unit := do
   if ← isEqFalse e then
-    let hNotE ← mkAppM ``not_of_eq_false #[← getEqFalseProof e]
+    let hNotE ← mkAppM ``of_eq_false #[← getEqFalseProof e]
     let (all, hAll) ← e.forallNot_of_notExists hNotE
     internalizeCore all none
     pushEq all (.const ``True []) (← mkEqTrue hAll)
@@ -2099,3 +2101,5 @@ def add (type : Expr) (proof : Expr) : CCM Unit := do
 end CCM
 
 end Mathlib.Tactic.CC
+
+set_option linter.style.longFile 2300
diff --git a/Mathlib/Tactic/CC/Datatypes.lean b/Mathlib/Tactic/CC/Datatypes.lean
index 1f883bb05ea2d..6c11094d2ff5a 100644
--- a/Mathlib/Tactic/CC/Datatypes.lean
+++ b/Mathlib/Tactic/CC/Datatypes.lean
@@ -5,9 +5,9 @@ Authors: Leonardo de Moura, Miyahara Kō
 -/
 import Lean.Meta.CongrTheorems
 import Lean.Meta.Tactic.Rfl
-import Batteries.Data.HashMap.Basic
 import Batteries.Data.RBMap.Basic
 import Mathlib.Lean.Meta.Basic
+import Std.Data.HashMap.Basic
 
 /-!
 # Datatypes for `cc`
@@ -110,7 +110,7 @@ structure CCCongrTheoremKey where
   deriving BEq, Hashable
 
 /-- Caches used to find corresponding `CCCongrTheorem`s. -/
-abbrev CCCongrTheoremCache := Batteries.HashMap CCCongrTheoremKey (Option CCCongrTheorem)
+abbrev CCCongrTheoremCache := Std.HashMap CCCongrTheoremKey (Option CCCongrTheorem)
 
 /-- Configs used in congruence closure modules. -/
 structure CCConfig where
@@ -415,7 +415,7 @@ inductive CongruencesKey
   deriving BEq, Hashable
 
 /-- Maps each expression (via `mkCongruenceKey`) to expressions it might be congruent to. -/
-abbrev Congruences := Batteries.HashMap CongruencesKey (List Expr)
+abbrev Congruences := Std.HashMap CongruencesKey (List Expr)
 
 structure SymmCongruencesKey where
   (h₁ h₂ : Expr)
@@ -426,7 +426,7 @@ structure SymmCongruencesKey where
 The `Name` identifies which relation the congruence is considered for.
 Note that this only works for two-argument relations: `ModEq n` and `ModEq m` are considered the
 same. -/
-abbrev SymmCongruences := Batteries.HashMap SymmCongruencesKey (List (Expr × Name))
+abbrev SymmCongruences := Std.HashMap SymmCongruencesKey (List (Expr × Name))
 
 /-- Stores the root representatives of subsingletons. -/
 abbrev SubsingletonReprs := RBExprMap Expr
@@ -667,7 +667,7 @@ end CCState
 
 /-- The congruence closure module (optionally) uses a normalizer.
     The idea is to use it (if available) to normalize auxiliary expressions
-    produced by internal propagation rules (e.g., subsingleton propagator).  -/
+    produced by internal propagation rules (e.g., subsingleton propagator). -/
 structure CCNormalizer where
   normalize : Expr → MetaM Expr
 
diff --git a/Mathlib/Tactic/CC/Lemmas.lean b/Mathlib/Tactic/CC/Lemmas.lean
index cd6fc4d14d2c8..12348c1e7b4cb 100644
--- a/Mathlib/Tactic/CC/Lemmas.lean
+++ b/Mathlib/Tactic/CC/Lemmas.lean
@@ -4,47 +4,45 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura
 -/
 
-import Mathlib.Init.Logic
-
 /-! Lemmas use by the congruence closure module -/
 
 namespace Mathlib.Tactic.CC
 
 theorem iff_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a ↔ b) = b :=
-  h.symm ▸ propext true_iff_iff
+  h.symm ▸ true_iff _
 
 theorem iff_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a ↔ b) = a :=
-  h.symm ▸ propext iff_true_iff
+  h.symm ▸ iff_true _
 
 theorem iff_eq_true_of_eq {a b : Prop} (h : a = b) : (a ↔ b) = True :=
-  h ▸ propext (iff_self_iff _)
+  h ▸ iff_self _
 
 theorem and_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a ∧ b) = b :=
-  h.symm ▸ propext (true_and_iff _)
+  h.symm ▸ true_and _
 
 theorem and_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a ∧ b) = a :=
-  h.symm ▸ propext (and_true_iff _)
+  h.symm ▸ and_true _
 
 theorem and_eq_of_eq_false_left {a b : Prop} (h : a = False) : (a ∧ b) = False :=
-  h.symm ▸ propext (false_and_iff _)
+  h.symm ▸ false_and _
 
 theorem and_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∧ b) = False :=
-  h.symm ▸ propext (and_false_iff _)
+  h.symm ▸ and_false _
 
 theorem and_eq_of_eq {a b : Prop} (h : a = b) : (a ∧ b) = a :=
   h ▸ propext and_self_iff
 
 theorem or_eq_of_eq_true_left {a b : Prop} (h : a = True) : (a ∨ b) = True :=
-  h.symm ▸ propext (true_or_iff _)
+  h.symm ▸ true_or _
 
 theorem or_eq_of_eq_true_right {a b : Prop} (h : b = True) : (a ∨ b) = True :=
-  h.symm ▸ propext (or_true_iff _)
+  h.symm ▸ or_true _
 
 theorem or_eq_of_eq_false_left {a b : Prop} (h : a = False) : (a ∨ b) = b :=
-  h.symm ▸ propext (false_or_iff _)
+  h.symm ▸ false_or _
 
 theorem or_eq_of_eq_false_right {a b : Prop} (h : b = False) : (a ∨ b) = a :=
-  h.symm ▸ propext (or_false_iff _)
+  h.symm ▸ or_false _
 
 theorem or_eq_of_eq {a b : Prop} (h : a = b) : (a ∨ b) = a :=
   h ▸ propext or_self_iff
@@ -88,7 +86,7 @@ theorem if_eq_of_eq_true {c : Prop} [d : Decidable c] {α : Sort u} (t e : α) (
 
 theorem if_eq_of_eq_false {c : Prop} [d : Decidable c] {α : Sort u} (t e : α) (h : c = False) :
     @ite α c d t e = e :=
-  if_neg (not_of_eq_false h)
+  if_neg (of_eq_false h)
 
 theorem if_eq_of_eq (c : Prop) [d : Decidable c] {α : Sort u} {t e : α} (h : t = e) :
     @ite α c d t e = t :=
diff --git a/Mathlib/Tactic/Cases.lean b/Mathlib/Tactic/Cases.lean
index 9c917e8d070eb..34ad80c027727 100644
--- a/Mathlib/Tactic/Cases.lean
+++ b/Mathlib/Tactic/Cases.lean
@@ -6,6 +6,7 @@ Authors: Mario Carneiro
 import Lean.Elab.Tactic.Induction
 import Batteries.Tactic.OpenPrivate
 import Mathlib.Lean.Expr.Basic
+import Batteries.Data.List.Basic
 
 /-!
 # Backward compatible implementation of lean 3 `cases` tactic
diff --git a/Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean b/Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean
index ec242f570e977..a43d5c5ea901b 100644
--- a/Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean
+++ b/Mathlib/Tactic/CategoryTheory/BicategoricalComp.lean
@@ -26,27 +26,22 @@ Used by the `⊗≫` bicategorical composition operator, and the `coherence` tac
 -/
 class BicategoricalCoherence (f g : a ⟶ b) where
   /-- The chosen structural isomorphism between to 1-morphisms. -/
-  hom : f ⟶ g
-  [isIso : IsIso hom]
+  iso : f ≅ g
 
 /-- Notation for identities up to unitors and associators. -/
 scoped[CategoryTheory.Bicategory] notation " ⊗𝟙 " =>
-  BicategoricalCoherence.hom -- type as \ot 𝟙
-
-attribute [instance] BicategoricalCoherence.isIso
-
-noncomputable section
+  BicategoricalCoherence.iso -- type as \ot 𝟙
 
 /-- Construct an isomorphism between two objects in a bicategorical category
 out of unitors and associators. -/
-def bicategoricalIso (f g : a ⟶ b) [BicategoricalCoherence f g] : f ≅ g :=
-  asIso ⊗𝟙
+abbrev bicategoricalIso (f g : a ⟶ b) [BicategoricalCoherence f g] : f ≅ g :=
+  ⊗𝟙
 
 /-- Compose two morphisms in a bicategorical category,
 inserting unitors and associators between as necessary. -/
 def bicategoricalComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
     (η : f ⟶ g) (θ : h ⟶ i) : f ⟶ i :=
-  η ≫ ⊗𝟙 ≫ θ
+  η ≫ ⊗𝟙.hom ≫ θ
 
 -- type as \ot \gg
 @[inherit_doc bicategoricalComp]
@@ -56,7 +51,7 @@ scoped[CategoryTheory.Bicategory] infixr:80 " ⊗≫ " => bicategoricalComp
 inserting unitors and associators between as necessary. -/
 def bicategoricalIsoComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
     (η : f ≅ g) (θ : h ≅ i) : f ≅ i :=
-  η ≪≫ asIso ⊗𝟙 ≪≫ θ
+  η ≪≫ ⊗𝟙 ≪≫ θ
 
 @[inherit_doc bicategoricalIsoComp]
 scoped[CategoryTheory.Bicategory] infixr:80 " ≪⊗≫ " =>
@@ -66,59 +61,59 @@ namespace BicategoricalCoherence
 
 @[simps]
 instance refl (f : a ⟶ b) : BicategoricalCoherence f f :=
-  ⟨𝟙 _⟩
+  ⟨Iso.refl _⟩
 
 @[simps]
 instance whiskerLeft (f : a ⟶ b) (g h : b ⟶ c)
     [BicategoricalCoherence g h] : BicategoricalCoherence (f ≫ g) (f ≫ h) :=
-  ⟨f ◁ ⊗𝟙⟩
+  ⟨whiskerLeftIso f ⊗𝟙⟩
 
 @[simps]
 instance whiskerRight (f g : a ⟶ b) (h : b ⟶ c)
     [BicategoricalCoherence f g] : BicategoricalCoherence (f ≫ h) (g ≫ h) :=
-  ⟨⊗𝟙 ▷ h⟩
+  ⟨whiskerRightIso ⊗𝟙 h⟩
 
 @[simps]
 instance tensorRight (f : a ⟶ b) (g : b ⟶ b)
     [BicategoricalCoherence (𝟙 b) g] : BicategoricalCoherence f (f ≫ g) :=
-  ⟨(ρ_ f).inv ≫ f ◁ ⊗𝟙⟩
+  ⟨(ρ_ f).symm ≪≫ (whiskerLeftIso f ⊗𝟙)⟩
 
 @[simps]
 instance tensorRight' (f : a ⟶ b) (g : b ⟶ b)
     [BicategoricalCoherence g (𝟙 b)] : BicategoricalCoherence (f ≫ g) f :=
-  ⟨f ◁ ⊗𝟙 ≫ (ρ_ f).hom⟩
+  ⟨whiskerLeftIso f ⊗𝟙 ≪≫ (ρ_ f)⟩
 
 @[simps]
 instance left (f g : a ⟶ b) [BicategoricalCoherence f g] :
     BicategoricalCoherence (𝟙 a ≫ f) g :=
-  ⟨(λ_ f).hom ≫ ⊗𝟙⟩
+  ⟨λ_ f ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance left' (f g : a ⟶ b) [BicategoricalCoherence f g] :
     BicategoricalCoherence f (𝟙 a ≫ g) :=
-  ⟨⊗𝟙 ≫ (λ_ g).inv⟩
+  ⟨⊗𝟙 ≪≫ (λ_ g).symm⟩
 
 @[simps]
 instance right (f g : a ⟶ b) [BicategoricalCoherence f g] :
     BicategoricalCoherence (f ≫ 𝟙 b) g :=
-  ⟨(ρ_ f).hom ≫ ⊗𝟙⟩
+  ⟨ρ_ f ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance right' (f g : a ⟶ b) [BicategoricalCoherence f g] :
     BicategoricalCoherence f (g ≫ 𝟙 b) :=
-  ⟨⊗𝟙 ≫ (ρ_ g).inv⟩
+  ⟨⊗𝟙 ≪≫ (ρ_ g).symm⟩
 
 @[simps]
 instance assoc (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
     [BicategoricalCoherence (f ≫ g ≫ h) i] :
     BicategoricalCoherence ((f ≫ g) ≫ h) i :=
-  ⟨(α_ f g h).hom ≫ ⊗𝟙⟩
+  ⟨α_ f g h ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance assoc' (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : a ⟶ d)
     [BicategoricalCoherence i (f ≫ g ≫ h)] :
     BicategoricalCoherence i ((f ≫ g) ≫ h) :=
-  ⟨⊗𝟙 ≫ (α_ f g h).inv⟩
+  ⟨⊗𝟙 ≪≫ (α_ f g h).symm⟩
 
 end BicategoricalCoherence
 
@@ -126,16 +121,4 @@ end BicategoricalCoherence
 theorem bicategoricalComp_refl {f g h : a ⟶ b} (η : f ⟶ g) (θ : g ⟶ h) : η ⊗≫ θ = η ≫ θ := by
   dsimp [bicategoricalComp]; simp
 
-example {f' : a ⟶ d} {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} {h' : a ⟶ d} (η : f' ⟶ f ≫ g ≫ h)
-    (θ : (f ≫ g) ≫ h ⟶ h') : f' ⟶ h' :=
-    η ⊗≫ θ
-
--- To automatically insert unitors/associators at the beginning or end,
--- you can use `η ⊗≫ 𝟙 _`
-example {f' : a ⟶ d} {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} (η : f' ⟶ (f ≫ g) ≫ h) :
-    f' ⟶ f ≫ g ≫ h :=
-  η ⊗≫ 𝟙 _
-
-end
-
 end CategoryTheory
diff --git a/Mathlib/Tactic/CategoryTheory/Bicategory/Basic.lean b/Mathlib/Tactic/CategoryTheory/Bicategory/Basic.lean
new file mode 100644
index 0000000000000..4a85534c5839e
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Bicategory/Basic.lean
@@ -0,0 +1,55 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Basic
+import Mathlib.Tactic.CategoryTheory.Bicategory.Normalize
+import Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence
+
+/-!
+# `bicategory` tactic
+
+This file provides `bicategory` tactic, which solves equations in a bicategory, where
+the two sides only differ by replacing strings of bicategory structural morphisms (that is,
+associators, unitors, and identities) with different strings of structural morphisms with the same
+source and target. In other words, `bicategory` solves equalities where both sides have the same
+string diagrams.
+
+The core function for the `bicategory` tactic is provided in
+`Mathlib.Tactic.CategoryTheory.Coherence.Basic`. See this file for more details about the
+implementation.
+
+-/
+
+open Lean Meta Elab Tactic
+open CategoryTheory Mathlib.Tactic.BicategoryLike
+
+namespace Mathlib.Tactic.Bicategory
+
+/-- Normalize the both sides of an equality. -/
+def bicategoryNf (mvarId : MVarId) : MetaM (List MVarId) := do
+  BicategoryLike.normalForm Bicategory.Context `bicategory mvarId
+
+@[inherit_doc bicategoryNf]
+elab "bicategory_nf" : tactic => withMainContext do
+  replaceMainGoal (← bicategoryNf (← getMainGoal))
+
+/--
+Use the coherence theorem for bicategories to solve equations in a bicategory,
+where the two sides only differ by replacing strings of bicategory structural morphisms
+(that is, associators, unitors, and identities)
+with different strings of structural morphisms with the same source and target.
+
+That is, `bicategory` can handle goals of the form
+`a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c'`
+where `a = a'`, `b = b'`, and `c = c'` can be proved using `bicategory_coherence`.
+-/
+def bicategory (mvarId : MVarId) : MetaM (List MVarId) :=
+  BicategoryLike.main  Bicategory.Context `bicategory mvarId
+
+@[inherit_doc bicategory]
+elab "bicategory" : tactic => withMainContext do
+  replaceMainGoal <| ← bicategory <| ← getMainGoal
+
+end Mathlib.Tactic.Bicategory
diff --git a/Mathlib/Tactic/CategoryTheory/Bicategory/Datatypes.lean b/Mathlib/Tactic/CategoryTheory/Bicategory/Datatypes.lean
new file mode 100644
index 0000000000000..42225947c25da
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Bicategory/Datatypes.lean
@@ -0,0 +1,511 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+import Mathlib.Tactic.CategoryTheory.BicategoricalComp
+
+/-!
+# Expressions for bicategories
+
+This file converts lean expressions representing 2-morphisms in bicategories into `Mor₂Iso`
+or `Mor` terms. The converted expressions are used in the coherence tactics and the string diagram
+widgets.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike Bicategory
+
+namespace Mathlib.Tactic.Bicategory
+
+/-- The domain of a morphism. -/
+def srcExpr (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, f, _]) => return f
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The codomain of a morphism. -/
+def tgtExpr (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, _, g]) => return g
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The domain of an isomorphism. -/
+def srcExprOfIso (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Iso, #[_, _, f, _]) => return f
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The codomain of an isomorphism. -/
+def tgtExprOfIso (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Iso, #[_, _, _, g]) => return g
+  | _ => throwError m!"{η} is not a morphism"
+
+initialize registerTraceClass `bicategory
+
+/-- The context for evaluating expressions. -/
+structure Context where
+  /-- The level for 2-morphisms. -/
+  level₂ : Level
+  /-- The level for 1-morphisms. -/
+  level₁ : Level
+  /-- The level for objects. -/
+  level₀ : Level
+  /-- The expression for the underlying category. -/
+  B : Q(Type level₀)
+  /-- The bicategory instance. -/
+  instBicategory : Q(Bicategory.{level₂, level₁} $B)
+
+/-- Populate a `context` object for evaluating `e`. -/
+def mkContext? (e : Expr) : MetaM (Option Context) := do
+  let e ← instantiateMVars e
+  let type ← instantiateMVars <| ← inferType e
+  match (← whnfR (← inferType e)).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, f, _]) =>
+    let fType ← instantiateMVars <| ← inferType f
+    match (← whnfR fType).getAppFnArgs with
+    | (``Quiver.Hom, #[_, _, a, _]) =>
+      let B ← inferType a
+      let .succ level₀ ← getLevel B | return none
+      let .succ level₁ ← getLevel fType | return none
+      let .succ level₂ ← getLevel type | return none
+      let .some instBicategory ← synthInstance?
+        (mkAppN (.const ``Bicategory [level₂, level₁, level₀]) #[B]) | return none
+      return some ⟨level₂, level₁, level₀, B, instBicategory⟩
+    | _ => return none
+  | _ => return none
+
+instance : BicategoryLike.Context Bicategory.Context where
+  mkContext? := Bicategory.mkContext?
+
+/-- The monad for the normalization of 2-morphisms. -/
+abbrev BicategoryM := CoherenceM Context
+
+instance : MonadMor₁ BicategoryM where
+  id₁M a := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a_e : Q($ctx.B) := a.e
+    return .id q(𝟙 $a_e) a
+  comp₁M f g := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($b ⟶ $c) := g.e
+    return .comp q($f_e ≫ $g_e) f g
+
+section
+
+universe w v u
+variable {B : Type u} [Bicategory.{w, v} B] {a b c : B}
+
+theorem structuralIso_inv {f g : a ⟶ b} (η : f ≅ g) :
+    η.symm.hom = η.inv := by
+  simp only [Iso.symm_hom]
+
+theorem structuralIsoOfExpr_comp {f g h : a ⟶ b}
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)
+    (θ : g ⟶ h) (θ' : g ≅ h) (ih_θ : θ'.hom = θ) :
+    (η' ≪≫ θ').hom  = η ≫ θ := by
+  simp [ih_η, ih_θ]
+
+theorem structuralIsoOfExpr_whiskerLeft (f : a ⟶ b) {g h : b ⟶ c}
+    (η : g ⟶ h) (η' : g ≅ h) (ih_η : η'.hom = η)  :
+    (whiskerLeftIso f η').hom = f ◁ η := by
+  simp [ih_η]
+
+theorem structuralIsoOfExpr_whiskerRight {f g : a ⟶ b} (h : b ⟶ c)
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)  :
+    (whiskerRightIso η' h).hom = η ▷ h := by
+  simp [ih_η]
+
+theorem StructuralOfExpr_bicategoricalComp {f g h i : a ⟶ b} [BicategoricalCoherence g h]
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) (θ : h ⟶ i) (θ' : h ≅ i) (ih_θ : θ'.hom = θ) :
+    (bicategoricalIsoComp η' θ').hom = η ⊗≫ θ := by
+  simp [ih_η, ih_θ, bicategoricalIsoComp, bicategoricalComp]
+
+end
+
+open MonadMor₁
+
+instance : MonadMor₂Iso BicategoryM where
+  associatorM f g h := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have d : Q($ctx.B) := h.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($b ⟶ $c) := g.e
+    have h_e : Q($c ⟶ $d) := h.e
+    return .associator q(α_ $f_e $g_e $h_e) f g h
+  leftUnitorM f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    return .leftUnitor q(λ_ $f_e) f
+  rightUnitorM f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    return .rightUnitor q(ρ_ $f_e) f
+  id₂M f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    return .id q(Iso.refl $f_e) f
+  coherenceHomM f g inst := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have inst : Q(BicategoricalCoherence $f_e $g_e) := inst
+    match (← whnfI inst).getAppFnArgs with
+    | (``BicategoricalCoherence.mk, #[_, _, _, _, _, _, α]) =>
+      let e : Q($f_e ≅ $g_e) := q(BicategoricalCoherence.iso)
+      return ⟨e, f, g, inst, α⟩
+    | _ => throwError m!"failed to unfold {inst}"
+  comp₂M η θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($a ⟶ $b) := h.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    have θ_e : Q($g_e ≅ $h_e) := θ.e
+    return .comp q($η_e ≪≫ $θ_e) f g h η θ
+  whiskerLeftM f η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← η.srcM
+    let h ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($b ⟶ $c) := g.e
+    have h_e : Q($b ⟶ $c) := h.e
+    have η_e : Q($g_e ≅ $h_e) := η.e
+    return .whiskerLeft q(whiskerLeftIso $f_e $η_e) f g h η
+  whiskerRightM η h := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := h.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($b ⟶ $c) := h.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .whiskerRight q(whiskerRightIso $η_e $h_e) f g η h
+  horizontalCompM _ _ := throwError "horizontal composition is not implemented"
+  symmM η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .inv q(Iso.symm $η_e) f g η
+  coherenceCompM α η θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($a ⟶ $b) := h.e
+    have i_e : Q($a ⟶ $b) := i.e
+    have _inst : Q(BicategoricalCoherence $g_e $h_e) := α.inst
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    have θ_e : Q($h_e ≅ $i_e) := θ.e
+    return .coherenceComp q($η_e ≪⊗≫ $θ_e) f g h i α η θ
+
+open MonadMor₂Iso
+
+instance : MonadMor₂ BicategoryM where
+  homM η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    let e : Q($f_e ⟶ $g_e) := q(Iso.hom $η_e)
+    have eq : Q(Iso.hom $η_e = $e) := q(rfl)
+    return .isoHom q(Iso.hom $η_e) ⟨η, eq⟩ η
+  atomHomM η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f := η.src
+    let g := η.tgt
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .mk q(Iso.hom $η_e) f g
+  invM η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    let e : Q($g_e ⟶ $f_e) := q(Iso.inv $η_e)
+    let η_inv ← symmM η
+    let eq : Q(Iso.inv $η_e = $e) := q(Iso.symm_hom $η_e)
+    return .isoInv e ⟨η_inv, eq⟩ η
+  atomInvM η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f := η.src
+    let g := η.tgt
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .mk q(Iso.inv $η_e) g f
+  id₂M f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    let e : Q($f_e ⟶ $f_e) := q(𝟙 $f_e)
+    let eq : Q(𝟙 $f_e = $e) := q(Iso.refl_hom $f_e)
+    return .id e ⟨.structuralAtom <| ← id₂M f, eq⟩ f
+  comp₂M η θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($a ⟶ $b) := h.e
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    have θ_e : Q($g_e ⟶ $h_e) := θ.e
+    let iso_lift? ← (match (η.isoLift?, θ.isoLift?) with
+      | (some η_iso, some θ_iso) =>
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have θ_iso_e : Q($g_e ≅ $h_e) := θ_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        have θ_iso_eq : Q(Iso.hom $θ_iso_e = $θ_e) := θ_iso.eq
+        let eq := q(structuralIsoOfExpr_comp _ _ $η_iso_eq _ _ $θ_iso_eq)
+        return .some ⟨← comp₂M η_iso.e θ_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⟶ $h_e) := q($η_e ≫ $θ_e)
+    return .comp e iso_lift? f g h η θ
+  whiskerLeftM f η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← η.srcM
+    let h ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($b ⟶ $c) := g.e
+    have h_e : Q($b ⟶ $c) := h.e
+    have η_e : Q($g_e ⟶ $h_e) := η.e
+    let iso_lift? ← (match η.isoLift? with
+      | some η_iso => do
+        have η_iso_e : Q($g_e ≅ $h_e) := η_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        let eq := q(structuralIsoOfExpr_whiskerLeft $f_e _ _ $η_iso_eq)
+        return .some ⟨← whiskerLeftM f η_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ≫ $g_e ⟶ $f_e ≫ $h_e) := q($f_e ◁ $η_e)
+    return .whiskerLeft e iso_lift? f g h η
+  whiskerRightM η h := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := h.src.e
+    have c : Q($ctx.B) := h.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($b ⟶ $c) := h.e
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    let iso_lift? ← (match η.isoLift? with
+      | some η_iso => do
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        let eq := q(structuralIsoOfExpr_whiskerRight $h_e _ _ $η_iso_eq)
+        return .some ⟨← whiskerRightM η_iso.e h, eq⟩
+      | _ => return none)
+    let e : Q($f_e ≫ $h_e ⟶ $g_e ≫ $h_e) := q($η_e ▷ $h_e)
+    return .whiskerRight e iso_lift? f g η h
+  horizontalCompM _ _ := throwError "horizontal composition is not implemented"
+  coherenceCompM α η θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f_e : Q($a ⟶ $b) := f.e
+    have g_e : Q($a ⟶ $b) := g.e
+    have h_e : Q($a ⟶ $b) := h.e
+    have i_e : Q($a ⟶ $b) := i.e
+    have _inst : Q(BicategoricalCoherence $g_e $h_e) := α.inst
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    have θ_e : Q($h_e ⟶ $i_e) := θ.e
+    let iso_lift? ← (match (η.isoLift?, θ.isoLift?) with
+      | (some η_iso, some θ_iso) => do
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have θ_iso_e : Q($h_e ≅ $i_e) := θ_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        have θ_iso_eq : Q(Iso.hom $θ_iso_e = $θ_e) := θ_iso.eq
+        let eq := q(StructuralOfExpr_bicategoricalComp _ _ $η_iso_eq _ _ $θ_iso_eq)
+        return .some ⟨← coherenceCompM α η_iso.e θ_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⟶ $i_e) := q($η_e ⊗≫ $θ_e)
+    return .coherenceComp e iso_lift? f g h i α η θ
+
+/-- Check that `e` is definitionally equal to `𝟙 a`. -/
+def id₁? (e : Expr) : BicategoryM (Option Obj) := do
+  let ctx ← read
+  let _bicat := ctx.instBicategory
+  let a : Q($ctx.B) ← mkFreshExprMVar ctx.B
+  if ← withDefault <| isDefEq e q(𝟙 $a) then
+    return .some ⟨← instantiateMVars a⟩
+  else
+    return none
+
+/-- Return `(f, g)` if `e` is definitionally equal to `f ≫ g`. -/
+def comp? (e : Expr) : BicategoryM (Option (Mor₁ × Mor₁)) := do
+  let ctx ← read
+  let _bicat := ctx.instBicategory
+  let a ← mkFreshExprMVarQ ctx.B
+  let b ← mkFreshExprMVarQ ctx.B
+  let c ← mkFreshExprMVarQ ctx.B
+  let f ← mkFreshExprMVarQ q($a ⟶ $b)
+  let g ← mkFreshExprMVarQ q($b ⟶ $c)
+  if ← withDefault <| isDefEq e q($f ≫ $g) then
+    let a ← instantiateMVars a
+    let b ← instantiateMVars b
+    let c ← instantiateMVars c
+    let f ← instantiateMVars f
+    let g ← instantiateMVars g
+    return some ((.of ⟨f, ⟨a⟩, ⟨b⟩⟩), .of ⟨g, ⟨b⟩, ⟨c⟩⟩)
+  else
+    return none
+
+/-- Construct a `Mor₁` expression from a Lean expression. -/
+partial def mor₁OfExpr (e : Expr) : BicategoryM Mor₁ := do
+  if let some f := (← get).cache.find? e then
+    return f
+  let f ←
+    if let some a ← id₁? e then
+      MonadMor₁.id₁M a
+    else if let some (f, g) ← comp? e then
+      MonadMor₁.comp₁M (← mor₁OfExpr f.e) (← mor₁OfExpr g.e)
+    else
+      return Mor₁.of ⟨e, ⟨← srcExpr e⟩, ⟨ ← tgtExpr e⟩⟩
+  modify fun s => { s with cache := s.cache.insert e f }
+  return f
+
+instance : MkMor₁ BicategoryM where
+  ofExpr := mor₁OfExpr
+
+/-- Construct a `Mor₂Iso` term from a Lean expression. -/
+partial def Mor₂IsoOfExpr (e : Expr) : BicategoryM Mor₂Iso := do
+  match (← whnfR e).getAppFnArgs with
+  | (``Bicategory.associator, #[_, _, _, _, _, _, f, g, h]) =>
+    associatorM' (← MkMor₁.ofExpr f) (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h)
+  | (``Bicategory.leftUnitor, #[_, _, _, _, f]) =>
+    leftUnitorM' (← MkMor₁.ofExpr f)
+  | (``Bicategory.rightUnitor, #[_, _, _, _, f]) =>
+    rightUnitorM' (← MkMor₁.ofExpr f)
+  | (``Iso.refl, #[_, _, f]) =>
+    id₂M' (← MkMor₁.ofExpr f)
+  | (``Iso.symm, #[_, _, _, _, η]) =>
+    symmM (← Mor₂IsoOfExpr η)
+  | (``Iso.trans, #[_, _, _, _, _, η, θ]) =>
+    comp₂M (← Mor₂IsoOfExpr η) (← Mor₂IsoOfExpr θ)
+  | (``Bicategory.whiskerLeftIso, #[_, _, _, _, _, f, _, _, η]) =>
+    whiskerLeftM (← MkMor₁.ofExpr f) (← Mor₂IsoOfExpr η)
+  | (``Bicategory.whiskerRightIso, #[_, _, _, _, _, _, _, η, h]) =>
+    whiskerRightM (← Mor₂IsoOfExpr η) (← MkMor₁.ofExpr h)
+  | (``bicategoricalIsoComp, #[_, _, _, _, _, g, h, _, inst, η, θ]) =>
+    let α ← coherenceHomM (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h) inst
+    coherenceCompM α (← Mor₂IsoOfExpr η) (← Mor₂IsoOfExpr θ)
+  | (``BicategoricalCoherence.iso, #[_, _, _, _, f, g, inst]) =>
+    coherenceHomM' (← MkMor₁.ofExpr f) (← MkMor₁.ofExpr g) inst
+  | _ =>
+    return .of ⟨e, ← MkMor₁.ofExpr (← srcExprOfIso e), ← MkMor₁.ofExpr (← tgtExprOfIso e)⟩
+
+open MonadMor₂ in
+/-- Construct a `Mor₂` term from a Lean expression. -/
+partial def Mor₂OfExpr (e : Expr) : BicategoryM Mor₂ := do
+  match ← whnfR e with
+  -- whnfR version of `Iso.hom η`
+  | .proj ``Iso 0 η => homM (← Mor₂IsoOfExpr η)
+  -- whnfR version of `Iso.inv η`
+  | .proj ``Iso 1 η => invM (← Mor₂IsoOfExpr η)
+  | .app .. => match (← whnfR e).getAppFnArgs with
+    | (``CategoryStruct.id, #[_, _, f]) => id₂M (← MkMor₁.ofExpr f)
+    | (``CategoryStruct.comp, #[_, _, _, _, _, η, θ]) =>
+      comp₂M (← Mor₂OfExpr η) (← Mor₂OfExpr θ)
+    | (``Bicategory.whiskerLeft, #[_, _, _, _, _, f, _, _, η]) =>
+      whiskerLeftM (← MkMor₁.ofExpr f) (← Mor₂OfExpr η)
+    | (``Bicategory.whiskerRight, #[_, _, _, _, _, _, _, η, h]) =>
+      whiskerRightM (← Mor₂OfExpr η) (← MkMor₁.ofExpr h)
+    | (``bicategoricalComp, #[_, _, _, _, _, g, h, _, inst, η, θ]) =>
+      let α ← coherenceHomM (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h) inst
+      coherenceCompM α (← Mor₂OfExpr η) (← Mor₂OfExpr θ)
+    | _ => return .of ⟨e, ← MkMor₁.ofExpr (← srcExpr e), ← MkMor₁.ofExpr (← tgtExpr e)⟩
+  | _ =>
+    return .of ⟨e, ← MkMor₁.ofExpr (← srcExpr e), ← MkMor₁.ofExpr (← tgtExpr e)⟩
+
+instance : BicategoryLike.MkMor₂ BicategoryM where
+  ofExpr := Mor₂OfExpr
+
+instance : MonadCoherehnceHom BicategoryM where
+  unfoldM α := Mor₂IsoOfExpr α.unfold
+
+end Mathlib.Tactic.Bicategory
diff --git a/Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.lean b/Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.lean
new file mode 100644
index 0000000000000..d53255094d5ee
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Bicategory/Normalize.lean
@@ -0,0 +1,548 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Normalize
+import Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes
+
+/-!
+# Normalization of 2-morphisms in bicategories
+
+This file provides the implementation of the normalization given in
+`Mathlib.Tactic.CategoryTheory.Coherence.Normalize`. See this file for more details.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike Bicategory
+
+namespace Mathlib.Tactic.Bicategory
+
+section
+
+universe w v u
+
+variable {B : Type u} [Bicategory.{w, v} B]
+
+variable {a b c d : B}
+variable {f f' g g' h i j : a ⟶ b}
+
+@[nolint synTaut]
+theorem evalComp_nil_nil (α : f ≅ g) (β : g ≅ h) :
+    (α ≪≫ β).hom = (α ≪≫ β).hom := by
+  simp
+
+theorem evalComp_nil_cons (α : f ≅ g) (β : g ≅ h) (η : h ⟶ i) (ηs : i ⟶ j) :
+    α.hom ≫ (β.hom ≫ η ≫ ηs) = (α ≪≫ β).hom ≫ η ≫ ηs := by
+  simp
+
+theorem evalComp_cons (α : f ≅ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : i ⟶ j} {ι : h ⟶ j}
+    (e_ι : ηs ≫ θ = ι)  :
+    (α.hom ≫ η ≫ ηs) ≫ θ = α.hom ≫ η ≫ ι := by
+  simp [e_ι]
+
+theorem eval_comp
+    {η η' : f ⟶ g} {θ θ' : g ⟶ h} {ι : f ⟶ h}
+    (e_η : η = η') (e_θ : θ = θ') (e_ηθ : η' ≫ θ' = ι) :
+    η ≫ θ = ι := by
+  simp [e_η, e_θ, e_ηθ]
+
+theorem eval_of (η : f ⟶ g) :
+    η = (Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom := by
+  simp
+
+theorem eval_monoidalComp
+    {η η' : f ⟶ g} {α : g ≅ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
+    (e_η : η = η') (e_θ : θ = θ') (e_αθ : α.hom ≫ θ' = αθ) (e_ηαθ : η' ≫ αθ = ηαθ) :
+    η ≫ α.hom ≫ θ = ηαθ := by
+  simp [e_η, e_θ, e_αθ, e_ηαθ]
+
+@[nolint synTaut]
+theorem evalWhiskerLeft_nil (f : a ⟶ b) {g h : b ⟶ c} (α : g ≅ h) :
+    (whiskerLeftIso f α).hom = (whiskerLeftIso f α).hom := by
+  simp
+
+theorem evalWhiskerLeft_of_cons
+    {f : a ⟶ b} {g h i j : b ⟶ c}
+    (α : g ≅ h) (η : h ⟶ i) {ηs : i ⟶ j} {θ : f ≫ i ⟶ f ≫ j} (e_θ : f ◁ ηs = θ) :
+    f ◁ (α.hom ≫ η ≫ ηs) = (whiskerLeftIso f α).hom ≫ f ◁ η ≫ θ := by
+  simp [e_θ]
+
+theorem evalWhiskerLeft_comp
+    {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d}
+    {η : h ⟶ i} {η₁ : g ≫ h ⟶ g ≫ i} {η₂ : f ≫ g ≫ h ⟶ f ≫ g ≫ i}
+    {η₃ : f ≫ g ≫ h ⟶ (f ≫ g) ≫ i} {η₄ : (f ≫ g) ≫ h ⟶ (f ≫ g) ≫ i}
+    (e_η₁ : g ◁ η = η₁) (e_η₂ : f ◁ η₁ = η₂)
+    (e_η₃ : η₂ ≫ (α_ _ _ _).inv = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄) :
+    (f ≫ g) ◁ η = η₄ := by
+  simp [e_η₁, e_η₂, e_η₃, e_η₄]
+
+theorem evalWhiskerLeft_id {η : f ⟶ g}
+    {η₁ : f ⟶ 𝟙 a ≫ g} {η₂ : 𝟙 a ≫ f ⟶ 𝟙 a ≫ g}
+    (e_η₁ : η ≫ (λ_ _).inv = η₁) (e_η₂ : (λ_ _).hom ≫ η₁ = η₂) :
+    𝟙 a ◁ η = η₂ := by
+  simp [e_η₁, e_η₂]
+
+theorem eval_whiskerLeft
+    {f : a ⟶ b} {g h : b ⟶ c}
+    {η η' : g ⟶ h} {θ : f ≫ g ⟶ f ≫ h}
+    (e_η : η = η') (e_θ : f ◁ η' = θ) :
+    f ◁ η = θ := by
+  simp [e_η, e_θ]
+
+theorem eval_whiskerRight
+    {f g : a ⟶ b} {h : b ⟶ c}
+    {η η' : f ⟶ g} {θ : f ≫ h ⟶ g ≫ h}
+    (e_η : η = η') (e_θ : η' ▷ h = θ) :
+    η ▷ h = θ := by
+  simp [e_η, e_θ]
+
+@[nolint synTaut]
+theorem evalWhiskerRight_nil (α : f ≅ g) (h : b ⟶ c) :
+    α.hom ▷ h = α.hom ▷ h := by
+  simp
+
+theorem evalWhiskerRightAux_of {f g : a ⟶ b} (η : f ⟶ g) (h : b ⟶ c) :
+    η ▷ h = (Iso.refl _).hom ≫ η ▷ h ≫ (Iso.refl _).hom := by
+  simp
+
+theorem evalWhiskerRight_cons_of_of
+    {f g h i : a ⟶ b} {j : b ⟶ c}
+    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {ηs₁ : h ≫ j ⟶ i ≫ j}
+    {η₁ : g ≫ j ⟶ h ≫ j} {η₂ : g ≫ j ⟶ i ≫ j} {η₃ : f ≫ j ⟶ i ≫ j}
+    (e_ηs₁ : ηs ▷ j = ηs₁) (e_η₁ : η ▷ j = η₁)
+    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃) :
+    (α.hom ≫ η ≫ ηs) ▷ j = η₃ := by
+  simp_all
+
+theorem evalWhiskerRight_cons_whisker
+    {f : a ⟶ b} {g : a ⟶ c} {h i : b ⟶ c} {j : a ⟶ c} {k : c ⟶ d}
+    {α : g ≅ f ≫ h} {η : h ⟶ i} {ηs : f ≫ i ⟶ j}
+    {η₁ : h ≫ k ⟶ i ≫ k} {η₂ : f ≫ (h ≫ k) ⟶ f ≫ (i ≫ k)} {ηs₁ : (f ≫ i) ≫ k ⟶ j ≫ k}
+    {ηs₂ : f ≫ (i ≫ k) ⟶ j ≫ k} {η₃ : f ≫ (h ≫ k) ⟶ j ≫ k} {η₄ : (f ≫ h) ≫ k ⟶ j ≫ k}
+    {η₅ : g ≫ k ⟶ j ≫ k}
+    (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ k = η₁) (e_η₂ : f ◁ η₁ = η₂)
+    (e_ηs₁ : ηs ▷ k = ηs₁) (e_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
+    (e_η₃ : η₂ ≫ ηs₂ = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄)
+    (e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅) :
+    (α.hom ≫ (f ◁ η) ≫ ηs) ▷ k = η₅ := by
+  simp at e_η₁ e_η₅
+  simp [e_η₁, e_η₂, e_ηs₁, e_ηs₂, e_η₃, e_η₄, e_η₅]
+
+theorem evalWhiskerRight_comp
+    {f f' : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d}
+    {η : f ⟶ f'} {η₁ : f ≫ g ⟶ f' ≫ g} {η₂ : (f ≫ g) ≫ h ⟶ (f' ≫ g) ≫ h}
+    {η₃ : (f ≫ g) ≫ h ⟶ f' ≫ (g ≫ h)} {η₄ : f ≫ (g ≫ h) ⟶ f' ≫ (g ≫ h)}
+    (e_η₁ : η ▷ g = η₁) (e_η₂ : η₁ ▷ h = η₂)
+    (e_η₃ : η₂ ≫ (α_ _ _ _).hom = η₃) (e_η₄ : (α_ _ _ _).inv ≫ η₃ = η₄) :
+    η ▷ (g ≫ h) = η₄ := by
+  simp [e_η₁, e_η₂, e_η₃, e_η₄]
+
+theorem evalWhiskerRight_id
+    {η : f ⟶ g} {η₁ : f ⟶ g ≫ 𝟙 b} {η₂ : f ≫ 𝟙 b ⟶ g ≫ 𝟙 b}
+    (e_η₁ : η ≫ (ρ_ _).inv = η₁) (e_η₂ : (ρ_ _).hom ≫ η₁ = η₂) :
+    η ▷ 𝟙 b = η₂ := by
+  simp [e_η₁, e_η₂]
+
+theorem eval_bicategoricalComp
+    {η η' : f ⟶ g} {α : g ≅ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
+    (e_η : η = η') (e_θ : θ = θ') (e_αθ : α.hom ≫ θ' = αθ) (e_ηαθ : η' ≫ αθ = ηαθ) :
+    η ≫ α.hom ≫ θ = ηαθ := by
+  simp [e_η, e_θ, e_αθ, e_ηαθ]
+
+end
+
+open Mor₂Iso
+
+instance : MkEvalComp BicategoryM where
+  mkEvalCompNilNil α β := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← β.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($g ≅ $h) := β.e
+    return q(evalComp_nil_nil $α $β)
+  mkEvalCompNilCons α β η ηs := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← β.tgtM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have i : Q($a ⟶ $b) := i.e
+    have j : Q($a ⟶ $b) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($g ≅ $h) := β.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($i ⟶ $j) := ηs.e.e
+    return q(evalComp_nil_cons $α $β $η $ηs)
+  mkEvalCompCons α η ηs θ ι e_ι := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    let j ← θ.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have i : Q($a ⟶ $b) := i.e
+    have j : Q($a ⟶ $b) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have θ : Q($i ⟶ $j) := θ.e.e
+    have ι : Q($h ⟶ $j) := ι.e.e
+    have e_ι : Q($ηs ≫ $θ = $ι) := e_ι
+    return q(evalComp_cons $α $η $e_ι)
+
+instance : MkEvalWhiskerLeft BicategoryM where
+  mkEvalWhiskerLeftNil f α := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← α.srcM
+    let h ← α.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have α : Q($g ≅ $h) := α.e
+    return q(evalWhiskerLeft_nil $f $α)
+  mkEvalWhiskerLeftOfCons f α η ηs θ e_θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← α.srcM
+    let h ← α.tgtM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have i : Q($b ⟶ $c) := i.e
+    have j : Q($b ⟶ $c) := j.e
+    have α : Q($g ≅ $h) := α.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($i ⟶ $j) := ηs.e.e
+    have θ : Q($f ≫ $i ⟶ $f ≫ $j) := θ.e.e
+    have e_θ : Q($f ◁ $ηs = $θ) := e_θ
+    return q(evalWhiskerLeft_of_cons $α $η $e_θ)
+  mkEvalWhiskerLeftComp f g η η₁ η₂ η₃ η₄ e_η₁ e_η₂ e_η₃ e_η₄ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let h ← η.srcM
+    let i ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have d : Q($ctx.B) := h.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($c ⟶ $d) := h.e
+    have i : Q($c ⟶ $d) := i.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have η₁ : Q($g ≫ $h ⟶ $g ≫ $i) := η₁.e.e
+    have η₂ : Q($f ≫ $g ≫ $h ⟶ $f ≫ $g ≫ $i) := η₂.e.e
+    have η₃ : Q($f ≫ $g ≫ $h ⟶ ($f ≫ $g) ≫ $i) := η₃.e.e
+    have η₄ : Q(($f ≫ $g) ≫ $h ⟶ ($f ≫ $g) ≫ $i) := η₄.e.e
+    have e_η₁ : Q($g ◁ $η = $η₁) := e_η₁
+    have e_η₂ : Q($f ◁ $η₁ = $η₂) := e_η₂
+    have e_η₃ : Q($η₂ ≫ (α_ _ _ _).inv = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).hom ≫ $η₃ = $η₄) := e_η₄
+    return q(evalWhiskerLeft_comp $e_η₁ $e_η₂ $e_η₃ $e_η₄)
+  mkEvalWhiskerLeftId η η₁ η₂ e_η₁ e_η₂ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have η₁ : Q($f ⟶ 𝟙 $a ≫ $g) := η₁.e.e
+    have η₂ : Q(𝟙 $a ≫ $f ⟶ 𝟙 $a ≫ $g) := η₂.e.e
+    have e_η₁ : Q($η ≫ (λ_ _).inv = $η₁) := e_η₁
+    have e_η₂ : Q((λ_ _).hom ≫ $η₁ = $η₂) := e_η₂
+    return q(evalWhiskerLeft_id $e_η₁ $e_η₂)
+
+instance : MkEvalWhiskerRight BicategoryM where
+  mkEvalWhiskerRightAuxOf η h := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := h.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have η : Q($f ⟶ $g) := η.e.e
+    return q(evalWhiskerRightAux_of $η $h)
+  mkEvalWhiskerRightAuxCons _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalWhiskerRightNil α h := do
+      let ctx ← read
+      let _bicat := ctx.instBicategory
+      let f ← α.srcM
+      let g ← α.tgtM
+      have a : Q($ctx.B) := f.src.e
+      have b : Q($ctx.B) := f.tgt.e
+      have c : Q($ctx.B) := h.tgt.e
+      have f : Q($a ⟶ $b) := f.e
+      have g : Q($a ⟶ $b) := g.e
+      have h : Q($b ⟶ $c) := h.e
+      have α : Q($f ≅ $g) := α.e
+      return q(evalWhiskerRight_nil $α $h)
+  mkEvalWhiskerRightConsOfOf j α η ηs ηs₁ η₁ η₂ η₃ e_ηs₁ e_η₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := j.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have i : Q($a ⟶ $b) := i.e
+    have j : Q($b ⟶ $c) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have ηs₁ : Q($h ≫ $j ⟶ $i ≫ $j) := ηs₁.e.e
+    have η₁ : Q($g ≫ $j ⟶ $h ≫ $j) := η₁.e.e
+    have η₂ : Q($g ≫ $j ⟶ $i ≫ $j) := η₂.e.e
+    have η₃ : Q($f ≫ $j ⟶ $i ≫ $j) := η₃.e.e
+    have e_ηs₁ : Q($ηs ▷ $j = $ηs₁) := e_ηs₁
+    have e_η₁ : Q($η ▷ $j = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ≫ $ηs₁ = $η₂) := e_η₂
+    have e_η₃ : Q((whiskerRightIso $α $j).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalWhiskerRight_cons_of_of $e_ηs₁ $e_η₁ $e_η₂ $e_η₃)
+  mkEvalWhiskerRightConsWhisker f k α η ηs η₁ η₂ ηs₁ ηs₂ η₃ η₄ η₅
+      e_η₁ e_η₂ e_ηs₁ e_ηs₂ e_η₃ e_η₄ e_η₅ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← α.srcM
+    let h ← η.srcM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := h.tgt.e
+    have d : Q($ctx.B) := k.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $c) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have i : Q($b ⟶ $c) := i.e
+    have j : Q($a ⟶ $c) := j.e
+    have k : Q($c ⟶ $d) := k.e
+    have α : Q($g ≅ $f ≫ $h) := α.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($f ≫ $i ⟶ $j) := ηs.e.e
+    have η₁ : Q($h ≫ $k ⟶ $i ≫ $k) := η₁.e.e
+    have η₂ : Q($f ≫ ($h ≫ $k) ⟶ $f ≫ ($i ≫ $k)) := η₂.e.e
+    have ηs₁ : Q(($f ≫ $i) ≫ $k ⟶ $j ≫ $k) := ηs₁.e.e
+    have ηs₂ : Q($f ≫ ($i ≫ $k) ⟶ $j ≫ $k) := ηs₂.e.e
+    have η₃ : Q($f ≫ ($h ≫ $k) ⟶ $j ≫ $k) := η₃.e.e
+    have η₄ : Q(($f ≫ $h) ≫ $k ⟶ $j ≫ $k) := η₄.e.e
+    have η₅ : Q($g ≫ $k ⟶ $j ≫ $k) := η₅.e.e
+    have e_η₁ : Q(((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) ▷ $k = $η₁) := e_η₁
+    have e_η₂ : Q($f ◁ $η₁ = $η₂) := e_η₂
+    have e_ηs₁ : Q($ηs ▷ $k = $ηs₁) := e_ηs₁
+    have e_ηs₂ : Q((α_ _ _ _).inv ≫ $ηs₁ = $ηs₂) := e_ηs₂
+    have e_η₃ : Q($η₂ ≫ $ηs₂ = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).hom ≫ $η₃ = $η₄) := e_η₄
+    have e_η₅ : Q((whiskerRightIso $α $k).hom ≫ $η₄ = $η₅) := e_η₅
+    return q(evalWhiskerRight_cons_whisker $e_η₁ $e_η₂ $e_ηs₁ $e_ηs₂ $e_η₃ $e_η₄ $e_η₅)
+  mkEvalWhiskerRightComp g h η η₁ η₂ η₃ η₄ e_η₁ e_η₂ e_η₃ e_η₄ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let f' ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have d : Q($ctx.B) := h.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have f' : Q($a ⟶ $b) := f'.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($c ⟶ $d) := h.e
+    have η : Q($f ⟶ $f') := η.e.e
+    have η₁ : Q($f ≫ $g ⟶ $f' ≫ $g) := η₁.e.e
+    have η₂ : Q(($f ≫ $g) ≫ $h ⟶ ($f' ≫ $g) ≫ $h) := η₂.e.e
+    have η₃ : Q(($f ≫ $g) ≫ $h ⟶ $f' ≫ ($g ≫ $h)) := η₃.e.e
+    have η₄ : Q($f ≫ ($g ≫ $h) ⟶ $f' ≫ ($g ≫ $h)) := η₄.e.e
+    have e_η₁ : Q($η ▷ $g = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ▷ $h = $η₂) := e_η₂
+    have e_η₃ : Q($η₂ ≫ (α_ _ _ _).hom = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).inv ≫ $η₃ = $η₄) := e_η₄
+    return q(evalWhiskerRight_comp $e_η₁ $e_η₂ $e_η₃ $e_η₄)
+  mkEvalWhiskerRightId η η₁ η₂ e_η₁ e_η₂ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η.srcM
+    let g ← η.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have η₁ : Q($f ⟶ $g ≫ 𝟙 $b) := η₁.e.e
+    have η₂ : Q($f ≫ 𝟙 $b ⟶ $g ≫ 𝟙 $b) := η₂.e.e
+    have e_η₁ : Q($η ≫ (ρ_ _).inv = $η₁) := e_η₁
+    have e_η₂ : Q((ρ_ _).hom ≫ $η₁ = $η₂) := e_η₂
+    return q(evalWhiskerRight_id $e_η₁ $e_η₂)
+
+instance : MkEvalHorizontalComp BicategoryM where
+  mkEvalHorizontalCompAuxOf _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompAuxCons _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompAux'Whisker _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompAux'OfWhisker _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompNilNil _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompNilCons _ _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompConsNil _ _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalHorizontalCompConsCons _ _ _ _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+
+instance : MkEval BicategoryM where
+  mkEvalComp η θ η' θ' ι e_η e_θ e_ηθ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let h ← θ'.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have θ : Q($g ⟶ $h) := θ.e
+    have θ' : Q($g ⟶ $h) := θ'.e.e
+    have ι : Q($f ⟶ $h) := ι.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($θ = $θ') := e_θ
+    have e_ηθ : Q($η' ≫ $θ' = $ι) := e_ηθ
+    return q(eval_comp $e_η $e_θ $e_ηθ)
+  mkEvalWhiskerLeft f η η' θ e_η e_θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let g ← η'.srcM
+    let h ← η'.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := g.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have η : Q($g ⟶ $h) := η.e
+    have η' : Q($g ⟶ $h) := η'.e.e
+    have θ : Q($f ≫ $g ⟶ $f ≫ $h) := θ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($f ◁ $η' = $θ) := e_θ
+    return q(eval_whiskerLeft $e_η $e_θ)
+  mkEvalWhiskerRight η h η' θ e_η e_θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have c : Q($ctx.B) := h.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have θ : Q($f ≫ $h ⟶ $g ≫ $h) := θ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($η' ▷ $h = $θ) := e_θ
+    return q(eval_whiskerRight $e_η $e_θ)
+  mkEvalHorizontalComp _ _ _ _ _ _ _ _ := do
+    throwError "not implemented"
+  mkEvalOf η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f := η.src
+    let g := η.tgt
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have η : Q($f ⟶ $g) := η.e
+    return q(eval_of $η)
+  mkEvalMonoidalComp η θ α η' θ' αθ ηαθ e_η e_θ e_αθ e_ηαθ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let h ← α.tgtM
+    let i ← θ'.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have h : Q($a ⟶ $b) := h.e
+    have i : Q($a ⟶ $b) := i.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have α : Q($g ≅ $h) := α.e
+    have θ : Q($h ⟶ $i) := θ.e
+    have θ' : Q($h ⟶ $i) := θ'.e.e
+    have αθ : Q($g ⟶ $i) := αθ.e.e
+    have ηαθ : Q($f ⟶ $i) := ηαθ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($θ = $θ') := e_θ
+    have e_αθ : Q(Iso.hom $α ≫ $θ' = $αθ) := e_αθ
+    have e_ηαθ : Q($η' ≫ $αθ = $ηαθ) := e_ηαθ
+    return q(eval_bicategoricalComp $e_η $e_θ $e_αθ $e_ηαθ)
+
+instance : MonadNormalExpr BicategoryM where
+  whiskerRightM η h := do
+    return .whisker (← MonadMor₂.whiskerRightM η.e (.of h)) η h
+  hConsM _ _ := do
+    throwError "not implemented"
+  whiskerLeftM f η := do
+    return .whisker (← MonadMor₂.whiskerLeftM (.of f) η.e) f η
+  nilM α := do
+    return .nil (← MonadMor₂.homM α) α
+  consM α η ηs := do
+    return .cons (← MonadMor₂.comp₂M (← MonadMor₂.homM α) (← MonadMor₂.comp₂M η.e ηs.e)) α η ηs
+
+instance : MkMor₂ BicategoryM where
+  ofExpr := Mor₂OfExpr
+
+end Mathlib.Tactic.Bicategory
diff --git a/Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean b/Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean
new file mode 100644
index 0000000000000..6185cb4adb483
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Bicategory/PureCoherence.lean
@@ -0,0 +1,280 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence
+import Mathlib.Tactic.CategoryTheory.Bicategory.Datatypes
+
+/-!
+# Coherence tactic for bicategories
+
+We provide a `bicategory_coherence` tactic,
+which proves that any two morphisms (with the same source and target)
+in a bicategory which are built out of associators and unitors
+are equal.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike Bicategory
+
+namespace Mathlib.Tactic.Bicategory
+
+section
+
+universe w v u
+
+variable {B : Type u} [Bicategory.{w, v} B] {a b c d e : B}
+
+local infixr:81 " ◁ " => Bicategory.whiskerLeftIso
+local infixl:81 " ▷ " => Bicategory.whiskerRightIso
+
+/-- The composition of the normalizing isomorphisms `η_f : p ≫ f ≅ pf` and `η_g : pf ≫ g ≅ pfg`. -/
+abbrev normalizeIsoComp {p : a ⟶ b} {f : b ⟶ c} {g : c ⟶ d} {pf : a ⟶ c} {pfg : a ⟶ d}
+    (η_f : p ≫ f ≅ pf) (η_g : pf ≫ g ≅ pfg) :=
+  (α_ _ _ _).symm ≪≫ whiskerRightIso η_f g ≪≫ η_g
+
+theorem naturality_associator
+    {p : a ⟶ b} {f : b ⟶ c} {g : c ⟶ d} {h : d ⟶ e} {pf : a ⟶ c} {pfg : a ⟶ d} {pfgh : a ⟶ e}
+    (η_f : p ≫ f ≅ pf) (η_g : pf ≫ g ≅ pfg) (η_h : pfg ≫ h ≅ pfgh) :
+    p ◁ (α_ f g h) ≪≫ (normalizeIsoComp η_f (normalizeIsoComp η_g η_h)) =
+    (normalizeIsoComp (normalizeIsoComp η_f η_g) η_h) :=
+  Iso.ext (by simp)
+
+theorem naturality_leftUnitor {p : a ⟶ b} {f : b ⟶ c} {pf : a ⟶ c} (η_f : p ≫ f ≅ pf) :
+    p ◁ (λ_ f) ≪≫ η_f = normalizeIsoComp (ρ_ p) η_f :=
+  Iso.ext (by simp)
+
+theorem naturality_rightUnitor {p : a ⟶ b} {f : b ⟶ c} {pf : a ⟶ c} (η_f : p ≫ f ≅ pf) :
+    p ◁ (ρ_ f) ≪≫ η_f = normalizeIsoComp η_f (ρ_ pf) :=
+  Iso.ext (by simp)
+
+theorem naturality_id {p : a ⟶ b} {f : b ⟶ c} {pf : a ⟶ c} (η_f : p ≫ f ≅ pf) :
+    p ◁ Iso.refl f ≪≫ η_f = η_f :=
+  Iso.ext (by simp)
+
+theorem naturality_comp {p : a ⟶ b} {f g h : b ⟶ c} {pf : a ⟶ c} {η : f ≅ g} {θ : g ≅ h}
+    (η_f : p ≫ f ≅ pf) (η_g : p ≫ g ≅ pf) (η_h : p ≫ h ≅ pf)
+    (ih_η : p ◁ η ≪≫ η_g = η_f) (ih_θ : p ◁ θ ≪≫ η_h = η_g) :
+    p ◁ (η ≪≫ θ) ≪≫ η_h = η_f := by
+  rw [← ih_η, ← ih_θ]
+  apply Iso.ext (by simp)
+
+theorem naturality_whiskerLeft {p : a ⟶ b} {f : b ⟶ c} {g h : c ⟶ d} {pf : a ⟶ c} {pfg : a ⟶ d}
+    {η : g ≅ h} (η_f : p ≫ f ≅ pf) (η_fg : pf ≫ g ≅ pfg) (η_fh : pf ≫ h ≅ pfg)
+    (ih_η : pf ◁ η ≪≫ η_fh = η_fg) :
+    p ◁ (f ◁ η) ≪≫ normalizeIsoComp η_f η_fh = normalizeIsoComp η_f η_fg := by
+  rw [← ih_η]
+  apply Iso.ext (by simp [← whisker_exchange_assoc])
+
+theorem naturality_whiskerRight {p : a ⟶ b} {f g : b ⟶ c} {h : c ⟶ d} {pf : a ⟶ c} {pfh : a ⟶ d}
+    {η : f ≅ g} (η_f : p ≫ f ≅ pf) (η_g : p ≫ g ≅ pf) (η_fh : pf ≫ h ≅ pfh)
+    (ih_η : p ◁ η ≪≫ η_g = η_f) :
+    p ◁ (η ▷ h) ≪≫ normalizeIsoComp η_g η_fh = normalizeIsoComp η_f η_fh := by
+  rw [← ih_η]
+  apply Iso.ext (by simp)
+
+theorem naturality_inv {p : a ⟶ b} {f g : b ⟶ c} {pf : a ⟶ c}
+    {η : f ≅ g} (η_f : p ≫ f ≅ pf) (η_g : p ≫ g ≅ pf) (ih : p ◁ η ≪≫ η_g = η_f) :
+    p ◁ η.symm ≪≫ η_f = η_g := by
+  rw [← ih]
+  apply Iso.ext (by simp)
+
+instance : MonadNormalizeNaturality BicategoryM where
+  mkNaturalityAssociator p pf pfg pfgh f g h η_f η_g η_h := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have d : Q($ctx.B) := g.tgt.e
+    have e : Q($ctx.B) := h.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have g : Q($c ⟶ $d) := g.e
+    have h : Q($d ⟶ $e) := h.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have pfg : Q($a ⟶ $d) := pfg.e.e
+    have pfgh : Q($a ⟶ $e) := pfgh.e.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    have η_g : Q($pf ≫ $g ≅ $pfg) := η_g.e
+    have η_h : Q($pfg ≫ $h ≅ $pfgh) := η_h.e
+    return q(naturality_associator $η_f $η_g $η_h)
+  mkNaturalityLeftUnitor p pf f η_f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    return q(naturality_leftUnitor $η_f)
+  mkNaturalityRightUnitor p pf f η_f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    return q(naturality_rightUnitor $η_f)
+  mkNaturalityId p pf f η_f := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    return q(naturality_id $η_f)
+  mkNaturalityComp p pf f g h η θ η_f η_g η_h ih_η ih_θ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($b ⟶ $c) := h.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have η : Q($f ≅ $g) := η.e
+    have θ : Q($g ≅ $h) := θ.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ≫ $g ≅ $pf) := η_g.e
+    have η_h : Q($p ≫ $h ≅ $pf) := η_h.e
+    have ih_η : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih_η
+    have ih_θ : Q($p ◁ $θ ≪≫ $η_h = $η_g) := ih_θ
+    return q(naturality_comp $η_f $η_g $η_h $ih_η $ih_θ)
+  mkNaturalityWhiskerLeft p pf pfg f g h η η_f η_fg η_fh ih_η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have d : Q($ctx.B) := g.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have g : Q($c ⟶ $d) := g.e
+    have h : Q($c ⟶ $d) := h.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have pfg : Q($a ⟶ $d) := pfg.e.e
+    have η : Q($g ≅ $h) := η.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    have η_fg : Q($pf ≫ $g ≅ $pfg) := η_fg.e
+    have η_fh : Q($pf ≫ $h ≅ $pfg) := η_fh.e
+    have ih_η : Q($pf ◁ $η ≪≫ $η_fh = $η_fg) := ih_η
+    return q(naturality_whiskerLeft $η_f $η_fg $η_fh $ih_η)
+  mkNaturalityWhiskerRight p pf pfh f g h η η_f η_g η_fh ih_η := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have d : Q($ctx.B) := h.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have h : Q($c ⟶ $d) := h.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have pfh : Q($a ⟶ $d) := pfh.e.e
+    have η : Q($f ≅ $g) := η.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ≫ $g ≅ $pf) := η_g.e
+    have η_fh : Q($pf ≫ $h ≅ $pfh) := η_fh.e
+    have ih_η : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih_η
+    return q(naturality_whiskerRight $η_f $η_g $η_fh $ih_η)
+  mkNaturalityHorizontalComp _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ := do
+    throwError "horizontal composition is not implemented"
+  mkNaturalityInv p pf f g η η_f η_g ih := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    have a : Q($ctx.B) := p.src.e
+    have b : Q($ctx.B) := p.tgt.e
+    have c : Q($ctx.B) := f.tgt.e
+    have p : Q($a ⟶ $b) := p.e.e
+    have f : Q($b ⟶ $c) := f.e
+    have g : Q($b ⟶ $c) := g.e
+    have pf : Q($a ⟶ $c) := pf.e.e
+    have η : Q($f ≅ $g) := η.e
+    have η_f : Q($p ≫ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ≫ $g ≅ $pf) := η_g.e
+    have ih : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih
+    return q(naturality_inv $η_f $η_g $ih)
+
+theorem of_normalize_eq {f g f' : a ⟶ b} {η θ : f ≅ g} (η_f : 𝟙 a ≫ f ≅ f') (η_g : 𝟙 a ≫ g ≅ f')
+    (h_η : 𝟙 a ◁ η ≪≫ η_g = η_f)
+    (h_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f) : η = θ := by
+  apply Iso.ext
+  calc
+    η.hom = (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom := by
+      simp [← reassoc_of% (congrArg Iso.hom h_η)]
+    _ = θ.hom := by
+      simp [← reassoc_of% (congrArg Iso.hom h_θ)]
+
+theorem mk_eq_of_naturality {f g f' : a ⟶ b} {η θ : f ⟶ g} {η' θ' : f ≅ g}
+    (η_f : 𝟙 a ≫ f ≅ f') (η_g : 𝟙 a ≫ g ≅ f')
+    (Hη : η'.hom = η) (Hθ : θ'.hom = θ)
+    (Hη' : whiskerLeftIso (𝟙 a) η' ≪≫ η_g = η_f)
+    (Hθ' : whiskerLeftIso (𝟙 a) θ' ≪≫ η_g = η_f) : η = θ :=
+  calc
+    η = η'.hom := Hη.symm
+    _ = (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom := by
+      simp [← reassoc_of% (congrArg Iso.hom Hη')]
+    _ = θ'.hom := by
+      simp [← reassoc_of% (congrArg Iso.hom Hθ')]
+    _ = θ := Hθ
+
+end
+
+instance : MkEqOfNaturality BicategoryM where
+  mkEqOfNaturality η θ ηIso θIso η_f η_g Hη Hθ := do
+    let ctx ← read
+    let _bicat := ctx.instBicategory
+    let η' := ηIso.e
+    let θ' := θIso.e
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let f' ← η_f.tgtM
+    have a : Q($ctx.B) := f.src.e
+    have b : Q($ctx.B) := f.tgt.e
+    have f : Q($a ⟶ $b) := f.e
+    have g : Q($a ⟶ $b) := g.e
+    have f' : Q($a ⟶ $b) := f'.e
+    have η : Q($f ⟶ $g) := η
+    have θ : Q($f ⟶ $g) := θ
+    have η'_e : Q($f ≅ $g) := η'.e
+    have θ'_e : Q($f ≅ $g) := θ'.e
+    have η_f : Q(𝟙 $a ≫ $f ≅ $f') := η_f.e
+    have η_g : Q(𝟙 $a ≫ $g ≅ $f') := η_g.e
+    have η_hom : Q(Iso.hom $η'_e = $η) := ηIso.eq
+    have Θ_hom : Q(Iso.hom $θ'_e = $θ) := θIso.eq
+    have Hη : Q(whiskerLeftIso (𝟙 $a) $η'_e ≪≫ $η_g = $η_f) := Hη
+    have Hθ : Q(whiskerLeftIso (𝟙 $a) $θ'_e ≪≫ $η_g = $η_f) := Hθ
+    return q(mk_eq_of_naturality $η_f $η_g $η_hom $Θ_hom $Hη $Hθ)
+
+open Elab.Tactic
+
+/-- Close the goal of the form `η = θ`, where `η` and `θ` are 2-isomorphisms made up only of
+associators, unitors, and identities.
+```lean
+example {B : Type} [Bicategory B] {a : B} :
+  (λ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom := by
+  bicategory_coherence
+```
+-/
+def pureCoherence (mvarId : MVarId) : MetaM (List MVarId) :=
+  BicategoryLike.pureCoherence Bicategory.Context `bicategory mvarId
+
+@[inherit_doc pureCoherence]
+elab "bicategory_coherence" : tactic => withMainContext do
+  replaceMainGoal <| ← Bicategory.pureCoherence <| ← getMainGoal
+
+end Mathlib.Tactic.Bicategory
diff --git a/Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean b/Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean
index 7f68ebf77c516..2e58492c79e90 100644
--- a/Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean
+++ b/Mathlib/Tactic/CategoryTheory/BicategoryCoherence.lean
@@ -112,8 +112,10 @@ def mkLiftMap₂LiftExpr (e : Expr) : TermElabM Expr := do
 def bicategory_coherence (g : MVarId) : TermElabM Unit := g.withContext do
   withOptions (fun opts => synthInstance.maxSize.set opts
     (max 256 (synthInstance.maxSize.get opts))) do
-  -- TODO: is this `dsimp only` step necessary? It doesn't appear to be in the tests below.
-  let (ty, _) ← dsimp (← g.getType) (← Simp.Context.ofNames [] true)
+  let thms := [``BicategoricalCoherence.iso, ``Iso.trans, ``Iso.symm, ``Iso.refl,
+    ``Bicategory.whiskerRightIso, ``Bicategory.whiskerLeftIso].foldl
+    (·.addDeclToUnfoldCore ·) {}
+  let (ty, _) ← dsimp (← g.getType) { simpTheorems := #[thms] }
   let some (_, lhs, rhs) := (← whnfR ty).eq? | exception g "Not an equation of morphisms."
   let lift_lhs ← mkLiftMap₂LiftExpr lhs
   let lift_rhs ← mkLiftMap₂LiftExpr rhs
diff --git a/Mathlib/Tactic/CategoryTheory/Coherence.lean b/Mathlib/Tactic/CategoryTheory/Coherence.lean
index 9aff001798481..a479585f2c608 100644
--- a/Mathlib/Tactic/CategoryTheory/Coherence.lean
+++ b/Mathlib/Tactic/CategoryTheory/Coherence.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Yuma Mizuno, Oleksandr Manzyuk
+Authors: Kim Morrison, Yuma Mizuno, Oleksandr Manzyuk
 -/
 import Mathlib.CategoryTheory.Monoidal.Free.Coherence
 import Mathlib.Lean.Meta
@@ -121,8 +121,10 @@ def mkProjectMapExpr (e : Expr) : TermElabM Expr := do
 def monoidal_coherence (g : MVarId) : TermElabM Unit := g.withContext do
   withOptions (fun opts => synthInstance.maxSize.set opts
     (max 512 (synthInstance.maxSize.get opts))) do
-  -- TODO: is this `dsimp only` step necessary? It doesn't appear to be in the tests below.
-  let (ty, _) ← dsimp (← g.getType) (← Simp.Context.ofNames [] true)
+  let thms := [``MonoidalCoherence.iso, ``Iso.trans, ``Iso.symm, ``Iso.refl,
+    ``MonoidalCategory.whiskerRightIso, ``MonoidalCategory.whiskerLeftIso].foldl
+    (·.addDeclToUnfoldCore ·) {}
+  let (ty, _) ← dsimp (← g.getType) { simpTheorems := #[thms] }
   let some (_, lhs, rhs) := (← whnfR ty).eq? | exception g "Not an equation of morphisms."
   let projectMap_lhs ← mkProjectMapExpr lhs
   let projectMap_rhs ← mkProjectMapExpr rhs
@@ -184,7 +186,10 @@ elab (name := liftable_prefixes) "liftable_prefixes" : tactic => do
     (max 256 (synthInstance.maxSize.get opts))) do
   evalTactic (← `(tactic|
     (simp (config := {failIfUnchanged := false}) only
-      [monoidalComp, Category.assoc, MonoidalCoherence.hom]) <;>
+      [monoidalComp, bicategoricalComp, Category.assoc, BicategoricalCoherence.iso,
+      MonoidalCoherence.iso, Iso.trans, Iso.symm, Iso.refl,
+      MonoidalCategory.whiskerRightIso, MonoidalCategory.whiskerLeftIso,
+      Bicategory.whiskerRightIso, Bicategory.whiskerLeftIso]) <;>
     (apply (cancel_epi (𝟙 _)).1 <;> try infer_instance) <;>
     (simp (config := {failIfUnchanged := false}) only
       [assoc_liftHom, Mathlib.Tactic.BicategoryCoherence.assoc_liftHom₂])))
@@ -284,9 +289,7 @@ syntax (name := coherence) "coherence" : tactic
 elab_rules : tactic
 | `(tactic| coherence) => do
   evalTactic (← `(tactic|
-    (simp (config := {failIfUnchanged := false}) only [bicategoricalComp,
-      BicategoricalCoherence.hom,
-      monoidalComp]);
+    (simp (config := {failIfUnchanged := false}) only [bicategoricalComp, monoidalComp]);
     whisker_simps (config := {failIfUnchanged := false});
     monoidal_simps (config := {failIfUnchanged := false})))
   coherence_loop
diff --git a/Mathlib/Tactic/CategoryTheory/Coherence/Basic.lean b/Mathlib/Tactic/CategoryTheory/Coherence/Basic.lean
new file mode 100644
index 0000000000000..53facfd616c82
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Coherence/Basic.lean
@@ -0,0 +1,107 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Normalize
+import Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence
+import Mathlib.CategoryTheory.Category.Basic
+
+/-!
+# The Core function for `monoidal` and `bicategory` tactics
+
+This file provides the function `BicategoryLike.main` for proving equalities in monoidal categories
+and bicategories. Using `main`, we will define the following tactics:
+- `monoidal` at `Mathlib.Tactic.CategoryTheory.Monoidal.Basic`
+- `bicategory` at `Mathlib.Tactic.CategoryTheory.Bicategory.Basic`
+
+The `main` first normalizes the both sides using `eval`, then compares the corresponding components.
+It closes the goal at non-structural parts with `rfl` and the goal at structural parts by
+`pureCoherence`.
+
+-/
+
+open Lean Meta Elab
+open CategoryTheory Mathlib.Tactic.BicategoryLike
+
+namespace Mathlib.Tactic.BicategoryLike
+
+theorem mk_eq {α : Type _} (a b a' b' : α) (ha : a = a') (hb : b = b') (h : a' = b') : a = b := by
+  simp [h, ha, hb]
+
+/-- Transform an equality between 2-morphisms into the equality between their normalizations. -/
+def normalForm (ρ : Type) [Context ρ]
+    [MonadMor₁ (CoherenceM ρ)]
+    [MonadMor₂Iso (CoherenceM ρ)]
+    [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)]
+    [MkMor₂ (CoherenceM ρ)]
+    [MonadMor₂ (CoherenceM ρ)]
+    (nm : Name) (mvarId : MVarId) : MetaM (List MVarId) := do
+  mvarId.withContext do
+    let e ← instantiateMVars <| ← mvarId.getType
+    withTraceNode nm (fun _ => return m!"normalize: {e}") do
+      let some (_, e₁, e₂) := (← whnfR <| ← instantiateMVars <| e).eq?
+        | throwError "{nm}_nf requires an equality goal"
+      let ctx : ρ ← mkContext e₁
+      CoherenceM.run (ctx := ctx) do
+        let e₁' ← MkMor₂.ofExpr e₁
+        let e₂' ← MkMor₂.ofExpr e₂
+        let e₁'' ← eval nm e₁'
+        let e₂'' ← eval nm e₂'
+        let H ← mkAppM ``mk_eq #[e₁, e₂, e₁''.expr.e.e, e₂''.expr.e.e, e₁''.proof, e₂''.proof]
+        mvarId.apply H
+
+universe v u
+
+theorem mk_eq_of_cons {C : Type u} [CategoryStruct.{v} C]
+    {f₁ f₂ f₃ f₄ : C}
+    (α α' : f₁ ⟶ f₂) (η η' : f₂ ⟶ f₃) (ηs ηs' : f₃ ⟶ f₄)
+    (e_α : α = α') (e_η : η = η') (e_ηs : ηs = ηs') :
+    α ≫ η ≫ ηs = α' ≫ η' ≫ ηs' := by
+  simp [e_α, e_η, e_ηs]
+
+/-- Split the goal `α ≫ η ≫ ηs = α' ≫ η' ≫ ηs'` into `α = α'`, `η = η'`, and `ηs = ηs'`. -/
+def ofNormalizedEq (mvarId : MVarId) : MetaM (List MVarId) := do
+  mvarId.withContext do
+    let e ← instantiateMVars <| ← mvarId.getType
+    let some (_, e₁, e₂) := (← whnfR e).eq? | throwError "requires an equality goal"
+    match (← whnfR e₁).getAppFnArgs, (← whnfR e₂).getAppFnArgs with
+    | (``CategoryStruct.comp, #[_, _, _, _, _, α, η]) ,
+      (``CategoryStruct.comp, #[_, _, _, _, _, α', η']) =>
+      match (← whnfR η).getAppFnArgs, (← whnfR η').getAppFnArgs with
+      | (``CategoryStruct.comp, #[_, _, _, _, _, η, ηs]),
+        (``CategoryStruct.comp, #[_, _, _, _, _, η', ηs']) =>
+        let e_α ← mkFreshExprMVar (← Meta.mkEq α α')
+        let e_η  ← mkFreshExprMVar (← Meta.mkEq η η')
+        let e_ηs ← mkFreshExprMVar (← Meta.mkEq ηs ηs')
+        let x ← mvarId.apply (← mkAppM ``mk_eq_of_cons #[α, α', η, η', ηs, ηs', e_α, e_η, e_ηs])
+        return x
+      | _, _ => throwError "failed to make a normalized equality for {e}"
+    | _, _ => throwError "failed to make a normalized equality for {e}"
+
+/-- List.splitEvenOdd [0, 1, 2, 3, 4] = ([0, 2, 4], [1, 3]) -/
+def List.splitEvenOdd {α : Type u} : List α → List α × List α
+  | [] => ([], [])
+  | [a] => ([a], [])
+  | a::b::xs =>
+    let (as, bs) := List.splitEvenOdd xs
+    (a::as, b::bs)
+
+/-- The core function for `monoidal` and `bicategory` tactics. -/
+def main (ρ : Type) [Context ρ] [MonadMor₁ (CoherenceM ρ)] [MonadMor₂Iso (CoherenceM ρ)]
+    [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)] [MkMor₂ (CoherenceM ρ)]
+    [MonadMor₂ (CoherenceM ρ)] [MonadCoherehnceHom (CoherenceM ρ)]
+    [MonadNormalizeNaturality (CoherenceM ρ)] [MkEqOfNaturality (CoherenceM ρ)]
+    (nm : Name) (mvarId : MVarId) : MetaM (List MVarId) :=
+  mvarId.withContext do
+    let mvarIds ← normalForm ρ nm mvarId
+    let (mvarIdsCoherence, mvarIdsRefl) := List.splitEvenOdd (← repeat' ofNormalizedEq mvarIds)
+    for mvarId in mvarIdsRefl do mvarId.refl
+    let mvarIds'' ← mvarIdsCoherence.mapM fun mvarId => do
+      withTraceNode nm (fun _ => do return m!"goal: {← mvarId.getType}") do
+        try
+          pureCoherence ρ nm mvarId
+        catch _ => return [mvarId]
+    return mvarIds''.join
+
+end Mathlib.Tactic.BicategoryLike
diff --git a/Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.lean b/Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.lean
new file mode 100644
index 0000000000000..52e90c1260e65
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Coherence/Datatypes.lean
@@ -0,0 +1,467 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Lean
+
+/-!
+# Datatypes for bicategory like structures
+
+This file defines the basic datatypes for bicategory like structures. We will use these datatypes
+to write tactics that can be applied to both monoidal categories and bicategories:
+- `Obj`: objects type
+- `Atom₁`: atomic 1-morphisms type
+- `Mor₁`: 1-morphisms type
+- `Atom`: atomic non-structural 2-morphisms type
+- `Mor₂`: 2-morphisms type
+- `AtomIso`: atomic non-structural 2-isomorphisms type
+- `Mor₂Iso`: 2-isomorphisms type
+- `NormalizedHom`: normalized 1-morphisms type
+
+A term of these datatypes wraps the corresponding `Expr` term, which can be extracted by
+e.g. `η.e` for `η : Mor₂`.
+
+The operations of these datatypes are defined in a monad `m` with the corresponding typeclasses:
+- `MonadMor₁`: operations on `Mor₁`
+- `MonadMor₂Iso`: operations on `Mor₂Iso`
+- `MonadMor₂`: operations on `Mor₂`
+
+For example, a monad `m` with `[MonadMor₂ m]` provides the operation
+`MonadMor₂.comp₂M : Mor₂Iso → Mor₂Iso → m Mor₂Iso`, which constructs the expression for the
+composition `η ≫ θ` of 2-morphisms `η` and `θ` in the monad `m`.
+
+-/
+
+open Lean Meta
+
+namespace Mathlib.Tactic
+
+namespace BicategoryLike
+
+/-- Expressions for objects. -/
+structure Obj where
+  /-- Extracts a lean expression from an `Obj` term. Return `none` in the monoidal
+  category context. -/
+  e? : Option Expr
+  deriving Inhabited
+
+/-- Extract a lean expression from an `Obj` term. -/
+def Obj.e (a : Obj) : Expr :=
+  a.e?.get!
+
+/-- Expressions for atomic 1-morphisms. -/
+structure Atom₁ : Type where
+  /-- Extract a lean expression from an `Atom₁` term. -/
+  e : Expr
+  /-- The domain of the 1-morphism. -/
+  src : Obj
+  /-- The codomain of the 1-morphism. -/
+  tgt : Obj
+  deriving Inhabited
+
+/-- A monad equipped with the ability to construct `Atom₁` terms. -/
+class MkAtom₁ (m : Type → Type) where
+  /-- Construct a `Atom₁` term from a lean expression. -/
+  ofExpr (e : Expr) : m Atom₁
+
+/-- Expressions for 1-morphisms. -/
+inductive Mor₁ : Type
+  /-- `id e a` is the expression for `𝟙 a`, where `e` is the underlying lean expression. -/
+  | id (e : Expr) (a : Obj) : Mor₁
+  /-- `comp e f g` is the expression for `f ≫ g`, where `e` is the underlying lean expression. -/
+  | comp (e : Expr) : Mor₁ → Mor₁ → Mor₁
+  /-- The expression for an atomic 1-morphism. -/
+  | of : Atom₁ → Mor₁
+  deriving Inhabited
+
+/-- A monad equipped with the ability to construct `Mor₁` terms. -/
+class MkMor₁ (m : Type → Type) where
+  /-- Construct a `Mor₁` term from a lean expression. -/
+  ofExpr (e : Expr) : m Mor₁
+
+/-- The underlying lean expression of a 1-morphism. -/
+def Mor₁.e : Mor₁ → Expr
+  | .id e _ => e
+  | .comp e _ _ => e
+  | .of a => a.e
+
+/-- The domain of a 1-morphism. -/
+def Mor₁.src : Mor₁ → Obj
+  | .id _ a => a
+  | .comp _ f _ => f.src
+  | .of f => f.src
+
+/-- The codomain of a 1-morphism. -/
+def Mor₁.tgt : Mor₁ → Obj
+  | .id _ a => a
+  | .comp _ _ g => g.tgt
+  | .of f => f.tgt
+
+/-- Converts a 1-morphism into a list of its components. -/
+def Mor₁.toList : Mor₁ → List Atom₁
+  | .id _ _ => []
+  | .comp _ f g => f.toList ++ g.toList
+  | .of f => [f]
+
+/-- A monad equipped with the ability to manipulate 1-morphisms. -/
+class MonadMor₁ (m : Type → Type) where
+  /-- The expression for `𝟙 a`. -/
+  id₁M (a : Obj) : m Mor₁
+  /-- The expression for `f ≫ g`. -/
+  comp₁M (f g : Mor₁) : m Mor₁
+
+/-- Expressions for coherence isomorphisms (i.e., structural 2-morphisms
+giveb by `BicategorycalCoherence.iso`). -/
+structure CoherenceHom where
+  /-- The underlying lean expression of a coherence isomorphism. -/
+  e : Expr
+  /-- The domain of a coherence isomorphism. -/
+  src : Mor₁
+  /-- The codomain of a coherence isomorphism. -/
+  tgt : Mor₁
+  /-- The `BicategoricalCoherence` instance. -/
+  inst : Expr
+  /-- Extract the structural 2-isomorphism. -/
+  unfold : Expr
+  deriving Inhabited
+
+/-- Expressions for atomic non-structural 2-isomorphisms. -/
+structure AtomIso where
+  /-- The underlying lean expression of an `AtomIso` term. -/
+  e : Expr
+  /-- The domain of a 2-isomorphism. -/
+  src : Mor₁
+  /-- The codomain of a 2-isomorphism. -/
+  tgt : Mor₁
+  deriving Inhabited
+
+/-- Expressions for atomic structural 2-morphisms. -/
+inductive StructuralAtom : Type
+  /-- The expression for the associator `α_ f g h`. -/
+  | associator (e : Expr) (f g h : Mor₁) : StructuralAtom
+  /-- The expression for the left unitor `λ_ f`. -/
+  | leftUnitor (e : Expr) (f : Mor₁) : StructuralAtom
+  /-- The expression for the right unitor `ρ_ f`. -/
+  | rightUnitor (e : Expr) (f : Mor₁) : StructuralAtom
+  | id (e : Expr) (f : Mor₁) : StructuralAtom
+  | coherenceHom (α : CoherenceHom) : StructuralAtom
+  deriving Inhabited
+
+/-- Expressions for 2-isomorphisms. -/
+inductive Mor₂Iso : Type where
+  | structuralAtom (α : StructuralAtom) : Mor₂Iso
+  | comp (e : Expr) (f g h : Mor₁) (η θ : Mor₂Iso) : Mor₂Iso
+  | whiskerLeft (e : Expr) (f g h : Mor₁) (η : Mor₂Iso) : Mor₂Iso
+  | whiskerRight (e : Expr) (f g : Mor₁) (η : Mor₂Iso) (h : Mor₁) : Mor₂Iso
+  | horizontalComp (e : Expr) (f₁ g₁ f₂ g₂ : Mor₁) (η θ : Mor₂Iso) : Mor₂Iso
+  | inv (e : Expr) (f g : Mor₁) (η : Mor₂Iso) : Mor₂Iso
+  | coherenceComp (e : Expr) (f g h i : Mor₁) (α : CoherenceHom) (η θ : Mor₂Iso) : Mor₂Iso
+  | of (η : AtomIso) : Mor₂Iso
+  deriving Inhabited
+
+/-- A monad equipped with the ability to unfold `BicategoricalCoherence.iso`. -/
+class MonadCoherehnceHom (m : Type → Type) where
+  /-- Unfold a coherence isomorphism. -/
+  unfoldM (α : CoherenceHom) : m Mor₂Iso
+
+/-- The underlying lean expression of a 2-isomorphism. -/
+def StructuralAtom.e : StructuralAtom → Expr
+  | .associator e .. => e
+  | .leftUnitor e .. => e
+  | .rightUnitor e .. => e
+  | .id e .. => e
+  | .coherenceHom α => α.e
+
+open MonadMor₁
+
+variable {m : Type → Type} [Monad m]
+
+/-- The domain of a 2-isomorphism. -/
+def StructuralAtom.srcM [MonadMor₁ m] : StructuralAtom → m Mor₁
+  | .associator _ f g h => do comp₁M (← comp₁M f g) h
+  | .leftUnitor _ f => do comp₁M (← id₁M f.src) f
+  | .rightUnitor _ f => do comp₁M f (← id₁M f.tgt)
+  | .id _ f => return f
+  | .coherenceHom α => return α.src
+
+/-- The codomain of a 2-isomorphism. -/
+def StructuralAtom.tgtM [MonadMor₁ m] : StructuralAtom → m Mor₁
+  | .associator _ f g h => do comp₁M f (← comp₁M g h)
+  | .leftUnitor _ f => return f
+  | .rightUnitor _ f => return f
+  | .id _ f => return f
+  | .coherenceHom α => return α.tgt
+
+/-- The underlying lean expression of a 2-isomorphism. -/
+def Mor₂Iso.e : Mor₂Iso → Expr
+  | .structuralAtom α => α.e
+  | .comp e .. => e
+  | .whiskerLeft e .. => e
+  | .whiskerRight e .. => e
+  | .horizontalComp e .. => e
+  | .inv e .. => e
+  | .coherenceComp e .. => e
+  | .of η => η.e
+
+/-- The domain of a 2-isomorphism. -/
+def Mor₂Iso.srcM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂Iso → m Mor₁
+  | .structuralAtom α => α.srcM
+  | .comp _ f .. => return f
+  | .whiskerLeft _ f g .. => do comp₁M f g
+  | .whiskerRight _ f _ _ h => do comp₁M f h
+  | .horizontalComp _ f₁ _ f₂ .. => do comp₁M f₁ f₂
+  | .inv _ _ g _ => return g
+  | .coherenceComp _ f .. => return f
+  | .of η => return η.src
+
+/-- The codomain of a 2-isomorphism. -/
+def Mor₂Iso.tgtM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂Iso → m Mor₁
+  | .structuralAtom α => α.tgtM
+  | .comp _ _ _ h .. => return h
+  | .whiskerLeft _ f _ h _ => do comp₁M f h
+  | .whiskerRight _ _ g _ h => do comp₁M g h
+  | .horizontalComp _ _ g₁ _ g₂ _ _ => do comp₁M g₁ g₂
+  | .inv _ f _ _ => return f
+  | .coherenceComp _ _ _ _ i .. => return i
+  | .of η => return η.tgt
+
+/-- A monad equipped with the ability to construct `Mor₂Iso` terms. -/
+class MonadMor₂Iso (m : Type → Type) where
+  /-- The expression for the associator `α_ f g h`. -/
+  associatorM (f g h : Mor₁) : m StructuralAtom
+  /-- The expression for the left unitor `λ_ f`. -/
+  leftUnitorM (f : Mor₁) : m StructuralAtom
+  /-- The expression for the right unitor `ρ_ f`. -/
+  rightUnitorM (f : Mor₁) : m StructuralAtom
+  /-- The expression for the identity `Iso.refl f`. -/
+  id₂M (f : Mor₁) : m StructuralAtom
+  /-- The expression for the coherence isomorphism `⊗𝟙 : f ⟶ g`. -/
+  coherenceHomM (f g : Mor₁) (inst : Expr) : m CoherenceHom
+  /-- The expression for the composition `η ≪≫ θ`. -/
+  comp₂M (η θ : Mor₂Iso) : m Mor₂Iso
+  /-- The expression for the left whiskering `whiskerLeftIso f η`. -/
+  whiskerLeftM (f : Mor₁) (η : Mor₂Iso) : m Mor₂Iso
+  /-- The expression for the right whiskering `whiskerRightIso η h`. -/
+  whiskerRightM (η : Mor₂Iso) (h : Mor₁) : m Mor₂Iso
+  /-- The expression for the horizontal composition `η ◫ θ`. -/
+  horizontalCompM (η θ : Mor₂Iso) : m Mor₂Iso
+  /-- The expression for the inverse `Iso.symm η`. -/
+  symmM (η : Mor₂Iso) : m Mor₂Iso
+  /-- The expression for the coherence composition `η ≪⊗≫ θ := η ≪≫ α ≪≫ θ`. -/
+  coherenceCompM (α : CoherenceHom) (η θ : Mor₂Iso) : m Mor₂Iso
+
+namespace MonadMor₂Iso
+
+variable {m : Type → Type} [Monad m] [MonadMor₂Iso m]
+
+/-- The expression for the associator `α_ f g h`. -/
+def associatorM' (f g h : Mor₁) : m Mor₂Iso := do
+  return .structuralAtom <| ← MonadMor₂Iso.associatorM f g h
+
+/-- The expression for the left unitor `λ_ f`. -/
+def leftUnitorM' (f : Mor₁) : m Mor₂Iso := do
+  return .structuralAtom <| ← MonadMor₂Iso.leftUnitorM f
+
+/-- The expression for the right unitor `ρ_ f`. -/
+def rightUnitorM' (f : Mor₁) : m Mor₂Iso := do
+  return .structuralAtom <| ← MonadMor₂Iso.rightUnitorM f
+
+/-- The expression for the identity `Iso.refl f`. -/
+def id₂M' (f : Mor₁) : m Mor₂Iso := do
+  return .structuralAtom <| ← MonadMor₂Iso.id₂M f
+
+/-- The expression for the coherence isomorphism `⊗𝟙 : f ⟶ g`. -/
+def coherenceHomM' (f g : Mor₁) (inst : Expr) : m Mor₂Iso := do
+  return .structuralAtom <| .coherenceHom <| ← MonadMor₂Iso.coherenceHomM f g inst
+
+end MonadMor₂Iso
+
+/-- Expressions for atomic non-structural 2-morphisms. -/
+structure Atom where
+  /-- Extract a lean expression from an `Atom` expression. -/
+  e : Expr
+  /-- The domain of a 2-morphism. -/
+  src : Mor₁
+  /-- The codomain of a 2-morphism. -/
+  tgt : Mor₁
+  deriving Inhabited
+
+/-- `Mor₂` expressions defined below will have the `isoLift? : Option IsoLift` field.
+For `η : Mor₂` such that `η.isoLift? = .some isoLift`, we have the following data:
+- `isoLift.e`: an expression for a 2-isomorphism `η'`, given as a `Mor₂Iso` term,
+- `isoLift.eq`: a lean expression for the proof that `η'.hom = η`.
+-/
+structure IsoLift where
+  /-- The expression for the 2-isomorphism. -/
+  e : Mor₂Iso
+  /-- The expression for the proof that the forward direction of the 2-isomorphism is equal to
+  the original 2-morphism. -/
+  eq : Expr
+
+/-- Expressions for 2-morphisms. -/
+inductive Mor₂ : Type where
+  /-- The expression for `Iso.hom`. -/
+  | isoHom (e : Expr) (isoLift : IsoLift) (iso : Mor₂Iso) : Mor₂
+  /-- The expression for `Iso.inv`. -/
+  | isoInv (e : Expr) (isoLift : IsoLift) (iso : Mor₂Iso) : Mor₂
+  /-- The expression for the identity `𝟙 f`. -/
+  | id (e : Expr) (isoLift : IsoLift) (f : Mor₁) : Mor₂
+  /-- The expression for the composition `η ≫ θ`. -/
+  | comp (e : Expr) (isoLift? : Option IsoLift) (f g h : Mor₁) (η θ : Mor₂) : Mor₂
+  /-- The expression for the left whiskering `f ◁ η` with `η : g ⟶ h`. -/
+  | whiskerLeft (e : Expr) (isoLift? : Option IsoLift) (f g h : Mor₁) (η : Mor₂) : Mor₂
+  /-- The expression for the right whiskering `η ▷ h` with `η : f ⟶ g`. -/
+  | whiskerRight (e : Expr) (isoLift? : Option IsoLift) (f g : Mor₁) (η : Mor₂) (h : Mor₁) : Mor₂
+  /-- The expression for the horizontal composition `η ◫ θ` with `η : f₁ ⟶ g₁` and `θ : f₂ ⟶ g₂`. -/
+  | horizontalComp (e : Expr) (isoLift? : Option IsoLift) (f₁ g₁ f₂ g₂ : Mor₁) (η θ : Mor₂) : Mor₂
+  /-- The expression for the coherence composition `η ⊗≫ θ := η ≫ α ≫ θ` with `η : f ⟶ g`
+  and `θ : h ⟶ i`. -/
+  | coherenceComp (e : Expr) (isoLift? : Option IsoLift) (f g h i : Mor₁)
+    (α : CoherenceHom) (η θ : Mor₂) : Mor₂
+  /-- The expression for an atomic non-structural 2-morphism. -/
+  | of (η : Atom) : Mor₂
+  deriving Inhabited
+
+/-- A monad equipped with the ability to construct `Mor₂` terms. -/
+class MkMor₂ (m : Type → Type) where
+  /-- Construct a `Mor₂` term from a lean expression. -/
+  ofExpr (e : Expr) : m Mor₂
+
+/-- The underlying lean expression of a 2-morphism. -/
+def Mor₂.e : Mor₂ → Expr
+  | .isoHom e .. => e
+  | .isoInv e .. => e
+  | .id e .. => e
+  | .comp e .. => e
+  | .whiskerLeft e .. => e
+  | .whiskerRight e .. => e
+  | .horizontalComp e .. => e
+  | .coherenceComp e .. => e
+  | .of η => η.e
+
+/-- `η.isoLift?` is a pair of a 2-isomorphism `η'` and a proof that `η'.hom = η`. If no such `η'`
+is found, returns `none`. This function does not seek `IsIso` instance. -/
+def Mor₂.isoLift? : Mor₂ → Option IsoLift
+  | .isoHom _ isoLift .. => some isoLift
+  | .isoInv _ isoLift .. => some isoLift
+  | .id _ isoLift .. => some isoLift
+  | .comp _ isoLift? .. => isoLift?
+  | .whiskerLeft _ isoLift? .. => isoLift?
+  | .whiskerRight _ isoLift? .. => isoLift?
+  | .horizontalComp _ isoLift? .. => isoLift?
+  | .coherenceComp _ isoLift? .. => isoLift?
+  | .of _ => none
+
+/-- The domain of a 2-morphism. -/
+def Mor₂.srcM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂ → m Mor₁
+  | .isoHom _ _ iso => iso.srcM
+  | .isoInv _ _ iso => iso.tgtM
+  | .id _ _ f => return f
+  | .comp _ _ f .. => return f
+  | .whiskerLeft _ _ f g .. => do comp₁M f g
+  | .whiskerRight _ _ f _ _ h => do comp₁M f h
+  | .horizontalComp _ _ f₁ _ f₂ .. => do comp₁M f₁ f₂
+  | .coherenceComp _ _ f .. => return f
+  | .of η => return η.src
+
+/-- The codomain of a 2-morphism. -/
+def Mor₂.tgtM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂ → m Mor₁
+  | .isoHom _ _ iso => iso.tgtM
+  | .isoInv _ _ iso => iso.srcM
+  | .id _ _ f => return f
+  | .comp _ _ _ _ h .. => return h
+  | .whiskerLeft _ _ f _ h _ => do comp₁M f h
+  | .whiskerRight _ _ _ g _ h => do comp₁M g h
+  | .horizontalComp _ _ _ g₁ _ g₂ _ _ => do comp₁M g₁ g₂
+  | .coherenceComp _ _ _ _ _ i .. => return i
+  | .of η => return η.tgt
+
+/-- A monad equipped with the ability to manipulate 2-morphisms. -/
+class MonadMor₂ (m : Type → Type) where
+  /-- The expression for `Iso.hom η`. -/
+  homM (η : Mor₂Iso) : m Mor₂
+  /-- The expression for `Iso.hom η`. -/
+  atomHomM (η : AtomIso) : m Atom
+  /-- The expression for `Iso.inv η`. -/
+  invM (η : Mor₂Iso) : m Mor₂
+  /-- The expression for `Iso.inv η`. -/
+  atomInvM (η : AtomIso) : m Atom
+  /-- The expression for the identity `𝟙 f`. -/
+  id₂M (f : Mor₁) : m Mor₂
+  /-- The expression for the composition `η ≫ θ`. -/
+  comp₂M (η θ : Mor₂) : m Mor₂
+  /-- The expression for the left whiskering `f ◁ η`. -/
+  whiskerLeftM (f : Mor₁) (η : Mor₂) : m Mor₂
+  /-- The expression for the right whiskering `η ▷ h`. -/
+  whiskerRightM (η : Mor₂) (h : Mor₁) : m Mor₂
+  /-- The expression for the horizontal composition `η ◫ θ`. -/
+  horizontalCompM (η θ : Mor₂) : m Mor₂
+  /-- The expression for the coherence composition `η ⊗≫ θ := η ≫ α ≫ θ`. -/
+  coherenceCompM (α : CoherenceHom) (η θ : Mor₂) : m Mor₂
+
+/-- Type of normalized 1-morphisms `((... ≫ h) ≫ g) ≫ f`. -/
+inductive NormalizedHom : Type
+  /-- The identity 1-morphism `𝟙 a`. -/
+  | nil (e : Mor₁) (a : Obj) : NormalizedHom
+  /-- The `cons` composes an atomic 1-morphism at the end of a normalized 1-morphism. -/
+  | cons (e : Mor₁) : NormalizedHom → Atom₁ → NormalizedHom
+  deriving Inhabited
+
+/-- The underlying expression of a normalized 1-morphism. -/
+def NormalizedHom.e : NormalizedHom → Mor₁
+  | NormalizedHom.nil e _ => e
+  | NormalizedHom.cons e _ _  => e
+
+/-- The domain of a normalized 1-morphism. -/
+def NormalizedHom.src : NormalizedHom → Obj
+  | NormalizedHom.nil _ a => a
+  | NormalizedHom.cons _ p _ => p.src
+
+/-- The codomain of a normalized 1-morphism. -/
+def NormalizedHom.tgt : NormalizedHom → Obj
+  | NormalizedHom.nil _ a => a
+  | NormalizedHom.cons _ _  f => f.tgt
+
+/-- Construct the `NormalizedHom.nil` term in `m`. -/
+def normalizedHom.nilM [MonadMor₁ m] (a : Obj) : m NormalizedHom := do
+  return NormalizedHom.nil (← id₁M a) a
+
+/-- Construct a `NormalizedHom.cons` term in `m`. -/
+def NormalizedHom.consM [MonadMor₁ m] (p : NormalizedHom) (f : Atom₁) :
+    m NormalizedHom := do
+  return NormalizedHom.cons (← comp₁M p.e (.of f)) p f
+
+/-- `Context ρ` provides the context for manipulating 2-morphisms in a monoidal category or
+bicategory. In particular, we will store `MonoidalCategory` or `Bicategory` instance in a context,
+and use this through a reader monad when we construct the lean expressions for 2-morphisms. -/
+class Context (ρ : Type) where
+  /-- Construct a context from a lean expression for a 2-morphism. -/
+  mkContext? : Expr → MetaM (Option ρ)
+
+export Context (mkContext?)
+
+/-- Construct a context from a lean expression for a 2-morphism. -/
+def mkContext {ρ  : Type} [Context ρ] (e : Expr) : MetaM ρ := do
+  match ← mkContext? e with
+  | some c => return c
+  | none => throwError "failed to construct a monoidal category or bicategory context from {e}"
+
+/-- The state for the `CoherenceM ρ` monad. -/
+structure State where
+  /-- The cache for evaluating lean expressions of 1-morphisms into `Mor₁` terms. -/
+  cache : PersistentExprMap Mor₁ := {}
+
+/-- The monad for manipulating 2-morphisms in a monoidal category or bicategory. -/
+abbrev CoherenceM (ρ : Type) := ReaderT ρ <| StateT State MetaM
+
+/-- Run the `CoherenceM ρ` monad. -/
+def CoherenceM.run {α ρ : Type} (x : CoherenceM ρ α) (ctx : ρ) (s : State := {}) :
+    MetaM α := do
+  Prod.fst <$> ReaderT.run x ctx s
+
+end BicategoryLike
+
+end Tactic
+
+end Mathlib
diff --git a/Mathlib/Tactic/CategoryTheory/Coherence/Normalize.lean b/Mathlib/Tactic/CategoryTheory/Coherence/Normalize.lean
new file mode 100644
index 0000000000000..63738eff6d0a9
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Coherence/Normalize.lean
@@ -0,0 +1,586 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+
+/-!
+# Normalization of 2-morphisms in bicategories
+
+This file provides a function that normalizes 2-morphisms in bicategories. The function also
+used to normalize morphisms in monoidal categories. This is used in the string diagram widget given
+in `Mathlib.Tactic.StringDiagram`, as well as `monoidal` and `bicategory` tactics.
+
+We say that the 2-morphism `η` in a bicategory is in normal form if
+1. `η` is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁` where each `αᵢ` is a
+  structural 2-morphism (consisting of associators and unitors),
+2. each `ηᵢ` is a non-structural 2-morphism of the form `f₁ ◁ ... ◁ fₙ ◁ θ`, and
+3. `θ` is of the form `ι₁ ◫ ... ◫ ιₗ`, and
+4. each `ιᵢ` is of the form `κ ▷ g₁ ▷ ... ▷ gₖ`.
+
+Note that the horizontal composition `◫` is not currently defined for bicategories. In the monoidal
+category setting, the horizontal composition is defined as the `tensorHom`, denoted by `⊗`.
+
+Note that the structural morphisms `αᵢ` are not necessarily normalized, as the main purpose
+is to get a list of the non-structural morphisms out.
+
+Currently, the primary application of the normalization tactic in mind is drawing string diagrams,
+which are graphical representations of morphisms in monoidal categories, in the infoview. When
+drawing string diagrams, we often ignore associators and unitors (i.e., drawing morphisms in
+strict monoidal categories). On the other hand, in Lean, it is considered difficult to formalize
+the concept of strict monoidal categories due to the feature of dependent type theory. The
+normalization tactic can remove associators and unitors from the expression, extracting the
+necessary data for drawing string diagrams.
+
+The string diagrams widget is to use Penrose (https://github.com/penrose) via ProofWidget.
+However, it should be noted that the normalization procedure in this file does not rely on specific
+settings, allowing for broader application. Future plans include the following. At least I (Yuma)
+would like to work on these in the future, but it might not be immediate. If anyone is interested,
+I would be happy to discuss.
+
+- Currently, the string diagrams widget only do drawing. It would be better they also generate
+  proofs. That is, by manipulating the string diagrams displayed in the infoview with a mouse to
+  generate proofs. In #10581, the string diagram widget only uses the morphisms generated by the
+  normalization tactic and does not use proof terms ensuring that the original morphism and the
+  normalized morphism are equal. Proof terms will be necessary for proof generation.
+
+- There is also the possibility of using homotopy.io (https://github.com/homotopy-io), a graphical
+  proof assistant for category theory, from Lean. At this point, I have very few ideas regarding
+  this approach.
+
+## Main definitions
+- `Tactic.BicategoryLike.eval`: Given a Lean expression `e` that represents a morphism in a monoidal
+category, this function returns a pair of `⟨e', pf⟩` where `e'` is the normalized expression of `e`
+and `pf` is a proof that `e = e'`.
+
+-/
+
+open Lean Meta
+
+namespace Mathlib.Tactic.BicategoryLike
+
+section
+
+/-- Expressions of the form `η ▷ f₁ ▷ ... ▷ fₙ`. -/
+inductive WhiskerRight : Type
+  /-- Construct the expression for an atomic 2-morphism. -/
+  | of (η : Atom) : WhiskerRight
+  /-- Construct the expression for `η ▷ f`. -/
+  | whisker (e : Mor₂) (η : WhiskerRight) (f : Atom₁) : WhiskerRight
+  deriving Inhabited
+
+/-- The underlying `Mor₂` term of a `WhiskerRight` term. -/
+def WhiskerRight.e : WhiskerRight → Mor₂
+  | .of η => .of η
+  | .whisker e .. => e
+
+/-- Expressions of the form `η₁ ⊗ ... ⊗ ηₙ`. -/
+inductive HorizontalComp : Type
+  | of (η : WhiskerRight) : HorizontalComp
+  | cons (e : Mor₂) (η : WhiskerRight) (ηs : HorizontalComp) :
+    HorizontalComp
+  deriving Inhabited
+
+/-- The underlying `Mor₂` term of a `HorizontalComp` term. -/
+def HorizontalComp.e : HorizontalComp → Mor₂
+  | .of η => η.e
+  | .cons e .. => e
+
+/-- Expressions of the form `f₁ ◁ ... ◁ fₙ ◁ η`. -/
+inductive WhiskerLeft : Type
+  /-- Construct the expression for a right-whiskered 2-morphism. -/
+  | of (η : HorizontalComp) : WhiskerLeft
+  /-- Construct the expression for `f ◁ η`. -/
+  | whisker (e : Mor₂) (f : Atom₁) (η : WhiskerLeft) : WhiskerLeft
+  deriving Inhabited
+
+/-- The underlying `Mor₂` term of a `WhiskerLeft` term. -/
+def WhiskerLeft.e : WhiskerLeft → Mor₂
+  | .of η => η.e
+  | .whisker e .. => e
+
+/-- Whether a given 2-isomorphism is structural or not. -/
+def Mor₂Iso.isStructural (α : Mor₂Iso) : Bool :=
+  match α with
+  | .structuralAtom _ => true
+  | .comp _ _ _ _ η θ => η.isStructural && θ.isStructural
+  | .whiskerLeft _ _ _ _ η => η.isStructural
+  | .whiskerRight _ _ _ η _ => η.isStructural
+  | .horizontalComp _ _ _ _ _ η θ => η.isStructural && θ.isStructural
+  | .inv _ _ _ η => η.isStructural
+  | .coherenceComp _ _ _ _ _ _ η θ => η.isStructural && θ.isStructural
+  | .of _ => false
+
+/-- Expressions for structural isomorphisms. We do not impose the condition `isStructural` since
+it is not needed to write the tactic. -/
+abbrev Structural := Mor₂Iso
+
+/-- Normalized expressions for 2-morphisms. -/
+inductive NormalExpr : Type
+  /-- Construct the expression for a structural 2-morphism. -/
+  | nil (e : Mor₂) (α : Structural) : NormalExpr
+  /-- Construct the normalized expression of a 2-morphism `α ≫ η ≫ ηs` recursively. -/
+  | cons (e : Mor₂) (α : Structural) (η : WhiskerLeft) (ηs : NormalExpr) : NormalExpr
+  deriving Inhabited
+
+/-- The underlying `Mor₂` term of a `NormalExpr` term. -/
+def NormalExpr.e : NormalExpr → Mor₂
+  | .nil e .. => e
+  | .cons e .. => e
+
+/-- A monad equipped with the ability to construct `WhiskerRight` terms. -/
+class MonadWhiskerRight (m : Type → Type) where
+  /-- The expression for the right whiskering `η ▷ f`. -/
+  whiskerRightM (η : WhiskerRight) (f : Atom₁) : m WhiskerRight
+
+/-- A monad equipped with the ability to construct `HorizontalComp` terms. -/
+class MonadHorizontalComp (m : Type → Type) extends MonadWhiskerRight m where
+  /-- The expression for the horizontal composition `η ◫ ηs`. -/
+  hConsM (η : WhiskerRight) (ηs : HorizontalComp) : m HorizontalComp
+
+/-- A monad equipped with the ability to construct `WhiskerLeft` terms. -/
+class MonadWhiskerLeft (m : Type → Type) extends MonadHorizontalComp m where
+  /-- The expression for the left whiskering `f ▷ η`. -/
+  whiskerLeftM (f : Atom₁) (η : WhiskerLeft) : m WhiskerLeft
+
+/-- A monad equipped with the ability to construct `NormalExpr` terms. -/
+class MonadNormalExpr (m : Type → Type) extends MonadWhiskerLeft m where
+  /-- The expression for the structural 2-morphism `α`. -/
+  nilM (α : Structural) : m NormalExpr
+  /-- The expression for the normalized 2-morphism `α ≫ η ≫ ηs`. -/
+  consM (headStructural : Structural) (η : WhiskerLeft) (ηs : NormalExpr) : m NormalExpr
+
+variable {m : Type → Type} [Monad m]
+
+open MonadMor₁
+
+/-- The domain of a 2-morphism. -/
+def WhiskerRight.srcM [MonadMor₁ m] : WhiskerRight → m Mor₁
+  | WhiskerRight.of η => return η.src
+  | WhiskerRight.whisker _ η f => do comp₁M (← η.srcM) (.of f)
+
+/-- The codomain of a 2-morphism. -/
+def WhiskerRight.tgtM [MonadMor₁ m] : WhiskerRight → m Mor₁
+  | WhiskerRight.of η => return η.tgt
+  | WhiskerRight.whisker _ η f => do comp₁M (← η.tgtM) (.of f)
+
+/-- The domain of a 2-morphism. -/
+def HorizontalComp.srcM [MonadMor₁ m] : HorizontalComp → m Mor₁
+  | HorizontalComp.of η => η.srcM
+  | HorizontalComp.cons _ η ηs => do comp₁M (← η.srcM) (← ηs.srcM)
+
+/-- The codomain of a 2-morphism. -/
+def HorizontalComp.tgtM [MonadMor₁ m] : HorizontalComp → m Mor₁
+  | HorizontalComp.of η => η.tgtM
+  | HorizontalComp.cons _ η ηs => do comp₁M (← η.tgtM) (← ηs.tgtM)
+
+/-- The domain of a 2-morphism. -/
+def WhiskerLeft.srcM [MonadMor₁ m] : WhiskerLeft → m Mor₁
+  | WhiskerLeft.of η => η.srcM
+  | WhiskerLeft.whisker _ f η => do comp₁M (.of f) (← η.srcM)
+
+/-- The codomain of a 2-morphism. -/
+def WhiskerLeft.tgtM [MonadMor₁ m] : WhiskerLeft → m Mor₁
+  | WhiskerLeft.of η => η.tgtM
+  | WhiskerLeft.whisker _ f η => do comp₁M (.of f) (← η.tgtM)
+
+/-- The domain of a 2-morphism. -/
+def NormalExpr.srcM [MonadMor₁ m] : NormalExpr → m Mor₁
+  | NormalExpr.nil _ η => η.srcM
+  | NormalExpr.cons _ α _ _ => α.srcM
+
+/-- The codomain of a 2-morphism. -/
+def NormalExpr.tgtM [MonadMor₁ m] : NormalExpr → m Mor₁
+  | NormalExpr.nil _ η => η.tgtM
+  | NormalExpr.cons _ _ _ ηs => ηs.tgtM
+
+namespace NormalExpr
+
+variable [MonadMor₂Iso m] [MonadNormalExpr m]
+
+/-- The identity 2-morphism as a term of `normalExpr`. -/
+def idM (f : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| .structuralAtom <| ← MonadMor₂Iso.id₂M f
+
+/-- The associator as a term of `normalExpr`. -/
+def associatorM (f g h : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| .structuralAtom <| ← MonadMor₂Iso.associatorM f g h
+
+/-- The inverse of the associator as a term of `normalExpr`. -/
+def associatorInvM (f g h : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| ← MonadMor₂Iso.symmM <|
+    .structuralAtom <| ← MonadMor₂Iso.associatorM f g h
+
+/-- The left unitor as a term of `normalExpr`. -/
+def leftUnitorM (f : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| .structuralAtom <| ← MonadMor₂Iso.leftUnitorM f
+
+/-- The inverse of the left unitor as a term of `normalExpr`. -/
+def leftUnitorInvM (f : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| ← MonadMor₂Iso.symmM <| .structuralAtom <| ← MonadMor₂Iso.leftUnitorM f
+
+/-- The right unitor as a term of `normalExpr`. -/
+def rightUnitorM (f : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| .structuralAtom <| ← MonadMor₂Iso.rightUnitorM f
+
+/-- The inverse of the right unitor as a term of `normalExpr`. -/
+def rightUnitorInvM (f : Mor₁) : m NormalExpr := do
+  MonadNormalExpr.nilM <| ← MonadMor₂Iso.symmM <| .structuralAtom <| ← MonadMor₂Iso.rightUnitorM f
+
+/-- Construct a `NormalExpr` expression from a `WhiskerLeft` expression. -/
+def ofM [MonadMor₁ m] (η : WhiskerLeft) : m NormalExpr := do
+  MonadNormalExpr.consM ((.structuralAtom <| ← MonadMor₂Iso.id₂M (← η.srcM))) η
+    (← MonadNormalExpr.nilM ((.structuralAtom <| ← MonadMor₂Iso.id₂M (← η.tgtM))))
+
+/-- Construct a `NormalExpr` expression from a Lean expression for an atomic 2-morphism. -/
+def ofAtomM [MonadMor₁ m] (η : Atom) : m NormalExpr :=
+  NormalExpr.ofM <| .of <| .of <| .of η
+
+end NormalExpr
+
+/-- Convert a `NormalExpr` expression into a list of `WhiskerLeft` expressions. -/
+def NormalExpr.toList : NormalExpr → List WhiskerLeft
+  | NormalExpr.nil _ _ => []
+  | NormalExpr.cons _ _ η ηs => η :: NormalExpr.toList ηs
+
+end
+
+section
+
+/-- The result of evaluating an expression into normal form. -/
+structure Eval.Result where
+  /-- The normalized expression of the 2-morphism. -/
+  expr : NormalExpr
+  /-- The proof that the normalized expression is equal to the original expression. -/
+  proof : Expr
+  deriving Inhabited
+
+variable {m : Type → Type} [Monad m]
+
+/-- Evaluate the expression `α ≫ β`. -/
+class MkEvalComp (m : Type → Type) where
+  /-- Evaluate `α ≫ β` -/
+  mkEvalCompNilNil (α β : Structural) : m Expr
+  /-- Evaluate `α ≫ (β ≫ η ≫ ηs)` -/
+  mkEvalCompNilCons (α β : Structural) (η : WhiskerLeft) (ηs : NormalExpr) : m Expr
+  /-- Evaluate `(α ≫ η ≫ ηs) ≫ θ` -/
+  mkEvalCompCons (α : Structural) (η : WhiskerLeft) (ηs θ ι : NormalExpr) (e_η : Expr) : m Expr
+
+/-- Evaluatte the expression `f ◁ η`. -/
+class MkEvalWhiskerLeft (m : Type → Type) where
+  /-- Evaluatte `f ◁ α` -/
+  mkEvalWhiskerLeftNil (f : Mor₁) (α : Structural) : m Expr
+  /-- Evaluate `f ◁ (α ≫ η ≫ ηs)`. -/
+  mkEvalWhiskerLeftOfCons (f : Atom₁) (α : Structural) (η : WhiskerLeft) (ηs θ : NormalExpr)
+    (e_θ : Expr) : m Expr
+  /-- Evaluate `(f ≫ g) ◁ η` -/
+  mkEvalWhiskerLeftComp (f g : Mor₁) (η η₁ η₂ η₃ η₄ : NormalExpr)
+    (e_η₁ e_η₂ e_η₃ e_η₄ : Expr) : m Expr
+  /-- Evaluate `𝟙 _ ◁ η` -/
+  mkEvalWhiskerLeftId (η η₁ η₂ : NormalExpr) (e_η₁ e_η₂ : Expr) : m Expr
+
+/-- Evaluate the expression `η ▷ f`. -/
+class MkEvalWhiskerRight (m : Type → Type) where
+  /-- Evaluate `η ▷ f` -/
+  mkEvalWhiskerRightAuxOf (η : WhiskerRight) (f : Atom₁) : m Expr
+  /-- Evaluate `(η ◫ ηs) ▷ f` -/
+  mkEvalWhiskerRightAuxCons (f : Atom₁) (η : WhiskerRight) (ηs : HorizontalComp)
+    (ηs' η₁ η₂ η₃ : NormalExpr) (e_ηs' e_η₁ e_η₂ e_η₃ : Expr) : m Expr
+  /-- Evaluate `α ▷ f` -/
+  mkEvalWhiskerRightNil (α : Structural) (f : Mor₁) : m Expr
+  /-- Evaluate ` (α ≫ η ≫ ηs) ▷ j` -/
+  mkEvalWhiskerRightConsOfOf (f : Atom₁) (α : Structural) (η : HorizontalComp)
+    (ηs ηs₁ η₁ η₂ η₃ : NormalExpr)
+    (e_ηs₁ e_η₁ e_η₂ e_η₃ : Expr) : m Expr
+  /-- Evaluate `(α ≫ (f ◁ η) ≫ ηs) ▷ g` -/
+  mkEvalWhiskerRightConsWhisker (f : Atom₁) (g : Mor₁) (α : Structural) (η : WhiskerLeft)
+    (ηs η₁ η₂ ηs₁ ηs₂ η₃ η₄ η₅ : NormalExpr) (e_η₁ e_η₂ e_ηs₁ e_ηs₂ e_η₃ e_η₄ e_η₅ : Expr) : m Expr
+  /-- Evaluate `η ▷ (g ⊗ h)` -/
+  mkEvalWhiskerRightComp (g h : Mor₁)
+    (η η₁ η₂ η₃ η₄ : NormalExpr) (e_η₁ e_η₂ e_η₃ e_η₄ : Expr) : m Expr
+  /-- Evaluate `η ▷ 𝟙 _` -/
+  mkEvalWhiskerRightId (η η₁ η₂ : NormalExpr) (e_η₁ e_η₂ : Expr) : m Expr
+
+/-- Evaluate the expression `η ◫ θ`. -/
+class MkEvalHorizontalComp (m : Type → Type) where
+  /-- Evaluate `η ◫ θ` -/
+  mkEvalHorizontalCompAuxOf (η : WhiskerRight) (θ : HorizontalComp) : m Expr
+  /-- Evaluate `(η ◫ ηs) ◫ θ` -/
+  mkEvalHorizontalCompAuxCons (η : WhiskerRight) (ηs θ : HorizontalComp)
+    (ηθ η₁ ηθ₁ ηθ₂ : NormalExpr) (e_ηθ e_η₁ e_ηθ₁ e_ηθ₂ : Expr) : m Expr
+  /-- Evaluate `(f ◁ η) ◫ θ` -/
+  mkEvalHorizontalCompAux'Whisker (f : Atom₁) (η θ : WhiskerLeft)
+    (ηθ ηθ₁ ηθ₂ ηθ₃ : NormalExpr) (e_ηθ e_ηθ₁ e_ηθ₂ e_ηθ₃ : Expr) : m Expr
+  /-- Evaluate `η ◫ (f ◁ θ)` -/
+  mkEvalHorizontalCompAux'OfWhisker (f : Atom₁) (η : HorizontalComp) (θ : WhiskerLeft)
+    (η₁ ηθ ηθ₁ ηθ₂ : NormalExpr) (e_ηθ e_η₁ e_ηθ₁ e_ηθ₂ : Expr) : m Expr
+  /-- Evaluate `α ◫ β` -/
+  mkEvalHorizontalCompNilNil (α β : Structural) : m Expr
+  /-- Evaluate `α ◫ (β ≫ η ≫ ηs)` -/
+  mkEvalHorizontalCompNilCons (α β : Structural) (η : WhiskerLeft)
+    (ηs η₁ ηs₁ η₂ η₃ : NormalExpr) (e_η₁ e_ηs₁ e_η₂ e_η₃ : Expr) : m Expr
+  /-- Evaluate `(α ≫ η ≫ ηs) ◫ β` -/
+  mkEvalHorizontalCompConsNil (α β : Structural) (η : WhiskerLeft) (ηs : NormalExpr)
+    (η₁ ηs₁ η₂ η₃ : NormalExpr) (e_η₁ e_ηs₁ e_η₂ e_η₃ : Expr) : m Expr
+  /-- Evaluate `(α ≫ η ≫ ηs) ◫ (β ≫ θ ≫ θs)` -/
+  mkEvalHorizontalCompConsCons (α β : Structural) (η θ : WhiskerLeft)
+    (ηs θs ηθ ηθs ηθ₁ ηθ₂ : NormalExpr) (e_ηθ e_ηθs e_ηθ₁ e_ηθ₂ : Expr) : m Expr
+
+/-- Evaluate the expression of a 2-morphism into a normalized form. -/
+class MkEval (m : Type → Type) extends
+    MkEvalComp m, MkEvalWhiskerLeft m, MkEvalWhiskerRight m, MkEvalHorizontalComp m where
+  /-- Evaluate the expression `η ≫ θ` into a normalized form. -/
+  mkEvalComp (η θ : Mor₂) (η' θ' ηθ : NormalExpr) (e_η e_θ e_ηθ : Expr) : m Expr
+  /-- Evaluate the expression `f ◁ η` into a normalized form. -/
+  mkEvalWhiskerLeft (f : Mor₁) (η : Mor₂) (η' θ : NormalExpr) (e_η e_θ : Expr) : m Expr
+  /-- Evaluate the expression `η ▷ f` into a normalized form. -/
+  mkEvalWhiskerRight (η : Mor₂) (h : Mor₁) (η' θ : NormalExpr) (e_η e_θ : Expr) : m Expr
+  /-- Evaluate the expression `η ◫ θ` into a normalized form. -/
+  mkEvalHorizontalComp (η θ : Mor₂) (η' θ' ι : NormalExpr) (e_η e_θ e_ι : Expr) : m Expr
+  /-- Evaluate the atomic 2-morphism `η` into a normalized form. -/
+  mkEvalOf (η : Atom) : m Expr
+  /-- Evaluate the expression `η ⊗≫ θ := η ≫ α ≫ θ` into a normalized form. -/
+  mkEvalMonoidalComp (η θ : Mor₂) (α : Structural) (η' θ' αθ ηαθ : NormalExpr)
+    (e_η e_θ e_αθ e_ηαθ : Expr) : m Expr
+
+variable {ρ : Type} [Context ρ]
+variable [MonadMor₂Iso (CoherenceM ρ)] [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)]
+
+open MkEvalComp MonadMor₂Iso MonadNormalExpr
+
+/-- Evaluate the expression `α ≫ η` into a normalized form. -/
+def evalCompNil (α : Structural) : NormalExpr → CoherenceM ρ Eval.Result
+  | .nil _ β => do return ⟨← nilM (← comp₂M α β), ← mkEvalCompNilNil α β⟩
+  | .cons _ β η ηs => do return ⟨← consM (← comp₂M α β) η ηs, ← mkEvalCompNilCons α β η ηs⟩
+
+/-- Evaluate the expression `η ≫ θ` into a normalized form. -/
+def evalComp : NormalExpr → NormalExpr → CoherenceM ρ Eval.Result
+  | .nil _ α, η => do evalCompNil α η
+  | .cons _ α η ηs, θ => do
+    let ⟨ι, e_ι⟩ ← evalComp ηs θ
+    return ⟨← consM α η ι, ← mkEvalCompCons α η ηs θ ι e_ι⟩
+
+open MkEvalWhiskerLeft
+
+variable [MonadMor₁ (CoherenceM ρ)] [MonadMor₂Iso (CoherenceM ρ)]
+
+/-- Evaluate the expression `f ◁ η` into a normalized form. -/
+def evalWhiskerLeft : Mor₁ → NormalExpr → CoherenceM ρ Eval.Result
+  | f, .nil _ α => do
+    return ⟨← nilM (← whiskerLeftM f α), ← mkEvalWhiskerLeftNil f α⟩
+  | .of f, .cons _ α η ηs => do
+    let η' ← MonadWhiskerLeft.whiskerLeftM f η
+    let ⟨θ, e_θ⟩ ← evalWhiskerLeft (.of f) ηs
+    let η'' ← consM (← whiskerLeftM (.of f) α) η' θ
+    return ⟨η'', ← mkEvalWhiskerLeftOfCons f α η ηs θ e_θ⟩
+  | .comp _ f g, η => do
+    let ⟨θ, e_θ⟩ ← evalWhiskerLeft g η
+    let ⟨ι, e_ι⟩ ← evalWhiskerLeft f θ
+    let h ← η.srcM
+    let h' ← η.tgtM
+    let ⟨ι', e_ι'⟩ ← evalComp ι (← NormalExpr.associatorInvM f g h')
+    let ⟨ι'', e_ι''⟩ ← evalComp (← NormalExpr.associatorM f g h) ι'
+    return ⟨ι'', ← mkEvalWhiskerLeftComp f g η θ ι ι' ι'' e_θ e_ι e_ι' e_ι''⟩
+  | .id _ _, η => do
+    let f ← η.srcM
+    let g ← η.tgtM
+    let ⟨η', e_η'⟩ ← evalComp η (← NormalExpr.leftUnitorInvM g)
+    let ⟨η'', e_η''⟩ ← evalComp (← NormalExpr.leftUnitorM f) η'
+    return ⟨η'', ← mkEvalWhiskerLeftId η η' η'' e_η' e_η''⟩
+
+open MkEvalWhiskerRight MkEvalHorizontalComp
+
+mutual
+
+/-- Evaluate the expression `η ▷ f` into a normalized form. -/
+partial def evalWhiskerRightAux : HorizontalComp → Atom₁ → CoherenceM ρ Eval.Result
+  | .of η, f => do
+    let η' ← NormalExpr.ofM <| .of <| .of <| ← MonadWhiskerRight.whiskerRightM η f
+    return ⟨η', ← mkEvalWhiskerRightAuxOf η f⟩
+  | .cons _ η ηs, f => do
+    let ⟨ηs', e_ηs'⟩ ← evalWhiskerRightAux ηs f
+    let ⟨η₁, e_η₁⟩ ← evalHorizontalComp (← NormalExpr.ofM <| .of <| .of η) ηs'
+    let ⟨η₂, e_η₂⟩ ← evalComp η₁ (← NormalExpr.associatorInvM (← η.tgtM) (← ηs.tgtM) (.of f))
+    let ⟨η₃, e_η₃⟩ ← evalComp (← NormalExpr.associatorM (← η.srcM) (← ηs.srcM) (.of f)) η₂
+    return ⟨η₃, ← mkEvalWhiskerRightAuxCons f η ηs ηs' η₁ η₂ η₃ e_ηs' e_η₁ e_η₂ e_η₃⟩
+
+/-- Evaluate the expression `η ▷ f` into a normalized form. -/
+partial def evalWhiskerRight : NormalExpr → Mor₁ → CoherenceM ρ Eval.Result
+  | .nil _ α, h => do
+    return ⟨← nilM (← whiskerRightM α h), ← mkEvalWhiskerRightNil α h⟩
+  | .cons _ α (.of η) ηs, .of f => do
+    let ⟨ηs₁, e_ηs₁⟩ ← evalWhiskerRight ηs (.of f)
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerRightAux η f
+    let ⟨η₂, e_η₂⟩ ← evalComp η₁ ηs₁
+    let ⟨η₃, e_η₃⟩ ← evalCompNil (← whiskerRightM α (.of f)) η₂
+    return ⟨η₃, ← mkEvalWhiskerRightConsOfOf f α η ηs ηs₁ η₁ η₂ η₃ e_ηs₁ e_η₁ e_η₂ e_η₃⟩
+  | .cons _ α (.whisker _ f η) ηs, h => do
+    let g ← η.srcM
+    let g' ← η.tgtM
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerRight (← consM (← id₂M' g) η (← NormalExpr.idM g')) h
+    let ⟨η₂, e_η₂⟩ ← evalWhiskerLeft (.of f) η₁
+    let ⟨ηs₁, e_ηs₁⟩ ← evalWhiskerRight ηs h
+    let α' ← whiskerRightM α h
+    let ⟨ηs₂, e_ηs₂⟩ ← evalComp (← NormalExpr.associatorInvM (.of f) g' h) ηs₁
+    let ⟨η₃, e_η₃⟩ ← evalComp η₂ ηs₂
+    let ⟨η₄, e_η₄⟩ ← evalComp (← NormalExpr.associatorM (.of f) g h) η₃
+    let ⟨η₅, e_η₅⟩ ← evalComp (← nilM α') η₄
+    return ⟨η₅, ← mkEvalWhiskerRightConsWhisker f h α η ηs η₁ η₂ ηs₁ ηs₂ η₃ η₄ η₅
+      e_η₁ e_η₂ e_ηs₁ e_ηs₂ e_η₃ e_η₄ e_η₅⟩
+  | η, .comp _ g h => do
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerRight η g
+    let ⟨η₂, e_η₂⟩ ← evalWhiskerRight η₁ h
+    let f ← η.srcM
+    let f' ← η.tgtM
+    let ⟨η₃, e_η₃⟩ ← evalComp η₂ (← NormalExpr.associatorM f' g h)
+    let ⟨η₄, e_η₄⟩ ← evalComp (← NormalExpr.associatorInvM f g h) η₃
+    return ⟨η₄, ← mkEvalWhiskerRightComp g h η η₁ η₂ η₃ η₄ e_η₁ e_η₂ e_η₃ e_η₄⟩
+  | η, .id _ _ => do
+    let f ← η.srcM
+    let g ← η.tgtM
+    let ⟨η₁, e_η₁⟩ ← evalComp η (← NormalExpr.rightUnitorInvM g)
+    let ⟨η₂, e_η₂⟩ ← evalComp (← NormalExpr.rightUnitorM f) η₁
+    return ⟨η₂, ← mkEvalWhiskerRightId η η₁ η₂ e_η₁ e_η₂⟩
+
+/-- Evaluate the expression `η ⊗ θ` into a normalized form. -/
+partial def evalHorizontalCompAux : HorizontalComp → HorizontalComp → CoherenceM ρ Eval.Result
+  | .of η, θ => do
+    return ⟨← NormalExpr.ofM <| .of <| ← MonadHorizontalComp.hConsM η θ,
+      ← mkEvalHorizontalCompAuxOf η θ⟩
+  | .cons _ η ηs, θ => do
+    let α ← NormalExpr.associatorM (← η.srcM) (← ηs.srcM) (← θ.srcM)
+    let α' ← NormalExpr.associatorInvM (← η.tgtM) (← ηs.tgtM) (← θ.tgtM)
+    let ⟨ηθ, e_ηθ⟩ ← evalHorizontalCompAux ηs θ
+    let ⟨η₁, e_η₁⟩ ← evalHorizontalComp (← NormalExpr.ofM <| .of <| .of η) ηθ
+    let ⟨ηθ₁, e_ηθ₁⟩ ← evalComp η₁ α'
+    let ⟨ηθ₂, e_ηθ₂⟩ ← evalComp α ηθ₁
+    return ⟨ηθ₂, ← mkEvalHorizontalCompAuxCons η ηs θ ηθ η₁ ηθ₁ ηθ₂ e_ηθ e_η₁ e_ηθ₁ e_ηθ₂⟩
+
+/-- Evaluate the expression `η ⊗ θ` into a normalized form. -/
+partial def evalHorizontalCompAux' : WhiskerLeft → WhiskerLeft → CoherenceM ρ Eval.Result
+  | .of η, .of θ => evalHorizontalCompAux η θ
+  | .whisker _ f η, θ => do
+    let ⟨ηθ, e_ηθ⟩ ← evalHorizontalCompAux' η θ
+    let ⟨ηθ₁, e_ηθ₁⟩ ← evalWhiskerLeft (.of f) ηθ
+    let ⟨ηθ₂, e_ηθ₂⟩ ← evalComp ηθ₁ (← NormalExpr.associatorInvM (.of f) (← η.tgtM) (← θ.tgtM))
+    let ⟨ηθ₃, e_ηθ₃⟩ ← evalComp (← NormalExpr.associatorM (.of f) (← η.srcM) (← θ.srcM)) ηθ₂
+    return ⟨ηθ₃, ← mkEvalHorizontalCompAux'Whisker f η θ ηθ ηθ₁ ηθ₂ ηθ₃ e_ηθ e_ηθ₁ e_ηθ₂ e_ηθ₃⟩
+  | .of η, .whisker _ f θ => do
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerRightAux η f
+    let ⟨ηθ, e_ηθ⟩ ← evalHorizontalComp η₁ (← NormalExpr.ofM θ)
+    let ⟨ηθ₁, e_ηθ₁⟩ ← evalComp ηθ (← NormalExpr.associatorM (← η.tgtM) (.of f) (← θ.tgtM))
+    let ⟨ηθ₂, e_ηθ₂⟩ ← evalComp (← NormalExpr.associatorInvM (← η.srcM) (.of f) (← θ.srcM)) ηθ₁
+    return ⟨ηθ₂, ← mkEvalHorizontalCompAux'OfWhisker f η θ ηθ η₁ ηθ₁ ηθ₂ e_η₁ e_ηθ e_ηθ₁ e_ηθ₂⟩
+
+/-- Evaluate the expression `η ⊗ θ` into a normalized form. -/
+partial def evalHorizontalComp : NormalExpr → NormalExpr → CoherenceM ρ Eval.Result
+  | .nil _ α, .nil _ β => do
+    return ⟨← nilM <| ← horizontalCompM α β, ← mkEvalHorizontalCompNilNil α β⟩
+  | .nil _ α, .cons _ β η ηs => do
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerLeft (← α.tgtM) (← NormalExpr.ofM η)
+    let ⟨ηs₁, e_ηs₁⟩ ← evalWhiskerLeft (← α.tgtM) ηs
+    let ⟨η₂, e_η₂⟩ ← evalComp η₁ ηs₁
+    let ⟨η₃, e_η₃⟩ ← evalCompNil (← horizontalCompM α β) η₂
+    return ⟨η₃, ← mkEvalHorizontalCompNilCons α β η ηs η₁ ηs₁ η₂ η₃ e_η₁ e_ηs₁ e_η₂ e_η₃⟩
+  | .cons _ α η ηs, .nil _ β => do
+    let ⟨η₁, e_η₁⟩ ← evalWhiskerRight (← NormalExpr.ofM η) (← β.tgtM)
+    let ⟨ηs₁, e_ηs₁⟩ ← evalWhiskerRight ηs (← β.tgtM)
+    let ⟨η₂, e_η₂⟩ ← evalComp η₁ ηs₁
+    let ⟨η₃, e_η₃⟩ ← evalCompNil (← horizontalCompM α β) η₂
+    return ⟨η₃, ← mkEvalHorizontalCompConsNil α β η ηs η₁ ηs₁ η₂ η₃ e_η₁ e_ηs₁ e_η₂ e_η₃⟩
+  | .cons _ α η ηs, .cons _ β θ θs => do
+    let ⟨ηθ, e_ηθ⟩ ← evalHorizontalCompAux' η θ
+    let ⟨ηθs, e_ηθs⟩ ← evalHorizontalComp ηs θs
+    let ⟨ηθ₁, e_ηθ₁⟩ ← evalComp ηθ ηθs
+    let ⟨ηθ₂, e_ηθ₂⟩ ← evalCompNil (← horizontalCompM α β) ηθ₁
+    return ⟨ηθ₂,
+      ← mkEvalHorizontalCompConsCons α β η θ ηs θs ηθ ηθs ηθ₁ ηθ₂ e_ηθ e_ηθs e_ηθ₁ e_ηθ₂⟩
+
+end
+
+open MkEval
+
+variable {ρ : Type} [Context ρ]
+    [MonadMor₁ (CoherenceM ρ)]
+    [MonadMor₂Iso (CoherenceM ρ)]
+    [MonadNormalExpr (CoherenceM ρ)] [MkEval (CoherenceM ρ)]
+    [MonadMor₂ (CoherenceM ρ)]
+    [MkMor₂ (CoherenceM ρ)]
+
+/-- Trace the proof of the normalization. -/
+def traceProof (nm : Name) (result : Expr) : CoherenceM ρ Unit := do
+  withTraceNode nm (fun _ => return m!"{checkEmoji} {← inferType result}") do
+    if ← isTracingEnabledFor nm then addTrace nm m!"proof: {result}"
+
+-- TODO: It takes a while to compile. Find out why.
+/-- Evaluate the expression of a 2-morphism into a normalized form. -/
+def eval (nm : Name) (e : Mor₂) : CoherenceM ρ Eval.Result := do
+  withTraceNode nm (fun _ => return m!"eval: {e.e}") do
+    match e with
+    | .isoHom _ _ α => withTraceNode nm (fun _ => return m!"Iso.hom") do match α with
+      | .structuralAtom α => return ⟨← nilM <| .structuralAtom α, ← mkEqRefl e.e⟩
+      | .of η =>
+        let η ← MonadMor₂.atomHomM η
+        let result ← mkEvalOf η
+        traceProof nm result
+        return ⟨← NormalExpr.ofAtomM η, result⟩
+      | _ => throwError "not implemented. try dsimp first."
+    | .isoInv _ _ α => withTraceNode nm (fun _ => return m!"Iso.inv") do match α with
+      | .structuralAtom α => return ⟨← nilM <| (← symmM (.structuralAtom α)), ← mkEqRefl e.e⟩
+      | .of η =>
+        let η ← MonadMor₂.atomInvM η
+        let result ← mkEvalOf η
+        traceProof nm result
+        return ⟨← NormalExpr.ofAtomM η, result⟩
+      | _ => throwError "not implemented. try dsimp first."
+    | .id _ _ f  =>
+      let α ← MonadMor₂Iso.id₂M f
+      return  ⟨← nilM <| .structuralAtom α, ← mkEqRefl e.e⟩
+    | .comp _ _ _ _ _ η θ => withTraceNode nm (fun _ => return m!"comp") do
+      let ⟨η', e_η⟩ ← eval nm η
+      let ⟨θ', e_θ⟩ ← eval nm θ
+      let ⟨ηθ, pf⟩ ← evalComp η' θ'
+      let result ← mkEvalComp η θ η' θ' ηθ e_η e_θ pf
+      traceProof nm result
+      return ⟨ηθ, result⟩
+    | .whiskerLeft _ _ f _ _ η => withTraceNode nm (fun _ => return m!"whiskerLeft") do
+      let ⟨η', e_η⟩ ← eval nm η
+      let ⟨θ, e_θ⟩ ← evalWhiskerLeft f η'
+      let result ← mkEvalWhiskerLeft f η η' θ e_η e_θ
+      traceProof nm result
+      return ⟨θ, result⟩
+    | .whiskerRight _ _ _ _ η h =>
+      withTraceNode nm (fun _ => return m!"whiskerRight") do
+        let ⟨η', e_η⟩ ← eval nm η
+        let ⟨θ, e_θ⟩ ← evalWhiskerRight η' h
+        let result ← mkEvalWhiskerRight η h η' θ e_η e_θ
+        traceProof nm result
+        return ⟨θ, result⟩
+    | .coherenceComp _ _ _ _ _ _ α₀ η θ =>
+      withTraceNode nm (fun _ => return m!"monoidalComp") do
+        let ⟨η', e_η⟩ ← eval nm η
+        let α₀ := .structuralAtom <| .coherenceHom α₀
+        let α ← nilM α₀
+        let ⟨θ', e_θ⟩ ← eval nm θ
+        let ⟨αθ, e_αθ⟩ ← evalComp α θ'
+        let ⟨ηαθ, e_ηαθ⟩ ← evalComp η' αθ
+        let result ← mkEvalMonoidalComp η θ α₀ η' θ' αθ ηαθ e_η e_θ e_αθ e_ηαθ
+        traceProof nm result
+        return ⟨ηαθ, result⟩
+    | .horizontalComp _ _ _ _ _ _ η θ =>
+      withTraceNode nm (fun _ => return m!"horizontalComp") do
+        let ⟨η', e_η⟩ ← eval nm η
+        let ⟨θ', e_θ⟩ ← eval nm θ
+        let ⟨ηθ, e_ηθ⟩ ← evalHorizontalComp η' θ'
+        let result ← mkEvalHorizontalComp η θ η' θ' ηθ e_η e_θ e_ηθ
+        traceProof nm result
+        return ⟨ηθ, result⟩
+    | .of η  =>
+      let result ← mkEvalOf η
+      traceProof nm result
+      return ⟨← NormalExpr.ofAtomM η, result⟩
+
+end
+
+end Mathlib.Tactic.BicategoryLike
diff --git a/Mathlib/Tactic/CategoryTheory/Coherence/PureCoherence.lean b/Mathlib/Tactic/CategoryTheory/Coherence/PureCoherence.lean
new file mode 100644
index 0000000000000..fb4cf07da5033
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Coherence/PureCoherence.lean
@@ -0,0 +1,197 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+
+/-!
+# Coherence tactic
+
+This file provides a meta framework for the coherence tactic, which solves goals of the form
+`η = θ`, where `η` and `θ` are 2-morphism in a bicategory or morphisms in a monoidal category
+made up only of associators, unitors, and identities.
+
+The function defined here is a meta reimplementation of the formalized coherence theorems provided
+in the following files:
+- Mathlib.CategoryTheory.Monoidal.Free.Coherence
+- Mathlib.CategoryTheory.Bicategory.Coherence
+See these files for a mathematical explanation of the proof of the coherence theorem.
+
+The actual tactics that users will use are given in
+- `Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence`
+- `Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence`
+
+-/
+
+open Lean Meta
+
+namespace Mathlib.Tactic
+
+namespace BicategoryLike
+
+/-- The result of normalizing a 1-morphism. -/
+structure Normalize.Result where
+  /-- The normalized 1-morphism. -/
+  normalizedHom : NormalizedHom
+  /-- The 2-morphism from the original 1-morphism to the normalized 1-morphism. -/
+  toNormalize : Mor₂Iso
+  deriving Inhabited
+
+open Mor₂Iso MonadMor₂Iso
+
+variable {ρ : Type} [Context ρ] [MonadMor₁ (CoherenceM ρ)] [MonadMor₂Iso (CoherenceM ρ)]
+
+/-- Meta version of `CategoryTheory.FreeBicategory.normalizeIso`. -/
+def normalize  (p : NormalizedHom) (f : Mor₁) :
+    CoherenceM ρ Normalize.Result := do
+  match f with
+  | .id _ _ =>
+    return ⟨p, ← rightUnitorM' p.e⟩
+  | .comp _ f g =>
+    let ⟨pf, η_f⟩ ← normalize p f
+    let η_f' ← whiskerRightM η_f g
+    let ⟨pfg, η_g⟩ ← normalize pf g
+    let η ← comp₂M η_f' η_g
+    let α ← symmM (← associatorM' p.e f g)
+    let η' ← comp₂M α η
+    return ⟨pfg, η'⟩
+  | .of f =>
+    let pf ← NormalizedHom.consM p f
+    let α ← id₂M' pf.e
+    return ⟨pf, α⟩
+
+/-- Lemmas to prove the meta version of `CategoryTheory.FreeBicategory.normalize_naturality`. -/
+class MonadNormalizeNaturality (m : Type → Type) where
+  /-- The naturality for the associator. -/
+  mkNaturalityAssociator (p pf pfg pfgh : NormalizedHom) (f g h : Mor₁)
+    (η_f η_g η_h : Mor₂Iso) : m Expr
+  /-- The naturality for the left unitor. -/
+  mkNaturalityLeftUnitor (p pf : NormalizedHom) (f : Mor₁) (η_f : Mor₂Iso) : m Expr
+  /-- The naturality for the right unitor. -/
+  mkNaturalityRightUnitor (p pf : NormalizedHom) (f : Mor₁) (η_f : Mor₂Iso) : m Expr
+  /-- The naturality for the identity. -/
+  mkNaturalityId (p pf : NormalizedHom) (f : Mor₁) (η_f : Mor₂Iso) : m Expr
+  /-- The naturality for the composition. -/
+  mkNaturalityComp (p pf : NormalizedHom) (f g h : Mor₁) (η θ η_f η_g η_h : Mor₂Iso)
+    (ih_η ih_θ : Expr) : m Expr
+  /-- The naturality for the left whiskering. -/
+  mkNaturalityWhiskerLeft (p pf pfg : NormalizedHom) (f g h : Mor₁)
+    (η η_f η_fg η_fh : Mor₂Iso) (ih_η : Expr) : m Expr
+  /-- The naturality for the right whiskering. -/
+  mkNaturalityWhiskerRight (p pf pfh : NormalizedHom) (f g h : Mor₁) (η η_f η_g η_fh : Mor₂Iso)
+    (ih_η : Expr) : m Expr
+  /-- The naturality for the horizontal composition. -/
+  mkNaturalityHorizontalComp (p pf₁ pf₁f₂ : NormalizedHom) (f₁ g₁ f₂ g₂ : Mor₁)
+    (η θ η_f₁ η_g₁ η_f₂ η_g₂ : Mor₂Iso) (ih_η ih_θ : Expr) : m Expr
+  /-- The naturality for the inverse. -/
+  mkNaturalityInv (p pf : NormalizedHom) (f g : Mor₁) (η η_f η_g : Mor₂Iso) (ih_η : Expr) : m Expr
+
+open MonadNormalizeNaturality
+
+variable [MonadCoherehnceHom (CoherenceM ρ)] [MonadNormalizeNaturality (CoherenceM ρ)]
+
+/-- Meta version of `CategoryTheory.FreeBicategory.normalize_naturality`. -/
+partial def naturality (nm : Name) (p : NormalizedHom) (η : Mor₂Iso) : CoherenceM ρ Expr := do
+  let result ← match η with
+  | .of _ => throwError m!"could not find a structural isomorphism, but {η.e}"
+  | .coherenceComp _ _ _ _ _ α η θ => withTraceNode nm (fun _ => return m!"monoidalComp") do
+    let α ← MonadCoherehnceHom.unfoldM α
+    let αθ ← comp₂M α θ
+    let ηαθ ← comp₂M η αθ
+    naturality nm p ηαθ
+  | .structuralAtom η => match η with
+    | .coherenceHom α => withTraceNode nm (fun _ => return m!"coherenceHom") do
+      let α ← MonadCoherehnceHom.unfoldM α
+      naturality nm p α
+    | .associator _ f g h => withTraceNode nm (fun _ => return m!"associator") do
+      let ⟨pf, η_f⟩ ← normalize p f
+      let ⟨pfg, η_g⟩ ← normalize pf g
+      let ⟨pfgh, η_h⟩ ← normalize pfg h
+      mkNaturalityAssociator p pf pfg pfgh f g h η_f η_g η_h
+    | .leftUnitor _ f => withTraceNode nm (fun _ => return m!"leftUnitor") do
+      let ⟨pf, η_f⟩ ← normalize p f
+      mkNaturalityLeftUnitor p pf f η_f
+    | .rightUnitor _ f => withTraceNode nm (fun _ => return m!"rightUnitor") do
+      let ⟨pf, η_f⟩ ← normalize p f
+      mkNaturalityRightUnitor p pf f η_f
+    | .id _ f => withTraceNode nm (fun _ => return m!"id") do
+    let ⟨pf, η_f⟩ ← normalize p f
+    mkNaturalityId p pf f η_f
+  | .comp _ f g h η θ => withTraceNode nm (fun _ => return m!"comp") do
+    let ⟨pf, η_f⟩ ← normalize p f
+    let ⟨_, η_g⟩ ← normalize p g
+    let ⟨_, η_h⟩ ← normalize p h
+    let ih_η ← naturality nm p η
+    let ih_θ ← naturality nm p θ
+    mkNaturalityComp p pf f g h η θ η_f η_g η_h ih_η ih_θ
+  | .whiskerLeft _ f g h η => withTraceNode nm (fun _ => return m!"whiskerLeft") do
+    let ⟨pf, η_f⟩ ← normalize p f
+    let ⟨pfg, η_fg⟩ ← normalize pf g
+    let ⟨_, η_fh⟩ ← normalize pf h
+    let ih ← naturality nm pf η
+    mkNaturalityWhiskerLeft p pf pfg f g h η η_f η_fg η_fh ih
+  | .whiskerRight _ f g η h => withTraceNode nm (fun _ => return m!"whiskerRight") do
+    let ⟨pf, η_f⟩ ← normalize p f
+    let ⟨_, η_g⟩ ← normalize p g
+    let ⟨pfh, η_fh⟩ ← normalize pf h
+    let ih ← naturality nm p η
+    mkNaturalityWhiskerRight p pf pfh f g h η η_f η_g η_fh ih
+  | .horizontalComp _ f₁ g₁ f₂ g₂ η θ => withTraceNode nm (fun _ => return m!"hComp") do
+    let ⟨pf₁, η_f₁⟩ ← normalize p f₁
+    let ⟨_, η_g₁⟩ ← normalize p g₁
+    let ⟨pf₁f₂, η_f₂⟩ ← normalize pf₁ f₂
+    let ⟨_, η_g₂⟩ ← normalize pf₁ g₂
+    let ih_η ← naturality nm p η
+    let ih_θ ← naturality nm pf₁ θ
+    mkNaturalityHorizontalComp p pf₁ pf₁f₂ f₁ g₁ f₂ g₂ η θ η_f₁ η_g₁ η_f₂ η_g₂ ih_η ih_θ
+  | .inv _ f g η => withTraceNode nm (fun _ => return m!"inv") do
+    let ⟨pf, η_f⟩ ← normalize p f
+    let ⟨_, η_g⟩ ← normalize p g
+    let ih_η ← naturality nm p η
+    mkNaturalityInv p pf f g η η_f η_g ih_η
+  withTraceNode nm (fun _ => return m!"{checkEmoji} {← inferType result}") do
+    if ← isTracingEnabledFor nm then addTrace nm m!"proof: {result}"
+  return result
+
+/-- Prove the equality between structural isomorphisms using the naturality of `normalize`. -/
+class MkEqOfNaturality (m : Type → Type) where
+  /-- Auxiliary function for `pureCoherence`. -/
+  mkEqOfNaturality (η θ : Expr) (η' θ' : IsoLift) (η_f η_g : Mor₂Iso) (Hη Hθ : Expr) : m Expr
+
+export MkEqOfNaturality (mkEqOfNaturality)
+
+/-- Close the goal of the form `η = θ`, where `η` and `θ` are 2-isomorphisms made up only of
+associators, unitors, and identities. -/
+def pureCoherence (ρ : Type) [Context ρ] [MkMor₂ (CoherenceM ρ)]
+    [MonadMor₁ (CoherenceM ρ)] [MonadMor₂Iso (CoherenceM ρ)]
+    [MonadCoherehnceHom (CoherenceM ρ)] [MonadNormalizeNaturality (CoherenceM ρ)]
+    [MkEqOfNaturality (CoherenceM ρ)]
+    (nm : Name) (mvarId : MVarId) : MetaM (List MVarId) :=
+  mvarId.withContext do
+    withTraceNode nm (fun ex => match ex with
+      | .ok _ => return m!"{checkEmoji} coherence equality: {← mvarId.getType}"
+      | .error err => return m!"{crossEmoji} {err.toMessageData}") do
+      let e ← instantiateMVars <| ← mvarId.getType
+      let some (_, η, θ) := (← whnfR e).eq?
+        | throwError "coherence requires an equality goal"
+      let ctx : ρ ← mkContext η
+      CoherenceM.run (ctx := ctx) do
+        let .some ηIso := (← MkMor₂.ofExpr η).isoLift? |
+          throwError "could not find a structural isomorphism, but {η}"
+        let .some θIso := (← MkMor₂.ofExpr θ).isoLift? |
+          throwError "could not find a structural isomorphism, but {θ}"
+        let f ← ηIso.e.srcM
+        let g ← ηIso.e.tgtM
+        let a := f.src
+        let nil ← normalizedHom.nilM a
+        let ⟨_, η_f⟩ ← normalize nil f
+        let ⟨_, η_g⟩ ← normalize nil g
+        let Hη ← withTraceNode nm (fun ex => do return m!"{exceptEmoji ex} LHS") do
+          naturality nm nil ηIso.e
+        let Hθ ← withTraceNode nm (fun ex => do return m!"{exceptEmoji ex} RHS") do
+          naturality nm nil θIso.e
+        let H ← mkEqOfNaturality η θ ηIso θIso η_f η_g Hη Hθ
+        mvarId.apply H
+
+end Mathlib.Tactic.BicategoryLike
diff --git a/Mathlib/Tactic/CategoryTheory/Elementwise.lean b/Mathlib/Tactic/CategoryTheory/Elementwise.lean
index d59751db664cf..4a108c00b9927 100644
--- a/Mathlib/Tactic/CategoryTheory/Elementwise.lean
+++ b/Mathlib/Tactic/CategoryTheory/Elementwise.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Kyle Miller
+Authors: Kim Morrison, Kyle Miller
 -/
 
 import Mathlib.CategoryTheory.ConcreteCategory.Basic
@@ -29,7 +29,7 @@ For more details, see the documentation attached to the `syntax` declaration.
 ## Implementation
 
 This closely follows the implementation of the `@[reassoc]` attribute, due to Simon Hudon and
-reimplemented by Scott Morrison in Lean 4.
+reimplemented by Kim Morrison in Lean 4.
 -/
 
 open Lean Meta Elab Tactic
@@ -93,7 +93,7 @@ def elementwiseExpr (src : Name) (type pf : Expr) (simpSides := true) :
         -- check that it's not a simp-trivial equality:
         forallTelescope ty' fun _ ty' => do
           if let some (_, lhs, rhs) := ty'.eq? then
-            if ← Std.Tactic.Lint.isSimpEq lhs rhs then
+            if ← Batteries.Tactic.Lint.isSimpEq lhs rhs then
               throwError "applying simp to both sides reduces elementwise lemma for {src} \
                 to the trivial equality {ty'}. \
                 Either add `nosimp` or remove the `elementwise` attribute."
diff --git a/Mathlib/Tactic/CategoryTheory/Monoidal.lean b/Mathlib/Tactic/CategoryTheory/Monoidal.lean
deleted file mode 100644
index 61bd308d9a8de..0000000000000
--- a/Mathlib/Tactic/CategoryTheory/Monoidal.lean
+++ /dev/null
@@ -1,705 +0,0 @@
-/-
-Copyright (c) 2024 Yuma Mizuno. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yuma Mizuno
--/
-import Mathlib.Tactic.CategoryTheory.Coherence
-
-/-!
-# Normalization of morphisms in monoidal categories
-This file provides a tactic that normalizes morphisms in monoidal categories. This is used in the
-string diagram widget given in `Mathlib.Tactic.StringDiagram`.
-We say that the morphism `η` in a monoidal category is in normal form if
-1. `η` is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁` where each `αᵢ` is a
-  structural 2-morphism (consisting of associators and unitors),
-2. each `ηᵢ` is a non-structural 2-morphism of the form `f₁ ◁ ... ◁ fₘ ◁ θ`, and
-3. `θ` is of the form `ι ▷ g₁ ▷ ... ▷ gₗ`
-
-Note that the structural morphisms `αᵢ` are not necessarily normalized, as the main purpose
-is to get a list of the non-structural morphisms out.
-
-Currently, the primary application of the normalization tactic in mind is drawing string diagrams,
-which are graphical representations of morphisms in monoidal categories, in the infoview. When
-drawing string diagrams, we often ignore associators and unitors (i.e., drawing morphisms in
-strict monoidal categories). On the other hand, in Lean, it is considered difficult to formalize
-the concept of strict monoidal categories due to the feature of dependent type theory. The
-normalization tactic can remove associators and unitors from the expression, extracting the
-necessary data for drawing string diagrams.
-
-The current plan on drawing string diagrams (#10581) is to use
-Penrose (https://github.com/penrose) via ProofWidget. However, it should be noted that the
-normalization procedure in this file does not rely on specific settings, allowing for broader
-application.
-
-Future plans include the following. At least I (Yuma) would like to work on these in the future,
-but it might not be immediate. If anyone is interested, I would be happy to discuss.
-
-- Currently (#10581), the string diagrams only do drawing. It would be better they also generate
-  proofs. That is, by manipulating the string diagrams displayed in the infoview with a mouse to
-  generate proofs. In #10581, the string diagram widget only uses the morphisms generated by the
-  normalization tactic and does not use proof terms ensuring that the original morphism and the
-  normalized morphism are equal. Proof terms will be necessary for proof generation.
-
-- There is also the possibility of using homotopy.io (https://github.com/homotopy-io), a graphical
-  proof assistant for category theory, from Lean. At this point, I have very few ideas regarding
-  this approach.
-
-- The normalization tactic allows for an alternative implementation of the coherent tactic.
-
-## Main definitions
-- `Tactic.Monoidal.eval`: Given a Lean expression `e` that represents a morphism in a monoidal
-category, this function returns a pair of `⟨e', pf⟩` where `e'` is the normalized expression of `e`
-and `pf` is a proof that `e = e'`.
-
--/
-
-namespace Mathlib.Tactic.Monoidal
-
-open Lean Meta Elab
-open CategoryTheory
-open Mathlib.Tactic.Coherence
-
-/-- The context for evaluating expressions. -/
-structure Context where
-  /-- The expression for the underlying category. -/
-  C : Expr
-
-/-- Populate a `context` object for evaluating `e`. -/
-def mkContext? (e : Expr) : MetaM (Option Context) := do
-  match (← inferType e).getAppFnArgs with
-  | (``Quiver.Hom, #[_, _, f, _]) =>
-    let C ← inferType f
-    return some ⟨C⟩
-  | _ => return none
-
-/-- The monad for the normalization of 2-morphisms. -/
-abbrev MonoidalM := ReaderT Context MetaM
-
-/-- Run a computation in the `M` monad. -/
-abbrev MonoidalM.run {α : Type} (c : Context) (m : MonoidalM α) : MetaM α :=
-  ReaderT.run m c
-
-/-- Expressions for atomic 1-morphisms. -/
-structure Atom₁ : Type where
-  /-- Extract a Lean expression from an `Atom₁` expression. -/
-  e : Expr
-
-/-- Expressions for 1-morphisms. -/
-inductive Mor₁ : Type
-  /-- `id` is the expression for `𝟙_ C`. -/
-  | id : Mor₁
-  /-- `comp X Y` is the expression for `X ⊗ Y` -/
-  | comp : Mor₁ → Mor₁ → Mor₁
-  /-- Construct the expression for an atomic 1-morphism. -/
-  | of : Atom₁ → Mor₁
-  deriving Inhabited
-
-/-- Converts a 1-morphism into a list of its components. -/
-def Mor₁.toList : Mor₁ → List Atom₁
-  | .id => []
-  | .comp f g => f.toList ++ g.toList
-  | .of f => [f]
-
-/-- Returns `𝟙_ C` if the expression `e` is of the form `𝟙_ C`. -/
-def isTensorUnit? (e : Expr) : MetaM (Option Expr) := do
-  let C ← mkFreshExprMVar none
-  let instC ← mkFreshExprMVar none
-  let instMC ← mkFreshExprMVar none
-  let unit := mkAppN (← mkConstWithFreshMVarLevels
-    ``MonoidalCategoryStruct.tensorUnit) #[C, instC, instMC]
-  if ← withDefault <| isDefEq e unit then
-    return ← instantiateMVars unit
-  else
-    return none
-
-/-- Returns `(f, g)` if the expression `e` is of the form `f ⊗ g`. -/
-def isTensorObj? (e : Expr) : MetaM (Option (Expr × Expr)) := do
-  let C ← mkFreshExprMVar none
-  let f ← mkFreshExprMVar C
-  let g ← mkFreshExprMVar C
-  let instC ← mkFreshExprMVar none
-  let instMC ← mkFreshExprMVar none
-  let fg := mkAppN (← mkConstWithFreshMVarLevels
-    ``MonoidalCategoryStruct.tensorObj) #[C, instC, instMC, f, g]
-  if ← withDefault <| isDefEq e fg then
-    return (← instantiateMVars f, ← instantiateMVars g)
-  else
-    return none
-
-/-- Construct a `Mor₁` expression from a Lean expression. -/
-partial def toMor₁ (e : Expr) : MetaM Mor₁ := do
-  if let some _ ← isTensorUnit? e then
-    return Mor₁.id
-  else if let some (f, g) ← isTensorObj? e then
-    return (← toMor₁ f).comp (← toMor₁ g)
-  else
-    return Mor₁.of ⟨e⟩
-
-/-- Expressions for atomic structural 2-morphisms. -/
-inductive StructuralAtom : Type
-  /-- The expression for the associator `(α_ f g h).hom`. -/
-  | associator (f g h : Mor₁) : StructuralAtom
-  /-- The expression for the inverse of the associator `(α_ f g h).inv`. -/
-  | associatorInv (f g h : Mor₁) : StructuralAtom
-  /-- The expression for the left unitor `(λ_ f).hom`. -/
-  | leftUnitor (f : Mor₁) : StructuralAtom
-  /-- The expression for the inverse of the left unitor `(λ_ f).inv`. -/
-  | leftUnitorInv (f : Mor₁) : StructuralAtom
-  /-- The expression for the right unitor `(ρ_ f).hom`. -/
-  | rightUnitor (f : Mor₁) : StructuralAtom
-  /-- The expression for the inverse of the right unitor `(ρ_ f).inv`. -/
-  | rightUnitorInv (f : Mor₁) : StructuralAtom
-  deriving Inhabited
-
-/-- Construct a `StructuralAtom` expression from a Lean expression. -/
-def structuralAtom? (e : Expr) : MetaM (Option StructuralAtom) := do
-  match e.getAppFnArgs with
-  | (``Iso.hom, #[_, _, _, _, η]) =>
-    match (← whnfR η).getAppFnArgs with
-    | (``MonoidalCategoryStruct.associator, #[_, _, _, f, g, h]) =>
-      return some <| .associator (← toMor₁ f) (← toMor₁ g) (← toMor₁ h)
-    | (``MonoidalCategoryStruct.leftUnitor, #[_, _, _, f]) =>
-      return some <| .leftUnitor (← toMor₁ f)
-    | (``MonoidalCategoryStruct.rightUnitor, #[_, _, _, f]) =>
-      return some <| .rightUnitor (← toMor₁ f)
-    | _ => return none
-  | (``Iso.inv, #[_, _, _, _, η]) =>
-    match (← whnfR η).getAppFnArgs with
-    | (``MonoidalCategoryStruct.associator, #[_, _, _, f, g, h]) =>
-      return some <| .associatorInv (← toMor₁ f) (← toMor₁ g) (← toMor₁ h)
-    | (``MonoidalCategoryStruct.leftUnitor, #[_, _, _, f]) =>
-      return some <| .leftUnitorInv (← toMor₁ f)
-    | (``MonoidalCategoryStruct.rightUnitor, #[_, _, _, f]) =>
-      return some <| .rightUnitorInv (← toMor₁ f)
-    | _ => return none
-  | _ => return none
-
-/-- Expressions for atomic non-structural 2-morphisms. -/
-structure Atom where
-  /-- Extract a Lean expression from an `Atom` expression. -/
-  e : Expr
-  deriving Inhabited
-
-/-- Expressions of the form `η ▷ f₁ ▷ ... ▷ fₙ`. -/
-inductive WhiskerRightExpr : Type
-  /-- Construct the expression for an atomic 2-morphism. -/
-  | of (η : Atom) : WhiskerRightExpr
-  /-- Construct the expression for `η ▷ f`. -/
-  | whisker (η : WhiskerRightExpr) (f : Atom₁) : WhiskerRightExpr
-  deriving Inhabited
-
-/-- Expressions of the form `f₁ ◁ ... ◁ fₙ ◁ η`. -/
-inductive WhiskerLeftExpr : Type
-  /-- Construct the expression for a right-whiskered 2-morphism. -/
-  | of (η : WhiskerRightExpr) : WhiskerLeftExpr
-  /-- Construct the expression for `f ◁ η`. -/
-  | whisker (f : Atom₁) (η : WhiskerLeftExpr) : WhiskerLeftExpr
-  deriving Inhabited
-
-/-- Expressions for structural 2-morphisms. -/
-inductive Structural : Type
-  /-- Expressions for atomic structural 2-morphisms. -/
-  | atom (η : StructuralAtom) : Structural
-  /-- Expressions for the identity `𝟙 f`. -/
-  | id (f : Mor₁) : Structural
-  /-- Expressions for the composition `η ≫ θ`. -/
-  | comp (α β : Structural) : Structural
-  /-- Expressions for the left whiskering `f ◁ η`. -/
-  | whiskerLeft (f : Mor₁) (η : Structural) : Structural
-  /-- Expressions for the right whiskering `η ▷ f`. -/
-  | whiskerRight (η : Structural) (f : Mor₁) : Structural
-  /-- Expressions for `α` in the monoidal composition `η ⊗≫ θ := η ≫ α ≫ θ`. -/
-  | monoidalCoherence (f g : Mor₁) (e : Expr) : Structural
-  deriving Inhabited
-
-/-- Normalized expressions for 2-morphisms. -/
-inductive NormalExpr : Type
-  /-- Construct the expression for a structural 2-morphism. -/
-  | nil (α : Structural) : NormalExpr
-  /-- Construct the normalized expression of 2-morphisms recursively. -/
-  | cons (head_structural : Structural) (head : WhiskerLeftExpr) (tail : NormalExpr) : NormalExpr
-  deriving Inhabited
-
-/-- The domain of a morphism. -/
-def src (η : Expr) : MetaM Mor₁ := do
-  match (← inferType η).getAppFnArgs with
-  | (``Quiver.Hom, #[_, _, f, _]) => toMor₁ f
-  | _ => throwError "{η} is not a morphism"
-
-/-- The codomain of a morphism. -/
-def tgt (η : Expr) : MetaM Mor₁ := do
-  match (← inferType η).getAppFnArgs with
-  | (``Quiver.Hom, #[_, _, _, g]) => toMor₁ g
-  | _ => throwError "{η} is not a morphism"
-
-/-- The domain of a 2-morphism. -/
-def Atom.src (η : Atom) : MetaM Mor₁ := do Monoidal.src η.e
-
-/-- The codomain of a 2-morphism. -/
-def Atom.tgt (η : Atom) : MetaM Mor₁ := do Monoidal.tgt η.e
-
-/-- The domain of a 2-morphism. -/
-def WhiskerRightExpr.src : WhiskerRightExpr → MetaM Mor₁
-  | WhiskerRightExpr.of η => η.src
-  | WhiskerRightExpr.whisker η f => return (← WhiskerRightExpr.src η).comp (Mor₁.of f)
-
-/-- The codomain of a 2-morphism. -/
-def WhiskerRightExpr.tgt : WhiskerRightExpr → MetaM Mor₁
-  | WhiskerRightExpr.of η => η.tgt
-  | WhiskerRightExpr.whisker η f => return (← WhiskerRightExpr.tgt η).comp (Mor₁.of f)
-
-/-- The domain of a 2-morphism. -/
-def WhiskerLeftExpr.src : WhiskerLeftExpr → MetaM Mor₁
-  | WhiskerLeftExpr.of η => WhiskerRightExpr.src η
-  | WhiskerLeftExpr.whisker f η => return (Mor₁.of f).comp (← WhiskerLeftExpr.src η)
-
-/-- The codomain of a 2-morphism. -/
-def WhiskerLeftExpr.tgt : WhiskerLeftExpr → MetaM Mor₁
-  | WhiskerLeftExpr.of η => WhiskerRightExpr.tgt η
-  | WhiskerLeftExpr.whisker f η => return (Mor₁.of f).comp (← WhiskerLeftExpr.tgt η)
-
-/-- The domain of a 2-morphism. -/
-def StructuralAtom.src : StructuralAtom → Mor₁
-  | .associator f g h => (f.comp g).comp h
-  | .associatorInv f g h => f.comp (g.comp h)
-  | .leftUnitor f => Mor₁.id.comp f
-  | .leftUnitorInv f => f
-  | .rightUnitor f => f.comp Mor₁.id
-  | .rightUnitorInv f => f
-
-/-- The codomain of a 2-morphism. -/
-def StructuralAtom.tgt : StructuralAtom → Mor₁
-  | .associator f g h => f.comp (g.comp h)
-  | .associatorInv f g h => (f.comp g).comp h
-  | .leftUnitor f => f
-  | .leftUnitorInv f => Mor₁.id.comp f
-  | .rightUnitor f => f
-  | .rightUnitorInv f => f.comp Mor₁.id
-
-/-- The domain of a 2-morphism. -/
-def Structural.src : Structural → Mor₁
-  | .atom η => η.src
-  | .id f => f
-  | .comp α _ => α.src
-  | .whiskerLeft f η => f.comp η.src
-  | .whiskerRight η f => η.src.comp f
-  | .monoidalCoherence f _ _ => f
-
-/-- The codomain of a 2-morphism. -/
-def Structural.tgt : Structural → Mor₁
-  | .atom η => η.tgt
-  | .id f => f
-  | .comp _ β => β.tgt
-  | .whiskerLeft f η => f.comp η.tgt
-  | .whiskerRight η f => η.tgt.comp f
-  | .monoidalCoherence _ g _ => g
-
-/-- The domain of a 2-morphism. -/
-def NormalExpr.src : NormalExpr → Mor₁
-  | NormalExpr.nil η => η.src
-  | NormalExpr.cons α _ _ => α.src
-
-/-- The codomain of a 2-morphism. -/
-def NormalExpr.tgt : NormalExpr → Mor₁
-  | NormalExpr.nil η => η.tgt
-  | NormalExpr.cons _ _ ηs => ηs.tgt
-
-/-- The associator as a term of `normalExpr`. -/
-def NormalExpr.associator (f g h : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .associator f g h
-
-/-- The inverse of the associator as a term of `normalExpr`. -/
-def NormalExpr.associatorInv (f g h : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .associatorInv f g h
-
-/-- The left unitor as a term of `normalExpr`. -/
-def NormalExpr.leftUnitor (f : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .leftUnitor f
-
-/-- The inverse of the left unitor as a term of `normalExpr`. -/
-def NormalExpr.leftUnitorInv (f : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .leftUnitorInv f
-
-/-- The right unitor as a term of `normalExpr`. -/
-def NormalExpr.rightUnitor (f : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .rightUnitor f
-
-/-- The inverse of the right unitor as a term of `normalExpr`. -/
-def NormalExpr.rightUnitorInv (f : Mor₁) : NormalExpr :=
-  .nil <| .atom <| .rightUnitorInv f
-
-/-- Return `η` for `η ▷ g₁ ▷ ... ▷ gₙ`. -/
-def WhiskerRightExpr.atom : WhiskerRightExpr → Atom
-  | WhiskerRightExpr.of η => η
-  | WhiskerRightExpr.whisker η _ => η.atom
-
-/-- Return `η` for `f₁ ◁ ... ◁ fₙ ◁ η ▷ g₁ ▷ ... ▷ gₙ`. -/
-def WhiskerLeftExpr.atom : WhiskerLeftExpr → Atom
-  | WhiskerLeftExpr.of η => η.atom
-  | WhiskerLeftExpr.whisker _ η => η.atom
-
-/-- Construct a `Structural` expression from a Lean expression for a structural 2-morphism. -/
-partial def structural? (e : Expr) : MetaM Structural := do
-  match (← whnfR e).getAppFnArgs with
-  | (``CategoryStruct.comp, #[_, _, _, α, β]) =>
-    return .comp (← structural? α) (← structural? β)
-  | (``CategoryStruct.id, #[_, f]) => return .id (← toMor₁ f)
-  | (``MonoidalCategoryStruct.whiskerLeft, #[f, η]) =>
-    return .whiskerLeft (← toMor₁ f) (← structural? η)
-  | (``MonoidalCategoryStruct.whiskerRight, #[η, f]) =>
-    return .whiskerRight (← structural? η) (← toMor₁ f)
-  | (``MonoidalCoherence.hom, #[_, _, f, g, inst]) =>
-    return .monoidalCoherence (← toMor₁ f) (← toMor₁ g) inst
-  | _ => match ← structuralAtom? e with
-    | some η => return .atom η
-    | none => throwError "not a structural 2-morphism"
-
-/-- Construct a `NormalExpr` expression from a `WhiskerLeftExpr` expression. -/
-def NormalExpr.of (η : WhiskerLeftExpr) : MetaM NormalExpr := do
-  return .cons (.id (← η.src)) η (.nil (.id (← η.tgt)))
-
-/-- Construct a `NormalExpr` expression from a Lean expression for an atomic 2-morphism. -/
-def NormalExpr.ofExpr (η : Expr) : MetaM NormalExpr :=
-  NormalExpr.of <| .of <| .of ⟨η⟩
-
-/-- If `e` is an expression of the form `η ⊗≫ θ := η ≫ α ≫ θ` in the monoidal category `C`,
-return the expression for `α` .-/
-def structuralOfMonoidalComp (C e : Expr) : MetaM Structural := do
-  let v ← mkFreshLevelMVar
-  let u ← mkFreshLevelMVar
-  _ ← isDefEq (.sort (.succ v)) (← inferType (← inferType e))
-  _ ← isDefEq (.sort (.succ u)) (← inferType C)
-  let W ← mkFreshExprMVar none
-  let X ← mkFreshExprMVar none
-  let Y ← mkFreshExprMVar none
-  let Z ← mkFreshExprMVar none
-  let f ← mkFreshExprMVar none
-  let g ← mkFreshExprMVar none
-  let α₀ ← mkFreshExprMVar none
-  let instC ← mkFreshExprMVar none
-  let αg := mkAppN (.const ``CategoryStruct.comp [v, u]) #[C, instC, X, Y, Z, α₀, g]
-  let fαg := mkAppN (.const ``CategoryStruct.comp [v, u]) #[C, instC, W, X, Z, f, αg]
-  _ ← isDefEq e fαg
-  structural? α₀
-
-section
-
-open scoped MonoidalCategory
-
-universe v u
-
-variable {C : Type u} [Category.{v} C]
-
-variable {f f' g g' h i j : C}
-
-theorem evalComp_nil_cons {f g h i j : C} (α : f ⟶ g) (β : g ⟶ h) (η : h ⟶ i) (ηs : i ⟶ j) :
-    α ≫ (β ≫ η ≫ ηs) = (α ≫ β) ≫ η ≫ ηs := by
-  simp
-
-@[nolint synTaut]
-theorem evalComp_nil_nil {f g h : C} (α : f ⟶ g) (β : g ⟶ h) :
-    α ≫ β = α ≫ β := by
-  simp
-
-theorem evalComp_cons {f g h i j : C} (α : f ⟶ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : i ⟶ j} {ι : h ⟶ j}
-    (pf_ι : ηs ≫ θ = ι)  :
-    (α ≫ η ≫ ηs) ≫ θ = α ≫ η ≫ ι := by
-  simp [pf_ι]
-
-theorem eval_comp
-    {η η' : f ⟶ g} {θ θ' : g ⟶ h} {ι : f ⟶ h}
-    (pf_η : η = η') (pf_θ : θ = θ') (pf_ηθ : η' ≫ θ' = ι) :
-    η ≫ θ = ι := by
-  simp [pf_η, pf_θ, pf_ηθ]
-
-theorem eval_of (η : f ⟶ g) :
-    η = 𝟙 _ ≫ η ≫ 𝟙 _ := by
-  simp
-
-theorem eval_monoidalComp
-    {η η' : f ⟶ g} {α : g ⟶ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
-    (pf_η : η = η') (pf_θ : θ = θ') (pf_αθ : α ≫ θ' = αθ) (pf_ηαθ : η' ≫ αθ = ηαθ) :
-    η ≫ α ≫ θ = ηαθ := by
-  simp [pf_η, pf_θ, pf_αθ, pf_ηαθ]
-
-variable [MonoidalCategory C]
-
-@[nolint synTaut]
-theorem evalWhiskerLeft_nil (f : C) (α : g ⟶ h) :
-    f ◁ α = f ◁ α := by
-  simp
-
-theorem evalWhiskerLeft_of_cons
-    (α : g ⟶ h) (η : h ⟶ i) {ηs : i ⟶ j} {θ : f ⊗ i ⟶ f ⊗ j} (pf_θ : f ◁ ηs = θ) :
-    f ◁ (α ≫ η ≫ ηs) = f ◁ α ≫ f ◁ η ≫ θ := by
-  simp [pf_θ]
-
-theorem evalWhiskerLeft_comp {η : h ⟶ i} {θ : g ⊗ h ⟶ g ⊗ i} {ι : f ⊗ g ⊗ h ⟶ f ⊗ g ⊗ i}
-    {ι' : f ⊗ g ⊗ h ⟶ (f ⊗ g) ⊗ i} {ι'' : (f ⊗ g) ⊗ h ⟶ (f ⊗ g) ⊗ i}
-    (pf_θ : g ◁ η = θ) (pf_ι : f ◁ θ = ι)
-    (pf_ι' : ι ≫ (α_ _ _ _).inv = ι') (pf_ι'' : (α_ _ _ _).hom ≫ ι' = ι'') :
-    (f ⊗ g) ◁ η = ι'' := by
-  simp [pf_θ, pf_ι, pf_ι', pf_ι'']
-
-theorem evalWhiskerLeft_id {f g : C} {η : f ⟶ g}
-    {η' : f ⟶ 𝟙_ C ⊗ g} {η'' : 𝟙_ C ⊗ f ⟶ 𝟙_ C ⊗ g}
-    (pf_η' : η ≫ (λ_ _).inv = η') (pf_η'' : (λ_ _).hom ≫ η' = η'') :
-    𝟙_ C ◁ η = η'' := by
-  simp [pf_η', pf_η'']
-
-theorem eval_whiskerLeft
-    {η η' : g ⟶ h} {θ : f ⊗ g ⟶ f ⊗ h}
-    (pf_η : η = η') (pf_θ : f ◁ η' = θ) :
-    f ◁ η = θ := by
-  simp [pf_η, pf_θ]
-
-theorem eval_whiskerRight
-    {η η' : f ⟶ g} {θ : f ⊗ h ⟶ g ⊗ h}
-    (pf_η : η = η') (pf_θ : η' ▷ h = θ) :
-    η ▷ h = θ := by
-  simp [pf_η, pf_θ]
-
-@[nolint synTaut]
-theorem evalWhiskerRight_nil (α : f ⟶ g) (h : C) :
-    α ▷ h = α ▷ h := by
-  simp
-
-theorem evalWhiskerRight_cons_of_of
-    (α : f ⟶ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : h ⊗ j ⟶ i ⊗ j}
-    (pf_θ : ηs ▷ j = θ) :
-    (α ≫ η ≫ ηs) ▷ j = α ▷ j ≫ η ▷ j ≫ θ := by
-  simp [pf_θ]
-
-theorem evalWhiskerRight_cons_whisker
-    {α : g ⟶ f ⊗ h} {η : h ⟶ i} {ηs : f ⊗ i ⟶ j} {k : C}
-    {η₁ : h ⊗ k ⟶ i ⊗ k} {η₂ : f ⊗ (h ⊗ k) ⟶ f ⊗ (i ⊗ k)} {ηs₁ : (f ⊗ i) ⊗ k ⟶ j ⊗ k}
-    {ηs₂ : f ⊗ (i ⊗ k) ⟶ j ⊗ k} {η₃ : f ⊗ (h ⊗ k) ⟶ j ⊗ k} {η₄ : (f ⊗ h) ⊗ k ⟶ j ⊗ k}
-    {η₅ : g ⊗ k ⟶ j ⊗ k}
-    (pf_η₁ : (𝟙 _ ≫ η ≫ 𝟙 _ ) ▷ k = η₁) (pf_η₂ : f ◁ η₁ = η₂)
-    (pf_ηs₁ : ηs ▷ k = ηs₁) (pf_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
-    (pf_η₃ : η₂ ≫ ηs₂ = η₃) (pf_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄) (pf_η₅ : α ▷ k ≫ η₄ = η₅) :
-    (α ≫ (f ◁ η) ≫ ηs) ▷ k = η₅ := by
-  simp at pf_η₁
-  simp [pf_η₁, pf_η₂, pf_ηs₁, pf_ηs₂, pf_η₃, pf_η₄, pf_η₅]
-
-theorem evalWhiskerRight_comp
-    {η : f ⟶ f'} {η₁ : f ⊗ g ⟶ f' ⊗ g} {η₂ : (f ⊗ g) ⊗ h ⟶ (f' ⊗ g) ⊗ h}
-    {η₃ : (f ⊗ g) ⊗ h ⟶ f' ⊗ (g ⊗ h)} {η₄ : f ⊗ (g ⊗ h) ⟶ f' ⊗ (g ⊗ h)}
-    (pf_η₁ : η ▷ g = η₁) (pf_η₂ : η₁ ▷ h = η₂)
-    (pf_η₃ : η₂ ≫ (α_ _ _ _).hom = η₃) (pf_η₄ : (α_ _ _ _).inv ≫ η₃ = η₄) :
-    η ▷ (g ⊗ h) = η₄ := by
-  simp [pf_η₁, pf_η₂, pf_η₃, pf_η₄]
-
-theorem evalWhiskerRight_id
-    {η : f ⟶ g} {η₁ : f ⟶ g ⊗ 𝟙_ C} {η₂ : f ⊗ 𝟙_ C ⟶ g ⊗ 𝟙_ C}
-    (pf_η₁ : η ≫ (ρ_ _).inv = η₁) (pf_η₂ : (ρ_ _).hom ≫ η₁ = η₂) :
-    η ▷ 𝟙_ C = η₂ := by
-  simp [pf_η₁, pf_η₂]
-
-end
-
-/-- Extract a Lean expression from a `Mor₁` expression. -/
-def Mor₁.e : Mor₁ → MonoidalM Expr
-  | .id => do
-    let ctx ← read
-    mkAppOptM ``MonoidalCategoryStruct.tensorUnit #[ctx.C, none, none]
-  | .comp f g => do
-    mkAppM ``MonoidalCategoryStruct.tensorObj #[← Mor₁.e f, ← Mor₁.e g]
-  | .of f => return f.e
-
-/-- Extract a Lean expression from a `StructuralAtom` expression. -/
-def StructuralAtom.e : StructuralAtom → MonoidalM Expr
-  | .associator f g h => do
-    mkAppM ``Iso.hom #[← mkAppM ``MonoidalCategoryStruct.associator #[← f.e, ← g.e, ← h.e]]
-  | .associatorInv f g h => do
-    mkAppM ``Iso.inv #[← mkAppM ``MonoidalCategoryStruct.associator #[← f.e, ← g.e, ← h.e]]
-  | .leftUnitor f => do
-    mkAppM ``Iso.hom #[← mkAppM ``MonoidalCategoryStruct.leftUnitor #[← f.e]]
-  | .leftUnitorInv f => do
-    mkAppM ``Iso.inv #[← mkAppM ``MonoidalCategoryStruct.leftUnitor #[← f.e]]
-  | .rightUnitor f => do
-    mkAppM ``Iso.hom #[← mkAppM ``MonoidalCategoryStruct.rightUnitor #[← f.e]]
-  | .rightUnitorInv f => do
-    mkAppM ``Iso.inv #[← mkAppM ``MonoidalCategoryStruct.rightUnitor #[← f.e]]
-
-/-- Extract a Lean expression from a `Structural` expression. -/
-partial def Structural.e : Structural → MonoidalM Expr
-  | .atom η => η.e
-  | .id f => do mkAppM ``CategoryStruct.id #[← f.e]
-  | .comp α β => do mkAppM ``CategoryStruct.comp #[← α.e, ← β.e]
-  | .whiskerLeft f η => do mkAppM ``MonoidalCategoryStruct.whiskerLeft #[← f.e, ← η.e]
-  | .whiskerRight η f => do mkAppM ``MonoidalCategoryStruct.whiskerRight #[← η.e, ← f.e]
-  | .monoidalCoherence _ _ e => do
-    mkAppOptM ``MonoidalCoherence.hom #[none, none, none, none, e]
-
-/-- Extract a Lean expression from a `WhiskerRightExpr` expression. -/
-def WhiskerRightExpr.e : WhiskerRightExpr → MonoidalM Expr
-  | WhiskerRightExpr.of η => return η.e
-  | WhiskerRightExpr.whisker η f => do
-    mkAppM ``MonoidalCategoryStruct.whiskerRight #[← η.e, f.e]
-
-/-- Extract a Lean expression from a `WhiskerLeftExpr` expression. -/
-def WhiskerLeftExpr.e : WhiskerLeftExpr → MonoidalM Expr
-  | WhiskerLeftExpr.of η => η.e
-  | WhiskerLeftExpr.whisker f η => do
-    mkAppM ``MonoidalCategoryStruct.whiskerLeft #[f.e, ← η.e]
-
-/-- Extract a Lean expression from a `NormalExpr` expression. -/
-def NormalExpr.e : NormalExpr → MonoidalM Expr
-  | NormalExpr.nil α => α.e
-  | NormalExpr.cons α η θ => do
-    mkAppM ``CategoryStruct.comp #[← α.e, ← mkAppM ``CategoryStruct.comp #[← η.e, ← θ.e]]
-
-/-- The result of evaluating an expression into normal form. -/
-structure Result where
-  /-- The normalized expression of the 2-morphism. -/
-  expr : NormalExpr
-  /-- The proof that the normalized expression is equal to the original expression. -/
-  proof : Expr
-
-/-- Evaluate the expression `η ≫ θ` into a normalized form. -/
-partial def evalComp : NormalExpr → NormalExpr → MonoidalM Result
-  | .nil α, .cons β η ηs => do
-    let η' := .cons (α.comp β) η ηs
-    return ⟨η', ← mkAppM ``evalComp_nil_cons #[← α.e, ← β.e, ← η.e, ← ηs.e]⟩
-  | .nil α, .nil α' => do
-    return ⟨.nil (α.comp α'), ← mkAppM ``evalComp_nil_nil #[← α.e, ← α'.e]⟩
-  | .cons α η ηs, θ => do
-    let ⟨ι, pf_ι⟩ ← evalComp ηs θ
-    let ι' := .cons α η ι
-    return ⟨ι', ← mkAppM ``evalComp_cons #[← α.e, ← η.e, pf_ι]⟩
-
-/-- Evaluate the expression `f ◁ η` into a normalized form. -/
-partial def evalWhiskerLeftExpr : Mor₁ → NormalExpr → MonoidalM Result
-  | f, .nil α => do
-    return ⟨.nil (.whiskerLeft f α), ← mkAppM ``evalWhiskerLeft_nil #[← f.e, ← α.e]⟩
-  | .of f, .cons α η ηs => do
-    let η' := WhiskerLeftExpr.whisker f η
-    let ⟨θ, pf_θ⟩ ← evalWhiskerLeftExpr (.of f) ηs
-    let η'' := .cons (.whiskerLeft (.of f) α) η' θ
-    return ⟨η'', ← mkAppM ``evalWhiskerLeft_of_cons #[← α.e, ← η.e, pf_θ]⟩
-  | .comp f g, η => do
-    let ⟨θ, pf_θ⟩ ← evalWhiskerLeftExpr g η
-    let ⟨ι, pf_ι⟩ ← evalWhiskerLeftExpr f θ
-    let h := η.src
-    let h' := η.tgt
-    let ⟨ι', pf_ι'⟩ ← evalComp ι (NormalExpr.associatorInv f g h')
-    let ⟨ι'', pf_ι''⟩ ← evalComp (NormalExpr.associator f g h) ι'
-    return ⟨ι'', ← mkAppM ``evalWhiskerLeft_comp #[pf_θ, pf_ι, pf_ι', pf_ι'']⟩
-  | .id, η => do
-    let f := η.src
-    let g := η.tgt
-    let ⟨η', pf_η'⟩ ← evalComp η (NormalExpr.leftUnitorInv g)
-    let ⟨η'', pf_η''⟩ ← evalComp (NormalExpr.leftUnitor f) η'
-    return ⟨η'', ← mkAppM ``evalWhiskerLeft_id #[pf_η', pf_η'']⟩
-
-/-- Evaluate the expression `η ▷ f` into a normalized form. -/
-partial def evalWhiskerRightExpr : NormalExpr → Mor₁ → MonoidalM Result
-  | .nil α, h => do
-    return ⟨.nil (.whiskerRight α h), ← mkAppM ``evalWhiskerRight_nil #[← α.e, ← h.e]⟩
-  | .cons α (.of η) ηs, .of f => do
-    let ⟨θ, pf_θ⟩ ← evalWhiskerRightExpr ηs (.of f)
-    let η' := .cons (.whiskerRight α (.of f)) (.of (.whisker η f)) θ
-    return ⟨η', ← mkAppM ``evalWhiskerRight_cons_of_of #[← α.e, ← η.e, pf_θ]⟩
-  | .cons α (.whisker f η) ηs, h => do
-    let g ← η.src
-    let g' ← η.tgt
-    let ⟨η₁, pf_η₁⟩ ← evalWhiskerRightExpr (.cons (.id g) η (.nil (.id g'))) h
-    let ⟨η₂, pf_η₂⟩ ← evalWhiskerLeftExpr (.of f) η₁
-    let ⟨ηs₁, pf_ηs₁⟩ ← evalWhiskerRightExpr ηs h
-    let α' := .whiskerRight α h
-    let ⟨ηs₂, pf_ηs₂⟩ ← evalComp (.associatorInv (.of f) g' h) ηs₁
-    let ⟨η₃, pf_η₃⟩ ← evalComp η₂ ηs₂
-    let ⟨η₄, pf_η₄⟩ ← evalComp (.associator (.of f) g h) η₃
-    let ⟨η₅, pf_η₅⟩ ← evalComp (.nil α') η₄
-    return ⟨η₅, ← mkAppM ``evalWhiskerRight_cons_whisker
-      #[pf_η₁, pf_η₂, pf_ηs₁, pf_ηs₂, pf_η₃, pf_η₄, pf_η₅]⟩
-  | η, .comp g h => do
-    let ⟨η₁, pf_η₁⟩ ← evalWhiskerRightExpr η g
-    let ⟨η₂, pf_η₂⟩ ← evalWhiskerRightExpr η₁ h
-    let f := η.src
-    let f' := η.tgt
-    let ⟨η₃, pf_η₃⟩ ← evalComp η₂ (.associator f' g h)
-    let ⟨η₄, pf_η₄⟩ ← evalComp (.associatorInv f g h) η₃
-    return ⟨η₄, ← mkAppM ``evalWhiskerRight_comp #[pf_η₁, pf_η₂, pf_η₃, pf_η₄]⟩
-  | η, .id => do
-    let f := η.src
-    let g := η.tgt
-    let ⟨η₁, pf_η₁⟩ ← evalComp η (.rightUnitorInv g)
-    let ⟨η₂, pf_η₂⟩ ← evalComp (.rightUnitor f) η₁
-    return ⟨η₂, ← mkAppM ``evalWhiskerRight_id #[pf_η₁, pf_η₂]⟩
-
-/-- Evaluate the expression of a 2-morphism into a normalized form. -/
-partial def eval (e : Expr) : MonoidalM Result := do
-  if let .some α ← structuralAtom? e then
-    return ⟨.nil <| .atom α, ← mkEqRefl (← α.e)⟩
-  else
-    match e.getAppFnArgs with
-    | (``CategoryStruct.id, #[_, _, f]) =>
-      return ⟨.nil (.id (← toMor₁ f)), ← mkEqRefl (← mkAppM ``CategoryStruct.id #[f])⟩
-    | (``CategoryStruct.comp, #[_, _, _, _, _, η, θ]) =>
-      let ⟨η_e, pf_η⟩ ← eval η
-      let ⟨θ_e, pf_θ⟩ ← eval θ
-      let ⟨ηθ, pf⟩ ← evalComp η_e θ_e
-      return ⟨ηθ, ← mkAppM ``eval_comp #[pf_η, pf_θ, pf]⟩
-    | (``MonoidalCategoryStruct.whiskerLeft, #[_, _, _, f, _, _, η]) =>
-      let ⟨η_e, pf_η⟩ ← eval η
-      let ⟨θ, pf_θ⟩ ← evalWhiskerLeftExpr (← toMor₁ f) η_e
-      return ⟨θ, ← mkAppM ``eval_whiskerLeft #[pf_η, pf_θ]⟩
-    | (``MonoidalCategoryStruct.whiskerRight, #[_, _, _, _, _, η, h]) =>
-      let ⟨η_e, pf_η⟩ ← eval η
-      let ⟨θ, pf_θ⟩ ← evalWhiskerRightExpr η_e (← toMor₁ h)
-      return ⟨θ, ← mkAppM ``eval_whiskerRight #[pf_η, pf_θ]⟩
-    | (``monoidalComp, #[C, _, _, _, _, _, _, η, θ]) =>
-      let ⟨η_e, pf_η⟩ ← eval η
-      let α₀ ← structuralOfMonoidalComp C e
-      let α := NormalExpr.nil α₀
-      let ⟨θ_e, pf_θ⟩ ← eval θ
-      let ⟨αθ, pf_θα⟩ ← evalComp α θ_e
-      let ⟨ηαθ, pf_ηαθ⟩ ← evalComp η_e αθ
-      return ⟨ηαθ, ← mkAppM ``eval_monoidalComp #[pf_η, pf_θ, pf_θα, pf_ηαθ]⟩
-    | _ =>
-      return ⟨← NormalExpr.ofExpr e, ← mkAppM ``eval_of #[e]⟩
-
-/-- Convert a `NormalExpr` expression into a list of `WhiskerLeftExpr` expressions. -/
-def NormalExpr.toList : NormalExpr → List WhiskerLeftExpr
-  | NormalExpr.nil _ => []
-  | NormalExpr.cons _ η ηs => η :: NormalExpr.toList ηs
-
-end Mathlib.Tactic.Monoidal
-
-open Mathlib.Tactic.Monoidal
-
-/-- `normalize% η` is the normalization of the 2-morphism `η`.
-1. The normalized 2-morphism is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁` where
-  each `αᵢ` is a structural 2-morphism (consisting of associators and unitors),
-2. each `ηᵢ` is a non-structural 2-morphism of the form `f₁ ◁ ... ◁ fₘ ◁ θ`, and
-3. `θ` is of the form `ι ▷ g₁ ▷ ... ▷ gₗ`
--/
-elab "normalize% " t:term:51 : term => do
-  let e ← Lean.Elab.Term.elabTerm t none
-  let some ctx ← mkContext? e
-    | throwError "not a morphism"
-  MonoidalM.run ctx do (← eval e).expr.e
-
-theorem mk_eq {α : Type _} (a b a' b' : α) (ha : a = a') (hb : b = b') (h : a' = b') : a = b := by
-  simp [h, ha, hb]
-
-open Lean Elab Meta Tactic in
-/-- Transform an equality between 2-morphisms into the equality between their normalizations. -/
-def mkEq (e : Expr) : MetaM Expr := do
-  let some (_, e₁, e₂) := (← whnfR <| e).eq?
-    | throwError "monoidal_nf requires an equality goal"
-  let some ctx ← mkContext? e₁
-    | throwError "the lhs and rhs must be morphisms"
-  MonoidalM.run ctx do
-    let ⟨e₁', p₁⟩ ← eval e₁
-    let ⟨e₂', p₂⟩ ← eval e₂
-    mkAppM ``mk_eq #[e₁, e₂, ← e₁'.e, ← e₂'.e, p₁, p₂]
-
-open Lean Elab Tactic in
-/-- Normalize the both sides of an equality. -/
-elab "monoidal_nf" : tactic => withMainContext do
-  let t ← getMainTarget
-  let mvarIds ← (← getMainGoal).apply (← mkEq t)
-  replaceMainGoal mvarIds
diff --git a/Mathlib/Tactic/CategoryTheory/Monoidal/Basic.lean b/Mathlib/Tactic/CategoryTheory/Monoidal/Basic.lean
new file mode 100644
index 0000000000000..6158728b44231
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Monoidal/Basic.lean
@@ -0,0 +1,55 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Basic
+import Mathlib.Tactic.CategoryTheory.Monoidal.Normalize
+import Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence
+
+/-!
+# `monoidal` tactic
+
+This file provides `monoidal` tactic, which solves equations in a monoidal category, where
+the two sides only differ by replacing strings of monoidal structural morphisms (that is,
+associators, unitors, and identities) with different strings of structural morphisms with the same
+source and target. In other words, `monoidal` solves equalities where both sides have the same
+string diagrams.
+
+The core function for the `monoidal` tactic is provided in
+`Mathlib.Tactic.CategoryTheory.Coherence.Basic`. See this file for more details about the
+implementation.
+
+-/
+
+open Lean Meta Elab Tactic
+open CategoryTheory Mathlib.Tactic.BicategoryLike
+
+namespace Mathlib.Tactic.Monoidal
+
+/-- Normalize the both sides of an equality. -/
+def monoidalNf (mvarId : MVarId) : MetaM (List MVarId) := do
+  BicategoryLike.normalForm Monoidal.Context `monoidal mvarId
+
+@[inherit_doc monoidalNf]
+elab "monoidal_nf" : tactic => withMainContext do
+  replaceMainGoal (← monoidalNf (← getMainGoal))
+
+/--
+Use the coherence theorem for monoidal categories to solve equations in a monoidal category,
+where the two sides only differ by replacing strings of monoidal structural morphisms
+(that is, associators, unitors, and identities)
+with different strings of structural morphisms with the same source and target.
+
+That is, `monoidal` can handle goals of the form
+`a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c'`
+where `a = a'`, `b = b'`, and `c = c'` can be proved using `monoidal_coherence`.
+-/
+def monoidal (mvarId : MVarId) : MetaM (List MVarId) :=
+  BicategoryLike.main Monoidal.Context `monoidal mvarId
+
+@[inherit_doc monoidal]
+elab "monoidal" : tactic => withMainContext do
+  replaceMainGoal <| ← monoidal <| ← getMainGoal
+
+end Mathlib.Tactic.Monoidal
diff --git a/Mathlib/Tactic/CategoryTheory/Monoidal/Datatypes.lean b/Mathlib/Tactic/CategoryTheory/Monoidal/Datatypes.lean
new file mode 100644
index 0000000000000..43c3ef94fc6cc
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Monoidal/Datatypes.lean
@@ -0,0 +1,505 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Datatypes
+import Mathlib.Tactic.CategoryTheory.MonoidalComp
+
+/-!
+# Expressions for monoidal categories
+
+This file converts lean expressions representing morphisms in monoidal categories into `Mor₂Iso`
+or `Mor` terms. The converted expressions are used in the coherence tactics and the string diagram
+widgets.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike MonoidalCategory
+
+namespace Mathlib.Tactic.Monoidal
+
+/-- The domain of a morphism. -/
+def srcExpr (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, f, _]) => return f
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The codomain of a morphism. -/
+def tgtExpr (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, _, g]) => return g
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The domain of an isomorphism. -/
+def srcExprOfIso (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Iso, #[_, _, f, _]) => return f
+  | _ => throwError m!"{η} is not a morphism"
+
+/-- The codomain of an isomorphism. -/
+def tgtExprOfIso (η : Expr) : MetaM Expr := do
+  match (← whnfR (← inferType η)).getAppFnArgs with
+  | (``Iso, #[_, _, _, g]) => return g
+  | _ => throwError m!"{η} is not a morphism"
+
+initialize registerTraceClass `monoidal
+
+/-- The context for evaluating expressions. -/
+structure Context where
+  /-- The level for morphisms. -/
+  level₂ : Level
+  /-- The level for objects. -/
+  level₁ : Level
+  /-- The expression for the underlying category. -/
+  C : Q(Type level₁)
+  /-- The category instance. -/
+  instCat : Q(Category.{level₂, level₁} $C)
+  /-- The monoidal category instance. -/
+  instMonoidal? : Option Q(MonoidalCategory.{level₂, level₁} $C)
+
+/-- Populate a `context` object for evaluating `e`. -/
+def mkContext? (e : Expr) : MetaM (Option Context) := do
+  let e ← instantiateMVars e
+  let type ← instantiateMVars <| ← inferType e
+  match (← whnfR type).getAppFnArgs with
+  | (``Quiver.Hom, #[_, _, f, _]) =>
+    let C ← instantiateMVars <| ← inferType f
+    let .succ level₁ ← getLevel C | return none
+    let .succ level₂ ← getLevel type | return none
+    let .some instCat ← synthInstance?
+      (mkAppN (.const ``Category [level₂, level₁]) #[C]) | return none
+    let instMonoidal? ← synthInstance?
+      (mkAppN (.const ``MonoidalCategory [level₂, level₁]) #[C, instCat])
+    return some ⟨level₂, level₁, C, instCat, instMonoidal?⟩
+  | _ => return none
+
+instance : BicategoryLike.Context Monoidal.Context where
+  mkContext? := Monoidal.mkContext?
+
+/-- The monad for the normalization of 2-morphisms. -/
+abbrev MonoidalM := CoherenceM Context
+
+/-- Throw an error if the monoidal category instance is not found. -/
+def synthMonoidalError {α : Type} : MetaM α := do
+  throwError "failed to find monoidal category instance"
+
+instance : MonadMor₁ MonoidalM where
+  id₁M a := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    return .id (q(MonoidalCategory.tensorUnit) : Q($ctx.C)) a
+  comp₁M f g := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f_e : Q($ctx.C) := f.e
+    let g_e : Q($ctx.C) := g.e
+    return .comp (q($f_e ⊗ $g_e)) f g
+
+section
+
+universe v u
+variable {C : Type u} [Category.{v} C]
+
+theorem structuralIsoOfExpr_comp {f g h : C}
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)
+    (θ : g ⟶ h) (θ' : g ≅ h) (ih_θ : θ'.hom = θ) :
+    (η' ≪≫ θ').hom  = η ≫ θ := by
+  simp [ih_η, ih_θ]
+
+theorem StructuralOfExpr_monoidalComp {f g h i : C} [MonoidalCoherence g h]
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η) (θ : h ⟶ i) (θ' : h ≅ i) (ih_θ : θ'.hom = θ) :
+    (η' ≪⊗≫ θ').hom = η ⊗≫ θ := by
+  simp [ih_η, ih_θ, monoidalIsoComp, monoidalComp, MonoidalCoherence.iso]
+
+variable [MonoidalCategory C]
+
+theorem structuralIsoOfExpr_whiskerLeft (f : C) {g h : C}
+    (η : g ⟶ h) (η' : g ≅ h) (ih_η : η'.hom = η)  :
+    (whiskerLeftIso f η').hom = f ◁ η := by
+  simp [ih_η]
+
+theorem structuralIsoOfExpr_whiskerRight {f g : C} (h : C)
+    (η : f ⟶ g) (η' : f ≅ g) (ih_η : η'.hom = η)  :
+    (whiskerRightIso η' h).hom = η ▷ h := by
+  simp [ih_η]
+
+theorem structuralIsoOfExpr_horizontalComp {f₁ g₁ f₂ g₂ : C}
+    (η : f₁ ⟶ g₁) (η' : f₁ ≅ g₁) (ih_η : η'.hom = η)
+    (θ : f₂ ⟶ g₂) (θ' : f₂ ≅ g₂) (ih_θ : θ'.hom = θ) :
+    (η' ⊗ θ').hom = η ⊗ θ := by
+  simp [ih_η, ih_θ]
+
+end
+
+open MonadMor₁
+
+instance : MonadMor₂Iso MonoidalM where
+  associatorM f g h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f_e : Q($ctx.C) := f.e
+    let g_e : Q($ctx.C) := g.e
+    let h_e : Q($ctx.C) := h.e
+    return .associator q(α_ $f_e $g_e $h_e) f g h
+  leftUnitorM f := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f_e : Q($ctx.C) := f.e
+    return .leftUnitor q(λ_ $f_e) f
+  rightUnitorM f := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f_e : Q($ctx.C) := f.e
+    return .rightUnitor q(ρ_ $f_e) f
+  id₂M f := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    have f_e : Q($ctx.C) := f.e
+    return .id q(Iso.refl $f_e) f
+  coherenceHomM f g inst := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have inst : Q(MonoidalCoherence $f_e $g_e) := inst
+    match (← whnfI inst).getAppFnArgs with
+    | (``MonoidalCoherence.mk, #[_, _, _, _, α]) =>
+      let e : Q($f_e ≅ $g_e) := q(MonoidalCoherence.iso)
+      return ⟨e, f, g, inst, α⟩
+    | _ => throwError m!"failed to unfold {inst}"
+  comp₂M η θ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    have θ_e : Q($g_e ≅ $h_e) := θ.e
+    return .comp q($η_e ≪≫ $θ_e) f g h η θ
+  whiskerLeftM f η := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η.srcM
+    let h ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($g_e ≅ $h_e) := η.e
+    return .whiskerLeft q(whiskerLeftIso $f_e $η_e) f g h η
+  whiskerRightM η h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .whiskerRight q(whiskerRightIso $η_e $h_e) f g η h
+  horizontalCompM η θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f₁ ← η.srcM
+    let g₁ ← η.tgtM
+    let f₂ ← θ.srcM
+    let g₂ ← θ.tgtM
+    have f₁_e : Q($ctx.C) := f₁.e
+    have g₁_e : Q($ctx.C) := g₁.e
+    have f₂_e : Q($ctx.C) := f₂.e
+    have g₂_e : Q($ctx.C) := g₂.e
+    have η_e : Q($f₁_e ≅ $g₁_e) := η.e
+    have θ_e : Q($f₂_e ≅ $g₂_e) := θ.e
+    return .horizontalComp q(tensorIso $η_e $θ_e) f₁ g₁ f₂ g₂ η θ
+  symmM η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .inv q(Iso.symm $η_e) f g η
+  coherenceCompM α η θ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have i_e : Q($ctx.C) := i.e
+    have _inst : Q(MonoidalCoherence $g_e $h_e) := α.inst
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    have θ_e : Q($h_e ≅ $i_e) := θ.e
+    return .coherenceComp q($η_e ≪⊗≫ $θ_e) f g h i α η θ
+
+open MonadMor₂Iso
+
+instance : MonadMor₂ MonoidalM where
+  homM η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    let e : Q($f_e ⟶ $g_e) := q(Iso.hom $η_e)
+    have eq : Q(Iso.hom $η_e = $e) := q(rfl)
+    return .isoHom e ⟨η, eq⟩ η
+  atomHomM η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f := η.src
+    let g := η.tgt
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .mk q(Iso.hom $η_e) f g
+  invM η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    let e : Q($g_e ⟶ $f_e) := q(Iso.inv $η_e)
+    let η_inv ← symmM η
+    let eq : Q(Iso.inv $η_e = $e) := q(Iso.symm_hom $η_e)
+    return .isoInv e ⟨η_inv, eq⟩ η
+  atomInvM η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f := η.src
+    let g := η.tgt
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have η_e : Q($f_e ≅ $g_e) := η.e
+    return .mk q(Iso.inv $η_e) g f
+  id₂M f := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    have f_e : Q($ctx.C) := f.e
+    let e : Q($f_e ⟶ $f_e) := q(𝟙 $f_e)
+    let eq : Q(𝟙 $f_e = $e) := q(Iso.refl_hom $f_e)
+    return .id e ⟨.structuralAtom <| ← id₂M f, eq⟩ f
+  comp₂M η θ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    have θ_e : Q($g_e ⟶ $h_e) := θ.e
+    let iso_lift? ← (match (η.isoLift?, θ.isoLift?) with
+      | (some η_iso, some θ_iso) =>
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have θ_iso_e : Q($g_e ≅ $h_e) := θ_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        have θ_iso_eq : Q(Iso.hom $θ_iso_e = $θ_e) := θ_iso.eq
+        let eq := q(structuralIsoOfExpr_comp _ _ $η_iso_eq _ _ $θ_iso_eq)
+        return .some ⟨← comp₂M η_iso.e θ_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⟶ $h_e) := q($η_e ≫ $θ_e)
+    return .comp e iso_lift? f g h η θ
+  whiskerLeftM f η := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η.srcM
+    let h ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($g_e ⟶ $h_e) := η.e
+    let iso_lift? ← (match η.isoLift? with
+      | some η_iso => do
+        have η_iso_e : Q($g_e ≅ $h_e) := η_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        let eq := q(structuralIsoOfExpr_whiskerLeft $f_e _ _ $η_iso_eq)
+        return .some ⟨← whiskerLeftM f η_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⊗ $g_e ⟶ $f_e ⊗ $h_e) := q($f_e ◁ $η_e)
+    return .whiskerLeft e iso_lift? f g h η
+  whiskerRightM η h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    let iso_lift? ← (match η.isoLift? with
+      | some η_iso => do
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        let eq := q(structuralIsoOfExpr_whiskerRight $h_e _ _ $η_iso_eq)
+        return .some ⟨← whiskerRightM η_iso.e h, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⊗ $h_e ⟶ $g_e ⊗ $h_e) := q($η_e ▷ $h_e)
+    return .whiskerRight e iso_lift? f g η h
+  horizontalCompM η θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f₁ ← η.srcM
+    let g₁ ← η.tgtM
+    let f₂ ← θ.srcM
+    let g₂ ← θ.tgtM
+    have f₁_e : Q($ctx.C) := f₁.e
+    have g₁_e : Q($ctx.C) := g₁.e
+    have f₂_e : Q($ctx.C) := f₂.e
+    have g₂_e : Q($ctx.C) := g₂.e
+    have η_e : Q($f₁_e ⟶ $g₁_e) := η.e
+    have θ_e : Q($f₂_e ⟶ $g₂_e) := θ.e
+    let iso_lift? ← (match (η.isoLift?, θ.isoLift?) with
+      | (some η_iso, some θ_iso) => do
+        have η_iso_e : Q($f₁_e ≅ $g₁_e) := η_iso.e.e
+        have θ_iso_e : Q($f₂_e ≅ $g₂_e) := θ_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        have θ_iso_eq : Q(Iso.hom $θ_iso_e = $θ_e) := θ_iso.eq
+        let eq := q(structuralIsoOfExpr_horizontalComp _ _ $η_iso_eq _ _ $θ_iso_eq)
+        return .some ⟨← horizontalCompM η_iso.e θ_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f₁_e ⊗ $f₂_e ⟶ $g₁_e ⊗ $g₂_e) := q($η_e ⊗ $θ_e)
+    return .horizontalComp e iso_lift? f₁ g₁ f₂ g₂ η θ
+  coherenceCompM α η θ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have i_e : Q($ctx.C) := i.e
+    have _inst : Q(MonoidalCoherence $g_e $h_e) := α.inst
+    have η_e : Q($f_e ⟶ $g_e) := η.e
+    have θ_e : Q($h_e ⟶ $i_e) := θ.e
+    let iso_lift? ← (match (η.isoLift?, θ.isoLift?) with
+      | (some η_iso, some θ_iso) => do
+        have η_iso_e : Q($f_e ≅ $g_e) := η_iso.e.e
+        have θ_iso_e : Q($h_e ≅ $i_e) := θ_iso.e.e
+        have η_iso_eq : Q(Iso.hom $η_iso_e = $η_e) := η_iso.eq
+        have θ_iso_eq : Q(Iso.hom $θ_iso_e = $θ_e) := θ_iso.eq
+        let eq := q(StructuralOfExpr_monoidalComp _ _ $η_iso_eq _ _ $θ_iso_eq)
+        return .some ⟨← coherenceCompM α η_iso.e θ_iso.e, eq⟩
+      | _ => return none)
+    let e : Q($f_e ⟶ $i_e) := q($η_e ⊗≫ $θ_e)
+    return .coherenceComp e iso_lift? f g h i α η θ
+
+/-- Check that `e` is definitionally equal to `𝟙_ C`. -/
+def id₁? (e : Expr) : MonoidalM (Option Obj) := do
+  let ctx ← read
+  match ctx.instMonoidal? with
+  | .some _monoidal => do
+    if ← withDefault <| isDefEq e (q(MonoidalCategory.tensorUnit) : Q($ctx.C)) then
+      return some ⟨none⟩
+    else
+      return none
+  | _ => return none
+
+/-- Return `(f, g)` if `e` is definitionally equal to `f ⊗ g`. -/
+def comp? (e : Expr) : MonoidalM (Option (Mor₁ × Mor₁)) := do
+  let ctx ← read
+  let f ← mkFreshExprMVarQ ctx.C
+  let g ← mkFreshExprMVarQ ctx.C
+  match ctx.instMonoidal? with
+    | .some _monoidal => do
+      if ← withDefault <| isDefEq e q($f ⊗ $g) then
+        let f ← instantiateMVars f
+        let g ← instantiateMVars g
+        return some ((.of ⟨f, ⟨none⟩, ⟨none⟩⟩ : Mor₁), (.of ⟨g, ⟨none⟩, ⟨none⟩⟩ : Mor₁))
+      else
+        return none
+    | _ => return none
+
+/-- Construct a `Mor₁` expression from a Lean expression. -/
+partial def mor₁OfExpr (e : Expr) : MonoidalM Mor₁ := do
+  if let some f := (← get).cache.find? e then
+    return f
+  let f ←
+    if let some a ← id₁? e then
+      MonadMor₁.id₁M a
+    else if let some (f, g) ← comp? e then
+      MonadMor₁.comp₁M (← mor₁OfExpr f.e) (← mor₁OfExpr g.e)
+    else
+      return Mor₁.of ⟨e, ⟨none⟩, ⟨none⟩⟩
+  modify fun s => { s with cache := s.cache.insert e f }
+  return f
+
+instance : MkMor₁ MonoidalM where
+  ofExpr := mor₁OfExpr
+
+/-- Construct a `Mor₂Iso` term from a Lean expression. -/
+partial def Mor₂IsoOfExpr (e : Expr) : MonoidalM Mor₂Iso := do
+  match (← whnfR e).getAppFnArgs with
+  | (``MonoidalCategoryStruct.associator, #[_, _, _, f, g, h]) =>
+    associatorM' (← MkMor₁.ofExpr f) (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h)
+  | (``MonoidalCategoryStruct.leftUnitor, #[_, _, _, f]) =>
+    leftUnitorM' (← MkMor₁.ofExpr f)
+  | (``MonoidalCategoryStruct.rightUnitor, #[_, _, _, f]) =>
+    rightUnitorM' (← MkMor₁.ofExpr f)
+  | (``Iso.refl, #[_, _, f]) =>
+    id₂M' (← MkMor₁.ofExpr f)
+  | (``Iso.symm, #[_, _, _, _, η]) =>
+    symmM (← Mor₂IsoOfExpr η)
+  | (``Iso.trans, #[_, _, _, _, _, η, θ]) =>
+    comp₂M (← Mor₂IsoOfExpr η) (← Mor₂IsoOfExpr θ)
+  | (``MonoidalCategory.whiskerLeftIso, #[_, _, _, f, _, _, η]) =>
+    whiskerLeftM (← MkMor₁.ofExpr f) (← Mor₂IsoOfExpr η)
+  | (``MonoidalCategory.whiskerRightIso, #[_, _, _, _, _, η, h]) =>
+    whiskerRightM (← Mor₂IsoOfExpr η) (← MkMor₁.ofExpr h)
+  | (``tensorIso, #[_, _, _, _, _, _, _, η, θ]) =>
+    horizontalCompM (← Mor₂IsoOfExpr η) (← Mor₂IsoOfExpr θ)
+  | (``monoidalIsoComp, #[_, _, _, g, h, _, inst, η, θ]) =>
+    let α ← coherenceHomM (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h) inst
+    coherenceCompM α (← Mor₂IsoOfExpr η) (← Mor₂IsoOfExpr θ)
+  | (``MonoidalCoherence.iso, #[_, _, f, g, inst]) =>
+    coherenceHomM' (← MkMor₁.ofExpr f) (← MkMor₁.ofExpr g) inst
+  | _ =>
+    return .of ⟨e, ← MkMor₁.ofExpr (← srcExprOfIso e), ← MkMor₁.ofExpr (← tgtExprOfIso e)⟩
+
+open MonadMor₂ in
+/-- Construct a `Mor₂` term from a Lean expression. -/
+partial def Mor₂OfExpr (e : Expr) : MonoidalM Mor₂ := do
+  match ← whnfR e with
+  -- whnfR version of `Iso.hom η`
+  | .proj ``Iso 0 η => homM (← Mor₂IsoOfExpr η)
+  -- whnfR version of `Iso.inv η`
+  | .proj ``Iso 1 η => invM (← Mor₂IsoOfExpr η)
+  | .app .. => match (← whnfR e).getAppFnArgs with
+    | (``CategoryStruct.id, #[_, _, f]) => id₂M (← MkMor₁.ofExpr f)
+    | (``CategoryStruct.comp, #[_, _, _, _, _, η, θ]) =>
+      comp₂M (← Mor₂OfExpr η) (← Mor₂OfExpr θ)
+    | (``MonoidalCategoryStruct.whiskerLeft, #[_, _, _, f, _, _, η]) =>
+      whiskerLeftM (← MkMor₁.ofExpr f) (← Mor₂OfExpr η)
+    | (``MonoidalCategoryStruct.whiskerRight, #[_, _, _, _, _, η, h]) =>
+      whiskerRightM (← Mor₂OfExpr η) (← MkMor₁.ofExpr h)
+    | (``MonoidalCategoryStruct.tensorHom, #[_, _, _, _, _, _, _, η, θ]) =>
+      horizontalCompM (← Mor₂OfExpr η) (← Mor₂OfExpr θ)
+    | (``monoidalComp, #[_, _, _, g, h, _, inst, η, θ]) =>
+      let α ← coherenceHomM (← MkMor₁.ofExpr g) (← MkMor₁.ofExpr h) inst
+      coherenceCompM α (← Mor₂OfExpr η) (← Mor₂OfExpr θ)
+    | _ => return .of ⟨e, ← MkMor₁.ofExpr (← srcExpr e), ← MkMor₁.ofExpr (← tgtExpr e)⟩
+  | _ =>
+    return .of ⟨e, ← MkMor₁.ofExpr (← srcExpr e), ← MkMor₁.ofExpr (← tgtExpr e)⟩
+
+instance : BicategoryLike.MkMor₂ MonoidalM where
+  ofExpr := Mor₂OfExpr
+
+instance : MonadCoherehnceHom MonoidalM where
+  unfoldM α := Mor₂IsoOfExpr α.unfold
+
+end Mathlib.Tactic.Monoidal
diff --git a/Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.lean b/Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.lean
new file mode 100644
index 0000000000000..f55e98f5088b0
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Monoidal/Normalize.lean
@@ -0,0 +1,787 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.Normalize
+import Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes
+
+/-!
+# Normalization of morphisms in monoidal categories
+
+This file provides the implementation of the normalization given in
+`Mathlib.Tactic.CategoryTheory.Coherence.Normalize`. See this file for more details.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike MonoidalCategory
+
+namespace Mathlib.Tactic.Monoidal
+
+section
+
+universe v u
+
+variable {C : Type u} [Category.{v} C]
+
+variable {f f' g g' h h' i i' j : C}
+
+@[nolint synTaut]
+theorem evalComp_nil_nil {f g h : C} (α : f ≅ g) (β : g ≅ h) :
+    (α ≪≫ β).hom = (α ≪≫ β).hom := by
+  simp
+
+theorem evalComp_nil_cons {f g h i j : C} (α : f ≅ g) (β : g ≅ h) (η : h ⟶ i) (ηs : i ⟶ j) :
+    α.hom ≫ (β.hom ≫ η ≫ ηs) = (α ≪≫ β).hom ≫ η ≫ ηs := by
+  simp
+
+theorem evalComp_cons {f g h i j : C} (α : f ≅ g) (η : g ⟶ h) {ηs : h ⟶ i} {θ : i ⟶ j} {ι : h ⟶ j}
+    (e_ι : ηs ≫ θ = ι)  :
+    (α.hom ≫ η ≫ ηs) ≫ θ = α.hom ≫ η ≫ ι := by
+  simp [e_ι]
+
+theorem eval_comp
+    {η η' : f ⟶ g} {θ θ' : g ⟶ h} {ι : f ⟶ h}
+    (e_η : η = η') (e_θ : θ = θ') (e_ηθ : η' ≫ θ' = ι) :
+    η ≫ θ = ι := by
+  simp [e_η, e_θ, e_ηθ]
+
+theorem eval_of (η : f ⟶ g) :
+    η = (Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom := by
+  simp
+
+theorem eval_monoidalComp
+    {η η' : f ⟶ g} {α : g ≅ h} {θ θ' : h ⟶ i} {αθ : g ⟶ i} {ηαθ : f ⟶ i}
+    (e_η : η = η') (e_θ : θ = θ') (e_αθ : α.hom ≫ θ' = αθ) (e_ηαθ : η' ≫ αθ = ηαθ) :
+    η ≫ α.hom ≫ θ = ηαθ := by
+  simp [e_η, e_θ, e_αθ, e_ηαθ]
+
+variable [MonoidalCategory C]
+
+@[nolint synTaut]
+theorem evalWhiskerLeft_nil (f : C) {g h : C} (α : g ≅ h) :
+    (whiskerLeftIso f α).hom = (whiskerLeftIso f α).hom := by
+  simp
+
+theorem evalWhiskerLeft_of_cons {f g h i j : C}
+    (α : g ≅ h) (η : h ⟶ i) {ηs : i ⟶ j} {θ : f ⊗ i ⟶ f ⊗ j} (e_θ : f ◁ ηs = θ) :
+    f ◁ (α.hom ≫ η ≫ ηs) = (whiskerLeftIso f α).hom ≫ f ◁ η ≫ θ := by
+  simp [e_θ]
+
+theorem evalWhiskerLeft_comp {f g h i : C}
+    {η : h ⟶ i} {η₁ : g ⊗ h ⟶ g ⊗ i} {η₂ : f ⊗ g ⊗ h ⟶ f ⊗ g ⊗ i}
+    {η₃ : f ⊗ g ⊗ h ⟶ (f ⊗ g) ⊗ i} {η₄ : (f ⊗ g) ⊗ h ⟶ (f ⊗ g) ⊗ i}
+    (e_η₁ : g ◁ η = η₁) (e_η₂ : f ◁ η₁ = η₂)
+    (e_η₃ : η₂ ≫ (α_ _ _ _).inv = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄) :
+    (f ⊗ g) ◁ η = η₄ := by
+  simp [e_η₁, e_η₂, e_η₃, e_η₄]
+
+theorem evalWhiskerLeft_id {f g : C} {η : f ⟶ g}
+    {η₁ : f ⟶ 𝟙_ C ⊗ g} {η₂ : 𝟙_ C ⊗ f ⟶ 𝟙_ C ⊗ g}
+    (e_η₁ : η ≫ (λ_ _).inv = η₁) (e_η₂ : (λ_ _).hom ≫ η₁ = η₂) :
+    𝟙_ C ◁ η = η₂ := by
+  simp [e_η₁, e_η₂]
+
+theorem eval_whiskerLeft {f g h : C}
+    {η η' : g ⟶ h} {θ : f ⊗ g ⟶ f ⊗ h}
+    (e_η : η = η') (e_θ : f ◁ η' = θ) :
+    f ◁ η = θ := by
+  simp [e_η, e_θ]
+
+theorem eval_whiskerRight {f g h : C}
+    {η η' : f ⟶ g} {θ : f ⊗ h ⟶ g ⊗ h}
+    (e_η : η = η') (e_θ : η' ▷ h = θ) :
+    η ▷ h = θ := by
+  simp [e_η, e_θ]
+
+theorem eval_tensorHom {f g h i : C}
+    {η η' : f ⟶ g} {θ θ' : h ⟶ i} {ι : f ⊗ h ⟶ g ⊗ i}
+    (e_η : η = η') (e_θ : θ = θ') (e_ι : η' ⊗ θ' = ι) :
+    η ⊗ θ = ι := by
+  simp [e_η, e_θ, e_ι]
+
+@[nolint synTaut]
+theorem evalWhiskerRight_nil {f g : C} (α : f ≅ g) (h : C) :
+    (whiskerRightIso α h).hom = (whiskerRightIso α h).hom := by
+  simp
+
+theorem evalWhiskerRight_cons_of_of {f g h i j : C}
+    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {ηs₁ : h ⊗ j ⟶ i ⊗ j}
+    {η₁ : g ⊗ j ⟶ h ⊗ j} {η₂ : g ⊗ j ⟶ i ⊗ j} {η₃ : f ⊗ j ⟶ i ⊗ j}
+    (e_ηs₁ : ηs ▷ j = ηs₁) (e_η₁ : η ▷ j = η₁)
+    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α j).hom ≫ η₂ = η₃) :
+    (α.hom ≫ η ≫ ηs) ▷ j = η₃ := by
+  simp_all
+
+theorem evalWhiskerRight_cons_whisker {f g h i j k : C}
+    {α : g ≅ f ⊗ h} {η : h ⟶ i} {ηs : f ⊗ i ⟶ j}
+    {η₁ : h ⊗ k ⟶ i ⊗ k} {η₂ : f ⊗ (h ⊗ k) ⟶ f ⊗ (i ⊗ k)} {ηs₁ : (f ⊗ i) ⊗ k ⟶ j ⊗ k}
+    {ηs₂ : f ⊗ (i ⊗ k) ⟶ j ⊗ k} {η₃ : f ⊗ (h ⊗ k) ⟶ j ⊗ k} {η₄ : (f ⊗ h) ⊗ k ⟶ j ⊗ k}
+    {η₅ : g ⊗ k ⟶ j ⊗ k}
+    (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ k = η₁) (e_η₂ : f ◁ η₁ = η₂)
+    (e_ηs₁ : ηs ▷ k = ηs₁) (e_ηs₂ : (α_ _ _ _).inv ≫ ηs₁ = ηs₂)
+    (e_η₃ : η₂ ≫ ηs₂ = η₃) (e_η₄ : (α_ _ _ _).hom ≫ η₃ = η₄)
+    (e_η₅ : (whiskerRightIso α k).hom ≫ η₄ = η₅) :
+    (α.hom ≫ (f ◁ η) ≫ ηs) ▷ k = η₅ := by
+  simp at e_η₁ e_η₅
+  simp [e_η₁, e_η₂, e_ηs₁, e_ηs₂, e_η₃, e_η₄, e_η₅]
+
+theorem evalWhiskerRight_comp {f f' g h : C}
+    {η : f ⟶ f'} {η₁ : f ⊗ g ⟶ f' ⊗ g} {η₂ : (f ⊗ g) ⊗ h ⟶ (f' ⊗ g) ⊗ h}
+    {η₃ : (f ⊗ g) ⊗ h ⟶ f' ⊗ (g ⊗ h)} {η₄ : f ⊗ (g ⊗ h) ⟶ f' ⊗ (g ⊗ h)}
+    (e_η₁ : η ▷ g = η₁) (e_η₂ : η₁ ▷ h = η₂)
+    (e_η₃ : η₂ ≫ (α_ _ _ _).hom = η₃) (e_η₄ : (α_ _ _ _).inv ≫ η₃ = η₄) :
+    η ▷ (g ⊗ h) = η₄ := by
+  simp [e_η₁, e_η₂, e_η₃, e_η₄]
+
+theorem evalWhiskerRight_id {f g : C}
+    {η : f ⟶ g} {η₁ : f ⟶ g ⊗ 𝟙_ C} {η₂ : f ⊗ 𝟙_ C ⟶ g ⊗ 𝟙_ C}
+    (e_η₁ : η ≫ (ρ_ _).inv = η₁) (e_η₂ : (ρ_ _).hom ≫ η₁ = η₂) :
+    η ▷ 𝟙_ C = η₂ := by
+  simp [e_η₁, e_η₂]
+
+theorem evalWhiskerRightAux_of {f g : C} (η : f ⟶ g) (h : C) :
+    η ▷ h = (Iso.refl _).hom ≫ η ▷ h ≫ (Iso.refl _).hom := by
+  simp
+
+theorem evalWhiskerRightAux_cons {f g h i j : C} {η : g ⟶ h} {ηs : i ⟶ j}
+    {ηs' : i ⊗ f ⟶ j ⊗ f} {η₁ : g ⊗ (i ⊗ f) ⟶ h ⊗ (j ⊗ f)}
+    {η₂ : g ⊗ (i ⊗ f) ⟶ (h ⊗ j) ⊗ f} {η₃ : (g ⊗ i) ⊗ f ⟶ (h ⊗ j) ⊗ f}
+    (e_ηs' : ηs ▷ f = ηs') (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ⊗ ηs' = η₁)
+    (e_η₂ : η₁ ≫ (α_ _ _ _).inv = η₂) (e_η₃ : (α_ _ _ _).hom ≫ η₂ = η₃) :
+    (η ⊗ ηs) ▷ f = η₃ := by
+  simp [← e_ηs', ← e_η₁, ← e_η₂, ← e_η₃, MonoidalCategory.tensorHom_def]
+
+theorem evalWhiskerRight_cons_of {f f' g h i : C} {α : f' ≅ g} {η : g ⟶ h} {ηs : h ⟶ i}
+    {ηs₁ : h ⊗ f ⟶ i ⊗ f} {η₁ : g ⊗ f ⟶ h ⊗ f} {η₂ : g ⊗ f ⟶ i ⊗ f}
+    {η₃ : f' ⊗ f ⟶ i ⊗ f}
+    (e_ηs₁ : ηs ▷ f = ηs₁) (e_η₁ : η ▷ f = η₁)
+    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (whiskerRightIso α f).hom ≫ η₂ = η₃) :
+    (α.hom ≫ η ≫ ηs) ▷ f = η₃ := by
+  simp_all
+
+theorem evalHorizontalCompAux_of {f g h i : C} (η : f ⟶ g) (θ : h ⟶ i) :
+    η ⊗ θ = (Iso.refl _).hom ≫ (η ⊗ θ) ≫ (Iso.refl _).hom := by
+  simp
+
+theorem evalHorizontalCompAux_cons {f f' g g' h i : C} {η : f ⟶ g} {ηs : f' ⟶ g'} {θ : h ⟶ i}
+    {ηθ : f' ⊗ h ⟶ g' ⊗ i} {η₁ : f ⊗ (f' ⊗ h) ⟶ g ⊗ (g' ⊗ i)}
+    {ηθ₁ : f ⊗ (f' ⊗ h) ⟶ (g ⊗ g') ⊗ i} {ηθ₂ : (f ⊗ f') ⊗ h ⟶ (g ⊗ g') ⊗ i}
+    (e_ηθ : ηs ⊗ θ = ηθ) (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ⊗ ηθ = η₁)
+    (e_ηθ₁ : η₁ ≫ (α_ _ _ _).inv = ηθ₁) (e_ηθ₂ : (α_ _ _ _).hom ≫ ηθ₁ = ηθ₂) :
+    (η ⊗ ηs) ⊗ θ = ηθ₂ := by
+  simp_all
+
+theorem evalHorizontalCompAux'_whisker {f f' g g' h : C} {η : g ⟶ h} {θ : f' ⟶ g'}
+    {ηθ : g ⊗ f' ⟶ h ⊗ g'} {η₁ : f ⊗ (g ⊗ f') ⟶ f ⊗ (h ⊗ g')}
+    {η₂ :  f ⊗ (g ⊗ f') ⟶ (f ⊗ h) ⊗ g'} {η₃ : (f ⊗ g) ⊗ f' ⟶ (f ⊗ h) ⊗ g'}
+    (e_ηθ : η ⊗ θ = ηθ) (e_η₁ : f ◁ ηθ = η₁)
+    (e_η₂ : η₁ ≫ (α_ _ _ _).inv = η₂) (e_η₃ : (α_ _ _ _).hom ≫ η₂ = η₃) :
+    (f ◁ η) ⊗ θ = η₃ := by
+  simp only [← e_ηθ, ← e_η₁, ← e_η₂, ← e_η₃]
+  simp [MonoidalCategory.tensorHom_def]
+
+theorem evalHorizontalCompAux'_of_whisker {f f' g g' h : C} {η : g ⟶ h} {θ : f' ⟶ g'}
+    {η₁ : g ⊗ f ⟶ h ⊗ f} {ηθ : (g ⊗ f) ⊗ f' ⟶ (h ⊗ f) ⊗ g'}
+    {ηθ₁ : (g ⊗ f) ⊗ f' ⟶ h ⊗ (f ⊗ g')}
+    {ηθ₂ : g ⊗ (f ⊗ f') ⟶ h ⊗ (f ⊗ g')}
+    (e_η₁ : η ▷ f = η₁) (e_ηθ : η₁ ⊗ ((Iso.refl _).hom ≫ θ ≫ (Iso.refl _).hom) = ηθ)
+    (e_ηθ₁ : ηθ ≫ (α_ _ _ _).hom = ηθ₁) (e_ηθ₂ : (α_ _ _ _).inv ≫ ηθ₁ = ηθ₂) :
+    η ⊗ (f ◁ θ) = ηθ₂ := by
+  simp only [← e_η₁, ← e_ηθ, ← e_ηθ₁, ← e_ηθ₂]
+  simp [MonoidalCategory.tensorHom_def]
+
+@[nolint synTaut]
+theorem evalHorizontalComp_nil_nil {f g h i : C} (α : f ≅ g) (β : h ≅ i) :
+    (α ⊗ β).hom = (α ⊗ β).hom := by
+  simp
+
+theorem evalHorizontalComp_nil_cons {f f' g g' h i : C}
+    {α : f ≅ g} {β : f' ≅ g'} {η : g' ⟶ h} {ηs : h ⟶ i}
+    {η₁ : g ⊗ g' ⟶ g ⊗ h} {ηs₁ : g ⊗ h ⟶ g ⊗ i}
+    {η₂ : g ⊗ g' ⟶ g ⊗ i} {η₃ : f ⊗ f' ⟶ g ⊗ i}
+    (e_η₁ : g ◁ ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) = η₁)
+    (e_ηs₁ : g ◁ ηs = ηs₁) (e_η₂ : η₁ ≫ ηs₁ = η₂)
+    (e_η₃ : (α ⊗ β).hom ≫ η₂ = η₃) :
+    α.hom ⊗ (β.hom ≫ η ≫ ηs) = η₃ := by
+  simp_all [MonoidalCategory.tensorHom_def]
+
+theorem evalHorizontalComp_cons_nil {f f' g g' h i : C}
+    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i} {β : f' ≅ g'}
+    {η₁ : g ⊗ g' ⟶ h ⊗ g'} {ηs₁ : h ⊗ g' ⟶ i ⊗ g'} {η₂ : g ⊗ g' ⟶ i ⊗ g'} {η₃ : f ⊗ f' ⟶ i ⊗ g'}
+    (e_η₁ : ((Iso.refl _).hom ≫ η ≫ (Iso.refl _).hom) ▷ g' = η₁) (e_ηs₁ : ηs ▷ g' = ηs₁)
+    (e_η₂ : η₁ ≫ ηs₁ = η₂) (e_η₃ : (α ⊗ β).hom ≫ η₂ = η₃) :
+    (α.hom ≫ η ≫ ηs) ⊗ β.hom = η₃ := by
+  simp_all [MonoidalCategory.tensorHom_def']
+
+theorem evalHorizontalComp_cons_cons {f f' g g' h h' i i' : C}
+    {α : f ≅ g} {η : g ⟶ h} {ηs : h ⟶ i}
+    {β : f' ≅ g'} {θ : g' ⟶ h'} {θs : h' ⟶ i'}
+    {ηθ : g ⊗ g' ⟶ h ⊗ h'} {ηθs : h ⊗ h' ⟶ i ⊗ i'}
+    {ηθ₁ : g ⊗ g' ⟶ i ⊗ i'} {ηθ₂ : f ⊗ f' ⟶ i ⊗ i'}
+    (e_ηθ : η ⊗ θ = ηθ) (e_ηθs : ηs ⊗ θs = ηθs)
+    (e_ηθ₁ : ηθ ≫ ηθs = ηθ₁) (e_ηθ₂ : (α ⊗ β).hom ≫ ηθ₁ = ηθ₂) :
+    (α.hom ≫ η ≫ ηs) ⊗ (β.hom ≫ θ ≫ θs) = ηθ₂ := by
+  simp [← e_ηθ , ← e_ηθs , ← e_ηθ₁, ← e_ηθ₂]
+
+end
+
+open Mor₂Iso
+
+instance : MkEvalComp MonoidalM where
+  mkEvalCompNilNil α β := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← β.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($g ≅ $h) := β.e
+    return q(evalComp_nil_nil $α $β)
+  mkEvalCompNilCons α β η ηs := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← β.tgtM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($g ≅ $h) := β.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($i ⟶ $j) := ηs.e.e
+    return q(evalComp_nil_cons $α $β $η $ηs)
+  mkEvalCompCons α η ηs θ ι e_ι := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    let j ← θ.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have θ : Q($i ⟶ $j) := θ.e.e
+    have ι : Q($h ⟶ $j) := ι.e.e
+    have e_ι : Q($ηs ≫ $θ = $ι) := e_ι
+    return q(evalComp_cons $α $η $e_ι)
+
+instance : MkEvalWhiskerLeft MonoidalM where
+  mkEvalWhiskerLeftNil f α := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← α.srcM
+    let h ← α.tgtM
+    have f_e : Q($ctx.C) := f.e
+    have g_e : Q($ctx.C) := g.e
+    have h_e : Q($ctx.C) := h.e
+    have α_e : Q($g_e ≅ $h_e) := α.e
+    return q(evalWhiskerLeft_nil $f_e $α_e)
+  mkEvalWhiskerLeftOfCons f α η ηs θ e_θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← α.srcM
+    let h ← α.tgtM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have α : Q($g ≅ $h) := α.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($i ⟶ $j) := ηs.e.e
+    have θ : Q($f ⊗ $i ⟶ $f ⊗ $j) := θ.e.e
+    have e_θ : Q($f ◁ $ηs = $θ) := e_θ
+    return q(evalWhiskerLeft_of_cons $α $η $e_θ)
+  mkEvalWhiskerLeftComp f g η η₁ η₂ η₃ η₄ e_η₁ e_η₂ e_η₃ e_η₄ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let h ← η.srcM
+    let i ← η.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have η₁ : Q($g ⊗ $h ⟶ $g ⊗ $i) := η₁.e.e
+    have η₂ : Q($f ⊗ $g ⊗ $h ⟶ $f ⊗ $g ⊗ $i) := η₂.e.e
+    have η₃ : Q($f ⊗ $g ⊗ $h ⟶ ($f ⊗ $g) ⊗ $i) := η₃.e.e
+    have η₄ : Q(($f ⊗ $g) ⊗ $h ⟶ ($f ⊗ $g) ⊗ $i) := η₄.e.e
+    have e_η₁ : Q($g ◁ $η = $η₁) := e_η₁
+    have e_η₂ : Q($f ◁ $η₁ = $η₂) := e_η₂
+    have e_η₃ : Q($η₂ ≫ (α_ _ _ _).inv = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).hom ≫ $η₃ = $η₄) := e_η₄
+    return q(evalWhiskerLeft_comp $e_η₁ $e_η₂ $e_η₃ $e_η₄)
+  mkEvalWhiskerLeftId η η₁ η₂ e_η₁ e_η₂ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have η₁ : Q($f ⟶ 𝟙_ _ ⊗ $g) := η₁.e.e
+    have η₂ : Q(𝟙_ _ ⊗ $f ⟶ 𝟙_ _ ⊗ $g) := η₂.e.e
+    have e_η₁ : Q($η ≫ (λ_ _).inv = $η₁) := e_η₁
+    have e_η₂ : Q((λ_ _).hom ≫ $η₁ = $η₂) := e_η₂
+    return q(evalWhiskerLeft_id $e_η₁ $e_η₂)
+
+instance : MkEvalWhiskerRight MonoidalM where
+  mkEvalWhiskerRightAuxOf η h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have h : Q($ctx.C) := h.e
+    return q(evalWhiskerRightAux_of $η $h)
+  mkEvalWhiskerRightAuxCons f η ηs ηs' η₁ η₂ η₃ e_ηs' e_η₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η.srcM
+    let h ← η.tgtM
+    let i ← ηs.srcM
+    let j ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($i ⟶ $j) := ηs.e.e
+    have ηs' : Q($i ⊗ $f ⟶ $j ⊗ $f) := ηs'.e.e
+    have η₁ : Q($g ⊗ ($i ⊗ $f) ⟶ $h ⊗ ($j ⊗ $f)) := η₁.e.e
+    have η₂ : Q($g ⊗ ($i ⊗ $f) ⟶ ($h ⊗ $j) ⊗ $f) := η₂.e.e
+    have η₃ : Q(($g ⊗ $i) ⊗ $f ⟶ ($h ⊗ $j) ⊗ $f) := η₃.e.e
+    have e_ηs' : Q($ηs ▷ $f = $ηs') := e_ηs'
+    have e_η₁ : Q(((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) ⊗ $ηs' = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ≫ (α_ _ _ _).inv = $η₂) := e_η₂
+    have e_η₃ : Q((α_ _ _ _).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalWhiskerRightAux_cons $e_ηs' $e_η₁ $e_η₂ $e_η₃)
+  mkEvalWhiskerRightNil α h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have α : Q($f ≅ $g) := α.e
+    return q(evalWhiskerRight_nil $α $h)
+  mkEvalWhiskerRightConsOfOf j α η ηs ηs₁ η₁ η₂ η₃ e_ηs₁ e_η₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have ηs₁ : Q($h ⊗ $j ⟶ $i ⊗ $j) := ηs₁.e.e
+    have η₁ : Q($g ⊗ $j ⟶ $h ⊗ $j) := η₁.e.e
+    have η₂ : Q($g ⊗ $j ⟶ $i ⊗ $j) := η₂.e.e
+    have η₃ : Q($f ⊗ $j ⟶ $i ⊗ $j) := η₃.e.e
+    have e_ηs₁ : Q($ηs ▷ $j = $ηs₁) := e_ηs₁
+    have e_η₁ : Q($η ▷ $j = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ≫ $ηs₁ = $η₂) := e_η₂
+    have e_η₃ : Q((whiskerRightIso $α $j).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalWhiskerRight_cons_of_of $e_ηs₁ $e_η₁ $e_η₂ $e_η₃)
+  mkEvalWhiskerRightConsWhisker f k α η ηs η₁ η₂ ηs₁ ηs₂ η₃ η₄ η₅
+      e_η₁ e_η₂ e_ηs₁ e_ηs₂ e_η₃ e_η₄ e_η₅ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← α.srcM
+    let h ← η.srcM
+    let i ← η.tgtM
+    let j ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have j : Q($ctx.C) := j.e
+    have k : Q($ctx.C) := k.e
+    have α : Q($g ≅ $f ⊗ $h) := α.e
+    have η : Q($h ⟶ $i) := η.e.e
+    have ηs : Q($f ⊗ $i ⟶ $j) := ηs.e.e
+    have η₁ : Q($h ⊗ $k ⟶ $i ⊗ $k) := η₁.e.e
+    have η₂ : Q($f ⊗ ($h ⊗ $k) ⟶ $f ⊗ ($i ⊗ $k)) := η₂.e.e
+    have ηs₁ : Q(($f ⊗ $i) ⊗ $k ⟶ $j ⊗ $k) := ηs₁.e.e
+    have ηs₂ : Q($f ⊗ ($i ⊗ $k) ⟶ $j ⊗ $k) := ηs₂.e.e
+    have η₃ : Q($f ⊗ ($h ⊗ $k) ⟶ $j ⊗ $k) := η₃.e.e
+    have η₄ : Q(($f ⊗ $h) ⊗ $k ⟶ $j ⊗ $k) := η₄.e.e
+    have η₅ : Q($g ⊗ $k ⟶ $j ⊗ $k) := η₅.e.e
+    have e_η₁ : Q(((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) ▷ $k = $η₁) := e_η₁
+    have e_η₂ : Q($f ◁ $η₁ = $η₂) := e_η₂
+    have e_ηs₁ : Q($ηs ▷ $k = $ηs₁) := e_ηs₁
+    have e_ηs₂ : Q((α_ _ _ _).inv ≫ $ηs₁ = $ηs₂) := e_ηs₂
+    have e_η₃ : Q($η₂ ≫ $ηs₂ = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).hom ≫ $η₃ = $η₄) := e_η₄
+    have e_η₅ : Q((whiskerRightIso $α $k).hom ≫ $η₄ = $η₅) := e_η₅
+    return q(evalWhiskerRight_cons_whisker $e_η₁ $e_η₂ $e_ηs₁ $e_ηs₂ $e_η₃ $e_η₄ $e_η₅)
+  mkEvalWhiskerRightComp g h η η₁ η₂ η₃ η₄ e_η₁ e_η₂ e_η₃ e_η₄ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let f' ← η.tgtM
+    have f : Q($ctx.C) := f.e
+    have f' : Q($ctx.C) := f'.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have η : Q($f ⟶ $f') := η.e.e
+    have η₁ : Q($f ⊗ $g ⟶ $f' ⊗ $g) := η₁.e.e
+    have η₂ : Q(($f ⊗ $g) ⊗ $h ⟶ ($f' ⊗ $g) ⊗ $h) := η₂.e.e
+    have η₃ : Q(($f ⊗ $g) ⊗ $h ⟶ $f' ⊗ ($g ⊗ $h)) := η₃.e.e
+    have η₄ : Q($f ⊗ ($g ⊗ $h) ⟶ $f' ⊗ ($g ⊗ $h)) := η₄.e.e
+    have e_η₁ : Q($η ▷ $g = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ▷ $h = $η₂) := e_η₂
+    have e_η₃ : Q($η₂ ≫ (α_ _ _ _).hom = $η₃) := e_η₃
+    have e_η₄ : Q((α_ _ _ _).inv ≫ $η₃ = $η₄) := e_η₄
+    return q(evalWhiskerRight_comp $e_η₁ $e_η₂ $e_η₃ $e_η₄)
+  mkEvalWhiskerRightId η η₁ η₂ e_η₁ e_η₂ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have η₁ : Q($f ⟶ $g ⊗ 𝟙_ _) := η₁.e.e
+    have η₂ : Q($f ⊗ 𝟙_ _ ⟶ $g ⊗ 𝟙_ _) := η₂.e.e
+    have e_η₁ : Q($η ≫ (ρ_ _).inv = $η₁) := e_η₁
+    have e_η₂ : Q((ρ_ _).hom ≫ $η₁ = $η₂) := e_η₂
+    return q(evalWhiskerRight_id $e_η₁ $e_η₂)
+
+instance : MkEvalHorizontalComp MonoidalM where
+  mkEvalHorizontalCompAuxOf η θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have θ : Q($h ⟶ $i) := θ.e.e
+    return q(evalHorizontalCompAux_of $η $θ)
+  mkEvalHorizontalCompAuxCons η ηs θ ηθ η₁ ηθ₁ ηθ₂ e_ηθ e_η₁ e_ηθ₁ e_ηθ₂ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η.srcM
+    let g ← η.tgtM
+    let f' ← ηs.srcM
+    let g' ← ηs.tgtM
+    let h ← θ.srcM
+    let i ← θ.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have η : Q($f ⟶ $g) := η.e.e
+    have ηs : Q($f' ⟶ $g') := ηs.e.e
+    have θ : Q($h ⟶ $i) := θ.e.e
+    have ηθ : Q($f' ⊗ $h ⟶ $g' ⊗ $i) := ηθ.e.e
+    have η₁ : Q($f ⊗ ($f' ⊗ $h) ⟶ $g ⊗ ($g' ⊗ $i)) := η₁.e.e
+    have ηθ₁ : Q($f ⊗ ($f' ⊗ $h) ⟶ ($g ⊗ $g') ⊗ $i) := ηθ₁.e.e
+    have ηθ₂ : Q(($f ⊗ $f') ⊗ $h ⟶ ($g ⊗ $g') ⊗ $i) := ηθ₂.e.e
+    have e_ηθ : Q($ηs ⊗ $θ = $ηθ) := e_ηθ
+    have e_η₁ : Q(((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) ⊗ $ηθ = $η₁) := e_η₁
+    have e_ηθ₁ : Q($η₁ ≫ (α_ _ _ _).inv = $ηθ₁) := e_ηθ₁
+    have e_ηθ₂ : Q((α_ _ _ _).hom ≫ $ηθ₁ = $ηθ₂) := e_ηθ₂
+    return q(evalHorizontalCompAux_cons $e_ηθ $e_η₁ $e_ηθ₁ $e_ηθ₂)
+  mkEvalHorizontalCompAux'Whisker f η θ ηθ η₁ η₂ η₃ e_ηθ e_η₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η.srcM
+    let h ← η.tgtM
+    let f' ← θ.srcM
+    let g' ← θ.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have θ : Q($f' ⟶ $g') := θ.e.e
+    have ηθ : Q($g ⊗ $f' ⟶ $h ⊗ $g') := ηθ.e.e
+    have η₁ : Q($f ⊗ ($g ⊗ $f') ⟶ $f ⊗ ($h ⊗ $g')) := η₁.e.e
+    have η₂ : Q($f ⊗ ($g ⊗ $f') ⟶ ($f ⊗ $h) ⊗ $g') := η₂.e.e
+    have η₃ : Q(($f ⊗ $g) ⊗ $f' ⟶ ($f ⊗ $h) ⊗ $g') := η₃.e.e
+    have e_ηθ : Q($η ⊗ $θ = $ηθ) := e_ηθ
+    have e_η₁ : Q($f ◁ $ηθ = $η₁) := e_η₁
+    have e_η₂ : Q($η₁ ≫ (α_ _ _ _).inv = $η₂) := e_η₂
+    have e_η₃ : Q((α_ _ _ _).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalHorizontalCompAux'_whisker $e_ηθ $e_η₁ $e_η₂ $e_η₃)
+  mkEvalHorizontalCompAux'OfWhisker f η θ η₁ ηθ ηθ₁ ηθ₂ e_η₁ e_ηθ e_ηθ₁ e_ηθ₂ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η.srcM
+    let h ← η.tgtM
+    let f' ← θ.srcM
+    let g' ← θ.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have θ : Q($f' ⟶ $g') := θ.e.e
+    have η₁ : Q($g ⊗ $f ⟶ $h ⊗ $f) := η₁.e.e
+    have ηθ : Q(($g ⊗ $f) ⊗ $f' ⟶ ($h ⊗ $f) ⊗ $g') := ηθ.e.e
+    have ηθ₁ : Q(($g ⊗ $f) ⊗ $f' ⟶ $h ⊗ ($f ⊗ $g')) := ηθ₁.e.e
+    have ηθ₂ : Q($g ⊗ ($f ⊗ $f') ⟶ $h ⊗ ($f ⊗ $g')) := ηθ₂.e.e
+    have e_η₁ : Q($η ▷ $f = $η₁) := e_η₁
+    have e_ηθ : Q($η₁ ⊗ ((Iso.refl _).hom ≫ $θ ≫ (Iso.refl _).hom) = $ηθ) := e_ηθ
+    have e_ηθ₁ : Q($ηθ ≫ (α_ _ _ _).hom = $ηθ₁) := e_ηθ₁
+    have e_ηθ₂ : Q((α_ _ _ _).inv ≫ $ηθ₁ = $ηθ₂) := e_ηθ₂
+    return q(evalHorizontalCompAux'_of_whisker $e_η₁ $e_ηθ $e_ηθ₁ $e_ηθ₂)
+  mkEvalHorizontalCompNilNil α β := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← β.srcM
+    let i ← β.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($h ≅ $i) := β.e
+    return q(evalHorizontalComp_nil_nil $α $β)
+  mkEvalHorizontalCompNilCons α β η ηs η₁ ηs₁ η₂ η₃ e_η₁ e_ηs₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    let f' ← β.srcM
+    let g' ← β.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have α : Q($f ≅ $g) := α.e
+    have β : Q($f' ≅ $g') := β.e
+    have η : Q($g' ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have η₁ : Q($g ⊗ $g' ⟶ $g ⊗ $h) := η₁.e.e
+    have ηs₁ : Q($g ⊗ $h ⟶ $g ⊗ $i) := ηs₁.e.e
+    have η₂ : Q($g ⊗ $g' ⟶ $g ⊗ $i) := η₂.e.e
+    have η₃ : Q($f ⊗ $f' ⟶ $g ⊗ $i) := η₃.e.e
+    have e_η₁ : Q($g ◁ ((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) = $η₁) := e_η₁
+    have e_ηs₁ : Q($g ◁ $ηs = $ηs₁) := e_ηs₁
+    have e_η₂ : Q($η₁ ≫ $ηs₁ = $η₂) := e_η₂
+    have e_η₃ : Q(($α ⊗ $β).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalHorizontalComp_nil_cons $e_η₁ $e_ηs₁ $e_η₂ $e_η₃)
+  mkEvalHorizontalCompConsNil α β η ηs η₁ ηs₁ η₂ η₃ e_η₁ e_ηs₁ e_η₂ e_η₃ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    let f' ← β.srcM
+    let g' ← β.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have β : Q($f' ≅ $g') := β.e
+    have η₁ : Q($g ⊗ $g' ⟶ $h ⊗ $g') := η₁.e.e
+    have ηs₁ : Q($h ⊗ $g' ⟶ $i ⊗ $g') := ηs₁.e.e
+    have η₂ : Q($g ⊗ $g' ⟶ $i ⊗ $g') := η₂.e.e
+    have η₃ : Q($f ⊗ $f' ⟶ $i ⊗ $g') := η₃.e.e
+    have e_η₁ : Q(((Iso.refl _).hom ≫ $η ≫ (Iso.refl _).hom) ▷ $g' = $η₁) := e_η₁
+    have e_ηs₁ : Q($ηs ▷ $g' = $ηs₁) := e_ηs₁
+    have e_η₂ : Q($η₁ ≫ $ηs₁ = $η₂) := e_η₂
+    have e_η₃ : Q(($α ⊗ $β).hom ≫ $η₂ = $η₃) := e_η₃
+    return q(evalHorizontalComp_cons_nil $e_η₁ $e_ηs₁ $e_η₂ $e_η₃)
+  mkEvalHorizontalCompConsCons α β η θ ηs θs ηθ ηθs ηθ₁ ηθ₂ e_ηθ e_ηθs e_ηθ₁ e_ηθ₂ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← α.srcM
+    let g ← α.tgtM
+    let h ← η.tgtM
+    let i ← ηs.tgtM
+    let f' ← β.srcM
+    let g' ← β.tgtM
+    let h' ← θ.tgtM
+    let i' ← θs.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have f' : Q($ctx.C) := f'.e
+    have g' : Q($ctx.C) := g'.e
+    have h' : Q($ctx.C) := h'.e
+    have i' : Q($ctx.C) := i'.e
+    have α : Q($f ≅ $g) := α.e
+    have η : Q($g ⟶ $h) := η.e.e
+    have ηs : Q($h ⟶ $i) := ηs.e.e
+    have β : Q($f' ≅ $g') := β.e
+    have θ : Q($g' ⟶ $h') := θ.e.e
+    have θs : Q($h' ⟶ $i') := θs.e.e
+    have ηθ : Q($g ⊗ $g' ⟶ $h ⊗ $h') := ηθ.e.e
+    have ηθs : Q($h ⊗ $h' ⟶ $i ⊗ $i') := ηθs.e.e
+    have ηθ₁ : Q($g ⊗ $g' ⟶ $i ⊗ $i') := ηθ₁.e.e
+    have ηθ₂ : Q($f ⊗ $f' ⟶ $i ⊗ $i') := ηθ₂.e.e
+    have e_ηθ : Q($η ⊗ $θ = $ηθ) := e_ηθ
+    have e_ηθs : Q($ηs ⊗ $θs = $ηθs) := e_ηθs
+    have e_ηθ₁ : Q($ηθ ≫ $ηθs = $ηθ₁) := e_ηθ₁
+    have e_ηθ₂ : Q(($α ⊗ $β).hom ≫ $ηθ₁ = $ηθ₂) := e_ηθ₂
+    return q(evalHorizontalComp_cons_cons $e_ηθ $e_ηθs $e_ηθ₁ $e_ηθ₂)
+
+instance : MkEval MonoidalM where
+  mkEvalComp η θ η' θ' ι e_η e_θ e_ηθ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let h ← θ'.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have θ : Q($g ⟶ $h) := θ.e
+    have θ' : Q($g ⟶ $h) := θ'.e.e
+    have ι : Q($f ⟶ $h) := ι.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($θ = $θ') := e_θ
+    have e_ηθ : Q($η' ≫ $θ' = $ι) := e_ηθ
+    return q(eval_comp $e_η $e_θ $e_ηθ)
+  mkEvalWhiskerLeft f η η' θ e_η e_θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let g ← η'.srcM
+    let h ← η'.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have η : Q($g ⟶ $h) := η.e
+    have η' : Q($g ⟶ $h) := η'.e.e
+    have θ : Q($f ⊗ $g ⟶ $f ⊗ $h) := θ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($f ◁ $η' = $θ) := e_θ
+    return q(eval_whiskerLeft $e_η $e_θ)
+  mkEvalWhiskerRight η h η' θ e_η e_θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have θ : Q($f ⊗ $h ⟶ $g ⊗ $h) := θ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($η' ▷ $h = $θ) := e_θ
+    return q(eval_whiskerRight $e_η $e_θ)
+  mkEvalHorizontalComp η θ η' θ' ι e_η e_θ e_ι := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let h ← θ'.srcM
+    let i ← θ'.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have θ : Q($h ⟶ $i) := θ.e
+    have θ' : Q($h ⟶ $i) := θ'.e.e
+    have ι : Q($f ⊗ $h ⟶ $g ⊗ $i) := ι.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($θ = $θ') := e_θ
+    have e_ι : Q($η' ⊗ $θ' = $ι) := e_ι
+    return q(eval_tensorHom $e_η $e_θ $e_ι)
+  mkEvalOf η := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f := η.src
+    let g := η.tgt
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have η : Q($f ⟶ $g) := η.e
+    return q(eval_of $η)
+  mkEvalMonoidalComp η θ α η' θ' αθ ηαθ e_η e_θ e_αθ e_ηαθ := do
+    let ctx ← read
+    let _cat := ctx.instCat
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let h ← α.tgtM
+    let i ← θ'.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have i : Q($ctx.C) := i.e
+    have η : Q($f ⟶ $g) := η.e
+    have η' : Q($f ⟶ $g) := η'.e.e
+    have α : Q($g ≅ $h) := α.e
+    have θ : Q($h ⟶ $i) := θ.e
+    have θ' : Q($h ⟶ $i) := θ'.e.e
+    have αθ : Q($g ⟶ $i) := αθ.e.e
+    have ηαθ : Q($f ⟶ $i) := ηαθ.e.e
+    have e_η : Q($η = $η') := e_η
+    have e_θ : Q($θ = $θ') := e_θ
+    have e_αθ : Q(Iso.hom $α ≫ $θ' = $αθ) := e_αθ
+    have e_ηαθ : Q($η' ≫ $αθ = $ηαθ) := e_ηαθ
+    return q(eval_monoidalComp $e_η $e_θ $e_αθ $e_ηαθ)
+
+instance : MonadNormalExpr MonoidalM where
+  whiskerRightM η h := do
+    return .whisker (← MonadMor₂.whiskerRightM η.e (.of h)) η h
+  hConsM η θ := do
+    return .cons (← MonadMor₂.horizontalCompM η.e θ.e) η θ
+  whiskerLeftM f η := do
+    return .whisker (← MonadMor₂.whiskerLeftM (.of f) η.e) f η
+  nilM α := do
+    return .nil (← MonadMor₂.homM α) α
+  consM α η ηs := do
+    return .cons (← MonadMor₂.comp₂M (← MonadMor₂.homM α) (← MonadMor₂.comp₂M η.e ηs.e)) α η ηs
+
+instance : MkMor₂ MonoidalM where
+  ofExpr := Mor₂OfExpr
+
+end Mathlib.Tactic.Monoidal
diff --git a/Mathlib/Tactic/CategoryTheory/Monoidal/PureCoherence.lean b/Mathlib/Tactic/CategoryTheory/Monoidal/PureCoherence.lean
new file mode 100644
index 0000000000000..56c83c25a1459
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/Monoidal/PureCoherence.lean
@@ -0,0 +1,277 @@
+/-
+Copyright (c) 2024 Yuma Mizuno. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuma Mizuno
+-/
+import Mathlib.Tactic.CategoryTheory.Coherence.PureCoherence
+import Mathlib.Tactic.CategoryTheory.Monoidal.Datatypes
+
+/-!
+# Coherence tactic for monoidal categories
+
+We provide a `monoidal_coherence` tactic,
+which proves that any two morphisms (with the same source and target)
+in a monoidal category which are built out of associators and unitors
+are equal.
+
+-/
+
+open Lean Meta Elab Qq
+open CategoryTheory Mathlib.Tactic.BicategoryLike MonoidalCategory
+
+namespace Mathlib.Tactic.Monoidal
+
+section
+
+universe v u
+
+variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
+
+local infixr:81 " ◁ " => MonoidalCategory.whiskerLeftIso
+local infixl:81 " ▷ " => MonoidalCategory.whiskerRightIso
+
+/-- The composition of the normalizing isomorphisms `η_f : p ⊗ f ≅ pf` and `η_g : pf ⊗ g ≅ pfg`. -/
+abbrev normalizeIsoComp {p f g pf pfg : C} (η_f : p ⊗ f ≅ pf) (η_g : pf ⊗ g ≅ pfg) :=
+  (α_ _ _ _).symm ≪≫ whiskerRightIso η_f g ≪≫ η_g
+
+theorem naturality_associator {p f g h pf pfg pfgh : C}
+    (η_f : p ⊗ f ≅ pf) (η_g : pf ⊗ g ≅ pfg) (η_h : pfg ⊗ h ≅ pfgh) :
+    p ◁ (α_ f g h) ≪≫ normalizeIsoComp η_f (normalizeIsoComp η_g η_h) =
+    normalizeIsoComp (normalizeIsoComp η_f η_g) η_h :=
+  Iso.ext (by simp)
+
+theorem naturality_leftUnitor {p f pf : C} (η_f : p ⊗ f ≅ pf) :
+    p ◁ (λ_ f) ≪≫ η_f = normalizeIsoComp (ρ_ p) η_f :=
+  Iso.ext (by simp)
+
+theorem naturality_rightUnitor {p f pf : C} (η_f : p ⊗ f ≅ pf) :
+    p ◁ (ρ_ f) ≪≫ η_f = normalizeIsoComp η_f (ρ_ pf) :=
+  Iso.ext (by simp)
+
+theorem naturality_id {p f pf : C} (η_f : p ⊗ f ≅ pf) :
+    p ◁ Iso.refl f ≪≫ η_f = η_f := by
+  simp
+
+theorem naturality_comp {p f g h pf : C} {η : f ≅ g} {θ : g ≅ h}
+    (η_f : p ⊗ f ≅ pf) (η_g : p ⊗ g ≅ pf) (η_h : p ⊗ h ≅ pf)
+    (ih_η : p ◁ η ≪≫ η_g = η_f) (ih_θ : p ◁ θ ≪≫ η_h = η_g) :
+    p ◁ (η ≪≫ θ) ≪≫ η_h = η_f := by
+  simp_all
+
+theorem naturality_whiskerLeft {p f g h pf pfg : C} {η : g ≅ h}
+    (η_f : p ⊗ f ≅ pf) (η_fg : pf ⊗ g ≅ pfg) (η_fh : (pf ⊗ h) ≅ pfg)
+    (ih_η : pf ◁ η ≪≫ η_fh = η_fg) :
+    p ◁ (f ◁ η) ≪≫ normalizeIsoComp η_f η_fh = normalizeIsoComp η_f η_fg := by
+  rw [← ih_η]
+  apply Iso.ext
+  simp [← whisker_exchange_assoc]
+
+theorem naturality_whiskerRight {p f g h pf pfh : C} {η : f ≅ g}
+    (η_f : p ⊗ f ≅ pf) (η_g : p ⊗ g ≅ pf) (η_fh : (pf ⊗ h) ≅ pfh)
+    (ih_η : p ◁ η ≪≫ η_g = η_f) :
+    p ◁ (η ▷ h) ≪≫ normalizeIsoComp η_g η_fh = normalizeIsoComp η_f η_fh := by
+  rw [← ih_η]
+  apply Iso.ext
+  simp
+
+theorem naturality_tensorHom {p f₁ g₁ f₂ g₂ pf₁ pf₁f₂ : C} {η : f₁ ≅ g₁} {θ : f₂ ≅ g₂}
+    (η_f₁ : p ⊗ f₁ ≅ pf₁) (η_g₁ : p ⊗ g₁ ≅ pf₁) (η_f₂ : pf₁ ⊗ f₂ ≅ pf₁f₂) (η_g₂ : pf₁ ⊗ g₂ ≅ pf₁f₂)
+    (ih_η : p ◁ η ≪≫ η_g₁ = η_f₁)
+    (ih_θ : pf₁ ◁ θ ≪≫ η_g₂ = η_f₂) :
+    p ◁ (η ⊗ θ) ≪≫ normalizeIsoComp η_g₁ η_g₂ = normalizeIsoComp η_f₁ η_f₂ := by
+  rw [tensorIso_def]
+  apply naturality_comp
+  · apply naturality_whiskerRight _ _ _ ih_η
+  · apply naturality_whiskerLeft _ _ _ ih_θ
+
+theorem naturality_inv {p f g pf : C} {η : f ≅ g}
+    (η_f : p ⊗ f ≅ pf) (η_g : p ⊗ g ≅ pf) (ih : p ◁ η ≪≫ η_g = η_f) :
+    p ◁ η.symm ≪≫ η_f = η_g := by
+  rw [← ih]
+  apply Iso.ext
+  simp
+
+instance : MonadNormalizeNaturality MonoidalM where
+  mkNaturalityAssociator p pf pfg pfgh f g h η_f η_g η_h := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have pf : Q($ctx.C) := pf.e.e
+    have pfg : Q($ctx.C) := pfg.e.e
+    have pfgh : Q($ctx.C) := pfgh.e.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    have η_g : Q($pf ⊗ $g ≅ $pfg) := η_g.e
+    have η_h : Q($pfg ⊗ $h ≅ $pfgh) := η_h.e
+    return q(naturality_associator $η_f $η_g $η_h)
+  mkNaturalityLeftUnitor p pf f η_f := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have pf : Q($ctx.C) := pf.e.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    return q(naturality_leftUnitor $η_f)
+  mkNaturalityRightUnitor p pf f η_f := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have pf : Q($ctx.C) := pf.e.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    return q(naturality_rightUnitor $η_f)
+  mkNaturalityId p pf f η_f := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have pf : Q($ctx.C) := pf.e.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    return q(naturality_id $η_f)
+  mkNaturalityComp p pf f g h η θ η_f η_g η_h ih_η ih_θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have pf : Q($ctx.C) := pf.e.e
+    have η : Q($f ≅ $g) := η.e
+    have θ : Q($g ≅ $h) := θ.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ⊗ $g ≅ $pf) := η_g.e
+    have η_h : Q($p ⊗ $h ≅ $pf) := η_h.e
+    have ih_η : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih_η
+    have ih_θ : Q($p ◁ $θ ≪≫ $η_h = $η_g) := ih_θ
+    return q(naturality_comp $η_f $η_g $η_h $ih_η $ih_θ)
+  mkNaturalityWhiskerLeft p pf pfg f g h η η_f η_fg η_fh ih_η := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have pf : Q($ctx.C) := pf.e.e
+    have pfg : Q($ctx.C) := pfg.e.e
+    have η : Q($g ≅ $h) := η.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    have η_fg : Q($pf ⊗ $g ≅ $pfg) := η_fg.e
+    have η_fh : Q($pf ⊗ $h ≅ $pfg) := η_fh.e
+    have ih_η : Q($pf ◁ $η ≪≫ $η_fh = $η_fg) := ih_η
+    return q(naturality_whiskerLeft $η_f $η_fg $η_fh $ih_η)
+  mkNaturalityWhiskerRight p pf pfh f g h η η_f η_g η_fh ih_η := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have h : Q($ctx.C) := h.e
+    have pf : Q($ctx.C) := pf.e.e
+    have pfh : Q($ctx.C) := pfh.e.e
+    have η : Q($f ≅ $g) := η.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ⊗ $g ≅ $pf) := η_g.e
+    have η_fh : Q($pf ⊗ $h ≅ $pfh) := η_fh.e
+    have ih_η : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih_η
+    return q(naturality_whiskerRight $η_f $η_g $η_fh $ih_η)
+  mkNaturalityHorizontalComp p pf₁ pf₁f₂ f₁ g₁ f₂ g₂ η θ η_f₁ η_g₁ η_f₂ η_g₂ ih_η ih_θ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f₁ : Q($ctx.C) := f₁.e
+    have g₁ : Q($ctx.C) := g₁.e
+    have f₂ : Q($ctx.C) := f₂.e
+    have g₂ : Q($ctx.C) := g₂.e
+    have pf₁ : Q($ctx.C) := pf₁.e.e
+    have pf₁f₂ : Q($ctx.C) := pf₁f₂.e.e
+    have η : Q($f₁ ≅ $g₁) := η.e
+    have θ : Q($f₂ ≅ $g₂) := θ.e
+    have η_f₁ : Q($p ⊗ $f₁ ≅ $pf₁) := η_f₁.e
+    have η_g₁ : Q($p ⊗ $g₁ ≅ $pf₁) := η_g₁.e
+    have η_f₂ : Q($pf₁ ⊗ $f₂ ≅ $pf₁f₂) := η_f₂.e
+    have η_g₂ : Q($pf₁ ⊗ $g₂ ≅ $pf₁f₂) := η_g₂.e
+    have ih_η : Q($p ◁ $η ≪≫ $η_g₁ = $η_f₁) := ih_η
+    have ih_θ : Q($pf₁ ◁ $θ ≪≫ $η_g₂ = $η_f₂) := ih_θ
+    return q(naturality_tensorHom $η_f₁ $η_g₁ $η_f₂ $η_g₂ $ih_η $ih_θ)
+  mkNaturalityInv p pf f g η η_f η_g ih := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    have p : Q($ctx.C) := p.e.e
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have pf : Q($ctx.C) := pf.e.e
+    have η : Q($f ≅ $g) := η.e
+    have η_f : Q($p ⊗ $f ≅ $pf) := η_f.e
+    have η_g : Q($p ⊗ $g ≅ $pf) := η_g.e
+    have ih : Q($p ◁ $η ≪≫ $η_g = $η_f) := ih
+    return q(naturality_inv $η_f $η_g $ih)
+
+theorem of_normalize_eq {f g f' : C} {η θ : f ≅ g} (η_f : 𝟙_ C ⊗ f ≅ f') (η_g : 𝟙_ C ⊗ g ≅ f')
+    (h_η : 𝟙_ C ◁ η ≪≫ η_g = η_f)
+    (h_θ : 𝟙_ C ◁ θ ≪≫ η_g = η_f) : η = θ := by
+  apply Iso.ext
+  calc
+    η.hom = (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom := by
+      simp [← reassoc_of% (congrArg Iso.hom h_η)]
+    _ = θ.hom := by
+      simp [← reassoc_of% (congrArg Iso.hom h_θ)]
+
+theorem mk_eq_of_naturality {f g f' : C} {η θ : f ⟶ g} {η' θ' : f ≅ g}
+    (η_f : 𝟙_ C ⊗ f ≅ f') (η_g : 𝟙_ C ⊗ g ≅ f')
+    (η_hom : η'.hom = η) (Θ_hom : θ'.hom = θ)
+    (Hη : whiskerLeftIso (𝟙_ C) η' ≪≫ η_g = η_f)
+    (Hθ : whiskerLeftIso (𝟙_ C) θ' ≪≫ η_g = η_f) : η = θ :=
+  calc
+    η = η'.hom := η_hom.symm
+    _ = (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom := by
+      simp [← reassoc_of% (congrArg Iso.hom Hη)]
+    _ = θ'.hom := by
+      simp [← reassoc_of% (congrArg Iso.hom Hθ)]
+    _ = θ := Θ_hom
+
+end
+
+instance : MkEqOfNaturality MonoidalM where
+  mkEqOfNaturality η θ ηIso θIso η_f η_g Hη Hθ := do
+    let ctx ← read
+    let .some _monoidal := ctx.instMonoidal? | synthMonoidalError
+    let η' := ηIso.e
+    let θ' := θIso.e
+    let f ← η'.srcM
+    let g ← η'.tgtM
+    let f' ← η_f.tgtM
+    have f : Q($ctx.C) := f.e
+    have g : Q($ctx.C) := g.e
+    have f' : Q($ctx.C) := f'.e
+    have η : Q($f ⟶ $g) := η
+    have θ : Q($f ⟶ $g) := θ
+    have η'_e : Q($f ≅ $g) := η'.e
+    have θ'_e : Q($f ≅ $g) := θ'.e
+    have η_f : Q(tensorUnit ⊗ $f ≅ $f') := η_f.e
+    have η_g : Q(tensorUnit ⊗ $g ≅ $f') := η_g.e
+    have η_hom : Q(Iso.hom $η'_e = $η) := ηIso.eq
+    have Θ_hom : Q(Iso.hom $θ'_e = $θ) := θIso.eq
+    have Hη : Q(whiskerLeftIso tensorUnit $η'_e ≪≫ $η_g = $η_f) := Hη
+    have Hθ : Q(whiskerLeftIso tensorUnit $θ'_e ≪≫ $η_g = $η_f) := Hθ
+    return q(mk_eq_of_naturality $η_f $η_g $η_hom $Θ_hom $Hη $Hθ)
+
+open Elab.Tactic
+
+/-- Close the goal of the form `η = θ`, where `η` and `θ` are 2-isomorphisms made up only of
+associators, unitors, and identities.
+```lean
+example {C : Type} [Category C] [MonoidalCategory C] :
+  (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by
+  monoidal_coherence
+```
+-/
+def pureCoherence (mvarId : MVarId) : MetaM (List MVarId) :=
+  BicategoryLike.pureCoherence Monoidal.Context `monoidal mvarId
+
+@[inherit_doc pureCoherence]
+elab "monoidal_coherence" : tactic => withMainContext do
+  replaceMainGoal <| ← Monoidal.pureCoherence <| ← getMainGoal
+
+end Mathlib.Tactic.Monoidal
diff --git a/Mathlib/Tactic/CategoryTheory/MonoidalComp.lean b/Mathlib/Tactic/CategoryTheory/MonoidalComp.lean
index e62ef8538919e..316faec90eedf 100644
--- a/Mathlib/Tactic/CategoryTheory/MonoidalComp.lean
+++ b/Mathlib/Tactic/CategoryTheory/MonoidalComp.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Yuma Mizuno, Oleksandr Manzyuk
+Authors: Kim Morrison, Yuma Mizuno, Oleksandr Manzyuk
 -/
 import Mathlib.CategoryTheory.Monoidal.Category
 
@@ -47,25 +47,20 @@ Used by the `⊗≫` monoidal composition operator, and the `coherence` tactic.
 -- We could likely turn this into a `Prop` valued existential if that proves useful.
 class MonoidalCoherence (X Y : C) where
   /-- A monoidal structural isomorphism between two objects. -/
-  hom : X ⟶ Y
-  [isIso : IsIso hom]
+  iso : X ≅ Y
 
 /-- Notation for identities up to unitors and associators. -/
 scoped[CategoryTheory.MonoidalCategory] notation " ⊗𝟙 " =>
-  MonoidalCoherence.hom -- type as \ot 𝟙
-
-attribute [instance] MonoidalCoherence.isIso
-
-noncomputable section
+  MonoidalCoherence.iso -- type as \ot 𝟙
 
 /-- Construct an isomorphism between two objects in a monoidal category
 out of unitors and associators. -/
-def monoidalIso (X Y : C) [MonoidalCoherence X Y] : X ≅ Y := asIso ⊗𝟙
+abbrev monoidalIso (X Y : C) [MonoidalCoherence X Y] : X ≅ Y := MonoidalCoherence.iso
 
 /-- Compose two morphisms in a monoidal category,
 inserting unitors and associators between as necessary. -/
 def monoidalComp {W X Y Z : C} [MonoidalCoherence X Y] (f : W ⟶ X) (g : Y ⟶ Z) : W ⟶ Z :=
-  f ≫ ⊗𝟙 ≫ g
+  f ≫ ⊗𝟙.hom ≫ g
 
 @[inherit_doc monoidalComp]
 scoped[CategoryTheory.MonoidalCategory] infixr:80 " ⊗≫ " =>
@@ -74,70 +69,68 @@ scoped[CategoryTheory.MonoidalCategory] infixr:80 " ⊗≫ " =>
 /-- Compose two isomorphisms in a monoidal category,
 inserting unitors and associators between as necessary. -/
 def monoidalIsoComp {W X Y Z : C} [MonoidalCoherence X Y] (f : W ≅ X) (g : Y ≅ Z) : W ≅ Z :=
-  f ≪≫ asIso ⊗𝟙 ≪≫ g
+  f ≪≫ ⊗𝟙 ≪≫ g
 
 @[inherit_doc monoidalIsoComp]
 scoped[CategoryTheory.MonoidalCategory] infixr:80 " ≪⊗≫ " =>
   monoidalIsoComp -- type as \ll \ot \gg
 
-end
-
 namespace MonoidalCoherence
 
 variable [MonoidalCategory C]
 
 @[simps]
-instance refl (X : C) : MonoidalCoherence X X := ⟨𝟙 _⟩
+instance refl (X : C) : MonoidalCoherence X X := ⟨Iso.refl _⟩
 
 @[simps]
 instance whiskerLeft (X Y Z : C) [MonoidalCoherence Y Z] :
     MonoidalCoherence (X ⊗ Y) (X ⊗ Z) :=
-  ⟨X ◁ ⊗𝟙⟩
+  ⟨whiskerLeftIso X ⊗𝟙⟩
 
 @[simps]
 instance whiskerRight (X Y Z : C) [MonoidalCoherence X Y] :
     MonoidalCoherence (X ⊗ Z) (Y ⊗ Z) :=
-  ⟨⊗𝟙 ▷ Z⟩
+  ⟨whiskerRightIso ⊗𝟙 Z⟩
 
 @[simps]
 instance tensor_right (X Y : C) [MonoidalCoherence (𝟙_ C) Y] :
     MonoidalCoherence X (X ⊗ Y) :=
-  ⟨(ρ_ X).inv ≫ X ◁  ⊗𝟙⟩
+  ⟨(ρ_ X).symm ≪≫ (whiskerLeftIso X ⊗𝟙)⟩
 
 @[simps]
 instance tensor_right' (X Y : C) [MonoidalCoherence Y (𝟙_ C)] :
     MonoidalCoherence (X ⊗ Y) X :=
-  ⟨X ◁ ⊗𝟙 ≫ (ρ_ X).hom⟩
+  ⟨whiskerLeftIso X ⊗𝟙 ≪≫ (ρ_ X)⟩
 
 @[simps]
 instance left (X Y : C) [MonoidalCoherence X Y] :
     MonoidalCoherence (𝟙_ C ⊗ X) Y :=
-  ⟨(λ_ X).hom ≫ ⊗𝟙⟩
+  ⟨λ_ X ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance left' (X Y : C) [MonoidalCoherence X Y] :
     MonoidalCoherence X (𝟙_ C ⊗ Y) :=
-  ⟨⊗𝟙 ≫ (λ_ Y).inv⟩
+  ⟨⊗𝟙 ≪≫ (λ_ Y).symm⟩
 
 @[simps]
 instance right (X Y : C) [MonoidalCoherence X Y] :
     MonoidalCoherence (X ⊗ 𝟙_ C) Y :=
-  ⟨(ρ_ X).hom ≫ ⊗𝟙⟩
+  ⟨ρ_ X ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance right' (X Y : C) [MonoidalCoherence X Y] :
     MonoidalCoherence X (Y ⊗ 𝟙_ C) :=
-  ⟨⊗𝟙 ≫ (ρ_ Y).inv⟩
+  ⟨⊗𝟙 ≪≫ (ρ_ Y).symm⟩
 
 @[simps]
 instance assoc (X Y Z W : C) [MonoidalCoherence (X ⊗ (Y ⊗ Z)) W] :
     MonoidalCoherence ((X ⊗ Y) ⊗ Z) W :=
-  ⟨(α_ X Y Z).hom ≫ ⊗𝟙⟩
+  ⟨α_ X Y Z ≪≫ ⊗𝟙⟩
 
 @[simps]
 instance assoc' (W X Y Z : C) [MonoidalCoherence W (X ⊗ (Y ⊗ Z))] :
     MonoidalCoherence W ((X ⊗ Y) ⊗ Z) :=
-  ⟨⊗𝟙 ≫ (α_ X Y Z).inv⟩
+  ⟨⊗𝟙 ≪≫ (α_ X Y Z).symm⟩
 
 end MonoidalCoherence
 
diff --git a/Mathlib/Tactic/CategoryTheory/Reassoc.lean b/Mathlib/Tactic/CategoryTheory/Reassoc.lean
index d83662642b472..f76eba34bf8a9 100644
--- a/Mathlib/Tactic/CategoryTheory/Reassoc.lean
+++ b/Mathlib/Tactic/CategoryTheory/Reassoc.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Functor.Basic
 import Mathlib.Util.AddRelatedDecl
@@ -29,7 +29,7 @@ namespace CategoryTheory
 
 variable {C : Type*} [Category C]
 
-/-- A variant of `eq_whisker` with a more convenient argument order for use in tactics.  -/
+/-- A variant of `eq_whisker` with a more convenient argument order for use in tactics. -/
 theorem eq_whisker' {X Y : C} {f g : X ⟶ Y} (w : f = g) {Z : C} (h : Y ⟶ Z) :
     f ≫ h = g ≫ h := by rw [w]
 
diff --git a/Mathlib/Tactic/CategoryTheory/Slice.lean b/Mathlib/Tactic/CategoryTheory/Slice.lean
index 60a77786bac0a..f3f88410158c6 100644
--- a/Mathlib/Tactic/CategoryTheory/Slice.lean
+++ b/Mathlib/Tactic/CategoryTheory/Slice.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 
 -/
 import Mathlib.CategoryTheory.Category.Basic
diff --git a/Mathlib/Tactic/CategoryTheory/ToApp.lean b/Mathlib/Tactic/CategoryTheory/ToApp.lean
new file mode 100644
index 0000000000000..93a6aebb3170d
--- /dev/null
+++ b/Mathlib/Tactic/CategoryTheory/ToApp.lean
@@ -0,0 +1,142 @@
+/-
+Copyright (c) 2024 Calle Sönne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Calle Sönne
+-/
+import Mathlib.CategoryTheory.Category.Cat
+import Mathlib.Util.AddRelatedDecl
+
+/-!
+# The `to_app` attribute
+
+Adding `@[to_app]` to a lemma named `F` of shape `∀ .., η = θ`, where `η θ : f ⟶ g` are 2-morphisms
+in some bicategory, create a new lemma named `F_app`. This lemma is obtained by first specializing
+the bicategory in which the equality is taking place to `Cat`, then applying `NatTrans.congr_app`
+to obtain a proof of `∀ ... (X : Cat), η.app X = θ.app X`, and finally simplifying the conclusion
+using some basic lemmas in the bicategory `Cat`:
+`Cat.whiskerLeft_app`, `Cat.whiskerRight_app`, `Cat.id_app`, `Cat.comp_app` and `Cat.eqToHom_app`
+
+So, for example, if the conclusion of `F` is `f ◁ η = θ` then the conclusion of `F_app` will be
+`η.app (f.obj X) = θ.app X`.
+
+This is useful for automatically generating lemmas that can be applied to expressions of 1-morphisms
+in `Cat` which contain components of 2-morphisms.
+
+There is also a term elaborator `to_app_of% t` for use within proofs.
+-/
+
+open Lean Meta Elab Tactic
+open Mathlib.Tactic
+
+namespace CategoryTheory
+
+/-- Simplify an expression in `Cat` using basic properties of `NatTrans.app`. -/
+def catAppSimp (e : Expr) : MetaM Simp.Result :=
+  simpOnlyNames [
+    ``Cat.whiskerLeft_app, ``Cat.whiskerRight_app, ``Cat.id_app, ``Cat.comp_app,
+    ``Cat.eqToHom_app] e
+    (config := { decide := false })
+
+/--
+Given a term of type `∀ ..., η = θ`, where `η θ : f ⟶ g` are 2-morphisms in some bicategory
+`B`, which is bound by the `∀` binder, get the corresponding equation in the bicategory `Cat`.
+
+It is important here that the levels in the term are level metavariables, as otherwise these will
+not be reassignable to the corresponding levels of `Cat`. -/
+def toCatExpr (e : Expr) : MetaM Expr := do
+  let (args, binderInfos, conclusion) ← forallMetaTelescope (← inferType e)
+  -- Find the expression corresponding to the bicategory, by anylizing `η = θ` (i.e. conclusion)
+  let B ←
+    match conclusion.getAppFnArgs with
+    | (`Eq, #[_, η, _]) =>
+      match (← inferType η).getAppFnArgs with
+      | (`Quiver.Hom, #[_, _, f, _]) =>
+        match (← inferType f).getAppFnArgs with
+        | (`Quiver.Hom, #[_, _, a, _]) => inferType a
+        | _ => throwError "The conclusion {conclusion} is not an equality of 2-morphisms!"
+      | _ => throwError "The conclusion {conclusion} is not an equality of 2-morphisms!"
+    | _ => throwError "The conclusion {conclusion} is not an equality!"
+  -- Create level metavariables to be used for `Cat.{v, u}`
+  let u ← mkFreshLevelMVar
+  let v ← mkFreshLevelMVar
+  -- Assign `B` to `Cat.{v, u}`
+  let _ ← isDefEq B (.const ``Cat [v, u])
+  -- Assign the right bicategory instance to `Cat.{v, u}`
+  let some inst ← args.findM? fun x => do
+      return (← inferType x).getAppFnArgs == (`CategoryTheory.Bicategory, #[B])
+    | throwError "Can not find the argument for the bicategory instance of the bicategory in which \
+      the equality is taking place."
+  let _ ← isDefEq inst (.const ``CategoryTheory.Cat.bicategory [v, u])
+  -- Construct the new expression
+  let value := mkAppN e args
+  let rec
+  /-- Recursive function which applies `mkLambdaFVars` stepwise
+  (so that each step can have different binderinfos) -/
+    apprec (i : Nat) (e : Expr) : MetaM Expr := do
+      if i < args.size then
+        let arg := args[i]!
+        let bi := binderInfos[i]!
+        let e' ← apprec (i + 1) e
+        unless arg != B && arg != inst do return e'
+        mkLambdaFVars #[arg] e' (binderInfoForMVars := bi)
+      else
+        return e
+  let value ← apprec 0 value
+  return value
+
+/--
+Given morphisms `f g : C ⟶ D` in the bicategory `Cat`, and an equation `η = θ` between 2-morphisms
+(possibly after a `∀` binder), produce the equation `∀ (X : C), f.app X = g.app X`, and simplify
+it using basic lemmas about `NatTrans.app`. -/
+def toAppExpr (e : Expr) : MetaM Expr := do
+  mapForallTelescope (fun e => do simpType catAppSimp (← mkAppM ``NatTrans.congr_app #[e])) e
+
+/--
+Adding `@[to_app]` to a lemma named `F` of shape `∀ .., η = θ`, where `η θ : f ⟶ g` are 2-morphisms
+in some bicategory, create a new lemma named `F_app`. This lemma is obtained by first specializing
+the bicategory in which the equality is taking place to `Cat`, then applying `NatTrans.congr_app`
+to obtain a proof of `∀ ... (X : Cat), η.app X = θ.app X`, and finally simplifying the conclusion
+using some basic lemmas in the bicategory `Cat`:
+`Cat.whiskerLeft_app`, `Cat.whiskerRight_app`, `Cat.id_app`, `Cat.comp_app` and `Cat.eqToHom_app`
+
+So, for example, if the conclusion of `F` is `f ◁ η = θ` then the conclusion of `F_app` will be
+`η.app (f.obj X) = θ.app X`.
+
+This is useful for automatically generating lemmas that can be applied to expressions of 1-morphisms
+in `Cat` which contain components of 2-morphisms.
+
+Note that if you want both the lemma and the new lemma to be `simp` lemmas, you should tag the lemma
+`@[to_app (attr := simp)]`. The variant `@[simp, to_app]` on a lemma `F` will tag `F` with
+`@[simp]`, but not `F_app` (this is sometimes useful).
+-/
+syntax (name := to_app) "to_app" (" (" &"attr" ":=" Parser.Term.attrInstance,* ")")? : attr
+
+initialize registerBuiltinAttribute {
+  name := `to_app
+  descr := ""
+  applicationTime := .afterCompilation
+  add := fun src ref kind => match ref with
+  | `(attr| to_app $[(attr := $stx?,*)]?) => MetaM.run' do
+    if (kind != AttributeKind.global) then
+      throwError "`to_app` can only be used as a global attribute"
+    addRelatedDecl src "_app" ref stx? fun type value levels => do
+      let levelMVars ← levels.mapM fun _ => mkFreshLevelMVar
+      let value ← mkExpectedTypeHint value type
+      let value := value.instantiateLevelParams levels levelMVars
+      let newValue ← toAppExpr (← toCatExpr value)
+      let r := (← getMCtx).levelMVarToParam (fun _ => false) (fun _ => false) newValue
+      let output := (r.expr, r.newParamNames.toList)
+      pure output
+  | _ => throwUnsupportedSyntax }
+
+open Term in
+/--
+Given an equation `t` of the form `η = θ` between 2-morphisms `f ⟶ g` with `f g : C ⟶ D` in the
+bicategory `Cat` (possibly after a `∀` binder), `to_app_of% t` produces the equation
+`∀ (X : C), η.app X = θ.app X` (where `X` is an object in the domain of `f` and `g`), and simplifies
+it suitably using basic lemmas about `NatTrans.app`.
+-/
+elab "to_app_of% " t:term : term => do
+  toAppExpr (← elabTerm t none)
+
+end CategoryTheory
diff --git a/Mathlib/Tactic/Common.lean b/Mathlib/Tactic/Common.lean
index eb2b980c73313..f9c7a25eb782c 100644
--- a/Mathlib/Tactic/Common.lean
+++ b/Mathlib/Tactic/Common.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 
 -- First import Aesop and Qq
@@ -15,6 +15,9 @@ import ImportGraph.Imports
 import Batteries.Tactic.Where
 import Batteries.Tactic.Basic
 
+-- Import syntax for leansearch
+import LeanSearchClient
+
 -- Import Mathlib-specific linters.
 import Mathlib.Tactic.Linter.Lint
 
diff --git a/Mathlib/Tactic/Constructor.lean b/Mathlib/Tactic/Constructor.lean
index 6ba7e4aee3325..64adccae545b7 100644
--- a/Mathlib/Tactic/Constructor.lean
+++ b/Mathlib/Tactic/Constructor.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Newell Jensen
+Authors: Kim Morrison, Newell Jensen
 -/
 import Mathlib.Init
 import Lean.Elab.SyntheticMVars
diff --git a/Mathlib/Tactic/ContinuousFunctionalCalculus.lean b/Mathlib/Tactic/ContinuousFunctionalCalculus.lean
index 0adbd41b52a2f..9eca49727383c 100644
--- a/Mathlib/Tactic/ContinuousFunctionalCalculus.lean
+++ b/Mathlib/Tactic/ContinuousFunctionalCalculus.lean
@@ -15,11 +15,17 @@ import Aesop
 At the moment, these tactics are just wrappers, but potentially they could be more sophisticated.
 -/
 
+declare_aesop_rule_sets [CStarAlgebra]
+
 /-- A tactic used to automatically discharge goals relating to the continuous functional calculus,
 specifically whether the element satisfies the predicate. -/
 syntax (name := cfcTac) "cfc_tac" : tactic
 macro_rules
-  | `(tactic| cfc_tac) => `(tactic| (try (first | assumption | infer_instance | aesop)))
+  | `(tactic| cfc_tac) => `(tactic|
+         try (first |
+              assumption |
+              infer_instance |
+              aesop (rule_sets := [$(Lean.mkIdent `CStarAlgebra):ident])))
 
 -- we may want to try using `fun_prop` directly in the future.
 /-- A tactic used to automatically discharge goals relating to the continuous functional calculus,
@@ -28,11 +34,14 @@ syntax (name := cfcContTac) "cfc_cont_tac" : tactic
 macro_rules
   | `(tactic| cfc_cont_tac) =>
     `(tactic| try (first
-      | fun_prop (disch := aesop (config := {warnOnNonterminal := false}))
+      | fun_prop (disch := aesop (config := {warnOnNonterminal := false})
+          (rule_sets := [$(Lean.mkIdent `CStarAlgebra):ident]))
       | assumption))
 
 /-- A tactic used to automatically discharge goals relating to the non-unital continuous functional
 calculus, specifically concerning whether `f 0 = 0`. -/
 syntax (name := cfcZeroTac) "cfc_zero_tac" : tactic
 macro_rules
-  | `(tactic| cfc_zero_tac) => `(tactic| try (first | aesop | assumption))
+  | `(tactic| cfc_zero_tac) =>
+      `(tactic| try
+          (first | aesop (rule_sets := [$(Lean.mkIdent `CStarAlgebra):ident]) | assumption))
diff --git a/Mathlib/Tactic/Convert.lean b/Mathlib/Tactic/Convert.lean
index aa2d570e1a2a9..c0dcc20b97221 100644
--- a/Mathlib/Tactic/Convert.lean
+++ b/Mathlib/Tactic/Convert.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Kyle Miller
+Authors: Kim Morrison, Kyle Miller
 -/
 import Mathlib.Tactic.CongrExclamation
 
diff --git a/Mathlib/Tactic/DefEqTransformations.lean b/Mathlib/Tactic/DefEqTransformations.lean
index f84e06c214a9f..1bd541cc82ae7 100644
--- a/Mathlib/Tactic/DefEqTransformations.lean
+++ b/Mathlib/Tactic/DefEqTransformations.lean
@@ -247,7 +247,7 @@ elab "eta_reduce" : conv => runDefEqConvTactic etaReduceAll
 
 /-- Eta expand every sub-expression in the given expression.
 
-As a side-effect, beta reduces any pre-existing instances of eta expanded terms.  -/
+As a side-effect, beta reduces any pre-existing instances of eta expanded terms. -/
 partial def etaExpandAll (e : Expr) : MetaM Expr := do
   let betaOrApp (f : Expr) (args : Array Expr) : Expr :=
     if f.etaExpanded?.isSome then f.beta args else mkAppN f args
diff --git a/Mathlib/Tactic/DeprecateMe.lean b/Mathlib/Tactic/DeprecateMe.lean
index 3fb56313bec00..b84b97f0b4c2c 100644
--- a/Mathlib/Tactic/DeprecateMe.lean
+++ b/Mathlib/Tactic/DeprecateMe.lean
@@ -107,7 +107,7 @@ Technically, the command also take an optional `String` argument to fill in the
 However, its use is mostly intended for debugging purposes, where having a variable date would
 make tests time-dependent.
 -/
-elab tk:"deprecate to " id:ident* dat:(str)? ppLine cmd:command : command => do
+elab tk:"deprecate to " id:ident* dat:(ppSpace str ppSpace)? ppLine cmd:command : command => do
   let oldEnv ← getEnv
   try
     elabCommand cmd
diff --git a/Mathlib/Tactic/Explode/Datatypes.lean b/Mathlib/Tactic/Explode/Datatypes.lean
index 2572429fcd5e5..1379a9f398989 100644
--- a/Mathlib/Tactic/Explode/Datatypes.lean
+++ b/Mathlib/Tactic/Explode/Datatypes.lean
@@ -66,7 +66,7 @@ structure Entries : Type where
 
 /-- Find a row where `Entry.expr` == `e`. -/
 def Entries.find? (es : Entries) (e : Expr) : Option Entry :=
-  es.s.find? e
+  es.s[e]?
 
 /-- Length of our entries. -/
 def Entries.size (es : Entries) : Nat :=
diff --git a/Mathlib/Tactic/FinCases.lean b/Mathlib/Tactic/FinCases.lean
index 90da0519ec7cb..5a59db53e67bd 100644
--- a/Mathlib/Tactic/FinCases.lean
+++ b/Mathlib/Tactic/FinCases.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2022 Hanting Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Hanting Zhang
+Authors: Kim Morrison, Hanting Zhang
 -/
 import Mathlib.Tactic.Core
 import Mathlib.Lean.Expr.Basic
@@ -60,7 +60,7 @@ partial def finCasesAt (g : MVarId) (hyp : FVarId) : MetaM (List MVarId) := g.wi
     -- Deal with `x : A`, where `[Fintype A]` is available:
     let inst ← synthInstance (← mkAppM ``Fintype #[type])
     let elems ← mkAppOptM ``Fintype.elems #[type, inst]
-    let t ← mkAppM ``Membership.mem #[.fvar hyp, elems]
+    let t ← mkAppM ``Membership.mem #[elems, .fvar hyp]
     let v ← mkAppOptM ``Fintype.complete #[type, inst, Expr.fvar hyp]
     let (fvar, g) ← (← g.assert `this t v).intro1P
     finCasesAt g fvar
diff --git a/Mathlib/Tactic/Find.lean b/Mathlib/Tactic/Find.lean
index 6d91737566f29..6e411fa824810 100644
--- a/Mathlib/Tactic/Find.lean
+++ b/Mathlib/Tactic/Find.lean
@@ -23,7 +23,7 @@ or the `find` tactic which looks for lemmas which are `apply`able against the cu
 
 -/
 
-open Lean
+open Lean Std
 open Lean.Meta
 open Lean.Elab
 open Lean.Elab
@@ -51,7 +51,7 @@ private def isBlackListed (declName : Name) : MetaM Bool := do
   <||> isRec declName
   <||> isMatcher declName
 
-initialize findDeclsPerHead : DeclCache (Lean.HashMap HeadIndex (Array Name)) ←
+initialize findDeclsPerHead : DeclCache (Std.HashMap HeadIndex (Array Name)) ←
   DeclCache.mk "#find: init cache" failure {} fun _ c headMap ↦ do
     if (← isBlackListed c.name) then
       return headMap
@@ -59,7 +59,7 @@ initialize findDeclsPerHead : DeclCache (Lean.HashMap HeadIndex (Array Name)) 
     -- to avoid leaking metavariables.
     let (_, _, ty) ← forallMetaTelescopeReducing c.type
     let head := ty.toHeadIndex
-    pure <| headMap.insert head (headMap.findD head #[] |>.push c.name)
+    pure <| headMap.insert head (headMap.getD head #[] |>.push c.name)
 
 def findType (t : Expr) : TermElabM Unit := withReducible do
   let t ← instantiateMVars t
@@ -68,7 +68,7 @@ def findType (t : Expr) : TermElabM Unit := withReducible do
 
   let env ← getEnv
   let mut numFound := 0
-  for n in (← findDeclsPerHead.get).findD head #[] do
+  for n in (← findDeclsPerHead.get).getD head #[] do
     let c := env.find? n |>.get!
     let cTy := c.instantiateTypeLevelParams (← mkFreshLevelMVars c.numLevelParams)
     let found ← forallTelescopeReducing cTy fun cParams cTy' ↦ do
diff --git a/Mathlib/Tactic/FunProp/ContDiff.lean b/Mathlib/Tactic/FunProp/ContDiff.lean
index 636f5fa0db5bf..037d1812296a5 100644
--- a/Mathlib/Tactic/FunProp/ContDiff.lean
+++ b/Mathlib/Tactic/FunProp/ContDiff.lean
@@ -23,8 +23,7 @@ variable {K : Type*} [NontriviallyNormedField K]
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace K F]
 variable {G : Type*} [NormedAddCommGroup G] [NormedSpace K G]
-variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace K G']
-variable {f f₀ f₁ g : E → F} {x} {s t} {n}
+variable {f : E → F} {x} {s} {n}
 
 theorem contDiff_id' : ContDiff K n (fun x : E => x) := contDiff_id
 
@@ -42,8 +41,8 @@ theorem ContDiffAt.comp' {f : E → F} {g : F → G} (hg : ContDiffAt K n g (f x
 -- theorem ContDiffOn.comp'' {g : F → G} {t : Set F} (hg : ContDiffOn K n g t)
 --     (hf : ContDiffOn K n f s) (st : Set.MapsTo f s t) : ContDiffOn K n (fun x => g (f x)) s :=
 
-variable {ι ι' : Type*} [Fintype ι] [Fintype ι'] {F' : ι → Type*} [∀ i, NormedAddCommGroup (F' i)]
-  [∀ i, NormedSpace K (F' i)] {φ : ∀ i, E → F' i} {Φ : E → ∀ i, F' i}
+variable {ι : Type*} [Fintype ι] {F' : ι → Type*} [∀ i, NormedAddCommGroup (F' i)]
+  [∀ i, NormedSpace K (F' i)] {Φ : E → ∀ i, F' i}
 
 theorem contDiff_pi' (hΦ : ∀ i, ContDiff K n fun x => Φ x i) : ContDiff K n Φ :=
   contDiff_pi.2 hΦ
@@ -60,8 +59,7 @@ section div
 
 variable {K : Type*} [NontriviallyNormedField K]
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
-variable {F : Type*} [NormedAddCommGroup F] [NormedSpace K F]
-variable {f f₀ f₁ g : E → F} {x} {s t} {n}
+variable {s}
 
 theorem ContDiffOn.div' [CompleteSpace K] {f g : E → K} {n} (hf : ContDiffOn K n f s)
     (hg : ContDiffOn K n g s) (h₀ : ∀ x ∈ s, g x ≠ 0) : ContDiffOn K n (fun x => f x / g x) s :=
@@ -74,14 +72,13 @@ end div
 section deriv
 
 variable {K : Type*} [NontriviallyNormedField K]
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace K E]
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace K F]
 
 /-- Original version `ContDiff.differentiable_iteratedDeriv` introduces a new variable `(n:ℕ∞)`
 and `funProp` can't work with such theorem. The theorem should be state where `n` is explicitly
 the smallest possible value i.e. `n=m+1`.
 
-In conjunction with `ContDiff.of_le` we can recover the full power of the original theorem.  -/
+In conjunction with `ContDiff.of_le` we can recover the full power of the original theorem. -/
 theorem ContDiff.differentiable_iteratedDeriv' {m : ℕ} {f : K → F}
     (hf : ContDiff K (m+1) f) : Differentiable K (iteratedDeriv m f) :=
   ContDiff.differentiable_iteratedDeriv m hf (Nat.cast_lt.mpr m.lt_succ_self)
diff --git a/Mathlib/Tactic/FunProp/Core.lean b/Mathlib/Tactic/FunProp/Core.lean
index 3e570580848bc..e337933dc09dc 100644
--- a/Mathlib/Tactic/FunProp/Core.lean
+++ b/Mathlib/Tactic/FunProp/Core.lean
@@ -39,28 +39,23 @@ synthesized value{indentExpr val}\nis not definitionally equal to{indentExpr x}"
     return false
 
 
-/-- Synthesize arguments `xs` either with typeclass synthesis,
-with `fun_prop` or with a discharger. -/
-def synthesizeArgs (thmId : Origin) (xs : Array Expr) (bis : Array BinderInfo)
+
+/-- Synthesize arguments `xs` either with typeclass synthesis, with `fun_prop` or with
+discharger. -/
+def synthesizeArgs (thmId : Origin) (xs : Array Expr)
     (funProp : Expr → FunPropM (Option Result)) :
     FunPropM Bool := do
   let mut postponed : Array Expr := #[]
-  for x in xs, bi in bis do
+  for x in xs do
     let type ← inferType x
-    if bi.isInstImplicit then
-      unless (← synthesizeInstance thmId x type) do
-        logError s!"Failed to synthesize instance `{← ppExpr type}` \
-        when applying theorem `{← ppOrigin' thmId}`."
-        return false
-    else if (← instantiateMVars x).isMVar then
+    if (← instantiateMVars x).isMVar then
 
       -- try type class
       if (← isClass? type).isSome then
         if (← synthesizeInstance thmId x type) then
           continue
-
-      -- try function property
-      if (← isFunProp type.getForallBody) then
+      else if (← isFunProp type.getForallBody) then
+        -- try function property
         if let .some ⟨proof⟩ ← funProp type then
           if (← isDefEq x proof) then
             continue
@@ -104,14 +99,14 @@ def synthesizeArgs (thmId : Origin) (xs : Array Expr) (bis : Array BinderInfo)
 
 
 /-- Try to apply theorem - core function -/
-def tryTheoremCore (xs : Array Expr) (bis : Array BinderInfo) (val : Expr) (type : Expr) (e : Expr)
+def tryTheoremCore (xs : Array Expr) (val : Expr) (type : Expr) (e : Expr)
     (thmId : Origin) (funProp : Expr → FunPropM (Option Result)) : FunPropM (Option Result) := do
   withTraceNode `Meta.Tactic.fun_prop
     (fun r => return s!"[{ExceptToEmoji.toEmoji r}] applying: {← ppOrigin' thmId}") do
 
   if (← isDefEq type e) then
 
-    if ¬(← synthesizeArgs thmId xs bis funProp) then
+    if ¬(← synthesizeArgs thmId xs funProp) then
       return none
     let proof ← instantiateMVars (mkAppN val xs)
 
@@ -129,17 +124,17 @@ def tryTheoremWithHint? (e : Expr) (thmOrigin : Origin)
   let go : FunPropM (Option Result) := do
     let thmProof ← thmOrigin.getValue
     let type ← inferType thmProof
-    let (xs, bis, type) ← forallMetaTelescope type
+    let (xs, _, type) ← forallMetaTelescope type
 
     for (i,x) in hint do
       try
         for (id,v) in hint do
           xs[id]!.mvarId!.assignIfDefeq v
       catch _ =>
-        trace[Meta.Tactic.fun_trans]
+        trace[Debug.Meta.Tactic.fun_prop]
           "failed to use hint {i} `{← ppExpr x} when applying theorem {← ppOrigin thmOrigin}"
 
-    tryTheoremCore xs bis thmProof type e thmOrigin funProp
+    tryTheoremCore xs thmProof type e thmOrigin funProp
 
   -- `simp` introduces new meta variable context depth for some reason
   -- This is probably to avoid mvar assignment when trying a theorem fails
@@ -199,7 +194,6 @@ def applyConstRule (funPropDecl : FunPropDecl) (e : Expr)
     logError msg
     trace[Meta.Tactic.fun_prop] msg
     return none
-
   for thm in thms do
     let .const := thm.thmArgs | return none
     if let .some r ← tryTheorem? e (.decl thm.thmName) funProp then
@@ -215,7 +209,6 @@ For example, `e = q(Continuous fun f => f x)` and `funPropDecl` is `FunPropDecl`
 def applyApplyRule (funPropDecl : FunPropDecl) (e : Expr)
     (funProp : Expr → FunPropM (Option Result)) : FunPropM (Option Result) := do
   let thms := (← getLambdaTheorems funPropDecl.funPropName .apply)
-
   for thm in thms do
     if let .some r ← tryTheoremWithHint? e (.decl thm.thmName) #[] funProp then
       return r
@@ -228,7 +221,7 @@ Try to prove `e` using *composition lambda theorem*.
 For example, `e = q(Continuous fun x => f (g x))` and `funPropDecl` is `FunPropDecl` for
 `Continuous`
 
-You also have to provide the functions `f` and `g`.  -/
+You also have to provide the functions `f` and `g`. -/
 def applyCompRule (funPropDecl : FunPropDecl) (e f g : Expr)
     (funProp : Expr → FunPropM (Option Result)) : FunPropM (Option Result) := do
 
@@ -346,7 +339,7 @@ def applyMorRules (funPropDecl : FunPropDecl) (e : Expr) (fData : FunctionData)
     trace[Debug.Meta.Tactic.fun_prop] "no theorem matched"
     return none
 
-/-- Prove function property of using *transition theorems*.  -/
+/-- Prove function property of using *transition theorems*. -/
 def applyTransitionRules (e : Expr) (funProp : Expr → FunPropM (Option Result)) :
     FunPropM (Option Result) := do
   withIncreasedTransitionDepth do
@@ -441,7 +434,8 @@ def getLocalTheorems (funPropDecl : FunPropDecl) (funOrigin : Origin)
       let .some (decl,f) ← getFunProp? b | return none
       unless decl.funPropName = funPropDecl.funPropName do return none
 
-      let .data fData ← getFunctionData? f (← unfoldNamePred) {zeta := false} | return none
+      let .data fData ← getFunctionData? f (← unfoldNamePred) {zeta := false, zetaDelta := false}
+        | return none
       unless (fData.getFnOrigin == funOrigin) do return none
 
       unless isOrderedSubsetOf mainArgs fData.mainArgs do return none
@@ -661,7 +655,7 @@ mutual
         let e' := e.setArg funPropDecl.funArgId b
         funProp (← mkLambdaFVars xs e')
 
-    match ← getFunctionData? f (← unfoldNamePred) {zeta := false} with
+    match ← getFunctionData? f (← unfoldNamePred) {zeta := false, zetaDelta := false} with
     | .letE f =>
       trace[Debug.Meta.Tactic.fun_prop] "let case on {← ppExpr f}"
       let e := e.setArg funPropDecl.funArgId f -- update e with reduced f
diff --git a/Mathlib/Tactic/FunProp/Differentiable.lean b/Mathlib/Tactic/FunProp/Differentiable.lean
index 5e20116dd53ac..1d81eec7d76f2 100644
--- a/Mathlib/Tactic/FunProp/Differentiable.lean
+++ b/Mathlib/Tactic/FunProp/Differentiable.lean
@@ -104,7 +104,6 @@ attribute [fun_prop]
   Differentiable.mul
   Differentiable.smul
   Differentiable.div
-  Differentiable.inv'
   Differentiable.inv
 
   DifferentiableAt.add
@@ -113,7 +112,6 @@ attribute [fun_prop]
   DifferentiableAt.mul
   DifferentiableAt.smul
   DifferentiableAt.div
-  DifferentiableAt.inv'
   DifferentiableAt.inv
 
   DifferentiableOn.add
@@ -122,7 +120,6 @@ attribute [fun_prop]
   DifferentiableOn.mul
   DifferentiableOn.smul
   DifferentiableOn.div
-  DifferentiableOn.inv'
   DifferentiableOn.inv
 
 
diff --git a/Mathlib/Tactic/FunProp/FunctionData.lean b/Mathlib/Tactic/FunProp/FunctionData.lean
index 88e61d9592bbd..04be35eced734 100644
--- a/Mathlib/Tactic/FunProp/FunctionData.lean
+++ b/Mathlib/Tactic/FunProp/FunctionData.lean
@@ -121,9 +121,9 @@ def getFunctionData? (f : Expr)
     (unfoldPred : Name → Bool := fun _ => false) (cfg : WhnfCoreConfig := {}) :
     MetaM MaybeFunctionData := do
 
-  let unfold := fun e : Expr =>
+  let unfold := fun e : Expr => do
     if let .some n := e.getAppFn'.constName? then
-      pure (unfoldPred n)
+      pure ((unfoldPred n) || (← isReducible n))
     else
       pure false
 
@@ -131,7 +131,7 @@ def getFunctionData? (f : Expr)
     | throwError m!"fun_prop bug: function expected, got `{f} : {← inferType f}, \
                     type ctor {(← inferType f).ctorName}"
   withLocalDeclD xName xType fun x => do
-    let fx' ← Mor.whnfPred (f.beta #[x]).eta unfold cfg
+    let fx' := (← Mor.whnfPred (f.beta #[x]).eta unfold cfg) |> headBetaThroughLet
     let f' ← mkLambdaFVars #[x] fx'
     match fx' with
     | .letE .. => return .letE f'
@@ -144,7 +144,7 @@ def FunctionData.unfoldHeadFVar? (fData : FunctionData) : MetaM (Option Expr) :=
   let .fvar id := fData.fn | return none
   let .some val ← id.getValue? | return none
   let f ← withLCtx fData.lctx fData.insts do
-    mkLambdaFVars #[fData.mainVar] (Mor.mkAppN val fData.args)
+    mkLambdaFVars #[fData.mainVar] (headBetaThroughLet (Mor.mkAppN val fData.args))
   return f
 
 /-- Type of morphism application. -/
diff --git a/Mathlib/Tactic/FunProp/Mor.lean b/Mathlib/Tactic/FunProp/Mor.lean
index 29a860cf1bdf2..73bf51afe47f0 100644
--- a/Mathlib/Tactic/FunProp/Mor.lean
+++ b/Mathlib/Tactic/FunProp/Mor.lean
@@ -89,7 +89,7 @@ Weak normal head form of an expression involving morphism applications.
 
 For example calling this on `coe (f a) b` will put `f` in weak normal head form instead of `coe`.
  -/
-def whnf (e : Expr)  (cfg : WhnfCoreConfig := {}) : MetaM Expr :=
+def whnf (e : Expr) (cfg : WhnfCoreConfig := {}) : MetaM Expr :=
   whnfPred e (fun _ => return false) cfg
 
 
@@ -120,6 +120,14 @@ where
         go f (as.push { coe := c, expr := x})
       else
         go (.app c f) (as.push { expr := x})
+    | .app (.proj n i f) x, as => do
+      -- convert proj back to function application
+      let env ← getEnv
+      let info := getStructureInfo? env n |>.get!
+      let projFn := getProjFnForField? env n (info.fieldNames[i]!) |>.get!
+      let .app c f ← mkAppM projFn #[f] | panic! "bug in Mor.withApp"
+
+      go (.app (.app c f) x) as
     | .app f a, as =>
       go f (as.push { expr := a })
     | f        , as => k f as.reverse
diff --git a/Mathlib/Tactic/FunProp/RefinedDiscrTree.lean b/Mathlib/Tactic/FunProp/RefinedDiscrTree.lean
index da93ffa29a9af..141a5bb819107 100644
--- a/Mathlib/Tactic/FunProp/RefinedDiscrTree.lean
+++ b/Mathlib/Tactic/FunProp/RefinedDiscrTree.lean
@@ -449,7 +449,7 @@ partial def reduce (e : Expr) (config : WhnfCoreConfig) : MetaM Expr := do
 /-- Repeatedly apply reduce while stripping lambda binders and introducing their variables -/
 @[specialize]
 partial def lambdaTelescopeReduce {m} [Monad m] [MonadLiftT MetaM m] [MonadControlT MetaM m]
-    [Inhabited α] (e : Expr) (fvars : List FVarId) (config : WhnfCoreConfig)
+    [Nonempty α] (e : Expr) (fvars : List FVarId) (config : WhnfCoreConfig)
     (k : Expr → List FVarId → m α) : m α := do
   match ← reduce e config with
   | .lam n d b bi =>
diff --git a/Mathlib/Tactic/FunProp/StateList.lean b/Mathlib/Tactic/FunProp/StateList.lean
index 57c9ac6bb2129..24bce5cf29fd1 100644
--- a/Mathlib/Tactic/FunProp/StateList.lean
+++ b/Mathlib/Tactic/FunProp/StateList.lean
@@ -2,7 +2,11 @@
 Copyright (c) 2023 J. W. Gerbscheid. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: J. W. Gerbscheid
+-/
+
+import Mathlib.Init
 
+/-!
 The combined state and list monad transformer.
 `StateListT σ α` is equivalent to `StateT σ (ListT α)` but more efficient.
 
@@ -21,7 +25,6 @@ in the do block are combined.
 
 -/
 
-import Mathlib.Init
 /-! StateList -/
 
 namespace Mathlib.Meta.FunProp
diff --git a/Mathlib/Tactic/FunProp/ToBatteries.lean b/Mathlib/Tactic/FunProp/ToBatteries.lean
index 0f46a33e449ee..11ce07dbc78c3 100644
--- a/Mathlib/Tactic/FunProp/ToBatteries.lean
+++ b/Mathlib/Tactic/FunProp/ToBatteries.lean
@@ -131,6 +131,43 @@ def etaExpand1 (f : Expr) : MetaM Expr := do
     withDefault do forallBoundedTelescope (← inferType f) (.some 1) fun xs _ => do
       mkLambdaFVars xs (mkAppN f xs)
 
+/-- Implementation of `betaThroughLet` -/
+private def betaThroughLetAux (f : Expr) (args : List Expr) : Expr :=
+  match f, args with
+  | f, [] => f
+  | .lam _ _ b _, a :: as => (betaThroughLetAux (b.instantiate1 a) as)
+  | .letE n t v b _, args => .letE n t v (betaThroughLetAux b args) false
+  | .mdata _ b, args => betaThroughLetAux b args
+  | f, args => mkAppN f args.toArray
+
+/-- Apply the given arguments to `f`, beta-reducing if `f` is a lambda expression. This variant
+does beta-reduction through let bindings without inlining them.
+
+Example
+```
+beta' (fun x => let y := x * x; fun z => x + y + z) #[a,b]
+==>
+let y := a * a; a + y + b
+```
+-/
+def betaThroughLet (f : Expr) (args : Array Expr) : Expr :=
+  betaThroughLetAux f args.toList
+
+/-- Beta reduces head of an expression, `(fun x => e) a` ==> `e[x/a]`. This version applies
+arguments through let bindings without inlining them.
+
+Example
+```
+headBeta' ((fun x => let y := x * x; fun z => x + y + z) a b)
+==>
+let y := a * a; a + y + b
+```
+-/
+def headBetaThroughLet (e : Expr) : Expr :=
+  let f := e.getAppFn
+  if f.isHeadBetaTargetFn true then betaThroughLet f e.getAppArgs else e
+
+
 end Meta.FunProp
 
 end Mathlib
diff --git a/Mathlib/Tactic/FunProp/Types.lean b/Mathlib/Tactic/FunProp/Types.lean
index e206436640a9e..6f12b788e1928 100644
--- a/Mathlib/Tactic/FunProp/Types.lean
+++ b/Mathlib/Tactic/FunProp/Types.lean
@@ -153,7 +153,7 @@ function property like continuity from another property like differentiability.
 The main reason is that if the user forgets to add a continuity theorem for function `foo` then
 `fun_prop` should report that there is a continuity theorem for `foo` missing. If we would log
 messages `transitionDepth > 0` then user will see messages saying that there is a missing theorem
-for differentiability, smoothness, ... for `foo`.  -/
+for differentiability, smoothness, ... for `foo`. -/
 def logError (msg : String) : FunPropM Unit := do
   if (← read).transitionDepth = 0 then
     modify fun s =>
diff --git a/Mathlib/Tactic/GCongr.lean b/Mathlib/Tactic/GCongr.lean
index dbee7afe65acd..ddb08c8e3e2dc 100644
--- a/Mathlib/Tactic/GCongr.lean
+++ b/Mathlib/Tactic/GCongr.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Heather Macbeth
 -/
 import Mathlib.Tactic.Positivity.Core
-import Mathlib.Tactic.GCongr.Core
+import Mathlib.Tactic.GCongr.CoreAttrs
 
 /-! # Setup for the `gcongr` tactic
 
diff --git a/Mathlib/Tactic/GCongr/Core.lean b/Mathlib/Tactic/GCongr/Core.lean
index 67a7dc94d4836..bf7982ef2748e 100644
--- a/Mathlib/Tactic/GCongr/Core.lean
+++ b/Mathlib/Tactic/GCongr/Core.lean
@@ -134,9 +134,9 @@ structure GCongrLemma where
 
 /-- Environment extension for "generalized congruence" (`gcongr`) lemmas. -/
 initialize gcongrExt : SimpleScopedEnvExtension ((Name × Name × Array Bool) × GCongrLemma)
-    (HashMap (Name × Name × Array Bool) (Array GCongrLemma)) ←
+    (Std.HashMap (Name × Name × Array Bool) (Array GCongrLemma)) ←
   registerSimpleScopedEnvExtension {
-    addEntry := fun m (n, lem) => m.insert n ((m.findD n #[]).push lem)
+    addEntry := fun m (n, lem) => m.insert n ((m.getD n #[]).push lem)
     initial := {}
   }
 
@@ -363,7 +363,7 @@ partial def _root_.Lean.MVarId.gcongr
   -- Look up the `@[gcongr]` lemmas whose conclusion has the same relation and head function as
   -- the goal and whether the boolean-array of varying/nonvarying arguments of such
   -- a lemma matches `varyingArgs`.
-  for lem in (gcongrExt.getState (← getEnv)).findD (relName, lhsHead, varyingArgs) #[] do
+  for lem in (gcongrExt.getState (← getEnv)).getD (relName, lhsHead, varyingArgs) #[] do
     let gs ← try
       -- Try `apply`-ing such a lemma to the goal.
       Except.ok <$> g.apply (← mkConstWithFreshMVarLevels lem.declName)
@@ -406,8 +406,13 @@ partial def _root_.Lean.MVarId.gcongr
       -- by the `apply`.
       for g in gs do
         if !(← g.isAssigned) && !subgoals.contains g then
-          try sideGoalDischarger g
-          catch _ => out := out.push g
+          let s ← saveState
+          try
+            let (_, g') ← g.intros
+            sideGoalDischarger g'
+          catch _ =>
+            s.restore
+            out := out.push g
       -- Return all unresolved subgoals, "main" or "side"
       return (true, names, out ++ subgoals)
   -- A. If there is no template, and there was no `@[gcongr]` lemma which matched the goal,
diff --git a/Mathlib/Tactic/GCongr/CoreAttrs.lean b/Mathlib/Tactic/GCongr/CoreAttrs.lean
new file mode 100644
index 0000000000000..423dbda223088
--- /dev/null
+++ b/Mathlib/Tactic/GCongr/CoreAttrs.lean
@@ -0,0 +1,16 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Tactic.GCongr.Core
+
+/-!
+# gcongr attributes for lemmas up in the import chain
+
+In this file we add `gcongr` attribute to lemmas in `Lean.Init`.
+We may add lemmas from other files imported by `Mathlib/Tactic/GCongr/Core` later.
+-/
+
+attribute [gcongr] List.Sublist.append List.Sublist.append_left List.Sublist.append_right
+  List.Sublist.reverse List.drop_sublist_drop_left List.Sublist.drop Nat.succ_le_succ
diff --git a/Mathlib/Tactic/Generalize.lean b/Mathlib/Tactic/Generalize.lean
index bcd97b1c34af3..20250ad5f6e41 100644
--- a/Mathlib/Tactic/Generalize.lean
+++ b/Mathlib/Tactic/Generalize.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2024 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Elab.Binders
diff --git a/Mathlib/Tactic/GeneralizeProofs.lean b/Mathlib/Tactic/GeneralizeProofs.lean
index 8bb825b5909f9..645637ae46a0f 100644
--- a/Mathlib/Tactic/GeneralizeProofs.lean
+++ b/Mathlib/Tactic/GeneralizeProofs.lean
@@ -109,10 +109,10 @@ def MGen.runMAbs {α : Type} (mx : MAbs α) : MGen (α × Array (Expr × Expr))
 Finds a proof of `prop` by looking at `propToFVar` and `propToProof`.
 -/
 def MAbs.findProof? (prop : Expr) : MAbs (Option Expr) := do
-  if let some pf := (← read).propToFVar.find? prop then
+  if let some pf := (← read).propToFVar[prop]? then
     return pf
   else
-    return (← get).propToProof.find? prop
+    return (← get).propToProof[prop]?
 
 /--
 Generalize `prop`, where `proof` is its proof.
@@ -152,7 +152,7 @@ def appArgExpectedTypes (f : Expr) (args : Array Expr) (ty? : Option Expr) :
     -- Try using the expected type, but (*) below might find a bad solution
     (guard ty?.isSome *> go f args ty?) <|> go f args none
 where
-  /-- Core implementation for `appArgExpectedTypes`.  -/
+  /-- Core implementation for `appArgExpectedTypes`. -/
   go (f : Expr) (args : Array Expr) (ty? : Option Expr) : MetaM (Array (Option Expr)) := do
     -- Metavariables for each argument to `f`:
     let mut margs := #[]
@@ -341,7 +341,7 @@ This continuation `k` is passed
 
 The `propToFVar` map is updated with the new proposition fvars.
 -/
-partial def withGeneralizedProofs {α : Type} [Inhabited α] (e : Expr) (ty? : Option Expr)
+partial def withGeneralizedProofs {α : Type} [Nonempty α] (e : Expr) (ty? : Option Expr)
     (k : Array Expr → Array Expr → Expr → MGen α) :
     MGen α := do
   let propToFVar := (← get).propToFVar
@@ -351,18 +351,18 @@ partial def withGeneralizedProofs {α : Type} [Inhabited α] (e : Expr) (ty? : O
     post-abstracted{indentD e}\nnew generalizations: {generalizations}"
   let rec
     /-- Core loop for `withGeneralizedProofs`, adds generalizations one at a time. -/
-    go [Inhabited α] (i : Nat) (fvars pfs : Array Expr)
+    go [Nonempty α] (i : Nat) (fvars pfs : Array Expr)
         (proofToFVar propToFVar : ExprMap Expr) : MGen α := do
       if h : i < generalizations.size then
         let (ty, pf) := generalizations[i]
-        let ty := (← instantiateMVars (ty.replace proofToFVar.find?)).cleanupAnnotations
+        let ty := (← instantiateMVars (ty.replace proofToFVar.get?)).cleanupAnnotations
         withLocalDeclD (← mkFreshUserName `pf) ty fun fvar => do
           go (i + 1) (fvars := fvars.push fvar) (pfs := pfs.push pf)
             (proofToFVar := proofToFVar.insert pf fvar)
             (propToFVar := propToFVar.insert ty fvar)
       else
         withNewLocalInstances fvars 0 do
-          let e' := e.replace proofToFVar.find?
+          let e' := e.replace proofToFVar.get?
           trace[Tactic.generalize_proofs] "after: e' = {e}"
           modify fun s => { s with propToFVar }
           k fvars pfs e'
@@ -395,7 +395,7 @@ where
           let g' ← mkFreshExprSyntheticOpaqueMVar tgt' tag
           g.assign <| .app g' tgt.letValue!
           return ← go g'.mvarId! i hs
-        if let some pf := (← get).propToFVar.find? ty then
+        if let some pf := (← get).propToFVar[ty]? then
           -- Eliminate this local hypothesis using the pre-existing proof, using proof irrelevance
           let tgt' := tgt.bindingBody!.instantiate1 pf
           let g' ← mkFreshExprSyntheticOpaqueMVar tgt' tag
diff --git a/Mathlib/Tactic/HelpCmd.lean b/Mathlib/Tactic/HelpCmd.lean
index 5587ad008455a..1dafab19653f5 100644
--- a/Mathlib/Tactic/HelpCmd.lean
+++ b/Mathlib/Tactic/HelpCmd.lean
@@ -69,7 +69,7 @@ private def elabHelpOption (id : Option Ident) : CommandElabM Unit := do
     | .ofInt val => s!"Int := {repr val}"
     | .ofSyntax val => s!"Syntax := {repr val}"
     if let some val := opts.find (.mkSimple name) then
-      msg1 := s!"{msg1}  (currently: {val})"
+      msg1 := s!"{msg1} (currently: {val})"
     msg := msg ++ .nest 2 (f!"option {name} : {msg1}" ++ .line ++ decl.descr) ++ .line ++ .line
   logInfo msg
 
diff --git a/Mathlib/Tactic/Hint.lean b/Mathlib/Tactic/Hint.lean
index 6194f2cbaaabb..8df08cd79a07e 100644
--- a/Mathlib/Tactic/Hint.lean
+++ b/Mathlib/Tactic/Hint.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Lean.Meta.Tactic.TryThis
 import Batteries.Linter.UnreachableTactic
diff --git a/Mathlib/Tactic/ITauto.lean b/Mathlib/Tactic/ITauto.lean
index 11fea885b2ddf..f0449d9828b14 100644
--- a/Mathlib/Tactic/ITauto.lean
+++ b/Mathlib/Tactic/ITauto.lean
@@ -3,13 +3,11 @@ Copyright (c) 2021 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
-import Batteries.Logic
 import Batteries.Tactic.Exact
 import Batteries.Tactic.Init
-import Mathlib.Tactic.Hint
+import Mathlib.Logic.Basic
 import Mathlib.Tactic.DeriveToExpr
 import Mathlib.Util.AtomM
-import Mathlib.Init.Logic
 import Qq
 
 /-!
diff --git a/Mathlib/Tactic/IntervalCases.lean b/Mathlib/Tactic/IntervalCases.lean
index 621a718ffea2c..37e90dfc742f4 100644
--- a/Mathlib/Tactic/IntervalCases.lean
+++ b/Mathlib/Tactic/IntervalCases.lean
@@ -1,10 +1,11 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Mario Carneiro
+Authors: Kim Morrison, Mario Carneiro
 -/
 import Mathlib.Tactic.NormNum
 import Mathlib.Tactic.FinCases
+import Mathlib.Control.Basic
 
 /-!
 # Case bash on variables in finite intervals
diff --git a/Mathlib/Tactic/IrreducibleDef.lean b/Mathlib/Tactic/IrreducibleDef.lean
index 2d3987da6409c..bfdf7272c85bd 100644
--- a/Mathlib/Tactic/IrreducibleDef.lean
+++ b/Mathlib/Tactic/IrreducibleDef.lean
@@ -50,7 +50,7 @@ local elab "eta_helper " t:term : term => do
     let lhs := (mkAppN lhs xs).headBeta
     mkForallFVars xs <|← mkEq lhs rhs
 
-/-- `val_proj x` elabs to the *primitive projection* `@x.val`.  -/
+/-- `val_proj x` elabs to the *primitive projection* `@x.val`. -/
 local elab "val_proj " e:term : term => do
   let e ← elabTerm (← `(($e : Subtype _))) none
   return mkProj ``Subtype 0 e
diff --git a/Mathlib/Tactic/Lemma.lean b/Mathlib/Tactic/Lemma.lean
index 901874a9f585a..10d49cd8d8a8f 100644
--- a/Mathlib/Tactic/Lemma.lean
+++ b/Mathlib/Tactic/Lemma.lean
@@ -12,8 +12,9 @@ import Lean.Parser.Command
 
 open Lean
 
+-- higher priority to override the one in Batteries
 /-- `lemma` means the same as `theorem`. It is used to denote "less important" theorems -/
-syntax (name := lemma) declModifiers
+syntax (name := lemma) (priority := default + 1) declModifiers
   group("lemma " declId ppIndent(declSig) declVal) : command
 
 /-- Implementation of the `lemma` command, by macro expansion to `theorem`. -/
diff --git a/Mathlib/Tactic/Linarith/Datatypes.lean b/Mathlib/Tactic/Linarith/Datatypes.lean
index 5e0832bb5836a..4b939583e90ec 100644
--- a/Mathlib/Tactic/Linarith/Datatypes.lean
+++ b/Mathlib/Tactic/Linarith/Datatypes.lean
@@ -311,7 +311,7 @@ structure CertificateOracle : Type where
   `hyps` by eliminating all variables ≤ `max_var`.
   If successful, it returns a map `coeff : Nat → Nat` as a certificate.
   This map represents that we can find a contradiction by taking the sum `∑ (coeff i) * hyps[i]`. -/
-  produceCertificate (hyps : List Comp) (max_var : Nat) : MetaM (Batteries.HashMap Nat Nat)
+  produceCertificate (hyps : List Comp) (max_var : Nat) : MetaM (Std.HashMap Nat Nat)
 
 /-!
 ### Auxiliary functions
diff --git a/Mathlib/Tactic/Linarith/Frontend.lean b/Mathlib/Tactic/Linarith/Frontend.lean
index 2143364d3f3ea..f23bfb3fa0d6e 100644
--- a/Mathlib/Tactic/Linarith/Frontend.lean
+++ b/Mathlib/Tactic/Linarith/Frontend.lean
@@ -92,7 +92,7 @@ disequality hypotheses, since this would lead to a number of runs exponential in
 disequalities in the context.
 
 The oracle is very modular. It can easily be replaced with another function of type
-`List Comp → ℕ → MetaM ((Batteries.HashMap ℕ ℕ))`,
+`List Comp → ℕ → MetaM ((Std.HashMap ℕ ℕ))`,
 which takes a list of comparisons and the largest variable
 index appearing in those comparisons, and returns a map from comparison indices to coefficients.
 An alternate oracle can be specified in the `LinarithConfig` object.
diff --git a/Mathlib/Tactic/Linarith/Oracle/FourierMotzkin.lean b/Mathlib/Tactic/Linarith/Oracle/FourierMotzkin.lean
index e65a8f3eba4cd..d1b3479c6c896 100644
--- a/Mathlib/Tactic/Linarith/Oracle/FourierMotzkin.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/FourierMotzkin.lean
@@ -3,8 +3,9 @@ Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Robert Y. Lewis
 -/
+import Mathlib.Std.Data.HashMap
+import Batteries.Lean.HashMap
 import Mathlib.Tactic.Linarith.Datatypes
-import Batteries.Data.HashMap.WF
 
 /-!
 # The Fourier-Motzkin elimination procedure
@@ -61,8 +62,8 @@ For example, suppose `cs` is produced by scaling assumption 2 by 5,
 and adding to that the sum of assumptions 1 and 2.
 `cs.flatten` maps `1 ↦ 1, 2 ↦ 6`.
  -/
-def CompSource.flatten : CompSource → HashMap Nat Nat
-  | (CompSource.assump n) => HashMap.empty.insert n 1
+def CompSource.flatten : CompSource → Std.HashMap Nat Nat
+  | (CompSource.assump n) => Std.HashMap.empty.insert n 1
   | (CompSource.add c1 c2) =>
       (CompSource.flatten c1).mergeWith (fun _ b b' => b + b') (CompSource.flatten c2)
   | (CompSource.scale n c) => (CompSource.flatten c).mapVal (fun _ v => v * n)
@@ -258,7 +259,7 @@ The linarith monad extends an exceptional monad with a `LinarithData` state.
 An exception produces a contradictory `PComp`.
 -/
 abbrev LinarithM : Type → Type :=
-  StateT LinarithData (ExceptT PComp Id)
+  StateT LinarithData (ExceptT PComp Lean.Core.CoreM)
 
 /-- Returns the current max variable. -/
 def getMaxVar : LinarithM ℕ :=
@@ -272,7 +273,7 @@ def getPCompSet : LinarithM PCompSet :=
 def validate : LinarithM Unit := do
   match (← getPCompSet).toList.find? (fun p : PComp => p.isContr) with
   | none => return ()
-  | some c => throw c
+  | some c => throwThe _ c
 
 /--
 Updates the current state with a new max variable and comparisons,
@@ -304,9 +305,12 @@ from the `linarith` state.
 -/
 def elimVarM (a : ℕ) : LinarithM Unit := do
   let vs ← getMaxVar
-  if (a ≤ vs) then (do
+  if (a ≤ vs) then
+    Lean.Core.checkSystem decl_name%.toString
     let ⟨pos, neg, notPresent⟩ := splitSetByVarSign a (← getPCompSet)
-    update (vs - 1) (pos.foldl (fun s p => s.union (elimWithSet a p neg)) notPresent))
+    update (vs - 1) (← pos.foldlM (fun s p => do
+      Lean.Core.checkSystem decl_name%.toString
+      pure (s.union (elimWithSet a p neg))) notPresent)
   else
     pure ()
 
@@ -327,9 +331,12 @@ def mkLinarithData (hyps : List Comp) (maxVar : ℕ) : LinarithData :=
 
 /-- An oracle that uses Fourier-Motzkin elimination. -/
 def CertificateOracle.fourierMotzkin : CertificateOracle where
-  produceCertificate hyps maxVar := match ExceptT.run
-      (StateT.run (do validate; elimAllVarsM : LinarithM Unit) (mkLinarithData hyps maxVar)) with
-  | (Except.ok _) => failure
-  | (Except.error contr) => return contr.src.flatten
+  produceCertificate hyps maxVar :=  do
+    let linarithData := mkLinarithData hyps maxVar
+    let result ←
+      (ExceptT.run (StateT.run (do validate; elimAllVarsM : LinarithM Unit) linarithData) : _)
+    match result with
+    | (Except.ok _) => failure
+    | (Except.error contr) => return contr.src.flatten
 
 end Linarith
diff --git a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm.lean b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm.lean
index e582844d49f0a..daeddd75376e8 100644
--- a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm.lean
@@ -14,8 +14,6 @@ The algorithm's entry point is the function `Linarith.SimplexAlgorithm.findPosit
 See the file `PositiveVector.lean` for details of how the procedure works.
 -/
 
-open Batteries
-
 namespace Linarith.SimplexAlgorithm
 
 /-- Preprocess the goal to pass it to `Linarith.SimplexAlgorithm.findPositiveVector`. -/
@@ -30,11 +28,10 @@ def preprocess (matType : ℕ → ℕ → Type) [UsableInSimplexAlgorithm matTyp
 /--
 Extract the certificate from the `vec` found by `Linarith.SimplexAlgorithm.findPositiveVector`.
 -/
-def postprocess (vec : Array ℚ) : HashMap ℕ ℕ :=
+def postprocess (vec : Array ℚ) : Std.HashMap ℕ ℕ :=
   let common_den : ℕ := vec.foldl (fun acc item => acc.lcm item.den) 1
   let vecNat : Array ℕ := vec.map (fun x : ℚ => (x * common_den).floor.toNat)
-  HashMap.ofList <| vecNat.toList.enum.filter (fun ⟨_, item⟩ => item != 0)
-
+  Std.HashMap.empty.insertMany <| vecNat.toList.enum.filter (fun ⟨_, item⟩ => item != 0)
 
 end SimplexAlgorithm
 
diff --git a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Datatypes.lean b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Datatypes.lean
index 986bb5ed40031..8a5a9488ae1a9 100644
--- a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Datatypes.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Datatypes.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Vasily Nesterov
 -/
 import Mathlib.Init
-import Lean.Data.HashMap
 import Batteries.Data.Rat.Basic
+import Std.Data.HashMap.Basic
 
 /-!
 # Datatypes for the Simplex Algorithm implementation
@@ -82,10 +82,10 @@ values.
 -/
 structure SparseMatrix (n m : Nat) where
   /-- The content of the matrix. -/
-  data : Array <| Lean.HashMap Nat Rat
+  data : Array <| Std.HashMap Nat Rat
 
 instance : UsableInSimplexAlgorithm SparseMatrix where
-  getElem mat i j := mat.data[i]!.findD j 0
+  getElem mat i j := mat.data[i]!.getD j 0
   setElem mat i j v :=
     if v == 0 then
       ⟨mat.data.modify i fun row => row.erase j⟩
@@ -96,7 +96,7 @@ instance : UsableInSimplexAlgorithm SparseMatrix where
       let rowVals := row.toList.map fun (j, v) => (i, j, v)
       rowVals ++ acc
   ofValues {n _ : Nat} vals := Id.run do
-    let mut data : Array (Lean.HashMap Nat Rat) := Array.mkArray n .empty
+    let mut data : Array (Std.HashMap Nat Rat) := Array.mkArray n .empty
     for ⟨i, j, v⟩ in vals do
       if v != 0 then
         data := data.modify i fun row => row.insert j v
@@ -105,12 +105,12 @@ instance : UsableInSimplexAlgorithm SparseMatrix where
   subtractRow mat i j coef :=
     let newData := mat.data.modify j fun row =>
       mat.data[i]!.fold (fun cur k val =>
-        let newVal := (cur.findD k 0) - coef * val
+        let newVal := (cur.getD k 0) - coef * val
         if newVal != 0 then cur.insert k newVal else cur.erase k
       ) row
     ⟨newData⟩
   divideRow mat i coef :=
-    let newData : Array (Lean.HashMap Nat Rat) := mat.data.modify i fun row =>
+    let newData : Array (Std.HashMap Nat Rat) := mat.data.modify i fun row =>
       row.fold (fun cur k v => cur.insert k (v / coef)) row
     ⟨newData⟩
 
diff --git a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Gauss.lean b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Gauss.lean
index ab46c5726b44f..93cad0a43aa77 100644
--- a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Gauss.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/Gauss.lean
@@ -15,7 +15,7 @@ solution which is done by standard Gaussian Elimination algorithm implemented in
 namespace Linarith.SimplexAlgorithm.Gauss
 
 /-- The monad for the Gaussian Elimination algorithm. -/
-abbrev GaussM (n m : Nat) (matType : Nat → Nat → Type) := StateM <| matType n m
+abbrev GaussM (n m : Nat) (matType : Nat → Nat → Type) := StateT (matType n m) Lean.CoreM
 
 variable {n m : Nat} {matType : Nat → Nat → Type} [UsableInSimplexAlgorithm matType]
 
@@ -35,6 +35,7 @@ def getTableauImp : GaussM n m matType <| Tableau matType := do
   let mut col : Nat := 0
 
   while row < n && col < m do
+    Lean.Core.checkSystem decl_name%.toString
     match ← findNonzeroRow row col with
     | .none =>
       free := free.push col
@@ -74,7 +75,7 @@ Given matrix `A`, solves the linear equation `A x = 0` and returns the solution
 some variables are free and others (basic) variable are expressed as linear combinations of the free
 ones.
 -/
-def getTableau (A : matType n m) : Tableau matType := Id.run do
+def getTableau (A : matType n m) : Lean.CoreM (Tableau matType) := do
   return (← getTableauImp.run A).fst
 
 end Linarith.SimplexAlgorithm.Gauss
diff --git a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/PositiveVector.lean b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/PositiveVector.lean
index c93e2e33f9b24..97f8d6d475622 100644
--- a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/PositiveVector.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/PositiveVector.lean
@@ -90,10 +90,10 @@ def findPositiveVector {n m : Nat} {matType : Nat → Nat → Type} [UsableInSim
 
   /- Using Gaussian elimination split variable into free and basic forming the tableau that will be
   operated by the Simplex Algorithm. -/
-  let initTableau := Gauss.getTableau B
+  let initTableau ← Gauss.getTableau B
 
   /- Run the Simplex Algorithm and extract the solution. -/
-  let res := runSimplexAlgorithm.run initTableau
+  let res ← runSimplexAlgorithm.run initTableau
   if res.fst.isOk then
     return extractSolution res.snd
   else
diff --git a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/SimplexAlgorithm.lean b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/SimplexAlgorithm.lean
index 924814fa69fce..9858e94c23f14 100644
--- a/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/SimplexAlgorithm.lean
+++ b/Mathlib/Tactic/Linarith/Oracle/SimplexAlgorithm/SimplexAlgorithm.lean
@@ -21,7 +21,7 @@ inductive SimplexAlgorithmException
 
 /-- The monad for the Simplex Algorithm. -/
 abbrev SimplexAlgorithmM (matType : Nat → Nat → Type) [UsableInSimplexAlgorithm matType] :=
-  ExceptT SimplexAlgorithmException <| StateM (Tableau matType)
+  ExceptT SimplexAlgorithmException <| StateT (Tableau matType) Lean.CoreM
 
 variable {matType : Nat → Nat → Type} [UsableInSimplexAlgorithm matType]
 
@@ -77,7 +77,7 @@ def chooseEnteringVar : SimplexAlgorithmM matType Nat := do
 
   /- If there is no such variable the solution does not exist for sure. -/
   match enterIdxOpt with
-  | .none => throw SimplexAlgorithmException.infeasible
+  | .none => throwThe SimplexAlgorithmException SimplexAlgorithmException.infeasible
   | .some enterIdx => return enterIdx
 
 /--
@@ -116,6 +116,7 @@ such exists.
 -/
 def runSimplexAlgorithm : SimplexAlgorithmM matType Unit := do
   while !(← checkSuccess) do
+    Lean.Core.checkSystem decl_name%.toString
     let ⟨exitIdx, enterIdx⟩ ← choosePivots
     doPivotOperation exitIdx enterIdx
 
diff --git a/Mathlib/Tactic/Linarith/Preprocessing.lean b/Mathlib/Tactic/Linarith/Preprocessing.lean
index 8d0a1bbd44edd..2a496d4cd5670 100644
--- a/Mathlib/Tactic/Linarith/Preprocessing.lean
+++ b/Mathlib/Tactic/Linarith/Preprocessing.lean
@@ -306,7 +306,7 @@ section nlinarith
 `findSquares s e` collects all terms of the form `a ^ 2` and `a * a` that appear in `e`
 and adds them to the set `s`.
 A pair `(i, true)` is added to `s` when `atoms[i]^2` appears in `e`,
-and `(i, false)` is added to `s` when `atoms[i]*atoms[i]` appears in `e`.  -/
+and `(i, false)` is added to `s` when `atoms[i]*atoms[i]` appears in `e`. -/
 partial def findSquares (s : RBSet (Nat × Bool) lexOrd.compare) (e : Expr) :
     AtomM (RBSet (Nat × Bool) lexOrd.compare) :=
   -- Completely traversing the expression is non-ideal,
diff --git a/Mathlib/Tactic/Linarith/Verification.lean b/Mathlib/Tactic/Linarith/Verification.lean
index 2439b1d8def02..b8702e24fc4f5 100644
--- a/Mathlib/Tactic/Linarith/Verification.lean
+++ b/Mathlib/Tactic/Linarith/Verification.lean
@@ -191,6 +191,7 @@ def proveFalseByLinarith (transparency : TransparencyMode) (oracle : Certificate
   | _, [] => throwError "no args to linarith"
   | g, l@(h::_) => do
       trace[linarith.detail] "Beginning work in `proveFalseByLinarith`."
+      Lean.Core.checkSystem decl_name%.toString
       -- for the elimination to work properly, we must add a proof of `-1 < 0` to the list,
       -- along with negated equality proofs.
       let l' ← addNegEqProofs l
@@ -202,7 +203,7 @@ def proveFalseByLinarith (transparency : TransparencyMode) (oracle : Certificate
       trace[linarith.detail] "... finished `linearFormsAndMaxVar`."
       trace[linarith.detail] "{comps}"
       -- perform the elimination and fail if no contradiction is found.
-      let certificate : Batteries.HashMap Nat Nat ← try
+      let certificate : Std.HashMap Nat Nat ← try
         oracle.produceCertificate comps max_var
       catch e =>
         trace[linarith] e.toMessageData
@@ -210,7 +211,7 @@ def proveFalseByLinarith (transparency : TransparencyMode) (oracle : Certificate
       trace[linarith] "linarith has found a contradiction: {certificate.toList}"
       let enum_inputs := inputs.enum
       -- construct a list pairing nonzero coeffs with the proof of their corresponding comparison
-      let zip := enum_inputs.filterMap fun ⟨n, e⟩ => (certificate.find? n).map (e, ·)
+      let zip := enum_inputs.filterMap fun ⟨n, e⟩ => (certificate[n]?).map (e, ·)
       let mls ← zip.mapM fun ⟨e, n⟩ => do mulExpr n (← leftOfIneqProof e)
       -- `sm` is the sum of input terms, scaled to cancel out all variables.
       let sm ← addExprs mls
diff --git a/Mathlib/Tactic/LinearCombination'.lean b/Mathlib/Tactic/LinearCombination'.lean
index e42ba57d7608c..c7965a7998ee9 100644
--- a/Mathlib/Tactic/LinearCombination'.lean
+++ b/Mathlib/Tactic/LinearCombination'.lean
@@ -174,12 +174,17 @@ syntax expStx := atomic(" (" &"exp" " := ") withoutPosition(num) ")"
   of a list of equalities and subtracting it from the target.
   The tactic will create a linear
   combination by adding the equalities together from left to right, so the order
-  of the input hypotheses does matter.  If the `normalize` field of the
-  configuration is set to false, then the tactic will simply set the user up to
+  of the input hypotheses does matter.  If the `norm` field of the
+  tactic is set to `skip`, then the tactic will simply set the user up to
   prove their target using the linear combination instead of normalizing the subtraction.
 
 Note: There is also a similar tactic `linear_combination` (no prime); this version is
-provided for backward compatibility.
+provided for backward compatibility.  Compared to this tactic, `linear_combination`:
+* drops the `←` syntax for reversing an equation, instead offering this operation using the `-`
+  syntax
+* does not support multiplication of two hypotheses (`h1 * h2`), division by a hypothesis (`3 / h`),
+  or inversion of a hypothesis (`h⁻¹`)
+* produces noisy output when the user adds or subtracts a constant to a hypothesis (`h + 3`)
 
 Note: The left and right sides of all the equalities should have the same
   type, and the coefficients should also have this type.  There must be
diff --git a/Mathlib/Tactic/LinearCombination.lean b/Mathlib/Tactic/LinearCombination.lean
index 532074909c984..ae81750d27b27 100644
--- a/Mathlib/Tactic/LinearCombination.lean
+++ b/Mathlib/Tactic/LinearCombination.lean
@@ -3,6 +3,7 @@ Copyright (c) 2022 Abby J. Goldberg. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Abby J. Goldberg, Mario Carneiro
 -/
+import Mathlib.Tactic.LinearCombination.Lemmas
 import Mathlib.Tactic.Ring
 
 /-!
@@ -33,23 +34,6 @@ namespace Mathlib.Tactic.LinearCombination
 open Lean hiding Rat
 open Elab Meta Term
 
-variable {α : Type*} {a a' a₁ a₂ b b' b₁ b₂ c : α}
-
-theorem pf_add_c [Add α] (p : a = b) (c : α) : a + c = b + c := p ▸ rfl
-theorem c_add_pf [Add α] (p : b = c) (a : α) : a + b = a + c := p ▸ rfl
-theorem add_pf [Add α] (p₁ : (a₁:α) = b₁) (p₂ : a₂ = b₂) : a₁ + a₂ = b₁ + b₂ := p₁ ▸ p₂ ▸ rfl
-theorem pf_sub_c [Sub α] (p : a = b) (c : α) : a - c = b - c := p ▸ rfl
-theorem c_sub_pf [Sub α] (p : b = c) (a : α) : a - b = a - c := p ▸ rfl
-theorem sub_pf [Sub α] (p₁ : (a₁:α) = b₁) (p₂ : a₂ = b₂) : a₁ - a₂ = b₁ - b₂ := p₁ ▸ p₂ ▸ rfl
-theorem neg_pf [Neg α] (p : (a:α) = b) : -a = -b := p ▸ rfl
-theorem pf_mul_c [Mul α] (p : a = b) (c : α) : a * c = b * c := p ▸ rfl
-theorem c_mul_pf [Mul α] (p : b = c) (a : α) : a * b = a * c := p ▸ rfl
-theorem mul_pf [Mul α] (p₁ : (a₁:α) = b₁) (p₂ : a₂ = b₂) : a₁ * a₂ = b₁ * b₂ := p₁ ▸ p₂ ▸ rfl
-theorem inv_pf [Inv α] (p : (a:α) = b) : a⁻¹ = b⁻¹ := p ▸ rfl
-theorem pf_div_c [Div α] (p : a = b) (c : α) : a / c = b / c := p ▸ rfl
-theorem c_div_pf [Div α] (p : b = c) (a : α) : a / b = a / c := p ▸ rfl
-theorem div_pf [Div α] (p₁ : (a₁:α) = b₁) (p₂ : a₂ = b₂) : a₁ / a₂ = b₁ / b₂ := p₁ ▸ p₂ ▸ rfl
-
 /-- Result of `expandLinearCombo`, either an equality proof or a value. -/
 inductive Expanded
   /-- A proof of `a = b`. -/
@@ -71,39 +55,40 @@ partial def expandLinearCombo (ty : Expr) (stx : Syntax.Term) : TermElabM Expand
   | `($e₁ + $e₂) => do
     match ← expandLinearCombo ty e₁, ← expandLinearCombo ty e₂ with
     | .const c₁, .const c₂ => .const <$> ``($c₁ + $c₂)
-    | .proof p₁, .const c₂ => .proof <$> ``(pf_add_c $p₁ $c₂)
-    | .const c₁, .proof p₂ => .proof <$> ``(c_add_pf $p₂ $c₁)
     | .proof p₁, .proof p₂ => .proof <$> ``(add_pf $p₁ $p₂)
+    | .proof p, .const c | .const c, .proof p =>
+      logWarningAt c "this constant has no effect on the linear combination; it can be dropped \
+        from the term"
+      pure (.proof p)
   | `($e₁ - $e₂) => do
     match ← expandLinearCombo ty e₁, ← expandLinearCombo ty e₂ with
     | .const c₁, .const c₂ => .const <$> ``($c₁ - $c₂)
-    | .proof p₁, .const c₂ => .proof <$> ``(pf_sub_c $p₁ $c₂)
-    | .const c₁, .proof p₂ => .proof <$> ``(c_sub_pf $p₂ $c₁)
-    | .proof p₁, .proof p₂ => .proof <$> ``(sub_pf $p₁ $p₂)
+    | .proof p, .const c =>
+      logWarningAt c "this constant has no effect on the linear combination; it can be dropped \
+        from the term"
+      pure (.proof p)
+    | .const c, .proof p =>
+      logWarningAt c "this constant has no effect on the linear combination; it can be dropped \
+        from the term"
+      .proof <$> ``(Eq.symm $p)
+    | .proof p₁, .proof p₂ => .proof <$> ``(add_pf $p₁ (Eq.symm $p₂))
   | `(-$e) => do
     match ← expandLinearCombo ty e with
     | .const c => .const <$> `(-$c)
-    | .proof p => .proof <$> ``(neg_pf $p)
-  | `(← $e) => do
-    match ← expandLinearCombo ty e with
-    | .const c => return .const c
     | .proof p => .proof <$> ``(Eq.symm $p)
-  | `($e₁ * $e₂) => do
+  | `($e₁ *%$tk $e₂) => do
     match ← expandLinearCombo ty e₁, ← expandLinearCombo ty e₂ with
     | .const c₁, .const c₂ => .const <$> ``($c₁ * $c₂)
     | .proof p₁, .const c₂ => .proof <$> ``(pf_mul_c $p₁ $c₂)
     | .const c₁, .proof p₂ => .proof <$> ``(c_mul_pf $p₂ $c₁)
-    | .proof p₁, .proof p₂ => .proof <$> ``(mul_pf $p₁ $p₂)
-  | `($e⁻¹) => do
-    match ← expandLinearCombo ty e with
-    | .const c => .const <$> `($c⁻¹)
-    | .proof p => .proof <$> ``(inv_pf $p)
-  | `($e₁ / $e₂) => do
+    | .proof _, .proof _ =>
+      throwErrorAt tk "'linear_combination' supports only linear operations"
+  | `($e₁ /%$tk $e₂) => do
     match ← expandLinearCombo ty e₁, ← expandLinearCombo ty e₂ with
     | .const c₁, .const c₂ => .const <$> ``($c₁ / $c₂)
     | .proof p₁, .const c₂ => .proof <$> ``(pf_div_c $p₁ $c₂)
-    | .const c₁, .proof p₂ => .proof <$> ``(c_div_pf $p₂ $c₁)
-    | .proof p₁, .proof p₂ => .proof <$> ``(div_pf $p₁ $p₂)
+    | _, .proof _ =>
+      throwErrorAt tk "'linear_combination' supports only linear operations"
   | e =>
     -- We have the expected type from the goal, so we can fully synthesize this leaf node.
     withSynthesize do
@@ -115,40 +100,39 @@ partial def expandLinearCombo (ty : Expr) (stx : Syntax.Term) : TermElabM Expand
       else
         .const <$> c.toSyntax
 
-theorem eq_trans₃ (p : (a:α) = b) (p₁ : a = a') (p₂ : b = b') : a' = b' := p₁ ▸ p₂ ▸ p
-
-theorem eq_of_add [AddGroup α] (p : (a:α) = b) (H : (a' - b') - (a - b) = 0) : a' = b' := by
-  rw [← sub_eq_zero] at p ⊢; rwa [sub_eq_zero, p] at H
-
-theorem eq_of_add_pow [Ring α] [NoZeroDivisors α] (n : ℕ) (p : (a:α) = b)
-    (H : (a' - b')^n - (a - b) = 0) : a' = b' := by
-  rw [← sub_eq_zero] at p ⊢; apply pow_eq_zero (n := n); rwa [sub_eq_zero, p] at H
-
-/-- Implementation of `linear_combination` and `linear_combination2`. -/
+/-- Implementation of `linear_combination`. -/
 def elabLinearCombination (tk : Syntax)
-    (norm? : Option Syntax.Tactic) (exp? : Option Syntax.NumLit) (input : Option Syntax.Term)
-    (twoGoals := false) : Tactic.TacticM Unit := Tactic.withMainContext do
+    (norm? : Option Syntax.Tactic) (exp? : Option Syntax.NumLit) (input : Option Syntax.Term) :
+    Tactic.TacticM Unit := Tactic.withMainContext do
   let some (ty, _) := (← (← Tactic.getMainGoal).getType').eq? |
     throwError "'linear_combination' only proves equalities"
   let p ← match input with
   | none => `(Eq.refl 0)
   | some e =>
     match ← expandLinearCombo ty e with
-    | .const c => `(Eq.refl $c)
+    | .const c =>
+      logWarningAt c "this constant has no effect on the linear combination; it can be dropped \
+        from the term"
+      `(Eq.refl 0)
     | .proof p => pure p
   let norm := norm?.getD (Unhygienic.run <| withRef tk `(tactic| ring1))
+  let lem : Ident ← mkIdent <$> do
+    try
+      -- if we are in a "true" ring, with well-behaved negation, it is better to present the
+      -- normalization tactic with a goal of the form `[stuff] = 0`, because this gives more useful
+      -- error messages on failure
+      let _ ← synthInstance (← mkAppM ``Neg #[ty])
+      pure ``eq_of_sub
+    catch _ =>
+      -- but otherwise (for example over `ℕ` or `ℝ≥0`) we can solve the problem by presenting the
+      -- normalization tactic with a goal of the form `[stuff] = [stuff]`
+      pure ``eq_of_add
   Term.withoutErrToSorry <| Tactic.evalTactic <| ← withFreshMacroScope <|
-  if twoGoals then
-    `(tactic| (
-      refine eq_trans₃ $p ?a ?b
-      case' a => $norm:tactic
-      case' b => $norm:tactic))
-  else
-    match exp? with
-    | some n =>
-      if n.getNat = 1 then `(tactic| (refine eq_of_add $p ?a; case' a => $norm:tactic))
-      else `(tactic| (refine eq_of_add_pow $n $p ?a; case' a => $norm:tactic))
-    | _ => `(tactic| (refine eq_of_add $p ?a; case' a => $norm:tactic))
+  match exp? with
+  | some n =>
+    if n.getNat = 1 then `(tactic| (refine $lem $p ?a; case' a => $norm:tactic))
+    else `(tactic| (refine eq_of_add_pow $n $p ?a; case' a => $norm:tactic))
+  | _ => `(tactic| (refine $lem $p ?a; case' a => $norm:tactic))
 
 /--
 The `(norm := $tac)` syntax says to use `tac` as a normalization postprocessor for
@@ -168,13 +152,17 @@ syntax expStx := atomic(" (" &"exp" " := ") withoutPosition(num) ")"
   of a list of equalities and subtracting it from the target.
   The tactic will create a linear
   combination by adding the equalities together from left to right, so the order
-  of the input hypotheses does matter.  If the `normalize` field of the
-  configuration is set to false, then the tactic will simply set the user up to
+  of the input hypotheses does matter.  If the `norm` field of the
+  tactic is set to `skip`, then the tactic will simply set the user up to
   prove their target using the linear combination instead of normalizing the subtraction.
 
-Note: The left and right sides of all the equalities should have the same
-  type, and the coefficients should also have this type.  There must be
-  instances of `Mul` and `AddGroup` for this type.
+Note: The left and right sides of all the equalities should have the same type `α`, and the
+coefficients should also have type `α`.  For full functionality `α` should be a commutative ring --
+strictly speaking, a commutative semiring with "cancellative" addition (in the semiring case,
+negation and subtraction will be handled "formally" as if operating in the enveloping ring). If a
+nonstandard normalization is used (for example `abel` or `skip`), the tactic will work over types
+`α` with less algebraic structure: the minimum is instances of `[Add α] [IsRightCancelAdd α]`
+together with instances of whatever operations are used in the tactic call.
 
 * The input `e` in `linear_combination e` is a linear combination of proofs of equalities,
   given as a sum/difference of coefficients multiplied by expressions.
@@ -190,17 +178,6 @@ Note: The left and right sides of all the equalities should have the same
 * `linear_combination (exp := n) e` will take the goal to the `n`th power before subtracting the
   combination `e`. In other words, if the goal is `t1 = t2`, `linear_combination (exp := n) e`
   will change the goal to `(t1 - t2)^n = 0` before proceeding as above.
-  This feature is not supported for `linear_combination2`.
-* `linear_combination2 e` is the same as `linear_combination e` but it produces two
-  subgoals instead of one: rather than proving that `(a - b) - (a' - b') = 0` where
-  `a' = b'` is the linear combination from `e` and `a = b` is the goal,
-  it instead attempts to prove `a = a'` and `b = b'`.
-  Because it does not use subtraction, this form is applicable also to semirings.
-  * Note that a goal which is provable by `linear_combination e` may not be provable
-    by `linear_combination2 e`; in general you may need to add a coefficient to `e`
-    to make both sides match, as in `linear_combination2 e + c`.
-  * You can also reverse equalities using `← h`, so for example if `h₁ : a = b`
-    then `2 * (← h)` is a proof of `2 * b = 2 * a`.
 
 Example Usage:
 ```
@@ -240,10 +217,4 @@ elab_rules : tactic
   | `(tactic| linear_combination%$tk $[(norm := $tac)]? $[(exp := $n)]? $(e)?) =>
     elabLinearCombination tk tac n e
 
-@[inherit_doc linearCombination]
-syntax "linear_combination2" (normStx)? (ppSpace colGt term)? : tactic
-elab_rules : tactic
-  | `(tactic| linear_combination2%$tk $[(norm := $tac)]? $(e)?) =>
-    elabLinearCombination tk tac none e true
-
 end Mathlib.Tactic.LinearCombination
diff --git a/Mathlib/Tactic/LinearCombination/Lemmas.lean b/Mathlib/Tactic/LinearCombination/Lemmas.lean
new file mode 100644
index 0000000000000..912fa19ffe828
--- /dev/null
+++ b/Mathlib/Tactic/LinearCombination/Lemmas.lean
@@ -0,0 +1,36 @@
+/-
+Copyright (c) 2022 Abby J. Goldberg. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Abby J. Goldberg, Mario Carneiro
+-/
+import Mathlib.Algebra.GroupWithZero.Basic
+import Mathlib.Algebra.Ring.Defs
+
+/-!
+# Lemmas for the linear_combination tactic
+
+These should not be used directly in user code.
+-/
+
+namespace Mathlib.Tactic.LinearCombination
+open Lean
+
+variable {α : Type*} {a a' a₁ a₂ b b' b₁ b₂ c : α}
+
+theorem add_pf [Add α] (p₁ : (a₁:α) = b₁) (p₂ : a₂ = b₂) : a₁ + a₂ = b₁ + b₂ := p₁ ▸ p₂ ▸ rfl
+theorem pf_mul_c [Mul α] (p : a = b) (c : α) : a * c = b * c := p ▸ rfl
+theorem c_mul_pf [Mul α] (p : b = c) (a : α) : a * b = a * c := p ▸ rfl
+theorem pf_div_c [Div α] (p : a = b) (c : α) : a / c = b / c := p ▸ rfl
+
+theorem eq_of_sub [AddGroup α] (p : (a:α) = b) (H : (a' - b') - (a - b) = 0) : a' = b' := by
+  rw [← sub_eq_zero] at p ⊢; rwa [sub_eq_zero, p] at H
+
+theorem eq_of_add [Add α] [IsRightCancelAdd α] (p : (a:α) = b) (H : a' + b = b' + a) : a' = b' := by
+  rw [p] at H
+  exact add_right_cancel H
+
+theorem eq_of_add_pow [Ring α] [NoZeroDivisors α] (n : ℕ) (p : (a:α) = b)
+    (H : (a' - b')^n - (a - b) = 0) : a' = b' := by
+  rw [← sub_eq_zero] at p ⊢; apply pow_eq_zero (n := n); rwa [sub_eq_zero, p] at H
+
+end Mathlib.Tactic.LinearCombination
diff --git a/Mathlib/Tactic/Linter.lean b/Mathlib/Tactic/Linter.lean
index 5ed156753cbd9..2f67da1bea192 100644
--- a/Mathlib/Tactic/Linter.lean
+++ b/Mathlib/Tactic/Linter.lean
@@ -1,16 +1,14 @@
 /-
-This is the `Linter`s file: it only imports files defining linters and is
-intended to be imported fairly early in `Mathlib`.
+This is the `Linter`s file: it imports files defining linters.
+All syntax linters enabled by default are imported in `Mathlib.Init`;
+this file contains all other linters.
 
 This file is ignored by `shake`:
 * it is in `ignoreAll`, meaning that all its imports are considered necessary;
 * it is in `ignoreImport`, meaning that where it is imported, it is considered necessary.
 -/
 
-import Mathlib.Tactic.Linter.GlobalAttributeIn
-import Mathlib.Tactic.Linter.HashCommandLinter
+import Mathlib.Tactic.Linter.FlexibleLinter
 import Mathlib.Tactic.Linter.HaveLetLinter
-import Mathlib.Tactic.Linter.Lint
-import Mathlib.Tactic.Linter.RefineLinter
-import Mathlib.Tactic.Linter.Style
-import Mathlib.Tactic.Linter.UnusedTactic
+import Mathlib.Tactic.Linter.MinImports
+import Mathlib.Tactic.Linter.PPRoundtrip
diff --git a/Mathlib/Tactic/Linter/AdmitLinter.lean b/Mathlib/Tactic/Linter/AdmitLinter.lean
index cc7fdbfb2ce62..602952c5a6692 100644
--- a/Mathlib/Tactic/Linter/AdmitLinter.lean
+++ b/Mathlib/Tactic/Linter/AdmitLinter.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Adomas Baliuka
 -/
 import Lean.Elab.Command
-import Lean.Linter.Util
 
 /-!
 # The "admit" linter
diff --git a/Mathlib/Tactic/Linter/DocPrime.lean b/Mathlib/Tactic/Linter/DocPrime.lean
new file mode 100644
index 0000000000000..79a08666bae3f
--- /dev/null
+++ b/Mathlib/Tactic/Linter/DocPrime.lean
@@ -0,0 +1,75 @@
+/-
+Copyright (c) 2024 Damiano Testa. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Damiano Testa
+-/
+import Lean.Elab.Command
+
+/-!
+# The "docPrime" linter
+
+The "docPrime" linter emits a warning on declarations that have no doc-string and whose
+name ends with a `'`. Such declarations are expected to have a documented explanation
+for the presence of a `'` in their name. This may consist of discussion of the difference relative
+to an unprimed version of that declaration, or an explanation as to why no better naming scheme
+is possible.
+-/
+
+open Lean Elab
+
+namespace Mathlib.Linter
+
+/--
+The "docPrime" linter emits a warning on declarations that have no doc-string and whose
+name ends with a `'`.
+
+The file `scripts/no_lints_prime_decls.txt` contains a list of temporary exceptions to this linter.
+This list should not be appended to, and become emptied over time.
+-/
+register_option linter.docPrime : Bool := {
+  defValue := false
+  descr := "enable the docPrime linter"
+}
+
+namespace DocPrime
+
+@[inherit_doc Mathlib.Linter.linter.docPrime]
+def docPrimeLinter : Linter where run := withSetOptionIn fun stx ↦ do
+  unless Linter.getLinterValue linter.docPrime (← getOptions) do
+    return
+  if (← get).messages.hasErrors then
+    return
+  unless [``Lean.Parser.Command.declaration, `lemma].contains stx.getKind do return
+  -- ignore private declarations
+  if (stx.find? (·.isOfKind ``Lean.Parser.Command.private)).isSome then return
+  let docstring := stx[0][0]
+  -- The current declaration's id, possibly followed by a list of universe names.
+  let declId :=
+    if stx[1].isOfKind ``Lean.Parser.Command.instance then
+      stx[1][3][0]
+    else
+      stx[1][1]
+  -- The name of the current declaration, with namespaces resolved.
+  let declName :=
+    if let `_root_ :: rest := declId[0].getId.components then
+      rest.foldl (· ++ ·) default
+    else (← getCurrNamespace) ++ declId[0].getId
+  let msg := m!"`{declName}` is missing a doc-string, please add one.\n\
+      Declarations whose name ends with a `'` are expected to contain an explanation for the \
+      presence of a `'` in their doc-string. This may consist of discussion of the difference \
+      relative to the unprimed version, or an explanation as to why no better naming scheme \
+      is possible."
+  if docstring[0][1].getAtomVal.isEmpty && declName.toString.back == '\'' then
+    if ← System.FilePath.pathExists "scripts/no_lints_prime_decls.txt" then
+      if (← IO.FS.lines "scripts/no_lints_prime_decls.txt").contains declName.toString then
+        return
+      else
+        Linter.logLint linter.docPrime declId msg
+    else
+      Linter.logLint linter.docPrime declId msg
+
+initialize addLinter docPrimeLinter
+
+end DocPrime
+
+end Mathlib.Linter
diff --git a/Mathlib/Tactic/Linter/FlexibleLinter.lean b/Mathlib/Tactic/Linter/FlexibleLinter.lean
new file mode 100644
index 0000000000000..525285a97bfef
--- /dev/null
+++ b/Mathlib/Tactic/Linter/FlexibleLinter.lean
@@ -0,0 +1,429 @@
+/-
+Copyright (c) 2024 Damiano Testa. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Damiano Testa
+-/
+import Lean.Elab.Command
+import Batteries.Data.Array.Basic
+import Batteries.Lean.HashSet
+
+/-!
+#  The "flexible" linter
+
+The "flexible" linter makes sure that a "rigid" tactic (such as `rw`) does not act on the
+output of a "flexible" tactic (such as `simp`).
+
+For example, this ensures that, if you want to use `simp [...]` in the middle of a proof,
+then you should replace `simp [...]` by one of
+* a `suffices \"expr after simp\" by simpa` line;
+* the output of `simp? [...]`, so that the final code contains `simp only [...]`;
+* something else that does not involve `simp`!
+
+Otherwise, the linter will complain.
+
+Simplifying and appealing to a geometric intuition, you can imagine a (tactic) proof like a
+directed graph, where
+* each node is a local hypothesis or a goal in some metavariable and
+* two hypotheses/goals are connected by an arrow if there is a tactic that modifies the source
+  of the arrow into the target (this does not apply well to all tactics, but it does apply to
+  a large number of them).
+With this in mind, a tactic like `rw [lemma]` takes a *very specific* input and return a
+*very predictable* output.
+Such a tactic is "rigid". Any tactic is rigid, unless it is in `flexible` or `stoppers`.
+Conversely, a tactic like `simp` acts on a wide variety of inputs and returns an output that
+is possibly unpredictable: if later modifications adds a `simp`-lemma or some internals of
+`simp` changes, the output of `simp` may change as well.
+Such a tactic is `flexible`. Other examples are `split`, `abel`, `norm_cast`,...
+Let's go back to the graph picture above.
+* ✅️ [`rigid` --> `flexible`]
+  A sequence `rw [lemma]; simp` is unlikely to break, since `rw [lemma]` produces the same output
+  unless some *really major* change happens!
+* ❌️ [`flexible` --> `rigid`]
+  A sequence `simp; rw [lemma]` is instead more likely to break, since the goal after `simp` is
+  subject to change by even a small, likely, modification of the `simp` set.
+* ✅️ [`flexible` --> `flexible`]
+  A sequence `simp; linarith` is also quite stable, since if `linarith` was able to close the
+  goal with a "weaker" `simp`, it will likely still be able to close the goal with a `simp`
+  that takes one further step.
+* ✅️ [`flexible` --> `stopper`]
+  Finally, a sequence `simp; ring_nf` is stable and, moreover, the output of `ring_nf` is a
+  "normal form", which means that it is likely to produce an unchanged result, even if the initial
+  input is different from the proof in its initial form.
+  A stopper can be followed by a rigid tactic, "stopping" the spread of the flexible reach.
+
+What the linter does is keeping track of nodes that are connected by `flexible` tactics and
+makes sure that only `flexible` or `stoppers` follow them.
+Such nodes are `Stained`.
+Whenever it reaches a `stopper` edge, the target node is no longer `Stained` and it is
+available again to `rigid` tactics.
+
+Currently, the only tactics that "start" the bookkeeping are most forms of non-`only` `simp`s.
+These are encoded by the `flexible?` predicate.
+Future modifications of the linter may increase the scope of the `flexible?` predicate and
+forbid a wider range of combinations.
+
+## TODO
+The example
+```lean
+example (h : 0 = 0) : True := by
+  simp at h
+  assumption
+```
+should trigger the linter, since `assumption` uses `h` that has been "stained" by `simp at h`.
+However, `assumption` contains no syntax information for the location `h`, so the linter in its
+current form does not catch this.
+
+## Implementation notes
+
+A large part of the code is devoted to tracking `FVar`s and `MVar`s between tactics.
+
+For the `FVar`s, this follows the following heuristic:
+* if the unique name of the `FVar` is preserved, then we use that;
+* otherwise, if the `userName` of the `FVar` is preserved, then we use that;
+* if neither is preserved, we drop the ball and stop tracking the `FVarId`.
+
+For the `MVar`s, we use the information of `Lean.Elab.TacticInfo.goalsBefore` and
+`Lean.Elab.TacticInfo.goalsAfter`.
+By looking at the `mvar`s that are either only "before" or only "after", we focus on the
+"active" goals.
+We then propagate all the `FVarId`s that were present in the "before" goals to the "after" goals,
+while leaving untouched the ones in the "inert" goals.
+-/
+
+open Lean Elab
+
+namespace Mathlib.Linter
+
+/-- The flexible linter makes sure that "rigid" tactics do not follow "flexible" tactics. -/
+register_option linter.flexible : Bool := {
+  defValue := false
+  descr := "enable the flexible linter"
+}
+
+/-- `flexible? stx` is `true` if `stx` is syntax for a tactic that takes a "wide" variety of
+inputs and modifies them in possibly unpredictable ways.
+
+The prototypical flexible tactic is `simp`.
+The prototypical non-flexible tactic `rw`.
+`simp only` is also non-flexible. -/
+--  TODO: adding more entries here, allows to consider more tactics to be flexible
+def flexible? : Syntax → Bool
+  | .node _ ``Lean.Parser.Tactic.simp #[_, _, _, only?, _, _] => only?[0].getAtomVal != "only"
+  | .node _ ``Lean.Parser.Tactic.simpAll #[_, _, _, only?, _] => only?[0].getAtomVal != "only"
+  | _ => false
+
+end Mathlib.Linter
+
+section goals_heuristic
+namespace Lean.Elab.TacticInfo
+
+/-!
+###  Heuristics for determining goals goals that a tactic modifies what they become
+
+The two definitions `goalsTargetedBy`, `goalsCreatedBy` extract a list of
+`MVarId`s attempting to determine on which goals the tactic `t` is acting and what are the
+resulting modified goals.
+This is mostly based on the heuristic that the tactic will "change" an `MVarId`.
+-/
+
+/-- `goalsTargetedBy t` are the `MVarId`s before the `TacticInfo` `t` that "disappear" after it.
+They should correspond to the goals in which the tactic `t` performs some action. -/
+def goalsTargetedBy (t : TacticInfo) : List MVarId :=
+  t.goalsBefore.filter (·.name ∉ t.goalsAfter.map (·.name))
+
+/-- `goalsCreatedBy t` are the `MVarId`s after the `TacticInfo` `t` that were not present before.
+They should correspond to the goals created or changed by the tactic `t`. -/
+def goalsCreatedBy (t : TacticInfo) : List MVarId :=
+  t.goalsAfter.filter (·.name ∉ t.goalsBefore.map (·.name))
+
+end Lean.Elab.TacticInfo
+end goals_heuristic
+
+namespace Mathlib.Linter.Flexible
+
+variable (take? : Syntax → Bool) in
+/-- `extractCtxAndGoals take? tree` takes as input a function `take? : Syntax → Bool` and
+an `InfoTree` and returns the array of pairs `(stx, mvars)`,
+where `stx` is a syntax node such that `take? stx` is `true` and
+`mvars` indicates the goal state:
+ * the context before `stx`
+ * the context after `stx`
+ * a list of metavariables closed by `stx`
+ * a list of metavariables created by `stx`
+
+A typical usage is to find the goals following a `simp` application.
+-/
+partial
+def extractCtxAndGoals : InfoTree →
+    Array (Syntax × MetavarContext × MetavarContext × List MVarId × List MVarId)
+  | .node k args =>
+    let kargs := (args.map extractCtxAndGoals).foldl (· ++ ·) #[]
+    if let .ofTacticInfo i := k then
+      if take? i.stx && (i.stx.getRange? true).isSome then
+        #[(i.stx, i.mctxBefore, i.mctxAfter, i.goalsTargetedBy, i.goalsCreatedBy)] ++ kargs
+      else kargs
+    else kargs
+  | .context _ t => extractCtxAndGoals t
+  | _ => default
+
+/-- `Stained` is the type of the stained locations: it can be
+* a `Name` (typically of associated to the `FVarId` of a local declaration);
+* the goal (`⊢`);
+* the "wildcard" -- all the declaration in context (`*`).
+-/
+inductive Stained
+  | name     : Name → Stained
+  | goal     : Stained
+  | wildcard : Stained
+  deriving Repr, Inhabited, DecidableEq, Hashable
+
+/-- Converting a `Stained` to a `String`:
+* a `Name` is represented by the corresponding string;
+* `goal` is represented by `⊢`;
+* `wildcard` is represented by `*`.
+-/
+instance : ToString Stained where
+  toString | .name n => n.toString | .goal => "⊢" | .wildcard => "*"
+
+/--
+`toStained stx` scans the input `Syntax` `stx` extracting identifiers and atoms, making an effort
+to convert them to `Stained`.
+The function is used to extract "location" information about `stx`: either explicit locations as in
+`rw [] at locations` or implicit ones as `rw [h]`.
+
+Whether or not what this function extracts really is a location will be determined by the linter
+using data embedded in the `InfoTree`s. -/
+partial
+def toStained : Syntax → Std.HashSet Stained
+  | .node _ _ arg => (arg.map toStained).foldl (.union) {}
+  | .ident _ _ val _ => {.name val}
+  | .atom _ val => match val with
+                  | "*" => {.wildcard}
+                  | "⊢" => {.goal}
+                  | "|" => {.goal}
+                  | _ => {}
+  | _ => {}
+
+/-- `getStained stx` expects `stx` to be an argument of a node of `SyntaxNodeKind`
+`Lean.Parser.Tactic.location`.
+Typically, we apply `getStained` to the output of `getLocs`.
+
+See `getStained!` for a similar function. -/
+partial
+def getStained (stx : Syntax) (all? : Syntax → Bool := fun _ ↦ false) : Std.HashSet Stained :=
+  match stx with
+    | stx@(.node _ ``Lean.Parser.Tactic.location loc) =>
+      if all? stx then {} else (loc.map toStained).foldl (·.union) {}
+    | .node _ _ args => (args.map (getStained · all?)).foldl (·.union) {}
+    | _ => default
+
+/-- `getStained! stx` expects `stx` to be an argument of a node of `SyntaxNodeKind`
+`Lean.Parser.Tactic.location`.
+Typically, we apply `getStained!` to the output of `getLocs`.
+
+It returns the `HashSet` of `Stained` determined by the locations in `stx`.
+
+The only difference with `getStained stx`, is that `getStained!` never returns `{}`:
+if `getStained stx = {}`, then `getStained' stx = {.goal}`.
+
+This means that tactics that do not have an explicit "`at`" in their syntax will be treated as
+acting on the main goal. -/
+def getStained! (stx : Syntax) (all? : Syntax → Bool := fun _ ↦ false) : Std.HashSet Stained :=
+  let out := getStained stx all?
+  if out.size == 0 then {.goal} else out
+
+/-- `Stained.toFMVarId mv lctx st` takes a metavariable `mv`, a local context `lctx` and
+a `Stained` `st` and returns the array of pairs `(FVarId, mv)`s that `lctx` assigns to `st`
+(the second component is always `mv`):
+* if `st` "is" a `Name`, returns the singleton of the `FVarId` with the name carried by `st`;
+* if `st` is `.goal`, returns the singleton `#[default]`;
+* if `st` is `.wildcard`, returns the array of all the `FVarId`s in `lctx` with also `default`
+  (to keep track of the `goal`).
+-/
+def Stained.toFMVarId (mv : MVarId) (lctx: LocalContext) : Stained → Array (FVarId × MVarId)
+  | name n   => match lctx.findFromUserName? n with
+                  | none => #[]
+                  | some decl => #[(decl.fvarId, mv)]
+  | goal     => #[(default, mv)]
+  | wildcard => (lctx.getFVarIds.push default).map (·, mv)
+
+/-- `SyntaxNodeKind`s that are mostly "formatting": mostly they are ignored
+because we do not want the linter to spend time on them.
+The nodes that they contain will be visited by the linter anyway.
+The nodes that *follow* them, though, will *not* be visited by the linter.
+-/
+def stoppers : Std.HashSet Name :=
+  { -- "properly stopper tactics": the effect of these tactics is to return a normal form
+    -- (or possibly be finishing tactics -- the ultimate normal form!
+    -- finishing tactics could equally well be considered as `flexible`, but as there is
+    -- no possibility of a follower anyway, it does not make a big difference.)
+    ``Lean.Parser.Tactic.tacticSorry,
+    ``Lean.Parser.Tactic.tacticRepeat_,
+    ``Lean.Parser.Tactic.tacticStop_,
+    `Mathlib.Tactic.Abel.abelNF,
+    `Mathlib.Tactic.RingNF.ringNF,
+    -- "continuators": the *effect* of these tactics is similar the "properly stoppers" above,
+    -- though they typically wrap other tactics inside them.
+    -- The linter ignores the wrapper, but does recurse into the enclosed tactics
+    ``Lean.Parser.Tactic.tacticSeq1Indented,
+    ``Lean.Parser.Tactic.tacticSeq,
+    ``Lean.Parser.Term.byTactic,
+    `by,
+    ``Lean.Parser.Tactic.tacticTry_,
+    `choice,  -- involved in `first`
+    ``Lean.Parser.Tactic.allGoals,
+    `Std.Tactic.«tacticOn_goal-_=>_»,
+    ``Lean.Parser.Tactic.«tactic_<;>_»,
+    ``cdotTk,
+    ``cdot }
+
+/-- `SyntaxNodeKind`s that are allowed to follow a flexible tactic:
+  `simp`, `simp_all`, `simpa`, `dsimp`, `constructor`, `congr`, `done`, `rfl`, `omega`, `abel`,
+  `ring`, `linarith`, `nlinarith`, `norm_cast`, `aesop`, `tauto`, `fun_prop`, `split`, `split_ifs`.
+-/
+def flexible : Std.HashSet Name :=
+  { ``Lean.Parser.Tactic.simp,
+    ``Lean.Parser.Tactic.simpAll,
+    ``Lean.Parser.Tactic.simpa,
+    ``Lean.Parser.Tactic.dsimp,
+    ``Lean.Parser.Tactic.constructor,
+    ``Lean.Parser.Tactic.congr,
+    ``Lean.Parser.Tactic.done,
+    ``Lean.Parser.Tactic.tacticRfl,
+    ``Lean.Parser.Tactic.omega,
+    `Mathlib.Tactic.Abel.abel,
+    `Mathlib.Tactic.RingNF.ring,
+    `Mathlib.Tactic.normNum,
+    `linarith,
+    `nlinarith,
+    ``Lean.Parser.Tactic.tacticNorm_cast_,
+    `Aesop.Frontend.Parser.aesopTactic,
+    `Mathlib.Tactic.Tauto.tauto,
+    `Mathlib.Meta.FunProp.funPropTacStx,
+    `Lean.Parser.Tactic.split,
+    `Mathlib.Tactic.splitIfs }
+
+/-- By default, if a `SyntaxNodeKind` is not special-cased here, then the linter assumes that
+the tactic will use the goal as well: this heuristic works well with `exact`, `refine`, `apply`.
+For tactics such as `cases` this is not true: for these tactics, `usesGoal?` yields `false. -/
+def usesGoal? : SyntaxNodeKind → Bool
+  | ``Lean.Parser.Tactic.cases => false
+  | `Mathlib.Tactic.cases' => false
+  | ``Lean.Parser.Tactic.obtain => false
+  | ``Lean.Parser.Tactic.tacticHave_ => false
+  | ``Lean.Parser.Tactic.rcases => false
+  | ``Lean.Parser.Tactic.specialize => false
+  | ``Lean.Parser.Tactic.subst => false
+  | ``«tacticBy_cases_:_» => false
+  | ``Lean.Parser.Tactic.induction => false
+  | _ => true
+
+/-- `getFVarIdCandidates fv name lctx` takes an input an `FVarId`, a `Name` and a `LocalContext`.
+It returns an array of guesses for a "best fit" `FVarId` in the given `LocalContext`.
+The first entry of the array is the input `FVarId` `fv`, if it is present.
+The next entry of the array is the `FVarId` with the given `Name`, if present.
+
+Usually, the first entry of the returned array is "the best approximation" to `(fv, name)`. -/
+def getFVarIdCandidates (fv : FVarId) (name : Name) (lctx : LocalContext) : Array FVarId :=
+  #[lctx.find? fv, lctx.findFromUserName? name].reduceOption.map (·.fvarId)
+
+/-!
+Tactics often change the name of the current `MVarId`, as well as the names of the `FVarId`s
+appearing in their local contexts.
+The function `reallyPersist` makes an attempt at "tracking" pairs `(fvar, mvar)` across a
+simultaneous change represented by an "old" list of `MVarId`s and the corresponding
+`MetavarContext` and a new one.
+
+This arises in the context of the information encoded in the `InfoTree`s when processing a
+tactic proof.
+-/
+
+/-- `persistFVars` is one step in persisting: track a single `FVarId` between two `LocalContext`s.
+If an `FVarId` with the same unique name exists in the new context, use it.
+Otherwise, if an `FVarId` with the same `userName` exists in the new context, use it.
+If both of these fail, return `default` (i.e. "fail"). -/
+def persistFVars (fv : FVarId) (before after : LocalContext) : FVarId :=
+  let ldecl := (before.find? fv).getD default
+  let name := ldecl.userName
+  (getFVarIdCandidates fv name after).getD 0 default
+
+/-- `reallyPersist` converts an array of pairs `(fvar, mvar)` to another array of the same type. -/
+def reallyPersist
+    (fmvars : Array (FVarId × MVarId)) (mvs0 mvs1 : List MVarId) (ctx0 ctx1 : MetavarContext) :
+    Array (FVarId × MVarId) := Id.run do
+  -- split the input `fmvars` into
+  -- * the `active` ones, whose `mvar` appears in `mvs0` and
+  -- * the `inert` ones, the rest.
+  -- `inert` gets copied unchanged, while we transform `active`
+  let (active, inert) := fmvars.partition fun (_, mv) => mvs0.contains mv
+  let mut new := #[]
+  for (fvar, mvar) in active do       -- for each `active` pair `(fvar, mvar)`
+    match ctx0.decls.find? mvar with  -- check if `mvar` is managed by `ctx0` (it should be)
+      | none => -- the `mvar` is not managed by `ctx0`: no change
+        new := new.push (fvar, mvar)
+      | some mvDecl0 => -- the `mvar` *is* managed by `ctx0`: push the pair `(fvar, mvar)` through
+        for mv1 in mvs1 do  -- for each new `MVarId` in `mvs1`
+          match ctx1.decls.find? mv1 with  -- check if `mv1` is managed by `ctx1` (it should be)
+            | none => dbg_trace "'really_persist' could this happen?" default -- ??? maybe `.push`?
+            | some mvDecl1 =>  -- we found a "new" declaration
+              let persisted_fv := persistFVars fvar mvDecl0.lctx mvDecl1.lctx  -- persist `fv`
+              new := new.push (persisted_fv, mv1)
+  return inert ++ new
+
+/-- The main implementation of the flexible linter. -/
+def flexibleLinter : Linter where run := withSetOptionIn fun _stx => do
+  unless Linter.getLinterValue linter.flexible (← getOptions) && (← getInfoState).enabled do
+    return
+  if (← MonadState.get).messages.hasErrors then
+    return
+  let trees ← getInfoTrees
+  let x := trees.toList.map (extractCtxAndGoals (fun _ => true))
+  -- `stains` records pairs `(location, mvar)`, where
+  -- * `location` is either a hypothesis or the main goal modified by a flexible tactic and
+  -- * `mvar` is the metavariable containing the modified location
+  let mut stains : Array ((FVarId × MVarId) × (Stained × Syntax)) := .empty
+  let mut msgs : Array (Syntax × Syntax × Stained) := #[]
+  for d in x do for (s, ctx0, ctx1, mvs0, mvs1) in d do
+    let skind := s.getKind
+    if stoppers.contains skind then continue
+    let shouldStain? := flexible? s && mvs1.length == mvs0.length
+    for d in getStained! s do
+      if shouldStain? then
+        for currMVar1 in mvs1 do
+          let lctx1 := ((ctx1.decls.find? currMVar1).getD default).lctx
+          let locsAfter := d.toFMVarId currMVar1 lctx1
+
+          for l in locsAfter do
+            stains := stains.push (l, (d, s))
+
+      else
+        let stained_in_syntax := if usesGoal? skind then (toStained s).insert d else toStained s
+        if !flexible.contains skind then
+          for currMv0 in mvs0 do
+            let lctx0 := ((ctx0.decls.find? currMv0).getD default).lctx
+            let mut foundFvs : Std.HashSet (FVarId × MVarId):= {}
+            for st in stained_in_syntax do
+              for d in st.toFMVarId currMv0 lctx0 do
+                if !foundFvs.contains d then foundFvs := foundFvs.insert d
+            for l in foundFvs do
+              if let some (_stdLoc, (st, kind)) := stains.find? (Prod.fst · == l) then
+                msgs := msgs.push (s, kind, st)
+
+      -- tactics often change the name of the current `MVarId`, so we migrate the `FvarId`s
+      -- in the "old" `mvars` to the "same" `FVarId` in the "new" `mvars`
+      let mut new : Array ((FVarId × MVarId) × (Stained × Syntax)) := .empty
+      for (fv, (stLoc, kd)) in stains do
+        let psisted := reallyPersist #[fv] mvs0 mvs1 ctx0 ctx1
+        if psisted == #[] && mvs1 != [] then
+          new := new.push (fv, (stLoc, kd))
+          dbg_trace "lost {((fv.1.name, fv.2.name), stLoc, kd)}"
+        for p in psisted do new := new.push (p, (stLoc, kd))
+      stains := new
+
+  for (s, stainStx, d) in msgs do
+    Linter.logLint linter.flexible stainStx m!"'{stainStx}' is a flexible tactic modifying '{d}'…"
+    logInfoAt s m!"… and '{s}' uses '{d}'!"
+
+initialize addLinter flexibleLinter
+
+end Mathlib.Linter.Flexible
diff --git a/Mathlib/Tactic/Linter/GlobalAttributeIn.lean b/Mathlib/Tactic/Linter/GlobalAttributeIn.lean
index b18e1defd647f..6a6c49e09f4ad 100644
--- a/Mathlib/Tactic/Linter/GlobalAttributeIn.lean
+++ b/Mathlib/Tactic/Linter/GlobalAttributeIn.lean
@@ -4,9 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Rothgang, Damiano Testa
 -/
 
-import Mathlib.Init
 import Lean.Elab.Command
-import Lean.Linter.Util
 
 /-!
 # Linter for `attribute [...] in` declarations
diff --git a/Mathlib/Tactic/Linter/HashCommandLinter.lean b/Mathlib/Tactic/Linter/HashCommandLinter.lean
index 3f490c47fbb36..66cac438ada34 100644
--- a/Mathlib/Tactic/Linter/HashCommandLinter.lean
+++ b/Mathlib/Tactic/Linter/HashCommandLinter.lean
@@ -4,9 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
 
-import Mathlib.Init
 import Lean.Elab.Command
-import Lean.Linter.Util
 import Batteries.Lean.HashSet
 
 /-!
@@ -54,7 +52,7 @@ private partial def withSetOptionIn' (cmd : CommandElab) : CommandElab := fun st
     cmd stx
 
 /-- `allowed_commands` is the `HashSet` of `#`-commands that are allowed in 'Mathlib'. -/
-private abbrev allowed_commands : HashSet String := { "#adaptation_note" }
+private abbrev allowed_commands : Std.HashSet String := { "#adaptation_note" }
 
 /-- Checks that no command beginning with `#` is present in 'Mathlib',
 except for the ones in `allowed_commands`.
diff --git a/Mathlib/Tactic/Linter/Lint.lean b/Mathlib/Tactic/Linter/Lint.lean
index 22f215b562485..6d00f05354dde 100644
--- a/Mathlib/Tactic/Linter/Lint.lean
+++ b/Mathlib/Tactic/Linter/Lint.lean
@@ -3,9 +3,6 @@ Copyright (c) 2023 Floris van Doorn. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
-import Mathlib.Init
-import Lean.Linter.Util
-import Batteries.Data.String.Matcher
 import Batteries.Tactic.Lint
 
 /-!
@@ -16,7 +13,7 @@ In this file we define additional linters for mathlib.
 Perhaps these should be moved to Batteries in the future.
 -/
 
-namespace Std.Tactic.Lint
+namespace Batteries.Tactic.Lint
 open Lean Meta
 
 /--
@@ -48,7 +45,7 @@ Linter that checks whether a structure should be in Prop.
     | some _ => return none -- TODO: enforce `YYYY-MM-DD` format
     | none => return m!"`deprecated` attribute without `since` date"
 
-end Std.Tactic.Lint
+end Batteries.Tactic.Lint
 
 namespace Mathlib.Linter
 
@@ -120,18 +117,18 @@ open Lean Elab Command
 /-- The "missing end" linter emits a warning on non-closed `section`s and `namespace`s.
 It allows the "outermost" `noncomputable section` to be left open (whether or not it is named).
 -/
-register_option linter.missingEnd : Bool := {
+register_option linter.style.missingEnd : Bool := {
   defValue := false
   descr := "enable the missing end linter"
 }
 
-namespace MissingEnd
+namespace Style.missingEnd
 
-@[inherit_doc Mathlib.Linter.linter.missingEnd]
+@[inherit_doc Mathlib.Linter.linter.style.missingEnd]
 def missingEndLinter : Linter where run := withSetOptionIn fun stx ↦ do
     -- Only run this linter at the end of a module.
     unless stx.isOfKind ``Lean.Parser.Command.eoi do return
-    if Linter.getLinterValue linter.missingEnd (← getOptions) &&
+    if Linter.getLinterValue linter.style.missingEnd (← getOptions) &&
         !(← MonadState.get).messages.hasErrors then
       let sc ← getScopes
       -- The last scope is always the "base scope", corresponding to no active `section`s or
@@ -144,12 +141,12 @@ def missingEndLinter : Linter where run := withSetOptionIn fun stx ↦ do
       if !ends.isEmpty then
         let ending := (ends.map Prod.fst).foldl (init := "") fun a b ↦
           a ++ s!"\n\nend{if b == "" then "" else " "}{b}"
-        Linter.logLint linter.missingEnd stx
+        Linter.logLint linter.style.missingEnd stx
          m!"unclosed sections or namespaces; expected: '{ending}'"
 
 initialize addLinter missingEndLinter
 
-end MissingEnd
+end Style.missingEnd
 
 /-!
 # The `cdot` linter
@@ -157,13 +154,15 @@ end MissingEnd
 The `cdot` linter is a syntax-linter that flags uses of the "cdot" `·` that are achieved
 by typing a character different from `·`.
 For instance, a "plain" dot `.` is allowed syntax, but is flagged by the linter.
+It also flags "isolated cdots", i.e. when the `·` is on its own line.
 -/
 
 /--
 The `cdot` linter flags uses of the "cdot" `·` that are achieved by typing a character
 different from `·`.
-For instance, a "plain" dot `.` is allowed syntax, but is flagged by the linter. -/
-register_option linter.cdot : Bool := {
+For instance, a "plain" dot `.` is allowed syntax, but is flagged by the linter.
+It also flags "isolated cdots", i.e. when the `·` is on its own line. -/
+register_option linter.style.cdot : Bool := {
   defValue := false
   descr := "enable the `cdot` linter"
 }
@@ -193,20 +192,29 @@ This is precisely what the `cdot` linter flags.
 def unwanted_cdot (stx : Syntax) : Array Syntax :=
   (findCDot stx).filter (!isCDot? ·)
 
-namespace CDotLinter
+namespace Style
 
-@[inherit_doc linter.cdot]
+@[inherit_doc linter.style.cdot]
 def cdotLinter : Linter where run := withSetOptionIn fun stx ↦ do
-    unless Linter.getLinterValue linter.cdot (← getOptions) do
+    unless Linter.getLinterValue linter.style.cdot (← getOptions) do
       return
     if (← MonadState.get).messages.hasErrors then
       return
     for s in unwanted_cdot stx do
-      Linter.logLint linter.cdot s m!"Please, use '·' (typed as `\\.`) instead of '{s}' as 'cdot'."
+      Linter.logLint linter.style.cdot s
+        m!"Please, use '·' (typed as `\\.`) instead of '{s}' as 'cdot'."
+    -- We also check for isolated cdot's, i.e. when the cdot is on its own line.
+    for cdot in Mathlib.Linter.findCDot stx do
+      match cdot.find? (·.isOfKind `token.«· ») with
+      | some (.node _ _ #[.atom (.original _ _ afterCDot _) _]) =>
+        if (afterCDot.takeWhile (·.isWhitespace)).contains '\n' then
+          logWarningAt cdot <| .tagged linter.style.cdot.name
+            m!"This central dot `·` is isolated; please merge it with the next line."
+      | _ => return
 
 initialize addLinter cdotLinter
 
-end CDotLinter
+end Style
 
 /-!
 # The `dollarSyntax` linter
@@ -217,7 +225,7 @@ These are disallowed by the mathlib style guide, as using `<|` pairs better with
 
 /-- The `dollarSyntax` linter flags uses of `<|` that are achieved by typing `$`.
 These are disallowed by the mathlib style guide, as using `<|` pairs better with `|>`. -/
-register_option linter.dollarSyntax : Bool := {
+register_option linter.style.dollarSyntax : Bool := {
   defValue := false
   descr := "enable the `dollarSyntax` linter"
 }
@@ -234,14 +242,15 @@ def findDollarSyntax : Syntax → Array Syntax
       | _ => dargs
   |_ => #[]
 
-@[inherit_doc linter.dollarSyntax]
+@[inherit_doc linter.style.dollarSyntax]
 def dollarSyntaxLinter : Linter where run := withSetOptionIn fun stx ↦ do
-    unless Linter.getLinterValue linter.dollarSyntax (← getOptions) do
+    unless Linter.getLinterValue linter.style.dollarSyntax (← getOptions) do
       return
     if (← MonadState.get).messages.hasErrors then
       return
     for s in findDollarSyntax stx do
-      Linter.logLint linter.dollarSyntax s m!"Please use '<|' instead of '$' for the pipe operator."
+      Linter.logLint linter.style.dollarSyntax s
+        m!"Please use '<|' instead of '$' for the pipe operator."
 
 initialize addLinter dollarSyntaxLinter
 
@@ -293,20 +302,94 @@ initialize addLinter lambdaSyntaxLinter
 
 end Style.lambdaSyntax
 
+/-!
+#  The "longFile" linter
+
+The "longFile" linter emits a warning on files which are longer than a certain number of lines
+(1500 by default).
+-/
+
+/--
+The "longFile" linter emits a warning on files which are longer than a certain number of lines
+(1500 by default on mathlib, no limit for downstream projects).
+If this option is set to `N` lines, the linter warns once a file has more than `N` lines.
+A value of `0` silences the linter entirely.
+-/
+register_option linter.style.longFile : Nat := {
+  defValue := 0
+  descr := "enable the longFile linter"
+}
+
+/-- The number of lines that the `longFile` linter considers the default. -/
+register_option linter.style.longFileDefValue : Nat := {
+  defValue := 1500
+  descr := "a soft upper bound on the number of lines of each file"
+}
+
+namespace Style.longFile
+
+@[inherit_doc Mathlib.Linter.linter.style.longFile]
+def longFileLinter : Linter where run := withSetOptionIn fun stx ↦ do
+  let linterBound := linter.style.longFile.get (← getOptions)
+  if linterBound == 0 then
+    return
+  let defValue := linter.style.longFileDefValue.get (← getOptions)
+  let smallOption := match stx with
+      | `(set_option linter.style.longFile $x) => TSyntax.getNat ⟨x.raw⟩ ≤ defValue
+      | _ => false
+  if smallOption then
+    logWarningAt stx <| .tagged linter.style.longFile.name
+      m!"The default value of the `longFile` linter is {defValue}.\n\
+        The current value of {linterBound} does not exceed the allowed bound.\n\
+        Please, remove the `set_option linter.style.longFile {linterBound}`."
+  else
+  -- Thanks to the above check, the linter option is either not set (and hence equal
+  -- to the default) or set to some value *larger* than the default.
+  -- `Parser.isTerminalCommand` allows `stx` to be `#exit`: this is useful for tests.
+  unless Parser.isTerminalCommand stx do return
+  -- We exclude `Mathlib.lean` from the linter: it exceeds linter's default number of allowed
+  -- lines, and it is an auto-generated import-only file.
+  -- TODO: if there are more such files, revise the implementation.
+  if (← getMainModule) == `Mathlib then return
+  if let some init := stx.getTailPos? then
+    -- the last line: we subtract 1, since the last line is expected to be empty
+    let lastLine := ((← getFileMap).toPosition init).line
+    if lastLine ≤ defValue && defValue < linterBound then
+      logWarningAt stx <| .tagged linter.style.longFile.name
+        m!"The default value of the `longFile` linter is {defValue}.\n\
+          This file is {lastLine} lines long which does not exceed the allowed bound.\n\
+          Please, remove the `set_option linter.style.longFile {linterBound}`."
+    else
+    -- `candidate` is divisible by `100` and satisfies `lastLine + 100 < candidate ≤ lastLine + 200`
+    -- note that either `lastLine ≤ defValue` and `defValue = linterBound` hold or
+    -- `candidate` is necessarily bigger than `lastLine` and hence bigger than `defValue`
+    let candidate := (lastLine / 100) * 100 + 200
+    let candidate := max candidate defValue
+    if linterBound < lastLine then
+      logWarningAt stx <| .tagged linter.style.longFile.name
+        m!"This file is {lastLine} lines long, but the limit is {linterBound}.\n\n\
+          You can extend the allowed length of the file using \
+          `set_option linter.style.longFile {candidate}`.\n\
+          You can completely disable this linter by setting the length limit to `0`."
+
+initialize addLinter longFileLinter
+
+end Style.longFile
+
 /-! # The "longLine linter" -/
 
 /-- The "longLine" linter emits a warning on lines longer than 100 characters.
 We allow lines containing URLs to be longer, though. -/
-register_option linter.longLine : Bool := {
+register_option linter.style.longLine : Bool := {
   defValue := false
   descr := "enable the longLine linter"
 }
 
-namespace LongLine
+namespace Style.longLine
 
-@[inherit_doc Mathlib.Linter.linter.longLine]
+@[inherit_doc Mathlib.Linter.linter.style.longLine]
 def longLineLinter : Linter where run := withSetOptionIn fun stx ↦ do
-    unless Linter.getLinterValue linter.longLine (← getOptions) do
+    unless Linter.getLinterValue linter.style.longLine (← getOptions) do
       return
     if (← MonadState.get).messages.hasErrors then
       return
@@ -328,12 +411,16 @@ def longLineLinter : Linter where run := withSetOptionIn fun stx ↦ do
     let longLines := ((sstr.getD default).splitOn "\n").filter fun line ↦
       (100 < (fm.toPosition line.stopPos).column)
     for line in longLines do
-      if !(line.containsSubstr "http") then
-        Linter.logLint linter.longLine (.ofRange ⟨line.startPos, line.stopPos⟩)
-          m!"This line exceeds the 100 character limit, please shorten it!"
-
+      if (line.splitOn "http").length ≤ 1 then
+        let stringMsg := if line.contains '"' then
+          "\nYou can use \"string gaps\" to format long strings: within a string quotation, \
+          using a '\' at the end of a line allows you to continue the string on the following \
+          line, removing all intervening whitespace."
+        else ""
+        Linter.logLint linter.style.longLine (.ofRange ⟨line.startPos, line.stopPos⟩)
+          m!"This line exceeds the 100 character limit, please shorten it!{stringMsg}"
 initialize addLinter longLineLinter
 
-end LongLine
+end Style.longLine
 
 end Mathlib.Linter
diff --git a/Mathlib/Tactic/Linter/OldObtain.lean b/Mathlib/Tactic/Linter/OldObtain.lean
index a540de9eb66cd..0fd427c77bbfe 100644
--- a/Mathlib/Tactic/Linter/OldObtain.lean
+++ b/Mathlib/Tactic/Linter/OldObtain.lean
@@ -4,9 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Rothgang
 -/
 
-import Mathlib.Init
 import Lean.Elab.Command
-import Lean.Linter.Util
 
 /-!
 # The `oldObtain` linter, against stream-of-conciousness `obtain`
diff --git a/Mathlib/Tactic/Linter/PPRoundtrip.lean b/Mathlib/Tactic/Linter/PPRoundtrip.lean
new file mode 100644
index 0000000000000..f6f25280cc338
--- /dev/null
+++ b/Mathlib/Tactic/Linter/PPRoundtrip.lean
@@ -0,0 +1,144 @@
+/-
+Copyright (c) 2024 Damiano Testa. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Damiano Testa
+-/
+
+import Lean.Elab.Command
+import Mathlib.Init
+
+/-!
+#  The "ppRoundtrip" linter
+
+The "ppRoundtrip" linter emits a warning when the syntax of a command differs substantially
+from the pretty-printed version of itself.
+-/
+open Lean Elab Command
+
+namespace Mathlib.Linter
+
+/--
+The "ppRoundtrip" linter emits a warning when the syntax of a command differs substantially
+from the pretty-printed version of itself.
+
+The linter makes an effort to start the highlighting at the first difference.
+However, it may not always be successful.
+It also prints both the source code and the "expected code" in a 5-character radius from
+the first difference.
+-/
+register_option linter.ppRoundtrip : Bool := {
+  defValue := false
+  descr := "enable the ppRoundtrip linter"
+}
+
+/-- `polishPP s` takes as input a `String` `s`, assuming that it is the output of
+pretty-printing a lean command.
+The main intent is to convert `s` to a reasonable candidate for a desirable source code format.
+The function first replaces consecutive whitespace sequences into a single space (` `), in an
+attempt to side-step line-break differences.
+After that, it applies some pre-emptive changes:
+* doc-module beginnings tend to have some whitespace following them, so we add a space back in;
+* name quotations such as ``` ``Nat``` get pretty-printed as ``` `` Nat```, so we remove a space
+  after double back-ticks, but take care of adding one more for triple (or more) back-ticks;
+* `notation3` is not followed by a pretty-printer space, so we add it here (#15515).
+-/
+def polishPP (s : String) : String :=
+  let s := s.split (·.isWhitespace)
+  (" ".intercalate (s.filter (!·.isEmpty)))
+    |>.replace "/-!" "/-! "
+    |>.replace "``` " "```  " -- avoid losing an existing space after the triple back-ticks
+                              -- as a consequence of the following replacement
+    |>.replace "`` " "``" -- weird pp ```#eval ``«Nat»``` pretty-prints as ```#eval `` «Nat»```
+    |>.replace "notation3(" "notation3 ("
+    |>.replace "notation3\"" "notation3 \""
+
+/-- `polishSource s` is similar to `polishPP s`, but expects the input to be actual source code.
+For this reason, `polishSource s` performs more conservative changes:
+it only replace all whitespace starting from a linebreak (`\n`) with a single whitespace. -/
+def polishSource (s : String) : String × Array Nat :=
+  let split := s.split (· == '\n')
+  let preWS := split.foldl (init := #[]) fun p q =>
+    let txt := q.trimLeft.length
+    (p.push (q.length - txt)).push txt
+  let preWS := preWS.eraseIdx 0
+  let s := (split.map .trimLeft).filter (· != "")
+  (" ".intercalate (s.filter (!·.isEmpty)), preWS)
+
+/-- `posToShiftedPos lths diff` takes as input an array `lths` of natural numbers,
+and one further natural number `diff`.
+It adds up the elements of `lths` occupying the odd positions, as long as the sum of the
+elements in the even positions does not exceed `diff`.
+It returns the sum of the accumulated odds and `diff`.
+This is useful to figure out the difference between the output of `polishSource s` and `s` itself.
+It plays a role similar to the `fileMap`. -/
+def posToShiftedPos (lths : Array Nat) (diff : Nat) : Nat := Id.run do
+  let mut (ws, noWS) := (diff, 0)
+  for con in [:lths.size / 2] do
+    let curr := lths[2 * con]!
+    if noWS + curr < diff then
+      noWS := noWS + curr
+      ws := ws + lths[2 * con + 1]!
+    else
+      break
+  return ws
+
+/-- `zoomString str centre offset` returns the substring of `str` consisting of the `offset`
+characters around the `centre`th character. -/
+def zoomString (str : String) (centre offset : Nat) : Substring :=
+  { str := str, startPos := ⟨centre - offset⟩, stopPos := ⟨centre + offset⟩ }
+
+/-- `capSourceInfo s p` "shortens" all end-position information in the `SourceInfo` `s` to be
+at most `p`, trimming down also the relevant substrings. -/
+def capSourceInfo (s : SourceInfo) (p : Nat) : SourceInfo :=
+  match s with
+    | .original leading pos trailing endPos =>
+      .original leading pos {trailing with stopPos := ⟨min endPos.1 p⟩} ⟨min endPos.1 p⟩
+    | .synthetic pos endPos canonical =>
+      .synthetic pos ⟨min endPos.1 p⟩ canonical
+    | .none => s
+
+/-- `capSyntax stx p` applies `capSourceInfo · s` to all `SourceInfo`s in all
+`node`s, `atom`s and `ident`s contained in `stx`.
+
+This is used to trim away all "fluff" that follows a command: comments and whitespace after
+a command get removed with `capSyntax stx stx.getTailPos?.get!`.
+-/
+partial
+def capSyntax (stx : Syntax) (p : Nat) : Syntax :=
+  match stx with
+    | .node si k args => .node (capSourceInfo si p) k (args.map (capSyntax · p))
+    | .atom si val => .atom (capSourceInfo si p) (val.take p)
+    | .ident si r v pr => .ident (capSourceInfo si p) { r with stopPos := ⟨min r.stopPos.1 p⟩ } v pr
+    | s => s
+
+namespace PPRoundtrip
+
+@[inherit_doc Mathlib.Linter.linter.ppRoundtrip]
+def ppRoundtrip : Linter where run := withSetOptionIn fun stx ↦ do
+    unless Linter.getLinterValue linter.ppRoundtrip (← getOptions) do
+      return
+    if (← MonadState.get).messages.hasErrors then
+      return
+    let stx := capSyntax stx (stx.getTailPos?.getD default).1
+    let origSubstring := stx.getSubstring?.getD default
+    let (real, lths) := polishSource origSubstring.toString
+    let fmt ← (liftCoreM do PrettyPrinter.ppCategory `command stx <|> (do
+      Linter.logLint linter.ppRoundtrip stx
+        m!"The ppRoundtrip linter had some parsing issues: \
+           feel free to silence it with `set_option linter.ppRoundtrip false in` \
+           and report this error!"
+      return real))
+    let st := polishPP fmt.pretty
+    if st != real then
+      let diff := real.firstDiffPos st
+      let pos := posToShiftedPos lths diff.1 + origSubstring.startPos.1
+      let f := origSubstring.str.drop (pos)
+      let extraLth := (f.takeWhile (· != st.get diff)).length
+      let srcCtxt := zoomString real diff.1 5
+      let ppCtxt  := zoomString st diff.1 5
+      Linter.logLint linter.ppRoundtrip (.ofRange ⟨⟨pos⟩, ⟨pos + extraLth + 1⟩⟩)
+        m!"source context\n'{srcCtxt}'\n'{ppCtxt}'\npretty-printed context"
+
+initialize addLinter ppRoundtrip
+
+end Mathlib.Linter.PPRoundtrip
diff --git a/Mathlib/Tactic/Linter/RefineLinter.lean b/Mathlib/Tactic/Linter/RefineLinter.lean
index 15e408efa9c0b..7bbc39eee11dc 100644
--- a/Mathlib/Tactic/Linter/RefineLinter.lean
+++ b/Mathlib/Tactic/Linter/RefineLinter.lean
@@ -3,9 +3,8 @@ Copyright (c) 2024 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathlib.Init
+
 import Lean.Elab.Command
-import Lean.Linter.Util
 
 /-!
 # The "refine" linter
diff --git a/Mathlib/Tactic/Linter/Style.lean b/Mathlib/Tactic/Linter/Style.lean
index 953fe2282015b..1f737feb7c3ef 100644
--- a/Mathlib/Tactic/Linter/Style.lean
+++ b/Mathlib/Tactic/Linter/Style.lean
@@ -4,9 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Rothgang
 -/
 
-import Mathlib.Init
 import Lean.Elab.Command
-import Lean.Linter.Util
 
 /-!
 ## Style linters
diff --git a/Mathlib/Tactic/Linter/TextBased.lean b/Mathlib/Tactic/Linter/TextBased.lean
index f49c1f7ea6c5c..b4556c694d081 100644
--- a/Mathlib/Tactic/Linter/TextBased.lean
+++ b/Mathlib/Tactic/Linter/TextBased.lean
@@ -6,6 +6,7 @@ Authors: Michael Rothgang
 
 import Batteries.Data.String.Matcher
 import Mathlib.Data.Nat.Notation
+import Std.Data.HashMap.Basic
 
 /-!
 ## Text-based linters
@@ -18,7 +19,6 @@ For now, this only contains linters checking
 - existence of module docstrings (in the right place)
 - for certain disallowed imports
 - if the string "adaptation note" is used instead of the command #adaptation_note
-- lines are at most 100 characters long (except for URLs)
 - files are at most 1500 lines long (unless specifically allowed).
 
 For historic reasons, some of these checks are still written in a Python script `lint-style.py`:
@@ -27,6 +27,12 @@ these are gradually being rewritten in Lean.
 This linter maintains a list of exceptions, for legacy reasons.
 Ideally, the length of the list of exceptions tends to 0.
 
+The `longFile` and the `longLine` *syntax* linter take care of flagging lines that exceed the
+100 character limit and files that exceed the 1500 line limit.
+The text-based versions of this file are still used for the files where the linter is not imported.
+This means that the exceptions for the text-based linters are shorter, as they do not need to
+include those handled with `set_option linter.style.longFile x`/`set_option linter.longLine false`.
+
 An executable running all these linters is defined in `scripts/lint-style.lean`.
 -/
 
@@ -57,13 +63,9 @@ inductive StyleError where
   /-- Lint against "too broad" imports, such as `Mathlib.Tactic` or any module in `Lake`
   (unless carefully measured) -/
   | broadImport (module : BroadImports)
-  /-- Line longer than 100 characters -/
-  | lineLength (actual : Int) : StyleError
-  /-- The current file was too large: this error contains the current number of lines
-  as well as a size limit (slightly larger). On future runs, this linter will allow this file
-  to grow up to this limit.
-  For diagnostic purposes, this may also contain a previous size limit, which is now exceeded. -/
-  | fileTooLong (numberLines : ℕ) (newSizeLimit : ℕ) (previousLimit : Option ℕ) : StyleError
+  /-- A line ends with windows line endings (\r\n) instead of unix ones (\n). -/
+  | windowsLineEnding
+  | duplicateImport (importStatement: String) (alreadyImportedLine: ℕ)
 deriving BEq
 
 /-- How to format style errors -/
@@ -79,7 +81,7 @@ inductive ErrorFormat
   deriving BEq
 
 /-- Create the underlying error message for a given `StyleError`. -/
-def StyleError.errorMessage (err : StyleError) (style : ErrorFormat) : String := match err with
+def StyleError.errorMessage (err : StyleError) : String := match err with
   | StyleError.copyright (some context) => s!"Malformed or missing copyright header: {context}"
   | StyleError.copyright none => "Malformed or missing copyright header"
   | StyleError.authors =>
@@ -92,17 +94,10 @@ def StyleError.errorMessage (err : StyleError) (style : ErrorFormat) : String :=
       "In the past, importing 'Lake' in mathlib has led to dramatic slow-downs of the linter (see \
       e.g. mathlib4#13779). Please consider carefully if this import is useful and make sure to \
       benchmark it. If this is fine, feel free to allow this linter."
-  | StyleError.lineLength n => s!"Line has {n} characters, which is more than 100"
-  | StyleError.fileTooLong currentSize sizeLimit previousLimit =>
-    match style with
-    | ErrorFormat.github =>
-      if let some n := previousLimit then
-        s!"file contains {currentSize} lines (at most {n} allowed), try to split it up"
-      else
-        s!"file contains {currentSize} lines, try to split it up"
-    | ErrorFormat.exceptionsFile =>
-        s!"{sizeLimit} file contains {currentSize} lines, try to split it up"
-    | ErrorFormat.humanReadable => s!"file contains {currentSize} lines, try to split it up"
+  | windowsLineEnding => "This line ends with a windows line ending (\r\n): please use Unix line\
+    endings (\n) instead"
+  | StyleError.duplicateImport (importStatement) (alreadyImportedLine) =>
+    s!"Duplicate imports: {importStatement} (already imported on line {alreadyImportedLine})"
 
 /-- The error code for a given style error. Keep this in sync with `parse?_errorContext` below! -/
 -- FUTURE: we're matching the old codes in `lint-style.py` for compatibility;
@@ -112,8 +107,8 @@ def StyleError.errorCode (err : StyleError) : String := match err with
   | StyleError.authors => "ERR_AUT"
   | StyleError.adaptationNote => "ERR_ADN"
   | StyleError.broadImport _ => "ERR_IMP"
-  | StyleError.lineLength _ => "ERR_LIN"
-  | StyleError.fileTooLong _ _ _ => "ERR_NUM_LIN"
+  | StyleError.windowsLineEnding => "ERR_WIN"
+  | StyleError.duplicateImport _ _ => "ERR_DIMP"
 
 /-- Context for a style error: the actual error, the line number in the file we're reading
 and the path to the file. -/
@@ -143,27 +138,17 @@ inductive ComparisonResult
 and, if it is, if we prefer replacing the new exception or keeping the previous one. -/
 def compare (existing new : ErrorContext) : ComparisonResult :=
   -- Two comparable error contexts must have the same path.
-  if existing.path != new.path then
-    ComparisonResult.Different
+  -- To avoid issues with different path separators across different operating systems,
+  -- we compare the set of path components instead.
+  if existing.path.components != new.path.components then ComparisonResult.Different
   -- We entirely ignore their line numbers: not sure if this is best.
 
   -- NB: keep the following in sync with `parse?_errorContext` below.
   -- Generally, comparable errors must have equal `StyleError`s, but there are some exceptions.
   else match (existing.error, new.error) with
-    -- File length errors are the biggest exceptions: generally, we prefer to keep the
-    -- existing entry, *except* when a newer entry is much shorter.
-  | (StyleError.fileTooLong n nLimit _, StyleError.fileTooLong m _mLimit _) =>
-    -- The only exception are "file too long" errors.
-    -- If a file got much longer, the existing exception does not apply;
-    if m > nLimit then ComparisonResult.Different
-    -- if it does apply, we prefer to keep the existing entry,
-    -- *unless* the newer entry is much shorter.
-    else if m + 200 <= n then ComparisonResult.Comparable false
-    else ComparisonResult.Comparable true
   -- We do *not* care about the *kind* of wrong copyright,
   -- nor about the particular length of a too long line.
   | (StyleError.copyright _, StyleError.copyright _) => ComparisonResult.Comparable true
-  | (StyleError.lineLength _, StyleError.lineLength _) => ComparisonResult.Comparable true
   -- In all other cases, `StyleErrors` must compare equal.
   | (a, b) => if a == b then ComparisonResult.Comparable true else ComparisonResult.Different
 
@@ -176,7 +161,7 @@ def ErrorContext.find?_comparable (e : ErrorContext) (exceptions : Array ErrorCo
 `style` specifies if the error should be formatted for humans to read, github problem matchers
 to consume, or for the style exceptions file. -/
 def outputMessage (errctx : ErrorContext) (style : ErrorFormat) : String :=
-  let errorMessage := errctx.error.errorMessage style
+  let errorMessage := errctx.error.errorMessage
   match style with
   | ErrorFormat.github =>
    -- We are outputting for github: duplicate file path, line number and error code,
@@ -207,29 +192,16 @@ def parse?_errorContext (line : String) : Option ErrorContext := Id.run do
         -- Use default values for parameters which are ignored for comparing style exceptions.
         -- NB: keep this in sync with `compare` above!
         | "ERR_COP" => some (StyleError.copyright none)
-        | "ERR_LIN" =>
-          if let some n := errorMessage.get? 2 then
-            match String.toNat? n with
-              | some n => return StyleError.lineLength n
-              | none => none
-          else none
         | "ERR_AUT" => some (StyleError.authors)
         | "ERR_ADN" => some (StyleError.adaptationNote)
+        | "ERR_WIN" => some (StyleError.windowsLineEnding)
+        | "ERR_DIMP" => some (StyleError.duplicateImport "" 0)
         | "ERR_IMP" =>
           -- XXX tweak exceptions messages to ease parsing?
           if (errorMessage.get! 0).containsSubstr "tactic" then
             some (StyleError.broadImport BroadImports.TacticFolder)
           else
             some (StyleError.broadImport BroadImports.Lake)
-        | "ERR_NUM_LIN" =>
-          -- Parse the error message in the script. `none` indicates invalid input.
-          match (errorMessage.get? 0, errorMessage.get? 3) with
-          | (some limit, some current) =>
-            match (String.toNat? limit, String.toNat? current) with
-            | (some sizeLimit, some currentSize) =>
-              some (StyleError.fileTooLong currentSize sizeLimit (some sizeLimit))
-            | _ => none
-          | _ => none
         | _ => none
       match String.toNat? lineNumber with
       | some n => err.map fun e ↦ (ErrorContext.mk e n path)
@@ -253,8 +225,11 @@ def formatErrors (errors : Array ErrorContext) (style : ErrorFormat) : IO Unit :
     IO.println (outputMessage e style)
 
 /-- Core logic of a text based linter: given a collection of lines,
-return an array of all style errors with line numbers. -/
-abbrev TextbasedLinter := Array String → Array (StyleError × ℕ)
+return an array of all style errors with line numbers. If possible,
+also return the collection of all lines, changed as needed to fix the linter errors.
+(Such automatic fixes are only possible for some kinds of `StyleError`s.)
+-/
+abbrev TextbasedLinter := Array String → Array (StyleError × ℕ) × (Option (Array String))
 
 /-! Definitions of the actual text-based linters. -/
 section
@@ -303,7 +278,7 @@ def copyrightHeaderLinter : TextbasedLinter := fun lines ↦ Id.run do
       -- If it does, we check the authors line is formatted correctly.
       if !isCorrectAuthorsLine line then
         output := output.push (StyleError.authors, 4)
-  return output
+  return (output, none)
 
 /-- Lint on any occurrences of the string "Adaptation note:" or variants thereof. -/
 def adaptationNoteLinter : TextbasedLinter := fun lines ↦ Id.run do
@@ -314,7 +289,27 @@ def adaptationNoteLinter : TextbasedLinter := fun lines ↦ Id.run do
     if line.containsSubstr "daptation note" then
       errors := errors.push (StyleError.adaptationNote, lineNumber)
     lineNumber := lineNumber + 1
-  return errors
+  return (errors, none)
+
+/-- Lint on a collection of input strings if one of the is a duplicate import statement. -/
+def duplicateImportsLinter : TextbasedLinter := fun lines ↦ Id.run do
+  let mut lineNumber := 1
+  let mut errors := Array.mkEmpty 0
+  let mut importStatements : Std.HashMap String ℕ := {}
+  for line in lines do
+    if line.startsWith "import " then
+      let lineWithoutComment := (line.splitOn "--")[0]!
+      let importStatement := lineWithoutComment.trim
+      if importStatements.contains importStatement then
+        let alreadyImportedLine := importStatements[importStatement]!
+        errors := errors.push (
+          (StyleError.duplicateImport importStatement alreadyImportedLine),
+          lineNumber
+        )
+      else
+        importStatements := importStatements.insert importStatement lineNumber
+    lineNumber := lineNumber + 1
+  return (errors, none)
 
 /-- Lint a collection of input strings if one of them contains an unnecessarily broad import. -/
 def broadImportsLinter : TextbasedLinter := fun lines ↦ Id.run do
@@ -342,17 +337,8 @@ def broadImportsLinter : TextbasedLinter := fun lines ↦ Id.run do
             else if name == "Lake" || name.startsWith "Lake." then
               errors := errors.push (StyleError.broadImport BroadImports.Lake, lineNumber)
       lineNumber := lineNumber + 1
-  return errors
+  return (errors, none)
 
-/-- Iterates over a collection of strings, finding all lines which are longer than 101 chars.
-We allow URLs to be longer, though.
--/
-def lineLengthLinter : TextbasedLinter := fun lines ↦ Id.run do
-  let errors := (lines.toList.enumFrom 1).filterMap (fun (lineNumber, line) ↦
-    if line.length > 101 && !line.containsSubstr "http" then
-      some (StyleError.lineLength line.length, lineNumber)
-    else none)
-  errors.toArray
 
 /-- Whether a collection of lines consists *only* of imports, blank lines and single-line comments.
 In practice, this means it's an imports-only file and exempt from almost all linting. -/
@@ -361,126 +347,94 @@ def isImportsOnlyFile (lines : Array String) : Bool :=
   -- this is in fact not necessary. (It is needed for `Tactic/Linter.lean`, though.)
   lines.all (fun line ↦ line.startsWith "import " || line == "" || line.startsWith "-- ")
 
-/-- Error if a collection of lines is too large. "Too large" means more than 1500 lines
-**and** longer than an optional previous limit.
-If the file is too large, return a matching `StyleError`, which includes a new size limit
-(which is somewhat larger than the current size). -/
-def checkFileLength (lines : Array String) (existingLimit : Option ℕ) : Option StyleError :=
-  Id.run do
-  if lines.size > 1500 then
-    let isLarger : Bool := match existingLimit with
-    | some mark => lines.size > mark
-    | none => true
-    if isLarger then
-      -- We add about 200 lines of slack to the current file size: small PRs will be unaffected,
-      -- but sufficiently large PRs will get nudged towards splitting up this file.
-      return some (StyleError.fileTooLong lines.size
-        ((Nat.div lines.size 100) * 100 + 200) existingLimit)
-  none
-
 end
 
 /-- All text-based linters registered in this file. -/
 def allLinters : Array TextbasedLinter := #[
-    copyrightHeaderLinter, adaptationNoteLinter, broadImportsLinter, lineLengthLinter
+    copyrightHeaderLinter, adaptationNoteLinter, broadImportsLinter, duplicateImportsLinter
   ]
 
-/-- Controls what kind of output this programme produces. -/
-inductive OutputSetting : Type
-  /-- Print any style error to standard output (the default) -/
-  | print (style : ErrorFormat)
-  /-- Update the style exceptions file (and still print style errors to standard output).
-  This adds entries for any new exceptions, removes any entries which are no longer necessary,
-  and tries to not modify exception entries unless necessary.
-  To fully regenerate the exceptions file, delete `style-exceptions.txt` and run again in this mode.
-  -/
-  | update
-  deriving BEq
 
-/-- Read a file and apply all text-based linters. Return a list of all unexpected errors.
-`sizeLimit` is any pre-existing limit on this file's size.
-`exceptions` are any other style exceptions. -/
-def lintFile (path : FilePath) (sizeLimit : Option ℕ) (exceptions : Array ErrorContext) :
-    IO (Array ErrorContext) := do
-  let lines ← IO.FS.lines path
-  -- We don't need to run any checks on imports-only files.
-  if isImportsOnlyFile lines then
-    return #[]
+/-- Read a file and apply all text-based linters.
+Return a list of all unexpected errors, and, if some errors could be fixed automatically,
+the collection of all lines with every automatic fix applied.
+`exceptions` are any pre-existing style exceptions for this file. -/
+def lintFile (path : FilePath) (exceptions : Array ErrorContext) :
+    IO (Array ErrorContext × Option (Array String)) := do
   let mut errors := #[]
-  if let some (StyleError.fileTooLong n limit ex) := checkFileLength lines sizeLimit then
-    errors := #[ErrorContext.mk (StyleError.fileTooLong n limit ex) 1 path]
-  let allOutput := (Array.map (fun lint ↦
-    (Array.map (fun (e, n) ↦ ErrorContext.mk e n path)) (lint lines))) allLinters
+  -- Whether any changes were made by auto-fixes.
+  let mut changes_made := false
+  -- Check for windows line endings first: as `FS.lines` treats Unix and Windows lines the same,
+  -- we need to analyse the actual file contents.
+  let contents ← IO.FS.readFile path
+  let replaced := contents.crlfToLf
+  if replaced != contents then
+    changes_made := true
+    errors := errors.push (ErrorContext.mk StyleError.windowsLineEnding 1 path)
+  let lines := (replaced.splitOn "\n").toArray
+
+  -- We don't need to run any further checks on imports-only files.
+  if isImportsOnlyFile lines then
+    return (errors, if changes_made then some lines else none)
+
+  -- All further style errors raised in this file.
+  let mut allOutput := #[]
+  -- A working copy of the lines in this file, modified by applying the auto-fixes.
+  let mut changed := lines
+
+  for lint in allLinters do
+    let (err, changes) := lint changed
+    allOutput := allOutput.append (Array.map (fun (e, n) ↦ #[(ErrorContext.mk e n path)]) err)
+    if let some c := changes then
+      changed := c
+      changes_made := true
   -- This list is not sorted: for github, this is fine.
   errors := errors.append
     (allOutput.flatten.filter (fun e ↦ (e.find?_comparable exceptions).isNone))
-  return errors
+  return (errors, if changes_made then some changed else none)
+
 
 /-- Lint a collection of modules for style violations.
 Print formatted errors for all unexpected style violations to standard output;
-update the list of style exceptions if configured so.
+correct automatically fixable style errors if configured so.
 Return the number of files which had new style errors.
 `moduleNames` are all the modules to lint,
 `mode` specifies what kind of output this script should produce,
 `fix` configures whether fixable errors should be corrected in-place. -/
-def lintModules (moduleNames : Array String) (mode : OutputSetting) (fix : Bool) : IO UInt32 := do
-  -- Read the style exceptions file.
-  -- We also have a `nolints` file with manual exceptions for the linter.
-  let exceptionsFilePath : FilePath := "scripts" / "style-exceptions.txt"
-  let exceptions ← IO.FS.lines exceptionsFilePath
-  let mut styleExceptions := parseStyleExceptions exceptions
+def lintModules (moduleNames : Array String) (style : ErrorFormat) (fix : Bool) : IO UInt32 := do
+  -- Read the `nolints` file, with manual exceptions for the linter.
   let nolints ← IO.FS.lines ("scripts" / "nolints-style.txt")
-  styleExceptions := styleExceptions.append (parseStyleExceptions nolints)
+  let styleExceptions := parseStyleExceptions nolints
 
   let mut numberErrorFiles : UInt32 := 0
   let mut allUnexpectedErrors := #[]
   for module in moduleNames do
     -- Convert the module name to a file name, then lint that file.
     let path := (mkFilePath (module.split (· == '.'))).addExtension "lean"
-    -- Find all size limits for this given file.
-    -- If several size limits are given (unlikely in practice), we use the first one.
-    let sizeLimits := (styleExceptions.filter (fun ex ↦ ex.path == path)).filterMap (fun errctx ↦
-      match errctx.error with
-      | StyleError.fileTooLong _ limit _ => some limit
-      | _ => none)
-    let errors :=
-    if let OutputSetting.print _ := mode then
-      ← lintFile path (sizeLimits.get? 0) styleExceptions
-    else
-      -- In "update" mode, we ignore the exceptions file (and only take `nolints` into account).
-      ← lintFile path none (parseStyleExceptions nolints)
+
+    let (errors, changed) := ← lintFile path styleExceptions
+    if let some c := changed then
+      if fix then
+        let _ := ← IO.FS.writeFile path ("\n".intercalate c.toList)
     if errors.size > 0 then
       allUnexpectedErrors := allUnexpectedErrors.append errors
       numberErrorFiles := numberErrorFiles + 1
-  match mode with
-  | OutputSetting.print style =>
-    -- Run the remaining python linters. It is easier to just run on all files.
-    -- If this poses an issue, I can either filter the output
-    -- or wait until lint-style.py is fully rewritten in Lean.
-    let args := if fix then #["--fix"] else #[]
-    let pythonOutput ← IO.Process.run { cmd := "./scripts/print-style-errors.sh", args := args }
-    if pythonOutput != "" then
-      numberErrorFiles := numberErrorFiles + 1
-      IO.print pythonOutput
-    formatErrors allUnexpectedErrors style
-    if allUnexpectedErrors.size > 0 && mode matches OutputSetting.print _ then
-      IO.println s!"error: found {allUnexpectedErrors.size} new style error(s)\n\
-        run `lake exe lint-style --update` to ignore all of them"
-  | OutputSetting.update =>
-    formatErrors allUnexpectedErrors ErrorFormat.humanReadable
-    -- Regenerate the style exceptions file, including the Python output.
-    IO.FS.writeFile exceptionsFilePath ""
-    let pythonOutput ← IO.Process.run { cmd := "./scripts/print-style-errors.sh" }
-    -- Combine style exception entries: for each new error, replace by a corresponding
-    -- previous exception if that is preferred.
-    let mut tweaked := allUnexpectedErrors.map fun err ↦
-      if let some existing := err.find?_comparable styleExceptions then
-        if let ComparisonResult.Comparable (true) := compare err existing then existing
-        else err
-      else err
-    let thisOutput := "\n".intercalate (tweaked.map
-        (fun err ↦ outputMessage err ErrorFormat.exceptionsFile)).toList
-    IO.FS.writeFile exceptionsFilePath s!"{pythonOutput}{thisOutput}\n"
+
+  -- Run the remaining python linters. It is easier to just run on all files.
+  -- If this poses an issue, I can either filter the output
+  -- or wait until lint-style.py is fully rewritten in Lean.
+  let args := if fix then #["--fix"] else #[]
+  let output ← IO.Process.output { cmd := "./scripts/print-style-errors.sh", args := args }
+  if output.exitCode != 0 then
+    numberErrorFiles := numberErrorFiles + 1
+    IO.eprintln s!"error: `print-style-error.sh` exited with code {output.exitCode}"
+    IO.eprint output.stderr
+  else if output.stdout != "" then
+    numberErrorFiles := numberErrorFiles + 1
+    IO.eprint output.stdout
+  formatErrors allUnexpectedErrors style
+  if allUnexpectedErrors.size > 0 then
+    IO.eprintln s!"error: found {allUnexpectedErrors.size} new style error(s)"
   return numberErrorFiles
 
 end Mathlib.Linter.TextBased
diff --git a/Mathlib/Tactic/Linter/UnusedTactic.lean b/Mathlib/Tactic/Linter/UnusedTactic.lean
index 1fa8b5c45c6b5..7528d2a53673f 100644
--- a/Mathlib/Tactic/Linter/UnusedTactic.lean
+++ b/Mathlib/Tactic/Linter/UnusedTactic.lean
@@ -3,9 +3,8 @@ Copyright (c) 2024 Damiano Testa. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa
 -/
-import Mathlib.Init
+
 import Lean.Elab.Command
-import Lean.Linter.Util
 import Batteries.Tactic.Unreachable
 
 /-!
@@ -51,7 +50,7 @@ before and after and see if there is some change.
 Yet another linter copied from the `unreachableTactic` linter!
 -/
 
-open Lean Elab
+open Lean Elab Std
 
 namespace Mathlib.Linter
 
@@ -64,13 +63,13 @@ register_option linter.unusedTactic : Bool := {
 namespace UnusedTactic
 
 /-- The monad for collecting the ranges of the syntaxes that do not modify any goal. -/
-abbrev M := StateRefT (HashMap String.Range Syntax) IO
+abbrev M := StateRefT (Std.HashMap String.Range Syntax) IO
 
 /-- `Parser`s allowed to not change the tactic state.
 This can be increased dynamically, using `#allow_unused_tactic`.
 -/
-initialize allowedRef : IO.Ref (HashSet SyntaxNodeKind) ←
-  IO.mkRef <| HashSet.empty
+initialize allowedRef : IO.Ref (Std.HashSet SyntaxNodeKind) ←
+  IO.mkRef <| Std.HashSet.empty
     |>.insert `Mathlib.Tactic.Says.says
     |>.insert `Batteries.Tactic.«tacticOn_goal-_=>_»
     -- attempt to speed up, by ignoring more tactics
@@ -114,7 +113,7 @@ A list of blacklisted syntax kinds, which are expected to have subterms that con
 unevaluated tactics.
 -/
 initialize ignoreTacticKindsRef : IO.Ref NameHashSet ←
-  IO.mkRef <| HashSet.empty
+  IO.mkRef <| Std.HashSet.empty
     |>.insert `Mathlib.Tactic.Says.says
     |>.insert ``Parser.Term.binderTactic
     |>.insert ``Lean.Parser.Term.dynamicQuot
@@ -219,7 +218,7 @@ def unusedTacticLinter : Linter where run := withSetOptionIn fun stx => do
   let key (r : String.Range) := (r.start.byteIdx, (-r.stop.byteIdx : Int))
   let mut last : String.Range := ⟨0, 0⟩
   for (r, stx) in let _ := @lexOrd; let _ := @ltOfOrd.{0}; unused.qsort (key ·.1 < key ·.1) do
-    if stx.getKind ∈ [``Batteries.Tactic.unreachable, ``Batteries.Tactic.unreachableConv] then
+    if stx.getKind ∈ [`Batteries.Tactic.unreachable, `Batteries.Tactic.unreachableConv] then
       continue
     if last.start ≤ r.start && r.stop ≤ last.stop then continue
     Linter.logLint linter.unusedTactic stx m!"'{stx}' tactic does nothing"
diff --git a/Mathlib/Tactic/MinImports.lean b/Mathlib/Tactic/MinImports.lean
index 788d4f9df7257..f079b40e3b93d 100644
--- a/Mathlib/Tactic/MinImports.lean
+++ b/Mathlib/Tactic/MinImports.lean
@@ -160,7 +160,7 @@ def getAllImports (cmd id : Syntax) (dbg? : Bool := false) :
               |>.append (getSyntaxNodeKinds cmd)
               |>.append (getAttrs env cmd)
   if dbg? then dbg_trace "{ts.toArray.qsort Name.lt}"
-  let mut hm : HashMap Nat Name := {}
+  let mut hm : Std.HashMap Nat Name := {}
   for imp in env.header.moduleNames do
     hm := hm.insert ((env.getModuleIdx? imp).getD default) imp
   let mut fins : NameSet := {}
@@ -168,7 +168,7 @@ def getAllImports (cmd id : Syntax) (dbg? : Bool := false) :
     let tns := t1::(← resolveGlobalName t1).map Prod.fst
     for t in tns do
       let new := match env.getModuleIdxFor? t with
-        | some t => (hm.find? t).get!
+        | some t => (hm.get? t).get!
         | none   => .anonymous -- instead of `getMainModule`, we omit the current module
       if !fins.contains new then fins := fins.insert new
   return fins.erase .anonymous
diff --git a/Mathlib/Tactic/ModCases.lean b/Mathlib/Tactic/ModCases.lean
index 9846cca299457..fd1a5afdd358f 100644
--- a/Mathlib/Tactic/ModCases.lean
+++ b/Mathlib/Tactic/ModCases.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro, Heather Macbeth
 -/
 import Mathlib.Data.Int.ModEq
+import Mathlib.Tactic.HaveI
 
 /-! # `mod_cases` tactic
 
diff --git a/Mathlib/Tactic/Module.lean b/Mathlib/Tactic/Module.lean
new file mode 100644
index 0000000000000..3d0d70d2126ca
--- /dev/null
+++ b/Mathlib/Tactic/Module.lean
@@ -0,0 +1,656 @@
+/-
+Copyright (c) 2024 Heather Macbeth. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Heather Macbeth
+-/
+import Mathlib.Algebra.Algebra.Tower
+import Mathlib.Algebra.BigOperators.GroupWithZero.Action
+import Mathlib.Tactic.Ring
+import Mathlib.Util.AtomM
+
+/-! # A tactic for normalization over modules
+
+This file provides the two tactics `match_scalars` and `module`.  Given a goal which is an equality
+in a type `M` (with `M` an `AddCommMonoid`), the `match_scalars` tactic parses the LHS and RHS of
+the goal as linear combinations of `M`-atoms over some semiring `R`, and reduces the goal to
+the respective equalities of the `R`-coefficients of each atom.  The `module` tactic does this and
+then runs the `ring` tactic on each of these coefficient-wise equalities, failing if this does not
+resolve them.
+
+The scalar type `R` is not pre-determined: instead it starts as `ℕ` (when each atom is initially
+given a scalar `(1:ℕ)`) and gets bumped up into bigger semirings when such semirings are
+encountered.  However, to permit this, it is assumed that there is a "linear order" on all the
+semirings which appear in the expression: for any two semirings `R` and `S` which occur, we have
+either `Algebra R S` or `Algebra S R`).
+-/
+
+open Lean hiding Module
+open Meta Elab Qq Mathlib.Tactic List
+
+namespace Mathlib.Tactic.Module
+
+/-! ### Theory of lists of pairs (scalar, vector)
+
+This section contains the lemmas which are orchestrated by the `match_scalars` and `module` tactics
+to prove goals in modules.  The basic object which these lemmas concern is `NF R M`, a type synonym
+for a list of ordered pairs in `R × M`, where typically `M` is an `R`-module.
+-/
+
+/-- Basic theoretical "normal form" object of the `match_scalars` and `module` tactics: a type
+synonym for a list of ordered pairs in `R × M`, where typically `M` is an `R`-module.  This is the
+form to which the tactics reduce module expressions.
+
+(It is not a full "normal form" because the scalars, i.e. `R` components, are not themselves
+ring-normalized.  But this partial normal form is more convenient for our purposes.)  -/
+def NF (R : Type*) (M : Type*) := List (R × M)
+
+namespace NF
+variable {S : Type*} {R : Type*} {M : Type*}
+
+/-- Augment a `Module.NF R M` object `l`, i.e. a list of pairs in `R × M`, by prepending another
+pair `p : R × M`. -/
+@[match_pattern]
+def cons (p : R × M) (l : NF R M) : NF R M := p :: l
+
+@[inherit_doc cons] infixl:100 " ::ᵣ " => cons
+
+/-- Evaluate a `Module.NF R M` object `l`, i.e. a list of pairs in `R × M`, to an element of `M`, by
+forming the "linear combination" it specifies: scalar-multiply each `R` term to the corresponding
+`M` term, then add them all up. -/
+def eval [Add M] [Zero M] [SMul R M] (l : NF R M) : M := (l.map (fun (⟨r, x⟩ : R × M) ↦ r • x)).sum
+
+@[simp] theorem eval_cons [AddMonoid M] [SMul R M] (p : R × M) (l : NF R M) :
+    (p ::ᵣ l).eval = p.1 • p.2 + l.eval := by
+  unfold eval cons
+  rw [List.map_cons]
+  rw [List.sum_cons]
+
+theorem atom_eq_eval [AddMonoid M] (x : M) : x = NF.eval [(1, x)] := by simp [eval]
+
+variable (M) in
+theorem zero_eq_eval [AddMonoid M] : (0:M) = NF.eval (R := ℕ) (M := M) [] := rfl
+
+theorem add_eq_eval₁ [AddMonoid M] [SMul R M] (a₁ : R × M) {a₂ : R × M} {l₁ l₂ l : NF R M}
+    (h : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval) :
+    (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval := by
+  simp only [eval_cons, ← h, add_assoc]
+
+theorem add_eq_eval₂ [Semiring R] [AddCommMonoid M] [Module R M] (r₁ r₂ : R) (x : M)
+    {l₁ l₂ l : NF R M} (h : l₁.eval + l₂.eval = l.eval) :
+    ((r₁, x) ::ᵣ l₁).eval + ((r₂, x) ::ᵣ l₂).eval = ((r₁ + r₂, x) ::ᵣ l).eval := by
+  simp only [← h, eval_cons, add_smul, add_assoc]
+  congr! 1
+  simp only [← add_assoc]
+  congr! 1
+  rw [add_comm]
+
+theorem add_eq_eval₃ [Semiring R] [AddCommMonoid M] [Module R M] {a₁ : R × M} (a₂ : R × M)
+    {l₁ l₂ l : NF R M} (h : (a₁ ::ᵣ l₁).eval + l₂.eval = l.eval) :
+    (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₂ ::ᵣ l).eval := by
+  simp only [eval_cons, ← h]
+  nth_rw 4 [add_comm]
+  simp only [add_assoc]
+  congr! 2
+  rw [add_comm]
+
+theorem add_eq_eval {R₁ R₂ : Type*} [AddCommMonoid M] [Semiring R] [Module R M] [Semiring R₁]
+    [Module R₁ M] [Semiring R₂] [Module R₂ M] {l₁ l₂ l : NF R M} {l₁' : NF R₁ M} {l₂' : NF R₂ M}
+    {x₁ x₂ : M} (hx₁ : x₁ = l₁'.eval) (hx₂ : x₂ = l₂'.eval) (h₁ : l₁.eval = l₁'.eval)
+    (h₂ : l₂.eval = l₂'.eval) (h : l₁.eval + l₂.eval = l.eval) :
+    x₁ + x₂ = l.eval := by
+  rw [hx₁, hx₂, ← h₁, ← h₂, h]
+
+theorem sub_eq_eval₁ [SMul R M] [AddGroup M] (a₁ : R × M) {a₂ : R × M} {l₁ l₂ l : NF R M}
+    (h : l₁.eval - (a₂ ::ᵣ l₂).eval = l.eval) :
+    (a₁ ::ᵣ l₁).eval - (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval := by
+  simp only [eval_cons, ← h, sub_eq_add_neg, add_assoc]
+
+theorem sub_eq_eval₂ [Ring R] [AddCommGroup M] [Module R M] (r₁ r₂ : R) (x : M) {l₁ l₂ l : NF R M}
+    (h : l₁.eval - l₂.eval = l.eval) :
+    ((r₁, x) ::ᵣ l₁).eval - ((r₂, x) ::ᵣ l₂).eval = ((r₁ - r₂, x) ::ᵣ l).eval := by
+  simp only [← h, eval_cons, sub_smul, sub_eq_add_neg, neg_add, add_smul, neg_smul, add_assoc]
+  congr! 1
+  simp only [← add_assoc]
+  congr! 1
+  rw [add_comm]
+
+theorem sub_eq_eval₃ [Ring R] [AddCommGroup M] [Module R M] {a₁ : R × M} (a₂ : R × M)
+    {l₁ l₂ l : NF R M} (h : (a₁ ::ᵣ l₁).eval - l₂.eval = l.eval) :
+    (a₁ ::ᵣ l₁).eval - (a₂ ::ᵣ l₂).eval = ((-a₂.1, a₂.2) ::ᵣ l).eval := by
+  simp only [eval_cons, neg_smul, neg_add, sub_eq_add_neg, ← h, ← add_assoc]
+  congr! 1
+  rw [add_comm, add_assoc]
+
+theorem sub_eq_eval {R₁ R₂ S₁ S₂ : Type*} [AddCommGroup M] [Ring R] [Module R M] [Semiring R₁]
+    [Module R₁ M] [Semiring R₂] [Module R₂ M] [Semiring S₁] [Module S₁ M] [Semiring S₂]
+    [Module S₂ M] {l₁ l₂ l : NF R M} {l₁' : NF R₁ M} {l₂' : NF R₂ M} {l₁'' : NF S₁ M}
+    {l₂'' : NF S₂ M} {x₁ x₂ : M} (hx₁ : x₁ = l₁''.eval) (hx₂ : x₂ = l₂''.eval)
+    (h₁' : l₁'.eval = l₁''.eval) (h₂' : l₂'.eval = l₂''.eval) (h₁ : l₁.eval = l₁'.eval)
+    (h₂ : l₂.eval = l₂'.eval) (h : l₁.eval - l₂.eval = l.eval) :
+    x₁ - x₂ = l.eval := by
+  rw [hx₁, hx₂, ← h₁', ← h₂', ← h₁, ← h₂, h]
+
+instance [Neg R] : Neg (NF R M) where
+  neg l := l.map fun (a, x) ↦ (-a, x)
+
+theorem eval_neg [AddCommGroup M] [Ring R] [Module R M] (l : NF R M) : (-l).eval = - l.eval := by
+  simp only [NF.eval, List.map_map, List.sum_neg, NF.instNeg]
+  congr
+  ext p
+  simp
+
+theorem zero_sub_eq_eval [AddCommGroup M] [Ring R] [Module R M] (l : NF R M) :
+    0 - l.eval = (-l).eval := by
+  simp [eval_neg]
+
+theorem neg_eq_eval [AddCommGroup M] [Semiring S] [Module S M] [Ring R] [Module R M] {l : NF R M}
+    {l₀ : NF S M} (hl : l.eval = l₀.eval) {x : M} (h : x = l₀.eval) :
+    - x = (-l).eval := by
+  rw [h, ← hl, eval_neg]
+
+instance [Mul R] : SMul R (NF R M) where
+  smul r l := l.map fun (a, x) ↦ (r * a, x)
+
+@[simp] theorem smul_apply [Mul R] (r : R) (l : NF R M) : r • l = l.map fun (a, x) ↦ (r * a, x) :=
+  rfl
+
+theorem eval_smul [AddCommMonoid M] [Semiring R] [Module R M] {l : NF R M} {x : M} (h : x = l.eval)
+    (r : R) : (r • l).eval = r • x := by
+  unfold NF.eval at h ⊢
+  simp only [h, smul_sum, map_map, NF.smul_apply]
+  congr
+  ext p
+  simp [mul_smul]
+
+theorem smul_eq_eval {R₀ : Type*} [AddCommMonoid M] [Semiring R] [Module R M] [Semiring R₀]
+    [Module R₀ M] [Semiring S] [Module S M]  {l : NF R M} {l₀ : NF R₀ M} {s : S} {r : R}
+    {x : M} (hx : x = l₀.eval) (hl : l.eval = l₀.eval) (hs : r • x = s • x) :
+    s • x = (r • l).eval := by
+  rw [← hs, hx, ← hl, eval_smul]
+  rfl
+
+theorem eq_cons_cons [AddMonoid M] [SMul R M] {r₁ r₂ : R} (m : M) {l₁ l₂ : NF R M} (h1 : r₁ = r₂)
+    (h2 : l₁.eval = l₂.eval) :
+    ((r₁, m) ::ᵣ l₁).eval = ((r₂, m) ::ᵣ l₂).eval := by
+  simp only [NF.eval, NF.cons] at *
+  simp [h1, h2]
+
+theorem eq_cons_const [AddCommMonoid M] [Semiring R] [Module R M] {r : R} (m : M) {n : M}
+    {l : NF R M} (h1 : r = 0) (h2 : l.eval = n) :
+    ((r, m) ::ᵣ l).eval = n := by
+  simp only [NF.eval, NF.cons] at *
+  simp [h1, h2]
+
+theorem eq_const_cons [AddCommMonoid M] [Semiring R] [Module R M] {r : R} (m : M) {n : M}
+    {l : NF R M} (h1 : 0 = r) (h2 : n = l.eval) :
+    n = ((r, m) ::ᵣ l).eval := by
+  simp only [NF.eval, NF.cons] at *
+  simp [← h1, h2]
+
+theorem eq_of_eval_eq_eval {R₁ R₂ : Type*} [AddCommMonoid M] [Semiring R] [Module R M] [Semiring R₁]
+    [Module R₁ M] [Semiring R₂] [Module R₂ M]  {l₁ l₂ : NF R M} {l₁' : NF R₁ M} {l₂' : NF R₂ M}
+    {x₁ x₂ : M} (hx₁ : x₁ = l₁'.eval) (hx₂ : x₂ = l₂'.eval) (h₁ : l₁.eval = l₁'.eval)
+    (h₂ : l₂.eval = l₂'.eval) (h : l₁.eval = l₂.eval) :
+    x₁ = x₂ := by
+  rw [hx₁, hx₂, ← h₁, ← h₂, h]
+
+variable (R)
+
+/-- Operate on a `Module.NF S M` object `l`, i.e. a list of pairs in `S × M`, where `S` is some
+commutative semiring, by applying to each `S`-component the algebra-map from `S` into a specified
+`S`-algebra `R`. -/
+def algebraMap [CommSemiring S] [Semiring R] [Algebra S R] (l : NF S M) : NF R M :=
+  l.map (fun ⟨s, x⟩ ↦ (_root_.algebraMap S R s, x))
+
+theorem eval_algebraMap [CommSemiring S] [Semiring R] [Algebra S R] [AddMonoid M] [SMul S M]
+    [MulAction R M] [IsScalarTower S R M] (l : NF S M) :
+    (l.algebraMap R).eval = l.eval := by
+  simp only [NF.eval, algebraMap, map_map]
+  congr
+  ext
+  simp [IsScalarTower.algebraMap_smul]
+
+end NF
+
+variable {u v : Level}
+
+/-! ### Lists of expressions representing scalars and vectors, and operations on such lists -/
+
+/-- Basic meta-code "normal form" object of the `match_scalars` and `module` tactics: a type synonym
+for a list of ordered triples comprising expressions representing terms of two types `R` and `M`
+(where typically `M` is an `R`-module), together with a natural number "index".
+
+The natural number represents the index of the `M` term in the `AtomM` monad: this is not enforced,
+but is sometimes assumed in operations.  Thus when items `((a₁, x₁), k)` and `((a₂, x₂), k)`
+appear in two different `Module.qNF` objects (i.e. with the same `ℕ`-index `k`), it is expected that
+the expressions `x₁` and `x₂` are the same.  It is also expected that the items in a `Module.qNF`
+list are in strictly increasing order by natural-number index.
+
+By forgetting the natural number indices, an expression representing a `Mathlib.Tactic.Module.NF`
+object can be built from a `Module.qNF` object; this construction is provided as
+`Mathlib.Tactic.Module.qNF.toNF`. -/
+abbrev qNF (R : Q(Type u)) (M : Q(Type v)) := List ((Q($R) × Q($M)) × ℕ)
+
+namespace qNF
+
+variable {M : Q(Type v)} {R : Q(Type u)}
+
+/-- Given `l` of type `qNF R M`, i.e. a list of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s and a natural
+number), build an `Expr` representing an object of type `NF R M` (i.e. `List (R × M)`) in the
+in the obvious way: by forgetting the natural numbers and gluing together the `Expr`s. -/
+def toNF (l : qNF R M) : Q(NF $R $M) :=
+  let l' : List Q($R × $M) := (l.map Prod.fst).map (fun (a, x) ↦ q(($a, $x)))
+  let qt : List Q($R × $M) → Q(List ($R × $M)) := List.rec q([]) (fun e _ l ↦ q($e ::ᵣ $l))
+  qt l'
+
+/-- Given `l` of type `qNF R₁ M`, i.e. a list of `(Q($R₁) × Q($M)) × ℕ`s (two `Expr`s and a natural
+number), apply an expression representing a function with domain `R₁` to each of the `Q($R₁)`
+components. -/
+def onScalar {u₁ u₂ : Level} {R₁ : Q(Type u₁)} {R₂ : Q(Type u₂)} (l : qNF R₁ M) (f : Q($R₁ → $R₂)) :
+    qNF R₂ M :=
+  l.map fun ((a, x), k) ↦ ((q($f $a), x), k)
+
+/-- Given two terms `l₁`, `l₂` of type `qNF R M`, i.e. lists of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s
+and a natural number), construct another such term `l`, which will have the property that in the
+`$R`-module `$M`, the sum of the "linear combinations" represented by `l₁` and `l₂` is the linear
+combination represented by `l`.
+
+The construction assumes, to be valid, that the lists `l₁` and `l₂` are in strictly increasing order
+by `ℕ`-component, and that if pairs `(a₁, x₁)` and `(a₂, x₂)` appear in `l₁`, `l₂` respectively with
+the same `ℕ`-component `k`, then the expressions `x₁` and `x₂` are equal.
+
+The construction is as follows: merge the two lists, except that if pairs `(a₁, x₁)` and `(a₂, x₂)`
+appear in `l₁`, `l₂` respectively with the same `ℕ`-component `k`, then contribute a term
+`(a₁ + a₂, x₁)` to the output list with `ℕ`-component `k`. -/
+def add (iR : Q(Semiring $R)) : qNF R M → qNF R M → qNF R M
+  | [], l => l
+  | l, [] => l
+  | ((a₁, x₁), k₁) ::ᵣ t₁, ((a₂, x₂), k₂) ::ᵣ t₂ =>
+    if k₁ < k₂ then
+      ((a₁, x₁), k₁) ::ᵣ add iR t₁ (((a₂, x₂), k₂) ::ᵣ t₂)
+    else if k₁ = k₂ then
+      ((q($a₁ + $a₂), x₁), k₁) ::ᵣ add iR t₁ t₂
+    else
+      ((a₂, x₂), k₂) ::ᵣ add iR (((a₁, x₁), k₁) ::ᵣ t₁) t₂
+
+/-- Given two terms `l₁`, `l₂` of type `qNF R M`, i.e. lists of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s
+and a natural number), recursively construct a proof that in the `$R`-module `$M`, the sum of the
+"linear combinations" represented by `l₁` and `l₂` is the linear combination represented by
+`Module.qNF.add iR l₁ l₁`.-/
+def mkAddProof {iR : Q(Semiring $R)} {iM : Q(AddCommMonoid $M)} (iRM : Q(Module $R $M))
+    (l₁ l₂ : qNF R M) :
+    Q(NF.eval $(l₁.toNF) + NF.eval $(l₂.toNF) = NF.eval $((qNF.add iR l₁ l₂).toNF)) :=
+  match l₁, l₂ with
+  | [], l => (q(zero_add (NF.eval $(l.toNF))):)
+  | l, [] => (q(add_zero (NF.eval $(l.toNF))):)
+  | ((a₁, x₁), k₁) ::ᵣ t₁, ((a₂, x₂), k₂) ::ᵣ t₂ =>
+    if k₁ < k₂ then
+      let pf := mkAddProof iRM t₁ (((a₂, x₂), k₂) ::ᵣ t₂)
+      (q(NF.add_eq_eval₁ ($a₁, $x₁) $pf):)
+    else if k₁ = k₂ then
+      let pf := mkAddProof iRM t₁ t₂
+      (q(NF.add_eq_eval₂ $a₁ $a₂ $x₁ $pf):)
+    else
+      let pf := mkAddProof iRM (((a₁, x₁), k₁) ::ᵣ t₁) t₂
+      (q(NF.add_eq_eval₃ ($a₂, $x₂) $pf):)
+
+/-- Given two terms `l₁`, `l₂` of type `qNF R M`, i.e. lists of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s
+and a natural number), construct another such term `l`, which will have the property that in the
+`$R`-module `$M`, the difference of the "linear combinations" represented by `l₁` and `l₂` is the
+linear combination represented by `l`.
+
+The construction assumes, to be valid, that the lists `l₁` and `l₂` are in strictly increasing order
+by `ℕ`-component, and that if pairs `(a₁, x₁)` and `(a₂, x₂)` appear in `l₁`, `l₂` respectively with
+the same `ℕ`-component `k`, then the expressions `x₁` and `x₂` are equal.
+
+The construction is as follows: merge the first list and the negation of the second list, except
+that if pairs `(a₁, x₁)` and `(a₂, x₂)` appear in `l₁`, `l₂` respectively with the same
+`ℕ`-component `k`, then contribute a term `(a₁ - a₂, x₁)` to the output list with `ℕ`-component `k`.
+-/
+def sub (iR : Q(Ring $R)) : qNF R M → qNF R M → qNF R M
+  | [], l => l.onScalar q(Neg.neg)
+  | l, [] => l
+  | ((a₁, x₁), k₁) ::ᵣ t₁, ((a₂, x₂), k₂) ::ᵣ t₂ =>
+    if k₁ < k₂ then
+      ((a₁, x₁), k₁) ::ᵣ sub iR t₁ (((a₂, x₂), k₂) ::ᵣ t₂)
+    else if k₁ = k₂ then
+      ((q($a₁ - $a₂), x₁), k₁) ::ᵣ sub iR t₁ t₂
+    else
+      ((q(-$a₂), x₂), k₂) ::ᵣ sub iR (((a₁, x₁), k₁) ::ᵣ t₁) t₂
+
+/-- Given two terms `l₁`, `l₂` of type `qNF R M`, i.e. lists of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s
+and a natural number), recursively construct a proof that in the `$R`-module `$M`, the difference
+of the "linear combinations" represented by `l₁` and `l₂` is the linear combination represented by
+`Module.qNF.sub iR l₁ l₁`.-/
+def mkSubProof (iR : Q(Ring $R)) (iM : Q(AddCommGroup $M)) (iRM : Q(Module $R $M))
+    (l₁ l₂ : qNF R M) :
+    Q(NF.eval $(l₁.toNF) - NF.eval $(l₂.toNF) = NF.eval $((qNF.sub iR l₁ l₂).toNF)) :=
+  match l₁, l₂ with
+  | [], l => (q(NF.zero_sub_eq_eval $(l.toNF)):)
+  | l, [] => (q(sub_zero (NF.eval $(l.toNF))):)
+  | ((a₁, x₁), k₁) ::ᵣ t₁, ((a₂, x₂), k₂) ::ᵣ t₂ =>
+    if k₁ < k₂ then
+      let pf := mkSubProof iR iM iRM t₁ (((a₂, x₂), k₂) ::ᵣ t₂)
+      (q(NF.sub_eq_eval₁ ($a₁, $x₁) $pf):)
+    else if k₁ = k₂ then
+      let pf := mkSubProof iR iM iRM t₁ t₂
+      (q(NF.sub_eq_eval₂ $a₁ $a₂ $x₁ $pf):)
+    else
+      let pf := mkSubProof iR iM iRM (((a₁, x₁), k₁) ::ᵣ t₁) t₂
+      (q(NF.sub_eq_eval₃ ($a₂, $x₂) $pf):)
+
+variable {iM : Q(AddCommMonoid $M)}
+  {u₁ : Level} {R₁ : Q(Type u₁)} {iR₁ : Q(Semiring $R₁)} (iRM₁ : Q(@Module $R₁ $M $iR₁ $iM))
+  {u₂ : Level} {R₂ : Q(Type u₂)} (iR₂ : Q(Semiring $R₂)) (iRM₂ : Q(@Module $R₂ $M $iR₂ $iM))
+
+/-- Given an expression `M` representing a type which is an `AddCommMonoid` and a module over *two*
+semirings `R₁` and `R₂`, find the "bigger" of the two semirings.  That is, we assume that it will
+turn out to be the case that either (1) `R₁` is an `R₂`-algebra and the `R₂` scalar action on `M` is
+induced from `R₁`'s scalar action on `M`, or (2) vice versa; we return the semiring `R₁` in the
+first case and `R₂` in the second case.
+
+Moreover, given expressions representing particular scalar multiplications of `R₁` and/or `R₂` on
+`M` (a `List (R₁ × M)`, a `List (R₂ × M)`, a pair `(r, x) : R₂ × M`), bump these up to the "big"
+ring by applying the algebra-map where needed. -/
+def matchRings (l₁ : qNF R₁ M) (l₂ : qNF R₂ M) (r : Q($R₂)) (x : Q($M)) :
+    MetaM <| Σ u : Level, Σ R : Q(Type u), Σ iR : Q(Semiring $R), Σ _ : Q(@Module $R $M $iR $iM),
+      (Σ l₁' : qNF R M, Q(NF.eval $(l₁'.toNF) = NF.eval $(l₁.toNF)))
+      × (Σ l₂' : qNF R M, Q(NF.eval $(l₂'.toNF) = NF.eval $(l₂.toNF)))
+      × (Σ r' : Q($R), Q($r' • $x = $r • $x)) := do
+  if ← withReducible <| isDefEq R₁ R₂ then
+  -- the case when `R₁ = R₂` is handled separately, so as not to require commutativity of that ring
+    pure ⟨u₁, R₁, iR₁, iRM₁, ⟨l₁, q(rfl)⟩, ⟨l₂, (q(@rfl _ (NF.eval $(l₂.toNF))):)⟩,
+      r, (q(@rfl _ ($r • $x)):)⟩
+  -- otherwise the "smaller" of the two rings must be commutative
+  else try
+    -- first try to exhibit `R₂` as an `R₁`-algebra
+    let _i₁ ← synthInstanceQ q(CommSemiring $R₁)
+    let _i₃ ← synthInstanceQ q(Algebra $R₁ $R₂)
+    let _i₄ ← synthInstanceQ q(IsScalarTower $R₁ $R₂ $M)
+    assumeInstancesCommute
+    let l₁' : qNF R₂ M := l₁.onScalar q(algebraMap $R₁ $R₂)
+    pure ⟨u₂, R₂, iR₂, iRM₂, ⟨l₁', (q(NF.eval_algebraMap $R₂ $(l₁.toNF)):)⟩, ⟨l₂, q(rfl)⟩,
+      r, q(rfl)⟩
+  catch _ => try
+    -- then if that fails, try to exhibit `R₁` as an `R₂`-algebra
+    let _i₁ ← synthInstanceQ q(CommSemiring $R₂)
+    let _i₃ ← synthInstanceQ q(Algebra $R₂ $R₁)
+    let _i₄ ← synthInstanceQ q(IsScalarTower $R₂ $R₁ $M)
+    assumeInstancesCommute
+    let l₂' : qNF R₁ M := l₂.onScalar q(algebraMap $R₂ $R₁)
+    let r' : Q($R₁) := q(algebraMap $R₂ $R₁ $r)
+    pure ⟨u₁, R₁, iR₁, iRM₁, ⟨l₁, q(rfl)⟩, ⟨l₂', (q(NF.eval_algebraMap $R₁ $(l₂.toNF)):)⟩,
+      r', (q(IsScalarTower.algebraMap_smul $R₁ $r $x):)⟩
+  catch _ =>
+    throwError "match_scalars failed: {R₁} is not an {R₂}-algebra and {R₂} is not an {R₁}-algebra"
+
+end qNF
+
+/-! ### Core of the `module` tactic -/
+
+variable {M : Q(Type v)}
+
+/-- The main algorithm behind the `match_scalars` and `module` tactics: partially-normalizing an
+expression in an additive commutative monoid `M` into the form c1 • x1 + c2 • x2 + ... c_k • x_k,
+where x1, x2, ... are distinct atoms in `M`, and c1, c2, ... are scalars. The scalar type of the
+expression is not pre-determined: instead it starts as `ℕ` (when each atom is initially given a
+scalar `(1:ℕ)`) and gets bumped up into bigger semirings when such semirings are encountered.
+
+It is assumed that there is a "linear order" on all the semirings which appear in the expression:
+for any two semirings `R` and `S` which occur, we have either `Algebra R S` or `Algebra S R`).
+
+TODO: implement a variant in which a semiring `R` is provided by the user, and the assumption is
+instead that for any semiring `S` which occurs, we have `Algebra S R`. The PR #16984 provides a
+proof-of-concept implementation of this variant, but it would need some polishing before joining
+Mathlib.
+
+Possible TODO, if poor performance on large problems is witnessed: switch the implementation from
+`AtomM` to `CanonM`, per the discussion
+https://github.com/leanprover-community/mathlib4/pull/16593/files#r1749623191 -/
+partial def parse (iM : Q(AddCommMonoid $M)) (x : Q($M)) :
+    AtomM (Σ u : Level, Σ R : Q(Type u), Σ iR : Q(Semiring $R), Σ _ : Q(@Module $R $M $iR $iM),
+      Σ l : qNF R M, Q($x = NF.eval $(l.toNF))) := do
+  match x with
+  /- parse an addition: `x₁ + x₂` -/
+  | ~q($x₁ + $x₂) =>
+    let ⟨_, _, _, iRM₁, l₁', pf₁'⟩ ← parse iM x₁
+    let ⟨_, _, _, iRM₂, l₂', pf₂'⟩ ← parse iM x₂
+    -- lift from the semirings of scalars parsed from `x₁`, `x₂` (say `R₁`, `R₂`) to `R₁ ⊗ R₂`
+    let ⟨u, R, iR, iRM, ⟨l₁, pf₁⟩, ⟨l₂, pf₂⟩, _⟩ ← qNF.matchRings iRM₁ _ iRM₂ l₁' l₂' q(0) q(0)
+    -- build the new list and proof
+    let pf := qNF.mkAddProof iRM l₁ l₂
+    pure ⟨u, R, iR, iRM, qNF.add iR l₁ l₂, (q(NF.add_eq_eval $pf₁' $pf₂' $pf₁ $pf₂ $pf):)⟩
+  /- parse a subtraction: `x₁ - x₂` -/
+  | ~q(@HSub.hSub _ _ _ (@instHSub _ $iM') $x₁ $x₂) =>
+    let ⟨_, _, _, iRM₁, l₁'', pf₁''⟩ ← parse iM x₁
+    let ⟨_, _, _, iRM₂, l₂'', pf₂''⟩ ← parse iM x₂
+    -- lift from the semirings of scalars parsed from `x₁`, `x₂` (say `R₁`, `R₂`) to `R₁ ⊗ R₂ ⊗ ℤ`
+    let iZ := q(Int.instSemiring)
+    let iMZ ← synthInstanceQ q(Module ℤ $M)
+    let ⟨_, _, _, iRM₁', ⟨l₁', pf₁'⟩, _, _⟩ ← qNF.matchRings iRM₁ iZ iMZ l₁'' [] q(0) q(0)
+    let ⟨_, _, _, iRM₂', ⟨l₂', pf₂'⟩, _, _⟩ ← qNF.matchRings iRM₂ iZ iMZ l₂'' [] q(0) q(0)
+    let ⟨u, R, iR, iRM, ⟨l₁, pf₁⟩, ⟨l₂, pf₂⟩, _⟩ ← qNF.matchRings iRM₁' _ iRM₂' l₁' l₂' q(0) q(0)
+    let iR' ← synthInstanceQ q(Ring $R)
+    let iM' ← synthInstanceQ q(AddCommGroup $M)
+    assumeInstancesCommute
+    -- build the new list and proof
+    let pf := qNF.mkSubProof iR' iM' iRM l₁ l₂
+    pure ⟨u, R, iR, iRM, qNF.sub iR' l₁ l₂,
+      q(NF.sub_eq_eval $pf₁'' $pf₂'' $pf₁' $pf₂' $pf₁ $pf₂ $pf)⟩
+  /- parse a negation: `-y` -/
+  | ~q(@Neg.neg _ $iM' $y) =>
+    let ⟨u₀, _, _, iRM₀, l₀, pf₀⟩ ← parse iM y
+    -- lift from original semiring of scalars (say `R₀`) to `R₀ ⊗ ℤ`
+    let _i ← synthInstanceQ q(AddCommGroup $M)
+    let iZ := q(Int.instSemiring)
+    let iMZ ← synthInstanceQ q(Module ℤ $M)
+    let ⟨u, R, iR, iRM, ⟨l, pf⟩, _, _⟩ ← qNF.matchRings iRM₀ iZ iMZ l₀ [] q(0) q(0)
+    let _i' ← synthInstanceQ q(Ring $R)
+    assumeInstancesCommute
+    -- build the new list and proof
+    pure ⟨u, R, iR, iRM, l.onScalar q(Neg.neg), (q(NF.neg_eq_eval $pf $pf₀):)⟩
+  /- parse a scalar multiplication: `(s₀ : S) • y` -/
+  | ~q(@HSMul.hSMul _ _ _ (@instHSMul $S _ $iS) $s₀ $y) =>
+    let ⟨_, _, _, iRM₀, l₀, pf₀⟩ ← parse iM y
+    let i₁ ← synthInstanceQ q(Semiring $S)
+    let i₂ ← synthInstanceQ q(Module $S $M)
+    assumeInstancesCommute
+    -- lift from original semiring of scalars (say `R₀`) to `R₀ ⊗ S`
+    let ⟨u, R, iR, iRM, ⟨l, pf_l⟩, _, ⟨s, pf_r⟩⟩ ← qNF.matchRings iRM₀ i₁ i₂ l₀ [] s₀ y
+    -- build the new list and proof
+    pure ⟨u, R, iR, iRM, l.onScalar q(HMul.hMul $s), (q(NF.smul_eq_eval $pf₀ $pf_l $pf_r):)⟩
+  /- parse a `(0:M)` -/
+  | ~q(0) =>
+    pure ⟨0, q(Nat), q(Nat.instSemiring), q(AddCommGroup.toNatModule), [], q(NF.zero_eq_eval $M)⟩
+  /- anything else should be treated as an atom -/
+  | _ =>
+    let k : ℕ ← AtomM.addAtom x
+    pure ⟨0, q(Nat), q(Nat.instSemiring), q(AddCommGroup.toNatModule), [((q(1), x), k)],
+      q(NF.atom_eq_eval $x)⟩
+
+/-- Given expressions `R` and `M` representing types such that `M`'s is a module over `R`'s, and
+given two terms `l₁`, `l₂` of type `qNF R M`, i.e. lists of `(Q($R) × Q($M)) × ℕ`s (two `Expr`s
+and a natural number), construct a list of new goals: that the `R`-coefficient of an `M`-atom which
+appears in only one list is zero, and that the `R`-coefficients of an `M`-atom which appears in both
+lists are equal.  Also construct (dependent on these new goals) a proof that the "linear
+combinations" represented by `l₁` and `l₂` are equal in `M`. -/
+partial def reduceCoefficientwise {R : Q(Type u)} {_ : Q(AddCommMonoid $M)} {_ : Q(Semiring $R)}
+    (iRM : Q(Module $R $M)) (l₁ l₂ : qNF R M) :
+    MetaM (List MVarId × Q(NF.eval $(l₁.toNF) = NF.eval $(l₂.toNF))) := do
+  match l₁, l₂ with
+  /- if both empty, return a `rfl` proof that `(0:M) = 0` -/
+  | [], [] =>
+    let pf : Q(NF.eval $(l₁.toNF) = NF.eval $(l₁.toNF)) := q(rfl)
+    pure ([], pf)
+  /- if one of the lists is empty and the other one is not, recurse down the nonempty one,
+    forming goals that each of the listed coefficents is equal to zero -/
+  | [], ((a, x), _) ::ᵣ L =>
+    let mvar : Q((0:$R) = $a) ← mkFreshExprMVar q((0:$R) = $a)
+    let (mvars, pf) ← reduceCoefficientwise iRM [] L
+    pure (mvar.mvarId! :: mvars, (q(NF.eq_const_cons $x $mvar $pf):))
+  | ((a, x), _) ::ᵣ L, [] =>
+    let mvar : Q($a = (0:$R)) ← mkFreshExprMVar q($a = (0:$R))
+    let (mvars, pf) ← reduceCoefficientwise iRM L []
+    pure (mvar.mvarId! :: mvars, (q(NF.eq_cons_const $x $mvar $pf):))
+  /- if both lists are nonempty, then deal with the numerically-smallest term in either list,
+    forming a goal that it is equal to zero (if it appears in only one list) or that its
+    coefficients in the two lists are the same (if it appears in both lists); then recurse -/
+  | ((a₁, x₁), k₁) ::ᵣ L₁, ((a₂, x₂), k₂) ::ᵣ L₂ =>
+    if k₁ < k₂ then
+      let mvar : Q($a₁ = (0:$R)) ← mkFreshExprMVar q($a₁ = (0:$R))
+      let (mvars, pf) ← reduceCoefficientwise iRM L₁ l₂
+      pure (mvar.mvarId! :: mvars, (q(NF.eq_cons_const $x₁ $mvar $pf):))
+    else if k₁ = k₂ then
+      let mvar : Q($a₁ = $a₂) ← mkFreshExprMVar q($a₁ = $a₂)
+      let (mvars, pf) ← reduceCoefficientwise iRM L₁ L₂
+      pure (mvar.mvarId! :: mvars, (q(NF.eq_cons_cons $x₁ $mvar $pf):))
+    else
+      let mvar : Q((0:$R) = $a₂) ← mkFreshExprMVar q((0:$R) = $a₂)
+      let (mvars, pf) ← reduceCoefficientwise iRM l₁ L₂
+      pure (mvar.mvarId! :: mvars, (q(NF.eq_const_cons $x₂ $mvar $pf):))
+
+/-- Given a goal which is an equality in a type `M` (with `M` an `AddCommMonoid`), parse the LHS and
+RHS of the goal as linear combinations of `M`-atoms over some semiring `R`, and reduce the goal to
+the respective equalities of the `R`-coefficients of each atom.
+
+This is an auxiliary function which produces slightly awkward goals in `R`; they are later cleaned
+up by the function `Mathlib.Tactic.Module.postprocess`. -/
+def matchScalarsAux (g : MVarId) : AtomM (List MVarId) := do
+  /- Parse the goal as an equality in a type `M` of two expressions `lhs` and `rhs`, with `M`
+  carrying an `AddCommMonoid` instance. -/
+  let eqData ← do
+    match (← g.getType').eq? with
+    | some e => pure e
+    | none => throwError "goal {← g.getType} is not an equality"
+  let .sort v₀ ← whnf (← inferType eqData.1) | unreachable!
+  let some v := v₀.dec | unreachable!
+  let ((M : Q(Type v)), (lhs : Q($M)), (rhs :Q($M))) := eqData
+  let iM ← synthInstanceQ q(AddCommMonoid.{v} $M)
+  /- Construct from the `lhs` expression a term `l₁` of type `qNF R₁ M` for some semiring `R₁` --
+  that is, a list of `(Q($R₁) × Q($M)) × ℕ`s (two `Expr`s and a natural number) -- together with a
+  proof that `lhs` is equal to the `R₁`-linear combination in `M` this represents. -/
+  let e₁ ← parse iM lhs
+  have u₁ : Level := e₁.fst
+  have R₁ : Q(Type u₁) := e₁.snd.fst
+  have _iR₁ : Q(Semiring.{u₁} $R₁) := e₁.snd.snd.fst
+  let iRM₁ ← synthInstanceQ q(Module $R₁ $M)
+  assumeInstancesCommute
+  have l₁ : qNF R₁ M := e₁.snd.snd.snd.snd.fst
+  let pf₁ : Q($lhs = NF.eval $(l₁.toNF)) := e₁.snd.snd.snd.snd.snd
+  /- Do the same for the `rhs` expression, obtaining a term `l₂` of type `qNF R₂ M` for some
+  semiring `R₂`. -/
+  let e₂ ← parse iM rhs
+  have u₂ : Level := e₂.fst
+  have R₂ : Q(Type u₂) := e₂.snd.fst
+  have _iR₂ : Q(Semiring.{u₂} $R₂) := e₂.snd.snd.fst
+  let iRM₂ ← synthInstanceQ q(Module $R₂ $M)
+  have l₂ : qNF R₂ M := e₂.snd.snd.snd.snd.fst
+  let pf₂ : Q($rhs = NF.eval $(l₂.toNF)) := e₂.snd.snd.snd.snd.snd
+  /- Lift everything to the same scalar ring, `R`. -/
+  let ⟨_, _, _, iRM, ⟨l₁', pf₁'⟩, ⟨l₂', pf₂'⟩, _⟩ ← qNF.matchRings iRM₁ _ iRM₂ l₁ l₂ q(0) q(0)
+  /- Construct a list of goals for the coefficientwise equality of these formal linear combinations,
+  and resolve our original goal (modulo these new goals). -/
+  let (mvars, pf) ← reduceCoefficientwise iRM l₁' l₂'
+  g.assign q(NF.eq_of_eval_eq_eval $pf₁ $pf₂ $pf₁' $pf₂' $pf)
+  return mvars
+
+/-- Lemmas used to post-process the result of the `match_scalars` and `module` tactics by converting
+the `algebraMap` operations which (which proliferate in the constructed scalar goals) to more
+familiar forms: `ℕ`, `ℤ` and `ℚ` casts. -/
+def algebraMapThms : Array Name := #[``eq_natCast, ``eq_intCast, ``eq_ratCast]
+
+/-- Postprocessing for the scalar goals constructed in the `match_scalars` and `module` tactics.
+These goals feature a proliferation of `algebraMap` operations (because the scalars start in `ℕ` and
+get successively bumped up by `algebraMap`s as new semirings are encountered), so we reinterpret the
+most commonly occuring `algebraMap`s (those out of `ℕ`, `ℤ` and `ℚ`) into their standard forms (`ℕ`,
+`ℤ` and `ℚ` casts) and then try to disperse the casts using the various `push_cast` lemmas. -/
+def postprocess (mvarId : MVarId) : MetaM MVarId := do
+  -- collect the available `push_cast` lemmas
+  let mut thms : SimpTheorems := ← NormCast.pushCastExt.getTheorems
+  -- augment this list with the `algebraMapThms` lemmas, which handle `algebraMap` operations
+  for thm in algebraMapThms do
+    let ⟨levelParams, _, proof⟩ ← abstractMVars (mkConst thm)
+    thms ← thms.add (.stx (← mkFreshId) Syntax.missing) levelParams proof
+  -- now run `simp` with these lemmas, and (importantly) *no* simprocs
+  let ctx : Simp.Context := {
+      config      := { failIfUnchanged := false }
+      simpTheorems := #[thms]
+    }
+  let (some r, _) ← simpTarget mvarId ctx (simprocs := #[]) |
+    throwError "internal error in match_scalars tactic: postprocessing should not close goals"
+  return r
+
+/-- Given a goal which is an equality in a type `M` (with `M` an `AddCommMonoid`), parse the LHS and
+RHS of the goal as linear combinations of `M`-atoms over some semiring `R`, and reduce the goal to
+the respective equalities of the `R`-coefficients of each atom. -/
+def matchScalars (g : MVarId) : MetaM (List MVarId) := do
+  let mvars ← AtomM.run .instances (matchScalarsAux g)
+  mvars.mapM postprocess
+
+/-- Given a goal which is an equality in a type `M` (with `M` an `AddCommMonoid`), parse the LHS and
+RHS of the goal as linear combinations of `M`-atoms over some semiring `R`, and reduce the goal to
+the respective equalities of the `R`-coefficients of each atom.
+
+For example, this produces the goal `⊢ a * 1 + b * 1 = (b + a) * 1`:
+```
+example [AddCommMonoid M] [Semiring R] [Module R M] (a b : R) (x : M) :
+    a • x + b • x = (b + a) • x := by
+  match_scalars
+```
+This produces the two goals `⊢ a * (a * 1) + b * (b * 1) = 1` (from the `x` atom) and
+`⊢ a * -(b * 1) + b * (a * 1) = 0` (from the `y` atom):
+```
+example [AddCommGroup M] [Ring R] [Module R M] (a b : R) (x : M) :
+    a • (a • x - b • y) + (b • a • y + b • b • x) = x := by
+  match_scalars
+```
+This produces the goal `⊢ -2 * (a * 1) = a * (-2 * 1)`:
+```
+example [AddCommGroup M] [Ring R] [Module R M] (a : R) (x : M) :
+    -(2:R) • a • x = a • (-2:ℤ) • x  := by
+  match_scalars
+```
+The scalar type for the goals produced by the `match_scalars` tactic is the largest scalar type
+encountered; for example, if `ℕ`, `ℚ` and a characteristic-zero field `K` all occur as scalars, then
+the goals produced are equalities in `K`.  A variant of `push_cast` is used internally in
+`match_scalars` to interpret scalars from the other types in this largest type.
+
+If the set of scalar types encountered is not totally ordered (in the sense that for all rings `R`,
+`S` encountered, it holds that either `Algebra R S` or `Algebra S R`), then the `match_scalars`
+tactic fails.
+-/
+elab "match_scalars" : tactic => Tactic.liftMetaTactic matchScalars
+
+/-- Given a goal which is an equality in a type `M` (with `M` an `AddCommMonoid`), parse the LHS and
+RHS of the goal as linear combinations of `M`-atoms over some commutative semiring `R`, and prove
+the goal by checking that the LHS- and RHS-coefficients of each atom are the same up to
+ring-normalization in `R`.
+
+(If the proofs of coefficient-wise equality will require more reasoning than just
+ring-normalization, use the tactic `match_scalars` instead, and then prove coefficient-wise equality
+by hand.)
+
+Example uses of the `module` tactic:
+```
+example [AddCommMonoid M] [CommSemiring R] [Module R M] (a b : R) (x : M) :
+    a • x + b • x = (b + a) • x := by
+  module
+
+example [AddCommMonoid M] [Field K] [CharZero K] [Module K M] (x : M) :
+    (2:K)⁻¹ • x + (3:K)⁻¹ • x + (6:K)⁻¹ • x = x := by
+  module
+
+example [AddCommGroup M] [CommRing R] [Module R M] (a : R) (v w : M) :
+    (1 + a ^ 2) • (v + w) - a • (a • v - w) = v + (1 + a + a ^ 2) • w := by
+  module
+
+example [AddCommGroup M] [CommRing R] [Module R M] (a b μ ν : R) (x y : M) :
+    (μ - ν) • a • x = (a • μ • x + b • ν • y) - ν • (a • x + b • y) := by
+  module
+```
+-/
+elab "module" : tactic => Tactic.liftMetaFinishingTactic fun g ↦ do
+  let l ← matchScalars g
+  discard <| l.mapM fun mvar ↦ AtomM.run .instances (Ring.proveEq mvar)
+
+end Mathlib.Tactic.Module
diff --git a/Mathlib/Tactic/NoncommRing.lean b/Mathlib/Tactic/NoncommRing.lean
index 46fd98f60bca4..f7c338b2c3b82 100644
--- a/Mathlib/Tactic/NoncommRing.lean
+++ b/Mathlib/Tactic/NoncommRing.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jireh Loreaux, Scott Morrison, Oliver Nash
+Authors: Jireh Loreaux, Kim Morrison, Oliver Nash
 -/
 import Mathlib.Algebra.Group.Action.Defs
 import Mathlib.Tactic.Abel
diff --git a/Mathlib/Tactic/NormNum/Basic.lean b/Mathlib/Tactic/NormNum/Basic.lean
index 5ab1887c9a653..f2921c511d2d3 100644
--- a/Mathlib/Tactic/NormNum/Basic.lean
+++ b/Mathlib/Tactic/NormNum/Basic.lean
@@ -196,11 +196,11 @@ theorem isRat_add {α} [Ring α] {f : α → α → α} {a b : α} {na nb nc : 
   have H := (Nat.cast_commute (α := α) da db).invOf_left.invOf_right.right_comm
   have h₁ := congr_arg (↑· * (⅟↑da * ⅟↑db : α)) h₁
   simp only [Int.cast_add, Int.cast_mul, Int.cast_natCast, ← mul_assoc,
-    add_mul, mul_mul_invOf_self_cancel] at h₁
+    add_mul, mul_invOf_cancel_right] at h₁
   have h₂ := congr_arg (↑nc * ↑· * (⅟↑da * ⅟↑db * ⅟↑dc : α)) h₂
-  simp only [H, mul_mul_invOf_self_cancel', Nat.cast_mul, ← mul_assoc] at h₁ h₂
+  simp only [H, mul_invOf_cancel_right', Nat.cast_mul, ← mul_assoc] at h₁ h₂
   rw [h₁, h₂, Nat.cast_commute]
-  simp only [mul_mul_invOf_self_cancel,
+  simp only [mul_invOf_cancel_right,
     (Nat.cast_commute (α := α) da dc).invOf_left.invOf_right.right_comm,
     (Nat.cast_commute (α := α) db dc).invOf_left.invOf_right.right_comm]
 
@@ -382,8 +382,8 @@ theorem isRat_mul {α} [Ring α] {f : α → α → α} {a b : α} {na nb nc : 
   simp only [← mul_assoc, (Nat.cast_commute (α := α) da nb).invOf_left.right_comm, h₁]
   have h₂ := congr_arg (↑nc * ↑· * (⅟↑da * ⅟↑db * ⅟↑dc : α)) h₂
   simp only [Nat.cast_mul, ← mul_assoc] at h₂; rw [H] at h₂
-  simp only [mul_mul_invOf_self_cancel'] at h₂; rw [h₂, Nat.cast_commute]
-  simp only [mul_mul_invOf_self_cancel,
+  simp only [mul_invOf_cancel_right'] at h₂; rw [h₂, Nat.cast_commute]
+  simp only [mul_invOf_cancel_right,
     (Nat.cast_commute (α := α) da dc).invOf_left.invOf_right.right_comm,
     (Nat.cast_commute (α := α) db dc).invOf_left.invOf_right.right_comm]
 
diff --git a/Mathlib/Tactic/NormNum/DivMod.lean b/Mathlib/Tactic/NormNum/DivMod.lean
index 3037be0ca7da7..821532050c549 100644
--- a/Mathlib/Tactic/NormNum/DivMod.lean
+++ b/Mathlib/Tactic/NormNum/DivMod.lean
@@ -147,8 +147,8 @@ theorem isInt_dvd_true : {a b : ℤ} → {a' b' c : ℤ} →
   | _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨_, rfl⟩
 
 theorem isInt_dvd_false : {a b : ℤ} → {a' b' : ℤ} →
-    IsInt a a' → IsInt b b' → Int.mod b' a' != 0 → ¬a ∣ b
-  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, e => mt Int.mod_eq_zero_of_dvd (by simpa using e)
+    IsInt a a' → IsInt b b' → Int.emod b' a' != 0 → ¬a ∣ b
+  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, e => mt Int.emod_eq_zero_of_dvd (by simpa using e)
 
 /-- The `norm_num` extension which identifies expressions of the form `(a : ℤ) ∣ b`,
 such that `norm_num` successfully recognises both `a` and `b`. -/
@@ -167,7 +167,7 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     haveI' : Int.mul $na $c =Q $nb := ⟨⟩
     return .isTrue q(isInt_dvd_true $pa $pb (.refl $nb))
   else
-    have : Q(Int.mod $nb $na != 0) := (q(Eq.refl true) : Expr)
+    have : Q(Int.emod $nb $na != 0) := (q(Eq.refl true) : Expr)
     return .isFalse q(isInt_dvd_false $pa $pb $this)
 
 end Mathlib.Meta.NormNum
diff --git a/Mathlib/Tactic/NormNum/Ineq.lean b/Mathlib/Tactic/NormNum/Ineq.lean
index 3a6b3579bea01..9c5a20797bcc2 100644
--- a/Mathlib/Tactic/NormNum/Ineq.lean
+++ b/Mathlib/Tactic/NormNum/Ineq.lean
@@ -62,7 +62,7 @@ theorem isRat_lt_true [LinearOrderedRing α] [Nontrivial α] : {a b : α} → {n
     have hb : 0 < ⅟(db : α) := pos_invOf_of_invertible_cast db
     have h := (mul_lt_mul_of_pos_left · hb) <| mul_lt_mul_of_pos_right h ha
     rw [← mul_assoc, Int.commute_cast] at h
-    simp? at h says simp only [Int.cast_mul, Int.cast_natCast, mul_mul_invOf_self_cancel'] at h
+    simp? at h says simp only [Int.cast_mul, Int.cast_natCast, mul_invOf_cancel_right'] at h
     rwa [Int.commute_cast] at h
 
 theorem isRat_le_false [LinearOrderedRing α] [Nontrivial α] {a b : α} {na nb : ℤ} {da db : ℕ}
@@ -108,12 +108,21 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
   let ⟨u, α, a⟩ ← inferTypeQ' a
   have b : Q($α) := b
   let ra ← derive a; let rb ← derive b
+  let lα ← synthInstanceQ q(LE $α)
+  guard <|← withNewMCtxDepth <| isDefEq f q(LE.le (α := $α))
+  core lα ra rb
+where
+  /-- Identify (as `true` or `false`) expressions of the form `a ≤ b`, where `a` and `b` are numeric
+  expressions whose evaluations to `NormNum.Result` have already been computed. -/
+  core {u : Level} {α : Q(Type u)} (lα : Q(LE $α)) {a b : Q($α)}
+    (ra : NormNum.Result a) (rb : NormNum.Result b) : MetaM (NormNum.Result q($a ≤ $b)) := do
+  let e := q($a ≤ $b)
   let rec intArm : MetaM (Result e) := do
     let _i ← inferOrderedRing α
-    guard <|← withNewMCtxDepth <| isDefEq f q(LE.le (α := $α))
     haveI' : $e =Q ($a ≤ $b) := ⟨⟩
     let ⟨za, na, pa⟩ ← ra.toInt q(OrderedRing.toRing)
     let ⟨zb, nb, pb⟩ ← rb.toInt q(OrderedRing.toRing)
+    assumeInstancesCommute
     if decide (za ≤ zb) then
       let r : Q(decide ($na ≤ $nb) = true) := (q(Eq.refl true) : Expr)
       return .isTrue q(isInt_le_true $pa $pb $r)
@@ -125,10 +134,10 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
   let rec ratArm : MetaM (Result e) := do
     -- We need a division ring with an order, and `LinearOrderedField` is the closest mathlib has.
     let _i ← inferLinearOrderedField α
-    guard <|← withNewMCtxDepth <| isDefEq f q(LE.le (α := $α))
     haveI' : $e =Q ($a ≤ $b) := ⟨⟩
     let ⟨qa, na, da, pa⟩ ← ra.toRat' q(Field.toDivisionRing)
     let ⟨qb, nb, db, pb⟩ ← rb.toRat' q(Field.toDivisionRing)
+    assumeInstancesCommute
     if decide (qa ≤ qb) then
       let r : Q(decide ($na * $db ≤ $nb * $da) = true) := (q(Eq.refl true) : Expr)
       return (.isTrue q(isRat_le_true $pa $pb $r))
@@ -144,8 +153,8 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     let _i ← inferOrderedSemiring α
     haveI' : $ra =Q by clear! $ra $rb; infer_instance := ⟨⟩
     haveI' : $rb =Q by clear! $ra $rb; infer_instance := ⟨⟩
-    guard <|← withNewMCtxDepth <| isDefEq f q(LE.le (α := $α))
     haveI' : $e =Q ($a ≤ $b) := ⟨⟩
+    assumeInstancesCommute
     if na.natLit! ≤ nb.natLit! then
       let r : Q(Nat.ble $na $nb = true) := (q(Eq.refl true) : Expr)
       return .isTrue q(isNat_le_true $pa $pb $r)
@@ -163,13 +172,21 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
   let ⟨u, α, a⟩ ← inferTypeQ' a
   have b : Q($α) := b
   let ra ← derive a; let rb ← derive b
+  let lα ← synthInstanceQ q(LT $α)
+  guard <|← withNewMCtxDepth <| isDefEq f q(LT.lt (α := $α))
+  core lα ra rb
+where
+  /-- Identify (as `true` or `false`) expressions of the form `a < b`, where `a` and `b` are numeric
+  expressions whose evaluations to `NormNum.Result` have already been computed. -/
+  core {u : Level} {α : Q(Type u)} (lα : Q(LT $α)) {a b : Q($α)}
+    (ra : NormNum.Result a) (rb : NormNum.Result b) : MetaM (NormNum.Result q($a < $b)) := do
+  let e := q($a < $b)
   let rec intArm : MetaM (Result e) := do
     let _i ← inferOrderedRing α
-    assumeInstancesCommute
-    guard <|← withNewMCtxDepth <| isDefEq f q(LT.lt (α := $α))
     haveI' : $e =Q ($a < $b) := ⟨⟩
     let ⟨za, na, pa⟩ ← ra.toInt q(OrderedRing.toRing)
     let ⟨zb, nb, pb⟩ ← rb.toInt q(OrderedRing.toRing)
+    assumeInstancesCommute
     if za < zb then
       if let .some _i ← trySynthInstanceQ (q(@Nontrivial $α) : Q(Prop)) then
         let r : Q(decide ($na < $nb) = true) := (q(Eq.refl true) : Expr)
@@ -184,7 +201,6 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     let _i ← inferLinearOrderedField α
     assumeInstancesCommute
     haveI' : $e =Q ($a < $b) := ⟨⟩
-    guard <|← withNewMCtxDepth <| isDefEq f q(LT.lt (α := $α))
     let ⟨qa, na, da, pa⟩ ← ra.toRat' q(Field.toDivisionRing)
     let ⟨qb, nb, db, pb⟩ ← rb.toRat' q(Field.toDivisionRing)
     if qa < qb then
@@ -202,7 +218,7 @@ such that `norm_num` successfully recognises both `a` and `b`. -/
     haveI' : $ra =Q by clear! $ra $rb; infer_instance := ⟨⟩
     haveI' : $rb =Q by clear! $ra $rb; infer_instance := ⟨⟩
     haveI' : $e =Q ($a < $b) := ⟨⟩
-    guard <|← withNewMCtxDepth <| isDefEq f q(LT.lt (α := $α))
+    assumeInstancesCommute
     if na.natLit! < nb.natLit! then
       if let .some _i ← trySynthInstanceQ q(CharZero $α) then
         let r : Q(Nat.ble $nb $na = false) := (q(Eq.refl false) : Expr)
diff --git a/Mathlib/Tactic/Peel.lean b/Mathlib/Tactic/Peel.lean
index b3ab2443710a5..0428aed375d7c 100644
--- a/Mathlib/Tactic/Peel.lean
+++ b/Mathlib/Tactic/Peel.lean
@@ -225,7 +225,7 @@ def peelArgsIff (l : List Name) : TacticM Unit := withMainContext do
 elab_rules : tactic
   | `(tactic| peel $[$num?:num]? $e:term $[with $l?* $n?]?) => withMainContext do
     /- we use `elabTermForApply` instead of `elabTerm` so that terms passed to `peel` can contain
-    quantifiers with implicit bound variables without causing errors or requiring `@`.  -/
+    quantifiers with implicit bound variables without causing errors or requiring `@`. -/
     let e ← elabTermForApply e false
     let n? := n?.bind fun n => if n.raw.isIdent then pure n.raw.getId else none
     let l := (l?.getD #[]).map getNameOfIdent' |>.toList
diff --git a/Mathlib/Tactic/Positivity/Core.lean b/Mathlib/Tactic/Positivity/Core.lean
index a03631e11fca8..5ea9541b1909c 100644
--- a/Mathlib/Tactic/Positivity/Core.lean
+++ b/Mathlib/Tactic/Positivity/Core.lean
@@ -122,7 +122,7 @@ variable {A : Type*} {e : A}
 lemma lt_of_le_of_ne' {a b : A} [PartialOrder A] :
     (a : A) ≤ b → b ≠ a → a < b := fun h₁ h₂ => lt_of_le_of_ne h₁ h₂.symm
 
-lemma pos_of_isNat {n : ℕ} [StrictOrderedSemiring A]
+lemma pos_of_isNat {n : ℕ} [OrderedSemiring A] [Nontrivial A]
     (h : NormNum.IsNat e n) (w : Nat.ble 1 n = true) : 0 < (e : A) := by
   rw [NormNum.IsNat.to_eq h rfl]
   apply Nat.cast_pos.2
@@ -184,11 +184,12 @@ def normNumPositivity (e : Q($α)) : MetaM (Strictness zα pα e) := catchNone d
   | .isBool .. => failure
   | .isNat _ lit p =>
     if 0 < lit.natLit! then
-      let _a ← synthInstanceQ q(StrictOrderedSemiring $α)
+      let _a ← synthInstanceQ q(OrderedSemiring $α)
+      let _a ← synthInstanceQ q(Nontrivial $α)
       assumeInstancesCommute
       have p : Q(NormNum.IsNat $e $lit) := p
       haveI' p' : Nat.ble 1 $lit =Q true := ⟨⟩
-      pure (.positive q(@pos_of_isNat $α _ _ _ $p $p'))
+      pure (.positive q(@pos_of_isNat $α _ _ _ _ $p $p'))
     else
       let _a ← synthInstanceQ q(OrderedSemiring $α)
       assumeInstancesCommute
diff --git a/Mathlib/Tactic/Propose.lean b/Mathlib/Tactic/Propose.lean
index 9ca99b07d8d03..4169210c6c5ce 100644
--- a/Mathlib/Tactic/Propose.lean
+++ b/Mathlib/Tactic/Propose.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Lean.Meta.Tactic.TryThis
 import Lean.Meta.Tactic.SolveByElim
diff --git a/Mathlib/Tactic/PushNeg.lean b/Mathlib/Tactic/PushNeg.lean
index ec451c4c090af..07699624b4fc5 100644
--- a/Mathlib/Tactic/PushNeg.lean
+++ b/Mathlib/Tactic/PushNeg.lean
@@ -5,11 +5,10 @@ Authors: Patrick Massot, Simon Hudon, Alice Laroche, Frédéric Dupuis, Jireh Lo
 -/
 
 import Lean.Elab.Tactic.Location
+import Mathlib.Data.Set.Defs
 import Mathlib.Logic.Basic
 import Mathlib.Order.Defs
 import Mathlib.Tactic.Conv
-import Mathlib.Data.Set.Defs
-import Lean.Elab.Tactic.Location
 
 /-!
 # The `push_neg` tactic
diff --git a/Mathlib/Tactic/RSuffices.lean b/Mathlib/Tactic/RSuffices.lean
index 22ca30eec31d5..ef0156c6581e3 100644
--- a/Mathlib/Tactic/RSuffices.lean
+++ b/Mathlib/Tactic/RSuffices.lean
@@ -13,6 +13,8 @@ of any syntax that would be valid in an `obtain` block. This tactic just calls `
 on the expression, and then `rotate_left`.
 -/
 
+namespace Mathlib.Tactic
+
 /--
 The `rsuffices` tactic is an alternative version of `suffices`, that allows the usage
 of any syntax that would be valid in an `obtain` block. This tactic just calls `obtain`
@@ -24,3 +26,5 @@ syntax (name := rsuffices) "rsuffices"
 macro_rules
 | `(tactic| rsuffices $[$pred]? $[: $foo]? $[:= $bar]?) =>
 `(tactic | (obtain $[$pred]? $[: $foo]? $[:= $bar]?; rotate_left))
+
+end Mathlib.Tactic
diff --git a/Mathlib/Tactic/Recover.lean b/Mathlib/Tactic/Recover.lean
index b330002cecfd5..abae8c5304d60 100644
--- a/Mathlib/Tactic/Recover.lean
+++ b/Mathlib/Tactic/Recover.lean
@@ -26,21 +26,21 @@ which occur in the target or local context or delayed assignment (if any) of
 `mvarId`, plus the metavariables which occur in these metavariables, etc.
 -/
 partial def getUnassignedGoalMVarDependencies (mvarId : MVarId) :
-    MetaM (HashSet MVarId) :=
+    MetaM (Std.HashSet MVarId) :=
   return (← go mvarId |>.run {}).snd
   where
     /-- auxiliary function for `getUnassignedGoalMVarDependencies` -/
-    addMVars (e : Expr) : StateRefT (HashSet MVarId) MetaM Unit := do
+    addMVars (e : Expr) : StateRefT (Std.HashSet MVarId) MetaM Unit := do
       let mvars ← getMVars e
       let mut s ← get
-      set ({} : HashSet MVarId) -- Ensure that `s` is not shared.
+      set ({} : Std.HashSet MVarId) -- Ensure that `s` is not shared.
       for mvarId in mvars do
         unless ← mvarId.isDelayedAssigned do
           s := s.insert mvarId
       set s
       mvars.forM go
     /-- auxiliary function for `getUnassignedGoalMVarDependencies` -/
-    go (mvarId : MVarId) : StateRefT (HashSet MVarId) MetaM Unit :=
+    go (mvarId : MVarId) : StateRefT (Std.HashSet MVarId) MetaM Unit :=
       withIncRecDepth do
         let mdecl ← mvarId.getDecl
         addMVars mdecl.type
@@ -60,7 +60,7 @@ that are not closed, starting from the original goal. -/
 elab "recover " tacs:tacticSeq : tactic => do
   let originalGoals ← getGoals
   evalTactic tacs
-  let mut unassigned : HashSet MVarId := {}
+  let mut unassigned : Std.HashSet MVarId := {}
   for mvarId in originalGoals do
     unless ← mvarId.isAssigned <||> mvarId.isDelayedAssigned do
       unassigned := unassigned.insert mvarId
diff --git a/Mathlib/Tactic/Relation/Symm.lean b/Mathlib/Tactic/Relation/Symm.lean
index 6057927cfb8b5..8d33854268714 100644
--- a/Mathlib/Tactic/Relation/Symm.lean
+++ b/Mathlib/Tactic/Relation/Symm.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Meta.Tactic.Symm
diff --git a/Mathlib/Tactic/RewriteSearch.lean b/Mathlib/Tactic/RewriteSearch.lean
index befd4ab7e0ee1..f91c389946783 100644
--- a/Mathlib/Tactic/RewriteSearch.lean
+++ b/Mathlib/Tactic/RewriteSearch.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Lean FRO, LLC. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Lean.Meta.Tactic.Rewrites
 import Mathlib.Algebra.Order.Group.Nat
diff --git a/Mathlib/Tactic/Ring/Basic.lean b/Mathlib/Tactic/Ring/Basic.lean
index bda62f7eef640..c5ac7fd426cd8 100644
--- a/Mathlib/Tactic/Ring/Basic.lean
+++ b/Mathlib/Tactic/Ring/Basic.lean
@@ -134,26 +134,28 @@ inductive ExSum : ∀ {u : Lean.Level} {α : Q(Type u)}, Q(CommSemiring $α) →
     ExProd sα a → ExSum sα b → ExSum sα q($a + $b)
 end
 
-
--- In this file, we would like to use multi-character auto-implicits.
-set_option autoImplicit true
-
 mutual -- partial only to speed up compilation
 
 /-- Equality test for expressions. This is not a `BEq` instance because it is heterogeneous. -/
-partial def ExBase.eq : ExBase sα a → ExBase sα b → Bool
+partial def ExBase.eq
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExBase sα a → ExBase sα b → Bool
   | .atom i, .atom j => i == j
   | .sum a, .sum b => a.eq b
   | _, _ => false
 
 @[inherit_doc ExBase.eq]
-partial def ExProd.eq : ExProd sα a → ExProd sα b → Bool
+partial def ExProd.eq
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExProd sα a → ExProd sα b → Bool
   | .const i _, .const j _ => i == j
   | .mul a₁ a₂ a₃, .mul b₁ b₂ b₃ => a₁.eq b₁ && a₂.eq b₂ && a₃.eq b₃
   | _, _ => false
 
 @[inherit_doc ExBase.eq]
-partial def ExSum.eq : ExSum sα a → ExSum sα b → Bool
+partial def ExSum.eq
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExSum sα a → ExSum sα b → Bool
   | .zero, .zero => true
   | .add a₁ a₂, .add b₁ b₂ => a₁.eq b₁ && a₂.eq b₂
   | _, _ => false
@@ -164,28 +166,34 @@ mutual -- partial only to speed up compilation
 A total order on normalized expressions.
 This is not an `Ord` instance because it is heterogeneous.
 -/
-partial def ExBase.cmp : ExBase sα a → ExBase sα b → Ordering
+partial def ExBase.cmp
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExBase sα a → ExBase sα b → Ordering
   | .atom i, .atom j => compare i j
   | .sum a, .sum b => a.cmp b
   | .atom .., .sum .. => .lt
   | .sum .., .atom .. => .gt
 
 @[inherit_doc ExBase.cmp]
-partial def ExProd.cmp : ExProd sα a → ExProd sα b → Ordering
+partial def ExProd.cmp
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExProd sα a → ExProd sα b → Ordering
   | .const i _, .const j _ => compare i j
   | .mul a₁ a₂ a₃, .mul b₁ b₂ b₃ => (a₁.cmp b₁).then (a₂.cmp b₂) |>.then (a₃.cmp b₃)
   | .const _ _, .mul .. => .lt
   | .mul .., .const _ _ => .gt
 
 @[inherit_doc ExBase.cmp]
-partial def ExSum.cmp : ExSum sα a → ExSum sα b → Ordering
+partial def ExSum.cmp
+    {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)} {a b : Q($α)} :
+    ExSum sα a → ExSum sα b → Ordering
   | .zero, .zero => .eq
   | .add a₁ a₂, .add b₁ b₂ => (a₁.cmp b₁).then (a₂.cmp b₂)
   | .zero, .add .. => .lt
   | .add .., .zero => .gt
 end
 
-variable {u : Lean.Level} {arg : Q(Type u)} {sα : Q(CommSemiring $arg)}
+variable {u : Lean.Level} {α : Q(Type u)} {sα : Q(CommSemiring $α)}
 
 instance : Inhabited (Σ e, (ExBase sα) e) := ⟨default, .atom 0⟩
 instance : Inhabited (Σ e, (ExSum sα) e) := ⟨_, .zero⟩
@@ -194,24 +202,28 @@ instance : Inhabited (Σ e, (ExProd sα) e) := ⟨default, .const 0 none⟩
 mutual
 
 /-- Converts `ExBase sα` to `ExBase sβ`, assuming `sα` and `sβ` are defeq. -/
-partial def ExBase.cast {a : Q($arg)} : ExBase sα a → Σ a, ExBase sβ a
+partial def ExBase.cast
+    {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring $β)} {a : Q($α)} :
+    ExBase sα a → Σ a, ExBase sβ a
   | .atom i => ⟨a, .atom i⟩
   | .sum a => let ⟨_, vb⟩ := a.cast; ⟨_, .sum vb⟩
 
 /-- Converts `ExProd sα` to `ExProd sβ`, assuming `sα` and `sβ` are defeq. -/
-partial def ExProd.cast {a : Q($arg)} : ExProd sα a → Σ a, ExProd sβ a
+partial def ExProd.cast
+    {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring $β)} {a : Q($α)} :
+    ExProd sα a → Σ a, ExProd sβ a
   | .const i h => ⟨a, .const i h⟩
   | .mul a₁ a₂ a₃ => ⟨_, .mul a₁.cast.2 a₂ a₃.cast.2⟩
 
 /-- Converts `ExSum sα` to `ExSum sβ`, assuming `sα` and `sβ` are defeq. -/
-partial def ExSum.cast {a : Q($arg)} : ExSum sα a → Σ a, ExSum sβ a
+partial def ExSum.cast
+    {v : Lean.Level} {β : Q(Type v)} {sβ : Q(CommSemiring $β)} {a : Q($α)} :
+    ExSum sα a → Σ a, ExSum sβ a
   | .zero => ⟨_, .zero⟩
   | .add a₁ a₂ => ⟨_, .add a₁.cast.2 a₂.cast.2⟩
 
 end
 
-set_option autoImplicit false
-
 variable {u : Lean.Level}
 
 /--
@@ -295,6 +307,11 @@ theorem add_overlap_pf_zero (x : R) (e) :
     IsNat (a + b) (nat_lit 0) → IsNat (x ^ e * a + x ^ e * b) (nat_lit 0)
   | ⟨h⟩ => ⟨by simp [h, ← mul_add]⟩
 
+-- TODO: decide if this is a good idea globally in
+-- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/.60MonadLift.20Option.20.28OptionT.20m.29.60/near/469097834
+private local instance {m} [Pure m] : MonadLift Option (OptionT m) where
+  monadLift f := .mk <| pure f
+
 /--
 Given monomials `va, vb`, attempts to add them together to get another monomial.
 If the monomials are not compatible, returns `none`.
@@ -302,7 +319,8 @@ For example, `xy + 2xy = 3xy` is a `.nonzero` overlap, while `xy + xz` returns `
 and `xy + -xy = 0` is a `.zero` overlap.
 -/
 def evalAddOverlap {a b : Q($α)} (va : ExProd sα a) (vb : ExProd sα b) :
-    Option (Overlap sα q($a + $b)) :=
+    OptionT Lean.Core.CoreM (Overlap sα q($a + $b)) := do
+  Lean.Core.checkSystem decl_name%.toString
   match va, vb with
   | .const za ha, .const zb hb => do
     let ra := Result.ofRawRat za a ha; let rb := Result.ofRawRat zb b hb
@@ -319,7 +337,7 @@ def evalAddOverlap {a b : Q($α)} (va : ExProd sα a) (vb : ExProd sα b) :
     | .zero p => pure <| .zero (q(add_overlap_pf_zero $a₁ $a₂ $p) : Expr)
     | .nonzero ⟨_, vc, p⟩ =>
       pure <| .nonzero ⟨_, .mul va₁ va₂ vc, (q(add_overlap_pf $a₁ $a₂ $p) : Expr)⟩
-  | _, _ => none
+  | _, _ => OptionT.fail
 
 theorem add_pf_zero_add (b : R) : 0 + b = b := by simp
 
@@ -347,25 +365,26 @@ theorem add_pf_add_gt (b₁ : R) (_ : a + b₂ = c) : a + (b₁ + b₂) = b₁ +
 * `(a₁ + a₂) + (b₁ + b₂) = b₁ + ((a₁ + a₂) + b₂)` (if not `a₁.lt b₁`)
 -/
 partial def evalAdd {a b : Q($α)} (va : ExSum sα a) (vb : ExSum sα b) :
-    Result (ExSum sα) q($a + $b) :=
+    Lean.Core.CoreM <| Result (ExSum sα) q($a + $b) := do
+  Lean.Core.checkSystem decl_name%.toString
   match va, vb with
-  | .zero, vb => ⟨b, vb, q(add_pf_zero_add $b)⟩
-  | va, .zero => ⟨a, va, q(add_pf_add_zero $a)⟩
+  | .zero, vb => return ⟨b, vb, q(add_pf_zero_add $b)⟩
+  | va, .zero => return ⟨a, va, q(add_pf_add_zero $a)⟩
   | .add (a := a₁) (b := _a₂) va₁ va₂, .add (a := b₁) (b := _b₂) vb₁ vb₂ =>
-    match evalAddOverlap sα va₁ vb₁ with
+    match ← (evalAddOverlap sα va₁ vb₁).run with
     | some (.nonzero ⟨_, vc₁, pc₁⟩) =>
-      let ⟨_, vc₂, pc₂⟩ := evalAdd va₂ vb₂
-      ⟨_, .add vc₁ vc₂, q(add_pf_add_overlap $pc₁ $pc₂)⟩
+      let ⟨_, vc₂, pc₂⟩ ← evalAdd va₂ vb₂
+      return ⟨_, .add vc₁ vc₂, q(add_pf_add_overlap $pc₁ $pc₂)⟩
     | some (.zero pc₁) =>
-      let ⟨c₂, vc₂, pc₂⟩ := evalAdd va₂ vb₂
-      ⟨c₂, vc₂, q(add_pf_add_overlap_zero $pc₁ $pc₂)⟩
+      let ⟨c₂, vc₂, pc₂⟩ ← evalAdd va₂ vb₂
+      return ⟨c₂, vc₂, q(add_pf_add_overlap_zero $pc₁ $pc₂)⟩
     | none =>
       if let .lt := va₁.cmp vb₁ then
-        let ⟨_c, vc, (pc : Q($_a₂ + ($b₁ + $_b₂) = $_c))⟩ := evalAdd va₂ vb
-        ⟨_, .add va₁ vc, q(add_pf_add_lt $a₁ $pc)⟩
+        let ⟨_c, vc, (pc : Q($_a₂ + ($b₁ + $_b₂) = $_c))⟩ ← evalAdd va₂ vb
+        return ⟨_, .add va₁ vc, q(add_pf_add_lt $a₁ $pc)⟩
       else
-        let ⟨_c, vc, (pc : Q($a₁ + $_a₂ + $_b₂ = $_c))⟩ := evalAdd va vb₂
-        ⟨_, .add vb₁ vc, q(add_pf_add_gt $b₁ $pc)⟩
+        let ⟨_c, vc, (pc : Q($a₁ + $_a₂ + $_b₂ = $_c))⟩ ← evalAdd va vb₂
+        return ⟨_, .add vb₁ vc, q(add_pf_add_gt $b₁ $pc)⟩
 
 theorem one_mul (a : R) : (nat_lit 1).rawCast * a = a := by simp [Nat.rawCast]
 
@@ -394,37 +413,38 @@ theorem mul_pp_pf_overlap {ea eb e : ℕ} (x : R) (_ : ea + eb = e) (_ : a₂ *
 * `(a₁ * a₂) * (b₁ * b₂) = b₁ * ((a₁ * a₂) * b₂)` (if not `a₁.lt b₁`)
 -/
 partial def evalMulProd {a b : Q($α)} (va : ExProd sα a) (vb : ExProd sα b) :
-    Result (ExProd sα) q($a * $b) :=
+    Lean.Core.CoreM <| Result (ExProd sα) q($a * $b) := do
+  Lean.Core.checkSystem decl_name%.toString
   match va, vb with
   | .const za ha, .const zb hb =>
     if za = 1 then
-      ⟨b, .const zb hb, (q(one_mul $b) : Expr)⟩
+      return ⟨b, .const zb hb, (q(one_mul $b) : Expr)⟩
     else if zb = 1 then
-      ⟨a, .const za ha, (q(mul_one $a) : Expr)⟩
+      return ⟨a, .const za ha, (q(mul_one $a) : Expr)⟩
     else
       let ra := Result.ofRawRat za a ha; let rb := Result.ofRawRat zb b hb
       let rc := (NormNum.evalMul.core q($a * $b) q(HMul.hMul) _ _
           q(CommSemiring.toSemiring) ra rb).get!
       let ⟨zc, hc⟩ := rc.toRatNZ.get!
       let ⟨c, pc⟩ := rc.toRawEq
-      ⟨c, .const zc hc, pc⟩
+      return ⟨c, .const zc hc, pc⟩
   | .mul (x := a₁) (e := a₂) va₁ va₂ va₃, .const _ _ =>
-    let ⟨_, vc, pc⟩ := evalMulProd va₃ vb
-    ⟨_, .mul va₁ va₂ vc, (q(mul_pf_left $a₁ $a₂ $pc) : Expr)⟩
+    let ⟨_, vc, pc⟩ ← evalMulProd va₃ vb
+    return ⟨_, .mul va₁ va₂ vc, (q(mul_pf_left $a₁ $a₂ $pc) : Expr)⟩
   | .const _ _, .mul (x := b₁) (e := b₂) vb₁ vb₂ vb₃ =>
-    let ⟨_, vc, pc⟩ := evalMulProd va vb₃
-    ⟨_, .mul vb₁ vb₂ vc, (q(mul_pf_right $b₁ $b₂ $pc) : Expr)⟩
-  | .mul (x := xa) (e := ea) vxa vea va₂, .mul (x := xb) (e := eb) vxb veb vb₂ => Id.run do
+    let ⟨_, vc, pc⟩ ← evalMulProd va vb₃
+    return ⟨_, .mul vb₁ vb₂ vc, (q(mul_pf_right $b₁ $b₂ $pc) : Expr)⟩
+  | .mul (x := xa) (e := ea) vxa vea va₂, .mul (x := xb) (e := eb) vxb veb vb₂ => do
     if vxa.eq vxb then
-      if let some (.nonzero ⟨_, ve, pe⟩) := evalAddOverlap sℕ vea veb then
-        let ⟨_, vc, pc⟩ := evalMulProd va₂ vb₂
+      if let some (.nonzero ⟨_, ve, pe⟩) ← (evalAddOverlap sℕ vea veb).run then
+        let ⟨_, vc, pc⟩ ← evalMulProd va₂ vb₂
         return ⟨_, .mul vxa ve vc, (q(mul_pp_pf_overlap $xa $pe $pc) : Expr)⟩
     if let .lt := (vxa.cmp vxb).then (vea.cmp veb) then
-      let ⟨_, vc, pc⟩ := evalMulProd va₂ vb
-      ⟨_, .mul vxa vea vc, (q(mul_pf_left $xa $ea $pc) : Expr)⟩
+      let ⟨_, vc, pc⟩ ← evalMulProd va₂ vb
+      return ⟨_, .mul vxa vea vc, (q(mul_pf_left $xa $ea $pc) : Expr)⟩
     else
-      let ⟨_, vc, pc⟩ := evalMulProd va vb₂
-      ⟨_, .mul vxb veb vc, (q(mul_pf_right $xb $eb $pc) : Expr)⟩
+      let ⟨_, vc, pc⟩ ← evalMulProd va vb₂
+      return ⟨_, .mul vxb veb vc, (q(mul_pf_right $xb $eb $pc) : Expr)⟩
 
 theorem mul_zero (a : R) : a * 0 = 0 := by simp
 
@@ -437,14 +457,15 @@ theorem mul_add {d : R} (_ : (a : R) * b₁ = c₁) (_ : a * b₂ = c₂) (_ : c
 * `a * 0 = 0`
 * `a * (b₁ + b₂) = (a * b₁) + (a * b₂)`
 -/
-def evalMul₁ {a b : Q($α)} (va : ExProd sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a * $b) :=
+def evalMul₁ {a b : Q($α)} (va : ExProd sα a) (vb : ExSum sα b) :
+    Lean.Core.CoreM <| Result (ExSum sα) q($a * $b) := do
   match vb with
-  | .zero => ⟨_, .zero, q(mul_zero $a)⟩
+  | .zero => return ⟨_, .zero, q(mul_zero $a)⟩
   | .add vb₁ vb₂ =>
-    let ⟨_, vc₁, pc₁⟩ := evalMulProd sα va vb₁
-    let ⟨_, vc₂, pc₂⟩ := evalMul₁ va vb₂
-    let ⟨_, vd, pd⟩ := evalAdd sα vc₁.toSum vc₂
-    ⟨_, vd, q(mul_add $pc₁ $pc₂ $pd)⟩
+    let ⟨_, vc₁, pc₁⟩ ← evalMulProd sα va vb₁
+    let ⟨_, vc₂, pc₂⟩ ← evalMul₁ va vb₂
+    let ⟨_, vd, pd⟩ ← evalAdd sα vc₁.toSum vc₂
+    return ⟨_, vd, q(mul_add $pc₁ $pc₂ $pd)⟩
 
 theorem zero_mul (b : R) : 0 * b = 0 := by simp
 
@@ -456,14 +477,15 @@ theorem add_mul {d : R} (_ : (a₁ : R) * b = c₁) (_ : a₂ * b = c₂) (_ : c
 * `0 * b = 0`
 * `(a₁ + a₂) * b = (a₁ * b) + (a₂ * b)`
 -/
-def evalMul {a b : Q($α)} (va : ExSum sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a * $b) :=
+def evalMul {a b : Q($α)} (va : ExSum sα a) (vb : ExSum sα b) :
+    Lean.Core.CoreM <| Result (ExSum sα) q($a * $b) := do
   match va with
-  | .zero => ⟨_, .zero, q(zero_mul $b)⟩
+  | .zero => return ⟨_, .zero, q(zero_mul $b)⟩
   | .add va₁ va₂ =>
-    let ⟨_, vc₁, pc₁⟩ := evalMul₁ sα va₁ vb
-    let ⟨_, vc₂, pc₂⟩ := evalMul va₂ vb
-    let ⟨_, vd, pd⟩ := evalAdd sα vc₁ vc₂
-    ⟨_, vd, q(add_mul $pc₁ $pc₂ $pd)⟩
+    let ⟨_, vc₁, pc₁⟩ ← evalMul₁ sα va₁ vb
+    let ⟨_, vc₂, pc₂⟩ ← evalMul va₂ vb
+    let ⟨_, vd, pd⟩ ← evalAdd sα vc₁ vc₂
+    return ⟨_, vd, q(add_mul $pc₁ $pc₂ $pd)⟩
 
 theorem natCast_nat (n) : ((Nat.rawCast n : ℕ) : R) = Nat.rawCast n := by simp
 
@@ -540,11 +562,11 @@ def evalNSMul {a : Q(ℕ)} {b : Q($α)} (va : ExSum sℕ a) (vb : ExSum sα b) :
   if ← isDefEq sα sℕ then
     let ⟨_, va'⟩ := va.cast
     have _b : Q(ℕ) := b
-    let ⟨(_c : Q(ℕ)), vc, (pc : Q($a * $_b = $_c))⟩ := evalMul sα va' vb
+    let ⟨(_c : Q(ℕ)), vc, (pc : Q($a * $_b = $_c))⟩ ← evalMul sα va' vb
     pure ⟨_, vc, (q(smul_nat $pc) : Expr)⟩
   else
     let ⟨_, va', pa'⟩ ← va.evalNatCast sα
-    let ⟨_, vc, pc⟩ := evalMul sα va' vb
+    let ⟨_, vc, pc⟩ ← evalMul sα va' vb
     pure ⟨_, vc, (q(smul_eq_cast $pa' $pc) : Expr)⟩
 
 theorem neg_one_mul {R} [Ring R] {a b : R} (_ : (Int.negOfNat (nat_lit 1)).rawCast * a = b) :
@@ -558,7 +580,9 @@ theorem neg_mul {R} [Ring R] (a₁ : R) (a₂) {a₃ b : R}
 * `-c = (-c)` (for `c` coefficient)
 * `-(a₁ * a₂) = a₁ * -a₂`
 -/
-def evalNegProd {a : Q($α)} (rα : Q(Ring $α)) (va : ExProd sα a) : Result (ExProd sα) q(-$a) :=
+def evalNegProd {a : Q($α)} (rα : Q(Ring $α)) (va : ExProd sα a) :
+    Lean.Core.CoreM <| Result (ExProd sα) q(-$a) := do
+  Lean.Core.checkSystem decl_name%.toString
   match va with
   | .const za ha =>
     let lit : Q(ℕ) := mkRawNatLit 1
@@ -569,10 +593,10 @@ def evalNegProd {a : Q($α)} (rα : Q(Ring $α)) (va : ExProd sα a) : Result (E
       q(CommSemiring.toSemiring) rm ra).get!
     let ⟨zb, hb⟩ := rb.toRatNZ.get!
     let ⟨b, (pb : Q((Int.negOfNat (nat_lit 1)).rawCast * $a = $b))⟩ := rb.toRawEq
-    ⟨b, .const zb hb, (q(neg_one_mul (R := $α) $pb) : Expr)⟩
+    return ⟨b, .const zb hb, (q(neg_one_mul (R := $α) $pb) : Expr)⟩
   | .mul (x := a₁) (e := a₂) va₁ va₂ va₃ =>
-    let ⟨_, vb, pb⟩ := evalNegProd rα va₃
-    ⟨_, .mul va₁ va₂ vb, (q(neg_mul $a₁ $a₂ $pb) : Expr)⟩
+    let ⟨_, vb, pb⟩ ← evalNegProd rα va₃
+    return ⟨_, .mul va₁ va₂ vb, (q(neg_mul $a₁ $a₂ $pb) : Expr)⟩
 
 theorem neg_zero {R} [Ring R] : -(0 : R) = 0 := by simp
 
@@ -585,13 +609,14 @@ theorem neg_add {R} [Ring R] {a₁ a₂ b₁ b₂ : R}
 * `-0 = 0` (for `c` coefficient)
 * `-(a₁ + a₂) = -a₁ + -a₂`
 -/
-def evalNeg {a : Q($α)} (rα : Q(Ring $α)) (va : ExSum sα a) : Result (ExSum sα) q(-$a) :=
+def evalNeg {a : Q($α)} (rα : Q(Ring $α)) (va : ExSum sα a) :
+    Lean.Core.CoreM <| Result (ExSum sα) q(-$a) := do
   match va with
-  | .zero => ⟨_, .zero, (q(neg_zero (R := $α)) : Expr)⟩
+  | .zero => return ⟨_, .zero, (q(neg_zero (R := $α)) : Expr)⟩
   | .add va₁ va₂ =>
-    let ⟨_, vb₁, pb₁⟩ := evalNegProd sα rα va₁
-    let ⟨_, vb₂, pb₂⟩ := evalNeg rα va₂
-    ⟨_, .add vb₁ vb₂, (q(neg_add $pb₁ $pb₂) : Expr)⟩
+    let ⟨_, vb₁, pb₁⟩ ← evalNegProd sα rα va₁
+    let ⟨_, vb₂, pb₂⟩ ← evalNeg rα va₂
+    return ⟨_, .add vb₁ vb₂, (q(neg_add $pb₁ $pb₂) : Expr)⟩
 
 theorem sub_pf {R} [Ring R] {a b c d : R}
     (_ : -b = c) (_ : a + c = d) : a - b = d := by subst_vars; simp [sub_eq_add_neg]
@@ -601,10 +626,11 @@ theorem sub_pf {R} [Ring R] {a b c d : R}
 * `a - b = a + -b`
 -/
 def evalSub {α : Q(Type u)} (sα : Q(CommSemiring $α)) {a b : Q($α)}
-    (rα : Q(Ring $α)) (va : ExSum sα a) (vb : ExSum sα b) : Result (ExSum sα) q($a - $b) :=
-  let ⟨_c, vc, pc⟩ := evalNeg sα rα vb
-  let ⟨d, vd, (pd : Q($a + $_c = $d))⟩ := evalAdd sα va vc
-  ⟨d, vd, (q(sub_pf $pc $pd) : Expr)⟩
+    (rα : Q(Ring $α)) (va : ExSum sα a) (vb : ExSum sα b) :
+    Lean.Core.CoreM <| Result (ExSum sα) q($a - $b) := do
+  let ⟨_c, vc, pc⟩ ← evalNeg sα rα vb
+  let ⟨d, vd, (pd : Q($a + $_c = $d))⟩ ← evalAdd sα va vc
+  return ⟨d, vd, (q(sub_pf $pc $pd) : Expr)⟩
 
 theorem pow_prod_atom (a : R) (b) : a ^ b = (a + 0) ^ b * (nat_lit 1).rawCast := by simp
 
@@ -706,22 +732,23 @@ into a sum of monomials.
 * `x ^ (2*n) = x ^ n * x ^ n`
 * `x ^ (2*n+1) = x ^ n * x ^ n * x`
 -/
-partial def evalPowNat {a : Q($α)} (va : ExSum sα a) (n : Q(ℕ)) : Result (ExSum sα) q($a ^ $n) :=
+partial def evalPowNat {a : Q($α)} (va : ExSum sα a) (n : Q(ℕ)) :
+    Lean.Core.CoreM <| Result (ExSum sα) q($a ^ $n) := do
   let nn := n.natLit!
   if nn = 1 then
-    ⟨_, va, (q(pow_one $a) : Expr)⟩
+    return ⟨_, va, (q(pow_one $a) : Expr)⟩
   else
     let nm := nn >>> 1
     have m : Q(ℕ) := mkRawNatLit nm
     if nn &&& 1 = 0 then
-      let ⟨_, vb, pb⟩ := evalPowNat va m
-      let ⟨_, vc, pc⟩ := evalMul sα vb vb
-      ⟨_, vc, (q(pow_bit0 $pb $pc) : Expr)⟩
+      let ⟨_, vb, pb⟩ ← evalPowNat va m
+      let ⟨_, vc, pc⟩ ← evalMul sα vb vb
+      return ⟨_, vc, (q(pow_bit0 $pb $pc) : Expr)⟩
     else
-      let ⟨_, vb, pb⟩ := evalPowNat va m
-      let ⟨_, vc, pc⟩ := evalMul sα vb vb
-      let ⟨_, vd, pd⟩ := evalMul sα vc va
-      ⟨_, vd, (q(pow_bit1 $pb $pc $pd) : Expr)⟩
+      let ⟨_, vb, pb⟩ ← evalPowNat va m
+      let ⟨_, vc, pc⟩ ← evalMul sα vb vb
+      let ⟨_, vd, pd⟩ ← evalMul sα vc va
+      return ⟨_, vd, (q(pow_bit1 $pb $pc $pd) : Expr)⟩
 
 theorem one_pow (b : ℕ) : ((nat_lit 1).rawCast : R) ^ b = (nat_lit 1).rawCast := by simp
 
@@ -738,10 +765,11 @@ theorem mul_pow {ea₁ b c₁ : ℕ} {xa₁ : R}
 In all other cases we use `evalPowProdAtom`.
 -/
 def evalPowProd {a : Q($α)} {b : Q(ℕ)} (va : ExProd sα a) (vb : ExProd sℕ b) :
-    Result (ExProd sα) q($a ^ $b) :=
-  let res : Option (Result (ExProd sα) q($a ^ $b)) := do
+    Lean.Core.CoreM <| Result (ExProd sα) q($a ^ $b) := do
+  Lean.Core.checkSystem decl_name%.toString
+  let res : OptionT Lean.Core.CoreM (Result (ExProd sα) q($a ^ $b)) := do
     match va, vb with
-    | .const 1, _ => some ⟨_, va, (q(one_pow (R := $α) $b) : Expr)⟩
+    | .const 1, _ => return ⟨_, va, (q(one_pow (R := $α) $b) : Expr)⟩
     | .const za ha, .const zb hb =>
       assert! 0 ≤ zb
       let ra := Result.ofRawRat za a ha
@@ -751,13 +779,13 @@ def evalPowProd {a : Q($α)} {b : Q(ℕ)} (va : ExProd sα a) (vb : ExProd sℕ
         q(CommSemiring.toSemiring) ra
       let ⟨zc, hc⟩ ← rc.toRatNZ
       let ⟨c, pc⟩ := rc.toRawEq
-      some ⟨c, .const zc hc, pc⟩
-    | .mul vxa₁ vea₁ va₂, vb => do
-      let ⟨_, vc₁, pc₁⟩ := evalMulProd sℕ vea₁ vb
-      let ⟨_, vc₂, pc₂⟩ := evalPowProd va₂ vb
-      some ⟨_, .mul vxa₁ vc₁ vc₂, q(mul_pow $pc₁ $pc₂)⟩
-    | _, _ => none
-  res.getD (evalPowProdAtom sα va vb)
+      return ⟨c, .const zc hc, pc⟩
+    | .mul vxa₁ vea₁ va₂, vb =>
+      let ⟨_, vc₁, pc₁⟩ ← evalMulProd sℕ vea₁ vb
+      let ⟨_, vc₂, pc₂⟩ ← evalPowProd va₂ vb
+      return ⟨_, .mul vxa₁ vc₁ vc₂, q(mul_pow $pc₁ $pc₂)⟩
+    | _, _ => OptionT.fail
+  return (← res.run).getD (evalPowProdAtom sα va vb)
 
 /--
 The result of `extractCoeff` is a numeral and a proof that the original expression
@@ -815,24 +843,25 @@ theorem pow_nat {b c k : ℕ} {d e : R} (_ : b = c * k) (_ : a ^ c = d) (_ : d ^
 Otherwise `a ^ b` is just encoded as `a ^ b * 1 + 0` using `evalPowAtom`.
 -/
 partial def evalPow₁ {a : Q($α)} {b : Q(ℕ)} (va : ExSum sα a) (vb : ExProd sℕ b) :
-    Result (ExSum sα) q($a ^ $b) :=
+    Lean.Core.CoreM <| Result (ExSum sα) q($a ^ $b) := do
   match va, vb with
   | va, .const 1 =>
     haveI : $b =Q Nat.rawCast (nat_lit 1) := ⟨⟩
-    ⟨_, va, q(pow_one_cast $a)⟩
+    return ⟨_, va, q(pow_one_cast $a)⟩
   | .zero, vb => match vb.evalPos with
-    | some p => ⟨_, .zero, q(zero_pow (R := $α) $p)⟩
-    | none => evalPowAtom sα (.sum .zero) vb
+    | some p => return ⟨_, .zero, q(zero_pow (R := $α) $p)⟩
+    | none => return evalPowAtom sα (.sum .zero) vb
   | ExSum.add va .zero, vb => -- TODO: using `.add` here takes a while to compile?
-    let ⟨_, vc, pc⟩ := evalPowProd sα va vb
-    ⟨_, vc.toSum, q(single_pow $pc)⟩
+    let ⟨_, vc, pc⟩ ← evalPowProd sα va vb
+    return ⟨_, vc.toSum, q(single_pow $pc)⟩
   | va, vb =>
     if vb.coeff > 1 then
       let ⟨k, _, vc, pc⟩ := extractCoeff vb
-      let ⟨_, vd, pd⟩ := evalPow₁ va vc
-      let ⟨_, ve, pe⟩ := evalPowNat sα vd k
-      ⟨_, ve, q(pow_nat $pc $pd $pe)⟩
-    else evalPowAtom sα (.sum va) vb
+      let ⟨_, vd, pd⟩ ← evalPow₁ va vc
+      let ⟨_, ve, pe⟩ ← evalPowNat sα vd k
+      return ⟨_, ve, q(pow_nat $pc $pd $pe)⟩
+    else
+      return evalPowAtom sα (.sum va) vb
 
 theorem pow_zero (a : R) : a ^ 0 = (nat_lit 1).rawCast + 0 := by simp
 
@@ -846,17 +875,17 @@ theorem pow_add {b₁ b₂ : ℕ} {d : R}
 * `a ^ (b₁ + b₂) = a ^ b₁ * a ^ b₂`
 -/
 def evalPow {a : Q($α)} {b : Q(ℕ)} (va : ExSum sα a) (vb : ExSum sℕ b) :
-    Result (ExSum sα) q($a ^ $b) :=
+    Lean.Core.CoreM <| Result (ExSum sα) q($a ^ $b) := do
   match vb with
-  | .zero => ⟨_, (ExProd.mkNat sα 1).2.toSum, q(pow_zero $a)⟩
+  | .zero => return ⟨_, (ExProd.mkNat sα 1).2.toSum, q(pow_zero $a)⟩
   | .add vb₁ vb₂ =>
-    let ⟨_, vc₁, pc₁⟩ := evalPow₁ sα va vb₁
-    let ⟨_, vc₂, pc₂⟩ := evalPow va vb₂
-    let ⟨_, vd, pd⟩ := evalMul sα vc₁ vc₂
-    ⟨_, vd, q(pow_add $pc₁ $pc₂ $pd)⟩
+    let ⟨_, vc₁, pc₁⟩ ← evalPow₁ sα va vb₁
+    let ⟨_, vc₂, pc₂⟩ ← evalPow va vb₂
+    let ⟨_, vd, pd⟩ ← evalMul sα vc₁ vc₂
+    return ⟨_, vd, q(pow_add $pc₁ $pc₂ $pd)⟩
 
 /-- This cache contains data required by the `ring` tactic during execution. -/
-structure Cache {α : Q(Type u)} (sα : Q(CommSemiring $α)) :=
+structure Cache {α : Q(Type u)} (sα : Q(CommSemiring $α)) where
   /-- A ring instance on `α`, if available. -/
   rα : Option Q(Ring $α)
   /-- A division ring instance on `α`, if available. -/
@@ -956,6 +985,7 @@ def evalInvAtom (a : Q($α)) : AtomM (Result (ExBase sα) q($a⁻¹)) := do
 -/
 def ExProd.evalInv {a : Q($α)} (czα : Option Q(CharZero $α)) (va : ExProd sα a) :
     AtomM (Result (ExProd sα) q($a⁻¹)) := do
+  Lean.Core.checkSystem decl_name%.toString
   match va with
   | .const c hc =>
     let ra := Result.ofRawRat c a hc
@@ -970,7 +1000,7 @@ def ExProd.evalInv {a : Q($α)} (czα : Option Q(CharZero $α)) (va : ExProd sα
   | .mul (x := a₁) (e := _a₂) _va₁ va₂ va₃ => do
     let ⟨_b₁, vb₁, pb₁⟩ ← evalInvAtom sα dα a₁
     let ⟨_b₃, vb₃, pb₃⟩ ← va₃.evalInv czα
-    let ⟨c, vc, (pc : Q($_b₃ * ($_b₁ ^ $_a₂ * Nat.rawCast 1) = $c))⟩ :=
+    let ⟨c, vc, (pc : Q($_b₃ * ($_b₁ ^ $_a₂ * Nat.rawCast 1) = $c))⟩ ←
       evalMulProd sα vb₃ (vb₁.toProd va₂)
     pure ⟨c, vc, (q(inv_mul $pb₁ $pb₃ $pc) : Expr)⟩
 
@@ -984,7 +1014,7 @@ def ExSum.evalInv {a : Q($α)} (czα : Option Q(CharZero $α)) (va : ExSum sα a
   match va with
   | ExSum.zero => pure ⟨_, .zero, (q(inv_zero (R := $α)) : Expr)⟩
   | ExSum.add va ExSum.zero => do
-    let ⟨_, vb, pb⟩ ← va.evalInv dα czα
+    let ⟨_, vb, pb⟩ ← va.evalInv sα dα czα
     pure ⟨_, vb.toSum, (q(inv_single $pb) : Expr)⟩
   | va => do
     let ⟨_, vb, pb⟩ ← evalInvAtom sα dα a
@@ -1002,7 +1032,7 @@ theorem div_pf {R} [DivisionRing R] {a b c d : R} (_ : b⁻¹ = c) (_ : a * c =
 def evalDiv {a b : Q($α)} (rα : Q(DivisionRing $α)) (czα : Option Q(CharZero $α)) (va : ExSum sα a)
     (vb : ExSum sα b) : AtomM (Result (ExSum sα) q($a / $b)) := do
   let ⟨_c, vc, pc⟩ ← vb.evalInv sα rα czα
-  let ⟨d, vd, (pd : Q($a * $_c = $d))⟩ := evalMul sα va vc
+  let ⟨d, vd, (pd : Q($a * $_c = $d))⟩ ← evalMul sα va vc
   pure ⟨d, vd, (q(div_pf $pc $pd) : Expr)⟩
 
 theorem add_congr (_ : a = a') (_ : b = b') (_ : a' + b' = c) : (a + b : R) = c := by
@@ -1077,14 +1107,14 @@ partial def eval {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring $α))
     | ~q($a + $b) =>
       let ⟨_, va, pa⟩ ← eval sα c a
       let ⟨_, vb, pb⟩ ← eval sα c b
-      let ⟨c, vc, p⟩ := evalAdd sα va vb
+      let ⟨c, vc, p⟩ ← evalAdd sα va vb
       pure ⟨c, vc, (q(add_congr $pa $pb $p) : Expr)⟩
     | _ => els
   | ``HMul.hMul, _, _ | ``Mul.mul, _, _ => match e with
     | ~q($a * $b) =>
       let ⟨_, va, pa⟩ ← eval sα c a
       let ⟨_, vb, pb⟩ ← eval sα c b
-      let ⟨c, vc, p⟩ := evalMul sα va vb
+      let ⟨c, vc, p⟩ ← evalMul sα va vb
       pure ⟨c, vc, (q(mul_congr $pa $pb $p) : Expr)⟩
     | _ => els
   | ``HSMul.hSMul, _, _ => match e with
@@ -1098,19 +1128,20 @@ partial def eval {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring $α))
     | ~q($a ^ $b) =>
       let ⟨_, va, pa⟩ ← eval sα c a
       let ⟨_, vb, pb⟩ ← eval sℕ .nat b
-      let ⟨c, vc, p⟩ := evalPow sα va vb
+      let ⟨c, vc, p⟩ ← evalPow sα va vb
       pure ⟨c, vc, (q(pow_congr $pa $pb $p) : Expr)⟩
     | _ => els
   | ``Neg.neg, some rα, _ => match e with
     | ~q(-$a) =>
       let ⟨_, va, pa⟩ ← eval sα c a
-      let ⟨b, vb, p⟩ := evalNeg sα rα va
+      let ⟨b, vb, p⟩ ← evalNeg sα rα va
       pure ⟨b, vb, (q(neg_congr $pa $p) : Expr)⟩
+    | _ => els
   | ``HSub.hSub, some rα, _ | ``Sub.sub, some rα, _ => match e with
     | ~q($a - $b) => do
       let ⟨_, va, pa⟩ ← eval sα c a
       let ⟨_, vb, pb⟩ ← eval sα c b
-      let ⟨c, vc, p⟩ := evalSub sα rα va vb
+      let ⟨c, vc, p⟩ ← evalSub sα rα va vb
       pure ⟨c, vc, (q(sub_congr $pa $pb $p) : Expr)⟩
     | _ => els
   | ``Inv.inv, _, some dα => match e with
@@ -1118,6 +1149,7 @@ partial def eval {u : Lean.Level} {α : Q(Type u)} (sα : Q(CommSemiring $α))
       let ⟨_, va, pa⟩ ← eval sα c a
       let ⟨b, vb, p⟩ ← va.evalInv sα dα c.czα
       pure ⟨b, vb, (q(inv_congr $pa $p) : Expr)⟩
+    | _ => els
   | ``HDiv.hDiv, _, some dα | ``Div.div, _, some dα => match e with
     | ~q($a / $b) => do
       let ⟨_, va, pa⟩ ← eval sα c a
diff --git a/Mathlib/Tactic/Ring/Compare.lean b/Mathlib/Tactic/Ring/Compare.lean
new file mode 100644
index 0000000000000..5868ee2ee1706
--- /dev/null
+++ b/Mathlib/Tactic/Ring/Compare.lean
@@ -0,0 +1,239 @@
+/-
+Copyright (c) 2024 Heather Macbeth. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Heather Macbeth
+-/
+import Mathlib.Tactic.Ring.Basic
+import Mathlib.Tactic.NormNum.Ineq
+
+/-!
+# Automation for proving inequalities in commutative (semi)rings
+
+This file provides automation for proving certain kinds of inequalities in commutative semirings:
+goals of the form `A ≤ B` and `A < B` for which the ring-normal forms of `A` and `B` differ by a
+nonnegative (resp. positive) constant.
+
+For example, `⊢ x + 3 + y < y + x + 4` is in scope because the normal forms of the LHS and RHS are,
+respectively, `3 + (x + y)` and `4 + (x + y)`, which differ by an additive constant.
+
+## Main declarations
+
+* `Mathlib.Tactic.Ring.proveLE`: prove goals of the form `A ≤ B` (subject to the scope constraints
+  described)
+* `Mathlib.Tactic.Ring.proveLT`: prove goals of the form `A < B` (subject to the scope constraints
+  described)
+
+## Implementation notes
+
+The automation is provided in the `MetaM` monad; that is, these functions are not user-facing. It
+would not be hard to provide user-facing versions (see the test file), but the scope of this
+automation is rather specialized and might be confusing to users. It is also subsumed by `linarith`.
+-/
+
+namespace Mathlib.Tactic.Ring
+
+open Lean Qq Meta
+
+/-! Rather than having the metaprograms `Mathlib.Tactic.Ring.evalLE` and
+`Mathlib.Tactic.Ring.evalLT` perform all type class inference at the point of use, we record in
+advance, as `abbrev`s, a few type class deductions which will certainly be necessary.  They add no
+new information (they can already be proved by `inferInstance`).
+
+This helps in speeding up the metaprograms in this file substantially -- about a 50% reduction in
+heartbeat count in representative test cases -- since otherwise a substantial fraction of their
+runtime is devoted to type class inference. -/
+
+section Typeclass
+
+/-- `OrderedCommSemiring` implies `CommSemiring`. -/
+abbrev cs_of_ocs (α : Type*) [OrderedCommSemiring α] : CommSemiring α := inferInstance
+
+/-- `OrderedCommSemiring` implies `AddMonoidWithOne`. -/
+abbrev amwo_of_ocs (α : Type*) [OrderedCommSemiring α] : AddMonoidWithOne α := inferInstance
+
+/-- `OrderedCommSemiring` implies `LE`. -/
+abbrev le_of_ocs (α : Type*) [OrderedCommSemiring α] : LE α := inferInstance
+
+/-- `StrictOrderedCommSemiring` implies `CommSemiring`. -/
+abbrev cs_of_socs (α : Type*) [StrictOrderedCommSemiring α] : CommSemiring α := inferInstance
+
+/-- `StrictOrderedCommSemiring` implies `AddMonoidWithOne`. -/
+abbrev amwo_of_socs (α : Type*) [StrictOrderedCommSemiring α] : AddMonoidWithOne α := inferInstance
+
+/-- `StrictOrderedCommSemiring` implies `LT`. -/
+abbrev lt_of_socs (α : Type*) [StrictOrderedCommSemiring α] : LT α := inferInstance
+
+end Typeclass
+
+/-! The lemmas like `add_le_add_right` in the root namespace are stated under minimal type classes,
+typically just `[CovariantClass α α (swap (· + ·)) (· ≤ ·)]` or similar.  Here we restate these
+lemmas under stronger type class assumptions (`[OrderedCommSemiring α]` or similar), which helps in
+speeding up the metaprograms in this file (`Mathlib.Tactic.Ring.proveLT` and
+`Mathlib.Tactic.Ring.proveLE`) substantially -- about a 50% reduction in heartbeat count in
+representative test cases -- since otherwise a substantial fraction of their runtime is devoted to
+type class inference.
+
+These metaprograms at least require `CommSemiring`, `LE`/`LT`, and all
+`CovariantClass`/`ContravariantClass` permutations for addition, and in their main use case (in
+`linear_combination`) the `Preorder` type class is also required, so it is rather little loss of
+generality simply to require `OrderedCommSemiring`/`StrictOrderedCommSemiring`. -/
+
+section Lemma
+
+theorem add_le_add_right {α : Type*} [OrderedCommSemiring α] {b c : α} (bc : b ≤ c) (a : α) :
+    b + a ≤ c + a :=
+  _root_.add_le_add_right bc a
+
+theorem add_le_of_nonpos_left {α : Type*} [OrderedCommSemiring α] (a : α) {b : α} (h : b ≤ 0) :
+    b + a ≤ a :=
+  _root_.add_le_of_nonpos_left h
+
+theorem le_add_of_nonneg_left {α : Type*} [OrderedCommSemiring α] (a : α) {b : α} (h : 0 ≤ b) :
+    a ≤ b + a :=
+  _root_.le_add_of_nonneg_left h
+
+theorem add_lt_add_right {α : Type*} [StrictOrderedCommSemiring α] {b c : α} (bc : b < c) (a : α) :
+    b + a < c + a :=
+  _root_.add_lt_add_right bc a
+
+theorem add_lt_of_neg_left {α : Type*} [StrictOrderedCommSemiring α] (a : α) {b : α} (h : b < 0) :
+    b + a < a :=
+  _root_.add_lt_of_neg_left a h
+
+theorem lt_add_of_pos_left {α : Type*} [StrictOrderedCommSemiring α] (a : α) {b : α} (h : 0 < b) :
+    a < b + a :=
+  _root_.lt_add_of_pos_left a h
+
+end Lemma
+
+/-- Inductive type carrying the two kinds of errors which can arise in the metaprograms
+`Mathlib.Tactic.Ring.evalLE` and `Mathlib.Tactic.Ring.evalLT`. -/
+inductive ExceptType | tooSmall | notComparable
+export ExceptType (tooSmall notComparable)
+
+/-- In a commutative semiring, given `Ring.ExSum` objects `va`, `vb` which differ by a positive
+(additive) constant, construct a proof of `$a < $b`, where `a` (resp. `b`) is the expression in the
+semiring to which `va` (resp. `vb`) evaluates. -/
+def evalLE {v : Level} {α : Q(Type v)} (_ : Q(OrderedCommSemiring $α)) {a b : Q($α)}
+    (va : Ring.ExSum q(cs_of_ocs $α) a) (vb : Ring.ExSum q(cs_of_ocs $α) b) :
+    MetaM (Except ExceptType Q($a ≤ $b)) := do
+  let lα : Q(LE $α) := q(le_of_ocs $α)
+  assumeInstancesCommute
+  let ⟨_, pz⟩ ← NormNum.mkOfNat α q(amwo_of_ocs $α) (mkRawNatLit 0)
+  let rz : NormNum.Result q((0:$α)) :=
+    NormNum.Result.isNat q(amwo_of_ocs $α) (mkRawNatLit 0) (q(NormNum.isNat_ofNat $α $pz):)
+  match va, vb with
+  /- `0 ≤ 0` -/
+  | .zero, .zero => pure <| .ok (q(le_refl (0:$α)):)
+  /- For numerals `ca` and `cb`, `ca + x ≤ cb + x` if `ca ≤ cb` -/
+  | .add (b := a') (.const (e := xa) ca hypa) va', .add (.const (e := xb) cb hypb) vb' => do
+    unless va'.eq vb' do return .error notComparable
+    let rxa := NormNum.Result.ofRawRat ca xa hypa
+    let rxb := NormNum.Result.ofRawRat cb xb hypb
+    let NormNum.Result.isTrue pf ← NormNum.evalLE.core lα rxa rxb | return .error tooSmall
+    pure <| .ok (q(add_le_add_right (a := $a') $pf):)
+  /- For a numeral `ca ≤ 0`, `ca + x ≤ x` -/
+  | .add (.const (e := xa) ca hypa) va', _ => do
+    unless va'.eq vb do return .error notComparable
+    let rxa := NormNum.Result.ofRawRat ca xa hypa
+    let NormNum.Result.isTrue pf ← NormNum.evalLE.core lα rxa rz | return .error tooSmall
+    pure <| .ok (q(add_le_of_nonpos_left (a := $b) $pf):)
+  /- For a numeral `0 ≤ cb`, `x ≤ cb + x` -/
+  | _, .add (.const (e := xb) cb hypb) vb' => do
+    unless va.eq vb' do return .error notComparable
+    let rxb := NormNum.Result.ofRawRat cb xb hypb
+    let NormNum.Result.isTrue pf ← NormNum.evalLE.core lα rz rxb | return .error tooSmall
+    pure <| .ok (q(le_add_of_nonneg_left (a := $a) $pf):)
+  | _, _ => return .error notComparable
+
+/-- In a commutative semiring, given `Ring.ExSum` objects `va`, `vb` which differ by a positive
+(additive) constant, construct a proof of `$a < $b`, where `a` (resp. `b`) is the expression in the
+semiring to which `va` (resp. `vb`) evaluates. -/
+def evalLT {v : Level} {α : Q(Type v)} (_ : Q(StrictOrderedCommSemiring $α)) {a b : Q($α)}
+    (va : Ring.ExSum q(cs_of_socs $α) a) (vb : Ring.ExSum q(cs_of_socs $α) b) :
+    MetaM (Except ExceptType Q($a < $b)) := do
+  let lα : Q(LT $α) := q(lt_of_socs $α)
+  assumeInstancesCommute
+  let ⟨_, pz⟩ ← NormNum.mkOfNat α q(amwo_of_socs $α) (mkRawNatLit 0)
+  let rz : NormNum.Result q((0:$α)) :=
+    NormNum.Result.isNat q(amwo_of_socs $α) (mkRawNatLit 0) (q(NormNum.isNat_ofNat $α $pz):)
+  match va, vb with
+  /- `0 < 0` -/
+  | .zero, .zero => return .error tooSmall
+  /- For numerals `ca` and `cb`, `ca + x < cb + x` if `ca < cb` -/
+  | .add (b := a') (.const (e := xa) ca hypa) va', .add (.const (e := xb) cb hypb) vb' => do
+    unless va'.eq vb' do return .error notComparable
+    let rxa := NormNum.Result.ofRawRat ca xa hypa
+    let rxb := NormNum.Result.ofRawRat cb xb hypb
+    let NormNum.Result.isTrue pf ← NormNum.evalLT.core lα rxa rxb | return .error tooSmall
+    pure <| .ok (q(add_lt_add_right $pf $a'):)
+  /- For a numeral `ca < 0`, `ca + x < x` -/
+  | .add (.const (e := xa) ca hypa) va', _ => do
+    unless va'.eq vb do return .error notComparable
+    let rxa := NormNum.Result.ofRawRat ca xa hypa
+    let NormNum.Result.isTrue pf ← NormNum.evalLT.core lα rxa rz | return .error tooSmall
+    have pf : Q($xa < 0) := pf
+    pure <| .ok (q(add_lt_of_neg_left $b $pf):)
+  /- For a numeral `0 < cb`, `x < cb + x` -/
+  | _, .add (.const (e := xb) cb hypb) vb' => do
+    unless va.eq vb' do return .error notComparable
+    let rxb := NormNum.Result.ofRawRat cb xb hypb
+    let NormNum.Result.isTrue pf ← NormNum.evalLT.core lα rz rxb | return .error tooSmall
+    pure <| .ok (q(lt_add_of_pos_left $a $pf):)
+  | _, _ => return .error notComparable
+
+theorem le_congr {α : Type*} [LE α] {a b c d : α} (h1 : a = b) (h2 : b ≤ c) (h3 : d = c) :
+    a ≤ d := by
+  rwa [h1, h3]
+
+theorem lt_congr {α : Type*} [LT α] {a b c d : α} (h1 : a = b) (h2 : b < c) (h3 : d = c) :
+    a < d := by
+  rwa [h1, h3]
+
+/-- Prove goals of the form `A ≤ B` in an ordered commutative semiring, if the ring-normal forms of
+`A` and `B` differ by a nonnegative (additive) constant. -/
+def proveLE (g : MVarId) : MetaM Unit := do
+  let some (α, e₁, e₂) := (← whnfR <|← instantiateMVars <|← g.getType).le?
+    | throwError "ring failed: not of the form `A ≤ B`"
+  let .sort u ← whnf (← inferType α) | unreachable!
+  let v ← try u.dec catch _ => throwError "not a type{indentExpr α}"
+  have α : Q(Type v) := α
+  let sα ← synthInstanceQ q(OrderedCommSemiring $α)
+  assumeInstancesCommute
+  have e₁ : Q($α) := e₁; have e₂ : Q($α) := e₂
+  let c ← mkCache q(cs_of_ocs $α)
+  let (⟨a, va, pa⟩, ⟨b, vb, pb⟩)
+    ← AtomM.run .instances do pure (← eval q(cs_of_ocs $α) c e₁, ← eval q(cs_of_ocs $α) c e₂)
+  match ← evalLE sα va vb with
+  | .ok p => g.assign q(le_congr $pa $p $pb)
+  | .error e =>
+    let g' ← mkFreshExprMVar (← (← ringCleanupRef.get) q($a ≤ $b))
+    match e with
+    | notComparable =>
+      throwError "ring failed, ring expressions not equal up to an additive constant\n{g'.mvarId!}"
+    | tooSmall => throwError "comparison failed, LHS is larger\n{g'.mvarId!}"
+
+/-- Prove goals of the form `A < B` in an ordered commutative semiring, if the ring-normal forms of
+`A` and `B` differ by a positive (additive) constant. -/
+def proveLT (g : MVarId) : MetaM Unit := do
+  let some (α, e₁, e₂) := (← whnfR <|← instantiateMVars <|← g.getType).lt?
+    | throwError "ring failed: not of the form `A < B`"
+  let .sort u ← whnf (← inferType α) | unreachable!
+  let v ← try u.dec catch _ => throwError "not a type{indentExpr α}"
+  have α : Q(Type v) := α
+  let sα ← synthInstanceQ q(StrictOrderedCommSemiring $α)
+  assumeInstancesCommute
+  have e₁ : Q($α) := e₁; have e₂ : Q($α) := e₂
+  let c ← mkCache q(cs_of_socs $α)
+  let (⟨a, va, pa⟩, ⟨b, vb, pb⟩)
+    ← AtomM.run .instances do pure (← eval q(cs_of_socs $α) c e₁, ← eval q(cs_of_socs $α) c e₂)
+  match ← evalLT sα va vb with
+  | .ok p => g.assign q(lt_congr $pa $p $pb)
+  | .error e =>
+    let g' ← mkFreshExprMVar (← (← ringCleanupRef.get) q($a < $b))
+    match e with
+    | notComparable =>
+      throwError "ring failed, ring expressions not equal up to an additive constant\n{g'.mvarId!}"
+    | tooSmall => throwError "comparison failed, LHS is at least as large\n{g'.mvarId!}"
+
+end Mathlib.Tactic.Ring
diff --git a/Mathlib/Tactic/Ring/RingNF.lean b/Mathlib/Tactic/Ring/RingNF.lean
index 4ac168ff6ec41..c538ee4c12311 100644
--- a/Mathlib/Tactic/Ring/RingNF.lean
+++ b/Mathlib/Tactic/Ring/RingNF.lean
@@ -147,7 +147,7 @@ partial def M.run
       ``rat_rawCast_neg, ``rat_rawCast_pos].foldlM (·.addConst · (post := false)) thms
     let ctx' := { ctx with simpTheorems := #[thms] }
     pure fun r' : Simp.Result ↦ do
-      r'.mkEqTrans (← Simp.main r'.expr ctx' (methods := ← Lean.Meta.Simp.mkDefaultMethods)).1
+      r'.mkEqTrans (← Simp.main r'.expr ctx' (methods := Lean.Meta.Simp.mkDefaultMethodsCore {})).1
   let nctx := { ctx, simp }
   let rec
     /-- The recursive context. -/
diff --git a/Mathlib/Tactic/Sat/FromLRAT.lean b/Mathlib/Tactic/Sat/FromLRAT.lean
index 6104d507b0039..b8d7f1a56e43c 100644
--- a/Mathlib/Tactic/Sat/FromLRAT.lean
+++ b/Mathlib/Tactic/Sat/FromLRAT.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
 import Mathlib.Algebra.Group.Nat
+import Batteries.Data.HashMap.Basic
 
 /-!
 # `lrat_proof` command
@@ -494,30 +495,30 @@ where
 open Lean
 
 namespace Parser
-open Lean Parsec
+open Lean Std.Internal.Parsec String
 
 /-- Parse a natural number -/
-def parseNat : Parsec Nat := Json.Parser.natMaybeZero
+def parseNat : String.Parser Nat := Json.Parser.natMaybeZero
 
 /-- Parse an integer -/
-def parseInt : Parsec Int := do
+def parseInt : String.Parser Int := do
   if (← peek!) = '-' then skip; pure <| -(← parseNat) else parseNat
 
 /-- Parse a list of integers terminated by 0 -/
-partial def parseInts (arr : Array Int := #[]) : Parsec (Array Int) := do
+partial def parseInts (arr : Array Int := #[]) : String.Parser (Array Int) := do
   match ← parseInt <* ws with
   | 0 => pure arr
   | n => parseInts (arr.push n)
 
 /-- Parse a list of natural numbers terminated by 0 -/
-partial def parseNats (arr : Array Nat := #[]) : Parsec (Array Nat) := do
+partial def parseNats (arr : Array Nat := #[]) : String.Parser (Array Nat) := do
   match ← parseNat <* ws with
   | 0 => pure arr
   | n => parseNats (arr.push n)
 
 /-- Parse a DIMACS format `.cnf` file.
 This is not very robust; we assume the file has had comments stripped. -/
-def parseDimacs : Parsec (Nat × Array (Array Int)) := do
+def parseDimacs : String.Parser (Nat × Array (Array Int)) := do
   pstring "p cnf" *> ws
   let nvars ← parseNat <* ws
   let nclauses ← parseNat <* ws
@@ -527,13 +528,15 @@ def parseDimacs : Parsec (Nat × Array (Array Int)) := do
   pure (nvars, clauses)
 
 /-- Parse an LRAT file into a list of steps. -/
-def parseLRAT : Parsec (Array LRATStep) := many do
+def parseLRAT : String.Parser (Array LRATStep) := many do
   let step ← parseNat <* ws
   if (← peek!) = 'd' then skip <* ws; pure <| LRATStep.del (← parseNats)
   else ws; pure <| LRATStep.add step (← parseInts) (← parseInts)
 
 end Parser
 
+open Std.Internal
+
 /-- Core of `fromLRAT`. Constructs the context and main proof definitions,
 but not the reification theorem. Returns:
 
diff --git a/Mathlib/Tactic/Says.lean b/Mathlib/Tactic/Says.lean
index 72b195ef15967..29bd33ed183cf 100644
--- a/Mathlib/Tactic/Says.lean
+++ b/Mathlib/Tactic/Says.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Kim Liesinger. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kim Liesinger
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Batteries.Data.String.Basic
@@ -40,7 +40,7 @@ register_option says.verify : Bool :=
 register_option says.no_verify_in_CI : Bool :=
   { defValue := false
     group := "says"
-    descr := "Disable reverification, even if `the `CI` environment variable is set." }
+    descr := "Disable reverification, even if the `CI` environment variable is set." }
 
 open Parser Tactic
 
diff --git a/Mathlib/Tactic/Simps/Basic.lean b/Mathlib/Tactic/Simps/Basic.lean
index 9d3ef0b63492b..1caba0e4825ab 100644
--- a/Mathlib/Tactic/Simps/Basic.lean
+++ b/Mathlib/Tactic/Simps/Basic.lean
@@ -269,31 +269,38 @@ syntax simpsRule := simpsRule.prefix <|> simpsRule.rename <|> simpsRule.erase <|
 syntax simpsProj := ppSpace ident (" (" simpsRule,+ ")")?
 
 /--
-This command specifies custom names and custom projections for the simp attribute `simpsAttr`.
-* You can specify custom names by writing e.g.
-  `initialize_simps_projections Equiv (toFun → apply, invFun → symm_apply)`.
-* See Note [custom simps projection] and the examples below for information how to declare custom
-  projections.
+This command allows customisation of the lemmas generated by `simps`.
+
+By default, tagging a definition of an element `myObj` of a structure `MyStruct` with `@[simps]`
+generates one `@[simp]` lemma `myObj_myProj` for each projection `myProj` of `MyStruct`. There are a
+few exceptions to this general rule:
 * For algebraic structures, we will automatically use the notation (like `Mul`)
   for the projections if such an instance is available.
 * By default, the projections to parent structures are not default projections,
   but all the data-carrying fields are (including those in parent structures).
+
+This default behavior is customisable as such:
 * You can disable a projection by default by running
-  `initialize_simps_projections Equiv (-invFun)`
+  `initialize_simps_projections MulEquiv (-invFun)`
   This will ensure that no simp lemmas are generated for this projection,
-  unless this projection is explicitly specified by the user.
+  unless this projection is explicitly specified by the user (as in
+  `@[simps invFun] def myEquiv : MulEquiv _ _ := _`).
 * Conversely, you can enable a projection by default by running
-  `initialize_simps_projections Equiv (+toEquiv)`.
+  `initialize_simps_projections MulEquiv (+toEquiv)`.
+* You can specify custom names by writing e.g.
+  `initialize_simps_projections MulEquiv (toFun → apply, invFun → symm_apply)`.
 * If you want the projection name added as a prefix in the generated lemma name, you can use
   `as_prefix fieldName`:
-  `initialize_simps_projections Equiv (toFun → coe, as_prefix coe)`
+  `initialize_simps_projections MulEquiv (toFun → coe, as_prefix coe)`
   Note that this does not influence the parsing of projection names: if you have a declaration
   `foo` and you want to apply the projections `snd`, `coe` (which is a prefix) and `fst`, in that
   order you can run `@[simps snd_coe_fst] def foo ...` and this will generate a lemma with the
   name `coe_foo_snd_fst`.
+
+Here are a few extra pieces of information:
   * Run `initialize_simps_projections?` (or `set_option trace.simps.verbose true`)
   to see the generated projections.
-* Running `initialize_simps_projections MyStruc` without arguments is not necessary, it has the
+* Running `initialize_simps_projections MyStruct` without arguments is not necessary, it has the
   same effect if you just add `@[simps]` to a declaration.
 * It is recommended to call `@[simps]` or `initialize_simps_projections` in the same file as the
   structure declaration. Otherwise, the projections could be generated multiple times in different
@@ -304,7 +311,7 @@ Some common uses:
   `initialize_simps_projections` after defining the `DFunLike` instance (or instance that implies
   a `DFunLike` instance).
   ```
-    instance {mM : Mul M} {mN : Mul N} : DFunLike (MulHom M N) M N := ...
+    instance {mM : Mul M} {mN : Mul N} : FunLike (MulHom M N) M N := ...
     initialize_simps_projections MulHom (toFun → apply)
   ```
   This will generate `foo_apply` lemmas for each declaration `foo`.
@@ -1172,7 +1179,7 @@ def simpsTac (ref : Syntax) (nm : Name) (cfg : Config := {})
   let env ← getEnv
   let some d := env.find? nm | throwError "Declaration {nm} doesn't exist."
   let lhs : Expr := mkConst d.name <| d.levelParams.map Level.param
-  let todo := todo.pwFilter (·.1 ≠ ·.1) |>.map fun (proj, stx) ↦ (proj ++ "_", stx)
+  let todo := todo.eraseDups |>.map fun (proj, stx) ↦ (proj ++ "_", stx)
   let mut cfg := cfg
   MetaM.run' <| addProjections ref d.levelParams
     nm d.type lhs (d.value?.getD default) #[] (mustBeStr := true) cfg todo []
diff --git a/Mathlib/Tactic/SlimCheck.lean b/Mathlib/Tactic/SlimCheck.lean
index e5088f231ead7..d30b24c1f4eb5 100644
--- a/Mathlib/Tactic/SlimCheck.lean
+++ b/Mathlib/Tactic/SlimCheck.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Simon Hudon. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Simon Hudon, Scott Morrison
+Authors: Simon Hudon, Kim Morrison
 -/
 
 import Mathlib.Testing.SlimCheck.Testable
@@ -181,7 +181,7 @@ elab_rules : tactic | `(tactic| slim_check $[$cfg]?) => withMainContext do
       Failed to create a `testable` instance for `{tgt}`.\
     \nWhat to do:\
     \n1. make sure that the types you are using have `SlimCheck.SampleableExt` instances\
-    \n  (you can use `#sample my_type` if you are unsure);\
+    \n (you can use `#sample my_type` if you are unsure);\
     \n2. make sure that the relations and predicates that your proposition use are decidable;\
     \n3. make sure that instances of `SlimCheck.Testable` exist that, when combined,\
     \n  apply to your decorated proposition:\
diff --git a/Mathlib/Tactic/SplitIfs.lean b/Mathlib/Tactic/SplitIfs.lean
index d648be56512c6..97bfe7da56045 100644
--- a/Mathlib/Tactic/SplitIfs.lean
+++ b/Mathlib/Tactic/SplitIfs.lean
@@ -74,7 +74,7 @@ private def discharge? (e : Expr) : SimpM (Option Expr) := do
 private def reduceIfsAt (loc : Location) : TacticM Unit := do
   let ctx ← SplitIf.getSimpContext
   let ctx := { ctx with config := { ctx.config with failIfUnchanged := false } }
-  let _ ← simpLocation ctx {} discharge? loc
+  let _ ← simpLocation ctx (← ({} : Simp.SimprocsArray).add `reduceCtorEq false) discharge? loc
   pure ()
 
 /-- Splits a single if-then-else expression and then reduces the resulting goals.
diff --git a/Mathlib/Tactic/StacksAttribute.lean b/Mathlib/Tactic/StacksAttribute.lean
index f023939846230..0dc60503a6780 100644
--- a/Mathlib/Tactic/StacksAttribute.lean
+++ b/Mathlib/Tactic/StacksAttribute.lean
@@ -6,22 +6,34 @@ Authors: Damiano Testa
 import Lean.Elab.Command
 
 /-!
-# The `stacks` attribute
+# The `stacks` and `kerodon` attributes
 
-This allows tagging of mathlib lemmas with the corresponding
-[Tags](https://stacks.math.columbia.edu/tags) from the Stacks Project.
+This allows tagging of mathlib results with the corresponding
+tags from the [Stacks Project](https://stacks.math.columbia.edu/tags) and
+[Kerodon](https://kerodon.net/tag/).
+
+While the Stacks Project is the main focus, because the tag format at Kerodon is
+compatible, the attribute can be used to tag results with Kerodon tags as well.
 -/
 
 open Lean Elab
 
-namespace Mathlib.Stacks
+namespace Mathlib.StacksTag
+
+/-- Web database users of projects tags -/
+inductive Database where
+  | kerodon
+  | stacks
+  deriving BEq, Hashable
 
-/-- `Tag` is the structure that carries the data of a Stacks Projects tag and a corresponding
+/-- `Tag` is the structure that carries the data of a project tag and a corresponding
 Mathlib declaration. -/
 structure Tag where
   /-- The name of the declaration with the given tag. -/
   declName : Name
-  /-- The Stacks Project tag. -/
+  /-- The online database where the tag is found. -/
+  database : Database
+  /-- The database tag. -/
   tag : String
   /-- The (optional) comment that comes with the given tag. -/
   comment : String
@@ -29,9 +41,9 @@ structure Tag where
 
 /-- Defines the `tagExt` extension for adding a `HashSet` of `Tag`s
 to the environment. -/
-initialize tagExt : SimplePersistentEnvExtension Tag (HashSet Tag) ←
+initialize tagExt : SimplePersistentEnvExtension Tag (Std.HashSet Tag) ←
   registerSimplePersistentEnvExtension {
-    addImportedFn := fun as => as.foldl HashSet.insertMany {}
+    addImportedFn := fun as => as.foldl Std.HashSet.insertMany {}
     addEntryFn := .insert
   }
 
@@ -40,78 +52,136 @@ initialize tagExt : SimplePersistentEnvExtension Tag (HashSet Tag) ←
 the `String`s `tag` and `comment` of the `stacks` attribute.
 It extends the `Tag` environment extension with the data `declName, tag, comment`.
 -/
-def addTagEntry {m : Type → Type} [MonadEnv m] (declName : Name) (tag comment : String) : m Unit :=
-  modifyEnv (tagExt.addEntry · { declName := declName, tag := tag, comment := comment })
+def addTagEntry {m : Type → Type} [MonadEnv m]
+    (declName : Name) (db : Database) (tag comment : String) : m Unit :=
+  modifyEnv (tagExt.addEntry ·
+    { declName := declName, database := db, tag := tag, comment := comment })
+
+open Parser
+
+/-- `stacksTag` is the node kind of Stacks Project Tags: a sequence of digits and
+uppercase letters. -/
+abbrev stacksTagKind : SyntaxNodeKind := `stacksTag
+
+/-- The main parser for Stacks Project Tags: it accepts any sequence of 4 digits or
+uppercase letters. -/
+def stacksTagFn : ParserFn := fun c s =>
+  let i := s.pos
+  let s := takeWhileFn (fun c => c.isAlphanum) c s
+  if s.hasError then
+    s
+  else if s.pos == i then
+    ParserState.mkError s "stacks tag"
+  else
+    let tag := Substring.mk c.input i s.pos |>.toString
+    if !tag.all fun c => c.isDigit || c.isUpper then
+      ParserState.mkUnexpectedError s
+        "Stacks tags must consist only of digits and uppercase letters."
+    else if tag.length != 4 then
+      ParserState.mkUnexpectedError s "Stacks tags must be exactly 4 characters"
+    else
+      mkNodeToken stacksTagKind i c s
+
+@[inherit_doc stacksTagFn]
+def stacksTagNoAntiquot : Parser := {
+  fn   := stacksTagFn
+  info := mkAtomicInfo "stacksTag"
+}
 
-/--
-The syntax for a Stacks tag: it is an optional number followed by an optional identifier.
-This allows `044Q3` and `GH3F6` as possibilities.
--/
-declare_syntax_cat stackTag
+@[inherit_doc stacksTagFn]
+def stacksTagParser : Parser :=
+  withAntiquot (mkAntiquot "stacksTag" stacksTagKind) stacksTagNoAntiquot
+
+end Mathlib.StacksTag
+
+open Mathlib.StacksTag
+
+/-- Extract the underlying tag as a string from a `stacksTag` node. -/
+def Lean.TSyntax.getStacksTag (stx : TSyntax stacksTagKind) : CoreM String := do
+  let some val := Syntax.isLit? stacksTagKind stx | throwError "Malformed Stacks tag"
+  return val
+
+namespace Lean.PrettyPrinter
+
+namespace Formatter
+
+/-- The formatter for Stacks Project Tags syntax. -/
+@[combinator_formatter stacksTagNoAntiquot] def stacksTagNoAntiquot.formatter :=
+  visitAtom stacksTagKind
 
-@[inherit_doc Parser.Category.stackTag]
-syntax (num)? (ident)? : stackTag
+end Formatter
 
-/-- The `stacks` attribute.
-Use it as `@[stacks TAG "Optional comment"]`.
-The `TAG` is mandatory.
+namespace Parenthesizer
 
-See the [Tags page](https://stacks.math.columbia.edu/tags) in the Stacks project for more details.
+/-- The parenthesizer for Stacks Project Tags syntax. -/
+@[combinator_parenthesizer stacksTagNoAntiquot] def stacksTagAntiquot.parenthesizer := visitToken
+
+end Lean.PrettyPrinter.Parenthesizer
+
+namespace Mathlib.StacksTag
+
+/-- The syntax category for the database name. -/
+declare_syntax_cat stacksTagDB
+
+/-- The syntax for a "kerodon" database identifier in a `@[kerodon]` attribute. -/
+syntax "kerodon" : stacksTagDB
+/-- The syntax for a "stacks" database identifier in a `@[stacks]` attribute. -/
+syntax "stacks" : stacksTagDB
+
+/-- The `stacksTag` attribute.
+Use it as `@[kerodon TAG "Optional comment"]` or `@[stacks TAG "Optional comment"]`
+depending on the database you are referencing.
+
+The `TAG` is mandatory and should be a sequence of 4 digits or uppercase letters.
+
+See the [Tags page](https://stacks.math.columbia.edu/tags) in the Stacks project or
+[Tags page](https://kerodon.net/tag/) in the Kerodon project for more details.
 -/
-syntax (name := stacks) "stacks " (stackTag)? (ppSpace str)? : attr
+syntax (name := stacksTag) stacksTagDB stacksTagParser (ppSpace str)? : attr
 
 initialize Lean.registerBuiltinAttribute {
-  name := `stacks
-  descr := "Apply a Stacks project tag to a theorem."
-  add := fun decl stx _attrKind => Lean.withRef stx do
-    -- check that the tag consists of 4 characters and
-    -- that only digits and uppercase letter are present
-    let tag := stx[1]
-    match tag.getSubstring? with
-      | none => logWarning "Please, enter a Tag after `stacks`."
-      | some str =>
-        let str := str.toString.trimRight
-        if str.length != 4 then
-          logWarningAt tag
-            m!"Tag '{str}' is {str.length} characters long, but it should be 4 characters long"
-        else if 2 ≤ (str.split (fun c => (!c.isUpper) && !c.isDigit)).length then
-          logWarningAt tag m!"Tag '{str}' should only consist of digits and uppercase letters"
-        else match stx with
-          | `(attr| stacks $_:stackTag $comment:str) => addTagEntry decl str comment.getString
-          | `(attr| stacks $_:stackTag) => addTagEntry decl str ""
-          | _ => throwUnsupportedSyntax
+  name := `stacksTag
+  descr := "Apply a Stacks or Kerodon project tag to a theorem."
+  add := fun decl stx _attrKind => match stx with
+    | `(attr| stacks $tag $[$comment]?) => do
+      addTagEntry decl .stacks (← tag.getStacksTag) <| (comment.map (·.getString)).getD ""
+    | `(attr| kerodon $tag $[$comment]?) => do
+      addTagEntry decl .kerodon (← tag.getStacksTag) <| (comment.map (·.getString)).getD ""
+    | _ => throwUnsupportedSyntax
 }
 
-end Mathlib.Stacks
+end Mathlib.StacksTag
 
-open Mathlib.Stacks
 /--
 `getSortedStackProjectTags env` returns the array of `Tags`, sorted by alphabetical order of tag.
 -/
-def Lean.Environment.getSortedStackProjectTags (env : Environment) : Array Tag :=
+private def Lean.Environment.getSortedStackProjectTags (env : Environment) : Array Tag :=
   tagExt.getState env |>.toArray.qsort (·.tag < ·.tag)
 
 /--
 `getSortedStackProjectDeclNames env tag` returns the array of declaration names of results
 with Stacks Project tag equal to `tag`.
 -/
-def Lean.Environment.getSortedStackProjectDeclNames (env : Environment) (tag : String) :
+private def Lean.Environment.getSortedStackProjectDeclNames (env : Environment) (tag : String) :
     Array Name :=
   let tags := env.getSortedStackProjectTags
   tags.filterMap fun d => if d.tag == tag then some d.declName else none
 
-/--
-`#stacks_tags` retrieves all declarations that have the `stacks` attribute.
+namespace Mathlib.StacksTag
 
-For each found declaration, it prints a line
-```
-'declaration_name' corresponds to tag 'declaration_tag'.
-```
-The variant `#stacks_tags!` also adds the theorem statement after each summary line.
+private def databaseURL (db : Database) : String :=
+  match db with
+  | .kerodon => "https://kerodon.net/tag/"
+  | .stacks => "https://stacks.math.columbia.edu/tag/"
+
+/--
+`traceStacksTags db verbose` prints the tags of the database `db` to the user and
+inlines the theorem statements if `verbose` is `true`.
 -/
-elab (name := Mathlib.Stacks.stacksTags) "#stacks_tags" tk:("!")?: command => do
+def traceStacksTags (db : Database) (verbose : Bool := false) :
+    Command.CommandElabM Unit := do
   let env ← getEnv
-  let entries := env.getSortedStackProjectTags
+  let entries := env.getSortedStackProjectTags |>.filter (·.database == db)
   if entries.isEmpty then logInfo "No tags found." else
   let mut msgs := #[m!""]
   for d in entries do
@@ -119,10 +189,35 @@ elab (name := Mathlib.Stacks.stacksTags) "#stacks_tags" tk:("!")?: command => do
     let (parL, parR) := if d.comment.isEmpty then ("", "") else (" (", ")")
     let cmt := parL ++ d.comment ++ parR
     msgs := msgs.push
-      m!"[Stacks Tag {d.tag}](https://stacks.math.columbia.edu/tag/{d.tag}) \
+      m!"[Stacks Tag {d.tag}]({databaseURL db ++ d.tag}) \
         corresponds to declaration '{dname}'.{cmt}"
-    if tk.isSome then
+    if verbose then
       let dType := ((env.find? dname).getD default).type
       msgs := (msgs.push m!"{dType}").push ""
   let msg := MessageData.joinSep msgs.toList "\n"
   logInfo msg
+
+/--
+`#stacks_tags` retrieves all declarations that have the `stacks` attribute.
+
+For each found declaration, it prints a line
+```
+'declaration_name' corresponds to tag 'declaration_tag'.
+```
+The variant `#stacks_tags!` also adds the theorem statement after each summary line.
+-/
+elab (name := stacksTags) "#stacks_tags" tk:("!")?: command =>
+  traceStacksTags .stacks (tk.isSome)
+
+/-- The `#kerodon_tags` command retrieves all declarations that have the `kerodon` attribute.
+
+For each found declaration, it prints a line
+```
+'declaration_name' corresponds to tag 'declaration_tag'.
+```
+The variant `#kerodon_tags!` also adds the theorem statement after each summary line.
+-/
+elab (name := kerodonTags) "#kerodon_tags" tk:("!")?: command =>
+  traceStacksTags .kerodon (tk.isSome)
+
+end Mathlib.StacksTag
diff --git a/Mathlib/Tactic/Subsingleton.lean b/Mathlib/Tactic/Subsingleton.lean
index 2a0bd07630091..b5b8b9b18f06a 100644
--- a/Mathlib/Tactic/Subsingleton.lean
+++ b/Mathlib/Tactic/Subsingleton.lean
@@ -97,7 +97,8 @@ def Lean.MVarId.subsingleton (g : MVarId) (insts : Array (Term × AbstractMVarsR
         g.assign <| mkApp4 (.const ``proof_irrel_heq []) xTy yTy x y
         return
       throwError "tactic 'subsingleton' could not prove heterogeneous equality"
-    throwError "tactic 'subsingleton' failed, goal is neither an equality nor heterogeneous equality"
+    throwError "tactic 'subsingleton' failed, goal is neither an equality nor a \
+      heterogeneous equality"
 
 namespace Mathlib.Tactic
 
diff --git a/Mathlib/Tactic/SuccessIfFailWithMsg.lean b/Mathlib/Tactic/SuccessIfFailWithMsg.lean
index 9d80c1c15e17a..70bb4e7e8468d 100644
--- a/Mathlib/Tactic/SuccessIfFailWithMsg.lean
+++ b/Mathlib/Tactic/SuccessIfFailWithMsg.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro, Simon Hudon, Sébastien Gouëzel, Scott Morrison, Thomas Murrills
+Authors: Mario Carneiro, Simon Hudon, Sébastien Gouëzel, Kim Morrison, Thomas Murrills
 -/
 import Mathlib.Init
 import Lean
diff --git a/Mathlib/Tactic/TFAE.lean b/Mathlib/Tactic/TFAE.lean
index a5508905dec20..2a4ec7918e278 100644
--- a/Mathlib/Tactic/TFAE.lean
+++ b/Mathlib/Tactic/TFAE.lean
@@ -15,48 +15,108 @@ This file provides the tactics `tfae_have` and `tfae_finish` for proving goals o
 `TFAE [P₁, P₂, ...]`.
 -/
 
-open List Lean Meta Expr Elab.Term Elab.Tactic Mathlib.Tactic Qq
-
 namespace Mathlib.Tactic.TFAE
 
-/-- An arrow of the form `←`, `→`, or `↔`. -/
-syntax impArrow := " → " <|> " ↔ " <|> " ← "
+/-! # Parsing and syntax
+
+We implement `tfae_have` in terms of a syntactic `have`. To support as much of the same syntax as
+possible, we recreate the parsers for `have`, except with the changes necessary for `tfae_have`.
+-/
+
+open Lean.Parser Term
+
+namespace Parser
+
+/- An arrow of the form `←`, `→`, or `↔`. -/
+private def impTo : Parser := leading_parser unicodeSymbol " → " " -> "
+private def impFrom : Parser := leading_parser unicodeSymbol " ← " " <- "
+private def impIff : Parser := leading_parser unicodeSymbol " ↔ " " <-> "
+private def impArrow : Parser := leading_parser impTo <|> impFrom <|> impIff
+
+/-- A `tfae_have` type specification, e.g. `1 ↔ 3` The numbers refer to the proposition at the
+corresponding position in the `TFAE` goal (starting at 1). -/
+private def tfaeType := leading_parser num >> impArrow >> num
+
+/-!
+The following parsers are similar to those for `have` in `Lean.Parser.Term`, but
+instead of `optType`, we use `tfaeType := num >> impArrow >> num` (as a `tfae_have` invocation must
+always include this specification). Also, we disallow including extra binders, as that makes no
+sense in this context; we also include `" : "` after the binder to avoid breaking `tfae_have 1 → 2`
+syntax (which, unlike `have`, omits `" : "`).
+-/
+
+/- See `haveIdLhs`.
+
+We omit `many (ppSpace >> letIdBinder)`, as it makes no sense to add extra arguments to a
+`tfae_have` decl.  -/
+private def tfaeHaveIdLhs := leading_parser
+  ((ppSpace >> binderIdent >> " : ") <|> hygieneInfo)  >> tfaeType
+/- See `haveIdDecl`. E.g. `h : 1 → 3 := term`. -/
+private def tfaeHaveIdDecl   := leading_parser (withAnonymousAntiquot := false)
+  atomic (tfaeHaveIdLhs >> " := ") >> termParser
+/- See `haveEqnsDecl`. E.g. `h : 1 → 3 | p => f p`. -/
+private def tfaeHaveEqnsDecl := leading_parser (withAnonymousAntiquot := false)
+  tfaeHaveIdLhs >> matchAlts
+/- See `letPatDecl`. E.g. `⟨mp, mpr⟩ : 1 ↔ 3 := term`. -/
+private def tfaeHavePatDecl  := leading_parser (withAnonymousAntiquot := false)
+  atomic (termParser >> pushNone >> " : " >> tfaeType >> " := ") >> termParser
+/- See `haveDecl`. Any of `tfaeHaveIdDecl`, `tfaeHavePatDecl`, or `tfaeHaveEqnsDecl`. -/
+private def tfaeHaveDecl     := leading_parser (withAnonymousAntiquot := false)
+  tfaeHaveIdDecl <|> (ppSpace >> tfaeHavePatDecl) <|> tfaeHaveEqnsDecl
+
+end Parser
+
+open Parser
 
 /--
 `tfae_have` introduces hypotheses for proving goals of the form `TFAE [P₁, P₂, ...]`. Specifically,
-`tfae_have i arrow j` introduces a hypothesis of type `Pᵢ arrow Pⱼ` to the local context,
-where `arrow` can be `→`, `←`, or `↔`. Note that `i` and `j` are natural number indices (beginning
-at 1) used to specify the propositions `P₁, P₂, ...` that appear in the `TFAE` goal list. A proof
-is required afterward, typically via a tactic block.
+`tfae_have i <arrow> j := ...` introduces a hypothesis of type `Pᵢ <arrow> Pⱼ` to the local
+context, where `<arrow>` can be `→`, `←`, or `↔`. Note that `i` and `j` are natural number indices
+(beginning at 1) used to specify the propositions `P₁, P₂, ...` that appear in the goal.
 
 ```lean
 example (h : P → R) : TFAE [P, Q, R] := by
-  tfae_have 1 → 3
-  · exact h
+  tfae_have 1 → 3 := h
   ...
 ```
 The resulting context now includes `tfae_1_to_3 : P → R`.
 
-The introduced hypothesis can be given a custom name, in analogy to `have` syntax:
+Once sufficient hypotheses have been introduced by `tfae_have`, `tfae_finish` can be used to close
+the goal. For example,
+
 ```lean
-tfae_have h : 2 ↔ 3
+example : TFAE [P, Q, R] := by
+  tfae_have 1 → 2 := sorry /- proof of P → Q -/
+  tfae_have 2 → 1 := sorry /- proof of Q → P -/
+  tfae_have 2 ↔ 3 := sorry /- proof of Q ↔ R -/
+  tfae_finish
 ```
 
-Once sufficient hypotheses have been introduced by `tfae_have`, `tfae_finish` can be used to close
-the goal.
+All relevant features of `have` are supported by `tfae_have`, including naming, destructuring, goal
+creation, and matching. These are demonstrated below.
 
 ```lean
-example : TFAE [P, Q, R] := by
+example : TFAE [P, Q] := by
+  -- `tfae_1_to_2 : P → Q`:
+  tfae_have 1 → 2 := sorry
+  -- `hpq : P → Q`:
+  tfae_have hpq : 1 → 2 := sorry
+  -- inaccessible `h✝ : P → Q`:
+  tfae_have _ : 1 → 2 := sorry
+  -- `tfae_1_to_2 : P → Q`, and `?a` is a new goal:
+  tfae_have 1 → 2 := f ?a
+  -- create a goal of type `P → Q`:
   tfae_have 1 → 2
-  · /- proof of P → Q -/
-  tfae_have 2 → 1
-  · /- proof of Q → P -/
-  tfae_have 2 ↔ 3
-  · /- proof of Q ↔ R -/
-  tfae_finish
+  · exact (sorry : P → Q)
+  -- match on `p : P` and prove `Q`:
+  tfae_have 1 → 2
+  | p => f p
+  -- introduces `pq : P → Q`, `qp : Q → P`:
+  tfae_have ⟨pq, qp⟩ : 1 ↔ 2 := sorry
+  ...
 ```
 -/
-syntax (name := tfaeHave) "tfae_have " (ident " : ")? num impArrow num : tactic
+syntax (name := tfaeHave) "tfae_have " tfaeHaveDecl : tactic
 
 /--
 `tfae_finish` is used to close goals of the form `TFAE [P₁, P₂, ...]` once a sufficient collection
@@ -67,19 +127,19 @@ of hypotheses of the form `Pᵢ → Pⱼ` or `Pᵢ ↔ Pⱼ` have been introduce
 Example:
 ```lean
 example : TFAE [P, Q, R] := by
-  tfae_have 1 → 2
-  · /- proof of P → Q -/
-  tfae_have 2 → 1
-  · /- proof of Q → P -/
-  tfae_have 2 ↔ 3
-  · /- proof of Q ↔ R -/
+  tfae_have 1 → 2 := sorry /- proof of P → Q -/
+  tfae_have 2 → 1 := sorry /- proof of Q → P -/
+  tfae_have 2 ↔ 3 := sorry /- proof of Q ↔ R -/
   tfae_finish
 ```
 -/
 syntax (name := tfaeFinish) "tfae_finish" : tactic
 
+
 /-! # Setup -/
 
+open List Lean Meta Expr Elab Tactic Mathlib.Tactic Qq
+
 /-- Extract a list of `Prop` expressions from an expression of the form `TFAE [P₁, P₂, ...]` as
 long as `[P₁, P₂, ...]` is an explicit list. -/
 partial def getTFAEList (t : Expr) : MetaM (Q(List Prop) × List Q(Prop)) := do
@@ -101,7 +161,7 @@ where
 variable (hyps : Array (ℕ × ℕ × Expr)) (atoms : Array Q(Prop))
 
 /-- Uses depth-first search to find a path from `P` to `P'`. -/
-partial def dfs (i j : ℕ) (P P' : Q(Prop)) (hP : Q($P)) : StateT (HashSet ℕ) MetaM Q($P') := do
+partial def dfs (i j : ℕ) (P P' : Q(Prop)) (hP : Q($P)) : StateT (Std.HashSet ℕ) MetaM Q($P') := do
   if i == j then
     return hP
   modify (·.insert i)
@@ -159,62 +219,72 @@ def proveTFAE (is : List ℕ) (l : Q(List Prop)) : MetaM Q(TFAE $l) := do
 /-! # `tfae_have` components -/
 
 /-- Construct a name for a hypothesis introduced by `tfae_have`. -/
-def mkTFAEHypName (i j : TSyntax `num) (arr : TSyntax ``impArrow) : MetaM Name := do
-  let arr ← match arr with
-  | `(impArrow| ← ) => pure "from"
-  | `(impArrow| → ) => pure "to"
-  | `(impArrow| ↔ ) => pure "iff"
-  | _ => throwErrorAt arr "expected '←', '→', or '↔'"
-  return .mkSimple <| String.intercalate "_" ["tfae", s!"{i.getNat}", arr, s!"{j.getNat}"]
-
-open Elab in
-/-- The core of `tfae_have`, which behaves like `haveLetCore` in `Mathlib.Tactic.Have`. -/
-def tfaeHaveCore (goal : MVarId) (name : Option (TSyntax `ident)) (i j : TSyntax `num)
-    (arrow : TSyntax ``impArrow) (t : Expr) : TermElabM (MVarId × MVarId) :=
-  goal.withContext do
-    let n := (Syntax.getId <$> name).getD <|← mkTFAEHypName i j arrow
-    let (goal1, t, p) ← do
-      let p ← mkFreshExprMVar t MetavarKind.syntheticOpaque n
-      pure (p.mvarId!, t, p)
-    let (fv, goal2) ← (← MVarId.assert goal n t p).intro1P
-    if let some stx := name then
-      goal2.withContext do
-        Term.addTermInfo' (isBinder := true) stx (mkFVar fv)
-    pure (goal1, goal2)
+def mkTFAEId : TSyntax ``tfaeType → MacroM Name
+  | `(tfaeType|$i:num $arr:impArrow $j:num) => do
+    let arr ← match arr with
+    | `(impArrow| ← ) => pure "from"
+    | `(impArrow| → ) => pure "to"
+    | `(impArrow| ↔ ) => pure "iff"
+    | _ => Macro.throwUnsupported
+    return .mkSimple <| String.intercalate "_" ["tfae", s!"{i.getNat}", arr, s!"{j.getNat}"]
+  | _ => Macro.throwUnsupported
 
 /-- Turn syntax for a given index into a natural number, as long as it lies between `1` and
 `maxIndex`. -/
-def elabIndex (i : TSyntax `num) (maxIndex : ℕ) : TacticM ℕ := do
+def elabIndex (i : TSyntax `num) (maxIndex : ℕ) : MetaM ℕ := do
   let i' := i.getNat
-  unless Nat.ble 1 i' && Nat.ble i' maxIndex do
-    throwError "{i} must be between 1 and {maxIndex}"
+  unless 1 ≤ i' && i' ≤ maxIndex do
+    throwErrorAt i "{i} must be between 1 and {maxIndex}"
   return i'
 
-/-- Construct an expression for the type `Pj → Pi`, `Pi → Pj`, or `Pi ↔ Pj` given expressions
-`Pi Pj : Q(Prop)` and `impArrow` syntax `arr`, depending on whether `arr` is `←`, `→`, or `↔`
-respectively. -/
-def mkImplType (Pi : Q(Prop)) (arr : TSyntax ``impArrow) (Pj : Q(Prop)) : MetaM Q(Prop) := do
-  match arr with
-  | `(impArrow| ← ) => pure q($Pj → $Pi)
-  | `(impArrow| → ) => pure q($Pi → $Pj)
-  | `(impArrow| ↔ ) => pure q($Pi ↔ $Pj)
-  | _ => throwErrorAt arr "expected '←', '→', or '↔'"
-
 /-! # Tactic implementation -/
 
-elab_rules : tactic
-| `(tactic| tfae_have $[$h:ident : ]? $i:num $arr:impArrow $j:num) => do
-  let goal ← getMainGoal
-  goal.withContext do
-    let (_, tfaeList) ← getTFAEList (← goal.getType)
-    let l₀ := tfaeList.length
-    let i' ← elabIndex i l₀
-    let j' ← elabIndex j l₀
+/-- Accesses the propositions at indices `i` and `j` of `tfaeList`, and constructs the expression
+`Pi <arr> Pj`, which will be the type of our `tfae_have` hypothesis -/
+def elabTFAEType (tfaeList : List Q(Prop)) : TSyntax ``tfaeType → TermElabM Expr
+  | stx@`(tfaeType|$i:num $arr:impArrow $j:num) => do
+    let l := tfaeList.length
+    let i' ← elabIndex i l
+    let j' ← elabIndex j l
     let Pi := tfaeList.get! (i'-1)
     let Pj := tfaeList.get! (j'-1)
-    let type ← mkImplType Pi arr Pj
-    let (goal1, goal2) ← tfaeHaveCore goal h i j arr type
-    replaceMainGoal [goal1, goal2]
+    Term.addTermInfo' i Pi q(Prop)
+    Term.addTermInfo' j Pj q(Prop)
+    match arr with
+    | `(impArrow| ← ) => Term.addTermInfo stx q($Pj → $Pi) q(Prop)
+    | `(impArrow| → ) => Term.addTermInfo stx q($Pi → $Pj) q(Prop)
+    | `(impArrow| ↔ ) => Term.addTermInfo stx q($Pi ↔ $Pj) q(Prop)
+    | _ => throwUnsupportedSyntax
+  | _ => throwUnsupportedSyntax
+
+/- Convert `tfae_have i <arr> j ...` to `tfae_have tfae_i_arr_j : i <arr> j ...`. See
+`expandHave`, which is responsible for inserting `this` in `have : A := ...`. -/
+macro_rules
+| `(tfaeHave|tfae_have $hy:hygieneInfo $t:tfaeType := $val) => do
+  let id := HygieneInfo.mkIdent hy (← mkTFAEId t) (canonical := true)
+  `(tfaeHave|tfae_have $id : $t := $val)
+| `(tfaeHave|tfae_have $hy:hygieneInfo $t:tfaeType $alts:matchAlts) => do
+  let id := HygieneInfo.mkIdent hy (← mkTFAEId t) (canonical := true)
+  `(tfaeHave|tfae_have $id : $t $alts)
+
+open Term
+
+elab_rules : tactic
+| `(tfaeHave|tfae_have $d:tfaeHaveDecl) => withMainContext do
+  let goal ← getMainGoal
+  let (_, tfaeList) ← getTFAEList (← goal.getType)
+  withRef d do
+    match d with
+    | `(tfaeHaveDecl| $b : $t:tfaeType := $pf:term) =>
+      let type ← elabTFAEType tfaeList t
+      evalTactic <|← `(tactic|have $b : $(← exprToSyntax type) := $pf)
+    | `(tfaeHaveDecl| $b : $t:tfaeType $alts:matchAlts) =>
+      let type ← elabTFAEType tfaeList t
+      evalTactic <|← `(tactic|have $b : $(← exprToSyntax type) $alts:matchAlts)
+    | `(tfaeHaveDecl| $pat:term : $t:tfaeType := $pf:term) =>
+      let type ← elabTFAEType tfaeList t
+      evalTactic <|← `(tactic|have $pat:term : $(← exprToSyntax type) := $pf)
+    | _ => throwUnsupportedSyntax
 
 elab_rules : tactic
 | `(tactic| tfae_finish) => do
@@ -225,7 +295,7 @@ elab_rules : tactic
       let is ← tfaeList.mapM AtomM.addAtom
       let mut hyps := #[]
       for hyp in ← getLocalHyps do
-        let ty ← inferType hyp
+        let ty ← whnfR <|← instantiateMVars <|← inferType hyp
         if let (``Iff, #[p1, p2]) := ty.getAppFnArgs then
           let q1 ← AtomM.addAtom p1
           let q2 ← AtomM.addAtom p2
@@ -237,6 +307,39 @@ elab_rules : tactic
           hyps := hyps.push (q1, q2, hyp)
       proveTFAE hyps (← get).atoms is tfaeListQ
 
+/-!
+
+# "Old-style" `tfae_have`
+
+We preserve the "old-style" `tfae_have` (which behaves like Mathlib `have`) for compatibility
+purposes.
+
+-/
+
+@[inherit_doc tfaeHave]
+syntax (name := tfaeHave') "tfae_have " tfaeHaveIdLhs : tactic
+
+macro_rules
+| `(tfaeHave'|tfae_have $hy:hygieneInfo $t:tfaeType) => do
+  let id := HygieneInfo.mkIdent hy (← mkTFAEId t) (canonical := true)
+  `(tfaeHave'|tfae_have $id : $t)
+
+elab_rules : tactic
+| `(tfaeHave'|tfae_have $d:tfaeHaveIdLhs) => withMainContext do
+  let goal ← getMainGoal
+  let (_, tfaeList) ← getTFAEList (← goal.getType)
+  -- Note that due to the macro above, the following match is exhaustive.
+  match d with
+  | `(tfaeHaveIdLhs| $b:ident : $t:tfaeType) =>
+    let n := b.getId
+    let type ← elabTFAEType tfaeList t
+    let p ← mkFreshExprMVar type MetavarKind.syntheticOpaque n
+    let (fv, mainGoal) ← (← MVarId.assert goal n type p).intro1P
+    mainGoal.withContext do
+      Term.addTermInfo' (isBinder := true) b (mkFVar fv)
+    replaceMainGoal [p.mvarId!, mainGoal]
+  | _ => throwUnsupportedSyntax
+
 end TFAE
 
 end Mathlib.Tactic
diff --git a/Mathlib/Tactic/TermCongr.lean b/Mathlib/Tactic/TermCongr.lean
index 11ff9166739ac..5ec0c3b92a6bf 100644
--- a/Mathlib/Tactic/TermCongr.lean
+++ b/Mathlib/Tactic/TermCongr.lean
@@ -139,7 +139,7 @@ def cHole? (e : Expr) (mvarCounterSaved? : Option Nat := none) : Option (Bool ×
     return (forLhs, val, pf)
   | _ => none
 
-/-- Returns any subexpression that is a recent congruence hole.  -/
+/-- Returns any subexpression that is a recent congruence hole. -/
 def hasCHole (mvarCounterSaved : Nat) (e : Expr) : Option Expr :=
   e.find? fun e' => (cHole? e' mvarCounterSaved).isSome
 
diff --git a/Mathlib/Tactic/ToAdditive/Frontend.lean b/Mathlib/Tactic/ToAdditive/Frontend.lean
index 31cfea1e8b165..bdc528a294073 100644
--- a/Mathlib/Tactic/ToAdditive/Frontend.lean
+++ b/Mathlib/Tactic/ToAdditive/Frontend.lean
@@ -818,9 +818,10 @@ partial def transformDeclAux
   if isProtected (← getEnv) src then
     setEnv <| addProtected (← getEnv) tgt
   if let some matcherInfo ← getMatcherInfo? src then
-    -- Use
-    --   Match.addMatcherInfo tgt matcherInfo
-    -- once on lean 4.13.
+    /-
+    Use `Match.addMatcherInfo tgt matcherInfo`
+    once https://github.com/leanprover/lean4/pull/5068 is in
+    -/
     modifyEnv fun env => Match.Extension.addMatcherInfo env tgt matcherInfo
 
 /-- Copy the instance attribute in a `to_additive`
@@ -964,6 +965,7 @@ def nameDict : String → List String
   | "zpowers"     => ["zmultiples"]
   | "powers"      => ["multiples"]
   | "multipliable"=> ["summable"]
+  | "gpfree"      => ["apfree"]
   | x             => [x]
 
 /--
@@ -1048,8 +1050,6 @@ def fixAbbreviation : List String → List String
                                       => "function" :: "_" :: "commute" :: fixAbbreviation s
   | "zero" :: "Le" :: "Part" :: s         => "posPart" :: fixAbbreviation s
   | "le" :: "Zero" :: "Part" :: s         => "negPart" :: fixAbbreviation s
-  | "three" :: "GPFree" :: s         => "three" :: "APFree" :: fixAbbreviation s
-  | "Three" :: "GPFree" :: s         => "Three" :: "APFree" :: fixAbbreviation s
   | "Division" :: "Add" :: "Monoid" :: s => "SubtractionMonoid" :: fixAbbreviation s
   | "division" :: "Add" :: "Monoid" :: s => "subtractionMonoid" :: fixAbbreviation s
   | "Sub" :: "Neg" :: "Zero" :: "Add" :: "Monoid" :: s => "SubNegZeroMonoid" :: fixAbbreviation s
@@ -1223,7 +1223,7 @@ partial def copyMetaData (cfg : Config) (src tgt : Name) : CoreM (Array Name) :=
     definitions. If we don't do that, the equation lemma for `src` might be generated later
     when doing a `rw`, but it won't be generated for `tgt`. -/
     additivizeLemmas #[src, tgt] "equation lemmas" fun nm ↦
-      (·.getD #[]) <$> MetaM.run' (getEqnsFor? nm true)
+      (·.getD #[]) <$> MetaM.run' (getEqnsFor? nm)
   MetaM.run' <| Elab.Term.TermElabM.run' <|
     applyAttributes cfg.ref cfg.attrs `to_additive src tgt
 
@@ -1506,3 +1506,5 @@ initialize registerBuiltinAttribute {
   }
 
 end ToAdditive
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Tactic/ToExpr.lean b/Mathlib/Tactic/ToExpr.lean
index 3c57fb543a3b6..b02e21d2b44bb 100644
--- a/Mathlib/Tactic/ToExpr.lean
+++ b/Mathlib/Tactic/ToExpr.lean
@@ -16,7 +16,7 @@ that come from core Lean 4 that do not handle universe polymorphism.
 In addition, we provide some additional `ToExpr` instances for core definitions.
 -/
 
-section override
+section override -- Note: this section uses `autoImplicit` pervasively
 namespace Lean
 
 attribute [-instance] Lean.instToExprOption
diff --git a/Mathlib/Tactic/Variable.lean b/Mathlib/Tactic/Variable.lean
index ae6e4f39f4ddb..ce1dd2966519a 100644
--- a/Mathlib/Tactic/Variable.lean
+++ b/Mathlib/Tactic/Variable.lean
@@ -250,7 +250,7 @@ def elabVariables : CommandElab := fun stx =>
 where
   extendScope (binders : TSyntaxArray ``bracketedBinder) : CommandElabM Unit := do
     for binder in binders do
-      let varUIds ← getBracketedBinderIds binder |>.mapM
+      let varUIds ← (← getBracketedBinderIds binder) |>.mapM
         (withFreshMacroScope ∘ MonadQuotation.addMacroScope)
       modifyScope fun scope =>
         { scope with varDecls := scope.varDecls.push binder, varUIds := scope.varUIds ++ varUIds }
@@ -270,7 +270,7 @@ where
       Term.withAutoBoundImplicit <| Term.elabBinders binders fun _ => pure ()
       -- Filter out omitted binders
       let binders' : TSyntaxArray ``bracketedBinder :=
-        (binders.zip toOmit).filterMap fun (b, omit) => if omit then none else some b
+        (binders.zip toOmit).filterMap fun (b, toOmit) => if toOmit then none else some b
       if let some expectedBinders := expectedBinders? then
         trace[«variable?»] "checking expected binders"
         /- We re-elaborate the binders to create an expression that represents the entire resulting
diff --git a/Mathlib/Tactic/WLOG.lean b/Mathlib/Tactic/WLOG.lean
index c996e5552b6df..461949d04ab7b 100644
--- a/Mathlib/Tactic/WLOG.lean
+++ b/Mathlib/Tactic/WLOG.lean
@@ -86,7 +86,8 @@ def _root_.Lean.MVarId.wlog (goal : MVarId) (h : Option Name) (P : Expr)
   let hGoal := HExpr.mvarId!
   /- Begin the "reduction goal" which will contain hypotheses `H` and `¬h`. For now, it only
   contains `H`. Keep track of that hypothesis' FVarId. -/
-  let (HFVarId, reductionGoal) ← goal.assertHypotheses #[⟨H, HType, HExpr⟩]
+  let (HFVarId, reductionGoal) ←
+    goal.assertHypotheses #[{ userName := H, type := HType, value := HExpr }]
   let HFVarId := HFVarId[0]!
   /- Clear the reverted fvars from the branch that will contain `h` as a hypothesis. -/
   let hGoal ← hGoal.tryClearMany revertedFVars
diff --git a/Mathlib/Tactic/Widget/Calc.lean b/Mathlib/Tactic/Widget/Calc.lean
index bbcba1e9996ae..5e39f560b44b2 100644
--- a/Mathlib/Tactic/Widget/Calc.lean
+++ b/Mathlib/Tactic/Widget/Calc.lean
@@ -17,7 +17,7 @@ new calc steps with holes specified by selected sub-expressions in the goal.
 -/
 
 section code_action
-open Std CodeAction
+open Batteries.CodeAction
 open Lean Server RequestM
 
 /-- Code action to create a `calc` tactic from the current goal. -/
diff --git a/Mathlib/Tactic/Widget/InteractiveUnfold.lean b/Mathlib/Tactic/Widget/InteractiveUnfold.lean
index b88ea0416d05d..7d3eae135546f 100644
--- a/Mathlib/Tactic/Widget/InteractiveUnfold.lean
+++ b/Mathlib/Tactic/Widget/InteractiveUnfold.lean
@@ -225,7 +225,7 @@ elab stx:"unfold?" : tactic => do
 
 /-- `#unfold? e` gives all unfolds of `e`.
 In tactic mode, use `unfold?` instead. -/
-syntax (name := unfoldCommand) "#unfold?" term : command
+syntax (name := unfoldCommand) "#unfold? " term : command
 
 open Elab
 /-- Elaborate a `#unfold?` command. -/
diff --git a/Mathlib/Tactic/Widget/StringDiagram.lean b/Mathlib/Tactic/Widget/StringDiagram.lean
index 5aca78961dfc3..9ba22761bcd5e 100644
--- a/Mathlib/Tactic/Widget/StringDiagram.lean
+++ b/Mathlib/Tactic/Widget/StringDiagram.lean
@@ -6,7 +6,9 @@ Authors: Yuma Mizuno
 import ProofWidgets.Component.PenroseDiagram
 import ProofWidgets.Component.Panel.Basic
 import ProofWidgets.Presentation.Expr
-import Mathlib.Tactic.CategoryTheory.Monoidal
+import ProofWidgets.Component.HtmlDisplay
+import Mathlib.Tactic.CategoryTheory.Bicategory.Normalize
+import Mathlib.Tactic.CategoryTheory.Monoidal.Normalize
 
 /-!
 # String Diagram Widget
@@ -21,6 +23,12 @@ show_panel_widgets [local StringDiagram]
 ```
 to enable the string diagram widget in the current section.
 
+We also have the `#string_diagram` command. For example,
+```lean
+#string_diagram MonoidalCategory.whisker_exchange
+```
+displays the string diagram for the exchange law of the left and right whiskerings.
+
 String diagrams are graphical representations of morphisms in monoidal categories, which are
 useful for rewriting computations. More precisely, objects in a monoidal category is represented
 by strings, and morphisms between two objects is represented by nodes connecting two strings
@@ -54,12 +62,13 @@ namespace Mathlib.Tactic
 
 open Lean Meta Elab
 open CategoryTheory
-open Mathlib.Tactic.Coherence
 
-open Mathlib.Tactic.Monoidal
+open BicategoryLike
 
 namespace Widget.StringDiagram
 
+initialize registerTraceClass `string_diagram
+
 /-! ## Objects in string diagrams -/
 
 /-- Nodes for 2-morphisms in a string diagram. -/
@@ -96,15 +105,15 @@ def Node.e : Node → Expr
 
 /-- The domain of the 2-morphism associated with a node as a list
 (the first component is the node itself). -/
-def Node.srcList : Node → MetaM (List (Node × Atom₁))
-  | Node.atom n => return (← n.atom.src).toList.map (fun f ↦ (.atom n, f))
-  | Node.id n => return [(.id n, n.id)]
+def Node.srcList : Node → List (Node × Atom₁)
+  | Node.atom n => n.atom.src.toList.map (fun f ↦ (.atom n, f))
+  | Node.id n => [(.id n, n.id)]
 
 /-- The codomain of the 2-morphism associated with a node as a list
 (the first component is the node itself). -/
-def Node.tarList : Node → MetaM (List (Node × Atom₁))
-  | Node.atom n => return (← n.atom.tgt).toList.map (fun f ↦ (.atom n, f))
-  | Node.id n => return [(.id n, n.id)]
+def Node.tarList : Node → List (Node × Atom₁)
+  | Node.atom n => n.atom.tgt.toList.map (fun f ↦ (.atom n, f))
+  | Node.id n => [(.id n, n.id)]
 
 /-- The vertical position of a node in a string diagram. -/
 def Node.vPos : Node → ℕ
@@ -121,10 +130,6 @@ def Node.hPosTar : Node → ℕ
   | Node.atom n => n.hPosTar
   | Node.id n => n.hPosTar
 
-/-- The list of nodes at the top of a string diagram. -/
-def topNodes (η : WhiskerLeftExpr) : MetaM (List Node) := do
-  return (← η.src).toList.enum.map (fun (i, f) => .id ⟨0, i, i, f⟩)
-
 /-- Strings in a string diagram. -/
 structure Strand : Type where
   /-- The horizontal position of the strand in the string diagram. -/
@@ -142,64 +147,79 @@ def Strand.vPos (s : Strand) : ℕ :=
 
 end Widget.StringDiagram
 
-namespace Monoidal
+namespace BicategoryLike
 
 open Widget.StringDiagram
 
 /-- The list of nodes associated with a 2-morphism. The position is counted from the
 specified natural numbers. -/
-def WhiskerRightExpr.nodes (v h₁ h₂ : ℕ) : WhiskerRightExpr → MetaM (List Node)
-  | WhiskerRightExpr.of η => do
-    return [.atom ⟨v, h₁, h₂, η⟩]
-  | WhiskerRightExpr.whisker η f => do
-    let ηs ← η.nodes v h₁ h₂
-    let k₁ := (← ηs.mapM (fun n ↦ n.srcList)).join.length
-    let k₂ := (← ηs.mapM (fun n ↦ n.tarList)).join.length
+def WhiskerRight.nodes (v h₁ h₂ : ℕ) : WhiskerRight → List Node
+  | WhiskerRight.of η => [.atom ⟨v, h₁, h₂, η⟩]
+  | WhiskerRight.whisker _ η f =>
+    let ηs := η.nodes v h₁ h₂
+    let k₁ := (ηs.map (fun n ↦ n.srcList)).join.length
+    let k₂ := (ηs.map (fun n ↦ n.tarList)).join.length
     let s : Node := .id ⟨v, h₁ + k₁, h₂ + k₂, f⟩
-    return ηs ++ [s]
+    ηs ++ [s]
 
 /-- The list of nodes associated with a 2-morphism. The position is counted from the
 specified natural numbers. -/
-def WhiskerLeftExpr.nodes (v h₁ h₂ : ℕ) : WhiskerLeftExpr → MetaM (List Node)
-  | WhiskerLeftExpr.of η => η.nodes v h₁ h₂
-  | WhiskerLeftExpr.whisker f η => do
+def HorizontalComp.nodes (v h₁ h₂ : ℕ) : HorizontalComp → List Node
+  | HorizontalComp.of η => η.nodes v h₁ h₂
+  | HorizontalComp.cons _ η ηs =>
+    let s₁ := η.nodes v h₁ h₂
+    let k₁ := (s₁.map (fun n ↦ n.srcList)).join.length
+    let k₂ := (s₁.map (fun n ↦ n.tarList)).join.length
+    let s₂ := ηs.nodes v (h₁ + k₁) (h₂ + k₂)
+    s₁ ++ s₂
+
+/-- The list of nodes associated with a 2-morphism. The position is counted from the
+specified natural numbers. -/
+def WhiskerLeft.nodes (v h₁ h₂ : ℕ) : WhiskerLeft → List Node
+  | WhiskerLeft.of η => η.nodes v h₁ h₂
+  | WhiskerLeft.whisker _ f η =>
     let s : Node := .id ⟨v, h₁, h₂, f⟩
-    let ss ← η.nodes v (h₁ + 1) (h₂ + 1)
-    return s :: ss
+    let ss := η.nodes v (h₁ + 1) (h₂ + 1)
+    s :: ss
+
+variable {ρ : Type} [Context ρ] [MonadMor₁ (CoherenceM ρ)]
+
+/-- The list of nodes at the top of a string diagram. -/
+def topNodes (η : WhiskerLeft) : CoherenceM ρ (List Node) := do
+  return (← η.srcM).toList.enum.map (fun (i, f) => .id ⟨0, i, i, f⟩)
 
 /-- The list of nodes at the top of a string diagram. The position is counted from the
 specified natural number. -/
-def NormalExpr.nodesAux (v : ℕ) : NormalExpr → MetaM (List (List Node))
-  | NormalExpr.nil α => return [(α.src).toList.enum.map (fun (i, f) => .id ⟨v, i, i, f⟩)]
-  | NormalExpr.cons _ η ηs => do
-    let s₁ ← η.nodes v 0 0
+def NormalExpr.nodesAux (v : ℕ) : NormalExpr → CoherenceM ρ (List (List Node))
+  | NormalExpr.nil _ α => return [(← α.srcM).toList.enum.map (fun (i, f) => .id ⟨v, i, i, f⟩)]
+  | NormalExpr.cons _ _ η ηs => do
+    let s₁ := η.nodes v 0 0
     let s₂ ← ηs.nodesAux (v + 1)
     return s₁ :: s₂
 
 /-- The list of nodes associated with a 2-morphism. -/
-def NormalExpr.nodes (e : NormalExpr) : MetaM (List (List Node)) := do
+def NormalExpr.nodes (e : NormalExpr) : CoherenceM ρ (List (List Node)) :=
   match e with
-  | NormalExpr.nil _ => return []
-  | NormalExpr.cons _ η _ => return (← topNodes η) :: (← e.nodesAux 1)
+  | NormalExpr.nil _ _ => return []
+  | NormalExpr.cons _ _ η _ => return (← topNodes η) :: (← e.nodesAux 1)
 
 /-- `pairs [a, b, c, d]` is `[(a, b), (b, c), (c, d)]`. -/
-def pairs {α : Type} : List α → List (α × α)
-  | [] => []
-  | [_] => []
-  | (x :: y :: ys) => (x, y) :: pairs (y :: ys)
+def pairs {α : Type} : List α → List (α × α) :=
+  fun l => l.zip (l.drop 1)
 
 /-- The list of strands associated with a 2-morphism. -/
-def NormalExpr.strands (e : NormalExpr) : MetaM (List (List Strand)) := do
+def NormalExpr.strands (e : NormalExpr) : CoherenceM ρ (List (List Strand)) := do
   let l ← e.nodes
   (pairs l).mapM fun (x, y) ↦ do
-    let xs := (← x.mapM (fun n ↦ n.tarList)).join
-    let ys := (← y.mapM (fun n ↦ n.srcList)).join
+    let xs := (x.map (fun n ↦ n.tarList)).join
+    let ys := (y.map (fun n ↦ n.srcList)).join
+    -- sanity check
     if xs.length ≠ ys.length then
       throwError "The number of the start and end points of a string does not match."
     (xs.zip ys).enum.mapM fun (k, (n₁, f₁), (n₂, _)) => do
       return ⟨n₁.hPosTar + k, n₁, n₂, f₁⟩
 
-end Monoidal
+end BicategoryLike
 
 namespace Widget.StringDiagram
 
@@ -243,9 +263,8 @@ def addConstructor (tp : String) (v : PenroseVar) (nm : String) (vs : List Penro
 open scoped Jsx in
 /-- Construct a string diagram from a Penrose `sub`stance program and expressions `embeds` to
 display as labels in the diagram. -/
-def mkStringDiagram (e : NormalExpr) : DiagramBuilderM PUnit := do
-  let nodes ← e.nodes
-  let strands ← e.strands
+def mkStringDiagram (nodes : List (List Node)) (strands : List (List Strand)) :
+    DiagramBuilderM PUnit := do
   /- Add 2-morphisms. -/
   for x in nodes.join do
     match x with
@@ -274,21 +293,59 @@ def dsl :=
 def sty :=
   include_str ".."/".."/".."/"widget"/"src"/"penrose"/"monoidal.sty"
 
-open scoped Jsx in
-/-- Construct a string diagram from the expression of a 2-morphism. -/
-def fromExpr (e : Expr) : MonoidalM Html := do
-  let e' := (← eval e).expr
-  DiagramBuilderM.run do
-    mkStringDiagram e'
-    match ← DiagramBuilderM.buildDiagram dsl sty with
-    | some html => return html
-    | none => return <span>No non-structural morphisms found.</span>
+/-- The kind of the context. -/
+inductive Kind where
+  | monoidal : Kind
+  | bicategory : Kind
+  | none : Kind
+
+/-- The name of the context. -/
+def Kind.name : Kind → Name
+  | Kind.monoidal => `monoidal
+  | Kind.bicategory => `bicategory
+  | Kind.none => default
+
+/-- Given an expression, return the kind of the context. -/
+def mkKind (e : Expr) : MetaM Kind := do
+  let e ← instantiateMVars e
+  let e ← (match (← whnfR e).eq? with
+    | some (_, lhs, _) => return lhs
+    | none => return e)
+  let ctx? ← BicategoryLike.mkContext? (ρ := Bicategory.Context) e
+  match ctx? with
+  | .some _ => return .bicategory
+  | .none =>
+    let ctx? ← BicategoryLike.mkContext? (ρ := Monoidal.Context) e
+    match ctx? with
+    | .some _ => return .monoidal
+    | .none => return .none
 
+open scoped Jsx in
 /-- Given a 2-morphism, return a string diagram. Otherwise `none`. -/
 def stringM? (e : Expr) : MetaM (Option Html) := do
   let e ← instantiateMVars e
-  let some ctx ← mkContext? e | return none
-  return some <| ← MonoidalM.run ctx <| fromExpr e
+  let k ← mkKind e
+  let x : Option (List (List Node) × List (List Strand)) ← (match k with
+    | .monoidal => do
+      let .some ctx ← BicategoryLike.mkContext? (ρ := Monoidal.Context) e | return .none
+      CoherenceM.run (ctx := ctx) do
+        let e' := (← BicategoryLike.eval k.name (← MkMor₂.ofExpr e)).expr
+        return .some (← e'.nodes, ← e'.strands)
+    | .bicategory => do
+      let .some ctx ← BicategoryLike.mkContext? (ρ := Bicategory.Context) e | return .none
+      CoherenceM.run (ctx := ctx) do
+        let e' := (← BicategoryLike.eval k.name (← MkMor₂.ofExpr e)).expr
+        return .some (← e'.nodes, ← e'.strands)
+    | .none => return .none)
+  match x with
+  | .none => return none
+  | .some (nodes, strands) => do
+    DiagramBuilderM.run do
+      mkStringDiagram nodes strands
+      trace[string_diagram] "Penrose substance: \n{(← get).sub}"
+      match ← DiagramBuilderM.buildDiagram dsl sty with
+      | some html => return html
+      | none => return <span>No non-structural morphisms found.</span>
 
 open scoped Jsx in
 /-- Help function for displaying two string diagrams in an equality. -/
@@ -318,12 +375,13 @@ def stringEqM? (e : Expr) : MetaM (Option Html) := do
 /-- Given an 2-morphism or equality between 2-morphisms, return a string diagram.
 Otherwise `none`. -/
 def stringMorOrEqM? (e : Expr) : MetaM (Option Html) := do
-  if let some html ← stringM? e then
-    return some html
-  else if let some html ← stringEqM? e then
-    return some html
-  else
-    return none
+  forallTelescopeReducing (← inferType e) fun xs a => do
+    if let some html ← stringM? (mkAppN e xs) then
+      return some html
+    else if let some html ← stringEqM? a then
+      return some html
+    else
+      return none
 
 /-- The `Expr` presenter for displaying string diagrams. -/
 @[expr_presenter]
@@ -361,4 +419,36 @@ open ProofWidgets
 def StringDiagram : Component PanelWidgetProps :=
   mk_rpc_widget% StringDiagram.rpc
 
+open Command
+
+/--
+Display the string diagram for a given term.
+
+Example usage:
+```
+/- String diagram for the equality theorem. -/
+#string_diagram MonoidalCategory.whisker_exchange
+
+/- String diagram for the morphism. -/
+variable {C : Type u} [Category.{v} C] [MonoidalCategory C] {X Y : C} (f : 𝟙_ C ⟶ X ⊗ Y) in
+#string_diagram f
+```
+-/
+syntax (name := stringDiagram) "#string_diagram " term : command
+
+@[command_elab stringDiagram, inherit_doc stringDiagram]
+def elabStringDiagramCmd : CommandElab := fun
+  | stx@`(#string_diagram $t:term) => do
+    let html ← runTermElabM fun _ => do
+      let e ← try mkConstWithFreshMVarLevels (← realizeGlobalConstNoOverloadWithInfo t)
+        catch _ => Term.levelMVarToParam (← instantiateMVars (← Term.elabTerm t none))
+      match ← StringDiagram.stringMorOrEqM? e with
+      | .some html => return html
+      | .none => throwError "could not find a morphism or equality: {e}"
+    liftCoreM <| Widget.savePanelWidgetInfo
+      (hash HtmlDisplay.javascript)
+      (return json% { html: $(← Server.RpcEncodable.rpcEncode html) })
+      stx
+  | stx => throwError "Unexpected syntax {stx}."
+
 end Mathlib.Tactic.Widget
diff --git a/Mathlib/Testing/SlimCheck/Functions.lean b/Mathlib/Testing/SlimCheck/Functions.lean
index 7f0f7d67f8cb1..55ef7dc3e20fa 100644
--- a/Mathlib/Testing/SlimCheck/Functions.lean
+++ b/Mathlib/Testing/SlimCheck/Functions.lean
@@ -177,8 +177,7 @@ def applyFinsupp (tf : TotalFunction α β) : α →₀ β where
     · intro h
       use (A.dlookup a).getD (0 : β)
       rw [← List.dlookup_dedupKeys] at h ⊢
-      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, and_true_iff, not_false_iff,
-        Option.mem_def]
+      simp only [h, ← List.mem_dlookup_iff A.nodupKeys_dedupKeys, not_false_iff, Option.mem_def]
       cases haA : List.dlookup a A.dedupKeys
       · simp [haA] at h
       · simp
@@ -324,18 +323,18 @@ theorem applyId_mem_iff [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup xs
       dsimp [List.dlookup] at h₃; split_ifs at h₃ with h
       · rw [Option.some_inj] at h₃
         subst x'; subst val
-        simp only [List.mem_cons, true_or_iff, eq_self_iff_true]
+        simp only [List.mem_cons, true_or, eq_self_iff_true]
       · cases' h₀ with _ _ h₀ h₅
         cases' h₂ with _ _ h₂ h₄
         have h₆ := Nat.succ.inj h₁
         specialize xs_ih h₅ h₃ h₄ h₆
-        simp only [Ne.symm h, xs_ih, List.mem_cons, false_or_iff]
+        simp only [Ne.symm h, xs_ih, List.mem_cons]
         suffices val ∈ ys by tauto
         erw [← Option.mem_def, List.mem_dlookup_iff] at h₃
         · simp only [Prod.toSigma, List.mem_map, heq_iff_eq, Prod.exists] at h₃
           rcases h₃ with ⟨a, b, h₃, h₄, h₅⟩
           apply (List.of_mem_zip h₃).2
-        simp only [List.NodupKeys, List.keys, comp, Prod.fst_toSigma, List.map_map]
+        simp only [List.NodupKeys, List.keys, comp_def, Prod.fst_toSigma, List.map_map]
         rwa [List.map_fst_zip _ _ (le_of_eq h₆)]
 
 theorem List.applyId_eq_self [DecidableEq α] {xs ys : List α} (x : α) :
@@ -364,7 +363,7 @@ theorem applyId_injective [DecidableEq α] {xs ys : List α} (h₀ : List.Nodup
     · symm; rw [h]
       rw [← List.applyId_zip_eq] <;> assumption
     · rw [← h₁.length_eq]
-      rw [List.getElem?_eq_some] at hx
+      rw [List.getElem?_eq_some_iff] at hx
       cases' hx with hx hx'
       exact hx
   · rw [← applyId_mem_iff h₀ h₁] at hx hy
@@ -430,9 +429,9 @@ protected def shrink {α : Type} [DecidableEq α] :
     have h₄ : ys'.length ≤ xs'.length := le_of_eq (List.Perm.length_eq h₀.symm)
     pure
       ⟨(List.zip xs' ys').map Prod.toSigma,
-        by simp only [comp, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma,
+        by simp only [comp_def, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma,
           Prod.snd_toSigma, List.map_map],
-        by simp only [comp, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map]⟩
+        by simp only [comp_def, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map]⟩
 
 /-- Create an injective function from one list and a permutation of that list. -/
 protected def mk (xs ys : List α) (h : xs ~ ys) (h' : ys.Nodup) : InjectiveFunction α :=
@@ -440,9 +439,9 @@ protected def mk (xs ys : List α) (h : xs ~ ys) (h' : ys.Nodup) : InjectiveFunc
   have h₁ : ys.length ≤ xs.length := le_of_eq h.length_eq.symm
   InjectiveFunction.mapToSelf (List.toFinmap' (xs.zip ys))
     (by
-      simp only [List.toFinmap', comp, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma,
+      simp only [List.toFinmap', comp_def, List.map_fst_zip, List.map_snd_zip, *, Prod.fst_toSigma,
         Prod.snd_toSigma, List.map_map])
-    (by simp only [List.toFinmap', comp, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map])
+    (by simp only [List.toFinmap', comp_def, List.map_snd_zip, *, Prod.snd_toSigma, List.map_map])
 
 protected theorem injective [DecidableEq α] (f : InjectiveFunction α) : Injective (apply f) := by
   cases' f with xs hperm hnodup
@@ -454,7 +453,7 @@ protected theorem injective [DecidableEq α] (f : InjectiveFunction α) : Inject
     induction xs with
     | nil => simp only [List.zip_nil_right, List.map_nil]
     | cons xs_hd xs_tl xs_ih =>
-      simp only [true_and_iff, Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons,
+      simp only [Prod.toSigma, eq_self_iff_true, Sigma.eta, List.zip_cons_cons,
         List.map, List.cons_inj_right]
       exact xs_ih
   revert hperm hnodup
diff --git a/Mathlib/Testing/SlimCheck/Gen.lean b/Mathlib/Testing/SlimCheck/Gen.lean
index da7ccace2053e..0f3d66842989e 100644
--- a/Mathlib/Testing/SlimCheck/Gen.lean
+++ b/Mathlib/Testing/SlimCheck/Gen.lean
@@ -111,7 +111,7 @@ def prodOf {α : Type u} {β : Type v} (x : Gen α) (y : Gen β) : Gen (α × β
 
 end Gen
 
-/-- Execute a `Gen` inside the `IO` monad using `size` as the example size-/
+/-- Execute a `Gen` inside the `IO` monad using `size` as the example size -/
 def Gen.run {α : Type} (x : Gen α) (size : Nat) : BaseIO α :=
   letI : MonadLift Id BaseIO := ⟨fun f => pure <| Id.run f⟩
   IO.runRand (ReaderT.run x ⟨size⟩:)
diff --git a/Mathlib/Testing/SlimCheck/Sampleable.lean b/Mathlib/Testing/SlimCheck/Sampleable.lean
index b2c011792955b..f5cedcf3cb680 100644
--- a/Mathlib/Testing/SlimCheck/Sampleable.lean
+++ b/Mathlib/Testing/SlimCheck/Sampleable.lean
@@ -220,7 +220,7 @@ instance Bool.sampleableExt : SampleableExt Bool :=
 
 /-- This can be specialized into customized `SampleableExt Char` instances.
 The resulting instance has `1 / length` chances of making an unrestricted choice of characters
-and it otherwise chooses a character from `chars` with uniform probabilities.  -/
+and it otherwise chooses a character from `chars` with uniform probabilities. -/
 def Char.sampleable (length : Nat) (chars : List Char) (pos : 0 < chars.length) :
     SampleableExt Char :=
   mkSelfContained do
@@ -256,7 +256,7 @@ instance List.sampleableExt [SampleableExt α] : SampleableExt (List α) where
 
 end Samplers
 
-/-- An annotation for values that should never get shrinked. -/
+/-- An annotation for values that should never get shrunk. -/
 def NoShrink (α : Type u) := α
 
 namespace NoShrink
diff --git a/Mathlib/Testing/SlimCheck/Testable.lean b/Mathlib/Testing/SlimCheck/Testable.lean
index e4ece33e569d8..af3278d4fc34d 100644
--- a/Mathlib/Testing/SlimCheck/Testable.lean
+++ b/Mathlib/Testing/SlimCheck/Testable.lean
@@ -423,7 +423,7 @@ end Testable
 
 section PrintableProp
 
-variable {α : Type*} {x y : α}
+variable {α : Type*}
 
 instance Eq.printableProp [Repr α] {x y : α} : PrintableProp (x = y) where
   printProp := s!"{repr x} = {repr y}"
diff --git a/Mathlib/Topology/AlexandrovDiscrete.lean b/Mathlib/Topology/AlexandrovDiscrete.lean
index a754e4fc9b10d..455d1c1792743 100644
--- a/Mathlib/Topology/AlexandrovDiscrete.lean
+++ b/Mathlib/Topology/AlexandrovDiscrete.lean
@@ -6,7 +6,7 @@ Authors: Yaël Dillies
 import Mathlib.Data.Set.Image
 import Mathlib.Topology.Bases
 import Mathlib.Topology.Inseparable
-import Mathlib.Topology.Compactness.Compact
+import Mathlib.Topology.Compactness.Exterior
 
 /-!
 # Alexandrov-discrete topological spaces
@@ -20,8 +20,6 @@ minimal neighborhood, which we call the *exterior* of the set.
 ## Main declarations
 
 * `AlexandrovDiscrete`: Prop-valued typeclass for a topological space to be Alexandrov-discrete
-* `exterior`: Intersection of all neighborhoods of a set. When the space is Alexandrov-discrete,
-  this is the minimal neighborhood of the set.
 
 ## Notes
 
@@ -116,45 +114,6 @@ lemma closure_sUnion (S : Set (Set α)) : closure (⋃₀ S) = ⋃ s ∈ S, clos
 
 end AlexandrovDiscrete
 
-variable {s t : Set α} {a x y : α}
-
-/-- The *exterior* of a set is the intersection of all its neighborhoods. In an Alexandrov-discrete
-space, this is the smallest neighborhood of the set.
-
-Note that this construction is unnamed in the literature. We choose the name in analogy to
-`interior`. -/
-def exterior (s : Set α) : Set α := (𝓝ˢ s).ker
-
-lemma exterior_singleton_eq_ker_nhds (a : α) : exterior {a} = (𝓝 a).ker := by simp [exterior]
-
-lemma exterior_def (s : Set α) : exterior s = ⋂₀ {t : Set α | IsOpen t ∧ s ⊆ t} :=
-  (hasBasis_nhdsSet _).ker.trans sInter_eq_biInter.symm
-
-lemma mem_exterior : a ∈ exterior s ↔ ∀ U, IsOpen U → s ⊆ U → a ∈ U := by simp [exterior_def]
-
-lemma subset_exterior_iff : s ⊆ exterior t ↔ ∀ U, IsOpen U → t ⊆ U → s ⊆ U := by
-  simp [exterior_def]
-
-lemma subset_exterior : s ⊆ exterior s := subset_exterior_iff.2 fun _ _ ↦ id
-
-lemma exterior_minimal (h₁ : s ⊆ t) (h₂ : IsOpen t) : exterior s ⊆ t := by
-  rw [exterior_def]; exact sInter_subset_of_mem ⟨h₂, h₁⟩
-
-lemma IsOpen.exterior_eq (h : IsOpen s) : exterior s = s :=
-  (exterior_minimal Subset.rfl h).antisymm subset_exterior
-
-lemma IsOpen.exterior_subset_iff (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t :=
-  ⟨subset_exterior.trans, fun h ↦ exterior_minimal h ht⟩
-
-@[mono] lemma exterior_mono : Monotone (exterior : Set α → Set α) :=
-  fun _s _t h ↦ ker_mono <| nhdsSet_mono h
-
-@[simp] lemma exterior_empty : exterior (∅ : Set α) = ∅ := isOpen_empty.exterior_eq
-@[simp] lemma exterior_univ : exterior (univ : Set α) = univ := isOpen_univ.exterior_eq
-
-@[simp] lemma exterior_eq_empty : exterior s = ∅ ↔ s = ∅ :=
-  ⟨eq_bot_mono subset_exterior, by rintro rfl; exact exterior_empty⟩
-
 lemma Inducing.alexandrovDiscrete [AlexandrovDiscrete α] {f : β → α} (h : Inducing f) :
     AlexandrovDiscrete β where
   isOpen_sInter S hS := by
@@ -163,20 +122,6 @@ lemma Inducing.alexandrovDiscrete [AlexandrovDiscrete α] {f : β → α} (h : I
     refine ⟨_, isOpen_iInter₂ hU, ?_⟩
     simp_rw [preimage_iInter, htU, sInter_eq_biInter]
 
-lemma IsOpen.exterior_subset (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t :=
-  ⟨subset_exterior.trans, fun h ↦ exterior_minimal h ht⟩
-
-lemma Set.Finite.isCompact_exterior (hs : s.Finite) : IsCompact (exterior s) := by
-  classical
-  refine isCompact_of_finite_subcover fun f hf hsf ↦ ?_
-  choose g hg using fun a (ha : a ∈ exterior s) ↦ mem_iUnion.1 (hsf ha)
-  refine ⟨hs.toFinset.attach.image fun a ↦
-    g a.1 <| subset_exterior <| (Finite.mem_toFinset _).1 a.2,
-    (isOpen_iUnion fun i ↦ isOpen_iUnion ?_).exterior_subset.2 ?_⟩
-  · exact fun _ ↦ hf _
-  refine fun a ha ↦ mem_iUnion₂.2 ⟨_, ?_, hg _ <| subset_exterior ha⟩
-  simp only [Finset.mem_image, Finset.mem_attach, true_and, Subtype.exists, Finite.mem_toFinset]
-  exact ⟨a, ha, rfl⟩
 end
 
 lemma AlexandrovDiscrete.sup {t₁ t₂ : TopologicalSpace α} (_ : @AlexandrovDiscrete α t₁)
@@ -194,7 +139,7 @@ lemma alexandrovDiscrete_iSup {t : ι → TopologicalSpace α} (_ : ∀ i, @Alex
 
 section
 variable [TopologicalSpace α] [TopologicalSpace β] [AlexandrovDiscrete α] [AlexandrovDiscrete β]
-  {s t : Set α} {a x y : α}
+  {s t : Set α} {a : α}
 
 @[simp] lemma isOpen_exterior : IsOpen (exterior s) := by
   rw [exterior_def]; exact isOpen_sInter fun _ ↦ And.left
@@ -220,47 +165,25 @@ lemma exterior_singleton_subset_iff_mem_nhds : exterior {a} ⊆ t ↔ t ∈ 𝓝
 lemma gc_exterior_interior : GaloisConnection (exterior : Set α → Set α) interior :=
   fun s t ↦ by simp [exterior_subset_iff, subset_interior_iff]
 
-@[simp] lemma exterior_exterior (s : Set α) : exterior (exterior s) = exterior s :=
-  isOpen_exterior.exterior_eq
-
-@[simp] lemma exterior_union (s t : Set α) : exterior (s ∪ t) = exterior s ∪ exterior t :=
-  gc_exterior_interior.l_sup
-
-@[simp] lemma nhdsSet_exterior (s : Set α) : 𝓝ˢ (exterior s) = 𝓝ˢ s := by
-  ext t; simp_rw [← exterior_subset_iff_mem_nhdsSet, exterior_exterior]
-
 @[simp] lemma principal_exterior (s : Set α) : 𝓟 (exterior s) = 𝓝ˢ s := by
   rw [← nhdsSet_exterior, isOpen_exterior.nhdsSet_eq]
 
-@[simp] lemma exterior_subset_exterior : exterior s ⊆ exterior t ↔ 𝓝ˢ s ≤ 𝓝ˢ t := by
-  refine ⟨?_, fun h ↦ ker_mono h⟩
-  simp_rw [le_def, ← exterior_subset_iff_mem_nhdsSet]
-  exact fun h u ↦ h.trans
-
-lemma specializes_iff_exterior_subset : x ⤳ y ↔ exterior {x} ⊆ exterior {y} := by
-  simp [Specializes]
-
 lemma isOpen_iff_forall_specializes : IsOpen s ↔ ∀ x y, x ⤳ y → y ∈ s → x ∈ s := by
-  refine ⟨fun hs x y hxy ↦ hxy.mem_open hs, fun hs ↦ ?_⟩
-  simp_rw [specializes_iff_exterior_subset] at hs
-  simp_rw [isOpen_iff_mem_nhds, mem_nhds_iff]
-  rintro a ha
-  refine ⟨_, fun b hb ↦ hs _ _ ?_ ha, isOpen_exterior, subset_exterior <| mem_singleton _⟩
-  rwa [isOpen_exterior.exterior_subset, singleton_subset_iff]
+  simp only [← exterior_subset_iff_isOpen, Set.subset_def, mem_exterior_iff_specializes, exists_imp,
+    and_imp, @forall_swap (_ ⤳ _)]
 
 lemma alexandrovDiscrete_coinduced {β : Type*} {f : α → β} :
     @AlexandrovDiscrete β (coinduced f ‹_›) :=
   @AlexandrovDiscrete.mk β (coinduced f ‹_›) fun S hS ↦ by
     rw [isOpen_coinduced, preimage_sInter]; exact isOpen_iInter₂ hS
 
-
 instance AlexandrovDiscrete.toFirstCountable : FirstCountableTopology α where
   nhds_generated_countable a := ⟨{exterior {a}}, countable_singleton _, by simp⟩
 
 instance AlexandrovDiscrete.toLocallyCompactSpace : LocallyCompactSpace α where
   local_compact_nhds a _U hU := ⟨exterior {a},
     isOpen_exterior.mem_nhds <| subset_exterior <| mem_singleton _,
-      exterior_singleton_subset_iff_mem_nhds.2 hU, (finite_singleton _).isCompact_exterior⟩
+      exterior_singleton_subset_iff_mem_nhds.2 hU, isCompact_singleton.exterior⟩
 
 instance Subtype.instAlexandrovDiscrete {p : α → Prop} : AlexandrovDiscrete {a // p a} :=
   inducing_subtype_val.alexandrovDiscrete
diff --git a/Mathlib/Topology/Algebra/Algebra.lean b/Mathlib/Topology/Algebra/Algebra.lean
index c229209194995..367ba0a65e42e 100644
--- a/Mathlib/Topology/Algebra/Algebra.lean
+++ b/Mathlib/Topology/Algebra/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Antoine Chambert-Loir, María Inés de Frutos-Fernández
+Authors: Kim Morrison, Antoine Chambert-Loir, María Inés de Frutos-Fernández
 -/
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Topology.Algebra.Module.Basic
@@ -241,7 +241,7 @@ whose `TopologicalClosure` is `⊤` is sent to another such submodule.
 That is, the image of a dense subalgebra under a map with dense range is dense.
 -/
 theorem _root_.DenseRange.topologicalClosure_map_subalgebra
-    [TopologicalSemiring B] {f : A →A[R] B}  (hf' : DenseRange f) {s : Subalgebra R A}
+    [TopologicalSemiring B] {f : A →A[R] B} (hf' : DenseRange f) {s : Subalgebra R A}
     (hs : s.topologicalClosure = ⊤) : (s.map (f : A →ₐ[R] B)).topologicalClosure = ⊤ := by
   rw [SetLike.ext'_iff] at hs ⊢
   simp only [Subalgebra.topologicalClosure_coe, coe_top, ← dense_iff_closure_eq, Subalgebra.coe_map,
@@ -408,7 +408,7 @@ theorem fst_comp_prod (f : A →A[R] B) (g : A →A[R] C) :
   ext fun _x => rfl
 
 @[simp]
-theorem snd_comp_prod  (f : A →A[R] B) (g : A →A[R] C) :
+theorem snd_comp_prod (f : A →A[R] B) (g : A →A[R] C) :
     (snd R B C).comp (f.prod g) = g :=
   ext fun _x => rfl
 
diff --git a/Mathlib/Topology/Algebra/Algebra/Rat.lean b/Mathlib/Topology/Algebra/Algebra/Rat.lean
index d63250277c31e..212c43adc23f5 100644
--- a/Mathlib/Topology/Algebra/Algebra/Rat.lean
+++ b/Mathlib/Topology/Algebra/Algebra/Rat.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Algebra.Rat
 import Mathlib.Topology.Algebra.Monoid
diff --git a/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean b/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean
new file mode 100644
index 0000000000000..803016fe59879
--- /dev/null
+++ b/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean
@@ -0,0 +1,261 @@
+/-
+Copyright (c) 2024 Jujian Zhang. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jujian Zhang, Nailin Guan, Yuyang Zhao
+-/
+import Mathlib.Algebra.Category.Grp.FiniteGrp
+import Mathlib.Topology.Algebra.ClosedSubgroup
+import Mathlib.Topology.Algebra.ContinuousMonoidHom
+import Mathlib.Topology.Category.Profinite.Basic
+/-!
+
+# Category of Profinite Groups
+
+We say `G` is a profinite group if it is a topological group which is compact and totally
+disconnected.
+
+## Main definitions and results
+
+* `ProfiniteGrp` is the category of profinite groups.
+
+* `ProfiniteGrp.pi` : The pi-type of profinite groups is also a profinite group.
+
+* `ofFiniteGrp` : A `FiniteGrp` when given the discrete topology can be considered as a
+  profinite group.
+
+* `ofClosedSubgroup` : A closed subgroup of a profinite group is profinite.
+
+-/
+
+universe u v
+
+open CategoryTheory Topology
+
+/--
+The category of profinite groups. A term of this type consists of a profinite
+set with a topological group structure.
+-/
+@[pp_with_univ]
+structure ProfiniteGrp where
+  /-- The underlying profinite topological space. -/
+  toProfinite : Profinite
+  /-- The group structure. -/
+  [group : Group toProfinite]
+  /-- The above data together form a topological group. -/
+  [topologicalGroup : TopologicalGroup toProfinite]
+
+/--
+The category of profinite additive groups. A term of this type consists of a profinite
+set with a topological additive group structure.
+-/
+@[pp_with_univ]
+structure ProfiniteAddGrp where
+  /-- The underlying profinite topological space. -/
+  toProfinite : Profinite
+  /-- The additive group structure. -/
+  [addGroup : AddGroup toProfinite]
+  /-- The above data together form a topological additive group. -/
+  [topologicalAddGroup : TopologicalAddGroup toProfinite]
+
+attribute [to_additive] ProfiniteGrp
+
+namespace ProfiniteGrp
+
+@[to_additive]
+instance : CoeSort ProfiniteGrp (Type u) where
+  coe G := G.toProfinite
+
+attribute [instance] group topologicalGroup
+    ProfiniteAddGrp.addGroup ProfiniteAddGrp.topologicalAddGroup
+
+@[to_additive]
+instance : Category ProfiniteGrp where
+  Hom A B := ContinuousMonoidHom A B
+  id A := ContinuousMonoidHom.id A
+  comp f g := ContinuousMonoidHom.comp g f
+
+@[to_additive]
+instance (G H : ProfiniteGrp) : FunLike (G ⟶ H) G H :=
+  inferInstanceAs <| FunLike (ContinuousMonoidHom G H) G H
+
+@[to_additive]
+instance (G H : ProfiniteGrp) : MonoidHomClass (G ⟶ H) G H :=
+  inferInstanceAs <| MonoidHomClass (ContinuousMonoidHom G H) G H
+
+@[to_additive]
+instance (G H : ProfiniteGrp) : ContinuousMapClass (G ⟶ H) G H :=
+  inferInstanceAs <| ContinuousMapClass (ContinuousMonoidHom G H) G H
+
+@[to_additive]
+instance : ConcreteCategory ProfiniteGrp where
+  forget :=
+  { obj := fun G => G
+    map := fun f => f }
+  forget_faithful :=
+    { map_injective := by
+        intro G H f g h
+        exact DFunLike.ext _ _ <| fun x => congr_fun h x }
+
+/-- Construct a term of `ProfiniteGrp` from a type endowed with the structure of a
+compact and totally disconnected topological group.
+(The condition of being Hausdorff can be omitted here because totally disconnected implies that {1}
+is a closed set, thus implying Hausdorff in a topological group.)-/
+@[to_additive "Construct a term of `ProfiniteAddGrp` from a type endowed with the structure of a
+compact and totally disconnected topological additive group.
+(The condition of being Hausdorff can be omitted here because totally disconnected implies that {0}
+is a closed set, thus implying Hausdorff in a topological additive group.)"]
+def of (G : Type u) [Group G] [TopologicalSpace G] [TopologicalGroup G]
+    [CompactSpace G] [TotallyDisconnectedSpace G] : ProfiniteGrp where
+  toProfinite := .of G
+  group := ‹_›
+  topologicalGroup := ‹_›
+
+@[to_additive (attr := simp)]
+theorem coe_of (X : ProfiniteGrp) : (of X : Type _) = X :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem coe_id (X : ProfiniteGrp) : (𝟙 ((forget ProfiniteGrp).obj X)) = id :=
+  rfl
+
+@[to_additive (attr := simp)]
+theorem coe_comp {X Y Z : ProfiniteGrp} (f : X ⟶ Y) (g : Y ⟶ Z) :
+    ((forget ProfiniteGrp).map f ≫ (forget ProfiniteGrp).map g) = g ∘ f :=
+  rfl
+
+/-- Construct a term of `ProfiniteGrp` from a type endowed with the structure of a
+profinite topological group. -/
+@[to_additive "Construct a term of `ProfiniteAddGrp` from a type endowed with the structure of a
+profinite topological additive group."]
+abbrev ofProfinite (G : Profinite) [Group G] [TopologicalGroup G] :
+    ProfiniteGrp := of G
+
+/-- The pi-type of profinite groups is a profinite group. -/
+@[to_additive "The pi-type of profinite additive groups is a
+profinite additive group."]
+def pi {α : Type u} (β : α → ProfiniteGrp) : ProfiniteGrp :=
+  let pitype := Profinite.pi fun (a : α) => (β a).toProfinite
+  letI (a : α): Group (β a).toProfinite := (β a).group
+  letI : Group pitype := Pi.group
+  letI : TopologicalGroup pitype := Pi.topologicalGroup
+  ofProfinite pitype
+
+/-- A `FiniteGrp` when given the discrete topology can be considered as a profinite group. -/
+@[to_additive "A `FiniteAddGrp` when given the discrete topology can be considered as a
+profinite additive group."]
+def ofFiniteGrp (G : FiniteGrp) : ProfiniteGrp :=
+  letI : TopologicalSpace G := ⊥
+  letI : DiscreteTopology G := ⟨rfl⟩
+  letI : TopologicalGroup G := {}
+  of G
+
+@[to_additive]
+instance : HasForget₂ FiniteGrp ProfiniteGrp where
+  forget₂ :=
+  { obj := ofFiniteGrp
+    map := fun f => ⟨f, by continuity⟩ }
+
+@[to_additive]
+instance : HasForget₂ ProfiniteGrp Grp where
+  forget₂ := {
+    obj := fun P => ⟨P, P.group⟩
+    map := fun f => f.toMonoidHom
+  }
+
+/-- A closed subgroup of a profinite group is profinite. -/
+def ofClosedSubgroup {G : ProfiniteGrp} (H : ClosedSubgroup G)  : ProfiniteGrp :=
+  letI : CompactSpace H := inferInstance
+  of H.1
+
+/-- The functor mapping a profinite group to its underlying profinite space. -/
+def profiniteGrpToProfinite : ProfiniteGrp ⥤ Profinite where
+  obj G := G.toProfinite
+  map f := ⟨f, by continuity⟩
+
+instance : profiniteGrpToProfinite.Faithful := {
+  map_injective := fun {_ _} _ _ h =>
+    ConcreteCategory.hom_ext_iff.mpr (congrFun (congrArg ContinuousMap.toFun h)) }
+
+end ProfiniteGrp
+
+/-!
+# Limits in the category of profinite groups
+
+In this section, we construct limits in the category of profinite groups.
+
+* `ProfiniteGrp.limitCone` : The explicit limit cone in `ProfiniteGrp`.
+
+* `ProfiniteGrp.limitConeIsLimit`: `ProfiniteGrp.limitCone` is a limit cone.
+
+-/
+
+section Limits
+
+namespace ProfiniteGrp
+
+section
+
+variable {J : Type v} [SmallCategory J] (F : J ⥤ ProfiniteGrp.{max v u})
+
+/-- Auxiliary construction to obtain the group structure on the limit of profinite groups. -/
+def limitConePtAux : Subgroup (Π j : J, F.obj j) where
+  carrier := {x | ∀ ⦃i j : J⦄ (π : i ⟶ j), F.map π (x i) = x j}
+  mul_mem' hx hy _ _ π := by simp only [Pi.mul_apply, map_mul, hx π, hy π]
+  one_mem' := by simp only [Set.mem_setOf_eq, Pi.one_apply, map_one, implies_true]
+  inv_mem' h _ _ π := by simp only [Pi.inv_apply, map_inv, h π]
+
+instance : Group (Profinite.limitCone (F ⋙ profiniteGrpToProfinite.{max v u})).pt :=
+  inferInstanceAs (Group (limitConePtAux F))
+
+instance : TopologicalGroup (Profinite.limitCone (F ⋙ profiniteGrpToProfinite.{max v u})).pt :=
+  inferInstanceAs (TopologicalGroup (limitConePtAux F))
+
+/-- The explicit limit cone in `ProfiniteGrp`. -/
+abbrev limitCone : Limits.Cone F where
+  pt := ofProfinite (Profinite.limitCone (F ⋙ profiniteGrpToProfinite.{max v u})).pt
+  π :=
+  { app := fun j => {
+      toFun := fun x => x.1 j
+      map_one' := rfl
+      map_mul' := fun x y => rfl
+      continuous_toFun := by
+        exact (continuous_apply j).comp (continuous_iff_le_induced.mpr fun U a => a) }
+    naturality := fun i j f => by
+      simp only [Functor.const_obj_obj, Functor.comp_obj,
+        Functor.const_obj_map, Category.id_comp, Functor.comp_map]
+      congr
+      exact funext fun x => (x.2 f).symm }
+
+/-- `ProfiniteGrp.limitCone` is a limit cone. -/
+def limitConeIsLimit : Limits.IsLimit (limitCone F) where
+  lift cone := {
+    ((Profinite.limitConeIsLimit (F ⋙ profiniteGrpToProfinite)).lift
+      (profiniteGrpToProfinite.mapCone cone)) with
+    map_one' := Subtype.ext (funext fun j ↦ map_one (cone.π.app j))
+    -- TODO: investigate whether it's possible to set up `ext` lemmas for the `TopCat`-related
+    -- categories so that `by ext j; exact map_one (cone.π.app j)` works here, similarly below.
+    map_mul' := fun _ _ ↦ Subtype.ext (funext fun j ↦ map_mul (cone.π.app j) _ _) }
+  uniq cone m h := by
+    apply profiniteGrpToProfinite.map_injective
+    simpa using (Profinite.limitConeIsLimit (F ⋙ profiniteGrpToProfinite)).uniq
+      (profiniteGrpToProfinite.mapCone cone) (profiniteGrpToProfinite.map m)
+      (fun j ↦ congrArg profiniteGrpToProfinite.map (h j))
+
+instance : Limits.HasLimit F where
+  exists_limit := Nonempty.intro
+    { cone := limitCone F
+      isLimit := limitConeIsLimit F }
+
+/-- The abbreviation for the limit of `ProfiniteGrp`s. -/
+abbrev limit : ProfiniteGrp := (ProfiniteGrp.limitCone F).pt
+
+end
+
+instance : Limits.PreservesLimits profiniteGrpToProfinite.{u} where
+  preservesLimitsOfShape := {
+    preservesLimit := fun {F} ↦ CategoryTheory.Limits.preservesLimitOfPreservesLimitCone
+      (limitConeIsLimit F) (Profinite.limitConeIsLimit (F ⋙ profiniteGrpToProfinite)) }
+
+end ProfiniteGrp
+
+end Limits
diff --git a/Mathlib/Topology/Algebra/ClosedSubgroup.lean b/Mathlib/Topology/Algebra/ClosedSubgroup.lean
new file mode 100644
index 0000000000000..302f09738270a
--- /dev/null
+++ b/Mathlib/Topology/Algebra/ClosedSubgroup.lean
@@ -0,0 +1,128 @@
+/-
+Copyright (c) 2024 Nailin Guan. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Nailin Guan
+-/
+
+import Mathlib.Topology.Algebra.Group.Basic
+import Mathlib.Algebra.Group.Subgroup.Basic
+import Mathlib.GroupTheory.Index
+
+/-!
+# Closed subgroups of a topological group
+
+This files builds the SemilatticeInf `ClosedSubgroup G` of closed subgroups in a
+topological group `G`, and its additive version `ClosedAddSubgroup`.
+
+# Main definitions and results
+
+* `normalCore_isClosed` : The `normalCore` of a closed subgroup is closed.
+
+* `finindex_closedSubgroup_isOpen` : A closed subgroup with finite index is open.
+
+-/
+
+section
+
+universe u v
+
+/-- The type of closed subgroups of a topological group. -/
+@[ext]
+structure ClosedSubgroup (G : Type u) [Group G] [TopologicalSpace G] extends Subgroup G where
+  isClosed' : IsClosed carrier
+
+/-- The type of closed subgroups of an additive topological group. -/
+@[ext]
+structure ClosedAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G] extends
+    AddSubgroup G where
+  isClosed' : IsClosed carrier
+
+attribute [to_additive] ClosedSubgroup
+
+attribute [coe] ClosedSubgroup.toSubgroup ClosedAddSubgroup.toAddSubgroup
+
+namespace ClosedSubgroup
+
+variable (G : Type u) [Group G] [TopologicalSpace G]
+
+variable {G} in
+@[to_additive]
+theorem toSubgroup_injective : Function.Injective
+    (ClosedSubgroup.toSubgroup : ClosedSubgroup G → Subgroup G) :=
+  fun A B h ↦ by
+  ext
+  rw [h]
+
+@[to_additive]
+instance : SetLike (ClosedSubgroup G) G where
+  coe U := U.1
+  coe_injective' _ _ h := toSubgroup_injective <| SetLike.ext' h
+
+@[to_additive]
+instance : SubgroupClass (ClosedSubgroup G) G where
+  mul_mem := Subsemigroup.mul_mem' _
+  one_mem U := U.one_mem'
+  inv_mem := Subgroup.inv_mem' _
+
+@[to_additive]
+instance : Coe (ClosedSubgroup G) (Subgroup G) where
+  coe := toSubgroup
+
+@[to_additive]
+instance instInfClosedSubgroup : Inf (ClosedSubgroup G) :=
+  ⟨fun U V ↦ ⟨U ⊓ V, U.isClosed'.inter V.isClosed'⟩⟩
+
+@[to_additive]
+instance instSemilatticeInfClosedSubgroup : SemilatticeInf (ClosedSubgroup G) :=
+  SetLike.coe_injective.semilatticeInf ((↑) : ClosedSubgroup G → Set G) fun _ _ ↦ rfl
+
+@[to_additive]
+instance [CompactSpace G] (H : ClosedSubgroup G) : CompactSpace H :=
+  isCompact_iff_compactSpace.mp (IsClosed.isCompact H.isClosed')
+
+end ClosedSubgroup
+
+open scoped Pointwise
+
+namespace Subgroup
+
+variable {G : Type u} [Group G] [TopologicalSpace G] [ContinuousMul G]
+
+lemma normalCore_isClosed (H : Subgroup G) (h : IsClosed (H : Set G)) :
+    IsClosed (H.normalCore : Set G) := by
+  rw [normalCore_eq_iInf_conjAct]
+  push_cast
+  apply isClosed_iInter
+  intro g
+  convert IsClosed.preimage (TopologicalGroup.continuous_conj (ConjAct.ofConjAct g⁻¹)) h
+  exact Set.ext (fun t ↦ Set.mem_smul_set_iff_inv_smul_mem)
+
+@[to_additive]
+lemma isOpen_of_isClosed_of_finiteIndex (H : Subgroup G) [H.FiniteIndex]
+    (h : IsClosed (H : Set G)) : IsOpen (H : Set G) := by
+  apply isClosed_compl_iff.mp
+  convert isClosed_iUnion_of_finite <| fun (x : {x : (G ⧸ H) // x ≠ QuotientGroup.mk 1})
+    ↦ IsClosed.smul h (Quotient.out' x.1)
+  ext x
+  refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
+  · have : QuotientGroup.mk 1 ≠ QuotientGroup.mk (s := H) x := by
+      apply QuotientGroup.eq.not.mpr
+      simpa only [inv_one, one_mul, ne_eq]
+    simp only [ne_eq, Set.mem_iUnion]
+    use ⟨QuotientGroup.mk (s := H) x, this.symm⟩,
+      (Quotient.out' (QuotientGroup.mk (s := H) x))⁻¹ * x
+    simp only [SetLike.mem_coe, smul_eq_mul, mul_inv_cancel_left, and_true]
+    exact QuotientGroup.eq.mp <| QuotientGroup.out_eq' (QuotientGroup.mk (s := H) x)
+  · rcases h with ⟨S,⟨y,hS⟩,mem⟩
+    simp only [← hS] at mem
+    rcases mem with ⟨h,hh,eq⟩
+    simp only [Set.mem_compl_iff, SetLike.mem_coe]
+    by_contra mH
+    simp only [← eq, ne_eq, smul_eq_mul] at mH
+    absurd y.2.symm
+    rw [← QuotientGroup.out_eq' y.1, QuotientGroup.eq]
+    simp only [inv_one, ne_eq, one_mul, (Subgroup.mul_mem_cancel_right H hh).mp mH]
+
+end Subgroup
+
+end
diff --git a/Mathlib/Topology/Algebra/ConstMulAction.lean b/Mathlib/Topology/Algebra/ConstMulAction.lean
index b76efc9f2c27f..39bd780335969 100644
--- a/Mathlib/Topology/Algebra/ConstMulAction.lean
+++ b/Mathlib/Topology/Algebra/ConstMulAction.lean
@@ -141,6 +141,13 @@ theorem Inseparable.const_smul {x y : α} (h : Inseparable x y) (c : M) :
     Inseparable (c • x) (c • y) :=
   h.map (continuous_const_smul c)
 
+@[to_additive]
+theorem Inducing.continuousConstSMul {N β : Type*} [SMul N β] [TopologicalSpace β]
+    {g : β → α} (hg : Inducing g) (f : N → M) (hf : ∀ {c : N} {x : β}, g (c • x) = f c • g x) :
+    ContinuousConstSMul N β where
+  continuous_const_smul c := by
+    simpa only [Function.comp_def, hf, hg.continuous_iff] using hg.continuous.const_smul (f c)
+
 end SMul
 
 section Monoid
@@ -444,6 +451,11 @@ theorem isOpenMap_quotient_mk'_mul [ContinuousConstSMul Γ T] :
   rw [isOpen_coinduced, MulAction.quotient_preimage_image_eq_union_mul U]
   exact isOpen_iUnion fun γ => isOpenMap_smul γ U hU
 
+@[to_additive]
+theorem MulAction.isOpenQuotientMap_quotientMk [ContinuousConstSMul Γ T] :
+    IsOpenQuotientMap (Quotient.mk (MulAction.orbitRel Γ T)) :=
+  ⟨surjective_quot_mk _, continuous_quot_mk, isOpenMap_quotient_mk'_mul⟩
+
 /-- The quotient by a discontinuous group action of a locally compact t2 space is t2. -/
 @[to_additive "The quotient by a discontinuous group action of a locally compact t2
 space is t2."]
diff --git a/Mathlib/Topology/Algebra/Constructions.lean b/Mathlib/Topology/Algebra/Constructions.lean
index 4ca7adfa8c318..96db33647784a 100644
--- a/Mathlib/Topology/Algebra/Constructions.lean
+++ b/Mathlib/Topology/Algebra/Constructions.lean
@@ -140,7 +140,7 @@ theorem continuous_val : Continuous ((↑) : Mˣ → M) :=
 @[to_additive]
 protected theorem continuous_iff {f : X → Mˣ} :
     Continuous f ↔ Continuous (val ∘ f) ∧ Continuous (fun x => ↑(f x)⁻¹ : X → M) := by
-  simp only [inducing_embedProduct.continuous_iff, embedProduct_apply, (· ∘ ·),
+  simp only [inducing_embedProduct.continuous_iff, embedProduct_apply, Function.comp_def,
     continuous_prod_mk, opHomeomorph.symm.inducing.continuous_iff, opHomeomorph_symm_apply,
     unop_op]
 
diff --git a/Mathlib/Topology/Algebra/ContinuousAffineMap.lean b/Mathlib/Topology/Algebra/ContinuousAffineMap.lean
index 39cb34e5d4264..0c94ea673275e 100644
--- a/Mathlib/Topology/Algebra/ContinuousAffineMap.lean
+++ b/Mathlib/Topology/Algebra/ContinuousAffineMap.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
 import Mathlib.LinearAlgebra.AffineSpace.AffineMap
-import Mathlib.Topology.ContinuousFunction.Basic
 import Mathlib.Topology.Algebra.Module.Basic
 
 /-!
diff --git a/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean b/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
index de9f0789c24cb..a753c9397ac13 100644
--- a/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
+++ b/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
@@ -5,7 +5,7 @@ Authors: Thomas Browning
 -/
 import Mathlib.Topology.Algebra.Equicontinuity
 import Mathlib.Topology.Algebra.Group.Compact
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Topology.UniformSpace.Ascoli
 
 /-!
@@ -131,16 +131,24 @@ def comp (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) : Continuou
   mk' (g.toMonoidHom.comp f.toMonoidHom) (g.continuous_toFun.comp f.continuous_toFun)
 
 /-- Product of two continuous homomorphisms on the same space. -/
-@[to_additive (attr := simps!) "Product of two continuous homomorphisms on the same space."]
+@[to_additive (attr := simps!) prod "Product of two continuous homomorphisms on the same space."]
 def prod (f : ContinuousMonoidHom A B) (g : ContinuousMonoidHom A C) :
     ContinuousMonoidHom A (B × C) :=
   mk' (f.toMonoidHom.prod g.toMonoidHom) (f.continuous_toFun.prod_mk g.continuous_toFun)
 
 /-- Product of two continuous homomorphisms on different spaces. -/
-@[to_additive (attr := simps!) "Product of two continuous homomorphisms on different spaces."]
-def prod_map (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) :
+@[to_additive (attr := simps!) prodMap
+  "Product of two continuous homomorphisms on different spaces."]
+def prodMap (f : ContinuousMonoidHom A C) (g : ContinuousMonoidHom B D) :
     ContinuousMonoidHom (A × B) (C × D) :=
-  mk' (f.toMonoidHom.prodMap g.toMonoidHom) (f.continuous_toFun.prod_map g.continuous_toFun)
+  mk' (f.toMonoidHom.prodMap g.toMonoidHom) (f.continuous_toFun.prodMap g.continuous_toFun)
+
+@[deprecated (since := "2024-10-05")] alias prod_map := prodMap
+@[deprecated (since := "2024-10-05")]
+alias _root_.ContinuousAddMonoidHom.sum_map := ContinuousAddMonoidHom.prodMap
+
+set_option linter.existingAttributeWarning false in
+attribute [to_additive existing] prod_map
 
 variable (A B C D E)
 
@@ -208,7 +216,7 @@ variable {A B C D E}
 @[to_additive (attr := simps!) "Coproduct of two continuous homomorphisms to the same space."]
 def coprod (f : ContinuousMonoidHom A E) (g : ContinuousMonoidHom B E) :
     ContinuousMonoidHom (A × B) E :=
-  (mul E).comp (f.prod_map g)
+  (mul E).comp (f.prodMap g)
 
 @[to_additive]
 instance : CommGroup (ContinuousMonoidHom A E) where
@@ -265,7 +273,7 @@ instance [T2Space B] : T2Space (ContinuousMonoidHom A B) :=
 instance : TopologicalGroup (ContinuousMonoidHom A E) :=
   let hi := inducing_toContinuousMap A E
   let hc := hi.continuous
-  { continuous_mul := hi.continuous_iff.mpr (continuous_mul.comp (Continuous.prod_map hc hc))
+  { continuous_mul := hi.continuous_iff.mpr (continuous_mul.comp (Continuous.prodMap hc hc))
     continuous_inv := hi.continuous_iff.mpr (continuous_inv.comp hc) }
 
 @[to_additive]
@@ -280,7 +288,7 @@ theorem continuous_comp [LocallyCompactSpace B] :
     Continuous fun f : ContinuousMonoidHom A B × ContinuousMonoidHom B C => f.2.comp f.1 :=
   (inducing_toContinuousMap A C).continuous_iff.2 <|
     ContinuousMap.continuous_comp'.comp
-      ((inducing_toContinuousMap A B).prod_map (inducing_toContinuousMap B C)).continuous
+      ((inducing_toContinuousMap A B).prodMap (inducing_toContinuousMap B C)).continuous
 
 @[to_additive]
 theorem continuous_comp_left (f : ContinuousMonoidHom A B) :
diff --git a/Mathlib/Topology/Algebra/Field.lean b/Mathlib/Topology/Algebra/Field.lean
index 6a6d37b9907fc..c897f0bb8fa18 100644
--- a/Mathlib/Topology/Algebra/Field.lean
+++ b/Mathlib/Topology/Algebra/Field.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2021 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison
+Authors: Patrick Massot, Kim Morrison
 -/
 import Mathlib.Algebra.Field.Subfield
 import Mathlib.Algebra.GroupWithZero.Divisibility
@@ -101,7 +101,7 @@ open Topology
 
 theorem IsLocalMin.inv {f : α → β} {a : α} (h1 : IsLocalMin f a) (h2 : ∀ᶠ z in 𝓝 a, 0 < f z) :
     IsLocalMax f⁻¹ a := by
-  filter_upwards [h1, h2] with z h3 h4 using(inv_le_inv h4 h2.self_of_nhds).mpr h3
+  filter_upwards [h1, h2] with z h3 h4 using(inv_le_inv₀ h4 h2.self_of_nhds).mpr h3
 
 end LocalExtr
 
@@ -132,7 +132,7 @@ theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 
     (hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) :
     EqOn f g S ∨ EqOn f (-g) S := by
   have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
-    simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx
+    simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx)), div_pow] using hsq hx
   simpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)]
     using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq
 
@@ -146,6 +146,6 @@ theorem IsPreconnected.eq_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [Cont
   rcases hS.eq_or_eq_neg_of_sq_eq hf hg @hsq @hg_ne with (h | h)
   · exact h hx
   · rw [h _, Pi.neg_apply, neg_eq_iff_add_eq_zero, ← two_mul, mul_eq_zero,
-      iff_false_iff.2 (hg_ne _)] at hy' ⊢ <;> assumption
+      (iff_of_eq (iff_false _)).2 (hg_ne _)] at hy' ⊢ <;> assumption
 
 end Preconnected
diff --git a/Mathlib/Topology/Algebra/FilterBasis.lean b/Mathlib/Topology/Algebra/FilterBasis.lean
index 63619d775c916..e2732d46cdbb7 100644
--- a/Mathlib/Topology/Algebra/FilterBasis.lean
+++ b/Mathlib/Topology/Algebra/FilterBasis.lean
@@ -84,7 +84,7 @@ variable {G : Type u} [Group G] {B : GroupFilterBasis G}
 
 @[to_additive]
 instance : Membership (Set G) (GroupFilterBasis G) :=
-  ⟨fun s f ↦ s ∈ f.sets⟩
+  ⟨fun f s ↦ s ∈ f.sets⟩
 
 @[to_additive]
 theorem one {U : Set G} : U ∈ B → (1 : G) ∈ U :=
@@ -149,10 +149,10 @@ theorem nhds_eq (B : GroupFilterBasis G) {x₀ : G} : @nhds G B.topology x₀ =
     filter_upwards [image_mem_map (B.mem_filter_of_mem V_in)]
     rintro _ ⟨x, hx, rfl⟩
     calc
-      a • U ⊇ a • (V * V) := smul_set_mono hVU
-      _ ⊇ a • x • V := smul_set_mono <| smul_set_subset_smul hx
-      _ = (a * x) • V := smul_smul ..
-      _ ∈ (a * x) • B.filter := smul_set_mem_smul_filter <| B.mem_filter_of_mem V_in
+      (a * x) • V ∈ (a * x) • B.filter := smul_set_mem_smul_filter <| B.mem_filter_of_mem V_in
+      _ = a • x • V := smul_smul .. |>.symm
+      _ ⊆ a • (V * V) := smul_set_mono <| smul_set_subset_smul hx
+      _ ⊆ a • U := smul_set_mono hVU
 
 @[to_additive]
 theorem nhds_one_eq (B : GroupFilterBasis G) :
@@ -224,7 +224,7 @@ namespace RingFilterBasis
 variable {R : Type u} [Ring R] (B : RingFilterBasis R)
 
 instance : Membership (Set R) (RingFilterBasis R) :=
-  ⟨fun s B ↦ s ∈ B.sets⟩
+  ⟨fun B s ↦ s ∈ B.sets⟩
 
 theorem mul {U : Set R} (hU : U ∈ B) : ∃ V ∈ B, V * V ⊆ U :=
   mul' hU
@@ -284,7 +284,7 @@ variable {R M : Type*} [CommRing R] [TopologicalSpace R] [AddCommGroup M] [Modul
   (B : ModuleFilterBasis R M)
 
 instance GroupFilterBasis.hasMem : Membership (Set M) (ModuleFilterBasis R M) :=
-  ⟨fun s B ↦ s ∈ B.sets⟩
+  ⟨fun B s ↦ s ∈ B.sets⟩
 
 theorem smul {U : Set M} (hU : U ∈ B) : ∃ V ∈ 𝓝 (0 : R), ∃ W ∈ B, V • W ⊆ U :=
   B.smul' hU
diff --git a/Mathlib/Topology/Algebra/Group/Basic.lean b/Mathlib/Topology/Algebra/Group/Basic.lean
index 36c67204cd37c..49c268a901a0c 100644
--- a/Mathlib/Topology/Algebra/Group/Basic.lean
+++ b/Mathlib/Topology/Algebra/Group/Basic.lean
@@ -7,6 +7,7 @@ import Mathlib.GroupTheory.GroupAction.ConjAct
 import Mathlib.GroupTheory.GroupAction.Quotient
 import Mathlib.Topology.Algebra.Monoid
 import Mathlib.Topology.Algebra.Constructions
+import Mathlib.Topology.Maps.OpenQuotient
 import Mathlib.Algebra.Order.Archimedean.Basic
 import Mathlib.GroupTheory.QuotientGroup.Basic
 
@@ -164,7 +165,7 @@ theorem ContinuousInv.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F 
     @ContinuousInv α (tβ.induced f) _ := by
   let _tα := tβ.induced f
   refine ⟨continuous_induced_rng.2 ?_⟩
-  simp only [Function.comp, map_inv]
+  simp only [Function.comp_def, map_inv]
   fun_prop
 
 @[to_additive]
@@ -304,6 +305,10 @@ protected def Homeomorph.inv (G : Type*) [TopologicalSpace G] [InvolutiveInv G]
 lemma Homeomorph.coe_inv {G : Type*} [TopologicalSpace G] [InvolutiveInv G] [ContinuousInv G] :
     ⇑(Homeomorph.inv G) = Inv.inv := rfl
 
+@[to_additive]
+theorem nhds_inv (a : G) : 𝓝 a⁻¹ = (𝓝 a)⁻¹ :=
+  ((Homeomorph.inv G).map_nhds_eq a).symm
+
 @[to_additive]
 theorem isOpenMap_inv : IsOpenMap (Inv.inv : G → G) :=
   (Homeomorph.inv _).isOpenMap
@@ -376,7 +381,7 @@ end LatticeOps
 theorem Inducing.continuousInv {G H : Type*} [Inv G] [Inv H] [TopologicalSpace G]
     [TopologicalSpace H] [ContinuousInv H] {f : G → H} (hf : Inducing f)
     (hf_inv : ∀ x, f x⁻¹ = (f x)⁻¹) : ContinuousInv G :=
-  ⟨hf.continuous_iff.2 <| by simpa only [(· ∘ ·), hf_inv] using hf.continuous.inv⟩
+  ⟨hf.continuous_iff.2 <| by simpa only [Function.comp_def, hf_inv] using hf.continuous.inv⟩
 
 section TopologicalGroup
 
@@ -531,7 +536,7 @@ end OrderedCommGroup
 
 @[to_additive]
 instance [TopologicalSpace H] [Group H] [TopologicalGroup H] : TopologicalGroup (G × H) where
-  continuous_inv := continuous_inv.prod_map continuous_inv
+  continuous_inv := continuous_inv.prodMap continuous_inv
 
 @[to_additive]
 instance Pi.topologicalGroup {C : β → Type*} [∀ b, TopologicalSpace (C b)] [∀ b, Group (C b)]
@@ -742,8 +747,8 @@ theorem continuous_of_continuousAt_one {M hom : Type*} [MulOneClass M] [Topologi
     (hf : ContinuousAt f 1) :
     Continuous f :=
   continuous_iff_continuousAt.2 fun x => by
-    simpa only [ContinuousAt, ← map_mul_left_nhds_one x, tendsto_map'_iff, (· ∘ ·), map_mul,
-      map_one, mul_one] using hf.tendsto.const_mul (f x)
+    simpa only [ContinuousAt, ← map_mul_left_nhds_one x, tendsto_map'_iff, Function.comp_def,
+      map_mul, map_one, mul_one] using hf.tendsto.const_mul (f x)
 
 @[to_additive continuous_of_continuousAt_zero₂]
 theorem continuous_of_continuousAt_one₂ {H M : Type*} [CommMonoid M] [TopologicalSpace M]
@@ -752,7 +757,7 @@ theorem continuous_of_continuousAt_one₂ {H M : Type*} [CommMonoid M] [Topologi
     (hl : ∀ x, ContinuousAt (f x) 1) (hr : ∀ y, ContinuousAt (f · y) 1) :
     Continuous (fun x : G × H ↦ f x.1 x.2) := continuous_iff_continuousAt.2 fun (x, y) => by
   simp only [ContinuousAt, nhds_prod_eq, ← map_mul_left_nhds_one x, ← map_mul_left_nhds_one y,
-    prod_map_map_eq, tendsto_map'_iff, (· ∘ ·), map_mul, MonoidHom.mul_apply] at *
+    prod_map_map_eq, tendsto_map'_iff, Function.comp_def, map_mul, MonoidHom.mul_apply] at *
   refine ((tendsto_const_nhds.mul ((hr y).comp tendsto_fst)).mul
     (((hl x).comp tendsto_snd).mul hf)).mono_right (le_of_eq ?_)
   simp only [map_one, mul_one, MonoidHom.one_apply]
@@ -778,8 +783,8 @@ theorem ContinuousInv.of_nhds_one {G : Type*} [Group G] [TopologicalSpace G]
   refine ⟨continuous_iff_continuousAt.2 fun x₀ => ?_⟩
   have : Tendsto (fun x => x₀⁻¹ * (x₀ * x⁻¹ * x₀⁻¹)) (𝓝 1) (map (x₀⁻¹ * ·) (𝓝 1)) :=
     (tendsto_map.comp <| hconj x₀).comp hinv
-  simpa only [ContinuousAt, hleft x₀, hleft x₀⁻¹, tendsto_map'_iff, (· ∘ ·), mul_assoc, mul_inv_rev,
-    inv_mul_cancel_left] using this
+  simpa only [ContinuousAt, hleft x₀, hleft x₀⁻¹, tendsto_map'_iff, Function.comp_def, mul_assoc,
+    mul_inv_rev, inv_mul_cancel_left] using this
 
 @[to_additive]
 theorem TopologicalGroup.of_nhds_one' {G : Type u} [Group G] [TopologicalSpace G]
@@ -806,7 +811,7 @@ theorem TopologicalGroup.of_nhds_one {G : Type u} [Group G] [TopologicalSpace G]
   replace hconj : ∀ x₀ : G, map (x₀ * · * x₀⁻¹) (𝓝 1) = 𝓝 1 :=
     fun x₀ => map_eq_of_inverse (x₀⁻¹ * · * x₀⁻¹⁻¹) (by ext; simp [mul_assoc]) (hconj _) (hconj _)
   rw [← hconj x₀]
-  simpa [(· ∘ ·)] using hleft _
+  simpa [Function.comp_def] using hleft _
 
 @[to_additive]
 theorem TopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [TopologicalSpace G]
@@ -815,52 +820,7 @@ theorem TopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [Topolog
     (hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1)) : TopologicalGroup G :=
   TopologicalGroup.of_nhds_one hmul hinv hleft (by simpa using tendsto_id)
 
-end TopologicalGroup
-
-section QuotientTopologicalGroup
-
-variable [TopologicalSpace G] [Group G] [TopologicalGroup G] (N : Subgroup G) (n : N.Normal)
-
-@[to_additive]
-instance QuotientGroup.Quotient.topologicalSpace {G : Type*} [Group G] [TopologicalSpace G]
-    (N : Subgroup G) : TopologicalSpace (G ⧸ N) :=
-  instTopologicalSpaceQuotient
-
-open QuotientGroup
-
-@[to_additive]
-theorem QuotientGroup.isOpenMap_coe : IsOpenMap ((↑) : G → G ⧸ N) := by
-  intro s s_op
-  change IsOpen (((↑) : G → G ⧸ N) ⁻¹' ((↑) '' s))
-  rw [QuotientGroup.preimage_image_mk N s]
-  exact isOpen_iUnion fun n => (continuous_mul_right _).isOpen_preimage s s_op
-
-@[to_additive]
-instance topologicalGroup_quotient [N.Normal] : TopologicalGroup (G ⧸ N) where
-  continuous_mul := by
-    have cont : Continuous (((↑) : G → G ⧸ N) ∘ fun p : G × G ↦ p.fst * p.snd) :=
-      continuous_quot_mk.comp continuous_mul
-    have quot : QuotientMap fun p : G × G ↦ ((p.1 : G ⧸ N), (p.2 : G ⧸ N)) := by
-      apply IsOpenMap.to_quotientMap
-      · exact (QuotientGroup.isOpenMap_coe N).prod (QuotientGroup.isOpenMap_coe N)
-      · exact continuous_quot_mk.prod_map continuous_quot_mk
-      · exact (surjective_quot_mk _).prodMap (surjective_quot_mk _)
-    exact quot.continuous_iff.2 cont
-  continuous_inv := by
-    have quot := IsOpenMap.to_quotientMap
-      (QuotientGroup.isOpenMap_coe N) continuous_quot_mk (surjective_quot_mk _)
-    rw [quot.continuous_iff]
-    exact continuous_quot_mk.comp continuous_inv
-
-/-- Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. -/
-@[to_additive
-  "Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient."]
-theorem QuotientGroup.nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = Filter.map (↑) (𝓝 x) :=
-  le_antisymm ((QuotientGroup.isOpenMap_coe N).nhds_le x) continuous_quot_mk.continuousAt
-
-variable (G)
-variable [FirstCountableTopology G]
-
+variable (G) in
 /-- Any first countable topological group has an antitone neighborhood basis `u : ℕ → Set G` for
 which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for
 `QuotientGroup.completeSpace` -/
@@ -868,7 +828,7 @@ which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a
   "Any first countable topological additive group has an antitone neighborhood basis
   `u : ℕ → set G` for which `u (n + 1) + u (n + 1) ⊆ u n`.
   The existence of such a neighborhood basis is a key tool for `QuotientAddGroup.completeSpace`"]
-theorem TopologicalGroup.exists_antitone_basis_nhds_one :
+theorem TopologicalGroup.exists_antitone_basis_nhds_one [FirstCountableTopology G] :
     ∃ u : ℕ → Set G, (𝓝 1).HasAntitoneBasis u ∧ ∀ n, u (n + 1) * u (n + 1) ⊆ u n := by
   rcases (𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩
   have :=
@@ -885,15 +845,88 @@ theorem TopologicalGroup.exists_antitone_basis_nhds_one :
   obtain ⟨φ, -, hφ, φ_anti_basis⟩ := HasAntitoneBasis.subbasis_with_rel ⟨hu, u_anti⟩ event_mul
   exact ⟨u ∘ φ, φ_anti_basis, fun n => hφ n.lt_succ_self⟩
 
-/-- In a first countable topological group `G` with normal subgroup `N`, `1 : G ⧸ N` has a
-countable neighborhood basis. -/
+end TopologicalGroup
+
+namespace QuotientGroup
+
+variable [TopologicalSpace G] [Group G]
+
+@[to_additive]
+instance instTopologicalSpace (N : Subgroup G) : TopologicalSpace (G ⧸ N) :=
+  instTopologicalSpaceQuotient
+
+@[to_additive]
+instance [CompactSpace G] (N : Subgroup G) : CompactSpace (G ⧸ N) :=
+  Quotient.compactSpace
+
+@[to_additive]
+theorem quotientMap_mk (N : Subgroup G) : QuotientMap (mk : G → G ⧸ N) :=
+  quotientMap_quot_mk
+
+@[to_additive]
+theorem continuous_mk {N : Subgroup G} : Continuous (mk : G → G ⧸ N) :=
+  continuous_quot_mk
+
+section ContinuousMul
+
+variable [ContinuousMul G] {N : Subgroup G}
+
+@[to_additive]
+theorem isOpenMap_coe : IsOpenMap ((↑) : G → G ⧸ N) := isOpenMap_quotient_mk'_mul
+
+@[to_additive]
+theorem isOpenQuotientMap_mk : IsOpenQuotientMap (mk : G → G ⧸ N) :=
+  MulAction.isOpenQuotientMap_quotientMk
+
+@[to_additive (attr := simp)]
+theorem dense_preimage_mk {s : Set (G ⧸ N)} : Dense ((↑) ⁻¹' s : Set G) ↔ Dense s :=
+  isOpenQuotientMap_mk.dense_preimage_iff
+
+@[to_additive]
+theorem dense_image_mk {s : Set G} :
+    Dense (mk '' s : Set (G ⧸ N)) ↔ Dense (s * (N : Set G)) := by
+  rw [← dense_preimage_mk, preimage_image_mk_eq_mul]
+
+@[to_additive]
+instance instContinuousSMul : ContinuousSMul G (G ⧸ N) where
+  continuous_smul := by
+    rw [← (IsOpenQuotientMap.id.prodMap isOpenQuotientMap_mk).continuous_comp_iff]
+    exact continuous_mk.comp continuous_mul
+
+variable (N)
+
+/-- Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient. -/
 @[to_additive
-  "In a first countable topological additive group `G` with normal additive subgroup
-  `N`, `0 : G ⧸ N` has a countable neighborhood basis."]
-instance QuotientGroup.nhds_one_isCountablyGenerated : (𝓝 (1 : G ⧸ N)).IsCountablyGenerated :=
-  (QuotientGroup.nhds_eq N 1).symm ▸ map.isCountablyGenerated _ _
+  "Neighborhoods in the quotient are precisely the map of neighborhoods in the prequotient."]
+theorem nhds_eq (x : G) : 𝓝 (x : G ⧸ N) = Filter.map (↑) (𝓝 x) :=
+  (isOpenQuotientMap_mk.map_nhds_eq _).symm
+
+@[to_additive]
+instance instFirstCountableTopology [FirstCountableTopology G] :
+    FirstCountableTopology (G ⧸ N) where
+  nhds_generated_countable := mk_surjective.forall.2 fun x ↦ nhds_eq N x ▸ inferInstance
+
+@[to_additive (attr := deprecated (since := "2024-08-05"))]
+theorem nhds_one_isCountablyGenerated [FirstCountableTopology G] [N.Normal] :
+    (𝓝 (1 : G ⧸ N)).IsCountablyGenerated :=
+  inferInstance
 
-end QuotientTopologicalGroup
+end ContinuousMul
+
+variable [TopologicalGroup G] (N : Subgroup G)
+
+@[to_additive]
+instance instTopologicalGroup [N.Normal] : TopologicalGroup (G ⧸ N) where
+  continuous_mul := by
+    rw [← (isOpenQuotientMap_mk.prodMap isOpenQuotientMap_mk).continuous_comp_iff]
+    exact continuous_mk.comp continuous_mul
+  continuous_inv := continuous_inv.quotient_map' _
+
+@[to_additive (attr := deprecated (since := "2024-08-05"))]
+theorem _root_.topologicalGroup_quotient [N.Normal] : TopologicalGroup (G ⧸ N) :=
+  instTopologicalGroup N
+
+end QuotientGroup
 
 /-- A typeclass saying that `p : G × G ↦ p.1 - p.2` is a continuous function. This property
 automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. -/
@@ -1221,11 +1254,8 @@ theorem IsClosed.mul_right_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
 theorem QuotientGroup.isClosedMap_coe {H : Subgroup G} (hH : IsCompact (H : Set G)) :
     IsClosedMap ((↑) : G → G ⧸ H) := by
   intro t ht
-  rw [← quotientMap_quotient_mk'.isClosed_preimage]
-  convert ht.mul_right_of_isCompact hH
-  refine (QuotientGroup.preimage_image_mk_eq_iUnion_image _ _).trans ?_
-  rw [iUnion_subtype, ← iUnion_mul_right_image]
-  rfl
+  rw [← (quotientMap_mk H).isClosed_preimage, preimage_image_mk_eq_mul]
+  exact ht.mul_right_of_isCompact hH
 
 @[to_additive]
 lemma subset_mul_closure_one {G} [MulOneClass G] [TopologicalSpace G] (s : Set G) :
@@ -1396,7 +1426,7 @@ theorem Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite (S : Sub
     (hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S.op G :=
   { finite_disjoint_inter_image := by
       intro K L hK hL
-      have : Continuous fun p : G × G => (p.1⁻¹, p.2) := continuous_inv.prod_map continuous_id
+      have : Continuous fun p : G × G => (p.1⁻¹, p.2) := continuous_inv.prodMap continuous_id
       have H : Set.Finite _ :=
         hS ((hK.prod hL).image (continuous_mul.comp this)).compl_mem_cocompact
       simp only [preimage_compl, compl_compl, coeSubtype, comp_apply] at H
@@ -1597,7 +1627,7 @@ instance [LocallyCompactSpace G] (N : Subgroup G) : LocallyCompactSpace (G ⧸ N
   obtain ⟨y, rfl⟩ : ∃ y, π y = x := Quot.exists_rep x
   have : π ⁻¹' n ∈ 𝓝 y := preimage_nhds_coinduced hn
   rcases local_compact_nhds this with ⟨s, s_mem, hs, s_comp⟩
-  exact ⟨π '' s, (QuotientGroup.isOpenMap_coe N).image_mem_nhds s_mem, mapsTo'.mp hs,
+  exact ⟨π '' s, QuotientGroup.isOpenMap_coe.image_mem_nhds s_mem, mapsTo'.mp hs,
     s_comp.image C⟩
 
 end
@@ -1846,8 +1876,6 @@ instance : SemilatticeInf (GroupTopology α) :=
 instance : Inhabited (GroupTopology α) :=
   ⟨⊤⟩
 
-local notation "cont" => @Continuous _ _
-
 /-- Infimum of a collection of group topologies. -/
 @[to_additive "Infimum of a collection of additive group topologies"]
 instance : InfSet (GroupTopology α) where
@@ -1915,3 +1943,5 @@ theorem coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Group β
   exact continuous_iff_coinduced_le.2 ht'
 
 end GroupTopology
+
+set_option linter.style.longFile 2100
diff --git a/Mathlib/Topology/Algebra/Group/Compact.lean b/Mathlib/Topology/Algebra/Group/Compact.lean
index 57b0155e2e6a8..b479c867ca91f 100644
--- a/Mathlib/Topology/Algebra/Group/Compact.lean
+++ b/Mathlib/Topology/Algebra/Group/Compact.lean
@@ -4,24 +4,18 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
 import Mathlib.Topology.Algebra.Group.Basic
-import Mathlib.Topology.CompactOpen
 import Mathlib.Topology.Sets.Compacts
 
 /-!
 # Additional results on topological groups
 
-Two results on topological groups that have been separated out as they require more substantial
-imports developing either positive compacts or the compact open topology.
-
+A result on topological groups that has been separated out
+as it requires more substantial imports developing positive compacts.
 -/
 
-universe u v w x
-
-variable {α : Type u} {β : Type v} {G : Type w} {H : Type x}
 
-section
-
-variable [TopologicalSpace G] [Group G] [TopologicalGroup G]
+universe u
+variable {G : Type u} [TopologicalSpace G] [Group G] [TopologicalGroup G]
 
 /-- Every topological group in which there exists a compact set with nonempty interior
 is locally compact. -/
@@ -32,21 +26,3 @@ theorem TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
     (K : PositiveCompacts G) : LocallyCompactSpace G :=
   let ⟨_x, hx⟩ := K.interior_nonempty
   K.isCompact.locallyCompactSpace_of_mem_nhds_of_group (mem_interior_iff_mem_nhds.1 hx)
-
-end
-
-section Quotient
-
-variable [Group G] [TopologicalSpace G] [TopologicalGroup G] {Γ : Subgroup G}
-
-@[to_additive]
-instance QuotientGroup.continuousSMul [LocallyCompactSpace G] : ContinuousSMul G (G ⧸ Γ) where
-  continuous_smul := by
-    let F : G × G ⧸ Γ → G ⧸ Γ := fun p => p.1 • p.2
-    change Continuous F
-    have H : Continuous (F ∘ fun p : G × G => (p.1, QuotientGroup.mk p.2)) := by
-      change Continuous fun p : G × G => QuotientGroup.mk (p.1 * p.2)
-      exact continuous_coinduced_rng.comp continuous_mul
-    exact QuotientMap.continuous_lift_prod_right quotientMap_quotient_mk' H
-
-end Quotient
diff --git a/Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean b/Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean
new file mode 100644
index 0000000000000..ae23b4d9c3b9b
--- /dev/null
+++ b/Mathlib/Topology/Algebra/Group/SubmonoidClosure.lean
@@ -0,0 +1,118 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+
+import Mathlib.Order.Filter.AtTopBot.Group
+import Mathlib.Topology.Algebra.Group.Basic
+
+/-!
+# Topological closure of the submonoid closure
+
+In this file we prove several versions of the following statement:
+if `G` is a compact topological group and `s : Set G`,
+then the topological closures of `Submonoid.closure s` and `Subgroup.closure s` are equal.
+
+The proof is based on the following observation, see `mapClusterPt_self_zpow_atTop_pow`:
+each `x^m`, `m : ℤ` is a limit point (`MapClusterPt`) of the sequence `x^n`, `n : ℕ`, as `n → ∞`.
+-/
+
+open Filter Function Set
+open scoped Topology
+
+variable {G : Type*}
+
+@[to_additive]
+theorem mapClusterPt_atTop_zpow_iff_pow [DivInvMonoid G] [TopologicalSpace G] {x y : G} :
+    MapClusterPt x atTop (y ^ · : ℤ → G) ↔ MapClusterPt x atTop (y ^ · : ℕ → G) := by
+  simp_rw [MapClusterPt, ← Nat.map_cast_int_atTop, map_map, comp_def, zpow_natCast]
+
+variable [Group G] [TopologicalSpace G] [CompactSpace G] [TopologicalGroup G]
+
+@[to_additive]
+theorem mapClusterPt_self_zpow_atTop_pow (x : G) (m : ℤ) :
+    MapClusterPt (x ^ m) atTop (x ^ · : ℕ → G) := by
+  obtain ⟨y, hy⟩ : ∃ y, MapClusterPt y atTop (x ^ · : ℤ → G) :=
+    exists_clusterPt_of_compactSpace _
+  rw [← mapClusterPt_atTop_zpow_iff_pow]
+  have H : MapClusterPt (x ^ m) (atTop.curry atTop) ↿(fun a b ↦ x ^ (m + b - a)) := by
+    have : ContinuousAt (fun yz ↦ x ^ m * yz.2 / yz.1) (y, y) := by fun_prop
+    simpa only [comp_def, ← zpow_sub, ← zpow_add, div_eq_mul_inv, Prod.map, mul_inv_cancel_right]
+      using (hy.curry_prodMap hy).continuousAt_comp this
+  suffices Tendsto ↿(fun a b ↦ m + b - a) (atTop.curry atTop) atTop from H.of_comp this
+  refine Tendsto.curry <| .of_forall fun a ↦ ?_
+  simp only [sub_eq_add_neg] -- TODO: add `Tendsto.atTop_sub_const` etc
+  exact tendsto_atTop_add_const_right _ _ (tendsto_atTop_add_const_left atTop m tendsto_id)
+
+@[to_additive]
+theorem mapClusterPt_one_atTop_pow (x : G) : MapClusterPt 1 atTop (x ^ · : ℕ → G) := by
+  simpa using mapClusterPt_self_zpow_atTop_pow x 0
+
+@[to_additive]
+theorem mapClusterPt_self_atTop_pow (x : G) : MapClusterPt x atTop (x ^ · : ℕ → G) := by
+  simpa using mapClusterPt_self_zpow_atTop_pow x 1
+
+@[to_additive]
+theorem mapClusterPt_atTop_pow_tfae (x y : G) :
+    List.TFAE [
+      MapClusterPt x atTop (y ^ · : ℕ → G),
+      MapClusterPt x atTop (y ^ · : ℤ → G),
+      x ∈ closure (range (y ^ · : ℕ → G)),
+      x ∈ closure (range (y ^ · : ℤ → G)),
+    ] := by
+  tfae_have 2 ↔ 1 := mapClusterPt_atTop_zpow_iff_pow
+  tfae_have 3 → 4 := by
+    refine fun h ↦ closure_mono (range_subset_iff.2 fun n ↦ ?_) h
+    exact ⟨n, zpow_natCast _ _⟩
+  tfae_have 4 → 1 := by
+    refine fun h ↦ closure_minimal ?_ isClosed_setOf_clusterPt h
+    exact range_subset_iff.2 (mapClusterPt_self_zpow_atTop_pow _)
+  tfae_have 1 → 3 := by
+    rw [mem_closure_iff_clusterPt]
+    exact (ClusterPt.mono · (le_principal_iff.2 range_mem_map))
+  tfae_finish
+
+@[to_additive]
+theorem mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers {x y : G} :
+    MapClusterPt x atTop (y ^ · : ℕ → G) ↔ x ∈ (Subgroup.zpowers y).topologicalClosure :=
+  (mapClusterPt_atTop_pow_tfae x y).out 0 3
+
+@[to_additive (attr := simp)]
+theorem mapClusterPt_inv_atTop_pow {x y : G} :
+    MapClusterPt x⁻¹ atTop (y ^ · : ℕ → G) ↔ MapClusterPt x atTop (y ^ · : ℕ → G) := by
+  simp only [mapClusterPt_atTop_pow_iff_mem_topologicalClosure_zpowers, inv_mem_iff]
+
+@[to_additive]
+theorem closure_range_zpow_eq_pow (x : G) :
+    closure (range (x ^ · : ℤ → G)) = closure (range (x ^ · : ℕ → G)) := by
+  ext y
+  exact (mapClusterPt_atTop_pow_tfae y x).out 3 2
+
+@[to_additive]
+theorem denseRange_zpow_iff_pow {x : G} :
+    DenseRange (x ^ · : ℤ → G) ↔ DenseRange (x ^ · : ℕ → G) := by
+  simp only [DenseRange, dense_iff_closure_eq, closure_range_zpow_eq_pow]
+
+@[to_additive]
+theorem topologicalClosure_subgroupClosure_toSubmonoid (s : Set G) :
+    (Subgroup.closure s).toSubmonoid.topologicalClosure =
+      (Submonoid.closure s).topologicalClosure := by
+  refine le_antisymm ?_ (closure_mono <| Subgroup.le_closure_toSubmonoid _)
+  refine Submonoid.topologicalClosure_minimal _ ?_ isClosed_closure
+  rw [Subgroup.closure_toSubmonoid, Submonoid.closure_le]
+  refine union_subset (Submonoid.subset_closure.trans subset_closure) fun x hx ↦ ?_
+  refine closure_mono (Submonoid.powers_le.2 (Submonoid.subset_closure <| Set.mem_inv.1 hx)) ?_
+  rw [Submonoid.coe_powers, ← closure_range_zpow_eq_pow, ← Subgroup.coe_zpowers,
+    ← Subgroup.topologicalClosure_coe, SetLike.mem_coe, ← inv_mem_iff]
+  exact subset_closure <| Subgroup.mem_zpowers _
+
+@[to_additive]
+theorem closure_submonoidClosure_eq_closure_subgroupClosure (s : Set G) :
+    closure (Submonoid.closure s : Set G) = closure (Subgroup.closure s) :=
+  congrArg SetLike.coe (topologicalClosure_subgroupClosure_toSubmonoid s).symm
+
+@[to_additive]
+theorem dense_submonoidClosure_iff_subgroupClosure {s : Set G} :
+    Dense (Submonoid.closure s : Set G) ↔ Dense (Subgroup.closure s : Set G) := by
+  simp only [dense_iff_closure_eq, closure_submonoidClosure_eq_closure_subgroupClosure]
diff --git a/Mathlib/Topology/Algebra/GroupCompletion.lean b/Mathlib/Topology/Algebra/GroupCompletion.lean
index 234486fbb1753..cadc887e6ae6e 100644
--- a/Mathlib/Topology/Algebra/GroupCompletion.lean
+++ b/Mathlib/Topology/Algebra/GroupCompletion.lean
@@ -181,8 +181,8 @@ theorem continuous_toCompl : Continuous (toCompl : α → Completion α) :=
 
 variable (α)
 
-theorem denseInducing_toCompl : DenseInducing (toCompl : α → Completion α) :=
-  denseInducing_coe
+theorem isDenseInducing_toCompl : IsDenseInducing (toCompl : α → Completion α) :=
+  isDenseInducing_coe
 
 variable {α}
 
diff --git a/Mathlib/Topology/Algebra/GroupWithZero.lean b/Mathlib/Topology/Algebra/GroupWithZero.lean
index a363dc902e11b..b601051b87dc5 100644
--- a/Mathlib/Topology/Algebra/GroupWithZero.lean
+++ b/Mathlib/Topology/Algebra/GroupWithZero.lean
@@ -142,7 +142,7 @@ lemma nhds_inv₀ (hx : x ≠ 0) : 𝓝 x⁻¹ = (𝓝 x)⁻¹ := by
 
 lemma tendsto_inv_iff₀ {l : Filter α} {f : α → G₀} (hx : x ≠ 0) :
     Tendsto (fun x ↦ (f x)⁻¹) l (𝓝 x⁻¹) ↔ Tendsto f l (𝓝 x) := by
-  simp only [nhds_inv₀ hx, ← Filter.comap_inv, tendsto_comap_iff, (· ∘ ·), inv_inv]
+  simp only [nhds_inv₀ hx, ← Filter.comap_inv, tendsto_comap_iff, Function.comp_def, inv_inv]
 
 end NhdsInv
 
@@ -300,7 +300,7 @@ theorem HasContinuousInv₀.of_nhds_one (h : Tendsto Inv.inv (𝓝 (1 : G₀)) (
     have hx' := inv_ne_zero hx
     rw [ContinuousAt, ← map_mul_left_nhds_one₀ hx, ← nhds_translation_mul_inv₀ hx',
       tendsto_map'_iff, tendsto_comap_iff]
-    simpa only [(· ∘ ·), mul_inv_rev, mul_inv_cancel_right₀ hx']
+    simpa only [Function.comp_def, mul_inv_rev, mul_inv_cancel_right₀ hx']
 
 end map_comap
 
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean b/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
index 9642c81b12ef4..bc17c8afdc322 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
@@ -204,7 +204,7 @@ protected theorem Multipliable.map_iff_of_leftInverse [CommMonoid γ] [Topologic
     Multipliable (g ∘ f) ↔ Multipliable f :=
   ⟨fun h ↦ by
     have := h.map _ hg'
-    rwa [← Function.comp.assoc, hinv.id] at this, fun h ↦ h.map _ hg⟩
+    rwa [← Function.comp_assoc, hinv.id] at this, fun h ↦ h.map _ hg⟩
 
 @[to_additive]
 theorem Multipliable.map_tprod [CommMonoid γ] [TopologicalSpace γ] [T2Space γ] (hf : Multipliable f)
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean b/Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean
index c68372a5f2ce0..b47e95b56f4ad 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean
@@ -146,7 +146,35 @@ end ContinuousMul
 
 section CompleteSpace
 
-variable [CommGroup α] [UniformSpace α] [UniformGroup α] [CompleteSpace α]
+variable [CommGroup α] [UniformSpace α] [UniformGroup α]
+
+@[to_additive]
+theorem HasProd.of_sigma {γ : β → Type*} {f : (Σ b : β, γ b) → α} {g : β → α} {a : α}
+    (hf : ∀ b, HasProd (fun c ↦ f ⟨b, c⟩) (g b)) (hg : HasProd g a)
+    (h : CauchySeq (fun (s : Finset (Σ b : β, γ b)) ↦ ∏ i ∈ s, f i)) :
+    HasProd f a := by
+  classical
+  apply le_nhds_of_cauchy_adhp h
+  simp only [← mapClusterPt_def, mapClusterPt_iff, frequently_atTop, ge_iff_le, le_eq_subset]
+  intro u hu s
+  rcases mem_nhds_iff.1 hu with ⟨v, vu, v_open, hv⟩
+  obtain ⟨t0, st0, ht0⟩ : ∃ t0, ∏ i ∈ t0, g i ∈ v ∧ s.image Sigma.fst ⊆ t0 := by
+    have A : ∀ᶠ t0 in (atTop : Filter (Finset β)), ∏ i ∈ t0, g i ∈ v := hg (v_open.mem_nhds hv)
+    exact (A.and (Ici_mem_atTop _)).exists
+  have L : Tendsto (fun t : Finset (Σb, γ b) ↦ ∏ p ∈ t.filter fun p ↦ p.1 ∈ t0, f p) atTop
+      (𝓝 <| ∏ b ∈ t0, g b) := by
+    simp only [← sigma_preimage_mk, prod_sigma]
+    refine tendsto_finset_prod _ fun b _ ↦ ?_
+    change
+      Tendsto (fun t ↦ (fun t ↦ ∏ s ∈ t, f ⟨b, s⟩) (preimage t (Sigma.mk b) _)) atTop (𝓝 (g b))
+    exact (hf b).comp (tendsto_finset_preimage_atTop_atTop (sigma_mk_injective))
+  have : ∃ t, ∏ p ∈ t.filter (fun p ↦ p.1 ∈ t0), f p ∈ v ∧ s ⊆ t :=
+    ((Tendsto.eventually_mem L (v_open.mem_nhds st0)).and (Ici_mem_atTop _)).exists
+  obtain ⟨t, tv, st⟩ := this
+  refine ⟨t.filter (fun p ↦ p.1 ∈ t0), fun x hx ↦ ?_, vu tv⟩
+  simpa only [mem_filter, st hx, true_and] using ht0 (mem_image_of_mem Sigma.fst hx)
+
+variable [CompleteSpace α]
 
 @[to_additive]
 theorem Multipliable.sigma_factor {γ : β → Type*} {f : (Σb : β, γ b) → α}
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Defs.lean b/Mathlib/Topology/Algebra/InfiniteSum/Defs.lean
index b4e52d99ca80c..92ae820ce40a0 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Defs.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Defs.lean
@@ -3,8 +3,9 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl
 -/
-import Mathlib.Topology.Separation
 import Mathlib.Algebra.BigOperators.Finprod
+import Mathlib.Order.Filter.AtTopBot.BigOperators
+import Mathlib.Topology.Separation
 
 /-!
 # Infinite sum and product over a topological monoid
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Group.lean b/Mathlib/Topology/Algebra/InfiniteSum/Group.lean
index 06be853f4ac9d..9532cd4e53ab9 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Group.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Group.lean
@@ -180,7 +180,7 @@ theorem cauchySeq_finset_iff_prod_vanishing :
       ∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint t s → (∏ b ∈ t, f b) ∈ e := by
   classical
   simp only [CauchySeq, cauchy_map_iff, and_iff_right atTop_neBot, prod_atTop_atTop_eq,
-    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, (· ∘ ·), atTop_neBot, true_and]
+    uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and]
   rw [tendsto_atTop']
   constructor
   · intro h e he
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean b/Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean
index 52b2b087f1a48..078edbe10ffe4 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean
@@ -17,13 +17,13 @@ variable {α β : Type*} [AddCommGroup α] [UniformSpace α] [UniformAddGroup α
 /-- A function `f` has a sum in an uniform additive group `α` if and only if it has that sum in the
 completion of `α`. -/
 theorem hasSum_iff_hasSum_compl (f : β → α) (a : α) :
-    HasSum (toCompl ∘ f) a ↔ HasSum f a := (denseInducing_toCompl α).hasSum_iff f a
+    HasSum (toCompl ∘ f) a ↔ HasSum f a := (isDenseInducing_toCompl α).hasSum_iff f a
 
 /-- A function `f` is summable in a uniform additive group `α` if and only if it is summable in
 `Completion α` and its sum in `Completion α` lies in the range of `toCompl : α →+ Completion α`. -/
 theorem summable_iff_summable_compl_and_tsum_mem (f : β → α) :
     Summable f ↔ Summable (toCompl ∘ f) ∧ ∑' i, toCompl (f i) ∈ Set.range toCompl :=
-  (denseInducing_toCompl α).summable_iff_tsum_comp_mem_range f
+  (isDenseInducing_toCompl α).summable_iff_tsum_comp_mem_range f
 
 /-- A function `f` is summable in a uniform additive group `α` if and only if the net of its partial
 sums is Cauchy and its sum in `Completion α` lies in the range of `toCompl : α →+ Completion α`.
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Module.lean b/Mathlib/Topology/Algebra/InfiniteSum/Module.lean
index 1201d78d799cc..e8ef8f1cd13dc 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Module.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Module.lean
@@ -240,7 +240,7 @@ variable {G : Type*} [Group G] {Γ : Subgroup G}
 @[to_additive "Given a subgroup `Γ` of an additive group `G`, and a function `f : G → M`, we
   automorphize `f` to a function `G ⧸ Γ → M` by summing over `Γ` orbits,
   `g ↦ ∑' (γ : Γ), f(γ • g)`."]
-noncomputable def QuotientGroup.automorphize  (f : G → M) : G ⧸ Γ → M := MulAction.automorphize f
+noncomputable def QuotientGroup.automorphize (f : G → M) : G ⧸ Γ → M := MulAction.automorphize f
 
 /-- Automorphization of a function into an `R`-`Module` distributes, that is, commutes with the
 `R`-scalar multiplication. -/
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean b/Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
index 09d1d25020184..688be889d23d8 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
@@ -74,7 +74,7 @@ theorem even_mul_odd {f : ℕ → M} (he : HasProd (fun k ↦ f (2 * k)) m)
   have := mul_right_injective₀ (two_ne_zero' ℕ)
   replace ho := ((add_left_injective 1).comp this).hasProd_range_iff.2 ho
   refine (this.hasProd_range_iff.2 he).mul_isCompl ?_ ho
-  simpa [(· ∘ ·)] using Nat.isCompl_even_odd
+  simpa [Function.comp_def] using Nat.isCompl_even_odd
 
 end ContinuousMul
 
@@ -324,7 +324,7 @@ lemma HasProd.nat_mul_neg_add_one {f : ℤ → M} (hf : HasProd f m) :
   · rw [prod_union, prod_image Nat.cast_injective.injOn, prod_image this.injOn,
       prod_mul_distrib]
     simp only [disjoint_iff_ne, mem_image, ne_eq, forall_exists_index, and_imp,
-      forall_apply_eq_imp_iff₂, not_false_eq_true, implies_true, forall_const]
+      forall_apply_eq_imp_iff₂, not_false_eq_true, implies_true, forall_const, reduceCtorEq]
 
 @[to_additive Summable.nat_add_neg_add_one]
 lemma Multipliable.nat_mul_neg_add_one {f : ℤ → M} (hf : Multipliable f) :
@@ -358,7 +358,7 @@ lemma HasProd.of_nat_of_neg_add_one {f : ℤ → M}
 
 @[to_additive Summable.of_nat_of_neg_add_one]
 lemma Multipliable.of_nat_of_neg_add_one {f : ℤ → M}
-    (hf₁ : Multipliable fun n : ℕ ↦ f n)  (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
+    (hf₁ : Multipliable fun n : ℕ ↦ f n) (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
     Multipliable f :=
   (hf₁.hasProd.of_nat_of_neg_add_one hf₂.hasProd).multipliable
 
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Order.lean b/Mathlib/Topology/Algebra/InfiniteSum/Order.lean
index 53773dace6081..88fad0d581136 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Order.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Order.lean
@@ -40,7 +40,7 @@ variable [OrderedCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {
 theorem hasProd_le (h : ∀ i, f i ≤ g i) (hf : HasProd f a₁) (hg : HasProd g a₂) : a₁ ≤ a₂ :=
   le_of_tendsto_of_tendsto' hf hg fun _ ↦ prod_le_prod' fun i _ ↦ h i
 
-@[to_additive (attr := mono)]
+@[to_additive]
 theorem hasProd_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f ≤ g) : a₁ ≤ a₂ :=
   hasProd_le h hf hg
 
diff --git a/Mathlib/Topology/Algebra/Module/Alternating/Basic.lean b/Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
index c6b9bc5f30347..1b43517bdbae1 100644
--- a/Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
+++ b/Mathlib/Topology/Algebra/Module/Alternating/Basic.lean
@@ -333,8 +333,8 @@ def _root_.ContinuousLinearEquiv.continuousAlternatingMapComp (e : M ≃L[R] M')
     M [⋀^ι]→L[R] N ≃ M' [⋀^ι]→L[R] N where
   toFun f := f.compContinuousLinearMap ↑e.symm
   invFun f := f.compContinuousLinearMap ↑e
-  left_inv f := by ext; simp [(· ∘ ·)]
-  right_inv f := by ext; simp [(· ∘ ·)]
+  left_inv f := by ext; simp [Function.comp_def]
+  right_inv f := by ext; simp [Function.comp_def]
 
 /-- A continuous linear equivalence of codomains defines an equivalence between continuous
 alternating maps. -/
@@ -447,9 +447,8 @@ end Semiring
 
 section Ring
 
-variable {R M M' N N' ι : Type*} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M]
-  [AddCommGroup M'] [Module R M'] [TopologicalSpace M'] [AddCommGroup N] [Module R N]
-  [TopologicalSpace N] [AddCommGroup N'] [Module R N'] [TopologicalSpace N'] {n : ℕ}
+variable {R M N ι : Type*} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M]
+  [AddCommGroup N] [Module R N] [TopologicalSpace N]
   (f g : M [⋀^ι]→L[R] N)
 
 @[simp]
@@ -489,10 +488,9 @@ end Ring
 
 section CommSemiring
 
-variable {R M M' N N' ι : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
-  [TopologicalSpace M] [AddCommMonoid M'] [Module R M'] [TopologicalSpace M'] [AddCommMonoid N]
-  [Module R N] [TopologicalSpace N] [AddCommMonoid N'] [Module R N'] [TopologicalSpace N'] {n : ℕ}
-  (f g : M [⋀^ι]→L[R] N)
+variable {R M N ι : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
+  [TopologicalSpace M] [AddCommMonoid N] [Module R N] [TopologicalSpace N]
+  (f : M [⋀^ι]→L[R] N)
 
 theorem map_piecewise_smul [DecidableEq ι] (c : ι → R) (m : ι → M) (s : Finset ι) :
     f (s.piecewise (fun i => c i • m i) m) = (∏ i ∈ s, c i) • f m :=
@@ -561,7 +559,7 @@ end Module
 
 section SMulRight
 
-variable {R A M N ι : Type*} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M]
+variable {R M N ι : Type*} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M]
   [Module R N] [TopologicalSpace R] [TopologicalSpace M] [TopologicalSpace N] [ContinuousSMul R N]
   (f : M [⋀^ι]→L[R] R) (z : N)
 
@@ -603,7 +601,7 @@ namespace ContinuousMultilinearMap
 
 variable {R M N ι : Type*} [Semiring R] [AddCommMonoid M] [Module R M] [TopologicalSpace M]
   [AddCommGroup N] [Module R N] [TopologicalSpace N] [TopologicalAddGroup N] [Fintype ι]
-  [DecidableEq ι] (f g : ContinuousMultilinearMap R (fun _ : ι => M) N)
+  [DecidableEq ι] (f : ContinuousMultilinearMap R (fun _ : ι => M) N)
 
 /-- Alternatization of a continuous multilinear map. -/
 @[simps (config := .lemmasOnly) apply_toContinuousMultilinearMap]
@@ -618,12 +616,12 @@ def alternatization : ContinuousMultilinearMap R (fun _ : ι => M) N →+ M [⋀
 
 theorem alternatization_apply_apply (v : ι → M) :
     alternatization f v = ∑ σ : Equiv.Perm ι, Equiv.Perm.sign σ • f (v ∘ σ) := by
-  simp [alternatization, (· ∘ ·)]
+  simp [alternatization, Function.comp_def]
 
 @[simp]
 theorem alternatization_apply_toAlternatingMap :
     (alternatization f).toAlternatingMap = MultilinearMap.alternatization f.1 := by
   ext v
-  simp [alternatization_apply_apply, MultilinearMap.alternatization_apply, (· ∘ ·)]
+  simp [alternatization_apply_apply, MultilinearMap.alternatization_apply, Function.comp_def]
 
 end ContinuousMultilinearMap
diff --git a/Mathlib/Topology/Algebra/Module/Alternating/Topology.lean b/Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
new file mode 100644
index 0000000000000..32efbbd5a39c7
--- /dev/null
+++ b/Mathlib/Topology/Algebra/Module/Alternating/Topology.lean
@@ -0,0 +1,230 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Topology.Algebra.Module.Multilinear.Topology
+import Mathlib.Topology.Algebra.Module.Alternating.Basic
+
+/-!
+# Topology on continuous alternating maps
+
+In this file we define `UniformSpace` and `TopologicalSpace` structures
+on the space of continuous alternating maps between topological vector spaces.
+
+The structures are induced by those on `ContinuousMultilinearMap`s,
+and most of the lemmas follow from the corresponding lemmas about `ContinuousMultilinearMap`s.
+-/
+
+open Bornology Function Set
+open scoped Topology UniformConvergence Filter
+
+namespace ContinuousAlternatingMap
+
+variable {𝕜 E F ι : Type*} [NormedField 𝕜]
+  [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [AddCommGroup F] [Module 𝕜 F]
+
+section IsClosedRange
+
+variable [TopologicalSpace F] [TopologicalAddGroup F]
+
+instance instTopologicalSpace : TopologicalSpace (E [⋀^ι]→L[𝕜] F) :=
+  .induced toContinuousMultilinearMap inferInstance
+
+lemma isClosed_range_toContinuousMultilinearMap [ContinuousSMul 𝕜 E] [T2Space F] :
+    IsClosed (Set.range (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) →
+      ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) F)) := by
+  simp only [range_toContinuousMultilinearMap, setOf_forall]
+  repeat refine isClosed_iInter fun _ ↦ ?_
+  exact isClosed_singleton.preimage (ContinuousMultilinearMap.continuous_eval_const _)
+
+end IsClosedRange
+
+section UniformAddGroup
+
+variable [UniformSpace F] [UniformAddGroup F]
+
+instance instUniformSpace : UniformSpace (E [⋀^ι]→L[𝕜] F) :=
+  .comap toContinuousMultilinearMap inferInstance
+
+lemma isUniformEmbedding_toContinuousMultilinearMap :
+    IsUniformEmbedding (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) → _) where
+  inj := toContinuousMultilinearMap_injective
+  comap_uniformity := rfl
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toContinuousMultilinearMap := isUniformEmbedding_toContinuousMultilinearMap
+
+lemma uniformContinuous_toContinuousMultilinearMap :
+    UniformContinuous (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F) → _) :=
+  isUniformEmbedding_toContinuousMultilinearMap.uniformContinuous
+
+theorem uniformContinuous_coe_fun [ContinuousSMul 𝕜 E] :
+    UniformContinuous (DFunLike.coe : (E [⋀^ι]→L[𝕜] F) → (ι → E) → F) :=
+  ContinuousMultilinearMap.uniformContinuous_coe_fun.comp
+    uniformContinuous_toContinuousMultilinearMap
+
+theorem uniformContinuous_eval_const [ContinuousSMul 𝕜 E] (x : ι → E) :
+    UniformContinuous fun f : E [⋀^ι]→L[𝕜] F ↦ f x :=
+  uniformContinuous_pi.1 uniformContinuous_coe_fun x
+
+instance instUniformAddGroup : UniformAddGroup (E [⋀^ι]→L[𝕜] F) :=
+  isUniformEmbedding_toContinuousMultilinearMap.uniformAddGroup
+    (toContinuousMultilinearMapLinear (R := ℕ))
+
+instance instUniformContinuousConstSMul {M : Type*}
+    [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
+    UniformContinuousConstSMul M (E [⋀^ι]→L[𝕜] F) :=
+  isUniformEmbedding_toContinuousMultilinearMap.uniformContinuousConstSMul fun _ _ ↦ rfl
+
+section CompleteSpace
+
+variable [ContinuousSMul 𝕜 E] [ContinuousConstSMul 𝕜 F] [CompleteSpace F] [T2Space F]
+
+open UniformOnFun in
+theorem completeSpace (h : RestrictGenTopology {s : Set (ι → E) | IsVonNBounded 𝕜 s}) :
+    CompleteSpace (E [⋀^ι]→L[𝕜] F) := by
+  have := ContinuousMultilinearMap.completeSpace (F := F) h
+  rw [completeSpace_iff_isComplete_range
+    isUniformEmbedding_toContinuousMultilinearMap.isUniformInducing]
+  apply isClosed_range_toContinuousMultilinearMap.isComplete
+
+instance instCompleteSpace [TopologicalAddGroup E] [SequentialSpace (ι → E)] :
+    CompleteSpace (E [⋀^ι]→L[𝕜] F) :=
+  completeSpace <| .of_seq fun _u x hux ↦ (hux.isVonNBounded_range 𝕜).insert x
+
+end CompleteSpace
+
+section RestrictScalars
+
+variable (𝕜' : Type*) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜]
+  [Module 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F] [ContinuousSMul 𝕜 E]
+
+theorem isUniformEmbedding_restrictScalars :
+    IsUniformEmbedding (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) := by
+  rw [← isUniformEmbedding_toContinuousMultilinearMap.of_comp_iff]
+  exact (ContinuousMultilinearMap.isUniformEmbedding_restrictScalars 𝕜').comp
+    isUniformEmbedding_toContinuousMultilinearMap
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_restrictScalars := isUniformEmbedding_restrictScalars
+
+theorem uniformContinuous_restrictScalars :
+    UniformContinuous (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
+  (isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
+
+end RestrictScalars
+
+end UniformAddGroup
+
+variable [TopologicalSpace F] [TopologicalAddGroup F]
+
+lemma embedding_toContinuousMultilinearMap :
+    Embedding (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F → _)) :=
+  letI := TopologicalAddGroup.toUniformSpace F
+  haveI := comm_topologicalAddGroup_is_uniform (G := F)
+  isUniformEmbedding_toContinuousMultilinearMap.embedding
+
+@[continuity, fun_prop]
+lemma continuous_toContinuousMultilinearMap :
+    Continuous (toContinuousMultilinearMap : (E [⋀^ι]→L[𝕜] F → _)) :=
+  embedding_toContinuousMultilinearMap.continuous
+
+instance instContinuousConstSMul
+    {M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
+    ContinuousConstSMul M (E [⋀^ι]→L[𝕜] F) :=
+  embedding_toContinuousMultilinearMap.continuousConstSMul id rfl
+
+instance instContinuousSMul [ContinuousSMul 𝕜 F] : ContinuousSMul 𝕜 (E [⋀^ι]→L[𝕜] F) :=
+  embedding_toContinuousMultilinearMap.continuousSMul continuous_id rfl
+
+theorem hasBasis_nhds_zero_of_basis {ι' : Type*} {p : ι' → Prop} {b : ι' → Set F}
+    (h : (𝓝 (0 : F)).HasBasis p b) :
+    (𝓝 (0 : E [⋀^ι]→L[𝕜] F)).HasBasis
+      (fun Si : Set (ι → E) × ι' => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
+      fun Si => { f | MapsTo f Si.1 (b Si.2) } := by
+  rw [nhds_induced]
+  exact (ContinuousMultilinearMap.hasBasis_nhds_zero_of_basis h).comap _
+
+theorem hasBasis_nhds_zero :
+    (𝓝 (0 : E [⋀^ι]→L[𝕜] F)).HasBasis
+      (fun SV : Set (ι → E) × Set F => IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0)
+      fun SV => { f | MapsTo f SV.1 SV.2 } :=
+  hasBasis_nhds_zero_of_basis (Filter.basis_sets _)
+
+variable [ContinuousSMul 𝕜 E]
+
+lemma closedEmbedding_toContinuousMultilinearMap [T2Space F] :
+    ClosedEmbedding (toContinuousMultilinearMap :
+      (E [⋀^ι]→L[𝕜] F) → ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) F) :=
+  ⟨embedding_toContinuousMultilinearMap, isClosed_range_toContinuousMultilinearMap⟩
+
+@[continuity, fun_prop]
+theorem continuous_eval_const (x : ι → E) :
+    Continuous fun p : E [⋀^ι]→L[𝕜] F ↦ p x :=
+  (ContinuousMultilinearMap.continuous_eval_const x).comp continuous_toContinuousMultilinearMap
+
+theorem continuous_coe_fun :
+    Continuous (DFunLike.coe : E [⋀^ι]→L[𝕜] F → (ι → E) → F) :=
+  continuous_pi continuous_eval_const
+
+instance instT2Space [T2Space F] : T2Space (E [⋀^ι]→L[𝕜] F) :=
+  .of_injective_continuous DFunLike.coe_injective continuous_coe_fun
+
+instance instT3Space [T2Space F] : T2Space (E [⋀^ι]→L[𝕜] F) :=
+  letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
+  haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
+  inferInstance
+
+section RestrictScalars
+
+variable {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜]
+  [Module 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F]
+
+theorem embedding_restrictScalars :
+    Embedding (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
+  letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
+  haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
+  (isUniformEmbedding_restrictScalars _).embedding
+
+@[continuity, fun_prop]
+theorem continuous_restrictScalars :
+    Continuous (restrictScalars 𝕜' : E [⋀^ι]→L[𝕜] F → E [⋀^ι]→L[𝕜'] F) :=
+  embedding_restrictScalars.continuous
+
+variable (𝕜') in
+/-- `ContinuousMultilinearMap.restrictScalars` as a `ContinuousLinearMap`. -/
+@[simps (config := .asFn) apply]
+def restrictScalarsCLM [ContinuousConstSMul 𝕜' F] :
+    E [⋀^ι]→L[𝕜] F →L[𝕜'] E [⋀^ι]→L[𝕜'] F where
+  toFun := restrictScalars 𝕜'
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+end RestrictScalars
+
+variable (𝕜 E F)
+
+/-- The application of a multilinear map as a `ContinuousLinearMap`. -/
+def apply [ContinuousConstSMul 𝕜 F] (m : ι → E) : E [⋀^ι]→L[𝕜] F →L[𝕜] F where
+  toFun c := c m
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+  cont := continuous_eval_const m
+
+variable {𝕜 E F}
+
+@[simp]
+lemma apply_apply [ContinuousConstSMul 𝕜 F] {m : ι → E} {c : E [⋀^ι]→L[𝕜] F} :
+    apply 𝕜 E F m c = c m := rfl
+
+theorem hasSum_eval {α : Type*} {p : α → E [⋀^ι]→L[𝕜] F}
+    {q : E [⋀^ι]→L[𝕜] F} (h : HasSum p q) (m : ι → E) :
+    HasSum (fun a => p a m) (q m) :=
+  h.map (applyAddHom m) (continuous_eval_const m)
+
+theorem tsum_eval [T2Space F] {α : Type*} {p : α → E [⋀^ι]→L[𝕜] F} (hp : Summable p)
+    (m : ι → E) : (∑' a, p a) m = ∑' a, p a m :=
+  (hasSum_eval hp.hasSum m).tsum_eq.symm
+
+end ContinuousAlternatingMap
diff --git a/Mathlib/Topology/Algebra/Module/Basic.lean b/Mathlib/Topology/Algebra/Module/Basic.lean
index b8f56f0dc9762..a1479380641d5 100644
--- a/Mathlib/Topology/Algebra/Module/Basic.lean
+++ b/Mathlib/Topology/Algebra/Module/Basic.lean
@@ -7,7 +7,6 @@ Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Fréd
 import Mathlib.Topology.Algebra.Ring.Basic
 import Mathlib.Topology.Algebra.MulAction
 import Mathlib.Topology.Algebra.UniformGroup
-import Mathlib.Topology.ContinuousFunction.Basic
 import Mathlib.Topology.UniformSpace.UniformEmbedding
 import Mathlib.Algebra.Algebra.Defs
 import Mathlib.LinearAlgebra.Projection
@@ -252,7 +251,7 @@ class ContinuousSemilinearMapClass (F : Type*) {R S : outParam Type*} [Semiring
 
 /-- `ContinuousLinearMapClass F R M M₂` asserts `F` is a type of bundled continuous
 `R`-linear maps `M → M₂`.  This is an abbreviation for
-`ContinuousSemilinearMapClass F (RingHom.id R) M M₂`.  -/
+`ContinuousSemilinearMapClass F (RingHom.id R) M M₂`. -/
 abbrev ContinuousLinearMapClass (F : Type*) (R : outParam Type*) [Semiring R]
     (M : outParam Type*) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam Type*)
     [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] [FunLike F M M₂] :=
@@ -727,6 +726,18 @@ theorem add_comp [ContinuousAdd M₃] (g₁ g₂ : M₂ →SL[σ₂₃] M₃) (f
   ext
   simp
 
+theorem comp_finset_sum {ι : Type*} {s : Finset ι}
+    [ContinuousAdd M₂] [ContinuousAdd M₃] (g : M₂ →SL[σ₂₃] M₃)
+    (f : ι → M₁ →SL[σ₁₂] M₂) : g.comp (∑ i ∈ s, f i) = ∑ i ∈ s, g.comp (f i) := by
+  ext
+  simp
+
+theorem finset_sum_comp {ι : Type*} {s : Finset ι}
+    [ContinuousAdd M₃] (g : ι → M₂ →SL[σ₂₃] M₃)
+    (f : M₁ →SL[σ₁₂] M₂) : (∑ i ∈ s, g i).comp f = ∑ i ∈ s, (g i).comp f := by
+  ext
+  simp only [coe_comp', coe_sum', Function.comp_apply, Finset.sum_apply]
+
 theorem comp_assoc {R₄ : Type*} [Semiring R₄] [Module R₄ M₄] {σ₁₄ : R₁ →+* R₄} {σ₂₄ : R₂ →+* R₄}
     {σ₃₄ : R₃ →+* R₄} [RingHomCompTriple σ₁₃ σ₃₄ σ₁₄] [RingHomCompTriple σ₂₃ σ₃₄ σ₂₄]
     [RingHomCompTriple σ₁₂ σ₂₄ σ₁₄] (h : M₃ →SL[σ₃₄] M₄) (g : M₂ →SL[σ₂₃] M₃) (f : M₁ →SL[σ₁₂] M₂) :
@@ -1172,12 +1183,10 @@ def iInfKerProjEquiv {I J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjo
     Submodule R (∀ i, φ i)) ≃L[R] ∀ i : I, φ i where
   toLinearEquiv := LinearMap.iInfKerProjEquiv R φ hd hu
   continuous_toFun :=
-    continuous_pi fun i => by
-      have :=
+    continuous_pi fun i =>
+      Continuous.comp (continuous_apply (π := φ) i) <|
         @continuous_subtype_val _ _ fun x =>
           x ∈ (⨅ i ∈ J, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i))
-      have := Continuous.comp (continuous_apply (π := φ) i) this
-      exact this
   continuous_invFun :=
     Continuous.subtype_mk
       (continuous_pi fun i => by
@@ -1760,7 +1769,7 @@ theorem coe_refl : ↑(ContinuousLinearEquiv.refl R₁ M₁) = ContinuousLinearM
 theorem coe_refl' : ⇑(ContinuousLinearEquiv.refl R₁ M₁) = id :=
   rfl
 
-/-- The inverse of a continuous linear equivalence as a continuous linear equivalence-/
+/-- The inverse of a continuous linear equivalence as a continuous linear equivalence -/
 @[symm]
 protected def symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ :=
   { e.toLinearEquiv.symm with
@@ -1807,8 +1816,8 @@ theorem trans_toLinearEquiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃S
 def prod [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) :
     (M₁ × M₃) ≃L[R₁] M₂ × M₄ :=
   { e.toLinearEquiv.prod e'.toLinearEquiv with
-    continuous_toFun := e.continuous_toFun.prod_map e'.continuous_toFun
-    continuous_invFun := e.continuous_invFun.prod_map e'.continuous_invFun }
+    continuous_toFun := e.continuous_toFun.prodMap e'.continuous_toFun
+    continuous_invFun := e.continuous_invFun.prodMap e'.continuous_invFun }
 
 @[simp, norm_cast]
 theorem prod_apply [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂)
@@ -1932,22 +1941,27 @@ protected theorem preimage_symm_preimage (e : M₁ ≃SL[σ₁₂] M₂) (s : Se
     e ⁻¹' (e.symm ⁻¹' s) = s :=
   e.symm.symm_preimage_preimage s
 
-protected theorem uniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
+lemma isUniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁] [UniformSpace E₂]
     [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂] [UniformAddGroup E₁]
-    [UniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂) : UniformEmbedding e :=
-  e.toLinearEquiv.toEquiv.uniformEmbedding e.toContinuousLinearMap.uniformContinuous
+    [UniformAddGroup E₂] (e : E₁ ≃SL[σ₁₂] E₂) : IsUniformEmbedding e :=
+  e.toLinearEquiv.toEquiv.isUniformEmbedding e.toContinuousLinearMap.uniformContinuous
     e.symm.toContinuousLinearMap.uniformContinuous
 
-protected theorem _root_.LinearEquiv.uniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁]
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
+
+protected theorem _root_.LinearEquiv.isUniformEmbedding {E₁ E₂ : Type*} [UniformSpace E₁]
     [UniformSpace E₂] [AddCommGroup E₁] [AddCommGroup E₂] [Module R₁ E₁] [Module R₂ E₂]
     [UniformAddGroup E₁] [UniformAddGroup E₂] (e : E₁ ≃ₛₗ[σ₁₂] E₂)
-    (h₁ : Continuous e) (h₂ : Continuous e.symm) : UniformEmbedding e :=
-  ContinuousLinearEquiv.uniformEmbedding
+    (h₁ : Continuous e) (h₂ : Continuous e.symm) : IsUniformEmbedding e :=
+  ContinuousLinearEquiv.isUniformEmbedding
     ({ e with
         continuous_toFun := h₁
         continuous_invFun := h₂ } :
       E₁ ≃SL[σ₁₂] E₂)
 
+@[deprecated (since := "2024-10-01")]
+alias _root_.LinearEquiv.uniformEmbedding := _root_.LinearEquiv.isUniformEmbedding
+
 /-- Create a `ContinuousLinearEquiv` from two `ContinuousLinearMap`s that are
 inverse of each other. -/
 def equivOfInverse (f₁ : M₁ →SL[σ₁₂] M₂) (f₂ : M₂ →SL[σ₂₁] M₁) (h₁ : Function.LeftInverse f₂ f₁)
@@ -2365,21 +2379,20 @@ variable {R M : Type*} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace
 instance _root_.QuotientModule.Quotient.topologicalSpace : TopologicalSpace (M ⧸ S) :=
   inferInstanceAs (TopologicalSpace (Quotient S.quotientRel))
 
-theorem isOpenMap_mkQ [TopologicalAddGroup M] : IsOpenMap S.mkQ :=
-  QuotientAddGroup.isOpenMap_coe S.toAddSubgroup
+theorem isOpenMap_mkQ [ContinuousAdd M] : IsOpenMap S.mkQ :=
+  QuotientAddGroup.isOpenMap_coe
+
+theorem isOpenQuotientMap_mkQ [ContinuousAdd M] : IsOpenQuotientMap S.mkQ :=
+  QuotientAddGroup.isOpenQuotientMap_mk
 
 instance topologicalAddGroup_quotient [TopologicalAddGroup M] : TopologicalAddGroup (M ⧸ S) :=
-  _root_.topologicalAddGroup_quotient S.toAddSubgroup
+  inferInstanceAs <| TopologicalAddGroup (M ⧸ S.toAddSubgroup)
 
 instance continuousSMul_quotient [TopologicalSpace R] [TopologicalAddGroup M] [ContinuousSMul R M] :
-    ContinuousSMul R (M ⧸ S) := by
-  constructor
-  have quot : QuotientMap fun au : R × M => (au.1, S.mkQ au.2) :=
-    IsOpenMap.to_quotientMap (IsOpenMap.id.prod S.isOpenMap_mkQ)
-      (continuous_id.prod_map continuous_quot_mk)
-      (Function.surjective_id.prodMap <| surjective_quot_mk _)
-  rw [quot.continuous_iff]
-  exact continuous_quot_mk.comp continuous_smul
+    ContinuousSMul R (M ⧸ S) where
+  continuous_smul := by
+    rw [← (IsOpenQuotientMap.id.prodMap S.isOpenQuotientMap_mkQ).continuous_comp_iff]
+    exact continuous_quot_mk.comp continuous_smul
 
 instance t3_quotient_of_isClosed [TopologicalAddGroup M] [IsClosed (S : Set M)] :
     T3Space (M ⧸ S) :=
@@ -2389,3 +2402,5 @@ instance t3_quotient_of_isClosed [TopologicalAddGroup M] [IsClosed (S : Set M)]
 end Submodule
 
 end Quotient
+
+set_option linter.style.longFile 2500
diff --git a/Mathlib/Topology/Algebra/Module/Cardinality.lean b/Mathlib/Topology/Algebra/Module/Cardinality.lean
index c45fb8205558f..b5c7c62b830cd 100644
--- a/Mathlib/Topology/Algebra/Module/Cardinality.lean
+++ b/Mathlib/Topology/Algebra/Module/Cardinality.lean
@@ -77,7 +77,7 @@ lemma cardinal_eq_of_mem_nhds_zero
         simp_rw [← inv_pow]
         apply tendsto_pow_atTop_nhds_zero_of_norm_lt_one
         rw [norm_inv]
-        exact inv_lt_one hc
+        exact inv_lt_one_of_one_lt₀ hc
       exact Tendsto.smul_const this x
     rw [zero_smul] at this
     filter_upwards [this hs] with n (hn : (c ^ n)⁻¹ • x ∈ s)
diff --git a/Mathlib/Topology/Algebra/Module/CharacterSpace.lean b/Mathlib/Topology/Algebra/Module/CharacterSpace.lean
index 98892800dddf3..39d280cac3ba7 100644
--- a/Mathlib/Topology/Algebra/Module/CharacterSpace.lean
+++ b/Mathlib/Topology/Algebra/Module/CharacterSpace.lean
@@ -5,7 +5,7 @@ Authors: Frédéric Dupuis
 -/
 import Mathlib.Topology.Algebra.Module.WeakDual
 import Mathlib.Algebra.Algebra.Spectrum
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Data.Set.Lattice
 
 /-!
@@ -212,7 +212,7 @@ variable (𝕜 A) [CommRing 𝕜] [NoZeroDivisors 𝕜] [TopologicalSpace 𝕜]
 
 /-- The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
 `A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `characterSpace 𝕜 A`.
-The character space itself consists of all algebra homomorphisms from `A` to `𝕜`.  -/
+The character space itself consists of all algebra homomorphisms from `A` to `𝕜`. -/
 @[simps]
 def gelfandTransform : A →ₐ[𝕜] C(characterSpace 𝕜 A, 𝕜) where
   toFun a :=
diff --git a/Mathlib/Topology/Algebra/Module/FiniteDimension.lean b/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
index 8b1c1ca19d66a..2cf817161d306 100644
--- a/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
+++ b/Mathlib/Topology/Algebra/Module/FiniteDimension.lean
@@ -47,7 +47,7 @@ universe u v w x
 
 noncomputable section
 
-open Set FiniteDimensional TopologicalSpace Filter
+open Filter Module Set TopologicalSpace
 
 section Field
 
@@ -106,7 +106,7 @@ theorem unique_topology_of_t2 {t : TopologicalSpace 𝕜} (h₁ : @TopologicalAd
       -- For that, we use that `𝓑` is balanced : since `‖ξ₀‖ < ε < ‖ξ‖`, we have `‖ξ₀ / ξ‖ ≤ 1`,
       -- hence `ξ₀ = (ξ₀ / ξ) • ξ ∈ 𝓑` because `ξ ∈ 𝓑`.
       refine (balancedCore_balanced _).smul_mem ?_ hξ
-      rw [norm_mul, norm_inv, mul_inv_le_iff (norm_pos_iff.mpr hξ0), mul_one]
+      rw [norm_mul, norm_inv, mul_inv_le_iff₀ (norm_pos_iff.mpr hξ0), one_mul]
       exact (hξ₀ε.trans h).le
   · -- Finally, to show `𝓣₀ ≤ 𝓣`, we simply argue that `id = (fun x ↦ x • 1)` is continuous from
     -- `(𝕜, 𝓣₀)` to `(𝕜, 𝓣)` because `(•) : (𝕜, 𝓣₀) × (𝕜, 𝓣) → (𝕜, 𝓣)` is continuous.
@@ -197,22 +197,22 @@ private theorem continuous_equivFun_basis_aux [T2Space E] {ι : Type v} [Fintype
   induction' hn : Fintype.card ι with n IH generalizing ι E
   · rw [Fintype.card_eq_zero_iff] at hn
     exact continuous_of_const fun x y => funext hn.elim
-  · haveI : FiniteDimensional 𝕜 E := of_fintype_basis ξ
+  · haveI : FiniteDimensional 𝕜 E := .of_fintype_basis ξ
     -- first step: thanks to the induction hypothesis, any n-dimensional subspace is equivalent
     -- to a standard space of dimension n, hence it is complete and therefore closed.
     have H₁ : ∀ s : Submodule 𝕜 E, finrank 𝕜 s = n → IsClosed (s : Set E) := by
       intro s s_dim
       letI : UniformAddGroup s := s.toAddSubgroup.uniformAddGroup
       let b := Basis.ofVectorSpace 𝕜 s
-      have U : UniformEmbedding b.equivFun.symm.toEquiv := by
+      have U : IsUniformEmbedding b.equivFun.symm.toEquiv := by
         have : Fintype.card (Basis.ofVectorSpaceIndex 𝕜 s) = n := by
           rw [← s_dim]
           exact (finrank_eq_card_basis b).symm
         have : Continuous b.equivFun := IH b this
         exact
-          b.equivFun.symm.uniformEmbedding b.equivFun.symm.toLinearMap.continuous_on_pi this
+          b.equivFun.symm.isUniformEmbedding b.equivFun.symm.toLinearMap.continuous_on_pi this
       have : IsComplete (s : Set E) :=
-        completeSpace_coe_iff_isComplete.1 ((completeSpace_congr U).1 (by infer_instance))
+        completeSpace_coe_iff_isComplete.1 ((completeSpace_congr U).1 inferInstance)
       exact this.isClosed
     -- second step: any linear form is continuous, as its kernel is closed by the first step
     have H₂ : ∀ f : E →ₗ[𝕜] 𝕜, Continuous f := by
@@ -264,7 +264,7 @@ continuous (see `LinearMap.continuous_of_finiteDimensional`), which in turn impl
 norms are equivalent in finite dimensions. -/
 theorem continuous_equivFun_basis [T2Space E] {ι : Type*} [Finite ι] (ξ : Basis ι 𝕜 E) :
     Continuous ξ.equivFun :=
-  haveI : FiniteDimensional 𝕜 E := of_fintype_basis ξ
+  haveI : FiniteDimensional 𝕜 E := .of_fintype_basis ξ
   ξ.equivFun.toLinearMap.continuous_of_finiteDimensional
 
 namespace LinearMap
@@ -490,8 +490,8 @@ variable (𝕜 E : Type*) [NontriviallyNormedField 𝕜]
 include 𝕜 in
 theorem FiniteDimensional.complete [FiniteDimensional 𝕜 E] : CompleteSpace E := by
   set e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun 𝕜 _ _ (finrank 𝕜 E)).symm
-  have : UniformEmbedding e.toLinearEquiv.toEquiv.symm := e.symm.uniformEmbedding
-  exact (completeSpace_congr this).1 (by infer_instance)
+  have : IsUniformEmbedding e.toEquiv.symm := e.symm.isUniformEmbedding
+  exact (completeSpace_congr this).1 inferInstance
 
 variable {𝕜 E}
 
diff --git a/Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean b/Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
index c4849ddb17994..ad413d2ba21ae 100644
--- a/Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
+++ b/Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean
@@ -202,7 +202,7 @@ end ContinuousAdd
 /-- If `f` is a continuous multilinear map, then `f.toContinuousLinearMap m i` is the continuous
 linear map obtained by fixing all coordinates but `i` equal to those of `m`, and varying the
 `i`-th coordinate. -/
-def toContinuousLinearMap [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) : M₁ i →L[R] M₂ :=
+@[simps!] def toContinuousLinearMap [DecidableEq ι] (m : ∀ i, M₁ i) (i : ι) : M₁ i →L[R] M₂ :=
   { f.toMultilinearMap.toLinearMap m i with
     cont := f.cont.comp (continuous_const.update i continuous_id) }
 
diff --git a/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean b/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
index 6b8d509bb8ec2..85af2419a28c0 100644
--- a/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
+++ b/Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
@@ -67,10 +67,7 @@ theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBoun
     let ⟨i₀⟩ := ‹Nonempty ι›
     set y := I.piecewise (fun i ↦ c i • x i) x
     calc
-      a • f x = f (update y i₀ ((a / ∏ i ∈ I, c i) • y i₀)) := by
-        rw [f.map_smul, update_eq_self, f.map_piecewise_smul, div_eq_mul_inv, mul_smul,
-          inv_smul_smul₀ hc₀']
-      _ ∈ V := hft fun i hi ↦ by
+      f (update y i₀ ((a / ∏ i ∈ I, c i) • y i₀)) ∈ V := hft fun i hi => by
         rcases eq_or_ne i i₀ with rfl | hne
         · simp_rw [update_same, y, I.piecewise_eq_of_mem _ _ hi, smul_smul]
           refine hc _ _ ?_ _ hx
@@ -81,6 +78,9 @@ theorem image_multilinear' [Nonempty ι] {s : Set (∀ i, E i)} (hs : IsVonNBoun
             _ = ‖c i‖ := one_mul _
         · simp_rw [update_noteq hne, y, I.piecewise_eq_of_mem _ _ hi]
           exact hc _ _ le_rfl _ hx
+      _ = a • f x := by
+        rw [f.map_smul, update_eq_self, f.map_piecewise_smul, div_eq_mul_inv, mul_smul,
+          inv_smul_smul₀ hc₀']
 
 /-- The image of a von Neumann bounded set under a continuous multilinear map
 is von Neumann bounded.
diff --git a/Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean b/Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
index fd88cf46b8e21..baf4b78201f9a 100644
--- a/Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
+++ b/Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean
@@ -65,18 +65,21 @@ section UniformAddGroup
 
 variable [UniformSpace F] [UniformAddGroup F]
 
-lemma uniformEmbedding_toUniformOnFun :
-    UniformEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) where
+lemma isUniformEmbedding_toUniformOnFun :
+    IsUniformEmbedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) where
   inj := DFunLike.coe_injective
   comap_uniformity := rfl
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toUniformOnFun := isUniformEmbedding_toUniformOnFun
+
 lemma embedding_toUniformOnFun : Embedding (toUniformOnFun : ContinuousMultilinearMap 𝕜 E F → _) :=
-  uniformEmbedding_toUniformOnFun.embedding
+  isUniformEmbedding_toUniformOnFun.embedding
 
 theorem uniformContinuous_coe_fun [∀ i, ContinuousSMul 𝕜 (E i)] :
     UniformContinuous (DFunLike.coe : ContinuousMultilinearMap 𝕜 E F → (Π i, E i) → F) :=
   (UniformOnFun.uniformContinuous_toFun isVonNBounded_covers).comp
-    uniformEmbedding_toUniformOnFun.uniformContinuous
+    isUniformEmbedding_toUniformOnFun.uniformContinuous
 
 theorem uniformContinuous_eval_const [∀ i, ContinuousSMul 𝕜 (E i)] (x : Π i, E i) :
     UniformContinuous fun f : ContinuousMultilinearMap 𝕜 E F ↦ f x :=
@@ -85,13 +88,15 @@ theorem uniformContinuous_eval_const [∀ i, ContinuousSMul 𝕜 (E i)] (x : Π
 instance instUniformAddGroup : UniformAddGroup (ContinuousMultilinearMap 𝕜 E F) :=
   let φ : ContinuousMultilinearMap 𝕜 E F →+ (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F :=
     { toFun := toUniformOnFun, map_add' := fun _ _ ↦ rfl, map_zero' := rfl }
-  uniformEmbedding_toUniformOnFun.uniformAddGroup φ
+  isUniformEmbedding_toUniformOnFun.uniformAddGroup φ
 
 instance instUniformContinuousConstSMul {M : Type*}
     [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜 M F] [ContinuousConstSMul M F] :
     UniformContinuousConstSMul M (ContinuousMultilinearMap 𝕜 E F) :=
   haveI := uniformContinuousConstSMul_of_continuousConstSMul M F
-  uniformEmbedding_toUniformOnFun.uniformContinuousConstSMul fun _ _ ↦ rfl
+  isUniformEmbedding_toUniformOnFun.uniformContinuousConstSMul fun _ _ ↦ rfl
+
+section CompleteSpace
 
 variable [∀ i, ContinuousSMul 𝕜 (E i)] [ContinuousConstSMul 𝕜 F] [CompleteSpace F] [T2Space F]
 
@@ -102,7 +107,7 @@ theorem completeSpace (h : RestrictGenTopology {s : Set (Π i, E i) | IsVonNBoun
   have H : ∀ {m : Π i, E i},
       Continuous fun f : (Π i, E i) →ᵤ[{s | IsVonNBounded 𝕜 s}] F ↦ toFun _ f m :=
     (uniformContinuous_eval (isVonNBounded_covers) _).continuous
-  rw [completeSpace_iff_isComplete_range uniformEmbedding_toUniformOnFun.toUniformInducing,
+  rw [completeSpace_iff_isComplete_range isUniformEmbedding_toUniformOnFun.isUniformInducing,
     range_toUniformOnFun]
   simp only [setOf_and, setOf_forall]
   apply_rules [IsClosed.isComplete, IsClosed.inter]
@@ -116,6 +121,33 @@ instance instCompleteSpace [∀ i, TopologicalAddGroup (E i)] [SequentialSpace (
     CompleteSpace (ContinuousMultilinearMap 𝕜 E F) :=
   completeSpace <| .of_seq fun _u x hux ↦ (hux.isVonNBounded_range 𝕜).insert x
 
+end CompleteSpace
+
+section RestrictScalars
+
+variable (𝕜' : Type*) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜]
+  [∀ i, Module 𝕜' (E i)] [∀ i, IsScalarTower 𝕜' 𝕜 (E i)] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F]
+  [∀ i, ContinuousSMul 𝕜 (E i)]
+
+theorem isUniformEmbedding_restrictScalars :
+    IsUniformEmbedding
+      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) := by
+  letI : NontriviallyNormedField 𝕜 :=
+    ⟨let ⟨x, hx⟩ := @NontriviallyNormedField.non_trivial 𝕜' _; ⟨algebraMap 𝕜' 𝕜 x, by simpa⟩⟩
+  rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff]
+  convert isUniformEmbedding_toUniformOnFun using 4 with s
+  exact ⟨fun h ↦ h.extend_scalars _, fun h ↦ h.restrict_scalars _⟩
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_restrictScalars := isUniformEmbedding_restrictScalars
+
+theorem uniformContinuous_restrictScalars :
+    UniformContinuous
+      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) :=
+  (isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
+
+end RestrictScalars
+
 end UniformAddGroup
 
 variable [TopologicalSpace F] [TopologicalAddGroup F]
@@ -171,6 +203,35 @@ theorem continuous_coe_fun :
 instance instT2Space [T2Space F] : T2Space (ContinuousMultilinearMap 𝕜 E F) :=
   .of_injective_continuous DFunLike.coe_injective continuous_coe_fun
 
+section RestrictScalars
+
+variable {𝕜' : Type*} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜]
+  [∀ i, Module 𝕜' (E i)] [∀ i, IsScalarTower 𝕜' 𝕜 (E i)] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F]
+
+theorem embedding_restrictScalars :
+    Embedding
+      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) :=
+  letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
+  haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
+  (isUniformEmbedding_restrictScalars _).embedding
+
+@[continuity, fun_prop]
+theorem continuous_restrictScalars :
+    Continuous
+      (restrictScalars 𝕜' : ContinuousMultilinearMap 𝕜 E F → ContinuousMultilinearMap 𝕜' E F) :=
+   embedding_restrictScalars.continuous
+
+variable (𝕜') in
+/-- `ContinuousMultilinearMap.restrictScalars` as a `ContinuousLinearMap`. -/
+@[simps (config := .asFn) apply]
+def restrictScalarsLinear [ContinuousConstSMul 𝕜' F] :
+    ContinuousMultilinearMap 𝕜 E F →L[𝕜'] ContinuousMultilinearMap 𝕜' E F where
+  toFun := restrictScalars 𝕜'
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+end RestrictScalars
+
 variable (𝕜 E F)
 
 /-- The application of a multilinear map as a `ContinuousLinearMap`. -/
diff --git a/Mathlib/Topology/Algebra/Module/StrongTopology.lean b/Mathlib/Topology/Algebra/Module/StrongTopology.lean
index a66e7b2cf135a..8e988265fd745 100644
--- a/Mathlib/Topology/Algebra/Module/StrongTopology.lean
+++ b/Mathlib/Topology/Algebra/Module/StrongTopology.lean
@@ -55,7 +55,8 @@ uniform convergence, bounded convergence
 -/
 
 
-open scoped Topology UniformConvergence
+open scoped Topology UniformConvergence Uniformity
+open Filter Set Function Bornology
 
 section General
 
@@ -63,7 +64,8 @@ section General
 
 variable {𝕜₁ 𝕜₂ : Type*} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) {E F : Type*}
   [AddCommGroup E] [Module 𝕜₁ E] [TopologicalSpace E]
-  [AddCommGroup F] [Module 𝕜₂ F] (F)
+  [AddCommGroup F] [Module 𝕜₂ F]
+variable (F)
 
 /-- Given `E` and `F` two topological vector spaces and `𝔖 : Set (Set E)`, then
 `UniformConvergenceCLM σ F 𝔖` is a type synonym of `E →SL[σ] F` equipped with the "topology of
@@ -91,7 +93,7 @@ instance instTopologicalSpace [TopologicalSpace F] [TopologicalAddGroup F] (𝔖
     (DFunLike.coe : (UniformConvergenceCLM σ F 𝔖) → (E →ᵤ[𝔖] F))
 
 theorem topologicalSpace_eq [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
-    instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced DFunLike.coe
+    instTopologicalSpace σ F 𝔖 = TopologicalSpace.induced (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe)
       (UniformOnFun.topologicalSpace E F 𝔖) := by
   rw [instTopologicalSpace]
   congr
@@ -102,12 +104,13 @@ that this has nice definitional properties. -/
 instance instUniformSpace [UniformSpace F] [UniformAddGroup F]
     (𝔖 : Set (Set E)) : UniformSpace (UniformConvergenceCLM σ F 𝔖) :=
   UniformSpace.replaceTopology
-    ((UniformOnFun.uniformSpace E F 𝔖).comap
-      (DFunLike.coe : (UniformConvergenceCLM σ F 𝔖) → (E →ᵤ[𝔖] F)))
+    ((UniformOnFun.uniformSpace E F 𝔖).comap (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe))
     (by rw [UniformConvergenceCLM.instTopologicalSpace, UniformAddGroup.toUniformSpace_eq]; rfl)
 
 theorem uniformSpace_eq [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
-    instUniformSpace σ F 𝔖 = UniformSpace.comap DFunLike.coe (UniformOnFun.uniformSpace E F 𝔖) := by
+    instUniformSpace σ F 𝔖 =
+      UniformSpace.comap (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe)
+        (UniformOnFun.uniformSpace E F 𝔖) := by
   rw [instUniformSpace, UniformSpace.replaceTopology_eq]
 
 @[simp]
@@ -116,23 +119,31 @@ theorem uniformity_toTopologicalSpace_eq [UniformSpace F] [UniformAddGroup F] (
       UniformConvergenceCLM.instTopologicalSpace σ F 𝔖 :=
   rfl
 
-theorem uniformEmbedding_coeFn [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
-    UniformEmbedding (α := UniformConvergenceCLM σ F 𝔖) (β := E →ᵤ[𝔖] F) DFunLike.coe :=
+theorem isUniformEmbedding_coeFn [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
+    IsUniformEmbedding (α := UniformConvergenceCLM σ F 𝔖) (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) :=
   ⟨⟨rfl⟩, DFunLike.coe_injective⟩
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coeFn := isUniformEmbedding_coeFn
+
 theorem embedding_coeFn [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
     Embedding (X := UniformConvergenceCLM σ F 𝔖) (Y := E →ᵤ[𝔖] F)
       (UniformOnFun.ofFun 𝔖 ∘ DFunLike.coe) :=
-  UniformEmbedding.embedding (uniformEmbedding_coeFn _ _ _)
+  IsUniformEmbedding.embedding (isUniformEmbedding_coeFn _ _ _)
 
 instance instAddCommGroup [TopologicalSpace F] [TopologicalAddGroup F] (𝔖 : Set (Set E)) :
     AddCommGroup (UniformConvergenceCLM σ F 𝔖) := ContinuousLinearMap.addCommGroup
 
+@[simp]
+theorem coe_zero [TopologicalSpace F] [TopologicalAddGroup F] (𝔖 : Set (Set E)) :
+    ⇑(0 : UniformConvergenceCLM σ F 𝔖) = 0 :=
+  rfl
+
 instance instUniformAddGroup [UniformSpace F] [UniformAddGroup F] (𝔖 : Set (Set E)) :
     UniformAddGroup (UniformConvergenceCLM σ F 𝔖) := by
   let φ : (UniformConvergenceCLM σ F 𝔖) →+ E →ᵤ[𝔖] F :=
     ⟨⟨(DFunLike.coe : (UniformConvergenceCLM σ F 𝔖) → E →ᵤ[𝔖] F), rfl⟩, fun _ _ => rfl⟩
-  exact (uniformEmbedding_coeFn _ _ _).uniformAddGroup φ
+  exact (isUniformEmbedding_coeFn _ _ _).uniformAddGroup φ
 
 instance instTopologicalAddGroup [TopologicalSpace F] [TopologicalAddGroup F]
     (𝔖 : Set (Set E)) : TopologicalAddGroup (UniformConvergenceCLM σ F 𝔖) := by
@@ -141,7 +152,7 @@ instance instTopologicalAddGroup [TopologicalSpace F] [TopologicalAddGroup F]
   infer_instance
 
 theorem t2Space [TopologicalSpace F] [TopologicalAddGroup F] [T2Space F]
-    (𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by
+    (𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by
   letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
   haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
   haveI : T2Space (E →ᵤ[𝔖] F) := UniformOnFun.t2Space_of_covering h𝔖
@@ -157,7 +168,7 @@ instance instModule (R : Type*) [Semiring R] [Module R F] [SMulCommClass 𝕜₂
 
 theorem continuousSMul [RingHomSurjective σ] [RingHomIsometric σ]
     [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] (𝔖 : Set (Set E))
-    (h𝔖₃ : ∀ S ∈ 𝔖, Bornology.IsVonNBounded 𝕜₁ S) :
+    (h𝔖₃ : ∀ S ∈ 𝔖, IsVonNBounded 𝕜₁ S) :
     ContinuousSMul 𝕜₂ (UniformConvergenceCLM σ F 𝔖) := by
   letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
   haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
@@ -184,11 +195,66 @@ theorem hasBasis_nhds_zero [TopologicalSpace F] [TopologicalAddGroup F]
       { f : UniformConvergenceCLM σ F 𝔖 | ∀ x ∈ SV.1, f x ∈ SV.2 } :=
   hasBasis_nhds_zero_of_basis σ F 𝔖 h𝔖₁ h𝔖₂ (𝓝 0).basis_sets
 
+theorem nhds_zero_eq_of_basis [TopologicalSpace F] [TopologicalAddGroup F] (𝔖 : Set (Set E))
+    {ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
+    𝓝 (0 : UniformConvergenceCLM σ F 𝔖) =
+      ⨅ (s : Set E) (_ : s ∈ 𝔖) (i : ι) (_ : p i),
+        𝓟 {f : UniformConvergenceCLM σ F 𝔖 | MapsTo f s (b i)} := by
+  letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
+  haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
+  rw [(embedding_coeFn σ F 𝔖).toInducing.nhds_eq_comap,
+    UniformOnFun.nhds_eq_of_basis _ _ h.uniformity_of_nhds_zero]
+  simp [MapsTo]
+
+theorem nhds_zero_eq [TopologicalSpace F] [TopologicalAddGroup F] (𝔖 : Set (Set E)) :
+    𝓝 (0 : UniformConvergenceCLM σ F 𝔖) =
+      ⨅ s ∈ 𝔖, ⨅ t ∈ 𝓝 (0 : F),
+        𝓟 {f : UniformConvergenceCLM σ F 𝔖 | MapsTo f s t} :=
+  nhds_zero_eq_of_basis _ _ _ (𝓝 0).basis_sets
+
+variable {F} in
+theorem eventually_nhds_zero_mapsTo [TopologicalSpace F] [TopologicalAddGroup F]
+    {𝔖 : Set (Set E)} {s : Set E} (hs : s ∈ 𝔖) {U : Set F} (hu : U ∈ 𝓝 0) :
+    ∀ᶠ f : UniformConvergenceCLM σ F 𝔖 in 𝓝 0, MapsTo f s U := by
+  rw [nhds_zero_eq]
+  apply_rules [mem_iInf_of_mem, mem_principal_self]
+
+variable {σ F} in
+theorem isVonNBounded_image2_apply {R : Type*} [SeminormedRing R]
+    [TopologicalSpace F] [TopologicalAddGroup F]
+    [Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
+    {𝔖 : Set (Set E)} {S : Set (UniformConvergenceCLM σ F 𝔖)} (hS : IsVonNBounded R S)
+    {s : Set E} (hs : s ∈ 𝔖) : IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) := by
+  intro U hU
+  filter_upwards [hS (eventually_nhds_zero_mapsTo σ hs hU)] with c hc
+  rw [image2_subset_iff]
+  intro f hf x hx
+  rcases hc hf with ⟨g, hg, rfl⟩
+  exact smul_mem_smul_set (hg hx)
+
+variable {σ F} in
+/-- A set `S` of continuous linear maps with topology of uniform convergence on sets `s ∈ 𝔖`
+is von Neumann bounded iff for any `s ∈ 𝔖`,
+the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded. -/
+theorem isVonNBounded_iff {R : Type*} [NormedDivisionRing R]
+    [TopologicalSpace F] [TopologicalAddGroup F]
+    [Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
+    {𝔖 : Set (Set E)} {S : Set (UniformConvergenceCLM σ F 𝔖)} :
+    IsVonNBounded R S ↔ ∀ s ∈ 𝔖, IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) := by
+  refine ⟨fun hS s hs ↦ isVonNBounded_image2_apply hS hs, fun h ↦ ?_⟩
+  simp_rw [isVonNBounded_iff_absorbing_le, nhds_zero_eq, le_iInf_iff, le_principal_iff]
+  intro s hs U hU
+  rw [Filter.mem_absorbing, Absorbs]
+  filter_upwards [h s hs hU, eventually_ne_cobounded 0] with c hc hc₀ f hf
+  rw [mem_smul_set_iff_inv_smul_mem₀ hc₀]
+  intro x hx
+  simpa only [mem_smul_set_iff_inv_smul_mem₀ hc₀] using hc (mem_image2_of_mem hf hx)
+
 instance instUniformContinuousConstSMul (M : Type*)
     [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
     [UniformSpace F] [UniformAddGroup F] [UniformContinuousConstSMul M F] (𝔖 : Set (Set E)) :
     UniformContinuousConstSMul M (UniformConvergenceCLM σ F 𝔖) :=
-  (uniformEmbedding_coeFn σ F 𝔖).toUniformInducing.uniformContinuousConstSMul fun _ _ ↦ by rfl
+  (isUniformEmbedding_coeFn σ F 𝔖).isUniformInducing.uniformContinuousConstSMul fun _ _ ↦ by rfl
 
 instance instContinuousConstSMul (M : Type*)
     [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
@@ -239,7 +305,7 @@ variable {𝕜₁ 𝕜₂ 𝕜₃ : Type*} [NormedField 𝕜₁] [NormedField 
 the operator norm when `E` and `F` are normed spaces. -/
 instance topologicalSpace [TopologicalSpace F] [TopologicalAddGroup F] :
     TopologicalSpace (E →SL[σ] F) :=
-  UniformConvergenceCLM.instTopologicalSpace σ F { S | Bornology.IsVonNBounded 𝕜₁ S }
+  UniformConvergenceCLM.instTopologicalSpace σ F { S | IsVonNBounded 𝕜₁ S }
 
 instance topologicalAddGroup [TopologicalSpace F] [TopologicalAddGroup F] :
     TopologicalAddGroup (E →SL[σ] F) :=
@@ -247,10 +313,10 @@ instance topologicalAddGroup [TopologicalSpace F] [TopologicalAddGroup F] :
 
 instance continuousSMul [RingHomSurjective σ] [RingHomIsometric σ] [TopologicalSpace F]
     [TopologicalAddGroup F] [ContinuousSMul 𝕜₂ F] : ContinuousSMul 𝕜₂ (E →SL[σ] F) :=
-  UniformConvergenceCLM.continuousSMul σ F { S | Bornology.IsVonNBounded 𝕜₁ S } fun _ hs => hs
+  UniformConvergenceCLM.continuousSMul σ F { S | IsVonNBounded 𝕜₁ S } fun _ hs => hs
 
 instance uniformSpace [UniformSpace F] [UniformAddGroup F] : UniformSpace (E →SL[σ] F) :=
-  UniformConvergenceCLM.instUniformSpace σ F { S | Bornology.IsVonNBounded 𝕜₁ S }
+  UniformConvergenceCLM.instUniformSpace σ F { S | IsVonNBounded 𝕜₁ S }
 
 instance uniformAddGroup [UniformSpace F] [UniformAddGroup F] : UniformAddGroup (E →SL[σ] F) :=
   UniformConvergenceCLM.instUniformAddGroup σ F _
@@ -259,25 +325,29 @@ instance [TopologicalSpace F] [TopologicalAddGroup F] [ContinuousSMul 𝕜₁ E]
     T2Space (E →SL[σ] F) :=
   UniformConvergenceCLM.t2Space σ F _
     (Set.eq_univ_of_forall fun x =>
-      Set.mem_sUnion_of_mem (Set.mem_singleton x) (Bornology.isVonNBounded_singleton x))
+      Set.mem_sUnion_of_mem (Set.mem_singleton x) (isVonNBounded_singleton x))
 
 protected theorem hasBasis_nhds_zero_of_basis [TopologicalSpace F] [TopologicalAddGroup F]
     {ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
-    (𝓝 (0 : E →SL[σ] F)).HasBasis (fun Si : Set E × ι => Bornology.IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2)
+    (𝓝 (0 : E →SL[σ] F)).HasBasis (fun Si : Set E × ι => IsVonNBounded 𝕜₁ Si.1 ∧ p Si.2)
       fun Si => { f : E →SL[σ] F | ∀ x ∈ Si.1, f x ∈ b Si.2 } :=
-  UniformConvergenceCLM.hasBasis_nhds_zero_of_basis σ F { S | Bornology.IsVonNBounded 𝕜₁ S }
-    ⟨∅, Bornology.isVonNBounded_empty 𝕜₁ E⟩
-    (directedOn_of_sup_mem fun _ _ => Bornology.IsVonNBounded.union) h
+  UniformConvergenceCLM.hasBasis_nhds_zero_of_basis σ F { S | IsVonNBounded 𝕜₁ S }
+    ⟨∅, isVonNBounded_empty 𝕜₁ E⟩
+    (directedOn_of_sup_mem fun _ _ => IsVonNBounded.union) h
 
 protected theorem hasBasis_nhds_zero [TopologicalSpace F] [TopologicalAddGroup F] :
     (𝓝 (0 : E →SL[σ] F)).HasBasis
-      (fun SV : Set E × Set F => Bornology.IsVonNBounded 𝕜₁ SV.1 ∧ SV.2 ∈ (𝓝 0 : Filter F))
+      (fun SV : Set E × Set F => IsVonNBounded 𝕜₁ SV.1 ∧ SV.2 ∈ (𝓝 0 : Filter F))
       fun SV => { f : E →SL[σ] F | ∀ x ∈ SV.1, f x ∈ SV.2 } :=
   ContinuousLinearMap.hasBasis_nhds_zero_of_basis (𝓝 0).basis_sets
 
-theorem uniformEmbedding_toUniformOnFun [UniformSpace F] [UniformAddGroup F] :
-    UniformEmbedding fun f : E →SL[σ] F ↦ UniformOnFun.ofFun {s | Bornology.IsVonNBounded 𝕜₁ s} f :=
-  UniformConvergenceCLM.uniformEmbedding_coeFn ..
+theorem isUniformEmbedding_toUniformOnFun [UniformSpace F] [UniformAddGroup F] :
+    IsUniformEmbedding
+      fun f : E →SL[σ] F ↦ UniformOnFun.ofFun {s | Bornology.IsVonNBounded 𝕜₁ s} f :=
+  UniformConvergenceCLM.isUniformEmbedding_coeFn ..
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toUniformOnFun := isUniformEmbedding_toUniformOnFun
 
 instance uniformContinuousConstSMul
     {M : Type*} [Monoid M] [DistribMulAction M F] [SMulCommClass 𝕜₂ M F]
@@ -290,6 +360,51 @@ instance continuousConstSMul {M : Type*} [Monoid M] [DistribMulAction M F] [SMul
     ContinuousConstSMul M (E →SL[σ] F) :=
   UniformConvergenceCLM.instContinuousConstSMul σ F _ _
 
+protected theorem nhds_zero_eq_of_basis [TopologicalSpace F] [TopologicalAddGroup F]
+    {ι : Type*} {p : ι → Prop} {b : ι → Set F} (h : (𝓝 0 : Filter F).HasBasis p b) :
+    𝓝 (0 : E →SL[σ] F) =
+      ⨅ (s : Set E) (_ : IsVonNBounded 𝕜₁ s) (i : ι) (_ : p i),
+        𝓟 {f : E →SL[σ] F | MapsTo f s (b i)} :=
+  UniformConvergenceCLM.nhds_zero_eq_of_basis _ _ _ h
+
+protected theorem nhds_zero_eq [TopologicalSpace F] [TopologicalAddGroup F] :
+    𝓝 (0 : E →SL[σ] F) =
+      ⨅ (s : Set E) (_ : IsVonNBounded 𝕜₁ s) (U : Set F) (_ : U ∈ 𝓝 0),
+        𝓟 {f : E →SL[σ] F | MapsTo f s U} :=
+  UniformConvergenceCLM.nhds_zero_eq ..
+
+/-- If `s` is a von Neumann bounded set and `U` is a neighbourhood of zero,
+then sufficiently small continuous linear maps map `s` to `U`. -/
+theorem eventually_nhds_zero_mapsTo [TopologicalSpace F] [TopologicalAddGroup F]
+    {s : Set E} (hs : IsVonNBounded 𝕜₁ s) {U : Set F} (hu : U ∈ 𝓝 0) :
+    ∀ᶠ f : E →SL[σ] F in 𝓝 0, MapsTo f s U :=
+  UniformConvergenceCLM.eventually_nhds_zero_mapsTo _ hs hu
+
+/-- If `S` is a von Neumann bounded set of continuous linear maps `f : E →SL[σ] F`
+and `s` is a von Neumann bounded set in the domain,
+then the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded.
+
+See also `isVonNBounded_iff` for an `Iff` version with stronger typeclass assumptions. -/
+theorem isVonNBounded_image2_apply {R : Type*} [SeminormedRing R]
+    [TopologicalSpace F] [TopologicalAddGroup F]
+    [Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
+    {S : Set (E →SL[σ] F)} (hS : IsVonNBounded R S) {s : Set E} (hs : IsVonNBounded 𝕜₁ s) :
+    IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) :=
+  UniformConvergenceCLM.isVonNBounded_image2_apply hS hs
+
+/-- A set `S` of continuous linear maps is von Neumann bounded
+iff for any von Neumann bounded set `s`,
+the set `{f x | (f ∈ S) (x ∈ s)}` is von Neumann bounded.
+
+For the forward implication with weaker typeclass assumptions, see `isVonNBounded_image2_apply`. -/
+theorem isVonNBounded_iff {R : Type*} [NormedDivisionRing R]
+    [TopologicalSpace F] [TopologicalAddGroup F]
+    [Module R F] [ContinuousConstSMul R F] [SMulCommClass 𝕜₂ R F]
+    {S : Set (E →SL[σ] F)} :
+    IsVonNBounded R S ↔
+      ∀ s, IsVonNBounded 𝕜₁ s → IsVonNBounded R (Set.image2 (fun f x ↦ f x) S s) :=
+  UniformConvergenceCLM.isVonNBounded_iff
+
 variable (G) [TopologicalSpace F] [TopologicalSpace G]
 
 /-- Pre-composition by a *fixed* continuous linear map as a continuous linear map.
@@ -361,15 +476,18 @@ variable [UniformSpace F] [UniformAddGroup F] [Module 𝕜 F]
   (𝕜' : Type*) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜]
   [Module 𝕜' E] [IsScalarTower 𝕜' 𝕜 E] [Module 𝕜' F] [IsScalarTower 𝕜' 𝕜 F]
 
-theorem uniformEmbedding_restrictScalars :
-    UniformEmbedding (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) := by
-  rw [← uniformEmbedding_toUniformOnFun.of_comp_iff]
-  convert uniformEmbedding_toUniformOnFun using 4 with s
+theorem isUniformEmbedding_restrictScalars :
+    IsUniformEmbedding (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) := by
+  rw [← isUniformEmbedding_toUniformOnFun.of_comp_iff]
+  convert isUniformEmbedding_toUniformOnFun using 4 with s
   exact ⟨fun h ↦ h.extend_scalars _, fun h ↦ h.restrict_scalars _⟩
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_restrictScalars := isUniformEmbedding_restrictScalars
+
 theorem uniformContinuous_restrictScalars :
     UniformContinuous (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) :=
-  (uniformEmbedding_restrictScalars 𝕜').uniformContinuous
+  (isUniformEmbedding_restrictScalars 𝕜').uniformContinuous
 
 end UniformSpace
 
@@ -381,7 +499,7 @@ theorem embedding_restrictScalars :
     Embedding (restrictScalars 𝕜' : (E →L[𝕜] F) → (E →L[𝕜'] F)) :=
   letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
   haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
-  (uniformEmbedding_restrictScalars _).embedding
+  (isUniformEmbedding_restrictScalars _).embedding
 
 @[continuity, fun_prop]
 theorem continuous_restrictScalars :
diff --git a/Mathlib/Topology/Algebra/Module/WeakBilin.lean b/Mathlib/Topology/Algebra/Module/WeakBilin.lean
new file mode 100644
index 0000000000000..916301730f678
--- /dev/null
+++ b/Mathlib/Topology/Algebra/Module/WeakBilin.lean
@@ -0,0 +1,161 @@
+/-
+Copyright (c) 2021 Kalle Kytölä. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Kalle Kytölä, Moritz Doll
+-/
+import Mathlib.Algebra.Algebra.Defs
+import Mathlib.Topology.Algebra.Group.Basic
+
+/-!
+# Weak dual topology
+
+This file defines the weak topology given two vector spaces `E` and `F` over a commutative semiring
+`𝕜` and a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`. The weak topology on `E` is the coarsest topology
+such that for all `y : F` every map `fun x => B x y` is continuous.
+
+## Main definitions
+
+The main definition is the type `WeakBilin B`.
+
+* Given `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`, the type `WeakBilin B` is a type synonym for `E`.
+* The instance `WeakBilin.instTopologicalSpace` is the weak topology induced by the bilinear form
+  `B`.
+
+## Main results
+
+We establish that `WeakBilin B` has the following structure:
+* `WeakBilin.instContinuousAdd`: The addition in `WeakBilin B` is continuous.
+* `WeakBilin.instContinuousSMul`: The scalar multiplication in `WeakBilin B` is continuous.
+
+We prove the following results characterizing the weak topology:
+* `eval_continuous`: For any `y : F`, the evaluation mapping `fun x => B x y` is continuous.
+* `continuous_of_continuous_eval`: For a mapping to `WeakBilin B` to be continuous,
+  it suffices that its compositions with pairing with `B` at all points `y : F` is continuous.
+* `tendsto_iff_forall_eval_tendsto`: Convergence in `WeakBilin B` can be characterized
+  in terms of convergence of the evaluations at all points `y : F`.
+
+## Notations
+
+No new notation is introduced.
+
+## References
+
+* [H. H. Schaefer, *Topological Vector Spaces*][schaefer1966]
+
+## Tags
+
+weak-star, weak dual, duality
+
+-/
+
+
+noncomputable section
+
+open Filter
+
+open Topology
+
+variable {α 𝕜 𝕝 E F : Type*}
+
+section WeakTopology
+
+/-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/
+@[nolint unusedArguments]
+def WeakBilin [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F]
+    (_ : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) := E
+
+namespace WeakBilin
+
+-- Porting note: the next two instances should be derived from the definition
+instance instAddCommMonoid [CommSemiring 𝕜] [a : AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F]
+    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommMonoid (WeakBilin B) := a
+
+instance instModule [CommSemiring 𝕜] [AddCommMonoid E] [m : Module 𝕜 E] [AddCommMonoid F]
+    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : Module 𝕜 (WeakBilin B) := m
+
+instance instAddCommGroup [CommSemiring 𝕜] [a : AddCommGroup E] [Module 𝕜 E] [AddCommMonoid F]
+    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommGroup (WeakBilin B) := a
+
+instance (priority := 100) instModule' [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E]
+    [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F] [m : Module 𝕝 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
+    Module 𝕝 (WeakBilin B) := m
+
+instance instIsScalarTower [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommMonoid E] [Module 𝕜 E]
+    [AddCommMonoid F] [Module 𝕜 F] [SMul 𝕝 𝕜] [Module 𝕝 E] [s : IsScalarTower 𝕝 𝕜 E]
+    (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : IsScalarTower 𝕝 𝕜 (WeakBilin B) := s
+
+section Semiring
+
+variable [TopologicalSpace 𝕜] [CommSemiring 𝕜]
+variable [AddCommMonoid E] [Module 𝕜 E]
+variable [AddCommMonoid F] [Module 𝕜 F]
+variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
+
+instance instTopologicalSpace : TopologicalSpace (WeakBilin B) :=
+  TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
+
+/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is continuous. -/
+theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
+  continuous_induced_dom
+
+theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
+  (continuous_pi_iff.mp (coeFn_continuous B)) y
+
+theorem continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakBilin B}
+    (h : ∀ y, Continuous fun a => B (g a) y) : Continuous g :=
+  continuous_induced_rng.2 (continuous_pi_iff.mpr h)
+
+/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is an embedding. -/
+theorem embedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
+    Embedding fun (x : WeakBilin B) y => B x y :=
+  Function.Injective.embedding_induced <| LinearMap.coe_injective.comp hB
+
+theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
+    (hB : Function.Injective B) :
+    Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by
+  rw [← tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]
+  rfl
+
+/-- Addition in `WeakBilin B` is continuous. -/
+instance instContinuousAdd [ContinuousAdd 𝕜] : ContinuousAdd (WeakBilin B) := by
+  refine ⟨continuous_induced_rng.2 ?_⟩
+  refine
+    cast (congr_arg _ ?_)
+      (((coeFn_continuous B).comp continuous_fst).add ((coeFn_continuous B).comp continuous_snd))
+  ext
+  simp only [Function.comp_apply, Pi.add_apply, map_add, LinearMap.add_apply]
+
+/-- Scalar multiplication by `𝕜` on `WeakBilin B` is continuous. -/
+instance instContinuousSMul [ContinuousSMul 𝕜 𝕜] : ContinuousSMul 𝕜 (WeakBilin B) := by
+  refine ⟨continuous_induced_rng.2 ?_⟩
+  refine cast (congr_arg _ ?_) (continuous_fst.smul ((coeFn_continuous B).comp continuous_snd))
+  ext
+  simp only [Function.comp_apply, Pi.smul_apply, LinearMap.map_smulₛₗ, RingHom.id_apply,
+    LinearMap.smul_apply]
+
+end Semiring
+
+section Ring
+
+variable [TopologicalSpace 𝕜] [CommRing 𝕜]
+variable [AddCommGroup E] [Module 𝕜 E]
+variable [AddCommGroup F] [Module 𝕜 F]
+
+
+variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
+
+/-- `WeakBilin B` is a `TopologicalAddGroup`, meaning that addition and negation are
+continuous. -/
+instance instTopologicalAddGroup [ContinuousAdd 𝕜] : TopologicalAddGroup (WeakBilin B) where
+  toContinuousAdd := by infer_instance
+  continuous_neg := by
+    refine continuous_induced_rng.2 (continuous_pi_iff.mpr fun y => ?_)
+    refine cast (congr_arg _ ?_) (eval_continuous B (-y))
+    ext x
+    simp only [map_neg, Function.comp_apply, LinearMap.neg_apply]
+
+end Ring
+
+end WeakBilin
+
+end WeakTopology
diff --git a/Mathlib/Topology/Algebra/Module/WeakDual.lean b/Mathlib/Topology/Algebra/Module/WeakDual.lean
index 8bc7353b3fce3..1a71efa833d83 100644
--- a/Mathlib/Topology/Algebra/Module/WeakDual.lean
+++ b/Mathlib/Topology/Algebra/Module/WeakDual.lean
@@ -3,28 +3,28 @@ Copyright (c) 2021 Kalle Kytölä. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä, Moritz Doll
 -/
-import Mathlib.Topology.Algebra.Module.Basic
 import Mathlib.LinearAlgebra.BilinearMap
+import Mathlib.Topology.Algebra.Module.Basic
+import Mathlib.Topology.Algebra.Module.WeakBilin
 
 /-!
 # Weak dual topology
 
-This file defines the weak topology given two vector spaces `E` and `F` over a commutative semiring
+We continue in the setting of `Mathlib.Topology.Algebra.Module.WeakBilin`,
+which defines the weak topology given two vector spaces `E` and `F` over a commutative semiring
 `𝕜` and a bilinear form `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`. The weak topology on `E` is the coarsest topology
 such that for all `y : F` every map `fun x => B x y` is continuous.
 
+In this file, we consider two special cases.
 In the case that `F = E →L[𝕜] 𝕜` and `B` being the canonical pairing, we obtain the weak-* topology,
 `WeakDual 𝕜 E := (E →L[𝕜] 𝕜)`. Interchanging the arguments in the bilinear form yields the
 weak topology `WeakSpace 𝕜 E := E`.
 
 ## Main definitions
 
-The main definitions are the types `WeakBilin B` for the general case and the two special cases
-`WeakDual 𝕜 E` and `WeakSpace 𝕜 E` with the respective topology instances on it.
+The main definitions are the types `WeakDual 𝕜 E` and `WeakSpace 𝕜 E`,
+with the respective topology instances on it.
 
-* Given `B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜`, the type `WeakBilin B` is a type synonym for `E`.
-* The instance `WeakBilin.instTopologicalSpace` is the weak topology induced by the bilinear form
-  `B`.
 * `WeakDual 𝕜 E` is a type synonym for `Dual 𝕜 E` (when the latter is defined): both are equal to
   the type `E →L[𝕜] 𝕜` of continuous linear maps from a module `E` over `𝕜` to the ring `𝕜`.
 * The instance `WeakDual.instTopologicalSpace` is the weak-* topology on `WeakDual 𝕜 E`, i.e., the
@@ -33,19 +33,6 @@ The main definitions are the types `WeakBilin B` for the general case and the tw
 * The instance `WeakSpace.instTopologicalSpace` is the weak topology on `E`, i.e., the
   coarsest topology such that all `v : dual 𝕜 E` remain continuous.
 
-## Main results
-
-We establish that `WeakBilin B` has the following structure:
-* `WeakBilin.instContinuousAdd`: The addition in `WeakBilin B` is continuous.
-* `WeakBilin.instContinuousSMul`: The scalar multiplication in `WeakBilin B` is continuous.
-
-We prove the following results characterizing the weak topology:
-* `eval_continuous`: For any `y : F`, the evaluation mapping `fun x => B x y` is continuous.
-* `continuous_of_continuous_eval`: For a mapping to `WeakBilin B` to be continuous,
-  it suffices that its compositions with pairing with `B` at all points `y : F` is continuous.
-* `tendsto_iff_forall_eval_tendsto`: Convergence in `WeakBilin B` can be characterized
-  in terms of convergence of the evaluations at all points `y : F`.
-
 ## Notations
 
 No new notation is introduced.
@@ -67,123 +54,16 @@ open Filter
 
 open Topology
 
-variable {α 𝕜 𝕝 R E F M : Type*}
-
-section WeakTopology
-
-/-- The space `E` equipped with the weak topology induced by the bilinear form `B`. -/
-@[nolint unusedArguments]
-def WeakBilin [CommSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F] [Module 𝕜 F]
-    (_ : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) := E
-
-namespace WeakBilin
-
--- Porting note: the next two instances should be derived from the definition
-instance instAddCommMonoid [CommSemiring 𝕜] [a : AddCommMonoid E] [Module 𝕜 E] [AddCommMonoid F]
-    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommMonoid (WeakBilin B) := a
-
-instance instModule [CommSemiring 𝕜] [AddCommMonoid E] [m : Module 𝕜 E] [AddCommMonoid F]
-    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : Module 𝕜 (WeakBilin B) := m
-
-instance instAddCommGroup [CommSemiring 𝕜] [a : AddCommGroup E] [Module 𝕜 E] [AddCommMonoid F]
-    [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : AddCommGroup (WeakBilin B) := a
-
-instance (priority := 100) instModule' [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommGroup E]
-    [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] [m : Module 𝕝 E] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
-    Module 𝕝 (WeakBilin B) := m
-
-instance instIsScalarTower [CommSemiring 𝕜] [CommSemiring 𝕝] [AddCommGroup E] [Module 𝕜 E]
-    [AddCommGroup F] [Module 𝕜 F] [SMul 𝕝 𝕜] [Module 𝕝 E] [s : IsScalarTower 𝕝 𝕜 E]
-    (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) : IsScalarTower 𝕝 𝕜 (WeakBilin B) := s
-
-section Semiring
-
-variable [TopologicalSpace 𝕜] [CommSemiring 𝕜]
-variable [AddCommMonoid E] [Module 𝕜 E]
-variable [AddCommMonoid F] [Module 𝕜 F]
-variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
-
-instance instTopologicalSpace : TopologicalSpace (WeakBilin B) :=
-  TopologicalSpace.induced (fun x y => B x y) Pi.topologicalSpace
-
-/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is continuous. -/
-theorem coeFn_continuous : Continuous fun (x : WeakBilin B) y => B x y :=
-  continuous_induced_dom
-
-theorem eval_continuous (y : F) : Continuous fun x : WeakBilin B => B x y :=
-  (continuous_pi_iff.mp (coeFn_continuous B)) y
-
-theorem continuous_of_continuous_eval [TopologicalSpace α] {g : α → WeakBilin B}
-    (h : ∀ y, Continuous fun a => B (g a) y) : Continuous g :=
-  continuous_induced_rng.2 (continuous_pi_iff.mpr h)
-
-/-- The coercion `(fun x y => B x y) : E → (F → 𝕜)` is an embedding. -/
-theorem embedding {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} (hB : Function.Injective B) :
-    Embedding fun (x : WeakBilin B) y => B x y :=
-  Function.Injective.embedding_induced <| LinearMap.coe_injective.comp hB
-
-theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
-    (hB : Function.Injective B) :
-    Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by
-  rw [← tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]
-  rfl
-
-/-- Addition in `WeakBilin B` is continuous. -/
-instance instContinuousAdd [ContinuousAdd 𝕜] : ContinuousAdd (WeakBilin B) := by
-  refine ⟨continuous_induced_rng.2 ?_⟩
-  refine
-    cast (congr_arg _ ?_)
-      (((coeFn_continuous B).comp continuous_fst).add ((coeFn_continuous B).comp continuous_snd))
-  ext
-  simp only [Function.comp_apply, Pi.add_apply, map_add, LinearMap.add_apply]
-
-/-- Scalar multiplication by `𝕜` on `WeakBilin B` is continuous. -/
-instance instContinuousSMul [ContinuousSMul 𝕜 𝕜] : ContinuousSMul 𝕜 (WeakBilin B) := by
-  refine ⟨continuous_induced_rng.2 ?_⟩
-  refine cast (congr_arg _ ?_) (continuous_fst.smul ((coeFn_continuous B).comp continuous_snd))
-  ext
-  simp only [Function.comp_apply, Pi.smul_apply, LinearMap.map_smulₛₗ, RingHom.id_apply,
-    LinearMap.smul_apply]
-
-end Semiring
-
-section Ring
-
-variable [TopologicalSpace 𝕜] [CommRing 𝕜]
-variable [AddCommGroup E] [Module 𝕜 E]
-variable [AddCommGroup F] [Module 𝕜 F]
-
-
-variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
-
-/-- `WeakBilin B` is a `TopologicalAddGroup`, meaning that addition and negation are
-continuous. -/
-instance instTopologicalAddGroup [ContinuousAdd 𝕜] : TopologicalAddGroup (WeakBilin B) where
-  toContinuousAdd := by infer_instance
-  continuous_neg := by
-    refine continuous_induced_rng.2 (continuous_pi_iff.mpr fun y => ?_)
-    refine cast (congr_arg _ ?_) (eval_continuous B (-y))
-    ext x
-    simp only [map_neg, Function.comp_apply, LinearMap.neg_apply]
-
-end Ring
-
-end WeakBilin
-
-end WeakTopology
-
-section WeakStarTopology
+variable {α 𝕜 𝕝 E F : Type*}
 
 /-- The canonical pairing of a vector space and its topological dual. -/
 def topDualPairing (𝕜 E) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜] [AddCommMonoid E]
     [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] : (E →L[𝕜] 𝕜) →ₗ[𝕜] E →ₗ[𝕜] 𝕜 :=
   ContinuousLinearMap.coeLM 𝕜
 
-variable [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
-variable [ContinuousConstSMul 𝕜 𝕜]
-variable [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
-
-theorem topDualPairing_apply (v : E →L[𝕜] 𝕜) (x : E) : topDualPairing 𝕜 E v x = v x :=
+theorem topDualPairing_apply [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
+    [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] [ContinuousConstSMul 𝕜 𝕜] (v : E →L[𝕜] 𝕜)
+    (x : E) : topDualPairing 𝕜 E v x = v x :=
   rfl
 
 /-- The weak star topology is the topology coarsest topology on `E →L[𝕜] 𝕜` such that all
@@ -194,6 +74,12 @@ def WeakDual (𝕜 E : Type*) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [Conti
 
 namespace WeakDual
 
+section Semiring
+
+variable [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
+variable [ContinuousConstSMul 𝕜 𝕜]
+variable [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
+
 -- Porting note: the next four instances should be derived from the definition
 instance instAddCommMonoid : AddCommMonoid (WeakDual 𝕜 E) :=
   WeakBilin.instAddCommMonoid (topDualPairing 𝕜 E)
@@ -266,6 +152,21 @@ instance instT2Space [T2Space 𝕜] : T2Space (WeakDual 𝕜 E) :=
     WeakBilin.embedding <|
       show Function.Injective (topDualPairing 𝕜 E) from ContinuousLinearMap.coe_injective
 
+end Semiring
+
+section Ring
+
+variable [CommRing 𝕜] [TopologicalSpace 𝕜] [TopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜]
+variable [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E]
+
+instance instAddCommGroup : AddCommGroup (WeakDual 𝕜 E) :=
+  WeakBilin.instAddCommGroup (topDualPairing 𝕜 E)
+
+instance instTopologicalAddGroup : TopologicalAddGroup (WeakDual 𝕜 E) :=
+  WeakBilin.instTopologicalAddGroup (topDualPairing 𝕜 E)
+
+end Ring
+
 end WeakDual
 
 /-- The weak topology is the topology coarsest topology on `E` such that all functionals
@@ -274,6 +175,12 @@ def WeakSpace (𝕜 E) [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAd
     [ContinuousConstSMul 𝕜 𝕜] [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E] :=
   WeakBilin (topDualPairing 𝕜 E).flip
 
+section Semiring
+
+variable [CommSemiring 𝕜] [TopologicalSpace 𝕜] [ContinuousAdd 𝕜]
+variable [ContinuousConstSMul 𝕜 𝕜]
+variable [AddCommMonoid E] [Module 𝕜 E] [TopologicalSpace E]
+
 namespace WeakSpace
 
 -- Porting note: the next four instances should be derived from the definition
@@ -289,6 +196,13 @@ instance instTopologicalSpace : TopologicalSpace (WeakSpace 𝕜 E) :=
 instance instContinuousAdd : ContinuousAdd (WeakSpace 𝕜 E) :=
   WeakBilin.instContinuousAdd (topDualPairing 𝕜 E).flip
 
+instance instModule' [CommSemiring 𝕝] [Module 𝕝 E] : Module 𝕝 (WeakSpace 𝕜 E) :=
+  WeakBilin.instModule' (topDualPairing 𝕜 E).flip
+
+instance instIsScalarTower [CommSemiring 𝕝] [Module 𝕝 𝕜] [Module 𝕝 E] [IsScalarTower 𝕝 𝕜 E] :
+    IsScalarTower 𝕝 𝕜 (WeakSpace 𝕜 E) :=
+  WeakBilin.instIsScalarTower (topDualPairing 𝕜 E).flip
+
 variable [AddCommMonoid F] [Module 𝕜 F] [TopologicalSpace F]
 
 /-- A continuous linear map from `E` to `F` is still continuous when `E` and `F` are equipped with
@@ -315,7 +229,7 @@ def toWeakSpace : E ≃ₗ[𝕜] WeakSpace 𝕜 E := LinearEquiv.refl 𝕜 E
 variable (𝕜 E) in
 /-- For a topological vector space `E`, "identity mapping" `E → WeakSpace 𝕜 E` is continuous.
 This definition implements it as a continuous linear map. -/
-def continuousLinearMapToWeakSpace : E →L[𝕜] WeakSpace 𝕜 E where
+def toWeakSpaceCLM : E →L[𝕜] WeakSpace 𝕜 E where
   __ := toWeakSpace 𝕜 E
   cont := by
     apply WeakBilin.continuous_of_continuous_eval
@@ -323,21 +237,21 @@ def continuousLinearMapToWeakSpace : E →L[𝕜] WeakSpace 𝕜 E where
 
 variable (𝕜 E) in
 @[simp]
-theorem continuousLinearMapToWeakSpace_eq_toWeakSpace (x : E) :
-    continuousLinearMapToWeakSpace 𝕜 E x = toWeakSpace 𝕜 E x := by rfl
+theorem toWeakSpaceCLM_eq_toWeakSpace (x : E) :
+    toWeakSpaceCLM 𝕜 E x = toWeakSpace 𝕜 E x := by rfl
 
-theorem continuousLinearMapToWeakSpace_bijective :
-    Function.Bijective (continuousLinearMapToWeakSpace 𝕜 E) :=
+theorem toWeakSpaceCLM_bijective :
+    Function.Bijective (toWeakSpaceCLM 𝕜 E) :=
   (toWeakSpace 𝕜 E).bijective
 
 /-- The canonical map from `WeakSpace 𝕜 E` to `E` is an open map. -/
 theorem isOpenMap_toWeakSpace_symm : IsOpenMap (toWeakSpace 𝕜 E).symm :=
-  IsOpenMap.of_inverse (continuousLinearMapToWeakSpace 𝕜 E).cont
+  IsOpenMap.of_inverse (toWeakSpaceCLM 𝕜 E).cont
     (toWeakSpace 𝕜 E).left_inv (toWeakSpace 𝕜 E).right_inv
 
 /-- A set in `E` which is open in the weak topology is open. -/
 theorem WeakSpace.isOpen_of_isOpen (V : Set E)
-    (hV : IsOpen ((continuousLinearMapToWeakSpace 𝕜 E) '' V : Set (WeakSpace 𝕜 E))) : IsOpen V := by
+    (hV : IsOpen ((toWeakSpaceCLM 𝕜 E) '' V : Set (WeakSpace 𝕜 E))) : IsOpen V := by
   simpa [Set.image_image] using isOpenMap_toWeakSpace_symm _ hV
 
 theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter α} {f : α → WeakDual 𝕜 E}
@@ -346,4 +260,21 @@ theorem tendsto_iff_forall_eval_tendsto_topDualPairing {l : Filter α} {f : α 
       ∀ y, Tendsto (fun i => topDualPairing 𝕜 E (f i) y) l (𝓝 (topDualPairing 𝕜 E x y)) :=
   WeakBilin.tendsto_iff_forall_eval_tendsto _ ContinuousLinearMap.coe_injective
 
-end WeakStarTopology
+end Semiring
+
+section Ring
+
+namespace WeakSpace
+
+variable [CommRing 𝕜] [TopologicalSpace 𝕜] [TopologicalAddGroup 𝕜] [ContinuousConstSMul 𝕜 𝕜]
+variable [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E] [TopologicalAddGroup E]
+
+instance instAddCommGroup : AddCommGroup (WeakSpace 𝕜 E) :=
+  WeakBilin.instAddCommGroup (topDualPairing 𝕜 E).flip
+
+instance instTopologicalAddGroup : TopologicalAddGroup (WeakSpace 𝕜 E) :=
+  WeakBilin.instTopologicalAddGroup (topDualPairing 𝕜 E).flip
+
+end WeakSpace
+
+end Ring
diff --git a/Mathlib/Topology/Algebra/Monoid.lean b/Mathlib/Topology/Algebra/Monoid.lean
index 27e929db7ae6a..126c6636dd313 100644
--- a/Mathlib/Topology/Algebra/Monoid.lean
+++ b/Mathlib/Topology/Algebra/Monoid.lean
@@ -7,8 +7,8 @@ import Mathlib.Algebra.BigOperators.Finprod
 import Mathlib.Order.Filter.Pointwise
 import Mathlib.Topology.Algebra.MulAction
 import Mathlib.Algebra.BigOperators.Pi
-import Mathlib.Topology.ContinuousFunction.Basic
 import Mathlib.Algebra.Group.ULift
+import Mathlib.Topology.ContinuousMap.Defs
 
 /-!
 # Theory of topological monoids
@@ -75,7 +75,7 @@ instance ContinuousMul.to_continuousSMul : ContinuousSMul M M :=
 instance ContinuousMul.to_continuousSMul_op : ContinuousSMul Mᵐᵒᵖ M :=
   ⟨show Continuous ((fun p : M × M => p.1 * p.2) ∘ Prod.swap ∘ Prod.map MulOpposite.unop id) from
       continuous_mul.comp <|
-        continuous_swap.comp <| Continuous.prod_map MulOpposite.continuous_unop continuous_id⟩
+        continuous_swap.comp <| Continuous.prodMap MulOpposite.continuous_unop continuous_id⟩
 
 @[to_additive]
 theorem ContinuousMul.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F α β] [MulOneClass α]
@@ -83,7 +83,7 @@ theorem ContinuousMul.induced {α : Type*} {β : Type*} {F : Type*} [FunLike F 
     @ContinuousMul α (tβ.induced f) _ := by
   let tα := tβ.induced f
   refine ⟨continuous_induced_rng.2 ?_⟩
-  simp only [Function.comp, map_mul]
+  simp only [Function.comp_def, map_mul]
   fun_prop
 
 @[to_additive (attr := continuity, fun_prop)]
diff --git a/Mathlib/Topology/Algebra/MulAction.lean b/Mathlib/Topology/Algebra/MulAction.lean
index 30ec72337a896..91dc843aa0507 100644
--- a/Mathlib/Topology/Algebra/MulAction.lean
+++ b/Mathlib/Topology/Algebra/MulAction.lean
@@ -71,6 +71,13 @@ section SMul
 
 variable [SMul M X] [ContinuousSMul M X]
 
+lemma IsScalarTower.continuousSMul {M : Type*} (N : Type*) {α : Type*} [Monoid N] [SMul M N]
+    [MulAction N α] [SMul M α] [IsScalarTower M N α] [TopologicalSpace M] [TopologicalSpace N]
+    [TopologicalSpace α] [ContinuousSMul M N] [ContinuousSMul N α] : ContinuousSMul M α :=
+  { continuous_smul := by
+      suffices Continuous (fun p : M × α ↦ (p.1 • (1 : N)) • p.2) by simpa
+      fun_prop }
+
 @[to_additive]
 instance : ContinuousSMul (ULift M) X :=
   ⟨(continuous_smul (M := M)).comp₂ (continuous_uLift_down.comp continuous_fst) continuous_snd⟩
@@ -85,7 +92,7 @@ theorem ContinuousSMul.induced {R : Type*} {α : Type*} {β : Type*} {F : Type*}
     (f : F) : @ContinuousSMul R α _ _ (tβ.induced f) := by
   let tα := tβ.induced f
   refine ⟨continuous_induced_rng.2 ?_⟩
-  simp only [Function.comp, map_smul]
+  simp only [Function.comp_def, map_smul]
   fun_prop
 
 @[to_additive]
@@ -125,13 +132,13 @@ action is."]
 instance ContinuousSMul.op [SMul Mᵐᵒᵖ X] [IsCentralScalar M X] : ContinuousSMul Mᵐᵒᵖ X :=
   ⟨by
     suffices Continuous fun p : M × X => MulOpposite.op p.fst • p.snd from
-      this.comp (MulOpposite.continuous_unop.prod_map continuous_id)
+      this.comp (MulOpposite.continuous_unop.prodMap continuous_id)
     simpa only [op_smul_eq_smul] using (continuous_smul : Continuous fun p : M × X => _)⟩
 
 @[to_additive]
 instance MulOpposite.continuousSMul : ContinuousSMul M Xᵐᵒᵖ :=
   ⟨MulOpposite.continuous_op.comp <|
-      continuous_smul.comp <| continuous_id.prod_map MulOpposite.continuous_unop⟩
+      continuous_smul.comp <| continuous_id.prodMap MulOpposite.continuous_unop⟩
 
 @[to_additive]
 protected theorem Specializes.smul {a b : M} {x y : X} (h₁ : a ⤳ b) (h₂ : x ⤳ y) :
@@ -216,6 +223,12 @@ variable [Group M] [MulAction M X] [ContinuousSMul M X]
 instance Subgroup.continuousSMul {S : Subgroup M} : ContinuousSMul S X :=
   S.toSubmonoid.continuousSMul
 
+variable (M)
+
+/-- The stabilizer of a continuous group action on a discrete space is an open subgroup. -/
+lemma stabilizer_isOpen [DiscreteTopology X] (x : X) : IsOpen (MulAction.stabilizer M x : Set M) :=
+  IsOpen.preimage (f := fun g ↦ g • x) (by fun_prop) (isOpen_discrete {x})
+
 end Group
 
 @[to_additive]
diff --git a/Mathlib/Topology/Algebra/Nonarchimedean/Completion.lean b/Mathlib/Topology/Algebra/Nonarchimedean/Completion.lean
index 8d88cf4ee0a0a..c5dd61852fecd 100644
--- a/Mathlib/Topology/Algebra/Nonarchimedean/Completion.lean
+++ b/Mathlib/Topology/Algebra/Nonarchimedean/Completion.lean
@@ -51,7 +51,7 @@ instance {G : Type*} [AddGroup G] [UniformSpace G] [UniformAddGroup G] [Nonarchi
       `0` in `Completion G`. This follows from the fact that `toCompl : G → Completion G` is dense
       inducing and `W` is a neighborhood of `0` in `G`. -/
       apply isOpen_of_mem_nhds (g := 0)
-      apply (denseInducing_toCompl _).closure_image_mem_nhds
+      apply (isDenseInducing_toCompl _).closure_image_mem_nhds
       exact mem_nhds_zero W
     use ⟨_, this⟩
     /- Finally, it remains to show that `V ⊆ U`. It suffices to show that `V ⊆ C`, which
diff --git a/Mathlib/Topology/Algebra/OpenSubgroup.lean b/Mathlib/Topology/Algebra/OpenSubgroup.lean
index 025b15e34f7a6..99c3a4c276f10 100644
--- a/Mathlib/Topology/Algebra/OpenSubgroup.lean
+++ b/Mathlib/Topology/Algebra/OpenSubgroup.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin
+Authors: Johan Commelin, Nailin Guan
 -/
 import Mathlib.RingTheory.Ideal.Basic
 import Mathlib.Topology.Algebra.Ring.Basic
@@ -10,7 +10,7 @@ import Mathlib.Topology.Sets.Opens
 /-!
 # Open subgroups of a topological groups
 
-This files builds the lattice `OpenSubgroup G` of open subgroups in a topological group `G`,
+This files builds the lattice `OpenSubgroup G` of open subgroups in a topological group `G`,
 and its additive version `OpenAddSubgroup`.  This lattice has a top element, the subgroup of all
 elements, but no bottom element in general. The trivial subgroup which is the natural candidate
 bottom has no reason to be open (this happens only in discrete groups).
@@ -135,8 +135,8 @@ instance : Inhabited (OpenSubgroup G) :=
 @[to_additive]
 theorem isClosed [ContinuousMul G] (U : OpenSubgroup G) : IsClosed (U : Set G) := by
   apply isOpen_compl_iff.1
-  refine isOpen_iff_forall_mem_open.2 fun x hx => ⟨(fun y => y * x⁻¹) ⁻¹' U, ?_, ?_, ?_⟩
-  · refine fun u hux hu => hx ?_
+  refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ⟨(fun y ↦ y * x⁻¹) ⁻¹' U, ?_, ?_, ?_⟩
+  · refine fun u hux hu ↦ hx ?_
     simp only [Set.mem_preimage, SetLike.mem_coe] at hux hu ⊢
     convert U.mul_mem (U.inv_mem hux) hu
     simp
@@ -170,7 +170,7 @@ end
 
 @[to_additive]
 instance instInfOpenSubgroup : Inf (OpenSubgroup G) :=
-  ⟨fun U V => ⟨U ⊓ V, U.isOpen.inter V.isOpen⟩⟩
+  ⟨fun U V ↦ ⟨U ⊓ V, U.isOpen.inter V.isOpen⟩⟩
 
 @[to_additive (attr := simp, norm_cast)]
 theorem coe_inf : (↑(U ⊓ V) : Set G) = (U : Set G) ∩ V :=
@@ -194,7 +194,7 @@ instance instPartialOrderOpenSubgroup : PartialOrder (OpenSubgroup G) := inferIn
 -- Porting note: we override `toPartialorder` to get better `le`
 @[to_additive]
 instance instSemilatticeInfOpenSubgroup : SemilatticeInf (OpenSubgroup G) :=
-  { SetLike.coe_injective.semilatticeInf ((↑) : OpenSubgroup G → Set G) fun _ _ => rfl with
+  { SetLike.coe_injective.semilatticeInf ((↑) : OpenSubgroup G → Set G) fun _ _ ↦ rfl with
     toInf := instInfOpenSubgroup
     toPartialOrder := instPartialOrderOpenSubgroup }
 
@@ -245,9 +245,9 @@ variable {G : Type*} [Group G] [TopologicalSpace G]
 @[to_additive]
 theorem isOpen_of_mem_nhds [ContinuousMul G] (H : Subgroup G) {g : G} (hg : (H : Set G) ∈ 𝓝 g) :
     IsOpen (H : Set G) := by
-  refine isOpen_iff_mem_nhds.2 fun x hx => ?_
+  refine isOpen_iff_mem_nhds.2 fun x hx ↦ ?_
   have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)
-  have : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=
+  have : Filter.Tendsto (fun y ↦ y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=
     (continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _)
   simpa only [SetLike.mem_coe, Filter.mem_map',
     H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg')] using this hg
@@ -269,6 +269,56 @@ theorem isOpen_of_one_mem_interior [ContinuousMul G] (H: Subgroup G)
     (h_1_int : (1 : G) ∈ interior (H : Set G)) : IsOpen (H : Set G) :=
   isOpen_of_mem_nhds H <| mem_interior_iff_mem_nhds.1 h_1_int
 
+@[to_additive]
+lemma isClosed_of_isOpen [ContinuousMul G] (U : Subgroup G) (h : IsOpen (U : Set G)) :
+    IsClosed (U : Set G) :=
+  OpenSubgroup.isClosed ⟨U, h⟩
+
+@[to_additive]
+lemma subgroupOf_isOpen (U K : Subgroup G) (h : IsOpen (K : Set G)) :
+    IsOpen (K.subgroupOf U : Set U) :=
+  Continuous.isOpen_preimage (continuous_iff_le_induced.mpr fun _ ↦ id) _ h
+
+@[to_additive]
+lemma discreteTopology [ContinuousMul G] (U : Subgroup G) (h : IsOpen (U : Set G)) :
+    DiscreteTopology (G ⧸ U) := by
+  refine singletons_open_iff_discrete.mp (fun g ↦ ?_)
+  induction' g using Quotient.inductionOn with g
+  show IsOpen (QuotientGroup.mk ⁻¹' {QuotientGroup.mk g})
+  convert_to IsOpen ((g * ·) '' U)
+  · ext g'
+    simp only [Set.mem_preimage, Set.mem_singleton_iff, QuotientGroup.eq, Set.image_mul_left]
+    rw [← U.inv_mem_iff]
+    simp
+  · exact Homeomorph.mulLeft g |>.isOpen_image |>.mpr h
+
+@[to_additive]
+instance [ContinuousMul G] (U : OpenSubgroup G) : DiscreteTopology (G ⧸ U.toSubgroup) :=
+  discreteTopology U.toSubgroup U.isOpen
+
+@[to_additive]
+lemma quotient_finite_of_isOpen [ContinuousMul G] [CompactSpace G] (U : Subgroup G)
+    (h : IsOpen (U : Set G)) : Finite (G ⧸ U) :=
+  have : DiscreteTopology (G ⧸ U) := U.discreteTopology h
+  finite_of_compact_of_discrete
+
+@[to_additive]
+instance [ContinuousMul G] [CompactSpace G] (U : OpenSubgroup G) : Finite (G ⧸ U.toSubgroup) :=
+  quotient_finite_of_isOpen U.toSubgroup U.isOpen
+
+@[to_additive]
+lemma quotient_finite_of_isOpen' [TopologicalGroup G] [CompactSpace G] (U : Subgroup G)
+    (K : Subgroup U) (hUopen : IsOpen (U : Set G)) (hKopen : IsOpen (K : Set U)) :
+    Finite (U ⧸ K) :=
+  have : CompactSpace U := isCompact_iff_compactSpace.mp <| IsClosed.isCompact <|
+    U.isClosed_of_isOpen hUopen
+  K.quotient_finite_of_isOpen hKopen
+
+@[to_additive]
+instance [TopologicalGroup G] [CompactSpace G] (U : OpenSubgroup G) (K : OpenSubgroup U) :
+    Finite (U ⧸ K.toSubgroup) :=
+  quotient_finite_of_isOpen' U.toSubgroup K.toSubgroup U.isOpen K.isOpen
+
 end Subgroup
 
 namespace OpenSubgroup
@@ -277,7 +327,7 @@ variable {G : Type*} [Group G] [TopologicalSpace G] [ContinuousMul G]
 
 @[to_additive]
 instance : Sup (OpenSubgroup G) :=
-  ⟨fun U V => ⟨U ⊔ V, Subgroup.isOpen_mono (le_sup_left : U.1 ≤ U.1 ⊔ V.1) U.isOpen⟩⟩
+  ⟨fun U V ↦ ⟨U ⊔ V, Subgroup.isOpen_mono (le_sup_left : U.1 ≤ U.1 ⊔ V.1) U.isOpen⟩⟩
 
 @[to_additive (attr := simp, norm_cast)]
 theorem toSubgroup_sup (U V : OpenSubgroup G) : (↑(U ⊔ V) : Subgroup G) = ↑U ⊔ ↑V := rfl
@@ -286,7 +336,7 @@ theorem toSubgroup_sup (U V : OpenSubgroup G) : (↑(U ⊔ V) : Subgroup G) = 
 @[to_additive]
 instance : Lattice (OpenSubgroup G) :=
   { instSemilatticeInfOpenSubgroup,
-    toSubgroup_injective.semilatticeSup ((↑) : OpenSubgroup G → Subgroup G) fun _ _ => rfl with
+    toSubgroup_injective.semilatticeSup ((↑) : OpenSubgroup G → Subgroup G) fun _ _ ↦ rfl with
     toPartialOrder := instPartialOrderOpenSubgroup }
 
 end OpenSubgroup
@@ -314,3 +364,92 @@ theorem isOpen_of_isOpen_subideal {U I : Ideal R} (h : U ≤ I) (hU : IsOpen (U
   @Submodule.isOpen_mono R R _ _ _ _ Semiring.toModule _ _ h hU
 
 end Ideal
+
+/-!
+# Open normal subgroups of a topological group
+
+This section builds the lattice `OpenNormalSubgroup G` of open subgroups in a topological group `G`,
+and its additive version `OpenNormalAddSubgroup`.
+
+-/
+
+section
+
+universe u
+
+/-- The type of open normal subgroups of a topological group. -/
+@[ext]
+structure OpenNormalSubgroup (G : Type u) [Group G] [TopologicalSpace G]
+  extends OpenSubgroup G where
+  isNormal' : toSubgroup.Normal := by infer_instance
+
+/-- The type of open normal subgroups of a topological additive group. -/
+@[ext]
+structure OpenNormalAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G]
+  extends OpenAddSubgroup G where
+  isNormal' : toAddSubgroup.Normal := by infer_instance
+
+attribute [to_additive] OpenNormalSubgroup
+
+namespace OpenNormalSubgroup
+
+variable {G : Type u} [Group G] [TopologicalSpace G]
+
+@[to_additive]
+instance (H : OpenNormalSubgroup G) : H.toSubgroup.Normal := H.isNormal'
+
+@[to_additive]
+theorem toSubgroup_injective : Function.Injective
+    (fun H ↦ H.toOpenSubgroup.toSubgroup : OpenNormalSubgroup G → Subgroup G) :=
+  fun A B h ↦ by
+  ext
+  dsimp at h
+  rw [h]
+
+@[to_additive]
+instance : SetLike (OpenNormalSubgroup G) G where
+  coe U := U.1
+  coe_injective' _ _ h := toSubgroup_injective <| SetLike.ext' h
+
+@[to_additive]
+instance : SubgroupClass (OpenNormalSubgroup G) G where
+  mul_mem := Subsemigroup.mul_mem' _
+  one_mem U := U.one_mem'
+  inv_mem := Subgroup.inv_mem' _
+
+@[to_additive]
+instance : Coe (OpenNormalSubgroup G) (Subgroup G) where
+  coe H := H.toOpenSubgroup.toSubgroup
+
+@[to_additive]
+instance instPartialOrderOpenNormalSubgroup : PartialOrder (OpenNormalSubgroup G) := inferInstance
+
+@[to_additive]
+instance instInfOpenNormalSubgroup : Inf (OpenNormalSubgroup G) :=
+  ⟨fun U V ↦ ⟨U.toOpenSubgroup ⊓ V.toOpenSubgroup,
+    Subgroup.normal_inf_normal U.toSubgroup V.toSubgroup⟩⟩
+
+@[to_additive]
+instance instSemilatticeInfOpenNormalSubgroup : SemilatticeInf (OpenNormalSubgroup G) :=
+  SetLike.coe_injective.semilatticeInf ((↑) : OpenNormalSubgroup G → Set G) fun _ _ ↦ rfl
+
+@[to_additive]
+instance [ContinuousMul G] : Sup (OpenNormalSubgroup G) :=
+  ⟨fun U V ↦ ⟨U.toOpenSubgroup ⊔ V.toOpenSubgroup,
+    Subgroup.sup_normal U.toOpenSubgroup.1 V.toOpenSubgroup.1⟩⟩
+
+@[to_additive]
+instance instSemilatticeSupOpenNormalSubgroup [ContinuousMul G] :
+    SemilatticeSup (OpenNormalSubgroup G) :=
+  toSubgroup_injective.semilatticeSup
+    (fun (H : OpenNormalSubgroup G) ↦ ↑H.toOpenSubgroup) (fun _ _ ↦ rfl)
+
+@[to_additive]
+instance [ContinuousMul G] : Lattice (OpenNormalSubgroup G) :=
+  { instSemilatticeInfOpenNormalSubgroup,
+    instSemilatticeSupOpenNormalSubgroup with
+    toPartialOrder := instPartialOrderOpenNormalSubgroup}
+
+end OpenNormalSubgroup
+
+end
diff --git a/Mathlib/Topology/Algebra/Order/Compact.lean b/Mathlib/Topology/Algebra/Order/Compact.lean
index 4bb1345749e01..1f1e14890b141 100644
--- a/Mathlib/Topology/Algebra/Order/Compact.lean
+++ b/Mathlib/Topology/Algebra/Order/Compact.lean
@@ -250,7 +250,8 @@ theorem cocompact_eq_atTop [NoMaxOrder α] [OrderBot α]
 theorem IsCompact.exists_isMinOn [ClosedIicTopology α] {s : Set β} (hs : IsCompact s)
     (ne_s : s.Nonempty) {f : β → α} (hf : ContinuousOn f s) : ∃ x ∈ s, IsMinOn f s x := by
   rcases (hs.image_of_continuousOn hf).exists_isLeast (ne_s.image f) with ⟨_, ⟨x, hxs, rfl⟩, hx⟩
-  exact ⟨x, hxs, forall_mem_image.1 hx⟩
+  refine ⟨x, hxs, forall_mem_image.1 (fun _ hb => hx <| mem_image_of_mem f ?_)⟩
+  rwa [(image_id' s).symm]
 
 /-- If a continuous function lies strictly above `a` on a compact set,
   it has a lower bound strictly above `a`. -/
diff --git a/Mathlib/Topology/Algebra/Order/Field.lean b/Mathlib/Topology/Algebra/Order/Field.lean
index 39438737e3443..b80ccdfcef320 100644
--- a/Mathlib/Topology/Algebra/Order/Field.lean
+++ b/Mathlib/Topology/Algebra/Order/Field.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
 -/
 import Mathlib.Data.Set.Pointwise.Interval
+import Mathlib.Order.Filter.AtTopBot.Field
 import Mathlib.Topology.Algebra.Field
 import Mathlib.Topology.Algebra.Order.Group
 
@@ -75,7 +76,8 @@ tends to a negative constant `C` then `f * g` tends to `Filter.atBot`. -/
 theorem Filter.Tendsto.atTop_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atTop)
     (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
   have := hf.atTop_mul (neg_pos.2 hC) hg.neg
-  simpa only [(· ∘ ·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this
+  simpa only [Function.comp_def, neg_mul_eq_mul_neg, neg_neg] using
+    tendsto_neg_atTop_atBot.comp this
 
 /-- In a linearly ordered field with the order topology, if `f` tends to a negative constant `C` and
 `g` tends to `Filter.atTop` then `f * g` tends to `Filter.atBot`. -/
@@ -88,14 +90,14 @@ tends to a positive constant `C` then `f * g` tends to `Filter.atBot`. -/
 theorem Filter.Tendsto.atBot_mul {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atBot)
     (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atBot := by
   have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul hC hg
-  simpa [(· ∘ ·)] using tendsto_neg_atTop_atBot.comp this
+  simpa [Function.comp_def] using tendsto_neg_atTop_atBot.comp this
 
 /-- In a linearly ordered field with the order topology, if `f` tends to `Filter.atBot` and `g`
 tends to a negative constant `C` then `f * g` tends to `Filter.atTop`. -/
 theorem Filter.Tendsto.atBot_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atBot)
     (hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by
   have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
-  simpa [(· ∘ ·)] using tendsto_neg_atBot_atTop.comp this
+  simpa [Function.comp_def] using tendsto_neg_atBot_atTop.comp this
 
 /-- In a linearly ordered field with the order topology, if `f` tends to a positive constant `C` and
 `g` tends to `Filter.atBot` then `f * g` tends to `Filter.atBot`. -/
@@ -193,10 +195,10 @@ instance (priority := 100) LinearOrderedSemifield.toHasContinuousInv₀ {𝕜}
   · obtain ⟨x', h₀, hxx', h₁⟩ : ∃ x', 0 < x' ∧ x ≤ x' ∧ x' < 1 :=
       ⟨max x (1 / 2), one_half_pos.trans_le (le_max_right _ _), le_max_left _ _,
         max_lt hx one_half_lt_one⟩
-    filter_upwards [Ioo_mem_nhds one_pos (one_lt_inv h₀ h₁)] with y hy
-    exact hxx'.trans_lt <| inv_inv x' ▸ inv_lt_inv_of_lt hy.1 hy.2
-  · filter_upwards [Ioi_mem_nhds (inv_lt_one hx)] with y hy
-    simpa only [inv_inv] using inv_lt_inv_of_lt (inv_pos.2 <| one_pos.trans hx) hy
+    filter_upwards [Ioo_mem_nhds one_pos ((one_lt_inv₀ h₀).2 h₁)] with y hy
+    exact hxx'.trans_lt <| lt_inv_of_lt_inv₀ hy.1 hy.2
+  · filter_upwards [Ioi_mem_nhds (inv_lt_one_of_one_lt₀ hx)] with y hy
+    exact inv_lt_of_inv_lt₀ (by positivity) hy
 
 instance (priority := 100) LinearOrderedField.toTopologicalDivisionRing :
     TopologicalDivisionRing 𝕜 := ⟨⟩
diff --git a/Mathlib/Topology/Algebra/Order/Floor.lean b/Mathlib/Topology/Algebra/Order/Floor.lean
index 3877c5331e7bf..bde8ae91b5715 100644
--- a/Mathlib/Topology/Algebra/Order/Floor.lean
+++ b/Mathlib/Topology/Algebra/Order/Floor.lean
@@ -3,9 +3,8 @@ Copyright (c) 2020 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 -/
-import Mathlib.Algebra.Order.Floor
+import Mathlib.Order.Filter.AtTopBot.Floor
 import Mathlib.Topology.Algebra.Order.Group
-import Mathlib.Topology.Order.Basic
 
 /-!
 # Topological facts about `Int.floor`, `Int.ceil` and `Int.fract`
@@ -27,8 +26,36 @@ This file proves statements about limits and continuity of functions involving `
 
 open Filter Function Int Set Topology
 
+namespace FloorSemiring
+
+open scoped Nat
+
+variable {K : Type*} [LinearOrderedField K] [FloorSemiring K] [TopologicalSpace K] [OrderTopology K]
+
+theorem tendsto_mul_pow_div_factorial_sub_atTop (a c : K) (d : ℕ) :
+    Tendsto (fun n ↦ a * c ^ n / (n - d)!) atTop (𝓝 0) := by
+  rw [tendsto_order]
+  constructor
+  all_goals
+    intro ε hε
+    filter_upwards [eventually_mul_pow_lt_factorial_sub (a * ε⁻¹) c d] with n h
+    rw [mul_right_comm, ← div_eq_mul_inv] at h
+  · rw [div_lt_iff_of_neg hε] at h
+    rwa [lt_div_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]
+  · rw [div_lt_iff₀ hε] at h
+    rwa [div_lt_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]
+
+theorem tendsto_pow_div_factorial_atTop (c : K) :
+    Tendsto (fun n ↦ c ^ n / n !) atTop (𝓝 0) := by
+  convert tendsto_mul_pow_div_factorial_sub_atTop 1 c 0
+  rw [one_mul]
+
+end FloorSemiring
+
 variable {α β γ : Type*} [LinearOrderedRing α] [FloorRing α]
 
+-- TODO: move to `Mathlib.Order.Filter.AtTopBot.Floor`
+
 theorem tendsto_floor_atTop : Tendsto (floor : α → ℤ) atTop atTop :=
   floor_mono.tendsto_atTop_atTop fun b =>
     ⟨(b + 1 : ℤ), by rw [floor_intCast]; exact (lt_add_one _).le⟩
@@ -179,7 +206,7 @@ theorem ContinuousOn.comp_fract' {f : β → α → γ} (h : ContinuousOn (uncur
         (tendsto_id.prod_map (tendsto_fract_right _))).mono_right (le_of_eq ?_) <;>
         simp [nhdsWithin_prod_eq, nhdsWithin_univ]
   · replace ht : t ≠ ⌊t⌋ := fun ht' => ht ⟨_, ht'⟩
-    refine (h.continuousAt ?_).comp (continuousAt_id.prod_map (continuousAt_fract ht))
+    refine (h.continuousAt ?_).comp (continuousAt_id.prodMap (continuousAt_fract ht))
     exact prod_mem_nhds univ_mem (Icc_mem_nhds (fract_pos.2 ht) (fract_lt_one _))
 
 theorem ContinuousOn.comp_fract {s : β → α} {f : β → α → γ}
diff --git a/Mathlib/Topology/Algebra/Order/Group.lean b/Mathlib/Topology/Algebra/Order/Group.lean
index ed113a88b9cf3..be3a1247cf1fa 100644
--- a/Mathlib/Topology/Algebra/Order/Group.lean
+++ b/Mathlib/Topology/Algebra/Order/Group.lean
@@ -15,9 +15,9 @@ topological group. We also prove continuity of `abs : G → G` and provide conve
 -/
 
 
-open Set Filter
+open Set Filter Function
 
-open Topology Filter
+open scoped Topology
 
 variable {α G : Type*} [TopologicalSpace G] [LinearOrderedAddCommGroup G] [OrderTopology G]
 variable {l : Filter α} {f g : α → G}
@@ -88,3 +88,39 @@ protected theorem ContinuousOn.abs (h : ContinuousOn f s) : ContinuousOn (fun x
 theorem tendsto_abs_nhdsWithin_zero : Tendsto (abs : G → G) (𝓝[≠] 0) (𝓝[>] 0) :=
   (continuous_abs.tendsto' (0 : G) 0 abs_zero).inf <|
     tendsto_principal_principal.2 fun _x => abs_pos.2
+
+/-- In a linearly ordered additive group, the integer multiples of an element are dense
+iff they are the whole group. -/
+theorem denseRange_zsmul_iff_surjective {a : G} :
+    DenseRange (· • a : ℤ → G) ↔ Surjective (· • a : ℤ → G) := by
+  refine ⟨fun h ↦ ?_, fun h ↦ h.denseRange⟩
+  wlog ha₀ : 0 < a generalizing a
+  · simp only [← range_iff_surjective, DenseRange] at *
+    rcases (not_lt.1 ha₀).eq_or_lt with rfl | hlt
+    · simpa only [smul_zero, range_const, dense_iff_closure_eq, closure_singleton] using h
+    · have H : range (· • -a : ℤ → G) = range (· • a : ℤ → G) := by
+        simpa only [smul_neg, ← neg_smul] using neg_surjective.range_comp (· • a)
+      rw [← H]
+      apply this <;> simpa only [H, neg_pos]
+  intro b
+  obtain ⟨m, hm, hm'⟩ : ∃ m : ℤ, m • a ∈ Ioo b (b + a + a) := by
+    have hne : (Ioo b (b + a + a)).Nonempty := ⟨b + a, by simpa⟩
+    simpa using h.exists_mem_open isOpen_Ioo hne
+  rcases eq_or_ne b ((m - 1) • a) with rfl | hne; · simp
+  suffices (Ioo (m • a) ((m + 1) • a)).Nonempty by
+    rcases h.exists_mem_open isOpen_Ioo this with ⟨l, hl⟩
+    have : m < l ∧ l < m + 1 := by simpa [zsmul_lt_zsmul_iff ha₀] using hl
+    omega
+  rcases hne.lt_or_lt with hlt | hlt
+  · refine ⟨b + a + a, hm', ?_⟩
+    simpa only [add_smul, sub_smul, one_smul, lt_sub_iff_add_lt, add_lt_add_iff_right] using hlt
+  · use b + a
+    simp only [mem_Ioo, add_smul, sub_smul, one_smul, add_lt_add_iff_right] at hlt ⊢
+    exact ⟨sub_lt_iff_lt_add.1 hlt, hm⟩
+
+/-- In a nontrivial densely linearly ordered additive group,
+the integer multiples of an element can't be dense. -/
+theorem not_denseRange_zsmul [Nontrivial G] [DenselyOrdered G] {a : G} :
+    ¬DenseRange (· • a : ℤ → G) :=
+  denseRange_zsmul_iff_surjective.not.mpr fun h ↦
+    not_isAddCyclic_of_denselyOrdered G ⟨⟨a, h⟩⟩
diff --git a/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean b/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
index 76d0241ae2ffb..2514169662c24 100644
--- a/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
+++ b/Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
@@ -7,7 +7,7 @@ import Mathlib.Algebra.BigOperators.Intervals
 import Mathlib.Algebra.BigOperators.Ring
 import Mathlib.Algebra.Order.Group.Indicator
 import Mathlib.Order.LiminfLimsup
-import Mathlib.Order.Filter.Archimedean
+import Mathlib.Order.Filter.AtTopBot.Archimedean
 import Mathlib.Order.Filter.CountableInter
 import Mathlib.Topology.Algebra.Group.Basic
 import Mathlib.Data.Set.Lattice
@@ -167,8 +167,8 @@ theorem limsSup_nhds (a : α) : limsSup (𝓝 a) = a :=
       | Or.inl ⟨c, hac, hcb⟩ => ⟨c, ge_mem_nhds hac, hcb⟩
       | Or.inr ⟨_, h⟩ => ⟨a, (𝓝 a).sets_of_superset (gt_mem_nhds hba) h, hba⟩
 
-theorem limsInf_nhds : ∀ a : α, limsInf (𝓝 a) = a :=
-  limsSup_nhds (α := αᵒᵈ)
+theorem limsInf_nhds (a : α) : limsInf (𝓝 a) = a :=
+  limsSup_nhds (α := αᵒᵈ) a
 
 /-- If a filter is converging, its limsup coincides with its limit. -/
 theorem limsInf_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsInf = a :=
@@ -184,8 +184,8 @@ theorem limsInf_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝
       _ ≤ f.limsInf := limsInf_le_limsInf_of_le h (isBounded_ge_nhds a) hb_le.isCobounded_flip)
 
 /-- If a filter is converging, its liminf coincides with its limit. -/
-theorem limsSup_eq_of_le_nhds : ∀ {f : Filter α} {a : α} [NeBot f], f ≤ 𝓝 a → f.limsSup = a :=
-  limsInf_eq_of_le_nhds (α := αᵒᵈ)
+theorem limsSup_eq_of_le_nhds {f : Filter α} {a : α} [NeBot f] (h : f ≤ 𝓝 a) : f.limsSup = a :=
+  limsInf_eq_of_le_nhds (α := αᵒᵈ) h
 
 /-- If a function has a limit, then its limsup coincides with its limit. -/
 theorem Filter.Tendsto.limsup_eq {f : Filter β} {u : β → α} {a : α} [NeBot f]
@@ -209,12 +209,9 @@ and is greater than or equal to the `limsup` of `f`, then `f` tends to `a` along
 theorem tendsto_of_le_liminf_of_limsup_le {f : Filter β} {u : β → α} {a : α} (hinf : a ≤ liminf u f)
     (hsup : limsup u f ≤ a) (h : f.IsBoundedUnder (· ≤ ·) u := by isBoundedDefault)
     (h' : f.IsBoundedUnder (· ≥ ·) u := by isBoundedDefault) : Tendsto u f (𝓝 a) := by
-  classical
-  by_cases hf : f = ⊥
-  · rw [hf]
-    exact tendsto_bot
-  · haveI : NeBot f := ⟨hf⟩
-    exact tendsto_of_liminf_eq_limsup (le_antisymm (le_trans (liminf_le_limsup h h') hsup) hinf)
+  rcases f.eq_or_neBot with rfl | _
+  · exact tendsto_bot
+  · exact tendsto_of_liminf_eq_limsup (le_antisymm (le_trans (liminf_le_limsup h h') hsup) hinf)
       (le_antisymm hsup (le_trans hinf (liminf_le_limsup h h'))) h h'
 
 /-- Assume that, for any `a < b`, a sequence can not be infinitely many times below `a` and
@@ -303,7 +300,7 @@ theorem Antitone.map_limsSup_of_continuousAt {F : Filter R} [NeBot F] {f : R →
     (cobdd : F.IsCobounded (· ≤ ·) := by isBoundedDefault) :
     f F.limsSup = F.liminf f := by
   apply le_antisymm
-  · rw [limsSup, f_decr.map_sInf_of_continuousAt' f_cont bdd_above cobdd]
+  · rw [limsSup, f_decr.map_csInf_of_continuousAt f_cont bdd_above cobdd]
     apply le_of_forall_lt
     intro c hc
     simp only [liminf, limsInf, eventually_map] at hc ⊢
@@ -559,20 +556,41 @@ lemma liminf_add_const (F : Filter ι) [NeBot F] [Add R] [ContinuousAdd R]
     (fun _ _ h ↦ add_le_add_right h c) (continuous_add_right c).continuousAt cobdd bdd_below).symm
 
 /-- `limsup (c - xᵢ) = c - liminf xᵢ`. -/
-lemma limsup_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
-    [OrderedSub R] [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
+lemma limsup_const_sub (F : Filter ι) [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R]
+    [CovariantClass R R (fun x y ↦ x + y) fun x y ↦ x ≤ y] (f : ι → R) (c : R)
     (cobdd : F.IsCoboundedUnder (· ≥ ·) f) (bdd_below : F.IsBoundedUnder (· ≥ ·) f) :
-    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F :=
-  (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ c - x)
+    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F := by
+  rcases F.eq_or_neBot with rfl | _
+  · simp only [liminf, limsInf, limsup, limsSup, map_bot, eventually_bot, Set.setOf_true]
+    simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
+    rcases cobdd with ⟨x, hx⟩
+    refine (csInf_le ?_ (Set.mem_univ _)).antisymm
+      (tsub_le_iff_tsub_le.1 (le_csSup ?_ (Set.mem_univ _)))
+    · refine ⟨x - x, mem_lowerBounds.2 fun y ↦ ?_⟩
+      simp only [Set.mem_univ, true_implies]
+      exact tsub_le_iff_tsub_le.1 (hx (x - y))
+    · refine ⟨x, mem_upperBounds.2 fun y ↦ ?_⟩
+      simp only [Set.mem_univ, hx y, implies_true]
+  · exact (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ c - x)
     (fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c).continuousAt cobdd bdd_below).symm
 
 /-- `limsup (xᵢ - c) = (limsup xᵢ) - c`. -/
-lemma limsup_sub_const (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
-    [OrderedSub R] (f : ι → R) (c : R)
+lemma limsup_sub_const (F : Filter ι) [AddCommSemigroup R] [Sub R] [ContinuousSub R] [OrderedSub R]
+    (f : ι → R) (c : R)
     (bdd_above : F.IsBoundedUnder (· ≤ ·) f) (cobdd : F.IsCoboundedUnder (· ≤ ·) f) :
-    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c :=
-  (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ x - c)
-    (fun _ _ h ↦ tsub_le_tsub_right h c) (continuous_sub_right c).continuousAt bdd_above cobdd).symm
+    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c := by
+  rcases F.eq_or_neBot with rfl | _
+  · have {a : R} : sInf Set.univ ≤ a := by
+      apply csInf_le _ (Set.mem_univ a)
+      simp only [IsCoboundedUnder, IsCobounded, map_bot, eventually_bot, true_implies] at cobdd
+      rcases cobdd with ⟨x, hx⟩
+      refine ⟨x, mem_lowerBounds.2 fun y ↦ ?_⟩
+      simp only [Set.mem_univ, hx y, implies_true]
+    simp only [limsup, limsSup, map_bot, eventually_bot, Set.setOf_true]
+    exact this.antisymm (tsub_le_iff_right.2 this)
+  · apply (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : R) ↦ x - c) _ _).symm
+    · exact fun _ _ h ↦ tsub_le_tsub_right h c
+    · exact (continuous_sub_right c).continuousAt
 
 /-- `liminf (c - xᵢ) = c - limsup xᵢ`. -/
 lemma liminf_const_sub (F : Filter ι) [NeBot F] [AddCommSemigroup R] [Sub R] [ContinuousSub R]
diff --git a/Mathlib/Topology/Algebra/Polynomial.lean b/Mathlib/Topology/Algebra/Polynomial.lean
index 82175ebb00e13..663b8bd81dc20 100644
--- a/Mathlib/Topology/Algebra/Polynomial.lean
+++ b/Mathlib/Topology/Algebra/Polynomial.lean
@@ -6,7 +6,7 @@ Authors: Robert Y. Lewis
 import Mathlib.Algebra.Polynomial.AlgebraMap
 import Mathlib.Algebra.Polynomial.Inductions
 import Mathlib.Algebra.Polynomial.Splits
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 import Mathlib.RingTheory.Polynomial.Vieta
 
 /-!
@@ -191,8 +191,7 @@ theorem coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]} (h1
       _ ≤ max B 1 ^ d * d.choose (d / 2) := by
         gcongr; exact (i.choose_mono h3).trans (i.choose_le_middle d)
   · rw [eq_one_of_roots_le hB h1 h2 h4, Polynomial.map_one, coeff_one]
-    refine _root_.trans ?_
-      (one_le_mul_of_one_le_of_one_le (one_le_pow_of_one_le (le_max_right B 1) d) ?_)
+    refine le_trans ?_ (one_le_mul_of_one_le_of_one_le (one_le_pow₀ (le_max_right B 1)) ?_)
     · split_ifs <;> norm_num
     · exact mod_cast Nat.succ_le_iff.mpr (Nat.choose_pos (d.div_le_self 2))
 
diff --git a/Mathlib/Topology/Algebra/PontryaginDual.lean b/Mathlib/Topology/Algebra/PontryaginDual.lean
index 80aa1eef70921..a5ba95512fe96 100644
--- a/Mathlib/Topology/Algebra/PontryaginDual.lean
+++ b/Mathlib/Topology/Algebra/PontryaginDual.lean
@@ -22,9 +22,9 @@ isomorphic to its double dual.
 
 open Pointwise Function
 
-variable (A B C D E G : Type*) [Monoid A] [Monoid B] [Monoid C] [Monoid D] [CommGroup E] [Group G]
-  [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C] [TopologicalSpace D]
-  [TopologicalSpace E] [TopologicalSpace G] [TopologicalGroup E] [TopologicalGroup G]
+variable (A B C G H : Type*) [Monoid A] [Monoid B] [Monoid C] [CommGroup G] [Group H]
+  [TopologicalSpace A] [TopologicalSpace B] [TopologicalSpace C]
+  [TopologicalSpace G] [TopologicalSpace H] [TopologicalGroup G] [TopologicalGroup H]
 
 /-- The Pontryagin dual of `A` is the group of continuous homomorphism `A → Circle`. -/
 def PontryaginDual :=
@@ -48,23 +48,23 @@ instance : TopologicalGroup (PontryaginDual A) :=
 noncomputable instance : Inhabited (PontryaginDual A) :=
   (inferInstance : Inhabited (ContinuousMonoidHom A Circle))
 
-instance [LocallyCompactSpace G] : LocallyCompactSpace (PontryaginDual G) := by
+instance [LocallyCompactSpace H] : LocallyCompactSpace (PontryaginDual H) := by
   let Vn : ℕ → Set Circle :=
     fun n ↦ Circle.exp '' { x | |x| < Real.pi / 2 ^ (n + 1)}
   have hVn : ∀ n x, x ∈ Vn n ↔ |Complex.arg x| < Real.pi / 2 ^ (n + 1) := by
     refine fun n x ↦ ⟨?_, fun hx ↦ ⟨Complex.arg x, hx, Circle.exp_arg x⟩⟩
     rintro ⟨t, ht : |t| < _, rfl⟩
-    have ht' := ht.trans_le (div_le_self Real.pi_nonneg (one_le_pow_of_one_le one_le_two (n + 1)))
+    have ht' := ht.trans_le (div_le_self Real.pi_nonneg (one_le_pow₀ one_le_two))
     rwa [Circle.arg_exp (neg_lt_of_abs_lt ht') (lt_of_abs_lt ht').le]
   refine ContinuousMonoidHom.locallyCompactSpace_of_hasBasis Vn ?_ ?_
   · intro n x h1 h2
     rw [hVn] at h1 h2 ⊢
     rwa [Circle.coe_mul, Complex.arg_mul x.coe_ne_zero x.coe_ne_zero,
-      ← two_mul, abs_mul, abs_two, ← lt_div_iff' two_pos, div_div, ← pow_succ] at h2
+      ← two_mul, abs_mul, abs_two, ← lt_div_iff₀' two_pos, div_div, ← pow_succ] at h2
     apply Set.Ioo_subset_Ioc_self
-    rw [← two_mul, Set.mem_Ioo, ← abs_lt, abs_mul, abs_two, ← lt_div_iff' two_pos]
+    rw [← two_mul, Set.mem_Ioo, ← abs_lt, abs_mul, abs_two, ← lt_div_iff₀' two_pos]
     exact h1.trans_le
-      (div_le_div_of_nonneg_left Real.pi_nonneg two_pos (le_self_pow one_le_two n.succ_ne_zero))
+      (div_le_div_of_nonneg_left Real.pi_nonneg two_pos (le_self_pow₀ one_le_two n.succ_ne_zero))
   · rw [← Circle.exp_zero, ← isLocalHomeomorph_circleExp.map_nhds_eq 0]
     refine ((nhds_basis_zero_abs_sub_lt ℝ).to_hasBasis
         (fun x hx ↦ ⟨Nat.ceil (Real.pi / x), trivial, fun t ht ↦ ?_⟩)
@@ -75,7 +75,7 @@ instance [LocallyCompactSpace G] : LocallyCompactSpace (PontryaginDual G) := by
     refine (Nat.le_ceil (Real.pi / x)).trans ?_
     exact_mod_cast (Nat.le_succ _).trans (Nat.lt_two_pow _).le
 
-variable {A B C D E}
+variable {A B C G}
 
 namespace PontryaginDual
 
@@ -107,15 +107,15 @@ theorem map_comp (g : ContinuousMonoidHom B C) (f : ContinuousMonoidHom A B) :
   ext fun _x => ext fun _y => rfl
 
 @[simp]
-nonrec theorem map_mul (f g : ContinuousMonoidHom A E) : map (f * g) = map f * map g :=
+nonrec theorem map_mul (f g : ContinuousMonoidHom A G) : map (f * g) = map f * map g :=
   ext fun x => ext fun y => map_mul x (f y) (g y)
 
-variable (A B C D E)
+variable (A B C G)
 
 /-- `ContinuousMonoidHom.dual` as a `ContinuousMonoidHom`. -/
-noncomputable def mapHom [LocallyCompactSpace E] :
-    ContinuousMonoidHom (ContinuousMonoidHom A E)
-      (ContinuousMonoidHom (PontryaginDual E) (PontryaginDual A)) where
+noncomputable def mapHom [LocallyCompactSpace G] :
+    ContinuousMonoidHom (ContinuousMonoidHom A G)
+      (ContinuousMonoidHom (PontryaginDual G) (PontryaginDual A)) where
   toFun := map
   map_one' := map_one
   map_mul' := map_mul
diff --git a/Mathlib/Topology/Algebra/ProperAction.lean b/Mathlib/Topology/Algebra/ProperAction.lean
index 2c645d7ab198d..4a337be7878ae 100644
--- a/Mathlib/Topology/Algebra/ProperAction.lean
+++ b/Mathlib/Topology/Algebra/ProperAction.lean
@@ -73,9 +73,8 @@ class ProperSMul (G X : Type*) [TopologicalSpace G] [TopologicalSpace X] [Group
 
 attribute [to_additive existing] properSMul_iff
 
-variable {G X Y Z : Type*} [Group G] [MulAction G X] [MulAction G Y]
-variable [TopologicalSpace G] [TopologicalSpace X] [TopologicalSpace Y]
-variable [TopologicalSpace Z]
+variable {G X : Type*} [Group G] [MulAction G X]
+variable [TopologicalSpace G] [TopologicalSpace X]
 
 /-- If a group acts properly then in particular it acts continuously. -/
 @[to_additive "If a group acts properly then in particular it acts continuously."]
@@ -86,7 +85,7 @@ instance (priority := 100) ProperSMul.toContinuousSMul [ProperSMul G X] : Contin
 /-- A group `G` acts properly on a topological space `X` if and only if for all ultrafilters
 `𝒰` on `X × G`, if `𝒰` converges to `(x₁, x₂)` along the map `(g, x) ↦ (g • x, x)`,
 then there exists `g : G` such that `g • x₂ = x₁` and `𝒰.fst` converges to `g`. -/
-@[to_additive "A group acts `G` properly on a topological space `X` if and only if
+@[to_additive "A group `G` acts properly on a topological space `X` if and only if
 for all ultrafilters `𝒰` on `X`, if `𝒰` converges to `(x₁, x₂)`
 along the map `(g, x) ↦ (g • x, x)`, then there exists `g : G` such that `g • x₂ = x₁`
 and `𝒰.fst` converges to `g`."]
@@ -129,11 +128,9 @@ theorem t2Space_quotient_mulAction_of_properSMul [ProperSMul G X] :
   rw [t2_iff_isClosed_diagonal]
   set R := MulAction.orbitRel G X
   let π : X → Quotient R := Quotient.mk'
-  have : QuotientMap (Prod.map π π) :=
-    (isOpenMap_quotient_mk'_mul.prod isOpenMap_quotient_mk'_mul).to_quotientMap
-      (continuous_quotient_mk'.prod_map continuous_quotient_mk')
-      ((surjective_quotient_mk' _).prodMap (surjective_quotient_mk' _))
-  rw [← this.isClosed_preimage]
+  have : IsOpenQuotientMap (Prod.map π π) :=
+    MulAction.isOpenQuotientMap_quotientMk.prodMap MulAction.isOpenQuotientMap_quotientMk
+  rw [← this.quotientMap.isClosed_preimage]
   convert ProperSMul.isProperMap_smul_pair.isClosedMap.isClosed_range
   · ext ⟨x₁, x₂⟩
     simp only [mem_preimage, map_apply, mem_diagonal_iff, mem_range, Prod.mk.injEq, Prod.exists,
@@ -175,7 +172,7 @@ theorem properSMul_of_closedEmbedding {H : Type*} [Group H] [MulAction H X] [Top
     (f_compat : ∀ (h : H) (x : X), f h • x = h • x) : ProperSMul H X where
   isProperMap_smul_pair := by
     have := isProperMap_of_closedEmbedding f_clemb
-    have h : IsProperMap (Prod.map f (fun x : X ↦ x)) := IsProperMap.prod_map this isProperMap_id
+    have h : IsProperMap (Prod.map f (fun x : X ↦ x)) := this.prodMap isProperMap_id
     have : (fun hx : H × X ↦ (hx.1 • hx.2, hx.2)) = (fun hx ↦ (f hx.1 • hx.2, hx.2)) := by
       simp [f_compat]
     rw [this]
@@ -247,7 +244,7 @@ theorem properlyDiscontinuousSMul_iff_properSMul [T2Space X] [DiscreteTopology G
     apply IsCompact.finite_of_discrete
     -- Now set `h : (g, x) ↦ (g⁻¹ • x, x)`, because `f` is proper by hypothesis, so is `h`.
     have : IsProperMap (fun gx : G × X ↦ (gx.1⁻¹ • gx.2, gx.2)) :=
-      (IsProperMap.prod_map (Homeomorph.isProperMap (Homeomorph.inv G)) isProperMap_id).comp <|
+      (IsProperMap.prodMap (Homeomorph.isProperMap (Homeomorph.inv G)) isProperMap_id).comp <|
         ProperSMul.isProperMap_smul_pair
     --But we also have that `{g | Set.Nonempty ((g • ·) '' K ∩ L)} = h ⁻¹ (K × L)`, which
     -- concludes the proof.
diff --git a/Mathlib/Topology/Algebra/ProperConstSMul.lean b/Mathlib/Topology/Algebra/ProperConstSMul.lean
index 30fbbb4b83bf7..7e3167890f5ee 100644
--- a/Mathlib/Topology/Algebra/ProperConstSMul.lean
+++ b/Mathlib/Topology/Algebra/ProperConstSMul.lean
@@ -61,7 +61,7 @@ instance {M X Y : Type*}
     [SMul M X] [TopologicalSpace X] [ProperConstSMul M X]
     [SMul M Y] [TopologicalSpace Y] [ProperConstSMul M Y] :
     ProperConstSMul M (X × Y) :=
-  ⟨fun c ↦ (isProperMap_smul c X).prod_map (isProperMap_smul c Y)⟩
+  ⟨fun c ↦ (isProperMap_smul c X).prodMap (isProperMap_smul c Y)⟩
 
 instance {M ι : Type*} {X : ι → Type*}
     [∀ i, SMul M (X i)] [∀ i, TopologicalSpace (X i)] [∀ i, ProperConstSMul M (X i)] :
diff --git a/Mathlib/Topology/Algebra/Ring/Basic.lean b/Mathlib/Topology/Algebra/Ring/Basic.lean
index 625d49d63a27c..94cf82277f373 100644
--- a/Mathlib/Topology/Algebra/Ring/Basic.lean
+++ b/Mathlib/Topology/Algebra/Ring/Basic.lean
@@ -40,7 +40,7 @@ The `TopologicalSemiring` class should *only* be instantiated in the presence of
 `NonUnitalNonAssocSemiring` instance; if there is an instance of `NonUnitalNonAssocRing`,
 then `TopologicalRing` should be used. Note: in the presence of `NonAssocRing`, these classes are
 mathematically equivalent (see `TopologicalSemiring.continuousNeg_of_mul` or
-`TopologicalSemiring.toTopologicalRing`).  -/
+`TopologicalSemiring.toTopologicalRing`). -/
 class TopologicalSemiring [TopologicalSpace α] [NonUnitalNonAssocSemiring α] extends
   ContinuousAdd α, ContinuousMul α : Prop
 
diff --git a/Mathlib/Topology/Algebra/Ring/Ideal.lean b/Mathlib/Topology/Algebra/Ring/Ideal.lean
index bd45ee761f821..aa3cf06eb57a1 100644
--- a/Mathlib/Topology/Algebra/Ring/Ideal.lean
+++ b/Mathlib/Topology/Algebra/Ring/Ideal.lean
@@ -51,26 +51,18 @@ instance topologicalRingQuotientTopology : TopologicalSpace (R ⧸ N) :=
 -- note for the reader: in the following, `mk` is `Ideal.Quotient.mk`, the canonical map `R → R/I`.
 variable [TopologicalRing R]
 
-theorem QuotientRing.isOpenMap_coe : IsOpenMap (mk N) := by
-  intro s s_op
-  change IsOpen (mk N ⁻¹' (mk N '' s))
-  rw [quotient_ring_saturate]
-  exact isOpen_iUnion fun ⟨n, _⟩ => isOpenMap_add_left n s s_op
+theorem QuotientRing.isOpenMap_coe : IsOpenMap (mk N) :=
+  QuotientAddGroup.isOpenMap_coe
+
+theorem QuotientRing.isOpenQuotientMap_mk : IsOpenQuotientMap (mk N) :=
+  QuotientAddGroup.isOpenQuotientMap_mk
 
 theorem QuotientRing.quotientMap_coe_coe : QuotientMap fun p : R × R => (mk N p.1, mk N p.2) :=
-  IsOpenMap.to_quotientMap ((QuotientRing.isOpenMap_coe N).prod (QuotientRing.isOpenMap_coe N))
-    ((continuous_quot_mk.comp continuous_fst).prod_mk (continuous_quot_mk.comp continuous_snd))
-    (by rintro ⟨⟨x⟩, ⟨y⟩⟩; exact ⟨(x, y), rfl⟩)
-
-instance topologicalRing_quotient : TopologicalRing (R ⧸ N) :=
-  TopologicalSemiring.toTopologicalRing
-    { continuous_add :=
-        have cont : Continuous (mk N ∘ fun p : R × R => p.fst + p.snd) :=
-          continuous_quot_mk.comp continuous_add
-        (QuotientMap.continuous_iff (QuotientRing.quotientMap_coe_coe N)).mpr cont
-      continuous_mul :=
-        have cont : Continuous (mk N ∘ fun p : R × R => p.fst * p.snd) :=
-          continuous_quot_mk.comp continuous_mul
-        (QuotientMap.continuous_iff (QuotientRing.quotientMap_coe_coe N)).mpr cont }
+  ((isOpenQuotientMap_mk N).prodMap (isOpenQuotientMap_mk N)).quotientMap
+
+instance topologicalRing_quotient : TopologicalRing (R ⧸ N) where
+  __ := QuotientAddGroup.instTopologicalAddGroup _
+  continuous_mul := (QuotientRing.quotientMap_coe_coe N).continuous_iff.2 <|
+    continuous_quot_mk.comp continuous_mul
 
 end CommRing
diff --git a/Mathlib/Topology/Algebra/SeparationQuotient.lean b/Mathlib/Topology/Algebra/SeparationQuotient.lean
index 140e089dc9563..83775bba6cbe3 100644
--- a/Mathlib/Topology/Algebra/SeparationQuotient.lean
+++ b/Mathlib/Topology/Algebra/SeparationQuotient.lean
@@ -3,8 +3,9 @@ Copyright (c) 2024 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Topology.Algebra.Module.Basic
 import Mathlib.LinearAlgebra.Basis.VectorSpace
+import Mathlib.Topology.Algebra.Module.Basic
+import Mathlib.Topology.Maps.OpenQuotient
 
 /-!
 # Algebraic operations on `SeparationQuotient`
@@ -63,6 +64,12 @@ instance instIsScalarTower [SMul M N] [ContinuousConstSMul N X] [IsScalarTower M
 
 end SMul
 
+instance instContinuousSMul {M X : Type*} [SMul M X] [TopologicalSpace M] [TopologicalSpace X]
+    [ContinuousSMul M X] : ContinuousSMul M (SeparationQuotient X) where
+  continuous_smul := by
+    rw [(IsOpenQuotientMap.id.prodMap isOpenQuotientMap_mk).quotientMap.continuous_iff]
+    exact continuous_mk.comp continuous_smul
+
 instance instSMulZeroClass {M X : Type*} [Zero X] [SMulZeroClass M X] [TopologicalSpace X]
     [ContinuousConstSMul M X] : SMulZeroClass M (SeparationQuotient X) :=
   ZeroHom.smulZeroClass ⟨mk, mk_zero⟩ mk_smul
@@ -190,6 +197,17 @@ instance instCommGroup [CommGroup G] [TopologicalGroup G] : CommGroup (Separatio
 
 end Group
 
+section UniformGroup
+
+@[to_additive]
+instance instUniformGroup {G : Type*} [Group G] [UniformSpace G] [UniformGroup G] :
+    UniformGroup (SeparationQuotient G) where
+  uniformContinuous_div := by
+    rw [uniformContinuous_dom₂]
+    exact uniformContinuous_mk.comp uniformContinuous_div
+
+end UniformGroup
+
 section MonoidWithZero
 
 variable {M₀ : Type*} [TopologicalSpace M₀]
@@ -315,6 +333,12 @@ instance instCommRing [CommRing R] [TopologicalRing R] :
   surjective_mk.commRing mk mk_zero mk_one mk_add mk_mul mk_neg mk_sub mk_smul mk_smul mk_pow
     mk_natCast mk_intCast
 
+/-- `SeparationQuotient.mk` as a `RingHom`. -/
+@[simps]
+def mkRingHom [NonAssocSemiring R] [TopologicalSemiring R] : R →+* SeparationQuotient R where
+  toFun := mk
+  map_one' := mk_one; map_zero' := mk_zero; map_add' := mk_add; map_mul' := mk_mul
+
 end Ring
 
 section DistribSMul
@@ -357,6 +381,21 @@ def mkCLM : M →L[R] SeparationQuotient M where
 
 end Module
 
+section Algebra
+variable {R A : Type*} [CommSemiring R] [Semiring A] [Algebra R A]
+    [TopologicalSpace A] [TopologicalSemiring A] [ContinuousConstSMul R A]
+
+instance instAlgebra : Algebra R (SeparationQuotient A) where
+  toRingHom := mkRingHom.comp (algebraMap R A)
+  commutes' r := Quotient.ind fun a => congrArg _ <| Algebra.commutes r a
+  smul_def' r := Quotient.ind fun a => congrArg _ <| Algebra.smul_def r a
+
+@[simp]
+theorem mk_algebraMap (r : R) : mk (algebraMap R A r) = algebraMap R (SeparationQuotient A) r :=
+  rfl
+
+end Algebra
+
 section VectorSpace
 
 variable (K E : Type*) [DivisionRing K] [AddCommGroup E] [Module K E]
@@ -391,6 +430,14 @@ theorem mk_outCLM (x : SeparationQuotient E) : mk (outCLM K E x) = x :=
 @[simp]
 theorem mk_comp_outCLM : mk ∘ outCLM K E = id := funext (mk_outCLM K)
 
+variable {K} in
+theorem postcomp_mkCLM_surjective {L : Type*} [Semiring L] (σ : L →+* K)
+    (F : Type*) [AddCommMonoid F] [Module L F] [TopologicalSpace F] :
+    Function.Surjective ((mkCLM K E).comp : (F →SL[σ] E) → (F →SL[σ] SeparationQuotient E)) := by
+  intro f
+  use (outCLM K E).comp f
+  rw [← ContinuousLinearMap.comp_assoc, mkCLM_comp_outCLM, ContinuousLinearMap.id_comp]
+
 /-- The `SeparationQuotient.outCLM K E` map is a topological embedding. -/
 theorem outCLM_embedding : Embedding (outCLM K E) :=
   Function.LeftInverse.embedding (mk_outCLM K) continuous_mk (map_continuous _)
@@ -405,16 +452,22 @@ section VectorSpaceUniform
 variable (K E : Type*) [DivisionRing K] [AddCommGroup E] [Module K E]
     [UniformSpace E] [UniformAddGroup E] [ContinuousConstSMul K E]
 
-theorem outCLM_uniformInducing : UniformInducing (outCLM K E) := by
-  rw [← uniformInducing_mk.uniformInducing_comp_iff, mk_comp_outCLM]
-  exact uniformInducing_id
+theorem outCLM_isUniformInducing : IsUniformInducing (outCLM K E) := by
+  rw [← isUniformInducing_mk.isUniformInducing_comp_iff, mk_comp_outCLM]
+  exact .id
 
-theorem outCLM_uniformEmbedding : UniformEmbedding (outCLM K E) where
+@[deprecated (since := "2024-10-05")]
+alias outCLM_uniformInducing := outCLM_isUniformInducing
+
+theorem outCLM_isUniformEmbedding : IsUniformEmbedding (outCLM K E) where
   inj := outCLM_injective K E
-  toUniformInducing := outCLM_uniformInducing K E
+  toIsUniformInducing := outCLM_isUniformInducing K E
+
+@[deprecated (since := "2024-10-01")]
+alias outCLM_uniformEmbedding := outCLM_isUniformEmbedding
 
 theorem outCLM_uniformContinuous : UniformContinuous (outCLM K E) :=
-  (outCLM_uniformInducing K E).uniformContinuous
+  (outCLM_isUniformInducing K E).uniformContinuous
 
 end VectorSpaceUniform
 
diff --git a/Mathlib/Topology/Algebra/Star.lean b/Mathlib/Topology/Algebra/Star.lean
index deb57a790041b..88af814e80c6e 100644
--- a/Mathlib/Topology/Algebra/Star.lean
+++ b/Mathlib/Topology/Algebra/Star.lean
@@ -6,7 +6,7 @@ Authors: Eric Wieser
 import Mathlib.Algebra.Star.Pi
 import Mathlib.Algebra.Star.Prod
 import Mathlib.Topology.Algebra.Constructions
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Defs
 
 /-!
 # Continuity of `star`
diff --git a/Mathlib/Topology/Algebra/StarSubalgebra.lean b/Mathlib/Topology/Algebra/StarSubalgebra.lean
index c535323f3084f..0703fc4753013 100644
--- a/Mathlib/Topology/Algebra/StarSubalgebra.lean
+++ b/Mathlib/Topology/Algebra/StarSubalgebra.lean
@@ -146,7 +146,7 @@ theorem _root_.StarAlgHom.ext_topologicalClosure [T2Space B] {S : StarSubalgebra
 
 theorem _root_.StarAlgHomClass.ext_topologicalClosure [T2Space B] {F : Type*}
     {S : StarSubalgebra R A} [FunLike F S.topologicalClosure B]
-    [AlgHomClass F R S.topologicalClosure B] [StarAlgHomClass F R S.topologicalClosure B] {φ ψ : F}
+    [AlgHomClass F R S.topologicalClosure B] [StarHomClass F S.topologicalClosure B] {φ ψ : F}
     (hφ : Continuous φ) (hψ : Continuous ψ) (h : ∀ x : S,
         φ (inclusion (le_topologicalClosure S) x) = ψ ((inclusion (le_topologicalClosure S)) x)) :
     φ = ψ := by
@@ -242,12 +242,12 @@ theorem induction_on {x y : A}
     exact mul u (subset_closure hu_mem) v (subset_closure hv_mem) (hu hu_mem) (hv hv_mem)
 
 theorem starAlgHomClass_ext [T2Space B] {F : Type*} {a : A}
-    [FunLike F (elementalStarAlgebra R a) B] [AlgHomClass F R _ B] [StarAlgHomClass F R _ B]
+    [FunLike F (elementalStarAlgebra R a) B] [AlgHomClass F R _ B] [StarHomClass F _ B]
     {φ ψ : F} (hφ : Continuous φ)
     (hψ : Continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) : φ = ψ := by
   -- Note: help with unfolding `elementalStarAlgebra`
-  have : StarAlgHomClass F R (↥(topologicalClosure (adjoin R {a}))) B :=
-    inferInstanceAs (StarAlgHomClass F R (elementalStarAlgebra R a) B)
+  have : StarHomClass F (↥(topologicalClosure (adjoin R {a}))) B :=
+    inferInstanceAs (StarHomClass F (elementalStarAlgebra R a) B)
   refine StarAlgHomClass.ext_topologicalClosure hφ hψ fun x => ?_
   refine adjoin_induction' x ?_ ?_ ?_ ?_ ?_
   exacts [fun y hy => by simpa only [Set.mem_singleton_iff.mp hy] using h, fun r => by
diff --git a/Mathlib/Topology/Algebra/UniformConvergence.lean b/Mathlib/Topology/Algebra/UniformConvergence.lean
index adf3afef18aff..beee906368d6e 100644
--- a/Mathlib/Topology/Algebra/UniformConvergence.lean
+++ b/Mathlib/Topology/Algebra/UniformConvergence.lean
@@ -252,8 +252,8 @@ protected theorem UniformOnFun.hasBasis_nhds_one_of_basis (𝔖 : Set <| Set α)
     (h : (𝓝 1 : Filter G).HasBasis p b) :
     (𝓝 1 : Filter (α →ᵤ[𝔖] G)).HasBasis (fun Si : Set α × ι => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si =>
       { f : α →ᵤ[𝔖] G | ∀ x ∈ Si.1, toFun 𝔖 f x ∈ b Si.2 } := by
-  have := h.uniformity_of_nhds_one_swapped
-  convert UniformOnFun.hasBasis_nhds_of_basis α _ 𝔖 (1 : α →ᵤ[𝔖] G) h𝔖₁ h𝔖₂ this
+  convert UniformOnFun.hasBasis_nhds_of_basis α _ 𝔖 (1 : α →ᵤ[𝔖] G) h𝔖₁ h𝔖₂ <|
+    h.uniformity_of_nhds_one_swapped
   simp [UniformOnFun.gen]
 
 @[to_additive]
diff --git a/Mathlib/Topology/Algebra/UniformField.lean b/Mathlib/Topology/Algebra/UniformField.lean
index 059b0ae0cacfb..da3fb454151fa 100644
--- a/Mathlib/Topology/Algebra/UniformField.lean
+++ b/Mathlib/Topology/Algebra/UniformField.lean
@@ -55,17 +55,17 @@ namespace UniformSpace
 namespace Completion
 
 instance (priority := 100) [T0Space K] : Nontrivial (hat K) :=
-  ⟨⟨0, 1, fun h => zero_ne_one <| (uniformEmbedding_coe K).inj h⟩⟩
+  ⟨⟨0, 1, fun h => zero_ne_one <| (isUniformEmbedding_coe K).inj h⟩⟩
 
 variable {K}
 
 /-- extension of inversion to the completion of a field. -/
 def hatInv : hat K → hat K :=
-  denseInducing_coe.extend fun x : K => (↑x⁻¹ : hat K)
+  isDenseInducing_coe.extend fun x : K => (↑x⁻¹ : hat K)
 
 theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
     ContinuousAt hatInv x := by
-  refine denseInducing_coe.continuousAt_extend ?_
+  refine isDenseInducing_coe.continuousAt_extend ?_
   apply mem_of_superset (compl_singleton_mem_nhds h)
   intro y y_ne
   rw [mem_compl_singleton_iff] at y_ne
@@ -77,13 +77,13 @@ theorem continuous_hatInv [CompletableTopField K] {x : hat K} (h : x ≠ 0) :
   rw [this, ← Filter.map_map]
   apply Cauchy.map _ (Completion.uniformContinuous_coe K)
   apply CompletableTopField.nice
-  · haveI := denseInducing_coe.comap_nhds_neBot y
+  · haveI := isDenseInducing_coe.comap_nhds_neBot y
     apply cauchy_nhds.comap
     rw [Completion.comap_coe_eq_uniformity]
   · have eq_bot : 𝓝 (0 : hat K) ⊓ 𝓝 y = ⊥ := by
       by_contra h
       exact y_ne (eq_of_nhds_neBot <| neBot_iff.mpr h).symm
-    erw [denseInducing_coe.nhds_eq_comap (0 : K), ← Filter.comap_inf, eq_bot]
+    erw [isDenseInducing_coe.nhds_eq_comap (0 : K), ← Filter.comap_inf, eq_bot]
     exact comap_bot
 
 open Classical in
@@ -97,7 +97,7 @@ instance instInvCompletion : Inv (hat K) :=
 variable [TopologicalDivisionRing K]
 
 theorem hatInv_extends {x : K} (h : x ≠ 0) : hatInv (x : hat K) = ↑(x⁻¹ : K) :=
-  denseInducing_coe.extend_eq_at ((continuous_coe K).continuousAt.comp (continuousAt_inv₀ h))
+  isDenseInducing_coe.extend_eq_at ((continuous_coe K).continuousAt.comp (continuousAt_inv₀ h))
 
 variable [CompletableTopField K]
 
@@ -111,7 +111,7 @@ theorem coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) := by
   · conv_lhs => dsimp [Inv.inv]
     rw [if_neg]
     · exact hatInv_extends h
-    · exact fun H => h (denseEmbedding_coe.inj H)
+    · exact fun H => h (isDenseEmbedding_coe.inj H)
 
 variable [UniformAddGroup K]
 
@@ -126,7 +126,7 @@ theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
       continuous_id.continuousAt.prod (continuous_hatInv x_ne)
     exact (_root_.continuous_mul.continuousAt.comp this : _)
   have clo : x ∈ closure (c '' {0}ᶜ) := by
-    have := denseInducing_coe.dense x
+    have := isDenseInducing_coe.dense x
     rw [← image_univ, show (univ : Set K) = {0} ∪ {0}ᶜ from (union_compl_self _).symm,
       image_union] at this
     apply mem_closure_of_mem_closure_union this
@@ -145,7 +145,7 @@ theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by
   rwa [closure_singleton, mem_singleton_iff] at fxclo
 
 instance instField : Field (hat K) where
-  exists_pair_ne := ⟨0, 1, fun h => zero_ne_one ((uniformEmbedding_coe K).inj h)⟩
+  exists_pair_ne := ⟨0, 1, fun h => zero_ne_one ((isUniformEmbedding_coe K).inj h)⟩
   mul_inv_cancel := fun x x_ne => by simp only [Inv.inv, if_neg x_ne, mul_hatInv_cancel x_ne]
   inv_zero := by simp only [Inv.inv, ite_true]
   -- TODO: use a better defeq
@@ -176,7 +176,7 @@ variable (L : Type*) [Field L] [UniformSpace L] [CompletableTopField L]
 instance Subfield.completableTopField (K : Subfield L) : CompletableTopField K where
   nice F F_cau inf_F := by
     let i : K →+* L := K.subtype
-    have hi : UniformInducing i := uniformEmbedding_subtype_val.toUniformInducing
+    have hi : IsUniformInducing i := isUniformEmbedding_subtype_val.isUniformInducing
     rw [← hi.cauchy_map_iff] at F_cau ⊢
     rw [map_comm (show (i ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ i by ext; rfl)]
     apply CompletableTopField.nice _ F_cau
@@ -201,12 +201,12 @@ variable {α β : Type*} [Field β] [b : UniformSpace β] [CompletableTopField 
 
 /-- The pullback of a completable topological field along a uniform inducing
 ring homomorphism is a completable topological field. -/
-theorem UniformInducing.completableTopField
+theorem IsUniformInducing.completableTopField
     [UniformSpace α] [T0Space α]
-    {f : α →+* β} (hf : UniformInducing f) :
+    {f : α →+* β} (hf : IsUniformInducing f) :
     CompletableTopField α := by
   refine CompletableTopField.mk (fun F F_cau inf_F => ?_)
-  rw [← UniformInducing.cauchy_map_iff hf] at F_cau ⊢
+  rw [← IsUniformInducing.cauchy_map_iff hf] at F_cau ⊢
   have h_comm : (f ∘ fun x => x⁻¹) = (fun x => x⁻¹) ∘ f := by
     ext; simp only [Function.comp_apply, map_inv₀, Subfield.coe_inv]
   rw [Filter.map_comm h_comm]
diff --git a/Mathlib/Topology/Algebra/UniformGroup.lean b/Mathlib/Topology/Algebra/UniformGroup.lean
index 80a4a5419140b..c05bd7340d340 100644
--- a/Mathlib/Topology/Algebra/UniformGroup.lean
+++ b/Mathlib/Topology/Algebra/UniformGroup.lean
@@ -93,6 +93,33 @@ theorem UniformContinuous.mul [UniformSpace β] {f : β → α} {g : β → α}
 theorem uniformContinuous_mul : UniformContinuous fun p : α × α => p.1 * p.2 :=
   uniformContinuous_fst.mul uniformContinuous_snd
 
+@[to_additive]
+theorem UniformContinuous.mul_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
+    (a : α) : UniformContinuous fun x ↦ f x * a :=
+  hf.mul uniformContinuous_const
+
+@[to_additive]
+theorem UniformContinuous.const_mul [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
+    (a : α) : UniformContinuous fun x ↦ a * f x :=
+  uniformContinuous_const.mul hf
+
+@[to_additive]
+theorem uniformContinuous_mul_left (a : α) : UniformContinuous fun b : α => a * b :=
+  uniformContinuous_id.const_mul _
+
+@[to_additive]
+theorem uniformContinuous_mul_right (a : α) : UniformContinuous fun b : α => b * a :=
+  uniformContinuous_id.mul_const _
+
+@[to_additive]
+theorem UniformContinuous.div_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f)
+    (a : α) : UniformContinuous fun x ↦ f x / a :=
+  hf.div uniformContinuous_const
+
+@[to_additive]
+theorem uniformContinuous_div_const (a : α) : UniformContinuous fun b : α => b / a :=
+  uniformContinuous_id.div_const _
+
 @[to_additive UniformContinuous.const_nsmul]
 theorem UniformContinuous.pow_const [UniformSpace β] {f : β → α} (hf : UniformContinuous f) :
     ∀ n : ℕ, UniformContinuous fun x => f x ^ n
@@ -146,19 +173,22 @@ theorem uniformity_translate_mul (a : α) : ((𝓤 α).map fun x : α × α => (
     (calc
       𝓤 α =
           ((𝓤 α).map fun x : α × α => (x.1 * a⁻¹, x.2 * a⁻¹)).map fun x : α × α =>
-            (x.1 * a, x.2 * a) := by simp [Filter.map_map, (· ∘ ·)]
+            (x.1 * a, x.2 * a) := by simp [Filter.map_map, Function.comp_def]
       _ ≤ (𝓤 α).map fun x : α × α => (x.1 * a, x.2 * a) :=
         Filter.map_mono (uniformContinuous_id.mul uniformContinuous_const)
       )
 
 @[to_additive]
-theorem uniformEmbedding_translate_mul (a : α) : UniformEmbedding fun x : α => x * a :=
+theorem isUniformEmbedding_translate_mul (a : α) : IsUniformEmbedding fun x : α => x * a :=
   { comap_uniformity := by
       nth_rw 1 [← uniformity_translate_mul a, comap_map]
       rintro ⟨p₁, p₂⟩ ⟨q₁, q₂⟩
       simp only [Prod.mk.injEq, mul_left_inj, imp_self]
     inj := mul_left_injective a }
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_translate_mul := isUniformEmbedding_translate_mul
+
 namespace MulOpposite
 
 @[to_additive]
@@ -195,18 +225,21 @@ theorem uniformGroup_inf {u₁ u₂ : UniformSpace β} (h₁ : @UniformGroup β
   cases b <;> assumption
 
 @[to_additive]
-lemma UniformInducing.uniformGroup {γ : Type*} [Group γ] [UniformSpace γ] [UniformGroup γ]
+lemma IsUniformInducing.uniformGroup {γ : Type*} [Group γ] [UniformSpace γ] [UniformGroup γ]
     [UniformSpace β] {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ]
-    (f : F) (hf : UniformInducing f) :
+    (f : F) (hf : IsUniformInducing f) :
     UniformGroup β where
   uniformContinuous_div := by
     simp_rw [hf.uniformContinuous_iff, Function.comp_def, map_div]
-    exact uniformContinuous_div.comp (hf.uniformContinuous.prod_map hf.uniformContinuous)
+    exact uniformContinuous_div.comp (hf.uniformContinuous.prodMap hf.uniformContinuous)
+
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformGroup := IsUniformInducing.uniformGroup
 
 @[to_additive]
 protected theorem UniformGroup.comap {γ : Type*} [Group γ] {u : UniformSpace γ} [UniformGroup γ]
     {F : Type*} [FunLike F β γ] [MonoidHomClass F β γ] (f : F) : @UniformGroup β (u.comap f) _ :=
-  letI : UniformSpace β := u.comap f; UniformInducing.uniformGroup f ⟨rfl⟩
+  letI : UniformSpace β := u.comap f; IsUniformInducing.uniformGroup f ⟨rfl⟩
 
 end LatticeOps
 
@@ -270,7 +303,7 @@ theorem uniformity_eq_comap_inv_mul_nhds_one :
     𝓤 α = comap (fun x : α × α => x.1⁻¹ * x.2) (𝓝 (1 : α)) := by
   rw [← comap_uniformity_mulOpposite, uniformity_eq_comap_nhds_one, ← op_one, ← comap_unop_nhds,
     comap_comap, comap_comap]
-  simp [(· ∘ ·)]
+  simp [Function.comp_def]
 
 @[to_additive]
 theorem uniformity_eq_comap_inv_mul_nhds_one_swapped :
@@ -464,7 +497,7 @@ def TopologicalGroup.toUniformSpace : UniformSpace G where
     refine mem_map.2 (mem_of_superset (mem_lift' <| preimage_mem_comap V_nhds) ?_)
     rintro ⟨x, y⟩ ⟨z, hz₁, hz₂⟩
     simpa using V_mul _ hz₂ _ hz₁
-  nhds_eq_comap_uniformity _ := by simp only [comap_comap, (· ∘ ·), nhds_translation_div]
+  nhds_eq_comap_uniformity _ := by simp only [comap_comap, Function.comp_def, nhds_translation_div]
 
 attribute [local instance] TopologicalGroup.toUniformSpace
 
@@ -568,7 +601,7 @@ theorem comm_topologicalGroup_is_uniform : UniformGroup G := by
   constructor
   rw [UniformContinuous, uniformity_prod_eq_prod, tendsto_map'_iff, uniformity_eq_comap_nhds_one' G,
     tendsto_comap_iff, prod_comap_comap_eq]
-  simp only [Function.comp, div_eq_mul_inv, mul_inv_rev, inv_inv, mul_comm, mul_left_comm] at *
+  simp only [Function.comp_def, div_eq_mul_inv, mul_inv_rev, inv_inv, mul_comm, mul_left_comm] at *
   simp only [inv_one, mul_one, ← mul_assoc] at this
   simp_rw [← mul_assoc, mul_comm]
   assumption
@@ -597,7 +630,7 @@ variable [TopologicalSpace β] [Group β]
 variable [FunLike hom β α] [MonoidHomClass hom β α] {e : hom}
 
 @[to_additive]
-theorem tendsto_div_comap_self (de : DenseInducing e) (x₀ : α) :
+theorem tendsto_div_comap_self (de : IsDenseInducing e) (x₀ : α) :
     Tendsto (fun t : β × β => t.2 / t.1) ((comap fun p : β × β => (e p.1, e p.2)) <| 𝓝 (x₀, x₀))
       (𝓝 1) := by
   have comm : ((fun x : α × α => x.2 / x.1) ∘ fun t : β × β => (e t.1, e t.2)) =
@@ -611,7 +644,7 @@ theorem tendsto_div_comap_self (de : DenseInducing e) (x₀ : α) :
 
 end
 
-namespace DenseInducing
+namespace IsDenseInducing
 
 variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
 variable {G : Type*}
@@ -623,8 +656,8 @@ variable [TopologicalSpace β] [AddCommGroup β]
 variable [TopologicalSpace γ] [AddCommGroup γ] [TopologicalAddGroup γ]
 variable [TopologicalSpace δ] [AddCommGroup δ]
 variable [UniformSpace G] [AddCommGroup G]
-variable {e : β →+ α} (de : DenseInducing e)
-variable {f : δ →+ γ} (df : DenseInducing f)
+variable {e : β →+ α} (de : IsDenseInducing e)
+variable {f : δ →+ γ} (df : IsDenseInducing f)
 variable {φ : β →+ δ →+ G}
 variable (hφ : Continuous (fun p : β × δ => φ p.1 p.2))
 variable {W' : Set G} (W'_nhd : W' ∈ 𝓝 (0 : G))
@@ -698,21 +731,21 @@ private theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (
   have h₄ := H x₁ x₁_in x xU₁ y yV₁ y' y'V₁
   exact W4 h₁ h₂ h₃ h₄
 
-open DenseInducing
+open IsDenseInducing
 
 variable [T0Space G] [CompleteSpace G]
 
 /-- Bourbaki GT III.6.5 Theorem I:
 ℤ-bilinear continuous maps from dense images into a complete Hausdorff group extend by continuity.
 Note: Bourbaki assumes that α and β are also complete Hausdorff, but this is not necessary. -/
-theorem extend_Z_bilin : Continuous (extend (de.prod df) (fun p : β × δ => φ p.1 p.2)) := by
+theorem extend_Z_bilin : Continuous (extend (de.prodMap df) (fun p : β × δ => φ p.1 p.2)) := by
   refine continuous_extend_of_cauchy _ ?_
   rintro ⟨x₀, y₀⟩
   constructor
   · apply NeBot.map
     apply comap_neBot
     intro U h
-    rcases mem_closure_iff_nhds.1 ((de.prod df).dense (x₀, y₀)) U h with ⟨x, x_in, ⟨z, z_x⟩⟩
+    rcases mem_closure_iff_nhds.1 ((de.prodMap df).dense (x₀, y₀)) U h with ⟨x, x_in, ⟨z, z_x⟩⟩
     exists z
     aesop
   · suffices map (fun p : (β × δ) × β × δ => (fun p : β × δ => φ p.1 p.2) p.2 -
@@ -740,7 +773,7 @@ theorem extend_Z_bilin : Continuous (extend (de.prod df) (fun p : β × δ => φ
       rcases p with ⟨⟨x, y⟩, ⟨x', y'⟩⟩
       apply h <;> tauto
 
-end DenseInducing
+end IsDenseInducing
 
 section CompleteQuotient
 
@@ -850,7 +883,7 @@ already equipped with a uniform structure.
 Even though `G` is equipped with a uniform structure, the quotient `G ⧸ N` does not inherit a
 uniform structure, so it is still provided manually via `TopologicalGroup.toUniformSpace`.
 In the most common use cases, this coincides (definitionally) with the uniform structure on the
-quotient obtained via other means.  -/
+quotient obtained via other means. -/
 @[to_additive "The quotient `G ⧸ N` of a complete first countable uniform additive group
 `G` by a normal additive subgroup is itself complete. Consequently, quotients of Banach spaces by
 subspaces are complete. In contrast to `QuotientAddGroup.completeSpace'`, in this version
diff --git a/Mathlib/Topology/Algebra/UniformMulAction.lean b/Mathlib/Topology/Algebra/UniformMulAction.lean
index 4c23eda8c9359..3f0de8b6107d6 100644
--- a/Mathlib/Topology/Algebra/UniformMulAction.lean
+++ b/Mathlib/Topology/Algebra/UniformMulAction.lean
@@ -88,13 +88,16 @@ theorem UniformContinuous.const_smul [UniformContinuousConstSMul M X] {f : Y →
   (uniformContinuous_const_smul c).comp hf
 
 @[to_additive]
-lemma UniformInducing.uniformContinuousConstSMul [SMul M Y] [UniformContinuousConstSMul M Y]
-    {f : X → Y} (hf : UniformInducing f) (hsmul : ∀ (c : M) x, f (c • x) = c • f x) :
+lemma IsUniformInducing.uniformContinuousConstSMul [SMul M Y] [UniformContinuousConstSMul M Y]
+    {f : X → Y} (hf : IsUniformInducing f) (hsmul : ∀ (c : M) x, f (c • x) = c • f x) :
     UniformContinuousConstSMul M X where
   uniformContinuous_const_smul c := by
     simpa only [hf.uniformContinuous_iff, Function.comp_def, hsmul]
       using hf.uniformContinuous.const_smul c
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformContinuousConstSMul := IsUniformInducing.uniformContinuousConstSMul
+
 /-- If a scalar action is central, then its right action is uniform continuous when its left action
 is. -/
 @[to_additive "If an additive action is central, then its right action is uniform
@@ -116,6 +119,39 @@ instance UniformGroup.to_uniformContinuousConstSMul {G : Type u} [Group G] [Unif
     [UniformGroup G] : UniformContinuousConstSMul G G :=
   ⟨fun _ => uniformContinuous_const.mul uniformContinuous_id⟩
 
+section Ring
+
+variable {R β : Type*} [Ring R] [UniformSpace R] [UniformSpace β]
+
+theorem UniformContinuous.const_mul' [UniformContinuousConstSMul R R] {f : β → R}
+    (hf : UniformContinuous f) (a : R) : UniformContinuous fun x ↦ a * f x :=
+  hf.const_smul a
+
+theorem UniformContinuous.mul_const' [UniformContinuousConstSMul Rᵐᵒᵖ R] {f : β → R}
+    (hf : UniformContinuous f) (a : R) : UniformContinuous fun x ↦ f x * a :=
+  hf.const_smul (MulOpposite.op a)
+
+theorem uniformContinuous_mul_left' [UniformContinuousConstSMul R R] (a : R) :
+    UniformContinuous fun b : R => a * b :=
+  uniformContinuous_id.const_mul' _
+
+theorem uniformContinuous_mul_right' [UniformContinuousConstSMul Rᵐᵒᵖ R] (a : R) :
+    UniformContinuous fun b : R => b * a :=
+  uniformContinuous_id.mul_const' _
+
+theorem UniformContinuous.div_const' {R β : Type*} [DivisionRing R] [UniformSpace R]
+    [UniformContinuousConstSMul Rᵐᵒᵖ R] [UniformSpace β] {f : β → R}
+    (hf : UniformContinuous f) (a : R) :
+    UniformContinuous fun x ↦ f x / a := by
+  simpa [div_eq_mul_inv] using hf.mul_const' a⁻¹
+
+theorem uniformContinuous_div_const' {R : Type*} [DivisionRing R] [UniformSpace R]
+    [UniformContinuousConstSMul Rᵐᵒᵖ R] (a : R) :
+    UniformContinuous fun b : R => b / a :=
+  uniformContinuous_id.div_const' _
+
+end Ring
+
 namespace UniformSpace
 
 namespace Completion
diff --git a/Mathlib/Topology/Algebra/UniformRing.lean b/Mathlib/Topology/Algebra/UniformRing.lean
index 8411e06c056e3..7ad0d76af8d94 100644
--- a/Mathlib/Topology/Algebra/UniformRing.lean
+++ b/Mathlib/Topology/Algebra/UniformRing.lean
@@ -38,7 +38,7 @@ noncomputable section
 universe u
 namespace UniformSpace.Completion
 
-open DenseInducing UniformSpace Function
+open IsDenseInducing UniformSpace Function
 
 section one_and_mul
 variable (α : Type*) [Ring α] [UniformSpace α]
@@ -47,7 +47,7 @@ instance one : One (Completion α) :=
   ⟨(1 : α)⟩
 
 instance mul : Mul (Completion α) :=
-  ⟨curry <| (denseInducing_coe.prod denseInducing_coe).extend ((↑) ∘ uncurry (· * ·))⟩
+  ⟨curry <| (isDenseInducing_coe.prodMap isDenseInducing_coe).extend ((↑) ∘ uncurry (· * ·))⟩
 
 @[norm_cast]
 theorem coe_one : ((1 : α) : Completion α) = 1 :=
@@ -59,7 +59,7 @@ variable {α : Type*} [Ring α] [UniformSpace α] [TopologicalRing α]
 
 @[norm_cast]
 theorem coe_mul (a b : α) : ((a * b : α) : Completion α) = a * b :=
-  ((denseInducing_coe.prod denseInducing_coe).extend_eq
+  ((isDenseInducing_coe.prodMap isDenseInducing_coe).extend_eq
       ((continuous_coe α).comp (@continuous_mul α _ _ _)) (a, b)).symm
 
 variable [UniformAddGroup α]
@@ -70,7 +70,7 @@ theorem continuous_mul : Continuous fun p : Completion α × Completion α => p.
     apply (continuous_coe α).comp _
     simp only [AddMonoidHom.coe_mul, AddMonoidHom.coe_mulLeft]
     exact _root_.continuous_mul
-  have di : DenseInducing (toCompl : α → Completion α) := denseInducing_coe
+  have di : IsDenseInducing (toCompl : α → Completion α) := isDenseInducing_coe
   convert di.extend_Z_bilin di this
 
 theorem Continuous.mul {β : Type*} [TopologicalSpace β] {f g : β → Completion α}
@@ -276,28 +276,28 @@ variable {γ : Type*} [UniformSpace γ] [Semiring γ] [TopologicalSemiring γ]
 variable [T2Space γ] [CompleteSpace γ]
 
 /-- The dense inducing extension as a ring homomorphism. -/
-noncomputable def DenseInducing.extendRingHom {i : α →+* β} {f : α →+* γ} (ue : UniformInducing i)
-    (dr : DenseRange i) (hf : UniformContinuous f) : β →+* γ where
-  toFun := (ue.denseInducing dr).extend f
+noncomputable def IsDenseInducing.extendRingHom {i : α →+* β} {f : α →+* γ}
+    (ue : IsUniformInducing i) (dr : DenseRange i) (hf : UniformContinuous f) : β →+* γ where
+  toFun := (ue.isDenseInducing dr).extend f
   map_one' := by
-    convert DenseInducing.extend_eq (ue.denseInducing dr) hf.continuous 1
+    convert IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 1
     exacts [i.map_one.symm, f.map_one.symm]
   map_zero' := by
-    convert DenseInducing.extend_eq (ue.denseInducing dr) hf.continuous 0 <;>
+    convert IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous 0 <;>
     simp only [map_zero]
   map_add' := by
     have h := (uniformContinuous_uniformly_extend ue dr hf).continuous
     refine fun x y => DenseRange.induction_on₂ dr ?_ (fun a b => ?_) x y
     · exact isClosed_eq (Continuous.comp h continuous_add)
         ((h.comp continuous_fst).add (h.comp continuous_snd))
-    · simp_rw [← i.map_add, DenseInducing.extend_eq (ue.denseInducing dr) hf.continuous _,
+    · simp_rw [← i.map_add, IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous _,
         ← f.map_add]
   map_mul' := by
     have h := (uniformContinuous_uniformly_extend ue dr hf).continuous
     refine fun x y => DenseRange.induction_on₂ dr ?_ (fun a b => ?_) x y
     · exact isClosed_eq (Continuous.comp h continuous_mul)
         ((h.comp continuous_fst).mul (h.comp continuous_snd))
-    · simp_rw [← i.map_mul, DenseInducing.extend_eq (ue.denseInducing dr) hf.continuous _,
+    · simp_rw [← i.map_mul, IsDenseInducing.extend_eq (ue.isDenseInducing dr) hf.continuous _,
         ← f.map_mul]
 
 end UniformExtension
diff --git a/Mathlib/Topology/Algebra/Valued/NormedValued.lean b/Mathlib/Topology/Algebra/Valued/NormedValued.lean
index a947a5a38370c..2c9ce7c08f2c8 100644
--- a/Mathlib/Topology/Algebra/Valued/NormedValued.lean
+++ b/Mathlib/Topology/Algebra/Valued/NormedValued.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: María Inés de Frutos-Fernández
 -/
 import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Group.Uniform
 import Mathlib.RingTheory.Valuation.RankOne
 import Mathlib.Topology.Algebra.Valued.ValuationTopology
 
@@ -96,12 +97,12 @@ def toNormedField : NormedField L :=
       haveI : Nonempty { ε : ℝ // ε > 0 } := nonempty_Ioi_subtype
       ext U
       rw [hasBasis_iff.mp (Valued.hasBasis_uniformity L Γ₀), iInf_subtype', mem_iInf_of_directed]
-      · simp only [true_and_iff, mem_principal, Subtype.exists, gt_iff_lt, exists_prop]
+      · simp only [true_and, mem_principal, Subtype.exists, gt_iff_lt, exists_prop]
         refine ⟨fun ⟨ε, hε⟩ => ?_, fun ⟨r, hr_pos, hr⟩ => ?_⟩
         · set δ : ℝ≥0 := hv.hom ε with hδ
           have hδ_pos : 0 < δ := by
             rw [hδ, ← _root_.map_zero hv.hom]
-            exact hv.strictMono (Units.zero_lt ε)
+            exact hv.strictMono _ (Units.zero_lt ε)
           use δ, hδ_pos
           apply subset_trans _ hε
           intro x hx
diff --git a/Mathlib/Topology/Algebra/Valued/ValuedField.lean b/Mathlib/Topology/Algebra/Valued/ValuedField.lean
index 2d7836874a275..99151442537d6 100644
--- a/Mathlib/Topology/Algebra/Valued/ValuedField.lean
+++ b/Mathlib/Topology/Algebra/Valued/ValuedField.lean
@@ -187,14 +187,14 @@ open WithZeroTopology
 
 /-- The extension of the valuation of a valued field to the completion of the field. -/
 noncomputable def extension : hat K → Γ₀ :=
-  Completion.denseInducing_coe.extend (v : K → Γ₀)
+  Completion.isDenseInducing_coe.extend (v : K → Γ₀)
 
 theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) := by
-  refine Completion.denseInducing_coe.continuous_extend ?_
+  refine Completion.isDenseInducing_coe.continuous_extend ?_
   intro x₀
   rcases eq_or_ne x₀ 0 with (rfl | h)
   · refine ⟨0, ?_⟩
-    erw [← Completion.denseInducing_coe.toInducing.nhds_eq_comap]
+    erw [← Completion.isDenseInducing_coe.toInducing.nhds_eq_comap]
     exact Valued.continuous_valuation.tendsto' 0 0 (map_zero v)
   · have preimage_one : v ⁻¹' {(1 : Γ₀)} ∈ 𝓝 (1 : K) := by
       have : (v (1 : K) : Γ₀) ≠ 0 := by
@@ -204,7 +204,7 @@ theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) :
       ext x
       rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]
     obtain ⟨V, V_in, hV⟩ : ∃ V ∈ 𝓝 (1 : hat K), ∀ x : K, (x : hat K) ∈ V → (v x : Γ₀) = 1 := by
-      rwa [Completion.denseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
+      rwa [Completion.isDenseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one
     have : ∃ V' ∈ 𝓝 (1 : hat K), (0 : hat K) ∉ V' ∧ ∀ (x) (_ : x ∈ V') (y) (_ : y ∈ V'),
       x * y⁻¹ ∈ V := by
       have : Tendsto (fun p : hat K × hat K => p.1 * p.2⁻¹) ((𝓝 1) ×ˢ (𝓝 1)) (𝓝 1) := by
@@ -265,8 +265,8 @@ theorem continuous_extension : Continuous (Valued.extension : hat K → Γ₀) :
 
 @[simp, norm_cast]
 theorem extension_extends (x : K) : extension (x : hat K) = v x := by
-  refine Completion.denseInducing_coe.extend_eq_of_tendsto ?_
-  rw [← Completion.denseInducing_coe.nhds_eq_comap]
+  refine Completion.isDenseInducing_coe.extend_eq_of_tendsto ?_
+  rw [← Completion.isDenseInducing_coe.nhds_eq_comap]
   exact Valued.continuous_valuation.continuousAt
 
 /-- the extension of a valuation on a division ring to its completion. -/
@@ -313,9 +313,9 @@ theorem closure_coe_completion_v_lt {γ : Γ₀ˣ} :
   suffices γ₀ ≠ 0 → (x ∈ closure ((↑) '' { x : K | v x < (γ : Γ₀) }) ↔ γ₀ < (γ : Γ₀)) by
     rcases eq_or_ne γ₀ 0 with h | h
     · simp only [h, (Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt,
-        iff_true_iff]
+        iff_true]
       apply subset_closure
-      exact ⟨0, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]; rfl⟩
+      exact ⟨0, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and]; rfl⟩
     · exact this h
   intro h
   have hγ₀ : extension ⁻¹' {γ₀} ∈ 𝓝 x :=
@@ -339,7 +339,7 @@ noncomputable instance valuedCompletion : Valued (hat K) Γ₀ where
       rw [this.mem_iff]
       exact exists_congr fun γ => by simp
     simp_rw [← closure_coe_completion_v_lt]
-    exact (hasBasis_nhds_zero K Γ₀).hasBasis_of_denseInducing Completion.denseInducing_coe
+    exact (hasBasis_nhds_zero K Γ₀).hasBasis_of_isDenseInducing Completion.isDenseInducing_coe
 
 -- Porting note: removed @[norm_cast] attribute due to error:
 -- norm_cast: badly shaped lemma, rhs can't start with coe
diff --git a/Mathlib/Topology/Algebra/WithZeroTopology.lean b/Mathlib/Topology/Algebra/WithZeroTopology.lean
index fad7aa486d3cf..8b290b9995546 100644
--- a/Mathlib/Topology/Algebra/WithZeroTopology.lean
+++ b/Mathlib/Topology/Algebra/WithZeroTopology.lean
@@ -159,7 +159,7 @@ scoped instance (priority := 100) : ContinuousMul Γ₀ where
     rintro ⟨x, y⟩
     wlog hle : x ≤ y generalizing x y
     · have := (this y x (le_of_not_le hle)).comp (continuous_swap.tendsto (x, y))
-      simpa only [mul_comm, Function.comp, Prod.swap] using this
+      simpa only [mul_comm, Function.comp_def, Prod.swap] using this
     rcases eq_or_ne x 0 with (rfl | hx) <;> [rcases eq_or_ne y 0 with (rfl | hy); skip]
     · rw [zero_mul]
       refine ((hasBasis_nhds_zero.prod_nhds hasBasis_nhds_zero).tendsto_iff hasBasis_nhds_zero).2
diff --git a/Mathlib/Topology/Baire/BaireMeasurable.lean b/Mathlib/Topology/Baire/BaireMeasurable.lean
new file mode 100644
index 0000000000000..02a160951b95f
--- /dev/null
+++ b/Mathlib/Topology/Baire/BaireMeasurable.lean
@@ -0,0 +1,181 @@
+/-
+Copyright (c) 2024 Felix Weilacher. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Felix Weilacher
+-/
+import Mathlib.Topology.GDelta
+import Mathlib.Topology.LocallyClosed
+import Mathlib.MeasureTheory.Constructions.EventuallyMeasurable
+import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
+
+/-!
+# Baire category and Baire measurable sets
+
+This file defines some of the basic notions of Baire category and Baire measurable sets.
+
+## Main definitions
+
+First, we define the notation `=ᵇ`. This denotes eventual equality with respect to the filter of
+`residual` sets in a topological space.
+
+A set `s` in a topological space `α` is called a `BaireMeasurableSet` or said to have the
+*property of Baire* if it satisfies either of the following equivalent conditions:
+
+* There is a *Borel* set `u` such that `s =ᵇ u`. (This is our definition)
+* There is an *open* set `u` such that `s =ᵇ u`. (See `BaireMeasurableSet.residual_eq_open`)
+
+-/
+
+variable (α : Type*) {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
+
+open Topology
+
+/-- Notation for `=ᶠ[residual _]`. That is, eventual equality with respect to
+the filter of residual sets.-/
+scoped[Topology] notation:50 f " =ᵇ " g:50 => Filter.EventuallyEq (residual _) f g
+
+/-- Notation to say that a property of points in a topological space holds
+almost everywhere in the sense of Baire category. That is, on a residual set. -/
+scoped[Topology] notation3 "∀ᵇ "(...)", "r:(scoped p => Filter.Eventually p <| residual _) => r
+
+/-- Notation to say that a property of points in a topological space holds on a non meager set. -/
+scoped[Topology] notation3 "∃ᵇ "(...)", "r:(scoped p => Filter.Frequently p <| residual _) => r
+
+variable {α}
+
+/-- We say a set is a `BaireMeasurableSet` if it differs from some Borel set by
+a meager set. This forms a σ-algebra.
+
+It is equivalent, and a more standard definition, to say that the set differs from
+some *open* set by a meager set. See `BaireMeasurableSet.iff_residualEq_isOpen` -/
+def BaireMeasurableSet (s : Set α) : Prop :=
+  @MeasurableSet _ (EventuallyMeasurableSpace (borel _) (residual _)) s
+
+variable {s t : Set α}
+
+namespace BaireMeasurableSet
+
+theorem of_mem_residual (h : s ∈ residual _) : BaireMeasurableSet s :=
+  eventuallyMeasurableSet_of_mem_filter (α := α) h
+
+theorem _root_.MeasurableSet.baireMeasurableSet [MeasurableSpace α] [BorelSpace α]
+    (h : MeasurableSet s) : BaireMeasurableSet s := by
+  borelize α
+  exact h.eventuallyMeasurableSet
+
+theorem _root_.IsOpen.baireMeasurableSet (h : IsOpen s) : BaireMeasurableSet s := by
+  borelize α
+  exact h.measurableSet.baireMeasurableSet
+
+theorem compl (h : BaireMeasurableSet s) : BaireMeasurableSet sᶜ := MeasurableSet.compl h
+
+theorem of_compl (h : BaireMeasurableSet sᶜ) : BaireMeasurableSet s := MeasurableSet.of_compl h
+
+theorem _root_.IsMeagre.baireMeasurableSet (h : IsMeagre s) : BaireMeasurableSet s :=
+  (of_mem_residual h).of_compl
+
+theorem iUnion {ι : Sort*} [Countable ι] {s : ι → Set α}
+    (h : ∀ i, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋃ i, s i) :=
+  MeasurableSet.iUnion h
+
+theorem biUnion {ι : Type*}  {s : ι → Set α} {t : Set ι} (ht : t.Countable)
+    (h : ∀ i ∈ t, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋃ i ∈ t, s i) :=
+  MeasurableSet.biUnion ht h
+
+theorem sUnion {s : Set (Set α)} (hs : s.Countable)
+    (h : ∀ t ∈ s, BaireMeasurableSet t) : BaireMeasurableSet (⋃₀ s) :=
+  MeasurableSet.sUnion hs h
+
+theorem iInter {ι : Sort*} [Countable ι] {s : ι → Set α}
+    (h : ∀ i, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋂ i, s i) :=
+  MeasurableSet.iInter h
+
+theorem biInter {ι : Type*}  {s : ι → Set α} {t : Set ι} (ht : t.Countable)
+    (h : ∀ i ∈ t, BaireMeasurableSet (s i)) : BaireMeasurableSet (⋂ i ∈ t, s i) :=
+  MeasurableSet.biInter ht h
+
+theorem sInter {s : Set (Set α)} (hs : s.Countable)
+    (h : ∀ t ∈ s, BaireMeasurableSet t) : BaireMeasurableSet (⋂₀ s) :=
+  MeasurableSet.sInter hs h
+
+theorem union (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
+    BaireMeasurableSet (s ∪ t) :=
+  MeasurableSet.union hs ht
+
+theorem inter (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
+    BaireMeasurableSet (s ∩ t) :=
+  MeasurableSet.inter hs ht
+
+theorem diff (hs : BaireMeasurableSet s) (ht : BaireMeasurableSet t) :
+    BaireMeasurableSet (s \ t) :=
+  MeasurableSet.diff hs ht
+
+theorem congr (hs : BaireMeasurableSet s) (h : s =ᵇ t) : BaireMeasurableSet t :=
+  EventuallyMeasurableSet.congr (α := α) hs h.symm
+
+end BaireMeasurableSet
+
+open Filter
+
+/--Any Borel set differs from some open set by a meager set. -/
+theorem MeasurableSet.residualEq_isOpen [MeasurableSpace α] [BorelSpace α] (h : MeasurableSet s) :
+    ∃ u : Set α, (IsOpen u) ∧ s =ᵇ u := by
+  apply h.induction_on_open (fun s hs => ⟨s, hs, EventuallyEq.rfl⟩)
+  · rintro s - ⟨u, uo, su⟩
+    refine ⟨(closure u)ᶜ, isClosed_closure.isOpen_compl,
+      EventuallyEq.compl (su.trans <| EventuallyLE.antisymm subset_closure.eventuallyLE ?_)⟩
+    have : (coborder u) ∈ residual _ :=
+      residual_of_dense_open uo.isLocallyClosed.isOpen_coborder dense_coborder
+    rw [coborder_eq_union_closure_compl] at this
+    rw [EventuallyLE]
+    convert this
+    simp only [le_Prop_eq, imp_iff_or_not]
+    rfl --terrible
+  rintro s - - hs
+  choose u uo su using hs
+  exact ⟨⋃ i, u i, isOpen_iUnion uo, EventuallyEq.countable_iUnion su⟩
+
+/--Any `BaireMeasurableSet` differs from some open set by a meager set. -/
+theorem BaireMeasurableSet.residualEq_isOpen (h : BaireMeasurableSet s) :
+    ∃ u : Set α, (IsOpen u) ∧ s =ᵇ u := by
+  borelize α
+  rcases h with ⟨t, ht, hst⟩
+  rcases ht.residualEq_isOpen with ⟨u, hu, htu⟩
+  exact ⟨u, hu, hst.trans htu⟩
+
+/--A set is Baire measurable if and only if it differs from some open set by a meager set. -/
+theorem BaireMeasurableSet.iff_residualEq_isOpen :
+    BaireMeasurableSet s ↔ ∃ u : Set α, (IsOpen u) ∧ s =ᵇ u :=
+  ⟨fun h => h.residualEq_isOpen , fun ⟨_, uo, ueq⟩ => uo.baireMeasurableSet.congr ueq.symm⟩
+
+section Map
+
+open Set
+
+variable {f : α → β}
+
+theorem tendsto_residual_of_isOpenMap (hc : Continuous f) (ho : IsOpenMap f) :
+    Tendsto f (residual α) (residual β) := by
+  apply le_countableGenerate_iff_of_countableInterFilter.mpr
+  rintro t ⟨ht, htd⟩
+  exact residual_of_dense_open (ht.preimage hc) (htd.preimage ho)
+
+/-- The preimage of a meager set under a continuous open map is meager. -/
+theorem IsMeagre.preimage_of_isOpenMap (hc : Continuous f) (ho : IsOpenMap f)
+    {s : Set β} (h : IsMeagre s) : IsMeagre (f ⁻¹' s) :=
+  tendsto_residual_of_isOpenMap hc ho h
+
+/-- The preimage of a `BaireMeasurableSet` under a continuous open map is Baire measurable. -/
+theorem BaireMeasurableSet.preimage (hc : Continuous f) (ho : IsOpenMap f)
+    {s : Set β} (h : BaireMeasurableSet s) : BaireMeasurableSet (f⁻¹' s) := by
+  rcases h with ⟨u, hu, hsu⟩
+  refine ⟨f ⁻¹' u, ?_, hsu.filter_mono <| tendsto_residual_of_isOpenMap hc ho⟩
+  borelize α β
+  exact hc.measurable hu
+
+theorem Homeomorph.residual_map_eq (h : α ≃ₜ β) : (residual α).map h = residual β := by
+  refine le_antisymm (tendsto_residual_of_isOpenMap h.continuous h.isOpenMap) (le_map ?_)
+  simp_rw [← preimage_symm]
+  exact tendsto_residual_of_isOpenMap h.symm.continuous h.symm.isOpenMap
+
+end Map
diff --git a/Mathlib/Topology/Bases.lean b/Mathlib/Topology/Bases.lean
index 7930817c3788d..bc50dd33b1af8 100644
--- a/Mathlib/Topology/Bases.lean
+++ b/Mathlib/Topology/Bases.lean
@@ -65,7 +65,7 @@ variable {α : Type u} {β : Type*} [t : TopologicalSpace α] {B : Set (Set α)}
   it suffices to take unions of the basis sets to get a topology (without taking
   finite intersections as well). -/
 structure IsTopologicalBasis (s : Set (Set α)) : Prop where
-  /-- For every point `x`, the set of `t ∈ s` such that `x ∈ t` is directed downwards.  -/
+  /-- For every point `x`, the set of `t ∈ s` such that `x ∈ t` is directed downwards. -/
   exists_subset_inter : ∀ t₁ ∈ s, ∀ t₂ ∈ s, ∀ x ∈ t₁ ∩ t₂, ∃ t₃ ∈ s, x ∈ t₃ ∧ t₃ ⊆ t₁ ∩ t₂
   /-- The sets from `s` cover the whole space. -/
   sUnion_eq : ⋃₀ s = univ
@@ -286,11 +286,6 @@ protected theorem IsTopologicalBasis.continuous_iff {β : Type*} [TopologicalSpa
     Continuous f ↔ ∀ s ∈ B, IsOpen (f ⁻¹' s) := by
   rw [hB.eq_generateFrom, continuous_generateFrom_iff]
 
-@[deprecated (since := "2023-12-24")]
-protected theorem IsTopologicalBasis.continuous {β : Type*} [TopologicalSpace β] {B : Set (Set β)}
-    (hB : IsTopologicalBasis B) (f : α → β) (hf : ∀ s ∈ B, IsOpen (f ⁻¹' s)) : Continuous f :=
-  hB.continuous_iff.2 hf
-
 variable (α)
 
 /-- A separable space is one with a countable dense subset, available through
@@ -382,9 +377,9 @@ instance {ι : Type*} {X : ι → Type*} [∀ i, TopologicalSpace (X i)] [∀ i,
       (htd i).exists_mem_open (huo i i.2).1 ⟨_, (huo i i.2).2⟩
     choose y hyt hyu using this
     lift y to ∀ i : I, t i using hyt
-    refine ⟨f ⟨I, y⟩, huU fun i (hi : i ∈ I) ↦ ?_, mem_range_self _⟩
+    refine ⟨f ⟨I, y⟩, huU fun i (hi : i ∈ I) ↦ ?_, mem_range_self ⟨I, y⟩⟩
     simp only [f, dif_pos hi]
-    exact hyu _
+    exact hyu ⟨i, _⟩
 
 instance [SeparableSpace α] {r : α → α → Prop} : SeparableSpace (Quot r) :=
   quotientMap_quot_mk.separableSpace
@@ -476,7 +471,7 @@ theorem IsSeparable.univ_pi {ι : Type*} [Countable ι] {X : ι → Type*} {s :
     refine ⟨range g, countable_range g, fun f hf ↦ mem_closure_iff.2 fun o ho hfo ↦ ?_⟩
     rcases isOpen_pi_iff.1 ho f hfo with ⟨I, u, huo, hI⟩
     rsuffices ⟨f, hf⟩ : ∃ f : (i : I) → c i, g ⟨I, f⟩ ∈ Set.pi I u
-    · exact ⟨g ⟨I, f⟩, hI hf, mem_range_self _⟩
+    · exact ⟨g ⟨I, f⟩, hI hf, mem_range_self ⟨I, f⟩⟩
     suffices H : ∀ i ∈ I, (u i ∩ c i).Nonempty by
       choose f hfu hfc using H
       refine ⟨fun i ↦ ⟨f i i.2, hfc i i.2⟩, fun i (hi : i ∈ I) ↦ ?_⟩
@@ -802,9 +797,8 @@ instance {ι : Type*} {π : ι → Type*} [Countable ι] [∀ a, TopologicalSpac
 instance (priority := 100) SecondCountableTopology.to_separableSpace [SecondCountableTopology α] :
     SeparableSpace α := by
   choose p hp using fun s : countableBasis α => nonempty_of_mem_countableBasis s.2
-  exact
-    ⟨⟨range p, countable_range _,
-        (isBasis_countableBasis α).dense_iff.2 fun o ho _ => ⟨p ⟨o, ho⟩, hp _, mem_range_self _⟩⟩⟩
+  exact ⟨⟨range p, countable_range _, (isBasis_countableBasis α).dense_iff.2 fun o ho _ =>
+          ⟨p ⟨o, ho⟩, hp ⟨o, _⟩, mem_range_self _⟩⟩⟩
 
 /-- A countable open cover induces a second-countable topology if all open covers
 are themselves second countable. -/
diff --git a/Mathlib/Topology/Basic.lean b/Mathlib/Topology/Basic.lean
index 6c821ade241e9..55a8aa4aec4ec 100644
--- a/Mathlib/Topology/Basic.lean
+++ b/Mathlib/Topology/Basic.lean
@@ -3,6 +3,7 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
 -/
+import Mathlib.Algebra.Group.Support
 import Mathlib.Order.Filter.Lift
 import Mathlib.Topology.Defs.Filter
 
@@ -128,7 +129,7 @@ theorem Set.Finite.isOpen_biInter {s : Set α} {f : α → Set X} (hs : s.Finite
 
 theorem isOpen_iInter_of_finite [Finite ι] {s : ι → Set X} (h : ∀ i, IsOpen (s i)) :
     IsOpen (⋂ i, s i) :=
-  (finite_range _).isOpen_sInter  (forall_mem_range.2 h)
+  (finite_range _).isOpen_sInter (forall_mem_range.2 h)
 
 theorem isOpen_biInter_finset {s : Finset α} {f : α → Set X} (h : ∀ i ∈ s, IsOpen (f i)) :
     IsOpen (⋂ i ∈ s, f i) :=
@@ -156,6 +157,12 @@ theorem isClosed_const {p : Prop} : IsClosed { _x : X | p } := ⟨isOpen_const (
 
 @[simp] theorem isClosed_univ : IsClosed (univ : Set X) := isClosed_const
 
+lemma IsOpen.isLocallyClosed (hs : IsOpen s) : IsLocallyClosed s :=
+  ⟨_, _, hs, isClosed_univ, (inter_univ _).symm⟩
+
+lemma IsClosed.isLocallyClosed (hs : IsClosed s) : IsLocallyClosed s :=
+  ⟨_, _, isOpen_univ, hs, (univ_inter _).symm⟩
+
 theorem IsClosed.union : IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∪ s₂) := by
   simpa only [← isOpen_compl_iff, compl_union] using IsOpen.inter
 
@@ -792,9 +799,11 @@ theorem frequently_frequently_nhds {p : X → Prop} :
   simp only [not_frequently, eventually_eventually_nhds]
 
 @[simp]
-theorem eventually_mem_nhds : (∀ᶠ x' in 𝓝 x, s ∈ 𝓝 x') ↔ s ∈ 𝓝 x :=
+theorem eventually_mem_nhds_iff : (∀ᶠ x' in 𝓝 x, s ∈ 𝓝 x') ↔ s ∈ 𝓝 x :=
   eventually_eventually_nhds
 
+@[deprecated (since := "2024-10-04")] alias eventually_mem_nhds := eventually_mem_nhds_iff
+
 @[simp]
 theorem nhds_bind_nhds : (𝓝 x).bind 𝓝 = 𝓝 x :=
   Filter.ext fun _ => eventually_eventually_nhds
@@ -839,18 +848,18 @@ theorem tendsto_nhds {f : α → X} {l : Filter α} :
 theorem tendsto_atTop_nhds [Nonempty α] [SemilatticeSup α] {f : α → X} :
     Tendsto f atTop (𝓝 x) ↔ ∀ U : Set X, x ∈ U → IsOpen U → ∃ N, ∀ n, N ≤ n → f n ∈ U :=
   (atTop_basis.tendsto_iff (nhds_basis_opens x)).trans <| by
-    simp only [and_imp, exists_prop, true_and_iff, mem_Ici]
+    simp only [and_imp, exists_prop, true_and, mem_Ici]
 
 theorem tendsto_const_nhds {f : Filter α} : Tendsto (fun _ : α => x) f (𝓝 x) :=
   tendsto_nhds.mpr fun _ _ ha => univ_mem' fun _ => ha
 
-theorem tendsto_atTop_of_eventually_const {ι : Type*} [SemilatticeSup ι] [Nonempty ι]
+theorem tendsto_atTop_of_eventually_const {ι : Type*} [Preorder ι]
     {u : ι → X} {i₀ : ι} (h : ∀ i ≥ i₀, u i = x) : Tendsto u atTop (𝓝 x) :=
-  Tendsto.congr' (EventuallyEq.symm (eventually_atTop.mpr ⟨i₀, h⟩)) tendsto_const_nhds
+  Tendsto.congr' (EventuallyEq.symm ((eventually_ge_atTop i₀).mono h)) tendsto_const_nhds
 
-theorem tendsto_atBot_of_eventually_const {ι : Type*} [SemilatticeInf ι] [Nonempty ι]
+theorem tendsto_atBot_of_eventually_const {ι : Type*} [Preorder ι]
     {u : ι → X} {i₀ : ι} (h : ∀ i ≤ i₀, u i = x) : Tendsto u atBot (𝓝 x) :=
-  Tendsto.congr' (EventuallyEq.symm (eventually_atBot.mpr ⟨i₀, h⟩)) tendsto_const_nhds
+  tendsto_atTop_of_eventually_const (ι := ιᵒᵈ) h
 
 theorem pure_le_nhds : pure ≤ (𝓝 : X → Filter X) := fun _ _ hs => mem_pure.2 <| mem_of_mem_nhds hs
 
@@ -889,6 +898,11 @@ theorem Filter.HasBasis.clusterPt_iff {ιX ιF} {pX : ιX → Prop} {sX : ιX 
     ClusterPt x F ↔ ∀ ⦃i⦄, pX i → ∀ ⦃j⦄, pF j → (sX i ∩ sF j).Nonempty :=
   hX.inf_basis_neBot_iff hF
 
+theorem Filter.HasBasis.clusterPt_iff_frequently {ι} {p : ι → Prop} {s : ι → Set X} {F : Filter X}
+    (hx : (𝓝 x).HasBasis p s) : ClusterPt x F ↔ ∀ i, p i → ∃ᶠ x in F, x ∈ s i := by
+  simp only [hx.clusterPt_iff F.basis_sets, Filter.frequently_iff, inter_comm (s _),
+    Set.Nonempty, id, mem_inter_iff]
+
 theorem clusterPt_iff {F : Filter X} :
     ClusterPt x F ↔ ∀ ⦃U : Set X⦄, U ∈ 𝓝 x → ∀ ⦃V⦄, V ∈ F → (U ∩ V).Nonempty :=
   inf_neBot_iff
@@ -934,31 +948,57 @@ theorem clusterPt_iff_ultrafilter {f : Filter X} : ClusterPt x f ↔
     ∃ u : Ultrafilter X, u ≤ f ∧ u ≤ 𝓝 x := by
   simp_rw [ClusterPt, ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
 
-theorem mapClusterPt_def {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
-    MapClusterPt x F u ↔ ClusterPt x (map u F) := Iff.rfl
+section MapClusterPt
 
-theorem mapClusterPt_iff {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
-    MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s := by
-  simp_rw [MapClusterPt, ClusterPt, inf_neBot_iff_frequently_left, frequently_map]
-  rfl
+variable {F : Filter α} {u : α → X} {x : X}
+
+theorem mapClusterPt_def : MapClusterPt x F u ↔ ClusterPt x (map u F) := Iff.rfl
+alias ⟨MapClusterPt.clusterPt, _⟩ := mapClusterPt_def
+
+theorem MapClusterPt.mono {G : Filter α} (h : MapClusterPt x F u) (hle : F ≤ G) :
+    MapClusterPt x G u :=
+  h.clusterPt.mono (map_mono hle)
+
+theorem MapClusterPt.tendsto_comp' [TopologicalSpace Y] {f : X → Y} {y : Y}
+    (hf : Tendsto f (𝓝 x ⊓ map u F) (𝓝 y)) (hu : MapClusterPt x F u) : MapClusterPt y F (f ∘ u) :=
+  (tendsto_inf.2 ⟨hf, tendsto_map.mono_left inf_le_right⟩).neBot (hx := hu)
 
-theorem mapClusterPt_iff_ultrafilter {ι : Type*} (x : X) (F : Filter ι) (u : ι → X) :
-    MapClusterPt x F u ↔ ∃ U : Ultrafilter ι, U ≤ F ∧ Tendsto u U (𝓝 x) := by
+theorem MapClusterPt.tendsto_comp [TopologicalSpace Y] {f : X → Y} {y : Y}
+    (hf : Tendsto f (𝓝 x) (𝓝 y)) (hu : MapClusterPt x F u) : MapClusterPt y F (f ∘ u) :=
+  hu.tendsto_comp' (hf.mono_left inf_le_left)
+
+theorem MapClusterPt.continuousAt_comp [TopologicalSpace Y] {f : X → Y} (hf : ContinuousAt f x)
+    (hu : MapClusterPt x F u) : MapClusterPt (f x) F (f ∘ u) :=
+  hu.tendsto_comp hf
+
+theorem Filter.HasBasis.mapClusterPt_iff_frequently {ι : Sort*} {p : ι → Prop} {s : ι → Set X}
+    (hx : (𝓝 x).HasBasis p s) : MapClusterPt x F u ↔ ∀ i, p i → ∃ᶠ a in F, u a ∈ s i := by
+  simp_rw [MapClusterPt, hx.clusterPt_iff_frequently, frequently_map]
+
+theorem mapClusterPt_iff : MapClusterPt x F u ↔ ∀ s ∈ 𝓝 x, ∃ᶠ a in F, u a ∈ s :=
+  (𝓝 x).basis_sets.mapClusterPt_iff_frequently
+
+theorem mapClusterPt_iff_ultrafilter :
+    MapClusterPt x F u ↔ ∃ U : Ultrafilter α, U ≤ F ∧ Tendsto u U (𝓝 x) := by
   simp_rw [MapClusterPt, ClusterPt, ← Filter.push_pull', map_neBot_iff, tendsto_iff_comap,
     ← le_inf_iff, exists_ultrafilter_iff, inf_comm]
 
-theorem mapClusterPt_comp {X α β : Type*} {x : X} [TopologicalSpace X] {F : Filter α} {φ : α → β}
-    {u : β → X} : MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (map φ F) u := Iff.rfl
+theorem mapClusterPt_comp {φ : α → β} {u : β → X} :
+    MapClusterPt x F (u ∘ φ) ↔ MapClusterPt x (map φ F) u := Iff.rfl
 
-theorem mapClusterPt_of_comp {F : Filter α} {φ : β → α} {p : Filter β}
-    {u : α → X} [NeBot p] (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) :
-    MapClusterPt x F u := by
-  have :=
-    calc
-      map (u ∘ φ) p = map u (map φ p) := map_map
-      _ ≤ map u F := map_mono h
-  have : map (u ∘ φ) p ≤ 𝓝 x ⊓ map u F := le_inf H this
-  exact neBot_of_le this
+theorem Filter.Tendsto.mapClusterPt [NeBot F] (h : Tendsto u F (𝓝 x)) : MapClusterPt x F u :=
+  .of_le_nhds h
+
+theorem MapClusterPt.of_comp {φ : β → α} {p : Filter β} (h : Tendsto φ p F)
+    (H : MapClusterPt x p (u ∘ φ)) : MapClusterPt x F u :=
+  H.clusterPt.mono <| map_mono h
+
+@[deprecated MapClusterPt.of_comp (since := "2024-09-07")]
+theorem mapClusterPt_of_comp {φ : β → α} {p : Filter β} [NeBot p]
+    (h : Tendsto φ p F) (H : Tendsto (u ∘ φ) p (𝓝 x)) : MapClusterPt x F u :=
+  .of_comp h H.mapClusterPt
+
+end MapClusterPt
 
 theorem accPt_sup (x : X) (F G : Filter X) :
     AccPt x (F ⊔ G) ↔ AccPt x F ∨ AccPt x G := by
@@ -1362,7 +1402,7 @@ theorem ContinuousAt.eventually_mem {f : X → Y} {x : X} (hf : ContinuousAt f x
     (hs : s ∈ 𝓝 (f x)) : ∀ᶠ y in 𝓝 x, f y ∈ s :=
   hf hs
 
-/-- If a function ``f` tends to somewhere other than `𝓝 (f x)` at `x`,
+/-- If a function `f` tends to somewhere other than `𝓝 (f x)` at `x`,
 then `f` is not continuous at `x`
 -/
 lemma not_continuousAt_of_tendsto {f : X → Y} {l₁ : Filter X} {l₂ : Filter Y} {x : X}
@@ -1738,3 +1778,5 @@ example [TopologicalSpace X] [TopologicalSpace Y] {x₀ : X} (f : X → X → Y)
   -- hf.comp_of_eq (continuousAt_id.prod continuousAt_id) rfl -- works
 ```
 -/
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/Topology/Bornology/Absorbs.lean b/Mathlib/Topology/Bornology/Absorbs.lean
index 186c664199d67..4afc9483a389f 100644
--- a/Mathlib/Topology/Bornology/Absorbs.lean
+++ b/Mathlib/Topology/Bornology/Absorbs.lean
@@ -22,6 +22,9 @@ We formulate it in a more general settings for two reasons:
 - some proofs look nicer with this definition than with something like
   `∃ r : ℝ, ∀ a : R, r ≤ ‖a‖ → B ⊆ a • A`.
 
+If `M` is a `GroupWithZero` (e.g., a division ring),
+the sets absorbing a given set form a filter, see `Filter.absorbing`.
+
 ## Implementation notes
 
 For now, all theorems assume that we deal with (a generalization of) a module over a division ring.
@@ -133,58 +136,70 @@ protected lemma zero [Zero E] [SMulZeroClass M E] {s : Set E} (hs : 0 ∈ s) : A
 
 end AddZero
 
+end Absorbs
+
 section GroupWithZero
 
 variable {G₀ α : Type*} [GroupWithZero G₀] [Bornology G₀] [MulAction G₀ α]
   {s t u : Set α} {S : Set (Set α)}
 
 @[simp]
-protected lemma univ : Absorbs G₀ univ s :=
+protected lemma Absorbs.univ : Absorbs G₀ univ s :=
   (eventually_ne_cobounded 0).mono fun a ha ↦ by rw [smul_set_univ₀ ha]; apply subset_univ
 
-lemma _root_.absorbs_iff_eventually_cobounded_mapsTo :
+lemma absorbs_iff_eventually_cobounded_mapsTo :
     Absorbs G₀ s t ↔ ∀ᶠ c in cobounded G₀, MapsTo (c⁻¹ • ·) t s :=
   eventually_congr <| (eventually_ne_cobounded 0).mono fun c hc ↦ by
     rw [← preimage_smul_inv₀ hc]; rfl
 
 alias ⟨eventually_cobounded_mapsTo, _⟩ := absorbs_iff_eventually_cobounded_mapsTo
 
-lemma _root_.Set.Finite.absorbs_sInter (hS : S.Finite) :
-    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t := by
-  simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_sInter, hS.eventually_all]
+@[simp]
+lemma absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u := by
+  simp only [absorbs_iff_eventually_cobounded_mapsTo, mapsTo_inter, eventually_and]
+
+protected lemma Absorbs.inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u :=
+  absorbs_inter.2 ⟨hs, ht⟩
 
-protected alias ⟨_, sInter⟩ := Set.Finite.absorbs_sInter
+variable (G₀ u) in
+/-- The filter of sets that absorb `u`. -/
+def Filter.absorbing : Filter α where
+  sets := {s | Absorbs G₀ s u}
+  univ_sets := .univ
+  sets_of_superset h := h.mono_left
+  inter_sets := .inter
 
 @[simp]
-lemma _root_.absorbs_inter : Absorbs G₀ (s ∩ t) u ↔ Absorbs G₀ s u ∧ Absorbs G₀ t u := by
-  simpa using ((finite_singleton t).insert s).absorbs_sInter
+lemma Filter.mem_absorbing : s ∈ absorbing G₀ u ↔ Absorbs G₀ s u := .rfl
 
-protected lemma inter (hs : Absorbs G₀ s u) (ht : Absorbs G₀ t u) : Absorbs G₀ (s ∩ t) u :=
-  absorbs_inter.2 ⟨hs, ht⟩
+lemma Set.Finite.absorbs_sInter (hS : S.Finite) :
+    Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t :=
+  sInter_mem (f := absorbing G₀ t) hS
+
+protected alias ⟨_, Absorbs.sInter⟩ := Set.Finite.absorbs_sInter
 
 @[simp]
-lemma _root_.absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :
+lemma absorbs_iInter {ι : Sort*} [Finite ι] {s : ι → Set α} :
     Absorbs G₀ (⋂ i, s i) t ↔ ∀ i, Absorbs G₀ (s i) t :=
-  (finite_range s).absorbs_sInter.trans forall_mem_range
+  iInter_mem (f := absorbing G₀ t)
 
-protected alias ⟨_, iInter⟩ := absorbs_iInter
+protected alias ⟨_, Absorbs.iInter⟩ := absorbs_iInter
 
-lemma _root_.Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
-    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t := by
-  simpa only [sInter_image, forall_mem_image] using (hI.image s).absorbs_sInter
+lemma Set.Finite.absorbs_biInter {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
+    Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t :=
+  biInter_mem (f := absorbing G₀ t) hI
 
-protected alias ⟨_, biInter⟩ := Set.Finite.absorbs_biInter
+protected alias ⟨_, Absorbs.biInter⟩ := Set.Finite.absorbs_biInter
 
 @[simp]
-lemma _root_.absorbs_zero_iff [NeBot (cobounded G₀)] {E : Type*} [AddMonoid E]
-    [DistribMulAction G₀ E] {s : Set E} : Absorbs G₀ s 0 ↔ 0 ∈ s := by
+lemma absorbs_zero_iff [NeBot (cobounded G₀)]
+    {E : Type*} [AddMonoid E] [DistribMulAction G₀ E] {s : Set E} :
+    Absorbs G₀ s 0 ↔ 0 ∈ s := by
   simp only [absorbs_iff_eventually_cobounded_mapsTo, ← singleton_zero,
     mapsTo_singleton, smul_zero, eventually_const]
 
 end GroupWithZero
 
-end Absorbs
-
 section AddGroup
 
 variable {M E : Type*} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M]
diff --git a/Mathlib/Topology/Bornology/BoundedOperation.lean b/Mathlib/Topology/Bornology/BoundedOperation.lean
index 9340172558c61..a322077983828 100644
--- a/Mathlib/Topology/Bornology/BoundedOperation.lean
+++ b/Mathlib/Topology/Bornology/BoundedOperation.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
 -/
 import Mathlib.Analysis.Normed.Group.Basic
-import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Analysis.Normed.Field.Lemmas
 
 /-!
 # Bounded operations
diff --git a/Mathlib/Topology/Bornology/Constructions.lean b/Mathlib/Topology/Bornology/Constructions.lean
index e53a16415f67c..c38d27051cf5b 100644
--- a/Mathlib/Topology/Bornology/Constructions.lean
+++ b/Mathlib/Topology/Bornology/Constructions.lean
@@ -77,7 +77,7 @@ theorem isBounded_prod_of_nonempty (hne : Set.Nonempty (s ×ˢ t)) :
 theorem isBounded_prod : IsBounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ IsBounded s ∧ IsBounded t := by
   rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
   rcases t.eq_empty_or_nonempty with (rfl | ht); · simp
-  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or_iff]
+  simp only [hs.ne_empty, ht.ne_empty, isBounded_prod_of_nonempty (hs.prod ht), false_or]
 
 theorem isBounded_prod_self : IsBounded (s ×ˢ s) ↔ IsBounded s := by
   rcases s.eq_empty_or_nonempty with (rfl | hs); · simp
@@ -109,7 +109,7 @@ theorem isBounded_pi_of_nonempty (hne : (pi univ S).Nonempty) :
 theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by
   by_cases hne : ∃ i, S i = ∅
   · simp [hne, univ_pi_eq_empty_iff.2 hne]
-  · simp only [hne, false_or_iff]
+  · simp only [hne, false_or]
     simp only [not_exists, ← Ne.eq_def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne
     exact isBounded_pi_of_nonempty hne
 
diff --git a/Mathlib/Topology/CWComplex.lean b/Mathlib/Topology/CWComplex.lean
new file mode 100644
index 0000000000000..8dae230580f73
--- /dev/null
+++ b/Mathlib/Topology/CWComplex.lean
@@ -0,0 +1,100 @@
+/-
+Copyright (c) 2024 Elliot Dean Young and Jiazhen Xia. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jiazhen Xia, Elliot Dean Young
+-/
+import Mathlib.Topology.Category.TopCat.Limits.Basic
+import Mathlib.Topology.Category.TopCat.Sphere
+import Mathlib.CategoryTheory.Limits.Shapes.Products
+import Mathlib.CategoryTheory.Functor.OfSequence
+
+/-!
+# CW-complexes
+
+This file defines (relative) CW-complexes.
+
+## Main definitions
+
+* `RelativeCWComplex`: A relative CW-complex is the colimit of an expanding sequence of subspaces
+  `sk i` (called the $(i-1)$-skeleton) for `i ≥ 0`, where `sk 0` (i.e., the $(-1)$-skeleton) is an
+  arbitrary topological space, and each `sk (n + 1)` (i.e., the $n$-skeleton) is obtained from
+  `sk n` (i.e., the $(n-1)$-skeleton) by attaching `n`-disks.
+
+* `CWComplex`: A CW-complex is a relative CW-complex whose `sk 0` (i.e., $(-1)$-skeleton) is empty.
+
+## References
+
+* [R. Fritsch and R. Piccinini, *Cellular Structures in Topology*][fritsch-piccinini1990]
+* The definition of CW-complexes follows David Wärn's suggestion on
+  [Zulip](https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there-code-for-X.3F/topic/Do.20we.20have.20CW.20complexes.3F/near/231769080).
+-/
+
+open CategoryTheory TopCat
+
+universe u
+
+namespace RelativeCWComplex
+
+/-- The inclusion map from the `n`-sphere to the `(n + 1)`-disk. (For `n = -1`, this
+involves the empty space `𝕊 (-1)`. This is the reason why `sphere` takes `n : ℤ` as
+an input rather than `n : ℕ`.) -/
+def sphereInclusion (n : ℤ) : 𝕊 n ⟶ 𝔻 (n + 1) where
+  toFun := fun ⟨p, hp⟩ ↦ ⟨p, le_of_eq hp⟩
+  continuous_toFun := ⟨fun t ⟨s, ⟨r, hro, hrs⟩, hst⟩ ↦ by
+    rw [isOpen_induced_iff, ← hst, ← hrs]
+    tauto⟩
+
+/-- A type witnessing that `X'` is obtained from `X` by attaching generalized cells `f : S ⟶ D` -/
+structure AttachGeneralizedCells {S D : TopCat.{u}} (f : S ⟶ D) (X X' : TopCat.{u}) where
+  /-- The index type over the generalized cells -/
+  cells : Type u
+  /-- An attaching map for each generalized cell -/
+  attachMaps : cells → (S ⟶ X)
+  /-- `X'` is the pushout of `∐ S ⟶ X` and `∐ S ⟶ ∐ D`. -/
+  iso_pushout : X' ≅ Limits.pushout (Limits.Sigma.desc attachMaps) (Limits.Sigma.map fun _ ↦ f)
+
+/-- A type witnessing that `X'` is obtained from `X` by attaching `(n + 1)`-disks -/
+def AttachCells (n : ℤ) := AttachGeneralizedCells (sphereInclusion n)
+
+end RelativeCWComplex
+
+/-- A relative CW-complex consists of an expanding sequence of subspaces `sk i` (called the
+$(i-1)$-skeleton) for `i ≥ 0`, where `sk 0` (i.e., the $(-1)$-skeleton) is an arbitrary topological
+space, and each `sk (n + 1)` (i.e., the `n`-skeleton) is obtained from `sk n` (i.e., the
+$(n-1)$-skeleton) by attaching `n`-disks. -/
+structure RelativeCWComplex where
+  /-- The skeletons. Note: `sk i` is usually called the $(i-1)$-skeleton in the math literature. -/
+  sk : ℕ → TopCat.{u}
+  /-- Each `sk (n + 1)` (i.e., the $n$-skeleton) is obtained from `sk n`
+  (i.e., the $(n-1)$-skeleton) by attaching `n`-disks. -/
+  attachCells (n : ℕ) : RelativeCWComplex.AttachCells ((n : ℤ) - 1) (sk n) (sk (n + 1))
+
+/-- A CW-complex is a relative CW-complex whose `sk 0` (i.e., $(-1)$-skeleton) is empty. -/
+structure CWComplex extends RelativeCWComplex.{u} where
+  /-- `sk 0` (i.e., the $(-1)$-skeleton) is empty. -/
+  isEmpty_sk_zero : IsEmpty (sk 0)
+
+namespace RelativeCWComplex
+
+noncomputable section Topology
+
+/-- The inclusion map from `X` to `X'`, when `X'` is obtained from `X`
+by attaching generalized cells `f : S ⟶ D`. -/
+def AttachGeneralizedCells.inclusion {S D : TopCat.{u}} {f : S ⟶ D} {X X' : TopCat.{u}}
+    (att : AttachGeneralizedCells f X X') : X ⟶ X' :=
+  Limits.pushout.inl _ _ ≫ att.iso_pushout.inv
+
+/-- The inclusion map from `sk n` (i.e., the $(n-1)$-skeleton) to `sk (n + 1)` (i.e., the
+$n$-skeleton) of a relative CW-complex -/
+def skInclusion (X : RelativeCWComplex.{u}) (n : ℕ) : X.sk n ⟶ X.sk (n + 1) :=
+  (X.attachCells n).inclusion
+
+/-- The topology on a relative CW-complex -/
+def toTopCat (X : RelativeCWComplex.{u}) : TopCat.{u} :=
+  Limits.colimit (Functor.ofSequence X.skInclusion)
+
+instance : Coe RelativeCWComplex TopCat where coe X := toTopCat X
+
+end Topology
+
+end RelativeCWComplex
diff --git a/Mathlib/Topology/Category/CompHaus/Basic.lean b/Mathlib/Topology/Category/CompHaus/Basic.lean
index 19e8edca81f80..0306bef4b8f7d 100644
--- a/Mathlib/Topology/Category/CompHaus/Basic.lean
+++ b/Mathlib/Topology/Category/CompHaus/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Adam Topaz, Bhavik Mehta
+Authors: Adam Topaz, Bhavik Mehta, Dagur Asgeirsson
 -/
 import Mathlib.CategoryTheory.Monad.Limits
 import Mathlib.Topology.StoneCech
@@ -24,6 +24,11 @@ equivalence of categories in `CompactumToCompHaus.isEquivalence`.
 See `Mathlib/Topology/Category/Compactum.lean` for a more detailed discussion where these
 definitions are introduced.
 
+## Implementation
+
+The category `CompHaus` is defined using the structure `CompHausLike`. See the file
+`CompHausLike.Basic` for more information.
+
 -/
 
 
@@ -219,6 +224,6 @@ theorem epi_iff_surjective {X Y : CompHaus.{u}} (f : X ⟶ Y) : Epi f ↔ Functi
 
 end CompHaus
 
-/-- Every `CompHausLike` admits a functor to `CompHaus`. -/
+/-- Every `CompHausLike` admits a functor to `CompHaus`. -/
 abbrev compHausLikeToCompHaus (P : TopCat → Prop) : CompHausLike P ⥤ CompHaus :=
   CompHausLike.toCompHausLike (by simp only [implies_true])
diff --git a/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean b/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
index c6517bf9f086a..11338ba0c53d7 100644
--- a/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
+++ b/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
@@ -41,12 +41,9 @@ theorem effectiveEpi_tfae
     , Epi π
     , Function.Surjective π
     ] := by
-  tfae_have 1 → 2
-  · intro; infer_instance
-  tfae_have 2 ↔ 3
-  · exact epi_iff_surjective π
-  tfae_have 3 → 1
-  · exact fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 2 ↔ 3 := epi_iff_surjective π
+  tfae_have 3 → 1 := fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
   tfae_finish
 
 instance : Preregular CompHaus :=
@@ -65,12 +62,12 @@ theorem effectiveEpiFamily_tfae
     , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
     ] := by
   tfae_have 2 → 1
-  · intro
+  | _ => by
     simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
   tfae_have 1 → 2
-  · intro; infer_instance
+  | _ => inferInstance
   tfae_have 3 → 2
-  · intro e
+  | e => by
     rw [epi_iff_surjective]
     intro b
     obtain ⟨t, x, h⟩ := e b
@@ -78,7 +75,8 @@ theorem effectiveEpiFamily_tfae
     change (Sigma.ι X t ≫ Sigma.desc π) x = _
     simpa using h
   tfae_have 2 → 3
-  · intro e; rw [epi_iff_surjective] at e
+  | e => by
+    rw [epi_iff_surjective] at e
     let i : ∐ X ≅ finiteCoproduct X :=
       (colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
     intro b
diff --git a/Mathlib/Topology/Category/CompHaus/Projective.lean b/Mathlib/Topology/Category/CompHaus/Projective.lean
index 17bdb7ef5c926..97532c9b45d96 100644
--- a/Mathlib/Topology/Category/CompHaus/Projective.lean
+++ b/Mathlib/Topology/Category/CompHaus/Projective.lean
@@ -52,7 +52,7 @@ instance projective_ultrafilter (X : Type*) : Projective (of <| Ultrafilter X) w
     let g'' : ContinuousMap Y Z := g
     have : g'' ∘ g' = id := hg'.comp_eq_id
     -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-    erw [comp.assoc, ultrafilter_extend_extends, ← comp.assoc, this, id_comp]
+    erw [comp_assoc, ultrafilter_extend_extends, ← comp_assoc, this, id_comp]
 
 /-- For any compact Hausdorff space `X`,
   the natural map `Ultrafilter X → X` is a projective presentation. -/
diff --git a/Mathlib/Topology/Category/CompHausLike/Basic.lean b/Mathlib/Topology/Category/CompHausLike/Basic.lean
index 05401767a1c2c..ca9de369a082e 100644
--- a/Mathlib/Topology/Category/CompHausLike/Basic.lean
+++ b/Mathlib/Topology/Category/CompHausLike/Basic.lean
@@ -10,6 +10,54 @@ import Mathlib.Topology.Category.TopCat.Basic
 # Categories of Compact Hausdorff Spaces
 
 We construct the category of compact Hausdorff spaces satisfying an additional property `P`.
+
+## Implementation
+
+We define a structure `CompHausLike` which takes as an argument a predicate `P` on topological
+spaces. It consists of the data of a topological space, satisfying the additional properties of
+being compact and Hausdorff, and satisfying `P`. We give a category structure to `CompHausLike P`
+induced by the forgetful functor to topological spaces.
+
+It used to be the case (before #12930 was merged) that several different categories of compact
+Hausdorff spaces, possibly satisfying some extra property, were defined from scratch in this way.
+For example, one would define a structure `CompHaus` as follows:
+
+```lean
+structure CompHaus where
+  toTop : TopCat
+  [is_compact : CompactSpace toTop]
+  [is_hausdorff : T2Space toTop]
+```
+
+and give it the category structure induced from topological spaces. Then the category of profinite
+spaces was defined as follows:
+
+```lean
+structure Profinite where
+  toCompHaus : CompHaus
+  [isTotallyDisconnected : TotallyDisconnectedSpace toCompHaus]
+```
+
+The categories `Stonean` consisting of extremally disconnected compact Hausdorff spaces and
+`LightProfinite` consisting of totally disconnected, second countable compact Hausdorff spaces were
+defined in a similar way. This resulted in code duplication, and reducing this duplication was part
+of the motivation for introducing `CompHausLike`.
+
+Using `CompHausLike`, we can now define
+`CompHaus := CompHausLike (fun _ ↦ True)`
+`Profinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X)`.
+`Stonean := CompHausLike (fun X ↦ ExtremallyDisconnected X)`.
+`LightProfinite := CompHausLike  (fun X ↦ TotallyDisconnectedSpace X ∧ SecondCountableTopology X)`.
+
+These four categories are important building blocks of condensed objects (see the files
+`Condensed.Basic` and `Condensed.Light.Basic`). These categories share many properties and often,
+one wants to argue about several of them simultaneously. This is the other part of the motivation
+for introducing `CompHausLike`. On paper, one would say "let `C` be on of the categories `CompHaus`
+or `Profinite`, then the following holds: ...". This was not possible in Lean using the old
+definitions. Using the new definitions, this becomes a matter of identifying what common property
+of `CompHaus` and `Profinite` is used in the proof in question, and then proving the theorem for
+`CompHausLike P` satisfying that property, and it will automatically apply to both `CompHaus` and
+`Profinite`.
 -/
 
 universe u
@@ -49,7 +97,7 @@ instance hasForget₂ : HasForget₂ (CompHausLike P) TopCat :=
 
 variable (X : Type u) [TopologicalSpace X] [CompactSpace X] [T2Space X]
 
-/-- This wraps the predicate `P : TopCat → Prop` in a typeclass. -/
+/-- This wraps the predicate `P : TopCat → Prop` in a typeclass. -/
 class HasProp : Prop where
   hasProp : P (TopCat.of X)
 
@@ -91,7 +139,7 @@ instance (X : CompHausLike.{u} P) : T2Space ((forget (CompHausLike P)).obj X) :=
 
 variable {P}
 
-/-- If `P` imples `P'`, then there is a functor from `CompHausLike P` to `CompHausLike P'`. -/
+/-- If `P` imples `P'`, then there is a functor from `CompHausLike P` to `CompHausLike P'`. -/
 @[simps]
 def toCompHausLike {P P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) :
     CompHausLike P ⥤ CompHausLike P' where
@@ -104,7 +152,7 @@ section
 
 variable {P P' : TopCat → Prop} (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop)
 
-/-- If `P` imples `P'`, then the functor from `CompHausLike P` to `CompHausLike P'` is fully
+/-- If `P` imples `P'`, then the functor from `CompHausLike P` to `CompHausLike P'` is fully
 faithful. -/
 def fullyFaithfulToCompHausLike : (toCompHausLike h).FullyFaithful :=
   fullyFaithfulInducedFunctor _
@@ -207,4 +255,13 @@ def isoEquivHomeo {X Y : CompHausLike.{u} P} : (X ≅ Y) ≃ (X ≃ₜ Y) where
   left_inv _ := rfl
   right_inv _ := rfl
 
+/-- A constant map as a morphism in `CompHausLike` -/
+def const {P : TopCat.{u} → Prop}
+    (T : CompHausLike.{u} P) {S : CompHausLike.{u} P} (s : S) : T ⟶ S :=
+  ContinuousMap.const _ s
+
+lemma const_comp {P : TopCat.{u} → Prop} {S T U : CompHausLike.{u} P}
+    (s : S) (g : S ⟶ U) : T.const s ≫ g = T.const (g s) :=
+  rfl
+
 end CompHausLike
diff --git a/Mathlib/Topology/Category/CompHausLike/Limits.lean b/Mathlib/Topology/Category/CompHausLike/Limits.lean
index c3705bfeab520..845abc60f56e7 100644
--- a/Mathlib/Topology/Category/CompHausLike/Limits.lean
+++ b/Mathlib/Topology/Category/CompHausLike/Limits.lean
@@ -18,14 +18,14 @@ which may be useful due to their definitional properties.
 * `HasExplicitFiniteCoproducts`: A typeclass describing the property that forming all finite
   disjoint unions is stable under the property `P`.
   - Given this property, we deduce that `CompHausLike P` has finite coproducts and the inclusion
-    functors to other `CompHausLike P'` and to `TopCat` preserve them.
+    functors to other `CompHausLike P'` and to `TopCat` preserve them.
 
 * `HasExplicitPullbacks`: A typeclass describing the property that forming all "explicit pullbacks"
   is stable under the property `P`. Here, explicit pullbacks are defined as a subset of the product.
   - Given this property, we deduce that `CompHausLike P` has pullbacks and the inclusion
-    functors to other `CompHausLike P'` and to `TopCat` preserve them.
+    functors to other `CompHausLike P'` and to `TopCat` preserve them.
   - We also define a variant `HasExplicitPullbacksOfInclusions` which is says that explicit
-    pullbacks along inclusion maps into finite disjoint unions exist. `Stonean` has this property
+    pullbacks along inclusion maps into finite disjoint unions exist. `Stonean` has this property
     but not the stronger one.
 
 ## Main results
@@ -127,7 +127,7 @@ class HasExplicitFiniteCoproducts : Prop where
   hasProp {α : Type w} [Finite α] (X : α → CompHausLike.{max u w} P) : HasExplicitFiniteCoproduct X
 
 /-
-This linter complains that the universes `u` and `w` only occur together, but `w` appears by itself
+This linter complains that the universes `u` and `w` only occur together, but `w` appears by itself
 in the indexing type of the coproduct. In almost all cases, `w` will be either `0` or `u`, but we
 want to allow both possibilities.
 -/
@@ -164,7 +164,7 @@ lemma Sigma.openEmbedding_ι (a : α) :
   change (Sigma.ι X a ≫ _) x = _
   simp
 
-/-- The functor to `TopCat` preserves finite coproducts if they exist. -/
+/-- The functor to `TopCat` preserves finite coproducts if they exist. -/
 instance (P) [HasExplicitFiniteCoproducts.{0} P] :
     PreservesFiniteCoproducts (compHausLikeToTop P) := by
   refine ⟨fun J hJ ↦ ⟨fun {F} ↦ ?_⟩⟩
@@ -173,7 +173,7 @@ instance (P) [HasExplicitFiniteCoproducts.{0} P] :
   apply preservesColimitOfPreservesColimitCocone (CompHausLike.finiteCoproduct.isColimit _)
   exact TopCat.sigmaCofanIsColimit _
 
-/-- The functor to another `CompHausLike` preserves finite coproducts if they exist. -/
+/-- The functor to another `CompHausLike` preserves finite coproducts if they exist. -/
 noncomputable instance {P' : TopCat.{u} → Prop}
     (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) :
     PreservesFiniteCoproducts (toCompHausLike h) := by
@@ -275,18 +275,18 @@ def pullback.isLimit : Limits.IsLimit (pullback.cone f g) :=
 instance : HasLimit (cospan f g) where
   exists_limit := ⟨⟨pullback.cone f g, pullback.isLimit f g⟩⟩
 
-/-- The functor to `TopCat` creates pullbacks if they exist. -/
+/-- The functor to `TopCat` creates pullbacks if they exist. -/
 noncomputable instance : CreatesLimit (cospan f g) (compHausLikeToTop P) := by
   refine createsLimitOfFullyFaithfulOfIso (pullback f g)
     (((TopCat.pullbackConeIsLimit f g).conePointUniqueUpToIso
         (limit.isLimit _)) ≪≫ Limits.lim.mapIso (?_ ≪≫ (diagramIsoCospan _).symm))
   exact Iso.refl _
 
-/-- The functor to `TopCat` preserves pullbacks. -/
+/-- The functor to `TopCat` preserves pullbacks. -/
 noncomputable instance : PreservesLimit (cospan f g) (compHausLikeToTop P) :=
   preservesLimitOfCreatesLimitAndHasLimit _ _
 
-/-- The functor to another `CompHausLike` preserves pullbacks. -/
+/-- The functor to another `CompHausLike` preserves pullbacks. -/
 noncomputable instance {P' : TopCat → Prop}
     (h : ∀ (X : CompHausLike P), P X.toTop → P' X.toTop) :
     PreservesLimit (cospan f g) (toCompHausLike h) := by
diff --git a/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean b/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean
new file mode 100644
index 0000000000000..fc574ef66ea35
--- /dev/null
+++ b/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean
@@ -0,0 +1,66 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Topology.Category.CompHausLike.Limits
+/-!
+
+# The sigma-comparison map
+
+This file defines the map `CompHausLike.sigmaComparison` associated to a presheaf `X` on
+`CompHausLike P`, and a finite family `S₁,...,Sₙ` of spaces in `CompHausLike P`, where `P` is
+stable under taking finite disjoint unions.
+
+The map `sigmaComparison` is the canonical map `X(S₁ ⊔ ... ⊔ Sₙ) ⟶ X(S₁) × ... × X(Sₙ)` induced by
+the inclusion maps `Sᵢ ⟶ S₁ ⊔ ... ⊔ Sₙ`, and it is an isomorphism when `X` preserves finite
+products.
+-/
+
+universe u w
+
+open CategoryTheory Limits
+
+namespace CompHausLike
+
+variable {P : TopCat.{u} → Prop} [HasExplicitFiniteCoproducts.{u} P]
+  (X : (CompHausLike.{u} P)ᵒᵖ ⥤ Type max u w) [PreservesFiniteProducts X]
+  {α : Type u} [Finite α] (σ : α → Type u)
+  [∀ a, TopologicalSpace (σ a)] [∀ a, CompactSpace (σ a)] [∀ a, T2Space (σ a)]
+  [∀ a, HasProp P (σ a)]
+
+instance : HasProp P (Σ (a : α), (σ a)) := HasExplicitFiniteCoproducts.hasProp (fun a ↦ of P (σ a))
+
+/--
+The comparison map from the value of a condensed set on a finite coproduct to the product of the
+values on the components.
+-/
+def sigmaComparison : X.obj ⟨(of P ((a : α) × σ a))⟩ ⟶ ((a : α) → X.obj ⟨of P (σ a)⟩) :=
+  fun x a ↦ X.map ⟨Sigma.mk a, continuous_sigmaMk⟩ x
+
+theorem sigmaComparison_eq_comp_isos : sigmaComparison X σ =
+    (X.mapIso (opCoproductIsoProduct'
+      (finiteCoproduct.isColimit.{u, u} (fun a ↦ of P (σ a)))
+      (productIsProduct fun x ↦ Opposite.op (of P (σ x))))).hom ≫
+    (PreservesProduct.iso X fun a ↦ ⟨of P (σ a)⟩).hom ≫
+    (Types.productIso.{u, max u w} fun a ↦ X.obj ⟨of P (σ a)⟩).hom := by
+  ext x a
+  simp only [Cofan.mk_pt, Fan.mk_pt, Functor.mapIso_hom,
+    PreservesProduct.iso_hom, types_comp_apply, Types.productIso_hom_comp_eval_apply]
+  have := congrFun (piComparison_comp_π X (fun a ↦ ⟨of P (σ a)⟩) a)
+  simp only [types_comp_apply] at this
+  rw [this, ← FunctorToTypes.map_comp_apply]
+  simp only [sigmaComparison]
+  apply congrFun
+  congr 2
+  rw [← opCoproductIsoProduct_inv_comp_ι]
+  simp only [coe_of, Opposite.unop_op, unop_comp, Quiver.Hom.unop_op, Category.assoc]
+  simp only [opCoproductIsoProduct, ← unop_comp, opCoproductIsoProduct'_comp_self]
+  erw [IsColimit.fac]
+  rfl
+
+instance isIsoSigmaComparison : IsIso <| sigmaComparison X σ := by
+  rw [sigmaComparison_eq_comp_isos]
+  infer_instance
+
+end CompHausLike
diff --git a/Mathlib/Topology/Category/CompactlyGenerated.lean b/Mathlib/Topology/Category/CompactlyGenerated.lean
index 200b869bfa079..2563c6dffb796 100644
--- a/Mathlib/Topology/Category/CompactlyGenerated.lean
+++ b/Mathlib/Topology/Category/CompactlyGenerated.lean
@@ -9,7 +9,7 @@ import Mathlib.CategoryTheory.Elementwise
 
 # Compactly generated topological spaces
 
-This file defines the category of compactly generated topological spaces. These are spaces `X` such
+This file defines the category of compactly generated topological spaces. These are spaces `X` such
 that a map `f : X → Y` is continuous whenever the composition `S → X → Y` is continuous for all
 compact Hausdorff spaces `S` mapping continuously to `X`.
 
diff --git a/Mathlib/Topology/Category/Compactum.lean b/Mathlib/Topology/Category/Compactum.lean
index 7cd821aa6d9d6..1bb55e129ab03 100644
--- a/Mathlib/Topology/Category/Compactum.lean
+++ b/Mathlib/Topology/Category/Compactum.lean
@@ -211,7 +211,7 @@ private theorem cl_cl {X : Compactum} (A : Set X) : cl (cl A) ⊆ cl A := by
   have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩ C ∈ C0 := by
     rintro B ⟨Q, hQ, rfl⟩ C ⟨R, hR, rfl⟩
     use Q ∩ R
-    simp only [and_true_iff, eq_self_iff_true, Set.preimage_inter]
+    simp only [and_true, eq_self_iff_true, Set.preimage_inter]
     exact inter_sets _ hQ hR
   -- All sets in C0 are nonempty.
   have claim2 : ∀ B ∈ C0, Set.Nonempty B := by
diff --git a/Mathlib/Topology/Category/LightProfinite/AsLimit.lean b/Mathlib/Topology/Category/LightProfinite/AsLimit.lean
index d5e2c6c5f4e62..48104258a6815 100644
--- a/Mathlib/Topology/Category/LightProfinite/AsLimit.lean
+++ b/Mathlib/Topology/Category/LightProfinite/AsLimit.lean
@@ -36,14 +36,14 @@ abbrev diagram : ℕᵒᵖ ⥤ LightProfinite := S.fintypeDiagram ⋙ FintypeCat
 
 /--
 A cone over `S.diagram` whose cone point is isomorphic to `S`.
-(Auxiliary definition, use `S.asLimitCone` instead.)
+(Auxiliary definition, use `S.asLimitCone` instead.)
 -/
 def asLimitConeAux : Cone S.diagram :=
   let c : Cone (S.diagram ⋙ lightToProfinite) := S.toLightDiagram.cone
   let hc : IsLimit c := S.toLightDiagram.isLimit
   liftLimit hc
 
-/-- An auxiliary isomorphism of cones used to prove that `S.asLimitConeAux` is a limit cone. -/
+/-- An auxiliary isomorphism of cones used to prove that `S.asLimitConeAux` is a limit cone. -/
 def isoMapCone : lightToProfinite.mapCone S.asLimitConeAux ≅ S.toLightDiagram.cone :=
   let c : Cone (S.diagram ⋙ lightToProfinite) := S.toLightDiagram.cone
   let hc : IsLimit c := S.toLightDiagram.isLimit
@@ -51,7 +51,7 @@ def isoMapCone : lightToProfinite.mapCone S.asLimitConeAux ≅ S.toLightDiagram.
 
 /--
 `S.asLimitConeAux` is indeed a limit cone.
-(Auxiliary definition, use `S.asLimit` instead.)
+(Auxiliary definition, use `S.asLimit` instead.)
 -/
 def asLimitAux : IsLimit S.asLimitConeAux :=
   let hc : IsLimit (lightToProfinite.mapCone S.asLimitConeAux) :=
diff --git a/Mathlib/Topology/Category/LightProfinite/Basic.lean b/Mathlib/Topology/Category/LightProfinite/Basic.lean
index 097f5cc942104..56a5871cbcd22 100644
--- a/Mathlib/Topology/Category/LightProfinite/Basic.lean
+++ b/Mathlib/Topology/Category/LightProfinite/Basic.lean
@@ -17,8 +17,14 @@ implemented as totally disconnected second countable compact Hausdorff spaces.
 This file also defines the category `LightDiagram`, which consists of those spaces that can be
 written as a sequential limit (in `Profinite`) of finite sets.
 
-We define an equivalence of categories `LightProfinite ≌ LightDiagram` and prove that these are
+We define an equivalence of categories `LightProfinite ≌ LightDiagram` and prove that these are
 essentially small categories.
+
+## Implementation
+
+The category `LightProfinite` is defined using the structure `CompHausLike`. See the file
+`CompHausLike.Basic` for more information.
+
 -/
 
 /- The basic API for `LightProfinite` is largely copied from the API of `Profinite`;
@@ -113,6 +119,11 @@ instance : FintypeCat.toLightProfinite.Faithful :=
 instance : FintypeCat.toLightProfinite.Full :=
   FintypeCat.toLightProfiniteFullyFaithful.full
 
+instance (X : FintypeCat.{u}) : Fintype (FintypeCat.toLightProfinite.obj X) :=
+  inferInstanceAs (Fintype X)
+
+instance (X : FintypeCat.{u}) : Fintype (LightProfinite.of X) :=  inferInstanceAs (Fintype X)
+
 end DiscreteTopology
 
 namespace LightProfinite
@@ -225,6 +236,9 @@ theorem epi_iff_surjective {X Y : LightProfinite.{u}} (f : X ⟶ Y) :
   · rw [← CategoryTheory.epi_iff_surjective]
     apply (forget LightProfinite).epi_of_epi_map
 
+instance : lightToProfinite.PreservesEpimorphisms where
+  preserves f _ := (Profinite.epi_iff_surjective _).mpr ((epi_iff_surjective f).mp inferInstance)
+
 end LightProfinite
 
 /-- A structure containing the data of sequential limit in `Profinite` of finite sets. -/
@@ -238,7 +252,7 @@ structure LightDiagram : Type (u+1) where
 
 namespace LightDiagram
 
-/-- The underlying `Profinite` of a `LightDiagram`. -/
+/-- The underlying `Profinite` of a `LightDiagram`. -/
 def toProfinite (S : LightDiagram) : Profinite := S.cone.pt
 
 @[simps!]
diff --git a/Mathlib/Topology/Category/LightProfinite/Extend.lean b/Mathlib/Topology/Category/LightProfinite/Extend.lean
new file mode 100644
index 0000000000000..4f7e5eb315b90
--- /dev/null
+++ b/Mathlib/Topology/Category/LightProfinite/Extend.lean
@@ -0,0 +1,198 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Topology.Category.LightProfinite.AsLimit
+import Mathlib.Topology.Category.Profinite.Extend
+
+/-!
+
+# Extending cones in `LightProfinite`
+
+Let `(Sₙ)_{n : ℕᵒᵖ}` be a sequential inverse system of finite sets and let `S` be
+its limit in `Profinite`. Let `G` be a functor from `LightProfinite` to a category `C` and suppose
+that `G` preserves the limit described above. Suppose further that the projection maps `S ⟶ Sₙ` are
+epimorphic for all `n`. Then `G.obj S` is isomorphic to a limit indexed by
+`StructuredArrow S toLightProfinite` (see `LightProfinite.Extend.isLimitCone`).
+
+We also provide the dual result for a functor of the form `G : LightProfiniteᵒᵖ ⥤ C`.
+
+We apply this to define `LightProfinite.diagram'`, `LightProfinite.asLimitCone'`, and
+`LightProfinite.asLimit'`, analogues to their unprimed versions in
+`Mathlib.Topology.Category.LightProfinite.AsLimit`, in which the
+indexing category is `StructuredArrow S toLightProfinite` instead of `ℕᵒᵖ`.
+-/
+
+universe u
+
+open CategoryTheory Limits FintypeCat Functor
+
+attribute [local instance] FintypeCat.discreteTopology ConcreteCategory.instFunLike
+
+namespace LightProfinite
+
+variable {F : ℕᵒᵖ ⥤ FintypeCat.{u}} (c : Cone <| F ⋙ toLightProfinite)
+
+namespace Extend
+
+/--
+Given a sequential cone in `LightProfinite` consisting of finite sets,
+we obtain a functor from the indexing category to `StructuredArrow c.pt toLightProfinite`.
+-/
+@[simps]
+def functor : ℕᵒᵖ ⥤ StructuredArrow c.pt toLightProfinite where
+  obj i := StructuredArrow.mk (c.π.app i)
+  map f := StructuredArrow.homMk (F.map f) (c.w f)
+
+-- We check that the original diagram factors through `LightProfinite.Extend.functor`.
+example : functor c ⋙ StructuredArrow.proj c.pt toLightProfinite ≅ F := Iso.refl _
+
+/--
+Given a sequential cone in `LightProfinite` consisting of finite sets,
+we obtain a functor from the opposite of the indexing category to
+`CostructuredArrow toProfinite.op ⟨c.pt⟩`.
+-/
+@[simps! obj map]
+def functorOp : ℕ ⥤ CostructuredArrow toLightProfinite.op ⟨c.pt⟩ :=
+  (functor c).rightOp ⋙ StructuredArrow.toCostructuredArrow _ _
+
+-- We check that the opposite of the original diagram factors through `Profinite.Extend.functorOp`.
+example : functorOp c ⋙ CostructuredArrow.proj toLightProfinite.op ⟨c.pt⟩ ≅ F.rightOp := Iso.refl _
+
+-- We check that `Profinite.Extend.functor` factors through `LightProfinite.Extend.functor`,
+-- via the equivalence `StructuredArrow.post _ _ lightToProfinite`.
+example : functor c ⋙ (StructuredArrow.post _ _ lightToProfinite) =
+  Profinite.Extend.functor (lightToProfinite.mapCone c) := rfl
+
+/--
+If the projection maps in the cone are epimorphic and the cone is limiting, then
+`LightProfinite.Extend.functor` is initial.
+-/
+theorem functor_initial (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Initial (functor c) := by
+  rw [initial_iff_comp_equivalence _ (StructuredArrow.post _ _ lightToProfinite)]
+  have : ∀ i, Epi ((lightToProfinite.mapCone c).π.app i) :=
+    fun i ↦ inferInstanceAs (Epi (lightToProfinite.map (c.π.app i)))
+  exact Profinite.Extend.functor_initial _ (isLimitOfPreserves lightToProfinite hc)
+
+/--
+If the projection maps in the cone are epimorphic and the cone is limiting, then
+`LightProfinite.Extend.functorOp` is final.
+-/
+theorem functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) := by
+  have := functor_initial c hc
+  have : ((StructuredArrow.toCostructuredArrow toLightProfinite c.pt)).IsEquivalence  :=
+    (inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence )
+  have : (functor c).rightOp.Final :=
+    inferInstanceAs ((opOpEquivalence ℕ).inverse ⋙ (functor c).op).Final
+  exact Functor.final_comp (functor c).rightOp _
+
+section Limit
+
+variable {C : Type*} [Category C] (G : LightProfinite ⥤ C)
+
+/--
+Given a functor `G` from `LightProfinite` and `S : LightProfinite`, we obtain a cone on
+`(StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G)` with cone point `G.obj S`.
+
+Whiskering this cone with `LightProfinite.Extend.functor c` gives `G.mapCone c` as we check in the
+example below.
+-/
+def cone (S : LightProfinite) :
+    Cone (StructuredArrow.proj S toLightProfinite ⋙ toLightProfinite ⋙ G) where
+  pt := G.obj S
+  π := {
+    app := fun i ↦ G.map i.hom
+    naturality := fun _ _ f ↦ (by
+      have := f.w
+      simp only [const_obj_obj, StructuredArrow.left_eq_id, const_obj_map, Category.id_comp,
+        StructuredArrow.w] at this
+      simp only [const_obj_obj, comp_obj, StructuredArrow.proj_obj, const_obj_map, Category.id_comp,
+        Functor.comp_map, StructuredArrow.proj_map, ← map_comp, StructuredArrow.w]) }
+
+example : G.mapCone c = (cone G c.pt).whisker (functor c) := rfl
+
+/--
+If `c` and `G.mapCone c` are limit cones and the projection maps in `c` are epimorphic,
+then `cone G c.pt` is a limit cone.
+-/
+noncomputable
+def isLimitCone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsLimit <| G.mapCone c) :
+    IsLimit (cone G c.pt) := (functor_initial c hc).isLimitWhiskerEquiv _ _ hc'
+
+end Limit
+
+section Colimit
+
+variable {C : Type*} [Category C] (G : LightProfiniteᵒᵖ ⥤ C)
+
+/--
+Given a functor `G` from `LightProfiniteᵒᵖ` and `S : LightProfinite`, we obtain a cocone on
+`(CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G)` with cocone point
+`G.obj ⟨S⟩`.
+
+Whiskering this cocone with `LightProfinite.Extend.functorOp c` gives `G.mapCocone c.op` as we
+check in the example below.
+-/
+@[simps]
+def cocone (S : LightProfinite) :
+    Cocone (CostructuredArrow.proj toLightProfinite.op ⟨S⟩ ⋙ toLightProfinite.op ⋙ G) where
+  pt := G.obj ⟨S⟩
+  ι := {
+    app := fun i ↦ G.map i.hom
+    naturality := fun _ _ f ↦ (by
+      have := f.w
+      simp only [op_obj, const_obj_obj, op_map, CostructuredArrow.right_eq_id, const_obj_map,
+        Category.comp_id] at this
+      simp only [comp_obj, CostructuredArrow.proj_obj, op_obj, const_obj_obj, Functor.comp_map,
+        CostructuredArrow.proj_map, op_map, ← map_comp, this, const_obj_map, Category.comp_id]) }
+
+example : G.mapCocone c.op = (cocone G c.pt).whisker
+  ((opOpEquivalence ℕ).functor ⋙ functorOp c) := rfl
+
+/--
+If `c` is a limit cone, `G.mapCocone c.op` is a colimit cone and the projection maps in `c`
+are epimorphic, then `cocone G c.pt` is a colimit cone.
+-/
+noncomputable
+def isColimitCocone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsColimit <| G.mapCocone c.op) :
+    IsColimit (cocone G c.pt) :=
+  haveI := functorOp_final c hc
+  (Functor.final_comp (opOpEquivalence ℕ).functor (functorOp c)).isColimitWhiskerEquiv _ _ hc'
+
+end Colimit
+
+end Extend
+
+open Extend
+
+section LightProfiniteAsLimit
+
+variable (S : LightProfinite.{u})
+
+/--
+A functor `StructuredArrow S toLightProfinite ⥤ FintypeCat` whose limit in `LightProfinite` is
+isomorphic to `S`.
+-/
+abbrev fintypeDiagram' : StructuredArrow S toLightProfinite ⥤ FintypeCat :=
+  StructuredArrow.proj S toLightProfinite
+
+/-- An abbreviation for `S.fintypeDiagram' ⋙ toLightProfinite`. -/
+abbrev diagram' : StructuredArrow S toLightProfinite ⥤ LightProfinite :=
+  S.fintypeDiagram' ⋙ toLightProfinite
+
+/-- A cone over `S.diagram'` whose cone point is `S`. -/
+def asLimitCone' : Cone (S.diagram') := cone (𝟭 _) S
+
+instance (i : ℕᵒᵖ) : Epi (S.asLimitCone.π.app i) :=
+  (epi_iff_surjective _).mpr (S.proj_surjective _)
+
+/-- `S.asLimitCone'` is a limit cone. -/
+noncomputable def asLimit' : IsLimit S.asLimitCone' := isLimitCone _ (𝟭 _) S.asLimit S.asLimit
+
+/-- A bundled version of `S.asLimitCone'` and `S.asLimit'`. -/
+noncomputable def lim' : LimitCone S.diagram' := ⟨S.asLimitCone', S.asLimit'⟩
+
+end LightProfiniteAsLimit
+
+end LightProfinite
diff --git a/Mathlib/Topology/Category/LightProfinite/Sequence.lean b/Mathlib/Topology/Category/LightProfinite/Sequence.lean
index 17d3590bd8020..ae729f14b1979 100644
--- a/Mathlib/Topology/Category/LightProfinite/Sequence.lean
+++ b/Mathlib/Topology/Category/LightProfinite/Sequence.lean
@@ -17,7 +17,7 @@ open CategoryTheory TopologicalSpace OnePoint
 
 namespace LightProfinite
 
-/-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. -/
+/-- The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0`. -/
 noncomputable def natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where
   toFun
     | ∞ => 0
@@ -26,7 +26,7 @@ noncomputable def natUnionInftyEmbedding : C(OnePoint ℕ, ℝ) where
     tendsto_one_div_add_atTop_nhds_zero_nat
 
 /--
-The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed
+The continuous map from `ℕ∪{∞}` to `ℝ` sending `n` to `1/(n+1)` and `∞` to `0` is a closed
 embedding.
 -/
 lemma closedEmbedding_natUnionInftyEmbedding : ClosedEmbedding natUnionInftyEmbedding := by
diff --git a/Mathlib/Topology/Category/Profinite/Basic.lean b/Mathlib/Topology/Category/Profinite/Basic.lean
index 2fcfa55ea7cfa..fdde9e65b27a8 100644
--- a/Mathlib/Topology/Category/Profinite/Basic.lean
+++ b/Mathlib/Topology/Category/Profinite/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Kevin Buzzard. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Kevin Buzzard, Calle Sönne
+Authors: Kevin Buzzard, Calle Sönne, Dagur Asgeirsson
 -/
 import Mathlib.CategoryTheory.FintypeCat
 import Mathlib.Topology.Category.CompHaus.Basic
@@ -23,6 +23,9 @@ is called `Profinite.toTop`.
 A profinite type is defined to be a topological space which is
 compact, Hausdorff and totally disconnected.
 
+The category `Profinite` is defined using the structure `CompHausLike`. See the file
+`CompHausLike.Basic` for more information.
+
 ## TODO
 
 * Define procategories and prove that `Profinite` is equivalent to `Pro (FintypeCat)`.
@@ -46,7 +49,7 @@ abbrev Profinite := CompHausLike (fun X ↦ TotallyDisconnectedSpace X)
 
 namespace Profinite
 
-instance  (X : Type*) [TopologicalSpace X]
+instance (X : Type*) [TopologicalSpace X]
     [TotallyDisconnectedSpace X] :  HasProp (fun Y ↦ TotallyDisconnectedSpace Y) X :=
   ⟨(inferInstance : TotallyDisconnectedSpace X)⟩
 
@@ -154,6 +157,10 @@ instance : FintypeCat.toProfinite.Faithful := FintypeCat.toProfiniteFullyFaithfu
 
 instance : FintypeCat.toProfinite.Full := FintypeCat.toProfiniteFullyFaithful.full
 
+instance (X : FintypeCat) : Fintype (FintypeCat.toProfinite.obj X) := inferInstanceAs (Fintype X)
+
+instance (X : FintypeCat) : Fintype (Profinite.of X) := inferInstanceAs (Fintype X)
+
 end DiscreteTopology
 
 end Profinite
@@ -247,4 +254,7 @@ theorem epi_iff_surjective {X Y : Profinite.{u}} (f : X ⟶ Y) : Epi f ↔ Funct
   · rw [← CategoryTheory.epi_iff_surjective]
     apply (forget Profinite).epi_of_epi_map
 
+/-- The pi-type of profinite spaces is profinite. -/
+def pi {α : Type u} (β : α → Profinite) : Profinite := .of (Π (a : α), β a)
+
 end Profinite
diff --git a/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean b/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean
index 2777ca0f06433..f6a2c77737320 100644
--- a/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean
+++ b/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean
@@ -37,12 +37,9 @@ theorem effectiveEpi_tfae
     , Epi π
     , Function.Surjective π
     ] := by
-  tfae_have 1 → 2
-  · intro; infer_instance
-  tfae_have 2 ↔ 3
-  · exact epi_iff_surjective π
-  tfae_have 3 → 1
-  · exact fun hπ ↦ ⟨⟨CompHausLike.effectiveEpiStruct π hπ⟩⟩
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 2 ↔ 3 := epi_iff_surjective π
+  tfae_have 3 → 1 := fun hπ ↦ ⟨⟨CompHausLike.effectiveEpiStruct π hπ⟩⟩
   tfae_finish
 
 instance : profiniteToCompHaus.PreservesEffectiveEpis where
@@ -54,7 +51,7 @@ instance : profiniteToCompHaus.ReflectsEffectiveEpis where
     ((Profinite.effectiveEpi_tfae f).out 0 2).mpr (((CompHaus.effectiveEpi_tfae _).out 0 2).mp h)
 
 /--
-An effective presentation of an `X : Profinite` with respect to the inclusion functor from `Stonean`
+An effective presentation of an `X : Profinite` with respect to the inclusion functor from `Stonean`
 -/
 noncomputable def profiniteToCompHausEffectivePresentation (X : CompHaus) :
     profiniteToCompHaus.EffectivePresentation X where
@@ -80,12 +77,11 @@ theorem effectiveEpiFamily_tfae
     , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
     ] := by
   tfae_have 2 → 1
-  · intro
+  | _ => by
     simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
-  tfae_have 1 → 2
-  · intro; infer_instance
-  tfae_have 3 ↔ 1
-  · erw [((CompHaus.effectiveEpiFamily_tfae
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 3 ↔ 1 := by
+    erw [((CompHaus.effectiveEpiFamily_tfae
       (fun a ↦ profiniteToCompHaus.obj (X a)) (fun a ↦ profiniteToCompHaus.map (π a))).out 2 0 : )]
     exact ⟨fun h ↦ profiniteToCompHaus.finite_effectiveEpiFamily_of_map _ _ h,
       fun _ ↦ inferInstance⟩
diff --git a/Mathlib/Topology/Category/Profinite/Extend.lean b/Mathlib/Topology/Category/Profinite/Extend.lean
new file mode 100644
index 0000000000000..2c00a5e5808e0
--- /dev/null
+++ b/Mathlib/Topology/Category/Profinite/Extend.lean
@@ -0,0 +1,217 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Topology.Category.Profinite.AsLimit
+import Mathlib.Topology.Category.Profinite.CofilteredLimit
+import Mathlib.CategoryTheory.Filtered.Final
+/-!
+
+# Extending cones in `Profinite`
+
+Let `(Sᵢ)_{i : I}` be a family of finite sets indexed by a cofiltered category `I` and let `S` be
+its limit in `Profinite`. Let `G` be a functor from `Profinite` to a category `C` and suppose that
+`G` preserves the limit described above. Suppose further that the projection maps `S ⟶ Sᵢ` are
+epimorphic for all `i`. Then `G.obj S` is isomorphic to a limit indexed by
+`StructuredArrow S toProfinite` (see `Profinite.Extend.isLimitCone`).
+
+We also provide the dual result for a functor of the form `G : Profiniteᵒᵖ ⥤ C`.
+
+We apply this to define `Profinite.diagram'`, `Profinite.asLimitCone'`, and `Profinite.asLimit'`,
+analogues to their unprimed versions in `Mathlib.Topology.Category.Profinite.AsLimit`, in which the
+indexing category is `StructuredArrow S toProfinite` instead of `DiscreteQuotient S`.
+-/
+
+universe u w
+
+open CategoryTheory Limits FintypeCat Functor
+
+attribute [local instance] ConcreteCategory.instFunLike
+
+namespace Profinite
+
+variable {I : Type u} [SmallCategory I] [IsCofiltered I]
+    {F : I ⥤ FintypeCat.{max u w}} (c : Cone <| F ⋙ toProfinite)
+
+/--
+A continuous map from a profinite set to a finite set factors through one of the components of
+the profinite set when written as a cofiltered limit of finite sets.
+-/
+lemma exists_hom (hc : IsLimit c) {X : FintypeCat} (f : c.pt ⟶ toProfinite.obj X) :
+    ∃ (i : I) (g : F.obj i ⟶ X), f = c.π.app i ≫ toProfinite.map g := by
+  let _ : TopologicalSpace X := ⊥
+  have : DiscreteTopology (toProfinite.obj X) := ⟨rfl⟩
+  let f' : LocallyConstant c.pt (toProfinite.obj X) :=
+    ⟨f, (IsLocallyConstant.iff_continuous _).mpr f.continuous⟩
+  obtain ⟨i, g, h⟩ := exists_locallyConstant.{_, u} c hc f'
+  refine ⟨i, (g : _ → _), ?_⟩
+  ext x
+  exact LocallyConstant.congr_fun h x
+
+namespace Extend
+
+/--
+Given a cone in `Profinite`, consisting of finite sets and indexed by a cofiltered category,
+we obtain a functor from the indexing category to `StructuredArrow c.pt toProfinite`.
+-/
+@[simps]
+def functor : I ⥤ StructuredArrow c.pt toProfinite where
+  obj i := StructuredArrow.mk (c.π.app i)
+  map f := StructuredArrow.homMk (F.map f) (c.w f)
+
+-- We check that the original diagram factors through `Profinite.Extend.functor`.
+example : functor c ⋙ StructuredArrow.proj c.pt toProfinite ≅ F := Iso.refl _
+
+/--
+Given a cone in `Profinite`, consisting of finite sets and indexed by a cofiltered category,
+we obtain a functor from the opposite of the indexing category to
+`CostructuredArrow toProfinite.op ⟨c.pt⟩`.
+-/
+@[simps! obj map]
+def functorOp : Iᵒᵖ ⥤ CostructuredArrow toProfinite.op ⟨c.pt⟩ :=
+  (functor c).op ⋙ StructuredArrow.toCostructuredArrow _ _
+
+-- We check that the opposite of the original diagram factors through `Profinite.Extend.functorOp`.
+example : functorOp c ⋙ CostructuredArrow.proj toProfinite.op ⟨c.pt⟩ ≅ F.op := Iso.refl _
+
+/--
+If the projection maps in the cone are epimorphic and the cone is limiting, then
+`Profinite.Extend.functor` is initial.
+
+TODO: investigate how to weaken the assumption `∀ i, Epi (c.π.app i)` to
+`∀ i, ∃ j (_ : j ⟶ i), Epi (c.π.app j)`.
+-/
+lemma functor_initial (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Initial (functor c) := by
+  let e : I ≌ ULiftHom.{w} (ULift.{w} I) := ULiftHomULiftCategory.equiv _
+  suffices (e.inverse ⋙ functor c).Initial from initial_of_equivalence_comp e.inverse (functor c)
+  rw [initial_iff_of_isCofiltered (F := e.inverse ⋙ functor c)]
+  constructor
+  · intro ⟨_, X, (f : c.pt ⟶ _)⟩
+    obtain ⟨i, g, h⟩ := exists_hom c hc f
+    refine ⟨⟨i⟩, ⟨StructuredArrow.homMk g h.symm⟩⟩
+  · intro ⟨_, X, (f : c.pt ⟶ _)⟩ ⟨i⟩ ⟨_, (s : F.obj i ⟶ X), (w : f = c.π.app i ≫ _)⟩
+      ⟨_, (s' : F.obj i ⟶ X), (w' : f = c.π.app i ≫ _)⟩
+    simp only [functor_obj, functor_map, StructuredArrow.hom_eq_iff, StructuredArrow.mk_right,
+      StructuredArrow.comp_right, StructuredArrow.homMk_right]
+    refine ⟨⟨i⟩, 𝟙 _, ?_⟩
+    simp only [CategoryTheory.Functor.map_id, Category.id_comp]
+    rw [w] at w'
+    exact toProfinite.map_injective <| Epi.left_cancellation _ _ w'
+
+/--
+If the projection maps in the cone are epimorphic and the cone is limiting, then
+`Profinite.Extend.functorOp` is final.
+-/
+lemma functorOp_final (hc : IsLimit c) [∀ i, Epi (c.π.app i)] : Final (functorOp c) := by
+  have := functor_initial c hc
+  have : ((StructuredArrow.toCostructuredArrow toProfinite c.pt)).IsEquivalence  :=
+    (inferInstance : (structuredArrowOpEquivalence _ _).functor.IsEquivalence )
+  exact Functor.final_comp (functor c).op _
+
+section Limit
+
+variable {C : Type*} [Category C] (G : Profinite ⥤ C)
+
+/--
+Given a functor `G` from `Profinite` and `S : Profinite`, we obtain a cone on
+`(StructuredArrow.proj S toProfinite ⋙ toProfinite ⋙ G)` with cone point `G.obj S`.
+
+Whiskering this cone with `Profinite.Extend.functor c` gives `G.mapCone c` as we check in the
+example below.
+-/
+@[simps]
+def cone (S : Profinite) :
+    Cone (StructuredArrow.proj S toProfinite ⋙ toProfinite ⋙ G) where
+  pt := G.obj S
+  π := {
+    app := fun i ↦ G.map i.hom
+    naturality := fun _ _ f ↦ (by
+      have := f.w
+      simp only [const_obj_obj, StructuredArrow.left_eq_id, const_obj_map, Category.id_comp,
+        StructuredArrow.w] at this
+      simp only [const_obj_obj, comp_obj, StructuredArrow.proj_obj, const_obj_map, Category.id_comp,
+        Functor.comp_map, StructuredArrow.proj_map, ← map_comp, StructuredArrow.w]) }
+
+example : G.mapCone c = (cone G c.pt).whisker (functor c) := rfl
+
+/--
+If `c` and `G.mapCone c` are limit cones and the projection maps in `c` are epimorphic,
+then `cone G c.pt` is a limit cone.
+-/
+noncomputable
+def isLimitCone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsLimit <| G.mapCone c) :
+    IsLimit (cone G c.pt) := (functor_initial c hc).isLimitWhiskerEquiv _ _ hc'
+
+end Limit
+
+section Colimit
+
+variable {C : Type*} [Category C] (G : Profiniteᵒᵖ ⥤ C)
+
+/--
+Given a functor `G` from `Profiniteᵒᵖ` and `S : Profinite`, we obtain a cocone on
+`(CostructuredArrow.proj toProfinite.op ⟨S⟩ ⋙ toProfinite.op ⋙ G)` with cocone point `G.obj ⟨S⟩`.
+
+Whiskering this cocone with `Profinite.Extend.functorOp c` gives `G.mapCocone c.op` as we check in
+the example below.
+-/
+@[simps]
+def cocone (S : Profinite) :
+    Cocone (CostructuredArrow.proj toProfinite.op ⟨S⟩ ⋙ toProfinite.op ⋙ G) where
+  pt := G.obj ⟨S⟩
+  ι := {
+    app := fun i ↦ G.map i.hom
+    naturality := fun _ _ f ↦ (by
+      have := f.w
+      simp only [op_obj, const_obj_obj, op_map, CostructuredArrow.right_eq_id, const_obj_map,
+        Category.comp_id] at this
+      simp only [comp_obj, CostructuredArrow.proj_obj, op_obj, const_obj_obj, Functor.comp_map,
+        CostructuredArrow.proj_map, op_map, ← map_comp, this, const_obj_map, Category.comp_id]) }
+
+example : G.mapCocone c.op = (cocone G c.pt).whisker (functorOp c) := rfl
+
+/--
+If `c` is a limit cone, `G.mapCocone c.op` is a colimit cone and the projection maps in `c`
+are epimorphic, then `cocone G c.pt` is a colimit cone.
+-/
+noncomputable
+def isColimitCocone (hc : IsLimit c) [∀ i, Epi (c.π.app i)] (hc' : IsColimit <| G.mapCocone c.op) :
+    IsColimit (cocone G c.pt) := (functorOp_final c hc).isColimitWhiskerEquiv _ _ hc'
+
+end Colimit
+
+end Extend
+
+open Extend
+
+section ProfiniteAsLimit
+
+variable (S : Profinite.{u})
+
+/--
+A functor `StructuredArrow S toProfinite ⥤ FintypeCat` whose limit in `Profinite` is isomorphic
+to `S`.
+-/
+abbrev fintypeDiagram' : StructuredArrow S toProfinite ⥤ FintypeCat :=
+  StructuredArrow.proj S toProfinite
+
+/-- An abbreviation for `S.fintypeDiagram' ⋙ toProfinite`. -/
+abbrev diagram' : StructuredArrow S toProfinite ⥤ Profinite :=
+  S.fintypeDiagram' ⋙ toProfinite
+
+/-- A cone over `S.diagram'` whose cone point is `S`. -/
+abbrev asLimitCone' : Cone (S.diagram') := cone (𝟭 _) S
+
+instance (i : DiscreteQuotient S) : Epi (S.asLimitCone.π.app i) :=
+  (epi_iff_surjective _).mpr i.proj_surjective
+
+/-- `S.asLimitCone'` is a limit cone. -/
+noncomputable def asLimit' : IsLimit S.asLimitCone' := isLimitCone _ (𝟭 _) S.asLimit S.asLimit
+
+/-- A bundled version of `S.asLimitCone'` and `S.asLimit'`. -/
+noncomputable def lim' : LimitCone S.diagram' := ⟨S.asLimitCone', S.asLimit'⟩
+
+end ProfiniteAsLimit
+
+end Profinite
diff --git a/Mathlib/Topology/Category/Profinite/Nobeling.lean b/Mathlib/Topology/Category/Profinite/Nobeling.lean
index 7eb3ad13ff892..3807d5656a53a 100644
--- a/Mathlib/Topology/Category/Profinite/Nobeling.lean
+++ b/Mathlib/Topology/Category/Profinite/Nobeling.lean
@@ -18,16 +18,16 @@ This file proves Nöbeling's theorem.
 ## Main result
 
 * `LocallyConstant.freeOfProfinite`: Nöbeling's theorem.
-  For `S : Profinite`, the `ℤ`-module `LocallyConstant S ℤ` is free.
+  For `S : Profinite`, the `ℤ`-module `LocallyConstant S ℤ` is free.
 
 ## Proof idea
 
 We follow the proof of theorem 5.4 in [scholze2019condensed], in which the idea is to embed `S` in
 a product of `I` copies of `Bool` for some sufficiently large `I`, and then to choose a
-well-ordering on `I` and use ordinal induction over that well-order. Here we can let `I` be
-the set of clopen subsets of `S` since `S` is totally separated.
+well-ordering on `I` and use ordinal induction over that well-order. Here we can let `I` be
+the set of clopen subsets of `S` since `S` is totally separated.
 
-The above means it suffices to prove the following statement: For a closed subset `C` of `I → Bool`,
+The above means it suffices to prove the following statement: For a closed subset `C` of `I → Bool`,
 the `ℤ`-module `LocallyConstant C ℤ` is free.
 
 For `i : I`, let `e C i : LocallyConstant C ℤ` denote the map `fun f ↦ (if f.val i then 1 else 0)`.
@@ -60,7 +60,7 @@ section Projections
 
 The purpose of this section is twofold.
 
-Firstly, in the proof that the set `GoodProducts C` spans the whole module `LocallyConstant C ℤ`,
+Firstly, in the proof that the set `GoodProducts C` spans the whole module `LocallyConstant C ℤ`,
 we need to project `C` down to finite discrete subsets and write `C` as a cofiltered limit of those.
 
 Secondly, in the inductive argument, we need to project `C` down to "smaller" sets satisfying the
@@ -73,11 +73,11 @@ In this section we define the relevant projection maps and prove some compatibil
 * Let `J : I → Prop`. Then `Proj J : (I → Bool) → (I → Bool)` is the projection mapping everything
   that satisfies `J i` to itself, and everything else to `false`.
 
-* The image of `C` under `Proj J` is denoted `π C J` and the corresponding map `C → π C J` is called
-  `ProjRestrict`. If `J` implies `K` we have a map `ProjRestricts : π C K → π C J`.
+* The image of `C` under `Proj J` is denoted `π C J` and the corresponding map `C → π C J` is called
+  `ProjRestrict`. If `J` implies `K` we have a map `ProjRestricts : π C K → π C J`.
 
 * `spanCone_isLimit` establishes that when `C` is compact, it can be written as a limit of its
-  images under the maps `Proj (· ∈ s)` where `s : Finset I`.
+  images under the maps `Proj (· ∈ s)` where `s : Finset I`.
 -/
 
 variable (J K L : I → Prop) [∀ i, Decidable (J i)] [∀ i, Decidable (K i)] [∀ i, Decidable (L i)]
@@ -171,7 +171,7 @@ theorem projRestricts_comp_projRestrict (h : ∀ i, J i → K i) :
 
 variable (J)
 
-/-- The objectwise map in the isomorphism `spanFunctor ≅ Profinite.indexFunctor`. -/
+/-- The objectwise map in the isomorphism `spanFunctor ≅ Profinite.indexFunctor`. -/
 def iso_map : C(π C J, (IndexFunctor.obj C J)) :=
   ⟨fun x ↦ ⟨fun i ↦ x.val i.val, by
     rcases x with ⟨x, y, hy, rfl⟩
@@ -264,7 +264,7 @@ of `e`.
 * `Products I` is the type of lists of decreasing elements of `I`, so a typical element is
   `[i₁, i₂,..., iᵣ]` with `i₁ > i₂ > ... > iᵣ`.
 
-* `Products.eval C` is the `C`-evaluation of a list. It takes a term `[i₁, i₂,..., iᵣ] : Products I`
+* `Products.eval C` is the `C`-evaluation of a list. It takes a term `[i₁, i₂,..., iᵣ] : Products I`
   and returns the actual product `e C i₁ ··· e C iᵣ : LocallyConstant C ℤ`.
 
 * `GoodProducts C` is the set of `Products I` such that their `C`-evaluation cannot be written as
@@ -272,7 +272,7 @@ of `e`.
 
 ### Main results
 
-* `Products.evalFacProp` and `Products.evalFacProps` establish the fact that `Products.eval` 
+* `Products.evalFacProp` and `Products.evalFacProps` establish the fact that `Products.eval`
   interacts nicely with the projection maps from the previous section.
 
 * `GoodProducts.span_iff_products`: the good products span `LocallyConstant C ℤ` iff all the
@@ -359,7 +359,7 @@ theorem injective : Function.Injective (eval C) := by
   · exfalso; apply ha; rw [h]
     exact Submodule.subset_span ⟨b, ⟨h',rfl⟩⟩
 
-/-- The image of the good products in the module `LocallyConstant C ℤ`. -/
+/-- The image of the good products in the module `LocallyConstant C ℤ`. -/
 def range := Set.range (GoodProducts.eval C)
 
 /-- The type of good products is equivalent to its image. -/
@@ -412,7 +412,7 @@ theorem evalFacProps {l : Products I} (J K : I → Prop)
   have : l.eval (π C J) ∘ Homeomorph.setCongr (proj_eq_of_subset C J K hJK) =
       l.eval (π (π C K) J) := by
     ext; simp [Homeomorph.setCongr, Products.eval_eq]
-  rw [ProjRestricts, ← Function.comp.assoc, this, ← evalFacProp (π C K) J h]
+  rw [ProjRestricts, ← Function.comp_assoc, this, ← evalFacProp (π C K) J h]
 
 theorem prop_of_isGood {l : Products I} (J : I → Prop) [∀ j, Decidable (J j)]
     (h : l.isGood (π C J)) : ∀ a, a ∈ l.val → J a := by
@@ -624,8 +624,8 @@ theorem GoodProducts.spanFin [IsWellOrder I (· < ·)] :
     rw [List.map_cons, List.prod_cons]
     intro ha
     specialize ih (by rw [List.chain'_cons'] at ha; exact ha.2)
-    rw [Finsupp.mem_span_image_iff_total] at ih
-    simp only [Finsupp.mem_supported, Finsupp.total_apply] at ih
+    rw [Finsupp.mem_span_image_iff_linearCombination] at ih
+    simp only [Finsupp.mem_supported, Finsupp.linearCombination_apply] at ih
     obtain ⟨c, hc, hc'⟩ := ih
     rw [← hc']; clear hc'
     have hmap := fun g ↦ map_finsupp_sum (LinearMap.mulLeft ℤ (e (π C (· ∈ s)) a)) c g
@@ -667,7 +667,7 @@ theorem fin_comap_jointlySurjective
     (spanCone_isLimit hC.isCompact) f
   exact ⟨(Opposite.unop J), g, h⟩
 
-/-- The good products span all of `LocallyConstant C ℤ` if `C` is closed. -/
+/-- The good products span all of `LocallyConstant C ℤ` if `C` is closed. -/
 theorem GoodProducts.span [IsWellOrder I (· < ·)] (hC : IsClosed C) :
     ⊤ ≤ Submodule.span ℤ (Set.range (eval C)) := by
   rw [span_iff_products]
@@ -692,14 +692,14 @@ can be regarded as the set of all strictly smaller ordinals, allowing to apply o
 
 ### Main definitions
 
-* `ord I i` is the term `i` of `I` regarded as an ordinal.
+* `ord I i` is the term `i` of `I` regarded as an ordinal.
 
-* `term I ho` is a sufficiently small ordinal regarded as a term of `I`.
+* `term I ho` is a sufficiently small ordinal regarded as a term of `I`.
 
-* `contained C o` is a predicate saying that `C` is "small" enough in relation to the ordinal `o`
+* `contained C o` is a predicate saying that `C` is "small" enough in relation to the ordinal `o`
   to satisfy the inductive hypothesis.
 
-* `P I` is the predicate on ordinals about linear independence of good products, which the rest of
+* `P I` is the predicate on ordinals about linear independence of good products, which the rest of
   this file is spent on proving by induction.
 -/
 
@@ -732,13 +732,13 @@ theorem ord_term {o : Ordinal} (ho : o < Ordinal.type ((·<·) : I → I → Pro
   · subst h
     exact ord_term_aux ho
 
-/-- A predicate saying that `C` is "small" enough to satisfy the inductive hypothesis. -/
+/-- A predicate saying that `C` is "small" enough to satisfy the inductive hypothesis. -/
 def contained (o : Ordinal) : Prop := ∀ f, f ∈ C → ∀ (i : I), f i = true → ord I i < o
 
 variable (I) in
 /--
 The predicate on ordinals which we prove by induction, see `GoodProducts.P0`,
-`GoodProducts.Plimit` and `GoodProducts.linearIndependentAux` in the section `Induction` below
+`GoodProducts.Plimit` and `GoodProducts.linearIndependentAux` in the section `Induction` below
 -/
 def P (o : Ordinal) : Prop :=
   o ≤ Ordinal.type (·<· : I → I → Prop) →
@@ -762,7 +762,7 @@ section Zero
 
 ## The zero case of the induction
 
-In this case, we have `contained C 0` which means that `C` is either empty or a singleton.
+In this case, we have `contained C 0` which means that `C` is either empty or a singleton.
 -/
 
 instance : Subsingleton (LocallyConstant (∅ : Set (I → Bool)) ℤ) :=
@@ -776,7 +776,7 @@ instance : IsEmpty { l // Products.isGood (∅ : Set (I → Bool)) l } :=
 theorem GoodProducts.linearIndependentEmpty {I} [LinearOrder I] :
     LinearIndependent ℤ (eval (∅ : Set (I → Bool))) := linearIndependent_empty_type
 
-/-- The empty list as a `Products` -/
+/-- The empty list as a `Products` -/
 def Products.nil : Products I := ⟨[], by simp only [List.chain'_nil]⟩
 
 theorem Products.lt_nil_empty {I} [LinearOrder I] : { m : Products I | m < Products.nil } = ∅ := by
@@ -848,13 +848,13 @@ precomposition with the projections defined in the section `Projections`.
 
 ### Main definitions
 
-* `πs` and `πs'` are the `ℤ`-linear maps corresponding to `ProjRestrict` and `ProjRestricts` 
+* `πs` and `πs'` are the `ℤ`-linear maps corresponding to `ProjRestrict` and `ProjRestricts`
   respectively.
 
 ### Main result
 
-* We prove that `πs` and `πs'` interact well with `Products.eval` and the main application is the
-  theorem `isGood_mono` which says that the property `isGood` is "monotone" on ordinals.
+* We prove that `πs` and `πs'` interact well with `Products.eval` and the main application is the
+  theorem `isGood_mono` which says that the property `isGood` is "monotone" on ordinals.
 -/
 
 theorem contained_eq_proj (o : Ordinal) (h : contained C o) :
@@ -963,7 +963,7 @@ section Limit
 We relate linear independence in `LocallyConstant (π C (ord I · < o')) ℤ` with linear independence
 in `LocallyConstant C ℤ`, where `contained C o` and `o' < o`.
 
-When `o` is a limit ordinal, we prove that the good products in `LocallyConstant C ℤ` are linearly
+When `o` is a limit ordinal, we prove that the good products in `LocallyConstant C ℤ` are linearly
 independent if and only if a certain directed union is linearly independent. Each term in this
 directed union is in bijection with the good products w.r.t. `π C (ord I · < o')` for an ordinal
 `o' < o`, and these are linearly independent by the inductive hypothesis.
@@ -972,13 +972,13 @@ directed union is in bijection with the good products w.r.t. `π C (ord I · < o
 
 * `GoodProducts.smaller` is the image of good products coming from a smaller ordinal.
 
-* `GoodProducts.range_equiv`: The image of the `GoodProducts` in `C` is equivalent to the union of
-  `smaller C o'` over all ordinals `o' < o`.
+* `GoodProducts.range_equiv`: The image of the `GoodProducts` in `C` is equivalent to the union of
+  `smaller C o'` over all ordinals `o' < o`.
 
 ### Main results
 
 * `Products.limitOrdinal`: for `o` a limit ordinal such that `contained C o`, a product `l` is good
-  w.r.t. `C` iff it there exists an ordinal `o' < o` such that `l` is good w.r.t.
+  w.r.t. `C` iff it there exists an ordinal `o' < o` such that `l` is good w.r.t.
   `π C (ord I · < o')`.
 
 * `GoodProducts.linearIndependent_iff_union_smaller` is the result mentioned above, that the good
@@ -988,8 +988,8 @@ directed union is in bijection with the good products w.r.t. `π C (ord I · < o
 namespace GoodProducts
 
 /--
-The image of the `GoodProducts` for `π C (ord I · < o)` in `LocallyConstant C ℤ`. The name `smaller`
-refers to the setting in which we will use this, when we are mapping in `GoodProducts` from a
+The image of the `GoodProducts` for `π C (ord I · < o)` in `LocallyConstant C ℤ`. The name `smaller`
+refers to the setting in which we will use this, when we are mapping in `GoodProducts` from a
 smaller set, i.e. when `o` is a smaller ordinal than the one `C` is "contained" in.
 -/
 def smaller (o : Ordinal) : Set (LocallyConstant C ℤ) :=
@@ -1043,7 +1043,7 @@ theorem smaller_mono {o₁ o₂ : Ordinal} (h : o₁ ≤ o₂) : smaller C o₁
     ext x
     rw [eval, ← Products.eval_πs' _ h (Products.prop_of_isGood  C _ gl), eval]
   · rw [← LocallyConstant.coe_inj, coe_πs C o₂, ← LocallyConstant.toFun_eq_coe, coe_πs',
-      Function.comp.assoc, projRestricts_comp_projRestrict C _, coe_πs]
+      Function.comp_assoc, projRestricts_comp_projRestrict C _, coe_πs]
     rfl
 
 end GoodProducts
@@ -1085,7 +1085,7 @@ theorem GoodProducts.union : range C = ⋃ (e : {o' // o' < o}), (smaller C e.va
     exact Products.isGood_mono C (le_of_lt h) hl
 
 /--
-The image of the `GoodProducts` in `C` is equivalent to the union of `smaller C o'` over all
+The image of the `GoodProducts` in `C` is equivalent to the union of `smaller C o'` over all
 ordinals `o' < o`.
 -/
 def GoodProducts.range_equiv : range C ≃ ⋃ (e : {o' // o' < o}), (smaller C e.val) :=
@@ -1108,8 +1108,8 @@ section Successor
 
 ## The successor case in the induction
 
-Here we assume that `o` is an ordinal such that `contained C (o+1)` and `o < I`. The element in `I`
-corresponding to `o` is called `term I ho`, but in this informal docstring we refer to it simply as
+Here we assume that `o` is an ordinal such that `contained C (o+1)` and `o < I`. The element in `I`
+corresponding to `o` is called `term I ho`, but in this informal docstring we refer to it simply as
 `o`.
 
 This section follows the proof in [scholze2019condensed] quite closely. A translation of the
@@ -1140,32 +1140,32 @@ corresponds to the last paragraph in the proof in [scholze2019condensed].
 The main definitions in the section `ExactSequence` are all just notation explained in the table
 above.
 
-The main definitions in the section `GoodProducts` are as follows:
+The main definitions in the section `GoodProducts` are as follows:
 
 * `MaxProducts`: the set of good products that contain the ordinal `o` (since we have
   `contained C (o+1)`, these all start with `o`).
 
 * `GoodProducts.sum_equiv`: the equivalence between `GoodProducts C` and the disjoint union of
-  `MaxProducts C` and `GoodProducts (π C (ord I · < o))`.
+  `MaxProducts C` and `GoodProducts (π C (ord I · < o))`.
 
 ### Main results
 
-* The main results in the section `ExactSequence` are `succ_mono` and `succ_exact` which together
-  say that the sequence given by `πs` and `Linear_CC'` is left exact:
+* The main results in the section `ExactSequence` are `succ_mono` and `succ_exact` which together
+  say that the sequence given by `πs` and `Linear_CC'` is left exact:
   ```
                                               f                        g
   0 --→ LocallyConstant (π C (ord I · < o)) ℤ --→ LocallyConstant C ℤ --→ LocallyConstant C' ℤ
   ```
-  where `f` is `πs` and `g` is `Linear_CC'`.
+  where `f` is `πs` and `g` is `Linear_CC'`.
 
-The main results in the section `GoodProducts` are as follows:
+The main results in the section `GoodProducts` are as follows:
 
 * `Products.max_eq_eval` says that the linear map on the right in the exact sequence, i.e.
-  `Linear_CC'`, takes the evaluation of a term of `MaxProducts` to the evaluation of the
+  `Linear_CC'`, takes the evaluation of a term of `MaxProducts` to the evaluation of the
   corresponding list with the leading `o` removed.
 
-* `GoodProducts.maxTail_isGood` says that removing the leading `o` from a term of `MaxProducts C` 
-  yields a list which `isGood` with respect to `C'`.
+* `GoodProducts.maxTail_isGood` says that removing the leading `o` from a term of `MaxProducts C`
+  yields a list which `isGood` with respect to `C'`.
 -/
 
 variable {o : Ordinal} (hC : IsClosed C) (hsC : contained C (Order.succ o))
@@ -1245,7 +1245,7 @@ theorem swapTrue_mem_C1 (f : π (C1 C ho) (ord I · < o)) :
     contrapose! hsC
     exact ⟨hsC, Order.succ_le_of_lt (h'.lt_of_ne' h)⟩
 
-/-- The first way to map `C'` into `C`. -/
+/-- The first way to map `C'` into `C`. -/
 def CC'₀ : C' C ho → C := fun g ↦ ⟨g.val,g.prop.1.1⟩
 
 /-- The second way to map `C'` into `C`. -/
@@ -1268,7 +1268,7 @@ noncomputable
 def Linear_CC'₁ : LocallyConstant C ℤ →ₗ[ℤ] LocallyConstant (C' C ho) ℤ :=
   LocallyConstant.comapₗ ℤ ⟨(CC'₁ C hsC ho), (continuous_CC'₁ C hsC ho)⟩
 
-/-- The difference between `Linear_CC'₁` and `Linear_CC'₀`. -/
+/-- The difference between `Linear_CC'₁` and `Linear_CC'₀`. -/
 noncomputable
 def Linear_CC' : LocallyConstant C ℤ →ₗ[ℤ] LocallyConstant (C' C ho) ℤ :=
   Linear_CC'₁ C hsC ho - Linear_CC'₀ C ho
@@ -1354,13 +1354,13 @@ theorem CC_exact {f : LocallyConstant C ℤ} (hf : Linear_CC' C hsC ho f = 0) :
       exact C1_projOrd C hsC ho hx₁
 
 variable (o) in
-theorem succ_mono : CategoryTheory.Mono (ModuleCat.ofHom (πs C o)) := by
+theorem succ_mono : CategoryTheory.Mono (ModuleCat.asHom (πs C o)) := by
   rw [ModuleCat.mono_iff_injective]
   exact injective_πs _ _
 
 include hC in
 theorem succ_exact :
-    (ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho))
+    (ShortComplex.mk (ModuleCat.asHom (πs C o)) (ModuleCat.asHom (Linear_CC' C hsC ho))
     (by ext; apply CC_comp_zero)).Exact := by
   rw [ShortComplex.moduleCat_exact_iff]
   intro f
@@ -1450,7 +1450,7 @@ theorem sum_equiv_comp_eval_eq_elim : eval C ∘ (sum_equiv C hsC ho).toFun =
 
 Then `SumEval C ho` is the map `u` in the diagram below. It is linearly independent if and only if
 `GoodProducts.eval C` is, see `linearIndependent_iff_sum`. The top row is the exact sequence given
-by `succ_exact` and `succ_mono`. The left square commutes by `GoodProducts.square_commutes`.
+by `succ_exact` and `succ_mono`. The left square commutes by `GoodProducts.square_commutes`.
 ```
 0 --→ N --→ M --→  P
       ↑     ↑      ↑
@@ -1478,7 +1478,7 @@ theorem span_sum : Set.range (eval C) = Set.range (Sum.elim
 
 
 theorem square_commutes : SumEval C ho ∘ Sum.inl =
-    ModuleCat.ofHom (πs C o) ∘ eval (π C (ord I · < o)) := by
+    ModuleCat.asHom (πs C o) ∘ eval (π C (ord I · < o)) := by
   ext l
   dsimp [SumEval]
   rw [← Products.eval_πs C (Products.prop_of_isGood  _ _ l.prop)]
@@ -1609,20 +1609,20 @@ theorem good_lt_maxProducts (q : GoodProducts (π C (ord I · < o)))
 
 include hC hsC in
 /--
-Removing the leading `o` from a term of `MaxProducts C` yields a list which `isGood` with respect to
+Removing the leading `o` from a term of `MaxProducts C` yields a list which `isGood` with respect to
 `C'`.
 -/
 theorem maxTail_isGood (l : MaxProducts C ho)
     (h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
     l.val.Tail.isGood (C' C ho) := by
   have : Inhabited I := ⟨term I ho⟩
-  -- Write `l.Tail` as a linear combination of smaller products:
+  -- Write `l.Tail` as a linear combination of smaller products:
   intro h
-  rw [Finsupp.mem_span_image_iff_total, ← max_eq_eval C hsC ho] at h
+  rw [Finsupp.mem_span_image_iff_linearCombination, ← max_eq_eval C hsC ho] at h
   obtain ⟨m, ⟨hmmem, hmsum⟩⟩ := h
-  rw [Finsupp.total_apply] at hmsum
+  rw [Finsupp.linearCombination_apply] at hmsum
 
-  -- Write the image of `l` under `Linear_CC'` as `Linear_CC'` applied to the linear combination
+  -- Write the image of `l` under `Linear_CC'` as `Linear_CC'` applied to the linear combination
   -- above, with leading `term I ho`'s added to each term:
   have : (Linear_CC' C hsC ho) (l.val.eval C) = (Linear_CC' C hsC ho)
       (Finsupp.sum m fun i a ↦ a • ((term I ho :: i.1).map (e C)).prod) := by
@@ -1644,9 +1644,9 @@ theorem maxTail_isGood (l : MaxProducts C ho)
     rfl
   have hse := succ_exact C hC hsC ho
   rw [ShortComplex.moduleCat_exact_iff_range_eq_ker] at hse
-  dsimp [ModuleCat.ofHom] at hse
+  dsimp [ModuleCat.asHom] at hse
 
-  -- Rewrite `this` using exact sequence manipulations to conclude that a term is in the range of
+  -- Rewrite `this` using exact sequence manipulations to conclude that a term is in the range of
   -- the linear map `πs`:
   rw [← LinearMap.sub_mem_ker_iff, ← hse] at this
   obtain ⟨(n : LocallyConstant (π C (ord I · < o)) ℤ), hn⟩ := this
@@ -1701,8 +1701,8 @@ include hC in
 theorem linearIndependent_comp_of_eval
     (h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
     LinearIndependent ℤ (eval (C' C ho)) →
-    LinearIndependent ℤ (ModuleCat.ofHom (Linear_CC' C hsC ho) ∘ SumEval C ho ∘ Sum.inr) := by
-  dsimp [SumEval, ModuleCat.ofHom]
+    LinearIndependent ℤ (ModuleCat.asHom (Linear_CC' C hsC ho) ∘ SumEval C ho ∘ Sum.inr) := by
+  dsimp [SumEval, ModuleCat.asHom]
   erw [max_eq_eval_unapply C hsC ho]
   intro h
   let f := MaxToGood C hC hsC ho h₁
@@ -1722,16 +1722,16 @@ section Induction
 
 ## The induction
 
-Here we put together the results of the sections `Zero`, `Limit` and `Successor` to prove the
-predicate `P I o` holds for all ordinals `o`, and conclude with the main result:
+Here we put together the results of the sections `Zero`, `Limit` and `Successor` to prove the
+predicate `P I o` holds for all ordinals `o`, and conclude with the main result:
 
-* `GoodProducts.linearIndependent` which says that `GoodProducts C` is linearly independent when `C`
+* `GoodProducts.linearIndependent` which says that `GoodProducts C` is linearly independent when `C`
   is closed.
 
 We also define
 
-* `GoodProducts.Basis` which uses `GoodProducts.linearIndependent` and `GoodProducts.span` to
-  define a basis for `LocallyConstant C ℤ` 
+* `GoodProducts.Basis` which uses `GoodProducts.linearIndependent` and `GoodProducts.span` to
+  define a basis for `LocallyConstant C ℤ`
 -/
 
 theorem GoodProducts.P0 : P I 0 := fun _ C _ hsC ↦ by
@@ -1771,7 +1771,7 @@ theorem GoodProducts.linearIndependent (hC : IsClosed C) :
   GoodProducts.linearIndependentAux (Ordinal.type (·<· : I → I → Prop)) (le_refl _)
     C hC (fun _ _ _ _ ↦ Ordinal.typein_lt_type _ _)
 
-/-- `GoodProducts C` as a `ℤ`-basis for `LocallyConstant C ℤ`.  -/
+/-- `GoodProducts C` as a `ℤ`-basis for `LocallyConstant C ℤ`. -/
 noncomputable
 def GoodProducts.Basis (hC : IsClosed C) :
     Basis (GoodProducts C) ℤ (LocallyConstant C ℤ) :=
@@ -1783,7 +1783,7 @@ variable {S : Profinite} {ι : S → I → Bool} (hι : ClosedEmbedding ι)
 include hι
 
 /--
-Given a profinite set `S` and a closed embedding `S → (I → Bool)`, the `ℤ`-module
+Given a profinite set `S` and a closed embedding `S → (I → Bool)`, the `ℤ`-module
 `LocallyConstant C ℤ` is free.
 -/
 theorem Nobeling_aux : Module.Free ℤ (LocallyConstant S ℤ) := Module.Free.of_equiv'
@@ -1800,7 +1800,7 @@ noncomputable
 def Nobeling.ι : S → ({C : Set S // IsClopen C} → Bool) := fun s C => decide (s ∈ C.1)
 
 open scoped Classical in
-/-- The map `Nobeling.ι` is a closed embedding. -/
+/-- The map `Nobeling.ι` is a closed embedding. -/
 theorem Nobeling.embedding : ClosedEmbedding (Nobeling.ι S) := by
   apply Continuous.closedEmbedding
   · dsimp (config := { unfoldPartialApp := true }) [ι]
@@ -1830,9 +1830,11 @@ end Profinite
 
 open Profinite NobelingProof
 
-/-- Nöbeling's theorem: the `ℤ`-module `LocallyConstant S ℤ` is free for every `S : Profinite` -/
+/-- Nöbeling's theorem: the `ℤ`-module `LocallyConstant S ℤ` is free for every `S : Profinite` -/
 instance LocallyConstant.freeOfProfinite (S : Profinite.{u}) :
     Module.Free ℤ (LocallyConstant S ℤ) :=
   @Nobeling_aux {C : Set S // IsClopen C}
     (IsWellOrder.linearOrder WellOrderingRel) WellOrderingRel.isWellOrder
     S (Nobeling.ι S) (Nobeling.embedding S)
+
+set_option linter.style.longFile 2000
diff --git a/Mathlib/Topology/Category/Profinite/Product.lean b/Mathlib/Topology/Category/Profinite/Product.lean
index 6760033fce6b9..0224264e33b98 100644
--- a/Mathlib/Topology/Category/Profinite/Product.lean
+++ b/Mathlib/Topology/Category/Profinite/Product.lean
@@ -13,7 +13,7 @@ Hausdorff spaces as a cofiltered limit in `Profinite` indexed by `Finset ι`.
 
 ## Main definitions
 
-- `Profinite.indexFunctor` is the functor `(Finset ι)ᵒᵖ ⥤ Profinite` indexing the limit. It maps
+- `Profinite.indexFunctor` is the functor `(Finset ι)ᵒᵖ ⥤ Profinite` indexing the limit. It maps
   `J` to the restriction of `C` to `J`
 - `Profinite.indexCone` is a cone on `Profinite.indexFunctor` with cone point `C`
 
diff --git a/Mathlib/Topology/Category/Profinite/Projective.lean b/Mathlib/Topology/Category/Profinite/Projective.lean
index becda45ececde..98eb2deeef7fc 100644
--- a/Mathlib/Topology/Category/Profinite/Projective.lean
+++ b/Mathlib/Topology/Category/Profinite/Projective.lean
@@ -51,7 +51,7 @@ instance projective_ultrafilter (X : Type u) : Projective (of <| Ultrafilter X)
     let g'' : ContinuousMap Y Z := g
     have : g'' ∘ g' = id := hg'.comp_eq_id
     -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
-    erw [comp.assoc, ultrafilter_extend_extends, ← comp.assoc, this, id_comp]
+    erw [comp_assoc, ultrafilter_extend_extends, ← comp_assoc, this, id_comp]
 
 /-- For any profinite `X`, the natural map `Ultrafilter X → X` is a projective presentation. -/
 def projectivePresentation (X : Profinite.{u}) : ProjectivePresentation X where
diff --git a/Mathlib/Topology/Category/Stonean/Adjunctions.lean b/Mathlib/Topology/Category/Stonean/Adjunctions.lean
index d20e34fdfc9fb..dd99b65695aed 100644
--- a/Mathlib/Topology/Category/Stonean/Adjunctions.lean
+++ b/Mathlib/Topology/Category/Stonean/Adjunctions.lean
@@ -12,7 +12,7 @@ import Mathlib.Topology.StoneCech
 
 This file constructs the left adjoint `typeToStonean` to the forgetful functor from Stonean spaces
 to sets, using the Stone-Cech compactification. This allows to conclude that the monomorphisms in
-`Stonean` are precisely the injective maps (see `Stonean.mono_iff_injective`).
+`Stonean` are precisely the injective maps (see `Stonean.mono_iff_injective`).
 -/
 
 universe u
diff --git a/Mathlib/Topology/Category/Stonean/Basic.lean b/Mathlib/Topology/Category/Stonean/Basic.lean
index 41c7eeabe0cde..47e784b8b878e 100644
--- a/Mathlib/Topology/Category/Stonean/Basic.lean
+++ b/Mathlib/Topology/Category/Stonean/Basic.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2023 Adam Topaz. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Adam Topaz
+Authors: Adam Topaz, Dagur Asgeirsson
 -/
 import Mathlib.Topology.ExtremallyDisconnected
 import Mathlib.Topology.Category.CompHaus.Projective
@@ -31,6 +31,11 @@ can be lifted along epimorphisms).
   spaces to compact Hausdorff spaces
 * `Stonean.toProfinite` : the functor from Stonean spaces to profinite spaces.
 
+## Implementation
+
+The category `Stonean` is defined using the structure `CompHausLike`. See the file
+`CompHausLike.Basic` for more information.
+
 -/
 universe u
 
@@ -83,7 +88,7 @@ abbrev fullyFaithfulToCompHaus : toCompHaus.FullyFaithful  :=
 
 open CompHausLike
 
-instance  (X : Type*) [TopologicalSpace X]
+instance (X : Type*) [TopologicalSpace X]
     [ExtremallyDisconnected X] : HasProp (fun Y ↦ ExtremallyDisconnected Y) X :=
   ⟨(inferInstance : ExtremallyDisconnected X)⟩
 
@@ -116,7 +121,7 @@ def mkFinite (X : Type*) [Finite X] [TopologicalSpace X] [DiscreteTopology X] :
     apply isOpen_discrete (closure U)
 
 /--
-A morphism in `Stonean` is an epi iff it is surjective.
+A morphism in `Stonean` is an epi iff it is surjective.
 -/
 lemma epi_iff_surjective {X Y : Stonean} (f : X ⟶ Y) :
     Epi f ↔ Function.Surjective f := by
@@ -186,7 +191,7 @@ end Stonean
 namespace CompHaus
 
 /-- If `X` is compact Hausdorff, `presentation X` is a Stonean space equipped with an epimorphism
-  down to `X` (see `CompHaus.presentation.π` and `CompHaus.presentation.epi_π`). It is a
+  down to `X` (see `CompHaus.presentation.π` and `CompHaus.presentation.epi_π`). It is a
   "constructive" witness to the fact that `CompHaus` has enough projectives. -/
 noncomputable
 def presentation (X : CompHaus) : Stonean where
@@ -253,7 +258,7 @@ end CompHaus
 namespace Profinite
 
 /-- If `X` is profinite, `presentation X` is a Stonean space equipped with an epimorphism down to
-    `X` (see `Profinite.presentation.π` and `Profinite.presentation.epi_π`). -/
+    `X` (see `Profinite.presentation.π` and `Profinite.presentation.epi_π`). -/
 noncomputable
 def presentation (X : Profinite) : Stonean where
   toTop := (profiniteToCompHaus.obj X).projectivePresentation.p.toTop
diff --git a/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean b/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
index 6f0dcd7b65055..1901963fcc1a9 100644
--- a/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
+++ b/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean
@@ -36,12 +36,9 @@ theorem effectiveEpi_tfae
     , Epi π
     , Function.Surjective π
     ] := by
-  tfae_have 1 → 2
-  · intro; infer_instance
-  tfae_have 2 ↔ 3
-  · exact epi_iff_surjective π
-  tfae_have 3 → 1
-  · exact fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 2 ↔ 3 := epi_iff_surjective π
+  tfae_have 3 → 1 := fun hπ ↦ ⟨⟨effectiveEpiStruct π hπ⟩⟩
   tfae_finish
 
 instance : Stonean.toCompHaus.PreservesEffectiveEpis where
@@ -55,7 +52,7 @@ instance : Stonean.toCompHaus.ReflectsEffectiveEpis where
       (((CompHaus.effectiveEpi_tfae (Stonean.toCompHaus.map f)).out 0 2).mp h)
 
 /--
-An effective presentation of an `X : CompHaus` with respect to the inclusion functor from `Stonean`
+An effective presentation of an `X : CompHaus` with respect to the inclusion functor from `Stonean`
 -/
 noncomputable def stoneanToCompHausEffectivePresentation (X : CompHaus) :
     Stonean.toCompHaus.EffectivePresentation X where
@@ -81,12 +78,11 @@ theorem effectiveEpiFamily_tfae
     , ∀ b : B, ∃ (a : α) (x : X a), π a x = b
     ] := by
   tfae_have 2 → 1
-  · intro
+  | _ => by
     simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
-  tfae_have 1 → 2
-  · intro; infer_instance
-  tfae_have 3 ↔ 1
-  · erw [((CompHaus.effectiveEpiFamily_tfae
+  tfae_have 1 → 2 := fun _ ↦ inferInstance
+  tfae_have 3 ↔ 1 := by
+    erw [((CompHaus.effectiveEpiFamily_tfae
       (fun a ↦ Stonean.toCompHaus.obj (X a)) (fun a ↦ Stonean.toCompHaus.map (π a))).out 2 0 : )]
     exact ⟨fun h ↦ Stonean.toCompHaus.finite_effectiveEpiFamily_of_map _ _ h,
       fun _ ↦ inferInstance⟩
diff --git a/Mathlib/Topology/Category/TopCat/Adjunctions.lean b/Mathlib/Topology/Category/TopCat/Adjunctions.lean
index db65a15d38f8a..e42823dc49ca5 100644
--- a/Mathlib/Topology/Category/TopCat/Adjunctions.lean
+++ b/Mathlib/Topology/Category/TopCat/Adjunctions.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Patrick Massot, Mario Carneiro
 -/
@@ -26,18 +26,16 @@ namespace TopCat
 /-- Equipping a type with the discrete topology is left adjoint to the forgetful functor
 `Top ⥤ Type`. -/
 @[simps! unit counit]
-def adj₁ : discrete ⊣ forget TopCat.{u} :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun X => id }
-      counit := { app := fun X => ⟨id, continuous_bot⟩ } }
+def adj₁ : discrete ⊣ forget TopCat.{u} where
+  unit := { app := fun X => id }
+  counit := { app := fun X => ⟨id, continuous_bot⟩ }
 
 /-- Equipping a type with the trivial topology is right adjoint to the forgetful functor
 `Top ⥤ Type`. -/
 @[simps! unit counit]
-def adj₂ : forget TopCat.{u} ⊣ trivial :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun X => ⟨id, continuous_top⟩ }
-      counit := { app := fun X => id } }
+def adj₂ : forget TopCat.{u} ⊣ trivial where
+  unit := { app := fun X => ⟨id, continuous_top⟩ }
+  counit := { app := fun X => id }
 
 instance : (forget TopCat.{u}).IsRightAdjoint :=
   ⟨_, ⟨adj₁⟩⟩
diff --git a/Mathlib/Topology/Category/TopCat/Basic.lean b/Mathlib/Topology/Category/TopCat/Basic.lean
index 5d001338b22c2..ab44aa713d2eb 100644
--- a/Mathlib/Topology/Category/TopCat/Basic.lean
+++ b/Mathlib/Topology/Category/TopCat/Basic.lean
@@ -1,10 +1,10 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison, Mario Carneiro
+Authors: Patrick Massot, Kim Morrison, Mario Carneiro
 -/
 import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # Category instance for topological spaces
@@ -56,7 +56,7 @@ instance topologicalSpaceUnbundled (X : TopCat) : TopologicalSpace X :=
 instance instFunLike (X Y : TopCat) : FunLike (X ⟶ Y) X Y :=
   inferInstanceAs <| FunLike C(X, Y) X Y
 
-instance instMonoidHomClass (X Y : TopCat) : ContinuousMapClass (X ⟶ Y) X Y :=
+instance instContinuousMapClass (X Y : TopCat) : ContinuousMapClass (X ⟶ Y) X Y :=
   inferInstanceAs <| ContinuousMapClass C(X, Y) X Y
 
 -- Porting note (#10618): simp can prove this; removed simp
diff --git a/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean b/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean
index 841f550bb606f..35693a3a1dedc 100644
--- a/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean
+++ b/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean
@@ -4,14 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Dagur Asgeirsson
 -/
 import Mathlib.CategoryTheory.EffectiveEpi.RegularEpi
-import Mathlib.CategoryTheory.EffectiveEpi.Comp
 import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
 /-!
 
 # Effective epimorphisms in `TopCat`
 
 This file proves the result `TopCat.effectiveEpi_iff_quotientMap`:
-The effective epimorphisms in `TopCat` are precisely the quotient maps.
+The effective epimorphisms in `TopCat` are precisely the quotient maps.
 
 -/
 
@@ -37,7 +36,7 @@ def effectiveEpiStructOfQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) (hπ : Quo
   fac e h := (hπ.lift_comp e
     fun a b hab ↦ DFunLike.congr_fun (h ⟨fun _ ↦ a, continuous_const⟩ ⟨fun _ ↦ b, continuous_const⟩
     (by ext; exact hab)) a)
-  /- Uniqueness follows from the fact that `QuotientMap.lift` is an equivalence (given by
+  /- Uniqueness follows from the fact that `QuotientMap.lift` is an equivalence (given by
   `QuotientMap.liftEquiv`). -/
   uniq e h g hm := by
     suffices g = hπ.liftEquiv ⟨e,
@@ -49,21 +48,21 @@ def effectiveEpiStructOfQuotientMap {B X : TopCat.{u}} (π : X ⟶ B) (hπ : Quo
     simp only [QuotientMap.liftEquiv_symm_apply_coe, ContinuousMap.comp_apply, ← hm]
     rfl
 
-/-- The effective epimorphisms in `TopCat` are precisely the quotient maps. -/
+/-- The effective epimorphisms in `TopCat` are precisely the quotient maps. -/
 theorem effectiveEpi_iff_quotientMap {B X : TopCat.{u}} (π : X ⟶ B) :
     EffectiveEpi π ↔ QuotientMap π := by
   /- The backward direction is given by `effectiveEpiStructOfQuotientMap` above. -/
   refine ⟨fun _ ↦ ?_, fun hπ ↦ ⟨⟨effectiveEpiStructOfQuotientMap π hπ⟩⟩⟩
-  /- Since `TopCat` has pullbacks, `π` is in fact a `RegularEpi`. This means that it exhibits `B` as
+  /- Since `TopCat` has pullbacks, `π` is in fact a `RegularEpi`. This means that it exhibits `B` as
     a coequalizer of two maps into `X`. It suffices to prove that `π` followed by the isomorphism to
     an arbitrary coequalizer is a quotient map. -/
   have hπ : RegularEpi π := inferInstance
   let F := parallelPair hπ.left hπ.right
   let i : B ≅ colimit F := hπ.isColimit.coconePointUniqueUpToIso (colimit.isColimit _)
   suffices QuotientMap (homeoOfIso i ∘ π) by
-    simpa [← Function.comp.assoc] using (homeoOfIso i).symm.quotientMap.comp this
+    simpa [← Function.comp_assoc] using (homeoOfIso i).symm.quotientMap.comp this
   constructor
-  /- Effective epimorphisms are epimorphisms and epimorphisms in `TopCat` are surjective. -/
+  /- Effective epimorphisms are epimorphisms and epimorphisms in `TopCat` are surjective. -/
   · change Function.Surjective (π ≫ i.hom)
     rw [← epi_iff_surjective]
     infer_instance
diff --git a/Mathlib/Topology/Category/TopCat/Limits/Basic.lean b/Mathlib/Topology/Category/TopCat/Limits/Basic.lean
index 2c9ab0122ae91..6cfe839d9f554 100644
--- a/Mathlib/Topology/Category/TopCat/Limits/Basic.lean
+++ b/Mathlib/Topology/Category/TopCat/Limits/Basic.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
+Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
 -/
 import Mathlib.Topology.Category.TopCat.Basic
 import Mathlib.CategoryTheory.Limits.Types
@@ -184,7 +184,7 @@ instance forgetPreservesColimits : PreservesColimits (forget : TopCat.{u} ⥤ Ty
 /-- The terminal object of `Top` is `PUnit`. -/
 def isTerminalPUnit : IsTerminal (TopCat.of PUnit.{u + 1}) :=
   haveI : ∀ X, Unique (X ⟶ TopCat.of PUnit.{u + 1}) := fun X =>
-    ⟨⟨⟨fun _ => PUnit.unit, by continuity⟩⟩, fun f => by ext; aesop⟩
+    ⟨⟨⟨fun _ => PUnit.unit, continuous_const⟩⟩, fun f => by ext; aesop⟩
   Limits.IsTerminal.ofUnique _
 
 /-- The terminal object of `Top` is `PUnit`. -/
diff --git a/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean b/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean
index a6ef38a679ede..7350fd1934dc2 100644
--- a/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean
+++ b/Mathlib/Topology/Category/TopCat/Limits/Cofiltered.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
+Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
 -/
 import Mathlib.Topology.Category.TopCat.Limits.Basic
 import Mathlib.CategoryTheory.Filtered.Basic
diff --git a/Mathlib/Topology/Category/TopCat/Limits/Konig.lean b/Mathlib/Topology/Category/TopCat/Limits/Konig.lean
index a8ca567c55146..0d6f7501547f2 100644
--- a/Mathlib/Topology/Category/TopCat/Limits/Konig.lean
+++ b/Mathlib/Topology/Category/TopCat/Limits/Konig.lean
@@ -133,8 +133,8 @@ theorem nonempty_limitCone_of_compact_t2_cofiltered_system (F : J ⥤ TopCat.{ma
   intro X Y f
   let G : FiniteDiagram J :=
     ⟨{X, Y},
-      {⟨X, Y, by simp only [true_or_iff, eq_self_iff_true, Finset.mem_insert], by
-          simp only [eq_self_iff_true, or_true_iff, Finset.mem_insert, Finset.mem_singleton], f⟩}⟩
+      {⟨X, Y, by simp only [true_or, eq_self_iff_true, Finset.mem_insert], by
+          simp only [eq_self_iff_true, or_true, Finset.mem_insert, Finset.mem_singleton], f⟩}⟩
   exact hu _ ⟨G, rfl⟩ (Finset.mem_singleton_self _)
 
 end TopologicalKonig
diff --git a/Mathlib/Topology/Category/TopCat/Limits/Products.lean b/Mathlib/Topology/Category/TopCat/Limits/Products.lean
index 43cd610adcbfb..12a0b862fc298 100644
--- a/Mathlib/Topology/Category/TopCat/Limits/Products.lean
+++ b/Mathlib/Topology/Category/TopCat/Limits/Products.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
+Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
 -/
 import Mathlib.Topology.Category.TopCat.EpiMono
 import Mathlib.Topology.Category.TopCat.Limits.Basic
@@ -58,7 +58,7 @@ equipped with the product topology.
 -/
 def piIsoPi {ι : Type v} (α : ι → TopCat.{max v u}) : ∏ᶜ α ≅ TopCat.of (∀ i, α i) :=
   (limit.isLimit _).conePointUniqueUpToIso (piFanIsLimit.{v, u} α)
-  -- Specifying the universes in `piFanIsLimit` wasn't necessary when we had `TopCatMax` 
+  -- Specifying the universes in `piFanIsLimit` wasn't necessary when we had `TopCatMax`
 
 @[reassoc (attr := simp)]
 theorem piIsoPi_inv_π {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
@@ -106,7 +106,7 @@ def sigmaCofanIsColimit {ι : Type v} (β : ι → TopCat.{max v u}) : IsColimit
 -/
 def sigmaIsoSigma {ι : Type v} (α : ι → TopCat.{max v u}) : ∐ α ≅ TopCat.of (Σi, α i) :=
   (colimit.isColimit _).coconePointUniqueUpToIso (sigmaCofanIsColimit.{v, u} α)
-  -- Specifying the universes in `sigmaCofanIsColimit` wasn't necessary when we had `TopCatMax` 
+  -- Specifying the universes in `sigmaCofanIsColimit` wasn't necessary when we had `TopCatMax`
 
 @[reassoc (attr := simp)]
 theorem sigmaIsoSigma_hom_ι {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι) :
diff --git a/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean b/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
index 80e0708f53c5a..7701c3fb0e265 100644
--- a/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
+++ b/Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2017 Scott Morrison. All rights reserved.
+Copyright (c) 2017 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
+Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
 -/
 import Mathlib.Topology.Category.TopCat.Limits.Products
 
diff --git a/Mathlib/Topology/Category/TopCat/OpenNhds.lean b/Mathlib/Topology/Category/TopCat/OpenNhds.lean
index c5465486c0688..d9325ee6922be 100644
--- a/Mathlib/Topology/Category/TopCat/OpenNhds.lean
+++ b/Mathlib/Topology/Category/TopCat/OpenNhds.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Category.TopCat.Opens
 import Mathlib.Data.Set.Subsingleton
@@ -10,12 +10,12 @@ import Mathlib.Data.Set.Subsingleton
 # The category of open neighborhoods of a point
 
 Given an object `X` of the category `TopCat` of topological spaces and a point `x : X`, this file
-builds the type `OpenNhds x` of open neighborhoods of `x` in `X` and endows it with the partial
+builds the type `OpenNhds x` of open neighborhoods of `x` in `X` and endows it with the partial
 order given by inclusion and the corresponding category structure (as a full subcategory of the
 poset category `Set X`). This is used in `Topology.Sheaves.Stalks` to build the stalk of a sheaf
 at `x` as a limit over `OpenNhds x`.
 
-## Main declarations
+## Main declarations
 
 Besides `OpenNhds`, the main constructions here are:
 
@@ -86,7 +86,7 @@ def inclusion (x : X) : OpenNhds x ⥤ Opens X :=
 theorem inclusion_obj (x : X) (U) (p) : (inclusion x).obj ⟨U, p⟩ = U :=
   rfl
 
-theorem openEmbedding {x : X} (U : OpenNhds x) : OpenEmbedding U.1.inclusion :=
+theorem openEmbedding {x : X} (U : OpenNhds x) : OpenEmbedding U.1.inclusion' :=
   U.1.openEmbedding
 
 /-- The preimage functor from neighborhoods of `f x` to neighborhoods of `x`. -/
@@ -144,9 +144,8 @@ def functorNhds (h : IsOpenMap f) (x : X) : OpenNhds x ⥤ OpenNhds (f x) where
   map i := h.functor.map i
 
 /-- An open map `f : X ⟶ Y` induces an adjunction between `OpenNhds x` and `OpenNhds (f x)`. -/
-def adjunctionNhds (h : IsOpenMap f) (x : X) : IsOpenMap.functorNhds h x ⊣ OpenNhds.map f x :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun U => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
-      counit := { app := fun V => homOfLE fun y ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV } }
+def adjunctionNhds (h : IsOpenMap f) (x : X) : IsOpenMap.functorNhds h x ⊣ OpenNhds.map f x where
+  unit := { app := fun U => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
+  counit := { app := fun V => homOfLE fun y ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV }
 
 end IsOpenMap
diff --git a/Mathlib/Topology/Category/TopCat/Opens.lean b/Mathlib/Topology/Category/TopCat/Opens.lean
index 5bebad78ee1ff..988c19dba4faa 100644
--- a/Mathlib/Topology/Category/TopCat/Opens.lean
+++ b/Mathlib/Topology/Category/TopCat/Opens.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Category.Preorder
 import Mathlib.CategoryTheory.EqToHom
@@ -111,21 +111,21 @@ theorem toTopCat_map (X : TopCat.{u}) {U V : Opens X} {f : U ⟶ V} {x} {h} :
 /-- The inclusion map from an open subset to the whole space, as a morphism in `TopCat`.
 -/
 @[simps (config := .asFn)]
-def inclusion {X : TopCat.{u}} (U : Opens X) : (toTopCat X).obj U ⟶ X where
+def inclusion' {X : TopCat.{u}} (U : Opens X) : (toTopCat X).obj U ⟶ X where
   toFun := _
   continuous_toFun := continuous_subtype_val
 
 @[simp]
-theorem coe_inclusion {X : TopCat} {U : Opens X} :
-    (inclusion U : U → X) = Subtype.val := rfl
+theorem coe_inclusion' {X : TopCat} {U : Opens X} :
+    (inclusion' U : U → X) = Subtype.val := rfl
 
-theorem openEmbedding {X : TopCat.{u}} (U : Opens X) : OpenEmbedding (inclusion U) :=
+theorem openEmbedding {X : TopCat.{u}} (U : Opens X) : OpenEmbedding (inclusion' U) :=
   IsOpen.openEmbedding_subtype_val U.2
 
 /-- The inclusion of the top open subset (i.e. the whole space) is an isomorphism.
 -/
 def inclusionTopIso (X : TopCat.{u}) : (toTopCat X).obj ⊤ ≅ X where
-  hom := inclusion ⊤
+  hom := inclusion' ⊤
   inv := ⟨fun x => ⟨x, trivial⟩, continuous_def.2 fun U ⟨_, hS, hSU⟩ => hSU ▸ hS⟩
 
 /-- `Opens.map f` gives the functor from open sets in Y to open set in X,
@@ -280,10 +280,9 @@ def IsOpenMap.functor {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) : Opens X
 /-- An open map `f : X ⟶ Y` induces an adjunction between `Opens X` and `Opens Y`.
 -/
 def IsOpenMap.adjunction {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) :
-    Adjunction hf.functor (TopologicalSpace.Opens.map f) :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun U => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
-      counit := { app := fun V => homOfLE fun y ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV } }
+    Adjunction hf.functor (TopologicalSpace.Opens.map f) where
+  unit := { app := fun U => homOfLE fun x hxU => ⟨x, hxU, rfl⟩ }
+  counit := { app := fun V => homOfLE fun y ⟨_, hfxV, hxy⟩ => hxy ▸ hfxV }
 
 instance IsOpenMap.functorFullOfMono {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) [H : Mono f] :
     hf.functor.Full where
@@ -310,7 +309,7 @@ theorem openEmbedding_obj_top {X : TopCat} (U : Opens X) :
   exact Set.image_univ.trans Subtype.range_coe
 
 @[simp]
-theorem inclusion_map_eq_top {X : TopCat} (U : Opens X) : (Opens.map U.inclusion).obj U = ⊤ := by
+theorem inclusion'_map_eq_top {X : TopCat} (U : Opens X) : (Opens.map U.inclusion').obj U = ⊤ := by
   ext1
   exact Subtype.coe_preimage_self _
 
@@ -318,7 +317,7 @@ theorem inclusion_map_eq_top {X : TopCat} (U : Opens X) : (Opens.map U.inclusion
 theorem adjunction_counit_app_self {X : TopCat} (U : Opens X) :
     U.openEmbedding.isOpenMap.adjunction.counit.app U = eqToHom (by simp) := Subsingleton.elim _ _
 
-theorem inclusion_top_functor (X : TopCat) :
+theorem inclusion'_top_functor (X : TopCat) :
     (@Opens.openEmbedding X ⊤).isOpenMap.functor = map (inclusionTopIso X).inv := by
   refine CategoryTheory.Functor.ext ?_ ?_
   · intro U
@@ -336,8 +335,8 @@ theorem functor_obj_map_obj {X Y : TopCat} {f : X ⟶ Y} (hf : IsOpenMap f) (U :
     exact ⟨x, hx, rfl⟩
 
 -- Porting note: added to ease the proof of `functor_map_eq_inf`
-lemma set_range_forget_map_inclusion {X : TopCat} (U : Opens X) :
-    Set.range ((forget TopCat).map (inclusion U)) = (U : Set X) := by
+lemma set_range_forget_map_inclusion' {X : TopCat} (U : Opens X) :
+    Set.range ((forget TopCat).map (inclusion' U)) = (U : Set X) := by
   ext x
   constructor
   · rintro ⟨x, rfl⟩
@@ -347,10 +346,10 @@ lemma set_range_forget_map_inclusion {X : TopCat} (U : Opens X) :
 
 @[simp]
 theorem functor_map_eq_inf {X : TopCat} (U V : Opens X) :
-    U.openEmbedding.isOpenMap.functor.obj ((Opens.map U.inclusion).obj V) = V ⊓ U := by
+    U.openEmbedding.isOpenMap.functor.obj ((Opens.map U.inclusion').obj V) = V ⊓ U := by
   ext1
   refine Set.image_preimage_eq_inter_range.trans ?_
-  erw [set_range_forget_map_inclusion U]
+  erw [set_range_forget_map_inclusion' U]
   rfl
 
 theorem map_functor_eq' {X U : TopCat} (f : U ⟶ X) (hf : OpenEmbedding f) (V) :
@@ -359,7 +358,7 @@ theorem map_functor_eq' {X U : TopCat} (f : U ⟶ X) (hf : OpenEmbedding f) (V)
 
 @[simp]
 theorem map_functor_eq {X : TopCat} {U : Opens X} (V : Opens U) :
-    ((Opens.map U.inclusion).obj <| U.openEmbedding.isOpenMap.functor.obj V) = V :=
+    ((Opens.map U.inclusion').obj <| U.openEmbedding.isOpenMap.functor.obj V) = V :=
   TopologicalSpace.Opens.map_functor_eq' _ U.openEmbedding V
 
 @[simp]
diff --git a/Mathlib/Topology/Category/TopCat/Sphere.lean b/Mathlib/Topology/Category/TopCat/Sphere.lean
new file mode 100644
index 0000000000000..b8f23e34fc643
--- /dev/null
+++ b/Mathlib/Topology/Category/TopCat/Sphere.lean
@@ -0,0 +1,39 @@
+/-
+Copyright (c) 2024 Elliot Dean Young and Jiazhen Xia. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jiazhen Xia, Elliot Dean Young
+-/
+
+import Mathlib.Analysis.InnerProductSpace.PiL2
+import Mathlib.Topology.Category.TopCat.Basic
+
+/-!
+# Euclidean spheres
+
+This files defines the `n`-sphere `𝕊 n` and the `n`-disk `𝔻` as objects in `TopCat`.
+The parameter `n` is in `ℤ` so as to facilitate the definition of
+CW-complexes (see the file `Topology.CWComplex`).
+
+-/
+
+universe u
+
+namespace TopCat
+
+/-- The `n`-sphere is the set of points in ℝⁿ⁺¹ whose norm equals `1`,
+endowed with the subspace topology. -/
+noncomputable def sphere (n : ℤ) : TopCat.{u} :=
+  TopCat.of <| ULift <| Metric.sphere (0 : EuclideanSpace ℝ <| Fin <| (n + 1).toNat) 1
+
+/-- The `n`-disk is the set of points in ℝⁿ whose norm is at most `1`,
+endowed with the subspace topology. -/
+noncomputable def disk (n : ℤ) : TopCat.{u} :=
+  TopCat.of <| ULift <| Metric.closedBall (0 : EuclideanSpace ℝ <| Fin <| n.toNat) 1
+
+/-- `𝕊 n` denotes the `n`-sphere. -/
+scoped prefix:arg "𝕊 " => sphere
+
+/-- `𝔻 n` denotes the `n`-disk. -/
+scoped prefix:arg "𝔻 " => disk
+
+end TopCat
diff --git a/Mathlib/Topology/Category/TopCat/Yoneda.lean b/Mathlib/Topology/Category/TopCat/Yoneda.lean
index 4875f93370077..65ed0d1e00d83 100644
--- a/Mathlib/Topology/Category/TopCat/Yoneda.lean
+++ b/Mathlib/Topology/Category/TopCat/Yoneda.lean
@@ -57,7 +57,7 @@ theorem piComparison_fac {α : Type} (X : α → TopCat) :
     Equiv.coe_fn_symm_mk, comp_assoc, sigmaMk_apply, ← opCoproductIsoProduct_inv_comp_ι]
   rfl
 
-/-- The universe polymorphic Yoneda presheaf on `TopCat` preserves finite products. -/
+/-- The universe polymorphic Yoneda presheaf on `TopCat` preserves finite products. -/
 noncomputable instance : PreservesFiniteProducts (yonedaPresheaf'.{w, w'} Y) where
   preserves J _ :=
     { preservesLimit := fun {K} =>
diff --git a/Mathlib/Topology/Category/TopCommRingCat.lean b/Mathlib/Topology/Category/TopCommRingCat.lean
index 153f2b41ff795..b6734c03f35c7 100644
--- a/Mathlib/Topology/Category/TopCommRingCat.lean
+++ b/Mathlib/Topology/Category/TopCommRingCat.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Ring.Basic
 import Mathlib.Topology.Category.TopCat.Basic
diff --git a/Mathlib/Topology/Category/UniformSpace.lean b/Mathlib/Topology/Category/UniformSpace.lean
index ebeee9ca6766e..0b3083f80e4e6 100644
--- a/Mathlib/Topology/Category/UniformSpace.lean
+++ b/Mathlib/Topology/Category/UniformSpace.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Reid Barton, Patrick Massot, Scott Morrison
+Authors: Reid Barton, Patrick Massot, Kim Morrison
 -/
 import Mathlib.CategoryTheory.Adjunction.Reflective
 import Mathlib.CategoryTheory.ConcreteCategory.UnbundledHom
diff --git a/Mathlib/Topology/Clopen.lean b/Mathlib/Topology/Clopen.lean
index c560e10009cd9..ff10a04698b33 100644
--- a/Mathlib/Topology/Clopen.lean
+++ b/Mathlib/Topology/Clopen.lean
@@ -102,6 +102,10 @@ theorem isClopen_inter_of_disjoint_cover_clopen {s a b : Set X} (h : IsClopen s)
   rintro x ⟨hx₁, hx₂⟩
   exact ⟨hx₁, by simpa [not_mem_of_mem_compl hx₂] using cover hx₁⟩
 
+theorem isClopen_of_disjoint_cover_open {a b : Set X} (cover : univ ⊆ a ∪ b)
+    (ha : IsOpen a) (hb : IsOpen b) (hab : Disjoint a b) : IsClopen a :=
+  univ_inter a ▸ isClopen_inter_of_disjoint_cover_clopen isClopen_univ cover ha hb hab
+
 @[simp]
 theorem isClopen_discrete [DiscreteTopology X] (s : Set X) : IsClopen s :=
   ⟨isClosed_discrete _, isOpen_discrete _⟩
diff --git a/Mathlib/Topology/CompactOpen.lean b/Mathlib/Topology/CompactOpen.lean
index 24a3372bc22ad..0338dd8e2d361 100644
--- a/Mathlib/Topology/CompactOpen.lean
+++ b/Mathlib/Topology/CompactOpen.lean
@@ -3,7 +3,7 @@ Copyright (c) 2018 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Reid Barton
 -/
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # The compact-open topology
@@ -93,6 +93,7 @@ theorem embedding_comp (g : C(Y, Z)) (hg : Embedding g) : Embedding (g.comp : C(
   ⟨inducing_comp g hg.1, fun _ _ ↦ (cancel_left hg.2).1⟩
 
 /-- `C(·, Z)` is a functor. -/
+@[fun_prop]
 theorem continuous_comp_left (f : C(X, Y)) : Continuous (fun g => g.comp f : C(Y, Z) → C(X, Z)) :=
   continuous_compactOpen.2 fun K hK U hU ↦ by
     simpa only [mapsTo_image_iff] using isOpen_setOf_mapsTo (hK.image f.2) hU
@@ -166,8 +167,6 @@ theorem continuous_eval [LocallyCompactPair X Y] : Continuous fun p : C(X, Y) ×
   rcases exists_mem_nhds_isCompact_mapsTo f.continuous (hU.mem_nhds hx) with ⟨K, hxK, hK, hKU⟩
   filter_upwards [prod_mem_nhds (eventually_mapsTo hK hU hKU) hxK] using fun _ h ↦ h.1 h.2
 
-@[deprecated (since := "2023-12-26")] alias continuous_eval' := continuous_eval
-
 /-- Evaluation of a continuous map `f` at a point `x` is continuous in `f`.
 
 Porting note: merged `continuous_eval_const` with `continuous_eval_const'` removing unneeded
@@ -277,7 +276,7 @@ theorem tendsto_compactOpen_iff_forall {ι : Type*} {l : Filter ι} (F : ι →
     Tendsto F l (𝓝 f) ↔
       ∀ K, IsCompact K → Tendsto (fun i => (F i).restrict K) l (𝓝 (f.restrict K)) := by
   rw [compactOpen_eq_iInf_induced]
-  simp [nhds_iInf, nhds_induced, Filter.tendsto_comap_iff, Function.comp]
+  simp [nhds_iInf, nhds_induced, Filter.tendsto_comap_iff, Function.comp_def]
 
 /-- A family `F` of functions in `C(X, Y)` converges in the compact-open topology, if and only if
 it converges in the compact-open topology on each compact subset of `X`. -/
@@ -375,7 +374,7 @@ theorem continuous_curry [LocallyCompactSpace (X × Y)] :
 /-- The uncurried form of a continuous map `X → C(Y, Z)` is a continuous map `X × Y → Z`. -/
 theorem continuous_uncurry_of_continuous [LocallyCompactSpace Y] (f : C(X, C(Y, Z))) :
     Continuous (Function.uncurry fun x y => f x y) :=
-  continuous_eval.comp <| f.continuous.prod_map continuous_id
+  continuous_eval.comp <| f.continuous.prodMap continuous_id
 
 /-- The uncurried form of a continuous map `X → C(Y, Z)` as a continuous map `X × Y → Z` (if `Y` is
     locally compact). If `X` is also locally compact, then this is a homeomorphism between the two
@@ -389,7 +388,7 @@ theorem continuous_uncurry [LocallyCompactSpace X] [LocallyCompactSpace Y] :
     Continuous (uncurry : C(X, C(Y, Z)) → C(X × Y, Z)) := by
   apply continuous_of_continuous_uncurry
   rw [← (Homeomorph.prodAssoc _ _ _).comp_continuous_iff']
-  apply continuous_eval.comp (continuous_eval.prod_map continuous_id)
+  apply continuous_eval.comp (continuous_eval.prodMap continuous_id)
 
 /-- The family of constant maps: `Y → C(X, Y)` as a continuous map. -/
 def const' : C(Y, C(X, Y)) :=
diff --git a/Mathlib/Topology/Compactification/OnePoint.lean b/Mathlib/Topology/Compactification/OnePoint.lean
index 0201f2ccbee78..ffe89b1a9a2a8 100644
--- a/Mathlib/Topology/Compactification/OnePoint.lean
+++ b/Mathlib/Topology/Compactification/OnePoint.lean
@@ -426,9 +426,12 @@ theorem denseRange_coe [NoncompactSpace X] : DenseRange ((↑) : X → OnePoint
   rw [DenseRange, ← compl_infty]
   exact dense_compl_singleton _
 
-theorem denseEmbedding_coe [NoncompactSpace X] : DenseEmbedding ((↑) : X → OnePoint X) :=
+theorem isDenseEmbedding_coe [NoncompactSpace X] : IsDenseEmbedding ((↑) : X → OnePoint X) :=
   { openEmbedding_coe with dense := denseRange_coe }
 
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_coe := isDenseEmbedding_coe
+
 @[simp, norm_cast]
 theorem specializes_coe {x y : X} : (x : OnePoint X) ⤳ y ↔ x ⤳ y :=
   openEmbedding_coe.toInducing.specializes_iff
@@ -507,7 +510,7 @@ example [WeaklyLocallyCompactSpace X] [T2Space X] : T4Space (OnePoint X) := infe
 
 /-- If `X` is not a compact space, then `OnePoint X` is a connected space. -/
 instance [PreconnectedSpace X] [NoncompactSpace X] : ConnectedSpace (OnePoint X) where
-  toPreconnectedSpace := denseEmbedding_coe.toDenseInducing.preconnectedSpace
+  toPreconnectedSpace := isDenseEmbedding_coe.toIsDenseInducing.preconnectedSpace
   toNonempty := inferInstance
 
 /-- If `X` is an infinite type with discrete topology (e.g., `ℕ`), then the identity map from
diff --git a/Mathlib/Topology/Compactness/Compact.lean b/Mathlib/Topology/Compactness/Compact.lean
index 1a468a8d2b236..b83613bdfbf38 100644
--- a/Mathlib/Topology/Compactness/Compact.lean
+++ b/Mathlib/Topology/Compactness/Compact.lean
@@ -229,11 +229,11 @@ theorem IsCompact.elim_directed_family_closed {ι : Type v} [hι : Nonempty ι]
     hs.elim_directed_cover (compl ∘ t) (fun i => (htc i).isOpen_compl)
       (by
         simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop,
-          mem_inter_iff, not_and, iff_self_iff, mem_iInter, mem_compl_iff] using hst)
+          mem_inter_iff, not_and, mem_iInter, mem_compl_iff] using hst)
       (hdt.mono_comp _ fun _ _ => compl_subset_compl.mpr)
   ⟨t, by
     simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_iUnion, exists_prop,
-      mem_inter_iff, not_and, iff_self_iff, mem_iInter, mem_compl_iff] using ht⟩
+      mem_inter_iff, not_and, mem_iInter, mem_compl_iff] using ht⟩
 
 -- Porting note (#11215): TODO: reformulate using `Disjoint`
 /-- For every family of closed sets whose intersection avoids a compact set,
@@ -301,7 +301,7 @@ theorem IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed (t : 
     (htcl : ∀ i, IsClosed (t i)) : (⋂ i, t i).Nonempty :=
   have tmono : Antitone t := antitone_nat_of_succ_le htd
   have htd : Directed (· ⊇ ·) t := tmono.directed_ge
-  have : ∀ i, t i ⊆ t 0 := fun i => tmono <| zero_le i
+  have : ∀ i, t i ⊆ t 0 := fun i => tmono <| Nat.zero_le i
   have htc : ∀ i, IsCompact (t i) := fun i => ht0.of_isClosed_subset (htcl i) (this i)
   IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed t htd htn htc htcl
 
diff --git a/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean b/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
index ebc17cc09ad5a..fbcd4585a405a 100644
--- a/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
+++ b/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
@@ -54,7 +54,7 @@ The compactly generated topology on a topological space `X`. This is the finest
 which makes all maps from compact Hausdorff spaces to `X`, which are continuous for the original
 topology, continuous.
 
-Note: this definition should be used with an explicit universe parameter `u` for the size of the
+Note: this definition should be used with an explicit universe parameter `u` for the size of the
 compact Hausdorff spaces mapping to `X`.
 -/
 def TopologicalSpace.compactlyGenerated (X : Type w) [TopologicalSpace X] : TopologicalSpace X :=
diff --git a/Mathlib/Topology/Compactness/Exterior.lean b/Mathlib/Topology/Compactness/Exterior.lean
new file mode 100644
index 0000000000000..986fcfe09e456
--- /dev/null
+++ b/Mathlib/Topology/Compactness/Exterior.lean
@@ -0,0 +1,30 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Tactic.Peel
+import Mathlib.Topology.Compactness.Compact
+import Mathlib.Topology.Exterior
+
+/-!
+# Compactness of the exterior of a set
+
+In this file we prove that the exterior of a set
+(defined as the intersection of all of its neighborhoods)
+is a compact set if and only if the original set is a compact set.
+-/
+
+variable {X : Type*} [TopologicalSpace X] {s : Set X}
+
+theorem IsCompact.exterior_iff : IsCompact (exterior s) ↔ IsCompact s := by
+  simp only [isCompact_iff_finite_subcover]
+  peel with ι U hUo
+  simp only [(isOpen_iUnion hUo).exterior_subset,
+    (isOpen_iUnion fun i ↦ isOpen_iUnion fun _ ↦ hUo i).exterior_subset]
+
+protected alias ⟨IsCompact.of_exterior, IsCompact.exterior⟩ := IsCompact.exterior_iff
+
+@[deprecated IsCompact.exterior (since := "2024-09-18")]
+lemma Set.Finite.isCompact_exterior (hs : s.Finite) : IsCompact (exterior s) :=
+  hs.isCompact.exterior
diff --git a/Mathlib/Topology/Compactness/Lindelof.lean b/Mathlib/Topology/Compactness/Lindelof.lean
index 4652fa61f3e11..6cbd814c5efbe 100644
--- a/Mathlib/Topology/Compactness/Lindelof.lean
+++ b/Mathlib/Topology/Compactness/Lindelof.lean
@@ -511,7 +511,7 @@ theorem IsClosed.isLindelof [LindelofSpace X] (h : IsClosed s) : IsLindelof s :=
 theorem IsCompact.isLindelof (hs : IsCompact s) :
     IsLindelof s := by tauto
 
-/-- A σ-compact set `s` is Lindelöf-/
+/-- A σ-compact set `s` is Lindelöf -/
 theorem IsSigmaCompact.isLindelof (hs : IsSigmaCompact s) :
     IsLindelof s := by
   rw [IsSigmaCompact] at hs
@@ -564,7 +564,7 @@ theorem Filter.coLindelof_neBot_iff : NeBot (Filter.coLindelof X) ↔ NonLindelo
 theorem not_LindelofSpace_iff : ¬LindelofSpace X ↔ NonLindelofSpace X :=
   ⟨fun h₁ => ⟨fun h₂ => h₁ ⟨h₂⟩⟩, fun ⟨h₁⟩ ⟨h₂⟩ => h₁ h₂⟩
 
-/-- A compact space `X` is Lindelöf.  -/
+/-- A compact space `X` is Lindelöf. -/
 instance (priority := 100) [CompactSpace X] : LindelofSpace X :=
   { isLindelof_univ := isCompact_univ.isLindelof}
 
@@ -692,7 +692,7 @@ for open sets in the definition, and then conclude that this holds for all sets
 def IsHereditarilyLindelof (s : Set X) :=
   ∀ t ⊆ s, IsLindelof t
 
-/-- Type class for Hereditarily Lindelöf spaces.  -/
+/-- Type class for Hereditarily Lindelöf spaces. -/
 class HereditarilyLindelofSpace (X : Type*) [TopologicalSpace X] : Prop where
   /-- In a Hereditarily Lindelöf space, `Set.univ` is a Hereditarily Lindelöf set. -/
   isHereditarilyLindelof_univ : IsHereditarilyLindelof (univ : Set X)
diff --git a/Mathlib/Topology/Compactness/LocallyCompact.lean b/Mathlib/Topology/Compactness/LocallyCompact.lean
index 1881270210e57..e0c44089ecd15 100644
--- a/Mathlib/Topology/Compactness/LocallyCompact.lean
+++ b/Mathlib/Topology/Compactness/LocallyCompact.lean
@@ -76,9 +76,6 @@ theorem LocallyCompactSpace.of_hasBasis {ι : X → Type*} {p : ∀ x, ι x →
     let ⟨i, hp, ht⟩ := (h x).mem_iff.1 ht
     ⟨s x i, (h x).mem_of_mem hp, ht, hc x i hp⟩⟩
 
-@[deprecated (since := "2023-12-29")]
-alias locallyCompactSpace_of_hasBasis := LocallyCompactSpace.of_hasBasis
-
 instance Prod.locallyCompactSpace (X : Type*) (Y : Type*) [TopologicalSpace X]
     [TopologicalSpace Y] [LocallyCompactSpace X] [LocallyCompactSpace Y] :
     LocallyCompactSpace (X × Y) :=
@@ -169,28 +166,37 @@ theorem exists_compact_between [LocallyCompactSpace X] {K U : Set X} (hK : IsCom
   let ⟨L, hKL, hL, hLU⟩ := exists_mem_nhdsSet_isCompact_mapsTo continuous_id hK hU h_KU
   ⟨L, hL, subset_interior_iff_mem_nhdsSet.2 hKL, hLU⟩
 
+/-- If `f` is a topology inducing map with a locally compact codomain and a locally closed range,
+then the domain of `f` is a locally compact space. -/
+theorem Inducing.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y} (hf : Inducing f)
+    (h : IsLocallyClosed (range f)) : LocallyCompactSpace X := by
+  rcases h with ⟨U, Z, hU, hZ, hUZ⟩
+  have (x : X) : (𝓝 x).HasBasis (fun s ↦ (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ U)
+      (fun s ↦ f ⁻¹' (s ∩ Z)) := by
+    have H : U ∈ 𝓝 (f x) := hU.mem_nhds (hUZ.subset <| mem_range_self _).1
+    rw [hf.nhds_eq_comap, ← comap_nhdsWithin_range, hUZ,
+      nhdsWithin_inter_of_mem (nhdsWithin_le_nhds H)]
+    exact (nhdsWithin_hasBasis ((compact_basis_nhds (f x)).restrict_subset H) _).comap _
+  refine .of_hasBasis this fun x s ⟨⟨_, hs⟩, hsU⟩ ↦ ?_
+  rw [hf.isCompact_preimage_iff]
+  exacts [hs.inter_right hZ, hUZ ▸ by gcongr]
+
 protected theorem ClosedEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
     (hf : ClosedEmbedding f) : LocallyCompactSpace X :=
-  haveI : ∀ x : X, (𝓝 x).HasBasis (fun s => s ∈ 𝓝 (f x) ∧ IsCompact s) (f ⁻¹' ·) := fun x ↦ by
-    rw [hf.toInducing.nhds_eq_comap]
-    exact (compact_basis_nhds _).comap _
-  .of_hasBasis this fun x s hs => hf.isCompact_preimage hs.2
+  hf.toInducing.locallyCompactSpace hf.isClosed_range.isLocallyClosed
+
+protected theorem OpenEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
+    (hf : OpenEmbedding f) : LocallyCompactSpace X :=
+  hf.toInducing.locallyCompactSpace hf.isOpen_range.isLocallyClosed
+
+protected theorem IsLocallyClosed.locallyCompactSpace [LocallyCompactSpace X] {s : Set X}
+    (hs : IsLocallyClosed s) : LocallyCompactSpace s :=
+  embedding_subtype_val.locallyCompactSpace <| by rwa [Subtype.range_val]
 
 protected theorem IsClosed.locallyCompactSpace [LocallyCompactSpace X] {s : Set X}
     (hs : IsClosed s) : LocallyCompactSpace s :=
-  (closedEmbedding_subtype_val hs).locallyCompactSpace
-
-protected theorem OpenEmbedding.locallyCompactSpace [LocallyCompactSpace Y] {f : X → Y}
-    (hf : OpenEmbedding f) : LocallyCompactSpace X := by
-  have : ∀ x : X,
-      (𝓝 x).HasBasis (fun s ↦ (s ∈ 𝓝 (f x) ∧ IsCompact s) ∧ s ⊆ range f) (f ⁻¹' ·) := fun x ↦ by
-    rw [hf.nhds_eq_comap]
-    exact ((compact_basis_nhds _).restrict_subset <| hf.isOpen_range.mem_nhds <|
-      mem_range_self _).comap _
-  refine .of_hasBasis this fun x s hs => ?_
-  rw [hf.toInducing.isCompact_iff, image_preimage_eq_of_subset hs.2]
-  exact hs.1.2
+  hs.isLocallyClosed.locallyCompactSpace
 
 protected theorem IsOpen.locallyCompactSpace [LocallyCompactSpace X] {s : Set X} (hs : IsOpen s) :
     LocallyCompactSpace s :=
-  hs.openEmbedding_subtype_val.locallyCompactSpace
+  hs.isLocallyClosed.locallyCompactSpace
diff --git a/Mathlib/Topology/Connected/Basic.lean b/Mathlib/Topology/Connected/Basic.lean
index 8d8d2a542f23d..54b61ed7b660c 100644
--- a/Mathlib/Topology/Connected/Basic.lean
+++ b/Mathlib/Topology/Connected/Basic.lean
@@ -506,7 +506,7 @@ theorem isConnected_connectedComponent {x : α} : IsConnected (connectedComponen
 theorem isConnected_connectedComponentIn_iff {x : α} {F : Set α} :
     IsConnected (connectedComponentIn F x) ↔ x ∈ F := by
   simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn,
-    and_true_iff]
+    and_true]
 
 theorem IsPreconnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsPreconnected s)
     (H2 : x ∈ s) : s ⊆ connectedComponent x := fun _z hz => mem_sUnion_of_mem hz ⟨H1, H2⟩
diff --git a/Mathlib/Topology/Connected/PathConnected.lean b/Mathlib/Topology/Connected/PathConnected.lean
index 07fea44bddd1e..644fcb694b3a8 100644
--- a/Mathlib/Topology/Connected/PathConnected.lean
+++ b/Mathlib/Topology/Connected/PathConnected.lean
@@ -33,7 +33,7 @@ Then there are corresponding relative notions for `F : Set X`.
 * `LocPathConnectedSpace X` is a predicate class asserting that `X` is locally path-connected:
   each point has a basis of path-connected neighborhoods (we do *not* ask these to be open).
 
-## Main theorems
+## Main theorems
 
 * `Joined` and `JoinedIn F` are transitive relations.
 
@@ -75,8 +75,8 @@ structure Path (x y : X) extends C(I, X) where
   target' : toFun 1 = y
 
 instance Path.funLike : FunLike (Path x y) I X where
-  coe := fun γ ↦ ⇑γ.toContinuousMap
-  coe_injective' := fun γ₁ γ₂ h => by
+  coe γ := ⇑γ.toContinuousMap
+  coe_injective' γ₁ γ₂ h := by
     simp only [DFunLike.coe_fn_eq] at h
     cases γ₁; cases γ₂; congr
 
@@ -200,7 +200,7 @@ instance topologicalSpace : TopologicalSpace (Path x y) :=
 
 theorem continuous_eval : Continuous fun p : Path x y × I => p.1 p.2 :=
   ContinuousMap.continuous_eval.comp <|
-    (continuous_induced_dom (α := Path x y)).prod_map continuous_id
+    (continuous_induced_dom (α := Path x y)).prodMap continuous_id
 
 @[continuity]
 theorem _root_.Continuous.path_eval {Y} [TopologicalSpace Y] {f : Y → Path x y} {g : Y → I}
@@ -247,7 +247,6 @@ theorem extend_zero : γ.extend 0 = x := by simp
 
 theorem extend_one : γ.extend 1 = y := by simp
 
-@[simp]
 theorem extend_extends' {a b : X} (γ : Path a b) (t : (Icc 0 1 : Set ℝ)) : γ.extend t = γ t :=
   IccExtend_val _ γ t
 
@@ -424,7 +423,7 @@ theorem cast_coe (γ : Path x y) {x' y'} (hx : x' = x) (hy : y' = y) : (γ.cast
 theorem symm_continuous_family {ι : Type*} [TopologicalSpace ι]
     {a b : ι → X} (γ : ∀ t : ι, Path (a t) (b t)) (h : Continuous ↿γ) :
     Continuous ↿fun t => (γ t).symm :=
-  h.comp (continuous_id.prod_map continuous_symm)
+  h.comp (continuous_id.prodMap continuous_symm)
 
 @[continuity]
 theorem continuous_symm : Continuous (symm : Path x y → Path y x) :=
@@ -434,7 +433,7 @@ theorem continuous_symm : Continuous (symm : Path x y → Path y x) :=
 theorem continuous_uncurry_extend_of_continuous_family {ι : Type*} [TopologicalSpace ι]
     {a b : ι → X} (γ : ∀ t : ι, Path (a t) (b t)) (h : Continuous ↿γ) :
     Continuous ↿fun t => (γ t).extend := by
-  apply h.comp (continuous_id.prod_map continuous_projIcc)
+  apply h.comp (continuous_id.prodMap continuous_projIcc)
   exact zero_le_one
 
 @[continuity]
@@ -448,12 +447,12 @@ theorem trans_continuous_family {ι : Type*} [TopologicalSpace ι]
   refine Continuous.if_le ?_ ?_ (continuous_subtype_val.comp continuous_snd) continuous_const ?_
   · change
       Continuous ((fun p : ι × ℝ => (γ₁ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x : I → ℝ))
-    exact h₁'.comp (continuous_id.prod_map <| continuous_const.mul continuous_subtype_val)
+    exact h₁'.comp (continuous_id.prodMap <| continuous_const.mul continuous_subtype_val)
   · change
       Continuous ((fun p : ι × ℝ => (γ₂ p.1).extend p.2) ∘ Prod.map id (fun x => 2 * x - 1 : I → ℝ))
     exact
       h₂'.comp
-        (continuous_id.prod_map <|
+        (continuous_id.prodMap <|
           (continuous_const.mul continuous_subtype_val).sub continuous_const)
   · rintro st hst
     simp [hst, mul_inv_cancel₀ (two_ne_zero' ℝ)]
diff --git a/Mathlib/Topology/Connected/Separation.lean b/Mathlib/Topology/Connected/Separation.lean
index 57db6948dc729..8ce21856914c0 100644
--- a/Mathlib/Topology/Connected/Separation.lean
+++ b/Mathlib/Topology/Connected/Separation.lean
@@ -11,7 +11,7 @@ import Mathlib.Topology.Separation
 This file provides an instance `T2Space X` given `TotallySeparatedSpace X`.
 
 ## TODO
-* Move the last part of `Topology/Separation` to this file.
+* Move the last part of `Topology/Separation` to this file.
 -/
 
 
diff --git a/Mathlib/Topology/Connected/TotallyDisconnected.lean b/Mathlib/Topology/Connected/TotallyDisconnected.lean
index 0da1004c037f4..b7f1ff02b4fb4 100644
--- a/Mathlib/Topology/Connected/TotallyDisconnected.lean
+++ b/Mathlib/Topology/Connected/TotallyDisconnected.lean
@@ -195,7 +195,7 @@ alias IsTotallySeparated.isTotallyDisconnected := isTotallyDisconnected_of_isTot
 
 /-- A space is totally separated if any two points can be separated by two disjoint open sets
 covering the whole space. -/
-class TotallySeparatedSpace (α : Type u) [TopologicalSpace α] : Prop where
+@[mk_iff] class TotallySeparatedSpace (α : Type u) [TopologicalSpace α] : Prop where
   /-- The universal set `Set.univ` in a totally separated space is totally separated. -/
   isTotallySeparated_univ : IsTotallySeparated (univ : Set α)
 
@@ -210,15 +210,19 @@ instance (priority := 100) TotallySeparatedSpace.of_discrete (α : Type*) [Topol
   ⟨fun _ _ b _ h => ⟨{b}ᶜ, {b}, isOpen_discrete _, isOpen_discrete _, h, rfl,
     (compl_union_self _).symm.subset, disjoint_compl_left⟩⟩
 
+theorem totallySeparatedSpace_iff_exists_isClopen {α : Type*} [TopologicalSpace α] :
+    TotallySeparatedSpace α ↔ ∀ x y : α, x ≠ y → ∃ U : Set α, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ := by
+  simp only [totallySeparatedSpace_iff, IsTotallySeparated, Set.Pairwise, mem_univ, true_implies]
+  refine forall₃_congr fun x y _ ↦
+    ⟨fun ⟨U, V, hU, hV, Ux, Vy, f, disj⟩ ↦ ?_, fun ⟨U, hU, Ux, Ucy⟩ ↦ ?_⟩
+  · exact ⟨U, isClopen_of_disjoint_cover_open f hU hV disj,
+      Ux, fun Uy ↦ Set.disjoint_iff.mp disj ⟨Uy, Vy⟩⟩
+  · exact ⟨U, Uᶜ, hU.2, hU.compl.2, Ux, Ucy, (Set.union_compl_self U).ge, disjoint_compl_right⟩
+
 theorem exists_isClopen_of_totally_separated {α : Type*} [TopologicalSpace α]
     [TotallySeparatedSpace α] {x y : α} (hxy : x ≠ y) :
-    ∃ U : Set α, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ := by
-  obtain ⟨U, V, hU, hV, Ux, Vy, f, disj⟩ :=
-    TotallySeparatedSpace.isTotallySeparated_univ (Set.mem_univ x) (Set.mem_univ y) hxy
-  have hU := isClopen_inter_of_disjoint_cover_clopen isClopen_univ f hU hV disj
-  rw [univ_inter _] at hU
-  rw [← Set.subset_compl_iff_disjoint_right, subset_compl_comm] at disj
-  exact ⟨U, hU, Ux, disj Vy⟩
+    ∃ U : Set α, IsClopen U ∧ x ∈ U ∧ y ∈ Uᶜ :=
+  totallySeparatedSpace_iff_exists_isClopen.mp ‹_› _ _ hxy
 
 end TotallySeparated
 
@@ -260,7 +264,6 @@ theorem Continuous.connectedComponentsLift_unique (h : Continuous f) (g : Connec
     (hg : g ∘ (↑) = f) : g = h.connectedComponentsLift :=
   connectedComponents_lift_unique' <| hg.trans h.connectedComponentsLift_comp_coe.symm
 
-
 instance ConnectedComponents.totallyDisconnectedSpace :
     TotallyDisconnectedSpace (ConnectedComponents α) := by
   rw [totallyDisconnectedSpace_iff_connectedComponent_singleton]
diff --git a/Mathlib/Topology/Constructions.lean b/Mathlib/Topology/Constructions.lean
index de021933eda18..b4bf95ad2f9e1 100644
--- a/Mathlib/Topology/Constructions.lean
+++ b/Mathlib/Topology/Constructions.lean
@@ -3,6 +3,8 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
+import Mathlib.Data.Finset.Piecewise
+import Mathlib.Order.Filter.Curry
 import Mathlib.Topology.Maps.Basic
 import Mathlib.Topology.NhdsSet
 
@@ -398,10 +400,12 @@ theorem Continuous.comp₄ {g : X × Y × Z × ζ → ε} (hg : Continuous g) {e
   hg.comp₃ he hf <| hk.prod_mk hl
 
 @[continuity]
-theorem Continuous.prod_map {f : Z → X} {g : W → Y} (hf : Continuous f) (hg : Continuous g) :
-    Continuous fun p : Z × W => (f p.1, g p.2) :=
+theorem Continuous.prodMap {f : Z → X} {g : W → Y} (hf : Continuous f) (hg : Continuous g) :
+    Continuous (Prod.map f g) :=
   hf.fst'.prod_mk hg.snd'
 
+@[deprecated (since := "2024-10-05")] alias Continuous.prod_map := Continuous.prodMap
+
 /-- A version of `continuous_inf_dom_left` for binary functions -/
 theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : TopologicalSpace X}
     {tb1 tb2 : TopologicalSpace Y} {tc1 : TopologicalSpace Z}
@@ -409,7 +413,7 @@ theorem continuous_inf_dom_left₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : Topolo
     haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
   have ha := @continuous_inf_dom_left _ _ id ta1 ta2 ta1 (@continuous_id _ (id _))
   have hb := @continuous_inf_dom_left _ _ id tb1 tb2 tb1 (@continuous_id _ (id _))
-  have h_continuous_id := @Continuous.prod_map _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
+  have h_continuous_id := @Continuous.prodMap _ _ _ _ ta1 tb1 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 
 /-- A version of `continuous_inf_dom_right` for binary functions -/
@@ -419,7 +423,7 @@ theorem continuous_inf_dom_right₂ {X Y Z} {f : X → Y → Z} {ta1 ta2 : Topol
     haveI := ta1 ⊓ ta2; haveI := tb1 ⊓ tb2; exact Continuous fun p : X × Y => f p.1 p.2 := by
   have ha := @continuous_inf_dom_right _ _ id ta1 ta2 ta2 (@continuous_id _ (id _))
   have hb := @continuous_inf_dom_right _ _ id tb1 tb2 tb2 (@continuous_id _ (id _))
-  have h_continuous_id := @Continuous.prod_map _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
+  have h_continuous_id := @Continuous.prodMap _ _ _ _ ta2 tb2 (ta1 ⊓ ta2) (tb1 ⊓ tb2) _ _ ha hb
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ h h_continuous_id
 
 /-- A version of `continuous_sInf_dom` for binary functions -/
@@ -431,7 +435,7 @@ theorem continuous_sInf_dom₂ {X Y Z} {f : X → Y → Z} {tas : Set (Topologic
     exact @Continuous _ _ _ tc fun p : X × Y => f p.1 p.2 := by
   have hX := continuous_sInf_dom hX continuous_id
   have hY := continuous_sInf_dom hY continuous_id
-  have h_continuous_id := @Continuous.prod_map _ _ _ _ tX tY (sInf tas) (sInf tbs) _ _ hX hY
+  have h_continuous_id := @Continuous.prodMap _ _ _ _ tX tY (sInf tas) (sInf tbs) _ _ hX hY
   exact @Continuous.comp _ _ _ (id _) (id _) _ _ _ hf h_continuous_id
 
 theorem Filter.Eventually.prod_inl_nhds {p : X → Prop} {x : X} (h : ∀ᶠ x in 𝓝 x, p x) (y : Y) :
@@ -514,6 +518,22 @@ theorem Filter.HasBasis.prod_nhds' {ιX ιY : Type*} {pX : ιX → Prop} {pY : 
     (𝓝 p).HasBasis (fun i : ιX × ιY => pX i.1 ∧ pY i.2) fun i => sx i.1 ×ˢ sy i.2 :=
   hx.prod_nhds hy
 
+theorem MapClusterPt.curry_prodMap {α β : Type*}
+    {f : α → X} {g : β → Y} {la : Filter α} {lb : Filter β} {x : X} {y : Y}
+    (hf : MapClusterPt x la f) (hg : MapClusterPt y lb g) :
+    MapClusterPt (x, y) (la.curry lb) (.map f g) := by
+  rw [mapClusterPt_iff] at hf hg
+  rw [((𝓝 x).basis_sets.prod_nhds (𝓝 y).basis_sets).mapClusterPt_iff_frequently]
+  rintro ⟨s, t⟩ ⟨hs, ht⟩
+  rw [frequently_curry_iff]
+  exact (hf s hs).mono fun x hx ↦ (hg t ht).mono fun y hy ↦ ⟨hx, hy⟩
+
+theorem MapClusterPt.prodMap {α β : Type*}
+    {f : α → X} {g : β → Y} {la : Filter α} {lb : Filter β} {x : X} {y : Y}
+    (hf : MapClusterPt x la f) (hg : MapClusterPt y lb g) :
+    MapClusterPt (x, y) (la ×ˢ lb) (.map f g) :=
+  (hf.curry_prodMap hg).mono <| map_mono curry_le_prod
+
 theorem mem_nhds_prod_iff' {x : X} {y : Y} {s : Set (X × Y)} :
     s ∈ 𝓝 (x, y) ↔ ∃ u v, IsOpen u ∧ x ∈ u ∧ IsOpen v ∧ y ∈ v ∧ u ×ˢ v ⊆ s :=
   ((nhds_basis_opens x).prod_nhds (nhds_basis_opens y)).mem_iff.trans <| by
@@ -564,13 +584,19 @@ theorem ContinuousAt.prod {f : X → Y} {g : X → Z} {x : X} (hf : ContinuousAt
     (hg : ContinuousAt g x) : ContinuousAt (fun x => (f x, g x)) x :=
   hf.prod_mk_nhds hg
 
-theorem ContinuousAt.prod_map {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.fst)
-    (hg : ContinuousAt g p.snd) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) p :=
+theorem ContinuousAt.prodMap {f : X → Z} {g : Y → W} {p : X × Y} (hf : ContinuousAt f p.fst)
+    (hg : ContinuousAt g p.snd) : ContinuousAt (Prod.map f g) p :=
   hf.fst''.prod hg.snd''
 
-theorem ContinuousAt.prod_map' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x)
-    (hg : ContinuousAt g y) : ContinuousAt (fun p : X × Y => (f p.1, g p.2)) (x, y) :=
-  hf.fst'.prod hg.snd'
+@[deprecated (since := "2024-10-05")] alias ContinuousAt.prod_map := ContinuousAt.prodMap
+
+/-- A version of `ContinuousAt.prodMap` that avoids `Prod.fst`/`Prod.snd`
+by assuming that the point is `(x, y)`. -/
+theorem ContinuousAt.prodMap' {f : X → Z} {g : Y → W} {x : X} {y : Y} (hf : ContinuousAt f x)
+    (hg : ContinuousAt g y) : ContinuousAt (Prod.map f g) (x, y) :=
+  hf.prodMap hg
+
+@[deprecated (since := "2024-10-05")] alias ContinuousAt.prod_map' := ContinuousAt.prodMap'
 
 theorem ContinuousAt.comp₂ {f : Y × Z → W} {g : X → Y} {h : X → Z} {x : X}
     (hf : ContinuousAt f (g x, h x)) (hg : ContinuousAt g x) (hh : ContinuousAt h x) :
@@ -600,7 +626,7 @@ theorem prod_generateFrom_generateFrom_eq {X Y : Type*} {s : Set (Set X)} {t : S
     (hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
     @instTopologicalSpaceProd X Y (generateFrom s) (generateFrom t) =
       generateFrom (image2 (·  ×ˢ ·) s t) :=
-  let G := generateFrom  (image2  (·  ×ˢ ·) s t)
+  let G := generateFrom (image2  (·  ×ˢ ·) s t)
   le_antisymm
     (le_generateFrom fun g ⟨u, hu, v, hv, g_eq⟩ =>
       g_eq.symm ▸
@@ -710,7 +736,7 @@ theorem isOpen_prod_iff' {s : Set X} {t : Set Y} :
         rw [← snd_image_prod st.1 t]
         exact isOpenMap_snd _ H
     · intro H
-      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false_iff] at H
+      simp only [st.1.ne_empty, st.2.ne_empty, not_false_iff, or_false] at H
       exact H.1.prod H.2
 
 theorem quotientMap_fst [Nonempty Y] : QuotientMap (Prod.fst : X × Y → X) :=
@@ -758,40 +784,50 @@ theorem Dense.prod {s : Set X} {t : Set Y} (hs : Dense s) (ht : Dense t) : Dense
   exact ⟨hs x.1, ht x.2⟩
 
 /-- If `f` and `g` are maps with dense range, then `Prod.map f g` has dense range. -/
-theorem DenseRange.prod_map {ι : Type*} {κ : Type*} {f : ι → Y} {g : κ → Z} (hf : DenseRange f)
+theorem DenseRange.prodMap {ι : Type*} {κ : Type*} {f : ι → Y} {g : κ → Z} (hf : DenseRange f)
     (hg : DenseRange g) : DenseRange (Prod.map f g) := by
   simpa only [DenseRange, prod_range_range_eq] using hf.prod hg
 
-theorem Inducing.prod_map {f : X → Y} {g : Z → W} (hf : Inducing f) (hg : Inducing g) :
+@[deprecated (since := "2024-10-05")] alias DenseRange.prod_map := DenseRange.prodMap
+
+theorem Inducing.prodMap {f : X → Y} {g : Z → W} (hf : Inducing f) (hg : Inducing g) :
     Inducing (Prod.map f g) :=
   inducing_iff_nhds.2 fun (x, z) => by simp_rw [Prod.map_def, nhds_prod_eq, hf.nhds_eq_comap,
     hg.nhds_eq_comap, prod_comap_comap_eq]
 
+@[deprecated (since := "2024-10-05")] alias Inducing.prod_map := Inducing.prodMap
+
 @[simp]
 theorem inducing_const_prod {x : X} {f : Y → Z} : (Inducing fun x' => (x, f x')) ↔ Inducing f := by
-  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
+  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp_def,
     induced_const, top_inf_eq]
 
 @[simp]
 theorem inducing_prod_const {y : Y} {f : X → Z} : (Inducing fun x => (f x, y)) ↔ Inducing f := by
-  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp,
+  simp_rw [inducing_iff, instTopologicalSpaceProd, induced_inf, induced_compose, Function.comp_def,
     induced_const, inf_top_eq]
 
-theorem Embedding.prod_map {f : X → Y} {g : Z → W} (hf : Embedding f) (hg : Embedding g) :
+theorem Embedding.prodMap {f : X → Y} {g : Z → W} (hf : Embedding f) (hg : Embedding g) :
     Embedding (Prod.map f g) :=
-  { hf.toInducing.prod_map hg.toInducing with
+  { hf.toInducing.prodMap hg.toInducing with
     inj := fun ⟨x₁, z₁⟩ ⟨x₂, z₂⟩ => by simp [hf.inj.eq_iff, hg.inj.eq_iff] }
 
-protected theorem IsOpenMap.prod {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
-    IsOpenMap fun p : X × Z => (f p.1, g p.2) := by
+@[deprecated (since := "2024-10-05")] alias Embedding.prod_map := Embedding.prodMap
+
+protected theorem IsOpenMap.prodMap {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
+    IsOpenMap (Prod.map f g) := by
   rw [isOpenMap_iff_nhds_le]
   rintro ⟨a, b⟩
-  rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq]
+  rw [nhds_prod_eq, nhds_prod_eq, ← Filter.prod_map_map_eq']
   exact Filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
 
-protected theorem OpenEmbedding.prod {f : X → Y} {g : Z → W} (hf : OpenEmbedding f)
-    (hg : OpenEmbedding g) : OpenEmbedding fun x : X × Z => (f x.1, g x.2) :=
-  openEmbedding_of_embedding_open (hf.1.prod_map hg.1) (hf.isOpenMap.prod hg.isOpenMap)
+@[deprecated (since := "2024-10-05")] alias IsOpenMap.prod := IsOpenMap.prodMap
+
+protected theorem OpenEmbedding.prodMap {f : X → Y} {g : Z → W} (hf : OpenEmbedding f)
+    (hg : OpenEmbedding g) : OpenEmbedding (Prod.map f g) :=
+  openEmbedding_of_embedding_open (hf.1.prodMap hg.1) (hf.isOpenMap.prodMap hg.isOpenMap)
+
+@[deprecated (since := "2024-10-05")] alias OpenEmbedding.prod := OpenEmbedding.prodMap
 
 theorem embedding_graph {f : X → Y} (hf : Continuous f) : Embedding fun x => (x, f x) :=
   embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id
@@ -799,6 +835,10 @@ theorem embedding_graph {f : X → Y} (hf : Continuous f) : Embedding fun x => (
 theorem embedding_prod_mk (x : X) : Embedding (Prod.mk x : Y → X × Y) :=
   embedding_of_embedding_compose (Continuous.Prod.mk x) continuous_snd embedding_id
 
+theorem IsOpenQuotientMap.prodMap {f : X → Y} {g : Z → W} (hf : IsOpenQuotientMap f)
+    (hg : IsOpenQuotientMap g) : IsOpenQuotientMap (Prod.map f g) :=
+  ⟨.prodMap hf.1 hg.1, .prodMap hf.2 hg.2, .prodMap hf.3 hg.3⟩
+
 end Prod
 
 section Bool
@@ -847,6 +887,10 @@ theorem continuous_inl : Continuous (@inl X Y) := ⟨fun _ => And.left⟩
 -- Porting note: the proof was `continuous_sup_rng_right continuous_coinduced_rng`
 theorem continuous_inr : Continuous (@inr X Y) := ⟨fun _ => And.right⟩
 
+@[fun_prop, continuity]
+lemma continuous_sum_swap : Continuous (@Sum.swap X Y) :=
+  Continuous.sum_elim continuous_inr continuous_inl
+
 theorem isOpen_sum_iff {s : Set (X ⊕ Y)} : IsOpen s ↔ IsOpen (inl ⁻¹' s) ∧ IsOpen (inr ⁻¹' s) :=
   Iff.rfl
 
@@ -911,7 +955,7 @@ theorem Continuous.sum_map {f : X → Y} {g : Z → W} (hf : Continuous f) (hg :
 
 theorem isOpenMap_sum {f : X ⊕ Y → Z} :
     IsOpenMap f ↔ (IsOpenMap fun a => f (inl a)) ∧ IsOpenMap fun b => f (inr b) := by
-  simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp]
+  simp only [isOpenMap_iff_nhds_le, Sum.forall, nhds_inl, nhds_inr, Filter.map_map, comp_def]
 
 theorem IsOpenMap.sumMap {f : X → Y} {g : Z → W} (hf : IsOpenMap f) (hg : IsOpenMap g) :
     IsOpenMap (Sum.map f g) := by
@@ -941,7 +985,7 @@ end Sum
 
 section Subtype
 
-variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {p : X → Prop}
+variable [TopologicalSpace X] [TopologicalSpace Y] {p : X → Prop}
 
 theorem inducing_subtype_val {t : Set Y} : Inducing ((↑) : t → Y) := ⟨rfl⟩
 
@@ -1063,7 +1107,7 @@ theorem embedding_inclusion {s t : Set X} (h : s ⊆ t) : Embedding (inclusion h
   embedding_subtype_val.codRestrict _ _
 
 /-- Let `s, t ⊆ X` be two subsets of a topological space `X`.  If `t ⊆ s` and the topology induced
-by `X`on `s` is discrete, then also the topology induces on `t` is discrete.  -/
+by `X`on `s` is discrete, then also the topology induces on `t` is discrete. -/
 theorem DiscreteTopology.of_subset {X : Type*} [TopologicalSpace X] {s t : Set X}
     (_ : DiscreteTopology s) (ts : t ⊆ s) : DiscreteTopology t :=
   (embedding_inclusion ts).discreteTopology
@@ -1096,7 +1140,7 @@ end Subtype
 
 section Quotient
 
-variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
+variable [TopologicalSpace X] [TopologicalSpace Y]
 variable {r : X → X → Prop} {s : Setoid X}
 
 theorem quotientMap_quot_mk : QuotientMap (@Quot.mk X r) :=
@@ -1139,7 +1183,7 @@ variable {ι : Type*} {π : ι → Type*} {κ : Type*} [TopologicalSpace X]
   [T : ∀ i, TopologicalSpace (π i)] {f : X → ∀ i : ι, π i}
 
 theorem continuous_pi_iff : Continuous f ↔ ∀ i, Continuous fun a => f a i := by
-  simp only [continuous_iInf_rng, continuous_induced_rng, comp]
+  simp only [continuous_iInf_rng, continuous_induced_rng, comp_def]
 
 @[continuity, fun_prop]
 theorem continuous_pi (h : ∀ i, Continuous fun a => f a i) : Continuous f :=
@@ -1197,17 +1241,51 @@ theorem Pi.continuous_postcomp [TopologicalSpace Y] {g : X → Y} (hg : Continuo
 lemma Pi.induced_precomp' {ι' : Type*} (φ : ι' → ι) :
     induced (fun (f : (∀ i, π i)) (j : ι') ↦ f (φ j)) Pi.topologicalSpace =
     ⨅ i', induced (eval (φ i')) (T (φ i')) := by
-  simp [Pi.topologicalSpace, induced_iInf, induced_compose, comp]
+  simp [Pi.topologicalSpace, induced_iInf, induced_compose, comp_def]
 
 lemma Pi.induced_precomp [TopologicalSpace Y] {ι' : Type*} (φ : ι' → ι) :
     induced (· ∘ φ) Pi.topologicalSpace =
     ⨅ i', induced (eval (φ i')) ‹TopologicalSpace Y› :=
   induced_precomp' φ
 
+@[continuity, fun_prop]
 lemma Pi.continuous_restrict (S : Set ι) :
     Continuous (S.restrict : (∀ i : ι, π i) → (∀ i : S, π i)) :=
   Pi.continuous_precomp' ((↑) : S → ι)
 
+@[continuity, fun_prop]
+lemma Pi.continuous_restrict₂ {s t : Set ι} (hst : s ⊆ t) : Continuous (restrict₂ (π := π) hst) :=
+  continuous_pi fun _ ↦ continuous_apply _
+
+@[continuity, fun_prop]
+theorem Finset.continuous_restrict (s : Finset ι) : Continuous (s.restrict (π := π)) :=
+  continuous_pi fun _ ↦ continuous_apply _
+
+@[continuity, fun_prop]
+theorem Finset.continuous_restrict₂ {s t : Finset ι} (hst : s ⊆ t) :
+    Continuous (Finset.restrict₂ (π := π) hst) :=
+  continuous_pi fun _ ↦ continuous_apply _
+
+variable [TopologicalSpace Z]
+
+@[continuity, fun_prop]
+theorem Pi.continuous_restrict_apply (s : Set X) {f : X → Z} (hf : Continuous f) :
+    Continuous (s.restrict f) := hf.comp continuous_subtype_val
+
+@[continuity, fun_prop]
+theorem Pi.continuous_restrict₂_apply {s t : Set X} (hst : s ⊆ t)
+    {f : t → Z} (hf : Continuous f) :
+    Continuous (restrict₂ (π := fun _ ↦ Z) hst f) := hf.comp (continuous_inclusion hst)
+
+@[continuity, fun_prop]
+theorem Finset.continuous_restrict_apply (s : Finset X) {f : X → Z} (hf : Continuous f) :
+    Continuous (s.restrict f) := hf.comp continuous_subtype_val
+
+@[continuity, fun_prop]
+theorem Finset.continuous_restrict₂_apply {s t : Finset X} (hst : s ⊆ t)
+    {f : t → Z} (hf : Continuous f) :
+    Continuous (restrict₂ (π := fun _ ↦ Z) hst f) := hf.comp (continuous_inclusion hst)
+
 lemma Pi.induced_restrict (S : Set ι) :
     induced (S.restrict) Pi.topologicalSpace =
     ⨅ i ∈ S, induced (eval i) (T i) := by
@@ -1382,7 +1460,7 @@ theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a
 
 theorem induced_to_pi {X : Type*} (f : X → ∀ i, π i) :
     induced f Pi.topologicalSpace = ⨅ i, induced (f · i) inferInstance := by
-  simp_rw [Pi.topologicalSpace, induced_iInf, induced_compose, Function.comp]
+  simp_rw [Pi.topologicalSpace, induced_iInf, induced_compose, Function.comp_def]
 
 /-- Suppose `π i` is a family of topological spaces indexed by `i : ι`, and `X` is a type
 endowed with a family of maps `f i : X → π i` for every `i : ι`, hence inducing a
@@ -1504,7 +1582,7 @@ theorem inducing_sigma {f : Sigma σ → X} :
 @[simp 1100]
 theorem continuous_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     Continuous (Sigma.map f₁ f₂) ↔ ∀ i, Continuous (f₂ i) :=
-  continuous_sigma_iff.trans <| by simp only [Sigma.map, embedding_sigmaMk.continuous_iff, comp]
+  continuous_sigma_iff.trans <| by simp only [Sigma.map, embedding_sigmaMk.continuous_iff, comp_def]
 
 @[continuity, fun_prop]
 theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (hf : ∀ i, Continuous (f₂ i)) :
@@ -1512,7 +1590,7 @@ theorem Continuous.sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁
   continuous_sigma_map.2 hf
 
 theorem isOpenMap_sigma {f : Sigma σ → X} : IsOpenMap f ↔ ∀ i, IsOpenMap fun a => f ⟨i, a⟩ := by
-  simp only [isOpenMap_iff_nhds_le, Sigma.forall, Sigma.nhds_eq, map_map, comp]
+  simp only [isOpenMap_iff_nhds_le, Sigma.forall, Sigma.nhds_eq, map_map, comp_def]
 
 theorem isOpenMap_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} :
     IsOpenMap (Sigma.map f₁ f₂) ↔ ∀ i, IsOpenMap (f₂ i) :=
@@ -1582,8 +1660,8 @@ theorem IsClosed.trans (ht : IsClosed t) (hs : IsClosed s) : IsClosed (t : Set X
 end Monad
 
 section NhdsSet
-variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
-  {s : Set X} {t : Set Y} {x : X}
+variable [TopologicalSpace X] [TopologicalSpace Y]
+  {s : Set X} {t : Set Y}
 
 /-- The product of a neighborhood of `s` and a neighborhood of `t` is a neighborhood of `s ×ˢ t`,
 formulated in terms of a filter inequality. -/
@@ -1604,3 +1682,5 @@ theorem Filter.Eventually.prod_nhdsSet {p : X × Y → Prop} {px : X → Prop} {
   nhdsSet_prod_le _ _ (mem_of_superset (prod_mem_prod hs ht) fun _ ⟨hx, hy⟩ ↦ hp hx hy)
 
 end NhdsSet
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Topology/ContinuousFunction/Algebra.lean b/Mathlib/Topology/ContinuousMap/Algebra.lean
similarity index 99%
rename from Mathlib/Topology/ContinuousFunction/Algebra.lean
rename to Mathlib/Topology/ContinuousMap/Algebra.lean
index c3143c043f066..4e62f17e049be 100644
--- a/Mathlib/Topology/ContinuousFunction/Algebra.lean
+++ b/Mathlib/Topology/ContinuousMap/Algebra.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Nicolò Cavalleri
+Authors: Kim Morrison, Nicolò Cavalleri
 -/
 import Mathlib.Algebra.Algebra.Pi
 import Mathlib.Algebra.Order.Group.Lattice
@@ -14,7 +14,7 @@ import Mathlib.Topology.Algebra.InfiniteSum.Basic
 import Mathlib.Topology.Algebra.Ring.Basic
 import Mathlib.Topology.Algebra.Star
 import Mathlib.Topology.Algebra.UniformGroup
-import Mathlib.Topology.ContinuousFunction.Ordered
+import Mathlib.Topology.ContinuousMap.Ordered
 import Mathlib.Topology.UniformSpace.CompactConvergence
 
 /-!
@@ -295,9 +295,9 @@ instance [LocallyCompactSpace α] [Mul β] [ContinuousMul β] : ContinuousMul C(
   ⟨by
     refine continuous_of_continuous_uncurry _ ?_
     have h1 : Continuous fun x : (C(α, β) × C(α, β)) × α => x.fst.fst x.snd :=
-      continuous_eval.comp (continuous_fst.prod_map continuous_id)
+      continuous_eval.comp (continuous_fst.prodMap continuous_id)
     have h2 : Continuous fun x : (C(α, β) × C(α, β)) × α => x.fst.snd x.snd :=
-      continuous_eval.comp (continuous_snd.prod_map continuous_id)
+      continuous_eval.comp (continuous_snd.prodMap continuous_id)
     exact h1.mul h2⟩
 
 /-- Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. -/
@@ -550,7 +550,7 @@ instance [LocallyCompactSpace α] [TopologicalSpace R] [SMul R M] [ContinuousSMu
   ⟨by
     refine continuous_of_continuous_uncurry _ ?_
     have h : Continuous fun x : (R × C(α, M)) × α => x.fst.snd x.snd :=
-      continuous_eval.comp (continuous_snd.prod_map continuous_id)
+      continuous_eval.comp (continuous_snd.prodMap continuous_id)
     exact (continuous_fst.comp continuous_fst).smul h⟩
 
 @[to_additive (attr := simp, norm_cast)]
diff --git a/Mathlib/Topology/ContinuousFunction/Basic.lean b/Mathlib/Topology/ContinuousMap/Basic.lean
similarity index 77%
rename from Mathlib/Topology/ContinuousFunction/Basic.lean
rename to Mathlib/Topology/ContinuousMap/Basic.lean
index b24e4f2ee999c..bd8aca0de948b 100644
--- a/Mathlib/Topology/ContinuousFunction/Basic.lean
+++ b/Mathlib/Topology/ContinuousMap/Basic.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Nicolò Cavalleri
 -/
 import Mathlib.Data.Set.UnionLift
+import Mathlib.Topology.ContinuousMap.Defs
 import Mathlib.Topology.Homeomorph
 
 /-!
@@ -19,37 +20,6 @@ be satisfied by itself and all stricter types.
 open Function
 open scoped Topology
 
-/-- The type of continuous maps from `α` to `β`.
-
-When possible, instead of parametrizing results over `(f : C(α, β))`,
-you should parametrize over `{F : Type*} [ContinuousMapClass F α β] (f : F)`.
-
-When you extend this structure, make sure to extend `ContinuousMapClass`. -/
-structure ContinuousMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
-  /-- The function `α → β` -/
-  protected toFun : α → β
-  /-- Proposition that `toFun` is continuous -/
-  protected continuous_toFun : Continuous toFun := by continuity
-
-/-- The type of continuous maps from `α` to `β`. -/
-notation "C(" α ", " β ")" => ContinuousMap α β
-
-section
-
-/-- `ContinuousMapClass F α β` states that `F` is a type of continuous maps.
-
-You should extend this class when you extend `ContinuousMap`. -/
-class ContinuousMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
-    [FunLike F α β] : Prop where
-  /-- Continuity -/
-  map_continuous (f : F) : Continuous f
-
-end
-
-export ContinuousMapClass (map_continuous)
-
-attribute [continuity, fun_prop] map_continuous
-
 section ContinuousMapClass
 
 variable {F α β : Type*} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β]
@@ -61,14 +31,9 @@ theorem map_continuousAt (f : F) (a : α) : ContinuousAt f a :=
 theorem map_continuousWithinAt (f : F) (s : Set α) (a : α) : ContinuousWithinAt f s a :=
   (map_continuous f).continuousWithinAt
 
-/-- Coerce a bundled morphism with a `ContinuousMapClass` instance to a `ContinuousMap`. -/
-@[coe] def toContinuousMap (f : F) : C(α, β) := ⟨f, map_continuous f⟩
-
-instance : CoeTC F C(α, β) := ⟨toContinuousMap⟩
-
 end ContinuousMapClass
 
-/-! ### Continuous maps-/
+/-! ### Continuous maps -/
 
 
 namespace ContinuousMap
@@ -76,75 +41,11 @@ namespace ContinuousMap
 variable {α β γ δ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
   [TopologicalSpace δ]
 
-instance funLike : FunLike C(α, β) α β where
-  coe := ContinuousMap.toFun
-  coe_injective' f g h := by cases f; cases g; congr
-
-instance toContinuousMapClass : ContinuousMapClass C(α, β) α β where
-  map_continuous := ContinuousMap.continuous_toFun
-
-@[simp]
-theorem toFun_eq_coe {f : C(α, β)} : f.toFun = (f : α → β) :=
-  rfl
-
-instance : CanLift (α → β) C(α, β) DFunLike.coe Continuous := ⟨fun f hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩
-
-/-- See note [custom simps projection]. -/
-def Simps.apply (f : C(α, β)) : α → β := f
-
--- this must come after the coe_to_fun definition
-initialize_simps_projections ContinuousMap (toFun → apply)
-
-@[simp] -- Porting note: removed `norm_cast` attribute
-protected theorem coe_coe {F : Type*} [FunLike F α β] [ContinuousMapClass F α β] (f : F) :
-    ⇑(f : C(α, β)) = f :=
-  rfl
-
-@[ext]
-theorem ext {f g : C(α, β)} (h : ∀ a, f a = g a) : f = g :=
-  DFunLike.ext _ _ h
-
-/-- Copy of a `ContinuousMap` with a new `toFun` equal to the old one. Useful to fix definitional
-equalities. -/
-protected def copy (f : C(α, β)) (f' : α → β) (h : f' = f) : C(α, β) where
-  toFun := f'
-  continuous_toFun := h.symm ▸ f.continuous_toFun
-
-@[simp]
-theorem coe_copy (f : C(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' :=
-  rfl
-
-theorem copy_eq (f : C(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f :=
-  DFunLike.ext' h
-
 variable {f g : C(α, β)}
 
-/-- Deprecated. Use `map_continuous` instead. -/
-protected theorem continuous (f : C(α, β)) : Continuous f :=
-  f.continuous_toFun
-
-@[continuity]
-theorem continuous_set_coe (s : Set C(α, β)) (f : s) : Continuous (f : α → β) :=
-  f.1.continuous
-
 /-- Deprecated. Use `map_continuousAt` instead. -/
 protected theorem continuousAt (f : C(α, β)) (x : α) : ContinuousAt f x :=
-  f.continuous.continuousAt
-
-/-- Deprecated. Use `DFunLike.congr_fun` instead. -/
-protected theorem congr_fun {f g : C(α, β)} (H : f = g) (x : α) : f x = g x :=
-  H ▸ rfl
-
-/-- Deprecated. Use `DFunLike.congr_arg` instead. -/
-protected theorem congr_arg (f : C(α, β)) {x y : α} (h : x = y) : f x = f y :=
-  h ▸ rfl
-
-theorem coe_injective : @Function.Injective C(α, β) (α → β) (↑) := fun f g h => by
-  cases f; cases g; congr
-
-@[simp]
-theorem coe_mk (f : α → β) (h : Continuous f) : ⇑(⟨f, h⟩ : C(α, β)) = f :=
-  rfl
+  map_continuousAt f x
 
 theorem map_specializes (f : C(α, β)) {x y : α} (h : x ⤳ y) : f x ⤳ f y :=
   h.map f.2
@@ -271,8 +172,6 @@ def prodMk (f : C(α, β₁)) (g : C(α, β₂)) : C(α, β₁ × β₂) where
 @[simps]
 def prodMap (f : C(α₁, α₂)) (g : C(β₁, β₂)) : C(α₁ × β₁, α₂ × β₂) where
   toFun := Prod.map f g
-  continuous_toFun := f.continuous.prod_map g.continuous
-  -- Porting note: proof was `continuity`
 
 @[simp]
 theorem prod_eval (f : C(α, β₁)) (g : C(α, β₂)) (a : α) : (prodMk f g) a = (f a, g a) :=
@@ -304,7 +203,7 @@ def sigma (f : ∀ i, C(X i, A)) : C((Σ i, X i), A) where
 variable (A X) in
 /--
 Giving a continuous map out of a disjoint union is the same as giving a continuous map out of
-each term. This is a version of `Equiv.piCurry` for continuous maps.
+each term. This is a version of `Equiv.piCurry` for continuous maps.
 -/
 @[simps]
 def sigmaEquiv : (∀ i, C(X i, A)) ≃ C((Σ i, X i), A) where
@@ -380,13 +279,14 @@ theorem restrict_apply_mk (f : C(α, β)) (s : Set α) (x : α) (hx : x ∈ s) :
 
 theorem injective_restrict [T2Space β] {s : Set α} (hs : Dense s) :
     Injective (restrict s : C(α, β) → C(s, β)) := fun f g h ↦
-  DFunLike.ext' <| f.continuous.ext_on hs g.continuous <| Set.restrict_eq_restrict_iff.1 <|
-    congr_arg DFunLike.coe h
+  DFunLike.ext' <| (map_continuous f).ext_on hs (map_continuous g) <|
+    Set.restrict_eq_restrict_iff.1 <| congr_arg DFunLike.coe h
 
 /-- The restriction of a continuous map to the preimage of a set. -/
 @[simps]
 def restrictPreimage (f : C(α, β)) (s : Set β) : C(f ⁻¹' s, s) :=
-  ⟨s.restrictPreimage f, continuous_iff_continuousAt.mpr fun _ => f.2.continuousAt.restrictPreimage⟩
+  ⟨s.restrictPreimage f, continuous_iff_continuousAt.mpr fun _ ↦
+    (map_continuousAt f _).restrictPreimage⟩
 
 end Restrict
 
@@ -404,8 +304,8 @@ noncomputable def liftCover : C(α, β) :=
     Set.iUnion_eq_univ_iff.2 fun x ↦ (hS x).imp fun _ ↦ mem_of_mem_nhds
   mk (Set.liftCover S (fun i ↦ φ i) hφ H) <| continuous_of_cover_nhds hS fun i ↦ by
     rw [continuousOn_iff_continuous_restrict]
-    simpa (config := { unfoldPartialApp := true }) only [Set.restrict, Set.liftCover_coe] using
-      (φ i).continuous
+    simpa (config := { unfoldPartialApp := true }) only [Set.restrict, Set.liftCover_coe]
+      using map_continuous (φ i)
 
 variable {S φ hφ hS}
 
@@ -463,8 +363,8 @@ variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalS
 def Function.RightInverse.homeomorph {f' : C(Y, X)} (hf : Function.RightInverse f' f) :
     Quotient (Setoid.ker f) ≃ₜ Y where
   toEquiv := Setoid.quotientKerEquivOfRightInverse _ _ hf
-  continuous_toFun := quotientMap_quot_mk.continuous_iff.mpr f.continuous
-  continuous_invFun := continuous_quotient_mk'.comp f'.continuous
+  continuous_toFun := quotientMap_quot_mk.continuous_iff.mpr (map_continuous f)
+  continuous_invFun := continuous_quotient_mk'.comp (map_continuous f')
 
 namespace QuotientMap
 
@@ -494,7 +394,7 @@ noncomputable def lift : C(Y, Z) where
   continuous_toFun := Continuous.comp (continuous_quot_lift _ g.2) (Homeomorph.continuous _)
 
 /--
-The obvious triangle induced by `QuotientMap.lift` commutes:
+The obvious triangle induced by `QuotientMap.lift` commutes:
 ```
      g
   X --→ Z
diff --git a/Mathlib/Topology/ContinuousFunction/Bounded.lean b/Mathlib/Topology/ContinuousMap/Bounded.lean
similarity index 99%
rename from Mathlib/Topology/ContinuousFunction/Bounded.lean
rename to Mathlib/Topology/ContinuousMap/Bounded.lean
index 80e9caca4ffa6..01840e85e6d01 100644
--- a/Mathlib/Topology/ContinuousFunction/Bounded.lean
+++ b/Mathlib/Topology/ContinuousMap/Bounded.lean
@@ -4,12 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Mario Carneiro, Yury Kudryashov, Heather Macbeth
 -/
 import Mathlib.Algebra.Module.MinimalAxioms
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Analysis.Normed.Order.Lattice
 import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
 import Mathlib.Analysis.CStarAlgebra.Basic
-import Mathlib.Analysis.Normed.Operator.ContinuousLinearMap
 import Mathlib.Topology.Bornology.BoundedOperation
+import Mathlib.Tactic.Monotonicity
 
 /-!
 # Bounded continuous functions
@@ -246,7 +246,7 @@ theorem inducing_coeFn : Inducing (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) 
     UniformFun.tendsto_iff_tendstoUniformly]
   simp [comp_def]
 
--- TODO: upgrade to a `UniformEmbedding`
+-- TODO: upgrade to a `IsUniformEmbedding`
 theorem embedding_coeFn : Embedding (UniformFun.ofFun ∘ (⇑) : (α →ᵇ β) → α →ᵤ β) :=
   ⟨inducing_coeFn, fun _ _ h => ext fun x => congr_fun h x⟩
 
@@ -524,7 +524,7 @@ theorem arzela_ascoli₂ (s : Set β) (hs : IsCompact s) (A : Set (α →ᵇ β)
       fun f hf => ?_
   · haveI : CompactSpace s := isCompact_iff_compactSpace.1 hs
     refine arzela_ascoli₁ _ (continuous_iff_isClosed.1 (continuous_comp M) _ closed) ?_
-    rw [uniformEmbedding_subtype_val.toUniformInducing.equicontinuous_iff]
+    rw [isUniformEmbedding_subtype_val.isUniformInducing.equicontinuous_iff]
     exact H.comp (A.restrictPreimage F)
   · let g := codRestrict s f fun x => in_s f x hf
     rw [show f = F g by ext; rfl] at hf ⊢
@@ -1513,3 +1513,5 @@ lemma norm_sub_nonneg (f : α →ᵇ ℝ) :
 end
 
 end BoundedContinuousFunction
+
+set_option linter.style.longFile 1700
diff --git a/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean b/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean
new file mode 100644
index 0000000000000..ff7efe16ef944
--- /dev/null
+++ b/Mathlib/Topology/ContinuousMap/BoundedCompactlySupported.lean
@@ -0,0 +1,104 @@
+/-
+Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yoh Tanimoto
+-/
+import Mathlib.Topology.ContinuousMap.Bounded
+import Mathlib.RingTheory.TwoSidedIdeal.Lattice
+
+/-!
+# Compactly supported bounded continuous functions
+
+The two-sided ideal of compactly supported bounded continuous functions taking values in a metric
+space, with the uniform distance.
+-/
+
+open Set BoundedContinuousFunction
+
+section CompactlySupported
+
+/-- The two-sided ideal of compactly supported functions. -/
+def compactlySupported (α γ : Type*) [TopologicalSpace α] [NonUnitalNormedRing γ] :
+    TwoSidedIdeal (α →ᵇ γ) :=
+  .mk' {z | HasCompactSupport z} .zero .add .neg' .mul_left .mul_right
+
+variable {α γ : Type*} [TopologicalSpace α] [NonUnitalNormedRing γ]
+
+@[inherit_doc]
+scoped[BoundedContinuousFunction] notation
+  "C_cb(" α ", " γ ")" => compactlySupported α γ
+
+lemma mem_compactlySupported {f : α →ᵇ γ} :
+    f ∈ C_cb(α, γ) ↔ HasCompactSupport f :=
+  TwoSidedIdeal.mem_mk' {z : α →ᵇ γ | HasCompactSupport z} .zero .add .neg' .mul_left .mul_right f
+
+lemma exist_norm_eq [c : Nonempty α] {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) : ∃ (x : α),
+    ‖f x‖ = ‖f‖ := by
+  by_cases hs : (tsupport f).Nonempty
+  · obtain ⟨x, _, hmax⟩ := mem_compactlySupported.mp h |>.exists_isMaxOn hs <|
+      (map_continuous f).norm.continuousOn
+    refine ⟨x, le_antisymm (norm_coe_le_norm f x) (norm_le (norm_nonneg _) |>.mpr fun y ↦ ?_)⟩
+    by_cases hy : y ∈ tsupport f
+    · exact hmax hy
+    · simp [image_eq_zero_of_nmem_tsupport hy]
+  · suffices f = 0 by simp [this]
+    rwa [not_nonempty_iff_eq_empty, tsupport_eq_empty_iff, ← coe_zero, ← DFunLike.ext'_iff] at hs
+
+theorem norm_lt_iff_of_compactlySupported {f : α →ᵇ γ} (h : f ∈ C_cb(α, γ)) {M : ℝ}
+    (M0 : 0 < M) : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M := by
+  refine ⟨fun hn x ↦ lt_of_le_of_lt (norm_coe_le_norm f x) hn, ?_⟩
+  · obtain (he | he) := isEmpty_or_nonempty α
+    · simpa
+    · obtain ⟨x, hx⟩ := exist_norm_eq h
+      exact fun h ↦ hx ▸ h x
+
+theorem norm_lt_iff_of_nonempty_compactlySupported [c : Nonempty α] {f : α →ᵇ γ}
+    (h : f ∈ C_cb(α, γ)) {M : ℝ} : ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M := by
+  obtain (hM | hM) := lt_or_le 0 M
+  · exact norm_lt_iff_of_compactlySupported h hM
+  · exact ⟨fun h ↦ False.elim <| (h.trans_le hM).not_le (by positivity),
+      fun h ↦ False.elim <| (h (Classical.arbitrary α) |>.trans_le hM).not_le (by positivity)⟩
+
+theorem compactlySupported_eq_top_of_isCompact (h : IsCompact (Set.univ : Set α)) :
+    C_cb(α, γ) = ⊤ :=
+  eq_top_iff.mpr fun _ _ ↦ h.of_isClosed_subset (isClosed_tsupport _) (subset_univ _)
+
+/- This is intentionally not marked `@[simp]` to prevent Lean looking for a `CompactSpace α`
+instance every time it sees `C_cb(α, γ)`. -/
+theorem compactlySupported_eq_top [CompactSpace α] : C_cb(α, γ) = ⊤ :=
+  compactlySupported_eq_top_of_isCompact CompactSpace.isCompact_univ
+
+theorem compactlySupported_eq_top_iff [Nontrivial γ] :
+    C_cb(α, γ) = ⊤ ↔ IsCompact (Set.univ : Set α) := by
+  refine ⟨fun h ↦ ?_, compactlySupported_eq_top_of_isCompact⟩
+  obtain ⟨x, hx⟩ := exists_ne (0 : γ)
+  simpa [tsupport, Function.support_const hx]
+    using (mem_compactlySupported (f := const α x).mp (by simp [h])).isCompact
+
+/-- A compactly supported continuous function is automatically bounded. This constructor gives
+an object of `α →ᵇ γ` from `g : α → γ` and these assumptions. -/
+def ofCompactSupport (g : α → γ) (hg₁ : Continuous g) (hg₂ : HasCompactSupport g) : α →ᵇ γ where
+  toFun := g
+  continuous_toFun := hg₁
+  map_bounded' := by
+    obtain (hs | hs) := (tsupport g).eq_empty_or_nonempty
+    · exact ⟨0, by simp [tsupport_eq_empty_iff.mp hs]⟩
+    · obtain ⟨z, _, hmax⟩ := hg₂.exists_isMaxOn hs <| hg₁.norm.continuousOn
+      refine ⟨2 * ‖g z‖, dist_le_two_norm' fun x ↦ ?_⟩
+      by_cases hx : x ∈ tsupport g
+      · exact isMaxOn_iff.mp hmax x hx
+      · simp [image_eq_zero_of_nmem_tsupport hx]
+
+lemma ofCompactSupport_mem (g : α → γ) (hg₁ : Continuous g) (hg₂ : HasCompactSupport g) :
+    ofCompactSupport g hg₁ hg₂ ∈ C_cb(α, γ) := mem_compactlySupported.mpr hg₂
+
+instance : SMul C(α, γ) C_cb(α, γ) where
+  smul := fun (g : C(α, γ)) => (fun (f : C_cb(α, γ)) =>
+    ⟨ofCompactSupport (g * (f : α →ᵇ γ) : α → γ) (Continuous.mul g.2 f.1.1.2)
+    (HasCompactSupport.mul_left (mem_compactlySupported.mp f.2)), by
+      apply mem_compactlySupported.mpr
+      rw [ofCompactSupport]
+      exact HasCompactSupport.mul_left <| mem_compactlySupported.mp f.2
+    ⟩)
+
+end CompactlySupported
diff --git a/Mathlib/Topology/ContinuousFunction/CocompactMap.lean b/Mathlib/Topology/ContinuousMap/CocompactMap.lean
similarity index 99%
rename from Mathlib/Topology/ContinuousFunction/CocompactMap.lean
rename to Mathlib/Topology/ContinuousMap/CocompactMap.lean
index 8794fe4397f20..d5662c0299843 100644
--- a/Mathlib/Topology/ContinuousFunction/CocompactMap.lean
+++ b/Mathlib/Topology/ContinuousMap/CocompactMap.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # Cocompact continuous maps
diff --git a/Mathlib/Topology/ContinuousFunction/Compact.lean b/Mathlib/Topology/ContinuousMap/Compact.lean
similarity index 78%
rename from Mathlib/Topology/ContinuousFunction/Compact.lean
rename to Mathlib/Topology/ContinuousMap/Compact.lean
index 4d0a46cb346b4..ca3851110e939 100644
--- a/Mathlib/Topology/ContinuousFunction/Compact.lean
+++ b/Mathlib/Topology/ContinuousMap/Compact.lean
@@ -1,9 +1,9 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 import Mathlib.Topology.UniformSpace.Compact
 import Mathlib.Topology.CompactOpen
 import Mathlib.Topology.Sets.Compacts
@@ -29,8 +29,8 @@ open NNReal BoundedContinuousFunction Set Metric
 
 namespace ContinuousMap
 
-variable {α β E : Type*} [TopologicalSpace α] [CompactSpace α] [MetricSpace β]
-  [NormedAddCommGroup E]
+variable {α β E : Type*}
+variable [TopologicalSpace α] [CompactSpace α] [PseudoMetricSpace β] [SeminormedAddCommGroup E]
 
 section
 
@@ -47,8 +47,8 @@ def equivBoundedOfCompact : C(α, β) ≃ (α →ᵇ β) :=
     ext
     rfl⟩
 
-theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfCompact α β) :=
-  UniformInducing.mk'
+theorem isUniformInducing_equivBoundedOfCompact : IsUniformInducing (equivBoundedOfCompact α β) :=
+  IsUniformInducing.mk'
     (by
       simp only [hasBasis_compactConvergenceUniformity.mem_iff, uniformity_basis_dist_le.mem_iff]
       exact fun s =>
@@ -59,8 +59,15 @@ theorem uniformInducing_equivBoundedOfCompact : UniformInducing (equivBoundedOfC
           ⟨⟨Set.univ, { p | dist p.1 p.2 ≤ ε }⟩, ⟨isCompact_univ, ⟨ε, hε, fun _ h => h⟩⟩,
             fun ⟨f, g⟩ h => hs _ _ (ht ((dist_le hε.le).mpr fun x => h x (mem_univ x)))⟩⟩)
 
-theorem uniformEmbedding_equivBoundedOfCompact : UniformEmbedding (equivBoundedOfCompact α β) :=
-  { uniformInducing_equivBoundedOfCompact α β with inj := (equivBoundedOfCompact α β).injective }
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_equivBoundedOfCompact := isUniformInducing_equivBoundedOfCompact
+
+theorem isUniformEmbedding_equivBoundedOfCompact : IsUniformEmbedding (equivBoundedOfCompact α β) :=
+  { isUniformInducing_equivBoundedOfCompact α β with
+    inj := (equivBoundedOfCompact α β).injective }
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_equivBoundedOfCompact := isUniformEmbedding_equivBoundedOfCompact
 
 /-- When `α` is compact, the bounded continuous maps `α →ᵇ 𝕜` are
 additively equivalent to `C(α, 𝕜)`.
@@ -82,8 +89,13 @@ theorem addEquivBoundedOfCompact_apply [AddMonoid β] [LipschitzAdd β] :
     ⇑(addEquivBoundedOfCompact α β) = mkOfCompact :=
   rfl
 
-instance metricSpace : MetricSpace C(α, β) :=
-  (uniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
+instance instPseudoMetricSpace : PseudoMetricSpace C(α, β) :=
+  (isUniformEmbedding_equivBoundedOfCompact α β).comapPseudoMetricSpace _
+
+instance instMetricSpace {β : Type*} [MetricSpace β] :
+    MetricSpace C(α, β) :=
+  (isUniformEmbedding_equivBoundedOfCompact α β).comapMetricSpace _
+
 
 /-- When `α` is compact, and `β` is a metric space, the bounded continuous maps `α →ᵇ β` are
 isometric to `C(α, β)`.
@@ -133,6 +145,13 @@ theorem dist_lt_iff (C0 : (0 : ℝ) < C) : dist f g < C ↔ ∀ x : α, dist (f
   rw [← dist_mkOfCompact, dist_lt_iff_of_compact C0]
   simp only [mkOfCompact_apply]
 
+instance {R} [Zero R] [Zero β] [PseudoMetricSpace R] [SMul R β] [BoundedSMul R β] :
+    BoundedSMul R C(α, β) where
+  dist_smul_pair' r f g := by
+    simpa only [← dist_mkOfCompact] using dist_smul_pair r (mkOfCompact f) (mkOfCompact g)
+  dist_pair_smul' r₁ r₂ f := by
+    simpa only [← dist_mkOfCompact] using dist_pair_smul r₁ r₂ (mkOfCompact f)
+
 end
 
 -- TODO at some point we will need lemmas characterising this norm!
@@ -150,13 +169,17 @@ theorem _root_.BoundedContinuousFunction.norm_toContinuousMap_eq (f : α →ᵇ
 
 open BoundedContinuousFunction
 
-instance : NormedAddCommGroup C(α, E) :=
-  { ContinuousMap.metricSpace _ _,
-    ContinuousMap.instAddCommGroupContinuousMap with
-    dist_eq := fun x y => by
-      rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
-    dist := dist
-    norm := norm }
+instance : SeminormedAddCommGroup C(α, E) where
+  __ := ContinuousMap.instPseudoMetricSpace _ _
+  __ := ContinuousMap.instAddCommGroupContinuousMap
+  dist_eq x y := by
+    rw [← norm_mkOfCompact, ← dist_mkOfCompact, dist_eq_norm, mkOfCompact_sub]
+  dist := dist
+  norm := norm
+
+instance {E : Type*} [NormedAddCommGroup E] : NormedAddCommGroup C(α, E) where
+  __ : SeminormedAddCommGroup C(α, E) := inferInstance
+  __ : MetricSpace C(α, E) := inferInstance
 
 instance [Nonempty α] [One E] [NormOneClass E] : NormOneClass C(α, E) where
   norm_one := by simp only [← norm_mkOfCompact, mkOfCompact_one, norm_one]
@@ -202,6 +225,11 @@ theorem neg_norm_le_apply (f : C(α, ℝ)) (x : α) : -‖f‖ ≤ f x :=
 theorem norm_eq_iSup_norm : ‖f‖ = ⨆ x : α, ‖f x‖ :=
   (mkOfCompact f).norm_eq_iSup_norm
 
+-- A version with better keys
+instance {X : Type*} [TopologicalSpace X] (K : TopologicalSpace.Compacts X) :
+    CompactSpace (K : Set X) :=
+  TopologicalSpace.Compacts.instCompactSpaceSubtypeMem ..
+
 theorem norm_restrict_mono_set {X : Type*} [TopologicalSpace X] (f : C(X, E))
     {K L : TopologicalSpace.Compacts X} (hKL : K ≤ L) : ‖f.restrict K‖ ≤ ‖f.restrict L‖ :=
   (norm_le _ (norm_nonneg _)).mpr fun x => norm_coe_le_norm (f.restrict L) <| Set.inclusion hKL x
@@ -210,11 +238,40 @@ end
 
 section
 
-variable {R : Type*} [NormedRing R]
+variable {R : Type*}
+
+instance [NonUnitalSeminormedRing R] : NonUnitalSeminormedRing C(α, R) where
+  __ : SeminormedAddCommGroup C(α, R) := inferInstance
+  __ : NonUnitalRing C(α, R) := inferInstance
+  norm_mul f g := norm_mul_le (mkOfCompact f) (mkOfCompact g)
+
+instance [NonUnitalSeminormedCommRing R] : NonUnitalSeminormedCommRing C(α, R) where
+  __ : NonUnitalSeminormedRing C(α, R) := inferInstance
+  __ : NonUnitalCommRing C(α, R) := inferInstance
+
+instance [SeminormedRing R] : SeminormedRing C(α, R) where
+  __ : NonUnitalSeminormedRing C(α, R) := inferInstance
+  __ : Ring C(α, R) := inferInstance
+
+instance [SeminormedCommRing R] : SeminormedCommRing C(α, R) where
+  __ : SeminormedRing C(α, R) := inferInstance
+  __ : CommRing C(α, R) := inferInstance
+
+instance [NonUnitalNormedRing R] : NonUnitalNormedRing C(α, R) where
+  __ : NormedAddCommGroup C(α, R) := inferInstance
+  __ : NonUnitalSeminormedRing C(α, R) := inferInstance
+
+instance [NonUnitalNormedCommRing R] : NonUnitalNormedCommRing C(α, R) where
+  __ : NonUnitalNormedRing C(α, R) := inferInstance
+  __ : NonUnitalCommRing C(α, R) := inferInstance
+
+instance [NormedRing R] : NormedRing C(α, R) where
+  __ : NormedAddCommGroup C(α, R) := inferInstance
+  __ : SeminormedRing C(α, R) := inferInstance
 
-instance : NormedRing C(α, R) :=
-  { (inferInstance : NormedAddCommGroup C(α, R)), ContinuousMap.instRing with
-    norm_mul := fun f g => norm_mul_le (mkOfCompact f) (mkOfCompact g) }
+instance [NormedCommRing R] : NormedCommRing C(α, R) where
+  __ : NormedRing C(α, R) := inferInstance
+  __ : CommRing C(α, R) := inferInstance
 
 end
 
@@ -223,7 +280,7 @@ section
 variable {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 E]
 
 instance normedSpace : NormedSpace 𝕜 C(α, E) where
-  norm_smul_le c f := (norm_smul_le c (mkOfCompact f) : _)
+  norm_smul_le := norm_smul_le
 
 section
 
@@ -282,7 +339,7 @@ end
 
 section
 
-variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [NormedRing γ] [NormedAlgebra 𝕜 γ]
+variable {𝕜 : Type*} {γ : Type*} [NormedField 𝕜] [SeminormedRing γ] [NormedAlgebra 𝕜 γ]
 
 instance : NormedAlgebra 𝕜 C(α, γ) :=
   { ContinuousMap.normedSpace, ContinuousMap.algebra with }
@@ -296,7 +353,7 @@ namespace ContinuousMap
 section UniformContinuity
 
 variable {α β : Type*}
-variable [MetricSpace α] [CompactSpace α] [MetricSpace β]
+variable [PseudoMetricSpace α] [CompactSpace α] [PseudoMetricSpace β]
 
 /-!
 We now set up some declarations making it convenient to use uniform continuity.
@@ -330,7 +387,7 @@ section CompLeft
 variable (X : Type*) {𝕜 β γ : Type*} [TopologicalSpace X] [CompactSpace X]
   [NontriviallyNormedField 𝕜]
 
-variable [NormedAddCommGroup β] [NormedSpace 𝕜 β] [NormedAddCommGroup γ] [NormedSpace 𝕜 γ]
+variable [SeminormedAddCommGroup β] [NormedSpace 𝕜 β] [SeminormedAddCommGroup γ] [NormedSpace 𝕜 γ]
 
 open ContinuousMap
 
@@ -377,7 +434,8 @@ section CompRight
 /-- Precomposition by a continuous map is itself a continuous map between spaces of continuous maps.
 -/
 def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
-    [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y)) : C(C(Y, T), C(X, T)) where
+    [TopologicalSpace Y] [CompactSpace Y] [PseudoMetricSpace T] (f : C(X, Y)) :
+    C(C(Y, T), C(X, T)) where
   toFun g := g.comp f
   continuous_toFun := by
     refine Metric.continuous_iff.mpr ?_
@@ -388,14 +446,15 @@ def compRightContinuousMap {X Y : Type*} (T : Type*) [TopologicalSpace X] [Compa
 
 @[simp]
 theorem compRightContinuousMap_apply {X Y : Type*} (T : Type*) [TopologicalSpace X]
-    [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : C(X, Y))
+    [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [PseudoMetricSpace T] (f : C(X, Y))
     (g : C(Y, T)) : (compRightContinuousMap T f) g = g.comp f :=
   rfl
 
 /-- Precomposition by a homeomorphism is itself a homeomorphism between spaces of continuous maps.
 -/
 def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactSpace X]
-    [TopologicalSpace Y] [CompactSpace Y] [MetricSpace T] (f : X ≃ₜ Y) : C(Y, T) ≃ₜ C(X, T) where
+    [TopologicalSpace Y] [CompactSpace Y] [PseudoMetricSpace T] (f : X ≃ₜ Y) :
+    C(Y, T) ≃ₜ C(X, T) where
   toFun := compRightContinuousMap T f.toContinuousMap
   invFun := compRightContinuousMap T f.symm.toContinuousMap
   left_inv g := ext fun _ => congr_arg g (f.apply_symm_apply _)
@@ -403,7 +462,7 @@ def compRightHomeomorph {X Y : Type*} (T : Type*) [TopologicalSpace X] [CompactS
 
 theorem compRightAlgHom_continuous {X Y : Type*} (R A : Type*) [TopologicalSpace X]
     [CompactSpace X] [TopologicalSpace Y] [CompactSpace Y] [CommSemiring R] [Semiring A]
-    [MetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
+    [PseudoMetricSpace A] [TopologicalSemiring A] [Algebra R A] (f : C(X, Y)) :
     Continuous (compRightAlgHom R A f) :=
   map_continuous (compRightContinuousMap A f)
 
@@ -452,7 +511,7 @@ Furthermore, if `α` is compact and `β` is a C⋆-ring, then `C(α, β)` is a C
 section NormedSpace
 
 variable {α : Type*} {β : Type*}
-variable [TopologicalSpace α] [NormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
+variable [TopologicalSpace α] [SeminormedAddCommGroup β] [StarAddMonoid β] [NormedStarGroup β]
 
 theorem _root_.BoundedContinuousFunction.mkOfCompact_star [CompactSpace α] (f : C(α, β)) :
     mkOfCompact (star f) = star (mkOfCompact f) :=
@@ -468,7 +527,7 @@ end NormedSpace
 section CStarRing
 
 variable {α : Type*} {β : Type*}
-variable [TopologicalSpace α] [NormedRing β] [StarRing β]
+variable [TopologicalSpace α] [NonUnitalNormedRing β] [StarRing β]
 
 instance [CompactSpace α] [CStarRing β] : CStarRing C(α, β) where
   norm_mul_self_le f := by
diff --git a/Mathlib/Topology/ContinuousFunction/CompactlySupported.lean b/Mathlib/Topology/ContinuousMap/CompactlySupported.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/CompactlySupported.lean
rename to Mathlib/Topology/ContinuousMap/CompactlySupported.lean
index d8d72123235af..cea48f4937a02 100644
--- a/Mathlib/Topology/ContinuousFunction/CompactlySupported.lean
+++ b/Mathlib/Topology/ContinuousMap/CompactlySupported.lean
@@ -3,8 +3,8 @@ Copyright (c) 2024 Yoh Tanimoto. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yoh Tanimoto
 -/
-import Mathlib.Topology.ContinuousFunction.CocompactMap
-import Mathlib.Topology.ContinuousFunction.ZeroAtInfty
+import Mathlib.Topology.ContinuousMap.CocompactMap
+import Mathlib.Topology.ContinuousMap.ZeroAtInfty
 import Mathlib.Topology.Support
 
 /-!
@@ -119,7 +119,6 @@ theorem eq_of_empty [IsEmpty α] (f g : C_c(α, β)) : f = g :=
 def ContinuousMap.liftCompactlySupported [CompactSpace α] : C(α, β) ≃ C_c(α, β) where
   toFun f :=
     { toFun := f
-      continuous_toFun := f.continuous
       hasCompactSupport' := HasCompactSupport.of_compactSpace f }
   invFun f := f
   left_inv _ := rfl
@@ -168,7 +167,7 @@ theorem mul_apply [MulZeroClass β] [ContinuousMul β] (f g : C_c(α, β)) : (f
 instance [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
     {F : Type*} [FunLike F α γ] [ContinuousMapClass F α γ] : SMul F C_c(α, β) where
   smul f g :=
-    ⟨⟨fun x ↦ f x • g x, (map_continuous f).smul g.continuous⟩, g.hasCompactSupport'.smul_left⟩
+    ⟨⟨fun x ↦ f x • g x, (map_continuous f).smul (map_continuous g)⟩, g.hasCompactSupport.smul_left⟩
 
 @[simp]
 theorem coe_smulc [Zero β] [TopologicalSpace γ] [SMulZeroClass γ β] [ContinuousSMul γ β]
@@ -209,7 +208,7 @@ def coeFnMonoidHom [AddMonoid β] [ContinuousAdd β] : C_c(α, β) →+ α → 
 
 instance [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] :
     SMul R C_c(α, β) :=
-  ⟨fun r f => ⟨⟨r • ⇑f, Continuous.const_smul f.continuous r⟩, HasCompactSupport.smul_left f.2⟩⟩
+  ⟨fun r f => ⟨⟨r • ⇑f, (map_continuous f).const_smul r⟩, HasCompactSupport.smul_left f.2⟩⟩
 
 @[simp, norm_cast]
 theorem coe_smul [Zero β] {R : Type*} [SMulZeroClass R β] [ContinuousConstSMul R β] (r : R)
diff --git a/Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean b/Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean
similarity index 89%
rename from Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean
rename to Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean
index 76f9a54f3d595..5fa65e8b7c342 100644
--- a/Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean
+++ b/Mathlib/Topology/ContinuousMap/ContinuousMapZero.lean
@@ -3,8 +3,8 @@ Copyright (c) 2024 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathlib.Topology.ContinuousFunction.Algebra
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.Algebra
+import Mathlib.Topology.ContinuousMap.Compact
 
 /-!
 # Continuous maps sending zero to zero
@@ -95,6 +95,14 @@ lemma closedEmbedding_toContinuousMap [T1Space R] :
     rw [range_toContinuousMap]
     exact isClosed_singleton.preimage <| ContinuousMap.continuous_eval_const 0
 
+@[fun_prop]
+lemma continuous_comp_left {X Y Z : Type*} [TopologicalSpace X]
+    [TopologicalSpace Y] [TopologicalSpace Z] [Zero X] [Zero Y] [Zero Z] (f : C(X, Y)₀) :
+    Continuous fun g : C(Y, Z)₀ ↦ g.comp f := by
+  rw [continuous_induced_rng]
+  show Continuous fun g : C(Y, Z)₀ ↦ (g : C(Y, Z)).comp (f : C(X, Y))
+  fun_prop
+
 /-- The identity function as an element of `C(s, R)₀` when `0 ∈ (s : Set R)`. -/
 @[simps!]
 protected def id {s : Set R} [Zero s] (h0 : ((0 : s) : R) = 0) : C(s, R)₀ :=
@@ -260,20 +268,26 @@ variable [Zero R] [UniformSpace R]
 
 protected instance instUniformSpace : UniformSpace C(X, R)₀ := .comap toContinuousMap inferInstance
 
-lemma uniformEmbedding_toContinuousMap :
-    UniformEmbedding ((↑) : C(X, R)₀ → C(X, R)) where
+lemma isUniformEmbedding_toContinuousMap :
+    IsUniformEmbedding ((↑) : C(X, R)₀ → C(X, R)) where
   comap_uniformity := rfl
   inj _ _ h := ext fun x ↦ congr($(h) x)
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toContinuousMap := isUniformEmbedding_toContinuousMap
+
 instance [T1Space R] [CompleteSpace C(X, R)] : CompleteSpace C(X, R)₀ :=
-  completeSpace_iff_isComplete_range uniformEmbedding_toContinuousMap.toUniformInducing
+  completeSpace_iff_isComplete_range isUniformEmbedding_toContinuousMap.isUniformInducing
     |>.mpr closedEmbedding_toContinuousMap.isClosed_range.isComplete
 
-lemma uniformEmbedding_comp {Y : Type*} [UniformSpace Y] [Zero Y] (g : C(Y, R)₀)
-    (hg : UniformEmbedding g) : UniformEmbedding (g.comp · : C(X, Y)₀ → C(X, R)₀) :=
-  uniformEmbedding_toContinuousMap.of_comp_iff.mp <|
-    ContinuousMap.uniformEmbedding_comp g.toContinuousMap hg |>.comp
-      uniformEmbedding_toContinuousMap
+lemma isUniformEmbedding_comp {Y : Type*} [UniformSpace Y] [Zero Y] (g : C(Y, R)₀)
+    (hg : IsUniformEmbedding g) : IsUniformEmbedding (g.comp · : C(X, Y)₀ → C(X, R)₀) :=
+  isUniformEmbedding_toContinuousMap.of_comp_iff.mp <|
+    ContinuousMap.isUniformEmbedding_comp g.toContinuousMap hg |>.comp
+      isUniformEmbedding_toContinuousMap
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_comp := isUniformEmbedding_comp
 
 /-- The uniform equivalence `C(X, R)₀ ≃ᵤ C(Y, R)₀` induced by a homeomorphism of the domains
 sending `0 : X` to `0 : Y`. -/
@@ -283,12 +297,12 @@ def _root_.UniformEquiv.arrowCongrLeft₀ {Y : Type*} [TopologicalSpace Y] [Zero
   invFun g := g.comp ⟨f.toContinuousMap, hf⟩
   left_inv g := ext fun _ ↦ congrArg g <| f.left_inv _
   right_inv g := ext fun _ ↦ congrArg g <| f.right_inv _
-  uniformContinuous_toFun := uniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
+  uniformContinuous_toFun := isUniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
     ContinuousMap.uniformContinuous_comp_left f.symm.toContinuousMap |>.comp
-    uniformEmbedding_toContinuousMap.uniformContinuous
-  uniformContinuous_invFun := uniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
+    isUniformEmbedding_toContinuousMap.uniformContinuous
+  uniformContinuous_invFun := isUniformEmbedding_toContinuousMap.uniformContinuous_iff.mpr <|
     ContinuousMap.uniformContinuous_comp_left f.toContinuousMap |>.comp
-    uniformEmbedding_toContinuousMap.uniformContinuous
+    isUniformEmbedding_toContinuousMap.uniformContinuous
 
 end UniformSpace
 
@@ -332,7 +346,7 @@ section Norm
 variable {α : Type*} {𝕜 : Type*} {R : Type*} [TopologicalSpace α] [CompactSpace α] [Zero α]
 
 noncomputable instance [MetricSpace R] [Zero R]: MetricSpace C(α, R)₀ :=
-  ContinuousMapZero.uniformEmbedding_toContinuousMap.comapMetricSpace _
+  ContinuousMapZero.isUniformEmbedding_toContinuousMap.comapMetricSpace _
 
 noncomputable instance [NormedAddCommGroup R] : Norm C(α, R)₀ where
   norm f := ‖(f : C(α, R))‖
diff --git a/Mathlib/Topology/ContinuousMap/Defs.lean b/Mathlib/Topology/ContinuousMap/Defs.lean
new file mode 100644
index 0000000000000..26742ca37a62e
--- /dev/null
+++ b/Mathlib/Topology/ContinuousMap/Defs.lean
@@ -0,0 +1,139 @@
+/-
+Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Nicolò Cavalleri, Yury Kudryashov
+-/
+import Mathlib.Tactic.Continuity
+import Mathlib.Tactic.Lift
+import Mathlib.Topology.Defs.Basic
+
+/-!
+# Continuous bundled maps
+
+In this file we define the type `ContinuousMap` of continuous bundled maps.
+
+We use the `DFunLike` design, so each type of morphisms has a companion typeclass
+which is meant to be satisfied by itself and all stricter types.
+-/
+
+open Function
+open scoped Topology
+
+/-- The type of continuous maps from `X` to `Y`.
+
+When possible, instead of parametrizing results over `(f : C(X, Y))`,
+you should parametrize over `{F : Type*} [ContinuousMapClass F X Y] (f : F)`.
+
+When you extend this structure, make sure to extend `ContinuousMapClass`. -/
+structure ContinuousMap (X Y : Type*) [TopologicalSpace X] [TopologicalSpace Y] where
+  /-- The function `X → Y` -/
+  protected toFun : X → Y
+  /-- Proposition that `toFun` is continuous -/
+  protected continuous_toFun : Continuous toFun := by continuity
+
+/-- The type of continuous maps from `X` to `Y`. -/
+notation "C(" X ", " Y ")" => ContinuousMap X Y
+
+section
+
+/-- `ContinuousMapClass F X Y` states that `F` is a type of continuous maps.
+
+You should extend this class when you extend `ContinuousMap`. -/
+class ContinuousMapClass (F : Type*) (X Y : outParam Type*)
+    [TopologicalSpace X] [TopologicalSpace Y] [FunLike F X Y] : Prop where
+  /-- Continuity -/
+  map_continuous (f : F) : Continuous f
+
+end
+
+export ContinuousMapClass (map_continuous)
+
+attribute [continuity, fun_prop] map_continuous
+
+section ContinuousMapClass
+
+variable {F X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] [FunLike F X Y]
+variable [ContinuousMapClass F X Y]
+
+/-- Coerce a bundled morphism with a `ContinuousMapClass` instance to a `ContinuousMap`. -/
+@[coe] def toContinuousMap (f : F) : C(X, Y) := ⟨f, map_continuous f⟩
+
+instance : CoeTC F C(X, Y) := ⟨toContinuousMap⟩
+
+end ContinuousMapClass
+
+/-! ### Continuous maps -/
+
+
+namespace ContinuousMap
+
+variable {X Y γ δ : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace γ]
+  [TopologicalSpace δ]
+
+instance instFunLike : FunLike C(X, Y) X Y where
+  coe := ContinuousMap.toFun
+  coe_injective' f g h := by cases f; cases g; congr
+
+instance instContinuousMapClass : ContinuousMapClass C(X, Y) X Y where
+  map_continuous := ContinuousMap.continuous_toFun
+
+@[simp]
+theorem toFun_eq_coe {f : C(X, Y)} : f.toFun = (f : X → Y) :=
+  rfl
+
+instance : CanLift (X → Y) C(X, Y) DFunLike.coe Continuous := ⟨fun f hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩
+
+/-- See note [custom simps projection]. -/
+def Simps.apply (f : C(X, Y)) : X → Y := f
+
+-- this must come after the coe_to_fun definition
+initialize_simps_projections ContinuousMap (toFun → apply)
+
+@[simp] -- Porting note: removed `norm_cast` attribute
+protected theorem coe_coe {F : Type*} [FunLike F X Y] [ContinuousMapClass F X Y] (f : F) :
+    ⇑(f : C(X, Y)) = f :=
+  rfl
+
+@[ext]
+theorem ext {f g : C(X, Y)} (h : ∀ a, f a = g a) : f = g :=
+  DFunLike.ext _ _ h
+
+/-- Copy of a `ContinuousMap` with a new `toFun` equal to the old one. Useful to fix definitional
+equalities. -/
+protected def copy (f : C(X, Y)) (f' : X → Y) (h : f' = f) : C(X, Y) where
+  toFun := f'
+  continuous_toFun := h.symm ▸ f.continuous_toFun
+
+@[simp]
+theorem coe_copy (f : C(X, Y)) (f' : X → Y) (h : f' = f) : ⇑(f.copy f' h) = f' :=
+  rfl
+
+theorem copy_eq (f : C(X, Y)) (f' : X → Y) (h : f' = f) : f.copy f' h = f :=
+  DFunLike.ext' h
+
+variable {f g : C(X, Y)}
+
+/-- Deprecated. Use `map_continuous` instead. -/
+protected theorem continuous (f : C(X, Y)) : Continuous f :=
+  f.continuous_toFun
+
+@[deprecated map_continuous (since := "2024-09-29")]
+theorem continuous_set_coe (s : Set C(X, Y)) (f : s) : Continuous (f : X → Y) :=
+  map_continuous _
+
+/-- Deprecated. Use `DFunLike.congr_fun` instead. -/
+protected theorem congr_fun {f g : C(X, Y)} (H : f = g) (x : X) : f x = g x :=
+  H ▸ rfl
+
+/-- Deprecated. Use `DFunLike.congr_arg` instead. -/
+protected theorem congr_arg (f : C(X, Y)) {x y : X} (h : x = y) : f x = f y :=
+  h ▸ rfl
+
+theorem coe_injective : Function.Injective (DFunLike.coe : C(X, Y) → (X → Y)) :=
+  DFunLike.coe_injective
+
+@[simp]
+theorem coe_mk (f : X → Y) (h : Continuous f) : ⇑(⟨f, h⟩ : C(X, Y)) = f :=
+  rfl
+
+end ContinuousMap
diff --git a/Mathlib/Topology/ContinuousFunction/Ideals.lean b/Mathlib/Topology/ContinuousMap/Ideals.lean
similarity index 99%
rename from Mathlib/Topology/ContinuousFunction/Ideals.lean
rename to Mathlib/Topology/ContinuousMap/Ideals.lean
index e40d94ca79034..5edb4350d5274 100644
--- a/Mathlib/Topology/ContinuousFunction/Ideals.lean
+++ b/Mathlib/Topology/ContinuousMap/Ideals.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
 import Mathlib.Topology.Algebra.Algebra
-import Mathlib.Topology.ContinuousFunction.Compact
+import Mathlib.Topology.ContinuousMap.Compact
 import Mathlib.Topology.UrysohnsLemma
 import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Analysis.Normed.Ring.Units
@@ -169,7 +169,7 @@ theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c)
       continuous_toFun :=
         ((map_continuous f).sup <| map_continuous _).inv₀ fun _ => (hc.trans_le le_sup_right).ne' },
     fun x =>
-    (inv_mul_le_iff (hc.trans_le le_sup_right)).mpr ((mul_one (f x ⊔ c)).symm ▸ le_sup_left),
+    (inv_mul_le_iff₀ (hc.trans_le le_sup_right)).mpr ((mul_one (f x ⊔ c)).symm ▸ le_sup_left),
     fun x hx => by
     simpa only [coe_const, mul_apply, coe_mk, Pi.inv_apply, Pi.sup_apply,
       Function.const_apply, sup_eq_left.mpr (Set.mem_setOf.mp hx), ne_eq, Pi.one_apply]
diff --git a/Mathlib/Topology/ContinuousFunction/LocallyConstant.lean b/Mathlib/Topology/ContinuousMap/LocallyConstant.lean
similarity index 92%
rename from Mathlib/Topology/ContinuousFunction/LocallyConstant.lean
rename to Mathlib/Topology/ContinuousMap/LocallyConstant.lean
index a9858942c825d..bbe26fe3ba211 100644
--- a/Mathlib/Topology/ContinuousFunction/LocallyConstant.lean
+++ b/Mathlib/Topology/ContinuousMap/LocallyConstant.lean
@@ -4,8 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
 import Mathlib.Topology.LocallyConstant.Algebra
-import Mathlib.Topology.ContinuousFunction.Basic
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Basic
+import Mathlib.Topology.ContinuousMap.Algebra
 
 /-!
 # The algebra morphism from locally constant functions to continuous functions.
@@ -15,7 +15,7 @@ import Mathlib.Topology.ContinuousFunction.Algebra
 
 namespace LocallyConstant
 
-variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] (f : LocallyConstant X Y)
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
 
 /-- The inclusion of locally-constant functions into continuous functions as a multiplicative
 monoid hom. -/
diff --git a/Mathlib/Topology/ContinuousFunction/Ordered.lean b/Mathlib/Topology/ContinuousMap/Ordered.lean
similarity index 94%
rename from Mathlib/Topology/ContinuousFunction/Ordered.lean
rename to Mathlib/Topology/ContinuousMap/Ordered.lean
index 0b4780410756f..14a4185687a12 100644
--- a/Mathlib/Topology/ContinuousFunction/Ordered.lean
+++ b/Mathlib/Topology/ContinuousMap/Ordered.lean
@@ -1,11 +1,11 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Shing Tak Lam
+Authors: Kim Morrison, Shing Tak Lam
 -/
-import Mathlib.Topology.ContinuousFunction.Basic
 import Mathlib.Topology.Order.Lattice
 import Mathlib.Topology.Order.ProjIcc
+import Mathlib.Topology.ContinuousMap.Defs
 
 /-!
 # Bundled continuous maps into orders, with order-compatible topology
@@ -85,7 +85,7 @@ section Extend
 
 variable [LinearOrder α] [OrderTopology α] {a b : α} (h : a ≤ b)
 
-/-- Extend a continuous function `f : C(Set.Icc a b, β)` to a function `f : C(α, β)`.  -/
+/-- Extend a continuous function `f : C(Set.Icc a b, β)` to a function `f : C(α, β)`. -/
 def IccExtend (f : C(Set.Icc a b, β)) : C(α, β) where
   toFun := Set.IccExtend h f
 
diff --git a/Mathlib/Topology/ContinuousFunction/Polynomial.lean b/Mathlib/Topology/ContinuousMap/Polynomial.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/Polynomial.lean
rename to Mathlib/Topology/ContinuousMap/Polynomial.lean
index 3fd434d44adc3..83a3dd5e6a0ed 100644
--- a/Mathlib/Topology/ContinuousFunction/Polynomial.lean
+++ b/Mathlib/Topology/ContinuousMap/Polynomial.lean
@@ -1,10 +1,10 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Algebra.Polynomial
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Topology.UnitInterval
 import Mathlib.Algebra.Star.Subalgebra
 
diff --git a/Mathlib/Topology/ContinuousFunction/Sigma.lean b/Mathlib/Topology/ContinuousMap/Sigma.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/Sigma.lean
rename to Mathlib/Topology/ContinuousMap/Sigma.lean
index a444781b712f5..320fd5cc4ab1f 100644
--- a/Mathlib/Topology/ContinuousFunction/Sigma.lean
+++ b/Mathlib/Topology/ContinuousMap/Sigma.lean
@@ -63,7 +63,7 @@ some `i` and a continuous map `g : C(X, Y i)`. See also `Continuous.exists_lift_
 with unbundled functions and `ContinuousMap.sigmaCodHomeomorph` for a homeomorphism defined using
 this fact. -/
 theorem exists_lift_sigma (f : C(X, Σ i, Y i)) : ∃ i g, f = (sigmaMk i).comp g :=
-  let ⟨i, g, hg, hfg⟩ := f.continuous.exists_lift_sigma
+  let ⟨i, g, hg, hfg⟩ := (map_continuous f).exists_lift_sigma
   ⟨i, ⟨g, hg⟩, DFunLike.ext' hfg⟩
 
 variable (X Y)
diff --git a/Mathlib/Topology/ContinuousFunction/StarOrdered.lean b/Mathlib/Topology/ContinuousMap/StarOrdered.lean
similarity index 97%
rename from Mathlib/Topology/ContinuousFunction/StarOrdered.lean
rename to Mathlib/Topology/ContinuousMap/StarOrdered.lean
index 18882fe259d74..b894be96cbeb1 100644
--- a/Mathlib/Topology/ContinuousFunction/StarOrdered.lean
+++ b/Mathlib/Topology/ContinuousMap/StarOrdered.lean
@@ -5,8 +5,8 @@ Authors: Jireh Loreaux
 -/
 import Mathlib.Analysis.Complex.Basic
 import Mathlib.Data.Real.StarOrdered
-import Mathlib.Topology.ContinuousFunction.Algebra
-import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
+import Mathlib.Topology.ContinuousMap.Algebra
+import Mathlib.Topology.ContinuousMap.ContinuousMapZero
 
 /-! # Continuous functions as a star-ordered ring -/
 
diff --git a/Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean b/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
rename to Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
index 57b51d1724771..643b6ad6595d6 100644
--- a/Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
+++ b/Mathlib/Topology/ContinuousMap/StoneWeierstrass.lean
@@ -1,13 +1,13 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Heather Macbeth
+Authors: Kim Morrison, Heather Macbeth
 -/
 import Mathlib.Algebra.Algebra.Subalgebra.Unitization
 import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Topology.Algebra.StarSubalgebra
-import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
-import Mathlib.Topology.ContinuousFunction.Weierstrass
+import Mathlib.Topology.ContinuousMap.ContinuousMapZero
+import Mathlib.Topology.ContinuousMap.Weierstrass
 
 /-!
 # The Stone-Weierstrass theorem
@@ -222,9 +222,7 @@ theorem sublattice_closure_eq_top (L : Set C(X, ℝ)) (nA : L.Nonempty)
   have W_nhd : ∀ x, W x ∈ 𝓝 x := by
     intro x
     refine IsOpen.mem_nhds ?_ ?_
-    · -- Porting note: mathlib3 `continuity` found `continuous_set_coe`
-      apply isOpen_lt (continuous_set_coe _ _)
-      continuity
+    · apply isOpen_lt <;> fun_prop
     · dsimp only [W, Set.mem_setOf_eq]
       rw [h_eq]
       exact lt_add_of_pos_right _ pos
diff --git a/Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean b/Mathlib/Topology/ContinuousMap/T0Sierpinski.lean
similarity index 97%
rename from Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean
rename to Mathlib/Topology/ContinuousMap/T0Sierpinski.lean
index 48655f0023ded..4e9903ccc7400 100644
--- a/Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean
+++ b/Mathlib/Topology/ContinuousMap/T0Sierpinski.lean
@@ -5,7 +5,7 @@ Authors: Ivan Sadofschi Costa
 -/
 import Mathlib.Topology.Order
 import Mathlib.Topology.Sets.Opens
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # Any T0 space embeds in a product of copies of the Sierpinski space.
diff --git a/Mathlib/Topology/ContinuousFunction/Units.lean b/Mathlib/Topology/ContinuousMap/Units.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/Units.lean
rename to Mathlib/Topology/ContinuousMap/Units.lean
index a8c880cf3a2bc..0337bcc434faa 100644
--- a/Mathlib/Topology/ContinuousFunction/Units.lean
+++ b/Mathlib/Topology/ContinuousMap/Units.lean
@@ -5,7 +5,7 @@ Authors: Jireh Loreaux
 -/
 import Mathlib.Analysis.Normed.Ring.Units
 import Mathlib.Algebra.Algebra.Spectrum
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 
 /-!
 # Units of continuous functions
diff --git a/Mathlib/Topology/ContinuousFunction/Weierstrass.lean b/Mathlib/Topology/ContinuousMap/Weierstrass.lean
similarity index 98%
rename from Mathlib/Topology/ContinuousFunction/Weierstrass.lean
rename to Mathlib/Topology/ContinuousMap/Weierstrass.lean
index 90730fc0e1d83..fbfdd66fb2023 100644
--- a/Mathlib/Topology/ContinuousFunction/Weierstrass.lean
+++ b/Mathlib/Topology/ContinuousMap/Weierstrass.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Analysis.SpecialFunctions.Bernstein
 import Mathlib.Topology.Algebra.Algebra
diff --git a/Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean b/Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean
similarity index 99%
rename from Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean
rename to Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean
index 932426106e17d..b9c5713bf06fd 100644
--- a/Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean
+++ b/Mathlib/Topology/ContinuousMap/ZeroAtInfty.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Jireh Loreaux. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jireh Loreaux
 -/
-import Mathlib.Topology.ContinuousFunction.Bounded
-import Mathlib.Topology.ContinuousFunction.CocompactMap
+import Mathlib.Topology.ContinuousMap.Bounded
+import Mathlib.Topology.ContinuousMap.CocompactMap
 
 /-!
 # Continuous functions vanishing at infinity
@@ -428,7 +428,7 @@ theorem isClosed_range_toBCF : IsClosed (range (toBCF : C₀(α, β) → α →
 /-- Continuous functions vanishing at infinity taking values in a complete space form a
 complete space. -/
 instance instCompleteSpace [CompleteSpace β] : CompleteSpace C₀(α, β) :=
-  (completeSpace_iff_isComplete_range isometry_toBCF.uniformInducing).mpr
+  (completeSpace_iff_isComplete_range isometry_toBCF.isUniformInducing).mpr
     isClosed_range_toBCF.isComplete
 
 end Metric
diff --git a/Mathlib/Topology/ContinuousOn.lean b/Mathlib/Topology/ContinuousOn.lean
index cc5cc3ae5b508..7608fd4158dbb 100644
--- a/Mathlib/Topology/ContinuousOn.lean
+++ b/Mathlib/Topology/ContinuousOn.lean
@@ -3,6 +3,7 @@ Copyright (c) 2019 Reid Barton. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
+import Mathlib.Algebra.Group.Indicator
 import Mathlib.Topology.Constructions
 
 /-!
@@ -53,12 +54,20 @@ theorem mem_closure_ne_iff_frequently_within {z : α} {s : Set α} :
   simp [mem_closure_iff_frequently, frequently_nhdsWithin_iff]
 
 @[simp]
-theorem eventually_nhdsWithin_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
+theorem eventually_eventually_nhdsWithin {a : α} {s : Set α} {p : α → Prop} :
     (∀ᶠ y in 𝓝[s] a, ∀ᶠ x in 𝓝[s] y, p x) ↔ ∀ᶠ x in 𝓝[s] a, p x := by
   refine ⟨fun h => ?_, fun h => (eventually_nhds_nhdsWithin.2 h).filter_mono inf_le_left⟩
   simp only [eventually_nhdsWithin_iff] at h ⊢
   exact h.mono fun x hx hxs => (hx hxs).self_of_nhds hxs
 
+@[deprecated (since := "2024-10-04")]
+alias eventually_nhdsWithin_nhdsWithin := eventually_eventually_nhdsWithin
+
+@[simp]
+theorem eventually_mem_nhdsWithin_iff {x : α} {s t : Set α} :
+    (∀ᶠ x' in 𝓝[s] x, t ∈ 𝓝[s] x') ↔ t ∈ 𝓝[s] x :=
+  eventually_eventually_nhdsWithin
+
 theorem nhdsWithin_eq (a : α) (s : Set α) :
     𝓝[s] a = ⨅ t ∈ { t : Set α | a ∈ t ∧ IsOpen t }, 𝓟 (t ∩ s) :=
   ((nhds_basis_opens a).inf_principal s).eq_biInf
@@ -608,7 +617,7 @@ theorem continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → Set α
     rw [ContinuousAt, ← nhdsWithin_eq_nhds.2 hi]
     exact hf _ _ (mem_of_mem_nhds hi)
 
-theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim
+@[simp] theorem continuousOn_empty (f : α → β) : ContinuousOn f ∅ := fun _ => False.elim
 
 @[simp]
 theorem continuousOn_singleton (f : α → β) (a : α) : ContinuousOn f {a} :=
@@ -684,8 +693,7 @@ theorem continuousWithinAt_singleton {f : α → β} {x : α} : ContinuousWithin
 @[simp]
 theorem continuousWithinAt_insert_self {f : α → β} {x : α} {s : Set α} :
     ContinuousWithinAt f (insert x s) x ↔ ContinuousWithinAt f s x := by
-  simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton,
-    true_and_iff]
+  simp only [← singleton_union, continuousWithinAt_union, continuousWithinAt_singleton, true_and]
 
 alias ⟨_, ContinuousWithinAt.insert_self⟩ := continuousWithinAt_insert_self
 
@@ -1138,7 +1146,8 @@ theorem IsOpen.ite' {s s' t : Set α} (hs : IsOpen s) (hs' : IsOpen s')
     (ht : ∀ x ∈ frontier t, x ∈ s ↔ x ∈ s') : IsOpen (t.ite s s') := by
   classical
     simp only [isOpen_iff_continuous_mem, Set.ite] at *
-    convert continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn
+    convert
+      continuous_piecewise (fun x hx => propext (ht x hx)) hs.continuousOn hs'.continuousOn using 2
     rename_i x
     by_cases hx : x ∈ t <;> simp [hx]
 
diff --git a/Mathlib/Topology/Defs/Basic.lean b/Mathlib/Topology/Defs/Basic.lean
index 607362753bace..e7526dd6e9024 100644
--- a/Mathlib/Topology/Defs/Basic.lean
+++ b/Mathlib/Topology/Defs/Basic.lean
@@ -151,6 +151,28 @@ def IsOpenMap (f : X → Y) : Prop := ∀ U : Set X, IsOpen U → IsOpen (f '' U
 if the image of any closed `U : Set X` is closed in `Y`. -/
 def IsClosedMap (f : X → Y) : Prop := ∀ U : Set X, IsClosed U → IsClosed (f '' U)
 
+/-- An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map.
+Equivalently, it is a surjective continuous open map.
+We use the latter characterization as a definition.
+
+Many important quotient maps are open quotient maps, including
+
+- the quotient map from a topological space to its quotient by the action of a group;
+- the quotient map from a topological group to its quotient by a normal subgroup;
+- the quotient map from a topological spaace to its separation quotient.
+
+Contrary to general quotient maps,
+the category of open quotient maps is closed under `Prod.map`.
+-/
+@[mk_iff]
+structure IsOpenQuotientMap (f : X → Y) : Prop where
+  /-- An open quotient map is surjective. -/
+  surjective : Function.Surjective f
+  /-- An open quotient map is continuous. -/
+  continuous : Continuous f
+  /-- An open quotient map is an open map. -/
+  isOpenMap : IsOpenMap f
+
 end Defs
 
 /-! ### Notation for non-standard topologies -/
diff --git a/Mathlib/Topology/Defs/Filter.lean b/Mathlib/Topology/Defs/Filter.lean
index 7266619675e3b..bf9b961755c3a 100644
--- a/Mathlib/Topology/Defs/Filter.lean
+++ b/Mathlib/Topology/Defs/Filter.lean
@@ -29,6 +29,12 @@ as well as other definitions that rely on `Filter`s.
   denoted by `𝓝ˢ s` in the `Topology` scope.
   A set `t` is called a neighborhood of `s`, if it includes an open set that includes `s`.
 
+* `exterior s`: The *exterior* of a set is the intersection of all its neighborhoods.
+  In an Alexandrov-discrete space, this is the smallest neighborhood of the set.
+
+  Note that this construction is unnamed in the literature.
+  We choose the name in analogy to `interior`.
+
 ### Continuity at a point
 
 * `ContinuousAt f x`: a function `f` is continuous at a point `x`,
@@ -146,6 +152,13 @@ def nhdsSet (s : Set X) : Filter X :=
 
 @[inherit_doc] scoped[Topology] notation "𝓝ˢ" => nhdsSet
 
+/-- The *exterior* of a set is the intersection of all its neighborhoods. In an Alexandrov-discrete
+space, this is the smallest neighborhood of the set.
+
+Note that this construction is unnamed in the literature. We choose the name in analogy to
+`interior`. -/
+def exterior (s : Set X) : Set X := (𝓝ˢ s).ker
+
 /-- A function between topological spaces is continuous at a point `x₀`
 if `f x` tends to `f x₀` when `x` tends to `x₀`. -/
 @[fun_prop]
diff --git a/Mathlib/Topology/DenseEmbedding.lean b/Mathlib/Topology/DenseEmbedding.lean
index 7957b2b180fe4..210caa3dca529 100644
--- a/Mathlib/Topology/DenseEmbedding.lean
+++ b/Mathlib/Topology/DenseEmbedding.lean
@@ -11,14 +11,14 @@ import Mathlib.Topology.Bases
 
 This file defines three properties of functions:
 
-* `DenseRange f`      means `f` has dense image;
-* `DenseInducing i`   means `i` is also `Inducing`, namely it induces the topology on its codomain;
-* `DenseEmbedding e`  means `e` is further an `Embedding`, namely it is injective and `Inducing`.
+* `DenseRange f`       means `f` has dense image;
+* `IsDenseInducing i`  means `i` is also `Inducing`, namely it induces the topology on its codomain;
+* `IsDenseEmbedding e` means `e` is further an `Embedding`, namely it is injective and `Inducing`.
 
 The main theorem `continuous_extend` gives a criterion for a function
 `f : X → Z` to a T₃ space Z to extend along a dense embedding
 `i : X → Y` to a continuous function `g : Y → Z`. Actually `i` only
-has to be `DenseInducing` (not necessarily injective).
+has to be `IsDenseInducing` (not necessarily injective).
 
 -/
 
@@ -32,30 +32,30 @@ variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
 
 /-- `i : α → β` is "dense inducing" if it has dense range and the topology on `α`
   is the one induced by `i` from the topology on `β`. -/
-structure DenseInducing [TopologicalSpace α] [TopologicalSpace β] (i : α → β)
+structure IsDenseInducing [TopologicalSpace α] [TopologicalSpace β] (i : α → β)
     extends Inducing i : Prop where
   /-- The range of a dense inducing map is a dense set. -/
   protected dense : DenseRange i
 
-namespace DenseInducing
+namespace IsDenseInducing
 
 variable [TopologicalSpace α] [TopologicalSpace β]
 variable {i : α → β}
 
-theorem nhds_eq_comap (di : DenseInducing i) : ∀ a : α, 𝓝 a = comap i (𝓝 <| i a) :=
+theorem nhds_eq_comap (di : IsDenseInducing i) : ∀ a : α, 𝓝 a = comap i (𝓝 <| i a) :=
   di.toInducing.nhds_eq_comap
 
-protected theorem continuous (di : DenseInducing i) : Continuous i :=
+protected theorem continuous (di : IsDenseInducing i) : Continuous i :=
   di.toInducing.continuous
 
-theorem closure_range (di : DenseInducing i) : closure (range i) = univ :=
+theorem closure_range (di : IsDenseInducing i) : closure (range i) = univ :=
   di.dense.closure_range
 
-protected theorem preconnectedSpace [PreconnectedSpace α] (di : DenseInducing i) :
+protected theorem preconnectedSpace [PreconnectedSpace α] (di : IsDenseInducing i) :
     PreconnectedSpace β :=
   di.dense.preconnectedSpace di.continuous
 
-theorem closure_image_mem_nhds {s : Set α} {a : α} (di : DenseInducing i) (hs : s ∈ 𝓝 a) :
+theorem closure_image_mem_nhds {s : Set α} {a : α} (di : IsDenseInducing i) (hs : s ∈ 𝓝 a) :
     closure (i '' s) ∈ 𝓝 (i a) := by
   rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs
   rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩
@@ -64,14 +64,14 @@ theorem closure_image_mem_nhds {s : Set α} {a : α} (di : DenseInducing i) (hs
     U ⊆ closure (i '' (i ⁻¹' U)) := di.dense.subset_closure_image_preimage_of_isOpen hUo
     _ ⊆ closure (i '' s) := closure_mono (image_subset i sub)
 
-theorem dense_image (di : DenseInducing i) {s : Set α} : Dense (i '' s) ↔ Dense s := by
+theorem dense_image (di : IsDenseInducing i) {s : Set α} : Dense (i '' s) ↔ Dense s := by
   refine ⟨fun H x => ?_, di.dense.dense_image di.continuous⟩
   rw [di.toInducing.closure_eq_preimage_closure_image, H.closure_eq, preimage_univ]
   trivial
 
 /-- If `i : α → β` is a dense embedding with dense complement of the range, then any compact set in
 `α` has empty interior. -/
-theorem interior_compact_eq_empty [T2Space β] (di : DenseInducing i) (hd : Dense (range i)ᶜ)
+theorem interior_compact_eq_empty [T2Space β] (di : IsDenseInducing i) (hd : Dense (range i)ᶜ)
     {s : Set α} (hs : IsCompact s) : interior s = ∅ := by
   refine eq_empty_iff_forall_not_mem.2 fun x hx => ?_
   rw [mem_interior_iff_mem_nhds] at hx
@@ -81,16 +81,19 @@ theorem interior_compact_eq_empty [T2Space β] (di : DenseInducing i) (hd : Dens
   exact hyi (image_subset_range _ _ hys)
 
 /-- The product of two dense inducings is a dense inducing -/
-protected theorem prod [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : α → β} {e₂ : γ → δ}
-    (de₁ : DenseInducing e₁) (de₂ : DenseInducing e₂) :
-    DenseInducing fun p : α × γ => (e₁ p.1, e₂ p.2) where
-  toInducing := de₁.toInducing.prod_map de₂.toInducing
-  dense := de₁.dense.prod_map de₂.dense
+protected theorem prodMap [TopologicalSpace γ] [TopologicalSpace δ] {e₁ : α → β} {e₂ : γ → δ}
+    (de₁ : IsDenseInducing e₁) (de₂ : IsDenseInducing e₂) :
+    IsDenseInducing (Prod.map e₁ e₂) where
+  toInducing := de₁.toInducing.prodMap de₂.toInducing
+  dense := de₁.dense.prodMap de₂.dense
+
+@[deprecated (since := "2024-10-06")]
+protected alias prod := IsDenseInducing.prodMap
 
 open TopologicalSpace
 
-/-- If the domain of a `DenseInducing` map is a separable space, then so is the codomain. -/
-protected theorem separableSpace [SeparableSpace α] (di : DenseInducing i) : SeparableSpace β :=
+/-- If the domain of a `IsDenseInducing` map is a separable space, then so is the codomain. -/
+protected theorem separableSpace [SeparableSpace α] (di : IsDenseInducing i) : SeparableSpace β :=
   di.dense.separableSpace di.continuous
 
 variable [TopologicalSpace δ] {f : γ → α} {g : γ → δ} {h : δ → β}
@@ -102,7 +105,7 @@ g↓     ↓e
  δ -h→ β
 ```
 -/
-theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : DenseInducing i)
+theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : IsDenseInducing i)
     (H : Tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f) : Tendsto f (comap g (𝓝 d)) (𝓝 a) := by
   have lim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d := map_comap_le
   replace lim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d) := map_mono lim1
@@ -111,10 +114,10 @@ theorem tendsto_comap_nhds_nhds {d : δ} {a : α} (di : DenseInducing i)
   rw [← di.nhds_eq_comap] at lim2
   exact le_trans lim1 lim2
 
-protected theorem nhdsWithin_neBot (di : DenseInducing i) (b : β) : NeBot (𝓝[range i] b) :=
+protected theorem nhdsWithin_neBot (di : IsDenseInducing i) (b : β) : NeBot (𝓝[range i] b) :=
   di.dense.nhdsWithin_neBot b
 
-theorem comap_nhds_neBot (di : DenseInducing i) (b : β) : NeBot (comap i (𝓝 b)) :=
+theorem comap_nhds_neBot (di : IsDenseInducing i) (b : β) : NeBot (comap i (𝓝 b)) :=
   comap_neBot fun s hs => by
     rcases mem_closure_iff_nhds.1 (di.dense b) s hs with ⟨_, ⟨ha, a, rfl⟩⟩
     exact ⟨a, ha⟩
@@ -122,38 +125,38 @@ theorem comap_nhds_neBot (di : DenseInducing i) (b : β) : NeBot (comap i (𝓝
 variable [TopologicalSpace γ]
 
 /-- If `i : α → β` is a dense inducing, then any function `f : α → γ` "extends" to a function `g =
-  DenseInducing.extend di f : β → γ`. If `γ` is Hausdorff and `f` has a continuous extension, then
+  IsDenseInducing.extend di f : β → γ`. If `γ` is Hausdorff and `f` has a continuous extension, then
   `g` is the unique such extension. In general, `g` might not be continuous or even extend `f`. -/
-def extend (di : DenseInducing i) (f : α → γ) (b : β) : γ :=
+def extend (di : IsDenseInducing i) (f : α → γ) (b : β) : γ :=
   @limUnder _ _ _ ⟨f (di.dense.some b)⟩ (comap i (𝓝 b)) f
 
-theorem extend_eq_of_tendsto [T2Space γ] (di : DenseInducing i) {b : β} {c : γ} {f : α → γ}
+theorem extend_eq_of_tendsto [T2Space γ] (di : IsDenseInducing i) {b : β} {c : γ} {f : α → γ}
     (hf : Tendsto f (comap i (𝓝 b)) (𝓝 c)) : di.extend f b = c :=
   haveI := di.comap_nhds_neBot
   hf.limUnder_eq
 
-theorem extend_eq_at [T2Space γ] (di : DenseInducing i) {f : α → γ} {a : α}
+theorem extend_eq_at [T2Space γ] (di : IsDenseInducing i) {f : α → γ} {a : α}
     (hf : ContinuousAt f a) : di.extend f (i a) = f a :=
   extend_eq_of_tendsto _ <| di.nhds_eq_comap a ▸ hf
 
-theorem extend_eq_at' [T2Space γ] (di : DenseInducing i) {f : α → γ} {a : α} (c : γ)
+theorem extend_eq_at' [T2Space γ] (di : IsDenseInducing i) {f : α → γ} {a : α} (c : γ)
     (hf : Tendsto f (𝓝 a) (𝓝 c)) : di.extend f (i a) = f a :=
   di.extend_eq_at (continuousAt_of_tendsto_nhds hf)
 
-theorem extend_eq [T2Space γ] (di : DenseInducing i) {f : α → γ} (hf : Continuous f) (a : α) :
+theorem extend_eq [T2Space γ] (di : IsDenseInducing i) {f : α → γ} (hf : Continuous f) (a : α) :
     di.extend f (i a) = f a :=
   di.extend_eq_at hf.continuousAt
 
 /-- Variation of `extend_eq` where we ask that `f` has a limit along `comap i (𝓝 b)` for each
 `b : β`. This is a strictly stronger assumption than continuity of `f`, but in a lot of cases
 you'd have to prove it anyway to use `continuous_extend`, so this avoids doing the work twice. -/
-theorem extend_eq' [T2Space γ] {f : α → γ} (di : DenseInducing i)
+theorem extend_eq' [T2Space γ] {f : α → γ} (di : IsDenseInducing i)
     (hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) (a : α) : di.extend f (i a) = f a := by
   rcases hf (i a) with ⟨b, hb⟩
   refine di.extend_eq_at' b ?_
   rwa [← di.toInducing.nhds_eq_comap] at hb
 
-theorem extend_unique_at [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : DenseInducing i)
+theorem extend_unique_at [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : IsDenseInducing i)
     (hf : ∀ᶠ x in comap i (𝓝 b), g (i x) = f x) (hg : ContinuousAt g b) : di.extend f b = g b := by
   refine di.extend_eq_of_tendsto fun s hs => mem_map.2 ?_
   suffices ∀ᶠ x : α in comap i (𝓝 b), g (i x) ∈ s from
@@ -163,11 +166,11 @@ theorem extend_unique_at [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (
   rintro _ hxs x rfl
   exact hxs
 
-theorem extend_unique [T2Space γ] {f : α → γ} {g : β → γ} (di : DenseInducing i)
+theorem extend_unique [T2Space γ] {f : α → γ} {g : β → γ} (di : IsDenseInducing i)
     (hf : ∀ x, g (i x) = f x) (hg : Continuous g) : di.extend f = g :=
   funext fun _ => extend_unique_at di (Eventually.of_forall hf) hg.continuousAt
 
-theorem continuousAt_extend [T3Space γ] {b : β} {f : α → γ} (di : DenseInducing i)
+theorem continuousAt_extend [T3Space γ] {b : β} {f : α → γ} (di : IsDenseInducing i)
     (hf : ∀ᶠ x in 𝓝 b, ∃ c, Tendsto f (comap i <| 𝓝 x) (𝓝 c)) : ContinuousAt (di.extend f) b := by
   set φ := di.extend f
   haveI := di.comap_nhds_neBot
@@ -189,61 +192,66 @@ theorem continuousAt_extend [T3Space γ] {b : β} {f : α → γ} (di : DenseInd
   use V₂
   tauto
 
-theorem continuous_extend [T3Space γ] {f : α → γ} (di : DenseInducing i)
+theorem continuous_extend [T3Space γ] {f : α → γ} (di : IsDenseInducing i)
     (hf : ∀ b, ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)) : Continuous (di.extend f) :=
   continuous_iff_continuousAt.mpr fun _ => di.continuousAt_extend <| univ_mem' hf
 
 theorem mk' (i : α → β) (c : Continuous i) (dense : ∀ x, x ∈ closure (range i))
-    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ b, i b ∈ t → b ∈ s) : DenseInducing i :=
+    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ b, i b ∈ t → b ∈ s) : IsDenseInducing i :=
   { toInducing := inducing_iff_nhds.2 fun a =>
       le_antisymm (c.tendsto _).le_comap (by simpa [Filter.le_def] using H a)
     dense }
 
-end DenseInducing
+end IsDenseInducing
 
 /-- A dense embedding is an embedding with dense image. -/
-structure DenseEmbedding [TopologicalSpace α] [TopologicalSpace β] (e : α → β) extends
-  DenseInducing e : Prop where
+structure IsDenseEmbedding [TopologicalSpace α] [TopologicalSpace β] (e : α → β) extends
+  IsDenseInducing e : Prop where
   /-- A dense embedding is injective. -/
   inj : Function.Injective e
 
-theorem DenseEmbedding.mk' [TopologicalSpace α] [TopologicalSpace β] (e : α → β) (c : Continuous e)
+lemma IsDenseEmbedding.mk' [TopologicalSpace α] [TopologicalSpace β] (e : α → β) (c : Continuous e)
     (dense : DenseRange e) (inj : Function.Injective e)
-    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (e a), ∀ b, e b ∈ t → b ∈ s) : DenseEmbedding e :=
-  { DenseInducing.mk' e c dense H with inj }
+    (H : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (e a), ∀ b, e b ∈ t → b ∈ s) : IsDenseEmbedding e :=
+  { IsDenseInducing.mk' e c dense H with inj }
+
+@[deprecated (since := "2024-09-30")]
+alias DenseEmbedding.mk' := IsDenseEmbedding.mk'
 
-namespace DenseEmbedding
+namespace IsDenseEmbedding
 
 open TopologicalSpace
 
 variable [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ]
 variable {e : α → β}
 
-theorem inj_iff (de : DenseEmbedding e) {x y} : e x = e y ↔ x = y :=
+theorem inj_iff (de : IsDenseEmbedding e) {x y} : e x = e y ↔ x = y :=
   de.inj.eq_iff
 
-theorem to_embedding (de : DenseEmbedding e) : Embedding e :=
+theorem to_embedding (de : IsDenseEmbedding e) : Embedding e :=
   { induced := de.induced
     inj := de.inj }
 
-/-- If the domain of a `DenseEmbedding` is a separable space, then so is its codomain. -/
-protected theorem separableSpace [SeparableSpace α] (de : DenseEmbedding e) : SeparableSpace β :=
-  de.toDenseInducing.separableSpace
+/-- If the domain of a `IsDenseEmbedding` is a separable space, then so is its codomain. -/
+protected theorem separableSpace [SeparableSpace α] (de : IsDenseEmbedding e) : SeparableSpace β :=
+  de.toIsDenseInducing.separableSpace
 
 /-- The product of two dense embeddings is a dense embedding. -/
-protected theorem prod {e₁ : α → β} {e₂ : γ → δ} (de₁ : DenseEmbedding e₁)
-    (de₂ : DenseEmbedding e₂) : DenseEmbedding fun p : α × γ => (e₁ p.1, e₂ p.2) :=
-  { de₁.toDenseInducing.prod de₂.toDenseInducing with
+protected theorem prodMap {e₁ : α → β} {e₂ : γ → δ} (de₁ : IsDenseEmbedding e₁)
+    (de₂ : IsDenseEmbedding e₂) : IsDenseEmbedding fun p : α × γ => (e₁ p.1, e₂ p.2) :=
+  { de₁.toIsDenseInducing.prodMap de₂.toIsDenseInducing with
     inj := de₁.inj.prodMap de₂.inj }
 
+@[deprecated (since := "2024-10-06")] protected alias prod := IsDenseEmbedding.prodMap
+
 /-- The dense embedding of a subtype inside its closure. -/
 @[simps]
 def subtypeEmb {α : Type*} (p : α → Prop) (e : α → β) (x : { x // p x }) :
     { x // x ∈ closure (e '' { x | p x }) } :=
   ⟨e x, subset_closure <| mem_image_of_mem e x.prop⟩
 
-protected theorem subtype (de : DenseEmbedding e) (p : α → Prop) :
-    DenseEmbedding (subtypeEmb p e) :=
+protected theorem subtype (de : IsDenseEmbedding e) (p : α → Prop) :
+    IsDenseEmbedding (subtypeEmb p e) :=
   { dense :=
       dense_iff_closure_eq.2 <| by
         ext ⟨x, hx⟩
@@ -253,20 +261,26 @@ protected theorem subtype (de : DenseEmbedding e) (p : α → Prop) :
     induced :=
       (induced_iff_nhds_eq _).2 fun ⟨x, hx⟩ => by
         simp [subtypeEmb, nhds_subtype_eq_comap, de.toInducing.nhds_eq_comap, comap_comap,
-          (· ∘ ·)] }
-
-theorem dense_image (de : DenseEmbedding e) {s : Set α} : Dense (e '' s) ↔ Dense s :=
-  de.toDenseInducing.dense_image
+          Function.comp_def] }
 
-end DenseEmbedding
+theorem dense_image (de : IsDenseEmbedding e) {s : Set α} : Dense (e '' s) ↔ Dense s :=
+  de.toIsDenseInducing.dense_image
 
-theorem denseEmbedding_id {α : Type*} [TopologicalSpace α] : DenseEmbedding (id : α → α) :=
+protected lemma id {α : Type*} [TopologicalSpace α] : IsDenseEmbedding (id : α → α) :=
   { embedding_id with dense := denseRange_id }
 
-theorem Dense.denseEmbedding_val [TopologicalSpace α] {s : Set α} (hs : Dense s) :
-    DenseEmbedding ((↑) : s → α) :=
+end IsDenseEmbedding
+
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_id := IsDenseEmbedding.id
+
+theorem Dense.isDenseEmbedding_val [TopologicalSpace α] {s : Set α} (hs : Dense s) :
+    IsDenseEmbedding ((↑) : s → α) :=
   { embedding_subtype_val with dense := hs.denseRange_val }
 
+@[deprecated (since := "2024-09-30")]
+alias Dense.denseEmbedding_val := Dense.isDenseEmbedding_val
+
 theorem isClosed_property [TopologicalSpace β] {e : α → β} {p : β → Prop} (he : DenseRange e)
     (hp : IsClosed { x | p x }) (h : ∀ a, p (e a)) : ∀ b, p b :=
   have : univ ⊆ { b | p b } :=
@@ -279,14 +293,14 @@ theorem isClosed_property [TopologicalSpace β] {e : α → β} {p : β → Prop
 
 theorem isClosed_property2 [TopologicalSpace β] {e : α → β} {p : β → β → Prop} (he : DenseRange e)
     (hp : IsClosed { q : β × β | p q.1 q.2 }) (h : ∀ a₁ a₂, p (e a₁) (e a₂)) : ∀ b₁ b₂, p b₁ b₂ :=
-  have : ∀ q : β × β, p q.1 q.2 := isClosed_property (he.prod_map he) hp fun _ => h _ _
+  have : ∀ q : β × β, p q.1 q.2 := isClosed_property (he.prodMap he) hp fun _ => h _ _
   fun b₁ b₂ => this ⟨b₁, b₂⟩
 
 theorem isClosed_property3 [TopologicalSpace β] {e : α → β} {p : β → β → β → Prop}
     (he : DenseRange e) (hp : IsClosed { q : β × β × β | p q.1 q.2.1 q.2.2 })
     (h : ∀ a₁ a₂ a₃, p (e a₁) (e a₂) (e a₃)) : ∀ b₁ b₂ b₃, p b₁ b₂ b₃ :=
   have : ∀ q : β × β × β, p q.1 q.2.1 q.2.2 :=
-    isClosed_property (he.prod_map <| he.prod_map he) hp fun _ => h _ _ _
+    isClosed_property (he.prodMap <| he.prodMap he) hp fun _ => h _ _ _
   fun b₁ b₂ b₃ => this ⟨b₁, b₂, b₃⟩
 
 @[elab_as_elim]
@@ -319,9 +333,9 @@ theorem DenseRange.equalizer (hfd : DenseRange f) {g h : β → γ} (hg : Contin
 end
 
 -- Bourbaki GT III §3 no.4 Proposition 7 (generalised to any dense-inducing map to a T₃ space)
-theorem Filter.HasBasis.hasBasis_of_denseInducing [TopologicalSpace α] [TopologicalSpace β]
+theorem Filter.HasBasis.hasBasis_of_isDenseInducing [TopologicalSpace α] [TopologicalSpace β]
     [T3Space β] {ι : Type*} {s : ι → Set α} {p : ι → Prop} {x : α} (h : (𝓝 x).HasBasis p s)
-    {f : α → β} (hf : DenseInducing f) : (𝓝 (f x)).HasBasis p fun i => closure <| f '' s i := by
+    {f : α → β} (hf : IsDenseInducing f) : (𝓝 (f x)).HasBasis p fun i => closure <| f '' s i := by
   rw [Filter.hasBasis_iff] at h ⊢
   intro T
   refine ⟨fun hT => ?_, fun hT => ?_⟩
diff --git a/Mathlib/Topology/DiscreteQuotient.lean b/Mathlib/Topology/DiscreteQuotient.lean
index 6c041809ef504..a6413350f7c54 100644
--- a/Mathlib/Topology/DiscreteQuotient.lean
+++ b/Mathlib/Topology/DiscreteQuotient.lean
@@ -348,7 +348,7 @@ open Classical in
 If `X` is a compact space, then we associate to any discrete quotient on `X` a finite set of
 clopen subsets of `X`, given by the fibers of `proj`.
 
-TODO: prove that these form a partition of `X` 
+TODO: prove that these form a partition of `X`
 -/
 noncomputable def finsetClopens [CompactSpace X]
     (d : DiscreteQuotient X) : Finset (Clopens X) := have : Fintype d := Fintype.ofFinite _
@@ -368,7 +368,7 @@ lemma comp_finsetClopens [CompactSpace X] :
     simpa [← h] using Quotient.mk_eq_iff_out (s := d.toSetoid)
   · exact fun ⟨y, h⟩ ↦ ⟨d.proj y, by ext; simp [h, proj]⟩
 
-/-- `finsetClopens X` is injective. -/
+/-- `finsetClopens X` is injective. -/
 theorem finsetClopens_inj [CompactSpace X] :
     (finsetClopens X).Injective := by
   apply Function.Injective.of_comp (f := Set.image (fun (t : Clopens X) ↦ t.carrier) ∘ Finset.toSet)
diff --git a/Mathlib/Topology/DiscreteSubset.lean b/Mathlib/Topology/DiscreteSubset.lean
index 834b2a12022fc..5238fbf4e8e1e 100644
--- a/Mathlib/Topology/DiscreteSubset.lean
+++ b/Mathlib/Topology/DiscreteSubset.lean
@@ -97,7 +97,7 @@ theorem isClosed_and_discrete_iff {S : Set X} :
     simpa [disjoint_iff, nhdsWithin, inf_assoc, hx] using H
 
 /-- The filter of sets with no accumulation points inside a set `S : Set X`, implemented
-as the supremum over all punctured neighborhoods within `S`.  -/
+as the supremum over all punctured neighborhoods within `S`. -/
 def Filter.codiscreteWithin (S : Set X) : Filter X := ⨆ x ∈ S, 𝓝[S \ {x}] x
 
 lemma mem_codiscreteWithin {S T : Set X} :
diff --git a/Mathlib/Topology/EMetricSpace/Basic.lean b/Mathlib/Topology/EMetricSpace/Basic.lean
index 79a2678c491d7..0aba00ba4c0d7 100644
--- a/Mathlib/Topology/EMetricSpace/Basic.lean
+++ b/Mathlib/Topology/EMetricSpace/Basic.lean
@@ -5,7 +5,6 @@ Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébas
 -/
 import Mathlib.Order.Interval.Finset.Nat
 import Mathlib.Topology.EMetricSpace.Defs
-import Mathlib.Topology.UniformSpace.Pi
 import Mathlib.Topology.UniformSpace.UniformConvergence
 import Mathlib.Topology.UniformSpace.UniformEmbedding
 
@@ -59,28 +58,38 @@ theorem edist_le_range_sum_of_edist_le {f : ℕ → α} (n : ℕ) {d : ℕ → 
 
 namespace EMetric
 
-theorem uniformInducing_iff [PseudoEMetricSpace β] {f : α → β} :
-    UniformInducing f ↔ UniformContinuous f ∧
+theorem isUniformInducing_iff [PseudoEMetricSpace β] {f : α → β} :
+    IsUniformInducing f ↔ UniformContinuous f ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ :=
-  uniformInducing_iff'.trans <| Iff.rfl.and <|
+  isUniformInducing_iff'.trans <| Iff.rfl.and <|
     ((uniformity_basis_edist.comap _).le_basis_iff uniformity_basis_edist).trans <| by
       simp only [subset_def, Prod.forall]; rfl
 
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_iff := isUniformInducing_iff
+
 /-- ε-δ characterization of uniform embeddings on pseudoemetric spaces -/
-nonrec theorem uniformEmbedding_iff [PseudoEMetricSpace β] {f : α → β} :
-    UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
+nonrec theorem isUniformEmbedding_iff [PseudoEMetricSpace β] {f : α → β} :
+    IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ :=
-  (uniformEmbedding_iff _).trans <| and_comm.trans <| Iff.rfl.and uniformInducing_iff
+  (isUniformEmbedding_iff _).trans <| and_comm.trans <| Iff.rfl.and isUniformInducing_iff
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_iff := isUniformEmbedding_iff
 
 /-- If a map between pseudoemetric spaces is a uniform embedding then the edistance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y`.
 
-In fact, this lemma holds for a `UniformInducing` map.
+In fact, this lemma holds for a `IsUniformInducing` map.
 TODO: generalize? -/
-theorem controlled_of_uniformEmbedding [PseudoEMetricSpace β] {f : α → β} (h : UniformEmbedding f) :
+theorem controlled_of_isUniformEmbedding [PseudoEMetricSpace β] {f : α → β}
+    (h : IsUniformEmbedding f) :
     (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, edist a b < δ → edist (f a) (f b) < ε) ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, edist (f a) (f b) < ε → edist a b < δ :=
-  ⟨uniformContinuous_iff.1 h.uniformContinuous, (uniformEmbedding_iff.1 h).2.2⟩
+  ⟨uniformContinuous_iff.1 h.uniformContinuous, (isUniformEmbedding_iff.1 h).2.2⟩
+
+@[deprecated (since := "2024-10-01")]
+alias controlled_of_uniformEmbedding := controlled_of_isUniformEmbedding
 
 /-- ε-δ characterization of Cauchy sequences on pseudoemetric spaces -/
 protected theorem cauchy_iff {f : Filter α} :
@@ -138,55 +147,6 @@ end EMetric
 
 open EMetric
 
-section Pi
-
-open Finset
-
-variable {π : β → Type*} [Fintype β]
-
--- Porting note: reordered instances
-instance [∀ b, EDist (π b)] : EDist (∀ b, π b) where
-  edist f g := Finset.sup univ fun b => edist (f b) (g b)
-
-theorem edist_pi_def [∀ b, EDist (π b)] (f g : ∀ b, π b) :
-    edist f g = Finset.sup univ fun b => edist (f b) (g b) :=
-  rfl
-
-theorem edist_le_pi_edist [∀ b, EDist (π b)] (f g : ∀ b, π b) (b : β) :
-    edist (f b) (g b) ≤ edist f g :=
-  le_sup (f := fun b => edist (f b) (g b)) (Finset.mem_univ b)
-
-theorem edist_pi_le_iff [∀ b, EDist (π b)] {f g : ∀ b, π b} {d : ℝ≥0∞} :
-    edist f g ≤ d ↔ ∀ b, edist (f b) (g b) ≤ d :=
-  Finset.sup_le_iff.trans <| by simp only [Finset.mem_univ, forall_const]
-
-theorem edist_pi_const_le (a b : α) : (edist (fun _ : β => a) fun _ => b) ≤ edist a b :=
-  edist_pi_le_iff.2 fun _ => le_rfl
-
-@[simp]
-theorem edist_pi_const [Nonempty β] (a b : α) : (edist (fun _ : β => a) fun _ => b) = edist a b :=
-  Finset.sup_const univ_nonempty (edist a b)
-
-/-- The product of a finite number of pseudoemetric spaces, with the max distance, is still
-a pseudoemetric space.
-This construction would also work for infinite products, but it would not give rise
-to the product topology. Hence, we only formalize it in the good situation of finitely many
-spaces. -/
-instance pseudoEMetricSpacePi [∀ b, PseudoEMetricSpace (π b)] : PseudoEMetricSpace (∀ b, π b) where
-  edist_self f := bot_unique <| Finset.sup_le <| by simp
-  edist_comm f g := by simp [edist_pi_def, edist_comm]
-  edist_triangle f g h := edist_pi_le_iff.2 fun b => le_trans (edist_triangle _ (g b) _)
-    (add_le_add (edist_le_pi_edist _ _ _) (edist_le_pi_edist _ _ _))
-  toUniformSpace := Pi.uniformSpace _
-  uniformity_edist := by
-    simp only [Pi.uniformity, PseudoEMetricSpace.uniformity_edist, comap_iInf, gt_iff_lt,
-      preimage_setOf_eq, comap_principal, edist_pi_def]
-    rw [iInf_comm]; congr; funext ε
-    rw [iInf_comm]; congr; funext εpos
-    simp [setOf_forall, εpos]
-
-end Pi
-
 namespace EMetric
 
 variable {x y z : α} {ε ε₁ ε₂ : ℝ≥0∞} {s t : Set α}
@@ -194,6 +154,8 @@ variable {x y z : α} {ε ε₁ ε₂ : ℝ≥0∞} {s t : Set α}
 theorem inseparable_iff : Inseparable x y ↔ edist x y = 0 := by
   simp [inseparable_iff_mem_closure, mem_closure_iff, edist_comm, forall_lt_iff_le']
 
+alias ⟨_root_.Inseparable.edist_eq_zero, _⟩ := EMetric.inseparable_iff
+
 -- see Note [nolint_ge]
 /-- In a pseudoemetric space, Cauchy sequences are characterized by the fact that, eventually,
 the pseudoedistance between its elements is arbitrarily small -/
@@ -230,7 +192,7 @@ section Compact
 
 -- Porting note (#11215): TODO: generalize to metrizable spaces
 /-- A compact set in a pseudo emetric space is separable, i.e., it is a subset of the closure of a
-countable set.  -/
+countable set. -/
 theorem subset_countable_closure_of_compact {s : Set α} (hs : IsCompact s) :
     ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by
   refine subset_countable_closure_of_almost_dense_set s fun ε hε => ?_
@@ -279,11 +241,14 @@ instance (priority := 100) EMetricSpace.instT0Space : T0Space γ where
 
 /-- A map between emetric spaces is a uniform embedding if and only if the edistance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
-theorem EMetric.uniformEmbedding_iff' [EMetricSpace β] {f : γ → β} :
-    UniformEmbedding f ↔
+theorem EMetric.isUniformEmbedding_iff' [EMetricSpace β] {f : γ → β} :
+    IsUniformEmbedding f ↔
       (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, edist a b < δ → edist (f a) (f b) < ε) ∧
         ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, edist (f a) (f b) < ε → edist a b < δ := by
-  rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]
+  rw [isUniformEmbedding_iff_isUniformInducing, isUniformInducing_iff, uniformContinuous_iff]
+
+@[deprecated (since := "2024-10-01")]
+alias EMetric.uniformEmbedding_iff' := EMetric.isUniformEmbedding_iff'
 
 /-- If a `PseudoEMetricSpace` is a T₀ space, then it is an `EMetricSpace`. -/
 -- Porting note: made `reducible`;
@@ -299,22 +264,6 @@ corresponding to the product of uniform spaces, to avoid diamond problems. -/
 instance Prod.emetricSpaceMax [EMetricSpace β] : EMetricSpace (γ × β) :=
   .ofT0PseudoEMetricSpace _
 
-section Pi
-
-open Finset
-
-variable {π : β → Type*} [Fintype β]
-
-/-- The product of a finite number of emetric spaces, with the max distance, is still
-an emetric space.
-This construction would also work for infinite products, but it would not give rise
-to the product topology. Hence, we only formalize it in the good situation of finitely many
-spaces. -/
-instance emetricSpacePi [∀ b, EMetricSpace (π b)] : EMetricSpace (∀ b, π b) :=
-  .ofT0PseudoEMetricSpace _
-
-end Pi
-
 namespace EMetric
 
 /-- A compact set in an emetric space is separable, i.e., it is the closure of a countable set. -/
diff --git a/Mathlib/Topology/EMetricSpace/Defs.lean b/Mathlib/Topology/EMetricSpace/Defs.lean
index 2d530cb9ad131..2b6096cad74f3 100644
--- a/Mathlib/Topology/EMetricSpace/Defs.lean
+++ b/Mathlib/Topology/EMetricSpace/Defs.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
 import Mathlib.Data.ENNReal.Inv
-import Mathlib.Topology.UniformSpace.Basic
+import Mathlib.Topology.UniformSpace.OfFun
 
 /-!
 # Extended metric spaces
@@ -25,7 +25,7 @@ to `EMetricSpace` at the end.
 -/
 
 assert_not_exists Nat.instLocallyFiniteOrder
-assert_not_exists UniformEmbedding
+assert_not_exists IsUniformEmbedding
 assert_not_exists TendstoUniformlyOnFilter
 
 open Set Filter
@@ -447,7 +447,7 @@ theorem tendsto_nhdsWithin_nhds {a b} :
     Tendsto f (𝓝[s] a) (𝓝 b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → edist x a < δ → edist (f x) b < ε := by
   rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
-  simp only [mem_univ, true_and_iff]
+  simp only [mem_univ, true_and]
 
 theorem tendsto_nhds_nhds {a b} :
     Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x⦄, edist x a < δ → edist (f x) b < ε :=
@@ -491,7 +491,7 @@ theorem tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
 theorem tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
     Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (u n) a < ε :=
   (atTop_basis.tendsto_iff nhds_basis_eball).trans <| by
-    simp only [exists_prop, true_and_iff, mem_Ici, mem_ball]
+    simp only [exists_prop, true_and, mem_Ici, mem_ball]
 
 section Compact
 
diff --git a/Mathlib/Topology/EMetricSpace/Diam.lean b/Mathlib/Topology/EMetricSpace/Diam.lean
index d1ff6d783af68..18895f5377d0d 100644
--- a/Mathlib/Topology/EMetricSpace/Diam.lean
+++ b/Mathlib/Topology/EMetricSpace/Diam.lean
@@ -3,8 +3,7 @@ Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
-import Mathlib.Topology.EMetricSpace.Basic
-import Mathlib.Data.ENNReal.Real
+import Mathlib.Topology.EMetricSpace.Pi
 
 /-!
 # Diameters of sets in extended metric spaces
@@ -74,7 +73,7 @@ theorem diam_iUnion_mem_option {ι : Type*} (o : Option ι) (s : ι → Set α)
 theorem diam_insert : diam (insert x s) = max (⨆ y ∈ s, edist x y) (diam s) :=
   eq_of_forall_ge_iff fun d => by
     simp only [diam_le_iff, forall_mem_insert, edist_self, edist_comm x, max_le_iff, iSup_le_iff,
-      zero_le, true_and_iff, forall_and, and_self_iff, ← and_assoc]
+      zero_le, true_and, forall_and, and_self_iff, ← and_assoc]
 
 theorem diam_pair : diam ({x, y} : Set α) = edist x y := by
   simp only [iSup_singleton, diam_insert, diam_singleton, ENNReal.max_zero_right]
diff --git a/Mathlib/Topology/EMetricSpace/Paracompact.lean b/Mathlib/Topology/EMetricSpace/Paracompact.lean
index dc2427907aac4..f7e7082bc036a 100644
--- a/Mathlib/Topology/EMetricSpace/Paracompact.lean
+++ b/Mathlib/Topology/EMetricSpace/Paracompact.lean
@@ -3,10 +3,10 @@ Copyright (c) 2021 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.SetTheory.Ordinal.Basic
 import Mathlib.Tactic.GCongr
 import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.EMetricSpace.Basic
+import Mathlib.SetTheory.Cardinal.Basic
 
 /-!
 # (Extended) metric spaces are paracompact
diff --git a/Mathlib/Topology/EMetricSpace/Pi.lean b/Mathlib/Topology/EMetricSpace/Pi.lean
new file mode 100644
index 0000000000000..bbf987884575b
--- /dev/null
+++ b/Mathlib/Topology/EMetricSpace/Pi.lean
@@ -0,0 +1,90 @@
+/-
+Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
+-/
+import Mathlib.Topology.EMetricSpace.Basic
+import Mathlib.Topology.UniformSpace.Pi
+
+/-!
+# Indexed product of extended metric spaces
+-/
+
+open Set Filter
+
+universe u v w
+
+variable {α : Type u} {β : Type v} {X : Type*}
+
+open scoped Uniformity Topology NNReal ENNReal Pointwise
+
+variable [PseudoEMetricSpace α]
+
+open EMetric
+
+section Pi
+
+open Finset
+
+variable {π : β → Type*} [Fintype β]
+
+-- Porting note: reordered instances
+instance [∀ b, EDist (π b)] : EDist (∀ b, π b) where
+  edist f g := Finset.sup univ fun b => edist (f b) (g b)
+
+theorem edist_pi_def [∀ b, EDist (π b)] (f g : ∀ b, π b) :
+    edist f g = Finset.sup univ fun b => edist (f b) (g b) :=
+  rfl
+
+theorem edist_le_pi_edist [∀ b, EDist (π b)] (f g : ∀ b, π b) (b : β) :
+    edist (f b) (g b) ≤ edist f g :=
+  le_sup (f := fun b => edist (f b) (g b)) (Finset.mem_univ b)
+
+theorem edist_pi_le_iff [∀ b, EDist (π b)] {f g : ∀ b, π b} {d : ℝ≥0∞} :
+    edist f g ≤ d ↔ ∀ b, edist (f b) (g b) ≤ d :=
+  Finset.sup_le_iff.trans <| by simp only [Finset.mem_univ, forall_const]
+
+theorem edist_pi_const_le (a b : α) : (edist (fun _ : β => a) fun _ => b) ≤ edist a b :=
+  edist_pi_le_iff.2 fun _ => le_rfl
+
+@[simp]
+theorem edist_pi_const [Nonempty β] (a b : α) : (edist (fun _ : β => a) fun _ => b) = edist a b :=
+  Finset.sup_const univ_nonempty (edist a b)
+
+/-- The product of a finite number of pseudoemetric spaces, with the max distance, is still
+a pseudoemetric space.
+This construction would also work for infinite products, but it would not give rise
+to the product topology. Hence, we only formalize it in the good situation of finitely many
+spaces. -/
+instance pseudoEMetricSpacePi [∀ b, PseudoEMetricSpace (π b)] : PseudoEMetricSpace (∀ b, π b) where
+  edist_self f := bot_unique <| Finset.sup_le <| by simp
+  edist_comm f g := by simp [edist_pi_def, edist_comm]
+  edist_triangle f g h := edist_pi_le_iff.2 fun b => le_trans (edist_triangle _ (g b) _)
+    (add_le_add (edist_le_pi_edist _ _ _) (edist_le_pi_edist _ _ _))
+  toUniformSpace := Pi.uniformSpace _
+  uniformity_edist := by
+    simp only [Pi.uniformity, PseudoEMetricSpace.uniformity_edist, comap_iInf, gt_iff_lt,
+      preimage_setOf_eq, comap_principal, edist_pi_def]
+    rw [iInf_comm]; congr; funext ε
+    rw [iInf_comm]; congr; funext εpos
+    simp [setOf_forall, εpos]
+
+end Pi
+
+variable {γ : Type w} [EMetricSpace γ]
+
+section Pi
+
+open Finset
+
+variable {π : β → Type*} [Fintype β]
+
+/-- The product of a finite number of emetric spaces, with the max distance, is still
+an emetric space.
+This construction would also work for infinite products, but it would not give rise
+to the product topology. Hence, we only formalize it in the good situation of finitely many
+spaces. -/
+instance emetricSpacePi [∀ b, EMetricSpace (π b)] : EMetricSpace (∀ b, π b) :=
+  .ofT0PseudoEMetricSpace _
+
+end Pi
diff --git a/Mathlib/Topology/ExtendFrom.lean b/Mathlib/Topology/ExtendFrom.lean
index e425d7ce55e1a..1c043eb2756ea 100644
--- a/Mathlib/Topology/ExtendFrom.lean
+++ b/Mathlib/Topology/ExtendFrom.lean
@@ -12,7 +12,7 @@ The main definition of this file is `extendFrom A f` where `f : X → Y`
 and `A : Set X`. This defines a new function `g : X → Y` which maps any
 `x₀ : X` to the limit of `f` as `x` tends to `x₀`, if such a limit exists.
 
-This is analogous to the way `DenseInducing.extend` "extends" a function
+This is analogous to the way `IsDenseInducing.extend` "extends" a function
 `f : X → Z` to a function `g : Y → Z` along a dense inducing `i : X → Y`.
 
 The main theorem we prove about this definition is `continuousOn_extendFrom`
diff --git a/Mathlib/Topology/Exterior.lean b/Mathlib/Topology/Exterior.lean
new file mode 100644
index 0000000000000..fcf995f4c441a
--- /dev/null
+++ b/Mathlib/Topology/Exterior.lean
@@ -0,0 +1,98 @@
+/-
+Copyright (c) 2023 Yaël Dillies. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yaël Dillies, Yury Kudryashov
+-/
+import Mathlib.Topology.NhdsSet
+import Mathlib.Topology.Inseparable
+
+/-!
+# Exterior of a set
+
+We define `exterior s` to be the intersection of all neighborhoods of `s`,
+see `Topology/Defs/Filter`.
+Note that this construction has no standard name in the literature.
+
+In this file we prove basic properties of this operation.
+-/
+
+open Set Filter
+open scoped Topology
+
+variable {X : Type*} [TopologicalSpace X] {s t : Set X} {x y : X}
+
+lemma exterior_singleton_eq_ker_nhds (x : X) : exterior {x} = (𝓝 x).ker := by simp [exterior]
+
+@[simp]
+theorem mem_exterior_singleton : x ∈ exterior {y} ↔ x ⤳ y := by
+  rw [exterior_singleton_eq_ker_nhds, ker_nhds_eq_specializes, mem_setOf]
+
+lemma exterior_def (s : Set X) : exterior s = ⋂₀ {t : Set X | IsOpen t ∧ s ⊆ t} :=
+  (hasBasis_nhdsSet _).ker.trans sInter_eq_biInter.symm
+
+lemma mem_exterior : x ∈ exterior s ↔ ∀ U, IsOpen U → s ⊆ U → x ∈ U := by simp [exterior_def]
+
+lemma subset_exterior_iff : s ⊆ exterior t ↔ ∀ U, IsOpen U → t ⊆ U → s ⊆ U := by
+  simp [exterior_def]
+
+lemma subset_exterior : s ⊆ exterior s := subset_exterior_iff.2 fun _ _ ↦ id
+
+lemma exterior_minimal (h₁ : s ⊆ t) (h₂ : IsOpen t) : exterior s ⊆ t := by
+  rw [exterior_def]; exact sInter_subset_of_mem ⟨h₂, h₁⟩
+
+lemma IsOpen.exterior_eq (h : IsOpen s) : exterior s = s :=
+  (exterior_minimal Subset.rfl h).antisymm subset_exterior
+
+lemma IsOpen.exterior_subset (ht : IsOpen t) : exterior s ⊆ t ↔ s ⊆ t :=
+  ⟨subset_exterior.trans, fun h ↦ exterior_minimal h ht⟩
+
+@[deprecated (since := "2024-09-18")] alias IsOpen.exterior_subset_iff := IsOpen.exterior_subset
+
+@[simp]
+theorem exterior_iUnion {ι : Sort*} (s : ι → Set X) :
+    exterior (⋃ i, s i) = ⋃ i, exterior (s i) := by
+  simp only [exterior, nhdsSet_iUnion, ker_iSup]
+
+@[simp]
+theorem exterior_union (s t : Set X) : exterior (s ∪ t) = exterior s ∪ exterior t := by
+  simp only [exterior, nhdsSet_union, ker_sup]
+
+@[simp]
+theorem exterior_sUnion (S : Set (Set X)) : exterior (⋃₀ S) = ⋃ s ∈ S, exterior s := by
+  simp only [sUnion_eq_biUnion, exterior_iUnion]
+
+theorem mem_exterior_iff_specializes : x ∈ exterior s ↔ ∃ y ∈ s, x ⤳ y := calc
+  x ∈ exterior s ↔ x ∈ exterior (⋃ y ∈ s, {y}) := by simp
+  _ ↔ ∃ y ∈ s, x ⤳ y := by
+    simp only [exterior_iUnion, mem_exterior_singleton, mem_iUnion₂, exists_prop]
+
+@[mono] lemma exterior_mono : Monotone (exterior : Set X → Set X) :=
+  fun _s _t h ↦ ker_mono <| nhdsSet_mono h
+
+/-- This name was used to be used for the `Iff` version,
+see `exterior_subset_exterior_iff_nhdsSet`.
+-/
+@[gcongr] lemma exterior_subset_exterior (h : s ⊆ t) : exterior s ⊆ exterior t := exterior_mono h
+
+@[simp] lemma exterior_subset_exterior_iff_nhdsSet : exterior s ⊆ exterior t ↔ 𝓝ˢ s ≤ 𝓝ˢ t := by
+  simp (config := {contextual := true}) only [subset_exterior_iff, (hasBasis_nhdsSet _).ge_iff,
+    and_imp, IsOpen.mem_nhdsSet, IsOpen.exterior_subset]
+
+theorem exterior_eq_exterior_iff_nhdsSet : exterior s = exterior t ↔ 𝓝ˢ s = 𝓝ˢ t := by
+  simp [le_antisymm_iff]
+
+lemma specializes_iff_exterior_subset : x ⤳ y ↔ exterior {x} ⊆ exterior {y} := by
+  simp [Specializes]
+
+@[simp] lemma exterior_empty : exterior (∅ : Set X) = ∅ := isOpen_empty.exterior_eq
+@[simp] lemma exterior_univ : exterior (univ : Set X) = univ := isOpen_univ.exterior_eq
+
+@[simp] lemma exterior_eq_empty : exterior s = ∅ ↔ s = ∅ :=
+  ⟨eq_bot_mono subset_exterior, by rintro rfl; exact exterior_empty⟩
+
+@[simp] lemma nhdsSet_exterior (s : Set X) : 𝓝ˢ (exterior s) = 𝓝ˢ s := by
+  refine le_antisymm ((hasBasis_nhdsSet _).ge_iff.2 ?_) (nhdsSet_mono subset_exterior)
+  exact fun U ⟨hUo, hsU⟩ ↦ hUo.mem_nhdsSet.2 <| hUo.exterior_subset.2 hsU
+
+@[simp] lemma exterior_exterior (s : Set X) : exterior (exterior s) = exterior s := by
+  simp only [exterior_eq_exterior_iff_nhdsSet, nhdsSet_exterior]
diff --git a/Mathlib/Topology/ExtremallyDisconnected.lean b/Mathlib/Topology/ExtremallyDisconnected.lean
index a5d3b08560a46..f81a6b4448fd2 100644
--- a/Mathlib/Topology/ExtremallyDisconnected.lean
+++ b/Mathlib/Topology/ExtremallyDisconnected.lean
@@ -88,7 +88,7 @@ theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneC
   let h : StoneCech X → Y := stoneCechExtend ht
   have hh : Continuous h := continuous_stoneCechExtend ht
   refine ⟨h, hh, denseRange_stoneCechUnit.equalizer (hg.comp hh) hf ?_⟩
-  rw [comp.assoc, stoneCechExtend_extends ht, ← comp.assoc, hs, id_comp]
+  rw [comp_assoc, stoneCechExtend_extends ht, ← comp_assoc, hs, id_comp]
 
 protected theorem CompactT2.Projective.extremallyDisconnected [CompactSpace X] [T2Space X]
     (h : CompactT2.Projective X) : ExtremallyDisconnected X := by
@@ -271,7 +271,7 @@ protected theorem CompactT2.ExtremallyDisconnected.projective [ExtremallyDisconn
   have π₂_cont : Continuous π₂ := continuous_snd.comp continuous_subtype_val
   refine ⟨E.restrict π₂ ∘ ρ'.symm, ⟨π₂_cont.continuousOn.restrict.comp ρ'.symm.continuous, ?_⟩⟩
   suffices f ∘ E.restrict π₂ = φ ∘ ρ' by
-    rw [← comp.assoc, this, comp.assoc, Homeomorph.self_comp_symm, comp_id]
+    rw [← comp_assoc, this, comp_assoc, Homeomorph.self_comp_symm, comp_id]
   ext x
   exact x.val.mem.symm
 
diff --git a/Mathlib/Topology/FiberBundle/Basic.lean b/Mathlib/Topology/FiberBundle/Basic.lean
index 4606e9604e100..39a1e8d2edc2b 100644
--- a/Mathlib/Topology/FiberBundle/Basic.lean
+++ b/Mathlib/Topology/FiberBundle/Basic.lean
@@ -439,7 +439,7 @@ def trivChange (i j : ι) : PartialHomeomorph (B × F) (B × F) where
     exacts [hx.1, ⟨⟨hx.1, hx.2⟩, hx.1⟩]
   right_inv' := by
     rintro ⟨x, v⟩ hx
-    simp only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ] at hx
+    simp only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true, mem_univ] at hx
     dsimp only
     rw [Z.coordChange_comp, Z.coordChange_self]
     · exact hx.2
@@ -469,9 +469,9 @@ def localTrivAsPartialEquiv (i : ι) : PartialEquiv Z.TotalSpace (B × F) where
   invFun p := ⟨p.1, Z.coordChange i (Z.indexAt p.1) p.1 p.2⟩
   toFun p := ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩
   map_source' p hp := by
-    simpa only [Set.mem_preimage, and_true_iff, Set.mem_univ, Set.prod_mk_mem_set_prod_eq] using hp
+    simpa only [Set.mem_preimage, and_true, Set.mem_univ, Set.prod_mk_mem_set_prod_eq] using hp
   map_target' p hp := by
-    simpa only [Set.mem_preimage, and_true_iff, Set.mem_univ, Set.mem_prod] using hp
+    simpa only [Set.mem_preimage, and_true, Set.mem_univ, Set.mem_prod] using hp
   left_inv' := by
     rintro ⟨x, v⟩ hx
     replace hx : x ∈ Z.baseSet i := hx
@@ -479,7 +479,7 @@ def localTrivAsPartialEquiv (i : ι) : PartialEquiv Z.TotalSpace (B × F) where
     rw [Z.coordChange_comp, Z.coordChange_self] <;> apply_rules [mem_baseSet_at, mem_inter]
   right_inv' := by
     rintro ⟨x, v⟩ hx
-    simp only [prod_mk_mem_set_prod_eq, and_true_iff, mem_univ] at hx
+    simp only [prod_mk_mem_set_prod_eq, and_true, mem_univ] at hx
     dsimp only
     rw [Z.coordChange_comp, Z.coordChange_self]
     exacts [hx, ⟨⟨hx, Z.mem_baseSet_at _⟩, hx⟩]
@@ -493,7 +493,7 @@ theorem mem_localTrivAsPartialEquiv_source (p : Z.TotalSpace) :
 theorem mem_localTrivAsPartialEquiv_target (p : B × F) :
     p ∈ (Z.localTrivAsPartialEquiv i).target ↔ p.1 ∈ Z.baseSet i := by
   erw [mem_prod]
-  simp only [and_true_iff, mem_univ]
+  simp only [and_true, mem_univ]
 
 theorem localTrivAsPartialEquiv_apply (p : Z.TotalSpace) :
     (Z.localTrivAsPartialEquiv i) p = ⟨p.1, Z.coordChange (Z.indexAt p.1) i p.1 p.2⟩ :=
@@ -508,9 +508,9 @@ theorem localTrivAsPartialEquiv_trans (i j : ι) :
     simp only [mem_localTrivAsPartialEquiv_target, mfld_simps]
     rfl
   · rintro ⟨x, v⟩ hx
-    simp only [trivChange, localTrivAsPartialEquiv, PartialEquiv.symm, true_and_iff,
+    simp only [trivChange, localTrivAsPartialEquiv, PartialEquiv.symm,
       Prod.mk.inj_iff, prod_mk_mem_set_prod_eq, PartialEquiv.trans_source, mem_inter_iff,
-      and_true_iff, mem_preimage, proj, mem_univ, eq_self_iff_true, (· ∘ ·),
+      mem_preimage, proj, mem_univ, eq_self_iff_true, (· ∘ ·),
       PartialEquiv.coe_trans, TotalSpace.proj] at hx ⊢
     simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at]
 
@@ -674,7 +674,7 @@ instance fiberBundle : FiberBundle F Z.Fiber where
   totalSpaceMk_inducing' b := inducing_iff_nhds.2 fun x ↦ by
     rw [(Z.localTrivAt b).nhds_eq_comap_inf_principal (mk_mem_localTrivAt_source _ _ _), comap_inf,
       comap_principal, comap_comap]
-    simp only [(· ∘ ·), localTrivAt_apply_mk, Trivialization.coe_coe,
+    simp only [Function.comp_def, localTrivAt_apply_mk, Trivialization.coe_coe,
       ← (embedding_prod_mk b).nhds_eq_comap]
     convert_to 𝓝 x = 𝓝 x ⊓ 𝓟 univ
     · congr
diff --git a/Mathlib/Topology/FiberBundle/Constructions.lean b/Mathlib/Topology/FiberBundle/Constructions.lean
index 01fe742ebee19..b5cac2629b8cf 100644
--- a/Mathlib/Topology/FiberBundle/Constructions.lean
+++ b/Mathlib/Topology/FiberBundle/Constructions.lean
@@ -138,13 +138,12 @@ theorem Prod.continuous_to_fun : ContinuousOn (Prod.toFun' e₁ e₂)
   have hf₁ : Continuous f₁ := (Prod.inducing_diag F₁ E₁ F₂ E₂).continuous
   have hf₂ : ContinuousOn f₂ (e₁.source ×ˢ e₂.source) :=
     e₁.toPartialHomeomorph.continuousOn.prod_map e₂.toPartialHomeomorph.continuousOn
-  have hf₃ : Continuous f₃ :=
-    (continuous_fst.comp continuous_fst).prod_mk (continuous_snd.prod_map continuous_snd)
+  have hf₃ : Continuous f₃ := by fun_prop
   refine ((hf₃.comp_continuousOn hf₂).comp hf₁.continuousOn ?_).congr ?_
   · rw [e₁.source_eq, e₂.source_eq]
     exact mapsTo_preimage _ _
   rintro ⟨b, v₁, v₂⟩ ⟨hb₁, _⟩
-  simp only [f₃, Prod.toFun', Prod.mk.inj_iff, Function.comp_apply, and_true_iff]
+  simp only [f₃, Prod.toFun', Prod.mk.inj_iff, Function.comp_apply, and_true]
   rw [e₁.coe_fst]
   rw [e₁.source_eq, mem_preimage]
   exact hb₁
@@ -176,8 +175,7 @@ theorem Prod.right_inv {x : B × F₁ × F₂}
 theorem Prod.continuous_inv_fun :
     ContinuousOn (Prod.invFun' e₁ e₂) ((e₁.baseSet ∩ e₂.baseSet) ×ˢ univ) := by
   rw [(Prod.inducing_diag F₁ E₁ F₂ E₂).continuousOn_iff]
-  have H₁ : Continuous fun p : B × F₁ × F₂ ↦ ((p.1, p.2.1), (p.1, p.2.2)) :=
-    (continuous_id.prod_map continuous_fst).prod_mk (continuous_id.prod_map continuous_snd)
+  have H₁ : Continuous fun p : B × F₁ × F₂ ↦ ((p.1, p.2.1), (p.1, p.2.2)) := by fun_prop
   refine (e₁.continuousOn_symm.prod_map e₂.continuousOn_symm).comp H₁.continuousOn ?_
   exact fun x h ↦ ⟨⟨h.1.1, mem_univ _⟩, ⟨h.1.2, mem_univ _⟩⟩
 
@@ -226,7 +224,7 @@ variable [∀ x, Zero (E₁ x)] [∀ x, Zero (E₂ x)] [∀ x : B, TopologicalSp
 noncomputable instance FiberBundle.prod : FiberBundle (F₁ × F₂) (E₁ ×ᵇ E₂) where
   totalSpaceMk_inducing' b := by
     rw [← (Prod.inducing_diag F₁ E₁ F₂ E₂).of_comp_iff]
-    exact (totalSpaceMk_inducing F₁ E₁ b).prod_map (totalSpaceMk_inducing F₂ E₂ b)
+    exact (totalSpaceMk_inducing F₁ E₁ b).prodMap (totalSpaceMk_inducing F₂ E₂ b)
   trivializationAtlas' := { e |
     ∃ (e₁ : Trivialization F₁ (π F₁ E₁)) (e₂ : Trivialization F₂ (π F₂ E₂))
       (_ : MemTrivializationAtlas e₁) (_ : MemTrivializationAtlas e₂),
@@ -292,7 +290,7 @@ variable [TopologicalSpace F] [TopologicalSpace B]
 theorem Pullback.continuous_totalSpaceMk [∀ x, TopologicalSpace (E x)] [FiberBundle F E]
     {f : B' → B} {x : B'} : Continuous (@TotalSpace.mk _ F (f *ᵖ E) x) := by
   simp only [continuous_iff_le_induced, Pullback.TotalSpace.topologicalSpace, induced_compose,
-    induced_inf, Function.comp, induced_const, top_inf_eq, pullbackTopology_def]
+    induced_inf, Function.comp_def, induced_const, top_inf_eq, pullbackTopology_def]
   exact le_of_eq (FiberBundle.totalSpaceMk_inducing F E (f x)).induced
 
 variable {E F}
@@ -309,7 +307,7 @@ noncomputable def Trivialization.pullback (e : Trivialization F (π F E)) (f : K
   target := (f ⁻¹' e.baseSet) ×ˢ univ
   map_source' x h := by
     simp_rw [e.source_eq, mem_preimage, Pullback.lift_proj] at h
-    simp_rw [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff, mem_preimage, h]
+    simp_rw [prod_mk_mem_set_prod_eq, mem_univ, and_true, mem_preimage, h]
   map_target' y h := by
     rw [mem_prod, mem_preimage] at h
     simp_rw [e.source_eq, mem_preimage, Pullback.lift_proj, h.1]
@@ -317,7 +315,7 @@ noncomputable def Trivialization.pullback (e : Trivialization F (π F E)) (f : K
     simp_rw [mem_preimage, e.mem_source, Pullback.lift_proj] at h
     simp_rw [Pullback.lift, e.symm_apply_apply_mk h]
   right_inv' x h := by
-    simp_rw [mem_prod, mem_preimage, mem_univ, and_true_iff] at h
+    simp_rw [mem_prod, mem_preimage, mem_univ, and_true] at h
     simp_rw [Pullback.lift_mk, e.apply_mk_symm h]
   open_source := by
     simp_rw [e.source_eq, ← preimage_comp]
@@ -330,11 +328,11 @@ noncomputable def Trivialization.pullback (e : Trivialization F (π F E)) (f : K
         e.continuousOn.comp (Pullback.continuous_lift F E f).continuousOn Subset.rfl)
   continuousOn_invFun := by
     dsimp only
-    simp_rw [(inducing_pullbackTotalSpaceEmbedding F E f).continuousOn_iff, Function.comp,
+    simp_rw [(inducing_pullbackTotalSpaceEmbedding F E f).continuousOn_iff, Function.comp_def,
       pullbackTotalSpaceEmbedding]
     refine
       continuousOn_fst.prod
-        (e.continuousOn_symm.comp ((map_continuous f).prod_map continuous_id).continuousOn
+        (e.continuousOn_symm.comp ((map_continuous f).prodMap continuous_id).continuousOn
           Subset.rfl)
   source_eq := by
     dsimp only
diff --git a/Mathlib/Topology/FiberBundle/Trivialization.lean b/Mathlib/Topology/FiberBundle/Trivialization.lean
index c1e63718a72cd..87a1362341a73 100644
--- a/Mathlib/Topology/FiberBundle/Trivialization.lean
+++ b/Mathlib/Topology/FiberBundle/Trivialization.lean
@@ -74,7 +74,7 @@ variable (e : Pretrivialization F proj) {x : Z}
 /-- Coercion of a pretrivialization to a function. We don't use `e.toFun` in the `CoeFun` instance
 because it is actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about
 `toPartialEquiv`. While we may want to switch to this behavior later, doing it mid-port will break a
-lot of proofs.  -/
+lot of proofs. -/
 @[coe] def toFun' : Z → (B × F) := e.toFun
 
 instance : CoeFun (Pretrivialization F proj) fun _ => Z → B × F := ⟨toFun'⟩
@@ -168,7 +168,7 @@ theorem preimage_symm_proj_inter (s : Set B) :
     e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ = (s ∩ e.baseSet) ×ˢ univ := by
   ext ⟨x, y⟩
   suffices x ∈ e.baseSet → (proj (e.toPartialEquiv.symm (x, y)) ∈ s ↔ x ∈ s) by
-    simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]
+    simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true, mem_univ, and_congr_left_iff]
   intro h
   rw [e.proj_symm_apply' h]
 
@@ -282,7 +282,7 @@ lemma ext' (e e' : Trivialization F proj) (h₁ : e.toPartialHomeomorph = e'.toP
 /-- Coercion of a trivialization to a function. We don't use `e.toFun` in the `CoeFun` instance
 because it is actually `e.toPartialEquiv.toFun`, so `simp` will apply lemmas about
 `toPartialEquiv`. While we may want to switch to this behavior later, doing it mid-port will break a
-lot of proofs.  -/
+lot of proofs. -/
 @[coe] def toFun' : Z → (B × F) := e.toFun
 
 /-- Natural identification as a `Pretrivialization`. -/
@@ -465,7 +465,7 @@ theorem preimageSingletonHomeomorph_symm_apply {b : B} (hb : b ∈ e.baseSet) (p
       ⟨e.symm (b, p), by rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]⟩ :=
   rfl
 
-/-- In the domain of a bundle trivialization, the projection is continuous-/
+/-- In the domain of a bundle trivialization, the projection is continuous -/
 theorem continuousAt_proj (ex : x ∈ e.source) : ContinuousAt proj x :=
   (e.map_proj_nhds ex).le
 
@@ -475,7 +475,7 @@ protected def compHomeomorph {Z' : Type*} [TopologicalSpace Z'] (h : Z' ≃ₜ Z
   toPartialHomeomorph := h.toPartialHomeomorph.trans e.toPartialHomeomorph
   baseSet := e.baseSet
   open_baseSet := e.open_baseSet
-  source_eq := by simp [source_eq, preimage_preimage, (· ∘ ·)]
+  source_eq := by simp [source_eq, preimage_preimage, Function.comp_def]
   target_eq := by simp [target_eq]
   proj_toFun p hp := by
     have hp : h p ∈ e.source := by simpa using hp
diff --git a/Mathlib/Topology/FiberPartition.lean b/Mathlib/Topology/FiberPartition.lean
new file mode 100644
index 0000000000000..8e21ee229b9e1
--- /dev/null
+++ b/Mathlib/Topology/FiberPartition.lean
@@ -0,0 +1,69 @@
+/-
+Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Dagur Asgeirsson
+-/
+import Mathlib.Topology.LocallyConstant.Basic
+import Mathlib.Logic.Function.FiberPartition
+/-!
+
+This file provides some API surrounding `Function.Fiber` (see
+`Mathlib.Logic.Function.FiberPartition`) in the presence of a topology on the domain of the
+function.
+
+Note: this API is designed to be useful when defining the counit of the adjunction between
+the functor which takes a set to the condensed set corresponding to locally constant maps to that
+set, and the forgetful functor from the category of condensed sets to the category of sets
+(see PR #14027).
+-/
+
+
+open Function
+
+variable {S Y : Type*} (f : S → Y)
+
+namespace TopologicalSpace.Fiber
+
+variable [TopologicalSpace S]
+
+/-- The canonical map from the disjoint union induced by `f` to `S`. -/
+@[simps apply]
+def sigmaIsoHom : C((x : Fiber f) × x.val, S) where
+  toFun | ⟨a, x⟩ => x.val
+
+lemma sigmaIsoHom_inj : Function.Injective (sigmaIsoHom f) := by
+  rintro ⟨⟨_, _, rfl⟩, ⟨_, hx⟩⟩ ⟨⟨_, _, rfl⟩, ⟨_, hy⟩⟩ h
+  refine Sigma.subtype_ext ?_ h
+  simp only [sigmaIsoHom_apply] at h
+  rw [Set.mem_preimage, Set.mem_singleton_iff] at hx hy
+  simp [← hx, ← hy, h]
+
+lemma sigmaIsoHom_surj : Function.Surjective (sigmaIsoHom f) :=
+  fun _ ↦ ⟨⟨⟨_, ⟨⟨_, Set.mem_range_self _⟩, rfl⟩⟩, ⟨_, rfl⟩⟩, rfl⟩
+
+/-- The inclusion map from a component of the disjoint union induced by `f` into `S`. -/
+def sigmaIncl (a : Fiber f) : C(a.val, S) where
+  toFun x := x.val
+
+/-- The inclusion map from a fiber of a composition into the intermediate fiber. -/
+def sigmaInclIncl {X : Type*} (g : Y → X) (a : Fiber (g ∘ f))
+    (b : Fiber (f ∘ (sigmaIncl (g ∘ f) a))) :
+    C(b.val, (Fiber.mk f (b.preimage).val).val) where
+  toFun x := ⟨x.val.val, by
+    have := x.prop
+    simp only [sigmaIncl, ContinuousMap.coe_mk, Fiber.mem_iff_eq_image, comp_apply] at this
+    rw [Fiber.mem_iff_eq_image, Fiber.mk_image, this, ← Fiber.map_preimage_eq_image]
+    simp [sigmaIncl]⟩
+
+variable (l : LocallyConstant S Y) [CompactSpace S]
+
+instance (x : Fiber l) : CompactSpace x.val := by
+  obtain ⟨y, hy⟩ := x.prop
+  rw [← isCompact_iff_compactSpace, ← hy]
+  exact (l.2.isClosed_fiber _).isCompact
+
+instance : Finite (Fiber l) :=
+  have : Finite (Set.range l) := l.range_finite
+  Finite.Set.finite_range _
+
+end TopologicalSpace.Fiber
diff --git a/Mathlib/Topology/Filter.lean b/Mathlib/Topology/Filter.lean
index abaddb408699f..17a22a272b238 100644
--- a/Mathlib/Topology/Filter.lean
+++ b/Mathlib/Topology/Filter.lean
@@ -71,7 +71,7 @@ theorem nhds_eq (l : Filter α) : 𝓝 l = l.lift' (Iic ∘ 𝓟) :=
       (· ∘ ·), mem_Iic, le_principal_iff]
 
 theorem nhds_eq' (l : Filter α) : 𝓝 l = l.lift' fun s => { l' | s ∈ l' } := by
-  simpa only [(· ∘ ·), Iic_principal] using nhds_eq l
+  simpa only [Function.comp_def, Iic_principal] using nhds_eq l
 
 protected theorem tendsto_nhds {la : Filter α} {lb : Filter β} {f : α → Filter β} :
     Tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ a in la, s ∈ f a := by
@@ -105,7 +105,7 @@ theorem mem_nhds_iff' {l : Filter α} {S : Set (Filter α)} :
 
 @[simp]
 theorem nhds_bot : 𝓝 (⊥ : Filter α) = pure ⊥ := by
-  simp [nhds_eq, (· ∘ ·), lift'_bot monotone_principal.Iic]
+  simp [nhds_eq, Function.comp_def, lift'_bot monotone_principal.Iic]
 
 @[simp]
 theorem nhds_top : 𝓝 (⊤ : Filter α) = ⊤ := by simp [nhds_eq]
@@ -131,8 +131,8 @@ theorem monotone_nhds : Monotone (𝓝 : Filter α → Filter (Filter α)) :=
   Monotone.of_map_inf nhds_inf
 
 theorem sInter_nhds (l : Filter α) : ⋂₀ { s | s ∈ 𝓝 l } = Iic l := by
-  simp_rw [nhds_eq, (· ∘ ·), sInter_lift'_sets monotone_principal.Iic, Iic, le_principal_iff,
-    ← setOf_forall, ← Filter.le_def]
+  simp_rw [nhds_eq, Function.comp_def, sInter_lift'_sets monotone_principal.Iic, Iic,
+    le_principal_iff, ← setOf_forall, ← Filter.le_def]
 
 @[simp]
 theorem nhds_mono {l₁ l₂ : Filter α} : 𝓝 l₁ ≤ 𝓝 l₂ ↔ l₁ ≤ l₂ := by
diff --git a/Mathlib/Topology/Germ.lean b/Mathlib/Topology/Germ.lean
index 5b1e7856f5d62..35d47bcfa9b8a 100644
--- a/Mathlib/Topology/Germ.lean
+++ b/Mathlib/Topology/Germ.lean
@@ -30,9 +30,6 @@ to the corresponding germ of functions `X → Z` at `x ∈ X` resp. `Y → Z` at
 `f` is constant.
 -/
 
-variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
-  [NormedAddCommGroup G] [NormedSpace ℝ G]
-
 open scoped Topology
 
 open Filter Set
diff --git a/Mathlib/Topology/Gluing.lean b/Mathlib/Topology/Gluing.lean
index 701ab09498e3c..6dc61fbb47e77 100644
--- a/Mathlib/Topology/Gluing.lean
+++ b/Mathlib/Topology/Gluing.lean
@@ -153,7 +153,7 @@ theorem eqvGen_of_π_eq
     -- Porting note: was `{x y : ∐ D.U} (h : 𝖣.π x = 𝖣.π y)`
     {x y : sigmaObj (β := D.toGlueData.J) (C := TopCat) D.toGlueData.U}
     (h : 𝖣.π x = 𝖣.π y) :
-    EqvGen
+    Relation.EqvGen
       -- Porting note: was (Types.CoequalizerRel 𝖣.diagram.fstSigmaMap 𝖣.diagram.sndSigmaMap)
       (Types.CoequalizerRel
         (X := sigmaObj (β := D.toGlueData.diagram.L) (C := TopCat) (D.toGlueData.diagram).left)
@@ -200,7 +200,7 @@ theorem ι_eq_iff_rel (i j : D.J) (x : D.U i) (y : D.U j) :
       show _ = Sigma.mk j y from ConcreteCategory.congr_hom (sigmaIsoSigma.{_, u} D.U).inv_hom_id _]
     change InvImage D.Rel (sigmaIsoSigma.{_, u} D.U).hom _ _
     rw [← (InvImage.equivalence _ _ D.rel_equiv).eqvGen_iff]
-    refine EqvGen.mono ?_ (D.eqvGen_of_π_eq h : _)
+    refine Relation.EqvGen.mono ?_ (D.eqvGen_of_π_eq h : _)
     rintro _ _ ⟨x⟩
     obtain ⟨⟨⟨i, j⟩, y⟩, rfl⟩ :=
       (ConcreteCategory.bijective_of_isIso (sigmaIsoSigma.{u, u} _).inv).2 x
@@ -342,17 +342,14 @@ instance (h : MkCore.{u}) (i j : h.J) : IsIso (h.t i j) := by
 
 /-- (Implementation) the restricted transition map to be fed into `TopCat.GlueData`. -/
 def MkCore.t' (h : MkCore.{u}) (i j k : h.J) :
-    pullback (h.V i j).inclusion (h.V i k).inclusion ⟶
-      pullback (h.V j k).inclusion (h.V j i).inclusion := by
+    pullback (h.V i j).inclusion' (h.V i k).inclusion' ⟶
+      pullback (h.V j k).inclusion' (h.V j i).inclusion' := by
   refine (pullbackIsoProdSubtype _ _).hom ≫ ⟨?_, ?_⟩ ≫ (pullbackIsoProdSubtype _ _).inv
   · intro x
     refine ⟨⟨⟨(h.t i j x.1.1).1, ?_⟩, h.t i j x.1.1⟩, rfl⟩
     rcases x with ⟨⟨⟨x, hx⟩, ⟨x', hx'⟩⟩, rfl : x = x'⟩
     exact h.t_inter _ ⟨x, hx⟩ hx'
-  -- Porting note: was `continuity`, see https://github.com/leanprover-community/mathlib4/issues/5030
-  have : Continuous (h.t i j) := map_continuous (self := ContinuousMap.toContinuousMapClass) _
-  set_option tactic.skipAssignedInstances false in
-  exact ((Continuous.subtype_mk (by fun_prop) _).prod_mk (by fun_prop)).subtype_mk _
+  fun_prop
 
 /-- This is a constructor of `TopCat.GlueData` whose arguments are in terms of elements and
 intersections rather than subobjects and pullbacks. Please refer to `TopCat.GlueData.MkCore` for
@@ -361,7 +358,7 @@ def mk' (h : MkCore.{u}) : TopCat.GlueData where
   J := h.J
   U := h.U
   V i := (Opens.toTopCat _).obj (h.V i.1 i.2)
-  f i j := (h.V i j).inclusion
+  f i j := (h.V i j).inclusion'
   f_id i := by
     -- Porting note (#12129): additional beta reduction needed
     beta_reduce
@@ -410,7 +407,7 @@ def ofOpenSubsets : TopCat.GlueData.{u} :=
   mk'.{u}
     { J
       U := fun i => (Opens.toTopCat <| TopCat.of α).obj (U i)
-      V := fun i j => (Opens.map <| Opens.inclusion _).obj (U j)
+      V := fun i j => (Opens.map <| Opens.inclusion' _).obj (U j)
       t := fun i j => ⟨fun x => ⟨⟨x.1.1, x.2⟩, x.1.2⟩, by
         -- Porting note: was `continuity`, see https://github.com/leanprover-community/mathlib4/issues/5030
         refine Continuous.subtype_mk ?_ ?_
@@ -429,13 +426,13 @@ This map is an open embedding (`fromOpenSubsetsGlue_openEmbedding`),
 and its range is `⋃ i, (U i : Set α)` (`range_fromOpenSubsetsGlue`).
 -/
 def fromOpenSubsetsGlue : (ofOpenSubsets U).toGlueData.glued ⟶ TopCat.of α :=
-  Multicoequalizer.desc _ _ (fun x => Opens.inclusion _) (by rintro ⟨i, j⟩; ext x; rfl)
+  Multicoequalizer.desc _ _ (fun x => Opens.inclusion' _) (by rintro ⟨i, j⟩; ext x; rfl)
 
 -- Porting note: `elementwise` here produces a bad lemma,
 -- where too much has been simplified, despite the `nosimp`.
 @[simp, elementwise nosimp]
 theorem ι_fromOpenSubsetsGlue (i : J) :
-    (ofOpenSubsets U).toGlueData.ι i ≫ fromOpenSubsetsGlue U = Opens.inclusion _ :=
+    (ofOpenSubsets U).toGlueData.ι i ≫ fromOpenSubsetsGlue U = Opens.inclusion' _ :=
   Multicoequalizer.π_desc _ _ _ _ _
 
 theorem fromOpenSubsetsGlue_injective : Function.Injective (fromOpenSubsetsGlue U) := by
@@ -457,7 +454,7 @@ theorem fromOpenSubsetsGlue_isOpenMap : IsOpenMap (fromOpenSubsetsGlue U) := by
   rw [isOpen_iff_forall_mem_open]
   rintro _ ⟨x, hx, rfl⟩
   obtain ⟨i, ⟨x, hx'⟩, rfl⟩ := (ofOpenSubsets U).ι_jointly_surjective x
-  use fromOpenSubsetsGlue U '' s ∩ Set.range (@Opens.inclusion (TopCat.of α) (U i))
+  use fromOpenSubsetsGlue U '' s ∩ Set.range (@Opens.inclusion' (TopCat.of α) (U i))
   use Set.inter_subset_left
   constructor
   · erw [← Set.image_preimage_eq_inter_range]
diff --git a/Mathlib/Topology/Hom/Open.lean b/Mathlib/Topology/Hom/Open.lean
index bec44e93eff1e..86051435726a8 100644
--- a/Mathlib/Topology/Hom/Open.lean
+++ b/Mathlib/Topology/Hom/Open.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # Continuous open maps
diff --git a/Mathlib/Topology/Homeomorph.lean b/Mathlib/Topology/Homeomorph.lean
index 8a42264219207..6d33a57317001 100644
--- a/Mathlib/Topology/Homeomorph.lean
+++ b/Mathlib/Topology/Homeomorph.lean
@@ -31,7 +31,7 @@ open Set Filter Function
 
 open Topology
 
-variable {X : Type*} {Y : Type*} {Z : Type*}
+variable {X Y W Z : Type*}
 
 -- not all spaces are homeomorphic to each other
 /-- Homeomorphism between `X` and `Y`, also called topological isomorphism -/
@@ -47,18 +47,18 @@ infixl:25 " ≃ₜ " => Homeomorph
 
 namespace Homeomorph
 
-variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
+variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace W] [TopologicalSpace Z]
   {X' Y' : Type*} [TopologicalSpace X'] [TopologicalSpace Y']
 
 theorem toEquiv_injective : Function.Injective (toEquiv : X ≃ₜ Y → X ≃ Y)
   | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
 
 instance : EquivLike (X ≃ₜ Y) X Y where
-  coe := fun h => h.toEquiv
-  inv := fun h => h.toEquiv.symm
-  left_inv := fun h => h.left_inv
-  right_inv := fun h => h.right_inv
-  coe_injective' := fun _ _ H _ => toEquiv_injective <| DFunLike.ext' H
+  coe h := h.toEquiv
+  inv h := h.toEquiv.symm
+  left_inv h := h.left_inv
+  right_inv h := h.right_inv
+  coe_injective' _ _ H _ := toEquiv_injective <| DFunLike.ext' H
 
 instance : CoeFun (X ≃ₜ Y) fun _ ↦ X → Y := ⟨DFunLike.coe⟩
 
@@ -308,9 +308,12 @@ protected theorem t2Space [T2Space X] (h : X ≃ₜ Y) : T2Space Y :=
 protected theorem t3Space [T3Space X] (h : X ≃ₜ Y) : T3Space Y :=
   h.symm.embedding.t3Space
 
-protected theorem denseEmbedding (h : X ≃ₜ Y) : DenseEmbedding h :=
+theorem isDenseEmbedding (h : X ≃ₜ Y) : IsDenseEmbedding h :=
   { h.embedding with dense := h.surjective.denseRange }
 
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding := isDenseEmbedding
+
 @[simp]
 theorem isOpen_preimage (h : X ≃ₜ Y) {s : Set Y} : IsOpen (h ⁻¹' s) ↔ IsOpen s :=
   h.quotientMap.isOpen_preimage
@@ -451,14 +454,14 @@ theorem comp_continuousWithinAt_iff (h : X ≃ₜ Y) (f : Z → X) (s : Set Z) (
 theorem comp_isOpenMap_iff (h : X ≃ₜ Y) {f : Z → X} : IsOpenMap (h ∘ f) ↔ IsOpenMap f := by
   refine ⟨?_, fun hf => h.isOpenMap.comp hf⟩
   intro hf
-  rw [← Function.id_comp f, ← h.symm_comp_self, Function.comp.assoc]
+  rw [← Function.id_comp f, ← h.symm_comp_self, Function.comp_assoc]
   exact h.symm.isOpenMap.comp hf
 
 @[simp]
 theorem comp_isOpenMap_iff' (h : X ≃ₜ Y) {f : Y → Z} : IsOpenMap (f ∘ h) ↔ IsOpenMap f := by
   refine ⟨?_, fun hf => hf.comp h.isOpenMap⟩
   intro hf
-  rw [← Function.comp_id f, ← h.self_comp_symm, ← Function.comp.assoc]
+  rw [← Function.comp_id f, ← h.self_comp_symm, ← Function.comp_assoc]
   exact hf.comp h.symm.isOpenMap
 
 /-- A homeomorphism `h : X ≃ₜ Y` lifts to a homeomorphism between subtypes corresponding to
@@ -495,8 +498,6 @@ def sumCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : X ⊕ Y ≃ₜ X' ⊕ Y
 
 /-- Product of two homeomorphisms. -/
 def prodCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : X × Y ≃ₜ X' × Y' where
-  continuous_toFun := h₁.continuous.prod_map h₂.continuous
-  continuous_invFun := h₁.symm.continuous.prod_map h₂.symm.continuous
   toEquiv := h₁.toEquiv.prodCongr h₂.toEquiv
 
 @[simp]
@@ -508,9 +509,81 @@ theorem prodCongr_symm (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') :
 theorem coe_prodCongr (h₁ : X ≃ₜ X') (h₂ : Y ≃ₜ Y') : ⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ :=
   rfl
 
-section
+-- Commutativity and associativity of the disjoint union of topological spaces,
+-- and the sum with an empty space.
+section sum
 
-variable (X Y Z)
+variable (X Y W Z)
+
+/-- `X ⊕ Y` is homeomorphic to `Y ⊕ X`. -/
+def sumComm : X ⊕ Y ≃ₜ Y ⊕ X where
+  toEquiv := Equiv.sumComm X Y
+  continuous_toFun := continuous_sum_swap
+  continuous_invFun := continuous_sum_swap
+
+@[simp]
+theorem sumComm_symm : (sumComm X Y).symm = sumComm Y X :=
+  rfl
+
+@[simp]
+theorem coe_sumComm : ⇑(sumComm X Y) = Sum.swap :=
+  rfl
+
+@[continuity, fun_prop]
+lemma continuous_sumAssoc : Continuous (Equiv.sumAssoc X Y Z) :=
+  Continuous.sum_elim (by fun_prop) (by fun_prop)
+
+@[continuity, fun_prop]
+lemma continuous_sumAssoc_symm : Continuous (Equiv.sumAssoc X Y Z).symm :=
+  Continuous.sum_elim (by fun_prop) (by fun_prop)
+
+/-- `(X ⊕ Y) ⊕ Z` is homeomorphic to `X ⊕ (Y ⊕ Z)`. -/
+def sumAssoc : (X ⊕ Y) ⊕ Z ≃ₜ X ⊕ Y ⊕ Z where
+  toEquiv := Equiv.sumAssoc X Y Z
+  continuous_toFun := continuous_sumAssoc X Y Z
+  continuous_invFun := continuous_sumAssoc_symm X Y Z
+
+@[simp]
+lemma sumAssoc_toEquiv : (sumAssoc X Y Z).toEquiv = Equiv.sumAssoc X Y Z := rfl
+
+/-- Four-way commutativity of the disjoint union. The name matches `add_add_add_comm`. -/
+def sumSumSumComm : (X ⊕ Y) ⊕ W ⊕ Z ≃ₜ (X ⊕ W) ⊕ Y ⊕ Z where
+  toEquiv := Equiv.sumSumSumComm X Y W Z
+  continuous_toFun := by
+    unfold Equiv.sumSumSumComm
+    dsimp only
+    have : Continuous (Sum.map (Sum.map (@id X) ⇑(Equiv.sumComm Y W)) (@id Z)) := by continuity
+    fun_prop
+  continuous_invFun := by
+    unfold Equiv.sumSumSumComm
+    dsimp only
+    have : Continuous (Sum.map (Sum.map (@id X) (Equiv.sumComm Y W).symm) (@id Z)) := by continuity
+    fun_prop
+
+@[simp]
+lemma sumSumSumComm_toEquiv : (sumSumSumComm X Y W Z).toEquiv = (Equiv.sumSumSumComm X Y W Z) := rfl
+
+@[simp]
+lemma sumSumSumComm_symm : (sumSumSumComm X Y W Z).symm = (sumSumSumComm X W Y Z) := rfl
+
+/-- The sum of `X` with any empty topological space is homeomorphic to `X`. -/
+@[simps! (config := .asFn) apply]
+def sumEmpty [IsEmpty Y] : X ⊕ Y ≃ₜ X where
+  toEquiv := Equiv.sumEmpty X Y
+  continuous_toFun := Continuous.sum_elim continuous_id (by fun_prop)
+  continuous_invFun := continuous_inl
+
+/-- The sum of `X` with any empty topological space is homeomorphic to `X`. -/
+def emptySum [IsEmpty Y] : Y ⊕ X ≃ₜ X := (sumComm Y X).trans (sumEmpty X Y)
+
+@[simp] theorem coe_emptySum [IsEmpty Y] : (emptySum X Y).toEquiv = Equiv.emptySum Y X := rfl
+
+end sum
+
+-- Commutativity and associativity of the product of top. spaces, and the product with `PUnit`.
+section prod
+
+variable (X Y W Z)
 
 /-- `X × Y` is homeomorphic to `Y × X`. -/
 def prodComm : X × Y ≃ₜ Y × X where
@@ -532,6 +605,25 @@ def prodAssoc : (X × Y) × Z ≃ₜ X × Y × Z where
   continuous_invFun := (continuous_fst.prod_mk continuous_snd.fst).prod_mk continuous_snd.snd
   toEquiv := Equiv.prodAssoc X Y Z
 
+@[simp]
+lemma prodAssoc_toEquiv : (prodAssoc X Y Z).toEquiv = Equiv.prodAssoc X Y Z := rfl
+
+/-- Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. -/
+def prodProdProdComm : (X × Y) × W × Z ≃ₜ (X × W) × Y × Z where
+  toEquiv := Equiv.prodProdProdComm X Y W Z
+  continuous_toFun := by
+    unfold Equiv.prodProdProdComm
+    dsimp only
+    fun_prop
+  continuous_invFun := by
+    unfold Equiv.prodProdProdComm
+    dsimp only
+    fun_prop
+
+@[simp]
+theorem prodProdProdComm_symm : (prodProdProdComm X Y W Z).symm = prodProdProdComm X W Y Z :=
+  rfl
+
 /-- `X × {*}` is homeomorphic to `X`. -/
 @[simps! (config := .asFn) apply]
 def prodPUnit : X × PUnit ≃ₜ X where
@@ -552,7 +644,7 @@ def homeomorphOfUnique [Unique X] [Unique Y] : X ≃ₜ Y :=
     continuous_toFun := continuous_const
     continuous_invFun := continuous_const }
 
-end
+end prod
 
 /-- `Equiv.piCongrLeft` as a homeomorphism: this is the natural homeomorphism
 `Π i, Y (e i) ≃ₜ Π j, Y j` obtained from a bijection `ι ≃ ι'`. -/
@@ -602,9 +694,9 @@ section Distrib
 def sumProdDistrib : (X ⊕ Y) × Z ≃ₜ (X × Z) ⊕ (Y × Z) :=
   Homeomorph.symm <|
     homeomorphOfContinuousOpen (Equiv.sumProdDistrib X Y Z).symm
-        ((continuous_inl.prod_map continuous_id).sum_elim
-          (continuous_inr.prod_map continuous_id)) <|
-      (isOpenMap_inl.prod IsOpenMap.id).sum_elim (isOpenMap_inr.prod IsOpenMap.id)
+        ((continuous_inl.prodMap continuous_id).sum_elim
+          (continuous_inr.prodMap continuous_id)) <|
+      (isOpenMap_inl.prodMap IsOpenMap.id).sum_elim (isOpenMap_inr.prodMap IsOpenMap.id)
 
 /-- `X × (Y ⊕ Z)` is homeomorphic to `X × Y ⊕ X × Z`. -/
 def prodSumDistrib : X × (Y ⊕ Z) ≃ₜ (X × Y) ⊕ (X × Z) :=
@@ -618,7 +710,7 @@ def sigmaProdDistrib : (Σ i, X i) × Y ≃ₜ Σ i, X i × Y :=
   Homeomorph.symm <|
     homeomorphOfContinuousOpen (Equiv.sigmaProdDistrib X Y).symm
       (continuous_sigma fun _ => continuous_sigmaMk.fst'.prod_mk continuous_snd)
-      (isOpenMap_sigma.2 fun _ => isOpenMap_sigmaMk.prod IsOpenMap.id)
+      (isOpenMap_sigma.2 fun _ => isOpenMap_sigmaMk.prodMap IsOpenMap.id)
 
 end Distrib
 
@@ -810,7 +902,10 @@ protected lemma quotientMap : QuotientMap f := (hf.homeomorph f).quotientMap
 protected lemma embedding : Embedding f := (hf.homeomorph f).embedding
 protected lemma openEmbedding : OpenEmbedding f := (hf.homeomorph f).openEmbedding
 protected lemma closedEmbedding : ClosedEmbedding f := (hf.homeomorph f).closedEmbedding
-protected lemma denseEmbedding : DenseEmbedding f := (hf.homeomorph f).denseEmbedding
+lemma isDenseEmbedding : IsDenseEmbedding f := (hf.homeomorph f).isDenseEmbedding
+
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding := isDenseEmbedding
 
 end IsHomeomorph
 
@@ -822,9 +917,9 @@ lemma isHomeomorph_iff_exists_homeomorph : IsHomeomorph f ↔ ∃ h : X ≃ₜ Y
 lemma isHomeomorph_iff_exists_inverse : IsHomeomorph f ↔ Continuous f ∧ ∃ g : Y → X,
     LeftInverse g f ∧ RightInverse g f ∧ Continuous g := by
   refine ⟨fun hf ↦ ⟨hf.continuous, ?_⟩, fun ⟨hf, g, hg⟩ ↦ ?_⟩
-  let h := hf.homeomorph f
-  exact ⟨h.symm, h.left_inv, h.right_inv, h.continuous_invFun⟩
-  exact (Homeomorph.mk ⟨f, g, hg.1, hg.2.1⟩ hf hg.2.2).isHomeomorph
+  · let h := hf.homeomorph f
+    exact ⟨h.symm, h.left_inv, h.right_inv, h.continuous_invFun⟩
+  · exact (Homeomorph.mk ⟨f, g, hg.1, hg.2.1⟩ hf hg.2.2).isHomeomorph
 
 /-- A map is a homeomorphism iff it is a surjective embedding. -/
 lemma isHomeomorph_iff_embedding_surjective : IsHomeomorph f ↔ Embedding f ∧ Surjective f where
@@ -855,7 +950,7 @@ lemma IsHomeomorph.sumMap {g : Z → W} (hf : IsHomeomorph f) (hg : IsHomeomorph
     IsHomeomorph (Sum.map f g) := ⟨hf.1.sum_map hg.1, hf.2.sumMap hg.2, hf.3.sum_map hg.3⟩
 
 lemma IsHomeomorph.prodMap {g : Z → W} (hf : IsHomeomorph f) (hg : IsHomeomorph g) :
-    IsHomeomorph (Prod.map f g) := ⟨hf.1.prod_map hg.1, hf.2.prod hg.2, hf.3.prodMap hg.3⟩
+    IsHomeomorph (Prod.map f g) := ⟨hf.1.prodMap hg.1, hf.2.prodMap hg.2, hf.3.prodMap hg.3⟩
 
 lemma IsHomeomorph.sigmaMap {ι κ : Type*} {X : ι → Type*} {Y : κ → Type*}
     [∀ i, TopologicalSpace (X i)] [∀ i, TopologicalSpace (Y i)] {f : ι → κ}
diff --git a/Mathlib/Topology/Homotopy/Basic.lean b/Mathlib/Topology/Homotopy/Basic.lean
index 75273bcf06755..e736983bd8031 100644
--- a/Mathlib/Topology/Homotopy/Basic.lean
+++ b/Mathlib/Topology/Homotopy/Basic.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Shing Tak Lam
 -/
 import Mathlib.Topology.Order.ProjIcc
-import Mathlib.Topology.ContinuousFunction.Ordered
+import Mathlib.Topology.ContinuousMap.Ordered
 import Mathlib.Topology.CompactOpen
 import Mathlib.Topology.UnitInterval
 
@@ -160,7 +160,6 @@ theorem extend_apply_of_one_le (F : Homotopy f₀ f₁) {t : ℝ} (ht : 1 ≤ t)
   rw [← F.apply_one]
   exact ContinuousMap.congr_fun (Set.IccExtend_of_right_le (zero_le_one' ℝ) F.curry ht) x
 
-@[simp]
 theorem extend_apply_coe (F : Homotopy f₀ f₁) (t : I) (x : X) : F.extend t x = F (t, x) :=
   ContinuousMap.congr_fun (Set.IccExtend_val (zero_le_one' ℝ) F.curry t) x
 
diff --git a/Mathlib/Topology/Homotopy/HomotopyGroup.lean b/Mathlib/Topology/Homotopy/HomotopyGroup.lean
index 93fad3a1199db..daaa41db02500 100644
--- a/Mathlib/Topology/Homotopy/HomotopyGroup.lean
+++ b/Mathlib/Topology/Homotopy/HomotopyGroup.lean
@@ -185,7 +185,7 @@ theorem continuous_toLoop (i : N) : Continuous (@toLoop N X _ x _ i) :=
   Path.continuous_uncurry_iff.1 <|
     Continuous.subtype_mk
       (ContinuousMap.continuous_eval.comp <|
-        Continuous.prod_map
+        Continuous.prodMap
           (ContinuousMap.continuous_curry.comp <|
             (ContinuousMap.continuous_comp_left _).comp continuous_subtype_val)
           continuous_id)
diff --git a/Mathlib/Topology/Inseparable.lean b/Mathlib/Topology/Inseparable.lean
index f4f888556c776..a179072f0d072 100644
--- a/Mathlib/Topology/Inseparable.lean
+++ b/Mathlib/Topology/Inseparable.lean
@@ -5,6 +5,7 @@ Authors: Andrew Yang, Yury Kudryashov
 -/
 import Mathlib.Tactic.TFAE
 import Mathlib.Topology.ContinuousOn
+import Mathlib.Topology.Maps.OpenQuotient
 
 /-!
 # Inseparable points in a topological space
@@ -53,20 +54,15 @@ theorem specializes_TFAE (x y : X) :
       y ∈ closure ({ x } : Set X),
       closure ({ y } : Set X) ⊆ closure { x },
       ClusterPt y (pure x)] := by
-  tfae_have 1 → 2
-  · exact (pure_le_nhds _).trans
-  tfae_have 2 → 3
-  · exact fun h s hso hy => h (hso.mem_nhds hy)
-  tfae_have 3 → 4
-  · exact fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
-  tfae_have 4 → 5
-  · exact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
-  tfae_have 6 ↔ 5
-  · exact isClosed_closure.closure_subset_iff.trans singleton_subset_iff
-  tfae_have 5 ↔ 7
-  · rw [mem_closure_iff_clusterPt, principal_singleton]
-  tfae_have 5 → 1
-  · refine fun h => (nhds_basis_opens _).ge_iff.2 ?_
+  tfae_have 1 → 2 := (pure_le_nhds _).trans
+  tfae_have 2 → 3 := fun h s hso hy => h (hso.mem_nhds hy)
+  tfae_have 3 → 4 := fun h s hsc hx => of_not_not fun hy => h sᶜ hsc.isOpen_compl hy hx
+  tfae_have 4 → 5 := fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)
+  tfae_have 6 ↔ 5 := isClosed_closure.closure_subset_iff.trans singleton_subset_iff
+  tfae_have 5 ↔ 7 := by
+    rw [mem_closure_iff_clusterPt, principal_singleton]
+  tfae_have 5 → 1 := by
+    refine fun h => (nhds_basis_opens _).ge_iff.2 ?_
     rintro s ⟨hy, ho⟩
     rcases mem_closure_iff.1 h s ho hy with ⟨z, hxs, rfl : z = x⟩
     exact ho.mem_nhds hxs
@@ -553,6 +549,9 @@ theorem preimage_image_mk_open (hs : IsOpen s) : mk ⁻¹' (mk '' s) = s := by
 theorem isOpenMap_mk : IsOpenMap (mk : X → SeparationQuotient X) := fun s hs =>
   quotientMap_mk.isOpen_preimage.1 <| by rwa [preimage_image_mk_open hs]
 
+theorem isOpenQuotientMap_mk : IsOpenQuotientMap (mk : X → SeparationQuotient X) :=
+  ⟨surjective_mk, continuous_mk, isOpenMap_mk⟩
+
 theorem preimage_image_mk_closed (hs : IsClosed s) : mk ⁻¹' (mk '' s) = s := by
   refine Subset.antisymm ?_ (subset_preimage_image _ _)
   rintro x ⟨y, hys, hxy⟩
@@ -604,14 +603,8 @@ theorem map_mk_nhdsWithin_preimage (s : Set (SeparationQuotient X)) (x : X) :
   rw [nhdsWithin, ← comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]
 
 /-- The map `(x, y) ↦ (mk x, mk y)` is a quotient map. -/
-theorem quotientMap_prodMap_mk : QuotientMap (Prod.map mk mk : X × Y → _) := by
-  have hsurj : Surjective (Prod.map mk mk : X × Y → _) := surjective_mk.prodMap surjective_mk
-  refine quotientMap_iff.2 ⟨hsurj, fun s ↦ ?_⟩
-  refine ⟨fun hs ↦ hs.preimage (continuous_mk.prod_map continuous_mk), fun hs ↦ ?_⟩
-  refine isOpen_iff_mem_nhds.2 <| hsurj.forall.2 fun (x, y) h ↦ ?_
-  rw [Prod.map_mk, nhds_prod_eq, ← map_mk_nhds, ← map_mk_nhds, Filter.prod_map_map_eq',
-    ← nhds_prod_eq, Filter.mem_map]
-  exact hs.mem_nhds h
+theorem quotientMap_prodMap_mk : QuotientMap (Prod.map mk mk : X × Y → _) :=
+  (isOpenQuotientMap_mk.prodMap isOpenQuotientMap_mk).quotientMap
 
 /-- Lift a map `f : X → α` such that `Inseparable x y → f x = f y` to a map
 `SeparationQuotient X → α`. -/
diff --git a/Mathlib/Topology/Instances/AddCircle.lean b/Mathlib/Topology/Instances/AddCircle.lean
index 988b0aced090f..3295ee564f992 100644
--- a/Mathlib/Topology/Instances/AddCircle.lean
+++ b/Mathlib/Topology/Instances/AddCircle.lean
@@ -112,7 +112,6 @@ theorem continuousAt_toIocMod (hx : (x : 𝕜 ⧸ zmultiples p) ≠ a) : Continu
 end Continuity
 
 /-- The "additive circle": `𝕜 ⧸ (ℤ ∙ p)`. See also `Circle` and `Real.angle`. -/
-@[nolint unusedArguments]
 abbrev AddCircle [LinearOrderedAddCommGroup 𝕜] (p : 𝕜) :=
   𝕜 ⧸ zmultiples p
 
@@ -462,7 +461,7 @@ def setAddOrderOfEquiv {n : ℕ} (hn : 0 < n) :
           obtain ⟨m, hm⟩ := h
           rw [← mul_div_right_comm, eq_div_iff, mul_comm, ← zsmul_eq_mul, mul_smul_comm, ←
             nsmul_eq_mul, ← natCast_zsmul, smul_smul,
-            (zsmul_strictMono_left hp.out).injective.eq_iff, mul_comm] at hm
+            zsmul_left_inj hp.out, mul_comm] at hm
           swap
           · exact Nat.cast_ne_zero.2 hn.ne'
           rw [← @Nat.cast_inj ℤ, ← sub_eq_zero]
diff --git a/Mathlib/Topology/Instances/Complex.lean b/Mathlib/Topology/Instances/Complex.lean
index 1bd41163d0387..4693c143277c8 100644
--- a/Mathlib/Topology/Instances/Complex.lean
+++ b/Mathlib/Topology/Instances/Complex.lean
@@ -39,7 +39,7 @@ theorem Complex.subfield_eq_of_closed {K : Subfield ℂ} (hc : IsClosed (K : Set
     simp only [Function.comp_apply, ofReal_ratCast, SetLike.mem_coe, SubfieldClass.ratCast_mem]
   nth_rw 1 [range_comp]
   refine subset_trans ?_ (image_closure_subset_closure_image continuous_ofReal)
-  rw [DenseRange.closure_range Rat.denseEmbedding_coe_real.dense]
+  rw [DenseRange.closure_range Rat.isDenseEmbedding_coe_real.dense]
   simp only [image_univ]
   rfl
 
@@ -52,13 +52,13 @@ theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ :
   letI : TopologicalRing K.topologicalClosure :=
     Subring.instTopologicalRing K.topologicalClosure.toSubring
   set ι : K → K.topologicalClosure := ⇑(Subfield.inclusion K.le_topologicalClosure)
-  have ui : UniformInducing ι :=
+  have ui : IsUniformInducing ι :=
     ⟨by
       erw [uniformity_subtype, uniformity_subtype, Filter.comap_comap]
       congr ⟩
-  let di := ui.denseInducing (?_ : DenseRange ι)
+  let di := ui.isDenseInducing (?_ : DenseRange ι)
   · -- extψ : closure(K) →+* ℂ is the extension of ψ : K →+* ℂ
-    let extψ := DenseInducing.extendRingHom ui di.dense hc
+    let extψ := IsDenseInducing.extendRingHom ui di.dense hc
     haveI hψ := (uniformContinuous_uniformly_extend ui di.dense hc).continuous
     cases' Complex.subfield_eq_of_closed (Subfield.isClosed_topologicalClosure K) with h h
     · left
@@ -76,7 +76,7 @@ theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ :
         -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
         erw [RingHom.comp_apply, RingHom.comp_apply, hr, RingEquiv.toRingHom_eq_coe] at this
         convert this using 1
-        · exact (DenseInducing.extend_eq di hc.continuous _).symm
+        · exact (IsDenseInducing.extend_eq di hc.continuous _).symm
         · rw [← ofReal.coe_rangeRestrict, hr]
           rfl
       obtain ⟨r, hr⟩ := SetLike.coe_mem (j (ι x))
@@ -94,11 +94,11 @@ theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ :
       · left
         ext1 z
         convert RingHom.congr_fun h z using 1
-        exact (DenseInducing.extend_eq di hc.continuous z).symm
+        exact (IsDenseInducing.extend_eq di hc.continuous z).symm
       · right
         ext1 z
         convert RingHom.congr_fun h z using 1
-        exact (DenseInducing.extend_eq di hc.continuous z).symm
+        exact (IsDenseInducing.extend_eq di hc.continuous z).symm
   · let j : { x // x ∈ closure (id '' { x | (K : Set ℂ) x }) } → (K.topologicalClosure : Set ℂ) :=
       fun x =>
       ⟨x, by
@@ -106,7 +106,7 @@ theorem Complex.uniformContinuous_ringHom_eq_id_or_conj (K : Subfield ℂ) {ψ :
         simp only [id, Set.image_id']
         rfl ⟩
     convert DenseRange.comp (Function.Surjective.denseRange _)
-        (DenseEmbedding.subtype denseEmbedding_id (K : Set ℂ)).dense (by continuity : Continuous j)
+      (IsDenseEmbedding.id.subtype (K : Set ℂ)).dense (by continuity : Continuous j)
     rintro ⟨y, hy⟩
     use
       ⟨y, by
diff --git a/Mathlib/Topology/Instances/ENNReal.lean b/Mathlib/Topology/Instances/ENNReal.lean
index e91af73104939..5623966b95601 100644
--- a/Mathlib/Topology/Instances/ENNReal.lean
+++ b/Mathlib/Topology/Instances/ENNReal.lean
@@ -91,7 +91,7 @@ theorem continuousAt_coe_iff {α : Type*} [TopologicalSpace α] {x : ℝ≥0} {f
 
 theorem nhds_coe_coe {r p : ℝ≥0} :
     𝓝 ((r : ℝ≥0∞), (p : ℝ≥0∞)) = (𝓝 (r, p)).map fun p : ℝ≥0 × ℝ≥0 => (↑p.1, ↑p.2) :=
-  ((openEmbedding_coe.prod openEmbedding_coe).map_nhds_eq (r, p)).symm
+  ((openEmbedding_coe.prodMap openEmbedding_coe).map_nhds_eq (r, p)).symm
 
 theorem continuous_ofReal : Continuous ENNReal.ofReal :=
   (continuous_coe_iff.2 continuous_id).comp continuous_real_toNNReal
@@ -267,8 +267,8 @@ instance : ContinuousAdd ℝ≥0∞ := by
   · exact tendsto_nhds_top_mono' continuousAt_fst fun p => le_add_right le_rfl
   rcases b with (_ | b)
   · exact tendsto_nhds_top_mono' continuousAt_snd fun p => le_add_left le_rfl
-  simp only [ContinuousAt, some_eq_coe, nhds_coe_coe, ← coe_add, tendsto_map'_iff, (· ∘ ·),
-    tendsto_coe, tendsto_add]
+  simp only [ContinuousAt, some_eq_coe, nhds_coe_coe, ← coe_add, tendsto_map'_iff,
+    Function.comp_def, tendsto_coe, tendsto_add]
 
 protected theorem tendsto_atTop_zero [Nonempty β] [SemilatticeSup β] {f : β → ℝ≥0∞} :
     Tendsto f atTop (𝓝 0) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, f n ≤ ε :=
@@ -291,7 +291,7 @@ theorem tendsto_sub : ∀ {a b : ℝ≥0∞}, (a ≠ ∞ ∨ b ≠ ∞) →
       (lt_mem_nhds <| @coe_lt_top (a + 1))).mono fun x hx =>
         tsub_eq_zero_iff_le.2 (hx.1.trans hx.2).le
   | (a : ℝ≥0), (b : ℝ≥0), _ => by
-    simp only [nhds_coe_coe, tendsto_map'_iff, ← ENNReal.coe_sub, (· ∘ ·), tendsto_coe]
+    simp only [nhds_coe_coe, tendsto_map'_iff, ← ENNReal.coe_sub, Function.comp_def, tendsto_coe]
     exact continuous_sub.tendsto (a, b)
 
 protected theorem Tendsto.sub {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
@@ -316,10 +316,11 @@ protected theorem tendsto_mul (ha : a ≠ 0 ∨ b ≠ ∞) (hb : b ≠ 0 ∨ a 
     induction b with
     | top =>
       simp only [ne_eq, or_false, not_true_eq_false] at ha
-      simpa [(· ∘ ·), mul_comm, mul_top ha]
+      simpa [Function.comp_def, mul_comm, mul_top ha]
         using (ht a ha).comp (continuous_swap.tendsto (ofNNReal a, ∞))
     | coe b =>
-      simp only [nhds_coe_coe, ← coe_mul, tendsto_coe, tendsto_map'_iff, (· ∘ ·), tendsto_mul]
+      simp only [nhds_coe_coe, ← coe_mul, tendsto_coe, tendsto_map'_iff, Function.comp_def,
+        tendsto_mul]
 
 protected theorem Tendsto.mul {f : Filter α} {ma : α → ℝ≥0∞} {mb : α → ℝ≥0∞} {a b : ℝ≥0∞}
     (hma : Tendsto ma f (𝓝 a)) (ha : a ≠ 0 ∨ b ≠ ∞) (hmb : Tendsto mb f (𝓝 b))
@@ -389,11 +390,11 @@ protected theorem continuous_pow (n : ℕ) : Continuous fun a : ℝ≥0∞ => a
   simp_rw [pow_add, pow_one, continuous_iff_continuousAt]
   intro x
   refine ENNReal.Tendsto.mul (IH.tendsto _) ?_ tendsto_id ?_ <;> by_cases H : x = 0
-  · simp only [H, zero_ne_top, Ne, or_true_iff, not_false_iff]
+  · simp only [H, zero_ne_top, Ne, or_true, not_false_iff]
   · exact Or.inl fun h => H (pow_eq_zero h)
-  · simp only [H, pow_eq_top_iff, zero_ne_top, false_or_iff, eq_self_iff_true, not_true, Ne,
-      not_false_iff, false_and_iff]
-  · simp only [H, true_or_iff, Ne, not_false_iff]
+  · simp only [H, pow_eq_top_iff, zero_ne_top, false_or, eq_self_iff_true, not_true, Ne,
+      not_false_iff, false_and]
+  · simp only [H, true_or, Ne, not_false_iff]
 
 theorem continuousOn_sub :
     ContinuousOn (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) { p : ℝ≥0∞ × ℝ≥0∞ | p ≠ ⟨∞, ∞⟩ } := by
@@ -405,7 +406,7 @@ theorem continuousOn_sub :
 theorem continuous_sub_left {a : ℝ≥0∞} (a_ne_top : a ≠ ∞) : Continuous (a - ·) := by
   change Continuous (Function.uncurry Sub.sub ∘ (a, ·))
   refine continuousOn_sub.comp_continuous (Continuous.Prod.mk a) fun x => ?_
-  simp only [a_ne_top, Ne, mem_setOf_eq, Prod.mk.inj_iff, false_and_iff, not_false_iff]
+  simp only [a_ne_top, Ne, mem_setOf_eq, Prod.mk.inj_iff, false_and, not_false_iff]
 
 theorem continuous_nnreal_sub {a : ℝ≥0} : Continuous fun x : ℝ≥0∞ => (a : ℝ≥0∞) - x :=
   continuous_sub_left coe_ne_top
@@ -422,7 +423,7 @@ theorem continuous_sub_right (a : ℝ≥0∞) : Continuous fun x : ℝ≥0∞ =>
   · rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
     apply ContinuousOn.comp_continuous continuousOn_sub (continuous_id'.prod_mk continuous_const)
     intro x
-    simp only [a_infty, Ne, mem_setOf_eq, Prod.mk.inj_iff, and_false_iff, not_false_iff]
+    simp only [a_infty, Ne, mem_setOf_eq, Prod.mk.inj_iff, and_false, not_false_iff]
 
 protected theorem Tendsto.pow {f : Filter α} {m : α → ℝ≥0∞} {a : ℝ≥0∞} {n : ℕ}
     (hm : Tendsto m f (𝓝 a)) : Tendsto (fun x => m x ^ n) f (𝓝 (a ^ n)) :=
@@ -434,34 +435,28 @@ theorem le_of_forall_lt_one_mul_le {x y : ℝ≥0∞} (h : ∀ a < 1, a * x ≤
   rw [one_mul] at this
   exact le_of_tendsto this (eventually_nhdsWithin_iff.2 <| Eventually.of_forall h)
 
+@[deprecated mul_iInf' (since := "2024-09-12")]
 theorem iInf_mul_left' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0)
-    (h0 : a = 0 → Nonempty ι) : ⨅ i, a * f i = a * ⨅ i, f i := by
-  by_cases H : a = ∞ ∧ ⨅ i, f i = 0
-  · rcases h H.1 H.2 with ⟨i, hi⟩
-    rw [H.2, mul_zero, ← bot_eq_zero, iInf_eq_bot]
-    exact fun b hb => ⟨i, by rwa [hi, mul_zero, ← bot_eq_zero]⟩
-  · rw [not_and_or] at H
-    cases isEmpty_or_nonempty ι
-    · rw [iInf_of_empty, iInf_of_empty, mul_top]
-      exact mt h0 (not_nonempty_iff.2 ‹_›)
-    · exact (ENNReal.mul_left_mono.map_iInf_of_continuousAt'
-        (ENNReal.continuousAt_const_mul H)).symm
+    (h0 : a = 0 → Nonempty ι) : ⨅ i, a * f i = a * ⨅ i, f i := .symm <| mul_iInf' h h0
 
+@[deprecated mul_iInf (since := "2024-09-12")]
 theorem iInf_mul_left {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
     (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) : ⨅ i, a * f i = a * ⨅ i, f i :=
-  iInf_mul_left' h fun _ => ‹Nonempty ι›
+  .symm <| mul_iInf h
 
+@[deprecated iInf_mul' (since := "2024-09-12")]
 theorem iInf_mul_right' {ι} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0)
-    (h0 : a = 0 → Nonempty ι) : ⨅ i, f i * a = (⨅ i, f i) * a := by
-  simpa only [mul_comm a] using iInf_mul_left' h h0
+    (h0 : a = 0 → Nonempty ι) : ⨅ i, f i * a = (⨅ i, f i) * a := .symm <| iInf_mul' h h0
 
+@[deprecated iInf_mul (since := "2024-09-12")]
 theorem iInf_mul_right {ι} [Nonempty ι] {f : ι → ℝ≥0∞} {a : ℝ≥0∞}
-    (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) : ⨅ i, f i * a = (⨅ i, f i) * a :=
-  iInf_mul_right' h fun _ => ‹Nonempty ι›
+    (h : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0) : ⨅ i, f i * a = (⨅ i, f i) * a := .symm <| iInf_mul h
 
+@[deprecated inv_iInf (since := "2024-09-12")]
 theorem inv_map_iInf {ι : Sort*} {x : ι → ℝ≥0∞} : (iInf x)⁻¹ = ⨆ i, (x i)⁻¹ :=
   OrderIso.invENNReal.map_iInf x
 
+@[deprecated inv_iSup (since := "2024-09-12")]
 theorem inv_map_iSup {ι : Sort*} {x : ι → ℝ≥0∞} : (iSup x)⁻¹ = ⨅ i, (x i)⁻¹ :=
   OrderIso.invENNReal.map_iSup x
 
@@ -503,115 +498,10 @@ protected theorem Tendsto.div_const {f : Filter α} {m : α → ℝ≥0∞} {a b
 protected theorem tendsto_inv_nat_nhds_zero : Tendsto (fun n : ℕ => (n : ℝ≥0∞)⁻¹) atTop (𝓝 0) :=
   ENNReal.inv_top ▸ ENNReal.tendsto_inv_iff.2 tendsto_nat_nhds_top
 
-theorem iSup_add {ι : Sort*} {s : ι → ℝ≥0∞} [Nonempty ι] : iSup s + a = ⨆ b, s b + a :=
-  Monotone.map_iSup_of_continuousAt' (continuousAt_id.add continuousAt_const) <|
-    monotone_id.add monotone_const
-
-theorem biSup_add' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
-    (⨆ (i) (_ : p i), f i) + a = ⨆ (i) (_ : p i), f i + a := by
-  haveI : Nonempty { i // p i } := nonempty_subtype.2 h
-  simp only [iSup_subtype', iSup_add]
-
-theorem add_biSup' {ι : Sort*} {p : ι → Prop} (h : ∃ i, p i) {f : ι → ℝ≥0∞} :
-    (a + ⨆ (i) (_ : p i), f i) = ⨆ (i) (_ : p i), a + f i := by
-  simp only [add_comm a, biSup_add' h]
-
-theorem biSup_add {ι} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞} :
-    (⨆ i ∈ s, f i) + a = ⨆ i ∈ s, f i + a :=
-  biSup_add' hs
-
-theorem add_biSup {ι} {s : Set ι} (hs : s.Nonempty) {f : ι → ℝ≥0∞} :
-    (a + ⨆ i ∈ s, f i) = ⨆ i ∈ s, a + f i :=
-  add_biSup' hs
-
-theorem sSup_add {s : Set ℝ≥0∞} (hs : s.Nonempty) : sSup s + a = ⨆ b ∈ s, b + a := by
-  rw [sSup_eq_iSup, biSup_add hs]
-
-theorem add_iSup {ι : Sort*} {s : ι → ℝ≥0∞} [Nonempty ι] : a + iSup s = ⨆ b, a + s b := by
-  rw [add_comm, iSup_add]; simp [add_comm]
-
-theorem iSup_add_iSup_le {ι ι' : Sort*} [Nonempty ι] [Nonempty ι'] {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞}
-    {a : ℝ≥0∞} (h : ∀ i j, f i + g j ≤ a) : iSup f + iSup g ≤ a := by
-  simp_rw [iSup_add, add_iSup]; exact iSup₂_le h
-
-theorem biSup_add_biSup_le' {ι ι'} {p : ι → Prop} {q : ι' → Prop} (hp : ∃ i, p i) (hq : ∃ j, q j)
-    {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i, p i → ∀ j, q j → f i + g j ≤ a) :
-    ((⨆ (i) (_ : p i), f i) + ⨆ (j) (_ : q j), g j) ≤ a := by
-  simp_rw [biSup_add' hp, add_biSup' hq]
-  exact iSup₂_le fun i hi => iSup₂_le (h i hi)
-
-theorem biSup_add_biSup_le {ι ι'} {s : Set ι} {t : Set ι'} (hs : s.Nonempty) (ht : t.Nonempty)
-    {f : ι → ℝ≥0∞} {g : ι' → ℝ≥0∞} {a : ℝ≥0∞} (h : ∀ i ∈ s, ∀ j ∈ t, f i + g j ≤ a) :
-    ((⨆ i ∈ s, f i) + ⨆ j ∈ t, g j) ≤ a :=
-  biSup_add_biSup_le' hs ht h
-
-theorem iSup_add_iSup {ι : Sort*} {f g : ι → ℝ≥0∞} (h : ∀ i j, ∃ k, f i + g j ≤ f k + g k) :
-    iSup f + iSup g = ⨆ a, f a + g a := by
-  cases isEmpty_or_nonempty ι
-  · simp only [iSup_of_empty, bot_eq_zero, zero_add]
-  · refine le_antisymm ?_ (iSup_le fun a => add_le_add (le_iSup _ _) (le_iSup _ _))
-    refine iSup_add_iSup_le fun i j => ?_
-    rcases h i j with ⟨k, hk⟩
-    exact le_iSup_of_le k hk
-
-theorem iSup_add_iSup_of_monotone {ι : Type*} [Preorder ι] [IsDirected ι (· ≤ ·)]
-    {f g : ι → ℝ≥0∞} (hf : Monotone f) (hg : Monotone g) : iSup f + iSup g = ⨆ a, f a + g a :=
-  iSup_add_iSup fun i j ↦ (exists_ge_ge i j).imp fun _k ⟨hi, hj⟩ ↦ by gcongr <;> apply_rules
-
-theorem finsetSum_iSup {α ι : Type*} {s : Finset α} {f : α → ι → ℝ≥0∞}
-    (hf : ∀ i j, ∃ k, ∀ a, f a i ≤ f a k ∧ f a j ≤ f a k) :
-    ∑ a ∈ s, ⨆ i, f a i = ⨆ i, ∑ a ∈ s, f a i := by
-  induction s using Finset.cons_induction with
-  | empty => simp
-  | cons a s ha ihs =>
-    simp_rw [Finset.sum_cons, ihs]
-    refine iSup_add_iSup fun i j ↦ (hf i j).imp fun k hk ↦ ?_
-    gcongr
-    exacts [(hk a).1, (hk _).2]
-
-theorem finsetSum_iSup_of_monotone {α} {ι} [Preorder ι] [IsDirected ι (· ≤ ·)]
-    {s : Finset α} {f : α → ι → ℝ≥0∞} (hf : ∀ a, Monotone (f a)) :
-    (∑ a ∈ s, iSup (f a)) = ⨆ n, ∑ a ∈ s, f a n :=
-  finsetSum_iSup fun i j ↦ (exists_ge_ge i j).imp fun _k ⟨hi, hj⟩ a ↦ ⟨hf a hi, hf a hj⟩
-
-@[deprecated (since := "2024-07-14")]
-alias finset_sum_iSup_nat := finsetSum_iSup_of_monotone
-
-theorem mul_iSup {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : a * iSup f = ⨆ i, a * f i := by
-  by_cases hf : ∀ i, f i = 0
-  · obtain rfl : f = fun _ => 0 := funext hf
-    simp only [iSup_zero_eq_zero, mul_zero]
-  · refine (monotone_id.const_mul' _).map_iSup_of_continuousAt ?_ (mul_zero a)
-    refine ENNReal.Tendsto.const_mul tendsto_id (Or.inl ?_)
-    exact mt iSup_eq_zero.1 hf
-
-theorem mul_sSup {s : Set ℝ≥0∞} {a : ℝ≥0∞} : a * sSup s = ⨆ i ∈ s, a * i := by
-  simp only [sSup_eq_iSup, mul_iSup]
-
-theorem iSup_mul {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : iSup f * a = ⨆ i, f i * a := by
-  rw [mul_comm, mul_iSup]; congr; funext; rw [mul_comm]
-
-theorem smul_iSup {ι : Sort*} {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (f : ι → ℝ≥0∞)
-    (c : R) : (c • ⨆ i, f i) = ⨆ i, c • f i := by
-  -- Porting note: replaced `iSup _` with `iSup f`
-  simp only [← smul_one_mul c (f _), ← smul_one_mul c (iSup f), ENNReal.mul_iSup]
-
-theorem smul_sSup {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (s : Set ℝ≥0∞) (c : R) :
-    c • sSup s = ⨆ i ∈ s, c • i := by
-  -- Porting note: replaced `_` with `s`
-  simp_rw [← smul_one_mul c (sSup s), ENNReal.mul_sSup, smul_one_mul]
-
-theorem iSup_div {ι : Sort*} {f : ι → ℝ≥0∞} {a : ℝ≥0∞} : iSup f / a = ⨆ i, f i / a :=
-  iSup_mul
-
 protected theorem tendsto_coe_sub {b : ℝ≥0∞} :
     Tendsto (fun b : ℝ≥0∞ => ↑r - b) (𝓝 b) (𝓝 (↑r - b)) :=
   continuous_nnreal_sub.tendsto _
 
-theorem sub_iSup {ι : Sort*} [Nonempty ι] {b : ι → ℝ≥0∞} (hr : a < ∞) :
-    (a - ⨆ i, b i) = ⨅ i, a - b i :=
-  antitone_const_tsub.map_iSup_of_continuousAt' (continuous_sub_left hr.ne).continuousAt
-
 theorem exists_countable_dense_no_zero_top :
     ∃ s : Set ℝ≥0∞, s.Countable ∧ Dense s ∧ 0 ∉ s ∧ ∞ ∉ s := by
   obtain ⟨s, s_count, s_dense, hs⟩ :
@@ -619,19 +509,7 @@ theorem exists_countable_dense_no_zero_top :
     exists_countable_dense_no_bot_top ℝ≥0∞
   exact ⟨s, s_count, s_dense, fun h => hs.1 0 (by simp) h, fun h => hs.2 ∞ (by simp) h⟩
 
-theorem exists_lt_add_of_lt_add {x y z : ℝ≥0∞} (h : x < y + z) (hy : y ≠ 0) (hz : z ≠ 0) :
-    ∃ y' z', y' < y ∧ z' < z ∧ x < y' + z' := by
-  have : NeZero y := ⟨hy⟩
-  have : NeZero z := ⟨hz⟩
-  have A : Tendsto (fun p : ℝ≥0∞ × ℝ≥0∞ => p.1 + p.2) (𝓝[<] y ×ˢ 𝓝[<] z) (𝓝 (y + z)) := by
-    apply Tendsto.mono_left _ (Filter.prod_mono nhdsWithin_le_nhds nhdsWithin_le_nhds)
-    rw [← nhds_prod_eq]
-    exact tendsto_add
-  rcases ((A.eventually (lt_mem_nhds h)).and
-      (Filter.prod_mem_prod self_mem_nhdsWithin self_mem_nhdsWithin)).exists with
-    ⟨⟨y', z'⟩, hx, hy', hz'⟩
-  exact ⟨y', z', hy', hz', hx⟩
-
+@[deprecated ofReal_iInf (since := "2024-09-12")]
 theorem ofReal_cinfi (f : α → ℝ) [Nonempty α] :
     ENNReal.ofReal (⨅ i, f i) = ⨅ i, ENNReal.ofReal (f i) := by
   by_cases hf : BddBelow (range f)
@@ -804,7 +682,7 @@ theorem tsum_const_eq_top_of_ne_zero {α : Type*} [Infinite α] {c : ℝ≥0∞}
     ∑' _ : α, c = ∞ := by
   have A : Tendsto (fun n : ℕ => (n : ℝ≥0∞) * c) atTop (𝓝 (∞ * c)) := by
     apply ENNReal.Tendsto.mul_const tendsto_nat_nhds_top
-    simp only [true_or_iff, top_ne_zero, Ne, not_false_iff]
+    simp only [true_or, top_ne_zero, Ne, not_false_iff]
   have B : ∀ n : ℕ, (n : ℝ≥0∞) * c ≤ ∑' _ : α, c := fun n => by
     rcases Infinite.exists_subset_card_eq α n with ⟨s, hs⟩
     simpa [hs] using @ENNReal.sum_le_tsum α (fun _ => c) s
@@ -1168,7 +1046,7 @@ theorem tsum_comp_le_tsum_of_inj {β : Type*} {f : α → ℝ} (hf : Summable f)
     {i : β → α} (hi : Function.Injective i) : tsum (f ∘ i) ≤ tsum f := by
   lift f to α → ℝ≥0 using hn
   rw [NNReal.summable_coe] at hf
-  simpa only [(· ∘ ·), ← NNReal.coe_tsum] using NNReal.tsum_comp_le_tsum_of_inj hf hi
+  simpa only [Function.comp_def, ← NNReal.coe_tsum] using NNReal.tsum_comp_le_tsum_of_inj hf hi
 
 /-- Comparison test of convergence of series of non-negative real numbers. -/
 theorem Summable.of_nonneg_of_le {f g : β → ℝ} (hg : ∀ b, 0 ≤ g b) (hgf : ∀ b, g b ≤ f b)
@@ -1235,7 +1113,7 @@ open EMetric
 theorem tendsto_iff_edist_tendsto_0 {l : Filter β} {f : β → α} {y : α} :
     Tendsto f l (𝓝 y) ↔ Tendsto (fun x => edist (f x) y) l (𝓝 0) := by
   simp only [EMetric.nhds_basis_eball.tendsto_right_iff, EMetric.mem_ball,
-    @tendsto_order ℝ≥0∞ β _ _, forall_prop_of_false ENNReal.not_lt_zero, forall_const, true_and_iff]
+    @tendsto_order ℝ≥0∞ β _ _, forall_prop_of_false ENNReal.not_lt_zero, forall_const, true_and]
 
 /-- Yet another metric characterization of Cauchy sequences on integers. This one is often the
 most efficient. -/
@@ -1459,9 +1337,11 @@ section LimsupLiminf
 
 variable {ι : Type*}
 
-lemma limsup_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
-    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c :=
-  (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ x - c)
+lemma limsup_sub_const (F : Filter ι) (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
+    Filter.limsup (fun i ↦ f i - c) F = Filter.limsup f F - c := by
+  rcases F.eq_or_neBot with rfl | _
+  · simp only [limsup_bot, bot_eq_zero', zero_le, tsub_eq_zero_of_le]
+  · exact (Monotone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ x - c)
     (fun _ _ h ↦ tsub_le_tsub_right h c) (continuous_sub_right c).continuousAt).symm
 
 lemma liminf_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : ℝ≥0∞) :
@@ -1469,14 +1349,14 @@ lemma liminf_sub_const (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) (c : 
   (Monotone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ x - c)
     (fun _ _ h ↦ tsub_le_tsub_right h c) (continuous_sub_right c).continuousAt).symm
 
-lemma limsup_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞)
-    {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
-    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F :=
-  (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ c - x)
+lemma limsup_const_sub (F : Filter ι) (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
+    Filter.limsup (fun i ↦ c - f i) F = c - Filter.liminf f F := by
+  rcases F.eq_or_neBot with rfl | _
+  · simp only [limsup_bot, bot_eq_zero', liminf_bot, le_top, tsub_eq_zero_of_le]
+  · exact (Antitone.map_limsInf_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ c - x)
     (fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c_ne_top).continuousAt).symm
 
-lemma liminf_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞)
-    {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
+lemma liminf_const_sub (F : Filter ι) [NeBot F] (f : ι → ℝ≥0∞) {c : ℝ≥0∞} (c_ne_top : c ≠ ∞) :
     Filter.liminf (fun i ↦ c - f i) F = c - Filter.limsup f F :=
   (Antitone.map_limsSup_of_continuousAt (F := F.map f) (f := fun (x : ℝ≥0∞) ↦ c - x)
     (fun _ _ h ↦ tsub_le_tsub_left h c) (continuous_sub_left c_ne_top).continuousAt).symm
diff --git a/Mathlib/Topology/Instances/ENat.lean b/Mathlib/Topology/Instances/ENat.lean
index 004d68f482d65..ee7773df5b64b 100644
--- a/Mathlib/Topology/Instances/ENat.lean
+++ b/Mathlib/Topology/Instances/ENat.lean
@@ -12,9 +12,8 @@ import Mathlib.Order.Interval.Set.WithBotTop
 # Topology on extended natural numbers
 -/
 
-open Set
-
-open scoped Topology Filter
+open Set Filter
+open scoped Topology
 
 namespace ENat
 
@@ -54,6 +53,11 @@ theorem mem_nhds_iff {x : ℕ∞} {s : Set ℕ∞} (hx : x ≠ ⊤) : s ∈ 𝓝
 theorem mem_nhds_natCast_iff (n : ℕ) {s : Set ℕ∞} : s ∈ 𝓝 (n : ℕ∞) ↔ (n : ℕ∞) ∈ s :=
   mem_nhds_iff (coe_ne_top _)
 
+theorem tendsto_nhds_top_iff_natCast_lt {α : Type*} {l : Filter α} {f : α → ℕ∞} :
+    Tendsto f l (𝓝 ⊤) ↔ ∀ n : ℕ, ∀ᶠ a in l, n < f a := by
+  simp_rw [nhds_top_order, lt_top_iff_ne_top, tendsto_iInf, tendsto_principal]
+  exact Option.ball_ne_none
+
 instance : ContinuousAdd ℕ∞ := by
   refine ⟨continuous_iff_continuousAt.2 fun (a, b) ↦ ?_⟩
   match a, b with
@@ -61,4 +65,62 @@ instance : ContinuousAdd ℕ∞ := by
   | (a : ℕ), ⊤ => exact tendsto_nhds_top_mono' continuousAt_snd fun p ↦ le_add_left le_rfl
   | (a : ℕ), (b : ℕ) => simp [ContinuousAt, nhds_prod_eq, tendsto_pure_nhds]
 
+instance : ContinuousMul ℕ∞ where
+  continuous_mul :=
+    have key (a : ℕ∞) : ContinuousAt (· * ·).uncurry (a, ⊤) := by
+      rcases (zero_le a).eq_or_gt with rfl | ha
+      · simp [ContinuousAt, nhds_prod_eq]
+      · simp only [ContinuousAt, Function.uncurry, mul_top ha.ne']
+        refine tendsto_nhds_top_mono continuousAt_snd ?_
+        filter_upwards [continuousAt_fst (lt_mem_nhds ha)] with (x, y) (hx : 0 < x)
+        exact le_mul_of_one_le_left (zero_le y) (Order.one_le_iff_pos.2 hx)
+    continuous_iff_continuousAt.2 <| Prod.forall.2 fun
+      | (a : ℕ∞), ⊤ => key a
+      | ⊤, (b : ℕ∞) =>
+        ((key b).comp_of_eq (continuous_swap.tendsto (⊤, b)) rfl).congr <|
+          .of_forall fun _ ↦ mul_comm ..
+      | (a : ℕ), (b : ℕ) => by
+        simp [ContinuousAt, nhds_prod_eq, tendsto_pure_nhds]
+
+protected theorem continuousAt_sub {a b : ℕ∞} (h : a ≠ ⊤ ∨ b ≠ ⊤) :
+    ContinuousAt (· - ·).uncurry (a, b) := by
+  match a, b, h with
+  | (a : ℕ), (b : ℕ), _ =>
+    simpa [ContinuousAt, nhds_prod_eq] using tendsto_pure_nhds _ _
+  | (a : ℕ), ⊤, _ =>
+    suffices ∀ᶠ b in 𝓝 ⊤, (a - b : ℕ∞) = 0 by
+      simpa [ContinuousAt, nhds_prod_eq]
+    filter_upwards [le_mem_nhds (WithTop.coe_lt_top a)] with b using tsub_eq_zero_of_le
+  | ⊤, (b : ℕ), _ =>
+    suffices ∀ n : ℕ, ∀ᶠ a : ℕ∞ in 𝓝 ⊤, b + n < a by
+      simpa [ContinuousAt, nhds_prod_eq, (· ∘ ·), lt_tsub_iff_left, tendsto_nhds_top_iff_natCast_lt]
+    exact fun n ↦ lt_mem_nhds <| WithTop.coe_lt_top (b + n)
+
 end ENat
+
+theorem Filter.Tendsto.enatSub {α : Type*} {l : Filter α} {f g : α → ℕ∞} {a b : ℕ∞}
+    (hf : Tendsto f l (𝓝 a)) (hg : Tendsto g l (𝓝 b)) (h : a ≠ ⊤ ∨ b ≠ ⊤) :
+    Tendsto (fun x ↦ f x - g x) l (𝓝 (a - b)) :=
+  (ENat.continuousAt_sub h).tendsto.comp (hf.prod_mk_nhds hg)
+
+variable {X : Type*} [TopologicalSpace X] {f g : X → ℕ∞} {s : Set X} {x : X}
+
+nonrec theorem ContinuousWithinAt.enatSub
+    (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) (h : f x ≠ ⊤ ∨ g x ≠ ⊤) :
+    ContinuousWithinAt (fun x ↦ f x - g x) s x :=
+  hf.enatSub hg h
+
+nonrec theorem ContinuousAt.enatSub
+    (hf : ContinuousAt f x) (hg : ContinuousAt g x) (h : f x ≠ ⊤ ∨ g x ≠ ⊤) :
+    ContinuousAt (fun x ↦ f x - g x) x :=
+  hf.enatSub hg h
+
+nonrec theorem ContinuousOn.enatSub
+    (hf : ContinuousOn f s) (hg : ContinuousOn g s) (h : ∀ x ∈ s, f x ≠ ⊤ ∨ g x ≠ ⊤) :
+    ContinuousOn (fun x ↦ f x - g x) s := fun x hx ↦
+  (hf x hx).enatSub (hg x hx) (h x hx)
+
+nonrec theorem Continuous.enatSub
+    (hf : Continuous f) (hg : Continuous g) (h : ∀ x, f x ≠ ⊤ ∨ g x ≠ ⊤) :
+    Continuous (fun x ↦ f x - g x) :=
+  continuous_iff_continuousAt.2 fun x ↦ hf.continuousAt.enatSub hg.continuousAt (h x)
diff --git a/Mathlib/Topology/Instances/EReal.lean b/Mathlib/Topology/Instances/EReal.lean
index a14e8778f0703..4d970aa153cb8 100644
--- a/Mathlib/Topology/Instances/EReal.lean
+++ b/Mathlib/Topology/Instances/EReal.lean
@@ -70,7 +70,7 @@ theorem nhds_coe {r : ℝ} : 𝓝 (r : EReal) = (𝓝 r).map (↑) :=
 
 theorem nhds_coe_coe {r p : ℝ} :
     𝓝 ((r : EReal), (p : EReal)) = (𝓝 (r, p)).map fun p : ℝ × ℝ => (↑p.1, ↑p.2) :=
-  ((openEmbedding_coe.prod openEmbedding_coe).map_nhds_eq (r, p)).symm
+  ((openEmbedding_coe.prodMap openEmbedding_coe).map_nhds_eq (r, p)).symm
 
 theorem tendsto_toReal {a : EReal} (ha : a ≠ ⊤) (h'a : a ≠ ⊥) :
     Tendsto EReal.toReal (𝓝 a) (𝓝 a.toReal) := by
@@ -145,84 +145,67 @@ theorem tendsto_nhds_bot_iff_real {α : Type*} {m : α → EReal} {f : Filter α
     Tendsto m f (𝓝 ⊥) ↔ ∀ x : ℝ, ∀ᶠ a in f, m a < x :=
   nhds_bot_basis.tendsto_right_iff.trans <| by simp only [true_implies, mem_Iio]
 
-/-! ### Liminfs and Limsups -/
+lemma nhdsWithin_top : 𝓝[≠] (⊤ : EReal) = (atTop).map Real.toEReal := by
+  apply (nhdsWithin_hasBasis nhds_top_basis_Ici _).ext (atTop_basis.map Real.toEReal)
+  · simp only [EReal.image_coe_Ici, true_and]
+    intro x hx
+    by_cases hx_bot : x = ⊥
+    · simp [hx_bot]
+    lift x to ℝ using ⟨hx.ne_top, hx_bot⟩
+    refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
+    simp [h1, h2.ne_top]
+  · simp only [EReal.image_coe_Ici, true_implies]
+    refine fun x ↦ ⟨x, ⟨EReal.coe_lt_top x, fun x ⟨(h1 : _ ≤ x), h2⟩ ↦ ?_⟩⟩
+    simp [h1, Ne.lt_top' fun a ↦ h2 a.symm]
+
+lemma nhdsWithin_bot : 𝓝[≠] (⊥ : EReal) = (atBot).map Real.toEReal := by
+  apply (nhdsWithin_hasBasis nhds_bot_basis_Iic _).ext (atBot_basis.map Real.toEReal)
+  · simp only [EReal.image_coe_Iic, Set.subset_compl_singleton_iff, Set.mem_Ioc, lt_self_iff_false,
+      bot_le, and_true, not_false_eq_true, true_and]
+    intro x hx
+    by_cases hx_top : x = ⊤
+    · simp [hx_top]
+    lift x to ℝ using ⟨hx_top, hx.ne_bot⟩
+    refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
+    simp [h2, h1.ne_bot]
+  · simp only [EReal.image_coe_Iic, true_implies]
+    refine fun x ↦ ⟨x, ⟨EReal.bot_lt_coe x, fun x ⟨(h1 : x ≤ _), h2⟩ ↦ ?_⟩⟩
+    simp [h1, Ne.bot_lt' fun a ↦ h2 a.symm]
+
+lemma tendsto_toReal_atTop : Tendsto EReal.toReal (𝓝[≠] ⊤) atTop := by
+  rw [nhdsWithin_top, tendsto_map'_iff]
+  exact tendsto_id
 
-section LimInfSup
+lemma tendsto_toReal_atBot : Tendsto EReal.toReal (𝓝[≠] ⊥) atBot := by
+  rw [nhdsWithin_bot, tendsto_map'_iff]
+  exact tendsto_id
 
-variable {α : Type*} {f : Filter α} {u v : α → EReal} {a b : EReal}
+/-! ### Infs and Sups -/
 
-lemma liminf_le_liminf (h : u ≤ᶠ[f] v) :
-    liminf u f ≤ liminf v f := Filter.liminf_le_liminf h
+variable {α : Type*} {u v : α → EReal}
 
-lemma limsup_le_limsup (h : u ≤ᶠ[f] v) :
-    limsup u f ≤ limsup v f := Filter.limsup_le_limsup h
+lemma add_iInf_le_iInf_add : (⨅ x, u x) + (⨅ x, v x) ≤ ⨅ x, (u + v) x := by
+  refine add_le_of_forall_add_le fun a a_u b b_v ↦ ?_
+  rw [lt_iInf_iff] at a_u b_v
+  rcases a_u with ⟨c, a_c, c_u⟩
+  rcases b_v with ⟨d, b_d, d_v⟩
+  simp only [Pi.add_apply, le_iInf_iff]
+  exact fun x ↦ add_le_add (lt_of_lt_of_le a_c (c_u x)).le (lt_of_lt_of_le b_d (d_v x)).le
 
-/-- This lemma is superseded by `limsup_add_le_of_le` (weaker hypothesis) and
-`limsup_add_lt_of_lt` (stronger thesis). -/
-private lemma limsup_add_le_of_lt (ha : limsup u f < a) (hb : limsup v f < b) :
-    limsup (u + v) f ≤ a + b := by
-  rcases eq_or_neBot f with (rfl | _)
-  · simp only [limsup_bot, bot_le]
-  rw [← @limsup_const EReal α _ f _ (a + b)]
-  apply limsup_le_limsup (Eventually.mp (Eventually.and (eventually_lt_of_limsup_lt ha)
-    (eventually_lt_of_limsup_lt hb)) (Eventually.of_forall _))
-  simp only [Pi.add_apply, and_imp]
-  intro x
-  exact fun ux_lt_a vx_lt_b ↦ add_le_add (le_of_lt ux_lt_a) (le_of_lt vx_lt_b)
+lemma iSup_add_le_add_iSup (h : ⨆ x, u x ≠ ⊥ ∨ ⨆ x, v x ≠ ⊤) (h' : ⨆ x, u x ≠ ⊤ ∨ ⨆ x, v x ≠ ⊥) :
+    ⨆ x, (u + v) x ≤ (⨆ x, u x) + (⨆ x, v x) := by
+  refine le_add_of_forall_le_add h h' fun a a_u b b_v ↦ ?_
+  rw [gt_iff_lt, iSup_lt_iff] at a_u b_v
+  rcases a_u with ⟨c, a_c, c_u⟩
+  rcases b_v with ⟨d, b_d, d_v⟩
+  simp only [Pi.add_apply, iSup_le_iff]
+  exact fun x ↦ add_le_add (lt_of_le_of_lt (c_u x) a_c).le (lt_of_le_of_lt (d_v x) b_d).le
 
-lemma limsup_add_lt_of_lt (ha : limsup u f < a) (hb : limsup v f < b) :
-    limsup (u + v) f < a + b := by
-  obtain ⟨c, hc, hca⟩ := DenselyOrdered.dense _ _ ha
-  obtain ⟨d, hd, hdb⟩ := DenselyOrdered.dense _ _ hb
-  exact (limsup_add_le_of_lt hc hd).trans_lt (add_lt_add hca hdb)
+/-! ### Liminfs and Limsups -/
 
-lemma limsup_add_bot_of_ne_top (h : limsup u f = ⊥) (h' : limsup v f ≠ ⊤) :
-    limsup (u + v) f = ⊥ := by
-  apply le_bot_iff.1
-  apply (le_iff_le_forall_real_gt ⊥ (limsup (u + v) f)).1
-  intro x
-  rcases exists_between_coe_real (h'.lt_top) with ⟨y, ⟨hy, _⟩⟩
-  rw [← sub_add_cancel x y, coe_add (x - y) y, coe_sub x y]
-  intro _
-  apply @limsup_add_le_of_lt α f u v (x - y) y _ hy
-  rw [h, ← coe_sub x y]
-  exact bot_lt_coe (x - y)
-
-lemma limsup_add_le_add_limsup
-    (h : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤) (h' : limsup u f ≠ ⊤ ∨ limsup v f ≠ ⊥) :
-    limsup (u + v) f ≤ (limsup u f) + (limsup v f) := by
-  rcases eq_bot_or_bot_lt (limsup u f) with (u_bot | u_nbot)
-  · have v_ntop := h.neg_resolve_left u_bot
-    rw [limsup_add_bot_of_ne_top u_bot v_ntop]; exact bot_le
-  rcases eq_bot_or_bot_lt (limsup v f) with (v_bot | v_nbot)
-  · have u_ntop := h'.neg_resolve_right v_bot
-    rw [add_comm, limsup_add_bot_of_ne_top v_bot u_ntop]; exact bot_le
-  rcases eq_top_or_lt_top (limsup v f) with (v_top | v_ntop)
-  · rw [v_top, add_top_of_ne_bot (ne_of_gt u_nbot)]; exact le_top
-  have limsup_v_real := coe_toReal (ne_of_lt v_ntop) (ne_of_gt v_nbot)
-  apply (le_iff_le_forall_real_gt _ _).1
-  intros x hx
-  rcases lt_iff_exists_real_btwn.1 hx with ⟨y, ⟨sum_lt_y, y_lt_x⟩⟩
-  have key₁ : limsup u f < (y - limsup v f) := by
-    apply lt_of_eq_of_lt _ (sub_lt_sub_of_lt_of_le sum_lt_y (le_of_eq (Eq.refl (limsup v f)))
-      (ne_of_gt v_nbot) (ne_of_lt v_ntop))
-    rw [← limsup_v_real, add_sub_cancel_right]
-  have key₂ : limsup v f < limsup v f + x - y := by
-    rw [← limsup_v_real]; norm_cast; norm_cast at y_lt_x; linarith
-  apply le_of_le_of_eq (limsup_add_le_of_lt key₁ key₂)
-  rw [← limsup_v_real]; norm_cast; linarith
+section LimInfSup
 
-lemma limsup_add_le_of_le (ha : limsup u f < a) (hb : limsup v f ≤ b) :
-    limsup (u + v) f ≤ a + b := by
-  rcases lt_or_eq_of_le hb with (hb | hb)
-  · exact limsup_add_le_of_lt ha hb
-  by_cases hb' : b = ⊤
-  · convert le_top
-    on_goal 1 => rw [hb']
-    -- This closes both remaining goals at once.
-    exact add_top_of_ne_bot ha.ne_bot
-  exact (limsup_add_le_add_limsup (hb ▸ Or.inr hb') (Or.inl ha.ne_top)).trans
-    (add_le_add ha.le hb.le)
+variable {α : Type*} {f : Filter α} {u v : α → EReal}
 
 lemma liminf_neg : liminf (- v) f = - limsup v f :=
   EReal.negOrderIso.limsup_apply.symm
@@ -230,58 +213,64 @@ lemma liminf_neg : liminf (- v) f = - limsup v f :=
 lemma limsup_neg : limsup (- v) f = - liminf v f :=
   EReal.negOrderIso.liminf_apply.symm
 
+lemma add_liminf_le_liminf_add : (liminf u f) + (liminf v f) ≤ liminf (u + v) f := by
+  refine add_le_of_forall_add_le fun a a_u b b_v ↦ (le_liminf_iff).2 fun c c_ab ↦ ?_
+  filter_upwards [eventually_lt_of_lt_liminf a_u, eventually_lt_of_lt_liminf b_v] with x a_x b_x
+  exact lt_trans c_ab (add_lt_add a_x b_x)
+
+lemma limsup_add_le_add_limsup (h : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤)
+    (h' : limsup u f ≠ ⊤ ∨ limsup v f ≠ ⊥) :
+    limsup (u + v) f ≤ (limsup u f) + (limsup v f) := by
+  refine le_add_of_forall_le_add h h' fun a a_u b b_v ↦ (limsup_le_iff).2 fun c c_ab ↦ ?_
+  filter_upwards [eventually_lt_of_limsup_lt a_u, eventually_lt_of_limsup_lt b_v] with x a_x b_x
+  exact (add_lt_add a_x b_x).trans c_ab
+
+lemma limsup_add_liminf_le_limsup_add : (limsup u f) + (liminf v f) ≤ limsup (u + v) f :=
+  add_le_of_forall_add_le fun a a_u b b_v ↦ (le_limsup_iff).2 fun c c_ab ↦
+    Frequently.mono (Frequently.and_eventually ((frequently_lt_of_lt_limsup) a_u)
+    ((eventually_lt_of_lt_liminf) b_v)) fun _ ab_x ↦ c_ab.trans (add_lt_add ab_x.1 ab_x.2)
+
+lemma liminf_add_le_limsup_add_liminf (h : limsup u f ≠ ⊥ ∨ liminf v f ≠ ⊤)
+    (h' : limsup u f ≠ ⊤ ∨ liminf v f ≠ ⊥) :
+    liminf (u + v) f ≤ (limsup u f) + (liminf v f) :=
+  le_add_of_forall_le_add h h' fun a a_u b b_v ↦ (liminf_le_iff).2 fun c c_ab ↦
+    Frequently.mono (Frequently.and_eventually ((frequently_lt_of_liminf_lt) b_v)
+    ((eventually_lt_of_limsup_lt) a_u)) fun _ ab_x ↦ (add_lt_add ab_x.2 ab_x.1).trans c_ab
+
+variable {a b : EReal}
+
+lemma limsup_add_bot_of_ne_top (h : limsup u f = ⊥) (h' : limsup v f ≠ ⊤) :
+    limsup (u + v) f = ⊥ := by
+  apply le_bot_iff.1 (le_trans (limsup_add_le_add_limsup (Or.inr h') _) _)
+  · rw [h]; exact Or.inl bot_ne_top
+  · rw [h, bot_add]
+
+lemma limsup_add_le_of_le (ha : limsup u f < a) (hb : limsup v f ≤ b) :
+    limsup (u + v) f ≤ a + b := by
+  rcases eq_top_or_lt_top b with (rfl | h)
+  · rw [add_top_of_ne_bot ha.ne_bot]; exact le_top
+  · exact le_trans (limsup_add_le_add_limsup (Or.inr (lt_of_le_of_lt hb h).ne) (Or.inl ha.ne_top))
+      (add_le_add ha.le hb)
+
 lemma liminf_add_gt_of_gt (ha : a < liminf u f) (hb : b < liminf v f) :
-    a + b < liminf (u + v) f := by
-  have ha' : a ≠ ⊤ := ha.ne_top
-  have hb' : b ≠ ⊤ := hb.ne_top
-  have h : limsup (-(u + v)) f = limsup (-u + -v) f := by
-    apply limsup_congr
-    filter_upwards [eventually_lt_of_lt_liminf ha, eventually_lt_of_lt_liminf hb] with x hax hbx
-    dsimp
-    rw [neg_add (Or.inl hax.ne_bot) (Or.inr hbx.ne_bot), sub_eq_add_neg]
-  rw [← neg_lt_neg_iff, ← limsup_neg] at ha hb ⊢
-  rw [neg_add (Or.inr hb') (Or.inl ha'), h]
-  exact limsup_add_lt_of_lt ha hb
+    a + b < liminf (u + v) f :=
+  lt_of_lt_of_le (add_lt_add ha hb) add_liminf_le_liminf_add
 
 lemma liminf_add_top_of_ne_bot (h : liminf u f = ⊤) (h' : liminf v f ≠ ⊥) :
     liminf (u + v) f = ⊤ := by
-  apply top_le_iff.1 ((ge_iff_le_forall_real_lt (liminf (u + v) f) ⊤).1 _)
-  intro x
-  rcases exists_between_coe_real (Ne.bot_lt h') with ⟨y, ⟨_, hy⟩⟩
-  intro _
-  rw [← sub_add_cancel x y, coe_add (x - y) y]
-  exact coe_sub x y ▸ @liminf_add_gt_of_gt α f u v (x - y) y
-    (h ▸ coe_sub x y ▸ coe_lt_top (x-y)) hy |>.le
-
-lemma add_liminf_le_liminf_add : (liminf u f) + (liminf v f) ≤ liminf (u + v) f := by
-  by_cases hu : liminf u f = ⊥
-  · simp_all
-  by_cases hv : liminf v f = ⊥
-  · simp_all
-  have h' : limsup (-(u + v)) f = limsup (-u + -v) f := by
-    apply limsup_congr
-    filter_upwards [eventually_lt_of_lt_liminf (bot_lt_iff_ne_bot.mpr hu),
-      eventually_lt_of_lt_liminf (bot_lt_iff_ne_bot.mpr hv)] with x hux hvx
-    dsimp
-    rw [neg_add (Or.inl hux.ne_bot) (Or.inr hvx.ne_bot), sub_eq_add_neg]
-  rw [← neg_le_neg_iff, neg_add (Or.inl hu) (Or.inr hv), sub_eq_add_neg]
-  rw [← neg_inj, neg_bot] at hu hv
-  simp_rw [← limsup_neg] at hu hv ⊢
-  exact h' ▸ limsup_add_le_add_limsup (Or.inr hv) (Or.inl hu)
+  apply top_le_iff.1 (le_trans _ (add_liminf_le_liminf_add))
+  rw [h, top_add_of_ne_bot h']
 
 lemma limsup_le_iff {b : EReal} : limsup u f ≤ b ↔ ∀ c : ℝ, b < c → ∀ᶠ a : α in f, u a ≤ c := by
-  rw [← le_iff_le_forall_real_gt]
-  refine ⟨?_, ?_⟩ <;> intro h c b_lt_c
-  · rcases exists_between_coe_real b_lt_c with ⟨d, b_lt_d, d_lt_c⟩
-    specialize h d b_lt_d
-    have key := Filter.eventually_lt_of_limsup_lt (lt_of_le_of_lt h d_lt_c)
-    apply Filter.mem_of_superset key
+  rw [← le_of_forall_lt_iff_le]
+  refine ⟨?_, ?_⟩ <;> intro h c b_c
+  · rcases exists_between_coe_real b_c with ⟨d, b_d, d_c⟩
+    apply mem_of_superset (eventually_lt_of_limsup_lt (lt_of_le_of_lt (h d b_d) d_c))
     rw [Set.setOf_subset_setOf]
-    exact fun a h' ↦ le_of_lt h'
-  · rcases eq_or_neBot f with (rfl | _)
+    exact fun _ h' ↦ h'.le
+  · rcases eq_or_neBot f with rfl | _
     · simp only [limsup_bot, bot_le]
-    · specialize h c b_lt_c
-      exact @Filter.limsup_const EReal α _ f _ (c : EReal) ▸ limsup_le_limsup h
+    · exact (limsup_le_of_le) (h c b_c)
 
 end LimInfSup
 
@@ -289,8 +278,8 @@ end LimInfSup
 
 theorem continuousAt_add_coe_coe (a b : ℝ) :
     ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, b) := by
-  simp only [ContinuousAt, nhds_coe_coe, ← coe_add, tendsto_map'_iff, (· ∘ ·), tendsto_coe,
-    tendsto_add]
+  simp only [ContinuousAt, nhds_coe_coe, ← coe_add, tendsto_map'_iff, Function.comp_def,
+    tendsto_coe, tendsto_add]
 
 theorem continuousAt_add_top_coe (a : ℝ) :
     ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, a) := by
@@ -301,7 +290,7 @@ theorem continuousAt_add_top_coe (a : ℝ) :
 
 theorem continuousAt_add_coe_top (a : ℝ) :
     ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊤) := by
-  simpa only [add_comm, (· ∘ ·), ContinuousAt, Prod.swap]
+  simpa only [add_comm, Function.comp_def, ContinuousAt, Prod.swap]
     using Tendsto.comp (continuousAt_add_top_coe a) (continuous_swap.tendsto ((a : EReal), ⊤))
 
 theorem continuousAt_add_top_top : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊤, ⊤) := by
@@ -319,7 +308,7 @@ theorem continuousAt_add_bot_coe (a : ℝ) :
 
 theorem continuousAt_add_coe_bot (a : ℝ) :
     ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (a, ⊥) := by
-  simpa only [add_comm, (· ∘ ·), ContinuousAt, Prod.swap]
+  simpa only [add_comm, Function.comp_def, ContinuousAt, Prod.swap]
     using Tendsto.comp (continuousAt_add_bot_coe a) (continuous_swap.tendsto ((a : EReal), ⊥))
 
 theorem continuousAt_add_bot_bot : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, ⊥) := by
@@ -369,7 +358,7 @@ private lemma continuousAt_mul_symm1 {a b : EReal}
     simp
   rw [this]
   apply ContinuousAt.comp (Continuous.continuousAt continuous_neg)
-    <| ContinuousAt.comp _ (ContinuousAt.prod_map (Continuous.continuousAt continuous_neg)
+    <| ContinuousAt.comp _ (ContinuousAt.prodMap (Continuous.continuousAt continuous_neg)
       (Continuous.continuousAt continuous_id))
   simp [h]
 
@@ -386,7 +375,7 @@ private lemma continuousAt_mul_symm3 {a b : EReal}
 private lemma continuousAt_mul_coe_coe (a b : ℝ) :
     ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (a, b) := by
   simp [ContinuousAt, EReal.nhds_coe_coe, ← EReal.coe_mul, Filter.tendsto_map'_iff,
-    (· ∘ ·), EReal.tendsto_coe, tendsto_mul]
+    Function.comp_def, EReal.tendsto_coe, tendsto_mul]
 
 private lemma continuousAt_mul_top_top :
     ContinuousAt (fun p : EReal × EReal ↦ p.1 * p.2) (⊤, ⊤) := by
diff --git a/Mathlib/Topology/Instances/Int.lean b/Mathlib/Topology/Instances/Int.lean
index eae87217dd5dc..5ee25d7b929f2 100644
--- a/Mathlib/Topology/Instances/Int.lean
+++ b/Mathlib/Topology/Instances/Int.lean
@@ -8,7 +8,7 @@ import Mathlib.Data.Int.SuccPred
 import Mathlib.Data.Int.ConditionallyCompleteOrder
 import Mathlib.Topology.Instances.Discrete
 import Mathlib.Topology.MetricSpace.Bounded
-import Mathlib.Order.Filter.Archimedean
+import Mathlib.Order.Filter.AtTopBot.Archimedean
 import Mathlib.Topology.MetricSpace.Basic
 
 /-!
@@ -39,13 +39,16 @@ theorem pairwise_one_le_dist : Pairwise fun m n : ℤ => 1 ≤ dist m n := by
   intro m n hne
   rw [dist_eq]; norm_cast; rwa [← zero_add (1 : ℤ), Int.add_one_le_iff, abs_pos, sub_ne_zero]
 
-theorem uniformEmbedding_coe_real : UniformEmbedding ((↑) : ℤ → ℝ) :=
-  uniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+theorem isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℤ → ℝ) :=
+  isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coe_real := isUniformEmbedding_coe_real
 
 theorem closedEmbedding_coe_real : ClosedEmbedding ((↑) : ℤ → ℝ) :=
   closedEmbedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
 
-instance : MetricSpace ℤ := Int.uniformEmbedding_coe_real.comapMetricSpace _
+instance : MetricSpace ℤ := Int.isUniformEmbedding_coe_real.comapMetricSpace _
 
 theorem preimage_ball (x : ℤ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r := rfl
 
diff --git a/Mathlib/Topology/Instances/NNReal.lean b/Mathlib/Topology/Instances/NNReal.lean
index f7759c47f7eea..6f0a140493505 100644
--- a/Mathlib/Topology/Instances/NNReal.lean
+++ b/Mathlib/Topology/Instances/NNReal.lean
@@ -6,6 +6,7 @@ Authors: Johan Commelin
 import Mathlib.Data.NNReal.Star
 import Mathlib.Topology.Algebra.InfiniteSum.Order
 import Mathlib.Topology.Algebra.InfiniteSum.Ring
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Topology.Instances.Real
 import Mathlib.Topology.MetricSpace.Isometry
 
@@ -93,6 +94,11 @@ noncomputable def _root_.ContinuousMap.realToNNReal : C(ℝ, ℝ≥0) :=
 theorem continuous_coe : Continuous ((↑) : ℝ≥0 → ℝ) :=
   continuous_subtype_val
 
+lemma _root_.ContinuousOn.ofReal_map_toNNReal {f : ℝ≥0 → ℝ≥0} {s : Set ℝ} {t : Set ℝ≥0}
+    (hf : ContinuousOn f t) (h : Set.MapsTo Real.toNNReal s t) :
+    ContinuousOn (fun x ↦ f x.toNNReal : ℝ → ℝ) s :=
+  continuous_subtype_val.comp_continuousOn <| hf.comp continuous_real_toNNReal.continuousOn h
+
 /-- Embedding of `ℝ≥0` to `ℝ` as a bundled continuous map. -/
 @[simps (config := .asFn)]
 def _root_.ContinuousMap.coeNNRealReal : C(ℝ≥0, ℝ) :=
@@ -257,7 +263,7 @@ section Monotone
 /-- A monotone, bounded above sequence `f : ℕ → ℝ` has a finite limit. -/
 theorem _root_.Real.tendsto_of_bddAbove_monotone {f : ℕ → ℝ} (h_bdd : BddAbove (Set.range f))
     (h_mon : Monotone f) : ∃ r : ℝ, Tendsto f atTop (𝓝 r) := by
-  obtain ⟨B, hB⟩ := Real.exists_isLUB  (Set.range_nonempty f) h_bdd
+  obtain ⟨B, hB⟩ := Real.exists_isLUB (Set.range_nonempty f) h_bdd
   exact ⟨B, tendsto_atTop_isLUB h_mon hB⟩
 
 /-- An antitone, bounded below sequence `f : ℕ → ℝ` has a finite limit. -/
diff --git a/Mathlib/Topology/Instances/Nat.lean b/Mathlib/Topology/Instances/Nat.lean
index 2e4857faf7c51..e90ce1cbe65b7 100644
--- a/Mathlib/Topology/Instances/Nat.lean
+++ b/Mathlib/Topology/Instances/Nat.lean
@@ -31,13 +31,16 @@ theorem dist_cast_real (x y : ℕ) : dist (x : ℝ) y = dist x y := rfl
 theorem pairwise_one_le_dist : Pairwise fun m n : ℕ => 1 ≤ dist m n := fun _ _ hne =>
   Int.pairwise_one_le_dist <| mod_cast hne
 
-theorem uniformEmbedding_coe_real : UniformEmbedding ((↑) : ℕ → ℝ) :=
-  uniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+theorem isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℕ → ℝ) :=
+  isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coe_real := isUniformEmbedding_coe_real
 
 theorem closedEmbedding_coe_real : ClosedEmbedding ((↑) : ℕ → ℝ) :=
   closedEmbedding_of_pairwise_le_dist zero_lt_one pairwise_one_le_dist
 
-instance : MetricSpace ℕ := Nat.uniformEmbedding_coe_real.comapMetricSpace _
+instance : MetricSpace ℕ := Nat.isUniformEmbedding_coe_real.comapMetricSpace _
 
 theorem preimage_ball (x : ℕ) (r : ℝ) : (↑) ⁻¹' ball (x : ℝ) r = ball x r := rfl
 
diff --git a/Mathlib/Topology/Instances/PNat.lean b/Mathlib/Topology/Instances/PNat.lean
index 3a05c74ac7076..dd79778f35d9b 100644
--- a/Mathlib/Topology/Instances/PNat.lean
+++ b/Mathlib/Topology/Instances/PNat.lean
@@ -24,7 +24,10 @@ theorem dist_eq (x y : ℕ+) : dist x y = |(↑x : ℝ) - ↑y| := rfl
 @[simp, norm_cast]
 theorem dist_coe (x y : ℕ+) : dist (↑x : ℕ) (↑y : ℕ) = dist x y := rfl
 
-theorem uniformEmbedding_coe : UniformEmbedding ((↑) : ℕ+ → ℕ) := uniformEmbedding_subtype_val
+theorem isUniformEmbedding_coe : IsUniformEmbedding ((↑) : ℕ+ → ℕ) := isUniformEmbedding_subtype_val
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coe := isUniformEmbedding_coe
 
 instance : DiscreteTopology ℕ+ := inferInstanceAs (DiscreteTopology { n : ℕ // 0 < n })
 
diff --git a/Mathlib/Topology/Instances/Rat.lean b/Mathlib/Topology/Instances/Rat.lean
index ce48d2b2344a3..715361b5de0d9 100644
--- a/Mathlib/Topology/Instances/Rat.lean
+++ b/Mathlib/Topology/Instances/Rat.lean
@@ -30,14 +30,20 @@ theorem dist_cast (x y : ℚ) : dist (x : ℝ) y = dist x y :=
 theorem uniformContinuous_coe_real : UniformContinuous ((↑) : ℚ → ℝ) :=
   uniformContinuous_comap
 
-theorem uniformEmbedding_coe_real : UniformEmbedding ((↑) : ℚ → ℝ) :=
-  uniformEmbedding_comap Rat.cast_injective
+theorem isUniformEmbedding_coe_real : IsUniformEmbedding ((↑) : ℚ → ℝ) :=
+  isUniformEmbedding_comap Rat.cast_injective
 
-theorem denseEmbedding_coe_real : DenseEmbedding ((↑) : ℚ → ℝ) :=
-  uniformEmbedding_coe_real.denseEmbedding Rat.denseRange_cast
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coe_real := isUniformEmbedding_coe_real
+
+theorem isDenseEmbedding_coe_real : IsDenseEmbedding ((↑) : ℚ → ℝ) :=
+  isUniformEmbedding_coe_real.isDenseEmbedding Rat.denseRange_cast
+
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_coe_real := isDenseEmbedding_coe_real
 
 theorem embedding_coe_real : Embedding ((↑) : ℚ → ℝ) :=
-  denseEmbedding_coe_real.to_embedding
+  isDenseEmbedding_coe_real.to_embedding
 
 theorem continuous_coe_real : Continuous ((↑) : ℚ → ℝ) :=
   uniformContinuous_coe_real.continuous
@@ -48,8 +54,11 @@ end Rat
 theorem Nat.dist_cast_rat (x y : ℕ) : dist (x : ℚ) y = dist x y := by
   rw [← Nat.dist_cast_real, ← Rat.dist_cast]; congr
 
-theorem Nat.uniformEmbedding_coe_rat : UniformEmbedding ((↑) : ℕ → ℚ) :=
-  uniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Nat.pairwise_one_le_dist
+theorem Nat.isUniformEmbedding_coe_rat : IsUniformEmbedding ((↑) : ℕ → ℚ) :=
+  isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Nat.pairwise_one_le_dist
+
+@[deprecated (since := "2024-10-01")]
+alias Nat.uniformEmbedding_coe_rat := Nat.isUniformEmbedding_coe_rat
 
 theorem Nat.closedEmbedding_coe_rat : ClosedEmbedding ((↑) : ℕ → ℚ) :=
   closedEmbedding_of_pairwise_le_dist zero_lt_one <| by simpa using Nat.pairwise_one_le_dist
@@ -58,8 +67,11 @@ theorem Nat.closedEmbedding_coe_rat : ClosedEmbedding ((↑) : ℕ → ℚ) :=
 theorem Int.dist_cast_rat (x y : ℤ) : dist (x : ℚ) y = dist x y := by
   rw [← Int.dist_cast_real, ← Rat.dist_cast]; congr
 
-theorem Int.uniformEmbedding_coe_rat : UniformEmbedding ((↑) : ℤ → ℚ) :=
-  uniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist
+theorem Int.isUniformEmbedding_coe_rat : IsUniformEmbedding ((↑) : ℤ → ℚ) :=
+  isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist
+
+@[deprecated (since := "2024-10-01")]
+alias Int.uniformEmbedding_coe_rat := Int.isUniformEmbedding_coe_rat
 
 theorem Int.closedEmbedding_coe_rat : ClosedEmbedding ((↑) : ℤ → ℚ) :=
   closedEmbedding_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist
@@ -69,10 +81,10 @@ namespace Rat
 instance : NoncompactSpace ℚ := Int.closedEmbedding_coe_rat.noncompactSpace
 
 theorem uniformContinuous_add : UniformContinuous fun p : ℚ × ℚ => p.1 + p.2 :=
-  Rat.uniformEmbedding_coe_real.toUniformInducing.uniformContinuous_iff.2 <| by
-    simp only [(· ∘ ·), Rat.cast_add]
+  Rat.isUniformEmbedding_coe_real.isUniformInducing.uniformContinuous_iff.2 <| by
+    simp only [Function.comp_def, Rat.cast_add]
     exact Real.uniformContinuous_add.comp
-      (Rat.uniformContinuous_coe_real.prod_map Rat.uniformContinuous_coe_real)
+      (Rat.uniformContinuous_coe_real.prodMap Rat.uniformContinuous_coe_real)
 
 theorem uniformContinuous_neg : UniformContinuous (@Neg.neg ℚ _) :=
   Metric.uniformContinuous_iff.2 fun ε ε0 =>
@@ -94,7 +106,7 @@ instance : TopologicalRing ℚ := inferInstance
 
 nonrec theorem totallyBounded_Icc (a b : ℚ) : TotallyBounded (Icc a b) := by
   simpa only [preimage_cast_Icc]
-    using totallyBounded_preimage Rat.uniformEmbedding_coe_real.toUniformInducing
+    using totallyBounded_preimage Rat.isUniformEmbedding_coe_real.isUniformInducing
       (totallyBounded_Icc (a : ℝ) b)
 
 end Rat
diff --git a/Mathlib/Topology/Instances/RatLemmas.lean b/Mathlib/Topology/Instances/RatLemmas.lean
index b94351980b416..1f633201490d0 100644
--- a/Mathlib/Topology/Instances/RatLemmas.lean
+++ b/Mathlib/Topology/Instances/RatLemmas.lean
@@ -36,10 +36,10 @@ local notation "ℚ∞" => OnePoint ℚ
 
 namespace Rat
 
-variable {p q : ℚ} {s t : Set ℚ}
+variable {p : ℚ} {s : Set ℚ}
 
 theorem interior_compact_eq_empty (hs : IsCompact s) : interior s = ∅ :=
-  denseEmbedding_coe_real.toDenseInducing.interior_compact_eq_empty dense_irrational hs
+  isDenseEmbedding_coe_real.toIsDenseInducing.interior_compact_eq_empty dense_irrational hs
 
 theorem dense_compl_compact (hs : IsCompact s) : Dense sᶜ :=
   interior_eq_empty_iff_dense_compl.1 (interior_compact_eq_empty hs)
@@ -72,7 +72,7 @@ theorem not_secondCountableTopology_opc : ¬SecondCountableTopology ℚ∞ := by
   exact not_firstCountableTopology_opc inferInstance
 
 instance : TotallyDisconnectedSpace ℚ := by
-  clear p q s t
+  clear p s
   refine ⟨fun s hsu hs x hx y hy => ?_⟩; clear hsu
   by_contra! H : x ≠ y
   wlog hlt : x < y
diff --git a/Mathlib/Topology/Instances/Real.lean b/Mathlib/Topology/Instances/Real.lean
index 5b895c2cdeb63..16325c2f3e0ab 100644
--- a/Mathlib/Topology/Instances/Real.lean
+++ b/Mathlib/Topology/Instances/Real.lean
@@ -3,16 +3,17 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathlib.Data.Real.Star
-import Mathlib.Algebra.Algebra.Basic
+import Mathlib.Algebra.Module.Rat
+import Mathlib.Algebra.Module.Submodule.Lattice
 import Mathlib.Algebra.Periodic
+import Mathlib.Data.Real.Star
 import Mathlib.Topology.Algebra.Order.Archimedean
 import Mathlib.Topology.Algebra.Order.Field
-import Mathlib.Topology.Algebra.UniformMulAction
 import Mathlib.Topology.Algebra.Star
+import Mathlib.Topology.Algebra.UniformMulAction
 import Mathlib.Topology.Instances.Int
-import Mathlib.Topology.Order.Bornology
 import Mathlib.Topology.Metrizable.Basic
+import Mathlib.Topology.Order.Bornology
 
 /-!
 # Topological properties of ℝ
diff --git a/Mathlib/Topology/Instances/RealVectorSpace.lean b/Mathlib/Topology/Instances/RealVectorSpace.lean
index a7eaa801c1d9f..5d4327d792021 100644
--- a/Mathlib/Topology/Instances/RealVectorSpace.lean
+++ b/Mathlib/Topology/Instances/RealVectorSpace.lean
@@ -23,7 +23,7 @@ theorem map_real_smul {G} [FunLike G E F] [AddMonoidHomClass G E F] (f : G) (hf
     (c : ℝ) (x : E) :
     f (c • x) = c • f x :=
   suffices (fun c : ℝ => f (c • x)) = fun c : ℝ => c • f x from congr_fun this c
-  Rat.denseEmbedding_coe_real.dense.equalizer (hf.comp <| continuous_id.smul continuous_const)
+  Rat.isDenseEmbedding_coe_real.dense.equalizer (hf.comp <| continuous_id.smul continuous_const)
     (continuous_id.smul continuous_const) (funext fun r => map_ratCast_smul f ℝ ℝ r x)
 
 namespace AddMonoidHom
diff --git a/Mathlib/Topology/KrullDimension.lean b/Mathlib/Topology/KrullDimension.lean
index b5387350eed0f..b5eee2e982ae4 100644
--- a/Mathlib/Topology/KrullDimension.lean
+++ b/Mathlib/Topology/KrullDimension.lean
@@ -12,17 +12,52 @@ import Mathlib.Topology.Sets.Closeds
 The Krull dimension of a topological space is the order theoretic Krull dimension applied to the
 collection of all its subsets that are closed and irreducible. Unfolding this definition, it is
 the length of longest series of closed irreducible subsets ordered by inclusion.
-
-TODO: The Krull dimension of `Spec(R)` equals the Krull dimension of `R`, for `R` a commutative
-  ring.
 -/
 
-open TopologicalSpace
+open TopologicalSpace Order
 
 /--
 The Krull dimension of a topological space is the supremum of lengths of chains of
 closed irreducible sets.
 -/
-noncomputable def topologicalKrullDim (T : Type _) [TopologicalSpace T] :
-    WithBot (WithTop ℕ) :=
+noncomputable def topologicalKrullDim (T : Type*) [TopologicalSpace T] : WithBot ℕ∞ :=
   krullDim (IrreducibleCloseds T)
+
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+
+/--
+Map induced on irreducible closed subsets by a closed continuous map `f`.
+This is just a wrapper around the image of `f` together with proofs that it
+preserves irreducibility (by continuity) and closedness (since `f` is closed).
+-/
+def IrreducibleCloseds.map {f : X → Y} (hf1 : Continuous f) (hf2 : IsClosedMap f)
+    (c : IrreducibleCloseds X) :
+    IrreducibleCloseds Y where
+  carrier := f '' c
+  is_irreducible' := c.is_irreducible'.image f hf1.continuousOn
+  is_closed' := hf2 c c.is_closed'
+
+/--
+Taking images under a closed embedding is strictly monotone on the preorder of irreducible closeds.
+-/
+lemma IrreducibleCloseds.map_strictMono {f : X → Y} (hf : ClosedEmbedding f) :
+    StrictMono (IrreducibleCloseds.map hf.continuous hf.isClosedMap) :=
+  fun ⦃_ _⦄ UltV ↦ hf.inj.image_strictMono UltV
+
+/--
+If `f : X → Y` is a closed embedding, then the Krull dimension of `X` is less than or equal
+to the Krull dimension of `Y`.
+-/
+theorem ClosedEmbedding.topologicalKrullDim_le (f : X → Y) (hf : ClosedEmbedding f) :
+    topologicalKrullDim X ≤ topologicalKrullDim Y :=
+  krullDim_le_of_strictMono _ (IrreducibleCloseds.map_strictMono hf)
+
+/-- The topological Krull dimension is invariant under homeomorphisms -/
+theorem IsHomeomorph.topologicalKrullDim_eq (f : X → Y) (h : IsHomeomorph f) :
+    topologicalKrullDim X = topologicalKrullDim Y :=
+  have fwd : topologicalKrullDim X ≤ topologicalKrullDim Y :=
+    ClosedEmbedding.topologicalKrullDim_le f h.closedEmbedding
+  have bwd : topologicalKrullDim Y ≤ topologicalKrullDim X :=
+    ClosedEmbedding.topologicalKrullDim_le (h.homeomorph f).symm
+    (h.homeomorph f).symm.closedEmbedding
+  le_antisymm fwd bwd
diff --git a/Mathlib/Topology/List.lean b/Mathlib/Topology/List.lean
index 563dc61834ff2..f9e3f9ddbdfc3 100644
--- a/Mathlib/Topology/List.lean
+++ b/Mathlib/Topology/List.lean
@@ -86,7 +86,7 @@ theorem tendsto_cons_iff {β : Type*} {f : List α → β} {b : Filter β} {a :
   have : 𝓝 (a::l) = (𝓝 a ×ˢ 𝓝 l).map fun p : α × List α => p.1::p.2 := by
     simp only [nhds_cons, Filter.prod_eq, (Filter.map_def _ _).symm,
       (Filter.seq_eq_filter_seq _ _).symm]
-    simp [-Filter.map_def, (· ∘ ·), functor_norm]
+    simp [-Filter.map_def, Function.comp_def, functor_norm]
   rw [this, Filter.tendsto_map'_iff]; rfl
 
 theorem continuous_cons : Continuous fun x : α × List α => (x.1::x.2 : List α) :=
@@ -124,7 +124,7 @@ theorem tendsto_insertNth' {a : α} :
     have : 𝓝 a ×ˢ 𝓝 (a'::l) =
         (𝓝 a ×ˢ (𝓝 a' ×ˢ 𝓝 l)).map fun p : α × α × List α => (p.1, p.2.1::p.2.2) := by
       simp only [nhds_cons, Filter.prod_eq, ← Filter.map_def, ← Filter.seq_eq_filter_seq]
-      simp [-Filter.map_def, (· ∘ ·), functor_norm]
+      simp [-Filter.map_def, Function.comp_def, functor_norm]
     rw [this, tendsto_map'_iff]
     exact
       (tendsto_fst.comp tendsto_snd).cons
diff --git a/Mathlib/Topology/LocalAtTarget.lean b/Mathlib/Topology/LocalAtTarget.lean
index 62a0b258ffe46..5970438a142e8 100644
--- a/Mathlib/Topology/LocalAtTarget.lean
+++ b/Mathlib/Topology/LocalAtTarget.lean
@@ -108,7 +108,10 @@ theorem isLocallyClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
     IsLocallyClosed s ↔ ∀ i, IsLocallyClosed ((↑) ⁻¹' s : Set (U i)) := by
   simp_rw [isLocallyClosed_iff_isOpen_coborder]
   rw [isOpen_iff_coe_preimage_of_iSup_eq_top hU]
-  exact forall_congr' fun i ↦ by rw [(U i).isOpen.openEmbedding_subtype_val.coborder_preimage]
+  exact forall_congr' fun i ↦ by
+    have : coborder ((↑) ⁻¹' s : Set (U i)) = Subtype.val ⁻¹' coborder s := by
+      exact (U i).isOpen.openEmbedding_subtype_val.coborder_preimage _
+    rw [this]
 
 theorem isOpenMap_iff_isOpenMap_of_iSup_eq_top :
     IsOpenMap f ↔ ∀ i, IsOpenMap ((U i).1.restrictPreimage f) := by
diff --git a/Mathlib/Topology/LocallyClosed.lean b/Mathlib/Topology/LocallyClosed.lean
index 90d7577eba4a9..fd260f0e11c03 100644
--- a/Mathlib/Topology/LocallyClosed.lean
+++ b/Mathlib/Topology/LocallyClosed.lean
@@ -61,6 +61,17 @@ lemma coborder_eq_compl_frontier_iff :
   simp_rw [coborder_eq_union_frontier_compl, union_eq_right, subset_compl_iff_disjoint_left,
     disjoint_frontier_iff_isOpen]
 
+theorem coborder_eq_union_closure_compl {s : Set X} : coborder s = s ∪ (closure s)ᶜ := by
+  rw [coborder, compl_eq_comm, compl_union, compl_compl, inter_comm]
+  rfl
+
+/-- The coborder of any set is dense -/
+theorem dense_coborder {s : Set X} :
+    Dense (coborder s) := by
+  rw [dense_iff_closure_eq, coborder_eq_union_closure_compl, closure_union, ← univ_subset_iff]
+  refine _root_.subset_trans ?_ (union_subset_union_right _ (subset_closure))
+  simp
+
 alias ⟨_, IsOpen.coborder_eq⟩ := coborder_eq_compl_frontier_iff
 
 lemma IsOpenMap.coborder_preimage_subset (hf : IsOpenMap f) (s : Set Y) :
@@ -88,12 +99,6 @@ lemma isClosed_preimage_val_coborder :
     IsClosed (coborder s ↓∩ s) := by
   rw [isClosed_preimage_val, inter_eq_right.mpr subset_coborder, coborder_inter_closure]
 
-lemma IsOpen.isLocallyClosed (hs : IsOpen s) : IsLocallyClosed s :=
-  ⟨_, _, hs, isClosed_univ, (inter_univ _).symm⟩
-
-lemma IsClosed.isLocallyClosed (hs : IsClosed s) : IsLocallyClosed s :=
-  ⟨_, _, isOpen_univ, hs, (univ_inter _).symm⟩
-
 lemma IsLocallyClosed.inter (hs : IsLocallyClosed s) (ht : IsLocallyClosed t) :
     IsLocallyClosed (s ∩ t) := by
   obtain ⟨U₁, Z₁, hU₁, hZ₁, rfl⟩ := hs
@@ -145,8 +150,8 @@ lemma isLocallyClosed_tfae (s : Set X) :
       ∀ x ∈ s, ∃ U ∈ 𝓝 x, IsClosed (U ↓∩ s),
       ∀ x ∈ s, ∃ U, x ∈ U ∧ IsOpen U ∧ U ∩ closure s ⊆ s,
       IsOpen (closure s ↓∩ s)] := by
-  tfae_have 1 → 2
-  · rintro ⟨U, Z, hU, hZ, rfl⟩
+  tfae_have 1 → 2 := by
+    rintro ⟨U, Z, hU, hZ, rfl⟩
     have : Z ∪ (frontier (U ∩ Z))ᶜ = univ := by
       nth_rw 1 [← hZ.closure_eq]
       rw [← compl_subset_iff_union, compl_subset_compl]
@@ -155,23 +160,23 @@ lemma isLocallyClosed_tfae (s : Set X) :
       inter_univ]
     exact hU.union isClosed_frontier.isOpen_compl
   tfae_have 2 → 3
-  · exact fun h x ↦ (⟨coborder s, h.mem_nhds <| subset_coborder ·, isClosed_preimage_val_coborder⟩)
+  | h, x => (⟨coborder s, h.mem_nhds <| subset_coborder ·, isClosed_preimage_val_coborder⟩)
   tfae_have 3 → 4
-  · intro h x hx
+  | h, x, hx => by
     obtain ⟨t, ht, ht'⟩ := h x hx
     obtain ⟨U, hUt, hU, hxU⟩ := mem_nhds_iff.mp ht
     rw [isClosed_preimage_val] at ht'
     exact ⟨U, hxU, hU, (subset_inter (inter_subset_left.trans hUt) (hU.inter_closure.trans
       (closure_mono <| inter_subset_inter hUt subset_rfl))).trans ht'⟩
   tfae_have 4 → 5
-  · intro H
+  | H => by
     choose U hxU hU e using H
     refine ⟨⋃ x ∈ s, U x ‹_›, isOpen_iUnion (isOpen_iUnion <| hU ·), ext fun x ↦ ⟨?_, ?_⟩⟩
     · rintro ⟨_, ⟨⟨y, rfl⟩, ⟨_, ⟨hy, rfl⟩, hxU⟩⟩⟩
       exact e y hy ⟨hxU, x.2⟩
     · exact (subset_iUnion₂ _ _ <| hxU x ·)
   tfae_have 5 → 1
-  · intro H
+  | H => by
     convert H.isLocallyClosed.image inducing_subtype_val
       (by simpa using isClosed_closure.isLocallyClosed)
     simpa using subset_closure
diff --git a/Mathlib/Topology/LocallyConstant/Algebra.lean b/Mathlib/Topology/LocallyConstant/Algebra.lean
index 1e70ce1a80c57..986f83f29a8ac 100644
--- a/Mathlib/Topology/LocallyConstant/Algebra.lean
+++ b/Mathlib/Topology/LocallyConstant/Algebra.lean
@@ -186,7 +186,7 @@ instance [NonAssocSemiring Y] : NonAssocSemiring (LocallyConstant X Y) :=
   Function.Injective.nonAssocSemiring DFunLike.coe DFunLike.coe_injective' rfl rfl
     (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) fun _ => rfl
 
-/-- The constant-function embedding, as a ring hom.  -/
+/-- The constant-function embedding, as a ring hom. -/
 @[simps]
 def constRingHom [NonAssocSemiring Y] : Y →+* LocallyConstant X Y :=
   { constMonoidHom, constAddMonoidHom with toFun := const X }
@@ -347,7 +347,7 @@ lemma ker_comapₗ [Semiring R] [AddCommMonoid Z] [Module R Z] (f : C(X, Y))
     LinearMap.ker (comapₗ R f : LocallyConstant Y Z →ₗ[R] LocallyConstant X Z) = ⊥ :=
   LinearMap.ker_eq_bot_of_injective <| comap_injective _ hfs
 
-/-- `LocallyConstant.congrLeft` as a linear equivalence. -/
+/-- `LocallyConstant.congrLeft` as a linear equivalence. -/
 @[simps!]
 def congrLeftₗ (R : Type*) [Semiring R] [AddCommMonoid Z] [Module R Z] (e : X ≃ₜ Y) :
     LocallyConstant X Z ≃ₗ[R] LocallyConstant Y Z where
@@ -362,7 +362,7 @@ def congrLeftRingEquiv [Semiring Z] (e : X ≃ₜ Y) :
   __ := comapMonoidHom ⟨_, e.symm.continuous⟩
   __ := comapAddMonoidHom ⟨_, e.symm.continuous⟩
 
-/-- `LocallyConstant.congrLeft` as an `AlgEquiv`. -/
+/-- `LocallyConstant.congrLeft` as an `AlgEquiv`. -/
 @[simps!]
 def congrLeftₐ (R : Type*) [CommSemiring R] [Semiring Z] [Algebra R Z] (e : X ≃ₜ Y) :
     LocallyConstant X Z ≃ₐ[R] LocallyConstant Y Z where
@@ -406,7 +406,7 @@ def mapₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z]
   toRingHom := mapRingHom f
   commutes' _ := by aesop
 
-/-- `LocallyConstant.congrRight` as a linear equivalence. -/
+/-- `LocallyConstant.congrRight` as a linear equivalence. -/
 @[simps!]
 def congrRightₗ (R : Type*) [Semiring R] [AddCommMonoid Y] [Module R Y]
     [AddCommMonoid Z] [Module R Z] (e : Y ≃ₗ[R] Z) :
@@ -422,7 +422,7 @@ def congrRightRingEquiv [Semiring Y] [Semiring Z] (e : Y ≃+* Z) :
   __ := mapMonoidHom e.toMonoidHom
   __ := mapAddMonoidHom e.toAddMonoidHom
 
-/-- `LocallyConstant.congrRight` as an `AlgEquiv`. -/
+/-- `LocallyConstant.congrRight` as an `AlgEquiv`. -/
 @[simps!]
 def congrRightₐ (R : Type*) [CommSemiring R] [Semiring Y] [Algebra R Y] [Semiring Z] [Algebra R Z]
     (e : Y ≃ₐ[R] Z) : LocallyConstant X Y ≃ₐ[R] LocallyConstant X Z where
diff --git a/Mathlib/Topology/LocallyConstant/Basic.lean b/Mathlib/Topology/LocallyConstant/Basic.lean
index 48f981df57cff..401cf78da5d19 100644
--- a/Mathlib/Topology/LocallyConstant/Basic.lean
+++ b/Mathlib/Topology/LocallyConstant/Basic.lean
@@ -39,18 +39,15 @@ protected theorem tfae (f : X → Y) :
       ∀ x, IsOpen { x' | f x' = f x },
       ∀ y, IsOpen (f ⁻¹' {y}),
       ∀ x, ∃ U : Set X, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x] := by
-  tfae_have 1 → 4
-  · exact fun h y => h {y}
-  tfae_have 4 → 3
-  · exact fun h x => h (f x)
-  tfae_have 3 → 2
-  · exact fun h x => IsOpen.mem_nhds (h x) rfl
+  tfae_have 1 → 4 := fun h y => h {y}
+  tfae_have 4 → 3 := fun h x => h (f x)
+  tfae_have 3 → 2 := fun h x => IsOpen.mem_nhds (h x) rfl
   tfae_have 2 → 5
-  · intro h x
+  | h, x => by
     rcases mem_nhds_iff.1 (h x) with ⟨U, eq, hU, hx⟩
     exact ⟨U, hU, hx, eq⟩
   tfae_have 5 → 1
-  · intro h s
+  | h, s => by
     refine isOpen_iff_forall_mem_open.2 fun x hx ↦ ?_
     rcases h x with ⟨U, hU, hxU, eq⟩
     exact ⟨U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm ▸ hx, hU, hxU⟩
@@ -465,7 +462,7 @@ end Indicator
 section Equiv
 
 /--
-The equivalence between `LocallyConstant X Z` and `LocallyConstant Y Z` given a
+The equivalence between `LocallyConstant X Z` and `LocallyConstant Y Z` given a
 homeomorphism `X ≃ₜ Y`
 -/
 @[simps]
@@ -480,7 +477,7 @@ def congrLeft [TopologicalSpace Y] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ Lo
     simp [comap_comap]
 
 /--
-The equivalence between `LocallyConstant X Y` and `LocallyConstant X Z` given an
+The equivalence between `LocallyConstant X Y` and `LocallyConstant X Z` given an
 equivalence `Y ≃ Z`
 -/
 @[simps]
@@ -556,7 +553,7 @@ lemma piecewise_apply_right {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : I
   · exact hfg x ⟨h, hx⟩
   · rfl
 
-/-- A variant of `LocallyConstant.piecewise` where the two closed sets cover a subset.
+/-- A variant of `LocallyConstant.piecewise` where the two closed sets cover a subset.
 
 TODO: Generalise this construction to `ContinuousMap`. -/
 def piecewise' {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
diff --git a/Mathlib/Topology/LocallyFinite.lean b/Mathlib/Topology/LocallyFinite.lean
index 62ae2d2ecfd21..fef472c111a08 100644
--- a/Mathlib/Topology/LocallyFinite.lean
+++ b/Mathlib/Topology/LocallyFinite.lean
@@ -183,7 +183,7 @@ end LocallyFinite
 
 @[simp]
 theorem Equiv.locallyFinite_comp_iff (e : ι' ≃ ι) : LocallyFinite (f ∘ e) ↔ LocallyFinite f :=
-  ⟨fun h => by simpa only [(· ∘ ·), e.apply_symm_apply] using h.comp_injective e.symm.injective,
+  ⟨fun h => by simpa only [comp_def, e.apply_symm_apply] using h.comp_injective e.symm.injective,
     fun h => h.comp_injective e.injective⟩
 
 theorem locallyFinite_sum {f : ι ⊕ ι' → Set X} :
@@ -197,7 +197,7 @@ theorem LocallyFinite.sum_elim {g : ι' → Set X} (hf : LocallyFinite f) (hg :
 
 theorem locallyFinite_option {f : Option ι → Set X} :
     LocallyFinite f ↔ LocallyFinite (f ∘ some) := by
-  rw [← (Equiv.optionEquivSumPUnit.{_, 0} ι).symm.locallyFinite_comp_iff, locallyFinite_sum]
+  rw [← (Equiv.optionEquivSumPUnit.{0, _} ι).symm.locallyFinite_comp_iff, locallyFinite_sum]
   simp only [locallyFinite_of_finite, and_true]
   rfl
 
diff --git a/Mathlib/Topology/Maps/Basic.lean b/Mathlib/Topology/Maps/Basic.lean
index e50f6343087cb..de7a4ffef4499 100644
--- a/Mathlib/Topology/Maps/Basic.lean
+++ b/Mathlib/Topology/Maps/Basic.lean
@@ -521,7 +521,7 @@ theorem openEmbedding_iff_embedding_open :
 
 theorem openEmbedding_of_continuous_injective_open
     (h₁ : Continuous f) (h₂ : Injective f) (h₃ : IsOpenMap f) : OpenEmbedding f := by
-  simp only [openEmbedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true_iff]
+  simp only [openEmbedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true]
   exact fun x =>
     le_antisymm (h₁.tendsto _).le_comap (@comap_map _ _ (𝓝 x) _ h₂ ▸ comap_mono (h₃.nhds_le _))
 
diff --git a/Mathlib/Topology/Maps/OpenQuotient.lean b/Mathlib/Topology/Maps/OpenQuotient.lean
new file mode 100644
index 0000000000000..88433f7a190fd
--- /dev/null
+++ b/Mathlib/Topology/Maps/OpenQuotient.lean
@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2024 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Topology.Maps.Basic
+
+/-!
+# Open quotient maps
+
+An open quotient map is an open map `f : X → Y` which is both an open map and a quotient map.
+Equivalently, it is a surjective continuous open map.
+We use the latter characterization as a definition.
+
+Many important quotient maps are open quotient maps, including
+
+- the quotient map from a topological space to its quotient by the action of a group;
+- the quotient map from a topological group to its quotient by a normal subgroup;
+- the quotient map from a topological spaace to its separation quotient.
+
+Contrary to general quotient maps,
+the category of open quotient maps is closed under `Prod.map`.
+-/
+
+open Function Set Filter
+open scoped Topology
+
+variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] {f : X → Y}
+
+namespace IsOpenQuotientMap
+
+protected theorem id : IsOpenQuotientMap (id : X → X) := ⟨surjective_id, continuous_id, .id⟩
+
+/-- An open quotient map is a quotient map. -/
+theorem quotientMap (h : IsOpenQuotientMap f) : QuotientMap f :=
+  h.isOpenMap.to_quotientMap h.continuous h.surjective
+
+theorem iff_isOpenMap_quotientMap : IsOpenQuotientMap f ↔ IsOpenMap f ∧ QuotientMap f :=
+  ⟨fun h ↦ ⟨h.isOpenMap, h.quotientMap⟩, fun ⟨ho, hq⟩ ↦ ⟨hq.surjective, hq.continuous, ho⟩⟩
+
+theorem of_isOpenMap_quotientMap (ho : IsOpenMap f) (hq : QuotientMap f) :
+    IsOpenQuotientMap f :=
+  iff_isOpenMap_quotientMap.2 ⟨ho, hq⟩
+
+theorem comp {g : Y → Z} (hg : IsOpenQuotientMap g) (hf : IsOpenQuotientMap f) :
+    IsOpenQuotientMap (g ∘ f) :=
+  ⟨.comp hg.1 hf.1, .comp hg.2 hf.2, .comp hg.3 hf.3⟩
+
+theorem map_nhds_eq (h : IsOpenQuotientMap f) (x : X) : map f (𝓝 x) = 𝓝 (f x) :=
+  le_antisymm h.continuous.continuousAt <| h.isOpenMap.nhds_le _
+
+theorem continuous_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} :
+    Continuous (g ∘ f) ↔ Continuous g :=
+  h.quotientMap.continuous_iff.symm
+
+theorem continuousAt_comp_iff (h : IsOpenQuotientMap f) {g : Y → Z} {x : X} :
+    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
+  simp only [ContinuousAt, ← h.map_nhds_eq, tendsto_map'_iff, comp_def]
+
+theorem dense_preimage_iff (h : IsOpenQuotientMap f) {s : Set Y} : Dense (f ⁻¹' s) ↔ Dense s :=
+  ⟨fun hs ↦ h.surjective.denseRange.dense_of_mapsTo h.continuous hs (mapsTo_preimage _ _),
+    fun hs ↦ hs.preimage h.isOpenMap⟩
+
+end IsOpenQuotientMap
diff --git a/Mathlib/Topology/Maps/Proper/Basic.lean b/Mathlib/Topology/Maps/Proper/Basic.lean
index 4132dd33262f6..00bd0a3bd8677 100644
--- a/Mathlib/Topology/Maps/Proper/Basic.lean
+++ b/Mathlib/Topology/Maps/Proper/Basic.lean
@@ -174,12 +174,12 @@ lemma isProperMap_of_comp_of_t2 [T2Space Y] (hf : Continuous f) (hg : Continuous
   exact ⟨x, hx⟩
 
 /-- A binary product of proper maps is proper. -/
-lemma IsProperMap.prod_map {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
+lemma IsProperMap.prodMap {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap g) :
     IsProperMap (Prod.map f g) := by
   simp_rw [isProperMap_iff_ultrafilter] at hf hg ⊢
   constructor
   -- Continuity is clear.
-  · exact hf.1.prod_map hg.1
+  · exact hf.1.prodMap hg.1
   -- Let `𝒰 : Ultrafilter (X × Z)`, and assume that `f × g` tends to some `(y, w) : Y × W`
   -- along `𝒰`.
   · intro 𝒰 ⟨y, w⟩ hyw
@@ -197,6 +197,8 @@ lemma IsProperMap.prod_map {g : Z → W} (hf : IsProperMap f) (hg : IsProperMap
     rw [nhds_prod_eq, le_prod]
     exact ⟨hx, hz⟩
 
+@[deprecated (since := "2024-10-06")] alias IsProperMap.prod_map := IsProperMap.prodMap
+
 /-- Any product of proper maps is proper. -/
 lemma IsProperMap.pi_map {X Y : ι → Type*} [∀ i, TopologicalSpace (X i)]
     [∀ i, TopologicalSpace (Y i)] {f : (i : ι) → X i → Y i} (h : ∀ i, IsProperMap (f i)) :
@@ -416,7 +418,7 @@ easier to use because it allows `Z` to live in any universe. -/
 theorem IsProperMap.universally_closed (Z) [TopologicalSpace Z] (h : IsProperMap f) :
     IsClosedMap (Prod.map f id : X × Z → Y × Z) :=
   -- `f × id` is proper as a product of proper maps, hence closed.
-  (h.prod_map isProperMap_id).isClosedMap
+  (h.prodMap isProperMap_id).isClosedMap
 
 /-- A map `f : X → Y` is proper if and only if it is continuous and the map
 `(Prod.map f id : X × Filter X → Y × Filter X)` is closed. This is stronger than
diff --git a/Mathlib/Topology/MetricSpace/Algebra.lean b/Mathlib/Topology/MetricSpace/Algebra.lean
index c2d744b24f070..e185b9abdc3ba 100644
--- a/Mathlib/Topology/MetricSpace/Algebra.lean
+++ b/Mathlib/Topology/MetricSpace/Algebra.lean
@@ -4,7 +4,9 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Heather Macbeth
 -/
 import Mathlib.Topology.Algebra.MulAction
+import Mathlib.Topology.Algebra.UniformMulAction
 import Mathlib.Topology.MetricSpace.Lipschitz
+import Mathlib.Topology.Algebra.SeparationQuotient
 
 /-!
 # Compatibility of algebraic operations with metric space structures
@@ -143,6 +145,11 @@ instance (priority := 100) BoundedSMul.continuousSMul : ContinuousSMul α β whe
           gcongr
       _ < ε := hδε
 
+instance (priority := 100) BoundedSMul.toUniformContinuousConstSMul :
+    UniformContinuousConstSMul α β :=
+  ⟨fun c => ((lipschitzWith_iff_dist_le_mul (K := nndist c 0)).2 fun _ _ =>
+    dist_smul_pair c _ _).uniformContinuous⟩
+
 -- this instance could be deduced from `NormedSpace.boundedSMul`, but we prove it separately
 -- here so that it is available earlier in the hierarchy
 instance Real.boundedSMul : BoundedSMul ℝ ℝ where
@@ -207,5 +214,11 @@ instance Prod.instBoundedSMul {α β γ : Type*} [PseudoMetricSpace α] [PseudoM
     max_le ((dist_pair_smul _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_left _ _) dist_nonneg)
       ((dist_pair_smul _ _ _).trans <| mul_le_mul_of_nonneg_left (le_max_right _ _) dist_nonneg)
 
+instance {α β : Type*}
+    [PseudoMetricSpace α] [PseudoMetricSpace β] [Zero α] [Zero β] [SMul α β] [BoundedSMul α β] :
+    BoundedSMul α (SeparationQuotient β) where
+  dist_smul_pair' _ := Quotient.ind₂ <| dist_smul_pair _
+  dist_pair_smul' _ _ := Quotient.ind <| dist_pair_smul _ _
+
 -- We don't have the `SMul α γ → SMul β δ → SMul (α × β) (γ × δ)` instance, but if we did, then
 -- `BoundedSMul α γ → BoundedSMul β δ → BoundedSMul (α × β) (γ × δ)` would hold
diff --git a/Mathlib/Topology/MetricSpace/Antilipschitz.lean b/Mathlib/Topology/MetricSpace/Antilipschitz.lean
index 88e3996a46af3..67c88ebcb3216 100644
--- a/Mathlib/Topology/MetricSpace/Antilipschitz.lean
+++ b/Mathlib/Topology/MetricSpace/Antilipschitz.lean
@@ -143,18 +143,22 @@ theorem comap_uniformity_le (hf : AntilipschitzWith K f) : (𝓤 β).comap (Prod
   rw [mul_comm]
   exact ENNReal.mul_lt_of_lt_div hx
 
-protected theorem uniformInducing (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
-    UniformInducing f :=
+theorem isUniformInducing (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
+    IsUniformInducing f :=
   ⟨le_antisymm hf.comap_uniformity_le hfc.le_comap⟩
 
-protected theorem uniformEmbedding {α : Type*} {β : Type*} [EMetricSpace α] [PseudoEMetricSpace β]
-    {K : ℝ≥0} {f : α → β} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
-    UniformEmbedding f :=
-  ⟨hf.uniformInducing hfc, hf.injective⟩
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing := isUniformInducing
+
+lemma isUniformEmbedding {α β : Type*} [EMetricSpace α] [PseudoEMetricSpace β] {K : ℝ≥0} {f : α → β}
+    (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) : IsUniformEmbedding f :=
+  ⟨hf.isUniformInducing hfc, hf.injective⟩
+
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
 
 theorem isComplete_range [CompleteSpace α] (hf : AntilipschitzWith K f)
     (hfc : UniformContinuous f) : IsComplete (range f) :=
-  (hf.uniformInducing hfc).isComplete_range
+  (hf.isUniformInducing hfc).isComplete_range
 
 theorem isClosed_range {α β : Type*} [PseudoEMetricSpace α] [EMetricSpace β] [CompleteSpace α]
     {f : α → β} {K : ℝ≥0} (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
@@ -164,7 +168,7 @@ theorem isClosed_range {α β : Type*} [PseudoEMetricSpace α] [EMetricSpace β]
 theorem closedEmbedding {α : Type*} {β : Type*} [EMetricSpace α] [EMetricSpace β] {K : ℝ≥0}
     {f : α → β} [CompleteSpace α] (hf : AntilipschitzWith K f) (hfc : UniformContinuous f) :
     ClosedEmbedding f :=
-  { (hf.uniformEmbedding hfc).embedding with isClosed_range := hf.isClosed_range hfc }
+  { (hf.isUniformEmbedding hfc).embedding with isClosed_range := hf.isClosed_range hfc }
 
 theorem subtype_coe (s : Set α) : AntilipschitzWith 1 ((↑) : s → α) :=
   AntilipschitzWith.id.restrict s
diff --git a/Mathlib/Topology/MetricSpace/Basic.lean b/Mathlib/Topology/MetricSpace/Basic.lean
index f3b2468b6e50b..185ec4912deba 100644
--- a/Mathlib/Topology/MetricSpace/Basic.lean
+++ b/Mathlib/Topology/MetricSpace/Basic.lean
@@ -32,11 +32,14 @@ instance (priority := 100) _root_.MetricSpace.instT0Space : T0Space γ where
 
 /-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/
-theorem uniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
-    UniformEmbedding f ↔
+theorem isUniformEmbedding_iff' [MetricSpace β] {f : γ → β} :
+    IsUniformEmbedding f ↔
       (∀ ε > 0, ∃ δ > 0, ∀ {a b : γ}, dist a b < δ → dist (f a) (f b) < ε) ∧
         ∀ δ > 0, ∃ ε > 0, ∀ {a b : γ}, dist (f a) (f b) < ε → dist a b < δ := by
-  rw [uniformEmbedding_iff_uniformInducing, uniformInducing_iff, uniformContinuous_iff]
+  rw [isUniformEmbedding_iff_isUniformInducing, isUniformInducing_iff, uniformContinuous_iff]
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_iff' := isUniformEmbedding_iff'
 
 /-- If a `PseudoMetricSpace` is a T₀ space, then it is a `MetricSpace`. -/
 abbrev _root_.MetricSpace.ofT0PseudoMetricSpace (α : Type*) [PseudoMetricSpace α] [T0Space α] :
@@ -60,10 +63,13 @@ theorem closedEmbedding_of_pairwise_le_dist {α : Type*} [TopologicalSpace α] [
 
 /-- If `f : β → α` sends any two distinct points to points at distance at least `ε > 0`, then
 `f` is a uniform embedding with respect to the discrete uniformity on `β`. -/
-theorem uniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
+theorem isUniformEmbedding_bot_of_pairwise_le_dist {β : Type*} {ε : ℝ} (hε : 0 < ε) {f : β → α}
     (hf : Pairwise fun x y => ε ≤ dist (f x) (f y)) :
-    @UniformEmbedding _ _ ⊥ (by infer_instance) f :=
-  uniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
+    @IsUniformEmbedding _ _ ⊥ (by infer_instance) f :=
+  isUniformEmbedding_of_spaced_out (dist_mem_uniformity hε) <| by simpa using hf
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_bot_of_pairwise_le_dist := isUniformEmbedding_bot_of_pairwise_le_dist
 
 end Metric
 
@@ -94,10 +100,13 @@ abbrev MetricSpace.induced {γ β} (f : γ → β) (hf : Function.Injective f) (
 
 /-- Pull back a metric space structure by a uniform embedding. This is a version of
 `MetricSpace.induced` useful in case if the domain already has a `UniformSpace` structure. -/
-abbrev UniformEmbedding.comapMetricSpace {α β} [UniformSpace α] [m : MetricSpace β] (f : α → β)
-    (h : UniformEmbedding f) : MetricSpace α :=
+abbrev IsUniformEmbedding.comapMetricSpace {α β} [UniformSpace α] [m : MetricSpace β] (f : α → β)
+    (h : IsUniformEmbedding f) : MetricSpace α :=
   .replaceUniformity (.induced f h.inj m) h.comap_uniformity.symm
 
+@[deprecated (since := "2024-10-03")]
+alias UniformEmbedding.comapMetricSpace := IsUniformEmbedding.comapMetricSpace
+
 /-- Pull back a metric space structure by an embedding. This is a version of
 `MetricSpace.induced` useful in case if the domain already has a `TopologicalSpace` structure. -/
 abbrev Embedding.comapMetricSpace {α β} [TopologicalSpace α] [m : MetricSpace β] (f : α → β)
diff --git a/Mathlib/Topology/MetricSpace/Bilipschitz.lean b/Mathlib/Topology/MetricSpace/Bilipschitz.lean
index 432fad174bbe3..0e69f60fc5dbc 100644
--- a/Mathlib/Topology/MetricSpace/Bilipschitz.lean
+++ b/Mathlib/Topology/MetricSpace/Bilipschitz.lean
@@ -55,7 +55,7 @@ instance : UniformSpace α := (inferInstance : UniformSpace β).comap f
 in order to avoid abuse of the definitional equality `α := β`. -/
 lemma uniformity_eq_of_bilipschitz (hf₁ : AntilipschitzWith K₁ f) (hf₂ : LipschitzWith K₂ f) :
     𝓤[(inferInstance : UniformSpace β).comap f] = 𝓤 α :=
-  hf₁.uniformInducing hf₂.uniformContinuous |>.comap_uniformity
+  hf₁.isUniformInducing hf₂.uniformContinuous |>.comap_uniformity
 
 end Uniformity
 
diff --git a/Mathlib/Topology/MetricSpace/CauSeqFilter.lean b/Mathlib/Topology/MetricSpace/CauSeqFilter.lean
index 8bf22f3af496b..c057c66bf50bb 100644
--- a/Mathlib/Topology/MetricSpace/CauSeqFilter.lean
+++ b/Mathlib/Topology/MetricSpace/CauSeqFilter.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Robert Y. Lewis, Sébastien Gouëzel
 -/
 import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Topology.MetricSpace.Cauchy
 
 /-!
 # Completeness in terms of `Cauchy` filters vs `isCauSeq` sequences
diff --git a/Mathlib/Topology/MetricSpace/Cauchy.lean b/Mathlib/Topology/MetricSpace/Cauchy.lean
index 7a51a0300765e..cec514c737140 100644
--- a/Mathlib/Topology/MetricSpace/Cauchy.lean
+++ b/Mathlib/Topology/MetricSpace/Cauchy.lean
@@ -88,7 +88,7 @@ theorem Metric.uniformCauchySeqOn_iff {γ : Type*} {F : β → γ → α} {s : S
     exact hab (hN b.fst hbl.ge b.snd hbr.ge x hx)
 
 /-- If the distance between `s n` and `s m`, `n ≤ m` is bounded above by `b n`
-and `b` converges to zero, then `s` is a Cauchy sequence.  -/
+and `b` converges to zero, then `s` is a Cauchy sequence. -/
 theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
     (h : ∀ n m : β, n ≤ m → dist (s n) (s m) ≤ b n) (h₀ : Tendsto b atTop (𝓝 0)) : CauchySeq s :=
   Metric.cauchySeq_iff'.2 fun ε ε0 => (h₀.eventually (gt_mem_nhds ε0)).exists.imp fun N hN n hn =>
@@ -97,7 +97,7 @@ theorem cauchySeq_of_le_tendsto_0' {s : β → α} (b : β → ℝ)
     _ < ε := hN
 
 /-- If the distance between `s n` and `s m`, `n, m ≥ N` is bounded above by `b N`
-and `b` converges to zero, then `s` is a Cauchy sequence.  -/
+and `b` converges to zero, then `s` is a Cauchy sequence. -/
 theorem cauchySeq_of_le_tendsto_0 {s : β → α} (b : β → ℝ)
     (h : ∀ n m N : β, N ≤ n → N ≤ m → dist (s n) (s m) ≤ b N) (h₀ : Tendsto b atTop (𝓝 0)) :
     CauchySeq s :=
diff --git a/Mathlib/Topology/MetricSpace/Closeds.lean b/Mathlib/Topology/MetricSpace/Closeds.lean
index 67b9d4cbf3244..2dd3375fbcd8e 100644
--- a/Mathlib/Topology/MetricSpace/Closeds.lean
+++ b/Mathlib/Topology/MetricSpace/Closeds.lean
@@ -232,9 +232,12 @@ instance NonemptyCompacts.emetricSpace : EMetricSpace (NonemptyCompacts α) wher
     rwa [s.isCompact.isClosed.closure_eq, t.isCompact.isClosed.closure_eq] at this
 
 /-- `NonemptyCompacts.toCloseds` is a uniform embedding (as it is an isometry) -/
-theorem NonemptyCompacts.ToCloseds.uniformEmbedding :
-    UniformEmbedding (@NonemptyCompacts.toCloseds α _ _) :=
-  Isometry.uniformEmbedding fun _ _ => rfl
+theorem NonemptyCompacts.ToCloseds.isUniformEmbedding :
+    IsUniformEmbedding (@NonemptyCompacts.toCloseds α _ _) :=
+  Isometry.isUniformEmbedding fun _ _ => rfl
+
+@[deprecated (since := "2024-10-01")]
+alias NonemptyCompacts.ToCloseds.uniformEmbedding := NonemptyCompacts.ToCloseds.isUniformEmbedding
 
 /-- The range of `NonemptyCompacts.toCloseds` is closed in a complete space -/
 theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
@@ -278,14 +281,14 @@ theorem NonemptyCompacts.isClosed_in_closeds [CompleteSpace α] :
 from the same statement for closed subsets -/
 instance NonemptyCompacts.completeSpace [CompleteSpace α] : CompleteSpace (NonemptyCompacts α) :=
   (completeSpace_iff_isComplete_range
-        NonemptyCompacts.ToCloseds.uniformEmbedding.toUniformInducing).2 <|
+        NonemptyCompacts.ToCloseds.isUniformEmbedding.isUniformInducing).2 <|
     NonemptyCompacts.isClosed_in_closeds.isComplete
 
 /-- In a compact space, the type of nonempty compact subsets is compact. This follows from
 the same statement for closed subsets -/
 instance NonemptyCompacts.compactSpace [CompactSpace α] : CompactSpace (NonemptyCompacts α) :=
   ⟨by
-    rw [NonemptyCompacts.ToCloseds.uniformEmbedding.embedding.isCompact_iff, image_univ]
+    rw [NonemptyCompacts.ToCloseds.isUniformEmbedding.embedding.isCompact_iff, image_univ]
     exact NonemptyCompacts.isClosed_in_closeds.isCompact⟩
 
 /-- In a second countable space, the type of nonempty compact subsets is second countable -/
diff --git a/Mathlib/Topology/MetricSpace/Completion.lean b/Mathlib/Topology/MetricSpace/Completion.lean
index 8d667f11c999c..97c5f07d31c6e 100644
--- a/Mathlib/Topology/MetricSpace/Completion.lean
+++ b/Mathlib/Topology/MetricSpace/Completion.lean
@@ -6,6 +6,8 @@ Authors: Sébastien Gouëzel
 import Mathlib.Topology.UniformSpace.Completion
 import Mathlib.Topology.MetricSpace.Isometry
 import Mathlib.Topology.MetricSpace.Lipschitz
+import Mathlib.Topology.MetricSpace.Algebra
+import Mathlib.Topology.Algebra.GroupCompletion
 import Mathlib.Topology.Instances.Real
 
 /-!
@@ -103,7 +105,7 @@ protected theorem mem_uniformity_dist (s : Set (Completion α × Completion α))
         · have Z := hε (not_le.1 h)
           simp only [Set.mem_setOf_eq] at Z
           exact Or.inr Z
-    simp only [not_le.mpr hxy, false_or_iff, not_le] at this
+    simp only [not_le.mpr hxy, false_or, not_le] at this
     exact ts this
   · /- Start from a set `s` containing an ε-neighborhood of the diagonal in `Completion α`. To show
         that it is an entourage, we use the fact that `dist` is uniformly continuous on
@@ -168,6 +170,27 @@ theorem coe_isometry : Isometry ((↑) : α → Completion α) :=
 protected theorem edist_eq (x y : α) : edist (x : Completion α) y = edist x y :=
   coe_isometry x y
 
+instance {M} [Zero M] [Zero α] [SMul M α] [PseudoMetricSpace M] [BoundedSMul M α] :
+    BoundedSMul M (Completion α) where
+  dist_smul_pair' c x₁ x₂ := by
+    induction x₁, x₂ using induction_on₂ with
+    | hp =>
+      exact isClosed_le
+        ((continuous_fst.const_smul _).dist (continuous_snd.const_smul _))
+        (continuous_const.mul (continuous_fst.dist continuous_snd))
+    | ih x₁ x₂ =>
+      rw [← coe_smul, ← coe_smul, Completion.dist_eq,  Completion.dist_eq]
+      exact dist_smul_pair c x₁ x₂
+  dist_pair_smul' c₁ c₂ x := by
+    induction x using induction_on with
+    | hp =>
+      exact isClosed_le
+        ((continuous_const_smul _).dist (continuous_const_smul _))
+        (continuous_const.mul (continuous_id.dist continuous_const))
+    | ih x =>
+      rw [← coe_smul, ← coe_smul, Completion.dist_eq, ← coe_zero, Completion.dist_eq]
+      exact dist_pair_smul c₁ c₂ x
+
 end UniformSpace.Completion
 
 open UniformSpace Completion NNReal
diff --git a/Mathlib/Topology/MetricSpace/Defs.lean b/Mathlib/Topology/MetricSpace/Defs.lean
index 600bc7f10f03d..3ce13e989e61a 100644
--- a/Mathlib/Topology/MetricSpace/Defs.lean
+++ b/Mathlib/Topology/MetricSpace/Defs.lean
@@ -80,11 +80,11 @@ theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by
 theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
   eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
 
-/-- Deduce the equality of points from the vanishing of the nonnegative distance-/
+/-- Deduce the equality of points from the vanishing of the nonnegative distance -/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
   simp only [NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
 
-/-- Characterize the equality of points as the vanishing of the nonnegative distance-/
+/-- Characterize the equality of points as the vanishing of the nonnegative distance -/
 @[simp]
 theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by
   simp only [NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
diff --git a/Mathlib/Topology/MetricSpace/Dilation.lean b/Mathlib/Topology/MetricSpace/Dilation.lean
index c68a245846375..a2b542947d55f 100644
--- a/Mathlib/Topology/MetricSpace/Dilation.lean
+++ b/Mathlib/Topology/MetricSpace/Dilation.lean
@@ -66,7 +66,7 @@ infixl:25 " →ᵈ " => Dilation
 
 /-- `DilationClass F α β r` states that `F` is a type of `r`-dilations.
 You should extend this typeclass when you extend `Dilation`. -/
-class DilationClass (F α β : Type*) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
+class DilationClass (F : Type*) (α β : outParam Type*) [PseudoEMetricSpace α] [PseudoEMetricSpace β]
     [FunLike F α β] : Prop where
   edist_eq' : ∀ f : F, ∃ r : ℝ≥0, r ≠ 0 ∧ ∀ x y : α, edist (f x) (f y) = r * edist x y
 
@@ -370,12 +370,15 @@ theorem cancel_left {g : β →ᵈ γ} {f₁ f₂ : α →ᵈ β} (hg : Injectiv
   ⟨fun h => Dilation.ext fun x => hg <| by rw [← comp_apply, h, comp_apply], fun h => h ▸ rfl⟩
 
 /-- A dilation from a metric space is a uniform inducing map -/
-protected theorem uniformInducing : UniformInducing (f : α → β) :=
-  (antilipschitz f).uniformInducing (lipschitz f).uniformContinuous
+theorem isUniformInducing : IsUniformInducing (f : α → β) :=
+  (antilipschitz f).isUniformInducing (lipschitz f).uniformContinuous
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing := isUniformInducing
 
 theorem tendsto_nhds_iff {ι : Type*} {g : ι → α} {a : Filter ι} {b : α} :
     Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto ((f : α → β) ∘ g) a (𝓝 (f b)) :=
-  (Dilation.uniformInducing f).inducing.tendsto_nhds_iff
+  (Dilation.isUniformInducing f).inducing.tendsto_nhds_iff
 
 /-- A dilation is continuous. -/
 theorem toContinuous : Continuous (f : α → β) :=
@@ -406,11 +409,11 @@ theorem mapsTo_emetric_closedBall (x : α) (r' : ℝ≥0∞) :
 
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] {g : γ → α} {s : Set γ} :
     ContinuousOn ((f : α → β) ∘ g) s ↔ ContinuousOn g s :=
-  (Dilation.uniformInducing f).inducing.continuousOn_iff.symm
+  (Dilation.isUniformInducing f).inducing.continuousOn_iff.symm
 
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] {g : γ → α} :
     Continuous ((f : α → β) ∘ g) ↔ Continuous g :=
-  (Dilation.uniformInducing f).inducing.continuous_iff.symm
+  (Dilation.isUniformInducing f).inducing.continuous_iff.symm
 
 end PseudoEmetricDilation
 
@@ -420,14 +423,16 @@ variable [EMetricSpace α]
 variable [FunLike F α β]
 
 /-- A dilation from a metric space is a uniform embedding -/
-protected theorem uniformEmbedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
-    UniformEmbedding f :=
-  (antilipschitz f).uniformEmbedding (lipschitz f).uniformContinuous
+lemma isUniformEmbedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
+    IsUniformEmbedding f :=
+  (antilipschitz f).isUniformEmbedding (lipschitz f).uniformContinuous
+
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
 
 /-- A dilation from a metric space is an embedding -/
 protected theorem embedding [PseudoEMetricSpace β] [DilationClass F α β] (f : F) :
     Embedding (f : α → β) :=
-  (Dilation.uniformEmbedding f).embedding
+  (Dilation.isUniformEmbedding f).embedding
 
 /-- A dilation from a complete emetric space is a closed embedding -/
 protected theorem closedEmbedding [CompleteSpace α] [EMetricSpace β] [DilationClass F α β] (f : F) :
diff --git a/Mathlib/Topology/MetricSpace/Gluing.lean b/Mathlib/Topology/MetricSpace/Gluing.lean
index 53a7579e013fa..e3f3f1233ca43 100644
--- a/Mathlib/Topology/MetricSpace/Gluing.lean
+++ b/Mathlib/Topology/MetricSpace/Gluing.lean
@@ -243,8 +243,7 @@ private theorem Sum.mem_uniformity (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
     · cases not_le_of_lt (lt_of_lt_of_le h (min_le_right _ _)) Sum.one_le_dist_inr_inl
     · exact hY (lt_of_lt_of_le h (le_trans (min_le_left _ _) (min_le_right _ _)))
   · rintro ⟨ε, ε0, H⟩
-    constructor <;> rw [Filter.mem_sets, Filter.mem_map, mem_uniformity_dist] <;>
-      exact ⟨ε, ε0, fun h => H _ _ h⟩
+    constructor <;> rw [Filter.mem_map, mem_uniformity_dist] <;> exact ⟨ε, ε0, fun h => H _ _ h⟩
 
 /-- The distance on the disjoint union indeed defines a metric space. All the distance properties
 follow from our choice of the distance. The harder work is to show that the uniform structure
@@ -432,7 +431,7 @@ protected theorem completeSpace [∀ i, CompleteSpace (E i)] : CompleteSpace (Σ
   set U := { p : (Σk, E k) × Σk, E k | dist p.1 p.2 < 1 }
   have hc : ∀ i, IsComplete (s i) := fun i => by
     simp only [s, ← range_sigmaMk]
-    exact (isometry_mk i).uniformInducing.isComplete_range
+    exact (isometry_mk i).isUniformInducing.isComplete_range
   have hd : ∀ (i j), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j := fun i j x hx y hy hxy =>
     (Eq.symm hx).trans ((fst_eq_of_dist_lt_one _ _ hxy).trans hy)
   refine completeSpace_of_isComplete_univ ?_
diff --git a/Mathlib/Topology/MetricSpace/GromovHausdorff.lean b/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
index fbeb7156e42d0..8b58e2adf2ada 100644
--- a/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
+++ b/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
@@ -255,7 +255,7 @@ theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Non
       rcases exists_mem_of_nonempty X with ⟨xX, _⟩
       have : ∃ y ∈ range Ψ, dist (Φ xX) y < diam (univ : Set X) + 1 + diam (univ : Set Y) := by
         rw [Ψrange]
-        have : Φ xX ∈ ↑p := Φrange.subst (mem_range_self _)
+        have : Φ xX ∈ (p : Set _) := Φrange ▸ (mem_range_self _)
         exact
           exists_dist_lt_of_hausdorffDist_lt this bound
             (hausdorffEdist_ne_top_of_nonempty_of_bounded p.nonempty q.nonempty
@@ -282,7 +282,7 @@ theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Non
     let F : (X ⊕ Y) × (X ⊕ Y) → ℝ := fun p => dist (f p.1) (f p.2)
     -- check that the induced "distance" is a candidate
     have Fgood : F ∈ candidates X Y := by
-      simp only [F, candidates, forall_const, and_true_iff, add_comm, eq_self_iff_true,
+      simp only [F, candidates, forall_const, add_comm, eq_self_iff_true,
         dist_eq_zero, and_self_iff, Set.mem_setOf_eq]
       repeat' constructor
       · exact fun x y =>
@@ -315,7 +315,7 @@ theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Non
     refine le_trans this (le_of_forall_le_of_dense fun r hr => ?_)
     have I1 : ∀ x : X, (⨅ y, Fb (inl x, inr y)) ≤ r := by
       intro x
-      have : f (inl x) ∈ ↑p := Φrange.subst (mem_range_self _)
+      have : f (inl x) ∈ (p : Set _) := Φrange ▸ (mem_range_self _)
       rcases exists_dist_lt_of_hausdorffDist_lt this hr
           (hausdorffEdist_ne_top_of_nonempty_of_bounded p.nonempty q.nonempty p.isCompact.isBounded
             q.isCompact.isBounded) with
@@ -331,7 +331,7 @@ theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Non
 
     have I2 : ∀ y : Y, (⨅ x, Fb (inl x, inr y)) ≤ r := by
       intro y
-      have : f (inr y) ∈ ↑q := Ψrange.subst (mem_range_self _)
+      have : f (inr y) ∈ (q : Set _) := Ψrange ▸ (mem_range_self _)
       rcases exists_dist_lt_of_hausdorffDist_lt' this hr
           (hausdorffEdist_ne_top_of_nonempty_of_bounded p.nonempty q.nonempty p.isCompact.isBounded
             q.isCompact.isBounded) with
@@ -398,7 +398,7 @@ instance : MetricSpace GHSpace where
       · exact ⟨0, by rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg⟩
       · simp only [mem_image, mem_prod, mem_setOf_eq, Prod.exists]
         exists y, y
-        simpa only [and_self_iff, hausdorffDist_self_zero, eq_self_iff_true, and_true_iff]
+        simpa only [and_self_iff, hausdorffDist_self_zero, eq_self_iff_true, and_true]
     · apply le_csInf
       · exact Set.Nonempty.image _ <| Set.Nonempty.prod ⟨y, hy⟩ ⟨y, hy⟩
       · rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg
@@ -940,7 +940,7 @@ limit of the `Y n`, and finally let `Z` be the completion of `Z0`.
 The images `X2 n` of `X n` in `Z` are at Hausdorff distance `< 1/2^n` by construction, hence they
 form a Cauchy sequence for the Hausdorff distance. By completeness (of `Z`, and therefore of its
 set of nonempty compact subsets), they converge to a limit `L`. This is the nonempty
-compact metric space we are looking for.  -/
+compact metric space we are looking for. -/
 variable (X : ℕ → Type) [∀ n, MetricSpace (X n)] [∀ n, CompactSpace (X n)] [∀ n, Nonempty (X n)]
 
 /-- Auxiliary structure used to glue metric spaces below, recording an isometric embedding
diff --git a/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean b/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
index f343b3d1c3901..9a6979a644bb1 100644
--- a/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
+++ b/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
@@ -5,7 +5,7 @@ Authors: Sébastien Gouëzel
 -/
 import Mathlib.Topology.MetricSpace.Gluing
 import Mathlib.Topology.MetricSpace.HausdorffDistance
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 
 /-!
 # The Gromov-Hausdorff distance is realized
diff --git a/Mathlib/Topology/MetricSpace/HausdorffDimension.lean b/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
index 98f606a3e4ad3..46a5798a92a07 100644
--- a/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
+++ b/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
@@ -84,7 +84,7 @@ Hausdorff measure, Hausdorff dimension, dimension
 
 open scoped MeasureTheory ENNReal NNReal Topology
 
-open MeasureTheory MeasureTheory.Measure Set TopologicalSpace FiniteDimensional Filter
+open MeasureTheory MeasureTheory.Measure Set TopologicalSpace Module Filter
 
 variable {ι X Y : Type*} [EMetricSpace X] [EMetricSpace Y]
 
@@ -262,7 +262,7 @@ theorem HolderOnWith.dimH_image_le (h : HolderOnWith C r f s) (hr : 0 < r) :
   borelize X Y
   refine dimH_le fun d hd => ?_
   have := h.hausdorffMeasure_image_le hr d.coe_nonneg
-  rw [hd, ENNReal.coe_rpow_of_nonneg _ d.coe_nonneg, top_le_iff] at this
+  rw [hd, ← ENNReal.coe_rpow_of_nonneg _ d.coe_nonneg, top_le_iff] at this
   have Hrd : μH[(r * d : ℝ≥0)] s = ⊤ := by
     contrapose this
     exact ENNReal.mul_ne_top ENNReal.coe_ne_top this
@@ -441,7 +441,7 @@ theorem dimH_univ_pi_fin (n : ℕ) : dimH (univ : Set (Fin n → ℝ)) = n := by
 
 theorem dimH_of_mem_nhds {x : E} {s : Set E} (h : s ∈ 𝓝 x) : dimH s = finrank ℝ E := by
   have e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ :=
-    ContinuousLinearEquiv.ofFinrankEq (FiniteDimensional.finrank_fin_fun ℝ).symm
+    ContinuousLinearEquiv.ofFinrankEq (Module.finrank_fin_fun ℝ).symm
   rw [← e.dimH_image]
   refine le_antisymm ?_ ?_
   · exact (dimH_mono (subset_univ _)).trans_eq (dimH_univ_pi_fin _)
@@ -459,7 +459,7 @@ theorem dimH_univ_eq_finrank : dimH (univ : Set E) = finrank ℝ E :=
   dimH_of_mem_nhds (@univ_mem _ (𝓝 0))
 
 theorem dimH_univ : dimH (univ : Set ℝ) = 1 := by
-  rw [dimH_univ_eq_finrank ℝ, FiniteDimensional.finrank_self, Nat.cast_one]
+  rw [dimH_univ_eq_finrank ℝ, Module.finrank_self, Nat.cast_one]
 
 variable {E}
 
diff --git a/Mathlib/Topology/MetricSpace/Holder.lean b/Mathlib/Topology/MetricSpace/Holder.lean
index c60629e1cc045..da0e2b14c4f81 100644
--- a/Mathlib/Topology/MetricSpace/Holder.lean
+++ b/Mathlib/Topology/MetricSpace/Holder.lean
@@ -106,7 +106,7 @@ theorem comp {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg r
     HolderOnWith (Cg * Cf ^ (rg : ℝ)) (rg * rf) (g ∘ f) s := by
   intro x hx y hy
   rw [ENNReal.coe_mul, mul_comm rg, NNReal.coe_mul, ENNReal.rpow_mul, mul_assoc,
-    ← ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg, ← ENNReal.mul_rpow_of_nonneg _ _ rg.coe_nonneg]
+    ENNReal.coe_rpow_of_nonneg _ rg.coe_nonneg, ← ENNReal.mul_rpow_of_nonneg _ _ rg.coe_nonneg]
   exact hg.edist_le_of_le (hst hx) (hst hy) (hf.edist_le hx hy)
 
 theorem comp_holderWith {Cg rg : ℝ≥0} {g : Y → Z} {t : Set Y} (hg : HolderOnWith Cg rg g t)
@@ -190,11 +190,23 @@ theorem ediam_image_le (hf : HolderWith C r f) (s : Set X) :
   EMetric.diam_image_le_iff.2 fun _ hx _ hy =>
     hf.edist_le_of_le <| EMetric.edist_le_diam_of_mem hx hy
 
+lemma const {y : Y} :
+    HolderWith C r (Function.const X y) := fun x₁ x₂ => by
+  simp only [Function.const_apply, edist_self, zero_le]
+
+lemma zero [Zero Y] : HolderWith C r (0 : X → Y) := .const
+
+lemma of_isEmpty [IsEmpty X] : HolderWith C r f := isEmptyElim
+
+lemma mono {C' : ℝ≥0} (hf : HolderWith C r f) (h : C ≤ C') :
+    HolderWith C' r f :=
+  fun x₁ x₂ ↦ (hf x₁ x₂).trans (mul_right_mono (coe_le_coe.2 h))
+
 end HolderWith
 
 end Emetric
 
-section Metric
+section PseudoMetric
 
 variable [PseudoMetricSpace X] [PseudoMetricSpace Y] {C r : ℝ≥0} {f : X → Y}
 
@@ -202,7 +214,7 @@ namespace HolderWith
 
 theorem nndist_le_of_le (hf : HolderWith C r f) {x y : X} {d : ℝ≥0} (hd : nndist x y ≤ d) :
     nndist (f x) (f y) ≤ C * d ^ (r : ℝ) := by
-  rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul, ←
+  rw [← ENNReal.coe_le_coe, ← edist_nndist, ENNReal.coe_mul,
     ENNReal.coe_rpow_of_nonneg _ r.coe_nonneg]
   apply hf.edist_le_of_le
   rwa [edist_nndist, ENNReal.coe_le_coe]
@@ -223,4 +235,40 @@ theorem dist_le (hf : HolderWith C r f) (x y : X) : dist (f x) (f y) ≤ C * dis
 
 end HolderWith
 
+end PseudoMetric
+
+section Metric
+
+variable [PseudoMetricSpace X] [MetricSpace Y] {C r : ℝ≥0} {f : X → Y}
+
+@[simp]
+lemma holderWith_zero_iff : HolderWith 0 r f ↔ ∀ x₁ x₂, f x₁ = f x₂ := by
+  refine ⟨fun h x₁ x₂ => ?_, fun h x₁ x₂ => h x₁ x₂ ▸ ?_⟩
+  · specialize h x₁ x₂
+    simp [ENNReal.coe_zero, zero_mul, nonpos_iff_eq_zero, edist_eq_zero] at h
+    assumption
+  · simp only [edist_self, ENNReal.coe_zero, zero_mul, le_refl]
+
 end Metric
+
+section SeminormedAddCommGroup
+
+variable [PseudoMetricSpace X] [SeminormedAddCommGroup Y] {C C' r : ℝ≥0} {f g : X → Y}
+
+namespace HolderWith
+
+lemma add (hf : HolderWith C r f) (hg : HolderWith C' r g) :
+    HolderWith (C + C') r (f + g) := fun x₁ x₂ => by
+  refine le_trans (edist_add_add_le _ _ _ _) <| le_trans (add_le_add (hf x₁ x₂) (hg x₁ x₂)) ?_
+  rw [coe_add, add_mul]
+
+lemma smul {α} [NormedDivisionRing α] [Module α Y] [BoundedSMul α Y] (a : α)
+    (hf : HolderWith C r f) : HolderWith (C * ‖a‖₊) r (a • f) := fun x₁ x₂ => by
+  rw [Pi.smul_apply, coe_mul, Pi.smul_apply, edist_smul₀, mul_comm (C : ℝ≥0∞),
+    ENNReal.smul_def, smul_eq_mul, mul_assoc]
+  gcongr
+  exact hf x₁ x₂
+
+end HolderWith
+
+end SeminormedAddCommGroup
diff --git a/Mathlib/Topology/MetricSpace/Infsep.lean b/Mathlib/Topology/MetricSpace/Infsep.lean
index 027a2e3139f9c..e451f9e7817b3 100644
--- a/Mathlib/Topology/MetricSpace/Infsep.lean
+++ b/Mathlib/Topology/MetricSpace/Infsep.lean
@@ -171,7 +171,7 @@ end EDist
 
 section PseudoEMetricSpace
 
-variable [PseudoEMetricSpace α] {x y z : α} {s t : Set α}
+variable [PseudoEMetricSpace α] {x y z : α} {s : Set α}
 
 theorem einfsep_pair (hxy : x ≠ y) : ({x, y} : Set α).einfsep = edist x y := by
   nth_rw 1 [← min_self (edist x y)]
@@ -238,7 +238,7 @@ end PseudoMetricSpace
 
 section EMetricSpace
 
-variable [EMetricSpace α] {x y z : α} {s t : Set α} {C : ℝ≥0∞} {sC : Set ℝ≥0∞}
+variable [EMetricSpace α] {s : Set α}
 
 theorem einfsep_pos_of_finite [Finite s] : 0 < s.einfsep := by
   cases nonempty_fintype s
@@ -312,7 +312,7 @@ end EDist
 
 section PseudoEMetricSpace
 
-variable [PseudoEMetricSpace α] {x y : α} {s : Set α}
+variable [PseudoEMetricSpace α] {x y : α}
 
 theorem infsep_pair_eq_toReal : ({x, y} : Set α).infsep = (edist x y).toReal := by
   by_cases hxy : x = y
diff --git a/Mathlib/Topology/MetricSpace/IsometricSMul.lean b/Mathlib/Topology/MetricSpace/IsometricSMul.lean
index 1d8b4a236200b..8bbc25f34f0ab 100644
--- a/Mathlib/Topology/MetricSpace/IsometricSMul.lean
+++ b/Mathlib/Topology/MetricSpace/IsometricSMul.lean
@@ -405,19 +405,19 @@ instance ULift.isometricSMul' : IsometricSMul M (ULift X) :=
 @[to_additive]
 instance {ι} {X : ι → Type*} [Fintype ι] [∀ i, SMul M (X i)] [∀ i, PseudoEMetricSpace (X i)]
     [∀ i, IsometricSMul M (X i)] : IsometricSMul M (∀ i, X i) :=
-  ⟨fun c => isometry_dcomp (fun _ => (c • ·)) fun i => isometry_smul (X i) c⟩
+  ⟨fun c => .piMap (fun _ => (c • ·)) fun i => isometry_smul (X i) c⟩
 
 @[to_additive]
 instance Pi.isometricSMul' {ι} {M X : ι → Type*} [Fintype ι] [∀ i, SMul (M i) (X i)]
     [∀ i, PseudoEMetricSpace (X i)] [∀ i, IsometricSMul (M i) (X i)] :
     IsometricSMul (∀ i, M i) (∀ i, X i) :=
-  ⟨fun c => isometry_dcomp (fun i => (c i • ·)) fun _ => isometry_smul _ _⟩
+  ⟨fun c => .piMap (fun i => (c i • ·)) fun _ => isometry_smul _ _⟩
 
 @[to_additive]
 instance Pi.isometricSMul'' {ι} {M : ι → Type*} [Fintype ι] [∀ i, Mul (M i)]
     [∀ i, PseudoEMetricSpace (M i)] [∀ i, IsometricSMul (M i)ᵐᵒᵖ (M i)] :
     IsometricSMul (∀ i, M i)ᵐᵒᵖ (∀ i, M i) :=
-  ⟨fun c => isometry_dcomp (fun i (x : M i) => x * c.unop i) fun _ => isometry_mul_right _⟩
+  ⟨fun c => .piMap (fun i (x : M i) => x * c.unop i) fun _ => isometry_mul_right _⟩
 
 instance Additive.isometricVAdd : IsometricVAdd (Additive M) X :=
   ⟨fun c => isometry_smul X (toMul c)⟩
diff --git a/Mathlib/Topology/MetricSpace/Isometry.lean b/Mathlib/Topology/MetricSpace/Isometry.lean
index b2464d0e4b65a..0640a68f619aa 100644
--- a/Mathlib/Topology/MetricSpace/Isometry.lean
+++ b/Mathlib/Topology/MetricSpace/Isometry.lean
@@ -28,7 +28,7 @@ open Function Set
 open scoped Topology ENNReal
 
 /-- An isometry (also known as isometric embedding) is a map preserving the edistance
-between pseudoemetric spaces, or equivalently the distance between pseudometric space.  -/
+between pseudoemetric spaces, or equivalently the distance between pseudometric space. -/
 def Isometry [PseudoEMetricSpace α] [PseudoEMetricSpace β] (f : α → β) : Prop :=
   ∀ x1 x2 : α, edist (f x1) (f x2) = edist x1 x2
 
@@ -84,10 +84,12 @@ theorem prod_map {δ} [PseudoEMetricSpace δ] {f : α → β} {g : γ → δ} (h
     (hg : Isometry g) : Isometry (Prod.map f g) := fun x y => by
   simp only [Prod.edist_eq, Prod.map_fst, hf.edist_eq, Prod.map_snd, hg.edist_eq]
 
-theorem _root_.isometry_dcomp {ι} [Fintype ι] {α β : ι → Type*} [∀ i, PseudoEMetricSpace (α i)]
+protected theorem piMap {ι} [Fintype ι] {α β : ι → Type*} [∀ i, PseudoEMetricSpace (α i)]
     [∀ i, PseudoEMetricSpace (β i)] (f : ∀ i, α i → β i) (hf : ∀ i, Isometry (f i)) :
-    Isometry (fun g : (i : ι) → α i => fun i => f i (g i)) := fun x y => by
-  simp only [edist_pi_def, (hf _).edist_eq]
+    Isometry (Pi.map f) := fun x y => by
+  simp only [edist_pi_def, (hf _).edist_eq, Pi.map_apply]
+
+@[deprecated (since := "2024-10-06")] alias _root_.isometry_dcomp := Isometry.piMap
 
 /-- The composition of isometries is an isometry. -/
 theorem comp {g : β → γ} {f : α → β} (hg : Isometry g) (hf : Isometry f) : Isometry (g ∘ f) :=
@@ -98,12 +100,15 @@ protected theorem uniformContinuous (hf : Isometry f) : UniformContinuous f :=
   hf.lipschitz.uniformContinuous
 
 /-- An isometry from a metric space is a uniform inducing map -/
-protected theorem uniformInducing (hf : Isometry f) : UniformInducing f :=
-  hf.antilipschitz.uniformInducing hf.uniformContinuous
+theorem isUniformInducing (hf : Isometry f) : IsUniformInducing f :=
+  hf.antilipschitz.isUniformInducing hf.uniformContinuous
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing := isUniformInducing
 
 theorem tendsto_nhds_iff {ι : Type*} {f : α → β} {g : ι → α} {a : Filter ι} {b : α}
     (hf : Isometry f) : Filter.Tendsto g a (𝓝 b) ↔ Filter.Tendsto (f ∘ g) a (𝓝 (f b)) :=
-  hf.uniformInducing.inducing.tendsto_nhds_iff
+  hf.isUniformInducing.inducing.tendsto_nhds_iff
 
 /-- An isometry is continuous. -/
 protected theorem continuous (hf : Isometry f) : Continuous f :=
@@ -144,11 +149,11 @@ theorem _root_.isometry_subtype_coe {s : Set α} : Isometry ((↑) : s → α) :
 
 theorem comp_continuousOn_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} {s : Set γ} :
     ContinuousOn (f ∘ g) s ↔ ContinuousOn g s :=
-  hf.uniformInducing.inducing.continuousOn_iff.symm
+  hf.isUniformInducing.inducing.continuousOn_iff.symm
 
 theorem comp_continuous_iff {γ} [TopologicalSpace γ] (hf : Isometry f) {g : γ → α} :
     Continuous (f ∘ g) ↔ Continuous g :=
-  hf.uniformInducing.inducing.continuous_iff.symm
+  hf.isUniformInducing.inducing.continuous_iff.symm
 
 end PseudoEmetricIsometry
 
@@ -162,12 +167,14 @@ protected theorem injective (h : Isometry f) : Injective f :=
   h.antilipschitz.injective
 
 /-- An isometry from an emetric space is a uniform embedding -/
-protected theorem uniformEmbedding (hf : Isometry f) : UniformEmbedding f :=
-  hf.antilipschitz.uniformEmbedding hf.lipschitz.uniformContinuous
+lemma isUniformEmbedding (hf : Isometry f) : IsUniformEmbedding f :=
+  hf.antilipschitz.isUniformEmbedding hf.lipschitz.uniformContinuous
+
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
 
 /-- An isometry from an emetric space is an embedding -/
 protected theorem embedding (hf : Isometry f) : Embedding f :=
-  hf.uniformEmbedding.embedding
+  hf.isUniformEmbedding.embedding
 
 /-- An isometry from a complete emetric space is a closed embedding -/
 theorem closedEmbedding [CompleteSpace α] [EMetricSpace γ] {f : α → γ} (hf : Isometry f) :
@@ -226,11 +233,14 @@ end Isometry
 -- namespace
 /-- A uniform embedding from a uniform space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
-theorem UniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β] {f : α → β}
-    (h : UniformEmbedding f) : (letI := h.comapMetricSpace f; Isometry f) :=
+theorem IsUniformEmbedding.to_isometry {α β} [UniformSpace α] [MetricSpace β] {f : α → β}
+    (h : IsUniformEmbedding f) : (letI := h.comapMetricSpace f; Isometry f) :=
   let _ := h.comapMetricSpace f
   Isometry.of_dist_eq fun _ _ => rfl
 
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.to_isometry := IsUniformEmbedding.to_isometry
+
 /-- An embedding from a topological space to a metric space is an isometry with respect to the
 induced metric space structure on the source space. -/
 theorem Embedding.to_isometry {α β} [TopologicalSpace α] [MetricSpace β] {f : α → β}
@@ -465,7 +475,7 @@ theorem mul_apply (e₁ e₂ : α ≃ᵢ α) (x : α) : (e₁ * e₂) x = e₁ (
 
 theorem completeSpace_iff (e : α ≃ᵢ β) : CompleteSpace α ↔ CompleteSpace β := by
   simp only [completeSpace_iff_isComplete_univ, ← e.range_eq_univ, ← image_univ,
-    isComplete_image_iff e.isometry.uniformInducing]
+    isComplete_image_iff e.isometry.isUniformInducing]
 
 protected theorem completeSpace [CompleteSpace β] (e : α ≃ᵢ β) : CompleteSpace α :=
   e.completeSpace_iff.2 ‹_›
diff --git a/Mathlib/Topology/MetricSpace/Kuratowski.lean b/Mathlib/Topology/MetricSpace/Kuratowski.lean
index 8063585c24d8e..5a21609d09845 100644
--- a/Mathlib/Topology/MetricSpace/Kuratowski.lean
+++ b/Mathlib/Topology/MetricSpace/Kuratowski.lean
@@ -19,16 +19,16 @@ noncomputable section
 
 open Set Metric TopologicalSpace NNReal ENNReal lp Function
 
-universe u v w
+universe u
 
-variable {α : Type u} {β : Type v} {γ : Type w}
+variable {α : Type u}
 
 namespace KuratowskiEmbedding
 
 /-! ### Any separable metric space can be embedded isometrically in ℓ^∞(ℕ, ℝ) -/
 
 
-variable {f g : ℓ^∞(ℕ)} {n : ℕ} {C : ℝ} [MetricSpace α] (x : ℕ → α) (a b : α)
+variable {n : ℕ} [MetricSpace α] (x : ℕ → α) (a : α)
 
 /-- A metric space can be embedded in `l^∞(ℝ)` via the distances to points in
 a fixed countable set, if this set is dense. This map is given in `kuratowskiEmbedding`,
diff --git a/Mathlib/Topology/MetricSpace/PiNat.lean b/Mathlib/Topology/MetricSpace/PiNat.lean
index b001d3c706ab2..bfd470b9972da 100644
--- a/Mathlib/Topology/MetricSpace/PiNat.lean
+++ b/Mathlib/Topology/MetricSpace/PiNat.lean
@@ -51,7 +51,7 @@ noncomputable section
 
 open Topology TopologicalSpace Set Metric Filter Function
 
-attribute [local simp] pow_le_pow_iff_right one_lt_two inv_le_inv zero_le_two zero_lt_two
+attribute [local simp] pow_le_pow_iff_right one_lt_two inv_le_inv₀ zero_le_two zero_lt_two
 
 variable {E : ℕ → Type*}
 
@@ -264,7 +264,7 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
   · simp
   rcases eq_or_ne y z with (rfl | hyz)
   · simp
-  simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne,
+  simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv₀, one_div, inv_pow, zero_lt_two, Ne,
     not_false_iff, le_max_iff, pow_le_pow_iff_right, one_lt_two, pow_pos,
     min_le_iff.1 (min_firstDiff_le x y z hxz)]
 
@@ -294,7 +294,7 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
   rcases eq_or_ne x y with (rfl | hne)
   · rfl
   have : n < firstDiff x y := by
-    simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff_right, one_lt_two] using h
+    simpa [dist_eq_of_ne hne, inv_lt_inv₀, pow_lt_pow_iff_right, one_lt_two] using h
   exact apply_eq_of_lt_firstDiff (hi.trans_lt this)
 
 /-- A function to a pseudo-metric-space is `1`-Lipschitz if and only if points in the same cylinder
@@ -786,7 +786,7 @@ theorem min_dist_le_dist_pi (x y : ∀ i, F i) (i : ι) :
 
 theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (1 / 2) ^ encode i) :
     dist (x i) (y i) ≤ dist x y := by
-  simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)
+  simpa only [not_le.2 h, false_or] using min_le_iff.1 (min_dist_le_dist_pi x y i)
 
 open Topology Filter NNReal
 
diff --git a/Mathlib/Topology/MetricSpace/Polish.lean b/Mathlib/Topology/MetricSpace/Polish.lean
index 55e0723b7dcbc..6db429aedccbf 100644
--- a/Mathlib/Topology/MetricSpace/Polish.lean
+++ b/Mathlib/Topology/MetricSpace/Polish.lean
@@ -3,11 +3,12 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
+import Mathlib.Analysis.Normed.Field.Basic
+import Mathlib.Topology.Instances.Nat
 import Mathlib.Topology.MetricSpace.PiNat
 import Mathlib.Topology.MetricSpace.Isometry
 import Mathlib.Topology.MetricSpace.Gluing
 import Mathlib.Topology.Sets.Opens
-import Mathlib.Analysis.Normed.Field.Basic
 
 /-!
 # Polish spaces
@@ -145,7 +146,7 @@ theorem _root_.ClosedEmbedding.polishSpace [TopologicalSpace α] [TopologicalSpa
   letI : MetricSpace α := hf.toEmbedding.comapMetricSpace f
   haveI : SecondCountableTopology α := hf.toEmbedding.secondCountableTopology
   have : CompleteSpace α := by
-    rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.uniformInducing]
+    rw [completeSpace_iff_isComplete_range hf.toEmbedding.to_isometry.isUniformInducing]
     exact hf.isClosed_range.isComplete
   infer_instance
 
@@ -255,6 +256,8 @@ theorem dist_val_le_dist (x y : CompleteCopy s) : dist x.1 y.1 ≤ dist x y :=
   le_add_of_nonneg_right (abs_nonneg _)
 
 instance : TopologicalSpace (CompleteCopy s) := inferInstanceAs (TopologicalSpace s)
+instance [SecondCountableTopology α] : SecondCountableTopology (CompleteCopy s) :=
+  inferInstanceAs (SecondCountableTopology s)
 instance : T0Space (CompleteCopy s) := inferInstanceAs (T0Space s)
 
 /-- A metric space structure on a subset `s` of a metric space, designed to make it complete
@@ -328,7 +331,6 @@ theorem _root_.IsOpen.polishSpace {α : Type*} [TopologicalSpace α] [PolishSpac
     (hs : IsOpen s) : PolishSpace s := by
   letI := upgradePolishSpace α
   lift s to Opens α using hs
-  have : SecondCountableTopology s.CompleteCopy := inferInstanceAs (SecondCountableTopology s)
   exact inferInstanceAs (PolishSpace s.CompleteCopy)
 
 end CompleteCopy
@@ -365,7 +367,13 @@ theorem _root_.IsClosed.isClopenable [TopologicalSpace α] [PolishSpace α] {s :
   · rw [← f.induced_symm]
     exact f.symm.polishSpace_induced
   · rw [isOpen_coinduced, isOpen_sum_iff]
-    simp [f, preimage_preimage]
+    simp only [preimage_preimage, f]
+    have inl (x : s) : (Equiv.Set.sumCompl s) (Sum.inl x) = x := Equiv.Set.sumCompl_apply_inl ..
+    have inr (x : ↑sᶜ) : (Equiv.Set.sumCompl s) (Sum.inr x) = x := Equiv.Set.sumCompl_apply_inr ..
+    simp_rw [inl, inr, Subtype.coe_preimage_self]
+    simp only [isOpen_univ, true_and]
+    rw [Subtype.preimage_coe_compl']
+    simp
 
 theorem IsClopenable.compl [TopologicalSpace α] {s : Set α} (hs : IsClopenable s) :
     IsClopenable sᶜ := by
diff --git a/Mathlib/Topology/MetricSpace/Pseudo/Basic.lean b/Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
index 5793a4509ed7c..550dee230352d 100644
--- a/Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
+++ b/Mathlib/Topology/MetricSpace/Pseudo/Basic.lean
@@ -61,24 +61,34 @@ namespace Metric
 -- instantiate pseudometric space as a topology
 variable {x y z : α} {δ ε ε₁ ε₂ : ℝ} {s : Set α}
 
-nonrec theorem uniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
-    UniformInducing f ↔ UniformContinuous f ∧
+nonrec theorem isUniformInducing_iff [PseudoMetricSpace β] {f : α → β} :
+    IsUniformInducing f ↔ UniformContinuous f ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
-  uniformInducing_iff'.trans <| Iff.rfl.and <|
+  isUniformInducing_iff'.trans <| Iff.rfl.and <|
     ((uniformity_basis_dist.comap _).le_basis_iff uniformity_basis_dist).trans <| by
       simp only [subset_def, Prod.forall, gt_iff_lt, preimage_setOf_eq, Prod.map_apply, mem_setOf]
 
-nonrec theorem uniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
-    UniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_iff := isUniformInducing_iff
+
+nonrec theorem isUniformEmbedding_iff [PseudoMetricSpace β] {f : α → β} :
+    IsUniformEmbedding f ↔ Function.Injective f ∧ UniformContinuous f ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ := by
-  rw [uniformEmbedding_iff, and_comm, uniformInducing_iff]
+  rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff]
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_iff := isUniformEmbedding_iff
 
 /-- If a map between pseudometric spaces is a uniform embedding then the distance between `f x`
 and `f y` is controlled in terms of the distance between `x` and `y`. -/
-theorem controlled_of_uniformEmbedding [PseudoMetricSpace β] {f : α → β} (h : UniformEmbedding f) :
+theorem controlled_of_isUniformEmbedding [PseudoMetricSpace β] {f : α → β}
+    (h : IsUniformEmbedding f) :
     (∀ ε > 0, ∃ δ > 0, ∀ {a b : α}, dist a b < δ → dist (f a) (f b) < ε) ∧
       ∀ δ > 0, ∃ ε > 0, ∀ {a b : α}, dist (f a) (f b) < ε → dist a b < δ :=
-  ⟨uniformContinuous_iff.1 h.uniformContinuous, (uniformEmbedding_iff.1 h).2.2⟩
+  ⟨uniformContinuous_iff.1 h.uniformContinuous, (isUniformEmbedding_iff.1 h).2.2⟩
+
+@[deprecated (since := "2024-10-01")]
+alias controlled_of_uniformEmbedding := controlled_of_isUniformEmbedding
 
 theorem totallyBounded_iff {s : Set α} :
     TotallyBounded s ↔ ∀ ε > 0, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, ball y ε :=
@@ -169,8 +179,20 @@ end Metric
 
 open Metric
 
+theorem Metric.inseparable_iff_nndist {x y : α} : Inseparable x y ↔ nndist x y = 0 := by
+  rw [EMetric.inseparable_iff, edist_nndist, ENNReal.coe_eq_zero]
+
+alias ⟨Inseparable.nndist_eq_zero, _⟩ := Metric.inseparable_iff_nndist
+
 theorem Metric.inseparable_iff {x y : α} : Inseparable x y ↔ dist x y = 0 := by
-  rw [EMetric.inseparable_iff, edist_nndist, dist_nndist, ENNReal.coe_eq_zero, NNReal.coe_eq_zero]
+  rw [Metric.inseparable_iff_nndist, dist_nndist, NNReal.coe_eq_zero]
+
+alias ⟨Inseparable.dist_eq_zero, _⟩ := Metric.inseparable_iff
+
+/-- A weaker version of `tendsto_nhds_unique` for `PseudoMetricSpace`. -/
+theorem tendsto_nhds_unique_dist {f : β → α} {l : Filter β} {x y : α} [NeBot l]
+    (ha : Tendsto f l (𝓝 x)) (hb : Tendsto f l (𝓝 y)) : dist x y = 0 :=
+  (tendsto_nhds_unique_inseparable ha hb).dist_eq_zero
 
 section Real
 
diff --git a/Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean b/Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
index 3ad6637ff4565..764f2c2b7f1f1 100644
--- a/Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
+++ b/Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean
@@ -44,10 +44,13 @@ def Inducing.comapPseudoMetricSpace {α β} [TopologicalSpace α] [m : PseudoMet
 /-- Pull back a pseudometric space structure by a uniform inducing map. This is a version of
 `PseudoMetricSpace.induced` useful in case if the domain already has a `UniformSpace`
 structure. -/
-def UniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β]
-    (f : α → β) (h : UniformInducing f) : PseudoMetricSpace α :=
+def IsUniformInducing.comapPseudoMetricSpace {α β} [UniformSpace α] [m : PseudoMetricSpace β]
+    (f : α → β) (h : IsUniformInducing f) : PseudoMetricSpace α :=
   .replaceUniformity (.induced f m) h.comap_uniformity.symm
 
+@[deprecated (since := "2024-10-08")] alias UniformInducing.comapPseudoMetricSpace :=
+  IsUniformInducing.comapPseudoMetricSpace
+
 instance Subtype.pseudoMetricSpace {p : α → Prop} : PseudoMetricSpace (Subtype p) :=
   PseudoMetricSpace.induced Subtype.val ‹_›
 
diff --git a/Mathlib/Topology/MetricSpace/Pseudo/Defs.lean b/Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
index 7a02659c96fe2..89df0d0b39ba3 100644
--- a/Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
+++ b/Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
@@ -263,15 +263,15 @@ theorem edist_lt_coe {x y : α} {c : ℝ≥0} : edist x y < c ↔ nndist x y < c
 theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by
   rw [edist_nndist, ENNReal.coe_le_coe]
 
-/-- In a pseudometric space, the extended distance is always finite-/
+/-- In a pseudometric space, the extended distance is always finite -/
 theorem edist_lt_top {α : Type*} [PseudoMetricSpace α] (x y : α) : edist x y < ⊤ :=
   (edist_dist x y).symm ▸ ENNReal.ofReal_lt_top
 
-/-- In a pseudometric space, the extended distance is always finite-/
+/-- In a pseudometric space, the extended distance is always finite -/
 theorem edist_ne_top (x y : α) : edist x y ≠ ⊤ :=
   (edist_lt_top x y).ne
 
-/-- `nndist x x` vanishes-/
+/-- `nndist x x` vanishes -/
 @[simp] theorem nndist_self (a : α) : nndist a a = 0 := NNReal.coe_eq_zero.1 (dist_self a)
 
 -- Porting note: `dist_nndist` and `coe_nndist` moved up
@@ -299,7 +299,7 @@ theorem nndist_dist (x y : α) : nndist x y = Real.toNNReal (dist x y) := by
 
 theorem nndist_comm (x y : α) : nndist x y = nndist y x := NNReal.eq <| dist_comm x y
 
-/-- Triangle inequality for the nonnegative distance-/
+/-- Triangle inequality for the nonnegative distance -/
 theorem nndist_triangle (x y z : α) : nndist x z ≤ nndist x y + nndist y z :=
   dist_triangle _ _ _
 
@@ -766,7 +766,7 @@ theorem tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝[s] a) (𝓝 b) ↔
       ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) b < ε := by
   rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
-  simp only [mem_univ, true_and_iff]
+  simp only [mem_univ, true_and]
 
 theorem tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
     Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, dist x a < δ → dist (f x) b < ε :=
@@ -1083,7 +1083,7 @@ theorem dist_dist_dist_le (x y x' y' : α) : dist (dist x y) (dist x' y') ≤ di
 
 theorem nhds_comap_dist (a : α) : ((𝓝 (0 : ℝ)).comap (dist · a)) = 𝓝 a := by
   simp only [@nhds_eq_comap_uniformity α, Metric.uniformity_eq_comap_nhds_zero, comap_comap,
-    (· ∘ ·), dist_comm]
+    Function.comp_def, dist_comm]
 
 theorem tendsto_iff_dist_tendsto_zero {f : β → α} {x : Filter β} {a : α} :
     Tendsto f x (𝓝 a) ↔ Tendsto (fun b => dist (f b) a) x (𝓝 0) := by
@@ -1096,7 +1096,7 @@ variable {x y z : α} {ε ε₁ ε₂ : ℝ} {s : Set α}
 theorem ball_subset_interior_closedBall : ball x ε ⊆ interior (closedBall x ε) :=
   interior_maximal ball_subset_closedBall isOpen_ball
 
-/-- ε-characterization of the closure in pseudometric spaces-/
+/-- ε-characterization of the closure in pseudometric spaces -/
 theorem mem_closure_iff {s : Set α} {a : α} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε :=
   (mem_closure_iff_nhds_basis nhds_basis_ball).trans <| by simp only [mem_ball, dist_comm]
 
diff --git a/Mathlib/Topology/MetricSpace/Pseudo/Pi.lean b/Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
index 3ace96bd69eb9..92a7c3925b2a1 100644
--- a/Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
+++ b/Mathlib/Topology/MetricSpace/Pseudo/Pi.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
 -/
 import Mathlib.Topology.Bornology.Constructions
-import Mathlib.Topology.EMetricSpace.Basic
+import Mathlib.Topology.EMetricSpace.Pi
 import Mathlib.Topology.MetricSpace.Pseudo.Defs
 
 /-!
diff --git a/Mathlib/Topology/MetricSpace/ShrinkingLemma.lean b/Mathlib/Topology/MetricSpace/ShrinkingLemma.lean
index 058f2115e3541..b7b081e3fb55d 100644
--- a/Mathlib/Topology/MetricSpace/ShrinkingLemma.lean
+++ b/Mathlib/Topology/MetricSpace/ShrinkingLemma.lean
@@ -27,7 +27,7 @@ open Set Metric
 open Topology
 
 variable {α : Type u} {ι : Type v} [MetricSpace α] [ProperSpace α] {c : ι → α}
-variable {x : α} {r : ℝ} {s : Set α}
+variable {s : Set α}
 
 /-- **Shrinking lemma** for coverings by open balls in a proper metric space. A point-finite open
 cover of a closed subset of a proper metric space by open balls can be shrunk to a new cover by
diff --git a/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean b/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
index c274fbbc8e376..50b3c349f536c 100644
--- a/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
+++ b/Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kalle Kytölä
 -/
 import Mathlib.Data.ENNReal.Basic
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 import Mathlib.Topology.MetricSpace.Thickening
 
 /-!
@@ -128,7 +128,7 @@ theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
     specialize δseq_lim ε ε_pos
     simp only [dist_zero_right, Real.norm_eq_abs, eventually_atTop] at δseq_lim
     rcases δseq_lim with ⟨N, hN⟩
-    apply @tendsto_atTop_of_eventually_const _ _ _ _ _ _ _ N
+    apply tendsto_atTop_of_eventually_const (i₀ := N)
     intro n n_large
     have key : x ∉ thickening ε E := by simpa only [thickening, mem_setOf_eq, not_lt] using ε_lt.le
     refine le_antisymm ?_ bot_le
diff --git a/Mathlib/Topology/MetricSpace/Ultra/Basic.lean b/Mathlib/Topology/MetricSpace/Ultra/Basic.lean
index 5b00adb8cfda9..ceccb0c003e73 100644
--- a/Mathlib/Topology/MetricSpace/Ultra/Basic.lean
+++ b/Mathlib/Topology/MetricSpace/Ultra/Basic.lean
@@ -50,6 +50,18 @@ lemma dist_triangle_max : dist x z ≤ max (dist x y) (dist y z) :=
 
 namespace IsUltrametricDist
 
+/-- All triangles are isosceles in an ultrametric space. -/
+lemma dist_eq_max_of_dist_ne_dist (h : dist x y ≠ dist y z) :
+    dist x z = max (dist x y) (dist y z) := by
+  apply le_antisymm (dist_triangle_max x y z)
+  rcases h.lt_or_lt with h | h
+  · rw [max_eq_right h.le]
+    apply (le_max_iff.mp <| dist_triangle_max y x z).resolve_left
+    simpa only [not_le, dist_comm x y] using h
+  · rw [max_eq_left h.le, dist_comm x y, dist_comm x z]
+    apply (le_max_iff.mp <| dist_triangle_max y z x).resolve_left
+    simpa only [not_le, dist_comm x y] using h
+
 instance subtype (p : X → Prop) : IsUltrametricDist (Subtype p) :=
   ⟨fun _ _ _ ↦ by simpa [Subtype.dist_eq] using dist_triangle_max _ _ _⟩
 
diff --git a/Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.lean b/Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.lean
new file mode 100644
index 0000000000000..62dd5bf7c5b27
--- /dev/null
+++ b/Mathlib/Topology/MetricSpace/Ultra/ContinuousMaps.lean
@@ -0,0 +1,22 @@
+/-
+Copyright (c) 2024 David Loeffler. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: David Loeffler
+-/
+
+import Mathlib.Topology.ContinuousMap.Compact
+import Mathlib.Topology.MetricSpace.Ultra.Basic
+
+/-!
+# Ultrametric structure on continuous maps
+-/
+
+/-- Continuous maps from a compact space to an ultrametric space are an ultrametric space. -/
+instance ContinuousMap.isUltrametricDist {X Y : Type*}
+    [TopologicalSpace X] [CompactSpace X] [MetricSpace Y] [IsUltrametricDist Y] :
+    IsUltrametricDist C(X, Y) := by
+  constructor
+  intro f g h
+  rw [ContinuousMap.dist_le (by positivity)]
+  refine fun x ↦ (dist_triangle_max (f x) (g x) (h x)).trans (max_le_max ?_ ?_) <;>
+  exact ContinuousMap.dist_apply_le_dist x
diff --git a/Mathlib/Topology/MetricSpace/Ultra/TotallySeparated.lean b/Mathlib/Topology/MetricSpace/Ultra/TotallySeparated.lean
new file mode 100644
index 0000000000000..09edc0330ec3e
--- /dev/null
+++ b/Mathlib/Topology/MetricSpace/Ultra/TotallySeparated.lean
@@ -0,0 +1,28 @@
+/-
+Copyright (c) 2024 Yakov Pechersky. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yakov Pechersky, David Loeffler
+-/
+import Mathlib.Topology.MetricSpace.Defs
+import Mathlib.Topology.MetricSpace.Ultra.Basic
+
+/-!
+# Ultrametric spaces are totally separated
+
+In a metric space with an ultrametric, the space is totally separated, hence totally disconnected.
+
+## Tags
+
+ultrametric, nonarchimedean, totally separated, totally disconnected
+-/
+open Metric IsUltrametricDist
+
+instance {X : Type*} [MetricSpace X] [IsUltrametricDist X] : TotallySeparatedSpace X :=
+  totallySeparatedSpace_iff_exists_isClopen.mpr fun x y h ↦ by
+    obtain ⟨r, hr, hr'⟩ := exists_between (dist_pos.mpr h)
+    refine ⟨_, IsUltrametricDist.isClopen_ball x r, ?_, ?_⟩
+    · simp only [mem_ball, dist_self, hr]
+    · simp only [Set.mem_compl, mem_ball, dist_comm, not_lt, hr'.le]
+
+example {X : Type*} [MetricSpace X] [IsUltrametricDist X] : TotallyDisconnectedSpace X :=
+  inferInstance
diff --git a/Mathlib/Topology/Metrizable/Uniformity.lean b/Mathlib/Topology/Metrizable/Uniformity.lean
index 2bef1df92ab23..3dc4176d75c21 100644
--- a/Mathlib/Topology/Metrizable/Uniformity.lean
+++ b/Mathlib/Topology/Metrizable/Uniformity.lean
@@ -154,12 +154,12 @@ theorem le_two_mul_dist_ofPreNNDist (d : X → X → ℝ≥0) (dist_self : ∀ x
       have hMl' : length (take M l) = M := (length_take _ _).trans (min_eq_left hMl.le)
       refine (ihn _ hMl _ _ _ hMl').trans ?_
       convert hMs.1.out
-      rw [take_zipWith, take, take_succ, getElem?_append hMl, getElem?_eq_getElem hMl,
+      rw [take_zipWith, take, take_succ, getElem?_append_left hMl, getElem?_eq_getElem hMl,
         ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
   · exact single_le_sum (fun x _ => zero_le x) _ (mem_iff_get.2 ⟨⟨M, hM_lt⟩, getElem_zipWith⟩)
   · rcases hMl.eq_or_lt with (rfl | hMl)
-    · simp only [getElem_append_right' le_rfl, sub_self, getElem_singleton, dist_self, zero_le]
-    rw [getElem_append _ hMl]
+    · simp only [getElem_append_right le_rfl, sub_self, getElem_singleton, dist_self, zero_le]
+    rw [getElem_append_left hMl]
     have hlen : length (drop (M + 1) l) = length l - (M + 1) := length_drop _ _
     have hlen_lt : length l - (M + 1) < length l := Nat.sub_lt_of_pos_le M.succ_pos hMl
     refine (ihn _ hlen_lt _ y _ hlen).trans ?_
@@ -206,7 +206,7 @@ protected theorem UniformSpace.metrizable_uniformity (X : Type*) [UniformSpace X
     split_ifs with h
     · rw [← not_forall] at h
       simp [h, pow_eq_zero_iff']
-    · simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff_iff] using h
+    · simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff] using h
   have hd_symm : ∀ x y, d x y = d y x := by
     intro x y
     simp only [d, @SymmetricRel.mk_mem_comm _ _ (hU_symm _) x y]
diff --git a/Mathlib/Topology/Metrizable/Urysohn.lean b/Mathlib/Topology/Metrizable/Urysohn.lean
index f136cbf094264..bddb6a6ead668 100644
--- a/Mathlib/Topology/Metrizable/Urysohn.lean
+++ b/Mathlib/Topology/Metrizable/Urysohn.lean
@@ -5,7 +5,7 @@ Authors: Yury Kudryashov
 -/
 import Mathlib.Analysis.SpecificLimits.Basic
 import Mathlib.Topology.UrysohnsLemma
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 import Mathlib.Topology.Metrizable.Basic
 /-!
 # Urysohn's Metrization Theorem
diff --git a/Mathlib/Topology/NhdsSet.lean b/Mathlib/Topology/NhdsSet.lean
index 5580d18fa47ef..3a3f45b2eeff7 100644
--- a/Mathlib/Topology/NhdsSet.lean
+++ b/Mathlib/Topology/NhdsSet.lean
@@ -144,7 +144,7 @@ theorem nhdsSet_insert (x : X) (s : Set X) : 𝓝ˢ (insert x s) = 𝓝 x ⊔ 
   rw [insert_eq, nhdsSet_union, nhdsSet_singleton]
 
 /-- Preimage of a set neighborhood of `t` under a continuous map `f` is a set neighborhood of `s`
-provided that `f` maps `s` to `t`.  -/
+provided that `f` maps `s` to `t`. -/
 theorem Continuous.tendsto_nhdsSet {f : X → Y} {t : Set Y} (hf : Continuous f)
     (hst : MapsTo f s t) : Tendsto f (𝓝ˢ s) (𝓝ˢ t) :=
   ((hasBasis_nhdsSet s).tendsto_iff (hasBasis_nhdsSet t)).mpr fun U hU =>
diff --git a/Mathlib/Topology/NoetherianSpace.lean b/Mathlib/Topology/NoetherianSpace.lean
index 74d4532b5976c..9c2e714b18849 100644
--- a/Mathlib/Topology/NoetherianSpace.lean
+++ b/Mathlib/Topology/NoetherianSpace.lean
@@ -43,12 +43,10 @@ variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
 namespace TopologicalSpace
 
 /-- Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. -/
-@[mk_iff]
-class NoetherianSpace : Prop where
-  wellFounded_opens : WellFounded ((· > ·) : Opens α → Opens α → Prop)
+abbrev NoetherianSpace : Prop := WellFoundedGT (Opens α)
 
 theorem noetherianSpace_iff_opens : NoetherianSpace α ↔ ∀ s : Opens α, IsCompact (s : Set α) := by
-  rw [noetherianSpace_iff, CompleteLattice.wellFounded_iff_isSupFiniteCompact,
+  rw [NoetherianSpace, CompleteLattice.wellFoundedGT_iff_isSupFiniteCompact,
     CompleteLattice.isSupFiniteCompact_iff_all_elements_compact]
   exact forall_congr' Opens.isCompactElement_iff
 
@@ -78,18 +76,15 @@ variable (α)
 open List in
 theorem noetherianSpace_TFAE :
     TFAE [NoetherianSpace α,
-      WellFounded fun s t : Closeds α => s < t,
+      WellFoundedLT (Closeds α),
       ∀ s : Set α, IsCompact s,
       ∀ s : Opens α, IsCompact (s : Set α)] := by
-  tfae_have 1 ↔ 2
-  · refine (noetherianSpace_iff α).trans (Opens.compl_bijective.2.wellFounded_iff ?_)
-    exact (@OrderIso.compl (Set α)).lt_iff_lt.symm
-  tfae_have 1 ↔ 4
-  · exact noetherianSpace_iff_opens α
-  tfae_have 1 → 3
-  · exact @NoetherianSpace.isCompact α _
-  tfae_have 3 → 4
-  · exact fun h s => h s
+  tfae_have 1 ↔ 2 := by
+    simp_rw [isWellFounded_iff]
+    exact Opens.compl_bijective.2.wellFounded_iff (@OrderIso.compl (Set α)).lt_iff_lt.symm
+  tfae_have 1 ↔ 4 := noetherianSpace_iff_opens α
+  tfae_have 1 → 3 := @NoetherianSpace.isCompact α _
+  tfae_have 3 → 4 := fun h s => h s
   tfae_finish
 
 variable {α}
@@ -97,9 +92,13 @@ variable {α}
 theorem noetherianSpace_iff_isCompact : NoetherianSpace α ↔ ∀ s : Set α, IsCompact s :=
   (noetherianSpace_TFAE α).out 0 2
 
+instance [NoetherianSpace α] : WellFoundedLT (Closeds α) :=
+  Iff.mp ((noetherianSpace_TFAE α).out 0 1) ‹_›
+
+@[deprecated (since := "2024-10-07")]
 theorem NoetherianSpace.wellFounded_closeds [NoetherianSpace α] :
     WellFounded fun s t : Closeds α => s < t :=
-  Iff.mp ((noetherianSpace_TFAE α).out 0 1) ‹_›
+  wellFounded_lt
 
 instance {α} : NoetherianSpace (CofiniteTopology α) := by
   simp only [noetherianSpace_iff_isCompact, isCompact_iff_ultrafilter_le_nhds,
@@ -156,7 +155,7 @@ instance (priority := 100) Finite.to_noetherianSpace [Finite α] : NoetherianSpa
 /-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
 theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace α] (s : Closeds α) :
     ∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = sSup S := by
-  apply wellFounded_closeds.induction s; clear s
+  apply wellFounded_lt.induction s; clear s
   intro s H
   rcases eq_or_ne s ⊥ with rfl | h₀
   · use ∅; simp
diff --git a/Mathlib/Topology/OmegaCompletePartialOrder.lean b/Mathlib/Topology/OmegaCompletePartialOrder.lean
index 7f45468c25312..89f1e3ee2fa58 100644
--- a/Mathlib/Topology/OmegaCompletePartialOrder.lean
+++ b/Mathlib/Topology/OmegaCompletePartialOrder.lean
@@ -39,21 +39,19 @@ variable (α : Type u) [OmegaCompletePartialOrder α]
 /-- The characteristic function of open sets is monotone and preserves
 the limits of chains. -/
 def IsOpen (s : Set α) : Prop :=
-  Continuous' fun x ↦ x ∈ s
+  ωScottContinuous fun x ↦ x ∈ s
 
-theorem isOpen_univ : IsOpen α univ :=
-  ⟨fun _ _ _ _ ↦ mem_univ _, @CompleteLattice.top_continuous α Prop _ _⟩
+theorem isOpen_univ : IsOpen α univ := @CompleteLattice.ωScottContinuous.top α Prop _ _
 
 theorem IsOpen.inter (s t : Set α) : IsOpen α s → IsOpen α t → IsOpen α (s ∩ t) :=
-  CompleteLattice.inf_continuous'
+  CompleteLattice.ωScottContinuous.inf
 
 theorem isOpen_sUnion (s : Set (Set α)) (hs : ∀ t ∈ s, IsOpen α t) : IsOpen α (⋃₀ s) := by
   simp only [IsOpen] at hs ⊢
-  convert CompleteLattice.sSup_continuous' (setOf ⁻¹' s) hs
-  simp only [sSup_apply, setOf_bijective.surjective.exists, exists_prop, mem_preimage,
-    SetCoe.exists, iSup_Prop_eq, mem_setOf_eq, mem_sUnion]
+  convert CompleteLattice.ωScottContinuous.sSup hs
+  aesop
 
-theorem IsOpen.isUpperSet {s : Set α} (hs : IsOpen α s) : IsUpperSet s := hs.fst
+theorem IsOpen.isUpperSet {s : Set α} (hs : IsOpen α s) : IsUpperSet s := hs.monotone
 
 end Scott
 
@@ -80,6 +78,8 @@ def notBelow :=
 
 theorem notBelow_isOpen : IsOpen (notBelow y) := by
   have h : Monotone (notBelow y) := fun x z hle ↦ mt hle.trans
+  dsimp only [IsOpen, TopologicalSpace.IsOpen, Scott.IsOpen]
+  rw [ωScottContinuous_iff_monotone_map_ωSup]
   refine ⟨h, fun c ↦ eq_of_forall_ge_iff fun z ↦ ?_⟩
   simp only [ωSup_le_iff, notBelow, mem_setOf_eq, le_Prop_eq, OrderHom.coe_mk, Chain.map_coe,
     Function.comp_apply, exists_imp, not_forall]
@@ -96,13 +96,15 @@ theorem isωSup_ωSup {α} [OmegaCompletePartialOrder α] (c : Chain α) : IsωS
   · apply ωSup_le
 
 theorem scottContinuous_of_continuous {α β} [OmegaCompletePartialOrder α]
-    [OmegaCompletePartialOrder β] (f : Scott α → Scott β) (hf : Continuous f) :
-    OmegaCompletePartialOrder.Continuous' f := by
+    [OmegaCompletePartialOrder β] (f : Scott α → Scott β) (hf : _root_.Continuous f) :
+    OmegaCompletePartialOrder.ωScottContinuous f := by
+  rw [ωScottContinuous_iff_monotone_map_ωSup]
   have h : Monotone f := fun x y h ↦ by
     have hf : IsUpperSet {x | ¬f x ≤ f y} := ((notBelow_isOpen (f y)).preimage hf).isUpperSet
     simpa only [mem_setOf_eq, le_refl, not_true, imp_false, not_not] using hf h
   refine ⟨h, fun c ↦ eq_of_forall_ge_iff fun z ↦ ?_⟩
-  rcases (notBelow_isOpen z).preimage hf with ⟨hf, hf'⟩
+  rcases (notBelow_isOpen z).preimage hf with hf''
+  let hf' := hf''.monotone_map_ωSup.2
   specialize hf' c
   simp only [OrderHom.coe_mk, mem_preimage, notBelow, mem_setOf_eq] at hf'
   rw [← not_iff_not]
@@ -112,11 +114,9 @@ theorem scottContinuous_of_continuous {α β} [OmegaCompletePartialOrder α]
 
 theorem continuous_of_scottContinuous {α β} [OmegaCompletePartialOrder α]
     [OmegaCompletePartialOrder β] (f : Scott α → Scott β)
-    (hf : OmegaCompletePartialOrder.Continuous' f) : Continuous f := by
+    (hf : ωScottContinuous f) : Continuous f := by
   rw [continuous_def]
   intro s hs
-  change Continuous' (s ∘ f)
-  cases' hs with hs hs'
-  cases' hf with hf hf'
-  apply Continuous.of_bundled
-  apply continuous_comp _ _ hf' hs'
+  dsimp only [IsOpen, TopologicalSpace.IsOpen, Scott.IsOpen]
+  simp_rw [mem_preimage, mem_def, ← Function.comp_def]
+  apply ωScottContinuous.comp hs hf
diff --git a/Mathlib/Topology/Order.lean b/Mathlib/Topology/Order.lean
index e533cd6e15a09..c1c9b629e8592 100644
--- a/Mathlib/Topology/Order.lean
+++ b/Mathlib/Topology/Order.lean
@@ -89,9 +89,6 @@ lemma tendsto_nhds_generateFrom_iff {β : Type*} {m : α → β} {f : Filter α}
   simp only [nhds_generateFrom, @forall_swap (b ∈ _), tendsto_iInf, mem_setOf_eq, and_imp,
     tendsto_principal]; rfl
 
-@[deprecated (since := "2023-12-24")]
-alias ⟨_, tendsto_nhds_generateFrom⟩ := tendsto_nhds_generateFrom_iff
-
 /-- Construct a topology on α given the filter of neighborhoods of each point of α. -/
 protected def mkOfNhds (n : α → Filter α) : TopologicalSpace α where
   IsOpen s := ∀ a ∈ s, s ∈ n a
@@ -520,6 +517,12 @@ lemma generateFrom_insert_univ {α : Type*} {s : Set (Set α)} :
     generateFrom (insert univ s) = generateFrom s :=
   generateFrom_insert_of_generateOpen .univ
 
+@[simp]
+lemma generateFrom_insert_empty {α : Type*} {s : Set (Set α)} :
+    generateFrom (insert ∅ s) = generateFrom s := by
+  rw [← sUnion_empty]
+  exact generateFrom_insert_of_generateOpen (.sUnion ∅ (fun s_1 a ↦ False.elim a))
+
 /-- This construction is left adjoint to the operation sending a topology on `α`
   to its neighborhood filter at a fixed point `a : α`. -/
 def nhdsAdjoint (a : α) (f : Filter α) : TopologicalSpace α where
@@ -615,9 +618,6 @@ lemma continuous_generateFrom_iff {t : TopologicalSpace α} {b : Set (Set β)} :
   rw [continuous_iff_coinduced_le, le_generateFrom_iff_subset_isOpen]
   simp only [isOpen_coinduced, preimage_id', subset_def, mem_setOf]
 
-@[deprecated (since := "2023-12-24")]
-alias ⟨_, continuous_generateFrom⟩ := continuous_generateFrom_iff
-
 @[continuity, fun_prop]
 theorem continuous_induced_dom {t : TopologicalSpace β} : Continuous[induced f t, t] f :=
   continuous_iff_le_induced.2 le_rfl
@@ -740,6 +740,15 @@ theorem map_nhds_induced_of_surjective [T : TopologicalSpace α] {f : β → α}
     (a : β) : map f (@nhds β (TopologicalSpace.induced f T) a) = 𝓝 (f a) := by
   rw [nhds_induced, map_comap_of_surjective hf]
 
+theorem continuous_nhdsAdjoint_dom [TopologicalSpace β] {f : α → β} {a : α} {l : Filter α} :
+    Continuous[nhdsAdjoint a l, _] f ↔ Tendsto f l (𝓝 (f a)) := by
+  simp_rw [continuous_iff_le_induced, gc_nhds _ _, nhds_induced, tendsto_iff_comap]
+
+theorem coinduced_nhdsAdjoint (f : α → β) (a : α) (l : Filter α) :
+    coinduced f (nhdsAdjoint a l) = nhdsAdjoint (f a) (map f l) :=
+  eq_of_forall_ge_iff fun _ ↦ by
+    rw [gc_nhds, ← continuous_iff_coinduced_le, continuous_nhdsAdjoint_dom, Tendsto]
+
 end Constructions
 
 section Induced
diff --git a/Mathlib/Topology/Order/Category/FrameAdjunction.lean b/Mathlib/Topology/Order/Category/FrameAdjunction.lean
index 66dd40b808326..e44b0f9de9062 100644
--- a/Mathlib/Topology/Order/Category/FrameAdjunction.lean
+++ b/Mathlib/Topology/Order/Category/FrameAdjunction.lean
@@ -109,11 +109,10 @@ def counitAppCont : FrameHom L (Opens <| PT L) where
   map_sSup' S := by ext; simp
 
 /-- The forgetful functor `topToLocale` is left adjoint to the functor `pt`. -/
-def adjunctionTopToLocalePT : topToLocale ⊣ pt :=
-  Adjunction.mkOfUnitCounit
-    { unit := { app := fun X ↦ ⟨localePointOfSpacePoint X, continuous_def.2 <|
+def adjunctionTopToLocalePT : topToLocale ⊣ pt where
+  unit := { app := fun X ↦ ⟨localePointOfSpacePoint X, continuous_def.2 <|
         by rintro _ ⟨u, rfl⟩; simpa using u.2⟩ }
-      counit := { app := fun L ↦ ⟨counitAppCont L⟩ } }
+  counit := { app := fun L ↦ ⟨counitAppCont L⟩ }
 
 end locale_top_adjunction
 
diff --git a/Mathlib/Topology/Order/Hom/Basic.lean b/Mathlib/Topology/Order/Hom/Basic.lean
index 11dc84c396d3c..b0e27200893c5 100644
--- a/Mathlib/Topology/Order/Hom/Basic.lean
+++ b/Mathlib/Topology/Order/Hom/Basic.lean
@@ -4,7 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
 import Mathlib.Order.Hom.Basic
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.Basic
+import Mathlib.Topology.ContinuousMap.Defs
 
 /-!
 # Continuous order homomorphisms
diff --git a/Mathlib/Topology/Order/IntermediateValue.lean b/Mathlib/Topology/Order/IntermediateValue.lean
index 06c7281cea152..9b86197c50a9e 100644
--- a/Mathlib/Topology/Order/IntermediateValue.lean
+++ b/Mathlib/Topology/Order/IntermediateValue.lean
@@ -443,7 +443,7 @@ theorem setOf_isPreconnected_eq_of_ordered :
       (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅}) := by
   refine Subset.antisymm setOf_isPreconnected_subset_of_ordered ?_
   simp only [subset_def, forall_mem_range, uncurry, or_imp, forall_and, mem_union,
-    mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true_iff,
+    mem_setOf_eq, insert_eq, mem_singleton_iff, forall_eq, forall_true_iff, and_true,
     isPreconnected_Icc, isPreconnected_Ico, isPreconnected_Ioc, isPreconnected_Ioo,
     isPreconnected_Ioi, isPreconnected_Iio, isPreconnected_Ici, isPreconnected_Iic,
     isPreconnected_univ, isPreconnected_empty]
diff --git a/Mathlib/Topology/Order/IsLUB.lean b/Mathlib/Topology/Order/IsLUB.lean
index 85085029dc118..2d6f0f82dc720 100644
--- a/Mathlib/Topology/Order/IsLUB.lean
+++ b/Mathlib/Topology/Order/IsLUB.lean
@@ -53,8 +53,9 @@ theorem IsLUB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.No
     NeBot (𝓝[s] a) :=
   mem_closure_iff_nhdsWithin_neBot.1 (ha.mem_closure hs)
 
-theorem IsGLB.nhdsWithin_neBot : ∀ {a : α} {s : Set α}, IsGLB s a → s.Nonempty → NeBot (𝓝[s] a) :=
-  IsLUB.nhdsWithin_neBot (α := αᵒᵈ)
+theorem IsGLB.nhdsWithin_neBot {a : α} {s : Set α} (ha : IsGLB s a) (hs : s.Nonempty) :
+    NeBot (𝓝[s] a) :=
+  IsLUB.nhdsWithin_neBot (α := αᵒᵈ) ha hs
 
 theorem isLUB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ upperBounds s) (hsf : s ∈ f)
     [NeBot (f ⊓ 𝓝 a)] : IsLUB s a :=
@@ -70,9 +71,10 @@ theorem isLUB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ upperBounds s) (
   rw [mem_closure_iff_clusterPt, ClusterPt, inf_comm] at hsf
   exact isLUB_of_mem_nhds hsa (mem_principal_self s)
 
-theorem isGLB_of_mem_nhds :
-    ∀ {s : Set α} {a : α} {f : Filter α}, a ∈ lowerBounds s → s ∈ f → NeBot (f ⊓ 𝓝 a) → IsGLB s a :=
-  isLUB_of_mem_nhds (α := αᵒᵈ)
+theorem isGLB_of_mem_nhds {s : Set α} {a : α} {f : Filter α} (hsa : a ∈ lowerBounds s) (hsf : s ∈ f)
+    [NeBot (f ⊓ 𝓝 a)] :
+    IsGLB s a :=
+  isLUB_of_mem_nhds (α := αᵒᵈ) hsa hsf
 
 theorem isGLB_of_mem_closure {s : Set α} {a : α} (hsa : a ∈ lowerBounds s) (hsf : a ∈ closure s) :
     IsGLB s a :=
@@ -114,20 +116,20 @@ theorem IsLUB.mem_lowerBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [Or
     (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ lowerBounds (f '' s) :=
   IsLUB.mem_upperBounds_of_tendsto (γ := γᵒᵈ) hf ha hb
 
-theorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :
-    ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},
-      AntitoneOn f s → IsLUB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsGLB (f '' s) b :=
-  IsLUB.isLUB_of_tendsto (γ := γᵒᵈ)
+theorem IsLUB.isGLB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
+    {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsLUB s a) (hs : s.Nonempty)
+    (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsGLB (f '' s) b :=
+  IsLUB.isLUB_of_tendsto (γ := γᵒᵈ) hf ha hs hb
 
 theorem IsGLB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
     {f : α → γ} {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a)
     (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) :=
   IsGLB.mem_lowerBounds_of_tendsto (γ := γᵒᵈ) hf ha hb
 
-theorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] :
-    ∀ {f : α → γ} {s : Set α} {a : α} {b : γ},
-      AntitoneOn f s → IsGLB s a → s.Nonempty → Tendsto f (𝓝[s] a) (𝓝 b) → IsLUB (f '' s) b :=
-  IsGLB.isGLB_of_tendsto (γ := γᵒᵈ)
+theorem IsGLB.isLUB_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ] {f : α → γ}
+    {s : Set α} {a : α} {b : γ} (hf : AntitoneOn f s) (ha : IsGLB s a) (hs : s.Nonempty)
+    (hb : Tendsto f (𝓝[s] a) (𝓝 b)) : IsLUB (f '' s) b :=
+  IsGLB.isGLB_of_tendsto (γ := γᵒᵈ) hf ha hs hb
 
 theorem IsLUB.mem_of_isClosed {a : α} {s : Set α} (ha : IsLUB s a) (hs : s.Nonempty)
     (sc : IsClosed s) : a ∈ s :=
diff --git a/Mathlib/Topology/Order/LawsonTopology.lean b/Mathlib/Topology/Order/LawsonTopology.lean
index 4c730963736e1..61c08e2a6fe21 100644
--- a/Mathlib/Topology/Order/LawsonTopology.lean
+++ b/Mathlib/Topology/Order/LawsonTopology.lean
@@ -144,7 +144,7 @@ instance instIsLawson : IsLawson (WithLawson α) := ⟨rfl⟩
 /-- If `α` is equipped with the Lawson topology, then it is homeomorphic to `WithLawson α`.
 -/
 def homeomorph [TopologicalSpace α] [IsLawson α] : WithLawson α ≃ₜ α :=
-  ofLawson.toHomeomorphOfInducing ⟨by erw [@IsLawson.topology_eq_lawson α _ _, induced_id]; rfl⟩
+  ofLawson.toHomeomorphOfInducing ⟨by erw [IsLawson.topology_eq_lawson (α := α), induced_id]; rfl⟩
 
 theorem isOpen_preimage_ofLawson {S : Set α} :
     IsOpen (ofLawson ⁻¹' S) ↔ (lawson α).IsOpen S := Iff.rfl
diff --git a/Mathlib/Topology/Order/LeftRightNhds.lean b/Mathlib/Topology/Order/LeftRightNhds.lean
index 626dbfd4cd1b9..c6f61ba355838 100644
--- a/Mathlib/Topology/Order/LeftRightNhds.lean
+++ b/Mathlib/Topology/Order/LeftRightNhds.lean
@@ -3,9 +3,9 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
 -/
-
+import Mathlib.Algebra.Ring.Pointwise.Set
+import Mathlib.Order.Filter.AtTopBot.Group
 import Mathlib.Topology.Order.Basic
-import Mathlib.Data.Set.Pointwise.Basic
 
 /-!
 # Neighborhoods to the left and to the right on an `OrderTopology`
@@ -43,17 +43,15 @@ theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
       s ∈ 𝓝[Ioo a b] a,
       ∃ u ∈ Ioc a b, Ioo a u ⊆ s,
       ∃ u ∈ Ioi a, Ioo a u ⊆ s] := by
-  tfae_have 1 ↔ 2
-  · rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]
-  tfae_have 1 ↔ 3
-  · rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
-  tfae_have 4 → 5
-  · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
+  tfae_have 1 ↔ 2 := by
+    rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]
+  tfae_have 1 ↔ 3 := by
+    rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
+  tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
   tfae_have 5 → 1
-  · rintro ⟨u, hau, hu⟩
-    exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu
+  | ⟨u, hau, hu⟩ => mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu
   tfae_have 1 → 4
-  · intro h
+  | h => by
     rcases mem_nhdsWithin_iff_exists_mem_nhds_inter.1 h with ⟨v, va, hv⟩
     rcases exists_Ico_subset_of_mem_nhds' va hab with ⟨u, au, hu⟩
     exact ⟨u, au, fun x hx => hv ⟨hu ⟨le_of_lt hx.1, hx.2⟩, hx.1⟩⟩
@@ -183,19 +181,15 @@ theorem TFAE_mem_nhdsWithin_Ici {a b : α} (hab : a < b) (s : Set α) :
       s ∈ 𝓝[Ico a b] a,
       ∃ u ∈ Ioc a b, Ico a u ⊆ s,
       ∃ u ∈ Ioi a , Ico a u ⊆ s] := by
-  tfae_have 1 ↔ 2
-  · rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]
-  tfae_have 1 ↔ 3
-  · rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]
-  tfae_have 1 ↔ 5
-  · exact (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff
-  tfae_have 4 → 5
-  · exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
+  tfae_have 1 ↔ 2 := by
+    rw [nhdsWithin_Icc_eq_nhdsWithin_Ici hab]
+  tfae_have 1 ↔ 3 := by
+    rw [nhdsWithin_Ico_eq_nhdsWithin_Ici hab]
+  tfae_have 1 ↔ 5 := (nhdsWithin_Ici_basis' ⟨b, hab⟩).mem_iff
+  tfae_have 4 → 5 := fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
   tfae_have 5 → 4
-  · rintro ⟨u, hua, hus⟩
-    exact
-      ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,
-        (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩
+  | ⟨u, hua, hus⟩ => ⟨min u b, ⟨lt_min hua hab, min_le_right _ _⟩,
+      (Ico_subset_Ico_right <| min_le_left _ _).trans hus⟩
   tfae_finish
 
 theorem mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset {a u' : α} {s : Set α} (hu' : a < u') :
diff --git a/Mathlib/Topology/Order/LowerUpperTopology.lean b/Mathlib/Topology/Order/LowerUpperTopology.lean
index efe6bf46b486b..95dc12c92c61f 100644
--- a/Mathlib/Topology/Order/LowerUpperTopology.lean
+++ b/Mathlib/Topology/Order/LowerUpperTopology.lean
@@ -263,10 +263,6 @@ lemma continuous_iff_Ici [TopologicalSpace β] {f : β → α} :
   obtain rfl := IsLower.topology_eq α
   simp [continuous_generateFrom_iff]
 
-/-- A function `f : β → α` with lower topology in the codomain is continuous provided that the
-preimage of every interval `Set.Ici a` is a closed set. -/
-@[deprecated (since := "2023-12-24")] alias ⟨_, continuous_of_Ici⟩ := continuous_iff_Ici
-
 end Preorder
 
 section PartialOrder
@@ -399,13 +395,6 @@ lemma continuous_iff_Iic [TopologicalSpace β] {f : β → α} :
     Continuous f ↔ ∀ a, IsClosed (f ⁻¹' (Iic a)) :=
   IsLower.continuous_iff_Ici (α := αᵒᵈ)
 
-/-- A function `f : β → α` with upper topology in the codomain is continuous
-provided that the preimage of every interval `Set.Iic a` is a closed set. -/
-@[deprecated (since := "2023-12-24")]
-lemma continuous_of_Iic [TopologicalSpace β] {f : β → α} (h : ∀ a, IsClosed (f ⁻¹' (Iic a))) :
-    Continuous f :=
-  continuous_iff_Iic.2 h
-
 end Preorder
 
 
@@ -509,3 +498,17 @@ lemma isLower_orderDual [Preorder α] [TopologicalSpace α] : IsLower αᵒᵈ 
   isUpper_orderDual.symm
 
 end Topology
+
+/-- The Sierpiński topology on `Prop` is the upper topology -/
+instance : IsUpper Prop where
+  topology_eq_upperTopology := by
+    rw [Topology.upper, sierpinskiSpace, ← generateFrom_insert_empty]
+    congr
+    exact le_antisymm
+      (fun h hs => by
+        simp only [compl_Iic, mem_setOf_eq]
+        rw [← Ioi_True, ← Ioi_False] at hs
+        rcases hs with (rfl | rfl)
+        · use True
+        · use False)
+      (by rintro _ ⟨a, rfl⟩; by_cases a <;> aesop (add simp [Ioi, lt_iff_le_not_le]))
diff --git a/Mathlib/Topology/Order/Monotone.lean b/Mathlib/Topology/Order/Monotone.lean
index e64473e08518c..c807d80f962d5 100644
--- a/Mathlib/Topology/Order/Monotone.lean
+++ b/Mathlib/Topology/Order/Monotone.lean
@@ -28,65 +28,122 @@ variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopol
 
 /-- A monotone function continuous at the supremum of a nonempty set sends this supremum to
 the supremum of the image of this set. -/
-theorem Monotone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
-    (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
+theorem MonotoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
+    (Cf : ContinuousWithinAt f A (sSup A))
+    (Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
     f (sSup A) = sSup (f '' A) :=
   --This is a particular case of the more general `IsLUB.isLUB_of_tendsto`
-  .symm <| ((isLUB_csSup A_nonemp A_bdd).isLUB_of_tendsto (Mf.monotoneOn _) A_nonemp <|
-    Cf.mono_left inf_le_left).csSup_eq (A_nonemp.image f)
+  .symm <| ((isLUB_csSup A_nonemp A_bdd).isLUB_of_tendsto Mf A_nonemp <|
+    Cf.mono_left fun ⦃_⦄ a ↦ a).csSup_eq (A_nonemp.image f)
+
+/-- A monotone function continuous at the supremum of a nonempty set sends this supremum to
+the supremum of the image of this set. -/
+theorem Monotone.map_csSup_of_continuousAt {f : α → β} {A : Set α}
+    (Cf : ContinuousAt f (sSup A)) (Mf : Monotone f) (A_nonemp : A.Nonempty)
+    (A_bdd : BddAbove A := by bddDefault) : f (sSup A) = sSup (f '' A) :=
+  MonotoneOn.map_csSup_of_continuousWithinAt Cf.continuousWithinAt
+    (Mf.monotoneOn _) A_nonemp A_bdd
+
+@[deprecated (since := "2024-08-26")] alias Monotone.map_sSup_of_continuousAt' :=
+  Monotone.map_csSup_of_continuousAt
 
 /-- A monotone function continuous at the indexed supremum over a nonempty `Sort` sends this indexed
 supremum to the indexed supremum of the composition. -/
-theorem Monotone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
+theorem Monotone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
     (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f)
     (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨆ i, f (g i) := by
-  rw [iSup, Monotone.map_sSup_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iSup]
-  rfl
+  rw [iSup, Monotone.map_csSup_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iSup,
+    comp_def]
+
+@[deprecated (since := "2024-08-26")] alias Monotone.map_iSup_of_continuousAt' :=
+  Monotone.map_ciSup_of_continuousAt
 
 /-- A monotone function continuous at the infimum of a nonempty set sends this infimum to
 the infimum of the image of this set. -/
-theorem Monotone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
+theorem MonotoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
+    (Cf : ContinuousWithinAt f A (sInf A))
+    (Mf : MonotoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
+    f (sInf A) = sInf (f '' A) :=
+  MonotoneOn.map_csSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd
+
+/-- A monotone function continuous at the infimum of a nonempty set sends this infimum to
+the infimum of the image of this set. -/
+theorem Monotone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
     (Mf : Monotone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
     f (sInf A) = sInf (f '' A) :=
-  Monotone.map_sSup_of_continuousAt' (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd
+  Monotone.map_csSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual A_nonemp A_bdd
+
+@[deprecated (since := "2024-08-26")] alias Monotone.map_sInf_of_continuousAt' :=
+  Monotone.map_csInf_of_continuousAt
 
 /-- A monotone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
 infimum to the indexed infimum of the composition. -/
-theorem Monotone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
+theorem Monotone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
     (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f)
     (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨅ i, f (g i) := by
-  rw [iInf, Monotone.map_sInf_of_continuousAt' Cf Mf (range_nonempty g) bdd, ← range_comp, iInf]
-  rfl
+  rw [iInf, Monotone.map_csInf_of_continuousAt Cf Mf (range_nonempty g) bdd, ← range_comp, iInf,
+    comp_def]
+
+@[deprecated (since := "2024-08-26")] alias Monotone.map_iInf_of_continuousAt' :=
+  Monotone.map_ciInf_of_continuousAt
+
+/-- An antitone function continuous at the infimum of a nonempty set sends this infimum to
+the supremum of the image of this set. -/
+theorem AntitoneOn.map_csInf_of_continuousWithinAt {f : α → β} {A : Set α}
+    (Cf : ContinuousWithinAt f A (sInf A))
+    (Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
+    f (sInf A) = sSup (f '' A) :=
+  MonotoneOn.map_csInf_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
 
 /-- An antitone function continuous at the infimum of a nonempty set sends this infimum to
 the supremum of the image of this set. -/
-theorem Antitone.map_sInf_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
+theorem Antitone.map_csInf_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sInf A))
     (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddBelow A := by bddDefault) :
     f (sInf A) = sSup (f '' A) :=
-  Monotone.map_sInf_of_continuousAt' (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
+  Monotone.map_csInf_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
+
+@[deprecated (since := "2024-08-26")] alias Antitone.map_sInf_of_continuousAt' :=
+  Antitone.map_csInf_of_continuousAt
 
 /-- An antitone function continuous at the indexed infimum over a nonempty `Sort` sends this indexed
 infimum to the indexed supremum of the composition. -/
-theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
+theorem Antitone.map_ciInf_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
     (Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
     (bdd : BddBelow (range g) := by bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) := by
-  rw [iInf, Antitone.map_sInf_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iSup]
-  rfl
+  rw [iInf, Antitone.map_csInf_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iSup,
+    comp_def]
+
+@[deprecated (since := "2024-08-26")] alias Antitone.map_iInf_of_continuousAt' :=
+  Antitone.map_ciInf_of_continuousAt
+
+/-- An antitone function continuous at the supremum of a nonempty set sends this supremum to
+the infimum of the image of this set. -/
+theorem AntitoneOn.map_csSup_of_continuousWithinAt {f : α → β} {A : Set α}
+    (Cf : ContinuousWithinAt f A (sSup A))
+    (Af : AntitoneOn f A) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
+    f (sSup A) = sInf (f '' A) :=
+  MonotoneOn.map_csSup_of_continuousWithinAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
 
 /-- An antitone function continuous at the supremum of a nonempty set sends this supremum to
 the infimum of the image of this set. -/
-theorem Antitone.map_sSup_of_continuousAt' {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
+theorem Antitone.map_csSup_of_continuousAt {f : α → β} {A : Set α} (Cf : ContinuousAt f (sSup A))
     (Af : Antitone f) (A_nonemp : A.Nonempty) (A_bdd : BddAbove A := by bddDefault) :
     f (sSup A) = sInf (f '' A) :=
-  Monotone.map_sSup_of_continuousAt' (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
+  Monotone.map_csSup_of_continuousAt (β := βᵒᵈ) Cf Af.dual_right A_nonemp A_bdd
+
+@[deprecated (since := "2024-08-26")] alias Antitone.map_sSup_of_continuousAt' :=
+  Antitone.map_csSup_of_continuousAt
 
 /-- An antitone function continuous at the indexed supremum over a nonempty `Sort` sends this
 indexed supremum to the indexed infimum of the composition. -/
-theorem Antitone.map_iSup_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
+theorem Antitone.map_ciSup_of_continuousAt {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
     (Cf : ContinuousAt f (iSup g)) (Af : Antitone f)
     (bdd : BddAbove (range g) := by bddDefault) : f (⨆ i, g i) = ⨅ i, f (g i) := by
-  rw [iSup, Antitone.map_sSup_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iInf]
-  rfl
+  rw [iSup, Antitone.map_csSup_of_continuousAt Cf Af (range_nonempty g) bdd, ← range_comp, iInf,
+    comp_def]
+
+@[deprecated (since := "2024-08-26")] alias Antitone.map_iSup_of_continuousAt' :=
+  Antitone.map_ciSup_of_continuousAt
 
 end ConditionallyCompleteLinearOrder
 
@@ -109,18 +166,32 @@ theorem IsClosed.sInf_mem {s : Set α} (hs : s.Nonempty) (hc : IsClosed s) : sIn
 
 /-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
 this supremum to the supremum of the image of this set. -/
-theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
-    (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) := by
+theorem MonotoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
+    (Cf : ContinuousWithinAt f s (sSup s))
+    (Mf : MonotoneOn f s) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) := by
   rcases s.eq_empty_or_nonempty with h | h
   · simp [h, fbot]
-  · exact Mf.map_sSup_of_continuousAt' Cf h
+  · exact Mf.map_csSup_of_continuousWithinAt Cf h
+
+/-- A monotone function `f` sending `bot` to `bot` and continuous at the supremum of a set sends
+this supremum to the supremum of the image of this set. -/
+theorem Monotone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
+    (Mf : Monotone f) (fbot : f ⊥ = ⊥) : f (sSup s) = sSup (f '' s) :=
+  MonotoneOn.map_sSup_of_continuousWithinAt Cf.continuousWithinAt (Mf.monotoneOn _) fbot
 
 /-- If a monotone function sending `bot` to `bot` is continuous at the indexed supremum over
 a `Sort`, then it sends this indexed supremum to the indexed supremum of the composition. -/
 theorem Monotone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι → α}
     (Cf : ContinuousAt f (iSup g)) (Mf : Monotone f) (fbot : f ⊥ = ⊥) :
     f (⨆ i, g i) = ⨆ i, f (g i) := by
-  rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup]; rfl
+  rw [iSup, Mf.map_sSup_of_continuousAt Cf fbot, ← range_comp, iSup, comp_def]
+
+/-- A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
+this infimum to the infimum of the image of this set. -/
+theorem MonotoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
+    (Cf : ContinuousWithinAt f s (sInf s)) (Mf : MonotoneOn f s) (ftop : f ⊤ = ⊤) :
+    f (sInf s) = sInf (f '' s) :=
+  MonotoneOn.map_sSup_of_continuousWithinAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop
 
 /-- A monotone function `f` sending `top` to `top` and continuous at the infimum of a set sends
 this infimum to the infimum of the image of this set. -/
@@ -134,6 +205,14 @@ theorem Monotone.map_iInf_of_continuousAt {ι : Sort*} {f : α → β} {g : ι 
     (Cf : ContinuousAt f (iInf g)) (Mf : Monotone f) (ftop : f ⊤ = ⊤) : f (iInf g) = iInf (f ∘ g) :=
   Monotone.map_iSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ftop
 
+/-- An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
+this supremum to the infimum of the image of this set. -/
+theorem AntitoneOn.map_sSup_of_continuousWithinAt {f : α → β} {s : Set α}
+    (Cf : ContinuousWithinAt f s (sSup s)) (Af : AntitoneOn f s) (fbot : f ⊥ = ⊤) :
+    f (sSup s) = sInf (f '' s) :=
+  MonotoneOn.map_sSup_of_continuousWithinAt
+    (show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sSup s) from Cf) Af fbot
+
 /-- An antitone function `f` sending `bot` to `top` and continuous at the supremum of a set sends
 this supremum to the infimum of the image of this set. -/
 theorem Antitone.map_sSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
@@ -149,6 +228,14 @@ theorem Antitone.map_iSup_of_continuousAt {ι : Sort*} {f : α → β} {g : ι 
   Monotone.map_iSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (iSup g) from Cf) Af
     fbot
 
+/-- An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
+this infimum to the supremum of the image of this set. -/
+theorem AntitoneOn.map_sInf_of_continuousWithinAt {f : α → β} {s : Set α}
+    (Cf : ContinuousWithinAt f s (sInf s)) (Af : AntitoneOn f s) (ftop : f ⊤ = ⊥) :
+    f (sInf s) = sSup (f '' s) :=
+  MonotoneOn.map_sInf_of_continuousWithinAt
+    (show ContinuousWithinAt (OrderDual.toDual ∘ f) s (sInf s) from Cf) Af ftop
+
 /-- An antitone function `f` sending `top` to `bot` and continuous at the infimum of a set sends
 this infimum to the supremum of the image of this set. -/
 theorem Antitone.map_sInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
@@ -192,60 +279,6 @@ theorem IsClosed.isGreatest_csSup {s : Set α} (hc : IsClosed s) (hs : s.Nonempt
     IsGreatest s (sSup s) :=
   IsClosed.isLeast_csInf (α := αᵒᵈ) hc hs B
 
-/-- If a monotone function is continuous at the supremum of a nonempty bounded above set `s`,
-then it sends this supremum to the supremum of the image of `s`. -/
-theorem Monotone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
-    (Mf : Monotone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sSup (f '' s) := by
-  refine ((isLUB_csSup (ne.image f) (Mf.map_bddAbove H)).unique ?_).symm
-  refine (isLUB_csSup ne H).isLUB_of_tendsto (fun x _ y _ xy => Mf xy) ne ?_
-  exact Cf.mono_left inf_le_left
-
-/-- If a monotone function is continuous at the indexed supremum of a bounded function on
-a nonempty `Sort`, then it sends this supremum to the supremum of the composition. -/
-theorem Monotone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
-    (Mf : Monotone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨆ i, f (g i) := by
-  rw [iSup, Mf.map_csSup_of_continuousAt Cf (range_nonempty _) H, ← range_comp, iSup]; rfl
-
-/-- If a monotone function is continuous at the infimum of a nonempty bounded below set `s`,
-then it sends this infimum to the infimum of the image of `s`. -/
-theorem Monotone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
-    (Mf : Monotone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sInf (f '' s) :=
-  Monotone.map_csSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual ne H
-
-/-- A continuous monotone function sends indexed infimum to indexed infimum in conditionally
-complete linear order, under a boundedness assumption. -/
-theorem Monotone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
-    (Mf : Monotone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨅ i, f (g i) :=
-  Monotone.map_ciSup_of_continuousAt (α := αᵒᵈ) (β := βᵒᵈ) Cf Mf.dual H
-
-/-- If an antitone function is continuous at the supremum of a nonempty bounded above set `s`,
-then it sends this supremum to the infimum of the image of `s`. -/
-theorem Antitone.map_csSup_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sSup s))
-    (Af : Antitone f) (ne : s.Nonempty) (H : BddAbove s) : f (sSup s) = sInf (f '' s) :=
-  Monotone.map_csSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sSup s) from Cf) Af
-    ne H
-
-/-- If an antitone function is continuous at the indexed supremum of a bounded function on
-a nonempty `Sort`, then it sends this supremum to the infimum of the composition. -/
-theorem Antitone.map_ciSup_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨆ i, g i))
-    (Af : Antitone f) (H : BddAbove (range g)) : f (⨆ i, g i) = ⨅ i, f (g i) :=
-  Monotone.map_ciSup_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (⨆ i, g i) from Cf)
-    Af H
-
-/-- If an antitone function is continuous at the infimum of a nonempty bounded below set `s`,
-then it sends this infimum to the supremum of the image of `s`. -/
-theorem Antitone.map_csInf_of_continuousAt {f : α → β} {s : Set α} (Cf : ContinuousAt f (sInf s))
-    (Af : Antitone f) (ne : s.Nonempty) (H : BddBelow s) : f (sInf s) = sSup (f '' s) :=
-  Monotone.map_csInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (sInf s) from Cf) Af
-    ne H
-
-/-- A continuous antitone function sends indexed infimum to indexed supremum in conditionally
-complete linear order, under a boundedness assumption. -/
-theorem Antitone.map_ciInf_of_continuousAt {f : α → β} {g : γ → α} (Cf : ContinuousAt f (⨅ i, g i))
-    (Af : Antitone f) (H : BddBelow (range g)) : f (⨅ i, g i) = ⨆ i, f (g i) :=
-  Monotone.map_ciInf_of_continuousAt (show ContinuousAt (OrderDual.toDual ∘ f) (⨅ i, g i) from Cf)
-    Af H
-
 lemma MonotoneOn.tendsto_nhdsWithin_Ioo_left {α β : Type*} [LinearOrder α] [TopologicalSpace α]
     [OrderTopology α] [ConditionallyCompleteLinearOrder β] [TopologicalSpace β] [OrderTopology β]
     {f : α → β} {x y : α} (h_nonempty : (Ioo y x).Nonempty) (Mf : MonotoneOn f (Ioo y x))
diff --git a/Mathlib/Topology/Order/OrderClosed.lean b/Mathlib/Topology/Order/OrderClosed.lean
index 7a476daeaf513..e2763d592bc24 100644
--- a/Mathlib/Topology/Order/OrderClosed.lean
+++ b/Mathlib/Topology/Order/OrderClosed.lean
@@ -543,7 +543,7 @@ namespace Subtype
 -- todo: add `OrderEmbedding.orderClosedTopology`
 instance {p : α → Prop} : OrderClosedTopology (Subtype p) :=
   have this : Continuous fun p : Subtype p × Subtype p => ((p.fst : α), (p.snd : α)) :=
-    continuous_subtype_val.prod_map continuous_subtype_val
+    continuous_subtype_val.prodMap continuous_subtype_val
   OrderClosedTopology.mk (t.isClosed_le'.preimage this)
 
 end Subtype
diff --git a/Mathlib/Topology/Order/ScottTopology.lean b/Mathlib/Topology/Order/ScottTopology.lean
index 8b5a06371adf8..6f26fd5939149 100644
--- a/Mathlib/Topology/Order/ScottTopology.lean
+++ b/Mathlib/Topology/Order/ScottTopology.lean
@@ -3,6 +3,7 @@ Copyright (c) 2023 Christopher Hoskin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
 -/
+import Mathlib.Order.ScottContinuity
 import Mathlib.Topology.Order.UpperLowerSetTopology
 
 /-!
@@ -333,8 +334,19 @@ lemma scott_eq_upper_of_completeLinearOrder : scott α = upper α := by
   letI := scott α
   rw [@isOpen_iff_Iic_compl_or_univ _ _ (scott α) ({ topology_eq_scott := rfl }) U]
 
+/- The upper topology on a complete linear order is the Scott topology -/
+instance [TopologicalSpace α] [IsUpper α] : IsScott α where
+  topology_eq_scott := by
+    rw [scott_eq_upper_of_completeLinearOrder]
+    exact IsUpper.topology_eq α
+
 end CompleteLinearOrder
 
+lemma isOpen_iff_scottContinuous_mem [Preorder α] {s : Set α} [TopologicalSpace α] [IsScott α] :
+    IsOpen s ↔ ScottContinuous fun x ↦ x ∈ s := by
+  rw [scottContinuous_iff_continuous]
+  exact isOpen_iff_continuous_mem
+
 end IsScott
 
 /--
diff --git a/Mathlib/Topology/Order/UpperLowerSetTopology.lean b/Mathlib/Topology/Order/UpperLowerSetTopology.lean
index ff3fc1e7a3de2..02681dbd9f86d 100644
--- a/Mathlib/Topology/Order/UpperLowerSetTopology.lean
+++ b/Mathlib/Topology/Order/UpperLowerSetTopology.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Christopher Hoskin
 -/
 import Mathlib.Topology.AlexandrovDiscrete
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Topology.Order.LowerUpperTopology
 
 /-!
@@ -74,10 +74,10 @@ def WithUpperSet (α : Type*) := α
 
 namespace WithUpperSet
 
-/-- `toUpperSet` is the identity function to the `WithUpperSet` of a type.  -/
+/-- `toUpperSet` is the identity function to the `WithUpperSet` of a type. -/
 @[match_pattern] def toUpperSet : α ≃ WithUpperSet α := Equiv.refl _
 
-/-- `ofUpperSet` is the identity function from the `WithUpperSet` of a type.  -/
+/-- `ofUpperSet` is the identity function from the `WithUpperSet` of a type. -/
 @[match_pattern] def ofUpperSet : WithUpperSet α ≃ α := Equiv.refl _
 
 @[simp] lemma to_WithUpperSet_symm_eq : (@toUpperSet α).symm = ofUpperSet := rfl
@@ -120,10 +120,10 @@ def WithLowerSet (α : Type*) := α
 
 namespace WithLowerSet
 
-/-- `toLowerSet` is the identity function to the `WithLowerSet` of a type.  -/
+/-- `toLowerSet` is the identity function to the `WithLowerSet` of a type. -/
 @[match_pattern] def toLowerSet : α ≃ WithLowerSet α := Equiv.refl _
 
-/-- `ofLowerSet` is the identity function from the `WithLowerSet` of a type.  -/
+/-- `ofLowerSet` is the identity function from the `WithLowerSet` of a type. -/
 @[match_pattern] def ofLowerSet : WithLowerSet α ≃ α := Equiv.refl _
 
 @[simp] lemma to_WithLowerSet_symm_eq : (@toLowerSet α).symm = ofLowerSet := rfl
diff --git a/Mathlib/Topology/PartialHomeomorph.lean b/Mathlib/Topology/PartialHomeomorph.lean
index 3c930a28e8556..98e7d2e0d5f74 100644
--- a/Mathlib/Topology/PartialHomeomorph.lean
+++ b/Mathlib/Topology/PartialHomeomorph.lean
@@ -361,7 +361,7 @@ theorem eventually_nhdsWithin' {x : X} (p : X → Prop) {s : Set X}
 
 /-- This lemma is useful in the manifold library in the case that `e` is a chart. It states that
   locally around `e x` the set `e.symm ⁻¹' s` is the same as the set intersected with the target
-  of `e` and some other neighborhood of `f x` (which will be the source of a chart on `Z`).  -/
+  of `e` and some other neighborhood of `f x` (which will be the source of a chart on `Z`). -/
 theorem preimage_eventuallyEq_target_inter_preimage_inter {e : PartialHomeomorph X Y} {s : Set X}
     {t : Set Z} {x : X} {f : X → Z} (hf : ContinuousWithinAt f s x) (hxe : x ∈ e.source)
     (ht : t ∈ 𝓝 (f x)) :
@@ -370,7 +370,7 @@ theorem preimage_eventuallyEq_target_inter_preimage_inter {e : PartialHomeomorph
   filter_upwards [e.open_source.mem_nhds hxe,
     mem_nhdsWithin_iff_eventually.mp (hf.preimage_mem_nhdsWithin ht)]
   intro y hy hyu
-  simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.mapsTo hy, true_and_iff, iff_self_and,
+  simp_rw [mem_inter_iff, mem_preimage, mem_inter_iff, e.mapsTo hy, true_and, iff_self_and,
     e.left_inv hy, iff_true_intro hyu]
 
 theorem isOpen_inter_preimage {s : Set Y} (hs : IsOpen s) : IsOpen (e.source ∩ e ⁻¹' s) :=
@@ -1002,7 +1002,7 @@ theorem continuousOn_iff_continuousOn_comp_right {f : Y → Z} {s : Set Y} (h :
 
 /-- Continuity within a set at a point can be read under left composition with a local
 homeomorphism if a neighborhood of the initial point is sent to the source of the local
-homeomorphism-/
+homeomorphism -/
 theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : Set Z} {x : Z}
     (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) :
     ContinuousWithinAt f s x ↔ ContinuousWithinAt (e ∘ f) s x := by
@@ -1015,7 +1015,7 @@ theorem continuousWithinAt_iff_continuousWithinAt_comp_left {f : Z → X} {s : S
   exact this.congr (fun y hy => by simp [e.left_inv hy.2]) (by simp [e.left_inv hx])
 
 /-- Continuity at a point can be read under left composition with a partial homeomorphism if a
-neighborhood of the initial point is sent to the source of the partial homeomorphism-/
+neighborhood of the initial point is sent to the source of the partial homeomorphism -/
 theorem continuousAt_iff_continuousAt_comp_left {f : Z → X} {x : Z} (h : f ⁻¹' e.source ∈ 𝓝 x) :
     ContinuousAt f x ↔ ContinuousAt (e ∘ f) x := by
   have hx : f x ∈ e.source := (mem_of_mem_nhds h : _)
@@ -1069,7 +1069,7 @@ theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) :
   lift x to e.source using hx
   rw [← e.open_source.nhdsWithin_eq x.2, ← map_nhds_subtype_val, ← map_comap_setCoe_val,
     e.toHomeomorphSourceTarget.nhds_eq_comap, nhds_subtype_eq_comap]
-  simp only [(· ∘ ·), toHomeomorphSourceTarget_apply_coe, comap_comap]
+  simp only [Function.comp_def, toHomeomorphSourceTarget_apply_coe, comap_comap]
 
 /-- If a partial homeomorphism has source and target equal to univ, then it induces a homeomorphism
 between the whole spaces, expressed in this definition. -/
diff --git a/Mathlib/Topology/PartitionOfUnity.lean b/Mathlib/Topology/PartitionOfUnity.lean
index a98ad48c12803..661a124cbc7d4 100644
--- a/Mathlib/Topology/PartitionOfUnity.lean
+++ b/Mathlib/Topology/PartitionOfUnity.lean
@@ -4,8 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.Algebra.BigOperators.Finprod
-import Mathlib.SetTheory.Ordinal.Basic
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.ShrinkingLemma
 import Mathlib.Topology.UrysohnsLemma
@@ -140,7 +139,7 @@ variable {E : Type*} [AddCommMonoid E] [SMulWithZero ℝ E] [TopologicalSpace E]
 
 instance : FunLike (PartitionOfUnity ι X s) ι C(X, ℝ) where
   coe := toFun
-  coe_injective' := fun f g h ↦ by cases f; cases g; congr
+  coe_injective' f g h := by cases f; cases g; congr
 
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locallyFinite'
@@ -312,7 +311,7 @@ variable {s : Set X} (f : BumpCovering ι X s)
 
 instance : FunLike (BumpCovering ι X s) ι C(X, ℝ) where
   coe := toFun
-  coe_injective' := fun f g h ↦ by cases f; cases g; congr
+  coe_injective' f g h := by cases f; cases g; congr
 
 protected theorem locallyFinite : LocallyFinite fun i => support (f i) :=
   f.locallyFinite'
diff --git a/Mathlib/Topology/Perfect.lean b/Mathlib/Topology/Perfect.lean
index 31622ec28846e..c390399234943 100644
--- a/Mathlib/Topology/Perfect.lean
+++ b/Mathlib/Topology/Perfect.lean
@@ -93,7 +93,7 @@ A topological space `X` is said to be perfect if its universe is a perfect set.
 Equivalently, this means that `𝓝[≠] x ≠ ⊥` for every point `x : X`.
 -/
 @[mk_iff perfectSpace_def]
-class PerfectSpace : Prop :=
+class PerfectSpace : Prop where
   univ_preperfect : Preperfect (Set.univ : Set α)
 
 theorem PerfectSpace.univ_perfect [PerfectSpace α] : Perfect (Set.univ : Set α) :=
diff --git a/Mathlib/Topology/PreorderRestrict.lean b/Mathlib/Topology/PreorderRestrict.lean
new file mode 100644
index 0000000000000..9d833618bfd40
--- /dev/null
+++ b/Mathlib/Topology/PreorderRestrict.lean
@@ -0,0 +1,39 @@
+/-
+Copyright (c) 2024 Etienne Marion. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Etienne Marion
+-/
+import Mathlib.Order.Restriction
+import Mathlib.Topology.Constructions
+
+/-!
+# Continuity of the restriction function for functions indexed by a preorder
+
+We prove that the map which restricts a function `f : (i : α) → X i` to elements `≤ a` is
+continuous.
+-/
+
+namespace Preorder
+
+variable {α : Type*} [Preorder α] {X : α  → Type*} [∀ i, TopologicalSpace (X i)]
+
+@[continuity, fun_prop]
+theorem continuous_restrictLe (a : α) : Continuous (restrictLe (π := X) a) :=
+  Pi.continuous_restrict _
+
+@[continuity, fun_prop]
+theorem continuous_restrictLe₂ {a b : α} (hab : a ≤ b) : Continuous (restrictLe₂ (π := X) hab) :=
+  Pi.continuous_restrict₂ _
+
+variable [LocallyFiniteOrderBot α]
+
+@[continuity, fun_prop]
+theorem continuous_frestrictLe (a : α) : Continuous (frestrictLe (π := X) a) :=
+  Finset.continuous_restrict _
+
+@[continuity, fun_prop]
+theorem continuous_frestrictLe₂ {a b : α} (hab : a ≤ b) :
+    Continuous (frestrictLe₂ (π := X) hab) :=
+  Finset.continuous_restrict₂ _
+
+end Preorder
diff --git a/Mathlib/Topology/RestrictGenTopology.lean b/Mathlib/Topology/RestrictGen.lean
similarity index 100%
rename from Mathlib/Topology/RestrictGenTopology.lean
rename to Mathlib/Topology/RestrictGen.lean
diff --git a/Mathlib/Topology/Separation.lean b/Mathlib/Topology/Separation.lean
index 1cc16c0294650..c2de323af42d9 100644
--- a/Mathlib/Topology/Separation.lean
+++ b/Mathlib/Topology/Separation.lean
@@ -259,7 +259,7 @@ end SeparatedNhds
 
 /-- A T₀ space, also known as a Kolmogorov space, is a topological space such that for every pair
 `x ≠ y`, there is an open set containing one but not the other. We formulate the definition in terms
-of the `Inseparable` relation.  -/
+of the `Inseparable` relation. -/
 class T0Space (X : Type u) [TopologicalSpace X] : Prop where
   /-- Two inseparable points in a T₀ space are equal. -/
   t0 : ∀ ⦃x y : X⦄, Inseparable x y → x = y
@@ -576,30 +576,28 @@ theorem t1Space_TFAE (X : Type u) [TopologicalSpace X] :
       ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y),
       ∀ ⦃x y : X⦄, x ≠ y → Disjoint (pure x) (𝓝 y),
       ∀ ⦃x y : X⦄, x ⤳ y → x = y] := by
-  tfae_have 1 ↔ 2
-  · exact ⟨fun h => h.1, fun h => ⟨h⟩⟩
-  tfae_have 2 ↔ 3
-  · simp only [isOpen_compl_iff]
-  tfae_have 5 ↔ 3
-  · refine forall_swap.trans ?_
+  tfae_have 1 ↔ 2 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
+  tfae_have 2 ↔ 3 := by
+    simp only [isOpen_compl_iff]
+  tfae_have 5 ↔ 3 := by
+    refine forall_swap.trans ?_
     simp only [isOpen_iff_mem_nhds, mem_compl_iff, mem_singleton_iff]
-  tfae_have 5 ↔ 6
-  · simp only [← subset_compl_singleton_iff, exists_mem_subset_iff]
-  tfae_have 5 ↔ 7
-  · simp only [(nhds_basis_opens _).mem_iff, subset_compl_singleton_iff, exists_prop, and_assoc,
+  tfae_have 5 ↔ 6 := by
+    simp only [← subset_compl_singleton_iff, exists_mem_subset_iff]
+  tfae_have 5 ↔ 7 := by
+    simp only [(nhds_basis_opens _).mem_iff, subset_compl_singleton_iff, exists_prop, and_assoc,
       and_left_comm]
-  tfae_have 5 ↔ 8
-  · simp only [← principal_singleton, disjoint_principal_right]
-  tfae_have 8 ↔ 9
-  · exact forall_swap.trans (by simp only [disjoint_comm, ne_comm])
-  tfae_have 1 → 4
-  · simp only [continuous_def, CofiniteTopology.isOpen_iff']
+  tfae_have 5 ↔ 8 := by
+    simp only [← principal_singleton, disjoint_principal_right]
+  tfae_have 8 ↔ 9 := forall_swap.trans (by simp only [disjoint_comm, ne_comm])
+  tfae_have 1 → 4 := by
+    simp only [continuous_def, CofiniteTopology.isOpen_iff']
     rintro H s (rfl | hs)
     exacts [isOpen_empty, compl_compl s ▸ (@Set.Finite.isClosed _ _ H _ hs).isOpen_compl]
-  tfae_have 4 → 2
-  · exact fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).preimage h
-  tfae_have 2 ↔ 10
-  · simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
+  tfae_have 4 → 2 :=
+    fun h x => (CofiniteTopology.isClosed_iff.2 <| Or.inr (finite_singleton _)).preimage h
+  tfae_have 2 ↔ 10 := by
+    simp only [← closure_subset_iff_isClosed, specializes_iff_mem_closure, subset_def,
       mem_singleton_iff, eq_comm]
   tfae_finish
 
@@ -974,13 +972,13 @@ theorem Filter.HasBasis.exists_inter_eq_singleton_of_mem_discrete {ι : Type*} {
   exact ⟨i, hi, hix.antisymm <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩
 
 /-- A point `x` in a discrete subset `s` of a topological space admits a neighbourhood
-that only meets `s` at `x`.  -/
+that only meets `s` at `x`. -/
 theorem nhds_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
     (hx : x ∈ s) : ∃ U ∈ 𝓝 x, U ∩ s = {x} := by
   simpa using (𝓝 x).basis_sets.exists_inter_eq_singleton_of_mem_discrete hx
 
 /-- Let `x` be a point in a discrete subset `s` of a topological space, then there exists an open
-set that only meets `s` at `x`.  -/
+set that only meets `s` at `x`. -/
 theorem isOpen_inter_eq_singleton_of_mem_discrete {s : Set X} [DiscreteTopology s] {x : X}
     (hx : x ∈ s) : ∃ U : Set X, IsOpen U ∧ U ∩ s = {x} := by
   obtain ⟨U, hU_nhds, hU_inter⟩ := nhds_inter_eq_singleton_of_mem_discrete hx
@@ -1045,6 +1043,17 @@ theorem r1Space_iff_inseparable_or_disjoint_nhds {X : Type*} [TopologicalSpace X
   ⟨fun _h x y ↦ (specializes_or_disjoint_nhds x y).imp_left Specializes.inseparable, fun h ↦
     ⟨fun x y ↦ (h x y).imp_left Inseparable.specializes⟩⟩
 
+theorem Inseparable.of_nhds_neBot {x y : X} (h : NeBot (𝓝 x ⊓ 𝓝 y)) :
+    Inseparable x y :=
+  (r1Space_iff_inseparable_or_disjoint_nhds.mp ‹_› _ _).resolve_right fun h' => h.ne h'.eq_bot
+
+/-- Limits are unique up to separability.
+
+A weaker version of `tendsto_nhds_unique` for `R1Space`. -/
+theorem tendsto_nhds_unique_inseparable {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
+    (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : Inseparable a b :=
+  .of_nhds_neBot <| neBot_of_le <| le_inf ha hb
+
 theorem isClosed_setOf_specializes : IsClosed { p : X × X | p.1 ⤳ p.2 } := by
   simp only [← isOpen_compl_iff, compl_setOf, ← disjoint_nhds_nhds_iff_not_specializes,
     isOpen_setOf_disjoint_nhds_nhds]
@@ -1103,7 +1112,7 @@ theorem exists_isCompact_superset_iff {s : Set X} :
 alias exists_compact_superset_iff := exists_isCompact_superset_iff
 
 /-- If `K` and `L` are disjoint compact sets in an R₁ topological space
-and `L` is also closed, then `K` and `L` have disjoint neighborhoods.  -/
+and `L` is also closed, then `K` and `L` have disjoint neighborhoods. -/
 theorem SeparatedNhds.of_isCompact_isCompact_isClosed {K L : Set X} (hK : IsCompact K)
     (hL : IsCompact L) (h'L : IsClosed L) (hd : Disjoint K L) : SeparatedNhds K L := by
   simp_rw [separatedNhds_iff_disjoint, hK.disjoint_nhdsSet_left, hL.disjoint_nhdsSet_right,
@@ -1382,11 +1391,11 @@ theorem isClosed_diagonal [T2Space X] : IsClosed (diagonal X) :=
 
 theorem tendsto_nhds_unique [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} [NeBot l]
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
-  eq_of_nhds_neBot <| neBot_of_le <| le_inf ha hb
+  (tendsto_nhds_unique_inseparable ha hb).eq
 
 theorem tendsto_nhds_unique' [T2Space X] {f : Y → X} {l : Filter Y} {a b : X} (_ : NeBot l)
     (ha : Tendsto f l (𝓝 a)) (hb : Tendsto f l (𝓝 b)) : a = b :=
-  eq_of_nhds_neBot <| neBot_of_le <| le_inf ha hb
+  tendsto_nhds_unique ha hb
 
 theorem tendsto_nhds_unique_of_eventuallyEq [T2Space X] {f g : Y → X} {l : Filter Y} {a b : X}
     [NeBot l] (ha : Tendsto f l (𝓝 a)) (hb : Tendsto g l (𝓝 b)) (hfg : f =ᶠ[l] g) : a = b :=
@@ -1442,7 +1451,7 @@ theorem Set.InjOn.exists_mem_nhdsSet {X Y : Type*} [TopologicalSpace X] [Topolog
     · rcases loc x hx with ⟨u, hu, hf⟩
       exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
     · suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun _ hne h ↦ absurd h hne
-      refine (fc x hx).prod_map' (fc y hy) <| isClosed_diagonal.isOpen_compl.mem_nhds ?_
+      refine (fc x hx).prodMap' (fc y hy) <| isClosed_diagonal.isOpen_compl.mem_nhds ?_
       exact inj.ne hx hy hne
   rw [← eventually_nhdsSet_iff_forall, sc.nhdsSet_prod_eq sc] at this
   exact eventually_prod_self_iff.1 this
@@ -1880,7 +1889,7 @@ of filters `𝓝ˢ s` and `𝓝 a`. -/
 @[mk_iff]
 class RegularSpace (X : Type u) [TopologicalSpace X] : Prop where
   /-- If `a` is a point that does not belong to a closed set `s`, then `a` and `s` admit disjoint
-  neighborhoods.  -/
+  neighborhoods. -/
   regular : ∀ {s : Set X} {a}, IsClosed s → a ∉ s → Disjoint (𝓝ˢ s) (𝓝 a)
 
 theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
@@ -1890,30 +1899,28 @@ theorem regularSpace_TFAE (X : Type u) [TopologicalSpace X] :
       ∀ (x : X) (s : Set X), s ∈ 𝓝 x → ∃ t ∈ 𝓝 x, IsClosed t ∧ t ⊆ s,
       ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x,
       ∀ x : X , (𝓝 x).lift' closure = 𝓝 x] := by
-  tfae_have 1 ↔ 5
-  · rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
+  tfae_have 1 ↔ 5 := by
+    rw [regularSpace_iff, (@compl_surjective (Set X) _).forall, forall_swap]
     simp only [isClosed_compl_iff, mem_compl_iff, Classical.not_not, @and_comm (_ ∈ _),
       (nhds_basis_opens _).lift'_closure.le_basis_iff (nhds_basis_opens _), and_imp,
       (nhds_basis_opens _).disjoint_iff_right, exists_prop, ← subset_interior_iff_mem_nhdsSet,
       interior_compl, compl_subset_compl]
-  tfae_have 5 → 6
-  · exact fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
+  tfae_have 5 → 6 := fun h a => (h a).antisymm (𝓝 _).le_lift'_closure
   tfae_have 6 → 4
-  · intro H a s hs
+  | H, a, s, hs => by
     rw [← H] at hs
     rcases (𝓝 a).basis_sets.lift'_closure.mem_iff.mp hs with ⟨U, hU, hUs⟩
     exact ⟨closure U, mem_of_superset hU subset_closure, isClosed_closure, hUs⟩
   tfae_have 4 → 2
-  · intro H s a ha
+  | H, s, a, ha => by
     have ha' : sᶜ ∈ 𝓝 a := by rwa [← mem_interior_iff_mem_nhds, interior_compl]
     rcases H _ _ ha' with ⟨U, hU, hUc, hUs⟩
     refine disjoint_of_disjoint_of_mem disjoint_compl_left ?_ hU
     rwa [← subset_interior_iff_mem_nhdsSet, hUc.isOpen_compl.interior_eq, subset_compl_comm]
-  tfae_have 2 → 3
-  · refine fun H a s => ⟨fun hd has => mem_closure_iff_nhds_ne_bot.mp has ?_, H s a⟩
+  tfae_have 2 → 3 := by
+    refine fun H a s => ⟨fun hd has => mem_closure_iff_nhds_ne_bot.mp has ?_, H s a⟩
     exact (hd.symm.mono_right <| @principal_le_nhdsSet _ _ s).eq_bot
-  tfae_have 3 → 1
-  · exact fun H => ⟨fun hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩
+  tfae_have 3 → 1 := fun H => ⟨fun hs ha => (H _ _).mpr <| hs.closure_eq.symm ▸ ha⟩
   tfae_finish
 
 theorem RegularSpace.of_lift'_closure_le (h : ∀ x : X, (𝓝 x).lift' closure ≤ 𝓝 x) :
@@ -1945,6 +1952,7 @@ alias RegularSpace.ofExistsMemNhdsIsClosedSubset := RegularSpace.of_exists_mem_n
 instance (priority := 100) [WeaklyLocallyCompactSpace X] [R1Space X] : RegularSpace X :=
   .of_hasBasis isCompact_isClosed_basis_nhds fun _ _ ⟨_, _, h⟩ ↦ h
 
+section
 variable [RegularSpace X] {x : X} {s : Set X}
 
 theorem disjoint_nhdsSet_nhds : Disjoint (𝓝ˢ s) (𝓝 x) ↔ x ∉ closure s := by
@@ -2057,9 +2065,11 @@ lemma SeparatedNhds.of_isCompact_isClosed {s t : Set X}
 @[deprecated (since := "2024-01-28")]
 alias separatedNhds_of_isCompact_isClosed := SeparatedNhds.of_isCompact_isClosed
 
+end
+
 /-- This technique to witness `HasSeparatingCover` in regular Lindelöf topological spaces
 will be used to prove regular Lindelöf spaces are normal. -/
-lemma IsClosed.HasSeparatingCover {s t : Set X} [LindelofSpace X]
+lemma IsClosed.HasSeparatingCover {s t : Set X} [LindelofSpace X] [RegularSpace X]
     (s_cl : IsClosed s) (t_cl : IsClosed t) (st_dis : Disjoint s t) : HasSeparatingCover s t := by
   -- `IsLindelof.indexed_countable_subcover` requires the space be Nonempty
   rcases isEmpty_or_nonempty X with empty_X | nonempty_X
@@ -2168,7 +2178,7 @@ end T25
 section T3
 
 /-- A T₃ space is a T₀ space which is a regular space. Any T₃ space is a T₁ space, a T₂ space, and
-a T₂.₅ space.  -/
+a T₂.₅ space. -/
 class T3Space (X : Type u) [TopologicalSpace X] extends T0Space X, RegularSpace X : Prop
 
 instance (priority := 90) instT3Space [T0Space X] [RegularSpace X] : T3Space X := ⟨⟩
@@ -2261,6 +2271,7 @@ instance (priority := 100) NormalSpace.of_compactSpace_r1Space [CompactSpace X]
     NormalSpace X where
   normal _s _t hs ht := .of_isCompact_isCompact_isClosed hs.isCompact ht.isCompact ht
 
+set_option pp.universes true in
 /-- A regular topological space with a Lindelöf topology is a normal space. A consequence of e.g.
 Corollaries 20.8 and 20.10 of [Willard's *General Topology*][zbMATH02107988] (without the
 assumption of Hausdorff). -/
@@ -2565,7 +2576,7 @@ theorem isTopologicalBasis_isClopen : IsTopologicalBasis { s : Set X | IsClopen
   tauto
 
 /-- Every member of an open set in a compact Hausdorff totally disconnected space
-  is contained in a clopen set contained in the open set.  -/
+  is contained in a clopen set contained in the open set. -/
 theorem compact_exists_isClopen_in_isOpen {x : X} {U : Set X} (is_open : IsOpen U) (memU : x ∈ U) :
     ∃ V : Set X, IsClopen V ∧ x ∈ V ∧ V ⊆ U :=
   isTopologicalBasis_isClopen.mem_nhds_iff.1 (is_open.mem_nhds memU)
@@ -2640,3 +2651,5 @@ instance ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (Connecte
   rw [ConnectedComponents.quotientMap_coe.isClopen_preimage] at hU
   refine ⟨Vᶜ, V, hU.compl.isOpen, hU.isOpen, ?_, hb mem_connectedComponent, disjoint_compl_left⟩
   exact fun h => flip Set.Nonempty.ne_empty ha ⟨a, mem_connectedComponent, h⟩
+
+set_option linter.style.longFile 2800
diff --git a/Mathlib/Topology/Sequences.lean b/Mathlib/Topology/Sequences.lean
index c3f13a33c72e3..2e0e2de87c365 100644
--- a/Mathlib/Topology/Sequences.lean
+++ b/Mathlib/Topology/Sequences.lean
@@ -158,7 +158,7 @@ theorem Inducing.frechetUrysohnSpace [FrechetUrysohnSpace Y] {f : X → Y} (hf :
   rcases hx with ⟨u, hus, hu⟩
   choose v hv hvu using hus
   refine ⟨v, hv, ?_⟩
-  simpa only [hf.tendsto_nhds_iff, (· ∘ ·), hvu]
+  simpa only [hf.tendsto_nhds_iff, Function.comp_def, hvu]
 
 /-- Subtype of a Fréchet-Urysohn space is a Fréchet-Urysohn space. -/
 instance Subtype.instFrechetUrysohnSpace [FrechetUrysohnSpace X] {p : X → Prop} :
@@ -328,7 +328,7 @@ protected theorem IsSeqCompact.totallyBounded (h : IsSeqCompact s) : TotallyBoun
   refine ⟨u, u_in, fun x _ φ hφ huφ => ?_⟩
   obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V :=
     huφ.cauchySeq.mem_entourage V_in
-  exact hu (φ <| N + 1) (φ N) (hφ <| lt_add_one N) (hN (N + 1) N N.le_succ le_rfl)
+  exact hu (φ <| N + 1) (φ N) (hφ <| Nat.lt_add_one N) (hN (N + 1) N N.le_succ le_rfl)
 
 variable [IsCountablyGenerated (𝓤 X)]
 
diff --git a/Mathlib/Topology/Sets/Closeds.lean b/Mathlib/Topology/Sets/Closeds.lean
index 5953aae8fe8e1..3a3de1ae74dbf 100644
--- a/Mathlib/Topology/Sets/Closeds.lean
+++ b/Mathlib/Topology/Sets/Closeds.lean
@@ -63,6 +63,9 @@ theorem coe_mk (s : Set α) (h) : (mk s h : Set α) = s :=
 protected def closure (s : Set α) : Closeds α :=
   ⟨closure s, isClosed_closure⟩
 
+@[simp]
+theorem mem_closure {s : Set α} {x : α} : x ∈ Closeds.closure s ↔ x ∈ closure s := .rfl
+
 theorem gc : GaloisConnection Closeds.closure ((↑) : Closeds α → Set α) := fun _ U =>
   ⟨subset_closure.trans, fun h => closure_minimal h U.closed⟩
 
diff --git a/Mathlib/Topology/Sets/Opens.lean b/Mathlib/Topology/Sets/Opens.lean
index fad80ae1e1cf9..cf7ecc4c6353c 100644
--- a/Mathlib/Topology/Sets/Opens.lean
+++ b/Mathlib/Topology/Sets/Opens.lean
@@ -6,7 +6,7 @@ Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
 import Mathlib.Order.Hom.CompleteLattice
 import Mathlib.Topology.Bases
 import Mathlib.Topology.Homeomorph
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Order.CompactlyGenerated.Basic
 import Mathlib.Order.Copy
 
@@ -75,6 +75,9 @@ instance : SetLike (Opens α) α where
 instance : CanLift (Set α) (Opens α) (↑) IsOpen :=
   ⟨fun s h => ⟨⟨s, h⟩, rfl⟩⟩
 
+instance instSecondCountableOpens [SecondCountableTopology α] (U : Opens α) :
+    SecondCountableTopology U := inferInstanceAs (SecondCountableTopology U.1)
+
 theorem «forall» {p : Opens α → Prop} : (∀ U, p U) ↔ ∀ (U : Set α) (hU : IsOpen U), p ⟨U, hU⟩ :=
   ⟨fun h _ _ => h _, fun h _ => h _ _⟩
 
@@ -104,6 +107,9 @@ theorem ext {U V : Opens α} (h : (U : Set α) = V) : U = V :=
 theorem coe_inj {U V : Opens α} : (U : Set α) = V ↔ U = V :=
   SetLike.ext'_iff.symm
 
+/-- A version of `Set.inclusion` not requiring definitional abuse -/
+abbrev inclusion {U V : Opens α} (h : U ≤ V) : U → V := Set.inclusion h
+
 protected theorem isOpen (U : Opens α) : IsOpen (U : Set α) :=
   U.is_open'
 
@@ -115,14 +121,18 @@ def Simps.coe (U : Opens α) : Set α := U
 initialize_simps_projections Opens (carrier → coe)
 
 /-- The interior of a set, as an element of `Opens`. -/
-nonrec def interior (s : Set α) : Opens α :=
+@[simps]
+protected def interior (s : Set α) : Opens α :=
   ⟨interior s, isOpen_interior⟩
 
-theorem gc : GaloisConnection ((↑) : Opens α → Set α) interior := fun U _ =>
+@[simp]
+theorem mem_interior {s : Set α} {x : α} : x ∈ Opens.interior s ↔ x ∈ _root_.interior s := .rfl
+
+theorem gc : GaloisConnection ((↑) : Opens α → Set α) Opens.interior := fun U _ =>
   ⟨fun h => interior_maximal h U.isOpen, fun h => le_trans h interior_subset⟩
 
 /-- The galois coinsertion between sets and opens. -/
-def gi : GaloisCoinsertion (↑) (@interior α _) where
+def gi : GaloisCoinsertion (↑) (@Opens.interior α _) where
   choice s hs := ⟨s, interior_eq_iff_isOpen.mp <| le_antisymm interior_subset hs⟩
   gc := gc
   u_l_le _ := interior_subset
@@ -347,6 +357,9 @@ theorem comap_mono (f : C(α, β)) {s t : Opens β} (h : s ≤ t) : comap f s 
 theorem coe_comap (f : C(α, β)) (U : Opens β) : ↑(comap f U) = f ⁻¹' U :=
   rfl
 
+@[simp]
+theorem mem_comap {f : C(α, β)} {U : Opens β} {x : α} : x ∈ comap f U ↔ f x ∈ U := .rfl
+
 protected theorem comap_comp (g : C(β, γ)) (f : C(α, β)) :
     comap (g.comp f) = (comap f).comp (comap g) :=
   rfl
diff --git a/Mathlib/Topology/Sheaves/Forget.lean b/Mathlib/Topology/Sheaves/Forget.lean
index 6783eda87f149..f92a32f660346 100644
--- a/Mathlib/Topology/Sheaves/Forget.lean
+++ b/Mathlib/Topology/Sheaves/Forget.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Category.Ring.Limits
 import Mathlib.Topology.Sheaves.Sheaf
diff --git a/Mathlib/Topology/Sheaves/Functors.lean b/Mathlib/Topology/Sheaves/Functors.lean
index ee393d3948d1a..d246fbdbe9fcc 100644
--- a/Mathlib/Topology/Sheaves/Functors.lean
+++ b/Mathlib/Topology/Sheaves/Functors.lean
@@ -34,6 +34,8 @@ open CategoryTheory.Limits
 
 open TopologicalSpace
 
+open scoped AlgebraicGeometry
+
 variable {C : Type u} [Category.{v} C]
 variable {X Y : TopCat.{w}} (f : X ⟶ Y)
 variable ⦃ι : Type w⦄ {U : ι → Opens Y}
diff --git a/Mathlib/Topology/Sheaves/Limits.lean b/Mathlib/Topology/Sheaves/Limits.lean
index 7309973470775..af880e9d7ae36 100644
--- a/Mathlib/Topology/Sheaves/Limits.lean
+++ b/Mathlib/Topology/Sheaves/Limits.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.Sheaf
 import Mathlib.CategoryTheory.Sites.Limits
diff --git a/Mathlib/Topology/Sheaves/LocalPredicate.lean b/Mathlib/Topology/Sheaves/LocalPredicate.lean
index 2c3ef5febbef1..c27257e4a8618 100644
--- a/Mathlib/Topology/Sheaves/LocalPredicate.lean
+++ b/Mathlib/Topology/Sheaves/LocalPredicate.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison, Adam Topaz
+Authors: Johan Commelin, Kim Morrison, Adam Topaz
 -/
 import Mathlib.Topology.Sheaves.SheafOfFunctions
 import Mathlib.Topology.Sheaves.Stalks
diff --git a/Mathlib/Topology/Sheaves/Operations.lean b/Mathlib/Topology/Sheaves/Operations.lean
index c45d513f7dc54..85cb01ea80e1a 100644
--- a/Mathlib/Topology/Sheaves/Operations.lean
+++ b/Mathlib/Topology/Sheaves/Operations.lean
@@ -88,7 +88,7 @@ noncomputable def submonoidPresheafOfStalk (S : ∀ x : X, Submonoid (F.stalk x)
     intro x
     change (F.map i.unop.op ≫ F.germ x) s ∈ _
     rw [F.germ_res]
-    exact hs _
+    exact hs ⟨_,_⟩
 
 noncomputable instance : Inhabited F.SubmonoidPresheaf :=
   ⟨F.submonoidPresheafOfStalk fun _ => ⊥⟩
diff --git a/Mathlib/Topology/Sheaves/Presheaf.lean b/Mathlib/Topology/Sheaves/Presheaf.lean
index 77a9b775aec7d..6b2b727509484 100644
--- a/Mathlib/Topology/Sheaves/Presheaf.lean
+++ b/Mathlib/Topology/Sheaves/Presheaf.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2018 Scott Morrison. All rights reserved.
+Copyright (c) 2018 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Mario Carneiro, Reid Barton, Andrew Yang
+Authors: Kim Morrison, Mario Carneiro, Reid Barton, Andrew Yang
 -/
 import Mathlib.Topology.Category.TopCat.Opens
 import Mathlib.CategoryTheory.Adjunction.Unique
@@ -153,9 +153,9 @@ variable (C)
 def pushforward {X Y : TopCat.{w}} (f : X ⟶ Y) : X.Presheaf C ⥤ Y.Presheaf C :=
   (whiskeringLeft _ _ _).obj (Opens.map f).op
 
-set_option quotPrecheck false in
-/-- push forward of a presheaf-/
-notation f:80 " _* " P:81 => (pushforward _ f).obj P
+/-- push forward of a presheaf -/
+scoped[AlgebraicGeometry] notation f:80 " _* " P:81 =>
+  Prefunctor.obj (Functor.toPrefunctor (TopCat.Presheaf.pushforward _ f)) P
 
 @[simp]
 theorem pushforward_map_app' {X Y : TopCat.{w}} (f : X ⟶ Y) {ℱ 𝒢 : X.Presheaf C} (α : ℱ ⟶ 𝒢)
@@ -291,7 +291,7 @@ def pullbackInvIsoPushforwardHom {X Y : TopCat.{v}} (H : X ≅ Y) :
 
 variable {C}
 
-/-- If `f '' U` is open, then `f⁻¹ℱ U ≅ ℱ (f '' U)`.  -/
+/-- If `f '' U` is open, then `f⁻¹ℱ U ≅ ℱ (f '' U)`. -/
 def pullbackObjObjOfImageOpen {X Y : TopCat.{v}} (f : X ⟶ Y) (ℱ : Y.Presheaf C) (U : Opens X)
     (H : IsOpen (f '' SetLike.coe U)) : ((pullback C f).obj ℱ).obj (op U) ≅ ℱ.obj (op ⟨_, H⟩) := by
   let x : CostructuredArrow (Opens.map f).op (op U) := CostructuredArrow.mk
diff --git a/Mathlib/Topology/Sheaves/PresheafOfFunctions.lean b/Mathlib/Topology/Sheaves/PresheafOfFunctions.lean
index 24ed94958851d..6049ed866f9d8 100644
--- a/Mathlib/Topology/Sheaves/PresheafOfFunctions.lean
+++ b/Mathlib/Topology/Sheaves/PresheafOfFunctions.lean
@@ -1,12 +1,12 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Yoneda
 import Mathlib.Topology.Sheaves.Presheaf
 import Mathlib.Topology.Category.TopCommRingCat
-import Mathlib.Topology.ContinuousFunction.Algebra
+import Mathlib.Topology.ContinuousMap.Algebra
 
 /-!
 # Presheaves of functions
diff --git a/Mathlib/Topology/Sheaves/Sheaf.lean b/Mathlib/Topology/Sheaves/Sheaf.lean
index c13307d1eeab1..6345a02e5c3c7 100644
--- a/Mathlib/Topology/Sheaves/Sheaf.lean
+++ b/Mathlib/Topology/Sheaves/Sheaf.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.Presheaf
 import Mathlib.CategoryTheory.Sites.Sheaf
diff --git a/Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean b/Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
index 25fd50eb55023..34b950b76fec8 100644
--- a/Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
+++ b/Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
 import Mathlib.CategoryTheory.Limits.Shapes.Products
diff --git a/Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean b/Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean
index ace2385ed29bd..318e4ad961054 100644
--- a/Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean
+++ b/Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.SheafCondition.Sites
 
diff --git a/Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean b/Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean
index 7dcee720fc546..9677da6bdf32e 100644
--- a/Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean
+++ b/Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover
 import Mathlib.CategoryTheory.Limits.Final
diff --git a/Mathlib/Topology/Sheaves/SheafOfFunctions.lean b/Mathlib/Topology/Sheaves/SheafOfFunctions.lean
index 4aea0cf255c09..e2c75974e54e3 100644
--- a/Mathlib/Topology/Sheaves/SheafOfFunctions.lean
+++ b/Mathlib/Topology/Sheaves/SheafOfFunctions.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin, Scott Morrison
+Authors: Johan Commelin, Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.PresheafOfFunctions
 import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
diff --git a/Mathlib/Topology/Sheaves/Sheafify.lean b/Mathlib/Topology/Sheaves/Sheafify.lean
index 14b6e48ce063c..423700fb192c9 100644
--- a/Mathlib/Topology/Sheaves/Sheafify.lean
+++ b/Mathlib/Topology/Sheaves/Sheafify.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2020 Scott Morrison. All rights reserved.
+Copyright (c) 2020 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Topology.Sheaves.LocalPredicate
 import Mathlib.Topology.Sheaves.Stalks
diff --git a/Mathlib/Topology/Sheaves/Skyscraper.lean b/Mathlib/Topology/Sheaves/Skyscraper.lean
index 542289c1577d7..9cdd092a962a9 100644
--- a/Mathlib/Topology/Sheaves/Skyscraper.lean
+++ b/Mathlib/Topology/Sheaves/Skyscraper.lean
@@ -34,6 +34,7 @@ TODO: generalize universe level when calculating stalks, after generalizing univ
 noncomputable section
 
 open TopologicalSpace TopCat CategoryTheory CategoryTheory.Limits Opposite
+open scoped AlgebraicGeometry
 
 universe u v w
 
@@ -259,42 +260,6 @@ def toSkyscraperPresheaf {𝓕 : Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶
     · split_ifs
       exact ((if_neg hV).symm.ndrec terminalIsTerminal).hom_ext ..
 
-/-- If `f : 𝓕 ⟶ skyscraperPresheaf p₀ c` is a natural transformation, then there is a morphism
-`𝓕.stalk p₀ ⟶ c` defined as the morphism from colimit to cocone at `c`.
--/
-def fromStalk {𝓕 : Presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraperPresheaf p₀ c) : 𝓕.stalk p₀ ⟶ c :=
-  let χ : Cocone ((OpenNhds.inclusion p₀).op ⋙ 𝓕) :=
-    Cocone.mk c <|
-      { app := fun U => f.app (op U.unop.1) ≫ eqToHom (if_pos U.unop.2)
-        naturality := fun U V inc => by
-          dsimp
-          erw [Category.comp_id, ← Category.assoc, comp_eqToHom_iff, Category.assoc,
-            eqToHom_trans, f.naturality, skyscraperPresheaf_map]
-          -- Porting note: added this `dsimp` and `rfl` in the end
-          dsimp only [skyscraperPresheaf_obj, unop_op, Eq.ndrec]
-          have hV : p₀ ∈ (OpenNhds.inclusion p₀).obj V.unop := V.unop.2; split_ifs <;>
-          simp only [comp_eqToHom_iff, Category.assoc, eqToHom_trans, eqToHom_refl,
-            Category.comp_id] <;> rfl }
-  colimit.desc _ χ
-
-theorem to_skyscraper_fromStalk {𝓕 : Presheaf C X} {c : C} (f : 𝓕 ⟶ skyscraperPresheaf p₀ c) :
-    toSkyscraperPresheaf p₀ (fromStalk _ f) = f := by
-  apply NatTrans.ext
-  ext U
-  dsimp
-  split_ifs with h
-  · erw [← Category.assoc, colimit.ι_desc, Category.assoc, eqToHom_trans, eqToHom_refl,
-      Category.comp_id]
-  · exact ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext ..
-
-theorem fromStalk_to_skyscraper {𝓕 : Presheaf C X} {c : C} (f : 𝓕.stalk p₀ ⟶ c) :
-    fromStalk p₀ (toSkyscraperPresheaf _ f) = f :=
-  colimit.hom_ext fun U => by
-    erw [colimit.ι_desc]; dsimp; rw [dif_pos U.unop.2]
-    rw [Category.assoc, Category.assoc, eqToHom_trans, eqToHom_refl, Category.comp_id,
-      Presheaf.germ]
-    congr 3
-
 /-- The unit in `Presheaf.stalkFunctor ⊣ skyscraperPresheafFunctor`
 -/
 @[simps]
@@ -334,30 +299,32 @@ open StalkSkyscraperPresheafAdjunctionAuxs
 -/
 def skyscraperPresheafStalkAdjunction [HasColimits C] :
     (Presheaf.stalkFunctor C p₀ : Presheaf C X ⥤ C) ⊣ skyscraperPresheafFunctor p₀ where
-  homEquiv c 𝓕 :=
-    { toFun := toSkyscraperPresheaf _
-      invFun := fromStalk _
-      left_inv := fromStalk_to_skyscraper _
-      right_inv := to_skyscraper_fromStalk _ }
   unit := StalkSkyscraperPresheafAdjunctionAuxs.unit _
   counit := StalkSkyscraperPresheafAdjunctionAuxs.counit _
-  homEquiv_unit {𝓕} c α := by
-    ext U
-    -- Porting note: `NatTrans.comp_app` is not picked up by `simp`
-    rw [NatTrans.comp_app]
-    simp only [Equiv.coe_fn_mk, toSkyscraperPresheaf_app, SkyscraperPresheafFunctor.map'_app,
-      skyscraperPresheafFunctor_map, unit_app]
+  left_triangle_components X := by
+    dsimp [Presheaf.stalkFunctor, toSkyscraperPresheaf]
+    ext
+    simp only [Functor.comp_obj, Functor.op_obj, ι_colimMap_assoc, skyscraperPresheaf_obj,
+      whiskerLeft_app, Category.comp_id]
     split_ifs with h
-    · erw [Category.id_comp, ← Category.assoc, comp_eqToHom_iff, Category.assoc, Category.assoc,
-        Category.assoc, Category.assoc, eqToHom_trans, eqToHom_refl, Category.comp_id, ←
-        Category.assoc _ _ α, eqToHom_trans, eqToHom_refl, Category.id_comp]
-    · apply ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext
-  homEquiv_counit {𝓕} c α := by
-    -- Porting note: added a `dsimp`
-    dsimp; ext U; simp only [Equiv.coe_fn_symm_mk, counit_app]
-    erw [colimit.ι_desc, ← Category.assoc, colimit.ι_map, whiskerLeft_app, Category.assoc,
-      colimit.ι_desc]
-    rfl
+    · simp [skyscraperPresheafStalkOfSpecializes]
+      rfl
+    · simp only [skyscraperPresheafStalkOfSpecializes, colimit.isoColimitCocone_ι_hom,
+        skyscraperPresheafCoconeOfSpecializes_pt, skyscraperPresheafCoconeOfSpecializes_ι_app,
+        Functor.comp_obj, Functor.op_obj, skyscraperPresheaf_obj, Functor.const_obj_obj]
+      rw [comp_eqToHom_iff]
+      apply ((if_neg h).symm.ndrec terminalIsTerminal).hom_ext
+  right_triangle_components Y := by
+    ext
+    simp only [skyscraperPresheafFunctor_obj, Functor.id_obj, skyscraperPresheaf_obj,
+      Functor.comp_obj, Presheaf.stalkFunctor_obj, unit_app, counit_app,
+      skyscraperPresheafStalkOfSpecializes, skyscraperPresheafFunctor_map, Presheaf.comp_app,
+      toSkyscraperPresheaf_app, Category.id_comp, SkyscraperPresheafFunctor.map'_app]
+    split_ifs with h
+    · simp [Presheaf.germ]
+      rfl
+    · simp
+      rfl
 
 instance [HasColimits C] : (skyscraperPresheafFunctor p₀ : C ⥤ Presheaf C X).IsRightAdjoint  :=
   (skyscraperPresheafStalkAdjunction _).isRightAdjoint
@@ -372,16 +339,15 @@ instance [HasColimits C] : (Presheaf.stalkFunctor C p₀).IsLeftAdjoint  :=
 def stalkSkyscraperSheafAdjunction [HasColimits C] :
     Sheaf.forget C X ⋙ Presheaf.stalkFunctor _ p₀ ⊣ skyscraperSheafFunctor p₀ where
   -- Porting note (#11041): `ext1` is changed to `Sheaf.Hom.ext`,
-  homEquiv 𝓕 c :=
-    ⟨fun f => ⟨toSkyscraperPresheaf p₀ f⟩, fun g => fromStalk p₀ g.1, fromStalk_to_skyscraper p₀,
-      fun g => Sheaf.Hom.ext <| to_skyscraper_fromStalk _ _⟩
   unit :=
     { app := fun 𝓕 => ⟨(StalkSkyscraperPresheafAdjunctionAuxs.unit p₀).app 𝓕.1⟩
       naturality := fun 𝓐 𝓑 f => Sheaf.Hom.ext <| by
         apply (StalkSkyscraperPresheafAdjunctionAuxs.unit p₀).naturality }
   counit := StalkSkyscraperPresheafAdjunctionAuxs.counit p₀
-  homEquiv_unit {𝓐} c f := Sheaf.Hom.ext (skyscraperPresheafStalkAdjunction p₀).homEquiv_unit
-  homEquiv_counit {𝓐} c f := (skyscraperPresheafStalkAdjunction p₀).homEquiv_counit
+  left_triangle_components X :=
+    ((skyscraperPresheafStalkAdjunction p₀).left_triangle_components X.val)
+  right_triangle_components Y :=
+    Sheaf.Hom.ext ((skyscraperPresheafStalkAdjunction p₀).right_triangle_components _)
 
 instance [HasColimits C] : (skyscraperSheafFunctor p₀ : C ⥤ Sheaf C X).IsRightAdjoint  :=
   (stalkSkyscraperSheafAdjunction _).isRightAdjoint
diff --git a/Mathlib/Topology/Sheaves/Stalks.lean b/Mathlib/Topology/Sheaves/Stalks.lean
index 36574cf04a2b3..d1e8867ba7f1a 100644
--- a/Mathlib/Topology/Sheaves/Stalks.lean
+++ b/Mathlib/Topology/Sheaves/Stalks.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Justus Springer
+Authors: Kim Morrison, Justus Springer
 -/
 import Mathlib.Topology.Category.TopCat.OpenNhds
 import Mathlib.Topology.Sheaves.Presheaf
@@ -60,6 +60,8 @@ open TopologicalSpace
 
 open Opposite
 
+open scoped AlgebraicGeometry
+
 variable {C : Type u} [Category.{v} C]
 variable [HasColimits.{v} C]
 variable {X Y Z : TopCat.{v}}
diff --git a/Mathlib/Topology/Sober.lean b/Mathlib/Topology/Sober.lean
index 05385bd2df372..a4717443959a5 100644
--- a/Mathlib/Topology/Sober.lean
+++ b/Mathlib/Topology/Sober.lean
@@ -91,7 +91,7 @@ end IsGenericPoint
 theorem isGenericPoint_iff_forall_closed (hS : IsClosed S) (hxS : x ∈ S) :
     IsGenericPoint x S ↔ ∀ Z : Set α, IsClosed Z → x ∈ Z → S ⊆ Z := by
   have : closure {x} ⊆ S := closure_minimal (singleton_subset_iff.2 hxS) hS
-  simp_rw [IsGenericPoint, subset_antisymm_iff, this, true_and_iff, closure, subset_sInter_iff,
+  simp_rw [IsGenericPoint, subset_antisymm_iff, this, true_and, closure, subset_sInter_iff,
     mem_setOf_eq, and_imp, singleton_subset_iff]
 
 end genericPoint
diff --git a/Mathlib/Topology/Specialization.lean b/Mathlib/Topology/Specialization.lean
index 53b1665a0beab..1c7ce45b4bb4b 100644
--- a/Mathlib/Topology/Specialization.lean
+++ b/Mathlib/Topology/Specialization.lean
@@ -5,7 +5,7 @@ Authors: Yaël Dillies
 -/
 import Mathlib.Order.Category.Preord
 import Mathlib.Topology.Category.TopCat.Basic
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Topology.Separation
 import Mathlib.Topology.Order.UpperLowerSetTopology
 
@@ -26,7 +26,7 @@ variable {α β γ : Type*}
 /-- `toEquiv` is the "identity" function to the `Specialization` of a type. -/
 @[match_pattern] def toEquiv : α ≃ Specialization α := Equiv.refl _
 
-/-- `ofEquiv` is the identity function from the `Specialization` of a type.  -/
+/-- `ofEquiv` is the identity function from the `Specialization` of a type. -/
 @[match_pattern] def ofEquiv : Specialization α ≃ α := Equiv.refl _
 
 @[simp] lemma toEquiv_symm : (@toEquiv α).symm = ofEquiv := rfl
@@ -63,7 +63,7 @@ instance instPartialOrder [T0Space α] : PartialOrder (Specialization α) := spe
 orders. -/
 def map (f : C(α, β)) : Specialization α →o Specialization β where
   toFun := toEquiv ∘ f ∘ ofEquiv
-  monotone' := f.continuous.specialization_monotone
+  monotone' := (map_continuous f).specialization_monotone
 
 @[simp] lemma map_id : map (ContinuousMap.id α) = OrderHom.id := rfl
 @[simp] lemma map_comp (g : C(β, γ)) (f : C(α, β)) : map (g.comp f) = (map g).comp (map f) := rfl
diff --git a/Mathlib/Topology/Spectral/Hom.lean b/Mathlib/Topology/Spectral/Hom.lean
index 695abf50017b4..30c704a2d0504 100644
--- a/Mathlib/Topology/Spectral/Hom.lean
+++ b/Mathlib/Topology/Spectral/Hom.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yaël Dillies. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies
 -/
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 /-!
 # Spectral maps
@@ -57,9 +57,9 @@ end Unbundled
 
 /-- The type of spectral maps from `α` to `β`. -/
 structure SpectralMap (α β : Type*) [TopologicalSpace α] [TopologicalSpace β] where
-  /-- function between topological spaces-/
+  /-- function between topological spaces -/
   toFun : α → β
-  /-- proof that `toFun` is a spectral map-/
+  /-- proof that `toFun` is a spectral map -/
   spectral' : IsSpectralMap toFun
 
 section
@@ -69,7 +69,7 @@ section
 You should extend this class when you extend `SpectralMap`. -/
 class SpectralMapClass (F α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
     [FunLike F α β] : Prop where
-  /-- statement that `F` is a type of spectral maps-/
+  /-- statement that `F` is a type of spectral maps -/
   map_spectral (f : F) : IsSpectralMap f
 
 end
diff --git a/Mathlib/Topology/StoneCech.lean b/Mathlib/Topology/StoneCech.lean
index 81b611c27866c..3beed85e1b698 100644
--- a/Mathlib/Topology/StoneCech.lean
+++ b/Mathlib/Topology/StoneCech.lean
@@ -13,7 +13,7 @@ Construction of the Stone-Čech compactification using ultrafilters.
 For any topological space `α`, we build a compact Hausdorff space `StoneCech α` and a continuous
 map `stoneCechUnit : α → StoneCech α` which is minimal in the sense of the following universal
 property: for any compact Hausdorff space `β` and every map `f : α → β` such that
-`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that
+`hf : Continuous f`, there is a unique map `stoneCechExtend hf : StoneCech α → β` such that
 `stoneCechExtend_extends : stoneCechExtend hf ∘ stoneCechUnit = f`.
 Continuity of this extension is asserted by `continuous_stoneCechExtend` and uniqueness by
 `stoneCech_hom_ext`.
@@ -100,7 +100,7 @@ instance Ultrafilter.t2Space : T2Space (Ultrafilter α) :=
 instance : TotallyDisconnectedSpace (Ultrafilter α) := by
   rw [totallyDisconnectedSpace_iff_connectedComponent_singleton]
   intro A
-  simp only [Set.eq_singleton_iff_unique_mem, mem_connectedComponent, true_and_iff]
+  simp only [Set.eq_singleton_iff_unique_mem, mem_connectedComponent, true_and]
   intro B hB
   rw [← Ultrafilter.coe_le_coe]
   intro s hs
@@ -148,15 +148,18 @@ theorem induced_topology_pure :
   simp
 
 /-- `pure : α → Ultrafilter α` defines a dense inducing of `α` in `Ultrafilter α`. -/
-theorem denseInducing_pure : @DenseInducing _ _ ⊥ _ (pure : α → Ultrafilter α) :=
+theorem isDenseInducing_pure : @IsDenseInducing _ _ ⊥ _ (pure : α → Ultrafilter α) :=
   letI : TopologicalSpace α := ⊥
   ⟨⟨induced_topology_pure.symm⟩, denseRange_pure⟩
 
 -- The following refined version will never be used
 /-- `pure : α → Ultrafilter α` defines a dense embedding of `α` in `Ultrafilter α`. -/
-theorem denseEmbedding_pure : @DenseEmbedding _ _ ⊥ _ (pure : α → Ultrafilter α) :=
+theorem isDenseEmbedding_pure : @IsDenseEmbedding _ _ ⊥ _ (pure : α → Ultrafilter α) :=
   letI : TopologicalSpace α := ⊥
-  { denseInducing_pure with inj := ultrafilter_pure_injective }
+  { isDenseInducing_pure with inj := ultrafilter_pure_injective }
+
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_pure := isDenseEmbedding_pure
 
 end Embedding
 
@@ -166,21 +169,21 @@ section Extension
   unique extension to a continuous function `Ultrafilter α → γ`. We
   already know it must be unique because `α → Ultrafilter α` is a
   dense embedding and `γ` is Hausdorff. For existence, we will invoke
-  `DenseInducing.continuous_extend`. -/
+  `IsDenseInducing.continuous_extend`. -/
 variable {γ : Type*} [TopologicalSpace γ]
 
 /-- The extension of a function `α → γ` to a function `Ultrafilter α → γ`.
   When `γ` is a compact Hausdorff space it will be continuous. -/
 def Ultrafilter.extend (f : α → γ) : Ultrafilter α → γ :=
   letI : TopologicalSpace α := ⊥
-  denseInducing_pure.extend f
+  isDenseInducing_pure.extend f
 
 variable [T2Space γ]
 
 theorem ultrafilter_extend_extends (f : α → γ) : Ultrafilter.extend f ∘ pure = f := by
   letI : TopologicalSpace α := ⊥
   haveI : DiscreteTopology α := ⟨rfl⟩
-  exact funext (denseInducing_pure.extend_eq continuous_of_discreteTopology)
+  exact funext (isDenseInducing_pure.extend_eq continuous_of_discreteTopology)
 
 variable [CompactSpace γ]
 
@@ -191,7 +194,7 @@ theorem continuous_ultrafilter_extend (f : α → γ) : Continuous (Ultrafilter.
       isCompact_univ.ultrafilter_le_nhds (b.map f) (by rw [le_principal_iff]; exact univ_mem)
     ⟨c, le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h'⟩
   let _ : TopologicalSpace α := ⊥
-  exact denseInducing_pure.continuous_extend h
+  exact isDenseInducing_pure.continuous_extend h
 
 /-- The value of `Ultrafilter.extend f` on an ultrafilter `b` is the
   unique limit of the ultrafilter `b.map f` in `γ`. -/
@@ -210,7 +213,7 @@ theorem ultrafilter_extend_eq_iff {f : α → γ} {b : Ultrafilter α} {c : γ}
      exact le_rfl,
    fun h ↦
     let _ : TopologicalSpace α := ⊥
-    denseInducing_pure.extend_eq_of_tendsto
+    isDenseInducing_pure.extend_eq_of_tendsto
       (le_trans (map_mono (ultrafilter_comap_pure_nhds _)) h)⟩
 
 end Extension
diff --git a/Mathlib/Topology/Support.lean b/Mathlib/Topology/Support.lean
index 3ddebbe31bfd6..e5e52b3d9982f 100644
--- a/Mathlib/Topology/Support.lean
+++ b/Mathlib/Topology/Support.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn, Patrick Massot
 -/
 import Mathlib.Algebra.GroupWithZero.Indicator
+import Mathlib.Algebra.Order.Group.Unbundled.Abs
 import Mathlib.Algebra.Module.Basic
 import Mathlib.Topology.Separation
 
@@ -299,8 +300,23 @@ variable {f f' : α → β} {x : α}
 theorem HasCompactMulSupport.mul (hf : HasCompactMulSupport f) (hf' : HasCompactMulSupport f') :
     HasCompactMulSupport (f * f') := hf.comp₂_left hf' (mul_one 1)
 
+@[to_additive, simp]
+protected lemma HasCompactMulSupport.one {α β : Type*} [TopologicalSpace α] [One β] :
+    HasCompactMulSupport (1 : α → β) := by
+  simp [HasCompactMulSupport, mulTSupport]
+
 end Monoid
 
+section DivisionMonoid
+
+@[to_additive]
+protected lemma HasCompactMulSupport.inv' {α β : Type*} [TopologicalSpace α] [DivisionMonoid β]
+    {f : α → β} (hf : HasCompactMulSupport f) :
+    HasCompactMulSupport (f⁻¹) := by
+  simpa only [HasCompactMulSupport, mulTSupport, mulSupport_inv'] using hf
+
+end DivisionMonoid
+
 section SMulZeroClass
 
 variable [TopologicalSpace α] [Zero M] [SMulZeroClass R M]
diff --git a/Mathlib/Topology/TietzeExtension.lean b/Mathlib/Topology/TietzeExtension.lean
index 704a18651fd7e..b1e175559e9fd 100644
--- a/Mathlib/Topology/TietzeExtension.lean
+++ b/Mathlib/Topology/TietzeExtension.lean
@@ -306,7 +306,7 @@ theorem exists_extension_forall_mem_Icc_of_closedEmbedding (f : X →ᵇ ℝ) {a
 embedding. Let `e` be a closed embedding of a nonempty topological space `X` into a normal
 topological space `Y`. Let `f` be a bounded continuous real-valued function on `X`. Then there
 exists a bounded continuous function `g : Y →ᵇ ℝ` such that `g ∘ e = f` and each value `g y` belongs
-to a closed interval `[f x₁, f x₂]` for some `x₁` and `x₂`.  -/
+to a closed interval `[f x₁, f x₂]` for some `x₁` and `x₂`. -/
 theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f : X →ᵇ ℝ)
     {e : X → Y} (he : ClosedEmbedding e) :
     ∃ g : Y →ᵇ ℝ, (∀ y, ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ g ∘ e = f := by
@@ -340,7 +340,7 @@ theorem exists_extension_forall_exists_le_ge_of_closedEmbedding [Nonempty X] (f
     · exact ⟨g, fun y => ⟨x, hg_mem _⟩, hgf⟩
     /- Otherwise, `g ⁻¹' {a}` is disjoint with `range e ∪ g ⁻¹' (Ici c)`, hence there exists a
         function `dg : Y → ℝ` such that `dg ∘ e = 0`, `dg y = 0` whenever `c ≤ g y`, `dg y = c - a`
-        whenever `g y = a`, and `0 ≤ dg y ≤ c - a` for all `y`.  -/
+        whenever `g y = a`, and `0 ≤ dg y ≤ c - a` for all `y`. -/
     have hd : Disjoint (range e ∪ g ⁻¹' Ici c) (g ⁻¹' {a}) := by
       refine disjoint_union_left.2 ⟨?_, Disjoint.preimage _ ?_⟩
       · rw [Set.disjoint_left]
diff --git a/Mathlib/Topology/UniformSpace/AbsoluteValue.lean b/Mathlib/Topology/UniformSpace/AbsoluteValue.lean
index fb21530f3f80a..55efcadaf7133 100644
--- a/Mathlib/Topology/UniformSpace/AbsoluteValue.lean
+++ b/Mathlib/Topology/UniformSpace/AbsoluteValue.lean
@@ -5,7 +5,7 @@ Authors: Patrick Massot
 -/
 import Mathlib.Algebra.Order.AbsoluteValue
 import Mathlib.Algebra.Order.Field.Basic
-import Mathlib.Topology.UniformSpace.Basic
+import Mathlib.Topology.UniformSpace.OfFun
 
 /-!
 # Uniform structure induced by an absolute value
diff --git a/Mathlib/Topology/UniformSpace/AbstractCompletion.lean b/Mathlib/Topology/UniformSpace/AbstractCompletion.lean
index 9f7d1efc35a9c..ce9024be56637 100644
--- a/Mathlib/Topology/UniformSpace/AbstractCompletion.lean
+++ b/Mathlib/Topology/UniformSpace/AbstractCompletion.lean
@@ -66,7 +66,7 @@ structure AbstractCompletion (α : Type u) [UniformSpace α] where
   /-- The completion is a T₀ space. -/
   separation : T0Space space
   /-- The map into the completion is uniform-inducing. -/
-  uniformInducing : UniformInducing coe
+  isUniformInducing : IsUniformInducing coe
   /-- The map into the completion has dense range. -/
   dense : DenseRange coe
 
@@ -81,18 +81,20 @@ local notation "hatα" => pkg.space
 
 local notation "ι" => pkg.coe
 
+@[deprecated (since := "2024-10-08")] alias uniformInducing := isUniformInducing
+
 /-- If `α` is complete, then it is an abstract completion of itself. -/
 def ofComplete [T0Space α] [CompleteSpace α] : AbstractCompletion α :=
-  mk α id inferInstance inferInstance inferInstance uniformInducing_id denseRange_id
+  mk α id inferInstance inferInstance inferInstance .id denseRange_id
 
 theorem closure_range : closure (range ι) = univ :=
   pkg.dense.closure_range
 
-theorem denseInducing : DenseInducing ι :=
-  ⟨pkg.uniformInducing.inducing, pkg.dense⟩
+theorem isDenseInducing : IsDenseInducing ι :=
+  ⟨pkg.isUniformInducing.inducing, pkg.dense⟩
 
 theorem uniformContinuous_coe : UniformContinuous ι :=
-  UniformInducing.uniformContinuous pkg.uniformInducing
+  IsUniformInducing.uniformContinuous pkg.isUniformInducing
 
 theorem continuous_coe : Continuous ι :=
   pkg.uniformContinuous_coe.continuous
@@ -114,23 +116,23 @@ section Extend
 
 /-- Extension of maps to completions -/
 protected def extend (f : α → β) : hatα → β :=
-  if UniformContinuous f then pkg.denseInducing.extend f else fun x => f (pkg.dense.some x)
+  if UniformContinuous f then pkg.isDenseInducing.extend f else fun x => f (pkg.dense.some x)
 
 variable {f : α → β}
 
-theorem extend_def (hf : UniformContinuous f) : pkg.extend f = pkg.denseInducing.extend f :=
+theorem extend_def (hf : UniformContinuous f) : pkg.extend f = pkg.isDenseInducing.extend f :=
   if_pos hf
 
 theorem extend_coe [T2Space β] (hf : UniformContinuous f) (a : α) : (pkg.extend f) (ι a) = f a := by
   rw [pkg.extend_def hf]
-  exact pkg.denseInducing.extend_eq hf.continuous a
+  exact pkg.isDenseInducing.extend_eq hf.continuous a
 
 variable [CompleteSpace β]
 
 theorem uniformContinuous_extend : UniformContinuous (pkg.extend f) := by
   by_cases hf : UniformContinuous f
   · rw [pkg.extend_def hf]
-    exact uniformContinuous_uniformly_extend pkg.uniformInducing pkg.dense hf
+    exact uniformContinuous_uniformly_extend pkg.isUniformInducing pkg.dense hf
   · change UniformContinuous (ite _ _ _)
     rw [if_neg hf]
     exact uniformContinuous_of_const fun a b => by congr 1
@@ -187,7 +189,7 @@ theorem map_unique {f : α → β} {g : hatα → hatβ} (hg : UniformContinuous
   pkg.funext (pkg.continuous_map _ _) hg.continuous <| by
     intro a
     change pkg.extend (ι' ∘ f) _ = _
-    simp_rw [(· ∘ ·), h, ← comp_apply (f := g)]
+    simp_rw [Function.comp_def, h, ← comp_apply (f := g)]
     rw [pkg.extend_coe (hg.comp pkg.uniformContinuous_coe)]
 
 @[simp]
@@ -276,17 +278,17 @@ theorem compare_comp_eq_compare (γ : Type*) [TopologicalSpace γ]
     letI := pkg.uniformStruct.toTopologicalSpace
     letI := pkg'.uniformStruct.toTopologicalSpace
     (∀ a : pkg.space,
-      Filter.Tendsto f (Filter.comap pkg.coe (𝓝 a)) (𝓝 ((pkg.denseInducing.extend f) a))) →
-      pkg.denseInducing.extend f ∘ pkg'.compare pkg = pkg'.denseInducing.extend f := by
+      Filter.Tendsto f (Filter.comap pkg.coe (𝓝 a)) (𝓝 ((pkg.isDenseInducing.extend f) a))) →
+      pkg.isDenseInducing.extend f ∘ pkg'.compare pkg = pkg'.isDenseInducing.extend f := by
   let _ := pkg'.uniformStruct
   let _ := pkg.uniformStruct
   intro h
-  have (x : α) : (pkg.denseInducing.extend f ∘ pkg'.compare pkg) (pkg'.coe x) = f x := by
-    simp only [Function.comp_apply, compare_coe, DenseInducing.extend_eq _ cont_f, implies_true]
-  apply (DenseInducing.extend_unique (AbstractCompletion.denseInducing _) this
+  have (x : α) : (pkg.isDenseInducing.extend f ∘ pkg'.compare pkg) (pkg'.coe x) = f x := by
+    simp only [Function.comp_apply, compare_coe, IsDenseInducing.extend_eq _ cont_f, implies_true]
+  apply (IsDenseInducing.extend_unique (AbstractCompletion.isDenseInducing _) this
     (Continuous.comp _ (uniformContinuous_compare pkg' pkg).continuous )).symm
-  apply DenseInducing.continuous_extend
-  exact fun a ↦ ⟨(pkg.denseInducing.extend f) a, h a⟩
+  apply IsDenseInducing.continuous_extend
+  exact fun a ↦ ⟨(pkg.isDenseInducing.extend f) a, h a⟩
 
 end Compare
 
@@ -305,8 +307,8 @@ protected def prod : AbstractCompletion (α × β) where
   uniformStruct := inferInstance
   complete := inferInstance
   separation := inferInstance
-  uniformInducing := UniformInducing.prod pkg.uniformInducing pkg'.uniformInducing
-  dense := DenseRange.prod_map pkg.dense pkg'.dense
+  isUniformInducing := IsUniformInducing.prod pkg.isUniformInducing pkg'.isUniformInducing
+  dense := pkg.dense.prodMap pkg'.dense
 
 end Prod
 
diff --git a/Mathlib/Topology/UniformSpace/Ascoli.lean b/Mathlib/Topology/UniformSpace/Ascoli.lean
index 3e5725b06a824..351606b63c56c 100644
--- a/Mathlib/Topology/UniformSpace/Ascoli.lean
+++ b/Mathlib/Topology/UniformSpace/Ascoli.lean
@@ -20,14 +20,14 @@ a family of compact subsets of `X`, and `α` is a uniform space.
   convergence coincide on equicontinuous subsets. This is the key fact that makes equicontinuity
   important in functional analysis. We state various versions of it:
   - as an equality of `UniformSpace`s: `Equicontinuous.comap_uniformFun_eq`
-  - in terms of `UniformInducing`: `Equicontinuous.uniformInducing_uniformFun_iff_pi`
+  - in terms of `IsUniformInducing`: `Equicontinuous.isUniformInducing_uniformFun_iff_pi`
   - in terms of `Inducing`: `Equicontinuous.inducing_uniformFun_iff_pi`
   - in terms of convergence along a filter: `Equicontinuous.tendsto_uniformFun_iff_pi`
 * As a consequence, if `𝔖` is a family of compact subsets of `X`, then the uniform structures of
   uniform convergence on `𝔖` and pointwise convergence on `⋃₀ 𝔖` coincide on equicontinuous
   subsets. Again, we prove multiple variations:
   - as an equality of `UniformSpace`s: `EquicontinuousOn.comap_uniformOnFun_eq`
-  - in terms of `UniformInducing`: `EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi'`
+  - in terms of `IsUniformInducing`: `EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'`
   - in terms of `Inducing`: `EquicontinuousOn.inducing_uniformOnFun_iff_pi'`
   - in terms of convergence along a filter: `EquicontinuousOn.tendsto_uniformOnFun_iff_pi'`
 * The **Arzela-Ascoli theorem** follows from the previous fact and Tykhonov's theorem.
@@ -70,8 +70,7 @@ equicontinuity, uniform convergence, ascoli
 
 open Set Filter Uniformity Topology Function UniformConvergence
 
-variable {ι X Y α β : Type*} [TopologicalSpace X] [UniformSpace α] [UniformSpace β]
-variable {F : ι → X → α} {G : ι → β → α}
+variable {ι X α : Type*} [TopologicalSpace X] [UniformSpace α] {F : ι → X → α}
 
 /-- Let `X` be a compact topological space, `α` a uniform space, and `F : ι → (X → α)` an
 equicontinuous family. Then, the uniform structures of uniform convergence and pointwise
@@ -80,7 +79,7 @@ convergence induce the same uniform structure on `ι`.
 In other words, pointwise convergence and uniform convergence coincide on an equicontinuous
 subset of `X → α`.
 
-Consider using `Equicontinuous.uniformInducing_uniformFun_iff_pi` and
+Consider using `Equicontinuous.isUniformInducing_uniformFun_iff_pi` and
 `Equicontinuous.inducing_uniformFun_iff_pi` instead, to avoid rewriting instances. -/
 theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicontinuous F) :
     (UniformFun.uniformSpace X α).comap F =
@@ -89,7 +88,7 @@ theorem Equicontinuous.comap_uniformFun_eq [CompactSpace X] (F_eqcont : Equicont
   refine le_antisymm (UniformSpace.comap_mono UniformFun.uniformContinuous_toFun) ?_
   -- A bit of rewriting to get a nice intermediate statement.
   change comap _ _ ≤ comap _ _
-  simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp]
+  simp_rw [Pi.uniformity, Filter.comap_iInf, comap_comap, Function.comp_def]
   refine ((UniformFun.hasBasis_uniformity X α).comap (Prod.map F F)).ge_iff.mpr ?_
   -- Core of the proof: we need to show that, for any entourage `U` in `α`,
   -- the set `𝐓(U) := {(i,j) : ι × ι | ∀ x : X, (F i x, F j x) ∈ U}` belongs to the filter
@@ -131,15 +130,19 @@ convergence induce the same uniform structure on `ι`.
 In other words, pointwise convergence and uniform convergence coincide on an equicontinuous
 subset of `X → α`.
 
-This is a version of `Equicontinuous.comap_uniformFun_eq` stated in terms of `UniformInducing`
+This is a version of `Equicontinuous.comap_uniformFun_eq` stated in terms of `IsUniformInducing`
 for convenuence. -/
-lemma Equicontinuous.uniformInducing_uniformFun_iff_pi [UniformSpace ι] [CompactSpace X]
+lemma Equicontinuous.isUniformInducing_uniformFun_iff_pi [UniformSpace ι] [CompactSpace X]
     (F_eqcont : Equicontinuous F) :
-    UniformInducing (UniformFun.ofFun ∘ F) ↔ UniformInducing F := by
-  rw [uniformInducing_iff_uniformSpace, uniformInducing_iff_uniformSpace,
+    IsUniformInducing (UniformFun.ofFun ∘ F) ↔ IsUniformInducing F := by
+  rw [isUniformInducing_iff_uniformSpace, isUniformInducing_iff_uniformSpace,
       ← F_eqcont.comap_uniformFun_eq]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias Equicontinuous.uniformInducing_uniformFun_iff_pi :=
+  Equicontinuous.isUniformInducing_uniformFun_iff_pi
+
 /-- Let `X` be a compact topological space, `α` a uniform space, and `F : ι → (X → α)` an
 equicontinuous family. Then, the topologies of uniform convergence and pointwise convergence induce
 the same topology on `ι`.
@@ -206,7 +209,7 @@ uniform structure on `ι`.
 In particular, pointwise convergence and compact convergence coincide on an equicontinuous
 subset of `X → α`.
 
-Consider using `EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi'` and
+Consider using `EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'` and
 `EquicontinuousOn.inducing_uniformOnFun_iff_pi'` instead to avoid rewriting instances,
 as well as their unprimed versions in case `𝔖` covers `X`. -/
 theorem EquicontinuousOn.comap_uniformOnFun_eq {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
@@ -220,7 +223,7 @@ theorem EquicontinuousOn.comap_uniformOnFun_eq {𝔖 : Set (Set X)} (𝔖_compac
   have H1 : (UniformOnFun.uniformSpace X α 𝔖).comap F =
       ⨅ (K ∈ 𝔖), (UniformFun.uniformSpace _ _).comap (K.restrict ∘ F) := by
     simp_rw [UniformOnFun.uniformSpace, UniformSpace.comap_iInf, ← UniformSpace.comap_comap,
-      UniformFun.ofFun, Equiv.coe_fn_mk, UniformOnFun.toFun, UniformOnFun.ofFun, Function.comp,
+      UniformFun.ofFun, Equiv.coe_fn_mk, UniformOnFun.toFun, UniformOnFun.ofFun, Function.comp_def,
       UniformFun, Equiv.coe_fn_symm_mk]
   -- Now, note that a similar fact is true for the uniform structure on `X → α` induced by
   -- the map `(⋃₀ 𝔖).restrict : (X → α) → ((⋃₀ 𝔖) → α)`: it is equal to the one induced by
@@ -247,37 +250,45 @@ uniform structure on `ι`.
 In particular, pointwise convergence and compact convergence coincide on an equicontinuous
 subset of `X → α`.
 
-This is a version of `EquicontinuousOn.comap_uniformOnFun_eq` stated in terms of `UniformInducing`
+This is a version of `EquicontinuousOn.comap_uniformOnFun_eq` stated in terms of `IsUniformInducing`
 for convenuence. -/
-lemma EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi' [UniformSpace ι]
+lemma EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi' [UniformSpace ι]
     {𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
     (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
-    UniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
-    UniformInducing ((⋃₀ 𝔖).restrict ∘ F) := by
-  rw [uniformInducing_iff_uniformSpace, uniformInducing_iff_uniformSpace,
+    IsUniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
+    IsUniformInducing ((⋃₀ 𝔖).restrict ∘ F) := by
+  rw [isUniformInducing_iff_uniformSpace, isUniformInducing_iff_uniformSpace,
       ← EquicontinuousOn.comap_uniformOnFun_eq 𝔖_compact F_eqcont]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi' :=
+  EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'
+
 /-- Let `X` be a topological space, `𝔖` a covering of `X` by compact subsets, `α` a uniform space,
 and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the uniform
 structures of uniform convergence on `𝔖` and pointwise convergence induce the same
 uniform structure on `ι`.
 
-This is a specialization of `EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi'` to
+This is a specialization of `EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi'` to
 the case where `𝔖` covers `X`. -/
-lemma EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi [UniformSpace ι]
+lemma EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi [UniformSpace ι]
     {𝔖 : Set (Set X)} (𝔖_covers : ⋃₀ 𝔖 = univ) (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
     (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
-    UniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
-    UniformInducing F := by
+    IsUniformInducing (UniformOnFun.ofFun 𝔖 ∘ F) ↔
+    IsUniformInducing F := by
   rw [eq_univ_iff_forall] at 𝔖_covers
   -- This obviously follows from the previous lemma, we formalize it by going through the
   -- isomorphism of uniform spaces between `(⋃₀ 𝔖) → α` and `X → α`.
   let φ : ((⋃₀ 𝔖) → α) ≃ᵤ (X → α) := UniformEquiv.piCongrLeft (β := fun _ ↦ α)
     (Equiv.subtypeUnivEquiv 𝔖_covers)
-  rw [EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi' 𝔖_compact F_eqcont,
+  rw [EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi' 𝔖_compact F_eqcont,
       show restrict (⋃₀ 𝔖) ∘ F = φ.symm ∘ F by rfl]
-  exact ⟨fun H ↦ φ.uniformInducing.comp H, fun H ↦ φ.symm.uniformInducing.comp H⟩
+  exact ⟨fun H ↦ φ.isUniformInducing.comp H, fun H ↦ φ.symm.isUniformInducing.comp H⟩
+
+@[deprecated (since := "2024-10-05")]
+alias EquicontinuousOn.uniformInducing_uniformOnFun_iff_pi :=
+  EquicontinuousOn.isUniformInducing_uniformOnFun_iff_pi
 
 /-- Let `X` be a topological space, `𝔖` a family of compact subsets of `X`, `α` a uniform space,
 and `F : ι → (X → α)` a family which is equicontinuous on each `K ∈ 𝔖`. Then, the topologies
@@ -495,7 +506,8 @@ theorem ArzelaAscoli.isCompact_of_equicontinuous
     rw [isCompact_iff_compactSpace] at hS1 ⊢
     exact (Equiv.toHomeomorphOfInducing _ h).symm.compactSpace
   rw [← inducing_subtype_val.of_comp_iff, ← EquicontinuousOn.inducing_uniformOnFun_iff_pi _ _ _]
-  · exact ContinuousMap.uniformEmbedding_toUniformOnFunIsCompact.inducing.comp inducing_subtype_val
+  · exact ContinuousMap.isUniformEmbedding_toUniformOnFunIsCompact.inducing.comp
+      inducing_subtype_val
   · exact eq_univ_iff_forall.mpr (fun x ↦ mem_sUnion_of_mem (mem_singleton x) isCompact_singleton)
   · exact fun _ ↦ id
   · exact fun K _ ↦ hS2.equicontinuousOn K
diff --git a/Mathlib/Topology/UniformSpace/Basic.lean b/Mathlib/Topology/UniformSpace/Basic.lean
index 5d50ef91ca453..caab9ab1db8cb 100644
--- a/Mathlib/Topology/UniformSpace/Basic.lean
+++ b/Mathlib/Topology/UniformSpace/Basic.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
 -/
 import Mathlib.Order.Filter.SmallSets
-import Mathlib.Tactic.Monotonicity
+import Mathlib.Tactic.Monotonicity.Basic
 import Mathlib.Topology.Compactness.Compact
 import Mathlib.Topology.NhdsSet
 import Mathlib.Algebra.Group.Defs
@@ -17,7 +17,7 @@ generalize to uniform spaces, e.g.
 
 * uniform continuity (in this file)
 * completeness (in `Cauchy.lean`)
-* extension of uniform continuous functions to complete spaces (in `UniformEmbedding.lean`)
+* extension of uniform continuous functions to complete spaces (in `IsUniformEmbedding.lean`)
 * totally bounded sets (in `Cauchy.lean`)
 * totally bounded complete sets are compact (in `Cauchy.lean`)
 
@@ -136,6 +136,31 @@ theorem mem_idRel {a b : α} : (a, b) ∈ @idRel α ↔ a = b :=
 theorem idRel_subset {s : Set (α × α)} : idRel ⊆ s ↔ ∀ a, (a, a) ∈ s := by
   simp [subset_def]
 
+theorem eq_singleton_left_of_prod_subset_idRel {X : Type _} {S T : Set X} (hS : S.Nonempty)
+    (hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, S = {x} := by
+  rcases hS, hT with ⟨⟨s, hs⟩, ⟨t, ht⟩⟩
+  refine ⟨s, eq_singleton_iff_nonempty_unique_mem.mpr ⟨⟨s, hs⟩, fun x hx ↦ ?_⟩⟩
+  rw [prod_subset_iff] at h_diag
+  replace hs := h_diag s hs t ht
+  replace hx := h_diag x hx t ht
+  simp only [idRel, mem_setOf_eq] at hx hs
+  rwa [← hs] at hx
+
+theorem eq_singleton_right_prod_subset_idRel {X : Type _} {S T : Set X} (hS : S.Nonempty)
+    (hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, T = {x} := by
+  rw [Set.prod_subset_iff] at h_diag
+  replace h_diag := fun x hx y hy => (h_diag y hy x hx).symm
+  exact eq_singleton_left_of_prod_subset_idRel hT hS (prod_subset_iff.mpr h_diag)
+
+theorem eq_singleton_prod_subset_idRel {X : Type _} {S T : Set X} (hS : S.Nonempty)
+    (hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, S = {x} ∧ T = {x} := by
+  obtain ⟨⟨x, hx⟩, ⟨y, hy⟩⟩ := eq_singleton_left_of_prod_subset_idRel hS hT h_diag,
+    eq_singleton_right_prod_subset_idRel hS hT h_diag
+  refine ⟨x, ⟨hx, ?_⟩⟩
+  rw [hy, Set.singleton_eq_singleton_iff]
+  exact (Set.prod_subset_iff.mp h_diag x (by simp only [hx, Set.mem_singleton]) y
+    (by simp only [hy, Set.mem_singleton])).symm
+
 /-- The composition of relations -/
 def compRel (r₁ r₂ : Set (α × α)) :=
   { p : α × α | ∃ z : α, (p.1, z) ∈ r₁ ∧ (z, p.2) ∈ r₂ }
@@ -367,33 +392,6 @@ theorem UniformSpace.replaceTopology_eq {α : Type*} [i : TopologicalSpace α] (
     (h : i = u.toTopologicalSpace) : u.replaceTopology h = u :=
   UniformSpace.ext rfl
 
--- Porting note: rfc: use `UniformSpace.Core.mkOfBasis`? This will change defeq here and there
-/-- Define a `UniformSpace` using a "distance" function. The function can be, e.g., the
-distance in a (usual or extended) metric space or an absolute value on a ring. -/
-def UniformSpace.ofFun {α : Type u} {β : Type v} [OrderedAddCommMonoid β]
-    (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
-    (triangle : ∀ x y z, d x z ≤ d x y + d y z)
-    (half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) :
-    UniformSpace α :=
-  .ofCore
-    { uniformity := ⨅ r > 0, 𝓟 { x | d x.1 x.2 < r }
-      refl := le_iInf₂ fun r hr => principal_mono.2 <| idRel_subset.2 fun x => by simpa [refl]
-      symm := tendsto_iInf_iInf fun r => tendsto_iInf_iInf fun _ => tendsto_principal_principal.2
-        fun x hx => by rwa [mem_setOf, symm]
-      comp := le_iInf₂ fun r hr => let ⟨δ, h0, hδr⟩ := half r hr; le_principal_iff.2 <|
-        mem_of_superset
-          (mem_lift' <| mem_iInf_of_mem δ <| mem_iInf_of_mem h0 <| mem_principal_self _)
-          fun (x, z) ⟨y, h₁, h₂⟩ => (triangle _ _ _).trans_lt (hδr _ h₁ _ h₂) }
-
-theorem UniformSpace.hasBasis_ofFun {α : Type u} {β : Type v} [LinearOrderedAddCommMonoid β]
-    (h₀ : ∃ x : β, 0 < x) (d : α → α → β) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
-    (triangle : ∀ x y z, d x z ≤ d x y + d y z)
-    (half : ∀ ε > (0 : β), ∃ δ > (0 : β), ∀ x < δ, ∀ y < δ, x + y < ε) :
-    𝓤[.ofFun d refl symm triangle half].HasBasis ((0 : β) < ·) (fun ε => { x | d x.1 x.2 < ε }) :=
-  hasBasis_biInf_principal'
-    (fun ε₁ h₁ ε₂ h₂ => ⟨min ε₁ ε₂, lt_min h₁ h₂, fun _x hx => lt_of_lt_of_le hx (min_le_left _ _),
-      fun _x hx => lt_of_lt_of_le hx (min_le_right _ _)⟩) h₀
-
 section UniformSpace
 
 variable [UniformSpace α]
@@ -564,16 +562,16 @@ theorem comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤
 ### Balls in uniform spaces
 -/
 
+namespace UniformSpace
+
 /-- The ball around `(x : β)` with respect to `(V : Set (β × β))`. Intended to be
 used for `V ∈ 𝓤 β`, but this is not needed for the definition. Recovers the
-notions of metric space ball when `V = {p | dist p.1 p.2 < r }`.  -/
-def UniformSpace.ball (x : β) (V : Set (β × β)) : Set β :=
-  Prod.mk x ⁻¹' V
+notions of metric space ball when `V = {p | dist p.1 p.2 < r }`. -/
+def ball (x : β) (V : Set (β × β)) : Set β := Prod.mk x ⁻¹' V
 
 open UniformSpace (ball)
 
-theorem UniformSpace.mem_ball_self (x : α) {V : Set (α × α)} (hV : V ∈ 𝓤 α) : x ∈ ball x V :=
-  refl_mem_uniformity hV
+lemma mem_ball_self (x : α) {V : Set (α × α)} : V ∈ 𝓤 α → x ∈ ball x V := refl_mem_uniformity
 
 /-- The triangle inequality for `UniformSpace.ball` -/
 theorem mem_ball_comp {V W : Set (β × β)} {x y z} (h : y ∈ ball x V) (h' : z ∈ ball y W) :
@@ -612,11 +610,10 @@ theorem mem_comp_of_mem_ball {V W : Set (β × β)} {x y z : β} (hV : Symmetric
   rw [mem_ball_symmetry hV] at hx
   exact ⟨z, hx, hy⟩
 
-theorem UniformSpace.isOpen_ball (x : α) {V : Set (α × α)} (hV : IsOpen V) : IsOpen (ball x V) :=
+lemma isOpen_ball (x : α) {V : Set (α × α)} (hV : IsOpen V) : IsOpen (ball x V) :=
   hV.preimage <| continuous_const.prod_mk continuous_id
 
-theorem UniformSpace.isClosed_ball (x : α) {V : Set (α × α)} (hV : IsClosed V) :
-    IsClosed (ball x V) :=
+lemma isClosed_ball (x : α) {V : Set (α × α)} (hV : IsClosed V) : IsClosed (ball x V) :=
   hV.preimage <| continuous_const.prod_mk continuous_id
 
 theorem mem_comp_comp {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β × β} :
@@ -629,10 +626,14 @@ theorem mem_comp_comp {V W M : Set (β × β)} (hW' : SymmetricRel W) {p : β ×
     rw [mem_ball_symmetry hW'] at z_in
     exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩
 
+end UniformSpace
+
 /-!
 ### Neighborhoods in uniform spaces
 -/
 
+open UniformSpace
+
 theorem mem_nhds_uniformity_iff_right {x : α} {s : Set α} :
     s ∈ 𝓝 x ↔ { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α := by
   simp only [nhds_eq_comap_uniformity, mem_comap_prod_mk]
@@ -740,12 +741,12 @@ theorem tendsto_left_nhds_uniformity {a : α} : Tendsto (fun a' => (a, a')) (
 theorem lift_nhds_left {x : α} {g : Set α → Filter β} (hg : Monotone g) :
     (𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g (ball x s) := by
   rw [nhds_eq_comap_uniformity, comap_lift_eq2 hg]
-  simp_rw [ball, Function.comp]
+  simp_rw [ball, Function.comp_def]
 
 theorem lift_nhds_right {x : α} {g : Set α → Filter β} (hg : Monotone g) :
     (𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g { y | (y, x) ∈ s } := by
   rw [nhds_eq_comap_uniformity', comap_lift_eq2 hg]
-  simp_rw [Function.comp, preimage]
+  simp_rw [Function.comp_def, preimage]
 
 theorem nhds_nhds_eq_uniformity_uniformity_prod {a b : α} :
     𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s : Set (α × α) =>
@@ -891,7 +892,6 @@ lemma DenseRange.iUnion_uniformity_ball {ι : Type*} {xs : ι → α}
 ### Uniformity bases
 -/
 
-
 /-- Open elements of `𝓤 α` form a basis of `𝓤 α`. -/
 theorem uniformity_hasBasis_open : HasBasis (𝓤 α) (fun V : Set (α × α) => V ∈ 𝓤 α ∧ IsOpen V) id :=
   hasBasis_self.2 fun s hs =>
@@ -1110,12 +1110,17 @@ abbrev UniformSpace.comap (f : α → β) (u : UniformSpace β) : UniformSpace 
     (comap_mono u.comp)
   toTopologicalSpace := u.toTopologicalSpace.induced f
   nhds_eq_comap_uniformity x := by
-    simp only [nhds_induced, nhds_eq_comap_uniformity, comap_comap, Function.comp]
+    simp only [nhds_induced, nhds_eq_comap_uniformity, comap_comap, Function.comp_def]
 
 theorem uniformity_comap {_ : UniformSpace β} (f : α → β) :
     𝓤[UniformSpace.comap f ‹_›] = comap (Prod.map f f) (𝓤 β) :=
   rfl
 
+lemma ball_preimage {f : α → β} {U : Set (β × β)} {x : α} :
+    UniformSpace.ball x (Prod.map f f ⁻¹' U) = f ⁻¹' UniformSpace.ball (f x) U := by
+  ext : 1
+  simp only [UniformSpace.ball, mem_preimage, Prod.map_apply]
+
 @[simp]
 theorem uniformSpace_comap_id {α : Type*} : UniformSpace.comap (id : α → α) = id := by
   ext : 2
@@ -1459,10 +1464,12 @@ theorem UniformContinuous.prod_mk_right {f : α × β → γ} (h : UniformContin
     UniformContinuous fun b => f (a, b) :=
   h.comp (uniformContinuous_const.prod_mk uniformContinuous_id)
 
-theorem UniformContinuous.prod_map [UniformSpace δ] {f : α → γ} {g : β → δ}
+theorem UniformContinuous.prodMap [UniformSpace δ] {f : α → γ} {g : β → δ}
     (hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous (Prod.map f g) :=
   (hf.comp uniformContinuous_fst).prod_mk (hg.comp uniformContinuous_snd)
 
+@[deprecated (since := "2024-10-06")] alias UniformContinuous.prod_map := UniformContinuous.prodMap
+
 theorem toTopologicalSpace_prod {α} {β} [u : UniformSpace α] [v : UniformSpace β] :
     @UniformSpace.toTopologicalSpace (α × β) instUniformSpaceProd =
       @instTopologicalSpaceProd α β u.toTopologicalSpace v.toTopologicalSpace :=
@@ -1478,7 +1485,7 @@ theorem uniformContinuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ua1
   have ha := @UniformContinuous.inf_dom_left _ _ id ua1 ua2 ua1 (@uniformContinuous_id _ (id _))
   have hb := @UniformContinuous.inf_dom_left _ _ id ub1 ub2 ub1 (@uniformContinuous_id _ (id _))
   have h_unif_cont_id :=
-    @UniformContinuous.prod_map _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua1 ub1 _ _ ha hb
+    @UniformContinuous.prodMap _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua1 ub1 _ _ ha hb
   exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ h h_unif_cont_id
 
 /-- A version of `UniformContinuous.inf_dom_right` for binary functions -/
@@ -1491,7 +1498,7 @@ theorem uniformContinuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ua
   have ha := @UniformContinuous.inf_dom_right _ _ id ua1 ua2 ua2 (@uniformContinuous_id _ (id _))
   have hb := @UniformContinuous.inf_dom_right _ _ id ub1 ub2 ub2 (@uniformContinuous_id _ (id _))
   have h_unif_cont_id :=
-    @UniformContinuous.prod_map _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua2 ub2 _ _ ha hb
+    @UniformContinuous.prodMap _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua2 ub2 _ _ ha hb
   exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ h h_unif_cont_id
 
 /-- A version of `uniformContinuous_sInf_dom` for binary functions -/
@@ -1504,7 +1511,7 @@ theorem uniformContinuous_sInf_dom₂ {α β γ} {f : α → β → γ} {uas : S
   let _ : UniformSpace (α × β) := instUniformSpaceProd
   have ha := uniformContinuous_sInf_dom ha uniformContinuous_id
   have hb := uniformContinuous_sInf_dom hb uniformContinuous_id
-  have h_unif_cont_id := @UniformContinuous.prod_map _ _ _ _ (sInf uas) (sInf ubs) ua ub _ _ ha hb
+  have h_unif_cont_id := @UniformContinuous.prodMap _ _ _ _ (sInf uas) (sInf ubs) ua ub _ _ ha hb
   exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ hf h_unif_cont_id
 
 end Prod
@@ -1540,7 +1547,7 @@ theorem UniformContinuous₂.comp {f : α → β → γ} {g : γ → δ} (hg : U
 theorem UniformContinuous₂.bicompl {f : α → β → γ} {ga : δ → α} {gb : δ' → β}
     (hf : UniformContinuous₂ f) (hga : UniformContinuous ga) (hgb : UniformContinuous gb) :
     UniformContinuous₂ (bicompl f ga gb) :=
-  hf.uniformContinuous.comp (hga.prod_map hgb)
+  hf.uniformContinuous.comp (hga.prodMap hgb)
 
 end
 
@@ -1732,7 +1739,7 @@ theorem continuousAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} :
 
 theorem continuousAt_iff_prod [TopologicalSpace β] {f : β → α} {b : β} :
     ContinuousAt f b ↔ Tendsto (fun x : β × β => (f x.1, f x.2)) (𝓝 (b, b)) (𝓤 α) :=
-  ⟨fun H => le_trans (H.prod_map' H) (nhds_le_uniformity _), fun H =>
+  ⟨fun H => le_trans (H.prodMap' H) (nhds_le_uniformity _), fun H =>
     continuousAt_iff'_left.2 <| H.comp <| tendsto_id.prod_mk_nhds tendsto_const_nhds⟩
 
 theorem continuousWithinAt_iff'_right [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
@@ -1793,3 +1800,5 @@ theorem Filter.Tendsto.congr_uniformity {α β} [UniformSpace β] {f g : α →
 theorem Uniform.tendsto_congr {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β}
     (hfg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto f l (𝓝 b) ↔ Tendsto g l (𝓝 b) :=
   ⟨fun h => h.congr_uniformity hfg, fun h => h.congr_uniformity hfg.uniformity_symm⟩
+
+set_option linter.style.longFile 1900
diff --git a/Mathlib/Topology/UniformSpace/Cauchy.lean b/Mathlib/Topology/UniformSpace/Cauchy.lean
index c22dc44a432b7..b2be7ed715fd7 100644
--- a/Mathlib/Topology/UniformSpace/Cauchy.lean
+++ b/Mathlib/Topology/UniformSpace/Cauchy.lean
@@ -6,6 +6,7 @@ Authors: Johannes Hölzl, Mario Carneiro
 import Mathlib.Topology.Algebra.Constructions
 import Mathlib.Topology.Bases
 import Mathlib.Topology.UniformSpace.Basic
+import Mathlib.Algebra.Order.Group.Nat
 
 /-!
 # Theory of Cauchy filters in uniform spaces. Complete uniform spaces. Totally bounded subsets.
@@ -207,7 +208,7 @@ theorem Function.Bijective.cauchySeq_comp_iff {f : ℕ → ℕ} (hf : Bijective
     CauchySeq (u ∘ f) ↔ CauchySeq u := by
   refine ⟨fun H => ?_, fun H => H.comp_injective hf.injective⟩
   lift f to ℕ ≃ ℕ using hf
-  simpa only [(· ∘ ·), f.apply_symm_apply] using H.comp_injective f.symm.injective
+  simpa only [Function.comp_def, f.apply_symm_apply] using H.comp_injective f.symm.injective
 
 theorem CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
     (hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
@@ -249,7 +250,7 @@ theorem CauchySeq.subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 
     exact ⟨N, fun k hk l hl => H _ (le_trans hk hl) _ hk⟩
   obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ∀ l ≥ φ n, (u l, u <| φ n) ∈ V n⟩ :=
     extraction_forall_of_eventually' this
-  exact ⟨φ, φ_extr, fun n => hφ _ _ (φ_extr <| lt_add_one n).le⟩
+  exact ⟨φ, φ_extr, fun n => hφ _ _ (φ_extr <| Nat.lt_add_one n).le⟩
 
 theorem Filter.Tendsto.subseq_mem_entourage {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
     {a : α} (hu : Tendsto u atTop (𝓝 a)) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧
@@ -263,7 +264,7 @@ theorem Filter.Tendsto.subseq_mem_entourage {V : ℕ → Set (α × α)} (hV : 
 theorem tendsto_nhds_of_cauchySeq_of_subseq [Preorder β] {u : β → α} (hu : CauchySeq u)
     {ι : Type*} {f : ι → β} {p : Filter ι} [NeBot p] (hf : Tendsto f p atTop) {a : α}
     (ha : Tendsto (u ∘ f) p (𝓝 a)) : Tendsto u atTop (𝓝 a) :=
-  le_nhds_of_cauchy_adhp hu (mapClusterPt_of_comp hf ha)
+  le_nhds_of_cauchy_adhp hu (ha.mapClusterPt.of_comp hf)
 
 /-- Any shift of a Cauchy sequence is also a Cauchy sequence. -/
 theorem cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u := by
@@ -281,7 +282,7 @@ theorem Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u
     CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i := by
   rw [cauchySeq_iff_tendsto, ← prod_atTop_atTop_eq]
   refine (atTop_basis.prod_self.tendsto_iff h).trans ?_
-  simp only [exists_prop, true_and_iff, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq,
+  simp only [exists_prop, true_and, MapsTo, preimage, subset_def, Prod.forall, mem_prod_eq,
     mem_setOf_eq, mem_Ici, and_imp, Prod.map, @forall_swap (_ ≤ _) β]
 
 theorem Filter.HasBasis.cauchySeq_iff' {γ} [Nonempty β] [SemilatticeSup β] {u : β → α}
@@ -786,4 +787,29 @@ theorem secondCountable_of_separable [SeparableSpace α] : SecondCountableTopolo
     refine ⟨_, ⟨y, hys, k, rfl⟩, (hts k).subset hxy, fun z hz => ?_⟩
     exact hUV (ball_subset_of_comp_subset (hk hxy) hUU' (hk hz))
 
+section DiscreteUniformity
+
+open Filter
+
+/-- A Cauchy filter in a discrete uniform space is contained in a principal filter-/
+theorem DiscreteUnif.cauchy_le_pure {X : Type _} {uX : UniformSpace X}
+    (hX : uX = ⊥) {α : Filter X} (hα : Cauchy α) : ∃ x : X, α = pure x := by
+  rcases hα with ⟨α_ne_bot, α_le⟩
+  rw [hX, bot_uniformity, le_principal_iff, mem_prod_iff] at α_le
+  obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := α_le
+  obtain ⟨x, rfl⟩ := eq_singleton_left_of_prod_subset_idRel (α_ne_bot.nonempty_of_mem hS)
+    (Filter.nonempty_of_mem hT) H
+  exact ⟨x, α_ne_bot.le_pure_iff.mp <| le_pure_iff.mpr hS⟩
+
+/-- A constant to which a Cauchy filter in a discrete uniform space converges. -/
+noncomputable def DiscreteUnif.cauchyConst {X : Type _} {uX : UniformSpace X}
+    (hX : uX = ⊥) {α : Filter X} (hα : Cauchy α) : X :=
+  (DiscreteUnif.cauchy_le_pure hX hα).choose
+
+theorem DiscreteUnif.eq_const_of_cauchy {X : Type _} {uX : UniformSpace X} (hX : uX = ⊥)
+    {α : Filter X} (hα : Cauchy α) : α = pure (DiscreteUnif.cauchyConst hX hα) :=
+  (DiscreteUnif.cauchy_le_pure hX hα).choose_spec
+
+end DiscreteUniformity
+
 end UniformSpace
diff --git a/Mathlib/Topology/UniformSpace/Compact.lean b/Mathlib/Topology/UniformSpace/Compact.lean
index 6c6f0ab1644d7..175d8d75ed946 100644
--- a/Mathlib/Topology/UniformSpace/Compact.lean
+++ b/Mathlib/Topology/UniformSpace/Compact.lean
@@ -134,14 +134,14 @@ def uniformSpaceOfCompactT2 [TopologicalSpace γ] [CompactSpace γ] [T2Space γ]
     -- So we have a contradiction
     exact hU₁₂.le_bot ⟨uw_in.2, wv_in.1⟩
   nhds_eq_comap_uniformity x := by
-    simp_rw [nhdsSet_diagonal, comap_iSup, nhds_prod_eq, comap_prod, (· ∘ ·), comap_id']
+    simp_rw [nhdsSet_diagonal, comap_iSup, nhds_prod_eq, comap_prod, Function.comp_def, comap_id']
     rw [iSup_split_single _ x, comap_const_of_mem fun V => mem_of_mem_nhds]
     suffices ∀ y ≠ x, comap (fun _ : γ ↦ x) (𝓝 y) ⊓ 𝓝 y ≤ 𝓝 x by simpa
     intro y hxy
     simp [comap_const_of_not_mem (compl_singleton_mem_nhds hxy) (not_not_intro rfl)]
 
 /-!
-### Heine-Cantor theorem
+### Heine-Cantor theorem
 -/
 
 
@@ -151,7 +151,7 @@ theorem CompactSpace.uniformContinuous_of_continuous [CompactSpace α] {f : α 
     (h : Continuous f) : UniformContinuous f :=
 calc map (Prod.map f f) (𝓤 α)
    = map (Prod.map f f) (𝓝ˢ (diagonal α)) := by rw [nhdsSet_diagonal_eq_uniformity]
- _ ≤ 𝓝ˢ (diagonal β)                      := (h.prod_map h).tendsto_nhdsSet mapsTo_prod_map_diagonal
+ _ ≤ 𝓝ˢ (diagonal β)                      := (h.prodMap h).tendsto_nhdsSet mapsTo_prod_map_diagonal
  _ ≤ 𝓤 β                                  := nhdsSet_diagonal_le_uniformity
 
 /-- Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly
diff --git a/Mathlib/Topology/UniformSpace/CompactConvergence.lean b/Mathlib/Topology/UniformSpace/CompactConvergence.lean
index dabba67db0cd2..f13aaeff613af 100644
--- a/Mathlib/Topology/UniformSpace/CompactConvergence.lean
+++ b/Mathlib/Topology/UniformSpace/CompactConvergence.lean
@@ -163,7 +163,7 @@ which defines topology of uniform convergence on compact sets.
 We use `ContinuousMap.tendsto_iff_forall_compact_tendstoUniformlyOn`
 to show that the induced topology agrees with the compact-open topology
 and replace the topology with `compactOpen` to avoid non-defeq diamonds,
-see Note [forgetful inheritance].  -/
+see Note [forgetful inheritance]. -/
 instance compactConvergenceUniformSpace : UniformSpace C(α, β) :=
   .replaceTopology (.comap toUniformOnFunIsCompact inferInstance) <| by
     refine TopologicalSpace.ext_nhds fun f ↦ eq_of_forall_le_iff fun l ↦ ?_
@@ -171,11 +171,14 @@ instance compactConvergenceUniformSpace : UniformSpace C(α, β) :=
       nhds_induced, tendsto_comap_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]
     rfl
 
-theorem uniformEmbedding_toUniformOnFunIsCompact :
-    UniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) where
+theorem isUniformEmbedding_toUniformOnFunIsCompact :
+    IsUniformEmbedding (toUniformOnFunIsCompact : C(α, β) → α →ᵤ[{K | IsCompact K}] β) where
   comap_uniformity := rfl
   inj := DFunLike.coe_injective
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_toUniformOnFunIsCompact := isUniformEmbedding_toUniformOnFunIsCompact
+
 -- The following definitions and theorems
 -- used to be a part of the construction of the `UniformSpace C(α, β)` structure
 -- before it was migrated to `UniformOnFun`
@@ -184,7 +187,7 @@ theorem _root_.Filter.HasBasis.compactConvergenceUniformity {ι : Type*} {pi : 
     {s : ι → Set (β × β)} (h : (𝓤 β).HasBasis pi s) :
     HasBasis (𝓤 C(α, β)) (fun p : Set α × ι => IsCompact p.1 ∧ pi p.2) fun p =>
       { fg : C(α, β) × C(α, β) | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ s p.2 } := by
-  rw [← uniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]
+  rw [← isUniformEmbedding_toUniformOnFunIsCompact.comap_uniformity]
   exact .comap _ <| UniformOnFun.hasBasis_uniformity_of_basis _ _ {K | IsCompact K}
     ⟨∅, isCompact_empty⟩ (directedOn_of_sup_mem fun _ _ ↦ IsCompact.union) h
 
@@ -260,27 +263,33 @@ variable {γ δ : Type*} [TopologicalSpace γ] [UniformSpace δ]
 
 theorem uniformContinuous_comp (g : C(β, δ)) (hg : UniformContinuous g) :
     UniformContinuous (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
-  uniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
+  isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
     UniformOnFun.postcomp_uniformContinuous hg |>.comp
-      uniformEmbedding_toUniformOnFunIsCompact.uniformContinuous
+      isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous
+
+theorem isUniformInducing_comp (g : C(β, δ)) (hg : IsUniformInducing g) :
+    IsUniformInducing (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
+  isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing.of_comp_iff.mp <|
+    UniformOnFun.postcomp_isUniformInducing hg |>.comp
+      isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_comp := isUniformInducing_comp
 
-theorem uniformInducing_comp (g : C(β, δ)) (hg : UniformInducing g) :
-    UniformInducing (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
-  uniformEmbedding_toUniformOnFunIsCompact.toUniformInducing.of_comp_iff.mp <|
-    UniformOnFun.postcomp_uniformInducing hg |>.comp
-      uniformEmbedding_toUniformOnFunIsCompact.toUniformInducing
+theorem isUniformEmbedding_comp (g : C(β, δ)) (hg : IsUniformEmbedding g) :
+    IsUniformEmbedding (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
+  isUniformEmbedding_toUniformOnFunIsCompact.of_comp_iff.mp <|
+    UniformOnFun.postcomp_isUniformEmbedding hg |>.comp
+      isUniformEmbedding_toUniformOnFunIsCompact
 
-theorem uniformEmbedding_comp (g : C(β, δ)) (hg : UniformEmbedding g) :
-    UniformEmbedding (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
-  uniformEmbedding_toUniformOnFunIsCompact.of_comp_iff.mp <|
-    UniformOnFun.postcomp_uniformEmbedding hg |>.comp
-      uniformEmbedding_toUniformOnFunIsCompact
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_comp := isUniformEmbedding_comp
 
 theorem uniformContinuous_comp_left (g : C(α, γ)) :
     UniformContinuous (fun f ↦ f.comp g : C(γ, β) → C(α, β)) :=
-  uniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
+  isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
     UniformOnFun.precomp_uniformContinuous (fun _ hK ↦ hK.image g.continuous) |>.comp
-      uniformEmbedding_toUniformOnFunIsCompact.uniformContinuous
+      isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous
 
 /-- Any pair of a homeomorphism `X ≃ₜ Z` and an isomorphism `Y ≃ᵤ T` of uniform spaces gives rise
 to an isomorphism `C(X, Y) ≃ᵤ C(Z, T)`. -/
@@ -334,7 +343,7 @@ theorem uniformSpace_eq_inf_precomp_of_cover {δ₁ δ₂ : Type*} [TopologicalS
   have h_preimage₂ : MapsTo (φ₂ ⁻¹' ·) 𝔖 𝔗₂ := fun K ↦ h_proper₂.isCompact_preimage
   have h_cover' : ∀ S ∈ 𝔖, S ⊆ range φ₁ ∪ range φ₂ := fun S _ ↦ h_cover ▸ subset_univ _
   -- ... and we just pull it back.
-  simp_rw [compactConvergenceUniformSpace, replaceTopology_eq, inferInstanceAs, inferInstance,
+  simp_rw [compactConvergenceUniformSpace, replaceTopology_eq,
     UniformOnFun.uniformSpace_eq_inf_precomp_of_cover _ _ _ _ _
       h_image₁ h_image₂ h_preimage₁ h_preimage₂ h_cover',
     UniformSpace.comap_inf, ← UniformSpace.comap_comap]
@@ -355,7 +364,7 @@ theorem uniformSpace_eq_iInf_precomp_of_cover {δ : ι → Type*} [∀ i, Topolo
       inter_eq_right.mp ?_⟩
     simp_rw [iUnion₂_inter, mem_setOf, iUnion_nonempty_self, ← iUnion_inter, h_cover, univ_inter]
   -- ... and we just pull it back.
-  simp_rw [compactConvergenceUniformSpace, replaceTopology_eq, inferInstanceAs, inferInstance,
+  simp_rw [compactConvergenceUniformSpace, replaceTopology_eq,
     UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover _ _ _ h_image h_preimage h_cover',
     UniformSpace.comap_iInf, ← UniformSpace.comap_comap]
   rfl
@@ -372,7 +381,8 @@ Sufficient conditions on `α` to satisfy this condition are (weak) local compact
 `ContinuousMap.instCompleteSpaceOfSequentialSpace`). -/
 lemma completeSpace_of_restrictGenTopology (h : RestrictGenTopology {K : Set α | IsCompact K}) :
     CompleteSpace C(α, β) := by
-  rw [completeSpace_iff_isComplete_range uniformEmbedding_toUniformOnFunIsCompact.toUniformInducing,
+  rw [completeSpace_iff_isComplete_range
+    isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing,
     range_toUniformOnFunIsCompact, ← completeSpace_coe_iff_isComplete]
   exact (UniformOnFun.isClosed_setOf_continuous h).completeSpace_coe
 
diff --git a/Mathlib/Topology/UniformSpace/CompareReals.lean b/Mathlib/Topology/UniformSpace/CompareReals.lean
index 709fc27902447..c443d0a470a42 100644
--- a/Mathlib/Topology/UniformSpace/CompareReals.lean
+++ b/Mathlib/Topology/UniformSpace/CompareReals.lean
@@ -70,10 +70,10 @@ def rationalCauSeqPkg : @AbstractCompletion ℚ <| (@AbsoluteValue.abs ℚ _).un
     (uniformStruct := by infer_instance)
     (complete := by infer_instance)
     (separation := by infer_instance)
-    (uniformInducing := by
+    (isUniformInducing := by
       rw [Rat.uniformSpace_eq]
-      exact Rat.uniformEmbedding_coe_real.toUniformInducing)
-    (dense := Rat.denseEmbedding_coe_real.dense)
+      exact Rat.isUniformEmbedding_coe_real.isUniformInducing)
+    (dense := Rat.isDenseEmbedding_coe_real.dense)
 
 namespace CompareReals
 
diff --git a/Mathlib/Topology/UniformSpace/CompleteSeparated.lean b/Mathlib/Topology/UniformSpace/CompleteSeparated.lean
index 79943bfbcdbcf..7444f5a467be3 100644
--- a/Mathlib/Topology/UniformSpace/CompleteSeparated.lean
+++ b/Mathlib/Topology/UniformSpace/CompleteSeparated.lean
@@ -27,20 +27,23 @@ theorem IsComplete.isClosed [UniformSpace α] [T0Space α] {s : Set α} (h : IsC
     rcases h f this inf_le_right with ⟨y, ys, fy⟩
     rwa [(tendsto_nhds_unique' ha inf_le_left fy : a = y)]
 
-theorem UniformEmbedding.toClosedEmbedding [UniformSpace α] [UniformSpace β] [CompleteSpace α]
-    [T0Space β] {f : α → β} (hf : UniformEmbedding f) :
+theorem IsUniformEmbedding.toClosedEmbedding [UniformSpace α] [UniformSpace β] [CompleteSpace α]
+    [T0Space β] {f : α → β} (hf : IsUniformEmbedding f) :
     ClosedEmbedding f :=
-  ⟨hf.embedding, hf.toUniformInducing.isComplete_range.isClosed⟩
+  ⟨hf.embedding, hf.isUniformInducing.isComplete_range.isClosed⟩
 
-namespace DenseInducing
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.toClosedEmbedding := IsUniformEmbedding.toClosedEmbedding
+
+namespace IsDenseInducing
 
 open Filter
 
 variable [TopologicalSpace α] {β : Type*} [TopologicalSpace β]
 variable {γ : Type*} [UniformSpace γ] [CompleteSpace γ] [T0Space γ]
 
-theorem continuous_extend_of_cauchy {e : α → β} {f : α → γ} (de : DenseInducing e)
+theorem continuous_extend_of_cauchy {e : α → β} {f : α → γ} (de : IsDenseInducing e)
     (h : ∀ b : β, Cauchy (map f (comap e <| 𝓝 b))) : Continuous (de.extend f) :=
   de.continuous_extend fun b => CompleteSpace.complete (h b)
 
-end DenseInducing
+end IsDenseInducing
diff --git a/Mathlib/Topology/UniformSpace/Completion.lean b/Mathlib/Topology/UniformSpace/Completion.lean
index 6e6085fcbee0d..4d1365e733ea3 100644
--- a/Mathlib/Topology/UniformSpace/Completion.lean
+++ b/Mathlib/Topology/UniformSpace/Completion.lean
@@ -80,7 +80,7 @@ private theorem symm_gen : map Prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lif
         { p : CauchyFilter α × CauchyFilter α | s ∈ (p.2.val ×ˢ p.1.val : Filter (α × α)) }
   have h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f := by
     delta gen
-    simp [map_lift'_eq, monotone_setOf, Filter.monotone_mem, Function.comp,
+    simp [map_lift'_eq, monotone_setOf, Filter.monotone_mem, Function.comp_def,
       image_swap_eq_preimage_swap]
   have h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen :=
     uniformity_lift_le_swap
@@ -143,7 +143,7 @@ theorem mem_uniformity' {s : Set (CauchyFilter α × CauchyFilter α)} :
 def pureCauchy (a : α) : CauchyFilter α :=
   ⟨pure a, cauchy_pure⟩
 
-theorem uniformInducing_pureCauchy : UniformInducing (pureCauchy : α → CauchyFilter α) :=
+theorem isUniformInducing_pureCauchy : IsUniformInducing (pureCauchy : α → CauchyFilter α) :=
   ⟨have : (preimage fun x : α × α => (pureCauchy x.fst, pureCauchy x.snd)) ∘ gen = id :=
       funext fun s =>
         Set.ext fun ⟨a₁, a₂⟩ => by simp [preimage, gen, pureCauchy, prod_principal_principal]
@@ -154,10 +154,16 @@ theorem uniformInducing_pureCauchy : UniformInducing (pureCauchy : α → Cauchy
       _ = 𝓤 α := by simp [this]
       ⟩
 
-theorem uniformEmbedding_pureCauchy : UniformEmbedding (pureCauchy : α → CauchyFilter α) :=
-  { uniformInducing_pureCauchy with
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_pureCauchy := isUniformInducing_pureCauchy
+
+theorem isUniformEmbedding_pureCauchy : IsUniformEmbedding (pureCauchy : α → CauchyFilter α) :=
+  { isUniformInducing_pureCauchy with
     inj := fun _a₁ _a₂ h => pure_injective <| Subtype.ext_iff_val.1 h }
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_pureCauchy := isUniformEmbedding_pureCauchy
+
 theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α) := fun f => by
   have h_ex : ∀ s ∈ 𝓤 (CauchyFilter α), ∃ y : α, (f, pureCauchy y) ∈ s := fun s hs =>
     let ⟨t'', ht''₁, (ht''₂ : gen t'' ⊆ s)⟩ := (mem_lift'_sets monotone_gen).mp hs
@@ -180,15 +186,18 @@ theorem denseRange_pureCauchy : DenseRange (pureCauchy : α → CauchyFilter α)
       ⟨mem_range_self y, hy⟩
     exact ⟨_, this⟩
 
-theorem denseInducing_pureCauchy : DenseInducing (pureCauchy : α → CauchyFilter α) :=
-  uniformInducing_pureCauchy.denseInducing denseRange_pureCauchy
+theorem isDenseInducing_pureCauchy : IsDenseInducing (pureCauchy : α → CauchyFilter α) :=
+  isUniformInducing_pureCauchy.isDenseInducing denseRange_pureCauchy
+
+theorem isDenseEmbedding_pureCauchy : IsDenseEmbedding (pureCauchy : α → CauchyFilter α) :=
+  isUniformEmbedding_pureCauchy.isDenseEmbedding denseRange_pureCauchy
 
-theorem denseEmbedding_pureCauchy : DenseEmbedding (pureCauchy : α → CauchyFilter α) :=
-  uniformEmbedding_pureCauchy.denseEmbedding denseRange_pureCauchy
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_pureCauchy := isDenseEmbedding_pureCauchy
 
 theorem nonempty_cauchyFilter_iff : Nonempty (CauchyFilter α) ↔ Nonempty α := by
   constructor <;> rintro ⟨c⟩
-  · have := eq_univ_iff_forall.1 denseEmbedding_pureCauchy.toDenseInducing.closure_range c
+  · have := eq_univ_iff_forall.1 isDenseEmbedding_pureCauchy.toIsDenseInducing.closure_range c
     obtain ⟨_, ⟨_, a, _⟩⟩ := mem_closure_iff.1 this _ isOpen_univ trivial
     exact ⟨a⟩
   · exact ⟨pureCauchy c⟩
@@ -199,7 +208,7 @@ section
 -- set_option eqn_compiler.zeta true
 
 instance : CompleteSpace (CauchyFilter α) :=
-  completeSpace_extension uniformInducing_pureCauchy denseRange_pureCauchy fun f hf =>
+  completeSpace_extension isUniformInducing_pureCauchy denseRange_pureCauchy fun f hf =>
     let f' : CauchyFilter α := ⟨f, hf⟩
     have : map pureCauchy f ≤ (𝓤 <| CauchyFilter α).lift' (preimage (Prod.mk f')) :=
       le_lift'.2 fun s hs =>
@@ -224,7 +233,7 @@ open Classical in
 /-- Extend a uniformly continuous function `α → β` to a function `CauchyFilter α → β`.
 Outputs junk when `f` is not uniformly continuous. -/
 def extend (f : α → β) : CauchyFilter α → β :=
-  if UniformContinuous f then denseInducing_pureCauchy.extend f
+  if UniformContinuous f then isDenseInducing_pureCauchy.extend f
   else fun x => f (nonempty_cauchyFilter_iff.1 ⟨x⟩).some
 
 section T0Space
@@ -234,7 +243,7 @@ variable [T0Space β]
 theorem extend_pureCauchy {f : α → β} (hf : UniformContinuous f) (a : α) :
     extend f (pureCauchy a) = f a := by
   rw [extend, if_pos hf]
-  exact uniformly_extend_of_ind uniformInducing_pureCauchy denseRange_pureCauchy hf _
+  exact uniformly_extend_of_ind isUniformInducing_pureCauchy denseRange_pureCauchy hf _
 
 end T0Space
 
@@ -243,7 +252,7 @@ variable [CompleteSpace β]
 theorem uniformContinuous_extend {f : α → β} : UniformContinuous (extend f) := by
   by_cases hf : UniformContinuous f
   · rw [extend, if_pos hf]
-    exact uniformContinuous_uniformly_extend uniformInducing_pureCauchy denseRange_pureCauchy hf
+    exact uniformContinuous_uniformly_extend isUniformInducing_pureCauchy denseRange_pureCauchy hf
   · rw [extend, if_neg hf]
     exact uniformContinuous_of_const fun a _b => by congr
 
@@ -316,12 +325,15 @@ instance : Coe α (Completion α) :=
 -- note [use has_coe_t]
 protected theorem coe_eq : ((↑) : α → Completion α) = SeparationQuotient.mk ∘ pureCauchy := rfl
 
-theorem uniformInducing_coe : UniformInducing ((↑) : α → Completion α) :=
-  SeparationQuotient.uniformInducing_mk.comp uniformInducing_pureCauchy
+theorem isUniformInducing_coe : IsUniformInducing ((↑) : α → Completion α) :=
+  SeparationQuotient.isUniformInducing_mk.comp isUniformInducing_pureCauchy
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_coe := isUniformInducing_coe
 
 theorem comap_coe_eq_uniformity :
     ((𝓤 _).comap fun p : α × α => ((p.1 : Completion α), (p.2 : Completion α))) = 𝓤 α :=
-  (uniformInducing_coe _).1
+  (isUniformInducing_coe _).1
 
 variable {α}
 
@@ -338,7 +350,7 @@ def cPkg {α : Type*} [UniformSpace α] : AbstractCompletion α where
   uniformStruct := by infer_instance
   complete := by infer_instance
   separation := by infer_instance
-  uniformInducing := Completion.uniformInducing_coe α
+  isUniformInducing := Completion.isUniformInducing_coe α
   dense := Completion.denseRange_coe
 
 instance AbstractCompletion.inhabited : Inhabited (AbstractCompletion α) :=
@@ -356,17 +368,20 @@ theorem uniformContinuous_coe : UniformContinuous ((↑) : α → Completion α)
 theorem continuous_coe : Continuous ((↑) : α → Completion α) :=
   cPkg.continuous_coe
 
-theorem uniformEmbedding_coe [T0Space α] : UniformEmbedding ((↑) : α → Completion α) :=
+theorem isUniformEmbedding_coe [T0Space α] : IsUniformEmbedding ((↑) : α → Completion α) :=
   { comap_uniformity := comap_coe_eq_uniformity α
     inj := separated_pureCauchy_injective }
 
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_coe := isUniformEmbedding_coe
+
 theorem coe_injective [T0Space α] : Function.Injective ((↑) : α → Completion α) :=
-  UniformEmbedding.inj (uniformEmbedding_coe _)
+  IsUniformEmbedding.inj (isUniformEmbedding_coe _)
 
 variable {α}
 
-theorem denseInducing_coe : DenseInducing ((↑) : α → Completion α) :=
-  { (uniformInducing_coe α).inducing with dense := denseRange_coe }
+theorem isDenseInducing_coe : IsDenseInducing ((↑) : α → Completion α) :=
+  { (isUniformInducing_coe α).inducing with dense := denseRange_coe }
 
 /-- The uniform bijection between a complete space and its uniform completion. -/
 def UniformCompletion.completeEquivSelf [CompleteSpace α] [T0Space α] : Completion α ≃ᵤ α :=
@@ -375,19 +390,22 @@ def UniformCompletion.completeEquivSelf [CompleteSpace α] [T0Space α] : Comple
 open TopologicalSpace
 
 instance separableSpace_completion [SeparableSpace α] : SeparableSpace (Completion α) :=
-  Completion.denseInducing_coe.separableSpace
+  Completion.isDenseInducing_coe.separableSpace
+
+theorem isDenseEmbedding_coe [T0Space α] : IsDenseEmbedding ((↑) : α → Completion α) :=
+  { isDenseInducing_coe with inj := separated_pureCauchy_injective }
 
-theorem denseEmbedding_coe [T0Space α] : DenseEmbedding ((↑) : α → Completion α) :=
-  { denseInducing_coe with inj := separated_pureCauchy_injective }
+@[deprecated (since := "2024-09-30")]
+alias denseEmbedding_coe := isDenseEmbedding_coe
 
 theorem denseRange_coe₂ :
     DenseRange fun x : α × β => ((x.1 : Completion α), (x.2 : Completion β)) :=
-  denseRange_coe.prod_map denseRange_coe
+  denseRange_coe.prodMap denseRange_coe
 
 theorem denseRange_coe₃ :
     DenseRange fun x : α × β × γ =>
       ((x.1 : Completion α), ((x.2.1 : Completion β), (x.2.2 : Completion γ))) :=
-  denseRange_coe.prod_map denseRange_coe₂
+  denseRange_coe.prodMap denseRange_coe₂
 
 @[elab_as_elim]
 theorem induction_on {p : Completion α → Prop} (a : Completion α) (hp : IsClosed { a | p a })
diff --git a/Mathlib/Topology/UniformSpace/Equicontinuity.lean b/Mathlib/Topology/UniformSpace/Equicontinuity.lean
index 5fdd4c239966d..807c30865e06d 100644
--- a/Mathlib/Topology/UniformSpace/Equicontinuity.lean
+++ b/Mathlib/Topology/UniformSpace/Equicontinuity.lean
@@ -704,56 +704,74 @@ theorem Filter.HasBasis.uniformEquicontinuousOn_iff {κ₁ κ₂ : Type*} {p₁
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
 equicontinuous at `x₀`. -/
-theorem UniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
-    (hu : UniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by
-  have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
+theorem IsUniformInducing.equicontinuousAt_iff {F : ι → X → α} {x₀ : X} {u : α → β}
+    (hu : IsUniformInducing u) : EquicontinuousAt F x₀ ↔ EquicontinuousAt ((u ∘ ·) ∘ F) x₀ := by
+  have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).inducing
   rw [equicontinuousAt_iff_continuousAt, equicontinuousAt_iff_continuousAt, this.continuousAt_iff]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.equicontinuousAt_iff := IsUniformInducing.equicontinuousAt_iff
+
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
 `x₀ : X` within a subset `S : Set X` iff the family `𝓕'`, obtained by composing each function
 of `𝓕` by `u`, is equicontinuous at `x₀` within `S`. -/
-theorem UniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
-    (hu : UniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
+lemma IsUniformInducing.equicontinuousWithinAt_iff {F : ι → X → α} {S : Set X} {x₀ : X} {u : α → β}
+    (hu : IsUniformInducing u) : EquicontinuousWithinAt F S x₀ ↔
       EquicontinuousWithinAt ((u ∘ ·) ∘ F) S x₀ := by
-  have := (UniformFun.postcomp_uniformInducing (α := ι) hu).inducing
+  have := (UniformFun.postcomp_isUniformInducing (α := ι) hu).inducing
   simp only [equicontinuousWithinAt_iff_continuousWithinAt, this.continuousWithinAt_iff]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.equicontinuousWithinAt_iff := IsUniformInducing.equicontinuousWithinAt_iff
+
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
 family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is equicontinuous. -/
-theorem UniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : UniformInducing u) :
+lemma IsUniformInducing.equicontinuous_iff {F : ι → X → α} {u : α → β} (hu : IsUniformInducing u) :
     Equicontinuous F ↔ Equicontinuous ((u ∘ ·) ∘ F) := by
   congrm ∀ x, ?_
   rw [hu.equicontinuousAt_iff]
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.equicontinuous_iff := IsUniformInducing.equicontinuous_iff
+
 /-- Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous on a
 subset `S : Set X` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is
 equicontinuous on `S`. -/
-theorem UniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β}
-    (hu : UniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S := by
+theorem IsUniformInducing.equicontinuousOn_iff {F : ι → X → α} {S : Set X} {u : α → β}
+    (hu : IsUniformInducing u) : EquicontinuousOn F S ↔ EquicontinuousOn ((u ∘ ·) ∘ F) S := by
   congrm ∀ x ∈ S, ?_
   rw [hu.equicontinuousWithinAt_iff]
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.equicontinuousOn_iff := IsUniformInducing.equicontinuousOn_iff
+
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`, is uniformly
 equicontinuous. -/
-theorem UniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
-    (hu : UniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by
-  have := UniformFun.postcomp_uniformInducing (α := ι) hu
+theorem IsUniformInducing.uniformEquicontinuous_iff {F : ι → β → α} {u : α → γ}
+    (hu : IsUniformInducing u) : UniformEquicontinuous F ↔ UniformEquicontinuous ((u ∘ ·) ∘ F) := by
+  have := UniformFun.postcomp_isUniformInducing (α := ι) hu
   simp only [uniformEquicontinuous_iff_uniformContinuous, this.uniformContinuous_iff]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformEquicontinuous_iff := IsUniformInducing.uniformEquicontinuous_iff
+
 /-- Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
 on a subset `S : Set β` iff the family `𝓕'`, obtained by composing each function of `𝓕` by `u`,
 is uniformly equicontinuous on `S`. -/
-theorem UniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
-    (hu : UniformInducing u) :
+theorem IsUniformInducing.uniformEquicontinuousOn_iff {F : ι → β → α} {S : Set β} {u : α → γ}
+    (hu : IsUniformInducing u) :
     UniformEquicontinuousOn F S ↔ UniformEquicontinuousOn ((u ∘ ·) ∘ F) S := by
-  have := UniformFun.postcomp_uniformInducing (α := ι) hu
+  have := UniformFun.postcomp_isUniformInducing (α := ι) hu
   simp only [uniformEquicontinuousOn_iff_uniformContinuousOn, this.uniformContinuousOn_iff]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformEquicontinuousOn_iff := IsUniformInducing.uniformEquicontinuousOn_iff
+
 /-- If a set of functions is equicontinuous at some `x₀` within a set `S`, the same is true for its
 closure in *any* topology for which evaluation at any `x ∈ S ∪ {x₀}` is continuous. Since
 this will be applied to `DFunLike` types, we state it for any topological space with a map
diff --git a/Mathlib/Topology/UniformSpace/Equiv.lean b/Mathlib/Topology/UniformSpace/Equiv.lean
index d4be20cb620ad..6fb396f7b0004 100644
--- a/Mathlib/Topology/UniformSpace/Equiv.lean
+++ b/Mathlib/Topology/UniformSpace/Equiv.lean
@@ -49,11 +49,11 @@ theorem toEquiv_injective : Function.Injective (toEquiv : α ≃ᵤ β → α 
   | ⟨e, h₁, h₂⟩, ⟨e', h₁', h₂'⟩, h => by simpa only [mk.injEq]
 
 instance : EquivLike (α ≃ᵤ β) α β where
-  coe := fun h => h.toEquiv
-  inv := fun h => h.toEquiv.symm
-  left_inv := fun h => h.left_inv
-  right_inv := fun h => h.right_inv
-  coe_injective' := fun _ _ H _ => toEquiv_injective <| DFunLike.ext' H
+  coe h := h.toEquiv
+  inv h := h.toEquiv.symm
+  left_inv h := h.left_inv
+  right_inv h := h.right_inv
+  coe_injective' _ _ H _ := toEquiv_injective <| DFunLike.ext' H
 
 @[simp]
 theorem uniformEquiv_mk_coe (a : Equiv α β) (b c) : (UniformEquiv.mk a b c : α → β) = a :=
@@ -196,28 +196,35 @@ theorem image_preimage (h : α ≃ᵤ β) (s : Set β) : h '' (h ⁻¹' s) = s :
 theorem preimage_image (h : α ≃ᵤ β) (s : Set α) : h ⁻¹' (h '' s) = s :=
   h.toEquiv.preimage_image s
 
-protected theorem uniformInducing (h : α ≃ᵤ β) : UniformInducing h :=
-  uniformInducing_of_compose h.uniformContinuous h.symm.uniformContinuous <| by
-    simp only [symm_comp_self, uniformInducing_id]
+theorem isUniformInducing (h : α ≃ᵤ β) : IsUniformInducing h :=
+  IsUniformInducing.of_comp h.uniformContinuous h.symm.uniformContinuous <| by
+    simp only [symm_comp_self, IsUniformInducing.id]
+
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing := isUniformInducing
 
 theorem comap_eq (h : α ≃ᵤ β) : UniformSpace.comap h ‹_› = ‹_› :=
-  h.uniformInducing.comap_uniformSpace
+  h.isUniformInducing.comap_uniformSpace
+
+lemma isUniformEmbedding (h : α ≃ᵤ β) : IsUniformEmbedding h := ⟨h.isUniformInducing, h.injective⟩
 
-protected theorem uniformEmbedding (h : α ≃ᵤ β) : UniformEmbedding h :=
-  ⟨h.uniformInducing, h.injective⟩
+@[deprecated (since := "2024-10-01")] alias uniformEmbedding := isUniformEmbedding
 
 theorem completeSpace_iff (h : α ≃ᵤ β) : CompleteSpace α ↔ CompleteSpace β :=
-  completeSpace_congr h.uniformEmbedding
+  completeSpace_congr h.isUniformEmbedding
 
 /-- Uniform equiv given a uniform embedding. -/
-noncomputable def ofUniformEmbedding (f : α → β) (hf : UniformEmbedding f) : α ≃ᵤ Set.range f where
-  uniformContinuous_toFun := hf.toUniformInducing.uniformContinuous.subtype_mk _
+noncomputable def ofIsUniformEmbedding (f : α → β) (hf : IsUniformEmbedding f) :
+    α ≃ᵤ Set.range f where
+  uniformContinuous_toFun := hf.isUniformInducing.uniformContinuous.subtype_mk _
   uniformContinuous_invFun := by
-    rw [hf.toUniformInducing.uniformContinuous_iff, Equiv.invFun_as_coe,
+    rw [hf.isUniformInducing.uniformContinuous_iff, Equiv.invFun_as_coe,
       Equiv.self_comp_ofInjective_symm]
     exact uniformContinuous_subtype_val
   toEquiv := Equiv.ofInjective f hf.inj
 
+@[deprecated (since := "2024-10-03")] alias ofUniformEmbedding := ofIsUniformEmbedding
+
 /-- If two sets are equal, then they are uniformly equivalent. -/
 def setCongr {s t : Set α} (h : s = t) : s ≃ᵤ t where
   uniformContinuous_toFun := uniformContinuous_subtype_val.subtype_mk _
@@ -327,7 +334,7 @@ def ulift : ULift.{v, u} α ≃ᵤ α :=
   { Equiv.ulift with
     uniformContinuous_toFun := uniformContinuous_comap
     uniformContinuous_invFun := by
-      have hf : UniformInducing (@Equiv.ulift.{v, u} α).toFun := ⟨rfl⟩
+      have hf : IsUniformInducing (@Equiv.ulift.{v, u} α).toFun := ⟨rfl⟩
       simp_rw [hf.uniformContinuous_iff]
       exact uniformContinuous_id }
 
@@ -366,8 +373,8 @@ end UniformEquiv
 
 /-- A uniform inducing equiv between uniform spaces is a uniform isomorphism. -/
 -- @[simps] -- Porting note: removed, `simps?` produced no `simp` lemmas
-def Equiv.toUniformEquivOfUniformInducing [UniformSpace α] [UniformSpace β] (f : α ≃ β)
-    (hf : UniformInducing f) : α ≃ᵤ β :=
+def Equiv.toUniformEquivOfIsUniformInducing [UniformSpace α] [UniformSpace β] (f : α ≃ β)
+    (hf : IsUniformInducing f) : α ≃ᵤ β :=
   { f with
     uniformContinuous_toFun := hf.uniformContinuous
     uniformContinuous_invFun := hf.uniformContinuous_iff.2 <| by simpa using uniformContinuous_id }
diff --git a/Mathlib/Topology/UniformSpace/OfFun.lean b/Mathlib/Topology/UniformSpace/OfFun.lean
new file mode 100644
index 0000000000000..6a1ff605f190e
--- /dev/null
+++ b/Mathlib/Topology/UniformSpace/OfFun.lean
@@ -0,0 +1,52 @@
+/-
+Copyright (c) 2023 Yury Kudryashov. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yury Kudryashov
+-/
+import Mathlib.Topology.UniformSpace.Basic
+import Mathlib.Algebra.Order.Monoid.Defs
+
+/-!
+# Construct a `UniformSpace` from a `dist`-like function
+
+In this file we provide a constructor for `UniformSpace`
+given a `dist`-like function
+
+## TODO
+
+RFC: use `UniformSpace.Core.mkOfBasis`? This will change defeq here and there
+-/
+
+open Filter Set
+open scoped Uniformity
+
+variable {X M : Type*}
+
+namespace UniformSpace
+
+/-- Define a `UniformSpace` using a "distance" function. The function can be, e.g., the
+distance in a (usual or extended) metric space or an absolute value on a ring. -/
+def ofFun [OrderedAddCommMonoid M] (d : X → X → M) (refl : ∀ x, d x x = 0)
+    (symm : ∀ x y, d x y = d y x) (triangle : ∀ x y z, d x z ≤ d x y + d y z)
+    (half : ∀ ε > (0 : M), ∃ δ > (0 : M), ∀ x < δ, ∀ y < δ, x + y < ε) :
+    UniformSpace X :=
+  .ofCore
+    { uniformity := ⨅ r > 0, 𝓟 { x | d x.1 x.2 < r }
+      refl := le_iInf₂ fun r hr => principal_mono.2 <| idRel_subset.2 fun x => by simpa [refl]
+      symm := tendsto_iInf_iInf fun r => tendsto_iInf_iInf fun _ => tendsto_principal_principal.2
+        fun x hx => by rwa [mem_setOf, symm]
+      comp := le_iInf₂ fun r hr => let ⟨δ, h0, hδr⟩ := half r hr; le_principal_iff.2 <|
+        mem_of_superset
+          (mem_lift' <| mem_iInf_of_mem δ <| mem_iInf_of_mem h0 <| mem_principal_self _)
+          fun (x, z) ⟨y, h₁, h₂⟩ => (triangle _ _ _).trans_lt (hδr _ h₁ _ h₂) }
+
+theorem hasBasis_ofFun [LinearOrderedAddCommMonoid M]
+    (h₀ : ∃ x : M, 0 < x) (d : X → X → M) (refl : ∀ x, d x x = 0) (symm : ∀ x y, d x y = d y x)
+    (triangle : ∀ x y z, d x z ≤ d x y + d y z)
+    (half : ∀ ε > (0 : M), ∃ δ > (0 : M), ∀ x < δ, ∀ y < δ, x + y < ε) :
+    𝓤[.ofFun d refl symm triangle half].HasBasis ((0 : M) < ·) (fun ε => { x | d x.1 x.2 < ε }) :=
+  hasBasis_biInf_principal'
+    (fun ε₁ h₁ ε₂ h₂ => ⟨min ε₁ ε₂, lt_min h₁ h₂, fun _x hx => lt_of_lt_of_le hx (min_le_left _ _),
+      fun _x hx => lt_of_lt_of_le hx (min_le_right _ _)⟩) h₀
+
+end UniformSpace
diff --git a/Mathlib/Topology/UniformSpace/Pi.lean b/Mathlib/Topology/UniformSpace/Pi.lean
index 6c6f45b6814fc..783300913d8d4 100644
--- a/Mathlib/Topology/UniformSpace/Pi.lean
+++ b/Mathlib/Topology/UniformSpace/Pi.lean
@@ -42,7 +42,7 @@ instance [Countable ι] [∀ i, IsCountablyGenerated (𝓤 (α i))] :
 theorem uniformContinuous_pi {β : Type*} [UniformSpace β] {f : β → ∀ i, α i} :
     UniformContinuous f ↔ ∀ i, UniformContinuous fun x => f x i := by
   -- Porting note: required `Function.comp` to close
-  simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp]
+  simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp_def]
 
 variable (α)
 
@@ -69,7 +69,7 @@ theorem Pi.uniformContinuous_postcomp {α : Type*} [UniformSpace α] {g : α →
 lemma Pi.uniformSpace_comap_precomp' (φ : ι' → ι) :
     UniformSpace.comap (fun g i' ↦ g (φ i')) (Pi.uniformSpace (fun i' ↦ α (φ i'))) =
     ⨅ i', UniformSpace.comap (eval (φ i')) (U (φ i')) := by
-  simp [Pi.uniformSpace_eq, UniformSpace.comap_iInf, ← UniformSpace.comap_comap, comp]
+  simp [Pi.uniformSpace_eq, UniformSpace.comap_iInf, ← UniformSpace.comap_comap, comp_def]
 
 lemma Pi.uniformSpace_comap_precomp (φ : ι' → ι) :
     UniformSpace.comap (· ∘ φ) (Pi.uniformSpace (fun _ ↦ β)) =
@@ -122,9 +122,9 @@ protected theorem CompleteSpace.iInf {ι X : Type*} {u : ι → UniformSpace X}
   nontriviality X
   rcases ht with ⟨t, ht, hut⟩
   -- The diagonal map `(X, ⨅ i, u i) → ∀ i, (X, u i)` is a uniform embedding.
-  have : @UniformInducing X (ι → X) (⨅ i, u i) (Pi.uniformSpace (U := u)) (const ι) := by
-    simp_rw [uniformInducing_iff, iInf_uniformity, Pi.uniformity, Filter.comap_iInf,
-      Filter.comap_comap, (· ∘ ·), const, Prod.eta, comap_id']
+  have : @IsUniformInducing X (ι → X) (⨅ i, u i) (Pi.uniformSpace (U := u)) (const ι) := by
+    simp_rw [isUniformInducing_iff, iInf_uniformity, Pi.uniformity, Filter.comap_iInf,
+      Filter.comap_comap, comp_def, const, Prod.eta, comap_id']
   -- Hence, it suffices to show that its range, the diagonal, is closed in `Π i, (X, u i)`.
   simp_rw [@completeSpace_iff_isComplete_range _ _ (_) (_) _ this, range_const_eq_diagonal,
     setOf_forall]
diff --git a/Mathlib/Topology/UniformSpace/Separation.lean b/Mathlib/Topology/UniformSpace/Separation.lean
index 9d11da63e1ac6..f50de8d5deebd 100644
--- a/Mathlib/Topology/UniformSpace/Separation.lean
+++ b/Mathlib/Topology/UniformSpace/Separation.lean
@@ -218,7 +218,7 @@ instance instUniformSpace : UniformSpace (SeparationQuotient α) where
     exact @hUt (x, z) ⟨y', this.mem_open (UniformSpace.isOpen_ball _ hUo) hxyU, hyzU⟩
   nhds_eq_comap_uniformity := surjective_mk.forall.2 fun x ↦ comap_injective surjective_mk <| by
     conv_lhs => rw [comap_mk_nhds_mk, nhds_eq_comap_uniformity, ← comap_map_mk_uniformity]
-    simp only [Filter.comap_comap, Function.comp, Prod.map_apply]
+    simp only [Filter.comap_comap, Function.comp_def, Prod.map_apply]
 
 theorem uniformity_eq : 𝓤 (SeparationQuotient α) = (𝓤 α).map (Prod.map mk mk) := rfl
 
@@ -285,6 +285,6 @@ theorem map_id : map (@id α) = id := map_unique uniformContinuous_id rfl
 
 theorem map_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f) (hg : UniformContinuous g) :
     map g ∘ map f = map (g ∘ f) :=
-  (map_unique (hg.comp hf) <| by simp only [Function.comp, map_mk, hf, hg]).symm
+  (map_unique (hg.comp hf) <| by simp only [Function.comp_def, map_mk, hf, hg]).symm
 
 end SeparationQuotient
diff --git a/Mathlib/Topology/UniformSpace/UniformConvergence.lean b/Mathlib/Topology/UniformSpace/UniformConvergence.lean
index a5dbd62933646..3d5593d33b3b1 100644
--- a/Mathlib/Topology/UniformSpace/UniformConvergence.lean
+++ b/Mathlib/Topology/UniformSpace/UniformConvergence.lean
@@ -62,7 +62,7 @@ Uniform limit, uniform convergence, tends uniformly to
 
 noncomputable section
 
-open Topology Uniformity Filter Set
+open Topology Uniformity Filter Set Uniform
 
 universe u v w x
 variable {α : Type u} {β : Type v} {γ : Type w} {ι : Type x} [UniformSpace β]
@@ -196,6 +196,12 @@ theorem TendstoUniformlyOn.congr {F' : ι → α → β} (hf : TendstoUniformlyO
   simp only [Set.EqOn] at hff'
   simp only [mem_prod_principal, hff', mem_setOf_eq]
 
+lemma tendstoUniformly_congr {F F' : ι → α → β} {f : α → β} (hF : F =ᶠ[p] F') :
+    TendstoUniformly F f p ↔ TendstoUniformly F' f p := by
+  simp_rw [← tendstoUniformlyOn_univ] at *
+  have HF := EventuallyEq.exists_mem hF
+  exact ⟨fun h => h.congr (by aesop), fun h => h.congr (by simp_rw [eqOn_comm]; aesop)⟩
+
 theorem TendstoUniformlyOn.congr_right {g : α → β} (hf : TendstoUniformlyOn F f p s)
     (hfg : EqOn f g s) : TendstoUniformlyOn F g p s := fun u hu => by
   filter_upwards [hf u hu] with i hi a ha using hfg ha ▸ hi a ha
@@ -383,10 +389,12 @@ theorem TendstoUniformlyOn.uniformCauchySeqOn (hF : TendstoUniformlyOn F f p s)
     hF.tendstoUniformlyOnFilter.uniformCauchySeqOnFilter
 
 /-- A uniformly Cauchy sequence converges uniformly to its limit -/
-theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto [NeBot p]
+theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto
     (hF : UniformCauchySeqOnFilter F p p')
     (hF' : ∀ᶠ x : α in p', Tendsto (fun n => F n x) p (𝓝 (f x))) :
     TendstoUniformlyOnFilter F f p p' := by
+  rcases p.eq_or_neBot with rfl | _
+  · simp only [TendstoUniformlyOnFilter, bot_prod, eventually_bot, implies_true]
   -- Proof idea: |f_n(x) - f(x)| ≤ |f_n(x) - f_m(x)| + |f_m(x) - f(x)|. We choose `n`
   -- so that |f_n(x) - f_m(x)| is uniformly small across `s` whenever `m ≥ n`. Then for
   -- a fixed `x`, we choose `m` sufficiently large such that |f_m(x) - f(x)| is small.
@@ -412,7 +420,7 @@ theorem UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto [NeBot p]
   exact ⟨F m x, ⟨hm.2, htsymm hm.1⟩⟩
 
 /-- A uniformly Cauchy sequence converges uniformly to its limit -/
-theorem UniformCauchySeqOn.tendstoUniformlyOn_of_tendsto [NeBot p] (hF : UniformCauchySeqOn F p s)
+theorem UniformCauchySeqOn.tendstoUniformlyOn_of_tendsto (hF : UniformCauchySeqOn F p s)
     (hF' : ∀ x : α, x ∈ s → Tendsto (fun n => F n x) p (𝓝 (f x))) : TendstoUniformlyOn F f p s :=
   tendstoUniformlyOn_iff_tendstoUniformlyOnFilter.mpr
     (hF.uniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto hF')
@@ -481,7 +489,7 @@ theorem UniformCauchySeqOn.prod' {β' : Type*} [UniformSpace β'] {F' : ι → 
 a Cauchy sequence. -/
 theorem UniformCauchySeqOn.cauchy_map [hp : NeBot p] (hf : UniformCauchySeqOn F p s) (hx : x ∈ s) :
     Cauchy (map (fun i => F i x) p) := by
-  simp only [cauchy_map_iff, hp, true_and_iff]
+  simp only [cauchy_map_iff, hp, true_and]
   intro u hu
   rw [mem_map]
   filter_upwards [hf u hu] with p hp using hp x hx
@@ -523,6 +531,31 @@ theorem tendstoUniformly_iff_seq_tendstoUniformly {l : Filter ι} [l.IsCountably
 
 end SeqTendsto
 
+section
+
+variable [NeBot p] {L : ι → β} {ℓ : β}
+
+theorem TendstoUniformlyOnFilter.tendsto_of_eventually_tendsto
+    (h1 : TendstoUniformlyOnFilter F f p p') (h2 : ∀ᶠ i in p, Tendsto (F i) p' (𝓝 (L i)))
+    (h3 : Tendsto L p (𝓝 ℓ)) : Tendsto f p' (𝓝 ℓ) := by
+  rw [tendsto_nhds_left]
+  intro s hs
+  rw [mem_map, Set.preimage, ← eventually_iff]
+  obtain ⟨t, ht, hts⟩ := comp3_mem_uniformity hs
+  have p1 : ∀ᶠ i in p, (L i, ℓ) ∈ t := tendsto_nhds_left.mp h3 ht
+  have p2 : ∀ᶠ i in p, ∀ᶠ x in p', (F i x, L i) ∈ t := by
+    filter_upwards [h2] with i h2 using tendsto_nhds_left.mp h2 ht
+  have p3 : ∀ᶠ i in p, ∀ᶠ x in p', (f x, F i x) ∈ t := (h1 t ht).curry
+  obtain ⟨i, p4, p5, p6⟩ := (p1.and (p2.and p3)).exists
+  filter_upwards [p5, p6] with x p5 p6 using hts ⟨F i x, p6, L i, p5, p4⟩
+
+theorem TendstoUniformly.tendsto_of_eventually_tendsto
+    (h1 : TendstoUniformly F f p) (h2 : ∀ᶠ i in p, Tendsto (F i) p' (𝓝 (L i)))
+    (h3 : Tendsto L p (𝓝 ℓ)) : Tendsto f p' (𝓝 ℓ) :=
+  (h1.tendstoUniformlyOnFilter.mono_right le_top).tendsto_of_eventually_tendsto h2 h3
+
+end
+
 variable [TopologicalSpace α]
 
 /-- A sequence of functions `Fₙ` converges locally uniformly on a set `s` to a limiting function
@@ -653,14 +686,14 @@ theorem tendstoLocallyUniformlyOn_TFAE [LocallyCompactSpace α] (G : ι → α 
       ∀ K, K ⊆ s → IsCompact K → TendstoUniformlyOn G g p K,
       ∀ x ∈ s, ∃ v ∈ 𝓝[s] x, TendstoUniformlyOn G g p v] := by
   tfae_have 1 → 2
-  · rintro h K hK1 hK2
-    exact (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK2).mp (h.mono hK1)
+  | h, K, hK1, hK2 =>
+    (tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK2).mp (h.mono hK1)
   tfae_have 2 → 3
-  · rintro h x hx
+  | h, x, hx => by
     obtain ⟨K, ⟨hK1, hK2⟩, hK3⟩ := (compact_basis_nhds x).mem_iff.mp (hs.mem_nhds hx)
     exact ⟨K, nhdsWithin_le_nhds hK1, h K hK3 hK2⟩
   tfae_have 3 → 1
-  · rintro h u hu x hx
+  | h, u, hu, x, hx => by
     obtain ⟨v, hv1, hv2⟩ := h x hx
     exact ⟨v, hv1, hv2 u hu⟩
   tfae_finish
diff --git a/Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean b/Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
index 60512c6535c4d..00cd2deafccea 100644
--- a/Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
+++ b/Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
@@ -6,7 +6,7 @@ Authors: Anatole Dedecker
 import Mathlib.Topology.UniformSpace.UniformConvergence
 import Mathlib.Topology.UniformSpace.Pi
 import Mathlib.Topology.UniformSpace.Equiv
-import Mathlib.Topology.RestrictGenTopology
+import Mathlib.Topology.RestrictGen
 
 /-!
 # Topology and uniform structure of uniform convergence
@@ -88,7 +88,7 @@ connection API to do most of the work.
 
 * `UniformOnFun.postcomp_uniformContinuous`: if `f : γ → β` is uniformly
   continuous, then `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is uniformly continuous.
-* `UniformOnFun.postcomp_uniformInducing`: if `f : γ → β` is a uniform
+* `UniformOnFun.postcomp_isUniformInducing`: if `f : γ → β` is a uniform
   inducing, then `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing.
 * `UniformOnFun.precomp_uniformContinuous`: let `f : γ → α`, `𝔖 : Set (Set α)`,
   `𝔗 : Set (Set γ)`, and assume that `∀ T ∈ 𝔗, f '' T ∈ 𝔖`. Then, the function
@@ -367,28 +367,35 @@ a uniform inducing function for the uniform structures of uniform convergence.
 
 More precisely, if `f : γ → β` is uniform inducing,
 then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is uniform inducing. -/
-protected theorem postcomp_uniformInducing [UniformSpace γ] {f : γ → β} (hf : UniformInducing f) :
-    UniformInducing (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) :=
+lemma postcomp_isUniformInducing [UniformSpace γ] {f : γ → β}
+    (hf : IsUniformInducing f) : IsUniformInducing (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) :=
   ⟨((UniformFun.hasBasis_uniformity _ _).comap _).eq_of_same_basis <|
     UniformFun.hasBasis_uniformity_of_basis _ _ (hf.basis_uniformity (𝓤 β).basis_sets)⟩
 
+@[deprecated (since := "2024-10-05")]
+alias postcomp_uniformInducing := postcomp_isUniformInducing
+
 /-- Post-composition by a uniform embedding is
 a uniform embedding for the uniform structures of uniform convergence.
 
 More precisely, if `f : γ → β` is a uniform embedding,
 then `(f ∘ ·) : (α →ᵤ γ) → (α →ᵤ β)` is a uniform embedding. -/
-protected theorem postcomp_uniformEmbedding [UniformSpace γ] {f : γ → β} (hf : UniformEmbedding f) :
-    UniformEmbedding (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) where
-  toUniformInducing := UniformFun.postcomp_uniformInducing hf.toUniformInducing
+protected theorem postcomp_isUniformEmbedding [UniformSpace γ] {f : γ → β}
+    (hf : IsUniformEmbedding f) :
+ IsUniformEmbedding (ofFun ∘ (f ∘ ·) ∘ toFun : (α →ᵤ γ) → α →ᵤ β) where
+  toIsUniformInducing := UniformFun.postcomp_isUniformInducing hf.isUniformInducing
   inj _ _ H := funext fun _ ↦ hf.inj (congrFun H _)
 
+@[deprecated (since := "2024-10-01")]
+alias postcomp_uniformEmbedding := UniformFun.postcomp_isUniformEmbedding
+
 -- Porting note: had to add a type annotation at `((f ∘ ·) : ((α → γ) → (α → β)))`
 /-- If `u` is a uniform structures on `β` and `f : γ → β`, then
 `𝒰(α, γ, comap f u) = comap (fun g ↦ f ∘ g) 𝒰(α, γ, u₁)`. -/
 protected theorem comap_eq {f : γ → β} :
     𝒰(α, γ, ‹UniformSpace β›.comap f) = 𝒰(α, β, _).comap (f ∘ ·) := by
   letI : UniformSpace γ := .comap f ‹_›
-  exact (UniformFun.postcomp_uniformInducing (f := f) ⟨rfl⟩).comap_uniformSpace.symm
+  exact (UniformFun.postcomp_isUniformInducing (f := f) ⟨rfl⟩).comap_uniformSpace.symm
 
 /-- Post-composition by a uniformly continuous function is uniformly continuous on `α →ᵤ β`.
 
@@ -461,7 +468,7 @@ protected def uniformEquivProdArrow [UniformSpace γ] : (α →ᵤ β × γ) ≃
   -- But `uβ × uγ` is defined as `comap fst uβ ⊓ comap snd uγ`, so we just have to apply
   -- `UniformFun.inf_eq` and `UniformFun.comap_eq`, which leaves us to check
   -- that some square commutes.
-  Equiv.toUniformEquivOfUniformInducing (Equiv.arrowProdEquivProdArrow _ _ _) <| by
+  Equiv.toUniformEquivOfIsUniformInducing (Equiv.arrowProdEquivProdArrow _ _ _) <| by
     constructor
     change
       comap (Prod.map (Equiv.arrowProdEquivProdArrow _ _ _) (Equiv.arrowProdEquivProdArrow _ _ _))
@@ -486,10 +493,10 @@ protected def uniformEquivPiComm : UniformEquiv (α →ᵤ ∀ i, δ i) (∀ i,
     -- But `Π i, uδ i` is defined as `⨅ i, comap (eval i) (uδ i)`, so we just have to apply
     -- `UniformFun.iInf_eq` and `UniformFun.comap_eq`, which leaves us to check
     -- that some square commutes.
-    @Equiv.toUniformEquivOfUniformInducing
+    @Equiv.toUniformEquivOfIsUniformInducing
     _ _ 𝒰(α, ∀ i, δ i, Pi.uniformSpace δ)
     (@Pi.uniformSpace ι (fun i => α → δ i) fun i => 𝒰(α, δ i, _)) (Equiv.piComm _) <| by
-      refine @UniformInducing.mk ?_ ?_ ?_ ?_ ?_ ?_
+      refine @IsUniformInducing.mk ?_ ?_ ?_ ?_ ?_ ?_
       change comap (Prod.map Function.swap Function.swap) _ = _
       rw [← uniformity_comap]
       congr
@@ -861,8 +868,8 @@ uniform structures of `𝔖`-convergence.
 
 More precisely, if `f : γ → β` is a uniform inducing, then
 `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform inducing. -/
-protected theorem postcomp_uniformInducing [UniformSpace γ] {f : γ → β} (hf : UniformInducing f) :
-    UniformInducing (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
+lemma postcomp_isUniformInducing [UniformSpace γ] {f : γ → β}
+    (hf : IsUniformInducing f) : IsUniformInducing (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) := by
   -- This is a direct consequence of `UniformOnFun.comap_eq`
   constructor
   replace hf : (𝓤 β).comap (Prod.map f f) = _ := hf.comap_uniformity
@@ -872,16 +879,22 @@ protected theorem postcomp_uniformInducing [UniformSpace γ] {f : γ → β} (hf
   rw [← UniformSpace.ext hf, UniformOnFun.comap_eq]
   rfl
 
+@[deprecated (since := "2024-10-05")]
+alias postcomp_uniformInducing := postcomp_isUniformInducing
+
 /-- Post-composition by a uniform embedding is a uniform embedding for the
 uniform structures of `𝔖`-convergence.
 
 More precisely, if `f : γ → β` is a uniform embedding, then
 `(fun g ↦ f ∘ g) : (α →ᵤ[𝔖] γ) → (α →ᵤ[𝔖] β)` is a uniform embedding. -/
-protected theorem postcomp_uniformEmbedding [UniformSpace γ] {f : γ → β} (hf : UniformEmbedding f) :
-    UniformEmbedding (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) where
-  toUniformInducing := UniformOnFun.postcomp_uniformInducing hf.toUniformInducing
+protected theorem postcomp_isUniformEmbedding [UniformSpace γ] {f : γ → β}
+    (hf : IsUniformEmbedding f) : IsUniformEmbedding (ofFun 𝔖 ∘ (f ∘ ·) ∘ toFun 𝔖) where
+  toIsUniformInducing := UniformOnFun.postcomp_isUniformInducing hf.isUniformInducing
   inj _ _ H := funext fun _ ↦ hf.inj (congrFun H _)
 
+@[deprecated (since := "2024-10-01")]
+alias postcomp_uniformEmbedding := UniformOnFun.postcomp_isUniformEmbedding
+
 /-- Turn a uniform isomorphism `γ ≃ᵤ β` into a uniform isomorphism `(α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β)`
 by post-composing. -/
 protected def congrRight [UniformSpace γ] (e : γ ≃ᵤ β) : (α →ᵤ[𝔖] γ) ≃ᵤ (α →ᵤ[𝔖] β) :=
@@ -1005,9 +1018,8 @@ protected def uniformEquivProdArrow [UniformSpace γ] :
   -- which leaves us to check that some square commutes.
   -- We could also deduce this from `UniformFun.uniformEquivProdArrow`,
   -- but it turns out to be more annoying.
-  ((UniformOnFun.ofFun 𝔖).symm.trans <|
-    (Equiv.arrowProdEquivProdArrow _ _ _).trans <|
-      (UniformOnFun.ofFun 𝔖).prodCongr (UniformOnFun.ofFun 𝔖)).toUniformEquivOfUniformInducing <| by
+  ((UniformOnFun.ofFun 𝔖).symm.trans <| (Equiv.arrowProdEquivProdArrow _ _ _).trans <|
+    (UniformOnFun.ofFun 𝔖).prodCongr (UniformOnFun.ofFun 𝔖)).toUniformEquivOfIsUniformInducing <| by
       constructor
       rw [uniformity_prod, comap_inf, comap_comap, comap_comap]
       have H := @UniformOnFun.inf_eq α (β × γ) 𝔖
@@ -1031,7 +1043,7 @@ protected def uniformEquivPiComm : (α →ᵤ[𝔖] ((i : ι) → δ i)) ≃ᵤ
   -- which leaves us to check that some square commutes.
   -- We could also deduce this from `UniformFun.uniformEquivPiComm`, but it turns out
   -- to be more annoying.
-  @Equiv.toUniformEquivOfUniformInducing (α →ᵤ[𝔖] ((i : ι) → δ i)) ((i : ι) → α →ᵤ[𝔖] δ i)
+  @Equiv.toUniformEquivOfIsUniformInducing (α →ᵤ[𝔖] ((i : ι) → δ i)) ((i : ι) → α →ᵤ[𝔖] δ i)
       _ _ (Equiv.piComm _) <| by
     constructor
     change comap (Prod.map Function.swap Function.swap) _ = _
@@ -1117,3 +1129,41 @@ instance {α β : Type*} [UniformSpace β] [CompleteSpace β] : CompleteSpace (
   (UniformOnFun.uniformEquivUniformFun β {univ} (mem_singleton _)).completeSpace_iff.1 inferInstance
 
 end UniformFun
+
+section UniformComposition
+
+variable {α β γ ι : Type*} [UniformSpace β] [UniformSpace γ] {p : Filter ι}
+
+/-- Composing on the left by a uniformly continuous function preserves uniform convergence -/
+theorem UniformContinuousOn.comp_tendstoUniformly (s : Set β) (F : ι → α → β) (f : α → β)
+    (hF : ∀ i x, F i x ∈ s) (hf : ∀ x, f x ∈ s)
+    {g : β → γ} (hg : UniformContinuousOn g s) (h : TendstoUniformly F f p) :
+    TendstoUniformly (fun i x => g (F i x)) (fun x => g (f x)) p := by
+  rw [uniformContinuousOn_iff_restrict] at hg
+  lift F to ι → α → s using hF with F' hF'
+  lift f to α → s using hf with f' hf'
+  rw [tendstoUniformly_iff_tendsto] at h
+  have : Tendsto (fun q : ι × α ↦ (f' q.2, (F' q.1 q.2))) (p ×ˢ ⊤) (𝓤 s) :=
+    h.of_tendsto_comp isUniformEmbedding_subtype_val.comap_uniformity.le
+  apply UniformContinuous.comp_tendstoUniformly hg ?_
+  rwa [← tendstoUniformly_iff_tendsto] at this
+
+theorem UniformContinuousOn.comp_tendstoUniformly_eventually (s : Set β) (F : ι → α → β) (f : α → β)
+    (hF : ∀ᶠ i in p, ∀ x, F i x ∈ s) (hf : ∀ x, f x ∈ s)
+    {g : β → γ} (hg : UniformContinuousOn g s) (h : TendstoUniformly F f p) :
+    TendstoUniformly (fun i => fun x => g (F i x)) (fun x => g (f x)) p := by
+  classical
+  rw [eventually_iff_exists_mem] at hF
+  obtain ⟨s', hs', hs⟩ := hF
+  let F' : ι → α → β := fun (i : ι) x => if i ∈ s' then F i x else f x
+  have hF : F =ᶠ[p] F' :=  by
+    rw [eventuallyEq_iff_exists_mem]
+    refine ⟨s', hs', fun y hy => by aesop⟩
+  have h' : TendstoUniformly F' f p := by
+    rwa [tendstoUniformly_congr hF] at h
+  apply (tendstoUniformly_congr _).mpr
+    (UniformContinuousOn.comp_tendstoUniformly s F' f (by aesop) hf hg h')
+  rw [eventuallyEq_iff_exists_mem]
+  refine ⟨s', hs', fun i hi => by aesop⟩
+
+end UniformComposition
diff --git a/Mathlib/Topology/UniformSpace/UniformEmbedding.lean b/Mathlib/Topology/UniformSpace/UniformEmbedding.lean
index b674ec1daf22e..d70f7fb955c8f 100644
--- a/Mathlib/Topology/UniformSpace/UniformEmbedding.lean
+++ b/Mathlib/Topology/UniformSpace/UniformEmbedding.lean
@@ -20,6 +20,7 @@ section
 
 universe u v w
 variable {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpace β] [UniformSpace γ]
+  {f : α → β}
 
 /-!
 ### Uniform inducing maps
@@ -27,102 +28,165 @@ variable {α : Type u} {β : Type v} {γ : Type w} [UniformSpace α] [UniformSpa
 
 /-- A map `f : α → β` between uniform spaces is called *uniform inducing* if the uniformity filter
 on `α` is the pullback of the uniformity filter on `β` under `Prod.map f f`. If `α` is a separated
-space, then this implies that `f` is injective, hence it is a `UniformEmbedding`. -/
+space, then this implies that `f` is injective, hence it is a `IsUniformEmbedding`. -/
 @[mk_iff]
-structure UniformInducing (f : α → β) : Prop where
+structure IsUniformInducing (f : α → β) : Prop where
   /-- The uniformity filter on the domain is the pullback of the uniformity filter on the codomain
   under `Prod.map f f`. -/
   comap_uniformity : comap (fun x : α × α => (f x.1, f x.2)) (𝓤 β) = 𝓤 α
 
-lemma uniformInducing_iff_uniformSpace {f : α → β} :
-    UniformInducing f ↔ ‹UniformSpace β›.comap f = ‹UniformSpace α› := by
-  rw [uniformInducing_iff, UniformSpace.ext_iff, Filter.ext_iff]
+@[deprecated (since := "2024-10-08")] alias UniformInducing := IsUniformInducing
+
+lemma isUniformInducing_iff_uniformSpace {f : α → β} :
+    IsUniformInducing f ↔ ‹UniformSpace β›.comap f = ‹UniformSpace α› := by
+  rw [isUniformInducing_iff, UniformSpace.ext_iff, Filter.ext_iff]
   rfl
 
-protected alias ⟨UniformInducing.comap_uniformSpace, _⟩ := uniformInducing_iff_uniformSpace
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_iff_uniformSpace := isUniformInducing_iff_uniformSpace
+
+protected alias ⟨IsUniformInducing.comap_uniformSpace, _⟩ := isUniformInducing_iff_uniformSpace
+
+@[deprecated (since := "2024-10-08")] alias UniformInducing.comap_uniformSpace :=
+  IsUniformInducing.comap_uniformSpace
+
+lemma isUniformInducing_iff' {f : α → β} :
+    IsUniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
+  rw [isUniformInducing_iff, UniformContinuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; rfl
 
-lemma uniformInducing_iff' {f : α → β} :
-    UniformInducing f ↔ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
-  rw [uniformInducing_iff, UniformContinuous, tendsto_iff_comap, le_antisymm_iff, and_comm]; rfl
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_iff' := isUniformInducing_iff'
 
-protected lemma Filter.HasBasis.uniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
+protected lemma Filter.HasBasis.isUniformInducing_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
     (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
-    UniformInducing f ↔
+    IsUniformInducing f ↔
       (∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
         (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by
-  simp [uniformInducing_iff', h.uniformContinuous_iff h', (h'.comap _).le_basis_iff h, subset_def]
+  simp [isUniformInducing_iff', h.uniformContinuous_iff h', (h'.comap _).le_basis_iff h, subset_def]
 
-theorem UniformInducing.mk' {f : α → β}
-    (h : ∀ s, s ∈ 𝓤 α ↔ ∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : UniformInducing f :=
+@[deprecated (since := "2024-10-05")]
+alias Filter.HasBasis.uniformInducing_iff := Filter.HasBasis.isUniformInducing_iff
+
+theorem IsUniformInducing.mk' {f : α → β}
+    (h : ∀ s, s ∈ 𝓤 α ↔ ∃ t ∈ 𝓤 β, ∀ x y : α, (f x, f y) ∈ t → (x, y) ∈ s) : IsUniformInducing f :=
   ⟨by simp [eq_comm, Filter.ext_iff, subset_def, h]⟩
 
-theorem uniformInducing_id : UniformInducing (@id α) :=
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.mk' := IsUniformInducing.mk'
+
+theorem IsUniformInducing.id : IsUniformInducing (@id α) :=
   ⟨by rw [← Prod.map_def, Prod.map_id, comap_id]⟩
 
-theorem UniformInducing.comp {g : β → γ} (hg : UniformInducing g) {f : α → β}
-    (hf : UniformInducing f) : UniformInducing (g ∘ f) :=
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_id := IsUniformInducing.id
+
+theorem IsUniformInducing.comp {g : β → γ} (hg : IsUniformInducing g) {f : α → β}
+    (hf : IsUniformInducing f) : IsUniformInducing (g ∘ f) :=
   ⟨by rw [← hf.1, ← hg.1, comap_comap]; rfl⟩
 
-theorem UniformInducing.of_comp_iff {g : β → γ} (hg : UniformInducing g) {f : α → β} :
-    UniformInducing (g ∘ f) ↔ UniformInducing f := by
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.comp := IsUniformInducing.comp
+
+theorem IsUniformInducing.of_comp_iff {g : β → γ} (hg : IsUniformInducing g) {f : α → β} :
+    IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
   refine ⟨fun h ↦ ?_, hg.comp⟩
-  rw [uniformInducing_iff, ← hg.comap_uniformity, comap_comap, ← h.comap_uniformity,
-    Function.comp, Function.comp]
+  rw [isUniformInducing_iff, ← hg.comap_uniformity, comap_comap, ← h.comap_uniformity,
+    Function.comp_def, Function.comp_def]
 
-theorem UniformInducing.basis_uniformity {f : α → β} (hf : UniformInducing f) {ι : Sort*}
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.of_comp_iff := IsUniformInducing.of_comp_iff
+
+theorem IsUniformInducing.basis_uniformity {f : α → β} (hf : IsUniformInducing f) {ι : Sort*}
     {p : ι → Prop} {s : ι → Set (β × β)} (H : (𝓤 β).HasBasis p s) :
     (𝓤 α).HasBasis p fun i => Prod.map f f ⁻¹' s i :=
   hf.1 ▸ H.comap _
 
-theorem UniformInducing.cauchy_map_iff {f : α → β} (hf : UniformInducing f) {F : Filter α} :
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.basis_uniformity := IsUniformInducing.basis_uniformity
+
+theorem IsUniformInducing.cauchy_map_iff {f : α → β} (hf : IsUniformInducing f) {F : Filter α} :
     Cauchy (map f F) ↔ Cauchy F := by
   simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity]
 
-theorem uniformInducing_of_compose {f : α → β} {g : β → γ} (hf : UniformContinuous f)
-    (hg : UniformContinuous g) (hgf : UniformInducing (g ∘ f)) : UniformInducing f := by
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.cauchy_map_iff := IsUniformInducing.cauchy_map_iff
+
+theorem IsUniformInducing.of_comp {f : α → β} {g : β → γ} (hf : UniformContinuous f)
+    (hg : UniformContinuous g) (hgf : IsUniformInducing (g ∘ f)) : IsUniformInducing f := by
   refine ⟨le_antisymm ?_ hf.le_comap⟩
   rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
   exact comap_mono hg.le_comap
 
-theorem UniformInducing.uniformContinuous {f : α → β} (hf : UniformInducing f) :
-    UniformContinuous f := (uniformInducing_iff'.1 hf).1
+@[deprecated (since := "2024-10-05")]
+alias uniformInducing_of_compose := IsUniformInducing.of_comp
+
+theorem IsUniformInducing.uniformContinuous {f : α → β} (hf : IsUniformInducing f) :
+    UniformContinuous f := (isUniformInducing_iff'.1 hf).1
 
-theorem UniformInducing.uniformContinuous_iff {f : α → β} {g : β → γ} (hg : UniformInducing g) :
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformContinuous := IsUniformInducing.uniformContinuous
+
+theorem IsUniformInducing.uniformContinuous_iff {f : α → β} {g : β → γ} (hg : IsUniformInducing g) :
     UniformContinuous f ↔ UniformContinuous (g ∘ f) := by
   dsimp only [UniformContinuous, Tendsto]
   simp only [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, Function.comp_def]
 
-protected theorem UniformInducing.uniformInducing_comp_iff {f : α → β} {g : β → γ}
-    (hg : UniformInducing g) : UniformInducing (g ∘ f) ↔ UniformInducing f := by
-  simp only [uniformInducing_iff, ← hg.comap_uniformity, comap_comap, Function.comp_def]
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformContinuous_iff := IsUniformInducing.uniformContinuous_iff
+
+protected theorem IsUniformInducing.isUniformInducing_comp_iff {f : α → β} {g : β → γ}
+    (hg : IsUniformInducing g) : IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
+  simp only [isUniformInducing_iff, ← hg.comap_uniformity, comap_comap, Function.comp_def]
+
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformInducing_comp_iff := IsUniformInducing.isUniformInducing_comp_iff
 
-theorem UniformInducing.uniformContinuousOn_iff {f : α → β} {g : β → γ} {S : Set α}
-    (hg : UniformInducing g) :
+theorem IsUniformInducing.uniformContinuousOn_iff {f : α → β} {g : β → γ} {S : Set α}
+    (hg : IsUniformInducing g) :
     UniformContinuousOn f S ↔ UniformContinuousOn (g ∘ f) S := by
   dsimp only [UniformContinuousOn, Tendsto]
   rw [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, comp_def, comp_def]
 
-theorem UniformInducing.inducing {f : α → β} (h : UniformInducing f) : Inducing f := by
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.uniformContinuousOn_iff := IsUniformInducing.uniformContinuousOn_iff
+
+theorem IsUniformInducing.inducing {f : α → β} (h : IsUniformInducing f) : Inducing f := by
   obtain rfl := h.comap_uniformSpace
   exact inducing_induced f
 
-theorem UniformInducing.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
-    {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformInducing e₁) (h₂ : UniformInducing e₂) :
-    UniformInducing fun p : α × β => (e₁ p.1, e₂ p.2) :=
-  ⟨by simp [(· ∘ ·), uniformity_prod, ← h₁.1, ← h₂.1, comap_inf, comap_comap]⟩
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.inducing := IsUniformInducing.inducing
+
+theorem IsUniformInducing.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
+    {e₁ : α → α'} {e₂ : β → β'} (h₁ : IsUniformInducing e₁) (h₂ : IsUniformInducing e₂) :
+    IsUniformInducing fun p : α × β => (e₁ p.1, e₂ p.2) :=
+  ⟨by simp [Function.comp_def, uniformity_prod, ← h₁.1, ← h₂.1, comap_inf, comap_comap]⟩
 
-theorem UniformInducing.denseInducing {f : α → β} (h : UniformInducing f) (hd : DenseRange f) :
-    DenseInducing f :=
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.prod := IsUniformInducing.prod
+
+lemma IsUniformInducing.isDenseInducing (h : IsUniformInducing f) (hd : DenseRange f) :
+    IsDenseInducing f :=
   { dense := hd
     induced := h.inducing.induced }
 
-theorem SeparationQuotient.uniformInducing_mk : UniformInducing (mk : α → SeparationQuotient α) :=
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.isDenseInducing := IsUniformInducing.isDenseInducing
+
+lemma SeparationQuotient.isUniformInducing_mk :
+    IsUniformInducing (mk : α → SeparationQuotient α) :=
   ⟨comap_mk_uniformity⟩
 
-protected theorem UniformInducing.injective [T0Space α] {f : α → β} (h : UniformInducing f) :
+@[deprecated (since := "2024-10-05")]
+alias SeparationQuotient.uniformInducing_mk := SeparationQuotient.isUniformInducing_mk
+
+protected theorem IsUniformInducing.injective [T0Space α] {f : α → β} (h : IsUniformInducing f) :
     Injective f :=
   h.inducing.injective
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.injective := IsUniformInducing.injective
+
 /-!
 ### Uniform embeddings
 -/
@@ -130,70 +194,118 @@ protected theorem UniformInducing.injective [T0Space α] {f : α → β} (h : Un
 /-- A map `f : α → β` between uniform spaces is a *uniform embedding* if it is uniform inducing and
 injective. If `α` is a separated space, then the latter assumption follows from the former. -/
 @[mk_iff]
-structure UniformEmbedding (f : α → β) extends UniformInducing f : Prop where
+structure IsUniformEmbedding (f : α → β) extends IsUniformInducing f : Prop where
   /-- A uniform embedding is injective. -/
   inj : Function.Injective f
 
-theorem uniformEmbedding_iff' {f : α → β} :
-    UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
-  rw [uniformEmbedding_iff, and_comm, uniformInducing_iff']
+lemma IsUniformEmbedding.isUniformInducing (hf : IsUniformEmbedding f) : IsUniformInducing f :=
+  hf.toIsUniformInducing
+
+@[deprecated (since := "2024-10-03")] alias UniformEmbedding := IsUniformEmbedding
 
-theorem Filter.HasBasis.uniformEmbedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
+theorem isUniformEmbedding_iff' {f : α → β} :
+    IsUniformEmbedding f ↔
+      Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by
+  rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff']
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_iff' := isUniformEmbedding_iff'
+
+theorem Filter.HasBasis.isUniformEmbedding_iff' {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
     (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
-    UniformEmbedding f ↔ Injective f ∧
+    IsUniformEmbedding f ↔ Injective f ∧
       (∀ i, p' i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ s' i) ∧
         (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by
-  rw [uniformEmbedding_iff, and_comm, h.uniformInducing_iff h']
+  rw [isUniformEmbedding_iff, and_comm, h.isUniformInducing_iff h']
 
-theorem Filter.HasBasis.uniformEmbedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
+@[deprecated (since := "2024-10-01")]
+alias Filter.HasBasis.uniformEmbedding_iff' := Filter.HasBasis.isUniformEmbedding_iff'
+
+theorem Filter.HasBasis.isUniformEmbedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
     (h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
-    UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
+    IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
       (∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by
-  simp only [h.uniformEmbedding_iff' h', h.uniformContinuous_iff h']
+  simp only [h.isUniformEmbedding_iff' h', h.uniformContinuous_iff h']
+
+@[deprecated (since := "2024-10-01")]
+alias Filter.HasBasis.uniformEmbedding_iff := Filter.HasBasis.isUniformEmbedding_iff
 
-theorem uniformEmbedding_subtype_val {p : α → Prop} :
-    UniformEmbedding (Subtype.val : Subtype p → α) :=
+theorem isUniformEmbedding_subtype_val {p : α → Prop} :
+    IsUniformEmbedding (Subtype.val : Subtype p → α) :=
   { comap_uniformity := rfl
     inj := Subtype.val_injective }
 
-theorem uniformEmbedding_set_inclusion {s t : Set α} (hst : s ⊆ t) :
-    UniformEmbedding (inclusion hst) where
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_subtype_val := isUniformEmbedding_subtype_val
+
+theorem isUniformEmbedding_set_inclusion {s t : Set α} (hst : s ⊆ t) :
+    IsUniformEmbedding (inclusion hst) where
   comap_uniformity := by rw [uniformity_subtype, uniformity_subtype, comap_comap]; rfl
   inj := inclusion_injective hst
 
-theorem UniformEmbedding.comp {g : β → γ} (hg : UniformEmbedding g) {f : α → β}
-    (hf : UniformEmbedding f) : UniformEmbedding (g ∘ f) :=
-  { hg.toUniformInducing.comp hf.toUniformInducing with inj := hg.inj.comp hf.inj }
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_set_inclusion := isUniformEmbedding_set_inclusion
+
+theorem IsUniformEmbedding.comp {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β}
+    (hf : IsUniformEmbedding f) : IsUniformEmbedding (g ∘ f) :=
+  { hg.isUniformInducing.comp hf.isUniformInducing with inj := hg.inj.comp hf.inj }
+
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.comp := IsUniformEmbedding.comp
 
-theorem UniformEmbedding.of_comp_iff {g : β → γ} (hg : UniformEmbedding g) {f : α → β} :
-    UniformEmbedding (g ∘ f) ↔ UniformEmbedding f := by
-  simp_rw [uniformEmbedding_iff, hg.toUniformInducing.of_comp_iff, hg.inj.of_comp_iff f]
+theorem IsUniformEmbedding.of_comp_iff {g : β → γ} (hg : IsUniformEmbedding g) {f : α → β} :
+    IsUniformEmbedding (g ∘ f) ↔ IsUniformEmbedding f := by
+  simp_rw [isUniformEmbedding_iff, hg.isUniformInducing.of_comp_iff, hg.inj.of_comp_iff f]
 
-theorem Equiv.uniformEmbedding {α β : Type*} [UniformSpace α] [UniformSpace β] (f : α ≃ β)
-    (h₁ : UniformContinuous f) (h₂ : UniformContinuous f.symm) : UniformEmbedding f :=
-  uniformEmbedding_iff'.2 ⟨f.injective, h₁, by rwa [← Equiv.prodCongr_apply, ← map_equiv_symm]⟩
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.of_comp_iff := IsUniformEmbedding.of_comp_iff
 
-theorem uniformEmbedding_inl : UniformEmbedding (Sum.inl : α → α ⊕ β) :=
-  uniformEmbedding_iff'.2 ⟨Sum.inl_injective, uniformContinuous_inl, fun s hs =>
+theorem Equiv.isUniformEmbedding {α β : Type*} [UniformSpace α] [UniformSpace β] (f : α ≃ β)
+    (h₁ : UniformContinuous f) (h₂ : UniformContinuous f.symm) : IsUniformEmbedding f :=
+  isUniformEmbedding_iff'.2 ⟨f.injective, h₁, by rwa [← Equiv.prodCongr_apply, ← map_equiv_symm]⟩
+
+@[deprecated (since := "2024-10-01")]
+alias Equiv.uniformEmbedding := Equiv.isUniformEmbedding
+
+theorem isUniformEmbedding_inl : IsUniformEmbedding (Sum.inl : α → α ⊕ β) :=
+  isUniformEmbedding_iff'.2 ⟨Sum.inl_injective, uniformContinuous_inl, fun s hs =>
     ⟨Prod.map Sum.inl Sum.inl '' s ∪ range (Prod.map Sum.inr Sum.inr),
       union_mem_sup (image_mem_map hs) range_mem_map,
       fun x h => by simpa [Prod.map_apply'] using h⟩⟩
 
-theorem uniformEmbedding_inr : UniformEmbedding (Sum.inr : β → α ⊕ β) :=
-  uniformEmbedding_iff'.2 ⟨Sum.inr_injective, uniformContinuous_inr, fun s hs =>
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_inl := isUniformEmbedding_inl
+
+theorem isUniformEmbedding_inr : IsUniformEmbedding (Sum.inr : β → α ⊕ β) :=
+  isUniformEmbedding_iff'.2 ⟨Sum.inr_injective, uniformContinuous_inr, fun s hs =>
     ⟨range (Prod.map Sum.inl Sum.inl) ∪ Prod.map Sum.inr Sum.inr '' s,
       union_mem_sup range_mem_map (image_mem_map hs),
       fun x h => by simpa [Prod.map_apply'] using h⟩⟩
 
-/-- If the domain of a `UniformInducing` map `f` is a T₀ space, then `f` is injective,
-hence it is a `UniformEmbedding`. -/
-protected theorem UniformInducing.uniformEmbedding [T0Space α] {f : α → β}
-    (hf : UniformInducing f) : UniformEmbedding f :=
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_inr := isUniformEmbedding_inr
+
+/-- If the domain of a `IsUniformInducing` map `f` is a T₀ space, then `f` is injective,
+hence it is a `IsUniformEmbedding`. -/
+protected theorem IsUniformInducing.isUniformEmbedding [T0Space α] {f : α → β}
+    (hf : IsUniformInducing f) : IsUniformEmbedding f :=
   ⟨hf, hf.inducing.injective⟩
 
-theorem uniformEmbedding_iff_uniformInducing [T0Space α] {f : α → β} :
-    UniformEmbedding f ↔ UniformInducing f :=
-  ⟨UniformEmbedding.toUniformInducing, UniformInducing.uniformEmbedding⟩
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.isUniformEmbedding := IsUniformInducing.isUniformEmbedding
+
+@[deprecated (since := "2024-10-01")]
+alias IsUniformInducing.uniformEmbedding := IsUniformInducing.isUniformEmbedding
+
+theorem isUniformEmbedding_iff_isUniformInducing [T0Space α] {f : α → β} :
+    IsUniformEmbedding f ↔ IsUniformInducing f :=
+  ⟨IsUniformEmbedding.isUniformInducing, IsUniformInducing.isUniformEmbedding⟩
+
+@[deprecated (since := "2024-10-05")]
+alias isUniformEmbedding_iff_uniformInducing := isUniformEmbedding_iff_isUniformInducing
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_iff_isUniformInducing := isUniformEmbedding_iff_isUniformInducing
 
 /-- If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
 `s ∈ 𝓤 β`, then `f` is uniform inducing with respect to the discrete uniformity on `α`:
@@ -210,88 +322,148 @@ theorem comap_uniformity_of_spaced_out {α} {f : α → β} {s : Set (β × β)}
 
 /-- If a map `f : α → β` sends any two distinct points to point that are **not** related by a fixed
 `s ∈ 𝓤 β`, then `f` is a uniform embedding with respect to the discrete uniformity on `α`. -/
-theorem uniformEmbedding_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
-    (hf : Pairwise fun x y => (f x, f y) ∉ s) : @UniformEmbedding α β ⊥ ‹_› f := by
+theorem isUniformEmbedding_of_spaced_out {α} {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
+    (hf : Pairwise fun x y => (f x, f y) ∉ s) : @IsUniformEmbedding α β ⊥ ‹_› f := by
   let _ : UniformSpace α := ⊥; have := discreteTopology_bot α
-  exact UniformInducing.uniformEmbedding ⟨comap_uniformity_of_spaced_out hs hf⟩
+  exact IsUniformInducing.isUniformEmbedding ⟨comap_uniformity_of_spaced_out hs hf⟩
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_of_spaced_out := isUniformEmbedding_of_spaced_out
 
-protected theorem UniformEmbedding.embedding {f : α → β} (h : UniformEmbedding f) : Embedding f :=
-  { toInducing := h.toUniformInducing.inducing
+protected lemma IsUniformEmbedding.embedding {f : α → β} (h : IsUniformEmbedding f) : Embedding f :=
+  { toInducing := h.isUniformInducing.inducing
     inj := h.inj }
 
-theorem UniformEmbedding.denseEmbedding {f : α → β} (h : UniformEmbedding f) (hd : DenseRange f) :
-    DenseEmbedding f :=
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.embedding := IsUniformEmbedding.embedding
+
+theorem IsUniformEmbedding.isDenseEmbedding {f : α → β} (h : IsUniformEmbedding f)
+    (hd : DenseRange f) : IsDenseEmbedding f :=
   { h.embedding with dense := hd }
 
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.isDenseEmbedding := IsUniformEmbedding.isDenseEmbedding
+
+@[deprecated (since := "2024-09-30")]
+alias IsUniformEmbedding.denseEmbedding := IsUniformEmbedding.isDenseEmbedding
+
 theorem closedEmbedding_of_spaced_out {α} [TopologicalSpace α] [DiscreteTopology α]
     [T0Space β] {f : α → β} {s : Set (β × β)} (hs : s ∈ 𝓤 β)
     (hf : Pairwise fun x y => (f x, f y) ∉ s) : ClosedEmbedding f := by
   rcases @DiscreteTopology.eq_bot α _ _ with rfl; let _ : UniformSpace α := ⊥
   exact
-    { (uniformEmbedding_of_spaced_out hs hf).embedding with
+    { (isUniformEmbedding_of_spaced_out hs hf).embedding with
       isClosed_range := isClosed_range_of_spaced_out hs hf }
 
-theorem closure_image_mem_nhds_of_uniformInducing {s : Set (α × α)} {e : α → β} (b : β)
-    (he₁ : UniformInducing e) (he₂ : DenseInducing e) (hs : s ∈ 𝓤 α) :
+theorem closure_image_mem_nhds_of_isUniformInducing {s : Set (α × α)} {e : α → β} (b : β)
+    (he₁ : IsUniformInducing e) (he₂ : IsDenseInducing e) (hs : s ∈ 𝓤 α) :
     ∃ a, closure (e '' { a' | (a, a') ∈ s }) ∈ 𝓝 b := by
   obtain ⟨U, ⟨hU, hUo, hsymm⟩, hs⟩ :
     ∃ U, (U ∈ 𝓤 β ∧ IsOpen U ∧ SymmetricRel U) ∧ Prod.map e e ⁻¹' U ⊆ s := by
       rwa [← he₁.comap_uniformity, (uniformity_hasBasis_open_symmetric.comap _).mem_iff] at hs
   rcases he₂.dense.mem_nhds (UniformSpace.ball_mem_nhds b hU) with ⟨a, ha⟩
-  refine ⟨a, mem_of_superset ?_ (closure_mono <| image_subset _ <| ball_mono hs a)⟩
+  refine ⟨a, mem_of_superset ?_ (closure_mono <| image_subset _ <| UniformSpace.ball_mono hs a)⟩
   have ho : IsOpen (UniformSpace.ball (e a) U) := UniformSpace.isOpen_ball (e a) hUo
-  refine mem_of_superset (ho.mem_nhds <| (mem_ball_symmetry hsymm).2 ha) fun y hy => ?_
+  refine mem_of_superset (ho.mem_nhds <| (UniformSpace.mem_ball_symmetry hsymm).2 ha) fun y hy => ?_
   refine mem_closure_iff_nhds.2 fun V hV => ?_
   rcases he₂.dense.mem_nhds (inter_mem hV (ho.mem_nhds hy)) with ⟨x, hxV, hxU⟩
   exact ⟨e x, hxV, mem_image_of_mem e hxU⟩
 
-theorem uniformEmbedding_subtypeEmb (p : α → Prop) {e : α → β} (ue : UniformEmbedding e)
-    (de : DenseEmbedding e) : UniformEmbedding (DenseEmbedding.subtypeEmb p e) :=
+@[deprecated (since := "2024-10-05")]
+alias closure_image_mem_nhds_of_uniformInducing := closure_image_mem_nhds_of_isUniformInducing
+
+theorem isUniformEmbedding_subtypeEmb (p : α → Prop) {e : α → β} (ue : IsUniformEmbedding e)
+    (de : IsDenseEmbedding e) : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) :=
   { comap_uniformity := by
-      simp [comap_comap, (· ∘ ·), DenseEmbedding.subtypeEmb, uniformity_subtype,
+      simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
         ue.comap_uniformity.symm]
     inj := (de.subtype p).inj }
 
-theorem UniformEmbedding.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
-    {e₁ : α → α'} {e₂ : β → β'} (h₁ : UniformEmbedding e₁) (h₂ : UniformEmbedding e₂) :
-    UniformEmbedding fun p : α × β => (e₁ p.1, e₂ p.2) :=
-  { h₁.toUniformInducing.prod h₂.toUniformInducing with inj := h₁.inj.prodMap h₂.inj }
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_subtypeEmb := isUniformEmbedding_subtypeEmb
+
+theorem IsUniformEmbedding.prod {α' : Type*} {β' : Type*} [UniformSpace α'] [UniformSpace β']
+    {e₁ : α → α'} {e₂ : β → β'} (h₁ : IsUniformEmbedding e₁) (h₂ : IsUniformEmbedding e₂) :
+    IsUniformEmbedding fun p : α × β => (e₁ p.1, e₂ p.2) :=
+  { h₁.isUniformInducing.prod h₂.isUniformInducing with inj := h₁.inj.prodMap h₂.inj }
+
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.prod := IsUniformEmbedding.prod
 
 /-- A set is complete iff its image under a uniform inducing map is complete. -/
-theorem isComplete_image_iff {m : α → β} {s : Set α} (hm : UniformInducing m) :
+theorem isComplete_image_iff {m : α → β} {s : Set α} (hm : IsUniformInducing m) :
     IsComplete (m '' s) ↔ IsComplete s := by
   have fact1 : SurjOn (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := surjOn_image .. |>.filter_map_Iic
   have fact2 : MapsTo (map m) (Iic <| 𝓟 s) (Iic <| 𝓟 <| m '' s) := mapsTo_image .. |>.filter_map_Iic
   simp_rw [IsComplete, imp.swap (a := Cauchy _), ← mem_Iic (b := 𝓟 _), fact1.forall fact2,
     hm.cauchy_map_iff, exists_mem_image, map_le_iff_le_comap, hm.inducing.nhds_eq_comap]
 
+/-- If `f : X → Y` is an `IsUniformInducing` map, the image `f '' s` of a set `s` is complete
+  if and only if `s` is complete. -/
+theorem IsUniformInducing.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
+    IsComplete (f '' s) ↔ IsComplete s := isComplete_image_iff hf
+
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.isComplete_iff := IsUniformInducing.isComplete_iff
+
+/-- If `f : X → Y` is an `IsUniformEmbedding`, the image `f '' s` of a set `s` is complete
+  if and only if `s` is complete. -/
+theorem IsUniformEmbedding.isComplete_iff {f : α → β} {s : Set α} (hf : IsUniformEmbedding f) :
+    IsComplete (f '' s) ↔ IsComplete s := hf.isUniformInducing.isComplete_iff
+
+@[deprecated (since := "2024-10-01")]
+alias UniformEmbedding.isComplete_iff := IsUniformEmbedding.isComplete_iff
+
+/-- Sets of a subtype are complete iff their image under the coercion is complete. -/
+theorem Subtype.isComplete_iff {p : α → Prop} {s : Set { x // p x }} :
+    IsComplete s ↔ IsComplete ((↑) '' s : Set α) :=
+  isUniformEmbedding_subtype_val.isComplete_iff.symm
+
 alias ⟨isComplete_of_complete_image, _⟩ := isComplete_image_iff
 
-theorem completeSpace_iff_isComplete_range {f : α → β} (hf : UniformInducing f) :
+theorem completeSpace_iff_isComplete_range {f : α → β} (hf : IsUniformInducing f) :
     CompleteSpace α ↔ IsComplete (range f) := by
   rw [completeSpace_iff_isComplete_univ, ← isComplete_image_iff hf, image_univ]
 
-theorem UniformInducing.isComplete_range [CompleteSpace α] {f : α → β} (hf : UniformInducing f) :
+alias ⟨_, IsUniformInducing.completeSpace⟩ := completeSpace_iff_isComplete_range
+
+@[deprecated (since := "2024-10-08")] alias UniformInducing.completeSpace :=
+  IsUniformInducing.completeSpace
+
+lemma IsUniformInducing.isComplete_range [CompleteSpace α] (hf : IsUniformInducing f) :
     IsComplete (range f) :=
   (completeSpace_iff_isComplete_range hf).1 ‹_›
 
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.isComplete_range := IsUniformInducing.isComplete_range
+
+/-- If `f` is a surjective uniform inducing map,
+then its domain is a complete space iff its codomain is a complete space.
+See also `_root_.completeSpace_congr` for a version that assumes `f` to be an equivalence. -/
+theorem IsUniformInducing.completeSpace_congr {f : α → β} (hf : IsUniformInducing f)
+    (hsurj : f.Surjective) : CompleteSpace α ↔ CompleteSpace β := by
+  rw [completeSpace_iff_isComplete_range hf, hsurj.range_eq, completeSpace_iff_isComplete_univ]
+
+@[deprecated (since := "2024-10-05")]
+alias UniformInducing.completeSpace_congr := IsUniformInducing.completeSpace_congr
+
 theorem SeparationQuotient.completeSpace_iff :
-    CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α := by
-  rw [completeSpace_iff_isComplete_univ, ← range_mk,
-    ← completeSpace_iff_isComplete_range uniformInducing_mk]
+    CompleteSpace (SeparationQuotient α) ↔ CompleteSpace α :=
+  .symm <| isUniformInducing_mk.completeSpace_congr surjective_mk
 
 instance SeparationQuotient.instCompleteSpace [CompleteSpace α] :
     CompleteSpace (SeparationQuotient α) :=
   completeSpace_iff.2 ‹_›
 
-theorem completeSpace_congr {e : α ≃ β} (he : UniformEmbedding e) :
-    CompleteSpace α ↔ CompleteSpace β := by
-  rw [completeSpace_iff_isComplete_range he.toUniformInducing, e.range_eq_univ,
-    completeSpace_iff_isComplete_univ]
+/-- See also `IsUniformInducing.completeSpace_congr`
+for a version that works for non-injective maps. -/
+theorem completeSpace_congr {e : α ≃ β} (he : IsUniformEmbedding e) :
+    CompleteSpace α ↔ CompleteSpace β :=
+  he.completeSpace_congr e.surjective
 
-theorem completeSpace_coe_iff_isComplete {s : Set α} : CompleteSpace s ↔ IsComplete s :=
-  (completeSpace_iff_isComplete_range uniformEmbedding_subtype_val.toUniformInducing).trans <| by
-    rw [Subtype.range_coe]
+theorem completeSpace_coe_iff_isComplete {s : Set α} : CompleteSpace s ↔ IsComplete s := by
+  rw [completeSpace_iff_isComplete_range isUniformEmbedding_subtype_val.isUniformInducing,
+    Subtype.range_coe]
 
 alias ⟨_, IsComplete.completeSpace_coe⟩ := completeSpace_coe_iff_isComplete
 
@@ -299,12 +471,14 @@ theorem IsClosed.completeSpace_coe [CompleteSpace α] {s : Set α} (hs : IsClose
     CompleteSpace s :=
   hs.isComplete.completeSpace_coe
 
+theorem completeSpace_ulift_iff : CompleteSpace (ULift α) ↔ CompleteSpace α :=
+  IsUniformInducing.completeSpace_congr ⟨rfl⟩ ULift.down_surjective
+
 /-- The lift of a complete space to another universe is still complete. -/
-instance ULift.completeSpace [h : CompleteSpace α] : CompleteSpace (ULift α) :=
-  haveI : UniformEmbedding (@Equiv.ulift α) := ⟨⟨rfl⟩, ULift.down_injective⟩
-  (completeSpace_congr this).2 h
+instance ULift.instCompleteSpace [CompleteSpace α] : CompleteSpace (ULift α) :=
+  completeSpace_ulift_iff.2 ‹_›
 
-theorem completeSpace_extension {m : β → α} (hm : UniformInducing m) (dense : DenseRange m)
+theorem completeSpace_extension {m : β → α} (hm : IsUniformInducing m) (dense : DenseRange m)
     (h : ∀ f : Filter β, Cauchy f → ∃ x : α, map m f ≤ 𝓝 x) : CompleteSpace α :=
   ⟨fun {f : Filter α} (hf : Cauchy f) =>
     let p : Set (α × α) → Set α → Set α := fun s t => { y : α | ∃ x : α, x ∈ t ∧ (x, y) ∈ s }
@@ -350,7 +524,7 @@ theorem completeSpace_extension {m : β → α} (hm : UniformInducing m) (dense
         _ ≤ 𝓝 x := le_nhds_of_cauchy_adhp ‹Cauchy g› this
         ⟩⟩
 
-lemma totallyBounded_image_iff {f : α → β} {s : Set α} (hf : UniformInducing f) :
+lemma totallyBounded_image_iff {f : α → β} {s : Set α} (hf : IsUniformInducing f) :
     TotallyBounded (f '' s) ↔ TotallyBounded s := by
   refine ⟨fun hs ↦ ?_, fun h ↦ h.image hf.uniformContinuous⟩
   simp_rw [(hf.basis_uniformity (basis_sets _)).totallyBounded_iff]
@@ -359,21 +533,24 @@ lemma totallyBounded_image_iff {f : α → β} {s : Set α} (hf : UniformInducin
   use u, hfin
   rwa [biUnion_image, image_subset_iff, preimage_iUnion₂] at h
 
-theorem totallyBounded_preimage {f : α → β} {s : Set β} (hf : UniformInducing f)
+theorem totallyBounded_preimage {f : α → β} {s : Set β} (hf : IsUniformInducing f)
     (hs : TotallyBounded s) : TotallyBounded (f ⁻¹' s) :=
   (totallyBounded_image_iff hf).1 <| hs.subset <| image_preimage_subset ..
 
 instance CompleteSpace.sum [CompleteSpace α] [CompleteSpace β] : CompleteSpace (α ⊕ β) := by
   rw [completeSpace_iff_isComplete_univ, ← range_inl_union_range_inr]
-  exact uniformEmbedding_inl.toUniformInducing.isComplete_range.union
-    uniformEmbedding_inr.toUniformInducing.isComplete_range
+  exact isUniformEmbedding_inl.isUniformInducing.isComplete_range.union
+    isUniformEmbedding_inr.isUniformInducing.isComplete_range
 
 end
 
-theorem uniformEmbedding_comap {α : Type*} {β : Type*} {f : α → β} [u : UniformSpace β]
-    (hf : Function.Injective f) : @UniformEmbedding α β (UniformSpace.comap f u) u f :=
-  @UniformEmbedding.mk _ _ (UniformSpace.comap f u) _ _
-    (@UniformInducing.mk _ _ (UniformSpace.comap f u) _ _ rfl) hf
+theorem isUniformEmbedding_comap {α : Type*} {β : Type*} {f : α → β} [u : UniformSpace β]
+    (hf : Function.Injective f) : @IsUniformEmbedding α β (UniformSpace.comap f u) u f :=
+  @IsUniformEmbedding.mk _ _ (UniformSpace.comap f u) _ _
+    (@IsUniformInducing.mk _ _ (UniformSpace.comap f u) _ _ rfl) hf
+
+@[deprecated (since := "2024-10-01")]
+alias uniformEmbedding_comap := isUniformEmbedding_comap
 
 /-- Pull back a uniform space structure by an embedding, adjusting the new uniform structure to
 make sure that its topology is defeq to the original one. -/
@@ -381,23 +558,26 @@ def Embedding.comapUniformSpace {α β} [TopologicalSpace α] [u : UniformSpace
     (h : Embedding f) : UniformSpace α :=
   (u.comap f).replaceTopology h.induced
 
-theorem Embedding.to_uniformEmbedding {α β} [TopologicalSpace α] [u : UniformSpace β] (f : α → β)
-    (h : Embedding f) : @UniformEmbedding α β (h.comapUniformSpace f) u f :=
+theorem Embedding.to_isUniformEmbedding {α β} [TopologicalSpace α] [u : UniformSpace β] (f : α → β)
+    (h : Embedding f) : @IsUniformEmbedding α β (h.comapUniformSpace f) u f :=
   let _ := h.comapUniformSpace f
   { comap_uniformity := rfl
     inj := h.inj }
 
+@[deprecated (since := "2024-10-01")]
+alias Embedding.to_uniformEmbedding := Embedding.to_isUniformEmbedding
+
 section UniformExtension
 
 variable {α : Type*} {β : Type*} {γ : Type*} [UniformSpace α] [UniformSpace β] [UniformSpace γ]
-  {e : β → α} (h_e : UniformInducing e) (h_dense : DenseRange e) {f : β → γ}
+  {e : β → α} (h_e : IsUniformInducing e) (h_dense : DenseRange e) {f : β → γ}
   (h_f : UniformContinuous f)
 
-local notation "ψ" => DenseInducing.extend (UniformInducing.denseInducing h_e h_dense) f
+local notation "ψ" => IsDenseInducing.extend (IsUniformInducing.isDenseInducing h_e h_dense) f
 
 include h_e h_dense h_f in
 theorem uniformly_extend_exists [CompleteSpace γ] (a : α) : ∃ c, Tendsto f (comap e (𝓝 a)) (𝓝 c) :=
-  let de := h_e.denseInducing h_dense
+  let de := h_e.isDenseInducing h_dense
   have : Cauchy (𝓝 a) := cauchy_nhds
   have : Cauchy (comap e (𝓝 a)) :=
     this.comap' (le_of_eq h_e.comap_uniformity) (de.comap_nhds_neBot _)
@@ -405,24 +585,25 @@ theorem uniformly_extend_exists [CompleteSpace γ] (a : α) : ∃ c, Tendsto f (
   CompleteSpace.complete this
 
 theorem uniform_extend_subtype [CompleteSpace γ] {p : α → Prop} {e : α → β} {f : α → γ} {b : β}
-    {s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : UniformEmbedding e)
+    {s : Set α} (hf : UniformContinuous fun x : Subtype p => f x.val) (he : IsUniformEmbedding e)
     (hd : ∀ x : β, x ∈ closure (range e)) (hb : closure (e '' s) ∈ 𝓝 b) (hs : IsClosed s)
     (hp : ∀ x ∈ s, p x) : ∃ c, Tendsto f (comap e (𝓝 b)) (𝓝 c) := by
-  have de : DenseEmbedding e := he.denseEmbedding hd
-  have de' : DenseEmbedding (DenseEmbedding.subtypeEmb p e) := de.subtype p
-  have ue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e) := uniformEmbedding_subtypeEmb _ he de
+  have de : IsDenseEmbedding e := he.isDenseEmbedding hd
+  have de' : IsDenseEmbedding (IsDenseEmbedding.subtypeEmb p e) := de.subtype p
+  have ue' : IsUniformEmbedding (IsDenseEmbedding.subtypeEmb p e) :=
+    isUniformEmbedding_subtypeEmb _ he de
   have : b ∈ closure (e '' { x | p x }) :=
     (closure_mono <| monotone_image <| hp) (mem_of_mem_nhds hb)
-  let ⟨c, hc⟩ := uniformly_extend_exists ue'.toUniformInducing de'.dense hf ⟨b, this⟩
+  let ⟨c, hc⟩ := uniformly_extend_exists ue'.isUniformInducing de'.dense hf ⟨b, this⟩
   replace hc : Tendsto (f ∘ Subtype.val (p := p)) (((𝓝 b).comap e).comap Subtype.val) (𝓝 c) := by
-    simpa only [nhds_subtype_eq_comap, comap_comap, DenseEmbedding.subtypeEmb_coe] using hc
+    simpa only [nhds_subtype_eq_comap, comap_comap, IsDenseEmbedding.subtypeEmb_coe] using hc
   refine ⟨c, (tendsto_comap'_iff ?_).1 hc⟩
   rw [Subtype.range_coe_subtype]
   exact ⟨_, hb, by rwa [← de.toInducing.closure_eq_preimage_closure_image, hs.closure_eq]⟩
 
 include h_e h_f in
 theorem uniformly_extend_spec [CompleteSpace γ] (a : α) : Tendsto f (comap e (𝓝 a)) (𝓝 (ψ a)) := by
-  simpa only [DenseInducing.extend] using
+  simpa only [IsDenseInducing.extend] using
     tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _)
 
 include h_f in
@@ -431,7 +612,7 @@ theorem uniformContinuous_uniformly_extend [CompleteSpace γ] : UniformContinuou
   have h_pnt : ∀ {a m}, m ∈ 𝓝 a → ∃ c ∈ f '' (e ⁻¹' m), (c, ψ a) ∈ s ∧ (ψ a, c) ∈ s :=
     fun {a m} hm =>
     have nb : NeBot (map f (comap e (𝓝 a))) :=
-      ((h_e.denseInducing h_dense).comap_nhds_neBot _).map _
+      ((h_e.isDenseInducing h_dense).comap_nhds_neBot _).map _
     have :
       f '' (e ⁻¹' m) ∩ ({ c | (c, ψ a) ∈ s } ∩ { c | (ψ a, c) ∈ s }) ∈ map f (comap e (𝓝 a)) :=
       inter_mem (image_mem_map <| preimage_mem_comap <| hm)
@@ -456,9 +637,9 @@ variable [T0Space γ]
 
 include h_f in
 theorem uniformly_extend_of_ind (b : β) : ψ (e b) = f b :=
-  DenseInducing.extend_eq_at _ h_f.continuous.continuousAt
+  IsDenseInducing.extend_eq_at _ h_f.continuous.continuousAt
 
 theorem uniformly_extend_unique {g : α → γ} (hg : ∀ b, g (e b) = f b) (hc : Continuous g) : ψ = g :=
-  DenseInducing.extend_unique _ hg hc
+  IsDenseInducing.extend_unique _ hg hc
 
 end UniformExtension
diff --git a/Mathlib/Topology/UnitInterval.lean b/Mathlib/Topology/UnitInterval.lean
index 52d5c0fbe4012..55c6405dda1b6 100644
--- a/Mathlib/Topology/UnitInterval.lean
+++ b/Mathlib/Topology/UnitInterval.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2020 Patrick Massot. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison
+Authors: Patrick Massot, Kim Morrison
 -/
 import Mathlib.Algebra.Order.Interval.Set.Instances
 import Mathlib.Order.Interval.Set.ProjIcc
@@ -40,10 +40,10 @@ theorem one_mem : (1 : ℝ) ∈ I :=
   ⟨zero_le_one, le_rfl⟩
 
 theorem mul_mem {x y : ℝ} (hx : x ∈ I) (hy : y ∈ I) : x * y ∈ I :=
-  ⟨mul_nonneg hx.1 hy.1, mul_le_one hx.2 hy.1 hy.2⟩
+  ⟨mul_nonneg hx.1 hy.1, mul_le_one₀ hx.2 hy.1 hy.2⟩
 
 theorem div_mem {x y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) (hxy : x ≤ y) : x / y ∈ I :=
-  ⟨div_nonneg hx hy, div_le_one_of_le hxy hy⟩
+  ⟨div_nonneg hx hy, div_le_one_of_le₀ hxy hy⟩
 
 theorem fract_mem (x : ℝ) : fract x ∈ I :=
   ⟨fract_nonneg _, (fract_lt_one _).le⟩
@@ -64,11 +64,10 @@ instance : BoundedOrder I := Set.Icc.boundedOrder zero_le_one
 
 lemma univ_eq_Icc : (univ : Set I) = Icc (0 : I) (1 : I) := Icc_bot_top.symm
 
-theorem coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 :=
-  not_iff_not.mpr coe_eq_zero
-
-theorem coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 :=
-  not_iff_not.mpr coe_eq_one
+@[norm_cast] theorem coe_ne_zero {x : I} : (x : ℝ) ≠ 0 ↔ x ≠ 0 := coe_eq_zero.not
+@[norm_cast] theorem coe_ne_one {x : I} : (x : ℝ) ≠ 1 ↔ x ≠ 1 := coe_eq_one.not
+@[simp, norm_cast] theorem coe_pos {x : I} : (0 : ℝ) < x ↔ 0 < x := Iff.rfl
+@[simp, norm_cast] theorem coe_lt_one {x : I} : (x : ℝ) < 1 ↔ x < 1 := Iff.rfl
 
 instance : Nonempty I :=
   ⟨0⟩
@@ -76,7 +75,6 @@ instance : Nonempty I :=
 instance : Mul I :=
   ⟨fun x y => ⟨x * y, mul_mem x.2 y.2⟩⟩
 
--- todo: we could set up a `LinearOrderedCommMonoidWithZero I` instance
 theorem mul_le_left {x y : I} : x * y ≤ x :=
   Subtype.coe_le_coe.mp <| mul_le_of_le_one_right x.2.1 y.2.2
 
@@ -127,9 +125,40 @@ theorem strictAnti_symm : StrictAnti σ := fun _ _ h ↦ sub_lt_sub_left (α :=
 @[deprecated (since := "2024-02-27")] alias involutive_symm := symm_involutive
 @[deprecated (since := "2024-02-27")] alias bijective_symm := symm_bijective
 
+@[simp]
+theorem symm_inj {i j : I} : σ i = σ j ↔ i = j := symm_bijective.injective.eq_iff
+
 theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 := by
   rw [coe_symm_eq, le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le, sub_half]
 
+@[simp]
+lemma symm_eq_one {i : I} : σ i = 1 ↔ i = 0 := by
+  rw [← symm_zero, symm_inj]
+
+@[simp]
+lemma symm_eq_zero {i : I} : σ i = 0 ↔ i = 1 := by
+  rw [← symm_one, symm_inj]
+
+@[simp]
+theorem symm_le_symm {i j : I} : σ i ≤ σ j ↔ j ≤ i := by
+  simp only [symm, Subtype.mk_le_mk, sub_le_sub_iff, add_le_add_iff_left, Subtype.coe_le_coe]
+
+theorem le_symm_comm {i j : I} : i ≤ σ j ↔ j ≤ σ i := by
+  rw [← symm_le_symm, symm_symm]
+
+theorem symm_le_comm {i j : I} : σ i ≤ j ↔ σ j ≤ i := by
+  rw [← symm_le_symm, symm_symm]
+
+@[simp]
+theorem symm_lt_symm {i j : I} : σ i < σ j ↔ j < i := by
+  simp only [symm, Subtype.mk_lt_mk, sub_lt_sub_iff_left, Subtype.coe_lt_coe]
+
+theorem lt_symm_comm {i j : I} : i < σ j ↔ j < σ i := by
+  rw [← symm_lt_symm, symm_symm]
+
+theorem symm_lt_comm {i j : I} : σ i < j ↔ σ j < i := by
+  rw [← symm_lt_symm, symm_symm]
+
 instance : ConnectedSpace I :=
   Subtype.connectedSpace ⟨nonempty_Icc.mpr zero_le_one, isPreconnected_Icc⟩
 
@@ -157,6 +186,17 @@ theorem nonneg' {t : I} : 0 ≤ t :=
 theorem le_one' {t : I} : t ≤ 1 :=
   t.2.2
 
+protected lemma pos_iff_ne_zero {x : I} : 0 < x ↔ x ≠ 0 := bot_lt_iff_ne_bot
+
+protected lemma lt_one_iff_ne_one {x : I} : x < 1 ↔ x ≠ 1 := lt_top_iff_ne_top
+
+lemma eq_one_or_eq_zero_of_le_mul {i j : I} (h : i ≤ j * i) : i = 0 ∨ j = 1 := by
+  contrapose! h
+  rw [← unitInterval.lt_one_iff_ne_one, ← coe_lt_one, ← unitInterval.pos_iff_ne_zero,
+    ← coe_pos] at h
+  rw [← Subtype.coe_lt_coe, coe_mul]
+  simpa using mul_lt_mul_of_pos_right h.right h.left
+
 instance : Nontrivial I := ⟨⟨1, 0, (one_ne_zero <| congrArg Subtype.val ·)⟩⟩
 
 theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc (0 : ℝ) (1 / a) := by
@@ -169,6 +209,16 @@ theorem mul_pos_mem_iff {a t : ℝ} (ha : 0 < a) : a * t ∈ I ↔ t ∈ Set.Icc
 theorem two_mul_sub_one_mem_iff {t : ℝ} : 2 * t - 1 ∈ I ↔ t ∈ Set.Icc (1 / 2 : ℝ) 1 := by
   constructor <;> rintro ⟨h₁, h₂⟩ <;> constructor <;> linarith
 
+instance : LinearOrderedCommMonoidWithZero I where
+  zero_mul i := zero_mul i
+  mul_zero i := mul_zero i
+  zero_le_one := nonneg'
+  mul_le_mul_left i j h_ij k := by
+    simp only [← Subtype.coe_le_coe, coe_mul]
+    apply mul_le_mul le_rfl ?_ (nonneg i) (nonneg k)
+    simp [h_ij]
+  __ := inferInstanceAs (LinearOrder I)
+
 end unitInterval
 
 section partition
diff --git a/Mathlib/Topology/UrysohnsBounded.lean b/Mathlib/Topology/UrysohnsBounded.lean
index fa436cb53dfbd..16835df30f181 100644
--- a/Mathlib/Topology/UrysohnsBounded.lean
+++ b/Mathlib/Topology/UrysohnsBounded.lean
@@ -4,14 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
 import Mathlib.Topology.UrysohnsLemma
-import Mathlib.Topology.ContinuousFunction.Bounded
+import Mathlib.Topology.ContinuousMap.Bounded
 
 /-!
 # Urysohn's lemma for bounded continuous functions
 
 In this file we reformulate Urysohn's lemma `exists_continuous_zero_one_of_isClosed` in terms of
 bounded continuous functions `X →ᵇ ℝ`. These lemmas live in a separate file because
-`Topology.ContinuousFunction.Bounded` imports too many other files.
+`Topology.ContinuousMap.Bounded` imports too many other files.
 
 ## Tags
 
diff --git a/Mathlib/Topology/UrysohnsLemma.lean b/Mathlib/Topology/UrysohnsLemma.lean
index 9d616a00838cc..705eb5ca08358 100644
--- a/Mathlib/Topology/UrysohnsLemma.lean
+++ b/Mathlib/Topology/UrysohnsLemma.lean
@@ -5,7 +5,7 @@ Authors: Yury Kudryashov
 -/
 import Mathlib.Analysis.Normed.Affine.AddTorsor
 import Mathlib.LinearAlgebra.AffineSpace.Ordered
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 import Mathlib.Topology.GDelta
 import Mathlib.Analysis.NormedSpace.FunctionSeries
 import Mathlib.Analysis.SpecificLimits.Basic
@@ -92,7 +92,7 @@ open neighborhood `U`, together with the assumption that `C` satisfies the prope
 latter assumption will make it possible to prove simultaneously both versions of Urysohn's lemma,
 in normal spaces (with `P` always true) and in locally compact spaces (with `P = IsCompact`).
 We put also in the structure the assumption that, for any such pair, one may find an intermediate
-pair inbetween satisfying `P`, to avoid carrying it around in the argument. -/
+pair in between satisfying `P`, to avoid carrying it around in the argument. -/
 structure CU {X : Type*} [TopologicalSpace X] (P : Set X → Prop) where
   /-- The inner set in the inductive construction towards Urysohn's lemma -/
   protected C : Set X
diff --git a/Mathlib/Topology/VectorBundle/Basic.lean b/Mathlib/Topology/VectorBundle/Basic.lean
index f7f9be5e65667..cf55b20054f31 100644
--- a/Mathlib/Topology/VectorBundle/Basic.lean
+++ b/Mathlib/Topology/VectorBundle/Basic.lean
@@ -538,7 +538,7 @@ theorem coordChange_linear_comp (i j k : ι) :
 @[nolint unusedArguments] -- Porting note(#5171): was `nolint has_nonempty_instance`
 def Index := ι
 
-/-- The base space of a vector bundle core, as a convenience function for dot notation-/
+/-- The base space of a vector bundle core, as a convenience function for dot notation -/
 @[nolint unusedArguments, reducible]
 def Base := B
 
diff --git a/Mathlib/Topology/VectorBundle/Hom.lean b/Mathlib/Topology/VectorBundle/Hom.lean
index 9f4c0e8288c7f..3b7a3339e4105 100644
--- a/Mathlib/Topology/VectorBundle/Hom.lean
+++ b/Mathlib/Topology/VectorBundle/Hom.lean
@@ -58,12 +58,6 @@ Porting note: after the port is done, we may want to remove this definition.
 protected abbrev Bundle.ContinuousLinearMap [∀ x, TopologicalSpace (E₁ x)]
     [∀ x, TopologicalSpace (E₂ x)] : B → Type _ := fun x => E₁ x →SL[σ] E₂ x
 
--- Porting note: possibly remove after the port
-instance Bundle.ContinuousLinearMap.module [∀ x, TopologicalSpace (E₁ x)]
-    [∀ x, TopologicalSpace (E₂ x)] [∀ x, TopologicalAddGroup (E₂ x)]
-    [∀ x, ContinuousConstSMul 𝕜₂ (E₂ x)] : ∀ x, Module 𝕜₂ (Bundle.ContinuousLinearMap σ E₁ E₂ x) :=
-  fun _ => inferInstance
-
 variable {E₁ E₂}
 variable [TopologicalSpace B] (e₁ e₁' : Trivialization F₁ (π F₁ E₁))
   (e₂ e₂' : Trivialization F₂ (π F₂ E₂))
diff --git a/Mathlib/Util/AddRelatedDecl.lean b/Mathlib/Util/AddRelatedDecl.lean
index f415c55828e22..ecea6a5c315b8 100644
--- a/Mathlib/Util/AddRelatedDecl.lean
+++ b/Mathlib/Util/AddRelatedDecl.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Floris van Doorn
+Authors: Kim Morrison, Floris van Doorn
 -/
 import Mathlib.Init
 import Lean.Elab.DeclarationRange
diff --git a/Mathlib/Util/AssertExists.lean b/Mathlib/Util/AssertExists.lean
index 2995ff633ca25..386f6f122afc9 100644
--- a/Mathlib/Util/AssertExists.lean
+++ b/Mathlib/Util/AssertExists.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Patrick Massot, Scott Morrison
+Authors: Patrick Massot, Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Elab.Command
diff --git a/Mathlib/Util/AssertExistsExt.lean b/Mathlib/Util/AssertExistsExt.lean
index 56e0aaad33a6f..e97a78563f4cf 100644
--- a/Mathlib/Util/AssertExistsExt.lean
+++ b/Mathlib/Util/AssertExistsExt.lean
@@ -30,9 +30,9 @@ structure AssertExists where
 
 /-- Defines the `assertExistsExt` extension for adding a `HashSet` of `AssertExists`s
 to the environment. -/
-initialize assertExistsExt : SimplePersistentEnvExtension AssertExists (HashSet AssertExists) ←
+initialize assertExistsExt : SimplePersistentEnvExtension AssertExists (Std.HashSet AssertExists) ←
   registerSimplePersistentEnvExtension {
-    addImportedFn := fun as => as.foldl HashSet.insertMany {}
+    addImportedFn := fun as => as.foldl Std.HashSet.insertMany {}
     addEntryFn := .insert
   }
 
diff --git a/Mathlib/Util/AtomM.lean b/Mathlib/Util/AtomM.lean
index 89f19cc7ea3ef..0b829028c0b11 100644
--- a/Mathlib/Util/AtomM.lean
+++ b/Mathlib/Util/AtomM.lean
@@ -11,13 +11,18 @@ import Lean.Meta.Tactic.Simp.Types
 
 This monad is used by tactics like `ring` and `abel` to keep uninterpreted atoms in a consistent
 order, and also to allow unifying atoms up to a specified transparency mode.
+
+Note: this can become very expensive because it is using `isDefEq`.
+For performance reasons, consider whether `Lean.Meta.Canonicalizer.canon` can be used instead.
+After canonicalizing, a `HashMap Expr Nat` suffices to keep track of previously seen atoms,
+and is much faster as it uses `Expr` equality rather than `isDefEq`.
 -/
 
 namespace Mathlib.Tactic
 open Lean Meta
 
 /-- The context (read-only state) of the `AtomM` monad. -/
-structure AtomM.Context :=
+structure AtomM.Context where
   /-- The reducibility setting for definitional equality of atoms -/
   red : TransparencyMode
   /-- A simplification to apply to atomic expressions when they are encountered,
@@ -26,7 +31,7 @@ structure AtomM.Context :=
   deriving Inhabited
 
 /-- The mutable state of the `AtomM` monad. -/
-structure AtomM.State :=
+structure AtomM.State where
   /-- The list of atoms-up-to-defeq encountered thus far, used for atom sorting. -/
   atoms : Array Expr := #[]
 
diff --git a/Mathlib/Util/CompileInductive.lean b/Mathlib/Util/CompileInductive.lean
index 421224f1fa26e..5c98f82d30a38 100644
--- a/Mathlib/Util/CompileInductive.lean
+++ b/Mathlib/Util/CompileInductive.lean
@@ -7,6 +7,7 @@ import Mathlib.Init
 import Lean.Elab.Command
 import Lean.Compiler.CSimpAttr
 import Lean.Util.FoldConsts
+import Lean.Data.AssocList
 
 /-!
 # Define the `compile_inductive%` command.
diff --git a/Mathlib/Util/CountHeartbeats.lean b/Mathlib/Util/CountHeartbeats.lean
index 795aac8b29aa1..bbc00430ba743 100644
--- a/Mathlib/Util/CountHeartbeats.lean
+++ b/Mathlib/Util/CountHeartbeats.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Util.Heartbeats
@@ -43,8 +43,8 @@ def runTacForHeartbeats (tac : TSyntax `Lean.Parser.Tactic.tacticSeq) (revert :
 Given a `List Nat`, return the minimum, maximum, and standard deviation.
 -/
 def variation (counts : List Nat) : List Nat :=
-  let min := counts.minimum?.getD 0
-  let max := counts.maximum?.getD 0
+  let min := counts.min?.getD 0
+  let max := counts.max?.getD 0
   let toFloat (n : Nat) := n.toUInt64.toFloat
   let toNat (f : Float) := f.toUInt64.toNat
   let counts' := counts.map toFloat
diff --git a/Mathlib/Util/Delaborators.lean b/Mathlib/Util/Delaborators.lean
index 561a55e9df20d..b45007dc4e2af 100644
--- a/Mathlib/Util/Delaborators.lean
+++ b/Mathlib/Util/Delaborators.lean
@@ -164,4 +164,4 @@ open Lean Lean.PrettyPrinter.Delaborator
   guard <| f.isAppOfArity ``Membership.mem 5
   let stx₁ ← SubExpr.withAppArg <| SubExpr.withNaryArg 3 delab
   let stx₂ ← SubExpr.withAppArg <| SubExpr.withNaryArg 4 delab
-  return ← `($stx₁ ∉ $stx₂)
+  return ← `($stx₂ ∉ $stx₁)
diff --git a/Mathlib/Util/Export.lean b/Mathlib/Util/Export.lean
index 503e077034a09..cf8ba9f835a02 100644
--- a/Mathlib/Util/Export.lean
+++ b/Mathlib/Util/Export.lean
@@ -13,7 +13,8 @@ A rudimentary export format, adapted from
 with support for lean 4 kernel primitives.
 -/
 
-open Lean (HashMap HashSet)
+open Lean
+open Std (HashMap HashSet)
 
 namespace Lean
 
@@ -36,15 +37,15 @@ instance : Coe Level Entry := ⟨Entry.level⟩
 instance : Coe Expr Entry := ⟨Entry.expr⟩
 
 structure Alloc (α) [BEq α] [Hashable α] where
-  map : HashMap α Nat
+  map : Std.HashMap α Nat
   next : Nat
 deriving Inhabited
 
 structure State where
-  names : Alloc Name := ⟨HashMap.empty.insert Name.anonymous 0, 1⟩
-  levels : Alloc Level := ⟨HashMap.empty.insert levelZero 0, 1⟩
+  names : Alloc Name := ⟨Std.HashMap.empty.insert Name.anonymous 0, 1⟩
+  levels : Alloc Level := ⟨Std.HashMap.empty.insert levelZero 0, 1⟩
   exprs : Alloc Expr
-  defs : HashSet Name
+  defs : Std.HashSet Name
   stk : Array (Bool × Entry)
 deriving Inhabited
 
@@ -76,7 +77,7 @@ def alloc {α} [BEq α] [Hashable α] [OfState α] (a : α) : ExportM Nat := do
   pure n
 
 def exportName (n : Name) : ExportM Nat := do
-  match (← get).names.map.find? n with
+  match (← get).names.map[n]? with
   | some i => pure i
   | none => match n with
     | .anonymous => pure 0
@@ -84,7 +85,7 @@ def exportName (n : Name) : ExportM Nat := do
     | .str p s => let i ← alloc n; IO.println s!"{i} #NS {← exportName p} {s}"; pure i
 
 def exportLevel (L : Level) : ExportM Nat := do
-  match (← get).levels.map.find? L with
+  match (← get).levels.map[L]? with
   | some i => pure i
   | none => match L with
     | .zero => pure 0
@@ -108,7 +109,7 @@ open ConstantInfo in
 mutual
 
 partial def exportExpr (E : Expr) : ExportM Nat := do
-  match (← get).exprs.map.find? E with
+  match (← get).exprs.map[E]? with
   | some i => pure i
   | none => match E with
     | .bvar n => let i ← alloc E; IO.println s!"{i} #EV {n}"; pure i
diff --git a/Mathlib/Util/LongNames.lean b/Mathlib/Util/LongNames.lean
index 1551a5e5ae21e..081b0dfe749d6 100644
--- a/Mathlib/Util/LongNames.lean
+++ b/Mathlib/Util/LongNames.lean
@@ -1,10 +1,11 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Lean.Name
 import Mathlib.Lean.Expr.Basic
+import Lean.Elab.Command
 
 /-!
 # Commands `#long_names` and `#long_instances`
diff --git a/Mathlib/Util/MemoFix.lean b/Mathlib/Util/MemoFix.lean
index 75a5d913d317d..0dead3fbcac55 100644
--- a/Mathlib/Util/MemoFix.lean
+++ b/Mathlib/Util/MemoFix.lean
@@ -3,8 +3,8 @@ Copyright (c) 2022 Gabriel Ebner. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Gabriel Ebner, Edward Ayers
 -/
+import Std.Data.HashMap.Basic
 import Mathlib.Init
-import Lean.Data.HashMap
 
 /-!
 # Fixpoint function with memoisation
@@ -12,15 +12,15 @@ import Lean.Data.HashMap
 -/
 
 universe u v
-open ShareCommon
+open ShareCommon Std
 
-private unsafe abbrev ObjectMap := @Lean.HashMap Object Object ⟨Object.ptrEq⟩ ⟨Object.hash⟩
+private unsafe abbrev ObjectMap := @Std.HashMap Object Object ⟨Object.ptrEq⟩ ⟨Object.hash⟩
 
 private unsafe def memoFixImplObj (f : (Object → Object) → (Object → Object)) (a : Object) :
     Object := unsafeBaseIO do
   let cache : IO.Ref ObjectMap ← ST.mkRef ∅
   let rec fix (a) := unsafeBaseIO do
-    if let some b := (← cache.get).find? a then
+    if let some b := (← cache.get)[a]? then
       return b
     let b := f fix a
     cache.modify (·.insert a b)
diff --git a/Mathlib/Util/Notation3.lean b/Mathlib/Util/Notation3.lean
index e7f520311181c..9ca6324227c12 100644
--- a/Mathlib/Util/Notation3.lean
+++ b/Mathlib/Util/Notation3.lean
@@ -97,12 +97,12 @@ structure MatchState where
   that have been found so far during the course of the matching algorithm.
   We store the contexts since we need to delaborate expressions after we leave
   scoping constructs. -/
-  vars : HashMap Name (SubExpr × LocalContext × LocalInstances)
+  vars : Std.HashMap Name (SubExpr × LocalContext × LocalInstances)
   /-- The binders accumulated while matching a `scoped` expression. -/
   scopeState : Option (Array (TSyntax ``extBinderParenthesized))
   /-- The arrays of delaborated `Term`s accumulated while matching
   `foldl` and `foldr` expressions. For `foldl`, the arrays are stored in reverse order. -/
-  foldState : HashMap Name (Array Term)
+  foldState : Std.HashMap Name (Array Term)
 
 /-- A matcher is a delaboration function that transforms `MatchState`s. -/
 def Matcher := MatchState → DelabM MatchState
@@ -118,7 +118,7 @@ def MatchState.empty : MatchState where
 saved context. Fails if the variable has no value. -/
 def MatchState.withVar {α : Type} (s : MatchState) (name : Name)
     (m : DelabM α) : DelabM α := do
-  let some (se, lctx, linsts) := s.vars.find? name | failure
+  let some (se, lctx, linsts) := s.vars[name]? | failure
   withLCtx lctx linsts <| withTheReader SubExpr (fun _ => se) <| m
 
 /-- Delaborate the given variable's value. Fails if the variable has no value.
@@ -138,7 +138,7 @@ def MatchState.captureSubexpr (s : MatchState) (name : Name) : DelabM MatchState
 /-- Get the accumulated array of delaborated terms for a given foldr/foldl.
 Returns `#[]` if nothing has been pushed yet. -/
 def MatchState.getFoldArray (s : MatchState) (name : Name) : Array Term :=
-  (s.foldState.find? name).getD #[]
+  s.foldState[name]?.getD #[]
 
 /-- Get the accumulated array of delaborated terms for a given foldr/foldl.
 Returns `#[]` if nothing has been pushed yet. -/
@@ -153,7 +153,7 @@ def MatchState.pushFold (s : MatchState) (name : Name) (t : Term) : MatchState :
 /-- Matcher that assigns the current `SubExpr` into the match state;
 if a value already exists, then it checks for equality. -/
 def matchVar (c : Name) : Matcher := fun s => do
-  if let some (se, _, _) := s.vars.find? c then
+  if let some (se, _, _) := s.vars[c]? then
     guard <| se.expr == (← getExpr)
     return s
   else
@@ -206,7 +206,7 @@ def matchLambda (matchDom : Matcher) (matchBody : Expr → Matcher) : Matcher :=
 with types that are fresh metavariables.
 This is used for example when initializing `p` in `(scoped p => ...)` when elaborating `...`. -/
 def setupLCtx (lctx : LocalContext) (boundNames : Array Name) :
-    MetaM (LocalContext × HashMap FVarId Name) := do
+    MetaM (LocalContext × Std.HashMap FVarId Name) := do
   let mut lctx := lctx
   let mut boundFVars := {}
   for name in boundNames do
@@ -224,18 +224,18 @@ If it succeeds generating a matcher, returns
 1. a list of keys that should be used for the `delab` attribute
    when defining the elaborator
 2. a `Term` that represents a `Matcher` for the given expression `e`. -/
-partial def exprToMatcher (boundFVars : HashMap FVarId Name) (localFVars : HashMap FVarId Term)
-      (e : Expr) :
+partial def exprToMatcher (boundFVars : Std.HashMap FVarId Name)
+    (localFVars : Std.HashMap FVarId Term) (e : Expr) :
     OptionT TermElabM (List Name × Term) := do
   match e with
   | .mvar .. => return ([], ← `(pure))
   | .const n _ => return ([`app ++ n], ← ``(matchExpr (Expr.isConstOf · $(quote n))))
   | .sort .. => return ([`sort], ← ``(matchExpr Expr.isSort))
   | .fvar fvarId =>
-    if let some n := boundFVars.find? fvarId then
+    if let some n := boundFVars[fvarId]? then
       -- This fvar is a pattern variable.
       return ([], ← ``(matchVar $(quote n)))
-    else if let some s := localFVars.find? fvarId then
+    else if let some s := localFVars[fvarId]? then
       -- This fvar is bound by a lambda or forall expression in the pattern itself
       return ([], ← ``(matchExpr (· == $s)))
     else
@@ -377,7 +377,7 @@ partial def matchFoldl (lit x y : Name) (smatcher : Matcher) (sinit : Matcher) :
     -- y gives the next element of the list
     let s := s.pushFold lit (← s.delabVar y expr)
     -- x gives the next lit
-    let some newLit := s.vars.find? x | failure
+    let some newLit := s.vars[x]? | failure
     -- If progress was not made, fail
     if newLit.1.expr == expr then failure
     -- Progress was made, so recurse
@@ -462,13 +462,13 @@ elab (name := notation3) doc:(docComment)? attrs?:(Parser.Term.attributes)? attr
     pp?:(ppSpace prettyPrintOpt)? items:(ppSpace notation3Item)+ " => " val:term : command => do
   -- We use raw `Name`s for variables. This maps variable names back to the
   -- identifiers that appear in `items`
-  let mut boundIdents : HashMap Name Ident := {}
+  let mut boundIdents : Std.HashMap Name Ident := {}
   -- Replacements to use for the `macro`
-  let mut boundValues : HashMap Name Syntax := {}
+  let mut boundValues : Std.HashMap Name Syntax := {}
   -- The names of the bound names in order, used when constructing patterns for delaboration.
   let mut boundNames : Array Name := #[]
   -- The normal/foldl/foldr type of each variable (for delaborator)
-  let mut boundType : HashMap Name BoundValueType := {}
+  let mut boundType : Std.HashMap Name BoundValueType := {}
   -- Function to update `syntaxArgs` and `pattArgs` using `macroArg` syntax
   let pushMacro (syntaxArgs : Array (TSyntax `stx)) (pattArgs : Array Syntax)
       (mac : TSyntax ``macroArg) := do
@@ -506,8 +506,8 @@ elab (name := notation3) doc:(docComment)? attrs?:(Parser.Term.attributes)? attr
       (syntaxArgs, pattArgs) ← pushMacro syntaxArgs pattArgs <| ←
         `(macroArg| $id:ident:sepBy(term $(prec?)?, $sep:str))
       -- N.B. `Syntax.getId` returns `.anonymous` for non-idents
-      let scopedTerm' ← scopedTerm.replaceM fun s => pure (boundValues.find? s.getId)
-      let init' ← init.replaceM fun s => pure (boundValues.find? s.getId)
+      let scopedTerm' ← scopedTerm.replaceM fun s => pure boundValues[s.getId]?
+      let init' ← init.replaceM fun s => pure boundValues[s.getId]?
       boundIdents := boundIdents.insert id.getId id
       match kind with
         | `(foldKind| foldl) =>
@@ -531,7 +531,7 @@ elab (name := notation3) doc:(docComment)? attrs?:(Parser.Term.attributes)? attr
         `(macroArg| $lit:ident:term $(prec?)?)
       matchers := matchers.push <|
         mkScopedMatcher lit.getId scopedId.getId scopedTerm boundNames
-      let scopedTerm' ← scopedTerm.replaceM fun s => pure (boundValues.find? s.getId)
+      let scopedTerm' ← scopedTerm.replaceM fun s => pure boundValues[s.getId]?
       boundIdents := boundIdents.insert lit.getId lit
       boundValues := boundValues.insert lit.getId <| ←
         `(expand_binders% ($scopedId => $scopedTerm') $$binders:extBinders,
@@ -560,7 +560,7 @@ elab (name := notation3) doc:(docComment)? attrs?:(Parser.Term.attributes)? attr
   let fullName := currNamespace ++ name
   trace[notation3] "syntax declaration has name {fullName}"
   let pat : Term := ⟨mkNode fullName pattArgs⟩
-  let val' ← val.replaceM fun s => pure (boundValues.find? s.getId)
+  let val' ← val.replaceM fun s => pure boundValues[s.getId]?
   let mut macroDecl ← `(macro_rules | `($pat) => `($val'))
   if isLocalAttrKind attrKind then
     -- For local notation, take section variables into account
@@ -584,7 +584,7 @@ elab (name := notation3) doc:(docComment)? attrs?:(Parser.Term.attributes)? attr
       trace[notation3] "matcher:{indentD matcher}"
       let mut result ← `(`($pat))
       for (name, id) in boundIdents.toArray do
-        match boundType.findD name .normal with
+        match boundType.getD name .normal with
         | .normal => result ← `(MatchState.delabVar s $(quote name) (some e) >>= fun $id => $result)
         | .foldl => result ←
           `(let $id := (MatchState.getFoldArray s $(quote name)).reverse; $result)
diff --git a/Mathlib/Util/Qq.lean b/Mathlib/Util/Qq.lean
index af1dbac4f7ad8..5232470c0d692 100644
--- a/Mathlib/Util/Qq.lean
+++ b/Mathlib/Util/Qq.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2023 Scott Morrison. All rights reserved.
+Copyright (c) 2023 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison, Alex J. Best
+Authors: Kim Morrison, Alex J. Best
 -/
 import Mathlib.Init
 import Qq
diff --git a/Mathlib/Util/SleepHeartbeats.lean b/Mathlib/Util/SleepHeartbeats.lean
index f6cb76c320e8d..7716e32f74a86 100644
--- a/Mathlib/Util/SleepHeartbeats.lean
+++ b/Mathlib/Util/SleepHeartbeats.lean
@@ -34,6 +34,7 @@ elab "sleep_heartbeats " n:num : tactic => do
      option -/
   | some m => sleepAtLeastHeartbeats (m * 1000)
 
+set_option linter.unusedTactic false in
 example : 1 = 1 := by
   sleep_heartbeats 1000
   rfl
diff --git a/Mathlib/Util/Superscript.lean b/Mathlib/Util/Superscript.lean
index 0bf0bdb933aa2..85d3b2fecae7f 100644
--- a/Mathlib/Util/Superscript.lean
+++ b/Mathlib/Util/Superscript.lean
@@ -29,7 +29,7 @@ universe u
 
 namespace Mathlib.Tactic
 
-open Lean Parser PrettyPrinter
+open Lean Parser PrettyPrinter Std
 
 namespace Superscript
 
@@ -38,9 +38,9 @@ instance : Hashable Char := ⟨fun c => hash c.1⟩
 /-- A bidirectional character mapping. -/
 structure Mapping where
   /-- Map from "special" (e.g. superscript) characters to "normal" characters. -/
-  toNormal : HashMap Char Char := {}
+  toNormal : Std.HashMap Char Char := {}
   /-- Map from "normal" text to "special" (e.g. superscript) characters. -/
-  toSpecial : HashMap Char Char := {}
+  toSpecial : Std.HashMap Char Char := {}
   deriving Inhabited
 
 /-- Constructs a mapping (intended for compile time use). Panics on violated invariants. -/
@@ -95,7 +95,8 @@ partial def satisfyTokensFn (p : Char → Bool) (errorMsg : String) (many := tru
 variable {α : Type u} [Inhabited α] (as : Array α) (leftOfPartition : α → Bool) in
 /-- Given a predicate `leftOfPartition` which is true for indexes `< i` and false for `≥ i`,
 returns `i`, by binary search. -/
-@[specialize] partial def partitionPoint (lo := 0) (hi := as.size) : Nat :=
+@[specialize]
+def partitionPoint (lo := 0) (hi := as.size) : Nat :=
   if lo < hi then
     let m := (lo + hi)/2
     let a := as.get! m
@@ -104,6 +105,7 @@ returns `i`, by binary search. -/
     else
       partitionPoint lo m
   else lo
+  termination_by hi - lo
 
 /-- The core function for super/subscript parsing. It consists of three stages:
 
@@ -128,7 +130,7 @@ partial def scriptFnNoAntiquot (m : Mapping) (errorMsg : String) (p : ParserFn)
       let mut pos := start
       while pos < stopTk do
         let c := input.get pos
-        let c' := m.toNormal.find! c
+        let c' := m.toNormal[c]!
         newStr := newStr.push c'
         pos := pos + c
         if c.utf8Size != c'.utf8Size then
@@ -213,7 +215,7 @@ def scriptParser.formatter (name : String) (m : Mapping) (k : SyntaxNodeKind) (p
   Formatter.node.formatter k p
   let st ← get
   let transformed : Except String _ := st.stack.mapM (·.mapStringsM fun s => do
-    let .some s := s.toList.mapM (m.toSpecial.insert ' ' ' ').find? | .error s
+    let .some s := s.toList.mapM (m.toSpecial.insert ' ' ' ').get? | .error s
     .ok ⟨s⟩)
   match transformed with
   | .error err =>
diff --git a/Mathlib/Util/SynthesizeUsing.lean b/Mathlib/Util/SynthesizeUsing.lean
index a9979d22f051f..6e65203680e84 100644
--- a/Mathlib/Util/SynthesizeUsing.lean
+++ b/Mathlib/Util/SynthesizeUsing.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2022 Scott Morrison. All rights reserved.
+Copyright (c) 2022 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Init
 import Lean.Elab.Tactic.Basic
diff --git a/Mathlib/Util/Time.lean b/Mathlib/Util/Time.lean
deleted file mode 100644
index 26da53c0c13ce..0000000000000
--- a/Mathlib/Util/Time.lean
+++ /dev/null
@@ -1,36 +0,0 @@
-/-
-Copyright (c) 2021 Mario Carneiro. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro
--/
-import Mathlib.Init
-import Lean
-
-/-!
-# Defines `#time` command.
-
-Time the elaboration of a command, and print the result (in milliseconds).
--/
-
-section
-open Lean Elab Command
-
-syntax (name := timeCmd) "#time " command : command
-
-/--
-Time the elaboration of a command, and print the result (in milliseconds).
-
-Example usage:
-```
-set_option maxRecDepth 100000 in
-#time example : (List.range 500).length = 500 := rfl
-```
--/
-@[command_elab timeCmd] def timeCmdElab : CommandElab
-  | `(#time%$tk $stx:command) => do
-    let start ← IO.monoMsNow
-    elabCommand stx
-    logInfoAt tk m!"time: {(← IO.monoMsNow) - start}ms"
-  | _ => throwUnsupportedSyntax
-
-end
diff --git a/README.md b/README.md
index 53d266e917989..dfe49c658f2b5 100644
--- a/README.md
+++ b/README.md
@@ -30,7 +30,7 @@ For more pointers, see [Learning Lean](https://leanprover-community.github.io/le
 ## Documentation
 
 Besides the installation guides above and [Lean's general
-documentation](https://leanprover.github.io/documentation/), the documentation
+documentation](https://docs.lean-lang.org/lean4/doc/), the documentation
 of mathlib consists of:
 
 - [The mathlib4 docs](https://leanprover-community.github.io/mathlib4_docs/index.html): documentation [generated
diff --git a/Shake/Main.lean b/Shake/Main.lean
index b6787a87fd6cf..e5b7645a31d81 100644
--- a/Shake/Main.lean
+++ b/Shake/Main.lean
@@ -81,7 +81,7 @@ abbrev Bitset := Nat
 /-- The main state of the checker, containing information on all loaded modules. -/
 structure State where
   /-- Maps a module name to its index in the module list. -/
-  toIdx : HashMap Name USize := {}
+  toIdx : Std.HashMap Name USize := {}
   /-- Maps a module index to the module name. -/
   modNames : Array Name := #[]
   /-- Maps a module index to the module data. -/
@@ -96,7 +96,7 @@ structure State where
   /-- Maps a constant name to the module index containing it.
   A value of `none` means the constant was found in multiple modules,
   in which case we do not track it. -/
-  constToIdx : HashMap Name (Option USize) := {}
+  constToIdx : Std.HashMap Name (Option USize) := {}
 
 /-- Returns `true` if this is a constant whose body should not be considered for dependency
 tracking purposes. -/
@@ -110,7 +110,7 @@ def isBlacklisted (name : Name) : Bool :=
 
 /-- Calculates the value of the `needs[i]` bitset for a given module `mod`.
 Bit `j` is set in the result if some constant from module `j` is used in this module. -/
-def calcNeeds (constToIdx : HashMap Name (Option USize)) (mod : ModuleData) : Bitset :=
+def calcNeeds (constToIdx : Std.HashMap Name (Option USize)) (mod : ModuleData) : Bitset :=
   mod.constants.foldl (init := 0) fun deps ci =>
     if isBlacklisted ci.name then deps else
     let deps := visitExpr ci.type deps
@@ -120,13 +120,13 @@ def calcNeeds (constToIdx : HashMap Name (Option USize)) (mod : ModuleData) : Bi
 where
   /-- Accumulate the results from expression `e` into `deps`. -/
   visitExpr e deps :=
-    Lean.Expr.foldConsts e deps fun c deps => match constToIdx.find? c with
+    Lean.Expr.foldConsts e deps fun c deps => match constToIdx[c]? with
       | some (some i) => deps ||| (1 <<< i.toNat)
       | _ => deps
 
 /-- Calculates the same as `calcNeeds` but tracing each module to a specific constant. -/
-def getExplanations (constToIdx : HashMap Name (Option USize)) (mod : ModuleData) :
-    HashMap USize (Name × Name) :=
+def getExplanations (constToIdx : Std.HashMap Name (Option USize)) (mod : ModuleData) :
+    Std.HashMap USize (Name × Name) :=
   mod.constants.foldl (init := {}) fun deps ci =>
     if isBlacklisted ci.name then deps else
     let deps := visitExpr ci.name ci.type deps
@@ -136,10 +136,10 @@ def getExplanations (constToIdx : HashMap Name (Option USize)) (mod : ModuleData
 where
   /-- Accumulate the results from expression `e` into `deps`. -/
   visitExpr name e deps :=
-    Lean.Expr.foldConsts e deps fun c deps => match constToIdx.find? c with
+    Lean.Expr.foldConsts e deps fun c deps => match constToIdx[c]? with
       | some (some i) =>
         if
-          if let some (name', _) := deps.find? i then
+          if let some (name', _) := deps[i]? then
             decide (name.toString.length < name'.toString.length)
           else true
         then
@@ -159,7 +159,7 @@ partial def loadModules (imports : Array Import) : StateT State IO (Array USize
   let mut transImps := 0
   for imp in imports do
     let s ← get
-    if let some i := s.toIdx.find? imp.module then
+    if let some i := s.toIdx[imp.module]? then
       imps := imps.push i
       transImps := transImps ||| s.transDeps[i]!
     else
@@ -181,14 +181,14 @@ partial def loadModules (imports : Array Import) : StateT State IO (Array USize
         transDeps := s.transDeps.push transDeps
         needs := s.needs
         constToIdx := mod.constNames.foldl (init := s.constToIdx) fun m a =>
-          match m.insertIfNew a n with
-          | (m, some (some _)) =>
+          match m.getThenInsertIfNew? a n with
+          | (some (some _), m) =>
             -- Note: If a constant is found in multiple modules, we assume it is an auto-generated
             -- definition which is created on demand, and therefore it is safe to ignore any
             -- dependencies via this definition because it will just be re-created in the current
             -- module if we don't import it.
             m.insert a none
-          | (m, _) => m
+          | (_, m) => m
       }
   return (imps, transImps)
 
@@ -198,17 +198,17 @@ partial def loadModules (imports : Array Import) : StateT State IO (Array USize
 * If `j ∈ added` then we want to add module index `j` to the imports of `i`.
   We keep this as a bitset because we will do transitive reduction before applying it
 -/
-def Edits := HashMap Name (NameSet × Bitset)
+def Edits := Std.HashMap Name (NameSet × Bitset)
 
 /-- Register that we want to remove `tgt` from the imports of `src`. -/
 def Edits.remove (ed : Edits) (src tgt : Name) : Edits :=
-  match ed.find? src with
+  match ed.get? src with
   | none => ed.insert src (RBTree.insert ∅ tgt, 0)
   | some (a, b) => ed.insert src (a.insert tgt, b)
 
 /-- Register that we want to add `tgt` to the imports of `src`. -/
 def Edits.add (ed : Edits) (src : Name) (tgt : Nat) : Edits :=
-  match ed.find? src with
+  match ed.get? src with
   | none => ed.insert src (∅, 1 <<< tgt)
   | some (a, b) => ed.insert src (a, b ||| (1 <<< tgt))
 
@@ -262,7 +262,7 @@ def visitModule (s : State) (srcSearchPath : SearchPath) (ignoreImps : Bitset)
   let mut toRemove := #[]
   let mut newDeps := 0
   for imp in s.mods[i]!.imports do
-    let j := s.toIdx.find! imp.module
+    let j := s.toIdx[imp.module]!
     if transDeps &&& (1 <<< j.toNat) == 0 then
       toRemove := toRemove.push j
     else
@@ -360,11 +360,11 @@ def visitModule (s : State) (srcSearchPath : SearchPath) (ignoreImps : Bitset)
   if explain then
     let explanation := getExplanations s.constToIdx s.mods[i]!
     let sanitize n := if n.hasMacroScopes then (sanitizeName n).run' { options := {} } else n
-    let run j := do
-      if let some (n, c) := explanation.find? j then
+    let run (j : USize) := do
+      if let some (n, c) := explanation[j]? then
         println! "  note: {s.modNames[i]!} requires {s.modNames[j]!}\
           \n    because {sanitize n} refers to {sanitize c}"
-    for imp in s.mods[i]!.imports do run <| s.toIdx.find! imp.module
+    for imp in s.mods[i]!.imports do run <| s.toIdx[imp.module]!
     for i in toAdd do run i.toUSize
 
   return edits
@@ -372,7 +372,7 @@ def visitModule (s : State) (srcSearchPath : SearchPath) (ignoreImps : Bitset)
 /-- Convert a list of module names to a bitset of module indexes -/
 def toBitset (s : State) (ns : List Name) : Bitset :=
   ns.foldl (init := 0) fun c name =>
-    match s.toIdx.find? name with
+    match s.toIdx[name]? with
     | some i => c ||| (1 <<< i.toNat)
     | none => c
 
@@ -470,7 +470,7 @@ def main (args : List String) : IO UInt32 := do
   -- Parse the config file
   let ignoreMods := toBitset s (cfg.ignoreAll?.getD [])
   let ignoreImps := toBitset s (cfg.ignoreImport?.getD [])
-  let ignore := (cfg.ignore?.getD {}).fold (init := mkHashMap) fun m a v =>
+  let ignore := (cfg.ignore?.getD {}).fold (init := Std.HashMap.empty) fun m a v =>
     m.insert a (toBitset s v.toList)
 
   let noIgnore (i : Nat) :=
@@ -493,11 +493,11 @@ def main (args : List String) : IO UInt32 := do
     println! "The following changes will be made automatically:"
 
   -- Check all selected modules
-  let mut edits : Edits := mkHashMap
+  let mut edits : Edits := Std.HashMap.empty
   for i in [0:s.mods.size], t in needs do
     if let some t := t then
       if noIgnore i then
-        let ignoreImps := ignoreImps ||| ignore.findD s.modNames[i]! 0
+        let ignoreImps := ignoreImps ||| ignore.getD s.modNames[i]! 0
         edits ← visitModule s srcSearchPath ignoreImps i t.get edits
           args.downstream args.githubStyle args.explain
 
diff --git a/docs/100.yaml b/docs/100.yaml
index d10c937e7c0eb..b1fb269096866 100644
--- a/docs/100.yaml
+++ b/docs/100.yaml
@@ -18,6 +18,12 @@
   title  : Prime Number Theorem
 6:
   title  : Gödel’s Incompleteness Theorem
+  author : Shogo Saito
+  links  :
+    results :
+    - First: https://github.com/FormalizedFormalLogic/Incompleteness/blob/master/Incompleteness/Arith/First.lean
+    - Second: https://github.com/FormalizedFormalLogic/Incompleteness/blob/master/Incompleteness/Arith/Second.lean
+    website: https://formalizedformallogic.github.io/Book/
 7:
   title  : Law of Quadratic Reciprocity
   decls  :
@@ -177,7 +183,7 @@
 49:
   title  : The Cayley-Hamilton Theorem
   decl   : Matrix.aeval_self_charpoly
-  author : Scott Morrison
+  author : Kim Morrison
 50:
   title  : The Number of Platonic Solids
 51:
@@ -339,7 +345,7 @@
   title  : Morley’s Theorem
 85:
   title  : Divisibility by 3 Rule
-  author : Scott Morrison
+  author : Kim Morrison
   decls  :
     - Nat.three_dvd_iff
 86:
diff --git a/docs/overview.yaml b/docs/overview.yaml
index be28612dbf810..d020bc5328e44 100644
--- a/docs/overview.yaml
+++ b/docs/overview.yaml
@@ -209,7 +209,7 @@ Topology:
     cluster point: 'ClusterPt'
     Hausdorff space: 'T2Space'
     sequential space: 'SequentialSpace'
-    extension by continuity: 'DenseInducing.extend'
+    extension by continuity: 'IsDenseInducing.extend'
     compactness in terms of filters: 'IsCompact'
     compactness in terms of open covers (Borel-Lebesgue): 'isCompact_iff_finite_subcover'
     connectedness: 'ConnectedSpace'
@@ -352,7 +352,7 @@ Analysis:
     Liouville theorem: 'Differentiable.apply_eq_apply_of_bounded'
     maximum modulus principle: 'Complex.eventually_eq_of_isLocalMax_norm'
     principle of isolated zeros: 'AnalyticAt.eventually_eq_zero_or_eventually_ne_zero'
-    principle of analytic continuation: 'AnalyticOn.eqOn_of_preconnected_of_frequently_eq'
+    principle of analytic continuation: 'AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq'
     analyticity of holomorphic functions: 'DifferentiableOn.analyticAt'
     Schwarz lemma: 'Complex.abs_le_abs_of_mapsTo_ball_self'
     removable singularity: 'Complex.differentiableOn_update_limUnder_insert_of_isLittleO'
diff --git a/docs/references.bib b/docs/references.bib
index 71178e6a2c7f3..2f63bf3e561a3 100644
--- a/docs/references.bib
+++ b/docs/references.bib
@@ -1396,6 +1396,18 @@ @Book{            friedmanscarr2005
   zbl           = {1080.46001}
 }
 
+@Book{            fritsch-piccinini1990,
+  place         = {Cambridge},
+  series        = {Cambridge Studies in Advanced Mathematics},
+  title         = {Cellular Structures in Topology},
+  publisher     = {Cambridge University Press},
+  author        = {Fritsch, Rudolf and Piccinini, Renzo},
+  year          = {1990},
+  collection    = {Cambridge Studies in Advanced Mathematics},
+  url           = {https://doi.org/10.1017/CBO9780511983948},
+  doi           = {10.1017/CBO9780511983948}
+}
+
 @Book{            fuchs1963,
   author        = {Fuchs, L.},
   title         = {Partially ordered algebraic systems},
@@ -2827,6 +2839,17 @@ @Misc{            ponton2020chebyshev
   primaryclass  = {math.NT}
 }
 
+@Article{         Prielipp1970,
+  author        = {Robert W. Prielipp},
+  title         = {PERFECT NUMBERS, ABUNDANT NUMBERS, AND DEFICIENT NUMBERS},
+  journal       = {The Mathematics Teacher},
+  volume        = {63},
+  year          = {1970},
+  pages         = {692--696},
+  issn          = {00255769},
+  url           = {http://www.jstor.org/stable/27958492}
+}
+
 @InCollection{    ribenboim1971,
   author        = {Ribenboim, Paulo},
   title         = {\'{E}pimorphismes de modules qui sont n\'{e}cessairement
diff --git a/docs/undergrad.yaml b/docs/undergrad.yaml
index 21619980559e5..4a3b5c75b8fd9 100644
--- a/docs/undergrad.yaml
+++ b/docs/undergrad.yaml
@@ -356,7 +356,9 @@ Single Variable Real Analysis:
     Weierstrass trigonometric approximation theorem: 'span_fourier_closure_eq_top'
   Convexity:
     convex functions of a real variable: 'ConvexOn'
-    continuity and differentiability of convex functions: 'https://en.wikipedia.org/wiki/Convex_function#Functions_of_one_variable'
+    continuity and differentiability of convex functions:
+      continuity: 'ConvexOn.continuousOn'
+      differentiability: 'https://en.wikipedia.org/wiki/Convex_function#Functions_of_one_variable'
     characterizations of convexity: 'convexOn_of_deriv2_nonneg'
     convexity inequalities: 'analysis/mean_inequalities.html'
 
@@ -387,7 +389,7 @@ Single Variable Complex Analysis:
     Cauchy formulas: 'Complex.two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable'
     analyticity of a holomorphic function: 'DifferentiableOn.analyticAt'
     principle of isolated zeros: 'AnalyticAt.eventually_eq_zero_or_eventually_ne_zero'
-    principle of analytic continuation: 'AnalyticOn.eqOn_of_preconnected_of_frequently_eq'
+    principle of analytic continuation: 'AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq'
     maximum principle: 'Complex.eventually_eq_of_isLocalMax_norm'
     isolated singularities: ''
     Laurent series: ''
diff --git a/lake-manifest.json b/lake-manifest.json
index df2052cdb65c8..4a49ec817f12a 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -5,7 +5,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "e6d3a32d66252a70fda1d56463e1da975b3b8f53",
+   "rev": "daf1ed91789811cf6bbb7bf2f4dad6b3bad8fbf4",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
@@ -15,7 +15,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "71f54425e6fe0fa75f3aef33a2813a7898392222",
+   "rev": "2b2f6d7fbe9d917fc010e9054c1ce11774c9088b",
    "name": "Qq",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -25,7 +25,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "f36af1a7011c102cdf3f5f6c31d2367de28da3a8",
+   "rev": "b20a88676fd00affb90cbc9f1ff004ae588103b3",
    "name": "aesop",
    "manifestFile": "lake-manifest.json",
    "inputRev": "master",
@@ -35,16 +35,16 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "a96aee5245720f588876021b6a0aa73efee49c76",
+   "rev": "eb08eee94098fe530ccd6d8751a86fe405473d4c",
    "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.41",
+   "inputRev": "v0.0.42",
    "inherited": false,
    "configFile": "lakefile.lean"},
   {"url": "https://github.com/leanprover/lean4-cli",
    "type": "git",
    "subDir": null,
-   "scope": "",
+   "scope": "leanprover",
    "rev": "2cf1030dc2ae6b3632c84a09350b675ef3e347d0",
    "name": "Cli",
    "manifestFile": "lake-manifest.json",
@@ -55,11 +55,21 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "57bd2065f1dbea5e9235646fb836c7cea9ab03b6",
+   "rev": "7376ac07aa2b0492372c056b7a2c3163b3026d1e",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": false,
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/LeanSearchClient",
+   "type": "git",
+   "subDir": null,
+   "scope": "leanprover-community",
+   "rev": "4b61d4abc1659f15ffda5ec24fdebc229d51d066",
+   "name": "LeanSearchClient",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
+   "inherited": false,
    "configFile": "lakefile.toml"}],
  "name": "mathlib",
  "lakeDir": ".lake"}
diff --git a/lakefile.lean b/lakefile.lean
index a5a6baf266f63..212047a97ac35 100644
--- a/lakefile.lean
+++ b/lakefile.lean
@@ -10,8 +10,10 @@ open Lake DSL
 require "leanprover-community" / "batteries" @ git "main"
 require "leanprover-community" / "Qq" @ git "master"
 require "leanprover-community" / "aesop" @ git "master"
-require "leanprover-community" / "proofwidgets" @ git "v0.0.41"
+require "leanprover-community" / "proofwidgets" @ git "v0.0.42"
 require "leanprover-community" / "importGraph" @ git "main"
+require "leanprover-community" / "LeanSearchClient" @ git "main"
+  from git "https://github.com/leanprover-community/LeanSearchClient" @ "main"
 
 /-!
 ## Options for building mathlib
@@ -23,15 +25,18 @@ require "leanprover-community" / "importGraph" @ git "main"
   (as well as `Archive`, `Counterexamples` and `test`).
 -/
 abbrev mathlibOnlyLinters : Array LeanOption := #[
+  ⟨`linter.docPrime, true⟩,
   ⟨`linter.hashCommand, true⟩,
-  ⟨`linter.missingEnd, true⟩,
-  ⟨`linter.cdot, true⟩,
-  ⟨`linter.dollarSyntax, true⟩,
-  ⟨`linter.style.lambdaSyntax, true⟩,
-  ⟨`linter.longLine, true⟩,
   ⟨`linter.oldObtain, true,⟩,
   ⟨`linter.refine, true⟩,
-  ⟨`linter.style.setOption, true⟩
+  ⟨`linter.style.cdot, true⟩,
+  ⟨`linter.style.dollarSyntax, true⟩,
+  ⟨`linter.style.lambdaSyntax, true⟩,
+  ⟨`linter.style.longLine, true⟩,
+  ⟨`linter.style.longFile, .ofNat 1500⟩,
+  ⟨`linter.style.missingEnd, true⟩,
+  ⟨`linter.style.setOption, true⟩,
+  ⟨`aesop.warn.applyIff, false⟩ -- This became a problem after https://github.com/leanprover-community/aesop/commit/29cf094e84ae9852f0011b47b6ddc684ffe4be5f
 ]
 
 /-- These options are passed as `leanOptions` to building mathlib, as well as the
@@ -80,6 +85,16 @@ lean_lib docs where
 ## Executables provided by Mathlib
 -/
 
+/--
+`lake exe autolabel 150100` adds a topic label to PR `150100` if there is a unique choice.
+This requires GitHub CLI `gh` to be installed!
+
+Calling `lake exe autolabel` without a PR number will print the result without applying
+any labels online.
+-/
+lean_exe autolabel where
+  srcDir := "scripts"
+
 /-- `lake exe cache get` retrieves precompiled `.olean` files from a central server. -/
 lean_exe cache where
   root := `Cache.Main
diff --git a/lean-toolchain b/lean-toolchain
index e7a4f40b892b4..eff86fd63de9e 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.11.0-rc2
+leanprover/lean4:v4.13.0-rc3
diff --git a/scripts/autolabel.lean b/scripts/autolabel.lean
new file mode 100644
index 0000000000000..922e98d1fee24
--- /dev/null
+++ b/scripts/autolabel.lean
@@ -0,0 +1,310 @@
+/-
+Copyright (c) 2024 Damiano Testa. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jon Eugster, Damiano Testa
+-/
+import Lean.Elab.Command
+
+/-!
+# Automatic labelling of PRs
+
+This file contains the script to automatically assign a GitHub label to a PR.
+
+## Label definition
+
+The mapping from GitHub labels to Mathlib folders is done in this file and
+needs to be updated here if necessary:
+
+* `AutoLabel.mathlibLabels` contains an assignment of GitHub labels to folders inside
+  the mathlib repository. If no folder is specified, a label like `t-set-theory` will be
+  interpreted as matching the folder `"Mathlib" / "SetTheory"`.
+* `AutoLabel.mathlibUnlabelled` contains subfolders of `Mathlib/` which are deliberately
+  left without topic label.
+
+## lake exe autolabel
+
+`lake exe autolabel` uses `git diff --name-only origin/master...HEAD` to determine which
+files have been modifed and then finds all labels which should be added based on these changes.
+These are printed for testing purposes.
+
+`lake exe autolabel [NUMBER]` will further try to add the applicable labels
+to the PR specified. This requires the **GitHub CLI** `gh` to be installed!
+Example: `lake exe autolabel 10402` for PR #10402.
+
+For the time being, the script only adds a label if it finds a **single unique label**
+which would apply. If multiple labels are found, nothing happens.
+
+## Workflow
+
+There is a mathlib workflow `.github/workflows/add_label_from_diff.yaml` which executes
+this script automatically.
+
+Currently it is set to run only one time when a PR is created.
+
+## Tests
+
+Additionally, the script does a few consistency checks:
+
+- it ensures all paths in specified in `AutoLabel.mathlibLabels` exist
+- It makes sure all subfolders of `Mathlib/` belong to at least one label.
+  There is `AutoLabel.mathlibUnlabelled` to add exceptions for this test.
+
+-/
+
+open Lean System
+
+namespace AutoLabel
+
+/--
+A `Label` consists of the
+* The `label` field is the actual GitHub label name.
+* The `dirs` field is the array of all "root paths" such that a modification in a file contained
+  in one of these paths should be labelled with `label`.
+* The `exclusions` field is the array of all "root paths" that are excluded, among the
+  ones that start with the ones in `dirs`.
+  Any modifications to a file in an excluded path is ignored for the purposes of labelling.
+-/
+structure Label where
+  /-- The label name as it appears on GitHub -/
+  label : String
+  /-- Array of paths which fall under this label. e.g. `"Mathlib" / "Algebra"`.
+
+  For a label of the form `t-set-theory` this defaults to `#["Mathlib" / "SetTheory"]`. -/
+  dirs : Array FilePath := if label.startsWith "t-" then
+      #["Mathlib" / ("".intercalate (label.splitOn "-" |>.drop 1 |>.map .capitalize))]
+    else #[]
+  /-- Array of paths which should be excluded.
+  Any modifications to a file in an excluded path are ignored for the purposes of labelling. -/
+  exclusions : Array FilePath := #[]
+  deriving BEq, Hashable
+
+/--
+Mathlib labels and their corresponding folders. Add new labels and folders here!
+-/
+def mathlibLabels : Array Label := #[
+  { label := "t-algebra",
+    dirs := #[
+      "Mathlib" / "Algebra",
+      "Mathlib" / "FieldTheory",
+      "Mathlib" / "RingTheory",
+      "Mathlib" / "GroupTheory",
+      "Mathlib" / "RepresentationTheory",
+      "Mathlib" / "LinearAlgebra"] },
+  { label := "t-algebraic-geometry",
+    dirs := #[
+      "Mathlib" / "AlgebraicGeometry",
+      "Mathlib" / "Geometry" / "RingedSpace"] },
+  { label := "t-analysis" },
+  { label := "t-category-theory" },
+  { label := "t-combinatorics" },
+  { label := "t-computability" },
+  { label := "t-condensed" },
+  { label := "t-data" },
+  { label := "t-differential-geometry",
+    dirs := #["Mathlib" / "Geometry" / "Manifold"] },
+  { label := "t-dynamics" },
+  { label := "t-euclidean-geometry",
+    dirs := #["Mathlib" / "Geometry" / "Euclidean"] },
+  { label := "t-linter",
+    dirs := #["Mathlib" / "Tactic" / "Linter"] },
+  { label := "t-logic",
+    dirs := #[
+      "Mathlib" / "Logic",
+      "Mathlib" / "ModelTheory"] },
+  { label := "t-measure-probability",
+    dirs := #[
+      "Mathlib" / "MeasureTheory",
+      "Mathlib" / "Probability",
+      "Mathlib" / "InformationTheory"] },
+  { label := "t-meta",
+    dirs := #[
+      "Mathlib" / "Control",
+      "Mathlib" / "Lean",
+      "Mathlib" / "Mathport",
+      "Mathlib" / "Tactic",
+      "Mathlib" / "Util"],
+    exclusions := #["Mathlib" / "Tactic" / "Linter"] },
+  { label := "t-number-theory" },
+  { label := "t-order" },
+  { label := "t-set-theory" },
+  { label := "t-topology",
+    dirs := #[
+      "Mathlib" / "Topology",
+      "Mathlib" / "AlgebraicTopology"] },
+  { label := "CI",
+    dirs := #[".github"] },
+  { label := "IMO",
+    dirs := #["Archive" / "Imo"] } ]
+
+/-- Exceptions inside `Mathlib/` which are not covered by any label. -/
+def mathlibUnlabelled : Array FilePath := #[
+    "Mathlib" / "Deprecated",
+    "Mathlib" / "Init",
+    "Mathlib" / "Testing",
+    "Mathlib" / "Std" ]
+
+/-- Checks if the folder `path` lies inside the folder `dir`. -/
+def _root_.System.FilePath.isPrefixOf (dir path : FilePath) : Bool :=
+  -- use `dir / ""` to prevent partial matching of folder names
+  (dir / "").normalize.toString.isPrefixOf (path / "").normalize.toString
+
+/--
+Return all names of labels in `mathlibLabels` which match
+at least one of the `files`.
+
+* `files`: array of relative paths starting from the mathlib root directory.
+-/
+def getMatchingLabels (files : Array FilePath) : Array String :=
+  let applicable := mathlibLabels.filter fun label ↦
+    -- first exclude all files the label excludes,
+    -- then see if any file remains included by the label
+    let notExcludedFiles := files.filter fun file ↦
+      label.exclusions.all (!·.isPrefixOf file)
+    label.dirs.any (fun dir ↦ notExcludedFiles.any (dir.isPrefixOf ·))
+  -- return sorted list of label names
+  applicable.map (·.label) |>.qsort (· < ·)
+
+/-!
+Testing the functionality of the declarations defined in this script
+-/
+section Tests
+
+-- Test `FilePath.isPrefixOf`
+#guard ("Mathlib" / "Algebra" : FilePath).isPrefixOf ("Mathlib" / "Algebra" / "Basic.lean")
+
+-- Test `FilePath.isPrefixOf` does not trigger on partial prefixes
+#guard ! ("Mathlib" / "Algebra" : FilePath).isPrefixOf ("Mathlib" / "AlgebraicGeometry")
+
+#guard getMatchingLabels #[] == #[]
+-- Test default value for `label.dirs` works
+#guard getMatchingLabels #["Mathlib" / "SetTheory" / "ZFC"] == #["t-set-theory"]
+-- Test exclusion
+#guard getMatchingLabels #["Mathlib" / "Tactic"/ "Abel.lean"] == #["t-meta"]
+#guard getMatchingLabels #["Mathlib" / "Tactic"/ "Linter" / "Lint.lean"] == #["t-linter"]
+#guard getMatchingLabels #[
+  "Mathlib" / "Tactic"/ "Linter" / "Lint.lean",
+  "Mathlib" / "Tactic" / "Abel.lean" ] == #["t-linter", "t-meta"]
+
+/-- Testing function to ensure the labels defined in `mathlibLabels` cover all
+subfolders of `Mathlib/`. -/
+partial def findUncoveredPaths (path : FilePath) (exceptions : Array FilePath := #[]) :
+    IO <| Array FilePath := do
+  let mut notMatched : Array FilePath := #[]
+  -- all directories inside `path`
+  let subDirs ← (← path.readDir).map (·.path) |>.filterM (do FilePath.isDir ·)
+  for dir in subDirs do
+    -- if the sub directory is not matched by a label,
+    -- we go recursively into it
+    if (getMatchingLabels #[dir]).size == 0 then
+      notMatched := notMatched ++ (← findUncoveredPaths dir exceptions)
+  -- a directory should be flagged if none of its sub-directories is matched by a label
+  -- note: we assume here the base directory, i.e. "Mathlib" is never matched by a label,
+  -- therefore we skip this test.
+  if notMatched.size == subDirs.size then
+    if exceptions.contains path then
+      return #[]
+    else
+      return #[path]
+  else
+    return notMatched
+
+end Tests
+
+/--
+Create a message which GitHub CI parses as annotation and displays at the specified file.
+
+Note: `file` is duplicated below so that it is also visible in the plain text output.
+
+* `type`: "error" or "warning"
+* `file`: file where the annotation should be displayed
+* `title`: title of the annotation
+* `message`: annotation message
+-/
+def githubAnnotation (type file title message : String) : String :=
+  s!"::{type} file={file},title={title}::{file}: {message}"
+
+end AutoLabel
+
+open IO AutoLabel in
+
+/-- `args` is expected to have length 0 or 1, where the first argument is the PR number.
+
+If a PR number is provided, the script requires GitHub CLI `gh` to be installed in order
+to add the label to the PR.
+
+## Exit codes:
+
+- `0`: success
+- `1`: invalid arguments provided
+- `2`: invalid labels defined
+- `3`: ~labels do not cover all of `Mathlib/`~ (unused; only emitting warning)
+-/
+unsafe def main (args : List String): IO UInt32 := do
+  if args.length > 1 then
+    println s!"::error:: autolabel: invalid number of arguments ({args.length}), \
+    expected at most 1. Please run without arguments or provide the target PR's \
+    number as a single argument!"
+    return 1
+  let prNumber? := args[0]?
+
+  -- test: validate that all paths in `mathlibLabels` actually exist
+  let mut valid := true
+  for label in mathlibLabels do
+    for dir in label.dirs do
+      unless ← FilePath.pathExists dir do
+        -- print github annotation error
+        println <| AutoLabel.githubAnnotation "error" "scripts/autolabel.lean"
+          s!"Misformatted `{ ``AutoLabel.mathlibLabels }`"
+          s!"directory '{dir}' does not exist but is included by label '{label.label}'. \
+          Please update `{ ``AutoLabel.mathlibLabels }`!"
+        valid := false
+    for dir in label.exclusions do
+      unless ← FilePath.pathExists dir do
+        -- print github annotation error
+        println <| AutoLabel.githubAnnotation "error" "scripts/autolabel.lean"
+          s!"Misformatted `{ ``AutoLabel.mathlibLabels }`"
+          s!"directory '{dir}' does not exist but is excluded by label '{label.label}'. \
+          Please update `{ ``AutoLabel.mathlibLabels }`!"
+        valid := false
+  unless valid do
+    return 2
+
+  -- test: validate that the labels cover all of the `Mathlib/` folder
+  let notMatchedPaths ← findUncoveredPaths "Mathlib" (exceptions := mathlibUnlabelled)
+  if notMatchedPaths.size > 0 then
+    -- print github annotation warning
+    -- note: only emitting a warning because the workflow is only triggered on the first commit
+    -- of a PR and could therefore lead to unexpected behaviour if a folder was created later.
+    println <| AutoLabel.githubAnnotation "warning" "scripts/autolabel.lean"
+      s!"Incomplete `{ ``AutoLabel.mathlibLabels }`"
+      s!"the following paths inside `Mathlib/` are not covered \
+      by any label: {notMatchedPaths} Please modify `AutoLabel.mathlibLabels` accordingly!"
+    -- return 3
+
+  -- get the modified files
+  let gitDiff ← IO.Process.run {
+    cmd := "git",
+    args := #["diff", "--name-only", "origin/master...HEAD"] }
+  let modifiedFiles : Array FilePath := (gitDiff.splitOn "\n").toArray.map (⟨·⟩)
+
+  -- find labels covering the modified files
+  let labels := getMatchingLabels modifiedFiles
+
+  println s!"::notice::Applicable labels: {labels}"
+
+  match labels with
+  | #[] =>
+    println s!"::warning::no label to add"
+  | #[label] =>
+    match prNumber? with
+    | some n =>
+      let _ ← IO.Process.run {
+        cmd := "gh",
+        args := #["pr", "edit", n, "--add-label", label] }
+      println s!"::notice::added label: {label}"
+    | none =>
+      println s!"::warning::no PR-number provided, not adding labels. \
+      (call `lake exe autolabel 150602` to add the labels to PR `150602`)"
+  | _ =>
+    println s!"::notice::not adding multiple labels: {labels}"
+  return 0
diff --git a/scripts/check-yaml.lean b/scripts/check-yaml.lean
index 23730a3fabb2e..44aca0b63758a 100644
--- a/scripts/check-yaml.lean
+++ b/scripts/check-yaml.lean
@@ -22,11 +22,11 @@ def readJsonFile (α) [FromJson α] (path : System.FilePath) : IO α := do
   let _ : MonadExceptOf String IO := ⟨throw ∘ IO.userError, fun x _ => x⟩
   liftExcept <| fromJson? <|← liftExcept <| Json.parse <|← IO.FS.readFile path
 
-def databases : List (String × String) := [
-  ("undergrad.json", "Entries in `docs/undergrad.yaml` refer to declarations that don't exist. Please correct the following:"),
-  ("overview.json", "Entries in `docs/overview.yaml` refer to declarations that don't exist. Please correct the following:"),
-  ("100.json", "Entries in `docs/100.yaml` refer to declarations that don't exist. Please correct the following:")
-]
+def databases : List (String × String) :=
+  ["undergrad", "overview", "100"].map fun dir =>
+    (dir ++ ".json",
+      s!"Entries in `docs/{dir}.yaml` refer to declarations that don't exist. \
+        Please correct the following:")
 
 def processDb (decls : ConstMap) : String × String → IO Bool
 | (file, msg) => do
diff --git a/scripts/create-adaptation-pr.sh b/scripts/create-adaptation-pr.sh
index 08db7705144c9..4fb326cfac9d6 100755
--- a/scripts/create-adaptation-pr.sh
+++ b/scripts/create-adaptation-pr.sh
@@ -14,15 +14,57 @@ set -e # abort whenever a command in the script fails
 # So please do not delete the following line, or the final two lines of this script.
 {
 
-if [ $# -ne 2 ]; then
+# Default values
+AUTO="no"
+
+# Function to display usage
+usage() {
   echo "Usage: $0 <BUMPVERSION> <NIGHTLYDATE>"
+  echo "       or"
+  echo "       $0 --bumpversion=<BUMPVERSION> --nightlydate=<NIGHTLYDATE> --nightlysha=<SHA> [--auto=<yes|no>]"
   echo "BUMPVERSION: The upcoming release that we are targeting, e.g., 'v4.10.0'"
   echo "NIGHTLYDATE: The date of the nightly toolchain currently used on 'nightly-testing'"
+  echo "NIGHTLYSHA: The SHA of the nightly toolchain that we want to adapt to"
+  echo "AUTO: Optional flag to specify automatic mode, default is 'no'"
   exit 1
+}
+
+# Parse arguments
+if [ $# -eq 2 ] && [[ $1 != --* ]] && [[ $2 != --* ]]; then
+  BUMPVERSION=$1
+  NIGHTLYDATE=$2
+elif [ $# -ge 2 ]; then
+  for arg in "$@"; do
+    case $arg in
+      --bumpversion=*)
+        BUMPVERSION="${arg#*=}"
+        shift
+        ;;
+      --nightlydate=*)
+        NIGHTLYDATE="${arg#*=}"
+        shift
+        ;;
+      --nightlysha=*)
+        NIGHTLYSHA="${arg#*=}"
+        shift
+        ;;
+      --auto=*)
+        AUTO="${arg#*=}"
+        shift
+        ;;
+      *)
+        usage
+        ;;
+    esac
+  done
+else
+  usage
 fi
 
-BUMPVERSION=$1 # "v4.10.0"
-NIGHTLYDATE=$2 # "2024-06-25"
+# Validate required arguments
+if [ -z "$BUMPVERSION" ] || [ -z "$NIGHTLYDATE" ]; then
+  usage
+fi
 
 # Check if 'gh' command is available
 if ! command -v gh &> /dev/null; then
@@ -38,8 +80,13 @@ if [ "$status" != "completed" ]; then
     gh run list --branch nightly-testing
     exit 1
   else
-    echo "The latest commit on 'nightly-testing' is still running CI."
-    read -p "Press enter to continue, or ctrl-C if you'd prefer to wait for CI."
+    if [ "$AUTO" = "yes" ]; then
+      echo "Auto mode enabled. Bailing out because the latest commit on 'nightly-testing' is still running CI."
+      exit 1
+    else
+      echo "The latest commit on 'nightly-testing' is still running CI."
+      read -p "Press enter to continue, or ctrl-C if you'd prefer to wait for CI."
+    fi
   fi
 fi
 
@@ -79,6 +126,13 @@ if git diff --name-only --diff-filter=U | grep -q .; then
   fi
 fi
 
+if git diff --name-only --diff-filter=U | grep -q . || ! git diff-index --quiet HEAD --; then
+  if [ "$AUTO" = "yes" ]; then
+    echo "Auto mode enabled. Bailing out due to unresolved conflicts or uncommitted changes."
+    exit 1
+  fi
+fi
+
 # Loop until all conflicts are resolved and committed
 while git diff --name-only --diff-filter=U | grep -q . || ! git diff-index --quiet HEAD --; do
   echo
@@ -99,23 +153,31 @@ echo
 echo "### [auto] create a new branch 'bump/nightly-$NIGHTLYDATE' and merge the latest changes from 'origin/nightly-testing'"
 
 git checkout -b "bump/nightly-$NIGHTLYDATE"
-git merge origin/nightly-testing || true # ignore error if there are conflicts
+git merge $NIGHTLYSHA || true # ignore error if there are conflicts
 
 # Check if there are merge conflicts
 if git diff --name-only --diff-filter=U | grep -q .; then
   echo
   echo "### [auto] Conflict resolution"
-  echo "### Automatically choosing 'lean-toolchain' and 'lake-manifest.json' from the newer branch"
-  echo "### In this case, the newer branch is 'origin/nightly-testing'"
-  git checkout origin/nightly-testing -- lean-toolchain lake-manifest.json
+  echo "### Automatically choosing 'lean-toolchain' and 'lake-manifest.json' from 'nightly-testing'"
+  git checkout $NIGHTLYSHA -- lean-toolchain lake-manifest.json
   git add lean-toolchain lake-manifest.json
 fi
 
+if git diff --name-only --diff-filter=U | grep -q .; then
+  if [ "$AUTO" = "yes" ]; then
+    echo "Auto mode enabled. Bailing out due to unresolved conflicts or uncommitted changes."
+    exit 1
+  fi
+fi
+
 # Check if there are more merge conflicts
 if git diff --name-only --diff-filter=U | grep -q .; then
   echo
   echo "### [user] Conflict resolution"
   echo "We are merging the latest changes from 'origin/nightly-testing' into 'bump/nightly-$NIGHTLYDATE'"
+  echo "Specifically, we are merging the following version of 'origin/nightly-testing':"
+  echo "$NIGHTLYSHA"
   echo "There seem to be conflicts: please resolve them"
   echo ""
   echo "  1) Open `pwd` in a new terminal and run 'git status'"
@@ -143,8 +205,13 @@ if git diff --name-only bump/$BUMPVERSION bump/nightly-$NIGHTLYDATE | grep -q .;
   echo "Here is a suggested 'gh' command to do this:"
   gh_command="gh pr create -t \"$pr_title\" -b '' -B bump/$BUMPVERSION"
   echo "> $gh_command"
-  echo "Shall I run this command for you? (y/n)"
-  read answer
+  if [ "$AUTO" = "yes" ]; then
+    echo "Auto mode enabled. Running the command..."
+    answer="y"
+  else
+    echo "Shall I run this command for you? (y/n)"
+    read answer
+  fi
   if [ "$answer" != "${answer#[Yy]}" ]; then
   	gh_output=$(eval $gh_command)
   	# Extract the PR number from the output
@@ -152,7 +219,7 @@ if git diff --name-only bump/$BUMPVERSION bump/nightly-$NIGHTLYDATE | grep -q .;
   fi
   
   echo
-  echo "### [user] post a link to the PR on Zulip"
+  echo "### [auto/user] post a link to the PR on Zulip"
   
   zulip_title="#$pr_number adaptations for nightly-$NIGHTLYDATE"
   zulip_body="> $pr_title #$pr_number"
@@ -161,7 +228,33 @@ if git diff --name-only bump/$BUMPVERSION bump/nightly-$NIGHTLYDATE | grep -q .;
   echo "Here is a suggested message:"
   echo "Title: $zulip_title"
   echo " Body: $zulip_body"
-  read -p "Press enter to continue"
+
+  if command -v zulip-send >/dev/null 2>&1; then
+    zulip_command="zulip-send --stream nightly-testing --subject \"$zulip_title\" --message \"$zulip_body\""
+    echo "Here is a suggested 'zulip-send' command to do this:"
+    echo "> $zulip_command"
+  
+    if [ "$AUTO" = "yes" ]; then
+      echo "Auto mode enabled. Running the command..."
+      answer="y"
+    else
+      echo "Shall I run this command for you? (y/n)"
+      read answer
+    fi
+  
+    if [ "$answer" != "${answer#[Yy]}" ]; then
+      eval $zulip_command
+    fi
+  else
+    echo "Zulip CLI is not installed. Please install it to send messages automatically."
+    if [ "$AUTO" = "yes" ]; then
+      exit 1
+    fi
+  fi
+  
+  if [ "$AUTO" != "yes" ]; then
+    read -p "Press enter to continue"
+  fi
 
 # else, let the user know that no PR is needed
 else
@@ -195,6 +288,15 @@ if git diff --name-only --diff-filter=U | grep -q .; then
   fi
 fi
 
+if git diff --name-only --diff-filter=U | grep -q . || ! git diff-index --quiet HEAD --; then
+  if [ "$AUTO" = "yes" ]; then
+    echo "Auto mode enabled. Bailing out due to unresolved conflicts or uncommitted changes."
+    echo "PR has been created, and message posted to Zulip."
+    echo "Error occured while merging the new branch into 'nightly-testing'."
+    exit 2
+  fi
+fi
+
 # Loop until all conflicts are resolved and committed
 while git diff --name-only --diff-filter=U | grep -q . || ! git diff-index --quiet HEAD --; do
   echo
diff --git a/scripts/fix-by-linebreaks.sh b/scripts/fix-by-linebreaks.sh
deleted file mode 100755
index f0fdd8aa12f08..0000000000000
--- a/scripts/fix-by-linebreaks.sh
+++ /dev/null
@@ -1,5 +0,0 @@
-#!/usr/bin/env bash
-# Modify all lean files in mathlib to put the "by" in lines that only contain "  by" at the end of the previous line,
-# when the previous line with " by" appended is not longer than 100 characters.
-
-grep -lr "^  by\$" Mathlib | xargs -n 1 awk -i inplace '{do {{if (match($0, "^  by$") && length(p) < 98 && (!(match(p, "^[ \t]*--.*$")))) {p=p " by";} else {if (NR!=1) {print p}; p=$0}}} while (getline == 1) if (getline==0) print p}'
diff --git a/scripts/get_tlabel.sh b/scripts/get_tlabel.sh
new file mode 100755
index 0000000000000..6e03a8f27d23f
--- /dev/null
+++ b/scripts/get_tlabel.sh
@@ -0,0 +1,32 @@
+#!/usr/bin/env bash
+
+ : <<'BASH_MODULE_DOCS'
+
+This script is used by the maintainer merge actions to extract
+* either `t-xxx` if `t-xxx` is the unique `t-`label of the PR;
+* or `generic` otherwise
+and stores it in `tlabels`.
+
+In turn, the string `tlabels` extracted above is converted into the
+topic name `maintainer merge: tlabels` for the post to the
+`maintainer merge` stream.
+
+BASH_MODULE_DOCS
+
+PR="${1}"
+
+>&2 printf $'Using PR: \'%s\'\n' "${PR}"
+
+tlabels="$(gh api --jq '.labels.[].name' "${PR}" | grep -- '^\(t-\|CI\|IMO\)' || printf 'generic\nlabel')"
+# print to error, since the stdout is captured into `GITHUB_OUTPUT
+>&2 printf 't-labels:\n---\n%s\n---\n' "${tlabels}"
+# if there is exactly 1 `t-xxx`, `CI`, or `IMO` label, use `maintainer merge: t-xxx`
+# if there isn't exactly 1 `t-xxx`, `CI`, or `IMO` label, use `maintainer merge`
+if [[ "$(wc -l <<<"${tlabels}")" -ne 1 ]]; then
+  topicName="maintainer merge"
+else
+  topicName="maintainer merge: ${tlabels}"
+fi
+
+>&2 printf $'Post to topic: \'%s\'\n' "${topicName}"
+echo "topic=${topicName}"
diff --git a/scripts/import-graph-report.py b/scripts/import-graph-report.py
index cf072b916c5e8..86775bf4c2fc1 100755
--- a/scripts/import-graph-report.py
+++ b/scripts/import-graph-report.py
@@ -10,6 +10,8 @@
 import json
 import sys
 
+high_import_threshold = 2
+
 def compare_counts(base_file, head_file, changed_files_txt):
     # Load the counts
     with open(head_file, 'r') as f:
@@ -29,6 +31,7 @@ def compare_counts(base_file, head_file, changed_files_txt):
 
     # Compare the counts
     changes = []
+    high_pct = []
     for file in changed_files:
         base_count = base_counts.get(file, 0)
         head_count = head_counts.get(file, 0)
@@ -36,6 +39,8 @@ def compare_counts(base_file, head_file, changed_files_txt):
             continue
         diff = head_count - base_count
         percent = (diff / base_count) * 100
+        if high_import_threshold < percent:
+            high_pct.append(f'| +{percent:.2f}% | `{file}` |')
         if diff < 0:  # Dependencies went down
             changes.append((file, base_count, head_count, diff, percent))
         elif diff > new_files:  # Dependencies went up by more than the number of new files
@@ -59,11 +64,19 @@ def compare_counts(base_file, head_file, changed_files_txt):
         message += '\n'.join(messages)
     else:
         message += 'No significant changes to the import graph'
-    return message
+
+    high_pct_report = ''
+    if high_pct:
+        high_pct_report += f'Import changes exceeding {high_import_threshold}%\n\n'
+        high_pct_report += '| %      | File |\n'
+        high_pct_report += '| -      | -    |\n'
+        high_pct_report += '\n'.join(high_pct)
+    return (message, high_pct_report)
 
 if __name__ == '__main__':
     base_file = sys.argv[1]
     head_file = sys.argv[2]
     changed_files_txt = sys.argv[3]
-    message = compare_counts(base_file, head_file, changed_files_txt)
+    (message, high_pct) = compare_counts(base_file, head_file, changed_files_txt)
     print(message)
+    print(high_pct)
diff --git a/scripts/init_creation.sh b/scripts/init_creation.sh
index f7469af5e39c6..7abca30241c43 100644
--- a/scripts/init_creation.sh
+++ b/scripts/init_creation.sh
@@ -2,11 +2,16 @@
 
  : <<'BASH_MODULE_DOC'
 
-These are the commands to generate a "root" `Mathlib/Init.lean` file, imported by all the
-`Mathlib` files that do not import any `Mathlib` file.
+These are the commands to add an import of `Mathlib/Init.lean` to all `Mathlib` files
+that do not import any `Mathlib` file.
 
 BASH_MODULE_DOC
 
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
 # `mathlibNonImportingFiles` generates the list of `Mathlib` files that do not have Mathlib imports.
 # The output of `lake exe graph` are many lines like
 # `  "Mathlib..." -> "Mathlib...";`
@@ -30,11 +35,3 @@ printf 'Adding `import Mathlib.Init` to all file that import no Mathlib file.\n'
 # The `sed` command appends the line `import Mathlib.Init` after the first
 # `-/[linebreaks]*` of each file printed by `mathlibNonImportingFiles`.
 sed -i -z 's=-/\n*=&import Mathlib.Init\n=' $(mathlibNonImportingFiles)
-
-printf 'Creating `Mathlib/Init.lean`.\n'
-
-# Creates the `Mathlib/Init.lean` files.
-echo '-- This is the root file in Mathlib: it is imported by virtually *all* Mathlib files' > Mathlib/Init.lean
-
-printf $'Don\'t forget to add `Mathlib.Init` to the `ignoreImport` field of `scripts/noshake.json`
-This ensures that `import Mathlib.Init` does not trigger a `shake` exception.\n'
diff --git a/scripts/install_debian.sh b/scripts/install_debian.sh
index d2085be2d8cfb..9d5caf69db239 100755
--- a/scripts/install_debian.sh
+++ b/scripts/install_debian.sh
@@ -1,6 +1,11 @@
 #!/usr/bin/env bash
 
-set -exo pipefail
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
+set -x
 
 sudo apt install -y git curl
 
diff --git a/scripts/install_macos.sh b/scripts/install_macos.sh
index ef46de9dc19ff..f45895008d4a0 100755
--- a/scripts/install_macos.sh
+++ b/scripts/install_macos.sh
@@ -1,6 +1,11 @@
 #!/usr/bin/env bash
 
-set -exo pipefail
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
+set -x
 
 # Install elan using the official script
 curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh
diff --git a/scripts/lean-pr-testing-comments.sh b/scripts/lean-pr-testing-comments.sh
index 189a6ea6ac536..ba5c9586b2e05 100755
--- a/scripts/lean-pr-testing-comments.sh
+++ b/scripts/lean-pr-testing-comments.sh
@@ -1,8 +1,13 @@
-## Create comments and labels on a Lean 4 PR after CI has finished on a `lean-pr-testing-NNNN` branch.
+## Create comments and labels on a Lean 4 or Batteries PR after CI has finished on a `*-pr-testing-NNNN` branch.
 ##
 ## See https://leanprover-community.github.io/contribute/tags_and_branches.html
 set -e
 
+# Ensure first argument is either 'lean' or 'batteries'.
+if [ -z "$1" ]; then
+  echo "The first argument must be either 'lean' or 'batteries'"
+  exit 1
+fi
 
 # TODO: The whole script ought to be rewritten in javascript, to avoid having to use curl for API calls.
 #
@@ -19,14 +24,29 @@ set -e
 #   LINT_OUTCOME: ${{ steps.lint.outcome }}
 #   TEST_OUTCOME: ${{ steps.test.outcome }}
 
+# Adjust the branch pattern and URLs based on the repository.
+if [ "$1" == "lean" ]; then
+  branch_prefix="lean-pr-testing"
+  repo_url="https://api.github.com/repos/leanprover/lean4"
+  base_branch="nightly-testing" # This really should be the relevant `nightly-testing-YYYY-MM-DD` tag.
+elif [ "$1" == "batteries" ]; then
+  branch_prefix="batteries-pr-testing"
+  repo_url="https://api.github.com/repos/leanprover-community/batteries"
+  base_branch="master"
+else
+  echo "Unknown repository: $1. Must be either 'lean' or 'batteries'."
+  exit 1
+fi
+
 # Extract branch name and check if it matches the pattern.
 branch_name=$(echo "$GITHUB_CONTEXT" | jq -r .ref | cut -d'/' -f3)
-if [[ "$branch_name" =~ ^lean-pr-testing-([0-9]+)$ ]]; then
+if [[ "$branch_name" =~ ^$branch_prefix-([0-9]+)$ ]]; then
   pr_number="${BASH_REMATCH[1]}"
   current_time=$(date "+%Y-%m-%d %H:%M:%S")
 
-  echo "This is a 'lean-pr-testing-$pr_number' branch, so we need to adjust labels and write a comment."
+  echo "This is a '$branch_prefix-$pr_number' branch, so we need to adjust labels and write a comment."
 
+  # Perform actions based on outcomes (same logic as before)
   if [ "$TEST_OUTCOME" == "success" ]; then
     echo "Removing label awaiting-mathlib"
     curl -L -s \
@@ -34,21 +54,21 @@ if [[ "$branch_name" =~ ^lean-pr-testing-([0-9]+)$ ]]; then
       -H "Accept: application/vnd.github+json" \
       -H "Authorization: Bearer $TOKEN" \
       -H "X-GitHub-Api-Version: 2022-11-28" \
-      https://api.github.com/repos/leanprover/lean4/issues/$pr_number/labels/awaiting-mathlib
+      $repo_url/issues/$pr_number/labels/awaiting-mathlib
     echo "Removing label breaks-mathlib"
     curl -L -s \
       -X DELETE \
       -H "Accept: application/vnd.github+json" \
       -H "Authorization: Bearer $TOKEN" \
       -H "X-GitHub-Api-Version: 2022-11-28" \
-      https://api.github.com/repos/leanprover/lean4/issues/$pr_number/labels/breaks-mathlib
+      $repo_url/issues/$pr_number/labels/breaks-mathlib
     echo "Adding label builds-mathlib"
     curl -L -s \
       -X POST \
       -H "Accept: application/vnd.github+json" \
       -H "Authorization: Bearer $TOKEN" \
       -H "X-GitHub-Api-Version: 2022-11-28" \
-      https://api.github.com/repos/leanprover/lean4/issues/$pr_number/labels \
+      $repo_url/issues/$pr_number/labels \
       -d '{"labels":["builds-mathlib"]}'
   elif [ "$LINT_OUTCOME" == "failure" ] || [ "$TEST_OUTCOME" == "failure" ] || [ "$COUNTEREXAMPLES_OUTCOME" == "failure" ] || [ "$ARCHIVE_OUTCOME" == "failure" ] || [ "$NOISY_OUTCOME" == "failure" ] || [ "$BUILD_OUTCOME" == "failure" ]; then
     echo "Removing label builds-mathlib"
@@ -57,32 +77,32 @@ if [[ "$branch_name" =~ ^lean-pr-testing-([0-9]+)$ ]]; then
       -H "Accept: application/vnd.github+json" \
       -H "Authorization: Bearer $TOKEN" \
       -H "X-GitHub-Api-Version: 2022-11-28" \
-      https://api.github.com/repos/leanprover/lean4/issues/$pr_number/labels/builds-mathlib
+      $repo_url/issues/$pr_number/labels/builds-mathlib
     echo "Adding label breaks-mathlib"
     curl -L -s \
       -X POST \
       -H "Accept: application/vnd.github+json" \
       -H "Authorization: Bearer $TOKEN" \
       -H "X-GitHub-Api-Version: 2022-11-28" \
-      https://api.github.com/repos/leanprover/lean4/issues/$pr_number/labels \
+      $repo_url/issues/$pr_number/labels \
       -d '{"labels":["breaks-mathlib"]}'
   fi
 
   # Use GitHub API to check if a comment already exists
   existing_comment=$(curl -L -s -H "Authorization: token $TOKEN" \
                           -H "Accept: application/vnd.github.v3+json" \
-                          "https://api.github.com/repos/leanprover/lean4/issues/$pr_number/comments" \
-                          | jq 'first(.[] | select(.body | test("^- . Mathlib") or startswith("Mathlib CI status")) | select(.user.login == "leanprover-community-mathlib4-bot"))')
+                          "$repo_url/issues/$pr_number/comments" \
+                          | jq 'first(.[] | select(.body | test("^- . Mathlib") or startswith("Mathlib CI status")) | select(.user.login == "leanprover-community-bot"))')
   existing_comment_id=$(echo "$existing_comment" | jq -r .id)
   existing_comment_body=$(echo "$existing_comment" | jq -r .body)
 
-  branch="[lean-pr-testing-$pr_number](https://github.com/leanprover-community/mathlib4/compare/nightly-testing...lean-pr-testing-$pr_number)"
+  branch="[$branch_prefix-$pr_number](https://github.com/leanprover-community/mathlib4/compare/$base_branch...$branch_prefix-$pr_number)"
   # Depending on the success/failure, set the appropriate message
   if [ "$LINT_OUTCOME" == "cancelled" ] || [ "$TEST_OUTCOME" == "cancelled" ] || [ "$COUNTEREXAMPLES_OUTCOME" == "cancelled" ] || [ "$ARCHIVE_OUTCOME" == "cancelled" ] || [ "$NOISY_OUTCOME" == "cancelled" ] || [ "$BUILD_OUTCOME" == "cancelled" ]; then
     message="- 🟡 Mathlib branch $branch build against this PR was cancelled. ($current_time) [View Log]($WORKFLOW_URL)"
   elif [ "$TEST_OUTCOME" == "success" ]; then
     message="- ✅ Mathlib branch $branch has successfully built against this PR. ($current_time) [View Log]($WORKFLOW_URL)"
-  elif [ "$BUILD_OUTCOME" == "failure" ] ; then
+  elif [ "$BUILD_OUTCOME" == "failure" ]; then
     message="- 💥 Mathlib branch $branch build failed against this PR. ($current_time) [View Log]($WORKFLOW_URL)"
   elif [ "$LINT_OUTCOME" == "failure" ]; then
     message="- ❌ Mathlib branch $branch built against this PR, but linting failed. ($current_time) [View Log]($WORKFLOW_URL)"
@@ -103,23 +123,22 @@ if [[ "$branch_name" =~ ^lean-pr-testing-([0-9]+)$ ]]; then
   # Append new result to the existing comment or post a new comment
   if [ -z "$existing_comment_id" ]; then
     # Post new comment with a bullet point
-    # Keep message in sync with https://github.com/leanprover/lean4/blob/master/.github/workflows/pr-release.yml
     intro="Mathlib CI status ([docs](https://leanprover-community.github.io/contribute/tags_and_branches.html)):"
-    echo "Posting as new comment at leanprover/lean4/issues/$pr_number/comments"
+    echo "Posting as new comment at $repo_url/issues/$pr_number/comments"
     curl -L -s \
       -X POST \
       -H "Authorization: token $TOKEN" \
       -H "Accept: application/vnd.github.v3+json" \
       -d "$(jq --null-input --arg intro "$intro" --arg val "$message" '{"body": ($intro + "\n" + $val)}')" \
-      "https://api.github.com/repos/leanprover/lean4/issues/$pr_number/comments"
+      "$repo_url/issues/$pr_number/comments"
   else
     # Append new result to the existing comment
-    echo "Appending to existing comment at leanprover/lean4/issues/$pr_number/comments"
+    echo "Appending to existing comment at $repo_url/issues/$pr_number/comments"
     curl -L -s \
       -X PATCH \
       -H "Authorization: token $TOKEN" \
       -H "Accept: application/vnd.github.v3+json" \
       -d "$(jq --null-input --arg existing "$existing_comment_body" --arg message "$message" '{"body":($existing + "\n" + $message)}')" \
-      "https://api.github.com/repos/leanprover/lean4/issues/comments/$existing_comment_id"
+      "$repo_url/issues/comments/$existing_comment_id"
   fi
 fi
diff --git a/scripts/lint-bib.sh b/scripts/lint-bib.sh
index 630f64c625db2..500f8c19947b9 100755
--- a/scripts/lint-bib.sh
+++ b/scripts/lint-bib.sh
@@ -1,6 +1,11 @@
 #!/usr/bin/env bash
 
-set -exo pipefail
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
+set -x
 # https://leanprover-community.github.io/contribute/doc.html#citing-other-works
 cp docs/references.bib docs/references.bib.old
 bibtool --preserve.key.case=on --preserve.keys=on --pass.comments=on --print.use.tab=off -s \
diff --git a/scripts/lint-style.lean b/scripts/lint-style.lean
index ed4ac6ebf23b0..a7eb1c953caa1 100644
--- a/scripts/lint-style.lean
+++ b/scripts/lint-style.lean
@@ -18,20 +18,23 @@ open Cli Mathlib.Linter.TextBased
 
 /-- Implementation of the `lint-style` command line program. -/
 def lintStyleCli (args : Cli.Parsed) : IO UInt32 := do
-  let mode : OutputSetting := match (args.hasFlag "update", args.hasFlag "github") with
-    | (true, _) => OutputSetting.update
-    | (false, true) => OutputSetting.print ErrorFormat.github
-    | (false, false) => OutputSetting.print ErrorFormat.humanReadable
+  let style : ErrorFormat := match args.hasFlag "github" with
+    | true => ErrorFormat.github
+    | false => ErrorFormat.humanReadable
+  let fix := args.hasFlag "fix"
   -- Read all module names to lint.
   let mut allModules := #[]
   for s in ["Archive.lean", "Counterexamples.lean", "Mathlib.lean"] do
     allModules := allModules.append ((← IO.FS.lines s).map (·.stripPrefix "import "))
   -- note: since we manually add "Batteries" to "Mathlib.lean", we remove it here manually
   allModules := allModules.erase "Batteries"
-  let numberErrorFiles ← lintModules allModules mode (args.hasFlag "fix")
-  -- Make sure to return an exit code of at most 125, so this return value can be used further
-  -- in shell scripts.
-  return min numberErrorFiles 125
+  let numberErrorFiles ← lintModules allModules style fix
+  -- If run with the `--fix` argument, return a zero exit code.
+  -- Otherwise, make sure to return an exit code of at most 125,
+  -- so this return value can be used further in shell scripts.
+  if args.hasFlag "fix" then
+    return 0
+  else return min numberErrorFiles 125
 
 /-- Setting up command line options and help text for `lake exe lint-style`. -/
 -- so far, no help options or so: perhaps that is fine?
@@ -43,11 +46,6 @@ def lintStyle : Cmd := `[Cli|
   FLAGS:
     github;     "Print errors in a format suitable for github problem matchers\n\
                  otherwise, produce human-readable output"
-    update;     "Also update the style exceptions file.\
-      This adds entries for any new exceptions, removes any entries which are no longer necessary,\
-      and tries to not modify exception entries unless necessary.
-      To fully regenerate the list of style exceptions, delete `style-exceptions.txt`
-      and run this script again with this flag."
     fix;        "Automatically fix the style error, if possible"
 ]
 
diff --git a/scripts/lint-style.py b/scripts/lint-style.py
index b8b0ed4779a5b..face057f0f464 100755
--- a/scripts/lint-style.py
+++ b/scripts/lint-style.py
@@ -26,8 +26,6 @@
 Do not add new linters here; please write them in Lean instead.
 
 To run all style linters, run `lake exe lint-style`.
-To update the list of allowed/ignored style exceptions, use
-    $ lake exe lint-style --update
 """
 
 # TODO: This is adapted from the linter for mathlib3. It should be rewritten in Lean.
@@ -40,9 +38,7 @@
 ERR_MOD = 2 # module docstring
 ERR_IBY = 11 # isolated by
 ERR_IWH = 22 # isolated where
-ERR_DOT = 12 # isolated or low focusing dot
 ERR_SEM = 13 # the substring " ;"
-ERR_WIN = 14 # Windows line endings "\r\n"
 ERR_TWS = 15 # trailing whitespace
 ERR_CLN = 16 # line starts with a colon
 ERR_IND = 17 # second line not correctly indented
@@ -50,25 +46,9 @@
 ERR_NSP = 20 # non-terminal simp
 
 exceptions = []
-
-SCRIPTS_DIR = Path(__file__).parent.resolve()
-ROOT_DIR = SCRIPTS_DIR.parent
-
-
-with SCRIPTS_DIR.joinpath("style-exceptions.txt").open(encoding="utf-8") as f:
-    for exline in f:
-        filename, _, _, _, _, errno, *extra = exline.split()
-        path = ROOT_DIR / filename
-        if errno == "ERR_MOD":
-            exceptions += [(ERR_MOD, path, None)]
-        elif errno in ["ERR_COP", "ERR_LIN", "ERR_ADN", "ERR_NUM_LIN"]:
-            pass # maintained by the Lean style linter now
-        else:
-            print(f"Error: unexpected errno in style-exceptions.txt: {errno}")
-            sys.exit(1)
-
 new_exceptions = False
 
+
 def annotate_comments(enumerate_lines):
     """
     Take a list of tuples of enumerated lines of the form
@@ -131,9 +111,6 @@ def line_endings_check(lines, path):
     errors = []
     newlines = []
     for line_nr, line in lines:
-        if "\r\n" in line:
-            errors += [(ERR_WIN, line_nr, path)]
-            line = line.replace("\r\n", "\n")
         if line.endswith(" \n"):
             errors += [(ERR_TWS, line_nr, path)]
             line = line.rstrip() + "\n"
@@ -264,11 +241,6 @@ def isolated_by_dot_semicolon_check(lines, path):
                     line = f"{indent}{line.lstrip()[3:]}"
         elif line.lstrip() == "where":
             errors += [(ERR_IWH, line_nr, path)]
-        if line.lstrip().startswith(". "):
-            errors += [(ERR_DOT, line_nr, path)]
-            line = line.replace(". ", "· ", 1)
-        if line.strip() in (".", "·"):
-            errors += [(ERR_DOT, line_nr, path)]
         if " ;" in line:
             errors += [(ERR_SEM, line_nr, path)]
             line = line.replace(" ;", ";")
@@ -293,18 +265,10 @@ def left_arrow_check(lines, path):
     return errors, newlines
 
 def output_message(path, line_nr, code, msg):
-    if len(exceptions) == 0:
-        # we are generating a new exceptions file
-        # filename first, then line so that we can call "sort" on the output
-        print(f"{path} : line {line_nr} : {code} : {msg}")
-    else:
-        if code.startswith("ERR"):
-            msg_type = "error"
-        if code.startswith("WRN"):
-            msg_type = "warning"
-        # We are outputting for github. We duplicate path, line_nr and code,
-        # so that they are also visible in the plaintext output.
-        print(f"::{msg_type} file={path},line={line_nr},code={code}::{path}:{line_nr} {code}: {msg}")
+    # We are outputting for github. We duplicate path, line_nr and code,
+    # so that they are also visible in the plaintext output.
+    print(f"::error file={path},line={line_nr},code={code}::{path}:{line_nr} {code}: {msg}")
+
 
 def format_errors(errors):
     global new_exceptions
@@ -318,12 +282,8 @@ def format_errors(errors):
             output_message(path, line_nr, "ERR_IBY", "Line is an isolated 'by'")
         if errno == ERR_IWH:
             output_message(path, line_nr, "ERR_IWH", "Line is an isolated where")
-        if errno == ERR_DOT:
-            output_message(path, line_nr, "ERR_DOT", "Line is an isolated focusing dot or uses . instead of ·")
         if errno == ERR_SEM:
             output_message(path, line_nr, "ERR_SEM", "Line contains a space before a semicolon")
-        if errno == ERR_WIN:
-            output_message(path, line_nr, "ERR_WIN", "Windows line endings (\\r\\n) detected")
         if errno == ERR_TWS:
             output_message(path, line_nr, "ERR_TWS", "Trailing whitespace detected on line")
         if errno == ERR_CLN:
diff --git a/scripts/mk_all.lean b/scripts/mk_all.lean
index c33a957e6149d..309a4bf49d2c4 100644
--- a/scripts/mk_all.lean
+++ b/scripts/mk_all.lean
@@ -64,7 +64,8 @@ def mkAllCLI (args : Parsed) : IO UInt32 := do
       updates := updates + 1
     else if (← IO.FS.readFile fileName) != fileContent then
       if check then
-        IO.println s!"The file '{fileName}' is out of date: run `lake exe mk_all{if git then " --git" else ""}` to update it"
+        IO.println s!"The file '{fileName}' is out of date: \
+          run `lake exe mk_all{if git then " --git" else ""}` to update it"
       else
         IO.println s!"Updating '{fileName}'"
         IO.FS.writeFile fileName fileContent
diff --git a/scripts/no_lints_prime_decls.txt b/scripts/no_lints_prime_decls.txt
new file mode 100644
index 0000000000000..16a703c2e1780
--- /dev/null
+++ b/scripts/no_lints_prime_decls.txt
@@ -0,0 +1,4879 @@
+AbelRuffini.not_solvable_by_rad'
+abs_add'
+abs_le_of_sq_le_sq'
+abs_lt_of_sq_lt_sq'
+abs_norm'
+abs_norm_sub_norm_le'
+Absorbent.zero_mem'
+ack_strict_mono_left'
+Action.inhabited'
+Action.tensorUnit_ρ'
+Action.tensor_ρ'
+AddAction.orbitZMultiplesEquiv_symm_apply'
+AddChar.div_apply'
+AddChar.inv_apply'
+AddChar.neg_apply'
+AddChar.sub_apply'
+AddChar.zmodChar_apply'
+AddCircle.addOrderOf_div_of_gcd_eq_one'
+AddCircle.continuous_mk'
+AddCircle.measurable_mk'
+AddCircle.norm_eq'
+AddCommGroup.intCast_modEq_intCast'
+AddCommGroup.ModEq.add_left_cancel'
+AddCommGroup.ModEq.add_right_cancel'
+AddCommGroup.modEq_sub_iff_add_modEq'
+AddCommGroup.ModEq.sub_left_cancel'
+AddCommGroup.ModEq.sub_right_cancel'
+AddCommGroup.sub_modEq_iff_modEq_add'
+AddConstMapClass.map_add_int'
+AddConstMapClass.map_add_nat'
+AddConstMapClass.map_add_ofNat'
+AddConstMapClass.map_int_add'
+AddConstMapClass.map_nat'
+AddConstMapClass.map_nat_add'
+AddConstMapClass.map_ofNat'
+AddConstMapClass.map_ofNat_add'
+AddConstMapClass.map_sub_int'
+AddConstMapClass.map_sub_nat'
+AddConstMapClass.map_sub_ofNat'
+add_div'
+AddGroup.int_smulCommClass'
+Additive.isometricVAdd'
+Additive.isometricVAdd''
+add_le_mul'
+AddMonoidAlgebra.lift_apply'
+AddMonoidAlgebra.lift_of'
+AddMonoidAlgebra.lift_unique'
+AddMonoidAlgebra.mem_grade_iff'
+AddMonoidHom.coe_smul'
+AddMonoidHom.coe_toMultiplicative'
+AddMonoidHom.coe_toMultiplicative''
+AddMonoid.nat_smulCommClass'
+add_sq'
+AddSubgroup.torsionBy.mod_self_nsmul'
+AddValuation.map_add'
+AddValuation.map_lt_sum'
+ADEInequality.admissible_A'
+ADEInequality.admissible_D'
+ADEInequality.admissible_of_one_lt_sumInv_aux'
+AdjoinRoot.algebraMap_eq'
+AdjoinRoot.coe_injective'
+AdjoinRoot.isIntegral_root'
+AdjoinRoot.Minpoly.toAdjoin_apply'
+AEMeasurable.comp_aemeasurable'
+aemeasurable_const'
+AEMeasurable.const_smul'
+AEMeasurable.div'
+aemeasurable_id'
+aemeasurable_id''
+AEMeasurable.inf'
+AEMeasurable.mono'
+AEMeasurable.mul'
+aemeasurable_of_tendsto_metrizable_ae'
+AEMeasurable.sup'
+AffineEquiv.coe_mk'
+AffineEquiv.linear_mk'
+AffineIsometryEquiv.coe_mk'
+AffineIsometryEquiv.coe_vaddConst'
+AffineIsometryEquiv.dist_pointReflection_self'
+AffineIsometryEquiv.linearIsometryEquiv_mk'
+AffineMap.coe_mk'
+AffineMap.lineMap_apply_module'
+AffineMap.lineMap_apply_ring'
+AffineSubspace.mem_perpBisector_iff_dist_eq'
+AkraBazziRecurrence.asympBound_def'
+AkraBazziRecurrence.dist_r_b'
+Algebra.adjoin_induction''
+Algebra.algebraMap_eq_smul_one'
+Algebra.fg_trans'
+Algebra.FormallyUnramified.ext'
+Algebra.FormallyUnramified.lift_unique'
+Algebra.Generators.Cotangent.module'
+Algebra.Generators.Cotangent.val_smul'
+Algebra.Generators.Cotangent.val_smul''
+Algebra.Generators.Cotangent.val_smul'''
+AlgebraicClosure.toStepOfLE'
+AlgebraicGeometry.basicOpen_eq_of_affine'
+AlgebraicGeometry.IsAffineOpen.fromSpec_preimage_basicOpen'
+AlgebraicGeometry.IsAffineOpen.isLocalization_stalk'
+AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left'
+AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right'
+AlgebraicGeometry.LocallyRingedSpace.Hom.ext'
+AlgebraicGeometry.LocallyRingedSpace.id_val'
+AlgebraicGeometry.LocallyRingedSpace.stalkMap_germ'
+AlgebraicGeometry.morphismRestrict_app'
+AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app'
+AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.c_iso'
+AlgebraicGeometry.PresheafedSpace.stalkMap_germ'
+AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.add_mem'
+AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.mul_mem'
+AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.neg_mem'
+AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.one_mem'
+AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.SectionSubring.zero_mem'
+AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'
+AlgebraicGeometry.Proj.stalkIso'_germ'
+AlgebraicGeometry.Scheme.Hom.appIso_hom'
+AlgebraicGeometry.Scheme.Hom.appLE_map'
+AlgebraicGeometry.Scheme.Hom.map_appLE'
+AlgebraicGeometry.Scheme.map_basicOpen'
+AlgebraicGeometry.Scheme.mem_basicOpen_top'
+AlgebraicGeometry.Scheme.Opens.germ_stalkIso_hom'
+AlgebraicGeometry.Scheme.stalkMap_germ'
+AlgebraicGeometry.SheafedSpace.comp_c_app'
+AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_left'
+AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_right'
+AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf'
+AlgebraicGeometry.Spec.locallyRingedSpaceObj_presheaf_map'
+AlgebraicGeometry.Spec.locallyRingedSpaceObj_sheaf'
+AlgebraicGeometry.stalkToFiberRingHom_germ'
+AlgebraicGeometry.StructureSheaf.comap_id'
+AlgebraicGeometry.StructureSheaf.const_apply'
+AlgebraicGeometry.StructureSheaf.const_mul_cancel'
+AlgebraicGeometry.StructureSheaf.IsFraction.eq_mk'
+AlgebraicGeometry.StructureSheaf.localizationToStalk_mk'
+AlgebraicGeometry.StructureSheaf.res_const'
+AlgebraicGeometry.StructureSheaf.stalkToFiberRingHom_germ'
+AlgebraicGeometry.StructureSheaf.toBasicOpen_mk'
+AlgebraicGeometry.ΓSpec.adjunction_counit_app'
+AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_counit_app'
+AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction_homEquiv_apply'
+AlgebraicGeometry.ΓSpec.toOpen_unit_app_val_c_app'
+algebraicIndependent_equiv'
+AlgebraicIndependent.map'
+AlgebraicIndependent.to_subtype_range'
+AlgebraicTopology.DoldKan.hσ'_eq'
+AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand'
+AlgebraicTopology.DoldKan.Γ₀.Obj.map_on_summand₀'
+AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono_δ₀'
+Algebra.IsAlgebraic.bijective_of_isScalarTower'
+Algebra.TensorProduct.algebraMap_apply'
+Algebra.TensorProduct.basis_repr_symm_apply'
+Algebra.TensorProduct.ext'
+Algebra.TensorProduct.intCast_def'
+Algebra.TensorProduct.natCast_def'
+Algebra.toMatrix_lmul'
+AlgEquiv.apply_smulCommClass'
+AlgEquiv.coe_restrictScalars'
+AlgEquiv.coe_ringEquiv'
+AlgEquiv.mk_coe'
+AlgHom.coe_mk'
+AlgHom.coe_restrictScalars'
+AlgHom.toAddMonoidHom'
+AlgHom.toMonoidHom'
+AlternatingMap.domCoprod.summand_mk''
+analyticOnNhd_congr'
+AnalyticOnNhd.congr'
+AnalyticOnNhd.eval_continuousLinearMap'
+AnalyticOnNhd.eval_linearMap'
+AntilipschitzWith.le_mul_nnnorm'
+AntilipschitzWith.le_mul_norm'
+AntilipschitzWith.to_rightInvOn'
+antisymm'
+Antitone.const_mul'
+Antitone.mul_const'
+AntitoneOn.const_mul'
+AntitoneOn.mul_const'
+Applicative.pure_seq_eq_map'
+ApplicativeTransformation.preserves_map'
+apply_abs_le_mul_of_one_le'
+ArithmeticFunction.mul_smul'
+ArithmeticFunction.one_smul'
+ArithmeticFunction.ppow_succ'
+ArithmeticFunction.sum_eq_iff_sum_smul_moebius_eq_on'
+Associated.dvd'
+Associated.of_pow_associated_of_prime'
+Associates.count_mul_of_coprime'
+Associates.dvd_of_mem_factors'
+Associates.map_subtype_coe_factors'
+Associates.mk_ne_zero'
+Associates.unique'
+Asymptotics.IsBigO.congr'
+Asymptotics.isBigO_const_mul_left_iff'
+Asymptotics.IsBigO.const_mul_right'
+Asymptotics.isBigO_const_mul_right_iff'
+Asymptotics.isBigO_fst_prod'
+Asymptotics.IsBigO.of_bound'
+Asymptotics.isBigO_of_le'
+Asymptotics.isBigO_self_const_mul'
+Asymptotics.isBigO_snd_prod'
+Asymptotics.IsBigOWith.congr'
+Asymptotics.IsBigOWith.const_mul_right'
+Asymptotics.isBigOWith_of_le'
+Asymptotics.IsBigOWith.pow'
+Asymptotics.isBigOWith_self_const_mul'
+Asymptotics.IsBigOWith.sup'
+Asymptotics.isBigOWith_zero'
+Asymptotics.isEquivalent_of_tendsto_one'
+Asymptotics.IsLittleO.congr'
+Asymptotics.isLittleO_const_mul_left_iff'
+Asymptotics.IsLittleO.const_mul_right'
+Asymptotics.isLittleO_const_mul_right_iff'
+Asymptotics.IsLittleO.def'
+Asymptotics.isLittleO_iff_nat_mul_le'
+Asymptotics.isLittleO_iff_tendsto'
+Asymptotics.isLittleO_irrefl'
+Asymptotics.IsLittleO.right_isBigO_add'
+Asymptotics.IsLittleO.right_isTheta_add'
+Asymptotics.isTheta_of_norm_eventuallyEq'
+Asymptotics.SuperpolynomialDecay.congr'
+ball_eq'
+Ballot.ballot_problem'
+Basis.det_map'
+Basis.det_reindex'
+Basis.mk_eq_rank'
+Basis.mk_eq_rank''
+Basis.reindexRange_repr'
+Basis.repr_eq_iff'
+Basis.tensorProduct_apply'
+Behrend.bound_aux'
+Behrend.lower_bound_le_one'
+Behrend.map_succ'
+bernoulli'_def'
+bernoulli'_spec'
+bernoulli_spec'
+bernsteinPolynomial.flip'
+Besicovitch.SatelliteConfig.hlast'
+Besicovitch.SatelliteConfig.inter'
+bihimp_eq'
+biInf_congr'
+biInf_finsetSigma'
+biInf_sigma'
+Bimod.AssociatorBimod.hom_left_act_hom'
+Bimod.AssociatorBimod.hom_right_act_hom'
+Bimod.comp_hom'
+Bimod.id_hom'
+Bimod.LeftUnitorBimod.hom_left_act_hom'
+Bimod.LeftUnitorBimod.hom_right_act_hom'
+Bimod.RightUnitorBimod.hom_left_act_hom'
+Bimod.RightUnitorBimod.hom_right_act_hom'
+Bimod.TensorBimod.actRight_one'
+Bimod.TensorBimod.left_assoc'
+Bimod.TensorBimod.middle_assoc'
+Bimod.TensorBimod.one_act_left'
+Bimod.TensorBimod.right_assoc'
+Bimon_.comp_hom'
+Bimon_.id_hom'
+birkhoffAverage_congr_ring'
+birkhoffAverage_one'
+birkhoffAverage_zero'
+birkhoffSum_one'
+birkhoffSum_succ'
+birkhoffSum_zero'
+biSup_congr'
+biSup_finsetSigma'
+biSup_sigma'
+BooleanRing.add_eq_zero'
+Bool.eq_false_of_not_eq_true'
+Bool.eq_true_of_not_eq_false'
+Bornology.ext_iff'
+Bornology.IsBounded.exists_pos_norm_le'
+Bornology.IsBounded.exists_pos_norm_lt'
+bound'
+BoundedContinuousFunction.const_apply'
+BoundedContinuousFunction.dist_le_two_norm'
+BoundedContinuousFunction.dist_nonneg'
+BoundedContinuousFunction.extend_apply'
+BoundedContinuousFunction.instModule'
+BoundedContinuousFunction.instSMul'
+BoundedLatticeHom.coe_comp_inf_hom'
+BoundedLatticeHom.coe_comp_lattice_hom'
+BoundedLatticeHom.coe_comp_sup_hom'
+BoxIntegral.Box.coe_mk'
+BoxIntegral.Box.volume_apply'
+BoxIntegral.IntegrationParams.toFilterDistortioniUnion_neBot'
+BoxIntegral.Prepartition.iUnion_def'
+BoxIntegral.Prepartition.mem_restrict'
+BoxIntegral.Prepartition.mem_split_iff'
+BoxIntegral.TaggedPrepartition.IsSubordinate.mono'
+Bundle.TotalSpace.mk'
+calc_eval_z'
+card_dvd_exponent_pow_rank'
+Cardinal.add_eq_max'
+Cardinal.add_le_add'
+Cardinal.add_mk_eq_max'
+Cardinal.aleph0_le_aleph'
+Cardinal.aleph_eq_aleph'
+Cardinal.alephIdx_aleph'
+Cardinal.cantor'
+Cardinal.lift_lt_univ'
+Cardinal.lift_mk_shrink'
+Cardinal.lift_mk_shrink''
+Cardinal.lt_univ'
+Cardinal.mk_eq_two_iff'
+Cardinal.mk_finsupp_lift_of_infinite'
+Cardinal.mk_finsupp_of_infinite'
+Cardinal.mul_comm'
+Cardinal.mul_eq_max'
+Cardinal.prod_const'
+Cardinal.sum_add_distrib'
+Cardinal.sum_const'
+Cardinal.two_le_iff'
+card_le_of_injective'
+card_le_of_surjective'
+catalan_succ'
+CategoryTheory.Abelian.coimageImageComparison_eq_coimageImageComparison'
+CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono'
+CategoryTheory.Abelian.epi_of_mono_of_epi_of_mono'
+CategoryTheory.Abelian.Ext.add_hom'
+CategoryTheory.Abelian.Ext.neg_hom'
+CategoryTheory.Abelian.FunctorCategory.coimageImageComparison_app'
+CategoryTheory.Abelian.mono_of_epi_of_epi_mono'
+CategoryTheory.Abelian.mono_of_epi_of_mono_of_mono'
+CategoryTheory.Abelian.OfCoimageImageComparisonIsIso.imageMonoFactorisation_e'
+CategoryTheory.Abelian.Pseudoelement.pseudoApply_mk'
+CategoryTheory.Abelian.Pseudoelement.zero_eq_zero'
+CategoryTheory.Abelian.Pseudoelement.zero_morphism_ext'
+CategoryTheory.ActionCategory.cases'
+CategoryTheory.additive_coyonedaObj'
+CategoryTheory.additive_yonedaObj'
+CategoryTheory.Adhesive.van_kampen'
+CategoryTheory.Adjunction.he''
+CategoryTheory.Arrow.iso_w'
+CategoryTheory.BicategoricalCoherence.assoc'
+CategoryTheory.BicategoricalCoherence.left'
+CategoryTheory.BicategoricalCoherence.right'
+CategoryTheory.BicategoricalCoherence.tensorRight'
+CategoryTheory.Bifunctor.diagonal'
+CategoryTheory.Biproduct.column_nonzero_of_iso'
+CategoryTheory.BraidedCategory.yang_baxter'
+CategoryTheory.BraidedFunctor.ext'
+CategoryTheory.CategoryOfElements.CreatesLimitsAux.π_liftedConeElement'
+CategoryTheory.CechNerveTerminalFrom.hasWidePullback'
+CategoryTheory.CommSq.HasLift.mk'
+CategoryTheory.Comonad.Coalgebra.Hom.ext'
+CategoryTheory.ComonadHom.ext'
+CategoryTheory.comp_apply'
+CategoryTheory.ComposableArrows.Exact.exact'
+CategoryTheory.ComposableArrows.IsComplex.zero'
+CategoryTheory.ComposableArrows.map'_inv_eq_inv_map'
+CategoryTheory.ComposableArrows.naturality'
+CategoryTheory.ComposableArrows.Precomp.map_zero_one'
+CategoryTheory.composePath_comp'
+CategoryTheory.conj_eqToHom_iff_heq'
+CategoryTheory.CosimplicialObject.δ_comp_δ'
+CategoryTheory.CosimplicialObject.δ_comp_δ''
+CategoryTheory.CosimplicialObject.δ_comp_δ_self'
+CategoryTheory.CosimplicialObject.δ_comp_σ_of_gt'
+CategoryTheory.CosimplicialObject.δ_comp_σ_self'
+CategoryTheory.CosimplicialObject.δ_comp_σ_succ'
+CategoryTheory.DifferentialObject.eqToHom_f'
+CategoryTheory.e_assoc'
+CategoryTheory.Endofunctor.Algebra.Initial.left_inv'
+CategoryTheory.eq_of_comp_left_eq'
+CategoryTheory.eq_of_comp_right_eq'
+CategoryTheory.Equivalence.cancel_counitInv_right_assoc'
+CategoryTheory.Equivalence.cancel_unit_right_assoc'
+CategoryTheory.ExactPairing.coevaluation_evaluation''
+CategoryTheory.ExactPairing.evaluation_coevaluation''
+CategoryTheory.exists_zigzag'
+CategoryTheory.forgetEnrichment_id'
+CategoryTheory.Functor.commShiftIso_add'
+CategoryTheory.Functor.coreflective'
+CategoryTheory.Functor.HasRightDerivedFunctor.mk'
+CategoryTheory.Functor.inl_biprodComparison'
+CategoryTheory.Functor.inr_biprodComparison'
+CategoryTheory.Functor.isContinuous_comp'
+CategoryTheory.Functor.IsHomological.mk'
+CategoryTheory.Functor.IsLocalization.mk'
+CategoryTheory.Functor.Iteration.Hom.ext'
+CategoryTheory.Functor.map_comp_heq'
+CategoryTheory.Functor.postcomp_map_heq'
+CategoryTheory.Functor.reflective'
+CategoryTheory.Functor.relativelyRepresentable.map_fst'
+CategoryTheory.Functor.relativelyRepresentable.w'
+CategoryTheory.Functor.shiftIso_add'
+CategoryTheory.Functor.shiftMap_comp'
+CategoryTheory.FunctorToTypes.jointly_surjective'
+CategoryTheory.FunctorToTypes.prod_ext'
+CategoryTheory.Functor.uncurry_obj_curry_obj_flip_flip'
+CategoryTheory.Functor.ι_biproductComparison'
+CategoryTheory.Grothendieck.comp_fiber'
+CategoryTheory.Grothendieck.id_fiber'
+CategoryTheory.GrothendieckTopology.OneHypercoverFamily.IsSheafIff.fac'
+CategoryTheory.GrothendieckTopology.OneHypercover.mem_sieve₁'
+CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.mk'
+CategoryTheory.Grpd.str'
+CategoryTheory.HasPullbacksOfInclusions.hasPullbackInr'
+CategoryTheory.HasPullbacksOfInclusions.preservesPullbackInl'
+CategoryTheory.HasSheafify.mk'
+CategoryTheory.Injective.injective_iff_preservesEpimorphisms_preadditive_yoneda_obj'
+CategoryTheory.IsCoreflexivePair.mk'
+CategoryTheory.IsHomLift.fac'
+CategoryTheory.IsHomLift.of_fac'
+CategoryTheory.Iso.inv_ext'
+CategoryTheory.IsPullback.inl_snd'
+CategoryTheory.IsPullback.inr_fst'
+CategoryTheory.IsPullback.of_hasBinaryProduct'
+CategoryTheory.IsPullback.of_is_bilimit'
+CategoryTheory.IsPushout.inl_snd'
+CategoryTheory.IsPushout.inr_fst'
+CategoryTheory.IsPushout.of_hasBinaryCoproduct'
+CategoryTheory.IsPushout.of_is_bilimit'
+CategoryTheory.IsReflexivePair.mk'
+CategoryTheory.isSheaf_yoneda'
+CategoryTheory.LaxBraidedFunctor.ext'
+CategoryTheory.Limits.biprod.hom_ext'
+CategoryTheory.Limits.biprod.map_eq_map'
+CategoryTheory.Limits.biprod.symmetry'
+CategoryTheory.Limits.biproduct.hom_ext'
+CategoryTheory.Limits.biproduct.map_eq_map'
+CategoryTheory.Limits.Cofork.IsColimit.π_desc'
+CategoryTheory.Limits.colimit.pre_map'
+CategoryTheory.Limits.colimMap_epi'
+CategoryTheory.Limits.Concrete.widePullback_ext'
+CategoryTheory.Limits.Concrete.widePushout_exists_rep'
+CategoryTheory.Limits.coprod.symmetry'
+CategoryTheory.Limits.equalizerSubobject_arrow'
+CategoryTheory.Limits.Fork.IsLimit.lift_ι'
+CategoryTheory.Limits.ImageMap.mk.injEq'
+CategoryTheory.Limits.imageSubobject_arrow'
+CategoryTheory.Limits.kernelSubobject_arrow'
+CategoryTheory.Limits.limit.map_pre'
+CategoryTheory.Limits.limLax_obj'
+CategoryTheory.Limits.limMap_mono'
+CategoryTheory.Limits.MonoCoprod.mk'
+CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inl'
+CategoryTheory.Limits.MonoCoprod.mono_binaryCofanSum_inr'
+CategoryTheory.Limits.MonoCoprod.mono_of_injective'
+CategoryTheory.Limits.Multicoequalizer.multicofork_ι_app_right'
+CategoryTheory.Limits.Multicofork.ofSigmaCofork_ι_app_right'
+CategoryTheory.Limits.parallelPair_initial_mk'
+CategoryTheory.Limits.Pi.map'_comp_map'
+CategoryTheory.Limits.Pi.map_comp_map'
+CategoryTheory.Limits.prod.symmetry'
+CategoryTheory.Limits.Sigma.map'_comp_map'
+CategoryTheory.Limits.Sigma.map_comp_map'
+CategoryTheory.Limits.Sigma.ι_comp_map'
+CategoryTheory.Limits.Types.Colimit.w_apply'
+CategoryTheory.Limits.Types.Colimit.ι_desc_apply'
+CategoryTheory.Limits.Types.Colimit.ι_map_apply'
+CategoryTheory.Limits.Types.limit_ext'
+CategoryTheory.Limits.Types.limit_ext_iff'
+CategoryTheory.Limits.Types.Limit.lift_π_apply'
+CategoryTheory.Limits.Types.Limit.map_π_apply'
+CategoryTheory.Limits.Types.Limit.w_apply'
+CategoryTheory.Limits.Types.Pushout.equivalence_rel'
+CategoryTheory.Limits.WalkingParallelPairHom.id.sizeOf_spec'
+CategoryTheory.Limits.zero_of_source_iso_zero'
+CategoryTheory.Limits.zero_of_target_iso_zero'
+CategoryTheory.Localization.Preadditive.comp_add'
+CategoryTheory.Localization.Preadditive.zero_add'
+CategoryTheory.Localization.SmallShiftedHom.equiv_shift'
+CategoryTheory.LocalizerMorphism.guitartExact_of_isRightDerivabilityStructure'
+CategoryTheory.LocalizerMorphism.IsLocalizedEquivalence.mk'
+CategoryTheory.Mat_.additiveObjIsoBiproduct_naturality'
+CategoryTheory.Monad.Algebra.Hom.ext'
+CategoryTheory.MonadHom.ext'
+CategoryTheory.MonoidalCategory.hom_inv_id_tensor'
+CategoryTheory.MonoidalCategory.hom_inv_whiskerRight'
+CategoryTheory.MonoidalCategory.inv_hom_id_tensor'
+CategoryTheory.MonoidalCategory.inv_hom_whiskerRight'
+CategoryTheory.MonoidalCategory.leftUnitor_tensor'
+CategoryTheory.MonoidalCategory.leftUnitor_tensor''
+CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv'
+CategoryTheory.MonoidalCategory.tensorHom_def'
+CategoryTheory.MonoidalCategory.tensor_hom_inv_id'
+CategoryTheory.MonoidalCategory.tensor_inv_hom_id'
+CategoryTheory.MonoidalCategory.tensorIso_def'
+CategoryTheory.MonoidalCategory.whiskerLeft_hom_inv'
+CategoryTheory.MonoidalCategory.whiskerLeft_inv_hom'
+CategoryTheory.MonoidalCoherence.assoc'
+CategoryTheory.MonoidalCoherence.left'
+CategoryTheory.MonoidalCoherence.right'
+CategoryTheory.MonoidalCoherence.tensor_right'
+CategoryTheory.MonoidalNatTrans.ext'
+CategoryTheory.MonoOver.mk'_coe'
+CategoryTheory.NatIso.naturality_1'
+CategoryTheory.NatIso.naturality_2'
+CategoryTheory.NatTrans.ext'
+CategoryTheory.NatTrans.id_app'
+CategoryTheory.NatTrans.vcomp_app'
+CategoryTheory.NonPreadditiveAbelian.neg_sub'
+CategoryTheory.OplaxNatTrans.Modification.comp_app'
+CategoryTheory.OplaxNatTrans.Modification.id_app'
+CategoryTheory.Preadditive.epi_iff_surjective'
+CategoryTheory.Preadditive.epi_of_isZero_cokernel'
+CategoryTheory.Preadditive.mono_iff_injective'
+CategoryTheory.Preadditive.mono_of_isZero_kernel'
+CategoryTheory.Prefunctor.mapPath_comp'
+CategoryTheory.PreOneHypercover.sieve₁_eq_pullback_sieve₁'
+CategoryTheory.PreservesPullbacksOfInclusions.preservesPullbackInl'
+CategoryTheory.PreservesPullbacksOfInclusions.preservesPullbackInr'
+CategoryTheory.Presheaf.isLocallyInjective_toSheafify'
+CategoryTheory.Presheaf.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top'
+CategoryTheory.Presheaf.isLocallySurjective_toSheafify'
+CategoryTheory.Pretriangulated.mem_distTriang_op_iff'
+CategoryTheory.Pretriangulated.Opposite.mem_distinguishedTriangles_iff'
+CategoryTheory.Projective.projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj'
+CategoryTheory.ProjectiveResolution.pOpcycles_comp_fromLeftDerivedZero'
+CategoryTheory.Quiv.str'
+CategoryTheory.Quotient.lift_unique'
+CategoryTheory.RanIsSheafOfIsCocontinuous.fac'
+CategoryTheory.RanIsSheafOfIsCocontinuous.liftAux_map'
+CategoryTheory.regularTopology.equalizerCondition_w'
+CategoryTheory.Sheaf.epi_of_isLocallySurjective'
+CategoryTheory.Sheaf.isLocallySurjective_iff_epi'
+CategoryTheory.SheafOfTypes.Hom.ext'
+CategoryTheory.shift_neg_shift'
+CategoryTheory.shift_shift'
+CategoryTheory.shift_shift_neg'
+CategoryTheory.shiftZero'
+CategoryTheory.ShortComplex.abLeftHomologyData_f'
+CategoryTheory.ShortComplex.epi_homologyMap_of_epi_cyclesMap'
+CategoryTheory.ShortComplex.Exact.desc'
+CategoryTheory.ShortComplex.Exact.epi_f'
+CategoryTheory.ShortComplex.exact_iff_epi_imageToKernel'
+CategoryTheory.ShortComplex.Exact.isIso_f'
+CategoryTheory.ShortComplex.Exact.isIso_g'
+CategoryTheory.ShortComplex.Exact.lift'
+CategoryTheory.ShortComplex.Exact.mono_g'
+CategoryTheory.ShortComplex.f'_cyclesMap'
+CategoryTheory.ShortComplex.HasHomology.mk'
+CategoryTheory.ShortComplex.hasHomology_of_epi_of_isIso_of_mono'
+CategoryTheory.ShortComplex.hasHomology_of_isIso_leftRightHomologyComparison'
+CategoryTheory.ShortComplex.hasHomology_of_preserves'
+CategoryTheory.ShortComplex.HasLeftHomology.mk'
+CategoryTheory.ShortComplex.hasLeftHomology_of_epi_of_isIso_of_mono'
+CategoryTheory.ShortComplex.hasLeftHomology_of_preserves'
+CategoryTheory.ShortComplex.HasRightHomology.mk'
+CategoryTheory.ShortComplex.hasRightHomology_of_epi_of_isIso_of_mono'
+CategoryTheory.ShortComplex.hasRightHomology_of_preserves'
+CategoryTheory.ShortComplex.HomologyData.exact_iff'
+CategoryTheory.ShortComplex.HomologyData.map_homologyMap'
+CategoryTheory.ShortComplex.isIso₂_of_shortExact_of_isIso₁₃'
+CategoryTheory.ShortComplex.isIso_cyclesMap_of_isIso_of_mono'
+CategoryTheory.ShortComplex.isIso_homologyMap_of_epi_of_isIso_of_mono'
+CategoryTheory.ShortComplex.isIso_leftRightHomologyComparison'
+CategoryTheory.ShortComplex.isIso_opcyclesMap_of_isIso_of_epi'
+CategoryTheory.ShortComplex.LeftHomologyData.exact_iff_epi_f'
+CategoryTheory.ShortComplex.LeftHomologyData.map_cyclesMap'
+CategoryTheory.ShortComplex.LeftHomologyData.map_f'
+CategoryTheory.ShortComplex.LeftHomologyData.map_leftHomologyMap'
+CategoryTheory.ShortComplex.LeftHomologyData.ofEpiOfIsIsoOfMono'_f'
+CategoryTheory.ShortComplex.LeftHomologyData.ofIsColimitCokernelCofork_f'
+CategoryTheory.ShortComplex.LeftHomologyData.ofIsLimitKernelFork_f'
+CategoryTheory.ShortComplex.LeftHomologyData.ofZeros_f'
+CategoryTheory.ShortComplex.LeftHomologyData.op_g'
+CategoryTheory.ShortComplex.LeftHomologyData.unop_g'
+CategoryTheory.ShortComplex.LeftHomologyData.τ₁_ofEpiOfIsIsoOfMono_f'
+CategoryTheory.ShortComplex.leftHomologyπ_naturality'
+CategoryTheory.ShortComplex.leftRightHomologyComparison'_eq_leftHomologpMap'_comp_iso_hom_comp_rightHomologyMap'
+CategoryTheory.ShortComplex.moduleCatLeftHomologyData_f'
+CategoryTheory.ShortComplex.mono_homologyMap_of_mono_opcyclesMap'
+CategoryTheory.ShortComplex.opcyclesMap'_g'
+CategoryTheory.ShortComplex.p_opcyclesMap'
+CategoryTheory.ShortComplex.quasiIso_iff_isIso_homologyMap'
+CategoryTheory.ShortComplex.quasiIso_iff_isIso_leftHomologyMap'
+CategoryTheory.ShortComplex.quasiIso_iff_isIso_rightHomologyMap'
+CategoryTheory.ShortComplex.RightHomologyData.exact_iff_mono_g'
+CategoryTheory.ShortComplex.RightHomologyData.map_g'
+CategoryTheory.ShortComplex.RightHomologyData.map_opcyclesMap'
+CategoryTheory.ShortComplex.RightHomologyData.map_rightHomologyMap'
+CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono_g'
+CategoryTheory.ShortComplex.RightHomologyData.ofIsColimitCokernelCofork_g'
+CategoryTheory.ShortComplex.RightHomologyData.ofIsLimitKernelFork_g'
+CategoryTheory.ShortComplex.RightHomologyData.ofZeros_g'
+CategoryTheory.ShortComplex.RightHomologyData.op_f'
+CategoryTheory.ShortComplex.RightHomologyData.p_g'
+CategoryTheory.ShortComplex.RightHomologyData.unop_f'
+CategoryTheory.ShortComplex.RightHomologyData.ι_g'
+CategoryTheory.ShortComplex.rightHomologyι_naturality'
+CategoryTheory.ShortComplex.ShortExact.mk'
+CategoryTheory.ShortComplex.ShortExact.δ_apply'
+CategoryTheory.ShortComplex.ShortExact.δ_eq'
+CategoryTheory.SimplicialObject.δ_comp_δ'
+CategoryTheory.SimplicialObject.δ_comp_δ''
+CategoryTheory.SimplicialObject.δ_comp_δ_self'
+CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'
+CategoryTheory.SimplicialObject.δ_comp_σ_self'
+CategoryTheory.SimplicialObject.δ_comp_σ_succ'
+CategoryTheory.SingleFunctors.shiftIso_add'
+CategoryTheory.StrongEpi.mk'
+CategoryTheory.StrongMono.mk'
+CategoryTheory.Subgroupoid.coe_inv_coe'
+CategoryTheory.Subgroupoid.IsNormal.conj'
+CategoryTheory.Subobject.inf_eq_map_pullback'
+CategoryTheory.Tor'_map_app'
+CategoryTheory.Triangulated.Subcategory.ext₁'
+CategoryTheory.Triangulated.Subcategory.ext₃'
+CategoryTheory.Triangulated.Subcategory.W_iff'
+CategoryTheory.Triangulated.Subcategory.W.mk'
+CategoryTheory.TwoSquare.GuitartExact.vComp'
+CategoryTheory.whiskerLeft_id'
+CategoryTheory.whiskerRight_id'
+CategoryTheory.yonedaEquiv_naturality'
+Cauchy.comap'
+CauchyFilter.mem_uniformity'
+cauchy_iff'
+cauchy_iInf_uniformSpace'
+cauchy_map_iff'
+Cauchy.mono'
+cauchy_pi_iff'
+cauchySeq_iff'
+CauSeq.bounded'
+CauSeq.mul_equiv_zero'
+cfc_comp'
+cfcₙ_comp'
+CharP.exists'
+CharP.natCast_eq_natCast'
+charP_of_injective_algebraMap'
+CharP.pi'
+CharP.subring'
+ChartedSpaceCore.open_source'
+CharTwo.neg_eq'
+CharTwo.sub_eq_add'
+ciInf_le'
+ciInf_le_of_le'
+ciInf_subtype'
+ciInf_subtype''
+CircleDeg1Lift.tendsto_translation_number'
+CircleDeg1Lift.tendsto_translation_number₀'
+CircleDeg1Lift.translationNumber_conj_eq'
+CircleDeg1Lift.translationNumber_eq_of_tendsto₀'
+circleIntegral.norm_integral_le_of_norm_le_const'
+circleMap_mem_sphere'
+ciSup_le'
+ciSup_le_iff'
+ciSup_mono'
+ciSup_or'
+ciSup_subtype'
+ciSup_subtype''
+Classical.choose_eq'
+CliffordAlgebra.instAlgebra'
+CliffordAlgebra.star_def'
+ClosedIciTopology.isClosed_ge'
+ClosedIicTopology.isClosed_le'
+closedUnderRestriction'
+closure_smul₀'
+clusterPt_iff_lift'_closure'
+ClusterPt.of_le_nhds'
+cmp_div_one'
+cmp_mul_left'
+cmp_mul_right'
+CochainComplex.HomComplex.Cochain.shift_v'
+CochainComplex.mappingCone.d_fst_v'
+CochainComplex.mappingCone.d_snd_v'
+CochainComplex.shiftFunctorAdd'_hom_app_f'
+CochainComplex.shiftFunctorAdd'_inv_app_f'
+CochainComplex.shiftFunctor_map_f'
+CochainComplex.shiftFunctor_obj_d'
+CochainComplex.shiftFunctor_obj_X'
+Codisjoint.ne_bot_of_ne_top'
+Codisjoint.of_codisjoint_sup_of_le'
+Codisjoint.sup_left'
+Codisjoint.sup_right'
+coe_comp_nnnorm'
+coe_nnnorm'
+CofiniteTopology.isOpen_iff'
+comap_norm_atTop'
+comap_norm_nhdsWithin_Ioi_zero'
+CommGrp.coe_comp'
+CommGrp.coe_id'
+CommMon_.comp'
+CommMon_.id'
+commProb_def'
+CommRingCat.equalizer_ι_is_local_ring_hom'
+CommRingCat.instCommRing'
+CommRingCat.instFunLike'
+CommRingCat.instFunLike''
+CommRingCat.instFunLike'''
+CommSemiRingCat.instCommSemiring'
+Commute.mul_self_sub_mul_self_eq'
+Comon_.comp_hom'
+Comon_.id_hom'
+CompactIccSpace.mk'
+CompactIccSpace.mk''
+CompHaus.toProfinite_obj'
+compl_beattySeq'
+CompleteLattice.Independent.comp'
+CompleteLattice.independent_def'
+CompleteLattice.independent_def''
+CompleteLattice.independent_of_dfinsupp_sumAddHom_injective'
+CompleteLattice.Independent.supIndep'
+CompleteLattice.inf_continuous'
+CompleteLattice.sSup_continuous'
+CompletelyDistribLattice.MinimalAxioms.iInf_iSup_eq'
+CompleteOrthogonalIdempotents.bijective_pi'
+CompleteSublattice.coe_sInf'
+CompleteSublattice.coe_sSup'
+Complex.abs_eq_one_iff'
+Complex.AbsTheory.abs_nonneg'
+Complex.affine_of_mapsTo_ball_of_exists_norm_dslope_eq_div'
+Complex.conj_mul'
+Complex.cos_eq_tsum'
+Complex.cos_sq'
+Complex.cos_two_mul'
+Complex.cpow_ofNat_mul'
+Complex.deriv_cos'
+Complex.equivRealProd_apply_le'
+Complex.exp_bound'
+Complex.hasStrictFDerivAt_cpow'
+Complex.hasSum_conj'
+Complex.hasSum_cos'
+Complex.hasSum_sin'
+Complex.mul_conj'
+Complex.ofReal_mul'
+Complex.rank_real_complex'
+Complex.restrictScalars_one_smulRight'
+ComplexShape.Embedding.not_boundaryGE_next'
+ComplexShape.Embedding.not_boundaryLE_prev'
+ComplexShape.next_add'
+ComplexShape.next_eq'
+ComplexShape.next_eq_self'
+ComplexShape.prev_eq'
+ComplexShape.prev_eq_self'
+Complex.sin_eq_tsum'
+Complex.stolzCone_subset_stolzSet_aux'
+Complex.tan_add'
+Complex.UnitDisc.instSMulCommClass_circle'
+Complex.UnitDisc.instSMulCommClass_closedBall'
+compl_sInf'
+compl_sSup'
+CompositionAsSet.lt_length'
+Composition.blocks_pos'
+Composition.mem_range_embedding_iff'
+Composition.one_le_blocks'
+Composition.sizeUpTo_succ'
+Computability.inhabitedΓ'
+ComputablePred.computable_iff_re_compl_re'
+Computable.vector_ofFn'
+Computation.bind_pure'
+Computation.eq_thinkN'
+Computation.map_pure'
+Computation.map_think'
+Computation.results_of_terminates'
+ConcaveOn.left_le_of_le_right'
+ConcaveOn.left_le_of_le_right''
+ConcaveOn.left_lt_of_lt_right'
+ConcaveOn.lt_right_of_left_lt'
+ConcaveOn.mul'
+ConcaveOn.mul_convexOn'
+ConcaveOn.right_le_of_le_left'
+ConcaveOn.right_le_of_le_left''
+ConcaveOn.smul'
+ConcaveOn.smul''
+ConcaveOn.smul_convexOn'
+Concept.ext'
+Con.coe_mk'
+conformalFactorAt_inner_eq_mul_inner'
+CongruenceSubgroup.Gamma1_mem'
+CongruenceSubgroup.Gamma_mem'
+ConjAct.smulCommClass'
+ConjAct.smulCommClass₀'
+ConjAct.unitsSMulCommClass'
+conjneg_neg'
+conjugate_le_conjugate'
+conjugate_lt_conjugate'
+conjugate_nonneg'
+conjugate_pos'
+Con.mrange_mk'
+ConnectedComponents.coe_eq_coe'
+connectedComponents_lift_unique'
+ContDiffAt.comp'
+contDiffAt_id'
+contDiffAt_pi'
+contDiffAt_prod'
+ContDiff.comp'
+contDiff_id'
+ContDiff.iterate_deriv'
+ContDiffOn.div'
+contDiffOn_id'
+contDiffOn_pi'
+contDiffOn_prod'
+contDiff_pi'
+contDiff_prod'
+ContDiffWithinAt.congr'
+ContDiffWithinAt.congr_of_eventually_eq'
+ContDiffWithinAt.contDiffOn'
+contDiffWithinAt_inter'
+contDiffWithinAt_prod'
+ContinuousAlgHom.coe_comp'
+ContinuousAlgHom.coe_fst'
+ContinuousAlgHom.coe_id'
+ContinuousAlgHom.coe_mk'
+ContinuousAlgHom.coe_prodMap'
+ContinuousAlgHom.coe_restrictScalars'
+ContinuousAlgHom.coe_snd'
+ContinuousAt.comp'
+continuousAt_const_cpow'
+ContinuousAt.div'
+continuousAt_extChartAt'
+continuousAt_extChartAt_symm'
+continuousAt_extChartAt_symm''
+ContinuousAt.finset_inf'
+ContinuousAt.finset_sup'
+continuousAt_id'
+continuousAt_iff_continuous_left'_right'
+ContinuousAt.inf'
+continuousAt_jacobiTheta₂'
+ContinuousAt.nnnorm'
+ContinuousAt.norm'
+continuousAt_pi'
+ContinuousAt.prod_map'
+ContinuousAt.sup'
+Continuous.comp'
+Continuous.comp_continuousOn'
+Continuous.div'
+continuous_div_left'
+continuous_div_right'
+Continuous.finset_inf'
+Continuous.finset_sup'
+continuous_id'
+continuous_if'
+Continuous.inf'
+ContinuousLinearEquiv.coe_refl'
+ContinuousLinearEquiv.comp_hasFDerivAt_iff'
+ContinuousLinearEquiv.comp_hasFDerivWithinAt_iff'
+ContinuousLinearEquiv.comp_right_hasFDerivAt_iff'
+ContinuousLinearEquiv.comp_right_hasFDerivWithinAt_iff'
+ContinuousLinearMap.apply_apply'
+ContinuousLinearMap.applySMulCommClass'
+ContinuousLinearMap.coe_add'
+ContinuousLinearMap.coe_comp'
+ContinuousLinearMap.coe_flipₗᵢ'
+ContinuousLinearMap.coeFn_compLp'
+ContinuousLinearMap.coe_fst'
+ContinuousLinearMap.coe_id'
+ContinuousLinearMap.coe_mk'
+ContinuousLinearMap.coe_neg'
+ContinuousLinearMap.coe_pi'
+ContinuousLinearMap.coe_prodMap'
+ContinuousLinearMap.coe_restrictScalars'
+ContinuousLinearMap.coe_restrict_scalarsL'
+ContinuousLinearMap.coe_smul'
+ContinuousLinearMap.coe_snd'
+ContinuousLinearMap.coe_sub'
+ContinuousLinearMap.coe_sum'
+ContinuousLinearMap.coe_zero'
+ContinuousLinearMap.compFormalMultilinearSeries_apply'
+ContinuousLinearMap.comp_memℒp'
+ContinuousLinearMap.integral_comp_comm'
+ContinuousLinearMap.measurable_apply'
+ContinuousLinearMap.mul_apply'
+ContinuousLinearMap.norm_extendTo𝕜'
+ContinuousLinearMap.opNorm_le_of_shell'
+ContinuousLinearMap.sub_apply'
+ContinuousLinearMap.toSpanSingleton_smul'
+ContinuousMap.coe_const'
+ContinuousMap.coe_inf'
+ContinuousMap.coe_sup'
+ContinuousMap.comp_yonedaPresheaf'
+ContinuousMap.continuous.comp'
+ContinuousMap.continuous_const'
+ContinuousMap.instSMul'
+ContinuousMap.liftCover_coe'
+ContinuousMap.liftCover_restrict'
+ContinuousMap.module'
+ContinuousMap.unitsLift_symm_apply_apply_inv'
+ContinuousMapZero.instIsScalarTower'
+ContinuousMapZero.instSMulCommClass'
+Continuous.matrix_blockDiag'
+Continuous.matrix_blockDiagonal'
+continuousMultilinearCurryRightEquiv_apply'
+continuousMultilinearCurryRightEquiv_symm_apply'
+continuous_nnnorm'
+Continuous.nnnorm'
+continuous_norm'
+Continuous.norm'
+ContinuousOn.circleIntegrable'
+ContinuousOn.comp'
+ContinuousOn.comp''
+ContinuousOn.div'
+ContinuousOn.finset_inf'
+ContinuousOn.finset_sup'
+continuousOn_id'
+ContinuousOn.if'
+continuousOn_iff'
+ContinuousOn.inf'
+ContinuousOn.nnnorm'
+ContinuousOn.norm'
+continuousOn_pi'
+ContinuousOn.piecewise'
+continuousOn_piecewise_ite'
+ContinuousOn.sup'
+Continuous.quotient_liftOn'
+Continuous.quotient_map'
+continuous_quotient_mk'
+Continuous.strictMono_of_inj_boundedOrder'
+Continuous.sup'
+ContinuousWithinAt.comp'
+ContinuousWithinAt.div'
+ContinuousWithinAt.finset_inf'
+ContinuousWithinAt.finset_sup'
+ContinuousWithinAt.inf'
+continuousWithinAt_inter'
+ContinuousWithinAt.nnnorm'
+ContinuousWithinAt.norm'
+ContinuousWithinAt.preimage_mem_nhdsWithin'
+ContinuousWithinAt.preimage_mem_nhdsWithin''
+ContinuousWithinAt.sup'
+contMDiffAt_extChartAt'
+contMDiffAt_finset_prod'
+ContMDiffAt.prod_map'
+contMDiff_finset_prod'
+ContMDiffMap.mdifferentiable'
+contMDiffOn_finset_prod'
+contMDiffOn_iff_of_mem_maximalAtlas'
+ContMDiffSection.mdifferentiable'
+contMDiffWithinAt_finset_prod'
+contMDiffWithinAt_iff_of_mem_source'
+contMDiffWithinAt_inter'
+ContractingWith.apriori_edist_iterate_efixedPoint_le'
+ContractingWith.edist_efixedPoint_le'
+ContractingWith.edist_efixedPoint_lt_top'
+ContractingWith.efixedPoint_isFixedPt'
+ContractingWith.efixedPoint_mem'
+ContractingWith.fixedPoint_unique'
+ContractingWith.one_sub_K_pos'
+ContractingWith.tendsto_iterate_efixedPoint'
+ConvexBody.coe_smul'
+Convex.mem_toCone'
+ConvexOn.le_left_of_right_le'
+ConvexOn.le_left_of_right_le''
+ConvexOn.le_right_of_left_le'
+ConvexOn.le_right_of_left_le''
+ConvexOn.lt_left_of_right_lt'
+ConvexOn.lt_right_of_left_lt'
+ConvexOn.mul'
+ConvexOn.mul_concaveOn'
+ConvexOn.smul'
+ConvexOn.smul''
+ConvexOn.smul_concaveOn'
+coord_norm'
+CovBy.ne'
+CoxeterSystem.alternatingWord_succ'
+CoxeterSystem.exists_reduced_word'
+CoxeterSystem.length_mul_ge_length_sub_length'
+CoxeterSystem.simple_mul_simple_pow'
+CPolynomialOn.congr'
+CPolynomialOn_congr'
+cpow_eq_nhds'
+cross_anticomm'
+csInf_le'
+csInf_le_csInf'
+csSup_le'
+csSup_le_csSup'
+csSup_le_iff'
+CStarAlgebra.conjugate_le_norm_smul'
+CStarAlgebra.instNonnegSpectrumClass'
+CStarRing.conjugate_le_norm_smul'
+CStarRing.instNonnegSpectrumClass'
+CStarRing.norm_star_mul_self'
+Ctop.Realizer.ext'
+Cubic.degree_of_a_eq_zero'
+Cubic.degree_of_a_ne_zero'
+Cubic.degree_of_b_eq_zero'
+Cubic.degree_of_b_ne_zero'
+Cubic.degree_of_c_eq_zero'
+Cubic.degree_of_c_ne_zero'
+Cubic.degree_of_d_eq_zero'
+Cubic.degree_of_d_ne_zero'
+Cubic.leadingCoeff_of_a_ne_zero'
+Cubic.leadingCoeff_of_b_ne_zero'
+Cubic.leadingCoeff_of_c_eq_zero'
+Cubic.leadingCoeff_of_c_ne_zero'
+Cubic.monic_of_a_eq_one'
+Cubic.monic_of_b_eq_one'
+Cubic.monic_of_c_eq_one'
+Cubic.monic_of_d_eq_one'
+Cubic.natDegree_of_a_eq_zero'
+Cubic.natDegree_of_a_ne_zero'
+Cubic.natDegree_of_b_eq_zero'
+Cubic.natDegree_of_b_ne_zero'
+Cubic.natDegree_of_c_eq_zero'
+Cubic.natDegree_of_c_ne_zero'
+Cubic.of_a_eq_zero'
+Cubic.of_b_eq_zero'
+Cubic.of_c_eq_zero'
+Cubic.of_d_eq_zero'
+Cycle.next_reverse_eq_prev'
+Cycle.prev_reverse_eq_next'
+CyclotomicField.algebra'
+dec_em'
+Decidable.mul_lt_mul''
+Decidable.Partrec.const'
+decide_False'
+decide_True'
+DedekindDomain.ProdAdicCompletions.algebra'
+DedekindDomain.ProdAdicCompletions.algebraMap_apply'
+DedekindDomain.ProdAdicCompletions.IsFiniteAdele.algebraMap'
+IsDenseEmbedding.mk'
+Dense.exists_ge'
+Dense.exists_le'
+IsDenseInducing.extend_eq_at'
+IsDenseInducing.mk'
+Denumerable.lower_raise'
+Denumerable.raise_lower'
+deriv_add_const'
+Derivation.apply_aeval_eq'
+Derivation.coe_mk'
+deriv_const'
+deriv_const_add'
+deriv_const_mul_field'
+deriv_id'
+deriv_id''
+deriv_inv'
+deriv_inv''
+deriv_mul_const_field'
+deriv.neg'
+deriv_neg'
+deriv_neg''
+deriv_pow'
+deriv_pow''
+deriv_sqrt_mul_log'
+deriv.star'
+derivWithin_congr_set'
+derivWithin_inv'
+derivWithin_pow'
+deriv_zpow'
+det_traceMatrix_ne_zero'
+DFinsupp.coe_mk'
+DFinsupp.filter_ne_eq_erase'
+DFinsupp.le_iff'
+DFinsupp.Lex.wellFounded'
+DFinsupp.wellFoundedLT'
+DFunLike.ext'
+DiffContOnCl.differentiableAt'
+Diffeomorph.symm_trans'
+DifferentiableAt.comp'
+differentiableAt_id'
+differentiableAt_inv'
+DifferentiableAt.inv'
+differentiableAt_pi''
+Differentiable.comp'
+differentiable_id'
+Differentiable.inv'
+DifferentiableOn.comp'
+differentiableOn_id'
+differentiableOn_inv'
+DifferentiableOn.inv'
+differentiableOn_pi''
+differentiable_pi''
+DifferentiableWithinAt.comp'
+differentiableWithinAt_congr_set'
+differentiableWithinAt_inter'
+differentiableWithinAt_inv'
+DifferentiableWithinAt.inv'
+differentiableWithinAt_pi''
+DirectedOn.mono'
+directedOn_pair'
+DirectSum.Gmodule.mul_smul'
+DirectSum.Gmodule.one_smul'
+DirichletCharacter.level_one'
+DirichletCharacter.toUnitHom_eq_char'
+discreteTopology_iff_orderTopology_of_pred_succ'
+DiscreteTopology.of_forall_le_norm'
+DiscreteTopology.orderTopology_of_pred_succ'
+DiscreteValuationRing.addVal_def'
+Disjoint.inf_left'
+Disjoint.inf_right'
+Disjoint.inter_left'
+Disjoint.inter_right'
+Disjoint.of_disjoint_inf_of_le'
+dist_eq_norm_div'
+dist_le_norm_add_norm'
+dist_midpoint_midpoint_le'
+dist_norm_norm_le'
+dist_partial_sum'
+dist_pi_le_iff'
+DistribMulActionHom.coe_fn_coe'
+dite_eq_iff'
+div_add'
+div_div_cancel'
+div_div_cancel_left'
+div_div_div_cancel_left'
+div_div_self'
+div_eq_iff_eq_mul'
+div_eq_of_eq_mul'
+div_eq_of_eq_mul''
+div_le_div''
+div_le_div_iff'
+div_le_div_left'
+div_le_div_right'
+div_left_inj'
+div_le_iff₀'
+div_le_iff_le_mul'
+div_le_iff_of_neg'
+div_le_one'
+div_lt_div'
+div_lt_div''
+div_lt_div_iff'
+div_lt_div_left'
+div_lt_div_right'
+div_lt_iff'
+div_lt_iff_lt_mul'
+div_lt_iff_of_neg'
+div_lt_one'
+div_mul_div_cancel'
+div_mul_div_cancel₀'
+div_self'
+div_self_mul_self'
+div_sub'
+Doset.disjoint_out'
+Doset.out_eq'
+DoubleCentralizer.nnnorm_def'
+DoubleCentralizer.norm_def'
+dvd_antisymm'
+dvd_geom_sum₂_iff_of_dvd_sub'
+edist_eq_coe_nnnorm'
+EllipticCurve.coe_inv_map_Δ'
+EllipticCurve.coe_inv_variableChange_Δ'
+EllipticCurve.coe_map_Δ'
+EllipticCurve.coe_variableChange_Δ'
+em'
+Embedding.mk'
+EMetric.diam_pos_iff'
+EMetric.diam_union'
+EMetric.mem_ball'
+EMetric.mem_closedBall'
+EMetric.totallyBounded_iff'
+ENat.sSup_eq_zero'
+Encodable.mem_decode₂'
+ENNReal.add_biSup'
+ENNReal.biSup_add'
+ENNReal.biSup_add_biSup_le'
+ENNReal.div_le_iff'
+ENNReal.div_le_of_le_mul'
+ENNReal.div_lt_of_lt_mul'
+ENNReal.exists_frequently_lt_of_liminf_ne_top'
+ENNReal.exists_pos_sum_of_countable'
+ENNReal.iInf_mul_left'
+ENNReal.iInf_mul_right'
+ENNReal.inv_le_inv'
+ENNReal.inv_lt_inv'
+ENNReal.log_pos_real'
+ENNReal.mul_div_cancel'
+ENNReal.mul_le_of_le_div'
+ENNReal.mul_lt_mul_left'
+ENNReal.mul_lt_mul_right'
+ENNReal.mul_lt_of_lt_div'
+ENNReal.mul_top'
+ENNReal.nhds_top'
+ENNReal.ofReal_le_ofReal_iff'
+ENNReal.ofReal_lt_ofReal_iff'
+ENNReal.ofReal_mul'
+ENNReal.range_coe'
+ENNReal.some_eq_coe'
+ENNReal.toNNReal_eq_toNNReal_iff'
+ENNReal.top_mul'
+ENNReal.toReal_eq_toReal_iff'
+ENNReal.toReal_mono'
+ENNReal.toReal_ofReal'
+ENNReal.tsum_eq_iSup_nat'
+ENNReal.tsum_eq_iSup_sum'
+ENNReal.tsum_prod'
+ENNReal.tsum_sigma'
+Eq.cmp_eq_eq'
+eq_div_iff_mul_eq'
+eq_div_iff_mul_eq''
+eq_div_of_mul_eq'
+eq_div_of_mul_eq''
+eq_intCast'
+eq_mul_of_div_eq'
+eq_natCast'
+eq_of_forall_dvd'
+eq_of_prime_pow_eq'
+eqOn_closure₂'
+eq_one_of_inv_eq'
+eq_one_of_mul_left'
+eq_one_of_mul_right'
+eqRec_heq'
+Equiv.bijOn'
+Equiv.coe_piCongr'
+Equiv.exists_congr'
+Equiv.existsUnique_congr'
+Equiv.forall₂_congr'
+Equiv.forall₃_congr'
+Equiv.forall_congr'
+Equiv.inhabited'
+Equiv.lawfulFunctor'
+Equiv.left_inv'
+Equiv.Perm.cycleType_eq'
+Equiv.Perm.exists_fixed_point_of_prime'
+Equiv.Perm.isCycle_of_prime_order'
+Equiv.Perm.isCycle_of_prime_order''
+Equiv.Perm.IsCycleOn.exists_pow_eq'
+Equiv.Perm.IsCycle.pow_eq_one_iff'
+Equiv.Perm.IsCycle.pow_eq_one_iff''
+Equiv.Perm.mem_support_cycleOf_iff'
+Equiv.Perm.prod_comp'
+Equiv.Perm.SameCycle.exists_pow_eq'
+Equiv.Perm.SameCycle.exists_pow_eq''
+Equiv.Perm.signAux_swap_zero_one'
+Equiv.Perm.sign_of_cycleType'
+Equiv.Perm.sign_swap'
+Equiv.right_inv'
+EReal.add_lt_add_of_lt_of_le'
+EReal.coe_neg'
+EReal.nhds_bot'
+EReal.nhds_top'
+EReal.sign_mul_inv_abs'
+essInf_const'
+essSup_const'
+essSup_mono_measure'
+estimator'
+EuclideanDomain.div_add_mod'
+EuclideanDomain.mod_add_div'
+EuclideanDomain.mul_div_cancel'
+EuclideanGeometry.center_eq_inversion'
+EuclideanGeometry.dist_center_eq_dist_center_of_mem_sphere'
+EuclideanGeometry.inversion_dist_center'
+EuclideanGeometry.inversion_eq_center'
+EuclideanGeometry.mem_sphere'
+EuclideanGeometry.Sphere.mem_coe'
+eventually_cobounded_le_norm'
+exists_apply_eq_apply'
+exists_apply_eq_apply2'
+exists_apply_eq_apply3'
+exists_associated_pow_of_mul_eq_pow'
+exists_Ico_subset_of_mem_nhds'
+exists_increasing_or_nonincreasing_subseq'
+exists_Ioc_subset_of_mem_nhds'
+exists_lt_of_lt_ciSup'
+exists_lt_of_lt_csSup'
+exists_maximal_independent'
+exists_one_lt'
+exists_one_lt_mul_of_lt'
+exists_reduced_fraction'
+exists_seq_strictAnti_tendsto'
+exists_seq_strictMono_tendsto'
+exists_square_le'
+exists_sum_eq_one_iff_pairwise_coprime'
+exists_unique_eq'
+existsUnique_zpow_near_of_one_lt'
+extChartAt_preimage_mem_nhds'
+extChartAt_source_mem_nhds'
+extChartAt_source_mem_nhdsWithin'
+extChartAt_target_mem_nhdsWithin'
+ext_nat'
+fderiv_continuousLinearEquiv_comp'
+fderiv_id'
+fderiv_list_prod'
+fderiv_mul'
+fderiv_mul_const'
+fderivWithin_congr'
+fderivWithin_congr_set'
+fderivWithin_eventually_congr_set'
+fderivWithin_id'
+fderivWithin_list_prod'
+fderivWithin_mul'
+fderivWithin_mul_const'
+FDRep.char_tensor'
+FermatLastTheoremWith.fermatLastTheoremWith'
+FiberBundleCore.open_source'
+Field.finInsepDegree_def'
+Field.primitive_element_iff_algHom_eq_of_eval'
+Filter.atBot_basis'
+Filter.atBot_basis_Iio'
+Filter.atTop_basis'
+Filter.atTop_basis_Ioi'
+Filter.bliminf_congr'
+Filter.blimsup_congr'
+Filter.comap_eq_lift'
+Filter.comap_eval_neBot_iff'
+Filter.comap_id'
+Filter.const_eventuallyEq'
+Filter.coprodᵢ_bot'
+Filter.coprodᵢ_eq_bot_iff'
+Filter.coprodᵢ_neBot_iff'
+Filter.countable_biInf_eq_iInf_seq'
+Filter.disjoint_comap_iff_map'
+Filter.eventually_atBot_prod_self'
+Filter.eventually_atTop_prod_self'
+Filter.eventuallyConst_pred'
+Filter.eventuallyConst_set'
+Filter.EventuallyEq.fderivWithin'
+Filter.EventuallyEq.iteratedFDerivWithin'
+Filter.EventuallyLE.mul_le_mul'
+Filter.eventually_smallSets'
+Filter.exists_forall_mem_of_hasBasis_mem_blimsup'
+Filter.ext'
+Filter.extraction_forall_of_eventually'
+Filter.extraction_of_frequently_atTop'
+Filter.frequently_atBot'
+Filter.frequently_atTop'
+Filter.Germ.coe_compTendsto'
+Filter.Germ.coe_smul'
+Filter.Germ.const_compTendsto'
+Filter.Germ.instDistribMulAction'
+Filter.Germ.instModule'
+Filter.Germ.instMulAction'
+Filter.Germ.instSMul'
+Filter.hasBasis_biInf_of_directed'
+Filter.hasBasis_biInf_principal'
+Filter.HasBasis.cauchySeq_iff'
+Filter.hasBasis_cobounded_norm'
+Filter.HasBasis.cobounded_of_norm'
+Filter.HasBasis.eventuallyConst_iff'
+Filter.hasBasis_iInf'
+Filter.hasBasis_iInf_of_directed'
+Filter.HasBasis.inf'
+Filter.HasBasis.lift'
+Filter.HasBasis.nhds'
+Filter.HasBasis.prod_nhds'
+Filter.HasBasis.sup'
+Filter.HasBasis.to_hasBasis'
+Filter.HasBasis.to_image_id'
+Filter.HasBasis.isUniformEmbedding_iff'
+Filter.iInf_neBot_iff_of_directed'
+Filter.iInf_sets_eq_finite'
+Filter.isScalarTower'
+Filter.isScalarTower''
+Filter.le_lift'
+Filter.le_limsup_of_frequently_le'
+Filter.le_pure_iff'
+Filter.lift_lift'_same_eq_lift'
+Filter.lift_lift'_same_le_lift'
+Filter.lift'_mono'
+Filter.lift_mono'
+Filter.liminf_eq_iSup_iInf_of_nat'
+Filter.liminf_le_of_frequently_le'
+Filter.limsup_eq_iInf_iSup_of_nat'
+Filter.map_id'
+Filter.map_inf'
+Filter.map_inv'
+Filter.map_one'
+Filter.map_prod_eq_map₂'
+Filter.mem_bind'
+Filter.mem_coclosed_compact'
+Filter.mem_cocompact'
+Filter.mem_comap'
+Filter.mem_comap''
+Filter.mem_iInf'
+Filter.mem_iInf_finite'
+Filter.mem_inf_principal'
+Filter.mem_lift'
+Filter.mem_map'
+Filter.mem_nhds_iff'
+Filter.mem_pi'
+Filter.mem_rcomap'
+Filter.mono_bliminf'
+Filter.mono_blimsup'
+Filter.monotone_lift'
+Filter.neBot_inf_comap_iff_map'
+Filter.nhds_eq'
+Filter.principal_le_lift'
+Filter.prod_comm'
+Filter.prod_lift'_lift'
+Filter.prod_map_map_eq'
+Filter.ptendsto_of_ptendsto'
+Filter.push_pull'
+Filter.rcomap'_rcomap'
+Filter.sInf_neBot_of_directed'
+Filter.smulCommClass_filter'
+Filter.smulCommClass_filter''
+Filter.tendsto_atBot'
+Filter.tendsto_atBot_add_left_of_ge'
+Filter.tendsto_atBot_add_nonpos_left'
+Filter.tendsto_atBot_add_nonpos_right'
+Filter.tendsto_atBot_add_right_of_ge'
+Filter.tendsto_atBot_mono'
+Filter.tendsto_atBot_of_add_bdd_below_left'
+Filter.tendsto_atBot_of_add_bdd_below_right'
+Filter.tendsto_atTop'
+Filter.tendsto_atTop_add_left_of_le'
+Filter.tendsto_atTop_add_nonneg_left'
+Filter.tendsto_atTop_add_nonneg_right'
+Filter.tendsto_atTop_add_right_of_le'
+Filter.tendsto_atTop_mono'
+Filter.tendsto_atTop_of_add_bdd_above_left'
+Filter.tendsto_atTop_of_add_bdd_above_right'
+Filter.tendsto_congr'
+Filter.Tendsto.congr'
+Filter.Tendsto.const_div'
+Filter.Tendsto.div'
+Filter.Tendsto.div_const'
+Filter.Tendsto.eventually_ne_atTop'
+Filter.tendsto_id'
+Filter.Tendsto.if'
+Filter.tendsto_iff_rtendsto'
+Filter.tendsto_iInf'
+Filter.Tendsto.inf_nhds'
+Filter.tendsto_inv₀_cobounded'
+Filter.tendsto_lift'
+Filter.Tendsto.nnnorm'
+Filter.Tendsto.norm'
+Filter.tendsto_prod_iff'
+Filter.Tendsto.sup_nhds'
+Filter.unbounded_of_tendsto_atBot'
+Filter.unbounded_of_tendsto_atTop'
+Filter.univ_mem'
+Fin.card_filter_univ_succ'
+Fin.castPred_zero'
+Fin.cycleRange_zero'
+Fin.exists_fin_succ'
+Fin.find_min'
+Fin.forall_fin_succ'
+Fin.insertNth_last'
+Fin.insertNth_zero'
+Fin.isEmpty'
+FiniteDimensional.finiteDimensional_pi'
+FiniteField.card'
+Finite.Set.finite_biUnion'
+Fin.last_pos'
+Finmap.ext_iff'
+Fin.mem_piFinset_succ'
+Fin.mul_one'
+Fin.mul_zero'
+Fin.one_mul'
+Fin.one_pos'
+Fin.orderIso_subsingleton'
+Fin.partialProd_succ'
+Finpartition.IsEquipartition.card_biUnion_offDiag_le'
+Finpartition.IsEquipartition.sum_nonUniforms_lt'
+Fin.pred_one'
+Fin.preimage_apply_01_prod'
+Fin.prod_congr'
+finprod_emb_domain'
+finprod_mem_inter_mul_diff'
+finprod_mem_inter_mulSupport_eq'
+Fin.prod_univ_get'
+Fin.prod_univ_two'
+finrank_real_complex_fact'
+finRotate_last'
+Finset.abs_sum_of_nonneg'
+Finset.aemeasurable_prod'
+Finset.aestronglyMeasurable_prod'
+Finset.card_le_card_of_forall_subsingleton'
+Finset.card_mul_le_card_mul'
+Finset.coe_inf'
+Finset.coe_max'
+Finset.coe_min'
+Finset.coe_sup'
+Finset.Colex.toColex_sdiff_le_toColex_sdiff'
+Finset.Colex.toColex_sdiff_lt_toColex_sdiff'
+Finset.decidableMem'
+Finset.disjoint_filter_filter'
+Finset.eq_of_mem_uIcc_of_mem_uIcc'
+Finset.eq_prod_range_div'
+Finset.erase_injOn'
+Finset.exists_le_of_prod_le'
+Finset.exists_lt_of_prod_lt'
+Finset.exists_mem_eq_inf'
+Finset.exists_mem_eq_sup'
+Finset.exists_one_lt_of_prod_one_of_exists_ne_one'
+Finset.expect_boole_mul'
+Finset.expect_dite_eq'
+Finset.expect_ite_eq'
+Finset.extract_gcd'
+Finset.filter_attach'
+Finset.filter_inj'
+Finset.filter_ne'
+Finset.forall_mem_not_eq'
+Finset.Icc_mul_Icc_subset'
+Finset.Icc_mul_Ico_subset'
+Finset.Icc_subset_uIcc'
+Finset.Ici_mul_Ici_subset'
+Finset.Ici_mul_Ioi_subset'
+Finset.Ico_mul_Icc_subset'
+Finset.Ico_mul_Ioc_subset'
+Finset.Ico_union_Ico'
+Finset.Iic_mul_Iic_subset'
+Finset.Iic_mul_Iio_subset'
+Finset.Iio_mul_Iic_subset'
+Finset.image₂_singleton_left'
+Finset.image_id'
+Finset.image_mul_left'
+Finset.image_mul_right'
+Finset.inf'_sup_inf'
+Finset.insert_inj_on'
+Finset.insert_sdiff_insert'
+Finset.insert_val'
+Finset.Ioc_mul_Ico_subset'
+Finset.Ioi_mul_Ici_subset'
+Finset.isGreatest_max'
+Finset.isLeast_min'
+Finset.isScalarTower'
+Finset.isScalarTower''
+Finset.le_inf'
+Finset.le_max'
+Finset.le_min'
+Finset.le_sum_condensed'
+Finset.le_sum_schlomilch'
+Finset.le_sup'
+Finset.lt_max'_of_mem_erase_max'
+Finset.map_filter'
+Finset.max'_eq_sup'
+Finset.measurable_prod'
+Finset.measurable_range_sup'
+Finset.measurable_range_sup''
+Finset.measurable_sup'
+Finset.mem_finsuppAntidiag'
+Finset.mem_inv'
+Finset.mem_map'
+Finset.mem_range_iff_mem_finset_range_of_mod_eq'
+Finset.mem_uIcc'
+Finset.min'_eq_inf'
+Finset.min'_lt_max'
+Finset.min'_lt_of_mem_erase_min'
+Finset.mulEnergy_eq_sum_sq'
+Finset.Nat.antidiagonal_eq_image'
+Finset.Nat.antidiagonal_eq_map'
+Finset.Nat.antidiagonal_succ'
+Finset.Nat.antidiagonal_succ_succ'
+Finset.Nat.prod_antidiagonal_succ'
+Finset.Nat.sum_antidiagonal_succ'
+Finset.nnnorm_prod_le'
+Finset.noncommProd_cons'
+Finset.noncommProd_insert_of_not_mem'
+Finset.Nonempty.csInf_eq_min'
+Finset.Nonempty.csSup_eq_max'
+Finset.norm_prod_le'
+Finset.nsmul_inf'
+Finset.nsmul_sup'
+Finset.ofDual_inf'
+Finset.ofDual_max'
+Finset.ofDual_min'
+Finset.ofDual_sup'
+Finset.one_le_prod'
+Finset.one_le_prod''
+Finset.one_lt_prod'
+Finset.pairwise_cons'
+Finset.pairwise_subtype_iff_pairwise_finset'
+Finset.piecewise_le_piecewise'
+Finset.piecewise_mem_Icc'
+Finset.PiFinsetCoe.canLift'
+Finset.preimage_mul_left_one'
+Finset.preimage_mul_right_one'
+Finset.prod_dite_eq'
+Finset.prod_eq_one_iff'
+Finset.prod_eq_one_iff_of_le_one'
+Finset.prod_eq_one_iff_of_one_le'
+Finset.prod_fiberwise'
+Finset.prod_fiberwise_eq_prod_filter'
+Finset.prod_fiberwise_le_prod_of_one_le_prod_fiber'
+Finset.prod_fiberwise_of_maps_to'
+Finset.prod_finset_product'
+Finset.prod_finset_product_right'
+Finset.prod_Ico_add'
+Finset.prod_image'
+Finset.prod_le_one'
+Finset.prod_le_prod_fiberwise_of_prod_fiber_le_one'
+Finset.prod_le_prod_of_ne_one'
+Finset.prod_le_prod_of_subset'
+Finset.prod_le_prod_of_subset_of_one_le'
+Finset.prod_le_univ_prod_of_one_le'
+Finset.prod_lt_one'
+Finset.prod_lt_prod'
+Finset.prod_lt_prod_of_subset'
+Finset.prod_mono_set'
+Finset.prod_mono_set_of_one_le'
+Finset.prod_pi_mulSingle'
+Finset.prod_preimage'
+Finset.prod_range_div'
+Finset.prod_range_succ'
+Finset.prod_sigma'
+Finset.range_add_one'
+Finset.sdiff_sdiff_left'
+Finset.single_le_prod'
+Finset.single_lt_prod'
+Finset.smulCommClass_finset'
+Finset.smulCommClass_finset''
+Finset.smul_prod'
+Finset.smul_univ₀'
+Finset.sorted_last_eq_max'
+Finset.sorted_zero_eq_min'
+Finset.stronglyMeasurable_prod'
+Finset.subset_singleton_iff'
+Finset.sum_apply'
+Finset.sum_condensed_le'
+Finset.sum_pow'
+Finset.sum_schlomilch_le'
+Finset.sup'_inf_sup'
+Finset.sup_singleton'
+Finset.sup_singleton''
+Finset.toDual_inf'
+Finset.toDual_max'
+Finset.toDual_min'
+Finset.toDual_sup'
+Finset.tprod_subtype'
+Finset.uIcc_subset_uIcc_iff_le'
+Finset.untrop_sum'
+Fin.size_positive'
+Fin.succ_zero_eq_one'
+Finsupp.apply_single'
+Finsupp.card_support_eq_one'
+Finsupp.card_support_le_one'
+Finsupp.equivMapDomain_refl'
+Finsupp.equivMapDomain_trans'
+Finsupp.ext_iff'
+Finsupp.le_iff'
+Finsupp.le_weight_of_ne_zero'
+Finsupp.Lex.wellFounded'
+Finsupp.mapDomain_apply'
+Finsupp.mapRange_add'
+Finsupp.mapRange_neg'
+Finsupp.mapRange_sub'
+Finsupp.mem_supported'
+Finsupp.mulHom_ext'
+Finsupp.smul_single'
+Finsupp.subtypeDomain_eq_zero_iff'
+Finsupp.sum_apply'
+Finsupp.sum_cons'
+Finsupp.sum_ite_self_eq'
+Finsupp.sum_smul_index'
+Finsupp.sum_smul_index_linearMap'
+Finsupp.sum_sum_index'
+Finsupp.support_eq_singleton'
+Finsupp.support_subset_singleton'
+Finsupp.univ_sum_single_apply'
+Finsupp.wellFoundedLT'
+Fintype.card_congr'
+Fintype.card_of_finset'
+Fintype.card_subtype_eq'
+Fintype.expect_dite_eq'
+Fintype.expect_ite_eq'
+Fintype.prod_fiberwise'
+Fintype.prod_mono'
+Fintype.prod_strictMono'
+Fin.univ_image_get'
+Fin.univ_image_getElem'
+Fin.val_one'
+Fin.val_one''
+Fin.zero_mul'
+Fin.zero_ne_one'
+FirstOrder.Language.addEmptyConstants_is_expansion_on'
+FirstOrder.Language.DirectLimit.cg'
+FirstOrder.Language.DirectLimit.funMap_quotient_mk'_sigma_mk'
+FirstOrder.Language.DirectLimit.lift_quotient_mk'_sigma_mk'
+FirstOrder.Language.DirectLimit.relMap_quotient_mk'_sigma_mk'
+FirstOrder.Language.Embedding.codRestrict_apply'
+FirstOrder.Language.funMap_quotient_mk'
+FirstOrder.Language.relMap_quotient_mk'
+FirstOrder.Language.Term.realize_quotient_mk'
+FixedPoints.minpoly.eval₂'
+FixedPoints.smulCommClass'
+forall_apply_eq_imp_iff'
+forall_eq_apply_imp_iff'
+forall_lt_iff_le'
+forall_prop_congr'
+forall_true_iff'
+FormalMultilinearSeries.apply_order_ne_zero'
+FormalMultilinearSeries.comp_coeff_zero'
+FormalMultilinearSeries.order_eq_find'
+FormalMultilinearSeries.order_eq_zero_iff'
+fourier_add'
+fourier_coe_apply'
+fourierIntegral_gaussian_innerProductSpace'
+fourierIntegral_gaussian_pi'
+fourier_neg'
+fourier_zero'
+four_ne_zero'
+FP.Float.sign'
+FractionalIdeal.absNorm_eq'
+FractionalIdeal.coeIdeal_eq_zero'
+FractionalIdeal.coeIdeal_inj'
+FractionalIdeal.coeIdeal_injective'
+FractionalIdeal.coeIdeal_le_coeIdeal'
+FractionalIdeal.coeIdeal_ne_zero'
+FractionalIdeal.inv_zero'
+FreeAbelianGroup.induction_on'
+FreeAbelianGroup.lift.add'
+FreeAbelianGroup.lift_neg'
+FreeGroup.map.id'
+FreeMagma.lift_comp_of'
+FreeMagma.map_mul'
+FreeMagma.traverse_mul'
+FreeMagma.traverse_pure'
+FreeMonoid.countP_of'
+FreeSemigroup.lift_comp_of'
+FreeSemigroup.map_mul'
+FreeSemigroup.traverse_mul'
+FreeSemigroup.traverse_pure'
+frontier_closedBall'
+frontier_Ici'
+frontier_Iic'
+frontier_Iio'
+frontier_Ioi'
+frontier_sphere'
+Function.Antiperiodic.funext'
+Function.Antiperiodic.mul_const'
+Function.Antiperiodic.sub_eq'
+Function.Bijective.of_comp_iff'
+Function.Commute.iterate_pos_le_iff_map_le'
+Function.Commute.iterate_pos_lt_iff_map_lt'
+Function.Commute.iterate_pos_lt_of_map_lt'
+Function.Exact.of_ladder_addEquiv_of_exact'
+Function.Exact.split_tfae'
+Function.extend_apply'
+FunctionField.InftyValuation.map_add_le_max'
+FunctionField.InftyValuation.map_mul'
+FunctionField.InftyValuation.map_one'
+FunctionField.InftyValuation.map_zero'
+Function.Injective.eq_iff'
+Function.Injective.ne_iff'
+Function.Injective.of_comp_iff'
+Function.Injective.surjective_comp_right'
+Function.iterate_succ'
+Function.iterate_succ_apply'
+Function.minimalPeriod_iterate_eq_div_gcd'
+Function.mulSupport_add_one'
+Function.mulSupport_curry'
+Function.mulSupport_inv'
+Function.mulSupport_one'
+Function.mulSupport_one_add'
+Function.mulSupport_one_sub'
+Function.mulSupport_prod_mk'
+Function.mulSupport_subset_iff'
+Function.Periodic.mul_const'
+Function.periodicOrbit_chain'
+Function.Periodic.sub_eq'
+Function.support_div'
+Function.support_inv'
+Function.support_mul'
+Function.support_pow'
+Function.Surjective.of_comp_iff'
+Function.update_comp_eq_of_forall_ne'
+Function.update_comp_eq_of_injective'
+Function.update_comp_eq_of_not_mem_range'
+GaloisCoinsertion.isCoatom_iff'
+GaloisConnection.l_csSup'
+GaloisConnection.l_u_l_eq_l'
+GaloisConnection.u_csInf'
+GaloisConnection.u_l_u_eq_u'
+GaloisInsertion.isAtom_iff'
+gauge_gaugeRescale'
+gauge_lt_eq'
+gauge_zero'
+GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I'
+GaussianInt.toComplex_def'
+gcd_assoc'
+gcd_comm'
+gcd_mul_left'
+gcd_mul_right'
+gcd_neg'
+gcd_one_left'
+gcd_one_right'
+gcd_zero_left'
+gcd_zero_right'
+GenContFract.of_convs_eq_convs'
+ge_of_tendsto'
+geom_sum_Ico'
+geom_sum_pos'
+geom_sum_succ'
+GradedTensorProduct.algebraMap_def'
+gradient_const'
+gradient_eq_deriv'
+gramSchmidt_def'
+gramSchmidt_def''
+gramSchmidtNormed_unit_length'
+gramSchmidtOrthonormalBasis_inv_triangular'
+Group.conjugatesOfSet_subset'
+Group.fg_iff'
+GroupTopology.ext'
+Grp.coe_comp'
+Grp.coe_id'
+Grp.SurjectiveOfEpiAuxs.h_apply_fromCoset'
+Grp.SurjectiveOfEpiAuxs.τ_apply_fromCoset'
+HahnModule.mul_smul'
+HahnModule.one_smul'
+HahnModule.support_smul_subset_vadd_support'
+HahnModule.zero_smul'
+HahnSeries.add_coeff'
+HahnSeries.algebraMap_apply'
+HahnSeries.mul_assoc'
+HahnSeries.mul_coeff_left'
+HahnSeries.mul_coeff_right'
+HahnSeries.neg_coeff'
+HahnSeries.sub_coeff'
+HasCompactMulSupport.intro'
+HasCompactMulSupport.inv'
+HasCompactMulSupport.mono'
+HasDerivAt.complexToReal_fderiv'
+hasDerivAt_exp_smul_const'
+hasDerivAt_exp_smul_const_of_mem_ball'
+HasDerivAtFilter.hasGradientAtFilter'
+HasDerivAt.hasGradientAt'
+hasDerivAt_id'
+hasDerivAt_neg'
+HasDerivWithinAt.complexToReal_fderiv'
+hasDerivWithinAt_congr_set'
+hasDerivWithinAt_iff_tendsto_slope'
+hasDerivWithinAt_inter'
+HasDerivWithinAt.limsup_slope_le'
+hasFDerivAt_exp_smul_const'
+hasFDerivAt_exp_smul_const_of_mem_ball'
+hasFDerivAtFilter_pi'
+hasFDerivAt_list_prod'
+hasFDerivAt_list_prod_attach'
+hasFDerivAt_list_prod_finRange'
+HasFDerivAt.mul'
+HasFDerivAt.mul_const'
+hasFDerivAt_pi'
+hasFDerivAt_pi''
+HasFDerivWithinAt.congr'
+hasFDerivWithinAt_congr_set'
+hasFDerivWithinAt_inter'
+HasFDerivWithinAt.list_prod'
+HasFDerivWithinAt.mul'
+HasFDerivWithinAt.mul_const'
+hasFDerivWithinAt_pi'
+hasFDerivWithinAt_pi''
+HasFiniteFPowerSeriesOnBall.mk'
+hasFPowerSeriesAt_iff'
+HasFPowerSeriesOnBall.factorial_smul'
+hasFTaylorSeriesUpToOn_pi'
+HasFTaylorSeriesUpToOn.zero_eq'
+HasFTaylorSeriesUpTo.zero_eq'
+HasGradientAtFilter.hasDerivAtFilter'
+HasGradientAt.hasDerivAt'
+hasGradientWithinAt_congr_set'
+HasLineDerivWithinAt.congr'
+HasLineDerivWithinAt.hasLineDerivAt'
+HasMFDerivAt.mul'
+hasMFDerivWithinAt_inter'
+HasMFDerivWithinAt.mul'
+HasOrthogonalProjection.map_linearIsometryEquiv'
+hasProd_nat_add_iff'
+HasStrictDerivAt.complexToReal_fderiv'
+hasStrictDerivAt_exp_smul_const'
+hasStrictDerivAt_exp_smul_const_of_mem_ball'
+hasStrictFDerivAt_exp_smul_const'
+hasStrictFDerivAt_exp_smul_const_of_mem_ball'
+hasStrictFDerivAt_list_prod'
+HasStrictFDerivAt.list_prod'
+hasStrictFDerivAt_list_prod_attach'
+hasStrictFDerivAt_list_prod_finRange'
+HasStrictFDerivAt.mul'
+HasStrictFDerivAt.mul_const'
+hasStrictFDerivAt_pi'
+hasStrictFDerivAt_pi''
+hasSum_choose_mul_geometric_of_norm_lt_one'
+hasSum_geometric_two'
+HasSum.matrix_blockDiag'
+HasSum.matrix_blockDiagonal'
+hasSum_sum_range_mul_of_summable_norm'
+Homeomorph.comp_continuousAt_iff'
+Homeomorph.comp_continuous_iff'
+Homeomorph.comp_isOpenMap_iff'
+HomogeneousIdeal.ext'
+HomologicalComplex₂.d₁_eq'
+HomologicalComplex₂.d₁_eq_zero'
+HomologicalComplex₂.d₂_eq'
+HomologicalComplex₂.d₂_eq_zero'
+HomologicalComplex₂.totalAux.d₁_eq'
+HomologicalComplex₂.totalAux.d₂_eq'
+HomologicalComplex.exactAt_iff'
+HomologicalComplex.extend.d_none_eq_zero'
+HomologicalComplex.homotopyCofiber.desc_f'
+HomologicalComplex.homotopyCofiber.ext_from_X'
+HomologicalComplex.homotopyCofiber.ext_to_X'
+HomologicalComplex.homotopyCofiber.inlX_d'
+HomologicalComplex.isZero_extend_X'
+HomologicalComplex.mapBifunctor.d₁_eq'
+HomologicalComplex.mapBifunctor.d₁_eq_zero'
+HomologicalComplex.mapBifunctor.d₂_eq'
+HomologicalComplex.mapBifunctor.d₂_eq_zero'
+HomologicalComplex.restrictionMap_f'
+HomotopyCategory.Pretriangulated.invRotate_distinguished_triangle'
+HomotopyCategory.Pretriangulated.rotate_distinguished_triangle'
+HurwitzZeta.jacobiTheta₂'_functional_equation'
+HurwitzZeta.oddKernel_def'
+Hyperreal.isSt_st'
+Icc_mem_nhdsWithin_Ici'
+Icc_mem_nhdsWithin_Iic'
+Icc_mem_nhdsWithin_Iio'
+Icc_mem_nhdsWithin_Ioi'
+Ico_mem_nhdsWithin_Ici'
+Ico_mem_nhdsWithin_Iio'
+Ico_mem_nhdsWithin_Ioi'
+Ideal.comap_map_of_surjective'
+Ideal.comap_sInf'
+Ideal.eq_jacobson_iff_sInf_maximal'
+Ideal.isJacobson_iff_sInf_maximal'
+Ideal.isJacobson_of_isIntegral'
+Ideal.isMaximal_comap_of_isIntegral_of_isMaximal'
+Ideal.IsMaximal.isPrime'
+Ideal.isMaximal_of_isIntegral_of_isMaximal_comap'
+Ideal.isPrime_ideal_prod_top'
+Ideal.IsPrime.inf_le'
+Ideal.isPrime_of_isPrime_prod_top'
+Ideal.mem_span_insert'
+Ideal.mem_span_singleton'
+Ideal.MvPolynomial.quotient_mk_comp_C_isIntegral_of_jacobson'
+Ideal.Polynomial.isMaximal_comap_C_of_isJacobson'
+Ideal.quotientInfToPiQuotient_mk'
+Ideal.Quotient.smulCommClass'
+Ideal.span_mul_span'
+Ideal.subset_union_prime'
+IfExpr.eval_ite_ite'
+iInf₂_mono'
+iInf_le'
+iInf_mono'
+iInf_prod'
+iInf_psigma'
+iInf_range'
+iInf_sigma'
+iInf_subtype'
+iInf_subtype''
+imageSubobjectIso_imageToKernel'
+Imo1962Q1.ProblemPredicate'
+imo1962_q4'
+Imo1969Q1.not_prime_of_int_mul'
+Imo2001Q2.imo2001_q2'
+imp_or'
+induced_orderTopology'
+Inducing.continuousAt_iff'
+Inducing.isClosed_iff'
+inf_compl_eq_bot'
+inf_eq_half_smul_add_sub_abs_sub'
+inner_map_polarization'
+InnerProductSpaceable.add_left_aux2'
+InnerProductSpaceable.add_left_aux4'
+Inseparable.specializes'
+Int.add_le_zero_iff_le_neg'
+Int.add_nonnneg_iff_neg_le'
+Int.ceil_eq_on_Ioc'
+Int.coprime_of_sq_sum'
+Int.dist_eq'
+integrable_cexp_quadratic'
+integrableOn_Icc_iff_integrableOn_Ico'
+integrableOn_Icc_iff_integrableOn_Ioc'
+integrableOn_Icc_iff_integrableOn_Ioo'
+integrableOn_Ici_iff_integrableOn_Ioi'
+integrableOn_Ico_iff_integrableOn_Ioo'
+integrableOn_Iic_iff_integrableOn_Iio'
+integrableOn_Ioc_iff_integrableOn_Ioo'
+Int.eq_one_or_neg_one_of_mul_eq_neg_one'
+Int.eq_one_or_neg_one_of_mul_eq_one'
+interior_closedBall'
+interior_eq_nhds'
+interior_Ici'
+interior_Iic'
+interior_sphere'
+IntermediateField.algebra'
+IntermediateField.charP'
+IntermediateField.eq_of_le_of_finrank_le''
+IntermediateField.exists_algHom_adjoin_of_splits''
+IntermediateField.exists_algHom_of_splits'
+IntermediateField.exists_finset_of_mem_supr'
+IntermediateField.exists_finset_of_mem_supr''
+IntermediateField.expChar'
+IntermediateField.finInsepDegree_bot'
+IntermediateField.finiteDimensional_iSup_of_finset'
+IntermediateField.finrank_bot'
+IntermediateField.finrank_top'
+IntermediateField.finSepDegree_bot'
+IntermediateField.insepDegree_bot'
+IntermediateField.lift_insepDegree_bot'
+IntermediateField.lift_sepDegree_bot'
+IntermediateField.module'
+IntermediateField.normalClosure_def'
+IntermediateField.normalClosure_def''
+IntermediateField.normal_iff_forall_map_eq'
+IntermediateField.normal_iff_forall_map_le'
+IntermediateField.rank_bot'
+IntermediateField.rank_top'
+IntermediateField.sepDegree_bot'
+intermediate_value_Ico'
+intermediate_value_Ioc'
+intermediate_value_Ioo'
+IntervalIntegrable.aestronglyMeasurable'
+intervalIntegrable_iff'
+IntervalIntegrable.mono_fun'
+IntervalIntegrable.mono_set'
+intervalIntegral.continuous_parametric_intervalIntegral_of_continuous'
+intervalIntegral.integral_congr_ae'
+intervalIntegral.integral_const'
+intervalIntegral.integral_deriv_comp_mul_deriv'
+intervalIntegral.integral_deriv_comp_smul_deriv'
+intervalIntegral.integral_deriv_eq_sub'
+intervalIntegral.integral_interval_sub_interval_comm'
+Int.even_add'
+Int.even_or_odd'
+Int.even_pow'
+Int.even_sub'
+Int.even_xor'_odd'
+Int.exists_gcd_one'
+Int.floor_eq_on_Ico'
+Int.Matrix.exists_ne_zero_int_vec_norm_le'
+Int.ModEq.add_left_cancel'
+Int.ModEq.add_right_cancel'
+Int.ModEq.mul_left'
+Int.ModEq.mul_right'
+Int.natAbs_ofNat'
+Int.odd_add'
+Int.odd_pow'
+Int.odd_sub'
+Int.Prime.dvd_mul'
+Int.Prime.dvd_pow'
+Int.toNat_lt'
+Int.two_pow_sub_pow'
+inv_div'
+inv_le'
+inv_le_div_iff_le_mul'
+inv_le_iff_one_le_mul'
+inv_le_inv'
+inv_lt'
+inv_lt_div_iff_lt_mul'
+inv_lt_iff_one_lt_mul'
+inv_lt_inv'
+inv_mul'
+inv_mul_le_iff'
+inv_mul_le_iff_le_mul'
+inv_mul_lt_iff'
+inv_mul_lt_iff_lt_mul'
+inv_neg'
+inv_neg''
+invOf_mul_cancel_left'
+invOf_mul_cancel_right'
+invOf_mul_self'
+invOf_one'
+inv_pos_le_iff_one_le_mul'
+inv_pos_lt_iff_one_lt_mul'
+inv_zpow'
+Ioc_mem_nhdsWithin_Iic'
+Ioc_mem_nhdsWithin_Iio'
+Ioc_mem_nhdsWithin_Ioi'
+Ioo_mem_nhdsWithin_Iio'
+Ioo_mem_nhdsWithin_Ioi'
+IsAbsoluteValue.abv_one'
+isAddFundamentalDomain_Ioc'
+isAdjointPair_toBilin'
+isAdjointPair_toLinearMap₂'
+IsAlgClosed.algebraMap_surjective_of_isIntegral'
+IsAntichain.eq'
+IsAntichain.interior_eq_empty'
+isArtinian_of_fg_of_artinian'
+isArtinian_submodule'
+IsBaseChange.algHom_ext'
+IsBoundedBilinearMap.isBigO'
+isBounded_iff_forall_norm_le'
+isBoundedUnder_ge_finset_inf'
+isBoundedUnder_le_finset_sup'
+IsCauSeq.bounded'
+isClosed_induced_iff'
+isCoboundedUnder_ge_finset_inf'
+isCoboundedUnder_le_finset_sup'
+IsCompact.elim_nhds_subcover'
+IsCompact.elim_nhds_subcover_nhdsSet'
+IsCompact.exists_bound_of_continuousOn'
+isCompact_iff_ultrafilter_le_nhds'
+IsCompact.tendsto_subseq'
+isComplete_iff_ultrafilter'
+IsCoprime.isUnit_of_dvd'
+IsCyclotomicExtension.neZero'
+IsCyclotomicExtension.Rat.discr_odd_prime'
+IsDedekindDomain.HeightOneSpectrum.adicCompletion.algebra'
+IsDedekindDomain.HeightOneSpectrum.adicCompletion.instIsScalarTower'
+IsDedekindDomain.HeightOneSpectrum.adicValued.has_uniform_continuous_const_smul'
+IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletion'
+isField_of_isIntegral_of_isField'
+IsFractionRing.mk'_num_den'
+IsFractionRing.num_mul_den_eq_num_iff_eq'
+IsGLB.exists_between'
+IsGLB.exists_between_self_add'
+isGLB_inv'
+IsGroupHom.inv_iff_ker'
+IsGroupHom.inv_ker_one'
+IsGroupHom.map_mul'
+IsGroupHom.one_iff_ker_inv'
+IsGroupHom.one_ker_inv'
+IsIntegralClosure.algebraMap_mk'
+isIntegral_localization'
+IsIntegral.minpoly_splits_tower_top'
+IsIntegral.of_mem_closure''
+IsInvariantSubring.coe_subtypeHom'
+IsKleinFour.card_four'
+IsLindelof.elim_nhds_subcover'
+IsLinearMap.isLinearMap_smul'
+IsLocalization.algebraMap_mk'
+IsLocalization.algEquiv_mk'
+IsLocalization.algEquiv_symm_mk'
+IsLocalization.map_id_mk'
+IsLocalization.map_mk'
+IsLocalization.mem_invSubmonoid_iff_exists_mk'
+IsLocalization.mk'_eq_iff_eq'
+IsLocalization.mk'_eq_of_eq'
+IsLocalization.mk'_mul_mk'_eq_one'
+IsLocalization.mk'_self'
+IsLocalization.mk'_self''
+IsLocalization.mk'_spec'
+IsLocalization.ringEquivOfRingEquiv_mk'
+IsLocalization.smul_mk'
+IsLocalization.surj''
+IsLocalization.toInvSubmonoid_eq_mk'
+isLocalizedModule_iff_isLocalization'
+IsLocalizedModule.iso_symm_apply'
+IsLocalizedModule.map_mk'
+IsLocalizedModule.mk'_add_mk'
+IsLocalizedModule.mk'_cancel'
+IsLocalizedModule.mk_eq_mk'
+IsLocalizedModule.mk'_eq_zero'
+IsLocalizedModule.mk'_mul_mk'
+IsLocalizedModule.mk'_sub_mk'
+IsLowerSet.cthickening'
+IsLowerSet.thickening'
+isLUB_csSup'
+IsLUB.exists_between'
+IsLUB.exists_between_sub_self'
+isLUB_hasProd'
+isLUB_inv'
+IsMax.not_isMin'
+IsMin.not_isMax'
+isNoetherian_iff'
+isNoetherian_submodule'
+IsometryEquiv.comp_continuous_iff'
+isOpen_extChartAt_preimage'
+isOpen_gt'
+isOpen_iff_ultrafilter'
+IsOpen.ite'
+isOpen_lt'
+isOpen_pi_iff'
+IsPathConnected.exists_path_through_family'
+IsPGroup.to_sup_of_normal_left'
+IsPGroup.to_sup_of_normal_right'
+IsPreconnected.union'
+IsPrimitiveRoot.card_rootsOfUnity'
+IsPrimitiveRoot.finite_quotient_span_sub_one'
+IsPrimitiveRoot.isPrimitiveRoot_iff'
+IsPrimitiveRoot.isUnit_unit'
+IsPrimitiveRoot.neZero'
+IsPrimitiveRoot.zmodEquivZPowers_symm_apply_pow'
+IsPrimitiveRoot.zmodEquivZPowers_symm_apply_zpow'
+isQuasiregular_iff_isUnit'
+isRegular_iff_ne_zero'
+isRegular_of_ne_zero'
+IsScalarTower.coe_toAlgHom'
+IsScalarTower.subalgebra'
+IsScalarTower.to_smulCommClass'
+IsSelfAdjoint.conjugate'
+isSemisimpleModule_of_isSemisimpleModule_submodule'
+IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'
+IsUnifLocDoublingMeasure.exists_measure_closedBall_le_mul'
+isUnit_iff_exists_inv'
+IsUnit.map'
+IsUnit.val_inv_unit'
+iSup₂_mono'
+iSup_mono'
+iSup_of_empty'
+IsUpperSet.cthickening'
+IsUpperSet.thickening'
+iSup_prod'
+iSup_psigma'
+iSup_range'
+iSup_sigma'
+iSup_subtype'
+iSup_subtype''
+ite_eq_iff'
+iteratedFDeriv_add_apply'
+iteratedFDeriv_const_smul_apply'
+iteratedFDerivWithin_eventually_congr_set'
+iter_deriv_inv'
+iter_deriv_pow'
+iter_deriv_zpow'
+jacobiTheta₂'_add_left'
+KaehlerDifferential.isScalarTower'
+KaehlerDifferential.module'
+LatticeHom.coe_comp_inf_hom'
+LatticeHom.coe_comp_sup_hom'
+LawfulFix.fix_eq'
+lcm_assoc'
+lcm_comm'
+le_abs'
+le_add_tsub'
+Lean.Elab.Tactic.TacticM.runCore'
+le_ciInf_iff'
+le_ciSup_iff'
+le_csInf_iff'
+le_csInf_iff''
+le_csSup_iff'
+le_div_iff₀'
+le_div_iff_mul_le'
+le_div_iff_of_neg'
+LeftOrdContinuous.map_sSup'
+Left.pow_lt_one_iff'
+legendreSym.eq_neg_one_iff'
+legendreSym.eq_one_iff'
+le_hasProd'
+le_iff_exists_mul'
+le_iff_forall_one_lt_lt_mul'
+le_inv'
+le_iSup'
+le_map_add_map_div'
+le_mul_iff_one_le_left'
+le_mul_iff_one_le_right'
+le_mul_of_le_of_one_le'
+le_mul_of_one_le_left'
+le_mul_of_one_le_right'
+le_nhdsAdjoint_iff'
+le_of_eq_of_le'
+le_of_forall_le'
+le_of_forall_lt'
+le_of_forall_one_lt_lt_mul'
+le_of_le_of_eq'
+le_of_mul_le_mul_left'
+le_of_mul_le_mul_right'
+le_of_pow_le_pow_left'
+le_of_tendsto'
+le_of_tendsto_of_tendsto'
+le_tprod'
+le_trans'
+Lex.instDistribMulAction'
+Lex.instDistribSMul'
+Lex.instIsScalarTower'
+Lex.instIsScalarTower''
+Lex.instModule'
+Lex.instMulAction'
+Lex.instMulActionWithZero'
+Lex.instPow'
+Lex.instSMulCommClass'
+Lex.instSMulCommClass''
+Lex.instSMulWithZero'
+LieAlgebra.IsKilling.apply_coroot_eq_cast'
+LieAlgebra.IsKilling.coe_corootSpace_eq_span_singleton'
+LieAlgebra.lieCharacter_apply_lie'
+LieAlgebra.mem_corootSpace'
+LieIdeal.map_sup_ker_eq_map'
+LieModule.chainTop_isNonZero'
+LieModule.coe_chainTop'
+LieModule.genWeightSpaceChain_def'
+LieModule.independent_genWeightSpace'
+LieModule.instIsTrivialOfSubsingleton'
+LieModule.isNilpotent_of_top_iff'
+LieModule.iSup_genWeightSpace_eq_top'
+LieModule.Weight.ext_iff'
+LieSubalgebra.coe_incl'
+LieSubalgebra.ext_iff'
+LieSubalgebra.mem_normalizer_iff'
+LieSubmodule.iSup_induction'
+LieSubmodule.lieIdeal_oper_eq_linear_span'
+LieSubmodule.mem_mk_iff'
+LieSubmodule.module'
+LieSubmodule.Quotient.mk_eq_zero'
+LieSubmodule.Quotient.module'
+LieSubmodule.Quotient.range_mk'
+LieSubmodule.Quotient.surjective_mk'
+LieSubmodule.Quotient.toEnd_comp_mk'
+LieSubmodule.sInf_coe_toSubmodule'
+LieSubmodule.sSup_coe_toSubmodule'
+liftOfDerivationToSquareZero_mk_apply'
+lift_rank_lt_rank_dual'
+LightProfinite.proj_comp_transitionMap'
+LightProfinite.proj_comp_transitionMapLE'
+liminf_finset_inf'
+limsup_finset_sup'
+linearDependent_comp_subtype'
+LinearEquiv.apply_smulCommClass'
+LinearEquiv.coe_toContinuousLinearEquiv'
+LinearEquiv.coe_toContinuousLinearEquiv_symm'
+LinearEquiv.isRegular_congr'
+LinearEquiv.isSMulRegular_congr'
+LinearEquiv.isWeaklyRegular_congr'
+LinearEquiv.mk_coe'
+linearIndependent_algHom_toLinearMap'
+LinearIndependent.cardinal_le_rank'
+linearIndependent_equiv'
+LinearIndependent.eq_zero_of_pair'
+linearIndependent_fin_succ'
+linearIndependent_iff'
+linearIndependent_iff''
+linearIndependent_inl_union_inr'
+linearIndependent_insert'
+linearIndependent_le_span_aux'
+linearIndependent_option'
+LinearIndependent.span_eq_top_of_card_eq_finrank'
+LinearIndependent.to_subtype_range'
+LinearIsometry.completeSpace_map'
+LinearIsometryEquiv.coe_coe''
+LinearIsometryEquiv.comp_fderiv'
+LinearIsometryEquiv.comp_hasFDerivAt_iff'
+LinearIsometryEquiv.comp_hasFDerivWithinAt_iff'
+LinearIsometry.isComplete_image_iff'
+LinearIsometry.isComplete_map_iff'
+LinearIsometry.map_orthogonalProjection'
+LinearMap.apply_smulCommClass'
+LinearMap.BilinForm.mul_toMatrix'
+LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero'
+LinearMap.BilinForm.Nondegenerate.toMatrix'
+LinearMap.BilinForm.toMatrix'_toBilin'
+LinearMap.coe_toContinuousLinearMap'
+LinearMap.detAux_def''
+LinearMap.det_toLin'
+LinearMap.det_toMatrix'
+LinearMap.det_zero'
+LinearMap.det_zero''
+LinearMap.disjoint_ker'
+LinearMap.dualMap_apply'
+LinearMap.extendScalarsOfIsLocalization_apply'
+LinearMap.IsProj.eq_conj_prod_map'
+LinearMap.IsScalarTower.compatibleSMul'
+LinearMap.IsSymmetric.orthogonalComplement_iSup_eigenspaces_eq_bot'
+LinearMap.IsSymmetric.orthogonalFamily_eigenspaces'
+LinearMap.ker_eq_bot'
+LinearMap.ker_smul'
+LinearMap.lcomp_apply'
+LinearMap.llcomp_apply'
+LinearMap.map_le_map_iff'
+LinearMap.minpoly_toMatrix'
+LinearMap.mkContinuous₂_norm_le'
+LinearMap.mul_apply'
+LinearMap.mul_toMatrix'
+LinearMap.ofIsCompl_eq'
+LinearMap.range_smul'
+LinearMap.separatingLeft_toLinearMap₂'_of_det_ne_zero'
+LinearMap.SeparatingLeft.toMatrix₂'
+LinearMap.stdBasis_apply'
+LinearMap.toMatrixAlgEquiv_apply'
+LinearMap.toMatrixAlgEquiv'_toLinAlgEquiv'
+LinearMap.toMatrixAlgEquiv_transpose_apply'
+LinearMap.toMatrix_apply'
+LinearMap.toMatrix'_toLin'
+LinearMap.toMatrix'_toLinearMap₂'
+LinearMap.toMatrix'_toLinearMapₛₗ₂'
+LinearMap.toMatrix_transpose_apply'
+LinearMap.trace_comp_comm'
+LinearMap.trace_conj'
+LinearMap.trace_eq_sum_trace_restrict'
+LinearMap.trace_mul_cycle'
+LinearMap.trace_prodMap'
+LinearMap.trace_tensorProduct'
+LinearMap.trace_transpose'
+LinearOrderedCommGroup.mul_lt_mul_left'
+LinearPMap.closure_def'
+LinearPMap.ext'
+LinearPMap.mem_graph_iff'
+LinearPMap.mem_graph_snd_inj'
+LinearPMap.toFun'
+lineDerivWithin_congr'
+LipschitzOnWith.of_dist_le'
+LipschitzWith.const'
+LipschitzWith.integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul'
+LipschitzWith.nnorm_le_mul'
+LipschitzWith.norm_le_mul'
+LipschitzWith.of_dist_le'
+lipschitzWith_one_nnnorm'
+lipschitzWith_one_norm'
+List.aemeasurable_prod'
+List.aestronglyMeasurable_prod'
+List.alternatingProd_cons'
+List.alternatingProd_cons_cons'
+list_casesOn'
+List.chain'_cons'
+List.Chain'.cons'
+List.chain'_map_of_chain'
+list_cons'
+List.cons_sublist_cons'
+List.count_cons'
+List.decidableChain'
+List.dedup_cons_of_mem'
+List.dedup_cons_of_not_mem'
+List.destutter_cons'
+List.destutter'_is_chain'
+List.destutter_is_chain'
+List.destutter_of_chain'
+List.drop_take_succ_join_eq_get'
+List.exists_le_of_prod_le'
+List.exists_lt_of_prod_lt'
+List.ext_get?'
+List.ext_get?_iff'
+List.filter_attach'
+List.filter_subset'
+list_foldl'
+List.foldl_eq_foldr'
+List.foldl_eq_of_comm'
+List.foldl_fixed'
+List.foldr_eq_of_comm'
+List.foldr_fixed'
+List.Forall₂.prod_le_prod'
+List.getLast_append'
+List.getLast_concat'
+List.getLast_singleton'
+List.get_reverse'
+List.get?_zipWith'
+List.inter_nil'
+List.isRotated_nil_iff'
+List.isRotated_singleton_iff'
+List.LE'
+List.left_unique_forall₂'
+List.le_maximum_of_mem'
+List.length_foldr_permutationsAux2'
+List.length_mergeSort'
+List.length_rotate'
+List.length_sublists'
+List.lookmap_id'
+List.LT'
+List.map₂Left_eq_map₂Left'
+List.map₂Right_eq_map₂Right'
+List.map_filter'
+List.map_mergeSort'
+List.map_permutations'
+List.map_permutationsAux2'
+List.measurable_prod'
+List.mem_destutter'
+List.mem_mergeSort'
+List.mem_permutations'
+List.mem_permutationsAux2'
+List.mem_sublists'
+List.minimum_le_of_mem'
+List.Nat.antidiagonal_succ'
+List.Nat.antidiagonal_succ_succ'
+List.next_cons_cons_eq'
+List.nnnorm_prod_le'
+List.nodup_sublists'
+List.norm_prod_le'
+List.not_lt_maximum_of_mem'
+List.not_lt_minimum_of_mem'
+List.ofFn_succ'
+List.Pairwise.chain'
+List.pairwise_map'
+List.Pairwise.sublists'
+List.perm_mergeSort'
+List.Perm.permutations'
+List.permutations_perm_permutations'
+List.prev_cons_cons_eq'
+List.prev_cons_cons_of_ne'
+List.prev_getLast_cons'
+List.prod_le_prod'
+List.prod_lt_prod'
+List.replicate_right_inj'
+List.replicate_succ'
+list_reverse'
+List.reverse_concat'
+List.reverse_cons'
+List.revzip_sublists'
+List.right_unique_forall₂'
+List.rotate_eq_rotate'
+List.rotate'_rotate'
+Lists'
+Lists.lt_sizeof_cons'
+Lists'.mem_of_subset'
+List.smul_prod'
+List.sorted_mergeSort'
+List.stronglyMeasurable_prod'
+List.SublistForall₂.prod_le_prod'
+List.sublists_eq_sublists'
+List.sublistsLen_sublist_sublists'
+List.sublists_perm_sublists'
+List.support_formPerm_le'
+List.support_formPerm_of_nodup'
+List.takeD_left'
+List.takeI_left'
+List.tendsto_insertNth'
+List.zipLeft_eq_zipLeft'
+List.zipRight_eq_zipRight'
+List.zipWith_swap_prod_support'
+localCohomology.moduleCat_enoughProjectives'
+Localization.algEquiv_mk'
+Localization.algEquiv_symm_mk'
+Localization.Away.mk_eq_monoidOf_mk'
+Localization.epi'
+Localization.liftOn₂_mk'
+Localization.liftOn_mk'
+Localization.localRingHom_mk'
+Localization.mk_eq_mk'
+Localization.mk_eq_mk_iff'
+Localization.mk_eq_monoidOf_mk'
+Localization.mulEquivOfQuotient_mk'
+Localization.mulEquivOfQuotient_symm_mk'
+localization_unit_isIso'
+LocalizedModule.add_assoc'
+LocalizedModule.add_comm'
+LocalizedModule.add_zero'
+LocalizedModule.algebra'
+LocalizedModule.algebraMap_mk'
+LocalizedModule.isModule'
+LocalizedModule.mul_smul'
+LocalizedModule.nsmul_succ'
+LocalizedModule.nsmul_zero'
+LocalizedModule.zero_add'
+LocallyFinite.continuous'
+LocallyFinite.continuousOn_iUnion'
+LocallyFinite.option_elim'
+LocalRing.of_surjective'
+logDeriv_id'
+lowerClosure_interior_subset'
+lp.eq_zero'
+lp.norm_le_of_forall_le'
+lp.norm_nonneg'
+lp.tsum_mul_le_mul_norm'
+LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable'
+lt_div_iff'
+lt_div_iff_mul_lt'
+lt_div_iff_of_neg'
+lt_iff_lt_of_le_iff_le'
+lt_inv'
+lt_inv_iff_mul_lt_one'
+LT.lt.ne'
+lt_mul_iff_one_lt_left'
+lt_mul_iff_one_lt_right'
+lt_mul_of_le_of_one_lt'
+lt_mul_of_lt_of_one_le'
+lt_mul_of_lt_of_one_lt'
+lt_mul_of_one_lt_left'
+lt_mul_of_one_lt_of_lt'
+lt_mul_of_one_lt_right'
+lt_of_eq_of_lt'
+lt_of_le_of_lt'
+lt_of_le_of_ne'
+lt_of_lt_of_eq'
+lt_of_lt_of_le'
+lt_of_mul_lt_mul_left'
+lt_of_mul_lt_mul_right'
+lt_of_pow_lt_pow_left'
+lt_trans'
+mabs_le'
+Magma.AssocQuotient.lift_comp_of'
+MapClusterPt.tendsto_comp'
+map_comp_div'
+map_comp_zpow'
+map_div'
+map_extChartAt_nhds'
+map_extChartAt_nhdsWithin'
+map_extChartAt_nhdsWithin_eq_image'
+map_extChartAt_symm_nhdsWithin'
+map_extChartAt_symm_nhdsWithin_range'
+map_finset_inf'
+map_finset_sup'
+map_natCast'
+map_ofNat'
+map_preNormEDS'
+mapsTo_omegaLimit'
+map_zpow'
+Mathlib.Meta.Finset.range_succ'
+Mathlib.Meta.Finset.range_zero'
+Mathlib.Meta.FunProp.StateList.toList'
+Mathlib.Meta.List.range_succ_eq_map'
+Mathlib.Meta.List.range_zero'
+Mathlib.Meta.Multiset.range_succ'
+Mathlib.Meta.Multiset.range_zero'
+Mathlib.Meta.NormNum.jacobiSymNat.qr₁'
+Mathlib.Meta.Positivity.lt_of_le_of_ne'
+Mathlib.Tactic.ComputeDegree.coeff_pow_of_natDegree_le_of_eq_ite'
+Mathlib.Tactic.ComputeDegree.degree_eq_of_le_of_coeff_ne_zero'
+Mathlib.Tactic.Group.zpow_trick_one'
+Mathlib.Tactic.Ring.atom_pf'
+Mathlib.Util.addAndCompile'
+Mathlib.Vector.eraseIdx_insertNth'
+Mathlib.Vector.prod_set'
+Mathlib.WhatsNew.mkHeader'
+Matrix.blockDiag'_blockDiagonal'
+Matrix.blockDiagonal'_apply'
+Matrix.blockDiagonal_apply'
+Matrix.blockTriangular_blockDiagonal'
+Matrix.blockTriangular_stdBasisMatrix'
+Matrix.blockTriangular_transvection'
+Matrix.cons_val'
+Matrix.cons_val_succ'
+Matrix.cons_val_zero'
+Matrix.det_apply'
+Matrix.det_units_conj'
+Matrix.det_updateColumn_smul'
+Matrix.det_updateRow_smul'
+Matrix.diagonal_apply_ne'
+Matrix.diagonal_intCast'
+Matrix.diagonal_mul_diagonal'
+Matrix.diagonal_natCast'
+Matrix.diagonal_ofNat'
+Matrix.diagonal_toLin'
+Matrix.dotProduct_diagonal'
+Matrix.dotProduct_zero'
+Matrix.empty_val'
+Matrix.exists_mulVec_eq_zero_iff'
+Matrix.exp_blockDiagonal'
+Matrix.exp_conj'
+Matrix.exp_units_conj'
+Matrix.head_val'
+Matrix.induction_on'
+Matrix.inv_pow'
+Matrix.inv_smul'
+Matrix.inv_zpow'
+Matrix.isAdjointPair_equiv'
+Matrix.ker_diagonal_toLin'
+Matrix.kronecker_assoc'
+Matrix.kroneckerTMul_assoc'
+Matrix.map_id'
+Matrix.mem_orthogonalGroup_iff'
+Matrix.mem_unitaryGroup_iff'
+Matrix.minpoly_toLin'
+Matrix.mul_apply'
+Matrix.Nondegenerate.toBilin'
+Matrix.Nondegenerate.toLinearMap₂'
+Matrix.one_apply_ne'
+Matrix.PosDef.of_toQuadraticForm'
+Matrix.PosDef.toQuadraticForm'
+Matrix.pow_inv_comm'
+Matrix.pow_sub'
+Matrix.range_toLin'
+Matrix.represents_iff'
+Matrix.tail_val'
+Matrix.toBilin'_apply'
+Matrix.toBilin'_toMatrix'
+Matrix.toLinAlgEquiv'_toMatrixAlgEquiv'
+Matrix.toLin'_apply'
+Matrix.toLinearMap₂'_apply'
+Matrix.toLinearMap₂'_toMatrix'
+Matrix.toLinearMapₛₗ₂'_toMatrix'
+Matrix.toLin'_toMatrix'
+Matrix.trace_blockDiagonal'
+Matrix.trace_mul_cycle'
+Matrix.twoBlockTriangular_det'
+Matrix.vec2_dotProduct'
+Matrix.vec3_dotProduct'
+Matrix.zero_dotProduct'
+Matrix.zpow_mul'
+Matroid.Base.exchange_base_of_indep'
+Matroid.base_restrict_iff'
+Matroid.Basis.basis'
+Matroid.basis_iff'
+Matroid.basis_iff_basis_closure_of_subset'
+Matroid.basis_restrict_iff'
+Matroid.closure_def'
+Matroid.coindep_iff_exists'
+Matroid.dual_base_iff'
+Matroid.dual_indep_iff_exists'
+Matroid.exists_basis'
+Matroid.Finitary.sum'
+Matroid.Indep.mem_closure_iff'
+Matroid.map_basis_iff'
+Matroid.mapSetEmbedding_indep_iff'
+Matroid.mem_closure_of_mem'
+Matroid.restrictSubtype_dual'
+Matroid.subset_closure_of_subset'
+Matroid.uniqueBaseOn_indep_iff'
+Matroid.uniqueBaseOn_restrict'
+max_def'
+max_div_div_left'
+max_div_div_right'
+max_div_min_eq_mabs'
+maximal_subset_iff'
+max_inv_inv'
+max_mul_mul_le_max_mul_max'
+max_rec'
+mdifferentiableWithinAt_iff'
+mdifferentiableWithinAt_inter'
+Measurable.comp'
+Measurable.comp_aemeasurable'
+Measurable.const_smul'
+Measurable.div'
+MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_deriv_mul'
+Measurable.ennreal_tsum'
+MeasurableEquiv.withDensity_ofReal_map_symm_apply_eq_integral_abs_deriv_mul'
+measurable_findGreatest'
+measurable_from_prod_countable'
+measurable_id'
+measurable_id''
+Measurable.inf'
+Measurable.iSup'
+Measurable.lintegral_kernel_prod_left'
+Measurable.lintegral_kernel_prod_right'
+Measurable.lintegral_kernel_prod_right''
+Measurable.mul'
+measurable_of_isClosed'
+measurable_quotient_mk'
+measurable_quotient_mk''
+measurableSet_eq_fun'
+MeasurableSet.image_inclusion'
+measurableSet_le'
+measurableSet_lt'
+Measurable.sup'
+measurable_to_countable'
+measurable_tProd_elim'
+MeasureTheory.abs_toReal_measure_sub_le_measure_symmDiff'
+MeasureTheory.adapted_predictablePart'
+MeasureTheory.addContent_union'
+MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded'
+MeasureTheory.AECover.integrable_of_lintegral_nnnorm_tendsto'
+MeasureTheory.ae_eq_comp'
+MeasureTheory.ae_eq_dirac'
+MeasureTheory.ae_eq_of_forall_setIntegral_eq_of_sigmaFinite'
+MeasureTheory.ae_eq_trim_iff_of_aeStronglyMeasurable'
+MeasureTheory.ae_lt_top'
+MeasureTheory.aemeasurable_withDensity_ennreal_iff'
+MeasureTheory.ae_restrict_iff'
+MeasureTheory.AEStronglyMeasurable.comp_ae_measurable'
+MeasureTheory.AEStronglyMeasurable.const_smul'
+MeasureTheory.AEStronglyMeasurable.convolution_integrand'
+MeasureTheory.AEStronglyMeasurable.convolution_integrand_snd'
+MeasureTheory.AEStronglyMeasurable.convolution_integrand_swap_snd'
+MeasureTheory.AEStronglyMeasurable'.of_subsingleton'
+MeasureTheory.ae_withDensity_iff'
+MeasureTheory.ae_withDensity_iff_ae_restrict'
+MeasureTheory.average_eq'
+MeasureTheory.condexp_bot'
+MeasureTheory.condexpIndL1Fin_smul'
+MeasureTheory.condexpIndL1_smul'
+MeasureTheory.condexpInd_smul'
+MeasureTheory.condexpIndSMul_smul'
+MeasureTheory.condexpL1CLM_of_aestronglyMeasurable'
+MeasureTheory.condexpL1_of_aestronglyMeasurable'
+MeasureTheory.condexp_of_aestronglyMeasurable'
+MeasureTheory.Content.innerContent_mono'
+MeasureTheory.diracProba_toMeasure_apply'
+MeasureTheory.eLpNorm_add_le'
+MeasureTheory.eLpNorm'_const'
+MeasureTheory.eLpNorm_const'
+MeasureTheory.eLpNorm_eq_eLpNorm'
+MeasureTheory.eLpNorm'_eq_zero_of_ae_zero'
+MeasureTheory.eLpNorm_indicator_const'
+MeasureTheory.eLpNorm'_le_eLpNorm'_mul_eLpNorm'
+MeasureTheory.eLpNorm_nnreal_eq_eLpNorm'
+MeasureTheory.eLpNorm_one_le_of_le'
+MeasureTheory.eLpNorm'_smul_le_mul_eLpNorm'
+MeasureTheory.eLpNorm_sub_le'
+MeasureTheory.eLpNorm'_zero'
+MeasureTheory.eLpNorm_zero'
+MeasureTheory.exp_llr_of_ac'
+MeasureTheory.exp_neg_llr'
+MeasureTheory.Filtration.stronglyMeasurable_limit_process'
+MeasureTheory.hasFiniteIntegral_congr'
+MeasureTheory.HasFiniteIntegral.congr'
+MeasureTheory.HasFiniteIntegral.mono'
+MeasureTheory.hasFiniteIntegral_prod_iff'
+MeasureTheory.HasPDF.congr'
+MeasureTheory.Ico_ae_eq_Icc'
+MeasureTheory.Ico_ae_eq_Ioc'
+MeasureTheory.Iio_ae_eq_Iic'
+MeasureTheory.inducedOuterMeasure_eq'
+MeasureTheory.inducedOuterMeasure_eq_extend'
+MeasureTheory.Integrable.add'
+MeasureTheory.Integrable.bdd_mul'
+MeasureTheory.Integrable.comp_mul_left'
+MeasureTheory.Integrable.comp_mul_right'
+MeasureTheory.integrable_congr'
+MeasureTheory.Integrable.congr'
+MeasureTheory.Integrable.const_mul'
+MeasureTheory.integrable_finset_sum'
+MeasureTheory.Integrable.mono'
+MeasureTheory.Integrable.mul_const'
+MeasureTheory.integrable_of_forall_fin_meas_le'
+MeasureTheory.Integrable.simpleFunc_mul'
+MeasureTheory.Integrable.toL1_smul'
+MeasureTheory.integrable_withDensity_iff_integrable_smul'
+MeasureTheory.integral_add'
+MeasureTheory.integral_countable'
+MeasureTheory.integral_dirac'
+MeasureTheory.integral_Icc_eq_integral_Ico'
+MeasureTheory.integral_Icc_eq_integral_Ioc'
+MeasureTheory.integral_Icc_eq_integral_Ioo'
+MeasureTheory.integral_Ici_eq_integral_Ioi'
+MeasureTheory.integral_Ico_eq_integral_Ioo'
+MeasureTheory.integral_Iic_eq_integral_Iio'
+MeasureTheory.integral_Ioc_eq_integral_Ioo'
+MeasureTheory.integral_neg'
+MeasureTheory.integral_singleton'
+MeasureTheory.integral_sub'
+MeasureTheory.integral_zero'
+MeasureTheory.Ioc_ae_eq_Icc'
+MeasureTheory.Ioi_ae_eq_Ici'
+MeasureTheory.Ioo_ae_eq_Icc'
+MeasureTheory.Ioo_ae_eq_Ico'
+MeasureTheory.Ioo_ae_eq_Ioc'
+MeasureTheory.isClosed_aeStronglyMeasurable'
+MeasureTheory.isComplete_aeStronglyMeasurable'
+MeasureTheory.IsFundamentalDomain.integral_eq_tsum'
+MeasureTheory.IsFundamentalDomain.integral_eq_tsum''
+MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum'
+MeasureTheory.IsFundamentalDomain.lintegral_eq_tsum''
+MeasureTheory.IsFundamentalDomain.measure_eq_tsum'
+MeasureTheory.IsFundamentalDomain.setIntegral_eq_tsum'
+MeasureTheory.IsFundamentalDomain.setLIntegral_eq_tsum'
+MeasureTheory.IsStoppingTime.measurableSet_eq'
+MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable'
+MeasureTheory.IsStoppingTime.measurableSet_eq_of_countable_range'
+MeasureTheory.IsStoppingTime.measurableSet_ge'
+MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable'
+MeasureTheory.IsStoppingTime.measurableSet_ge_of_countable_range'
+MeasureTheory.IsStoppingTime.measurableSet_gt'
+MeasureTheory.IsStoppingTime.measurableSet_le'
+MeasureTheory.IsStoppingTime.measurableSet_lt'
+MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable'
+MeasureTheory.IsStoppingTime.measurableSet_lt_of_countable_range'
+MeasureTheory.IsStoppingTime.measurableSpace_le'
+MeasureTheory.L1.norm_setToL1_le'
+MeasureTheory.L1.norm_setToL1_le_mul_norm'
+MeasureTheory.L1.setToL1_add_left'
+MeasureTheory.L1.setToL1_congr_left'
+MeasureTheory.L1.setToL1_eq_setToL1'
+MeasureTheory.L1.setToL1_mono_left'
+MeasureTheory.L1.setToL1_smul_left'
+MeasureTheory.L1.setToL1_zero_left'
+MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le'
+MeasureTheory.L1.SimpleFunc.setToL1S_add_left'
+MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left'
+MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left'
+MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left'
+MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left'
+MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left'
+MeasureTheory.L1.SimpleFunc.setToL1S_mono_left'
+MeasureTheory.L1.SimpleFunc.setToL1S_smul_left'
+MeasureTheory.L1.SimpleFunc.setToL1S_zero_left'
+MeasureTheory.L2.add_left'
+MeasureTheory.L2.norm_sq_eq_inner'
+MeasureTheory.L2.smul_left'
+MeasureTheory.laverage_eq'
+MeasureTheory.lintegral_add_left'
+MeasureTheory.lintegral_add_right'
+MeasureTheory.lintegral_const_mul'
+MeasureTheory.lintegral_const_mul''
+MeasureTheory.lintegral_count'
+MeasureTheory.lintegral_countable'
+MeasureTheory.lintegral_dirac'
+MeasureTheory.lintegral_eq_zero_iff'
+MeasureTheory.lintegral_finset_sum'
+MeasureTheory.lintegral_iInf'
+MeasureTheory.lintegral_map'
+MeasureTheory.lintegral_mono'
+MeasureTheory.lintegral_mono_fn'
+MeasureTheory.lintegral_mono_set'
+MeasureTheory.lintegral_mul_const'
+MeasureTheory.lintegral_mul_const''
+MeasureTheory.lintegral_rpow_nnnorm_eq_rpow_eLpNorm'
+MeasureTheory.lintegral_singleton'
+MeasureTheory.lintegral_sub'
+MeasureTheory.lintegral_sub_le'
+MeasureTheory.lmarginal_union'
+MeasureTheory.locallyIntegrable_finset_sum'
+MeasureTheory.lowerCrossingTime_stabilize'
+MeasureTheory.Lp.ae_tendsto_of_cauchy_eLpNorm'
+MeasureTheory.Lp.eLpNorm'_lim_le_liminf_eLpNorm'
+MeasureTheory.Lp.eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm'
+MeasureTheory.lpMeas.aeStronglyMeasurable'
+MeasureTheory.Lp.norm_const'
+MeasureTheory.Lp.simpleFunc.eq'
+MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp'
+MeasureTheory.Lp.tendsto_Lp_iff_tendsto_ℒp''
+MeasureTheory.measurableSet_filtrationOfSet'
+MeasureTheory.measurableSet_sigmaFiniteSetWRT'
+MeasureTheory.Measure.ae_sum_iff'
+MeasureTheory.Measure.bind_zero_right'
+MeasureTheory.Measure.count_apply_eq_top'
+MeasureTheory.Measure.count_apply_finite'
+MeasureTheory.Measure.count_apply_finset'
+MeasureTheory.Measure.count_apply_lt_top'
+MeasureTheory.Measure.count_eq_zero_iff'
+MeasureTheory.Measure.count_injective_image'
+MeasureTheory.Measure.count_ne_zero'
+MeasureTheory.Measure.count_ne_zero''
+MeasureTheory.Measure.count_singleton'
+MeasureTheory.measure_diff'
+MeasureTheory.measure_diff_null'
+MeasureTheory.Measure.dirac_apply'
+MeasureTheory.Measure.empty_of_count_eq_zero'
+MeasureTheory.Measure.ext_iff'
+MeasureTheory.Measure.haveLebesgueDecompositionSMul'
+MeasureTheory.Measure.InnerRegularWRT.map'
+MeasureTheory.Measure.integral_toReal_rnDeriv'
+MeasureTheory.measure_inter_conull'
+MeasureTheory.Measure.inv_rnDeriv'
+MeasureTheory.measure_iUnion_null_iff'
+MeasureTheory.Measure.LebesgueDecomposition.iSup_mem_measurableLE'
+MeasureTheory.Measure.LebesgueDecomposition.iSup_monotone'
+MeasureTheory.Measure.le_iff'
+MeasureTheory.Measure.lt_iff'
+MeasureTheory.Measure.map_id'
+MeasureTheory.Measure.measurable_bind'
+MeasureTheory.Measure.MeasureDense.nonempty'
+MeasureTheory.Measure.nonpos_iff_eq_zero'
+MeasureTheory.Measure.pi_noAtoms'
+MeasureTheory.MeasurePreserving.integral_comp'
+MeasureTheory.Measure.restrict_apply₀'
+MeasureTheory.Measure.restrict_apply_eq_zero'
+MeasureTheory.Measure.restrict_restrict'
+MeasureTheory.Measure.restrict_restrict₀'
+MeasureTheory.Measure.restrict_singleton'
+MeasureTheory.Measure.restrict_union'
+MeasureTheory.Measure.restrict_union_add_inter'
+MeasureTheory.Measure.rnDeriv_mul_rnDeriv'
+MeasureTheory.Measure.rnDeriv_pos'
+MeasureTheory.Measure.setIntegral_toReal_rnDeriv'
+MeasureTheory.Measure.setIntegral_toReal_rnDeriv_eq_withDensity'
+MeasureTheory.Measure.setLIntegral_rnDeriv'
+MeasureTheory.Measure.sum_apply_eq_zero'
+MeasureTheory.Measure.toSphere_apply'
+MeasureTheory.Measure.toSphere_apply_univ'
+MeasureTheory.measure_union'
+MeasureTheory.measure_union₀'
+MeasureTheory.measure_union_add_inter'
+MeasureTheory.measure_union_add_inter₀'
+MeasureTheory.memℒp_finset_sum'
+MeasureTheory.Memℒp.integrable_norm_rpow'
+MeasureTheory.Memℒp.meas_ge_lt_top'
+MeasureTheory.mem_lpMeas_iff_aeStronglyMeasurable'
+MeasureTheory.mem_lpMeasSubgroup_iff_aeStronglyMeasurable'
+MeasureTheory.Memℒp.mono'
+MeasureTheory.norm_indicatorConstLp'
+MeasureTheory.norm_setIntegral_le_of_norm_le_const'
+MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae'
+MeasureTheory.norm_setIntegral_le_of_norm_le_const_ae''
+MeasureTheory.norm_setToFun_le'
+MeasureTheory.norm_setToFun_le_mul_norm'
+MeasureTheory.NullMeasurable.measurable'
+MeasureTheory.OuterMeasure.empty'
+MeasureTheory.OuterMeasure.isCaratheodory_iff_le'
+MeasureTheory.OuterMeasure.iUnion_null_iff'
+MeasureTheory.OuterMeasure.le_boundedBy'
+MeasureTheory.OuterMeasure.mono'
+MeasureTheory.OuterMeasure.mono''
+MeasureTheory.OuterMeasure.top_apply'
+MeasureTheory.OuterMeasure.trim_eq_iInf'
+MeasureTheory.pdf.eq_of_map_eq_withDensity'
+MeasureTheory.pdf.quasiMeasurePreserving_hasPDF'
+MeasureTheory.piPremeasure_pi'
+MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto'
+MeasureTheory.ProgMeasurable.finset_prod'
+MeasureTheory.progMeasurable_of_tendsto'
+MeasureTheory.restrict_dirac'
+MeasureTheory.restrict_withDensity'
+MeasureTheory.setAverage_eq'
+MeasureTheory.setIntegral_dirac'
+MeasureTheory.setIntegral_tilted'
+MeasureTheory.setLaverage_eq'
+MeasureTheory.setLIntegral_dirac'
+MeasureTheory.setLIntegral_eq_zero_iff'
+MeasureTheory.setLIntegral_mono'
+MeasureTheory.setLIntegral_mono_ae'
+MeasureTheory.setLIntegral_tilted'
+MeasureTheory.setLIntegral_withDensity_eq_lintegral_mul₀'
+MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀'
+MeasureTheory.setToFun_add_left'
+MeasureTheory.setToFun_congr_left'
+MeasureTheory.setToFun_finset_sum'
+MeasureTheory.setToFun_measure_zero'
+MeasureTheory.setToFun_mono_left'
+MeasureTheory.setToFun_smul_left'
+MeasureTheory.setToFun_zero_left'
+MeasureTheory.sigmaFinite_restrict_sigmaFiniteSetWRT'
+MeasureTheory.SigmaFinite.withDensity_of_ne_top'
+MeasureTheory.SignedMeasure.eq_singularPart'
+MeasureTheory.SignedMeasure.exists_subset_restrict_nonpos'
+MeasureTheory.SignedMeasure.haveLebesgueDecomposition_mk'
+MeasureTheory.SignedMeasure.restrictNonposSeq_disjoint'
+MeasureTheory.SignedMeasure.someExistsOneDivLT_subset'
+MeasureTheory.SimpleFunc.extend_apply'
+MeasureTheory.SimpleFunc.extend_comp_eq'
+MeasureTheory.SimpleFunc.lintegral_eq_of_subset'
+MeasureTheory.SimpleFunc.lintegral_map'
+MeasureTheory.SimpleFunc.setToSimpleFunc_add_left'
+MeasureTheory.SimpleFunc.setToSimpleFunc_congr'
+MeasureTheory.SimpleFunc.setToSimpleFunc_const'
+MeasureTheory.SimpleFunc.setToSimpleFunc_mono_left'
+MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg'
+MeasureTheory.SimpleFunc.setToSimpleFunc_smul_left'
+MeasureTheory.SimpleFunc.setToSimpleFunc_zero'
+MeasureTheory.SimpleFunc.simpleFunc_bot'
+MeasureTheory.stoppedProcess_eq'
+MeasureTheory.stoppedProcess_eq''
+MeasureTheory.stoppedValue_eq'
+MeasureTheory.stoppedValue_piecewise_const'
+MeasureTheory.stoppedValue_sub_eq_sum'
+MeasureTheory.StronglyMeasurable.aeStronglyMeasurable'
+MeasureTheory.StronglyMeasurable.const_smul'
+MeasureTheory.StronglyMeasurable.integral_kernel_prod_left'
+MeasureTheory.StronglyMeasurable.integral_kernel_prod_left''
+MeasureTheory.StronglyMeasurable.integral_kernel_prod_right'
+MeasureTheory.StronglyMeasurable.integral_kernel_prod_right''
+MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le'
+MeasureTheory.Subsingleton.aestronglyMeasurable'
+MeasureTheory.Subsingleton.stronglyMeasurable'
+MeasureTheory.TendstoInMeasure.congr'
+MeasureTheory.TendstoInMeasure.exists_seq_tendsto_ae'
+MeasureTheory.tendsto_sum_indicator_atTop_iff'
+MeasureTheory.tilted_apply'
+MeasureTheory.tilted_apply_eq_ofReal_integral'
+MeasureTheory.tilted_const'
+MeasureTheory.tilted_neg_same'
+MeasureTheory.tilted_zero'
+MeasureTheory.upcrossingsBefore_zero'
+MeasureTheory.upperCrossingTime_stabilize'
+MeasureTheory.upperCrossingTime_zero'
+MeasureTheory.VectorMeasure.ext_iff'
+MeasureTheory.VectorMeasure.le_iff'
+MeasureTheory.weightedSMul_union'
+MeasureTheory.withDensity_apply'
+MeasureTheory.withDensity_apply_eq_zero'
+MeasureTheory.withDensity_smul'
+MeasureTheory.withDensityᵥ_add'
+MeasureTheory.withDensityᵥ_neg'
+MeasureTheory.withDensityᵥ_smul'
+MeasureTheory.withDensityᵥ_smul_eq_withDensityᵥ_withDensity'
+MeasureTheory.withDensityᵥ_sub'
+MeasureTheory.zero_mem_ℒp'
+mem_ball_iff_norm''
+mem_ball_iff_norm'''
+mem_closedBall_iff_norm''
+mem_closedBall_iff_norm'''
+mem_closure_iff_nhds'
+mem_closure_iff_nhds_basis'
+mem_coclosed_Lindelof'
+mem_codiscrete'
+mem_coLindelof'
+memℓp_gen'
+mem_nhds_prod_iff'
+mem_pairSelfAdjointMatricesSubmodule'
+mem_rootsOfUnity'
+mem_rootsOfUnity_prime_pow_mul_iff'
+mem_selfAdjointMatricesSubmodule'
+mem_skewAdjointMatricesSubmodule'
+mem_sphere_iff_norm'
+Metric.ball_eq_ball'
+Metric.ball_subset_ball'
+Metric.closedBall_subset_ball'
+Metric.closedBall_subset_closedBall'
+Metric.closedBall_zero'
+Metric.continuousAt_iff'
+Metric.continuous_iff'
+Metric.continuousOn_iff'
+Metric.continuousWithinAt_iff'
+Metric.cthickening_eq_iInter_cthickening'
+Metric.cthickening_eq_iInter_thickening'
+Metric.cthickening_eq_iInter_thickening''
+Metric.mem_ball'
+Metric.mem_closedBall'
+Metric.mem_of_closed'
+Metric.mem_sphere'
+midpoint_eq_iff'
+min_def'
+min_div_div_left'
+min_div_div_right'
+minimal_subset_iff'
+min_inv_inv'
+min_mul_distrib'
+min_mul_min_le_min_mul_mul'
+minpoly.dvd_map_of_isScalarTower'
+minpoly.eq_X_sub_C'
+minpoly.unique'
+min_rec'
+Miu.le_pow2_and_pow2_eq_mod3'
+mk_eq_mk_of_basis'
+Mod_.comp_hom'
+Mod_.id_hom'
+ModularCyclotomicCharacter.toFun_spec'
+ModularCyclotomicCharacter.toFun_spec''
+ModularCyclotomicCharacter.toFun_unique'
+Module.Baer.ExtensionOfMaxAdjoin.extendIdealTo_wd'
+ModuleCat.CoextendScalars.smul_apply'
+ModuleCat.hasLimits'
+ModuleCat.restrictScalars.smul_def'
+Module.End_algebraMap_isUnit_inv_apply_eq_iff'
+Module.End.smulCommClass'
+Module.free_of_finite_type_torsion_free'
+Module.Free.of_subsingleton'
+Module.mem_support_iff'
+Module.not_mem_support_iff'
+Module.projective_def'
+Monad.mapM'
+Monad.sequence'
+Mon_.comp_hom'
+Mon_.id_hom'
+MonoidAlgebra.lift_apply'
+MonoidAlgebra.lift_unique'
+Monoid.CoprodI.lift_comp_of'
+Monoid.CoprodI.lift_of'
+Monoid.Coprod.induction_on'
+Monoid.exponent_eq_iSup_orderOf'
+Monoid.exponent_min'
+MonoidHom.coe_toAdditive'
+MonoidHom.coe_toAdditive''
+MonoidHom.comap_bot'
+MonoidHom.map_zpow'
+MonoidHom.prod_map_comap_prod'
+Monoid.PushoutI.NormalWord.base_smul_def'
+Monoid.PushoutI.NormalWord.summand_smul_def'
+Monotone.const_mul'
+Monotone.mul_const'
+MonotoneOn.const_mul'
+MonotoneOn.mul_const'
+MulActionHom.comp_inverse'
+MulActionHom.inverse_eq_inverse'
+MulActionHom.inverse'_inverse'
+MulAction.mem_fixedPoints'
+MulAction.mem_stabilizer_finset'
+MulAction.mem_stabilizer_set'
+MulAction.orbitRel.quotient_eq_of_quotient_subgroup_eq'
+MulAction.orbitRel.Quotient.mem_subgroup_orbit_iff'
+MulAction.orbitZPowersEquiv_symm_apply'
+MulAction.Quotient.coe_smul_out'
+MulAction.Quotient.mk_smul_out'
+MulAction.right_quotientAction'
+MulChar.star_apply'
+mul_div_assoc'
+mul_div_cancel_of_imp'
+mul_eq_mul_iff_eq_and_eq_of_pos'
+mul_eq_of_eq_div'
+mul_eq_one'
+MulEquiv.mk_coe'
+MulHom.prod_map_comap_prod'
+mul_inv_le_iff'
+mul_inv_le_iff_le_mul'
+mul_inv_le_mul_inv_iff'
+mul_inv_lt_iff'
+mul_inv_lt_iff_le_mul'
+mul_inv_lt_mul_inv_iff'
+mul_invOf_cancel_left'
+mul_invOf_cancel_right'
+mul_invOf_self'
+mul_left_cancel''
+mul_left_inj'
+mul_le_iff_le_one_left'
+mul_le_iff_le_one_right'
+mul_le_mul'
+mul_le_mul_left'
+mul_le_mul_of_nonneg'
+mul_le_mul_of_nonneg_of_nonpos'
+mul_le_mul_of_nonpos_of_nonneg'
+mul_le_mul_of_nonpos_of_nonpos'
+mul_le_mul_right'
+mul_le_of_le_of_le_one'
+mul_le_of_le_one_left'
+mul_le_of_le_one_of_le'
+mul_le_of_le_one_right'
+mul_lt_iff_lt_one_left'
+mul_lt_iff_lt_one_right'
+mul_lt_mul_left'
+mul_lt_mul_of_pos'
+mul_lt_mul_right'
+mul_lt_of_le_of_lt_one'
+mul_lt_of_le_one_of_lt'
+mul_lt_of_lt_of_le_one'
+mul_lt_of_lt_of_lt_one'
+mul_lt_of_lt_one_left'
+mul_lt_of_lt_one_of_le'
+mul_lt_of_lt_one_of_lt'
+mul_lt_of_lt_one_right'
+mul_ne_one'
+mul_right_cancel''
+mul_right_inj'
+mul_rotate'
+MulSemiringActionHom.coe_fn_coe'
+MultilinearMap.mkContinuousLinear_norm_le'
+MultilinearMap.mkContinuousMultilinear_norm_le'
+Multipliable.sigma'
+Multiplicative.isometricSMul'
+Multiplicative.isometricVAdd''
+multiplicity.is_greatest'
+multiplicity.mul'
+multiplicity.pow'
+multiplicity.unique'
+Multiset.add_le_add_iff_left'
+Multiset.aemeasurable_prod'
+Multiset.aestronglyMeasurable_prod'
+Multiset.antidiagonal_coe'
+Multiset.attach_map_val'
+Multiset.count_sum'
+Multiset.dedup_subset'
+Multiset.ext'
+Multiset.extract_gcd'
+Multiset.filter_attach'
+Multiset.filter_eq'
+Multiset.foldl_induction'
+Multiset.foldr_induction'
+Multiset.induction_on'
+Multiset.map_const'
+Multiset.map_filter'
+Multiset.map_id'
+Multiset.measurable_prod'
+Multiset.Nat.antidiagonal_succ'
+Multiset.Nat.antidiagonal_succ_succ'
+Multiset.noncommProd_cons'
+Multiset.powersetAux_perm_powersetAux'
+Multiset.powersetCard_coe'
+Multiset.powerset_coe'
+Multiset.prod_hom'
+Multiset.prod_lt_prod'
+Multiset.prod_lt_prod_of_nonempty'
+Multiset.prod_map_inv'
+Multiset.prod_X_add_C_coeff'
+Multiset.quot_mk_to_coe'
+Multiset.quot_mk_to_coe''
+Multiset.revzip_powersetAux'
+Multiset.revzip_powersetAux_perm_aux'
+Multiset.smul_prod'
+Multiset.stronglyMeasurable_prod'
+Multiset.subset_dedup'
+MvFunctor.f'
+MvFunctor.g'
+MvFunctor.id_map'
+MvPFunctor.liftP_iff'
+MvPFunctor.M.bisim'
+MvPFunctor.M.dest_corec'
+MvPFunctor.M.dest'_eq_dest'
+MvPFunctor.M.dest_eq_dest'
+MvPFunctor.wDest'_wMk'
+MvPolynomial.aeval_zero'
+MvPolynomial.algHom_ext'
+MvPolynomial.C_mul'
+MvPolynomial.coeff_monomial_mul'
+MvPolynomial.coeff_mul_monomial'
+MvPolynomial.coeff_mul_X'
+MvPolynomial.coeff_X'
+MvPolynomial.coeff_X_mul'
+MvPolynomial.degrees_X'
+MvPolynomial.eval₂_eq'
+MvPolynomial.eval₂Hom_congr'
+MvPolynomial.eval₂Hom_X'
+MvPolynomial.eval₂Hom_zero'
+MvPolynomial.eval_eq'
+MvPolynomial.eval_eq_eval_mv_eval'
+MvPolynomial.eval_zero'
+MvPolynomial.finSuccEquiv_support'
+MvPolynomial.homogeneousComponent_eq_zero'
+MvPolynomial.isLocalization_C_mk'
+MvPolynomial.monomial_zero'
+MvPolynomial.support_esymm'
+MvPolynomial.support_esymm''
+MvPolynomial.weightedHomogeneousComponent_eq_zero'
+MvPowerSeries.algebraMap_apply'
+MvPowerSeries.algebraMap_apply''
+MvPowerSeries.invOfUnit_eq'
+MvQPF.Cofix.dest_corec'
+MvQPF.liftR_map_last'
+MvQPF.recF_eq'
+MvQPF.wEquiv.abs'
+Nat.add_descFactorial_eq_ascFactorial'
+Nat.ascFactorial_eq_factorial_mul_choose'
+Nat.bit_add'
+Nat.card_eq_two_iff'
+Nat.cauchy_induction'
+Nat.choose_eq_asc_factorial_div_factorial'
+Nat.choose_succ_succ'
+Nat.coprime_of_dvd'
+Nat.count_add'
+Nat.count_succ'
+Nat.decreasingInduction_succ'
+Nat.digits_def'
+Nat.digits_zero_succ'
+Nat.dist_tri_left'
+Nat.dist_tri_right'
+Nat.div_add_mod'
+Nat.div_le_of_le_mul'
+Nat.div_le_self'
+Nat.div_lt_iff_lt_mul'
+Nat.dvd_sub'
+Nat.eq_sqrt'
+Nat.eq_sub_of_add_eq'
+Nat.equivProdNatFactoredNumbers_apply'
+Nat.equivProdNatSmoothNumbers_apply'
+Nat.even_add'
+Nat.even_or_odd'
+Nat.even_pow'
+Nat.even_sub'
+Nat.even_xor_odd'
+Nat.exists_mul_self'
+Nat.factorial_inj'
+Nat.find_min'
+Nat.floor_eq_iff'
+Nat.floor_eq_on_Ico'
+Nat.floor_lt'
+Nat.Icc_eq_range'
+Nat.Ico_eq_range'
+Nat.iInf_le_succ'
+Nat.iInf_lt_succ'
+Nat.Ioc_eq_range'
+Nat.Ioo_eq_range'
+Nat.iSup_le_succ'
+Nat.iSup_lt_succ'
+Nat.le_div_iff_mul_le'
+Nat.le_floor_iff'
+Nat.le_minFac'
+Nat.le_nth_count'
+Nat.leRecOn_succ'
+Nat.leRec_succ'
+Nat.le_sqrt'
+Nat.log_eq_one_iff'
+Nat.lt_sub_iff_add_lt'
+Nat.lt_succ_sqrt'
+Nat.mem_primeFactorsList'
+Nat.mod_add_div'
+Nat.ModEq.add_left_cancel'
+Nat.ModEq.add_right_cancel'
+Nat.ModEq.cancel_left_div_gcd'
+Nat.ModEq.cancel_right_div_gcd'
+Nat.modEq_list_prod_iff'
+Nat.ModEq.mul_left'
+Nat.ModEq.mul_left_cancel_iff'
+Nat.ModEq.mul_right'
+Nat.ModEq.mul_right_cancel_iff'
+Nat.monotone_primeCounting'
+Nat.mul_add_mod'
+Nat.mul_div_cancel_left'
+nat_mul_inj'
+Nat.mul_lt_mul''
+Nat.not_exists_sq'
+Nat.not_prime_mul'
+Nat.nth_le_nth'
+Nat.nth_lt_nth'
+Nat.odd_add'
+Nat.odd_sub'
+Nat.ofDigits_modEq'
+Nat.ofDigits_zmodeq'
+Nat.one_le_pow'
+Nat.one_lt_pow'
+Nat.one_lt_two_pow'
+Nat.pair_unpair'
+Nat.Partrec.Code.encode_lt_rfind'
+Nat.Partrec.Code.rec_prim'
+Nat.Partrec'.comp'
+Nat.Partrec.merge'
+Nat.Partrec.prec'
+Nat.Partrec.rfind'
+Nat.pow_lt_ascFactorial'
+Nat.pow_sub_lt_descFactorial'
+Nat.prime_def_lt'
+Nat.prime_def_lt''
+Nat.Prime.eq_two_or_odd'
+Nat.primeFactorsList_chain'
+Nat.Prime.not_prime_pow'
+Nat.Prime.one_lt'
+Nat.Primrec.casesOn'
+Nat.Primrec'.comp'
+Nat.Primrec'.prec'
+Nat.Primrec.swap'
+Nat.prod_divisorsAntidiagonal'
+Nat.rfind_dom'
+Nat.rfind_min'
+Nat.sInf_add'
+Nat.size_shiftLeft'
+Nat.sq_mul_squarefree_of_pos'
+Nat.sqrt_add_eq'
+Nat.sqrt_eq'
+Nat.sqrt_le'
+Nat.sqrt_lt'
+Nat.sqrt_mul_sqrt_lt_succ'
+Nat.sub_eq_of_eq_add'
+Nat.sub_lt_iff_lt_add'
+Nat.succ_le_succ_sqrt'
+Nat.succ_pos'
+Nat.sum_totient'
+Nat.surjective_primeCounting'
+Nat.tendsto_primeCounting'
+Nat.uIcc_eq_range'
+Ne.bot_lt'
+neg_div'
+neg_gcd'
+neg_of_smul_neg_left'
+neg_of_smul_neg_right'
+neg_pow'
+Ne.lt_of_le'
+Ne.lt_top'
+ne_of_irrefl'
+ne_of_ne_of_eq'
+newton_seq_dist_tendsto'
+NeZero.ne'
+NeZero.of_gt'
+ne_zero_of_irreducible_X_pow_sub_C'
+nhds_basis_Ioo'
+nhds_basis_uniformity'
+nhds_def'
+nhds_eq_comap_uniformity'
+nhds_eq_uniformity'
+nhds_left'_sup_nhds_right'
+nhds_left_sup_nhds_right'
+nhds_one_symm'
+nhdsWithin_eq_nhdsWithin'
+nhdsWithin_extChartAt_target_eq'
+nhdsWithin_Ici_basis'
+nhdsWithin_Ici_eq'
+nhdsWithin_Ici_eq''
+nhdsWithin_Iic_basis'
+nhdsWithin_Iic_eq'
+nhdsWithin_Iic_eq''
+nhdsWithin_Iio_basis'
+nhdsWithin_Iio_neBot'
+nhdsWithin_Iio_self_neBot'
+nhdsWithin_inter'
+nhdsWithin_inter_of_mem'
+nhdsWithin_Ioi_basis'
+nhdsWithin_Ioi_neBot'
+nhdsWithin_Ioi_self_neBot'
+nhdsWithin_pi_eq'
+nhdsWithin_restrict'
+nhdsWithin_restrict''
+nndist_eq_nnnorm_vsub'
+nndist_midpoint_midpoint_le'
+nndist_nnnorm_nnnorm_le'
+nnnorm_algebraMap'
+nnnorm_eq_zero'
+nnnorm_inv'
+nnnorm_le_nnnorm_add_nnnorm_div'
+nnnorm_le_pi_nnnorm'
+nnnorm_mul_le'
+nnnorm_ne_zero_iff'
+nnnorm_one'
+nnnorm_pos'
+NNRat.instSMulCommClass'
+NNReal.ball_zero_eq_Ico'
+NNReal.closedBall_zero_eq_Icc'
+NNReal.div_le_iff'
+NNReal.div_le_of_le_mul'
+NNReal.div_lt_iff'
+NNReal.inner_le_Lp_mul_Lq_tsum'
+NNReal.le_div_iff'
+NNReal.list_prod_map_rpow'
+NNReal.Lp_add_le_tsum'
+NNReal.lt_div_iff'
+NNReal.nndist_zero_eq_val'
+NNReal.rpow_add'
+NNReal.rpow_add_intCast'
+NNReal.rpow_add_natCast'
+NNReal.rpow_add_one'
+NNReal.rpow_one_add'
+NNReal.rpow_one_sub'
+NNReal.rpow_sub'
+NNReal.rpow_sub_intCast'
+NNReal.rpow_sub_natCast'
+NNReal.rpow_sub_one'
+NNReal.tendsto_coe'
+NonUnitalAlgHom.coe_inverse'
+NonUnitalAlgHom.coe_restrictScalars'
+NonUnitalStarAlgebra.adjoin_induction'
+NonUnitalStarAlgHom.coe_mk'
+NonUnitalStarAlgHom.coe_restrictScalars'
+NonUnitalStarSubalgebra.instIsScalarTower'
+NonUnitalStarSubalgebra.instSMulCommClass'
+NonUnitalStarSubalgebra.module'
+NonUnitalSubalgebra.instIsScalarTower'
+NonUnitalSubalgebra.instModule'
+NonUnitalSubalgebra.instSMulCommClass'
+NonUnitalSubring.coe_mk'
+NonUnitalSubring.eq_top_iff'
+NonUnitalSubring.mem_mk'
+NonUnitalSubsemiring.coe_mk'
+NonUnitalSubsemiring.eq_top_iff'
+NonUnitalSubsemiring.mem_mk'
+normalClosure_eq_iSup_adjoin'
+norm_algebraMap'
+NormedAddCommGroup.cauchy_series_of_le_geometric'
+NormedAddCommGroup.cauchy_series_of_le_geometric''
+NormedAddGroupHom.coe_mkNormedAddGroupHom'
+NormedAddGroupHom.completion_coe'
+NormedAddGroupHom.norm_comp_le_of_le'
+NormedRing.inverse_one_sub_nth_order'
+NormedSpace.exp_conj'
+NormedSpace.expSeries_apply_eq'
+NormedSpace.expSeries_apply_eq_div'
+NormedSpace.exp_series_hasSum_exp'
+NormedSpace.expSeries_hasSum_exp_of_mem_ball'
+NormedSpace.expSeries_summable'
+NormedSpace.expSeries_summable_of_mem_ball'
+NormedSpace.exp_units_conj'
+NormedSpace.isVonNBounded_iff'
+NormedSpace.norm_expSeries_summable'
+NormedSpace.norm_expSeries_summable_of_mem_ball'
+norm_eq_of_mem_sphere'
+norm_eq_zero''
+norm_eq_zero'''
+norm_inv'
+norm_le_norm_add_const_of_dist_le'
+norm_le_norm_add_norm_div'
+norm_le_of_mem_closedBall'
+norm_le_pi_norm'
+norm_le_zero_iff''
+norm_le_zero_iff'''
+norm_lt_of_mem_ball'
+norm_ne_zero_iff'
+norm_nonneg'
+norm_of_subsingleton'
+norm_one'
+norm_pos_iff''
+norm_pos_iff'''
+norm_sub_norm_le'
+norm_toNNReal'
+not_dvd_index_sylow'
+not_lt_zero'
+not_mem_of_lt_csInf'
+npow_mul'
+nsmul_eq_mul'
+nullMeasurableSet_lt'
+Num.add_ofNat'
+NumberField.InfinitePlace.orbitRelEquiv_apply_mk''
+NumberField.mixedEmbedding.convexBodySumFun_apply'
+NumberField.mixedEmbedding.norm_eq_zero_iff'
+NumberField.Units.regulator_eq_det'
+Num.cast_sub'
+Num.cast_succ'
+Num.cast_zero'
+Num.mem_ofZNum'
+Num.of_to_nat'
+Num.succ_ofInt'
+odd_add_one_self'
+odd_add_self_one'
+ofReal_norm_eq_coe_nnnorm'
+OmegaCompletePartialOrder.const_continuous'
+OmegaCompletePartialOrder.ContinuousHom.bind_continuous'
+OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge'
+OmegaCompletePartialOrder.ContinuousHom.ite_continuous'
+OmegaCompletePartialOrder.ContinuousHom.map_continuous'
+OmegaCompletePartialOrder.ContinuousHom.seq_continuous'
+OmegaCompletePartialOrder.Continuous.of_bundled'
+OmegaCompletePartialOrder.flip₁_continuous'
+OmegaCompletePartialOrder.flip₂_continuous'
+OmegaCompletePartialOrder.id_continuous'
+OmegaCompletePartialOrder.ScottContinuous.continuous'
+one_le_div'
+one_le_finprod'
+one_le_pow_of_one_le'
+one_le_thickenedIndicator_apply'
+one_le_two'
+one_lt_div'
+one_lt_finprod'
+one_lt_pow'
+one_lt_zpow'
+one_ne_zero'
+OnePoint.continuousAt_infty'
+OnePoint.isOpen_iff_of_mem'
+OnePoint.tendsto_nhds_infty'
+ONote.exists_lt_mul_omega0'
+ONote.exists_lt_omega0_opow'
+ONote.fastGrowing_zero'
+ONote.NF.below_of_lt'
+ONote.nf_repr_split'
+ONote.NF.snd'
+ONote.split_eq_scale_split'
+OpenEmbedding.tendsto_nhds_iff'
+openSegment_eq_image'
+openSegment_eq_Ioo'
+Option.bind_congr'
+Option.bind_eq_bind'
+Option.bind_eq_some'
+Option.guard_eq_some'
+Option.map_bind'
+Option.map_coe'
+Option.none_bind'
+Option.none_orElse'
+Option.orElse_eq_none'
+Option.orElse_eq_some'
+Option.orElse_none'
+Option.some_bind'
+Option.some_orElse'
+or_congr_left'
+or_congr_right'
+OrderDual.continuousConstSMul'
+OrderDual.instDistribMulAction'
+OrderDual.instDistribSMul'
+OrderDual.instIsScalarTower'
+OrderDual.instIsScalarTower''
+OrderDual.instModule'
+OrderDual.instMulAction'
+OrderDual.instMulActionWithZero'
+OrderDual.instPow'
+OrderDual.instSMulCommClass'
+OrderDual.instSMulCommClass''
+OrderDual.instSMulWithZero'
+Order.height_le_iff'
+Order.Ideal.IsMaximal.isCoatom'
+OrderIso.isGLB_image'
+OrderIso.isGLB_preimage'
+OrderIso.isLUB_image'
+OrderIso.isLUB_preimage'
+OrderIso.map_bot'
+OrderIso.map_csInf'
+OrderIso.map_csSup'
+OrderIso.map_top'
+OrderIso.subsingleton_of_wellFoundedGT'
+OrderIso.subsingleton_of_wellFoundedLT'
+Order.isPredPrelimitRecOn_pred'
+Order.isSuccPrelimitRecOn_succ'
+Order.not_isPredPrelimit_iff'
+Order.not_isSuccPrelimit_iff'
+orderOf_eq_zero_iff'
+orderOf_pow'
+Ordinal.add_lt_add_iff_left'
+Ordinal.blsub_eq_lsub'
+Ordinal.brange_bfamilyOfFamily'
+Ordinal.bsup_eq_sup'
+Ordinal.cof_eq'
+Ordinal.comp_bfamilyOfFamily'
+Ordinal.comp_familyOfBFamily'
+Ordinal.enum_le_enum'
+Ordinal.enum_zero_le'
+Ordinal.IsNormal.le_set'
+Ordinal.lift_down'
+Ordinal.lift.principalSeg_top'
+Ordinal.liftPrincipalSeg_top'
+Ordinal.lsub_eq_blsub'
+Ordinal.lt_nmul_iff₃'
+Ordinal.mul_eq_zero'
+Ordinal.nhds_right'
+Ordinal.nmul_le_iff₃'
+Ordinal.nmul_nadd_le₃'
+Ordinal.nmul_nadd_lt₃'
+Ordinal.pred_eq_iff_not_succ'
+Ordinal.range_familyOfBFamily'
+Ordinal.relIso_enum'
+Ordinal.succ_le_iff'
+Ordinal.sup_eq_bsup'
+Ordinal.toPGame_moveLeft'
+Ordinal.type_def'
+Ordinal.typein_le_typein'
+Ordinal.type_le_iff'
+Ordinal.zero_opow'
+Ordnode.all_balance'
+Ordnode.all_node'
+Ordnode.balance_eq_balance'
+Ordnode.balanceL_eq_balance'
+Ordnode.balanceR_eq_balance'
+Ordnode.dual_balance'
+Ordnode.dual_node'
+Ordnode.length_toList'
+Ordnode.Raised.dist_le'
+Ordnode.size_balance'
+Ordnode.Sized.balance'
+Ordnode.Sized.eq_node'
+Ordnode.Sized.node'
+Ordnode.Valid'.balance'
+Ordnode.Valid'.node'
+OreLocalization.add'
+OreLocalization.add''
+OreLocalization.div_eq_one'
+OreLocalization.inv'
+OreLocalization.mul_cancel'
+OreLocalization.oreDiv_add_char'
+OreLocalization.smul'
+OreLocalization.smul_cancel'
+OreLocalization.zero_oreDiv'
+Orientation.inner_rightAngleRotation_swap'
+Orientation.kahler_comp_rightAngleRotation'
+Orientation.rightAngleRotation_map'
+Orientation.volumeForm_robust'
+Padic.complete'
+Padic.complete''
+Padic.lim'
+padicNormE.eq_padic_norm'
+padicNormE.image'
+padicNorm.sum_le'
+padicNorm.sum_lt'
+Padic.rat_dense'
+padicValNat_def'
+padicValNat.div'
+PartENat.casesOn'
+PartENat.get_natCast'
+PartENat.get_ofNat'
+PartENat.toWithTop_natCast'
+PartENat.toWithTop_one'
+PartENat.toWithTop_top'
+PartENat.toWithTop_zero'
+Part.eq_none_iff'
+Part.Fix.approx_mono'
+Part.fix_def'
+PartialEquiv.image_source_inter_eq'
+PartialEquiv.symm_image_target_inter_eq'
+PartialEquiv.trans_refl_restr'
+PartialEquiv.trans_source'
+PartialEquiv.trans_source''
+PartialEquiv.trans_target'
+PartialEquiv.trans_target''
+PartialHomeomorph.contDiffWithinAt_extend_coord_change'
+PartialHomeomorph.continuousAt_extend_symm'
+PartialHomeomorph.eventually_left_inverse'
+PartialHomeomorph.eventually_nhds'
+PartialHomeomorph.eventually_nhdsWithin'
+PartialHomeomorph.eventually_right_inverse'
+PartialHomeomorph.extend_coord_change_source_mem_nhdsWithin'
+PartialHomeomorph.extend_target'
+PartialHomeomorph.image_source_inter_eq'
+PartialHomeomorph.IsImage.iff_preimage_eq'
+PartialHomeomorph.IsImage.iff_symm_preimage_eq'
+PartialHomeomorph.isOpen_extend_preimage'
+PartialHomeomorph.ofSet_trans'
+PartialHomeomorph.prod_eq_prod_of_nonempty'
+PartialHomeomorph.restr_source'
+PartialHomeomorph.restr_toPartialEquiv'
+PartialHomeomorph.trans_of_set'
+PartialHomeomorph.trans_source'
+PartialHomeomorph.trans_source''
+PartialHomeomorph.trans_target'
+PartialHomeomorph.trans_target''
+PartitionOfUnity.exists_finset_nhd'
+PartitionOfUnity.sum_finsupport'
+Part.map_id'
+Partrec₂.unpaired'
+Partrec.const'
+Partrec.merge'
+PathConnectedSpace.exists_path_through_family'
+Path.extend_extends'
+pcontinuous_iff'
+Pell.eq_of_xn_modEq'
+Perfection.coeff_iterate_frobenius'
+Perfection.coeff_pow_p'
+PerfectionMap.comp_equiv'
+PerfectionMap.comp_symm_equiv'
+PFunctor.Approx.head_succ'
+PFunctor.liftp_iff'
+PFunctor.M.agree_iff_agree'
+PFunctor.M.bisim'
+PFunctor.M.casesOn_mk'
+PFunctor.M.ext'
+PFunctor.M.head_eq_head'
+PFunctor.M.isPath_cons'
+Pi.compact_Icc_space'
+Pi.continuous_postcomp'
+Pi.continuous_precomp'
+Pi.cstarRing'
+Pi.distribMulAction'
+Pi.distribSMul'
+pi_Icc_mem_nhds'
+pi_Ici_mem_nhds'
+pi_Ico_mem_nhds'
+pi_Iic_mem_nhds'
+pi_Iio_mem_nhds'
+Pi.induced_precomp'
+Pi.infConvergenceClass'
+Pi.instBoundedSMul'
+pi_Ioc_mem_nhds'
+pi_Ioi_mem_nhds'
+pi_Ioo_mem_nhds'
+Pi.isometricSMul'
+Pi.isometricSMul''
+Pi.isScalarTower'
+Pi.isScalarTower''
+Pi.lawfulFix'
+Pi.Lex.noMaxOrder'
+Pi.module'
+Pi.mulAction'
+Pi.mulActionWithZero'
+Pi.mulDistribMulAction'
+pinGroup.star_eq_inv'
+pi_nnnorm_const'
+pi_nnnorm_const_le'
+Pi.nnnorm_def'
+pi_nnnorm_le_iff'
+pi_nnnorm_lt_iff'
+pi_norm_const'
+pi_norm_const_le'
+Pi.norm_def'
+pi_norm_le_iff_of_nonempty'
+Pi.orderClosedTopology'
+Pi.smul'
+Pi.smul_apply'
+Pi.smulCommClass'
+Pi.smulCommClass''
+Pi.smul_def'
+Pi.smulWithZero'
+Pi.smulZeroClass'
+PiSubtype.canLift'
+Pi.supConvergenceClass'
+PiTensorProduct.add_tprodCoeff'
+PiTensorProduct.distribMulAction'
+PiTensorProduct.hasSMul'
+PiTensorProduct.isScalarTower'
+PiTensorProduct.lift.unique'
+PiTensorProduct.module'
+PiTensorProduct.smulCommClass'
+PiTensorProduct.smul_tprodCoeff'
+PiTensorProduct.zero_tprodCoeff'
+Pi.uniformContinuous_postcomp'
+Pi.uniformContinuous_precomp'
+Pi.uniformSpace_comap_precomp'
+PNat.coe_toPNat'
+PNat.div_add_mod'
+PNat.dvd_iff'
+PNat.factorMultiset_le_iff'
+PNat.find_min'
+PNat.gcd_rel_left'
+PNat.gcd_rel_right'
+PNat.mod_add_div'
+PNat.XgcdType.reduce_isReduced'
+PNat.XgcdType.reduce_isSpecial'
+pNilradical_eq_bot'
+Pointed.Hom.comp_toFun'
+Pointed.Hom.id_toFun'
+Polynomial.add'
+Polynomial.addHom_ext'
+Polynomial.aeval_apply_smul_mem_of_le_comap'
+Polynomial.aeval_eq_sum_range'
+Polynomial.as_sum_range'
+Polynomial.card_roots'
+Polynomial.card_roots_sub_C'
+Polynomial.card_support_eq'
+Polynomial.card_support_eraseLead'
+Polynomial.C_mul'
+Polynomial.coeff_expand_mul'
+Polynomial.coeff_mul_X_pow'
+Polynomial.coeff_restriction'
+Polynomial.coeff_toSubring'
+Polynomial.coeff_X_pow_mul'
+Polynomial.coeff_zero_eq_aeval_zero'
+Polynomial.degree_eq_card_roots'
+Polynomial.degree_mul'
+Polynomial.degree_pow'
+Polynomial.div_tendsto_atBot_of_degree_gt'
+Polynomial.div_tendsto_atTop_of_degree_gt'
+Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero'
+Polynomial.eval₂_comp'
+Polynomial.eval₂_eq_sum_range'
+Polynomial.eval₂_mul'
+Polynomial.eval₂_mul_C'
+Polynomial.eval₂_pow'
+Polynomial.eval_eq_sum_range'
+Polynomial.eval_smul'
+Polynomial.exists_root_of_splits'
+Polynomial.expand_contract'
+Polynomial.hasseDeriv_one'
+Polynomial.hasseDeriv_zero'
+Polynomial.HasSeparableContraction.dvd_degree'
+Polynomial.hermite_eq_deriv_gaussian'
+Polynomial.isRoot_cyclotomic_iff'
+Polynomial.isUnit_iff'
+Polynomial.isUnitTrinomial_iff'
+Polynomial.isUnitTrinomial_iff''
+Polynomial.leadingCoeff_add_of_degree_lt'
+Polynomial.leadingCoeff_map'
+Polynomial.leadingCoeff_mul'
+Polynomial.leadingCoeff_pow'
+Polynomial.leadingCoeff_sub_of_degree_lt'
+Polynomial.lhom_ext'
+Polynomial.lt_rootMultiplicity_iff_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
+Polynomial.lt_rootMultiplicity_of_isRoot_iterate_derivative_of_mem_nonZeroDivisors'
+Polynomial.map_dvd_map'
+Polynomial.map_rootOfSplits'
+Polynomial.mem_aroots'
+Polynomial.mem_roots'
+Polynomial.mem_rootSet'
+Polynomial.mem_roots_sub_C'
+Polynomial.mkDerivation_one_eq_derivative'
+PolynomialModule.eval_map'
+PolynomialModule.isScalarTower'
+Polynomial.Monic.geom_sum'
+Polynomial.Monic.irreducible_iff_natDegree'
+Polynomial.Monic.natDegree_mul'
+Polynomial.monic_zero_iff_subsingleton'
+Polynomial.mul'
+Polynomial.mul_scaleRoots'
+Polynomial.natDegree_eq_card_roots'
+Polynomial.natDegree_eq_support_max'
+Polynomial.natDegree_mul'
+Polynomial.natDegree_pow'
+Polynomial.natDegree_removeFactor'
+Polynomial.natTrailingDegree_eq_support_min'
+Polynomial.natTrailingDegree_mul'
+Polynomial.neg'
+Polynomial.ringHom_ext'
+Polynomial.rootMultiplicity_eq_natTrailingDegree'
+Polynomial.rootMultiplicity_mul'
+Polynomial.rootMultiplicity_pos'
+Polynomial.rootSet_maps_to'
+Polynomial.roots_ne_zero_of_splits'
+Polynomial.scaleRoots_dvd'
+Polynomial.separable_def'
+Polynomial.Separable.of_pow'
+Polynomial.separable_prod'
+Polynomial.separable_prod_X_sub_C_iff'
+polynomial_smul_apply'
+Polynomial.splits_of_splits_mul'
+Polynomial.SplittingField.algebra'
+Polynomial.SplittingFieldAux.algebra'
+Polynomial.SplittingFieldAux.algebra''
+Polynomial.SplittingFieldAux.algebra'''
+Polynomial.SplittingFieldAux.scalar_tower'
+Polynomial.sum_add'
+Polynomial.sum_smul_index'
+Polynomial.support_binomial'
+Polynomial.support_C_mul_X'
+Polynomial.support_C_mul_X_pow'
+Polynomial.support_monomial'
+Polynomial.support_trinomial'
+Polynomial.taylor_zero'
+Polynomial.trailingDegree_mul'
+Polynomial.trinomial_leading_coeff'
+Polynomial.trinomial_trailing_coeff'
+PosNum.cast_one'
+PosNum.cast_sub'
+PosNum.of_to_nat'
+PosNum.one_sub'
+PosNum.pred'_succ'
+PosNum.succ'_pred'
+pow_add_pow_le'
+pow_card_eq_one'
+pow_eq_zero_iff'
+PowerBasis.exists_eq_aeval'
+PowerBasis.mem_span_pow'
+PowerSeries.algebraMap_apply'
+PowerSeries.algebraMap_apply''
+PowerSeries.algebraPolynomial'
+PowerSeries.coeff_mul_X_pow'
+PowerSeries.coeff_X_pow_mul'
+PowerSeries.derivative_inv'
+PowerSeries.invOfUnit_eq'
+PowerSeries.trunc_derivative'
+PowerSeries.trunc_zero'
+pow_le_one'
+pow_le_pow_iff_right'
+pow_le_pow_left'
+pow_le_pow_right'
+pow_le_pow_right_of_le_one'
+pow_lt_one'
+pow_lt_pow_iff_right'
+pow_lt_pow_left'
+pow_lt_pow_right'
+pow_mul'
+pow_mul_comm'
+pow_right_strictMono'
+pow_succ'
+pow_three'
+ppow_mul'
+PProd.exists'
+PProd.forall'
+PredOrder.prelimitRecOn_pred'
+preimage_nhdsWithin_coinduced'
+PresheafOfModules.sheafificationHomEquiv_hom'
+Pretrivialization.apply_symm_apply'
+Pretrivialization.coe_fst'
+Pretrivialization.continuousLinearMap_symm_apply'
+Pretrivialization.ext'
+Pretrivialization.mk_proj_snd'
+Pretrivialization.proj_symm_apply'
+PrimeMultiset.prod_dvd_iff'
+PrimeSpectrum.iSup_basicOpen_eq_top_iff'
+Primrec₂.nat_iff'
+Primrec₂.unpaired'
+Primrec.nat_casesOn'
+Primrec.nat_omega_rec'
+Primrec.nat_rec'
+Primrec.vector_get'
+Primrec.vector_ofFn'
+PrincipalSeg.coe_coe_fn'
+ProbabilityTheory.centralMoment_one'
+ProbabilityTheory.cgf_const'
+ProbabilityTheory.cgf_zero'
+ProbabilityTheory.cond_apply'
+ProbabilityTheory.cond_cond_eq_cond_inter'
+ProbabilityTheory.condCount_inter'
+ProbabilityTheory.condexp_ae_eq_integral_condexpKernel'
+ProbabilityTheory.condexpKernel_ae_eq_condexp'
+ProbabilityTheory.CondIndepSets.condIndep'
+ProbabilityTheory.cond_mul_eq_inter'
+ProbabilityTheory.evariance_def'
+ProbabilityTheory.gaussianReal_absolutelyContinuous'
+ProbabilityTheory.hasFiniteIntegral_compProd_iff'
+ProbabilityTheory.iIndep.iIndepSets'
+ProbabilityTheory.IndepFun.integral_mul'
+ProbabilityTheory.IndepFun.mgf_add'
+ProbabilityTheory.IndepSets.indep'
+ProbabilityTheory.IsMarkovKernel.is_probability_measure'
+ProbabilityTheory.IsMeasurableRatCDF.stieltjesFunctionAux_def'
+ProbabilityTheory.Kernel.borelMarkovFromReal_apply'
+ProbabilityTheory.Kernel.comap_apply'
+ProbabilityTheory.Kernel.comap_id'
+ProbabilityTheory.Kernel.comapRight_apply'
+ProbabilityTheory.Kernel.comp_apply'
+ProbabilityTheory.Kernel.const_comp'
+ProbabilityTheory.Kernel.deterministic_apply'
+ProbabilityTheory.Kernel.ext_iff'
+ProbabilityTheory.Kernel.finset_sum_apply'
+ProbabilityTheory.Kernel.fst_apply'
+ProbabilityTheory.Kernel.iIndep.iIndepSets'
+ProbabilityTheory.Kernel.IndepSets.indep'
+ProbabilityTheory.Kernel.integral_deterministic'
+ProbabilityTheory.Kernel.integral_integral_add'
+ProbabilityTheory.Kernel.integral_integral_sub'
+ProbabilityTheory.Kernel.lintegral_deterministic'
+ProbabilityTheory.Kernel.map_apply'
+ProbabilityTheory.Kernel.map_id'
+ProbabilityTheory.Kernel.measurable_kernel_prod_mk_left'
+ProbabilityTheory.Kernel.measure_eq_zero_or_one_of_indepSet_self'
+ProbabilityTheory.Kernel.piecewise_apply'
+ProbabilityTheory.Kernel.prod_apply'
+ProbabilityTheory.Kernel.prodMkLeft_apply'
+ProbabilityTheory.Kernel.prodMkRight_apply'
+ProbabilityTheory.Kernel.restrict_apply'
+ProbabilityTheory.Kernel.rnDeriv_def'
+ProbabilityTheory.Kernel.rnDeriv_eq_top_iff'
+ProbabilityTheory.Kernel.setIntegral_deterministic'
+ProbabilityTheory.Kernel.setLIntegral_deterministic'
+ProbabilityTheory.Kernel.snd_apply'
+ProbabilityTheory.Kernel.sum_apply'
+ProbabilityTheory.Kernel.swapLeft_apply'
+ProbabilityTheory.Kernel.swapRight_apply'
+ProbabilityTheory.Kernel.withDensity_apply'
+ProbabilityTheory.Kernel.withDensity_one'
+ProbabilityTheory.Kernel.withDensity_zero'
+ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun''
+ProbabilityTheory.measurable_preCDF'
+ProbabilityTheory.mgf_const'
+ProbabilityTheory.mgf_pos'
+ProbabilityTheory.mgf_zero'
+ProbabilityTheory.variance_def'
+ProbabilityTheory.variance_smul'
+Prod.exists'
+Prod.forall'
+Prod.isometricSMul'
+Prod.isometricSMul''
+Prod.map_apply'
+Prod.map_fst'
+Prod.map_id'
+Prod.map_snd'
+prod_mul_tprod_nat_mul'
+Profinite.NobelingProof.coe_πs'
+Profinite.NobelingProof.contained_C'
+Profinite.NobelingProof.injective_πs'
+Profinite.NobelingProof.Products.eval_πs'
+Profinite.NobelingProof.Products.eval_πs_image'
+Profinite.NobelingProof.Products.max_eq_o_cons_tail'
+Projectivization.submodule_mk''
+Prop.countable'
+QPF.Cofix.bisim'
+QPF.liftp_iff'
+QPF.recF_eq'
+QPF.Wequiv.abs'
+quadraticChar_eq_pow_of_char_ne_two'
+QuadraticForm.equivalent_weightedSumSquares_units_of_nondegenerate'
+QuadraticForm.posDef_of_toMatrix'
+QuadraticForm.posDef_toMatrix'
+QuadraticMap.isSymm_toMatrix'
+QuadraticMap.map_sum'
+quasiIsoAt_iff'
+quasiIsoAt_iff_exactAt'
+QuaternionAlgebra.self_add_star'
+QuaternionAlgebra.star_add_self'
+Quaternion.normSq_def'
+Quaternion.self_add_star'
+Quaternion.star_add_self'
+Quiver.Hom.unop_op'
+Quiver.Path.comp_inj'
+QuotientAddGroup.btw_coe_iff'
+Quotient.eq'
+Quotient.eq''
+Quotient.exact'
+QuotientGroup.coe_mk'
+QuotientGroup.congr_mk'
+QuotientGroup.kerLift_mk'
+QuotientGroup.ker_mk'
+QuotientGroup.lift_mk'
+QuotientGroup.map_mk'
+QuotientGroup.mk'_eq_mk'
+QuotientGroup.out_eq'
+Quotient.hrecOn₂'_mk''
+Quotient.hrecOn'_mk''
+Quotient.liftOn₂'_mk''
+Quotient.liftOn'_mk''
+Quotient.map₂'_mk''
+Quotient.map'_mk''
+quotientMap_quotient_mk'
+Quotient.mk_out'
+Quotient.out_eq'
+Quotient.sound'
+Quotient.surjective_liftOn'
+range_pow_padicValNat_subset_divisors'
+rank_finsupp'
+rank_fun'
+rank_lt_rank_dual'
+Rat.add_def''
+Rat.add_num_den'
+Rat.cast_mk'
+Rat.div_def'
+Rat.divInt_mul_divInt'
+Rat.divInt_self'
+Rat.floor_def'
+RatFunc.liftAlgHom_apply_div'
+RatFunc.liftMonoidWithZeroHom_apply_div'
+RatFunc.liftRingHom_apply_div'
+RatFunc.mk_eq_div'
+RatFunc.mk_eq_mk'
+RatFunc.mk_one'
+RatFunc.num_div'
+RatFunc.ofFractionRing_mk'
+Rat.instSMulCommClass'
+Rat.inv_def'
+Rat.inv_divInt'
+Rat.le_toNNRat_iff_coe_le'
+Rat.mk'_mul_mk'
+Rat.mul_num_den'
+Rat.normalize_eq_mk'
+Rat.sub_def''
+Rat.substr_num_den'
+Rat.toNNRat_div'
+Rat.toNNRat_lt_toNNRat_iff'
+RCLike.hasSum_conj'
+RCLike.I_im'
+RCLike.normSq_eq_def'
+RCLike.zero_re'
+Real.arcsin_le_iff_le_sin'
+Real.arcsin_lt_iff_lt_sin'
+Real.arcsin_sin'
+Real.binEntropy_eq_negMulLog_add_negMulLog_one_sub'
+Real.b_ne_one'
+Real.coe_toNNReal'
+Real.continuousAt_const_rpow'
+Real.continuous_log'
+Real.cosh_sq'
+Real.cos_sq'
+Real.cos_two_mul'
+Real.deriv_cos'
+Real.deriv_log'
+Real.deriv_rpow_const'
+Real.eulerMascheroniConstant_lt_eulerMascheroniSeq'
+Real.eulerMascheroniSeq_lt_eulerMascheroniSeq'
+Real.exp_approx_end'
+Real.exp_bound'
+Real.exp_bound_div_one_sub_of_interval'
+Real.fourierIntegral_continuousLinearMap_apply'
+Real.fourierIntegral_continuousMultilinearMap_apply'
+Real.fourierIntegral_eq'
+Real.fourierIntegralInv_eq'
+Real.hasDerivAt_arctan'
+Real.inner_le_Lp_mul_Lq_tsum_of_nonneg'
+Real.le_arcsin_iff_sin_le'
+Real.le_def'
+Real.le_sqrt'
+Real.le_toNNReal_iff_coe_le'
+Real.list_prod_map_rpow'
+Real.logb_nonpos_iff'
+Real.log_nonpos_iff'
+Real.Lp_add_le_tsum_of_nonneg'
+Real.lt_arcsin_iff_sin_lt'
+Real.natCastle_toNNReal'
+Real.nndist_eq'
+Real.rpow_add'
+Real.rpow_add_intCast'
+Real.rpow_add_natCast'
+Real.rpow_add_one'
+Real.rpow_le_rpow_of_exponent_ge'
+Real.rpow_lt_one_iff'
+Real.rpow_one_add'
+Real.rpow_one_sub'
+Real.rpow_sub'
+Real.rpow_sub_intCast'
+Real.rpow_sub_natCast'
+Real.rpow_sub_one'
+Real.sin_arcsin'
+Real.sqrt_div'
+Real.sqrt_div_self'
+Real.sqrt_eq_zero'
+Real.sqrt_le_sqrt_iff'
+Real.sqrt_lt'
+Real.sqrt_mul'
+Real.sqrt_ne_zero'
+Real.strictAnti_eulerMascheroniSeq'
+Real.surjOn_log'
+Real.surjOn_logb'
+Real.tan_add'
+Real.tan_eq_zero_iff'
+Real.tendsto_eulerMascheroniSeq'
+Real.tendsto_integral_gaussian_smul'
+Real.toNNReal_div'
+Real.toNNReal_le_toNNReal_iff'
+Real.toNNReal_lt_natCast'
+Real.toNNReal_lt_toNNReal_iff'
+RegularExpression.rmatch_iff_matches'
+Relation.ReflTransGen.lift'
+Relation.TransGen.closed'
+Relation.TransGen.head'
+Relation.TransGen.lift'
+Relation.TransGen.tail'
+RelSeries.last_snoc'
+RelSeries.toList_chain'
+RightOrdContinuous.map_sInf'
+Ring.choose_one_right'
+Ring.choose_zero_right'
+RingCon.smulCommClass'
+RingEquiv.mk_coe'
+RingHom.eq_intCast'
+RingHom.surjectiveOnStalks_iff_forall_maximal'
+Ring.inverse_eq_inv'
+Ring.mul_inverse_rev'
+Ring.multichoose_one_right'
+Ring.multichoose_zero_right'
+RingQuot.ringQuot_ext'
+RingTheory.Sequence.IsRegular.cons'
+RingTheory.Sequence.isRegular_cons_iff'
+RingTheory.Sequence.isWeaklyRegular_append_iff'
+RingTheory.Sequence.IsWeaklyRegular.cons'
+RingTheory.Sequence.isWeaklyRegular_cons_iff'
+RootPairing.coroot_eq_coreflection_of_root_eq'
+RootPairing.ne_zero'
+rootsOfUnity.integer_power_of_ringEquiv'
+root_X_pow_sub_C_ne_zero'
+SameRay.of_subsingleton'
+schnirelmannDensity_congr'
+sdiff_eq_self_iff_disjoint'
+sdiff_le'
+sdiff_le_iff'
+sdiff_sdiff_left'
+sdiff_sdiff_right'
+sdiff_sdiff_sup_sdiff'
+sdiff_sup_self'
+sdiff_symmDiff'
+segment_eq_Icc'
+segment_eq_image'
+Semigroup.opposite_smulCommClass'
+Seminorm.ball_finset_sup'
+Seminorm.ball_zero'
+Seminorm.closedBall_finset_sup'
+Seminorm.closedBall_zero'
+Seminorm.coe_sSup_eq'
+Seminorm.continuous'
+Seminorm.continuousAt_zero'
+Seminorm.uniformContinuous'
+Semiquot.blur_eq_blur'
+Semiquot.mem_blur'
+Semiquot.mem_pure'
+SeparationQuotient.uniformContinuous_lift'
+Set.biInter_and'
+Set.biInter_finsetSigma'
+Set.biInter_le_succ'
+Set.biInter_lt_succ'
+Set.biInter_sigma'
+Set.bijOn_of_subsingleton'
+Set.biUnion_and'
+Set.biUnion_finsetSigma'
+Set.biUnion_finsetSigma_univ'
+Set.biUnion_le_succ'
+Set.biUnion_lt_succ'
+Set.biUnion_sigma'
+SetCoe.exists'
+SetCoe.forall'
+Set.empty_card'
+Set.encard_exchange'
+Set.eq_of_mem_uIcc_of_mem_uIcc'
+Set.eq_of_mem_uIoc_of_mem_uIoc'
+Set.eq_of_nonempty_of_subsingleton'
+Set.EqOn.piecewise_ite'
+Set.eval_preimage'
+Set.exists_intermediate_set'
+Set.finite'
+Set.finite_diff_iUnion_Ioo'
+Set.Finite.eq_of_subset_of_encard_le'
+Set.Finite.preimage'
+Set.Finite.seq'
+Set.Finite.toFinset_insert'
+Set.fintypeBind'
+Set.fintypeBiUnion'
+Set.fintypeSeq'
+Set.Icc_mul_Icc_subset'
+Set.Icc_mul_Ico_subset'
+Set.Icc_subset_uIcc'
+Set.Icc_union_Icc'
+Set.Icc_union_Ici'
+Set.Ici_mul_Ici_subset'
+Set.Ici_mul_Ioi_subset'
+Set.Ico_mul_Icc_subset'
+Set.Ico_mul_Ioc_subset'
+Set.Ico_union_Ici'
+Set.Ico_union_Ico'
+Set.Iic_mul_Iic_subset'
+Set.Iic_mul_Iio_subset'
+Set.Iic_union_Icc'
+Set.Iic_union_Ioc'
+Set.iInter₂_mono'
+Set.iInter_iInter_eq'
+Set.iInter_mono'
+Set.iInter_mono''
+Set.iInter_sigma'
+Set.Iio_mul_Iic_subset'
+Set.Iio_union_Ico'
+Set.Iio_union_Ioo'
+Set.image_affine_Icc'
+Set.image_mul_left'
+Set.image_mul_left_Icc'
+Set.image_mul_right'
+Set.image_mul_right_Icc'
+Set.Infinite.preimage'
+setIntegral_withDensity_eq_setIntegral_smul₀'
+Set.Ioc_mul_Ico_subset'
+Set.Ioc_subset_uIoc'
+Set.Ioc_union_Ioc'
+Set.Ioc_union_Ioi'
+Set.Ioi_mul_Ici_subset'
+Set.Ioo_union_Ioi'
+Set.Ioo_union_Ioo'
+Set.isScalarTower'
+Set.isScalarTower''
+Set.iUnion₂_mono'
+Set.iUnion_iUnion_eq'
+Set.iUnion_mono'
+Set.iUnion_mono''
+Set.iUnion_sigma'
+Set.LeftInvOn.image_image'
+Set.LeftInvOn.image_inter'
+SetLike.ext'
+Set.mapsTo'
+Set.mapsTo_of_subsingleton'
+Set.mulIndicator_apply_le'
+Set.mulIndicator_compl'
+Set.mulIndicator_diff'
+Set.mulIndicator_div'
+Set.mulIndicator_empty'
+Set.mulIndicator_eq_one'
+Set.mulIndicator_inv'
+Set.mulIndicator_le'
+Set.mulIndicator_le_mulIndicator'
+Set.mulIndicator_le_self'
+Set.mulIndicator_mul'
+Set.mulIndicator_one'
+Set.ncard_eq_toFinset_card'
+Set.ncard_exchange'
+Set.nonempty_of_ssubset'
+Set.Nonempty.preimage'
+Setoid.comm'
+Setoid.eqv_class_mem'
+Setoid.ext'
+Setoid.ker_apply_mk_out'
+Setoid.refl'
+Setoid.symm'
+Setoid.trans'
+Set.ordConnected_iInter'
+Set.OrdConnected.inter'
+Set.ordConnected_pi'
+Set.PairwiseDisjoint.elim'
+Set.Pairwise.mono'
+Set.piecewise_mem_Icc'
+Set.pi_eq_empty_iff'
+Set.PiSetCoe.canLift'
+Set.preimage_eq_preimage'
+Set.preimage_id'
+Set.preimage_mul_left_one'
+Set.preimage_mul_right_one'
+Set.Quotient.range_mk''
+Set.range_id'
+Set.range_ite_subset'
+Set.range_quotient_lift_on'
+Set.range_quotient_mk'
+Set.setOf_eq_eq_singleton'
+Set.singleton_pi'
+Set.Sized.subsingleton'
+Set.smulCommClass_set'
+Set.smulCommClass_set''
+Set.smul_inter_ne_empty_iff'
+Set.smul_univ₀'
+Set.star_inv'
+Set.star_mem_centralizer'
+Set.surjOn_of_subsingleton'
+SetTheory.Game.birthday_neg'
+SetTheory.PGame.add_le_add_right'
+SetTheory.PGame.Equiv.not_fuzzy'
+SetTheory.PGame.Fuzzy.not_equiv'
+SetTheory.PGame.LF.not_equiv'
+SetTheory.PGame.moveLeft_neg'
+SetTheory.PGame.moveLeft_neg_symm'
+SetTheory.PGame.moveLeft_nim'
+SetTheory.PGame.moveRight_neg'
+SetTheory.PGame.moveRight_neg_symm'
+SetTheory.PGame.moveRight_nim'
+SetTheory.PGame.ofLists_moveLeft'
+SetTheory.PGame.ofLists_moveRight'
+SetTheory.PGame.relabel_moveLeft'
+SetTheory.PGame.relabel_moveRight'
+SetTheory.PGame.Subsequent.mk_right'
+SetTheory.PGame.zero_lf_inv'
+Set.uIcc_subset_uIcc_iff_le'
+Set.union_diff_cancel'
+Set.WellFoundedOn.mono'
+Sigma.exists'
+Sigma.forall'
+sigma_mk_preimage_image'
+SimpleGraph.Adj.ne'
+SimpleGraph.cliqueSet_mono'
+SimpleGraph.cycleGraph_adj'
+SimpleGraph.dart_edge_eq_mk'_iff'
+SimpleGraph.FarFromTriangleFree.cliqueFinset_nonempty'
+SimpleGraph.Subgraph.connected_iff'
+SimpleGraph.Subgraph.Connected.mono'
+SimpleGraph.Subgraph.degree_le'
+SimpleGraph.TripartiteFromTriangles.Graph.in₀₁_iff'
+SimpleGraph.TripartiteFromTriangles.Graph.in₀₂_iff'
+SimpleGraph.TripartiteFromTriangles.Graph.in₁₀_iff'
+SimpleGraph.TripartiteFromTriangles.Graph.in₁₂_iff'
+SimpleGraph.TripartiteFromTriangles.Graph.in₂₀_iff'
+SimpleGraph.TripartiteFromTriangles.Graph.in₂₁_iff'
+SimpleGraph.Walk.coe_support_append'
+SimpleGraph.Walk.IsPath.mk'
+simple_iff_isSimpleModule'
+SimplexCategory.eq_comp_δ_of_not_surjective'
+SimplexCategory.eq_σ_comp_of_not_injective'
+SimplexCategory.Hom.ext'
+SimplexCategory.δ_comp_δ'
+SimplexCategory.δ_comp_δ''
+SimplexCategory.δ_comp_δ_self'
+SimplexCategory.δ_comp_σ_of_gt'
+SimplexCategory.δ_comp_σ_self'
+SimplexCategory.δ_comp_σ_succ'
+SimplicialObject.Splitting.hom_ext'
+SimplicialObject.Splitting.IndexSet.ext'
+sInf_eq_iInf'
+sInf_image'
+skewAdjoint.conjugate'
+SlashInvariantForm.slash_action_eqn'
+small_biInter'
+small_iInter'
+small_sInter'
+smoothAt_finset_prod'
+smooth_finset_prod'
+SmoothManifoldWithCorners.mk'
+SmoothMap.instSMul'
+SmoothMap.module'
+SmoothMap.smul_comp'
+smoothOn_finset_prod'
+SmoothPartitionOfUnity.sum_finsupport'
+smoothWithinAt_finset_prod'
+smul_ball''
+smul_closedBall'
+smul_closedBall''
+SMulCommClass.nnrat'
+SMulCommClass.rat'
+smul_div'
+smul_eq_smul_iff_eq_and_eq_of_pos'
+smul_finprod'
+smul_inv'
+smul_left_injective'
+smul_le_smul'
+smul_lt_smul'
+smul_lt_smul_of_le_of_lt'
+smul_lt_smul_of_lt_of_le'
+smul_mul'
+smul_nonneg'
+smul_pos'
+smul_pow'
+smul_sphere'
+spec'
+SpectralMap.coe_comp_continuousMap'
+spinGroup.star_eq_inv'
+sq_le_sq'
+sq_lt_sq'
+sSup_eq_bot'
+sSup_eq_iSup'
+sSup_image'
+StarAlgHom.coe_mk'
+star_comm_self'
+StarConvex.sub'
+star_inv'
+Stream'
+Stream'.drop_tail'
+Stream'.get_succ_iterate'
+Stream'.Seq1.map_join'
+Stream'.tail_drop'
+Stream'.take_succ'
+StrictAnti.const_mul'
+StrictAnti.ite'
+StrictAnti.mul_const'
+StrictAntiOn.const_mul'
+StrictAntiOn.mul_const'
+StrictMono.const_mul'
+StrictMono.ite'
+StrictMono.mul_const'
+StrictMonoOn.const_mul'
+StrictMonoOn.mul_const'
+StrictWeakOrder.not_lt_of_equiv'
+String.LT'
+StructureGroupoid.LocalInvariantProp.congr'
+StructureGroupoid.LocalInvariantProp.congr_nhdsWithin'
+StructureGroupoid.LocalInvariantProp.liftPropWithinAt_inter'
+Subalgebra.algebra'
+Subalgebra.coe_valA'
+Subalgebra.module'
+Subbimodule.smul_mem'
+sub_div'
+Subgroup.center_eq_infi'
+Subgroup.comap_equiv_eq_map_symm'
+Subgroup.commutator_def'
+Subgroup.disjoint_def'
+Subgroup.eq_top_iff'
+Subgroup.finiteIndex_iInf'
+Subgroup.map_equiv_eq_comap_symm'
+Subgroup.map_le_map_iff'
+Subgroup.mem_normalizer_iff'
+Subgroup.mem_normalizer_iff''
+Subgroup.mem_sup'
+Subgroup.Normal.conj_mem'
+Subgroup.quotient_finite_of_isOpen'
+Subgroup.smul_diff'
+Subgroup.smul_diff_smul'
+Subgroup.smul_opposite_image_mul_preimage'
+Subgroup.transferTransversal_apply'
+Subgroup.transferTransversal_apply''
+Sublattice.coe_inf'
+SubmoduleClass.module'
+Submodule.coe_continuous_linearProjOfClosedCompl'
+Submodule.coe_prodEquivOfIsCompl'
+Submodule.coe_subtypeL'
+Submodule.comap_smul'
+Submodule.disjoint_def'
+Submodule.disjoint_span_singleton'
+Submodule.eq_top_iff'
+Submodule.hasSMul'
+Submodule.inhabited'
+Submodule.isScalarTower'
+Submodule.ker_liftQ_eq_bot'
+Submodule.le_sInf'
+Submodule.linearProjOfIsCompl_apply_right'
+Submodule.map_smul'
+Submodule.map_smul''
+Submodule.map_toAddSubmonoid'
+Submodule.mem_annihilator'
+Submodule.mem_colon'
+Submodule.mem_ideal_smul_span_iff_exists_sum'
+Submodule.mem_localized'
+Submodule.mem_span_insert'
+Submodule.mem_sup'
+Submodule.module'
+Submodule.orderIsoMapComap_apply'
+Submodule.orderIsoMapComap_symm_apply'
+Submodule.Quotient.distribMulAction'
+Submodule.Quotient.distribSMul'
+Submodule.Quotient.eq'
+Submodule.Quotient.instSMul'
+Submodule.Quotient.mk'_eq_mk'
+Submodule.Quotient.module'
+Submodule.Quotient.mulAction'
+Submodule.Quotient.smulZeroClass'
+Submodule.sInf_le'
+Submodule.smul_mem_iff'
+Submodule.smul_mem_span_smul'
+Submodule.span_image'
+Submodule.unique'
+Submonoid.disjoint_def'
+Submonoid.eq_top_iff'
+Submonoid.LocalizationMap.eq'
+Submonoid.LocalizationMap.map_mk'
+Submonoid.LocalizationMap.mk'_eq_iff_eq'
+Submonoid.LocalizationMap.mk'_eq_of_eq'
+Submonoid.LocalizationMap.mk'_self'
+Submonoid.LocalizationMap.mk'_spec'
+Submonoid.LocalizationMap.mulEquivOfMulEquiv_mk'
+Submonoid.LocalizationMap.mul_mk'_one_eq_mk'
+Submonoid.LocalizationMap.sec_spec'
+Submonoid.LocalizationMap.symm_comp_ofMulEquivOfLocalizations_apply'
+Submonoid.mrange_inl'
+Submonoid.mrange_inr'
+SubMulAction.isScalarTower'
+SubMulAction.mem_one'
+SubMulAction.smul'
+SubMulAction.smul_mem_iff'
+Subring.closure_induction'
+Subring.coe_mk'
+Subring.eq_top_iff'
+Subring.mem_mk'
+Subsemigroup.eq_top_iff'
+Subsemiring.closure_induction'
+Subsemiring.coe_mk'
+Subsemiring.eq_top_iff'
+Subsemiring.mem_mk'
+subset_interior_mul'
+Subsingleton.antitone'
+Subsingleton.monotone'
+sub_sq'
+Subtype.preimage_coe_compl'
+SuccOrder.prelimitRecOn_succ'
+suffixLevenshtein_nil'
+sum_bernoulli'
+summable_geometric_two'
+Summable.matrix_blockDiag'
+summable_matrix_blockDiagonal'
+Summable.matrix_blockDiagonal'
+summable_mul_of_summable_norm'
+summable_of_isBigO'
+summable_of_isBigO_nat'
+summable_star_iff'
+summable_sum_mul_antidiagonal_of_summable_norm'
+summable_sum_mul_range_of_summable_norm'
+sup_eq_half_smul_add_add_abs_sub'
+sup_sdiff_cancel'
+Surreal.dyadicMap_apply_pow'
+Surreal.nsmul_pow_two_powHalf'
+Sym2.instDecidableRel'
+Sym2.mem_iff'
+Sym2.other_eq_other'
+Sym2.other_invol'
+Sym2.other_mem'
+Sym2.other_spec'
+Sym2.rel_iff'
+Sym.inhabitedSym'
+symmDiff_eq'
+symmDiff_eq_Xor'
+symmDiff_symmDiff_right'
+symmDiff_symmDiff_self'
+symmDiff_top'
+SymplecticGroup.coe_inv'
+SymplecticGroup.mem_iff'
+t0Space_iff_uniformity'
+Tactic.NormNum.int_gcd_helper'
+Tactic.NormNum.nat_gcd_helper_1'
+Tactic.NormNum.nat_gcd_helper_2'
+tendsto_ceil_left'
+tendsto_ceil_right'
+tendsto_const_mul_pow_nhds_iff'
+tendsto_floor_left'
+tendsto_floor_right'
+tendsto_fract_left'
+tendsto_fract_right'
+tendsto_gauge_nhds_zero'
+tendsto_indicator_const_apply_iff_eventually'
+tendsto_indicator_const_iff_forall_eventually'
+tendsto_indicator_const_iff_tendsto_pi_pure'
+tendsto_measure_Icc_nhdsWithin_right'
+tendsto_nhds_bot_mono'
+tendsto_nhds_top_mono'
+tendsto_nhds_unique'
+tendsto_norm'
+tendsto_norm_atTop_iff_cobounded'
+tendsto_norm_cobounded_atTop'
+tendsto_norm_cocompact_atTop'
+tendsto_norm_zero'
+TensorProduct.ext'
+TensorProduct.finsuppLeft_smul'
+TensorProduct.isPushout'
+TensorProduct.lift.tmul'
+TensorProduct.smul_tmul'
+Theorems100.«82».Cube.hw'
+Theorems100.num_series'
+three_ne_zero'
+toIcoDiv_add_left'
+toIcoDiv_add_right'
+toIcoDiv_add_zsmul'
+toIcoDiv_neg'
+toIcoDiv_sub'
+toIcoDiv_sub_eq_toIcoDiv_add'
+toIcoDiv_sub_zsmul'
+toIcoMod_add_left'
+toIcoMod_add_right'
+toIcoMod_add_zsmul'
+toIcoMod_mem_Ico'
+toIcoMod_neg'
+toIcoMod_sub'
+toIcoMod_sub_zsmul'
+toIcoMod_zsmul_add'
+toIocDiv_add_left'
+toIocDiv_add_right'
+toIocDiv_add_zsmul'
+toIocDiv_neg'
+toIocDiv_sub'
+toIocDiv_sub_eq_toIocDiv_add'
+toIocDiv_sub_zsmul'
+toIocMod_add_left'
+toIocMod_add_right'
+toIocMod_add_zsmul'
+toIocMod_neg'
+toIocMod_sub'
+toIocMod_sub_zsmul'
+toIocMod_zsmul_add'
+toIxxMod_total'
+TopCat.GlueData.preimage_image_eq_image'
+TopCat.openEmbedding_iff_comp_isIso'
+TopCat.openEmbedding_iff_isIso_comp'
+TopCat.Presheaf.germ_stalkSpecializes'
+TopCat.Presheaf.pushforward_eq'
+TopCat.Presheaf.pushforward_map_app'
+TopologicalGroup.of_nhds_one'
+TopologicalSpace.OpenNhds.map_id_obj'
+TopologicalSpace.Opens.coe_inclusion'
+TopologicalSpace.Opens.map_comp_obj'
+TopologicalSpace.Opens.map_functor_eq'
+TopologicalSpace.Opens.map_id_obj'
+TopologicalSpace.Opens.openEmbedding'
+TopologicalSpace.Opens.set_range_forget_map_inclusion'
+TopologicalSpace.SecondCountableTopology.mk'
+Topology.WithScott.isOpen_iff_isUpperSet_and_scottHausdorff_open'
+top_sdiff'
+top_symmDiff'
+toSubalgebra_toIntermediateField'
+T_pow'
+tprod_comm'
+tprod_eq_prod'
+tprod_eq_zero_mul'
+tprod_le_of_prod_le'
+tprod_prod'
+tprod_sigma'
+Traversable.map_traverse'
+Traversable.naturality'
+Traversable.traverse_eq_map_id'
+Traversable.traverse_map'
+Trivialization.apply_symm_apply'
+Trivialization.coe_coordChangeL'
+Trivialization.coe_fst'
+Trivialization.coe_fst_eventuallyEq_proj'
+Trivialization.continuousLinearEquivAt_apply'
+Trivialization.ext'
+Trivialization.mk_proj_snd'
+Trivialization.proj_symm_apply'
+TrivSqZeroExt.algebra'
+TrivSqZeroExt.algebraMap_eq_inl'
+TrivSqZeroExt.algHom_ext'
+TrivSqZeroExt.snd_pow_of_smul_comm'
+TruncatedWittVector.commutes'
+TruncatedWittVector.commutes_symm'
+tsum_choose_mul_geometric_of_norm_lt_one'
+tsum_geometric_two'
+tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm'
+tsum_mul_tsum_eq_tsum_sum_range_of_summable_norm'
+tsum_mul_tsum_of_summable_norm'
+Tuple.proj_equiv₁'
+Turing.PartrecToTM2.trStmts₁_supports'
+Turing.Reaches₀.tail'
+Turing.Tape.exists_mk'
+Turing.Tape.map_mk'
+Turing.Tape.move_left_mk'
+Turing.Tape.move_right_mk'
+Turing.Tape.write_mk'
+Turing.TM1to1.trTape_mk'
+Turing.tr_eval'
+two_ne_zero'
+TwoSidedIdeal.mem_mk'
+TypeVec.appendFun_comp'
+TypeVec.drop_append1'
+TypeVec.dropFun_RelLast'
+TypeVec.subtypeVal_toSubtype'
+TypeVec.toSubtype'_of_subtype'
+ULift.distribMulAction'
+ULift.distribSMul'
+ULift.isometricSMul'
+ULift.isScalarTower'
+ULift.isScalarTower''
+ULift.module'
+ULift.mulAction'
+ULift.mulActionWithZero'
+ULift.mulDistribMulAction'
+ULift.smulWithZero'
+ULift.smulZeroClass'
+Ultrafilter.le_of_inf_neBot'
+Ultrafilter.map_id'
+UniformCauchySeqOn.prod'
+uniformContinuous_comap'
+UniformContinuous.const_mul'
+uniformContinuous_div_const'
+UniformContinuous.div_const'
+UniformContinuous.mul_const'
+uniformContinuous_mul_left'
+uniformContinuous_mul_right'
+uniformContinuous_nnnorm'
+uniformContinuous_norm'
+isUniformEmbedding_iff'
+UniformGroup.mk'
+isUniformInducing_iff'
+IsUniformInducing.mk'
+uniformity_basis_edist'
+uniformity_basis_edist_le'
+uniformity_eq_comap_nhds_one'
+UniformSpace.Completion.ext'
+unique'
+uniqueDiffWithinAt_inter'
+UniqueDiffWithinAt.inter'
+UniqueFactorizationMonoid.exists_reduced_factors'
+UniqueMDiffWithinAt.inter'
+UniqueMDiffWithinAt.smooth_bundle_preimage'
+Unique.subsingleton_unique'
+Unique.subtypeEq'
+unitary.star_eq_inv'
+Unitization.algHom_ext''
+Unitization.quasispectrum_eq_spectrum_inr'
+Units.coe_map'
+Units.conj_pow'
+Units.inv_mul'
+Units.mul_inv'
+UniversalEnvelopingAlgebra.lift_ι_apply'
+update_le_update_iff'
+upperClosure_interior_subset'
+UpperHalfPlane.cosh_dist'
+UpperHalfPlane.ext_iff'
+UpperHalfPlane.ModularGroup.det_coe'
+UpperHalfPlane.mul_smul'
+UV.compress_of_disjoint_of_le'
+Valuation.Integers.one_of_isUnit'
+Valuation.map_add'
+Valuation.map_sum_lt'
+ValuationSubring.isIntegral_of_mem_ringOfIntegers'
+Vector.continuous_insertNth'
+VitaliFamily.ae_tendsto_lintegral_div'
+volume_regionBetween_eq_integral'
+volume_regionBetween_eq_lintegral'
+WCovBy.of_le_of_le'
+WeakBilin.instModule'
+WeakSpace.instModule'
+WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal'
+WeierstrassCurve.Affine.equation_iff'
+WeierstrassCurve.Affine.nonsingular_iff'
+WeierstrassCurve.Affine.Point.add_of_X_ne'
+WeierstrassCurve.Affine.Point.add_of_Y_ne'
+WeierstrassCurve.Affine.Point.add_self_of_Y_ne'
+WeierstrassCurve.baseChange_preΨ'
+WeierstrassCurve.coeff_preΨ'
+WeierstrassCurve.Jacobian.add_of_Y_ne'
+WeierstrassCurve.Jacobian.addX_eq'
+WeierstrassCurve.Jacobian.addX_of_X_eq'
+WeierstrassCurve.Jacobian.addY_of_X_eq'
+WeierstrassCurve.Jacobian.dblXYZ_of_Y_eq'
+WeierstrassCurve.Jacobian.dblZ_ne_zero_of_Y_ne'
+WeierstrassCurve.Jacobian.equiv_iff_eq_of_Z_eq'
+WeierstrassCurve.Jacobian.isUnit_dblZ_of_Y_ne'
+WeierstrassCurve.Jacobian.negAddY_eq'
+WeierstrassCurve.Jacobian.negAddY_of_X_eq'
+WeierstrassCurve.Jacobian.neg_of_Z_eq_zero'
+WeierstrassCurve.Jacobian.Y_eq_iff'
+WeierstrassCurve.Jacobian.Y_eq_of_Y_ne'
+WeierstrassCurve.Jacobian.Y_ne_negY_of_Y_ne'
+WeierstrassCurve.leadingCoeff_preΨ'
+WeierstrassCurve.map_preΨ'
+WeierstrassCurve.natDegree_coeff_preΨ'
+WeierstrassCurve.natDegree_preΨ'
+WeierstrassCurve.Projective.addX_eq'
+WeierstrassCurve.Projective.addY_of_X_eq'
+WeierstrassCurve.Projective.addZ_eq'
+WeierstrassCurve.Projective.dblX_eq'
+WeierstrassCurve.Projective.dblY_of_Y_eq'
+WeierstrassCurve.Projective.dblZ_ne_zero_of_Y_ne'
+WeierstrassCurve.Projective.equiv_iff_eq_of_Z_eq'
+WeierstrassCurve.Projective.isUnit_dblZ_of_Y_ne'
+WeierstrassCurve.Projective.negAddY_eq'
+WeierstrassCurve.Projective.negAddY_of_X_eq'
+WeierstrassCurve.Projective.negDblY_eq'
+WeierstrassCurve.Projective.negDblY_of_Y_eq'
+WeierstrassCurve.Projective.Y_eq_iff'
+WeierstrassCurve.Projective.Y_eq_of_Y_ne'
+WeierstrassCurve.Projective.Y_ne_negY_of_Y_ne'
+WellFounded.monotone_chain_condition'
+WfDvdMonoid.max_power_factor'
+WithBot.bot_mul'
+WithBot.coe_sInf'
+WithBot.coe_sSup'
+WithBot.le_coe_unbot'
+WithBot.mul_bot'
+WithBot.unbot_one'
+WithTop.coe_sInf'
+WithTop.coe_sSup'
+WithTop.distrib'
+WithTop.mul_top'
+WithTop.top_mul'
+WithTop.untop_one'
+WithZero.map'_map'
+WittVector.aeval_verschiebung_poly'
+WittVector.exists_eq_pow_p_mul'
+WittVector.idIsPolyI'
+WittVector.nth_mul_coeff'
+WittVector.poly_eq_of_wittPolynomial_bind_eq'
+WittVector.RecursionBase.solution_spec'
+WittVector.RecursionMain.succNthVal_spec'
+WittVector.truncate_mk'
+WriterT.callCC'
+WriterT.goto_mkLabel'
+WriterT.mkLabel'
+WType.cardinal_mk_eq_sum'
+WType.WType'
+Xor'
+xor_iff_not_iff'
+X_pow_sub_C_eq_prod'
+zero_le'
+zero_lt_one_add_norm_sq'
+zero_mem_ℓp'
+zero_ne_one'
+ZFSet.IsTransitive.sUnion'
+ZMod.cast_add'
+ZMod.cast_id'
+ZMod.cast_intCast'
+ZMod.cast_mul'
+ZMod.cast_natCast'
+ZMod.cast_one'
+ZMod.cast_pow'
+ZMod.cast_sub'
+ZMod.intCast_eq_intCast_iff'
+ZMod.invDFT_apply'
+ZMod.invDFT_def'
+ZMod.natCast_eq_natCast_iff'
+ZMod.natCast_self'
+ZMod.neg_val'
+ZMod.nontrivial'
+ZMod.val_mul'
+ZMod.val_neg'
+ZMod.val_one'
+ZMod.val_one''
+ZMod.val_unit'
+ZNum.cast_zero'
+ZNum.of_to_int'
+zpow_add'
+zpow_eq_zpow_emod'
+zpow_le_zpow'
+zpow_le_zpow_iff'
+zpow_lt_zpow'
+zpow_lt_zpow_iff'
+zpow_mul'
+zsmul_eq_mul'
+Zsqrtd.norm_eq_one_iff'
diff --git a/scripts/nolints-style.txt b/scripts/nolints-style.txt
index 2b0bbbe992a8d..9467ae24e5650 100644
--- a/scripts/nolints-style.txt
+++ b/scripts/nolints-style.txt
@@ -1,9 +1,6 @@
 -- Manual exceptions for the text-based linters.
--- This file is to `style-exceptions.txt` what `nolints.json` is to `@nolint` attributes:
--- The latter is supposed to become and stay mostly empty over time (though files longer than
--- 1500 lines can be transient exceptions for some longer time period),
--- the former could be necessary in the long term.
--- In this case, it's a side-effect of making the linter stricter than its Python ancestor.
+-- The entries in this file could be necessary in the long term.
+-- In some cases, entries are a side effect of making the linter stricter than its Python ancestor.
 
 -- The `Mathlib/Init.lean` files does not have a copyright header
 Mathlib/Init.lean : line 2 : ERR_COP : Malformed or missing copyright header: Copyright line is malformed
@@ -29,8 +26,9 @@ Mathlib/Tactic/AdaptationNote.lean : line 21 : ERR_ADN : Found the string "Adapt
 Mathlib/Tactic/AdaptationNote.lean : line 27 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
 Mathlib/Tactic/AdaptationNote.lean : line 39 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
 Mathlib/Tactic/AdaptationNote.lean : line 52 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
-Mathlib/Tactic/Linter/TextBased.lean : line 33 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
-Mathlib/Tactic/Linter/TextBased.lean : line 60 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
-Mathlib/Tactic/Linter/TextBased.lean : line 222 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
-Mathlib/Tactic/Linter/TextBased.lean : line 227 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
-Mathlib/Tactic/Linter/TextBased.lean : line 228 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 20 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 49 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 84 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 274 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 279 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
+Mathlib/Tactic/Linter/TextBased.lean : line 280 : ERR_ADN : Found the string "Adaptation note:", please use the #adaptation_note command instead
diff --git a/scripts/nolints.json b/scripts/nolints.json
index b2ed89639f44c..8ea41fbf362a9 100644
--- a/scripts/nolints.json
+++ b/scripts/nolints.json
@@ -6,32 +6,23 @@
  ["docBlame", "CongrState"],
  ["docBlame", "Cont"],
  ["docBlame", "ContT"],
- ["docBlame", "ExistsUnique"],
  ["docBlame", "IsDecEq"],
  ["docBlame", "IsDecRefl"],
- ["docBlame", "IsIdempotent"],
  ["docBlame", "IsLeftCancel"],
  ["docBlame", "IsRightCancel"],
- ["docBlame", "IsSymmOp"],
  ["docBlame", "LawfulMonadCont"],
  ["docBlame", "LeftCancelative"],
- ["docBlame", "LeftCommutative"],
  ["docBlame", "LeftDistributive"],
  ["docBlame", "LeftIdentity"],
  ["docBlame", "MonadCont"],
  ["docBlame", "MonadWriter"],
  ["docBlame", "One"],
  ["docBlame", "RightCancelative"],
- ["docBlame", "RightCommutative"],
  ["docBlame", "RightDistributive"],
  ["docBlame", "RightIdentity"],
  ["docBlame", "RightInverse"],
  ["docBlame", "Writer"],
  ["docBlame", "WriterT"],
- ["docBlame", "Xor'"],
- ["docBlame", "Zero"],
- ["docBlame", "bit0"],
- ["docBlame", "bit1"],
  ["docBlame", "cancelDenominators"],
  ["docBlame", "cancelDenominatorsAt"],
  ["docBlame", "cancelDenominatorsTarget"],
@@ -51,13 +42,11 @@
  ["docBlame", "«term_≃ᵃⁱ[_]_»"],
  ["docBlame", "«term_≃ᵈ_»"],
  ["docBlame", "«term_≡_[SMOD_]»"],
- ["docBlame", "termℤ"],
  ["docBlame", "«termℤ√_»"],
  ["docBlame", "«term∫_In_.._,_»"],
  ["docBlame", "«term∫_In_.._,_∂_»"],
  ["docBlame", "«term∮_InC(_,_),_»"],
  ["docBlame", "«term⨍_In_.._,_»"],
- ["docBlame", "timeCmd"],
  ["docBlame", "when"],
  ["docBlame", "whenM"],
  ["docBlame", "Action.V"],
@@ -80,7 +69,6 @@
  ["docBlame", "Bitraversable.bitraverse"],
  ["docBlame", "BoundedContinuousFunction.«term_→ᵇ_»"],
  ["docBlame", "BoundedRandom.randomR"],
- ["docBlame", "BumpCovering.toFun"],
  ["docBlame", "ByteSlice.arr"],
  ["docBlame", "ByteSlice.len"],
  ["docBlame", "ByteSlice.off"],
@@ -279,11 +267,9 @@
  ["docBlame", "Nat.subInduction"],
  ["docBlame", "Nat.«termOrd_compl[_]_»"],
  ["docBlame", "Nat.«termOrd_proj[_]_»"],
- ["docBlame", "Nat.twoStepInduction"],
  ["docBlame", "One.one"],
  ["docBlame", "OptionT.callCC"],
  ["docBlame", "OptionT.mkLabel"],
- ["docBlame", "PartitionOfUnity.toFun"],
  ["docBlame", "PicardLindelof.C"],
  ["docBlame", "PicardLindelof.L"],
  ["docBlame", "PicardLindelof.R"],
@@ -298,7 +284,6 @@
  ["docBlame", "PowerBasis.basis"],
  ["docBlame", "PowerBasis.dim"],
  ["docBlame", "PowerBasis.gen"],
- ["docBlame", "PresheafOfModules.presheaf"],
  ["docBlame", "Pretrivialization.baseSet"],
  ["docBlame", "PrimeSpectrum.asIdeal"],
  ["docBlame", "ProbabilityTheory.«termEVar[_]»"],
@@ -306,9 +291,7 @@
  ["docBlame", "ProbabilityTheory.«term_=ₐₛ_»"],
  ["docBlame", "ProbabilityTheory.«term_[_]»"],
  ["docBlame", "ProbabilityTheory.«term_×ₖ_»"],
- ["docBlame", "ProbabilityTheory.«term_∘ₖ_»"],
  ["docBlame", "ProbabilityTheory.«term_≤ₐₛ_»"],
- ["docBlame", "ProbabilityTheory.«term_⊗ₖ_»"],
  ["docBlame", "ProbabilityTheory.«term_⟦_|_⟧»"],
  ["docBlame", "ProbabilityTheory.termℙ"],
  ["docBlame", "ProbabilityTheory.«term∂_/∂_»"],
@@ -337,7 +320,6 @@
  ["docBlame", "SchwartzMap.toFun"],
  ["docBlame", "SemiRingCat.forget_obj_eq_coe"],
  ["docBlame", "Semigrp.forget_obj_eq_coe"],
- ["docBlame", "Set.«term{_|_}»"],
  ["docBlame", "Shrink.rec"],
  ["docBlame", "SlashAction.map"],
  ["docBlame", "SlashInvariantForm.toFun"],
@@ -391,7 +373,6 @@
  ["docBlame", "WriterT.mkLabel'"],
  ["docBlame", "WriterT.run"],
  ["docBlame", "WriterT.runThe"],
- ["docBlame", "Zero.zero"],
  ["docBlame", "Zsqrtd.im"],
  ["docBlame", "Zsqrtd.re"],
  ["docBlame", "algebraMap.coeHTCT"],
@@ -427,8 +408,6 @@
  ["docBlame", "CategoryTheory.Comma.right"],
  ["docBlame", "CategoryTheory.CommaMorphism.left"],
  ["docBlame", "CategoryTheory.CommaMorphism.right"],
- ["docBlame", "CategoryTheory.Comonad.δ'"],
- ["docBlame", "CategoryTheory.Comonad.ε'"],
  ["docBlame", "CategoryTheory.EnrichedCategory.Hom"],
  ["docBlame", "CategoryTheory.EnrichedCategory.comp"],
  ["docBlame", "CategoryTheory.EnrichedCategory.id"],
@@ -441,8 +420,6 @@
  ["docBlame", "CategoryTheory.GlueData.t"],
  ["docBlame", "CategoryTheory.GlueData.t'"],
  ["docBlame", "CategoryTheory.GradedNatTrans.app"],
- ["docBlame",
-  "CategoryTheory.GrothendieckTopology.sheafificationIsoPresheafToSheafCompSheafToPreasheaf"],
  ["docBlame", "CategoryTheory.HalfBraiding.β"],
  ["docBlame", "CategoryTheory.Mat_.X"],
  ["docBlame", "CategoryTheory.Mat_.ι"],
@@ -451,8 +428,6 @@
  ["docBlame", "CategoryTheory.Monad.PreservesColimitOfIsReflexivePair"],
  ["docBlame", "CategoryTheory.Monad.PreservesColimitOfIsSplitPair"],
  ["docBlame", "CategoryTheory.Monad.ReflectsColimitOfIsSplitPair"],
- ["docBlame", "CategoryTheory.Monad.η'"],
- ["docBlame", "CategoryTheory.Monad.μ'"],
  ["docBlame", "CategoryTheory.Presieve.yonedaFamilyOfElements_fromCocone"],
  ["docBlame", "CategoryTheory.Pretopology.coverings"],
  ["docBlame", "CategoryTheory.ProjectivePresentation.f"],
@@ -489,7 +464,6 @@
  ["docBlame", "GromovHausdorff.AuxGluingStruct.Space"],
  ["docBlame", "GromovHausdorff.AuxGluingStruct.embed"],
  ["docBlame", "GromovHausdorff.AuxGluingStruct.metric"],
- ["docBlame", "HahnSeries.SummableFamily.toFun"],
  ["docBlame", "HomogeneousLocalization.NumDenSameDeg.deg"],
  ["docBlame", "HomogeneousLocalization.NumDenSameDeg.den"],
  ["docBlame", "HomogeneousLocalization.NumDenSameDeg.num"],
@@ -497,7 +471,6 @@
  ["docBlame", "Ideal.Filtration.N"],
  ["docBlame", "IntermediateField.delabAdjoinNotation.delabInsertArray"],
  ["docBlame", "IsDedekindDomain.HeightOneSpectrum.asIdeal"],
- ["docBlame", "Lean.Attr.substAttr"],
  ["docBlame", "Lean.Export.Alloc"],
  ["docBlame", "Lean.Export.Entry"],
  ["docBlame", "Lean.Export.OfState"],
@@ -515,10 +488,6 @@
  ["docBlame", "Lean.Expr.modifyRevArg"],
  ["docBlame", "Lean.MVarId.casesType"],
  ["docBlame", "Lean.MVarId.congrCore!"],
- ["docBlame", "Lean.Meta.checkTypeIsProp"],
- ["docBlame", "Lean.Meta.mkSimpTheoremCore"],
- ["docBlame", "Lean.Meta.preprocess"],
- ["docBlame", "Lean.Meta.shouldPreprocess"],
  ["docBlame", "Lean.Name.isBlackListed"],
  ["docBlame", "Lean.PHashSet.toList"],
  ["docBlame", "LocallyFinite.Realizer.bas"],
@@ -571,7 +540,6 @@
  ["docBlame", "Order.PFilter.dual"],
  ["docBlame", "PProd.mk.injArrow"],
  ["docBlame", "PicardLindelof.FunSpace.toFun"],
- ["docBlame", "PresheafOfModules.Hom.hom"],
  ["docBlame", "Prod.mk.injArrow"],
  ["docBlame", "QuaternionAlgebra.Basis.i"],
  ["docBlame", "QuaternionAlgebra.Basis.j"],
@@ -602,7 +570,6 @@
  ["docBlame", "SlimCheck.Testable.run"],
  ["docBlame", "SlimCheck.Testable.runProp"],
  ["docBlame", "Stream'.WSeq.«term_~ʷ_»"],
- ["docBlame", "String.toAsciiByteArray.loop"],
  ["docBlame", "Submodule.quotientPi_aux.invFun"],
  ["docBlame", "Submodule.quotientPi_aux.toFun"],
  ["docBlame", "Tactic.Elementwise.tacticElementwise!___"],
@@ -660,7 +627,6 @@
  ["docBlame", "CategoryTheory.Triangulated.Octahedron.m₃"],
  ["docBlame", "FirstOrder.Language.ElementaryEmbedding.toFun"],
  ["docBlame", "FirstOrder.Language.ElementarySubstructure.toSubstructure"],
- ["docBlame", "FirstOrder.Language.IsOrdered.leSymb"],
  ["docBlame", "FirstOrder.Language.LEquiv.invLHom"],
  ["docBlame", "FirstOrder.Language.LEquiv.toLHom"],
  ["docBlame", "FirstOrder.Language.LHom.onFunction"],
@@ -746,10 +712,10 @@
  ["docBlame", "Mathlib.Command.Variable.variable?.maxSteps"],
  ["docBlame", "Mathlib.Meta.NormNum.evalEq.intArm"],
  ["docBlame", "Mathlib.Meta.NormNum.evalEq.ratArm"],
- ["docBlame", "Mathlib.Meta.NormNum.evalLE.intArm"],
- ["docBlame", "Mathlib.Meta.NormNum.evalLE.ratArm"],
- ["docBlame", "Mathlib.Meta.NormNum.evalLT.intArm"],
- ["docBlame", "Mathlib.Meta.NormNum.evalLT.ratArm"],
+ ["docBlame", "Mathlib.Meta.NormNum.evalLE.core.intArm"],
+ ["docBlame", "Mathlib.Meta.NormNum.evalLE.core.ratArm"],
+ ["docBlame", "Mathlib.Meta.NormNum.evalLT.core.intArm"],
+ ["docBlame", "Mathlib.Meta.NormNum.evalLT.core.ratArm"],
  ["docBlame", "Mathlib.Meta.NormNum.evalMinFac.aux"],
  ["docBlame", "Mathlib.Meta.NormNum.evalMinFac.core"],
  ["docBlame", "Mathlib.Meta.NormNum.evalNatPrime.core"],
@@ -785,4 +751,4 @@
  ["docBlame", "Mathlib.Meta.NormNum.evalAdd.core.ratArm"],
  ["docBlame", "Mathlib.Meta.NormNum.evalMul.core.intArm"],
  ["docBlame", "Mathlib.Meta.NormNum.evalMul.core.ratArm"],
-  ["unusedArguments", "Combinator.K"]]
+ ["unusedArguments", "Combinator.K"]]
diff --git a/scripts/noshake.json b/scripts/noshake.json
index 565c7fd2a7c5f..6448419d87171 100644
--- a/scripts/noshake.json
+++ b/scripts/noshake.json
@@ -39,6 +39,7 @@
   "Mathlib.CategoryTheory.Category.Init",
   "Mathlib.Combinatorics.SimpleGraph.Init",
   "Mathlib.Control.Traversable.Lemmas",
+  "Mathlib.Data.Finsupp.Notation",
   "Mathlib.Data.Int.Defs",
   "Mathlib.Data.Matrix.Notation",
   "Mathlib.Data.Matroid.Init",
@@ -101,7 +102,7 @@
   "Mathlib.Tactic.FinCases",
   "Mathlib.Tactic.Find",
   "Mathlib.Tactic.GCongr",
-  "Mathlib.Tactic.GCongr.Core",
+  "Mathlib.Tactic.GCongr.CoreAttrs",
   "Mathlib.Tactic.Generalize",
   "Mathlib.Tactic.GeneralizeProofs",
   "Mathlib.Tactic.Group",
@@ -193,8 +194,10 @@
  "ignoreAll":
  ["Batteries.Tactic.Basic", "Mathlib.Mathport.Syntax", "Mathlib.Tactic.Linter"],
  "ignore":
- {"Mathlib.Topology.Sheaves.Forget": ["Mathlib.Algebra.Category.Ring.Limits"],
-  "Mathlib.Topology.Order.LeftRightNhds": ["Mathlib.Data.Set.Pointwise.Basic"],
+ {"Mathlib.Topology.UniformSpace.Basic": ["Mathlib.Tactic.Monotonicity.Basic"],
+  "Mathlib.Topology.Sheaves.Forget": ["Mathlib.Algebra.Category.Ring.Limits"],
+  "Mathlib.Topology.Order.LeftRightNhds":
+  ["Mathlib.Algebra.Ring.Pointwise.Set"],
   "Mathlib.Topology.Germ": ["Mathlib.Analysis.Normed.Module.Basic"],
   "Mathlib.Topology.Defs.Basic": ["Mathlib.Tactic.FunProp"],
   "Mathlib.Topology.Category.UniformSpace":
@@ -209,7 +212,7 @@
   "Mathlib.Tactic.Use": ["Batteries.Logic"],
   "Mathlib.Tactic.TermCongr": ["Mathlib.Logic.Basic"],
   "Mathlib.Tactic.Tauto": ["Mathlib.Logic.Basic"],
-  "Mathlib.Tactic.TFAE": ["Mathlib.Data.List.TFAE"],
+  "Mathlib.Tactic.TFAE": ["Mathlib.Data.List.TFAE", "Mathlib.Tactic.Have"],
   "Mathlib.Tactic.Subsingleton": ["Mathlib.Logic.Basic", "Std.Logic"],
   "Mathlib.Tactic.ReduceModChar":
   ["Mathlib.Data.ZMod.Basic", "Mathlib.RingTheory.Polynomial.Basic"],
@@ -234,10 +237,11 @@
   "Mathlib.Tactic.Measurability":
   ["Mathlib.Algebra.Group.Defs", "Mathlib.Tactic.Measurability.Init"],
   "Mathlib.Tactic.Linter.UnusedTactic": ["Batteries.Tactic.Unreachable"],
+  "Mathlib.Tactic.LinearCombination":
+  ["Mathlib.Tactic.LinearCombination.Lemmas"],
   "Mathlib.Tactic.Lemma": ["Lean.Parser.Command"],
   "Mathlib.Tactic.IrreducibleDef": ["Mathlib.Data.Subtype"],
-  "Mathlib.Tactic.ITauto":
-  ["Batteries.Logic", "Batteries.Tactic.Init", "Mathlib.Init.Logic"],
+  "Mathlib.Tactic.ITauto": ["Batteries.Tactic.Init", "Mathlib.Logic.Basic"],
   "Mathlib.Tactic.Group": ["Mathlib.Algebra.Group.Commutator"],
   "Mathlib.Tactic.GCongr.Core": ["Mathlib.Order.Defs"],
   "Mathlib.Tactic.GCongr": ["Mathlib.Tactic.Positivity.Core"],
@@ -269,10 +273,14 @@
   "Mathlib.Tactic.Continuity": ["Mathlib.Tactic.Continuity.Init"],
   "Mathlib.Tactic.CongrExclamation": ["Mathlib.Logic.Basic"],
   "Mathlib.Tactic.Choose": ["Mathlib.Logic.Function.Basic"],
+  "Mathlib.Tactic.CategoryTheory.ToApp":
+  ["Mathlib.CategoryTheory.Category.Cat"],
   "Mathlib.Tactic.CategoryTheory.Slice":
   ["Mathlib.CategoryTheory.Category.Basic"],
   "Mathlib.Tactic.CategoryTheory.Reassoc":
   ["Mathlib.CategoryTheory.Functor.Basic"],
+  "Mathlib.Tactic.CategoryTheory.Monoidal":
+  ["Mathlib.Tactic.CategoryTheory.MonoidalComp"],
   "Mathlib.Tactic.CategoryTheory.Coherence":
   ["Mathlib.CategoryTheory.Monoidal.Free.Coherence",
    "Mathlib.Tactic.CategoryTheory.MonoidalComp"],
@@ -294,11 +302,18 @@
   "Mathlib.Tactic.Basic": ["Mathlib.Tactic.Linter.OldObtain"],
   "Mathlib.Tactic.Attr.Register": ["Lean.Meta.Tactic.Simp.SimpTheorems"],
   "Mathlib.Tactic.ArithMult": ["Mathlib.Tactic.ArithMult.Init"],
+  "Mathlib.Tactic.Algebraize": ["Mathlib.Algebra.Algebra.Tower"],
   "Mathlib.RingTheory.PowerSeries.Basic":
   ["Mathlib.Algebra.CharP.Defs", "Mathlib.Tactic.MoveAdd"],
   "Mathlib.RingTheory.PolynomialAlgebra": ["Mathlib.Data.Matrix.DMatrix"],
   "Mathlib.RingTheory.MvPolynomial.Homogeneous":
   ["Mathlib.Algebra.DirectSum.Internal"],
+  "Mathlib.RingTheory.KrullDimension.Basic":
+  ["Mathlib.Algebra.MvPolynomial.CommRing", "Mathlib.Algebra.Polynomial.Basic"],
+  "Mathlib.RingTheory.IntegralClosure.IsIntegral.Defs":
+  ["Mathlib.Tactic.Algebraize"],
+  "Mathlib.RingTheory.Finiteness":
+  ["Mathlib.Algebra.Algebra.RestrictScalars", "Mathlib.Tactic.Algebraize"],
   "Mathlib.RingTheory.Binomial": ["Mathlib.Algebra.Order.Floor"],
   "Mathlib.RepresentationTheory.FdRep":
   ["Mathlib.CategoryTheory.Monoidal.Rigid.Braided"],
@@ -330,10 +345,6 @@
   "Mathlib.Lean.Expr.ExtraRecognizers": ["Mathlib.Data.Set.Operations"],
   "Mathlib.Lean.Expr.Basic": ["Batteries.Logic"],
   "Mathlib.Init.Data.Nat.Lemmas": ["Batteries.Data.Nat.Lemmas", "Batteries.WF"],
-  "Mathlib.Init.Data.List.Lemmas": ["Batteries.Data.List.Lemmas"],
-  "Mathlib.Init.Data.List.Basic": ["Batteries.Data.List.Basic"],
-  "Mathlib.Init.Data.Int.Basic": ["Batteries.Data.Int.Order"],
-  "Mathlib.Init.Data.Bool.Lemmas": ["Mathlib.Tactic.AdaptationNote"],
   "Mathlib.GroupTheory.MonoidLocalization.Basic": ["Mathlib.Init.Data.Prod"],
   "Mathlib.Geometry.Manifold.Sheaf.Smooth":
   ["Mathlib.CategoryTheory.Sites.Whiskering"],
@@ -354,13 +365,13 @@
   ["Batteries.Data.Nat.Lemmas", "Mathlib.Data.List.Basic"],
   "Mathlib.Data.List.Lemmas": ["Mathlib.Data.List.InsertNth"],
   "Mathlib.Data.List.Defs": ["Batteries.Data.RBMap.Basic"],
-  "Mathlib.Data.List.Basic": ["Mathlib.Data.Option.Basic"],
+  "Mathlib.Data.List.Basic":
+  ["Mathlib.Control.Basic", "Mathlib.Data.Option.Basic"],
   "Mathlib.Data.LazyList.Basic": ["Mathlib.Lean.Thunk"],
   "Mathlib.Data.Int.Order.Basic": ["Mathlib.Data.Int.Notation"],
   "Mathlib.Data.Int.Defs": ["Batteries.Data.Int.Order"],
   "Mathlib.Data.FunLike.Basic": ["Mathlib.Logic.Function.Basic"],
   "Mathlib.Data.Finset.Basic": ["Mathlib.Data.Finset.Attr"],
-  "Mathlib.Data.DFinsupp.Notation": ["Mathlib.Data.Finsupp.Notation"],
   "Mathlib.Data.ByteArray": ["Batteries.Data.ByteSubarray"],
   "Mathlib.Data.Bool.Basic": ["Batteries.Tactic.Init"],
   "Mathlib.Control.Traversable.Instances": ["Mathlib.Control.Applicative"],
@@ -370,10 +381,21 @@
   ["Mathlib.Algebra.Order.Field.Basic"],
   "Mathlib.CategoryTheory.Sites.IsSheafFor":
   ["Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono"],
+  "Mathlib.CategoryTheory.Monoidal.Rigid.Basic":
+  ["Mathlib.Tactic.CategoryTheory.Monoidal.Basic"],
+  "Mathlib.CategoryTheory.Monoidal.Braided.Basic":
+  ["Mathlib.Tactic.CategoryTheory.Monoidal.Basic"],
   "Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits":
   ["Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback",
    "Mathlib.CategoryTheory.Limits.Shapes.Pullbacks"],
   "Mathlib.CategoryTheory.Limits.IsLimit": ["Batteries.Tactic.Congr"],
+  "Mathlib.CategoryTheory.Category.Basic": ["Mathlib.Tactic.StacksAttribute"],
+  "Mathlib.CategoryTheory.Bicategory.Functor.Oplax":
+  ["Mathlib.Tactic.CategoryTheory.ToApp"],
+  "Mathlib.CategoryTheory.Bicategory.Functor.Lax":
+  ["Mathlib.Tactic.CategoryTheory.ToApp"],
+  "Mathlib.CategoryTheory.Bicategory.Adjunction":
+  ["Mathlib.Tactic.CategoryTheory.Bicategory.Basic"],
   "Mathlib.Analysis.Normed.Operator.LinearIsometry":
   ["Mathlib.Algebra.Star.Basic"],
   "Mathlib.Analysis.InnerProductSpace.Basic":
diff --git a/scripts/print-style-errors.sh b/scripts/print-style-errors.sh
index 0f0ca588e598c..f48fdd08c79a1 100755
--- a/scripts/print-style-errors.sh
+++ b/scripts/print-style-errors.sh
@@ -1,11 +1,16 @@
 #!/usr/bin/env bash
 
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
 # Print all errors of the python style linter. This script is temporary and should be removed
 # once the Python style linters have been rewritten in Lean.
-# Humans should never run this directly, but at most through `lean exe lint-style --update`
+# Humans should never run this directly, but at most through `lean exe lint-style --fix`
 
 # use C locale so that sorting is the same on macOS and Linux
 # see https://unix.stackexchange.com/questions/362728/why-does-gnu-sort-sort-differently-on-my-osx-machine-and-linux-machine
-find Mathlib -name '*.lean' | xargs ./scripts/lint-style.py "$@" | LC_ALL=C sort
-find Archive -name '*.lean' | xargs ./scripts/lint-style.py "$@" | LC_ALL=C sort
-find Counterexamples -name '*.lean' | xargs ./scripts/lint-style.py "$@" | LC_ALL=C sort
+find Mathlib -name '*.lean' -print0 | xargs --null ./scripts/lint-style.py "$@" | LC_ALL=C sort || true
+find Archive -name '*.lean' -print0 | xargs --null ./scripts/lint-style.py "$@" | LC_ALL=C sort || true
+find Counterexamples -name '*.lean' -print0 | xargs --null ./scripts/lint-style.py "$@" | LC_ALL=C sort || true
diff --git a/scripts/style-exceptions.txt b/scripts/style-exceptions.txt
deleted file mode 100644
index ef92d5ed1d830..0000000000000
--- a/scripts/style-exceptions.txt
+++ /dev/null
@@ -1,68 +0,0 @@
-Mathlib/Algebra/BigOperators/Group/Finset.lean : line 1 : ERR_NUM_LIN : 2400 file contains 2245 lines, try to split it up
-Mathlib/Algebra/Group/Subgroup/Basic.lean : line 1 : ERR_NUM_LIN : 3000 file contains 2893 lines, try to split it up
-Mathlib/Algebra/MonoidAlgebra/Basic.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1965 lines, try to split it up
-Mathlib/Algebra/MvPolynomial/Basic.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1544 lines, try to split it up
-Mathlib/Algebra/Order/Floor.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1593 lines, try to split it up
-Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1511 lines, try to split it up
-Mathlib/Analysis/Asymptotics/Asymptotics.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1955 lines, try to split it up
-Mathlib/Analysis/Calculus/ContDiff/Basic.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1952 lines, try to split it up
-Mathlib/Analysis/Calculus/ContDiff/Defs.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1595 lines, try to split it up
-Mathlib/Analysis/InnerProductSpace/Basic.lean : line 1 : ERR_NUM_LIN : 2500 file contains 2433 lines, try to split it up
-Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1998 lines, try to split it up
-Mathlib/Computability/TMToPartrec.lean : line 1 : ERR_NUM_LIN : 2200 file contains 2071 lines, try to split it up
-Mathlib/Computability/TuringMachine.lean : line 1 : ERR_NUM_LIN : 2700 file contains 2561 lines, try to split it up
-Mathlib/Data/DFinsupp/Basic.lean : line 1 : ERR_NUM_LIN : 2200 file contains 2078 lines, try to split it up
-Mathlib/Data/Fin/Basic.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1524 lines, try to split it up
-Mathlib/Data/Finset/Basic.lean : line 1 : ERR_NUM_LIN : 3200 file contains 3057 lines, try to split it up
-Mathlib/Data/Finset/Lattice.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1968 lines, try to split it up
-Mathlib/Data/Finset/Pointwise/Basic.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1997 lines, try to split it up
-Mathlib/Data/Finsupp/Basic.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1703 lines, try to split it up
-Mathlib/Data/List/Basic.lean : line 1 : ERR_NUM_LIN : 2700 file contains 2598 lines, try to split it up
-Mathlib/Data/Matrix/Basic.lean : line 1 : ERR_NUM_LIN : 2700 file contains 2519 lines, try to split it up
-Mathlib/Data/Multiset/Basic.lean : line 1 : ERR_NUM_LIN : 2900 file contains 2725 lines, try to split it up
-Mathlib/Data/Num/Lemmas.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1576 lines, try to split it up
-Mathlib/Data/Ordmap/Ordset.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1580 lines, try to split it up
-Mathlib/Data/QPF/Multivariate/Basic.lean : line 76 : ERR_LIN : Line has 127 characters, which is more than 100
-Mathlib/Data/Real/EReal.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1778 lines, try to split it up
-Mathlib/Data/Seq/WSeq.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1622 lines, try to split it up
-Mathlib/Data/Set/Basic.lean : line 1 : ERR_NUM_LIN : 2300 file contains 2170 lines, try to split it up
-Mathlib/Data/Set/Finite.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1535 lines, try to split it up
-Mathlib/Data/Set/Function.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1740 lines, try to split it up
-Mathlib/Data/Set/Lattice.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1932 lines, try to split it up
-Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1693 lines, try to split it up
-Mathlib/LinearAlgebra/Dual.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1738 lines, try to split it up
-Mathlib/LinearAlgebra/Multilinear/Basic.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1775 lines, try to split it up
-Mathlib/Logic/Equiv/Basic.lean : line 1 : ERR_NUM_LIN : 2000 file contains 1806 lines, try to split it up
-Mathlib/MeasureTheory/Function/LpSpace.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1978 lines, try to split it up
-Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean : line 1 : ERR_NUM_LIN : 2000 file contains 1879 lines, try to split it up
-Mathlib/MeasureTheory/Integral/Bochner.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1947 lines, try to split it up
-Mathlib/MeasureTheory/Integral/FundThmCalculus.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1647 lines, try to split it up
-Mathlib/MeasureTheory/Integral/Lebesgue.lean : line 1 : ERR_NUM_LIN : 2200 file contains 2051 lines, try to split it up
-Mathlib/MeasureTheory/Integral/SetIntegral.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1579 lines, try to split it up
-Mathlib/MeasureTheory/Integral/SetToL1.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1640 lines, try to split it up
-Mathlib/MeasureTheory/Measure/MeasureSpace.lean : line 1 : ERR_NUM_LIN : 2200 file contains 2008 lines, try to split it up
-Mathlib/MeasureTheory/Measure/Typeclasses.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1531 lines, try to split it up
-Mathlib/Order/CompleteLattice.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1709 lines, try to split it up
-Mathlib/Order/ConditionallyCompleteLattice/Basic.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1549 lines, try to split it up
-Mathlib/Order/Filter/AtTopBot.lean : line 1 : ERR_NUM_LIN : 2000 file contains 1828 lines, try to split it up
-Mathlib/Order/Filter/Basic.lean : line 1 : ERR_NUM_LIN : 3000 file contains 2898 lines, try to split it up
-Mathlib/Order/Hom/Lattice.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1661 lines, try to split it up
-Mathlib/Order/Interval/Set/Basic.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1652 lines, try to split it up
-Mathlib/Order/LiminfLimsup.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1791 lines, try to split it up
-Mathlib/Order/UpperLower/Basic.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1795 lines, try to split it up
-Mathlib/RingTheory/UniqueFactorizationDomain.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1982 lines, try to split it up
-Mathlib/SetTheory/Cardinal/Basic.lean : line 1 : ERR_NUM_LIN : 2200 file contains 2004 lines, try to split it up
-Mathlib/SetTheory/Game/PGame.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1753 lines, try to split it up
-Mathlib/SetTheory/Ordinal/Arithmetic.lean : line 1 : ERR_NUM_LIN : 2400 file contains 2269 lines, try to split it up
-Mathlib/SetTheory/ZFC/Basic.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1549 lines, try to split it up
-Mathlib/Tactic/CC/Addition.lean : line 1 : ERR_NUM_LIN : 2300 file contains 2100 lines, try to split it up
-Mathlib/Tactic/ToAdditive/Frontend.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1502 lines, try to split it up
-Mathlib/Topology/Algebra/Group/Basic.lean : line 1 : ERR_NUM_LIN : 2100 file contains 1932 lines, try to split it up
-Mathlib/Topology/Algebra/Module/Basic.lean : line 1 : ERR_NUM_LIN : 2600 file contains 2405 lines, try to split it up
-Mathlib/Topology/Basic.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1731 lines, try to split it up
-Mathlib/Topology/Category/Profinite/Nobeling.lean : line 1 : ERR_NUM_LIN : 2000 file contains 1806 lines, try to split it up
-Mathlib/Topology/Constructions.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1591 lines, try to split it up
-Mathlib/Topology/ContinuousFunction/Bounded.lean : line 1 : ERR_NUM_LIN : 1800 file contains 1682 lines, try to split it up
-Mathlib/Topology/Instances/ENNReal.lean : line 1 : ERR_NUM_LIN : 1700 file contains 1513 lines, try to split it up
-Mathlib/Topology/Separation.lean : line 1 : ERR_NUM_LIN : 2900 file contains 2716 lines, try to split it up
-Mathlib/Topology/UniformSpace/Basic.lean : line 1 : ERR_NUM_LIN : 1900 file contains 1768 lines, try to split it up
diff --git a/scripts/technical-debt-metrics.sh b/scripts/technical-debt-metrics.sh
index cd49fd69bfaa1..ce9a3a5202b16 100755
--- a/scripts/technical-debt-metrics.sh
+++ b/scripts/technical-debt-metrics.sh
@@ -1,5 +1,10 @@
 #!/usr/bin/env bash
 
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
 # `./scripts/technical-debt-metrics.sh` returns a tally of some technical debts in current Mathlib,
 # reporting also the change with respect to the same counts in
 # Mathlib from last week.
@@ -7,18 +12,8 @@
 # the script takes two optional arguments `<optCurrCommit> <optReferenceCommit>`
 # and tallies the same technical debts on `<optCurrCommit>` using `<optReferenceCommit>`
 # as a reference.
-
-if [ -n "${1}" ]; then
-  currCommit="${1}"
-else
-  currCommit="$(git rev-parse HEAD)"
-fi
-
-if [ -n "${2}" ]; then
-  refCommit="${2}"
-else
-  refCommit="$(git log --pretty=%H --since="$(date -I -d 'last week')" | tail -n -1)"
-fi
+currCommit="${1:-"$(git rev-parse HEAD)"}"
+refCommit="${2:-"$(git log --pretty=%H --since="$(date -I -d 'last week')" | tail -n -1)"}"
 
 # `tdc` produces a semi-formatted output of the form
 # ...
@@ -54,10 +49,11 @@ for i in ${!titlesAndRegexes[@]}; do
 done
 
 printf '%s|%s\n' "$(grep -c 'docBlame' scripts/nolints.json)" "documentation nolint entries"
-printf '%s|%s\n' "$(grep -c 'ERR_NUM_LIN' scripts/style-exceptions.txt)" "large files"
+# We count the number of large files, making sure to avoid counting the test file `test/Lint.lean`.
+printf '%s|%s\n' "$(git grep '^set_option linter.style.longFile [0-9]*' Mathlib | wc -l)" "large files"
 printf '%s|%s\n' "$(git grep "^open .*Classical" | grep -v " in$" -c)" "bare open (scoped) Classical"
-# We print the number of files, not the number of matches --- hence, the nested grep.
-printf '%s|%s\n' "$(git grep -c 'autoImplicit true' | grep -c -v 'test')" "non-test files with autoImplicit true"
+
+printf '%s|%s\n' "$(wc -l < scripts/no_lints_prime_decls.txt)" "exceptions for the docPrime linter"
 
 deprecatedFiles="$(git ls-files '**/Deprecated/*.lean' | xargs wc -l | sed 's=^ *==')"
 
diff --git a/scripts/update_nolints_CI.sh b/scripts/update_nolints_CI.sh
index 42c50b9c2074e..551b6664fa1e5 100755
--- a/scripts/update_nolints_CI.sh
+++ b/scripts/update_nolints_CI.sh
@@ -1,10 +1,13 @@
 #!/usr/bin/env bash
 
-# Check if there are changes to `nolints.json` or `style-exceptions.txt`,
-# and file a PR updating these files otherwise.
+# Check if there are changes to `nolints.json` and file a PR updating it if necessary.
 # DO NOT run this as a human; this is meant only for automation usage!
 
-set -e
+# Make this script robust against unintentional errors.
+# See e.g. http://redsymbol.net/articles/unofficial-bash-strict-mode/ for explanation.
+set -euo pipefail
+IFS=$'\n\t'
+
 set -x
 
 remote_name=origin-bot
@@ -12,17 +15,17 @@ branch_name=nolints
 owner_name=leanprover-community
 
 # Exit if the branch already exists
-git fetch "$remote_name"
+git fetch --quiet "$remote_name"
 git rev-parse --verify --quiet "refs/remotes/${remote_name}/${branch_name}" && exit 0
 
 # Exit if there are no changes relative to master
-git diff-index --quiet "refs/remotes/${remote_name}/master" -- scripts/nolints.json scripts/style-exceptions.txt && exit 0
+git diff-index --quiet "refs/remotes/${remote_name}/master" -- scripts/nolints.json && exit 0
 
-pr_title='chore(scripts): update nolints.json and style-exceptions.txt'
+pr_title='chore(scripts): update nolints.json'
 pr_body='I am happy to remove some nolints for you!'
 
 git checkout -b "$branch_name"
-git add scripts/nolints.json scripts/style-exceptions.txt
+git add scripts/nolints.json
 git commit -m "$pr_title"
 
 gh_api() {
@@ -34,7 +37,7 @@ gh_api() {
 
 git push "${remote_name}" "HEAD:$branch_name"
 
-pr_id=$(gh_api "repos/${owner_name}/mathlib/pulls" -X POST -d @- <<EOF | jq -r .number
+pr_id=$(gh_api "repos/${owner_name}/mathlib4/pulls" -X POST -d @- <<EOF | jq -r .number
 {
   "title": "$pr_title",
   "head": "$branch_name",
@@ -44,6 +47,6 @@ pr_id=$(gh_api "repos/${owner_name}/mathlib/pulls" -X POST -d @- <<EOF | jq -r .
 EOF
 )
 
-gh_api "repos/${owner_name}/mathlib/issues/${pr_id}/comments" -X POST -d @- <<EOF
+gh_api "repos/${owner_name}/mathlib4/issues/${pr_id}/comments" -X POST -d @- <<EOF
 { "body": "bors r+" }
 EOF
diff --git a/test/AssertExists.lean b/test/AssertExists.lean
index 0be0b5c0a3a13..1b12f4b92ea92 100644
--- a/test/AssertExists.lean
+++ b/test/AssertExists.lean
@@ -12,10 +12,10 @@ theorem Nats : True := .intro
 
 /--
 info:
-✅ 'Nats' (declaration) asserted in 'test.AssertExists'.
+✅️ 'Nats' (declaration) asserted in 'test.AssertExists'.
 ---
-✅ means the declaration or import exists.
-❌ means the declaration or import does not exist.
+✅️ means the declaration or import exists.
+❌️ means the declaration or import does not exist.
 -/
 #guard_msgs in
 #check_assertions
@@ -31,23 +31,23 @@ assert_not_imported
 
 /--
 warning:
-✅ 'Nats' (declaration) asserted in 'test.AssertExists'.
-❌ 'I_do_not_exist' (module) asserted in 'test.AssertExists'.
-❌ 'Mathlib.Tactic.Common' (module) asserted in 'test.AssertExists'.
+✅️ 'Nats' (declaration) asserted in 'test.AssertExists'.
+❌️ 'I_do_not_exist' (module) asserted in 'test.AssertExists'.
+❌️ 'Mathlib.Tactic.Common' (module) asserted in 'test.AssertExists'.
 ---
-✅ means the declaration or import exists.
-❌ means the declaration or import does not exist.
+✅️ means the declaration or import exists.
+❌️ means the declaration or import does not exist.
 -/
 #guard_msgs in
 #check_assertions
 
 /--
 warning:
-❌ 'I_do_not_exist' (module) asserted in 'test.AssertExists'.
-❌ 'Mathlib.Tactic.Common' (module) asserted in 'test.AssertExists'.
+❌️ 'I_do_not_exist' (module) asserted in 'test.AssertExists'.
+❌️ 'Mathlib.Tactic.Common' (module) asserted in 'test.AssertExists'.
 ---
-✅ means the declaration or import exists.
-❌ means the declaration or import does not exist.
+✅️ means the declaration or import exists.
+❌️ means the declaration or import does not exist.
 -/
 #guard_msgs in
 #check_assertions!
diff --git a/test/Bound/bound.lean b/test/Bound/bound.lean
index cdb633309e96a..4891a0313efff 100644
--- a/test/Bound/bound.lean
+++ b/test/Bound/bound.lean
@@ -17,6 +17,7 @@ open scoped NNReal
 -- Tests that work with `bound`, but not `positivity`, `gcongr`, or `norm_num`
 section bound_only
 variable {a b c x y : ℝ} {z : ℂ} {n : ℕ}
+
 example (h : x < y) : y - x > 0 := by bound
 example (h : x < y) : Real.exp (y - x) > 1 := by bound
 example (h : x < y) (y0 : 0 < y) : x / y < 1 := by bound
@@ -42,16 +43,17 @@ section guess_tests
 variable {a b c : ℝ} {n m : ℕ}
 example (h : a ≤ b) : a ≤ max b c := by bound
 example (h : a ≤ c) : a ≤ max b c := by bound
-example (h : a < b) : a < max b c := by bound
-example (h : a < c) : a < max b c := by bound
 example (h : a ≤ c) : min a b ≤ c := by bound
 example (h : b ≤ c) : min a b ≤ c := by bound
+example (h : a < b) : a < max b c := by bound
+example (h : a < c) : a < max b c := by bound
 example (h : a < c) : min a b < c := by bound
 example (h : b < c) : min a b < c := by bound
 example (a1 : 1 ≤ a) (h : m ≤ n) : a^m ≤ a^n := by bound
 example (a0 : 0 ≤ a) (a1 : a ≤ 1) (h : n ≤ m) : a^m ≤ a^n := by bound
 example (a1 : 1 ≤ a) (h : b ≤ c) : a^b ≤ a^c := by bound
 example (a0 : 0 < a) (a1 : a ≤ 1) (h : c ≤ b) : a^b ≤ a^c := by bound
+
 end guess_tests
 
 section positive_tests
@@ -108,6 +110,7 @@ example (h : x > 0) : x ≥ 0 := by bound
 example (hc : c ≥ 0) (h : a ≤ b) : a / c ≤ b / c := by bound
 example (ha : a ≥ 0) (hc : c > 0) (h : b ≥ c) : a / b ≤ a / c := by bound
 example (x y : ℝ) (x0 : 0 < x) (h : x ≤ y) : x.log ≤ y.log := by bound
+
 end bound_tests
 
 /-- This broke without appropriate `g.withContext` use in an older implementation of `bound`.
diff --git a/test/CategoryTheory/Bicategory/Basic.lean b/test/CategoryTheory/Bicategory/Basic.lean
new file mode 100644
index 0000000000000..14c74e2ad760c
--- /dev/null
+++ b/test/CategoryTheory/Bicategory/Basic.lean
@@ -0,0 +1,21 @@
+import Mathlib.Tactic.CategoryTheory.Bicategory.Basic
+
+open CategoryTheory Mathlib.Tactic BicategoryLike
+open Bicategory
+
+universe w v u
+
+variable {B : Type u} [Bicategory.{w, v} B]
+variable {a b c d : B}
+
+example {f j : a ⟶ d} {g : a ⟶ b} {h : b ⟶ c} {i : c ⟶ d}
+    (η : f ⟶ g ≫ (h ≫ i)) (θ : (g ≫ h) ≫ i ⟶ j) :
+    η ⊗≫ θ = η ≫ (α_ _ _ _).inv ≫ θ := by
+  bicategory
+
+example {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d} (η : h ⟶ i) :
+    (f ≫ g) ◁ η = (α_ _ _ _).hom ≫ f ◁ g ◁ η ≫ (α_ _ _ _).inv := by
+  bicategory
+
+example {f g h : a ⟶ b} {η : f ⟶ g} {θ : g ⟶ h} : η ≫ θ = η ≫ θ := by
+  bicategory
diff --git a/test/CategoryTheory/Bicategory/Normalize.lean b/test/CategoryTheory/Bicategory/Normalize.lean
new file mode 100644
index 0000000000000..7e6b171a32ec0
--- /dev/null
+++ b/test/CategoryTheory/Bicategory/Normalize.lean
@@ -0,0 +1,54 @@
+import Mathlib.Tactic.CategoryTheory.Bicategory.Normalize
+
+open CategoryTheory Mathlib.Tactic BicategoryLike
+open Bicategory
+
+/-- `normalize% η` is the normalization of the 2-morphism `η`.
+1. The normalized 2-morphism is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁` where
+  each `αᵢ` is a structural 2-morphism (consisting of associators and unitors),
+2. each `ηᵢ` is a non-structural 2-morphism of the form `f₁ ◁ ... ◁ fₘ ◁ θ`, and
+3. `θ` is of the form `ι ▷ g₁ ▷ ... ▷ gₗ`
+-/
+elab "normalize% " t:term:51 : term => do
+  let e ← Lean.Elab.Term.elabTerm t none
+  let ctx : Bicategory.Context ← BicategoryLike.mkContext e
+  CoherenceM.run (ctx := ctx) do
+    return (← BicategoryLike.eval `bicategory (← MkMor₂.ofExpr e)).expr.e.e
+
+universe w v u
+
+variable {B : Type u} [Bicategory.{w, v} B]
+variable {a b c d e : B}
+
+variable {f : a ⟶ b} {g : b ⟶ c} in
+#guard_expr normalize% f ◁ 𝟙 g = (whiskerLeftIso f (Iso.refl g)).hom
+variable {f : a ⟶ b} {g : b ⟶ c} in
+#guard_expr normalize% 𝟙 f ▷ g = (whiskerRightIso (Iso.refl f) g).hom
+variable {f : a ⟶ b} {g h i : b ⟶ c} {η : g ⟶ h} {θ : h ⟶ i} in
+#guard_expr normalize% f ◁ (η ≫ θ) = _ ≫ f ◁ η ≫ _ ≫ f ◁ θ ≫ _
+variable {f g h : a ⟶ b} {i : b ⟶ c} {η : f ⟶ g} {θ : g ⟶ h} in
+#guard_expr normalize% (η ≫ θ) ▷ i = _ ≫ η ▷ i ≫ _ ≫ θ ▷ i ≫ _
+variable {η : 𝟙 a ⟶ 𝟙 a} in
+#guard_expr normalize% 𝟙 a ◁ η = _ ≫ η ≫ _
+variable {f : a ⟶ b} {g : b ⟶ c} {h i : c ⟶ d} {η : h ⟶ i} in
+#guard_expr normalize% (f ≫ g) ◁ η = _ ≫ f ◁ g ◁ η ≫ _
+variable {η : 𝟙 a ⟶ 𝟙 a} in
+#guard_expr normalize% η ▷ 𝟙 a = _ ≫ η ≫ _
+variable {f g : a ⟶ b} {h : b ⟶ c} {i : c ⟶ d} {η : f ⟶ g} in
+#guard_expr normalize% η ▷ (h ≫ i) = _ ≫ η ▷ h ▷ i ≫ _
+variable {f : a ⟶ b} {g h : b ⟶ c} {i : c ⟶ d} {η : g ⟶ h} in
+#guard_expr normalize% (f ◁ η) ▷ i = _ ≫ f ◁ η ▷ i ≫ _
+variable {f : a ⟶ b} in
+#guard_expr normalize% (λ_ f).hom = (λ_ f).hom
+variable {f : a ⟶ b} in
+#guard_expr normalize% (λ_ f).inv = ((λ_ f).symm).hom
+variable {f : a ⟶ b} in
+#guard_expr normalize% (ρ_ f).hom = (ρ_ f).hom
+variable {f : a ⟶ b} in
+#guard_expr normalize% (ρ_ f).inv = ((ρ_ f).symm).hom
+variable {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} in
+#guard_expr normalize% (α_ f g h).hom = (α_ _ _ _).hom
+variable {f : a ⟶ b} {g : b ⟶ c} {h : c ⟶ d} in
+#guard_expr normalize% (α_ f g h).inv = ((α_ f g h).symm).hom
+variable {f : a ⟶ b} {g : b ⟶ c} in
+#guard_expr normalize% 𝟙 (f ≫ g) = (Iso.refl (f ≫ g)).hom
diff --git a/test/CategoryTheory/Coherence.lean b/test/CategoryTheory/Coherence.lean
index 14f04ea1f2efb..35f0cdf3b3148 100644
--- a/test/CategoryTheory/Coherence.lean
+++ b/test/CategoryTheory/Coherence.lean
@@ -4,7 +4,7 @@ open CategoryTheory
 
 universe w v u
 
-section monoidal
+section Monoidal
 variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
 open scoped MonoidalCategory
 
@@ -79,7 +79,7 @@ example (X₁ X₂ : C) :
     (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).inv) := by
   coherence
 
-end monoidal
+end Monoidal
 
 section Bicategory
 
@@ -106,6 +106,7 @@ example (f : a ⟶ b) (g : b ⟶ c) :
 example : 𝟙 (𝟙 a ≫ 𝟙 a) ≫ (λ_ (𝟙 a)).hom = 𝟙 (𝟙 a ≫ 𝟙 a) ≫ (ρ_ (𝟙 a)).hom := by
   bicategory_coherence
 
+set_option linter.unusedVariables false in
 example (f g : a ⟶ a) (η : 𝟙 a ⟶ f) (θ : f ⟶ g) (w : false) :
   (λ_ (𝟙 a)).hom ≫ η ≫ θ = (ρ_ (𝟙 a)).hom ≫ η ≫ θ := by
   coherence
diff --git a/test/CategoryTheory/Monoidal.lean b/test/CategoryTheory/Monoidal.lean
deleted file mode 100644
index fd521a53e7c5d..0000000000000
--- a/test/CategoryTheory/Monoidal.lean
+++ /dev/null
@@ -1,44 +0,0 @@
-import Mathlib.Tactic.CategoryTheory.Monoidal
-
-open CategoryTheory
-open scoped MonoidalCategory
-
-universe v u
-
-variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
-variable {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
-
-#guard_expr normalize% X ◁ 𝟙 Y = X ◁ 𝟙 Y
-#guard_expr normalize% 𝟙 X ▷ Y = 𝟙 X ▷ Y
-#guard_expr normalize% X ◁ (f ≫ g) = _ ≫ X ◁ f ≫ _ ≫ X ◁ g ≫ _
-#guard_expr normalize% (f ≫ g) ▷ Y = _ ≫ f ▷ Y ≫ _ ≫ g ▷ Y ≫ _
-#guard_expr normalize% 𝟙_ C ◁ f = _ ≫ f ≫ _
-#guard_expr normalize% (X ⊗ Y) ◁ f = _ ≫ X ◁ Y ◁ f ≫ _
-#guard_expr normalize% f ▷ 𝟙_ C = _ ≫ f ≫ _
-#guard_expr normalize% f ▷ (X ⊗ Y) = _ ≫ f ▷ X ▷ Y ≫ _
-#guard_expr normalize% (X ◁ f) ▷ Y = _ ≫ X ◁ f ▷ Y ≫ _
-#guard_expr normalize% (λ_ X).hom = (λ_ X).hom
-#guard_expr normalize% (λ_ X).inv = (λ_ X).inv
-#guard_expr normalize% (ρ_ X).hom = (ρ_ X).hom
-#guard_expr normalize% (ρ_ X).inv = (ρ_ X).inv
-#guard_expr normalize% (α_ X Y Z).hom = (α_ _ _ _).hom
-#guard_expr normalize% (α_ X Y Z).inv = (α_ _ _ _).inv
-#guard_expr normalize% 𝟙 (X ⊗ Y) = 𝟙 (X ⊗ Y)
-variable {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) in
-#guard_expr normalize% R V₁ V₂ ▷ V₃ ⊗≫ V₂ ◁ R V₁ V₃ = _ ≫ R V₁ V₂ ▷ V₃ ≫ _ ≫ V₂ ◁ R V₁ V₃ ≫ _
-
-example (f : U ⟶ V ⊗ (W ⊗ X)) (g : (V ⊗ W) ⊗ X ⟶ Y) :
-    f ⊗≫ g = f ≫ 𝟙 _ ≫ (α_ _ _ _).inv ≫ g := by
-  monoidal_nf
-  repeat' apply congrArg₂ (· ≫ ·) ?_ <| congrArg₂ (· ≫ ·) rfl ?_
-  all_goals simp
-
-example : (X ⊗ Y) ◁ f = (α_ _ _ _).hom ≫ X ◁ Y ◁ f ≫ (α_ _ _ _).inv := by
-  monoidal_nf
-  repeat' apply congrArg₂ (· ≫ ·) ?_ <| congrArg₂ (· ≫ ·) rfl ?_
-  all_goals simp
-
-example : f ≫ g = f ≫ g := by
-  monoidal_nf
-  repeat' apply congrArg₂ (· ≫ ·) ?_ <| congrArg₂ (· ≫ ·) rfl ?_
-  all_goals simp
diff --git a/test/CategoryTheory/Monoidal/Basic.lean b/test/CategoryTheory/Monoidal/Basic.lean
new file mode 100644
index 0000000000000..c003140e47a35
--- /dev/null
+++ b/test/CategoryTheory/Monoidal/Basic.lean
@@ -0,0 +1,27 @@
+import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
+
+open CategoryTheory Mathlib.Tactic BicategoryLike
+open MonoidalCategory
+
+universe v u
+
+variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
+variable {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
+
+example (f : U ⟶ V ⊗ (W ⊗ X)) (g : (V ⊗ W) ⊗ X ⟶ Y) :
+    f ⊗≫ g = f ≫ (α_ _ _ _).inv ≫ g := by
+  monoidal
+
+example (f : Z ⟶ W) : (X ⊗ Y) ◁ f = (α_ _ _ _).hom ≫ X ◁ Y ◁ f ≫ (α_ _ _ _).inv := by
+  monoidal
+
+example : f ≫ g = f ≫ g := by
+  monoidal
+
+example : (f ⊗ g) ▷ X = (α_ _ _ _).hom ≫ (f ⊗ g ▷ X) ≫ (α_ _ _ _).inv := by
+  monoidal
+
+example {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) :
+    R V₁ V₂ ▷ V₃ ⊗≫ V₂ ◁ R V₁ V₃ =
+      R V₁ V₂ ▷ V₃ ≫ (α_ _ _ _).hom ⊗≫ 𝟙 _ ≫ V₂ ◁ R V₁ V₃ := by
+  monoidal
diff --git a/test/CategoryTheory/Monoidal/Normalize.lean b/test/CategoryTheory/Monoidal/Normalize.lean
new file mode 100644
index 0000000000000..9ae14f44ac833
--- /dev/null
+++ b/test/CategoryTheory/Monoidal/Normalize.lean
@@ -0,0 +1,41 @@
+import Mathlib.Tactic.CategoryTheory.Monoidal.Normalize
+
+open CategoryTheory Mathlib.Tactic BicategoryLike
+open MonoidalCategory
+
+/-- `normalize% η` is the normalization of the 2-morphism `η`.
+1. The normalized 2-morphism is of the form `α₀ ≫ η₀ ≫ α₁ ≫ η₁ ≫ ... αₘ ≫ ηₘ ≫ αₘ₊₁` where
+  each `αᵢ` is a structural 2-morphism (consisting of associators and unitors),
+2. each `ηᵢ` is a non-structural 2-morphism of the form `f₁ ◁ ... ◁ fₘ ◁ θ`, and
+3. `θ` is of the form `ι ▷ g₁ ▷ ... ▷ gₗ`
+-/
+elab "normalize% " t:term:51 : term => do
+  let e ← Lean.Elab.Term.elabTerm t none
+  let ctx : Monoidal.Context ← BicategoryLike.mkContext e
+  CoherenceM.run (ctx := ctx) do
+    return (← BicategoryLike.eval `monoidal (← MkMor₂.ofExpr e)).expr.e.e
+
+universe v u
+
+variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
+variable {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z)
+
+#guard_expr normalize% X ◁ 𝟙 Y = (whiskerLeftIso X (Iso.refl Y)).hom
+#guard_expr normalize% 𝟙 X ▷ Y = (whiskerRightIso (Iso.refl X) Y).hom
+#guard_expr normalize% X ◁ (f ≫ g) = _ ≫ X ◁ f ≫ _ ≫ X ◁ g ≫ _
+#guard_expr normalize% (f ≫ g) ▷ Y = _ ≫ f ▷ Y ≫ _ ≫ g ▷ Y ≫ _
+#guard_expr normalize% 𝟙_ C ◁ f = _ ≫ f ≫ _
+#guard_expr normalize% (X ⊗ Y) ◁ f = _ ≫ X ◁ Y ◁ f ≫ _
+#guard_expr normalize% f ▷ 𝟙_ C = _ ≫ f ≫ _
+#guard_expr normalize% f ▷ (X ⊗ Y) = _ ≫ f ▷ X ▷ Y ≫ _
+#guard_expr normalize% (X ◁ f) ▷ Y = _ ≫ X ◁ f ▷ Y ≫ _
+#guard_expr normalize% (λ_ X).hom = (λ_ X).hom
+#guard_expr normalize% (λ_ X).inv = ((λ_ X).symm).hom
+#guard_expr normalize% (ρ_ X).hom = (ρ_ X).hom
+#guard_expr normalize% (ρ_ X).inv = ((ρ_ X).symm).hom
+#guard_expr normalize% (α_ X Y Z).hom = (α_ _ _ _).hom
+#guard_expr normalize% (α_ X Y Z).inv = ((α_ X Y Z).symm).hom
+#guard_expr normalize% 𝟙 (X ⊗ Y) = (Iso.refl (X ⊗ Y)).hom
+#guard_expr normalize% f ⊗ g = _ ≫ (f ⊗ g) ≫ _
+variable {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) in
+#guard_expr normalize% R V₁ V₂ ▷ V₃ ⊗≫ V₂ ◁ R V₁ V₃ = _ ≫ R V₁ V₂ ▷ V₃ ≫ _ ≫ V₂ ◁ R V₁ V₃ ≫ _
diff --git a/test/CategoryTheory/SubstHomLift.lean b/test/CategoryTheory/SubstHomLift.lean
new file mode 100644
index 0000000000000..fc07db561cd82
--- /dev/null
+++ b/test/CategoryTheory/SubstHomLift.lean
@@ -0,0 +1,25 @@
+import Mathlib.CategoryTheory.FiberedCategory.HomLift
+
+universe u₁ v₁ u₂ v₂
+
+open CategoryTheory Category
+
+variable {𝒮 : Type u₁} {𝒳 : Type u₂} [Category.{v₁} 𝒳] [Category.{v₂} 𝒮] (p : 𝒳 ⥤ 𝒮)
+
+
+/-- Testing simple substitution -/
+example {R S : 𝒮} {a b : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) [p.IsHomLift f φ] : f = f := by
+  subst_hom_lift p f φ
+  rename_i h
+  guard_hyp h : p.IsHomLift (p.map φ) φ
+  guard_target = p.map φ = p.map φ
+  trivial
+
+/-- Test substitution with more complicated expression -/
+example {R S T : 𝒮} {a b c : 𝒳} (f : R ⟶ S) (g : S ⟶ T) (φ : a ⟶ b) (ψ : b ⟶ c)
+    [p.IsHomLift f (φ ≫ ψ)] : f = f := by
+  subst_hom_lift p f (φ ≫ ψ)
+  rename_i h
+  guard_hyp h : p.IsHomLift (p.map (φ ≫ ψ)) (φ ≫ ψ)
+  guard_target = p.map (φ ≫ ψ) = p.map (φ ≫ ψ)
+  trivial
diff --git a/test/CategoryTheory/ToApp.lean b/test/CategoryTheory/ToApp.lean
new file mode 100644
index 0000000000000..6a4da3643397e
--- /dev/null
+++ b/test/CategoryTheory/ToApp.lean
@@ -0,0 +1,41 @@
+import Mathlib.Tactic.CategoryTheory.ToApp
+import Mathlib.CategoryTheory.Bicategory.Functor.Prelax
+
+universe w v u
+
+namespace CategoryTheory.ToAppTest
+
+open Bicategory Category
+
+variable {B : Type u} [Bicategory.{w, v} B] {a b c d e : B}
+
+@[to_app]
+theorem whiskerLeft_hom_inv (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) :
+    f ◁ η.hom ≫ f ◁ η.inv = 𝟙 (f ≫ g) := by
+  rw [← Bicategory.whiskerLeft_comp, Iso.hom_inv_id, Bicategory.whiskerLeft_id]
+
+example {a b c : Cat} (f : a ⟶ b) {g h : b ⟶ c} (η : g ≅ h) (X : a) :
+    η.hom.app (f.obj X) ≫ η.inv.app (f.obj X) = 𝟙 ((f ≫ g).obj X) :=
+  whiskerLeft_hom_inv_app f η X
+
+@[to_app]
+theorem pentagon_hom_hom_inv_inv_hom (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
+    (α_ f (g ≫ h) i).hom ≫ f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv =
+      (α_ f g h).inv ▷ i ≫ (α_ (f ≫ g) h i).hom :=
+  eq_of_inv_eq_inv (by simp)
+
+example {a b c d e : Cat} (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) (X : ↑a) :
+    (α_ f (g ≫ h) i).hom.app X ≫ (α_ g h i).hom.app (f.obj X) ≫ (α_ f g (h ≫ i)).inv.app X =
+      i.map ((α_ f g h).inv.app X) ≫ (α_ (f ≫ g) h i).hom.app X :=
+  pentagon_hom_hom_inv_inv_hom_app f g h i X
+
+@[to_app]
+theorem testThm {C : Type*} [Bicategory C] (F : PrelaxFunctor B C) {a b : B} {f g : a ⟶ b}
+    (η : f ⟶ g) : F.map₂ η ≫ F.map₂ (𝟙 g) = F.map₂ η := by simp
+
+example {B : Type u_1} [Bicategory B] (F : PrelaxFunctor B Cat)
+    {a b : B} {f g : a ⟶ b} (η : f ⟶ g) (X : ↑(F.obj a)) :
+    (F.map₂ η).app X ≫ (F.map₂ (𝟙 g)).app X = (F.map₂ η).app X :=
+  testThm_app F η X
+
+end CategoryTheory.ToAppTest
diff --git a/test/Change.lean b/test/Change.lean
index babc274c86b5c..a221341553ada 100644
--- a/test/Change.lean
+++ b/test/Change.lean
@@ -1,5 +1,6 @@
 import Mathlib.Tactic.Change
 
+set_option linter.style.setOption false
 set_option pp.unicode.fun true
 set_option autoImplicit true
 
diff --git a/test/Check.lean b/test/Check.lean
index 5deb44a3b1e48..baa5c95f009a5 100644
--- a/test/Check.lean
+++ b/test/Check.lean
@@ -6,6 +6,8 @@ open Lean PrettyPrinter Delaborator in @[delab mvar] def delabMVar : Delab := do
   unless kind.isNatural do failure
   `(?m)
 
+set_option linter.unusedTactic false
+
 /-!
 Basic check of `#check`
 -/
diff --git a/test/Clear!.lean b/test/Clear!.lean
index 51a427231d9c3..cf2817c2edc17 100644
--- a/test/Clear!.lean
+++ b/test/Clear!.lean
@@ -13,7 +13,8 @@ example [delete_this : Inhabited Nat] : Inhabited Nat := by
   infer_instance
 
 -- Confirms clear! can clear the dependencies of multiple hypotheses
-example (delete_this : Nat) (delete_this2 : Nat) (_delete_this_dep : delete_this = delete_this2) : Nat := by
+example (delete_this : Nat) (delete_this2 : Nat) (_delete_this_dep : delete_this = delete_this2) :
+    Nat := by
   clear! delete_this delete_this2
   fail_if_success assumption
   exact 0
diff --git a/test/ClearExcept.lean b/test/ClearExcept.lean
index eb1c41a293f92..63bdec1685482 100644
--- a/test/ClearExcept.lean
+++ b/test/ClearExcept.lean
@@ -1,5 +1,7 @@
 import Mathlib.Tactic.ClearExcept
 
+set_option linter.unusedTactic false
+
 -- Most basic test
 example (_delete_this : Nat) (dont_delete_this : Int) : Nat := by
   clear * - dont_delete_this
@@ -12,13 +14,16 @@ example [dont_delete_this : Inhabited Nat] (dont_delete_this2 : Prop) : Inhabite
   assumption
 
 -- Confirms that clearExcept can clear hypotheses even when they have dependencies
-example (delete_this : Nat) (_delete_this2 : delete_this = delete_this) (dont_delete_this : Int) : Nat := by
+example (delete_this : Nat) (_delete_this2 : delete_this = delete_this) (dont_delete_this : Int) :
+    Nat := by
   clear * - dont_delete_this
   fail_if_success assumption
   exact dont_delete_this.toNat
 
--- Confirms that clearExcept does not clear hypotheses when they have dependencies that should not be cleared
-example (dont_delete_this : Nat) (dont_delete_this2 : dont_delete_this = dont_delete_this) : Nat := by
+-- Confirms that clearExcept does not clear hypotheses
+-- when they have dependencies that should not be cleared
+example (dont_delete_this : Nat) (dont_delete_this2 : dont_delete_this = dont_delete_this) :
+    Nat := by
   clear * - dont_delete_this2
   exact dont_delete_this
 
diff --git a/test/Clear_.lean b/test/Clear_.lean
index c3dfeefe0a60c..163790d5fb052 100644
--- a/test/Clear_.lean
+++ b/test/Clear_.lean
@@ -1,6 +1,8 @@
 import Mathlib.Tactic.Clear_
 import Mathlib.Tactic.Replace
 
+set_option linter.unusedTactic false
+
 -- Most basic test
 example (_delete_this : Nat) : Nat := by
   clear_
@@ -19,14 +21,16 @@ example (_delete_this : Nat) (dont_delete_this : Int) : Nat := by
   exact dont_delete_this.toNat
 
 -- Confirms that clear_ can clear hypotheses even when they have dependencies
-example (_delete_this : Type) (_delete_this_dep : _delete_this) (_delete_this_rw : _delete_this = Nat)
-  (_delete_this_dep_dep : _delete_this_dep = _delete_this_dep) : Nat := by
+example (_delete_this : Type) (_delete_this_dep : _delete_this)
+    (_delete_this_rw : _delete_this = Nat)
+    (_delete_this_dep_dep : _delete_this_dep = _delete_this_dep) : Nat := by
   clear_
   fail_if_success
     rw [← _delete_this_rw]
   exact 0
 
--- Confirms that clear_ does not clear hypotheses when they have dependencies that should not be cleared
+-- Confirms that clear_ does not clear hypotheses
+-- when they have dependencies that should not be cleared
 example (_dont_delete_this : Type) (dep : _dont_delete_this) : _dont_delete_this := by
   clear_
   assumption
diff --git a/test/Continuity.lean b/test/Continuity.lean
index 7df66051c1b4e..bf0d389a95e2a 100644
--- a/test/Continuity.lean
+++ b/test/Continuity.lean
@@ -1,6 +1,6 @@
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
 import Mathlib.Topology.Basic
-import Mathlib.Topology.ContinuousFunction.Basic
+import Mathlib.Topology.ContinuousMap.Basic
 
 set_option autoImplicit true
 section basic
@@ -45,7 +45,7 @@ example : Continuous (fun x : ℝ => exp ((max x (-x)) + sin x)^2) := by
 example : Continuous (fun x : ℝ => exp ((max x (-x)) + sin (cos x))^2) := by
   continuity
 
--- Examples taken from `Topology.ContinuousFunction.Basic`:
+-- Examples taken from `Topology.ContinuousMap.Basic`:
 
 example (b : Y) : Continuous (fun _ : X => b) := by continuity
 
@@ -53,8 +53,7 @@ example (f : C(X, Y)) (g : C(Y, Z)) : Continuous (g ∘ f) := by continuity
 
 example (f : C(X, Y)) (g : C(X, Z)) : Continuous (fun x => (f x, g x)) := by continuity
 
-example (f : C(X, Y)) (g : C(W, Z)) : Continuous (Prod.map f g) := --by continuity
-  f.continuous.prod_map g.continuous
+example (f : C(X, Y)) (g : C(W, Z)) : Continuous (Prod.map f g) := by continuity
 
 example (f : ∀ i, C(X, X' i)) : Continuous (fun a i => f i a) := by continuity
 
diff --git a/test/DefEqTransformations.lean b/test/DefEqTransformations.lean
index dc01755e7f38c..496e641a680f8 100644
--- a/test/DefEqTransformations.lean
+++ b/test/DefEqTransformations.lean
@@ -1,11 +1,12 @@
 import Mathlib.Tactic.DefEqTransformations
-import Mathlib.Init.Logic
 
 set_option autoImplicit true
 
 private axiom test_sorry : ∀ {α}, α
 namespace Tests
 
+set_option linter.unusedTactic false
+
 example : id (1 = 1) := by
   with_reducible whnf
   guard_target =ₛ id (1 = 1)
diff --git a/test/DeriveToExpr.lean b/test/DeriveToExpr.lean
index 7780eef44e28c..bed70062a0acb 100644
--- a/test/DeriveToExpr.lean
+++ b/test/DeriveToExpr.lean
@@ -57,6 +57,7 @@ instance {α : Type u} [ToExpr α] [ToLevel.{u+1}] : ToExpr (Bool → α) where
 
 deriving instance ToExpr for Bar
 
+set_option linter.unusedTactic false in
 example : True := by
   run_tac do
     let f : Bool → Nat | false => 0 | true => 1
diff --git a/test/DocPrime.lean b/test/DocPrime.lean
new file mode 100644
index 0000000000000..3809fa9675e65
--- /dev/null
+++ b/test/DocPrime.lean
@@ -0,0 +1,80 @@
+import Mathlib.Tactic.Linter.DocPrime
+import Mathlib.Tactic.Lemma
+
+set_option linter.docPrime true
+
+-- no warning on a primed-declaration with a doc-string containing `'`
+/-- X' has a doc-string -/
+def X' := 0
+
+-- no warning on a declaration whose name contains a `'` *and does not end with it*
+def X'X := 0
+
+-- A list of universe names in the declaration is handled correctly, i.e. warns.
+/--
+warning: `Y'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+def Y'.{u} := ULift.{u} Nat
+
+namespace X
+/--
+warning: `ABC.thm_no_doc1'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+theorem _root_.ABC.thm_no_doc1' : True := .intro
+
+/--
+warning: `X.thm_no_doc2'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+theorem thm_no_doc2' : True := .intro
+
+end X
+
+/--
+warning: `thm_no_doc'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+theorem thm_no_doc' : True := .intro
+
+/--
+warning: `thm_with_attr_no_doc'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+@[simp]
+theorem thm_with_attr_no_doc' : True := .intro
+
+/--
+warning: `inst_no_doc'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+instance inst_no_doc' : True := .intro
+
+/--
+warning: `abbrev_no_doc'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+abbrev abbrev_no_doc' : True := .intro
+
+/--
+warning: `def_no_doc'` is missing a doc-string, please add one.
+Declarations whose name ends with a `'` are expected to contain an explanation for the presence of a `'` in their doc-string. This may consist of discussion of the difference relative to the unprimed version, or an explanation as to why no better naming scheme is possible.
+note: this linter can be disabled with `set_option linter.docPrime false`
+-/
+#guard_msgs in
+def def_no_doc' : True := .intro
diff --git a/test/ExtractGoal.lean b/test/ExtractGoal.lean
index a69a4e97db20b..c4bb2a70c7fb6 100644
--- a/test/ExtractGoal.lean
+++ b/test/ExtractGoal.lean
@@ -1,8 +1,8 @@
 import Mathlib.Tactic.ExtractGoal
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Order.Basic
 import Mathlib.Data.Nat.Defs
 
+set_option linter.style.setOption false
 set_option pp.unicode.fun true
 set_option autoImplicit true
 set_option linter.unusedVariables false
diff --git a/test/ExtractLets.lean b/test/ExtractLets.lean
index 1cbe0dcc7678c..b2ff5f1b6ecb4 100644
--- a/test/ExtractLets.lean
+++ b/test/ExtractLets.lean
@@ -32,7 +32,6 @@ example (h : let x := 1; let y := 2; x + 1 = y) : True := by
 
 example (h : let x := 1; let y := 2; x + 1 = y) : True := by
   extract_lets x at h
-  intros
   guard_hyp x : Nat := 1
   guard_hyp h :ₛ let y := 2; x + 1 = y
   trivial
diff --git a/test/FlexibleLinter.lean b/test/FlexibleLinter.lean
new file mode 100644
index 0000000000000..75ff835bdc5a3
--- /dev/null
+++ b/test/FlexibleLinter.lean
@@ -0,0 +1,342 @@
+import Batteries.Tactic.PermuteGoals
+import Mathlib.Tactic.Linter.FlexibleLinter
+import Mathlib.Tactic.Abel
+import Mathlib.Tactic.Ring
+
+set_option linter.flexible false
+
+/--
+warning: 'simp at h' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'exact h' uses 'h'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example (h : 0 + 0 = 0) : True := by
+  simp at h
+  try exact h
+
+-- `subst` does not use the goal
+#guard_msgs in
+example {a b : Nat} (h : a = b) : a + 0 = b := by
+  simp
+  subst h
+  rfl
+
+-- `by_cases` does not use the goal
+#guard_msgs in
+example {a b : Nat} (h : a = b) : a + 0 = b := by
+  simp
+  by_cases a = b
+  subst h; rfl
+  subst h; rfl
+
+-- `induction` does not use the goal
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'assumption' uses '⊢'!
+---
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'assumption' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example {a b : Nat} (h : a = b) : a + 0 = b := by
+  simp
+  induction a <;> assumption
+
+
+/--
+warning: 'simp at h' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'exact h' uses 'h'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example (h : 0 = 0 ∨ 0 = 0) : True := by
+  cases h <;>
+    rename_i h <;>
+    simp at h
+  · exact h
+  · assumption --exact h
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'on_goal 2 => · contradiction' uses '⊢'!
+---
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'contradiction' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example (h : 0 = 1 ∨ 0 = 1) : 0 = 1 ∧ 0 = 1 := by
+  cases h <;> simp
+  on_goal 2 => · contradiction
+  · contradiction
+
+-- `omega` is a follower and `all_goals` is a `combinatorLike`
+#guard_msgs in
+example {a : Nat} : a + 1 + 0 = 1 + a := by simp; all_goals omega
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'contradiction' uses '⊢'!
+---
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'contradiction' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example (h : 0 = 1 ∨ 0 = 1) : 0 = 1 ∧ 0 = 1 := by
+  cases h <;> simp
+  · contradiction
+  · contradiction
+
+/--
+warning: 'simp at h k' is a flexible tactic modifying 'k'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at k' uses 'k'!
+---
+warning: 'simp at h k' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at h' uses 'h'!
+-/
+#guard_msgs in
+-- `simp at h` stains `h` but not other locations
+set_option linter.flexible true in
+example {h : 0 = 0} {k : 1 = 1} : True := by
+  simp at h k;
+  rw [← Classical.not_not (a := True)]
+  -- flag the two below vvv do not above ^^^
+  rw [← Classical.not_not (a := True)] at k
+  rw [← Classical.not_not (a := True)] at h
+  assumption
+
+-- `specialize` does not touch, by default, the target
+#guard_msgs in
+example {a b : Nat} (h : ∀ c, c + a + b = a + c) : (0 + 2 + 1 + a + b) = a + 3 := by
+  simp
+  specialize h 3
+  simp_all
+
+-- `norm_num` is allowed after `simp`.
+#guard_msgs in
+example : (0 + 2 : Rat) + 1 = 3 := by
+  simp
+  norm_num
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [add_comm]' uses '⊢'!
+-/
+#guard_msgs in
+-- `norm_num` is allowed after `simp`, but "passes along the stain".
+set_option linter.flexible true in
+example {a : Rat} : a + (0 + 2 + 1 : Rat) = 3 + a := by
+  simp
+  norm_num
+  rw [add_comm]
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'exact h.symm' uses '⊢'!
+-/
+#guard_msgs in
+-- `congr` is allowed after `simp`, but "passes along the stain".
+set_option linter.flexible true in
+example {a b : Nat} (h : a = b) : a + b + 0 = b + a := by
+  simp
+  congr
+  exact h.symm
+
+-- `done` is an allowed follower
+#guard_msgs in
+example (h : False) : 0 ≠ 0 := by
+  try (simp; done)
+  exact h.elim
+
+--  `abel_nf` is a `rigidifier`: the "stain" of `simp` does not continue past `abel_nf`.
+#guard_msgs in
+example {a b : Nat} (h : a + b = a + (b + 1)) : a + b = b + a + 0 + 1 := by
+  simp
+  abel_nf
+  assumption
+
+--  `abel` is an allowed `simp`-follower.
+#guard_msgs in
+example {a b : Nat} : a + b = b + a + 0 := by
+  simp
+  abel
+
+--  `ring_nf` is a `rigidifier`: the "stain" of `simp` does not continue past `ring_nf`.
+#guard_msgs in
+example {a b : Nat} (h : a + b = 1 + a + b) : a + b = b + a + 0 + 1 := by
+  simp
+  ring_nf
+  assumption
+
+--  `ring` is an allowed `simp`-follower.
+#guard_msgs in
+example {a b : Nat} : a + b = b + a + 0 := by
+  simp
+  ring
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'contradiction' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example (h : 0 = 1 ∨ 0 = 1) : 0 = 1 ∧ 0 = 1 := by
+  cases h <;> simp
+  · simp_all
+  · contradiction
+
+-- forget stained locations, once the corresponding goal is closed
+#guard_msgs in
+example (n : Nat) : n + 1 = 1 + n := by
+  by_cases 0 = 0
+  · simp_all
+    omega
+  · have : 0 ≠ 1 := by
+      intro h
+      -- should not flag `cases`!
+      cases h
+    -- should not flag `exact`!
+    exact Nat.add_comm ..
+
+/--
+warning: 'simp at h' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at h' uses 'h'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+-- `simp at h` stains `h` but not other locations
+example {h : 0 = 0} {k : 1 = 1} : ¬ ¬ True := by
+  simp at h
+  rw [← Nat.add_zero 1] at k
+  -- flag below vvv do not flag above ^^^
+  rw [← Classical.not_not (a := True)] at h
+  --exact h -- <-- flagged
+  assumption
+
+/--
+warning: 'simp at h k' is a flexible tactic modifying 'k'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at k' uses 'k'!
+---
+warning: 'simp at h k' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at h' uses 'h'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+-- `simp at h` stains `h` but not other locations
+example {h : 0 = 0} {k : 1 = 1} : True := by
+  simp at h k
+  rw [← Classical.not_not (a := True)]
+  -- flag the two below vvv do not above ^^^
+  rw [← Classical.not_not (a := True)] at k
+  rw [← Classical.not_not (a := True)] at h
+  assumption
+
+/--
+warning: 'simp at h' is a flexible tactic modifying 'h'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rw [← Classical.not_not (a := True)] at h' uses 'h'!
+-/
+#guard_msgs in
+-- `simp at h` stains `h` but not other locations
+set_option linter.flexible true in
+example {h : 0 = 0} : True := by
+  simp at h
+  rw [← Classical.not_not (a := True)]
+  -- flag below vvv do not flag above ^^^
+  rw [← Classical.not_not (a := True)] at h
+  assumption
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rwa [← Classical.not_not (a := False)]' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example {h : False} : 0 = 1 := by
+  simp
+  rw [← Classical.not_not (a := False)] at h
+  -- flag below vvv do not flag above ^^^
+  rwa [← Classical.not_not (a := False)]
+
+/--
+warning: 'simp' is a flexible tactic modifying '⊢'…
+note: this linter can be disabled with `set_option linter.flexible false`
+---
+info: … and 'rwa [← Classical.not_not (a := False)]' uses '⊢'!
+-/
+#guard_msgs in
+set_option linter.flexible true in
+example {h : False} : 0 = 1 ∧ 0 = 1 := by
+  constructor
+  · simpa
+  . simp
+    rw [← Classical.not_not (a := False)] at h
+    rwa [← Classical.not_not (a := False)]
+
+section test_internals
+open Lean Mathlib.Linter Flexible
+
+/-- `flex? tac` logs an info `true` if the tactic is flexible, logs a warning `false` otherwise. -/
+elab "flex? " tac:tactic : command => do
+  match flexible? tac with
+    | true  => logWarningAt tac m!"{flexible? tac}"
+    | false => logInfoAt tac m!"{flexible? tac}"
+
+section
+set_option linter.unusedTactic false
+set_option linter.unreachableTactic false
+/-- info: false -/#guard_msgs in
+flex? done
+/-- info: false -/#guard_msgs in
+flex? simp only
+/-- info: false -/#guard_msgs in
+flex? simp_all only
+/-- warning: true -/#guard_msgs in
+flex? simp
+/-- warning: true -/#guard_msgs in
+flex? simp_all
+end
+
+/-- info: #[h] -/ #guard_msgs in
+#eval show CoreM _ from do
+  let h := mkIdent `h
+  let hc : TSyntax `Lean.Parser.Tactic.casesTarget := ⟨h⟩
+  IO.println s!"{(toStained (← `(tactic| cases $hc))).toArray}"
diff --git a/test/GCongr/inequalities.lean b/test/GCongr/inequalities.lean
index da96f38939562..02f0a2850793c 100644
--- a/test/GCongr/inequalities.lean
+++ b/test/GCongr/inequalities.lean
@@ -228,3 +228,13 @@ example {α β : Type*}  [SemilatticeSup α] (f : β → α)
     {s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty) :
     s₁.sup' h₁ f ≤ s₂.sup' h₂ f := by
   gcongr
+
+/-! Test that `gcongr` can solve side goals of the form `∀ i, f i` when `f i` is in scope for
+`positivity` -/
+
+example {ι : Type*} [Fintype ι] {f g : ι → ℝ} : ∏ i, f i ^ 2 ≤ ∏ i, g i ^ 2 := by
+  gcongr with i _ i _
+  · guard_target = 0 ≤ f i
+    exact test_sorry
+  · guard_target = f i ≤ g i
+    exact test_sorry
diff --git a/test/GuardGoalNums.lean b/test/GuardGoalNums.lean
index 89e219de17f11..143a75fa193c8 100644
--- a/test/GuardGoalNums.lean
+++ b/test/GuardGoalNums.lean
@@ -1,5 +1,7 @@
 import Mathlib.Tactic.GuardGoalNums
 
+set_option linter.unusedTactic false
+
 example : true ∧ true := by
   constructor
   guard_goal_nums 2
diff --git a/test/GuardHypNums.lean b/test/GuardHypNums.lean
index cafdfbdc05042..3498547c21238 100644
--- a/test/GuardHypNums.lean
+++ b/test/GuardHypNums.lean
@@ -1,5 +1,7 @@
 import Mathlib.Tactic.GuardHypNums
 
+set_option linter.unusedTactic false
+
 example (a b c : Nat) (_ : a = b) (_ : c = 3) : true := by
   guard_hyp_nums 6
   trivial
diff --git a/test/HashCommandLinter.lean b/test/HashCommandLinter.lean
index 76be1fc3d6a6e..203db4657db9e 100644
--- a/test/HashCommandLinter.lean
+++ b/test/HashCommandLinter.lean
@@ -49,6 +49,7 @@ note: this linter can be disabled with `set_option linter.hashCommand false`
 #guard_msgs in
 #guard true
 
+set_option linter.unusedTactic false in
 /--
 warning: `#`-commands, such as '#check_tactic', are not allowed in 'Mathlib'
 note: this linter can be disabled with `set_option linter.hashCommand false`
diff --git a/test/Lint.lean b/test/Lint.lean
index 2b08cae6de29b..25bf9c9147d13 100644
--- a/test/Lint.lean
+++ b/test/Lint.lean
@@ -59,19 +59,23 @@ export Nat (add)
 
 end add
 
-set_option linter.cdot false in
+section cdotLinter
+
+set_option linter.style.cdot false
+
+set_option linter.globalAttributeIn false in
 /--
 warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
-note: this linter can be disabled with `set_option linter.cdot false`
+note: this linter can be disabled with `set_option linter.style.cdot false`
 ---
 warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
-note: this linter can be disabled with `set_option linter.cdot false`
+note: this linter can be disabled with `set_option linter.style.cdot false`
 ---
 warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
-note: this linter can be disabled with `set_option linter.cdot false`
+note: this linter can be disabled with `set_option linter.style.cdot false`
 -/
 #guard_msgs in
-set_option linter.cdot true in
+set_option linter.style.cdot true in
 attribute [instance] Int.add in
 instance : Inhabited Nat where
   default := by
@@ -79,25 +83,67 @@ instance : Inhabited Nat where
       · have : Nat → Nat → Nat := (· + .)
         . exact 0
 
-set_option linter.cdot false in
+set_option linter.style.cdot false in
+/--
+warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
+note: this linter can be disabled with `set_option linter.style.cdot false`
+-/
+#guard_msgs in
+set_option linter.style.cdot true in
+example : Add Nat where add := (. + ·)
+
 /--
 warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
-note: this linter can be disabled with `set_option linter.cdot false`
+note: this linter can be disabled with `set_option linter.style.cdot false`
 -/
 #guard_msgs in
-set_option linter.cdot true in
+set_option linter.style.cdot true in
 example : Add Nat where add := (. + ·)
 
-set_option linter.dollarSyntax false in
+/--
+warning: Please, use '·' (typed as `\.`) instead of '.' as 'cdot'.
+note: this linter can be disabled with `set_option linter.style.cdot false`
+---
+warning: This central dot `·` is isolated; please merge it with the next line.
+---
+warning: This central dot `·` is isolated; please merge it with the next line.
+-/
+#guard_msgs in
+set_option linter.style.cdot true in
+example : Nat := by
+  have : Nat := by
+    ·
+      -- some empty have
+      have := 0
+      ·
+
+        -- another
+        have := 1
+        . exact 2
+  exact 0
+
+#guard_msgs in
+set_option linter.style.cdot true in
+example : True := by
+  have : Nat := by
+    -- This is how code should look: no error.
+    · -- comment
+      exact 37
+  trivial
+
+end cdotLinter
+
+set_option linter.globalAttributeIn false in
+set_option linter.style.dollarSyntax false in
 /--
 warning: Please use '<|' instead of '$' for the pipe operator.
-note: this linter can be disabled with `set_option linter.dollarSyntax false`
+note: this linter can be disabled with `set_option linter.style.dollarSyntax false`
 ---
 warning: Please use '<|' instead of '$' for the pipe operator.
-note: this linter can be disabled with `set_option linter.dollarSyntax false`
+note: this linter can be disabled with `set_option linter.style.dollarSyntax false`
 -/
 #guard_msgs in
-set_option linter.dollarSyntax true in
+set_option linter.style.dollarSyntax true in
 attribute [instance] Int.add in
 instance (f g : Nat → Nat) : Inhabited Nat where
   default := by
@@ -208,42 +254,25 @@ example : ℕ → ℕ := set_option linter.style.lambdaSyntax true in λ _ ↦ 0
 
 end lambdaSyntaxLinter
 
-set_option linter.dollarSyntax false in
-/--
-warning: Please use '<|' instead of '$' for the pipe operator.
-note: this linter can be disabled with `set_option linter.dollarSyntax false`
----
-warning: Please use '<|' instead of '$' for the pipe operator.
-note: this linter can be disabled with `set_option linter.dollarSyntax false`
--/
-#guard_msgs in
-set_option linter.dollarSyntax true in
-attribute [instance] Int.add in
-instance (f g : Nat → Nat) : Inhabited Nat where
-  default := by
-    · have := 0
-      · have : Nat := f $ g $ 0
-        · exact 0
-
-set_option linter.longLine false
+set_option linter.style.longLine false
 /--
 warning: This line exceeds the 100 character limit, please shorten it!
-note: this linter can be disabled with `set_option linter.longLine false`
+note: this linter can be disabled with `set_option linter.style.longLine false`
 -/
 #guard_msgs in
-set_option linter.longLine true in
+set_option linter.style.longLine true in
 /-!                                                                                                -/
 
 #guard_msgs in
 -- Lines with more than 100 characters containing URLs are allowed.
-set_option linter.longLine true in
+set_option linter.style.longLine true in
 /-!  http                                                                                          -/
 
-set_option linter.longLine true
+set_option linter.style.longLine true
 -- The *argument* of `#guard_msgs` is *not* exempt from the linter.
 /--
 warning: This line exceeds the 100 character limit, please shorten it!
-note: this linter can be disabled with `set_option linter.longLine false`
+note: this linter can be disabled with `set_option linter.style.longLine false`
 -/
 #guard_msgs in                                                                            #guard true
 
@@ -254,3 +283,14 @@ info: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
 -/
 #guard_msgs in
 #eval List.range 27
+
+/--
+info: "                              \"                                                            " : String
+---
+warning: This line exceeds the 100 character limit, please shorten it!
+You can use "string gaps" to format long strings: within a string quotation, using a '' at the end of a line allows you to continue the string on the following line, removing all intervening whitespace.
+note: this linter can be disabled with `set_option linter.style.longLine false`
+-/
+#guard_msgs in
+set_option linter.style.longLine true in
+#check "                              \"                                                            "
diff --git a/test/LongFile.lean b/test/LongFile.lean
new file mode 100644
index 0000000000000..faec16c404806
--- /dev/null
+++ b/test/LongFile.lean
@@ -0,0 +1,68 @@
+import Mathlib.Tactic.Linter.Lint
+
+/-
+# Testing the `longFile` linter
+
+Things to note:
+* `set_option linter.style.longFile 0` disables the linter, allowing us to set a value smaller than
+  `1500` without triggering the warning for setting a small value for the option;
+* `guard_msgs ... in #exit` and `set_option ... in #exit` allow processing of the file *beyond*
+  `#exit`, since they wrap `#exit` inside an anonymous section,
+  making Lean active again *after* that anonymous section.
+
+-/
+
+section longFile
+
+/--
+warning: The default value of the `longFile` linter is 1500.
+The current value of 1500 does not exceed the allowed bound.
+Please, remove the `set_option linter.style.longFile 1500`.
+-/
+#guard_msgs in
+-- Do not allow setting a "small" `longFile` linter option
+set_option linter.style.longFile 1500
+
+/--
+warning: using 'exit' to interrupt Lean
+---
+warning: The default value of the `longFile` linter is 1500.
+This file is 36 lines long which does not exceed the allowed bound.
+Please, remove the `set_option linter.style.longFile 1600`.
+-/
+#guard_msgs in
+-- Do not allow unnecessarily increasing the `longFile` linter option
+set_option linter.style.longFile 1600 in
+#exit
+
+/--
+warning: using 'exit' to interrupt Lean
+---
+warning: This file is 51 lines long, but the limit is 10.
+
+You can extend the allowed length of the file using `set_option linter.style.longFile 1500`.
+You can completely disable this linter by setting the length limit to `0`.
+-/
+#guard_msgs in
+-- First, we silence the linter, so that we can set a default value smaller than 1500.
+set_option linter.style.longFile 0 in
+-- Next, we test that the `longFile` linter warns when a file exceeds the allowed value.
+set_option linter.style.longFile 10 in
+#exit
+
+/--
+warning: using 'exit' to interrupt Lean
+---
+warning: The default value of the `longFile` linter is 1500.
+This file is 66 lines long which does not exceed the allowed bound.
+Please, remove the `set_option linter.style.longFile 1700`.
+-/
+#guard_msgs in
+-- First, we silence the linter, so that we can set a default value smaller than 1500.
+set_option linter.style.longFile 0 in
+-- If we set the allowed bound for the `longFile` linter that is too large,
+-- the linter tells us to use a smaller bound.
+set_option linter.style.longFile 1700 in
+#exit
+
+end longFile
diff --git a/test/MLList.lean b/test/MLList.lean
deleted file mode 100644
index 48aa924f5d021..0000000000000
--- a/test/MLList.lean
+++ /dev/null
@@ -1,68 +0,0 @@
-/-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
--/
-import Batteries.Data.MLList.Basic
-import Mathlib.Control.Basic
-
-abbrev S (α : Type) := StateT (List Nat) Option α
-def append (x : Nat) : S Unit :=
-  fun s => some ((), x :: s)
-
-def F : Nat → S Nat
-  | 0 => failure
-  | (n+1) => do
-      append (n+1)
-      pure n
-
-open Lean
-
-run_cmd Lean.Elab.Command.liftTermElabM do
-  -- Note that `fix` fails if any invocation of `F` fails.
-  -- This is different from previous behaviour, where it just terminated the lazy list.
-  -- Hence we must use `.takeAsList 11` here rather than `.force`.
-  let x := ((MLList.fix F 10).takeAsList 11).run []
-  guard <| x = some ([10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], [1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
-
-example : ((MLList.fix F 10).takeAsList 4).run [] = some ([10, 9, 8, 7], [8, 9, 10]) := by
-  native_decide
-example :
-    (((MLList.fix F 10).map fun n => n*n).takeAsList 3).run [] =
-      some ([100, 81, 64], [9, 10]) := by
-  native_decide
-
-def l1 : MLList S Nat := MLList.ofList [0,1,2]
-def l2 : MLList S Nat := MLList.ofList [3,4,5]
-def ll : MLList S Nat := (MLList.ofList [l1, l2]).join
-
-run_cmd Lean.Elab.Command.liftTermElabM do
-  let x := ll.force.run []
-  guard <| x = some ([0, 1, 2, 3, 4, 5], [])
-
-def half_or_fail (n : Nat) : MetaM Nat :=
-do guard (n % 2 = 0)
-   pure (n / 2)
-
-run_cmd Lean.Elab.Command.liftTermElabM do
-  let x : MLList MetaM Nat := MLList.range
-  let y := x.filterMapM fun n => try? <| half_or_fail n
-  let z ← y.takeAsList 10
-  guard <| z.length = 10
-
-run_cmd Lean.Elab.Command.liftTermElabM do
-  let R : MLList MetaM Nat := MLList.range
-  let S : MLList MetaM Nat := R.filterMapM fun n => try? do
-    guard (n % 5 = 0)
-    pure n
-  let n ← R.takeAsList 5
-  let m ← S.head
-  guard <| n = [0,1,2,3,4]
-  guard <| m = 0
-
-run_cmd Lean.Elab.Command.liftTermElabM do
-  let R : MLList MetaM Nat := MLList.range
-  let n ← R.firstM fun n => try? do
-    guard (n = 5)
-    pure n
-  guard <| n = 5
diff --git a/test/MfldSetTac.lean b/test/MfldSetTac.lean
index ae6bf752c33cf..ad921934de097 100644
--- a/test/MfldSetTac.lean
+++ b/test/MfldSetTac.lean
@@ -24,7 +24,8 @@ section stub_lemmas
 structure PartialHomeomorph (α : Type u) (β : Type u) extends PartialEquiv α β
 
 noncomputable
-instance PartialHomeomorph.has_coe_to_fun : CoeFun (PartialHomeomorph α β) (fun _ ↦ α → β) := test_sorry
+instance PartialHomeomorph.has_coe_to_fun : CoeFun (PartialHomeomorph α β) (fun _ ↦ α → β) :=
+  test_sorry
 
 noncomputable
 def PartialHomeomorph.symm (_e : PartialHomeomorph α β) : PartialHomeomorph β α := test_sorry
@@ -46,7 +47,7 @@ test_sorry
   (e.toPartialEquiv.symm : β → α) = (e.symm : β → α) :=
 test_sorry
 
-structure ModelWithCorners (𝕜 E H : Type u) extends PartialEquiv H E :=
+structure ModelWithCorners (𝕜 E H : Type u) extends PartialEquiv H E where
   (source_eq : source = Set.univ)
 
 attribute [mfld_simps] ModelWithCorners.source_eq
@@ -55,7 +56,8 @@ noncomputable
 def ModelWithCorners.symm (_I : ModelWithCorners 𝕜 E H) : PartialEquiv E H := test_sorry
 
 noncomputable
-instance ModelWithCorners.has_coe_to_fun : CoeFun (ModelWithCorners 𝕜 E H) (fun _ ↦ H → E) := test_sorry
+instance ModelWithCorners.has_coe_to_fun : CoeFun (ModelWithCorners 𝕜 E H) (fun _ ↦ H → E) :=
+  test_sorry
 
 @[mfld_simps] lemma ModelWithCorners.left_inv (I : ModelWithCorners 𝕜 E H) (x : H) :
   I.symm (I x) = x :=
diff --git a/test/MinImports.lean b/test/MinImports.lean
index 174943a1933b9..abf1594c8f565 100644
--- a/test/MinImports.lean
+++ b/test/MinImports.lean
@@ -1,4 +1,3 @@
-import Mathlib.Tactic.Linter.MinImports
 import Mathlib.Tactic.NormNum.Basic
 import Mathlib.Tactic.FunProp.Attr
 
@@ -37,8 +36,7 @@ noncomputable instance : Semiring Nat := inferInstance
 /--
 info: ℤ : Type
 ---
-info: import Lean.Parser.Command
-import Mathlib.Data.Int.Notation
+info: import Mathlib.Data.Int.Notation
 -/
 #guard_msgs in
 #min_imports in #check ℤ
@@ -78,7 +76,7 @@ lemma hi (n : ℕ) : n = n := by extract_goal; rfl
 
 /--
 warning: Imports increased to
-[Init.Guard, Lean.Parser.Term, Mathlib.Data.Int.Notation]
+[Init.Guard, Mathlib.Data.Int.Notation]
 note: this linter can be disabled with `set_option linter.minImports false`
 -/
 #guard_msgs in
@@ -95,7 +93,7 @@ set_option linter.minImports false in
 
 /--
 warning: Imports increased to
-[Init.Guard, Lean.Parser.Term, Mathlib.Data.Int.Notation]
+[Init.Guard, Mathlib.Data.Int.Notation]
 note: this linter can be disabled with `set_option linter.minImports false`
 -/
 #guard_msgs in
@@ -118,7 +116,7 @@ note: this linter can be disabled with `set_option linter.minImports false`
 
 /--
 warning: Imports increased to
-[Mathlib.Tactic.FunProp.Attr, Mathlib.Tactic.Linter.MinImports, Mathlib.Tactic.NormNum.Basic]
+[Mathlib.Tactic.FunProp.Attr, Mathlib.Tactic.NormNum.Basic]
 note: this linter can be disabled with `set_option linter.minImports false`
 -/
 #guard_msgs in
diff --git a/test/MoveAdd.lean b/test/MoveAdd.lean
index e115ab0e6a48c..74d7a9fd44d70 100644
--- a/test/MoveAdd.lean
+++ b/test/MoveAdd.lean
@@ -1,5 +1,4 @@
 import Mathlib.Tactic.MoveAdd
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Algebra.Ring.Nat
 
 universe u
diff --git a/test/NthRewrite.lean b/test/NthRewrite.lean
index 608d18fdb6bf8..287039007dce6 100644
--- a/test/NthRewrite.lean
+++ b/test/NthRewrite.lean
@@ -14,7 +14,7 @@ example [AddZeroClass G] {a : G} (h : a = a): a = (a + 0) := by
 example [AddZeroClass G] {a : G} : a + a = a + (a + 0) := by
   nth_rw 2 [← add_zero a]
 
-structure F :=
+structure F where
   (a : ℕ)
   (v : Vector ℕ a)
   (p : v.val = [])
@@ -22,7 +22,7 @@ structure F :=
 example (f : F) : f.v.val = [] := by
   nth_rw 1 [f.p]
 
-structure Cat :=
+structure Cat where
   (O : Type)
   (H : O → O → Type)
   (i : (o : O) → H o o)
diff --git a/test/PPRoundtrip.lean b/test/PPRoundtrip.lean
new file mode 100644
index 0000000000000..752a26277923b
--- /dev/null
+++ b/test/PPRoundtrip.lean
@@ -0,0 +1,66 @@
+import Mathlib.Tactic.Linter.PPRoundtrip
+
+/--
+info: "a   a"
+---
+warning: source context
+'al "    a '
+'al " a a\n'
+pretty-printed context
+note: this linter can be disabled with `set_option linter.ppRoundtrip false`
+-/
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+#eval "    a   a\n       "    |>.trim
+
+/--
+warning: source context
+'rd ¬   fa'
+'rd ¬false'
+pretty-printed context
+note: this linter can be disabled with `set_option linter.ppRoundtrip false`
+-/
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+#guard ¬   false
+
+/--
+warning: source context
+'le {a: Nat'
+'le {a : Na'
+pretty-printed context
+note: this linter can be disabled with `set_option linter.ppRoundtrip false`
+-/
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+variable {a: Nat}
+
+/--
+warning: source context
+' {a :Nat}'
+' {a : Nat}'
+pretty-printed context
+note: this linter can be disabled with `set_option linter.ppRoundtrip false`
+-/
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+variable {a :Nat}
+
+/--
+info: (fun x1 x2 => x1 + x2) 0 1 : Nat
+---
+warning: source context
+'k (·+·) '
+'k (· + ·'
+pretty-printed context
+note: this linter can be disabled with `set_option linter.ppRoundtrip false`
+-/
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+#check (·+·) 0 1
+
+#guard_msgs in
+set_option linter.ppRoundtrip true in
+-- check that trailing comments do not trigger the linter
+example : 0 = 0 := by
+  rw [] -- this goal is closed by the `rfl` implied by `rw`
diff --git a/test/Polynomial.lean b/test/Polynomial.lean
index dde09d7984a81..e56b3b9a2c5b5 100644
--- a/test/Polynomial.lean
+++ b/test/Polynomial.lean
@@ -1,4 +1,5 @@
 import Mathlib.Algebra.Polynomial.Basic
+import Mathlib.Algebra.Module.ULift
 
 open Polynomial
 
diff --git a/test/ProdAssoc.lean b/test/ProdAssoc.lean
index 7802ba1458acd..38927efb92ffa 100644
--- a/test/ProdAssoc.lean
+++ b/test/ProdAssoc.lean
@@ -3,8 +3,7 @@ import Mathlib.Tactic.ProdAssoc
 variable {α β γ δ : Type*}
 
 example : (α × β) × (γ × δ) ≃ α × (β × γ) × δ := by
-  have := (prod_assoc% : (α × β) × (γ × δ) ≃ α × (β × γ) × δ)
-  exact this
+  exact (prod_assoc% : (α × β) × (γ × δ) ≃ α × (β × γ) × δ)
 
 example : (α × β) × (γ × δ) ≃ α × (β × γ) × δ := prod_assoc%
 
diff --git a/test/RewriteSearch/Basic.lean b/test/RewriteSearch/Basic.lean
index ef4fb08df014d..8b70141718832 100644
--- a/test/RewriteSearch/Basic.lean
+++ b/test/RewriteSearch/Basic.lean
@@ -17,7 +17,7 @@ example (xs ys : List α) : (xs ++ ys).length = ys.length + xs.length := by
 -- This worked in previous versions, but for now doesn't.
 -- There are of course better tools for AC rewriting, but it would be nice if `rw_search`
 -- could do a little of it in the course of a longer rewrite.
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 /-!
 -- /-
 -- info: Try this: rw [← add_assoc, add_right_comm, add_assoc, add_add_add_comm, ← add_assoc, add_right_comm]
@@ -35,7 +35,7 @@ example (xs ys : List α) :
     (xs ++ ys ++ ys).length = 2 * ys.length + xs.length := by
   rw_search
 
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 /-!
 info: Try this: rw [List.length_append, List.length_append, Nat.two_mul, Nat.add_assoc, Nat.add_left_comm, Nat.add_right_comm, Nat.add_assoc]
 -/
@@ -44,7 +44,7 @@ example (xs ys : List α) :
     (xs ++ ys ++ ys).length = 2 * ys.length + xs.length := by
   rw_search [-add_rotate]
 
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 /-!
 info: Try this: rw [Int.add_right_comm, add_right_cancel_iff, add_sub_left_comm, add_sub, Int.add_sub_cancel]
 -/
diff --git a/test/RewriteSearch/Polynomial.lean b/test/RewriteSearch/Polynomial.lean
index 2fcec787de1f2..cf1cab0e57a5f 100644
--- a/test/RewriteSearch/Polynomial.lean
+++ b/test/RewriteSearch/Polynomial.lean
@@ -174,7 +174,7 @@ example {R : Type u} [Ring R] [Nontrivial R] (x : R) :
 #guard_msgs(drop info) in
 example {S : Type v} [Ring S] (c : S) :
     Polynomial.nextCoeff (Polynomial.X - Polynomial.C c) = -c := by
-  rw_search [-Polynomial.nextCoeff_X_sub_C]
+  rw_search
   -- Mathlib proof:
   -- rw [sub_eq_add_neg, ← map_neg C c, nextCoeff_X_add_C]
   done
diff --git a/test/SimpRw.lean b/test/SimpRw.lean
index bf5c80caee23a..466c3c32025e1 100644
--- a/test/SimpRw.lean
+++ b/test/SimpRw.lean
@@ -34,6 +34,7 @@ example {a : Nat}
   (∀ b, a - 1 ≤ b) = ∀ b c : Nat, c < a → c < b + 1 := by
   simp_rw [h1, h2]
 
+set_option linter.unusedTactic false in
 -- `simp_rw` respects config options
 example : 1 = 2 := by
   let a := 2
diff --git a/test/Simps.lean b/test/Simps.lean
index 4fb569642e899..af1bc4b6f4a5e 100644
--- a/test/Simps.lean
+++ b/test/Simps.lean
@@ -91,7 +91,7 @@ initialize_simps_projections Something
 
 universe v u w
 
-structure Equiv' (α : Sort _) (β : Sort _) :=
+structure Equiv' (α : Sort _) (β : Sort _) where
   (toFun     : α → β)
   (invFun    : β → α)
   (left_inv  : invFun.LeftInverse toFun)
@@ -262,7 +262,7 @@ run_cmd liftTermElabM <| do
   guard <| env.find? `rflWithData'_toEquiv_toFun |>.isNone
   guard <| env.find? `test_sneaky_extra |>.isNone
 
-structure PartiallyAppliedStr :=
+structure PartiallyAppliedStr where
   (data : ℕ → MyProd ℕ ℕ)
 
 /- if we have a partially applied constructor, we treat it as if it were eta-expanded -/
@@ -279,7 +279,7 @@ run_cmd liftTermElabM <| do
   guard <| simpsAttr.getParam? env `partially_applied_term ==
     #[`partially_applied_term_data_fst, `partially_applied_term_data_snd]
 
-structure VeryPartiallyAppliedStr :=
+structure VeryPartiallyAppliedStr where
   (data : ∀β, ℕ → β → MyProd ℕ β)
 
 /- if we have a partially applied constructor, we treat it as if it were eta-expanded.
@@ -424,12 +424,12 @@ run_cmd liftTermElabM <| do
   guard <| env.find? `pprodEquivProd22_invFun_snd |>.isSome
 
 /- Tests with universe levels -/
-class has_hom (obj : Type u) : Type (max u (v+1)) :=
+class has_hom (obj : Type u) : Type (max u (v+1)) where
   (hom : obj → obj → Type v)
 
 infixr:10 " ⟶ " => has_hom.hom -- type as \h
 
-class CategoryStruct (obj : Type u) extends has_hom.{v} obj : Type (max u (v+1)) :=
+class CategoryStruct (obj : Type u) extends has_hom.{v} obj : Type (max u (v+1)) where
   (id   : ∀ X : obj, hom X X)
   (comp : ∀ {X Y Z : obj}, (X ⟶ Y) → (Y ⟶ Z) → (X ⟶ Z))
 
@@ -450,7 +450,7 @@ example (X Y Z : Type u) (f : X ⟶ Y) (g : Y ⟶ Z) {k : X → Z} (h : ∀ x, g
 
 namespace coercing
 
-structure FooStr :=
+structure FooStr where
  (c : Type)
  (x : c)
 
@@ -462,7 +462,7 @@ instance : CoeSort FooStr Type := ⟨FooStr.c⟩
 example {x : Type} (h : ℕ = x) : foo = x := by simp only [foo_c]; rw [h]
 example {x : ℕ} (h : (3 : ℕ) = x) : foo.x = x := by simp only [foo_x]; rw [h]
 
-structure VooStr (n : ℕ) :=
+structure VooStr (n : ℕ) where
  (c : Type)
  (x : c)
 
@@ -474,7 +474,7 @@ instance (n : ℕ) : CoeSort (VooStr n) Type := ⟨VooStr.c⟩
 example {x : Type} (h : ℕ = x) : voo = x := by simp only [voo_c]; rw [h]
 example {x : ℕ} (h : (3 : ℕ) = x) : voo.x = x := by simp only [voo_x]; rw [h]
 
-structure Equiv2 (α : Sort _) (β : Sort _) :=
+structure Equiv2 (α : Sort _) (β : Sort _) where
   (toFun     : α → β)
   (invFun    : β → α)
   (left_inv  : invFun.LeftInverse toFun)
@@ -515,7 +515,7 @@ class Semigroup (G : Type u) extends Mul G where
 example {α β} [Semigroup α] [Semigroup β] (x y : α × β) : x * y = (x.1 * y.1, x.2 * y.2) := by simp
 example {α β} [Semigroup α] [Semigroup β] (x y : α × β) : (x * y).1 = x.1 * y.1 := by simp
 
-structure BSemigroup :=
+structure BSemigroup where
   (G : Type _)
   (op : G → G → G)
   -- (infix:60 " * " => op) -- this seems to be removed
@@ -535,8 +535,8 @@ protected def prod (G H : BSemigroup) : BSemigroup :=
 
 end BSemigroup
 
-class ExtendingStuff (G : Type u) extends Mul G, Zero G, Neg G, HasSubset G :=
-  (new_axiom : ∀ x : G, x * - 0 ⊆ - x)
+class ExtendingStuff (G : Type u) extends Mul G, Zero G, Neg G, HasSubset G where
+  new_axiom : ∀ x : G, x * - 0 ⊆ - x
 
 @[simps] def bar : ExtendingStuff ℕ :=
   { mul := (·*·)
@@ -550,8 +550,8 @@ attribute [local instance] bar
 example (x : ℕ) : x * - 0 ⊆ - x := by simp
 end
 
-class new_ExtendingStuff (G : Type u) extends Mul G, Zero G, Neg G, HasSubset G :=
-  (new_axiom : ∀ x : G, x * - 0 ⊆ - x)
+class new_ExtendingStuff (G : Type u) extends Mul G, Zero G, Neg G, HasSubset G where
+  new_axiom : ∀ x : G, x * - 0 ⊆ - x
 
 @[simps] def new_bar : new_ExtendingStuff ℕ :=
   { mul := (·*·)
@@ -570,7 +570,7 @@ end coercing
 
 namespace ManualCoercion
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -598,7 +598,7 @@ end ManualCoercion
 
 namespace FaultyManualCoercion
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -622,7 +622,7 @@ namespace ManualInitialize
 /- defining a manual coercion. -/
 variable {α β γ : Sort _}
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -654,7 +654,7 @@ namespace FaultyUniverses
 
 variable {α β γ : Sort _}
 
-structure Equiv (α : Sort u) (β : Sort v) :=
+structure Equiv (α : Sort u) (β : Sort v) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -683,7 +683,7 @@ namespace ManualUniverses
 
 variable {α β γ : Sort _}
 
-structure Equiv (α : Sort u) (β : Sort v) :=
+structure Equiv (α : Sort u) (β : Sort v) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -704,7 +704,7 @@ end ManualUniverses
 
 namespace ManualProjectionNames
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -744,7 +744,7 @@ end ManualProjectionNames
 
 namespace PrefixProjectionNames
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -791,7 +791,7 @@ end PrefixProjectionNames
 
 
 -- test transparency setting
-structure SetPlus (α : Type) :=
+structure SetPlus (α : Type) where
   (s : Set α)
   (x : α)
   (h : x ∈ s)
@@ -818,7 +818,7 @@ example {x : Set ℕ} (h : Set.univ = x) : Nat.SetPlus3.s = x := by
 
 namespace NestedNonFullyApplied
 
-structure Equiv (α : Sort _) (β : Sort _) :=
+structure Equiv (α : Sort _) (β : Sort _) where
   (toFun  : α → β)
   (invFun : β → α)
 
@@ -854,19 +854,19 @@ example (e : α ≃ β) {x : β → α} (h : e.invFun = x) : (Equiv.symm2.invFun
 end NestedNonFullyApplied
 
 -- test that type classes which are props work
-class PropClass (n : ℕ) : Prop :=
-  (has_true : True)
+class PropClass (n : ℕ) : Prop where
+  has_true : True
 
 instance has_PropClass (n : ℕ) : PropClass n := ⟨trivial⟩
 
-structure NeedsPropClass (n : ℕ) [PropClass n] :=
+structure NeedsPropClass (n : ℕ) [PropClass n] where
   (t : True)
 
 @[simps] def test_PropClass : NeedsPropClass 1 :=
   { t := trivial }
 
 /- check that when the coercion is given in eta-expanded form, we can also find the coercion. -/
-structure AlgHom (R A B : Type _) :=
+structure AlgHom (R A B : Type _) where
   (toFun : A → B)
 
 instance (R A B : Type _) : CoeFun (AlgHom R A B) (fun _ ↦ A → B) := ⟨fun f ↦ f.toFun⟩
@@ -931,7 +931,7 @@ section
 
 attribute [local simp] Nat.add
 
-structure MyType :=
+structure MyType where
   (A : Type)
 
 @[simps (config := {simpRhs := true})] def myTypeDef : MyType :=
@@ -972,7 +972,7 @@ instance {α β} : CoeFun (α ≃ β) (fun _ ↦ α → β) := ⟨Equiv'.toFun
 @[simps] protected def Equiv'.symm {α β} (f : α ≃ β) : β ≃ α :=
   ⟨f.invFun, f, f.right_inv, f.left_inv⟩
 
-structure DecoratedEquiv (α : Sort _) (β : Sort _) extends Equiv' α β :=
+structure DecoratedEquiv (α : Sort _) (β : Sort _) extends Equiv' α β where
   (P_toFun  : Function.Injective toFun )
   (P_invFun : Function.Injective invFun)
 
@@ -1022,7 +1022,7 @@ example {α : Type} (x z : α) (h : x = z) : foo2 α x = z := by
   guard_target = x = z
   rw [h]
 
-structure FurtherDecoratedEquiv (α : Sort _) (β : Sort _) extends DecoratedEquiv α β :=
+structure FurtherDecoratedEquiv (α : Sort _) (β : Sort _) extends DecoratedEquiv α β where
   (Q_toFun  : Function.Surjective toFun )
   (Q_invFun : Function.Surjective invFun )
 
@@ -1097,11 +1097,11 @@ def fffoo2 (α : Type) : OneMore α α := fffoo α
 /- test the case where a projection takes additional arguments. -/
 variable {ι : Type _} [DecidableEq ι] (A : ι → Type _)
 
-structure ZeroHom (M N : Type _) [Zero M] [Zero N] :=
+structure ZeroHom (M N : Type _) [Zero M] [Zero N] where
   (toFun : M → N)
   (map_zero' : toFun 0 = 0)
 
-structure AddHom (M N : Type _) [Add M] [Add N] :=
+structure AddHom (M N : Type _) [Add M] [Add N] where
   (toFun : M → N)
   (map_add' : ∀ x y, toFun (x + y) = toFun x + toFun y)
 
@@ -1112,7 +1112,7 @@ infixr:25 " →+ " => AddMonoidHom
 
 instance (M N : Type _) [AddMonoid M] [AddMonoid N] : CoeFun (M →+ N) (fun _ ↦ M → N) := ⟨(·.toFun)⟩
 
-class AddHomPlus [Add ι] [∀ i, AddCommMonoid (A i)] :=
+class AddHomPlus [Add ι] [∀ i, AddCommMonoid (A i)] where
   (myMul {i} : A i →+ A i)
 
 def AddHomPlus.Simps.apply [Add ι] [∀ i, AddCommMonoid (A i)] [AddHomPlus A] {i : ι} (x : A i) :
@@ -1121,7 +1121,7 @@ def AddHomPlus.Simps.apply [Add ι] [∀ i, AddCommMonoid (A i)] [AddHomPlus A]
 
 initialize_simps_projections AddHomPlus (myMul_toFun → apply, -myMul)
 
-class AddHomPlus2 [Add ι] :=
+class AddHomPlus2 [Add ι] where
   (myMul {i j} : A i ≃ (A j ≃ A (i + j)))
 
 def AddHomPlus2.Simps.mul [Add ι] [AddHomPlus2 A] {i j : ι} (x : A i) (y : A j) : A (i + j) :=
@@ -1153,7 +1153,7 @@ end comp_projs
 section
 /-! Check that the tactic also works if the elaborated type of `type` reduces to `Sort _`, but is
   not `Sort _` itself. -/
-structure MyFunctor (C D : Type _) :=
+structure MyFunctor (C D : Type _) where
   (obj : C → D)
 local infixr:26 " ⥤ " => MyFunctor
 
diff --git a/test/SplitIfs.lean b/test/SplitIfs.lean
index e8dbcd8c69649..1809a04cb709e 100644
--- a/test/SplitIfs.lean
+++ b/test/SplitIfs.lean
@@ -73,6 +73,7 @@ example (P Q : Prop) (w : if P then (if Q then true else true) else true = true)
   · trivial
   · trivial
 
+set_option linter.unusedTactic false in
 example (u : Nat) : (if u = u then 0 else 1) = 0 := by
   have h : u = u := by rfl
   split_ifs
diff --git a/test/StacksAttribute.lean b/test/StacksAttribute.lean
index 0734beada247f..92fe5efb40fcd 100644
--- a/test/StacksAttribute.lean
+++ b/test/StacksAttribute.lean
@@ -5,26 +5,20 @@ import Mathlib.Tactic.StacksAttribute
 #stacks_tags
 
 namespace X
-/--
-warning: Tag '04 Q' should only consist of digits and uppercase letters
----
-warning: Tag '044QQ' is 5 characters long, but it should be 4 characters long
----
-warning: Tag 'loA1' should only consist of digits and uppercase letters
--/
-#guard_msgs in
-@[stacks 04 Q "", stacks A04Q "A comment", stacks 044QQ, stacks loA1]
+
+@[stacks A04Q "A comment", kerodon B15R "Also a comment"]
 theorem tagged : True := .intro
 
 end X
 
-/--
-warning: Please, enter a Tag after `stacks`.
----
-warning: Please, enter a Tag after `stacks`.
--/
 #guard_msgs in
-@[stacks "", stacks]
+@[stacks 0BR2, kerodon 0X12]
+example : True := .intro
+
+@[stacks 0BR2, stacks 0X14 "I can also have a comment"]
+example : True := .intro
+
+@[stacks 0X14 "I can also have a comment"]
 example : True := .intro
 
 /--
@@ -41,3 +35,43 @@ True
 -/
 #guard_msgs in
 #stacks_tags!
+
+/--
+info:
+[Stacks Tag B15R](https://kerodon.net/tag/B15R) corresponds to declaration 'X.tagged'. (Also a comment)
+True
+-/
+#guard_msgs in
+#kerodon_tags!
+
+section errors
+
+open Lean Parser Mathlib.StacksTag
+
+def captureException (env : Environment) (s : ParserFn) (input : String) : Except String Syntax :=
+  let ictx := mkInputContext input "<input>"
+  let s := s.run ictx { env, options := {} } (getTokenTable env) (mkParserState input)
+  if !s.allErrors.isEmpty then
+    .error (s.toErrorMsg ictx)
+  else if ictx.input.atEnd s.pos then
+    .ok s.stxStack.back
+  else
+    .error ((s.mkError "end of input").toErrorMsg ictx)
+
+/-- error: <input>:1:3: Stacks tags must be exactly 4 characters -/
+#guard_msgs in
+run_cmd do
+  let _ ← Lean.ofExcept <| captureException (← getEnv) stacksTagFn "A05"
+
+/-- error: <input>:1:4: Stacks tags must consist only of digits and uppercase letters. -/
+#guard_msgs in
+run_cmd do
+  let _ ← Lean.ofExcept <| captureException (← getEnv) stacksTagFn "aaaa"
+
+/-- error: <input>:1:0: expected stacks tag -/
+#guard_msgs in
+run_cmd do
+  let env ← getEnv
+  let _ ← Lean.ofExcept <| captureException env stacksTagFn "\"A04Q\""
+
+end errors
diff --git a/test/StringDiagram.lean b/test/StringDiagram.lean
index ec654cf09ce73..f6c65dc77c848 100644
--- a/test/StringDiagram.lean
+++ b/test/StringDiagram.lean
@@ -3,10 +3,10 @@ import ProofWidgets.Component.Panel.SelectionPanel
 
 /-! ## Example use of string diagram widgets -/
 
-section MonoidalCategory
-
 open ProofWidgets Mathlib.Tactic.Widget
 
+section MonoidalCategory
+
 open CategoryTheory
 open scoped MonoidalCategory
 
@@ -14,6 +14,8 @@ universe v u
 
 variable {C : Type u} [Category.{v} C] [MonoidalCategory C]
 
+section
+
 lemma left_triangle {X Y : C} (η : 𝟙_ _ ⟶ X ⊗ Y) (ε : Y ⊗ X ⟶ 𝟙_ _) (w : False) :
     η ▷ X ≫ (α_ _ _ _).hom ≫ X ◁ ε = (λ_ _).hom ≫ (ρ_ _).inv := by
   /- Displays string diagrams for the both sides of the goal. -/
@@ -24,7 +26,7 @@ lemma left_triangle {X Y : C} (η : 𝟙_ _ ⟶ X ⊗ Y) (ε : Y ⊗ X ⟶ 𝟙_
       /- Place the cursor here and shift-click the 2-morphisms in the tactic state. -/
       exact w.elim
 
-/- Instead of writing `with_panel_widgets` everywhere, you can also use this command.  -/
+/- Instead of writing `with_panel_widgets` everywhere, you can also use this command. -/
 show_panel_widgets [local StringDiagram, local SelectionPanel]
 
 lemma yang_baxter {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) (w : False) :
@@ -57,4 +59,432 @@ example {X₁ Y₁ X₂ Y₂ : C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) : f 
   rw [MonoidalCategory.whisker_exchange]
   rw [MonoidalCategory.tensorHom_def]
 
+end
+
+set_option trace.string_diagram true
+
+variable {C : Type u} [Category.{v} C] [i : MonoidalCategory C] {X Y : C}
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_1_1)
+
+[string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_1_1)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram MonoidalCategory.whisker_exchange
+
+/-- info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+
+[string_diagram] Penrose substance: Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram MonoidalCategory.whisker_exchange_assoc
+
+/--
+info: [string_diagram] Penrose substance:
+
+[string_diagram] Penrose substance:
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram MonoidalCategory.pentagon
+
+/--
+info: [string_diagram] Penrose substance:
+
+[string_diagram] Penrose substance:
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram MonoidalCategory.whiskerLeft_id
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_1_0_0, E_1_0_2)
+    Left(E_2_0_0, E_2_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_2)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+    Mor1 f_1_4 := MakeString (E_1_0_2, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+
+[string_diagram] Penrose substance:
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram left_triangle
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_2_2)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_2_2)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_0_0)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_2_2)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+    Mor1 f_1_4 := MakeString (E_1_2_2, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_2_3 := MakeString (E_2_1_1, E_3_2_2)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_1 := MakeString (E_3_0_0, E_4_1_1)
+    Mor1 f_3_4 := MakeString (E_3_2_2, E_4_2_2)
+
+[string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_2_2)
+    Left(E_3_0_0, E_3_1_1)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_0_0)
+    Mor1 f_1_3 := MakeString (E_1_1_1, E_2_2_2)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_1 := MakeString (E_2_0_0, E_3_1_1)
+    Mor1 f_2_4 := MakeString (E_2_2_2, E_3_1_1)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_2 := MakeString (E_3_1_1, E_4_1_1)
+    Mor1 f_3_3 := MakeString (E_3_1_1, E_4_2_2)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram yang_baxter
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_2_2)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_2_2)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_0_0)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_2_2)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+    Mor1 f_1_4 := MakeString (E_1_2_2, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_2_3 := MakeString (E_2_1_1, E_3_2_2)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_1 := MakeString (E_3_0_0, E_4_1_1)
+    Mor1 f_3_4 := MakeString (E_3_2_2, E_4_2_2)
+
+[string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_2_2)
+    Left(E_3_0_0, E_3_1_1)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_0_0)
+    Mor1 f_1_3 := MakeString (E_1_1_1, E_2_2_2)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_1 := MakeString (E_2_0_0, E_3_1_1)
+    Mor1 f_2_4 := MakeString (E_2_2_2, E_3_1_1)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_2 := MakeString (E_3_1_1, E_4_1_1)
+    Mor1 f_3_3 := MakeString (E_3_1_1, E_4_2_2)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram yang_baxter'
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_2_2)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_2_2)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_0_0)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_2_2)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+    Mor1 f_1_4 := MakeString (E_1_2_2, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_2_3 := MakeString (E_2_1_1, E_3_2_2)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_1 := MakeString (E_3_0_0, E_4_1_1)
+    Mor1 f_3_4 := MakeString (E_3_2_2, E_4_2_2)
+
+[string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_2_2)
+    Left(E_3_0_0, E_3_1_1)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_0_0)
+    Mor1 f_1_3 := MakeString (E_1_1_1, E_2_2_2)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_1 := MakeString (E_2_0_0, E_3_1_1)
+    Mor1 f_2_4 := MakeString (E_2_2_2, E_3_1_1)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_2 := MakeString (E_3_1_1, E_4_1_1)
+    Mor1 f_3_3 := MakeString (E_3_1_1, E_4_2_2)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram yang_baxter''
+
+/--
+info: [string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+
+[string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram Category.assoc
+
+/--
+info: [string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+
+[string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram Functor.map_comp
+
+/--
+info: [string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+
+[string_diagram] Penrose substance:
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram NatTrans.naturality
+
+variable (f : 𝟙_ _ ⟶ X ⊗ Y) in
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_2_0_0, E_2_1_1)
+    Above(E_1_0_0, E_2_0_0)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram f
+
+variable (g : Y ⊗ X ⟶ 𝟙_ _) in
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_0_0)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram g
+
+abbrev yangBaxterLhs {V₁ V₂ V₃ : C} (R : ∀ V₁ V₂ : C, V₁ ⊗ V₂ ⟶ V₂ ⊗ V₁) :=
+  R V₁ V₂ ▷ V₃ ≫ (α_ _ ..).hom ≫ _ ◁ R _ _ ≫ (α_ _ ..).inv ≫ R _ _ ▷ _ ≫ (α_ _ ..).hom
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_0_1_1, E_0_2_2)
+    Left(E_1_0_0, E_1_2_2)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_2_2)
+    Left(E_4_0_0, E_4_1_1)
+    Left(E_4_1_1, E_4_2_2)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Above(E_3_0_0, E_4_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_0_0)
+    Mor1 f_0_4 := MakeString (E_0_2_2, E_1_2_2)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_1 := MakeString (E_1_0_0, E_2_1_1)
+    Mor1 f_1_4 := MakeString (E_1_2_2, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_0_0)
+    Mor1 f_2_3 := MakeString (E_2_1_1, E_3_2_2)
+    Mor1 f_3_0 := MakeString (E_3_0_0, E_4_0_0)
+    Mor1 f_3_1 := MakeString (E_3_0_0, E_4_1_1)
+    Mor1 f_3_4 := MakeString (E_3_2_2, E_4_2_2)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram yangBaxterLhs
+
 end MonoidalCategory
+
+section Bicategory
+
+open CategoryTheory
+
+set_option trace.string_diagram true
+
+/--
+info: [string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_1_1)
+
+[string_diagram] Penrose substance:
+    Left(E_0_0_0, E_0_1_1)
+    Left(E_1_0_0, E_1_1_1)
+    Left(E_2_0_0, E_2_1_1)
+    Left(E_3_0_0, E_3_1_1)
+    Above(E_0_0_0, E_1_0_0)
+    Above(E_1_0_0, E_2_0_0)
+    Above(E_2_0_0, E_3_0_0)
+    Mor1 f_0_0 := MakeString (E_0_0_0, E_1_0_0)
+    Mor1 f_0_2 := MakeString (E_0_1_1, E_1_1_1)
+    Mor1 f_1_0 := MakeString (E_1_0_0, E_2_0_0)
+    Mor1 f_1_2 := MakeString (E_1_1_1, E_2_1_1)
+    Mor1 f_2_0 := MakeString (E_2_0_0, E_3_0_0)
+    Mor1 f_2_2 := MakeString (E_2_1_1, E_3_1_1)
+-/
+#guard_msgs (whitespace := lax) in
+#string_diagram Bicategory.whisker_exchange
+
+end Bicategory
diff --git a/test/TCSynth.lean b/test/TCSynth.lean
index f26afeaae9055..2371d40e8e971 100644
--- a/test/TCSynth.lean
+++ b/test/TCSynth.lean
@@ -84,9 +84,10 @@ end
 section
 
 -- Initial issue: https://github.com/leanprover-community/mathlib4/issues/12232
+-- reduced from 9000 to 1000 after `@[simp low] map_zero` in #16679 (only 10 needed)
 
 open Equiv in
-set_option synthInstance.maxHeartbeats 9000 in
+set_option synthInstance.maxHeartbeats 1000 in
 example {n : ℕ} (p : Fin (n + 1)) (e : Perm (Fin n)) :
     Equiv.Perm.decomposeFin.symm (p, e) 0 = p := by simp
 
diff --git a/test/TermBeta.lean b/test/TermBeta.lean
index 30a3b74742f04..f54d02ecde615 100644
--- a/test/TermBeta.lean
+++ b/test/TermBeta.lean
@@ -1,5 +1,6 @@
 import Mathlib.Util.TermBeta
 -- On command line, tests format functions with => rather than ↦ without this.
+set_option linter.style.setOption false
 set_option pp.unicode.fun true
 
 /-- info: (fun x ↦ x) true : Bool -/
@@ -23,7 +24,7 @@ set_option pp.unicode.fun true
 /-- info: ∀ (i : Nat), 0 ≤ i : Prop -/
 #guard_msgs in #check ∀ i : Nat, beta% (fun j => 0 ≤ j) i
 
-/-- info: (fun x x_1 ↦ x && x_1) true false : Bool -/
+/-- info: (fun x1 x2 ↦ x1 && x2) true false : Bool -/
 #guard_msgs in #check (· && ·) true false
 
 /-- info: true && false : Bool -/
diff --git a/test/TypeCheck.lean b/test/TypeCheck.lean
index c707c493a6bd8..9d38a614053df 100644
--- a/test/TypeCheck.lean
+++ b/test/TypeCheck.lean
@@ -1,5 +1,7 @@
 import Mathlib.Tactic.TypeCheck
 
+set_option linter.unusedTactic false
+
 /-- A term where `inferType` returns `Prop`, but which does not type check. -/
 elab "wrong" : term =>
   return Lean.mkApp2 (.const ``id [.zero]) (.sort .zero) (.app (.sort .zero) (.sort .zero))
diff --git a/test/UnsetOption.lean b/test/UnsetOption.lean
index b6cbbe719b387..796a26191f37b 100644
--- a/test/UnsetOption.lean
+++ b/test/UnsetOption.lean
@@ -1,5 +1,7 @@
 import Mathlib.Tactic.UnsetOption
 
+set_option linter.style.setOption false
+set_option linter.unusedTactic false
 set_option pp.all true
 
 example : True := by
diff --git a/test/UnusedTactic.lean b/test/UnusedTactic.lean
index 5dd2b6a81221b..e12d30af91f30 100644
--- a/test/UnusedTactic.lean
+++ b/test/UnusedTactic.lean
@@ -4,7 +4,7 @@ import Mathlib.Tactic.AdaptationNote
 def why2 : True → True := (by refine ·)
 
 example : True := by
-  #adaptation_note /--hi-/
+  #adaptation_note /-- hi -/
   exact .intro
 
 -- both `;` and `<;>` are unseen by the linter
diff --git a/test/Use.lean b/test/Use.lean
index a82114dfd2b85..94acfd37571ab 100644
--- a/test/Use.lean
+++ b/test/Use.lean
@@ -205,8 +205,8 @@ example (α : Type u) : Embedding α α × Unit := by
 -- Note(kmill): mathlib3 `use` would try to rewrite any lingering existentials with
 -- `exists_prop` to turn them into conjunctions. It did not do this recursively.
 
--- example : ∃ (n : Nat) (h : n > 0), n = n :=
--- by
+set_option linter.style.longLine false in
+-- example : ∃ (n : Nat) (h : n > 0), n = n := by
 --   use 1
 --   -- goal should now be `1 > 0 ∧ 1 = 1`, whereas it would be `∃ (H : 1 > 0), 1 = 1` after existsi 1.
 --   guard_target = 1 > 0 ∧ 1 = 1
diff --git a/test/ValuedCSP.lean b/test/ValuedCSP.lean
index f60e3aed3e421..48b89612eacdc 100644
--- a/test/ValuedCSP.lean
+++ b/test/ValuedCSP.lean
@@ -40,6 +40,8 @@ private def exampleFiniteValuedInstance : exampleFiniteValuedCSP.Instance (Fin 2
 example : exampleFiniteValuedInstance.IsOptimumSolution ![(0 : ℚ), (0 : ℚ)] := by
   intro s
   convert_to 0 ≤ exampleFiniteValuedInstance.evalSolution s
+  · simp [exampleFiniteValuedInstance, ValuedCSP.Instance.evalSolution]
+    exact Rat.zero_add 0
   rw [ValuedCSP.Instance.evalSolution, exampleFiniteValuedInstance]
   convert_to 0 ≤ |s 0| + |s 1|
   · simp [ValuedCSP.unaryTerm, ValuedCSP.Term.evalSolution, Function.OfArity.uncurry]
diff --git a/test/Zify.lean b/test/Zify.lean
index 0ff0f60de8ab5..a61537de310b2 100644
--- a/test/Zify.lean
+++ b/test/Zify.lean
@@ -29,11 +29,11 @@ example (a b : ℕ) (h : (a : ℤ) ≤ b) : a ≤ b := by
   guard_target = (a : ℤ) ≤ b
   exact h
 
-/-example (a b : ℕ) (h : a = b ∧ b < a) : False := by
+/- example (a b : ℕ) (h : a = b ∧ b < a) : False := by
   zify at h
   rcases h with ⟨ha, hb⟩
   -- Preorder for `ℤ` is missing
-  exact ne_of_lt hb ha-/
+  exact ne_of_lt hb ha -/
 
 example (a b c : ℕ) (h : a - b < c) (hab : b ≤ a) : True := by
   zify [hab] at h
diff --git a/test/aesop_cat.lean b/test/aesop_cat.lean
index 7790b9c1b983c..549cd69b92adb 100644
--- a/test/aesop_cat.lean
+++ b/test/aesop_cat.lean
@@ -10,6 +10,8 @@ example : Foo where
   x := sorry
 
 /--
+error: could not synthesize default value for field 'w' of 'Foo' using tactics
+---
 error: tactic 'aesop' failed, failed to prove the goal after exhaustive search.
 Initial goal:
   ⊢ 35 = 37
diff --git a/test/algebraize.lean b/test/algebraize.lean
new file mode 100644
index 0000000000000..7cebc3123da0a
--- /dev/null
+++ b/test/algebraize.lean
@@ -0,0 +1,96 @@
+import Mathlib.Tactic.Algebraize
+
+section example_definitions
+
+/-- Test property for when `RingHom` and `Algebra` properties are definitionally the same,
+see e.g. `RingHom.FiniteType` for a concrete example of this. -/
+class Algebra.testProperty1 (A B : Type*) [CommRing A] [CommRing B] [Algebra A B] : Prop where
+  out : ∀ x : A, algebraMap A B x = 0
+
+/-- Test property for when `RingHom` and `Algebra` properties are definitionally the same,
+see e.g. `RingHom.FiniteType` for a concrete example of this. -/
+@[algebraize]
+def RingHom.testProperty1 {A B : Type*} [CommRing A] [CommRing B] (f : A →+* B) : Prop :=
+  @Algebra.testProperty1 A B _ _ f.toAlgebra
+
+/-- Test property for when the `RingHom` porperty corresponds to a `Module` property (that is
+definitionally the same). See e.g. `Module.Finite` for a concrete example of this. -/
+class Module.testProperty2 (A M : Type*) [Semiring A] [AddCommMonoid M] [Module A M] : Prop where
+  out : ∀ x : A, ∀ M : M, x • M = 0
+
+/-- Test property for when the `RingHom` porperty corresponds to a `Module` property (that is
+definitionally the same). See e.g. `Module.Finite` for a concrete example of this. -/
+@[algebraize Module.testProperty2]
+def RingHom.testProperty2 {A B : Type*} [CommRing A] [CommRing B] (f : A →+* B) : Prop :=
+  letI : Algebra A B := f.toAlgebra
+  Module.testProperty2 A B
+
+/-- Test property for when the `RingHom` porperty corresponds to a `Algebra` property that is not
+definitionally the same, and needs to be created through a lemma. See e.g. `Algebra.IsIntegral` for
+an example. -/
+class Algebra.testProperty3 (A B : Type*) [CommRing A] [CommRing B] [Algebra A B] : Prop where
+  out : Algebra.testProperty1 A B
+
+/- Test property for when the `RingHom` porperty corresponds to a `Algebra` property that is not
+definitionally the same, and needs to be created through a lemma. See e.g. `Algebra.IsIntegral` for
+an example. -/
+@[algebraize Algebra.testProperty3.mk]
+def RingHom.testProperty3 {A B : Type*} [CommRing A] [CommRing B] (f : A →+* B) : Prop :=
+  f.testProperty1
+
+end example_definitions
+
+set_option tactic.hygienic false
+
+/-- Synthesize algebra instance from ring hom. -/
+example (A B : Type*) [CommRing A] [CommRing B] (f : A →+* B) : True := by
+  fail_if_success -- Check that this instance is not available by default
+    have h : Algebra A B := inferInstance
+  algebraize [f]
+  guard_hyp algInst := f.toAlgebra
+  trivial
+
+/-- Synthesize algebra instance from a composition -/
+example (A B C : Type*) [CommRing A] [CommRing B] [CommRing C] (f : A →+* B) (g : B →+* C) :
+    True := by
+  fail_if_success -- Check that this instance is not available by default
+    have h : Algebra A C := inferInstance
+  algebraize [g.comp f]
+  guard_hyp algInst := (g.comp f).toAlgebra
+  trivial
+
+/-- Synthesize algebra instance and scalar tower instance from a composition -/
+example (A B C : Type*) [CommRing A] [CommRing B] [CommRing C] (f : A →+* B) (g : B →+* C) :
+    True := by
+  fail_if_success -- Check that this instance is not available by default
+    have h : IsScalarTower A B C := inferInstance
+  algebraize [f, g, g.comp f]
+  guard_hyp scalarTowerInst := IsScalarTower.of_algebraMap_eq' rfl
+  trivial
+
+example (A B : Type*) [CommRing A] [CommRing B] (f : A →+* B) (hf : f.testProperty1) : True := by
+  algebraize [f]
+  guard_hyp algebraizeInst : Algebra.testProperty1 A B := hf
+  trivial
+
+example (A B : Type*) [CommRing A] [CommRing B] (f : A →+* B) (hf : f.testProperty2) : True := by
+  algebraize [f]
+  guard_hyp algebraizeInst : Module.testProperty2 A B := hf
+  trivial
+
+example (A B : Type*) [CommRing A] [CommRing B] (f : A →+* B) (hf : f.testProperty3) : True := by
+  algebraize [f]
+  guard_hyp algebraizeInst : Algebra.testProperty3 A B := ⟨hf⟩
+  trivial
+
+/-- Synthesize from morphism property of a composition (and check that tower is also synthesized). -/
+example (A B C : Type*) [CommRing A] [CommRing B] [CommRing C] (f : A →+* B) (g : B →+* C)
+    (hfg : (g.comp f).testProperty1) : True := by
+  fail_if_success -- Check that this instance is not available by default
+    have h : Algebra.Flat A C := inferInstance
+  fail_if_success
+    have h : IsScalarTower A B C := inferInstance
+  algebraize [f, g, g.comp f]
+  guard_hyp algebraizeInst : Algebra.testProperty1 A C := hfg
+  guard_hyp scalarTowerInst := IsScalarTower.of_algebraMap_eq' rfl
+  trivial
diff --git a/test/basicTactics.lean b/test/basicTactics.lean
index 87c0505900a57..2eaa3aee8ce7e 100644
--- a/test/basicTactics.lean
+++ b/test/basicTactics.lean
@@ -5,6 +5,7 @@ example : ∀ a b : Nat, a = b → b = a := by
   introv h
   exact h.symm
 
+set_option linter.unusedTactic false in
 example (n : Nat) : n = n := by
   induction n
   exacts [rfl, rfl]
@@ -45,6 +46,7 @@ example (n m : Nat) : Unit := by
   cases m
   iterate exact ()
 
+set_option linter.unusedTactic false in
 example (n : Nat) : Nat := by
   iterate exact () -- silently succeeds, after iterating 0 times
   iterate exact n
diff --git a/test/byContra.lean b/test/byContra.lean
index dd0d781abe597..56a378b97d1e6 100644
--- a/test/byContra.lean
+++ b/test/byContra.lean
@@ -2,7 +2,6 @@
 import Mathlib.Tactic.ByContra
 import Mathlib.Tactic.Rename
 import Mathlib.Tactic.Set
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Order.Basic
 import Mathlib.Data.Nat.Defs
 
diff --git a/test/cases.lean b/test/cases.lean
index b199a9b8748e9..664d0db8f51f8 100644
--- a/test/cases.lean
+++ b/test/cases.lean
@@ -1,6 +1,5 @@
 import Batteries.Logic
 import Mathlib.Tactic.Cases
-import Mathlib.Init.Logic
 import Mathlib.Data.Nat.Notation
 
 set_option autoImplicit true
diff --git a/test/casesm.lean b/test/casesm.lean
index 5af4572b66196..327f1e0bda163 100644
--- a/test/casesm.lean
+++ b/test/casesm.lean
@@ -2,6 +2,7 @@ import Mathlib.Tactic.CasesM
 
 set_option autoImplicit true
 
+set_option linter.unusedTactic false in
 example (h : a ∧ b ∨ c ∧ d) (h2 : e ∧ f) : True := by
   casesm* _∨_, _∧_
   · clear ‹a› ‹b› ‹e› ‹f›; (fail_if_success clear ‹c›); trivial
diff --git a/test/congr.lean b/test/congr.lean
index 7dd402f4cb5dc..d1e2254a6dd4a 100644
--- a/test/congr.lean
+++ b/test/congr.lean
@@ -343,3 +343,16 @@ example
     {α : Type} (inst1 : BEq α) [LawfulBEq α] (inst2 : BEq α) [LawfulBEq α] (xs : List α) (x : α) :
     @List.erase _ inst1 xs x = @List.erase _ inst2 xs x := by
   congr! (config := { beqEq := false })
+
+
+/-!
+Check that congruence theorem generator operates at default transparency.
+Fixes error reported on Zulip:
+https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/congr!.20internal.20error/near/464820779
+-/
+
+def F := ∀ x : ℕ, x = 0 → ℕ
+def F.A (_ : F) : ℕ := 0
+def F.B (_ : F) : ℕ := 0
+theorem bug (H : F) (hp : H.A = 0) (hp' : H.B = 0) :
+  H H.A hp = H H.B hp' := by with_reducible congr!
diff --git a/test/convert.lean b/test/convert.lean
index f00e2c3f65a52..4f404da7265ba 100644
--- a/test/convert.lean
+++ b/test/convert.lean
@@ -110,6 +110,7 @@ example : True := by
 -- Prior to https://github.com/leanprover/lean4/pull/4493 it did,
 -- because previously bodies of `example`s were (confusingly!) allowed to
 -- affect the elaboration of the signature!
+set_option linter.unusedTactic false in
 example {α β : Type u} [Fintype α] [Fintype β] : Fintype.card α = Fintype.card β := by
   congr!
   guard_target = Fintype.card α = Fintype.card β
diff --git a/test/delabLinearIndependent.lean b/test/delabLinearIndependent.lean
index f9755826577ee..3dbfbf8b3fda9 100644
--- a/test/delabLinearIndependent.lean
+++ b/test/delabLinearIndependent.lean
@@ -10,7 +10,7 @@ variable (h : LinearIndependent K (fun b => b : s → V)) in
 #guard_msgs in #check h
 
 variable (h : LinearIndependent K (Subtype.val : s → V)) in
-/-- info: h : LinearIndependent (ι := { x // x ∈ s }) K Subtype.val -/
+/-- info: h : LinearIndependent K Subtype.val -/
 #guard_msgs in #check h
 
 variable (h : LinearIndependent K (by exact Subtype.val : s → V)) in
diff --git a/test/fail_if_no_progress.lean b/test/fail_if_no_progress.lean
index a4284da8a7c3a..3a69c7344ef13 100644
--- a/test/fail_if_no_progress.lean
+++ b/test/fail_if_no_progress.lean
@@ -2,6 +2,7 @@ import Mathlib.Tactic.FailIfNoProgress
 import Mathlib.Tactic.Basic
 
 set_option linter.unusedVariables false
+set_option linter.style.setOption false
 set_option pp.unicode.fun true
 
 section success
diff --git a/test/finsupp_notation.lean b/test/finsupp_notation.lean
index e713fedca935a..e042b67a8903f 100644
--- a/test/finsupp_notation.lean
+++ b/test/finsupp_notation.lean
@@ -21,7 +21,7 @@ info:
     reprStr (Finsupp.mk {1, 2} (fun | 1 | 2 => 3 | _ => 0) (fun x => by aesop))
       = "fun₀ | 1 => 3 | 2 => 3"
 
-/-! ## (computable) number theory examples-/
+/-! ## (computable) number theory examples -/
 
 /-- info: fun₀ | 2 => 2 | 7 => 1 -/
 #guard_msgs in
diff --git a/test/fun_prop.lean b/test/fun_prop.lean
index 5b7a7d504d31b..953fa071dd9a3 100644
--- a/test/fun_prop.lean
+++ b/test/fun_prop.lean
@@ -188,7 +188,7 @@ The theorem `Measurable.apply_continuousLinearMap` states measurability in `f` i
 form.
 -/
 
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 attribute [fun_prop]
   ContinuousLinearMap.measurable       -- Measurable fun (x : E) => DFunLike.coe L x
   ContinuousLinearMap.measurable_comp  -- Measurable φ → Measurable fun (x : E) => DFunLike.coe L (φ x)
@@ -203,7 +203,7 @@ A silly example that everything together works as expected
 example (f : ℝ → ℝ → (ℝ →L[ℝ] ℝ)) (hf : Continuous (fun (x,y) => f x y)) :
     Measurable fun x => (f (x / x) (x * x) 1 + x) := by fun_prop
 
-set_option linter.longLine false in
+set_option linter.style.longLine false in
 /-!
 In the current state of `fun_prop`, morphism theorems **have to** be stated in compositional form.
 Sometimes they might work in uncurried form but `fun_prop` is not designed that way right now.
diff --git a/test/fun_prop_dev.lean b/test/fun_prop_dev.lean
index ea11cb4c71bad..f78abae43617a 100644
--- a/test/fun_prop_dev.lean
+++ b/test/fun_prop_dev.lean
@@ -6,6 +6,7 @@ Authors: Tomáš Skřivan
 import Mathlib.Tactic.FunProp
 import Mathlib.Logic.Function.Basic
 import Mathlib.Data.FunLike.Basic
+import Aesop
 
 /-! # Tests for the `fun_prop` tactic
 
@@ -13,6 +14,8 @@ This file is designed for development of fun_prop and does not depend on most of
 two function properties `Con` and `Lin` which roughly correspond to `Continuity` and `IsLinearMap`.
 -/
 
+set_option linter.style.longLine false
+
 open Function
 
 variable {α β γ δ ι : Type _} {E : α → Type _}
@@ -25,18 +28,13 @@ set_option linter.unusedVariables false
 -- define function propositions --
 ----------------------------------
 
-class Obj (α : Type _) : Type where
-
-instance [Obj α] [Obj β] : Obj (α × β) := ⟨⟩
-instance [∀ x, Obj (E x)] : Obj ((x' : α) → E x') := ⟨⟩
-instance : Obj Nat := ⟨⟩
-
 @[fun_prop] opaque Con {α β} (f : α → β) : Prop
 @[fun_prop] opaque Lin {α β} (f : α → β) : Prop
 
 -- state basic lambda calculus rules --
 ---------------------------------------
 
+-- variable [Obj α] [Obj β] [Obj γ] [Obj δ] [∀ x, Obj (E x)]
 
 @[fun_prop] theorem Con_id : Con (id : α → α) := silentSorry
 @[fun_prop] theorem Con_const (y : β) : Con (fun x : α => y) := silentSorry
@@ -48,7 +46,7 @@ instance : Obj Nat := ⟨⟩
 
 -- Lin is missing `const` theorem
 @[fun_prop] theorem Lin_id : Lin (fun x : α => x) := silentSorry
-@[fun_prop] theorem Lin_const {β} [Obj β] [Zero β] : Lin (fun x : α => (0 : β)) := silentSorry
+@[fun_prop] theorem Lin_const {β} [Zero β] : Lin (fun x : α => (0 : β)) := silentSorry
 @[fun_prop] theorem Lin_apply (x : α) : Lin (fun f : α → β => f x) := silentSorry
 @[fun_prop] theorem Lin_applyDep (x : α) : Lin (fun f : (x' : α) → E x' => f x) := silentSorry
 @[fun_prop] theorem Lin_comp (f : β → γ) (g : α → β) (hf : Lin f) (hg : Lin g) : Lin (f ∘ g) := silentSorry
@@ -109,10 +107,10 @@ structure LinHom (α β) where
 infixr:25 " -o " => LinHom
 
 instance : CoeFun (α ->> β) (fun _ => α → β) where
-  coe := fun f => f.toFun
+  coe f := f.toFun
 
 instance : FunLike (α -o β) α β where
-  coe := fun f => f.toFun
+  coe f := f.toFun
   coe_injective' := silentSorry
 
 #eval Lean.Elab.Command.liftTermElabM do
@@ -130,9 +128,6 @@ instance [HasUncurry β γ δ] : HasUncurry (α -o β) (α × γ) δ :=
   ⟨fun f p ↦ (↿(f p.1)) p.2⟩
 
 
-instance : Obj (α ->> β) := ⟨⟩
-instance : Obj (α -o β) := ⟨⟩
-
 -- morphism theorems i.e. theorems about `FunLike.coe` --
 ---------------------------------------------------------
 
@@ -239,6 +234,7 @@ example (f : α → β ->> γ) (hf : Con fun (x,y) => f x y) (y) : Con fun x =>
 example (f : α → β ->> γ) (hf : Con fun (x,y) => f x y) : Con fun x y => f x y := by fun_prop
 example (f : α → β ->> γ) (hf : Con fun (x,y) => f x y) (x) : Con fun y => f x y := by fun_prop
 example (f : α → α ->> (α → α)) (hf : Con fun (x,y,z) => f x y z) (x) : Con fun y => f x y := by fun_prop
+example (f : α → α ->> (α → α)) (y : α) (hf : Con fun (x,y,z) => f x y z) : Con fun x => f y x x := by fun_prop
 example (f : α → α ->> (α → α)) (hf : Con fun (x,y,z) => f x y z) : Con fun x y => f y x x := by fun_prop
 
 example (f : α → β ->> γ) (hf : Con ↿f) (y) : Con fun x => f x y := by fun_prop
@@ -317,7 +313,7 @@ example (x) : Con fun (f : α ->> α) => f (f x) := by fun_prop
 example (x) : Con fun (f : α ->> α) => f (f (f x)) := by fun_prop
 
 
-example [Zero α] [Obj α] [Add α] : Lin (fun x : α => (0 : α) + x + (0 : α) + (0 : α) + x) := by fun_prop
+example [Zero α] [Add α] : Lin (fun x : α => (0 : α) + x + (0 : α) + (0 : α) + x) := by fun_prop
 
 noncomputable
 def foo : α ->> α ->> α := silentSorry
@@ -463,3 +459,97 @@ Issues:
 -/
 #guard_msgs in
 example : Con (fun x : α => f3 x) := by fun_prop (config:={maxTransitionDepth:=0})
+
+@[fun_prop] opaque Dif (𝕜:Type) [Add 𝕜] {α β} (f : α → β) : Prop
+
+variable {𝕜 : Type}
+@[fun_prop] theorem Dif_id [Add 𝕜] : Dif 𝕜 (id : α → α) := silentSorry
+@[fun_prop] theorem Dif_const [Add 𝕜] (y : β) : Dif 𝕜 (fun x : α => y) := silentSorry
+@[fun_prop] theorem Dif_apply [Add 𝕜] (x : α) : Dif 𝕜 (fun f : α → β => f x) := silentSorry
+@[fun_prop] theorem Dif_applyDep [Add 𝕜] (x : α) : Dif 𝕜 (fun f : (x' : α) → E x' => f x) := silentSorry
+@[fun_prop] theorem Dif_comp [Add 𝕜] (f : β → γ) (g : α → β) (hf : Dif 𝕜 f) (hg : Dif 𝕜 g) : Dif 𝕜 (fun x => f (g x)) := silentSorry
+@[fun_prop] theorem Dif_pi [Add 𝕜] (f : β → (i : α) → (E i)) (hf : ∀ i, Dif 𝕜 (fun x => f x i)) : Dif 𝕜 (fun x i => f x i) := silentSorry
+
+@[fun_prop]
+theorem Dif_Con [Add 𝕜] (f : α → β) (hf : Dif 𝕜 f) : Con f := silentSorry
+
+def f4 (a : α) := a
+
+example (hf : Dif Nat (f4 : α → α)) : Con (f4 : α → α) := by fun_prop (disch:=aesop)
+
+@[fun_prop]
+theorem f4_dif : Dif Nat (f4 : α → α) := silentSorry
+
+example (hf : Dif Nat (f4 : α → α)) : Con (f4 : α → α) := by fun_prop (disch:=aesop)
+
+
+-- Test abbrev transparency
+abbrev my_id {α} (a : α) := a
+example : Con (fun x : α => my_id x) := by fun_prop
+example (f : α → β) (hf : Con (my_id f)) : Con f := by fun_prop
+
+-- Testing some issues with bundled morphisms of multiple arguments
+structure Mor where
+  toFun : Int → Int → Int
+  hcon : Con (fun (x,y) => toFun x y)
+
+@[fun_prop]
+theorem Mor.toFun_Con (m : Mor) (f g : α → Int) (hf : Con f) (g : α → Int) (hg : Con g) :
+    Con (fun x => m.toFun (f x) (g x)) := by
+  have := m.hcon
+  fun_prop
+
+-- Test improved beta reduction of the head function when we interleave lambdas and lets
+example [Add α] (a : α) : Con (fun x0 : α =>
+  (fun x =>
+    let y := x + x
+    fun z : α =>
+      x + y + z) x0 a) := by fun_prop
+
+example [Add α] (a : α) :
+  let f := (fun x : α =>
+    let y := x + x
+    fun z : α =>
+      x + y + z)
+  Con (fun x => f x a) := by fun_prop
+
+example [Add α] (a a' : α) : Con (fun x0 : α =>
+  (fun x =>
+    let y := x + x
+    fun z : α =>
+      let h := x + y + z
+      fun w =>
+        w + x + y + z + h) x0 a a') := by fun_prop
+
+
+-- test that local function is being properly unfolded
+example [Add α] (a : α) :
+  let f := (fun x : α =>
+    let y := x + x
+    fun z : α =>
+      x + y + z)
+  Con (fun x =>
+    f x a) := by
+  fun_prop
+
+
+-- Test that local theorem is being used
+/--
+info: [Meta.Tactic.fun_prop] [✅️] Con fun x => f x y
+  [Meta.Tactic.fun_prop] candidate local theorems for f #[this : Con f]
+  [Meta.Tactic.fun_prop] removing argument to later use this : Con f
+  [Meta.Tactic.fun_prop] [✅️] applying: Con_comp
+    [Meta.Tactic.fun_prop] [✅️] Con fun f => f y
+      [Meta.Tactic.fun_prop] [✅️] applying: Con_apply
+    [Meta.Tactic.fun_prop] [✅️] Con fun x => f x
+      [Meta.Tactic.fun_prop] candidate local theorems for f #[this : Con f]
+      [Meta.Tactic.fun_prop] [✅️] applying: this : Con f
+-/
+#guard_msgs in
+example [Add α] (y : α):
+  let f := (fun x y : α => x+x+y)
+  Con (fun x => f x y) := by
+  intro f
+  have : Con f := by fun_prop
+  set_option trace.Meta.Tactic.fun_prop true in
+  fun_prop
diff --git a/test/instance_diamonds.lean b/test/instance_diamonds.lean
index 7b746593fdb05..f26a6b719c7ff 100644
--- a/test/instance_diamonds.lean
+++ b/test/instance_diamonds.lean
@@ -141,7 +141,7 @@ example : @Monoid.toMulOneClass (Multiplicative ℕ) CommMonoid.toMonoid =
 
 end Multiplicative
 
-/-! ## `Finsupp` instances-/
+/-! ## `Finsupp` instances -/
 
 
 section Finsupp
diff --git a/test/interactiveUnfold.lean b/test/interactiveUnfold.lean
index 3aec6b16ad15e..b17d3daa3cae8 100644
--- a/test/interactiveUnfold.lean
+++ b/test/interactiveUnfold.lean
@@ -65,7 +65,7 @@ info: Unfolds for 1 + 1:
 
 /--
 info: Unfolds for 3 ∈ {1, 2, 3}:
-· Set.Mem 3 {1, 2, 3}
+· {1, 2, 3}.Mem 3
 · {1, 2, 3} 3
 · Set.insert 1 {2, 3} 3
 · {b | b = 1 ∨ b ∈ {2, 3}} 3
@@ -86,7 +86,7 @@ variable (A B : Set Nat) (n : Nat)
 
 /--
 info: Unfolds for 1 ∈ A ∪ B:
-· Set.Mem 1 (A ∪ B)
+· (A ∪ B).Mem 1
 · (A ∪ B) 1
 · A.union B 1
 · {a | a ∈ A ∨ a ∈ B} 1
diff --git a/test/interval_cases.lean b/test/interval_cases.lean
index bf08cde4046dd..02cbfc56c324a 100644
--- a/test/interval_cases.lean
+++ b/test/interval_cases.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2019 Scott Morrison. All rights reserved.
+Copyright (c) 2019 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 
 import Mathlib.Tactic.IntervalCases
diff --git a/test/itauto.lean b/test/itauto.lean
index b628906eca815..94353f251dd30 100644
--- a/test/itauto.lean
+++ b/test/itauto.lean
@@ -55,6 +55,7 @@ example (b : Bool) : ¬b ∨ b := by itauto *
 example (p : Prop) : ¬p ∨ p := by itauto! [p]
 example (p : Prop) : ¬p ∨ p := by itauto! *
 
+set_option linter.unusedTactic false in
 -- failure tests
 example (p q r : Prop) : True := by
   haveI : p ∨ ¬p := by (fail_if_success itauto); sorry
diff --git a/test/linear_combination.lean b/test/linear_combination.lean
index c9e02f7fa847e..9f652a8186a63 100644
--- a/test/linear_combination.lean
+++ b/test/linear_combination.lean
@@ -120,10 +120,53 @@ example {α} [h : CommRing α] {a b c d e f : α} (h1 : a * d = b * c) (h2 : c *
 example (x y z w : ℚ) (hzw : z = w) : x * z + 2 * y * z = x * w + 2 * y * w := by
   linear_combination (x + 2 * y) * hzw
 
-example (x : ℤ) : x ^ 2 = x ^ 2 := by linear_combination x ^ 2
-
 example (x y : ℤ) (h : x = 0) : y ^ 2 * x = 0 := by linear_combination y ^ 2 * h
 
+/-! ### Tests in semirings -/
+
+example (a _b : ℕ) (h1 : a = 3) : a = 3 := by
+  linear_combination h1
+
+example {a b : ℕ} (h1 : a = b + 4) (h2 : b = 2) : a = 6 := by
+  linear_combination h1 + h2
+
+example {a : ℕ} (h : a = 3) : 3 = a := by linear_combination -h
+
+example {a b : ℕ} (h1 : 3 * a = b + 5) (h2 : 2 * a = b + 3) : a = 2 := by
+  linear_combination h1 - h2
+
+/- Note: currently negation/subtraction is handled differently in "constants" than in "proofs", so
+in particular negation/subtraction does not "distribute". The following four tests record the
+current behaviour, without taking a stance on whether this should be considered a feature or a bug.
+-/
+
+example {a : ℕ}  (h : a = 3) : a ^ 2 + 3 = 4 * a := by
+  linear_combination a * h - h
+
+/--
+error: ring failed, ring expressions not equal
+a b : ℕ
+h : a = 3
+⊢ 3 + a ^ 2 + (a - 1) * 3 = a * 4 + a * (a - 1)
+-/
+#guard_msgs in
+example {a b : ℕ}  (h : a = 3) : a ^ 2 + 3 = 4 * a := by
+  linear_combination (a - 1) * h
+
+example {a b c : ℕ} (h1 : c = 1) (h2 : a - b = 4) : (a - b) * c = 4 := by
+  linear_combination (a - b) * h1 + h2
+
+/--
+error: ring failed, ring expressions not equal
+a b c : ℕ
+h1 : c = 1
+h2 : a - b = 4
+⊢ 4 + (a - b) * c + c * b + a = 4 + (a - b) + c * a + b
+-/
+#guard_msgs in
+example {a b c : ℕ} (h1 : c = 1) (h2 : a - b = 4) : (a - b) * c = 4 := by
+  linear_combination a * h1 - b * h1 + h2
+
 /-! ### Cases that explicitly use a config -/
 
 example (x y : ℚ) (h1 : 3 * x + 2 * y = 10) (h2 : 2 * x + 5 * y = 3) : -11 * y + 1 = 11 + 1 := by
@@ -154,6 +197,8 @@ example (x y : ℤ) (h1 : x = -3) (_h2 : y = 10) : 2 * x = -6 := by
 example {x y z w : ℤ} (_h₁ : 3 * x = 4 + y) (_h₂ : x + 2 * y = 1) : z + w = w + z := by
   linear_combination
 
+example (x : ℤ) : x ^ 2 = x ^ 2 := by linear_combination
+
 -- this interacts as expected with options
 example {x y z w : ℤ} (_h₁ : 3 * x = 4 + y) (_h₂ : x + 2 * y = 1) : z + w = w + z := by
   linear_combination (norm := skip)
@@ -184,6 +229,15 @@ example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y = -2 * y + 1 :
 
 /-! ### Cases that should fail -/
 
+/--
+error: ring failed, ring expressions not equal
+a : ℤ
+ha : a = 1
+⊢ -1 = 0
+-/
+#guard_msgs in
+example (a : ℤ) (ha : a = 1) : a = 2 := by linear_combination ha
+
 /--
 error: ring failed, ring expressions not equal
 a : ℚ
@@ -211,17 +265,31 @@ but is expected to have type
 example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) : x * y + 2 * x = 1 := by
   linear_combination h1 + (0 : ℝ) * h2
 
--- This fails because the linear_combination tactic requires the equations
---   and coefficients to use a type that fulfills the add_group condition,
---   and ℕ does not.
-example (a _b : ℕ) (h1 : a = 3) : a = 3 := by
-  fail_if_success linear_combination h1
-  linear_combination2 h1
-
 example (a b : ℤ) (x y : ℝ) (hab : a = b) (hxy : x = y) : 2 * x = 2 * y := by
   fail_if_success linear_combination 2 * hab
   linear_combination 2 * hxy
 
+/--
+warning: this constant has no effect on the linear combination; it can be dropped from the term
+-/
+#guard_msgs in
+example (x y : ℤ) (h1 : 3 * x + 2 * y = 10) : 3 * x + 2 * y = 10 := by
+  linear_combination h1 + 3
+
+/--
+warning: this constant has no effect on the linear combination; it can be dropped from the term
+-/
+#guard_msgs in
+example (x : ℤ) : x ^ 2 = x ^ 2 := by linear_combination x ^ 2
+
+/-- error: 'linear_combination' supports only linear operations -/
+#guard_msgs in
+example {x y : ℤ} (h : x = y) : x ^ 2 = y ^ 2 := by linear_combination h * h
+
+/-- error: 'linear_combination' supports only linear operations -/
+#guard_msgs in
+example {x y : ℤ} (h : x = y) : 3 / x = 3 / y := by linear_combination 3 / h
+
 /-! ### Cases with exponent -/
 
 example (x y z : ℚ) (h : x = y) (h2 : x * y = 0) : x + y*z = 0 := by
diff --git a/test/matrix.lean b/test/matrix.lean
index 962e193b1ef97..522acbe923c7d 100644
--- a/test/matrix.lean
+++ b/test/matrix.lean
@@ -154,7 +154,7 @@ example {α : Type _} [CommRing α] {a b c d : α} :
       Fin.isValue, of_apply, cons_val', empty_val', cons_val_fin_one, cons_val_zero, det_unique,
       Fin.default_eq_zero, submatrix_apply, Fin.succ_zero_eq_one, cons_val_one, head_fin_const,
       Fin.sum_univ_succ, Fin.val_zero, pow_zero, one_mul, Fin.zero_succAbove, head_cons,
-      Finset.univ_unique, Fin.val_succ, Fin.coe_fin_one, zero_add, pow_one, cons_val_succ, neg_mul,
+      Finset.univ_unique, Fin.val_succ, Fin.val_eq_zero, zero_add, pow_one, cons_val_succ, neg_mul,
       Fin.succ_succAbove_zero, Finset.sum_const, Finset.card_singleton, smul_neg, one_smul]
   ring
 
@@ -167,7 +167,7 @@ example {α : Type _} [CommRing α] {a b c d e f g h i : α} :
       submatrix_apply, Fin.succ_zero_eq_one, cons_val_one, head_cons, submatrix_submatrix,
       det_unique, Fin.default_eq_zero, Function.comp_apply, Fin.succ_one_eq_two, cons_val_two,
       tail_cons, head_fin_const, Fin.sum_univ_succ, Fin.val_zero, pow_zero, one_mul,
-      Fin.zero_succAbove, Finset.univ_unique, Fin.val_succ, Fin.coe_fin_one, zero_add, pow_one,
+      Fin.zero_succAbove, Finset.univ_unique, Fin.val_succ, Fin.val_eq_zero, zero_add, pow_one,
       neg_mul, Fin.succ_succAbove_zero, Finset.sum_neg_distrib, Finset.sum_singleton, cons_val_succ,
       Fin.succ_succAbove_one, even_two, Even.neg_pow, one_pow, Finset.sum_const,
       Finset.card_singleton, one_smul]
diff --git a/test/measurability.lean b/test/measurability.lean
index 2ebf1a799ee68..d373a1d2d8576 100644
--- a/test/measurability.lean
+++ b/test/measurability.lean
@@ -82,7 +82,7 @@ example [Div β] [MeasurableDiv₂ β] (hf : Measurable f) (hg : Measurable g)
 
 example [AddCommMonoid β] [MeasurableAdd₂ β] {s : Finset ℕ} {F : ℕ → α → β}
     (hF : ∀ i, Measurable (F i)) : Measurable (∑ i ∈ s, (fun x => F (i+1) x + F i x)) := by
-  measurability
+  fun_prop
 
 example [AddCommMonoid β] [MeasurableAdd₂ β] {s : Finset ℕ} {F : ℕ → α → β}
     (hF : ∀ i, AEMeasurable (F i) μ) : AEMeasurable (∑ i ∈ s, (fun x => F (i+1) x + F i x)) μ := by
diff --git a/test/meta.lean b/test/meta.lean
index 5cd408d0b195a..3d5620d3af41c 100644
--- a/test/meta.lean
+++ b/test/meta.lean
@@ -11,6 +11,7 @@ namespace Tests
 open Lean Meta
 
 private axiom test_sorry : ∀ {α}, α
+set_option linter.style.setOption false in
 set_option pp.unicode.fun true
 
 def eTrue := Expr.const ``True []
@@ -59,6 +60,8 @@ elab "test_forallNot_of_notExists" t:term : tactic => do
   unless ← isDefEq ety' (← inferType e') do throwError "bad proof"
   logInfo m!"{ety'}"
 
+set_option linter.unusedTactic false
+
 /-- info: ∀ (x : Nat), ¬0 < x -/
 #guard_msgs in
 example (h : ¬ ∃ x, 0 < x) : False := by
diff --git a/test/module.lean b/test/module.lean
new file mode 100644
index 0000000000000..ee59dd02d0d9f
--- /dev/null
+++ b/test/module.lean
@@ -0,0 +1,308 @@
+/-
+Copyright (c) 2024 Heather Macbeth. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Heather Macbeth
+-/
+import Mathlib.Tactic.FieldSimp
+import Mathlib.Tactic.LinearCombination
+import Mathlib.Tactic.Module
+import Mathlib.Tactic.NoncommRing
+import Mathlib.Tactic.Positivity
+
+/-! # Tests for the module-normalization tactic -/
+
+open Mathlib.Tactic.LinearCombination
+
+variable {V : Type*} {K : Type*} {t u v w x y z : V} {a b c d e f μ ν ρ : K}
+
+/-! ### `ℕ` (most tests copied from the `abel` tactic) -/
+
+section Nat
+variable [AddCommMonoid V]
+
+example : x + (y + x) = x + x + y := by module
+example : (3 : ℕ) • x = x + (2 : ℕ) • x := by module
+example : 0 + x = x := by module
+example (n : ℕ) : n • x = n • x := by module
+example (n : ℕ) : 0 + n • x = n • x := by module
+example : x + (y + (x + (z + (x + (u + (x + v)))))) = v + u + z + y + 4 • x := by module
+example : x + y = y + x := by module
+example : x + 2 • x = 2 • x + x := by module
+
+example : x + (y + x) = x + x + y ∨ False := by
+  left
+  module
+
+/--
+error: unsolved goals
+V : Type u_1
+K : Type u_2
+t u v w x y z : V
+a b c d e f μ ν ρ : K
+inst✝ : AddCommMonoid V
+⊢ 1 = 1
+
+V : Type u_1
+K : Type u_2
+t u v w x y z : V
+a b c d e f μ ν ρ : K
+inst✝ : AddCommMonoid V
+⊢ 1 = 2 * 1
+-/
+#guard_msgs in
+example : x + y = x + 2 • y := by match_scalars
+
+/--
+error: ring failed, ring expressions not equal
+V : Type u_1
+K : Type u_2
+t u v w x y z : V
+a b c d e f μ ν ρ : K
+inst✝ : AddCommMonoid V
+⊢ 1 = 2
+-/
+#guard_msgs in
+example : x + y = x + 2 • y := by module
+
+/-- error: goal x ≠ y is not an equality -/
+#guard_msgs in
+example : x ≠ y := by module
+
+end Nat
+
+/-! ### `ℤ` (most tests copied from the `abel` tactic) -/
+
+variable [AddCommGroup V]
+
+example : (x + y) - ((y + x) + x) = -x := by module
+example : x - 0 = x := by module
+example : (3 : ℤ) • x = x + (2 : ℤ) • x := by module
+example : x - 2 • y = x - 2 • y := by module
+example : (x + y) - ((y + x) + x) = -x := by module
+example : x + y + (z + w - x) = y + z + w := by module
+example : x + y + z + (z - x - x) = (-1) • x + y + 2 • z := by module
+example : -x + x = 0 := by module
+example : x - (0 - 0) = x := by module
+example : x + (y - x) = y  := by module
+example : -y + (z - x) = z - y - x := by module
+
+example : x + y = y + x ∧ (↑((1:ℕ) + 1) : ℚ) = 2 := by
+  constructor
+  module -- do not focus this tactic: the double goal is the point of the test
+  guard_target =ₐ (↑((1:ℕ) + 1) : ℚ) = 2
+  norm_cast
+
+-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Interaction.20of.20abel.20with.20casting/near/319895001
+example : True := by
+  have : ∀ (p q r s : V), s + p - q = s - r - (q - r - p) := by
+    intro p q r s
+    module
+  trivial
+
+example : True := by
+  have : ∀ (p q r s : V), s + p - q = s - r - (q - r - p) := by
+    intro p q r s
+    match_scalars
+    · decide
+    · decide
+    · decide
+    · decide
+  trivial
+
+-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Interaction.20of.20abel.20with.20casting/near/319897374
+example : y = x + z - (x - y + z) := by
+  have : True := trivial
+  module
+
+example : y = x + z - (x - y + z) := by
+  have : True := trivial
+  match_scalars <;> decide
+
+/--
+error: unsolved goals
+V : Type u_1
+K : Type u_2
+t u v w x y z : V
+a b c d e f μ ν ρ : K
+inst✝ : AddCommGroup V
+⊢ -1 + 1 = 0
+-/
+#guard_msgs in
+example : -x + x = 0 := by
+  match_scalars
+
+/-! ### Commutative ring -/
+
+section CommRing
+variable [CommRing K] [Module K V]
+
+example : a • x + b • x = (a + b) • x := by module
+example : a • x - b • x = (a - b) • x := by module
+example : a • x - b • y = a • x + (-b) • y := by module
+example : 2 • a • x = a • 2 • x := by module
+example : a • x - b • y = a • x + (-b) • y := by module
+example : (μ - ν) • a • x = (a • μ • x + b • ν • y) - ν • (a • x + b • y) := by module
+example : (μ - ν) • b • y = μ • (a • x + b • y) - (a • μ • x + b • ν • y) := by module
+
+-- from https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/smul.20diamond/near/457163013
+example : (4 : ℤ) • v = (4 : K) • v := by module
+example : (4 : ℕ) • v = (4 : K) • v := by module
+
+-- from https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/linear_combination.20for.20groups/near/437042918
+example : (1 + a ^ 2) • (v + w) - a • (a • v - w) = v + (1 + a + a ^ 2) • w := by module
+
+example (h : a = b) : a • x = b • x := by
+  match_scalars
+  linear_combination h
+
+/- `linear_combination` does not currently handle `•`.  The following mimics what should eventually
+be performed by a `linear_combination` call, with exact syntax TBD -- maybe
+`linear_combination (norm := module) h • x` or `module_combination h • x`. -/
+example (h : a = b) : a • x = b • x := by
+  apply eq_of_add (congr($h • x):)
+  module
+
+example (h : a ^ 2 + b ^ 2 = 1) : a • (a • x - b • y) + (b • a • y + b • b • x) = x := by
+  match_scalars
+  · linear_combination h
+  · ring
+
+example (h : a ^ 2 + b ^ 2 = 1) : a • (a • x - b • y) + (b • a • y + b • b • x) = x := by
+  -- `linear_combination (norm := module) h • x`
+  apply eq_of_add (congr($h • x):)
+  module
+
+example (h1 : a • x + b • y = 0) (h2 : a • μ • x + b • ν • y = 0) :
+    (μ - ν) • a • x = 0 ∧ (μ - ν) • b • y = 0 := by
+  constructor
+  · -- `linear_combination (norm := module) h2 - ν • h1`
+    apply eq_of_add (congr($h2 - ν • $h1):)
+    module
+  · -- `linear_combination (norm := module) μ • h1 + h2`
+    apply eq_of_add (congr(μ • $h1 - $h2):)
+    module
+
+example (h1 : 0 • z + a • x + b • y = 0) (h2 : 0 • ρ • z + a • μ • x + b • ν • y = 0) :
+    (μ - ν) • a • x = 0 := by
+  -- `linear_combination (norm := module) h2 - ν • h1`
+  apply eq_of_add (congr($h2 - ν • $h1):)
+  module
+
+example
+    (h1 : a • x + b • y + c • z = 0)
+    (h2 : a • μ • x + b • ν • y + c • ρ • z = 0)
+    (h3 : a • μ • μ • x + b • ν • ν • y + c • ρ • ρ • z = 0) :
+    (μ - ν) • (μ - ρ) • a • x = 0 ∧ (μ - ν) • (ν - ρ) • b • y = 0
+      ∧ (μ - ρ) • (ν - ρ) • c • z = 0 := by
+  refine ⟨?_, ?_, ?_⟩
+  · -- `linear_combination (norm := module) h3 - (ν + ρ) • h2 + ν • ρ • h1`
+    apply eq_of_add (congr($h3 - (ν + ρ) • $h2 + ν • ρ • $h1):)
+    module
+  · -- `linear_combination (norm := module) - h3 + (μ + ρ) • h2 - μ • ρ • h1`
+    apply eq_of_add (congr(- $h3 + (μ + ρ) • $h2 - μ • ρ • $h1):)
+    module
+  · -- `linear_combination (norm := module) h3 - (μ + ν) • h2 + μ • ν • h1`
+    apply eq_of_add (congr($h3 - (μ + ν) • $h2 + μ • ν • $h1):)
+    module
+
+/--
+error: ring failed, ring expressions not equal
+V : Type u_1
+K : Type u_2
+t u v w x y z : V
+a b c d e f μ ν ρ : K
+inst✝² : AddCommGroup V
+inst✝¹ : CommRing K
+inst✝ : Module K V
+⊢ a * 2 = 2
+-/
+#guard_msgs in
+example : 2 • a • x = 2 • x := by module
+
+end CommRing
+
+/-! ### (Noncommutative) ring -/
+
+section Ring
+variable [Ring K] [Module K V]
+
+example : a • x + b • x = (b + a) • x := by
+  match_scalars
+  noncomm_ring
+
+example : 2 • a • x = a • (2:ℤ) • x := by
+  match_scalars
+  noncomm_ring
+
+example (h : a = b) : a • x = b • x := by
+  match_scalars
+  simp [h]
+
+example : (a - b) • a • x + b • b • x = a • a • x + b • (-a + b) • x := by
+  match_scalars
+  noncomm_ring
+
+end Ring
+
+/-! ### Characteristic-zero field -/
+
+section CharZeroField
+variable [Field K] [CharZero K] [Module K V]
+
+example : (2:K)⁻¹ • x + (3:K)⁻¹ • x + (6:K)⁻¹ • x = x := by module
+
+example (h₁ : t - u = -(v - w)) (h₂ : t + u = v + w) : t = w := by
+  -- `linear_combination (norm := module) 2⁻¹ • h₁ + 2⁻¹ • h₂`
+  apply eq_of_add (congr((2:K)⁻¹ • $h₁ + (2:K)⁻¹ • $h₂):)
+  module
+
+end CharZeroField
+
+/-! ### Linearly ordered field -/
+
+section LinearOrderedField
+variable [LinearOrderedField K] [Module K V]
+
+example (ha : 0 ≤ a) (hb : 0 < b) :
+    x = (a / (a + b)) • y + (b / (a + b)) • (x + (a / b) • (x - y)) := by
+  match_scalars
+  · field_simp
+    ring
+  · field_simp
+    ring
+
+-- From Analysis.Convex.StoneSeparation
+example (hab : 0 < a * b + c * d) :
+    (a * b / (a * b + c * d) * e) • u + (c * d / (a * b + c * d) * f) • v +
+      ((a * b / (a * b + c * d)) • d • x + (c * d / (a * b + c * d)) • b • y) =
+      (a * b + c * d)⁻¹ • ((a * b * e) • u + ((c * d * f) • v +
+        ((a * b) • d • x + (c * d) • b • y))) := by
+  match_scalars
+  · field_simp
+  · field_simp
+  · field_simp
+  · field_simp
+
+example (h₁ : 1 = a ^ 2 + b ^ 2) (h₂ : 1 - a ≠ 0) :
+    ((2 / (1 - a)) ^ 2 * b ^ 2 + 4)⁻¹ • (4:K) • ((2 / (1 - a)) • y)
+    + ((2 / (1 - a)) ^ 2 * b ^ 2 + 4)⁻¹ • ((2 / (1 - a)) ^ 2 * b ^ 2 - 4) • x
+    = a • x + y := by
+  -- `linear_combination (norm := skip) (h₁ * (b ^ 2 + (1 - a) ^ 2)⁻¹) • (y + (a - 1) • x)`
+  apply eq_of_add (congr(($h₁ * (b ^ 2 + (1 - a) ^ 2)⁻¹) • (y + (a - 1) • x)):)
+  match_scalars
+  · field_simp
+    ring
+  · field_simp
+    ring
+
+example (h₁ : 1 = a ^ 2 + b ^ 2) (h₂ : 1 - a ≠ 0) :
+    ((2 / (1 - a)) ^ 2 * b ^ 2 + 4)⁻¹ • (4:K) • ((2 / (1 - a)) • y)
+    + ((2 / (1 - a)) ^ 2 * b ^ 2 + 4)⁻¹ • ((2 / (1 - a)) ^ 2 * b ^ 2 - 4) • x
+    = a • x + y := by
+  match_scalars
+  · field_simp
+    linear_combination 4 * (1 - a) * h₁
+  · field_simp
+    linear_combination 4 * (a - 1) ^ 3 * h₁
+
+end LinearOrderedField
diff --git a/test/notation3.lean b/test/notation3.lean
index eecf73b2e31a5..c4473b94e10e7 100644
--- a/test/notation3.lean
+++ b/test/notation3.lean
@@ -1,7 +1,7 @@
 import Mathlib.Util.Notation3
-import Mathlib.Init.Data.Nat.Lemmas
 import Mathlib.Data.Nat.Defs
 
+set_option linter.style.setOption false
 set_option pp.unicode.fun true
 set_option autoImplicit true
 
@@ -151,6 +151,7 @@ matcher from the expansion. (Use `set_option trace.notation3 true` to get some d
 end
 
 section
+set_option linter.unusedTactic false
 local notation3 (prettyPrint := false) "#" n => Fin.mk n (by decide)
 
 example : Fin 5 := #1
diff --git a/test/polyrith.lean b/test/polyrith.lean
index a0bd8b137e1f3..58c192b53bdc6 100644
--- a/test/polyrith.lean
+++ b/test/polyrith.lean
@@ -7,7 +7,7 @@ import Mathlib.Tactic.Polyrith
 
 -- Except for the `import`, the doc-modules and the following `set_option`, this file is just
 -- comments and whitespace.  Once the file gets revived, the linting can start!
-set_option linter.longLine false
+set_option linter.style.longLine false
 
 /-!
 
@@ -183,7 +183,7 @@ A full test suite is provided at the bottom of the file.
 --   "(((((2 * var0) - 3) + (9 * var1)) + (3 * var2)) - (((8 - var3) + (3 * var2)) - (3 * var0)))"]
 --   "linear_combination 2 * h1 - h2 + 3 * h3 - 3 * h4"
 
--- /-! ### Case with ambiguous identifiers-/
+-- /-! ### Case with ambiguous identifiers -/
 
 -- example («def evil» y : ℤ) (h1 : 3*«def evil» + 2*y = 10) :
 --   3*«def evil» + 2*y = 10 :=
@@ -481,7 +481,7 @@ example (a b c d : ℚ) (h1 : a = 4) (h2 : 3 = b) (h3 : c*3 = d) (h4 : -d = a) :
     2*a - 3 + 9*c + 3*d = 8 - b + 3*d - 3*a := by
   polyrith
 
-/-! ### Case with ambiguous identifiers-/
+/-! ### Case with ambiguous identifiers -/
 -- set_option trace.Meta.Tactic.polyrith true
 example («def evil» y : ℤ) (h1 : 3*«def evil» + 2*y = 10) :
     3*«def evil» + 2*y = 10 := by
@@ -631,7 +631,7 @@ example (a b c d : ℚ) (h1 : a = 4) (h2 : 3 = b) (h3 : c*3 = d) (h4 : -d = a) :
     2*a - 3 + 9*c + 3*d = 8 - b + 3*d - 3*a := by
   create_polyrith_test
 
-/-! ### Case with ambiguous identifiers-/
+/-! ### Case with ambiguous identifiers -/
 
 example («def evil» y : ℤ) (h1 : 3*«def evil» + 2*y = 10) :
     3*«def evil» + 2*y = 10 := by
diff --git a/test/positivity.lean b/test/positivity.lean
index d75fda862bd69..fc773c8abd962 100644
--- a/test/positivity.lean
+++ b/test/positivity.lean
@@ -23,6 +23,9 @@ example : 0 ≤ 3 := by positivity
 
 example : 0 < 3 := by positivity
 
+example : (0 : ℝ≥0∞) < 1 := by positivity
+example : (0 : ℝ≥0∞) < 2 := by positivity
+
 /- ## Goals working directly from a hypothesis -/
 -- set_option trace.Meta.debug true
 -- sudo set_option trace.Tactic.positivity true
diff --git a/test/propose.lean b/test/propose.lean
index bb39586ed5ccd..04d97bb804156 100644
--- a/test/propose.lean
+++ b/test/propose.lean
@@ -2,6 +2,7 @@ import Mathlib.Tactic.Propose
 import Mathlib.Tactic.GuardHypNums
 import Mathlib.Algebra.Associated.Basic
 import Mathlib.Data.Set.Subsingleton
+import Batteries.Data.List.Lemmas
 
 -- For debugging, you may find these options useful:
 -- set_option trace.Tactic.propose true
@@ -11,9 +12,9 @@ set_option autoImplicit true
 theorem foo (L M : List α) (w : L.Disjoint M) (m : a ∈ L) : a ∉ M := fun h => w m h
 
 /--
-info: Try this: have : K.Disjoint M := List.disjoint_of_subset_left m w
----
 info: Try this: have : M.Disjoint L := List.disjoint_symm w
+---
+info: Try this: have : K.Disjoint M := List.disjoint_of_subset_left m w
 -/
 #guard_msgs in
 example (K L M : List α) (w : L.Disjoint M) (m : K ⊆ L) : True := by
@@ -70,16 +71,16 @@ info: Try this: have : IsUnit p := isUnit_of_dvd_one h
 ---
 info: Try this: have : ¬IsUnit p := not_unit hp
 ---
-info: Try this: have : ¬p ∣ 1 := not_dvd_one hp
+info: Try this: have : p ∣ p * p ↔ p ∣ p ∨ p ∣ p := Prime.dvd_mul hp
 ---
 info: Try this: have : p ∣ p ∨ p ∣ p := dvd_or_dvd hp (Exists.intro p (Eq.refl (p * p)))
 ---
-info: Try this: have : p ≠ 0 := ne_zero hp
----
-info: Try this: have : p ∣ p * p ↔ p ∣ p ∨ p ∣ p := Prime.dvd_mul hp
+info: Try this: have : ¬p ∣ 1 := not_dvd_one hp
 ---
 info: Try this: have : IsPrimal p := isPrimal hp
 ---
+info: Try this: have : p ≠ 0 := ne_zero hp
+---
 info: Try this: have : p ≠ 1 := ne_one hp
 -/
 #guard_msgs in
diff --git a/test/recover.lean b/test/recover.lean
index f31c6d95e8cca..72d81332f21b0 100644
--- a/test/recover.lean
+++ b/test/recover.lean
@@ -1,10 +1,12 @@
 import Mathlib.Tactic.Recover
 
+set_option linter.unusedTactic false
+
 /-- problematic tactic for testing recovery -/
 elab "this" "is" "a" "problem" : tactic =>
   Lean.Elab.Tactic.setGoals []
 
-/- The main test-/
+/- The main test -/
 example : 1 = 1 := by
   recover this is a problem
   rfl
diff --git a/test/ring_compare.lean b/test/ring_compare.lean
new file mode 100644
index 0000000000000..3e282d9fbfe30
--- /dev/null
+++ b/test/ring_compare.lean
@@ -0,0 +1,114 @@
+import Mathlib.Tactic.NormNum.OfScientific
+import Mathlib.Tactic.Ring.Compare
+import Mathlib.Tactic.Ring.RingNF
+
+open Lean Elab Tactic
+
+elab "ring_le" : tactic => liftMetaFinishingTactic Mathlib.Tactic.Ring.proveLE
+elab "ring_lt" : tactic => liftMetaFinishingTactic Mathlib.Tactic.Ring.proveLT
+
+section Nat
+variable {x y : ℕ}
+
+example : 3 ≤ (3:ℕ) := by ring_le
+example : 1 ≤ (3:ℕ) := by ring_le
+example : 0 ≤ (3:ℕ) + 1 := by ring_le
+example : x ≤ x + 3 := by ring_le
+example : x ≤ 1 + x := by ring_le
+example : x + y + 1 ≤ y + x + 3 := by ring_le
+example : x + y ≤ y + x + 3 := by ring_le
+example : x + y + 1 ≤ y + 4 + x := by ring_le
+
+example : 1 < (3:ℕ) := by ring_lt
+example : 0 < (3:ℕ) + 1 := by ring_lt
+example : x < x + 3 := by ring_lt
+example : x < 1 + x := by ring_lt
+example : x + y + 1 < y + x + 3 := by ring_lt
+example : x + y < y + x + 3 := by ring_lt
+example : x + y + 1 < y + 4 + x := by ring_lt
+
+end Nat
+
+section LinearOrderedField
+variable {K : Type*} [LinearOrderedField K] {x y : K}
+
+example : (0:K) ≤ 0 := by ring_le
+example : 3 ≤ (3:K) := by ring_le
+example : 1 ≤ (3:K) := by ring_le
+example : -1 ≤ (3:K) := by ring_le
+example : 1.5 ≤ (3:K) := by ring_le
+example : 0 ≤ x + 3 - x := by ring_le
+example : -1 ≤ x - x := by ring_le
+example : x + y + 1 ≤ y + x + 3 := by ring_le
+example : x + y + 1 ≤ y + x + 1 := by ring_le
+example : x + y ≤ y + x + 3 := by ring_le
+example : x + y - 3 ≤ y + x := by ring_le
+example : x + y - x + 1 ≤ y + (4:K) := by ring_le
+
+example : 1 < (3:K) := by ring_lt
+example : -1 < (3:K) := by ring_lt
+example : 1.5 < (3:K) := by ring_lt
+example : 0 < x + 3 - x := by ring_lt
+example : -1 < x - x := by ring_lt
+example : x + y + 1 < y + x + 3 := by ring_lt
+example : x + y < y + x + 3 := by ring_lt
+example : x + y - 3 < y + x := by ring_lt
+example : x + y - x + 1 < y + (4:K) := by ring_lt
+
+/- The speed of `Mathlib.Tactic.Ring.proveLE` is very sensitive to how much typeclass inference is
+demanded by the lemmas it orchestrates.  This example took 1112 heartbeats (and 40 ms on a good
+laptop) on an implementation with "minimal" typeclasses everywhere, e.g. lots of
+`CovariantClass`/`ContravariantClass`, and takes 662 heartbeats (28 ms on a good laptop) on the
+implementation at the time of joining Mathlib (October 2024). -/
+set_option maxHeartbeats 750 in
+example : x + y - x + 1 ≤ y + (4:K) := by ring_le
+
+/- The speed of `Mathlib.Tactic.Ring.proveLT` is very sensitive to how much typeclass inference is
+demanded by the lemmas it orchestrates.  This example took 1410 heartbeats (and 48 ms on a good
+laptop) on an implementation with "minimal" typeclasses everywhere, e.g. lots of
+`CovariantClass`/`ContravariantClass`, and takes 676 heartbeats (28 ms on a good laptop) on the
+implementation at the time of joining Mathlib (October 2024). -/
+set_option maxHeartbeats 750 in
+example : x + y - x + 1 < y + (4:K) := by ring_lt
+
+/--
+error: ring failed, ring expressions not equal up to an additive constant
+K : Type u_1
+inst✝ : LinearOrderedField K
+x y : K
+⊢ 1 + x + y ≤ 3 + y
+-/
+#guard_msgs in
+example : x + y + 1 ≤ y + 3 := by ring_le
+
+/--
+error: comparison failed, LHS is larger
+K : Type u_1
+inst✝ : LinearOrderedField K
+x y : K
+⊢ 4 + x + y ≤ 3 + x + y
+-/
+#guard_msgs in
+example : x + y + 4 ≤ y + x + 3 := by ring_le
+
+/--
+error: ring failed, ring expressions not equal up to an additive constant
+K : Type u_1
+inst✝ : LinearOrderedField K
+x y : K
+⊢ 1 + x + y < 3 + y
+-/
+#guard_msgs in
+example : x + y + 1 < y + 3 := by ring_lt
+
+/--
+error: comparison failed, LHS is at least as large
+K : Type u_1
+inst✝ : LinearOrderedField K
+x y : K
+⊢ 4 + x + y < 4 + x + y
+-/
+#guard_msgs in
+example : x + y + 4 < y + x + 4 := by ring_lt
+
+end LinearOrderedField
diff --git a/test/says.lean b/test/says.lean
index 0124fb55fd94c..4315e51cb870d 100644
--- a/test/says.lean
+++ b/test/says.lean
@@ -57,6 +57,7 @@ example (x y : List α) : (x ++ y).length = x.length + y.length := by
   simp? says simp only []
 
 set_option linter.unreachableTactic false
+set_option linter.unusedTactic false in
 -- Now we check that `says` does not consume following tactics unless they are indented.
 /--
 error: Tactic `simp` did not produce any messages.
@@ -100,7 +101,8 @@ def very_long_lemma_name_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa : Q → P := fun _
 @[simp]
 def very_long_lemma_name_bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb : Q := trivial
 /--
-info: Try this: aesop? says simp_all only [very_long_lemma_name_bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb,
+info: Try this: aesop? says
+  simp_all only [very_long_lemma_name_bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb,
     very_long_lemma_name_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa]
 -/
 #guard_msgs in
diff --git a/test/search/DepthFirst.lean b/test/search/DepthFirst.lean
deleted file mode 100644
index fb20922be969c..0000000000000
--- a/test/search/DepthFirst.lean
+++ /dev/null
@@ -1,11 +0,0 @@
-import Lean.Meta.Basic
-import Mathlib.Data.MLList.DepthFirst
-
-/--
-info:
--/
-#guard_msgs in
--- We perform a depth first search of the "proper divisors in descending order" tree.
-#eval show Lean.MetaM Unit from do
-  let r := depthFirstRemovingDuplicates' (fun n => List.range n |>.filter (n % · = 0) |>.reverse) 24
-  guard <| r = [24, 12, 6, 3, 1, 2, 4, 8]
diff --git a/test/set_like.lean b/test/set_like.lean
index 284818ca75cd9..2847a594eb156 100644
--- a/test/set_like.lean
+++ b/test/set_like.lean
@@ -51,7 +51,7 @@ example [Monoid M] (x y z : M) (S₁ S₂ : Submonoid M) (h : S₁ ≤ S₂) (hx
     x * y * z ∈ S₁ ⊔ S₂ := by
   aesop
 
-example [Monoid M] (x y z : M) (S : Submonoid M) (hxy : x * y ∈ S)  (hz : z ∈ S) :
+example [Monoid M] (x y z : M) (S : Submonoid M) (hxy : x * y ∈ S) (hz : z ∈ S) :
     z * (x * y) ∈ S := by
   aesop
 
diff --git a/test/slow_simp.lean b/test/slow_simp.lean
index fa6de3b23f5b9..651d0e10e6014 100644
--- a/test/slow_simp.lean
+++ b/test/slow_simp.lean
@@ -61,7 +61,7 @@ def PointedSpaceEquiv_inverse : Under (TopCat.of Unit) ⥤ PointedSpace where
     base := by
       have := f.w
       replace this := DFunLike.congr_fun this ()
-      simp [- Under.w] at this
+      simp [-Under.w] at this
       simp
       exact this.symm }
   map_comp := by intros; simp_all; rfl -- This is the slow step.
diff --git a/test/solve_by_elim/basic.lean b/test/solve_by_elim/basic.lean
index 100df3cf25fd5..ce407f3089f28 100644
--- a/test/solve_by_elim/basic.lean
+++ b/test/solve_by_elim/basic.lean
@@ -1,9 +1,9 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
-import Mathlib.Init.Logic
+import Lean.Meta.Tactic.SolveByElim
 import Mathlib.Tactic.Constructor
 import Batteries.Tactic.PermuteGoals
 import Batteries.Test.Internal.DummyLabelAttr
diff --git a/test/solve_by_elim/instances.lean b/test/solve_by_elim/instances.lean
index f92a260c4d24d..6c060e07c9f27 100644
--- a/test/solve_by_elim/instances.lean
+++ b/test/solve_by_elim/instances.lean
@@ -1,7 +1,7 @@
 /-
-Copyright (c) 2021 Scott Morrison. All rights reserved.
+Copyright (c) 2021 Kim Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Scott Morrison
+Authors: Kim Morrison
 -/
 import Mathlib.Algebra.Order.GroupWithZero.Synonym
 import Mathlib.Algebra.Order.Ring.Nat
diff --git a/test/symm.lean b/test/symm.lean
index ac2bf3a8d6830..7d88a90868362 100644
--- a/test/symm.lean
+++ b/test/symm.lean
@@ -15,6 +15,7 @@ def sameParity : Nat → Nat → Prop
 
 example (a b : Nat) : sameParity a b → sameParity b a := by intros; symm; assumption
 
+set_option linter.unusedTactic false in
 example (a b c : Nat) (ab : a = b) (bc : b = c) : c = a := by
   symm_saturate
   -- Run twice to check that we don't add repeated copies.
@@ -43,7 +44,8 @@ infixl:25 " ≃* " => MulEquiv
 
 @[symm]
 def foo_symm {M N : Type _} [Mul M] [Mul N] (h : M ≃* N) : N ≃* M :=
-  { h.toEquiv.symm with map_mul' := (h.toMulHom.inverse h.toEquiv.symm h.left_inv h.right_inv).map_mul }
+  { h.toEquiv.symm with map_mul' :=
+    (h.toMulHom.inverse h.toEquiv.symm h.left_inv h.right_inv).map_mul }
 
 def MyEq (n m : Nat) := ∃ k, n + k = m ∧ m + k = n
 
diff --git a/test/tactic_timeout.lean b/test/tactic_timeout.lean
new file mode 100644
index 0000000000000..455b2ff5e2399
--- /dev/null
+++ b/test/tactic_timeout.lean
@@ -0,0 +1,98 @@
+import Mathlib.Tactic.Linarith
+
+/-!
+# Test that tactics respond to a cancellation request
+-/
+
+
+variable {α}
+
+open Lean Elab Tactic
+
+/-! versions of try/catch that catch `interrupted` too  -/
+section catch_interrupted
+attribute [-instance]
+  Lean.instMonadExceptOfExceptionCoreM Lean.Elab.Tactic.instMonadExceptExceptionTacticM
+
+def Meta.tryCatchAll (m : MetaM α) (h : Exception → MetaM α) : MetaM α := tryCatch m h
+def Term.tryCatchAll (m : TermElabM α) (h : Exception → TermElabM α) : TermElabM α := tryCatch m h
+def Tactic.tryCatchAll (x : TacticM α) (h : Exception → TacticM α) : TacticM α := do
+  let b ← saveState
+  try x catch ex => b.restore; h ex
+
+end catch_interrupted
+
+section test_infra
+
+def Tactic.withTimeout (ms : UInt32) (t : TacticM α) : TacticM (α ⊕ Nat) := do
+  let tk ← IO.CancelToken.new
+  withTheReader Core.Context (fun s => { s with cancelTk? := some tk }) do
+    let t0 ← IO.monoMsNow
+    let watchdog ← IO.asTask do
+      IO.sleep ms
+      tk.set
+    let r ← Tactic.tryCatchAll (.inl <$> t)
+      (fun e => do
+        IO.cancel watchdog
+        if !e.isInterrupt || !(← tk.isSet) then
+          throw e
+        else
+          let duration := (← IO.monoMsNow) - t0
+          return .inr duration)
+    IO.cancel watchdog
+    return r
+
+/-- `with_timeout 100 => tac` allows `tac` only 100ms to run. -/
+elab "with_timeout " ms:num "=>" tac:tacticSeq : tactic => do
+  let ms := ms.getNat.toUInt32
+  if let .inr _duration ← Tactic.withTimeout ms (evalTactic tac) then
+    throwError f!"Tactic took more than {ms}ms"
+
+set_option linter.unusedTactic false
+
+/-- error: Tactic took more than 500ms -/
+#guard_msgs in
+example : True := by
+  with_timeout 500 =>
+    sleep 1000
+    trivial
+
+example: True := by
+  with_timeout 500 =>
+    sleep 100
+    trivial
+
+end test_infra
+
+/-- `check_timeouts 100 => tac` checks that `tac` never goes longer than `100ms` without checking
+for cancellation. -/
+elab "check_timeouts " tol_ms:num "=>" tac:tacticSeq : tactic => do
+  let mut t := 0
+  let tol_ms := tol_ms.getNat
+  repeat do
+    if let .inr duration ← Tactic.withTimeout t.toUInt32 (evalTactic tac) then
+      if duration > t + tol_ms then
+        logError f!"Tactic took much more than {t}ms ({duration}ms)"
+      trace[debug] "Tactic overran from {t}ms to {duration}ms"
+    else
+      break
+    t := t + tol_ms
+
+set_option maxHeartbeats 0
+set_option linter.unusedTactic false
+set_option linter.unusedVariables false
+
+theorem linear_combination_with_10_terms
+    (a b c d e f g h i j : Int)
+    (h0 : -e + g + -h + i = 0)
+    (h1 : b + -d + -e + f + g + i = 0)
+    (h2 : -b + j = 0)
+    (h3 : c + d + -f + -i = 0)
+    (h4 : b + c + e + -g + -h + i + j = 0)
+    (h5 : -a + b + d + f + -h + -i = 0)
+    (h6 : a + d + e + -g + -h = 0)
+    (h7 : -a + d + -f + -h + j = 0)
+    (h8 : a + -d + e + f + g + h + -i + j = 0)
+    (h9 : -a + b + c + -e + -f + h + j = 0) :
+      -2*a + b + 2*c + d + -3*f + -g + 3*h + -3*i = 0 := by
+  check_timeouts 250 => nlinarith
diff --git a/test/tfae.lean b/test/tfae.lean
index 71bab17d358db..677908af74af9 100644
--- a/test/tfae.lean
+++ b/test/tfae.lean
@@ -1,4 +1,5 @@
 import Mathlib.Tactic.TFAE
+import Mathlib.Tactic.SuccessIfFailWithMsg
 
 open List
 set_option autoImplicit true
@@ -121,3 +122,47 @@ example (h₁ : P → Q) (h₂ : Q → P) : TFAE [P, Q] := by
   tfae_finish
 
 end context
+
+section term
+
+axiom P : Prop
+axiom Q : Prop
+axiom pq : P → Q
+axiom qp : Q → P
+
+example : TFAE [P, Q] := by
+  tfae_have h : 1 → 2 := pq
+  guard_hyp h : P → Q
+  tfae_have _ : 1 ← 2 := qp
+  tfae_finish
+
+example : TFAE [P, Q] := by
+  have n : ℕ := 4
+  tfae_have 1 → 2 := by
+    guard_hyp n : ℕ -- hypotheses are accessible (context is correct)
+    guard_target =ₛ P → Q -- expected type is known
+    exact pq
+  tfae_have 1 ← 2 := qp
+  tfae_finish
+
+example : TFAE [P, Q] := by
+  have n : ℕ := 3
+  tfae_have 2 ← 1 := fun p => ?Qgoal
+  case Qgoal => exact pq p
+  refine ?a
+  fail_if_success (tfae_have 1 ← 2 := ((?a).out 1 2 sorry sorry).mpr)
+  tfae_have 2 → 1 := qp
+  tfae_finish
+
+example : TFAE [P, Q] := by
+  tfae_have 1 → 2
+  | p => pq p
+  tfae_have 2 → 1
+  | q => qp q
+  tfae_finish
+
+example : TFAE [P, Q] := by
+  tfae_have ⟨mp, mpr⟩ : 1 ↔ 2 := ⟨pq, qp⟩
+  tfae_finish
+
+end term
diff --git a/test/toAdditive.lean b/test/toAdditive.lean
index 13390631ccf6c..bd2ad86c8e087 100644
--- a/test/toAdditive.lean
+++ b/test/toAdditive.lean
@@ -1,6 +1,5 @@
 import Mathlib.Algebra.Group.Defs
 import Mathlib.Lean.Exception
-import Mathlib.Util.Time
 import Qq.MetaM
 open Qq Lean Meta Elab Command ToAdditive
 
@@ -20,12 +19,12 @@ def foo0 {α} [Mul α] [One α] (x y : α) : α := x * y * 1
 
 theorem bar0_works : bar0 3 4 = 7 := by decide
 
-class my_has_pow (α : Type u) (β : Type v) :=
+class my_has_pow (α : Type u) (β : Type v) where
   (pow : α → β → α)
 
 instance : my_has_pow Nat Nat := ⟨fun a b => a ^ b⟩
 
-class my_has_scalar (M : Type u) (α : Type v) :=
+class my_has_scalar (M : Type u) (α : Type v) where
   (smul : M → α → α)
 
 instance : my_has_scalar Nat Nat := ⟨fun a b => a * b⟩
diff --git a/test/trace.lean b/test/trace.lean
index 1ba2fcaed0693..829fd7ee45918 100644
--- a/test/trace.lean
+++ b/test/trace.lean
@@ -1,4 +1,7 @@
 import Mathlib.Tactic.Trace
+
+set_option linter.unusedTactic false
+
 /--
 info: 7
 -/
diff --git a/test/trans.lean b/test/trans.lean
index 31f9e5506fb08..50b5bc01e7b20 100644
--- a/test/trans.lean
+++ b/test/trans.lean
@@ -90,6 +90,7 @@ example {A B C : Prop} (h : A → B) (g : B → C) : A → C := by
   · guard_target =ₛ B → C
     exact g
 
+set_option linter.unusedTactic false in
 /-- `trans` for arrows between types. -/
 example {A B C : Type} (h : A → B) (g : B → C) : A → C := by
   trans
@@ -99,6 +100,7 @@ example {A B C : Type} (h : A → B) (g : B → C) : A → C := by
 
 universe u v w
 
+set_option linter.unusedTactic false in
 /-- `trans` for arrows between types. -/
 example {A : Type u} {B : Type v} {C : Type w} (h : A → B) (g : B → C) : A → C := by
   trans